topology.metric_space.basic ⟷ Mathlib.Topology.MetricSpace.Basic

mathlib commit https://github.com/leanprover-community/mathlib/commit/65a1391a0106c9204fe45bc73a039f056558cb83

@@ -2384,7 +2384,7 @@ theorem sphere_prod (x : α × β) (r : ℝ) :
   · ext ⟨x', y'⟩
     simp_rw [Set.mem_union, Set.mem_prod, Metric.mem_closedBall, Metric.mem_sphere, Prod.dist_eq,
       max_eq_iff]
-    refine' or_congr (and_congr_right _) ((and_comm' _ _).trans (and_congr_left _))
+    refine' or_congr (and_congr_right _) ((and_comm _ _).trans (and_congr_left _))
     all_goals rintro rfl; rfl
 #align sphere_prod sphere_prod
 -/
@@ -2592,7 +2592,7 @@ theorem closedBall_zero' (x : α) : closedBall x 0 = closure {x} :=
 #print Metric.dense_iff /-
 theorem dense_iff {s : Set α} : Dense s ↔ ∀ x, ∀ r > 0, (ball x r ∩ s).Nonempty :=
   forall_congr' fun x => by
-    simp only [mem_closure_iff, Set.Nonempty, exists_prop, mem_inter_iff, mem_ball', and_comm']
+    simp only [mem_closure_iff, Set.Nonempty, exists_prop, mem_inter_iff, mem_ball', and_comm]
 #align metric.dense_iff Metric.dense_iff
 -/
 
@@ -2741,7 +2741,7 @@ theorem nndist_pi_lt_iff {f g : ∀ b, π b} {r : ℝ≥0} (hr : 0 < r) :
 theorem nndist_pi_eq_iff {f g : ∀ b, π b} {r : ℝ≥0} (hr : 0 < r) :
     nndist f g = r ↔ (∃ i, nndist (f i) (g i) = r) ∧ ∀ b, nndist (f b) (g b) ≤ r :=
   by
-  rw [eq_iff_le_not_lt, nndist_pi_lt_iff hr, nndist_pi_le_iff, Classical.not_forall, and_comm']
+  rw [eq_iff_le_not_lt, nndist_pi_lt_iff hr, nndist_pi_le_iff, Classical.not_forall, and_comm]
   simp_rw [not_lt, and_congr_left_iff, le_antisymm_iff]
   intro h
   refine' exists_congr fun b => _

mathlib commit https://github.com/leanprover-community/mathlib/commit/65a1391a0106c9204fe45bc73a039f056558cb83

@@ -5,7 +5,7 @@ Authors: Jeremy Avigad, Robert Y. Lewis, Johannes Hölzl, Mario Carneiro, Sébas
 -/
 import Tactic.Positivity
 import Topology.Algebra.Order.Compact
-import Topology.MetricSpace.EmetricSpace
+import Topology.EMetricSpace.Basic
 import Topology.Bornology.Constructions
 
 #align_import topology.metric_space.basic from "leanprover-community/mathlib"@"c8f305514e0d47dfaa710f5a52f0d21b588e6328"
@@ -911,7 +911,7 @@ theorem iUnion_inter_closedBall_nat (s : Set α) (x : α) : (⋃ n : ℕ, s ∩
 
 #print Metric.ball_subset /-
 theorem ball_subset (h : dist x y ≤ ε₂ - ε₁) : ball x ε₁ ⊆ ball y ε₂ := fun z zx => by
-  rw [← add_sub_cancel'_right ε₁ ε₂] <;>
+  rw [← add_sub_cancel ε₁ ε₂] <;>
     exact lt_of_le_of_lt (dist_triangle z x y) (add_lt_add_of_lt_of_le zx h)
 #align metric.ball_subset Metric.ball_subset
 -/
@@ -1125,7 +1125,7 @@ theorem uniformContinuous_iff [PseudoMetricSpace β] {f : α → β} :
 #align metric.uniform_continuous_iff Metric.uniformContinuous_iff
 -/
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:641:2: warning: expanding binder collection (x y «expr ∈ » s) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:642:2: warning: expanding binder collection (x y «expr ∈ » s) -/
 #print Metric.uniformContinuousOn_iff /-
 theorem uniformContinuousOn_iff [PseudoMetricSpace β] {f : α → β} {s : Set α} :
     UniformContinuousOn f s ↔
@@ -1134,7 +1134,7 @@ theorem uniformContinuousOn_iff [PseudoMetricSpace β] {f : α → β} {s : Set
 #align metric.uniform_continuous_on_iff Metric.uniformContinuousOn_iff
 -/
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:641:2: warning: expanding binder collection (x y «expr ∈ » s) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:642:2: warning: expanding binder collection (x y «expr ∈ » s) -/
 #print Metric.uniformContinuousOn_iff_le /-
 theorem uniformContinuousOn_iff_le [PseudoMetricSpace β] {f : α → β} {s : Set α} :
     UniformContinuousOn f s ↔
@@ -1201,7 +1201,7 @@ theorem totallyBounded_of_finite_discretization {s : Set α}
 #align metric.totally_bounded_of_finite_discretization Metric.totallyBounded_of_finite_discretization
 -/
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:641:2: warning: expanding binder collection (t «expr ⊆ » s) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:642:2: warning: expanding binder collection (t «expr ⊆ » s) -/
 #print Metric.finite_approx_of_totallyBounded /-
 theorem finite_approx_of_totallyBounded {s : Set α} (hs : TotallyBounded s) :
     ∀ ε > 0, ∃ (t : _) (_ : t ⊆ s), Set.Finite t ∧ s ⊆ ⋃ y ∈ t, ball y ε :=
@@ -1270,7 +1270,7 @@ theorem tendstoUniformly_iff {ι : Type _} {F : ι → β → α} {f : β → α
 #align metric.tendsto_uniformly_iff Metric.tendstoUniformly_iff
 -/
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:641:2: warning: expanding binder collection (x y «expr ∈ » t) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:642:2: warning: expanding binder collection (x y «expr ∈ » t) -/
 #print Metric.cauchy_iff /-
 protected theorem cauchy_iff {f : Filter α} :
     Cauchy f ↔ NeBot f ∧ ∀ ε > 0, ∃ t ∈ f, ∀ (x) (_ : x ∈ t) (y) (_ : y ∈ t), dist x y < ε :=
@@ -1882,15 +1882,15 @@ theorem Real.closedBall_eq_Icc {x r : ℝ} : closedBall x r = Icc (x - r) (x + r
 
 #print Real.Ioo_eq_ball /-
 theorem Real.Ioo_eq_ball (x y : ℝ) : Ioo x y = ball ((x + y) / 2) ((y - x) / 2) := by
-  rw [Real.ball_eq_Ioo, ← sub_div, add_comm, ← sub_add, add_sub_cancel', add_self_div_two, ←
-    add_div, add_assoc, add_sub_cancel'_right, add_self_div_two]
+  rw [Real.ball_eq_Ioo, ← sub_div, add_comm, ← sub_add, add_sub_cancel_left, add_self_div_two, ←
+    add_div, add_assoc, add_sub_cancel, add_self_div_two]
 #align real.Ioo_eq_ball Real.Ioo_eq_ball
 -/
 
 #print Real.Icc_eq_closedBall /-
 theorem Real.Icc_eq_closedBall (x y : ℝ) : Icc x y = closedBall ((x + y) / 2) ((y - x) / 2) := by
-  rw [Real.closedBall_eq_Icc, ← sub_div, add_comm, ← sub_add, add_sub_cancel', add_self_div_two, ←
-    add_div, add_assoc, add_sub_cancel'_right, add_self_div_two]
+  rw [Real.closedBall_eq_Icc, ← sub_div, add_comm, ← sub_add, add_sub_cancel_left, add_self_div_two,
+    ← add_div, add_assoc, add_sub_cancel, add_self_div_two]
 #align real.Icc_eq_closed_ball Real.Icc_eq_closedBall
 -/
 
@@ -2002,7 +2002,7 @@ section CauchySeq
 
 variable [Nonempty β] [SemilatticeSup β]
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:641:2: warning: expanding binder collection (m n «expr ≥ » N) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:642:2: warning: expanding binder collection (m n «expr ≥ » N) -/
 #print Metric.cauchySeq_iff /-
 -- see Note [nolint_ge]
 /-- In a pseudometric space, Cauchy sequences are characterized by the fact that, eventually,
@@ -2917,7 +2917,7 @@ end Pi
 
 section Compact
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:641:2: warning: expanding binder collection (t «expr ⊆ » s) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:642:2: warning: expanding binder collection (t «expr ⊆ » s) -/
 #print finite_cover_balls_of_compact /-
 /-- Any compact set in a pseudometric space can be covered by finitely many balls of a given
 positive radius -/
@@ -3029,7 +3029,7 @@ instance (priority := 100) locally_compact_of_proper [ProperSpace α] : LocallyC
 #align locally_compact_of_proper locally_compact_of_proper
 -/
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:641:2: warning: expanding binder collection (x y «expr ∈ » t) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:642:2: warning: expanding binder collection (x y «expr ∈ » t) -/
 #print complete_of_proper /-
 -- see Note [lower instance priority]
 /-- A proper space is complete -/
@@ -3161,7 +3161,7 @@ namespace Metric
 
 /- warning: metric.bounded clashes with bornology.is_bounded -> Bornology.IsBounded
 Case conversion may be inaccurate. Consider using '#align metric.bounded Bornology.IsBoundedₓ'. -/
-/- ./././Mathport/Syntax/Translate/Basic.lean:641:2: warning: expanding binder collection (x y «expr ∈ » s) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:642:2: warning: expanding binder collection (x y «expr ∈ » s) -/
 #print Bornology.IsBounded /-
 /-- Boundedness of a subset of a pseudometric space. We formulate the definition to work
 even in the empty space. -/

mathlib commit https://github.com/leanprover-community/mathlib/commit/65a1391a0106c9204fe45bc73a039f056558cb83

@@ -172,18 +172,18 @@ theorem PseudoMetricSpace.ext {α : Type _} {m m' : PseudoMetricSpace α}
     (h : m.toHasDist = m'.toHasDist) : m = m' :=
   by
   rcases m with ⟨⟩; rcases m' with ⟨⟩
-  dsimp at h 
+  dsimp at h
   subst h
   congr
   · ext x y : 2
-    dsimp at m_edist_dist m'_edist_dist 
+    dsimp at m_edist_dist m'_edist_dist
     simp [m_edist_dist, m'_edist_dist]
-  · dsimp at m_uniformity_dist m'_uniformity_dist 
-    rw [← m'_uniformity_dist] at m_uniformity_dist 
+  · dsimp at m_uniformity_dist m'_uniformity_dist
+    rw [← m'_uniformity_dist] at m_uniformity_dist
     exact UniformSpace.ext m_uniformity_dist
   · ext1
-    dsimp at m_cobounded_sets m'_cobounded_sets 
-    rw [← m'_cobounded_sets] at m_cobounded_sets 
+    dsimp at m_cobounded_sets m'_cobounded_sets
+    rw [← m'_cobounded_sets] at m_cobounded_sets
     exact filter_eq m_cobounded_sets
 #align pseudo_metric_space.ext PseudoMetricSpace.ext
 -/
@@ -882,7 +882,7 @@ theorem dist_lt_add_of_nonempty_closedBall_inter_ball (h : (closedBall x ε₁ 
 #print Metric.dist_lt_add_of_nonempty_ball_inter_closedBall /-
 theorem dist_lt_add_of_nonempty_ball_inter_closedBall (h : (ball x ε₁ ∩ closedBall y ε₂).Nonempty) :
     dist x y < ε₁ + ε₂ := by
-  rw [inter_comm] at h 
+  rw [inter_comm] at h
   rw [add_comm, dist_comm]
   exact dist_lt_add_of_nonempty_closed_ball_inter_ball h
 #align metric.dist_lt_add_of_nonempty_ball_inter_closed_ball Metric.dist_lt_add_of_nonempty_ball_inter_closedBall
@@ -1207,7 +1207,7 @@ theorem finite_approx_of_totallyBounded {s : Set α} (hs : TotallyBounded s) :
     ∀ ε > 0, ∃ (t : _) (_ : t ⊆ s), Set.Finite t ∧ s ⊆ ⋃ y ∈ t, ball y ε :=
   by
   intro ε ε_pos
-  rw [totallyBounded_iff_subset] at hs 
+  rw [totallyBounded_iff_subset] at hs
   exact hs _ (dist_mem_uniformity ε_pos)
 #align metric.finite_approx_of_totally_bounded Metric.finite_approx_of_totallyBounded
 -/
@@ -1580,7 +1580,7 @@ protected theorem Metric.uniformity_basis_edist :
       simp only [edist_dist, ENNReal.ofReal_lt_ofReal_iff ε0]
       exact Hε
     · rcases ENNReal.lt_iff_exists_real_btwn.1 ε0 with ⟨ε', _, ε0', hε⟩
-      rw [ENNReal.ofReal_pos] at ε0' 
+      rw [ENNReal.ofReal_pos] at ε0'
       refine' ⟨ε', ε0', fun a b h => Hε (lt_trans _ hε)⟩
       rwa [edist_dist, ENNReal.ofReal_lt_ofReal_iff ε0']⟩
 #align pseudo_metric.uniformity_basis_edist Metric.uniformity_basis_edist
@@ -2041,7 +2041,7 @@ theorem Metric.uniformCauchySeqOn_iff {γ : Type _} {F : β → γ → α} {s :
       @Filter.eventually_atTop_prod_self' _ _ _ fun m =>
         ∀ x : γ, x ∈ s → dist (F m.fst x) (F m.snd x) < ε]
     specialize h u hu
-    rw [prod_at_top_at_top_eq] at h 
+    rw [prod_at_top_at_top_eq] at h
     exact h.mono fun n h x hx => set.mem_set_of_eq.mp (h x hx)
   · intro h u hu
     rcases metric.mem_uniformity_dist.mp hu with ⟨ε, hε, hab⟩
@@ -2407,7 +2407,7 @@ theorem uniformContinuous_dist : UniformContinuous fun p : α × α => dist p.1
       have :=
         add_lt_add (abs_sub_lt_iff.1 (lt_of_le_of_lt (abs_dist_sub_le p₁ q₁ p₂) h₁)).1
           (abs_sub_lt_iff.1 (lt_of_le_of_lt (abs_dist_sub_le p₂ q₂ q₁) h₂)).1
-      rwa [add_halves, dist_comm p₂, sub_add_sub_cancel, dist_comm q₂] at this ⟩
+      rwa [add_halves, dist_comm p₂, sub_add_sub_cancel, dist_comm q₂] at this⟩
 #align uniform_continuous_dist uniformContinuous_dist
 -/
 
@@ -2876,7 +2876,7 @@ theorem sphere_pi (x : ∀ b, π b) {r : ℝ} (h : 0 < r ∨ Nonempty β) :
     refine' subset_Union_of_subset default _
     intro x hx
     replace hx := hx.le
-    rw [dist_pi_le_iff le_rfl] at hx 
+    rw [dist_pi_le_iff le_rfl] at hx
     exact le_antisymm (hx default) dist_nonneg
   · ext
     simp [dist_pi_eq_iff hr, dist_pi_le_iff hr.le]
@@ -3099,7 +3099,7 @@ center and a strictly smaller radius that includes `s`. -/
 theorem exists_lt_subset_ball (hs : IsClosed s) (h : s ⊆ ball x r) : ∃ r' < r, s ⊆ ball x r' :=
   by
   cases' le_or_lt r 0 with hr hr
-  · rw [ball_eq_empty.2 hr, subset_empty_iff] at h ; subst s
+  · rw [ball_eq_empty.2 hr, subset_empty_iff] at h; subst s
     exact (exists_lt r).imp fun r' hr' => ⟨hr', empty_subset _⟩
   · exact (exists_pos_lt_subset_ball hr hs h).imp fun r' hr' => ⟨hr'.fst.2, hr'.snd⟩
 #align exists_lt_subset_ball exists_lt_subset_ball
@@ -3153,7 +3153,7 @@ theorem lebesgue_number_lemma_of_metric {s : Set α} {ι} {c : ι → Set α} (h
 #print lebesgue_number_lemma_of_metric_sUnion /-
 theorem lebesgue_number_lemma_of_metric_sUnion {s : Set α} {c : Set (Set α)} (hs : IsCompact s)
     (hc₁ : ∀ t ∈ c, IsOpen t) (hc₂ : s ⊆ ⋃₀ c) : ∃ δ > 0, ∀ x ∈ s, ∃ t ∈ c, ball x δ ⊆ t := by
-  rw [sUnion_eq_Union] at hc₂  <;> simpa using lebesgue_number_lemma_of_metric hs (by simpa) hc₂
+  rw [sUnion_eq_Union] at hc₂ <;> simpa using lebesgue_number_lemma_of_metric hs (by simpa) hc₂
 #align lebesgue_number_lemma_of_metric_sUnion lebesgue_number_lemma_of_metric_sUnion
 -/
 
@@ -3478,7 +3478,7 @@ theorem exists_isOpen_isBounded_image_of_isCompact_of_forall_continuousAt [Topol
     {k : Set β} {f : β → α} (hk : IsCompact k) (hf : ∀ x ∈ k, ContinuousAt f x) :
     ∃ t, k ⊆ t ∧ IsOpen t ∧ IsBounded (f '' t) :=
   by
-  simp_rw [← continuousWithinAt_univ] at hf 
+  simp_rw [← continuousWithinAt_univ] at hf
   simpa only [inter_univ] using
     exists_is_open_bounded_image_inter_of_is_compact_of_forall_continuous_within_at hk hf
 #align metric.exists_is_open_bounded_image_of_is_compact_of_forall_continuous_at Metric.exists_isOpen_isBounded_image_of_isCompact_of_forall_continuousAt
@@ -3736,7 +3736,7 @@ theorem dist_le_diam_of_mem (h : IsBounded s) (hx : x ∈ s) (hy : y ∈ s) : di
 
 #print Metric.ediam_of_unbounded /-
 theorem ediam_of_unbounded (h : ¬IsBounded s) : EMetric.diam s = ∞ := by
-  rwa [bounded_iff_ediam_ne_top, Classical.not_not] at h 
+  rwa [bounded_iff_ediam_ne_top, Classical.not_not] at h
 #align metric.ediam_of_unbounded Metric.ediam_of_unbounded
 -/
 
@@ -3768,7 +3768,7 @@ theorem diam_union {t : Set α} (xs : x ∈ s) (yt : y ∈ t) :
   by_cases H : bounded (s ∪ t)
   · have hs : bounded s := H.mono (subset_union_left _ _)
     have ht : bounded t := H.mono (subset_union_right _ _)
-    rw [bounded_iff_ediam_ne_top] at H hs ht 
+    rw [bounded_iff_ediam_ne_top] at H hs ht
     rw [dist_edist, diam, diam, diam, ← ENNReal.toReal_add, ← ENNReal.toReal_add,
             ENNReal.toReal_le_toReal] <;>
           repeat' apply ENNReal.add_ne_top.2 <;> constructor <;>
@@ -3826,7 +3826,7 @@ theorem IsComplete.nonempty_iInter_of_nonempty_biInter {s : ℕ → Set α} (h0
     intro n N hn
     apply mem_of_subset_of_mem _ (h N).choose_spec
     intro x hx
-    simp only [mem_Inter] at hx 
+    simp only [mem_Inter] at hx
     exact hx n hn
   have : ∀ n, u n ∈ s 0 := fun n => I 0 n (zero_le _)
   have : CauchySeq u := by
@@ -3860,7 +3860,7 @@ theorem exists_isLocalMin_mem_ball [ProperSpace α] [TopologicalSpace β]
     (hf : ContinuousOn f (closedBall a r)) (hz : z ∈ closedBall a r)
     (hf1 : ∀ z' ∈ sphere a r, f z < f z') : ∃ z ∈ ball a r, IsLocalMin f z :=
   by
-  simp_rw [← closed_ball_diff_ball] at hf1 
+  simp_rw [← closed_ball_diff_ball] at hf1
   exact
     (is_compact_closed_ball a r).exists_isLocalMin_mem_open ball_subset_closed_ball hf hz hf1
       is_open_ball
@@ -3933,7 +3933,7 @@ theorem MetricSpace.ext {α : Type _} {m m' : MetricSpace α} (h : m.toHasDist =
   by
   have h' : m.to_pseudo_metric_space = m'.to_pseudo_metric_space := PseudoMetricSpace.ext h
   rcases m with ⟨⟩; rcases m' with ⟨⟩
-  dsimp at h' 
+  dsimp at h'
   subst h'
 #align metric_space.ext MetricSpace.ext
 -/
@@ -4054,12 +4054,12 @@ theorem subsingleton_sphere (x : γ) {r : ℝ} (hr : r ≤ 0) : (sphere x r).Sub
 #align metric.subsingleton_sphere Metric.subsingleton_sphere
 -/
 
-#print MetricSpace.to_separated /-
+#print MetricSpace.instT0Space /-
 -- see Note [lower instance priority]
-instance (priority := 100) MetricSpace.to_separated : SeparatedSpace γ :=
-  separated_def.2 fun x y h =>
+instance (priority := 100) MetricSpace.instT0Space : T0Space γ :=
+  t0Space_iff_uniformity.2 fun x y h =>
     eq_of_forall_dist_le fun ε ε0 => le_of_lt (h _ (dist_mem_uniformity ε0))
-#align metric_space.to_separated MetricSpace.to_separated
+#align metric_space.to_separated MetricSpace.instT0Space
 -/
 
 #print Metric.uniformEmbedding_iff' /-
@@ -4321,7 +4321,7 @@ instance metricSpacePi : MetricSpace (∀ b, π b) :=
       by
       have eq1 : edist f g = 0 := by simp only [edist_dist, eq0, ENNReal.ofReal_zero]
       have eq2 : (sup univ fun b : β => edist (f b) (g b)) ≤ 0 := le_of_eq eq1
-      simp only [Finset.sup_le_iff] at eq2 
+      simp only [Finset.sup_le_iff] at eq2
       exact funext fun b => edist_le_zero.1 <| eq2 b <| mem_univ b }
 #align metric_space_pi metricSpacePi
 -/
@@ -4364,20 +4364,20 @@ end Metric
 
 section EqRel
 
-instance {α : Type u} [PseudoMetricSpace α] : Dist (UniformSpace.SeparationQuotient α)
+instance {α : Type u} [PseudoMetricSpace α] : Dist (SeparationQuotient α)
     where dist p q :=
     Quotient.liftOn₂' p q dist fun x y x' y' hx hy => by
-      rw [dist_edist, dist_edist, ← UniformSpace.SeparationQuotient.edist_mk x, ←
-        UniformSpace.SeparationQuotient.edist_mk x', Quot.sound hx, Quot.sound hy]
+      rw [dist_edist, dist_edist, ← SeparationQuotient.edist_mk x, ← SeparationQuotient.edist_mk x',
+        Quot.sound hx, Quot.sound hy]
 
-#print UniformSpace.SeparationQuotient.dist_mk /-
-theorem UniformSpace.SeparationQuotient.dist_mk {α : Type u} [PseudoMetricSpace α] (p q : α) :
-    @dist (UniformSpace.SeparationQuotient α) _ (Quot.mk _ p) (Quot.mk _ q) = dist p q :=
+#print SeparationQuotient.dist_mk /-
+theorem SeparationQuotient.dist_mk {α : Type u} [PseudoMetricSpace α] (p q : α) :
+    @dist (SeparationQuotient α) _ (Quot.mk _ p) (Quot.mk _ q) = dist p q :=
   rfl
-#align uniform_space.separation_quotient.dist_mk UniformSpace.SeparationQuotient.dist_mk
+#align uniform_space.separation_quotient.dist_mk SeparationQuotient.dist_mk
 -/
 
-instance {α : Type u} [PseudoMetricSpace α] : MetricSpace (UniformSpace.SeparationQuotient α) :=
+instance {α : Type u} [PseudoMetricSpace α] : MetricSpace (SeparationQuotient α) :=
   EMetricSpace.toMetricSpaceOfDist dist (fun x y => Quotient.inductionOn₂' x y edist_ne_top)
     fun x y => Quotient.inductionOn₂' x y dist_edist
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/65a1391a0106c9204fe45bc73a039f056558cb83

@@ -3275,8 +3275,7 @@ theorem IsBounded.subset_closedBall_lt (h : IsBounded s) (a : ℝ) (c : α) :
 #print Metric.isBounded_closure_of_isBounded /-
 theorem isBounded_closure_of_isBounded (h : IsBounded s) : IsBounded (closure s) :=
   let ⟨C, h⟩ := h
-  ⟨C, fun a ha b hb =>
-    (ClosedIicTopology.isClosed_le' C).closure_subset <| map_mem_closure₂ continuous_dist ha hb h⟩
+  ⟨C, fun a ha b hb => (isClosed_Iic C).closure_subset <| map_mem_closure₂ continuous_dist ha hb h⟩
 #align metric.bounded_closure_of_bounded Metric.isBounded_closure_of_isBounded
 -/
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/65a1391a0106c9204fe45bc73a039f056558cb83

@@ -497,7 +497,7 @@ theorem nndist_dist (x y : α) : nndist x y = Real.toNNReal (dist x y) := by
 
 #print nndist_comm /-
 theorem nndist_comm (x y : α) : nndist x y = nndist y x := by
-  simpa only [dist_nndist, NNReal.coe_eq] using dist_comm x y
+  simpa only [dist_nndist, NNReal.coe_inj] using dist_comm x y
 #align nndist_comm nndist_comm
 -/
 
@@ -2254,7 +2254,7 @@ theorem NNReal.nndist_eq (a b : ℝ≥0) : nndist a b = max (a - b) (b - a) :=
   by
   wlog h : b ≤ a
   · rw [nndist_comm, max_comm]; exact this b a (le_of_not_le h)
-  rw [← NNReal.coe_eq, ← dist_nndist, NNReal.dist_eq, tsub_eq_zero_iff_le.2 h,
+  rw [← NNReal.coe_inj, ← dist_nndist, NNReal.dist_eq, tsub_eq_zero_iff_le.2 h,
     max_eq_left (zero_le <| a - b), ← NNReal.coe_sub h, abs_of_nonneg (a - b).coe_nonneg]
 #align nnreal.nndist_eq NNReal.nndist_eq
 -/
@@ -2772,7 +2772,7 @@ theorem dist_pi_eq_iff {f g : ∀ b, π b} {r : ℝ} (hr : 0 < r) :
     dist f g = r ↔ (∃ i, dist (f i) (g i) = r) ∧ ∀ b, dist (f b) (g b) ≤ r :=
   by
   lift r to ℝ≥0 using hr.le
-  simp_rw [← coe_nndist, NNReal.coe_eq, nndist_pi_eq_iff hr, NNReal.coe_le_coe]
+  simp_rw [← coe_nndist, NNReal.coe_inj, nndist_pi_eq_iff hr, NNReal.coe_le_coe]
 #align dist_pi_eq_iff dist_pi_eq_iff
 -/
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/65a1391a0106c9204fe45bc73a039f056558cb83

@@ -2993,10 +2993,10 @@ theorem tendsto_dist_left_cocompact_atTop [ProperSpace α] (x : α) :
 #align tendsto_dist_left_cocompact_at_top tendsto_dist_left_cocompact_atTop
 -/
 
-#print properSpace_of_compact_closedBall_of_le /-
+#print ProperSpace.of_isCompact_closedBall_of_le /-
 /-- If all closed balls of large enough radius are compact, then the space is proper. Especially
 useful when the lower bound for the radius is 0. -/
-theorem properSpace_of_compact_closedBall_of_le (R : ℝ)
+theorem ProperSpace.of_isCompact_closedBall_of_le (R : ℝ)
     (h : ∀ x : α, ∀ r, R ≤ r → IsCompact (closedBall x r)) : ProperSpace α :=
   ⟨by
     intro x r
@@ -3009,7 +3009,7 @@ theorem properSpace_of_compact_closedBall_of_le (R : ℝ)
         exact closed_ball_subset_closed_ball (le_of_lt (not_le.1 hr))
       rw [this]
       exact (h x R le_rfl).inter_right is_closed_ball⟩
-#align proper_space_of_compact_closed_ball_of_le properSpace_of_compact_closedBall_of_le
+#align proper_space_of_compact_closed_ball_of_le ProperSpace.of_isCompact_closedBall_of_le
 -/
 
 #print proper_of_compact /-
@@ -3065,7 +3065,7 @@ instance prod_properSpace {α : Type _} {β : Type _} [PseudoMetricSpace α] [Ps
 instance pi_properSpace {π : β → Type _} [Fintype β] [∀ b, PseudoMetricSpace (π b)]
     [h : ∀ b, ProperSpace (π b)] : ProperSpace (∀ b, π b) :=
   by
-  refine' properSpace_of_compact_closedBall_of_le 0 fun x r hr => _
+  refine' ProperSpace.of_isCompact_closedBall_of_le 0 fun x r hr => _
   rw [closedBall_pi _ hr]
   apply isCompact_univ_pi fun b => _
   apply (h b).isCompact_closedBall

mathlib commit https://github.com/leanprover-community/mathlib/commit/65a1391a0106c9204fe45bc73a039f056558cb83

@@ -2607,7 +2607,37 @@ theorem denseRange_iff {f : β → α} : DenseRange f ↔ ∀ x, ∀ r > 0, ∃
 This is not obvious, as the countable set whose closure covers `s` does not need in general to
 be contained in `s`. -/
 theorem TopologicalSpace.IsSeparable.separableSpace {s : Set α} (hs : IsSeparable s) :
-    SeparableSpace s := by classical
+    SeparableSpace s := by
+  classical
+  rcases eq_empty_or_nonempty s with (rfl | ⟨⟨x₀, x₀s⟩⟩)
+  · infer_instance
+  rcases hs with ⟨c, hc, h'c⟩
+  haveI : Encodable c := hc.to_encodable
+  obtain ⟨u, -, u_pos, u_lim⟩ :
+    ∃ u : ℕ → ℝ, StrictAnti u ∧ (∀ n : ℕ, 0 < u n) ∧ tendsto u at_top (𝓝 0) :=
+    exists_seq_strictAnti_tendsto (0 : ℝ)
+  let f : c × ℕ → α := fun p =>
+    if h : (Metric.ball (p.1 : α) (u p.2) ∩ s).Nonempty then h.some else x₀
+  have fs : ∀ p, f p ∈ s := by
+    rintro ⟨y, n⟩
+    by_cases h : (ball (y : α) (u n) ∩ s).Nonempty
+    · simpa only [f, h, dif_pos] using h.some_spec.2
+    · simpa only [f, h, not_false_iff, dif_neg]
+  let g : c × ℕ → s := fun p => ⟨f p, fs p⟩
+  apply separable_space_of_dense_range g
+  apply Metric.denseRange_iff.2
+  rintro ⟨x, xs⟩ r (rpos : 0 < r)
+  obtain ⟨n, hn⟩ : ∃ n, u n < r / 2 := ((tendsto_order.1 u_lim).2 _ (half_pos rpos)).exists
+  obtain ⟨z, zc, hz⟩ : ∃ z ∈ c, dist x z < u n := Metric.mem_closure_iff.1 (h'c xs) _ (u_pos n)
+  refine' ⟨(⟨z, zc⟩, n), _⟩
+  change dist x (f (⟨z, zc⟩, n)) < r
+  have A : (Metric.ball z (u n) ∩ s).Nonempty := ⟨x, hz, xs⟩
+  dsimp [f]
+  simp only [A, dif_pos]
+  calc
+    dist x A.some ≤ dist x z + dist z A.some := dist_triangle _ _ _
+    _ < r / 2 + r / 2 := (add_lt_add (hz.trans hn) ((Metric.mem_ball'.1 A.some_spec.1).trans hn))
+    _ = r := add_halves _
 #align topological_space.is_separable.separable_space TopologicalSpace.IsSeparable.separableSpace
 -/
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/65a1391a0106c9204fe45bc73a039f056558cb83

@@ -2607,37 +2607,7 @@ theorem denseRange_iff {f : β → α} : DenseRange f ↔ ∀ x, ∀ r > 0, ∃
 This is not obvious, as the countable set whose closure covers `s` does not need in general to
 be contained in `s`. -/
 theorem TopologicalSpace.IsSeparable.separableSpace {s : Set α} (hs : IsSeparable s) :
-    SeparableSpace s := by
-  classical
-  rcases eq_empty_or_nonempty s with (rfl | ⟨⟨x₀, x₀s⟩⟩)
-  · infer_instance
-  rcases hs with ⟨c, hc, h'c⟩
-  haveI : Encodable c := hc.to_encodable
-  obtain ⟨u, -, u_pos, u_lim⟩ :
-    ∃ u : ℕ → ℝ, StrictAnti u ∧ (∀ n : ℕ, 0 < u n) ∧ tendsto u at_top (𝓝 0) :=
-    exists_seq_strictAnti_tendsto (0 : ℝ)
-  let f : c × ℕ → α := fun p =>
-    if h : (Metric.ball (p.1 : α) (u p.2) ∩ s).Nonempty then h.some else x₀
-  have fs : ∀ p, f p ∈ s := by
-    rintro ⟨y, n⟩
-    by_cases h : (ball (y : α) (u n) ∩ s).Nonempty
-    · simpa only [f, h, dif_pos] using h.some_spec.2
-    · simpa only [f, h, not_false_iff, dif_neg]
-  let g : c × ℕ → s := fun p => ⟨f p, fs p⟩
-  apply separable_space_of_dense_range g
-  apply Metric.denseRange_iff.2
-  rintro ⟨x, xs⟩ r (rpos : 0 < r)
-  obtain ⟨n, hn⟩ : ∃ n, u n < r / 2 := ((tendsto_order.1 u_lim).2 _ (half_pos rpos)).exists
-  obtain ⟨z, zc, hz⟩ : ∃ z ∈ c, dist x z < u n := Metric.mem_closure_iff.1 (h'c xs) _ (u_pos n)
-  refine' ⟨(⟨z, zc⟩, n), _⟩
-  change dist x (f (⟨z, zc⟩, n)) < r
-  have A : (Metric.ball z (u n) ∩ s).Nonempty := ⟨x, hz, xs⟩
-  dsimp [f]
-  simp only [A, dif_pos]
-  calc
-    dist x A.some ≤ dist x z + dist z A.some := dist_triangle _ _ _
-    _ < r / 2 + r / 2 := (add_lt_add (hz.trans hn) ((Metric.mem_ball'.1 A.some_spec.1).trans hn))
-    _ = r := add_halves _
+    SeparableSpace s := by classical
 #align topological_space.is_separable.separable_space TopologicalSpace.IsSeparable.separableSpace
 -/
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/65a1391a0106c9204fe45bc73a039f056558cb83

@@ -3024,7 +3024,7 @@ instance (priority := 100) proper_of_compact [CompactSpace α] : ProperSpace α
 -- see Note [lower instance priority]
 /-- A proper space is locally compact -/
 instance (priority := 100) locally_compact_of_proper [ProperSpace α] : LocallyCompactSpace α :=
-  locallyCompactSpace_of_hasBasis (fun x => nhds_basis_closedBall) fun x ε ε0 =>
+  LocallyCompactSpace.of_hasBasis (fun x => nhds_basis_closedBall) fun x ε ε0 =>
     isCompact_closedBall _ _
 #align locally_compact_of_proper locally_compact_of_proper
 -/

mathlib commit https://github.com/leanprover-community/mathlib/commit/65a1391a0106c9204fe45bc73a039f056558cb83

@@ -3189,12 +3189,14 @@ theorem isBounded_empty : IsBounded (∅ : Set α) :=
 #align metric.bounded_empty Bornology.isBounded_empty
 -/
 
-theorem nonempty_of_unbounded (h : ¬IsBounded s) : s.Nonempty :=
+#print Bornology.nonempty_of_not_isBounded /-
+theorem nonempty_of_not_isBounded (h : ¬IsBounded s) : s.Nonempty :=
   by
   rw [nonempty_iff_ne_empty]
   rintro rfl
   exact h bounded_empty
-#align metric.nonempty_of_unbounded Metric.nonempty_of_unbounded
+#align metric.nonempty_of_unbounded Bornology.nonempty_of_not_isBounded
+-/
 
 #print Bornology.isBounded_iff_forall_mem /-
 theorem isBounded_iff_forall_mem : IsBounded s ↔ ∀ x ∈ s, IsBounded s :=
@@ -3623,11 +3625,13 @@ theorem diam_singleton : diam ({x} : Set α) = 0 :=
 #align metric.diam_singleton Metric.diam_singleton
 -/
 
+#print Metric.diam_one /-
 @[simp, to_additive]
 theorem diam_one [One α] : diam (1 : Set α) = 0 :=
   diam_singleton
 #align metric.diam_one Metric.diam_one
 #align metric.diam_zero Metric.diam_zero
+-/
 
 #print Metric.diam_pair /-
 -- Does not work as a simp-lemma, since {x, y} reduces to (insert y {x})

mathlib commit https://github.com/leanprover-community/mathlib/commit/b1abe23ae96fef89ad30d9f4362c307f72a55010

@@ -2741,7 +2741,7 @@ theorem nndist_pi_lt_iff {f g : ∀ b, π b} {r : ℝ≥0} (hr : 0 < r) :
 theorem nndist_pi_eq_iff {f g : ∀ b, π b} {r : ℝ≥0} (hr : 0 < r) :
     nndist f g = r ↔ (∃ i, nndist (f i) (g i) = r) ∧ ∀ b, nndist (f b) (g b) ≤ r :=
   by
-  rw [eq_iff_le_not_lt, nndist_pi_lt_iff hr, nndist_pi_le_iff, not_forall, and_comm']
+  rw [eq_iff_le_not_lt, nndist_pi_lt_iff hr, nndist_pi_le_iff, Classical.not_forall, and_comm']
   simp_rw [not_lt, and_congr_left_iff, le_antisymm_iff]
   intro h
   refine' exists_congr fun b => _

mathlib commit https://github.com/leanprover-community/mathlib/commit/c8f305514e0d47dfaa710f5a52f0d21b588e6328

@@ -8,7 +8,7 @@ import Topology.Algebra.Order.Compact
 import Topology.MetricSpace.EmetricSpace
 import Topology.Bornology.Constructions
 
-#align_import topology.metric_space.basic from "leanprover-community/mathlib"@"8047de4d911cdef39c2d646165eea972f7f9f539"
+#align_import topology.metric_space.basic from "leanprover-community/mathlib"@"c8f305514e0d47dfaa710f5a52f0d21b588e6328"
 
 /-!
 # Metric spaces
@@ -54,7 +54,7 @@ metric, pseudo_metric, dist
 
 open Set Filter TopologicalSpace Bornology
 
-open scoped uniformity Topology BigOperators Filter NNReal ENNReal
+open scoped uniformity Topology BigOperators Filter NNReal ENNReal Pointwise
 
 universe u v w
 
@@ -3189,6 +3189,13 @@ theorem isBounded_empty : IsBounded (∅ : Set α) :=
 #align metric.bounded_empty Bornology.isBounded_empty
 -/
 
+theorem nonempty_of_unbounded (h : ¬IsBounded s) : s.Nonempty :=
+  by
+  rw [nonempty_iff_ne_empty]
+  rintro rfl
+  exact h bounded_empty
+#align metric.nonempty_of_unbounded Metric.nonempty_of_unbounded
+
 #print Bornology.isBounded_iff_forall_mem /-
 theorem isBounded_iff_forall_mem : IsBounded s ↔ ∀ x ∈ s, IsBounded s :=
   ⟨fun h _ _ => h, fun H =>
@@ -3616,6 +3623,12 @@ theorem diam_singleton : diam ({x} : Set α) = 0 :=
 #align metric.diam_singleton Metric.diam_singleton
 -/
 
+@[simp, to_additive]
+theorem diam_one [One α] : diam (1 : Set α) = 0 :=
+  diam_singleton
+#align metric.diam_one Metric.diam_one
+#align metric.diam_zero Metric.diam_zero
+
 #print Metric.diam_pair /-
 -- Does not work as a simp-lemma, since {x, y} reduces to (insert y {x})
 theorem diam_pair : diam ({x, y} : Set α) = dist x y := by

mathlib commit https://github.com/leanprover-community/mathlib/commit/ce64cd319bb6b3e82f31c2d38e79080d377be451

@@ -3866,41 +3866,39 @@ unsafe def positivity_diam : expr → tactic strictness
 
 end Tactic
 
-#print comap_dist_right_atTop_le_cocompact /-
-theorem comap_dist_right_atTop_le_cocompact (x : α) :
-    comap (fun y => dist y x) atTop ≤ cocompact α :=
+theorem Metric.cobounded_le_cocompact (x : α) : comap (fun y => dist y x) atTop ≤ cocompact α :=
   by
   refine' filter.has_basis_cocompact.ge_iff.2 fun s hs => mem_comap.2 _
   rcases hs.bounded.subset_ball x with ⟨r, hr⟩
   exact ⟨Ioi r, Ioi_mem_at_top r, fun y hy hys => (mem_closed_ball.1 <| hr hys).not_lt hy⟩
-#align comap_dist_right_at_top_le_cocompact comap_dist_right_atTop_le_cocompact
--/
+#align comap_dist_right_at_top_le_cocompact Metric.cobounded_le_cocompactₓ
 
-#print comap_dist_left_atTop_le_cocompact /-
-theorem comap_dist_left_atTop_le_cocompact (x : α) : comap (dist x) atTop ≤ cocompact α := by
-  simpa only [dist_comm _ x] using comap_dist_right_atTop_le_cocompact x
-#align comap_dist_left_at_top_le_cocompact comap_dist_left_atTop_le_cocompact
+/- warning: comap_dist_left_at_top_le_cocompact clashes with comap_dist_right_at_top_le_cocompact -> Metric.cobounded_le_cocompactₓ
+Case conversion may be inaccurate. Consider using '#align comap_dist_left_at_top_le_cocompact Metric.cobounded_le_cocompactₓₓ'. -/
+#print Metric.cobounded_le_cocompactₓ /-
+theorem Metric.cobounded_le_cocompact (x : α) : comap (dist x) atTop ≤ cocompact α := by
+  simpa only [dist_comm _ x] using Metric.cobounded_le_cocompact x
+#align comap_dist_left_at_top_le_cocompact Metric.cobounded_le_cocompactₓ
 -/
 
-#print comap_dist_right_atTop_eq_cocompact /-
-theorem comap_dist_right_atTop_eq_cocompact [ProperSpace α] (x : α) :
+#print Metric.cobounded_eq_cocompact /-
+theorem Metric.cobounded_eq_cocompact [ProperSpace α] (x : α) :
     comap (fun y => dist y x) atTop = cocompact α :=
-  (comap_dist_right_atTop_le_cocompact x).antisymm <|
-    (tendsto_dist_right_cocompact_atTop x).le_comap
-#align comap_dist_right_at_top_eq_cocompact comap_dist_right_atTop_eq_cocompact
+  (Metric.cobounded_le_cocompact x).antisymm <| (tendsto_dist_right_cocompact_atTop x).le_comap
+#align comap_dist_right_at_top_eq_cocompact Metric.cobounded_eq_cocompact
 -/
 
 #print comap_dist_left_atTop_eq_cocompact /-
 theorem comap_dist_left_atTop_eq_cocompact [ProperSpace α] (x : α) :
     comap (dist x) atTop = cocompact α :=
-  (comap_dist_left_atTop_le_cocompact x).antisymm <| (tendsto_dist_left_cocompact_atTop x).le_comap
+  (Metric.cobounded_le_cocompact x).antisymm <| (tendsto_dist_left_cocompact_atTop x).le_comap
 #align comap_dist_left_at_top_eq_cocompact comap_dist_left_atTop_eq_cocompact
 -/
 
 #print tendsto_cocompact_of_tendsto_dist_comp_atTop /-
 theorem tendsto_cocompact_of_tendsto_dist_comp_atTop {f : β → α} {l : Filter β} (x : α)
     (h : Tendsto (fun y => dist (f y) x) l atTop) : Tendsto f l (cocompact α) := by
-  refine' tendsto.mono_right _ (comap_dist_right_atTop_le_cocompact x); rwa [tendsto_comap_iff]
+  refine' tendsto.mono_right _ (Metric.cobounded_le_cocompact x); rwa [tendsto_comap_iff]
 #align tendsto_cocompact_of_tendsto_dist_comp_at_top tendsto_cocompact_of_tendsto_dist_comp_atTop
 -/
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/ce64cd319bb6b3e82f31c2d38e79080d377be451

@@ -2870,7 +2870,7 @@ theorem sphere_pi (x : ∀ b, π b) {r : ℝ} (h : 0 < r ∨ Nonempty β) :
   by
   obtain hr | rfl | hr := lt_trichotomy r 0
   · simp [hr]
-  · rw [closed_ball_eq_sphere_of_nonpos le_rfl, eq_comm, Set.inter_eq_right_iff_subset]
+  · rw [closed_ball_eq_sphere_of_nonpos le_rfl, eq_comm, Set.inter_eq_right]
     letI := h.resolve_left (lt_irrefl _)
     inhabit β
     refine' subset_Union_of_subset default _

mathlib commit https://github.com/leanprover-community/mathlib/commit/ce64cd319bb6b3e82f31c2d38e79080d377be451

@@ -3,10 +3,10 @@ Copyright (c) 2015, 2017 Jeremy Avigad. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Jeremy Avigad, Robert Y. Lewis, Johannes Hölzl, Mario Carneiro, Sébastien Gouëzel
 -/
-import Mathbin.Tactic.Positivity
-import Mathbin.Topology.Algebra.Order.Compact
-import Mathbin.Topology.MetricSpace.EmetricSpace
-import Mathbin.Topology.Bornology.Constructions
+import Tactic.Positivity
+import Topology.Algebra.Order.Compact
+import Topology.MetricSpace.EmetricSpace
+import Topology.Bornology.Constructions
 
 #align_import topology.metric_space.basic from "leanprover-community/mathlib"@"8047de4d911cdef39c2d646165eea972f7f9f539"
 
@@ -128,7 +128,7 @@ private theorem pseudo_metric_space.dist_nonneg' {α} {x y : α} (dist : α →
       _ ≥ 0 := by rw [← dist_self x] <;> apply dist_triangle
   nonneg_of_mul_nonneg_right this zero_lt_two
 
-/- ./././Mathport/Syntax/Translate/Expr.lean:336:4: warning: unsupported (TODO): `[tacs] -/
+/- ./././Mathport/Syntax/Translate/Expr.lean:337:4: warning: unsupported (TODO): `[tacs] -/
 /-- This tactic is used to populate `pseudo_metric_space.edist_dist` when the default `edist` is
 used. -/
 protected unsafe def pseudo_metric_space.edist_dist_tac : tactic Unit :=
@@ -1125,7 +1125,7 @@ theorem uniformContinuous_iff [PseudoMetricSpace β] {f : α → β} :
 #align metric.uniform_continuous_iff Metric.uniformContinuous_iff
 -/
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (x y «expr ∈ » s) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:641:2: warning: expanding binder collection (x y «expr ∈ » s) -/
 #print Metric.uniformContinuousOn_iff /-
 theorem uniformContinuousOn_iff [PseudoMetricSpace β] {f : α → β} {s : Set α} :
     UniformContinuousOn f s ↔
@@ -1134,7 +1134,7 @@ theorem uniformContinuousOn_iff [PseudoMetricSpace β] {f : α → β} {s : Set
 #align metric.uniform_continuous_on_iff Metric.uniformContinuousOn_iff
 -/
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (x y «expr ∈ » s) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:641:2: warning: expanding binder collection (x y «expr ∈ » s) -/
 #print Metric.uniformContinuousOn_iff_le /-
 theorem uniformContinuousOn_iff_le [PseudoMetricSpace β] {f : α → β} {s : Set α} :
     UniformContinuousOn f s ↔
@@ -1201,7 +1201,7 @@ theorem totallyBounded_of_finite_discretization {s : Set α}
 #align metric.totally_bounded_of_finite_discretization Metric.totallyBounded_of_finite_discretization
 -/
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (t «expr ⊆ » s) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:641:2: warning: expanding binder collection (t «expr ⊆ » s) -/
 #print Metric.finite_approx_of_totallyBounded /-
 theorem finite_approx_of_totallyBounded {s : Set α} (hs : TotallyBounded s) :
     ∀ ε > 0, ∃ (t : _) (_ : t ⊆ s), Set.Finite t ∧ s ⊆ ⋃ y ∈ t, ball y ε :=
@@ -1270,7 +1270,7 @@ theorem tendstoUniformly_iff {ι : Type _} {F : ι → β → α} {f : β → α
 #align metric.tendsto_uniformly_iff Metric.tendstoUniformly_iff
 -/
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (x y «expr ∈ » t) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:641:2: warning: expanding binder collection (x y «expr ∈ » t) -/
 #print Metric.cauchy_iff /-
 protected theorem cauchy_iff {f : Filter α} :
     Cauchy f ↔ NeBot f ∧ ∀ ε > 0, ∃ t ∈ f, ∀ (x) (_ : x ∈ t) (y) (_ : y ∈ t), dist x y < ε :=
@@ -2002,7 +2002,7 @@ section CauchySeq
 
 variable [Nonempty β] [SemilatticeSup β]
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (m n «expr ≥ » N) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:641:2: warning: expanding binder collection (m n «expr ≥ » N) -/
 #print Metric.cauchySeq_iff /-
 -- see Note [nolint_ge]
 /-- In a pseudometric space, Cauchy sequences are characterized by the fact that, eventually,
@@ -2917,7 +2917,7 @@ end Pi
 
 section Compact
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (t «expr ⊆ » s) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:641:2: warning: expanding binder collection (t «expr ⊆ » s) -/
 #print finite_cover_balls_of_compact /-
 /-- Any compact set in a pseudometric space can be covered by finitely many balls of a given
 positive radius -/
@@ -2955,7 +2955,7 @@ export ProperSpace (isCompact_closedBall)
 /-- In a proper pseudometric space, all spheres are compact. -/
 theorem isCompact_sphere {α : Type _} [PseudoMetricSpace α] [ProperSpace α] (x : α) (r : ℝ) :
     IsCompact (sphere x r) :=
-  isCompact_of_isClosed_subset (isCompact_closedBall x r) isClosed_sphere sphere_subset_closedBall
+  IsCompact.of_isClosed_subset (isCompact_closedBall x r) isClosed_sphere sphere_subset_closedBall
 #align is_compact_sphere isCompact_sphere
 -/
 
@@ -3029,7 +3029,7 @@ instance (priority := 100) locally_compact_of_proper [ProperSpace α] : LocallyC
 #align locally_compact_of_proper locally_compact_of_proper
 -/
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (x y «expr ∈ » t) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:641:2: warning: expanding binder collection (x y «expr ∈ » t) -/
 #print complete_of_proper /-
 -- see Note [lower instance priority]
 /-- A proper space is complete -/
@@ -3083,7 +3083,7 @@ theorem exists_pos_lt_subset_ball (hr : 0 < r) (hs : IsClosed s) (h : s ⊆ ball
   rcases eq_empty_or_nonempty s with (rfl | hne)
   · exact ⟨r / 2, ⟨half_pos hr, half_lt_self hr⟩, empty_subset _⟩
   have : IsCompact s :=
-    isCompact_of_isClosed_subset (is_compact_closed_ball x r) hs
+    IsCompact.of_isClosed_subset (is_compact_closed_ball x r) hs
       (subset.trans h ball_subset_closed_ball)
   obtain ⟨y, hys, hy⟩ : ∃ y ∈ s, s ⊆ closed_ball x (dist y x)
   exact this.exists_forall_ge hne (continuous_id.dist continuous_const).ContinuousOn
@@ -3161,7 +3161,7 @@ namespace Metric
 
 /- warning: metric.bounded clashes with bornology.is_bounded -> Bornology.IsBounded
 Case conversion may be inaccurate. Consider using '#align metric.bounded Bornology.IsBoundedₓ'. -/
-/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (x y «expr ∈ » s) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:641:2: warning: expanding binder collection (x y «expr ∈ » s) -/
 #print Bornology.IsBounded /-
 /-- Boundedness of a subset of a pseudometric space. We formulate the definition to work
 even in the empty space. -/
@@ -3505,7 +3505,7 @@ theorem isCompact_of_isClosed_isBounded [ProperSpace α] (hc : IsClosed s) (hb :
   rcases eq_empty_or_nonempty s with (rfl | ⟨x, hx⟩)
   · exact isCompact_empty
   · rcases hb.subset_ball x with ⟨r, hr⟩
-    exact isCompact_of_isClosed_subset (is_compact_closed_ball x r) hc hr
+    exact IsCompact.of_isClosed_subset (is_compact_closed_ball x r) hc hr
 #align metric.is_compact_of_is_closed_bounded Metric.isCompact_of_isClosed_isBounded
 -/
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/ce64cd319bb6b3e82f31c2d38e79080d377be451

@@ -3159,76 +3159,80 @@ theorem lebesgue_number_lemma_of_metric_sUnion {s : Set α} {c : Set (Set α)} (
 
 namespace Metric
 
+/- warning: metric.bounded clashes with bornology.is_bounded -> Bornology.IsBounded
+Case conversion may be inaccurate. Consider using '#align metric.bounded Bornology.IsBoundedₓ'. -/
 /- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (x y «expr ∈ » s) -/
-#print Metric.Bounded /-
+#print Bornology.IsBounded /-
 /-- Boundedness of a subset of a pseudometric space. We formulate the definition to work
 even in the empty space. -/
-def Bounded (s : Set α) : Prop :=
+def IsBounded (s : Set α) : Prop :=
   ∃ C, ∀ (x) (_ : x ∈ s) (y) (_ : y ∈ s), dist x y ≤ C
-#align metric.bounded Metric.Bounded
+#align metric.bounded Bornology.IsBounded
 -/
 
 section Bounded
 
 variable {x : α} {s t : Set α} {r : ℝ}
 
-#print Metric.bounded_iff_isBounded /-
-theorem bounded_iff_isBounded (s : Set α) : Bounded s ↔ IsBounded s :=
+theorem isBounded_iff_isBounded (s : Set α) : IsBounded s ↔ IsBounded s :=
   by
   change bounded s ↔ sᶜ ∈ (cobounded α).sets
-  simp [PseudoMetricSpace.cobounded_sets, Metric.Bounded]
-#align metric.bounded_iff_is_bounded Metric.bounded_iff_isBounded
--/
+  simp [PseudoMetricSpace.cobounded_sets, Bornology.IsBounded]
+#align metric.bounded_iff_is_bounded Metric.isBounded_iff_isBounded
 
-#print Metric.bounded_empty /-
+/- warning: metric.bounded_empty clashes with bornology.is_bounded_empty -> Bornology.isBounded_empty
+Case conversion may be inaccurate. Consider using '#align metric.bounded_empty Bornology.isBounded_emptyₓ'. -/
+#print Bornology.isBounded_empty /-
 @[simp]
-theorem bounded_empty : Bounded (∅ : Set α) :=
+theorem isBounded_empty : IsBounded (∅ : Set α) :=
   ⟨0, by simp⟩
-#align metric.bounded_empty Metric.bounded_empty
+#align metric.bounded_empty Bornology.isBounded_empty
 -/
 
-#print Metric.bounded_iff_mem_bounded /-
-theorem bounded_iff_mem_bounded : Bounded s ↔ ∀ x ∈ s, Bounded s :=
+#print Bornology.isBounded_iff_forall_mem /-
+theorem isBounded_iff_forall_mem : IsBounded s ↔ ∀ x ∈ s, IsBounded s :=
   ⟨fun h _ _ => h, fun H =>
-    s.eq_empty_or_nonempty.elim (fun hs => hs.symm ▸ bounded_empty) fun ⟨x, hx⟩ => H x hx⟩
-#align metric.bounded_iff_mem_bounded Metric.bounded_iff_mem_bounded
+    s.eq_empty_or_nonempty.elim (fun hs => hs.symm ▸ isBounded_empty) fun ⟨x, hx⟩ => H x hx⟩
+#align metric.bounded_iff_mem_bounded Bornology.isBounded_iff_forall_mem
 -/
 
-#print Metric.Bounded.mono /-
+/- warning: metric.bounded.mono clashes with bornology.is_bounded.subset -> Bornology.IsBounded.subset
+Case conversion may be inaccurate. Consider using '#align metric.bounded.mono Bornology.IsBounded.subsetₓ'. -/
+#print Bornology.IsBounded.subset /-
 /-- Subsets of a bounded set are also bounded -/
-theorem Bounded.mono (incl : s ⊆ t) : Bounded t → Bounded s :=
+theorem IsBounded.subset (incl : s ⊆ t) : IsBounded t → IsBounded s :=
   Exists.imp fun C hC x hx y hy => hC x (incl hx) y (incl hy)
-#align metric.bounded.mono Metric.Bounded.mono
+#align metric.bounded.mono Bornology.IsBounded.subset
 -/
 
-#print Metric.bounded_closedBall /-
+#print Metric.isBounded_closedBall /-
 /-- Closed balls are bounded -/
-theorem bounded_closedBall : Bounded (closedBall x r) :=
+theorem isBounded_closedBall : IsBounded (closedBall x r) :=
   ⟨r + r, fun y hy z hz => by
     simp only [mem_closed_ball] at *
     calc
       dist y z ≤ dist y x + dist z x := dist_triangle_right _ _ _
       _ ≤ r + r := add_le_add hy hz⟩
-#align metric.bounded_closed_ball Metric.bounded_closedBall
+#align metric.bounded_closed_ball Metric.isBounded_closedBall
 -/
 
-#print Metric.bounded_ball /-
+#print Metric.isBounded_ball /-
 /-- Open balls are bounded -/
-theorem bounded_ball : Bounded (ball x r) :=
-  bounded_closedBall.mono ball_subset_closedBall
-#align metric.bounded_ball Metric.bounded_ball
+theorem isBounded_ball : IsBounded (ball x r) :=
+  isBounded_closedBall.mono ball_subset_closedBall
+#align metric.bounded_ball Metric.isBounded_ball
 -/
 
-#print Metric.bounded_sphere /-
+#print Metric.isBounded_sphere /-
 /-- Spheres are bounded -/
-theorem bounded_sphere : Bounded (sphere x r) :=
-  bounded_closedBall.mono sphere_subset_closedBall
-#align metric.bounded_sphere Metric.bounded_sphere
+theorem isBounded_sphere : IsBounded (sphere x r) :=
+  isBounded_closedBall.mono sphere_subset_closedBall
+#align metric.bounded_sphere Metric.isBounded_sphere
 -/
 
-#print Metric.bounded_iff_subset_ball /-
+#print Metric.isBounded_iff_subset_closedBall /-
 /-- Given a point, a bounded subset is included in some ball around this point -/
-theorem bounded_iff_subset_ball (c : α) : Bounded s ↔ ∃ r, s ⊆ closedBall c r :=
+theorem isBounded_iff_subset_closedBall (c : α) : IsBounded s ↔ ∃ r, s ⊆ closedBall c r :=
   by
   constructor <;> rintro ⟨C, hC⟩
   · cases' s.eq_empty_or_nonempty with h h
@@ -3240,45 +3244,48 @@ theorem bounded_iff_subset_ball (c : α) : Bounded s ↔ ∃ r, s ⊆ closedBall
             dist y c ≤ dist y x + dist x c := dist_triangle _ _ _
             _ ≤ C + dist x c := add_le_add_right (hC y hy x hx) _⟩
   · exact bounded_closed_ball.mono hC
-#align metric.bounded_iff_subset_ball Metric.bounded_iff_subset_ball
+#align metric.bounded_iff_subset_ball Metric.isBounded_iff_subset_closedBall
 -/
 
-#print Metric.Bounded.subset_ball /-
-theorem Bounded.subset_ball (h : Bounded s) (c : α) : ∃ r, s ⊆ closedBall c r :=
-  (bounded_iff_subset_ball c).1 h
-#align metric.bounded.subset_ball Metric.Bounded.subset_ball
+#print Bornology.IsBounded.subset_closedBall /-
+theorem IsBounded.subset_closedBall (h : IsBounded s) (c : α) : ∃ r, s ⊆ closedBall c r :=
+  (isBounded_iff_subset_closedBall c).1 h
+#align metric.bounded.subset_ball Bornology.IsBounded.subset_closedBall
 -/
 
-#print Metric.Bounded.subset_ball_lt /-
-theorem Bounded.subset_ball_lt (h : Bounded s) (a : ℝ) (c : α) : ∃ r, a < r ∧ s ⊆ closedBall c r :=
+#print Bornology.IsBounded.subset_closedBall_lt /-
+theorem IsBounded.subset_closedBall_lt (h : IsBounded s) (a : ℝ) (c : α) :
+    ∃ r, a < r ∧ s ⊆ closedBall c r :=
   by
   rcases h.subset_ball c with ⟨r, hr⟩
   refine' ⟨max r (a + 1), lt_of_lt_of_le (by linarith) (le_max_right _ _), _⟩
   exact subset.trans hr (closed_ball_subset_closed_ball (le_max_left _ _))
-#align metric.bounded.subset_ball_lt Metric.Bounded.subset_ball_lt
+#align metric.bounded.subset_ball_lt Bornology.IsBounded.subset_closedBall_lt
 -/
 
-#print Metric.bounded_closure_of_bounded /-
-theorem bounded_closure_of_bounded (h : Bounded s) : Bounded (closure s) :=
+#print Metric.isBounded_closure_of_isBounded /-
+theorem isBounded_closure_of_isBounded (h : IsBounded s) : IsBounded (closure s) :=
   let ⟨C, h⟩ := h
   ⟨C, fun a ha b hb =>
     (ClosedIicTopology.isClosed_le' C).closure_subset <| map_mem_closure₂ continuous_dist ha hb h⟩
-#align metric.bounded_closure_of_bounded Metric.bounded_closure_of_bounded
+#align metric.bounded_closure_of_bounded Metric.isBounded_closure_of_isBounded
 -/
 
 alias bounded.closure := bounded_closure_of_bounded
-#align metric.bounded.closure Metric.Bounded.closure
+#align metric.bounded.closure Bornology.IsBounded.closure
 
-#print Metric.bounded_closure_iff /-
+#print Metric.isBounded_closure_iff /-
 @[simp]
-theorem bounded_closure_iff : Bounded (closure s) ↔ Bounded s :=
+theorem isBounded_closure_iff : IsBounded (closure s) ↔ IsBounded s :=
   ⟨fun h => h.mono subset_closure, fun h => h.closure⟩
-#align metric.bounded_closure_iff Metric.bounded_closure_iff
+#align metric.bounded_closure_iff Metric.isBounded_closure_iff
 -/
 
-#print Metric.Bounded.union /-
+/- warning: metric.bounded.union clashes with bornology.is_bounded.union -> Bornology.IsBounded.union
+Case conversion may be inaccurate. Consider using '#align metric.bounded.union Bornology.IsBounded.unionₓ'. -/
+#print Bornology.IsBounded.union /-
 /-- The union of two bounded sets is bounded. -/
-theorem Bounded.union (hs : Bounded s) (ht : Bounded t) : Bounded (s ∪ t) :=
+theorem IsBounded.union (hs : IsBounded s) (ht : IsBounded t) : IsBounded (s ∪ t) :=
   by
   refine' bounded_iff_mem_bounded.2 fun x _ => _
   rw [bounded_iff_subset_ball x] at hs ht ⊢
@@ -3287,30 +3294,36 @@ theorem Bounded.union (hs : Bounded s) (ht : Bounded t) : Bounded (s ∪ t) :=
     ⟨max Cs Ct,
       union_subset (subset.trans hCs <| closed_ball_subset_closed_ball <| le_max_left _ _)
         (subset.trans hCt <| closed_ball_subset_closed_ball <| le_max_right _ _)⟩
-#align metric.bounded.union Metric.Bounded.union
+#align metric.bounded.union Bornology.IsBounded.union
 -/
 
-#print Metric.bounded_union /-
+/- warning: metric.bounded_union clashes with bornology.is_bounded_union -> Bornology.isBounded_union
+Case conversion may be inaccurate. Consider using '#align metric.bounded_union Bornology.isBounded_unionₓ'. -/
+#print Bornology.isBounded_union /-
 /-- The union of two sets is bounded iff each of the sets is bounded. -/
 @[simp]
-theorem bounded_union : Bounded (s ∪ t) ↔ Bounded s ∧ Bounded t :=
+theorem isBounded_union : IsBounded (s ∪ t) ↔ IsBounded s ∧ IsBounded t :=
   ⟨fun h => ⟨h.mono (by simp), h.mono (by simp)⟩, fun h => h.1.union h.2⟩
-#align metric.bounded_union Metric.bounded_union
+#align metric.bounded_union Bornology.isBounded_union
 -/
 
-#print Metric.bounded_biUnion /-
+/- warning: metric.bounded_bUnion clashes with bornology.is_bounded_bUnion -> Bornology.isBounded_biUnion
+Case conversion may be inaccurate. Consider using '#align metric.bounded_bUnion Bornology.isBounded_biUnionₓ'. -/
+#print Bornology.isBounded_biUnion /-
 /-- A finite union of bounded sets is bounded -/
-theorem bounded_biUnion {I : Set β} {s : β → Set α} (H : I.Finite) :
-    Bounded (⋃ i ∈ I, s i) ↔ ∀ i ∈ I, Bounded (s i) :=
+theorem isBounded_biUnion {I : Set β} {s : β → Set α} (H : I.Finite) :
+    IsBounded (⋃ i ∈ I, s i) ↔ ∀ i ∈ I, IsBounded (s i) :=
   Finite.induction_on H (by simp) fun x I _ _ IH => by simp [or_imp, forall_and, IH]
-#align metric.bounded_bUnion Metric.bounded_biUnion
+#align metric.bounded_bUnion Bornology.isBounded_biUnion
 -/
 
+/- warning: metric.bounded.prod clashes with bornology.is_bounded.prod -> Bornology.IsBounded.prod
+Case conversion may be inaccurate. Consider using '#align metric.bounded.prod Bornology.IsBounded.prodₓ'. -/
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
-#print Metric.Bounded.prod /-
-protected theorem Bounded.prod [PseudoMetricSpace β] {s : Set α} {t : Set β} (hs : Bounded s)
-    (ht : Bounded t) : Bounded (s ×ˢ t) :=
+#print Bornology.IsBounded.prod /-
+protected theorem IsBounded.prod [PseudoMetricSpace β] {s : Set α} {t : Set β} (hs : IsBounded s)
+    (ht : IsBounded t) : IsBounded (s ×ˢ t) :=
   by
   refine' bounded_iff_mem_bounded.mpr fun x hx => _
   rcases hs.subset_ball x.1 with ⟨rs, hrs⟩
@@ -3321,56 +3334,62 @@ protected theorem Bounded.prod [PseudoMetricSpace β] {s : Set α} {t : Set β}
   exact
     prod_mono (hrs.trans <| closed_ball_subset_closed_ball <| le_max_left _ _)
       (hrt.trans <| closed_ball_subset_closed_ball <| le_max_right _ _)
-#align metric.bounded.prod Metric.Bounded.prod
+#align metric.bounded.prod Bornology.IsBounded.prod
 -/
 
-#print TotallyBounded.bounded /-
+#print TotallyBounded.isBounded /-
 /-- A totally bounded set is bounded -/
-theorem TotallyBounded.bounded {s : Set α} (h : TotallyBounded s) : Bounded s :=
+theorem TotallyBounded.isBounded {s : Set α} (h : TotallyBounded s) : IsBounded s :=
   let-- We cover the totally bounded set by finitely many balls of radius 1,
     -- and then argue that a finite union of bounded sets is bounded
     ⟨t, fint, subs⟩ :=
     (totallyBounded_iff.mp h) 1 zero_lt_one
-  Bounded.mono subs <| (bounded_biUnion fint).2 fun i hi => bounded_ball
-#align totally_bounded.bounded TotallyBounded.bounded
+  IsBounded.subset subs <| (isBounded_biUnion fint).2 fun i hi => isBounded_ball
+#align totally_bounded.bounded TotallyBounded.isBounded
 -/
 
-#print IsCompact.bounded /-
+#print IsCompact.isBounded /-
 /-- A compact set is bounded -/
-theorem IsCompact.bounded {s : Set α} (h : IsCompact s) : Bounded s :=
+theorem IsCompact.isBounded {s : Set α} (h : IsCompact s) : IsBounded s :=
   -- A compact set is totally bounded, thus bounded
       h.TotallyBounded.Bounded
-#align is_compact.bounded IsCompact.bounded
+#align is_compact.bounded IsCompact.isBounded
 -/
 
-#print Metric.bounded_of_finite /-
+/- warning: metric.bounded_of_finite clashes with set.finite.is_bounded -> Set.Finite.isBounded
+Case conversion may be inaccurate. Consider using '#align metric.bounded_of_finite Set.Finite.isBoundedₓ'. -/
+#print Set.Finite.isBounded /-
 /-- A finite set is bounded -/
-theorem bounded_of_finite {s : Set α} (h : s.Finite) : Bounded s :=
+theorem Set.Finite.isBounded {s : Set α} (h : s.Finite) : IsBounded s :=
   h.IsCompact.Bounded
-#align metric.bounded_of_finite Metric.bounded_of_finite
+#align metric.bounded_of_finite Set.Finite.isBounded
 -/
 
+/- warning: set.finite.bounded clashes with set.finite.is_bounded -> Set.Finite.isBounded
+Case conversion may be inaccurate. Consider using '#align set.finite.bounded Set.Finite.isBoundedₓ'. -/
 alias _root_.set.finite.bounded := bounded_of_finite
-#align set.finite.bounded Set.Finite.bounded
+#align set.finite.bounded Set.Finite.isBounded
 
-#print Metric.bounded_singleton /-
+/- warning: metric.bounded_singleton clashes with bornology.is_bounded_singleton -> Bornology.isBounded_singleton
+Case conversion may be inaccurate. Consider using '#align metric.bounded_singleton Bornology.isBounded_singletonₓ'. -/
+#print Bornology.isBounded_singleton /-
 /-- A singleton is bounded -/
-theorem bounded_singleton {x : α} : Bounded ({x} : Set α) :=
-  bounded_of_finite <| finite_singleton _
-#align metric.bounded_singleton Metric.bounded_singleton
+theorem isBounded_singleton {x : α} : IsBounded ({x} : Set α) :=
+  Set.Finite.isBounded <| finite_singleton _
+#align metric.bounded_singleton Bornology.isBounded_singleton
 -/
 
-#print Metric.bounded_range_iff /-
+#print Metric.isBounded_range_iff /-
 /-- Characterization of the boundedness of the range of a function -/
-theorem bounded_range_iff {f : β → α} : Bounded (range f) ↔ ∃ C, ∀ x y, dist (f x) (f y) ≤ C :=
+theorem isBounded_range_iff {f : β → α} : IsBounded (range f) ↔ ∃ C, ∀ x y, dist (f x) (f y) ≤ C :=
   exists_congr fun C =>
     ⟨fun H x y => H _ ⟨x, rfl⟩ _ ⟨y, rfl⟩, by rintro H _ ⟨x, rfl⟩ _ ⟨y, rfl⟩ <;> exact H x y⟩
-#align metric.bounded_range_iff Metric.bounded_range_iff
+#align metric.bounded_range_iff Metric.isBounded_range_iff
 -/
 
-#print Metric.bounded_range_of_tendsto_cofinite_uniformity /-
-theorem bounded_range_of_tendsto_cofinite_uniformity {f : β → α}
-    (hf : Tendsto (Prod.map f f) (cofinite ×ᶠ cofinite) (𝓤 α)) : Bounded (range f) :=
+#print Metric.isBounded_range_of_tendsto_cofinite_uniformity /-
+theorem isBounded_range_of_tendsto_cofinite_uniformity {f : β → α}
+    (hf : Tendsto (Prod.map f f) (cofinite ×ᶠ cofinite) (𝓤 α)) : IsBounded (range f) :=
   by
   rcases(has_basis_cofinite.prod_self.tendsto_iff uniformity_basis_dist).1 hf 1 zero_lt_one with
     ⟨s, hsf, hs1⟩
@@ -3378,50 +3397,50 @@ theorem bounded_range_of_tendsto_cofinite_uniformity {f : β → α}
   use(hsf.image f).Bounded, 1
   rintro _ ⟨x, hx, rfl⟩ _ ⟨y, hy, rfl⟩
   exact le_of_lt (hs1 (x, y) ⟨hx, hy⟩)
-#align metric.bounded_range_of_tendsto_cofinite_uniformity Metric.bounded_range_of_tendsto_cofinite_uniformity
+#align metric.bounded_range_of_tendsto_cofinite_uniformity Metric.isBounded_range_of_tendsto_cofinite_uniformity
 -/
 
-#print Metric.bounded_range_of_cauchy_map_cofinite /-
-theorem bounded_range_of_cauchy_map_cofinite {f : β → α} (hf : Cauchy (map f cofinite)) :
-    Bounded (range f) :=
-  bounded_range_of_tendsto_cofinite_uniformity <| (cauchy_map_iff.1 hf).2
-#align metric.bounded_range_of_cauchy_map_cofinite Metric.bounded_range_of_cauchy_map_cofinite
+#print Metric.isBounded_range_of_cauchy_map_cofinite /-
+theorem isBounded_range_of_cauchy_map_cofinite {f : β → α} (hf : Cauchy (map f cofinite)) :
+    IsBounded (range f) :=
+  isBounded_range_of_tendsto_cofinite_uniformity <| (cauchy_map_iff.1 hf).2
+#align metric.bounded_range_of_cauchy_map_cofinite Metric.isBounded_range_of_cauchy_map_cofinite
 -/
 
-#print CauchySeq.bounded_range /-
-theorem CauchySeq.bounded_range {f : ℕ → α} (hf : CauchySeq f) : Bounded (range f) :=
-  bounded_range_of_cauchy_map_cofinite <| by rwa [Nat.cofinite_eq_atTop]
-#align cauchy_seq.bounded_range CauchySeq.bounded_range
+#print CauchySeq.isBounded_range /-
+theorem CauchySeq.isBounded_range {f : ℕ → α} (hf : CauchySeq f) : IsBounded (range f) :=
+  isBounded_range_of_cauchy_map_cofinite <| by rwa [Nat.cofinite_eq_atTop]
+#align cauchy_seq.bounded_range CauchySeq.isBounded_range
 -/
 
-#print Metric.bounded_range_of_tendsto_cofinite /-
-theorem bounded_range_of_tendsto_cofinite {f : β → α} {a : α} (hf : Tendsto f cofinite (𝓝 a)) :
-    Bounded (range f) :=
-  bounded_range_of_tendsto_cofinite_uniformity <|
+#print Metric.isBounded_range_of_tendsto_cofinite /-
+theorem isBounded_range_of_tendsto_cofinite {f : β → α} {a : α} (hf : Tendsto f cofinite (𝓝 a)) :
+    IsBounded (range f) :=
+  isBounded_range_of_tendsto_cofinite_uniformity <|
     (hf.Prod_map hf).mono_right <| nhds_prod_eq.symm.trans_le (nhds_le_uniformity a)
-#align metric.bounded_range_of_tendsto_cofinite Metric.bounded_range_of_tendsto_cofinite
+#align metric.bounded_range_of_tendsto_cofinite Metric.isBounded_range_of_tendsto_cofinite
 -/
 
-#print Metric.bounded_of_compactSpace /-
+#print Metric.isBounded_of_compactSpace /-
 /-- In a compact space, all sets are bounded -/
-theorem bounded_of_compactSpace [CompactSpace α] : Bounded s :=
+theorem isBounded_of_compactSpace [CompactSpace α] : IsBounded s :=
   isCompact_univ.Bounded.mono (subset_univ _)
-#align metric.bounded_of_compact_space Metric.bounded_of_compactSpace
+#align metric.bounded_of_compact_space Metric.isBounded_of_compactSpace
 -/
 
-#print Metric.bounded_range_of_tendsto /-
-theorem bounded_range_of_tendsto (u : ℕ → α) {x : α} (hu : Tendsto u atTop (𝓝 x)) :
-    Bounded (range u) :=
-  hu.CauchySeq.bounded_range
-#align metric.bounded_range_of_tendsto Metric.bounded_range_of_tendsto
+#print Metric.isBounded_range_of_tendsto /-
+theorem isBounded_range_of_tendsto (u : ℕ → α) {x : α} (hu : Tendsto u atTop (𝓝 x)) :
+    IsBounded (range u) :=
+  hu.CauchySeq.isBounded_range
+#align metric.bounded_range_of_tendsto Metric.isBounded_range_of_tendsto
 -/
 
-#print Metric.exists_isOpen_bounded_image_inter_of_isCompact_of_forall_continuousWithinAt /-
+#print Metric.exists_isOpen_isBounded_image_inter_of_isCompact_of_forall_continuousWithinAt /-
 /-- If a function is continuous within a set `s` at every point of a compact set `k`, then it is
 bounded on some open neighborhood of `k` in `s`. -/
-theorem exists_isOpen_bounded_image_inter_of_isCompact_of_forall_continuousWithinAt
+theorem exists_isOpen_isBounded_image_inter_of_isCompact_of_forall_continuousWithinAt
     [TopologicalSpace β] {k s : Set β} {f : β → α} (hk : IsCompact k)
-    (hf : ∀ x ∈ k, ContinuousWithinAt f s x) : ∃ t, k ⊆ t ∧ IsOpen t ∧ Bounded (f '' (t ∩ s)) :=
+    (hf : ∀ x ∈ k, ContinuousWithinAt f s x) : ∃ t, k ⊆ t ∧ IsOpen t ∧ IsBounded (f '' (t ∩ s)) :=
   by
   apply hk.induction_on
   · exact ⟨∅, subset.refl _, isOpen_empty, by simp only [image_empty, bounded_empty, empty_inter]⟩
@@ -3441,114 +3460,115 @@ theorem exists_isOpen_bounded_image_inter_of_isCompact_of_forall_continuousWithi
       exact u_open.mem_nhds xu
     · apply bounded.mono (image_subset _ uf)
       exact bounded_ball.mono (image_preimage_subset _ _)
-#align metric.exists_is_open_bounded_image_inter_of_is_compact_of_forall_continuous_within_at Metric.exists_isOpen_bounded_image_inter_of_isCompact_of_forall_continuousWithinAt
+#align metric.exists_is_open_bounded_image_inter_of_is_compact_of_forall_continuous_within_at Metric.exists_isOpen_isBounded_image_inter_of_isCompact_of_forall_continuousWithinAt
 -/
 
-#print Metric.exists_isOpen_bounded_image_of_isCompact_of_forall_continuousAt /-
+#print Metric.exists_isOpen_isBounded_image_of_isCompact_of_forall_continuousAt /-
 /-- If a function is continuous at every point of a compact set `k`, then it is bounded on
 some open neighborhood of `k`. -/
-theorem exists_isOpen_bounded_image_of_isCompact_of_forall_continuousAt [TopologicalSpace β]
+theorem exists_isOpen_isBounded_image_of_isCompact_of_forall_continuousAt [TopologicalSpace β]
     {k : Set β} {f : β → α} (hk : IsCompact k) (hf : ∀ x ∈ k, ContinuousAt f x) :
-    ∃ t, k ⊆ t ∧ IsOpen t ∧ Bounded (f '' t) :=
+    ∃ t, k ⊆ t ∧ IsOpen t ∧ IsBounded (f '' t) :=
   by
   simp_rw [← continuousWithinAt_univ] at hf 
   simpa only [inter_univ] using
     exists_is_open_bounded_image_inter_of_is_compact_of_forall_continuous_within_at hk hf
-#align metric.exists_is_open_bounded_image_of_is_compact_of_forall_continuous_at Metric.exists_isOpen_bounded_image_of_isCompact_of_forall_continuousAt
+#align metric.exists_is_open_bounded_image_of_is_compact_of_forall_continuous_at Metric.exists_isOpen_isBounded_image_of_isCompact_of_forall_continuousAt
 -/
 
-#print Metric.exists_isOpen_bounded_image_inter_of_isCompact_of_continuousOn /-
+#print Metric.exists_isOpen_isBounded_image_inter_of_isCompact_of_continuousOn /-
 /-- If a function is continuous on a set `s` containing a compact set `k`, then it is bounded on
 some open neighborhood of `k` in `s`. -/
-theorem exists_isOpen_bounded_image_inter_of_isCompact_of_continuousOn [TopologicalSpace β]
+theorem exists_isOpen_isBounded_image_inter_of_isCompact_of_continuousOn [TopologicalSpace β]
     {k s : Set β} {f : β → α} (hk : IsCompact k) (hks : k ⊆ s) (hf : ContinuousOn f s) :
-    ∃ t, k ⊆ t ∧ IsOpen t ∧ Bounded (f '' (t ∩ s)) :=
-  exists_isOpen_bounded_image_inter_of_isCompact_of_forall_continuousWithinAt hk fun x hx =>
+    ∃ t, k ⊆ t ∧ IsOpen t ∧ IsBounded (f '' (t ∩ s)) :=
+  exists_isOpen_isBounded_image_inter_of_isCompact_of_forall_continuousWithinAt hk fun x hx =>
     hf x (hks hx)
-#align metric.exists_is_open_bounded_image_inter_of_is_compact_of_continuous_on Metric.exists_isOpen_bounded_image_inter_of_isCompact_of_continuousOn
+#align metric.exists_is_open_bounded_image_inter_of_is_compact_of_continuous_on Metric.exists_isOpen_isBounded_image_inter_of_isCompact_of_continuousOn
 -/
 
-#print Metric.exists_isOpen_bounded_image_of_isCompact_of_continuousOn /-
+#print Metric.exists_isOpen_isBounded_image_of_isCompact_of_continuousOn /-
 /-- If a function is continuous on a neighborhood of a compact set `k`, then it is bounded on
 some open neighborhood of `k`. -/
-theorem exists_isOpen_bounded_image_of_isCompact_of_continuousOn [TopologicalSpace β] {k s : Set β}
-    {f : β → α} (hk : IsCompact k) (hs : IsOpen s) (hks : k ⊆ s) (hf : ContinuousOn f s) :
-    ∃ t, k ⊆ t ∧ IsOpen t ∧ Bounded (f '' t) :=
-  exists_isOpen_bounded_image_of_isCompact_of_forall_continuousAt hk fun x hx =>
+theorem exists_isOpen_isBounded_image_of_isCompact_of_continuousOn [TopologicalSpace β]
+    {k s : Set β} {f : β → α} (hk : IsCompact k) (hs : IsOpen s) (hks : k ⊆ s)
+    (hf : ContinuousOn f s) : ∃ t, k ⊆ t ∧ IsOpen t ∧ IsBounded (f '' t) :=
+  exists_isOpen_isBounded_image_of_isCompact_of_forall_continuousAt hk fun x hx =>
     hf.ContinuousAt (hs.mem_nhds (hks hx))
-#align metric.exists_is_open_bounded_image_of_is_compact_of_continuous_on Metric.exists_isOpen_bounded_image_of_isCompact_of_continuousOn
+#align metric.exists_is_open_bounded_image_of_is_compact_of_continuous_on Metric.exists_isOpen_isBounded_image_of_isCompact_of_continuousOn
 -/
 
-#print Metric.isCompact_of_isClosed_bounded /-
+#print Metric.isCompact_of_isClosed_isBounded /-
 /-- The **Heine–Borel theorem**: In a proper space, a closed bounded set is compact. -/
-theorem isCompact_of_isClosed_bounded [ProperSpace α] (hc : IsClosed s) (hb : Bounded s) :
+theorem isCompact_of_isClosed_isBounded [ProperSpace α] (hc : IsClosed s) (hb : IsBounded s) :
     IsCompact s := by
   rcases eq_empty_or_nonempty s with (rfl | ⟨x, hx⟩)
   · exact isCompact_empty
   · rcases hb.subset_ball x with ⟨r, hr⟩
     exact isCompact_of_isClosed_subset (is_compact_closed_ball x r) hc hr
-#align metric.is_compact_of_is_closed_bounded Metric.isCompact_of_isClosed_bounded
+#align metric.is_compact_of_is_closed_bounded Metric.isCompact_of_isClosed_isBounded
 -/
 
-#print Metric.Bounded.isCompact_closure /-
+#print Bornology.IsBounded.isCompact_closure /-
 /-- The **Heine–Borel theorem**: In a proper space, the closure of a bounded set is compact. -/
-theorem Bounded.isCompact_closure [ProperSpace α] (h : Bounded s) : IsCompact (closure s) :=
-  isCompact_of_isClosed_bounded isClosed_closure h.closure
-#align metric.bounded.is_compact_closure Metric.Bounded.isCompact_closure
+theorem IsBounded.isCompact_closure [ProperSpace α] (h : IsBounded s) : IsCompact (closure s) :=
+  isCompact_of_isClosed_isBounded isClosed_closure h.closure
+#align metric.bounded.is_compact_closure Bornology.IsBounded.isCompact_closure
 -/
 
 #print Metric.isCompact_iff_isClosed_bounded /-
 /-- The **Heine–Borel theorem**:
 In a proper Hausdorff space, a set is compact if and only if it is closed and bounded. -/
 theorem isCompact_iff_isClosed_bounded [T2Space α] [ProperSpace α] :
-    IsCompact s ↔ IsClosed s ∧ Bounded s :=
-  ⟨fun h => ⟨h.IsClosed, h.Bounded⟩, fun h => isCompact_of_isClosed_bounded h.1 h.2⟩
+    IsCompact s ↔ IsClosed s ∧ IsBounded s :=
+  ⟨fun h => ⟨h.IsClosed, h.Bounded⟩, fun h => isCompact_of_isClosed_isBounded h.1 h.2⟩
 #align metric.is_compact_iff_is_closed_bounded Metric.isCompact_iff_isClosed_bounded
 -/
 
-#print Metric.compactSpace_iff_bounded_univ /-
-theorem compactSpace_iff_bounded_univ [ProperSpace α] : CompactSpace α ↔ Bounded (univ : Set α) :=
-  ⟨@bounded_of_compactSpace α _ _, fun hb => ⟨isCompact_of_isClosed_bounded isClosed_univ hb⟩⟩
-#align metric.compact_space_iff_bounded_univ Metric.compactSpace_iff_bounded_univ
+#print Metric.compactSpace_iff_isBounded_univ /-
+theorem compactSpace_iff_isBounded_univ [ProperSpace α] :
+    CompactSpace α ↔ IsBounded (univ : Set α) :=
+  ⟨@isBounded_of_compactSpace α _ _, fun hb => ⟨isCompact_of_isClosed_isBounded isClosed_univ hb⟩⟩
+#align metric.compact_space_iff_bounded_univ Metric.compactSpace_iff_isBounded_univ
 -/
 
 section ConditionallyCompleteLinearOrder
 
 variable [Preorder α] [CompactIccSpace α]
 
-#print Metric.bounded_Icc /-
-theorem bounded_Icc (a b : α) : Bounded (Icc a b) :=
+#print Metric.isBounded_Icc /-
+theorem isBounded_Icc (a b : α) : IsBounded (Icc a b) :=
   (totallyBounded_Icc a b).Bounded
-#align metric.bounded_Icc Metric.bounded_Icc
+#align metric.bounded_Icc Metric.isBounded_Icc
 -/
 
-#print Metric.bounded_Ico /-
-theorem bounded_Ico (a b : α) : Bounded (Ico a b) :=
+#print Metric.isBounded_Ico /-
+theorem isBounded_Ico (a b : α) : IsBounded (Ico a b) :=
   (totallyBounded_Ico a b).Bounded
-#align metric.bounded_Ico Metric.bounded_Ico
+#align metric.bounded_Ico Metric.isBounded_Ico
 -/
 
-#print Metric.bounded_Ioc /-
-theorem bounded_Ioc (a b : α) : Bounded (Ioc a b) :=
+#print Metric.isBounded_Ioc /-
+theorem isBounded_Ioc (a b : α) : IsBounded (Ioc a b) :=
   (totallyBounded_Ioc a b).Bounded
-#align metric.bounded_Ioc Metric.bounded_Ioc
+#align metric.bounded_Ioc Metric.isBounded_Ioc
 -/
 
-#print Metric.bounded_Ioo /-
-theorem bounded_Ioo (a b : α) : Bounded (Ioo a b) :=
+#print Metric.isBounded_Ioo /-
+theorem isBounded_Ioo (a b : α) : IsBounded (Ioo a b) :=
   (totallyBounded_Ioo a b).Bounded
-#align metric.bounded_Ioo Metric.bounded_Ioo
+#align metric.bounded_Ioo Metric.isBounded_Ioo
 -/
 
-#print Metric.bounded_of_bddAbove_of_bddBelow /-
+#print Metric.isBounded_of_bddAbove_of_bddBelow /-
 /-- In a pseudo metric space with a conditionally complete linear order such that the order and the
     metric structure give the same topology, any order-bounded set is metric-bounded. -/
-theorem bounded_of_bddAbove_of_bddBelow {s : Set α} (h₁ : BddAbove s) (h₂ : BddBelow s) :
-    Bounded s :=
+theorem isBounded_of_bddAbove_of_bddBelow {s : Set α} (h₁ : BddAbove s) (h₂ : BddBelow s) :
+    IsBounded s :=
   let ⟨u, hu⟩ := h₁
   let ⟨l, hl⟩ := h₂
-  Bounded.mono (fun x hx => mem_Icc.mpr ⟨hl hx, hu hx⟩) (bounded_Icc l u)
-#align metric.bounded_of_bdd_above_of_bdd_below Metric.bounded_of_bddAbove_of_bddBelow
+  IsBounded.subset (fun x hx => mem_Icc.mpr ⟨hl hx, hu hx⟩) (isBounded_Icc l u)
+#align metric.bounded_of_bdd_above_of_bdd_below Metric.isBounded_of_bddAbove_of_bddBelow
 -/
 
 end ConditionallyCompleteLinearOrder
@@ -3653,19 +3673,19 @@ theorem dist_le_diam_of_mem' (h : EMetric.diam s ≠ ⊤) (hx : x ∈ s) (hy : y
 #align metric.dist_le_diam_of_mem' Metric.dist_le_diam_of_mem'
 -/
 
-#print Metric.bounded_iff_ediam_ne_top /-
+#print Metric.isBounded_iff_ediam_ne_top /-
 /-- Characterize the boundedness of a set in terms of the finiteness of its emetric.diameter. -/
-theorem bounded_iff_ediam_ne_top : Bounded s ↔ EMetric.diam s ≠ ⊤ :=
+theorem isBounded_iff_ediam_ne_top : IsBounded s ↔ EMetric.diam s ≠ ⊤ :=
   Iff.intro
     (fun ⟨C, hC⟩ => ne_top_of_le_ne_top ENNReal.ofReal_ne_top <| ediam_le_of_forall_dist_le hC)
     fun h => ⟨diam s, fun x hx y hy => dist_le_diam_of_mem' h hx hy⟩
-#align metric.bounded_iff_ediam_ne_top Metric.bounded_iff_ediam_ne_top
+#align metric.bounded_iff_ediam_ne_top Metric.isBounded_iff_ediam_ne_top
 -/
 
-#print Metric.Bounded.ediam_ne_top /-
-theorem Bounded.ediam_ne_top (h : Bounded s) : EMetric.diam s ≠ ⊤ :=
-  bounded_iff_ediam_ne_top.1 h
-#align metric.bounded.ediam_ne_top Metric.Bounded.ediam_ne_top
+#print Bornology.IsBounded.ediam_ne_top /-
+theorem IsBounded.ediam_ne_top (h : IsBounded s) : EMetric.diam s ≠ ⊤ :=
+  isBounded_iff_ediam_ne_top.1 h
+#align metric.bounded.ediam_ne_top Bornology.IsBounded.ediam_ne_top
 -/
 
 #print Metric.ediam_univ_eq_top_iff_noncompact /-
@@ -3693,13 +3713,13 @@ theorem diam_univ_of_noncompact [ProperSpace α] [NoncompactSpace α] : diam (un
 
 #print Metric.dist_le_diam_of_mem /-
 /-- The distance between two points in a set is controlled by the diameter of the set. -/
-theorem dist_le_diam_of_mem (h : Bounded s) (hx : x ∈ s) (hy : y ∈ s) : dist x y ≤ diam s :=
+theorem dist_le_diam_of_mem (h : IsBounded s) (hx : x ∈ s) (hy : y ∈ s) : dist x y ≤ diam s :=
   dist_le_diam_of_mem' h.ediam_ne_top hx hy
 #align metric.dist_le_diam_of_mem Metric.dist_le_diam_of_mem
 -/
 
 #print Metric.ediam_of_unbounded /-
-theorem ediam_of_unbounded (h : ¬Bounded s) : EMetric.diam s = ∞ := by
+theorem ediam_of_unbounded (h : ¬IsBounded s) : EMetric.diam s = ∞ := by
   rwa [bounded_iff_ediam_ne_top, Classical.not_not] at h 
 #align metric.ediam_of_unbounded Metric.ediam_of_unbounded
 -/
@@ -3707,14 +3727,14 @@ theorem ediam_of_unbounded (h : ¬Bounded s) : EMetric.diam s = ∞ := by
 #print Metric.diam_eq_zero_of_unbounded /-
 /-- An unbounded set has zero diameter. If you would prefer to get the value ∞, use `emetric.diam`.
 This lemma makes it possible to avoid side conditions in some situations -/
-theorem diam_eq_zero_of_unbounded (h : ¬Bounded s) : diam s = 0 := by
+theorem diam_eq_zero_of_unbounded (h : ¬IsBounded s) : diam s = 0 := by
   rw [diam, ediam_of_unbounded h, ENNReal.top_toReal]
 #align metric.diam_eq_zero_of_unbounded Metric.diam_eq_zero_of_unbounded
 -/
 
 #print Metric.diam_mono /-
 /-- If `s ⊆ t`, then the diameter of `s` is bounded by that of `t`, provided `t` is bounded. -/
-theorem diam_mono {s t : Set α} (h : s ⊆ t) (ht : Bounded t) : diam s ≤ diam t :=
+theorem diam_mono {s t : Set α} (h : s ⊆ t) (ht : IsBounded t) : diam s ≤ diam t :=
   by
   unfold diam
   rw [ENNReal.toReal_le_toReal (bounded.mono h ht).ediam_ne_top ht.ediam_ne_top]
@@ -3782,7 +3802,7 @@ theorem diam_ball {r : ℝ} (h : 0 ≤ r) : diam (ball x r) ≤ 2 * r :=
 /-- If a family of complete sets with diameter tending to `0` is such that each finite intersection
 is nonempty, then the total intersection is also nonempty. -/
 theorem IsComplete.nonempty_iInter_of_nonempty_biInter {s : ℕ → Set α} (h0 : IsComplete (s 0))
-    (hs : ∀ n, IsClosed (s n)) (h's : ∀ n, Bounded (s n)) (h : ∀ N, (⋂ n ≤ N, s n).Nonempty)
+    (hs : ∀ n, IsClosed (s n)) (h's : ∀ n, IsBounded (s n)) (h : ∀ N, (⋂ n ≤ N, s n).Nonempty)
     (h' : Tendsto (fun n => diam (s n)) atTop (𝓝 0)) : (⋂ n, s n).Nonempty :=
   by
   let u N := (h N).some
@@ -3810,7 +3830,7 @@ theorem IsComplete.nonempty_iInter_of_nonempty_biInter {s : ℕ → Set α} (h0
 /-- In a complete space, if a family of closed sets with diameter tending to `0` is such that each
 finite intersection is nonempty, then the total intersection is also nonempty. -/
 theorem nonempty_iInter_of_nonempty_biInter [CompleteSpace α] {s : ℕ → Set α}
-    (hs : ∀ n, IsClosed (s n)) (h's : ∀ n, Bounded (s n)) (h : ∀ N, (⋂ n ≤ N, s n).Nonempty)
+    (hs : ∀ n, IsClosed (s n)) (h's : ∀ n, IsBounded (s n)) (h : ∀ N, (⋂ n ≤ N, s n).Nonempty)
     (h' : Tendsto (fun n => diam (s n)) atTop (𝓝 0)) : (⋂ n, s n).Nonempty :=
   (hs 0).IsComplete.nonempty_iInter_of_nonempty_biInter hs h's h h'
 #align metric.nonempty_Inter_of_nonempty_bInter Metric.nonempty_iInter_of_nonempty_biInter

mathlib commit https://github.com/leanprover-community/mathlib/commit/32a7e535287f9c73f2e4d2aef306a39190f0b504

@@ -180,7 +180,7 @@ theorem PseudoMetricSpace.ext {α : Type _} {m m' : PseudoMetricSpace α}
     simp [m_edist_dist, m'_edist_dist]
   · dsimp at m_uniformity_dist m'_uniformity_dist 
     rw [← m'_uniformity_dist] at m_uniformity_dist 
-    exact uniformSpace_eq m_uniformity_dist
+    exact UniformSpace.ext m_uniformity_dist
   · ext1
     dsimp at m_cobounded_sets m'_cobounded_sets 
     rw [← m'_cobounded_sets] at m_cobounded_sets 
@@ -989,7 +989,7 @@ theorem isBounded_iff_nndist {s : Set α} :
 theorem toUniformSpace_eq :
     ‹PseudoMetricSpace α›.toUniformSpace =
       UniformSpace.ofDist dist dist_self dist_comm dist_triangle :=
-  uniformSpace_eq PseudoMetricSpace.uniformity_dist
+  UniformSpace.ext PseudoMetricSpace.uniformity_dist
 #align metric.to_uniform_space_eq Metric.toUniformSpace_eq
 -/
 
@@ -1973,7 +1973,7 @@ theorem Filter.Tendsto.congr_dist {ι : Type _} {f₁ f₂ : ι → α} {p : Fil
 #align filter.tendsto.congr_dist Filter.Tendsto.congr_dist
 -/
 
-alias Filter.Tendsto.congr_dist ← tendsto_of_tendsto_of_dist
+alias tendsto_of_tendsto_of_dist := Filter.Tendsto.congr_dist
 #align tendsto_of_tendsto_of_dist tendsto_of_tendsto_of_dist
 
 #print tendsto_iff_of_dist /-
@@ -2933,7 +2933,7 @@ theorem finite_cover_balls_of_compact {α : Type u} [PseudoMetricSpace α] {s :
 #align finite_cover_balls_of_compact finite_cover_balls_of_compact
 -/
 
-alias finite_cover_balls_of_compact ← IsCompact.finite_cover_balls
+alias IsCompact.finite_cover_balls := finite_cover_balls_of_compact
 #align is_compact.finite_cover_balls IsCompact.finite_cover_balls
 
 end Compact
@@ -3266,7 +3266,7 @@ theorem bounded_closure_of_bounded (h : Bounded s) : Bounded (closure s) :=
 #align metric.bounded_closure_of_bounded Metric.bounded_closure_of_bounded
 -/
 
-alias bounded_closure_of_bounded ← bounded.closure
+alias bounded.closure := bounded_closure_of_bounded
 #align metric.bounded.closure Metric.Bounded.closure
 
 #print Metric.bounded_closure_iff /-
@@ -3350,7 +3350,7 @@ theorem bounded_of_finite {s : Set α} (h : s.Finite) : Bounded s :=
 #align metric.bounded_of_finite Metric.bounded_of_finite
 -/
 
-alias bounded_of_finite ← _root_.set.finite.bounded
+alias _root_.set.finite.bounded := bounded_of_finite
 #align set.finite.bounded Set.Finite.bounded
 
 #print Metric.bounded_singleton /-

mathlib commit https://github.com/leanprover-community/mathlib/commit/32a7e535287f9c73f2e4d2aef306a39190f0b504

@@ -3261,7 +3261,8 @@ theorem Bounded.subset_ball_lt (h : Bounded s) (a : ℝ) (c : α) : ∃ r, a < r
 #print Metric.bounded_closure_of_bounded /-
 theorem bounded_closure_of_bounded (h : Bounded s) : Bounded (closure s) :=
   let ⟨C, h⟩ := h
-  ⟨C, fun a ha b hb => (isClosed_le' C).closure_subset <| map_mem_closure₂ continuous_dist ha hb h⟩
+  ⟨C, fun a ha b hb =>
+    (ClosedIicTopology.isClosed_le' C).closure_subset <| map_mem_closure₂ continuous_dist ha hb h⟩
 #align metric.bounded_closure_of_bounded Metric.bounded_closure_of_bounded
 -/
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/63721b2c3eba6c325ecf8ae8cca27155a4f6306f

@@ -2047,7 +2047,7 @@ theorem Metric.uniformCauchySeqOn_iff {γ : Type _} {F : β → γ → α} {s :
     rcases metric.mem_uniformity_dist.mp hu with ⟨ε, hε, hab⟩
     rcases h ε hε with ⟨N, hN⟩
     rw [prod_at_top_at_top_eq, eventually_at_top]
-    use (N, N)
+    use(N, N)
     intro b hb x hx
     rcases hb with ⟨hbl, hbr⟩
     exact hab (hN b.fst hbl.ge b.snd hbr.ge x hx)
@@ -3374,7 +3374,7 @@ theorem bounded_range_of_tendsto_cofinite_uniformity {f : β → α}
   rcases(has_basis_cofinite.prod_self.tendsto_iff uniformity_basis_dist).1 hf 1 zero_lt_one with
     ⟨s, hsf, hs1⟩
   rw [← image_univ, ← union_compl_self s, image_union, bounded_union]
-  use (hsf.image f).Bounded, 1
+  use(hsf.image f).Bounded, 1
   rintro _ ⟨x, hx, rfl⟩ _ ⟨y, hy, rfl⟩
   exact le_of_lt (hs1 (x, y) ⟨hx, hy⟩)
 #align metric.bounded_range_of_tendsto_cofinite_uniformity Metric.bounded_range_of_tendsto_cofinite_uniformity

mathlib commit https://github.com/leanprover-community/mathlib/commit/c0c52abb75074ed8b73a948341f50521fbf43b4c

@@ -667,14 +667,18 @@ theorem ne_of_mem_sphere (h : y ∈ sphere x ε) (hε : ε ≠ 0) : y ≠ x := b
 #align metric.ne_of_mem_sphere Metric.ne_of_mem_sphere
 -/
 
+#print Metric.nonneg_of_mem_sphere /-
 theorem nonneg_of_mem_sphere (hy : y ∈ sphere x ε) : 0 ≤ ε :=
   dist_nonneg.trans_eq hy
 #align metric.nonneg_of_mem_sphere Metric.nonneg_of_mem_sphere
+-/
 
+#print Metric.sphere_eq_empty_of_neg /-
 @[simp]
 theorem sphere_eq_empty_of_neg (hε : ε < 0) : sphere x ε = ∅ :=
   Set.eq_empty_iff_forall_not_mem.mpr fun y hy => (nonneg_of_mem_sphere hy).not_lt hε
 #align metric.sphere_eq_empty_of_neg Metric.sphere_eq_empty_of_neg
+-/
 
 #print Metric.sphere_eq_empty_of_subsingleton /-
 theorem sphere_eq_empty_of_subsingleton [Subsingleton α] (hε : ε ≠ 0) : sphere x ε = ∅ :=
@@ -708,10 +712,12 @@ theorem closedBall_eq_empty : closedBall x ε = ∅ ↔ ε < 0 := by
 #align metric.closed_ball_eq_empty Metric.closedBall_eq_empty
 -/
 
+#print Metric.closedBall_eq_sphere_of_nonpos /-
 /-- Closed balls and spheres coincide when the radius is non-positive -/
 theorem closedBall_eq_sphere_of_nonpos (hε : ε ≤ 0) : closedBall x ε = sphere x ε :=
   Set.ext fun _ => (hε.trans dist_nonneg).le_iff_eq
 #align metric.closed_ball_eq_sphere_of_nonpos Metric.closedBall_eq_sphere_of_nonpos
+-/
 
 #print Metric.ball_subset_closedBall /-
 theorem ball_subset_closedBall : ball x ε ⊆ closedBall x ε := fun y (hy : _ < _) => le_of_lt hy
@@ -2367,6 +2373,7 @@ theorem closedBall_prod_same (x : α) (y : β) (r : ℝ) :
 
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
+#print sphere_prod /-
 theorem sphere_prod (x : α × β) (r : ℝ) :
     sphere x r = sphere x.1 r ×ˢ closedBall x.2 r ∪ closedBall x.1 r ×ˢ sphere x.2 r :=
   by
@@ -2380,6 +2387,7 @@ theorem sphere_prod (x : α × β) (r : ℝ) :
     refine' or_congr (and_congr_right _) ((and_comm' _ _).trans (and_congr_left _))
     all_goals rintro rfl; rfl
 #align sphere_prod sphere_prod
+-/
 
 end Prod
 
@@ -2722,11 +2730,14 @@ theorem nndist_pi_le_iff {f g : ∀ b, π b} {r : ℝ≥0} :
 #align nndist_pi_le_iff nndist_pi_le_iff
 -/
 
+#print nndist_pi_lt_iff /-
 theorem nndist_pi_lt_iff {f g : ∀ b, π b} {r : ℝ≥0} (hr : 0 < r) :
     nndist f g < r ↔ ∀ b, nndist (f b) (g b) < r := by
   simp [nndist_pi_def, Finset.sup_lt_iff (show ⊥ < r from hr)]
 #align nndist_pi_lt_iff nndist_pi_lt_iff
+-/
 
+#print nndist_pi_eq_iff /-
 theorem nndist_pi_eq_iff {f g : ∀ b, π b} {r : ℝ≥0} (hr : 0 < r) :
     nndist f g = r ↔ (∃ i, nndist (f i) (g i) = r) ∧ ∀ b, nndist (f b) (g b) ≤ r :=
   by
@@ -2736,6 +2747,7 @@ theorem nndist_pi_eq_iff {f g : ∀ b, π b} {r : ℝ≥0} (hr : 0 < r) :
   refine' exists_congr fun b => _
   apply (and_iff_right <| h _).symm
 #align nndist_pi_eq_iff nndist_pi_eq_iff
+-/
 
 #print dist_pi_lt_iff /-
 theorem dist_pi_lt_iff {f g : ∀ b, π b} {r : ℝ} (hr : 0 < r) :
@@ -2755,12 +2767,14 @@ theorem dist_pi_le_iff {f g : ∀ b, π b} {r : ℝ} (hr : 0 ≤ r) :
 #align dist_pi_le_iff dist_pi_le_iff
 -/
 
+#print dist_pi_eq_iff /-
 theorem dist_pi_eq_iff {f g : ∀ b, π b} {r : ℝ} (hr : 0 < r) :
     dist f g = r ↔ (∃ i, dist (f i) (g i) = r) ∧ ∀ b, dist (f b) (g b) ≤ r :=
   by
   lift r to ℝ≥0 using hr.le
   simp_rw [← coe_nndist, NNReal.coe_eq, nndist_pi_eq_iff hr, NNReal.coe_le_coe]
 #align dist_pi_eq_iff dist_pi_eq_iff
+-/
 
 #print dist_pi_le_iff' /-
 theorem dist_pi_le_iff' [Nonempty β] {f g : ∀ b, π b} {r : ℝ} :
@@ -2848,6 +2862,7 @@ theorem closedBall_pi' [Nonempty β] (x : ∀ b, π b) (r : ℝ) :
 #align closed_ball_pi' closedBall_pi'
 -/
 
+#print sphere_pi /-
 /-- A sphere in a product space is a union of spheres on each component restricted to the closed
 ball. -/
 theorem sphere_pi (x : ∀ b, π b) {r : ℝ} (h : 0 < r ∨ Nonempty β) :
@@ -2866,6 +2881,7 @@ theorem sphere_pi (x : ∀ b, π b) {r : ℝ} (h : 0 < r ∨ Nonempty β) :
   · ext
     simp [dist_pi_eq_iff hr, dist_pi_le_iff hr.le]
 #align sphere_pi sphere_pi
+-/
 
 #print Fin.nndist_insertNth_insertNth /-
 @[simp]

mathlib commit https://github.com/leanprover-community/mathlib/commit/8047de4d911cdef39c2d646165eea972f7f9f539

@@ -8,7 +8,7 @@ import Mathbin.Topology.Algebra.Order.Compact
 import Mathbin.Topology.MetricSpace.EmetricSpace
 import Mathbin.Topology.Bornology.Constructions
 
-#align_import topology.metric_space.basic from "leanprover-community/mathlib"@"8000bbbe2e9d39b84edb993d88781f536a8a3fa8"
+#align_import topology.metric_space.basic from "leanprover-community/mathlib"@"8047de4d911cdef39c2d646165eea972f7f9f539"
 
 /-!
 # Metric spaces
@@ -667,6 +667,15 @@ theorem ne_of_mem_sphere (h : y ∈ sphere x ε) (hε : ε ≠ 0) : y ≠ x := b
 #align metric.ne_of_mem_sphere Metric.ne_of_mem_sphere
 -/
 
+theorem nonneg_of_mem_sphere (hy : y ∈ sphere x ε) : 0 ≤ ε :=
+  dist_nonneg.trans_eq hy
+#align metric.nonneg_of_mem_sphere Metric.nonneg_of_mem_sphere
+
+@[simp]
+theorem sphere_eq_empty_of_neg (hε : ε < 0) : sphere x ε = ∅ :=
+  Set.eq_empty_iff_forall_not_mem.mpr fun y hy => (nonneg_of_mem_sphere hy).not_lt hε
+#align metric.sphere_eq_empty_of_neg Metric.sphere_eq_empty_of_neg
+
 #print Metric.sphere_eq_empty_of_subsingleton /-
 theorem sphere_eq_empty_of_subsingleton [Subsingleton α] (hε : ε ≠ 0) : sphere x ε = ∅ :=
   Set.eq_empty_iff_forall_not_mem.mpr fun y hy => ne_of_mem_sphere hy hε (Subsingleton.elim _ _)
@@ -699,6 +708,11 @@ theorem closedBall_eq_empty : closedBall x ε = ∅ ↔ ε < 0 := by
 #align metric.closed_ball_eq_empty Metric.closedBall_eq_empty
 -/
 
+/-- Closed balls and spheres coincide when the radius is non-positive -/
+theorem closedBall_eq_sphere_of_nonpos (hε : ε ≤ 0) : closedBall x ε = sphere x ε :=
+  Set.ext fun _ => (hε.trans dist_nonneg).le_iff_eq
+#align metric.closed_ball_eq_sphere_of_nonpos Metric.closedBall_eq_sphere_of_nonpos
+
 #print Metric.ball_subset_closedBall /-
 theorem ball_subset_closedBall : ball x ε ⊆ closedBall x ε := fun y (hy : _ < _) => le_of_lt hy
 #align metric.ball_subset_closed_ball Metric.ball_subset_closedBall
@@ -2351,6 +2365,22 @@ theorem closedBall_prod_same (x : α) (y : β) (r : ℝ) :
 #align closed_ball_prod_same closedBall_prod_same
 -/
 
+/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
+/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
+theorem sphere_prod (x : α × β) (r : ℝ) :
+    sphere x r = sphere x.1 r ×ˢ closedBall x.2 r ∪ closedBall x.1 r ×ˢ sphere x.2 r :=
+  by
+  obtain hr | rfl | hr := lt_trichotomy r 0
+  · simp [hr]
+  · cases x
+    simp_rw [← closed_ball_eq_sphere_of_nonpos le_rfl, union_self, closedBall_prod_same]
+  · ext ⟨x', y'⟩
+    simp_rw [Set.mem_union, Set.mem_prod, Metric.mem_closedBall, Metric.mem_sphere, Prod.dist_eq,
+      max_eq_iff]
+    refine' or_congr (and_congr_right _) ((and_comm' _ _).trans (and_congr_left _))
+    all_goals rintro rfl; rfl
+#align sphere_prod sphere_prod
+
 end Prod
 
 #print uniformContinuous_dist /-
@@ -2692,12 +2722,27 @@ theorem nndist_pi_le_iff {f g : ∀ b, π b} {r : ℝ≥0} :
 #align nndist_pi_le_iff nndist_pi_le_iff
 -/
 
+theorem nndist_pi_lt_iff {f g : ∀ b, π b} {r : ℝ≥0} (hr : 0 < r) :
+    nndist f g < r ↔ ∀ b, nndist (f b) (g b) < r := by
+  simp [nndist_pi_def, Finset.sup_lt_iff (show ⊥ < r from hr)]
+#align nndist_pi_lt_iff nndist_pi_lt_iff
+
+theorem nndist_pi_eq_iff {f g : ∀ b, π b} {r : ℝ≥0} (hr : 0 < r) :
+    nndist f g = r ↔ (∃ i, nndist (f i) (g i) = r) ∧ ∀ b, nndist (f b) (g b) ≤ r :=
+  by
+  rw [eq_iff_le_not_lt, nndist_pi_lt_iff hr, nndist_pi_le_iff, not_forall, and_comm']
+  simp_rw [not_lt, and_congr_left_iff, le_antisymm_iff]
+  intro h
+  refine' exists_congr fun b => _
+  apply (and_iff_right <| h _).symm
+#align nndist_pi_eq_iff nndist_pi_eq_iff
+
 #print dist_pi_lt_iff /-
 theorem dist_pi_lt_iff {f g : ∀ b, π b} {r : ℝ} (hr : 0 < r) :
     dist f g < r ↔ ∀ b, dist (f b) (g b) < r :=
   by
   lift r to ℝ≥0 using hr.le
-  simp [dist_pi_def, Finset.sup_lt_iff (show ⊥ < r from hr)]
+  exact nndist_pi_lt_iff hr
 #align dist_pi_lt_iff dist_pi_lt_iff
 -/
 
@@ -2710,6 +2755,13 @@ theorem dist_pi_le_iff {f g : ∀ b, π b} {r : ℝ} (hr : 0 ≤ r) :
 #align dist_pi_le_iff dist_pi_le_iff
 -/
 
+theorem dist_pi_eq_iff {f g : ∀ b, π b} {r : ℝ} (hr : 0 < r) :
+    dist f g = r ↔ (∃ i, dist (f i) (g i) = r) ∧ ∀ b, dist (f b) (g b) ≤ r :=
+  by
+  lift r to ℝ≥0 using hr.le
+  simp_rw [← coe_nndist, NNReal.coe_eq, nndist_pi_eq_iff hr, NNReal.coe_le_coe]
+#align dist_pi_eq_iff dist_pi_eq_iff
+
 #print dist_pi_le_iff' /-
 theorem dist_pi_le_iff' [Nonempty β] {f g : ∀ b, π b} {r : ℝ} :
     dist f g ≤ r ↔ ∀ b, dist (f b) (g b) ≤ r :=
@@ -2796,6 +2848,25 @@ theorem closedBall_pi' [Nonempty β] (x : ∀ b, π b) (r : ℝ) :
 #align closed_ball_pi' closedBall_pi'
 -/
 
+/-- A sphere in a product space is a union of spheres on each component restricted to the closed
+ball. -/
+theorem sphere_pi (x : ∀ b, π b) {r : ℝ} (h : 0 < r ∨ Nonempty β) :
+    sphere x r = (⋃ i : β, Function.eval i ⁻¹' sphere (x i) r) ∩ closedBall x r :=
+  by
+  obtain hr | rfl | hr := lt_trichotomy r 0
+  · simp [hr]
+  · rw [closed_ball_eq_sphere_of_nonpos le_rfl, eq_comm, Set.inter_eq_right_iff_subset]
+    letI := h.resolve_left (lt_irrefl _)
+    inhabit β
+    refine' subset_Union_of_subset default _
+    intro x hx
+    replace hx := hx.le
+    rw [dist_pi_le_iff le_rfl] at hx 
+    exact le_antisymm (hx default) dist_nonneg
+  · ext
+    simp [dist_pi_eq_iff hr, dist_pi_le_iff hr.le]
+#align sphere_pi sphere_pi
+
 #print Fin.nndist_insertNth_insertNth /-
 @[simp]
 theorem Fin.nndist_insertNth_insertNth {n : ℕ} {α : Fin (n + 1) → Type _}

mathlib commit https://github.com/leanprover-community/mathlib/commit/8ea5598db6caeddde6cb734aa179cc2408dbd345

@@ -2,17 +2,14 @@
 Copyright (c) 2015, 2017 Jeremy Avigad. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Jeremy Avigad, Robert Y. Lewis, Johannes Hölzl, Mario Carneiro, Sébastien Gouëzel
-
-! This file was ported from Lean 3 source module topology.metric_space.basic
-! leanprover-community/mathlib commit 8000bbbe2e9d39b84edb993d88781f536a8a3fa8
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
 -/
 import Mathbin.Tactic.Positivity
 import Mathbin.Topology.Algebra.Order.Compact
 import Mathbin.Topology.MetricSpace.EmetricSpace
 import Mathbin.Topology.Bornology.Constructions
 
+#align_import topology.metric_space.basic from "leanprover-community/mathlib"@"8000bbbe2e9d39b84edb993d88781f536a8a3fa8"
+
 /-!
 # Metric spaces
 
@@ -1108,7 +1105,7 @@ theorem uniformContinuous_iff [PseudoMetricSpace β] {f : α → β} :
 #align metric.uniform_continuous_iff Metric.uniformContinuous_iff
 -/
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:638:2: warning: expanding binder collection (x y «expr ∈ » s) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (x y «expr ∈ » s) -/
 #print Metric.uniformContinuousOn_iff /-
 theorem uniformContinuousOn_iff [PseudoMetricSpace β] {f : α → β} {s : Set α} :
     UniformContinuousOn f s ↔
@@ -1117,7 +1114,7 @@ theorem uniformContinuousOn_iff [PseudoMetricSpace β] {f : α → β} {s : Set
 #align metric.uniform_continuous_on_iff Metric.uniformContinuousOn_iff
 -/
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:638:2: warning: expanding binder collection (x y «expr ∈ » s) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (x y «expr ∈ » s) -/
 #print Metric.uniformContinuousOn_iff_le /-
 theorem uniformContinuousOn_iff_le [PseudoMetricSpace β] {f : α → β} {s : Set α} :
     UniformContinuousOn f s ↔
@@ -1184,7 +1181,7 @@ theorem totallyBounded_of_finite_discretization {s : Set α}
 #align metric.totally_bounded_of_finite_discretization Metric.totallyBounded_of_finite_discretization
 -/
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:638:2: warning: expanding binder collection (t «expr ⊆ » s) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (t «expr ⊆ » s) -/
 #print Metric.finite_approx_of_totallyBounded /-
 theorem finite_approx_of_totallyBounded {s : Set α} (hs : TotallyBounded s) :
     ∀ ε > 0, ∃ (t : _) (_ : t ⊆ s), Set.Finite t ∧ s ⊆ ⋃ y ∈ t, ball y ε :=
@@ -1253,7 +1250,7 @@ theorem tendstoUniformly_iff {ι : Type _} {F : ι → β → α} {f : β → α
 #align metric.tendsto_uniformly_iff Metric.tendstoUniformly_iff
 -/
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:638:2: warning: expanding binder collection (x y «expr ∈ » t) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (x y «expr ∈ » t) -/
 #print Metric.cauchy_iff /-
 protected theorem cauchy_iff {f : Filter α} :
     Cauchy f ↔ NeBot f ∧ ∀ ε > 0, ∃ t ∈ f, ∀ (x) (_ : x ∈ t) (y) (_ : y ∈ t), dist x y < ε :=
@@ -1985,7 +1982,7 @@ section CauchySeq
 
 variable [Nonempty β] [SemilatticeSup β]
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:638:2: warning: expanding binder collection (m n «expr ≥ » N) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (m n «expr ≥ » N) -/
 #print Metric.cauchySeq_iff /-
 -- see Note [nolint_ge]
 /-- In a pseudometric space, Cauchy sequences are characterized by the fact that, eventually,
@@ -2833,7 +2830,7 @@ end Pi
 
 section Compact
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:638:2: warning: expanding binder collection (t «expr ⊆ » s) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (t «expr ⊆ » s) -/
 #print finite_cover_balls_of_compact /-
 /-- Any compact set in a pseudometric space can be covered by finitely many balls of a given
 positive radius -/
@@ -2945,7 +2942,7 @@ instance (priority := 100) locally_compact_of_proper [ProperSpace α] : LocallyC
 #align locally_compact_of_proper locally_compact_of_proper
 -/
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:638:2: warning: expanding binder collection (x y «expr ∈ » t) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (x y «expr ∈ » t) -/
 #print complete_of_proper /-
 -- see Note [lower instance priority]
 /-- A proper space is complete -/
@@ -3075,7 +3072,7 @@ theorem lebesgue_number_lemma_of_metric_sUnion {s : Set α} {c : Set (Set α)} (
 
 namespace Metric
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:638:2: warning: expanding binder collection (x y «expr ∈ » s) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (x y «expr ∈ » s) -/
 #print Metric.Bounded /-
 /-- Boundedness of a subset of a pseudometric space. We formulate the definition to work
 even in the empty space. -/

mathlib commit https://github.com/leanprover-community/mathlib/commit/2fe465deb81bcd7ccafa065bb686888a82f15372

@@ -2802,7 +2802,7 @@ theorem closedBall_pi' [Nonempty β] (x : ∀ b, π b) (r : ℝ) :
 #print Fin.nndist_insertNth_insertNth /-
 @[simp]
 theorem Fin.nndist_insertNth_insertNth {n : ℕ} {α : Fin (n + 1) → Type _}
-    [∀ i, PseudoMetricSpace (α i)] (i : Fin (n + 1)) (x y : α i) (f g : ∀ j, α (i.succAbove j)) :
+    [∀ i, PseudoMetricSpace (α i)] (i : Fin (n + 1)) (x y : α i) (f g : ∀ j, α (i.succAboveEmb j)) :
     nndist (i.insertNth x f) (i.insertNth y g) = max (nndist x y) (nndist f g) :=
   eq_of_forall_ge_iff fun c => by simp [nndist_pi_le_iff, i.forall_iff_succ_above]
 #align fin.nndist_insert_nth_insert_nth Fin.nndist_insertNth_insertNth
@@ -2811,7 +2811,7 @@ theorem Fin.nndist_insertNth_insertNth {n : ℕ} {α : Fin (n + 1) → Type _}
 #print Fin.dist_insertNth_insertNth /-
 @[simp]
 theorem Fin.dist_insertNth_insertNth {n : ℕ} {α : Fin (n + 1) → Type _}
-    [∀ i, PseudoMetricSpace (α i)] (i : Fin (n + 1)) (x y : α i) (f g : ∀ j, α (i.succAbove j)) :
+    [∀ i, PseudoMetricSpace (α i)] (i : Fin (n + 1)) (x y : α i) (f g : ∀ j, α (i.succAboveEmb j)) :
     dist (i.insertNth x f) (i.insertNth y g) = max (dist x y) (dist f g) := by
   simp only [dist_nndist, Fin.nndist_insertNth_insertNth, NNReal.coe_max]
 #align fin.dist_insert_nth_insert_nth Fin.dist_insertNth_insertNth

mathlib commit https://github.com/leanprover-community/mathlib/commit/d30d31261cdb4d2f5e612eabc3c4bf45556350d5

@@ -3733,17 +3733,17 @@ theorem nonempty_iInter_of_nonempty_biInter [CompleteSpace α] {s : ℕ → Set
 
 end Diam
 
-#print Metric.exists_local_min_mem_ball /-
-theorem exists_local_min_mem_ball [ProperSpace α] [TopologicalSpace β]
+#print Metric.exists_isLocalMin_mem_ball /-
+theorem exists_isLocalMin_mem_ball [ProperSpace α] [TopologicalSpace β]
     [ConditionallyCompleteLinearOrder β] [OrderTopology β] {f : α → β} {a z : α} {r : ℝ}
     (hf : ContinuousOn f (closedBall a r)) (hz : z ∈ closedBall a r)
     (hf1 : ∀ z' ∈ sphere a r, f z < f z') : ∃ z ∈ ball a r, IsLocalMin f z :=
   by
   simp_rw [← closed_ball_diff_ball] at hf1 
   exact
-    (is_compact_closed_ball a r).exists_local_min_mem_open ball_subset_closed_ball hf hz hf1
+    (is_compact_closed_ball a r).exists_isLocalMin_mem_open ball_subset_closed_ball hf hz hf1
       is_open_ball
-#align metric.exists_local_min_mem_ball Metric.exists_local_min_mem_ball
+#align metric.exists_local_min_mem_ball Metric.exists_isLocalMin_mem_ball
 -/
 
 end Metric

mathlib commit https://github.com/leanprover-community/mathlib/commit/2a0ce625dbb0ffbc7d1316597de0b25c1ec75303

@@ -178,7 +178,7 @@ theorem PseudoMetricSpace.ext {α : Type _} {m m' : PseudoMetricSpace α}
   dsimp at h 
   subst h
   congr
-  · ext (x y) : 2
+  · ext x y : 2
     dsimp at m_edist_dist m'_edist_dist 
     simp [m_edist_dist, m'_edist_dist]
   · dsimp at m_uniformity_dist m'_uniformity_dist 

mathlib commit https://github.com/leanprover-community/mathlib/commit/9fb8964792b4237dac6200193a0d533f1b3f7423

@@ -63,6 +63,7 @@ universe u v w
 
 variable {α : Type u} {β : Type v} {X ι : Type _}
 
+#print UniformSpace.ofDist /-
 /-- Construct a uniform structure from a distance function and metric space axioms -/
 def UniformSpace.ofDist (dist : α → α → ℝ) (dist_self : ∀ x : α, dist x x = 0)
     (dist_comm : ∀ x y : α, dist x y = dist y x)
@@ -70,6 +71,7 @@ def UniformSpace.ofDist (dist : α → α → ℝ) (dist_self : ∀ x : α, dist
   UniformSpace.ofFun dist dist_self dist_comm dist_triangle fun ε ε0 =>
     ⟨ε / 2, half_pos ε0, fun x hx y hy => add_halves ε ▸ add_lt_add hx hy⟩
 #align uniform_space_of_dist UniformSpace.ofDist
+-/
 
 /-- This is an internal lemma used to construct a bornology from a metric in `bornology.of_dist`. -/
 private theorem bounded_iff_aux {α : Type _} (dist : α → α → ℝ)
@@ -202,6 +204,7 @@ instance (priority := 200) PseudoMetricSpace.toEDist : EDist α :=
 #align pseudo_metric_space.to_has_edist PseudoMetricSpace.toEDist
 -/
 
+#print PseudoMetricSpace.ofDistTopology /-
 /-- Construct a pseudo-metric space structure whose underlying topological space structure
 (definitionally) agrees which a pre-existing topology which is compatible with a given distance
 function. -/
@@ -226,11 +229,14 @@ def PseudoMetricSpace.ofDistTopology {α : Type u} [TopologicalSpace α] (dist :
     toBornology := Bornology.ofDist dist dist_self dist_comm dist_triangle
     cobounded_sets := rfl }
 #align pseudo_metric_space.of_dist_topology PseudoMetricSpace.ofDistTopology
+-/
 
+#print dist_self /-
 @[simp]
 theorem dist_self (x : α) : dist x x = 0 :=
   PseudoMetricSpace.dist_self x
 #align dist_self dist_self
+-/
 
 #print dist_comm /-
 theorem dist_comm (x y : α) : dist x y = dist y x :=
@@ -244,34 +250,47 @@ theorem edist_dist (x y : α) : edist x y = ENNReal.ofReal (dist x y) :=
 #align edist_dist edist_dist
 -/
 
+#print dist_triangle /-
 theorem dist_triangle (x y z : α) : dist x z ≤ dist x y + dist y z :=
   PseudoMetricSpace.dist_triangle x y z
 #align dist_triangle dist_triangle
+-/
 
+#print dist_triangle_left /-
 theorem dist_triangle_left (x y z : α) : dist x y ≤ dist z x + dist z y := by
   rw [dist_comm z] <;> apply dist_triangle
 #align dist_triangle_left dist_triangle_left
+-/
 
+#print dist_triangle_right /-
 theorem dist_triangle_right (x y z : α) : dist x y ≤ dist x z + dist y z := by
   rw [dist_comm y] <;> apply dist_triangle
 #align dist_triangle_right dist_triangle_right
+-/
 
+#print dist_triangle4 /-
 theorem dist_triangle4 (x y z w : α) : dist x w ≤ dist x y + dist y z + dist z w :=
   calc
     dist x w ≤ dist x z + dist z w := dist_triangle x z w
     _ ≤ dist x y + dist y z + dist z w := add_le_add_right (dist_triangle x y z) _
 #align dist_triangle4 dist_triangle4
+-/
 
+#print dist_triangle4_left /-
 theorem dist_triangle4_left (x₁ y₁ x₂ y₂ : α) :
     dist x₂ y₂ ≤ dist x₁ y₁ + (dist x₁ x₂ + dist y₁ y₂) := by
   rw [add_left_comm, dist_comm x₁, ← add_assoc]; apply dist_triangle4
 #align dist_triangle4_left dist_triangle4_left
+-/
 
+#print dist_triangle4_right /-
 theorem dist_triangle4_right (x₁ y₁ x₂ y₂ : α) :
     dist x₁ y₁ ≤ dist x₁ x₂ + dist y₁ y₂ + dist x₂ y₂ := by rw [add_right_comm, dist_comm y₁];
   apply dist_triangle4
 #align dist_triangle4_right dist_triangle4_right
+-/
 
+#print dist_le_Ico_sum_dist /-
 /-- The triangle (polygon) inequality for sequences of points; `finset.Ico` version. -/
 theorem dist_le_Ico_sum_dist (f : ℕ → α) {m n} (h : m ≤ n) :
     dist (f m) (f n) ≤ ∑ i in Finset.Ico m n, dist (f i) (f (i + 1)) :=
@@ -286,13 +305,17 @@ theorem dist_le_Ico_sum_dist (f : ℕ → α) {m n} (h : m ≤ n) :
       _ = ∑ i in Finset.Ico m (n + 1), _ := by
         rw [Nat.Ico_succ_right_eq_insert_Ico hn, Finset.sum_insert, add_comm] <;> simp
 #align dist_le_Ico_sum_dist dist_le_Ico_sum_dist
+-/
 
+#print dist_le_range_sum_dist /-
 /-- The triangle (polygon) inequality for sequences of points; `finset.range` version. -/
 theorem dist_le_range_sum_dist (f : ℕ → α) (n : ℕ) :
     dist (f 0) (f n) ≤ ∑ i in Finset.range n, dist (f i) (f (i + 1)) :=
   Nat.Ico_zero_eq_range ▸ dist_le_Ico_sum_dist f (Nat.zero_le n)
 #align dist_le_range_sum_dist dist_le_range_sum_dist
+-/
 
+#print dist_le_Ico_sum_of_dist_le /-
 /-- A version of `dist_le_Ico_sum_dist` with each intermediate distance replaced
 with an upper estimate. -/
 theorem dist_le_Ico_sum_of_dist_le {f : ℕ → α} {m n} (hmn : m ≤ n) {d : ℕ → ℝ}
@@ -301,7 +324,9 @@ theorem dist_le_Ico_sum_of_dist_le {f : ℕ → α} {m n} (hmn : m ≤ n) {d : 
   le_trans (dist_le_Ico_sum_dist f hmn) <|
     Finset.sum_le_sum fun k hk => hd (Finset.mem_Ico.1 hk).1 (Finset.mem_Ico.1 hk).2
 #align dist_le_Ico_sum_of_dist_le dist_le_Ico_sum_of_dist_le
+-/
 
+#print dist_le_range_sum_of_dist_le /-
 /-- A version of `dist_le_range_sum_dist` with each intermediate distance replaced
 with an upper estimate. -/
 theorem dist_le_range_sum_of_dist_le {f : ℕ → α} (n : ℕ) {d : ℕ → ℝ}
@@ -309,20 +334,25 @@ theorem dist_le_range_sum_of_dist_le {f : ℕ → α} (n : ℕ) {d : ℕ → ℝ
     dist (f 0) (f n) ≤ ∑ i in Finset.range n, d i :=
   Nat.Ico_zero_eq_range ▸ dist_le_Ico_sum_of_dist_le (zero_le n) fun _ _ => hd
 #align dist_le_range_sum_of_dist_le dist_le_range_sum_of_dist_le
+-/
 
 #print swap_dist /-
 theorem swap_dist : Function.swap (@dist α _) = dist := by funext x y <;> exact dist_comm _ _
 #align swap_dist swap_dist
 -/
 
+#print abs_dist_sub_le /-
 theorem abs_dist_sub_le (x y z : α) : |dist x z - dist y z| ≤ dist x y :=
   abs_sub_le_iff.2
     ⟨sub_le_iff_le_add.2 (dist_triangle _ _ _), sub_le_iff_le_add.2 (dist_triangle_left _ _ _)⟩
 #align abs_dist_sub_le abs_dist_sub_le
+-/
 
+#print dist_nonneg /-
 theorem dist_nonneg {x y : α} : 0 ≤ dist x y :=
   PseudoMetricSpace.dist_nonneg' dist dist_self dist_comm dist_triangle
 #align dist_nonneg dist_nonneg
+-/
 
 section
 
@@ -337,10 +367,12 @@ unsafe def _root_.tactic.positivity_dist : expr → tactic strictness
 
 end
 
+#print abs_dist /-
 @[simp]
 theorem abs_dist {a b : α} : |dist a b| = dist a b :=
   abs_of_nonneg dist_nonneg
 #align abs_dist abs_dist
+-/
 
 #print NNDist /-
 /-- A version of `has_dist` that takes value in `ℝ≥0`. -/
@@ -380,62 +412,84 @@ theorem coe_nnreal_ennreal_nndist (x y : α) : ↑(nndist x y) = edist x y :=
 #align coe_nnreal_ennreal_nndist coe_nnreal_ennreal_nndist
 -/
 
+#print edist_lt_coe /-
 @[simp, norm_cast]
 theorem edist_lt_coe {x y : α} {c : ℝ≥0} : edist x y < c ↔ nndist x y < c := by
   rw [edist_nndist, ENNReal.coe_lt_coe]
 #align edist_lt_coe edist_lt_coe
+-/
 
+#print edist_le_coe /-
 @[simp, norm_cast]
 theorem edist_le_coe {x y : α} {c : ℝ≥0} : edist x y ≤ c ↔ nndist x y ≤ c := by
   rw [edist_nndist, ENNReal.coe_le_coe]
 #align edist_le_coe edist_le_coe
+-/
 
+#print edist_lt_top /-
 /-- In a pseudometric space, the extended distance is always finite-/
 theorem edist_lt_top {α : Type _} [PseudoMetricSpace α] (x y : α) : edist x y < ⊤ :=
   (edist_dist x y).symm ▸ ENNReal.ofReal_lt_top
 #align edist_lt_top edist_lt_top
+-/
 
+#print edist_ne_top /-
 /-- In a pseudometric space, the extended distance is always finite-/
 theorem edist_ne_top (x y : α) : edist x y ≠ ⊤ :=
   (edist_lt_top x y).Ne
 #align edist_ne_top edist_ne_top
+-/
 
+#print nndist_self /-
 /-- `nndist x x` vanishes-/
 @[simp]
 theorem nndist_self (a : α) : nndist a a = 0 :=
   (NNReal.coe_eq_zero _).1 (dist_self a)
 #align nndist_self nndist_self
+-/
 
+#print dist_nndist /-
 /-- Express `dist` in terms of `nndist`-/
 theorem dist_nndist (x y : α) : dist x y = ↑(nndist x y) :=
   rfl
 #align dist_nndist dist_nndist
+-/
 
+#print coe_nndist /-
 @[simp, norm_cast]
 theorem coe_nndist (x y : α) : ↑(nndist x y) = dist x y :=
   (dist_nndist x y).symm
 #align coe_nndist coe_nndist
+-/
 
+#print dist_lt_coe /-
 @[simp, norm_cast]
 theorem dist_lt_coe {x y : α} {c : ℝ≥0} : dist x y < c ↔ nndist x y < c :=
   Iff.rfl
 #align dist_lt_coe dist_lt_coe
+-/
 
+#print dist_le_coe /-
 @[simp, norm_cast]
 theorem dist_le_coe {x y : α} {c : ℝ≥0} : dist x y ≤ c ↔ nndist x y ≤ c :=
   Iff.rfl
 #align dist_le_coe dist_le_coe
+-/
 
+#print edist_lt_ofReal /-
 @[simp]
 theorem edist_lt_ofReal {x y : α} {r : ℝ} : edist x y < ENNReal.ofReal r ↔ dist x y < r := by
   rw [edist_dist, ENNReal.ofReal_lt_ofReal_iff_of_nonneg dist_nonneg]
 #align edist_lt_of_real edist_lt_ofReal
+-/
 
+#print edist_le_ofReal /-
 @[simp]
 theorem edist_le_ofReal {x y : α} {r : ℝ} (hr : 0 ≤ r) :
     edist x y ≤ ENNReal.ofReal r ↔ dist x y ≤ r := by
   rw [edist_dist, ENNReal.ofReal_le_ofReal_iff hr]
 #align edist_le_of_real edist_le_ofReal
+-/
 
 #print nndist_dist /-
 /-- Express `nndist` in terms of `dist`-/
@@ -450,18 +504,24 @@ theorem nndist_comm (x y : α) : nndist x y = nndist y x := by
 #align nndist_comm nndist_comm
 -/
 
+#print nndist_triangle /-
 /-- Triangle inequality for the nonnegative distance-/
 theorem nndist_triangle (x y z : α) : nndist x z ≤ nndist x y + nndist y z :=
   dist_triangle _ _ _
 #align nndist_triangle nndist_triangle
+-/
 
+#print nndist_triangle_left /-
 theorem nndist_triangle_left (x y z : α) : nndist x y ≤ nndist z x + nndist z y :=
   dist_triangle_left _ _ _
 #align nndist_triangle_left nndist_triangle_left
+-/
 
+#print nndist_triangle_right /-
 theorem nndist_triangle_right (x y z : α) : nndist x y ≤ nndist x z + nndist y z :=
   dist_triangle_right _ _ _
 #align nndist_triangle_right nndist_triangle_right
+-/
 
 #print dist_edist /-
 /-- Express `dist` in terms of `edist`-/
@@ -482,36 +542,51 @@ def ball (x : α) (ε : ℝ) : Set α :=
 #align metric.ball Metric.ball
 -/
 
+#print Metric.mem_ball /-
 @[simp]
 theorem mem_ball : y ∈ ball x ε ↔ dist y x < ε :=
   Iff.rfl
 #align metric.mem_ball Metric.mem_ball
+-/
 
+#print Metric.mem_ball' /-
 theorem mem_ball' : y ∈ ball x ε ↔ dist x y < ε := by rw [dist_comm, mem_ball]
 #align metric.mem_ball' Metric.mem_ball'
+-/
 
+#print Metric.pos_of_mem_ball /-
 theorem pos_of_mem_ball (hy : y ∈ ball x ε) : 0 < ε :=
   dist_nonneg.trans_lt hy
 #align metric.pos_of_mem_ball Metric.pos_of_mem_ball
+-/
 
+#print Metric.mem_ball_self /-
 theorem mem_ball_self (h : 0 < ε) : x ∈ ball x ε :=
   show dist x x < ε by rw [dist_self] <;> assumption
 #align metric.mem_ball_self Metric.mem_ball_self
+-/
 
+#print Metric.nonempty_ball /-
 @[simp]
 theorem nonempty_ball : (ball x ε).Nonempty ↔ 0 < ε :=
   ⟨fun ⟨x, hx⟩ => pos_of_mem_ball hx, fun h => ⟨x, mem_ball_self h⟩⟩
 #align metric.nonempty_ball Metric.nonempty_ball
+-/
 
+#print Metric.ball_eq_empty /-
 @[simp]
 theorem ball_eq_empty : ball x ε = ∅ ↔ ε ≤ 0 := by
   rw [← not_nonempty_iff_eq_empty, nonempty_ball, not_lt]
 #align metric.ball_eq_empty Metric.ball_eq_empty
+-/
 
+#print Metric.ball_zero /-
 @[simp]
 theorem ball_zero : ball x 0 = ∅ := by rw [ball_eq_empty]
 #align metric.ball_zero Metric.ball_zero
+-/
 
+#print Metric.exists_lt_mem_ball_of_mem_ball /-
 /-- If a point belongs to an open ball, then there is a strictly smaller radius whose ball also
 contains it.
 
@@ -521,26 +596,35 @@ theorem exists_lt_mem_ball_of_mem_ball (h : x ∈ ball y ε) : ∃ ε' < ε, x 
   simp only [mem_ball] at h ⊢
   exact ⟨(ε + dist x y) / 2, by linarith, by linarith⟩
 #align metric.exists_lt_mem_ball_of_mem_ball Metric.exists_lt_mem_ball_of_mem_ball
+-/
 
+#print Metric.ball_eq_ball /-
 theorem ball_eq_ball (ε : ℝ) (x : α) :
     UniformSpace.ball x {p | dist p.2 p.1 < ε} = Metric.ball x ε :=
   rfl
 #align metric.ball_eq_ball Metric.ball_eq_ball
+-/
 
+#print Metric.ball_eq_ball' /-
 theorem ball_eq_ball' (ε : ℝ) (x : α) :
     UniformSpace.ball x {p | dist p.1 p.2 < ε} = Metric.ball x ε := by ext;
   simp [dist_comm, UniformSpace.ball]
 #align metric.ball_eq_ball' Metric.ball_eq_ball'
+-/
 
+#print Metric.iUnion_ball_nat /-
 @[simp]
 theorem iUnion_ball_nat (x : α) : (⋃ n : ℕ, ball x n) = univ :=
   iUnion_eq_univ_iff.2 fun y => exists_nat_gt (dist y x)
 #align metric.Union_ball_nat Metric.iUnion_ball_nat
+-/
 
+#print Metric.iUnion_ball_nat_succ /-
 @[simp]
 theorem iUnion_ball_nat_succ (x : α) : (⋃ n : ℕ, ball x (n + 1)) = univ :=
   iUnion_eq_univ_iff.2 fun y => (exists_nat_gt (dist y x)).imp fun n hn => hn.trans (lt_add_one _)
 #align metric.Union_ball_nat_succ Metric.iUnion_ball_nat_succ
+-/
 
 #print Metric.closedBall /-
 /-- `closed_ball x ε` is the set of all points `y` with `dist y x ≤ ε` -/
@@ -549,13 +633,17 @@ def closedBall (x : α) (ε : ℝ) :=
 #align metric.closed_ball Metric.closedBall
 -/
 
+#print Metric.mem_closedBall /-
 @[simp]
 theorem mem_closedBall : y ∈ closedBall x ε ↔ dist y x ≤ ε :=
   Iff.rfl
 #align metric.mem_closed_ball Metric.mem_closedBall
+-/
 
+#print Metric.mem_closedBall' /-
 theorem mem_closedBall' : y ∈ closedBall x ε ↔ dist x y ≤ ε := by rw [dist_comm, mem_closed_ball]
 #align metric.mem_closed_ball' Metric.mem_closedBall'
+-/
 
 #print Metric.sphere /-
 /-- `sphere x ε` is the set of all points `y` with `dist y x = ε` -/
@@ -576,31 +664,43 @@ theorem mem_sphere' : y ∈ sphere x ε ↔ dist x y = ε := by rw [dist_comm, m
 #align metric.mem_sphere' Metric.mem_sphere'
 -/
 
+#print Metric.ne_of_mem_sphere /-
 theorem ne_of_mem_sphere (h : y ∈ sphere x ε) (hε : ε ≠ 0) : y ≠ x := by contrapose! hε; symm;
   simpa [hε] using h
 #align metric.ne_of_mem_sphere Metric.ne_of_mem_sphere
+-/
 
+#print Metric.sphere_eq_empty_of_subsingleton /-
 theorem sphere_eq_empty_of_subsingleton [Subsingleton α] (hε : ε ≠ 0) : sphere x ε = ∅ :=
   Set.eq_empty_iff_forall_not_mem.mpr fun y hy => ne_of_mem_sphere hy hε (Subsingleton.elim _ _)
 #align metric.sphere_eq_empty_of_subsingleton Metric.sphere_eq_empty_of_subsingleton
+-/
 
+#print Metric.sphere_isEmpty_of_subsingleton /-
 theorem sphere_isEmpty_of_subsingleton [Subsingleton α] (hε : ε ≠ 0) : IsEmpty (sphere x ε) := by
   simp only [sphere_eq_empty_of_subsingleton hε, Set.hasEmptyc.Emptyc.isEmpty α]
 #align metric.sphere_is_empty_of_subsingleton Metric.sphere_isEmpty_of_subsingleton
+-/
 
+#print Metric.mem_closedBall_self /-
 theorem mem_closedBall_self (h : 0 ≤ ε) : x ∈ closedBall x ε :=
   show dist x x ≤ ε by rw [dist_self] <;> assumption
 #align metric.mem_closed_ball_self Metric.mem_closedBall_self
+-/
 
+#print Metric.nonempty_closedBall /-
 @[simp]
 theorem nonempty_closedBall : (closedBall x ε).Nonempty ↔ 0 ≤ ε :=
   ⟨fun ⟨x, hx⟩ => dist_nonneg.trans hx, fun h => ⟨x, mem_closedBall_self h⟩⟩
 #align metric.nonempty_closed_ball Metric.nonempty_closedBall
+-/
 
+#print Metric.closedBall_eq_empty /-
 @[simp]
 theorem closedBall_eq_empty : closedBall x ε = ∅ ↔ ε < 0 := by
   rw [← not_nonempty_iff_eq_empty, nonempty_closed_ball, not_le]
 #align metric.closed_ball_eq_empty Metric.closedBall_eq_empty
+-/
 
 #print Metric.ball_subset_closedBall /-
 theorem ball_subset_closedBall : ball x ε ⊆ closedBall x ε := fun y (hy : _ < _) => le_of_lt hy
@@ -612,48 +712,66 @@ theorem sphere_subset_closedBall : sphere x ε ⊆ closedBall x ε := fun y => l
 #align metric.sphere_subset_closed_ball Metric.sphere_subset_closedBall
 -/
 
+#print Metric.closedBall_disjoint_ball /-
 theorem closedBall_disjoint_ball (h : δ + ε ≤ dist x y) : Disjoint (closedBall x δ) (ball y ε) :=
   Set.disjoint_left.mpr fun a ha1 ha2 =>
     (h.trans <| dist_triangle_left _ _ _).not_lt <| add_lt_add_of_le_of_lt ha1 ha2
 #align metric.closed_ball_disjoint_ball Metric.closedBall_disjoint_ball
+-/
 
+#print Metric.ball_disjoint_closedBall /-
 theorem ball_disjoint_closedBall (h : δ + ε ≤ dist x y) : Disjoint (ball x δ) (closedBall y ε) :=
   (closedBall_disjoint_ball <| by rwa [add_comm, dist_comm]).symm
 #align metric.ball_disjoint_closed_ball Metric.ball_disjoint_closedBall
+-/
 
+#print Metric.ball_disjoint_ball /-
 theorem ball_disjoint_ball (h : δ + ε ≤ dist x y) : Disjoint (ball x δ) (ball y ε) :=
   (closedBall_disjoint_ball h).mono_left ball_subset_closedBall
 #align metric.ball_disjoint_ball Metric.ball_disjoint_ball
+-/
 
+#print Metric.closedBall_disjoint_closedBall /-
 theorem closedBall_disjoint_closedBall (h : δ + ε < dist x y) :
     Disjoint (closedBall x δ) (closedBall y ε) :=
   Set.disjoint_left.mpr fun a ha1 ha2 =>
     h.not_le <| (dist_triangle_left _ _ _).trans <| add_le_add ha1 ha2
 #align metric.closed_ball_disjoint_closed_ball Metric.closedBall_disjoint_closedBall
+-/
 
+#print Metric.sphere_disjoint_ball /-
 theorem sphere_disjoint_ball : Disjoint (sphere x ε) (ball x ε) :=
   Set.disjoint_left.mpr fun y hy₁ hy₂ => absurd hy₁ <| ne_of_lt hy₂
 #align metric.sphere_disjoint_ball Metric.sphere_disjoint_ball
+-/
 
+#print Metric.ball_union_sphere /-
 @[simp]
 theorem ball_union_sphere : ball x ε ∪ sphere x ε = closedBall x ε :=
   Set.ext fun y => (@le_iff_lt_or_eq ℝ _ _ _).symm
 #align metric.ball_union_sphere Metric.ball_union_sphere
+-/
 
+#print Metric.sphere_union_ball /-
 @[simp]
 theorem sphere_union_ball : sphere x ε ∪ ball x ε = closedBall x ε := by
   rw [union_comm, ball_union_sphere]
 #align metric.sphere_union_ball Metric.sphere_union_ball
+-/
 
+#print Metric.closedBall_diff_sphere /-
 @[simp]
 theorem closedBall_diff_sphere : closedBall x ε \ sphere x ε = ball x ε := by
   rw [← ball_union_sphere, Set.union_diff_cancel_right sphere_disjoint_ball.symm.le_bot]
 #align metric.closed_ball_diff_sphere Metric.closedBall_diff_sphere
+-/
 
+#print Metric.closedBall_diff_ball /-
 @[simp]
 theorem closedBall_diff_ball : closedBall x ε \ ball x ε = sphere x ε := by
   rw [← ball_union_sphere, Set.union_diff_cancel_left sphere_disjoint_ball.symm.le_bot]
 #align metric.closed_ball_diff_ball Metric.closedBall_diff_ball
+-/
 
 #print Metric.mem_ball_comm /-
 theorem mem_ball_comm : x ∈ ball y ε ↔ y ∈ ball x ε := by rw [mem_ball', mem_ball]
@@ -671,25 +789,34 @@ theorem mem_sphere_comm : x ∈ sphere y ε ↔ y ∈ sphere x ε := by rw [mem_
 #align metric.mem_sphere_comm Metric.mem_sphere_comm
 -/
 
+#print Metric.ball_subset_ball /-
 theorem ball_subset_ball (h : ε₁ ≤ ε₂) : ball x ε₁ ⊆ ball x ε₂ := fun y (yx : _ < ε₁) =>
   lt_of_lt_of_le yx h
 #align metric.ball_subset_ball Metric.ball_subset_ball
+-/
 
+#print Metric.closedBall_eq_bInter_ball /-
 theorem closedBall_eq_bInter_ball : closedBall x ε = ⋂ δ > ε, ball x δ := by
   ext y <;> rw [mem_closed_ball, ← forall_lt_iff_le', mem_Inter₂] <;> rfl
 #align metric.closed_ball_eq_bInter_ball Metric.closedBall_eq_bInter_ball
+-/
 
+#print Metric.ball_subset_ball' /-
 theorem ball_subset_ball' (h : ε₁ + dist x y ≤ ε₂) : ball x ε₁ ⊆ ball y ε₂ := fun z hz =>
   calc
     dist z y ≤ dist z x + dist x y := dist_triangle _ _ _
     _ < ε₁ + dist x y := (add_lt_add_right hz _)
     _ ≤ ε₂ := h
 #align metric.ball_subset_ball' Metric.ball_subset_ball'
+-/
 
+#print Metric.closedBall_subset_closedBall /-
 theorem closedBall_subset_closedBall (h : ε₁ ≤ ε₂) : closedBall x ε₁ ⊆ closedBall x ε₂ :=
   fun y (yx : _ ≤ ε₁) => le_trans yx h
 #align metric.closed_ball_subset_closed_ball Metric.closedBall_subset_closedBall
+-/
 
+#print Metric.closedBall_subset_closedBall' /-
 theorem closedBall_subset_closedBall' (h : ε₁ + dist x y ≤ ε₂) :
     closedBall x ε₁ ⊆ closedBall y ε₂ := fun z hz =>
   calc
@@ -697,11 +824,15 @@ theorem closedBall_subset_closedBall' (h : ε₁ + dist x y ≤ ε₂) :
     _ ≤ ε₁ + dist x y := (add_le_add_right hz _)
     _ ≤ ε₂ := h
 #align metric.closed_ball_subset_closed_ball' Metric.closedBall_subset_closedBall'
+-/
 
+#print Metric.closedBall_subset_ball /-
 theorem closedBall_subset_ball (h : ε₁ < ε₂) : closedBall x ε₁ ⊆ ball x ε₂ :=
   fun y (yh : dist y x ≤ ε₁) => lt_of_le_of_lt yh h
 #align metric.closed_ball_subset_ball Metric.closedBall_subset_ball
+-/
 
+#print Metric.closedBall_subset_ball' /-
 theorem closedBall_subset_ball' (h : ε₁ + dist x y < ε₂) : closedBall x ε₁ ⊆ ball y ε₂ :=
   fun z hz =>
   calc
@@ -709,7 +840,9 @@ theorem closedBall_subset_ball' (h : ε₁ + dist x y < ε₂) : closedBall x ε
     _ ≤ ε₁ + dist x y := (add_le_add_right hz _)
     _ < ε₂ := h
 #align metric.closed_ball_subset_ball' Metric.closedBall_subset_ball'
+-/
 
+#print Metric.dist_le_add_of_nonempty_closedBall_inter_closedBall /-
 theorem dist_le_add_of_nonempty_closedBall_inter_closedBall
     (h : (closedBall x ε₁ ∩ closedBall y ε₂).Nonempty) : dist x y ≤ ε₁ + ε₂ :=
   let ⟨z, hz⟩ := h
@@ -717,7 +850,9 @@ theorem dist_le_add_of_nonempty_closedBall_inter_closedBall
     dist x y ≤ dist z x + dist z y := dist_triangle_left _ _ _
     _ ≤ ε₁ + ε₂ := add_le_add hz.1 hz.2
 #align metric.dist_le_add_of_nonempty_closed_ball_inter_closed_ball Metric.dist_le_add_of_nonempty_closedBall_inter_closedBall
+-/
 
+#print Metric.dist_lt_add_of_nonempty_closedBall_inter_ball /-
 theorem dist_lt_add_of_nonempty_closedBall_inter_ball (h : (closedBall x ε₁ ∩ ball y ε₂).Nonempty) :
     dist x y < ε₁ + ε₂ :=
   let ⟨z, hz⟩ := h
@@ -725,41 +860,56 @@ theorem dist_lt_add_of_nonempty_closedBall_inter_ball (h : (closedBall x ε₁ 
     dist x y ≤ dist z x + dist z y := dist_triangle_left _ _ _
     _ < ε₁ + ε₂ := add_lt_add_of_le_of_lt hz.1 hz.2
 #align metric.dist_lt_add_of_nonempty_closed_ball_inter_ball Metric.dist_lt_add_of_nonempty_closedBall_inter_ball
+-/
 
+#print Metric.dist_lt_add_of_nonempty_ball_inter_closedBall /-
 theorem dist_lt_add_of_nonempty_ball_inter_closedBall (h : (ball x ε₁ ∩ closedBall y ε₂).Nonempty) :
     dist x y < ε₁ + ε₂ := by
   rw [inter_comm] at h 
   rw [add_comm, dist_comm]
   exact dist_lt_add_of_nonempty_closed_ball_inter_ball h
 #align metric.dist_lt_add_of_nonempty_ball_inter_closed_ball Metric.dist_lt_add_of_nonempty_ball_inter_closedBall
+-/
 
+#print Metric.dist_lt_add_of_nonempty_ball_inter_ball /-
 theorem dist_lt_add_of_nonempty_ball_inter_ball (h : (ball x ε₁ ∩ ball y ε₂).Nonempty) :
     dist x y < ε₁ + ε₂ :=
   dist_lt_add_of_nonempty_closedBall_inter_ball <|
     h.mono (inter_subset_inter ball_subset_closedBall Subset.rfl)
 #align metric.dist_lt_add_of_nonempty_ball_inter_ball Metric.dist_lt_add_of_nonempty_ball_inter_ball
+-/
 
+#print Metric.iUnion_closedBall_nat /-
 @[simp]
 theorem iUnion_closedBall_nat (x : α) : (⋃ n : ℕ, closedBall x n) = univ :=
   iUnion_eq_univ_iff.2 fun y => exists_nat_ge (dist y x)
 #align metric.Union_closed_ball_nat Metric.iUnion_closedBall_nat
+-/
 
+#print Metric.iUnion_inter_closedBall_nat /-
 theorem iUnion_inter_closedBall_nat (s : Set α) (x : α) : (⋃ n : ℕ, s ∩ closedBall x n) = s := by
   rw [← inter_Union, Union_closed_ball_nat, inter_univ]
 #align metric.Union_inter_closed_ball_nat Metric.iUnion_inter_closedBall_nat
+-/
 
+#print Metric.ball_subset /-
 theorem ball_subset (h : dist x y ≤ ε₂ - ε₁) : ball x ε₁ ⊆ ball y ε₂ := fun z zx => by
   rw [← add_sub_cancel'_right ε₁ ε₂] <;>
     exact lt_of_le_of_lt (dist_triangle z x y) (add_lt_add_of_lt_of_le zx h)
 #align metric.ball_subset Metric.ball_subset
+-/
 
+#print Metric.ball_half_subset /-
 theorem ball_half_subset (y) (h : y ∈ ball x (ε / 2)) : ball y (ε / 2) ⊆ ball x ε :=
   ball_subset <| by rw [sub_self_div_two] <;> exact le_of_lt h
 #align metric.ball_half_subset Metric.ball_half_subset
+-/
 
+#print Metric.exists_ball_subset_ball /-
 theorem exists_ball_subset_ball (h : y ∈ ball x ε) : ∃ ε' > 0, ball y ε' ⊆ ball x ε :=
   ⟨_, sub_pos.2 h, ball_subset <| by rw [sub_sub_self]⟩
 #align metric.exists_ball_subset_ball Metric.exists_ball_subset_ball
+-/
 
 #print Metric.forall_of_forall_mem_closedBall /-
 /-- If a property holds for all points in closed balls of arbitrarily large radii, then it holds for
@@ -785,24 +935,30 @@ theorem forall_of_forall_mem_ball (p : α → Prop) (x : α)
 #align metric.forall_of_forall_mem_ball Metric.forall_of_forall_mem_ball
 -/
 
+#print Metric.isBounded_iff /-
 theorem isBounded_iff {s : Set α} :
     IsBounded s ↔ ∃ C : ℝ, ∀ ⦃x⦄, x ∈ s → ∀ ⦃y⦄, y ∈ s → dist x y ≤ C := by
   rw [is_bounded_def, ← Filter.mem_sets, (@PseudoMetricSpace.cobounded_sets α _).out, mem_set_of_eq,
     compl_compl]
 #align metric.is_bounded_iff Metric.isBounded_iff
+-/
 
+#print Metric.isBounded_iff_eventually /-
 theorem isBounded_iff_eventually {s : Set α} :
     IsBounded s ↔ ∀ᶠ C in atTop, ∀ ⦃x⦄, x ∈ s → ∀ ⦃y⦄, y ∈ s → dist x y ≤ C :=
   isBounded_iff.trans
     ⟨fun ⟨C, h⟩ => eventually_atTop.2 ⟨C, fun C' hC' x hx y hy => (h hx hy).trans hC'⟩,
       Eventually.exists⟩
 #align metric.is_bounded_iff_eventually Metric.isBounded_iff_eventually
+-/
 
+#print Metric.isBounded_iff_exists_ge /-
 theorem isBounded_iff_exists_ge {s : Set α} (c : ℝ) :
     IsBounded s ↔ ∃ C, c ≤ C ∧ ∀ ⦃x⦄, x ∈ s → ∀ ⦃y⦄, y ∈ s → dist x y ≤ C :=
   ⟨fun h => ((eventually_ge_atTop c).And (isBounded_iff_eventually.1 h)).exists, fun h =>
     isBounded_iff.2 <| h.imp fun _ => And.right⟩
 #align metric.is_bounded_iff_exists_ge Metric.isBounded_iff_exists_ge
+-/
 
 #print Metric.isBounded_iff_nndist /-
 theorem isBounded_iff_nndist {s : Set α} :
@@ -820,13 +976,16 @@ theorem toUniformSpace_eq :
 #align metric.to_uniform_space_eq Metric.toUniformSpace_eq
 -/
 
+#print Metric.uniformity_basis_dist /-
 theorem uniformity_basis_dist :
     (𝓤 α).HasBasis (fun ε : ℝ => 0 < ε) fun ε => {p : α × α | dist p.1 p.2 < ε} :=
   by
   rw [to_uniform_space_eq]
   exact UniformSpace.hasBasis_ofFun (exists_gt _) _ _ _ _ _
 #align metric.uniformity_basis_dist Metric.uniformity_basis_dist
+-/
 
+#print Metric.mk_uniformity_basis /-
 /-- Given `f : β → ℝ`, if `f` sends `{i | p i}` to a set of positive numbers
 accumulating to zero, then `f i`-neighborhoods of the diagonal form a basis of `𝓤 α`.
 
@@ -843,41 +1002,53 @@ protected theorem mk_uniformity_basis {β : Type _} {p : β → Prop} {f : β 
     exact ⟨i, hi, fun x (hx : _ < _) => hε <| lt_of_lt_of_le hx H⟩
   · exact fun ⟨i, hi, H⟩ => ⟨f i, hf₀ i hi, H⟩
 #align metric.mk_uniformity_basis Metric.mk_uniformity_basis
+-/
 
+#print Metric.uniformity_basis_dist_rat /-
 theorem uniformity_basis_dist_rat :
     (𝓤 α).HasBasis (fun r : ℚ => 0 < r) fun r => {p : α × α | dist p.1 p.2 < r} :=
   Metric.mk_uniformity_basis (fun _ => Rat.cast_pos.2) fun ε hε =>
     let ⟨r, hr0, hrε⟩ := exists_rat_btwn hε
     ⟨r, Rat.cast_pos.1 hr0, hrε.le⟩
 #align metric.uniformity_basis_dist_rat Metric.uniformity_basis_dist_rat
+-/
 
+#print Metric.uniformity_basis_dist_inv_nat_succ /-
 theorem uniformity_basis_dist_inv_nat_succ :
     (𝓤 α).HasBasis (fun _ => True) fun n : ℕ => {p : α × α | dist p.1 p.2 < 1 / (↑n + 1)} :=
   Metric.mk_uniformity_basis (fun n _ => div_pos zero_lt_one <| Nat.cast_add_one_pos n) fun ε ε0 =>
     (exists_nat_one_div_lt ε0).imp fun n hn => ⟨trivial, le_of_lt hn⟩
 #align metric.uniformity_basis_dist_inv_nat_succ Metric.uniformity_basis_dist_inv_nat_succ
+-/
 
+#print Metric.uniformity_basis_dist_inv_nat_pos /-
 theorem uniformity_basis_dist_inv_nat_pos :
     (𝓤 α).HasBasis (fun n : ℕ => 0 < n) fun n : ℕ => {p : α × α | dist p.1 p.2 < 1 / ↑n} :=
   Metric.mk_uniformity_basis (fun n hn => div_pos zero_lt_one <| Nat.cast_pos.2 hn) fun ε ε0 =>
     let ⟨n, hn⟩ := exists_nat_one_div_lt ε0
     ⟨n + 1, Nat.succ_pos n, by exact_mod_cast hn.le⟩
 #align metric.uniformity_basis_dist_inv_nat_pos Metric.uniformity_basis_dist_inv_nat_pos
+-/
 
+#print Metric.uniformity_basis_dist_pow /-
 theorem uniformity_basis_dist_pow {r : ℝ} (h0 : 0 < r) (h1 : r < 1) :
     (𝓤 α).HasBasis (fun n : ℕ => True) fun n : ℕ => {p : α × α | dist p.1 p.2 < r ^ n} :=
   Metric.mk_uniformity_basis (fun n hn => pow_pos h0 _) fun ε ε0 =>
     let ⟨n, hn⟩ := exists_pow_lt_of_lt_one ε0 h1
     ⟨n, trivial, hn.le⟩
 #align metric.uniformity_basis_dist_pow Metric.uniformity_basis_dist_pow
+-/
 
+#print Metric.uniformity_basis_dist_lt /-
 theorem uniformity_basis_dist_lt {R : ℝ} (hR : 0 < R) :
     (𝓤 α).HasBasis (fun r : ℝ => 0 < r ∧ r < R) fun r => {p : α × α | dist p.1 p.2 < r} :=
   Metric.mk_uniformity_basis (fun r => And.left) fun r hr =>
     ⟨min r (R / 2), ⟨lt_min hr (half_pos hR), min_lt_iff.2 <| Or.inr (half_lt_self hR)⟩,
       min_le_left _ _⟩
 #align metric.uniformity_basis_dist_lt Metric.uniformity_basis_dist_lt
+-/
 
+#print Metric.mk_uniformity_basis_le /-
 /-- Given `f : β → ℝ`, if `f` sends `{i | p i}` to a set of positive numbers
 accumulating to zero, then closed neighborhoods of the diagonal of sizes `{f i | p i}`
 form a basis of `𝓤 α`.
@@ -896,50 +1067,66 @@ protected theorem mk_uniformity_basis_le {β : Type _} {p : β → Prop} {f : β
     exact ⟨i, hi, fun x (hx : _ ≤ _) => hε <| lt_of_le_of_lt (le_trans hx H) hε'.2⟩
   · exact fun ⟨i, hi, H⟩ => ⟨f i, hf₀ i hi, fun x (hx : _ < _) => H (le_of_lt hx)⟩
 #align metric.mk_uniformity_basis_le Metric.mk_uniformity_basis_le
+-/
 
+#print Metric.uniformity_basis_dist_le /-
 /-- Contant size closed neighborhoods of the diagonal form a basis
 of the uniformity filter. -/
 theorem uniformity_basis_dist_le :
     (𝓤 α).HasBasis (fun ε : ℝ => 0 < ε) fun ε => {p : α × α | dist p.1 p.2 ≤ ε} :=
   Metric.mk_uniformity_basis_le (fun _ => id) fun ε ε₀ => ⟨ε, ε₀, le_refl ε⟩
 #align metric.uniformity_basis_dist_le Metric.uniformity_basis_dist_le
+-/
 
+#print Metric.uniformity_basis_dist_le_pow /-
 theorem uniformity_basis_dist_le_pow {r : ℝ} (h0 : 0 < r) (h1 : r < 1) :
     (𝓤 α).HasBasis (fun n : ℕ => True) fun n : ℕ => {p : α × α | dist p.1 p.2 ≤ r ^ n} :=
   Metric.mk_uniformity_basis_le (fun n hn => pow_pos h0 _) fun ε ε0 =>
     let ⟨n, hn⟩ := exists_pow_lt_of_lt_one ε0 h1
     ⟨n, trivial, hn.le⟩
 #align metric.uniformity_basis_dist_le_pow Metric.uniformity_basis_dist_le_pow
+-/
 
+#print Metric.mem_uniformity_dist /-
 theorem mem_uniformity_dist {s : Set (α × α)} :
     s ∈ 𝓤 α ↔ ∃ ε > 0, ∀ {a b : α}, dist a b < ε → (a, b) ∈ s :=
   uniformity_basis_dist.mem_uniformity_iff
 #align metric.mem_uniformity_dist Metric.mem_uniformity_dist
+-/
 
+#print Metric.dist_mem_uniformity /-
 /-- A constant size neighborhood of the diagonal is an entourage. -/
 theorem dist_mem_uniformity {ε : ℝ} (ε0 : 0 < ε) : {p : α × α | dist p.1 p.2 < ε} ∈ 𝓤 α :=
   mem_uniformity_dist.2 ⟨ε, ε0, fun a b => id⟩
 #align metric.dist_mem_uniformity Metric.dist_mem_uniformity
+-/
 
+#print Metric.uniformContinuous_iff /-
 theorem uniformContinuous_iff [PseudoMetricSpace β] {f : α → β} :
     UniformContinuous f ↔ ∀ ε > 0, ∃ δ > 0, ∀ {a b : α}, dist a b < δ → dist (f a) (f b) < ε :=
   uniformity_basis_dist.uniformContinuous_iff uniformity_basis_dist
 #align metric.uniform_continuous_iff Metric.uniformContinuous_iff
+-/
 
 /- ./././Mathport/Syntax/Translate/Basic.lean:638:2: warning: expanding binder collection (x y «expr ∈ » s) -/
+#print Metric.uniformContinuousOn_iff /-
 theorem uniformContinuousOn_iff [PseudoMetricSpace β] {f : α → β} {s : Set α} :
     UniformContinuousOn f s ↔
       ∀ ε > 0, ∃ δ > 0, ∀ (x) (_ : x ∈ s) (y) (_ : y ∈ s), dist x y < δ → dist (f x) (f y) < ε :=
   Metric.uniformity_basis_dist.uniformContinuousOn_iff Metric.uniformity_basis_dist
 #align metric.uniform_continuous_on_iff Metric.uniformContinuousOn_iff
+-/
 
 /- ./././Mathport/Syntax/Translate/Basic.lean:638:2: warning: expanding binder collection (x y «expr ∈ » s) -/
+#print Metric.uniformContinuousOn_iff_le /-
 theorem uniformContinuousOn_iff_le [PseudoMetricSpace β] {f : α → β} {s : Set α} :
     UniformContinuousOn f s ↔
       ∀ ε > 0, ∃ δ > 0, ∀ (x) (_ : x ∈ s) (y) (_ : y ∈ s), dist x y ≤ δ → dist (f x) (f y) ≤ ε :=
   Metric.uniformity_basis_dist_le.uniformContinuousOn_iff Metric.uniformity_basis_dist_le
 #align metric.uniform_continuous_on_iff_le Metric.uniformContinuousOn_iff_le
+-/
 
+#print Metric.uniformEmbedding_iff /-
 theorem uniformEmbedding_iff [PseudoMetricSpace β] {f : α → β} :
     UniformEmbedding f ↔
       Function.Injective f ∧
@@ -948,7 +1135,9 @@ theorem uniformEmbedding_iff [PseudoMetricSpace β] {f : α → β} :
   simp only [uniformity_basis_dist.uniform_embedding_iff uniformity_basis_dist, exists_prop]
   rfl
 #align metric.uniform_embedding_iff Metric.uniformEmbedding_iff
+-/
 
+#print Metric.controlled_of_uniformEmbedding /-
 /-- If a map between pseudometric spaces is a uniform embedding then the distance between `f x`
 and `f y` is controlled in terms of the distance between `x` and `y`. -/
 theorem controlled_of_uniformEmbedding [PseudoMetricSpace β] {f : α → β} :
@@ -959,7 +1148,9 @@ theorem controlled_of_uniformEmbedding [PseudoMetricSpace β] {f : α → β} :
   intro h
   exact ⟨uniformContinuous_iff.1 (uniformEmbedding_iff.1 h).2.1, (uniformEmbedding_iff.1 h).2.2⟩
 #align metric.controlled_of_uniform_embedding Metric.controlled_of_uniformEmbedding
+-/
 
+#print Metric.totallyBounded_iff /-
 theorem totallyBounded_iff {s : Set α} :
     TotallyBounded s ↔ ∀ ε > 0, ∃ t : Set α, t.Finite ∧ s ⊆ ⋃ y ∈ t, ball y ε :=
   ⟨fun H ε ε0 => H _ (dist_mem_uniformity ε0), fun H r ru =>
@@ -967,7 +1158,9 @@ theorem totallyBounded_iff {s : Set α} :
     let ⟨t, ft, h⟩ := H ε ε0
     ⟨t, ft, h.trans <| iUnion₂_mono fun y yt z => hε⟩⟩
 #align metric.totally_bounded_iff Metric.totallyBounded_iff
+-/
 
+#print Metric.totallyBounded_of_finite_discretization /-
 /-- A pseudometric space is totally bounded if one can reconstruct up to any ε>0 any element of the
 space from finitely many data. -/
 theorem totallyBounded_of_finite_discretization {s : Set α}
@@ -989,8 +1182,10 @@ theorem totallyBounded_of_finite_discretization {s : Set α}
   simp only [Set.mem_iUnion, Set.mem_range]
   exact ⟨_, ⟨F ⟨x, xs⟩, rfl⟩, hF _ _ this.symm⟩
 #align metric.totally_bounded_of_finite_discretization Metric.totallyBounded_of_finite_discretization
+-/
 
 /- ./././Mathport/Syntax/Translate/Basic.lean:638:2: warning: expanding binder collection (t «expr ⊆ » s) -/
+#print Metric.finite_approx_of_totallyBounded /-
 theorem finite_approx_of_totallyBounded {s : Set α} (hs : TotallyBounded s) :
     ∀ ε > 0, ∃ (t : _) (_ : t ⊆ s), Set.Finite t ∧ s ⊆ ⋃ y ∈ t, ball y ε :=
   by
@@ -998,7 +1193,9 @@ theorem finite_approx_of_totallyBounded {s : Set α} (hs : TotallyBounded s) :
   rw [totallyBounded_iff_subset] at hs 
   exact hs _ (dist_mem_uniformity ε_pos)
 #align metric.finite_approx_of_totally_bounded Metric.finite_approx_of_totallyBounded
+-/
 
+#print Metric.tendstoUniformlyOnFilter_iff /-
 /-- Expressing uniform convergence using `dist` -/
 theorem tendstoUniformlyOnFilter_iff {ι : Type _} {F : ι → β → α} {f : β → α} {p : Filter ι}
     {p' : Filter β} :
@@ -1009,7 +1206,9 @@ theorem tendstoUniformlyOnFilter_iff {ι : Type _} {F : ι → β → α} {f : 
   rcases mem_uniformity_dist.1 hu with ⟨ε, εpos, hε⟩
   refine' (H ε εpos).mono fun n hn => hε hn
 #align metric.tendsto_uniformly_on_filter_iff Metric.tendstoUniformlyOnFilter_iff
+-/
 
+#print Metric.tendstoLocallyUniformlyOn_iff /-
 /-- Expressing locally uniform convergence on a set using `dist`. -/
 theorem tendstoLocallyUniformlyOn_iff {ι : Type _} [TopologicalSpace β] {F : ι → β → α} {f : β → α}
     {p : Filter ι} {s : Set β} :
@@ -1021,7 +1220,9 @@ theorem tendstoLocallyUniformlyOn_iff {ι : Type _} [TopologicalSpace β] {F : 
   rcases H ε εpos x hx with ⟨t, ht, Ht⟩
   exact ⟨t, ht, Ht.mono fun n hs x hx => hε (hs x hx)⟩
 #align metric.tendsto_locally_uniformly_on_iff Metric.tendstoLocallyUniformlyOn_iff
+-/
 
+#print Metric.tendstoUniformlyOn_iff /-
 /-- Expressing uniform convergence on a set using `dist`. -/
 theorem tendstoUniformlyOn_iff {ι : Type _} {F : ι → β → α} {f : β → α} {p : Filter ι} {s : Set β} :
     TendstoUniformlyOn F f p s ↔ ∀ ε > 0, ∀ᶠ n in p, ∀ x ∈ s, dist (f x) (F n x) < ε :=
@@ -1030,7 +1231,9 @@ theorem tendstoUniformlyOn_iff {ι : Type _} {F : ι → β → α} {f : β →
   rcases mem_uniformity_dist.1 hu with ⟨ε, εpos, hε⟩
   exact (H ε εpos).mono fun n hs x hx => hε (hs x hx)
 #align metric.tendsto_uniformly_on_iff Metric.tendstoUniformlyOn_iff
+-/
 
+#print Metric.tendstoLocallyUniformly_iff /-
 /-- Expressing locally uniform convergence using `dist`. -/
 theorem tendstoLocallyUniformly_iff {ι : Type _} [TopologicalSpace β] {F : ι → β → α} {f : β → α}
     {p : Filter ι} :
@@ -1040,37 +1243,51 @@ theorem tendstoLocallyUniformly_iff {ι : Type _} [TopologicalSpace β] {F : ι
   simp only [← tendstoLocallyUniformlyOn_univ, tendsto_locally_uniformly_on_iff, nhdsWithin_univ,
     mem_univ, forall_const, exists_prop]
 #align metric.tendsto_locally_uniformly_iff Metric.tendstoLocallyUniformly_iff
+-/
 
+#print Metric.tendstoUniformly_iff /-
 /-- Expressing uniform convergence using `dist`. -/
 theorem tendstoUniformly_iff {ι : Type _} {F : ι → β → α} {f : β → α} {p : Filter ι} :
     TendstoUniformly F f p ↔ ∀ ε > 0, ∀ᶠ n in p, ∀ x, dist (f x) (F n x) < ε := by
   rw [← tendstoUniformlyOn_univ, tendsto_uniformly_on_iff]; simp
 #align metric.tendsto_uniformly_iff Metric.tendstoUniformly_iff
+-/
 
 /- ./././Mathport/Syntax/Translate/Basic.lean:638:2: warning: expanding binder collection (x y «expr ∈ » t) -/
+#print Metric.cauchy_iff /-
 protected theorem cauchy_iff {f : Filter α} :
     Cauchy f ↔ NeBot f ∧ ∀ ε > 0, ∃ t ∈ f, ∀ (x) (_ : x ∈ t) (y) (_ : y ∈ t), dist x y < ε :=
   uniformity_basis_dist.cauchy_iff
 #align metric.cauchy_iff Metric.cauchy_iff
+-/
 
+#print Metric.nhds_basis_ball /-
 theorem nhds_basis_ball : (𝓝 x).HasBasis (fun ε : ℝ => 0 < ε) (ball x) :=
   nhds_basis_uniformity uniformity_basis_dist
 #align metric.nhds_basis_ball Metric.nhds_basis_ball
+-/
 
+#print Metric.mem_nhds_iff /-
 theorem mem_nhds_iff : s ∈ 𝓝 x ↔ ∃ ε > 0, ball x ε ⊆ s :=
   nhds_basis_ball.mem_iff
 #align metric.mem_nhds_iff Metric.mem_nhds_iff
+-/
 
+#print Metric.eventually_nhds_iff /-
 theorem eventually_nhds_iff {p : α → Prop} :
     (∀ᶠ y in 𝓝 x, p y) ↔ ∃ ε > 0, ∀ ⦃y⦄, dist y x < ε → p y :=
   mem_nhds_iff
 #align metric.eventually_nhds_iff Metric.eventually_nhds_iff
+-/
 
+#print Metric.eventually_nhds_iff_ball /-
 theorem eventually_nhds_iff_ball {p : α → Prop} :
     (∀ᶠ y in 𝓝 x, p y) ↔ ∃ ε > 0, ∀ y ∈ ball x ε, p y :=
   mem_nhds_iff
 #align metric.eventually_nhds_iff_ball Metric.eventually_nhds_iff_ball
+-/
 
+#print Metric.eventually_prod_nhds_iff /-
 /-- A version of `filter.eventually_prod_iff` where the second filter consists of neighborhoods
 in a pseudo-metric space.-/
 theorem eventually_prod_nhds_iff {f : Filter ι} {x₀ : α} {p : ι × α → Prop} :
@@ -1084,7 +1301,9 @@ theorem eventually_prod_nhds_iff {f : Filter ι} {x₀ : α} {p : ι × α → P
   · rintro ⟨r, ⟨ε, hε, hεr⟩, hp⟩; exact ⟨ε, hε, fun i hi x hx => hp hi <| hεr hx⟩
   · rintro ⟨ε, hε, hp⟩; exact ⟨fun x => dist x x₀ < ε, ⟨ε, hε, fun y => id⟩, @hp⟩
 #align metric.eventually_prod_nhds_iff Metric.eventually_prod_nhds_iff
+-/
 
+#print Metric.eventually_nhds_prod_iff /-
 /-- A version of `filter.eventually_prod_iff` where the first filter consists of neighborhoods
 in a pseudo-metric space.-/
 theorem eventually_nhds_prod_iff {ι α} [PseudoMetricSpace α] {f : Filter ι} {x₀ : α}
@@ -1097,34 +1316,47 @@ theorem eventually_nhds_prod_iff {ι α} [PseudoMetricSpace α] {f : Filter ι}
   constructor <;>
     · rintro ⟨a1, a2, a3, a4, a5⟩; refine' ⟨a3, a4, a1, a2, fun b1 b2 b3 b4 => a5 b4 b2⟩
 #align metric.eventually_nhds_prod_iff Metric.eventually_nhds_prod_iff
+-/
 
+#print Metric.nhds_basis_closedBall /-
 theorem nhds_basis_closedBall : (𝓝 x).HasBasis (fun ε : ℝ => 0 < ε) (closedBall x) :=
   nhds_basis_uniformity uniformity_basis_dist_le
 #align metric.nhds_basis_closed_ball Metric.nhds_basis_closedBall
+-/
 
+#print Metric.nhds_basis_ball_inv_nat_succ /-
 theorem nhds_basis_ball_inv_nat_succ :
     (𝓝 x).HasBasis (fun _ => True) fun n : ℕ => ball x (1 / (↑n + 1)) :=
   nhds_basis_uniformity uniformity_basis_dist_inv_nat_succ
 #align metric.nhds_basis_ball_inv_nat_succ Metric.nhds_basis_ball_inv_nat_succ
+-/
 
+#print Metric.nhds_basis_ball_inv_nat_pos /-
 theorem nhds_basis_ball_inv_nat_pos :
     (𝓝 x).HasBasis (fun n => 0 < n) fun n : ℕ => ball x (1 / ↑n) :=
   nhds_basis_uniformity uniformity_basis_dist_inv_nat_pos
 #align metric.nhds_basis_ball_inv_nat_pos Metric.nhds_basis_ball_inv_nat_pos
+-/
 
+#print Metric.nhds_basis_ball_pow /-
 theorem nhds_basis_ball_pow {r : ℝ} (h0 : 0 < r) (h1 : r < 1) :
     (𝓝 x).HasBasis (fun n => True) fun n : ℕ => ball x (r ^ n) :=
   nhds_basis_uniformity (uniformity_basis_dist_pow h0 h1)
 #align metric.nhds_basis_ball_pow Metric.nhds_basis_ball_pow
+-/
 
+#print Metric.nhds_basis_closedBall_pow /-
 theorem nhds_basis_closedBall_pow {r : ℝ} (h0 : 0 < r) (h1 : r < 1) :
     (𝓝 x).HasBasis (fun n => True) fun n : ℕ => closedBall x (r ^ n) :=
   nhds_basis_uniformity (uniformity_basis_dist_le_pow h0 h1)
 #align metric.nhds_basis_closed_ball_pow Metric.nhds_basis_closedBall_pow
+-/
 
+#print Metric.isOpen_iff /-
 theorem isOpen_iff : IsOpen s ↔ ∀ x ∈ s, ∃ ε > 0, ball x ε ⊆ s := by
   simp only [isOpen_iff_mem_nhds, mem_nhds_iff]
 #align metric.is_open_iff Metric.isOpen_iff
+-/
 
 #print Metric.isOpen_ball /-
 theorem isOpen_ball : IsOpen (ball x ε) :=
@@ -1132,13 +1364,17 @@ theorem isOpen_ball : IsOpen (ball x ε) :=
 #align metric.is_open_ball Metric.isOpen_ball
 -/
 
+#print Metric.ball_mem_nhds /-
 theorem ball_mem_nhds (x : α) {ε : ℝ} (ε0 : 0 < ε) : ball x ε ∈ 𝓝 x :=
   isOpen_ball.mem_nhds (mem_ball_self ε0)
 #align metric.ball_mem_nhds Metric.ball_mem_nhds
+-/
 
+#print Metric.closedBall_mem_nhds /-
 theorem closedBall_mem_nhds (x : α) {ε : ℝ} (ε0 : 0 < ε) : closedBall x ε ∈ 𝓝 x :=
   mem_of_superset (ball_mem_nhds x ε0) ball_subset_closedBall
 #align metric.closed_ball_mem_nhds Metric.closedBall_mem_nhds
+-/
 
 #print Metric.closedBall_mem_nhds_of_mem /-
 theorem closedBall_mem_nhds_of_mem {x c : α} {ε : ℝ} (h : x ∈ ball c ε) : closedBall c ε ∈ 𝓝 x :=
@@ -1146,22 +1382,29 @@ theorem closedBall_mem_nhds_of_mem {x c : α} {ε : ℝ} (h : x ∈ ball c ε) :
 #align metric.closed_ball_mem_nhds_of_mem Metric.closedBall_mem_nhds_of_mem
 -/
 
+#print Metric.nhdsWithin_basis_ball /-
 theorem nhdsWithin_basis_ball {s : Set α} :
     (𝓝[s] x).HasBasis (fun ε : ℝ => 0 < ε) fun ε => ball x ε ∩ s :=
   nhdsWithin_hasBasis nhds_basis_ball s
 #align metric.nhds_within_basis_ball Metric.nhdsWithin_basis_ball
+-/
 
+#print Metric.mem_nhdsWithin_iff /-
 theorem mem_nhdsWithin_iff {t : Set α} : s ∈ 𝓝[t] x ↔ ∃ ε > 0, ball x ε ∩ t ⊆ s :=
   nhdsWithin_basis_ball.mem_iff
 #align metric.mem_nhds_within_iff Metric.mem_nhdsWithin_iff
+-/
 
+#print Metric.tendsto_nhdsWithin_nhdsWithin /-
 theorem tendsto_nhdsWithin_nhdsWithin [PseudoMetricSpace β] {t : Set β} {f : α → β} {a b} :
     Tendsto f (𝓝[s] a) (𝓝[t] b) ↔
       ∀ ε > 0, ∃ δ > 0, ∀ {x : α}, x ∈ s → dist x a < δ → f x ∈ t ∧ dist (f x) b < ε :=
   (nhdsWithin_basis_ball.tendsto_iffₓ nhdsWithin_basis_ball).trans <|
     forall₂_congr fun ε hε => exists₂_congr fun δ hδ => forall_congr' fun x => by simp <;> itauto
 #align metric.tendsto_nhds_within_nhds_within Metric.tendsto_nhdsWithin_nhdsWithin
+-/
 
+#print Metric.tendsto_nhdsWithin_nhds /-
 theorem tendsto_nhdsWithin_nhds [PseudoMetricSpace β] {f : α → β} {a b} :
     Tendsto f (𝓝[s] a) (𝓝 b) ↔
       ∀ ε > 0, ∃ δ > 0, ∀ {x : α}, x ∈ s → dist x a < δ → dist (f x) b < ε :=
@@ -1169,63 +1412,87 @@ theorem tendsto_nhdsWithin_nhds [PseudoMetricSpace β] {f : α → β} {a b} :
   rw [← nhdsWithin_univ b, tendsto_nhds_within_nhds_within]
   simp only [mem_univ, true_and_iff]
 #align metric.tendsto_nhds_within_nhds Metric.tendsto_nhdsWithin_nhds
+-/
 
+#print Metric.tendsto_nhds_nhds /-
 theorem tendsto_nhds_nhds [PseudoMetricSpace β] {f : α → β} {a b} :
     Tendsto f (𝓝 a) (𝓝 b) ↔ ∀ ε > 0, ∃ δ > 0, ∀ {x : α}, dist x a < δ → dist (f x) b < ε :=
   nhds_basis_ball.tendsto_iffₓ nhds_basis_ball
 #align metric.tendsto_nhds_nhds Metric.tendsto_nhds_nhds
+-/
 
+#print Metric.continuousAt_iff /-
 theorem continuousAt_iff [PseudoMetricSpace β] {f : α → β} {a : α} :
     ContinuousAt f a ↔ ∀ ε > 0, ∃ δ > 0, ∀ {x : α}, dist x a < δ → dist (f x) (f a) < ε := by
   rw [ContinuousAt, tendsto_nhds_nhds]
 #align metric.continuous_at_iff Metric.continuousAt_iff
+-/
 
+#print Metric.continuousWithinAt_iff /-
 theorem continuousWithinAt_iff [PseudoMetricSpace β] {f : α → β} {a : α} {s : Set α} :
     ContinuousWithinAt f s a ↔
       ∀ ε > 0, ∃ δ > 0, ∀ {x : α}, x ∈ s → dist x a < δ → dist (f x) (f a) < ε :=
   by rw [ContinuousWithinAt, tendsto_nhds_within_nhds]
 #align metric.continuous_within_at_iff Metric.continuousWithinAt_iff
+-/
 
+#print Metric.continuousOn_iff /-
 theorem continuousOn_iff [PseudoMetricSpace β] {f : α → β} {s : Set α} :
     ContinuousOn f s ↔ ∀ b ∈ s, ∀ ε > 0, ∃ δ > 0, ∀ a ∈ s, dist a b < δ → dist (f a) (f b) < ε := by
   simp [ContinuousOn, continuous_within_at_iff]
 #align metric.continuous_on_iff Metric.continuousOn_iff
+-/
 
+#print Metric.continuous_iff /-
 theorem continuous_iff [PseudoMetricSpace β] {f : α → β} :
     Continuous f ↔ ∀ (b), ∀ ε > 0, ∃ δ > 0, ∀ a, dist a b < δ → dist (f a) (f b) < ε :=
   continuous_iff_continuousAt.trans <| forall_congr' fun b => tendsto_nhds_nhds
 #align metric.continuous_iff Metric.continuous_iff
+-/
 
+#print Metric.tendsto_nhds /-
 theorem tendsto_nhds {f : Filter β} {u : β → α} {a : α} :
     Tendsto u f (𝓝 a) ↔ ∀ ε > 0, ∀ᶠ x in f, dist (u x) a < ε :=
   nhds_basis_ball.tendsto_right_iff
 #align metric.tendsto_nhds Metric.tendsto_nhds
+-/
 
+#print Metric.continuousAt_iff' /-
 theorem continuousAt_iff' [TopologicalSpace β] {f : β → α} {b : β} :
     ContinuousAt f b ↔ ∀ ε > 0, ∀ᶠ x in 𝓝 b, dist (f x) (f b) < ε := by
   rw [ContinuousAt, tendsto_nhds]
 #align metric.continuous_at_iff' Metric.continuousAt_iff'
+-/
 
+#print Metric.continuousWithinAt_iff' /-
 theorem continuousWithinAt_iff' [TopologicalSpace β] {f : β → α} {b : β} {s : Set β} :
     ContinuousWithinAt f s b ↔ ∀ ε > 0, ∀ᶠ x in 𝓝[s] b, dist (f x) (f b) < ε := by
   rw [ContinuousWithinAt, tendsto_nhds]
 #align metric.continuous_within_at_iff' Metric.continuousWithinAt_iff'
+-/
 
+#print Metric.continuousOn_iff' /-
 theorem continuousOn_iff' [TopologicalSpace β] {f : β → α} {s : Set β} :
     ContinuousOn f s ↔ ∀ b ∈ s, ∀ ε > 0, ∀ᶠ x in 𝓝[s] b, dist (f x) (f b) < ε := by
   simp [ContinuousOn, continuous_within_at_iff']
 #align metric.continuous_on_iff' Metric.continuousOn_iff'
+-/
 
+#print Metric.continuous_iff' /-
 theorem continuous_iff' [TopologicalSpace β] {f : β → α} :
     Continuous f ↔ ∀ (a), ∀ ε > 0, ∀ᶠ x in 𝓝 a, dist (f x) (f a) < ε :=
   continuous_iff_continuousAt.trans <| forall_congr' fun b => tendsto_nhds
 #align metric.continuous_iff' Metric.continuous_iff'
+-/
 
+#print Metric.tendsto_atTop /-
 theorem tendsto_atTop [Nonempty β] [SemilatticeSup β] {u : β → α} {a : α} :
     Tendsto u atTop (𝓝 a) ↔ ∀ ε > 0, ∃ N, ∀ n ≥ N, dist (u n) a < ε :=
   (atTop_basis.tendsto_iffₓ nhds_basis_ball).trans <| by simp only [exists_prop, true_and_iff]; rfl
 #align metric.tendsto_at_top Metric.tendsto_atTop
+-/
 
+#print Metric.tendsto_atTop' /-
 /-- A variant of `tendsto_at_top` that
 uses `∃ N, ∀ n > N, ...` rather than `∃ N, ∀ n ≥ N, ...`
 -/
@@ -1234,42 +1501,54 @@ theorem tendsto_atTop' [Nonempty β] [SemilatticeSup β] [NoMaxOrder β] {u : β
   (atTop_basis_Ioi.tendsto_iffₓ nhds_basis_ball).trans <| by simp only [exists_prop, true_and_iff];
     rfl
 #align metric.tendsto_at_top' Metric.tendsto_atTop'
+-/
 
+#print Metric.isOpen_singleton_iff /-
 theorem isOpen_singleton_iff {α : Type _} [PseudoMetricSpace α] {x : α} :
     IsOpen ({x} : Set α) ↔ ∃ ε > 0, ∀ y, dist y x < ε → y = x := by
   simp [is_open_iff, subset_singleton_iff, mem_ball]
 #align metric.is_open_singleton_iff Metric.isOpen_singleton_iff
+-/
 
+#print Metric.exists_ball_inter_eq_singleton_of_mem_discrete /-
 /-- Given a point `x` in a discrete subset `s` of a pseudometric space, there is an open ball
 centered at `x` and intersecting `s` only at `x`. -/
 theorem exists_ball_inter_eq_singleton_of_mem_discrete [DiscreteTopology s] {x : α} (hx : x ∈ s) :
     ∃ ε > 0, Metric.ball x ε ∩ s = {x} :=
   nhds_basis_ball.exists_inter_eq_singleton_of_mem_discrete hx
 #align metric.exists_ball_inter_eq_singleton_of_mem_discrete Metric.exists_ball_inter_eq_singleton_of_mem_discrete
+-/
 
+#print Metric.exists_closedBall_inter_eq_singleton_of_discrete /-
 /-- Given a point `x` in a discrete subset `s` of a pseudometric space, there is a closed ball
 of positive radius centered at `x` and intersecting `s` only at `x`. -/
 theorem exists_closedBall_inter_eq_singleton_of_discrete [DiscreteTopology s] {x : α} (hx : x ∈ s) :
     ∃ ε > 0, Metric.closedBall x ε ∩ s = {x} :=
   nhds_basis_closedBall.exists_inter_eq_singleton_of_mem_discrete hx
 #align metric.exists_closed_ball_inter_eq_singleton_of_discrete Metric.exists_closedBall_inter_eq_singleton_of_discrete
+-/
 
+#print Dense.exists_dist_lt /-
 theorem Dense.exists_dist_lt {s : Set α} (hs : Dense s) (x : α) {ε : ℝ} (hε : 0 < ε) :
     ∃ y ∈ s, dist x y < ε :=
   by
   have : (ball x ε).Nonempty := by simp [hε]
   simpa only [mem_ball'] using hs.exists_mem_open is_open_ball this
 #align dense.exists_dist_lt Dense.exists_dist_lt
+-/
 
+#print DenseRange.exists_dist_lt /-
 theorem DenseRange.exists_dist_lt {β : Type _} {f : β → α} (hf : DenseRange f) (x : α) {ε : ℝ}
     (hε : 0 < ε) : ∃ y, dist x (f y) < ε :=
   exists_range_iff.1 (hf.exists_dist_lt x hε)
 #align dense_range.exists_dist_lt DenseRange.exists_dist_lt
+-/
 
 end Metric
 
 open Metric
 
+#print Metric.uniformity_basis_edist /-
 /-Instantiate a pseudometric space as a pseudoemetric space. Before we can state the instance,
 we need to show that the uniform structure coming from the edistance and the
 distance coincide. -/
@@ -1288,10 +1567,13 @@ protected theorem Metric.uniformity_basis_edist :
       refine' ⟨ε', ε0', fun a b h => Hε (lt_trans _ hε)⟩
       rwa [edist_dist, ENNReal.ofReal_lt_ofReal_iff ε0']⟩
 #align pseudo_metric.uniformity_basis_edist Metric.uniformity_basis_edist
+-/
 
+#print Metric.uniformity_edist /-
 theorem Metric.uniformity_edist : 𝓤 α = ⨅ ε > 0, 𝓟 {p : α × α | edist p.1 p.2 < ε} :=
   Metric.uniformity_basis_edist.eq_biInf
 #align metric.uniformity_edist Metric.uniformity_edist
+-/
 
 #print PseudoMetricSpace.toPseudoEMetricSpace /-
 -- see Note [lower instance priority]
@@ -1311,10 +1593,12 @@ instance (priority := 100) PseudoMetricSpace.toPseudoEMetricSpace : PseudoEMetri
 #align pseudo_metric_space.to_pseudo_emetric_space PseudoMetricSpace.toPseudoEMetricSpace
 -/
 
+#print Metric.eball_top_eq_univ /-
 /-- In a pseudometric space, an open ball of infinite radius is the whole space -/
 theorem Metric.eball_top_eq_univ (x : α) : EMetric.ball x ∞ = Set.univ :=
   Set.eq_univ_iff_forall.mpr fun y => edist_lt_top y x
 #align metric.eball_top_eq_univ Metric.eball_top_eq_univ
+-/
 
 #print Metric.emetric_ball /-
 /-- Balls defined using the distance or the edistance coincide -/
@@ -1327,32 +1611,42 @@ theorem Metric.emetric_ball {x : α} {ε : ℝ} : EMetric.ball x (ENNReal.ofReal
 #align metric.emetric_ball Metric.emetric_ball
 -/
 
+#print Metric.emetric_ball_nnreal /-
 /-- Balls defined using the distance or the edistance coincide -/
 @[simp]
 theorem Metric.emetric_ball_nnreal {x : α} {ε : ℝ≥0} : EMetric.ball x ε = ball x ε := by
   convert Metric.emetric_ball; simp
 #align metric.emetric_ball_nnreal Metric.emetric_ball_nnreal
+-/
 
+#print Metric.emetric_closedBall /-
 /-- Closed balls defined using the distance or the edistance coincide -/
 theorem Metric.emetric_closedBall {x : α} {ε : ℝ} (h : 0 ≤ ε) :
     EMetric.closedBall x (ENNReal.ofReal ε) = closedBall x ε := by
   ext y <;> simp [edist_dist] <;> rw [ENNReal.ofReal_le_ofReal_iff h]
 #align metric.emetric_closed_ball Metric.emetric_closedBall
+-/
 
+#print Metric.emetric_closedBall_nnreal /-
 /-- Closed balls defined using the distance or the edistance coincide -/
 @[simp]
 theorem Metric.emetric_closedBall_nnreal {x : α} {ε : ℝ≥0} :
     EMetric.closedBall x ε = closedBall x ε := by convert Metric.emetric_closedBall ε.2; simp
 #align metric.emetric_closed_ball_nnreal Metric.emetric_closedBall_nnreal
+-/
 
+#print Metric.emetric_ball_top /-
 @[simp]
 theorem Metric.emetric_ball_top (x : α) : EMetric.ball x ⊤ = univ :=
   eq_univ_of_forall fun y => edist_lt_top _ _
 #align metric.emetric_ball_top Metric.emetric_ball_top
+-/
 
+#print Metric.inseparable_iff /-
 theorem Metric.inseparable_iff {x y : α} : Inseparable x y ↔ dist x y = 0 := by
   rw [EMetric.inseparable_iff, edist_nndist, dist_nndist, ENNReal.coe_eq_zero, NNReal.coe_eq_zero]
 #align metric.inseparable_iff Metric.inseparable_iff
+-/
 
 #print PseudoMetricSpace.replaceUniformity /-
 /-- Build a new pseudometric space from an old one where the bundled uniform structure is provably
@@ -1397,6 +1691,7 @@ theorem PseudoMetricSpace.replaceTopology_eq {γ} [U : TopologicalSpace γ] (m :
 #align pseudo_metric_space.replace_topology_eq PseudoMetricSpace.replaceTopology_eq
 -/
 
+#print PseudoEMetricSpace.toPseudoMetricSpaceOfDist /-
 /-- One gets a pseudometric space from an emetric space if the edistance
 is everywhere finite, by pushing the edistance to reals. We set it up so that the edist and the
 uniformity are defeq in the pseudometric space and the pseudoemetric space. In this definition, the
@@ -1419,7 +1714,9 @@ def PseudoEMetricSpace.toPseudoMetricSpaceOfDist {α : Type u} [e : PseudoEMetri
       edist_dist := fun x y => by simp [h, ENNReal.ofReal_toReal, edist_ne_top] }
   m.replaceUniformity <| by rw [uniformity_pseudoedist, Metric.uniformity_edist]; rfl
 #align pseudo_emetric_space.to_pseudo_metric_space_of_dist PseudoEMetricSpace.toPseudoMetricSpaceOfDist
+-/
 
+#print PseudoEMetricSpace.toPseudoMetricSpace /-
 /-- One gets a pseudometric space from an emetric space if the edistance
 is everywhere finite, by pushing the edistance to reals. We set it up so that the edist and the
 uniformity are defeq in the pseudometric space and the emetric space. -/
@@ -1428,6 +1725,7 @@ def PseudoEMetricSpace.toPseudoMetricSpace {α : Type u} [e : PseudoEMetricSpace
   PseudoEMetricSpace.toPseudoMetricSpaceOfDist (fun x y => ENNReal.toReal (edist x y)) h fun x y =>
     rfl
 #align pseudo_emetric_space.to_pseudo_metric_space PseudoEMetricSpace.toPseudoMetricSpace
+-/
 
 #print PseudoMetricSpace.replaceBornology /-
 /-- Build a new pseudometric space from an old one where the bundled bornology structure is provably
@@ -1453,6 +1751,7 @@ theorem PseudoMetricSpace.replaceBornology_eq {α} [m : PseudoMetricSpace α] [B
 #align pseudo_metric_space.replace_bornology_eq PseudoMetricSpace.replaceBornology_eq
 -/
 
+#print Metric.complete_of_convergent_controlled_sequences /-
 /-- A very useful criterion to show that a space is complete is to show that all sequences
 which satisfy a bound of the form `dist (u n) (u m) < B N` for all `n m ≥ N` are
 converging. This is often applied for `B N = 2^{-N}`, i.e., with a very fast convergence to
@@ -1466,6 +1765,7 @@ theorem Metric.complete_of_convergent_controlled_sequences (B : ℕ → Real) (h
   UniformSpace.complete_of_convergent_controlled_sequences
     (fun n => {p : α × α | dist p.1 p.2 < B n}) (fun n => dist_mem_uniformity <| hB n) H
 #align metric.complete_of_convergent_controlled_sequences Metric.complete_of_convergent_controlled_sequences
+-/
 
 #print Metric.complete_of_cauchySeq_tendsto /-
 theorem Metric.complete_of_cauchySeq_tendsto :
@@ -1487,69 +1787,95 @@ instance Real.pseudoMetricSpace : PseudoMetricSpace ℝ
 #align real.pseudo_metric_space Real.pseudoMetricSpace
 -/
 
+#print Real.dist_eq /-
 theorem Real.dist_eq (x y : ℝ) : dist x y = |x - y| :=
   rfl
 #align real.dist_eq Real.dist_eq
+-/
 
+#print Real.nndist_eq /-
 theorem Real.nndist_eq (x y : ℝ) : nndist x y = Real.nnabs (x - y) :=
   rfl
 #align real.nndist_eq Real.nndist_eq
+-/
 
+#print Real.nndist_eq' /-
 theorem Real.nndist_eq' (x y : ℝ) : nndist x y = Real.nnabs (y - x) :=
   nndist_comm _ _
 #align real.nndist_eq' Real.nndist_eq'
+-/
 
+#print Real.dist_0_eq_abs /-
 theorem Real.dist_0_eq_abs (x : ℝ) : dist x 0 = |x| := by simp [Real.dist_eq]
 #align real.dist_0_eq_abs Real.dist_0_eq_abs
+-/
 
+#print Real.dist_left_le_of_mem_uIcc /-
 theorem Real.dist_left_le_of_mem_uIcc {x y z : ℝ} (h : y ∈ uIcc x z) : dist x y ≤ dist x z := by
   simpa only [dist_comm x] using abs_sub_left_of_mem_uIcc h
 #align real.dist_left_le_of_mem_uIcc Real.dist_left_le_of_mem_uIcc
+-/
 
+#print Real.dist_right_le_of_mem_uIcc /-
 theorem Real.dist_right_le_of_mem_uIcc {x y z : ℝ} (h : y ∈ uIcc x z) : dist y z ≤ dist x z := by
   simpa only [dist_comm _ z] using abs_sub_right_of_mem_uIcc h
 #align real.dist_right_le_of_mem_uIcc Real.dist_right_le_of_mem_uIcc
+-/
 
+#print Real.dist_le_of_mem_uIcc /-
 theorem Real.dist_le_of_mem_uIcc {x y x' y' : ℝ} (hx : x ∈ uIcc x' y') (hy : y ∈ uIcc x' y') :
     dist x y ≤ dist x' y' :=
   abs_sub_le_of_uIcc_subset_uIcc <| uIcc_subset_uIcc (by rwa [uIcc_comm]) (by rwa [uIcc_comm])
 #align real.dist_le_of_mem_uIcc Real.dist_le_of_mem_uIcc
+-/
 
+#print Real.dist_le_of_mem_Icc /-
 theorem Real.dist_le_of_mem_Icc {x y x' y' : ℝ} (hx : x ∈ Icc x' y') (hy : y ∈ Icc x' y') :
     dist x y ≤ y' - x' := by
   simpa only [Real.dist_eq, abs_of_nonpos (sub_nonpos.2 <| hx.1.trans hx.2), neg_sub] using
     Real.dist_le_of_mem_uIcc (Icc_subset_uIcc hx) (Icc_subset_uIcc hy)
 #align real.dist_le_of_mem_Icc Real.dist_le_of_mem_Icc
+-/
 
+#print Real.dist_le_of_mem_Icc_01 /-
 theorem Real.dist_le_of_mem_Icc_01 {x y : ℝ} (hx : x ∈ Icc (0 : ℝ) 1) (hy : y ∈ Icc (0 : ℝ) 1) :
     dist x y ≤ 1 := by simpa only [sub_zero] using Real.dist_le_of_mem_Icc hx hy
 #align real.dist_le_of_mem_Icc_01 Real.dist_le_of_mem_Icc_01
+-/
 
 instance : OrderTopology ℝ :=
   orderTopology_of_nhds_abs fun x => by
     simp only [nhds_basis_ball.eq_binfi, ball, Real.dist_eq, abs_sub_comm]
 
+#print Real.ball_eq_Ioo /-
 theorem Real.ball_eq_Ioo (x r : ℝ) : ball x r = Ioo (x - r) (x + r) :=
   Set.ext fun y => by
     rw [mem_ball, dist_comm, Real.dist_eq, abs_sub_lt_iff, mem_Ioo, ← sub_lt_iff_lt_add',
       sub_lt_comm]
 #align real.ball_eq_Ioo Real.ball_eq_Ioo
+-/
 
+#print Real.closedBall_eq_Icc /-
 theorem Real.closedBall_eq_Icc {x r : ℝ} : closedBall x r = Icc (x - r) (x + r) := by
   ext y <;>
     rw [mem_closed_ball, dist_comm, Real.dist_eq, abs_sub_le_iff, mem_Icc, ← sub_le_iff_le_add',
       sub_le_comm]
 #align real.closed_ball_eq_Icc Real.closedBall_eq_Icc
+-/
 
+#print Real.Ioo_eq_ball /-
 theorem Real.Ioo_eq_ball (x y : ℝ) : Ioo x y = ball ((x + y) / 2) ((y - x) / 2) := by
   rw [Real.ball_eq_Ioo, ← sub_div, add_comm, ← sub_add, add_sub_cancel', add_self_div_two, ←
     add_div, add_assoc, add_sub_cancel'_right, add_self_div_two]
 #align real.Ioo_eq_ball Real.Ioo_eq_ball
+-/
 
+#print Real.Icc_eq_closedBall /-
 theorem Real.Icc_eq_closedBall (x y : ℝ) : Icc x y = closedBall ((x + y) / 2) ((y - x) / 2) := by
   rw [Real.closedBall_eq_Icc, ← sub_div, add_comm, ← sub_add, add_sub_cancel', add_self_div_two, ←
     add_div, add_assoc, add_sub_cancel'_right, add_self_div_two]
 #align real.Icc_eq_closed_ball Real.Icc_eq_closedBall
+-/
 
 section MetricOrdered
 
@@ -1581,51 +1907,66 @@ theorem totallyBounded_Ioo (a b : α) : TotallyBounded (Ioo a b) :=
 
 end MetricOrdered
 
+#print squeeze_zero' /-
 /-- Special case of the sandwich theorem; see `tendsto_of_tendsto_of_tendsto_of_le_of_le'` for the
 general case. -/
 theorem squeeze_zero' {α} {f g : α → ℝ} {t₀ : Filter α} (hf : ∀ᶠ t in t₀, 0 ≤ f t)
     (hft : ∀ᶠ t in t₀, f t ≤ g t) (g0 : Tendsto g t₀ (nhds 0)) : Tendsto f t₀ (𝓝 0) :=
   tendsto_of_tendsto_of_tendsto_of_le_of_le' tendsto_const_nhds g0 hf hft
 #align squeeze_zero' squeeze_zero'
+-/
 
+#print squeeze_zero /-
 /-- Special case of the sandwich theorem; see `tendsto_of_tendsto_of_tendsto_of_le_of_le`
 and  `tendsto_of_tendsto_of_tendsto_of_le_of_le'` for the general case. -/
 theorem squeeze_zero {α} {f g : α → ℝ} {t₀ : Filter α} (hf : ∀ t, 0 ≤ f t) (hft : ∀ t, f t ≤ g t)
     (g0 : Tendsto g t₀ (𝓝 0)) : Tendsto f t₀ (𝓝 0) :=
   squeeze_zero' (eventually_of_forall hf) (eventually_of_forall hft) g0
 #align squeeze_zero squeeze_zero
+-/
 
+#print Metric.uniformity_eq_comap_nhds_zero /-
 theorem Metric.uniformity_eq_comap_nhds_zero :
     𝓤 α = comap (fun p : α × α => dist p.1 p.2) (𝓝 (0 : ℝ)) :=
   by
   ext s
   simp [mem_uniformity_dist, (nhds_basis_ball.comap _).mem_iff, subset_def, Real.dist_0_eq_abs]
 #align metric.uniformity_eq_comap_nhds_zero Metric.uniformity_eq_comap_nhds_zero
+-/
 
+#print cauchySeq_iff_tendsto_dist_atTop_0 /-
 theorem cauchySeq_iff_tendsto_dist_atTop_0 [Nonempty β] [SemilatticeSup β] {u : β → α} :
     CauchySeq u ↔ Tendsto (fun n : β × β => dist (u n.1) (u n.2)) atTop (𝓝 0) := by
   rw [cauchySeq_iff_tendsto, Metric.uniformity_eq_comap_nhds_zero, tendsto_comap_iff, Prod.map_def]
 #align cauchy_seq_iff_tendsto_dist_at_top_0 cauchySeq_iff_tendsto_dist_atTop_0
+-/
 
+#print tendsto_uniformity_iff_dist_tendsto_zero /-
 theorem tendsto_uniformity_iff_dist_tendsto_zero {ι : Type _} {f : ι → α × α} {p : Filter ι} :
     Tendsto f p (𝓤 α) ↔ Tendsto (fun x => dist (f x).1 (f x).2) p (𝓝 0) := by
   rw [Metric.uniformity_eq_comap_nhds_zero, tendsto_comap_iff]
 #align tendsto_uniformity_iff_dist_tendsto_zero tendsto_uniformity_iff_dist_tendsto_zero
+-/
 
+#print Filter.Tendsto.congr_dist /-
 theorem Filter.Tendsto.congr_dist {ι : Type _} {f₁ f₂ : ι → α} {p : Filter ι} {a : α}
     (h₁ : Tendsto f₁ p (𝓝 a)) (h : Tendsto (fun x => dist (f₁ x) (f₂ x)) p (𝓝 0)) :
     Tendsto f₂ p (𝓝 a) :=
   h₁.congr_uniformity <| tendsto_uniformity_iff_dist_tendsto_zero.2 h
 #align filter.tendsto.congr_dist Filter.Tendsto.congr_dist
+-/
 
 alias Filter.Tendsto.congr_dist ← tendsto_of_tendsto_of_dist
 #align tendsto_of_tendsto_of_dist tendsto_of_tendsto_of_dist
 
+#print tendsto_iff_of_dist /-
 theorem tendsto_iff_of_dist {ι : Type _} {f₁ f₂ : ι → α} {p : Filter ι} {a : α}
     (h : Tendsto (fun x => dist (f₁ x) (f₂ x)) p (𝓝 0)) : Tendsto f₁ p (𝓝 a) ↔ Tendsto f₂ p (𝓝 a) :=
   Uniform.tendsto_congr <| tendsto_uniformity_iff_dist_tendsto_zero.2 h
 #align tendsto_iff_of_dist tendsto_iff_of_dist
+-/
 
+#print eventually_closedBall_subset /-
 /-- If `u` is a neighborhood of `x`, then for small enough `r`, the closed ball
 `closed_ball x r` is contained in `u`. -/
 theorem eventually_closedBall_subset {x : α} {u : Set α} (hu : u ∈ 𝓝 x) :
@@ -1636,6 +1977,7 @@ theorem eventually_closedBall_subset {x : α} {u : Set α} (hu : u ∈ 𝓝 x) :
   have : Iic ε ∈ 𝓝 (0 : ℝ) := Iic_mem_nhds εpos
   filter_upwards [this] with _ hr using subset.trans (closed_ball_subset_closed_ball hr) hε
 #align eventually_closed_ball_subset eventually_closedBall_subset
+-/
 
 end Real
 
@@ -1644,6 +1986,7 @@ section CauchySeq
 variable [Nonempty β] [SemilatticeSup β]
 
 /- ./././Mathport/Syntax/Translate/Basic.lean:638:2: warning: expanding binder collection (m n «expr ≥ » N) -/
+#print Metric.cauchySeq_iff /-
 -- see Note [nolint_ge]
 /-- In a pseudometric space, Cauchy sequences are characterized by the fact that, eventually,
 the distance between its elements is arbitrarily small -/
@@ -1652,13 +1995,17 @@ theorem Metric.cauchySeq_iff {u : β → α} :
     CauchySeq u ↔ ∀ ε > 0, ∃ N, ∀ (m) (_ : m ≥ N) (n) (_ : n ≥ N), dist (u m) (u n) < ε :=
   uniformity_basis_dist.cauchySeq_iff
 #align metric.cauchy_seq_iff Metric.cauchySeq_iff
+-/
 
+#print Metric.cauchySeq_iff' /-
 /-- A variation around the pseudometric characterization of Cauchy sequences -/
 theorem Metric.cauchySeq_iff' {u : β → α} :
     CauchySeq u ↔ ∀ ε > 0, ∃ N, ∀ n ≥ N, dist (u n) (u N) < ε :=
   uniformity_basis_dist.cauchySeq_iff'
 #align metric.cauchy_seq_iff' Metric.cauchySeq_iff'
+-/
 
+#print Metric.uniformCauchySeqOn_iff /-
 -- see Note [nolint_ge]
 /-- In a pseudometric space, unifom Cauchy sequences are characterized by the fact that, eventually,
 the distance between all its elements is uniformly, arbitrarily small -/
@@ -1688,7 +2035,9 @@ theorem Metric.uniformCauchySeqOn_iff {γ : Type _} {F : β → γ → α} {s :
     rcases hb with ⟨hbl, hbr⟩
     exact hab (hN b.fst hbl.ge b.snd hbr.ge x hx)
 #align metric.uniform_cauchy_seq_on_iff Metric.uniformCauchySeqOn_iff
+-/
 
+#print cauchySeq_of_le_tendsto_0' /-
 /-- If the distance between `s n` and `s m`, `n ≤ m` is bounded above by `b n`
 and `b` converges to zero, then `s` is a Cauchy sequence.  -/
 theorem cauchySeq_of_le_tendsto_0' {s : β → α} (b : β → ℝ)
@@ -1700,7 +2049,9 @@ theorem cauchySeq_of_le_tendsto_0' {s : β → α} (b : β → ℝ)
         _ ≤ b N := (h _ _ hn)
         _ < ε := hN
 #align cauchy_seq_of_le_tendsto_0' cauchySeq_of_le_tendsto_0'
+-/
 
+#print cauchySeq_of_le_tendsto_0 /-
 /-- If the distance between `s n` and `s m`, `n, m ≥ N` is bounded above by `b N`
 and `b` converges to zero, then `s` is a Cauchy sequence.  -/
 theorem cauchySeq_of_le_tendsto_0 {s : β → α} (b : β → ℝ)
@@ -1708,7 +2059,9 @@ theorem cauchySeq_of_le_tendsto_0 {s : β → α} (b : β → ℝ)
     CauchySeq s :=
   cauchySeq_of_le_tendsto_0' b (fun n m hnm => h _ _ _ le_rfl hnm) h₀
 #align cauchy_seq_of_le_tendsto_0 cauchySeq_of_le_tendsto_0
+-/
 
+#print cauchySeq_bdd /-
 /-- A Cauchy sequence on the natural numbers is bounded. -/
 theorem cauchySeq_bdd {u : ℕ → α} (hu : CauchySeq u) : ∃ R > 0, ∀ m n, dist (u m) (u n) < R :=
   by
@@ -1725,7 +2078,9 @@ theorem cauchySeq_bdd {u : ℕ → α} (hu : CauchySeq u) : ∃ R > 0, ∀ m n,
   · have : _ ≤ R := Finset.le_sup (Finset.mem_range.2 h)
     exact lt_of_le_of_lt this (lt_add_of_pos_right _ zero_lt_one)
 #align cauchy_seq_bdd cauchySeq_bdd
+-/
 
+#print cauchySeq_iff_le_tendsto_0 /-
 /-- Yet another metric characterization of Cauchy sequences on integers. This one is often the
 most efficient. -/
 theorem cauchySeq_iff_le_tendsto_0 {s : ℕ → α} :
@@ -1763,6 +2118,7 @@ theorem cauchySeq_iff_le_tendsto_0 {s : ℕ → α} :
     exact le_of_lt (hN _ (le_trans hn hm') _ (le_trans hn hn')), fun ⟨b, _, b_bound, b_lim⟩ =>
     cauchySeq_of_le_tendsto_0 b b_bound b_lim⟩
 #align cauchy_seq_iff_le_tendsto_0 cauchySeq_iff_le_tendsto_0
+-/
 
 end CauchySeq
 
@@ -1870,10 +2226,13 @@ section NNReal
 instance : PseudoMetricSpace ℝ≥0 :=
   Subtype.pseudoMetricSpace
 
+#print NNReal.dist_eq /-
 theorem NNReal.dist_eq (a b : ℝ≥0) : dist a b = |(a : ℝ) - b| :=
   rfl
 #align nnreal.dist_eq NNReal.dist_eq
+-/
 
+#print NNReal.nndist_eq /-
 theorem NNReal.nndist_eq (a b : ℝ≥0) : nndist a b = max (a - b) (b - a) :=
   by
   wlog h : b ≤ a
@@ -1881,22 +2240,29 @@ theorem NNReal.nndist_eq (a b : ℝ≥0) : nndist a b = max (a - b) (b - a) :=
   rw [← NNReal.coe_eq, ← dist_nndist, NNReal.dist_eq, tsub_eq_zero_iff_le.2 h,
     max_eq_left (zero_le <| a - b), ← NNReal.coe_sub h, abs_of_nonneg (a - b).coe_nonneg]
 #align nnreal.nndist_eq NNReal.nndist_eq
+-/
 
+#print NNReal.nndist_zero_eq_val /-
 @[simp]
 theorem NNReal.nndist_zero_eq_val (z : ℝ≥0) : nndist 0 z = z := by
   simp only [NNReal.nndist_eq, max_eq_right, tsub_zero, zero_tsub, zero_le']
 #align nnreal.nndist_zero_eq_val NNReal.nndist_zero_eq_val
+-/
 
+#print NNReal.nndist_zero_eq_val' /-
 @[simp]
 theorem NNReal.nndist_zero_eq_val' (z : ℝ≥0) : nndist z 0 = z := by rw [nndist_comm];
   exact NNReal.nndist_zero_eq_val z
 #align nnreal.nndist_zero_eq_val' NNReal.nndist_zero_eq_val'
+-/
 
+#print NNReal.le_add_nndist /-
 theorem NNReal.le_add_nndist (a b : ℝ≥0) : a ≤ b + nndist a b :=
   by
   suffices (a : ℝ) ≤ (b : ℝ) + dist a b by exact nnreal.coe_le_coe.mp this
   linarith [le_of_abs_le (by rfl : abs (a - b : ℝ) ≤ dist a b)]
 #align nnreal.le_add_nndist NNReal.le_add_nndist
+-/
 
 end NNReal
 
@@ -1907,23 +2273,31 @@ variable [PseudoMetricSpace β]
 instance : PseudoMetricSpace (ULift β) :=
   PseudoMetricSpace.induced ULift.down ‹_›
 
+#print ULift.dist_eq /-
 theorem ULift.dist_eq (x y : ULift β) : dist x y = dist x.down y.down :=
   rfl
 #align ulift.dist_eq ULift.dist_eq
+-/
 
+#print ULift.nndist_eq /-
 theorem ULift.nndist_eq (x y : ULift β) : nndist x y = nndist x.down y.down :=
   rfl
 #align ulift.nndist_eq ULift.nndist_eq
+-/
 
+#print ULift.dist_up_up /-
 @[simp]
 theorem ULift.dist_up_up (x y : β) : dist (ULift.up x) (ULift.up y) = dist x y :=
   rfl
 #align ulift.dist_up_up ULift.dist_up_up
+-/
 
+#print ULift.nndist_up_up /-
 @[simp]
 theorem ULift.nndist_up_up (x y : β) : nndist (ULift.up x) (ULift.up y) = nndist x y :=
   rfl
 #align ulift.nndist_up_up ULift.nndist_up_up
+-/
 
 end ULift
 
@@ -1945,9 +2319,11 @@ instance Prod.pseudoMetricSpaceMax : PseudoMetricSpace (α × β) :=
 #align prod.pseudo_metric_space_max Prod.pseudoMetricSpaceMax
 -/
 
+#print Prod.dist_eq /-
 theorem Prod.dist_eq {x y : α × β} : dist x y = max (dist x.1 y.1) (dist x.2 y.2) :=
   rfl
 #align prod.dist_eq Prod.dist_eq
+-/
 
 #print dist_prod_same_left /-
 @[simp]
@@ -1980,6 +2356,7 @@ theorem closedBall_prod_same (x : α) (y : β) (r : ℝ) :
 
 end Prod
 
+#print uniformContinuous_dist /-
 theorem uniformContinuous_dist : UniformContinuous fun p : α × α => dist p.1 p.2 :=
   Metric.uniformContinuous_iff.2 fun ε ε0 =>
     ⟨ε / 2, half_pos ε0, by
@@ -1997,6 +2374,7 @@ theorem uniformContinuous_dist : UniformContinuous fun p : α × α => dist p.1
           (abs_sub_lt_iff.1 (lt_of_le_of_lt (abs_dist_sub_le p₂ q₂ q₁) h₂)).1
       rwa [add_halves, dist_comm p₂, sub_add_sub_cancel, dist_comm q₂] at this ⟩
 #align uniform_continuous_dist uniformContinuous_dist
+-/
 
 #print UniformContinuous.dist /-
 theorem UniformContinuous.dist [UniformSpace β] {f g : β → α} (hf : UniformContinuous f)
@@ -2005,10 +2383,12 @@ theorem UniformContinuous.dist [UniformSpace β] {f g : β → α} (hf : Uniform
 #align uniform_continuous.dist UniformContinuous.dist
 -/
 
+#print continuous_dist /-
 @[continuity]
 theorem continuous_dist : Continuous fun p : α × α => dist p.1 p.2 :=
   uniformContinuous_dist.Continuous
 #align continuous_dist continuous_dist
+-/
 
 #print Continuous.dist /-
 @[continuity]
@@ -2025,16 +2405,21 @@ theorem Filter.Tendsto.dist {f g : β → α} {x : Filter β} {a b : α} (hf : T
 #align filter.tendsto.dist Filter.Tendsto.dist
 -/
 
+#print nhds_comap_dist /-
 theorem nhds_comap_dist (a : α) : ((𝓝 (0 : ℝ)).comap fun a' => dist a' a) = 𝓝 a := by
   simp only [@nhds_eq_comap_uniformity α, Metric.uniformity_eq_comap_nhds_zero, comap_comap,
     (· ∘ ·), dist_comm]
 #align nhds_comap_dist nhds_comap_dist
+-/
 
+#print tendsto_iff_dist_tendsto_zero /-
 theorem tendsto_iff_dist_tendsto_zero {f : β → α} {x : Filter β} {a : α} :
     Tendsto f x (𝓝 a) ↔ Tendsto (fun b => dist (f b) a) x (𝓝 0) := by
   rw [← nhds_comap_dist a, tendsto_comap_iff]
 #align tendsto_iff_dist_tendsto_zero tendsto_iff_dist_tendsto_zero
+-/
 
+#print continuous_iff_continuous_dist /-
 theorem continuous_iff_continuous_dist [TopologicalSpace β] {f : β → α} :
     Continuous f ↔ Continuous fun x : β × β => dist (f x.1) (f x.2) :=
   ⟨fun h => (h.comp continuous_fst).dist (h.comp continuous_snd), fun h =>
@@ -2042,29 +2427,40 @@ theorem continuous_iff_continuous_dist [TopologicalSpace β] {f : β → α} :
       tendsto_iff_dist_tendsto_zero.2 <|
         (h.comp (continuous_id.prod_mk continuous_const)).tendsto' _ _ <| dist_self _⟩
 #align continuous_iff_continuous_dist continuous_iff_continuous_dist
+-/
 
+#print uniformContinuous_nndist /-
 theorem uniformContinuous_nndist : UniformContinuous fun p : α × α => nndist p.1 p.2 :=
   uniformContinuous_dist.subtype_mk _
 #align uniform_continuous_nndist uniformContinuous_nndist
+-/
 
+#print UniformContinuous.nndist /-
 theorem UniformContinuous.nndist [UniformSpace β] {f g : β → α} (hf : UniformContinuous f)
     (hg : UniformContinuous g) : UniformContinuous fun b => nndist (f b) (g b) :=
   uniformContinuous_nndist.comp (hf.prod_mk hg)
 #align uniform_continuous.nndist UniformContinuous.nndist
+-/
 
+#print continuous_nndist /-
 theorem continuous_nndist : Continuous fun p : α × α => nndist p.1 p.2 :=
   uniformContinuous_nndist.Continuous
 #align continuous_nndist continuous_nndist
+-/
 
+#print Continuous.nndist /-
 theorem Continuous.nndist [TopologicalSpace β] {f g : β → α} (hf : Continuous f)
     (hg : Continuous g) : Continuous fun b => nndist (f b) (g b) :=
   continuous_nndist.comp (hf.prod_mk hg : _)
 #align continuous.nndist Continuous.nndist
+-/
 
+#print Filter.Tendsto.nndist /-
 theorem Filter.Tendsto.nndist {f g : β → α} {x : Filter β} {a b : α} (hf : Tendsto f x (𝓝 a))
     (hg : Tendsto g x (𝓝 b)) : Tendsto (fun x => nndist (f x) (g x)) x (𝓝 (nndist a b)) :=
   (continuous_nndist.Tendsto (a, b)).comp (hf.prod_mk_nhds hg)
 #align filter.tendsto.nndist Filter.Tendsto.nndist
+-/
 
 namespace Metric
 
@@ -2120,43 +2516,58 @@ theorem ball_subset_interior_closedBall : ball x ε ⊆ interior (closedBall x 
 #align metric.ball_subset_interior_closed_ball Metric.ball_subset_interior_closedBall
 -/
 
+#print Metric.mem_closure_iff /-
 /-- ε-characterization of the closure in pseudometric spaces-/
 theorem mem_closure_iff {s : Set α} {a : α} : a ∈ closure s ↔ ∀ ε > 0, ∃ b ∈ s, dist a b < ε :=
   (mem_closure_iff_nhds_basis nhds_basis_ball).trans <| by simp only [mem_ball, dist_comm]
 #align metric.mem_closure_iff Metric.mem_closure_iff
+-/
 
+#print Metric.mem_closure_range_iff /-
 theorem mem_closure_range_iff {e : β → α} {a : α} :
     a ∈ closure (range e) ↔ ∀ ε > 0, ∃ k : β, dist a (e k) < ε := by
   simp only [mem_closure_iff, exists_range_iff]
 #align metric.mem_closure_range_iff Metric.mem_closure_range_iff
+-/
 
+#print Metric.mem_closure_range_iff_nat /-
 theorem mem_closure_range_iff_nat {e : β → α} {a : α} :
     a ∈ closure (range e) ↔ ∀ n : ℕ, ∃ k : β, dist a (e k) < 1 / ((n : ℝ) + 1) :=
   (mem_closure_iff_nhds_basis nhds_basis_ball_inv_nat_succ).trans <| by
     simp only [mem_ball, dist_comm, exists_range_iff, forall_const]
 #align metric.mem_closure_range_iff_nat Metric.mem_closure_range_iff_nat
+-/
 
+#print Metric.mem_of_closed' /-
 theorem mem_of_closed' {s : Set α} (hs : IsClosed s) {a : α} :
     a ∈ s ↔ ∀ ε > 0, ∃ b ∈ s, dist a b < ε := by
   simpa only [hs.closure_eq] using @mem_closure_iff _ _ s a
 #align metric.mem_of_closed' Metric.mem_of_closed'
+-/
 
+#print Metric.closedBall_zero' /-
 theorem closedBall_zero' (x : α) : closedBall x 0 = closure {x} :=
   Subset.antisymm
     (fun y hy =>
       mem_closure_iff.2 fun ε ε0 => ⟨x, mem_singleton x, (mem_closedBall.1 hy).trans_lt ε0⟩)
     (closure_minimal (singleton_subset_iff.2 (dist_self x).le) isClosed_ball)
 #align metric.closed_ball_zero' Metric.closedBall_zero'
+-/
 
+#print Metric.dense_iff /-
 theorem dense_iff {s : Set α} : Dense s ↔ ∀ x, ∀ r > 0, (ball x r ∩ s).Nonempty :=
   forall_congr' fun x => by
     simp only [mem_closure_iff, Set.Nonempty, exists_prop, mem_inter_iff, mem_ball', and_comm']
 #align metric.dense_iff Metric.dense_iff
+-/
 
+#print Metric.denseRange_iff /-
 theorem denseRange_iff {f : β → α} : DenseRange f ↔ ∀ x, ∀ r > 0, ∃ y, dist x (f y) < r :=
   forall_congr' fun x => by simp only [mem_closure_iff, exists_range_iff]
 #align metric.dense_range_iff Metric.denseRange_iff
+-/
 
+#print TopologicalSpace.IsSeparable.separableSpace /-
 /-- If a set `s` is separable, then the corresponding subtype is separable in a metric space.
 This is not obvious, as the countable set whose closure covers `s` does not need in general to
 be contained in `s`. -/
@@ -2193,6 +2604,7 @@ theorem TopologicalSpace.IsSeparable.separableSpace {s : Set α} (hs : IsSeparab
     _ < r / 2 + r / 2 := (add_lt_add (hz.trans hn) ((Metric.mem_ball'.1 A.some_spec.1).trans hn))
     _ = r := add_halves _
 #align topological_space.is_separable.separable_space TopologicalSpace.IsSeparable.separableSpace
+-/
 
 #print Inducing.isSeparable_preimage /-
 /-- The preimage of a separable set by an inducing map is separable. -/
@@ -2265,32 +2677,43 @@ instance pseudoMetricSpacePi : PseudoMetricSpace (∀ b, π b) :=
 #align pseudo_metric_space_pi pseudoMetricSpacePi
 -/
 
+#print nndist_pi_def /-
 theorem nndist_pi_def (f g : ∀ b, π b) : nndist f g = sup univ fun b => nndist (f b) (g b) :=
   NNReal.eq rfl
 #align nndist_pi_def nndist_pi_def
+-/
 
+#print dist_pi_def /-
 theorem dist_pi_def (f g : ∀ b, π b) : dist f g = (sup univ fun b => nndist (f b) (g b) : ℝ≥0) :=
   rfl
 #align dist_pi_def dist_pi_def
+-/
 
+#print nndist_pi_le_iff /-
 theorem nndist_pi_le_iff {f g : ∀ b, π b} {r : ℝ≥0} :
     nndist f g ≤ r ↔ ∀ b, nndist (f b) (g b) ≤ r := by simp [nndist_pi_def]
 #align nndist_pi_le_iff nndist_pi_le_iff
+-/
 
+#print dist_pi_lt_iff /-
 theorem dist_pi_lt_iff {f g : ∀ b, π b} {r : ℝ} (hr : 0 < r) :
     dist f g < r ↔ ∀ b, dist (f b) (g b) < r :=
   by
   lift r to ℝ≥0 using hr.le
   simp [dist_pi_def, Finset.sup_lt_iff (show ⊥ < r from hr)]
 #align dist_pi_lt_iff dist_pi_lt_iff
+-/
 
+#print dist_pi_le_iff /-
 theorem dist_pi_le_iff {f g : ∀ b, π b} {r : ℝ} (hr : 0 ≤ r) :
     dist f g ≤ r ↔ ∀ b, dist (f b) (g b) ≤ r :=
   by
   lift r to ℝ≥0 using hr
   exact nndist_pi_le_iff
 #align dist_pi_le_iff dist_pi_le_iff
+-/
 
+#print dist_pi_le_iff' /-
 theorem dist_pi_le_iff' [Nonempty β] {f g : ∀ b, π b} {r : ℝ} :
     dist f g ≤ r ↔ ∀ b, dist (f b) (g b) ≤ r :=
   by
@@ -2301,10 +2724,13 @@ theorem dist_pi_le_iff' [Nonempty β] {f g : ∀ b, π b} {r : ℝ} :
       iff_of_false (fun h => hr <| dist_nonneg.trans h) fun h =>
         hr <| dist_nonneg.trans <| h <| Classical.arbitrary _
 #align dist_pi_le_iff' dist_pi_le_iff'
+-/
 
+#print dist_pi_const_le /-
 theorem dist_pi_const_le (a b : α) : (dist (fun _ : β => a) fun _ => b) ≤ dist a b :=
   (dist_pi_le_iff dist_nonneg).2 fun _ => le_rfl
 #align dist_pi_const_le dist_pi_const_le
+-/
 
 #print nndist_pi_const_le /-
 theorem nndist_pi_const_le (a b : α) : (nndist (fun _ : β => a) fun _ => b) ≤ nndist a b :=
@@ -2327,54 +2753,71 @@ theorem nndist_pi_const [Nonempty β] (a b : α) :
 #align nndist_pi_const nndist_pi_const
 -/
 
+#print nndist_le_pi_nndist /-
 theorem nndist_le_pi_nndist (f g : ∀ b, π b) (b : β) : nndist (f b) (g b) ≤ nndist f g := by
   rw [nndist_pi_def]; exact Finset.le_sup (Finset.mem_univ b)
 #align nndist_le_pi_nndist nndist_le_pi_nndist
+-/
 
+#print dist_le_pi_dist /-
 theorem dist_le_pi_dist (f g : ∀ b, π b) (b : β) : dist (f b) (g b) ≤ dist f g := by
   simp only [dist_nndist, NNReal.coe_le_coe, nndist_le_pi_nndist f g b]
 #align dist_le_pi_dist dist_le_pi_dist
+-/
 
+#print ball_pi /-
 /-- An open ball in a product space is a product of open balls. See also `metric.ball_pi'`
 for a version assuming `nonempty β` instead of `0 < r`. -/
 theorem ball_pi (x : ∀ b, π b) {r : ℝ} (hr : 0 < r) :
     ball x r = Set.pi univ fun b => ball (x b) r := by ext p; simp [dist_pi_lt_iff hr]
 #align ball_pi ball_pi
+-/
 
+#print ball_pi' /-
 /-- An open ball in a product space is a product of open balls. See also `metric.ball_pi`
 for a version assuming `0 < r` instead of `nonempty β`. -/
 theorem ball_pi' [Nonempty β] (x : ∀ b, π b) (r : ℝ) :
     ball x r = Set.pi univ fun b => ball (x b) r :=
   (lt_or_le 0 r).elim (ball_pi x) fun hr => by simp [ball_eq_empty.2 hr]
 #align ball_pi' ball_pi'
+-/
 
+#print closedBall_pi /-
 /-- A closed ball in a product space is a product of closed balls. See also `metric.closed_ball_pi'`
 for a version assuming `nonempty β` instead of `0 ≤ r`. -/
 theorem closedBall_pi (x : ∀ b, π b) {r : ℝ} (hr : 0 ≤ r) :
     closedBall x r = Set.pi univ fun b => closedBall (x b) r := by ext p; simp [dist_pi_le_iff hr]
 #align closed_ball_pi closedBall_pi
+-/
 
+#print closedBall_pi' /-
 /-- A closed ball in a product space is a product of closed balls. See also `metric.closed_ball_pi`
 for a version assuming `0 ≤ r` instead of `nonempty β`. -/
 theorem closedBall_pi' [Nonempty β] (x : ∀ b, π b) (r : ℝ) :
     closedBall x r = Set.pi univ fun b => closedBall (x b) r :=
   (le_or_lt 0 r).elim (closedBall_pi x) fun hr => by simp [closed_ball_eq_empty.2 hr]
 #align closed_ball_pi' closedBall_pi'
+-/
 
+#print Fin.nndist_insertNth_insertNth /-
 @[simp]
 theorem Fin.nndist_insertNth_insertNth {n : ℕ} {α : Fin (n + 1) → Type _}
     [∀ i, PseudoMetricSpace (α i)] (i : Fin (n + 1)) (x y : α i) (f g : ∀ j, α (i.succAbove j)) :
     nndist (i.insertNth x f) (i.insertNth y g) = max (nndist x y) (nndist f g) :=
   eq_of_forall_ge_iff fun c => by simp [nndist_pi_le_iff, i.forall_iff_succ_above]
 #align fin.nndist_insert_nth_insert_nth Fin.nndist_insertNth_insertNth
+-/
 
+#print Fin.dist_insertNth_insertNth /-
 @[simp]
 theorem Fin.dist_insertNth_insertNth {n : ℕ} {α : Fin (n + 1) → Type _}
     [∀ i, PseudoMetricSpace (α i)] (i : Fin (n + 1)) (x y : α i) (f g : ∀ j, α (i.succAbove j)) :
     dist (i.insertNth x f) (i.insertNth y g) = max (dist x y) (dist f g) := by
   simp only [dist_nndist, Fin.nndist_insertNth_insertNth, NNReal.coe_max]
 #align fin.dist_insert_nth_insert_nth Fin.dist_insertNth_insertNth
+-/
 
+#print Real.dist_le_of_mem_pi_Icc /-
 theorem Real.dist_le_of_mem_pi_Icc {x y x' y' : β → ℝ} (hx : x ∈ Icc x' y') (hy : y ∈ Icc x' y') :
     dist x y ≤ dist x' y' :=
   by
@@ -2384,12 +2827,14 @@ theorem Real.dist_le_of_mem_pi_Icc {x y x' y' : β → ℝ} (hx : x ∈ Icc x' y
     refine' Icc_subset_uIcc _
   exacts [⟨hx.1 _, hx.2 _⟩, ⟨hy.1 _, hy.2 _⟩]
 #align real.dist_le_of_mem_pi_Icc Real.dist_le_of_mem_pi_Icc
+-/
 
 end Pi
 
 section Compact
 
 /- ./././Mathport/Syntax/Translate/Basic.lean:638:2: warning: expanding binder collection (t «expr ⊆ » s) -/
+#print finite_cover_balls_of_compact /-
 /-- Any compact set in a pseudometric space can be covered by finitely many balls of a given
 positive radius -/
 theorem finite_cover_balls_of_compact {α : Type u} [PseudoMetricSpace α] {s : Set α}
@@ -2402,6 +2847,7 @@ theorem finite_cover_balls_of_compact {α : Type u} [PseudoMetricSpace α] {s :
     simp
     exact ⟨x, ⟨xs, by simpa⟩⟩
 #align finite_cover_balls_of_compact finite_cover_balls_of_compact
+-/
 
 alias finite_cover_balls_of_compact ← IsCompact.finite_cover_balls
 #align is_compact.finite_cover_balls IsCompact.finite_cover_balls
@@ -2463,6 +2909,7 @@ theorem tendsto_dist_left_cocompact_atTop [ProperSpace α] (x : α) :
 #align tendsto_dist_left_cocompact_at_top tendsto_dist_left_cocompact_atTop
 -/
 
+#print properSpace_of_compact_closedBall_of_le /-
 /-- If all closed balls of large enough radius are compact, then the space is proper. Especially
 useful when the lower bound for the radius is 0. -/
 theorem properSpace_of_compact_closedBall_of_le (R : ℝ)
@@ -2479,6 +2926,7 @@ theorem properSpace_of_compact_closedBall_of_le (R : ℝ)
       rw [this]
       exact (h x R le_rfl).inter_right is_closed_ball⟩
 #align proper_space_of_compact_closed_ball_of_le properSpace_of_compact_closedBall_of_le
+-/
 
 #print proper_of_compact /-
 -- A compact pseudometric space is proper 
@@ -2542,6 +2990,7 @@ instance pi_properSpace {π : β → Type _} [Fintype β] [∀ b, PseudoMetricSp
 
 variable [ProperSpace α] {x : α} {r : ℝ} {s : Set α}
 
+#print exists_pos_lt_subset_ball /-
 /-- If a nonempty ball in a proper space includes a closed set `s`, then there exists a nonempty
 ball with the same center and a strictly smaller radius that includes `s`. -/
 theorem exists_pos_lt_subset_ball (hr : 0 < r) (hs : IsClosed s) (h : s ⊆ ball x r) :
@@ -2558,7 +3007,9 @@ theorem exists_pos_lt_subset_ball (hr : 0 < r) (hs : IsClosed s) (h : s ⊆ ball
   rcases exists_between hyr with ⟨r', hyr', hrr'⟩
   exact ⟨r', ⟨dist_nonneg.trans_lt hyr', hrr'⟩, subset.trans hy <| closed_ball_subset_ball hyr'⟩
 #align exists_pos_lt_subset_ball exists_pos_lt_subset_ball
+-/
 
+#print exists_lt_subset_ball /-
 /-- If a ball in a proper space includes a closed set `s`, then there exists a ball with the same
 center and a strictly smaller radius that includes `s`. -/
 theorem exists_lt_subset_ball (hs : IsClosed s) (h : s ⊆ ball x r) : ∃ r' < r, s ⊆ ball x r' :=
@@ -2568,6 +3019,7 @@ theorem exists_lt_subset_ball (hs : IsClosed s) (h : s ⊆ ball x r) : ∃ r' <
     exact (exists_lt r).imp fun r' hr' => ⟨hr', empty_subset _⟩
   · exact (exists_pos_lt_subset_ball hr hs h).imp fun r' hr' => ⟨hr'.fst.2, hr'.snd⟩
 #align exists_lt_subset_ball exists_lt_subset_ball
+-/
 
 end ProperSpace
 
@@ -2584,6 +3036,7 @@ section SecondCountable
 
 open TopologicalSpace
 
+#print Metric.secondCountable_of_almost_dense_set /-
 /-- A pseudometric space is second countable if, for every `ε > 0`, there is a countable set which
 is `ε`-dense. -/
 theorem secondCountable_of_almost_dense_set
@@ -2596,11 +3049,13 @@ theorem secondCountable_of_almost_dense_set
   refine' ⟨s, hsc, Union₂_eq_univ_iff.2 fun x => ⟨y x, hys _, le_trans _ ε'ε.le⟩⟩
   exact_mod_cast hyx x
 #align metric.second_countable_of_almost_dense_set Metric.secondCountable_of_almost_dense_set
+-/
 
 end SecondCountable
 
 end Metric
 
+#print lebesgue_number_lemma_of_metric /-
 theorem lebesgue_number_lemma_of_metric {s : Set α} {ι} {c : ι → Set α} (hs : IsCompact s)
     (hc₁ : ∀ i, IsOpen (c i)) (hc₂ : s ⊆ ⋃ i, c i) : ∃ δ > 0, ∀ x ∈ s, ∃ i, ball x δ ⊆ c i :=
   let ⟨n, en, hn⟩ := lebesgue_number_lemma hs hc₁ hc₂
@@ -2609,11 +3064,14 @@ theorem lebesgue_number_lemma_of_metric {s : Set α} {ι} {c : ι → Set α} (h
     let ⟨i, hi⟩ := hn x hx
     ⟨i, fun y hy => hi (hδ (mem_ball'.mp hy))⟩⟩
 #align lebesgue_number_lemma_of_metric lebesgue_number_lemma_of_metric
+-/
 
+#print lebesgue_number_lemma_of_metric_sUnion /-
 theorem lebesgue_number_lemma_of_metric_sUnion {s : Set α} {c : Set (Set α)} (hs : IsCompact s)
     (hc₁ : ∀ t ∈ c, IsOpen t) (hc₂ : s ⊆ ⋃₀ c) : ∃ δ > 0, ∀ x ∈ s, ∃ t ∈ c, ball x δ ⊆ t := by
   rw [sUnion_eq_Union] at hc₂  <;> simpa using lebesgue_number_lemma_of_metric hs (by simpa) hc₂
 #align lebesgue_number_lemma_of_metric_sUnion lebesgue_number_lemma_of_metric_sUnion
+-/
 
 namespace Metric
 
@@ -2707,12 +3165,14 @@ theorem Bounded.subset_ball (h : Bounded s) (c : α) : ∃ r, s ⊆ closedBall c
 #align metric.bounded.subset_ball Metric.Bounded.subset_ball
 -/
 
+#print Metric.Bounded.subset_ball_lt /-
 theorem Bounded.subset_ball_lt (h : Bounded s) (a : ℝ) (c : α) : ∃ r, a < r ∧ s ⊆ closedBall c r :=
   by
   rcases h.subset_ball c with ⟨r, hr⟩
   refine' ⟨max r (a + 1), lt_of_lt_of_le (by linarith) (le_max_right _ _), _⟩
   exact subset.trans hr (closed_ball_subset_closed_ball (le_max_left _ _))
 #align metric.bounded.subset_ball_lt Metric.Bounded.subset_ball_lt
+-/
 
 #print Metric.bounded_closure_of_bounded /-
 theorem bounded_closure_of_bounded (h : Bounded s) : Bounded (closure s) :=
@@ -2731,6 +3191,7 @@ theorem bounded_closure_iff : Bounded (closure s) ↔ Bounded s :=
 #align metric.bounded_closure_iff Metric.bounded_closure_iff
 -/
 
+#print Metric.Bounded.union /-
 /-- The union of two bounded sets is bounded. -/
 theorem Bounded.union (hs : Bounded s) (ht : Bounded t) : Bounded (s ∪ t) :=
   by
@@ -2742,12 +3203,15 @@ theorem Bounded.union (hs : Bounded s) (ht : Bounded t) : Bounded (s ∪ t) :=
       union_subset (subset.trans hCs <| closed_ball_subset_closed_ball <| le_max_left _ _)
         (subset.trans hCt <| closed_ball_subset_closed_ball <| le_max_right _ _)⟩
 #align metric.bounded.union Metric.Bounded.union
+-/
 
+#print Metric.bounded_union /-
 /-- The union of two sets is bounded iff each of the sets is bounded. -/
 @[simp]
 theorem bounded_union : Bounded (s ∪ t) ↔ Bounded s ∧ Bounded t :=
   ⟨fun h => ⟨h.mono (by simp), h.mono (by simp)⟩, fun h => h.1.union h.2⟩
 #align metric.bounded_union Metric.bounded_union
+-/
 
 #print Metric.bounded_biUnion /-
 /-- A finite union of bounded sets is bounded -/
@@ -2811,11 +3275,13 @@ theorem bounded_singleton {x : α} : Bounded ({x} : Set α) :=
 #align metric.bounded_singleton Metric.bounded_singleton
 -/
 
+#print Metric.bounded_range_iff /-
 /-- Characterization of the boundedness of the range of a function -/
 theorem bounded_range_iff {f : β → α} : Bounded (range f) ↔ ∃ C, ∀ x y, dist (f x) (f y) ≤ C :=
   exists_congr fun C =>
     ⟨fun H x y => H _ ⟨x, rfl⟩ _ ⟨y, rfl⟩, by rintro H _ ⟨x, rfl⟩ _ ⟨y, rfl⟩ <;> exact H x y⟩
 #align metric.bounded_range_iff Metric.bounded_range_iff
+-/
 
 #print Metric.bounded_range_of_tendsto_cofinite_uniformity /-
 theorem bounded_range_of_tendsto_cofinite_uniformity {f : β → α}
@@ -2865,6 +3331,7 @@ theorem bounded_range_of_tendsto (u : ℕ → α) {x : α} (hu : Tendsto u atTop
 #align metric.bounded_range_of_tendsto Metric.bounded_range_of_tendsto
 -/
 
+#print Metric.exists_isOpen_bounded_image_inter_of_isCompact_of_forall_continuousWithinAt /-
 /-- If a function is continuous within a set `s` at every point of a compact set `k`, then it is
 bounded on some open neighborhood of `k` in `s`. -/
 theorem exists_isOpen_bounded_image_inter_of_isCompact_of_forall_continuousWithinAt
@@ -2890,6 +3357,7 @@ theorem exists_isOpen_bounded_image_inter_of_isCompact_of_forall_continuousWithi
     · apply bounded.mono (image_subset _ uf)
       exact bounded_ball.mono (image_preimage_subset _ _)
 #align metric.exists_is_open_bounded_image_inter_of_is_compact_of_forall_continuous_within_at Metric.exists_isOpen_bounded_image_inter_of_isCompact_of_forall_continuousWithinAt
+-/
 
 #print Metric.exists_isOpen_bounded_image_of_isCompact_of_forall_continuousAt /-
 /-- If a function is continuous at every point of a compact set `k`, then it is bounded on
@@ -2904,6 +3372,7 @@ theorem exists_isOpen_bounded_image_of_isCompact_of_forall_continuousAt [Topolog
 #align metric.exists_is_open_bounded_image_of_is_compact_of_forall_continuous_at Metric.exists_isOpen_bounded_image_of_isCompact_of_forall_continuousAt
 -/
 
+#print Metric.exists_isOpen_bounded_image_inter_of_isCompact_of_continuousOn /-
 /-- If a function is continuous on a set `s` containing a compact set `k`, then it is bounded on
 some open neighborhood of `k` in `s`. -/
 theorem exists_isOpen_bounded_image_inter_of_isCompact_of_continuousOn [TopologicalSpace β]
@@ -2912,6 +3381,7 @@ theorem exists_isOpen_bounded_image_inter_of_isCompact_of_continuousOn [Topologi
   exists_isOpen_bounded_image_inter_of_isCompact_of_forall_continuousWithinAt hk fun x hx =>
     hf x (hks hx)
 #align metric.exists_is_open_bounded_image_inter_of_is_compact_of_continuous_on Metric.exists_isOpen_bounded_image_inter_of_isCompact_of_continuousOn
+-/
 
 #print Metric.exists_isOpen_bounded_image_of_isCompact_of_continuousOn /-
 /-- If a function is continuous on a neighborhood of a compact set `k`, then it is bounded on
@@ -3012,26 +3482,34 @@ noncomputable def diam (s : Set α) : ℝ :=
 #align metric.diam Metric.diam
 -/
 
+#print Metric.diam_nonneg /-
 /-- The diameter of a set is always nonnegative -/
 theorem diam_nonneg : 0 ≤ diam s :=
   ENNReal.toReal_nonneg
 #align metric.diam_nonneg Metric.diam_nonneg
+-/
 
+#print Metric.diam_subsingleton /-
 theorem diam_subsingleton (hs : s.Subsingleton) : diam s = 0 := by
   simp only [diam, EMetric.diam_subsingleton hs, ENNReal.zero_toReal]
 #align metric.diam_subsingleton Metric.diam_subsingleton
+-/
 
+#print Metric.diam_empty /-
 /-- The empty set has zero diameter -/
 @[simp]
 theorem diam_empty : diam (∅ : Set α) = 0 :=
   diam_subsingleton subsingleton_empty
 #align metric.diam_empty Metric.diam_empty
+-/
 
+#print Metric.diam_singleton /-
 /-- A singleton has zero diameter -/
 @[simp]
 theorem diam_singleton : diam ({x} : Set α) = 0 :=
   diam_subsingleton subsingleton_singleton
 #align metric.diam_singleton Metric.diam_singleton
+-/
 
 #print Metric.diam_pair /-
 -- Does not work as a simp-lemma, since {x, y} reduces to (insert y {x})
@@ -3040,6 +3518,7 @@ theorem diam_pair : diam ({x, y} : Set α) = dist x y := by
 #align metric.diam_pair Metric.diam_pair
 -/
 
+#print Metric.diam_triple /-
 -- Does not work as a simp-lemma, since {x, y, z} reduces to (insert z (insert y {x}))
 theorem diam_triple :
     Metric.diam ({x, y, z} : Set α) = max (max (dist x y) (dist x z)) (dist y z) :=
@@ -3047,21 +3526,27 @@ theorem diam_triple :
   simp only [Metric.diam, EMetric.diam_triple, dist_edist]
   rw [ENNReal.toReal_max, ENNReal.toReal_max] <;> apply_rules [ne_of_lt, edist_lt_top, max_lt]
 #align metric.diam_triple Metric.diam_triple
+-/
 
+#print Metric.ediam_le_of_forall_dist_le /-
 /-- If the distance between any two points in a set is bounded by some constant `C`,
 then `ennreal.of_real C`  bounds the emetric diameter of this set. -/
 theorem ediam_le_of_forall_dist_le {C : ℝ} (h : ∀ x ∈ s, ∀ y ∈ s, dist x y ≤ C) :
     EMetric.diam s ≤ ENNReal.ofReal C :=
   EMetric.diam_le fun x hx y hy => (edist_dist x y).symm ▸ ENNReal.ofReal_le_ofReal (h x hx y hy)
 #align metric.ediam_le_of_forall_dist_le Metric.ediam_le_of_forall_dist_le
+-/
 
+#print Metric.diam_le_of_forall_dist_le /-
 /-- If the distance between any two points in a set is bounded by some non-negative constant,
 this constant bounds the diameter. -/
 theorem diam_le_of_forall_dist_le {C : ℝ} (h₀ : 0 ≤ C) (h : ∀ x ∈ s, ∀ y ∈ s, dist x y ≤ C) :
     diam s ≤ C :=
   ENNReal.toReal_le_of_le_ofReal h₀ (ediam_le_of_forall_dist_le h)
 #align metric.diam_le_of_forall_dist_le Metric.diam_le_of_forall_dist_le
+-/
 
+#print Metric.diam_le_of_forall_dist_le_of_nonempty /-
 /-- If the distance between any two points in a nonempty set is bounded by some constant,
 this constant bounds the diameter. -/
 theorem diam_le_of_forall_dist_le_of_nonempty (hs : s.Nonempty) {C : ℝ}
@@ -3071,7 +3556,9 @@ theorem diam_le_of_forall_dist_le_of_nonempty (hs : s.Nonempty) {C : ℝ}
     le_trans dist_nonneg (h x hx x hx)
   diam_le_of_forall_dist_le h₀ h
 #align metric.diam_le_of_forall_dist_le_of_nonempty Metric.diam_le_of_forall_dist_le_of_nonempty
+-/
 
+#print Metric.dist_le_diam_of_mem' /-
 /-- The distance between two points in a set is controlled by the diameter of the set. -/
 theorem dist_le_diam_of_mem' (h : EMetric.diam s ≠ ⊤) (hx : x ∈ s) (hy : y ∈ s) :
     dist x y ≤ diam s := by
@@ -3079,50 +3566,68 @@ theorem dist_le_diam_of_mem' (h : EMetric.diam s ≠ ⊤) (hx : x ∈ s) (hy : y
   rw [ENNReal.toReal_le_toReal (edist_ne_top _ _) h]
   exact EMetric.edist_le_diam_of_mem hx hy
 #align metric.dist_le_diam_of_mem' Metric.dist_le_diam_of_mem'
+-/
 
+#print Metric.bounded_iff_ediam_ne_top /-
 /-- Characterize the boundedness of a set in terms of the finiteness of its emetric.diameter. -/
 theorem bounded_iff_ediam_ne_top : Bounded s ↔ EMetric.diam s ≠ ⊤ :=
   Iff.intro
     (fun ⟨C, hC⟩ => ne_top_of_le_ne_top ENNReal.ofReal_ne_top <| ediam_le_of_forall_dist_le hC)
     fun h => ⟨diam s, fun x hx y hy => dist_le_diam_of_mem' h hx hy⟩
 #align metric.bounded_iff_ediam_ne_top Metric.bounded_iff_ediam_ne_top
+-/
 
+#print Metric.Bounded.ediam_ne_top /-
 theorem Bounded.ediam_ne_top (h : Bounded s) : EMetric.diam s ≠ ⊤ :=
   bounded_iff_ediam_ne_top.1 h
 #align metric.bounded.ediam_ne_top Metric.Bounded.ediam_ne_top
+-/
 
+#print Metric.ediam_univ_eq_top_iff_noncompact /-
 theorem ediam_univ_eq_top_iff_noncompact [ProperSpace α] :
     EMetric.diam (univ : Set α) = ∞ ↔ NoncompactSpace α := by
   rw [← not_compactSpace_iff, compact_space_iff_bounded_univ, bounded_iff_ediam_ne_top,
     Classical.not_not]
 #align metric.ediam_univ_eq_top_iff_noncompact Metric.ediam_univ_eq_top_iff_noncompact
+-/
 
+#print Metric.ediam_univ_of_noncompact /-
 @[simp]
 theorem ediam_univ_of_noncompact [ProperSpace α] [NoncompactSpace α] :
     EMetric.diam (univ : Set α) = ∞ :=
   ediam_univ_eq_top_iff_noncompact.mpr ‹_›
 #align metric.ediam_univ_of_noncompact Metric.ediam_univ_of_noncompact
+-/
 
+#print Metric.diam_univ_of_noncompact /-
 @[simp]
 theorem diam_univ_of_noncompact [ProperSpace α] [NoncompactSpace α] : diam (univ : Set α) = 0 := by
   simp [diam]
 #align metric.diam_univ_of_noncompact Metric.diam_univ_of_noncompact
+-/
 
+#print Metric.dist_le_diam_of_mem /-
 /-- The distance between two points in a set is controlled by the diameter of the set. -/
 theorem dist_le_diam_of_mem (h : Bounded s) (hx : x ∈ s) (hy : y ∈ s) : dist x y ≤ diam s :=
   dist_le_diam_of_mem' h.ediam_ne_top hx hy
 #align metric.dist_le_diam_of_mem Metric.dist_le_diam_of_mem
+-/
 
+#print Metric.ediam_of_unbounded /-
 theorem ediam_of_unbounded (h : ¬Bounded s) : EMetric.diam s = ∞ := by
   rwa [bounded_iff_ediam_ne_top, Classical.not_not] at h 
 #align metric.ediam_of_unbounded Metric.ediam_of_unbounded
+-/
 
+#print Metric.diam_eq_zero_of_unbounded /-
 /-- An unbounded set has zero diameter. If you would prefer to get the value ∞, use `emetric.diam`.
 This lemma makes it possible to avoid side conditions in some situations -/
 theorem diam_eq_zero_of_unbounded (h : ¬Bounded s) : diam s = 0 := by
   rw [diam, ediam_of_unbounded h, ENNReal.top_toReal]
 #align metric.diam_eq_zero_of_unbounded Metric.diam_eq_zero_of_unbounded
+-/
 
+#print Metric.diam_mono /-
 /-- If `s ⊆ t`, then the diameter of `s` is bounded by that of `t`, provided `t` is bounded. -/
 theorem diam_mono {s t : Set α} (h : s ⊆ t) (ht : Bounded t) : diam s ≤ diam t :=
   by
@@ -3130,7 +3635,9 @@ theorem diam_mono {s t : Set α} (h : s ⊆ t) (ht : Bounded t) : diam s ≤ dia
   rw [ENNReal.toReal_le_toReal (bounded.mono h ht).ediam_ne_top ht.ediam_ne_top]
   exact EMetric.diam_mono h
 #align metric.diam_mono Metric.diam_mono
+-/
 
+#print Metric.diam_union /-
 /-- The diameter of a union is controlled by the sum of the diameters, and the distance between
 any two points in each of the sets. This lemma is true without any side condition, since it is
 obviously true if `s ∪ t` is unbounded. -/
@@ -3150,14 +3657,18 @@ theorem diam_union {t : Set α} (xs : x ∈ s) (yt : y ∈ t) :
   · rw [diam_eq_zero_of_unbounded H]
     apply_rules [add_nonneg, diam_nonneg, dist_nonneg]
 #align metric.diam_union Metric.diam_union
+-/
 
+#print Metric.diam_union' /-
 /-- If two sets intersect, the diameter of the union is bounded by the sum of the diameters. -/
 theorem diam_union' {t : Set α} (h : (s ∩ t).Nonempty) : diam (s ∪ t) ≤ diam s + diam t :=
   by
   rcases h with ⟨x, ⟨xs, xt⟩⟩
   simpa using diam_union xs xt
 #align metric.diam_union' Metric.diam_union'
+-/
 
+#print Metric.diam_le_of_subset_closedBall /-
 theorem diam_le_of_subset_closedBall {r : ℝ} (hr : 0 ≤ r) (h : s ⊆ closedBall x r) :
     diam s ≤ 2 * r :=
   diam_le_of_forall_dist_le (mul_nonneg zero_le_two hr) fun a ha b hb =>
@@ -3166,17 +3677,23 @@ theorem diam_le_of_subset_closedBall {r : ℝ} (hr : 0 ≤ r) (h : s ⊆ closedB
       _ ≤ r + r := (add_le_add (h ha) (h hb))
       _ = 2 * r := by simp [mul_two, mul_comm]
 #align metric.diam_le_of_subset_closed_ball Metric.diam_le_of_subset_closedBall
+-/
 
+#print Metric.diam_closedBall /-
 /-- The diameter of a closed ball of radius `r` is at most `2 r`. -/
 theorem diam_closedBall {r : ℝ} (h : 0 ≤ r) : diam (closedBall x r) ≤ 2 * r :=
   diam_le_of_subset_closedBall h Subset.rfl
 #align metric.diam_closed_ball Metric.diam_closedBall
+-/
 
+#print Metric.diam_ball /-
 /-- The diameter of a ball of radius `r` is at most `2 r`. -/
 theorem diam_ball {r : ℝ} (h : 0 ≤ r) : diam (ball x r) ≤ 2 * r :=
   diam_le_of_subset_closedBall h ball_subset_closedBall
 #align metric.diam_ball Metric.diam_ball
+-/
 
+#print IsComplete.nonempty_iInter_of_nonempty_biInter /-
 /-- If a family of complete sets with diameter tending to `0` is such that each finite intersection
 is nonempty, then the total intersection is also nonempty. -/
 theorem IsComplete.nonempty_iInter_of_nonempty_biInter {s : ℕ → Set α} (h0 : IsComplete (s 0))
@@ -3202,7 +3719,9 @@ theorem IsComplete.nonempty_iInter_of_nonempty_biInter {s : ℕ → Set α} (h0
   filter_upwards [Ici_mem_at_top n] with p hp
   exact I n p hp
 #align is_complete.nonempty_Inter_of_nonempty_bInter IsComplete.nonempty_iInter_of_nonempty_biInter
+-/
 
+#print Metric.nonempty_iInter_of_nonempty_biInter /-
 /-- In a complete space, if a family of closed sets with diameter tending to `0` is such that each
 finite intersection is nonempty, then the total intersection is also nonempty. -/
 theorem nonempty_iInter_of_nonempty_biInter [CompleteSpace α] {s : ℕ → Set α}
@@ -3210,9 +3729,11 @@ theorem nonempty_iInter_of_nonempty_biInter [CompleteSpace α] {s : ℕ → Set
     (h' : Tendsto (fun n => diam (s n)) atTop (𝓝 0)) : (⋂ n, s n).Nonempty :=
   (hs 0).IsComplete.nonempty_iInter_of_nonempty_biInter hs h's h h'
 #align metric.nonempty_Inter_of_nonempty_bInter Metric.nonempty_iInter_of_nonempty_biInter
+-/
 
 end Diam
 
+#print Metric.exists_local_min_mem_ball /-
 theorem exists_local_min_mem_ball [ProperSpace α] [TopologicalSpace β]
     [ConditionallyCompleteLinearOrder β] [OrderTopology β] {f : α → β} {a z : α} {r : ℝ}
     (hf : ContinuousOn f (closedBall a r)) (hz : z ∈ closedBall a r)
@@ -3223,6 +3744,7 @@ theorem exists_local_min_mem_ball [ProperSpace α] [TopologicalSpace β]
     (is_compact_closed_ball a r).exists_local_min_mem_open ball_subset_closed_ball hf hz hf1
       is_open_ball
 #align metric.exists_local_min_mem_ball Metric.exists_local_min_mem_ball
+-/
 
 end Metric
 
@@ -3239,6 +3761,7 @@ unsafe def positivity_diam : expr → tactic strictness
 
 end Tactic
 
+#print comap_dist_right_atTop_le_cocompact /-
 theorem comap_dist_right_atTop_le_cocompact (x : α) :
     comap (fun y => dist y x) atTop ≤ cocompact α :=
   by
@@ -3246,10 +3769,13 @@ theorem comap_dist_right_atTop_le_cocompact (x : α) :
   rcases hs.bounded.subset_ball x with ⟨r, hr⟩
   exact ⟨Ioi r, Ioi_mem_at_top r, fun y hy hys => (mem_closed_ball.1 <| hr hys).not_lt hy⟩
 #align comap_dist_right_at_top_le_cocompact comap_dist_right_atTop_le_cocompact
+-/
 
+#print comap_dist_left_atTop_le_cocompact /-
 theorem comap_dist_left_atTop_le_cocompact (x : α) : comap (dist x) atTop ≤ cocompact α := by
   simpa only [dist_comm _ x] using comap_dist_right_atTop_le_cocompact x
 #align comap_dist_left_at_top_le_cocompact comap_dist_left_atTop_le_cocompact
+-/
 
 #print comap_dist_right_atTop_eq_cocompact /-
 theorem comap_dist_right_atTop_eq_cocompact [ProperSpace α] (x : α) :
@@ -3293,6 +3819,7 @@ theorem MetricSpace.ext {α : Type _} {m m' : MetricSpace α} (h : m.toHasDist =
 #align metric_space.ext MetricSpace.ext
 -/
 
+#print MetricSpace.ofDistTopology /-
 /-- Construct a metric space structure whose underlying topological space structure
 (definitionally) agrees which a pre-existing topology which is compatible with a given distance
 function. -/
@@ -3304,80 +3831,109 @@ def MetricSpace.ofDistTopology {α : Type u} [TopologicalSpace α] (dist : α 
   { PseudoMetricSpace.ofDistTopology dist dist_self dist_comm dist_triangle H with
     eq_of_dist_eq_zero }
 #align metric_space.of_dist_topology MetricSpace.ofDistTopology
+-/
 
 variable {γ : Type w} [MetricSpace γ]
 
+#print eq_of_dist_eq_zero /-
 theorem eq_of_dist_eq_zero {x y : γ} : dist x y = 0 → x = y :=
   MetricSpace.eq_of_dist_eq_zero
 #align eq_of_dist_eq_zero eq_of_dist_eq_zero
+-/
 
+#print dist_eq_zero /-
 @[simp]
 theorem dist_eq_zero {x y : γ} : dist x y = 0 ↔ x = y :=
   Iff.intro eq_of_dist_eq_zero fun this : x = y => this ▸ dist_self _
 #align dist_eq_zero dist_eq_zero
+-/
 
+#print zero_eq_dist /-
 @[simp]
 theorem zero_eq_dist {x y : γ} : 0 = dist x y ↔ x = y := by rw [eq_comm, dist_eq_zero]
 #align zero_eq_dist zero_eq_dist
+-/
 
+#print dist_ne_zero /-
 theorem dist_ne_zero {x y : γ} : dist x y ≠ 0 ↔ x ≠ y := by
   simpa only [not_iff_not] using dist_eq_zero
 #align dist_ne_zero dist_ne_zero
+-/
 
+#print dist_le_zero /-
 @[simp]
 theorem dist_le_zero {x y : γ} : dist x y ≤ 0 ↔ x = y := by
   simpa [le_antisymm_iff, dist_nonneg] using @dist_eq_zero _ _ x y
 #align dist_le_zero dist_le_zero
+-/
 
+#print dist_pos /-
 @[simp]
 theorem dist_pos {x y : γ} : 0 < dist x y ↔ x ≠ y := by
   simpa only [not_le] using not_congr dist_le_zero
 #align dist_pos dist_pos
+-/
 
+#print eq_of_forall_dist_le /-
 theorem eq_of_forall_dist_le {x y : γ} (h : ∀ ε > 0, dist x y ≤ ε) : x = y :=
   eq_of_dist_eq_zero (eq_of_le_of_forall_le_of_dense dist_nonneg h)
 #align eq_of_forall_dist_le eq_of_forall_dist_le
+-/
 
+#print eq_of_nndist_eq_zero /-
 /-- Deduce the equality of points with the vanishing of the nonnegative distance-/
 theorem eq_of_nndist_eq_zero {x y : γ} : nndist x y = 0 → x = y := by
   simp only [← NNReal.eq_iff, ← dist_nndist, imp_self, NNReal.coe_zero, dist_eq_zero]
 #align eq_of_nndist_eq_zero eq_of_nndist_eq_zero
+-/
 
+#print nndist_eq_zero /-
 /-- Characterize the equality of points with the vanishing of the nonnegative distance-/
 @[simp]
 theorem nndist_eq_zero {x y : γ} : nndist x y = 0 ↔ x = y := by
   simp only [← NNReal.eq_iff, ← dist_nndist, imp_self, NNReal.coe_zero, dist_eq_zero]
 #align nndist_eq_zero nndist_eq_zero
+-/
 
+#print zero_eq_nndist /-
 @[simp]
 theorem zero_eq_nndist {x y : γ} : 0 = nndist x y ↔ x = y := by
   simp only [← NNReal.eq_iff, ← dist_nndist, imp_self, NNReal.coe_zero, zero_eq_dist]
 #align zero_eq_nndist zero_eq_nndist
+-/
 
 namespace Metric
 
 variable {x : γ} {s : Set γ}
 
+#print Metric.closedBall_zero /-
 @[simp]
 theorem closedBall_zero : closedBall x 0 = {x} :=
   Set.ext fun y => dist_le_zero
 #align metric.closed_ball_zero Metric.closedBall_zero
+-/
 
+#print Metric.sphere_zero /-
 @[simp]
 theorem sphere_zero : sphere x 0 = {x} :=
   Set.ext fun y => dist_eq_zero
 #align metric.sphere_zero Metric.sphere_zero
+-/
 
+#print Metric.subsingleton_closedBall /-
 theorem subsingleton_closedBall (x : γ) {r : ℝ} (hr : r ≤ 0) : (closedBall x r).Subsingleton :=
   by
   rcases hr.lt_or_eq with (hr | rfl)
   · rw [closed_ball_eq_empty.2 hr]; exact subsingleton_empty
   · rw [closed_ball_zero]; exact subsingleton_singleton
 #align metric.subsingleton_closed_ball Metric.subsingleton_closedBall
+-/
 
+#print Metric.subsingleton_sphere /-
 theorem subsingleton_sphere (x : γ) {r : ℝ} (hr : r ≤ 0) : (sphere x r).Subsingleton :=
   (subsingleton_closedBall x hr).anti sphere_subset_closedBall
 #align metric.subsingleton_sphere Metric.subsingleton_sphere
+-/
 
 #print MetricSpace.to_separated /-
 -- see Note [lower instance priority]
@@ -3387,6 +3943,7 @@ instance (priority := 100) MetricSpace.to_separated : SeparatedSpace γ :=
 #align metric_space.to_separated MetricSpace.to_separated
 -/
 
+#print Metric.uniformEmbedding_iff' /-
 /-- A map between metric spaces is a uniform embedding if and only if the distance between `f x`
 and `f y` is controlled in terms of the distance between `x` and `y` and conversely. -/
 theorem uniformEmbedding_iff' [MetricSpace β] {f : γ → β} :
@@ -3398,6 +3955,7 @@ theorem uniformEmbedding_iff' [MetricSpace β] {f : γ → β} :
     uniformity_basis_dist.uniform_inducing_iff uniformity_basis_dist, exists_prop]
   rfl
 #align metric.uniform_embedding_iff' Metric.uniformEmbedding_iff'
+-/
 
 #print MetricSpace.ofT0PseudoMetricSpace /-
 /-- If a `pseudo_metric_space` is a T₀ space, then it is a `metric_space`. -/
@@ -3416,17 +3974,22 @@ instance (priority := 100) MetricSpace.toEMetricSpace : EMetricSpace γ :=
 #align metric_space.to_emetric_space MetricSpace.toEMetricSpace
 -/
 
+#print Metric.isClosed_of_pairwise_le_dist /-
 theorem isClosed_of_pairwise_le_dist {s : Set γ} {ε : ℝ} (hε : 0 < ε)
     (hs : s.Pairwise fun x y => ε ≤ dist x y) : IsClosed s :=
   isClosed_of_spaced_out (dist_mem_uniformity hε) <| by simpa using hs
 #align metric.is_closed_of_pairwise_le_dist Metric.isClosed_of_pairwise_le_dist
+-/
 
+#print Metric.closedEmbedding_of_pairwise_le_dist /-
 theorem closedEmbedding_of_pairwise_le_dist {α : Type _} [TopologicalSpace α] [DiscreteTopology α]
     {ε : ℝ} (hε : 0 < ε) {f : α → γ} (hf : Pairwise fun x y => ε ≤ dist (f x) (f y)) :
     ClosedEmbedding f :=
   closedEmbedding_of_spaced_out (dist_mem_uniformity hε) <| by simpa using hf
 #align metric.closed_embedding_of_pairwise_le_dist Metric.closedEmbedding_of_pairwise_le_dist
+-/
 
+#print Metric.uniformEmbedding_bot_of_pairwise_le_dist /-
 /-- If `f : β → α` sends any two distinct points to points at distance at least `ε > 0`, then
 `f` is a uniform embedding with respect to the discrete uniformity on `β`. -/
 theorem uniformEmbedding_bot_of_pairwise_le_dist {β : Type _} {ε : ℝ} (hε : 0 < ε) {f : β → α}
@@ -3434,6 +3997,7 @@ theorem uniformEmbedding_bot_of_pairwise_le_dist {β : Type _} {ε : ℝ} (hε :
     @UniformEmbedding _ _ ⊥ (by infer_instance) f :=
   uniformEmbedding_of_spaced_out (dist_mem_uniformity hε) <| by simpa using hf
 #align metric.uniform_embedding_bot_of_pairwise_le_dist Metric.uniformEmbedding_bot_of_pairwise_le_dist
+-/
 
 end Metric
 
@@ -3474,6 +4038,7 @@ theorem MetricSpace.replaceTopology_eq {γ} [U : TopologicalSpace γ] (m : Metri
 #align metric_space.replace_topology_eq MetricSpace.replaceTopology_eq
 -/
 
+#print EMetricSpace.toMetricSpaceOfDist /-
 /-- One gets a metric space from an emetric space if the edistance
 is everywhere finite, by pushing the edistance to reals. We set it up so that the edist and the
 uniformity are defeq in the metric space and the emetric space. In this definition, the distance
@@ -3485,7 +4050,9 @@ def EMetricSpace.toMetricSpaceOfDist {α : Type u} [e : EMetricSpace α] (dist :
   @MetricSpace.ofT0PseudoMetricSpace α
     (PseudoEMetricSpace.toPseudoMetricSpaceOfDist dist edist_ne_top h) _
 #align emetric_space.to_metric_space_of_dist EMetricSpace.toMetricSpaceOfDist
+-/
 
+#print EMetricSpace.toMetricSpace /-
 /-- One gets a metric space from an emetric space if the edistance
 is everywhere finite, by pushing the edistance to reals. We set it up so that the edist and the
 uniformity are defeq in the metric space and the emetric space. -/
@@ -3493,6 +4060,7 @@ def EMetricSpace.toMetricSpace {α : Type u} [EMetricSpace α] (h : ∀ x y : α
     MetricSpace α :=
   EMetricSpace.toMetricSpaceOfDist (fun x y => ENNReal.toReal (edist x y)) h fun x y => rfl
 #align emetric_space.to_metric_space EMetricSpace.toMetricSpace
+-/
 
 #print MetricSpace.replaceBornology /-
 /-- Build a new metric space from an old one where the bundled bornology structure is provably
@@ -3647,6 +4215,7 @@ section SecondCountable
 
 open TopologicalSpace
 
+#print Metric.secondCountable_of_countable_discretization /-
 /-- A metric space is second countable if one can reconstruct up to any `ε>0` any element of the
 space from countably many data. -/
 theorem secondCountable_of_countable_discretization {α : Type u} [MetricSpace α]
@@ -3668,6 +4237,7 @@ theorem secondCountable_of_countable_discretization {α : Type u} [MetricSpace 
   have : F x' = F x := Function.invFun_eq ⟨x, rfl⟩
   exact ⟨x', mem_range_self _, hF _ _ this.symm⟩
 #align metric.second_countable_of_countable_discretization Metric.secondCountable_of_countable_discretization
+-/
 
 end SecondCountable
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/a3e83f0fa4391c8740f7d773a7a9b74e311ae2a3

@@ -282,7 +282,7 @@ theorem dist_le_Ico_sum_dist (f : ℕ → α) {m n} (h : m ≤ n) :
   · intro n hn hrec
     calc
       dist (f m) (f (n + 1)) ≤ dist (f m) (f n) + dist _ _ := dist_triangle _ _ _
-      _ ≤ (∑ i in Finset.Ico m n, _) + _ := (add_le_add hrec le_rfl)
+      _ ≤ ∑ i in Finset.Ico m n, _ + _ := (add_le_add hrec le_rfl)
       _ = ∑ i in Finset.Ico m (n + 1), _ := by
         rw [Nat.Ico_succ_right_eq_insert_Ico hn, Finset.sum_insert, add_comm] <;> simp
 #align dist_le_Ico_sum_dist dist_le_Ico_sum_dist

mathlib commit https://github.com/leanprover-community/mathlib/commit/7e5137f579de09a059a5ce98f364a04e221aabf0

@@ -127,10 +127,9 @@ private theorem pseudo_metric_space.dist_nonneg' {α} {x y : α} (dist : α →
     calc
       2 * dist x y = dist x y + dist y x := by rw [dist_comm x y, two_mul]
       _ ≥ 0 := by rw [← dist_self x] <;> apply dist_triangle
-      
   nonneg_of_mul_nonneg_right this zero_lt_two
 
-/- ./././Mathport/Syntax/Translate/Expr.lean:330:4: warning: unsupported (TODO): `[tacs] -/
+/- ./././Mathport/Syntax/Translate/Expr.lean:336:4: warning: unsupported (TODO): `[tacs] -/
 /-- This tactic is used to populate `pseudo_metric_space.edist_dist` when the default `edist` is
 used. -/
 protected unsafe def pseudo_metric_space.edist_dist_tac : tactic Unit :=
@@ -261,7 +260,6 @@ theorem dist_triangle4 (x y z w : α) : dist x w ≤ dist x y + dist y z + dist
   calc
     dist x w ≤ dist x z + dist z w := dist_triangle x z w
     _ ≤ dist x y + dist y z + dist z w := add_le_add_right (dist_triangle x y z) _
-    
 #align dist_triangle4 dist_triangle4
 
 theorem dist_triangle4_left (x₁ y₁ x₂ y₂ : α) :
@@ -287,7 +285,6 @@ theorem dist_le_Ico_sum_dist (f : ℕ → α) {m n} (h : m ≤ n) :
       _ ≤ (∑ i in Finset.Ico m n, _) + _ := (add_le_add hrec le_rfl)
       _ = ∑ i in Finset.Ico m (n + 1), _ := by
         rw [Nat.Ico_succ_right_eq_insert_Ico hn, Finset.sum_insert, add_comm] <;> simp
-      
 #align dist_le_Ico_sum_dist dist_le_Ico_sum_dist
 
 /-- The triangle (polygon) inequality for sequences of points; `finset.range` version. -/
@@ -687,7 +684,6 @@ theorem ball_subset_ball' (h : ε₁ + dist x y ≤ ε₂) : ball x ε₁ ⊆ ba
     dist z y ≤ dist z x + dist x y := dist_triangle _ _ _
     _ < ε₁ + dist x y := (add_lt_add_right hz _)
     _ ≤ ε₂ := h
-    
 #align metric.ball_subset_ball' Metric.ball_subset_ball'
 
 theorem closedBall_subset_closedBall (h : ε₁ ≤ ε₂) : closedBall x ε₁ ⊆ closedBall x ε₂ :=
@@ -700,7 +696,6 @@ theorem closedBall_subset_closedBall' (h : ε₁ + dist x y ≤ ε₂) :
     dist z y ≤ dist z x + dist x y := dist_triangle _ _ _
     _ ≤ ε₁ + dist x y := (add_le_add_right hz _)
     _ ≤ ε₂ := h
-    
 #align metric.closed_ball_subset_closed_ball' Metric.closedBall_subset_closedBall'
 
 theorem closedBall_subset_ball (h : ε₁ < ε₂) : closedBall x ε₁ ⊆ ball x ε₂ :=
@@ -713,7 +708,6 @@ theorem closedBall_subset_ball' (h : ε₁ + dist x y < ε₂) : closedBall x ε
     dist z y ≤ dist z x + dist x y := dist_triangle _ _ _
     _ ≤ ε₁ + dist x y := (add_le_add_right hz _)
     _ < ε₂ := h
-    
 #align metric.closed_ball_subset_ball' Metric.closedBall_subset_ball'
 
 theorem dist_le_add_of_nonempty_closedBall_inter_closedBall
@@ -722,7 +716,6 @@ theorem dist_le_add_of_nonempty_closedBall_inter_closedBall
   calc
     dist x y ≤ dist z x + dist z y := dist_triangle_left _ _ _
     _ ≤ ε₁ + ε₂ := add_le_add hz.1 hz.2
-    
 #align metric.dist_le_add_of_nonempty_closed_ball_inter_closed_ball Metric.dist_le_add_of_nonempty_closedBall_inter_closedBall
 
 theorem dist_lt_add_of_nonempty_closedBall_inter_ball (h : (closedBall x ε₁ ∩ ball y ε₂).Nonempty) :
@@ -731,7 +724,6 @@ theorem dist_lt_add_of_nonempty_closedBall_inter_ball (h : (closedBall x ε₁ 
   calc
     dist x y ≤ dist z x + dist z y := dist_triangle_left _ _ _
     _ < ε₁ + ε₂ := add_lt_add_of_le_of_lt hz.1 hz.2
-    
 #align metric.dist_lt_add_of_nonempty_closed_ball_inter_ball Metric.dist_lt_add_of_nonempty_closedBall_inter_ball
 
 theorem dist_lt_add_of_nonempty_ball_inter_closedBall (h : (ball x ε₁ ∩ closedBall y ε₂).Nonempty) :
@@ -1707,7 +1699,6 @@ theorem cauchySeq_of_le_tendsto_0' {s : β → α} (b : β → ℝ)
         dist (s n) (s N) = dist (s N) (s n) := dist_comm _ _
         _ ≤ b N := (h _ _ hn)
         _ < ε := hN
-        
 #align cauchy_seq_of_le_tendsto_0' cauchySeq_of_le_tendsto_0'
 
 /-- If the distance between `s n` and `s m`, `n, m ≥ N` is bounded above by `b N`
@@ -2201,7 +2192,6 @@ theorem TopologicalSpace.IsSeparable.separableSpace {s : Set α} (hs : IsSeparab
     dist x A.some ≤ dist x z + dist z A.some := dist_triangle _ _ _
     _ < r / 2 + r / 2 := (add_lt_add (hz.trans hn) ((Metric.mem_ball'.1 A.some_spec.1).trans hn))
     _ = r := add_halves _
-    
 #align topological_space.is_separable.separable_space TopologicalSpace.IsSeparable.separableSpace
 
 #print Inducing.isSeparable_preimage /-
@@ -2676,8 +2666,7 @@ theorem bounded_closedBall : Bounded (closedBall x r) :=
     simp only [mem_closed_ball] at *
     calc
       dist y z ≤ dist y x + dist z x := dist_triangle_right _ _ _
-      _ ≤ r + r := add_le_add hy hz
-      ⟩
+      _ ≤ r + r := add_le_add hy hz⟩
 #align metric.bounded_closed_ball Metric.bounded_closedBall
 -/
 
@@ -2707,8 +2696,7 @@ theorem bounded_iff_subset_ball (c : α) : Bounded s ↔ ∃ r, s ⊆ closedBall
         ⟨C + dist x c, fun y hy =>
           calc
             dist y c ≤ dist y x + dist x c := dist_triangle _ _ _
-            _ ≤ C + dist x c := add_le_add_right (hC y hy x hx) _
-            ⟩
+            _ ≤ C + dist x c := add_le_add_right (hC y hy x hx) _⟩
   · exact bounded_closed_ball.mono hC
 #align metric.bounded_iff_subset_ball Metric.bounded_iff_subset_ball
 -/
@@ -3177,7 +3165,6 @@ theorem diam_le_of_subset_closedBall {r : ℝ} (hr : 0 ≤ r) (h : s ⊆ closedB
       dist a b ≤ dist a x + dist b x := dist_triangle_right _ _ _
       _ ≤ r + r := (add_le_add (h ha) (h hb))
       _ = 2 * r := by simp [mul_two, mul_comm]
-      
 #align metric.diam_le_of_subset_closed_ball Metric.diam_le_of_subset_closedBall
 
 /-- The diameter of a closed ball of radius `r` is at most `2 r`. -/

mathlib commit https://github.com/leanprover-community/mathlib/commit/31c24aa72e7b3e5ed97a8412470e904f82b81004

@@ -934,14 +934,14 @@ theorem uniformContinuous_iff [PseudoMetricSpace β] {f : α → β} :
   uniformity_basis_dist.uniformContinuous_iff uniformity_basis_dist
 #align metric.uniform_continuous_iff Metric.uniformContinuous_iff
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (x y «expr ∈ » s) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:638:2: warning: expanding binder collection (x y «expr ∈ » s) -/
 theorem uniformContinuousOn_iff [PseudoMetricSpace β] {f : α → β} {s : Set α} :
     UniformContinuousOn f s ↔
       ∀ ε > 0, ∃ δ > 0, ∀ (x) (_ : x ∈ s) (y) (_ : y ∈ s), dist x y < δ → dist (f x) (f y) < ε :=
   Metric.uniformity_basis_dist.uniformContinuousOn_iff Metric.uniformity_basis_dist
 #align metric.uniform_continuous_on_iff Metric.uniformContinuousOn_iff
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (x y «expr ∈ » s) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:638:2: warning: expanding binder collection (x y «expr ∈ » s) -/
 theorem uniformContinuousOn_iff_le [PseudoMetricSpace β] {f : α → β} {s : Set α} :
     UniformContinuousOn f s ↔
       ∀ ε > 0, ∃ δ > 0, ∀ (x) (_ : x ∈ s) (y) (_ : y ∈ s), dist x y ≤ δ → dist (f x) (f y) ≤ ε :=
@@ -998,7 +998,7 @@ theorem totallyBounded_of_finite_discretization {s : Set α}
   exact ⟨_, ⟨F ⟨x, xs⟩, rfl⟩, hF _ _ this.symm⟩
 #align metric.totally_bounded_of_finite_discretization Metric.totallyBounded_of_finite_discretization
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (t «expr ⊆ » s) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:638:2: warning: expanding binder collection (t «expr ⊆ » s) -/
 theorem finite_approx_of_totallyBounded {s : Set α} (hs : TotallyBounded s) :
     ∀ ε > 0, ∃ (t : _) (_ : t ⊆ s), Set.Finite t ∧ s ⊆ ⋃ y ∈ t, ball y ε :=
   by
@@ -1055,7 +1055,7 @@ theorem tendstoUniformly_iff {ι : Type _} {F : ι → β → α} {f : β → α
   rw [← tendstoUniformlyOn_univ, tendsto_uniformly_on_iff]; simp
 #align metric.tendsto_uniformly_iff Metric.tendstoUniformly_iff
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (x y «expr ∈ » t) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:638:2: warning: expanding binder collection (x y «expr ∈ » t) -/
 protected theorem cauchy_iff {f : Filter α} :
     Cauchy f ↔ NeBot f ∧ ∀ ε > 0, ∃ t ∈ f, ∀ (x) (_ : x ∈ t) (y) (_ : y ∈ t), dist x y < ε :=
   uniformity_basis_dist.cauchy_iff
@@ -1651,7 +1651,7 @@ section CauchySeq
 
 variable [Nonempty β] [SemilatticeSup β]
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (m n «expr ≥ » N) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:638:2: warning: expanding binder collection (m n «expr ≥ » N) -/
 -- see Note [nolint_ge]
 /-- In a pseudometric space, Cauchy sequences are characterized by the fact that, eventually,
 the distance between its elements is arbitrarily small -/
@@ -2399,7 +2399,7 @@ end Pi
 
 section Compact
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (t «expr ⊆ » s) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:638:2: warning: expanding binder collection (t «expr ⊆ » s) -/
 /-- Any compact set in a pseudometric space can be covered by finitely many balls of a given
 positive radius -/
 theorem finite_cover_balls_of_compact {α : Type u} [PseudoMetricSpace α] {s : Set α}
@@ -2507,7 +2507,7 @@ instance (priority := 100) locally_compact_of_proper [ProperSpace α] : LocallyC
 #align locally_compact_of_proper locally_compact_of_proper
 -/
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (x y «expr ∈ » t) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:638:2: warning: expanding binder collection (x y «expr ∈ » t) -/
 #print complete_of_proper /-
 -- see Note [lower instance priority]
 /-- A proper space is complete -/
@@ -2627,7 +2627,7 @@ theorem lebesgue_number_lemma_of_metric_sUnion {s : Set α} {c : Set (Set α)} (
 
 namespace Metric
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (x y «expr ∈ » s) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:638:2: warning: expanding binder collection (x y «expr ∈ » s) -/
 #print Metric.Bounded /-
 /-- Boundedness of a subset of a pseudometric space. We formulate the definition to work
 even in the empty space. -/

mathlib commit https://github.com/leanprover-community/mathlib/commit/5f25c089cb34db4db112556f23c50d12da81b297

@@ -90,7 +90,7 @@ private theorem bounded_iff_aux {α : Type _} (dist : α → α → ℝ)
 def Bornology.ofDist {α : Type _} (dist : α → α → ℝ) (dist_self : ∀ x : α, dist x x = 0)
     (dist_comm : ∀ x y : α, dist x y = dist y x)
     (dist_triangle : ∀ x y z : α, dist x z ≤ dist x y + dist y z) : Bornology α :=
-  Bornology.ofBounded { s : Set α | ∃ C, ∀ ⦃x⦄, x ∈ s → ∀ ⦃y⦄, y ∈ s → dist x y ≤ C }
+  Bornology.ofBounded {s : Set α | ∃ C, ∀ ⦃x⦄, x ∈ s → ∀ ⦃y⦄, y ∈ s → dist x y ≤ C}
     ⟨0, fun x hx y => hx.elim⟩ (fun s ⟨c, hc⟩ t h => ⟨c, fun x hx y hy => hc (h hx) (h hy)⟩)
     (fun s hs t ht => by
       rcases s.eq_empty_or_nonempty with (rfl | ⟨z, hz⟩)
@@ -159,10 +159,10 @@ class PseudoMetricSpace (α : Type u) extends Dist α : Type u where
     run_tac
       pseudo_metric_space.edist_dist_tac
   toUniformSpace : UniformSpace α := UniformSpace.ofDist dist dist_self dist_comm dist_triangle
-  uniformity_dist : 𝓤 α = ⨅ ε > 0, 𝓟 { p : α × α | dist p.1 p.2 < ε } := by intros; rfl
+  uniformity_dist : 𝓤 α = ⨅ ε > 0, 𝓟 {p : α × α | dist p.1 p.2 < ε} := by intros; rfl
   toBornology : Bornology α := Bornology.ofDist dist dist_self dist_comm dist_triangle
   cobounded_sets :
-    (Bornology.cobounded α).sets = { s | ∃ C, ∀ ⦃x⦄, x ∈ sᶜ → ∀ ⦃y⦄, y ∈ sᶜ → dist x y ≤ C } := by
+    (Bornology.cobounded α).sets = {s | ∃ C, ∀ ⦃x⦄, x ∈ sᶜ → ∀ ⦃y⦄, y ∈ sᶜ → dist x y ≤ C} := by
     intros; rfl
 #align pseudo_metric_space PseudoMetricSpace
 -/
@@ -481,7 +481,7 @@ variable {x y z : α} {δ ε ε₁ ε₂ : ℝ} {s : Set α}
 #print Metric.ball /-
 /-- `ball x ε` is the set of all points `y` with `dist y x < ε` -/
 def ball (x : α) (ε : ℝ) : Set α :=
-  { y | dist y x < ε }
+  {y | dist y x < ε}
 #align metric.ball Metric.ball
 -/
 
@@ -526,12 +526,12 @@ theorem exists_lt_mem_ball_of_mem_ball (h : x ∈ ball y ε) : ∃ ε' < ε, x 
 #align metric.exists_lt_mem_ball_of_mem_ball Metric.exists_lt_mem_ball_of_mem_ball
 
 theorem ball_eq_ball (ε : ℝ) (x : α) :
-    UniformSpace.ball x { p | dist p.2 p.1 < ε } = Metric.ball x ε :=
+    UniformSpace.ball x {p | dist p.2 p.1 < ε} = Metric.ball x ε :=
   rfl
 #align metric.ball_eq_ball Metric.ball_eq_ball
 
 theorem ball_eq_ball' (ε : ℝ) (x : α) :
-    UniformSpace.ball x { p | dist p.1 p.2 < ε } = Metric.ball x ε := by ext;
+    UniformSpace.ball x {p | dist p.1 p.2 < ε} = Metric.ball x ε := by ext;
   simp [dist_comm, UniformSpace.ball]
 #align metric.ball_eq_ball' Metric.ball_eq_ball'
 
@@ -548,7 +548,7 @@ theorem iUnion_ball_nat_succ (x : α) : (⋃ n : ℕ, ball x (n + 1)) = univ :=
 #print Metric.closedBall /-
 /-- `closed_ball x ε` is the set of all points `y` with `dist y x ≤ ε` -/
 def closedBall (x : α) (ε : ℝ) :=
-  { y | dist y x ≤ ε }
+  {y | dist y x ≤ ε}
 #align metric.closed_ball Metric.closedBall
 -/
 
@@ -563,7 +563,7 @@ theorem mem_closedBall' : y ∈ closedBall x ε ↔ dist x y ≤ ε := by rw [di
 #print Metric.sphere /-
 /-- `sphere x ε` is the set of all points `y` with `dist y x = ε` -/
 def sphere (x : α) (ε : ℝ) :=
-  { y | dist y x = ε }
+  {y | dist y x = ε}
 #align metric.sphere Metric.sphere
 -/
 
@@ -829,7 +829,7 @@ theorem toUniformSpace_eq :
 -/
 
 theorem uniformity_basis_dist :
-    (𝓤 α).HasBasis (fun ε : ℝ => 0 < ε) fun ε => { p : α × α | dist p.1 p.2 < ε } :=
+    (𝓤 α).HasBasis (fun ε : ℝ => 0 < ε) fun ε => {p : α × α | dist p.1 p.2 < ε} :=
   by
   rw [to_uniform_space_eq]
   exact UniformSpace.hasBasis_ofFun (exists_gt _) _ _ _ _ _
@@ -842,7 +842,7 @@ For specific bases see `uniformity_basis_dist`, `uniformity_basis_dist_inv_nat_s
 and `uniformity_basis_dist_inv_nat_pos`. -/
 protected theorem mk_uniformity_basis {β : Type _} {p : β → Prop} {f : β → ℝ}
     (hf₀ : ∀ i, p i → 0 < f i) (hf : ∀ ⦃ε⦄, 0 < ε → ∃ (i : _) (hi : p i), f i ≤ ε) :
-    (𝓤 α).HasBasis p fun i => { p : α × α | dist p.1 p.2 < f i } :=
+    (𝓤 α).HasBasis p fun i => {p : α × α | dist p.1 p.2 < f i} :=
   by
   refine' ⟨fun s => uniformity_basis_dist.mem_iff.trans _⟩
   constructor
@@ -853,34 +853,34 @@ protected theorem mk_uniformity_basis {β : Type _} {p : β → Prop} {f : β 
 #align metric.mk_uniformity_basis Metric.mk_uniformity_basis
 
 theorem uniformity_basis_dist_rat :
-    (𝓤 α).HasBasis (fun r : ℚ => 0 < r) fun r => { p : α × α | dist p.1 p.2 < r } :=
+    (𝓤 α).HasBasis (fun r : ℚ => 0 < r) fun r => {p : α × α | dist p.1 p.2 < r} :=
   Metric.mk_uniformity_basis (fun _ => Rat.cast_pos.2) fun ε hε =>
     let ⟨r, hr0, hrε⟩ := exists_rat_btwn hε
     ⟨r, Rat.cast_pos.1 hr0, hrε.le⟩
 #align metric.uniformity_basis_dist_rat Metric.uniformity_basis_dist_rat
 
 theorem uniformity_basis_dist_inv_nat_succ :
-    (𝓤 α).HasBasis (fun _ => True) fun n : ℕ => { p : α × α | dist p.1 p.2 < 1 / (↑n + 1) } :=
+    (𝓤 α).HasBasis (fun _ => True) fun n : ℕ => {p : α × α | dist p.1 p.2 < 1 / (↑n + 1)} :=
   Metric.mk_uniformity_basis (fun n _ => div_pos zero_lt_one <| Nat.cast_add_one_pos n) fun ε ε0 =>
     (exists_nat_one_div_lt ε0).imp fun n hn => ⟨trivial, le_of_lt hn⟩
 #align metric.uniformity_basis_dist_inv_nat_succ Metric.uniformity_basis_dist_inv_nat_succ
 
 theorem uniformity_basis_dist_inv_nat_pos :
-    (𝓤 α).HasBasis (fun n : ℕ => 0 < n) fun n : ℕ => { p : α × α | dist p.1 p.2 < 1 / ↑n } :=
+    (𝓤 α).HasBasis (fun n : ℕ => 0 < n) fun n : ℕ => {p : α × α | dist p.1 p.2 < 1 / ↑n} :=
   Metric.mk_uniformity_basis (fun n hn => div_pos zero_lt_one <| Nat.cast_pos.2 hn) fun ε ε0 =>
     let ⟨n, hn⟩ := exists_nat_one_div_lt ε0
     ⟨n + 1, Nat.succ_pos n, by exact_mod_cast hn.le⟩
 #align metric.uniformity_basis_dist_inv_nat_pos Metric.uniformity_basis_dist_inv_nat_pos
 
 theorem uniformity_basis_dist_pow {r : ℝ} (h0 : 0 < r) (h1 : r < 1) :
-    (𝓤 α).HasBasis (fun n : ℕ => True) fun n : ℕ => { p : α × α | dist p.1 p.2 < r ^ n } :=
+    (𝓤 α).HasBasis (fun n : ℕ => True) fun n : ℕ => {p : α × α | dist p.1 p.2 < r ^ n} :=
   Metric.mk_uniformity_basis (fun n hn => pow_pos h0 _) fun ε ε0 =>
     let ⟨n, hn⟩ := exists_pow_lt_of_lt_one ε0 h1
     ⟨n, trivial, hn.le⟩
 #align metric.uniformity_basis_dist_pow Metric.uniformity_basis_dist_pow
 
 theorem uniformity_basis_dist_lt {R : ℝ} (hR : 0 < R) :
-    (𝓤 α).HasBasis (fun r : ℝ => 0 < r ∧ r < R) fun r => { p : α × α | dist p.1 p.2 < r } :=
+    (𝓤 α).HasBasis (fun r : ℝ => 0 < r ∧ r < R) fun r => {p : α × α | dist p.1 p.2 < r} :=
   Metric.mk_uniformity_basis (fun r => And.left) fun r hr =>
     ⟨min r (R / 2), ⟨lt_min hr (half_pos hR), min_lt_iff.2 <| Or.inr (half_lt_self hR)⟩,
       min_le_left _ _⟩
@@ -894,7 +894,7 @@ Currently we have only one specific basis `uniformity_basis_dist_le` based on th
 More can be easily added if needed in the future. -/
 protected theorem mk_uniformity_basis_le {β : Type _} {p : β → Prop} {f : β → ℝ}
     (hf₀ : ∀ x, p x → 0 < f x) (hf : ∀ ε, 0 < ε → ∃ (x : _) (hx : p x), f x ≤ ε) :
-    (𝓤 α).HasBasis p fun x => { p : α × α | dist p.1 p.2 ≤ f x } :=
+    (𝓤 α).HasBasis p fun x => {p : α × α | dist p.1 p.2 ≤ f x} :=
   by
   refine' ⟨fun s => uniformity_basis_dist.mem_iff.trans _⟩
   constructor
@@ -908,12 +908,12 @@ protected theorem mk_uniformity_basis_le {β : Type _} {p : β → Prop} {f : β
 /-- Contant size closed neighborhoods of the diagonal form a basis
 of the uniformity filter. -/
 theorem uniformity_basis_dist_le :
-    (𝓤 α).HasBasis (fun ε : ℝ => 0 < ε) fun ε => { p : α × α | dist p.1 p.2 ≤ ε } :=
+    (𝓤 α).HasBasis (fun ε : ℝ => 0 < ε) fun ε => {p : α × α | dist p.1 p.2 ≤ ε} :=
   Metric.mk_uniformity_basis_le (fun _ => id) fun ε ε₀ => ⟨ε, ε₀, le_refl ε⟩
 #align metric.uniformity_basis_dist_le Metric.uniformity_basis_dist_le
 
 theorem uniformity_basis_dist_le_pow {r : ℝ} (h0 : 0 < r) (h1 : r < 1) :
-    (𝓤 α).HasBasis (fun n : ℕ => True) fun n : ℕ => { p : α × α | dist p.1 p.2 ≤ r ^ n } :=
+    (𝓤 α).HasBasis (fun n : ℕ => True) fun n : ℕ => {p : α × α | dist p.1 p.2 ≤ r ^ n} :=
   Metric.mk_uniformity_basis_le (fun n hn => pow_pos h0 _) fun ε ε0 =>
     let ⟨n, hn⟩ := exists_pow_lt_of_lt_one ε0 h1
     ⟨n, trivial, hn.le⟩
@@ -925,7 +925,7 @@ theorem mem_uniformity_dist {s : Set (α × α)} :
 #align metric.mem_uniformity_dist Metric.mem_uniformity_dist
 
 /-- A constant size neighborhood of the diagonal is an entourage. -/
-theorem dist_mem_uniformity {ε : ℝ} (ε0 : 0 < ε) : { p : α × α | dist p.1 p.2 < ε } ∈ 𝓤 α :=
+theorem dist_mem_uniformity {ε : ℝ} (ε0 : 0 < ε) : {p : α × α | dist p.1 p.2 < ε} ∈ 𝓤 α :=
   mem_uniformity_dist.2 ⟨ε, ε0, fun a b => id⟩
 #align metric.dist_mem_uniformity Metric.dist_mem_uniformity
 
@@ -1283,7 +1283,7 @@ we need to show that the uniform structure coming from the edistance and the
 distance coincide. -/
 /-- Expressing the uniformity in terms of `edist` -/
 protected theorem Metric.uniformity_basis_edist :
-    (𝓤 α).HasBasis (fun ε : ℝ≥0∞ => 0 < ε) fun ε => { p | edist p.1 p.2 < ε } :=
+    (𝓤 α).HasBasis (fun ε : ℝ≥0∞ => 0 < ε) fun ε => {p | edist p.1 p.2 < ε} :=
   ⟨by
     intro t
     refine' mem_uniformity_dist.trans ⟨_, _⟩ <;> rintro ⟨ε, ε0, Hε⟩
@@ -1297,7 +1297,7 @@ protected theorem Metric.uniformity_basis_edist :
       rwa [edist_dist, ENNReal.ofReal_lt_ofReal_iff ε0']⟩
 #align pseudo_metric.uniformity_basis_edist Metric.uniformity_basis_edist
 
-theorem Metric.uniformity_edist : 𝓤 α = ⨅ ε > 0, 𝓟 { p : α × α | edist p.1 p.2 < ε } :=
+theorem Metric.uniformity_edist : 𝓤 α = ⨅ ε > 0, 𝓟 {p : α × α | edist p.1 p.2 < ε} :=
   Metric.uniformity_basis_edist.eq_biInf
 #align metric.uniformity_edist Metric.uniformity_edist
 
@@ -1472,7 +1472,7 @@ theorem Metric.complete_of_convergent_controlled_sequences (B : ℕ → Real) (h
         (∀ N n m : ℕ, N ≤ n → N ≤ m → dist (u n) (u m) < B N) → ∃ x, Tendsto u atTop (𝓝 x)) :
     CompleteSpace α :=
   UniformSpace.complete_of_convergent_controlled_sequences
-    (fun n => { p : α × α | dist p.1 p.2 < B n }) (fun n => dist_mem_uniformity <| hB n) H
+    (fun n => {p : α × α | dist p.1 p.2 < B n}) (fun n => dist_mem_uniformity <| hB n) H
 #align metric.complete_of_convergent_controlled_sequences Metric.complete_of_convergent_controlled_sequences
 
 #print Metric.complete_of_cauchySeq_tendsto /-
@@ -1642,7 +1642,7 @@ theorem eventually_closedBall_subset {x : α} {u : Set α} (hu : u ∈ 𝓝 x) :
   obtain ⟨ε, εpos, hε⟩ : ∃ (ε : _) (hε : 0 < ε), closed_ball x ε ⊆ u :=
     nhds_basis_closed_ball.mem_iff.1 hu
   have : Iic ε ∈ 𝓝 (0 : ℝ) := Iic_mem_nhds εpos
-  filter_upwards [this]with _ hr using subset.trans (closed_ball_subset_closed_ball hr) hε
+  filter_upwards [this] with _ hr using subset.trans (closed_ball_subset_closed_ball hr) hε
 #align eventually_closed_ball_subset eventually_closedBall_subset
 
 end Real
@@ -1679,7 +1679,7 @@ theorem Metric.uniformCauchySeqOn_iff {γ : Type _} {F : β → γ → α} {s :
   by
   constructor
   · intro h ε hε
-    let u := { a : α × α | dist a.fst a.snd < ε }
+    let u := {a : α × α | dist a.fst a.snd < ε}
     have hu : u ∈ 𝓤 α := metric.mem_uniformity_dist.mpr ⟨ε, hε, fun a b => by simp⟩
     rw [←
       @Filter.eventually_atTop_prod_self' _ _ _ fun m =>
@@ -1748,7 +1748,7 @@ theorem cauchySeq_iff_le_tendsto_0 {s : ℕ → α} :
       the supremum of the distances between `s n` and `s m` for `n m ≥ N`.
       First, we prove that all these distances are bounded, as otherwise the Sup
       would not make sense. -/
-    let S N := (fun p : ℕ × ℕ => dist (s p.1) (s p.2)) '' { p | p.1 ≥ N ∧ p.2 ≥ N }
+    let S N := (fun p : ℕ × ℕ => dist (s p.1) (s p.2)) '' {p | p.1 ≥ N ∧ p.2 ≥ N}
     have hS : ∀ N, ∃ x, ∀ y ∈ S N, y ≤ x :=
       by
       rcases cauchySeq_bdd hs with ⟨R, R0, hR⟩
@@ -2172,36 +2172,36 @@ be contained in `s`. -/
 theorem TopologicalSpace.IsSeparable.separableSpace {s : Set α} (hs : IsSeparable s) :
     SeparableSpace s := by
   classical
-    rcases eq_empty_or_nonempty s with (rfl | ⟨⟨x₀, x₀s⟩⟩)
-    · infer_instance
-    rcases hs with ⟨c, hc, h'c⟩
-    haveI : Encodable c := hc.to_encodable
-    obtain ⟨u, -, u_pos, u_lim⟩ :
-      ∃ u : ℕ → ℝ, StrictAnti u ∧ (∀ n : ℕ, 0 < u n) ∧ tendsto u at_top (𝓝 0) :=
-      exists_seq_strictAnti_tendsto (0 : ℝ)
-    let f : c × ℕ → α := fun p =>
-      if h : (Metric.ball (p.1 : α) (u p.2) ∩ s).Nonempty then h.some else x₀
-    have fs : ∀ p, f p ∈ s := by
-      rintro ⟨y, n⟩
-      by_cases h : (ball (y : α) (u n) ∩ s).Nonempty
-      · simpa only [f, h, dif_pos] using h.some_spec.2
-      · simpa only [f, h, not_false_iff, dif_neg]
-    let g : c × ℕ → s := fun p => ⟨f p, fs p⟩
-    apply separable_space_of_dense_range g
-    apply Metric.denseRange_iff.2
-    rintro ⟨x, xs⟩ r (rpos : 0 < r)
-    obtain ⟨n, hn⟩ : ∃ n, u n < r / 2 := ((tendsto_order.1 u_lim).2 _ (half_pos rpos)).exists
-    obtain ⟨z, zc, hz⟩ : ∃ z ∈ c, dist x z < u n := Metric.mem_closure_iff.1 (h'c xs) _ (u_pos n)
-    refine' ⟨(⟨z, zc⟩, n), _⟩
-    change dist x (f (⟨z, zc⟩, n)) < r
-    have A : (Metric.ball z (u n) ∩ s).Nonempty := ⟨x, hz, xs⟩
-    dsimp [f]
-    simp only [A, dif_pos]
-    calc
-      dist x A.some ≤ dist x z + dist z A.some := dist_triangle _ _ _
-      _ < r / 2 + r / 2 := (add_lt_add (hz.trans hn) ((Metric.mem_ball'.1 A.some_spec.1).trans hn))
-      _ = r := add_halves _
-      
+  rcases eq_empty_or_nonempty s with (rfl | ⟨⟨x₀, x₀s⟩⟩)
+  · infer_instance
+  rcases hs with ⟨c, hc, h'c⟩
+  haveI : Encodable c := hc.to_encodable
+  obtain ⟨u, -, u_pos, u_lim⟩ :
+    ∃ u : ℕ → ℝ, StrictAnti u ∧ (∀ n : ℕ, 0 < u n) ∧ tendsto u at_top (𝓝 0) :=
+    exists_seq_strictAnti_tendsto (0 : ℝ)
+  let f : c × ℕ → α := fun p =>
+    if h : (Metric.ball (p.1 : α) (u p.2) ∩ s).Nonempty then h.some else x₀
+  have fs : ∀ p, f p ∈ s := by
+    rintro ⟨y, n⟩
+    by_cases h : (ball (y : α) (u n) ∩ s).Nonempty
+    · simpa only [f, h, dif_pos] using h.some_spec.2
+    · simpa only [f, h, not_false_iff, dif_neg]
+  let g : c × ℕ → s := fun p => ⟨f p, fs p⟩
+  apply separable_space_of_dense_range g
+  apply Metric.denseRange_iff.2
+  rintro ⟨x, xs⟩ r (rpos : 0 < r)
+  obtain ⟨n, hn⟩ : ∃ n, u n < r / 2 := ((tendsto_order.1 u_lim).2 _ (half_pos rpos)).exists
+  obtain ⟨z, zc, hz⟩ : ∃ z ∈ c, dist x z < u n := Metric.mem_closure_iff.1 (h'c xs) _ (u_pos n)
+  refine' ⟨(⟨z, zc⟩, n), _⟩
+  change dist x (f (⟨z, zc⟩, n)) < r
+  have A : (Metric.ball z (u n) ∩ s).Nonempty := ⟨x, hz, xs⟩
+  dsimp [f]
+  simp only [A, dif_pos]
+  calc
+    dist x A.some ≤ dist x z + dist z A.some := dist_triangle _ _ _
+    _ < r / 2 + r / 2 := (add_lt_add (hz.trans hn) ((Metric.mem_ball'.1 A.some_spec.1).trans hn))
+    _ = r := add_halves _
+    
 #align topological_space.is_separable.separable_space TopologicalSpace.IsSeparable.separableSpace
 
 #print Inducing.isSeparable_preimage /-
@@ -3212,7 +3212,7 @@ theorem IsComplete.nonempty_iInter_of_nonempty_biInter {s : ℕ → Set α} (h0
     cauchySeq_tendsto_of_isComplete h0 (fun n => I 0 n (zero_le _)) this
   refine' ⟨x, mem_Inter.2 fun n => _⟩
   apply (hs n).mem_of_tendsto xlim
-  filter_upwards [Ici_mem_at_top n]with p hp
+  filter_upwards [Ici_mem_at_top n] with p hp
   exact I n p hp
 #align is_complete.nonempty_Inter_of_nonempty_bInter IsComplete.nonempty_iInter_of_nonempty_biInter
 
@@ -3587,7 +3587,7 @@ instance : MetricSpace PUnit.{u + 1} where
   toUniformSpace := PUnit.uniformSpace
   uniformity_dist := by
     simp only
-    have : ne_bot (⨅ ε > (0 : ℝ), 𝓟 { p : PUnit.{u + 1} × PUnit.{u + 1} | 0 < ε }) :=
+    have : ne_bot (⨅ ε > (0 : ℝ), 𝓟 {p : PUnit.{u + 1} × PUnit.{u + 1} | 0 < ε}) :=
       @uniformity.neBot _
         (UniformSpace.ofDist (fun _ _ => 0) (fun _ => rfl) (fun _ _ => rfl) fun _ _ _ => by
           rw [zero_add])

mathlib commit https://github.com/leanprover-community/mathlib/commit/cca40788df1b8755d5baf17ab2f27dacc2e17acb

@@ -95,7 +95,7 @@ def Bornology.ofDist {α : Type _} (dist : α → α → ℝ) (dist_self : ∀ x
     (fun s hs t ht => by
       rcases s.eq_empty_or_nonempty with (rfl | ⟨z, hz⟩)
       · exact (empty_union t).symm ▸ ht
-      · simp only [fun u => bounded_iff_aux dist dist_comm dist_triangle u z] at hs ht⊢
+      · simp only [fun u => bounded_iff_aux dist dist_comm dist_triangle u z] at hs ht ⊢
         rcases hs, ht with ⟨⟨r₁, hr₁⟩, ⟨r₂, hr₂⟩⟩
         exact
           ⟨max r₁ r₂, fun x hx =>
@@ -159,11 +159,11 @@ class PseudoMetricSpace (α : Type u) extends Dist α : Type u where
     run_tac
       pseudo_metric_space.edist_dist_tac
   toUniformSpace : UniformSpace α := UniformSpace.ofDist dist dist_self dist_comm dist_triangle
-  uniformity_dist : 𝓤 α = ⨅ ε > 0, 𝓟 { p : α × α | dist p.1 p.2 < ε } := by intros ; rfl
+  uniformity_dist : 𝓤 α = ⨅ ε > 0, 𝓟 { p : α × α | dist p.1 p.2 < ε } := by intros; rfl
   toBornology : Bornology α := Bornology.ofDist dist dist_self dist_comm dist_triangle
   cobounded_sets :
     (Bornology.cobounded α).sets = { s | ∃ C, ∀ ⦃x⦄, x ∈ sᶜ → ∀ ⦃y⦄, y ∈ sᶜ → dist x y ≤ C } := by
-    intros ; rfl
+    intros; rfl
 #align pseudo_metric_space PseudoMetricSpace
 -/
 
@@ -174,18 +174,18 @@ theorem PseudoMetricSpace.ext {α : Type _} {m m' : PseudoMetricSpace α}
     (h : m.toHasDist = m'.toHasDist) : m = m' :=
   by
   rcases m with ⟨⟩; rcases m' with ⟨⟩
-  dsimp at h
+  dsimp at h 
   subst h
   congr
   · ext (x y) : 2
-    dsimp at m_edist_dist m'_edist_dist
+    dsimp at m_edist_dist m'_edist_dist 
     simp [m_edist_dist, m'_edist_dist]
-  · dsimp at m_uniformity_dist m'_uniformity_dist
-    rw [← m'_uniformity_dist] at m_uniformity_dist
+  · dsimp at m_uniformity_dist m'_uniformity_dist 
+    rw [← m'_uniformity_dist] at m_uniformity_dist 
     exact uniformSpace_eq m_uniformity_dist
   · ext1
-    dsimp at m_cobounded_sets m'_cobounded_sets
-    rw [← m'_cobounded_sets] at m_cobounded_sets
+    dsimp at m_cobounded_sets m'_cobounded_sets 
+    rw [← m'_cobounded_sets] at m_cobounded_sets 
     exact filter_eq m_cobounded_sets
 #align pseudo_metric_space.ext PseudoMetricSpace.ext
 -/
@@ -521,7 +521,7 @@ contains it.
 See also `exists_lt_subset_ball`. -/
 theorem exists_lt_mem_ball_of_mem_ball (h : x ∈ ball y ε) : ∃ ε' < ε, x ∈ ball y ε' :=
   by
-  simp only [mem_ball] at h⊢
+  simp only [mem_ball] at h ⊢
   exact ⟨(ε + dist x y) / 2, by linarith, by linarith⟩
 #align metric.exists_lt_mem_ball_of_mem_ball Metric.exists_lt_mem_ball_of_mem_ball
 
@@ -736,7 +736,7 @@ theorem dist_lt_add_of_nonempty_closedBall_inter_ball (h : (closedBall x ε₁ 
 
 theorem dist_lt_add_of_nonempty_ball_inter_closedBall (h : (ball x ε₁ ∩ closedBall y ε₂).Nonempty) :
     dist x y < ε₁ + ε₂ := by
-  rw [inter_comm] at h
+  rw [inter_comm] at h 
   rw [add_comm, dist_comm]
   exact dist_lt_add_of_nonempty_closed_ball_inter_ball h
 #align metric.dist_lt_add_of_nonempty_ball_inter_closed_ball Metric.dist_lt_add_of_nonempty_ball_inter_closedBall
@@ -775,7 +775,7 @@ all points. -/
 theorem forall_of_forall_mem_closedBall (p : α → Prop) (x : α)
     (H : ∃ᶠ R : ℝ in atTop, ∀ y ∈ closedBall x R, p y) (y : α) : p y :=
   by
-  obtain ⟨R, hR, h⟩ : ∃ (R : ℝ)(H : dist y x ≤ R), ∀ z : α, z ∈ closed_ball x R → p z :=
+  obtain ⟨R, hR, h⟩ : ∃ (R : ℝ) (H : dist y x ≤ R), ∀ z : α, z ∈ closed_ball x R → p z :=
     frequently_iff.1 H (Ici_mem_at_top (dist y x))
   exact h _ hR
 #align metric.forall_of_forall_mem_closed_ball Metric.forall_of_forall_mem_closedBall
@@ -787,7 +787,7 @@ points. -/
 theorem forall_of_forall_mem_ball (p : α → Prop) (x : α)
     (H : ∃ᶠ R : ℝ in atTop, ∀ y ∈ ball x R, p y) (y : α) : p y :=
   by
-  obtain ⟨R, hR, h⟩ : ∃ (R : ℝ)(H : dist y x < R), ∀ z : α, z ∈ ball x R → p z :=
+  obtain ⟨R, hR, h⟩ : ∃ (R : ℝ) (H : dist y x < R), ∀ z : α, z ∈ ball x R → p z :=
     frequently_iff.1 H (Ioi_mem_at_top (dist y x))
   exact h _ hR
 #align metric.forall_of_forall_mem_ball Metric.forall_of_forall_mem_ball
@@ -841,13 +841,13 @@ accumulating to zero, then `f i`-neighborhoods of the diagonal form a basis of `
 For specific bases see `uniformity_basis_dist`, `uniformity_basis_dist_inv_nat_succ`,
 and `uniformity_basis_dist_inv_nat_pos`. -/
 protected theorem mk_uniformity_basis {β : Type _} {p : β → Prop} {f : β → ℝ}
-    (hf₀ : ∀ i, p i → 0 < f i) (hf : ∀ ⦃ε⦄, 0 < ε → ∃ (i : _)(hi : p i), f i ≤ ε) :
+    (hf₀ : ∀ i, p i → 0 < f i) (hf : ∀ ⦃ε⦄, 0 < ε → ∃ (i : _) (hi : p i), f i ≤ ε) :
     (𝓤 α).HasBasis p fun i => { p : α × α | dist p.1 p.2 < f i } :=
   by
   refine' ⟨fun s => uniformity_basis_dist.mem_iff.trans _⟩
   constructor
   · rintro ⟨ε, ε₀, hε⟩
-    obtain ⟨i, hi, H⟩ : ∃ (i : _)(hi : p i), f i ≤ ε; exact hf ε₀
+    obtain ⟨i, hi, H⟩ : ∃ (i : _) (hi : p i), f i ≤ ε; exact hf ε₀
     exact ⟨i, hi, fun x (hx : _ < _) => hε <| lt_of_lt_of_le hx H⟩
   · exact fun ⟨i, hi, H⟩ => ⟨f i, hf₀ i hi, H⟩
 #align metric.mk_uniformity_basis Metric.mk_uniformity_basis
@@ -893,7 +893,7 @@ form a basis of `𝓤 α`.
 Currently we have only one specific basis `uniformity_basis_dist_le` based on this constructor.
 More can be easily added if needed in the future. -/
 protected theorem mk_uniformity_basis_le {β : Type _} {p : β → Prop} {f : β → ℝ}
-    (hf₀ : ∀ x, p x → 0 < f x) (hf : ∀ ε, 0 < ε → ∃ (x : _)(hx : p x), f x ≤ ε) :
+    (hf₀ : ∀ x, p x → 0 < f x) (hf : ∀ ε, 0 < ε → ∃ (x : _) (hx : p x), f x ≤ ε) :
     (𝓤 α).HasBasis p fun x => { p : α × α | dist p.1 p.2 ≤ f x } :=
   by
   refine' ⟨fun s => uniformity_basis_dist.mem_iff.trans _⟩
@@ -981,7 +981,7 @@ space from finitely many data. -/
 theorem totallyBounded_of_finite_discretization {s : Set α}
     (H :
       ∀ ε > (0 : ℝ),
-        ∃ (β : Type u)(_ : Fintype β)(F : s → β), ∀ x y, F x = F y → dist (x : α) y < ε) :
+        ∃ (β : Type u) (_ : Fintype β) (F : s → β), ∀ x y, F x = F y → dist (x : α) y < ε) :
     TotallyBounded s := by
   cases' s.eq_empty_or_nonempty with hs hs
   · rw [hs]; exact totallyBounded_empty
@@ -1000,10 +1000,10 @@ theorem totallyBounded_of_finite_discretization {s : Set α}
 
 /- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (t «expr ⊆ » s) -/
 theorem finite_approx_of_totallyBounded {s : Set α} (hs : TotallyBounded s) :
-    ∀ ε > 0, ∃ (t : _)(_ : t ⊆ s), Set.Finite t ∧ s ⊆ ⋃ y ∈ t, ball y ε :=
+    ∀ ε > 0, ∃ (t : _) (_ : t ⊆ s), Set.Finite t ∧ s ⊆ ⋃ y ∈ t, ball y ε :=
   by
   intro ε ε_pos
-  rw [totallyBounded_iff_subset] at hs
+  rw [totallyBounded_iff_subset] at hs 
   exact hs _ (dist_mem_uniformity ε_pos)
 #align metric.finite_approx_of_totally_bounded Metric.finite_approx_of_totallyBounded
 
@@ -1083,7 +1083,7 @@ theorem eventually_nhds_iff_ball {p : α → Prop} :
 in a pseudo-metric space.-/
 theorem eventually_prod_nhds_iff {f : Filter ι} {x₀ : α} {p : ι × α → Prop} :
     (∀ᶠ x in f ×ᶠ 𝓝 x₀, p x) ↔
-      ∃ (pa : ι → Prop)(ha : ∀ᶠ i in f, pa i),
+      ∃ (pa : ι → Prop) (ha : ∀ᶠ i in f, pa i),
         ∃ ε > 0, ∀ {i}, pa i → ∀ {x}, dist x x₀ < ε → p (i, x) :=
   by
   simp_rw [eventually_prod_iff, Metric.eventually_nhds_iff]
@@ -1099,7 +1099,7 @@ theorem eventually_nhds_prod_iff {ι α} [PseudoMetricSpace α] {f : Filter ι}
     {p : α × ι → Prop} :
     (∀ᶠ x in 𝓝 x₀ ×ᶠ f, p x) ↔
       ∃ ε > (0 : ℝ),
-        ∃ (pa : ι → Prop)(ha : ∀ᶠ i in f, pa i), ∀ {x}, dist x x₀ < ε → ∀ {i}, pa i → p (x, i) :=
+        ∃ (pa : ι → Prop) (ha : ∀ᶠ i in f, pa i), ∀ {x}, dist x x₀ < ε → ∀ {i}, pa i → p (x, i) :=
   by
   rw [eventually_swap_iff, Metric.eventually_prod_nhds_iff]
   constructor <;>
@@ -1292,7 +1292,7 @@ protected theorem Metric.uniformity_basis_edist :
       simp only [edist_dist, ENNReal.ofReal_lt_ofReal_iff ε0]
       exact Hε
     · rcases ENNReal.lt_iff_exists_real_btwn.1 ε0 with ⟨ε', _, ε0', hε⟩
-      rw [ENNReal.ofReal_pos] at ε0'
+      rw [ENNReal.ofReal_pos] at ε0' 
       refine' ⟨ε', ε0', fun a b h => Hε (lt_trans _ hε)⟩
       rwa [edist_dist, ENNReal.ofReal_lt_ofReal_iff ε0']⟩
 #align pseudo_metric.uniformity_basis_edist Metric.uniformity_basis_edist
@@ -1639,7 +1639,7 @@ theorem tendsto_iff_of_dist {ι : Type _} {f₁ f₂ : ι → α} {p : Filter ι
 theorem eventually_closedBall_subset {x : α} {u : Set α} (hu : u ∈ 𝓝 x) :
     ∀ᶠ r in 𝓝 (0 : ℝ), closedBall x r ⊆ u :=
   by
-  obtain ⟨ε, εpos, hε⟩ : ∃ (ε : _)(hε : 0 < ε), closed_ball x ε ⊆ u :=
+  obtain ⟨ε, εpos, hε⟩ : ∃ (ε : _) (hε : 0 < ε), closed_ball x ε ⊆ u :=
     nhds_basis_closed_ball.mem_iff.1 hu
   have : Iic ε ∈ 𝓝 (0 : ℝ) := Iic_mem_nhds εpos
   filter_upwards [this]with _ hr using subset.trans (closed_ball_subset_closed_ball hr) hε
@@ -1685,7 +1685,7 @@ theorem Metric.uniformCauchySeqOn_iff {γ : Type _} {F : β → γ → α} {s :
       @Filter.eventually_atTop_prod_self' _ _ _ fun m =>
         ∀ x : γ, x ∈ s → dist (F m.fst x) (F m.snd x) < ε]
     specialize h u hu
-    rw [prod_at_top_at_top_eq] at h
+    rw [prod_at_top_at_top_eq] at h 
     exact h.mono fun n h x hx => set.mem_set_of_eq.mp (h x hx)
   · intro h u hu
     rcases metric.mem_uniformity_dist.mp hu with ⟨ε, hε, hab⟩
@@ -1995,7 +1995,7 @@ theorem uniformContinuous_dist : UniformContinuous fun p : α × α => dist p.1
       suffices
       · intro p q h; cases' p with p₁ p₂; cases' q with q₁ q₂
         cases' max_lt_iff.1 h with h₁ h₂; clear h
-        dsimp at h₁ h₂⊢
+        dsimp at h₁ h₂ ⊢
         rw [Real.dist_eq]
         refine' abs_sub_lt_iff.2 ⟨_, _⟩
         · revert p₁ p₂ q₁ q₂ h₁ h₂; exact this
@@ -2004,7 +2004,7 @@ theorem uniformContinuous_dist : UniformContinuous fun p : α × α => dist p.1
       have :=
         add_lt_add (abs_sub_lt_iff.1 (lt_of_le_of_lt (abs_dist_sub_le p₁ q₁ p₂) h₁)).1
           (abs_sub_lt_iff.1 (lt_of_le_of_lt (abs_dist_sub_le p₂ q₂ q₁) h₂)).1
-      rwa [add_halves, dist_comm p₂, sub_add_sub_cancel, dist_comm q₂] at this⟩
+      rwa [add_halves, dist_comm p₂, sub_add_sub_cancel, dist_comm q₂] at this ⟩
 #align uniform_continuous_dist uniformContinuous_dist
 
 #print UniformContinuous.dist /-
@@ -2216,7 +2216,7 @@ protected theorem Inducing.isSeparable_preimage {f : β → α} [TopologicalSpac
   have : Inducing g := (hf.comp inducing_subtype_val).codRestrict _
   haveI : second_countable_topology (f ⁻¹' s) := this.second_countable_topology
   rw [show f ⁻¹' s = coe '' (univ : Set (f ⁻¹' s)) by
-      simpa only [image_univ, Subtype.range_coe_subtype] ]
+      simpa only [image_univ, Subtype.range_coe_subtype]]
   exact (is_separable_of_separable_space _).image continuous_subtype_val
 #align inducing.is_separable_preimage Inducing.isSeparable_preimage
 -/
@@ -2392,7 +2392,7 @@ theorem Real.dist_le_of_mem_pi_Icc {x y x' y' : β → ℝ} (hx : x ∈ Icc x' y
       (dist_pi_le_iff dist_nonneg).2 fun b =>
         (Real.dist_le_of_mem_uIcc _ _).trans (dist_le_pi_dist _ _ b) <;>
     refine' Icc_subset_uIcc _
-  exacts[⟨hx.1 _, hx.2 _⟩, ⟨hy.1 _, hy.2 _⟩]
+  exacts [⟨hx.1 _, hx.2 _⟩, ⟨hy.1 _, hy.2 _⟩]
 #align real.dist_le_of_mem_pi_Icc Real.dist_le_of_mem_pi_Icc
 
 end Pi
@@ -2404,7 +2404,7 @@ section Compact
 positive radius -/
 theorem finite_cover_balls_of_compact {α : Type u} [PseudoMetricSpace α] {s : Set α}
     (hs : IsCompact s) {e : ℝ} (he : 0 < e) :
-    ∃ (t : _)(_ : t ⊆ s), Set.Finite t ∧ s ⊆ ⋃ x ∈ t, ball x e :=
+    ∃ (t : _) (_ : t ⊆ s), Set.Finite t ∧ s ⊆ ⋃ x ∈ t, ball x e :=
   by
   apply hs.elim_finite_subcover_image
   · simp [is_open_ball]
@@ -2574,7 +2574,7 @@ center and a strictly smaller radius that includes `s`. -/
 theorem exists_lt_subset_ball (hs : IsClosed s) (h : s ⊆ ball x r) : ∃ r' < r, s ⊆ ball x r' :=
   by
   cases' le_or_lt r 0 with hr hr
-  · rw [ball_eq_empty.2 hr, subset_empty_iff] at h; subst s
+  · rw [ball_eq_empty.2 hr, subset_empty_iff] at h ; subst s
     exact (exists_lt r).imp fun r' hr' => ⟨hr', empty_subset _⟩
   · exact (exists_pos_lt_subset_ball hr hs h).imp fun r' hr' => ⟨hr'.fst.2, hr'.snd⟩
 #align exists_lt_subset_ball exists_lt_subset_ball
@@ -2622,7 +2622,7 @@ theorem lebesgue_number_lemma_of_metric {s : Set α} {ι} {c : ι → Set α} (h
 
 theorem lebesgue_number_lemma_of_metric_sUnion {s : Set α} {c : Set (Set α)} (hs : IsCompact s)
     (hc₁ : ∀ t ∈ c, IsOpen t) (hc₂ : s ⊆ ⋃₀ c) : ∃ δ > 0, ∀ x ∈ s, ∃ t ∈ c, ball x δ ⊆ t := by
-  rw [sUnion_eq_Union] at hc₂ <;> simpa using lebesgue_number_lemma_of_metric hs (by simpa) hc₂
+  rw [sUnion_eq_Union] at hc₂  <;> simpa using lebesgue_number_lemma_of_metric hs (by simpa) hc₂
 #align lebesgue_number_lemma_of_metric_sUnion lebesgue_number_lemma_of_metric_sUnion
 
 namespace Metric
@@ -2747,7 +2747,7 @@ theorem bounded_closure_iff : Bounded (closure s) ↔ Bounded s :=
 theorem Bounded.union (hs : Bounded s) (ht : Bounded t) : Bounded (s ∪ t) :=
   by
   refine' bounded_iff_mem_bounded.2 fun x _ => _
-  rw [bounded_iff_subset_ball x] at hs ht⊢
+  rw [bounded_iff_subset_ball x] at hs ht ⊢
   rcases hs with ⟨Cs, hCs⟩; rcases ht with ⟨Ct, hCt⟩
   exact
     ⟨max Cs Ct,
@@ -2910,7 +2910,7 @@ theorem exists_isOpen_bounded_image_of_isCompact_of_forall_continuousAt [Topolog
     {k : Set β} {f : β → α} (hk : IsCompact k) (hf : ∀ x ∈ k, ContinuousAt f x) :
     ∃ t, k ⊆ t ∧ IsOpen t ∧ Bounded (f '' t) :=
   by
-  simp_rw [← continuousWithinAt_univ] at hf
+  simp_rw [← continuousWithinAt_univ] at hf 
   simpa only [inter_univ] using
     exists_is_open_bounded_image_inter_of_is_compact_of_forall_continuous_within_at hk hf
 #align metric.exists_is_open_bounded_image_of_is_compact_of_forall_continuous_at Metric.exists_isOpen_bounded_image_of_isCompact_of_forall_continuousAt
@@ -3126,7 +3126,7 @@ theorem dist_le_diam_of_mem (h : Bounded s) (hx : x ∈ s) (hy : y ∈ s) : dist
 #align metric.dist_le_diam_of_mem Metric.dist_le_diam_of_mem
 
 theorem ediam_of_unbounded (h : ¬Bounded s) : EMetric.diam s = ∞ := by
-  rwa [bounded_iff_ediam_ne_top, Classical.not_not] at h
+  rwa [bounded_iff_ediam_ne_top, Classical.not_not] at h 
 #align metric.ediam_of_unbounded Metric.ediam_of_unbounded
 
 /-- An unbounded set has zero diameter. If you would prefer to get the value ∞, use `emetric.diam`.
@@ -3152,7 +3152,7 @@ theorem diam_union {t : Set α} (xs : x ∈ s) (yt : y ∈ t) :
   by_cases H : bounded (s ∪ t)
   · have hs : bounded s := H.mono (subset_union_left _ _)
     have ht : bounded t := H.mono (subset_union_right _ _)
-    rw [bounded_iff_ediam_ne_top] at H hs ht
+    rw [bounded_iff_ediam_ne_top] at H hs ht 
     rw [dist_edist, diam, diam, diam, ← ENNReal.toReal_add, ← ENNReal.toReal_add,
             ENNReal.toReal_le_toReal] <;>
           repeat' apply ENNReal.add_ne_top.2 <;> constructor <;>
@@ -3201,14 +3201,14 @@ theorem IsComplete.nonempty_iInter_of_nonempty_biInter {s : ℕ → Set α} (h0
     intro n N hn
     apply mem_of_subset_of_mem _ (h N).choose_spec
     intro x hx
-    simp only [mem_Inter] at hx
+    simp only [mem_Inter] at hx 
     exact hx n hn
   have : ∀ n, u n ∈ s 0 := fun n => I 0 n (zero_le _)
   have : CauchySeq u := by
     apply cauchySeq_of_le_tendsto_0 _ _ h'
     intro m n N hm hn
     exact dist_le_diam_of_mem (h's N) (I _ _ hm) (I _ _ hn)
-  obtain ⟨x, hx, xlim⟩ : ∃ (x : α)(H : x ∈ s 0), tendsto (fun n : ℕ => u n) at_top (𝓝 x) :=
+  obtain ⟨x, hx, xlim⟩ : ∃ (x : α) (H : x ∈ s 0), tendsto (fun n : ℕ => u n) at_top (𝓝 x) :=
     cauchySeq_tendsto_of_isComplete h0 (fun n => I 0 n (zero_le _)) this
   refine' ⟨x, mem_Inter.2 fun n => _⟩
   apply (hs n).mem_of_tendsto xlim
@@ -3231,7 +3231,7 @@ theorem exists_local_min_mem_ball [ProperSpace α] [TopologicalSpace β]
     (hf : ContinuousOn f (closedBall a r)) (hz : z ∈ closedBall a r)
     (hf1 : ∀ z' ∈ sphere a r, f z < f z') : ∃ z ∈ ball a r, IsLocalMin f z :=
   by
-  simp_rw [← closed_ball_diff_ball] at hf1
+  simp_rw [← closed_ball_diff_ball] at hf1 
   exact
     (is_compact_closed_ball a r).exists_local_min_mem_open ball_subset_closed_ball hf hz hf1
       is_open_ball
@@ -3301,7 +3301,7 @@ theorem MetricSpace.ext {α : Type _} {m m' : MetricSpace α} (h : m.toHasDist =
   by
   have h' : m.to_pseudo_metric_space = m'.to_pseudo_metric_space := PseudoMetricSpace.ext h
   rcases m with ⟨⟩; rcases m' with ⟨⟩
-  dsimp at h'
+  dsimp at h' 
   subst h'
 #align metric_space.ext MetricSpace.ext
 -/
@@ -3647,7 +3647,7 @@ instance metricSpacePi : MetricSpace (∀ b, π b) :=
       by
       have eq1 : edist f g = 0 := by simp only [edist_dist, eq0, ENNReal.ofReal_zero]
       have eq2 : (sup univ fun b : β => edist (f b) (g b)) ≤ 0 := le_of_eq eq1
-      simp only [Finset.sup_le_iff] at eq2
+      simp only [Finset.sup_le_iff] at eq2 
       exact funext fun b => edist_le_zero.1 <| eq2 b <| mem_univ b }
 #align metric_space_pi metricSpacePi
 -/
@@ -3664,7 +3664,8 @@ open TopologicalSpace
 space from countably many data. -/
 theorem secondCountable_of_countable_discretization {α : Type u} [MetricSpace α]
     (H :
-      ∀ ε > (0 : ℝ), ∃ (β : Type _)(_ : Encodable β)(F : α → β), ∀ x y, F x = F y → dist x y ≤ ε) :
+      ∀ ε > (0 : ℝ),
+        ∃ (β : Type _) (_ : Encodable β) (F : α → β), ∀ x y, F x = F y → dist x y ≤ ε) :
     SecondCountableTopology α :=
   by
   cases' (univ : Set α).eq_empty_or_nonempty with hs hs

mathlib commit https://github.com/leanprover-community/mathlib/commit/cca40788df1b8755d5baf17ab2f27dacc2e17acb

@@ -130,16 +130,12 @@ private theorem pseudo_metric_space.dist_nonneg' {α} {x y : α} (dist : α →
       
   nonneg_of_mul_nonneg_right this zero_lt_two
 
-/- warning: pseudo_metric_space.edist_dist_tac clashes with [anonymous] -> [anonymous]
-Case conversion may be inaccurate. Consider using '#align pseudo_metric_space.edist_dist_tac [anonymous]ₓ'. -/
 /- ./././Mathport/Syntax/Translate/Expr.lean:330:4: warning: unsupported (TODO): `[tacs] -/
-#print [anonymous] /-
 /-- This tactic is used to populate `pseudo_metric_space.edist_dist` when the default `edist` is
 used. -/
-protected unsafe def [anonymous] : tactic Unit :=
+protected unsafe def pseudo_metric_space.edist_dist_tac : tactic Unit :=
   tactic.intros >> sorry
-#align pseudo_metric_space.edist_dist_tac [anonymous]
--/
+#align pseudo_metric_space.edist_dist_tac pseudo_metric_space.edist_dist_tac
 
 #print PseudoMetricSpace /-
 /- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic pseudo_metric_space.edist_dist_tac -/
@@ -161,7 +157,7 @@ class PseudoMetricSpace (α : Type u) extends Dist α : Type u where
     @coe ℝ≥0 _ _ ⟨dist x y, PseudoMetricSpace.dist_nonneg' _ ‹_› ‹_› ‹_›⟩
   edist_dist : ∀ x y : α, edist x y = ENNReal.ofReal (dist x y) := by
     run_tac
-      [anonymous]
+      pseudo_metric_space.edist_dist_tac
   toUniformSpace : UniformSpace α := UniformSpace.ofDist dist dist_self dist_comm dist_triangle
   uniformity_dist : 𝓤 α = ⨅ ε > 0, 𝓟 { p : α × α | dist p.1 p.2 < ε } := by intros ; rfl
   toBornology : Bornology α := Bornology.ofDist dist dist_self dist_comm dist_triangle

mathlib commit https://github.com/leanprover-community/mathlib/commit/917c3c072e487b3cccdbfeff17e75b40e45f66cb

@@ -57,7 +57,7 @@ metric, pseudo_metric, dist
 
 open Set Filter TopologicalSpace Bornology
 
-open uniformity Topology BigOperators Filter NNReal ENNReal
+open scoped uniformity Topology BigOperators Filter NNReal ENNReal
 
 universe u v w
 
@@ -816,11 +816,13 @@ theorem isBounded_iff_exists_ge {s : Set α} (c : ℝ) :
     isBounded_iff.2 <| h.imp fun _ => And.right⟩
 #align metric.is_bounded_iff_exists_ge Metric.isBounded_iff_exists_ge
 
+#print Metric.isBounded_iff_nndist /-
 theorem isBounded_iff_nndist {s : Set α} :
     IsBounded s ↔ ∃ C : ℝ≥0, ∀ ⦃x⦄, x ∈ s → ∀ ⦃y⦄, y ∈ s → nndist x y ≤ C := by
   simp only [is_bounded_iff_exists_ge 0, NNReal.exists, ← NNReal.coe_le_coe, ← dist_nndist,
     NNReal.coe_mk, exists_prop]
 #align metric.is_bounded_iff_nndist Metric.isBounded_iff_nndist
+-/
 
 #print Metric.toUniformSpace_eq /-
 theorem toUniformSpace_eq :
@@ -2318,9 +2320,11 @@ theorem dist_pi_const_le (a b : α) : (dist (fun _ : β => a) fun _ => b) ≤ di
   (dist_pi_le_iff dist_nonneg).2 fun _ => le_rfl
 #align dist_pi_const_le dist_pi_const_le
 
+#print nndist_pi_const_le /-
 theorem nndist_pi_const_le (a b : α) : (nndist (fun _ : β => a) fun _ => b) ≤ nndist a b :=
   nndist_pi_le_iff.2 fun _ => le_rfl
 #align nndist_pi_const_le nndist_pi_const_le
+-/
 
 #print dist_pi_const /-
 @[simp]

mathlib commit https://github.com/leanprover-community/mathlib/commit/917c3c072e487b3cccdbfeff17e75b40e45f66cb

@@ -63,12 +63,6 @@ universe u v w
 
 variable {α : Type u} {β : Type v} {X ι : Type _}
 
-/- warning: uniform_space_of_dist -> UniformSpace.ofDist is a dubious translation:
-lean 3 declaration is
-  forall {α : Type.{u1}} (dist : α -> α -> Real), (forall (x : α), Eq.{1} Real (dist x x) (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero)))) -> (forall (x : α) (y : α), Eq.{1} Real (dist x y) (dist y x)) -> (forall (x : α) (y : α) (z : α), LE.le.{0} Real Real.hasLe (dist x z) (HAdd.hAdd.{0, 0, 0} Real Real Real (instHAdd.{0} Real Real.hasAdd) (dist x y) (dist y z))) -> (UniformSpace.{u1} α)
-but is expected to have type
-  forall {α : Type.{u1}} (dist : α -> α -> Real), (forall (x : α), Eq.{1} Real (dist x x) (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal))) -> (forall (x : α) (y : α), Eq.{1} Real (dist x y) (dist y x)) -> (forall (x : α) (y : α) (z : α), LE.le.{0} Real Real.instLEReal (dist x z) (HAdd.hAdd.{0, 0, 0} Real Real Real (instHAdd.{0} Real Real.instAddReal) (dist x y) (dist y z))) -> (UniformSpace.{u1} α)
-Case conversion may be inaccurate. Consider using '#align uniform_space_of_dist UniformSpace.ofDistₓ'. -/
 /-- Construct a uniform structure from a distance function and metric space axioms -/
 def UniformSpace.ofDist (dist : α → α → ℝ) (dist_self : ∀ x : α, dist x x = 0)
     (dist_comm : ∀ x y : α, dist x y = dist y x)
@@ -213,12 +207,6 @@ instance (priority := 200) PseudoMetricSpace.toEDist : EDist α :=
 #align pseudo_metric_space.to_has_edist PseudoMetricSpace.toEDist
 -/
 
-/- warning: pseudo_metric_space.of_dist_topology -> PseudoMetricSpace.ofDistTopology is a dubious translation:
-lean 3 declaration is
-  forall {α : Type.{u1}} [_inst_2 : TopologicalSpace.{u1} α] (dist : α -> α -> Real), (forall (x : α), Eq.{1} Real (dist x x) (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero)))) -> (forall (x : α) (y : α), Eq.{1} Real (dist x y) (dist y x)) -> (forall (x : α) (y : α) (z : α), LE.le.{0} Real Real.hasLe (dist x z) (HAdd.hAdd.{0, 0, 0} Real Real Real (instHAdd.{0} Real Real.hasAdd) (dist x y) (dist y z))) -> (forall (s : Set.{u1} α), Iff (IsOpen.{u1} α _inst_2 s) (forall (x : α), (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) x s) -> (Exists.{1} Real (fun (ε : Real) => Exists.{0} (GT.gt.{0} Real Real.hasLt ε (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero)))) (fun (H : GT.gt.{0} Real Real.hasLt ε (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero)))) => forall (y : α), (LT.lt.{0} Real Real.hasLt (dist x y) ε) -> (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) y s)))))) -> (PseudoMetricSpace.{u1} α)
-but is expected to have type
-  forall {α : Type.{u1}} [_inst_2 : TopologicalSpace.{u1} α] (dist : α -> α -> Real), (forall (x : α), Eq.{1} Real (dist x x) (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal))) -> (forall (x : α) (y : α), Eq.{1} Real (dist x y) (dist y x)) -> (forall (x : α) (y : α) (z : α), LE.le.{0} Real Real.instLEReal (dist x z) (HAdd.hAdd.{0, 0, 0} Real Real Real (instHAdd.{0} Real Real.instAddReal) (dist x y) (dist y z))) -> (forall (s : Set.{u1} α), Iff (IsOpen.{u1} α _inst_2 s) (forall (x : α), (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) x s) -> (Exists.{1} Real (fun (ε : Real) => And (GT.gt.{0} Real Real.instLTReal ε (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal))) (forall (y : α), (LT.lt.{0} Real Real.instLTReal (dist x y) ε) -> (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) y s)))))) -> (PseudoMetricSpace.{u1} α)
-Case conversion may be inaccurate. Consider using '#align pseudo_metric_space.of_dist_topology PseudoMetricSpace.ofDistTopologyₓ'. -/
 /-- Construct a pseudo-metric space structure whose underlying topological space structure
 (definitionally) agrees which a pre-existing topology which is compatible with a given distance
 function. -/
@@ -244,12 +232,6 @@ def PseudoMetricSpace.ofDistTopology {α : Type u} [TopologicalSpace α] (dist :
     cobounded_sets := rfl }
 #align pseudo_metric_space.of_dist_topology PseudoMetricSpace.ofDistTopology
 
-/- warning: dist_self -> dist_self is a dubious translation:
-lean 3 declaration is
-  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] (x : α), Eq.{1} Real (Dist.dist.{u1} α (PseudoMetricSpace.toHasDist.{u1} α _inst_1) x x) (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero)))
-but is expected to have type
-  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] (x : α), Eq.{1} Real (Dist.dist.{u1} α (PseudoMetricSpace.toDist.{u1} α _inst_1) x x) (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal))
-Case conversion may be inaccurate. Consider using '#align dist_self dist_selfₓ'. -/
 @[simp]
 theorem dist_self (x : α) : dist x x = 0 :=
   PseudoMetricSpace.dist_self x
@@ -267,42 +249,18 @@ theorem edist_dist (x y : α) : edist x y = ENNReal.ofReal (dist x y) :=
 #align edist_dist edist_dist
 -/
 
-/- warning: dist_triangle -> dist_triangle is a dubious translation:
-lean 3 declaration is
-  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] (x : α) (y : α) (z : α), LE.le.{0} Real Real.hasLe (Dist.dist.{u1} α (PseudoMetricSpace.toHasDist.{u1} α _inst_1) x z) (HAdd.hAdd.{0, 0, 0} Real Real Real (instHAdd.{0} Real Real.hasAdd) (Dist.dist.{u1} α (PseudoMetricSpace.toHasDist.{u1} α _inst_1) x y) (Dist.dist.{u1} α (PseudoMetricSpace.toHasDist.{u1} α _inst_1) y z))
-but is expected to have type
-  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] (x : α) (y : α) (z : α), LE.le.{0} Real Real.instLEReal (Dist.dist.{u1} α (PseudoMetricSpace.toDist.{u1} α _inst_1) x z) (HAdd.hAdd.{0, 0, 0} Real Real Real (instHAdd.{0} Real Real.instAddReal) (Dist.dist.{u1} α (PseudoMetricSpace.toDist.{u1} α _inst_1) x y) (Dist.dist.{u1} α (PseudoMetricSpace.toDist.{u1} α _inst_1) y z))
-Case conversion may be inaccurate. Consider using '#align dist_triangle dist_triangleₓ'. -/
 theorem dist_triangle (x y z : α) : dist x z ≤ dist x y + dist y z :=
   PseudoMetricSpace.dist_triangle x y z
 #align dist_triangle dist_triangle
 
-/- warning: dist_triangle_left -> dist_triangle_left is a dubious translation:
-lean 3 declaration is
-  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] (x : α) (y : α) (z : α), LE.le.{0} Real Real.hasLe (Dist.dist.{u1} α (PseudoMetricSpace.toHasDist.{u1} α _inst_1) x y) (HAdd.hAdd.{0, 0, 0} Real Real Real (instHAdd.{0} Real Real.hasAdd) (Dist.dist.{u1} α (PseudoMetricSpace.toHasDist.{u1} α _inst_1) z x) (Dist.dist.{u1} α (PseudoMetricSpace.toHasDist.{u1} α _inst_1) z y))
-but is expected to have type
-  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] (x : α) (y : α) (z : α), LE.le.{0} Real Real.instLEReal (Dist.dist.{u1} α (PseudoMetricSpace.toDist.{u1} α _inst_1) x y) (HAdd.hAdd.{0, 0, 0} Real Real Real (instHAdd.{0} Real Real.instAddReal) (Dist.dist.{u1} α (PseudoMetricSpace.toDist.{u1} α _inst_1) z x) (Dist.dist.{u1} α (PseudoMetricSpace.toDist.{u1} α _inst_1) z y))
-Case conversion may be inaccurate. Consider using '#align dist_triangle_left dist_triangle_leftₓ'. -/
 theorem dist_triangle_left (x y z : α) : dist x y ≤ dist z x + dist z y := by
   rw [dist_comm z] <;> apply dist_triangle
 #align dist_triangle_left dist_triangle_left
 
-/- warning: dist_triangle_right -> dist_triangle_right is a dubious translation:
-lean 3 declaration is
-  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] (x : α) (y : α) (z : α), LE.le.{0} Real Real.hasLe (Dist.dist.{u1} α (PseudoMetricSpace.toHasDist.{u1} α _inst_1) x y) (HAdd.hAdd.{0, 0, 0} Real Real Real (instHAdd.{0} Real Real.hasAdd) (Dist.dist.{u1} α (PseudoMetricSpace.toHasDist.{u1} α _inst_1) x z) (Dist.dist.{u1} α (PseudoMetricSpace.toHasDist.{u1} α _inst_1) y z))
-but is expected to have type
-  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] (x : α) (y : α) (z : α), LE.le.{0} Real Real.instLEReal (Dist.dist.{u1} α (PseudoMetricSpace.toDist.{u1} α _inst_1) x y) (HAdd.hAdd.{0, 0, 0} Real Real Real (instHAdd.{0} Real Real.instAddReal) (Dist.dist.{u1} α (PseudoMetricSpace.toDist.{u1} α _inst_1) x z) (Dist.dist.{u1} α (PseudoMetricSpace.toDist.{u1} α _inst_1) y z))
-Case conversion may be inaccurate. Consider using '#align dist_triangle_right dist_triangle_rightₓ'. -/
 theorem dist_triangle_right (x y z : α) : dist x y ≤ dist x z + dist y z := by
   rw [dist_comm y] <;> apply dist_triangle
 #align dist_triangle_right dist_triangle_right
 
-/- warning: dist_triangle4 -> dist_triangle4 is a dubious translation:
-lean 3 declaration is
-  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] (x : α) (y : α) (z : α) (w : α), LE.le.{0} Real Real.hasLe (Dist.dist.{u1} α (PseudoMetricSpace.toHasDist.{u1} α _inst_1) x w) (HAdd.hAdd.{0, 0, 0} Real Real Real (instHAdd.{0} Real Real.hasAdd) (HAdd.hAdd.{0, 0, 0} Real Real Real (instHAdd.{0} Real Real.hasAdd) (Dist.dist.{u1} α (PseudoMetricSpace.toHasDist.{u1} α _inst_1) x y) (Dist.dist.{u1} α (PseudoMetricSpace.toHasDist.{u1} α _inst_1) y z)) (Dist.dist.{u1} α (PseudoMetricSpace.toHasDist.{u1} α _inst_1) z w))
-but is expected to have type
-  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] (x : α) (y : α) (z : α) (w : α), LE.le.{0} Real Real.instLEReal (Dist.dist.{u1} α (PseudoMetricSpace.toDist.{u1} α _inst_1) x w) (HAdd.hAdd.{0, 0, 0} Real Real Real (instHAdd.{0} Real Real.instAddReal) (HAdd.hAdd.{0, 0, 0} Real Real Real (instHAdd.{0} Real Real.instAddReal) (Dist.dist.{u1} α (PseudoMetricSpace.toDist.{u1} α _inst_1) x y) (Dist.dist.{u1} α (PseudoMetricSpace.toDist.{u1} α _inst_1) y z)) (Dist.dist.{u1} α (PseudoMetricSpace.toDist.{u1} α _inst_1) z w))
-Case conversion may be inaccurate. Consider using '#align dist_triangle4 dist_triangle4ₓ'. -/
 theorem dist_triangle4 (x y z w : α) : dist x w ≤ dist x y + dist y z + dist z w :=
   calc
     dist x w ≤ dist x z + dist z w := dist_triangle x z w
@@ -310,34 +268,16 @@ theorem dist_triangle4 (x y z w : α) : dist x w ≤ dist x y + dist y z + dist
     
 #align dist_triangle4 dist_triangle4
 
-/- warning: dist_triangle4_left -> dist_triangle4_left is a dubious translation:
-lean 3 declaration is
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-but is expected to have type
-  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] (x₁ : α) (y₁ : α) (x₂ : α) (y₂ : α), LE.le.{0} Real Real.instLEReal (Dist.dist.{u1} α (PseudoMetricSpace.toDist.{u1} α _inst_1) x₂ y₂) (HAdd.hAdd.{0, 0, 0} Real Real Real (instHAdd.{0} Real Real.instAddReal) (Dist.dist.{u1} α (PseudoMetricSpace.toDist.{u1} α _inst_1) x₁ y₁) (HAdd.hAdd.{0, 0, 0} Real Real Real (instHAdd.{0} Real Real.instAddReal) (Dist.dist.{u1} α (PseudoMetricSpace.toDist.{u1} α _inst_1) x₁ x₂) (Dist.dist.{u1} α (PseudoMetricSpace.toDist.{u1} α _inst_1) y₁ y₂)))
-Case conversion may be inaccurate. Consider using '#align dist_triangle4_left dist_triangle4_leftₓ'. -/
 theorem dist_triangle4_left (x₁ y₁ x₂ y₂ : α) :
     dist x₂ y₂ ≤ dist x₁ y₁ + (dist x₁ x₂ + dist y₁ y₂) := by
   rw [add_left_comm, dist_comm x₁, ← add_assoc]; apply dist_triangle4
 #align dist_triangle4_left dist_triangle4_left
 
-/- warning: dist_triangle4_right -> dist_triangle4_right is a dubious translation:
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-but is expected to have type
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-Case conversion may be inaccurate. Consider using '#align dist_triangle4_right dist_triangle4_rightₓ'. -/
 theorem dist_triangle4_right (x₁ y₁ x₂ y₂ : α) :
     dist x₁ y₁ ≤ dist x₁ x₂ + dist y₁ y₂ + dist x₂ y₂ := by rw [add_right_comm, dist_comm y₁];
   apply dist_triangle4
 #align dist_triangle4_right dist_triangle4_right
 
-/- warning: dist_le_Ico_sum_dist -> dist_le_Ico_sum_dist is a dubious translation:
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-Case conversion may be inaccurate. Consider using '#align dist_le_Ico_sum_dist dist_le_Ico_sum_distₓ'. -/
 /-- The triangle (polygon) inequality for sequences of points; `finset.Ico` version. -/
 theorem dist_le_Ico_sum_dist (f : ℕ → α) {m n} (h : m ≤ n) :
     dist (f m) (f n) ≤ ∑ i in Finset.Ico m n, dist (f i) (f (i + 1)) :=
@@ -354,24 +294,12 @@ theorem dist_le_Ico_sum_dist (f : ℕ → α) {m n} (h : m ≤ n) :
       
 #align dist_le_Ico_sum_dist dist_le_Ico_sum_dist
 
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-Case conversion may be inaccurate. Consider using '#align dist_le_range_sum_dist dist_le_range_sum_distₓ'. -/
 /-- The triangle (polygon) inequality for sequences of points; `finset.range` version. -/
 theorem dist_le_range_sum_dist (f : ℕ → α) (n : ℕ) :
     dist (f 0) (f n) ≤ ∑ i in Finset.range n, dist (f i) (f (i + 1)) :=
   Nat.Ico_zero_eq_range ▸ dist_le_Ico_sum_dist f (Nat.zero_le n)
 #align dist_le_range_sum_dist dist_le_range_sum_dist
 
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 /-- A version of `dist_le_Ico_sum_dist` with each intermediate distance replaced
 with an upper estimate. -/
 theorem dist_le_Ico_sum_of_dist_le {f : ℕ → α} {m n} (hmn : m ≤ n) {d : ℕ → ℝ}
@@ -381,12 +309,6 @@ theorem dist_le_Ico_sum_of_dist_le {f : ℕ → α} {m n} (hmn : m ≤ n) {d : 
     Finset.sum_le_sum fun k hk => hd (Finset.mem_Ico.1 hk).1 (Finset.mem_Ico.1 hk).2
 #align dist_le_Ico_sum_of_dist_le dist_le_Ico_sum_of_dist_le
 
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-Case conversion may be inaccurate. Consider using '#align dist_le_range_sum_of_dist_le dist_le_range_sum_of_dist_leₓ'. -/
 /-- A version of `dist_le_range_sum_dist` with each intermediate distance replaced
 with an upper estimate. -/
 theorem dist_le_range_sum_of_dist_le {f : ℕ → α} (n : ℕ) {d : ℕ → ℝ}
@@ -400,23 +322,11 @@ theorem swap_dist : Function.swap (@dist α _) = dist := by funext x y <;> exact
 #align swap_dist swap_dist
 -/
 
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-Case conversion may be inaccurate. Consider using '#align abs_dist_sub_le abs_dist_sub_leₓ'. -/
 theorem abs_dist_sub_le (x y z : α) : |dist x z - dist y z| ≤ dist x y :=
   abs_sub_le_iff.2
     ⟨sub_le_iff_le_add.2 (dist_triangle _ _ _), sub_le_iff_le_add.2 (dist_triangle_left _ _ _)⟩
 #align abs_dist_sub_le abs_dist_sub_le
 
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 theorem dist_nonneg {x y : α} : 0 ≤ dist x y :=
   PseudoMetricSpace.dist_nonneg' dist dist_self dist_comm dist_triangle
 #align dist_nonneg dist_nonneg
@@ -434,12 +344,6 @@ unsafe def _root_.tactic.positivity_dist : expr → tactic strictness
 
 end
 
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 @[simp]
 theorem abs_dist {a b : α} : |dist a b| = dist a b :=
   abs_of_nonneg dist_nonneg
@@ -483,123 +387,57 @@ theorem coe_nnreal_ennreal_nndist (x y : α) : ↑(nndist x y) = edist x y :=
 #align coe_nnreal_ennreal_nndist coe_nnreal_ennreal_nndist
 -/
 
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 @[simp, norm_cast]
 theorem edist_lt_coe {x y : α} {c : ℝ≥0} : edist x y < c ↔ nndist x y < c := by
   rw [edist_nndist, ENNReal.coe_lt_coe]
 #align edist_lt_coe edist_lt_coe
 
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 @[simp, norm_cast]
 theorem edist_le_coe {x y : α} {c : ℝ≥0} : edist x y ≤ c ↔ nndist x y ≤ c := by
   rw [edist_nndist, ENNReal.coe_le_coe]
 #align edist_le_coe edist_le_coe
 
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 /-- In a pseudometric space, the extended distance is always finite-/
 theorem edist_lt_top {α : Type _} [PseudoMetricSpace α] (x y : α) : edist x y < ⊤ :=
   (edist_dist x y).symm ▸ ENNReal.ofReal_lt_top
 #align edist_lt_top edist_lt_top
 
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 /-- In a pseudometric space, the extended distance is always finite-/
 theorem edist_ne_top (x y : α) : edist x y ≠ ⊤ :=
   (edist_lt_top x y).Ne
 #align edist_ne_top edist_ne_top
 
-/- warning: nndist_self -> nndist_self is a dubious translation:
-lean 3 declaration is
-  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] (a : α), Eq.{1} NNReal (NNDist.nndist.{u1} α (PseudoMetricSpace.toNNDist.{u1} α _inst_1) a a) (OfNat.ofNat.{0} NNReal 0 (OfNat.mk.{0} NNReal 0 (Zero.zero.{0} NNReal (MulZeroClass.toHasZero.{0} NNReal (NonUnitalNonAssocSemiring.toMulZeroClass.{0} NNReal (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} NNReal (Semiring.toNonAssocSemiring.{0} NNReal NNReal.semiring)))))))
-but is expected to have type
-  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] (a : α), Eq.{1} NNReal (NNDist.nndist.{u1} α (PseudoMetricSpace.toNNDist.{u1} α _inst_1) a a) (OfNat.ofNat.{0} NNReal 0 (Zero.toOfNat0.{0} NNReal instNNRealZero))
-Case conversion may be inaccurate. Consider using '#align nndist_self nndist_selfₓ'. -/
 /-- `nndist x x` vanishes-/
 @[simp]
 theorem nndist_self (a : α) : nndist a a = 0 :=
   (NNReal.coe_eq_zero _).1 (dist_self a)
 #align nndist_self nndist_self
 
-/- warning: dist_nndist -> dist_nndist is a dubious translation:
-lean 3 declaration is
-  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] (x : α) (y : α), Eq.{1} Real (Dist.dist.{u1} α (PseudoMetricSpace.toHasDist.{u1} α _inst_1) x y) ((fun (a : Type) (b : Type) [self : HasLiftT.{1, 1} a b] => self.0) NNReal Real (HasLiftT.mk.{1, 1} NNReal Real (CoeTCₓ.coe.{1, 1} NNReal Real (coeBase.{1, 1} NNReal Real NNReal.Real.hasCoe))) (NNDist.nndist.{u1} α (PseudoMetricSpace.toNNDist.{u1} α _inst_1) x y))
-but is expected to have type
-  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] (x : α) (y : α), Eq.{1} Real (Dist.dist.{u1} α (PseudoMetricSpace.toDist.{u1} α _inst_1) x y) (NNReal.toReal (NNDist.nndist.{u1} α (PseudoMetricSpace.toNNDist.{u1} α _inst_1) x y))
-Case conversion may be inaccurate. Consider using '#align dist_nndist dist_nndistₓ'. -/
 /-- Express `dist` in terms of `nndist`-/
 theorem dist_nndist (x y : α) : dist x y = ↑(nndist x y) :=
   rfl
 #align dist_nndist dist_nndist
 
-/- warning: coe_nndist -> coe_nndist is a dubious translation:
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-  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] (x : α) (y : α), Eq.{1} Real ((fun (a : Type) (b : Type) [self : HasLiftT.{1, 1} a b] => self.0) NNReal Real (HasLiftT.mk.{1, 1} NNReal Real (CoeTCₓ.coe.{1, 1} NNReal Real (coeBase.{1, 1} NNReal Real NNReal.Real.hasCoe))) (NNDist.nndist.{u1} α (PseudoMetricSpace.toNNDist.{u1} α _inst_1) x y)) (Dist.dist.{u1} α (PseudoMetricSpace.toHasDist.{u1} α _inst_1) x y)
-but is expected to have type
-  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] (x : α) (y : α), Eq.{1} Real (NNReal.toReal (NNDist.nndist.{u1} α (PseudoMetricSpace.toNNDist.{u1} α _inst_1) x y)) (Dist.dist.{u1} α (PseudoMetricSpace.toDist.{u1} α _inst_1) x y)
-Case conversion may be inaccurate. Consider using '#align coe_nndist coe_nndistₓ'. -/
 @[simp, norm_cast]
 theorem coe_nndist (x y : α) : ↑(nndist x y) = dist x y :=
   (dist_nndist x y).symm
 #align coe_nndist coe_nndist
 
-/- warning: dist_lt_coe -> dist_lt_coe is a dubious translation:
-lean 3 declaration is
-  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {x : α} {y : α} {c : NNReal}, Iff (LT.lt.{0} Real Real.hasLt (Dist.dist.{u1} α (PseudoMetricSpace.toHasDist.{u1} α _inst_1) x y) ((fun (a : Type) (b : Type) [self : HasLiftT.{1, 1} a b] => self.0) NNReal Real (HasLiftT.mk.{1, 1} NNReal Real (CoeTCₓ.coe.{1, 1} NNReal Real (coeBase.{1, 1} NNReal Real NNReal.Real.hasCoe))) c)) (LT.lt.{0} NNReal (Preorder.toHasLt.{0} NNReal (PartialOrder.toPreorder.{0} NNReal (OrderedCancelAddCommMonoid.toPartialOrder.{0} NNReal (StrictOrderedSemiring.toOrderedCancelAddCommMonoid.{0} NNReal NNReal.strictOrderedSemiring)))) (NNDist.nndist.{u1} α (PseudoMetricSpace.toNNDist.{u1} α _inst_1) x y) c)
-but is expected to have type
-  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {x : α} {y : α} {c : NNReal}, Iff (LT.lt.{0} Real Real.instLTReal (Dist.dist.{u1} α (PseudoMetricSpace.toDist.{u1} α _inst_1) x y) (NNReal.toReal c)) (LT.lt.{0} NNReal (Preorder.toLT.{0} NNReal (PartialOrder.toPreorder.{0} NNReal (StrictOrderedSemiring.toPartialOrder.{0} NNReal instNNRealStrictOrderedSemiring))) (NNDist.nndist.{u1} α (PseudoMetricSpace.toNNDist.{u1} α _inst_1) x y) c)
-Case conversion may be inaccurate. Consider using '#align dist_lt_coe dist_lt_coeₓ'. -/
 @[simp, norm_cast]
 theorem dist_lt_coe {x y : α} {c : ℝ≥0} : dist x y < c ↔ nndist x y < c :=
   Iff.rfl
 #align dist_lt_coe dist_lt_coe
 
-/- warning: dist_le_coe -> dist_le_coe is a dubious translation:
-lean 3 declaration is
-  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {x : α} {y : α} {c : NNReal}, Iff (LE.le.{0} Real Real.hasLe (Dist.dist.{u1} α (PseudoMetricSpace.toHasDist.{u1} α _inst_1) x y) ((fun (a : Type) (b : Type) [self : HasLiftT.{1, 1} a b] => self.0) NNReal Real (HasLiftT.mk.{1, 1} NNReal Real (CoeTCₓ.coe.{1, 1} NNReal Real (coeBase.{1, 1} NNReal Real NNReal.Real.hasCoe))) c)) (LE.le.{0} NNReal (Preorder.toHasLe.{0} NNReal (PartialOrder.toPreorder.{0} NNReal (OrderedCancelAddCommMonoid.toPartialOrder.{0} NNReal (StrictOrderedSemiring.toOrderedCancelAddCommMonoid.{0} NNReal NNReal.strictOrderedSemiring)))) (NNDist.nndist.{u1} α (PseudoMetricSpace.toNNDist.{u1} α _inst_1) x y) c)
-but is expected to have type
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-Case conversion may be inaccurate. Consider using '#align dist_le_coe dist_le_coeₓ'. -/
 @[simp, norm_cast]
 theorem dist_le_coe {x y : α} {c : ℝ≥0} : dist x y ≤ c ↔ nndist x y ≤ c :=
   Iff.rfl
 #align dist_le_coe dist_le_coe
 
-/- warning: edist_lt_of_real -> edist_lt_ofReal is a dubious translation:
-lean 3 declaration is
-  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {x : α} {y : α} {r : Real}, Iff (LT.lt.{0} ENNReal (Preorder.toHasLt.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (CompleteSemilatticeInf.toPartialOrder.{0} ENNReal (CompleteLattice.toCompleteSemilatticeInf.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))))) (EDist.edist.{u1} α (PseudoMetricSpace.toEDist.{u1} α _inst_1) x y) (ENNReal.ofReal r)) (LT.lt.{0} Real Real.hasLt (Dist.dist.{u1} α (PseudoMetricSpace.toHasDist.{u1} α _inst_1) x y) r)
-but is expected to have type
-  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {x : α} {y : α} {r : Real}, Iff (LT.lt.{0} ENNReal (Preorder.toLT.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (CompleteSemilatticeInf.toPartialOrder.{0} ENNReal (CompleteLattice.toCompleteSemilatticeInf.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal))))) (EDist.edist.{u1} α (PseudoMetricSpace.toEDist.{u1} α _inst_1) x y) (ENNReal.ofReal r)) (LT.lt.{0} Real Real.instLTReal (Dist.dist.{u1} α (PseudoMetricSpace.toDist.{u1} α _inst_1) x y) r)
-Case conversion may be inaccurate. Consider using '#align edist_lt_of_real edist_lt_ofRealₓ'. -/
 @[simp]
 theorem edist_lt_ofReal {x y : α} {r : ℝ} : edist x y < ENNReal.ofReal r ↔ dist x y < r := by
   rw [edist_dist, ENNReal.ofReal_lt_ofReal_iff_of_nonneg dist_nonneg]
 #align edist_lt_of_real edist_lt_ofReal
 
-/- warning: edist_le_of_real -> edist_le_ofReal is a dubious translation:
-lean 3 declaration is
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-but is expected to have type
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-Case conversion may be inaccurate. Consider using '#align edist_le_of_real edist_le_ofRealₓ'. -/
 @[simp]
 theorem edist_le_ofReal {x y : α} {r : ℝ} (hr : 0 ≤ r) :
     edist x y ≤ ENNReal.ofReal r ↔ dist x y ≤ r := by
@@ -619,33 +457,15 @@ theorem nndist_comm (x y : α) : nndist x y = nndist y x := by
 #align nndist_comm nndist_comm
 -/
 
-/- warning: nndist_triangle -> nndist_triangle is a dubious translation:
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-Case conversion may be inaccurate. Consider using '#align nndist_triangle nndist_triangleₓ'. -/
 /-- Triangle inequality for the nonnegative distance-/
 theorem nndist_triangle (x y z : α) : nndist x z ≤ nndist x y + nndist y z :=
   dist_triangle _ _ _
 #align nndist_triangle nndist_triangle
 
-/- warning: nndist_triangle_left -> nndist_triangle_left is a dubious translation:
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-but is expected to have type
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-Case conversion may be inaccurate. Consider using '#align nndist_triangle_left nndist_triangle_leftₓ'. -/
 theorem nndist_triangle_left (x y z : α) : nndist x y ≤ nndist z x + nndist z y :=
   dist_triangle_left _ _ _
 #align nndist_triangle_left nndist_triangle_left
 
-/- warning: nndist_triangle_right -> nndist_triangle_right is a dubious translation:
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-but is expected to have type
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-Case conversion may be inaccurate. Consider using '#align nndist_triangle_right nndist_triangle_rightₓ'. -/
 theorem nndist_triangle_right (x y z : α) : nndist x y ≤ nndist x z + nndist y z :=
   dist_triangle_right _ _ _
 #align nndist_triangle_right nndist_triangle_right
@@ -669,84 +489,36 @@ def ball (x : α) (ε : ℝ) : Set α :=
 #align metric.ball Metric.ball
 -/
 
-/- warning: metric.mem_ball -> Metric.mem_ball is a dubious translation:
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-but is expected to have type
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-Case conversion may be inaccurate. Consider using '#align metric.mem_ball Metric.mem_ballₓ'. -/
 @[simp]
 theorem mem_ball : y ∈ ball x ε ↔ dist y x < ε :=
   Iff.rfl
 #align metric.mem_ball Metric.mem_ball
 
-/- warning: metric.mem_ball' -> Metric.mem_ball' is a dubious translation:
-lean 3 declaration is
-  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {x : α} {y : α} {ε : Real}, Iff (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) y (Metric.ball.{u1} α _inst_1 x ε)) (LT.lt.{0} Real Real.hasLt (Dist.dist.{u1} α (PseudoMetricSpace.toHasDist.{u1} α _inst_1) x y) ε)
-but is expected to have type
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-Case conversion may be inaccurate. Consider using '#align metric.mem_ball' Metric.mem_ball'ₓ'. -/
 theorem mem_ball' : y ∈ ball x ε ↔ dist x y < ε := by rw [dist_comm, mem_ball]
 #align metric.mem_ball' Metric.mem_ball'
 
-/- warning: metric.pos_of_mem_ball -> Metric.pos_of_mem_ball is a dubious translation:
-lean 3 declaration is
-  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {x : α} {y : α} {ε : Real}, (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) y (Metric.ball.{u1} α _inst_1 x ε)) -> (LT.lt.{0} Real Real.hasLt (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero))) ε)
-but is expected to have type
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-Case conversion may be inaccurate. Consider using '#align metric.pos_of_mem_ball Metric.pos_of_mem_ballₓ'. -/
 theorem pos_of_mem_ball (hy : y ∈ ball x ε) : 0 < ε :=
   dist_nonneg.trans_lt hy
 #align metric.pos_of_mem_ball Metric.pos_of_mem_ball
 
-/- warning: metric.mem_ball_self -> Metric.mem_ball_self is a dubious translation:
-lean 3 declaration is
-  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {x : α} {ε : Real}, (LT.lt.{0} Real Real.hasLt (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero))) ε) -> (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) x (Metric.ball.{u1} α _inst_1 x ε))
-but is expected to have type
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-Case conversion may be inaccurate. Consider using '#align metric.mem_ball_self Metric.mem_ball_selfₓ'. -/
 theorem mem_ball_self (h : 0 < ε) : x ∈ ball x ε :=
   show dist x x < ε by rw [dist_self] <;> assumption
 #align metric.mem_ball_self Metric.mem_ball_self
 
-/- warning: metric.nonempty_ball -> Metric.nonempty_ball is a dubious translation:
-lean 3 declaration is
-  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {x : α} {ε : Real}, Iff (Set.Nonempty.{u1} α (Metric.ball.{u1} α _inst_1 x ε)) (LT.lt.{0} Real Real.hasLt (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero))) ε)
-but is expected to have type
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-Case conversion may be inaccurate. Consider using '#align metric.nonempty_ball Metric.nonempty_ballₓ'. -/
 @[simp]
 theorem nonempty_ball : (ball x ε).Nonempty ↔ 0 < ε :=
   ⟨fun ⟨x, hx⟩ => pos_of_mem_ball hx, fun h => ⟨x, mem_ball_self h⟩⟩
 #align metric.nonempty_ball Metric.nonempty_ball
 
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-lean 3 declaration is
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-but is expected to have type
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-Case conversion may be inaccurate. Consider using '#align metric.ball_eq_empty Metric.ball_eq_emptyₓ'. -/
 @[simp]
 theorem ball_eq_empty : ball x ε = ∅ ↔ ε ≤ 0 := by
   rw [← not_nonempty_iff_eq_empty, nonempty_ball, not_lt]
 #align metric.ball_eq_empty Metric.ball_eq_empty
 
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-but is expected to have type
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-Case conversion may be inaccurate. Consider using '#align metric.ball_zero Metric.ball_zeroₓ'. -/
 @[simp]
 theorem ball_zero : ball x 0 = ∅ := by rw [ball_eq_empty]
 #align metric.ball_zero Metric.ball_zero
 
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-but is expected to have type
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-Case conversion may be inaccurate. Consider using '#align metric.exists_lt_mem_ball_of_mem_ball Metric.exists_lt_mem_ball_of_mem_ballₓ'. -/
 /-- If a point belongs to an open ball, then there is a strictly smaller radius whose ball also
 contains it.
 
@@ -757,45 +529,21 @@ theorem exists_lt_mem_ball_of_mem_ball (h : x ∈ ball y ε) : ∃ ε' < ε, x 
   exact ⟨(ε + dist x y) / 2, by linarith, by linarith⟩
 #align metric.exists_lt_mem_ball_of_mem_ball Metric.exists_lt_mem_ball_of_mem_ball
 
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-but is expected to have type
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-Case conversion may be inaccurate. Consider using '#align metric.ball_eq_ball Metric.ball_eq_ballₓ'. -/
 theorem ball_eq_ball (ε : ℝ) (x : α) :
     UniformSpace.ball x { p | dist p.2 p.1 < ε } = Metric.ball x ε :=
   rfl
 #align metric.ball_eq_ball Metric.ball_eq_ball
 
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-Case conversion may be inaccurate. Consider using '#align metric.ball_eq_ball' Metric.ball_eq_ball'ₓ'. -/
 theorem ball_eq_ball' (ε : ℝ) (x : α) :
     UniformSpace.ball x { p | dist p.1 p.2 < ε } = Metric.ball x ε := by ext;
   simp [dist_comm, UniformSpace.ball]
 #align metric.ball_eq_ball' Metric.ball_eq_ball'
 
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-lean 3 declaration is
-  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] (x : α), Eq.{succ u1} (Set.{u1} α) (Set.iUnion.{u1, 1} α Nat (fun (n : Nat) => Metric.ball.{u1} α _inst_1 x ((fun (a : Type) (b : Type) [self : HasLiftT.{1, 1} a b] => self.0) Nat Real (HasLiftT.mk.{1, 1} Nat Real (CoeTCₓ.coe.{1, 1} Nat Real (Nat.castCoe.{0} Real Real.hasNatCast))) n))) (Set.univ.{u1} α)
-but is expected to have type
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 @[simp]
 theorem iUnion_ball_nat (x : α) : (⋃ n : ℕ, ball x n) = univ :=
   iUnion_eq_univ_iff.2 fun y => exists_nat_gt (dist y x)
 #align metric.Union_ball_nat Metric.iUnion_ball_nat
 
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 @[simp]
 theorem iUnion_ball_nat_succ (x : α) : (⋃ n : ℕ, ball x (n + 1)) = univ :=
   iUnion_eq_univ_iff.2 fun y => (exists_nat_gt (dist y x)).imp fun n hn => hn.trans (lt_add_one _)
@@ -808,23 +556,11 @@ def closedBall (x : α) (ε : ℝ) :=
 #align metric.closed_ball Metric.closedBall
 -/
 
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 @[simp]
 theorem mem_closedBall : y ∈ closedBall x ε ↔ dist y x ≤ ε :=
   Iff.rfl
 #align metric.mem_closed_ball Metric.mem_closedBall
 
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 theorem mem_closedBall' : y ∈ closedBall x ε ↔ dist x y ≤ ε := by rw [dist_comm, mem_closed_ball]
 #align metric.mem_closed_ball' Metric.mem_closedBall'
 
@@ -847,63 +583,27 @@ theorem mem_sphere' : y ∈ sphere x ε ↔ dist x y = ε := by rw [dist_comm, m
 #align metric.mem_sphere' Metric.mem_sphere'
 -/
 
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-Case conversion may be inaccurate. Consider using '#align metric.ne_of_mem_sphere Metric.ne_of_mem_sphereₓ'. -/
 theorem ne_of_mem_sphere (h : y ∈ sphere x ε) (hε : ε ≠ 0) : y ≠ x := by contrapose! hε; symm;
   simpa [hε] using h
 #align metric.ne_of_mem_sphere Metric.ne_of_mem_sphere
 
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 theorem sphere_eq_empty_of_subsingleton [Subsingleton α] (hε : ε ≠ 0) : sphere x ε = ∅ :=
   Set.eq_empty_iff_forall_not_mem.mpr fun y hy => ne_of_mem_sphere hy hε (Subsingleton.elim _ _)
 #align metric.sphere_eq_empty_of_subsingleton Metric.sphere_eq_empty_of_subsingleton
 
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 theorem sphere_isEmpty_of_subsingleton [Subsingleton α] (hε : ε ≠ 0) : IsEmpty (sphere x ε) := by
   simp only [sphere_eq_empty_of_subsingleton hε, Set.hasEmptyc.Emptyc.isEmpty α]
 #align metric.sphere_is_empty_of_subsingleton Metric.sphere_isEmpty_of_subsingleton
 
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-Case conversion may be inaccurate. Consider using '#align metric.mem_closed_ball_self Metric.mem_closedBall_selfₓ'. -/
 theorem mem_closedBall_self (h : 0 ≤ ε) : x ∈ closedBall x ε :=
   show dist x x ≤ ε by rw [dist_self] <;> assumption
 #align metric.mem_closed_ball_self Metric.mem_closedBall_self
 
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 @[simp]
 theorem nonempty_closedBall : (closedBall x ε).Nonempty ↔ 0 ≤ ε :=
   ⟨fun ⟨x, hx⟩ => dist_nonneg.trans hx, fun h => ⟨x, mem_closedBall_self h⟩⟩
 #align metric.nonempty_closed_ball Metric.nonempty_closedBall
 
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 @[simp]
 theorem closedBall_eq_empty : closedBall x ε = ∅ ↔ ε < 0 := by
   rw [← not_nonempty_iff_eq_empty, nonempty_closed_ball, not_le]
@@ -919,98 +619,44 @@ theorem sphere_subset_closedBall : sphere x ε ⊆ closedBall x ε := fun y => l
 #align metric.sphere_subset_closed_ball Metric.sphere_subset_closedBall
 -/
 
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 theorem closedBall_disjoint_ball (h : δ + ε ≤ dist x y) : Disjoint (closedBall x δ) (ball y ε) :=
   Set.disjoint_left.mpr fun a ha1 ha2 =>
     (h.trans <| dist_triangle_left _ _ _).not_lt <| add_lt_add_of_le_of_lt ha1 ha2
 #align metric.closed_ball_disjoint_ball Metric.closedBall_disjoint_ball
 
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 theorem ball_disjoint_closedBall (h : δ + ε ≤ dist x y) : Disjoint (ball x δ) (closedBall y ε) :=
   (closedBall_disjoint_ball <| by rwa [add_comm, dist_comm]).symm
 #align metric.ball_disjoint_closed_ball Metric.ball_disjoint_closedBall
 
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 theorem ball_disjoint_ball (h : δ + ε ≤ dist x y) : Disjoint (ball x δ) (ball y ε) :=
   (closedBall_disjoint_ball h).mono_left ball_subset_closedBall
 #align metric.ball_disjoint_ball Metric.ball_disjoint_ball
 
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-Case conversion may be inaccurate. Consider using '#align metric.closed_ball_disjoint_closed_ball Metric.closedBall_disjoint_closedBallₓ'. -/
 theorem closedBall_disjoint_closedBall (h : δ + ε < dist x y) :
     Disjoint (closedBall x δ) (closedBall y ε) :=
   Set.disjoint_left.mpr fun a ha1 ha2 =>
     h.not_le <| (dist_triangle_left _ _ _).trans <| add_le_add ha1 ha2
 #align metric.closed_ball_disjoint_closed_ball Metric.closedBall_disjoint_closedBall
 
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 theorem sphere_disjoint_ball : Disjoint (sphere x ε) (ball x ε) :=
   Set.disjoint_left.mpr fun y hy₁ hy₂ => absurd hy₁ <| ne_of_lt hy₂
 #align metric.sphere_disjoint_ball Metric.sphere_disjoint_ball
 
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 @[simp]
 theorem ball_union_sphere : ball x ε ∪ sphere x ε = closedBall x ε :=
   Set.ext fun y => (@le_iff_lt_or_eq ℝ _ _ _).symm
 #align metric.ball_union_sphere Metric.ball_union_sphere
 
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-Case conversion may be inaccurate. Consider using '#align metric.sphere_union_ball Metric.sphere_union_ballₓ'. -/
 @[simp]
 theorem sphere_union_ball : sphere x ε ∪ ball x ε = closedBall x ε := by
   rw [union_comm, ball_union_sphere]
 #align metric.sphere_union_ball Metric.sphere_union_ball
 
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 @[simp]
 theorem closedBall_diff_sphere : closedBall x ε \ sphere x ε = ball x ε := by
   rw [← ball_union_sphere, Set.union_diff_cancel_right sphere_disjoint_ball.symm.le_bot]
 #align metric.closed_ball_diff_sphere Metric.closedBall_diff_sphere
 
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-Case conversion may be inaccurate. Consider using '#align metric.closed_ball_diff_ball Metric.closedBall_diff_ballₓ'. -/
 @[simp]
 theorem closedBall_diff_ball : closedBall x ε \ ball x ε = sphere x ε := by
   rw [← ball_union_sphere, Set.union_diff_cancel_left sphere_disjoint_ball.symm.le_bot]
@@ -1032,32 +678,14 @@ theorem mem_sphere_comm : x ∈ sphere y ε ↔ y ∈ sphere x ε := by rw [mem_
 #align metric.mem_sphere_comm Metric.mem_sphere_comm
 -/
 
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-but is expected to have type
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-Case conversion may be inaccurate. Consider using '#align metric.ball_subset_ball Metric.ball_subset_ballₓ'. -/
 theorem ball_subset_ball (h : ε₁ ≤ ε₂) : ball x ε₁ ⊆ ball x ε₂ := fun y (yx : _ < ε₁) =>
   lt_of_lt_of_le yx h
 #align metric.ball_subset_ball Metric.ball_subset_ball
 
-/- warning: metric.closed_ball_eq_bInter_ball -> Metric.closedBall_eq_bInter_ball is a dubious translation:
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-  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {x : α} {ε : Real}, Eq.{succ u1} (Set.{u1} α) (Metric.closedBall.{u1} α _inst_1 x ε) (Set.iInter.{u1, 1} α Real (fun (δ : Real) => Set.iInter.{u1, 0} α (GT.gt.{0} Real Real.hasLt δ ε) (fun (H : GT.gt.{0} Real Real.hasLt δ ε) => Metric.ball.{u1} α _inst_1 x δ)))
-but is expected to have type
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-Case conversion may be inaccurate. Consider using '#align metric.closed_ball_eq_bInter_ball Metric.closedBall_eq_bInter_ballₓ'. -/
 theorem closedBall_eq_bInter_ball : closedBall x ε = ⋂ δ > ε, ball x δ := by
   ext y <;> rw [mem_closed_ball, ← forall_lt_iff_le', mem_Inter₂] <;> rfl
 #align metric.closed_ball_eq_bInter_ball Metric.closedBall_eq_bInter_ball
 
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-but is expected to have type
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-Case conversion may be inaccurate. Consider using '#align metric.ball_subset_ball' Metric.ball_subset_ball'ₓ'. -/
 theorem ball_subset_ball' (h : ε₁ + dist x y ≤ ε₂) : ball x ε₁ ⊆ ball y ε₂ := fun z hz =>
   calc
     dist z y ≤ dist z x + dist x y := dist_triangle _ _ _
@@ -1066,22 +694,10 @@ theorem ball_subset_ball' (h : ε₁ + dist x y ≤ ε₂) : ball x ε₁ ⊆ ba
     
 #align metric.ball_subset_ball' Metric.ball_subset_ball'
 
-/- warning: metric.closed_ball_subset_closed_ball -> Metric.closedBall_subset_closedBall is a dubious translation:
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-but is expected to have type
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-Case conversion may be inaccurate. Consider using '#align metric.closed_ball_subset_closed_ball Metric.closedBall_subset_closedBallₓ'. -/
 theorem closedBall_subset_closedBall (h : ε₁ ≤ ε₂) : closedBall x ε₁ ⊆ closedBall x ε₂ :=
   fun y (yx : _ ≤ ε₁) => le_trans yx h
 #align metric.closed_ball_subset_closed_ball Metric.closedBall_subset_closedBall
 
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-Case conversion may be inaccurate. Consider using '#align metric.closed_ball_subset_closed_ball' Metric.closedBall_subset_closedBall'ₓ'. -/
 theorem closedBall_subset_closedBall' (h : ε₁ + dist x y ≤ ε₂) :
     closedBall x ε₁ ⊆ closedBall y ε₂ := fun z hz =>
   calc
@@ -1091,22 +707,10 @@ theorem closedBall_subset_closedBall' (h : ε₁ + dist x y ≤ ε₂) :
     
 #align metric.closed_ball_subset_closed_ball' Metric.closedBall_subset_closedBall'
 
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-Case conversion may be inaccurate. Consider using '#align metric.closed_ball_subset_ball Metric.closedBall_subset_ballₓ'. -/
 theorem closedBall_subset_ball (h : ε₁ < ε₂) : closedBall x ε₁ ⊆ ball x ε₂ :=
   fun y (yh : dist y x ≤ ε₁) => lt_of_le_of_lt yh h
 #align metric.closed_ball_subset_ball Metric.closedBall_subset_ball
 
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-Case conversion may be inaccurate. Consider using '#align metric.closed_ball_subset_ball' Metric.closedBall_subset_ball'ₓ'. -/
 theorem closedBall_subset_ball' (h : ε₁ + dist x y < ε₂) : closedBall x ε₁ ⊆ ball y ε₂ :=
   fun z hz =>
   calc
@@ -1116,12 +720,6 @@ theorem closedBall_subset_ball' (h : ε₁ + dist x y < ε₂) : closedBall x ε
     
 #align metric.closed_ball_subset_ball' Metric.closedBall_subset_ball'
 
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-Case conversion may be inaccurate. Consider using '#align metric.dist_le_add_of_nonempty_closed_ball_inter_closed_ball Metric.dist_le_add_of_nonempty_closedBall_inter_closedBallₓ'. -/
 theorem dist_le_add_of_nonempty_closedBall_inter_closedBall
     (h : (closedBall x ε₁ ∩ closedBall y ε₂).Nonempty) : dist x y ≤ ε₁ + ε₂ :=
   let ⟨z, hz⟩ := h
@@ -1131,12 +729,6 @@ theorem dist_le_add_of_nonempty_closedBall_inter_closedBall
     
 #align metric.dist_le_add_of_nonempty_closed_ball_inter_closed_ball Metric.dist_le_add_of_nonempty_closedBall_inter_closedBall
 
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-Case conversion may be inaccurate. Consider using '#align metric.dist_lt_add_of_nonempty_closed_ball_inter_ball Metric.dist_lt_add_of_nonempty_closedBall_inter_ballₓ'. -/
 theorem dist_lt_add_of_nonempty_closedBall_inter_ball (h : (closedBall x ε₁ ∩ ball y ε₂).Nonempty) :
     dist x y < ε₁ + ε₂ :=
   let ⟨z, hz⟩ := h
@@ -1146,12 +738,6 @@ theorem dist_lt_add_of_nonempty_closedBall_inter_ball (h : (closedBall x ε₁ 
     
 #align metric.dist_lt_add_of_nonempty_closed_ball_inter_ball Metric.dist_lt_add_of_nonempty_closedBall_inter_ball
 
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-Case conversion may be inaccurate. Consider using '#align metric.dist_lt_add_of_nonempty_ball_inter_closed_ball Metric.dist_lt_add_of_nonempty_ball_inter_closedBallₓ'. -/
 theorem dist_lt_add_of_nonempty_ball_inter_closedBall (h : (ball x ε₁ ∩ closedBall y ε₂).Nonempty) :
     dist x y < ε₁ + ε₂ := by
   rw [inter_comm] at h
@@ -1159,66 +745,30 @@ theorem dist_lt_add_of_nonempty_ball_inter_closedBall (h : (ball x ε₁ ∩ clo
   exact dist_lt_add_of_nonempty_closed_ball_inter_ball h
 #align metric.dist_lt_add_of_nonempty_ball_inter_closed_ball Metric.dist_lt_add_of_nonempty_ball_inter_closedBall
 
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-Case conversion may be inaccurate. Consider using '#align metric.dist_lt_add_of_nonempty_ball_inter_ball Metric.dist_lt_add_of_nonempty_ball_inter_ballₓ'. -/
 theorem dist_lt_add_of_nonempty_ball_inter_ball (h : (ball x ε₁ ∩ ball y ε₂).Nonempty) :
     dist x y < ε₁ + ε₂ :=
   dist_lt_add_of_nonempty_closedBall_inter_ball <|
     h.mono (inter_subset_inter ball_subset_closedBall Subset.rfl)
 #align metric.dist_lt_add_of_nonempty_ball_inter_ball Metric.dist_lt_add_of_nonempty_ball_inter_ball
 
-/- warning: metric.Union_closed_ball_nat -> Metric.iUnion_closedBall_nat is a dubious translation:
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-Case conversion may be inaccurate. Consider using '#align metric.Union_closed_ball_nat Metric.iUnion_closedBall_natₓ'. -/
 @[simp]
 theorem iUnion_closedBall_nat (x : α) : (⋃ n : ℕ, closedBall x n) = univ :=
   iUnion_eq_univ_iff.2 fun y => exists_nat_ge (dist y x)
 #align metric.Union_closed_ball_nat Metric.iUnion_closedBall_nat
 
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-Case conversion may be inaccurate. Consider using '#align metric.Union_inter_closed_ball_nat Metric.iUnion_inter_closedBall_natₓ'. -/
 theorem iUnion_inter_closedBall_nat (s : Set α) (x : α) : (⋃ n : ℕ, s ∩ closedBall x n) = s := by
   rw [← inter_Union, Union_closed_ball_nat, inter_univ]
 #align metric.Union_inter_closed_ball_nat Metric.iUnion_inter_closedBall_nat
 
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-but is expected to have type
-  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {x : α} {y : α} {ε₁ : Real} {ε₂ : Real}, (LE.le.{0} Real Real.instLEReal (Dist.dist.{u1} α (PseudoMetricSpace.toDist.{u1} α _inst_1) x y) (HSub.hSub.{0, 0, 0} Real Real Real (instHSub.{0} Real Real.instSubReal) ε₂ ε₁)) -> (HasSubset.Subset.{u1} (Set.{u1} α) (Set.instHasSubsetSet.{u1} α) (Metric.ball.{u1} α _inst_1 x ε₁) (Metric.ball.{u1} α _inst_1 y ε₂))
-Case conversion may be inaccurate. Consider using '#align metric.ball_subset Metric.ball_subsetₓ'. -/
 theorem ball_subset (h : dist x y ≤ ε₂ - ε₁) : ball x ε₁ ⊆ ball y ε₂ := fun z zx => by
   rw [← add_sub_cancel'_right ε₁ ε₂] <;>
     exact lt_of_le_of_lt (dist_triangle z x y) (add_lt_add_of_lt_of_le zx h)
 #align metric.ball_subset Metric.ball_subset
 
-/- warning: metric.ball_half_subset -> Metric.ball_half_subset is a dubious translation:
-lean 3 declaration is
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-but is expected to have type
-  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {x : α} {ε : Real} (y : α), (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) y (Metric.ball.{u1} α _inst_1 x (HDiv.hDiv.{0, 0, 0} Real Real Real (instHDiv.{0} Real (LinearOrderedField.toDiv.{0} Real Real.instLinearOrderedFieldReal)) ε (OfNat.ofNat.{0} Real 2 (instOfNat.{0} Real 2 Real.natCast (instAtLeastTwoHAddNatInstHAddInstAddNatOfNat (OfNat.ofNat.{0} Nat 0 (instOfNatNat 0)))))))) -> (HasSubset.Subset.{u1} (Set.{u1} α) (Set.instHasSubsetSet.{u1} α) (Metric.ball.{u1} α _inst_1 y (HDiv.hDiv.{0, 0, 0} Real Real Real (instHDiv.{0} Real (LinearOrderedField.toDiv.{0} Real Real.instLinearOrderedFieldReal)) ε (OfNat.ofNat.{0} Real 2 (instOfNat.{0} Real 2 Real.natCast (instAtLeastTwoHAddNatInstHAddInstAddNatOfNat (OfNat.ofNat.{0} Nat 0 (instOfNatNat 0))))))) (Metric.ball.{u1} α _inst_1 x ε))
-Case conversion may be inaccurate. Consider using '#align metric.ball_half_subset Metric.ball_half_subsetₓ'. -/
 theorem ball_half_subset (y) (h : y ∈ ball x (ε / 2)) : ball y (ε / 2) ⊆ ball x ε :=
   ball_subset <| by rw [sub_self_div_two] <;> exact le_of_lt h
 #align metric.ball_half_subset Metric.ball_half_subset
 
-/- warning: metric.exists_ball_subset_ball -> Metric.exists_ball_subset_ball is a dubious translation:
-lean 3 declaration is
-  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {x : α} {y : α} {ε : Real}, (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) y (Metric.ball.{u1} α _inst_1 x ε)) -> (Exists.{1} Real (fun (ε' : Real) => Exists.{0} (GT.gt.{0} Real Real.hasLt ε' (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero)))) (fun (H : GT.gt.{0} Real Real.hasLt ε' (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero)))) => HasSubset.Subset.{u1} (Set.{u1} α) (Set.hasSubset.{u1} α) (Metric.ball.{u1} α _inst_1 y ε') (Metric.ball.{u1} α _inst_1 x ε))))
-but is expected to have type
-  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {x : α} {y : α} {ε : Real}, (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) y (Metric.ball.{u1} α _inst_1 x ε)) -> (Exists.{1} Real (fun (ε' : Real) => And (GT.gt.{0} Real Real.instLTReal ε' (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal))) (HasSubset.Subset.{u1} (Set.{u1} α) (Set.instHasSubsetSet.{u1} α) (Metric.ball.{u1} α _inst_1 y ε') (Metric.ball.{u1} α _inst_1 x ε))))
-Case conversion may be inaccurate. Consider using '#align metric.exists_ball_subset_ball Metric.exists_ball_subset_ballₓ'. -/
 theorem exists_ball_subset_ball (h : y ∈ ball x ε) : ∃ ε' > 0, ball y ε' ⊆ ball x ε :=
   ⟨_, sub_pos.2 h, ball_subset <| by rw [sub_sub_self]⟩
 #align metric.exists_ball_subset_ball Metric.exists_ball_subset_ball
@@ -1247,24 +797,12 @@ theorem forall_of_forall_mem_ball (p : α → Prop) (x : α)
 #align metric.forall_of_forall_mem_ball Metric.forall_of_forall_mem_ball
 -/
 
-/- warning: metric.is_bounded_iff -> Metric.isBounded_iff is a dubious translation:
-lean 3 declaration is
-  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {s : Set.{u1} α}, Iff (Bornology.IsBounded.{u1} α (PseudoMetricSpace.toBornology.{u1} α _inst_1) s) (Exists.{1} Real (fun (C : Real) => forall {{x : α}}, (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) x s) -> (forall {{y : α}}, (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) y s) -> (LE.le.{0} Real Real.hasLe (Dist.dist.{u1} α (PseudoMetricSpace.toHasDist.{u1} α _inst_1) x y) C))))
-but is expected to have type
-  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {s : Set.{u1} α}, Iff (Bornology.IsBounded.{u1} α (PseudoMetricSpace.toBornology.{u1} α _inst_1) s) (Exists.{1} Real (fun (C : Real) => forall {{x : α}}, (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) x s) -> (forall {{y : α}}, (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) y s) -> (LE.le.{0} Real Real.instLEReal (Dist.dist.{u1} α (PseudoMetricSpace.toDist.{u1} α _inst_1) x y) C))))
-Case conversion may be inaccurate. Consider using '#align metric.is_bounded_iff Metric.isBounded_iffₓ'. -/
 theorem isBounded_iff {s : Set α} :
     IsBounded s ↔ ∃ C : ℝ, ∀ ⦃x⦄, x ∈ s → ∀ ⦃y⦄, y ∈ s → dist x y ≤ C := by
   rw [is_bounded_def, ← Filter.mem_sets, (@PseudoMetricSpace.cobounded_sets α _).out, mem_set_of_eq,
     compl_compl]
 #align metric.is_bounded_iff Metric.isBounded_iff
 
-/- warning: metric.is_bounded_iff_eventually -> Metric.isBounded_iff_eventually is a dubious translation:
-lean 3 declaration is
-  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {s : Set.{u1} α}, Iff (Bornology.IsBounded.{u1} α (PseudoMetricSpace.toBornology.{u1} α _inst_1) s) (Filter.Eventually.{0} Real (fun (C : Real) => forall {{x : α}}, (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) x s) -> (forall {{y : α}}, (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) y s) -> (LE.le.{0} Real Real.hasLe (Dist.dist.{u1} α (PseudoMetricSpace.toHasDist.{u1} α _inst_1) x y) C))) (Filter.atTop.{0} Real Real.preorder))
-but is expected to have type
-  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {s : Set.{u1} α}, Iff (Bornology.IsBounded.{u1} α (PseudoMetricSpace.toBornology.{u1} α _inst_1) s) (Filter.Eventually.{0} Real (fun (C : Real) => forall {{x : α}}, (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) x s) -> (forall {{y : α}}, (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) y s) -> (LE.le.{0} Real Real.instLEReal (Dist.dist.{u1} α (PseudoMetricSpace.toDist.{u1} α _inst_1) x y) C))) (Filter.atTop.{0} Real Real.instPreorderReal))
-Case conversion may be inaccurate. Consider using '#align metric.is_bounded_iff_eventually Metric.isBounded_iff_eventuallyₓ'. -/
 theorem isBounded_iff_eventually {s : Set α} :
     IsBounded s ↔ ∀ᶠ C in atTop, ∀ ⦃x⦄, x ∈ s → ∀ ⦃y⦄, y ∈ s → dist x y ≤ C :=
   isBounded_iff.trans
@@ -1272,24 +810,12 @@ theorem isBounded_iff_eventually {s : Set α} :
       Eventually.exists⟩
 #align metric.is_bounded_iff_eventually Metric.isBounded_iff_eventually
 
-/- warning: metric.is_bounded_iff_exists_ge -> Metric.isBounded_iff_exists_ge is a dubious translation:
-lean 3 declaration is
-  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {s : Set.{u1} α} (c : Real), Iff (Bornology.IsBounded.{u1} α (PseudoMetricSpace.toBornology.{u1} α _inst_1) s) (Exists.{1} Real (fun (C : Real) => And (LE.le.{0} Real Real.hasLe c C) (forall {{x : α}}, (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) x s) -> (forall {{y : α}}, (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) y s) -> (LE.le.{0} Real Real.hasLe (Dist.dist.{u1} α (PseudoMetricSpace.toHasDist.{u1} α _inst_1) x y) C)))))
-but is expected to have type
-  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {s : Set.{u1} α} (c : Real), Iff (Bornology.IsBounded.{u1} α (PseudoMetricSpace.toBornology.{u1} α _inst_1) s) (Exists.{1} Real (fun (C : Real) => And (LE.le.{0} Real Real.instLEReal c C) (forall {{x : α}}, (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) x s) -> (forall {{y : α}}, (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) y s) -> (LE.le.{0} Real Real.instLEReal (Dist.dist.{u1} α (PseudoMetricSpace.toDist.{u1} α _inst_1) x y) C)))))
-Case conversion may be inaccurate. Consider using '#align metric.is_bounded_iff_exists_ge Metric.isBounded_iff_exists_geₓ'. -/
 theorem isBounded_iff_exists_ge {s : Set α} (c : ℝ) :
     IsBounded s ↔ ∃ C, c ≤ C ∧ ∀ ⦃x⦄, x ∈ s → ∀ ⦃y⦄, y ∈ s → dist x y ≤ C :=
   ⟨fun h => ((eventually_ge_atTop c).And (isBounded_iff_eventually.1 h)).exists, fun h =>
     isBounded_iff.2 <| h.imp fun _ => And.right⟩
 #align metric.is_bounded_iff_exists_ge Metric.isBounded_iff_exists_ge
 
-/- warning: metric.is_bounded_iff_nndist -> Metric.isBounded_iff_nndist is a dubious translation:
-lean 3 declaration is
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-but is expected to have type
-  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {s : Set.{u1} α}, Iff (Bornology.IsBounded.{u1} α (PseudoMetricSpace.toBornology.{u1} α _inst_1) s) (Exists.{1} NNReal (fun (C : NNReal) => forall {{x : α}}, (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) x s) -> (forall {{y : α}}, (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) y s) -> (LE.le.{0} NNReal (Preorder.toLE.{0} NNReal (PartialOrder.toPreorder.{0} NNReal (StrictOrderedSemiring.toPartialOrder.{0} NNReal instNNRealStrictOrderedSemiring))) (NNDist.nndist.{u1} α (PseudoMetricSpace.toNNDist.{u1} α _inst_1) x y) C))))
-Case conversion may be inaccurate. Consider using '#align metric.is_bounded_iff_nndist Metric.isBounded_iff_nndistₓ'. -/
 theorem isBounded_iff_nndist {s : Set α} :
     IsBounded s ↔ ∃ C : ℝ≥0, ∀ ⦃x⦄, x ∈ s → ∀ ⦃y⦄, y ∈ s → nndist x y ≤ C := by
   simp only [is_bounded_iff_exists_ge 0, NNReal.exists, ← NNReal.coe_le_coe, ← dist_nndist,
@@ -1304,12 +830,6 @@ theorem toUniformSpace_eq :
 #align metric.to_uniform_space_eq Metric.toUniformSpace_eq
 -/
 
-/- warning: metric.uniformity_basis_dist -> Metric.uniformity_basis_dist is a dubious translation:
-lean 3 declaration is
-  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α], Filter.HasBasis.{u1, 1} (Prod.{u1, u1} α α) Real (uniformity.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) (fun (ε : Real) => LT.lt.{0} Real Real.hasLt (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero))) ε) (fun (ε : Real) => setOf.{u1} (Prod.{u1, u1} α α) (fun (p : Prod.{u1, u1} α α) => LT.lt.{0} Real Real.hasLt (Dist.dist.{u1} α (PseudoMetricSpace.toHasDist.{u1} α _inst_1) (Prod.fst.{u1, u1} α α p) (Prod.snd.{u1, u1} α α p)) ε))
-but is expected to have type
-  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α], Filter.HasBasis.{u1, 1} (Prod.{u1, u1} α α) Real (uniformity.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) (fun (ε : Real) => LT.lt.{0} Real Real.instLTReal (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal)) ε) (fun (ε : Real) => setOf.{u1} (Prod.{u1, u1} α α) (fun (p : Prod.{u1, u1} α α) => LT.lt.{0} Real Real.instLTReal (Dist.dist.{u1} α (PseudoMetricSpace.toDist.{u1} α _inst_1) (Prod.fst.{u1, u1} α α p) (Prod.snd.{u1, u1} α α p)) ε))
-Case conversion may be inaccurate. Consider using '#align metric.uniformity_basis_dist Metric.uniformity_basis_distₓ'. -/
 theorem uniformity_basis_dist :
     (𝓤 α).HasBasis (fun ε : ℝ => 0 < ε) fun ε => { p : α × α | dist p.1 p.2 < ε } :=
   by
@@ -1317,12 +837,6 @@ theorem uniformity_basis_dist :
   exact UniformSpace.hasBasis_ofFun (exists_gt _) _ _ _ _ _
 #align metric.uniformity_basis_dist Metric.uniformity_basis_dist
 
-/- warning: metric.mk_uniformity_basis -> Metric.mk_uniformity_basis is a dubious translation:
-lean 3 declaration is
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-Case conversion may be inaccurate. Consider using '#align metric.mk_uniformity_basis Metric.mk_uniformity_basisₓ'. -/
 /-- Given `f : β → ℝ`, if `f` sends `{i | p i}` to a set of positive numbers
 accumulating to zero, then `f i`-neighborhoods of the diagonal form a basis of `𝓤 α`.
 
@@ -1340,12 +854,6 @@ protected theorem mk_uniformity_basis {β : Type _} {p : β → Prop} {f : β 
   · exact fun ⟨i, hi, H⟩ => ⟨f i, hf₀ i hi, H⟩
 #align metric.mk_uniformity_basis Metric.mk_uniformity_basis
 
-/- warning: metric.uniformity_basis_dist_rat -> Metric.uniformity_basis_dist_rat is a dubious translation:
-lean 3 declaration is
-  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α], Filter.HasBasis.{u1, 1} (Prod.{u1, u1} α α) Rat (uniformity.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) (fun (r : Rat) => LT.lt.{0} Rat Rat.hasLt (OfNat.ofNat.{0} Rat 0 (OfNat.mk.{0} Rat 0 (Zero.zero.{0} Rat Rat.hasZero))) r) (fun (r : Rat) => setOf.{u1} (Prod.{u1, u1} α α) (fun (p : Prod.{u1, u1} α α) => LT.lt.{0} Real Real.hasLt (Dist.dist.{u1} α (PseudoMetricSpace.toHasDist.{u1} α _inst_1) (Prod.fst.{u1, u1} α α p) (Prod.snd.{u1, u1} α α p)) ((fun (a : Type) (b : Type) [self : HasLiftT.{1, 1} a b] => self.0) Rat Real (HasLiftT.mk.{1, 1} Rat Real (CoeTCₓ.coe.{1, 1} Rat Real (Rat.castCoe.{0} Real Real.hasRatCast))) r)))
-but is expected to have type
-  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α], Filter.HasBasis.{u1, 1} (Prod.{u1, u1} α α) Rat (uniformity.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) (fun (r : Rat) => LT.lt.{0} Rat Rat.instLTRat_1 (OfNat.ofNat.{0} Rat 0 (Rat.instOfNatRat 0)) r) (fun (r : Rat) => setOf.{u1} (Prod.{u1, u1} α α) (fun (p : Prod.{u1, u1} α α) => LT.lt.{0} Real Real.instLTReal (Dist.dist.{u1} α (PseudoMetricSpace.toDist.{u1} α _inst_1) (Prod.fst.{u1, u1} α α p) (Prod.snd.{u1, u1} α α p)) (Rat.cast.{0} Real Real.ratCast r)))
-Case conversion may be inaccurate. Consider using '#align metric.uniformity_basis_dist_rat Metric.uniformity_basis_dist_ratₓ'. -/
 theorem uniformity_basis_dist_rat :
     (𝓤 α).HasBasis (fun r : ℚ => 0 < r) fun r => { p : α × α | dist p.1 p.2 < r } :=
   Metric.mk_uniformity_basis (fun _ => Rat.cast_pos.2) fun ε hε =>
@@ -1353,24 +861,12 @@ theorem uniformity_basis_dist_rat :
     ⟨r, Rat.cast_pos.1 hr0, hrε.le⟩
 #align metric.uniformity_basis_dist_rat Metric.uniformity_basis_dist_rat
 
-/- warning: metric.uniformity_basis_dist_inv_nat_succ -> Metric.uniformity_basis_dist_inv_nat_succ is a dubious translation:
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-Case conversion may be inaccurate. Consider using '#align metric.uniformity_basis_dist_inv_nat_succ Metric.uniformity_basis_dist_inv_nat_succₓ'. -/
 theorem uniformity_basis_dist_inv_nat_succ :
     (𝓤 α).HasBasis (fun _ => True) fun n : ℕ => { p : α × α | dist p.1 p.2 < 1 / (↑n + 1) } :=
   Metric.mk_uniformity_basis (fun n _ => div_pos zero_lt_one <| Nat.cast_add_one_pos n) fun ε ε0 =>
     (exists_nat_one_div_lt ε0).imp fun n hn => ⟨trivial, le_of_lt hn⟩
 #align metric.uniformity_basis_dist_inv_nat_succ Metric.uniformity_basis_dist_inv_nat_succ
 
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-Case conversion may be inaccurate. Consider using '#align metric.uniformity_basis_dist_inv_nat_pos Metric.uniformity_basis_dist_inv_nat_posₓ'. -/
 theorem uniformity_basis_dist_inv_nat_pos :
     (𝓤 α).HasBasis (fun n : ℕ => 0 < n) fun n : ℕ => { p : α × α | dist p.1 p.2 < 1 / ↑n } :=
   Metric.mk_uniformity_basis (fun n hn => div_pos zero_lt_one <| Nat.cast_pos.2 hn) fun ε ε0 =>
@@ -1378,12 +874,6 @@ theorem uniformity_basis_dist_inv_nat_pos :
     ⟨n + 1, Nat.succ_pos n, by exact_mod_cast hn.le⟩
 #align metric.uniformity_basis_dist_inv_nat_pos Metric.uniformity_basis_dist_inv_nat_pos
 
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-Case conversion may be inaccurate. Consider using '#align metric.uniformity_basis_dist_pow Metric.uniformity_basis_dist_powₓ'. -/
 theorem uniformity_basis_dist_pow {r : ℝ} (h0 : 0 < r) (h1 : r < 1) :
     (𝓤 α).HasBasis (fun n : ℕ => True) fun n : ℕ => { p : α × α | dist p.1 p.2 < r ^ n } :=
   Metric.mk_uniformity_basis (fun n hn => pow_pos h0 _) fun ε ε0 =>
@@ -1391,12 +881,6 @@ theorem uniformity_basis_dist_pow {r : ℝ} (h0 : 0 < r) (h1 : r < 1) :
     ⟨n, trivial, hn.le⟩
 #align metric.uniformity_basis_dist_pow Metric.uniformity_basis_dist_pow
 
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-Case conversion may be inaccurate. Consider using '#align metric.uniformity_basis_dist_lt Metric.uniformity_basis_dist_ltₓ'. -/
 theorem uniformity_basis_dist_lt {R : ℝ} (hR : 0 < R) :
     (𝓤 α).HasBasis (fun r : ℝ => 0 < r ∧ r < R) fun r => { p : α × α | dist p.1 p.2 < r } :=
   Metric.mk_uniformity_basis (fun r => And.left) fun r hr =>
@@ -1404,12 +888,6 @@ theorem uniformity_basis_dist_lt {R : ℝ} (hR : 0 < R) :
       min_le_left _ _⟩
 #align metric.uniformity_basis_dist_lt Metric.uniformity_basis_dist_lt
 
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-Case conversion may be inaccurate. Consider using '#align metric.mk_uniformity_basis_le Metric.mk_uniformity_basis_leₓ'. -/
 /-- Given `f : β → ℝ`, if `f` sends `{i | p i}` to a set of positive numbers
 accumulating to zero, then closed neighborhoods of the diagonal of sizes `{f i | p i}`
 form a basis of `𝓤 α`.
@@ -1429,12 +907,6 @@ protected theorem mk_uniformity_basis_le {β : Type _} {p : β → Prop} {f : β
   · exact fun ⟨i, hi, H⟩ => ⟨f i, hf₀ i hi, fun x (hx : _ < _) => H (le_of_lt hx)⟩
 #align metric.mk_uniformity_basis_le Metric.mk_uniformity_basis_le
 
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-Case conversion may be inaccurate. Consider using '#align metric.uniformity_basis_dist_le Metric.uniformity_basis_dist_leₓ'. -/
 /-- Contant size closed neighborhoods of the diagonal form a basis
 of the uniformity filter. -/
 theorem uniformity_basis_dist_le :
@@ -1442,12 +914,6 @@ theorem uniformity_basis_dist_le :
   Metric.mk_uniformity_basis_le (fun _ => id) fun ε ε₀ => ⟨ε, ε₀, le_refl ε⟩
 #align metric.uniformity_basis_dist_le Metric.uniformity_basis_dist_le
 
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-Case conversion may be inaccurate. Consider using '#align metric.uniformity_basis_dist_le_pow Metric.uniformity_basis_dist_le_powₓ'. -/
 theorem uniformity_basis_dist_le_pow {r : ℝ} (h0 : 0 < r) (h1 : r < 1) :
     (𝓤 α).HasBasis (fun n : ℕ => True) fun n : ℕ => { p : α × α | dist p.1 p.2 ≤ r ^ n } :=
   Metric.mk_uniformity_basis_le (fun n hn => pow_pos h0 _) fun ε ε0 =>
@@ -1455,45 +921,21 @@ theorem uniformity_basis_dist_le_pow {r : ℝ} (h0 : 0 < r) (h1 : r < 1) :
     ⟨n, trivial, hn.le⟩
 #align metric.uniformity_basis_dist_le_pow Metric.uniformity_basis_dist_le_pow
 
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-Case conversion may be inaccurate. Consider using '#align metric.mem_uniformity_dist Metric.mem_uniformity_distₓ'. -/
 theorem mem_uniformity_dist {s : Set (α × α)} :
     s ∈ 𝓤 α ↔ ∃ ε > 0, ∀ {a b : α}, dist a b < ε → (a, b) ∈ s :=
   uniformity_basis_dist.mem_uniformity_iff
 #align metric.mem_uniformity_dist Metric.mem_uniformity_dist
 
-/- warning: metric.dist_mem_uniformity -> Metric.dist_mem_uniformity is a dubious translation:
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-  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {ε : Real}, (LT.lt.{0} Real Real.hasLt (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero))) ε) -> (Membership.Mem.{u1, u1} (Set.{u1} (Prod.{u1, u1} α α)) (Filter.{u1} (Prod.{u1, u1} α α)) (Filter.hasMem.{u1} (Prod.{u1, u1} α α)) (setOf.{u1} (Prod.{u1, u1} α α) (fun (p : Prod.{u1, u1} α α) => LT.lt.{0} Real Real.hasLt (Dist.dist.{u1} α (PseudoMetricSpace.toHasDist.{u1} α _inst_1) (Prod.fst.{u1, u1} α α p) (Prod.snd.{u1, u1} α α p)) ε)) (uniformity.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)))
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-Case conversion may be inaccurate. Consider using '#align metric.dist_mem_uniformity Metric.dist_mem_uniformityₓ'. -/
 /-- A constant size neighborhood of the diagonal is an entourage. -/
 theorem dist_mem_uniformity {ε : ℝ} (ε0 : 0 < ε) : { p : α × α | dist p.1 p.2 < ε } ∈ 𝓤 α :=
   mem_uniformity_dist.2 ⟨ε, ε0, fun a b => id⟩
 #align metric.dist_mem_uniformity Metric.dist_mem_uniformity
 
-/- warning: metric.uniform_continuous_iff -> Metric.uniformContinuous_iff is a dubious translation:
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-Case conversion may be inaccurate. Consider using '#align metric.uniform_continuous_iff Metric.uniformContinuous_iffₓ'. -/
 theorem uniformContinuous_iff [PseudoMetricSpace β] {f : α → β} :
     UniformContinuous f ↔ ∀ ε > 0, ∃ δ > 0, ∀ {a b : α}, dist a b < δ → dist (f a) (f b) < ε :=
   uniformity_basis_dist.uniformContinuous_iff uniformity_basis_dist
 #align metric.uniform_continuous_iff Metric.uniformContinuous_iff
 
-/- warning: metric.uniform_continuous_on_iff -> Metric.uniformContinuousOn_iff is a dubious translation:
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-Case conversion may be inaccurate. Consider using '#align metric.uniform_continuous_on_iff Metric.uniformContinuousOn_iffₓ'. -/
 /- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (x y «expr ∈ » s) -/
 theorem uniformContinuousOn_iff [PseudoMetricSpace β] {f : α → β} {s : Set α} :
     UniformContinuousOn f s ↔
@@ -1501,12 +943,6 @@ theorem uniformContinuousOn_iff [PseudoMetricSpace β] {f : α → β} {s : Set
   Metric.uniformity_basis_dist.uniformContinuousOn_iff Metric.uniformity_basis_dist
 #align metric.uniform_continuous_on_iff Metric.uniformContinuousOn_iff
 
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-Case conversion may be inaccurate. Consider using '#align metric.uniform_continuous_on_iff_le Metric.uniformContinuousOn_iff_leₓ'. -/
 /- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (x y «expr ∈ » s) -/
 theorem uniformContinuousOn_iff_le [PseudoMetricSpace β] {f : α → β} {s : Set α} :
     UniformContinuousOn f s ↔
@@ -1514,12 +950,6 @@ theorem uniformContinuousOn_iff_le [PseudoMetricSpace β] {f : α → β} {s : S
   Metric.uniformity_basis_dist_le.uniformContinuousOn_iff Metric.uniformity_basis_dist_le
 #align metric.uniform_continuous_on_iff_le Metric.uniformContinuousOn_iff_le
 
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-Case conversion may be inaccurate. Consider using '#align metric.uniform_embedding_iff Metric.uniformEmbedding_iffₓ'. -/
 theorem uniformEmbedding_iff [PseudoMetricSpace β] {f : α → β} :
     UniformEmbedding f ↔
       Function.Injective f ∧
@@ -1529,12 +959,6 @@ theorem uniformEmbedding_iff [PseudoMetricSpace β] {f : α → β} :
   rfl
 #align metric.uniform_embedding_iff Metric.uniformEmbedding_iff
 
-/- warning: metric.controlled_of_uniform_embedding -> Metric.controlled_of_uniformEmbedding is a dubious translation:
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-Case conversion may be inaccurate. Consider using '#align metric.controlled_of_uniform_embedding Metric.controlled_of_uniformEmbeddingₓ'. -/
 /-- If a map between pseudometric spaces is a uniform embedding then the distance between `f x`
 and `f y` is controlled in terms of the distance between `x` and `y`. -/
 theorem controlled_of_uniformEmbedding [PseudoMetricSpace β] {f : α → β} :
@@ -1546,12 +970,6 @@ theorem controlled_of_uniformEmbedding [PseudoMetricSpace β] {f : α → β} :
   exact ⟨uniformContinuous_iff.1 (uniformEmbedding_iff.1 h).2.1, (uniformEmbedding_iff.1 h).2.2⟩
 #align metric.controlled_of_uniform_embedding Metric.controlled_of_uniformEmbedding
 
-/- warning: metric.totally_bounded_iff -> Metric.totallyBounded_iff is a dubious translation:
-lean 3 declaration is
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-Case conversion may be inaccurate. Consider using '#align metric.totally_bounded_iff Metric.totallyBounded_iffₓ'. -/
 theorem totallyBounded_iff {s : Set α} :
     TotallyBounded s ↔ ∀ ε > 0, ∃ t : Set α, t.Finite ∧ s ⊆ ⋃ y ∈ t, ball y ε :=
   ⟨fun H ε ε0 => H _ (dist_mem_uniformity ε0), fun H r ru =>
@@ -1560,9 +978,6 @@ theorem totallyBounded_iff {s : Set α} :
     ⟨t, ft, h.trans <| iUnion₂_mono fun y yt z => hε⟩⟩
 #align metric.totally_bounded_iff Metric.totallyBounded_iff
 
-/- warning: metric.totally_bounded_of_finite_discretization -> Metric.totallyBounded_of_finite_discretization is a dubious translation:
-<too large>
-Case conversion may be inaccurate. Consider using '#align metric.totally_bounded_of_finite_discretization Metric.totallyBounded_of_finite_discretizationₓ'. -/
 /-- A pseudometric space is totally bounded if one can reconstruct up to any ε>0 any element of the
 space from finitely many data. -/
 theorem totallyBounded_of_finite_discretization {s : Set α}
@@ -1585,12 +1000,6 @@ theorem totallyBounded_of_finite_discretization {s : Set α}
   exact ⟨_, ⟨F ⟨x, xs⟩, rfl⟩, hF _ _ this.symm⟩
 #align metric.totally_bounded_of_finite_discretization Metric.totallyBounded_of_finite_discretization
 
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 /- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (t «expr ⊆ » s) -/
 theorem finite_approx_of_totallyBounded {s : Set α} (hs : TotallyBounded s) :
     ∀ ε > 0, ∃ (t : _)(_ : t ⊆ s), Set.Finite t ∧ s ⊆ ⋃ y ∈ t, ball y ε :=
@@ -1600,12 +1009,6 @@ theorem finite_approx_of_totallyBounded {s : Set α} (hs : TotallyBounded s) :
   exact hs _ (dist_mem_uniformity ε_pos)
 #align metric.finite_approx_of_totally_bounded Metric.finite_approx_of_totallyBounded
 
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 /-- Expressing uniform convergence using `dist` -/
 theorem tendstoUniformlyOnFilter_iff {ι : Type _} {F : ι → β → α} {f : β → α} {p : Filter ι}
     {p' : Filter β} :
@@ -1617,12 +1020,6 @@ theorem tendstoUniformlyOnFilter_iff {ι : Type _} {F : ι → β → α} {f : 
   refine' (H ε εpos).mono fun n hn => hε hn
 #align metric.tendsto_uniformly_on_filter_iff Metric.tendstoUniformlyOnFilter_iff
 
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 /-- Expressing locally uniform convergence on a set using `dist`. -/
 theorem tendstoLocallyUniformlyOn_iff {ι : Type _} [TopologicalSpace β] {F : ι → β → α} {f : β → α}
     {p : Filter ι} {s : Set β} :
@@ -1635,12 +1032,6 @@ theorem tendstoLocallyUniformlyOn_iff {ι : Type _} [TopologicalSpace β] {F : 
   exact ⟨t, ht, Ht.mono fun n hs x hx => hε (hs x hx)⟩
 #align metric.tendsto_locally_uniformly_on_iff Metric.tendstoLocallyUniformlyOn_iff
 
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 /-- Expressing uniform convergence on a set using `dist`. -/
 theorem tendstoUniformlyOn_iff {ι : Type _} {F : ι → β → α} {f : β → α} {p : Filter ι} {s : Set β} :
     TendstoUniformlyOn F f p s ↔ ∀ ε > 0, ∀ᶠ n in p, ∀ x ∈ s, dist (f x) (F n x) < ε :=
@@ -1650,12 +1041,6 @@ theorem tendstoUniformlyOn_iff {ι : Type _} {F : ι → β → α} {f : β →
   exact (H ε εpos).mono fun n hs x hx => hε (hs x hx)
 #align metric.tendsto_uniformly_on_iff Metric.tendstoUniformlyOn_iff
 
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 /-- Expressing locally uniform convergence using `dist`. -/
 theorem tendstoLocallyUniformly_iff {ι : Type _} [TopologicalSpace β] {F : ι → β → α} {f : β → α}
     {p : Filter ι} :
@@ -1666,78 +1051,36 @@ theorem tendstoLocallyUniformly_iff {ι : Type _} [TopologicalSpace β] {F : ι
     mem_univ, forall_const, exists_prop]
 #align metric.tendsto_locally_uniformly_iff Metric.tendstoLocallyUniformly_iff
 
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 /-- Expressing uniform convergence using `dist`. -/
 theorem tendstoUniformly_iff {ι : Type _} {F : ι → β → α} {f : β → α} {p : Filter ι} :
     TendstoUniformly F f p ↔ ∀ ε > 0, ∀ᶠ n in p, ∀ x, dist (f x) (F n x) < ε := by
   rw [← tendstoUniformlyOn_univ, tendsto_uniformly_on_iff]; simp
 #align metric.tendsto_uniformly_iff Metric.tendstoUniformly_iff
 
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 /- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (x y «expr ∈ » t) -/
 protected theorem cauchy_iff {f : Filter α} :
     Cauchy f ↔ NeBot f ∧ ∀ ε > 0, ∃ t ∈ f, ∀ (x) (_ : x ∈ t) (y) (_ : y ∈ t), dist x y < ε :=
   uniformity_basis_dist.cauchy_iff
 #align metric.cauchy_iff Metric.cauchy_iff
 
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 theorem nhds_basis_ball : (𝓝 x).HasBasis (fun ε : ℝ => 0 < ε) (ball x) :=
   nhds_basis_uniformity uniformity_basis_dist
 #align metric.nhds_basis_ball Metric.nhds_basis_ball
 
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-Case conversion may be inaccurate. Consider using '#align metric.mem_nhds_iff Metric.mem_nhds_iffₓ'. -/
 theorem mem_nhds_iff : s ∈ 𝓝 x ↔ ∃ ε > 0, ball x ε ⊆ s :=
   nhds_basis_ball.mem_iff
 #align metric.mem_nhds_iff Metric.mem_nhds_iff
 
-/- warning: metric.eventually_nhds_iff -> Metric.eventually_nhds_iff is a dubious translation:
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-Case conversion may be inaccurate. Consider using '#align metric.eventually_nhds_iff Metric.eventually_nhds_iffₓ'. -/
 theorem eventually_nhds_iff {p : α → Prop} :
     (∀ᶠ y in 𝓝 x, p y) ↔ ∃ ε > 0, ∀ ⦃y⦄, dist y x < ε → p y :=
   mem_nhds_iff
 #align metric.eventually_nhds_iff Metric.eventually_nhds_iff
 
-/- warning: metric.eventually_nhds_iff_ball -> Metric.eventually_nhds_iff_ball is a dubious translation:
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-Case conversion may be inaccurate. Consider using '#align metric.eventually_nhds_iff_ball Metric.eventually_nhds_iff_ballₓ'. -/
 theorem eventually_nhds_iff_ball {p : α → Prop} :
     (∀ᶠ y in 𝓝 x, p y) ↔ ∃ ε > 0, ∀ y ∈ ball x ε, p y :=
   mem_nhds_iff
 #align metric.eventually_nhds_iff_ball Metric.eventually_nhds_iff_ball
 
-/- warning: metric.eventually_prod_nhds_iff -> Metric.eventually_prod_nhds_iff is a dubious translation:
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-Case conversion may be inaccurate. Consider using '#align metric.eventually_prod_nhds_iff Metric.eventually_prod_nhds_iffₓ'. -/
 /-- A version of `filter.eventually_prod_iff` where the second filter consists of neighborhoods
 in a pseudo-metric space.-/
 theorem eventually_prod_nhds_iff {f : Filter ι} {x₀ : α} {p : ι × α → Prop} :
@@ -1752,12 +1095,6 @@ theorem eventually_prod_nhds_iff {f : Filter ι} {x₀ : α} {p : ι × α → P
   · rintro ⟨ε, hε, hp⟩; exact ⟨fun x => dist x x₀ < ε, ⟨ε, hε, fun y => id⟩, @hp⟩
 #align metric.eventually_prod_nhds_iff Metric.eventually_prod_nhds_iff
 
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 /-- A version of `filter.eventually_prod_iff` where the first filter consists of neighborhoods
 in a pseudo-metric space.-/
 theorem eventually_nhds_prod_iff {ι α} [PseudoMetricSpace α] {f : Filter ι} {x₀ : α}
@@ -1771,66 +1108,30 @@ theorem eventually_nhds_prod_iff {ι α} [PseudoMetricSpace α] {f : Filter ι}
     · rintro ⟨a1, a2, a3, a4, a5⟩; refine' ⟨a3, a4, a1, a2, fun b1 b2 b3 b4 => a5 b4 b2⟩
 #align metric.eventually_nhds_prod_iff Metric.eventually_nhds_prod_iff
 
-/- warning: metric.nhds_basis_closed_ball -> Metric.nhds_basis_closedBall is a dubious translation:
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-Case conversion may be inaccurate. Consider using '#align metric.nhds_basis_closed_ball Metric.nhds_basis_closedBallₓ'. -/
 theorem nhds_basis_closedBall : (𝓝 x).HasBasis (fun ε : ℝ => 0 < ε) (closedBall x) :=
   nhds_basis_uniformity uniformity_basis_dist_le
 #align metric.nhds_basis_closed_ball Metric.nhds_basis_closedBall
 
-/- warning: metric.nhds_basis_ball_inv_nat_succ -> Metric.nhds_basis_ball_inv_nat_succ is a dubious translation:
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-Case conversion may be inaccurate. Consider using '#align metric.nhds_basis_ball_inv_nat_succ Metric.nhds_basis_ball_inv_nat_succₓ'. -/
 theorem nhds_basis_ball_inv_nat_succ :
     (𝓝 x).HasBasis (fun _ => True) fun n : ℕ => ball x (1 / (↑n + 1)) :=
   nhds_basis_uniformity uniformity_basis_dist_inv_nat_succ
 #align metric.nhds_basis_ball_inv_nat_succ Metric.nhds_basis_ball_inv_nat_succ
 
-/- warning: metric.nhds_basis_ball_inv_nat_pos -> Metric.nhds_basis_ball_inv_nat_pos is a dubious translation:
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-Case conversion may be inaccurate. Consider using '#align metric.nhds_basis_ball_inv_nat_pos Metric.nhds_basis_ball_inv_nat_posₓ'. -/
 theorem nhds_basis_ball_inv_nat_pos :
     (𝓝 x).HasBasis (fun n => 0 < n) fun n : ℕ => ball x (1 / ↑n) :=
   nhds_basis_uniformity uniformity_basis_dist_inv_nat_pos
 #align metric.nhds_basis_ball_inv_nat_pos Metric.nhds_basis_ball_inv_nat_pos
 
-/- warning: metric.nhds_basis_ball_pow -> Metric.nhds_basis_ball_pow is a dubious translation:
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-Case conversion may be inaccurate. Consider using '#align metric.nhds_basis_ball_pow Metric.nhds_basis_ball_powₓ'. -/
 theorem nhds_basis_ball_pow {r : ℝ} (h0 : 0 < r) (h1 : r < 1) :
     (𝓝 x).HasBasis (fun n => True) fun n : ℕ => ball x (r ^ n) :=
   nhds_basis_uniformity (uniformity_basis_dist_pow h0 h1)
 #align metric.nhds_basis_ball_pow Metric.nhds_basis_ball_pow
 
-/- warning: metric.nhds_basis_closed_ball_pow -> Metric.nhds_basis_closedBall_pow is a dubious translation:
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-  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {x : α} {r : Real}, (LT.lt.{0} Real Real.instLTReal (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal)) r) -> (LT.lt.{0} Real Real.instLTReal r (OfNat.ofNat.{0} Real 1 (One.toOfNat1.{0} Real Real.instOneReal))) -> (Filter.HasBasis.{u1, 1} α Nat (nhds.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) x) (fun (n : Nat) => True) (fun (n : Nat) => Metric.closedBall.{u1} α _inst_1 x (HPow.hPow.{0, 0, 0} Real Nat Real (instHPow.{0, 0} Real Nat (Monoid.Pow.{0} Real Real.instMonoidReal)) r n)))
-Case conversion may be inaccurate. Consider using '#align metric.nhds_basis_closed_ball_pow Metric.nhds_basis_closedBall_powₓ'. -/
 theorem nhds_basis_closedBall_pow {r : ℝ} (h0 : 0 < r) (h1 : r < 1) :
     (𝓝 x).HasBasis (fun n => True) fun n : ℕ => closedBall x (r ^ n) :=
   nhds_basis_uniformity (uniformity_basis_dist_le_pow h0 h1)
 #align metric.nhds_basis_closed_ball_pow Metric.nhds_basis_closedBall_pow
 
-/- warning: metric.is_open_iff -> Metric.isOpen_iff is a dubious translation:
-lean 3 declaration is
-  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {s : Set.{u1} α}, Iff (IsOpen.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) s) (forall (x : α), (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) x s) -> (Exists.{1} Real (fun (ε : Real) => Exists.{0} (GT.gt.{0} Real Real.hasLt ε (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero)))) (fun (H : GT.gt.{0} Real Real.hasLt ε (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero)))) => HasSubset.Subset.{u1} (Set.{u1} α) (Set.hasSubset.{u1} α) (Metric.ball.{u1} α _inst_1 x ε) s))))
-but is expected to have type
-  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {s : Set.{u1} α}, Iff (IsOpen.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) s) (forall (x : α), (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) x s) -> (Exists.{1} Real (fun (ε : Real) => And (GT.gt.{0} Real Real.instLTReal ε (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal))) (HasSubset.Subset.{u1} (Set.{u1} α) (Set.instHasSubsetSet.{u1} α) (Metric.ball.{u1} α _inst_1 x ε) s))))
-Case conversion may be inaccurate. Consider using '#align metric.is_open_iff Metric.isOpen_iffₓ'. -/
 theorem isOpen_iff : IsOpen s ↔ ∀ x ∈ s, ∃ ε > 0, ball x ε ⊆ s := by
   simp only [isOpen_iff_mem_nhds, mem_nhds_iff]
 #align metric.is_open_iff Metric.isOpen_iff
@@ -1841,22 +1142,10 @@ theorem isOpen_ball : IsOpen (ball x ε) :=
 #align metric.is_open_ball Metric.isOpen_ball
 -/
 
-/- warning: metric.ball_mem_nhds -> Metric.ball_mem_nhds is a dubious translation:
-lean 3 declaration is
-  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] (x : α) {ε : Real}, (LT.lt.{0} Real Real.hasLt (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero))) ε) -> (Membership.Mem.{u1, u1} (Set.{u1} α) (Filter.{u1} α) (Filter.hasMem.{u1} α) (Metric.ball.{u1} α _inst_1 x ε) (nhds.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) x))
-but is expected to have type
-  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] (x : α) {ε : Real}, (LT.lt.{0} Real Real.instLTReal (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal)) ε) -> (Membership.mem.{u1, u1} (Set.{u1} α) (Filter.{u1} α) (instMembershipSetFilter.{u1} α) (Metric.ball.{u1} α _inst_1 x ε) (nhds.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) x))
-Case conversion may be inaccurate. Consider using '#align metric.ball_mem_nhds Metric.ball_mem_nhdsₓ'. -/
 theorem ball_mem_nhds (x : α) {ε : ℝ} (ε0 : 0 < ε) : ball x ε ∈ 𝓝 x :=
   isOpen_ball.mem_nhds (mem_ball_self ε0)
 #align metric.ball_mem_nhds Metric.ball_mem_nhds
 
-/- warning: metric.closed_ball_mem_nhds -> Metric.closedBall_mem_nhds is a dubious translation:
-lean 3 declaration is
-  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] (x : α) {ε : Real}, (LT.lt.{0} Real Real.hasLt (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero))) ε) -> (Membership.Mem.{u1, u1} (Set.{u1} α) (Filter.{u1} α) (Filter.hasMem.{u1} α) (Metric.closedBall.{u1} α _inst_1 x ε) (nhds.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) x))
-but is expected to have type
-  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] (x : α) {ε : Real}, (LT.lt.{0} Real Real.instLTReal (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal)) ε) -> (Membership.mem.{u1, u1} (Set.{u1} α) (Filter.{u1} α) (instMembershipSetFilter.{u1} α) (Metric.closedBall.{u1} α _inst_1 x ε) (nhds.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) x))
-Case conversion may be inaccurate. Consider using '#align metric.closed_ball_mem_nhds Metric.closedBall_mem_nhdsₓ'. -/
 theorem closedBall_mem_nhds (x : α) {ε : ℝ} (ε0 : 0 < ε) : closedBall x ε ∈ 𝓝 x :=
   mem_of_superset (ball_mem_nhds x ε0) ball_subset_closedBall
 #align metric.closed_ball_mem_nhds Metric.closedBall_mem_nhds
@@ -1867,33 +1156,15 @@ theorem closedBall_mem_nhds_of_mem {x c : α} {ε : ℝ} (h : x ∈ ball c ε) :
 #align metric.closed_ball_mem_nhds_of_mem Metric.closedBall_mem_nhds_of_mem
 -/
 
-/- warning: metric.nhds_within_basis_ball -> Metric.nhdsWithin_basis_ball is a dubious translation:
-lean 3 declaration is
-  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {x : α} {s : Set.{u1} α}, Filter.HasBasis.{u1, 1} α Real (nhdsWithin.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) x s) (fun (ε : Real) => LT.lt.{0} Real Real.hasLt (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero))) ε) (fun (ε : Real) => Inter.inter.{u1} (Set.{u1} α) (Set.hasInter.{u1} α) (Metric.ball.{u1} α _inst_1 x ε) s)
-but is expected to have type
-  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {x : α} {s : Set.{u1} α}, Filter.HasBasis.{u1, 1} α Real (nhdsWithin.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) x s) (fun (ε : Real) => LT.lt.{0} Real Real.instLTReal (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal)) ε) (fun (ε : Real) => Inter.inter.{u1} (Set.{u1} α) (Set.instInterSet.{u1} α) (Metric.ball.{u1} α _inst_1 x ε) s)
-Case conversion may be inaccurate. Consider using '#align metric.nhds_within_basis_ball Metric.nhdsWithin_basis_ballₓ'. -/
 theorem nhdsWithin_basis_ball {s : Set α} :
     (𝓝[s] x).HasBasis (fun ε : ℝ => 0 < ε) fun ε => ball x ε ∩ s :=
   nhdsWithin_hasBasis nhds_basis_ball s
 #align metric.nhds_within_basis_ball Metric.nhdsWithin_basis_ball
 
-/- warning: metric.mem_nhds_within_iff -> Metric.mem_nhdsWithin_iff is a dubious translation:
-lean 3 declaration is
-  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {x : α} {s : Set.{u1} α} {t : Set.{u1} α}, Iff (Membership.Mem.{u1, u1} (Set.{u1} α) (Filter.{u1} α) (Filter.hasMem.{u1} α) s (nhdsWithin.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) x t)) (Exists.{1} Real (fun (ε : Real) => Exists.{0} (GT.gt.{0} Real Real.hasLt ε (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero)))) (fun (H : GT.gt.{0} Real Real.hasLt ε (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero)))) => HasSubset.Subset.{u1} (Set.{u1} α) (Set.hasSubset.{u1} α) (Inter.inter.{u1} (Set.{u1} α) (Set.hasInter.{u1} α) (Metric.ball.{u1} α _inst_1 x ε) t) s)))
-but is expected to have type
-  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {x : α} {s : Set.{u1} α} {t : Set.{u1} α}, Iff (Membership.mem.{u1, u1} (Set.{u1} α) (Filter.{u1} α) (instMembershipSetFilter.{u1} α) s (nhdsWithin.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) x t)) (Exists.{1} Real (fun (ε : Real) => And (GT.gt.{0} Real Real.instLTReal ε (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal))) (HasSubset.Subset.{u1} (Set.{u1} α) (Set.instHasSubsetSet.{u1} α) (Inter.inter.{u1} (Set.{u1} α) (Set.instInterSet.{u1} α) (Metric.ball.{u1} α _inst_1 x ε) t) s)))
-Case conversion may be inaccurate. Consider using '#align metric.mem_nhds_within_iff Metric.mem_nhdsWithin_iffₓ'. -/
 theorem mem_nhdsWithin_iff {t : Set α} : s ∈ 𝓝[t] x ↔ ∃ ε > 0, ball x ε ∩ t ⊆ s :=
   nhdsWithin_basis_ball.mem_iff
 #align metric.mem_nhds_within_iff Metric.mem_nhdsWithin_iff
 
-/- warning: metric.tendsto_nhds_within_nhds_within -> Metric.tendsto_nhdsWithin_nhdsWithin is a dubious translation:
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-  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : PseudoMetricSpace.{u1} α] {s : Set.{u1} α} [_inst_2 : PseudoMetricSpace.{u2} β] {t : Set.{u2} β} {f : α -> β} {a : α} {b : β}, Iff (Filter.Tendsto.{u1, u2} α β f (nhdsWithin.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) a s) (nhdsWithin.{u2} β (UniformSpace.toTopologicalSpace.{u2} β (PseudoMetricSpace.toUniformSpace.{u2} β _inst_2)) b t)) (forall (ε : Real), (GT.gt.{0} Real Real.instLTReal ε (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal))) -> (Exists.{1} Real (fun (δ : Real) => And (GT.gt.{0} Real Real.instLTReal δ (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal))) (forall {x : α}, (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) x s) -> (LT.lt.{0} Real Real.instLTReal (Dist.dist.{u1} α (PseudoMetricSpace.toDist.{u1} α _inst_1) x a) δ) -> (And (Membership.mem.{u2, u2} β (Set.{u2} β) (Set.instMembershipSet.{u2} β) (f x) t) (LT.lt.{0} Real Real.instLTReal (Dist.dist.{u2} β (PseudoMetricSpace.toDist.{u2} β _inst_2) (f x) b) ε))))))
-Case conversion may be inaccurate. Consider using '#align metric.tendsto_nhds_within_nhds_within Metric.tendsto_nhdsWithin_nhdsWithinₓ'. -/
 theorem tendsto_nhdsWithin_nhdsWithin [PseudoMetricSpace β] {t : Set β} {f : α → β} {a b} :
     Tendsto f (𝓝[s] a) (𝓝[t] b) ↔
       ∀ ε > 0, ∃ δ > 0, ∀ {x : α}, x ∈ s → dist x a < δ → f x ∈ t ∧ dist (f x) b < ε :=
@@ -1901,12 +1172,6 @@ theorem tendsto_nhdsWithin_nhdsWithin [PseudoMetricSpace β] {t : Set β} {f : 
     forall₂_congr fun ε hε => exists₂_congr fun δ hδ => forall_congr' fun x => by simp <;> itauto
 #align metric.tendsto_nhds_within_nhds_within Metric.tendsto_nhdsWithin_nhdsWithin
 
-/- warning: metric.tendsto_nhds_within_nhds -> Metric.tendsto_nhdsWithin_nhds is a dubious translation:
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-Case conversion may be inaccurate. Consider using '#align metric.tendsto_nhds_within_nhds Metric.tendsto_nhdsWithin_nhdsₓ'. -/
 theorem tendsto_nhdsWithin_nhds [PseudoMetricSpace β] {f : α → β} {a b} :
     Tendsto f (𝓝[s] a) (𝓝 b) ↔
       ∀ ε > 0, ∃ δ > 0, ∀ {x : α}, x ∈ s → dist x a < δ → dist (f x) b < ε :=
@@ -1915,134 +1180,62 @@ theorem tendsto_nhdsWithin_nhds [PseudoMetricSpace β] {f : α → β} {a b} :
   simp only [mem_univ, true_and_iff]
 #align metric.tendsto_nhds_within_nhds Metric.tendsto_nhdsWithin_nhds
 
-/- warning: metric.tendsto_nhds_nhds -> Metric.tendsto_nhds_nhds is a dubious translation:
-lean 3 declaration is
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-but is expected to have type
-  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : PseudoMetricSpace.{u1} α] [_inst_2 : PseudoMetricSpace.{u2} β] {f : α -> β} {a : α} {b : β}, Iff (Filter.Tendsto.{u1, u2} α β f (nhds.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) a) (nhds.{u2} β (UniformSpace.toTopologicalSpace.{u2} β (PseudoMetricSpace.toUniformSpace.{u2} β _inst_2)) b)) (forall (ε : Real), (GT.gt.{0} Real Real.instLTReal ε (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal))) -> (Exists.{1} Real (fun (δ : Real) => And (GT.gt.{0} Real Real.instLTReal δ (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal))) (forall {x : α}, (LT.lt.{0} Real Real.instLTReal (Dist.dist.{u1} α (PseudoMetricSpace.toDist.{u1} α _inst_1) x a) δ) -> (LT.lt.{0} Real Real.instLTReal (Dist.dist.{u2} β (PseudoMetricSpace.toDist.{u2} β _inst_2) (f x) b) ε)))))
-Case conversion may be inaccurate. Consider using '#align metric.tendsto_nhds_nhds Metric.tendsto_nhds_nhdsₓ'. -/
 theorem tendsto_nhds_nhds [PseudoMetricSpace β] {f : α → β} {a b} :
     Tendsto f (𝓝 a) (𝓝 b) ↔ ∀ ε > 0, ∃ δ > 0, ∀ {x : α}, dist x a < δ → dist (f x) b < ε :=
   nhds_basis_ball.tendsto_iffₓ nhds_basis_ball
 #align metric.tendsto_nhds_nhds Metric.tendsto_nhds_nhds
 
-/- warning: metric.continuous_at_iff -> Metric.continuousAt_iff is a dubious translation:
-lean 3 declaration is
-  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : PseudoMetricSpace.{u1} α] [_inst_2 : PseudoMetricSpace.{u2} β] {f : α -> β} {a : α}, Iff (ContinuousAt.{u1, u2} α β (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) (UniformSpace.toTopologicalSpace.{u2} β (PseudoMetricSpace.toUniformSpace.{u2} β _inst_2)) f a) (forall (ε : Real), (GT.gt.{0} Real Real.hasLt ε (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero)))) -> (Exists.{1} Real (fun (δ : Real) => Exists.{0} (GT.gt.{0} Real Real.hasLt δ (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero)))) (fun (H : GT.gt.{0} Real Real.hasLt δ (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero)))) => forall {x : α}, (LT.lt.{0} Real Real.hasLt (Dist.dist.{u1} α (PseudoMetricSpace.toHasDist.{u1} α _inst_1) x a) δ) -> (LT.lt.{0} Real Real.hasLt (Dist.dist.{u2} β (PseudoMetricSpace.toHasDist.{u2} β _inst_2) (f x) (f a)) ε)))))
-but is expected to have type
-  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : PseudoMetricSpace.{u1} α] [_inst_2 : PseudoMetricSpace.{u2} β] {f : α -> β} {a : α}, Iff (ContinuousAt.{u1, u2} α β (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) (UniformSpace.toTopologicalSpace.{u2} β (PseudoMetricSpace.toUniformSpace.{u2} β _inst_2)) f a) (forall (ε : Real), (GT.gt.{0} Real Real.instLTReal ε (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal))) -> (Exists.{1} Real (fun (δ : Real) => And (GT.gt.{0} Real Real.instLTReal δ (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal))) (forall {x : α}, (LT.lt.{0} Real Real.instLTReal (Dist.dist.{u1} α (PseudoMetricSpace.toDist.{u1} α _inst_1) x a) δ) -> (LT.lt.{0} Real Real.instLTReal (Dist.dist.{u2} β (PseudoMetricSpace.toDist.{u2} β _inst_2) (f x) (f a)) ε)))))
-Case conversion may be inaccurate. Consider using '#align metric.continuous_at_iff Metric.continuousAt_iffₓ'. -/
 theorem continuousAt_iff [PseudoMetricSpace β] {f : α → β} {a : α} :
     ContinuousAt f a ↔ ∀ ε > 0, ∃ δ > 0, ∀ {x : α}, dist x a < δ → dist (f x) (f a) < ε := by
   rw [ContinuousAt, tendsto_nhds_nhds]
 #align metric.continuous_at_iff Metric.continuousAt_iff
 
-/- warning: metric.continuous_within_at_iff -> Metric.continuousWithinAt_iff is a dubious translation:
-lean 3 declaration is
-  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : PseudoMetricSpace.{u1} α] [_inst_2 : PseudoMetricSpace.{u2} β] {f : α -> β} {a : α} {s : Set.{u1} α}, Iff (ContinuousWithinAt.{u1, u2} α β (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) (UniformSpace.toTopologicalSpace.{u2} β (PseudoMetricSpace.toUniformSpace.{u2} β _inst_2)) f s a) (forall (ε : Real), (GT.gt.{0} Real Real.hasLt ε (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero)))) -> (Exists.{1} Real (fun (δ : Real) => Exists.{0} (GT.gt.{0} Real Real.hasLt δ (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero)))) (fun (H : GT.gt.{0} Real Real.hasLt δ (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero)))) => forall {x : α}, (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) x s) -> (LT.lt.{0} Real Real.hasLt (Dist.dist.{u1} α (PseudoMetricSpace.toHasDist.{u1} α _inst_1) x a) δ) -> (LT.lt.{0} Real Real.hasLt (Dist.dist.{u2} β (PseudoMetricSpace.toHasDist.{u2} β _inst_2) (f x) (f a)) ε)))))
-but is expected to have type
-  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : PseudoMetricSpace.{u1} α] [_inst_2 : PseudoMetricSpace.{u2} β] {f : α -> β} {a : α} {s : Set.{u1} α}, Iff (ContinuousWithinAt.{u1, u2} α β (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) (UniformSpace.toTopologicalSpace.{u2} β (PseudoMetricSpace.toUniformSpace.{u2} β _inst_2)) f s a) (forall (ε : Real), (GT.gt.{0} Real Real.instLTReal ε (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal))) -> (Exists.{1} Real (fun (δ : Real) => And (GT.gt.{0} Real Real.instLTReal δ (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal))) (forall {x : α}, (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) x s) -> (LT.lt.{0} Real Real.instLTReal (Dist.dist.{u1} α (PseudoMetricSpace.toDist.{u1} α _inst_1) x a) δ) -> (LT.lt.{0} Real Real.instLTReal (Dist.dist.{u2} β (PseudoMetricSpace.toDist.{u2} β _inst_2) (f x) (f a)) ε)))))
-Case conversion may be inaccurate. Consider using '#align metric.continuous_within_at_iff Metric.continuousWithinAt_iffₓ'. -/
 theorem continuousWithinAt_iff [PseudoMetricSpace β] {f : α → β} {a : α} {s : Set α} :
     ContinuousWithinAt f s a ↔
       ∀ ε > 0, ∃ δ > 0, ∀ {x : α}, x ∈ s → dist x a < δ → dist (f x) (f a) < ε :=
   by rw [ContinuousWithinAt, tendsto_nhds_within_nhds]
 #align metric.continuous_within_at_iff Metric.continuousWithinAt_iff
 
-/- warning: metric.continuous_on_iff -> Metric.continuousOn_iff is a dubious translation:
-lean 3 declaration is
-  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : PseudoMetricSpace.{u1} α] [_inst_2 : PseudoMetricSpace.{u2} β] {f : α -> β} {s : Set.{u1} α}, Iff (ContinuousOn.{u1, u2} α β (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) (UniformSpace.toTopologicalSpace.{u2} β (PseudoMetricSpace.toUniformSpace.{u2} β _inst_2)) f s) (forall (b : α), (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) b s) -> (forall (ε : Real), (GT.gt.{0} Real Real.hasLt ε (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero)))) -> (Exists.{1} Real (fun (δ : Real) => Exists.{0} (GT.gt.{0} Real Real.hasLt δ (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero)))) (fun (H : GT.gt.{0} Real Real.hasLt δ (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero)))) => forall (a : α), (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) a s) -> (LT.lt.{0} Real Real.hasLt (Dist.dist.{u1} α (PseudoMetricSpace.toHasDist.{u1} α _inst_1) a b) δ) -> (LT.lt.{0} Real Real.hasLt (Dist.dist.{u2} β (PseudoMetricSpace.toHasDist.{u2} β _inst_2) (f a) (f b)) ε))))))
-but is expected to have type
-  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : PseudoMetricSpace.{u1} α] [_inst_2 : PseudoMetricSpace.{u2} β] {f : α -> β} {s : Set.{u1} α}, Iff (ContinuousOn.{u1, u2} α β (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) (UniformSpace.toTopologicalSpace.{u2} β (PseudoMetricSpace.toUniformSpace.{u2} β _inst_2)) f s) (forall (b : α), (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) b s) -> (forall (ε : Real), (GT.gt.{0} Real Real.instLTReal ε (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal))) -> (Exists.{1} Real (fun (δ : Real) => And (GT.gt.{0} Real Real.instLTReal δ (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal))) (forall (a : α), (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) a s) -> (LT.lt.{0} Real Real.instLTReal (Dist.dist.{u1} α (PseudoMetricSpace.toDist.{u1} α _inst_1) a b) δ) -> (LT.lt.{0} Real Real.instLTReal (Dist.dist.{u2} β (PseudoMetricSpace.toDist.{u2} β _inst_2) (f a) (f b)) ε))))))
-Case conversion may be inaccurate. Consider using '#align metric.continuous_on_iff Metric.continuousOn_iffₓ'. -/
 theorem continuousOn_iff [PseudoMetricSpace β] {f : α → β} {s : Set α} :
     ContinuousOn f s ↔ ∀ b ∈ s, ∀ ε > 0, ∃ δ > 0, ∀ a ∈ s, dist a b < δ → dist (f a) (f b) < ε := by
   simp [ContinuousOn, continuous_within_at_iff]
 #align metric.continuous_on_iff Metric.continuousOn_iff
 
-/- warning: metric.continuous_iff -> Metric.continuous_iff is a dubious translation:
-lean 3 declaration is
-  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : PseudoMetricSpace.{u1} α] [_inst_2 : PseudoMetricSpace.{u2} β] {f : α -> β}, Iff (Continuous.{u1, u2} α β (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) (UniformSpace.toTopologicalSpace.{u2} β (PseudoMetricSpace.toUniformSpace.{u2} β _inst_2)) f) (forall (b : α) (ε : Real), (GT.gt.{0} Real Real.hasLt ε (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero)))) -> (Exists.{1} Real (fun (δ : Real) => Exists.{0} (GT.gt.{0} Real Real.hasLt δ (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero)))) (fun (H : GT.gt.{0} Real Real.hasLt δ (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero)))) => forall (a : α), (LT.lt.{0} Real Real.hasLt (Dist.dist.{u1} α (PseudoMetricSpace.toHasDist.{u1} α _inst_1) a b) δ) -> (LT.lt.{0} Real Real.hasLt (Dist.dist.{u2} β (PseudoMetricSpace.toHasDist.{u2} β _inst_2) (f a) (f b)) ε)))))
-but is expected to have type
-  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : PseudoMetricSpace.{u1} α] [_inst_2 : PseudoMetricSpace.{u2} β] {f : α -> β}, Iff (Continuous.{u1, u2} α β (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) (UniformSpace.toTopologicalSpace.{u2} β (PseudoMetricSpace.toUniformSpace.{u2} β _inst_2)) f) (forall (b : α) (ε : Real), (GT.gt.{0} Real Real.instLTReal ε (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal))) -> (Exists.{1} Real (fun (δ : Real) => And (GT.gt.{0} Real Real.instLTReal δ (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal))) (forall (a : α), (LT.lt.{0} Real Real.instLTReal (Dist.dist.{u1} α (PseudoMetricSpace.toDist.{u1} α _inst_1) a b) δ) -> (LT.lt.{0} Real Real.instLTReal (Dist.dist.{u2} β (PseudoMetricSpace.toDist.{u2} β _inst_2) (f a) (f b)) ε)))))
-Case conversion may be inaccurate. Consider using '#align metric.continuous_iff Metric.continuous_iffₓ'. -/
 theorem continuous_iff [PseudoMetricSpace β] {f : α → β} :
     Continuous f ↔ ∀ (b), ∀ ε > 0, ∃ δ > 0, ∀ a, dist a b < δ → dist (f a) (f b) < ε :=
   continuous_iff_continuousAt.trans <| forall_congr' fun b => tendsto_nhds_nhds
 #align metric.continuous_iff Metric.continuous_iff
 
-/- warning: metric.tendsto_nhds -> Metric.tendsto_nhds is a dubious translation:
-lean 3 declaration is
-  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : PseudoMetricSpace.{u1} α] {f : Filter.{u2} β} {u : β -> α} {a : α}, Iff (Filter.Tendsto.{u2, u1} β α u f (nhds.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) a)) (forall (ε : Real), (GT.gt.{0} Real Real.hasLt ε (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero)))) -> (Filter.Eventually.{u2} β (fun (x : β) => LT.lt.{0} Real Real.hasLt (Dist.dist.{u1} α (PseudoMetricSpace.toHasDist.{u1} α _inst_1) (u x) a) ε) f))
-but is expected to have type
-  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : PseudoMetricSpace.{u1} α] {f : Filter.{u2} β} {u : β -> α} {a : α}, Iff (Filter.Tendsto.{u2, u1} β α u f (nhds.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) a)) (forall (ε : Real), (GT.gt.{0} Real Real.instLTReal ε (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal))) -> (Filter.Eventually.{u2} β (fun (x : β) => LT.lt.{0} Real Real.instLTReal (Dist.dist.{u1} α (PseudoMetricSpace.toDist.{u1} α _inst_1) (u x) a) ε) f))
-Case conversion may be inaccurate. Consider using '#align metric.tendsto_nhds Metric.tendsto_nhdsₓ'. -/
 theorem tendsto_nhds {f : Filter β} {u : β → α} {a : α} :
     Tendsto u f (𝓝 a) ↔ ∀ ε > 0, ∀ᶠ x in f, dist (u x) a < ε :=
   nhds_basis_ball.tendsto_right_iff
 #align metric.tendsto_nhds Metric.tendsto_nhds
 
-/- warning: metric.continuous_at_iff' -> Metric.continuousAt_iff' is a dubious translation:
-lean 3 declaration is
-  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : PseudoMetricSpace.{u1} α] [_inst_2 : TopologicalSpace.{u2} β] {f : β -> α} {b : β}, Iff (ContinuousAt.{u2, u1} β α _inst_2 (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) f b) (forall (ε : Real), (GT.gt.{0} Real Real.hasLt ε (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero)))) -> (Filter.Eventually.{u2} β (fun (x : β) => LT.lt.{0} Real Real.hasLt (Dist.dist.{u1} α (PseudoMetricSpace.toHasDist.{u1} α _inst_1) (f x) (f b)) ε) (nhds.{u2} β _inst_2 b)))
-but is expected to have type
-  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : PseudoMetricSpace.{u1} α] [_inst_2 : TopologicalSpace.{u2} β] {f : β -> α} {b : β}, Iff (ContinuousAt.{u2, u1} β α _inst_2 (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) f b) (forall (ε : Real), (GT.gt.{0} Real Real.instLTReal ε (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal))) -> (Filter.Eventually.{u2} β (fun (x : β) => LT.lt.{0} Real Real.instLTReal (Dist.dist.{u1} α (PseudoMetricSpace.toDist.{u1} α _inst_1) (f x) (f b)) ε) (nhds.{u2} β _inst_2 b)))
-Case conversion may be inaccurate. Consider using '#align metric.continuous_at_iff' Metric.continuousAt_iff'ₓ'. -/
 theorem continuousAt_iff' [TopologicalSpace β] {f : β → α} {b : β} :
     ContinuousAt f b ↔ ∀ ε > 0, ∀ᶠ x in 𝓝 b, dist (f x) (f b) < ε := by
   rw [ContinuousAt, tendsto_nhds]
 #align metric.continuous_at_iff' Metric.continuousAt_iff'
 
-/- warning: metric.continuous_within_at_iff' -> Metric.continuousWithinAt_iff' is a dubious translation:
-lean 3 declaration is
-  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : PseudoMetricSpace.{u1} α] [_inst_2 : TopologicalSpace.{u2} β] {f : β -> α} {b : β} {s : Set.{u2} β}, Iff (ContinuousWithinAt.{u2, u1} β α _inst_2 (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) f s b) (forall (ε : Real), (GT.gt.{0} Real Real.hasLt ε (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero)))) -> (Filter.Eventually.{u2} β (fun (x : β) => LT.lt.{0} Real Real.hasLt (Dist.dist.{u1} α (PseudoMetricSpace.toHasDist.{u1} α _inst_1) (f x) (f b)) ε) (nhdsWithin.{u2} β _inst_2 b s)))
-but is expected to have type
-  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : PseudoMetricSpace.{u1} α] [_inst_2 : TopologicalSpace.{u2} β] {f : β -> α} {b : β} {s : Set.{u2} β}, Iff (ContinuousWithinAt.{u2, u1} β α _inst_2 (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) f s b) (forall (ε : Real), (GT.gt.{0} Real Real.instLTReal ε (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal))) -> (Filter.Eventually.{u2} β (fun (x : β) => LT.lt.{0} Real Real.instLTReal (Dist.dist.{u1} α (PseudoMetricSpace.toDist.{u1} α _inst_1) (f x) (f b)) ε) (nhdsWithin.{u2} β _inst_2 b s)))
-Case conversion may be inaccurate. Consider using '#align metric.continuous_within_at_iff' Metric.continuousWithinAt_iff'ₓ'. -/
 theorem continuousWithinAt_iff' [TopologicalSpace β] {f : β → α} {b : β} {s : Set β} :
     ContinuousWithinAt f s b ↔ ∀ ε > 0, ∀ᶠ x in 𝓝[s] b, dist (f x) (f b) < ε := by
   rw [ContinuousWithinAt, tendsto_nhds]
 #align metric.continuous_within_at_iff' Metric.continuousWithinAt_iff'
 
-/- warning: metric.continuous_on_iff' -> Metric.continuousOn_iff' is a dubious translation:
-lean 3 declaration is
-  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : PseudoMetricSpace.{u1} α] [_inst_2 : TopologicalSpace.{u2} β] {f : β -> α} {s : Set.{u2} β}, Iff (ContinuousOn.{u2, u1} β α _inst_2 (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) f s) (forall (b : β), (Membership.Mem.{u2, u2} β (Set.{u2} β) (Set.hasMem.{u2} β) b s) -> (forall (ε : Real), (GT.gt.{0} Real Real.hasLt ε (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero)))) -> (Filter.Eventually.{u2} β (fun (x : β) => LT.lt.{0} Real Real.hasLt (Dist.dist.{u1} α (PseudoMetricSpace.toHasDist.{u1} α _inst_1) (f x) (f b)) ε) (nhdsWithin.{u2} β _inst_2 b s))))
-but is expected to have type
-  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : PseudoMetricSpace.{u1} α] [_inst_2 : TopologicalSpace.{u2} β] {f : β -> α} {s : Set.{u2} β}, Iff (ContinuousOn.{u2, u1} β α _inst_2 (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) f s) (forall (b : β), (Membership.mem.{u2, u2} β (Set.{u2} β) (Set.instMembershipSet.{u2} β) b s) -> (forall (ε : Real), (GT.gt.{0} Real Real.instLTReal ε (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal))) -> (Filter.Eventually.{u2} β (fun (x : β) => LT.lt.{0} Real Real.instLTReal (Dist.dist.{u1} α (PseudoMetricSpace.toDist.{u1} α _inst_1) (f x) (f b)) ε) (nhdsWithin.{u2} β _inst_2 b s))))
-Case conversion may be inaccurate. Consider using '#align metric.continuous_on_iff' Metric.continuousOn_iff'ₓ'. -/
 theorem continuousOn_iff' [TopologicalSpace β] {f : β → α} {s : Set β} :
     ContinuousOn f s ↔ ∀ b ∈ s, ∀ ε > 0, ∀ᶠ x in 𝓝[s] b, dist (f x) (f b) < ε := by
   simp [ContinuousOn, continuous_within_at_iff']
 #align metric.continuous_on_iff' Metric.continuousOn_iff'
 
-/- warning: metric.continuous_iff' -> Metric.continuous_iff' is a dubious translation:
-lean 3 declaration is
-  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : PseudoMetricSpace.{u1} α] [_inst_2 : TopologicalSpace.{u2} β] {f : β -> α}, Iff (Continuous.{u2, u1} β α _inst_2 (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) f) (forall (a : β) (ε : Real), (GT.gt.{0} Real Real.hasLt ε (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero)))) -> (Filter.Eventually.{u2} β (fun (x : β) => LT.lt.{0} Real Real.hasLt (Dist.dist.{u1} α (PseudoMetricSpace.toHasDist.{u1} α _inst_1) (f x) (f a)) ε) (nhds.{u2} β _inst_2 a)))
-but is expected to have type
-  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : PseudoMetricSpace.{u1} α] [_inst_2 : TopologicalSpace.{u2} β] {f : β -> α}, Iff (Continuous.{u2, u1} β α _inst_2 (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) f) (forall (a : β) (ε : Real), (GT.gt.{0} Real Real.instLTReal ε (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal))) -> (Filter.Eventually.{u2} β (fun (x : β) => LT.lt.{0} Real Real.instLTReal (Dist.dist.{u1} α (PseudoMetricSpace.toDist.{u1} α _inst_1) (f x) (f a)) ε) (nhds.{u2} β _inst_2 a)))
-Case conversion may be inaccurate. Consider using '#align metric.continuous_iff' Metric.continuous_iff'ₓ'. -/
 theorem continuous_iff' [TopologicalSpace β] {f : β → α} :
     Continuous f ↔ ∀ (a), ∀ ε > 0, ∀ᶠ x in 𝓝 a, dist (f x) (f a) < ε :=
   continuous_iff_continuousAt.trans <| forall_congr' fun b => tendsto_nhds
 #align metric.continuous_iff' Metric.continuous_iff'
 
-/- warning: metric.tendsto_at_top -> Metric.tendsto_atTop is a dubious translation:
-lean 3 declaration is
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 theorem tendsto_atTop [Nonempty β] [SemilatticeSup β] {u : β → α} {a : α} :
     Tendsto u atTop (𝓝 a) ↔ ∀ ε > 0, ∃ N, ∀ n ≥ N, dist (u n) a < ε :=
   (atTop_basis.tendsto_iffₓ nhds_basis_ball).trans <| by simp only [exists_prop, true_and_iff]; rfl
 #align metric.tendsto_at_top Metric.tendsto_atTop
 
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 /-- A variant of `tendsto_at_top` that
 uses `∃ N, ∀ n > N, ...` rather than `∃ N, ∀ n ≥ N, ...`
 -/
@@ -2052,23 +1245,11 @@ theorem tendsto_atTop' [Nonempty β] [SemilatticeSup β] [NoMaxOrder β] {u : β
     rfl
 #align metric.tendsto_at_top' Metric.tendsto_atTop'
 
-/- warning: metric.is_open_singleton_iff -> Metric.isOpen_singleton_iff is a dubious translation:
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 theorem isOpen_singleton_iff {α : Type _} [PseudoMetricSpace α] {x : α} :
     IsOpen ({x} : Set α) ↔ ∃ ε > 0, ∀ y, dist y x < ε → y = x := by
   simp [is_open_iff, subset_singleton_iff, mem_ball]
 #align metric.is_open_singleton_iff Metric.isOpen_singleton_iff
 
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 /-- Given a point `x` in a discrete subset `s` of a pseudometric space, there is an open ball
 centered at `x` and intersecting `s` only at `x`. -/
 theorem exists_ball_inter_eq_singleton_of_mem_discrete [DiscreteTopology s] {x : α} (hx : x ∈ s) :
@@ -2076,12 +1257,6 @@ theorem exists_ball_inter_eq_singleton_of_mem_discrete [DiscreteTopology s] {x :
   nhds_basis_ball.exists_inter_eq_singleton_of_mem_discrete hx
 #align metric.exists_ball_inter_eq_singleton_of_mem_discrete Metric.exists_ball_inter_eq_singleton_of_mem_discrete
 
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 /-- Given a point `x` in a discrete subset `s` of a pseudometric space, there is a closed ball
 of positive radius centered at `x` and intersecting `s` only at `x`. -/
 theorem exists_closedBall_inter_eq_singleton_of_discrete [DiscreteTopology s] {x : α} (hx : x ∈ s) :
@@ -2089,12 +1264,6 @@ theorem exists_closedBall_inter_eq_singleton_of_discrete [DiscreteTopology s] {x
   nhds_basis_closedBall.exists_inter_eq_singleton_of_mem_discrete hx
 #align metric.exists_closed_ball_inter_eq_singleton_of_discrete Metric.exists_closedBall_inter_eq_singleton_of_discrete
 
-/- warning: dense.exists_dist_lt -> Dense.exists_dist_lt is a dubious translation:
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 theorem Dense.exists_dist_lt {s : Set α} (hs : Dense s) (x : α) {ε : ℝ} (hε : 0 < ε) :
     ∃ y ∈ s, dist x y < ε :=
   by
@@ -2102,12 +1271,6 @@ theorem Dense.exists_dist_lt {s : Set α} (hs : Dense s) (x : α) {ε : ℝ} (h
   simpa only [mem_ball'] using hs.exists_mem_open is_open_ball this
 #align dense.exists_dist_lt Dense.exists_dist_lt
 
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-Case conversion may be inaccurate. Consider using '#align dense_range.exists_dist_lt DenseRange.exists_dist_ltₓ'. -/
 theorem DenseRange.exists_dist_lt {β : Type _} {f : β → α} (hf : DenseRange f) (x : α) {ε : ℝ}
     (hε : 0 < ε) : ∃ y, dist x (f y) < ε :=
   exists_range_iff.1 (hf.exists_dist_lt x hε)
@@ -2117,12 +1280,6 @@ end Metric
 
 open Metric
 
-/- warning: pseudo_metric.uniformity_basis_edist -> Metric.uniformity_basis_edist is a dubious translation:
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-Case conversion may be inaccurate. Consider using '#align pseudo_metric.uniformity_basis_edist Metric.uniformity_basis_edistₓ'. -/
 /-Instantiate a pseudometric space as a pseudoemetric space. Before we can state the instance,
 we need to show that the uniform structure coming from the edistance and the
 distance coincide. -/
@@ -2142,12 +1299,6 @@ protected theorem Metric.uniformity_basis_edist :
       rwa [edist_dist, ENNReal.ofReal_lt_ofReal_iff ε0']⟩
 #align pseudo_metric.uniformity_basis_edist Metric.uniformity_basis_edist
 
-/- warning: metric.uniformity_edist -> Metric.uniformity_edist is a dubious translation:
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-Case conversion may be inaccurate. Consider using '#align metric.uniformity_edist Metric.uniformity_edistₓ'. -/
 theorem Metric.uniformity_edist : 𝓤 α = ⨅ ε > 0, 𝓟 { p : α × α | edist p.1 p.2 < ε } :=
   Metric.uniformity_basis_edist.eq_biInf
 #align metric.uniformity_edist Metric.uniformity_edist
@@ -2170,12 +1321,6 @@ instance (priority := 100) PseudoMetricSpace.toPseudoEMetricSpace : PseudoEMetri
 #align pseudo_metric_space.to_pseudo_emetric_space PseudoMetricSpace.toPseudoEMetricSpace
 -/
 
-/- warning: metric.eball_top_eq_univ -> Metric.eball_top_eq_univ is a dubious translation:
-lean 3 declaration is
-  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] (x : α), Eq.{succ u1} (Set.{u1} α) (EMetric.ball.{u1} α (PseudoMetricSpace.toPseudoEMetricSpace.{u1} α _inst_1) x (Top.top.{0} ENNReal (CompleteLattice.toHasTop.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder)))) (Set.univ.{u1} α)
-but is expected to have type
-  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] (x : α), Eq.{succ u1} (Set.{u1} α) (EMetric.ball.{u1} α (PseudoMetricSpace.toPseudoEMetricSpace.{u1} α _inst_1) x (Top.top.{0} ENNReal (CompleteLattice.toTop.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal)))) (Set.univ.{u1} α)
-Case conversion may be inaccurate. Consider using '#align metric.eball_top_eq_univ Metric.eball_top_eq_univₓ'. -/
 /-- In a pseudometric space, an open ball of infinite radius is the whole space -/
 theorem Metric.eball_top_eq_univ (x : α) : EMetric.ball x ∞ = Set.univ :=
   Set.eq_univ_iff_forall.mpr fun y => edist_lt_top y x
@@ -2192,59 +1337,29 @@ theorem Metric.emetric_ball {x : α} {ε : ℝ} : EMetric.ball x (ENNReal.ofReal
 #align metric.emetric_ball Metric.emetric_ball
 -/
 
-/- warning: metric.emetric_ball_nnreal -> Metric.emetric_ball_nnreal is a dubious translation:
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-  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {x : α} {ε : NNReal}, Eq.{succ u1} (Set.{u1} α) (EMetric.ball.{u1} α (PseudoMetricSpace.toPseudoEMetricSpace.{u1} α _inst_1) x ((fun (a : Type) (b : Type) [self : HasLiftT.{1, 1} a b] => self.0) NNReal ENNReal (HasLiftT.mk.{1, 1} NNReal ENNReal (CoeTCₓ.coe.{1, 1} NNReal ENNReal (coeBase.{1, 1} NNReal ENNReal ENNReal.hasCoe))) ε)) (Metric.ball.{u1} α _inst_1 x ((fun (a : Type) (b : Type) [self : HasLiftT.{1, 1} a b] => self.0) NNReal Real (HasLiftT.mk.{1, 1} NNReal Real (CoeTCₓ.coe.{1, 1} NNReal Real (coeBase.{1, 1} NNReal Real NNReal.Real.hasCoe))) ε))
-but is expected to have type
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-Case conversion may be inaccurate. Consider using '#align metric.emetric_ball_nnreal Metric.emetric_ball_nnrealₓ'. -/
 /-- Balls defined using the distance or the edistance coincide -/
 @[simp]
 theorem Metric.emetric_ball_nnreal {x : α} {ε : ℝ≥0} : EMetric.ball x ε = ball x ε := by
   convert Metric.emetric_ball; simp
 #align metric.emetric_ball_nnreal Metric.emetric_ball_nnreal
 
-/- warning: metric.emetric_closed_ball -> Metric.emetric_closedBall is a dubious translation:
-lean 3 declaration is
-  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {x : α} {ε : Real}, (LE.le.{0} Real Real.hasLe (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero))) ε) -> (Eq.{succ u1} (Set.{u1} α) (EMetric.closedBall.{u1} α (PseudoMetricSpace.toPseudoEMetricSpace.{u1} α _inst_1) x (ENNReal.ofReal ε)) (Metric.closedBall.{u1} α _inst_1 x ε))
-but is expected to have type
-  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {x : α} {ε : Real}, (LE.le.{0} Real Real.instLEReal (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal)) ε) -> (Eq.{succ u1} (Set.{u1} α) (EMetric.closedBall.{u1} α (PseudoMetricSpace.toPseudoEMetricSpace.{u1} α _inst_1) x (ENNReal.ofReal ε)) (Metric.closedBall.{u1} α _inst_1 x ε))
-Case conversion may be inaccurate. Consider using '#align metric.emetric_closed_ball Metric.emetric_closedBallₓ'. -/
 /-- Closed balls defined using the distance or the edistance coincide -/
 theorem Metric.emetric_closedBall {x : α} {ε : ℝ} (h : 0 ≤ ε) :
     EMetric.closedBall x (ENNReal.ofReal ε) = closedBall x ε := by
   ext y <;> simp [edist_dist] <;> rw [ENNReal.ofReal_le_ofReal_iff h]
 #align metric.emetric_closed_ball Metric.emetric_closedBall
 
-/- warning: metric.emetric_closed_ball_nnreal -> Metric.emetric_closedBall_nnreal is a dubious translation:
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-but is expected to have type
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-Case conversion may be inaccurate. Consider using '#align metric.emetric_closed_ball_nnreal Metric.emetric_closedBall_nnrealₓ'. -/
 /-- Closed balls defined using the distance or the edistance coincide -/
 @[simp]
 theorem Metric.emetric_closedBall_nnreal {x : α} {ε : ℝ≥0} :
     EMetric.closedBall x ε = closedBall x ε := by convert Metric.emetric_closedBall ε.2; simp
 #align metric.emetric_closed_ball_nnreal Metric.emetric_closedBall_nnreal
 
-/- warning: metric.emetric_ball_top -> Metric.emetric_ball_top is a dubious translation:
-lean 3 declaration is
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-but is expected to have type
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-Case conversion may be inaccurate. Consider using '#align metric.emetric_ball_top Metric.emetric_ball_topₓ'. -/
 @[simp]
 theorem Metric.emetric_ball_top (x : α) : EMetric.ball x ⊤ = univ :=
   eq_univ_of_forall fun y => edist_lt_top _ _
 #align metric.emetric_ball_top Metric.emetric_ball_top
 
-/- warning: metric.inseparable_iff -> Metric.inseparable_iff is a dubious translation:
-lean 3 declaration is
-  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {x : α} {y : α}, Iff (Inseparable.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) x y) (Eq.{1} Real (Dist.dist.{u1} α (PseudoMetricSpace.toHasDist.{u1} α _inst_1) x y) (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero))))
-but is expected to have type
-  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {x : α} {y : α}, Iff (Inseparable.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) x y) (Eq.{1} Real (Dist.dist.{u1} α (PseudoMetricSpace.toDist.{u1} α _inst_1) x y) (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal)))
-Case conversion may be inaccurate. Consider using '#align metric.inseparable_iff Metric.inseparable_iffₓ'. -/
 theorem Metric.inseparable_iff {x y : α} : Inseparable x y ↔ dist x y = 0 := by
   rw [EMetric.inseparable_iff, edist_nndist, dist_nndist, ENNReal.coe_eq_zero, NNReal.coe_eq_zero]
 #align metric.inseparable_iff Metric.inseparable_iff
@@ -2292,12 +1407,6 @@ theorem PseudoMetricSpace.replaceTopology_eq {γ} [U : TopologicalSpace γ] (m :
 #align pseudo_metric_space.replace_topology_eq PseudoMetricSpace.replaceTopology_eq
 -/
 
-/- warning: pseudo_emetric_space.to_pseudo_metric_space_of_dist -> PseudoEMetricSpace.toPseudoMetricSpaceOfDist is a dubious translation:
-lean 3 declaration is
-  forall {α : Type.{u1}} [e : PseudoEMetricSpace.{u1} α] (dist : α -> α -> Real), (forall (x : α) (y : α), Ne.{1} ENNReal (EDist.edist.{u1} α (PseudoEMetricSpace.toHasEdist.{u1} α e) x y) (Top.top.{0} ENNReal (CompleteLattice.toHasTop.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder)))) -> (forall (x : α) (y : α), Eq.{1} Real (dist x y) (ENNReal.toReal (EDist.edist.{u1} α (PseudoEMetricSpace.toHasEdist.{u1} α e) x y))) -> (PseudoMetricSpace.{u1} α)
-but is expected to have type
-  forall {α : Type.{u1}} [e : PseudoEMetricSpace.{u1} α] (dist : α -> α -> Real), (forall (x : α) (y : α), Ne.{1} ENNReal (EDist.edist.{u1} α (PseudoEMetricSpace.toEDist.{u1} α e) x y) (Top.top.{0} ENNReal (CompleteLattice.toTop.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal)))) -> (forall (x : α) (y : α), Eq.{1} Real (dist x y) (ENNReal.toReal (EDist.edist.{u1} α (PseudoEMetricSpace.toEDist.{u1} α e) x y))) -> (PseudoMetricSpace.{u1} α)
-Case conversion may be inaccurate. Consider using '#align pseudo_emetric_space.to_pseudo_metric_space_of_dist PseudoEMetricSpace.toPseudoMetricSpaceOfDistₓ'. -/
 /-- One gets a pseudometric space from an emetric space if the edistance
 is everywhere finite, by pushing the edistance to reals. We set it up so that the edist and the
 uniformity are defeq in the pseudometric space and the pseudoemetric space. In this definition, the
@@ -2321,12 +1430,6 @@ def PseudoEMetricSpace.toPseudoMetricSpaceOfDist {α : Type u} [e : PseudoEMetri
   m.replaceUniformity <| by rw [uniformity_pseudoedist, Metric.uniformity_edist]; rfl
 #align pseudo_emetric_space.to_pseudo_metric_space_of_dist PseudoEMetricSpace.toPseudoMetricSpaceOfDist
 
-/- warning: pseudo_emetric_space.to_pseudo_metric_space -> PseudoEMetricSpace.toPseudoMetricSpace is a dubious translation:
-lean 3 declaration is
-  forall {α : Type.{u1}} [e : PseudoEMetricSpace.{u1} α], (forall (x : α) (y : α), Ne.{1} ENNReal (EDist.edist.{u1} α (PseudoEMetricSpace.toHasEdist.{u1} α e) x y) (Top.top.{0} ENNReal (CompleteLattice.toHasTop.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder)))) -> (PseudoMetricSpace.{u1} α)
-but is expected to have type
-  forall {α : Type.{u1}} [e : PseudoEMetricSpace.{u1} α], (forall (x : α) (y : α), Ne.{1} ENNReal (EDist.edist.{u1} α (PseudoEMetricSpace.toEDist.{u1} α e) x y) (Top.top.{0} ENNReal (CompleteLattice.toTop.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal)))) -> (PseudoMetricSpace.{u1} α)
-Case conversion may be inaccurate. Consider using '#align pseudo_emetric_space.to_pseudo_metric_space PseudoEMetricSpace.toPseudoMetricSpaceₓ'. -/
 /-- One gets a pseudometric space from an emetric space if the edistance
 is everywhere finite, by pushing the edistance to reals. We set it up so that the edist and the
 uniformity are defeq in the pseudometric space and the emetric space. -/
@@ -2360,12 +1463,6 @@ theorem PseudoMetricSpace.replaceBornology_eq {α} [m : PseudoMetricSpace α] [B
 #align pseudo_metric_space.replace_bornology_eq PseudoMetricSpace.replaceBornology_eq
 -/
 
-/- warning: metric.complete_of_convergent_controlled_sequences -> Metric.complete_of_convergent_controlled_sequences is a dubious translation:
-lean 3 declaration is
-  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] (B : Nat -> Real), (forall (n : Nat), LT.lt.{0} Real Real.hasLt (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero))) (B n)) -> (forall (u : Nat -> α), (forall (N : Nat) (n : Nat) (m : Nat), (LE.le.{0} Nat Nat.hasLe N n) -> (LE.le.{0} Nat Nat.hasLe N m) -> (LT.lt.{0} Real Real.hasLt (Dist.dist.{u1} α (PseudoMetricSpace.toHasDist.{u1} α _inst_1) (u n) (u m)) (B N))) -> (Exists.{succ u1} α (fun (x : α) => Filter.Tendsto.{0, u1} Nat α u (Filter.atTop.{0} Nat (PartialOrder.toPreorder.{0} Nat (OrderedCancelAddCommMonoid.toPartialOrder.{0} Nat (StrictOrderedSemiring.toOrderedCancelAddCommMonoid.{0} Nat Nat.strictOrderedSemiring)))) (nhds.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) x)))) -> (CompleteSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1))
-but is expected to have type
-  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] (B : Nat -> Real), (forall (n : Nat), LT.lt.{0} Real Real.instLTReal (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal)) (B n)) -> (forall (u : Nat -> α), (forall (N : Nat) (n : Nat) (m : Nat), (LE.le.{0} Nat instLENat N n) -> (LE.le.{0} Nat instLENat N m) -> (LT.lt.{0} Real Real.instLTReal (Dist.dist.{u1} α (PseudoMetricSpace.toDist.{u1} α _inst_1) (u n) (u m)) (B N))) -> (Exists.{succ u1} α (fun (x : α) => Filter.Tendsto.{0, u1} Nat α u (Filter.atTop.{0} Nat (PartialOrder.toPreorder.{0} Nat (StrictOrderedSemiring.toPartialOrder.{0} Nat Nat.strictOrderedSemiring))) (nhds.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) x)))) -> (CompleteSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1))
-Case conversion may be inaccurate. Consider using '#align metric.complete_of_convergent_controlled_sequences Metric.complete_of_convergent_controlled_sequencesₓ'. -/
 /-- A very useful criterion to show that a space is complete is to show that all sequences
 which satisfy a bound of the form `dist (u n) (u m) < B N` for all `n m ≥ N` are
 converging. This is often applied for `B N = 2^{-N}`, i.e., with a very fast convergence to
@@ -2400,94 +1497,40 @@ instance Real.pseudoMetricSpace : PseudoMetricSpace ℝ
 #align real.pseudo_metric_space Real.pseudoMetricSpace
 -/
 
-/- warning: real.dist_eq -> Real.dist_eq is a dubious translation:
-lean 3 declaration is
-  forall (x : Real) (y : Real), Eq.{1} Real (Dist.dist.{0} Real (PseudoMetricSpace.toHasDist.{0} Real Real.pseudoMetricSpace) x y) (Abs.abs.{0} Real (Neg.toHasAbs.{0} Real Real.hasNeg Real.hasSup) (HSub.hSub.{0, 0, 0} Real Real Real (instHSub.{0} Real Real.hasSub) x y))
-but is expected to have type
-  forall (x : Real) (y : Real), Eq.{1} Real (Dist.dist.{0} Real (PseudoMetricSpace.toDist.{0} Real Real.pseudoMetricSpace) x y) (Abs.abs.{0} Real (Neg.toHasAbs.{0} Real Real.instNegReal Real.instSupReal) (HSub.hSub.{0, 0, 0} Real Real Real (instHSub.{0} Real Real.instSubReal) x y))
-Case conversion may be inaccurate. Consider using '#align real.dist_eq Real.dist_eqₓ'. -/
 theorem Real.dist_eq (x y : ℝ) : dist x y = |x - y| :=
   rfl
 #align real.dist_eq Real.dist_eq
 
-/- warning: real.nndist_eq -> Real.nndist_eq is a dubious translation:
-lean 3 declaration is
-  forall (x : Real) (y : Real), Eq.{1} NNReal (NNDist.nndist.{0} Real (PseudoMetricSpace.toNNDist.{0} Real Real.pseudoMetricSpace) x y) (coeFn.{1, 1} (MonoidWithZeroHom.{0, 0} Real NNReal (NonAssocSemiring.toMulZeroOneClass.{0} Real (NonAssocRing.toNonAssocSemiring.{0} Real (Ring.toNonAssocRing.{0} Real Real.ring))) (NonAssocSemiring.toMulZeroOneClass.{0} NNReal (Semiring.toNonAssocSemiring.{0} NNReal NNReal.semiring))) (fun (_x : MonoidWithZeroHom.{0, 0} Real NNReal (NonAssocSemiring.toMulZeroOneClass.{0} Real (NonAssocRing.toNonAssocSemiring.{0} Real (Ring.toNonAssocRing.{0} Real Real.ring))) (NonAssocSemiring.toMulZeroOneClass.{0} NNReal (Semiring.toNonAssocSemiring.{0} NNReal NNReal.semiring))) => Real -> NNReal) (MonoidWithZeroHom.hasCoeToFun.{0, 0} Real NNReal (NonAssocSemiring.toMulZeroOneClass.{0} Real (NonAssocRing.toNonAssocSemiring.{0} Real (Ring.toNonAssocRing.{0} Real Real.ring))) (NonAssocSemiring.toMulZeroOneClass.{0} NNReal (Semiring.toNonAssocSemiring.{0} NNReal NNReal.semiring))) Real.nnabs (HSub.hSub.{0, 0, 0} Real Real Real (instHSub.{0} Real Real.hasSub) x y))
-but is expected to have type
-  forall (x : Real) (y : Real), Eq.{1} NNReal (NNDist.nndist.{0} Real (PseudoMetricSpace.toNNDist.{0} Real Real.pseudoMetricSpace) x y) (FunLike.coe.{1, 1, 1} (MonoidWithZeroHom.{0, 0} Real NNReal (NonAssocSemiring.toMulZeroOneClass.{0} Real (Semiring.toNonAssocSemiring.{0} Real Real.semiring)) (NonAssocSemiring.toMulZeroOneClass.{0} NNReal (Semiring.toNonAssocSemiring.{0} NNReal instNNRealSemiring))) Real (fun (_x : Real) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : Real) => NNReal) _x) (MulHomClass.toFunLike.{0, 0, 0} (MonoidWithZeroHom.{0, 0} Real NNReal (NonAssocSemiring.toMulZeroOneClass.{0} Real (Semiring.toNonAssocSemiring.{0} Real Real.semiring)) (NonAssocSemiring.toMulZeroOneClass.{0} NNReal (Semiring.toNonAssocSemiring.{0} NNReal instNNRealSemiring))) Real NNReal (MulOneClass.toMul.{0} Real (MulZeroOneClass.toMulOneClass.{0} Real (NonAssocSemiring.toMulZeroOneClass.{0} Real (Semiring.toNonAssocSemiring.{0} Real Real.semiring)))) (MulOneClass.toMul.{0} NNReal (MulZeroOneClass.toMulOneClass.{0} NNReal (NonAssocSemiring.toMulZeroOneClass.{0} NNReal (Semiring.toNonAssocSemiring.{0} NNReal instNNRealSemiring)))) (MonoidHomClass.toMulHomClass.{0, 0, 0} (MonoidWithZeroHom.{0, 0} Real NNReal (NonAssocSemiring.toMulZeroOneClass.{0} Real (Semiring.toNonAssocSemiring.{0} Real Real.semiring)) (NonAssocSemiring.toMulZeroOneClass.{0} NNReal (Semiring.toNonAssocSemiring.{0} NNReal instNNRealSemiring))) Real NNReal (MulZeroOneClass.toMulOneClass.{0} Real (NonAssocSemiring.toMulZeroOneClass.{0} Real (Semiring.toNonAssocSemiring.{0} Real Real.semiring))) (MulZeroOneClass.toMulOneClass.{0} NNReal (NonAssocSemiring.toMulZeroOneClass.{0} NNReal (Semiring.toNonAssocSemiring.{0} NNReal instNNRealSemiring))) (MonoidWithZeroHomClass.toMonoidHomClass.{0, 0, 0} (MonoidWithZeroHom.{0, 0} Real NNReal (NonAssocSemiring.toMulZeroOneClass.{0} Real (Semiring.toNonAssocSemiring.{0} Real Real.semiring)) (NonAssocSemiring.toMulZeroOneClass.{0} NNReal (Semiring.toNonAssocSemiring.{0} NNReal instNNRealSemiring))) Real NNReal (NonAssocSemiring.toMulZeroOneClass.{0} Real (Semiring.toNonAssocSemiring.{0} Real Real.semiring)) (NonAssocSemiring.toMulZeroOneClass.{0} NNReal (Semiring.toNonAssocSemiring.{0} NNReal instNNRealSemiring)) (MonoidWithZeroHom.monoidWithZeroHomClass.{0, 0} Real NNReal (NonAssocSemiring.toMulZeroOneClass.{0} Real (Semiring.toNonAssocSemiring.{0} Real Real.semiring)) (NonAssocSemiring.toMulZeroOneClass.{0} NNReal (Semiring.toNonAssocSemiring.{0} NNReal instNNRealSemiring)))))) Real.nnabs (HSub.hSub.{0, 0, 0} Real Real Real (instHSub.{0} Real Real.instSubReal) x y))
-Case conversion may be inaccurate. Consider using '#align real.nndist_eq Real.nndist_eqₓ'. -/
 theorem Real.nndist_eq (x y : ℝ) : nndist x y = Real.nnabs (x - y) :=
   rfl
 #align real.nndist_eq Real.nndist_eq
 
-/- warning: real.nndist_eq' -> Real.nndist_eq' is a dubious translation:
-lean 3 declaration is
-  forall (x : Real) (y : Real), Eq.{1} NNReal (NNDist.nndist.{0} Real (PseudoMetricSpace.toNNDist.{0} Real Real.pseudoMetricSpace) x y) (coeFn.{1, 1} (MonoidWithZeroHom.{0, 0} Real NNReal (NonAssocSemiring.toMulZeroOneClass.{0} Real (NonAssocRing.toNonAssocSemiring.{0} Real (Ring.toNonAssocRing.{0} Real Real.ring))) (NonAssocSemiring.toMulZeroOneClass.{0} NNReal (Semiring.toNonAssocSemiring.{0} NNReal NNReal.semiring))) (fun (_x : MonoidWithZeroHom.{0, 0} Real NNReal (NonAssocSemiring.toMulZeroOneClass.{0} Real (NonAssocRing.toNonAssocSemiring.{0} Real (Ring.toNonAssocRing.{0} Real Real.ring))) (NonAssocSemiring.toMulZeroOneClass.{0} NNReal (Semiring.toNonAssocSemiring.{0} NNReal NNReal.semiring))) => Real -> NNReal) (MonoidWithZeroHom.hasCoeToFun.{0, 0} Real NNReal (NonAssocSemiring.toMulZeroOneClass.{0} Real (NonAssocRing.toNonAssocSemiring.{0} Real (Ring.toNonAssocRing.{0} Real Real.ring))) (NonAssocSemiring.toMulZeroOneClass.{0} NNReal (Semiring.toNonAssocSemiring.{0} NNReal NNReal.semiring))) Real.nnabs (HSub.hSub.{0, 0, 0} Real Real Real (instHSub.{0} Real Real.hasSub) y x))
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-Case conversion may be inaccurate. Consider using '#align real.nndist_eq' Real.nndist_eq'ₓ'. -/
 theorem Real.nndist_eq' (x y : ℝ) : nndist x y = Real.nnabs (y - x) :=
   nndist_comm _ _
 #align real.nndist_eq' Real.nndist_eq'
 
-/- warning: real.dist_0_eq_abs -> Real.dist_0_eq_abs is a dubious translation:
-lean 3 declaration is
-  forall (x : Real), Eq.{1} Real (Dist.dist.{0} Real (PseudoMetricSpace.toHasDist.{0} Real Real.pseudoMetricSpace) x (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero)))) (Abs.abs.{0} Real (Neg.toHasAbs.{0} Real Real.hasNeg Real.hasSup) x)
-but is expected to have type
-  forall (x : Real), Eq.{1} Real (Dist.dist.{0} Real (PseudoMetricSpace.toDist.{0} Real Real.pseudoMetricSpace) x (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal))) (Abs.abs.{0} Real (Neg.toHasAbs.{0} Real Real.instNegReal Real.instSupReal) x)
-Case conversion may be inaccurate. Consider using '#align real.dist_0_eq_abs Real.dist_0_eq_absₓ'. -/
 theorem Real.dist_0_eq_abs (x : ℝ) : dist x 0 = |x| := by simp [Real.dist_eq]
 #align real.dist_0_eq_abs Real.dist_0_eq_abs
 
-/- warning: real.dist_left_le_of_mem_uIcc -> Real.dist_left_le_of_mem_uIcc is a dubious translation:
-lean 3 declaration is
-  forall {x : Real} {y : Real} {z : Real}, (Membership.Mem.{0, 0} Real (Set.{0} Real) (Set.hasMem.{0} Real) y (Set.uIcc.{0} Real Real.lattice x z)) -> (LE.le.{0} Real Real.hasLe (Dist.dist.{0} Real (PseudoMetricSpace.toHasDist.{0} Real Real.pseudoMetricSpace) x y) (Dist.dist.{0} Real (PseudoMetricSpace.toHasDist.{0} Real Real.pseudoMetricSpace) x z))
-but is expected to have type
-  forall {x : Real} {y : Real} {z : Real}, (Membership.mem.{0, 0} Real (Set.{0} Real) (Set.instMembershipSet.{0} Real) y (Set.uIcc.{0} Real Real.lattice x z)) -> (LE.le.{0} Real Real.instLEReal (Dist.dist.{0} Real (PseudoMetricSpace.toDist.{0} Real Real.pseudoMetricSpace) x y) (Dist.dist.{0} Real (PseudoMetricSpace.toDist.{0} Real Real.pseudoMetricSpace) x z))
-Case conversion may be inaccurate. Consider using '#align real.dist_left_le_of_mem_uIcc Real.dist_left_le_of_mem_uIccₓ'. -/
 theorem Real.dist_left_le_of_mem_uIcc {x y z : ℝ} (h : y ∈ uIcc x z) : dist x y ≤ dist x z := by
   simpa only [dist_comm x] using abs_sub_left_of_mem_uIcc h
 #align real.dist_left_le_of_mem_uIcc Real.dist_left_le_of_mem_uIcc
 
-/- warning: real.dist_right_le_of_mem_uIcc -> Real.dist_right_le_of_mem_uIcc is a dubious translation:
-lean 3 declaration is
-  forall {x : Real} {y : Real} {z : Real}, (Membership.Mem.{0, 0} Real (Set.{0} Real) (Set.hasMem.{0} Real) y (Set.uIcc.{0} Real Real.lattice x z)) -> (LE.le.{0} Real Real.hasLe (Dist.dist.{0} Real (PseudoMetricSpace.toHasDist.{0} Real Real.pseudoMetricSpace) y z) (Dist.dist.{0} Real (PseudoMetricSpace.toHasDist.{0} Real Real.pseudoMetricSpace) x z))
-but is expected to have type
-  forall {x : Real} {y : Real} {z : Real}, (Membership.mem.{0, 0} Real (Set.{0} Real) (Set.instMembershipSet.{0} Real) y (Set.uIcc.{0} Real Real.lattice x z)) -> (LE.le.{0} Real Real.instLEReal (Dist.dist.{0} Real (PseudoMetricSpace.toDist.{0} Real Real.pseudoMetricSpace) y z) (Dist.dist.{0} Real (PseudoMetricSpace.toDist.{0} Real Real.pseudoMetricSpace) x z))
-Case conversion may be inaccurate. Consider using '#align real.dist_right_le_of_mem_uIcc Real.dist_right_le_of_mem_uIccₓ'. -/
 theorem Real.dist_right_le_of_mem_uIcc {x y z : ℝ} (h : y ∈ uIcc x z) : dist y z ≤ dist x z := by
   simpa only [dist_comm _ z] using abs_sub_right_of_mem_uIcc h
 #align real.dist_right_le_of_mem_uIcc Real.dist_right_le_of_mem_uIcc
 
-/- warning: real.dist_le_of_mem_uIcc -> Real.dist_le_of_mem_uIcc is a dubious translation:
-lean 3 declaration is
-  forall {x : Real} {y : Real} {x' : Real} {y' : Real}, (Membership.Mem.{0, 0} Real (Set.{0} Real) (Set.hasMem.{0} Real) x (Set.uIcc.{0} Real Real.lattice x' y')) -> (Membership.Mem.{0, 0} Real (Set.{0} Real) (Set.hasMem.{0} Real) y (Set.uIcc.{0} Real Real.lattice x' y')) -> (LE.le.{0} Real Real.hasLe (Dist.dist.{0} Real (PseudoMetricSpace.toHasDist.{0} Real Real.pseudoMetricSpace) x y) (Dist.dist.{0} Real (PseudoMetricSpace.toHasDist.{0} Real Real.pseudoMetricSpace) x' y'))
-but is expected to have type
-  forall {x : Real} {y : Real} {x' : Real} {y' : Real}, (Membership.mem.{0, 0} Real (Set.{0} Real) (Set.instMembershipSet.{0} Real) x (Set.uIcc.{0} Real Real.lattice x' y')) -> (Membership.mem.{0, 0} Real (Set.{0} Real) (Set.instMembershipSet.{0} Real) y (Set.uIcc.{0} Real Real.lattice x' y')) -> (LE.le.{0} Real Real.instLEReal (Dist.dist.{0} Real (PseudoMetricSpace.toDist.{0} Real Real.pseudoMetricSpace) x y) (Dist.dist.{0} Real (PseudoMetricSpace.toDist.{0} Real Real.pseudoMetricSpace) x' y'))
-Case conversion may be inaccurate. Consider using '#align real.dist_le_of_mem_uIcc Real.dist_le_of_mem_uIccₓ'. -/
 theorem Real.dist_le_of_mem_uIcc {x y x' y' : ℝ} (hx : x ∈ uIcc x' y') (hy : y ∈ uIcc x' y') :
     dist x y ≤ dist x' y' :=
   abs_sub_le_of_uIcc_subset_uIcc <| uIcc_subset_uIcc (by rwa [uIcc_comm]) (by rwa [uIcc_comm])
 #align real.dist_le_of_mem_uIcc Real.dist_le_of_mem_uIcc
 
-/- warning: real.dist_le_of_mem_Icc -> Real.dist_le_of_mem_Icc is a dubious translation:
-lean 3 declaration is
-  forall {x : Real} {y : Real} {x' : Real} {y' : Real}, (Membership.Mem.{0, 0} Real (Set.{0} Real) (Set.hasMem.{0} Real) x (Set.Icc.{0} Real Real.preorder x' y')) -> (Membership.Mem.{0, 0} Real (Set.{0} Real) (Set.hasMem.{0} Real) y (Set.Icc.{0} Real Real.preorder x' y')) -> (LE.le.{0} Real Real.hasLe (Dist.dist.{0} Real (PseudoMetricSpace.toHasDist.{0} Real Real.pseudoMetricSpace) x y) (HSub.hSub.{0, 0, 0} Real Real Real (instHSub.{0} Real Real.hasSub) y' x'))
-but is expected to have type
-  forall {x : Real} {y : Real} {x' : Real} {y' : Real}, (Membership.mem.{0, 0} Real (Set.{0} Real) (Set.instMembershipSet.{0} Real) x (Set.Icc.{0} Real Real.instPreorderReal x' y')) -> (Membership.mem.{0, 0} Real (Set.{0} Real) (Set.instMembershipSet.{0} Real) y (Set.Icc.{0} Real Real.instPreorderReal x' y')) -> (LE.le.{0} Real Real.instLEReal (Dist.dist.{0} Real (PseudoMetricSpace.toDist.{0} Real Real.pseudoMetricSpace) x y) (HSub.hSub.{0, 0, 0} Real Real Real (instHSub.{0} Real Real.instSubReal) y' x'))
-Case conversion may be inaccurate. Consider using '#align real.dist_le_of_mem_Icc Real.dist_le_of_mem_Iccₓ'. -/
 theorem Real.dist_le_of_mem_Icc {x y x' y' : ℝ} (hx : x ∈ Icc x' y') (hy : y ∈ Icc x' y') :
     dist x y ≤ y' - x' := by
   simpa only [Real.dist_eq, abs_of_nonpos (sub_nonpos.2 <| hx.1.trans hx.2), neg_sub] using
     Real.dist_le_of_mem_uIcc (Icc_subset_uIcc hx) (Icc_subset_uIcc hy)
 #align real.dist_le_of_mem_Icc Real.dist_le_of_mem_Icc
 
-/- warning: real.dist_le_of_mem_Icc_01 -> Real.dist_le_of_mem_Icc_01 is a dubious translation:
-lean 3 declaration is
-  forall {x : Real} {y : Real}, (Membership.Mem.{0, 0} Real (Set.{0} Real) (Set.hasMem.{0} Real) x (Set.Icc.{0} Real Real.preorder (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero))) (OfNat.ofNat.{0} Real 1 (OfNat.mk.{0} Real 1 (One.one.{0} Real Real.hasOne))))) -> (Membership.Mem.{0, 0} Real (Set.{0} Real) (Set.hasMem.{0} Real) y (Set.Icc.{0} Real Real.preorder (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero))) (OfNat.ofNat.{0} Real 1 (OfNat.mk.{0} Real 1 (One.one.{0} Real Real.hasOne))))) -> (LE.le.{0} Real Real.hasLe (Dist.dist.{0} Real (PseudoMetricSpace.toHasDist.{0} Real Real.pseudoMetricSpace) x y) (OfNat.ofNat.{0} Real 1 (OfNat.mk.{0} Real 1 (One.one.{0} Real Real.hasOne))))
-but is expected to have type
-  forall {x : Real} {y : Real}, (Membership.mem.{0, 0} Real (Set.{0} Real) (Set.instMembershipSet.{0} Real) x (Set.Icc.{0} Real Real.instPreorderReal (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal)) (OfNat.ofNat.{0} Real 1 (One.toOfNat1.{0} Real Real.instOneReal)))) -> (Membership.mem.{0, 0} Real (Set.{0} Real) (Set.instMembershipSet.{0} Real) y (Set.Icc.{0} Real Real.instPreorderReal (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal)) (OfNat.ofNat.{0} Real 1 (One.toOfNat1.{0} Real Real.instOneReal)))) -> (LE.le.{0} Real Real.instLEReal (Dist.dist.{0} Real (PseudoMetricSpace.toDist.{0} Real Real.pseudoMetricSpace) x y) (OfNat.ofNat.{0} Real 1 (One.toOfNat1.{0} Real Real.instOneReal)))
-Case conversion may be inaccurate. Consider using '#align real.dist_le_of_mem_Icc_01 Real.dist_le_of_mem_Icc_01ₓ'. -/
 theorem Real.dist_le_of_mem_Icc_01 {x y : ℝ} (hx : x ∈ Icc (0 : ℝ) 1) (hy : y ∈ Icc (0 : ℝ) 1) :
     dist x y ≤ 1 := by simpa only [sub_zero] using Real.dist_le_of_mem_Icc hx hy
 #align real.dist_le_of_mem_Icc_01 Real.dist_le_of_mem_Icc_01
@@ -2496,47 +1539,23 @@ instance : OrderTopology ℝ :=
   orderTopology_of_nhds_abs fun x => by
     simp only [nhds_basis_ball.eq_binfi, ball, Real.dist_eq, abs_sub_comm]
 
-/- warning: real.ball_eq_Ioo -> Real.ball_eq_Ioo is a dubious translation:
-lean 3 declaration is
-  forall (x : Real) (r : Real), Eq.{1} (Set.{0} Real) (Metric.ball.{0} Real Real.pseudoMetricSpace x r) (Set.Ioo.{0} Real Real.preorder (HSub.hSub.{0, 0, 0} Real Real Real (instHSub.{0} Real Real.hasSub) x r) (HAdd.hAdd.{0, 0, 0} Real Real Real (instHAdd.{0} Real Real.hasAdd) x r))
-but is expected to have type
-  forall (x : Real) (r : Real), Eq.{1} (Set.{0} Real) (Metric.ball.{0} Real Real.pseudoMetricSpace x r) (Set.Ioo.{0} Real Real.instPreorderReal (HSub.hSub.{0, 0, 0} Real Real Real (instHSub.{0} Real Real.instSubReal) x r) (HAdd.hAdd.{0, 0, 0} Real Real Real (instHAdd.{0} Real Real.instAddReal) x r))
-Case conversion may be inaccurate. Consider using '#align real.ball_eq_Ioo Real.ball_eq_Iooₓ'. -/
 theorem Real.ball_eq_Ioo (x r : ℝ) : ball x r = Ioo (x - r) (x + r) :=
   Set.ext fun y => by
     rw [mem_ball, dist_comm, Real.dist_eq, abs_sub_lt_iff, mem_Ioo, ← sub_lt_iff_lt_add',
       sub_lt_comm]
 #align real.ball_eq_Ioo Real.ball_eq_Ioo
 
-/- warning: real.closed_ball_eq_Icc -> Real.closedBall_eq_Icc is a dubious translation:
-lean 3 declaration is
-  forall {x : Real} {r : Real}, Eq.{1} (Set.{0} Real) (Metric.closedBall.{0} Real Real.pseudoMetricSpace x r) (Set.Icc.{0} Real Real.preorder (HSub.hSub.{0, 0, 0} Real Real Real (instHSub.{0} Real Real.hasSub) x r) (HAdd.hAdd.{0, 0, 0} Real Real Real (instHAdd.{0} Real Real.hasAdd) x r))
-but is expected to have type
-  forall {x : Real} {r : Real}, Eq.{1} (Set.{0} Real) (Metric.closedBall.{0} Real Real.pseudoMetricSpace x r) (Set.Icc.{0} Real Real.instPreorderReal (HSub.hSub.{0, 0, 0} Real Real Real (instHSub.{0} Real Real.instSubReal) x r) (HAdd.hAdd.{0, 0, 0} Real Real Real (instHAdd.{0} Real Real.instAddReal) x r))
-Case conversion may be inaccurate. Consider using '#align real.closed_ball_eq_Icc Real.closedBall_eq_Iccₓ'. -/
 theorem Real.closedBall_eq_Icc {x r : ℝ} : closedBall x r = Icc (x - r) (x + r) := by
   ext y <;>
     rw [mem_closed_ball, dist_comm, Real.dist_eq, abs_sub_le_iff, mem_Icc, ← sub_le_iff_le_add',
       sub_le_comm]
 #align real.closed_ball_eq_Icc Real.closedBall_eq_Icc
 
-/- warning: real.Ioo_eq_ball -> Real.Ioo_eq_ball is a dubious translation:
-lean 3 declaration is
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-Case conversion may be inaccurate. Consider using '#align real.Ioo_eq_ball Real.Ioo_eq_ballₓ'. -/
 theorem Real.Ioo_eq_ball (x y : ℝ) : Ioo x y = ball ((x + y) / 2) ((y - x) / 2) := by
   rw [Real.ball_eq_Ioo, ← sub_div, add_comm, ← sub_add, add_sub_cancel', add_self_div_two, ←
     add_div, add_assoc, add_sub_cancel'_right, add_self_div_two]
 #align real.Ioo_eq_ball Real.Ioo_eq_ball
 
-/- warning: real.Icc_eq_closed_ball -> Real.Icc_eq_closedBall is a dubious translation:
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-Case conversion may be inaccurate. Consider using '#align real.Icc_eq_closed_ball Real.Icc_eq_closedBallₓ'. -/
 theorem Real.Icc_eq_closedBall (x y : ℝ) : Icc x y = closedBall ((x + y) / 2) ((y - x) / 2) := by
   rw [Real.closedBall_eq_Icc, ← sub_div, add_comm, ← sub_add, add_sub_cancel', add_self_div_two, ←
     add_div, add_assoc, add_sub_cancel'_right, add_self_div_two]
@@ -2572,12 +1591,6 @@ theorem totallyBounded_Ioo (a b : α) : TotallyBounded (Ioo a b) :=
 
 end MetricOrdered
 
-/- warning: squeeze_zero' -> squeeze_zero' is a dubious translation:
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-Case conversion may be inaccurate. Consider using '#align squeeze_zero' squeeze_zero'ₓ'. -/
 /-- Special case of the sandwich theorem; see `tendsto_of_tendsto_of_tendsto_of_le_of_le'` for the
 general case. -/
 theorem squeeze_zero' {α} {f g : α → ℝ} {t₀ : Filter α} (hf : ∀ᶠ t in t₀, 0 ≤ f t)
@@ -2585,12 +1598,6 @@ theorem squeeze_zero' {α} {f g : α → ℝ} {t₀ : Filter α} (hf : ∀ᶠ t
   tendsto_of_tendsto_of_tendsto_of_le_of_le' tendsto_const_nhds g0 hf hft
 #align squeeze_zero' squeeze_zero'
 
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-Case conversion may be inaccurate. Consider using '#align squeeze_zero squeeze_zeroₓ'. -/
 /-- Special case of the sandwich theorem; see `tendsto_of_tendsto_of_tendsto_of_le_of_le`
 and  `tendsto_of_tendsto_of_tendsto_of_le_of_le'` for the general case. -/
 theorem squeeze_zero {α} {f g : α → ℝ} {t₀ : Filter α} (hf : ∀ t, 0 ≤ f t) (hft : ∀ t, f t ≤ g t)
@@ -2598,12 +1605,6 @@ theorem squeeze_zero {α} {f g : α → ℝ} {t₀ : Filter α} (hf : ∀ t, 0 
   squeeze_zero' (eventually_of_forall hf) (eventually_of_forall hft) g0
 #align squeeze_zero squeeze_zero
 
-/- warning: metric.uniformity_eq_comap_nhds_zero -> Metric.uniformity_eq_comap_nhds_zero is a dubious translation:
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-Case conversion may be inaccurate. Consider using '#align metric.uniformity_eq_comap_nhds_zero Metric.uniformity_eq_comap_nhds_zeroₓ'. -/
 theorem Metric.uniformity_eq_comap_nhds_zero :
     𝓤 α = comap (fun p : α × α => dist p.1 p.2) (𝓝 (0 : ℝ)) :=
   by
@@ -2611,66 +1612,30 @@ theorem Metric.uniformity_eq_comap_nhds_zero :
   simp [mem_uniformity_dist, (nhds_basis_ball.comap _).mem_iff, subset_def, Real.dist_0_eq_abs]
 #align metric.uniformity_eq_comap_nhds_zero Metric.uniformity_eq_comap_nhds_zero
 
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 theorem cauchySeq_iff_tendsto_dist_atTop_0 [Nonempty β] [SemilatticeSup β] {u : β → α} :
     CauchySeq u ↔ Tendsto (fun n : β × β => dist (u n.1) (u n.2)) atTop (𝓝 0) := by
   rw [cauchySeq_iff_tendsto, Metric.uniformity_eq_comap_nhds_zero, tendsto_comap_iff, Prod.map_def]
 #align cauchy_seq_iff_tendsto_dist_at_top_0 cauchySeq_iff_tendsto_dist_atTop_0
 
-/- warning: tendsto_uniformity_iff_dist_tendsto_zero -> tendsto_uniformity_iff_dist_tendsto_zero is a dubious translation:
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-Case conversion may be inaccurate. Consider using '#align tendsto_uniformity_iff_dist_tendsto_zero tendsto_uniformity_iff_dist_tendsto_zeroₓ'. -/
 theorem tendsto_uniformity_iff_dist_tendsto_zero {ι : Type _} {f : ι → α × α} {p : Filter ι} :
     Tendsto f p (𝓤 α) ↔ Tendsto (fun x => dist (f x).1 (f x).2) p (𝓝 0) := by
   rw [Metric.uniformity_eq_comap_nhds_zero, tendsto_comap_iff]
 #align tendsto_uniformity_iff_dist_tendsto_zero tendsto_uniformity_iff_dist_tendsto_zero
 
-/- warning: filter.tendsto.congr_dist -> Filter.Tendsto.congr_dist is a dubious translation:
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-Case conversion may be inaccurate. Consider using '#align filter.tendsto.congr_dist Filter.Tendsto.congr_distₓ'. -/
 theorem Filter.Tendsto.congr_dist {ι : Type _} {f₁ f₂ : ι → α} {p : Filter ι} {a : α}
     (h₁ : Tendsto f₁ p (𝓝 a)) (h : Tendsto (fun x => dist (f₁ x) (f₂ x)) p (𝓝 0)) :
     Tendsto f₂ p (𝓝 a) :=
   h₁.congr_uniformity <| tendsto_uniformity_iff_dist_tendsto_zero.2 h
 #align filter.tendsto.congr_dist Filter.Tendsto.congr_dist
 
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-Case conversion may be inaccurate. Consider using '#align tendsto_of_tendsto_of_dist tendsto_of_tendsto_of_distₓ'. -/
 alias Filter.Tendsto.congr_dist ← tendsto_of_tendsto_of_dist
 #align tendsto_of_tendsto_of_dist tendsto_of_tendsto_of_dist
 
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 theorem tendsto_iff_of_dist {ι : Type _} {f₁ f₂ : ι → α} {p : Filter ι} {a : α}
     (h : Tendsto (fun x => dist (f₁ x) (f₂ x)) p (𝓝 0)) : Tendsto f₁ p (𝓝 a) ↔ Tendsto f₂ p (𝓝 a) :=
   Uniform.tendsto_congr <| tendsto_uniformity_iff_dist_tendsto_zero.2 h
 #align tendsto_iff_of_dist tendsto_iff_of_dist
 
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 /-- If `u` is a neighborhood of `x`, then for small enough `r`, the closed ball
 `closed_ball x r` is contained in `u`. -/
 theorem eventually_closedBall_subset {x : α} {u : Set α} (hu : u ∈ 𝓝 x) :
@@ -2688,12 +1653,6 @@ section CauchySeq
 
 variable [Nonempty β] [SemilatticeSup β]
 
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 /- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (m n «expr ≥ » N) -/
 -- see Note [nolint_ge]
 /-- In a pseudometric space, Cauchy sequences are characterized by the fact that, eventually,
@@ -2704,24 +1663,12 @@ theorem Metric.cauchySeq_iff {u : β → α} :
   uniformity_basis_dist.cauchySeq_iff
 #align metric.cauchy_seq_iff Metric.cauchySeq_iff
 
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 /-- A variation around the pseudometric characterization of Cauchy sequences -/
 theorem Metric.cauchySeq_iff' {u : β → α} :
     CauchySeq u ↔ ∀ ε > 0, ∃ N, ∀ n ≥ N, dist (u n) (u N) < ε :=
   uniformity_basis_dist.cauchySeq_iff'
 #align metric.cauchy_seq_iff' Metric.cauchySeq_iff'
 
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 -- see Note [nolint_ge]
 /-- In a pseudometric space, unifom Cauchy sequences are characterized by the fact that, eventually,
 the distance between all its elements is uniformly, arbitrarily small -/
@@ -2752,12 +1699,6 @@ theorem Metric.uniformCauchySeqOn_iff {γ : Type _} {F : β → γ → α} {s :
     exact hab (hN b.fst hbl.ge b.snd hbr.ge x hx)
 #align metric.uniform_cauchy_seq_on_iff Metric.uniformCauchySeqOn_iff
 
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 /-- If the distance between `s n` and `s m`, `n ≤ m` is bounded above by `b n`
 and `b` converges to zero, then `s` is a Cauchy sequence.  -/
 theorem cauchySeq_of_le_tendsto_0' {s : β → α} (b : β → ℝ)
@@ -2771,12 +1712,6 @@ theorem cauchySeq_of_le_tendsto_0' {s : β → α} (b : β → ℝ)
         
 #align cauchy_seq_of_le_tendsto_0' cauchySeq_of_le_tendsto_0'
 
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 /-- If the distance between `s n` and `s m`, `n, m ≥ N` is bounded above by `b N`
 and `b` converges to zero, then `s` is a Cauchy sequence.  -/
 theorem cauchySeq_of_le_tendsto_0 {s : β → α} (b : β → ℝ)
@@ -2785,12 +1720,6 @@ theorem cauchySeq_of_le_tendsto_0 {s : β → α} (b : β → ℝ)
   cauchySeq_of_le_tendsto_0' b (fun n m hnm => h _ _ _ le_rfl hnm) h₀
 #align cauchy_seq_of_le_tendsto_0 cauchySeq_of_le_tendsto_0
 
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-  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {u : Nat -> α}, (CauchySeq.{u1, 0} α Nat (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1) (Lattice.toSemilatticeSup.{0} Nat (DistribLattice.toLattice.{0} Nat instDistribLatticeNat)) u) -> (Exists.{1} Real (fun (R : Real) => And (GT.gt.{0} Real Real.instLTReal R (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal))) (forall (m : Nat) (n : Nat), LT.lt.{0} Real Real.instLTReal (Dist.dist.{u1} α (PseudoMetricSpace.toDist.{u1} α _inst_1) (u m) (u n)) R)))
-Case conversion may be inaccurate. Consider using '#align cauchy_seq_bdd cauchySeq_bddₓ'. -/
 /-- A Cauchy sequence on the natural numbers is bounded. -/
 theorem cauchySeq_bdd {u : ℕ → α} (hu : CauchySeq u) : ∃ R > 0, ∀ m n, dist (u m) (u n) < R :=
   by
@@ -2808,12 +1737,6 @@ theorem cauchySeq_bdd {u : ℕ → α} (hu : CauchySeq u) : ∃ R > 0, ∀ m n,
     exact lt_of_le_of_lt this (lt_add_of_pos_right _ zero_lt_one)
 #align cauchy_seq_bdd cauchySeq_bdd
 
-/- warning: cauchy_seq_iff_le_tendsto_0 -> cauchySeq_iff_le_tendsto_0 is a dubious translation:
-lean 3 declaration is
-  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {s : Nat -> α}, Iff (CauchySeq.{u1, 0} α Nat (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1) (CanonicallyLinearOrderedAddMonoid.semilatticeSup.{0} Nat Nat.canonicallyLinearOrderedAddMonoid) s) (Exists.{1} (Nat -> Real) (fun (b : Nat -> Real) => And (forall (n : Nat), LE.le.{0} Real Real.hasLe (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero))) (b n)) (And (forall (n : Nat) (m : Nat) (N : Nat), (LE.le.{0} Nat Nat.hasLe N n) -> (LE.le.{0} Nat Nat.hasLe N m) -> (LE.le.{0} Real Real.hasLe (Dist.dist.{u1} α (PseudoMetricSpace.toHasDist.{u1} α _inst_1) (s n) (s m)) (b N))) (Filter.Tendsto.{0, 0} Nat Real b (Filter.atTop.{0} Nat (PartialOrder.toPreorder.{0} Nat (OrderedCancelAddCommMonoid.toPartialOrder.{0} Nat (StrictOrderedSemiring.toOrderedCancelAddCommMonoid.{0} Nat Nat.strictOrderedSemiring)))) (nhds.{0} Real (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero))))))))
-but is expected to have type
-  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {s : Nat -> α}, Iff (CauchySeq.{u1, 0} α Nat (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1) (Lattice.toSemilatticeSup.{0} Nat (DistribLattice.toLattice.{0} Nat instDistribLatticeNat)) s) (Exists.{1} (Nat -> Real) (fun (b : Nat -> Real) => And (forall (n : Nat), LE.le.{0} Real Real.instLEReal (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal)) (b n)) (And (forall (n : Nat) (m : Nat) (N : Nat), (LE.le.{0} Nat instLENat N n) -> (LE.le.{0} Nat instLENat N m) -> (LE.le.{0} Real Real.instLEReal (Dist.dist.{u1} α (PseudoMetricSpace.toDist.{u1} α _inst_1) (s n) (s m)) (b N))) (Filter.Tendsto.{0, 0} Nat Real b (Filter.atTop.{0} Nat (PartialOrder.toPreorder.{0} Nat (StrictOrderedSemiring.toPartialOrder.{0} Nat Nat.strictOrderedSemiring))) (nhds.{0} Real (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal)))))))
-Case conversion may be inaccurate. Consider using '#align cauchy_seq_iff_le_tendsto_0 cauchySeq_iff_le_tendsto_0ₓ'. -/
 /-- Yet another metric characterization of Cauchy sequences on integers. This one is often the
 most efficient. -/
 theorem cauchySeq_iff_le_tendsto_0 {s : ℕ → α} :
@@ -2958,22 +1881,10 @@ section NNReal
 instance : PseudoMetricSpace ℝ≥0 :=
   Subtype.pseudoMetricSpace
 
-/- warning: nnreal.dist_eq -> NNReal.dist_eq is a dubious translation:
-lean 3 declaration is
-  forall (a : NNReal) (b : NNReal), Eq.{1} Real (Dist.dist.{0} NNReal (PseudoMetricSpace.toHasDist.{0} NNReal NNReal.pseudoMetricSpace) a b) (Abs.abs.{0} Real (Neg.toHasAbs.{0} Real Real.hasNeg Real.hasSup) (HSub.hSub.{0, 0, 0} Real Real Real (instHSub.{0} Real Real.hasSub) ((fun (a : Type) (b : Type) [self : HasLiftT.{1, 1} a b] => self.0) NNReal Real (HasLiftT.mk.{1, 1} NNReal Real (CoeTCₓ.coe.{1, 1} NNReal Real (coeBase.{1, 1} NNReal Real NNReal.Real.hasCoe))) a) ((fun (a : Type) (b : Type) [self : HasLiftT.{1, 1} a b] => self.0) NNReal Real (HasLiftT.mk.{1, 1} NNReal Real (CoeTCₓ.coe.{1, 1} NNReal Real (coeBase.{1, 1} NNReal Real NNReal.Real.hasCoe))) b)))
-but is expected to have type
-  forall (a : NNReal) (b : NNReal), Eq.{1} Real (Dist.dist.{0} NNReal (PseudoMetricSpace.toDist.{0} NNReal instPseudoMetricSpaceNNReal) a b) (Abs.abs.{0} Real (Neg.toHasAbs.{0} Real Real.instNegReal Real.instSupReal) (HSub.hSub.{0, 0, 0} Real Real Real (instHSub.{0} Real Real.instSubReal) (NNReal.toReal a) (NNReal.toReal b)))
-Case conversion may be inaccurate. Consider using '#align nnreal.dist_eq NNReal.dist_eqₓ'. -/
 theorem NNReal.dist_eq (a b : ℝ≥0) : dist a b = |(a : ℝ) - b| :=
   rfl
 #align nnreal.dist_eq NNReal.dist_eq
 
-/- warning: nnreal.nndist_eq -> NNReal.nndist_eq is a dubious translation:
-lean 3 declaration is
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-but is expected to have type
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-Case conversion may be inaccurate. Consider using '#align nnreal.nndist_eq NNReal.nndist_eqₓ'. -/
 theorem NNReal.nndist_eq (a b : ℝ≥0) : nndist a b = max (a - b) (b - a) :=
   by
   wlog h : b ≤ a
@@ -2982,34 +1893,16 @@ theorem NNReal.nndist_eq (a b : ℝ≥0) : nndist a b = max (a - b) (b - a) :=
     max_eq_left (zero_le <| a - b), ← NNReal.coe_sub h, abs_of_nonneg (a - b).coe_nonneg]
 #align nnreal.nndist_eq NNReal.nndist_eq
 
-/- warning: nnreal.nndist_zero_eq_val -> NNReal.nndist_zero_eq_val is a dubious translation:
-lean 3 declaration is
-  forall (z : NNReal), Eq.{1} NNReal (NNDist.nndist.{0} NNReal (PseudoMetricSpace.toNNDist.{0} NNReal NNReal.pseudoMetricSpace) (OfNat.ofNat.{0} NNReal 0 (OfNat.mk.{0} NNReal 0 (Zero.zero.{0} NNReal (MulZeroClass.toHasZero.{0} NNReal (NonUnitalNonAssocSemiring.toMulZeroClass.{0} NNReal (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} NNReal (Semiring.toNonAssocSemiring.{0} NNReal NNReal.semiring))))))) z) z
-but is expected to have type
-  forall (z : NNReal), Eq.{1} NNReal (NNDist.nndist.{0} NNReal (PseudoMetricSpace.toNNDist.{0} NNReal instPseudoMetricSpaceNNReal) (OfNat.ofNat.{0} NNReal 0 (Zero.toOfNat0.{0} NNReal instNNRealZero)) z) z
-Case conversion may be inaccurate. Consider using '#align nnreal.nndist_zero_eq_val NNReal.nndist_zero_eq_valₓ'. -/
 @[simp]
 theorem NNReal.nndist_zero_eq_val (z : ℝ≥0) : nndist 0 z = z := by
   simp only [NNReal.nndist_eq, max_eq_right, tsub_zero, zero_tsub, zero_le']
 #align nnreal.nndist_zero_eq_val NNReal.nndist_zero_eq_val
 
-/- warning: nnreal.nndist_zero_eq_val' -> NNReal.nndist_zero_eq_val' is a dubious translation:
-lean 3 declaration is
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-but is expected to have type
-  forall (z : NNReal), Eq.{1} NNReal (NNDist.nndist.{0} NNReal (PseudoMetricSpace.toNNDist.{0} NNReal instPseudoMetricSpaceNNReal) z (OfNat.ofNat.{0} NNReal 0 (Zero.toOfNat0.{0} NNReal instNNRealZero))) z
-Case conversion may be inaccurate. Consider using '#align nnreal.nndist_zero_eq_val' NNReal.nndist_zero_eq_val'ₓ'. -/
 @[simp]
 theorem NNReal.nndist_zero_eq_val' (z : ℝ≥0) : nndist z 0 = z := by rw [nndist_comm];
   exact NNReal.nndist_zero_eq_val z
 #align nnreal.nndist_zero_eq_val' NNReal.nndist_zero_eq_val'
 
-/- warning: nnreal.le_add_nndist -> NNReal.le_add_nndist is a dubious translation:
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-but is expected to have type
-  forall (a : NNReal) (b : NNReal), LE.le.{0} NNReal (Preorder.toLE.{0} NNReal (PartialOrder.toPreorder.{0} NNReal (StrictOrderedSemiring.toPartialOrder.{0} NNReal instNNRealStrictOrderedSemiring))) a (HAdd.hAdd.{0, 0, 0} NNReal NNReal NNReal (instHAdd.{0} NNReal (Distrib.toAdd.{0} NNReal (NonUnitalNonAssocSemiring.toDistrib.{0} NNReal (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} NNReal (Semiring.toNonAssocSemiring.{0} NNReal instNNRealSemiring))))) b (NNDist.nndist.{0} NNReal (PseudoMetricSpace.toNNDist.{0} NNReal instPseudoMetricSpaceNNReal) a b))
-Case conversion may be inaccurate. Consider using '#align nnreal.le_add_nndist NNReal.le_add_nndistₓ'. -/
 theorem NNReal.le_add_nndist (a b : ℝ≥0) : a ≤ b + nndist a b :=
   by
   suffices (a : ℝ) ≤ (b : ℝ) + dist a b by exact nnreal.coe_le_coe.mp this
@@ -3025,43 +1918,19 @@ variable [PseudoMetricSpace β]
 instance : PseudoMetricSpace (ULift β) :=
   PseudoMetricSpace.induced ULift.down ‹_›
 
-/- warning: ulift.dist_eq -> ULift.dist_eq is a dubious translation:
-lean 3 declaration is
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-but is expected to have type
-  forall {β : Type.{u2}} [_inst_2 : PseudoMetricSpace.{u2} β] (x : ULift.{u1, u2} β) (y : ULift.{u1, u2} β), Eq.{1} Real (Dist.dist.{max u2 u1} (ULift.{u1, u2} β) (PseudoMetricSpace.toDist.{max u2 u1} (ULift.{u1, u2} β) (instPseudoMetricSpaceULift.{u2, u1} β _inst_2)) x y) (Dist.dist.{u2} β (PseudoMetricSpace.toDist.{u2} β _inst_2) (ULift.down.{u1, u2} β x) (ULift.down.{u1, u2} β y))
-Case conversion may be inaccurate. Consider using '#align ulift.dist_eq ULift.dist_eqₓ'. -/
 theorem ULift.dist_eq (x y : ULift β) : dist x y = dist x.down y.down :=
   rfl
 #align ulift.dist_eq ULift.dist_eq
 
-/- warning: ulift.nndist_eq -> ULift.nndist_eq is a dubious translation:
-lean 3 declaration is
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-but is expected to have type
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-Case conversion may be inaccurate. Consider using '#align ulift.nndist_eq ULift.nndist_eqₓ'. -/
 theorem ULift.nndist_eq (x y : ULift β) : nndist x y = nndist x.down y.down :=
   rfl
 #align ulift.nndist_eq ULift.nndist_eq
 
-/- warning: ulift.dist_up_up -> ULift.dist_up_up is a dubious translation:
-lean 3 declaration is
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-but is expected to have type
-  forall {β : Type.{u2}} [_inst_2 : PseudoMetricSpace.{u2} β] (x : β) (y : β), Eq.{1} Real (Dist.dist.{max u2 u1} (ULift.{u1, u2} β) (PseudoMetricSpace.toDist.{max u2 u1} (ULift.{u1, u2} β) (instPseudoMetricSpaceULift.{u2, u1} β _inst_2)) (ULift.up.{u1, u2} β x) (ULift.up.{u1, u2} β y)) (Dist.dist.{u2} β (PseudoMetricSpace.toDist.{u2} β _inst_2) x y)
-Case conversion may be inaccurate. Consider using '#align ulift.dist_up_up ULift.dist_up_upₓ'. -/
 @[simp]
 theorem ULift.dist_up_up (x y : β) : dist (ULift.up x) (ULift.up y) = dist x y :=
   rfl
 #align ulift.dist_up_up ULift.dist_up_up
 
-/- warning: ulift.nndist_up_up -> ULift.nndist_up_up is a dubious translation:
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-but is expected to have type
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-Case conversion may be inaccurate. Consider using '#align ulift.nndist_up_up ULift.nndist_up_upₓ'. -/
 @[simp]
 theorem ULift.nndist_up_up (x y : β) : nndist (ULift.up x) (ULift.up y) = nndist x y :=
   rfl
@@ -3087,12 +1956,6 @@ instance Prod.pseudoMetricSpaceMax : PseudoMetricSpace (α × β) :=
 #align prod.pseudo_metric_space_max Prod.pseudoMetricSpaceMax
 -/
 
-/- warning: prod.dist_eq -> Prod.dist_eq is a dubious translation:
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-but is expected to have type
-  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : PseudoMetricSpace.{u1} α] [_inst_2 : PseudoMetricSpace.{u2} β] {x : Prod.{u1, u2} α β} {y : Prod.{u1, u2} α β}, Eq.{1} Real (Dist.dist.{max u1 u2} (Prod.{u1, u2} α β) (PseudoMetricSpace.toDist.{max u1 u2} (Prod.{u1, u2} α β) (Prod.pseudoMetricSpaceMax.{u1, u2} α β _inst_1 _inst_2)) x y) (Max.max.{0} Real (LinearOrderedRing.toMax.{0} Real Real.instLinearOrderedRingReal) (Dist.dist.{u1} α (PseudoMetricSpace.toDist.{u1} α _inst_1) (Prod.fst.{u1, u2} α β x) (Prod.fst.{u1, u2} α β y)) (Dist.dist.{u2} β (PseudoMetricSpace.toDist.{u2} β _inst_2) (Prod.snd.{u1, u2} α β x) (Prod.snd.{u1, u2} α β y)))
-Case conversion may be inaccurate. Consider using '#align prod.dist_eq Prod.dist_eqₓ'. -/
 theorem Prod.dist_eq {x y : α × β} : dist x y = max (dist x.1 y.1) (dist x.2 y.2) :=
   rfl
 #align prod.dist_eq Prod.dist_eq
@@ -3128,12 +1991,6 @@ theorem closedBall_prod_same (x : α) (y : β) (r : ℝ) :
 
 end Prod
 
-/- warning: uniform_continuous_dist -> uniformContinuous_dist is a dubious translation:
-lean 3 declaration is
-  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α], UniformContinuous.{u1, 0} (Prod.{u1, u1} α α) Real (Prod.uniformSpace.{u1, u1} α α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1) (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace) (fun (p : Prod.{u1, u1} α α) => Dist.dist.{u1} α (PseudoMetricSpace.toHasDist.{u1} α _inst_1) (Prod.fst.{u1, u1} α α p) (Prod.snd.{u1, u1} α α p))
-but is expected to have type
-  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α], UniformContinuous.{u1, 0} (Prod.{u1, u1} α α) Real (instUniformSpaceProd.{u1, u1} α α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1) (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace) (fun (p : Prod.{u1, u1} α α) => Dist.dist.{u1} α (PseudoMetricSpace.toDist.{u1} α _inst_1) (Prod.fst.{u1, u1} α α p) (Prod.snd.{u1, u1} α α p))
-Case conversion may be inaccurate. Consider using '#align uniform_continuous_dist uniformContinuous_distₓ'. -/
 theorem uniformContinuous_dist : UniformContinuous fun p : α × α => dist p.1 p.2 :=
   Metric.uniformContinuous_iff.2 fun ε ε0 =>
     ⟨ε / 2, half_pos ε0, by
@@ -3159,12 +2016,6 @@ theorem UniformContinuous.dist [UniformSpace β] {f g : β → α} (hf : Uniform
 #align uniform_continuous.dist UniformContinuous.dist
 -/
 
-/- warning: continuous_dist -> continuous_dist is a dubious translation:
-lean 3 declaration is
-  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α], Continuous.{u1, 0} (Prod.{u1, u1} α α) Real (Prod.topologicalSpace.{u1, u1} α α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1))) (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) (fun (p : Prod.{u1, u1} α α) => Dist.dist.{u1} α (PseudoMetricSpace.toHasDist.{u1} α _inst_1) (Prod.fst.{u1, u1} α α p) (Prod.snd.{u1, u1} α α p))
-but is expected to have type
-  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α], Continuous.{u1, 0} (Prod.{u1, u1} α α) Real (instTopologicalSpaceProd.{u1, u1} α α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1))) (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) (fun (p : Prod.{u1, u1} α α) => Dist.dist.{u1} α (PseudoMetricSpace.toDist.{u1} α _inst_1) (Prod.fst.{u1, u1} α α p) (Prod.snd.{u1, u1} α α p))
-Case conversion may be inaccurate. Consider using '#align continuous_dist continuous_distₓ'. -/
 @[continuity]
 theorem continuous_dist : Continuous fun p : α × α => dist p.1 p.2 :=
   uniformContinuous_dist.Continuous
@@ -3185,34 +2036,16 @@ theorem Filter.Tendsto.dist {f g : β → α} {x : Filter β} {a b : α} (hf : T
 #align filter.tendsto.dist Filter.Tendsto.dist
 -/
 
-/- warning: nhds_comap_dist -> nhds_comap_dist is a dubious translation:
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-but is expected to have type
-  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] (a : α), Eq.{succ u1} (Filter.{u1} α) (Filter.comap.{u1, 0} α Real (fun (a' : α) => Dist.dist.{u1} α (PseudoMetricSpace.toDist.{u1} α _inst_1) a' a) (nhds.{0} Real (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal)))) (nhds.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) a)
-Case conversion may be inaccurate. Consider using '#align nhds_comap_dist nhds_comap_distₓ'. -/
 theorem nhds_comap_dist (a : α) : ((𝓝 (0 : ℝ)).comap fun a' => dist a' a) = 𝓝 a := by
   simp only [@nhds_eq_comap_uniformity α, Metric.uniformity_eq_comap_nhds_zero, comap_comap,
     (· ∘ ·), dist_comm]
 #align nhds_comap_dist nhds_comap_dist
 
-/- warning: tendsto_iff_dist_tendsto_zero -> tendsto_iff_dist_tendsto_zero is a dubious translation:
-lean 3 declaration is
-  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : PseudoMetricSpace.{u1} α] {f : β -> α} {x : Filter.{u2} β} {a : α}, Iff (Filter.Tendsto.{u2, u1} β α f x (nhds.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) a)) (Filter.Tendsto.{u2, 0} β Real (fun (b : β) => Dist.dist.{u1} α (PseudoMetricSpace.toHasDist.{u1} α _inst_1) (f b) a) x (nhds.{0} Real (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero)))))
-but is expected to have type
-  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : PseudoMetricSpace.{u1} α] {f : β -> α} {x : Filter.{u2} β} {a : α}, Iff (Filter.Tendsto.{u2, u1} β α f x (nhds.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) a)) (Filter.Tendsto.{u2, 0} β Real (fun (b : β) => Dist.dist.{u1} α (PseudoMetricSpace.toDist.{u1} α _inst_1) (f b) a) x (nhds.{0} Real (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal))))
-Case conversion may be inaccurate. Consider using '#align tendsto_iff_dist_tendsto_zero tendsto_iff_dist_tendsto_zeroₓ'. -/
 theorem tendsto_iff_dist_tendsto_zero {f : β → α} {x : Filter β} {a : α} :
     Tendsto f x (𝓝 a) ↔ Tendsto (fun b => dist (f b) a) x (𝓝 0) := by
   rw [← nhds_comap_dist a, tendsto_comap_iff]
 #align tendsto_iff_dist_tendsto_zero tendsto_iff_dist_tendsto_zero
 
-/- warning: continuous_iff_continuous_dist -> continuous_iff_continuous_dist is a dubious translation:
-lean 3 declaration is
-  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : PseudoMetricSpace.{u1} α] [_inst_2 : TopologicalSpace.{u2} β] {f : β -> α}, Iff (Continuous.{u2, u1} β α _inst_2 (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) f) (Continuous.{u2, 0} (Prod.{u2, u2} β β) Real (Prod.topologicalSpace.{u2, u2} β β _inst_2 _inst_2) (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) (fun (x : Prod.{u2, u2} β β) => Dist.dist.{u1} α (PseudoMetricSpace.toHasDist.{u1} α _inst_1) (f (Prod.fst.{u2, u2} β β x)) (f (Prod.snd.{u2, u2} β β x))))
-but is expected to have type
-  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : PseudoMetricSpace.{u1} α] [_inst_2 : TopologicalSpace.{u2} β] {f : β -> α}, Iff (Continuous.{u2, u1} β α _inst_2 (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) f) (Continuous.{u2, 0} (Prod.{u2, u2} β β) Real (instTopologicalSpaceProd.{u2, u2} β β _inst_2 _inst_2) (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) (fun (x : Prod.{u2, u2} β β) => Dist.dist.{u1} α (PseudoMetricSpace.toDist.{u1} α _inst_1) (f (Prod.fst.{u2, u2} β β x)) (f (Prod.snd.{u2, u2} β β x))))
-Case conversion may be inaccurate. Consider using '#align continuous_iff_continuous_dist continuous_iff_continuous_distₓ'. -/
 theorem continuous_iff_continuous_dist [TopologicalSpace β] {f : β → α} :
     Continuous f ↔ Continuous fun x : β × β => dist (f x.1) (f x.2) :=
   ⟨fun h => (h.comp continuous_fst).dist (h.comp continuous_snd), fun h =>
@@ -3221,54 +2054,24 @@ theorem continuous_iff_continuous_dist [TopologicalSpace β] {f : β → α} :
         (h.comp (continuous_id.prod_mk continuous_const)).tendsto' _ _ <| dist_self _⟩
 #align continuous_iff_continuous_dist continuous_iff_continuous_dist
 
-/- warning: uniform_continuous_nndist -> uniformContinuous_nndist is a dubious translation:
-lean 3 declaration is
-  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α], UniformContinuous.{u1, 0} (Prod.{u1, u1} α α) NNReal (Prod.uniformSpace.{u1, u1} α α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1) (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) (PseudoMetricSpace.toUniformSpace.{0} NNReal NNReal.pseudoMetricSpace) (fun (p : Prod.{u1, u1} α α) => NNDist.nndist.{u1} α (PseudoMetricSpace.toNNDist.{u1} α _inst_1) (Prod.fst.{u1, u1} α α p) (Prod.snd.{u1, u1} α α p))
-but is expected to have type
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-Case conversion may be inaccurate. Consider using '#align uniform_continuous_nndist uniformContinuous_nndistₓ'. -/
 theorem uniformContinuous_nndist : UniformContinuous fun p : α × α => nndist p.1 p.2 :=
   uniformContinuous_dist.subtype_mk _
 #align uniform_continuous_nndist uniformContinuous_nndist
 
-/- warning: uniform_continuous.nndist -> UniformContinuous.nndist is a dubious translation:
-lean 3 declaration is
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-but is expected to have type
-  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : PseudoMetricSpace.{u1} α] [_inst_2 : UniformSpace.{u2} β] {f : β -> α} {g : β -> α}, (UniformContinuous.{u2, u1} β α _inst_2 (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1) f) -> (UniformContinuous.{u2, u1} β α _inst_2 (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1) g) -> (UniformContinuous.{u2, 0} β NNReal _inst_2 (PseudoMetricSpace.toUniformSpace.{0} NNReal instPseudoMetricSpaceNNReal) (fun (b : β) => NNDist.nndist.{u1} α (PseudoMetricSpace.toNNDist.{u1} α _inst_1) (f b) (g b)))
-Case conversion may be inaccurate. Consider using '#align uniform_continuous.nndist UniformContinuous.nndistₓ'. -/
 theorem UniformContinuous.nndist [UniformSpace β] {f g : β → α} (hf : UniformContinuous f)
     (hg : UniformContinuous g) : UniformContinuous fun b => nndist (f b) (g b) :=
   uniformContinuous_nndist.comp (hf.prod_mk hg)
 #align uniform_continuous.nndist UniformContinuous.nndist
 
-/- warning: continuous_nndist -> continuous_nndist is a dubious translation:
-lean 3 declaration is
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-but is expected to have type
-  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α], Continuous.{u1, 0} (Prod.{u1, u1} α α) NNReal (instTopologicalSpaceProd.{u1, u1} α α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1))) (UniformSpace.toTopologicalSpace.{0} NNReal (PseudoMetricSpace.toUniformSpace.{0} NNReal instPseudoMetricSpaceNNReal)) (fun (p : Prod.{u1, u1} α α) => NNDist.nndist.{u1} α (PseudoMetricSpace.toNNDist.{u1} α _inst_1) (Prod.fst.{u1, u1} α α p) (Prod.snd.{u1, u1} α α p))
-Case conversion may be inaccurate. Consider using '#align continuous_nndist continuous_nndistₓ'. -/
 theorem continuous_nndist : Continuous fun p : α × α => nndist p.1 p.2 :=
   uniformContinuous_nndist.Continuous
 #align continuous_nndist continuous_nndist
 
-/- warning: continuous.nndist -> Continuous.nndist is a dubious translation:
-lean 3 declaration is
-  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : PseudoMetricSpace.{u1} α] [_inst_2 : TopologicalSpace.{u2} β] {f : β -> α} {g : β -> α}, (Continuous.{u2, u1} β α _inst_2 (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) f) -> (Continuous.{u2, u1} β α _inst_2 (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) g) -> (Continuous.{u2, 0} β NNReal _inst_2 (UniformSpace.toTopologicalSpace.{0} NNReal (PseudoMetricSpace.toUniformSpace.{0} NNReal NNReal.pseudoMetricSpace)) (fun (b : β) => NNDist.nndist.{u1} α (PseudoMetricSpace.toNNDist.{u1} α _inst_1) (f b) (g b)))
-but is expected to have type
-  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : PseudoMetricSpace.{u1} α] [_inst_2 : TopologicalSpace.{u2} β] {f : β -> α} {g : β -> α}, (Continuous.{u2, u1} β α _inst_2 (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) f) -> (Continuous.{u2, u1} β α _inst_2 (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) g) -> (Continuous.{u2, 0} β NNReal _inst_2 (UniformSpace.toTopologicalSpace.{0} NNReal (PseudoMetricSpace.toUniformSpace.{0} NNReal instPseudoMetricSpaceNNReal)) (fun (b : β) => NNDist.nndist.{u1} α (PseudoMetricSpace.toNNDist.{u1} α _inst_1) (f b) (g b)))
-Case conversion may be inaccurate. Consider using '#align continuous.nndist Continuous.nndistₓ'. -/
 theorem Continuous.nndist [TopologicalSpace β] {f g : β → α} (hf : Continuous f)
     (hg : Continuous g) : Continuous fun b => nndist (f b) (g b) :=
   continuous_nndist.comp (hf.prod_mk hg : _)
 #align continuous.nndist Continuous.nndist
 
-/- warning: filter.tendsto.nndist -> Filter.Tendsto.nndist is a dubious translation:
-lean 3 declaration is
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-but is expected to have type
-  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : PseudoMetricSpace.{u1} α] {f : β -> α} {g : β -> α} {x : Filter.{u2} β} {a : α} {b : α}, (Filter.Tendsto.{u2, u1} β α f x (nhds.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) a)) -> (Filter.Tendsto.{u2, u1} β α g x (nhds.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) b)) -> (Filter.Tendsto.{u2, 0} β NNReal (fun (x : β) => NNDist.nndist.{u1} α (PseudoMetricSpace.toNNDist.{u1} α _inst_1) (f x) (g x)) x (nhds.{0} NNReal (UniformSpace.toTopologicalSpace.{0} NNReal (PseudoMetricSpace.toUniformSpace.{0} NNReal instPseudoMetricSpaceNNReal)) (NNDist.nndist.{u1} α (PseudoMetricSpace.toNNDist.{u1} α _inst_1) a b)))
-Case conversion may be inaccurate. Consider using '#align filter.tendsto.nndist Filter.Tendsto.nndistₓ'. -/
 theorem Filter.Tendsto.nndist {f g : β → α} {x : Filter β} {a b : α} (hf : Tendsto f x (𝓝 a))
     (hg : Tendsto g x (𝓝 b)) : Tendsto (fun x => nndist (f x) (g x)) x (𝓝 (nndist a b)) :=
   (continuous_nndist.Tendsto (a, b)).comp (hf.prod_mk_nhds hg)
@@ -3328,57 +2131,27 @@ theorem ball_subset_interior_closedBall : ball x ε ⊆ interior (closedBall x 
 #align metric.ball_subset_interior_closed_ball Metric.ball_subset_interior_closedBall
 -/
 
-/- warning: metric.mem_closure_iff -> Metric.mem_closure_iff is a dubious translation:
-lean 3 declaration is
-  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {s : Set.{u1} α} {a : α}, Iff (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) a (closure.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) s)) (forall (ε : Real), (GT.gt.{0} Real Real.hasLt ε (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero)))) -> (Exists.{succ u1} α (fun (b : α) => Exists.{0} (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) b s) (fun (H : Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) b s) => LT.lt.{0} Real Real.hasLt (Dist.dist.{u1} α (PseudoMetricSpace.toHasDist.{u1} α _inst_1) a b) ε))))
-but is expected to have type
-  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {s : Set.{u1} α} {a : α}, Iff (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) a (closure.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) s)) (forall (ε : Real), (GT.gt.{0} Real Real.instLTReal ε (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal))) -> (Exists.{succ u1} α (fun (b : α) => And (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) b s) (LT.lt.{0} Real Real.instLTReal (Dist.dist.{u1} α (PseudoMetricSpace.toDist.{u1} α _inst_1) a b) ε))))
-Case conversion may be inaccurate. Consider using '#align metric.mem_closure_iff Metric.mem_closure_iffₓ'. -/
 /-- ε-characterization of the closure in pseudometric spaces-/
 theorem mem_closure_iff {s : Set α} {a : α} : a ∈ closure s ↔ ∀ ε > 0, ∃ b ∈ s, dist a b < ε :=
   (mem_closure_iff_nhds_basis nhds_basis_ball).trans <| by simp only [mem_ball, dist_comm]
 #align metric.mem_closure_iff Metric.mem_closure_iff
 
-/- warning: metric.mem_closure_range_iff -> Metric.mem_closure_range_iff is a dubious translation:
-lean 3 declaration is
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-Case conversion may be inaccurate. Consider using '#align metric.mem_closure_range_iff Metric.mem_closure_range_iffₓ'. -/
 theorem mem_closure_range_iff {e : β → α} {a : α} :
     a ∈ closure (range e) ↔ ∀ ε > 0, ∃ k : β, dist a (e k) < ε := by
   simp only [mem_closure_iff, exists_range_iff]
 #align metric.mem_closure_range_iff Metric.mem_closure_range_iff
 
-/- warning: metric.mem_closure_range_iff_nat -> Metric.mem_closure_range_iff_nat is a dubious translation:
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-Case conversion may be inaccurate. Consider using '#align metric.mem_closure_range_iff_nat Metric.mem_closure_range_iff_natₓ'. -/
 theorem mem_closure_range_iff_nat {e : β → α} {a : α} :
     a ∈ closure (range e) ↔ ∀ n : ℕ, ∃ k : β, dist a (e k) < 1 / ((n : ℝ) + 1) :=
   (mem_closure_iff_nhds_basis nhds_basis_ball_inv_nat_succ).trans <| by
     simp only [mem_ball, dist_comm, exists_range_iff, forall_const]
 #align metric.mem_closure_range_iff_nat Metric.mem_closure_range_iff_nat
 
-/- warning: metric.mem_of_closed' -> Metric.mem_of_closed' is a dubious translation:
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-Case conversion may be inaccurate. Consider using '#align metric.mem_of_closed' Metric.mem_of_closed'ₓ'. -/
 theorem mem_of_closed' {s : Set α} (hs : IsClosed s) {a : α} :
     a ∈ s ↔ ∀ ε > 0, ∃ b ∈ s, dist a b < ε := by
   simpa only [hs.closure_eq] using @mem_closure_iff _ _ s a
 #align metric.mem_of_closed' Metric.mem_of_closed'
 
-/- warning: metric.closed_ball_zero' -> Metric.closedBall_zero' is a dubious translation:
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-Case conversion may be inaccurate. Consider using '#align metric.closed_ball_zero' Metric.closedBall_zero'ₓ'. -/
 theorem closedBall_zero' (x : α) : closedBall x 0 = closure {x} :=
   Subset.antisymm
     (fun y hy =>
@@ -3386,33 +2159,15 @@ theorem closedBall_zero' (x : α) : closedBall x 0 = closure {x} :=
     (closure_minimal (singleton_subset_iff.2 (dist_self x).le) isClosed_ball)
 #align metric.closed_ball_zero' Metric.closedBall_zero'
 
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-Case conversion may be inaccurate. Consider using '#align metric.dense_iff Metric.dense_iffₓ'. -/
 theorem dense_iff {s : Set α} : Dense s ↔ ∀ x, ∀ r > 0, (ball x r ∩ s).Nonempty :=
   forall_congr' fun x => by
     simp only [mem_closure_iff, Set.Nonempty, exists_prop, mem_inter_iff, mem_ball', and_comm']
 #align metric.dense_iff Metric.dense_iff
 
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-Case conversion may be inaccurate. Consider using '#align metric.dense_range_iff Metric.denseRange_iffₓ'. -/
 theorem denseRange_iff {f : β → α} : DenseRange f ↔ ∀ x, ∀ r > 0, ∃ y, dist x (f y) < r :=
   forall_congr' fun x => by simp only [mem_closure_iff, exists_range_iff]
 #align metric.dense_range_iff Metric.denseRange_iff
 
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-Case conversion may be inaccurate. Consider using '#align topological_space.is_separable.separable_space TopologicalSpace.IsSeparable.separableSpaceₓ'. -/
 /-- If a set `s` is separable, then the corresponding subtype is separable in a metric space.
 This is not obvious, as the countable set whose closure covers `s` does not need in general to
 be contained in `s`. -/
@@ -3522,42 +2277,18 @@ instance pseudoMetricSpacePi : PseudoMetricSpace (∀ b, π b) :=
 #align pseudo_metric_space_pi pseudoMetricSpacePi
 -/
 
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-Case conversion may be inaccurate. Consider using '#align nndist_pi_def nndist_pi_defₓ'. -/
 theorem nndist_pi_def (f g : ∀ b, π b) : nndist f g = sup univ fun b => nndist (f b) (g b) :=
   NNReal.eq rfl
 #align nndist_pi_def nndist_pi_def
 
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-Case conversion may be inaccurate. Consider using '#align dist_pi_def dist_pi_defₓ'. -/
 theorem dist_pi_def (f g : ∀ b, π b) : dist f g = (sup univ fun b => nndist (f b) (g b) : ℝ≥0) :=
   rfl
 #align dist_pi_def dist_pi_def
 
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 theorem nndist_pi_le_iff {f g : ∀ b, π b} {r : ℝ≥0} :
     nndist f g ≤ r ↔ ∀ b, nndist (f b) (g b) ≤ r := by simp [nndist_pi_def]
 #align nndist_pi_le_iff nndist_pi_le_iff
 
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 theorem dist_pi_lt_iff {f g : ∀ b, π b} {r : ℝ} (hr : 0 < r) :
     dist f g < r ↔ ∀ b, dist (f b) (g b) < r :=
   by
@@ -3565,12 +2296,6 @@ theorem dist_pi_lt_iff {f g : ∀ b, π b} {r : ℝ} (hr : 0 < r) :
   simp [dist_pi_def, Finset.sup_lt_iff (show ⊥ < r from hr)]
 #align dist_pi_lt_iff dist_pi_lt_iff
 
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 theorem dist_pi_le_iff {f g : ∀ b, π b} {r : ℝ} (hr : 0 ≤ r) :
     dist f g ≤ r ↔ ∀ b, dist (f b) (g b) ≤ r :=
   by
@@ -3578,12 +2303,6 @@ theorem dist_pi_le_iff {f g : ∀ b, π b} {r : ℝ} (hr : 0 ≤ r) :
   exact nndist_pi_le_iff
 #align dist_pi_le_iff dist_pi_le_iff
 
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 theorem dist_pi_le_iff' [Nonempty β] {f g : ∀ b, π b} {r : ℝ} :
     dist f g ≤ r ↔ ∀ b, dist (f b) (g b) ≤ r :=
   by
@@ -3595,22 +2314,10 @@ theorem dist_pi_le_iff' [Nonempty β] {f g : ∀ b, π b} {r : ℝ} :
         hr <| dist_nonneg.trans <| h <| Classical.arbitrary _
 #align dist_pi_le_iff' dist_pi_le_iff'
 
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 theorem dist_pi_const_le (a b : α) : (dist (fun _ : β => a) fun _ => b) ≤ dist a b :=
   (dist_pi_le_iff dist_nonneg).2 fun _ => le_rfl
 #align dist_pi_const_le dist_pi_const_le
 
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 theorem nndist_pi_const_le (a b : α) : (nndist (fun _ : β => a) fun _ => b) ≤ nndist a b :=
   nndist_pi_le_iff.2 fun _ => le_rfl
 #align nndist_pi_const_le nndist_pi_const_le
@@ -3630,44 +2337,20 @@ theorem nndist_pi_const [Nonempty β] (a b : α) :
 #align nndist_pi_const nndist_pi_const
 -/
 
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 theorem nndist_le_pi_nndist (f g : ∀ b, π b) (b : β) : nndist (f b) (g b) ≤ nndist f g := by
   rw [nndist_pi_def]; exact Finset.le_sup (Finset.mem_univ b)
 #align nndist_le_pi_nndist nndist_le_pi_nndist
 
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 theorem dist_le_pi_dist (f g : ∀ b, π b) (b : β) : dist (f b) (g b) ≤ dist f g := by
   simp only [dist_nndist, NNReal.coe_le_coe, nndist_le_pi_nndist f g b]
 #align dist_le_pi_dist dist_le_pi_dist
 
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 /-- An open ball in a product space is a product of open balls. See also `metric.ball_pi'`
 for a version assuming `nonempty β` instead of `0 < r`. -/
 theorem ball_pi (x : ∀ b, π b) {r : ℝ} (hr : 0 < r) :
     ball x r = Set.pi univ fun b => ball (x b) r := by ext p; simp [dist_pi_lt_iff hr]
 #align ball_pi ball_pi
 
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 /-- An open ball in a product space is a product of open balls. See also `metric.ball_pi`
 for a version assuming `0 < r` instead of `nonempty β`. -/
 theorem ball_pi' [Nonempty β] (x : ∀ b, π b) (r : ℝ) :
@@ -3675,24 +2358,12 @@ theorem ball_pi' [Nonempty β] (x : ∀ b, π b) (r : ℝ) :
   (lt_or_le 0 r).elim (ball_pi x) fun hr => by simp [ball_eq_empty.2 hr]
 #align ball_pi' ball_pi'
 
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 /-- A closed ball in a product space is a product of closed balls. See also `metric.closed_ball_pi'`
 for a version assuming `nonempty β` instead of `0 ≤ r`. -/
 theorem closedBall_pi (x : ∀ b, π b) {r : ℝ} (hr : 0 ≤ r) :
     closedBall x r = Set.pi univ fun b => closedBall (x b) r := by ext p; simp [dist_pi_le_iff hr]
 #align closed_ball_pi closedBall_pi
 
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-Case conversion may be inaccurate. Consider using '#align closed_ball_pi' closedBall_pi'ₓ'. -/
 /-- A closed ball in a product space is a product of closed balls. See also `metric.closed_ball_pi`
 for a version assuming `0 ≤ r` instead of `nonempty β`. -/
 theorem closedBall_pi' [Nonempty β] (x : ∀ b, π b) (r : ℝ) :
@@ -3700,9 +2371,6 @@ theorem closedBall_pi' [Nonempty β] (x : ∀ b, π b) (r : ℝ) :
   (le_or_lt 0 r).elim (closedBall_pi x) fun hr => by simp [closed_ball_eq_empty.2 hr]
 #align closed_ball_pi' closedBall_pi'
 
-/- warning: fin.nndist_insert_nth_insert_nth -> Fin.nndist_insertNth_insertNth is a dubious translation:
-<too large>
-Case conversion may be inaccurate. Consider using '#align fin.nndist_insert_nth_insert_nth Fin.nndist_insertNth_insertNthₓ'. -/
 @[simp]
 theorem Fin.nndist_insertNth_insertNth {n : ℕ} {α : Fin (n + 1) → Type _}
     [∀ i, PseudoMetricSpace (α i)] (i : Fin (n + 1)) (x y : α i) (f g : ∀ j, α (i.succAbove j)) :
@@ -3710,9 +2378,6 @@ theorem Fin.nndist_insertNth_insertNth {n : ℕ} {α : Fin (n + 1) → Type _}
   eq_of_forall_ge_iff fun c => by simp [nndist_pi_le_iff, i.forall_iff_succ_above]
 #align fin.nndist_insert_nth_insert_nth Fin.nndist_insertNth_insertNth
 
-/- warning: fin.dist_insert_nth_insert_nth -> Fin.dist_insertNth_insertNth is a dubious translation:
-<too large>
-Case conversion may be inaccurate. Consider using '#align fin.dist_insert_nth_insert_nth Fin.dist_insertNth_insertNthₓ'. -/
 @[simp]
 theorem Fin.dist_insertNth_insertNth {n : ℕ} {α : Fin (n + 1) → Type _}
     [∀ i, PseudoMetricSpace (α i)] (i : Fin (n + 1)) (x y : α i) (f g : ∀ j, α (i.succAbove j)) :
@@ -3720,12 +2385,6 @@ theorem Fin.dist_insertNth_insertNth {n : ℕ} {α : Fin (n + 1) → Type _}
   simp only [dist_nndist, Fin.nndist_insertNth_insertNth, NNReal.coe_max]
 #align fin.dist_insert_nth_insert_nth Fin.dist_insertNth_insertNth
 
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 theorem Real.dist_le_of_mem_pi_Icc {x y x' y' : β → ℝ} (hx : x ∈ Icc x' y') (hy : y ∈ Icc x' y') :
     dist x y ≤ dist x' y' :=
   by
@@ -3740,12 +2399,6 @@ end Pi
 
 section Compact
 
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-Case conversion may be inaccurate. Consider using '#align finite_cover_balls_of_compact finite_cover_balls_of_compactₓ'. -/
 /- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (t «expr ⊆ » s) -/
 /-- Any compact set in a pseudometric space can be covered by finitely many balls of a given
 positive radius -/
@@ -3760,12 +2413,6 @@ theorem finite_cover_balls_of_compact {α : Type u} [PseudoMetricSpace α] {s :
     exact ⟨x, ⟨xs, by simpa⟩⟩
 #align finite_cover_balls_of_compact finite_cover_balls_of_compact
 
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-Case conversion may be inaccurate. Consider using '#align is_compact.finite_cover_balls IsCompact.finite_cover_ballsₓ'. -/
 alias finite_cover_balls_of_compact ← IsCompact.finite_cover_balls
 #align is_compact.finite_cover_balls IsCompact.finite_cover_balls
 
@@ -3826,12 +2473,6 @@ theorem tendsto_dist_left_cocompact_atTop [ProperSpace α] (x : α) :
 #align tendsto_dist_left_cocompact_at_top tendsto_dist_left_cocompact_atTop
 -/
 
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-Case conversion may be inaccurate. Consider using '#align proper_space_of_compact_closed_ball_of_le properSpace_of_compact_closedBall_of_leₓ'. -/
 /-- If all closed balls of large enough radius are compact, then the space is proper. Especially
 useful when the lower bound for the radius is 0. -/
 theorem properSpace_of_compact_closedBall_of_le (R : ℝ)
@@ -3911,12 +2552,6 @@ instance pi_properSpace {π : β → Type _} [Fintype β] [∀ b, PseudoMetricSp
 
 variable [ProperSpace α] {x : α} {r : ℝ} {s : Set α}
 
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 /-- If a nonempty ball in a proper space includes a closed set `s`, then there exists a nonempty
 ball with the same center and a strictly smaller radius that includes `s`. -/
 theorem exists_pos_lt_subset_ball (hr : 0 < r) (hs : IsClosed s) (h : s ⊆ ball x r) :
@@ -3934,12 +2569,6 @@ theorem exists_pos_lt_subset_ball (hr : 0 < r) (hs : IsClosed s) (h : s ⊆ ball
   exact ⟨r', ⟨dist_nonneg.trans_lt hyr', hrr'⟩, subset.trans hy <| closed_ball_subset_ball hyr'⟩
 #align exists_pos_lt_subset_ball exists_pos_lt_subset_ball
 
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-Case conversion may be inaccurate. Consider using '#align exists_lt_subset_ball exists_lt_subset_ballₓ'. -/
 /-- If a ball in a proper space includes a closed set `s`, then there exists a ball with the same
 center and a strictly smaller radius that includes `s`. -/
 theorem exists_lt_subset_ball (hs : IsClosed s) (h : s ⊆ ball x r) : ∃ r' < r, s ⊆ ball x r' :=
@@ -3965,12 +2594,6 @@ section SecondCountable
 
 open TopologicalSpace
 
-/- warning: metric.second_countable_of_almost_dense_set -> Metric.secondCountable_of_almost_dense_set is a dubious translation:
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-but is expected to have type
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-Case conversion may be inaccurate. Consider using '#align metric.second_countable_of_almost_dense_set Metric.secondCountable_of_almost_dense_setₓ'. -/
 /-- A pseudometric space is second countable if, for every `ε > 0`, there is a countable set which
 is `ε`-dense. -/
 theorem secondCountable_of_almost_dense_set
@@ -3988,12 +2611,6 @@ end SecondCountable
 
 end Metric
 
-/- warning: lebesgue_number_lemma_of_metric -> lebesgue_number_lemma_of_metric is a dubious translation:
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-Case conversion may be inaccurate. Consider using '#align lebesgue_number_lemma_of_metric lebesgue_number_lemma_of_metricₓ'. -/
 theorem lebesgue_number_lemma_of_metric {s : Set α} {ι} {c : ι → Set α} (hs : IsCompact s)
     (hc₁ : ∀ i, IsOpen (c i)) (hc₂ : s ⊆ ⋃ i, c i) : ∃ δ > 0, ∀ x ∈ s, ∃ i, ball x δ ⊆ c i :=
   let ⟨n, en, hn⟩ := lebesgue_number_lemma hs hc₁ hc₂
@@ -4003,12 +2620,6 @@ theorem lebesgue_number_lemma_of_metric {s : Set α} {ι} {c : ι → Set α} (h
     ⟨i, fun y hy => hi (hδ (mem_ball'.mp hy))⟩⟩
 #align lebesgue_number_lemma_of_metric lebesgue_number_lemma_of_metric
 
-/- warning: lebesgue_number_lemma_of_metric_sUnion -> lebesgue_number_lemma_of_metric_sUnion is a dubious translation:
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-Case conversion may be inaccurate. Consider using '#align lebesgue_number_lemma_of_metric_sUnion lebesgue_number_lemma_of_metric_sUnionₓ'. -/
 theorem lebesgue_number_lemma_of_metric_sUnion {s : Set α} {c : Set (Set α)} (hs : IsCompact s)
     (hc₁ : ∀ t ∈ c, IsOpen t) (hc₂ : s ⊆ ⋃₀ c) : ∃ δ > 0, ∀ x ∈ s, ∃ t ∈ c, ball x δ ⊆ t := by
   rw [sUnion_eq_Union] at hc₂ <;> simpa using lebesgue_number_lemma_of_metric hs (by simpa) hc₂
@@ -4108,12 +2719,6 @@ theorem Bounded.subset_ball (h : Bounded s) (c : α) : ∃ r, s ⊆ closedBall c
 #align metric.bounded.subset_ball Metric.Bounded.subset_ball
 -/
 
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-but is expected to have type
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-Case conversion may be inaccurate. Consider using '#align metric.bounded.subset_ball_lt Metric.Bounded.subset_ball_ltₓ'. -/
 theorem Bounded.subset_ball_lt (h : Bounded s) (a : ℝ) (c : α) : ∃ r, a < r ∧ s ⊆ closedBall c r :=
   by
   rcases h.subset_ball c with ⟨r, hr⟩
@@ -4138,12 +2743,6 @@ theorem bounded_closure_iff : Bounded (closure s) ↔ Bounded s :=
 #align metric.bounded_closure_iff Metric.bounded_closure_iff
 -/
 
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-but is expected to have type
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-Case conversion may be inaccurate. Consider using '#align metric.bounded.union Metric.Bounded.unionₓ'. -/
 /-- The union of two bounded sets is bounded. -/
 theorem Bounded.union (hs : Bounded s) (ht : Bounded t) : Bounded (s ∪ t) :=
   by
@@ -4156,12 +2755,6 @@ theorem Bounded.union (hs : Bounded s) (ht : Bounded t) : Bounded (s ∪ t) :=
         (subset.trans hCt <| closed_ball_subset_closed_ball <| le_max_right _ _)⟩
 #align metric.bounded.union Metric.Bounded.union
 
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-but is expected to have type
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-Case conversion may be inaccurate. Consider using '#align metric.bounded_union Metric.bounded_unionₓ'. -/
 /-- The union of two sets is bounded iff each of the sets is bounded. -/
 @[simp]
 theorem bounded_union : Bounded (s ∪ t) ↔ Bounded s ∧ Bounded t :=
@@ -4230,12 +2823,6 @@ theorem bounded_singleton {x : α} : Bounded ({x} : Set α) :=
 #align metric.bounded_singleton Metric.bounded_singleton
 -/
 
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-but is expected to have type
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-Case conversion may be inaccurate. Consider using '#align metric.bounded_range_iff Metric.bounded_range_iffₓ'. -/
 /-- Characterization of the boundedness of the range of a function -/
 theorem bounded_range_iff {f : β → α} : Bounded (range f) ↔ ∃ C, ∀ x y, dist (f x) (f y) ≤ C :=
   exists_congr fun C =>
@@ -4290,12 +2877,6 @@ theorem bounded_range_of_tendsto (u : ℕ → α) {x : α} (hu : Tendsto u atTop
 #align metric.bounded_range_of_tendsto Metric.bounded_range_of_tendsto
 -/
 
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-Case conversion may be inaccurate. Consider using '#align metric.exists_is_open_bounded_image_inter_of_is_compact_of_forall_continuous_within_at Metric.exists_isOpen_bounded_image_inter_of_isCompact_of_forall_continuousWithinAtₓ'. -/
 /-- If a function is continuous within a set `s` at every point of a compact set `k`, then it is
 bounded on some open neighborhood of `k` in `s`. -/
 theorem exists_isOpen_bounded_image_inter_of_isCompact_of_forall_continuousWithinAt
@@ -4335,12 +2916,6 @@ theorem exists_isOpen_bounded_image_of_isCompact_of_forall_continuousAt [Topolog
 #align metric.exists_is_open_bounded_image_of_is_compact_of_forall_continuous_at Metric.exists_isOpen_bounded_image_of_isCompact_of_forall_continuousAt
 -/
 
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-Case conversion may be inaccurate. Consider using '#align metric.exists_is_open_bounded_image_inter_of_is_compact_of_continuous_on Metric.exists_isOpen_bounded_image_inter_of_isCompact_of_continuousOnₓ'. -/
 /-- If a function is continuous on a set `s` containing a compact set `k`, then it is bounded on
 some open neighborhood of `k` in `s`. -/
 theorem exists_isOpen_bounded_image_inter_of_isCompact_of_continuousOn [TopologicalSpace β]
@@ -4449,45 +3024,21 @@ noncomputable def diam (s : Set α) : ℝ :=
 #align metric.diam Metric.diam
 -/
 
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-but is expected to have type
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-Case conversion may be inaccurate. Consider using '#align metric.diam_nonneg Metric.diam_nonnegₓ'. -/
 /-- The diameter of a set is always nonnegative -/
 theorem diam_nonneg : 0 ≤ diam s :=
   ENNReal.toReal_nonneg
 #align metric.diam_nonneg Metric.diam_nonneg
 
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-but is expected to have type
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-Case conversion may be inaccurate. Consider using '#align metric.diam_subsingleton Metric.diam_subsingletonₓ'. -/
 theorem diam_subsingleton (hs : s.Subsingleton) : diam s = 0 := by
   simp only [diam, EMetric.diam_subsingleton hs, ENNReal.zero_toReal]
 #align metric.diam_subsingleton Metric.diam_subsingleton
 
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-but is expected to have type
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-Case conversion may be inaccurate. Consider using '#align metric.diam_empty Metric.diam_emptyₓ'. -/
 /-- The empty set has zero diameter -/
 @[simp]
 theorem diam_empty : diam (∅ : Set α) = 0 :=
   diam_subsingleton subsingleton_empty
 #align metric.diam_empty Metric.diam_empty
 
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-Case conversion may be inaccurate. Consider using '#align metric.diam_singleton Metric.diam_singletonₓ'. -/
 /-- A singleton has zero diameter -/
 @[simp]
 theorem diam_singleton : diam ({x} : Set α) = 0 :=
@@ -4501,12 +3052,6 @@ theorem diam_pair : diam ({x, y} : Set α) = dist x y := by
 #align metric.diam_pair Metric.diam_pair
 -/
 
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-lean 3 declaration is
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-Case conversion may be inaccurate. Consider using '#align metric.diam_triple Metric.diam_tripleₓ'. -/
 -- Does not work as a simp-lemma, since {x, y, z} reduces to (insert z (insert y {x}))
 theorem diam_triple :
     Metric.diam ({x, y, z} : Set α) = max (max (dist x y) (dist x z)) (dist y z) :=
@@ -4515,12 +3060,6 @@ theorem diam_triple :
   rw [ENNReal.toReal_max, ENNReal.toReal_max] <;> apply_rules [ne_of_lt, edist_lt_top, max_lt]
 #align metric.diam_triple Metric.diam_triple
 
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-  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {s : Set.{u1} α} {C : Real}, (forall (x : α), (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) x s) -> (forall (y : α), (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) y s) -> (LE.le.{0} Real Real.hasLe (Dist.dist.{u1} α (PseudoMetricSpace.toHasDist.{u1} α _inst_1) x y) C))) -> (LE.le.{0} ENNReal (Preorder.toHasLe.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (CompleteSemilatticeInf.toPartialOrder.{0} ENNReal (CompleteLattice.toCompleteSemilatticeInf.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))))) (EMetric.diam.{u1} α (PseudoMetricSpace.toPseudoEMetricSpace.{u1} α _inst_1) s) (ENNReal.ofReal C))
-but is expected to have type
-  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {s : Set.{u1} α} {C : Real}, (forall (x : α), (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) x s) -> (forall (y : α), (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) y s) -> (LE.le.{0} Real Real.instLEReal (Dist.dist.{u1} α (PseudoMetricSpace.toDist.{u1} α _inst_1) x y) C))) -> (LE.le.{0} ENNReal (Preorder.toLE.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (CompleteSemilatticeInf.toPartialOrder.{0} ENNReal (CompleteLattice.toCompleteSemilatticeInf.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal))))) (EMetric.diam.{u1} α (PseudoMetricSpace.toPseudoEMetricSpace.{u1} α _inst_1) s) (ENNReal.ofReal C))
-Case conversion may be inaccurate. Consider using '#align metric.ediam_le_of_forall_dist_le Metric.ediam_le_of_forall_dist_leₓ'. -/
 /-- If the distance between any two points in a set is bounded by some constant `C`,
 then `ennreal.of_real C`  bounds the emetric diameter of this set. -/
 theorem ediam_le_of_forall_dist_le {C : ℝ} (h : ∀ x ∈ s, ∀ y ∈ s, dist x y ≤ C) :
@@ -4528,12 +3067,6 @@ theorem ediam_le_of_forall_dist_le {C : ℝ} (h : ∀ x ∈ s, ∀ y ∈ s, dist
   EMetric.diam_le fun x hx y hy => (edist_dist x y).symm ▸ ENNReal.ofReal_le_ofReal (h x hx y hy)
 #align metric.ediam_le_of_forall_dist_le Metric.ediam_le_of_forall_dist_le
 
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-  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {s : Set.{u1} α} {C : Real}, (LE.le.{0} Real Real.hasLe (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero))) C) -> (forall (x : α), (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) x s) -> (forall (y : α), (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) y s) -> (LE.le.{0} Real Real.hasLe (Dist.dist.{u1} α (PseudoMetricSpace.toHasDist.{u1} α _inst_1) x y) C))) -> (LE.le.{0} Real Real.hasLe (Metric.diam.{u1} α _inst_1 s) C)
-but is expected to have type
-  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {s : Set.{u1} α} {C : Real}, (LE.le.{0} Real Real.instLEReal (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal)) C) -> (forall (x : α), (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) x s) -> (forall (y : α), (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) y s) -> (LE.le.{0} Real Real.instLEReal (Dist.dist.{u1} α (PseudoMetricSpace.toDist.{u1} α _inst_1) x y) C))) -> (LE.le.{0} Real Real.instLEReal (Metric.diam.{u1} α _inst_1 s) C)
-Case conversion may be inaccurate. Consider using '#align metric.diam_le_of_forall_dist_le Metric.diam_le_of_forall_dist_leₓ'. -/
 /-- If the distance between any two points in a set is bounded by some non-negative constant,
 this constant bounds the diameter. -/
 theorem diam_le_of_forall_dist_le {C : ℝ} (h₀ : 0 ≤ C) (h : ∀ x ∈ s, ∀ y ∈ s, dist x y ≤ C) :
@@ -4541,12 +3074,6 @@ theorem diam_le_of_forall_dist_le {C : ℝ} (h₀ : 0 ≤ C) (h : ∀ x ∈ s, 
   ENNReal.toReal_le_of_le_ofReal h₀ (ediam_le_of_forall_dist_le h)
 #align metric.diam_le_of_forall_dist_le Metric.diam_le_of_forall_dist_le
 
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-but is expected to have type
-  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {s : Set.{u1} α}, (Set.Nonempty.{u1} α s) -> (forall {C : Real}, (forall (x : α), (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) x s) -> (forall (y : α), (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) y s) -> (LE.le.{0} Real Real.instLEReal (Dist.dist.{u1} α (PseudoMetricSpace.toDist.{u1} α _inst_1) x y) C))) -> (LE.le.{0} Real Real.instLEReal (Metric.diam.{u1} α _inst_1 s) C))
-Case conversion may be inaccurate. Consider using '#align metric.diam_le_of_forall_dist_le_of_nonempty Metric.diam_le_of_forall_dist_le_of_nonemptyₓ'. -/
 /-- If the distance between any two points in a nonempty set is bounded by some constant,
 this constant bounds the diameter. -/
 theorem diam_le_of_forall_dist_le_of_nonempty (hs : s.Nonempty) {C : ℝ}
@@ -4557,12 +3084,6 @@ theorem diam_le_of_forall_dist_le_of_nonempty (hs : s.Nonempty) {C : ℝ}
   diam_le_of_forall_dist_le h₀ h
 #align metric.diam_le_of_forall_dist_le_of_nonempty Metric.diam_le_of_forall_dist_le_of_nonempty
 
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-but is expected to have type
-  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {s : Set.{u1} α} {x : α} {y : α}, (Ne.{1} ENNReal (EMetric.diam.{u1} α (PseudoMetricSpace.toPseudoEMetricSpace.{u1} α _inst_1) s) (Top.top.{0} ENNReal (CompleteLattice.toTop.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal)))) -> (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) x s) -> (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) y s) -> (LE.le.{0} Real Real.instLEReal (Dist.dist.{u1} α (PseudoMetricSpace.toDist.{u1} α _inst_1) x y) (Metric.diam.{u1} α _inst_1 s))
-Case conversion may be inaccurate. Consider using '#align metric.dist_le_diam_of_mem' Metric.dist_le_diam_of_mem'ₓ'. -/
 /-- The distance between two points in a set is controlled by the diameter of the set. -/
 theorem dist_le_diam_of_mem' (h : EMetric.diam s ≠ ⊤) (hx : x ∈ s) (hy : y ∈ s) :
     dist x y ≤ diam s := by
@@ -4571,12 +3092,6 @@ theorem dist_le_diam_of_mem' (h : EMetric.diam s ≠ ⊤) (hx : x ∈ s) (hy : y
   exact EMetric.edist_le_diam_of_mem hx hy
 #align metric.dist_le_diam_of_mem' Metric.dist_le_diam_of_mem'
 
-/- warning: metric.bounded_iff_ediam_ne_top -> Metric.bounded_iff_ediam_ne_top is a dubious translation:
-lean 3 declaration is
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-but is expected to have type
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-Case conversion may be inaccurate. Consider using '#align metric.bounded_iff_ediam_ne_top Metric.bounded_iff_ediam_ne_topₓ'. -/
 /-- Characterize the boundedness of a set in terms of the finiteness of its emetric.diameter. -/
 theorem bounded_iff_ediam_ne_top : Bounded s ↔ EMetric.diam s ≠ ⊤ :=
   Iff.intro
@@ -4584,90 +3099,42 @@ theorem bounded_iff_ediam_ne_top : Bounded s ↔ EMetric.diam s ≠ ⊤ :=
     fun h => ⟨diam s, fun x hx y hy => dist_le_diam_of_mem' h hx hy⟩
 #align metric.bounded_iff_ediam_ne_top Metric.bounded_iff_ediam_ne_top
 
-/- warning: metric.bounded.ediam_ne_top -> Metric.Bounded.ediam_ne_top is a dubious translation:
-lean 3 declaration is
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-but is expected to have type
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-Case conversion may be inaccurate. Consider using '#align metric.bounded.ediam_ne_top Metric.Bounded.ediam_ne_topₓ'. -/
 theorem Bounded.ediam_ne_top (h : Bounded s) : EMetric.diam s ≠ ⊤ :=
   bounded_iff_ediam_ne_top.1 h
 #align metric.bounded.ediam_ne_top Metric.Bounded.ediam_ne_top
 
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-but is expected to have type
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-Case conversion may be inaccurate. Consider using '#align metric.ediam_univ_eq_top_iff_noncompact Metric.ediam_univ_eq_top_iff_noncompactₓ'. -/
 theorem ediam_univ_eq_top_iff_noncompact [ProperSpace α] :
     EMetric.diam (univ : Set α) = ∞ ↔ NoncompactSpace α := by
   rw [← not_compactSpace_iff, compact_space_iff_bounded_univ, bounded_iff_ediam_ne_top,
     Classical.not_not]
 #align metric.ediam_univ_eq_top_iff_noncompact Metric.ediam_univ_eq_top_iff_noncompact
 
-/- warning: metric.ediam_univ_of_noncompact -> Metric.ediam_univ_of_noncompact is a dubious translation:
-lean 3 declaration is
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-Case conversion may be inaccurate. Consider using '#align metric.ediam_univ_of_noncompact Metric.ediam_univ_of_noncompactₓ'. -/
 @[simp]
 theorem ediam_univ_of_noncompact [ProperSpace α] [NoncompactSpace α] :
     EMetric.diam (univ : Set α) = ∞ :=
   ediam_univ_eq_top_iff_noncompact.mpr ‹_›
 #align metric.ediam_univ_of_noncompact Metric.ediam_univ_of_noncompact
 
-/- warning: metric.diam_univ_of_noncompact -> Metric.diam_univ_of_noncompact is a dubious translation:
-lean 3 declaration is
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-Case conversion may be inaccurate. Consider using '#align metric.diam_univ_of_noncompact Metric.diam_univ_of_noncompactₓ'. -/
 @[simp]
 theorem diam_univ_of_noncompact [ProperSpace α] [NoncompactSpace α] : diam (univ : Set α) = 0 := by
   simp [diam]
 #align metric.diam_univ_of_noncompact Metric.diam_univ_of_noncompact
 
-/- warning: metric.dist_le_diam_of_mem -> Metric.dist_le_diam_of_mem is a dubious translation:
-lean 3 declaration is
-  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {s : Set.{u1} α} {x : α} {y : α}, (Metric.Bounded.{u1} α _inst_1 s) -> (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) x s) -> (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) y s) -> (LE.le.{0} Real Real.hasLe (Dist.dist.{u1} α (PseudoMetricSpace.toHasDist.{u1} α _inst_1) x y) (Metric.diam.{u1} α _inst_1 s))
-but is expected to have type
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-Case conversion may be inaccurate. Consider using '#align metric.dist_le_diam_of_mem Metric.dist_le_diam_of_memₓ'. -/
 /-- The distance between two points in a set is controlled by the diameter of the set. -/
 theorem dist_le_diam_of_mem (h : Bounded s) (hx : x ∈ s) (hy : y ∈ s) : dist x y ≤ diam s :=
   dist_le_diam_of_mem' h.ediam_ne_top hx hy
 #align metric.dist_le_diam_of_mem Metric.dist_le_diam_of_mem
 
-/- warning: metric.ediam_of_unbounded -> Metric.ediam_of_unbounded is a dubious translation:
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-Case conversion may be inaccurate. Consider using '#align metric.ediam_of_unbounded Metric.ediam_of_unboundedₓ'. -/
 theorem ediam_of_unbounded (h : ¬Bounded s) : EMetric.diam s = ∞ := by
   rwa [bounded_iff_ediam_ne_top, Classical.not_not] at h
 #align metric.ediam_of_unbounded Metric.ediam_of_unbounded
 
-/- warning: metric.diam_eq_zero_of_unbounded -> Metric.diam_eq_zero_of_unbounded is a dubious translation:
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-Case conversion may be inaccurate. Consider using '#align metric.diam_eq_zero_of_unbounded Metric.diam_eq_zero_of_unboundedₓ'. -/
 /-- An unbounded set has zero diameter. If you would prefer to get the value ∞, use `emetric.diam`.
 This lemma makes it possible to avoid side conditions in some situations -/
 theorem diam_eq_zero_of_unbounded (h : ¬Bounded s) : diam s = 0 := by
   rw [diam, ediam_of_unbounded h, ENNReal.top_toReal]
 #align metric.diam_eq_zero_of_unbounded Metric.diam_eq_zero_of_unbounded
 
-/- warning: metric.diam_mono -> Metric.diam_mono is a dubious translation:
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-Case conversion may be inaccurate. Consider using '#align metric.diam_mono Metric.diam_monoₓ'. -/
 /-- If `s ⊆ t`, then the diameter of `s` is bounded by that of `t`, provided `t` is bounded. -/
 theorem diam_mono {s t : Set α} (h : s ⊆ t) (ht : Bounded t) : diam s ≤ diam t :=
   by
@@ -4676,12 +3143,6 @@ theorem diam_mono {s t : Set α} (h : s ⊆ t) (ht : Bounded t) : diam s ≤ dia
   exact EMetric.diam_mono h
 #align metric.diam_mono Metric.diam_mono
 
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 /-- The diameter of a union is controlled by the sum of the diameters, and the distance between
 any two points in each of the sets. This lemma is true without any side condition, since it is
 obviously true if `s ∪ t` is unbounded. -/
@@ -4702,12 +3163,6 @@ theorem diam_union {t : Set α} (xs : x ∈ s) (yt : y ∈ t) :
     apply_rules [add_nonneg, diam_nonneg, dist_nonneg]
 #align metric.diam_union Metric.diam_union
 
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 /-- If two sets intersect, the diameter of the union is bounded by the sum of the diameters. -/
 theorem diam_union' {t : Set α} (h : (s ∩ t).Nonempty) : diam (s ∪ t) ≤ diam s + diam t :=
   by
@@ -4715,12 +3170,6 @@ theorem diam_union' {t : Set α} (h : (s ∩ t).Nonempty) : diam (s ∪ t) ≤ d
   simpa using diam_union xs xt
 #align metric.diam_union' Metric.diam_union'
 
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 theorem diam_le_of_subset_closedBall {r : ℝ} (hr : 0 ≤ r) (h : s ⊆ closedBall x r) :
     diam s ≤ 2 * r :=
   diam_le_of_forall_dist_le (mul_nonneg zero_le_two hr) fun a ha b hb =>
@@ -4731,34 +3180,16 @@ theorem diam_le_of_subset_closedBall {r : ℝ} (hr : 0 ≤ r) (h : s ⊆ closedB
       
 #align metric.diam_le_of_subset_closed_ball Metric.diam_le_of_subset_closedBall
 
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-Case conversion may be inaccurate. Consider using '#align metric.diam_closed_ball Metric.diam_closedBallₓ'. -/
 /-- The diameter of a closed ball of radius `r` is at most `2 r`. -/
 theorem diam_closedBall {r : ℝ} (h : 0 ≤ r) : diam (closedBall x r) ≤ 2 * r :=
   diam_le_of_subset_closedBall h Subset.rfl
 #align metric.diam_closed_ball Metric.diam_closedBall
 
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 /-- The diameter of a ball of radius `r` is at most `2 r`. -/
 theorem diam_ball {r : ℝ} (h : 0 ≤ r) : diam (ball x r) ≤ 2 * r :=
   diam_le_of_subset_closedBall h ball_subset_closedBall
 #align metric.diam_ball Metric.diam_ball
 
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 /-- If a family of complete sets with diameter tending to `0` is such that each finite intersection
 is nonempty, then the total intersection is also nonempty. -/
 theorem IsComplete.nonempty_iInter_of_nonempty_biInter {s : ℕ → Set α} (h0 : IsComplete (s 0))
@@ -4785,12 +3216,6 @@ theorem IsComplete.nonempty_iInter_of_nonempty_biInter {s : ℕ → Set α} (h0
   exact I n p hp
 #align is_complete.nonempty_Inter_of_nonempty_bInter IsComplete.nonempty_iInter_of_nonempty_biInter
 
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-Case conversion may be inaccurate. Consider using '#align metric.nonempty_Inter_of_nonempty_bInter Metric.nonempty_iInter_of_nonempty_biInterₓ'. -/
 /-- In a complete space, if a family of closed sets with diameter tending to `0` is such that each
 finite intersection is nonempty, then the total intersection is also nonempty. -/
 theorem nonempty_iInter_of_nonempty_biInter [CompleteSpace α] {s : ℕ → Set α}
@@ -4801,12 +3226,6 @@ theorem nonempty_iInter_of_nonempty_biInter [CompleteSpace α] {s : ℕ → Set
 
 end Diam
 
-/- warning: metric.exists_local_min_mem_ball -> Metric.exists_local_min_mem_ball is a dubious translation:
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-  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : PseudoMetricSpace.{u1} α] [_inst_2 : ProperSpace.{u1} α _inst_1] [_inst_3 : TopologicalSpace.{u2} β] [_inst_4 : ConditionallyCompleteLinearOrder.{u2} β] [_inst_5 : OrderTopology.{u2} β _inst_3 (PartialOrder.toPreorder.{u2} β (SemilatticeInf.toPartialOrder.{u2} β (Lattice.toSemilatticeInf.{u2} β (ConditionallyCompleteLattice.toLattice.{u2} β (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u2} β _inst_4)))))] {f : α -> β} {a : α} {z : α} {r : Real}, (ContinuousOn.{u1, u2} α β (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) _inst_3 f (Metric.closedBall.{u1} α _inst_1 a r)) -> (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) z (Metric.closedBall.{u1} α _inst_1 a r)) -> (forall (z' : α), (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) z' (Metric.sphere.{u1} α _inst_1 a r)) -> (LT.lt.{u2} β (Preorder.toHasLt.{u2} β (PartialOrder.toPreorder.{u2} β (SemilatticeInf.toPartialOrder.{u2} β (Lattice.toSemilatticeInf.{u2} β (ConditionallyCompleteLattice.toLattice.{u2} β (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u2} β _inst_4)))))) (f z) (f z'))) -> (Exists.{succ u1} α (fun (z : α) => Exists.{0} (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) z (Metric.ball.{u1} α _inst_1 a r)) (fun (H : Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) z (Metric.ball.{u1} α _inst_1 a r)) => IsLocalMin.{u1, u2} α β (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) (PartialOrder.toPreorder.{u2} β (SemilatticeInf.toPartialOrder.{u2} β (Lattice.toSemilatticeInf.{u2} β (ConditionallyCompleteLattice.toLattice.{u2} β (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u2} β _inst_4))))) f z)))
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-  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : PseudoMetricSpace.{u1} α] [_inst_2 : ProperSpace.{u1} α _inst_1] [_inst_3 : TopologicalSpace.{u2} β] [_inst_4 : ConditionallyCompleteLinearOrder.{u2} β] [_inst_5 : OrderTopology.{u2} β _inst_3 (PartialOrder.toPreorder.{u2} β (SemilatticeInf.toPartialOrder.{u2} β (Lattice.toSemilatticeInf.{u2} β (ConditionallyCompleteLattice.toLattice.{u2} β (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u2} β _inst_4)))))] {f : α -> β} {a : α} {z : α} {r : Real}, (ContinuousOn.{u1, u2} α β (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) _inst_3 f (Metric.closedBall.{u1} α _inst_1 a r)) -> (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) z (Metric.closedBall.{u1} α _inst_1 a r)) -> (forall (z' : α), (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) z' (Metric.sphere.{u1} α _inst_1 a r)) -> (LT.lt.{u2} β (Preorder.toLT.{u2} β (PartialOrder.toPreorder.{u2} β (SemilatticeInf.toPartialOrder.{u2} β (Lattice.toSemilatticeInf.{u2} β (ConditionallyCompleteLattice.toLattice.{u2} β (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u2} β _inst_4)))))) (f z) (f z'))) -> (Exists.{succ u1} α (fun (z : α) => And (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) z (Metric.ball.{u1} α _inst_1 a r)) (IsLocalMin.{u1, u2} α β (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) (PartialOrder.toPreorder.{u2} β (SemilatticeInf.toPartialOrder.{u2} β (Lattice.toSemilatticeInf.{u2} β (ConditionallyCompleteLattice.toLattice.{u2} β (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u2} β _inst_4))))) f z)))
-Case conversion may be inaccurate. Consider using '#align metric.exists_local_min_mem_ball Metric.exists_local_min_mem_ballₓ'. -/
 theorem exists_local_min_mem_ball [ProperSpace α] [TopologicalSpace β]
     [ConditionallyCompleteLinearOrder β] [OrderTopology β] {f : α → β} {a z : α} {r : ℝ}
     (hf : ContinuousOn f (closedBall a r)) (hz : z ∈ closedBall a r)
@@ -4833,12 +3252,6 @@ unsafe def positivity_diam : expr → tactic strictness
 
 end Tactic
 
-/- warning: comap_dist_right_at_top_le_cocompact -> comap_dist_right_atTop_le_cocompact is a dubious translation:
-lean 3 declaration is
-  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] (x : α), LE.le.{u1} (Filter.{u1} α) (Preorder.toHasLe.{u1} (Filter.{u1} α) (PartialOrder.toPreorder.{u1} (Filter.{u1} α) (Filter.partialOrder.{u1} α))) (Filter.comap.{u1, 0} α Real (fun (y : α) => Dist.dist.{u1} α (PseudoMetricSpace.toHasDist.{u1} α _inst_1) y x) (Filter.atTop.{0} Real Real.preorder)) (Filter.cocompact.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)))
-but is expected to have type
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-Case conversion may be inaccurate. Consider using '#align comap_dist_right_at_top_le_cocompact comap_dist_right_atTop_le_cocompactₓ'. -/
 theorem comap_dist_right_atTop_le_cocompact (x : α) :
     comap (fun y => dist y x) atTop ≤ cocompact α :=
   by
@@ -4847,12 +3260,6 @@ theorem comap_dist_right_atTop_le_cocompact (x : α) :
   exact ⟨Ioi r, Ioi_mem_at_top r, fun y hy hys => (mem_closed_ball.1 <| hr hys).not_lt hy⟩
 #align comap_dist_right_at_top_le_cocompact comap_dist_right_atTop_le_cocompact
 
-/- warning: comap_dist_left_at_top_le_cocompact -> comap_dist_left_atTop_le_cocompact is a dubious translation:
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-  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] (x : α), LE.le.{u1} (Filter.{u1} α) (Preorder.toHasLe.{u1} (Filter.{u1} α) (PartialOrder.toPreorder.{u1} (Filter.{u1} α) (Filter.partialOrder.{u1} α))) (Filter.comap.{u1, 0} α Real (Dist.dist.{u1} α (PseudoMetricSpace.toHasDist.{u1} α _inst_1) x) (Filter.atTop.{0} Real Real.preorder)) (Filter.cocompact.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)))
-but is expected to have type
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-Case conversion may be inaccurate. Consider using '#align comap_dist_left_at_top_le_cocompact comap_dist_left_atTop_le_cocompactₓ'. -/
 theorem comap_dist_left_atTop_le_cocompact (x : α) : comap (dist x) atTop ≤ cocompact α := by
   simpa only [dist_comm _ x] using comap_dist_right_atTop_le_cocompact x
 #align comap_dist_left_at_top_le_cocompact comap_dist_left_atTop_le_cocompact
@@ -4899,12 +3306,6 @@ theorem MetricSpace.ext {α : Type _} {m m' : MetricSpace α} (h : m.toHasDist =
 #align metric_space.ext MetricSpace.ext
 -/
 
-/- warning: metric_space.of_dist_topology -> MetricSpace.ofDistTopology is a dubious translation:
-lean 3 declaration is
-  forall {α : Type.{u1}} [_inst_2 : TopologicalSpace.{u1} α] (dist : α -> α -> Real), (forall (x : α), Eq.{1} Real (dist x x) (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero)))) -> (forall (x : α) (y : α), Eq.{1} Real (dist x y) (dist y x)) -> (forall (x : α) (y : α) (z : α), LE.le.{0} Real Real.hasLe (dist x z) (HAdd.hAdd.{0, 0, 0} Real Real Real (instHAdd.{0} Real Real.hasAdd) (dist x y) (dist y z))) -> (forall (s : Set.{u1} α), Iff (IsOpen.{u1} α _inst_2 s) (forall (x : α), (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) x s) -> (Exists.{1} Real (fun (ε : Real) => Exists.{0} (GT.gt.{0} Real Real.hasLt ε (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero)))) (fun (H : GT.gt.{0} Real Real.hasLt ε (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero)))) => forall (y : α), (LT.lt.{0} Real Real.hasLt (dist x y) ε) -> (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) y s)))))) -> (forall (x : α) (y : α), (Eq.{1} Real (dist x y) (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero)))) -> (Eq.{succ u1} α x y)) -> (MetricSpace.{u1} α)
-but is expected to have type
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-Case conversion may be inaccurate. Consider using '#align metric_space.of_dist_topology MetricSpace.ofDistTopologyₓ'. -/
 /-- Construct a metric space structure whose underlying topological space structure
 (definitionally) agrees which a pre-existing topology which is compatible with a given distance
 function. -/
@@ -4919,108 +3320,48 @@ def MetricSpace.ofDistTopology {α : Type u} [TopologicalSpace α] (dist : α 
 
 variable {γ : Type w} [MetricSpace γ]
 
-/- warning: eq_of_dist_eq_zero -> eq_of_dist_eq_zero is a dubious translation:
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-  forall {γ : Type.{u1}} [_inst_2 : MetricSpace.{u1} γ] {x : γ} {y : γ}, (Eq.{1} Real (Dist.dist.{u1} γ (PseudoMetricSpace.toHasDist.{u1} γ (MetricSpace.toPseudoMetricSpace.{u1} γ _inst_2)) x y) (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero)))) -> (Eq.{succ u1} γ x y)
-but is expected to have type
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-Case conversion may be inaccurate. Consider using '#align eq_of_dist_eq_zero eq_of_dist_eq_zeroₓ'. -/
 theorem eq_of_dist_eq_zero {x y : γ} : dist x y = 0 → x = y :=
   MetricSpace.eq_of_dist_eq_zero
 #align eq_of_dist_eq_zero eq_of_dist_eq_zero
 
-/- warning: dist_eq_zero -> dist_eq_zero is a dubious translation:
-lean 3 declaration is
-  forall {γ : Type.{u1}} [_inst_2 : MetricSpace.{u1} γ] {x : γ} {y : γ}, Iff (Eq.{1} Real (Dist.dist.{u1} γ (PseudoMetricSpace.toHasDist.{u1} γ (MetricSpace.toPseudoMetricSpace.{u1} γ _inst_2)) x y) (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero)))) (Eq.{succ u1} γ x y)
-but is expected to have type
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-Case conversion may be inaccurate. Consider using '#align dist_eq_zero dist_eq_zeroₓ'. -/
 @[simp]
 theorem dist_eq_zero {x y : γ} : dist x y = 0 ↔ x = y :=
   Iff.intro eq_of_dist_eq_zero fun this : x = y => this ▸ dist_self _
 #align dist_eq_zero dist_eq_zero
 
-/- warning: zero_eq_dist -> zero_eq_dist is a dubious translation:
-lean 3 declaration is
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-but is expected to have type
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-Case conversion may be inaccurate. Consider using '#align zero_eq_dist zero_eq_distₓ'. -/
 @[simp]
 theorem zero_eq_dist {x y : γ} : 0 = dist x y ↔ x = y := by rw [eq_comm, dist_eq_zero]
 #align zero_eq_dist zero_eq_dist
 
-/- warning: dist_ne_zero -> dist_ne_zero is a dubious translation:
-lean 3 declaration is
-  forall {γ : Type.{u1}} [_inst_2 : MetricSpace.{u1} γ] {x : γ} {y : γ}, Iff (Ne.{1} Real (Dist.dist.{u1} γ (PseudoMetricSpace.toHasDist.{u1} γ (MetricSpace.toPseudoMetricSpace.{u1} γ _inst_2)) x y) (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero)))) (Ne.{succ u1} γ x y)
-but is expected to have type
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-Case conversion may be inaccurate. Consider using '#align dist_ne_zero dist_ne_zeroₓ'. -/
 theorem dist_ne_zero {x y : γ} : dist x y ≠ 0 ↔ x ≠ y := by
   simpa only [not_iff_not] using dist_eq_zero
 #align dist_ne_zero dist_ne_zero
 
-/- warning: dist_le_zero -> dist_le_zero is a dubious translation:
-lean 3 declaration is
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-but is expected to have type
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-Case conversion may be inaccurate. Consider using '#align dist_le_zero dist_le_zeroₓ'. -/
 @[simp]
 theorem dist_le_zero {x y : γ} : dist x y ≤ 0 ↔ x = y := by
   simpa [le_antisymm_iff, dist_nonneg] using @dist_eq_zero _ _ x y
 #align dist_le_zero dist_le_zero
 
-/- warning: dist_pos -> dist_pos is a dubious translation:
-lean 3 declaration is
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-but is expected to have type
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-Case conversion may be inaccurate. Consider using '#align dist_pos dist_posₓ'. -/
 @[simp]
 theorem dist_pos {x y : γ} : 0 < dist x y ↔ x ≠ y := by
   simpa only [not_le] using not_congr dist_le_zero
 #align dist_pos dist_pos
 
-/- warning: eq_of_forall_dist_le -> eq_of_forall_dist_le is a dubious translation:
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-Case conversion may be inaccurate. Consider using '#align eq_of_forall_dist_le eq_of_forall_dist_leₓ'. -/
 theorem eq_of_forall_dist_le {x y : γ} (h : ∀ ε > 0, dist x y ≤ ε) : x = y :=
   eq_of_dist_eq_zero (eq_of_le_of_forall_le_of_dense dist_nonneg h)
 #align eq_of_forall_dist_le eq_of_forall_dist_le
 
-/- warning: eq_of_nndist_eq_zero -> eq_of_nndist_eq_zero is a dubious translation:
-lean 3 declaration is
-  forall {γ : Type.{u1}} [_inst_2 : MetricSpace.{u1} γ] {x : γ} {y : γ}, (Eq.{1} NNReal (NNDist.nndist.{u1} γ (PseudoMetricSpace.toNNDist.{u1} γ (MetricSpace.toPseudoMetricSpace.{u1} γ _inst_2)) x y) (OfNat.ofNat.{0} NNReal 0 (OfNat.mk.{0} NNReal 0 (Zero.zero.{0} NNReal (MulZeroClass.toHasZero.{0} NNReal (NonUnitalNonAssocSemiring.toMulZeroClass.{0} NNReal (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} NNReal (Semiring.toNonAssocSemiring.{0} NNReal NNReal.semiring)))))))) -> (Eq.{succ u1} γ x y)
-but is expected to have type
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-Case conversion may be inaccurate. Consider using '#align eq_of_nndist_eq_zero eq_of_nndist_eq_zeroₓ'. -/
 /-- Deduce the equality of points with the vanishing of the nonnegative distance-/
 theorem eq_of_nndist_eq_zero {x y : γ} : nndist x y = 0 → x = y := by
   simp only [← NNReal.eq_iff, ← dist_nndist, imp_self, NNReal.coe_zero, dist_eq_zero]
 #align eq_of_nndist_eq_zero eq_of_nndist_eq_zero
 
-/- warning: nndist_eq_zero -> nndist_eq_zero is a dubious translation:
-lean 3 declaration is
-  forall {γ : Type.{u1}} [_inst_2 : MetricSpace.{u1} γ] {x : γ} {y : γ}, Iff (Eq.{1} NNReal (NNDist.nndist.{u1} γ (PseudoMetricSpace.toNNDist.{u1} γ (MetricSpace.toPseudoMetricSpace.{u1} γ _inst_2)) x y) (OfNat.ofNat.{0} NNReal 0 (OfNat.mk.{0} NNReal 0 (Zero.zero.{0} NNReal (MulZeroClass.toHasZero.{0} NNReal (NonUnitalNonAssocSemiring.toMulZeroClass.{0} NNReal (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} NNReal (Semiring.toNonAssocSemiring.{0} NNReal NNReal.semiring)))))))) (Eq.{succ u1} γ x y)
-but is expected to have type
-  forall {γ : Type.{u1}} [_inst_2 : MetricSpace.{u1} γ] {x : γ} {y : γ}, Iff (Eq.{1} NNReal (NNDist.nndist.{u1} γ (PseudoMetricSpace.toNNDist.{u1} γ (MetricSpace.toPseudoMetricSpace.{u1} γ _inst_2)) x y) (OfNat.ofNat.{0} NNReal 0 (Zero.toOfNat0.{0} NNReal instNNRealZero))) (Eq.{succ u1} γ x y)
-Case conversion may be inaccurate. Consider using '#align nndist_eq_zero nndist_eq_zeroₓ'. -/
 /-- Characterize the equality of points with the vanishing of the nonnegative distance-/
 @[simp]
 theorem nndist_eq_zero {x y : γ} : nndist x y = 0 ↔ x = y := by
   simp only [← NNReal.eq_iff, ← dist_nndist, imp_self, NNReal.coe_zero, dist_eq_zero]
 #align nndist_eq_zero nndist_eq_zero
 
-/- warning: zero_eq_nndist -> zero_eq_nndist is a dubious translation:
-lean 3 declaration is
-  forall {γ : Type.{u1}} [_inst_2 : MetricSpace.{u1} γ] {x : γ} {y : γ}, Iff (Eq.{1} NNReal (OfNat.ofNat.{0} NNReal 0 (OfNat.mk.{0} NNReal 0 (Zero.zero.{0} NNReal (MulZeroClass.toHasZero.{0} NNReal (NonUnitalNonAssocSemiring.toMulZeroClass.{0} NNReal (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} NNReal (Semiring.toNonAssocSemiring.{0} NNReal NNReal.semiring))))))) (NNDist.nndist.{u1} γ (PseudoMetricSpace.toNNDist.{u1} γ (MetricSpace.toPseudoMetricSpace.{u1} γ _inst_2)) x y)) (Eq.{succ u1} γ x y)
-but is expected to have type
-  forall {γ : Type.{u1}} [_inst_2 : MetricSpace.{u1} γ] {x : γ} {y : γ}, Iff (Eq.{1} NNReal (OfNat.ofNat.{0} NNReal 0 (Zero.toOfNat0.{0} NNReal instNNRealZero)) (NNDist.nndist.{u1} γ (PseudoMetricSpace.toNNDist.{u1} γ (MetricSpace.toPseudoMetricSpace.{u1} γ _inst_2)) x y)) (Eq.{succ u1} γ x y)
-Case conversion may be inaccurate. Consider using '#align zero_eq_nndist zero_eq_nndistₓ'. -/
 @[simp]
 theorem zero_eq_nndist {x y : γ} : 0 = nndist x y ↔ x = y := by
   simp only [← NNReal.eq_iff, ← dist_nndist, imp_self, NNReal.coe_zero, zero_eq_dist]
@@ -5030,34 +3371,16 @@ namespace Metric
 
 variable {x : γ} {s : Set γ}
 
-/- warning: metric.closed_ball_zero -> Metric.closedBall_zero is a dubious translation:
-lean 3 declaration is
-  forall {γ : Type.{u1}} [_inst_2 : MetricSpace.{u1} γ] {x : γ}, Eq.{succ u1} (Set.{u1} γ) (Metric.closedBall.{u1} γ (MetricSpace.toPseudoMetricSpace.{u1} γ _inst_2) x (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero)))) (Singleton.singleton.{u1, u1} γ (Set.{u1} γ) (Set.hasSingleton.{u1} γ) x)
-but is expected to have type
-  forall {γ : Type.{u1}} [_inst_2 : MetricSpace.{u1} γ] {x : γ}, Eq.{succ u1} (Set.{u1} γ) (Metric.closedBall.{u1} γ (MetricSpace.toPseudoMetricSpace.{u1} γ _inst_2) x (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal))) (Singleton.singleton.{u1, u1} γ (Set.{u1} γ) (Set.instSingletonSet.{u1} γ) x)
-Case conversion may be inaccurate. Consider using '#align metric.closed_ball_zero Metric.closedBall_zeroₓ'. -/
 @[simp]
 theorem closedBall_zero : closedBall x 0 = {x} :=
   Set.ext fun y => dist_le_zero
 #align metric.closed_ball_zero Metric.closedBall_zero
 
-/- warning: metric.sphere_zero -> Metric.sphere_zero is a dubious translation:
-lean 3 declaration is
-  forall {γ : Type.{u1}} [_inst_2 : MetricSpace.{u1} γ] {x : γ}, Eq.{succ u1} (Set.{u1} γ) (Metric.sphere.{u1} γ (MetricSpace.toPseudoMetricSpace.{u1} γ _inst_2) x (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero)))) (Singleton.singleton.{u1, u1} γ (Set.{u1} γ) (Set.hasSingleton.{u1} γ) x)
-but is expected to have type
-  forall {γ : Type.{u1}} [_inst_2 : MetricSpace.{u1} γ] {x : γ}, Eq.{succ u1} (Set.{u1} γ) (Metric.sphere.{u1} γ (MetricSpace.toPseudoMetricSpace.{u1} γ _inst_2) x (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal))) (Singleton.singleton.{u1, u1} γ (Set.{u1} γ) (Set.instSingletonSet.{u1} γ) x)
-Case conversion may be inaccurate. Consider using '#align metric.sphere_zero Metric.sphere_zeroₓ'. -/
 @[simp]
 theorem sphere_zero : sphere x 0 = {x} :=
   Set.ext fun y => dist_eq_zero
 #align metric.sphere_zero Metric.sphere_zero
 
-/- warning: metric.subsingleton_closed_ball -> Metric.subsingleton_closedBall is a dubious translation:
-lean 3 declaration is
-  forall {γ : Type.{u1}} [_inst_2 : MetricSpace.{u1} γ] (x : γ) {r : Real}, (LE.le.{0} Real Real.hasLe r (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero)))) -> (Set.Subsingleton.{u1} γ (Metric.closedBall.{u1} γ (MetricSpace.toPseudoMetricSpace.{u1} γ _inst_2) x r))
-but is expected to have type
-  forall {γ : Type.{u1}} [_inst_2 : MetricSpace.{u1} γ] (x : γ) {r : Real}, (LE.le.{0} Real Real.instLEReal r (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal))) -> (Set.Subsingleton.{u1} γ (Metric.closedBall.{u1} γ (MetricSpace.toPseudoMetricSpace.{u1} γ _inst_2) x r))
-Case conversion may be inaccurate. Consider using '#align metric.subsingleton_closed_ball Metric.subsingleton_closedBallₓ'. -/
 theorem subsingleton_closedBall (x : γ) {r : ℝ} (hr : r ≤ 0) : (closedBall x r).Subsingleton :=
   by
   rcases hr.lt_or_eq with (hr | rfl)
@@ -5065,12 +3388,6 @@ theorem subsingleton_closedBall (x : γ) {r : ℝ} (hr : r ≤ 0) : (closedBall
   · rw [closed_ball_zero]; exact subsingleton_singleton
 #align metric.subsingleton_closed_ball Metric.subsingleton_closedBall
 
-/- warning: metric.subsingleton_sphere -> Metric.subsingleton_sphere is a dubious translation:
-lean 3 declaration is
-  forall {γ : Type.{u1}} [_inst_2 : MetricSpace.{u1} γ] (x : γ) {r : Real}, (LE.le.{0} Real Real.hasLe r (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero)))) -> (Set.Subsingleton.{u1} γ (Metric.sphere.{u1} γ (MetricSpace.toPseudoMetricSpace.{u1} γ _inst_2) x r))
-but is expected to have type
-  forall {γ : Type.{u1}} [_inst_2 : MetricSpace.{u1} γ] (x : γ) {r : Real}, (LE.le.{0} Real Real.instLEReal r (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal))) -> (Set.Subsingleton.{u1} γ (Metric.sphere.{u1} γ (MetricSpace.toPseudoMetricSpace.{u1} γ _inst_2) x r))
-Case conversion may be inaccurate. Consider using '#align metric.subsingleton_sphere Metric.subsingleton_sphereₓ'. -/
 theorem subsingleton_sphere (x : γ) {r : ℝ} (hr : r ≤ 0) : (sphere x r).Subsingleton :=
   (subsingleton_closedBall x hr).anti sphere_subset_closedBall
 #align metric.subsingleton_sphere Metric.subsingleton_sphere
@@ -5083,12 +3400,6 @@ instance (priority := 100) MetricSpace.to_separated : SeparatedSpace γ :=
 #align metric_space.to_separated MetricSpace.to_separated
 -/
 
-/- warning: metric.uniform_embedding_iff' -> Metric.uniformEmbedding_iff' is a dubious translation:
-lean 3 declaration is
-  forall {β : Type.{u1}} {γ : Type.{u2}} [_inst_2 : MetricSpace.{u2} γ] [_inst_3 : MetricSpace.{u1} β] {f : γ -> β}, Iff (UniformEmbedding.{u2, u1} γ β (PseudoMetricSpace.toUniformSpace.{u2} γ (MetricSpace.toPseudoMetricSpace.{u2} γ _inst_2)) (PseudoMetricSpace.toUniformSpace.{u1} β (MetricSpace.toPseudoMetricSpace.{u1} β _inst_3)) f) (And (forall (ε : Real), (GT.gt.{0} Real Real.hasLt ε (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero)))) -> (Exists.{1} Real (fun (δ : Real) => Exists.{0} (GT.gt.{0} Real Real.hasLt δ (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero)))) (fun (H : GT.gt.{0} Real Real.hasLt δ (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero)))) => forall {a : γ} {b : γ}, (LT.lt.{0} Real Real.hasLt (Dist.dist.{u2} γ (PseudoMetricSpace.toHasDist.{u2} γ (MetricSpace.toPseudoMetricSpace.{u2} γ _inst_2)) a b) δ) -> (LT.lt.{0} Real Real.hasLt (Dist.dist.{u1} β (PseudoMetricSpace.toHasDist.{u1} β (MetricSpace.toPseudoMetricSpace.{u1} β _inst_3)) (f a) (f b)) ε))))) (forall (δ : Real), (GT.gt.{0} Real Real.hasLt δ (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero)))) -> (Exists.{1} Real (fun (ε : Real) => Exists.{0} (GT.gt.{0} Real Real.hasLt ε (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero)))) (fun (H : GT.gt.{0} Real Real.hasLt ε (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero)))) => forall {a : γ} {b : γ}, (LT.lt.{0} Real Real.hasLt (Dist.dist.{u1} β (PseudoMetricSpace.toHasDist.{u1} β (MetricSpace.toPseudoMetricSpace.{u1} β _inst_3)) (f a) (f b)) ε) -> (LT.lt.{0} Real Real.hasLt (Dist.dist.{u2} γ (PseudoMetricSpace.toHasDist.{u2} γ (MetricSpace.toPseudoMetricSpace.{u2} γ _inst_2)) a b) δ))))))
-but is expected to have type
-  forall {β : Type.{u1}} {γ : Type.{u2}} [_inst_2 : MetricSpace.{u2} γ] [_inst_3 : MetricSpace.{u1} β] {f : γ -> β}, Iff (UniformEmbedding.{u2, u1} γ β (PseudoMetricSpace.toUniformSpace.{u2} γ (MetricSpace.toPseudoMetricSpace.{u2} γ _inst_2)) (PseudoMetricSpace.toUniformSpace.{u1} β (MetricSpace.toPseudoMetricSpace.{u1} β _inst_3)) f) (And (forall (ε : Real), (GT.gt.{0} Real Real.instLTReal ε (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal))) -> (Exists.{1} Real (fun (δ : Real) => And (GT.gt.{0} Real Real.instLTReal δ (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal))) (forall {a : γ} {b : γ}, (LT.lt.{0} Real Real.instLTReal (Dist.dist.{u2} γ (PseudoMetricSpace.toDist.{u2} γ (MetricSpace.toPseudoMetricSpace.{u2} γ _inst_2)) a b) δ) -> (LT.lt.{0} Real Real.instLTReal (Dist.dist.{u1} β (PseudoMetricSpace.toDist.{u1} β (MetricSpace.toPseudoMetricSpace.{u1} β _inst_3)) (f a) (f b)) ε))))) (forall (δ : Real), (GT.gt.{0} Real Real.instLTReal δ (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal))) -> (Exists.{1} Real (fun (ε : Real) => And (GT.gt.{0} Real Real.instLTReal ε (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal))) (forall {a : γ} {b : γ}, (LT.lt.{0} Real Real.instLTReal (Dist.dist.{u1} β (PseudoMetricSpace.toDist.{u1} β (MetricSpace.toPseudoMetricSpace.{u1} β _inst_3)) (f a) (f b)) ε) -> (LT.lt.{0} Real Real.instLTReal (Dist.dist.{u2} γ (PseudoMetricSpace.toDist.{u2} γ (MetricSpace.toPseudoMetricSpace.{u2} γ _inst_2)) a b) δ))))))
-Case conversion may be inaccurate. Consider using '#align metric.uniform_embedding_iff' Metric.uniformEmbedding_iff'ₓ'. -/
 /-- A map between metric spaces is a uniform embedding if and only if the distance between `f x`
 and `f y` is controlled in terms of the distance between `x` and `y` and conversely. -/
 theorem uniformEmbedding_iff' [MetricSpace β] {f : γ → β} :
@@ -5118,35 +3429,17 @@ instance (priority := 100) MetricSpace.toEMetricSpace : EMetricSpace γ :=
 #align metric_space.to_emetric_space MetricSpace.toEMetricSpace
 -/
 
-/- warning: metric.is_closed_of_pairwise_le_dist -> Metric.isClosed_of_pairwise_le_dist is a dubious translation:
-lean 3 declaration is
-  forall {γ : Type.{u1}} [_inst_2 : MetricSpace.{u1} γ] {s : Set.{u1} γ} {ε : Real}, (LT.lt.{0} Real Real.hasLt (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero))) ε) -> (Set.Pairwise.{u1} γ s (fun (x : γ) (y : γ) => LE.le.{0} Real Real.hasLe ε (Dist.dist.{u1} γ (PseudoMetricSpace.toHasDist.{u1} γ (MetricSpace.toPseudoMetricSpace.{u1} γ _inst_2)) x y))) -> (IsClosed.{u1} γ (UniformSpace.toTopologicalSpace.{u1} γ (PseudoMetricSpace.toUniformSpace.{u1} γ (MetricSpace.toPseudoMetricSpace.{u1} γ _inst_2))) s)
-but is expected to have type
-  forall {γ : Type.{u1}} [_inst_2 : MetricSpace.{u1} γ] {s : Set.{u1} γ} {ε : Real}, (LT.lt.{0} Real Real.instLTReal (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal)) ε) -> (Set.Pairwise.{u1} γ s (fun (x : γ) (y : γ) => LE.le.{0} Real Real.instLEReal ε (Dist.dist.{u1} γ (PseudoMetricSpace.toDist.{u1} γ (MetricSpace.toPseudoMetricSpace.{u1} γ _inst_2)) x y))) -> (IsClosed.{u1} γ (UniformSpace.toTopologicalSpace.{u1} γ (PseudoMetricSpace.toUniformSpace.{u1} γ (MetricSpace.toPseudoMetricSpace.{u1} γ _inst_2))) s)
-Case conversion may be inaccurate. Consider using '#align metric.is_closed_of_pairwise_le_dist Metric.isClosed_of_pairwise_le_distₓ'. -/
 theorem isClosed_of_pairwise_le_dist {s : Set γ} {ε : ℝ} (hε : 0 < ε)
     (hs : s.Pairwise fun x y => ε ≤ dist x y) : IsClosed s :=
   isClosed_of_spaced_out (dist_mem_uniformity hε) <| by simpa using hs
 #align metric.is_closed_of_pairwise_le_dist Metric.isClosed_of_pairwise_le_dist
 
-/- warning: metric.closed_embedding_of_pairwise_le_dist -> Metric.closedEmbedding_of_pairwise_le_dist is a dubious translation:
-lean 3 declaration is
-  forall {γ : Type.{u1}} [_inst_2 : MetricSpace.{u1} γ] {α : Type.{u2}} [_inst_3 : TopologicalSpace.{u2} α] [_inst_4 : DiscreteTopology.{u2} α _inst_3] {ε : Real}, (LT.lt.{0} Real Real.hasLt (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero))) ε) -> (forall {f : α -> γ}, (Pairwise.{u2} α (fun (x : α) (y : α) => LE.le.{0} Real Real.hasLe ε (Dist.dist.{u1} γ (PseudoMetricSpace.toHasDist.{u1} γ (MetricSpace.toPseudoMetricSpace.{u1} γ _inst_2)) (f x) (f y)))) -> (ClosedEmbedding.{u2, u1} α γ _inst_3 (UniformSpace.toTopologicalSpace.{u1} γ (PseudoMetricSpace.toUniformSpace.{u1} γ (MetricSpace.toPseudoMetricSpace.{u1} γ _inst_2))) f))
-but is expected to have type
-  forall {γ : Type.{u2}} [_inst_2 : MetricSpace.{u2} γ] {α : Type.{u1}} [_inst_3 : TopologicalSpace.{u1} α] [_inst_4 : DiscreteTopology.{u1} α _inst_3] {ε : Real}, (LT.lt.{0} Real Real.instLTReal (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal)) ε) -> (forall {f : α -> γ}, (Pairwise.{u1} α (fun (x : α) (y : α) => LE.le.{0} Real Real.instLEReal ε (Dist.dist.{u2} γ (PseudoMetricSpace.toDist.{u2} γ (MetricSpace.toPseudoMetricSpace.{u2} γ _inst_2)) (f x) (f y)))) -> (ClosedEmbedding.{u1, u2} α γ _inst_3 (UniformSpace.toTopologicalSpace.{u2} γ (PseudoMetricSpace.toUniformSpace.{u2} γ (MetricSpace.toPseudoMetricSpace.{u2} γ _inst_2))) f))
-Case conversion may be inaccurate. Consider using '#align metric.closed_embedding_of_pairwise_le_dist Metric.closedEmbedding_of_pairwise_le_distₓ'. -/
 theorem closedEmbedding_of_pairwise_le_dist {α : Type _} [TopologicalSpace α] [DiscreteTopology α]
     {ε : ℝ} (hε : 0 < ε) {f : α → γ} (hf : Pairwise fun x y => ε ≤ dist (f x) (f y)) :
     ClosedEmbedding f :=
   closedEmbedding_of_spaced_out (dist_mem_uniformity hε) <| by simpa using hf
 #align metric.closed_embedding_of_pairwise_le_dist Metric.closedEmbedding_of_pairwise_le_dist
 
-/- warning: metric.uniform_embedding_bot_of_pairwise_le_dist -> Metric.uniformEmbedding_bot_of_pairwise_le_dist is a dubious translation:
-lean 3 declaration is
-  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {β : Type.{u2}} {ε : Real}, (LT.lt.{0} Real Real.hasLt (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero))) ε) -> (forall {f : β -> α}, (Pairwise.{u2} β (fun (x : β) (y : β) => LE.le.{0} Real Real.hasLe ε (Dist.dist.{u1} α (PseudoMetricSpace.toHasDist.{u1} α _inst_1) (f x) (f y)))) -> (UniformEmbedding.{u2, u1} β α (Bot.bot.{u2} (UniformSpace.{u2} β) (UniformSpace.hasBot.{u2} β)) (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1) f))
-but is expected to have type
-  forall {α : Type.{u2}} [_inst_1 : PseudoMetricSpace.{u2} α] {β : Type.{u1}} {ε : Real}, (LT.lt.{0} Real Real.instLTReal (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal)) ε) -> (forall {f : β -> α}, (Pairwise.{u1} β (fun (x : β) (y : β) => LE.le.{0} Real Real.instLEReal ε (Dist.dist.{u2} α (PseudoMetricSpace.toDist.{u2} α _inst_1) (f x) (f y)))) -> (UniformEmbedding.{u1, u2} β α (Bot.bot.{u1} (UniformSpace.{u1} β) (instBotUniformSpace.{u1} β)) (inferInstance.{succ u2} (UniformSpace.{u2} α) (PseudoMetricSpace.toUniformSpace.{u2} α _inst_1)) f))
-Case conversion may be inaccurate. Consider using '#align metric.uniform_embedding_bot_of_pairwise_le_dist Metric.uniformEmbedding_bot_of_pairwise_le_distₓ'. -/
 /-- If `f : β → α` sends any two distinct points to points at distance at least `ε > 0`, then
 `f` is a uniform embedding with respect to the discrete uniformity on `β`. -/
 theorem uniformEmbedding_bot_of_pairwise_le_dist {β : Type _} {ε : ℝ} (hε : 0 < ε) {f : β → α}
@@ -5194,12 +3487,6 @@ theorem MetricSpace.replaceTopology_eq {γ} [U : TopologicalSpace γ] (m : Metri
 #align metric_space.replace_topology_eq MetricSpace.replaceTopology_eq
 -/
 
-/- warning: emetric_space.to_metric_space_of_dist -> EMetricSpace.toMetricSpaceOfDist is a dubious translation:
-lean 3 declaration is
-  forall {α : Type.{u1}} [e : EMetricSpace.{u1} α] (dist : α -> α -> Real), (forall (x : α) (y : α), Ne.{1} ENNReal (EDist.edist.{u1} α (PseudoEMetricSpace.toHasEdist.{u1} α (EMetricSpace.toPseudoEmetricSpace.{u1} α e)) x y) (Top.top.{0} ENNReal (CompleteLattice.toHasTop.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder)))) -> (forall (x : α) (y : α), Eq.{1} Real (dist x y) (ENNReal.toReal (EDist.edist.{u1} α (PseudoEMetricSpace.toHasEdist.{u1} α (EMetricSpace.toPseudoEmetricSpace.{u1} α e)) x y))) -> (MetricSpace.{u1} α)
-but is expected to have type
-  forall {α : Type.{u1}} [e : EMetricSpace.{u1} α] (dist : α -> α -> Real), (forall (x : α) (y : α), Ne.{1} ENNReal (EDist.edist.{u1} α (PseudoEMetricSpace.toEDist.{u1} α (EMetricSpace.toPseudoEMetricSpace.{u1} α e)) x y) (Top.top.{0} ENNReal (CompleteLattice.toTop.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal)))) -> (forall (x : α) (y : α), Eq.{1} Real (dist x y) (ENNReal.toReal (EDist.edist.{u1} α (PseudoEMetricSpace.toEDist.{u1} α (EMetricSpace.toPseudoEMetricSpace.{u1} α e)) x y))) -> (MetricSpace.{u1} α)
-Case conversion may be inaccurate. Consider using '#align emetric_space.to_metric_space_of_dist EMetricSpace.toMetricSpaceOfDistₓ'. -/
 /-- One gets a metric space from an emetric space if the edistance
 is everywhere finite, by pushing the edistance to reals. We set it up so that the edist and the
 uniformity are defeq in the metric space and the emetric space. In this definition, the distance
@@ -5212,12 +3499,6 @@ def EMetricSpace.toMetricSpaceOfDist {α : Type u} [e : EMetricSpace α] (dist :
     (PseudoEMetricSpace.toPseudoMetricSpaceOfDist dist edist_ne_top h) _
 #align emetric_space.to_metric_space_of_dist EMetricSpace.toMetricSpaceOfDist
 
-/- warning: emetric_space.to_metric_space -> EMetricSpace.toMetricSpace is a dubious translation:
-lean 3 declaration is
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-but is expected to have type
-  forall {α : Type.{u1}} [_inst_3 : EMetricSpace.{u1} α], (forall (x : α) (y : α), Ne.{1} ENNReal (EDist.edist.{u1} α (PseudoEMetricSpace.toEDist.{u1} α (EMetricSpace.toPseudoEMetricSpace.{u1} α _inst_3)) x y) (Top.top.{0} ENNReal (CompleteLattice.toTop.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal)))) -> (MetricSpace.{u1} α)
-Case conversion may be inaccurate. Consider using '#align emetric_space.to_metric_space EMetricSpace.toMetricSpaceₓ'. -/
 /-- One gets a metric space from an emetric space if the edistance
 is everywhere finite, by pushing the edistance to reals. We set it up so that the edist and the
 uniformity are defeq in the metric space and the emetric space. -/
@@ -5379,12 +3660,6 @@ section SecondCountable
 
 open TopologicalSpace
 
-/- warning: metric.second_countable_of_countable_discretization -> Metric.secondCountable_of_countable_discretization is a dubious translation:
-lean 3 declaration is
-  forall {α : Type.{u1}} [_inst_3 : MetricSpace.{u1} α], (forall (ε : Real), (GT.gt.{0} Real Real.hasLt ε (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero)))) -> (Exists.{succ (succ u2)} Type.{u2} (fun (β : Type.{u2}) => Exists.{succ u2} (Encodable.{u2} β) (fun (_x : Encodable.{u2} β) => Exists.{max (succ u1) (succ u2)} (α -> β) (fun (F : α -> β) => forall (x : α) (y : α), (Eq.{succ u2} β (F x) (F y)) -> (LE.le.{0} Real Real.hasLe (Dist.dist.{u1} α (PseudoMetricSpace.toHasDist.{u1} α (MetricSpace.toPseudoMetricSpace.{u1} α _inst_3)) x y) ε)))))) -> (TopologicalSpace.SecondCountableTopology.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α (MetricSpace.toPseudoMetricSpace.{u1} α _inst_3))))
-but is expected to have type
-  forall {α : Type.{u2}} [_inst_3 : MetricSpace.{u2} α], (forall (ε : Real), (GT.gt.{0} Real Real.instLTReal ε (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal))) -> (Exists.{succ (succ u1)} Type.{u1} (fun (β : Type.{u1}) => Exists.{succ u1} (Encodable.{u1} β) (fun (_x : Encodable.{u1} β) => Exists.{max (succ u2) (succ u1)} (α -> β) (fun (F : α -> β) => forall (x : α) (y : α), (Eq.{succ u1} β (F x) (F y)) -> (LE.le.{0} Real Real.instLEReal (Dist.dist.{u2} α (PseudoMetricSpace.toDist.{u2} α (MetricSpace.toPseudoMetricSpace.{u2} α _inst_3)) x y) ε)))))) -> (TopologicalSpace.SecondCountableTopology.{u2} α (UniformSpace.toTopologicalSpace.{u2} α (PseudoMetricSpace.toUniformSpace.{u2} α (MetricSpace.toPseudoMetricSpace.{u2} α _inst_3))))
-Case conversion may be inaccurate. Consider using '#align metric.second_countable_of_countable_discretization Metric.secondCountable_of_countable_discretizationₓ'. -/
 /-- A metric space is second countable if one can reconstruct up to any `ε>0` any element of the
 space from countably many data. -/
 theorem secondCountable_of_countable_discretization {α : Type u} [MetricSpace α]

mathlib commit https://github.com/leanprover-community/mathlib/commit/917c3c072e487b3cccdbfeff17e75b40e45f66cb

@@ -90,11 +90,7 @@ private theorem bounded_iff_aux {α : Type _} (dist : α → α → ℝ)
   ·
     exact
       ⟨C + C, fun x hx y hy =>
-        (dist_triangle x a y).trans
-          (add_le_add (hC hx)
-            (by
-              rw [dist_comm]
-              exact hC hy))⟩
+        (dist_triangle x a y).trans (add_le_add (hC hx) (by rw [dist_comm]; exact hC hy))⟩
 
 /-- Construct a bornology from a distance function and metric space axioms. -/
 def Bornology.ofDist {α : Type _} (dist : α → α → ℝ) (dist_self : ∀ x : α, dist x x = 0)
@@ -112,8 +108,7 @@ def Bornology.ofDist {α : Type _} (dist : α → α → ℝ) (dist_self : ∀ x
             Or.elim hx (fun hx' => (hr₁ hx').trans (le_max_left _ _)) fun hx' =>
               (hr₂ hx').trans (le_max_right _ _)⟩)
     fun z =>
-    ⟨0, fun x hx y hy => by
-      rw [eq_of_mem_singleton hx, eq_of_mem_singleton hy]
+    ⟨0, fun x hx y hy => by rw [eq_of_mem_singleton hx, eq_of_mem_singleton hy];
       exact (dist_self z).le⟩
 #align bornology.of_dist Bornology.ofDistₓ
 
@@ -174,18 +169,11 @@ class PseudoMetricSpace (α : Type u) extends Dist α : Type u where
     run_tac
       [anonymous]
   toUniformSpace : UniformSpace α := UniformSpace.ofDist dist dist_self dist_comm dist_triangle
-  uniformity_dist :
-    𝓤 α = ⨅ ε > 0, 𝓟 { p : α × α |
-            dist p.1 p.2 < ε } := by
-    intros
-    rfl
+  uniformity_dist : 𝓤 α = ⨅ ε > 0, 𝓟 { p : α × α | dist p.1 p.2 < ε } := by intros ; rfl
   toBornology : Bornology α := Bornology.ofDist dist dist_self dist_comm dist_triangle
   cobounded_sets :
-    (Bornology.cobounded α).sets =
-      { s | ∃ C, ∀ ⦃x⦄, x ∈ sᶜ →
-              ∀ ⦃y⦄, y ∈ sᶜ → dist x y ≤ C } := by
-    intros
-    rfl
+    (Bornology.cobounded α).sets = { s | ∃ C, ∀ ⦃x⦄, x ∈ sᶜ → ∀ ⦃y⦄, y ∈ sᶜ → dist x y ≤ C } := by
+    intros ; rfl
 #align pseudo_metric_space PseudoMetricSpace
 -/
 
@@ -195,8 +183,7 @@ class PseudoMetricSpace (α : Type u) extends Dist α : Type u where
 theorem PseudoMetricSpace.ext {α : Type _} {m m' : PseudoMetricSpace α}
     (h : m.toHasDist = m'.toHasDist) : m = m' :=
   by
-  rcases m with ⟨⟩
-  rcases m' with ⟨⟩
+  rcases m with ⟨⟩; rcases m' with ⟨⟩
   dsimp at h
   subst h
   congr
@@ -330,10 +317,8 @@ but is expected to have type
   forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] (x₁ : α) (y₁ : α) (x₂ : α) (y₂ : α), LE.le.{0} Real Real.instLEReal (Dist.dist.{u1} α (PseudoMetricSpace.toDist.{u1} α _inst_1) x₂ y₂) (HAdd.hAdd.{0, 0, 0} Real Real Real (instHAdd.{0} Real Real.instAddReal) (Dist.dist.{u1} α (PseudoMetricSpace.toDist.{u1} α _inst_1) x₁ y₁) (HAdd.hAdd.{0, 0, 0} Real Real Real (instHAdd.{0} Real Real.instAddReal) (Dist.dist.{u1} α (PseudoMetricSpace.toDist.{u1} α _inst_1) x₁ x₂) (Dist.dist.{u1} α (PseudoMetricSpace.toDist.{u1} α _inst_1) y₁ y₂)))
 Case conversion may be inaccurate. Consider using '#align dist_triangle4_left dist_triangle4_leftₓ'. -/
 theorem dist_triangle4_left (x₁ y₁ x₂ y₂ : α) :
-    dist x₂ y₂ ≤ dist x₁ y₁ + (dist x₁ x₂ + dist y₁ y₂) :=
-  by
-  rw [add_left_comm, dist_comm x₁, ← add_assoc]
-  apply dist_triangle4
+    dist x₂ y₂ ≤ dist x₁ y₁ + (dist x₁ x₂ + dist y₁ y₂) := by
+  rw [add_left_comm, dist_comm x₁, ← add_assoc]; apply dist_triangle4
 #align dist_triangle4_left dist_triangle4_left
 
 /- warning: dist_triangle4_right -> dist_triangle4_right is a dubious translation:
@@ -343,9 +328,7 @@ but is expected to have type
   forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] (x₁ : α) (y₁ : α) (x₂ : α) (y₂ : α), LE.le.{0} Real Real.instLEReal (Dist.dist.{u1} α (PseudoMetricSpace.toDist.{u1} α _inst_1) x₁ y₁) (HAdd.hAdd.{0, 0, 0} Real Real Real (instHAdd.{0} Real Real.instAddReal) (HAdd.hAdd.{0, 0, 0} Real Real Real (instHAdd.{0} Real Real.instAddReal) (Dist.dist.{u1} α (PseudoMetricSpace.toDist.{u1} α _inst_1) x₁ x₂) (Dist.dist.{u1} α (PseudoMetricSpace.toDist.{u1} α _inst_1) y₁ y₂)) (Dist.dist.{u1} α (PseudoMetricSpace.toDist.{u1} α _inst_1) x₂ y₂))
 Case conversion may be inaccurate. Consider using '#align dist_triangle4_right dist_triangle4_rightₓ'. -/
 theorem dist_triangle4_right (x₁ y₁ x₂ y₂ : α) :
-    dist x₁ y₁ ≤ dist x₁ x₂ + dist y₁ y₂ + dist x₂ y₂ :=
-  by
-  rw [add_right_comm, dist_comm y₁]
+    dist x₁ y₁ ≤ dist x₁ x₂ + dist y₁ y₂ + dist x₂ y₂ := by rw [add_right_comm, dist_comm y₁];
   apply dist_triangle4
 #align dist_triangle4_right dist_triangle4_right
 
@@ -792,9 +775,7 @@ but is expected to have type
   forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] (ε : Real) (x : α), Eq.{succ u1} (Set.{u1} α) (UniformSpace.ball.{u1} α x (setOf.{u1} (Prod.{u1, u1} α α) (fun (p : Prod.{u1, u1} α α) => LT.lt.{0} Real Real.instLTReal (Dist.dist.{u1} α (PseudoMetricSpace.toDist.{u1} α _inst_1) (Prod.fst.{u1, u1} α α p) (Prod.snd.{u1, u1} α α p)) ε))) (Metric.ball.{u1} α _inst_1 x ε)
 Case conversion may be inaccurate. Consider using '#align metric.ball_eq_ball' Metric.ball_eq_ball'ₓ'. -/
 theorem ball_eq_ball' (ε : ℝ) (x : α) :
-    UniformSpace.ball x { p | dist p.1 p.2 < ε } = Metric.ball x ε :=
-  by
-  ext
+    UniformSpace.ball x { p | dist p.1 p.2 < ε } = Metric.ball x ε := by ext;
   simp [dist_comm, UniformSpace.ball]
 #align metric.ball_eq_ball' Metric.ball_eq_ball'
 
@@ -872,10 +853,7 @@ lean 3 declaration is
 but is expected to have type
   forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {x : α} {y : α} {ε : Real}, (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) y (Metric.sphere.{u1} α _inst_1 x ε)) -> (Ne.{1} Real ε (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal))) -> (Ne.{succ u1} α y x)
 Case conversion may be inaccurate. Consider using '#align metric.ne_of_mem_sphere Metric.ne_of_mem_sphereₓ'. -/
-theorem ne_of_mem_sphere (h : y ∈ sphere x ε) (hε : ε ≠ 0) : y ≠ x :=
-  by
-  contrapose! hε
-  symm
+theorem ne_of_mem_sphere (h : y ∈ sphere x ε) (hε : ε ≠ 0) : y ≠ x := by contrapose! hε; symm;
   simpa [hε] using h
 #align metric.ne_of_mem_sphere Metric.ne_of_mem_sphere
 
@@ -1357,8 +1335,7 @@ protected theorem mk_uniformity_basis {β : Type _} {p : β → Prop} {f : β 
   refine' ⟨fun s => uniformity_basis_dist.mem_iff.trans _⟩
   constructor
   · rintro ⟨ε, ε₀, hε⟩
-    obtain ⟨i, hi, H⟩ : ∃ (i : _)(hi : p i), f i ≤ ε
-    exact hf ε₀
+    obtain ⟨i, hi, H⟩ : ∃ (i : _)(hi : p i), f i ≤ ε; exact hf ε₀
     exact ⟨i, hi, fun x (hx : _ < _) => hε <| lt_of_lt_of_le hx H⟩
   · exact fun ⟨i, hi, H⟩ => ⟨f i, hf₀ i hi, H⟩
 #align metric.mk_uniformity_basis Metric.mk_uniformity_basis
@@ -1594,8 +1571,7 @@ theorem totallyBounded_of_finite_discretization {s : Set α}
         ∃ (β : Type u)(_ : Fintype β)(F : s → β), ∀ x y, F x = F y → dist (x : α) y < ε) :
     TotallyBounded s := by
   cases' s.eq_empty_or_nonempty with hs hs
-  · rw [hs]
-    exact totallyBounded_empty
+  · rw [hs]; exact totallyBounded_empty
   rcases hs with ⟨x0, hx0⟩
   haveI : Inhabited s := ⟨⟨x0, hx0⟩⟩
   refine' totally_bounded_iff.2 fun ε ε0 => _
@@ -1698,10 +1674,8 @@ but is expected to have type
 Case conversion may be inaccurate. Consider using '#align metric.tendsto_uniformly_iff Metric.tendstoUniformly_iffₓ'. -/
 /-- Expressing uniform convergence using `dist`. -/
 theorem tendstoUniformly_iff {ι : Type _} {F : ι → β → α} {f : β → α} {p : Filter ι} :
-    TendstoUniformly F f p ↔ ∀ ε > 0, ∀ᶠ n in p, ∀ x, dist (f x) (F n x) < ε :=
-  by
-  rw [← tendstoUniformlyOn_univ, tendsto_uniformly_on_iff]
-  simp
+    TendstoUniformly F f p ↔ ∀ ε > 0, ∀ᶠ n in p, ∀ x, dist (f x) (F n x) < ε := by
+  rw [← tendstoUniformlyOn_univ, tendsto_uniformly_on_iff]; simp
 #align metric.tendsto_uniformly_iff Metric.tendstoUniformly_iff
 
 /- warning: metric.cauchy_iff -> Metric.cauchy_iff is a dubious translation:
@@ -1774,10 +1748,8 @@ theorem eventually_prod_nhds_iff {f : Filter ι} {x₀ : α} {p : ι × α → P
   simp_rw [eventually_prod_iff, Metric.eventually_nhds_iff]
   refine' exists_congr fun q => exists_congr fun hq => _
   constructor
-  · rintro ⟨r, ⟨ε, hε, hεr⟩, hp⟩
-    exact ⟨ε, hε, fun i hi x hx => hp hi <| hεr hx⟩
-  · rintro ⟨ε, hε, hp⟩
-    exact ⟨fun x => dist x x₀ < ε, ⟨ε, hε, fun y => id⟩, @hp⟩
+  · rintro ⟨r, ⟨ε, hε, hεr⟩, hp⟩; exact ⟨ε, hε, fun i hi x hx => hp hi <| hεr hx⟩
+  · rintro ⟨ε, hε, hp⟩; exact ⟨fun x => dist x x₀ < ε, ⟨ε, hε, fun y => id⟩, @hp⟩
 #align metric.eventually_prod_nhds_iff Metric.eventually_prod_nhds_iff
 
 /- warning: metric.eventually_nhds_prod_iff -> Metric.eventually_nhds_prod_iff is a dubious translation:
@@ -1796,8 +1768,7 @@ theorem eventually_nhds_prod_iff {ι α} [PseudoMetricSpace α] {f : Filter ι}
   by
   rw [eventually_swap_iff, Metric.eventually_prod_nhds_iff]
   constructor <;>
-    · rintro ⟨a1, a2, a3, a4, a5⟩
-      refine' ⟨a3, a4, a1, a2, fun b1 b2 b3 b4 => a5 b4 b2⟩
+    · rintro ⟨a1, a2, a3, a4, a5⟩; refine' ⟨a3, a4, a1, a2, fun b1 b2 b3 b4 => a5 b4 b2⟩
 #align metric.eventually_nhds_prod_iff Metric.eventually_nhds_prod_iff
 
 /- warning: metric.nhds_basis_closed_ball -> Metric.nhds_basis_closedBall is a dubious translation:
@@ -2063,10 +2034,7 @@ but is expected to have type
 Case conversion may be inaccurate. Consider using '#align metric.tendsto_at_top Metric.tendsto_atTopₓ'. -/
 theorem tendsto_atTop [Nonempty β] [SemilatticeSup β] {u : β → α} {a : α} :
     Tendsto u atTop (𝓝 a) ↔ ∀ ε > 0, ∃ N, ∀ n ≥ N, dist (u n) a < ε :=
-  (atTop_basis.tendsto_iffₓ nhds_basis_ball).trans <|
-    by
-    simp only [exists_prop, true_and_iff]
-    rfl
+  (atTop_basis.tendsto_iffₓ nhds_basis_ball).trans <| by simp only [exists_prop, true_and_iff]; rfl
 #align metric.tendsto_at_top Metric.tendsto_atTop
 
 /- warning: metric.tendsto_at_top' -> Metric.tendsto_atTop' is a dubious translation:
@@ -2080,9 +2048,7 @@ uses `∃ N, ∀ n > N, ...` rather than `∃ N, ∀ n ≥ N, ...`
 -/
 theorem tendsto_atTop' [Nonempty β] [SemilatticeSup β] [NoMaxOrder β] {u : β → α} {a : α} :
     Tendsto u atTop (𝓝 a) ↔ ∀ ε > 0, ∃ N, ∀ n > N, dist (u n) a < ε :=
-  (atTop_basis_Ioi.tendsto_iffₓ nhds_basis_ball).trans <|
-    by
-    simp only [exists_prop, true_and_iff]
+  (atTop_basis_Ioi.tendsto_iffₓ nhds_basis_ball).trans <| by simp only [exists_prop, true_and_iff];
     rfl
 #align metric.tendsto_at_top' Metric.tendsto_atTop'
 
@@ -2234,10 +2200,8 @@ but is expected to have type
 Case conversion may be inaccurate. Consider using '#align metric.emetric_ball_nnreal Metric.emetric_ball_nnrealₓ'. -/
 /-- Balls defined using the distance or the edistance coincide -/
 @[simp]
-theorem Metric.emetric_ball_nnreal {x : α} {ε : ℝ≥0} : EMetric.ball x ε = ball x ε :=
-  by
-  convert Metric.emetric_ball
-  simp
+theorem Metric.emetric_ball_nnreal {x : α} {ε : ℝ≥0} : EMetric.ball x ε = ball x ε := by
+  convert Metric.emetric_ball; simp
 #align metric.emetric_ball_nnreal Metric.emetric_ball_nnreal
 
 /- warning: metric.emetric_closed_ball -> Metric.emetric_closedBall is a dubious translation:
@@ -2261,10 +2225,7 @@ Case conversion may be inaccurate. Consider using '#align metric.emetric_closed_
 /-- Closed balls defined using the distance or the edistance coincide -/
 @[simp]
 theorem Metric.emetric_closedBall_nnreal {x : α} {ε : ℝ≥0} :
-    EMetric.closedBall x ε = closedBall x ε :=
-  by
-  convert Metric.emetric_closedBall ε.2
-  simp
+    EMetric.closedBall x ε = closedBall x ε := by convert Metric.emetric_closedBall ε.2; simp
 #align metric.emetric_closed_ball_nnreal Metric.emetric_closedBall_nnreal
 
 /- warning: metric.emetric_ball_top -> Metric.emetric_ball_top is a dubious translation:
@@ -2309,10 +2270,7 @@ def PseudoMetricSpace.replaceUniformity {α} [U : UniformSpace α] (m : PseudoMe
 
 #print PseudoMetricSpace.replaceUniformity_eq /-
 theorem PseudoMetricSpace.replaceUniformity_eq {α} [U : UniformSpace α] (m : PseudoMetricSpace α)
-    (H : 𝓤[U] = 𝓤[PseudoEMetricSpace.toUniformSpace]) : m.replaceUniformity H = m :=
-  by
-  ext
-  rfl
+    (H : 𝓤[U] = 𝓤[PseudoEMetricSpace.toUniformSpace]) : m.replaceUniformity H = m := by ext; rfl
 #align pseudo_metric_space.replace_uniformity_eq PseudoMetricSpace.replaceUniformity_eq
 -/
 
@@ -2330,10 +2288,7 @@ def PseudoMetricSpace.replaceTopology {γ} [U : TopologicalSpace γ] (m : Pseudo
 
 #print PseudoMetricSpace.replaceTopology_eq /-
 theorem PseudoMetricSpace.replaceTopology_eq {γ} [U : TopologicalSpace γ] (m : PseudoMetricSpace γ)
-    (H : U = m.toUniformSpace.toTopologicalSpace) : m.replaceTopology H = m :=
-  by
-  ext
-  rfl
+    (H : U = m.toUniformSpace.toTopologicalSpace) : m.replaceTopology H = m := by ext; rfl
 #align pseudo_metric_space.replace_topology_eq PseudoMetricSpace.replaceTopology_eq
 -/
 
@@ -2363,10 +2318,7 @@ def PseudoEMetricSpace.toPseudoMetricSpaceOfDist {α : Type u} [e : PseudoEMetri
         · simp [ENNReal.add_eq_top, edist_ne_top]
       edist := edist
       edist_dist := fun x y => by simp [h, ENNReal.ofReal_toReal, edist_ne_top] }
-  m.replaceUniformity <|
-    by
-    rw [uniformity_pseudoedist, Metric.uniformity_edist]
-    rfl
+  m.replaceUniformity <| by rw [uniformity_pseudoedist, Metric.uniformity_edist]; rfl
 #align pseudo_emetric_space.to_pseudo_metric_space_of_dist PseudoEMetricSpace.toPseudoMetricSpaceOfDist
 
 /- warning: pseudo_emetric_space.to_pseudo_metric_space -> PseudoEMetricSpace.toPseudoMetricSpace is a dubious translation:
@@ -2404,10 +2356,7 @@ def PseudoMetricSpace.replaceBornology {α} [B : Bornology α] (m : PseudoMetric
 #print PseudoMetricSpace.replaceBornology_eq /-
 theorem PseudoMetricSpace.replaceBornology_eq {α} [m : PseudoMetricSpace α] [B : Bornology α]
     (H : ∀ s, @IsBounded _ B s ↔ @IsBounded _ PseudoMetricSpace.toBornology s) :
-    PseudoMetricSpace.replaceBornology _ H = m :=
-  by
-  ext
-  rfl
+    PseudoMetricSpace.replaceBornology _ H = m := by ext; rfl
 #align pseudo_metric_space.replace_bornology_eq PseudoMetricSpace.replaceBornology_eq
 -/
 
@@ -2882,15 +2831,12 @@ theorem cauchySeq_iff_le_tendsto_0 {s : ℕ → α} :
     have hS : ∀ N, ∃ x, ∀ y ∈ S N, y ≤ x :=
       by
       rcases cauchySeq_bdd hs with ⟨R, R0, hR⟩
-      refine' fun N => ⟨R, _⟩
-      rintro _ ⟨⟨m, n⟩, _, rfl⟩
+      refine' fun N => ⟨R, _⟩; rintro _ ⟨⟨m, n⟩, _, rfl⟩
       exact le_of_lt (hR m n)
     have bdd : BddAbove (range fun p : ℕ × ℕ => dist (s p.1) (s p.2)) :=
       by
       rcases cauchySeq_bdd hs with ⟨R, R0, hR⟩
-      use R
-      rintro _ ⟨⟨m, n⟩, rfl⟩
-      exact le_of_lt (hR m n)
+      use R; rintro _ ⟨⟨m, n⟩, rfl⟩; exact le_of_lt (hR m n)
     -- Prove that it bounds the distances of points in the Cauchy sequence
     have ub : ∀ m n N, N ≤ m → N ≤ n → dist (s m) (s n) ≤ Sup (S N) := fun m n N hm hn =>
       le_csSup (hS N) ⟨⟨_, _⟩, ⟨hm, hn⟩, rfl⟩
@@ -3031,8 +2977,7 @@ Case conversion may be inaccurate. Consider using '#align nnreal.nndist_eq NNRea
 theorem NNReal.nndist_eq (a b : ℝ≥0) : nndist a b = max (a - b) (b - a) :=
   by
   wlog h : b ≤ a
-  · rw [nndist_comm, max_comm]
-    exact this b a (le_of_not_le h)
+  · rw [nndist_comm, max_comm]; exact this b a (le_of_not_le h)
   rw [← NNReal.coe_eq, ← dist_nndist, NNReal.dist_eq, tsub_eq_zero_iff_le.2 h,
     max_eq_left (zero_le <| a - b), ← NNReal.coe_sub h, abs_of_nonneg (a - b).coe_nonneg]
 #align nnreal.nndist_eq NNReal.nndist_eq
@@ -3055,9 +3000,7 @@ but is expected to have type
   forall (z : NNReal), Eq.{1} NNReal (NNDist.nndist.{0} NNReal (PseudoMetricSpace.toNNDist.{0} NNReal instPseudoMetricSpaceNNReal) z (OfNat.ofNat.{0} NNReal 0 (Zero.toOfNat0.{0} NNReal instNNRealZero))) z
 Case conversion may be inaccurate. Consider using '#align nnreal.nndist_zero_eq_val' NNReal.nndist_zero_eq_val'ₓ'. -/
 @[simp]
-theorem NNReal.nndist_zero_eq_val' (z : ℝ≥0) : nndist z 0 = z :=
-  by
-  rw [nndist_comm]
+theorem NNReal.nndist_zero_eq_val' (z : ℝ≥0) : nndist z 0 = z := by rw [nndist_comm];
   exact NNReal.nndist_zero_eq_val z
 #align nnreal.nndist_zero_eq_val' NNReal.nndist_zero_eq_val'
 
@@ -3195,16 +3138,12 @@ theorem uniformContinuous_dist : UniformContinuous fun p : α × α => dist p.1
   Metric.uniformContinuous_iff.2 fun ε ε0 =>
     ⟨ε / 2, half_pos ε0, by
       suffices
-      · intro p q h
-        cases' p with p₁ p₂
-        cases' q with q₁ q₂
-        cases' max_lt_iff.1 h with h₁ h₂
-        clear h
+      · intro p q h; cases' p with p₁ p₂; cases' q with q₁ q₂
+        cases' max_lt_iff.1 h with h₁ h₂; clear h
         dsimp at h₁ h₂⊢
         rw [Real.dist_eq]
         refine' abs_sub_lt_iff.2 ⟨_, _⟩
-        · revert p₁ p₂ q₁ q₂ h₁ h₂
-          exact this
+        · revert p₁ p₂ q₁ q₂ h₁ h₂; exact this
         · apply this <;> rwa [dist_comm]
       intro p₁ p₂ q₁ q₂ h₁ h₂
       have :=
@@ -3697,10 +3636,8 @@ lean 3 declaration is
 but is expected to have type
   forall {β : Type.{u2}} {π : β -> Type.{u1}} [_inst_2 : Fintype.{u2} β] [_inst_3 : forall (b : β), PseudoMetricSpace.{u1} (π b)] (f : forall (b : β), π b) (g : forall (b : β), π b) (b : β), LE.le.{0} NNReal (Preorder.toLE.{0} NNReal (PartialOrder.toPreorder.{0} NNReal (StrictOrderedSemiring.toPartialOrder.{0} NNReal instNNRealStrictOrderedSemiring))) (NNDist.nndist.{u1} (π b) (PseudoMetricSpace.toNNDist.{u1} (π b) (_inst_3 b)) (f b) (g b)) (NNDist.nndist.{max u2 u1} (forall (b : β), π b) (PseudoMetricSpace.toNNDist.{max u2 u1} (forall (b : β), π b) (pseudoMetricSpacePi.{u2, u1} β (fun (b : β) => π b) _inst_2 (fun (b : β) => _inst_3 b))) f g)
 Case conversion may be inaccurate. Consider using '#align nndist_le_pi_nndist nndist_le_pi_nndistₓ'. -/
-theorem nndist_le_pi_nndist (f g : ∀ b, π b) (b : β) : nndist (f b) (g b) ≤ nndist f g :=
-  by
-  rw [nndist_pi_def]
-  exact Finset.le_sup (Finset.mem_univ b)
+theorem nndist_le_pi_nndist (f g : ∀ b, π b) (b : β) : nndist (f b) (g b) ≤ nndist f g := by
+  rw [nndist_pi_def]; exact Finset.le_sup (Finset.mem_univ b)
 #align nndist_le_pi_nndist nndist_le_pi_nndist
 
 /- warning: dist_le_pi_dist -> dist_le_pi_dist is a dubious translation:
@@ -3722,10 +3659,7 @@ Case conversion may be inaccurate. Consider using '#align ball_pi ball_piₓ'. -
 /-- An open ball in a product space is a product of open balls. See also `metric.ball_pi'`
 for a version assuming `nonempty β` instead of `0 < r`. -/
 theorem ball_pi (x : ∀ b, π b) {r : ℝ} (hr : 0 < r) :
-    ball x r = Set.pi univ fun b => ball (x b) r :=
-  by
-  ext p
-  simp [dist_pi_lt_iff hr]
+    ball x r = Set.pi univ fun b => ball (x b) r := by ext p; simp [dist_pi_lt_iff hr]
 #align ball_pi ball_pi
 
 /- warning: ball_pi' -> ball_pi' is a dubious translation:
@@ -3750,10 +3684,7 @@ Case conversion may be inaccurate. Consider using '#align closed_ball_pi closedB
 /-- A closed ball in a product space is a product of closed balls. See also `metric.closed_ball_pi'`
 for a version assuming `nonempty β` instead of `0 ≤ r`. -/
 theorem closedBall_pi (x : ∀ b, π b) {r : ℝ} (hr : 0 ≤ r) :
-    closedBall x r = Set.pi univ fun b => closedBall (x b) r :=
-  by
-  ext p
-  simp [dist_pi_le_iff hr]
+    closedBall x r = Set.pi univ fun b => closedBall (x b) r := by ext p; simp [dist_pi_le_iff hr]
 #align closed_ball_pi closedBall_pi
 
 /- warning: closed_ball_pi' -> closedBall_pi' is a dubious translation:
@@ -4014,8 +3945,7 @@ center and a strictly smaller radius that includes `s`. -/
 theorem exists_lt_subset_ball (hs : IsClosed s) (h : s ⊆ ball x r) : ∃ r' < r, s ⊆ ball x r' :=
   by
   cases' le_or_lt r 0 with hr hr
-  · rw [ball_eq_empty.2 hr, subset_empty_iff] at h
-    subst s
+  · rw [ball_eq_empty.2 hr, subset_empty_iff] at h; subst s
     exact (exists_lt r).imp fun r' hr' => ⟨hr', empty_subset _⟩
   · exact (exists_pos_lt_subset_ball hr hs h).imp fun r' hr' => ⟨hr'.fst.2, hr'.snd⟩
 #align exists_lt_subset_ball exists_lt_subset_ball
@@ -4160,8 +4090,7 @@ theorem bounded_iff_subset_ball (c : α) : Bounded s ↔ ∃ r, s ⊆ closedBall
   by
   constructor <;> rintro ⟨C, hC⟩
   · cases' s.eq_empty_or_nonempty with h h
-    · subst s
-      exact ⟨0, by simp⟩
+    · subst s; exact ⟨0, by simp⟩
     · rcases h with ⟨x, hx⟩
       exact
         ⟨C + dist x c, fun y hy =>
@@ -4945,10 +4874,8 @@ theorem comap_dist_left_atTop_eq_cocompact [ProperSpace α] (x : α) :
 
 #print tendsto_cocompact_of_tendsto_dist_comp_atTop /-
 theorem tendsto_cocompact_of_tendsto_dist_comp_atTop {f : β → α} {l : Filter β} (x : α)
-    (h : Tendsto (fun y => dist (f y) x) l atTop) : Tendsto f l (cocompact α) :=
-  by
-  refine' tendsto.mono_right _ (comap_dist_right_atTop_le_cocompact x)
-  rwa [tendsto_comap_iff]
+    (h : Tendsto (fun y => dist (f y) x) l atTop) : Tendsto f l (cocompact α) := by
+  refine' tendsto.mono_right _ (comap_dist_right_atTop_le_cocompact x); rwa [tendsto_comap_iff]
 #align tendsto_cocompact_of_tendsto_dist_comp_at_top tendsto_cocompact_of_tendsto_dist_comp_atTop
 -/
 
@@ -4966,8 +4893,7 @@ theorem MetricSpace.ext {α : Type _} {m m' : MetricSpace α} (h : m.toHasDist =
     m = m' :=
   by
   have h' : m.to_pseudo_metric_space = m'.to_pseudo_metric_space := PseudoMetricSpace.ext h
-  rcases m with ⟨⟩
-  rcases m' with ⟨⟩
+  rcases m with ⟨⟩; rcases m' with ⟨⟩
   dsimp at h'
   subst h'
 #align metric_space.ext MetricSpace.ext
@@ -5135,10 +5061,8 @@ Case conversion may be inaccurate. Consider using '#align metric.subsingleton_cl
 theorem subsingleton_closedBall (x : γ) {r : ℝ} (hr : r ≤ 0) : (closedBall x r).Subsingleton :=
   by
   rcases hr.lt_or_eq with (hr | rfl)
-  · rw [closed_ball_eq_empty.2 hr]
-    exact subsingleton_empty
-  · rw [closed_ball_zero]
-    exact subsingleton_singleton
+  · rw [closed_ball_eq_empty.2 hr]; exact subsingleton_empty
+  · rw [closed_ball_zero]; exact subsingleton_singleton
 #align metric.subsingleton_closed_ball Metric.subsingleton_closedBall
 
 /- warning: metric.subsingleton_sphere -> Metric.subsingleton_sphere is a dubious translation:
@@ -5247,10 +5171,7 @@ def MetricSpace.replaceUniformity {γ} [U : UniformSpace γ] (m : MetricSpace γ
 
 #print MetricSpace.replaceUniformity_eq /-
 theorem MetricSpace.replaceUniformity_eq {γ} [U : UniformSpace γ] (m : MetricSpace γ)
-    (H : 𝓤[U] = 𝓤[PseudoEMetricSpace.toUniformSpace]) : m.replaceUniformity H = m :=
-  by
-  ext
-  rfl
+    (H : 𝓤[U] = 𝓤[PseudoEMetricSpace.toUniformSpace]) : m.replaceUniformity H = m := by ext; rfl
 #align metric_space.replace_uniformity_eq MetricSpace.replaceUniformity_eq
 -/
 
@@ -5269,9 +5190,7 @@ def MetricSpace.replaceTopology {γ} [U : TopologicalSpace γ] (m : MetricSpace
 #print MetricSpace.replaceTopology_eq /-
 theorem MetricSpace.replaceTopology_eq {γ} [U : TopologicalSpace γ] (m : MetricSpace γ)
     (H : U = m.toPseudoMetricSpace.toUniformSpace.toTopologicalSpace) : m.replaceTopology H = m :=
-  by
-  ext
-  rfl
+  by ext; rfl
 #align metric_space.replace_topology_eq MetricSpace.replaceTopology_eq
 -/
 
@@ -5321,9 +5240,7 @@ def MetricSpace.replaceBornology {α} [B : Bornology α] (m : MetricSpace α)
 #print MetricSpace.replaceBornology_eq /-
 theorem MetricSpace.replaceBornology_eq {α} [m : MetricSpace α] [B : Bornology α]
     (H : ∀ s, @IsBounded _ B s ↔ @IsBounded _ PseudoMetricSpace.toBornology s) :
-    MetricSpace.replaceBornology _ H = m := by
-  ext
-  rfl
+    MetricSpace.replaceBornology _ H = m := by ext; rfl
 #align metric_space.replace_bornology_eq MetricSpace.replaceBornology_eq
 -/
 
@@ -5476,8 +5393,7 @@ theorem secondCountable_of_countable_discretization {α : Type u} [MetricSpace 
     SecondCountableTopology α :=
   by
   cases' (univ : Set α).eq_empty_or_nonempty with hs hs
-  · haveI : CompactSpace α := ⟨by rw [hs] <;> exact isCompact_empty⟩
-    · infer_instance
+  · haveI : CompactSpace α := ⟨by rw [hs] <;> exact isCompact_empty⟩; · infer_instance
   rcases hs with ⟨x0, hx0⟩
   letI : Inhabited α := ⟨x0⟩
   refine' second_countable_of_almost_dense_set fun ε ε0 => _

mathlib commit https://github.com/leanprover-community/mathlib/commit/917c3c072e487b3cccdbfeff17e75b40e45f66cb

@@ -95,7 +95,6 @@ private theorem bounded_iff_aux {α : Type _} (dist : α → α → ℝ)
             (by
               rw [dist_comm]
               exact hC hy))⟩
-#align bounded_iff_aux bounded_iff_aux
 
 /-- Construct a bornology from a distance function and metric space axioms. -/
 def Bornology.ofDist {α : Type _} (dist : α → α → ℝ) (dist_self : ∀ x : α, dist x x = 0)
@@ -141,7 +140,6 @@ private theorem pseudo_metric_space.dist_nonneg' {α} {x y : α} (dist : α →
       _ ≥ 0 := by rw [← dist_self x] <;> apply dist_triangle
       
   nonneg_of_mul_nonneg_right this zero_lt_two
-#align pseudo_metric_space.dist_nonneg' pseudo_metric_space.dist_nonneg'
 
 /- warning: pseudo_metric_space.edist_dist_tac clashes with [anonymous] -> [anonymous]
 Case conversion may be inaccurate. Consider using '#align pseudo_metric_space.edist_dist_tac [anonymous]ₓ'. -/
@@ -1586,10 +1584,7 @@ theorem totallyBounded_iff {s : Set α} :
 #align metric.totally_bounded_iff Metric.totallyBounded_iff
 
 /- warning: metric.totally_bounded_of_finite_discretization -> Metric.totallyBounded_of_finite_discretization is a dubious translation:
-lean 3 declaration is
-  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {s : Set.{u1} α}, (forall (ε : Real), (GT.gt.{0} Real Real.hasLt ε (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero)))) -> (Exists.{succ (succ u1)} Type.{u1} (fun (β : Type.{u1}) => Exists.{succ u1} (Fintype.{u1} β) (fun (_x : Fintype.{u1} β) => Exists.{succ u1} ((coeSort.{succ u1, succ (succ u1)} (Set.{u1} α) Type.{u1} (Set.hasCoeToSort.{u1} α) s) -> β) (fun (F : (coeSort.{succ u1, succ (succ u1)} (Set.{u1} α) Type.{u1} (Set.hasCoeToSort.{u1} α) s) -> β) => forall (x : coeSort.{succ u1, succ (succ u1)} (Set.{u1} α) Type.{u1} (Set.hasCoeToSort.{u1} α) s) (y : coeSort.{succ u1, succ (succ u1)} (Set.{u1} α) Type.{u1} (Set.hasCoeToSort.{u1} α) s), (Eq.{succ u1} β (F x) (F y)) -> (LT.lt.{0} Real Real.hasLt (Dist.dist.{u1} α (PseudoMetricSpace.toHasDist.{u1} α _inst_1) ((fun (a : Type.{u1}) (b : Type.{u1}) [self : HasLiftT.{succ u1, succ u1} a b] => self.0) (coeSort.{succ u1, succ (succ u1)} (Set.{u1} α) Type.{u1} (Set.hasCoeToSort.{u1} α) s) α (HasLiftT.mk.{succ u1, succ u1} (coeSort.{succ u1, succ (succ u1)} (Set.{u1} α) Type.{u1} (Set.hasCoeToSort.{u1} α) s) α (CoeTCₓ.coe.{succ u1, succ u1} (coeSort.{succ u1, succ (succ u1)} (Set.{u1} α) Type.{u1} (Set.hasCoeToSort.{u1} α) s) α (coeBase.{succ u1, succ u1} (coeSort.{succ u1, succ (succ u1)} (Set.{u1} α) Type.{u1} (Set.hasCoeToSort.{u1} α) s) α (coeSubtype.{succ u1} α (fun (x : α) => Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) x s))))) x) ((fun (a : Type.{u1}) (b : Type.{u1}) [self : HasLiftT.{succ u1, succ u1} a b] => self.0) (coeSort.{succ u1, succ (succ u1)} (Set.{u1} α) Type.{u1} (Set.hasCoeToSort.{u1} α) s) α (HasLiftT.mk.{succ u1, succ u1} (coeSort.{succ u1, succ (succ u1)} (Set.{u1} α) Type.{u1} (Set.hasCoeToSort.{u1} α) s) α (CoeTCₓ.coe.{succ u1, succ u1} (coeSort.{succ u1, succ (succ u1)} (Set.{u1} α) Type.{u1} (Set.hasCoeToSort.{u1} α) s) α (coeBase.{succ u1, succ u1} (coeSort.{succ u1, succ (succ u1)} (Set.{u1} α) Type.{u1} (Set.hasCoeToSort.{u1} α) s) α (coeSubtype.{succ u1} α (fun (x : α) => Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) x s))))) y)) ε)))))) -> (TotallyBounded.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1) s)
-but is expected to have type
-  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {s : Set.{u1} α}, (forall (ε : Real), (GT.gt.{0} Real Real.instLTReal ε (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal))) -> (Exists.{succ (succ u1)} Type.{u1} (fun (β : Type.{u1}) => Exists.{succ u1} (Fintype.{u1} β) (fun (_x : Fintype.{u1} β) => Exists.{succ u1} ((Set.Elem.{u1} α s) -> β) (fun (F : (Set.Elem.{u1} α s) -> β) => forall (x : Set.Elem.{u1} α s) (y : Set.Elem.{u1} α s), (Eq.{succ u1} β (F x) (F y)) -> (LT.lt.{0} Real Real.instLTReal (Dist.dist.{u1} α (PseudoMetricSpace.toDist.{u1} α _inst_1) (Subtype.val.{succ u1} α (fun (x : α) => Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) x s) x) (Subtype.val.{succ u1} α (fun (x : α) => Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) x s) y)) ε)))))) -> (TotallyBounded.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1) s)
+<too large>
 Case conversion may be inaccurate. Consider using '#align metric.totally_bounded_of_finite_discretization Metric.totallyBounded_of_finite_discretizationₓ'. -/
 /-- A pseudometric space is totally bounded if one can reconstruct up to any ε>0 any element of the
 space from finitely many data. -/
@@ -3576,7 +3571,7 @@ instance pseudoMetricSpacePi : PseudoMetricSpace (∀ b, π b) :=
   exact ne_of_lt ((Finset.sup_lt_iff bot_lt_top).2 fun b hb => edist_lt_top _ _)
   show ↑(sup univ fun b => nndist (f b) (g b)) = (sup univ fun b => edist (f b) (g b)).toReal
   · simp only [edist_nndist, ← ENNReal.coe_finset_sup, ENNReal.coe_toReal]
-  show @is_bounded _ Pi.bornology s ↔ @is_bounded _ PseudoMetricSpace.toBornology _
+  show @is_bounded _ Pi.instBornology s ↔ @is_bounded _ PseudoMetricSpace.toBornology _
   · simp only [← is_bounded_def, is_bounded_iff_eventually, ← forall_is_bounded_image_eval_iff,
       ball_image_iff, ← eventually_all, Function.eval_apply, @dist_nndist (π _)]
     refine' eventually_congr ((eventually_ge_at_top 0).mono fun C hC => _)
@@ -3775,10 +3770,7 @@ theorem closedBall_pi' [Nonempty β] (x : ∀ b, π b) (r : ℝ) :
 #align closed_ball_pi' closedBall_pi'
 
 /- warning: fin.nndist_insert_nth_insert_nth -> Fin.nndist_insertNth_insertNth is a dubious translation:
-lean 3 declaration is
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+<too large>
 Case conversion may be inaccurate. Consider using '#align fin.nndist_insert_nth_insert_nth Fin.nndist_insertNth_insertNthₓ'. -/
 @[simp]
 theorem Fin.nndist_insertNth_insertNth {n : ℕ} {α : Fin (n + 1) → Type _}
@@ -3788,10 +3780,7 @@ theorem Fin.nndist_insertNth_insertNth {n : ℕ} {α : Fin (n + 1) → Type _}
 #align fin.nndist_insert_nth_insert_nth Fin.nndist_insertNth_insertNth
 
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(instOfNatNat 1))))) (Fin n) (fun (_x : Fin n) => (fun (x._@.Mathlib.Order.Hom.Lattice._hyg.494 : Fin n) => Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) _x) (InfHomClass.toFunLike.{0, 0, 0} (OrderEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin n) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))))) (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (Lattice.toInf.{0} (Fin n) (DistribLattice.toLattice.{0} (Fin n) (instDistribLattice.{0} (Fin n) (Fin.instLinearOrderFin n)))) (Lattice.toInf.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (Fin.instLatticeFinHAddNatInstHAddInstAddNatOfNat n)) 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(OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (Fin.instLinearOrderFin n) (Fin.instLatticeFinHAddNatInstHAddInstAddNatOfNat n) (RelEmbedding.instRelHomClassRelEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.682 : Fin n) (x._@.Mathlib.Order.Hom.Basic._hyg.684 : Fin n) => LE.le.{0} (Fin n) (instLEFin n) x._@.Mathlib.Order.Hom.Basic._hyg.682 x._@.Mathlib.Order.Hom.Basic._hyg.684) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.697 : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (x._@.Mathlib.Order.Hom.Basic._hyg.699 : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) => LE.le.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) x._@.Mathlib.Order.Hom.Basic._hyg.697 x._@.Mathlib.Order.Hom.Basic._hyg.699))))) (Fin.succAbove n i) j)) (Fin.fintype n) (fun (b : Fin n) => _inst_4 (FunLike.coe.{1, 1, 1} (OrderEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin n) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))))) (Fin n) (fun (_x : Fin n) => (fun (x._@.Mathlib.Order.Hom.Lattice._hyg.494 : Fin n) => Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) _x) (InfHomClass.toFunLike.{0, 0, 0} (OrderEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin n) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))))) (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (Lattice.toInf.{0} (Fin n) (DistribLattice.toLattice.{0} (Fin n) (instDistribLattice.{0} (Fin n) (Fin.instLinearOrderFin n)))) (Lattice.toInf.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (Fin.instLatticeFinHAddNatInstHAddInstAddNatOfNat n)) (LatticeHomClass.toInfHomClass.{0, 0, 0} (OrderEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin n) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))))) (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (DistribLattice.toLattice.{0} (Fin n) (instDistribLattice.{0} (Fin n) (Fin.instLinearOrderFin n))) (Fin.instLatticeFinHAddNatInstHAddInstAddNatOfNat n) (OrderHomClass.toLatticeHomClass.{0, 0, 0} (OrderEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin n) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))))) (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (Fin.instLinearOrderFin n) (Fin.instLatticeFinHAddNatInstHAddInstAddNatOfNat n) (RelEmbedding.instRelHomClassRelEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.682 : Fin n) (x._@.Mathlib.Order.Hom.Basic._hyg.684 : Fin n) => LE.le.{0} (Fin n) (instLEFin n) x._@.Mathlib.Order.Hom.Basic._hyg.682 x._@.Mathlib.Order.Hom.Basic._hyg.684) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.697 : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (x._@.Mathlib.Order.Hom.Basic._hyg.699 : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) => LE.le.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) x._@.Mathlib.Order.Hom.Basic._hyg.697 x._@.Mathlib.Order.Hom.Basic._hyg.699))))) (Fin.succAbove n i) b)))) f g))
+<too large>
 Case conversion may be inaccurate. Consider using '#align fin.dist_insert_nth_insert_nth Fin.dist_insertNth_insertNthₓ'. -/
 @[simp]
 theorem Fin.dist_insertNth_insertNth {n : ℕ} {α : Fin (n + 1) → Type _}

mathlib commit https://github.com/leanprover-community/mathlib/commit/ef95945cd48c932c9e034872bd25c3c220d9c946

@@ -1618,7 +1618,7 @@ theorem totallyBounded_of_finite_discretization {s : Set α}
 lean 3 declaration is
   forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {s : Set.{u1} α}, (TotallyBounded.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1) s) -> (forall (ε : Real), (GT.gt.{0} Real Real.hasLt ε (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero)))) -> (Exists.{succ u1} (Set.{u1} α) (fun (t : Set.{u1} α) => Exists.{0} (HasSubset.Subset.{u1} (Set.{u1} α) (Set.hasSubset.{u1} α) t s) (fun (H : HasSubset.Subset.{u1} (Set.{u1} α) (Set.hasSubset.{u1} α) t s) => And (Set.Finite.{u1} α t) (HasSubset.Subset.{u1} (Set.{u1} α) (Set.hasSubset.{u1} α) s (Set.iUnion.{u1, succ u1} α α (fun (y : α) => Set.iUnion.{u1, 0} α (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) y t) (fun (H : Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) y t) => Metric.ball.{u1} α _inst_1 y ε))))))))
 but is expected to have type
-  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {s : Set.{u1} α}, (TotallyBounded.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1) s) -> (forall (ε : Real), (GT.gt.{0} Real Real.instLTReal ε (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal))) -> (Exists.{succ u1} (Set.{u1} α) (fun (t : Set.{u1} α) => And (HasSubset.Subset.{u1} (Set.{u1} α) (Set.instHasSubsetSet.{u1} α) t s) (And (Set.Finite.{u1} α t) (HasSubset.Subset.{u1} (Set.{u1} α) (Set.instHasSubsetSet.{u1} α) s (Set.iUnion.{u1, succ u1} α α (fun (y : α) => Set.iUnion.{u1, 0} α (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) y t) (fun (h._@.Mathlib.Topology.MetricSpace.Basic._hyg.10758 : Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) y t) => Metric.ball.{u1} α _inst_1 y ε))))))))
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {s : Set.{u1} α}, (TotallyBounded.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1) s) -> (forall (ε : Real), (GT.gt.{0} Real Real.instLTReal ε (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal))) -> (Exists.{succ u1} (Set.{u1} α) (fun (t : Set.{u1} α) => And (HasSubset.Subset.{u1} (Set.{u1} α) (Set.instHasSubsetSet.{u1} α) t s) (And (Set.Finite.{u1} α t) (HasSubset.Subset.{u1} (Set.{u1} α) (Set.instHasSubsetSet.{u1} α) s (Set.iUnion.{u1, succ u1} α α (fun (y : α) => Set.iUnion.{u1, 0} α (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) y t) (fun (h._@.Mathlib.Topology.MetricSpace.Basic._hyg.11021 : Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) y t) => Metric.ball.{u1} α _inst_1 y ε))))))))
 Case conversion may be inaccurate. Consider using '#align metric.finite_approx_of_totally_bounded Metric.finite_approx_of_totallyBoundedₓ'. -/
 /- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (t «expr ⊆ » s) -/
 theorem finite_approx_of_totallyBounded {s : Set α} (hs : TotallyBounded s) :
@@ -3824,7 +3824,7 @@ section Compact
 lean 3 declaration is
   forall {α : Type.{u1}} [_inst_2 : PseudoMetricSpace.{u1} α] {s : Set.{u1} α}, (IsCompact.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_2)) s) -> (forall {e : Real}, (LT.lt.{0} Real Real.hasLt (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero))) e) -> (Exists.{succ u1} (Set.{u1} α) (fun (t : Set.{u1} α) => Exists.{0} (HasSubset.Subset.{u1} (Set.{u1} α) (Set.hasSubset.{u1} α) t s) (fun (H : HasSubset.Subset.{u1} (Set.{u1} α) (Set.hasSubset.{u1} α) t s) => And (Set.Finite.{u1} α t) (HasSubset.Subset.{u1} (Set.{u1} α) (Set.hasSubset.{u1} α) s (Set.iUnion.{u1, succ u1} α α (fun (x : α) => Set.iUnion.{u1, 0} α (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) x t) (fun (H : Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) x t) => Metric.ball.{u1} α _inst_2 x e))))))))
 but is expected to have type
-  forall {α : Type.{u1}} [_inst_2 : PseudoMetricSpace.{u1} α] {s : Set.{u1} α}, (IsCompact.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_2)) s) -> (forall {e : Real}, (LT.lt.{0} Real Real.instLTReal (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal)) e) -> (Exists.{succ u1} (Set.{u1} α) (fun (t : Set.{u1} α) => And (HasSubset.Subset.{u1} (Set.{u1} α) (Set.instHasSubsetSet.{u1} α) t s) (And (Set.Finite.{u1} α t) (HasSubset.Subset.{u1} (Set.{u1} α) (Set.instHasSubsetSet.{u1} α) s (Set.iUnion.{u1, succ u1} α α (fun (x : α) => Set.iUnion.{u1, 0} α (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) x t) (fun (h._@.Mathlib.Topology.MetricSpace.Basic._hyg.25536 : Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) x t) => Metric.ball.{u1} α _inst_2 x e))))))))
+  forall {α : Type.{u1}} [_inst_2 : PseudoMetricSpace.{u1} α] {s : Set.{u1} α}, (IsCompact.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_2)) s) -> (forall {e : Real}, (LT.lt.{0} Real Real.instLTReal (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal)) e) -> (Exists.{succ u1} (Set.{u1} α) (fun (t : Set.{u1} α) => And (HasSubset.Subset.{u1} (Set.{u1} α) (Set.instHasSubsetSet.{u1} α) t s) (And (Set.Finite.{u1} α t) (HasSubset.Subset.{u1} (Set.{u1} α) (Set.instHasSubsetSet.{u1} α) s (Set.iUnion.{u1, succ u1} α α (fun (x : α) => Set.iUnion.{u1, 0} α (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) x t) (fun (h._@.Mathlib.Topology.MetricSpace.Basic._hyg.25799 : Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) x t) => Metric.ball.{u1} α _inst_2 x e))))))))
 Case conversion may be inaccurate. Consider using '#align finite_cover_balls_of_compact finite_cover_balls_of_compactₓ'. -/
 /- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (t «expr ⊆ » s) -/
 /-- Any compact set in a pseudometric space can be covered by finitely many balls of a given
@@ -3844,7 +3844,7 @@ theorem finite_cover_balls_of_compact {α : Type u} [PseudoMetricSpace α] {s :
 lean 3 declaration is
   forall {α : Type.{u1}} [_inst_2 : PseudoMetricSpace.{u1} α] {s : Set.{u1} α}, (IsCompact.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_2)) s) -> (forall {e : Real}, (LT.lt.{0} Real Real.hasLt (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero))) e) -> (Exists.{succ u1} (Set.{u1} α) (fun (t : Set.{u1} α) => Exists.{0} (HasSubset.Subset.{u1} (Set.{u1} α) (Set.hasSubset.{u1} α) t s) (fun (H : HasSubset.Subset.{u1} (Set.{u1} α) (Set.hasSubset.{u1} α) t s) => And (Set.Finite.{u1} α t) (HasSubset.Subset.{u1} (Set.{u1} α) (Set.hasSubset.{u1} α) s (Set.iUnion.{u1, succ u1} α α (fun (x : α) => Set.iUnion.{u1, 0} α (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) x t) (fun (H : Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) x t) => Metric.ball.{u1} α _inst_2 x e))))))))
 but is expected to have type
-  forall {α : Type.{u1}} [_inst_2 : PseudoMetricSpace.{u1} α] {s : Set.{u1} α}, (IsCompact.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_2)) s) -> (forall {e : Real}, (LT.lt.{0} Real Real.instLTReal (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal)) e) -> (Exists.{succ u1} (Set.{u1} α) (fun (t : Set.{u1} α) => And (HasSubset.Subset.{u1} (Set.{u1} α) (Set.instHasSubsetSet.{u1} α) t s) (And (Set.Finite.{u1} α t) (HasSubset.Subset.{u1} (Set.{u1} α) (Set.instHasSubsetSet.{u1} α) s (Set.iUnion.{u1, succ u1} α α (fun (x : α) => Set.iUnion.{u1, 0} α (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) x t) (fun (h._@.Mathlib.Topology.MetricSpace.Basic._hyg.25536 : Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) x t) => Metric.ball.{u1} α _inst_2 x e))))))))
+  forall {α : Type.{u1}} [_inst_2 : PseudoMetricSpace.{u1} α] {s : Set.{u1} α}, (IsCompact.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_2)) s) -> (forall {e : Real}, (LT.lt.{0} Real Real.instLTReal (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal)) e) -> (Exists.{succ u1} (Set.{u1} α) (fun (t : Set.{u1} α) => And (HasSubset.Subset.{u1} (Set.{u1} α) (Set.instHasSubsetSet.{u1} α) t s) (And (Set.Finite.{u1} α t) (HasSubset.Subset.{u1} (Set.{u1} α) (Set.instHasSubsetSet.{u1} α) s (Set.iUnion.{u1, succ u1} α α (fun (x : α) => Set.iUnion.{u1, 0} α (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) x t) (fun (h._@.Mathlib.Topology.MetricSpace.Basic._hyg.25799 : Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) x t) => Metric.ball.{u1} α _inst_2 x e))))))))
 Case conversion may be inaccurate. Consider using '#align is_compact.finite_cover_balls IsCompact.finite_cover_ballsₓ'. -/
 alias finite_cover_balls_of_compact ← IsCompact.finite_cover_balls
 #align is_compact.finite_cover_balls IsCompact.finite_cover_balls

mathlib commit https://github.com/leanprover-community/mathlib/commit/95a87616d63b3cb49d3fe678d416fbe9c4217bf4

@@ -2470,7 +2470,7 @@ theorem Real.dist_eq (x y : ℝ) : dist x y = |x - y| :=
 lean 3 declaration is
   forall (x : Real) (y : Real), Eq.{1} NNReal (NNDist.nndist.{0} Real (PseudoMetricSpace.toNNDist.{0} Real Real.pseudoMetricSpace) x y) (coeFn.{1, 1} (MonoidWithZeroHom.{0, 0} Real NNReal (NonAssocSemiring.toMulZeroOneClass.{0} Real (NonAssocRing.toNonAssocSemiring.{0} Real (Ring.toNonAssocRing.{0} Real Real.ring))) (NonAssocSemiring.toMulZeroOneClass.{0} NNReal (Semiring.toNonAssocSemiring.{0} NNReal NNReal.semiring))) (fun (_x : MonoidWithZeroHom.{0, 0} Real NNReal (NonAssocSemiring.toMulZeroOneClass.{0} Real (NonAssocRing.toNonAssocSemiring.{0} Real (Ring.toNonAssocRing.{0} Real Real.ring))) (NonAssocSemiring.toMulZeroOneClass.{0} NNReal (Semiring.toNonAssocSemiring.{0} NNReal NNReal.semiring))) => Real -> NNReal) (MonoidWithZeroHom.hasCoeToFun.{0, 0} Real NNReal (NonAssocSemiring.toMulZeroOneClass.{0} Real (NonAssocRing.toNonAssocSemiring.{0} Real (Ring.toNonAssocRing.{0} Real Real.ring))) (NonAssocSemiring.toMulZeroOneClass.{0} NNReal (Semiring.toNonAssocSemiring.{0} NNReal NNReal.semiring))) Real.nnabs (HSub.hSub.{0, 0, 0} Real Real Real (instHSub.{0} Real Real.hasSub) x y))
 but is expected to have type
-  forall (x : Real) (y : Real), Eq.{1} NNReal (NNDist.nndist.{0} Real (PseudoMetricSpace.toNNDist.{0} Real Real.pseudoMetricSpace) x y) (FunLike.coe.{1, 1, 1} (MonoidWithZeroHom.{0, 0} Real NNReal (NonAssocSemiring.toMulZeroOneClass.{0} Real (Semiring.toNonAssocSemiring.{0} Real Real.semiring)) (NonAssocSemiring.toMulZeroOneClass.{0} NNReal (Semiring.toNonAssocSemiring.{0} NNReal instNNRealSemiring))) Real (fun (_x : Real) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : Real) => NNReal) _x) (MulHomClass.toFunLike.{0, 0, 0} (MonoidWithZeroHom.{0, 0} Real NNReal (NonAssocSemiring.toMulZeroOneClass.{0} Real (Semiring.toNonAssocSemiring.{0} Real Real.semiring)) (NonAssocSemiring.toMulZeroOneClass.{0} NNReal (Semiring.toNonAssocSemiring.{0} NNReal instNNRealSemiring))) Real NNReal (MulOneClass.toMul.{0} Real (MulZeroOneClass.toMulOneClass.{0} Real (NonAssocSemiring.toMulZeroOneClass.{0} Real (Semiring.toNonAssocSemiring.{0} Real Real.semiring)))) (MulOneClass.toMul.{0} NNReal (MulZeroOneClass.toMulOneClass.{0} NNReal (NonAssocSemiring.toMulZeroOneClass.{0} NNReal (Semiring.toNonAssocSemiring.{0} NNReal instNNRealSemiring)))) (MonoidHomClass.toMulHomClass.{0, 0, 0} (MonoidWithZeroHom.{0, 0} Real NNReal (NonAssocSemiring.toMulZeroOneClass.{0} Real (Semiring.toNonAssocSemiring.{0} Real Real.semiring)) (NonAssocSemiring.toMulZeroOneClass.{0} NNReal (Semiring.toNonAssocSemiring.{0} NNReal instNNRealSemiring))) Real NNReal (MulZeroOneClass.toMulOneClass.{0} Real (NonAssocSemiring.toMulZeroOneClass.{0} Real (Semiring.toNonAssocSemiring.{0} Real Real.semiring))) (MulZeroOneClass.toMulOneClass.{0} NNReal (NonAssocSemiring.toMulZeroOneClass.{0} NNReal (Semiring.toNonAssocSemiring.{0} NNReal instNNRealSemiring))) (MonoidWithZeroHomClass.toMonoidHomClass.{0, 0, 0} (MonoidWithZeroHom.{0, 0} Real NNReal (NonAssocSemiring.toMulZeroOneClass.{0} Real (Semiring.toNonAssocSemiring.{0} Real Real.semiring)) (NonAssocSemiring.toMulZeroOneClass.{0} NNReal (Semiring.toNonAssocSemiring.{0} NNReal instNNRealSemiring))) Real NNReal (NonAssocSemiring.toMulZeroOneClass.{0} Real (Semiring.toNonAssocSemiring.{0} Real Real.semiring)) (NonAssocSemiring.toMulZeroOneClass.{0} NNReal (Semiring.toNonAssocSemiring.{0} NNReal instNNRealSemiring)) (MonoidWithZeroHom.monoidWithZeroHomClass.{0, 0} Real NNReal (NonAssocSemiring.toMulZeroOneClass.{0} Real (Semiring.toNonAssocSemiring.{0} Real Real.semiring)) (NonAssocSemiring.toMulZeroOneClass.{0} NNReal (Semiring.toNonAssocSemiring.{0} NNReal instNNRealSemiring)))))) Real.nnabs (HSub.hSub.{0, 0, 0} Real Real Real (instHSub.{0} Real Real.instSubReal) x y))
+  forall (x : Real) (y : Real), Eq.{1} NNReal (NNDist.nndist.{0} Real (PseudoMetricSpace.toNNDist.{0} Real Real.pseudoMetricSpace) x y) (FunLike.coe.{1, 1, 1} (MonoidWithZeroHom.{0, 0} Real NNReal (NonAssocSemiring.toMulZeroOneClass.{0} Real (Semiring.toNonAssocSemiring.{0} Real Real.semiring)) (NonAssocSemiring.toMulZeroOneClass.{0} NNReal (Semiring.toNonAssocSemiring.{0} NNReal instNNRealSemiring))) Real (fun (_x : Real) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : Real) => NNReal) _x) (MulHomClass.toFunLike.{0, 0, 0} (MonoidWithZeroHom.{0, 0} Real NNReal (NonAssocSemiring.toMulZeroOneClass.{0} Real (Semiring.toNonAssocSemiring.{0} Real Real.semiring)) (NonAssocSemiring.toMulZeroOneClass.{0} NNReal (Semiring.toNonAssocSemiring.{0} NNReal instNNRealSemiring))) Real NNReal (MulOneClass.toMul.{0} Real (MulZeroOneClass.toMulOneClass.{0} Real (NonAssocSemiring.toMulZeroOneClass.{0} Real (Semiring.toNonAssocSemiring.{0} Real Real.semiring)))) (MulOneClass.toMul.{0} NNReal (MulZeroOneClass.toMulOneClass.{0} NNReal (NonAssocSemiring.toMulZeroOneClass.{0} NNReal (Semiring.toNonAssocSemiring.{0} NNReal instNNRealSemiring)))) (MonoidHomClass.toMulHomClass.{0, 0, 0} (MonoidWithZeroHom.{0, 0} Real NNReal (NonAssocSemiring.toMulZeroOneClass.{0} Real (Semiring.toNonAssocSemiring.{0} Real Real.semiring)) (NonAssocSemiring.toMulZeroOneClass.{0} NNReal (Semiring.toNonAssocSemiring.{0} NNReal instNNRealSemiring))) Real NNReal (MulZeroOneClass.toMulOneClass.{0} Real (NonAssocSemiring.toMulZeroOneClass.{0} Real (Semiring.toNonAssocSemiring.{0} Real Real.semiring))) (MulZeroOneClass.toMulOneClass.{0} NNReal (NonAssocSemiring.toMulZeroOneClass.{0} NNReal (Semiring.toNonAssocSemiring.{0} NNReal instNNRealSemiring))) (MonoidWithZeroHomClass.toMonoidHomClass.{0, 0, 0} (MonoidWithZeroHom.{0, 0} Real NNReal (NonAssocSemiring.toMulZeroOneClass.{0} Real (Semiring.toNonAssocSemiring.{0} Real Real.semiring)) (NonAssocSemiring.toMulZeroOneClass.{0} NNReal (Semiring.toNonAssocSemiring.{0} NNReal instNNRealSemiring))) Real NNReal (NonAssocSemiring.toMulZeroOneClass.{0} Real (Semiring.toNonAssocSemiring.{0} Real Real.semiring)) (NonAssocSemiring.toMulZeroOneClass.{0} NNReal (Semiring.toNonAssocSemiring.{0} NNReal instNNRealSemiring)) (MonoidWithZeroHom.monoidWithZeroHomClass.{0, 0} Real NNReal (NonAssocSemiring.toMulZeroOneClass.{0} Real (Semiring.toNonAssocSemiring.{0} Real Real.semiring)) (NonAssocSemiring.toMulZeroOneClass.{0} NNReal (Semiring.toNonAssocSemiring.{0} NNReal instNNRealSemiring)))))) Real.nnabs (HSub.hSub.{0, 0, 0} Real Real Real (instHSub.{0} Real Real.instSubReal) x y))
 Case conversion may be inaccurate. Consider using '#align real.nndist_eq Real.nndist_eqₓ'. -/
 theorem Real.nndist_eq (x y : ℝ) : nndist x y = Real.nnabs (x - y) :=
   rfl
@@ -2480,7 +2480,7 @@ theorem Real.nndist_eq (x y : ℝ) : nndist x y = Real.nnabs (x - y) :=
 lean 3 declaration is
   forall (x : Real) (y : Real), Eq.{1} NNReal (NNDist.nndist.{0} Real (PseudoMetricSpace.toNNDist.{0} Real Real.pseudoMetricSpace) x y) (coeFn.{1, 1} (MonoidWithZeroHom.{0, 0} Real NNReal (NonAssocSemiring.toMulZeroOneClass.{0} Real (NonAssocRing.toNonAssocSemiring.{0} Real (Ring.toNonAssocRing.{0} Real Real.ring))) (NonAssocSemiring.toMulZeroOneClass.{0} NNReal (Semiring.toNonAssocSemiring.{0} NNReal NNReal.semiring))) (fun (_x : MonoidWithZeroHom.{0, 0} Real NNReal (NonAssocSemiring.toMulZeroOneClass.{0} Real (NonAssocRing.toNonAssocSemiring.{0} Real (Ring.toNonAssocRing.{0} Real Real.ring))) (NonAssocSemiring.toMulZeroOneClass.{0} NNReal (Semiring.toNonAssocSemiring.{0} NNReal NNReal.semiring))) => Real -> NNReal) (MonoidWithZeroHom.hasCoeToFun.{0, 0} Real NNReal (NonAssocSemiring.toMulZeroOneClass.{0} Real (NonAssocRing.toNonAssocSemiring.{0} Real (Ring.toNonAssocRing.{0} Real Real.ring))) (NonAssocSemiring.toMulZeroOneClass.{0} NNReal (Semiring.toNonAssocSemiring.{0} NNReal NNReal.semiring))) Real.nnabs (HSub.hSub.{0, 0, 0} Real Real Real (instHSub.{0} Real Real.hasSub) y x))
 but is expected to have type
-  forall (x : Real) (y : Real), Eq.{1} NNReal (NNDist.nndist.{0} Real (PseudoMetricSpace.toNNDist.{0} Real Real.pseudoMetricSpace) x y) (FunLike.coe.{1, 1, 1} (MonoidWithZeroHom.{0, 0} Real NNReal (NonAssocSemiring.toMulZeroOneClass.{0} Real (Semiring.toNonAssocSemiring.{0} Real Real.semiring)) (NonAssocSemiring.toMulZeroOneClass.{0} NNReal (Semiring.toNonAssocSemiring.{0} NNReal instNNRealSemiring))) Real (fun (_x : Real) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : Real) => NNReal) _x) (MulHomClass.toFunLike.{0, 0, 0} (MonoidWithZeroHom.{0, 0} Real NNReal (NonAssocSemiring.toMulZeroOneClass.{0} Real (Semiring.toNonAssocSemiring.{0} Real Real.semiring)) (NonAssocSemiring.toMulZeroOneClass.{0} NNReal (Semiring.toNonAssocSemiring.{0} NNReal instNNRealSemiring))) Real NNReal (MulOneClass.toMul.{0} Real (MulZeroOneClass.toMulOneClass.{0} Real (NonAssocSemiring.toMulZeroOneClass.{0} Real (Semiring.toNonAssocSemiring.{0} Real Real.semiring)))) (MulOneClass.toMul.{0} NNReal (MulZeroOneClass.toMulOneClass.{0} NNReal (NonAssocSemiring.toMulZeroOneClass.{0} NNReal (Semiring.toNonAssocSemiring.{0} NNReal instNNRealSemiring)))) (MonoidHomClass.toMulHomClass.{0, 0, 0} (MonoidWithZeroHom.{0, 0} Real NNReal (NonAssocSemiring.toMulZeroOneClass.{0} Real (Semiring.toNonAssocSemiring.{0} Real Real.semiring)) (NonAssocSemiring.toMulZeroOneClass.{0} NNReal (Semiring.toNonAssocSemiring.{0} NNReal instNNRealSemiring))) Real NNReal (MulZeroOneClass.toMulOneClass.{0} Real (NonAssocSemiring.toMulZeroOneClass.{0} Real (Semiring.toNonAssocSemiring.{0} Real Real.semiring))) (MulZeroOneClass.toMulOneClass.{0} NNReal (NonAssocSemiring.toMulZeroOneClass.{0} NNReal (Semiring.toNonAssocSemiring.{0} NNReal instNNRealSemiring))) (MonoidWithZeroHomClass.toMonoidHomClass.{0, 0, 0} (MonoidWithZeroHom.{0, 0} Real NNReal (NonAssocSemiring.toMulZeroOneClass.{0} Real (Semiring.toNonAssocSemiring.{0} Real Real.semiring)) (NonAssocSemiring.toMulZeroOneClass.{0} NNReal (Semiring.toNonAssocSemiring.{0} NNReal instNNRealSemiring))) Real NNReal (NonAssocSemiring.toMulZeroOneClass.{0} Real (Semiring.toNonAssocSemiring.{0} Real Real.semiring)) (NonAssocSemiring.toMulZeroOneClass.{0} NNReal (Semiring.toNonAssocSemiring.{0} NNReal instNNRealSemiring)) (MonoidWithZeroHom.monoidWithZeroHomClass.{0, 0} Real NNReal (NonAssocSemiring.toMulZeroOneClass.{0} Real (Semiring.toNonAssocSemiring.{0} Real Real.semiring)) (NonAssocSemiring.toMulZeroOneClass.{0} NNReal (Semiring.toNonAssocSemiring.{0} NNReal instNNRealSemiring)))))) Real.nnabs (HSub.hSub.{0, 0, 0} Real Real Real (instHSub.{0} Real Real.instSubReal) y x))
+  forall (x : Real) (y : Real), Eq.{1} NNReal (NNDist.nndist.{0} Real (PseudoMetricSpace.toNNDist.{0} Real Real.pseudoMetricSpace) x y) (FunLike.coe.{1, 1, 1} (MonoidWithZeroHom.{0, 0} Real NNReal (NonAssocSemiring.toMulZeroOneClass.{0} Real (Semiring.toNonAssocSemiring.{0} Real Real.semiring)) (NonAssocSemiring.toMulZeroOneClass.{0} NNReal (Semiring.toNonAssocSemiring.{0} NNReal instNNRealSemiring))) Real (fun (_x : Real) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : Real) => NNReal) _x) (MulHomClass.toFunLike.{0, 0, 0} (MonoidWithZeroHom.{0, 0} Real NNReal (NonAssocSemiring.toMulZeroOneClass.{0} Real (Semiring.toNonAssocSemiring.{0} Real Real.semiring)) (NonAssocSemiring.toMulZeroOneClass.{0} NNReal (Semiring.toNonAssocSemiring.{0} NNReal instNNRealSemiring))) Real NNReal (MulOneClass.toMul.{0} Real (MulZeroOneClass.toMulOneClass.{0} Real (NonAssocSemiring.toMulZeroOneClass.{0} Real (Semiring.toNonAssocSemiring.{0} Real Real.semiring)))) (MulOneClass.toMul.{0} NNReal (MulZeroOneClass.toMulOneClass.{0} NNReal (NonAssocSemiring.toMulZeroOneClass.{0} NNReal (Semiring.toNonAssocSemiring.{0} NNReal instNNRealSemiring)))) (MonoidHomClass.toMulHomClass.{0, 0, 0} (MonoidWithZeroHom.{0, 0} Real NNReal (NonAssocSemiring.toMulZeroOneClass.{0} Real (Semiring.toNonAssocSemiring.{0} Real Real.semiring)) (NonAssocSemiring.toMulZeroOneClass.{0} NNReal (Semiring.toNonAssocSemiring.{0} NNReal instNNRealSemiring))) Real NNReal (MulZeroOneClass.toMulOneClass.{0} Real (NonAssocSemiring.toMulZeroOneClass.{0} Real (Semiring.toNonAssocSemiring.{0} Real Real.semiring))) (MulZeroOneClass.toMulOneClass.{0} NNReal (NonAssocSemiring.toMulZeroOneClass.{0} NNReal (Semiring.toNonAssocSemiring.{0} NNReal instNNRealSemiring))) (MonoidWithZeroHomClass.toMonoidHomClass.{0, 0, 0} (MonoidWithZeroHom.{0, 0} Real NNReal (NonAssocSemiring.toMulZeroOneClass.{0} Real (Semiring.toNonAssocSemiring.{0} Real Real.semiring)) (NonAssocSemiring.toMulZeroOneClass.{0} NNReal (Semiring.toNonAssocSemiring.{0} NNReal instNNRealSemiring))) Real NNReal (NonAssocSemiring.toMulZeroOneClass.{0} Real (Semiring.toNonAssocSemiring.{0} Real Real.semiring)) (NonAssocSemiring.toMulZeroOneClass.{0} NNReal (Semiring.toNonAssocSemiring.{0} NNReal instNNRealSemiring)) (MonoidWithZeroHom.monoidWithZeroHomClass.{0, 0} Real NNReal (NonAssocSemiring.toMulZeroOneClass.{0} Real (Semiring.toNonAssocSemiring.{0} Real Real.semiring)) (NonAssocSemiring.toMulZeroOneClass.{0} NNReal (Semiring.toNonAssocSemiring.{0} NNReal instNNRealSemiring)))))) Real.nnabs (HSub.hSub.{0, 0, 0} Real Real Real (instHSub.{0} Real Real.instSubReal) y x))
 Case conversion may be inaccurate. Consider using '#align real.nndist_eq' Real.nndist_eq'ₓ'. -/
 theorem Real.nndist_eq' (x y : ℝ) : nndist x y = Real.nnabs (y - x) :=
   nndist_comm _ _
@@ -3778,7 +3778,7 @@ theorem closedBall_pi' [Nonempty β] (x : ∀ b, π b) (r : ℝ) :
 lean 3 declaration is
   forall {n : Nat} {α : (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) -> Type.{u1}} [_inst_4 : forall (i : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))), PseudoMetricSpace.{u1} (α i)] (i : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (x : α i) (y : α i) (f : forall (j : Fin n), α (coeFn.{1, 1} (OrderEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (Fin.hasLe n) (Preorder.toHasLe.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (PartialOrder.toPreorder.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (Fin.partialOrder (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne)))))))) (fun (_x : RelEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (LE.le.{0} (Fin n) (Fin.hasLe n)) (LE.le.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (Preorder.toHasLe.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (PartialOrder.toPreorder.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (Fin.partialOrder (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))))))) => (Fin n) -> (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne)))))) (RelEmbedding.hasCoeToFun.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (LE.le.{0} (Fin n) (Fin.hasLe n)) (LE.le.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (Preorder.toHasLe.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (PartialOrder.toPreorder.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (Fin.partialOrder (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))))))) (Fin.succAbove n i) j)) (g : forall (j : Fin n), α (coeFn.{1, 1} (OrderEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (Fin.hasLe n) (Preorder.toHasLe.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (PartialOrder.toPreorder.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (Fin.partialOrder (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne)))))))) (fun (_x : RelEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (LE.le.{0} (Fin n) (Fin.hasLe n)) (LE.le.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (Preorder.toHasLe.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (PartialOrder.toPreorder.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (Fin.partialOrder (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))))))) => (Fin n) -> (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n 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 but is expected to have type
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+  forall {n : Nat} {α : (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) -> Type.{u1}} [_inst_4 : forall (i : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))), PseudoMetricSpace.{u1} (α i)] (i : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (x : α i) (y : α i) (f : forall (j : Fin n), α (FunLike.coe.{1, 1, 1} (OrderEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin n) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))))) (Fin n) (fun (_x : Fin n) => (fun (x._@.Mathlib.Order.Hom.Lattice._hyg.494 : Fin n) => Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) _x) 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(Fin.instLinearOrderFin n) (Fin.instLatticeFinHAddNatInstHAddInstAddNatOfNat n) (RelEmbedding.instRelHomClassRelEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.682 : Fin n) (x._@.Mathlib.Order.Hom.Basic._hyg.684 : Fin n) => LE.le.{0} (Fin n) (instLEFin n) x._@.Mathlib.Order.Hom.Basic._hyg.682 x._@.Mathlib.Order.Hom.Basic._hyg.684) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.697 : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (x._@.Mathlib.Order.Hom.Basic._hyg.699 : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) => LE.le.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) x._@.Mathlib.Order.Hom.Basic._hyg.697 x._@.Mathlib.Order.Hom.Basic._hyg.699))))) (Fin.succAbove n i) j)) (PseudoMetricSpace.toNNDist.{u1} (forall (j : Fin n), α (FunLike.coe.{1, 1, 1} (OrderEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin n) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))))) (Fin n) (fun (_x : Fin n) => (fun (x._@.Mathlib.Order.Hom.Lattice._hyg.494 : Fin n) => Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) _x) (InfHomClass.toFunLike.{0, 0, 0} (OrderEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin n) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))))) (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (Lattice.toInf.{0} (Fin n) (DistribLattice.toLattice.{0} (Fin n) (instDistribLattice.{0} (Fin n) (Fin.instLinearOrderFin n)))) (Lattice.toInf.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (Fin.instLatticeFinHAddNatInstHAddInstAddNatOfNat n)) (LatticeHomClass.toInfHomClass.{0, 0, 0} (OrderEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin n) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))))) (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (DistribLattice.toLattice.{0} (Fin n) (instDistribLattice.{0} (Fin n) (Fin.instLinearOrderFin n))) (Fin.instLatticeFinHAddNatInstHAddInstAddNatOfNat n) (OrderHomClass.toLatticeHomClass.{0, 0, 0} (OrderEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin n) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))))) (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (Fin.instLinearOrderFin n) (Fin.instLatticeFinHAddNatInstHAddInstAddNatOfNat n) (RelEmbedding.instRelHomClassRelEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.682 : Fin n) (x._@.Mathlib.Order.Hom.Basic._hyg.684 : Fin n) => LE.le.{0} (Fin n) (instLEFin n) x._@.Mathlib.Order.Hom.Basic._hyg.682 x._@.Mathlib.Order.Hom.Basic._hyg.684) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.697 : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (x._@.Mathlib.Order.Hom.Basic._hyg.699 : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) => LE.le.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) x._@.Mathlib.Order.Hom.Basic._hyg.697 x._@.Mathlib.Order.Hom.Basic._hyg.699))))) (Fin.succAbove n i) j)) (pseudoMetricSpacePi.{0, u1} (Fin n) (fun (j : Fin n) => α (FunLike.coe.{1, 1, 1} (OrderEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin n) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))))) (Fin n) (fun (_x : Fin n) => (fun (x._@.Mathlib.Order.Hom.Lattice._hyg.494 : Fin n) => Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) _x) (InfHomClass.toFunLike.{0, 0, 0} (OrderEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin n) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))))) (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (Lattice.toInf.{0} (Fin n) (DistribLattice.toLattice.{0} (Fin n) (instDistribLattice.{0} (Fin n) (Fin.instLinearOrderFin n)))) (Lattice.toInf.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (Fin.instLatticeFinHAddNatInstHAddInstAddNatOfNat n)) (LatticeHomClass.toInfHomClass.{0, 0, 0} (OrderEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin n) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))))) (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (DistribLattice.toLattice.{0} (Fin n) (instDistribLattice.{0} (Fin n) (Fin.instLinearOrderFin n))) (Fin.instLatticeFinHAddNatInstHAddInstAddNatOfNat n) (OrderHomClass.toLatticeHomClass.{0, 0, 0} (OrderEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin n) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))))) (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (Fin.instLinearOrderFin n) (Fin.instLatticeFinHAddNatInstHAddInstAddNatOfNat n) (RelEmbedding.instRelHomClassRelEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.682 : Fin n) (x._@.Mathlib.Order.Hom.Basic._hyg.684 : Fin n) => LE.le.{0} (Fin n) (instLEFin n) x._@.Mathlib.Order.Hom.Basic._hyg.682 x._@.Mathlib.Order.Hom.Basic._hyg.684) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.697 : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (x._@.Mathlib.Order.Hom.Basic._hyg.699 : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) => LE.le.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) x._@.Mathlib.Order.Hom.Basic._hyg.697 x._@.Mathlib.Order.Hom.Basic._hyg.699))))) (Fin.succAbove n i) j)) (Fin.fintype n) (fun (b : Fin n) => _inst_4 (FunLike.coe.{1, 1, 1} (OrderEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin n) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))))) (Fin n) (fun (_x : Fin n) => (fun (x._@.Mathlib.Order.Hom.Lattice._hyg.494 : Fin n) => Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) _x) (InfHomClass.toFunLike.{0, 0, 0} (OrderEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin n) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))))) (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (Lattice.toInf.{0} (Fin n) (DistribLattice.toLattice.{0} (Fin n) (instDistribLattice.{0} (Fin n) (Fin.instLinearOrderFin n)))) (Lattice.toInf.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (Fin.instLatticeFinHAddNatInstHAddInstAddNatOfNat n)) (LatticeHomClass.toInfHomClass.{0, 0, 0} (OrderEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin n) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))))) (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (DistribLattice.toLattice.{0} (Fin n) (instDistribLattice.{0} (Fin n) (Fin.instLinearOrderFin n))) (Fin.instLatticeFinHAddNatInstHAddInstAddNatOfNat n) (OrderHomClass.toLatticeHomClass.{0, 0, 0} (OrderEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin n) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))))) (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (Fin.instLinearOrderFin n) (Fin.instLatticeFinHAddNatInstHAddInstAddNatOfNat n) (RelEmbedding.instRelHomClassRelEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.682 : Fin n) (x._@.Mathlib.Order.Hom.Basic._hyg.684 : Fin n) => LE.le.{0} (Fin n) (instLEFin n) x._@.Mathlib.Order.Hom.Basic._hyg.682 x._@.Mathlib.Order.Hom.Basic._hyg.684) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.697 : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (x._@.Mathlib.Order.Hom.Basic._hyg.699 : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) => LE.le.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) x._@.Mathlib.Order.Hom.Basic._hyg.697 x._@.Mathlib.Order.Hom.Basic._hyg.699))))) (Fin.succAbove n i) b)))) f g))
 Case conversion may be inaccurate. Consider using '#align fin.nndist_insert_nth_insert_nth Fin.nndist_insertNth_insertNthₓ'. -/
 @[simp]
 theorem Fin.nndist_insertNth_insertNth {n : ℕ} {α : Fin (n + 1) → Type _}
@@ -3791,7 +3791,7 @@ theorem Fin.nndist_insertNth_insertNth {n : ℕ} {α : Fin (n + 1) → Type _}
 lean 3 declaration is
   forall {n : Nat} {α : (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) -> Type.{u1}} [_inst_4 : forall (i : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))), PseudoMetricSpace.{u1} (α i)] (i : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (x : α i) (y : α i) (f : forall (j : Fin n), α (coeFn.{1, 1} (OrderEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (Fin.hasLe n) (Preorder.toHasLe.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (PartialOrder.toPreorder.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (Fin.partialOrder (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne)))))))) (fun (_x : RelEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (LE.le.{0} (Fin n) (Fin.hasLe n)) (LE.le.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (Preorder.toHasLe.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (PartialOrder.toPreorder.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 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 but is expected to have type
-  forall {n : Nat} {α : (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) -> Type.{u1}} [_inst_4 : forall (i : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))), PseudoMetricSpace.{u1} (α i)] (i : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (x : α i) (y : α i) (f : forall (j : Fin n), α (FunLike.coe.{1, 1, 1} (OrderEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin n) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))))) (Fin n) (fun (_x : Fin n) => (fun (x._@.Mathlib.Order.Hom.Lattice._hyg.494 : Fin n) => Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) _x) 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+  forall {n : Nat} {α : (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) -> Type.{u1}} [_inst_4 : forall (i : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))), PseudoMetricSpace.{u1} (α i)] (i : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (x : α i) (y : α i) (f : forall (j : Fin n), α (FunLike.coe.{1, 1, 1} (OrderEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin n) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))))) (Fin n) (fun (_x : Fin n) => (fun (x._@.Mathlib.Order.Hom.Lattice._hyg.494 : Fin n) => Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) _x) 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(x._@.Mathlib.Order.Hom.Basic._hyg.697 : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (x._@.Mathlib.Order.Hom.Basic._hyg.699 : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) => LE.le.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) x._@.Mathlib.Order.Hom.Basic._hyg.697 x._@.Mathlib.Order.Hom.Basic._hyg.699))))) (Fin.succAbove n i) j)) (pseudoMetricSpacePi.{0, u1} (Fin n) (fun (j : Fin n) => α (FunLike.coe.{1, 1, 1} (OrderEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin n) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 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Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) x._@.Mathlib.Order.Hom.Basic._hyg.697 x._@.Mathlib.Order.Hom.Basic._hyg.699))))) (Fin.succAbove n i) j)) (Fin.fintype n) (fun (b : Fin n) => _inst_4 (FunLike.coe.{1, 1, 1} (OrderEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin n) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))))) (Fin n) (fun (_x : Fin n) => (fun (x._@.Mathlib.Order.Hom.Lattice._hyg.494 : Fin n) => Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) _x) (InfHomClass.toFunLike.{0, 0, 0} (OrderEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin n) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} 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(Fin.instLatticeFinHAddNatInstHAddInstAddNatOfNat n) (OrderHomClass.toLatticeHomClass.{0, 0, 0} (OrderEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin n) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))))) (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (Fin.instLinearOrderFin n) (Fin.instLatticeFinHAddNatInstHAddInstAddNatOfNat n) (RelEmbedding.instRelHomClassRelEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.682 : Fin n) (x._@.Mathlib.Order.Hom.Basic._hyg.684 : Fin n) => LE.le.{0} (Fin n) (instLEFin n) x._@.Mathlib.Order.Hom.Basic._hyg.682 x._@.Mathlib.Order.Hom.Basic._hyg.684) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.697 : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (x._@.Mathlib.Order.Hom.Basic._hyg.699 : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) => LE.le.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) x._@.Mathlib.Order.Hom.Basic._hyg.697 x._@.Mathlib.Order.Hom.Basic._hyg.699))))) (Fin.succAbove n i) b)))) f g))
 Case conversion may be inaccurate. Consider using '#align fin.dist_insert_nth_insert_nth Fin.dist_insertNth_insertNthₓ'. -/
 @[simp]
 theorem Fin.dist_insertNth_insertNth {n : ℕ} {α : Fin (n + 1) → Type _}

mathlib commit https://github.com/leanprover-community/mathlib/commit/0b9eaaa7686280fad8cce467f5c3c57ee6ce77f8

@@ -504,7 +504,7 @@ theorem coe_nnreal_ennreal_nndist (x y : α) : ↑(nndist x y) = edist x y :=
 
 /- warning: edist_lt_coe -> edist_lt_coe is a dubious translation:
 lean 3 declaration is
-  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {x : α} {y : α} {c : NNReal}, Iff (LT.lt.{0} ENNReal (Preorder.toLT.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (CompleteSemilatticeInf.toPartialOrder.{0} ENNReal (CompleteLattice.toCompleteSemilatticeInf.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))))) (EDist.edist.{u1} α (PseudoMetricSpace.toEDist.{u1} α _inst_1) x y) ((fun (a : Type) (b : Type) [self : HasLiftT.{1, 1} a b] => self.0) NNReal ENNReal (HasLiftT.mk.{1, 1} NNReal ENNReal (CoeTCₓ.coe.{1, 1} NNReal ENNReal (coeBase.{1, 1} NNReal ENNReal ENNReal.hasCoe))) c)) (LT.lt.{0} NNReal (Preorder.toLT.{0} NNReal (PartialOrder.toPreorder.{0} NNReal (OrderedCancelAddCommMonoid.toPartialOrder.{0} NNReal (StrictOrderedSemiring.toOrderedCancelAddCommMonoid.{0} NNReal NNReal.strictOrderedSemiring)))) (NNDist.nndist.{u1} α (PseudoMetricSpace.toNNDist.{u1} α _inst_1) x y) c)
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {x : α} {y : α} {c : NNReal}, Iff (LT.lt.{0} ENNReal (Preorder.toHasLt.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (CompleteSemilatticeInf.toPartialOrder.{0} ENNReal (CompleteLattice.toCompleteSemilatticeInf.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))))) (EDist.edist.{u1} α (PseudoMetricSpace.toEDist.{u1} α _inst_1) x y) ((fun (a : Type) (b : Type) [self : HasLiftT.{1, 1} a b] => self.0) NNReal ENNReal (HasLiftT.mk.{1, 1} NNReal ENNReal (CoeTCₓ.coe.{1, 1} NNReal ENNReal (coeBase.{1, 1} NNReal ENNReal ENNReal.hasCoe))) c)) (LT.lt.{0} NNReal (Preorder.toHasLt.{0} NNReal (PartialOrder.toPreorder.{0} NNReal (OrderedCancelAddCommMonoid.toPartialOrder.{0} NNReal (StrictOrderedSemiring.toOrderedCancelAddCommMonoid.{0} NNReal NNReal.strictOrderedSemiring)))) (NNDist.nndist.{u1} α (PseudoMetricSpace.toNNDist.{u1} α _inst_1) x y) c)
 but is expected to have type
   forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {x : α} {y : α} {c : NNReal}, Iff (LT.lt.{0} ENNReal (Preorder.toLT.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (CompleteSemilatticeInf.toPartialOrder.{0} ENNReal (CompleteLattice.toCompleteSemilatticeInf.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal))))) (EDist.edist.{u1} α (PseudoMetricSpace.toEDist.{u1} α _inst_1) x y) (ENNReal.some c)) (LT.lt.{0} NNReal (Preorder.toLT.{0} NNReal (PartialOrder.toPreorder.{0} NNReal (StrictOrderedSemiring.toPartialOrder.{0} NNReal instNNRealStrictOrderedSemiring))) (NNDist.nndist.{u1} α (PseudoMetricSpace.toNNDist.{u1} α _inst_1) x y) c)
 Case conversion may be inaccurate. Consider using '#align edist_lt_coe edist_lt_coeₓ'. -/
@@ -515,7 +515,7 @@ theorem edist_lt_coe {x y : α} {c : ℝ≥0} : edist x y < c ↔ nndist x y < c
 
 /- warning: edist_le_coe -> edist_le_coe is a dubious translation:
 lean 3 declaration is
-  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {x : α} {y : α} {c : NNReal}, Iff (LE.le.{0} ENNReal (Preorder.toLE.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (CompleteSemilatticeInf.toPartialOrder.{0} ENNReal (CompleteLattice.toCompleteSemilatticeInf.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))))) (EDist.edist.{u1} α (PseudoMetricSpace.toEDist.{u1} α _inst_1) x y) ((fun (a : Type) (b : Type) [self : HasLiftT.{1, 1} a b] => self.0) NNReal ENNReal (HasLiftT.mk.{1, 1} NNReal ENNReal (CoeTCₓ.coe.{1, 1} NNReal ENNReal (coeBase.{1, 1} NNReal ENNReal ENNReal.hasCoe))) c)) (LE.le.{0} NNReal (Preorder.toLE.{0} NNReal (PartialOrder.toPreorder.{0} NNReal (OrderedCancelAddCommMonoid.toPartialOrder.{0} NNReal (StrictOrderedSemiring.toOrderedCancelAddCommMonoid.{0} NNReal NNReal.strictOrderedSemiring)))) (NNDist.nndist.{u1} α (PseudoMetricSpace.toNNDist.{u1} α _inst_1) x y) c)
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {x : α} {y : α} {c : NNReal}, Iff (LE.le.{0} ENNReal (Preorder.toHasLe.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (CompleteSemilatticeInf.toPartialOrder.{0} ENNReal (CompleteLattice.toCompleteSemilatticeInf.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))))) (EDist.edist.{u1} α (PseudoMetricSpace.toEDist.{u1} α _inst_1) x y) ((fun (a : Type) (b : Type) [self : HasLiftT.{1, 1} a b] => self.0) NNReal ENNReal (HasLiftT.mk.{1, 1} NNReal ENNReal (CoeTCₓ.coe.{1, 1} NNReal ENNReal (coeBase.{1, 1} NNReal ENNReal ENNReal.hasCoe))) c)) (LE.le.{0} NNReal (Preorder.toHasLe.{0} NNReal (PartialOrder.toPreorder.{0} NNReal (OrderedCancelAddCommMonoid.toPartialOrder.{0} NNReal (StrictOrderedSemiring.toOrderedCancelAddCommMonoid.{0} NNReal NNReal.strictOrderedSemiring)))) (NNDist.nndist.{u1} α (PseudoMetricSpace.toNNDist.{u1} α _inst_1) x y) c)
 but is expected to have type
   forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {x : α} {y : α} {c : NNReal}, Iff (LE.le.{0} ENNReal (Preorder.toLE.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (CompleteSemilatticeInf.toPartialOrder.{0} ENNReal (CompleteLattice.toCompleteSemilatticeInf.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal))))) (EDist.edist.{u1} α (PseudoMetricSpace.toEDist.{u1} α _inst_1) x y) (ENNReal.some c)) (LE.le.{0} NNReal (Preorder.toLE.{0} NNReal (PartialOrder.toPreorder.{0} NNReal (StrictOrderedSemiring.toPartialOrder.{0} NNReal instNNRealStrictOrderedSemiring))) (NNDist.nndist.{u1} α (PseudoMetricSpace.toNNDist.{u1} α _inst_1) x y) c)
 Case conversion may be inaccurate. Consider using '#align edist_le_coe edist_le_coeₓ'. -/
@@ -526,7 +526,7 @@ theorem edist_le_coe {x y : α} {c : ℝ≥0} : edist x y ≤ c ↔ nndist x y 
 
 /- warning: edist_lt_top -> edist_lt_top is a dubious translation:
 lean 3 declaration is
-  forall {α : Type.{u1}} [_inst_2 : PseudoMetricSpace.{u1} α] (x : α) (y : α), LT.lt.{0} ENNReal (Preorder.toLT.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (CompleteSemilatticeInf.toPartialOrder.{0} ENNReal (CompleteLattice.toCompleteSemilatticeInf.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))))) (EDist.edist.{u1} α (PseudoMetricSpace.toEDist.{u1} α _inst_2) x y) (Top.top.{0} ENNReal (CompleteLattice.toHasTop.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder)))
+  forall {α : Type.{u1}} [_inst_2 : PseudoMetricSpace.{u1} α] (x : α) (y : α), LT.lt.{0} ENNReal (Preorder.toHasLt.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (CompleteSemilatticeInf.toPartialOrder.{0} ENNReal (CompleteLattice.toCompleteSemilatticeInf.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))))) (EDist.edist.{u1} α (PseudoMetricSpace.toEDist.{u1} α _inst_2) x y) (Top.top.{0} ENNReal (CompleteLattice.toHasTop.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder)))
 but is expected to have type
   forall {α : Type.{u1}} [_inst_2 : PseudoMetricSpace.{u1} α] (x : α) (y : α), LT.lt.{0} ENNReal (Preorder.toLT.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (CompleteSemilatticeInf.toPartialOrder.{0} ENNReal (CompleteLattice.toCompleteSemilatticeInf.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal))))) (EDist.edist.{u1} α (PseudoMetricSpace.toEDist.{u1} α _inst_2) x y) (Top.top.{0} ENNReal (CompleteLattice.toTop.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal)))
 Case conversion may be inaccurate. Consider using '#align edist_lt_top edist_lt_topₓ'. -/
@@ -582,7 +582,7 @@ theorem coe_nndist (x y : α) : ↑(nndist x y) = dist x y :=
 
 /- warning: dist_lt_coe -> dist_lt_coe is a dubious translation:
 lean 3 declaration is
-  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {x : α} {y : α} {c : NNReal}, Iff (LT.lt.{0} Real Real.hasLt (Dist.dist.{u1} α (PseudoMetricSpace.toHasDist.{u1} α _inst_1) x y) ((fun (a : Type) (b : Type) [self : HasLiftT.{1, 1} a b] => self.0) NNReal Real (HasLiftT.mk.{1, 1} NNReal Real (CoeTCₓ.coe.{1, 1} NNReal Real (coeBase.{1, 1} NNReal Real NNReal.Real.hasCoe))) c)) (LT.lt.{0} NNReal (Preorder.toLT.{0} NNReal (PartialOrder.toPreorder.{0} NNReal (OrderedCancelAddCommMonoid.toPartialOrder.{0} NNReal (StrictOrderedSemiring.toOrderedCancelAddCommMonoid.{0} NNReal NNReal.strictOrderedSemiring)))) (NNDist.nndist.{u1} α (PseudoMetricSpace.toNNDist.{u1} α _inst_1) x y) c)
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {x : α} {y : α} {c : NNReal}, Iff (LT.lt.{0} Real Real.hasLt (Dist.dist.{u1} α (PseudoMetricSpace.toHasDist.{u1} α _inst_1) x y) ((fun (a : Type) (b : Type) [self : HasLiftT.{1, 1} a b] => self.0) NNReal Real (HasLiftT.mk.{1, 1} NNReal Real (CoeTCₓ.coe.{1, 1} NNReal Real (coeBase.{1, 1} NNReal Real NNReal.Real.hasCoe))) c)) (LT.lt.{0} NNReal (Preorder.toHasLt.{0} NNReal (PartialOrder.toPreorder.{0} NNReal (OrderedCancelAddCommMonoid.toPartialOrder.{0} NNReal (StrictOrderedSemiring.toOrderedCancelAddCommMonoid.{0} NNReal NNReal.strictOrderedSemiring)))) (NNDist.nndist.{u1} α (PseudoMetricSpace.toNNDist.{u1} α _inst_1) x y) c)
 but is expected to have type
   forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {x : α} {y : α} {c : NNReal}, Iff (LT.lt.{0} Real Real.instLTReal (Dist.dist.{u1} α (PseudoMetricSpace.toDist.{u1} α _inst_1) x y) (NNReal.toReal c)) (LT.lt.{0} NNReal (Preorder.toLT.{0} NNReal (PartialOrder.toPreorder.{0} NNReal (StrictOrderedSemiring.toPartialOrder.{0} NNReal instNNRealStrictOrderedSemiring))) (NNDist.nndist.{u1} α (PseudoMetricSpace.toNNDist.{u1} α _inst_1) x y) c)
 Case conversion may be inaccurate. Consider using '#align dist_lt_coe dist_lt_coeₓ'. -/
@@ -593,7 +593,7 @@ theorem dist_lt_coe {x y : α} {c : ℝ≥0} : dist x y < c ↔ nndist x y < c :
 
 /- warning: dist_le_coe -> dist_le_coe is a dubious translation:
 lean 3 declaration is
-  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {x : α} {y : α} {c : NNReal}, Iff (LE.le.{0} Real Real.hasLe (Dist.dist.{u1} α (PseudoMetricSpace.toHasDist.{u1} α _inst_1) x y) ((fun (a : Type) (b : Type) [self : HasLiftT.{1, 1} a b] => self.0) NNReal Real (HasLiftT.mk.{1, 1} NNReal Real (CoeTCₓ.coe.{1, 1} NNReal Real (coeBase.{1, 1} NNReal Real NNReal.Real.hasCoe))) c)) (LE.le.{0} NNReal (Preorder.toLE.{0} NNReal (PartialOrder.toPreorder.{0} NNReal (OrderedCancelAddCommMonoid.toPartialOrder.{0} NNReal (StrictOrderedSemiring.toOrderedCancelAddCommMonoid.{0} NNReal NNReal.strictOrderedSemiring)))) (NNDist.nndist.{u1} α (PseudoMetricSpace.toNNDist.{u1} α _inst_1) x y) c)
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {x : α} {y : α} {c : NNReal}, Iff (LE.le.{0} Real Real.hasLe (Dist.dist.{u1} α (PseudoMetricSpace.toHasDist.{u1} α _inst_1) x y) ((fun (a : Type) (b : Type) [self : HasLiftT.{1, 1} a b] => self.0) NNReal Real (HasLiftT.mk.{1, 1} NNReal Real (CoeTCₓ.coe.{1, 1} NNReal Real (coeBase.{1, 1} NNReal Real NNReal.Real.hasCoe))) c)) (LE.le.{0} NNReal (Preorder.toHasLe.{0} NNReal (PartialOrder.toPreorder.{0} NNReal (OrderedCancelAddCommMonoid.toPartialOrder.{0} NNReal (StrictOrderedSemiring.toOrderedCancelAddCommMonoid.{0} NNReal NNReal.strictOrderedSemiring)))) (NNDist.nndist.{u1} α (PseudoMetricSpace.toNNDist.{u1} α _inst_1) x y) c)
 but is expected to have type
   forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {x : α} {y : α} {c : NNReal}, Iff (LE.le.{0} Real Real.instLEReal (Dist.dist.{u1} α (PseudoMetricSpace.toDist.{u1} α _inst_1) x y) (NNReal.toReal c)) (LE.le.{0} NNReal (Preorder.toLE.{0} NNReal (PartialOrder.toPreorder.{0} NNReal (StrictOrderedSemiring.toPartialOrder.{0} NNReal instNNRealStrictOrderedSemiring))) (NNDist.nndist.{u1} α (PseudoMetricSpace.toNNDist.{u1} α _inst_1) x y) c)
 Case conversion may be inaccurate. Consider using '#align dist_le_coe dist_le_coeₓ'. -/
@@ -604,7 +604,7 @@ theorem dist_le_coe {x y : α} {c : ℝ≥0} : dist x y ≤ c ↔ nndist x y ≤
 
 /- warning: edist_lt_of_real -> edist_lt_ofReal is a dubious translation:
 lean 3 declaration is
-  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {x : α} {y : α} {r : Real}, Iff (LT.lt.{0} ENNReal (Preorder.toLT.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (CompleteSemilatticeInf.toPartialOrder.{0} ENNReal (CompleteLattice.toCompleteSemilatticeInf.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))))) (EDist.edist.{u1} α (PseudoMetricSpace.toEDist.{u1} α _inst_1) x y) (ENNReal.ofReal r)) (LT.lt.{0} Real Real.hasLt (Dist.dist.{u1} α (PseudoMetricSpace.toHasDist.{u1} α _inst_1) x y) r)
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {x : α} {y : α} {r : Real}, Iff (LT.lt.{0} ENNReal (Preorder.toHasLt.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (CompleteSemilatticeInf.toPartialOrder.{0} ENNReal (CompleteLattice.toCompleteSemilatticeInf.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))))) (EDist.edist.{u1} α (PseudoMetricSpace.toEDist.{u1} α _inst_1) x y) (ENNReal.ofReal r)) (LT.lt.{0} Real Real.hasLt (Dist.dist.{u1} α (PseudoMetricSpace.toHasDist.{u1} α _inst_1) x y) r)
 but is expected to have type
   forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {x : α} {y : α} {r : Real}, Iff (LT.lt.{0} ENNReal (Preorder.toLT.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (CompleteSemilatticeInf.toPartialOrder.{0} ENNReal (CompleteLattice.toCompleteSemilatticeInf.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal))))) (EDist.edist.{u1} α (PseudoMetricSpace.toEDist.{u1} α _inst_1) x y) (ENNReal.ofReal r)) (LT.lt.{0} Real Real.instLTReal (Dist.dist.{u1} α (PseudoMetricSpace.toDist.{u1} α _inst_1) x y) r)
 Case conversion may be inaccurate. Consider using '#align edist_lt_of_real edist_lt_ofRealₓ'. -/
@@ -615,7 +615,7 @@ theorem edist_lt_ofReal {x y : α} {r : ℝ} : edist x y < ENNReal.ofReal r ↔
 
 /- warning: edist_le_of_real -> edist_le_ofReal is a dubious translation:
 lean 3 declaration is
-  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {x : α} {y : α} {r : Real}, (LE.le.{0} Real Real.hasLe (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero))) r) -> (Iff (LE.le.{0} ENNReal (Preorder.toLE.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (CompleteSemilatticeInf.toPartialOrder.{0} ENNReal (CompleteLattice.toCompleteSemilatticeInf.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))))) (EDist.edist.{u1} α (PseudoMetricSpace.toEDist.{u1} α _inst_1) x y) (ENNReal.ofReal r)) (LE.le.{0} Real Real.hasLe (Dist.dist.{u1} α (PseudoMetricSpace.toHasDist.{u1} α _inst_1) x y) r))
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {x : α} {y : α} {r : Real}, (LE.le.{0} Real Real.hasLe (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero))) r) -> (Iff (LE.le.{0} ENNReal (Preorder.toHasLe.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (CompleteSemilatticeInf.toPartialOrder.{0} ENNReal (CompleteLattice.toCompleteSemilatticeInf.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))))) (EDist.edist.{u1} α (PseudoMetricSpace.toEDist.{u1} α _inst_1) x y) (ENNReal.ofReal r)) (LE.le.{0} Real Real.hasLe (Dist.dist.{u1} α (PseudoMetricSpace.toHasDist.{u1} α _inst_1) x y) r))
 but is expected to have type
   forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {x : α} {y : α} {r : Real}, (LE.le.{0} Real Real.instLEReal (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal)) r) -> (Iff (LE.le.{0} ENNReal (Preorder.toLE.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (CompleteSemilatticeInf.toPartialOrder.{0} ENNReal (CompleteLattice.toCompleteSemilatticeInf.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal))))) (EDist.edist.{u1} α (PseudoMetricSpace.toEDist.{u1} α _inst_1) x y) (ENNReal.ofReal r)) (LE.le.{0} Real Real.instLEReal (Dist.dist.{u1} α (PseudoMetricSpace.toDist.{u1} α _inst_1) x y) r))
 Case conversion may be inaccurate. Consider using '#align edist_le_of_real edist_le_ofRealₓ'. -/
@@ -640,7 +640,7 @@ theorem nndist_comm (x y : α) : nndist x y = nndist y x := by
 
 /- warning: nndist_triangle -> nndist_triangle is a dubious translation:
 lean 3 declaration is
-  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] (x : α) (y : α) (z : α), LE.le.{0} NNReal (Preorder.toLE.{0} NNReal (PartialOrder.toPreorder.{0} NNReal (OrderedCancelAddCommMonoid.toPartialOrder.{0} NNReal (StrictOrderedSemiring.toOrderedCancelAddCommMonoid.{0} NNReal NNReal.strictOrderedSemiring)))) (NNDist.nndist.{u1} α (PseudoMetricSpace.toNNDist.{u1} α _inst_1) x z) (HAdd.hAdd.{0, 0, 0} NNReal NNReal NNReal (instHAdd.{0} NNReal (Distrib.toHasAdd.{0} NNReal (NonUnitalNonAssocSemiring.toDistrib.{0} NNReal (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} NNReal (Semiring.toNonAssocSemiring.{0} NNReal NNReal.semiring))))) (NNDist.nndist.{u1} α (PseudoMetricSpace.toNNDist.{u1} α _inst_1) x y) (NNDist.nndist.{u1} α (PseudoMetricSpace.toNNDist.{u1} α _inst_1) y z))
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] (x : α) (y : α) (z : α), LE.le.{0} NNReal (Preorder.toHasLe.{0} NNReal (PartialOrder.toPreorder.{0} NNReal (OrderedCancelAddCommMonoid.toPartialOrder.{0} NNReal (StrictOrderedSemiring.toOrderedCancelAddCommMonoid.{0} NNReal NNReal.strictOrderedSemiring)))) (NNDist.nndist.{u1} α (PseudoMetricSpace.toNNDist.{u1} α _inst_1) x z) (HAdd.hAdd.{0, 0, 0} NNReal NNReal NNReal (instHAdd.{0} NNReal (Distrib.toHasAdd.{0} NNReal (NonUnitalNonAssocSemiring.toDistrib.{0} NNReal (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} NNReal (Semiring.toNonAssocSemiring.{0} NNReal NNReal.semiring))))) (NNDist.nndist.{u1} α (PseudoMetricSpace.toNNDist.{u1} α _inst_1) x y) (NNDist.nndist.{u1} α (PseudoMetricSpace.toNNDist.{u1} α _inst_1) y z))
 but is expected to have type
   forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] (x : α) (y : α) (z : α), LE.le.{0} NNReal (Preorder.toLE.{0} NNReal (PartialOrder.toPreorder.{0} NNReal (StrictOrderedSemiring.toPartialOrder.{0} NNReal instNNRealStrictOrderedSemiring))) (NNDist.nndist.{u1} α (PseudoMetricSpace.toNNDist.{u1} α _inst_1) x z) (HAdd.hAdd.{0, 0, 0} NNReal NNReal NNReal (instHAdd.{0} NNReal (Distrib.toAdd.{0} NNReal (NonUnitalNonAssocSemiring.toDistrib.{0} NNReal (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} NNReal (Semiring.toNonAssocSemiring.{0} NNReal instNNRealSemiring))))) (NNDist.nndist.{u1} α (PseudoMetricSpace.toNNDist.{u1} α _inst_1) x y) (NNDist.nndist.{u1} α (PseudoMetricSpace.toNNDist.{u1} α _inst_1) y z))
 Case conversion may be inaccurate. Consider using '#align nndist_triangle nndist_triangleₓ'. -/
@@ -651,7 +651,7 @@ theorem nndist_triangle (x y z : α) : nndist x z ≤ nndist x y + nndist y z :=
 
 /- warning: nndist_triangle_left -> nndist_triangle_left is a dubious translation:
 lean 3 declaration is
-  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] (x : α) (y : α) (z : α), LE.le.{0} NNReal (Preorder.toLE.{0} NNReal (PartialOrder.toPreorder.{0} NNReal (OrderedCancelAddCommMonoid.toPartialOrder.{0} NNReal (StrictOrderedSemiring.toOrderedCancelAddCommMonoid.{0} NNReal NNReal.strictOrderedSemiring)))) (NNDist.nndist.{u1} α (PseudoMetricSpace.toNNDist.{u1} α _inst_1) x y) (HAdd.hAdd.{0, 0, 0} NNReal NNReal NNReal (instHAdd.{0} NNReal (Distrib.toHasAdd.{0} NNReal (NonUnitalNonAssocSemiring.toDistrib.{0} NNReal (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} NNReal (Semiring.toNonAssocSemiring.{0} NNReal NNReal.semiring))))) (NNDist.nndist.{u1} α (PseudoMetricSpace.toNNDist.{u1} α _inst_1) z x) (NNDist.nndist.{u1} α (PseudoMetricSpace.toNNDist.{u1} α _inst_1) z y))
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] (x : α) (y : α) (z : α), LE.le.{0} NNReal (Preorder.toHasLe.{0} NNReal (PartialOrder.toPreorder.{0} NNReal (OrderedCancelAddCommMonoid.toPartialOrder.{0} NNReal (StrictOrderedSemiring.toOrderedCancelAddCommMonoid.{0} NNReal NNReal.strictOrderedSemiring)))) (NNDist.nndist.{u1} α (PseudoMetricSpace.toNNDist.{u1} α _inst_1) x y) (HAdd.hAdd.{0, 0, 0} NNReal NNReal NNReal (instHAdd.{0} NNReal (Distrib.toHasAdd.{0} NNReal (NonUnitalNonAssocSemiring.toDistrib.{0} NNReal (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} NNReal (Semiring.toNonAssocSemiring.{0} NNReal NNReal.semiring))))) (NNDist.nndist.{u1} α (PseudoMetricSpace.toNNDist.{u1} α _inst_1) z x) (NNDist.nndist.{u1} α (PseudoMetricSpace.toNNDist.{u1} α _inst_1) z y))
 but is expected to have type
   forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] (x : α) (y : α) (z : α), LE.le.{0} NNReal (Preorder.toLE.{0} NNReal (PartialOrder.toPreorder.{0} NNReal (StrictOrderedSemiring.toPartialOrder.{0} NNReal instNNRealStrictOrderedSemiring))) (NNDist.nndist.{u1} α (PseudoMetricSpace.toNNDist.{u1} α _inst_1) x y) (HAdd.hAdd.{0, 0, 0} NNReal NNReal NNReal (instHAdd.{0} NNReal (Distrib.toAdd.{0} NNReal (NonUnitalNonAssocSemiring.toDistrib.{0} NNReal (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} NNReal (Semiring.toNonAssocSemiring.{0} NNReal instNNRealSemiring))))) (NNDist.nndist.{u1} α (PseudoMetricSpace.toNNDist.{u1} α _inst_1) z x) (NNDist.nndist.{u1} α (PseudoMetricSpace.toNNDist.{u1} α _inst_1) z y))
 Case conversion may be inaccurate. Consider using '#align nndist_triangle_left nndist_triangle_leftₓ'. -/
@@ -661,7 +661,7 @@ theorem nndist_triangle_left (x y z : α) : nndist x y ≤ nndist z x + nndist z
 
 /- warning: nndist_triangle_right -> nndist_triangle_right is a dubious translation:
 lean 3 declaration is
-  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] (x : α) (y : α) (z : α), LE.le.{0} NNReal (Preorder.toLE.{0} NNReal (PartialOrder.toPreorder.{0} NNReal (OrderedCancelAddCommMonoid.toPartialOrder.{0} NNReal (StrictOrderedSemiring.toOrderedCancelAddCommMonoid.{0} NNReal NNReal.strictOrderedSemiring)))) (NNDist.nndist.{u1} α (PseudoMetricSpace.toNNDist.{u1} α _inst_1) x y) (HAdd.hAdd.{0, 0, 0} NNReal NNReal NNReal (instHAdd.{0} NNReal (Distrib.toHasAdd.{0} NNReal (NonUnitalNonAssocSemiring.toDistrib.{0} NNReal (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} NNReal (Semiring.toNonAssocSemiring.{0} NNReal NNReal.semiring))))) (NNDist.nndist.{u1} α (PseudoMetricSpace.toNNDist.{u1} α _inst_1) x z) (NNDist.nndist.{u1} α (PseudoMetricSpace.toNNDist.{u1} α _inst_1) y z))
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] (x : α) (y : α) (z : α), LE.le.{0} NNReal (Preorder.toHasLe.{0} NNReal (PartialOrder.toPreorder.{0} NNReal (OrderedCancelAddCommMonoid.toPartialOrder.{0} NNReal (StrictOrderedSemiring.toOrderedCancelAddCommMonoid.{0} NNReal NNReal.strictOrderedSemiring)))) (NNDist.nndist.{u1} α (PseudoMetricSpace.toNNDist.{u1} α _inst_1) x y) (HAdd.hAdd.{0, 0, 0} NNReal NNReal NNReal (instHAdd.{0} NNReal (Distrib.toHasAdd.{0} NNReal (NonUnitalNonAssocSemiring.toDistrib.{0} NNReal (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} NNReal (Semiring.toNonAssocSemiring.{0} NNReal NNReal.semiring))))) (NNDist.nndist.{u1} α (PseudoMetricSpace.toNNDist.{u1} α _inst_1) x z) (NNDist.nndist.{u1} α (PseudoMetricSpace.toNNDist.{u1} α _inst_1) y z))
 but is expected to have type
   forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] (x : α) (y : α) (z : α), LE.le.{0} NNReal (Preorder.toLE.{0} NNReal (PartialOrder.toPreorder.{0} NNReal (StrictOrderedSemiring.toPartialOrder.{0} NNReal instNNRealStrictOrderedSemiring))) (NNDist.nndist.{u1} α (PseudoMetricSpace.toNNDist.{u1} α _inst_1) x y) (HAdd.hAdd.{0, 0, 0} NNReal NNReal NNReal (instHAdd.{0} NNReal (Distrib.toAdd.{0} NNReal (NonUnitalNonAssocSemiring.toDistrib.{0} NNReal (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} NNReal (Semiring.toNonAssocSemiring.{0} NNReal instNNRealSemiring))))) (NNDist.nndist.{u1} α (PseudoMetricSpace.toNNDist.{u1} α _inst_1) x z) (NNDist.nndist.{u1} α (PseudoMetricSpace.toNNDist.{u1} α _inst_1) y z))
 Case conversion may be inaccurate. Consider using '#align nndist_triangle_right nndist_triangle_rightₓ'. -/
@@ -1308,13 +1308,17 @@ theorem isBounded_iff_exists_ge {s : Set α} (c : ℝ) :
     isBounded_iff.2 <| h.imp fun _ => And.right⟩
 #align metric.is_bounded_iff_exists_ge Metric.isBounded_iff_exists_ge
 
-#print Metric.isBounded_iff_nndist /-
+/- warning: metric.is_bounded_iff_nndist -> Metric.isBounded_iff_nndist is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {s : Set.{u1} α}, Iff (Bornology.IsBounded.{u1} α (PseudoMetricSpace.toBornology.{u1} α _inst_1) s) (Exists.{1} NNReal (fun (C : NNReal) => forall {{x : α}}, (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) x s) -> (forall {{y : α}}, (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) y s) -> (LE.le.{0} NNReal (Preorder.toHasLe.{0} NNReal (PartialOrder.toPreorder.{0} NNReal (OrderedCancelAddCommMonoid.toPartialOrder.{0} NNReal (StrictOrderedSemiring.toOrderedCancelAddCommMonoid.{0} NNReal NNReal.strictOrderedSemiring)))) (NNDist.nndist.{u1} α (PseudoMetricSpace.toNNDist.{u1} α _inst_1) x y) C))))
+but is expected to have type
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {s : Set.{u1} α}, Iff (Bornology.IsBounded.{u1} α (PseudoMetricSpace.toBornology.{u1} α _inst_1) s) (Exists.{1} NNReal (fun (C : NNReal) => forall {{x : α}}, (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) x s) -> (forall {{y : α}}, (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) y s) -> (LE.le.{0} NNReal (Preorder.toLE.{0} NNReal (PartialOrder.toPreorder.{0} NNReal (StrictOrderedSemiring.toPartialOrder.{0} NNReal instNNRealStrictOrderedSemiring))) (NNDist.nndist.{u1} α (PseudoMetricSpace.toNNDist.{u1} α _inst_1) x y) C))))
+Case conversion may be inaccurate. Consider using '#align metric.is_bounded_iff_nndist Metric.isBounded_iff_nndistₓ'. -/
 theorem isBounded_iff_nndist {s : Set α} :
     IsBounded s ↔ ∃ C : ℝ≥0, ∀ ⦃x⦄, x ∈ s → ∀ ⦃y⦄, y ∈ s → nndist x y ≤ C := by
   simp only [is_bounded_iff_exists_ge 0, NNReal.exists, ← NNReal.coe_le_coe, ← dist_nndist,
     NNReal.coe_mk, exists_prop]
 #align metric.is_bounded_iff_nndist Metric.isBounded_iff_nndist
--/
 
 #print Metric.toUniformSpace_eq /-
 theorem toUniformSpace_eq :
@@ -2058,7 +2062,7 @@ theorem continuous_iff' [TopologicalSpace β] {f : β → α} :
 
 /- warning: metric.tendsto_at_top -> Metric.tendsto_atTop is a dubious translation:
 lean 3 declaration is
-  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : PseudoMetricSpace.{u1} α] [_inst_2 : Nonempty.{succ u2} β] [_inst_3 : SemilatticeSup.{u2} β] {u : β -> α} {a : α}, Iff (Filter.Tendsto.{u2, u1} β α u (Filter.atTop.{u2} β (PartialOrder.toPreorder.{u2} β (SemilatticeSup.toPartialOrder.{u2} β _inst_3))) (nhds.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) a)) (forall (ε : Real), (GT.gt.{0} Real Real.hasLt ε (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero)))) -> (Exists.{succ u2} β (fun (N : β) => forall (n : β), (GE.ge.{u2} β (Preorder.toLE.{u2} β (PartialOrder.toPreorder.{u2} β (SemilatticeSup.toPartialOrder.{u2} β _inst_3))) n N) -> (LT.lt.{0} Real Real.hasLt (Dist.dist.{u1} α (PseudoMetricSpace.toHasDist.{u1} α _inst_1) (u n) a) ε))))
+  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : PseudoMetricSpace.{u1} α] [_inst_2 : Nonempty.{succ u2} β] [_inst_3 : SemilatticeSup.{u2} β] {u : β -> α} {a : α}, Iff (Filter.Tendsto.{u2, u1} β α u (Filter.atTop.{u2} β (PartialOrder.toPreorder.{u2} β (SemilatticeSup.toPartialOrder.{u2} β _inst_3))) (nhds.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) a)) (forall (ε : Real), (GT.gt.{0} Real Real.hasLt ε (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero)))) -> (Exists.{succ u2} β (fun (N : β) => forall (n : β), (GE.ge.{u2} β (Preorder.toHasLe.{u2} β (PartialOrder.toPreorder.{u2} β (SemilatticeSup.toPartialOrder.{u2} β _inst_3))) n N) -> (LT.lt.{0} Real Real.hasLt (Dist.dist.{u1} α (PseudoMetricSpace.toHasDist.{u1} α _inst_1) (u n) a) ε))))
 but is expected to have type
   forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : PseudoMetricSpace.{u1} α] [_inst_2 : Nonempty.{succ u2} β] [_inst_3 : SemilatticeSup.{u2} β] {u : β -> α} {a : α}, Iff (Filter.Tendsto.{u2, u1} β α u (Filter.atTop.{u2} β (PartialOrder.toPreorder.{u2} β (SemilatticeSup.toPartialOrder.{u2} β _inst_3))) (nhds.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) a)) (forall (ε : Real), (GT.gt.{0} Real Real.instLTReal ε (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal))) -> (Exists.{succ u2} β (fun (N : β) => forall (n : β), (GE.ge.{u2} β (Preorder.toLE.{u2} β (PartialOrder.toPreorder.{u2} β (SemilatticeSup.toPartialOrder.{u2} β _inst_3))) n N) -> (LT.lt.{0} Real Real.instLTReal (Dist.dist.{u1} α (PseudoMetricSpace.toDist.{u1} α _inst_1) (u n) a) ε))))
 Case conversion may be inaccurate. Consider using '#align metric.tendsto_at_top Metric.tendsto_atTopₓ'. -/
@@ -2072,7 +2076,7 @@ theorem tendsto_atTop [Nonempty β] [SemilatticeSup β] {u : β → α} {a : α}
 
 /- warning: metric.tendsto_at_top' -> Metric.tendsto_atTop' is a dubious translation:
 lean 3 declaration is
-  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : PseudoMetricSpace.{u1} α] [_inst_2 : Nonempty.{succ u2} β] [_inst_3 : SemilatticeSup.{u2} β] [_inst_4 : NoMaxOrder.{u2} β (Preorder.toLT.{u2} β (PartialOrder.toPreorder.{u2} β (SemilatticeSup.toPartialOrder.{u2} β _inst_3)))] {u : β -> α} {a : α}, Iff (Filter.Tendsto.{u2, u1} β α u (Filter.atTop.{u2} β (PartialOrder.toPreorder.{u2} β (SemilatticeSup.toPartialOrder.{u2} β _inst_3))) (nhds.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) a)) (forall (ε : Real), (GT.gt.{0} Real Real.hasLt ε (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero)))) -> (Exists.{succ u2} β (fun (N : β) => forall (n : β), (GT.gt.{u2} β (Preorder.toLT.{u2} β (PartialOrder.toPreorder.{u2} β (SemilatticeSup.toPartialOrder.{u2} β _inst_3))) n N) -> (LT.lt.{0} Real Real.hasLt (Dist.dist.{u1} α (PseudoMetricSpace.toHasDist.{u1} α _inst_1) (u n) a) ε))))
+  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : PseudoMetricSpace.{u1} α] [_inst_2 : Nonempty.{succ u2} β] [_inst_3 : SemilatticeSup.{u2} β] [_inst_4 : NoMaxOrder.{u2} β (Preorder.toHasLt.{u2} β (PartialOrder.toPreorder.{u2} β (SemilatticeSup.toPartialOrder.{u2} β _inst_3)))] {u : β -> α} {a : α}, Iff (Filter.Tendsto.{u2, u1} β α u (Filter.atTop.{u2} β (PartialOrder.toPreorder.{u2} β (SemilatticeSup.toPartialOrder.{u2} β _inst_3))) (nhds.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) a)) (forall (ε : Real), (GT.gt.{0} Real Real.hasLt ε (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero)))) -> (Exists.{succ u2} β (fun (N : β) => forall (n : β), (GT.gt.{u2} β (Preorder.toHasLt.{u2} β (PartialOrder.toPreorder.{u2} β (SemilatticeSup.toPartialOrder.{u2} β _inst_3))) n N) -> (LT.lt.{0} Real Real.hasLt (Dist.dist.{u1} α (PseudoMetricSpace.toHasDist.{u1} α _inst_1) (u n) a) ε))))
 but is expected to have type
   forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : PseudoMetricSpace.{u1} α] [_inst_2 : Nonempty.{succ u2} β] [_inst_3 : SemilatticeSup.{u2} β] [_inst_4 : NoMaxOrder.{u2} β (Preorder.toLT.{u2} β (PartialOrder.toPreorder.{u2} β (SemilatticeSup.toPartialOrder.{u2} β _inst_3)))] {u : β -> α} {a : α}, Iff (Filter.Tendsto.{u2, u1} β α u (Filter.atTop.{u2} β (PartialOrder.toPreorder.{u2} β (SemilatticeSup.toPartialOrder.{u2} β _inst_3))) (nhds.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) a)) (forall (ε : Real), (GT.gt.{0} Real Real.instLTReal ε (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal))) -> (Exists.{succ u2} β (fun (N : β) => forall (n : β), (GT.gt.{u2} β (Preorder.toLT.{u2} β (PartialOrder.toPreorder.{u2} β (SemilatticeSup.toPartialOrder.{u2} β _inst_3))) n N) -> (LT.lt.{0} Real Real.instLTReal (Dist.dist.{u1} α (PseudoMetricSpace.toDist.{u1} α _inst_1) (u n) a) ε))))
 Case conversion may be inaccurate. Consider using '#align metric.tendsto_at_top' Metric.tendsto_atTop'ₓ'. -/
@@ -2154,7 +2158,7 @@ open Metric
 
 /- warning: pseudo_metric.uniformity_basis_edist -> Metric.uniformity_basis_edist is a dubious translation:
 lean 3 declaration is
-  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α], Filter.HasBasis.{u1, 1} (Prod.{u1, u1} α α) ENNReal (uniformity.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) (fun (ε : ENNReal) => LT.lt.{0} ENNReal (Preorder.toLT.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (CompleteSemilatticeInf.toPartialOrder.{0} ENNReal (CompleteLattice.toCompleteSemilatticeInf.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))))) (OfNat.ofNat.{0} ENNReal 0 (OfNat.mk.{0} ENNReal 0 (Zero.zero.{0} ENNReal ENNReal.hasZero))) ε) (fun (ε : ENNReal) => setOf.{u1} (Prod.{u1, u1} α α) (fun (p : Prod.{u1, u1} α α) => LT.lt.{0} ENNReal (Preorder.toLT.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (CompleteSemilatticeInf.toPartialOrder.{0} ENNReal (CompleteLattice.toCompleteSemilatticeInf.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))))) (EDist.edist.{u1} α (PseudoMetricSpace.toEDist.{u1} α _inst_1) (Prod.fst.{u1, u1} α α p) (Prod.snd.{u1, u1} α α p)) ε))
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α], Filter.HasBasis.{u1, 1} (Prod.{u1, u1} α α) ENNReal (uniformity.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) (fun (ε : ENNReal) => LT.lt.{0} ENNReal (Preorder.toHasLt.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (CompleteSemilatticeInf.toPartialOrder.{0} ENNReal (CompleteLattice.toCompleteSemilatticeInf.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))))) (OfNat.ofNat.{0} ENNReal 0 (OfNat.mk.{0} ENNReal 0 (Zero.zero.{0} ENNReal ENNReal.hasZero))) ε) (fun (ε : ENNReal) => setOf.{u1} (Prod.{u1, u1} α α) (fun (p : Prod.{u1, u1} α α) => LT.lt.{0} ENNReal (Preorder.toHasLt.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (CompleteSemilatticeInf.toPartialOrder.{0} ENNReal (CompleteLattice.toCompleteSemilatticeInf.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))))) (EDist.edist.{u1} α (PseudoMetricSpace.toEDist.{u1} α _inst_1) (Prod.fst.{u1, u1} α α p) (Prod.snd.{u1, u1} α α p)) ε))
 but is expected to have type
   forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α], Filter.HasBasis.{u1, 1} (Prod.{u1, u1} α α) ENNReal (uniformity.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) (fun (ε : ENNReal) => LT.lt.{0} ENNReal (Preorder.toLT.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (CompleteSemilatticeInf.toPartialOrder.{0} ENNReal (CompleteLattice.toCompleteSemilatticeInf.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal))))) (OfNat.ofNat.{0} ENNReal 0 (Zero.toOfNat0.{0} ENNReal instENNRealZero)) ε) (fun (ε : ENNReal) => setOf.{u1} (Prod.{u1, u1} α α) (fun (p : Prod.{u1, u1} α α) => LT.lt.{0} ENNReal (Preorder.toLT.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (CompleteSemilatticeInf.toPartialOrder.{0} ENNReal (CompleteLattice.toCompleteSemilatticeInf.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal))))) (EDist.edist.{u1} α (PseudoEMetricSpace.toEDist.{u1} α (PseudoMetricSpace.toPseudoEMetricSpace.{u1} α _inst_1)) (Prod.fst.{u1, u1} α α p) (Prod.snd.{u1, u1} α α p)) ε))
 Case conversion may be inaccurate. Consider using '#align pseudo_metric.uniformity_basis_edist Metric.uniformity_basis_edistₓ'. -/
@@ -2179,7 +2183,7 @@ protected theorem Metric.uniformity_basis_edist :
 
 /- warning: metric.uniformity_edist -> Metric.uniformity_edist is a dubious translation:
 lean 3 declaration is
-  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α], Eq.{succ u1} (Filter.{u1} (Prod.{u1, u1} α α)) (uniformity.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) (iInf.{u1, 1} (Filter.{u1} (Prod.{u1, u1} α α)) (ConditionallyCompleteLattice.toHasInf.{u1} (Filter.{u1} (Prod.{u1, u1} α α)) (CompleteLattice.toConditionallyCompleteLattice.{u1} (Filter.{u1} (Prod.{u1, u1} α α)) (Filter.completeLattice.{u1} (Prod.{u1, u1} α α)))) ENNReal (fun (ε : ENNReal) => iInf.{u1, 0} (Filter.{u1} (Prod.{u1, u1} α α)) (ConditionallyCompleteLattice.toHasInf.{u1} (Filter.{u1} (Prod.{u1, u1} α α)) (CompleteLattice.toConditionallyCompleteLattice.{u1} (Filter.{u1} (Prod.{u1, u1} α α)) (Filter.completeLattice.{u1} (Prod.{u1, u1} α α)))) (GT.gt.{0} ENNReal (Preorder.toLT.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (CompleteSemilatticeInf.toPartialOrder.{0} ENNReal (CompleteLattice.toCompleteSemilatticeInf.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))))) ε (OfNat.ofNat.{0} ENNReal 0 (OfNat.mk.{0} ENNReal 0 (Zero.zero.{0} ENNReal ENNReal.hasZero)))) (fun (H : GT.gt.{0} ENNReal (Preorder.toLT.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (CompleteSemilatticeInf.toPartialOrder.{0} ENNReal (CompleteLattice.toCompleteSemilatticeInf.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))))) ε (OfNat.ofNat.{0} ENNReal 0 (OfNat.mk.{0} ENNReal 0 (Zero.zero.{0} ENNReal ENNReal.hasZero)))) => Filter.principal.{u1} (Prod.{u1, u1} α α) (setOf.{u1} (Prod.{u1, u1} α α) (fun (p : Prod.{u1, u1} α α) => LT.lt.{0} ENNReal (Preorder.toLT.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (CompleteSemilatticeInf.toPartialOrder.{0} ENNReal (CompleteLattice.toCompleteSemilatticeInf.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))))) (EDist.edist.{u1} α (PseudoMetricSpace.toEDist.{u1} α _inst_1) (Prod.fst.{u1, u1} α α p) (Prod.snd.{u1, u1} α α p)) ε)))))
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α], Eq.{succ u1} (Filter.{u1} (Prod.{u1, u1} α α)) (uniformity.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) (iInf.{u1, 1} (Filter.{u1} (Prod.{u1, u1} α α)) (ConditionallyCompleteLattice.toHasInf.{u1} (Filter.{u1} (Prod.{u1, u1} α α)) (CompleteLattice.toConditionallyCompleteLattice.{u1} (Filter.{u1} (Prod.{u1, u1} α α)) (Filter.completeLattice.{u1} (Prod.{u1, u1} α α)))) ENNReal (fun (ε : ENNReal) => iInf.{u1, 0} (Filter.{u1} (Prod.{u1, u1} α α)) (ConditionallyCompleteLattice.toHasInf.{u1} (Filter.{u1} (Prod.{u1, u1} α α)) (CompleteLattice.toConditionallyCompleteLattice.{u1} (Filter.{u1} (Prod.{u1, u1} α α)) (Filter.completeLattice.{u1} (Prod.{u1, u1} α α)))) (GT.gt.{0} ENNReal (Preorder.toHasLt.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (CompleteSemilatticeInf.toPartialOrder.{0} ENNReal (CompleteLattice.toCompleteSemilatticeInf.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))))) ε (OfNat.ofNat.{0} ENNReal 0 (OfNat.mk.{0} ENNReal 0 (Zero.zero.{0} ENNReal ENNReal.hasZero)))) (fun (H : GT.gt.{0} ENNReal (Preorder.toHasLt.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (CompleteSemilatticeInf.toPartialOrder.{0} ENNReal (CompleteLattice.toCompleteSemilatticeInf.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))))) ε (OfNat.ofNat.{0} ENNReal 0 (OfNat.mk.{0} ENNReal 0 (Zero.zero.{0} ENNReal ENNReal.hasZero)))) => Filter.principal.{u1} (Prod.{u1, u1} α α) (setOf.{u1} (Prod.{u1, u1} α α) (fun (p : Prod.{u1, u1} α α) => LT.lt.{0} ENNReal (Preorder.toHasLt.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (CompleteSemilatticeInf.toPartialOrder.{0} ENNReal (CompleteLattice.toCompleteSemilatticeInf.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))))) (EDist.edist.{u1} α (PseudoMetricSpace.toEDist.{u1} α _inst_1) (Prod.fst.{u1, u1} α α p) (Prod.snd.{u1, u1} α α p)) ε)))))
 but is expected to have type
   forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α], Eq.{succ u1} (Filter.{u1} (Prod.{u1, u1} α α)) (uniformity.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) (iInf.{u1, 1} (Filter.{u1} (Prod.{u1, u1} α α)) (ConditionallyCompleteLattice.toInfSet.{u1} (Filter.{u1} (Prod.{u1, u1} α α)) (CompleteLattice.toConditionallyCompleteLattice.{u1} (Filter.{u1} (Prod.{u1, u1} α α)) (Filter.instCompleteLatticeFilter.{u1} (Prod.{u1, u1} α α)))) ENNReal (fun (ε : ENNReal) => iInf.{u1, 0} (Filter.{u1} (Prod.{u1, u1} α α)) (ConditionallyCompleteLattice.toInfSet.{u1} (Filter.{u1} (Prod.{u1, u1} α α)) (CompleteLattice.toConditionallyCompleteLattice.{u1} (Filter.{u1} (Prod.{u1, u1} α α)) (Filter.instCompleteLatticeFilter.{u1} (Prod.{u1, u1} α α)))) (GT.gt.{0} ENNReal (Preorder.toLT.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (CompleteSemilatticeInf.toPartialOrder.{0} ENNReal (CompleteLattice.toCompleteSemilatticeInf.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal))))) ε (OfNat.ofNat.{0} ENNReal 0 (Zero.toOfNat0.{0} ENNReal instENNRealZero))) (fun (H : GT.gt.{0} ENNReal (Preorder.toLT.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (CompleteSemilatticeInf.toPartialOrder.{0} ENNReal (CompleteLattice.toCompleteSemilatticeInf.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal))))) ε (OfNat.ofNat.{0} ENNReal 0 (Zero.toOfNat0.{0} ENNReal instENNRealZero))) => Filter.principal.{u1} (Prod.{u1, u1} α α) (setOf.{u1} (Prod.{u1, u1} α α) (fun (p : Prod.{u1, u1} α α) => LT.lt.{0} ENNReal (Preorder.toLT.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (CompleteSemilatticeInf.toPartialOrder.{0} ENNReal (CompleteLattice.toCompleteSemilatticeInf.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal))))) (EDist.edist.{u1} α (PseudoMetricSpace.toEDist.{u1} α _inst_1) (Prod.fst.{u1, u1} α α p) (Prod.snd.{u1, u1} α α p)) ε)))))
 Case conversion may be inaccurate. Consider using '#align metric.uniformity_edist Metric.uniformity_edistₓ'. -/
@@ -2742,7 +2746,7 @@ variable [Nonempty β] [SemilatticeSup β]
 
 /- warning: metric.cauchy_seq_iff -> Metric.cauchySeq_iff is a dubious translation:
 lean 3 declaration is
-  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : PseudoMetricSpace.{u1} α] [_inst_2 : Nonempty.{succ u2} β] [_inst_3 : SemilatticeSup.{u2} β] {u : β -> α}, Iff (CauchySeq.{u1, u2} α β (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1) _inst_3 u) (forall (ε : Real), (GT.gt.{0} Real Real.hasLt ε (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero)))) -> (Exists.{succ u2} β (fun (N : β) => forall (m : β), (GE.ge.{u2} β (Preorder.toLE.{u2} β (PartialOrder.toPreorder.{u2} β (SemilatticeSup.toPartialOrder.{u2} β _inst_3))) m N) -> (forall (n : β), (GE.ge.{u2} β (Preorder.toLE.{u2} β (PartialOrder.toPreorder.{u2} β (SemilatticeSup.toPartialOrder.{u2} β _inst_3))) n N) -> (LT.lt.{0} Real Real.hasLt (Dist.dist.{u1} α (PseudoMetricSpace.toHasDist.{u1} α _inst_1) (u m) (u n)) ε)))))
+  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : PseudoMetricSpace.{u1} α] [_inst_2 : Nonempty.{succ u2} β] [_inst_3 : SemilatticeSup.{u2} β] {u : β -> α}, Iff (CauchySeq.{u1, u2} α β (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1) _inst_3 u) (forall (ε : Real), (GT.gt.{0} Real Real.hasLt ε (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero)))) -> (Exists.{succ u2} β (fun (N : β) => forall (m : β), (GE.ge.{u2} β (Preorder.toHasLe.{u2} β (PartialOrder.toPreorder.{u2} β (SemilatticeSup.toPartialOrder.{u2} β _inst_3))) m N) -> (forall (n : β), (GE.ge.{u2} β (Preorder.toHasLe.{u2} β (PartialOrder.toPreorder.{u2} β (SemilatticeSup.toPartialOrder.{u2} β _inst_3))) n N) -> (LT.lt.{0} Real Real.hasLt (Dist.dist.{u1} α (PseudoMetricSpace.toHasDist.{u1} α _inst_1) (u m) (u n)) ε)))))
 but is expected to have type
   forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : PseudoMetricSpace.{u1} α] [_inst_2 : Nonempty.{succ u2} β] [_inst_3 : SemilatticeSup.{u2} β] {u : β -> α}, Iff (CauchySeq.{u1, u2} α β (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1) _inst_3 u) (forall (ε : Real), (GT.gt.{0} Real Real.instLTReal ε (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal))) -> (Exists.{succ u2} β (fun (N : β) => forall (m : β), (GE.ge.{u2} β (Preorder.toLE.{u2} β (PartialOrder.toPreorder.{u2} β (SemilatticeSup.toPartialOrder.{u2} β _inst_3))) m N) -> (forall (n : β), (GE.ge.{u2} β (Preorder.toLE.{u2} β (PartialOrder.toPreorder.{u2} β (SemilatticeSup.toPartialOrder.{u2} β _inst_3))) n N) -> (LT.lt.{0} Real Real.instLTReal (Dist.dist.{u1} α (PseudoMetricSpace.toDist.{u1} α _inst_1) (u m) (u n)) ε)))))
 Case conversion may be inaccurate. Consider using '#align metric.cauchy_seq_iff Metric.cauchySeq_iffₓ'. -/
@@ -2758,7 +2762,7 @@ theorem Metric.cauchySeq_iff {u : β → α} :
 
 /- warning: metric.cauchy_seq_iff' -> Metric.cauchySeq_iff' is a dubious translation:
 lean 3 declaration is
-  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : PseudoMetricSpace.{u1} α] [_inst_2 : Nonempty.{succ u2} β] [_inst_3 : SemilatticeSup.{u2} β] {u : β -> α}, Iff (CauchySeq.{u1, u2} α β (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1) _inst_3 u) (forall (ε : Real), (GT.gt.{0} Real Real.hasLt ε (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero)))) -> (Exists.{succ u2} β (fun (N : β) => forall (n : β), (GE.ge.{u2} β (Preorder.toLE.{u2} β (PartialOrder.toPreorder.{u2} β (SemilatticeSup.toPartialOrder.{u2} β _inst_3))) n N) -> (LT.lt.{0} Real Real.hasLt (Dist.dist.{u1} α (PseudoMetricSpace.toHasDist.{u1} α _inst_1) (u n) (u N)) ε))))
+  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : PseudoMetricSpace.{u1} α] [_inst_2 : Nonempty.{succ u2} β] [_inst_3 : SemilatticeSup.{u2} β] {u : β -> α}, Iff (CauchySeq.{u1, u2} α β (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1) _inst_3 u) (forall (ε : Real), (GT.gt.{0} Real Real.hasLt ε (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero)))) -> (Exists.{succ u2} β (fun (N : β) => forall (n : β), (GE.ge.{u2} β (Preorder.toHasLe.{u2} β (PartialOrder.toPreorder.{u2} β (SemilatticeSup.toPartialOrder.{u2} β _inst_3))) n N) -> (LT.lt.{0} Real Real.hasLt (Dist.dist.{u1} α (PseudoMetricSpace.toHasDist.{u1} α _inst_1) (u n) (u N)) ε))))
 but is expected to have type
   forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : PseudoMetricSpace.{u1} α] [_inst_2 : Nonempty.{succ u2} β] [_inst_3 : SemilatticeSup.{u2} β] {u : β -> α}, Iff (CauchySeq.{u1, u2} α β (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1) _inst_3 u) (forall (ε : Real), (GT.gt.{0} Real Real.instLTReal ε (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal))) -> (Exists.{succ u2} β (fun (N : β) => forall (n : β), (GE.ge.{u2} β (Preorder.toLE.{u2} β (PartialOrder.toPreorder.{u2} β (SemilatticeSup.toPartialOrder.{u2} β _inst_3))) n N) -> (LT.lt.{0} Real Real.instLTReal (Dist.dist.{u1} α (PseudoMetricSpace.toDist.{u1} α _inst_1) (u n) (u N)) ε))))
 Case conversion may be inaccurate. Consider using '#align metric.cauchy_seq_iff' Metric.cauchySeq_iff'ₓ'. -/
@@ -2770,7 +2774,7 @@ theorem Metric.cauchySeq_iff' {u : β → α} :
 
 /- warning: metric.uniform_cauchy_seq_on_iff -> Metric.uniformCauchySeqOn_iff is a dubious translation:
 lean 3 declaration is
-  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : PseudoMetricSpace.{u1} α] [_inst_2 : Nonempty.{succ u2} β] [_inst_3 : SemilatticeSup.{u2} β] {γ : Type.{u3}} {F : β -> γ -> α} {s : Set.{u3} γ}, Iff (UniformCauchySeqOn.{u3, u1, u2} γ α β (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1) F (Filter.atTop.{u2} β (PartialOrder.toPreorder.{u2} β (SemilatticeSup.toPartialOrder.{u2} β _inst_3))) s) (forall (ε : Real), (GT.gt.{0} Real Real.hasLt ε (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero)))) -> (Exists.{succ u2} β (fun (N : β) => forall (m : β), (GE.ge.{u2} β (Preorder.toLE.{u2} β (PartialOrder.toPreorder.{u2} β (SemilatticeSup.toPartialOrder.{u2} β _inst_3))) m N) -> (forall (n : β), (GE.ge.{u2} β (Preorder.toLE.{u2} β (PartialOrder.toPreorder.{u2} β (SemilatticeSup.toPartialOrder.{u2} β _inst_3))) n N) -> (forall (x : γ), (Membership.Mem.{u3, u3} γ (Set.{u3} γ) (Set.hasMem.{u3} γ) x s) -> (LT.lt.{0} Real Real.hasLt (Dist.dist.{u1} α (PseudoMetricSpace.toHasDist.{u1} α _inst_1) (F m x) (F n x)) ε))))))
+  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : PseudoMetricSpace.{u1} α] [_inst_2 : Nonempty.{succ u2} β] [_inst_3 : SemilatticeSup.{u2} β] {γ : Type.{u3}} {F : β -> γ -> α} {s : Set.{u3} γ}, Iff (UniformCauchySeqOn.{u3, u1, u2} γ α β (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1) F (Filter.atTop.{u2} β (PartialOrder.toPreorder.{u2} β (SemilatticeSup.toPartialOrder.{u2} β _inst_3))) s) (forall (ε : Real), (GT.gt.{0} Real Real.hasLt ε (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero)))) -> (Exists.{succ u2} β (fun (N : β) => forall (m : β), (GE.ge.{u2} β (Preorder.toHasLe.{u2} β (PartialOrder.toPreorder.{u2} β (SemilatticeSup.toPartialOrder.{u2} β _inst_3))) m N) -> (forall (n : β), (GE.ge.{u2} β (Preorder.toHasLe.{u2} β (PartialOrder.toPreorder.{u2} β (SemilatticeSup.toPartialOrder.{u2} β _inst_3))) n N) -> (forall (x : γ), (Membership.Mem.{u3, u3} γ (Set.{u3} γ) (Set.hasMem.{u3} γ) x s) -> (LT.lt.{0} Real Real.hasLt (Dist.dist.{u1} α (PseudoMetricSpace.toHasDist.{u1} α _inst_1) (F m x) (F n x)) ε))))))
 but is expected to have type
   forall {α : Type.{u2}} {β : Type.{u3}} [_inst_1 : PseudoMetricSpace.{u2} α] [_inst_2 : Nonempty.{succ u3} β] [_inst_3 : SemilatticeSup.{u3} β] {γ : Type.{u1}} {F : β -> γ -> α} {s : Set.{u1} γ}, Iff (UniformCauchySeqOn.{u1, u2, u3} γ α β (PseudoMetricSpace.toUniformSpace.{u2} α _inst_1) F (Filter.atTop.{u3} β (PartialOrder.toPreorder.{u3} β (SemilatticeSup.toPartialOrder.{u3} β _inst_3))) s) (forall (ε : Real), (GT.gt.{0} Real Real.instLTReal ε (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal))) -> (Exists.{succ u3} β (fun (N : β) => forall (m : β), (GE.ge.{u3} β (Preorder.toLE.{u3} β (PartialOrder.toPreorder.{u3} β (SemilatticeSup.toPartialOrder.{u3} β _inst_3))) m N) -> (forall (n : β), (GE.ge.{u3} β (Preorder.toLE.{u3} β (PartialOrder.toPreorder.{u3} β (SemilatticeSup.toPartialOrder.{u3} β _inst_3))) n N) -> (forall (x : γ), (Membership.mem.{u1, u1} γ (Set.{u1} γ) (Set.instMembershipSet.{u1} γ) x s) -> (LT.lt.{0} Real Real.instLTReal (Dist.dist.{u2} α (PseudoMetricSpace.toDist.{u2} α _inst_1) (F m x) (F n x)) ε))))))
 Case conversion may be inaccurate. Consider using '#align metric.uniform_cauchy_seq_on_iff Metric.uniformCauchySeqOn_iffₓ'. -/
@@ -2806,7 +2810,7 @@ theorem Metric.uniformCauchySeqOn_iff {γ : Type _} {F : β → γ → α} {s :
 
 /- warning: cauchy_seq_of_le_tendsto_0' -> cauchySeq_of_le_tendsto_0' is a dubious translation:
 lean 3 declaration is
-  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : PseudoMetricSpace.{u1} α] [_inst_2 : Nonempty.{succ u2} β] [_inst_3 : SemilatticeSup.{u2} β] {s : β -> α} (b : β -> Real), (forall (n : β) (m : β), (LE.le.{u2} β (Preorder.toLE.{u2} β (PartialOrder.toPreorder.{u2} β (SemilatticeSup.toPartialOrder.{u2} β _inst_3))) n m) -> (LE.le.{0} Real Real.hasLe (Dist.dist.{u1} α (PseudoMetricSpace.toHasDist.{u1} α _inst_1) (s n) (s m)) (b n))) -> (Filter.Tendsto.{u2, 0} β Real b (Filter.atTop.{u2} β (PartialOrder.toPreorder.{u2} β (SemilatticeSup.toPartialOrder.{u2} β _inst_3))) (nhds.{0} Real (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero))))) -> (CauchySeq.{u1, u2} α β (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1) _inst_3 s)
+  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : PseudoMetricSpace.{u1} α] [_inst_2 : Nonempty.{succ u2} β] [_inst_3 : SemilatticeSup.{u2} β] {s : β -> α} (b : β -> Real), (forall (n : β) (m : β), (LE.le.{u2} β (Preorder.toHasLe.{u2} β (PartialOrder.toPreorder.{u2} β (SemilatticeSup.toPartialOrder.{u2} β _inst_3))) n m) -> (LE.le.{0} Real Real.hasLe (Dist.dist.{u1} α (PseudoMetricSpace.toHasDist.{u1} α _inst_1) (s n) (s m)) (b n))) -> (Filter.Tendsto.{u2, 0} β Real b (Filter.atTop.{u2} β (PartialOrder.toPreorder.{u2} β (SemilatticeSup.toPartialOrder.{u2} β _inst_3))) (nhds.{0} Real (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero))))) -> (CauchySeq.{u1, u2} α β (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1) _inst_3 s)
 but is expected to have type
   forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : PseudoMetricSpace.{u1} α] [_inst_2 : Nonempty.{succ u2} β] [_inst_3 : SemilatticeSup.{u2} β] {s : β -> α} (b : β -> Real), (forall (n : β) (m : β), (LE.le.{u2} β (Preorder.toLE.{u2} β (PartialOrder.toPreorder.{u2} β (SemilatticeSup.toPartialOrder.{u2} β _inst_3))) n m) -> (LE.le.{0} Real Real.instLEReal (Dist.dist.{u1} α (PseudoMetricSpace.toDist.{u1} α _inst_1) (s n) (s m)) (b n))) -> (Filter.Tendsto.{u2, 0} β Real b (Filter.atTop.{u2} β (PartialOrder.toPreorder.{u2} β (SemilatticeSup.toPartialOrder.{u2} β _inst_3))) (nhds.{0} Real (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal)))) -> (CauchySeq.{u1, u2} α β (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1) _inst_3 s)
 Case conversion may be inaccurate. Consider using '#align cauchy_seq_of_le_tendsto_0' cauchySeq_of_le_tendsto_0'ₓ'. -/
@@ -2825,7 +2829,7 @@ theorem cauchySeq_of_le_tendsto_0' {s : β → α} (b : β → ℝ)
 
 /- warning: cauchy_seq_of_le_tendsto_0 -> cauchySeq_of_le_tendsto_0 is a dubious translation:
 lean 3 declaration is
-  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : PseudoMetricSpace.{u1} α] [_inst_2 : Nonempty.{succ u2} β] [_inst_3 : SemilatticeSup.{u2} β] {s : β -> α} (b : β -> Real), (forall (n : β) (m : β) (N : β), (LE.le.{u2} β (Preorder.toLE.{u2} β (PartialOrder.toPreorder.{u2} β (SemilatticeSup.toPartialOrder.{u2} β _inst_3))) N n) -> (LE.le.{u2} β (Preorder.toLE.{u2} β (PartialOrder.toPreorder.{u2} β (SemilatticeSup.toPartialOrder.{u2} β _inst_3))) N m) -> (LE.le.{0} Real Real.hasLe (Dist.dist.{u1} α (PseudoMetricSpace.toHasDist.{u1} α _inst_1) (s n) (s m)) (b N))) -> (Filter.Tendsto.{u2, 0} β Real b (Filter.atTop.{u2} β (PartialOrder.toPreorder.{u2} β (SemilatticeSup.toPartialOrder.{u2} β _inst_3))) (nhds.{0} Real (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero))))) -> (CauchySeq.{u1, u2} α β (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1) _inst_3 s)
+  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : PseudoMetricSpace.{u1} α] [_inst_2 : Nonempty.{succ u2} β] [_inst_3 : SemilatticeSup.{u2} β] {s : β -> α} (b : β -> Real), (forall (n : β) (m : β) (N : β), (LE.le.{u2} β (Preorder.toHasLe.{u2} β (PartialOrder.toPreorder.{u2} β (SemilatticeSup.toPartialOrder.{u2} β _inst_3))) N n) -> (LE.le.{u2} β (Preorder.toHasLe.{u2} β (PartialOrder.toPreorder.{u2} β (SemilatticeSup.toPartialOrder.{u2} β _inst_3))) N m) -> (LE.le.{0} Real Real.hasLe (Dist.dist.{u1} α (PseudoMetricSpace.toHasDist.{u1} α _inst_1) (s n) (s m)) (b N))) -> (Filter.Tendsto.{u2, 0} β Real b (Filter.atTop.{u2} β (PartialOrder.toPreorder.{u2} β (SemilatticeSup.toPartialOrder.{u2} β _inst_3))) (nhds.{0} Real (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero))))) -> (CauchySeq.{u1, u2} α β (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1) _inst_3 s)
 but is expected to have type
   forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : PseudoMetricSpace.{u1} α] [_inst_2 : Nonempty.{succ u2} β] [_inst_3 : SemilatticeSup.{u2} β] {s : β -> α} (b : β -> Real), (forall (n : β) (m : β) (N : β), (LE.le.{u2} β (Preorder.toLE.{u2} β (PartialOrder.toPreorder.{u2} β (SemilatticeSup.toPartialOrder.{u2} β _inst_3))) N n) -> (LE.le.{u2} β (Preorder.toLE.{u2} β (PartialOrder.toPreorder.{u2} β (SemilatticeSup.toPartialOrder.{u2} β _inst_3))) N m) -> (LE.le.{0} Real Real.instLEReal (Dist.dist.{u1} α (PseudoMetricSpace.toDist.{u1} α _inst_1) (s n) (s m)) (b N))) -> (Filter.Tendsto.{u2, 0} β Real b (Filter.atTop.{u2} β (PartialOrder.toPreorder.{u2} β (SemilatticeSup.toPartialOrder.{u2} β _inst_3))) (nhds.{0} Real (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal)))) -> (CauchySeq.{u1, u2} α β (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1) _inst_3 s)
 Case conversion may be inaccurate. Consider using '#align cauchy_seq_of_le_tendsto_0 cauchySeq_of_le_tendsto_0ₓ'. -/
@@ -3064,7 +3068,7 @@ theorem NNReal.nndist_zero_eq_val' (z : ℝ≥0) : nndist z 0 = z :=
 
 /- warning: nnreal.le_add_nndist -> NNReal.le_add_nndist is a dubious translation:
 lean 3 declaration is
-  forall (a : NNReal) (b : NNReal), LE.le.{0} NNReal (Preorder.toLE.{0} NNReal (PartialOrder.toPreorder.{0} NNReal (OrderedCancelAddCommMonoid.toPartialOrder.{0} NNReal (StrictOrderedSemiring.toOrderedCancelAddCommMonoid.{0} NNReal NNReal.strictOrderedSemiring)))) a (HAdd.hAdd.{0, 0, 0} NNReal NNReal NNReal (instHAdd.{0} NNReal (Distrib.toHasAdd.{0} NNReal (NonUnitalNonAssocSemiring.toDistrib.{0} NNReal (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} NNReal (Semiring.toNonAssocSemiring.{0} NNReal NNReal.semiring))))) b (NNDist.nndist.{0} NNReal (PseudoMetricSpace.toNNDist.{0} NNReal NNReal.pseudoMetricSpace) a b))
+  forall (a : NNReal) (b : NNReal), LE.le.{0} NNReal (Preorder.toHasLe.{0} NNReal (PartialOrder.toPreorder.{0} NNReal (OrderedCancelAddCommMonoid.toPartialOrder.{0} NNReal (StrictOrderedSemiring.toOrderedCancelAddCommMonoid.{0} NNReal NNReal.strictOrderedSemiring)))) a (HAdd.hAdd.{0, 0, 0} NNReal NNReal NNReal (instHAdd.{0} NNReal (Distrib.toHasAdd.{0} NNReal (NonUnitalNonAssocSemiring.toDistrib.{0} NNReal (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} NNReal (Semiring.toNonAssocSemiring.{0} NNReal NNReal.semiring))))) b (NNDist.nndist.{0} NNReal (PseudoMetricSpace.toNNDist.{0} NNReal NNReal.pseudoMetricSpace) a b))
 but is expected to have type
   forall (a : NNReal) (b : NNReal), LE.le.{0} NNReal (Preorder.toLE.{0} NNReal (PartialOrder.toPreorder.{0} NNReal (StrictOrderedSemiring.toPartialOrder.{0} NNReal instNNRealStrictOrderedSemiring))) a (HAdd.hAdd.{0, 0, 0} NNReal NNReal NNReal (instHAdd.{0} NNReal (Distrib.toAdd.{0} NNReal (NonUnitalNonAssocSemiring.toDistrib.{0} NNReal (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} NNReal (Semiring.toNonAssocSemiring.{0} NNReal instNNRealSemiring))))) b (NNDist.nndist.{0} NNReal (PseudoMetricSpace.toNNDist.{0} NNReal instPseudoMetricSpaceNNReal) a b))
 Case conversion may be inaccurate. Consider using '#align nnreal.le_add_nndist NNReal.le_add_nndistₓ'. -/
@@ -3606,7 +3610,7 @@ theorem dist_pi_def (f g : ∀ b, π b) : dist f g = (sup univ fun b => nndist (
 
 /- warning: nndist_pi_le_iff -> nndist_pi_le_iff is a dubious translation:
 lean 3 declaration is
-  forall {β : Type.{u1}} {π : β -> Type.{u2}} [_inst_2 : Fintype.{u1} β] [_inst_3 : forall (b : β), PseudoMetricSpace.{u2} (π b)] {f : forall (b : β), π b} {g : forall (b : β), π b} {r : NNReal}, Iff (LE.le.{0} NNReal (Preorder.toLE.{0} NNReal (PartialOrder.toPreorder.{0} NNReal (OrderedCancelAddCommMonoid.toPartialOrder.{0} NNReal (StrictOrderedSemiring.toOrderedCancelAddCommMonoid.{0} NNReal NNReal.strictOrderedSemiring)))) (NNDist.nndist.{max u1 u2} (forall (b : β), π b) (PseudoMetricSpace.toNNDist.{max u1 u2} (forall (b : β), π b) (pseudoMetricSpacePi.{u1, u2} β (fun (b : β) => π b) _inst_2 (fun (b : β) => _inst_3 b))) f g) r) (forall (b : β), LE.le.{0} NNReal (Preorder.toLE.{0} NNReal (PartialOrder.toPreorder.{0} NNReal (OrderedCancelAddCommMonoid.toPartialOrder.{0} NNReal (StrictOrderedSemiring.toOrderedCancelAddCommMonoid.{0} NNReal NNReal.strictOrderedSemiring)))) (NNDist.nndist.{u2} (π b) (PseudoMetricSpace.toNNDist.{u2} (π b) (_inst_3 b)) (f b) (g b)) r)
+  forall {β : Type.{u1}} {π : β -> Type.{u2}} [_inst_2 : Fintype.{u1} β] [_inst_3 : forall (b : β), PseudoMetricSpace.{u2} (π b)] {f : forall (b : β), π b} {g : forall (b : β), π b} {r : NNReal}, Iff (LE.le.{0} NNReal (Preorder.toHasLe.{0} NNReal (PartialOrder.toPreorder.{0} NNReal (OrderedCancelAddCommMonoid.toPartialOrder.{0} NNReal (StrictOrderedSemiring.toOrderedCancelAddCommMonoid.{0} NNReal NNReal.strictOrderedSemiring)))) (NNDist.nndist.{max u1 u2} (forall (b : β), π b) (PseudoMetricSpace.toNNDist.{max u1 u2} (forall (b : β), π b) (pseudoMetricSpacePi.{u1, u2} β (fun (b : β) => π b) _inst_2 (fun (b : β) => _inst_3 b))) f g) r) (forall (b : β), LE.le.{0} NNReal (Preorder.toHasLe.{0} NNReal (PartialOrder.toPreorder.{0} NNReal (OrderedCancelAddCommMonoid.toPartialOrder.{0} NNReal (StrictOrderedSemiring.toOrderedCancelAddCommMonoid.{0} NNReal NNReal.strictOrderedSemiring)))) (NNDist.nndist.{u2} (π b) (PseudoMetricSpace.toNNDist.{u2} (π b) (_inst_3 b)) (f b) (g b)) r)
 but is expected to have type
   forall {β : Type.{u2}} {π : β -> Type.{u1}} [_inst_2 : Fintype.{u2} β] [_inst_3 : forall (b : β), PseudoMetricSpace.{u1} (π b)] {f : forall (b : β), π b} {g : forall (b : β), π b} {r : NNReal}, Iff (LE.le.{0} NNReal (Preorder.toLE.{0} NNReal (PartialOrder.toPreorder.{0} NNReal (StrictOrderedSemiring.toPartialOrder.{0} NNReal instNNRealStrictOrderedSemiring))) (NNDist.nndist.{max u2 u1} (forall (b : β), π b) (PseudoMetricSpace.toNNDist.{max u2 u1} (forall (b : β), π b) (pseudoMetricSpacePi.{u2, u1} β (fun (b : β) => π b) _inst_2 (fun (b : β) => _inst_3 b))) f g) r) (forall (b : β), LE.le.{0} NNReal (Preorder.toLE.{0} NNReal (PartialOrder.toPreorder.{0} NNReal (StrictOrderedSemiring.toPartialOrder.{0} NNReal instNNRealStrictOrderedSemiring))) (NNDist.nndist.{u1} (π b) (PseudoMetricSpace.toNNDist.{u1} (π b) (_inst_3 b)) (f b) (g b)) r)
 Case conversion may be inaccurate. Consider using '#align nndist_pi_le_iff nndist_pi_le_iffₓ'. -/
@@ -3667,11 +3671,15 @@ theorem dist_pi_const_le (a b : α) : (dist (fun _ : β => a) fun _ => b) ≤ di
   (dist_pi_le_iff dist_nonneg).2 fun _ => le_rfl
 #align dist_pi_const_le dist_pi_const_le
 
-#print nndist_pi_const_le /-
+/- warning: nndist_pi_const_le -> nndist_pi_const_le is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : PseudoMetricSpace.{u1} α] [_inst_2 : Fintype.{u2} β] (a : α) (b : α), LE.le.{0} NNReal (Preorder.toHasLe.{0} NNReal (PartialOrder.toPreorder.{0} NNReal (OrderedCancelAddCommMonoid.toPartialOrder.{0} NNReal (StrictOrderedSemiring.toOrderedCancelAddCommMonoid.{0} NNReal NNReal.strictOrderedSemiring)))) (NNDist.nndist.{max u2 u1} (β -> α) (PseudoMetricSpace.toNNDist.{max u2 u1} (β -> α) (pseudoMetricSpacePi.{u2, u1} β (fun (_x : β) => α) _inst_2 (fun (b : β) => _inst_1))) (fun (_x : β) => a) (fun (_x : β) => b)) (NNDist.nndist.{u1} α (PseudoMetricSpace.toNNDist.{u1} α _inst_1) a b)
+but is expected to have type
+  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : PseudoMetricSpace.{u1} α] [_inst_2 : Fintype.{u2} β] (a : α) (b : α), LE.le.{0} NNReal (Preorder.toLE.{0} NNReal (PartialOrder.toPreorder.{0} NNReal (StrictOrderedSemiring.toPartialOrder.{0} NNReal instNNRealStrictOrderedSemiring))) (NNDist.nndist.{max u1 u2} (β -> α) (PseudoMetricSpace.toNNDist.{max u1 u2} (β -> α) (pseudoMetricSpacePi.{u2, u1} β (fun (_x : β) => α) _inst_2 (fun (b : β) => _inst_1))) (fun (_x : β) => a) (fun (_x : β) => b)) (NNDist.nndist.{u1} α (PseudoMetricSpace.toNNDist.{u1} α _inst_1) a b)
+Case conversion may be inaccurate. Consider using '#align nndist_pi_const_le nndist_pi_const_leₓ'. -/
 theorem nndist_pi_const_le (a b : α) : (nndist (fun _ : β => a) fun _ => b) ≤ nndist a b :=
   nndist_pi_le_iff.2 fun _ => le_rfl
 #align nndist_pi_const_le nndist_pi_const_le
--/
 
 #print dist_pi_const /-
 @[simp]
@@ -3690,7 +3698,7 @@ theorem nndist_pi_const [Nonempty β] (a b : α) :
 
 /- warning: nndist_le_pi_nndist -> nndist_le_pi_nndist is a dubious translation:
 lean 3 declaration is
-  forall {β : Type.{u1}} {π : β -> Type.{u2}} [_inst_2 : Fintype.{u1} β] [_inst_3 : forall (b : β), PseudoMetricSpace.{u2} (π b)] (f : forall (b : β), π b) (g : forall (b : β), π b) (b : β), LE.le.{0} NNReal (Preorder.toLE.{0} NNReal (PartialOrder.toPreorder.{0} NNReal (OrderedCancelAddCommMonoid.toPartialOrder.{0} NNReal (StrictOrderedSemiring.toOrderedCancelAddCommMonoid.{0} NNReal NNReal.strictOrderedSemiring)))) (NNDist.nndist.{u2} (π b) (PseudoMetricSpace.toNNDist.{u2} (π b) (_inst_3 b)) (f b) (g b)) (NNDist.nndist.{max u1 u2} (forall (b : β), π b) (PseudoMetricSpace.toNNDist.{max u1 u2} (forall (b : β), π b) (pseudoMetricSpacePi.{u1, u2} β (fun (b : β) => π b) _inst_2 (fun (b : β) => _inst_3 b))) f g)
+  forall {β : Type.{u1}} {π : β -> Type.{u2}} [_inst_2 : Fintype.{u1} β] [_inst_3 : forall (b : β), PseudoMetricSpace.{u2} (π b)] (f : forall (b : β), π b) (g : forall (b : β), π b) (b : β), LE.le.{0} NNReal (Preorder.toHasLe.{0} NNReal (PartialOrder.toPreorder.{0} NNReal (OrderedCancelAddCommMonoid.toPartialOrder.{0} NNReal (StrictOrderedSemiring.toOrderedCancelAddCommMonoid.{0} NNReal NNReal.strictOrderedSemiring)))) (NNDist.nndist.{u2} (π b) (PseudoMetricSpace.toNNDist.{u2} (π b) (_inst_3 b)) (f b) (g b)) (NNDist.nndist.{max u1 u2} (forall (b : β), π b) (PseudoMetricSpace.toNNDist.{max u1 u2} (forall (b : β), π b) (pseudoMetricSpacePi.{u1, u2} β (fun (b : β) => π b) _inst_2 (fun (b : β) => _inst_3 b))) f g)
 but is expected to have type
   forall {β : Type.{u2}} {π : β -> Type.{u1}} [_inst_2 : Fintype.{u2} β] [_inst_3 : forall (b : β), PseudoMetricSpace.{u1} (π b)] (f : forall (b : β), π b) (g : forall (b : β), π b) (b : β), LE.le.{0} NNReal (Preorder.toLE.{0} NNReal (PartialOrder.toPreorder.{0} NNReal (StrictOrderedSemiring.toPartialOrder.{0} NNReal instNNRealStrictOrderedSemiring))) (NNDist.nndist.{u1} (π b) (PseudoMetricSpace.toNNDist.{u1} (π b) (_inst_3 b)) (f b) (g b)) (NNDist.nndist.{max u2 u1} (forall (b : β), π b) (PseudoMetricSpace.toNNDist.{max u2 u1} (forall (b : β), π b) (pseudoMetricSpacePi.{u2, u1} β (fun (b : β) => π b) _inst_2 (fun (b : β) => _inst_3 b))) f g)
 Case conversion may be inaccurate. Consider using '#align nndist_le_pi_nndist nndist_le_pi_nndistₓ'. -/
@@ -3768,7 +3776,7 @@ theorem closedBall_pi' [Nonempty β] (x : ∀ b, π b) (r : ℝ) :
 
 /- warning: fin.nndist_insert_nth_insert_nth -> Fin.nndist_insertNth_insertNth is a dubious translation:
 lean 3 declaration is
-  forall {n : Nat} {α : (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) -> Type.{u1}} [_inst_4 : forall (i : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))), PseudoMetricSpace.{u1} (α i)] (i : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (x : α i) (y : α i) (f : forall (j : Fin n), α (coeFn.{1, 1} (OrderEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (Fin.hasLe n) (Preorder.toLE.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (PartialOrder.toPreorder.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (Fin.partialOrder (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne)))))))) (fun (_x : RelEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (LE.le.{0} (Fin n) (Fin.hasLe n)) (LE.le.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (Preorder.toLE.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (PartialOrder.toPreorder.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (Fin.partialOrder (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))))))) => (Fin n) -> (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne)))))) (RelEmbedding.hasCoeToFun.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (LE.le.{0} (Fin n) (Fin.hasLe n)) (LE.le.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (Preorder.toLE.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (PartialOrder.toPreorder.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (Fin.partialOrder (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))))))) (Fin.succAbove n i) j)) (g : forall (j : Fin n), α (coeFn.{1, 1} (OrderEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (Fin.hasLe n) (Preorder.toLE.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (PartialOrder.toPreorder.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (Fin.partialOrder (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne)))))))) (fun (_x : 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+  forall {n : Nat} {α : (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) -> Type.{u1}} [_inst_4 : forall (i : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))), PseudoMetricSpace.{u1} (α i)] (i : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (x : α i) (y : α i) (f : forall (j : Fin n), α (coeFn.{1, 1} (OrderEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (Fin.hasLe n) (Preorder.toHasLe.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (PartialOrder.toPreorder.{0} (Fin 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 but is expected to have type
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n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (Lattice.toInf.{0} (Fin n) (DistribLattice.toLattice.{0} (Fin n) (instDistribLattice.{0} (Fin n) (Fin.instLinearOrderFin n)))) (Lattice.toInf.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (Fin.instLatticeFinHAddNatInstHAddInstAddNatOfNat n)) (LatticeHomClass.toInfHomClass.{0, 0, 0} (OrderEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin n) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))))) (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (DistribLattice.toLattice.{0} (Fin n) (instDistribLattice.{0} (Fin n) (Fin.instLinearOrderFin n))) 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(x._@.Mathlib.Order.Hom.Basic._hyg.695 : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (x._@.Mathlib.Order.Hom.Basic._hyg.697 : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) => LE.le.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) x._@.Mathlib.Order.Hom.Basic._hyg.695 x._@.Mathlib.Order.Hom.Basic._hyg.697))))) (Fin.succAbove n i) j)) (pseudoMetricSpacePi.{0, u1} (Fin n) (fun (j : Fin n) => α (FunLike.coe.{1, 1, 1} (OrderEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin n) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 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Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) x._@.Mathlib.Order.Hom.Basic._hyg.695 x._@.Mathlib.Order.Hom.Basic._hyg.697))))) (Fin.succAbove n i) j)) (Fin.fintype n) (fun (b : Fin n) => _inst_4 (FunLike.coe.{1, 1, 1} (OrderEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin n) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))))) (Fin n) (fun (_x : Fin n) => (fun (x._@.Mathlib.Order.Hom.Lattice._hyg.494 : Fin n) => Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) _x) (InfHomClass.toFunLike.{0, 0, 0} (OrderEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin n) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} 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(Fin.instLatticeFinHAddNatInstHAddInstAddNatOfNat n) (OrderHomClass.toLatticeHomClass.{0, 0, 0} (OrderEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin n) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))))) (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (Fin.instLinearOrderFin n) (Fin.instLatticeFinHAddNatInstHAddInstAddNatOfNat n) (RelEmbedding.instRelHomClassRelEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.680 : Fin n) (x._@.Mathlib.Order.Hom.Basic._hyg.682 : Fin n) => LE.le.{0} (Fin n) (instLEFin n) x._@.Mathlib.Order.Hom.Basic._hyg.680 x._@.Mathlib.Order.Hom.Basic._hyg.682) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.695 : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (x._@.Mathlib.Order.Hom.Basic._hyg.697 : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) => LE.le.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) x._@.Mathlib.Order.Hom.Basic._hyg.695 x._@.Mathlib.Order.Hom.Basic._hyg.697))))) (Fin.succAbove n i) b)))) f g))
 Case conversion may be inaccurate. Consider using '#align fin.nndist_insert_nth_insert_nth Fin.nndist_insertNth_insertNthₓ'. -/
@@ -3781,7 +3789,7 @@ theorem Fin.nndist_insertNth_insertNth {n : ℕ} {α : Fin (n + 1) → Type _}
 
 /- warning: fin.dist_insert_nth_insert_nth -> Fin.dist_insertNth_insertNth is a dubious translation:
 lean 3 declaration is
-  forall {n : Nat} {α : (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) -> Type.{u1}} [_inst_4 : forall (i : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))), PseudoMetricSpace.{u1} (α i)] (i : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (x : α i) (y : α i) (f : forall (j : Fin n), α (coeFn.{1, 1} (OrderEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (Fin.hasLe n) (Preorder.toLE.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (PartialOrder.toPreorder.{0} (Fin (HAdd.hAdd.{0, 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 but is expected to have type
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(HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (Lattice.toInf.{0} (Fin n) (DistribLattice.toLattice.{0} (Fin n) (instDistribLattice.{0} (Fin n) (Fin.instLinearOrderFin n)))) (Lattice.toInf.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (Fin.instLatticeFinHAddNatInstHAddInstAddNatOfNat n)) (LatticeHomClass.toInfHomClass.{0, 0, 0} (OrderEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin n) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))))) (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (DistribLattice.toLattice.{0} (Fin n) (instDistribLattice.{0} (Fin n) (Fin.instLinearOrderFin n))) (Fin.instLatticeFinHAddNatInstHAddInstAddNatOfNat n) (OrderHomClass.toLatticeHomClass.{0, 0, 0} (OrderEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin n) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))))) (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (Fin.instLinearOrderFin n) (Fin.instLatticeFinHAddNatInstHAddInstAddNatOfNat n) (RelEmbedding.instRelHomClassRelEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.680 : Fin n) (x._@.Mathlib.Order.Hom.Basic._hyg.682 : Fin n) => LE.le.{0} (Fin n) (instLEFin n) x._@.Mathlib.Order.Hom.Basic._hyg.680 x._@.Mathlib.Order.Hom.Basic._hyg.682) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.695 : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (x._@.Mathlib.Order.Hom.Basic._hyg.697 : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) => LE.le.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) x._@.Mathlib.Order.Hom.Basic._hyg.695 x._@.Mathlib.Order.Hom.Basic._hyg.697))))) (Fin.succAbove n i) j)) (pseudoMetricSpacePi.{0, u1} (Fin n) (fun (j : Fin n) => α (FunLike.coe.{1, 1, 1} (OrderEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin n) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))))) (Fin n) (fun (_x : Fin n) => (fun (x._@.Mathlib.Order.Hom.Lattice._hyg.494 : Fin n) => Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) _x) (InfHomClass.toFunLike.{0, 0, 0} (OrderEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin n) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))))) (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (Lattice.toInf.{0} (Fin n) (DistribLattice.toLattice.{0} (Fin n) (instDistribLattice.{0} (Fin n) (Fin.instLinearOrderFin n)))) (Lattice.toInf.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (Fin.instLatticeFinHAddNatInstHAddInstAddNatOfNat n)) (LatticeHomClass.toInfHomClass.{0, 0, 0} (OrderEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin n) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))))) (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (DistribLattice.toLattice.{0} (Fin n) (instDistribLattice.{0} (Fin n) (Fin.instLinearOrderFin n))) (Fin.instLatticeFinHAddNatInstHAddInstAddNatOfNat n) (OrderHomClass.toLatticeHomClass.{0, 0, 0} (OrderEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin n) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))))) (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (Fin.instLinearOrderFin n) (Fin.instLatticeFinHAddNatInstHAddInstAddNatOfNat n) (RelEmbedding.instRelHomClassRelEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.680 : Fin n) (x._@.Mathlib.Order.Hom.Basic._hyg.682 : Fin n) => LE.le.{0} (Fin n) (instLEFin n) x._@.Mathlib.Order.Hom.Basic._hyg.680 x._@.Mathlib.Order.Hom.Basic._hyg.682) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.695 : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (x._@.Mathlib.Order.Hom.Basic._hyg.697 : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) => LE.le.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) x._@.Mathlib.Order.Hom.Basic._hyg.695 x._@.Mathlib.Order.Hom.Basic._hyg.697))))) (Fin.succAbove n i) j)) (Fin.fintype n) (fun (b : Fin n) => _inst_4 (FunLike.coe.{1, 1, 1} (OrderEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin n) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))))) (Fin n) (fun (_x : Fin n) => (fun (x._@.Mathlib.Order.Hom.Lattice._hyg.494 : Fin n) => Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) _x) (InfHomClass.toFunLike.{0, 0, 0} (OrderEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin n) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))))) (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (Lattice.toInf.{0} (Fin n) (DistribLattice.toLattice.{0} (Fin n) (instDistribLattice.{0} (Fin n) (Fin.instLinearOrderFin n)))) (Lattice.toInf.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (Fin.instLatticeFinHAddNatInstHAddInstAddNatOfNat n)) (LatticeHomClass.toInfHomClass.{0, 0, 0} (OrderEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin n) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))))) (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (DistribLattice.toLattice.{0} (Fin n) (instDistribLattice.{0} (Fin n) (Fin.instLinearOrderFin n))) (Fin.instLatticeFinHAddNatInstHAddInstAddNatOfNat n) (OrderHomClass.toLatticeHomClass.{0, 0, 0} (OrderEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin n) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))))) (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (Fin.instLinearOrderFin n) (Fin.instLatticeFinHAddNatInstHAddInstAddNatOfNat n) (RelEmbedding.instRelHomClassRelEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.680 : Fin n) (x._@.Mathlib.Order.Hom.Basic._hyg.682 : Fin n) => LE.le.{0} (Fin n) (instLEFin n) x._@.Mathlib.Order.Hom.Basic._hyg.680 x._@.Mathlib.Order.Hom.Basic._hyg.682) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.695 : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (x._@.Mathlib.Order.Hom.Basic._hyg.697 : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) => LE.le.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) x._@.Mathlib.Order.Hom.Basic._hyg.695 x._@.Mathlib.Order.Hom.Basic._hyg.697))))) (Fin.succAbove n i) b)))) f g))
 Case conversion may be inaccurate. Consider using '#align fin.dist_insert_nth_insert_nth Fin.dist_insertNth_insertNthₓ'. -/
@@ -4591,7 +4599,7 @@ theorem diam_triple :
 
 /- warning: metric.ediam_le_of_forall_dist_le -> Metric.ediam_le_of_forall_dist_le is a dubious translation:
 lean 3 declaration is
-  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {s : Set.{u1} α} {C : Real}, (forall (x : α), (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) x s) -> (forall (y : α), (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) y s) -> (LE.le.{0} Real Real.hasLe (Dist.dist.{u1} α (PseudoMetricSpace.toHasDist.{u1} α _inst_1) x y) C))) -> (LE.le.{0} ENNReal (Preorder.toLE.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (CompleteSemilatticeInf.toPartialOrder.{0} ENNReal (CompleteLattice.toCompleteSemilatticeInf.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))))) (EMetric.diam.{u1} α (PseudoMetricSpace.toPseudoEMetricSpace.{u1} α _inst_1) s) (ENNReal.ofReal C))
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {s : Set.{u1} α} {C : Real}, (forall (x : α), (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) x s) -> (forall (y : α), (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) y s) -> (LE.le.{0} Real Real.hasLe (Dist.dist.{u1} α (PseudoMetricSpace.toHasDist.{u1} α _inst_1) x y) C))) -> (LE.le.{0} ENNReal (Preorder.toHasLe.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (CompleteSemilatticeInf.toPartialOrder.{0} ENNReal (CompleteLattice.toCompleteSemilatticeInf.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))))) (EMetric.diam.{u1} α (PseudoMetricSpace.toPseudoEMetricSpace.{u1} α _inst_1) s) (ENNReal.ofReal C))
 but is expected to have type
   forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {s : Set.{u1} α} {C : Real}, (forall (x : α), (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) x s) -> (forall (y : α), (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) y s) -> (LE.le.{0} Real Real.instLEReal (Dist.dist.{u1} α (PseudoMetricSpace.toDist.{u1} α _inst_1) x y) C))) -> (LE.le.{0} ENNReal (Preorder.toLE.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (CompleteSemilatticeInf.toPartialOrder.{0} ENNReal (CompleteLattice.toCompleteSemilatticeInf.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal))))) (EMetric.diam.{u1} α (PseudoMetricSpace.toPseudoEMetricSpace.{u1} α _inst_1) s) (ENNReal.ofReal C))
 Case conversion may be inaccurate. Consider using '#align metric.ediam_le_of_forall_dist_le Metric.ediam_le_of_forall_dist_leₓ'. -/
@@ -4877,7 +4885,7 @@ end Diam
 
 /- warning: metric.exists_local_min_mem_ball -> Metric.exists_local_min_mem_ball is a dubious translation:
 lean 3 declaration is
-  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : PseudoMetricSpace.{u1} α] [_inst_2 : ProperSpace.{u1} α _inst_1] [_inst_3 : TopologicalSpace.{u2} β] [_inst_4 : ConditionallyCompleteLinearOrder.{u2} β] [_inst_5 : OrderTopology.{u2} β _inst_3 (PartialOrder.toPreorder.{u2} β (SemilatticeInf.toPartialOrder.{u2} β (Lattice.toSemilatticeInf.{u2} β (ConditionallyCompleteLattice.toLattice.{u2} β (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u2} β _inst_4)))))] {f : α -> β} {a : α} {z : α} {r : Real}, (ContinuousOn.{u1, u2} α β (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) _inst_3 f (Metric.closedBall.{u1} α _inst_1 a r)) -> (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) z (Metric.closedBall.{u1} α _inst_1 a r)) -> (forall (z' : α), (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) z' (Metric.sphere.{u1} α _inst_1 a r)) -> (LT.lt.{u2} β (Preorder.toLT.{u2} β (PartialOrder.toPreorder.{u2} β (SemilatticeInf.toPartialOrder.{u2} β (Lattice.toSemilatticeInf.{u2} β (ConditionallyCompleteLattice.toLattice.{u2} β (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u2} β _inst_4)))))) (f z) (f z'))) -> (Exists.{succ u1} α (fun (z : α) => Exists.{0} (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) z (Metric.ball.{u1} α _inst_1 a r)) (fun (H : Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) z (Metric.ball.{u1} α _inst_1 a r)) => IsLocalMin.{u1, u2} α β (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) (PartialOrder.toPreorder.{u2} β (SemilatticeInf.toPartialOrder.{u2} β (Lattice.toSemilatticeInf.{u2} β (ConditionallyCompleteLattice.toLattice.{u2} β (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u2} β _inst_4))))) f z)))
+  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : PseudoMetricSpace.{u1} α] [_inst_2 : ProperSpace.{u1} α _inst_1] [_inst_3 : TopologicalSpace.{u2} β] [_inst_4 : ConditionallyCompleteLinearOrder.{u2} β] [_inst_5 : OrderTopology.{u2} β _inst_3 (PartialOrder.toPreorder.{u2} β (SemilatticeInf.toPartialOrder.{u2} β (Lattice.toSemilatticeInf.{u2} β (ConditionallyCompleteLattice.toLattice.{u2} β (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u2} β _inst_4)))))] {f : α -> β} {a : α} {z : α} {r : Real}, (ContinuousOn.{u1, u2} α β (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) _inst_3 f (Metric.closedBall.{u1} α _inst_1 a r)) -> (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) z (Metric.closedBall.{u1} α _inst_1 a r)) -> (forall (z' : α), (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) z' (Metric.sphere.{u1} α _inst_1 a r)) -> (LT.lt.{u2} β (Preorder.toHasLt.{u2} β (PartialOrder.toPreorder.{u2} β (SemilatticeInf.toPartialOrder.{u2} β (Lattice.toSemilatticeInf.{u2} β (ConditionallyCompleteLattice.toLattice.{u2} β (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u2} β _inst_4)))))) (f z) (f z'))) -> (Exists.{succ u1} α (fun (z : α) => Exists.{0} (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) z (Metric.ball.{u1} α _inst_1 a r)) (fun (H : Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) z (Metric.ball.{u1} α _inst_1 a r)) => IsLocalMin.{u1, u2} α β (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) (PartialOrder.toPreorder.{u2} β (SemilatticeInf.toPartialOrder.{u2} β (Lattice.toSemilatticeInf.{u2} β (ConditionallyCompleteLattice.toLattice.{u2} β (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u2} β _inst_4))))) f z)))
 but is expected to have type
   forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : PseudoMetricSpace.{u1} α] [_inst_2 : ProperSpace.{u1} α _inst_1] [_inst_3 : TopologicalSpace.{u2} β] [_inst_4 : ConditionallyCompleteLinearOrder.{u2} β] [_inst_5 : OrderTopology.{u2} β _inst_3 (PartialOrder.toPreorder.{u2} β (SemilatticeInf.toPartialOrder.{u2} β (Lattice.toSemilatticeInf.{u2} β (ConditionallyCompleteLattice.toLattice.{u2} β (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u2} β _inst_4)))))] {f : α -> β} {a : α} {z : α} {r : Real}, (ContinuousOn.{u1, u2} α β (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) _inst_3 f (Metric.closedBall.{u1} α _inst_1 a r)) -> (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) z (Metric.closedBall.{u1} α _inst_1 a r)) -> (forall (z' : α), (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) z' (Metric.sphere.{u1} α _inst_1 a r)) -> (LT.lt.{u2} β (Preorder.toLT.{u2} β (PartialOrder.toPreorder.{u2} β (SemilatticeInf.toPartialOrder.{u2} β (Lattice.toSemilatticeInf.{u2} β (ConditionallyCompleteLattice.toLattice.{u2} β (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u2} β _inst_4)))))) (f z) (f z'))) -> (Exists.{succ u1} α (fun (z : α) => And (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) z (Metric.ball.{u1} α _inst_1 a r)) (IsLocalMin.{u1, u2} α β (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) (PartialOrder.toPreorder.{u2} β (SemilatticeInf.toPartialOrder.{u2} β (Lattice.toSemilatticeInf.{u2} β (ConditionallyCompleteLattice.toLattice.{u2} β (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u2} β _inst_4))))) f z)))
 Case conversion may be inaccurate. Consider using '#align metric.exists_local_min_mem_ball Metric.exists_local_min_mem_ballₓ'. -/
@@ -4909,7 +4917,7 @@ end Tactic
 
 /- warning: comap_dist_right_at_top_le_cocompact -> comap_dist_right_atTop_le_cocompact is a dubious translation:
 lean 3 declaration is
-  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] (x : α), LE.le.{u1} (Filter.{u1} α) (Preorder.toLE.{u1} (Filter.{u1} α) (PartialOrder.toPreorder.{u1} (Filter.{u1} α) (Filter.partialOrder.{u1} α))) (Filter.comap.{u1, 0} α Real (fun (y : α) => Dist.dist.{u1} α (PseudoMetricSpace.toHasDist.{u1} α _inst_1) y x) (Filter.atTop.{0} Real Real.preorder)) (Filter.cocompact.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)))
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] (x : α), LE.le.{u1} (Filter.{u1} α) (Preorder.toHasLe.{u1} (Filter.{u1} α) (PartialOrder.toPreorder.{u1} (Filter.{u1} α) (Filter.partialOrder.{u1} α))) (Filter.comap.{u1, 0} α Real (fun (y : α) => Dist.dist.{u1} α (PseudoMetricSpace.toHasDist.{u1} α _inst_1) y x) (Filter.atTop.{0} Real Real.preorder)) (Filter.cocompact.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)))
 but is expected to have type
   forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] (x : α), LE.le.{u1} (Filter.{u1} α) (Preorder.toLE.{u1} (Filter.{u1} α) (PartialOrder.toPreorder.{u1} (Filter.{u1} α) (Filter.instPartialOrderFilter.{u1} α))) (Filter.comap.{u1, 0} α Real (fun (y : α) => Dist.dist.{u1} α (PseudoMetricSpace.toDist.{u1} α _inst_1) y x) (Filter.atTop.{0} Real Real.instPreorderReal)) (Filter.cocompact.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)))
 Case conversion may be inaccurate. Consider using '#align comap_dist_right_at_top_le_cocompact comap_dist_right_atTop_le_cocompactₓ'. -/
@@ -4923,7 +4931,7 @@ theorem comap_dist_right_atTop_le_cocompact (x : α) :
 
 /- warning: comap_dist_left_at_top_le_cocompact -> comap_dist_left_atTop_le_cocompact is a dubious translation:
 lean 3 declaration is
-  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] (x : α), LE.le.{u1} (Filter.{u1} α) (Preorder.toLE.{u1} (Filter.{u1} α) (PartialOrder.toPreorder.{u1} (Filter.{u1} α) (Filter.partialOrder.{u1} α))) (Filter.comap.{u1, 0} α Real (Dist.dist.{u1} α (PseudoMetricSpace.toHasDist.{u1} α _inst_1) x) (Filter.atTop.{0} Real Real.preorder)) (Filter.cocompact.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)))
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] (x : α), LE.le.{u1} (Filter.{u1} α) (Preorder.toHasLe.{u1} (Filter.{u1} α) (PartialOrder.toPreorder.{u1} (Filter.{u1} α) (Filter.partialOrder.{u1} α))) (Filter.comap.{u1, 0} α Real (Dist.dist.{u1} α (PseudoMetricSpace.toHasDist.{u1} α _inst_1) x) (Filter.atTop.{0} Real Real.preorder)) (Filter.cocompact.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)))
 but is expected to have type
   forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] (x : α), LE.le.{u1} (Filter.{u1} α) (Preorder.toLE.{u1} (Filter.{u1} α) (PartialOrder.toPreorder.{u1} (Filter.{u1} α) (Filter.instPartialOrderFilter.{u1} α))) (Filter.comap.{u1, 0} α Real (Dist.dist.{u1} α (PseudoMetricSpace.toDist.{u1} α _inst_1) x) (Filter.atTop.{0} Real Real.instPreorderReal)) (Filter.cocompact.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)))
 Case conversion may be inaccurate. Consider using '#align comap_dist_left_at_top_le_cocompact comap_dist_left_atTop_le_cocompactₓ'. -/

mathlib commit https://github.com/leanprover-community/mathlib/commit/7ad820c4997738e2f542f8a20f32911f52020e26

@@ -1614,7 +1614,7 @@ theorem totallyBounded_of_finite_discretization {s : Set α}
 lean 3 declaration is
   forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {s : Set.{u1} α}, (TotallyBounded.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1) s) -> (forall (ε : Real), (GT.gt.{0} Real Real.hasLt ε (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero)))) -> (Exists.{succ u1} (Set.{u1} α) (fun (t : Set.{u1} α) => Exists.{0} (HasSubset.Subset.{u1} (Set.{u1} α) (Set.hasSubset.{u1} α) t s) (fun (H : HasSubset.Subset.{u1} (Set.{u1} α) (Set.hasSubset.{u1} α) t s) => And (Set.Finite.{u1} α t) (HasSubset.Subset.{u1} (Set.{u1} α) (Set.hasSubset.{u1} α) s (Set.iUnion.{u1, succ u1} α α (fun (y : α) => Set.iUnion.{u1, 0} α (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) y t) (fun (H : Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) y t) => Metric.ball.{u1} α _inst_1 y ε))))))))
 but is expected to have type
-  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {s : Set.{u1} α}, (TotallyBounded.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1) s) -> (forall (ε : Real), (GT.gt.{0} Real Real.instLTReal ε (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal))) -> (Exists.{succ u1} (Set.{u1} α) (fun (t : Set.{u1} α) => And (HasSubset.Subset.{u1} (Set.{u1} α) (Set.instHasSubsetSet.{u1} α) t s) (And (Set.Finite.{u1} α t) (HasSubset.Subset.{u1} (Set.{u1} α) (Set.instHasSubsetSet.{u1} α) s (Set.iUnion.{u1, succ u1} α α (fun (y : α) => Set.iUnion.{u1, 0} α (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) y t) (fun (h._@.Mathlib.Topology.MetricSpace.Basic._hyg.10766 : Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) y t) => Metric.ball.{u1} α _inst_1 y ε))))))))
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {s : Set.{u1} α}, (TotallyBounded.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1) s) -> (forall (ε : Real), (GT.gt.{0} Real Real.instLTReal ε (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal))) -> (Exists.{succ u1} (Set.{u1} α) (fun (t : Set.{u1} α) => And (HasSubset.Subset.{u1} (Set.{u1} α) (Set.instHasSubsetSet.{u1} α) t s) (And (Set.Finite.{u1} α t) (HasSubset.Subset.{u1} (Set.{u1} α) (Set.instHasSubsetSet.{u1} α) s (Set.iUnion.{u1, succ u1} α α (fun (y : α) => Set.iUnion.{u1, 0} α (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) y t) (fun (h._@.Mathlib.Topology.MetricSpace.Basic._hyg.10758 : Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) y t) => Metric.ball.{u1} α _inst_1 y ε))))))))
 Case conversion may be inaccurate. Consider using '#align metric.finite_approx_of_totally_bounded Metric.finite_approx_of_totallyBoundedₓ'. -/
 /- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (t «expr ⊆ » s) -/
 theorem finite_approx_of_totallyBounded {s : Set α} (hs : TotallyBounded s) :
@@ -2070,22 +2070,22 @@ theorem tendsto_atTop [Nonempty β] [SemilatticeSup β] {u : β → α} {a : α}
     rfl
 #align metric.tendsto_at_top Metric.tendsto_atTop
 
-/- warning: metric.tendsto_at_top' -> Metric.tendsto_at_top' is a dubious translation:
+/- warning: metric.tendsto_at_top' -> Metric.tendsto_atTop' is a dubious translation:
 lean 3 declaration is
   forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : PseudoMetricSpace.{u1} α] [_inst_2 : Nonempty.{succ u2} β] [_inst_3 : SemilatticeSup.{u2} β] [_inst_4 : NoMaxOrder.{u2} β (Preorder.toLT.{u2} β (PartialOrder.toPreorder.{u2} β (SemilatticeSup.toPartialOrder.{u2} β _inst_3)))] {u : β -> α} {a : α}, Iff (Filter.Tendsto.{u2, u1} β α u (Filter.atTop.{u2} β (PartialOrder.toPreorder.{u2} β (SemilatticeSup.toPartialOrder.{u2} β _inst_3))) (nhds.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) a)) (forall (ε : Real), (GT.gt.{0} Real Real.hasLt ε (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero)))) -> (Exists.{succ u2} β (fun (N : β) => forall (n : β), (GT.gt.{u2} β (Preorder.toLT.{u2} β (PartialOrder.toPreorder.{u2} β (SemilatticeSup.toPartialOrder.{u2} β _inst_3))) n N) -> (LT.lt.{0} Real Real.hasLt (Dist.dist.{u1} α (PseudoMetricSpace.toHasDist.{u1} α _inst_1) (u n) a) ε))))
 but is expected to have type
   forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : PseudoMetricSpace.{u1} α] [_inst_2 : Nonempty.{succ u2} β] [_inst_3 : SemilatticeSup.{u2} β] [_inst_4 : NoMaxOrder.{u2} β (Preorder.toLT.{u2} β (PartialOrder.toPreorder.{u2} β (SemilatticeSup.toPartialOrder.{u2} β _inst_3)))] {u : β -> α} {a : α}, Iff (Filter.Tendsto.{u2, u1} β α u (Filter.atTop.{u2} β (PartialOrder.toPreorder.{u2} β (SemilatticeSup.toPartialOrder.{u2} β _inst_3))) (nhds.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) a)) (forall (ε : Real), (GT.gt.{0} Real Real.instLTReal ε (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal))) -> (Exists.{succ u2} β (fun (N : β) => forall (n : β), (GT.gt.{u2} β (Preorder.toLT.{u2} β (PartialOrder.toPreorder.{u2} β (SemilatticeSup.toPartialOrder.{u2} β _inst_3))) n N) -> (LT.lt.{0} Real Real.instLTReal (Dist.dist.{u1} α (PseudoMetricSpace.toDist.{u1} α _inst_1) (u n) a) ε))))
-Case conversion may be inaccurate. Consider using '#align metric.tendsto_at_top' Metric.tendsto_at_top'ₓ'. -/
+Case conversion may be inaccurate. Consider using '#align metric.tendsto_at_top' Metric.tendsto_atTop'ₓ'. -/
 /-- A variant of `tendsto_at_top` that
 uses `∃ N, ∀ n > N, ...` rather than `∃ N, ∀ n ≥ N, ...`
 -/
-theorem tendsto_at_top' [Nonempty β] [SemilatticeSup β] [NoMaxOrder β] {u : β → α} {a : α} :
+theorem tendsto_atTop' [Nonempty β] [SemilatticeSup β] [NoMaxOrder β] {u : β → α} {a : α} :
     Tendsto u atTop (𝓝 a) ↔ ∀ ε > 0, ∃ N, ∀ n > N, dist (u n) a < ε :=
   (atTop_basis_Ioi.tendsto_iffₓ nhds_basis_ball).trans <|
     by
     simp only [exists_prop, true_and_iff]
     rfl
-#align metric.tendsto_at_top' Metric.tendsto_at_top'
+#align metric.tendsto_at_top' Metric.tendsto_atTop'
 
 /- warning: metric.is_open_singleton_iff -> Metric.isOpen_singleton_iff is a dubious translation:
 lean 3 declaration is
@@ -3816,7 +3816,7 @@ section Compact
 lean 3 declaration is
   forall {α : Type.{u1}} [_inst_2 : PseudoMetricSpace.{u1} α] {s : Set.{u1} α}, (IsCompact.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_2)) s) -> (forall {e : Real}, (LT.lt.{0} Real Real.hasLt (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero))) e) -> (Exists.{succ u1} (Set.{u1} α) (fun (t : Set.{u1} α) => Exists.{0} (HasSubset.Subset.{u1} (Set.{u1} α) (Set.hasSubset.{u1} α) t s) (fun (H : HasSubset.Subset.{u1} (Set.{u1} α) (Set.hasSubset.{u1} α) t s) => And (Set.Finite.{u1} α t) (HasSubset.Subset.{u1} (Set.{u1} α) (Set.hasSubset.{u1} α) s (Set.iUnion.{u1, succ u1} α α (fun (x : α) => Set.iUnion.{u1, 0} α (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) x t) (fun (H : Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) x t) => Metric.ball.{u1} α _inst_2 x e))))))))
 but is expected to have type
-  forall {α : Type.{u1}} [_inst_2 : PseudoMetricSpace.{u1} α] {s : Set.{u1} α}, (IsCompact.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_2)) s) -> (forall {e : Real}, (LT.lt.{0} Real Real.instLTReal (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal)) e) -> (Exists.{succ u1} (Set.{u1} α) (fun (t : Set.{u1} α) => And (HasSubset.Subset.{u1} (Set.{u1} α) (Set.instHasSubsetSet.{u1} α) t s) (And (Set.Finite.{u1} α t) (HasSubset.Subset.{u1} (Set.{u1} α) (Set.instHasSubsetSet.{u1} α) s (Set.iUnion.{u1, succ u1} α α (fun (x : α) => Set.iUnion.{u1, 0} α (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) x t) (fun (h._@.Mathlib.Topology.MetricSpace.Basic._hyg.25516 : Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) x t) => Metric.ball.{u1} α _inst_2 x e))))))))
+  forall {α : Type.{u1}} [_inst_2 : PseudoMetricSpace.{u1} α] {s : Set.{u1} α}, (IsCompact.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_2)) s) -> (forall {e : Real}, (LT.lt.{0} Real Real.instLTReal (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal)) e) -> (Exists.{succ u1} (Set.{u1} α) (fun (t : Set.{u1} α) => And (HasSubset.Subset.{u1} (Set.{u1} α) (Set.instHasSubsetSet.{u1} α) t s) (And (Set.Finite.{u1} α t) (HasSubset.Subset.{u1} (Set.{u1} α) (Set.instHasSubsetSet.{u1} α) s (Set.iUnion.{u1, succ u1} α α (fun (x : α) => Set.iUnion.{u1, 0} α (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) x t) (fun (h._@.Mathlib.Topology.MetricSpace.Basic._hyg.25536 : Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) x t) => Metric.ball.{u1} α _inst_2 x e))))))))
 Case conversion may be inaccurate. Consider using '#align finite_cover_balls_of_compact finite_cover_balls_of_compactₓ'. -/
 /- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (t «expr ⊆ » s) -/
 /-- Any compact set in a pseudometric space can be covered by finitely many balls of a given
@@ -3836,7 +3836,7 @@ theorem finite_cover_balls_of_compact {α : Type u} [PseudoMetricSpace α] {s :
 lean 3 declaration is
   forall {α : Type.{u1}} [_inst_2 : PseudoMetricSpace.{u1} α] {s : Set.{u1} α}, (IsCompact.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_2)) s) -> (forall {e : Real}, (LT.lt.{0} Real Real.hasLt (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero))) e) -> (Exists.{succ u1} (Set.{u1} α) (fun (t : Set.{u1} α) => Exists.{0} (HasSubset.Subset.{u1} (Set.{u1} α) (Set.hasSubset.{u1} α) t s) (fun (H : HasSubset.Subset.{u1} (Set.{u1} α) (Set.hasSubset.{u1} α) t s) => And (Set.Finite.{u1} α t) (HasSubset.Subset.{u1} (Set.{u1} α) (Set.hasSubset.{u1} α) s (Set.iUnion.{u1, succ u1} α α (fun (x : α) => Set.iUnion.{u1, 0} α (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) x t) (fun (H : Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) x t) => Metric.ball.{u1} α _inst_2 x e))))))))
 but is expected to have type
-  forall {α : Type.{u1}} [_inst_2 : PseudoMetricSpace.{u1} α] {s : Set.{u1} α}, (IsCompact.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_2)) s) -> (forall {e : Real}, (LT.lt.{0} Real Real.instLTReal (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal)) e) -> (Exists.{succ u1} (Set.{u1} α) (fun (t : Set.{u1} α) => And (HasSubset.Subset.{u1} (Set.{u1} α) (Set.instHasSubsetSet.{u1} α) t s) (And (Set.Finite.{u1} α t) (HasSubset.Subset.{u1} (Set.{u1} α) (Set.instHasSubsetSet.{u1} α) s (Set.iUnion.{u1, succ u1} α α (fun (x : α) => Set.iUnion.{u1, 0} α (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) x t) (fun (h._@.Mathlib.Topology.MetricSpace.Basic._hyg.25516 : Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) x t) => Metric.ball.{u1} α _inst_2 x e))))))))
+  forall {α : Type.{u1}} [_inst_2 : PseudoMetricSpace.{u1} α] {s : Set.{u1} α}, (IsCompact.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_2)) s) -> (forall {e : Real}, (LT.lt.{0} Real Real.instLTReal (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal)) e) -> (Exists.{succ u1} (Set.{u1} α) (fun (t : Set.{u1} α) => And (HasSubset.Subset.{u1} (Set.{u1} α) (Set.instHasSubsetSet.{u1} α) t s) (And (Set.Finite.{u1} α t) (HasSubset.Subset.{u1} (Set.{u1} α) (Set.instHasSubsetSet.{u1} α) s (Set.iUnion.{u1, succ u1} α α (fun (x : α) => Set.iUnion.{u1, 0} α (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) x t) (fun (h._@.Mathlib.Topology.MetricSpace.Basic._hyg.25536 : Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) x t) => Metric.ball.{u1} α _inst_2 x e))))))))
 Case conversion may be inaccurate. Consider using '#align is_compact.finite_cover_balls IsCompact.finite_cover_ballsₓ'. -/
 alias finite_cover_balls_of_compact ← IsCompact.finite_cover_balls
 #align is_compact.finite_cover_balls IsCompact.finite_cover_balls

mathlib commit https://github.com/leanprover-community/mathlib/commit/e3fb84046afd187b710170887195d50bada934ee

@@ -800,27 +800,27 @@ theorem ball_eq_ball' (ε : ℝ) (x : α) :
   simp [dist_comm, UniformSpace.ball]
 #align metric.ball_eq_ball' Metric.ball_eq_ball'
 
-/- warning: metric.Union_ball_nat -> Metric.unionᵢ_ball_nat is a dubious translation:
+/- warning: metric.Union_ball_nat -> Metric.iUnion_ball_nat is a dubious translation:
 lean 3 declaration is
-  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] (x : α), Eq.{succ u1} (Set.{u1} α) (Set.unionᵢ.{u1, 1} α Nat (fun (n : Nat) => Metric.ball.{u1} α _inst_1 x ((fun (a : Type) (b : Type) [self : HasLiftT.{1, 1} a b] => self.0) Nat Real (HasLiftT.mk.{1, 1} Nat Real (CoeTCₓ.coe.{1, 1} Nat Real (Nat.castCoe.{0} Real Real.hasNatCast))) n))) (Set.univ.{u1} α)
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] (x : α), Eq.{succ u1} (Set.{u1} α) (Set.iUnion.{u1, 1} α Nat (fun (n : Nat) => Metric.ball.{u1} α _inst_1 x ((fun (a : Type) (b : Type) [self : HasLiftT.{1, 1} a b] => self.0) Nat Real (HasLiftT.mk.{1, 1} Nat Real (CoeTCₓ.coe.{1, 1} Nat Real (Nat.castCoe.{0} Real Real.hasNatCast))) n))) (Set.univ.{u1} α)
 but is expected to have type
-  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] (x : α), Eq.{succ u1} (Set.{u1} α) (Set.unionᵢ.{u1, 1} α Nat (fun (n : Nat) => Metric.ball.{u1} α _inst_1 x (Nat.cast.{0} Real Real.natCast n))) (Set.univ.{u1} α)
-Case conversion may be inaccurate. Consider using '#align metric.Union_ball_nat Metric.unionᵢ_ball_natₓ'. -/
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] (x : α), Eq.{succ u1} (Set.{u1} α) (Set.iUnion.{u1, 1} α Nat (fun (n : Nat) => Metric.ball.{u1} α _inst_1 x (Nat.cast.{0} Real Real.natCast n))) (Set.univ.{u1} α)
+Case conversion may be inaccurate. Consider using '#align metric.Union_ball_nat Metric.iUnion_ball_natₓ'. -/
 @[simp]
-theorem unionᵢ_ball_nat (x : α) : (⋃ n : ℕ, ball x n) = univ :=
-  unionᵢ_eq_univ_iff.2 fun y => exists_nat_gt (dist y x)
-#align metric.Union_ball_nat Metric.unionᵢ_ball_nat
+theorem iUnion_ball_nat (x : α) : (⋃ n : ℕ, ball x n) = univ :=
+  iUnion_eq_univ_iff.2 fun y => exists_nat_gt (dist y x)
+#align metric.Union_ball_nat Metric.iUnion_ball_nat
 
-/- warning: metric.Union_ball_nat_succ -> Metric.unionᵢ_ball_nat_succ is a dubious translation:
+/- warning: metric.Union_ball_nat_succ -> Metric.iUnion_ball_nat_succ is a dubious translation:
 lean 3 declaration is
-  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] (x : α), Eq.{succ u1} (Set.{u1} α) (Set.unionᵢ.{u1, 1} α Nat (fun (n : Nat) => Metric.ball.{u1} α _inst_1 x (HAdd.hAdd.{0, 0, 0} Real Real Real (instHAdd.{0} Real Real.hasAdd) ((fun (a : Type) (b : Type) [self : HasLiftT.{1, 1} a b] => self.0) Nat Real (HasLiftT.mk.{1, 1} Nat Real (CoeTCₓ.coe.{1, 1} Nat Real (Nat.castCoe.{0} Real Real.hasNatCast))) n) (OfNat.ofNat.{0} Real 1 (OfNat.mk.{0} Real 1 (One.one.{0} Real Real.hasOne)))))) (Set.univ.{u1} α)
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] (x : α), Eq.{succ u1} (Set.{u1} α) (Set.iUnion.{u1, 1} α Nat (fun (n : Nat) => Metric.ball.{u1} α _inst_1 x (HAdd.hAdd.{0, 0, 0} Real Real Real (instHAdd.{0} Real Real.hasAdd) ((fun (a : Type) (b : Type) [self : HasLiftT.{1, 1} a b] => self.0) Nat Real (HasLiftT.mk.{1, 1} Nat Real (CoeTCₓ.coe.{1, 1} Nat Real (Nat.castCoe.{0} Real Real.hasNatCast))) n) (OfNat.ofNat.{0} Real 1 (OfNat.mk.{0} Real 1 (One.one.{0} Real Real.hasOne)))))) (Set.univ.{u1} α)
 but is expected to have type
-  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] (x : α), Eq.{succ u1} (Set.{u1} α) (Set.unionᵢ.{u1, 1} α Nat (fun (n : Nat) => Metric.ball.{u1} α _inst_1 x (HAdd.hAdd.{0, 0, 0} Real Real Real (instHAdd.{0} Real Real.instAddReal) (Nat.cast.{0} Real Real.natCast n) (OfNat.ofNat.{0} Real 1 (One.toOfNat1.{0} Real Real.instOneReal))))) (Set.univ.{u1} α)
-Case conversion may be inaccurate. Consider using '#align metric.Union_ball_nat_succ Metric.unionᵢ_ball_nat_succₓ'. -/
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] (x : α), Eq.{succ u1} (Set.{u1} α) (Set.iUnion.{u1, 1} α Nat (fun (n : Nat) => Metric.ball.{u1} α _inst_1 x (HAdd.hAdd.{0, 0, 0} Real Real Real (instHAdd.{0} Real Real.instAddReal) (Nat.cast.{0} Real Real.natCast n) (OfNat.ofNat.{0} Real 1 (One.toOfNat1.{0} Real Real.instOneReal))))) (Set.univ.{u1} α)
+Case conversion may be inaccurate. Consider using '#align metric.Union_ball_nat_succ Metric.iUnion_ball_nat_succₓ'. -/
 @[simp]
-theorem unionᵢ_ball_nat_succ (x : α) : (⋃ n : ℕ, ball x (n + 1)) = univ :=
-  unionᵢ_eq_univ_iff.2 fun y => (exists_nat_gt (dist y x)).imp fun n hn => hn.trans (lt_add_one _)
-#align metric.Union_ball_nat_succ Metric.unionᵢ_ball_nat_succ
+theorem iUnion_ball_nat_succ (x : α) : (⋃ n : ℕ, ball x (n + 1)) = univ :=
+  iUnion_eq_univ_iff.2 fun y => (exists_nat_gt (dist y x)).imp fun n hn => hn.trans (lt_add_one _)
+#align metric.Union_ball_nat_succ Metric.iUnion_ball_nat_succ
 
 #print Metric.closedBall /-
 /-- `closed_ball x ε` is the set of all points `y` with `dist y x ≤ ε` -/
@@ -1068,9 +1068,9 @@ theorem ball_subset_ball (h : ε₁ ≤ ε₂) : ball x ε₁ ⊆ ball x ε₂ :
 
 /- warning: metric.closed_ball_eq_bInter_ball -> Metric.closedBall_eq_bInter_ball is a dubious translation:
 lean 3 declaration is
-  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {x : α} {ε : Real}, Eq.{succ u1} (Set.{u1} α) (Metric.closedBall.{u1} α _inst_1 x ε) (Set.interᵢ.{u1, 1} α Real (fun (δ : Real) => Set.interᵢ.{u1, 0} α (GT.gt.{0} Real Real.hasLt δ ε) (fun (H : GT.gt.{0} Real Real.hasLt δ ε) => Metric.ball.{u1} α _inst_1 x δ)))
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {x : α} {ε : Real}, Eq.{succ u1} (Set.{u1} α) (Metric.closedBall.{u1} α _inst_1 x ε) (Set.iInter.{u1, 1} α Real (fun (δ : Real) => Set.iInter.{u1, 0} α (GT.gt.{0} Real Real.hasLt δ ε) (fun (H : GT.gt.{0} Real Real.hasLt δ ε) => Metric.ball.{u1} α _inst_1 x δ)))
 but is expected to have type
-  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {x : α} {ε : Real}, Eq.{succ u1} (Set.{u1} α) (Metric.closedBall.{u1} α _inst_1 x ε) (Set.interᵢ.{u1, 1} α Real (fun (δ : Real) => Set.interᵢ.{u1, 0} α (GT.gt.{0} Real Real.instLTReal δ ε) (fun (H : GT.gt.{0} Real Real.instLTReal δ ε) => Metric.ball.{u1} α _inst_1 x δ)))
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {x : α} {ε : Real}, Eq.{succ u1} (Set.{u1} α) (Metric.closedBall.{u1} α _inst_1 x ε) (Set.iInter.{u1, 1} α Real (fun (δ : Real) => Set.iInter.{u1, 0} α (GT.gt.{0} Real Real.instLTReal δ ε) (fun (H : GT.gt.{0} Real Real.instLTReal δ ε) => Metric.ball.{u1} α _inst_1 x δ)))
 Case conversion may be inaccurate. Consider using '#align metric.closed_ball_eq_bInter_ball Metric.closedBall_eq_bInter_ballₓ'. -/
 theorem closedBall_eq_bInter_ball : closedBall x ε = ⋂ δ > ε, ball x δ := by
   ext y <;> rw [mem_closed_ball, ← forall_lt_iff_le', mem_Inter₂] <;> rfl
@@ -1195,26 +1195,26 @@ theorem dist_lt_add_of_nonempty_ball_inter_ball (h : (ball x ε₁ ∩ ball y ε
     h.mono (inter_subset_inter ball_subset_closedBall Subset.rfl)
 #align metric.dist_lt_add_of_nonempty_ball_inter_ball Metric.dist_lt_add_of_nonempty_ball_inter_ball
 
-/- warning: metric.Union_closed_ball_nat -> Metric.unionᵢ_closedBall_nat is a dubious translation:
+/- warning: metric.Union_closed_ball_nat -> Metric.iUnion_closedBall_nat is a dubious translation:
 lean 3 declaration is
-  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] (x : α), Eq.{succ u1} (Set.{u1} α) (Set.unionᵢ.{u1, 1} α Nat (fun (n : Nat) => Metric.closedBall.{u1} α _inst_1 x ((fun (a : Type) (b : Type) [self : HasLiftT.{1, 1} a b] => self.0) Nat Real (HasLiftT.mk.{1, 1} Nat Real (CoeTCₓ.coe.{1, 1} Nat Real (Nat.castCoe.{0} Real Real.hasNatCast))) n))) (Set.univ.{u1} α)
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] (x : α), Eq.{succ u1} (Set.{u1} α) (Set.iUnion.{u1, 1} α Nat (fun (n : Nat) => Metric.closedBall.{u1} α _inst_1 x ((fun (a : Type) (b : Type) [self : HasLiftT.{1, 1} a b] => self.0) Nat Real (HasLiftT.mk.{1, 1} Nat Real (CoeTCₓ.coe.{1, 1} Nat Real (Nat.castCoe.{0} Real Real.hasNatCast))) n))) (Set.univ.{u1} α)
 but is expected to have type
-  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] (x : α), Eq.{succ u1} (Set.{u1} α) (Set.unionᵢ.{u1, 1} α Nat (fun (n : Nat) => Metric.closedBall.{u1} α _inst_1 x (Nat.cast.{0} Real Real.natCast n))) (Set.univ.{u1} α)
-Case conversion may be inaccurate. Consider using '#align metric.Union_closed_ball_nat Metric.unionᵢ_closedBall_natₓ'. -/
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] (x : α), Eq.{succ u1} (Set.{u1} α) (Set.iUnion.{u1, 1} α Nat (fun (n : Nat) => Metric.closedBall.{u1} α _inst_1 x (Nat.cast.{0} Real Real.natCast n))) (Set.univ.{u1} α)
+Case conversion may be inaccurate. Consider using '#align metric.Union_closed_ball_nat Metric.iUnion_closedBall_natₓ'. -/
 @[simp]
-theorem unionᵢ_closedBall_nat (x : α) : (⋃ n : ℕ, closedBall x n) = univ :=
-  unionᵢ_eq_univ_iff.2 fun y => exists_nat_ge (dist y x)
-#align metric.Union_closed_ball_nat Metric.unionᵢ_closedBall_nat
+theorem iUnion_closedBall_nat (x : α) : (⋃ n : ℕ, closedBall x n) = univ :=
+  iUnion_eq_univ_iff.2 fun y => exists_nat_ge (dist y x)
+#align metric.Union_closed_ball_nat Metric.iUnion_closedBall_nat
 
-/- warning: metric.Union_inter_closed_ball_nat -> Metric.unionᵢ_inter_closedBall_nat is a dubious translation:
+/- warning: metric.Union_inter_closed_ball_nat -> Metric.iUnion_inter_closedBall_nat is a dubious translation:
 lean 3 declaration is
-  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] (s : Set.{u1} α) (x : α), Eq.{succ u1} (Set.{u1} α) (Set.unionᵢ.{u1, 1} α Nat (fun (n : Nat) => Inter.inter.{u1} (Set.{u1} α) (Set.hasInter.{u1} α) s (Metric.closedBall.{u1} α _inst_1 x ((fun (a : Type) (b : Type) [self : HasLiftT.{1, 1} a b] => self.0) Nat Real (HasLiftT.mk.{1, 1} Nat Real (CoeTCₓ.coe.{1, 1} Nat Real (Nat.castCoe.{0} Real Real.hasNatCast))) n)))) s
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] (s : Set.{u1} α) (x : α), Eq.{succ u1} (Set.{u1} α) (Set.iUnion.{u1, 1} α Nat (fun (n : Nat) => Inter.inter.{u1} (Set.{u1} α) (Set.hasInter.{u1} α) s (Metric.closedBall.{u1} α _inst_1 x ((fun (a : Type) (b : Type) [self : HasLiftT.{1, 1} a b] => self.0) Nat Real (HasLiftT.mk.{1, 1} Nat Real (CoeTCₓ.coe.{1, 1} Nat Real (Nat.castCoe.{0} Real Real.hasNatCast))) n)))) s
 but is expected to have type
-  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] (s : Set.{u1} α) (x : α), Eq.{succ u1} (Set.{u1} α) (Set.unionᵢ.{u1, 1} α Nat (fun (n : Nat) => Inter.inter.{u1} (Set.{u1} α) (Set.instInterSet.{u1} α) s (Metric.closedBall.{u1} α _inst_1 x (Nat.cast.{0} Real Real.natCast n)))) s
-Case conversion may be inaccurate. Consider using '#align metric.Union_inter_closed_ball_nat Metric.unionᵢ_inter_closedBall_natₓ'. -/
-theorem unionᵢ_inter_closedBall_nat (s : Set α) (x : α) : (⋃ n : ℕ, s ∩ closedBall x n) = s := by
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] (s : Set.{u1} α) (x : α), Eq.{succ u1} (Set.{u1} α) (Set.iUnion.{u1, 1} α Nat (fun (n : Nat) => Inter.inter.{u1} (Set.{u1} α) (Set.instInterSet.{u1} α) s (Metric.closedBall.{u1} α _inst_1 x (Nat.cast.{0} Real Real.natCast n)))) s
+Case conversion may be inaccurate. Consider using '#align metric.Union_inter_closed_ball_nat Metric.iUnion_inter_closedBall_natₓ'. -/
+theorem iUnion_inter_closedBall_nat (s : Set α) (x : α) : (⋃ n : ℕ, s ∩ closedBall x n) = s := by
   rw [← inter_Union, Union_closed_ball_nat, inter_univ]
-#align metric.Union_inter_closed_ball_nat Metric.unionᵢ_inter_closedBall_nat
+#align metric.Union_inter_closed_ball_nat Metric.iUnion_inter_closedBall_nat
 
 /- warning: metric.ball_subset -> Metric.ball_subset is a dubious translation:
 lean 3 declaration is
@@ -1569,16 +1569,16 @@ theorem controlled_of_uniformEmbedding [PseudoMetricSpace β] {f : α → β} :
 
 /- warning: metric.totally_bounded_iff -> Metric.totallyBounded_iff is a dubious translation:
 lean 3 declaration is
-  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {s : Set.{u1} α}, Iff (TotallyBounded.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1) s) (forall (ε : Real), (GT.gt.{0} Real Real.hasLt ε (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero)))) -> (Exists.{succ u1} (Set.{u1} α) (fun (t : Set.{u1} α) => And (Set.Finite.{u1} α t) (HasSubset.Subset.{u1} (Set.{u1} α) (Set.hasSubset.{u1} α) s (Set.unionᵢ.{u1, succ u1} α α (fun (y : α) => Set.unionᵢ.{u1, 0} α (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) y t) (fun (H : Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) y t) => Metric.ball.{u1} α _inst_1 y ε)))))))
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {s : Set.{u1} α}, Iff (TotallyBounded.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1) s) (forall (ε : Real), (GT.gt.{0} Real Real.hasLt ε (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero)))) -> (Exists.{succ u1} (Set.{u1} α) (fun (t : Set.{u1} α) => And (Set.Finite.{u1} α t) (HasSubset.Subset.{u1} (Set.{u1} α) (Set.hasSubset.{u1} α) s (Set.iUnion.{u1, succ u1} α α (fun (y : α) => Set.iUnion.{u1, 0} α (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) y t) (fun (H : Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) y t) => Metric.ball.{u1} α _inst_1 y ε)))))))
 but is expected to have type
-  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {s : Set.{u1} α}, Iff (TotallyBounded.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1) s) (forall (ε : Real), (GT.gt.{0} Real Real.instLTReal ε (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal))) -> (Exists.{succ u1} (Set.{u1} α) (fun (t : Set.{u1} α) => And (Set.Finite.{u1} α t) (HasSubset.Subset.{u1} (Set.{u1} α) (Set.instHasSubsetSet.{u1} α) s (Set.unionᵢ.{u1, succ u1} α α (fun (y : α) => Set.unionᵢ.{u1, 0} α (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) y t) (fun (H : Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) y t) => Metric.ball.{u1} α _inst_1 y ε)))))))
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {s : Set.{u1} α}, Iff (TotallyBounded.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1) s) (forall (ε : Real), (GT.gt.{0} Real Real.instLTReal ε (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal))) -> (Exists.{succ u1} (Set.{u1} α) (fun (t : Set.{u1} α) => And (Set.Finite.{u1} α t) (HasSubset.Subset.{u1} (Set.{u1} α) (Set.instHasSubsetSet.{u1} α) s (Set.iUnion.{u1, succ u1} α α (fun (y : α) => Set.iUnion.{u1, 0} α (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) y t) (fun (H : Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) y t) => Metric.ball.{u1} α _inst_1 y ε)))))))
 Case conversion may be inaccurate. Consider using '#align metric.totally_bounded_iff Metric.totallyBounded_iffₓ'. -/
 theorem totallyBounded_iff {s : Set α} :
     TotallyBounded s ↔ ∀ ε > 0, ∃ t : Set α, t.Finite ∧ s ⊆ ⋃ y ∈ t, ball y ε :=
   ⟨fun H ε ε0 => H _ (dist_mem_uniformity ε0), fun H r ru =>
     let ⟨ε, ε0, hε⟩ := mem_uniformity_dist.1 ru
     let ⟨t, ft, h⟩ := H ε ε0
-    ⟨t, ft, h.trans <| unionᵢ₂_mono fun y yt z => hε⟩⟩
+    ⟨t, ft, h.trans <| iUnion₂_mono fun y yt z => hε⟩⟩
 #align metric.totally_bounded_iff Metric.totallyBounded_iff
 
 /- warning: metric.totally_bounded_of_finite_discretization -> Metric.totallyBounded_of_finite_discretization is a dubious translation:
@@ -1606,15 +1606,15 @@ theorem totallyBounded_of_finite_discretization {s : Set α}
   refine' ⟨range (Subtype.val ∘ Finv), finite_range _, fun x xs => _⟩
   let x' := Finv (F ⟨x, xs⟩)
   have : F x' = F ⟨x, xs⟩ := Function.invFun_eq ⟨⟨x, xs⟩, rfl⟩
-  simp only [Set.mem_unionᵢ, Set.mem_range]
+  simp only [Set.mem_iUnion, Set.mem_range]
   exact ⟨_, ⟨F ⟨x, xs⟩, rfl⟩, hF _ _ this.symm⟩
 #align metric.totally_bounded_of_finite_discretization Metric.totallyBounded_of_finite_discretization
 
 /- warning: metric.finite_approx_of_totally_bounded -> Metric.finite_approx_of_totallyBounded is a dubious translation:
 lean 3 declaration is
-  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {s : Set.{u1} α}, (TotallyBounded.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1) s) -> (forall (ε : Real), (GT.gt.{0} Real Real.hasLt ε (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero)))) -> (Exists.{succ u1} (Set.{u1} α) (fun (t : Set.{u1} α) => Exists.{0} (HasSubset.Subset.{u1} (Set.{u1} α) (Set.hasSubset.{u1} α) t s) (fun (H : HasSubset.Subset.{u1} (Set.{u1} α) (Set.hasSubset.{u1} α) t s) => And (Set.Finite.{u1} α t) (HasSubset.Subset.{u1} (Set.{u1} α) (Set.hasSubset.{u1} α) s (Set.unionᵢ.{u1, succ u1} α α (fun (y : α) => Set.unionᵢ.{u1, 0} α (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) y t) (fun (H : Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) y t) => Metric.ball.{u1} α _inst_1 y ε))))))))
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {s : Set.{u1} α}, (TotallyBounded.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1) s) -> (forall (ε : Real), (GT.gt.{0} Real Real.hasLt ε (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero)))) -> (Exists.{succ u1} (Set.{u1} α) (fun (t : Set.{u1} α) => Exists.{0} (HasSubset.Subset.{u1} (Set.{u1} α) (Set.hasSubset.{u1} α) t s) (fun (H : HasSubset.Subset.{u1} (Set.{u1} α) (Set.hasSubset.{u1} α) t s) => And (Set.Finite.{u1} α t) (HasSubset.Subset.{u1} (Set.{u1} α) (Set.hasSubset.{u1} α) s (Set.iUnion.{u1, succ u1} α α (fun (y : α) => Set.iUnion.{u1, 0} α (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) y t) (fun (H : Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) y t) => Metric.ball.{u1} α _inst_1 y ε))))))))
 but is expected to have type
-  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {s : Set.{u1} α}, (TotallyBounded.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1) s) -> (forall (ε : Real), (GT.gt.{0} Real Real.instLTReal ε (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal))) -> (Exists.{succ u1} (Set.{u1} α) (fun (t : Set.{u1} α) => And (HasSubset.Subset.{u1} (Set.{u1} α) (Set.instHasSubsetSet.{u1} α) t s) (And (Set.Finite.{u1} α t) (HasSubset.Subset.{u1} (Set.{u1} α) (Set.instHasSubsetSet.{u1} α) s (Set.unionᵢ.{u1, succ u1} α α (fun (y : α) => Set.unionᵢ.{u1, 0} α (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) y t) (fun (h._@.Mathlib.Topology.MetricSpace.Basic._hyg.10766 : Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) y t) => Metric.ball.{u1} α _inst_1 y ε))))))))
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {s : Set.{u1} α}, (TotallyBounded.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1) s) -> (forall (ε : Real), (GT.gt.{0} Real Real.instLTReal ε (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal))) -> (Exists.{succ u1} (Set.{u1} α) (fun (t : Set.{u1} α) => And (HasSubset.Subset.{u1} (Set.{u1} α) (Set.instHasSubsetSet.{u1} α) t s) (And (Set.Finite.{u1} α t) (HasSubset.Subset.{u1} (Set.{u1} α) (Set.instHasSubsetSet.{u1} α) s (Set.iUnion.{u1, succ u1} α α (fun (y : α) => Set.iUnion.{u1, 0} α (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) y t) (fun (h._@.Mathlib.Topology.MetricSpace.Basic._hyg.10766 : Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) y t) => Metric.ball.{u1} α _inst_1 y ε))))))))
 Case conversion may be inaccurate. Consider using '#align metric.finite_approx_of_totally_bounded Metric.finite_approx_of_totallyBoundedₓ'. -/
 /- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (t «expr ⊆ » s) -/
 theorem finite_approx_of_totallyBounded {s : Set α} (hs : TotallyBounded s) :
@@ -2179,12 +2179,12 @@ protected theorem Metric.uniformity_basis_edist :
 
 /- warning: metric.uniformity_edist -> Metric.uniformity_edist is a dubious translation:
 lean 3 declaration is
-  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α], Eq.{succ u1} (Filter.{u1} (Prod.{u1, u1} α α)) (uniformity.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) (infᵢ.{u1, 1} (Filter.{u1} (Prod.{u1, u1} α α)) (ConditionallyCompleteLattice.toHasInf.{u1} (Filter.{u1} (Prod.{u1, u1} α α)) (CompleteLattice.toConditionallyCompleteLattice.{u1} (Filter.{u1} (Prod.{u1, u1} α α)) (Filter.completeLattice.{u1} (Prod.{u1, u1} α α)))) ENNReal (fun (ε : ENNReal) => infᵢ.{u1, 0} (Filter.{u1} (Prod.{u1, u1} α α)) (ConditionallyCompleteLattice.toHasInf.{u1} (Filter.{u1} (Prod.{u1, u1} α α)) (CompleteLattice.toConditionallyCompleteLattice.{u1} (Filter.{u1} (Prod.{u1, u1} α α)) (Filter.completeLattice.{u1} (Prod.{u1, u1} α α)))) (GT.gt.{0} ENNReal (Preorder.toLT.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (CompleteSemilatticeInf.toPartialOrder.{0} ENNReal (CompleteLattice.toCompleteSemilatticeInf.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))))) ε (OfNat.ofNat.{0} ENNReal 0 (OfNat.mk.{0} ENNReal 0 (Zero.zero.{0} ENNReal ENNReal.hasZero)))) (fun (H : GT.gt.{0} ENNReal (Preorder.toLT.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (CompleteSemilatticeInf.toPartialOrder.{0} ENNReal (CompleteLattice.toCompleteSemilatticeInf.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))))) ε (OfNat.ofNat.{0} ENNReal 0 (OfNat.mk.{0} ENNReal 0 (Zero.zero.{0} ENNReal ENNReal.hasZero)))) => Filter.principal.{u1} (Prod.{u1, u1} α α) (setOf.{u1} (Prod.{u1, u1} α α) (fun (p : Prod.{u1, u1} α α) => LT.lt.{0} ENNReal (Preorder.toLT.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (CompleteSemilatticeInf.toPartialOrder.{0} ENNReal (CompleteLattice.toCompleteSemilatticeInf.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))))) (EDist.edist.{u1} α (PseudoMetricSpace.toEDist.{u1} α _inst_1) (Prod.fst.{u1, u1} α α p) (Prod.snd.{u1, u1} α α p)) ε)))))
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α], Eq.{succ u1} (Filter.{u1} (Prod.{u1, u1} α α)) (uniformity.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) (iInf.{u1, 1} (Filter.{u1} (Prod.{u1, u1} α α)) (ConditionallyCompleteLattice.toHasInf.{u1} (Filter.{u1} (Prod.{u1, u1} α α)) (CompleteLattice.toConditionallyCompleteLattice.{u1} (Filter.{u1} (Prod.{u1, u1} α α)) (Filter.completeLattice.{u1} (Prod.{u1, u1} α α)))) ENNReal (fun (ε : ENNReal) => iInf.{u1, 0} (Filter.{u1} (Prod.{u1, u1} α α)) (ConditionallyCompleteLattice.toHasInf.{u1} (Filter.{u1} (Prod.{u1, u1} α α)) (CompleteLattice.toConditionallyCompleteLattice.{u1} (Filter.{u1} (Prod.{u1, u1} α α)) (Filter.completeLattice.{u1} (Prod.{u1, u1} α α)))) (GT.gt.{0} ENNReal (Preorder.toLT.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (CompleteSemilatticeInf.toPartialOrder.{0} ENNReal (CompleteLattice.toCompleteSemilatticeInf.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))))) ε (OfNat.ofNat.{0} ENNReal 0 (OfNat.mk.{0} ENNReal 0 (Zero.zero.{0} ENNReal ENNReal.hasZero)))) (fun (H : GT.gt.{0} ENNReal (Preorder.toLT.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (CompleteSemilatticeInf.toPartialOrder.{0} ENNReal (CompleteLattice.toCompleteSemilatticeInf.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))))) ε (OfNat.ofNat.{0} ENNReal 0 (OfNat.mk.{0} ENNReal 0 (Zero.zero.{0} ENNReal ENNReal.hasZero)))) => Filter.principal.{u1} (Prod.{u1, u1} α α) (setOf.{u1} (Prod.{u1, u1} α α) (fun (p : Prod.{u1, u1} α α) => LT.lt.{0} ENNReal (Preorder.toLT.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (CompleteSemilatticeInf.toPartialOrder.{0} ENNReal (CompleteLattice.toCompleteSemilatticeInf.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))))) (EDist.edist.{u1} α (PseudoMetricSpace.toEDist.{u1} α _inst_1) (Prod.fst.{u1, u1} α α p) (Prod.snd.{u1, u1} α α p)) ε)))))
 but is expected to have type
-  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α], Eq.{succ u1} (Filter.{u1} (Prod.{u1, u1} α α)) (uniformity.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) (infᵢ.{u1, 1} (Filter.{u1} (Prod.{u1, u1} α α)) (ConditionallyCompleteLattice.toInfSet.{u1} (Filter.{u1} (Prod.{u1, u1} α α)) (CompleteLattice.toConditionallyCompleteLattice.{u1} (Filter.{u1} (Prod.{u1, u1} α α)) (Filter.instCompleteLatticeFilter.{u1} (Prod.{u1, u1} α α)))) ENNReal (fun (ε : ENNReal) => infᵢ.{u1, 0} (Filter.{u1} (Prod.{u1, u1} α α)) (ConditionallyCompleteLattice.toInfSet.{u1} (Filter.{u1} (Prod.{u1, u1} α α)) (CompleteLattice.toConditionallyCompleteLattice.{u1} (Filter.{u1} (Prod.{u1, u1} α α)) (Filter.instCompleteLatticeFilter.{u1} (Prod.{u1, u1} α α)))) (GT.gt.{0} ENNReal (Preorder.toLT.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (CompleteSemilatticeInf.toPartialOrder.{0} ENNReal (CompleteLattice.toCompleteSemilatticeInf.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal))))) ε (OfNat.ofNat.{0} ENNReal 0 (Zero.toOfNat0.{0} ENNReal instENNRealZero))) (fun (H : GT.gt.{0} ENNReal (Preorder.toLT.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (CompleteSemilatticeInf.toPartialOrder.{0} ENNReal (CompleteLattice.toCompleteSemilatticeInf.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal))))) ε (OfNat.ofNat.{0} ENNReal 0 (Zero.toOfNat0.{0} ENNReal instENNRealZero))) => Filter.principal.{u1} (Prod.{u1, u1} α α) (setOf.{u1} (Prod.{u1, u1} α α) (fun (p : Prod.{u1, u1} α α) => LT.lt.{0} ENNReal (Preorder.toLT.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (CompleteSemilatticeInf.toPartialOrder.{0} ENNReal (CompleteLattice.toCompleteSemilatticeInf.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal))))) (EDist.edist.{u1} α (PseudoMetricSpace.toEDist.{u1} α _inst_1) (Prod.fst.{u1, u1} α α p) (Prod.snd.{u1, u1} α α p)) ε)))))
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α], Eq.{succ u1} (Filter.{u1} (Prod.{u1, u1} α α)) (uniformity.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) (iInf.{u1, 1} (Filter.{u1} (Prod.{u1, u1} α α)) (ConditionallyCompleteLattice.toInfSet.{u1} (Filter.{u1} (Prod.{u1, u1} α α)) (CompleteLattice.toConditionallyCompleteLattice.{u1} (Filter.{u1} (Prod.{u1, u1} α α)) (Filter.instCompleteLatticeFilter.{u1} (Prod.{u1, u1} α α)))) ENNReal (fun (ε : ENNReal) => iInf.{u1, 0} (Filter.{u1} (Prod.{u1, u1} α α)) (ConditionallyCompleteLattice.toInfSet.{u1} (Filter.{u1} (Prod.{u1, u1} α α)) (CompleteLattice.toConditionallyCompleteLattice.{u1} (Filter.{u1} (Prod.{u1, u1} α α)) (Filter.instCompleteLatticeFilter.{u1} (Prod.{u1, u1} α α)))) (GT.gt.{0} ENNReal (Preorder.toLT.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (CompleteSemilatticeInf.toPartialOrder.{0} ENNReal (CompleteLattice.toCompleteSemilatticeInf.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal))))) ε (OfNat.ofNat.{0} ENNReal 0 (Zero.toOfNat0.{0} ENNReal instENNRealZero))) (fun (H : GT.gt.{0} ENNReal (Preorder.toLT.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (CompleteSemilatticeInf.toPartialOrder.{0} ENNReal (CompleteLattice.toCompleteSemilatticeInf.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal))))) ε (OfNat.ofNat.{0} ENNReal 0 (Zero.toOfNat0.{0} ENNReal instENNRealZero))) => Filter.principal.{u1} (Prod.{u1, u1} α α) (setOf.{u1} (Prod.{u1, u1} α α) (fun (p : Prod.{u1, u1} α α) => LT.lt.{0} ENNReal (Preorder.toLT.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (CompleteSemilatticeInf.toPartialOrder.{0} ENNReal (CompleteLattice.toCompleteSemilatticeInf.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal))))) (EDist.edist.{u1} α (PseudoMetricSpace.toEDist.{u1} α _inst_1) (Prod.fst.{u1, u1} α α p) (Prod.snd.{u1, u1} α α p)) ε)))))
 Case conversion may be inaccurate. Consider using '#align metric.uniformity_edist Metric.uniformity_edistₓ'. -/
 theorem Metric.uniformity_edist : 𝓤 α = ⨅ ε > 0, 𝓟 { p : α × α | edist p.1 p.2 < ε } :=
-  Metric.uniformity_basis_edist.eq_binfᵢ
+  Metric.uniformity_basis_edist.eq_biInf
 #align metric.uniformity_edist Metric.uniformity_edist
 
 #print PseudoMetricSpace.toPseudoEMetricSpace /-
@@ -2894,14 +2894,14 @@ theorem cauchySeq_iff_le_tendsto_0 {s : ℕ → α} :
       exact le_of_lt (hR m n)
     -- Prove that it bounds the distances of points in the Cauchy sequence
     have ub : ∀ m n N, N ≤ m → N ≤ n → dist (s m) (s n) ≤ Sup (S N) := fun m n N hm hn =>
-      le_csupₛ (hS N) ⟨⟨_, _⟩, ⟨hm, hn⟩, rfl⟩
+      le_csSup (hS N) ⟨⟨_, _⟩, ⟨hm, hn⟩, rfl⟩
     have S0m : ∀ n, (0 : ℝ) ∈ S n := fun n => ⟨⟨n, n⟩, ⟨le_rfl, le_rfl⟩, dist_self _⟩
-    have S0 := fun n => le_csupₛ (hS n) (S0m n)
+    have S0 := fun n => le_csSup (hS n) (S0m n)
     -- Prove that it tends to `0`, by using the Cauchy property of `s`
     refine' ⟨fun N => Sup (S N), S0, ub, Metric.tendsto_atTop.2 fun ε ε0 => _⟩
     refine' (Metric.cauchySeq_iff.1 hs (ε / 2) (half_pos ε0)).imp fun N hN n hn => _
     rw [Real.dist_0_eq_abs, abs_of_nonneg (S0 n)]
-    refine' lt_of_le_of_lt (csupₛ_le ⟨_, S0m _⟩ _) (half_lt_self ε0)
+    refine' lt_of_le_of_lt (csSup_le ⟨_, S0m _⟩ _) (half_lt_self ε0)
     rintro _ ⟨⟨m', n'⟩, ⟨hm', hn'⟩, rfl⟩
     exact le_of_lt (hN _ (le_trans hn hm') _ (le_trans hn hn')), fun ⟨b, _, b_bound, b_lim⟩ =>
     cauchySeq_of_le_tendsto_0 b b_bound b_lim⟩
@@ -2920,7 +2920,7 @@ def PseudoMetricSpace.induced {α β} (f : α → β) (m : PseudoMetricSpace β)
   edist x y := edist (f x) (f y)
   edist_dist x y := edist_dist _ _
   toUniformSpace := UniformSpace.comap f m.toUniformSpace
-  uniformity_dist := (uniformity_basis_dist.comap _).eq_binfᵢ
+  uniformity_dist := (uniformity_basis_dist.comap _).eq_biInf
   toBornology := Bornology.induced f
   cobounded_sets :=
     Set.ext <|
@@ -3814,9 +3814,9 @@ section Compact
 
 /- warning: finite_cover_balls_of_compact -> finite_cover_balls_of_compact is a dubious translation:
 lean 3 declaration is
-  forall {α : Type.{u1}} [_inst_2 : PseudoMetricSpace.{u1} α] {s : Set.{u1} α}, (IsCompact.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_2)) s) -> (forall {e : Real}, (LT.lt.{0} Real Real.hasLt (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero))) e) -> (Exists.{succ u1} (Set.{u1} α) (fun (t : Set.{u1} α) => Exists.{0} (HasSubset.Subset.{u1} (Set.{u1} α) (Set.hasSubset.{u1} α) t s) (fun (H : HasSubset.Subset.{u1} (Set.{u1} α) (Set.hasSubset.{u1} α) t s) => And (Set.Finite.{u1} α t) (HasSubset.Subset.{u1} (Set.{u1} α) (Set.hasSubset.{u1} α) s (Set.unionᵢ.{u1, succ u1} α α (fun (x : α) => Set.unionᵢ.{u1, 0} α (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) x t) (fun (H : Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) x t) => Metric.ball.{u1} α _inst_2 x e))))))))
+  forall {α : Type.{u1}} [_inst_2 : PseudoMetricSpace.{u1} α] {s : Set.{u1} α}, (IsCompact.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_2)) s) -> (forall {e : Real}, (LT.lt.{0} Real Real.hasLt (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero))) e) -> (Exists.{succ u1} (Set.{u1} α) (fun (t : Set.{u1} α) => Exists.{0} (HasSubset.Subset.{u1} (Set.{u1} α) (Set.hasSubset.{u1} α) t s) (fun (H : HasSubset.Subset.{u1} (Set.{u1} α) (Set.hasSubset.{u1} α) t s) => And (Set.Finite.{u1} α t) (HasSubset.Subset.{u1} (Set.{u1} α) (Set.hasSubset.{u1} α) s (Set.iUnion.{u1, succ u1} α α (fun (x : α) => Set.iUnion.{u1, 0} α (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) x t) (fun (H : Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) x t) => Metric.ball.{u1} α _inst_2 x e))))))))
 but is expected to have type
-  forall {α : Type.{u1}} [_inst_2 : PseudoMetricSpace.{u1} α] {s : Set.{u1} α}, (IsCompact.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_2)) s) -> (forall {e : Real}, (LT.lt.{0} Real Real.instLTReal (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal)) e) -> (Exists.{succ u1} (Set.{u1} α) (fun (t : Set.{u1} α) => And (HasSubset.Subset.{u1} (Set.{u1} α) (Set.instHasSubsetSet.{u1} α) t s) (And (Set.Finite.{u1} α t) (HasSubset.Subset.{u1} (Set.{u1} α) (Set.instHasSubsetSet.{u1} α) s (Set.unionᵢ.{u1, succ u1} α α (fun (x : α) => Set.unionᵢ.{u1, 0} α (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) x t) (fun (h._@.Mathlib.Topology.MetricSpace.Basic._hyg.25516 : Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) x t) => Metric.ball.{u1} α _inst_2 x e))))))))
+  forall {α : Type.{u1}} [_inst_2 : PseudoMetricSpace.{u1} α] {s : Set.{u1} α}, (IsCompact.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_2)) s) -> (forall {e : Real}, (LT.lt.{0} Real Real.instLTReal (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal)) e) -> (Exists.{succ u1} (Set.{u1} α) (fun (t : Set.{u1} α) => And (HasSubset.Subset.{u1} (Set.{u1} α) (Set.instHasSubsetSet.{u1} α) t s) (And (Set.Finite.{u1} α t) (HasSubset.Subset.{u1} (Set.{u1} α) (Set.instHasSubsetSet.{u1} α) s (Set.iUnion.{u1, succ u1} α α (fun (x : α) => Set.iUnion.{u1, 0} α (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) x t) (fun (h._@.Mathlib.Topology.MetricSpace.Basic._hyg.25516 : Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) x t) => Metric.ball.{u1} α _inst_2 x e))))))))
 Case conversion may be inaccurate. Consider using '#align finite_cover_balls_of_compact finite_cover_balls_of_compactₓ'. -/
 /- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (t «expr ⊆ » s) -/
 /-- Any compact set in a pseudometric space can be covered by finitely many balls of a given
@@ -3834,9 +3834,9 @@ theorem finite_cover_balls_of_compact {α : Type u} [PseudoMetricSpace α] {s :
 
 /- warning: is_compact.finite_cover_balls -> IsCompact.finite_cover_balls is a dubious translation:
 lean 3 declaration is
-  forall {α : Type.{u1}} [_inst_2 : PseudoMetricSpace.{u1} α] {s : Set.{u1} α}, (IsCompact.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_2)) s) -> (forall {e : Real}, (LT.lt.{0} Real Real.hasLt (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero))) e) -> (Exists.{succ u1} (Set.{u1} α) (fun (t : Set.{u1} α) => Exists.{0} (HasSubset.Subset.{u1} (Set.{u1} α) (Set.hasSubset.{u1} α) t s) (fun (H : HasSubset.Subset.{u1} (Set.{u1} α) (Set.hasSubset.{u1} α) t s) => And (Set.Finite.{u1} α t) (HasSubset.Subset.{u1} (Set.{u1} α) (Set.hasSubset.{u1} α) s (Set.unionᵢ.{u1, succ u1} α α (fun (x : α) => Set.unionᵢ.{u1, 0} α (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) x t) (fun (H : Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) x t) => Metric.ball.{u1} α _inst_2 x e))))))))
+  forall {α : Type.{u1}} [_inst_2 : PseudoMetricSpace.{u1} α] {s : Set.{u1} α}, (IsCompact.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_2)) s) -> (forall {e : Real}, (LT.lt.{0} Real Real.hasLt (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero))) e) -> (Exists.{succ u1} (Set.{u1} α) (fun (t : Set.{u1} α) => Exists.{0} (HasSubset.Subset.{u1} (Set.{u1} α) (Set.hasSubset.{u1} α) t s) (fun (H : HasSubset.Subset.{u1} (Set.{u1} α) (Set.hasSubset.{u1} α) t s) => And (Set.Finite.{u1} α t) (HasSubset.Subset.{u1} (Set.{u1} α) (Set.hasSubset.{u1} α) s (Set.iUnion.{u1, succ u1} α α (fun (x : α) => Set.iUnion.{u1, 0} α (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) x t) (fun (H : Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) x t) => Metric.ball.{u1} α _inst_2 x e))))))))
 but is expected to have type
-  forall {α : Type.{u1}} [_inst_2 : PseudoMetricSpace.{u1} α] {s : Set.{u1} α}, (IsCompact.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_2)) s) -> (forall {e : Real}, (LT.lt.{0} Real Real.instLTReal (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal)) e) -> (Exists.{succ u1} (Set.{u1} α) (fun (t : Set.{u1} α) => And (HasSubset.Subset.{u1} (Set.{u1} α) (Set.instHasSubsetSet.{u1} α) t s) (And (Set.Finite.{u1} α t) (HasSubset.Subset.{u1} (Set.{u1} α) (Set.instHasSubsetSet.{u1} α) s (Set.unionᵢ.{u1, succ u1} α α (fun (x : α) => Set.unionᵢ.{u1, 0} α (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) x t) (fun (h._@.Mathlib.Topology.MetricSpace.Basic._hyg.25516 : Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) x t) => Metric.ball.{u1} α _inst_2 x e))))))))
+  forall {α : Type.{u1}} [_inst_2 : PseudoMetricSpace.{u1} α] {s : Set.{u1} α}, (IsCompact.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_2)) s) -> (forall {e : Real}, (LT.lt.{0} Real Real.instLTReal (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal)) e) -> (Exists.{succ u1} (Set.{u1} α) (fun (t : Set.{u1} α) => And (HasSubset.Subset.{u1} (Set.{u1} α) (Set.instHasSubsetSet.{u1} α) t s) (And (Set.Finite.{u1} α t) (HasSubset.Subset.{u1} (Set.{u1} α) (Set.instHasSubsetSet.{u1} α) s (Set.iUnion.{u1, succ u1} α α (fun (x : α) => Set.iUnion.{u1, 0} α (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) x t) (fun (h._@.Mathlib.Topology.MetricSpace.Basic._hyg.25516 : Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) x t) => Metric.ball.{u1} α _inst_2 x e))))))))
 Case conversion may be inaccurate. Consider using '#align is_compact.finite_cover_balls IsCompact.finite_cover_ballsₓ'. -/
 alias finite_cover_balls_of_compact ← IsCompact.finite_cover_balls
 #align is_compact.finite_cover_balls IsCompact.finite_cover_balls
@@ -4063,9 +4063,9 @@ end Metric
 
 /- warning: lebesgue_number_lemma_of_metric -> lebesgue_number_lemma_of_metric is a dubious translation:
 lean 3 declaration is
-  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {s : Set.{u1} α} {ι : Sort.{u2}} {c : ι -> (Set.{u1} α)}, (IsCompact.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) s) -> (forall (i : ι), IsOpen.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) (c i)) -> (HasSubset.Subset.{u1} (Set.{u1} α) (Set.hasSubset.{u1} α) s (Set.unionᵢ.{u1, u2} α ι (fun (i : ι) => c i))) -> (Exists.{1} Real (fun (δ : Real) => Exists.{0} (GT.gt.{0} Real Real.hasLt δ (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero)))) (fun (H : GT.gt.{0} Real Real.hasLt δ (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero)))) => forall (x : α), (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) x s) -> (Exists.{u2} ι (fun (i : ι) => HasSubset.Subset.{u1} (Set.{u1} α) (Set.hasSubset.{u1} α) (Metric.ball.{u1} α _inst_1 x δ) (c i))))))
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {s : Set.{u1} α} {ι : Sort.{u2}} {c : ι -> (Set.{u1} α)}, (IsCompact.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) s) -> (forall (i : ι), IsOpen.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) (c i)) -> (HasSubset.Subset.{u1} (Set.{u1} α) (Set.hasSubset.{u1} α) s (Set.iUnion.{u1, u2} α ι (fun (i : ι) => c i))) -> (Exists.{1} Real (fun (δ : Real) => Exists.{0} (GT.gt.{0} Real Real.hasLt δ (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero)))) (fun (H : GT.gt.{0} Real Real.hasLt δ (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero)))) => forall (x : α), (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) x s) -> (Exists.{u2} ι (fun (i : ι) => HasSubset.Subset.{u1} (Set.{u1} α) (Set.hasSubset.{u1} α) (Metric.ball.{u1} α _inst_1 x δ) (c i))))))
 but is expected to have type
-  forall {α : Type.{u2}} [_inst_1 : PseudoMetricSpace.{u2} α] {s : Set.{u2} α} {ι : Sort.{u1}} {c : ι -> (Set.{u2} α)}, (IsCompact.{u2} α (UniformSpace.toTopologicalSpace.{u2} α (PseudoMetricSpace.toUniformSpace.{u2} α _inst_1)) s) -> (forall (i : ι), IsOpen.{u2} α (UniformSpace.toTopologicalSpace.{u2} α (PseudoMetricSpace.toUniformSpace.{u2} α _inst_1)) (c i)) -> (HasSubset.Subset.{u2} (Set.{u2} α) (Set.instHasSubsetSet.{u2} α) s (Set.unionᵢ.{u2, u1} α ι (fun (i : ι) => c i))) -> (Exists.{1} Real (fun (δ : Real) => And (GT.gt.{0} Real Real.instLTReal δ (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal))) (forall (x : α), (Membership.mem.{u2, u2} α (Set.{u2} α) (Set.instMembershipSet.{u2} α) x s) -> (Exists.{u1} ι (fun (i : ι) => HasSubset.Subset.{u2} (Set.{u2} α) (Set.instHasSubsetSet.{u2} α) (Metric.ball.{u2} α _inst_1 x δ) (c i))))))
+  forall {α : Type.{u2}} [_inst_1 : PseudoMetricSpace.{u2} α] {s : Set.{u2} α} {ι : Sort.{u1}} {c : ι -> (Set.{u2} α)}, (IsCompact.{u2} α (UniformSpace.toTopologicalSpace.{u2} α (PseudoMetricSpace.toUniformSpace.{u2} α _inst_1)) s) -> (forall (i : ι), IsOpen.{u2} α (UniformSpace.toTopologicalSpace.{u2} α (PseudoMetricSpace.toUniformSpace.{u2} α _inst_1)) (c i)) -> (HasSubset.Subset.{u2} (Set.{u2} α) (Set.instHasSubsetSet.{u2} α) s (Set.iUnion.{u2, u1} α ι (fun (i : ι) => c i))) -> (Exists.{1} Real (fun (δ : Real) => And (GT.gt.{0} Real Real.instLTReal δ (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal))) (forall (x : α), (Membership.mem.{u2, u2} α (Set.{u2} α) (Set.instMembershipSet.{u2} α) x s) -> (Exists.{u1} ι (fun (i : ι) => HasSubset.Subset.{u2} (Set.{u2} α) (Set.instHasSubsetSet.{u2} α) (Metric.ball.{u2} α _inst_1 x δ) (c i))))))
 Case conversion may be inaccurate. Consider using '#align lebesgue_number_lemma_of_metric lebesgue_number_lemma_of_metricₓ'. -/
 theorem lebesgue_number_lemma_of_metric {s : Set α} {ι} {c : ι → Set α} (hs : IsCompact s)
     (hc₁ : ∀ i, IsOpen (c i)) (hc₂ : s ⊆ ⋃ i, c i) : ∃ δ > 0, ∀ x ∈ s, ∃ i, ball x δ ⊆ c i :=
@@ -4076,16 +4076,16 @@ theorem lebesgue_number_lemma_of_metric {s : Set α} {ι} {c : ι → Set α} (h
     ⟨i, fun y hy => hi (hδ (mem_ball'.mp hy))⟩⟩
 #align lebesgue_number_lemma_of_metric lebesgue_number_lemma_of_metric
 
-/- warning: lebesgue_number_lemma_of_metric_sUnion -> lebesgue_number_lemma_of_metric_unionₛ is a dubious translation:
+/- warning: lebesgue_number_lemma_of_metric_sUnion -> lebesgue_number_lemma_of_metric_sUnion is a dubious translation:
 lean 3 declaration is
-  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {s : Set.{u1} α} {c : Set.{u1} (Set.{u1} α)}, (IsCompact.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) s) -> (forall (t : Set.{u1} α), (Membership.Mem.{u1, u1} (Set.{u1} α) (Set.{u1} (Set.{u1} α)) (Set.hasMem.{u1} (Set.{u1} α)) t c) -> (IsOpen.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) t)) -> (HasSubset.Subset.{u1} (Set.{u1} α) (Set.hasSubset.{u1} α) s (Set.unionₛ.{u1} α c)) -> (Exists.{1} Real (fun (δ : Real) => Exists.{0} (GT.gt.{0} Real Real.hasLt δ (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero)))) (fun (H : GT.gt.{0} Real Real.hasLt δ (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero)))) => forall (x : α), (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) x s) -> (Exists.{succ u1} (Set.{u1} α) (fun (t : Set.{u1} α) => Exists.{0} (Membership.Mem.{u1, u1} (Set.{u1} α) (Set.{u1} (Set.{u1} α)) (Set.hasMem.{u1} (Set.{u1} α)) t c) (fun (H : Membership.Mem.{u1, u1} (Set.{u1} α) (Set.{u1} (Set.{u1} α)) (Set.hasMem.{u1} (Set.{u1} α)) t c) => HasSubset.Subset.{u1} (Set.{u1} α) (Set.hasSubset.{u1} α) (Metric.ball.{u1} α _inst_1 x δ) t))))))
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {s : Set.{u1} α} {c : Set.{u1} (Set.{u1} α)}, (IsCompact.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) s) -> (forall (t : Set.{u1} α), (Membership.Mem.{u1, u1} (Set.{u1} α) (Set.{u1} (Set.{u1} α)) (Set.hasMem.{u1} (Set.{u1} α)) t c) -> (IsOpen.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) t)) -> (HasSubset.Subset.{u1} (Set.{u1} α) (Set.hasSubset.{u1} α) s (Set.sUnion.{u1} α c)) -> (Exists.{1} Real (fun (δ : Real) => Exists.{0} (GT.gt.{0} Real Real.hasLt δ (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero)))) (fun (H : GT.gt.{0} Real Real.hasLt δ (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero)))) => forall (x : α), (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) x s) -> (Exists.{succ u1} (Set.{u1} α) (fun (t : Set.{u1} α) => Exists.{0} (Membership.Mem.{u1, u1} (Set.{u1} α) (Set.{u1} (Set.{u1} α)) (Set.hasMem.{u1} (Set.{u1} α)) t c) (fun (H : Membership.Mem.{u1, u1} (Set.{u1} α) (Set.{u1} (Set.{u1} α)) (Set.hasMem.{u1} (Set.{u1} α)) t c) => HasSubset.Subset.{u1} (Set.{u1} α) (Set.hasSubset.{u1} α) (Metric.ball.{u1} α _inst_1 x δ) t))))))
 but is expected to have type
-  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {s : Set.{u1} α} {c : Set.{u1} (Set.{u1} α)}, (IsCompact.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) s) -> (forall (t : Set.{u1} α), (Membership.mem.{u1, u1} (Set.{u1} α) (Set.{u1} (Set.{u1} α)) (Set.instMembershipSet.{u1} (Set.{u1} α)) t c) -> (IsOpen.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) t)) -> (HasSubset.Subset.{u1} (Set.{u1} α) (Set.instHasSubsetSet.{u1} α) s (Set.unionₛ.{u1} α c)) -> (Exists.{1} Real (fun (δ : Real) => And (GT.gt.{0} Real Real.instLTReal δ (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal))) (forall (x : α), (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) x s) -> (Exists.{succ u1} (Set.{u1} α) (fun (t : Set.{u1} α) => And (Membership.mem.{u1, u1} (Set.{u1} α) (Set.{u1} (Set.{u1} α)) (Set.instMembershipSet.{u1} (Set.{u1} α)) t c) (HasSubset.Subset.{u1} (Set.{u1} α) (Set.instHasSubsetSet.{u1} α) (Metric.ball.{u1} α _inst_1 x δ) t))))))
-Case conversion may be inaccurate. Consider using '#align lebesgue_number_lemma_of_metric_sUnion lebesgue_number_lemma_of_metric_unionₛₓ'. -/
-theorem lebesgue_number_lemma_of_metric_unionₛ {s : Set α} {c : Set (Set α)} (hs : IsCompact s)
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {s : Set.{u1} α} {c : Set.{u1} (Set.{u1} α)}, (IsCompact.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) s) -> (forall (t : Set.{u1} α), (Membership.mem.{u1, u1} (Set.{u1} α) (Set.{u1} (Set.{u1} α)) (Set.instMembershipSet.{u1} (Set.{u1} α)) t c) -> (IsOpen.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) t)) -> (HasSubset.Subset.{u1} (Set.{u1} α) (Set.instHasSubsetSet.{u1} α) s (Set.sUnion.{u1} α c)) -> (Exists.{1} Real (fun (δ : Real) => And (GT.gt.{0} Real Real.instLTReal δ (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal))) (forall (x : α), (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) x s) -> (Exists.{succ u1} (Set.{u1} α) (fun (t : Set.{u1} α) => And (Membership.mem.{u1, u1} (Set.{u1} α) (Set.{u1} (Set.{u1} α)) (Set.instMembershipSet.{u1} (Set.{u1} α)) t c) (HasSubset.Subset.{u1} (Set.{u1} α) (Set.instHasSubsetSet.{u1} α) (Metric.ball.{u1} α _inst_1 x δ) t))))))
+Case conversion may be inaccurate. Consider using '#align lebesgue_number_lemma_of_metric_sUnion lebesgue_number_lemma_of_metric_sUnionₓ'. -/
+theorem lebesgue_number_lemma_of_metric_sUnion {s : Set α} {c : Set (Set α)} (hs : IsCompact s)
     (hc₁ : ∀ t ∈ c, IsOpen t) (hc₂ : s ⊆ ⋃₀ c) : ∃ δ > 0, ∀ x ∈ s, ∃ t ∈ c, ball x δ ⊆ t := by
   rw [sUnion_eq_Union] at hc₂ <;> simpa using lebesgue_number_lemma_of_metric hs (by simpa) hc₂
-#align lebesgue_number_lemma_of_metric_sUnion lebesgue_number_lemma_of_metric_unionₛ
+#align lebesgue_number_lemma_of_metric_sUnion lebesgue_number_lemma_of_metric_sUnion
 
 namespace Metric
 
@@ -4242,12 +4242,12 @@ theorem bounded_union : Bounded (s ∪ t) ↔ Bounded s ∧ Bounded t :=
   ⟨fun h => ⟨h.mono (by simp), h.mono (by simp)⟩, fun h => h.1.union h.2⟩
 #align metric.bounded_union Metric.bounded_union
 
-#print Metric.bounded_bunionᵢ /-
+#print Metric.bounded_biUnion /-
 /-- A finite union of bounded sets is bounded -/
-theorem bounded_bunionᵢ {I : Set β} {s : β → Set α} (H : I.Finite) :
+theorem bounded_biUnion {I : Set β} {s : β → Set α} (H : I.Finite) :
     Bounded (⋃ i ∈ I, s i) ↔ ∀ i ∈ I, Bounded (s i) :=
   Finite.induction_on H (by simp) fun x I _ _ IH => by simp [or_imp, forall_and, IH]
-#align metric.bounded_bUnion Metric.bounded_bunionᵢ
+#align metric.bounded_bUnion Metric.bounded_biUnion
 -/
 
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
@@ -4275,7 +4275,7 @@ theorem TotallyBounded.bounded {s : Set α} (h : TotallyBounded s) : Bounded s :
     -- and then argue that a finite union of bounded sets is bounded
     ⟨t, fint, subs⟩ :=
     (totallyBounded_iff.mp h) 1 zero_lt_one
-  Bounded.mono subs <| (bounded_bunionᵢ fint).2 fun i hi => bounded_ball
+  Bounded.mono subs <| (bounded_biUnion fint).2 fun i hi => bounded_ball
 #align totally_bounded.bounded TotallyBounded.bounded
 -/
 
@@ -4827,15 +4827,15 @@ theorem diam_ball {r : ℝ} (h : 0 ≤ r) : diam (ball x r) ≤ 2 * r :=
   diam_le_of_subset_closedBall h ball_subset_closedBall
 #align metric.diam_ball Metric.diam_ball
 
-/- warning: is_complete.nonempty_Inter_of_nonempty_bInter -> IsComplete.nonempty_interᵢ_of_nonempty_binterᵢ is a dubious translation:
+/- warning: is_complete.nonempty_Inter_of_nonempty_bInter -> IsComplete.nonempty_iInter_of_nonempty_biInter is a dubious translation:
 lean 3 declaration is
-  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {s : Nat -> (Set.{u1} α)}, (IsComplete.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1) (s (OfNat.ofNat.{0} Nat 0 (OfNat.mk.{0} Nat 0 (Zero.zero.{0} Nat Nat.hasZero))))) -> (forall (n : Nat), IsClosed.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) (s n)) -> (forall (n : Nat), Metric.Bounded.{u1} α _inst_1 (s n)) -> (forall (N : Nat), Set.Nonempty.{u1} α (Set.interᵢ.{u1, 1} α Nat (fun (n : Nat) => Set.interᵢ.{u1, 0} α (LE.le.{0} Nat Nat.hasLe n N) (fun (H : LE.le.{0} Nat Nat.hasLe n N) => s n)))) -> (Filter.Tendsto.{0, 0} Nat Real (fun (n : Nat) => Metric.diam.{u1} α _inst_1 (s n)) (Filter.atTop.{0} Nat (PartialOrder.toPreorder.{0} Nat (OrderedCancelAddCommMonoid.toPartialOrder.{0} Nat (StrictOrderedSemiring.toOrderedCancelAddCommMonoid.{0} Nat Nat.strictOrderedSemiring)))) (nhds.{0} Real (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero))))) -> (Set.Nonempty.{u1} α (Set.interᵢ.{u1, 1} α Nat (fun (n : Nat) => s n)))
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {s : Nat -> (Set.{u1} α)}, (IsComplete.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1) (s (OfNat.ofNat.{0} Nat 0 (OfNat.mk.{0} Nat 0 (Zero.zero.{0} Nat Nat.hasZero))))) -> (forall (n : Nat), IsClosed.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) (s n)) -> (forall (n : Nat), Metric.Bounded.{u1} α _inst_1 (s n)) -> (forall (N : Nat), Set.Nonempty.{u1} α (Set.iInter.{u1, 1} α Nat (fun (n : Nat) => Set.iInter.{u1, 0} α (LE.le.{0} Nat Nat.hasLe n N) (fun (H : LE.le.{0} Nat Nat.hasLe n N) => s n)))) -> (Filter.Tendsto.{0, 0} Nat Real (fun (n : Nat) => Metric.diam.{u1} α _inst_1 (s n)) (Filter.atTop.{0} Nat (PartialOrder.toPreorder.{0} Nat (OrderedCancelAddCommMonoid.toPartialOrder.{0} Nat (StrictOrderedSemiring.toOrderedCancelAddCommMonoid.{0} Nat Nat.strictOrderedSemiring)))) (nhds.{0} Real (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero))))) -> (Set.Nonempty.{u1} α (Set.iInter.{u1, 1} α Nat (fun (n : Nat) => s n)))
 but is expected to have type
-  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {s : Nat -> (Set.{u1} α)}, (IsComplete.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1) (s (OfNat.ofNat.{0} Nat 0 (instOfNatNat 0)))) -> (forall (n : Nat), IsClosed.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) (s n)) -> (forall (n : Nat), Metric.Bounded.{u1} α _inst_1 (s n)) -> (forall (N : Nat), Set.Nonempty.{u1} α (Set.interᵢ.{u1, 1} α Nat (fun (n : Nat) => Set.interᵢ.{u1, 0} α (LE.le.{0} Nat instLENat n N) (fun (H : LE.le.{0} Nat instLENat n N) => s n)))) -> (Filter.Tendsto.{0, 0} Nat Real (fun (n : Nat) => Metric.diam.{u1} α _inst_1 (s n)) (Filter.atTop.{0} Nat (PartialOrder.toPreorder.{0} Nat (StrictOrderedSemiring.toPartialOrder.{0} Nat Nat.strictOrderedSemiring))) (nhds.{0} Real (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal)))) -> (Set.Nonempty.{u1} α (Set.interᵢ.{u1, 1} α Nat (fun (n : Nat) => s n)))
-Case conversion may be inaccurate. Consider using '#align is_complete.nonempty_Inter_of_nonempty_bInter IsComplete.nonempty_interᵢ_of_nonempty_binterᵢₓ'. -/
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {s : Nat -> (Set.{u1} α)}, (IsComplete.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1) (s (OfNat.ofNat.{0} Nat 0 (instOfNatNat 0)))) -> (forall (n : Nat), IsClosed.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) (s n)) -> (forall (n : Nat), Metric.Bounded.{u1} α _inst_1 (s n)) -> (forall (N : Nat), Set.Nonempty.{u1} α (Set.iInter.{u1, 1} α Nat (fun (n : Nat) => Set.iInter.{u1, 0} α (LE.le.{0} Nat instLENat n N) (fun (H : LE.le.{0} Nat instLENat n N) => s n)))) -> (Filter.Tendsto.{0, 0} Nat Real (fun (n : Nat) => Metric.diam.{u1} α _inst_1 (s n)) (Filter.atTop.{0} Nat (PartialOrder.toPreorder.{0} Nat (StrictOrderedSemiring.toPartialOrder.{0} Nat Nat.strictOrderedSemiring))) (nhds.{0} Real (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal)))) -> (Set.Nonempty.{u1} α (Set.iInter.{u1, 1} α Nat (fun (n : Nat) => s n)))
+Case conversion may be inaccurate. Consider using '#align is_complete.nonempty_Inter_of_nonempty_bInter IsComplete.nonempty_iInter_of_nonempty_biInterₓ'. -/
 /-- If a family of complete sets with diameter tending to `0` is such that each finite intersection
 is nonempty, then the total intersection is also nonempty. -/
-theorem IsComplete.nonempty_interᵢ_of_nonempty_binterᵢ {s : ℕ → Set α} (h0 : IsComplete (s 0))
+theorem IsComplete.nonempty_iInter_of_nonempty_biInter {s : ℕ → Set α} (h0 : IsComplete (s 0))
     (hs : ∀ n, IsClosed (s n)) (h's : ∀ n, Bounded (s n)) (h : ∀ N, (⋂ n ≤ N, s n).Nonempty)
     (h' : Tendsto (fun n => diam (s n)) atTop (𝓝 0)) : (⋂ n, s n).Nonempty :=
   by
@@ -4857,21 +4857,21 @@ theorem IsComplete.nonempty_interᵢ_of_nonempty_binterᵢ {s : ℕ → Set α}
   apply (hs n).mem_of_tendsto xlim
   filter_upwards [Ici_mem_at_top n]with p hp
   exact I n p hp
-#align is_complete.nonempty_Inter_of_nonempty_bInter IsComplete.nonempty_interᵢ_of_nonempty_binterᵢ
+#align is_complete.nonempty_Inter_of_nonempty_bInter IsComplete.nonempty_iInter_of_nonempty_biInter
 
-/- warning: metric.nonempty_Inter_of_nonempty_bInter -> Metric.nonempty_interᵢ_of_nonempty_binterᵢ is a dubious translation:
+/- warning: metric.nonempty_Inter_of_nonempty_bInter -> Metric.nonempty_iInter_of_nonempty_biInter is a dubious translation:
 lean 3 declaration is
-  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] [_inst_2 : CompleteSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)] {s : Nat -> (Set.{u1} α)}, (forall (n : Nat), IsClosed.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) (s n)) -> (forall (n : Nat), Metric.Bounded.{u1} α _inst_1 (s n)) -> (forall (N : Nat), Set.Nonempty.{u1} α (Set.interᵢ.{u1, 1} α Nat (fun (n : Nat) => Set.interᵢ.{u1, 0} α (LE.le.{0} Nat Nat.hasLe n N) (fun (H : LE.le.{0} Nat Nat.hasLe n N) => s n)))) -> (Filter.Tendsto.{0, 0} Nat Real (fun (n : Nat) => Metric.diam.{u1} α _inst_1 (s n)) (Filter.atTop.{0} Nat (PartialOrder.toPreorder.{0} Nat (OrderedCancelAddCommMonoid.toPartialOrder.{0} Nat (StrictOrderedSemiring.toOrderedCancelAddCommMonoid.{0} Nat Nat.strictOrderedSemiring)))) (nhds.{0} Real (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero))))) -> (Set.Nonempty.{u1} α (Set.interᵢ.{u1, 1} α Nat (fun (n : Nat) => s n)))
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] [_inst_2 : CompleteSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)] {s : Nat -> (Set.{u1} α)}, (forall (n : Nat), IsClosed.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) (s n)) -> (forall (n : Nat), Metric.Bounded.{u1} α _inst_1 (s n)) -> (forall (N : Nat), Set.Nonempty.{u1} α (Set.iInter.{u1, 1} α Nat (fun (n : Nat) => Set.iInter.{u1, 0} α (LE.le.{0} Nat Nat.hasLe n N) (fun (H : LE.le.{0} Nat Nat.hasLe n N) => s n)))) -> (Filter.Tendsto.{0, 0} Nat Real (fun (n : Nat) => Metric.diam.{u1} α _inst_1 (s n)) (Filter.atTop.{0} Nat (PartialOrder.toPreorder.{0} Nat (OrderedCancelAddCommMonoid.toPartialOrder.{0} Nat (StrictOrderedSemiring.toOrderedCancelAddCommMonoid.{0} Nat Nat.strictOrderedSemiring)))) (nhds.{0} Real (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero))))) -> (Set.Nonempty.{u1} α (Set.iInter.{u1, 1} α Nat (fun (n : Nat) => s n)))
 but is expected to have type
-  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] [_inst_2 : CompleteSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)] {s : Nat -> (Set.{u1} α)}, (forall (n : Nat), IsClosed.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) (s n)) -> (forall (n : Nat), Metric.Bounded.{u1} α _inst_1 (s n)) -> (forall (N : Nat), Set.Nonempty.{u1} α (Set.interᵢ.{u1, 1} α Nat (fun (n : Nat) => Set.interᵢ.{u1, 0} α (LE.le.{0} Nat instLENat n N) (fun (H : LE.le.{0} Nat instLENat n N) => s n)))) -> (Filter.Tendsto.{0, 0} Nat Real (fun (n : Nat) => Metric.diam.{u1} α _inst_1 (s n)) (Filter.atTop.{0} Nat (PartialOrder.toPreorder.{0} Nat (StrictOrderedSemiring.toPartialOrder.{0} Nat Nat.strictOrderedSemiring))) (nhds.{0} Real (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal)))) -> (Set.Nonempty.{u1} α (Set.interᵢ.{u1, 1} α Nat (fun (n : Nat) => s n)))
-Case conversion may be inaccurate. Consider using '#align metric.nonempty_Inter_of_nonempty_bInter Metric.nonempty_interᵢ_of_nonempty_binterᵢₓ'. -/
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] [_inst_2 : CompleteSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)] {s : Nat -> (Set.{u1} α)}, (forall (n : Nat), IsClosed.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) (s n)) -> (forall (n : Nat), Metric.Bounded.{u1} α _inst_1 (s n)) -> (forall (N : Nat), Set.Nonempty.{u1} α (Set.iInter.{u1, 1} α Nat (fun (n : Nat) => Set.iInter.{u1, 0} α (LE.le.{0} Nat instLENat n N) (fun (H : LE.le.{0} Nat instLENat n N) => s n)))) -> (Filter.Tendsto.{0, 0} Nat Real (fun (n : Nat) => Metric.diam.{u1} α _inst_1 (s n)) (Filter.atTop.{0} Nat (PartialOrder.toPreorder.{0} Nat (StrictOrderedSemiring.toPartialOrder.{0} Nat Nat.strictOrderedSemiring))) (nhds.{0} Real (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal)))) -> (Set.Nonempty.{u1} α (Set.iInter.{u1, 1} α Nat (fun (n : Nat) => s n)))
+Case conversion may be inaccurate. Consider using '#align metric.nonempty_Inter_of_nonempty_bInter Metric.nonempty_iInter_of_nonempty_biInterₓ'. -/
 /-- In a complete space, if a family of closed sets with diameter tending to `0` is such that each
 finite intersection is nonempty, then the total intersection is also nonempty. -/
-theorem nonempty_interᵢ_of_nonempty_binterᵢ [CompleteSpace α] {s : ℕ → Set α}
+theorem nonempty_iInter_of_nonempty_biInter [CompleteSpace α] {s : ℕ → Set α}
     (hs : ∀ n, IsClosed (s n)) (h's : ∀ n, Bounded (s n)) (h : ∀ N, (⋂ n ≤ N, s n).Nonempty)
     (h' : Tendsto (fun n => diam (s n)) atTop (𝓝 0)) : (⋂ n, s n).Nonempty :=
-  (hs 0).IsComplete.nonempty_interᵢ_of_nonempty_binterᵢ hs h's h h'
-#align metric.nonempty_Inter_of_nonempty_bInter Metric.nonempty_interᵢ_of_nonempty_binterᵢ
+  (hs 0).IsComplete.nonempty_iInter_of_nonempty_biInter hs h's h h'
+#align metric.nonempty_Inter_of_nonempty_bInter Metric.nonempty_iInter_of_nonempty_biInter
 
 end Diam
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/08e1d8d4d989df3a6df86f385e9053ec8a372cc1

@@ -2466,7 +2466,7 @@ theorem Real.dist_eq (x y : ℝ) : dist x y = |x - y| :=
 lean 3 declaration is
   forall (x : Real) (y : Real), Eq.{1} NNReal (NNDist.nndist.{0} Real (PseudoMetricSpace.toNNDist.{0} Real Real.pseudoMetricSpace) x y) (coeFn.{1, 1} (MonoidWithZeroHom.{0, 0} Real NNReal (NonAssocSemiring.toMulZeroOneClass.{0} Real (NonAssocRing.toNonAssocSemiring.{0} Real (Ring.toNonAssocRing.{0} Real Real.ring))) (NonAssocSemiring.toMulZeroOneClass.{0} NNReal (Semiring.toNonAssocSemiring.{0} NNReal NNReal.semiring))) (fun (_x : MonoidWithZeroHom.{0, 0} Real NNReal (NonAssocSemiring.toMulZeroOneClass.{0} Real (NonAssocRing.toNonAssocSemiring.{0} Real (Ring.toNonAssocRing.{0} Real Real.ring))) (NonAssocSemiring.toMulZeroOneClass.{0} NNReal (Semiring.toNonAssocSemiring.{0} NNReal NNReal.semiring))) => Real -> NNReal) (MonoidWithZeroHom.hasCoeToFun.{0, 0} Real NNReal (NonAssocSemiring.toMulZeroOneClass.{0} Real (NonAssocRing.toNonAssocSemiring.{0} Real (Ring.toNonAssocRing.{0} Real Real.ring))) (NonAssocSemiring.toMulZeroOneClass.{0} NNReal (Semiring.toNonAssocSemiring.{0} NNReal NNReal.semiring))) Real.nnabs (HSub.hSub.{0, 0, 0} Real Real Real (instHSub.{0} Real Real.hasSub) x y))
 but is expected to have type
-  forall (x : Real) (y : Real), Eq.{1} NNReal (NNDist.nndist.{0} Real (PseudoMetricSpace.toNNDist.{0} Real Real.pseudoMetricSpace) x y) (FunLike.coe.{1, 1, 1} (MonoidWithZeroHom.{0, 0} Real NNReal (NonAssocSemiring.toMulZeroOneClass.{0} Real (NonAssocRing.toNonAssocSemiring.{0} Real (Ring.toNonAssocRing.{0} Real Real.instRingReal))) (NonAssocSemiring.toMulZeroOneClass.{0} NNReal (Semiring.toNonAssocSemiring.{0} NNReal instNNRealSemiring))) Real (fun (_x : Real) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : Real) => NNReal) _x) (MulHomClass.toFunLike.{0, 0, 0} (MonoidWithZeroHom.{0, 0} Real NNReal (NonAssocSemiring.toMulZeroOneClass.{0} Real (NonAssocRing.toNonAssocSemiring.{0} Real (Ring.toNonAssocRing.{0} Real Real.instRingReal))) (NonAssocSemiring.toMulZeroOneClass.{0} NNReal (Semiring.toNonAssocSemiring.{0} NNReal instNNRealSemiring))) Real NNReal (MulOneClass.toMul.{0} Real (MulZeroOneClass.toMulOneClass.{0} Real (NonAssocSemiring.toMulZeroOneClass.{0} Real (NonAssocRing.toNonAssocSemiring.{0} Real (Ring.toNonAssocRing.{0} Real Real.instRingReal))))) (MulOneClass.toMul.{0} NNReal (MulZeroOneClass.toMulOneClass.{0} NNReal (NonAssocSemiring.toMulZeroOneClass.{0} NNReal (Semiring.toNonAssocSemiring.{0} NNReal instNNRealSemiring)))) (MonoidHomClass.toMulHomClass.{0, 0, 0} (MonoidWithZeroHom.{0, 0} Real NNReal (NonAssocSemiring.toMulZeroOneClass.{0} Real (NonAssocRing.toNonAssocSemiring.{0} Real (Ring.toNonAssocRing.{0} Real Real.instRingReal))) (NonAssocSemiring.toMulZeroOneClass.{0} NNReal (Semiring.toNonAssocSemiring.{0} NNReal instNNRealSemiring))) Real NNReal (MulZeroOneClass.toMulOneClass.{0} Real (NonAssocSemiring.toMulZeroOneClass.{0} Real (NonAssocRing.toNonAssocSemiring.{0} Real (Ring.toNonAssocRing.{0} Real Real.instRingReal)))) (MulZeroOneClass.toMulOneClass.{0} NNReal (NonAssocSemiring.toMulZeroOneClass.{0} NNReal (Semiring.toNonAssocSemiring.{0} NNReal instNNRealSemiring))) (MonoidWithZeroHomClass.toMonoidHomClass.{0, 0, 0} (MonoidWithZeroHom.{0, 0} Real NNReal (NonAssocSemiring.toMulZeroOneClass.{0} Real (NonAssocRing.toNonAssocSemiring.{0} Real (Ring.toNonAssocRing.{0} Real Real.instRingReal))) (NonAssocSemiring.toMulZeroOneClass.{0} NNReal (Semiring.toNonAssocSemiring.{0} NNReal instNNRealSemiring))) Real NNReal (NonAssocSemiring.toMulZeroOneClass.{0} Real (NonAssocRing.toNonAssocSemiring.{0} Real (Ring.toNonAssocRing.{0} Real Real.instRingReal))) (NonAssocSemiring.toMulZeroOneClass.{0} NNReal (Semiring.toNonAssocSemiring.{0} NNReal instNNRealSemiring)) (MonoidWithZeroHom.monoidWithZeroHomClass.{0, 0} Real NNReal (NonAssocSemiring.toMulZeroOneClass.{0} Real (NonAssocRing.toNonAssocSemiring.{0} Real (Ring.toNonAssocRing.{0} Real Real.instRingReal))) (NonAssocSemiring.toMulZeroOneClass.{0} NNReal (Semiring.toNonAssocSemiring.{0} NNReal instNNRealSemiring)))))) Real.nnabs (HSub.hSub.{0, 0, 0} Real Real Real (instHSub.{0} Real Real.instSubReal) x y))
+  forall (x : Real) (y : Real), Eq.{1} NNReal (NNDist.nndist.{0} Real (PseudoMetricSpace.toNNDist.{0} Real Real.pseudoMetricSpace) x y) (FunLike.coe.{1, 1, 1} (MonoidWithZeroHom.{0, 0} Real NNReal (NonAssocSemiring.toMulZeroOneClass.{0} Real (Semiring.toNonAssocSemiring.{0} Real Real.semiring)) (NonAssocSemiring.toMulZeroOneClass.{0} NNReal (Semiring.toNonAssocSemiring.{0} NNReal instNNRealSemiring))) Real (fun (_x : Real) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : Real) => NNReal) _x) (MulHomClass.toFunLike.{0, 0, 0} (MonoidWithZeroHom.{0, 0} Real NNReal (NonAssocSemiring.toMulZeroOneClass.{0} Real (Semiring.toNonAssocSemiring.{0} Real Real.semiring)) (NonAssocSemiring.toMulZeroOneClass.{0} NNReal (Semiring.toNonAssocSemiring.{0} NNReal instNNRealSemiring))) Real NNReal (MulOneClass.toMul.{0} Real (MulZeroOneClass.toMulOneClass.{0} Real (NonAssocSemiring.toMulZeroOneClass.{0} Real (Semiring.toNonAssocSemiring.{0} Real Real.semiring)))) (MulOneClass.toMul.{0} NNReal (MulZeroOneClass.toMulOneClass.{0} NNReal (NonAssocSemiring.toMulZeroOneClass.{0} NNReal (Semiring.toNonAssocSemiring.{0} NNReal instNNRealSemiring)))) (MonoidHomClass.toMulHomClass.{0, 0, 0} (MonoidWithZeroHom.{0, 0} Real NNReal (NonAssocSemiring.toMulZeroOneClass.{0} Real (Semiring.toNonAssocSemiring.{0} Real Real.semiring)) (NonAssocSemiring.toMulZeroOneClass.{0} NNReal (Semiring.toNonAssocSemiring.{0} NNReal instNNRealSemiring))) Real NNReal (MulZeroOneClass.toMulOneClass.{0} Real (NonAssocSemiring.toMulZeroOneClass.{0} Real (Semiring.toNonAssocSemiring.{0} Real Real.semiring))) (MulZeroOneClass.toMulOneClass.{0} NNReal (NonAssocSemiring.toMulZeroOneClass.{0} NNReal (Semiring.toNonAssocSemiring.{0} NNReal instNNRealSemiring))) (MonoidWithZeroHomClass.toMonoidHomClass.{0, 0, 0} (MonoidWithZeroHom.{0, 0} Real NNReal (NonAssocSemiring.toMulZeroOneClass.{0} Real (Semiring.toNonAssocSemiring.{0} Real Real.semiring)) (NonAssocSemiring.toMulZeroOneClass.{0} NNReal (Semiring.toNonAssocSemiring.{0} NNReal instNNRealSemiring))) Real NNReal (NonAssocSemiring.toMulZeroOneClass.{0} Real (Semiring.toNonAssocSemiring.{0} Real Real.semiring)) (NonAssocSemiring.toMulZeroOneClass.{0} NNReal (Semiring.toNonAssocSemiring.{0} NNReal instNNRealSemiring)) (MonoidWithZeroHom.monoidWithZeroHomClass.{0, 0} Real NNReal (NonAssocSemiring.toMulZeroOneClass.{0} Real (Semiring.toNonAssocSemiring.{0} Real Real.semiring)) (NonAssocSemiring.toMulZeroOneClass.{0} NNReal (Semiring.toNonAssocSemiring.{0} NNReal instNNRealSemiring)))))) Real.nnabs (HSub.hSub.{0, 0, 0} Real Real Real (instHSub.{0} Real Real.instSubReal) x y))
 Case conversion may be inaccurate. Consider using '#align real.nndist_eq Real.nndist_eqₓ'. -/
 theorem Real.nndist_eq (x y : ℝ) : nndist x y = Real.nnabs (x - y) :=
   rfl
@@ -2476,7 +2476,7 @@ theorem Real.nndist_eq (x y : ℝ) : nndist x y = Real.nnabs (x - y) :=
 lean 3 declaration is
   forall (x : Real) (y : Real), Eq.{1} NNReal (NNDist.nndist.{0} Real (PseudoMetricSpace.toNNDist.{0} Real Real.pseudoMetricSpace) x y) (coeFn.{1, 1} (MonoidWithZeroHom.{0, 0} Real NNReal (NonAssocSemiring.toMulZeroOneClass.{0} Real (NonAssocRing.toNonAssocSemiring.{0} Real (Ring.toNonAssocRing.{0} Real Real.ring))) (NonAssocSemiring.toMulZeroOneClass.{0} NNReal (Semiring.toNonAssocSemiring.{0} NNReal NNReal.semiring))) (fun (_x : MonoidWithZeroHom.{0, 0} Real NNReal (NonAssocSemiring.toMulZeroOneClass.{0} Real (NonAssocRing.toNonAssocSemiring.{0} Real (Ring.toNonAssocRing.{0} Real Real.ring))) (NonAssocSemiring.toMulZeroOneClass.{0} NNReal (Semiring.toNonAssocSemiring.{0} NNReal NNReal.semiring))) => Real -> NNReal) (MonoidWithZeroHom.hasCoeToFun.{0, 0} Real NNReal (NonAssocSemiring.toMulZeroOneClass.{0} Real (NonAssocRing.toNonAssocSemiring.{0} Real (Ring.toNonAssocRing.{0} Real Real.ring))) (NonAssocSemiring.toMulZeroOneClass.{0} NNReal (Semiring.toNonAssocSemiring.{0} NNReal NNReal.semiring))) Real.nnabs (HSub.hSub.{0, 0, 0} Real Real Real (instHSub.{0} Real Real.hasSub) y x))
 but is expected to have type
-  forall (x : Real) (y : Real), Eq.{1} NNReal (NNDist.nndist.{0} Real (PseudoMetricSpace.toNNDist.{0} Real Real.pseudoMetricSpace) x y) (FunLike.coe.{1, 1, 1} (MonoidWithZeroHom.{0, 0} Real NNReal (NonAssocSemiring.toMulZeroOneClass.{0} Real (NonAssocRing.toNonAssocSemiring.{0} Real (Ring.toNonAssocRing.{0} Real Real.instRingReal))) (NonAssocSemiring.toMulZeroOneClass.{0} NNReal (Semiring.toNonAssocSemiring.{0} NNReal instNNRealSemiring))) Real (fun (_x : Real) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : Real) => NNReal) _x) (MulHomClass.toFunLike.{0, 0, 0} (MonoidWithZeroHom.{0, 0} Real NNReal (NonAssocSemiring.toMulZeroOneClass.{0} Real (NonAssocRing.toNonAssocSemiring.{0} Real (Ring.toNonAssocRing.{0} Real Real.instRingReal))) (NonAssocSemiring.toMulZeroOneClass.{0} NNReal (Semiring.toNonAssocSemiring.{0} NNReal instNNRealSemiring))) Real NNReal (MulOneClass.toMul.{0} Real (MulZeroOneClass.toMulOneClass.{0} Real (NonAssocSemiring.toMulZeroOneClass.{0} Real (NonAssocRing.toNonAssocSemiring.{0} Real (Ring.toNonAssocRing.{0} Real Real.instRingReal))))) (MulOneClass.toMul.{0} NNReal (MulZeroOneClass.toMulOneClass.{0} NNReal (NonAssocSemiring.toMulZeroOneClass.{0} NNReal (Semiring.toNonAssocSemiring.{0} NNReal instNNRealSemiring)))) (MonoidHomClass.toMulHomClass.{0, 0, 0} (MonoidWithZeroHom.{0, 0} Real NNReal (NonAssocSemiring.toMulZeroOneClass.{0} Real (NonAssocRing.toNonAssocSemiring.{0} Real (Ring.toNonAssocRing.{0} Real Real.instRingReal))) (NonAssocSemiring.toMulZeroOneClass.{0} NNReal (Semiring.toNonAssocSemiring.{0} NNReal instNNRealSemiring))) Real NNReal (MulZeroOneClass.toMulOneClass.{0} Real (NonAssocSemiring.toMulZeroOneClass.{0} Real (NonAssocRing.toNonAssocSemiring.{0} Real (Ring.toNonAssocRing.{0} Real Real.instRingReal)))) (MulZeroOneClass.toMulOneClass.{0} NNReal (NonAssocSemiring.toMulZeroOneClass.{0} NNReal (Semiring.toNonAssocSemiring.{0} NNReal instNNRealSemiring))) (MonoidWithZeroHomClass.toMonoidHomClass.{0, 0, 0} (MonoidWithZeroHom.{0, 0} Real NNReal (NonAssocSemiring.toMulZeroOneClass.{0} Real (NonAssocRing.toNonAssocSemiring.{0} Real (Ring.toNonAssocRing.{0} Real Real.instRingReal))) (NonAssocSemiring.toMulZeroOneClass.{0} NNReal (Semiring.toNonAssocSemiring.{0} NNReal instNNRealSemiring))) Real NNReal (NonAssocSemiring.toMulZeroOneClass.{0} Real (NonAssocRing.toNonAssocSemiring.{0} Real (Ring.toNonAssocRing.{0} Real Real.instRingReal))) (NonAssocSemiring.toMulZeroOneClass.{0} NNReal (Semiring.toNonAssocSemiring.{0} NNReal instNNRealSemiring)) (MonoidWithZeroHom.monoidWithZeroHomClass.{0, 0} Real NNReal (NonAssocSemiring.toMulZeroOneClass.{0} Real (NonAssocRing.toNonAssocSemiring.{0} Real (Ring.toNonAssocRing.{0} Real Real.instRingReal))) (NonAssocSemiring.toMulZeroOneClass.{0} NNReal (Semiring.toNonAssocSemiring.{0} NNReal instNNRealSemiring)))))) Real.nnabs (HSub.hSub.{0, 0, 0} Real Real Real (instHSub.{0} Real Real.instSubReal) y x))
+  forall (x : Real) (y : Real), Eq.{1} NNReal (NNDist.nndist.{0} Real (PseudoMetricSpace.toNNDist.{0} Real Real.pseudoMetricSpace) x y) (FunLike.coe.{1, 1, 1} (MonoidWithZeroHom.{0, 0} Real NNReal (NonAssocSemiring.toMulZeroOneClass.{0} Real (Semiring.toNonAssocSemiring.{0} Real Real.semiring)) (NonAssocSemiring.toMulZeroOneClass.{0} NNReal (Semiring.toNonAssocSemiring.{0} NNReal instNNRealSemiring))) Real (fun (_x : Real) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : Real) => NNReal) _x) (MulHomClass.toFunLike.{0, 0, 0} (MonoidWithZeroHom.{0, 0} Real NNReal (NonAssocSemiring.toMulZeroOneClass.{0} Real (Semiring.toNonAssocSemiring.{0} Real Real.semiring)) (NonAssocSemiring.toMulZeroOneClass.{0} NNReal (Semiring.toNonAssocSemiring.{0} NNReal instNNRealSemiring))) Real NNReal (MulOneClass.toMul.{0} Real (MulZeroOneClass.toMulOneClass.{0} Real (NonAssocSemiring.toMulZeroOneClass.{0} Real (Semiring.toNonAssocSemiring.{0} Real Real.semiring)))) (MulOneClass.toMul.{0} NNReal (MulZeroOneClass.toMulOneClass.{0} NNReal (NonAssocSemiring.toMulZeroOneClass.{0} NNReal (Semiring.toNonAssocSemiring.{0} NNReal instNNRealSemiring)))) (MonoidHomClass.toMulHomClass.{0, 0, 0} (MonoidWithZeroHom.{0, 0} Real NNReal (NonAssocSemiring.toMulZeroOneClass.{0} Real (Semiring.toNonAssocSemiring.{0} Real Real.semiring)) (NonAssocSemiring.toMulZeroOneClass.{0} NNReal (Semiring.toNonAssocSemiring.{0} NNReal instNNRealSemiring))) Real NNReal (MulZeroOneClass.toMulOneClass.{0} Real (NonAssocSemiring.toMulZeroOneClass.{0} Real (Semiring.toNonAssocSemiring.{0} Real Real.semiring))) (MulZeroOneClass.toMulOneClass.{0} NNReal (NonAssocSemiring.toMulZeroOneClass.{0} NNReal (Semiring.toNonAssocSemiring.{0} NNReal instNNRealSemiring))) (MonoidWithZeroHomClass.toMonoidHomClass.{0, 0, 0} (MonoidWithZeroHom.{0, 0} Real NNReal (NonAssocSemiring.toMulZeroOneClass.{0} Real (Semiring.toNonAssocSemiring.{0} Real Real.semiring)) (NonAssocSemiring.toMulZeroOneClass.{0} NNReal (Semiring.toNonAssocSemiring.{0} NNReal instNNRealSemiring))) Real NNReal (NonAssocSemiring.toMulZeroOneClass.{0} Real (Semiring.toNonAssocSemiring.{0} Real Real.semiring)) (NonAssocSemiring.toMulZeroOneClass.{0} NNReal (Semiring.toNonAssocSemiring.{0} NNReal instNNRealSemiring)) (MonoidWithZeroHom.monoidWithZeroHomClass.{0, 0} Real NNReal (NonAssocSemiring.toMulZeroOneClass.{0} Real (Semiring.toNonAssocSemiring.{0} Real Real.semiring)) (NonAssocSemiring.toMulZeroOneClass.{0} NNReal (Semiring.toNonAssocSemiring.{0} NNReal instNNRealSemiring)))))) Real.nnabs (HSub.hSub.{0, 0, 0} Real Real Real (instHSub.{0} Real Real.instSubReal) y x))
 Case conversion may be inaccurate. Consider using '#align real.nndist_eq' Real.nndist_eq'ₓ'. -/
 theorem Real.nndist_eq' (x y : ℝ) : nndist x y = Real.nnabs (y - x) :=
   nndist_comm _ _

mathlib commit https://github.com/leanprover-community/mathlib/commit/3905fa80e62c0898131285baab35559fbc4e5cda

@@ -1614,7 +1614,7 @@ theorem totallyBounded_of_finite_discretization {s : Set α}
 lean 3 declaration is
   forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {s : Set.{u1} α}, (TotallyBounded.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1) s) -> (forall (ε : Real), (GT.gt.{0} Real Real.hasLt ε (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero)))) -> (Exists.{succ u1} (Set.{u1} α) (fun (t : Set.{u1} α) => Exists.{0} (HasSubset.Subset.{u1} (Set.{u1} α) (Set.hasSubset.{u1} α) t s) (fun (H : HasSubset.Subset.{u1} (Set.{u1} α) (Set.hasSubset.{u1} α) t s) => And (Set.Finite.{u1} α t) (HasSubset.Subset.{u1} (Set.{u1} α) (Set.hasSubset.{u1} α) s (Set.unionᵢ.{u1, succ u1} α α (fun (y : α) => Set.unionᵢ.{u1, 0} α (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) y t) (fun (H : Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) y t) => Metric.ball.{u1} α _inst_1 y ε))))))))
 but is expected to have type
-  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {s : Set.{u1} α}, (TotallyBounded.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1) s) -> (forall (ε : Real), (GT.gt.{0} Real Real.instLTReal ε (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal))) -> (Exists.{succ u1} (Set.{u1} α) (fun (t : Set.{u1} α) => And (HasSubset.Subset.{u1} (Set.{u1} α) (Set.instHasSubsetSet.{u1} α) t s) (And (Set.Finite.{u1} α t) (HasSubset.Subset.{u1} (Set.{u1} α) (Set.instHasSubsetSet.{u1} α) s (Set.unionᵢ.{u1, succ u1} α α (fun (y : α) => Set.unionᵢ.{u1, 0} α (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) y t) (fun (h._@.Mathlib.Topology.MetricSpace.Basic._hyg.10746 : Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) y t) => Metric.ball.{u1} α _inst_1 y ε))))))))
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {s : Set.{u1} α}, (TotallyBounded.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1) s) -> (forall (ε : Real), (GT.gt.{0} Real Real.instLTReal ε (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal))) -> (Exists.{succ u1} (Set.{u1} α) (fun (t : Set.{u1} α) => And (HasSubset.Subset.{u1} (Set.{u1} α) (Set.instHasSubsetSet.{u1} α) t s) (And (Set.Finite.{u1} α t) (HasSubset.Subset.{u1} (Set.{u1} α) (Set.instHasSubsetSet.{u1} α) s (Set.unionᵢ.{u1, succ u1} α α (fun (y : α) => Set.unionᵢ.{u1, 0} α (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) y t) (fun (h._@.Mathlib.Topology.MetricSpace.Basic._hyg.10766 : Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) y t) => Metric.ball.{u1} α _inst_1 y ε))))))))
 Case conversion may be inaccurate. Consider using '#align metric.finite_approx_of_totally_bounded Metric.finite_approx_of_totallyBoundedₓ'. -/
 /- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (t «expr ⊆ » s) -/
 theorem finite_approx_of_totallyBounded {s : Set α} (hs : TotallyBounded s) :
@@ -3816,7 +3816,7 @@ section Compact
 lean 3 declaration is
   forall {α : Type.{u1}} [_inst_2 : PseudoMetricSpace.{u1} α] {s : Set.{u1} α}, (IsCompact.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_2)) s) -> (forall {e : Real}, (LT.lt.{0} Real Real.hasLt (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero))) e) -> (Exists.{succ u1} (Set.{u1} α) (fun (t : Set.{u1} α) => Exists.{0} (HasSubset.Subset.{u1} (Set.{u1} α) (Set.hasSubset.{u1} α) t s) (fun (H : HasSubset.Subset.{u1} (Set.{u1} α) (Set.hasSubset.{u1} α) t s) => And (Set.Finite.{u1} α t) (HasSubset.Subset.{u1} (Set.{u1} α) (Set.hasSubset.{u1} α) s (Set.unionᵢ.{u1, succ u1} α α (fun (x : α) => Set.unionᵢ.{u1, 0} α (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) x t) (fun (H : Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) x t) => Metric.ball.{u1} α _inst_2 x e))))))))
 but is expected to have type
-  forall {α : Type.{u1}} [_inst_2 : PseudoMetricSpace.{u1} α] {s : Set.{u1} α}, (IsCompact.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_2)) s) -> (forall {e : Real}, (LT.lt.{0} Real Real.instLTReal (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal)) e) -> (Exists.{succ u1} (Set.{u1} α) (fun (t : Set.{u1} α) => And (HasSubset.Subset.{u1} (Set.{u1} α) (Set.instHasSubsetSet.{u1} α) t s) (And (Set.Finite.{u1} α t) (HasSubset.Subset.{u1} (Set.{u1} α) (Set.instHasSubsetSet.{u1} α) s (Set.unionᵢ.{u1, succ u1} α α (fun (x : α) => Set.unionᵢ.{u1, 0} α (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) x t) (fun (h._@.Mathlib.Topology.MetricSpace.Basic._hyg.25496 : Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) x t) => Metric.ball.{u1} α _inst_2 x e))))))))
+  forall {α : Type.{u1}} [_inst_2 : PseudoMetricSpace.{u1} α] {s : Set.{u1} α}, (IsCompact.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_2)) s) -> (forall {e : Real}, (LT.lt.{0} Real Real.instLTReal (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal)) e) -> (Exists.{succ u1} (Set.{u1} α) (fun (t : Set.{u1} α) => And (HasSubset.Subset.{u1} (Set.{u1} α) (Set.instHasSubsetSet.{u1} α) t s) (And (Set.Finite.{u1} α t) (HasSubset.Subset.{u1} (Set.{u1} α) (Set.instHasSubsetSet.{u1} α) s (Set.unionᵢ.{u1, succ u1} α α (fun (x : α) => Set.unionᵢ.{u1, 0} α (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) x t) (fun (h._@.Mathlib.Topology.MetricSpace.Basic._hyg.25516 : Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) x t) => Metric.ball.{u1} α _inst_2 x e))))))))
 Case conversion may be inaccurate. Consider using '#align finite_cover_balls_of_compact finite_cover_balls_of_compactₓ'. -/
 /- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (t «expr ⊆ » s) -/
 /-- Any compact set in a pseudometric space can be covered by finitely many balls of a given
@@ -3836,7 +3836,7 @@ theorem finite_cover_balls_of_compact {α : Type u} [PseudoMetricSpace α] {s :
 lean 3 declaration is
   forall {α : Type.{u1}} [_inst_2 : PseudoMetricSpace.{u1} α] {s : Set.{u1} α}, (IsCompact.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_2)) s) -> (forall {e : Real}, (LT.lt.{0} Real Real.hasLt (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero))) e) -> (Exists.{succ u1} (Set.{u1} α) (fun (t : Set.{u1} α) => Exists.{0} (HasSubset.Subset.{u1} (Set.{u1} α) (Set.hasSubset.{u1} α) t s) (fun (H : HasSubset.Subset.{u1} (Set.{u1} α) (Set.hasSubset.{u1} α) t s) => And (Set.Finite.{u1} α t) (HasSubset.Subset.{u1} (Set.{u1} α) (Set.hasSubset.{u1} α) s (Set.unionᵢ.{u1, succ u1} α α (fun (x : α) => Set.unionᵢ.{u1, 0} α (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) x t) (fun (H : Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) x t) => Metric.ball.{u1} α _inst_2 x e))))))))
 but is expected to have type
-  forall {α : Type.{u1}} [_inst_2 : PseudoMetricSpace.{u1} α] {s : Set.{u1} α}, (IsCompact.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_2)) s) -> (forall {e : Real}, (LT.lt.{0} Real Real.instLTReal (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal)) e) -> (Exists.{succ u1} (Set.{u1} α) (fun (t : Set.{u1} α) => And (HasSubset.Subset.{u1} (Set.{u1} α) (Set.instHasSubsetSet.{u1} α) t s) (And (Set.Finite.{u1} α t) (HasSubset.Subset.{u1} (Set.{u1} α) (Set.instHasSubsetSet.{u1} α) s (Set.unionᵢ.{u1, succ u1} α α (fun (x : α) => Set.unionᵢ.{u1, 0} α (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) x t) (fun (h._@.Mathlib.Topology.MetricSpace.Basic._hyg.25496 : Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) x t) => Metric.ball.{u1} α _inst_2 x e))))))))
+  forall {α : Type.{u1}} [_inst_2 : PseudoMetricSpace.{u1} α] {s : Set.{u1} α}, (IsCompact.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_2)) s) -> (forall {e : Real}, (LT.lt.{0} Real Real.instLTReal (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal)) e) -> (Exists.{succ u1} (Set.{u1} α) (fun (t : Set.{u1} α) => And (HasSubset.Subset.{u1} (Set.{u1} α) (Set.instHasSubsetSet.{u1} α) t s) (And (Set.Finite.{u1} α t) (HasSubset.Subset.{u1} (Set.{u1} α) (Set.instHasSubsetSet.{u1} α) s (Set.unionᵢ.{u1, succ u1} α α (fun (x : α) => Set.unionᵢ.{u1, 0} α (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) x t) (fun (h._@.Mathlib.Topology.MetricSpace.Basic._hyg.25516 : Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) x t) => Metric.ball.{u1} α _inst_2 x e))))))))
 Case conversion may be inaccurate. Consider using '#align is_compact.finite_cover_balls IsCompact.finite_cover_ballsₓ'. -/
 alias finite_cover_balls_of_compact ← IsCompact.finite_cover_balls
 #align is_compact.finite_cover_balls IsCompact.finite_cover_balls

mathlib commit https://github.com/leanprover-community/mathlib/commit/8b8ba04e2f326f3f7cf24ad129beda58531ada61

@@ -3770,7 +3770,7 @@ theorem closedBall_pi' [Nonempty β] (x : ∀ b, π b) (r : ℝ) :
 lean 3 declaration is
   forall {n : Nat} {α : (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) -> Type.{u1}} [_inst_4 : forall (i : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))), PseudoMetricSpace.{u1} (α i)] (i : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (x : α i) (y : α i) (f : forall (j : Fin n), α (coeFn.{1, 1} (OrderEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (Fin.hasLe n) (Preorder.toLE.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (PartialOrder.toPreorder.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (Fin.partialOrder (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne)))))))) (fun (_x : RelEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (LE.le.{0} (Fin n) (Fin.hasLe n)) (LE.le.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (Preorder.toLE.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (PartialOrder.toPreorder.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (Fin.partialOrder (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))))))) => (Fin n) -> (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne)))))) (RelEmbedding.hasCoeToFun.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (LE.le.{0} (Fin n) (Fin.hasLe n)) (LE.le.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (Preorder.toLE.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (PartialOrder.toPreorder.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} 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 but is expected to have type
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(Fin.instLatticeFinHAddNatInstHAddInstAddNatOfNat n) (OrderHomClass.toLatticeHomClass.{0, 0, 0} (OrderEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin n) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))))) (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (Fin.instLinearOrderFin n) (Fin.instLatticeFinHAddNatInstHAddInstAddNatOfNat n) (RelEmbedding.instRelHomClassRelEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.680 : Fin n) (x._@.Mathlib.Order.Hom.Basic._hyg.682 : Fin n) => LE.le.{0} (Fin n) (instLEFin n) x._@.Mathlib.Order.Hom.Basic._hyg.680 x._@.Mathlib.Order.Hom.Basic._hyg.682) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.695 : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (x._@.Mathlib.Order.Hom.Basic._hyg.697 : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) => LE.le.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) x._@.Mathlib.Order.Hom.Basic._hyg.695 x._@.Mathlib.Order.Hom.Basic._hyg.697))))) (Fin.succAbove n i) b)))) f g))
 Case conversion may be inaccurate. Consider using '#align fin.nndist_insert_nth_insert_nth Fin.nndist_insertNth_insertNthₓ'. -/
 @[simp]
 theorem Fin.nndist_insertNth_insertNth {n : ℕ} {α : Fin (n + 1) → Type _}
@@ -3783,7 +3783,7 @@ theorem Fin.nndist_insertNth_insertNth {n : ℕ} {α : Fin (n + 1) → Type _}
 lean 3 declaration is
   forall {n : Nat} {α : (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) -> Type.{u1}} [_inst_4 : forall (i : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))), PseudoMetricSpace.{u1} (α i)] (i : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (x : α i) (y : α i) (f : forall (j : Fin n), α (coeFn.{1, 1} (OrderEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (Fin.hasLe n) (Preorder.toLE.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (PartialOrder.toPreorder.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (Fin.partialOrder (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne)))))))) (fun (_x : RelEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (LE.le.{0} (Fin n) (Fin.hasLe n)) (LE.le.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (Preorder.toLE.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (PartialOrder.toPreorder.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (Fin.partialOrder (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))))))) => (Fin n) -> (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne)))))) (RelEmbedding.hasCoeToFun.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (LE.le.{0} (Fin n) (Fin.hasLe n)) (LE.le.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (Preorder.toLE.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (PartialOrder.toPreorder.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (Fin.partialOrder (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))))))) (Fin.succAbove n i) j)) (g : forall (j : Fin n), α (coeFn.{1, 1} (OrderEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (Fin.hasLe n) (Preorder.toLE.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (PartialOrder.toPreorder.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (Fin.partialOrder (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne)))))))) (fun (_x : RelEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (LE.le.{0} (Fin n) (Fin.hasLe n)) (LE.le.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (Preorder.toLE.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (PartialOrder.toPreorder.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (Fin.partialOrder (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))))))) => (Fin n) -> (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne)))))) (RelEmbedding.hasCoeToFun.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (LE.le.{0} (Fin n) (Fin.hasLe n)) (LE.le.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (Preorder.toLE.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (PartialOrder.toPreorder.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (Fin.partialOrder (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))))))) (Fin.succAbove n i) j)), Eq.{1} Real (Dist.dist.{u1} (forall (j : Fin (HAdd.hAdd.{0, 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 but is expected to have type
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(Fin.instLatticeFinHAddNatInstHAddInstAddNatOfNat n) (OrderHomClass.toLatticeHomClass.{0, 0, 0} (OrderEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin n) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))))) (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (Fin.instLinearOrderFin n) (Fin.instLatticeFinHAddNatInstHAddInstAddNatOfNat n) (RelEmbedding.instRelHomClassRelEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.680 : Fin n) (x._@.Mathlib.Order.Hom.Basic._hyg.682 : Fin n) => LE.le.{0} (Fin n) (instLEFin n) x._@.Mathlib.Order.Hom.Basic._hyg.680 x._@.Mathlib.Order.Hom.Basic._hyg.682) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.695 : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (x._@.Mathlib.Order.Hom.Basic._hyg.697 : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) => LE.le.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) x._@.Mathlib.Order.Hom.Basic._hyg.695 x._@.Mathlib.Order.Hom.Basic._hyg.697))))) (Fin.succAbove n i) b)))) f g))
 Case conversion may be inaccurate. Consider using '#align fin.dist_insert_nth_insert_nth Fin.dist_insertNth_insertNthₓ'. -/
 @[simp]
 theorem Fin.dist_insertNth_insertNth {n : ℕ} {α : Fin (n + 1) → Type _}

mathlib commit https://github.com/leanprover-community/mathlib/commit/fa78268d4d77cb2b2fbc89f0527e2e7807763780

@@ -3359,10 +3359,12 @@ theorem closure_closedBall : closure (closedBall x ε) = closedBall x ε :=
 #align metric.closure_closed_ball Metric.closure_closedBall
 -/
 
+#print Metric.closure_sphere /-
 @[simp]
 theorem closure_sphere : closure (sphere x ε) = sphere x ε :=
   isClosed_sphere.closure_eq
 #align metric.closure_sphere Metric.closure_sphere
+-/
 
 #print Metric.closure_ball_subset_closedBall /-
 theorem closure_ball_subset_closedBall : closure (ball x ε) ⊆ closedBall x ε :=
@@ -3814,7 +3816,7 @@ section Compact
 lean 3 declaration is
   forall {α : Type.{u1}} [_inst_2 : PseudoMetricSpace.{u1} α] {s : Set.{u1} α}, (IsCompact.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_2)) s) -> (forall {e : Real}, (LT.lt.{0} Real Real.hasLt (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero))) e) -> (Exists.{succ u1} (Set.{u1} α) (fun (t : Set.{u1} α) => Exists.{0} (HasSubset.Subset.{u1} (Set.{u1} α) (Set.hasSubset.{u1} α) t s) (fun (H : HasSubset.Subset.{u1} (Set.{u1} α) (Set.hasSubset.{u1} α) t s) => And (Set.Finite.{u1} α t) (HasSubset.Subset.{u1} (Set.{u1} α) (Set.hasSubset.{u1} α) s (Set.unionᵢ.{u1, succ u1} α α (fun (x : α) => Set.unionᵢ.{u1, 0} α (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) x t) (fun (H : Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) x t) => Metric.ball.{u1} α _inst_2 x e))))))))
 but is expected to have type
-  forall {α : Type.{u1}} [_inst_2 : PseudoMetricSpace.{u1} α] {s : Set.{u1} α}, (IsCompact.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_2)) s) -> (forall {e : Real}, (LT.lt.{0} Real Real.instLTReal (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal)) e) -> (Exists.{succ u1} (Set.{u1} α) (fun (t : Set.{u1} α) => And (HasSubset.Subset.{u1} (Set.{u1} α) (Set.instHasSubsetSet.{u1} α) t s) (And (Set.Finite.{u1} α t) (HasSubset.Subset.{u1} (Set.{u1} α) (Set.instHasSubsetSet.{u1} α) s (Set.unionᵢ.{u1, succ u1} α α (fun (x : α) => Set.unionᵢ.{u1, 0} α (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) x t) (fun (h._@.Mathlib.Topology.MetricSpace.Basic._hyg.25460 : Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) x t) => Metric.ball.{u1} α _inst_2 x e))))))))
+  forall {α : Type.{u1}} [_inst_2 : PseudoMetricSpace.{u1} α] {s : Set.{u1} α}, (IsCompact.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_2)) s) -> (forall {e : Real}, (LT.lt.{0} Real Real.instLTReal (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal)) e) -> (Exists.{succ u1} (Set.{u1} α) (fun (t : Set.{u1} α) => And (HasSubset.Subset.{u1} (Set.{u1} α) (Set.instHasSubsetSet.{u1} α) t s) (And (Set.Finite.{u1} α t) (HasSubset.Subset.{u1} (Set.{u1} α) (Set.instHasSubsetSet.{u1} α) s (Set.unionᵢ.{u1, succ u1} α α (fun (x : α) => Set.unionᵢ.{u1, 0} α (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) x t) (fun (h._@.Mathlib.Topology.MetricSpace.Basic._hyg.25496 : Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) x t) => Metric.ball.{u1} α _inst_2 x e))))))))
 Case conversion may be inaccurate. Consider using '#align finite_cover_balls_of_compact finite_cover_balls_of_compactₓ'. -/
 /- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (t «expr ⊆ » s) -/
 /-- Any compact set in a pseudometric space can be covered by finitely many balls of a given
@@ -3834,7 +3836,7 @@ theorem finite_cover_balls_of_compact {α : Type u} [PseudoMetricSpace α] {s :
 lean 3 declaration is
   forall {α : Type.{u1}} [_inst_2 : PseudoMetricSpace.{u1} α] {s : Set.{u1} α}, (IsCompact.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_2)) s) -> (forall {e : Real}, (LT.lt.{0} Real Real.hasLt (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero))) e) -> (Exists.{succ u1} (Set.{u1} α) (fun (t : Set.{u1} α) => Exists.{0} (HasSubset.Subset.{u1} (Set.{u1} α) (Set.hasSubset.{u1} α) t s) (fun (H : HasSubset.Subset.{u1} (Set.{u1} α) (Set.hasSubset.{u1} α) t s) => And (Set.Finite.{u1} α t) (HasSubset.Subset.{u1} (Set.{u1} α) (Set.hasSubset.{u1} α) s (Set.unionᵢ.{u1, succ u1} α α (fun (x : α) => Set.unionᵢ.{u1, 0} α (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) x t) (fun (H : Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) x t) => Metric.ball.{u1} α _inst_2 x e))))))))
 but is expected to have type
-  forall {α : Type.{u1}} [_inst_2 : PseudoMetricSpace.{u1} α] {s : Set.{u1} α}, (IsCompact.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_2)) s) -> (forall {e : Real}, (LT.lt.{0} Real Real.instLTReal (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal)) e) -> (Exists.{succ u1} (Set.{u1} α) (fun (t : Set.{u1} α) => And (HasSubset.Subset.{u1} (Set.{u1} α) (Set.instHasSubsetSet.{u1} α) t s) (And (Set.Finite.{u1} α t) (HasSubset.Subset.{u1} (Set.{u1} α) (Set.instHasSubsetSet.{u1} α) s (Set.unionᵢ.{u1, succ u1} α α (fun (x : α) => Set.unionᵢ.{u1, 0} α (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) x t) (fun (h._@.Mathlib.Topology.MetricSpace.Basic._hyg.25460 : Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) x t) => Metric.ball.{u1} α _inst_2 x e))))))))
+  forall {α : Type.{u1}} [_inst_2 : PseudoMetricSpace.{u1} α] {s : Set.{u1} α}, (IsCompact.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_2)) s) -> (forall {e : Real}, (LT.lt.{0} Real Real.instLTReal (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal)) e) -> (Exists.{succ u1} (Set.{u1} α) (fun (t : Set.{u1} α) => And (HasSubset.Subset.{u1} (Set.{u1} α) (Set.instHasSubsetSet.{u1} α) t s) (And (Set.Finite.{u1} α t) (HasSubset.Subset.{u1} (Set.{u1} α) (Set.instHasSubsetSet.{u1} α) s (Set.unionᵢ.{u1, succ u1} α α (fun (x : α) => Set.unionᵢ.{u1, 0} α (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) x t) (fun (h._@.Mathlib.Topology.MetricSpace.Basic._hyg.25496 : Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) x t) => Metric.ball.{u1} α _inst_2 x e))))))))
 Case conversion may be inaccurate. Consider using '#align is_compact.finite_cover_balls IsCompact.finite_cover_ballsₓ'. -/
 alias finite_cover_balls_of_compact ← IsCompact.finite_cover_balls
 #align is_compact.finite_cover_balls IsCompact.finite_cover_balls

mathlib commit https://github.com/leanprover-community/mathlib/commit/9b2b58d6b14b895b2f375108e765cb47de71aebd

@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Jeremy Avigad, Robert Y. Lewis, Johannes Hölzl, Mario Carneiro, Sébastien Gouëzel
 
 ! This file was ported from Lean 3 source module topology.metric_space.basic
-! leanprover-community/mathlib commit f47581155c818e6361af4e4fda60d27d020c226b
+! leanprover-community/mathlib commit 8000bbbe2e9d39b84edb993d88781f536a8a3fa8
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
@@ -3359,6 +3359,11 @@ theorem closure_closedBall : closure (closedBall x ε) = closedBall x ε :=
 #align metric.closure_closed_ball Metric.closure_closedBall
 -/
 
+@[simp]
+theorem closure_sphere : closure (sphere x ε) = sphere x ε :=
+  isClosed_sphere.closure_eq
+#align metric.closure_sphere Metric.closure_sphere
+
 #print Metric.closure_ball_subset_closedBall /-
 theorem closure_ball_subset_closedBall : closure (ball x ε) ⊆ closedBall x ε :=
   closure_minimal ball_subset_closedBall isClosed_ball

mathlib commit https://github.com/leanprover-community/mathlib/commit/730c6d4cab72b9d84fcfb9e95e8796e9cd8f40ba

@@ -3763,7 +3763,7 @@ theorem closedBall_pi' [Nonempty β] (x : ∀ b, π b) (r : ℝ) :
 lean 3 declaration is
   forall {n : Nat} {α : (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) -> Type.{u1}} [_inst_4 : forall (i : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))), PseudoMetricSpace.{u1} (α i)] (i : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (x : α i) (y : α i) (f : forall (j : Fin n), α (coeFn.{1, 1} (OrderEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (Fin.hasLe n) (Preorder.toLE.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (PartialOrder.toPreorder.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (Fin.partialOrder (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne)))))))) (fun (_x : RelEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (LE.le.{0} (Fin n) (Fin.hasLe n)) (LE.le.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (Preorder.toLE.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (PartialOrder.toPreorder.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (Fin.partialOrder (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))))))) => (Fin n) -> (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne)))))) (RelEmbedding.hasCoeToFun.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (LE.le.{0} (Fin n) (Fin.hasLe n)) (LE.le.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (Preorder.toLE.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (PartialOrder.toPreorder.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} 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 but is expected to have type
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: Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) => LE.le.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) x._@.Mathlib.Order.Hom.Basic._hyg.695 x._@.Mathlib.Order.Hom.Basic._hyg.697) (Fin.succAbove n i)) j)) (PseudoMetricSpace.toNNDist.{u1} (forall (j : Fin n), α (FunLike.coe.{1, 1, 1} (Function.Embedding.{1, 1} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))))) (Fin n) (fun (_x : Fin n) => (fun (x._@.Mathlib.Data.FunLike.Embedding._hyg.19 : Fin n) => Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) _x) (EmbeddingLike.toFunLike.{1, 1, 1} (Function.Embedding.{1, 1} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat 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(Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) x._@.Mathlib.Order.Hom.Basic._hyg.695 x._@.Mathlib.Order.Hom.Basic._hyg.697) (Fin.succAbove n i)) j)) (Fin.fintype n) (fun (b : Fin n) => _inst_4 (FunLike.coe.{1, 1, 1} (Function.Embedding.{1, 1} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))))) (Fin n) (fun (_x : Fin n) => (fun (x._@.Mathlib.Data.FunLike.Embedding._hyg.19 : Fin n) => Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) _x) (EmbeddingLike.toFunLike.{1, 1, 1} (Function.Embedding.{1, 1} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))))) (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} 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(OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) x._@.Mathlib.Order.Hom.Basic._hyg.695 x._@.Mathlib.Order.Hom.Basic._hyg.697) (Fin.succAbove n i)) b)))) f g))
+  forall {n : Nat} {α : (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) -> Type.{u1}} [_inst_4 : forall (i : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))), PseudoMetricSpace.{u1} (α i)] (i : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (x : α i) (y : α i) (f : forall (j : Fin n), α (FunLike.coe.{1, 1, 1} (OrderEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin n) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))))) (Fin n) (fun (_x : Fin n) => (fun (x._@.Mathlib.Order.RelIso.Basic._hyg.867 : Fin n) => Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) _x) 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g))
 Case conversion may be inaccurate. Consider using '#align fin.nndist_insert_nth_insert_nth Fin.nndist_insertNth_insertNthₓ'. -/
 @[simp]
 theorem Fin.nndist_insertNth_insertNth {n : ℕ} {α : Fin (n + 1) → Type _}
@@ -3776,7 +3776,7 @@ theorem Fin.nndist_insertNth_insertNth {n : ℕ} {α : Fin (n + 1) → Type _}
 lean 3 declaration is
   forall {n : Nat} {α : (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) -> Type.{u1}} [_inst_4 : forall (i : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))), PseudoMetricSpace.{u1} (α i)] (i : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (x : α i) (y : α i) (f : forall (j : Fin n), α (coeFn.{1, 1} (OrderEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (Fin.hasLe n) (Preorder.toLE.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (PartialOrder.toPreorder.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (Fin.partialOrder (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne)))))))) (fun (_x : RelEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (LE.le.{0} (Fin n) (Fin.hasLe n)) (LE.le.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (Preorder.toLE.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (PartialOrder.toPreorder.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (Fin.partialOrder (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))))))) => (Fin n) -> (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne)))))) (RelEmbedding.hasCoeToFun.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (LE.le.{0} (Fin n) (Fin.hasLe n)) (LE.le.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (Preorder.toLE.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (PartialOrder.toPreorder.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (Fin.partialOrder (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))))))) (Fin.succAbove n i) j)) (g : forall (j : Fin n), α (coeFn.{1, 1} (OrderEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (Fin.hasLe n) (Preorder.toLE.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (PartialOrder.toPreorder.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (Fin.partialOrder (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne)))))))) (fun (_x : RelEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (LE.le.{0} (Fin n) (Fin.hasLe n)) (LE.le.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (Preorder.toLE.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (PartialOrder.toPreorder.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (Fin.partialOrder (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))))))) => (Fin n) -> (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne)))))) (RelEmbedding.hasCoeToFun.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (LE.le.{0} (Fin n) (Fin.hasLe n)) (LE.le.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (Preorder.toLE.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (PartialOrder.toPreorder.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (Fin.partialOrder (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))))))) (Fin.succAbove n i) j)), Eq.{1} Real (Dist.dist.{u1} (forall (j : Fin (HAdd.hAdd.{0, 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 but is expected to have type
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(instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))))) (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.680 : Fin n) (x._@.Mathlib.Order.Hom.Basic._hyg.682 : Fin n) => LE.le.{0} (Fin n) (instLEFin n) x._@.Mathlib.Order.Hom.Basic._hyg.680 x._@.Mathlib.Order.Hom.Basic._hyg.682) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.695 : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (x._@.Mathlib.Order.Hom.Basic._hyg.697 : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) => LE.le.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) x._@.Mathlib.Order.Hom.Basic._hyg.695 x._@.Mathlib.Order.Hom.Basic._hyg.697) (RelEmbedding.instRelHomClassRelEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.680 : Fin n) (x._@.Mathlib.Order.Hom.Basic._hyg.682 : Fin n) => LE.le.{0} (Fin n) (instLEFin n) x._@.Mathlib.Order.Hom.Basic._hyg.680 x._@.Mathlib.Order.Hom.Basic._hyg.682) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.695 : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (x._@.Mathlib.Order.Hom.Basic._hyg.697 : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) => LE.le.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) x._@.Mathlib.Order.Hom.Basic._hyg.695 x._@.Mathlib.Order.Hom.Basic._hyg.697))) (Fin.succAbove n i) j)) (Fin.fintype n) (fun (b : Fin n) => _inst_4 (FunLike.coe.{1, 1, 1} (OrderEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin n) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))))) (Fin n) (fun (_x : Fin n) => (fun (x._@.Mathlib.Order.RelIso.Basic._hyg.867 : Fin n) => Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) _x) (RelHomClass.toFunLike.{0, 0, 0} (OrderEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin n) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))))) (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.680 : Fin n) (x._@.Mathlib.Order.Hom.Basic._hyg.682 : Fin n) => LE.le.{0} (Fin n) (instLEFin n) x._@.Mathlib.Order.Hom.Basic._hyg.680 x._@.Mathlib.Order.Hom.Basic._hyg.682) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.695 : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (x._@.Mathlib.Order.Hom.Basic._hyg.697 : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) => LE.le.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) x._@.Mathlib.Order.Hom.Basic._hyg.695 x._@.Mathlib.Order.Hom.Basic._hyg.697) (RelEmbedding.instRelHomClassRelEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.680 : Fin n) (x._@.Mathlib.Order.Hom.Basic._hyg.682 : Fin n) => LE.le.{0} (Fin n) (instLEFin n) x._@.Mathlib.Order.Hom.Basic._hyg.680 x._@.Mathlib.Order.Hom.Basic._hyg.682) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.695 : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (x._@.Mathlib.Order.Hom.Basic._hyg.697 : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) => LE.le.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) x._@.Mathlib.Order.Hom.Basic._hyg.695 x._@.Mathlib.Order.Hom.Basic._hyg.697))) (Fin.succAbove n i) b)))) f g))
 Case conversion may be inaccurate. Consider using '#align fin.dist_insert_nth_insert_nth Fin.dist_insertNth_insertNthₓ'. -/
 @[simp]
 theorem Fin.dist_insertNth_insertNth {n : ℕ} {α : Fin (n + 1) → Type _}

mathlib commit https://github.com/leanprover-community/mathlib/commit/e05ead7993520a432bec94ac504842d90707ad63

@@ -4820,15 +4820,15 @@ theorem diam_ball {r : ℝ} (h : 0 ≤ r) : diam (ball x r) ≤ 2 * r :=
   diam_le_of_subset_closedBall h ball_subset_closedBall
 #align metric.diam_ball Metric.diam_ball
 
-/- warning: is_complete.nonempty_Inter_of_nonempty_bInter -> IsComplete.nonempty_interᵢ_of_nonempty_bInter is a dubious translation:
+/- warning: is_complete.nonempty_Inter_of_nonempty_bInter -> IsComplete.nonempty_interᵢ_of_nonempty_binterᵢ is a dubious translation:
 lean 3 declaration is
   forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {s : Nat -> (Set.{u1} α)}, (IsComplete.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1) (s (OfNat.ofNat.{0} Nat 0 (OfNat.mk.{0} Nat 0 (Zero.zero.{0} Nat Nat.hasZero))))) -> (forall (n : Nat), IsClosed.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) (s n)) -> (forall (n : Nat), Metric.Bounded.{u1} α _inst_1 (s n)) -> (forall (N : Nat), Set.Nonempty.{u1} α (Set.interᵢ.{u1, 1} α Nat (fun (n : Nat) => Set.interᵢ.{u1, 0} α (LE.le.{0} Nat Nat.hasLe n N) (fun (H : LE.le.{0} Nat Nat.hasLe n N) => s n)))) -> (Filter.Tendsto.{0, 0} Nat Real (fun (n : Nat) => Metric.diam.{u1} α _inst_1 (s n)) (Filter.atTop.{0} Nat (PartialOrder.toPreorder.{0} Nat (OrderedCancelAddCommMonoid.toPartialOrder.{0} Nat (StrictOrderedSemiring.toOrderedCancelAddCommMonoid.{0} Nat Nat.strictOrderedSemiring)))) (nhds.{0} Real (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero))))) -> (Set.Nonempty.{u1} α (Set.interᵢ.{u1, 1} α Nat (fun (n : Nat) => s n)))
 but is expected to have type
   forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {s : Nat -> (Set.{u1} α)}, (IsComplete.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1) (s (OfNat.ofNat.{0} Nat 0 (instOfNatNat 0)))) -> (forall (n : Nat), IsClosed.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) (s n)) -> (forall (n : Nat), Metric.Bounded.{u1} α _inst_1 (s n)) -> (forall (N : Nat), Set.Nonempty.{u1} α (Set.interᵢ.{u1, 1} α Nat (fun (n : Nat) => Set.interᵢ.{u1, 0} α (LE.le.{0} Nat instLENat n N) (fun (H : LE.le.{0} Nat instLENat n N) => s n)))) -> (Filter.Tendsto.{0, 0} Nat Real (fun (n : Nat) => Metric.diam.{u1} α _inst_1 (s n)) (Filter.atTop.{0} Nat (PartialOrder.toPreorder.{0} Nat (StrictOrderedSemiring.toPartialOrder.{0} Nat Nat.strictOrderedSemiring))) (nhds.{0} Real (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal)))) -> (Set.Nonempty.{u1} α (Set.interᵢ.{u1, 1} α Nat (fun (n : Nat) => s n)))
-Case conversion may be inaccurate. Consider using '#align is_complete.nonempty_Inter_of_nonempty_bInter IsComplete.nonempty_interᵢ_of_nonempty_bInterₓ'. -/
+Case conversion may be inaccurate. Consider using '#align is_complete.nonempty_Inter_of_nonempty_bInter IsComplete.nonempty_interᵢ_of_nonempty_binterᵢₓ'. -/
 /-- If a family of complete sets with diameter tending to `0` is such that each finite intersection
 is nonempty, then the total intersection is also nonempty. -/
-theorem IsComplete.nonempty_interᵢ_of_nonempty_bInter {s : ℕ → Set α} (h0 : IsComplete (s 0))
+theorem IsComplete.nonempty_interᵢ_of_nonempty_binterᵢ {s : ℕ → Set α} (h0 : IsComplete (s 0))
     (hs : ∀ n, IsClosed (s n)) (h's : ∀ n, Bounded (s n)) (h : ∀ N, (⋂ n ≤ N, s n).Nonempty)
     (h' : Tendsto (fun n => diam (s n)) atTop (𝓝 0)) : (⋂ n, s n).Nonempty :=
   by
@@ -4850,21 +4850,21 @@ theorem IsComplete.nonempty_interᵢ_of_nonempty_bInter {s : ℕ → Set α} (h0
   apply (hs n).mem_of_tendsto xlim
   filter_upwards [Ici_mem_at_top n]with p hp
   exact I n p hp
-#align is_complete.nonempty_Inter_of_nonempty_bInter IsComplete.nonempty_interᵢ_of_nonempty_bInter
+#align is_complete.nonempty_Inter_of_nonempty_bInter IsComplete.nonempty_interᵢ_of_nonempty_binterᵢ
 
-/- warning: metric.nonempty_Inter_of_nonempty_bInter -> Metric.nonempty_interᵢ_of_nonempty_bInter is a dubious translation:
+/- warning: metric.nonempty_Inter_of_nonempty_bInter -> Metric.nonempty_interᵢ_of_nonempty_binterᵢ is a dubious translation:
 lean 3 declaration is
   forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] [_inst_2 : CompleteSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)] {s : Nat -> (Set.{u1} α)}, (forall (n : Nat), IsClosed.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) (s n)) -> (forall (n : Nat), Metric.Bounded.{u1} α _inst_1 (s n)) -> (forall (N : Nat), Set.Nonempty.{u1} α (Set.interᵢ.{u1, 1} α Nat (fun (n : Nat) => Set.interᵢ.{u1, 0} α (LE.le.{0} Nat Nat.hasLe n N) (fun (H : LE.le.{0} Nat Nat.hasLe n N) => s n)))) -> (Filter.Tendsto.{0, 0} Nat Real (fun (n : Nat) => Metric.diam.{u1} α _inst_1 (s n)) (Filter.atTop.{0} Nat (PartialOrder.toPreorder.{0} Nat (OrderedCancelAddCommMonoid.toPartialOrder.{0} Nat (StrictOrderedSemiring.toOrderedCancelAddCommMonoid.{0} Nat Nat.strictOrderedSemiring)))) (nhds.{0} Real (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero))))) -> (Set.Nonempty.{u1} α (Set.interᵢ.{u1, 1} α Nat (fun (n : Nat) => s n)))
 but is expected to have type
   forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] [_inst_2 : CompleteSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)] {s : Nat -> (Set.{u1} α)}, (forall (n : Nat), IsClosed.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) (s n)) -> (forall (n : Nat), Metric.Bounded.{u1} α _inst_1 (s n)) -> (forall (N : Nat), Set.Nonempty.{u1} α (Set.interᵢ.{u1, 1} α Nat (fun (n : Nat) => Set.interᵢ.{u1, 0} α (LE.le.{0} Nat instLENat n N) (fun (H : LE.le.{0} Nat instLENat n N) => s n)))) -> (Filter.Tendsto.{0, 0} Nat Real (fun (n : Nat) => Metric.diam.{u1} α _inst_1 (s n)) (Filter.atTop.{0} Nat (PartialOrder.toPreorder.{0} Nat (StrictOrderedSemiring.toPartialOrder.{0} Nat Nat.strictOrderedSemiring))) (nhds.{0} Real (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal)))) -> (Set.Nonempty.{u1} α (Set.interᵢ.{u1, 1} α Nat (fun (n : Nat) => s n)))
-Case conversion may be inaccurate. Consider using '#align metric.nonempty_Inter_of_nonempty_bInter Metric.nonempty_interᵢ_of_nonempty_bInterₓ'. -/
+Case conversion may be inaccurate. Consider using '#align metric.nonempty_Inter_of_nonempty_bInter Metric.nonempty_interᵢ_of_nonempty_binterᵢₓ'. -/
 /-- In a complete space, if a family of closed sets with diameter tending to `0` is such that each
 finite intersection is nonempty, then the total intersection is also nonempty. -/
-theorem nonempty_interᵢ_of_nonempty_bInter [CompleteSpace α] {s : ℕ → Set α}
+theorem nonempty_interᵢ_of_nonempty_binterᵢ [CompleteSpace α] {s : ℕ → Set α}
     (hs : ∀ n, IsClosed (s n)) (h's : ∀ n, Bounded (s n)) (h : ∀ N, (⋂ n ≤ N, s n).Nonempty)
     (h' : Tendsto (fun n => diam (s n)) atTop (𝓝 0)) : (⋂ n, s n).Nonempty :=
-  (hs 0).IsComplete.nonempty_interᵢ_of_nonempty_bInter hs h's h h'
-#align metric.nonempty_Inter_of_nonempty_bInter Metric.nonempty_interᵢ_of_nonempty_bInter
+  (hs 0).IsComplete.nonempty_interᵢ_of_nonempty_binterᵢ hs h's h h'
+#align metric.nonempty_Inter_of_nonempty_bInter Metric.nonempty_interᵢ_of_nonempty_binterᵢ
 
 end Diam
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/5ec62c8106221a3f9160e4e4fcc3eed79fe213e9

@@ -3809,7 +3809,7 @@ section Compact
 lean 3 declaration is
   forall {α : Type.{u1}} [_inst_2 : PseudoMetricSpace.{u1} α] {s : Set.{u1} α}, (IsCompact.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_2)) s) -> (forall {e : Real}, (LT.lt.{0} Real Real.hasLt (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero))) e) -> (Exists.{succ u1} (Set.{u1} α) (fun (t : Set.{u1} α) => Exists.{0} (HasSubset.Subset.{u1} (Set.{u1} α) (Set.hasSubset.{u1} α) t s) (fun (H : HasSubset.Subset.{u1} (Set.{u1} α) (Set.hasSubset.{u1} α) t s) => And (Set.Finite.{u1} α t) (HasSubset.Subset.{u1} (Set.{u1} α) (Set.hasSubset.{u1} α) s (Set.unionᵢ.{u1, succ u1} α α (fun (x : α) => Set.unionᵢ.{u1, 0} α (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) x t) (fun (H : Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) x t) => Metric.ball.{u1} α _inst_2 x e))))))))
 but is expected to have type
-  forall {α : Type.{u1}} [_inst_2 : PseudoMetricSpace.{u1} α] {s : Set.{u1} α}, (IsCompact.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_2)) s) -> (forall {e : Real}, (LT.lt.{0} Real Real.instLTReal (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal)) e) -> (Exists.{succ u1} (Set.{u1} α) (fun (t : Set.{u1} α) => And (HasSubset.Subset.{u1} (Set.{u1} α) (Set.instHasSubsetSet.{u1} α) t s) (And (Set.Finite.{u1} α t) (HasSubset.Subset.{u1} (Set.{u1} α) (Set.instHasSubsetSet.{u1} α) s (Set.unionᵢ.{u1, succ u1} α α (fun (x : α) => Set.unionᵢ.{u1, 0} α (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) x t) (fun (h._@.Mathlib.Topology.MetricSpace.Basic._hyg.25465 : Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) x t) => Metric.ball.{u1} α _inst_2 x e))))))))
+  forall {α : Type.{u1}} [_inst_2 : PseudoMetricSpace.{u1} α] {s : Set.{u1} α}, (IsCompact.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_2)) s) -> (forall {e : Real}, (LT.lt.{0} Real Real.instLTReal (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal)) e) -> (Exists.{succ u1} (Set.{u1} α) (fun (t : Set.{u1} α) => And (HasSubset.Subset.{u1} (Set.{u1} α) (Set.instHasSubsetSet.{u1} α) t s) (And (Set.Finite.{u1} α t) (HasSubset.Subset.{u1} (Set.{u1} α) (Set.instHasSubsetSet.{u1} α) s (Set.unionᵢ.{u1, succ u1} α α (fun (x : α) => Set.unionᵢ.{u1, 0} α (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) x t) (fun (h._@.Mathlib.Topology.MetricSpace.Basic._hyg.25460 : Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) x t) => Metric.ball.{u1} α _inst_2 x e))))))))
 Case conversion may be inaccurate. Consider using '#align finite_cover_balls_of_compact finite_cover_balls_of_compactₓ'. -/
 /- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (t «expr ⊆ » s) -/
 /-- Any compact set in a pseudometric space can be covered by finitely many balls of a given
@@ -3829,7 +3829,7 @@ theorem finite_cover_balls_of_compact {α : Type u} [PseudoMetricSpace α] {s :
 lean 3 declaration is
   forall {α : Type.{u1}} [_inst_2 : PseudoMetricSpace.{u1} α] {s : Set.{u1} α}, (IsCompact.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_2)) s) -> (forall {e : Real}, (LT.lt.{0} Real Real.hasLt (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero))) e) -> (Exists.{succ u1} (Set.{u1} α) (fun (t : Set.{u1} α) => Exists.{0} (HasSubset.Subset.{u1} (Set.{u1} α) (Set.hasSubset.{u1} α) t s) (fun (H : HasSubset.Subset.{u1} (Set.{u1} α) (Set.hasSubset.{u1} α) t s) => And (Set.Finite.{u1} α t) (HasSubset.Subset.{u1} (Set.{u1} α) (Set.hasSubset.{u1} α) s (Set.unionᵢ.{u1, succ u1} α α (fun (x : α) => Set.unionᵢ.{u1, 0} α (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) x t) (fun (H : Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) x t) => Metric.ball.{u1} α _inst_2 x e))))))))
 but is expected to have type
-  forall {α : Type.{u1}} [_inst_2 : PseudoMetricSpace.{u1} α] {s : Set.{u1} α}, (IsCompact.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_2)) s) -> (forall {e : Real}, (LT.lt.{0} Real Real.instLTReal (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal)) e) -> (Exists.{succ u1} (Set.{u1} α) (fun (t : Set.{u1} α) => And (HasSubset.Subset.{u1} (Set.{u1} α) (Set.instHasSubsetSet.{u1} α) t s) (And (Set.Finite.{u1} α t) (HasSubset.Subset.{u1} (Set.{u1} α) (Set.instHasSubsetSet.{u1} α) s (Set.unionᵢ.{u1, succ u1} α α (fun (x : α) => Set.unionᵢ.{u1, 0} α (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) x t) (fun (h._@.Mathlib.Topology.MetricSpace.Basic._hyg.25465 : Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) x t) => Metric.ball.{u1} α _inst_2 x e))))))))
+  forall {α : Type.{u1}} [_inst_2 : PseudoMetricSpace.{u1} α] {s : Set.{u1} α}, (IsCompact.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_2)) s) -> (forall {e : Real}, (LT.lt.{0} Real Real.instLTReal (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal)) e) -> (Exists.{succ u1} (Set.{u1} α) (fun (t : Set.{u1} α) => And (HasSubset.Subset.{u1} (Set.{u1} α) (Set.instHasSubsetSet.{u1} α) t s) (And (Set.Finite.{u1} α t) (HasSubset.Subset.{u1} (Set.{u1} α) (Set.instHasSubsetSet.{u1} α) s (Set.unionᵢ.{u1, succ u1} α α (fun (x : α) => Set.unionᵢ.{u1, 0} α (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) x t) (fun (h._@.Mathlib.Topology.MetricSpace.Basic._hyg.25460 : Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) x t) => Metric.ball.{u1} α _inst_2 x e))))))))
 Case conversion may be inaccurate. Consider using '#align is_compact.finite_cover_balls IsCompact.finite_cover_ballsₓ'. -/
 alias finite_cover_balls_of_compact ← IsCompact.finite_cover_balls
 #align is_compact.finite_cover_balls IsCompact.finite_cover_balls

mathlib commit https://github.com/leanprover-community/mathlib/commit/3180fab693e2cee3bff62675571264cb8778b212

@@ -1365,7 +1365,7 @@ protected theorem mk_uniformity_basis {β : Type _} {p : β → Prop} {f : β 
 lean 3 declaration is
   forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α], Filter.HasBasis.{u1, 1} (Prod.{u1, u1} α α) Rat (uniformity.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) (fun (r : Rat) => LT.lt.{0} Rat Rat.hasLt (OfNat.ofNat.{0} Rat 0 (OfNat.mk.{0} Rat 0 (Zero.zero.{0} Rat Rat.hasZero))) r) (fun (r : Rat) => setOf.{u1} (Prod.{u1, u1} α α) (fun (p : Prod.{u1, u1} α α) => LT.lt.{0} Real Real.hasLt (Dist.dist.{u1} α (PseudoMetricSpace.toHasDist.{u1} α _inst_1) (Prod.fst.{u1, u1} α α p) (Prod.snd.{u1, u1} α α p)) ((fun (a : Type) (b : Type) [self : HasLiftT.{1, 1} a b] => self.0) Rat Real (HasLiftT.mk.{1, 1} Rat Real (CoeTCₓ.coe.{1, 1} Rat Real (Rat.castCoe.{0} Real Real.hasRatCast))) r)))
 but is expected to have type
-  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α], Filter.HasBasis.{u1, 1} (Prod.{u1, u1} α α) Rat (uniformity.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) (fun (r : Rat) => LT.lt.{0} Rat Rat.instLTRat_1 (OfNat.ofNat.{0} Rat 0 (Rat.instOfNatRat 0)) r) (fun (r : Rat) => setOf.{u1} (Prod.{u1, u1} α α) (fun (p : Prod.{u1, u1} α α) => LT.lt.{0} Real Real.instLTReal (Dist.dist.{u1} α (PseudoMetricSpace.toDist.{u1} α _inst_1) (Prod.fst.{u1, u1} α α p) (Prod.snd.{u1, u1} α α p)) (RatCast.ratCast.{0} Real Real.ratCast r)))
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α], Filter.HasBasis.{u1, 1} (Prod.{u1, u1} α α) Rat (uniformity.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) (fun (r : Rat) => LT.lt.{0} Rat Rat.instLTRat_1 (OfNat.ofNat.{0} Rat 0 (Rat.instOfNatRat 0)) r) (fun (r : Rat) => setOf.{u1} (Prod.{u1, u1} α α) (fun (p : Prod.{u1, u1} α α) => LT.lt.{0} Real Real.instLTReal (Dist.dist.{u1} α (PseudoMetricSpace.toDist.{u1} α _inst_1) (Prod.fst.{u1, u1} α α p) (Prod.snd.{u1, u1} α α p)) (Rat.cast.{0} Real Real.ratCast r)))
 Case conversion may be inaccurate. Consider using '#align metric.uniformity_basis_dist_rat Metric.uniformity_basis_dist_ratₓ'. -/
 theorem uniformity_basis_dist_rat :
     (𝓤 α).HasBasis (fun r : ℚ => 0 < r) fun r => { p : α × α | dist p.1 p.2 < r } :=
@@ -2466,7 +2466,7 @@ theorem Real.dist_eq (x y : ℝ) : dist x y = |x - y| :=
 lean 3 declaration is
   forall (x : Real) (y : Real), Eq.{1} NNReal (NNDist.nndist.{0} Real (PseudoMetricSpace.toNNDist.{0} Real Real.pseudoMetricSpace) x y) (coeFn.{1, 1} (MonoidWithZeroHom.{0, 0} Real NNReal (NonAssocSemiring.toMulZeroOneClass.{0} Real (NonAssocRing.toNonAssocSemiring.{0} Real (Ring.toNonAssocRing.{0} Real Real.ring))) (NonAssocSemiring.toMulZeroOneClass.{0} NNReal (Semiring.toNonAssocSemiring.{0} NNReal NNReal.semiring))) (fun (_x : MonoidWithZeroHom.{0, 0} Real NNReal (NonAssocSemiring.toMulZeroOneClass.{0} Real (NonAssocRing.toNonAssocSemiring.{0} Real (Ring.toNonAssocRing.{0} Real Real.ring))) (NonAssocSemiring.toMulZeroOneClass.{0} NNReal (Semiring.toNonAssocSemiring.{0} NNReal NNReal.semiring))) => Real -> NNReal) (MonoidWithZeroHom.hasCoeToFun.{0, 0} Real NNReal (NonAssocSemiring.toMulZeroOneClass.{0} Real (NonAssocRing.toNonAssocSemiring.{0} Real (Ring.toNonAssocRing.{0} Real Real.ring))) (NonAssocSemiring.toMulZeroOneClass.{0} NNReal (Semiring.toNonAssocSemiring.{0} NNReal NNReal.semiring))) Real.nnabs (HSub.hSub.{0, 0, 0} Real Real Real (instHSub.{0} Real Real.hasSub) x y))
 but is expected to have type
-  forall (x : Real) (y : Real), Eq.{1} NNReal (NNDist.nndist.{0} Real (PseudoMetricSpace.toNNDist.{0} Real Real.pseudoMetricSpace) x y) (FunLike.coe.{1, 1, 1} (MonoidWithZeroHom.{0, 0} Real NNReal (NonAssocSemiring.toMulZeroOneClass.{0} Real (NonAssocRing.toNonAssocSemiring.{0} Real (Ring.toNonAssocRing.{0} Real Real.instRingReal))) (NonAssocSemiring.toMulZeroOneClass.{0} NNReal (Semiring.toNonAssocSemiring.{0} NNReal instNNRealSemiring))) Real (fun (_x : Real) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2372 : Real) => NNReal) _x) (MulHomClass.toFunLike.{0, 0, 0} (MonoidWithZeroHom.{0, 0} Real NNReal (NonAssocSemiring.toMulZeroOneClass.{0} Real (NonAssocRing.toNonAssocSemiring.{0} Real (Ring.toNonAssocRing.{0} Real Real.instRingReal))) (NonAssocSemiring.toMulZeroOneClass.{0} NNReal (Semiring.toNonAssocSemiring.{0} NNReal instNNRealSemiring))) Real NNReal (MulOneClass.toMul.{0} Real (MulZeroOneClass.toMulOneClass.{0} Real (NonAssocSemiring.toMulZeroOneClass.{0} Real (NonAssocRing.toNonAssocSemiring.{0} Real (Ring.toNonAssocRing.{0} Real Real.instRingReal))))) (MulOneClass.toMul.{0} NNReal (MulZeroOneClass.toMulOneClass.{0} NNReal (NonAssocSemiring.toMulZeroOneClass.{0} NNReal (Semiring.toNonAssocSemiring.{0} NNReal instNNRealSemiring)))) (MonoidHomClass.toMulHomClass.{0, 0, 0} (MonoidWithZeroHom.{0, 0} Real NNReal (NonAssocSemiring.toMulZeroOneClass.{0} Real (NonAssocRing.toNonAssocSemiring.{0} Real (Ring.toNonAssocRing.{0} Real Real.instRingReal))) (NonAssocSemiring.toMulZeroOneClass.{0} NNReal (Semiring.toNonAssocSemiring.{0} NNReal instNNRealSemiring))) Real NNReal (MulZeroOneClass.toMulOneClass.{0} Real (NonAssocSemiring.toMulZeroOneClass.{0} Real (NonAssocRing.toNonAssocSemiring.{0} Real (Ring.toNonAssocRing.{0} Real Real.instRingReal)))) (MulZeroOneClass.toMulOneClass.{0} NNReal (NonAssocSemiring.toMulZeroOneClass.{0} NNReal (Semiring.toNonAssocSemiring.{0} NNReal instNNRealSemiring))) (MonoidWithZeroHomClass.toMonoidHomClass.{0, 0, 0} (MonoidWithZeroHom.{0, 0} Real NNReal (NonAssocSemiring.toMulZeroOneClass.{0} Real (NonAssocRing.toNonAssocSemiring.{0} Real (Ring.toNonAssocRing.{0} Real Real.instRingReal))) (NonAssocSemiring.toMulZeroOneClass.{0} NNReal (Semiring.toNonAssocSemiring.{0} NNReal instNNRealSemiring))) Real NNReal (NonAssocSemiring.toMulZeroOneClass.{0} Real (NonAssocRing.toNonAssocSemiring.{0} Real (Ring.toNonAssocRing.{0} Real Real.instRingReal))) (NonAssocSemiring.toMulZeroOneClass.{0} NNReal (Semiring.toNonAssocSemiring.{0} NNReal instNNRealSemiring)) (MonoidWithZeroHom.monoidWithZeroHomClass.{0, 0} Real NNReal (NonAssocSemiring.toMulZeroOneClass.{0} Real (NonAssocRing.toNonAssocSemiring.{0} Real (Ring.toNonAssocRing.{0} Real Real.instRingReal))) (NonAssocSemiring.toMulZeroOneClass.{0} NNReal (Semiring.toNonAssocSemiring.{0} NNReal instNNRealSemiring)))))) Real.nnabs (HSub.hSub.{0, 0, 0} Real Real Real (instHSub.{0} Real Real.instSubReal) x y))
+  forall (x : Real) (y : Real), Eq.{1} NNReal (NNDist.nndist.{0} Real (PseudoMetricSpace.toNNDist.{0} Real Real.pseudoMetricSpace) x y) (FunLike.coe.{1, 1, 1} (MonoidWithZeroHom.{0, 0} Real NNReal (NonAssocSemiring.toMulZeroOneClass.{0} Real (NonAssocRing.toNonAssocSemiring.{0} Real (Ring.toNonAssocRing.{0} Real Real.instRingReal))) (NonAssocSemiring.toMulZeroOneClass.{0} NNReal (Semiring.toNonAssocSemiring.{0} NNReal instNNRealSemiring))) Real (fun (_x : Real) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : Real) => NNReal) _x) (MulHomClass.toFunLike.{0, 0, 0} (MonoidWithZeroHom.{0, 0} Real NNReal (NonAssocSemiring.toMulZeroOneClass.{0} Real (NonAssocRing.toNonAssocSemiring.{0} Real (Ring.toNonAssocRing.{0} Real Real.instRingReal))) (NonAssocSemiring.toMulZeroOneClass.{0} NNReal (Semiring.toNonAssocSemiring.{0} NNReal instNNRealSemiring))) Real NNReal (MulOneClass.toMul.{0} Real (MulZeroOneClass.toMulOneClass.{0} Real (NonAssocSemiring.toMulZeroOneClass.{0} Real (NonAssocRing.toNonAssocSemiring.{0} Real (Ring.toNonAssocRing.{0} Real Real.instRingReal))))) (MulOneClass.toMul.{0} NNReal (MulZeroOneClass.toMulOneClass.{0} NNReal (NonAssocSemiring.toMulZeroOneClass.{0} NNReal (Semiring.toNonAssocSemiring.{0} NNReal instNNRealSemiring)))) (MonoidHomClass.toMulHomClass.{0, 0, 0} (MonoidWithZeroHom.{0, 0} Real NNReal (NonAssocSemiring.toMulZeroOneClass.{0} Real (NonAssocRing.toNonAssocSemiring.{0} Real (Ring.toNonAssocRing.{0} Real Real.instRingReal))) (NonAssocSemiring.toMulZeroOneClass.{0} NNReal (Semiring.toNonAssocSemiring.{0} NNReal instNNRealSemiring))) Real NNReal (MulZeroOneClass.toMulOneClass.{0} Real (NonAssocSemiring.toMulZeroOneClass.{0} Real (NonAssocRing.toNonAssocSemiring.{0} Real (Ring.toNonAssocRing.{0} Real Real.instRingReal)))) (MulZeroOneClass.toMulOneClass.{0} NNReal (NonAssocSemiring.toMulZeroOneClass.{0} NNReal (Semiring.toNonAssocSemiring.{0} NNReal instNNRealSemiring))) (MonoidWithZeroHomClass.toMonoidHomClass.{0, 0, 0} (MonoidWithZeroHom.{0, 0} Real NNReal (NonAssocSemiring.toMulZeroOneClass.{0} Real (NonAssocRing.toNonAssocSemiring.{0} Real (Ring.toNonAssocRing.{0} Real Real.instRingReal))) (NonAssocSemiring.toMulZeroOneClass.{0} NNReal (Semiring.toNonAssocSemiring.{0} NNReal instNNRealSemiring))) Real NNReal (NonAssocSemiring.toMulZeroOneClass.{0} Real (NonAssocRing.toNonAssocSemiring.{0} Real (Ring.toNonAssocRing.{0} Real Real.instRingReal))) (NonAssocSemiring.toMulZeroOneClass.{0} NNReal (Semiring.toNonAssocSemiring.{0} NNReal instNNRealSemiring)) (MonoidWithZeroHom.monoidWithZeroHomClass.{0, 0} Real NNReal (NonAssocSemiring.toMulZeroOneClass.{0} Real (NonAssocRing.toNonAssocSemiring.{0} Real (Ring.toNonAssocRing.{0} Real Real.instRingReal))) (NonAssocSemiring.toMulZeroOneClass.{0} NNReal (Semiring.toNonAssocSemiring.{0} NNReal instNNRealSemiring)))))) Real.nnabs (HSub.hSub.{0, 0, 0} Real Real Real (instHSub.{0} Real Real.instSubReal) x y))
 Case conversion may be inaccurate. Consider using '#align real.nndist_eq Real.nndist_eqₓ'. -/
 theorem Real.nndist_eq (x y : ℝ) : nndist x y = Real.nnabs (x - y) :=
   rfl
@@ -2476,7 +2476,7 @@ theorem Real.nndist_eq (x y : ℝ) : nndist x y = Real.nnabs (x - y) :=
 lean 3 declaration is
   forall (x : Real) (y : Real), Eq.{1} NNReal (NNDist.nndist.{0} Real (PseudoMetricSpace.toNNDist.{0} Real Real.pseudoMetricSpace) x y) (coeFn.{1, 1} (MonoidWithZeroHom.{0, 0} Real NNReal (NonAssocSemiring.toMulZeroOneClass.{0} Real (NonAssocRing.toNonAssocSemiring.{0} Real (Ring.toNonAssocRing.{0} Real Real.ring))) (NonAssocSemiring.toMulZeroOneClass.{0} NNReal (Semiring.toNonAssocSemiring.{0} NNReal NNReal.semiring))) (fun (_x : MonoidWithZeroHom.{0, 0} Real NNReal (NonAssocSemiring.toMulZeroOneClass.{0} Real (NonAssocRing.toNonAssocSemiring.{0} Real (Ring.toNonAssocRing.{0} Real Real.ring))) (NonAssocSemiring.toMulZeroOneClass.{0} NNReal (Semiring.toNonAssocSemiring.{0} NNReal NNReal.semiring))) => Real -> NNReal) (MonoidWithZeroHom.hasCoeToFun.{0, 0} Real NNReal (NonAssocSemiring.toMulZeroOneClass.{0} Real (NonAssocRing.toNonAssocSemiring.{0} Real (Ring.toNonAssocRing.{0} Real Real.ring))) (NonAssocSemiring.toMulZeroOneClass.{0} NNReal (Semiring.toNonAssocSemiring.{0} NNReal NNReal.semiring))) Real.nnabs (HSub.hSub.{0, 0, 0} Real Real Real (instHSub.{0} Real Real.hasSub) y x))
 but is expected to have type
-  forall (x : Real) (y : Real), Eq.{1} NNReal (NNDist.nndist.{0} Real (PseudoMetricSpace.toNNDist.{0} Real Real.pseudoMetricSpace) x y) (FunLike.coe.{1, 1, 1} (MonoidWithZeroHom.{0, 0} Real NNReal (NonAssocSemiring.toMulZeroOneClass.{0} Real (NonAssocRing.toNonAssocSemiring.{0} Real (Ring.toNonAssocRing.{0} Real Real.instRingReal))) (NonAssocSemiring.toMulZeroOneClass.{0} NNReal (Semiring.toNonAssocSemiring.{0} NNReal instNNRealSemiring))) Real (fun (_x : Real) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2372 : Real) => NNReal) _x) (MulHomClass.toFunLike.{0, 0, 0} (MonoidWithZeroHom.{0, 0} Real NNReal (NonAssocSemiring.toMulZeroOneClass.{0} Real (NonAssocRing.toNonAssocSemiring.{0} Real (Ring.toNonAssocRing.{0} Real Real.instRingReal))) (NonAssocSemiring.toMulZeroOneClass.{0} NNReal (Semiring.toNonAssocSemiring.{0} NNReal instNNRealSemiring))) Real NNReal (MulOneClass.toMul.{0} Real (MulZeroOneClass.toMulOneClass.{0} Real (NonAssocSemiring.toMulZeroOneClass.{0} Real (NonAssocRing.toNonAssocSemiring.{0} Real (Ring.toNonAssocRing.{0} Real Real.instRingReal))))) (MulOneClass.toMul.{0} NNReal (MulZeroOneClass.toMulOneClass.{0} NNReal (NonAssocSemiring.toMulZeroOneClass.{0} NNReal (Semiring.toNonAssocSemiring.{0} NNReal instNNRealSemiring)))) (MonoidHomClass.toMulHomClass.{0, 0, 0} (MonoidWithZeroHom.{0, 0} Real NNReal (NonAssocSemiring.toMulZeroOneClass.{0} Real (NonAssocRing.toNonAssocSemiring.{0} Real (Ring.toNonAssocRing.{0} Real Real.instRingReal))) (NonAssocSemiring.toMulZeroOneClass.{0} NNReal (Semiring.toNonAssocSemiring.{0} NNReal instNNRealSemiring))) Real NNReal (MulZeroOneClass.toMulOneClass.{0} Real (NonAssocSemiring.toMulZeroOneClass.{0} Real (NonAssocRing.toNonAssocSemiring.{0} Real (Ring.toNonAssocRing.{0} Real Real.instRingReal)))) (MulZeroOneClass.toMulOneClass.{0} NNReal (NonAssocSemiring.toMulZeroOneClass.{0} NNReal (Semiring.toNonAssocSemiring.{0} NNReal instNNRealSemiring))) (MonoidWithZeroHomClass.toMonoidHomClass.{0, 0, 0} (MonoidWithZeroHom.{0, 0} Real NNReal (NonAssocSemiring.toMulZeroOneClass.{0} Real (NonAssocRing.toNonAssocSemiring.{0} Real (Ring.toNonAssocRing.{0} Real Real.instRingReal))) (NonAssocSemiring.toMulZeroOneClass.{0} NNReal (Semiring.toNonAssocSemiring.{0} NNReal instNNRealSemiring))) Real NNReal (NonAssocSemiring.toMulZeroOneClass.{0} Real (NonAssocRing.toNonAssocSemiring.{0} Real (Ring.toNonAssocRing.{0} Real Real.instRingReal))) (NonAssocSemiring.toMulZeroOneClass.{0} NNReal (Semiring.toNonAssocSemiring.{0} NNReal instNNRealSemiring)) (MonoidWithZeroHom.monoidWithZeroHomClass.{0, 0} Real NNReal (NonAssocSemiring.toMulZeroOneClass.{0} Real (NonAssocRing.toNonAssocSemiring.{0} Real (Ring.toNonAssocRing.{0} Real Real.instRingReal))) (NonAssocSemiring.toMulZeroOneClass.{0} NNReal (Semiring.toNonAssocSemiring.{0} NNReal instNNRealSemiring)))))) Real.nnabs (HSub.hSub.{0, 0, 0} Real Real Real (instHSub.{0} Real Real.instSubReal) y x))
+  forall (x : Real) (y : Real), Eq.{1} NNReal (NNDist.nndist.{0} Real (PseudoMetricSpace.toNNDist.{0} Real Real.pseudoMetricSpace) x y) (FunLike.coe.{1, 1, 1} (MonoidWithZeroHom.{0, 0} Real NNReal (NonAssocSemiring.toMulZeroOneClass.{0} Real (NonAssocRing.toNonAssocSemiring.{0} Real (Ring.toNonAssocRing.{0} Real Real.instRingReal))) (NonAssocSemiring.toMulZeroOneClass.{0} NNReal (Semiring.toNonAssocSemiring.{0} NNReal instNNRealSemiring))) Real (fun (_x : Real) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : Real) => NNReal) _x) (MulHomClass.toFunLike.{0, 0, 0} (MonoidWithZeroHom.{0, 0} Real NNReal (NonAssocSemiring.toMulZeroOneClass.{0} Real (NonAssocRing.toNonAssocSemiring.{0} Real (Ring.toNonAssocRing.{0} Real Real.instRingReal))) (NonAssocSemiring.toMulZeroOneClass.{0} NNReal (Semiring.toNonAssocSemiring.{0} NNReal instNNRealSemiring))) Real NNReal (MulOneClass.toMul.{0} Real (MulZeroOneClass.toMulOneClass.{0} Real (NonAssocSemiring.toMulZeroOneClass.{0} Real (NonAssocRing.toNonAssocSemiring.{0} Real (Ring.toNonAssocRing.{0} Real Real.instRingReal))))) (MulOneClass.toMul.{0} NNReal (MulZeroOneClass.toMulOneClass.{0} NNReal (NonAssocSemiring.toMulZeroOneClass.{0} NNReal (Semiring.toNonAssocSemiring.{0} NNReal instNNRealSemiring)))) (MonoidHomClass.toMulHomClass.{0, 0, 0} (MonoidWithZeroHom.{0, 0} Real NNReal (NonAssocSemiring.toMulZeroOneClass.{0} Real (NonAssocRing.toNonAssocSemiring.{0} Real (Ring.toNonAssocRing.{0} Real Real.instRingReal))) (NonAssocSemiring.toMulZeroOneClass.{0} NNReal (Semiring.toNonAssocSemiring.{0} NNReal instNNRealSemiring))) Real NNReal (MulZeroOneClass.toMulOneClass.{0} Real (NonAssocSemiring.toMulZeroOneClass.{0} Real (NonAssocRing.toNonAssocSemiring.{0} Real (Ring.toNonAssocRing.{0} Real Real.instRingReal)))) (MulZeroOneClass.toMulOneClass.{0} NNReal (NonAssocSemiring.toMulZeroOneClass.{0} NNReal (Semiring.toNonAssocSemiring.{0} NNReal instNNRealSemiring))) (MonoidWithZeroHomClass.toMonoidHomClass.{0, 0, 0} (MonoidWithZeroHom.{0, 0} Real NNReal (NonAssocSemiring.toMulZeroOneClass.{0} Real (NonAssocRing.toNonAssocSemiring.{0} Real (Ring.toNonAssocRing.{0} Real Real.instRingReal))) (NonAssocSemiring.toMulZeroOneClass.{0} NNReal (Semiring.toNonAssocSemiring.{0} NNReal instNNRealSemiring))) Real NNReal (NonAssocSemiring.toMulZeroOneClass.{0} Real (NonAssocRing.toNonAssocSemiring.{0} Real (Ring.toNonAssocRing.{0} Real Real.instRingReal))) (NonAssocSemiring.toMulZeroOneClass.{0} NNReal (Semiring.toNonAssocSemiring.{0} NNReal instNNRealSemiring)) (MonoidWithZeroHom.monoidWithZeroHomClass.{0, 0} Real NNReal (NonAssocSemiring.toMulZeroOneClass.{0} Real (NonAssocRing.toNonAssocSemiring.{0} Real (Ring.toNonAssocRing.{0} Real Real.instRingReal))) (NonAssocSemiring.toMulZeroOneClass.{0} NNReal (Semiring.toNonAssocSemiring.{0} NNReal instNNRealSemiring)))))) Real.nnabs (HSub.hSub.{0, 0, 0} Real Real Real (instHSub.{0} Real Real.instSubReal) y x))
 Case conversion may be inaccurate. Consider using '#align real.nndist_eq' Real.nndist_eq'ₓ'. -/
 theorem Real.nndist_eq' (x y : ℝ) : nndist x y = Real.nnabs (y - x) :=
   nndist_comm _ _
@@ -3809,7 +3809,7 @@ section Compact
 lean 3 declaration is
   forall {α : Type.{u1}} [_inst_2 : PseudoMetricSpace.{u1} α] {s : Set.{u1} α}, (IsCompact.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_2)) s) -> (forall {e : Real}, (LT.lt.{0} Real Real.hasLt (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero))) e) -> (Exists.{succ u1} (Set.{u1} α) (fun (t : Set.{u1} α) => Exists.{0} (HasSubset.Subset.{u1} (Set.{u1} α) (Set.hasSubset.{u1} α) t s) (fun (H : HasSubset.Subset.{u1} (Set.{u1} α) (Set.hasSubset.{u1} α) t s) => And (Set.Finite.{u1} α t) (HasSubset.Subset.{u1} (Set.{u1} α) (Set.hasSubset.{u1} α) s (Set.unionᵢ.{u1, succ u1} α α (fun (x : α) => Set.unionᵢ.{u1, 0} α (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) x t) (fun (H : Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) x t) => Metric.ball.{u1} α _inst_2 x e))))))))
 but is expected to have type
-  forall {α : Type.{u1}} [_inst_2 : PseudoMetricSpace.{u1} α] {s : Set.{u1} α}, (IsCompact.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_2)) s) -> (forall {e : Real}, (LT.lt.{0} Real Real.instLTReal (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal)) e) -> (Exists.{succ u1} (Set.{u1} α) (fun (t : Set.{u1} α) => And (HasSubset.Subset.{u1} (Set.{u1} α) (Set.instHasSubsetSet.{u1} α) t s) (And (Set.Finite.{u1} α t) (HasSubset.Subset.{u1} (Set.{u1} α) (Set.instHasSubsetSet.{u1} α) s (Set.unionᵢ.{u1, succ u1} α α (fun (x : α) => Set.unionᵢ.{u1, 0} α (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) x t) (fun (h._@.Mathlib.Topology.MetricSpace.Basic._hyg.25459 : Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) x t) => Metric.ball.{u1} α _inst_2 x e))))))))
+  forall {α : Type.{u1}} [_inst_2 : PseudoMetricSpace.{u1} α] {s : Set.{u1} α}, (IsCompact.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_2)) s) -> (forall {e : Real}, (LT.lt.{0} Real Real.instLTReal (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal)) e) -> (Exists.{succ u1} (Set.{u1} α) (fun (t : Set.{u1} α) => And (HasSubset.Subset.{u1} (Set.{u1} α) (Set.instHasSubsetSet.{u1} α) t s) (And (Set.Finite.{u1} α t) (HasSubset.Subset.{u1} (Set.{u1} α) (Set.instHasSubsetSet.{u1} α) s (Set.unionᵢ.{u1, succ u1} α α (fun (x : α) => Set.unionᵢ.{u1, 0} α (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) x t) (fun (h._@.Mathlib.Topology.MetricSpace.Basic._hyg.25465 : Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) x t) => Metric.ball.{u1} α _inst_2 x e))))))))
 Case conversion may be inaccurate. Consider using '#align finite_cover_balls_of_compact finite_cover_balls_of_compactₓ'. -/
 /- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (t «expr ⊆ » s) -/
 /-- Any compact set in a pseudometric space can be covered by finitely many balls of a given
@@ -3829,7 +3829,7 @@ theorem finite_cover_balls_of_compact {α : Type u} [PseudoMetricSpace α] {s :
 lean 3 declaration is
   forall {α : Type.{u1}} [_inst_2 : PseudoMetricSpace.{u1} α] {s : Set.{u1} α}, (IsCompact.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_2)) s) -> (forall {e : Real}, (LT.lt.{0} Real Real.hasLt (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero))) e) -> (Exists.{succ u1} (Set.{u1} α) (fun (t : Set.{u1} α) => Exists.{0} (HasSubset.Subset.{u1} (Set.{u1} α) (Set.hasSubset.{u1} α) t s) (fun (H : HasSubset.Subset.{u1} (Set.{u1} α) (Set.hasSubset.{u1} α) t s) => And (Set.Finite.{u1} α t) (HasSubset.Subset.{u1} (Set.{u1} α) (Set.hasSubset.{u1} α) s (Set.unionᵢ.{u1, succ u1} α α (fun (x : α) => Set.unionᵢ.{u1, 0} α (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) x t) (fun (H : Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) x t) => Metric.ball.{u1} α _inst_2 x e))))))))
 but is expected to have type
-  forall {α : Type.{u1}} [_inst_2 : PseudoMetricSpace.{u1} α] {s : Set.{u1} α}, (IsCompact.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_2)) s) -> (forall {e : Real}, (LT.lt.{0} Real Real.instLTReal (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal)) e) -> (Exists.{succ u1} (Set.{u1} α) (fun (t : Set.{u1} α) => And (HasSubset.Subset.{u1} (Set.{u1} α) (Set.instHasSubsetSet.{u1} α) t s) (And (Set.Finite.{u1} α t) (HasSubset.Subset.{u1} (Set.{u1} α) (Set.instHasSubsetSet.{u1} α) s (Set.unionᵢ.{u1, succ u1} α α (fun (x : α) => Set.unionᵢ.{u1, 0} α (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) x t) (fun (h._@.Mathlib.Topology.MetricSpace.Basic._hyg.25459 : Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) x t) => Metric.ball.{u1} α _inst_2 x e))))))))
+  forall {α : Type.{u1}} [_inst_2 : PseudoMetricSpace.{u1} α] {s : Set.{u1} α}, (IsCompact.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_2)) s) -> (forall {e : Real}, (LT.lt.{0} Real Real.instLTReal (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal)) e) -> (Exists.{succ u1} (Set.{u1} α) (fun (t : Set.{u1} α) => And (HasSubset.Subset.{u1} (Set.{u1} α) (Set.instHasSubsetSet.{u1} α) t s) (And (Set.Finite.{u1} α t) (HasSubset.Subset.{u1} (Set.{u1} α) (Set.instHasSubsetSet.{u1} α) s (Set.unionᵢ.{u1, succ u1} α α (fun (x : α) => Set.unionᵢ.{u1, 0} α (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) x t) (fun (h._@.Mathlib.Topology.MetricSpace.Basic._hyg.25465 : Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) x t) => Metric.ball.{u1} α _inst_2 x e))))))))
 Case conversion may be inaccurate. Consider using '#align is_compact.finite_cover_balls IsCompact.finite_cover_ballsₓ'. -/
 alias finite_cover_balls_of_compact ← IsCompact.finite_cover_balls
 #align is_compact.finite_cover_balls IsCompact.finite_cover_balls

mathlib commit https://github.com/leanprover-community/mathlib/commit/21e3562c5e12d846c7def5eff8cdbc520d7d4936

@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Jeremy Avigad, Robert Y. Lewis, Johannes Hölzl, Mario Carneiro, Sébastien Gouëzel
 
 ! This file was ported from Lean 3 source module topology.metric_space.basic
-! leanprover-community/mathlib commit 195fcd60ff2bfe392543bceb0ec2adcdb472db4c
+! leanprover-community/mathlib commit f47581155c818e6361af4e4fda60d27d020c226b
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
@@ -16,6 +16,9 @@ import Mathbin.Topology.Bornology.Constructions
 /-!
 # Metric spaces
 
+> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.
+> Any changes to this file require a corresponding PR to mathlib4.
+
 This file defines metric spaces. Many definitions and theorems expected
 on metric spaces are already introduced on uniform spaces and topological spaces.
 For example: open and closed sets, compactness, completeness, continuity and uniform continuity

mathlib commit https://github.com/leanprover-community/mathlib/commit/38f16f960f5006c6c0c2bac7b0aba5273188f4e5

@@ -2463,7 +2463,7 @@ theorem Real.dist_eq (x y : ℝ) : dist x y = |x - y| :=
 lean 3 declaration is
   forall (x : Real) (y : Real), Eq.{1} NNReal (NNDist.nndist.{0} Real (PseudoMetricSpace.toNNDist.{0} Real Real.pseudoMetricSpace) x y) (coeFn.{1, 1} (MonoidWithZeroHom.{0, 0} Real NNReal (NonAssocSemiring.toMulZeroOneClass.{0} Real (NonAssocRing.toNonAssocSemiring.{0} Real (Ring.toNonAssocRing.{0} Real Real.ring))) (NonAssocSemiring.toMulZeroOneClass.{0} NNReal (Semiring.toNonAssocSemiring.{0} NNReal NNReal.semiring))) (fun (_x : MonoidWithZeroHom.{0, 0} Real NNReal (NonAssocSemiring.toMulZeroOneClass.{0} Real (NonAssocRing.toNonAssocSemiring.{0} Real (Ring.toNonAssocRing.{0} Real Real.ring))) (NonAssocSemiring.toMulZeroOneClass.{0} NNReal (Semiring.toNonAssocSemiring.{0} NNReal NNReal.semiring))) => Real -> NNReal) (MonoidWithZeroHom.hasCoeToFun.{0, 0} Real NNReal (NonAssocSemiring.toMulZeroOneClass.{0} Real (NonAssocRing.toNonAssocSemiring.{0} Real (Ring.toNonAssocRing.{0} Real Real.ring))) (NonAssocSemiring.toMulZeroOneClass.{0} NNReal (Semiring.toNonAssocSemiring.{0} NNReal NNReal.semiring))) Real.nnabs (HSub.hSub.{0, 0, 0} Real Real Real (instHSub.{0} Real Real.hasSub) x y))
 but is expected to have type
-  forall (x : Real) (y : Real), Eq.{1} NNReal (NNDist.nndist.{0} Real (PseudoMetricSpace.toNNDist.{0} Real Real.pseudoMetricSpace) x y) (FunLike.coe.{1, 1, 1} (MonoidWithZeroHom.{0, 0} Real NNReal (NonAssocSemiring.toMulZeroOneClass.{0} Real (NonAssocRing.toNonAssocSemiring.{0} Real (Ring.toNonAssocRing.{0} Real Real.instRingReal))) (NonAssocSemiring.toMulZeroOneClass.{0} NNReal (Semiring.toNonAssocSemiring.{0} NNReal instNNRealSemiring))) Real (fun (_x : Real) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2398 : Real) => NNReal) _x) (MulHomClass.toFunLike.{0, 0, 0} (MonoidWithZeroHom.{0, 0} Real NNReal (NonAssocSemiring.toMulZeroOneClass.{0} Real (NonAssocRing.toNonAssocSemiring.{0} Real (Ring.toNonAssocRing.{0} Real Real.instRingReal))) (NonAssocSemiring.toMulZeroOneClass.{0} NNReal (Semiring.toNonAssocSemiring.{0} NNReal instNNRealSemiring))) Real NNReal (MulOneClass.toMul.{0} Real (MulZeroOneClass.toMulOneClass.{0} Real (NonAssocSemiring.toMulZeroOneClass.{0} Real (NonAssocRing.toNonAssocSemiring.{0} Real (Ring.toNonAssocRing.{0} Real Real.instRingReal))))) (MulOneClass.toMul.{0} NNReal (MulZeroOneClass.toMulOneClass.{0} NNReal (NonAssocSemiring.toMulZeroOneClass.{0} NNReal (Semiring.toNonAssocSemiring.{0} NNReal instNNRealSemiring)))) (MonoidHomClass.toMulHomClass.{0, 0, 0} (MonoidWithZeroHom.{0, 0} Real NNReal (NonAssocSemiring.toMulZeroOneClass.{0} Real (NonAssocRing.toNonAssocSemiring.{0} Real (Ring.toNonAssocRing.{0} Real Real.instRingReal))) (NonAssocSemiring.toMulZeroOneClass.{0} NNReal (Semiring.toNonAssocSemiring.{0} NNReal instNNRealSemiring))) Real NNReal (MulZeroOneClass.toMulOneClass.{0} Real (NonAssocSemiring.toMulZeroOneClass.{0} Real (NonAssocRing.toNonAssocSemiring.{0} Real (Ring.toNonAssocRing.{0} Real Real.instRingReal)))) (MulZeroOneClass.toMulOneClass.{0} NNReal (NonAssocSemiring.toMulZeroOneClass.{0} NNReal (Semiring.toNonAssocSemiring.{0} NNReal instNNRealSemiring))) (MonoidWithZeroHomClass.toMonoidHomClass.{0, 0, 0} (MonoidWithZeroHom.{0, 0} Real NNReal (NonAssocSemiring.toMulZeroOneClass.{0} Real (NonAssocRing.toNonAssocSemiring.{0} Real (Ring.toNonAssocRing.{0} Real Real.instRingReal))) (NonAssocSemiring.toMulZeroOneClass.{0} NNReal (Semiring.toNonAssocSemiring.{0} NNReal instNNRealSemiring))) Real NNReal (NonAssocSemiring.toMulZeroOneClass.{0} Real (NonAssocRing.toNonAssocSemiring.{0} Real (Ring.toNonAssocRing.{0} Real Real.instRingReal))) (NonAssocSemiring.toMulZeroOneClass.{0} NNReal (Semiring.toNonAssocSemiring.{0} NNReal instNNRealSemiring)) (MonoidWithZeroHom.monoidWithZeroHomClass.{0, 0} Real NNReal (NonAssocSemiring.toMulZeroOneClass.{0} Real (NonAssocRing.toNonAssocSemiring.{0} Real (Ring.toNonAssocRing.{0} Real Real.instRingReal))) (NonAssocSemiring.toMulZeroOneClass.{0} NNReal (Semiring.toNonAssocSemiring.{0} NNReal instNNRealSemiring)))))) Real.nnabs (HSub.hSub.{0, 0, 0} Real Real Real (instHSub.{0} Real Real.instSubReal) x y))
+  forall (x : Real) (y : Real), Eq.{1} NNReal (NNDist.nndist.{0} Real (PseudoMetricSpace.toNNDist.{0} Real Real.pseudoMetricSpace) x y) (FunLike.coe.{1, 1, 1} (MonoidWithZeroHom.{0, 0} Real NNReal (NonAssocSemiring.toMulZeroOneClass.{0} Real (NonAssocRing.toNonAssocSemiring.{0} Real (Ring.toNonAssocRing.{0} Real Real.instRingReal))) (NonAssocSemiring.toMulZeroOneClass.{0} NNReal (Semiring.toNonAssocSemiring.{0} NNReal instNNRealSemiring))) Real (fun (_x : Real) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2372 : Real) => NNReal) _x) (MulHomClass.toFunLike.{0, 0, 0} (MonoidWithZeroHom.{0, 0} Real NNReal (NonAssocSemiring.toMulZeroOneClass.{0} Real (NonAssocRing.toNonAssocSemiring.{0} Real (Ring.toNonAssocRing.{0} Real Real.instRingReal))) (NonAssocSemiring.toMulZeroOneClass.{0} NNReal (Semiring.toNonAssocSemiring.{0} NNReal instNNRealSemiring))) Real NNReal (MulOneClass.toMul.{0} Real (MulZeroOneClass.toMulOneClass.{0} Real (NonAssocSemiring.toMulZeroOneClass.{0} Real (NonAssocRing.toNonAssocSemiring.{0} Real (Ring.toNonAssocRing.{0} Real Real.instRingReal))))) (MulOneClass.toMul.{0} NNReal (MulZeroOneClass.toMulOneClass.{0} NNReal (NonAssocSemiring.toMulZeroOneClass.{0} NNReal (Semiring.toNonAssocSemiring.{0} NNReal instNNRealSemiring)))) (MonoidHomClass.toMulHomClass.{0, 0, 0} (MonoidWithZeroHom.{0, 0} Real NNReal (NonAssocSemiring.toMulZeroOneClass.{0} Real (NonAssocRing.toNonAssocSemiring.{0} Real (Ring.toNonAssocRing.{0} Real Real.instRingReal))) (NonAssocSemiring.toMulZeroOneClass.{0} NNReal (Semiring.toNonAssocSemiring.{0} NNReal instNNRealSemiring))) Real NNReal (MulZeroOneClass.toMulOneClass.{0} Real (NonAssocSemiring.toMulZeroOneClass.{0} Real (NonAssocRing.toNonAssocSemiring.{0} Real (Ring.toNonAssocRing.{0} Real Real.instRingReal)))) (MulZeroOneClass.toMulOneClass.{0} NNReal (NonAssocSemiring.toMulZeroOneClass.{0} NNReal (Semiring.toNonAssocSemiring.{0} NNReal instNNRealSemiring))) (MonoidWithZeroHomClass.toMonoidHomClass.{0, 0, 0} (MonoidWithZeroHom.{0, 0} Real NNReal (NonAssocSemiring.toMulZeroOneClass.{0} Real (NonAssocRing.toNonAssocSemiring.{0} Real (Ring.toNonAssocRing.{0} Real Real.instRingReal))) (NonAssocSemiring.toMulZeroOneClass.{0} NNReal (Semiring.toNonAssocSemiring.{0} NNReal instNNRealSemiring))) Real NNReal (NonAssocSemiring.toMulZeroOneClass.{0} Real (NonAssocRing.toNonAssocSemiring.{0} Real (Ring.toNonAssocRing.{0} Real Real.instRingReal))) (NonAssocSemiring.toMulZeroOneClass.{0} NNReal (Semiring.toNonAssocSemiring.{0} NNReal instNNRealSemiring)) (MonoidWithZeroHom.monoidWithZeroHomClass.{0, 0} Real NNReal (NonAssocSemiring.toMulZeroOneClass.{0} Real (NonAssocRing.toNonAssocSemiring.{0} Real (Ring.toNonAssocRing.{0} Real Real.instRingReal))) (NonAssocSemiring.toMulZeroOneClass.{0} NNReal (Semiring.toNonAssocSemiring.{0} NNReal instNNRealSemiring)))))) Real.nnabs (HSub.hSub.{0, 0, 0} Real Real Real (instHSub.{0} Real Real.instSubReal) x y))
 Case conversion may be inaccurate. Consider using '#align real.nndist_eq Real.nndist_eqₓ'. -/
 theorem Real.nndist_eq (x y : ℝ) : nndist x y = Real.nnabs (x - y) :=
   rfl
@@ -2473,7 +2473,7 @@ theorem Real.nndist_eq (x y : ℝ) : nndist x y = Real.nnabs (x - y) :=
 lean 3 declaration is
   forall (x : Real) (y : Real), Eq.{1} NNReal (NNDist.nndist.{0} Real (PseudoMetricSpace.toNNDist.{0} Real Real.pseudoMetricSpace) x y) (coeFn.{1, 1} (MonoidWithZeroHom.{0, 0} Real NNReal (NonAssocSemiring.toMulZeroOneClass.{0} Real (NonAssocRing.toNonAssocSemiring.{0} Real (Ring.toNonAssocRing.{0} Real Real.ring))) (NonAssocSemiring.toMulZeroOneClass.{0} NNReal (Semiring.toNonAssocSemiring.{0} NNReal NNReal.semiring))) (fun (_x : MonoidWithZeroHom.{0, 0} Real NNReal (NonAssocSemiring.toMulZeroOneClass.{0} Real (NonAssocRing.toNonAssocSemiring.{0} Real (Ring.toNonAssocRing.{0} Real Real.ring))) (NonAssocSemiring.toMulZeroOneClass.{0} NNReal (Semiring.toNonAssocSemiring.{0} NNReal NNReal.semiring))) => Real -> NNReal) (MonoidWithZeroHom.hasCoeToFun.{0, 0} Real NNReal (NonAssocSemiring.toMulZeroOneClass.{0} Real (NonAssocRing.toNonAssocSemiring.{0} Real (Ring.toNonAssocRing.{0} Real Real.ring))) (NonAssocSemiring.toMulZeroOneClass.{0} NNReal (Semiring.toNonAssocSemiring.{0} NNReal NNReal.semiring))) Real.nnabs (HSub.hSub.{0, 0, 0} Real Real Real (instHSub.{0} Real Real.hasSub) y x))
 but is expected to have type
-  forall (x : Real) (y : Real), Eq.{1} NNReal (NNDist.nndist.{0} Real (PseudoMetricSpace.toNNDist.{0} Real Real.pseudoMetricSpace) x y) (FunLike.coe.{1, 1, 1} (MonoidWithZeroHom.{0, 0} Real NNReal (NonAssocSemiring.toMulZeroOneClass.{0} Real (NonAssocRing.toNonAssocSemiring.{0} Real (Ring.toNonAssocRing.{0} Real Real.instRingReal))) (NonAssocSemiring.toMulZeroOneClass.{0} NNReal (Semiring.toNonAssocSemiring.{0} NNReal instNNRealSemiring))) Real (fun (_x : Real) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2398 : Real) => NNReal) _x) (MulHomClass.toFunLike.{0, 0, 0} (MonoidWithZeroHom.{0, 0} Real NNReal (NonAssocSemiring.toMulZeroOneClass.{0} Real (NonAssocRing.toNonAssocSemiring.{0} Real (Ring.toNonAssocRing.{0} Real Real.instRingReal))) (NonAssocSemiring.toMulZeroOneClass.{0} NNReal (Semiring.toNonAssocSemiring.{0} NNReal instNNRealSemiring))) Real NNReal (MulOneClass.toMul.{0} Real (MulZeroOneClass.toMulOneClass.{0} Real (NonAssocSemiring.toMulZeroOneClass.{0} Real (NonAssocRing.toNonAssocSemiring.{0} Real (Ring.toNonAssocRing.{0} Real Real.instRingReal))))) (MulOneClass.toMul.{0} NNReal (MulZeroOneClass.toMulOneClass.{0} NNReal (NonAssocSemiring.toMulZeroOneClass.{0} NNReal (Semiring.toNonAssocSemiring.{0} NNReal instNNRealSemiring)))) (MonoidHomClass.toMulHomClass.{0, 0, 0} (MonoidWithZeroHom.{0, 0} Real NNReal (NonAssocSemiring.toMulZeroOneClass.{0} Real (NonAssocRing.toNonAssocSemiring.{0} Real (Ring.toNonAssocRing.{0} Real Real.instRingReal))) (NonAssocSemiring.toMulZeroOneClass.{0} NNReal (Semiring.toNonAssocSemiring.{0} NNReal instNNRealSemiring))) Real NNReal (MulZeroOneClass.toMulOneClass.{0} Real (NonAssocSemiring.toMulZeroOneClass.{0} Real (NonAssocRing.toNonAssocSemiring.{0} Real (Ring.toNonAssocRing.{0} Real Real.instRingReal)))) (MulZeroOneClass.toMulOneClass.{0} NNReal (NonAssocSemiring.toMulZeroOneClass.{0} NNReal (Semiring.toNonAssocSemiring.{0} NNReal instNNRealSemiring))) (MonoidWithZeroHomClass.toMonoidHomClass.{0, 0, 0} (MonoidWithZeroHom.{0, 0} Real NNReal (NonAssocSemiring.toMulZeroOneClass.{0} Real (NonAssocRing.toNonAssocSemiring.{0} Real (Ring.toNonAssocRing.{0} Real Real.instRingReal))) (NonAssocSemiring.toMulZeroOneClass.{0} NNReal (Semiring.toNonAssocSemiring.{0} NNReal instNNRealSemiring))) Real NNReal (NonAssocSemiring.toMulZeroOneClass.{0} Real (NonAssocRing.toNonAssocSemiring.{0} Real (Ring.toNonAssocRing.{0} Real Real.instRingReal))) (NonAssocSemiring.toMulZeroOneClass.{0} NNReal (Semiring.toNonAssocSemiring.{0} NNReal instNNRealSemiring)) (MonoidWithZeroHom.monoidWithZeroHomClass.{0, 0} Real NNReal (NonAssocSemiring.toMulZeroOneClass.{0} Real (NonAssocRing.toNonAssocSemiring.{0} Real (Ring.toNonAssocRing.{0} Real Real.instRingReal))) (NonAssocSemiring.toMulZeroOneClass.{0} NNReal (Semiring.toNonAssocSemiring.{0} NNReal instNNRealSemiring)))))) Real.nnabs (HSub.hSub.{0, 0, 0} Real Real Real (instHSub.{0} Real Real.instSubReal) y x))
+  forall (x : Real) (y : Real), Eq.{1} NNReal (NNDist.nndist.{0} Real (PseudoMetricSpace.toNNDist.{0} Real Real.pseudoMetricSpace) x y) (FunLike.coe.{1, 1, 1} (MonoidWithZeroHom.{0, 0} Real NNReal (NonAssocSemiring.toMulZeroOneClass.{0} Real (NonAssocRing.toNonAssocSemiring.{0} Real (Ring.toNonAssocRing.{0} Real Real.instRingReal))) (NonAssocSemiring.toMulZeroOneClass.{0} NNReal (Semiring.toNonAssocSemiring.{0} NNReal instNNRealSemiring))) Real (fun (_x : Real) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2372 : Real) => NNReal) _x) (MulHomClass.toFunLike.{0, 0, 0} (MonoidWithZeroHom.{0, 0} Real NNReal (NonAssocSemiring.toMulZeroOneClass.{0} Real (NonAssocRing.toNonAssocSemiring.{0} Real (Ring.toNonAssocRing.{0} Real Real.instRingReal))) (NonAssocSemiring.toMulZeroOneClass.{0} NNReal (Semiring.toNonAssocSemiring.{0} NNReal instNNRealSemiring))) Real NNReal (MulOneClass.toMul.{0} Real (MulZeroOneClass.toMulOneClass.{0} Real (NonAssocSemiring.toMulZeroOneClass.{0} Real (NonAssocRing.toNonAssocSemiring.{0} Real (Ring.toNonAssocRing.{0} Real Real.instRingReal))))) (MulOneClass.toMul.{0} NNReal (MulZeroOneClass.toMulOneClass.{0} NNReal (NonAssocSemiring.toMulZeroOneClass.{0} NNReal (Semiring.toNonAssocSemiring.{0} NNReal instNNRealSemiring)))) (MonoidHomClass.toMulHomClass.{0, 0, 0} (MonoidWithZeroHom.{0, 0} Real NNReal (NonAssocSemiring.toMulZeroOneClass.{0} Real (NonAssocRing.toNonAssocSemiring.{0} Real (Ring.toNonAssocRing.{0} Real Real.instRingReal))) (NonAssocSemiring.toMulZeroOneClass.{0} NNReal (Semiring.toNonAssocSemiring.{0} NNReal instNNRealSemiring))) Real NNReal (MulZeroOneClass.toMulOneClass.{0} Real (NonAssocSemiring.toMulZeroOneClass.{0} Real (NonAssocRing.toNonAssocSemiring.{0} Real (Ring.toNonAssocRing.{0} Real Real.instRingReal)))) (MulZeroOneClass.toMulOneClass.{0} NNReal (NonAssocSemiring.toMulZeroOneClass.{0} NNReal (Semiring.toNonAssocSemiring.{0} NNReal instNNRealSemiring))) (MonoidWithZeroHomClass.toMonoidHomClass.{0, 0, 0} (MonoidWithZeroHom.{0, 0} Real NNReal (NonAssocSemiring.toMulZeroOneClass.{0} Real (NonAssocRing.toNonAssocSemiring.{0} Real (Ring.toNonAssocRing.{0} Real Real.instRingReal))) (NonAssocSemiring.toMulZeroOneClass.{0} NNReal (Semiring.toNonAssocSemiring.{0} NNReal instNNRealSemiring))) Real NNReal (NonAssocSemiring.toMulZeroOneClass.{0} Real (NonAssocRing.toNonAssocSemiring.{0} Real (Ring.toNonAssocRing.{0} Real Real.instRingReal))) (NonAssocSemiring.toMulZeroOneClass.{0} NNReal (Semiring.toNonAssocSemiring.{0} NNReal instNNRealSemiring)) (MonoidWithZeroHom.monoidWithZeroHomClass.{0, 0} Real NNReal (NonAssocSemiring.toMulZeroOneClass.{0} Real (NonAssocRing.toNonAssocSemiring.{0} Real (Ring.toNonAssocRing.{0} Real Real.instRingReal))) (NonAssocSemiring.toMulZeroOneClass.{0} NNReal (Semiring.toNonAssocSemiring.{0} NNReal instNNRealSemiring)))))) Real.nnabs (HSub.hSub.{0, 0, 0} Real Real Real (instHSub.{0} Real Real.instSubReal) y x))
 Case conversion may be inaccurate. Consider using '#align real.nndist_eq' Real.nndist_eq'ₓ'. -/
 theorem Real.nndist_eq' (x y : ℝ) : nndist x y = Real.nnabs (y - x) :=
   nndist_comm _ _

mathlib commit https://github.com/leanprover-community/mathlib/commit/195fcd60ff2bfe392543bceb0ec2adcdb472db4c

@@ -60,13 +60,19 @@ universe u v w
 
 variable {α : Type u} {β : Type v} {X ι : Type _}
 
+/- warning: uniform_space_of_dist -> UniformSpace.ofDist is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} (dist : α -> α -> Real), (forall (x : α), Eq.{1} Real (dist x x) (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero)))) -> (forall (x : α) (y : α), Eq.{1} Real (dist x y) (dist y x)) -> (forall (x : α) (y : α) (z : α), LE.le.{0} Real Real.hasLe (dist x z) (HAdd.hAdd.{0, 0, 0} Real Real Real (instHAdd.{0} Real Real.hasAdd) (dist x y) (dist y z))) -> (UniformSpace.{u1} α)
+but is expected to have type
+  forall {α : Type.{u1}} (dist : α -> α -> Real), (forall (x : α), Eq.{1} Real (dist x x) (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal))) -> (forall (x : α) (y : α), Eq.{1} Real (dist x y) (dist y x)) -> (forall (x : α) (y : α) (z : α), LE.le.{0} Real Real.instLEReal (dist x z) (HAdd.hAdd.{0, 0, 0} Real Real Real (instHAdd.{0} Real Real.instAddReal) (dist x y) (dist y z))) -> (UniformSpace.{u1} α)
+Case conversion may be inaccurate. Consider using '#align uniform_space_of_dist UniformSpace.ofDistₓ'. -/
 /-- Construct a uniform structure from a distance function and metric space axioms -/
-def uniformSpaceOfDist (dist : α → α → ℝ) (dist_self : ∀ x : α, dist x x = 0)
+def UniformSpace.ofDist (dist : α → α → ℝ) (dist_self : ∀ x : α, dist x x = 0)
     (dist_comm : ∀ x y : α, dist x y = dist y x)
     (dist_triangle : ∀ x y z : α, dist x z ≤ dist x y + dist y z) : UniformSpace α :=
   UniformSpace.ofFun dist dist_self dist_comm dist_triangle fun ε ε0 =>
     ⟨ε / 2, half_pos ε0, fun x hx y hy => add_halves ε ▸ add_lt_add hx hy⟩
-#align uniform_space_of_dist uniformSpaceOfDist
+#align uniform_space_of_dist UniformSpace.ofDist
 
 /-- This is an internal lemma used to construct a bornology from a metric in `bornology.of_dist`. -/
 private theorem bounded_iff_aux {α : Type _} (dist : α → α → ℝ)
@@ -107,16 +113,18 @@ def Bornology.ofDist {α : Type _} (dist : α → α → ℝ) (dist_self : ∀ x
     ⟨0, fun x hx y hy => by
       rw [eq_of_mem_singleton hx, eq_of_mem_singleton hy]
       exact (dist_self z).le⟩
-#align bornology.of_dist Bornology.ofDist
+#align bornology.of_dist Bornology.ofDistₓ
 
+#print Dist /-
 /-- The distance function (given an ambient metric space on `α`), which returns
   a nonnegative real number `dist x y` given `x y : α`. -/
 @[ext]
-class HasDist (α : Type _) where
+class Dist (α : Type _) where
   dist : α → α → ℝ
-#align has_dist HasDist
+#align has_dist Dist
+-/
 
-export HasDist (dist)
+export Dist (dist)
 
 -- the uniform structure and the emetric space structure are embedded in the metric space structure
 -- to avoid instance diamond issues. See Note [forgetful inheritance].
@@ -132,13 +140,18 @@ private theorem pseudo_metric_space.dist_nonneg' {α} {x y : α} (dist : α →
   nonneg_of_mul_nonneg_right this zero_lt_two
 #align pseudo_metric_space.dist_nonneg' pseudo_metric_space.dist_nonneg'
 
+/- warning: pseudo_metric_space.edist_dist_tac clashes with [anonymous] -> [anonymous]
+Case conversion may be inaccurate. Consider using '#align pseudo_metric_space.edist_dist_tac [anonymous]ₓ'. -/
 /- ./././Mathport/Syntax/Translate/Expr.lean:330:4: warning: unsupported (TODO): `[tacs] -/
+#print [anonymous] /-
 /-- This tactic is used to populate `pseudo_metric_space.edist_dist` when the default `edist` is
 used. -/
-protected unsafe def pseudo_metric_space.edist_dist_tac : tactic Unit :=
+protected unsafe def [anonymous] : tactic Unit :=
   tactic.intros >> sorry
-#align pseudo_metric_space.edist_dist_tac pseudo_metric_space.edist_dist_tac
+#align pseudo_metric_space.edist_dist_tac [anonymous]
+-/
 
+#print PseudoMetricSpace /-
 /- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic pseudo_metric_space.edist_dist_tac -/
 /-- Pseudo metric and Metric spaces
 
@@ -150,7 +163,7 @@ structure. When instantiating a `pseudo_metric_space` structure, the uniformity
 necessary, they will be filled in by default. In the same way, each (pseudo) metric space induces a
 (pseudo) emetric space structure. It is included in the structure, but filled in by default.
 -/
-class PseudoMetricSpace (α : Type u) extends HasDist α : Type u where
+class PseudoMetricSpace (α : Type u) extends Dist α : Type u where
   dist_self : ∀ x : α, dist x x = 0
   dist_comm : ∀ x y : α, dist x y = dist y x
   dist_triangle : ∀ x y z : α, dist x z ≤ dist x y + dist y z
@@ -158,8 +171,8 @@ class PseudoMetricSpace (α : Type u) extends HasDist α : Type u where
     @coe ℝ≥0 _ _ ⟨dist x y, PseudoMetricSpace.dist_nonneg' _ ‹_› ‹_› ‹_›⟩
   edist_dist : ∀ x y : α, edist x y = ENNReal.ofReal (dist x y) := by
     run_tac
-      pseudo_metric_space.edist_dist_tac
-  toUniformSpace : UniformSpace α := uniformSpaceOfDist dist dist_self dist_comm dist_triangle
+      [anonymous]
+  toUniformSpace : UniformSpace α := UniformSpace.ofDist dist dist_self dist_comm dist_triangle
   uniformity_dist :
     𝓤 α = ⨅ ε > 0, 𝓟 { p : α × α |
             dist p.1 p.2 < ε } := by
@@ -173,7 +186,9 @@ class PseudoMetricSpace (α : Type u) extends HasDist α : Type u where
     intros
     rfl
 #align pseudo_metric_space PseudoMetricSpace
+-/
 
+#print PseudoMetricSpace.ext /-
 /-- Two pseudo metric space structures with the same distance function coincide. -/
 @[ext]
 theorem PseudoMetricSpace.ext {α : Type _} {m m' : PseudoMetricSpace α}
@@ -195,6 +210,7 @@ theorem PseudoMetricSpace.ext {α : Type _} {m m' : PseudoMetricSpace α}
     rw [← m'_cobounded_sets] at m_cobounded_sets
     exact filter_eq m_cobounded_sets
 #align pseudo_metric_space.ext PseudoMetricSpace.ext
+-/
 
 variable [PseudoMetricSpace α]
 
@@ -202,11 +218,19 @@ attribute [instance] PseudoMetricSpace.toUniformSpace
 
 attribute [instance] PseudoMetricSpace.toBornology
 
+#print PseudoMetricSpace.toEDist /-
 -- see Note [lower instance priority]
-instance (priority := 200) PseudoMetricSpace.toHasEdist : EDist α :=
+instance (priority := 200) PseudoMetricSpace.toEDist : EDist α :=
   ⟨PseudoMetricSpace.edist⟩
-#align pseudo_metric_space.to_has_edist PseudoMetricSpace.toHasEdist
+#align pseudo_metric_space.to_has_edist PseudoMetricSpace.toEDist
+-/
 
+/- warning: pseudo_metric_space.of_dist_topology -> PseudoMetricSpace.ofDistTopology is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} [_inst_2 : TopologicalSpace.{u1} α] (dist : α -> α -> Real), (forall (x : α), Eq.{1} Real (dist x x) (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero)))) -> (forall (x : α) (y : α), Eq.{1} Real (dist x y) (dist y x)) -> (forall (x : α) (y : α) (z : α), LE.le.{0} Real Real.hasLe (dist x z) (HAdd.hAdd.{0, 0, 0} Real Real Real (instHAdd.{0} Real Real.hasAdd) (dist x y) (dist y z))) -> (forall (s : Set.{u1} α), Iff (IsOpen.{u1} α _inst_2 s) (forall (x : α), (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) x s) -> (Exists.{1} Real (fun (ε : Real) => Exists.{0} (GT.gt.{0} Real Real.hasLt ε (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero)))) (fun (H : GT.gt.{0} Real Real.hasLt ε (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero)))) => forall (y : α), (LT.lt.{0} Real Real.hasLt (dist x y) ε) -> (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) y s)))))) -> (PseudoMetricSpace.{u1} α)
+but is expected to have type
+  forall {α : Type.{u1}} [_inst_2 : TopologicalSpace.{u1} α] (dist : α -> α -> Real), (forall (x : α), Eq.{1} Real (dist x x) (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal))) -> (forall (x : α) (y : α), Eq.{1} Real (dist x y) (dist y x)) -> (forall (x : α) (y : α) (z : α), LE.le.{0} Real Real.instLEReal (dist x z) (HAdd.hAdd.{0, 0, 0} Real Real Real (instHAdd.{0} Real Real.instAddReal) (dist x y) (dist y z))) -> (forall (s : Set.{u1} α), Iff (IsOpen.{u1} α _inst_2 s) (forall (x : α), (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) x s) -> (Exists.{1} Real (fun (ε : Real) => And (GT.gt.{0} Real Real.instLTReal ε (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal))) (forall (y : α), (LT.lt.{0} Real Real.instLTReal (dist x y) ε) -> (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) y s)))))) -> (PseudoMetricSpace.{u1} α)
+Case conversion may be inaccurate. Consider using '#align pseudo_metric_space.of_dist_topology PseudoMetricSpace.ofDistTopologyₓ'. -/
 /-- Construct a pseudo-metric space structure whose underlying topological space structure
 (definitionally) agrees which a pre-existing topology which is compatible with a given distance
 function. -/
@@ -226,37 +250,71 @@ def PseudoMetricSpace.ofDistTopology {α : Type u} [TopologicalSpace α] (dist :
               ((UniformSpace.hasBasis_ofFun (exists_gt (0 : ℝ)) dist _ _ _ _).comap
                         (Prod.mk x)).mem_iff.symm.trans
                 mem_comap_prod_mk
-        toCore := (uniformSpaceOfDist dist dist_self dist_comm dist_triangle).toCore }
+        toCore := (UniformSpace.ofDist dist dist_self dist_comm dist_triangle).toCore }
     uniformity_dist := rfl
     toBornology := Bornology.ofDist dist dist_self dist_comm dist_triangle
     cobounded_sets := rfl }
 #align pseudo_metric_space.of_dist_topology PseudoMetricSpace.ofDistTopology
 
+/- warning: dist_self -> dist_self is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] (x : α), Eq.{1} Real (Dist.dist.{u1} α (PseudoMetricSpace.toHasDist.{u1} α _inst_1) x x) (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero)))
+but is expected to have type
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] (x : α), Eq.{1} Real (Dist.dist.{u1} α (PseudoMetricSpace.toDist.{u1} α _inst_1) x x) (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal))
+Case conversion may be inaccurate. Consider using '#align dist_self dist_selfₓ'. -/
 @[simp]
 theorem dist_self (x : α) : dist x x = 0 :=
   PseudoMetricSpace.dist_self x
 #align dist_self dist_self
 
+#print dist_comm /-
 theorem dist_comm (x y : α) : dist x y = dist y x :=
   PseudoMetricSpace.dist_comm x y
 #align dist_comm dist_comm
+-/
 
+#print edist_dist /-
 theorem edist_dist (x y : α) : edist x y = ENNReal.ofReal (dist x y) :=
   PseudoMetricSpace.edist_dist x y
 #align edist_dist edist_dist
+-/
 
+/- warning: dist_triangle -> dist_triangle is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] (x : α) (y : α) (z : α), LE.le.{0} Real Real.hasLe (Dist.dist.{u1} α (PseudoMetricSpace.toHasDist.{u1} α _inst_1) x z) (HAdd.hAdd.{0, 0, 0} Real Real Real (instHAdd.{0} Real Real.hasAdd) (Dist.dist.{u1} α (PseudoMetricSpace.toHasDist.{u1} α _inst_1) x y) (Dist.dist.{u1} α (PseudoMetricSpace.toHasDist.{u1} α _inst_1) y z))
+but is expected to have type
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] (x : α) (y : α) (z : α), LE.le.{0} Real Real.instLEReal (Dist.dist.{u1} α (PseudoMetricSpace.toDist.{u1} α _inst_1) x z) (HAdd.hAdd.{0, 0, 0} Real Real Real (instHAdd.{0} Real Real.instAddReal) (Dist.dist.{u1} α (PseudoMetricSpace.toDist.{u1} α _inst_1) x y) (Dist.dist.{u1} α (PseudoMetricSpace.toDist.{u1} α _inst_1) y z))
+Case conversion may be inaccurate. Consider using '#align dist_triangle dist_triangleₓ'. -/
 theorem dist_triangle (x y z : α) : dist x z ≤ dist x y + dist y z :=
   PseudoMetricSpace.dist_triangle x y z
 #align dist_triangle dist_triangle
 
+/- warning: dist_triangle_left -> dist_triangle_left is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] (x : α) (y : α) (z : α), LE.le.{0} Real Real.hasLe (Dist.dist.{u1} α (PseudoMetricSpace.toHasDist.{u1} α _inst_1) x y) (HAdd.hAdd.{0, 0, 0} Real Real Real (instHAdd.{0} Real Real.hasAdd) (Dist.dist.{u1} α (PseudoMetricSpace.toHasDist.{u1} α _inst_1) z x) (Dist.dist.{u1} α (PseudoMetricSpace.toHasDist.{u1} α _inst_1) z y))
+but is expected to have type
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] (x : α) (y : α) (z : α), LE.le.{0} Real Real.instLEReal (Dist.dist.{u1} α (PseudoMetricSpace.toDist.{u1} α _inst_1) x y) (HAdd.hAdd.{0, 0, 0} Real Real Real (instHAdd.{0} Real Real.instAddReal) (Dist.dist.{u1} α (PseudoMetricSpace.toDist.{u1} α _inst_1) z x) (Dist.dist.{u1} α (PseudoMetricSpace.toDist.{u1} α _inst_1) z y))
+Case conversion may be inaccurate. Consider using '#align dist_triangle_left dist_triangle_leftₓ'. -/
 theorem dist_triangle_left (x y z : α) : dist x y ≤ dist z x + dist z y := by
   rw [dist_comm z] <;> apply dist_triangle
 #align dist_triangle_left dist_triangle_left
 
+/- warning: dist_triangle_right -> dist_triangle_right is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] (x : α) (y : α) (z : α), LE.le.{0} Real Real.hasLe (Dist.dist.{u1} α (PseudoMetricSpace.toHasDist.{u1} α _inst_1) x y) (HAdd.hAdd.{0, 0, 0} Real Real Real (instHAdd.{0} Real Real.hasAdd) (Dist.dist.{u1} α (PseudoMetricSpace.toHasDist.{u1} α _inst_1) x z) (Dist.dist.{u1} α (PseudoMetricSpace.toHasDist.{u1} α _inst_1) y z))
+but is expected to have type
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] (x : α) (y : α) (z : α), LE.le.{0} Real Real.instLEReal (Dist.dist.{u1} α (PseudoMetricSpace.toDist.{u1} α _inst_1) x y) (HAdd.hAdd.{0, 0, 0} Real Real Real (instHAdd.{0} Real Real.instAddReal) (Dist.dist.{u1} α (PseudoMetricSpace.toDist.{u1} α _inst_1) x z) (Dist.dist.{u1} α (PseudoMetricSpace.toDist.{u1} α _inst_1) y z))
+Case conversion may be inaccurate. Consider using '#align dist_triangle_right dist_triangle_rightₓ'. -/
 theorem dist_triangle_right (x y z : α) : dist x y ≤ dist x z + dist y z := by
   rw [dist_comm y] <;> apply dist_triangle
 #align dist_triangle_right dist_triangle_right
 
+/- warning: dist_triangle4 -> dist_triangle4 is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] (x : α) (y : α) (z : α) (w : α), LE.le.{0} Real Real.hasLe (Dist.dist.{u1} α (PseudoMetricSpace.toHasDist.{u1} α _inst_1) x w) (HAdd.hAdd.{0, 0, 0} Real Real Real (instHAdd.{0} Real Real.hasAdd) (HAdd.hAdd.{0, 0, 0} Real Real Real (instHAdd.{0} Real Real.hasAdd) (Dist.dist.{u1} α (PseudoMetricSpace.toHasDist.{u1} α _inst_1) x y) (Dist.dist.{u1} α (PseudoMetricSpace.toHasDist.{u1} α _inst_1) y z)) (Dist.dist.{u1} α (PseudoMetricSpace.toHasDist.{u1} α _inst_1) z w))
+but is expected to have type
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] (x : α) (y : α) (z : α) (w : α), LE.le.{0} Real Real.instLEReal (Dist.dist.{u1} α (PseudoMetricSpace.toDist.{u1} α _inst_1) x w) (HAdd.hAdd.{0, 0, 0} Real Real Real (instHAdd.{0} Real Real.instAddReal) (HAdd.hAdd.{0, 0, 0} Real Real Real (instHAdd.{0} Real Real.instAddReal) (Dist.dist.{u1} α (PseudoMetricSpace.toDist.{u1} α _inst_1) x y) (Dist.dist.{u1} α (PseudoMetricSpace.toDist.{u1} α _inst_1) y z)) (Dist.dist.{u1} α (PseudoMetricSpace.toDist.{u1} α _inst_1) z w))
+Case conversion may be inaccurate. Consider using '#align dist_triangle4 dist_triangle4ₓ'. -/
 theorem dist_triangle4 (x y z w : α) : dist x w ≤ dist x y + dist y z + dist z w :=
   calc
     dist x w ≤ dist x z + dist z w := dist_triangle x z w
@@ -264,6 +322,12 @@ theorem dist_triangle4 (x y z w : α) : dist x w ≤ dist x y + dist y z + dist
     
 #align dist_triangle4 dist_triangle4
 
+/- warning: dist_triangle4_left -> dist_triangle4_left is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] (x₁ : α) (y₁ : α) (x₂ : α) (y₂ : α), LE.le.{0} Real Real.hasLe (Dist.dist.{u1} α (PseudoMetricSpace.toHasDist.{u1} α _inst_1) x₂ y₂) (HAdd.hAdd.{0, 0, 0} Real Real Real (instHAdd.{0} Real Real.hasAdd) (Dist.dist.{u1} α (PseudoMetricSpace.toHasDist.{u1} α _inst_1) x₁ y₁) (HAdd.hAdd.{0, 0, 0} Real Real Real (instHAdd.{0} Real Real.hasAdd) (Dist.dist.{u1} α (PseudoMetricSpace.toHasDist.{u1} α _inst_1) x₁ x₂) (Dist.dist.{u1} α (PseudoMetricSpace.toHasDist.{u1} α _inst_1) y₁ y₂)))
+but is expected to have type
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] (x₁ : α) (y₁ : α) (x₂ : α) (y₂ : α), LE.le.{0} Real Real.instLEReal (Dist.dist.{u1} α (PseudoMetricSpace.toDist.{u1} α _inst_1) x₂ y₂) (HAdd.hAdd.{0, 0, 0} Real Real Real (instHAdd.{0} Real Real.instAddReal) (Dist.dist.{u1} α (PseudoMetricSpace.toDist.{u1} α _inst_1) x₁ y₁) (HAdd.hAdd.{0, 0, 0} Real Real Real (instHAdd.{0} Real Real.instAddReal) (Dist.dist.{u1} α (PseudoMetricSpace.toDist.{u1} α _inst_1) x₁ x₂) (Dist.dist.{u1} α (PseudoMetricSpace.toDist.{u1} α _inst_1) y₁ y₂)))
+Case conversion may be inaccurate. Consider using '#align dist_triangle4_left dist_triangle4_leftₓ'. -/
 theorem dist_triangle4_left (x₁ y₁ x₂ y₂ : α) :
     dist x₂ y₂ ≤ dist x₁ y₁ + (dist x₁ x₂ + dist y₁ y₂) :=
   by
@@ -271,6 +335,12 @@ theorem dist_triangle4_left (x₁ y₁ x₂ y₂ : α) :
   apply dist_triangle4
 #align dist_triangle4_left dist_triangle4_left
 
+/- warning: dist_triangle4_right -> dist_triangle4_right is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] (x₁ : α) (y₁ : α) (x₂ : α) (y₂ : α), LE.le.{0} Real Real.hasLe (Dist.dist.{u1} α (PseudoMetricSpace.toHasDist.{u1} α _inst_1) x₁ y₁) (HAdd.hAdd.{0, 0, 0} Real Real Real (instHAdd.{0} Real Real.hasAdd) (HAdd.hAdd.{0, 0, 0} Real Real Real (instHAdd.{0} Real Real.hasAdd) (Dist.dist.{u1} α (PseudoMetricSpace.toHasDist.{u1} α _inst_1) x₁ x₂) (Dist.dist.{u1} α (PseudoMetricSpace.toHasDist.{u1} α _inst_1) y₁ y₂)) (Dist.dist.{u1} α (PseudoMetricSpace.toHasDist.{u1} α _inst_1) x₂ y₂))
+but is expected to have type
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] (x₁ : α) (y₁ : α) (x₂ : α) (y₂ : α), LE.le.{0} Real Real.instLEReal (Dist.dist.{u1} α (PseudoMetricSpace.toDist.{u1} α _inst_1) x₁ y₁) (HAdd.hAdd.{0, 0, 0} Real Real Real (instHAdd.{0} Real Real.instAddReal) (HAdd.hAdd.{0, 0, 0} Real Real Real (instHAdd.{0} Real Real.instAddReal) (Dist.dist.{u1} α (PseudoMetricSpace.toDist.{u1} α _inst_1) x₁ x₂) (Dist.dist.{u1} α (PseudoMetricSpace.toDist.{u1} α _inst_1) y₁ y₂)) (Dist.dist.{u1} α (PseudoMetricSpace.toDist.{u1} α _inst_1) x₂ y₂))
+Case conversion may be inaccurate. Consider using '#align dist_triangle4_right dist_triangle4_rightₓ'. -/
 theorem dist_triangle4_right (x₁ y₁ x₂ y₂ : α) :
     dist x₁ y₁ ≤ dist x₁ x₂ + dist y₁ y₂ + dist x₂ y₂ :=
   by
@@ -278,6 +348,12 @@ theorem dist_triangle4_right (x₁ y₁ x₂ y₂ : α) :
   apply dist_triangle4
 #align dist_triangle4_right dist_triangle4_right
 
+/- warning: dist_le_Ico_sum_dist -> dist_le_Ico_sum_dist is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] (f : Nat -> α) {m : Nat} {n : Nat}, (LE.le.{0} Nat Nat.hasLe m n) -> (LE.le.{0} Real Real.hasLe (Dist.dist.{u1} α (PseudoMetricSpace.toHasDist.{u1} α _inst_1) (f m) (f n)) (Finset.sum.{0, 0} Real Nat Real.addCommMonoid (Finset.Ico.{0} Nat (PartialOrder.toPreorder.{0} Nat (OrderedCancelAddCommMonoid.toPartialOrder.{0} Nat (StrictOrderedSemiring.toOrderedCancelAddCommMonoid.{0} Nat Nat.strictOrderedSemiring))) Nat.locallyFiniteOrder m n) (fun (i : Nat) => Dist.dist.{u1} α (PseudoMetricSpace.toHasDist.{u1} α _inst_1) (f i) (f (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) i (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))))))
+but is expected to have type
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] (f : Nat -> α) {m : Nat} {n : Nat}, (LE.le.{0} Nat instLENat m n) -> (LE.le.{0} Real Real.instLEReal (Dist.dist.{u1} α (PseudoMetricSpace.toDist.{u1} α _inst_1) (f m) (f n)) (Finset.sum.{0, 0} Real Nat Real.instAddCommMonoidReal (Finset.Ico.{0} Nat (PartialOrder.toPreorder.{0} Nat (StrictOrderedSemiring.toPartialOrder.{0} Nat Nat.strictOrderedSemiring)) instLocallyFiniteOrderNatToPreorderToPartialOrderStrictOrderedSemiring m n) (fun (i : Nat) => Dist.dist.{u1} α (PseudoMetricSpace.toDist.{u1} α _inst_1) (f i) (f (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) i (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))))))
+Case conversion may be inaccurate. Consider using '#align dist_le_Ico_sum_dist dist_le_Ico_sum_distₓ'. -/
 /-- The triangle (polygon) inequality for sequences of points; `finset.Ico` version. -/
 theorem dist_le_Ico_sum_dist (f : ℕ → α) {m n} (h : m ≤ n) :
     dist (f m) (f n) ≤ ∑ i in Finset.Ico m n, dist (f i) (f (i + 1)) :=
@@ -294,12 +370,24 @@ theorem dist_le_Ico_sum_dist (f : ℕ → α) {m n} (h : m ≤ n) :
       
 #align dist_le_Ico_sum_dist dist_le_Ico_sum_dist
 
+/- warning: dist_le_range_sum_dist -> dist_le_range_sum_dist is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] (f : Nat -> α) (n : Nat), LE.le.{0} Real Real.hasLe (Dist.dist.{u1} α (PseudoMetricSpace.toHasDist.{u1} α _inst_1) (f (OfNat.ofNat.{0} Nat 0 (OfNat.mk.{0} Nat 0 (Zero.zero.{0} Nat Nat.hasZero)))) (f n)) (Finset.sum.{0, 0} Real Nat Real.addCommMonoid (Finset.range n) (fun (i : Nat) => Dist.dist.{u1} α (PseudoMetricSpace.toHasDist.{u1} α _inst_1) (f i) (f (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) i (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne)))))))
+but is expected to have type
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] (f : Nat -> α) (n : Nat), LE.le.{0} Real Real.instLEReal (Dist.dist.{u1} α (PseudoMetricSpace.toDist.{u1} α _inst_1) (f (OfNat.ofNat.{0} Nat 0 (instOfNatNat 0))) (f n)) (Finset.sum.{0, 0} Real Nat Real.instAddCommMonoidReal (Finset.range n) (fun (i : Nat) => Dist.dist.{u1} α (PseudoMetricSpace.toDist.{u1} α _inst_1) (f i) (f (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) i (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))))))
+Case conversion may be inaccurate. Consider using '#align dist_le_range_sum_dist dist_le_range_sum_distₓ'. -/
 /-- The triangle (polygon) inequality for sequences of points; `finset.range` version. -/
 theorem dist_le_range_sum_dist (f : ℕ → α) (n : ℕ) :
     dist (f 0) (f n) ≤ ∑ i in Finset.range n, dist (f i) (f (i + 1)) :=
   Nat.Ico_zero_eq_range ▸ dist_le_Ico_sum_dist f (Nat.zero_le n)
 #align dist_le_range_sum_dist dist_le_range_sum_dist
 
+/- warning: dist_le_Ico_sum_of_dist_le -> dist_le_Ico_sum_of_dist_le is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {f : Nat -> α} {m : Nat} {n : Nat}, (LE.le.{0} Nat Nat.hasLe m n) -> (forall {d : Nat -> Real}, (forall {k : Nat}, (LE.le.{0} Nat Nat.hasLe m k) -> (LT.lt.{0} Nat Nat.hasLt k n) -> (LE.le.{0} Real Real.hasLe (Dist.dist.{u1} α (PseudoMetricSpace.toHasDist.{u1} α _inst_1) (f k) (f (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) k (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne)))))) (d k))) -> (LE.le.{0} Real Real.hasLe (Dist.dist.{u1} α (PseudoMetricSpace.toHasDist.{u1} α _inst_1) (f m) (f n)) (Finset.sum.{0, 0} Real Nat Real.addCommMonoid (Finset.Ico.{0} Nat (PartialOrder.toPreorder.{0} Nat (OrderedCancelAddCommMonoid.toPartialOrder.{0} Nat (StrictOrderedSemiring.toOrderedCancelAddCommMonoid.{0} Nat Nat.strictOrderedSemiring))) Nat.locallyFiniteOrder m n) (fun (i : Nat) => d i))))
+but is expected to have type
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {f : Nat -> α} {m : Nat} {n : Nat}, (LE.le.{0} Nat instLENat m n) -> (forall {d : Nat -> Real}, (forall {k : Nat}, (LE.le.{0} Nat instLENat m k) -> (LT.lt.{0} Nat instLTNat k n) -> (LE.le.{0} Real Real.instLEReal (Dist.dist.{u1} α (PseudoMetricSpace.toDist.{u1} α _inst_1) (f k) (f (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) k (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))))) (d k))) -> (LE.le.{0} Real Real.instLEReal (Dist.dist.{u1} α (PseudoMetricSpace.toDist.{u1} α _inst_1) (f m) (f n)) (Finset.sum.{0, 0} Real Nat Real.instAddCommMonoidReal (Finset.Ico.{0} Nat (PartialOrder.toPreorder.{0} Nat (StrictOrderedSemiring.toPartialOrder.{0} Nat Nat.strictOrderedSemiring)) instLocallyFiniteOrderNatToPreorderToPartialOrderStrictOrderedSemiring m n) (fun (i : Nat) => d i))))
+Case conversion may be inaccurate. Consider using '#align dist_le_Ico_sum_of_dist_le dist_le_Ico_sum_of_dist_leₓ'. -/
 /-- A version of `dist_le_Ico_sum_dist` with each intermediate distance replaced
 with an upper estimate. -/
 theorem dist_le_Ico_sum_of_dist_le {f : ℕ → α} {m n} (hmn : m ≤ n) {d : ℕ → ℝ}
@@ -309,6 +397,12 @@ theorem dist_le_Ico_sum_of_dist_le {f : ℕ → α} {m n} (hmn : m ≤ n) {d : 
     Finset.sum_le_sum fun k hk => hd (Finset.mem_Ico.1 hk).1 (Finset.mem_Ico.1 hk).2
 #align dist_le_Ico_sum_of_dist_le dist_le_Ico_sum_of_dist_le
 
+/- warning: dist_le_range_sum_of_dist_le -> dist_le_range_sum_of_dist_le is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {f : Nat -> α} (n : Nat) {d : Nat -> Real}, (forall {k : Nat}, (LT.lt.{0} Nat Nat.hasLt k n) -> (LE.le.{0} Real Real.hasLe (Dist.dist.{u1} α (PseudoMetricSpace.toHasDist.{u1} α _inst_1) (f k) (f (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) k (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne)))))) (d k))) -> (LE.le.{0} Real Real.hasLe (Dist.dist.{u1} α (PseudoMetricSpace.toHasDist.{u1} α _inst_1) (f (OfNat.ofNat.{0} Nat 0 (OfNat.mk.{0} Nat 0 (Zero.zero.{0} Nat Nat.hasZero)))) (f n)) (Finset.sum.{0, 0} Real Nat Real.addCommMonoid (Finset.range n) (fun (i : Nat) => d i)))
+but is expected to have type
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {f : Nat -> α} (n : Nat) {d : Nat -> Real}, (forall {k : Nat}, (LT.lt.{0} Nat instLTNat k n) -> (LE.le.{0} Real Real.instLEReal (Dist.dist.{u1} α (PseudoMetricSpace.toDist.{u1} α _inst_1) (f k) (f (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) k (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))))) (d k))) -> (LE.le.{0} Real Real.instLEReal (Dist.dist.{u1} α (PseudoMetricSpace.toDist.{u1} α _inst_1) (f (OfNat.ofNat.{0} Nat 0 (instOfNatNat 0))) (f n)) (Finset.sum.{0, 0} Real Nat Real.instAddCommMonoidReal (Finset.range n) (fun (i : Nat) => d i)))
+Case conversion may be inaccurate. Consider using '#align dist_le_range_sum_of_dist_le dist_le_range_sum_of_dist_leₓ'. -/
 /-- A version of `dist_le_range_sum_dist` with each intermediate distance replaced
 with an upper estimate. -/
 theorem dist_le_range_sum_of_dist_le {f : ℕ → α} (n : ℕ) {d : ℕ → ℝ}
@@ -317,14 +411,28 @@ theorem dist_le_range_sum_of_dist_le {f : ℕ → α} (n : ℕ) {d : ℕ → ℝ
   Nat.Ico_zero_eq_range ▸ dist_le_Ico_sum_of_dist_le (zero_le n) fun _ _ => hd
 #align dist_le_range_sum_of_dist_le dist_le_range_sum_of_dist_le
 
+#print swap_dist /-
 theorem swap_dist : Function.swap (@dist α _) = dist := by funext x y <;> exact dist_comm _ _
 #align swap_dist swap_dist
+-/
 
+/- warning: abs_dist_sub_le -> abs_dist_sub_le is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] (x : α) (y : α) (z : α), LE.le.{0} Real Real.hasLe (Abs.abs.{0} Real (Neg.toHasAbs.{0} Real Real.hasNeg Real.hasSup) (HSub.hSub.{0, 0, 0} Real Real Real (instHSub.{0} Real Real.hasSub) (Dist.dist.{u1} α (PseudoMetricSpace.toHasDist.{u1} α _inst_1) x z) (Dist.dist.{u1} α (PseudoMetricSpace.toHasDist.{u1} α _inst_1) y z))) (Dist.dist.{u1} α (PseudoMetricSpace.toHasDist.{u1} α _inst_1) x y)
+but is expected to have type
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] (x : α) (y : α) (z : α), LE.le.{0} Real Real.instLEReal (Abs.abs.{0} Real (Neg.toHasAbs.{0} Real Real.instNegReal Real.instSupReal) (HSub.hSub.{0, 0, 0} Real Real Real (instHSub.{0} Real Real.instSubReal) (Dist.dist.{u1} α (PseudoMetricSpace.toDist.{u1} α _inst_1) x z) (Dist.dist.{u1} α (PseudoMetricSpace.toDist.{u1} α _inst_1) y z))) (Dist.dist.{u1} α (PseudoMetricSpace.toDist.{u1} α _inst_1) x y)
+Case conversion may be inaccurate. Consider using '#align abs_dist_sub_le abs_dist_sub_leₓ'. -/
 theorem abs_dist_sub_le (x y z : α) : |dist x z - dist y z| ≤ dist x y :=
   abs_sub_le_iff.2
     ⟨sub_le_iff_le_add.2 (dist_triangle _ _ _), sub_le_iff_le_add.2 (dist_triangle_left _ _ _)⟩
 #align abs_dist_sub_le abs_dist_sub_le
 
+/- warning: dist_nonneg -> dist_nonneg is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {x : α} {y : α}, LE.le.{0} Real Real.hasLe (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero))) (Dist.dist.{u1} α (PseudoMetricSpace.toHasDist.{u1} α _inst_1) x y)
+but is expected to have type
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {x : α} {y : α}, LE.le.{0} Real Real.instLEReal (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal)) (Dist.dist.{u1} α (PseudoMetricSpace.toDist.{u1} α _inst_1) x y)
+Case conversion may be inaccurate. Consider using '#align dist_nonneg dist_nonnegₓ'. -/
 theorem dist_nonneg {x y : α} : 0 ≤ dist x y :=
   PseudoMetricSpace.dist_nonneg' dist dist_self dist_comm dist_triangle
 #align dist_nonneg dist_nonneg
@@ -342,163 +450,319 @@ unsafe def _root_.tactic.positivity_dist : expr → tactic strictness
 
 end
 
+/- warning: abs_dist -> abs_dist is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {a : α} {b : α}, Eq.{1} Real (Abs.abs.{0} Real (Neg.toHasAbs.{0} Real Real.hasNeg Real.hasSup) (Dist.dist.{u1} α (PseudoMetricSpace.toHasDist.{u1} α _inst_1) a b)) (Dist.dist.{u1} α (PseudoMetricSpace.toHasDist.{u1} α _inst_1) a b)
+but is expected to have type
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {a : α} {b : α}, Eq.{1} Real (Abs.abs.{0} Real (Neg.toHasAbs.{0} Real Real.instNegReal Real.instSupReal) (Dist.dist.{u1} α (PseudoMetricSpace.toDist.{u1} α _inst_1) a b)) (Dist.dist.{u1} α (PseudoMetricSpace.toDist.{u1} α _inst_1) a b)
+Case conversion may be inaccurate. Consider using '#align abs_dist abs_distₓ'. -/
 @[simp]
 theorem abs_dist {a b : α} : |dist a b| = dist a b :=
   abs_of_nonneg dist_nonneg
 #align abs_dist abs_dist
 
+#print NNDist /-
 /-- A version of `has_dist` that takes value in `ℝ≥0`. -/
-class HasNndist (α : Type _) where
+class NNDist (α : Type _) where
   nndist : α → α → ℝ≥0
-#align has_nndist HasNndist
+#align has_nndist NNDist
+-/
 
-export HasNndist (nndist)
+export NNDist (nndist)
 
+#print PseudoMetricSpace.toNNDist /-
 -- see Note [lower instance priority]
 /-- Distance as a nonnegative real number. -/
-instance (priority := 100) PseudoMetricSpace.toHasNndist : HasNndist α :=
+instance (priority := 100) PseudoMetricSpace.toNNDist : NNDist α :=
   ⟨fun a b => ⟨dist a b, dist_nonneg⟩⟩
-#align pseudo_metric_space.to_has_nndist PseudoMetricSpace.toHasNndist
+#align pseudo_metric_space.to_has_nndist PseudoMetricSpace.toNNDist
+-/
 
+#print nndist_edist /-
 /-- Express `nndist` in terms of `edist`-/
 theorem nndist_edist (x y : α) : nndist x y = (edist x y).toNNReal := by
   simp [nndist, edist_dist, Real.toNNReal, max_eq_left dist_nonneg, ENNReal.ofReal]
 #align nndist_edist nndist_edist
+-/
 
+#print edist_nndist /-
 /-- Express `edist` in terms of `nndist`-/
 theorem edist_nndist (x y : α) : edist x y = ↑(nndist x y) := by
   simpa only [edist_dist, ENNReal.ofReal_eq_coe_nnreal dist_nonneg]
 #align edist_nndist edist_nndist
+-/
 
+#print coe_nnreal_ennreal_nndist /-
 @[simp, norm_cast]
-theorem coe_nNReal_eNNReal_nndist (x y : α) : ↑(nndist x y) = edist x y :=
+theorem coe_nnreal_ennreal_nndist (x y : α) : ↑(nndist x y) = edist x y :=
   (edist_nndist x y).symm
-#align coe_nnreal_ennreal_nndist coe_nNReal_eNNReal_nndist
+#align coe_nnreal_ennreal_nndist coe_nnreal_ennreal_nndist
+-/
 
+/- warning: edist_lt_coe -> edist_lt_coe is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {x : α} {y : α} {c : NNReal}, Iff (LT.lt.{0} ENNReal (Preorder.toLT.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (CompleteSemilatticeInf.toPartialOrder.{0} ENNReal (CompleteLattice.toCompleteSemilatticeInf.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))))) (EDist.edist.{u1} α (PseudoMetricSpace.toEDist.{u1} α _inst_1) x y) ((fun (a : Type) (b : Type) [self : HasLiftT.{1, 1} a b] => self.0) NNReal ENNReal (HasLiftT.mk.{1, 1} NNReal ENNReal (CoeTCₓ.coe.{1, 1} NNReal ENNReal (coeBase.{1, 1} NNReal ENNReal ENNReal.hasCoe))) c)) (LT.lt.{0} NNReal (Preorder.toLT.{0} NNReal (PartialOrder.toPreorder.{0} NNReal (OrderedCancelAddCommMonoid.toPartialOrder.{0} NNReal (StrictOrderedSemiring.toOrderedCancelAddCommMonoid.{0} NNReal NNReal.strictOrderedSemiring)))) (NNDist.nndist.{u1} α (PseudoMetricSpace.toNNDist.{u1} α _inst_1) x y) c)
+but is expected to have type
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {x : α} {y : α} {c : NNReal}, Iff (LT.lt.{0} ENNReal (Preorder.toLT.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (CompleteSemilatticeInf.toPartialOrder.{0} ENNReal (CompleteLattice.toCompleteSemilatticeInf.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal))))) (EDist.edist.{u1} α (PseudoMetricSpace.toEDist.{u1} α _inst_1) x y) (ENNReal.some c)) (LT.lt.{0} NNReal (Preorder.toLT.{0} NNReal (PartialOrder.toPreorder.{0} NNReal (StrictOrderedSemiring.toPartialOrder.{0} NNReal instNNRealStrictOrderedSemiring))) (NNDist.nndist.{u1} α (PseudoMetricSpace.toNNDist.{u1} α _inst_1) x y) c)
+Case conversion may be inaccurate. Consider using '#align edist_lt_coe edist_lt_coeₓ'. -/
 @[simp, norm_cast]
 theorem edist_lt_coe {x y : α} {c : ℝ≥0} : edist x y < c ↔ nndist x y < c := by
   rw [edist_nndist, ENNReal.coe_lt_coe]
 #align edist_lt_coe edist_lt_coe
 
+/- warning: edist_le_coe -> edist_le_coe is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {x : α} {y : α} {c : NNReal}, Iff (LE.le.{0} ENNReal (Preorder.toLE.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (CompleteSemilatticeInf.toPartialOrder.{0} ENNReal (CompleteLattice.toCompleteSemilatticeInf.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))))) (EDist.edist.{u1} α (PseudoMetricSpace.toEDist.{u1} α _inst_1) x y) ((fun (a : Type) (b : Type) [self : HasLiftT.{1, 1} a b] => self.0) NNReal ENNReal (HasLiftT.mk.{1, 1} NNReal ENNReal (CoeTCₓ.coe.{1, 1} NNReal ENNReal (coeBase.{1, 1} NNReal ENNReal ENNReal.hasCoe))) c)) (LE.le.{0} NNReal (Preorder.toLE.{0} NNReal (PartialOrder.toPreorder.{0} NNReal (OrderedCancelAddCommMonoid.toPartialOrder.{0} NNReal (StrictOrderedSemiring.toOrderedCancelAddCommMonoid.{0} NNReal NNReal.strictOrderedSemiring)))) (NNDist.nndist.{u1} α (PseudoMetricSpace.toNNDist.{u1} α _inst_1) x y) c)
+but is expected to have type
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {x : α} {y : α} {c : NNReal}, Iff (LE.le.{0} ENNReal (Preorder.toLE.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (CompleteSemilatticeInf.toPartialOrder.{0} ENNReal (CompleteLattice.toCompleteSemilatticeInf.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal))))) (EDist.edist.{u1} α (PseudoMetricSpace.toEDist.{u1} α _inst_1) x y) (ENNReal.some c)) (LE.le.{0} NNReal (Preorder.toLE.{0} NNReal (PartialOrder.toPreorder.{0} NNReal (StrictOrderedSemiring.toPartialOrder.{0} NNReal instNNRealStrictOrderedSemiring))) (NNDist.nndist.{u1} α (PseudoMetricSpace.toNNDist.{u1} α _inst_1) x y) c)
+Case conversion may be inaccurate. Consider using '#align edist_le_coe edist_le_coeₓ'. -/
 @[simp, norm_cast]
 theorem edist_le_coe {x y : α} {c : ℝ≥0} : edist x y ≤ c ↔ nndist x y ≤ c := by
   rw [edist_nndist, ENNReal.coe_le_coe]
 #align edist_le_coe edist_le_coe
 
+/- warning: edist_lt_top -> edist_lt_top is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} [_inst_2 : PseudoMetricSpace.{u1} α] (x : α) (y : α), LT.lt.{0} ENNReal (Preorder.toLT.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (CompleteSemilatticeInf.toPartialOrder.{0} ENNReal (CompleteLattice.toCompleteSemilatticeInf.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))))) (EDist.edist.{u1} α (PseudoMetricSpace.toEDist.{u1} α _inst_2) x y) (Top.top.{0} ENNReal (CompleteLattice.toHasTop.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder)))
+but is expected to have type
+  forall {α : Type.{u1}} [_inst_2 : PseudoMetricSpace.{u1} α] (x : α) (y : α), LT.lt.{0} ENNReal (Preorder.toLT.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (CompleteSemilatticeInf.toPartialOrder.{0} ENNReal (CompleteLattice.toCompleteSemilatticeInf.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal))))) (EDist.edist.{u1} α (PseudoMetricSpace.toEDist.{u1} α _inst_2) x y) (Top.top.{0} ENNReal (CompleteLattice.toTop.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal)))
+Case conversion may be inaccurate. Consider using '#align edist_lt_top edist_lt_topₓ'. -/
 /-- In a pseudometric space, the extended distance is always finite-/
 theorem edist_lt_top {α : Type _} [PseudoMetricSpace α] (x y : α) : edist x y < ⊤ :=
   (edist_dist x y).symm ▸ ENNReal.ofReal_lt_top
 #align edist_lt_top edist_lt_top
 
+/- warning: edist_ne_top -> edist_ne_top is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] (x : α) (y : α), Ne.{1} ENNReal (EDist.edist.{u1} α (PseudoMetricSpace.toEDist.{u1} α _inst_1) x y) (Top.top.{0} ENNReal (CompleteLattice.toHasTop.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder)))
+but is expected to have type
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] (x : α) (y : α), Ne.{1} ENNReal (EDist.edist.{u1} α (PseudoMetricSpace.toEDist.{u1} α _inst_1) x y) (Top.top.{0} ENNReal (CompleteLattice.toTop.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal)))
+Case conversion may be inaccurate. Consider using '#align edist_ne_top edist_ne_topₓ'. -/
 /-- In a pseudometric space, the extended distance is always finite-/
 theorem edist_ne_top (x y : α) : edist x y ≠ ⊤ :=
   (edist_lt_top x y).Ne
 #align edist_ne_top edist_ne_top
 
+/- warning: nndist_self -> nndist_self is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] (a : α), Eq.{1} NNReal (NNDist.nndist.{u1} α (PseudoMetricSpace.toNNDist.{u1} α _inst_1) a a) (OfNat.ofNat.{0} NNReal 0 (OfNat.mk.{0} NNReal 0 (Zero.zero.{0} NNReal (MulZeroClass.toHasZero.{0} NNReal (NonUnitalNonAssocSemiring.toMulZeroClass.{0} NNReal (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} NNReal (Semiring.toNonAssocSemiring.{0} NNReal NNReal.semiring)))))))
+but is expected to have type
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] (a : α), Eq.{1} NNReal (NNDist.nndist.{u1} α (PseudoMetricSpace.toNNDist.{u1} α _inst_1) a a) (OfNat.ofNat.{0} NNReal 0 (Zero.toOfNat0.{0} NNReal instNNRealZero))
+Case conversion may be inaccurate. Consider using '#align nndist_self nndist_selfₓ'. -/
 /-- `nndist x x` vanishes-/
 @[simp]
 theorem nndist_self (a : α) : nndist a a = 0 :=
   (NNReal.coe_eq_zero _).1 (dist_self a)
 #align nndist_self nndist_self
 
+/- warning: dist_nndist -> dist_nndist is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] (x : α) (y : α), Eq.{1} Real (Dist.dist.{u1} α (PseudoMetricSpace.toHasDist.{u1} α _inst_1) x y) ((fun (a : Type) (b : Type) [self : HasLiftT.{1, 1} a b] => self.0) NNReal Real (HasLiftT.mk.{1, 1} NNReal Real (CoeTCₓ.coe.{1, 1} NNReal Real (coeBase.{1, 1} NNReal Real NNReal.Real.hasCoe))) (NNDist.nndist.{u1} α (PseudoMetricSpace.toNNDist.{u1} α _inst_1) x y))
+but is expected to have type
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] (x : α) (y : α), Eq.{1} Real (Dist.dist.{u1} α (PseudoMetricSpace.toDist.{u1} α _inst_1) x y) (NNReal.toReal (NNDist.nndist.{u1} α (PseudoMetricSpace.toNNDist.{u1} α _inst_1) x y))
+Case conversion may be inaccurate. Consider using '#align dist_nndist dist_nndistₓ'. -/
 /-- Express `dist` in terms of `nndist`-/
 theorem dist_nndist (x y : α) : dist x y = ↑(nndist x y) :=
   rfl
 #align dist_nndist dist_nndist
 
+/- warning: coe_nndist -> coe_nndist is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] (x : α) (y : α), Eq.{1} Real ((fun (a : Type) (b : Type) [self : HasLiftT.{1, 1} a b] => self.0) NNReal Real (HasLiftT.mk.{1, 1} NNReal Real (CoeTCₓ.coe.{1, 1} NNReal Real (coeBase.{1, 1} NNReal Real NNReal.Real.hasCoe))) (NNDist.nndist.{u1} α (PseudoMetricSpace.toNNDist.{u1} α _inst_1) x y)) (Dist.dist.{u1} α (PseudoMetricSpace.toHasDist.{u1} α _inst_1) x y)
+but is expected to have type
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] (x : α) (y : α), Eq.{1} Real (NNReal.toReal (NNDist.nndist.{u1} α (PseudoMetricSpace.toNNDist.{u1} α _inst_1) x y)) (Dist.dist.{u1} α (PseudoMetricSpace.toDist.{u1} α _inst_1) x y)
+Case conversion may be inaccurate. Consider using '#align coe_nndist coe_nndistₓ'. -/
 @[simp, norm_cast]
 theorem coe_nndist (x y : α) : ↑(nndist x y) = dist x y :=
   (dist_nndist x y).symm
 #align coe_nndist coe_nndist
 
+/- warning: dist_lt_coe -> dist_lt_coe is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {x : α} {y : α} {c : NNReal}, Iff (LT.lt.{0} Real Real.hasLt (Dist.dist.{u1} α (PseudoMetricSpace.toHasDist.{u1} α _inst_1) x y) ((fun (a : Type) (b : Type) [self : HasLiftT.{1, 1} a b] => self.0) NNReal Real (HasLiftT.mk.{1, 1} NNReal Real (CoeTCₓ.coe.{1, 1} NNReal Real (coeBase.{1, 1} NNReal Real NNReal.Real.hasCoe))) c)) (LT.lt.{0} NNReal (Preorder.toLT.{0} NNReal (PartialOrder.toPreorder.{0} NNReal (OrderedCancelAddCommMonoid.toPartialOrder.{0} NNReal (StrictOrderedSemiring.toOrderedCancelAddCommMonoid.{0} NNReal NNReal.strictOrderedSemiring)))) (NNDist.nndist.{u1} α (PseudoMetricSpace.toNNDist.{u1} α _inst_1) x y) c)
+but is expected to have type
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {x : α} {y : α} {c : NNReal}, Iff (LT.lt.{0} Real Real.instLTReal (Dist.dist.{u1} α (PseudoMetricSpace.toDist.{u1} α _inst_1) x y) (NNReal.toReal c)) (LT.lt.{0} NNReal (Preorder.toLT.{0} NNReal (PartialOrder.toPreorder.{0} NNReal (StrictOrderedSemiring.toPartialOrder.{0} NNReal instNNRealStrictOrderedSemiring))) (NNDist.nndist.{u1} α (PseudoMetricSpace.toNNDist.{u1} α _inst_1) x y) c)
+Case conversion may be inaccurate. Consider using '#align dist_lt_coe dist_lt_coeₓ'. -/
 @[simp, norm_cast]
 theorem dist_lt_coe {x y : α} {c : ℝ≥0} : dist x y < c ↔ nndist x y < c :=
   Iff.rfl
 #align dist_lt_coe dist_lt_coe
 
+/- warning: dist_le_coe -> dist_le_coe is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {x : α} {y : α} {c : NNReal}, Iff (LE.le.{0} Real Real.hasLe (Dist.dist.{u1} α (PseudoMetricSpace.toHasDist.{u1} α _inst_1) x y) ((fun (a : Type) (b : Type) [self : HasLiftT.{1, 1} a b] => self.0) NNReal Real (HasLiftT.mk.{1, 1} NNReal Real (CoeTCₓ.coe.{1, 1} NNReal Real (coeBase.{1, 1} NNReal Real NNReal.Real.hasCoe))) c)) (LE.le.{0} NNReal (Preorder.toLE.{0} NNReal (PartialOrder.toPreorder.{0} NNReal (OrderedCancelAddCommMonoid.toPartialOrder.{0} NNReal (StrictOrderedSemiring.toOrderedCancelAddCommMonoid.{0} NNReal NNReal.strictOrderedSemiring)))) (NNDist.nndist.{u1} α (PseudoMetricSpace.toNNDist.{u1} α _inst_1) x y) c)
+but is expected to have type
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {x : α} {y : α} {c : NNReal}, Iff (LE.le.{0} Real Real.instLEReal (Dist.dist.{u1} α (PseudoMetricSpace.toDist.{u1} α _inst_1) x y) (NNReal.toReal c)) (LE.le.{0} NNReal (Preorder.toLE.{0} NNReal (PartialOrder.toPreorder.{0} NNReal (StrictOrderedSemiring.toPartialOrder.{0} NNReal instNNRealStrictOrderedSemiring))) (NNDist.nndist.{u1} α (PseudoMetricSpace.toNNDist.{u1} α _inst_1) x y) c)
+Case conversion may be inaccurate. Consider using '#align dist_le_coe dist_le_coeₓ'. -/
 @[simp, norm_cast]
 theorem dist_le_coe {x y : α} {c : ℝ≥0} : dist x y ≤ c ↔ nndist x y ≤ c :=
   Iff.rfl
 #align dist_le_coe dist_le_coe
 
+/- warning: edist_lt_of_real -> edist_lt_ofReal is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {x : α} {y : α} {r : Real}, Iff (LT.lt.{0} ENNReal (Preorder.toLT.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (CompleteSemilatticeInf.toPartialOrder.{0} ENNReal (CompleteLattice.toCompleteSemilatticeInf.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))))) (EDist.edist.{u1} α (PseudoMetricSpace.toEDist.{u1} α _inst_1) x y) (ENNReal.ofReal r)) (LT.lt.{0} Real Real.hasLt (Dist.dist.{u1} α (PseudoMetricSpace.toHasDist.{u1} α _inst_1) x y) r)
+but is expected to have type
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {x : α} {y : α} {r : Real}, Iff (LT.lt.{0} ENNReal (Preorder.toLT.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (CompleteSemilatticeInf.toPartialOrder.{0} ENNReal (CompleteLattice.toCompleteSemilatticeInf.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal))))) (EDist.edist.{u1} α (PseudoMetricSpace.toEDist.{u1} α _inst_1) x y) (ENNReal.ofReal r)) (LT.lt.{0} Real Real.instLTReal (Dist.dist.{u1} α (PseudoMetricSpace.toDist.{u1} α _inst_1) x y) r)
+Case conversion may be inaccurate. Consider using '#align edist_lt_of_real edist_lt_ofRealₓ'. -/
 @[simp]
 theorem edist_lt_ofReal {x y : α} {r : ℝ} : edist x y < ENNReal.ofReal r ↔ dist x y < r := by
   rw [edist_dist, ENNReal.ofReal_lt_ofReal_iff_of_nonneg dist_nonneg]
 #align edist_lt_of_real edist_lt_ofReal
 
+/- warning: edist_le_of_real -> edist_le_ofReal is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {x : α} {y : α} {r : Real}, (LE.le.{0} Real Real.hasLe (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero))) r) -> (Iff (LE.le.{0} ENNReal (Preorder.toLE.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (CompleteSemilatticeInf.toPartialOrder.{0} ENNReal (CompleteLattice.toCompleteSemilatticeInf.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))))) (EDist.edist.{u1} α (PseudoMetricSpace.toEDist.{u1} α _inst_1) x y) (ENNReal.ofReal r)) (LE.le.{0} Real Real.hasLe (Dist.dist.{u1} α (PseudoMetricSpace.toHasDist.{u1} α _inst_1) x y) r))
+but is expected to have type
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {x : α} {y : α} {r : Real}, (LE.le.{0} Real Real.instLEReal (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal)) r) -> (Iff (LE.le.{0} ENNReal (Preorder.toLE.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (CompleteSemilatticeInf.toPartialOrder.{0} ENNReal (CompleteLattice.toCompleteSemilatticeInf.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal))))) (EDist.edist.{u1} α (PseudoMetricSpace.toEDist.{u1} α _inst_1) x y) (ENNReal.ofReal r)) (LE.le.{0} Real Real.instLEReal (Dist.dist.{u1} α (PseudoMetricSpace.toDist.{u1} α _inst_1) x y) r))
+Case conversion may be inaccurate. Consider using '#align edist_le_of_real edist_le_ofRealₓ'. -/
 @[simp]
 theorem edist_le_ofReal {x y : α} {r : ℝ} (hr : 0 ≤ r) :
     edist x y ≤ ENNReal.ofReal r ↔ dist x y ≤ r := by
   rw [edist_dist, ENNReal.ofReal_le_ofReal_iff hr]
 #align edist_le_of_real edist_le_ofReal
 
+#print nndist_dist /-
 /-- Express `nndist` in terms of `dist`-/
 theorem nndist_dist (x y : α) : nndist x y = Real.toNNReal (dist x y) := by
   rw [dist_nndist, Real.toNNReal_coe]
 #align nndist_dist nndist_dist
+-/
 
+#print nndist_comm /-
 theorem nndist_comm (x y : α) : nndist x y = nndist y x := by
   simpa only [dist_nndist, NNReal.coe_eq] using dist_comm x y
 #align nndist_comm nndist_comm
+-/
 
+/- warning: nndist_triangle -> nndist_triangle is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] (x : α) (y : α) (z : α), LE.le.{0} NNReal (Preorder.toLE.{0} NNReal (PartialOrder.toPreorder.{0} NNReal (OrderedCancelAddCommMonoid.toPartialOrder.{0} NNReal (StrictOrderedSemiring.toOrderedCancelAddCommMonoid.{0} NNReal NNReal.strictOrderedSemiring)))) (NNDist.nndist.{u1} α (PseudoMetricSpace.toNNDist.{u1} α _inst_1) x z) (HAdd.hAdd.{0, 0, 0} NNReal NNReal NNReal (instHAdd.{0} NNReal (Distrib.toHasAdd.{0} NNReal (NonUnitalNonAssocSemiring.toDistrib.{0} NNReal (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} NNReal (Semiring.toNonAssocSemiring.{0} NNReal NNReal.semiring))))) (NNDist.nndist.{u1} α (PseudoMetricSpace.toNNDist.{u1} α _inst_1) x y) (NNDist.nndist.{u1} α (PseudoMetricSpace.toNNDist.{u1} α _inst_1) y z))
+but is expected to have type
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] (x : α) (y : α) (z : α), LE.le.{0} NNReal (Preorder.toLE.{0} NNReal (PartialOrder.toPreorder.{0} NNReal (StrictOrderedSemiring.toPartialOrder.{0} NNReal instNNRealStrictOrderedSemiring))) (NNDist.nndist.{u1} α (PseudoMetricSpace.toNNDist.{u1} α _inst_1) x z) (HAdd.hAdd.{0, 0, 0} NNReal NNReal NNReal (instHAdd.{0} NNReal (Distrib.toAdd.{0} NNReal (NonUnitalNonAssocSemiring.toDistrib.{0} NNReal (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} NNReal (Semiring.toNonAssocSemiring.{0} NNReal instNNRealSemiring))))) (NNDist.nndist.{u1} α (PseudoMetricSpace.toNNDist.{u1} α _inst_1) x y) (NNDist.nndist.{u1} α (PseudoMetricSpace.toNNDist.{u1} α _inst_1) y z))
+Case conversion may be inaccurate. Consider using '#align nndist_triangle nndist_triangleₓ'. -/
 /-- Triangle inequality for the nonnegative distance-/
 theorem nndist_triangle (x y z : α) : nndist x z ≤ nndist x y + nndist y z :=
   dist_triangle _ _ _
 #align nndist_triangle nndist_triangle
 
+/- warning: nndist_triangle_left -> nndist_triangle_left is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] (x : α) (y : α) (z : α), LE.le.{0} NNReal (Preorder.toLE.{0} NNReal (PartialOrder.toPreorder.{0} NNReal (OrderedCancelAddCommMonoid.toPartialOrder.{0} NNReal (StrictOrderedSemiring.toOrderedCancelAddCommMonoid.{0} NNReal NNReal.strictOrderedSemiring)))) (NNDist.nndist.{u1} α (PseudoMetricSpace.toNNDist.{u1} α _inst_1) x y) (HAdd.hAdd.{0, 0, 0} NNReal NNReal NNReal (instHAdd.{0} NNReal (Distrib.toHasAdd.{0} NNReal (NonUnitalNonAssocSemiring.toDistrib.{0} NNReal (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} NNReal (Semiring.toNonAssocSemiring.{0} NNReal NNReal.semiring))))) (NNDist.nndist.{u1} α (PseudoMetricSpace.toNNDist.{u1} α _inst_1) z x) (NNDist.nndist.{u1} α (PseudoMetricSpace.toNNDist.{u1} α _inst_1) z y))
+but is expected to have type
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] (x : α) (y : α) (z : α), LE.le.{0} NNReal (Preorder.toLE.{0} NNReal (PartialOrder.toPreorder.{0} NNReal (StrictOrderedSemiring.toPartialOrder.{0} NNReal instNNRealStrictOrderedSemiring))) (NNDist.nndist.{u1} α (PseudoMetricSpace.toNNDist.{u1} α _inst_1) x y) (HAdd.hAdd.{0, 0, 0} NNReal NNReal NNReal (instHAdd.{0} NNReal (Distrib.toAdd.{0} NNReal (NonUnitalNonAssocSemiring.toDistrib.{0} NNReal (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} NNReal (Semiring.toNonAssocSemiring.{0} NNReal instNNRealSemiring))))) (NNDist.nndist.{u1} α (PseudoMetricSpace.toNNDist.{u1} α _inst_1) z x) (NNDist.nndist.{u1} α (PseudoMetricSpace.toNNDist.{u1} α _inst_1) z y))
+Case conversion may be inaccurate. Consider using '#align nndist_triangle_left nndist_triangle_leftₓ'. -/
 theorem nndist_triangle_left (x y z : α) : nndist x y ≤ nndist z x + nndist z y :=
   dist_triangle_left _ _ _
 #align nndist_triangle_left nndist_triangle_left
 
+/- warning: nndist_triangle_right -> nndist_triangle_right is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] (x : α) (y : α) (z : α), LE.le.{0} NNReal (Preorder.toLE.{0} NNReal (PartialOrder.toPreorder.{0} NNReal (OrderedCancelAddCommMonoid.toPartialOrder.{0} NNReal (StrictOrderedSemiring.toOrderedCancelAddCommMonoid.{0} NNReal NNReal.strictOrderedSemiring)))) (NNDist.nndist.{u1} α (PseudoMetricSpace.toNNDist.{u1} α _inst_1) x y) (HAdd.hAdd.{0, 0, 0} NNReal NNReal NNReal (instHAdd.{0} NNReal (Distrib.toHasAdd.{0} NNReal (NonUnitalNonAssocSemiring.toDistrib.{0} NNReal (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} NNReal (Semiring.toNonAssocSemiring.{0} NNReal NNReal.semiring))))) (NNDist.nndist.{u1} α (PseudoMetricSpace.toNNDist.{u1} α _inst_1) x z) (NNDist.nndist.{u1} α (PseudoMetricSpace.toNNDist.{u1} α _inst_1) y z))
+but is expected to have type
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] (x : α) (y : α) (z : α), LE.le.{0} NNReal (Preorder.toLE.{0} NNReal (PartialOrder.toPreorder.{0} NNReal (StrictOrderedSemiring.toPartialOrder.{0} NNReal instNNRealStrictOrderedSemiring))) (NNDist.nndist.{u1} α (PseudoMetricSpace.toNNDist.{u1} α _inst_1) x y) (HAdd.hAdd.{0, 0, 0} NNReal NNReal NNReal (instHAdd.{0} NNReal (Distrib.toAdd.{0} NNReal (NonUnitalNonAssocSemiring.toDistrib.{0} NNReal (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} NNReal (Semiring.toNonAssocSemiring.{0} NNReal instNNRealSemiring))))) (NNDist.nndist.{u1} α (PseudoMetricSpace.toNNDist.{u1} α _inst_1) x z) (NNDist.nndist.{u1} α (PseudoMetricSpace.toNNDist.{u1} α _inst_1) y z))
+Case conversion may be inaccurate. Consider using '#align nndist_triangle_right nndist_triangle_rightₓ'. -/
 theorem nndist_triangle_right (x y z : α) : nndist x y ≤ nndist x z + nndist y z :=
   dist_triangle_right _ _ _
 #align nndist_triangle_right nndist_triangle_right
 
+#print dist_edist /-
 /-- Express `dist` in terms of `edist`-/
 theorem dist_edist (x y : α) : dist x y = (edist x y).toReal := by
   rw [edist_dist, ENNReal.toReal_ofReal dist_nonneg]
 #align dist_edist dist_edist
+-/
 
 namespace Metric
 
 -- instantiate pseudometric space as a topology
 variable {x y z : α} {δ ε ε₁ ε₂ : ℝ} {s : Set α}
 
+#print Metric.ball /-
 /-- `ball x ε` is the set of all points `y` with `dist y x < ε` -/
 def ball (x : α) (ε : ℝ) : Set α :=
   { y | dist y x < ε }
 #align metric.ball Metric.ball
+-/
 
+/- warning: metric.mem_ball -> Metric.mem_ball is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {x : α} {y : α} {ε : Real}, Iff (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) y (Metric.ball.{u1} α _inst_1 x ε)) (LT.lt.{0} Real Real.hasLt (Dist.dist.{u1} α (PseudoMetricSpace.toHasDist.{u1} α _inst_1) y x) ε)
+but is expected to have type
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {x : α} {y : α} {ε : Real}, Iff (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) y (Metric.ball.{u1} α _inst_1 x ε)) (LT.lt.{0} Real Real.instLTReal (Dist.dist.{u1} α (PseudoMetricSpace.toDist.{u1} α _inst_1) y x) ε)
+Case conversion may be inaccurate. Consider using '#align metric.mem_ball Metric.mem_ballₓ'. -/
 @[simp]
 theorem mem_ball : y ∈ ball x ε ↔ dist y x < ε :=
   Iff.rfl
 #align metric.mem_ball Metric.mem_ball
 
+/- warning: metric.mem_ball' -> Metric.mem_ball' is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {x : α} {y : α} {ε : Real}, Iff (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) y (Metric.ball.{u1} α _inst_1 x ε)) (LT.lt.{0} Real Real.hasLt (Dist.dist.{u1} α (PseudoMetricSpace.toHasDist.{u1} α _inst_1) x y) ε)
+but is expected to have type
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {x : α} {y : α} {ε : Real}, Iff (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) y (Metric.ball.{u1} α _inst_1 x ε)) (LT.lt.{0} Real Real.instLTReal (Dist.dist.{u1} α (PseudoMetricSpace.toDist.{u1} α _inst_1) x y) ε)
+Case conversion may be inaccurate. Consider using '#align metric.mem_ball' Metric.mem_ball'ₓ'. -/
 theorem mem_ball' : y ∈ ball x ε ↔ dist x y < ε := by rw [dist_comm, mem_ball]
 #align metric.mem_ball' Metric.mem_ball'
 
+/- warning: metric.pos_of_mem_ball -> Metric.pos_of_mem_ball is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {x : α} {y : α} {ε : Real}, (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) y (Metric.ball.{u1} α _inst_1 x ε)) -> (LT.lt.{0} Real Real.hasLt (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero))) ε)
+but is expected to have type
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {x : α} {y : α} {ε : Real}, (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) y (Metric.ball.{u1} α _inst_1 x ε)) -> (LT.lt.{0} Real Real.instLTReal (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal)) ε)
+Case conversion may be inaccurate. Consider using '#align metric.pos_of_mem_ball Metric.pos_of_mem_ballₓ'. -/
 theorem pos_of_mem_ball (hy : y ∈ ball x ε) : 0 < ε :=
   dist_nonneg.trans_lt hy
 #align metric.pos_of_mem_ball Metric.pos_of_mem_ball
 
+/- warning: metric.mem_ball_self -> Metric.mem_ball_self is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {x : α} {ε : Real}, (LT.lt.{0} Real Real.hasLt (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero))) ε) -> (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) x (Metric.ball.{u1} α _inst_1 x ε))
+but is expected to have type
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {x : α} {ε : Real}, (LT.lt.{0} Real Real.instLTReal (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal)) ε) -> (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) x (Metric.ball.{u1} α _inst_1 x ε))
+Case conversion may be inaccurate. Consider using '#align metric.mem_ball_self Metric.mem_ball_selfₓ'. -/
 theorem mem_ball_self (h : 0 < ε) : x ∈ ball x ε :=
   show dist x x < ε by rw [dist_self] <;> assumption
 #align metric.mem_ball_self Metric.mem_ball_self
 
+/- warning: metric.nonempty_ball -> Metric.nonempty_ball is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {x : α} {ε : Real}, Iff (Set.Nonempty.{u1} α (Metric.ball.{u1} α _inst_1 x ε)) (LT.lt.{0} Real Real.hasLt (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero))) ε)
+but is expected to have type
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {x : α} {ε : Real}, Iff (Set.Nonempty.{u1} α (Metric.ball.{u1} α _inst_1 x ε)) (LT.lt.{0} Real Real.instLTReal (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal)) ε)
+Case conversion may be inaccurate. Consider using '#align metric.nonempty_ball Metric.nonempty_ballₓ'. -/
 @[simp]
 theorem nonempty_ball : (ball x ε).Nonempty ↔ 0 < ε :=
   ⟨fun ⟨x, hx⟩ => pos_of_mem_ball hx, fun h => ⟨x, mem_ball_self h⟩⟩
 #align metric.nonempty_ball Metric.nonempty_ball
 
+/- warning: metric.ball_eq_empty -> Metric.ball_eq_empty is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {x : α} {ε : Real}, Iff (Eq.{succ u1} (Set.{u1} α) (Metric.ball.{u1} α _inst_1 x ε) (EmptyCollection.emptyCollection.{u1} (Set.{u1} α) (Set.hasEmptyc.{u1} α))) (LE.le.{0} Real Real.hasLe ε (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero))))
+but is expected to have type
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {x : α} {ε : Real}, Iff (Eq.{succ u1} (Set.{u1} α) (Metric.ball.{u1} α _inst_1 x ε) (EmptyCollection.emptyCollection.{u1} (Set.{u1} α) (Set.instEmptyCollectionSet.{u1} α))) (LE.le.{0} Real Real.instLEReal ε (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal)))
+Case conversion may be inaccurate. Consider using '#align metric.ball_eq_empty Metric.ball_eq_emptyₓ'. -/
 @[simp]
 theorem ball_eq_empty : ball x ε = ∅ ↔ ε ≤ 0 := by
   rw [← not_nonempty_iff_eq_empty, nonempty_ball, not_lt]
 #align metric.ball_eq_empty Metric.ball_eq_empty
 
+/- warning: metric.ball_zero -> Metric.ball_zero is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {x : α}, Eq.{succ u1} (Set.{u1} α) (Metric.ball.{u1} α _inst_1 x (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero)))) (EmptyCollection.emptyCollection.{u1} (Set.{u1} α) (Set.hasEmptyc.{u1} α))
+but is expected to have type
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {x : α}, Eq.{succ u1} (Set.{u1} α) (Metric.ball.{u1} α _inst_1 x (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal))) (EmptyCollection.emptyCollection.{u1} (Set.{u1} α) (Set.instEmptyCollectionSet.{u1} α))
+Case conversion may be inaccurate. Consider using '#align metric.ball_zero Metric.ball_zeroₓ'. -/
 @[simp]
 theorem ball_zero : ball x 0 = ∅ := by rw [ball_eq_empty]
 #align metric.ball_zero Metric.ball_zero
 
+/- warning: metric.exists_lt_mem_ball_of_mem_ball -> Metric.exists_lt_mem_ball_of_mem_ball is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {x : α} {y : α} {ε : Real}, (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) x (Metric.ball.{u1} α _inst_1 y ε)) -> (Exists.{1} Real (fun (ε' : Real) => Exists.{0} (LT.lt.{0} Real Real.hasLt ε' ε) (fun (H : LT.lt.{0} Real Real.hasLt ε' ε) => Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) x (Metric.ball.{u1} α _inst_1 y ε'))))
+but is expected to have type
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {x : α} {y : α} {ε : Real}, (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) x (Metric.ball.{u1} α _inst_1 y ε)) -> (Exists.{1} Real (fun (ε' : Real) => And (LT.lt.{0} Real Real.instLTReal ε' ε) (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) x (Metric.ball.{u1} α _inst_1 y ε'))))
+Case conversion may be inaccurate. Consider using '#align metric.exists_lt_mem_ball_of_mem_ball Metric.exists_lt_mem_ball_of_mem_ballₓ'. -/
 /-- If a point belongs to an open ball, then there is a strictly smaller radius whose ball also
 contains it.
 
@@ -509,11 +773,23 @@ theorem exists_lt_mem_ball_of_mem_ball (h : x ∈ ball y ε) : ∃ ε' < ε, x 
   exact ⟨(ε + dist x y) / 2, by linarith, by linarith⟩
 #align metric.exists_lt_mem_ball_of_mem_ball Metric.exists_lt_mem_ball_of_mem_ball
 
+/- warning: metric.ball_eq_ball -> Metric.ball_eq_ball is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] (ε : Real) (x : α), Eq.{succ u1} (Set.{u1} α) (UniformSpace.ball.{u1} α x (setOf.{u1} (Prod.{u1, u1} α α) (fun (p : Prod.{u1, u1} α α) => LT.lt.{0} Real Real.hasLt (Dist.dist.{u1} α (PseudoMetricSpace.toHasDist.{u1} α _inst_1) (Prod.snd.{u1, u1} α α p) (Prod.fst.{u1, u1} α α p)) ε))) (Metric.ball.{u1} α _inst_1 x ε)
+but is expected to have type
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] (ε : Real) (x : α), Eq.{succ u1} (Set.{u1} α) (UniformSpace.ball.{u1} α x (setOf.{u1} (Prod.{u1, u1} α α) (fun (p : Prod.{u1, u1} α α) => LT.lt.{0} Real Real.instLTReal (Dist.dist.{u1} α (PseudoMetricSpace.toDist.{u1} α _inst_1) (Prod.snd.{u1, u1} α α p) (Prod.fst.{u1, u1} α α p)) ε))) (Metric.ball.{u1} α _inst_1 x ε)
+Case conversion may be inaccurate. Consider using '#align metric.ball_eq_ball Metric.ball_eq_ballₓ'. -/
 theorem ball_eq_ball (ε : ℝ) (x : α) :
     UniformSpace.ball x { p | dist p.2 p.1 < ε } = Metric.ball x ε :=
   rfl
 #align metric.ball_eq_ball Metric.ball_eq_ball
 
+/- warning: metric.ball_eq_ball' -> Metric.ball_eq_ball' is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] (ε : Real) (x : α), Eq.{succ u1} (Set.{u1} α) (UniformSpace.ball.{u1} α x (setOf.{u1} (Prod.{u1, u1} α α) (fun (p : Prod.{u1, u1} α α) => LT.lt.{0} Real Real.hasLt (Dist.dist.{u1} α (PseudoMetricSpace.toHasDist.{u1} α _inst_1) (Prod.fst.{u1, u1} α α p) (Prod.snd.{u1, u1} α α p)) ε))) (Metric.ball.{u1} α _inst_1 x ε)
+but is expected to have type
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] (ε : Real) (x : α), Eq.{succ u1} (Set.{u1} α) (UniformSpace.ball.{u1} α x (setOf.{u1} (Prod.{u1, u1} α α) (fun (p : Prod.{u1, u1} α α) => LT.lt.{0} Real Real.instLTReal (Dist.dist.{u1} α (PseudoMetricSpace.toDist.{u1} α _inst_1) (Prod.fst.{u1, u1} α α p) (Prod.snd.{u1, u1} α α p)) ε))) (Metric.ball.{u1} α _inst_1 x ε)
+Case conversion may be inaccurate. Consider using '#align metric.ball_eq_ball' Metric.ball_eq_ball'ₓ'. -/
 theorem ball_eq_ball' (ε : ℝ) (x : α) :
     UniformSpace.ball x { p | dist p.1 p.2 < ε } = Metric.ball x ε :=
   by
@@ -521,42 +797,80 @@ theorem ball_eq_ball' (ε : ℝ) (x : α) :
   simp [dist_comm, UniformSpace.ball]
 #align metric.ball_eq_ball' Metric.ball_eq_ball'
 
+/- warning: metric.Union_ball_nat -> Metric.unionᵢ_ball_nat is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] (x : α), Eq.{succ u1} (Set.{u1} α) (Set.unionᵢ.{u1, 1} α Nat (fun (n : Nat) => Metric.ball.{u1} α _inst_1 x ((fun (a : Type) (b : Type) [self : HasLiftT.{1, 1} a b] => self.0) Nat Real (HasLiftT.mk.{1, 1} Nat Real (CoeTCₓ.coe.{1, 1} Nat Real (Nat.castCoe.{0} Real Real.hasNatCast))) n))) (Set.univ.{u1} α)
+but is expected to have type
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] (x : α), Eq.{succ u1} (Set.{u1} α) (Set.unionᵢ.{u1, 1} α Nat (fun (n : Nat) => Metric.ball.{u1} α _inst_1 x (Nat.cast.{0} Real Real.natCast n))) (Set.univ.{u1} α)
+Case conversion may be inaccurate. Consider using '#align metric.Union_ball_nat Metric.unionᵢ_ball_natₓ'. -/
 @[simp]
 theorem unionᵢ_ball_nat (x : α) : (⋃ n : ℕ, ball x n) = univ :=
   unionᵢ_eq_univ_iff.2 fun y => exists_nat_gt (dist y x)
 #align metric.Union_ball_nat Metric.unionᵢ_ball_nat
 
+/- warning: metric.Union_ball_nat_succ -> Metric.unionᵢ_ball_nat_succ is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] (x : α), Eq.{succ u1} (Set.{u1} α) (Set.unionᵢ.{u1, 1} α Nat (fun (n : Nat) => Metric.ball.{u1} α _inst_1 x (HAdd.hAdd.{0, 0, 0} Real Real Real (instHAdd.{0} Real Real.hasAdd) ((fun (a : Type) (b : Type) [self : HasLiftT.{1, 1} a b] => self.0) Nat Real (HasLiftT.mk.{1, 1} Nat Real (CoeTCₓ.coe.{1, 1} Nat Real (Nat.castCoe.{0} Real Real.hasNatCast))) n) (OfNat.ofNat.{0} Real 1 (OfNat.mk.{0} Real 1 (One.one.{0} Real Real.hasOne)))))) (Set.univ.{u1} α)
+but is expected to have type
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] (x : α), Eq.{succ u1} (Set.{u1} α) (Set.unionᵢ.{u1, 1} α Nat (fun (n : Nat) => Metric.ball.{u1} α _inst_1 x (HAdd.hAdd.{0, 0, 0} Real Real Real (instHAdd.{0} Real Real.instAddReal) (Nat.cast.{0} Real Real.natCast n) (OfNat.ofNat.{0} Real 1 (One.toOfNat1.{0} Real Real.instOneReal))))) (Set.univ.{u1} α)
+Case conversion may be inaccurate. Consider using '#align metric.Union_ball_nat_succ Metric.unionᵢ_ball_nat_succₓ'. -/
 @[simp]
 theorem unionᵢ_ball_nat_succ (x : α) : (⋃ n : ℕ, ball x (n + 1)) = univ :=
   unionᵢ_eq_univ_iff.2 fun y => (exists_nat_gt (dist y x)).imp fun n hn => hn.trans (lt_add_one _)
 #align metric.Union_ball_nat_succ Metric.unionᵢ_ball_nat_succ
 
+#print Metric.closedBall /-
 /-- `closed_ball x ε` is the set of all points `y` with `dist y x ≤ ε` -/
 def closedBall (x : α) (ε : ℝ) :=
   { y | dist y x ≤ ε }
 #align metric.closed_ball Metric.closedBall
+-/
 
+/- warning: metric.mem_closed_ball -> Metric.mem_closedBall is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {x : α} {y : α} {ε : Real}, Iff (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) y (Metric.closedBall.{u1} α _inst_1 x ε)) (LE.le.{0} Real Real.hasLe (Dist.dist.{u1} α (PseudoMetricSpace.toHasDist.{u1} α _inst_1) y x) ε)
+but is expected to have type
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {x : α} {y : α} {ε : Real}, Iff (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) y (Metric.closedBall.{u1} α _inst_1 x ε)) (LE.le.{0} Real Real.instLEReal (Dist.dist.{u1} α (PseudoMetricSpace.toDist.{u1} α _inst_1) y x) ε)
+Case conversion may be inaccurate. Consider using '#align metric.mem_closed_ball Metric.mem_closedBallₓ'. -/
 @[simp]
 theorem mem_closedBall : y ∈ closedBall x ε ↔ dist y x ≤ ε :=
   Iff.rfl
 #align metric.mem_closed_ball Metric.mem_closedBall
 
-theorem mem_closed_ball' : y ∈ closedBall x ε ↔ dist x y ≤ ε := by rw [dist_comm, mem_closed_ball]
-#align metric.mem_closed_ball' Metric.mem_closed_ball'
+/- warning: metric.mem_closed_ball' -> Metric.mem_closedBall' is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {x : α} {y : α} {ε : Real}, Iff (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) y (Metric.closedBall.{u1} α _inst_1 x ε)) (LE.le.{0} Real Real.hasLe (Dist.dist.{u1} α (PseudoMetricSpace.toHasDist.{u1} α _inst_1) x y) ε)
+but is expected to have type
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {x : α} {y : α} {ε : Real}, Iff (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) y (Metric.closedBall.{u1} α _inst_1 x ε)) (LE.le.{0} Real Real.instLEReal (Dist.dist.{u1} α (PseudoMetricSpace.toDist.{u1} α _inst_1) x y) ε)
+Case conversion may be inaccurate. Consider using '#align metric.mem_closed_ball' Metric.mem_closedBall'ₓ'. -/
+theorem mem_closedBall' : y ∈ closedBall x ε ↔ dist x y ≤ ε := by rw [dist_comm, mem_closed_ball]
+#align metric.mem_closed_ball' Metric.mem_closedBall'
 
+#print Metric.sphere /-
 /-- `sphere x ε` is the set of all points `y` with `dist y x = ε` -/
 def sphere (x : α) (ε : ℝ) :=
   { y | dist y x = ε }
 #align metric.sphere Metric.sphere
+-/
 
+#print Metric.mem_sphere /-
 @[simp]
 theorem mem_sphere : y ∈ sphere x ε ↔ dist y x = ε :=
   Iff.rfl
 #align metric.mem_sphere Metric.mem_sphere
+-/
 
+#print Metric.mem_sphere' /-
 theorem mem_sphere' : y ∈ sphere x ε ↔ dist x y = ε := by rw [dist_comm, mem_sphere]
 #align metric.mem_sphere' Metric.mem_sphere'
+-/
 
+/- warning: metric.ne_of_mem_sphere -> Metric.ne_of_mem_sphere is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {x : α} {y : α} {ε : Real}, (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) y (Metric.sphere.{u1} α _inst_1 x ε)) -> (Ne.{1} Real ε (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero)))) -> (Ne.{succ u1} α y x)
+but is expected to have type
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {x : α} {y : α} {ε : Real}, (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) y (Metric.sphere.{u1} α _inst_1 x ε)) -> (Ne.{1} Real ε (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal))) -> (Ne.{succ u1} α y x)
+Case conversion may be inaccurate. Consider using '#align metric.ne_of_mem_sphere Metric.ne_of_mem_sphereₓ'. -/
 theorem ne_of_mem_sphere (h : y ∈ sphere x ε) (hε : ε ≠ 0) : y ≠ x :=
   by
   contrapose! hε
@@ -564,95 +878,207 @@ theorem ne_of_mem_sphere (h : y ∈ sphere x ε) (hε : ε ≠ 0) : y ≠ x :=
   simpa [hε] using h
 #align metric.ne_of_mem_sphere Metric.ne_of_mem_sphere
 
+/- warning: metric.sphere_eq_empty_of_subsingleton -> Metric.sphere_eq_empty_of_subsingleton is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {x : α} {ε : Real} [_inst_2 : Subsingleton.{succ u1} α], (Ne.{1} Real ε (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero)))) -> (Eq.{succ u1} (Set.{u1} α) (Metric.sphere.{u1} α _inst_1 x ε) (EmptyCollection.emptyCollection.{u1} (Set.{u1} α) (Set.hasEmptyc.{u1} α)))
+but is expected to have type
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {x : α} {ε : Real} [_inst_2 : Subsingleton.{succ u1} α], (Ne.{1} Real ε (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal))) -> (Eq.{succ u1} (Set.{u1} α) (Metric.sphere.{u1} α _inst_1 x ε) (EmptyCollection.emptyCollection.{u1} (Set.{u1} α) (Set.instEmptyCollectionSet.{u1} α)))
+Case conversion may be inaccurate. Consider using '#align metric.sphere_eq_empty_of_subsingleton Metric.sphere_eq_empty_of_subsingletonₓ'. -/
 theorem sphere_eq_empty_of_subsingleton [Subsingleton α] (hε : ε ≠ 0) : sphere x ε = ∅ :=
   Set.eq_empty_iff_forall_not_mem.mpr fun y hy => ne_of_mem_sphere hy hε (Subsingleton.elim _ _)
 #align metric.sphere_eq_empty_of_subsingleton Metric.sphere_eq_empty_of_subsingleton
 
+/- warning: metric.sphere_is_empty_of_subsingleton -> Metric.sphere_isEmpty_of_subsingleton is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {x : α} {ε : Real} [_inst_2 : Subsingleton.{succ u1} α], (Ne.{1} Real ε (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero)))) -> (IsEmpty.{succ u1} (coeSort.{succ u1, succ (succ u1)} (Set.{u1} α) Type.{u1} (Set.hasCoeToSort.{u1} α) (Metric.sphere.{u1} α _inst_1 x ε)))
+but is expected to have type
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {x : α} {ε : Real} [_inst_2 : Subsingleton.{succ u1} α], (Ne.{1} Real ε (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal))) -> (IsEmpty.{succ u1} (Set.Elem.{u1} α (Metric.sphere.{u1} α _inst_1 x ε)))
+Case conversion may be inaccurate. Consider using '#align metric.sphere_is_empty_of_subsingleton Metric.sphere_isEmpty_of_subsingletonₓ'. -/
 theorem sphere_isEmpty_of_subsingleton [Subsingleton α] (hε : ε ≠ 0) : IsEmpty (sphere x ε) := by
   simp only [sphere_eq_empty_of_subsingleton hε, Set.hasEmptyc.Emptyc.isEmpty α]
 #align metric.sphere_is_empty_of_subsingleton Metric.sphere_isEmpty_of_subsingleton
 
+/- warning: metric.mem_closed_ball_self -> Metric.mem_closedBall_self is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {x : α} {ε : Real}, (LE.le.{0} Real Real.hasLe (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero))) ε) -> (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) x (Metric.closedBall.{u1} α _inst_1 x ε))
+but is expected to have type
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {x : α} {ε : Real}, (LE.le.{0} Real Real.instLEReal (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal)) ε) -> (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) x (Metric.closedBall.{u1} α _inst_1 x ε))
+Case conversion may be inaccurate. Consider using '#align metric.mem_closed_ball_self Metric.mem_closedBall_selfₓ'. -/
 theorem mem_closedBall_self (h : 0 ≤ ε) : x ∈ closedBall x ε :=
   show dist x x ≤ ε by rw [dist_self] <;> assumption
 #align metric.mem_closed_ball_self Metric.mem_closedBall_self
 
+/- warning: metric.nonempty_closed_ball -> Metric.nonempty_closedBall is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {x : α} {ε : Real}, Iff (Set.Nonempty.{u1} α (Metric.closedBall.{u1} α _inst_1 x ε)) (LE.le.{0} Real Real.hasLe (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero))) ε)
+but is expected to have type
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {x : α} {ε : Real}, Iff (Set.Nonempty.{u1} α (Metric.closedBall.{u1} α _inst_1 x ε)) (LE.le.{0} Real Real.instLEReal (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal)) ε)
+Case conversion may be inaccurate. Consider using '#align metric.nonempty_closed_ball Metric.nonempty_closedBallₓ'. -/
 @[simp]
 theorem nonempty_closedBall : (closedBall x ε).Nonempty ↔ 0 ≤ ε :=
   ⟨fun ⟨x, hx⟩ => dist_nonneg.trans hx, fun h => ⟨x, mem_closedBall_self h⟩⟩
 #align metric.nonempty_closed_ball Metric.nonempty_closedBall
 
+/- warning: metric.closed_ball_eq_empty -> Metric.closedBall_eq_empty is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {x : α} {ε : Real}, Iff (Eq.{succ u1} (Set.{u1} α) (Metric.closedBall.{u1} α _inst_1 x ε) (EmptyCollection.emptyCollection.{u1} (Set.{u1} α) (Set.hasEmptyc.{u1} α))) (LT.lt.{0} Real Real.hasLt ε (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero))))
+but is expected to have type
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {x : α} {ε : Real}, Iff (Eq.{succ u1} (Set.{u1} α) (Metric.closedBall.{u1} α _inst_1 x ε) (EmptyCollection.emptyCollection.{u1} (Set.{u1} α) (Set.instEmptyCollectionSet.{u1} α))) (LT.lt.{0} Real Real.instLTReal ε (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal)))
+Case conversion may be inaccurate. Consider using '#align metric.closed_ball_eq_empty Metric.closedBall_eq_emptyₓ'. -/
 @[simp]
 theorem closedBall_eq_empty : closedBall x ε = ∅ ↔ ε < 0 := by
   rw [← not_nonempty_iff_eq_empty, nonempty_closed_ball, not_le]
 #align metric.closed_ball_eq_empty Metric.closedBall_eq_empty
 
+#print Metric.ball_subset_closedBall /-
 theorem ball_subset_closedBall : ball x ε ⊆ closedBall x ε := fun y (hy : _ < _) => le_of_lt hy
 #align metric.ball_subset_closed_ball Metric.ball_subset_closedBall
+-/
 
+#print Metric.sphere_subset_closedBall /-
 theorem sphere_subset_closedBall : sphere x ε ⊆ closedBall x ε := fun y => le_of_eq
 #align metric.sphere_subset_closed_ball Metric.sphere_subset_closedBall
+-/
 
+/- warning: metric.closed_ball_disjoint_ball -> Metric.closedBall_disjoint_ball is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {x : α} {y : α} {δ : Real} {ε : Real}, (LE.le.{0} Real Real.hasLe (HAdd.hAdd.{0, 0, 0} Real Real Real (instHAdd.{0} Real Real.hasAdd) δ ε) (Dist.dist.{u1} α (PseudoMetricSpace.toHasDist.{u1} α _inst_1) x y)) -> (Disjoint.{u1} (Set.{u1} α) (CompleteSemilatticeInf.toPartialOrder.{u1} (Set.{u1} α) (CompleteLattice.toCompleteSemilatticeInf.{u1} (Set.{u1} α) (Order.Coframe.toCompleteLattice.{u1} (Set.{u1} α) (CompleteDistribLattice.toCoframe.{u1} (Set.{u1} α) (CompleteBooleanAlgebra.toCompleteDistribLattice.{u1} (Set.{u1} α) (Set.completeBooleanAlgebra.{u1} α)))))) (GeneralizedBooleanAlgebra.toOrderBot.{u1} (Set.{u1} α) (BooleanAlgebra.toGeneralizedBooleanAlgebra.{u1} (Set.{u1} α) (Set.booleanAlgebra.{u1} α))) (Metric.closedBall.{u1} α _inst_1 x δ) (Metric.ball.{u1} α _inst_1 y ε))
+but is expected to have type
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {x : α} {y : α} {δ : Real} {ε : Real}, (LE.le.{0} Real Real.instLEReal (HAdd.hAdd.{0, 0, 0} Real Real Real (instHAdd.{0} Real Real.instAddReal) δ ε) (Dist.dist.{u1} α (PseudoMetricSpace.toDist.{u1} α _inst_1) x y)) -> (Disjoint.{u1} (Set.{u1} α) (CompleteSemilatticeInf.toPartialOrder.{u1} (Set.{u1} α) (CompleteLattice.toCompleteSemilatticeInf.{u1} (Set.{u1} α) (Order.Coframe.toCompleteLattice.{u1} (Set.{u1} α) (CompleteDistribLattice.toCoframe.{u1} (Set.{u1} α) (CompleteBooleanAlgebra.toCompleteDistribLattice.{u1} (Set.{u1} α) (Set.instCompleteBooleanAlgebraSet.{u1} α)))))) (BoundedOrder.toOrderBot.{u1} (Set.{u1} α) (Preorder.toLE.{u1} (Set.{u1} α) (PartialOrder.toPreorder.{u1} (Set.{u1} α) (CompleteSemilatticeInf.toPartialOrder.{u1} (Set.{u1} α) (CompleteLattice.toCompleteSemilatticeInf.{u1} (Set.{u1} α) (Order.Coframe.toCompleteLattice.{u1} (Set.{u1} α) (CompleteDistribLattice.toCoframe.{u1} (Set.{u1} α) (CompleteBooleanAlgebra.toCompleteDistribLattice.{u1} (Set.{u1} α) (Set.instCompleteBooleanAlgebraSet.{u1} α)))))))) (CompleteLattice.toBoundedOrder.{u1} (Set.{u1} α) (Order.Coframe.toCompleteLattice.{u1} (Set.{u1} α) (CompleteDistribLattice.toCoframe.{u1} (Set.{u1} α) (CompleteBooleanAlgebra.toCompleteDistribLattice.{u1} (Set.{u1} α) (Set.instCompleteBooleanAlgebraSet.{u1} α)))))) (Metric.closedBall.{u1} α _inst_1 x δ) (Metric.ball.{u1} α _inst_1 y ε))
+Case conversion may be inaccurate. Consider using '#align metric.closed_ball_disjoint_ball Metric.closedBall_disjoint_ballₓ'. -/
 theorem closedBall_disjoint_ball (h : δ + ε ≤ dist x y) : Disjoint (closedBall x δ) (ball y ε) :=
   Set.disjoint_left.mpr fun a ha1 ha2 =>
     (h.trans <| dist_triangle_left _ _ _).not_lt <| add_lt_add_of_le_of_lt ha1 ha2
 #align metric.closed_ball_disjoint_ball Metric.closedBall_disjoint_ball
 
+/- warning: metric.ball_disjoint_closed_ball -> Metric.ball_disjoint_closedBall is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {x : α} {y : α} {δ : Real} {ε : Real}, (LE.le.{0} Real Real.hasLe (HAdd.hAdd.{0, 0, 0} Real Real Real (instHAdd.{0} Real Real.hasAdd) δ ε) (Dist.dist.{u1} α (PseudoMetricSpace.toHasDist.{u1} α _inst_1) x y)) -> (Disjoint.{u1} (Set.{u1} α) (CompleteSemilatticeInf.toPartialOrder.{u1} (Set.{u1} α) (CompleteLattice.toCompleteSemilatticeInf.{u1} (Set.{u1} α) (Order.Coframe.toCompleteLattice.{u1} (Set.{u1} α) (CompleteDistribLattice.toCoframe.{u1} (Set.{u1} α) (CompleteBooleanAlgebra.toCompleteDistribLattice.{u1} (Set.{u1} α) (Set.completeBooleanAlgebra.{u1} α)))))) (GeneralizedBooleanAlgebra.toOrderBot.{u1} (Set.{u1} α) (BooleanAlgebra.toGeneralizedBooleanAlgebra.{u1} (Set.{u1} α) (Set.booleanAlgebra.{u1} α))) (Metric.ball.{u1} α _inst_1 x δ) (Metric.closedBall.{u1} α _inst_1 y ε))
+but is expected to have type
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {x : α} {y : α} {δ : Real} {ε : Real}, (LE.le.{0} Real Real.instLEReal (HAdd.hAdd.{0, 0, 0} Real Real Real (instHAdd.{0} Real Real.instAddReal) δ ε) (Dist.dist.{u1} α (PseudoMetricSpace.toDist.{u1} α _inst_1) x y)) -> (Disjoint.{u1} (Set.{u1} α) (CompleteSemilatticeInf.toPartialOrder.{u1} (Set.{u1} α) (CompleteLattice.toCompleteSemilatticeInf.{u1} (Set.{u1} α) (Order.Coframe.toCompleteLattice.{u1} (Set.{u1} α) (CompleteDistribLattice.toCoframe.{u1} (Set.{u1} α) (CompleteBooleanAlgebra.toCompleteDistribLattice.{u1} (Set.{u1} α) (Set.instCompleteBooleanAlgebraSet.{u1} α)))))) (BoundedOrder.toOrderBot.{u1} (Set.{u1} α) (Preorder.toLE.{u1} (Set.{u1} α) (PartialOrder.toPreorder.{u1} (Set.{u1} α) (CompleteSemilatticeInf.toPartialOrder.{u1} (Set.{u1} α) (CompleteLattice.toCompleteSemilatticeInf.{u1} (Set.{u1} α) (Order.Coframe.toCompleteLattice.{u1} (Set.{u1} α) (CompleteDistribLattice.toCoframe.{u1} (Set.{u1} α) (CompleteBooleanAlgebra.toCompleteDistribLattice.{u1} (Set.{u1} α) (Set.instCompleteBooleanAlgebraSet.{u1} α)))))))) (CompleteLattice.toBoundedOrder.{u1} (Set.{u1} α) (Order.Coframe.toCompleteLattice.{u1} (Set.{u1} α) (CompleteDistribLattice.toCoframe.{u1} (Set.{u1} α) (CompleteBooleanAlgebra.toCompleteDistribLattice.{u1} (Set.{u1} α) (Set.instCompleteBooleanAlgebraSet.{u1} α)))))) (Metric.ball.{u1} α _inst_1 x δ) (Metric.closedBall.{u1} α _inst_1 y ε))
+Case conversion may be inaccurate. Consider using '#align metric.ball_disjoint_closed_ball Metric.ball_disjoint_closedBallₓ'. -/
 theorem ball_disjoint_closedBall (h : δ + ε ≤ dist x y) : Disjoint (ball x δ) (closedBall y ε) :=
   (closedBall_disjoint_ball <| by rwa [add_comm, dist_comm]).symm
 #align metric.ball_disjoint_closed_ball Metric.ball_disjoint_closedBall
 
+/- warning: metric.ball_disjoint_ball -> Metric.ball_disjoint_ball is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {x : α} {y : α} {δ : Real} {ε : Real}, (LE.le.{0} Real Real.hasLe (HAdd.hAdd.{0, 0, 0} Real Real Real (instHAdd.{0} Real Real.hasAdd) δ ε) (Dist.dist.{u1} α (PseudoMetricSpace.toHasDist.{u1} α _inst_1) x y)) -> (Disjoint.{u1} (Set.{u1} α) (CompleteSemilatticeInf.toPartialOrder.{u1} (Set.{u1} α) (CompleteLattice.toCompleteSemilatticeInf.{u1} (Set.{u1} α) (Order.Coframe.toCompleteLattice.{u1} (Set.{u1} α) (CompleteDistribLattice.toCoframe.{u1} (Set.{u1} α) (CompleteBooleanAlgebra.toCompleteDistribLattice.{u1} (Set.{u1} α) (Set.completeBooleanAlgebra.{u1} α)))))) (GeneralizedBooleanAlgebra.toOrderBot.{u1} (Set.{u1} α) (BooleanAlgebra.toGeneralizedBooleanAlgebra.{u1} (Set.{u1} α) (Set.booleanAlgebra.{u1} α))) (Metric.ball.{u1} α _inst_1 x δ) (Metric.ball.{u1} α _inst_1 y ε))
+but is expected to have type
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {x : α} {y : α} {δ : Real} {ε : Real}, (LE.le.{0} Real Real.instLEReal (HAdd.hAdd.{0, 0, 0} Real Real Real (instHAdd.{0} Real Real.instAddReal) δ ε) (Dist.dist.{u1} α (PseudoMetricSpace.toDist.{u1} α _inst_1) x y)) -> (Disjoint.{u1} (Set.{u1} α) (CompleteSemilatticeInf.toPartialOrder.{u1} (Set.{u1} α) (CompleteLattice.toCompleteSemilatticeInf.{u1} (Set.{u1} α) (Order.Coframe.toCompleteLattice.{u1} (Set.{u1} α) (CompleteDistribLattice.toCoframe.{u1} (Set.{u1} α) (CompleteBooleanAlgebra.toCompleteDistribLattice.{u1} (Set.{u1} α) (Set.instCompleteBooleanAlgebraSet.{u1} α)))))) (BoundedOrder.toOrderBot.{u1} (Set.{u1} α) (Preorder.toLE.{u1} (Set.{u1} α) (PartialOrder.toPreorder.{u1} (Set.{u1} α) (CompleteSemilatticeInf.toPartialOrder.{u1} (Set.{u1} α) (CompleteLattice.toCompleteSemilatticeInf.{u1} (Set.{u1} α) (Order.Coframe.toCompleteLattice.{u1} (Set.{u1} α) (CompleteDistribLattice.toCoframe.{u1} (Set.{u1} α) (CompleteBooleanAlgebra.toCompleteDistribLattice.{u1} (Set.{u1} α) (Set.instCompleteBooleanAlgebraSet.{u1} α)))))))) (CompleteLattice.toBoundedOrder.{u1} (Set.{u1} α) (Order.Coframe.toCompleteLattice.{u1} (Set.{u1} α) (CompleteDistribLattice.toCoframe.{u1} (Set.{u1} α) (CompleteBooleanAlgebra.toCompleteDistribLattice.{u1} (Set.{u1} α) (Set.instCompleteBooleanAlgebraSet.{u1} α)))))) (Metric.ball.{u1} α _inst_1 x δ) (Metric.ball.{u1} α _inst_1 y ε))
+Case conversion may be inaccurate. Consider using '#align metric.ball_disjoint_ball Metric.ball_disjoint_ballₓ'. -/
 theorem ball_disjoint_ball (h : δ + ε ≤ dist x y) : Disjoint (ball x δ) (ball y ε) :=
   (closedBall_disjoint_ball h).mono_left ball_subset_closedBall
 #align metric.ball_disjoint_ball Metric.ball_disjoint_ball
 
+/- warning: metric.closed_ball_disjoint_closed_ball -> Metric.closedBall_disjoint_closedBall is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {x : α} {y : α} {δ : Real} {ε : Real}, (LT.lt.{0} Real Real.hasLt (HAdd.hAdd.{0, 0, 0} Real Real Real (instHAdd.{0} Real Real.hasAdd) δ ε) (Dist.dist.{u1} α (PseudoMetricSpace.toHasDist.{u1} α _inst_1) x y)) -> (Disjoint.{u1} (Set.{u1} α) (CompleteSemilatticeInf.toPartialOrder.{u1} (Set.{u1} α) (CompleteLattice.toCompleteSemilatticeInf.{u1} (Set.{u1} α) (Order.Coframe.toCompleteLattice.{u1} (Set.{u1} α) (CompleteDistribLattice.toCoframe.{u1} (Set.{u1} α) (CompleteBooleanAlgebra.toCompleteDistribLattice.{u1} (Set.{u1} α) (Set.completeBooleanAlgebra.{u1} α)))))) (GeneralizedBooleanAlgebra.toOrderBot.{u1} (Set.{u1} α) (BooleanAlgebra.toGeneralizedBooleanAlgebra.{u1} (Set.{u1} α) (Set.booleanAlgebra.{u1} α))) (Metric.closedBall.{u1} α _inst_1 x δ) (Metric.closedBall.{u1} α _inst_1 y ε))
+but is expected to have type
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {x : α} {y : α} {δ : Real} {ε : Real}, (LT.lt.{0} Real Real.instLTReal (HAdd.hAdd.{0, 0, 0} Real Real Real (instHAdd.{0} Real Real.instAddReal) δ ε) (Dist.dist.{u1} α (PseudoMetricSpace.toDist.{u1} α _inst_1) x y)) -> (Disjoint.{u1} (Set.{u1} α) (CompleteSemilatticeInf.toPartialOrder.{u1} (Set.{u1} α) (CompleteLattice.toCompleteSemilatticeInf.{u1} (Set.{u1} α) (Order.Coframe.toCompleteLattice.{u1} (Set.{u1} α) (CompleteDistribLattice.toCoframe.{u1} (Set.{u1} α) (CompleteBooleanAlgebra.toCompleteDistribLattice.{u1} (Set.{u1} α) (Set.instCompleteBooleanAlgebraSet.{u1} α)))))) (BoundedOrder.toOrderBot.{u1} (Set.{u1} α) (Preorder.toLE.{u1} (Set.{u1} α) (PartialOrder.toPreorder.{u1} (Set.{u1} α) (CompleteSemilatticeInf.toPartialOrder.{u1} (Set.{u1} α) (CompleteLattice.toCompleteSemilatticeInf.{u1} (Set.{u1} α) (Order.Coframe.toCompleteLattice.{u1} (Set.{u1} α) (CompleteDistribLattice.toCoframe.{u1} (Set.{u1} α) (CompleteBooleanAlgebra.toCompleteDistribLattice.{u1} (Set.{u1} α) (Set.instCompleteBooleanAlgebraSet.{u1} α)))))))) (CompleteLattice.toBoundedOrder.{u1} (Set.{u1} α) (Order.Coframe.toCompleteLattice.{u1} (Set.{u1} α) (CompleteDistribLattice.toCoframe.{u1} (Set.{u1} α) (CompleteBooleanAlgebra.toCompleteDistribLattice.{u1} (Set.{u1} α) (Set.instCompleteBooleanAlgebraSet.{u1} α)))))) (Metric.closedBall.{u1} α _inst_1 x δ) (Metric.closedBall.{u1} α _inst_1 y ε))
+Case conversion may be inaccurate. Consider using '#align metric.closed_ball_disjoint_closed_ball Metric.closedBall_disjoint_closedBallₓ'. -/
 theorem closedBall_disjoint_closedBall (h : δ + ε < dist x y) :
     Disjoint (closedBall x δ) (closedBall y ε) :=
   Set.disjoint_left.mpr fun a ha1 ha2 =>
     h.not_le <| (dist_triangle_left _ _ _).trans <| add_le_add ha1 ha2
 #align metric.closed_ball_disjoint_closed_ball Metric.closedBall_disjoint_closedBall
 
+/- warning: metric.sphere_disjoint_ball -> Metric.sphere_disjoint_ball is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {x : α} {ε : Real}, Disjoint.{u1} (Set.{u1} α) (CompleteSemilatticeInf.toPartialOrder.{u1} (Set.{u1} α) (CompleteLattice.toCompleteSemilatticeInf.{u1} (Set.{u1} α) (Order.Coframe.toCompleteLattice.{u1} (Set.{u1} α) (CompleteDistribLattice.toCoframe.{u1} (Set.{u1} α) (CompleteBooleanAlgebra.toCompleteDistribLattice.{u1} (Set.{u1} α) (Set.completeBooleanAlgebra.{u1} α)))))) (GeneralizedBooleanAlgebra.toOrderBot.{u1} (Set.{u1} α) (BooleanAlgebra.toGeneralizedBooleanAlgebra.{u1} (Set.{u1} α) (Set.booleanAlgebra.{u1} α))) (Metric.sphere.{u1} α _inst_1 x ε) (Metric.ball.{u1} α _inst_1 x ε)
+but is expected to have type
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {x : α} {ε : Real}, Disjoint.{u1} (Set.{u1} α) (CompleteSemilatticeInf.toPartialOrder.{u1} (Set.{u1} α) (CompleteLattice.toCompleteSemilatticeInf.{u1} (Set.{u1} α) (Order.Coframe.toCompleteLattice.{u1} (Set.{u1} α) (CompleteDistribLattice.toCoframe.{u1} (Set.{u1} α) (CompleteBooleanAlgebra.toCompleteDistribLattice.{u1} (Set.{u1} α) (Set.instCompleteBooleanAlgebraSet.{u1} α)))))) (BoundedOrder.toOrderBot.{u1} (Set.{u1} α) (Preorder.toLE.{u1} (Set.{u1} α) (PartialOrder.toPreorder.{u1} (Set.{u1} α) (CompleteSemilatticeInf.toPartialOrder.{u1} (Set.{u1} α) (CompleteLattice.toCompleteSemilatticeInf.{u1} (Set.{u1} α) (Order.Coframe.toCompleteLattice.{u1} (Set.{u1} α) (CompleteDistribLattice.toCoframe.{u1} (Set.{u1} α) (CompleteBooleanAlgebra.toCompleteDistribLattice.{u1} (Set.{u1} α) (Set.instCompleteBooleanAlgebraSet.{u1} α)))))))) (CompleteLattice.toBoundedOrder.{u1} (Set.{u1} α) (Order.Coframe.toCompleteLattice.{u1} (Set.{u1} α) (CompleteDistribLattice.toCoframe.{u1} (Set.{u1} α) (CompleteBooleanAlgebra.toCompleteDistribLattice.{u1} (Set.{u1} α) (Set.instCompleteBooleanAlgebraSet.{u1} α)))))) (Metric.sphere.{u1} α _inst_1 x ε) (Metric.ball.{u1} α _inst_1 x ε)
+Case conversion may be inaccurate. Consider using '#align metric.sphere_disjoint_ball Metric.sphere_disjoint_ballₓ'. -/
 theorem sphere_disjoint_ball : Disjoint (sphere x ε) (ball x ε) :=
   Set.disjoint_left.mpr fun y hy₁ hy₂ => absurd hy₁ <| ne_of_lt hy₂
 #align metric.sphere_disjoint_ball Metric.sphere_disjoint_ball
 
+/- warning: metric.ball_union_sphere -> Metric.ball_union_sphere is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {x : α} {ε : Real}, Eq.{succ u1} (Set.{u1} α) (Union.union.{u1} (Set.{u1} α) (Set.hasUnion.{u1} α) (Metric.ball.{u1} α _inst_1 x ε) (Metric.sphere.{u1} α _inst_1 x ε)) (Metric.closedBall.{u1} α _inst_1 x ε)
+but is expected to have type
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {x : α} {ε : Real}, Eq.{succ u1} (Set.{u1} α) (Union.union.{u1} (Set.{u1} α) (Set.instUnionSet.{u1} α) (Metric.ball.{u1} α _inst_1 x ε) (Metric.sphere.{u1} α _inst_1 x ε)) (Metric.closedBall.{u1} α _inst_1 x ε)
+Case conversion may be inaccurate. Consider using '#align metric.ball_union_sphere Metric.ball_union_sphereₓ'. -/
 @[simp]
 theorem ball_union_sphere : ball x ε ∪ sphere x ε = closedBall x ε :=
   Set.ext fun y => (@le_iff_lt_or_eq ℝ _ _ _).symm
 #align metric.ball_union_sphere Metric.ball_union_sphere
 
+/- warning: metric.sphere_union_ball -> Metric.sphere_union_ball is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {x : α} {ε : Real}, Eq.{succ u1} (Set.{u1} α) (Union.union.{u1} (Set.{u1} α) (Set.hasUnion.{u1} α) (Metric.sphere.{u1} α _inst_1 x ε) (Metric.ball.{u1} α _inst_1 x ε)) (Metric.closedBall.{u1} α _inst_1 x ε)
+but is expected to have type
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {x : α} {ε : Real}, Eq.{succ u1} (Set.{u1} α) (Union.union.{u1} (Set.{u1} α) (Set.instUnionSet.{u1} α) (Metric.sphere.{u1} α _inst_1 x ε) (Metric.ball.{u1} α _inst_1 x ε)) (Metric.closedBall.{u1} α _inst_1 x ε)
+Case conversion may be inaccurate. Consider using '#align metric.sphere_union_ball Metric.sphere_union_ballₓ'. -/
 @[simp]
 theorem sphere_union_ball : sphere x ε ∪ ball x ε = closedBall x ε := by
   rw [union_comm, ball_union_sphere]
 #align metric.sphere_union_ball Metric.sphere_union_ball
 
+/- warning: metric.closed_ball_diff_sphere -> Metric.closedBall_diff_sphere is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {x : α} {ε : Real}, Eq.{succ u1} (Set.{u1} α) (SDiff.sdiff.{u1} (Set.{u1} α) (BooleanAlgebra.toHasSdiff.{u1} (Set.{u1} α) (Set.booleanAlgebra.{u1} α)) (Metric.closedBall.{u1} α _inst_1 x ε) (Metric.sphere.{u1} α _inst_1 x ε)) (Metric.ball.{u1} α _inst_1 x ε)
+but is expected to have type
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {x : α} {ε : Real}, Eq.{succ u1} (Set.{u1} α) (SDiff.sdiff.{u1} (Set.{u1} α) (Set.instSDiffSet.{u1} α) (Metric.closedBall.{u1} α _inst_1 x ε) (Metric.sphere.{u1} α _inst_1 x ε)) (Metric.ball.{u1} α _inst_1 x ε)
+Case conversion may be inaccurate. Consider using '#align metric.closed_ball_diff_sphere Metric.closedBall_diff_sphereₓ'. -/
 @[simp]
 theorem closedBall_diff_sphere : closedBall x ε \ sphere x ε = ball x ε := by
   rw [← ball_union_sphere, Set.union_diff_cancel_right sphere_disjoint_ball.symm.le_bot]
 #align metric.closed_ball_diff_sphere Metric.closedBall_diff_sphere
 
+/- warning: metric.closed_ball_diff_ball -> Metric.closedBall_diff_ball is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {x : α} {ε : Real}, Eq.{succ u1} (Set.{u1} α) (SDiff.sdiff.{u1} (Set.{u1} α) (BooleanAlgebra.toHasSdiff.{u1} (Set.{u1} α) (Set.booleanAlgebra.{u1} α)) (Metric.closedBall.{u1} α _inst_1 x ε) (Metric.ball.{u1} α _inst_1 x ε)) (Metric.sphere.{u1} α _inst_1 x ε)
+but is expected to have type
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {x : α} {ε : Real}, Eq.{succ u1} (Set.{u1} α) (SDiff.sdiff.{u1} (Set.{u1} α) (Set.instSDiffSet.{u1} α) (Metric.closedBall.{u1} α _inst_1 x ε) (Metric.ball.{u1} α _inst_1 x ε)) (Metric.sphere.{u1} α _inst_1 x ε)
+Case conversion may be inaccurate. Consider using '#align metric.closed_ball_diff_ball Metric.closedBall_diff_ballₓ'. -/
 @[simp]
 theorem closedBall_diff_ball : closedBall x ε \ ball x ε = sphere x ε := by
   rw [← ball_union_sphere, Set.union_diff_cancel_left sphere_disjoint_ball.symm.le_bot]
 #align metric.closed_ball_diff_ball Metric.closedBall_diff_ball
 
+#print Metric.mem_ball_comm /-
 theorem mem_ball_comm : x ∈ ball y ε ↔ y ∈ ball x ε := by rw [mem_ball', mem_ball]
 #align metric.mem_ball_comm Metric.mem_ball_comm
+-/
 
+#print Metric.mem_closedBall_comm /-
 theorem mem_closedBall_comm : x ∈ closedBall y ε ↔ y ∈ closedBall x ε := by
   rw [mem_closed_ball', mem_closed_ball]
 #align metric.mem_closed_ball_comm Metric.mem_closedBall_comm
+-/
 
+#print Metric.mem_sphere_comm /-
 theorem mem_sphere_comm : x ∈ sphere y ε ↔ y ∈ sphere x ε := by rw [mem_sphere', mem_sphere]
 #align metric.mem_sphere_comm Metric.mem_sphere_comm
+-/
 
+/- warning: metric.ball_subset_ball -> Metric.ball_subset_ball is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {x : α} {ε₁ : Real} {ε₂ : Real}, (LE.le.{0} Real Real.hasLe ε₁ ε₂) -> (HasSubset.Subset.{u1} (Set.{u1} α) (Set.hasSubset.{u1} α) (Metric.ball.{u1} α _inst_1 x ε₁) (Metric.ball.{u1} α _inst_1 x ε₂))
+but is expected to have type
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {x : α} {ε₁ : Real} {ε₂ : Real}, (LE.le.{0} Real Real.instLEReal ε₁ ε₂) -> (HasSubset.Subset.{u1} (Set.{u1} α) (Set.instHasSubsetSet.{u1} α) (Metric.ball.{u1} α _inst_1 x ε₁) (Metric.ball.{u1} α _inst_1 x ε₂))
+Case conversion may be inaccurate. Consider using '#align metric.ball_subset_ball Metric.ball_subset_ballₓ'. -/
 theorem ball_subset_ball (h : ε₁ ≤ ε₂) : ball x ε₁ ⊆ ball x ε₂ := fun y (yx : _ < ε₁) =>
   lt_of_lt_of_le yx h
 #align metric.ball_subset_ball Metric.ball_subset_ball
 
+/- warning: metric.closed_ball_eq_bInter_ball -> Metric.closedBall_eq_bInter_ball is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {x : α} {ε : Real}, Eq.{succ u1} (Set.{u1} α) (Metric.closedBall.{u1} α _inst_1 x ε) (Set.interᵢ.{u1, 1} α Real (fun (δ : Real) => Set.interᵢ.{u1, 0} α (GT.gt.{0} Real Real.hasLt δ ε) (fun (H : GT.gt.{0} Real Real.hasLt δ ε) => Metric.ball.{u1} α _inst_1 x δ)))
+but is expected to have type
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {x : α} {ε : Real}, Eq.{succ u1} (Set.{u1} α) (Metric.closedBall.{u1} α _inst_1 x ε) (Set.interᵢ.{u1, 1} α Real (fun (δ : Real) => Set.interᵢ.{u1, 0} α (GT.gt.{0} Real Real.instLTReal δ ε) (fun (H : GT.gt.{0} Real Real.instLTReal δ ε) => Metric.ball.{u1} α _inst_1 x δ)))
+Case conversion may be inaccurate. Consider using '#align metric.closed_ball_eq_bInter_ball Metric.closedBall_eq_bInter_ballₓ'. -/
 theorem closedBall_eq_bInter_ball : closedBall x ε = ⋂ δ > ε, ball x δ := by
   ext y <;> rw [mem_closed_ball, ← forall_lt_iff_le', mem_Inter₂] <;> rfl
 #align metric.closed_ball_eq_bInter_ball Metric.closedBall_eq_bInter_ball
 
+/- warning: metric.ball_subset_ball' -> Metric.ball_subset_ball' is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {x : α} {y : α} {ε₁ : Real} {ε₂ : Real}, (LE.le.{0} Real Real.hasLe (HAdd.hAdd.{0, 0, 0} Real Real Real (instHAdd.{0} Real Real.hasAdd) ε₁ (Dist.dist.{u1} α (PseudoMetricSpace.toHasDist.{u1} α _inst_1) x y)) ε₂) -> (HasSubset.Subset.{u1} (Set.{u1} α) (Set.hasSubset.{u1} α) (Metric.ball.{u1} α _inst_1 x ε₁) (Metric.ball.{u1} α _inst_1 y ε₂))
+but is expected to have type
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {x : α} {y : α} {ε₁ : Real} {ε₂ : Real}, (LE.le.{0} Real Real.instLEReal (HAdd.hAdd.{0, 0, 0} Real Real Real (instHAdd.{0} Real Real.instAddReal) ε₁ (Dist.dist.{u1} α (PseudoMetricSpace.toDist.{u1} α _inst_1) x y)) ε₂) -> (HasSubset.Subset.{u1} (Set.{u1} α) (Set.instHasSubsetSet.{u1} α) (Metric.ball.{u1} α _inst_1 x ε₁) (Metric.ball.{u1} α _inst_1 y ε₂))
+Case conversion may be inaccurate. Consider using '#align metric.ball_subset_ball' Metric.ball_subset_ball'ₓ'. -/
 theorem ball_subset_ball' (h : ε₁ + dist x y ≤ ε₂) : ball x ε₁ ⊆ ball y ε₂ := fun z hz =>
   calc
     dist z y ≤ dist z x + dist x y := dist_triangle _ _ _
@@ -661,23 +1087,47 @@ theorem ball_subset_ball' (h : ε₁ + dist x y ≤ ε₂) : ball x ε₁ ⊆ ba
     
 #align metric.ball_subset_ball' Metric.ball_subset_ball'
 
+/- warning: metric.closed_ball_subset_closed_ball -> Metric.closedBall_subset_closedBall is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {x : α} {ε₁ : Real} {ε₂ : Real}, (LE.le.{0} Real Real.hasLe ε₁ ε₂) -> (HasSubset.Subset.{u1} (Set.{u1} α) (Set.hasSubset.{u1} α) (Metric.closedBall.{u1} α _inst_1 x ε₁) (Metric.closedBall.{u1} α _inst_1 x ε₂))
+but is expected to have type
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {x : α} {ε₁ : Real} {ε₂ : Real}, (LE.le.{0} Real Real.instLEReal ε₁ ε₂) -> (HasSubset.Subset.{u1} (Set.{u1} α) (Set.instHasSubsetSet.{u1} α) (Metric.closedBall.{u1} α _inst_1 x ε₁) (Metric.closedBall.{u1} α _inst_1 x ε₂))
+Case conversion may be inaccurate. Consider using '#align metric.closed_ball_subset_closed_ball Metric.closedBall_subset_closedBallₓ'. -/
 theorem closedBall_subset_closedBall (h : ε₁ ≤ ε₂) : closedBall x ε₁ ⊆ closedBall x ε₂ :=
   fun y (yx : _ ≤ ε₁) => le_trans yx h
 #align metric.closed_ball_subset_closed_ball Metric.closedBall_subset_closedBall
 
-theorem closedBall_subset_closed_ball' (h : ε₁ + dist x y ≤ ε₂) :
+/- warning: metric.closed_ball_subset_closed_ball' -> Metric.closedBall_subset_closedBall' is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {x : α} {y : α} {ε₁ : Real} {ε₂ : Real}, (LE.le.{0} Real Real.hasLe (HAdd.hAdd.{0, 0, 0} Real Real Real (instHAdd.{0} Real Real.hasAdd) ε₁ (Dist.dist.{u1} α (PseudoMetricSpace.toHasDist.{u1} α _inst_1) x y)) ε₂) -> (HasSubset.Subset.{u1} (Set.{u1} α) (Set.hasSubset.{u1} α) (Metric.closedBall.{u1} α _inst_1 x ε₁) (Metric.closedBall.{u1} α _inst_1 y ε₂))
+but is expected to have type
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {x : α} {y : α} {ε₁ : Real} {ε₂ : Real}, (LE.le.{0} Real Real.instLEReal (HAdd.hAdd.{0, 0, 0} Real Real Real (instHAdd.{0} Real Real.instAddReal) ε₁ (Dist.dist.{u1} α (PseudoMetricSpace.toDist.{u1} α _inst_1) x y)) ε₂) -> (HasSubset.Subset.{u1} (Set.{u1} α) (Set.instHasSubsetSet.{u1} α) (Metric.closedBall.{u1} α _inst_1 x ε₁) (Metric.closedBall.{u1} α _inst_1 y ε₂))
+Case conversion may be inaccurate. Consider using '#align metric.closed_ball_subset_closed_ball' Metric.closedBall_subset_closedBall'ₓ'. -/
+theorem closedBall_subset_closedBall' (h : ε₁ + dist x y ≤ ε₂) :
     closedBall x ε₁ ⊆ closedBall y ε₂ := fun z hz =>
   calc
     dist z y ≤ dist z x + dist x y := dist_triangle _ _ _
     _ ≤ ε₁ + dist x y := (add_le_add_right hz _)
     _ ≤ ε₂ := h
     
-#align metric.closed_ball_subset_closed_ball' Metric.closedBall_subset_closed_ball'
-
+#align metric.closed_ball_subset_closed_ball' Metric.closedBall_subset_closedBall'
+
+/- warning: metric.closed_ball_subset_ball -> Metric.closedBall_subset_ball is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {x : α} {ε₁ : Real} {ε₂ : Real}, (LT.lt.{0} Real Real.hasLt ε₁ ε₂) -> (HasSubset.Subset.{u1} (Set.{u1} α) (Set.hasSubset.{u1} α) (Metric.closedBall.{u1} α _inst_1 x ε₁) (Metric.ball.{u1} α _inst_1 x ε₂))
+but is expected to have type
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {x : α} {ε₁ : Real} {ε₂ : Real}, (LT.lt.{0} Real Real.instLTReal ε₁ ε₂) -> (HasSubset.Subset.{u1} (Set.{u1} α) (Set.instHasSubsetSet.{u1} α) (Metric.closedBall.{u1} α _inst_1 x ε₁) (Metric.ball.{u1} α _inst_1 x ε₂))
+Case conversion may be inaccurate. Consider using '#align metric.closed_ball_subset_ball Metric.closedBall_subset_ballₓ'. -/
 theorem closedBall_subset_ball (h : ε₁ < ε₂) : closedBall x ε₁ ⊆ ball x ε₂ :=
   fun y (yh : dist y x ≤ ε₁) => lt_of_le_of_lt yh h
 #align metric.closed_ball_subset_ball Metric.closedBall_subset_ball
 
+/- warning: metric.closed_ball_subset_ball' -> Metric.closedBall_subset_ball' is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {x : α} {y : α} {ε₁ : Real} {ε₂ : Real}, (LT.lt.{0} Real Real.hasLt (HAdd.hAdd.{0, 0, 0} Real Real Real (instHAdd.{0} Real Real.hasAdd) ε₁ (Dist.dist.{u1} α (PseudoMetricSpace.toHasDist.{u1} α _inst_1) x y)) ε₂) -> (HasSubset.Subset.{u1} (Set.{u1} α) (Set.hasSubset.{u1} α) (Metric.closedBall.{u1} α _inst_1 x ε₁) (Metric.ball.{u1} α _inst_1 y ε₂))
+but is expected to have type
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {x : α} {y : α} {ε₁ : Real} {ε₂ : Real}, (LT.lt.{0} Real Real.instLTReal (HAdd.hAdd.{0, 0, 0} Real Real Real (instHAdd.{0} Real Real.instAddReal) ε₁ (Dist.dist.{u1} α (PseudoMetricSpace.toDist.{u1} α _inst_1) x y)) ε₂) -> (HasSubset.Subset.{u1} (Set.{u1} α) (Set.instHasSubsetSet.{u1} α) (Metric.closedBall.{u1} α _inst_1 x ε₁) (Metric.ball.{u1} α _inst_1 y ε₂))
+Case conversion may be inaccurate. Consider using '#align metric.closed_ball_subset_ball' Metric.closedBall_subset_ball'ₓ'. -/
 theorem closedBall_subset_ball' (h : ε₁ + dist x y < ε₂) : closedBall x ε₁ ⊆ ball y ε₂ :=
   fun z hz =>
   calc
@@ -687,6 +1137,12 @@ theorem closedBall_subset_ball' (h : ε₁ + dist x y < ε₂) : closedBall x ε
     
 #align metric.closed_ball_subset_ball' Metric.closedBall_subset_ball'
 
+/- warning: metric.dist_le_add_of_nonempty_closed_ball_inter_closed_ball -> Metric.dist_le_add_of_nonempty_closedBall_inter_closedBall is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {x : α} {y : α} {ε₁ : Real} {ε₂ : Real}, (Set.Nonempty.{u1} α (Inter.inter.{u1} (Set.{u1} α) (Set.hasInter.{u1} α) (Metric.closedBall.{u1} α _inst_1 x ε₁) (Metric.closedBall.{u1} α _inst_1 y ε₂))) -> (LE.le.{0} Real Real.hasLe (Dist.dist.{u1} α (PseudoMetricSpace.toHasDist.{u1} α _inst_1) x y) (HAdd.hAdd.{0, 0, 0} Real Real Real (instHAdd.{0} Real Real.hasAdd) ε₁ ε₂))
+but is expected to have type
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {x : α} {y : α} {ε₁ : Real} {ε₂ : Real}, (Set.Nonempty.{u1} α (Inter.inter.{u1} (Set.{u1} α) (Set.instInterSet.{u1} α) (Metric.closedBall.{u1} α _inst_1 x ε₁) (Metric.closedBall.{u1} α _inst_1 y ε₂))) -> (LE.le.{0} Real Real.instLEReal (Dist.dist.{u1} α (PseudoMetricSpace.toDist.{u1} α _inst_1) x y) (HAdd.hAdd.{0, 0, 0} Real Real Real (instHAdd.{0} Real Real.instAddReal) ε₁ ε₂))
+Case conversion may be inaccurate. Consider using '#align metric.dist_le_add_of_nonempty_closed_ball_inter_closed_ball Metric.dist_le_add_of_nonempty_closedBall_inter_closedBallₓ'. -/
 theorem dist_le_add_of_nonempty_closedBall_inter_closedBall
     (h : (closedBall x ε₁ ∩ closedBall y ε₂).Nonempty) : dist x y ≤ ε₁ + ε₂ :=
   let ⟨z, hz⟩ := h
@@ -696,6 +1152,12 @@ theorem dist_le_add_of_nonempty_closedBall_inter_closedBall
     
 #align metric.dist_le_add_of_nonempty_closed_ball_inter_closed_ball Metric.dist_le_add_of_nonempty_closedBall_inter_closedBall
 
+/- warning: metric.dist_lt_add_of_nonempty_closed_ball_inter_ball -> Metric.dist_lt_add_of_nonempty_closedBall_inter_ball is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {x : α} {y : α} {ε₁ : Real} {ε₂ : Real}, (Set.Nonempty.{u1} α (Inter.inter.{u1} (Set.{u1} α) (Set.hasInter.{u1} α) (Metric.closedBall.{u1} α _inst_1 x ε₁) (Metric.ball.{u1} α _inst_1 y ε₂))) -> (LT.lt.{0} Real Real.hasLt (Dist.dist.{u1} α (PseudoMetricSpace.toHasDist.{u1} α _inst_1) x y) (HAdd.hAdd.{0, 0, 0} Real Real Real (instHAdd.{0} Real Real.hasAdd) ε₁ ε₂))
+but is expected to have type
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {x : α} {y : α} {ε₁ : Real} {ε₂ : Real}, (Set.Nonempty.{u1} α (Inter.inter.{u1} (Set.{u1} α) (Set.instInterSet.{u1} α) (Metric.closedBall.{u1} α _inst_1 x ε₁) (Metric.ball.{u1} α _inst_1 y ε₂))) -> (LT.lt.{0} Real Real.instLTReal (Dist.dist.{u1} α (PseudoMetricSpace.toDist.{u1} α _inst_1) x y) (HAdd.hAdd.{0, 0, 0} Real Real Real (instHAdd.{0} Real Real.instAddReal) ε₁ ε₂))
+Case conversion may be inaccurate. Consider using '#align metric.dist_lt_add_of_nonempty_closed_ball_inter_ball Metric.dist_lt_add_of_nonempty_closedBall_inter_ballₓ'. -/
 theorem dist_lt_add_of_nonempty_closedBall_inter_ball (h : (closedBall x ε₁ ∩ ball y ε₂).Nonempty) :
     dist x y < ε₁ + ε₂ :=
   let ⟨z, hz⟩ := h
@@ -705,6 +1167,12 @@ theorem dist_lt_add_of_nonempty_closedBall_inter_ball (h : (closedBall x ε₁ 
     
 #align metric.dist_lt_add_of_nonempty_closed_ball_inter_ball Metric.dist_lt_add_of_nonempty_closedBall_inter_ball
 
+/- warning: metric.dist_lt_add_of_nonempty_ball_inter_closed_ball -> Metric.dist_lt_add_of_nonempty_ball_inter_closedBall is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {x : α} {y : α} {ε₁ : Real} {ε₂ : Real}, (Set.Nonempty.{u1} α (Inter.inter.{u1} (Set.{u1} α) (Set.hasInter.{u1} α) (Metric.ball.{u1} α _inst_1 x ε₁) (Metric.closedBall.{u1} α _inst_1 y ε₂))) -> (LT.lt.{0} Real Real.hasLt (Dist.dist.{u1} α (PseudoMetricSpace.toHasDist.{u1} α _inst_1) x y) (HAdd.hAdd.{0, 0, 0} Real Real Real (instHAdd.{0} Real Real.hasAdd) ε₁ ε₂))
+but is expected to have type
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {x : α} {y : α} {ε₁ : Real} {ε₂ : Real}, (Set.Nonempty.{u1} α (Inter.inter.{u1} (Set.{u1} α) (Set.instInterSet.{u1} α) (Metric.ball.{u1} α _inst_1 x ε₁) (Metric.closedBall.{u1} α _inst_1 y ε₂))) -> (LT.lt.{0} Real Real.instLTReal (Dist.dist.{u1} α (PseudoMetricSpace.toDist.{u1} α _inst_1) x y) (HAdd.hAdd.{0, 0, 0} Real Real Real (instHAdd.{0} Real Real.instAddReal) ε₁ ε₂))
+Case conversion may be inaccurate. Consider using '#align metric.dist_lt_add_of_nonempty_ball_inter_closed_ball Metric.dist_lt_add_of_nonempty_ball_inter_closedBallₓ'. -/
 theorem dist_lt_add_of_nonempty_ball_inter_closedBall (h : (ball x ε₁ ∩ closedBall y ε₂).Nonempty) :
     dist x y < ε₁ + ε₂ := by
   rw [inter_comm] at h
@@ -712,34 +1180,71 @@ theorem dist_lt_add_of_nonempty_ball_inter_closedBall (h : (ball x ε₁ ∩ clo
   exact dist_lt_add_of_nonempty_closed_ball_inter_ball h
 #align metric.dist_lt_add_of_nonempty_ball_inter_closed_ball Metric.dist_lt_add_of_nonempty_ball_inter_closedBall
 
+/- warning: metric.dist_lt_add_of_nonempty_ball_inter_ball -> Metric.dist_lt_add_of_nonempty_ball_inter_ball is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {x : α} {y : α} {ε₁ : Real} {ε₂ : Real}, (Set.Nonempty.{u1} α (Inter.inter.{u1} (Set.{u1} α) (Set.hasInter.{u1} α) (Metric.ball.{u1} α _inst_1 x ε₁) (Metric.ball.{u1} α _inst_1 y ε₂))) -> (LT.lt.{0} Real Real.hasLt (Dist.dist.{u1} α (PseudoMetricSpace.toHasDist.{u1} α _inst_1) x y) (HAdd.hAdd.{0, 0, 0} Real Real Real (instHAdd.{0} Real Real.hasAdd) ε₁ ε₂))
+but is expected to have type
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {x : α} {y : α} {ε₁ : Real} {ε₂ : Real}, (Set.Nonempty.{u1} α (Inter.inter.{u1} (Set.{u1} α) (Set.instInterSet.{u1} α) (Metric.ball.{u1} α _inst_1 x ε₁) (Metric.ball.{u1} α _inst_1 y ε₂))) -> (LT.lt.{0} Real Real.instLTReal (Dist.dist.{u1} α (PseudoMetricSpace.toDist.{u1} α _inst_1) x y) (HAdd.hAdd.{0, 0, 0} Real Real Real (instHAdd.{0} Real Real.instAddReal) ε₁ ε₂))
+Case conversion may be inaccurate. Consider using '#align metric.dist_lt_add_of_nonempty_ball_inter_ball Metric.dist_lt_add_of_nonempty_ball_inter_ballₓ'. -/
 theorem dist_lt_add_of_nonempty_ball_inter_ball (h : (ball x ε₁ ∩ ball y ε₂).Nonempty) :
     dist x y < ε₁ + ε₂ :=
   dist_lt_add_of_nonempty_closedBall_inter_ball <|
     h.mono (inter_subset_inter ball_subset_closedBall Subset.rfl)
 #align metric.dist_lt_add_of_nonempty_ball_inter_ball Metric.dist_lt_add_of_nonempty_ball_inter_ball
 
+/- warning: metric.Union_closed_ball_nat -> Metric.unionᵢ_closedBall_nat is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] (x : α), Eq.{succ u1} (Set.{u1} α) (Set.unionᵢ.{u1, 1} α Nat (fun (n : Nat) => Metric.closedBall.{u1} α _inst_1 x ((fun (a : Type) (b : Type) [self : HasLiftT.{1, 1} a b] => self.0) Nat Real (HasLiftT.mk.{1, 1} Nat Real (CoeTCₓ.coe.{1, 1} Nat Real (Nat.castCoe.{0} Real Real.hasNatCast))) n))) (Set.univ.{u1} α)
+but is expected to have type
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] (x : α), Eq.{succ u1} (Set.{u1} α) (Set.unionᵢ.{u1, 1} α Nat (fun (n : Nat) => Metric.closedBall.{u1} α _inst_1 x (Nat.cast.{0} Real Real.natCast n))) (Set.univ.{u1} α)
+Case conversion may be inaccurate. Consider using '#align metric.Union_closed_ball_nat Metric.unionᵢ_closedBall_natₓ'. -/
 @[simp]
 theorem unionᵢ_closedBall_nat (x : α) : (⋃ n : ℕ, closedBall x n) = univ :=
   unionᵢ_eq_univ_iff.2 fun y => exists_nat_ge (dist y x)
 #align metric.Union_closed_ball_nat Metric.unionᵢ_closedBall_nat
 
+/- warning: metric.Union_inter_closed_ball_nat -> Metric.unionᵢ_inter_closedBall_nat is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] (s : Set.{u1} α) (x : α), Eq.{succ u1} (Set.{u1} α) (Set.unionᵢ.{u1, 1} α Nat (fun (n : Nat) => Inter.inter.{u1} (Set.{u1} α) (Set.hasInter.{u1} α) s (Metric.closedBall.{u1} α _inst_1 x ((fun (a : Type) (b : Type) [self : HasLiftT.{1, 1} a b] => self.0) Nat Real (HasLiftT.mk.{1, 1} Nat Real (CoeTCₓ.coe.{1, 1} Nat Real (Nat.castCoe.{0} Real Real.hasNatCast))) n)))) s
+but is expected to have type
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] (s : Set.{u1} α) (x : α), Eq.{succ u1} (Set.{u1} α) (Set.unionᵢ.{u1, 1} α Nat (fun (n : Nat) => Inter.inter.{u1} (Set.{u1} α) (Set.instInterSet.{u1} α) s (Metric.closedBall.{u1} α _inst_1 x (Nat.cast.{0} Real Real.natCast n)))) s
+Case conversion may be inaccurate. Consider using '#align metric.Union_inter_closed_ball_nat Metric.unionᵢ_inter_closedBall_natₓ'. -/
 theorem unionᵢ_inter_closedBall_nat (s : Set α) (x : α) : (⋃ n : ℕ, s ∩ closedBall x n) = s := by
   rw [← inter_Union, Union_closed_ball_nat, inter_univ]
 #align metric.Union_inter_closed_ball_nat Metric.unionᵢ_inter_closedBall_nat
 
+/- warning: metric.ball_subset -> Metric.ball_subset is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {x : α} {y : α} {ε₁ : Real} {ε₂ : Real}, (LE.le.{0} Real Real.hasLe (Dist.dist.{u1} α (PseudoMetricSpace.toHasDist.{u1} α _inst_1) x y) (HSub.hSub.{0, 0, 0} Real Real Real (instHSub.{0} Real Real.hasSub) ε₂ ε₁)) -> (HasSubset.Subset.{u1} (Set.{u1} α) (Set.hasSubset.{u1} α) (Metric.ball.{u1} α _inst_1 x ε₁) (Metric.ball.{u1} α _inst_1 y ε₂))
+but is expected to have type
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {x : α} {y : α} {ε₁ : Real} {ε₂ : Real}, (LE.le.{0} Real Real.instLEReal (Dist.dist.{u1} α (PseudoMetricSpace.toDist.{u1} α _inst_1) x y) (HSub.hSub.{0, 0, 0} Real Real Real (instHSub.{0} Real Real.instSubReal) ε₂ ε₁)) -> (HasSubset.Subset.{u1} (Set.{u1} α) (Set.instHasSubsetSet.{u1} α) (Metric.ball.{u1} α _inst_1 x ε₁) (Metric.ball.{u1} α _inst_1 y ε₂))
+Case conversion may be inaccurate. Consider using '#align metric.ball_subset Metric.ball_subsetₓ'. -/
 theorem ball_subset (h : dist x y ≤ ε₂ - ε₁) : ball x ε₁ ⊆ ball y ε₂ := fun z zx => by
   rw [← add_sub_cancel'_right ε₁ ε₂] <;>
     exact lt_of_le_of_lt (dist_triangle z x y) (add_lt_add_of_lt_of_le zx h)
 #align metric.ball_subset Metric.ball_subset
 
+/- warning: metric.ball_half_subset -> Metric.ball_half_subset is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {x : α} {ε : Real} (y : α), (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) y (Metric.ball.{u1} α _inst_1 x (HDiv.hDiv.{0, 0, 0} Real Real Real (instHDiv.{0} Real (DivInvMonoid.toHasDiv.{0} Real (DivisionRing.toDivInvMonoid.{0} Real Real.divisionRing))) ε (OfNat.ofNat.{0} Real 2 (OfNat.mk.{0} Real 2 (bit0.{0} Real Real.hasAdd (One.one.{0} Real Real.hasOne))))))) -> (HasSubset.Subset.{u1} (Set.{u1} α) (Set.hasSubset.{u1} α) (Metric.ball.{u1} α _inst_1 y (HDiv.hDiv.{0, 0, 0} Real Real Real (instHDiv.{0} Real (DivInvMonoid.toHasDiv.{0} Real (DivisionRing.toDivInvMonoid.{0} Real Real.divisionRing))) ε (OfNat.ofNat.{0} Real 2 (OfNat.mk.{0} Real 2 (bit0.{0} Real Real.hasAdd (One.one.{0} Real Real.hasOne)))))) (Metric.ball.{u1} α _inst_1 x ε))
+but is expected to have type
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {x : α} {ε : Real} (y : α), (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) y (Metric.ball.{u1} α _inst_1 x (HDiv.hDiv.{0, 0, 0} Real Real Real (instHDiv.{0} Real (LinearOrderedField.toDiv.{0} Real Real.instLinearOrderedFieldReal)) ε (OfNat.ofNat.{0} Real 2 (instOfNat.{0} Real 2 Real.natCast (instAtLeastTwoHAddNatInstHAddInstAddNatOfNat (OfNat.ofNat.{0} Nat 0 (instOfNatNat 0)))))))) -> (HasSubset.Subset.{u1} (Set.{u1} α) (Set.instHasSubsetSet.{u1} α) (Metric.ball.{u1} α _inst_1 y (HDiv.hDiv.{0, 0, 0} Real Real Real (instHDiv.{0} Real (LinearOrderedField.toDiv.{0} Real Real.instLinearOrderedFieldReal)) ε (OfNat.ofNat.{0} Real 2 (instOfNat.{0} Real 2 Real.natCast (instAtLeastTwoHAddNatInstHAddInstAddNatOfNat (OfNat.ofNat.{0} Nat 0 (instOfNatNat 0))))))) (Metric.ball.{u1} α _inst_1 x ε))
+Case conversion may be inaccurate. Consider using '#align metric.ball_half_subset Metric.ball_half_subsetₓ'. -/
 theorem ball_half_subset (y) (h : y ∈ ball x (ε / 2)) : ball y (ε / 2) ⊆ ball x ε :=
   ball_subset <| by rw [sub_self_div_two] <;> exact le_of_lt h
 #align metric.ball_half_subset Metric.ball_half_subset
 
+/- warning: metric.exists_ball_subset_ball -> Metric.exists_ball_subset_ball is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {x : α} {y : α} {ε : Real}, (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) y (Metric.ball.{u1} α _inst_1 x ε)) -> (Exists.{1} Real (fun (ε' : Real) => Exists.{0} (GT.gt.{0} Real Real.hasLt ε' (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero)))) (fun (H : GT.gt.{0} Real Real.hasLt ε' (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero)))) => HasSubset.Subset.{u1} (Set.{u1} α) (Set.hasSubset.{u1} α) (Metric.ball.{u1} α _inst_1 y ε') (Metric.ball.{u1} α _inst_1 x ε))))
+but is expected to have type
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {x : α} {y : α} {ε : Real}, (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) y (Metric.ball.{u1} α _inst_1 x ε)) -> (Exists.{1} Real (fun (ε' : Real) => And (GT.gt.{0} Real Real.instLTReal ε' (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal))) (HasSubset.Subset.{u1} (Set.{u1} α) (Set.instHasSubsetSet.{u1} α) (Metric.ball.{u1} α _inst_1 y ε') (Metric.ball.{u1} α _inst_1 x ε))))
+Case conversion may be inaccurate. Consider using '#align metric.exists_ball_subset_ball Metric.exists_ball_subset_ballₓ'. -/
 theorem exists_ball_subset_ball (h : y ∈ ball x ε) : ∃ ε' > 0, ball y ε' ⊆ ball x ε :=
   ⟨_, sub_pos.2 h, ball_subset <| by rw [sub_sub_self]⟩
 #align metric.exists_ball_subset_ball Metric.exists_ball_subset_ball
 
+#print Metric.forall_of_forall_mem_closedBall /-
 /-- If a property holds for all points in closed balls of arbitrarily large radii, then it holds for
 all points. -/
 theorem forall_of_forall_mem_closedBall (p : α → Prop) (x : α)
@@ -749,7 +1254,9 @@ theorem forall_of_forall_mem_closedBall (p : α → Prop) (x : α)
     frequently_iff.1 H (Ici_mem_at_top (dist y x))
   exact h _ hR
 #align metric.forall_of_forall_mem_closed_ball Metric.forall_of_forall_mem_closedBall
+-/
 
+#print Metric.forall_of_forall_mem_ball /-
 /-- If a property holds for all points in balls of arbitrarily large radii, then it holds for all
 points. -/
 theorem forall_of_forall_mem_ball (p : α → Prop) (x : α)
@@ -759,13 +1266,26 @@ theorem forall_of_forall_mem_ball (p : α → Prop) (x : α)
     frequently_iff.1 H (Ioi_mem_at_top (dist y x))
   exact h _ hR
 #align metric.forall_of_forall_mem_ball Metric.forall_of_forall_mem_ball
+-/
 
+/- warning: metric.is_bounded_iff -> Metric.isBounded_iff is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {s : Set.{u1} α}, Iff (Bornology.IsBounded.{u1} α (PseudoMetricSpace.toBornology.{u1} α _inst_1) s) (Exists.{1} Real (fun (C : Real) => forall {{x : α}}, (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) x s) -> (forall {{y : α}}, (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) y s) -> (LE.le.{0} Real Real.hasLe (Dist.dist.{u1} α (PseudoMetricSpace.toHasDist.{u1} α _inst_1) x y) C))))
+but is expected to have type
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {s : Set.{u1} α}, Iff (Bornology.IsBounded.{u1} α (PseudoMetricSpace.toBornology.{u1} α _inst_1) s) (Exists.{1} Real (fun (C : Real) => forall {{x : α}}, (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) x s) -> (forall {{y : α}}, (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) y s) -> (LE.le.{0} Real Real.instLEReal (Dist.dist.{u1} α (PseudoMetricSpace.toDist.{u1} α _inst_1) x y) C))))
+Case conversion may be inaccurate. Consider using '#align metric.is_bounded_iff Metric.isBounded_iffₓ'. -/
 theorem isBounded_iff {s : Set α} :
     IsBounded s ↔ ∃ C : ℝ, ∀ ⦃x⦄, x ∈ s → ∀ ⦃y⦄, y ∈ s → dist x y ≤ C := by
   rw [is_bounded_def, ← Filter.mem_sets, (@PseudoMetricSpace.cobounded_sets α _).out, mem_set_of_eq,
     compl_compl]
 #align metric.is_bounded_iff Metric.isBounded_iff
 
+/- warning: metric.is_bounded_iff_eventually -> Metric.isBounded_iff_eventually is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {s : Set.{u1} α}, Iff (Bornology.IsBounded.{u1} α (PseudoMetricSpace.toBornology.{u1} α _inst_1) s) (Filter.Eventually.{0} Real (fun (C : Real) => forall {{x : α}}, (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) x s) -> (forall {{y : α}}, (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) y s) -> (LE.le.{0} Real Real.hasLe (Dist.dist.{u1} α (PseudoMetricSpace.toHasDist.{u1} α _inst_1) x y) C))) (Filter.atTop.{0} Real Real.preorder))
+but is expected to have type
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {s : Set.{u1} α}, Iff (Bornology.IsBounded.{u1} α (PseudoMetricSpace.toBornology.{u1} α _inst_1) s) (Filter.Eventually.{0} Real (fun (C : Real) => forall {{x : α}}, (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) x s) -> (forall {{y : α}}, (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) y s) -> (LE.le.{0} Real Real.instLEReal (Dist.dist.{u1} α (PseudoMetricSpace.toDist.{u1} α _inst_1) x y) C))) (Filter.atTop.{0} Real Real.instPreorderReal))
+Case conversion may be inaccurate. Consider using '#align metric.is_bounded_iff_eventually Metric.isBounded_iff_eventuallyₓ'. -/
 theorem isBounded_iff_eventually {s : Set α} :
     IsBounded s ↔ ∀ᶠ C in atTop, ∀ ⦃x⦄, x ∈ s → ∀ ⦃y⦄, y ∈ s → dist x y ≤ C :=
   isBounded_iff.trans
@@ -773,24 +1293,40 @@ theorem isBounded_iff_eventually {s : Set α} :
       Eventually.exists⟩
 #align metric.is_bounded_iff_eventually Metric.isBounded_iff_eventually
 
+/- warning: metric.is_bounded_iff_exists_ge -> Metric.isBounded_iff_exists_ge is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {s : Set.{u1} α} (c : Real), Iff (Bornology.IsBounded.{u1} α (PseudoMetricSpace.toBornology.{u1} α _inst_1) s) (Exists.{1} Real (fun (C : Real) => And (LE.le.{0} Real Real.hasLe c C) (forall {{x : α}}, (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) x s) -> (forall {{y : α}}, (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) y s) -> (LE.le.{0} Real Real.hasLe (Dist.dist.{u1} α (PseudoMetricSpace.toHasDist.{u1} α _inst_1) x y) C)))))
+but is expected to have type
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {s : Set.{u1} α} (c : Real), Iff (Bornology.IsBounded.{u1} α (PseudoMetricSpace.toBornology.{u1} α _inst_1) s) (Exists.{1} Real (fun (C : Real) => And (LE.le.{0} Real Real.instLEReal c C) (forall {{x : α}}, (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) x s) -> (forall {{y : α}}, (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) y s) -> (LE.le.{0} Real Real.instLEReal (Dist.dist.{u1} α (PseudoMetricSpace.toDist.{u1} α _inst_1) x y) C)))))
+Case conversion may be inaccurate. Consider using '#align metric.is_bounded_iff_exists_ge Metric.isBounded_iff_exists_geₓ'. -/
 theorem isBounded_iff_exists_ge {s : Set α} (c : ℝ) :
     IsBounded s ↔ ∃ C, c ≤ C ∧ ∀ ⦃x⦄, x ∈ s → ∀ ⦃y⦄, y ∈ s → dist x y ≤ C :=
   ⟨fun h => ((eventually_ge_atTop c).And (isBounded_iff_eventually.1 h)).exists, fun h =>
     isBounded_iff.2 <| h.imp fun _ => And.right⟩
 #align metric.is_bounded_iff_exists_ge Metric.isBounded_iff_exists_ge
 
+#print Metric.isBounded_iff_nndist /-
 theorem isBounded_iff_nndist {s : Set α} :
     IsBounded s ↔ ∃ C : ℝ≥0, ∀ ⦃x⦄, x ∈ s → ∀ ⦃y⦄, y ∈ s → nndist x y ≤ C := by
   simp only [is_bounded_iff_exists_ge 0, NNReal.exists, ← NNReal.coe_le_coe, ← dist_nndist,
     NNReal.coe_mk, exists_prop]
 #align metric.is_bounded_iff_nndist Metric.isBounded_iff_nndist
+-/
 
+#print Metric.toUniformSpace_eq /-
 theorem toUniformSpace_eq :
     ‹PseudoMetricSpace α›.toUniformSpace =
-      uniformSpaceOfDist dist dist_self dist_comm dist_triangle :=
+      UniformSpace.ofDist dist dist_self dist_comm dist_triangle :=
   uniformSpace_eq PseudoMetricSpace.uniformity_dist
 #align metric.to_uniform_space_eq Metric.toUniformSpace_eq
+-/
 
+/- warning: metric.uniformity_basis_dist -> Metric.uniformity_basis_dist is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α], Filter.HasBasis.{u1, 1} (Prod.{u1, u1} α α) Real (uniformity.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) (fun (ε : Real) => LT.lt.{0} Real Real.hasLt (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero))) ε) (fun (ε : Real) => setOf.{u1} (Prod.{u1, u1} α α) (fun (p : Prod.{u1, u1} α α) => LT.lt.{0} Real Real.hasLt (Dist.dist.{u1} α (PseudoMetricSpace.toHasDist.{u1} α _inst_1) (Prod.fst.{u1, u1} α α p) (Prod.snd.{u1, u1} α α p)) ε))
+but is expected to have type
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α], Filter.HasBasis.{u1, 1} (Prod.{u1, u1} α α) Real (uniformity.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) (fun (ε : Real) => LT.lt.{0} Real Real.instLTReal (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal)) ε) (fun (ε : Real) => setOf.{u1} (Prod.{u1, u1} α α) (fun (p : Prod.{u1, u1} α α) => LT.lt.{0} Real Real.instLTReal (Dist.dist.{u1} α (PseudoMetricSpace.toDist.{u1} α _inst_1) (Prod.fst.{u1, u1} α α p) (Prod.snd.{u1, u1} α α p)) ε))
+Case conversion may be inaccurate. Consider using '#align metric.uniformity_basis_dist Metric.uniformity_basis_distₓ'. -/
 theorem uniformity_basis_dist :
     (𝓤 α).HasBasis (fun ε : ℝ => 0 < ε) fun ε => { p : α × α | dist p.1 p.2 < ε } :=
   by
@@ -798,6 +1334,12 @@ theorem uniformity_basis_dist :
   exact UniformSpace.hasBasis_ofFun (exists_gt _) _ _ _ _ _
 #align metric.uniformity_basis_dist Metric.uniformity_basis_dist
 
+/- warning: metric.mk_uniformity_basis -> Metric.mk_uniformity_basis is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {β : Type.{u2}} {p : β -> Prop} {f : β -> Real}, (forall (i : β), (p i) -> (LT.lt.{0} Real Real.hasLt (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero))) (f i))) -> (forall {{ε : Real}}, (LT.lt.{0} Real Real.hasLt (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero))) ε) -> (Exists.{succ u2} β (fun (i : β) => Exists.{0} (p i) (fun (hi : p i) => LE.le.{0} Real Real.hasLe (f i) ε)))) -> (Filter.HasBasis.{u1, succ u2} (Prod.{u1, u1} α α) β (uniformity.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) p (fun (i : β) => setOf.{u1} (Prod.{u1, u1} α α) (fun (p : Prod.{u1, u1} α α) => LT.lt.{0} Real Real.hasLt (Dist.dist.{u1} α (PseudoMetricSpace.toHasDist.{u1} α _inst_1) (Prod.fst.{u1, u1} α α p) (Prod.snd.{u1, u1} α α p)) (f i))))
+but is expected to have type
+  forall {α : Type.{u2}} [_inst_1 : PseudoMetricSpace.{u2} α] {β : Type.{u1}} {p : β -> Prop} {f : β -> Real}, (forall (i : β), (p i) -> (LT.lt.{0} Real Real.instLTReal (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal)) (f i))) -> (forall {{ε : Real}}, (LT.lt.{0} Real Real.instLTReal (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal)) ε) -> (Exists.{succ u1} β (fun (i : β) => And (p i) (LE.le.{0} Real Real.instLEReal (f i) ε)))) -> (Filter.HasBasis.{u2, succ u1} (Prod.{u2, u2} α α) β (uniformity.{u2} α (PseudoMetricSpace.toUniformSpace.{u2} α _inst_1)) p (fun (i : β) => setOf.{u2} (Prod.{u2, u2} α α) (fun (p : Prod.{u2, u2} α α) => LT.lt.{0} Real Real.instLTReal (Dist.dist.{u2} α (PseudoMetricSpace.toDist.{u2} α _inst_1) (Prod.fst.{u2, u2} α α p) (Prod.snd.{u2, u2} α α p)) (f i))))
+Case conversion may be inaccurate. Consider using '#align metric.mk_uniformity_basis Metric.mk_uniformity_basisₓ'. -/
 /-- Given `f : β → ℝ`, if `f` sends `{i | p i}` to a set of positive numbers
 accumulating to zero, then `f i`-neighborhoods of the diagonal form a basis of `𝓤 α`.
 
@@ -816,6 +1358,12 @@ protected theorem mk_uniformity_basis {β : Type _} {p : β → Prop} {f : β 
   · exact fun ⟨i, hi, H⟩ => ⟨f i, hf₀ i hi, H⟩
 #align metric.mk_uniformity_basis Metric.mk_uniformity_basis
 
+/- warning: metric.uniformity_basis_dist_rat -> Metric.uniformity_basis_dist_rat is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α], Filter.HasBasis.{u1, 1} (Prod.{u1, u1} α α) Rat (uniformity.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) (fun (r : Rat) => LT.lt.{0} Rat Rat.hasLt (OfNat.ofNat.{0} Rat 0 (OfNat.mk.{0} Rat 0 (Zero.zero.{0} Rat Rat.hasZero))) r) (fun (r : Rat) => setOf.{u1} (Prod.{u1, u1} α α) (fun (p : Prod.{u1, u1} α α) => LT.lt.{0} Real Real.hasLt (Dist.dist.{u1} α (PseudoMetricSpace.toHasDist.{u1} α _inst_1) (Prod.fst.{u1, u1} α α p) (Prod.snd.{u1, u1} α α p)) ((fun (a : Type) (b : Type) [self : HasLiftT.{1, 1} a b] => self.0) Rat Real (HasLiftT.mk.{1, 1} Rat Real (CoeTCₓ.coe.{1, 1} Rat Real (Rat.castCoe.{0} Real Real.hasRatCast))) r)))
+but is expected to have type
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α], Filter.HasBasis.{u1, 1} (Prod.{u1, u1} α α) Rat (uniformity.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) (fun (r : Rat) => LT.lt.{0} Rat Rat.instLTRat_1 (OfNat.ofNat.{0} Rat 0 (Rat.instOfNatRat 0)) r) (fun (r : Rat) => setOf.{u1} (Prod.{u1, u1} α α) (fun (p : Prod.{u1, u1} α α) => LT.lt.{0} Real Real.instLTReal (Dist.dist.{u1} α (PseudoMetricSpace.toDist.{u1} α _inst_1) (Prod.fst.{u1, u1} α α p) (Prod.snd.{u1, u1} α α p)) (RatCast.ratCast.{0} Real Real.ratCast r)))
+Case conversion may be inaccurate. Consider using '#align metric.uniformity_basis_dist_rat Metric.uniformity_basis_dist_ratₓ'. -/
 theorem uniformity_basis_dist_rat :
     (𝓤 α).HasBasis (fun r : ℚ => 0 < r) fun r => { p : α × α | dist p.1 p.2 < r } :=
   Metric.mk_uniformity_basis (fun _ => Rat.cast_pos.2) fun ε hε =>
@@ -823,12 +1371,24 @@ theorem uniformity_basis_dist_rat :
     ⟨r, Rat.cast_pos.1 hr0, hrε.le⟩
 #align metric.uniformity_basis_dist_rat Metric.uniformity_basis_dist_rat
 
+/- warning: metric.uniformity_basis_dist_inv_nat_succ -> Metric.uniformity_basis_dist_inv_nat_succ is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α], Filter.HasBasis.{u1, 1} (Prod.{u1, u1} α α) Nat (uniformity.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) (fun (_x : Nat) => True) (fun (n : Nat) => setOf.{u1} (Prod.{u1, u1} α α) (fun (p : Prod.{u1, u1} α α) => LT.lt.{0} Real Real.hasLt (Dist.dist.{u1} α (PseudoMetricSpace.toHasDist.{u1} α _inst_1) (Prod.fst.{u1, u1} α α p) (Prod.snd.{u1, u1} α α p)) (HDiv.hDiv.{0, 0, 0} Real Real Real (instHDiv.{0} Real (DivInvMonoid.toHasDiv.{0} Real (DivisionRing.toDivInvMonoid.{0} Real Real.divisionRing))) (OfNat.ofNat.{0} Real 1 (OfNat.mk.{0} Real 1 (One.one.{0} Real Real.hasOne))) (HAdd.hAdd.{0, 0, 0} Real Real Real (instHAdd.{0} Real Real.hasAdd) ((fun (a : Type) (b : Type) [self : HasLiftT.{1, 1} a b] => self.0) Nat Real (HasLiftT.mk.{1, 1} Nat Real (CoeTCₓ.coe.{1, 1} Nat Real (Nat.castCoe.{0} Real Real.hasNatCast))) n) (OfNat.ofNat.{0} Real 1 (OfNat.mk.{0} Real 1 (One.one.{0} Real Real.hasOne)))))))
+but is expected to have type
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α], Filter.HasBasis.{u1, 1} (Prod.{u1, u1} α α) Nat (uniformity.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) (fun (_x : Nat) => True) (fun (n : Nat) => setOf.{u1} (Prod.{u1, u1} α α) (fun (p : Prod.{u1, u1} α α) => LT.lt.{0} Real Real.instLTReal (Dist.dist.{u1} α (PseudoMetricSpace.toDist.{u1} α _inst_1) (Prod.fst.{u1, u1} α α p) (Prod.snd.{u1, u1} α α p)) (HDiv.hDiv.{0, 0, 0} Real Real Real (instHDiv.{0} Real (LinearOrderedField.toDiv.{0} Real Real.instLinearOrderedFieldReal)) (OfNat.ofNat.{0} Real 1 (One.toOfNat1.{0} Real Real.instOneReal)) (HAdd.hAdd.{0, 0, 0} Real Real Real (instHAdd.{0} Real Real.instAddReal) (Nat.cast.{0} Real Real.natCast n) (OfNat.ofNat.{0} Real 1 (One.toOfNat1.{0} Real Real.instOneReal))))))
+Case conversion may be inaccurate. Consider using '#align metric.uniformity_basis_dist_inv_nat_succ Metric.uniformity_basis_dist_inv_nat_succₓ'. -/
 theorem uniformity_basis_dist_inv_nat_succ :
     (𝓤 α).HasBasis (fun _ => True) fun n : ℕ => { p : α × α | dist p.1 p.2 < 1 / (↑n + 1) } :=
   Metric.mk_uniformity_basis (fun n _ => div_pos zero_lt_one <| Nat.cast_add_one_pos n) fun ε ε0 =>
     (exists_nat_one_div_lt ε0).imp fun n hn => ⟨trivial, le_of_lt hn⟩
 #align metric.uniformity_basis_dist_inv_nat_succ Metric.uniformity_basis_dist_inv_nat_succ
 
+/- warning: metric.uniformity_basis_dist_inv_nat_pos -> Metric.uniformity_basis_dist_inv_nat_pos is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α], Filter.HasBasis.{u1, 1} (Prod.{u1, u1} α α) Nat (uniformity.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) (fun (n : Nat) => LT.lt.{0} Nat Nat.hasLt (OfNat.ofNat.{0} Nat 0 (OfNat.mk.{0} Nat 0 (Zero.zero.{0} Nat Nat.hasZero))) n) (fun (n : Nat) => setOf.{u1} (Prod.{u1, u1} α α) (fun (p : Prod.{u1, u1} α α) => LT.lt.{0} Real Real.hasLt (Dist.dist.{u1} α (PseudoMetricSpace.toHasDist.{u1} α _inst_1) (Prod.fst.{u1, u1} α α p) (Prod.snd.{u1, u1} α α p)) (HDiv.hDiv.{0, 0, 0} Real Real Real (instHDiv.{0} Real (DivInvMonoid.toHasDiv.{0} Real (DivisionRing.toDivInvMonoid.{0} Real Real.divisionRing))) (OfNat.ofNat.{0} Real 1 (OfNat.mk.{0} Real 1 (One.one.{0} Real Real.hasOne))) ((fun (a : Type) (b : Type) [self : HasLiftT.{1, 1} a b] => self.0) Nat Real (HasLiftT.mk.{1, 1} Nat Real (CoeTCₓ.coe.{1, 1} Nat Real (Nat.castCoe.{0} Real Real.hasNatCast))) n))))
+but is expected to have type
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α], Filter.HasBasis.{u1, 1} (Prod.{u1, u1} α α) Nat (uniformity.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) (fun (n : Nat) => LT.lt.{0} Nat instLTNat (OfNat.ofNat.{0} Nat 0 (instOfNatNat 0)) n) (fun (n : Nat) => setOf.{u1} (Prod.{u1, u1} α α) (fun (p : Prod.{u1, u1} α α) => LT.lt.{0} Real Real.instLTReal (Dist.dist.{u1} α (PseudoMetricSpace.toDist.{u1} α _inst_1) (Prod.fst.{u1, u1} α α p) (Prod.snd.{u1, u1} α α p)) (HDiv.hDiv.{0, 0, 0} Real Real Real (instHDiv.{0} Real (LinearOrderedField.toDiv.{0} Real Real.instLinearOrderedFieldReal)) (OfNat.ofNat.{0} Real 1 (One.toOfNat1.{0} Real Real.instOneReal)) (Nat.cast.{0} Real Real.natCast n))))
+Case conversion may be inaccurate. Consider using '#align metric.uniformity_basis_dist_inv_nat_pos Metric.uniformity_basis_dist_inv_nat_posₓ'. -/
 theorem uniformity_basis_dist_inv_nat_pos :
     (𝓤 α).HasBasis (fun n : ℕ => 0 < n) fun n : ℕ => { p : α × α | dist p.1 p.2 < 1 / ↑n } :=
   Metric.mk_uniformity_basis (fun n hn => div_pos zero_lt_one <| Nat.cast_pos.2 hn) fun ε ε0 =>
@@ -836,6 +1396,12 @@ theorem uniformity_basis_dist_inv_nat_pos :
     ⟨n + 1, Nat.succ_pos n, by exact_mod_cast hn.le⟩
 #align metric.uniformity_basis_dist_inv_nat_pos Metric.uniformity_basis_dist_inv_nat_pos
 
+/- warning: metric.uniformity_basis_dist_pow -> Metric.uniformity_basis_dist_pow is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {r : Real}, (LT.lt.{0} Real Real.hasLt (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero))) r) -> (LT.lt.{0} Real Real.hasLt r (OfNat.ofNat.{0} Real 1 (OfNat.mk.{0} Real 1 (One.one.{0} Real Real.hasOne)))) -> (Filter.HasBasis.{u1, 1} (Prod.{u1, u1} α α) Nat (uniformity.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) (fun (n : Nat) => True) (fun (n : Nat) => setOf.{u1} (Prod.{u1, u1} α α) (fun (p : Prod.{u1, u1} α α) => LT.lt.{0} Real Real.hasLt (Dist.dist.{u1} α (PseudoMetricSpace.toHasDist.{u1} α _inst_1) (Prod.fst.{u1, u1} α α p) (Prod.snd.{u1, u1} α α p)) (HPow.hPow.{0, 0, 0} Real Nat Real (instHPow.{0, 0} Real Nat (Monoid.Pow.{0} Real Real.monoid)) r n))))
+but is expected to have type
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {r : Real}, (LT.lt.{0} Real Real.instLTReal (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal)) r) -> (LT.lt.{0} Real Real.instLTReal r (OfNat.ofNat.{0} Real 1 (One.toOfNat1.{0} Real Real.instOneReal))) -> (Filter.HasBasis.{u1, 1} (Prod.{u1, u1} α α) Nat (uniformity.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) (fun (n : Nat) => True) (fun (n : Nat) => setOf.{u1} (Prod.{u1, u1} α α) (fun (p : Prod.{u1, u1} α α) => LT.lt.{0} Real Real.instLTReal (Dist.dist.{u1} α (PseudoMetricSpace.toDist.{u1} α _inst_1) (Prod.fst.{u1, u1} α α p) (Prod.snd.{u1, u1} α α p)) (HPow.hPow.{0, 0, 0} Real Nat Real (instHPow.{0, 0} Real Nat (Monoid.Pow.{0} Real Real.instMonoidReal)) r n))))
+Case conversion may be inaccurate. Consider using '#align metric.uniformity_basis_dist_pow Metric.uniformity_basis_dist_powₓ'. -/
 theorem uniformity_basis_dist_pow {r : ℝ} (h0 : 0 < r) (h1 : r < 1) :
     (𝓤 α).HasBasis (fun n : ℕ => True) fun n : ℕ => { p : α × α | dist p.1 p.2 < r ^ n } :=
   Metric.mk_uniformity_basis (fun n hn => pow_pos h0 _) fun ε ε0 =>
@@ -843,6 +1409,12 @@ theorem uniformity_basis_dist_pow {r : ℝ} (h0 : 0 < r) (h1 : r < 1) :
     ⟨n, trivial, hn.le⟩
 #align metric.uniformity_basis_dist_pow Metric.uniformity_basis_dist_pow
 
+/- warning: metric.uniformity_basis_dist_lt -> Metric.uniformity_basis_dist_lt is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {R : Real}, (LT.lt.{0} Real Real.hasLt (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero))) R) -> (Filter.HasBasis.{u1, 1} (Prod.{u1, u1} α α) Real (uniformity.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) (fun (r : Real) => And (LT.lt.{0} Real Real.hasLt (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero))) r) (LT.lt.{0} Real Real.hasLt r R)) (fun (r : Real) => setOf.{u1} (Prod.{u1, u1} α α) (fun (p : Prod.{u1, u1} α α) => LT.lt.{0} Real Real.hasLt (Dist.dist.{u1} α (PseudoMetricSpace.toHasDist.{u1} α _inst_1) (Prod.fst.{u1, u1} α α p) (Prod.snd.{u1, u1} α α p)) r)))
+but is expected to have type
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {R : Real}, (LT.lt.{0} Real Real.instLTReal (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal)) R) -> (Filter.HasBasis.{u1, 1} (Prod.{u1, u1} α α) Real (uniformity.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) (fun (r : Real) => And (LT.lt.{0} Real Real.instLTReal (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal)) r) (LT.lt.{0} Real Real.instLTReal r R)) (fun (r : Real) => setOf.{u1} (Prod.{u1, u1} α α) (fun (p : Prod.{u1, u1} α α) => LT.lt.{0} Real Real.instLTReal (Dist.dist.{u1} α (PseudoMetricSpace.toDist.{u1} α _inst_1) (Prod.fst.{u1, u1} α α p) (Prod.snd.{u1, u1} α α p)) r)))
+Case conversion may be inaccurate. Consider using '#align metric.uniformity_basis_dist_lt Metric.uniformity_basis_dist_ltₓ'. -/
 theorem uniformity_basis_dist_lt {R : ℝ} (hR : 0 < R) :
     (𝓤 α).HasBasis (fun r : ℝ => 0 < r ∧ r < R) fun r => { p : α × α | dist p.1 p.2 < r } :=
   Metric.mk_uniformity_basis (fun r => And.left) fun r hr =>
@@ -850,6 +1422,12 @@ theorem uniformity_basis_dist_lt {R : ℝ} (hR : 0 < R) :
       min_le_left _ _⟩
 #align metric.uniformity_basis_dist_lt Metric.uniformity_basis_dist_lt
 
+/- warning: metric.mk_uniformity_basis_le -> Metric.mk_uniformity_basis_le is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {β : Type.{u2}} {p : β -> Prop} {f : β -> Real}, (forall (x : β), (p x) -> (LT.lt.{0} Real Real.hasLt (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero))) (f x))) -> (forall (ε : Real), (LT.lt.{0} Real Real.hasLt (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero))) ε) -> (Exists.{succ u2} β (fun (x : β) => Exists.{0} (p x) (fun (hx : p x) => LE.le.{0} Real Real.hasLe (f x) ε)))) -> (Filter.HasBasis.{u1, succ u2} (Prod.{u1, u1} α α) β (uniformity.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) p (fun (x : β) => setOf.{u1} (Prod.{u1, u1} α α) (fun (p : Prod.{u1, u1} α α) => LE.le.{0} Real Real.hasLe (Dist.dist.{u1} α (PseudoMetricSpace.toHasDist.{u1} α _inst_1) (Prod.fst.{u1, u1} α α p) (Prod.snd.{u1, u1} α α p)) (f x))))
+but is expected to have type
+  forall {α : Type.{u2}} [_inst_1 : PseudoMetricSpace.{u2} α] {β : Type.{u1}} {p : β -> Prop} {f : β -> Real}, (forall (x : β), (p x) -> (LT.lt.{0} Real Real.instLTReal (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal)) (f x))) -> (forall (ε : Real), (LT.lt.{0} Real Real.instLTReal (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal)) ε) -> (Exists.{succ u1} β (fun (x : β) => And (p x) (LE.le.{0} Real Real.instLEReal (f x) ε)))) -> (Filter.HasBasis.{u2, succ u1} (Prod.{u2, u2} α α) β (uniformity.{u2} α (PseudoMetricSpace.toUniformSpace.{u2} α _inst_1)) p (fun (x : β) => setOf.{u2} (Prod.{u2, u2} α α) (fun (p : Prod.{u2, u2} α α) => LE.le.{0} Real Real.instLEReal (Dist.dist.{u2} α (PseudoMetricSpace.toDist.{u2} α _inst_1) (Prod.fst.{u2, u2} α α p) (Prod.snd.{u2, u2} α α p)) (f x))))
+Case conversion may be inaccurate. Consider using '#align metric.mk_uniformity_basis_le Metric.mk_uniformity_basis_leₓ'. -/
 /-- Given `f : β → ℝ`, if `f` sends `{i | p i}` to a set of positive numbers
 accumulating to zero, then closed neighborhoods of the diagonal of sizes `{f i | p i}`
 form a basis of `𝓤 α`.
@@ -869,6 +1447,12 @@ protected theorem mk_uniformity_basis_le {β : Type _} {p : β → Prop} {f : β
   · exact fun ⟨i, hi, H⟩ => ⟨f i, hf₀ i hi, fun x (hx : _ < _) => H (le_of_lt hx)⟩
 #align metric.mk_uniformity_basis_le Metric.mk_uniformity_basis_le
 
+/- warning: metric.uniformity_basis_dist_le -> Metric.uniformity_basis_dist_le is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α], Filter.HasBasis.{u1, 1} (Prod.{u1, u1} α α) Real (uniformity.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) (fun (ε : Real) => LT.lt.{0} Real Real.hasLt (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero))) ε) (fun (ε : Real) => setOf.{u1} (Prod.{u1, u1} α α) (fun (p : Prod.{u1, u1} α α) => LE.le.{0} Real Real.hasLe (Dist.dist.{u1} α (PseudoMetricSpace.toHasDist.{u1} α _inst_1) (Prod.fst.{u1, u1} α α p) (Prod.snd.{u1, u1} α α p)) ε))
+but is expected to have type
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α], Filter.HasBasis.{u1, 1} (Prod.{u1, u1} α α) Real (uniformity.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) (fun (ε : Real) => LT.lt.{0} Real Real.instLTReal (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal)) ε) (fun (ε : Real) => setOf.{u1} (Prod.{u1, u1} α α) (fun (p : Prod.{u1, u1} α α) => LE.le.{0} Real Real.instLEReal (Dist.dist.{u1} α (PseudoMetricSpace.toDist.{u1} α _inst_1) (Prod.fst.{u1, u1} α α p) (Prod.snd.{u1, u1} α α p)) ε))
+Case conversion may be inaccurate. Consider using '#align metric.uniformity_basis_dist_le Metric.uniformity_basis_dist_leₓ'. -/
 /-- Contant size closed neighborhoods of the diagonal form a basis
 of the uniformity filter. -/
 theorem uniformity_basis_dist_le :
@@ -876,6 +1460,12 @@ theorem uniformity_basis_dist_le :
   Metric.mk_uniformity_basis_le (fun _ => id) fun ε ε₀ => ⟨ε, ε₀, le_refl ε⟩
 #align metric.uniformity_basis_dist_le Metric.uniformity_basis_dist_le
 
+/- warning: metric.uniformity_basis_dist_le_pow -> Metric.uniformity_basis_dist_le_pow is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {r : Real}, (LT.lt.{0} Real Real.hasLt (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero))) r) -> (LT.lt.{0} Real Real.hasLt r (OfNat.ofNat.{0} Real 1 (OfNat.mk.{0} Real 1 (One.one.{0} Real Real.hasOne)))) -> (Filter.HasBasis.{u1, 1} (Prod.{u1, u1} α α) Nat (uniformity.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) (fun (n : Nat) => True) (fun (n : Nat) => setOf.{u1} (Prod.{u1, u1} α α) (fun (p : Prod.{u1, u1} α α) => LE.le.{0} Real Real.hasLe (Dist.dist.{u1} α (PseudoMetricSpace.toHasDist.{u1} α _inst_1) (Prod.fst.{u1, u1} α α p) (Prod.snd.{u1, u1} α α p)) (HPow.hPow.{0, 0, 0} Real Nat Real (instHPow.{0, 0} Real Nat (Monoid.Pow.{0} Real Real.monoid)) r n))))
+but is expected to have type
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {r : Real}, (LT.lt.{0} Real Real.instLTReal (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal)) r) -> (LT.lt.{0} Real Real.instLTReal r (OfNat.ofNat.{0} Real 1 (One.toOfNat1.{0} Real Real.instOneReal))) -> (Filter.HasBasis.{u1, 1} (Prod.{u1, u1} α α) Nat (uniformity.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) (fun (n : Nat) => True) (fun (n : Nat) => setOf.{u1} (Prod.{u1, u1} α α) (fun (p : Prod.{u1, u1} α α) => LE.le.{0} Real Real.instLEReal (Dist.dist.{u1} α (PseudoMetricSpace.toDist.{u1} α _inst_1) (Prod.fst.{u1, u1} α α p) (Prod.snd.{u1, u1} α α p)) (HPow.hPow.{0, 0, 0} Real Nat Real (instHPow.{0, 0} Real Nat (Monoid.Pow.{0} Real Real.instMonoidReal)) r n))))
+Case conversion may be inaccurate. Consider using '#align metric.uniformity_basis_dist_le_pow Metric.uniformity_basis_dist_le_powₓ'. -/
 theorem uniformity_basis_dist_le_pow {r : ℝ} (h0 : 0 < r) (h1 : r < 1) :
     (𝓤 α).HasBasis (fun n : ℕ => True) fun n : ℕ => { p : α × α | dist p.1 p.2 ≤ r ^ n } :=
   Metric.mk_uniformity_basis_le (fun n hn => pow_pos h0 _) fun ε ε0 =>
@@ -883,21 +1473,45 @@ theorem uniformity_basis_dist_le_pow {r : ℝ} (h0 : 0 < r) (h1 : r < 1) :
     ⟨n, trivial, hn.le⟩
 #align metric.uniformity_basis_dist_le_pow Metric.uniformity_basis_dist_le_pow
 
+/- warning: metric.mem_uniformity_dist -> Metric.mem_uniformity_dist is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {s : Set.{u1} (Prod.{u1, u1} α α)}, Iff (Membership.Mem.{u1, u1} (Set.{u1} (Prod.{u1, u1} α α)) (Filter.{u1} (Prod.{u1, u1} α α)) (Filter.hasMem.{u1} (Prod.{u1, u1} α α)) s (uniformity.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1))) (Exists.{1} Real (fun (ε : Real) => Exists.{0} (GT.gt.{0} Real Real.hasLt ε (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero)))) (fun (H : GT.gt.{0} Real Real.hasLt ε (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero)))) => forall {a : α} {b : α}, (LT.lt.{0} Real Real.hasLt (Dist.dist.{u1} α (PseudoMetricSpace.toHasDist.{u1} α _inst_1) a b) ε) -> (Membership.Mem.{u1, u1} (Prod.{u1, u1} α α) (Set.{u1} (Prod.{u1, u1} α α)) (Set.hasMem.{u1} (Prod.{u1, u1} α α)) (Prod.mk.{u1, u1} α α a b) s))))
+but is expected to have type
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {s : Set.{u1} (Prod.{u1, u1} α α)}, Iff (Membership.mem.{u1, u1} (Set.{u1} (Prod.{u1, u1} α α)) (Filter.{u1} (Prod.{u1, u1} α α)) (instMembershipSetFilter.{u1} (Prod.{u1, u1} α α)) s (uniformity.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1))) (Exists.{1} Real (fun (ε : Real) => And (GT.gt.{0} Real Real.instLTReal ε (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal))) (forall {a : α} {b : α}, (LT.lt.{0} Real Real.instLTReal (Dist.dist.{u1} α (PseudoMetricSpace.toDist.{u1} α _inst_1) a b) ε) -> (Membership.mem.{u1, u1} (Prod.{u1, u1} α α) (Set.{u1} (Prod.{u1, u1} α α)) (Set.instMembershipSet.{u1} (Prod.{u1, u1} α α)) (Prod.mk.{u1, u1} α α a b) s))))
+Case conversion may be inaccurate. Consider using '#align metric.mem_uniformity_dist Metric.mem_uniformity_distₓ'. -/
 theorem mem_uniformity_dist {s : Set (α × α)} :
     s ∈ 𝓤 α ↔ ∃ ε > 0, ∀ {a b : α}, dist a b < ε → (a, b) ∈ s :=
   uniformity_basis_dist.mem_uniformity_iff
 #align metric.mem_uniformity_dist Metric.mem_uniformity_dist
 
+/- warning: metric.dist_mem_uniformity -> Metric.dist_mem_uniformity is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {ε : Real}, (LT.lt.{0} Real Real.hasLt (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero))) ε) -> (Membership.Mem.{u1, u1} (Set.{u1} (Prod.{u1, u1} α α)) (Filter.{u1} (Prod.{u1, u1} α α)) (Filter.hasMem.{u1} (Prod.{u1, u1} α α)) (setOf.{u1} (Prod.{u1, u1} α α) (fun (p : Prod.{u1, u1} α α) => LT.lt.{0} Real Real.hasLt (Dist.dist.{u1} α (PseudoMetricSpace.toHasDist.{u1} α _inst_1) (Prod.fst.{u1, u1} α α p) (Prod.snd.{u1, u1} α α p)) ε)) (uniformity.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)))
+but is expected to have type
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {ε : Real}, (LT.lt.{0} Real Real.instLTReal (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal)) ε) -> (Membership.mem.{u1, u1} (Set.{u1} (Prod.{u1, u1} α α)) (Filter.{u1} (Prod.{u1, u1} α α)) (instMembershipSetFilter.{u1} (Prod.{u1, u1} α α)) (setOf.{u1} (Prod.{u1, u1} α α) (fun (p : Prod.{u1, u1} α α) => LT.lt.{0} Real Real.instLTReal (Dist.dist.{u1} α (PseudoMetricSpace.toDist.{u1} α _inst_1) (Prod.fst.{u1, u1} α α p) (Prod.snd.{u1, u1} α α p)) ε)) (uniformity.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)))
+Case conversion may be inaccurate. Consider using '#align metric.dist_mem_uniformity Metric.dist_mem_uniformityₓ'. -/
 /-- A constant size neighborhood of the diagonal is an entourage. -/
 theorem dist_mem_uniformity {ε : ℝ} (ε0 : 0 < ε) : { p : α × α | dist p.1 p.2 < ε } ∈ 𝓤 α :=
   mem_uniformity_dist.2 ⟨ε, ε0, fun a b => id⟩
 #align metric.dist_mem_uniformity Metric.dist_mem_uniformity
 
+/- warning: metric.uniform_continuous_iff -> Metric.uniformContinuous_iff is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : PseudoMetricSpace.{u1} α] [_inst_2 : PseudoMetricSpace.{u2} β] {f : α -> β}, Iff (UniformContinuous.{u1, u2} α β (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1) (PseudoMetricSpace.toUniformSpace.{u2} β _inst_2) f) (forall (ε : Real), (GT.gt.{0} Real Real.hasLt ε (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero)))) -> (Exists.{1} Real (fun (δ : Real) => Exists.{0} (GT.gt.{0} Real Real.hasLt δ (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero)))) (fun (H : GT.gt.{0} Real Real.hasLt δ (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero)))) => forall {a : α} {b : α}, (LT.lt.{0} Real Real.hasLt (Dist.dist.{u1} α (PseudoMetricSpace.toHasDist.{u1} α _inst_1) a b) δ) -> (LT.lt.{0} Real Real.hasLt (Dist.dist.{u2} β (PseudoMetricSpace.toHasDist.{u2} β _inst_2) (f a) (f b)) ε)))))
+but is expected to have type
+  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : PseudoMetricSpace.{u1} α] [_inst_2 : PseudoMetricSpace.{u2} β] {f : α -> β}, Iff (UniformContinuous.{u1, u2} α β (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1) (PseudoMetricSpace.toUniformSpace.{u2} β _inst_2) f) (forall (ε : Real), (GT.gt.{0} Real Real.instLTReal ε (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal))) -> (Exists.{1} Real (fun (δ : Real) => And (GT.gt.{0} Real Real.instLTReal δ (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal))) (forall {a : α} {b : α}, (LT.lt.{0} Real Real.instLTReal (Dist.dist.{u1} α (PseudoMetricSpace.toDist.{u1} α _inst_1) a b) δ) -> (LT.lt.{0} Real Real.instLTReal (Dist.dist.{u2} β (PseudoMetricSpace.toDist.{u2} β _inst_2) (f a) (f b)) ε)))))
+Case conversion may be inaccurate. Consider using '#align metric.uniform_continuous_iff Metric.uniformContinuous_iffₓ'. -/
 theorem uniformContinuous_iff [PseudoMetricSpace β] {f : α → β} :
     UniformContinuous f ↔ ∀ ε > 0, ∃ δ > 0, ∀ {a b : α}, dist a b < δ → dist (f a) (f b) < ε :=
   uniformity_basis_dist.uniformContinuous_iff uniformity_basis_dist
 #align metric.uniform_continuous_iff Metric.uniformContinuous_iff
 
+/- warning: metric.uniform_continuous_on_iff -> Metric.uniformContinuousOn_iff is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : PseudoMetricSpace.{u1} α] [_inst_2 : PseudoMetricSpace.{u2} β] {f : α -> β} {s : Set.{u1} α}, Iff (UniformContinuousOn.{u1, u2} α β (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1) (PseudoMetricSpace.toUniformSpace.{u2} β _inst_2) f s) (forall (ε : Real), (GT.gt.{0} Real Real.hasLt ε (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero)))) -> (Exists.{1} Real (fun (δ : Real) => Exists.{0} (GT.gt.{0} Real Real.hasLt δ (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero)))) (fun (H : GT.gt.{0} Real Real.hasLt δ (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero)))) => forall (x : α), (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) x s) -> (forall (y : α), (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) y s) -> (LT.lt.{0} Real Real.hasLt (Dist.dist.{u1} α (PseudoMetricSpace.toHasDist.{u1} α _inst_1) x y) δ) -> (LT.lt.{0} Real Real.hasLt (Dist.dist.{u2} β (PseudoMetricSpace.toHasDist.{u2} β _inst_2) (f x) (f y)) ε))))))
+but is expected to have type
+  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : PseudoMetricSpace.{u1} α] [_inst_2 : PseudoMetricSpace.{u2} β] {f : α -> β} {s : Set.{u1} α}, Iff (UniformContinuousOn.{u1, u2} α β (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1) (PseudoMetricSpace.toUniformSpace.{u2} β _inst_2) f s) (forall (ε : Real), (GT.gt.{0} Real Real.instLTReal ε (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal))) -> (Exists.{1} Real (fun (δ : Real) => And (GT.gt.{0} Real Real.instLTReal δ (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal))) (forall (x : α), (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) x s) -> (forall (y : α), (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) y s) -> (LT.lt.{0} Real Real.instLTReal (Dist.dist.{u1} α (PseudoMetricSpace.toDist.{u1} α _inst_1) x y) δ) -> (LT.lt.{0} Real Real.instLTReal (Dist.dist.{u2} β (PseudoMetricSpace.toDist.{u2} β _inst_2) (f x) (f y)) ε))))))
+Case conversion may be inaccurate. Consider using '#align metric.uniform_continuous_on_iff Metric.uniformContinuousOn_iffₓ'. -/
 /- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (x y «expr ∈ » s) -/
 theorem uniformContinuousOn_iff [PseudoMetricSpace β] {f : α → β} {s : Set α} :
     UniformContinuousOn f s ↔
@@ -905,6 +1519,12 @@ theorem uniformContinuousOn_iff [PseudoMetricSpace β] {f : α → β} {s : Set
   Metric.uniformity_basis_dist.uniformContinuousOn_iff Metric.uniformity_basis_dist
 #align metric.uniform_continuous_on_iff Metric.uniformContinuousOn_iff
 
+/- warning: metric.uniform_continuous_on_iff_le -> Metric.uniformContinuousOn_iff_le is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : PseudoMetricSpace.{u1} α] [_inst_2 : PseudoMetricSpace.{u2} β] {f : α -> β} {s : Set.{u1} α}, Iff (UniformContinuousOn.{u1, u2} α β (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1) (PseudoMetricSpace.toUniformSpace.{u2} β _inst_2) f s) (forall (ε : Real), (GT.gt.{0} Real Real.hasLt ε (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero)))) -> (Exists.{1} Real (fun (δ : Real) => Exists.{0} (GT.gt.{0} Real Real.hasLt δ (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero)))) (fun (H : GT.gt.{0} Real Real.hasLt δ (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero)))) => forall (x : α), (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) x s) -> (forall (y : α), (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) y s) -> (LE.le.{0} Real Real.hasLe (Dist.dist.{u1} α (PseudoMetricSpace.toHasDist.{u1} α _inst_1) x y) δ) -> (LE.le.{0} Real Real.hasLe (Dist.dist.{u2} β (PseudoMetricSpace.toHasDist.{u2} β _inst_2) (f x) (f y)) ε))))))
+but is expected to have type
+  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : PseudoMetricSpace.{u1} α] [_inst_2 : PseudoMetricSpace.{u2} β] {f : α -> β} {s : Set.{u1} α}, Iff (UniformContinuousOn.{u1, u2} α β (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1) (PseudoMetricSpace.toUniformSpace.{u2} β _inst_2) f s) (forall (ε : Real), (GT.gt.{0} Real Real.instLTReal ε (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal))) -> (Exists.{1} Real (fun (δ : Real) => And (GT.gt.{0} Real Real.instLTReal δ (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal))) (forall (x : α), (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) x s) -> (forall (y : α), (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) y s) -> (LE.le.{0} Real Real.instLEReal (Dist.dist.{u1} α (PseudoMetricSpace.toDist.{u1} α _inst_1) x y) δ) -> (LE.le.{0} Real Real.instLEReal (Dist.dist.{u2} β (PseudoMetricSpace.toDist.{u2} β _inst_2) (f x) (f y)) ε))))))
+Case conversion may be inaccurate. Consider using '#align metric.uniform_continuous_on_iff_le Metric.uniformContinuousOn_iff_leₓ'. -/
 /- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (x y «expr ∈ » s) -/
 theorem uniformContinuousOn_iff_le [PseudoMetricSpace β] {f : α → β} {s : Set α} :
     UniformContinuousOn f s ↔
@@ -912,6 +1532,12 @@ theorem uniformContinuousOn_iff_le [PseudoMetricSpace β] {f : α → β} {s : S
   Metric.uniformity_basis_dist_le.uniformContinuousOn_iff Metric.uniformity_basis_dist_le
 #align metric.uniform_continuous_on_iff_le Metric.uniformContinuousOn_iff_le
 
+/- warning: metric.uniform_embedding_iff -> Metric.uniformEmbedding_iff is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : PseudoMetricSpace.{u1} α] [_inst_2 : PseudoMetricSpace.{u2} β] {f : α -> β}, Iff (UniformEmbedding.{u1, u2} α β (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1) (PseudoMetricSpace.toUniformSpace.{u2} β _inst_2) f) (And (Function.Injective.{succ u1, succ u2} α β f) (And (UniformContinuous.{u1, u2} α β (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1) (PseudoMetricSpace.toUniformSpace.{u2} β _inst_2) f) (forall (δ : Real), (GT.gt.{0} Real Real.hasLt δ (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero)))) -> (Exists.{1} Real (fun (ε : Real) => Exists.{0} (GT.gt.{0} Real Real.hasLt ε (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero)))) (fun (H : GT.gt.{0} Real Real.hasLt ε (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero)))) => forall {a : α} {b : α}, (LT.lt.{0} Real Real.hasLt (Dist.dist.{u2} β (PseudoMetricSpace.toHasDist.{u2} β _inst_2) (f a) (f b)) ε) -> (LT.lt.{0} Real Real.hasLt (Dist.dist.{u1} α (PseudoMetricSpace.toHasDist.{u1} α _inst_1) a b) δ)))))))
+but is expected to have type
+  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : PseudoMetricSpace.{u1} α] [_inst_2 : PseudoMetricSpace.{u2} β] {f : α -> β}, Iff (UniformEmbedding.{u1, u2} α β (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1) (PseudoMetricSpace.toUniformSpace.{u2} β _inst_2) f) (And (Function.Injective.{succ u1, succ u2} α β f) (And (UniformContinuous.{u1, u2} α β (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1) (PseudoMetricSpace.toUniformSpace.{u2} β _inst_2) f) (forall (δ : Real), (GT.gt.{0} Real Real.instLTReal δ (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal))) -> (Exists.{1} Real (fun (ε : Real) => And (GT.gt.{0} Real Real.instLTReal ε (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal))) (forall {a : α} {b : α}, (LT.lt.{0} Real Real.instLTReal (Dist.dist.{u2} β (PseudoMetricSpace.toDist.{u2} β _inst_2) (f a) (f b)) ε) -> (LT.lt.{0} Real Real.instLTReal (Dist.dist.{u1} α (PseudoMetricSpace.toDist.{u1} α _inst_1) a b) δ)))))))
+Case conversion may be inaccurate. Consider using '#align metric.uniform_embedding_iff Metric.uniformEmbedding_iffₓ'. -/
 theorem uniformEmbedding_iff [PseudoMetricSpace β] {f : α → β} :
     UniformEmbedding f ↔
       Function.Injective f ∧
@@ -921,6 +1547,12 @@ theorem uniformEmbedding_iff [PseudoMetricSpace β] {f : α → β} :
   rfl
 #align metric.uniform_embedding_iff Metric.uniformEmbedding_iff
 
+/- warning: metric.controlled_of_uniform_embedding -> Metric.controlled_of_uniformEmbedding is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : PseudoMetricSpace.{u1} α] [_inst_2 : PseudoMetricSpace.{u2} β] {f : α -> β}, (UniformEmbedding.{u1, u2} α β (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1) (PseudoMetricSpace.toUniformSpace.{u2} β _inst_2) f) -> (And (forall (ε : Real), (GT.gt.{0} Real Real.hasLt ε (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero)))) -> (Exists.{1} Real (fun (δ : Real) => Exists.{0} (GT.gt.{0} Real Real.hasLt δ (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero)))) (fun (H : GT.gt.{0} Real Real.hasLt δ (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero)))) => forall {a : α} {b : α}, (LT.lt.{0} Real Real.hasLt (Dist.dist.{u1} α (PseudoMetricSpace.toHasDist.{u1} α _inst_1) a b) δ) -> (LT.lt.{0} Real Real.hasLt (Dist.dist.{u2} β (PseudoMetricSpace.toHasDist.{u2} β _inst_2) (f a) (f b)) ε))))) (forall (δ : Real), (GT.gt.{0} Real Real.hasLt δ (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero)))) -> (Exists.{1} Real (fun (ε : Real) => Exists.{0} (GT.gt.{0} Real Real.hasLt ε (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero)))) (fun (H : GT.gt.{0} Real Real.hasLt ε (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero)))) => forall {a : α} {b : α}, (LT.lt.{0} Real Real.hasLt (Dist.dist.{u2} β (PseudoMetricSpace.toHasDist.{u2} β _inst_2) (f a) (f b)) ε) -> (LT.lt.{0} Real Real.hasLt (Dist.dist.{u1} α (PseudoMetricSpace.toHasDist.{u1} α _inst_1) a b) δ))))))
+but is expected to have type
+  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : PseudoMetricSpace.{u1} α] [_inst_2 : PseudoMetricSpace.{u2} β] {f : α -> β}, (UniformEmbedding.{u1, u2} α β (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1) (PseudoMetricSpace.toUniformSpace.{u2} β _inst_2) f) -> (And (forall (ε : Real), (GT.gt.{0} Real Real.instLTReal ε (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal))) -> (Exists.{1} Real (fun (δ : Real) => And (GT.gt.{0} Real Real.instLTReal δ (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal))) (forall {a : α} {b : α}, (LT.lt.{0} Real Real.instLTReal (Dist.dist.{u1} α (PseudoMetricSpace.toDist.{u1} α _inst_1) a b) δ) -> (LT.lt.{0} Real Real.instLTReal (Dist.dist.{u2} β (PseudoMetricSpace.toDist.{u2} β _inst_2) (f a) (f b)) ε))))) (forall (δ : Real), (GT.gt.{0} Real Real.instLTReal δ (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal))) -> (Exists.{1} Real (fun (ε : Real) => And (GT.gt.{0} Real Real.instLTReal ε (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal))) (forall {a : α} {b : α}, (LT.lt.{0} Real Real.instLTReal (Dist.dist.{u2} β (PseudoMetricSpace.toDist.{u2} β _inst_2) (f a) (f b)) ε) -> (LT.lt.{0} Real Real.instLTReal (Dist.dist.{u1} α (PseudoMetricSpace.toDist.{u1} α _inst_1) a b) δ))))))
+Case conversion may be inaccurate. Consider using '#align metric.controlled_of_uniform_embedding Metric.controlled_of_uniformEmbeddingₓ'. -/
 /-- If a map between pseudometric spaces is a uniform embedding then the distance between `f x`
 and `f y` is controlled in terms of the distance between `x` and `y`. -/
 theorem controlled_of_uniformEmbedding [PseudoMetricSpace β] {f : α → β} :
@@ -932,6 +1564,12 @@ theorem controlled_of_uniformEmbedding [PseudoMetricSpace β] {f : α → β} :
   exact ⟨uniformContinuous_iff.1 (uniformEmbedding_iff.1 h).2.1, (uniformEmbedding_iff.1 h).2.2⟩
 #align metric.controlled_of_uniform_embedding Metric.controlled_of_uniformEmbedding
 
+/- warning: metric.totally_bounded_iff -> Metric.totallyBounded_iff is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {s : Set.{u1} α}, Iff (TotallyBounded.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1) s) (forall (ε : Real), (GT.gt.{0} Real Real.hasLt ε (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero)))) -> (Exists.{succ u1} (Set.{u1} α) (fun (t : Set.{u1} α) => And (Set.Finite.{u1} α t) (HasSubset.Subset.{u1} (Set.{u1} α) (Set.hasSubset.{u1} α) s (Set.unionᵢ.{u1, succ u1} α α (fun (y : α) => Set.unionᵢ.{u1, 0} α (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) y t) (fun (H : Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) y t) => Metric.ball.{u1} α _inst_1 y ε)))))))
+but is expected to have type
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {s : Set.{u1} α}, Iff (TotallyBounded.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1) s) (forall (ε : Real), (GT.gt.{0} Real Real.instLTReal ε (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal))) -> (Exists.{succ u1} (Set.{u1} α) (fun (t : Set.{u1} α) => And (Set.Finite.{u1} α t) (HasSubset.Subset.{u1} (Set.{u1} α) (Set.instHasSubsetSet.{u1} α) s (Set.unionᵢ.{u1, succ u1} α α (fun (y : α) => Set.unionᵢ.{u1, 0} α (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) y t) (fun (H : Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) y t) => Metric.ball.{u1} α _inst_1 y ε)))))))
+Case conversion may be inaccurate. Consider using '#align metric.totally_bounded_iff Metric.totallyBounded_iffₓ'. -/
 theorem totallyBounded_iff {s : Set α} :
     TotallyBounded s ↔ ∀ ε > 0, ∃ t : Set α, t.Finite ∧ s ⊆ ⋃ y ∈ t, ball y ε :=
   ⟨fun H ε ε0 => H _ (dist_mem_uniformity ε0), fun H r ru =>
@@ -940,6 +1578,12 @@ theorem totallyBounded_iff {s : Set α} :
     ⟨t, ft, h.trans <| unionᵢ₂_mono fun y yt z => hε⟩⟩
 #align metric.totally_bounded_iff Metric.totallyBounded_iff
 
+/- warning: metric.totally_bounded_of_finite_discretization -> Metric.totallyBounded_of_finite_discretization is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {s : Set.{u1} α}, (forall (ε : Real), (GT.gt.{0} Real Real.hasLt ε (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero)))) -> (Exists.{succ (succ u1)} Type.{u1} (fun (β : Type.{u1}) => Exists.{succ u1} (Fintype.{u1} β) (fun (_x : Fintype.{u1} β) => Exists.{succ u1} ((coeSort.{succ u1, succ (succ u1)} (Set.{u1} α) Type.{u1} (Set.hasCoeToSort.{u1} α) s) -> β) (fun (F : (coeSort.{succ u1, succ (succ u1)} (Set.{u1} α) Type.{u1} (Set.hasCoeToSort.{u1} α) s) -> β) => forall (x : coeSort.{succ u1, succ (succ u1)} (Set.{u1} α) Type.{u1} (Set.hasCoeToSort.{u1} α) s) (y : coeSort.{succ u1, succ (succ u1)} (Set.{u1} α) Type.{u1} (Set.hasCoeToSort.{u1} α) s), (Eq.{succ u1} β (F x) (F y)) -> (LT.lt.{0} Real Real.hasLt (Dist.dist.{u1} α (PseudoMetricSpace.toHasDist.{u1} α _inst_1) ((fun (a : Type.{u1}) (b : Type.{u1}) [self : HasLiftT.{succ u1, succ u1} a b] => self.0) (coeSort.{succ u1, succ (succ u1)} (Set.{u1} α) Type.{u1} (Set.hasCoeToSort.{u1} α) s) α (HasLiftT.mk.{succ u1, succ u1} (coeSort.{succ u1, succ (succ u1)} (Set.{u1} α) Type.{u1} (Set.hasCoeToSort.{u1} α) s) α (CoeTCₓ.coe.{succ u1, succ u1} (coeSort.{succ u1, succ (succ u1)} (Set.{u1} α) Type.{u1} (Set.hasCoeToSort.{u1} α) s) α (coeBase.{succ u1, succ u1} (coeSort.{succ u1, succ (succ u1)} (Set.{u1} α) Type.{u1} (Set.hasCoeToSort.{u1} α) s) α (coeSubtype.{succ u1} α (fun (x : α) => Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) x s))))) x) ((fun (a : Type.{u1}) (b : Type.{u1}) [self : HasLiftT.{succ u1, succ u1} a b] => self.0) (coeSort.{succ u1, succ (succ u1)} (Set.{u1} α) Type.{u1} (Set.hasCoeToSort.{u1} α) s) α (HasLiftT.mk.{succ u1, succ u1} (coeSort.{succ u1, succ (succ u1)} (Set.{u1} α) Type.{u1} (Set.hasCoeToSort.{u1} α) s) α (CoeTCₓ.coe.{succ u1, succ u1} (coeSort.{succ u1, succ (succ u1)} (Set.{u1} α) Type.{u1} (Set.hasCoeToSort.{u1} α) s) α (coeBase.{succ u1, succ u1} (coeSort.{succ u1, succ (succ u1)} (Set.{u1} α) Type.{u1} (Set.hasCoeToSort.{u1} α) s) α (coeSubtype.{succ u1} α (fun (x : α) => Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) x s))))) y)) ε)))))) -> (TotallyBounded.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1) s)
+but is expected to have type
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {s : Set.{u1} α}, (forall (ε : Real), (GT.gt.{0} Real Real.instLTReal ε (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal))) -> (Exists.{succ (succ u1)} Type.{u1} (fun (β : Type.{u1}) => Exists.{succ u1} (Fintype.{u1} β) (fun (_x : Fintype.{u1} β) => Exists.{succ u1} ((Set.Elem.{u1} α s) -> β) (fun (F : (Set.Elem.{u1} α s) -> β) => forall (x : Set.Elem.{u1} α s) (y : Set.Elem.{u1} α s), (Eq.{succ u1} β (F x) (F y)) -> (LT.lt.{0} Real Real.instLTReal (Dist.dist.{u1} α (PseudoMetricSpace.toDist.{u1} α _inst_1) (Subtype.val.{succ u1} α (fun (x : α) => Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) x s) x) (Subtype.val.{succ u1} α (fun (x : α) => Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) x s) y)) ε)))))) -> (TotallyBounded.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1) s)
+Case conversion may be inaccurate. Consider using '#align metric.totally_bounded_of_finite_discretization Metric.totallyBounded_of_finite_discretizationₓ'. -/
 /-- A pseudometric space is totally bounded if one can reconstruct up to any ε>0 any element of the
 space from finitely many data. -/
 theorem totallyBounded_of_finite_discretization {s : Set α}
@@ -963,6 +1607,12 @@ theorem totallyBounded_of_finite_discretization {s : Set α}
   exact ⟨_, ⟨F ⟨x, xs⟩, rfl⟩, hF _ _ this.symm⟩
 #align metric.totally_bounded_of_finite_discretization Metric.totallyBounded_of_finite_discretization
 
+/- warning: metric.finite_approx_of_totally_bounded -> Metric.finite_approx_of_totallyBounded is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {s : Set.{u1} α}, (TotallyBounded.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1) s) -> (forall (ε : Real), (GT.gt.{0} Real Real.hasLt ε (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero)))) -> (Exists.{succ u1} (Set.{u1} α) (fun (t : Set.{u1} α) => Exists.{0} (HasSubset.Subset.{u1} (Set.{u1} α) (Set.hasSubset.{u1} α) t s) (fun (H : HasSubset.Subset.{u1} (Set.{u1} α) (Set.hasSubset.{u1} α) t s) => And (Set.Finite.{u1} α t) (HasSubset.Subset.{u1} (Set.{u1} α) (Set.hasSubset.{u1} α) s (Set.unionᵢ.{u1, succ u1} α α (fun (y : α) => Set.unionᵢ.{u1, 0} α (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) y t) (fun (H : Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) y t) => Metric.ball.{u1} α _inst_1 y ε))))))))
+but is expected to have type
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {s : Set.{u1} α}, (TotallyBounded.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1) s) -> (forall (ε : Real), (GT.gt.{0} Real Real.instLTReal ε (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal))) -> (Exists.{succ u1} (Set.{u1} α) (fun (t : Set.{u1} α) => And (HasSubset.Subset.{u1} (Set.{u1} α) (Set.instHasSubsetSet.{u1} α) t s) (And (Set.Finite.{u1} α t) (HasSubset.Subset.{u1} (Set.{u1} α) (Set.instHasSubsetSet.{u1} α) s (Set.unionᵢ.{u1, succ u1} α α (fun (y : α) => Set.unionᵢ.{u1, 0} α (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) y t) (fun (h._@.Mathlib.Topology.MetricSpace.Basic._hyg.10746 : Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) y t) => Metric.ball.{u1} α _inst_1 y ε))))))))
+Case conversion may be inaccurate. Consider using '#align metric.finite_approx_of_totally_bounded Metric.finite_approx_of_totallyBoundedₓ'. -/
 /- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (t «expr ⊆ » s) -/
 theorem finite_approx_of_totallyBounded {s : Set α} (hs : TotallyBounded s) :
     ∀ ε > 0, ∃ (t : _)(_ : t ⊆ s), Set.Finite t ∧ s ⊆ ⋃ y ∈ t, ball y ε :=
@@ -972,6 +1622,12 @@ theorem finite_approx_of_totallyBounded {s : Set α} (hs : TotallyBounded s) :
   exact hs _ (dist_mem_uniformity ε_pos)
 #align metric.finite_approx_of_totally_bounded Metric.finite_approx_of_totallyBounded
 
+/- warning: metric.tendsto_uniformly_on_filter_iff -> Metric.tendstoUniformlyOnFilter_iff is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : PseudoMetricSpace.{u1} α] {ι : Type.{u3}} {F : ι -> β -> α} {f : β -> α} {p : Filter.{u3} ι} {p' : Filter.{u2} β}, Iff (TendstoUniformlyOnFilter.{u2, u1, u3} β α ι (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1) F f p p') (forall (ε : Real), (GT.gt.{0} Real Real.hasLt ε (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero)))) -> (Filter.Eventually.{max u3 u2} (Prod.{u3, u2} ι β) (fun (n : Prod.{u3, u2} ι β) => LT.lt.{0} Real Real.hasLt (Dist.dist.{u1} α (PseudoMetricSpace.toHasDist.{u1} α _inst_1) (f (Prod.snd.{u3, u2} ι β n)) (F (Prod.fst.{u3, u2} ι β n) (Prod.snd.{u3, u2} ι β n))) ε) (Filter.prod.{u3, u2} ι β p p')))
+but is expected to have type
+  forall {α : Type.{u2}} {β : Type.{u3}} [_inst_1 : PseudoMetricSpace.{u2} α] {ι : Type.{u1}} {F : ι -> β -> α} {f : β -> α} {p : Filter.{u1} ι} {p' : Filter.{u3} β}, Iff (TendstoUniformlyOnFilter.{u3, u2, u1} β α ι (PseudoMetricSpace.toUniformSpace.{u2} α _inst_1) F f p p') (forall (ε : Real), (GT.gt.{0} Real Real.instLTReal ε (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal))) -> (Filter.Eventually.{max u3 u1} (Prod.{u1, u3} ι β) (fun (n : Prod.{u1, u3} ι β) => LT.lt.{0} Real Real.instLTReal (Dist.dist.{u2} α (PseudoMetricSpace.toDist.{u2} α _inst_1) (f (Prod.snd.{u1, u3} ι β n)) (F (Prod.fst.{u1, u3} ι β n) (Prod.snd.{u1, u3} ι β n))) ε) (Filter.prod.{u1, u3} ι β p p')))
+Case conversion may be inaccurate. Consider using '#align metric.tendsto_uniformly_on_filter_iff Metric.tendstoUniformlyOnFilter_iffₓ'. -/
 /-- Expressing uniform convergence using `dist` -/
 theorem tendstoUniformlyOnFilter_iff {ι : Type _} {F : ι → β → α} {f : β → α} {p : Filter ι}
     {p' : Filter β} :
@@ -983,6 +1639,12 @@ theorem tendstoUniformlyOnFilter_iff {ι : Type _} {F : ι → β → α} {f : 
   refine' (H ε εpos).mono fun n hn => hε hn
 #align metric.tendsto_uniformly_on_filter_iff Metric.tendstoUniformlyOnFilter_iff
 
+/- warning: metric.tendsto_locally_uniformly_on_iff -> Metric.tendstoLocallyUniformlyOn_iff is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : PseudoMetricSpace.{u1} α] {ι : Type.{u3}} [_inst_2 : TopologicalSpace.{u2} β] {F : ι -> β -> α} {f : β -> α} {p : Filter.{u3} ι} {s : Set.{u2} β}, Iff (TendstoLocallyUniformlyOn.{u2, u1, u3} β α ι (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1) _inst_2 F f p s) (forall (ε : Real), (GT.gt.{0} Real Real.hasLt ε (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero)))) -> (forall (x : β), (Membership.Mem.{u2, u2} β (Set.{u2} β) (Set.hasMem.{u2} β) x s) -> (Exists.{succ u2} (Set.{u2} β) (fun (t : Set.{u2} β) => Exists.{0} (Membership.Mem.{u2, u2} (Set.{u2} β) (Filter.{u2} β) (Filter.hasMem.{u2} β) t (nhdsWithin.{u2} β _inst_2 x s)) (fun (H : Membership.Mem.{u2, u2} (Set.{u2} β) (Filter.{u2} β) (Filter.hasMem.{u2} β) t (nhdsWithin.{u2} β _inst_2 x s)) => Filter.Eventually.{u3} ι (fun (n : ι) => forall (y : β), (Membership.Mem.{u2, u2} β (Set.{u2} β) (Set.hasMem.{u2} β) y t) -> (LT.lt.{0} Real Real.hasLt (Dist.dist.{u1} α (PseudoMetricSpace.toHasDist.{u1} α _inst_1) (f y) (F n y)) ε)) p)))))
+but is expected to have type
+  forall {α : Type.{u2}} {β : Type.{u3}} [_inst_1 : PseudoMetricSpace.{u2} α] {ι : Type.{u1}} [_inst_2 : TopologicalSpace.{u3} β] {F : ι -> β -> α} {f : β -> α} {p : Filter.{u1} ι} {s : Set.{u3} β}, Iff (TendstoLocallyUniformlyOn.{u3, u2, u1} β α ι (PseudoMetricSpace.toUniformSpace.{u2} α _inst_1) _inst_2 F f p s) (forall (ε : Real), (GT.gt.{0} Real Real.instLTReal ε (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal))) -> (forall (x : β), (Membership.mem.{u3, u3} β (Set.{u3} β) (Set.instMembershipSet.{u3} β) x s) -> (Exists.{succ u3} (Set.{u3} β) (fun (t : Set.{u3} β) => And (Membership.mem.{u3, u3} (Set.{u3} β) (Filter.{u3} β) (instMembershipSetFilter.{u3} β) t (nhdsWithin.{u3} β _inst_2 x s)) (Filter.Eventually.{u1} ι (fun (n : ι) => forall (y : β), (Membership.mem.{u3, u3} β (Set.{u3} β) (Set.instMembershipSet.{u3} β) y t) -> (LT.lt.{0} Real Real.instLTReal (Dist.dist.{u2} α (PseudoMetricSpace.toDist.{u2} α _inst_1) (f y) (F n y)) ε)) p)))))
+Case conversion may be inaccurate. Consider using '#align metric.tendsto_locally_uniformly_on_iff Metric.tendstoLocallyUniformlyOn_iffₓ'. -/
 /-- Expressing locally uniform convergence on a set using `dist`. -/
 theorem tendstoLocallyUniformlyOn_iff {ι : Type _} [TopologicalSpace β] {F : ι → β → α} {f : β → α}
     {p : Filter ι} {s : Set β} :
@@ -995,6 +1657,12 @@ theorem tendstoLocallyUniformlyOn_iff {ι : Type _} [TopologicalSpace β] {F : 
   exact ⟨t, ht, Ht.mono fun n hs x hx => hε (hs x hx)⟩
 #align metric.tendsto_locally_uniformly_on_iff Metric.tendstoLocallyUniformlyOn_iff
 
+/- warning: metric.tendsto_uniformly_on_iff -> Metric.tendstoUniformlyOn_iff is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : PseudoMetricSpace.{u1} α] {ι : Type.{u3}} {F : ι -> β -> α} {f : β -> α} {p : Filter.{u3} ι} {s : Set.{u2} β}, Iff (TendstoUniformlyOn.{u2, u1, u3} β α ι (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1) F f p s) (forall (ε : Real), (GT.gt.{0} Real Real.hasLt ε (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero)))) -> (Filter.Eventually.{u3} ι (fun (n : ι) => forall (x : β), (Membership.Mem.{u2, u2} β (Set.{u2} β) (Set.hasMem.{u2} β) x s) -> (LT.lt.{0} Real Real.hasLt (Dist.dist.{u1} α (PseudoMetricSpace.toHasDist.{u1} α _inst_1) (f x) (F n x)) ε)) p))
+but is expected to have type
+  forall {α : Type.{u2}} {β : Type.{u3}} [_inst_1 : PseudoMetricSpace.{u2} α] {ι : Type.{u1}} {F : ι -> β -> α} {f : β -> α} {p : Filter.{u1} ι} {s : Set.{u3} β}, Iff (TendstoUniformlyOn.{u3, u2, u1} β α ι (PseudoMetricSpace.toUniformSpace.{u2} α _inst_1) F f p s) (forall (ε : Real), (GT.gt.{0} Real Real.instLTReal ε (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal))) -> (Filter.Eventually.{u1} ι (fun (n : ι) => forall (x : β), (Membership.mem.{u3, u3} β (Set.{u3} β) (Set.instMembershipSet.{u3} β) x s) -> (LT.lt.{0} Real Real.instLTReal (Dist.dist.{u2} α (PseudoMetricSpace.toDist.{u2} α _inst_1) (f x) (F n x)) ε)) p))
+Case conversion may be inaccurate. Consider using '#align metric.tendsto_uniformly_on_iff Metric.tendstoUniformlyOn_iffₓ'. -/
 /-- Expressing uniform convergence on a set using `dist`. -/
 theorem tendstoUniformlyOn_iff {ι : Type _} {F : ι → β → α} {f : β → α} {p : Filter ι} {s : Set β} :
     TendstoUniformlyOn F f p s ↔ ∀ ε > 0, ∀ᶠ n in p, ∀ x ∈ s, dist (f x) (F n x) < ε :=
@@ -1004,6 +1672,12 @@ theorem tendstoUniformlyOn_iff {ι : Type _} {F : ι → β → α} {f : β →
   exact (H ε εpos).mono fun n hs x hx => hε (hs x hx)
 #align metric.tendsto_uniformly_on_iff Metric.tendstoUniformlyOn_iff
 
+/- warning: metric.tendsto_locally_uniformly_iff -> Metric.tendstoLocallyUniformly_iff is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : PseudoMetricSpace.{u1} α] {ι : Type.{u3}} [_inst_2 : TopologicalSpace.{u2} β] {F : ι -> β -> α} {f : β -> α} {p : Filter.{u3} ι}, Iff (TendstoLocallyUniformly.{u2, u1, u3} β α ι (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1) _inst_2 F f p) (forall (ε : Real), (GT.gt.{0} Real Real.hasLt ε (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero)))) -> (forall (x : β), Exists.{succ u2} (Set.{u2} β) (fun (t : Set.{u2} β) => Exists.{0} (Membership.Mem.{u2, u2} (Set.{u2} β) (Filter.{u2} β) (Filter.hasMem.{u2} β) t (nhds.{u2} β _inst_2 x)) (fun (H : Membership.Mem.{u2, u2} (Set.{u2} β) (Filter.{u2} β) (Filter.hasMem.{u2} β) t (nhds.{u2} β _inst_2 x)) => Filter.Eventually.{u3} ι (fun (n : ι) => forall (y : β), (Membership.Mem.{u2, u2} β (Set.{u2} β) (Set.hasMem.{u2} β) y t) -> (LT.lt.{0} Real Real.hasLt (Dist.dist.{u1} α (PseudoMetricSpace.toHasDist.{u1} α _inst_1) (f y) (F n y)) ε)) p))))
+but is expected to have type
+  forall {α : Type.{u2}} {β : Type.{u3}} [_inst_1 : PseudoMetricSpace.{u2} α] {ι : Type.{u1}} [_inst_2 : TopologicalSpace.{u3} β] {F : ι -> β -> α} {f : β -> α} {p : Filter.{u1} ι}, Iff (TendstoLocallyUniformly.{u3, u2, u1} β α ι (PseudoMetricSpace.toUniformSpace.{u2} α _inst_1) _inst_2 F f p) (forall (ε : Real), (GT.gt.{0} Real Real.instLTReal ε (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal))) -> (forall (x : β), Exists.{succ u3} (Set.{u3} β) (fun (t : Set.{u3} β) => And (Membership.mem.{u3, u3} (Set.{u3} β) (Filter.{u3} β) (instMembershipSetFilter.{u3} β) t (nhds.{u3} β _inst_2 x)) (Filter.Eventually.{u1} ι (fun (n : ι) => forall (y : β), (Membership.mem.{u3, u3} β (Set.{u3} β) (Set.instMembershipSet.{u3} β) y t) -> (LT.lt.{0} Real Real.instLTReal (Dist.dist.{u2} α (PseudoMetricSpace.toDist.{u2} α _inst_1) (f y) (F n y)) ε)) p))))
+Case conversion may be inaccurate. Consider using '#align metric.tendsto_locally_uniformly_iff Metric.tendstoLocallyUniformly_iffₓ'. -/
 /-- Expressing locally uniform convergence using `dist`. -/
 theorem tendstoLocallyUniformly_iff {ι : Type _} [TopologicalSpace β] {F : ι → β → α} {f : β → α}
     {p : Filter ι} :
@@ -1014,6 +1688,12 @@ theorem tendstoLocallyUniformly_iff {ι : Type _} [TopologicalSpace β] {F : ι
     mem_univ, forall_const, exists_prop]
 #align metric.tendsto_locally_uniformly_iff Metric.tendstoLocallyUniformly_iff
 
+/- warning: metric.tendsto_uniformly_iff -> Metric.tendstoUniformly_iff is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : PseudoMetricSpace.{u1} α] {ι : Type.{u3}} {F : ι -> β -> α} {f : β -> α} {p : Filter.{u3} ι}, Iff (TendstoUniformly.{u2, u1, u3} β α ι (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1) F f p) (forall (ε : Real), (GT.gt.{0} Real Real.hasLt ε (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero)))) -> (Filter.Eventually.{u3} ι (fun (n : ι) => forall (x : β), LT.lt.{0} Real Real.hasLt (Dist.dist.{u1} α (PseudoMetricSpace.toHasDist.{u1} α _inst_1) (f x) (F n x)) ε) p))
+but is expected to have type
+  forall {α : Type.{u2}} {β : Type.{u3}} [_inst_1 : PseudoMetricSpace.{u2} α] {ι : Type.{u1}} {F : ι -> β -> α} {f : β -> α} {p : Filter.{u1} ι}, Iff (TendstoUniformly.{u3, u2, u1} β α ι (PseudoMetricSpace.toUniformSpace.{u2} α _inst_1) F f p) (forall (ε : Real), (GT.gt.{0} Real Real.instLTReal ε (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal))) -> (Filter.Eventually.{u1} ι (fun (n : ι) => forall (x : β), LT.lt.{0} Real Real.instLTReal (Dist.dist.{u2} α (PseudoMetricSpace.toDist.{u2} α _inst_1) (f x) (F n x)) ε) p))
+Case conversion may be inaccurate. Consider using '#align metric.tendsto_uniformly_iff Metric.tendstoUniformly_iffₓ'. -/
 /-- Expressing uniform convergence using `dist`. -/
 theorem tendstoUniformly_iff {ι : Type _} {F : ι → β → α} {f : β → α} {p : Filter ι} :
     TendstoUniformly F f p ↔ ∀ ε > 0, ∀ᶠ n in p, ∀ x, dist (f x) (F n x) < ε :=
@@ -1022,30 +1702,66 @@ theorem tendstoUniformly_iff {ι : Type _} {F : ι → β → α} {f : β → α
   simp
 #align metric.tendsto_uniformly_iff Metric.tendstoUniformly_iff
 
+/- warning: metric.cauchy_iff -> Metric.cauchy_iff is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {f : Filter.{u1} α}, Iff (Cauchy.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1) f) (And (Filter.NeBot.{u1} α f) (forall (ε : Real), (GT.gt.{0} Real Real.hasLt ε (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero)))) -> (Exists.{succ u1} (Set.{u1} α) (fun (t : Set.{u1} α) => Exists.{0} (Membership.Mem.{u1, u1} (Set.{u1} α) (Filter.{u1} α) (Filter.hasMem.{u1} α) t f) (fun (H : Membership.Mem.{u1, u1} (Set.{u1} α) (Filter.{u1} α) (Filter.hasMem.{u1} α) t f) => forall (x : α), (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) x t) -> (forall (y : α), (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) y t) -> (LT.lt.{0} Real Real.hasLt (Dist.dist.{u1} α (PseudoMetricSpace.toHasDist.{u1} α _inst_1) x y) ε)))))))
+but is expected to have type
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {f : Filter.{u1} α}, Iff (Cauchy.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1) f) (And (Filter.NeBot.{u1} α f) (forall (ε : Real), (GT.gt.{0} Real Real.instLTReal ε (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal))) -> (Exists.{succ u1} (Set.{u1} α) (fun (t : Set.{u1} α) => And (Membership.mem.{u1, u1} (Set.{u1} α) (Filter.{u1} α) (instMembershipSetFilter.{u1} α) t f) (forall (x : α), (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) x t) -> (forall (y : α), (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) y t) -> (LT.lt.{0} Real Real.instLTReal (Dist.dist.{u1} α (PseudoMetricSpace.toDist.{u1} α _inst_1) x y) ε)))))))
+Case conversion may be inaccurate. Consider using '#align metric.cauchy_iff Metric.cauchy_iffₓ'. -/
 /- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (x y «expr ∈ » t) -/
 protected theorem cauchy_iff {f : Filter α} :
     Cauchy f ↔ NeBot f ∧ ∀ ε > 0, ∃ t ∈ f, ∀ (x) (_ : x ∈ t) (y) (_ : y ∈ t), dist x y < ε :=
   uniformity_basis_dist.cauchy_iff
 #align metric.cauchy_iff Metric.cauchy_iff
 
+/- warning: metric.nhds_basis_ball -> Metric.nhds_basis_ball is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {x : α}, Filter.HasBasis.{u1, 1} α Real (nhds.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) x) (fun (ε : Real) => LT.lt.{0} Real Real.hasLt (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero))) ε) (Metric.ball.{u1} α _inst_1 x)
+but is expected to have type
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {x : α}, Filter.HasBasis.{u1, 1} α Real (nhds.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) x) (fun (ε : Real) => LT.lt.{0} Real Real.instLTReal (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal)) ε) (Metric.ball.{u1} α _inst_1 x)
+Case conversion may be inaccurate. Consider using '#align metric.nhds_basis_ball Metric.nhds_basis_ballₓ'. -/
 theorem nhds_basis_ball : (𝓝 x).HasBasis (fun ε : ℝ => 0 < ε) (ball x) :=
   nhds_basis_uniformity uniformity_basis_dist
 #align metric.nhds_basis_ball Metric.nhds_basis_ball
 
+/- warning: metric.mem_nhds_iff -> Metric.mem_nhds_iff is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {x : α} {s : Set.{u1} α}, Iff (Membership.Mem.{u1, u1} (Set.{u1} α) (Filter.{u1} α) (Filter.hasMem.{u1} α) s (nhds.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) x)) (Exists.{1} Real (fun (ε : Real) => Exists.{0} (GT.gt.{0} Real Real.hasLt ε (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero)))) (fun (H : GT.gt.{0} Real Real.hasLt ε (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero)))) => HasSubset.Subset.{u1} (Set.{u1} α) (Set.hasSubset.{u1} α) (Metric.ball.{u1} α _inst_1 x ε) s)))
+but is expected to have type
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {x : α} {s : Set.{u1} α}, Iff (Membership.mem.{u1, u1} (Set.{u1} α) (Filter.{u1} α) (instMembershipSetFilter.{u1} α) s (nhds.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) x)) (Exists.{1} Real (fun (ε : Real) => And (GT.gt.{0} Real Real.instLTReal ε (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal))) (HasSubset.Subset.{u1} (Set.{u1} α) (Set.instHasSubsetSet.{u1} α) (Metric.ball.{u1} α _inst_1 x ε) s)))
+Case conversion may be inaccurate. Consider using '#align metric.mem_nhds_iff Metric.mem_nhds_iffₓ'. -/
 theorem mem_nhds_iff : s ∈ 𝓝 x ↔ ∃ ε > 0, ball x ε ⊆ s :=
   nhds_basis_ball.mem_iff
 #align metric.mem_nhds_iff Metric.mem_nhds_iff
 
+/- warning: metric.eventually_nhds_iff -> Metric.eventually_nhds_iff is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {x : α} {p : α -> Prop}, Iff (Filter.Eventually.{u1} α (fun (y : α) => p y) (nhds.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) x)) (Exists.{1} Real (fun (ε : Real) => Exists.{0} (GT.gt.{0} Real Real.hasLt ε (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero)))) (fun (H : GT.gt.{0} Real Real.hasLt ε (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero)))) => forall {{y : α}}, (LT.lt.{0} Real Real.hasLt (Dist.dist.{u1} α (PseudoMetricSpace.toHasDist.{u1} α _inst_1) y x) ε) -> (p y))))
+but is expected to have type
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {x : α} {p : α -> Prop}, Iff (Filter.Eventually.{u1} α (fun (y : α) => p y) (nhds.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) x)) (Exists.{1} Real (fun (ε : Real) => And (GT.gt.{0} Real Real.instLTReal ε (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal))) (forall {{y : α}}, (LT.lt.{0} Real Real.instLTReal (Dist.dist.{u1} α (PseudoMetricSpace.toDist.{u1} α _inst_1) y x) ε) -> (p y))))
+Case conversion may be inaccurate. Consider using '#align metric.eventually_nhds_iff Metric.eventually_nhds_iffₓ'. -/
 theorem eventually_nhds_iff {p : α → Prop} :
     (∀ᶠ y in 𝓝 x, p y) ↔ ∃ ε > 0, ∀ ⦃y⦄, dist y x < ε → p y :=
   mem_nhds_iff
 #align metric.eventually_nhds_iff Metric.eventually_nhds_iff
 
+/- warning: metric.eventually_nhds_iff_ball -> Metric.eventually_nhds_iff_ball is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {x : α} {p : α -> Prop}, Iff (Filter.Eventually.{u1} α (fun (y : α) => p y) (nhds.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) x)) (Exists.{1} Real (fun (ε : Real) => Exists.{0} (GT.gt.{0} Real Real.hasLt ε (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero)))) (fun (H : GT.gt.{0} Real Real.hasLt ε (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero)))) => forall (y : α), (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) y (Metric.ball.{u1} α _inst_1 x ε)) -> (p y))))
+but is expected to have type
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {x : α} {p : α -> Prop}, Iff (Filter.Eventually.{u1} α (fun (y : α) => p y) (nhds.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) x)) (Exists.{1} Real (fun (ε : Real) => And (GT.gt.{0} Real Real.instLTReal ε (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal))) (forall (y : α), (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) y (Metric.ball.{u1} α _inst_1 x ε)) -> (p y))))
+Case conversion may be inaccurate. Consider using '#align metric.eventually_nhds_iff_ball Metric.eventually_nhds_iff_ballₓ'. -/
 theorem eventually_nhds_iff_ball {p : α → Prop} :
     (∀ᶠ y in 𝓝 x, p y) ↔ ∃ ε > 0, ∀ y ∈ ball x ε, p y :=
   mem_nhds_iff
 #align metric.eventually_nhds_iff_ball Metric.eventually_nhds_iff_ball
 
+/- warning: metric.eventually_prod_nhds_iff -> Metric.eventually_prod_nhds_iff is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} {ι : Type.{u2}} [_inst_1 : PseudoMetricSpace.{u1} α] {f : Filter.{u2} ι} {x₀ : α} {p : (Prod.{u2, u1} ι α) -> Prop}, Iff (Filter.Eventually.{max u2 u1} (Prod.{u2, u1} ι α) (fun (x : Prod.{u2, u1} ι α) => p x) (Filter.prod.{u2, u1} ι α f (nhds.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) x₀))) (Exists.{succ u2} (ι -> Prop) (fun (pa : ι -> Prop) => Exists.{0} (Filter.Eventually.{u2} ι (fun (i : ι) => pa i) f) (fun (ha : Filter.Eventually.{u2} ι (fun (i : ι) => pa i) f) => Exists.{1} Real (fun (ε : Real) => Exists.{0} (GT.gt.{0} Real Real.hasLt ε (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero)))) (fun (H : GT.gt.{0} Real Real.hasLt ε (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero)))) => forall {i : ι}, (pa i) -> (forall {x : α}, (LT.lt.{0} Real Real.hasLt (Dist.dist.{u1} α (PseudoMetricSpace.toHasDist.{u1} α _inst_1) x x₀) ε) -> (p (Prod.mk.{u2, u1} ι α i x))))))))
+but is expected to have type
+  forall {α : Type.{u2}} {ι : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u2} α] {f : Filter.{u1} ι} {x₀ : α} {p : (Prod.{u1, u2} ι α) -> Prop}, Iff (Filter.Eventually.{max u2 u1} (Prod.{u1, u2} ι α) (fun (x : Prod.{u1, u2} ι α) => p x) (Filter.prod.{u1, u2} ι α f (nhds.{u2} α (UniformSpace.toTopologicalSpace.{u2} α (PseudoMetricSpace.toUniformSpace.{u2} α _inst_1)) x₀))) (Exists.{succ u1} (ι -> Prop) (fun (pa : ι -> Prop) => And (Filter.Eventually.{u1} ι (fun (i : ι) => pa i) f) (Exists.{1} Real (fun (ε : Real) => And (GT.gt.{0} Real Real.instLTReal ε (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal))) (forall {i : ι}, (pa i) -> (forall {x : α}, (LT.lt.{0} Real Real.instLTReal (Dist.dist.{u2} α (PseudoMetricSpace.toDist.{u2} α _inst_1) x x₀) ε) -> (p (Prod.mk.{u1, u2} ι α i x))))))))
+Case conversion may be inaccurate. Consider using '#align metric.eventually_prod_nhds_iff Metric.eventually_prod_nhds_iffₓ'. -/
 /-- A version of `filter.eventually_prod_iff` where the second filter consists of neighborhoods
 in a pseudo-metric space.-/
 theorem eventually_prod_nhds_iff {f : Filter ι} {x₀ : α} {p : ι × α → Prop} :
@@ -1062,6 +1778,12 @@ theorem eventually_prod_nhds_iff {f : Filter ι} {x₀ : α} {p : ι × α → P
     exact ⟨fun x => dist x x₀ < ε, ⟨ε, hε, fun y => id⟩, @hp⟩
 #align metric.eventually_prod_nhds_iff Metric.eventually_prod_nhds_iff
 
+/- warning: metric.eventually_nhds_prod_iff -> Metric.eventually_nhds_prod_iff is a dubious translation:
+lean 3 declaration is
+  forall {ι : Type.{u1}} {α : Type.{u2}} [_inst_2 : PseudoMetricSpace.{u2} α] {f : Filter.{u1} ι} {x₀ : α} {p : (Prod.{u2, u1} α ι) -> Prop}, Iff (Filter.Eventually.{max u2 u1} (Prod.{u2, u1} α ι) (fun (x : Prod.{u2, u1} α ι) => p x) (Filter.prod.{u2, u1} α ι (nhds.{u2} α (UniformSpace.toTopologicalSpace.{u2} α (PseudoMetricSpace.toUniformSpace.{u2} α _inst_2)) x₀) f)) (Exists.{1} Real (fun (ε : Real) => Exists.{0} (GT.gt.{0} Real Real.hasLt ε (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero)))) (fun (H : GT.gt.{0} Real Real.hasLt ε (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero)))) => Exists.{succ u1} (ι -> Prop) (fun (pa : ι -> Prop) => Exists.{0} (Filter.Eventually.{u1} ι (fun (i : ι) => pa i) f) (fun (ha : Filter.Eventually.{u1} ι (fun (i : ι) => pa i) f) => forall {x : α}, (LT.lt.{0} Real Real.hasLt (Dist.dist.{u2} α (PseudoMetricSpace.toHasDist.{u2} α _inst_2) x x₀) ε) -> (forall {i : ι}, (pa i) -> (p (Prod.mk.{u2, u1} α ι x i))))))))
+but is expected to have type
+  forall {ι : Type.{u2}} {α : Type.{u1}} [_inst_2 : PseudoMetricSpace.{u1} α] {f : Filter.{u2} ι} {x₀ : α} {p : (Prod.{u1, u2} α ι) -> Prop}, Iff (Filter.Eventually.{max u1 u2} (Prod.{u1, u2} α ι) (fun (x : Prod.{u1, u2} α ι) => p x) (Filter.prod.{u1, u2} α ι (nhds.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_2)) x₀) f)) (Exists.{1} Real (fun (ε : Real) => And (GT.gt.{0} Real Real.instLTReal ε (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal))) (Exists.{succ u2} (ι -> Prop) (fun (pa : ι -> Prop) => And (Filter.Eventually.{u2} ι (fun (i : ι) => pa i) f) (forall {x : α}, (LT.lt.{0} Real Real.instLTReal (Dist.dist.{u1} α (PseudoMetricSpace.toDist.{u1} α _inst_2) x x₀) ε) -> (forall {i : ι}, (pa i) -> (p (Prod.mk.{u1, u2} α ι x i))))))))
+Case conversion may be inaccurate. Consider using '#align metric.eventually_nhds_prod_iff Metric.eventually_nhds_prod_iffₓ'. -/
 /-- A version of `filter.eventually_prod_iff` where the first filter consists of neighborhoods
 in a pseudo-metric space.-/
 theorem eventually_nhds_prod_iff {ι α} [PseudoMetricSpace α] {f : Filter ι} {x₀ : α}
@@ -1076,59 +1798,129 @@ theorem eventually_nhds_prod_iff {ι α} [PseudoMetricSpace α] {f : Filter ι}
       refine' ⟨a3, a4, a1, a2, fun b1 b2 b3 b4 => a5 b4 b2⟩
 #align metric.eventually_nhds_prod_iff Metric.eventually_nhds_prod_iff
 
+/- warning: metric.nhds_basis_closed_ball -> Metric.nhds_basis_closedBall is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {x : α}, Filter.HasBasis.{u1, 1} α Real (nhds.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) x) (fun (ε : Real) => LT.lt.{0} Real Real.hasLt (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero))) ε) (Metric.closedBall.{u1} α _inst_1 x)
+but is expected to have type
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {x : α}, Filter.HasBasis.{u1, 1} α Real (nhds.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) x) (fun (ε : Real) => LT.lt.{0} Real Real.instLTReal (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal)) ε) (Metric.closedBall.{u1} α _inst_1 x)
+Case conversion may be inaccurate. Consider using '#align metric.nhds_basis_closed_ball Metric.nhds_basis_closedBallₓ'. -/
 theorem nhds_basis_closedBall : (𝓝 x).HasBasis (fun ε : ℝ => 0 < ε) (closedBall x) :=
   nhds_basis_uniformity uniformity_basis_dist_le
 #align metric.nhds_basis_closed_ball Metric.nhds_basis_closedBall
 
+/- warning: metric.nhds_basis_ball_inv_nat_succ -> Metric.nhds_basis_ball_inv_nat_succ is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {x : α}, Filter.HasBasis.{u1, 1} α Nat (nhds.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) x) (fun (_x : Nat) => True) (fun (n : Nat) => Metric.ball.{u1} α _inst_1 x (HDiv.hDiv.{0, 0, 0} Real Real Real (instHDiv.{0} Real (DivInvMonoid.toHasDiv.{0} Real (DivisionRing.toDivInvMonoid.{0} Real Real.divisionRing))) (OfNat.ofNat.{0} Real 1 (OfNat.mk.{0} Real 1 (One.one.{0} Real Real.hasOne))) (HAdd.hAdd.{0, 0, 0} Real Real Real (instHAdd.{0} Real Real.hasAdd) ((fun (a : Type) (b : Type) [self : HasLiftT.{1, 1} a b] => self.0) Nat Real (HasLiftT.mk.{1, 1} Nat Real (CoeTCₓ.coe.{1, 1} Nat Real (Nat.castCoe.{0} Real Real.hasNatCast))) n) (OfNat.ofNat.{0} Real 1 (OfNat.mk.{0} Real 1 (One.one.{0} Real Real.hasOne))))))
+but is expected to have type
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {x : α}, Filter.HasBasis.{u1, 1} α Nat (nhds.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) x) (fun (_x : Nat) => True) (fun (n : Nat) => Metric.ball.{u1} α _inst_1 x (HDiv.hDiv.{0, 0, 0} Real Real Real (instHDiv.{0} Real (LinearOrderedField.toDiv.{0} Real Real.instLinearOrderedFieldReal)) (OfNat.ofNat.{0} Real 1 (One.toOfNat1.{0} Real Real.instOneReal)) (HAdd.hAdd.{0, 0, 0} Real Real Real (instHAdd.{0} Real Real.instAddReal) (Nat.cast.{0} Real Real.natCast n) (OfNat.ofNat.{0} Real 1 (One.toOfNat1.{0} Real Real.instOneReal)))))
+Case conversion may be inaccurate. Consider using '#align metric.nhds_basis_ball_inv_nat_succ Metric.nhds_basis_ball_inv_nat_succₓ'. -/
 theorem nhds_basis_ball_inv_nat_succ :
     (𝓝 x).HasBasis (fun _ => True) fun n : ℕ => ball x (1 / (↑n + 1)) :=
   nhds_basis_uniformity uniformity_basis_dist_inv_nat_succ
 #align metric.nhds_basis_ball_inv_nat_succ Metric.nhds_basis_ball_inv_nat_succ
 
+/- warning: metric.nhds_basis_ball_inv_nat_pos -> Metric.nhds_basis_ball_inv_nat_pos is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {x : α}, Filter.HasBasis.{u1, 1} α Nat (nhds.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) x) (fun (n : Nat) => LT.lt.{0} Nat Nat.hasLt (OfNat.ofNat.{0} Nat 0 (OfNat.mk.{0} Nat 0 (Zero.zero.{0} Nat Nat.hasZero))) n) (fun (n : Nat) => Metric.ball.{u1} α _inst_1 x (HDiv.hDiv.{0, 0, 0} Real Real Real (instHDiv.{0} Real (DivInvMonoid.toHasDiv.{0} Real (DivisionRing.toDivInvMonoid.{0} Real Real.divisionRing))) (OfNat.ofNat.{0} Real 1 (OfNat.mk.{0} Real 1 (One.one.{0} Real Real.hasOne))) ((fun (a : Type) (b : Type) [self : HasLiftT.{1, 1} a b] => self.0) Nat Real (HasLiftT.mk.{1, 1} Nat Real (CoeTCₓ.coe.{1, 1} Nat Real (Nat.castCoe.{0} Real Real.hasNatCast))) n)))
+but is expected to have type
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {x : α}, Filter.HasBasis.{u1, 1} α Nat (nhds.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) x) (fun (n : Nat) => LT.lt.{0} Nat instLTNat (OfNat.ofNat.{0} Nat 0 (instOfNatNat 0)) n) (fun (n : Nat) => Metric.ball.{u1} α _inst_1 x (HDiv.hDiv.{0, 0, 0} Real Real Real (instHDiv.{0} Real (LinearOrderedField.toDiv.{0} Real Real.instLinearOrderedFieldReal)) (OfNat.ofNat.{0} Real 1 (One.toOfNat1.{0} Real Real.instOneReal)) (Nat.cast.{0} Real Real.natCast n)))
+Case conversion may be inaccurate. Consider using '#align metric.nhds_basis_ball_inv_nat_pos Metric.nhds_basis_ball_inv_nat_posₓ'. -/
 theorem nhds_basis_ball_inv_nat_pos :
     (𝓝 x).HasBasis (fun n => 0 < n) fun n : ℕ => ball x (1 / ↑n) :=
   nhds_basis_uniformity uniformity_basis_dist_inv_nat_pos
 #align metric.nhds_basis_ball_inv_nat_pos Metric.nhds_basis_ball_inv_nat_pos
 
+/- warning: metric.nhds_basis_ball_pow -> Metric.nhds_basis_ball_pow is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {x : α} {r : Real}, (LT.lt.{0} Real Real.hasLt (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero))) r) -> (LT.lt.{0} Real Real.hasLt r (OfNat.ofNat.{0} Real 1 (OfNat.mk.{0} Real 1 (One.one.{0} Real Real.hasOne)))) -> (Filter.HasBasis.{u1, 1} α Nat (nhds.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) x) (fun (n : Nat) => True) (fun (n : Nat) => Metric.ball.{u1} α _inst_1 x (HPow.hPow.{0, 0, 0} Real Nat Real (instHPow.{0, 0} Real Nat (Monoid.Pow.{0} Real Real.monoid)) r n)))
+but is expected to have type
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {x : α} {r : Real}, (LT.lt.{0} Real Real.instLTReal (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal)) r) -> (LT.lt.{0} Real Real.instLTReal r (OfNat.ofNat.{0} Real 1 (One.toOfNat1.{0} Real Real.instOneReal))) -> (Filter.HasBasis.{u1, 1} α Nat (nhds.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) x) (fun (n : Nat) => True) (fun (n : Nat) => Metric.ball.{u1} α _inst_1 x (HPow.hPow.{0, 0, 0} Real Nat Real (instHPow.{0, 0} Real Nat (Monoid.Pow.{0} Real Real.instMonoidReal)) r n)))
+Case conversion may be inaccurate. Consider using '#align metric.nhds_basis_ball_pow Metric.nhds_basis_ball_powₓ'. -/
 theorem nhds_basis_ball_pow {r : ℝ} (h0 : 0 < r) (h1 : r < 1) :
     (𝓝 x).HasBasis (fun n => True) fun n : ℕ => ball x (r ^ n) :=
   nhds_basis_uniformity (uniformity_basis_dist_pow h0 h1)
 #align metric.nhds_basis_ball_pow Metric.nhds_basis_ball_pow
 
+/- warning: metric.nhds_basis_closed_ball_pow -> Metric.nhds_basis_closedBall_pow is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {x : α} {r : Real}, (LT.lt.{0} Real Real.hasLt (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero))) r) -> (LT.lt.{0} Real Real.hasLt r (OfNat.ofNat.{0} Real 1 (OfNat.mk.{0} Real 1 (One.one.{0} Real Real.hasOne)))) -> (Filter.HasBasis.{u1, 1} α Nat (nhds.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) x) (fun (n : Nat) => True) (fun (n : Nat) => Metric.closedBall.{u1} α _inst_1 x (HPow.hPow.{0, 0, 0} Real Nat Real (instHPow.{0, 0} Real Nat (Monoid.Pow.{0} Real Real.monoid)) r n)))
+but is expected to have type
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {x : α} {r : Real}, (LT.lt.{0} Real Real.instLTReal (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal)) r) -> (LT.lt.{0} Real Real.instLTReal r (OfNat.ofNat.{0} Real 1 (One.toOfNat1.{0} Real Real.instOneReal))) -> (Filter.HasBasis.{u1, 1} α Nat (nhds.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) x) (fun (n : Nat) => True) (fun (n : Nat) => Metric.closedBall.{u1} α _inst_1 x (HPow.hPow.{0, 0, 0} Real Nat Real (instHPow.{0, 0} Real Nat (Monoid.Pow.{0} Real Real.instMonoidReal)) r n)))
+Case conversion may be inaccurate. Consider using '#align metric.nhds_basis_closed_ball_pow Metric.nhds_basis_closedBall_powₓ'. -/
 theorem nhds_basis_closedBall_pow {r : ℝ} (h0 : 0 < r) (h1 : r < 1) :
     (𝓝 x).HasBasis (fun n => True) fun n : ℕ => closedBall x (r ^ n) :=
   nhds_basis_uniformity (uniformity_basis_dist_le_pow h0 h1)
 #align metric.nhds_basis_closed_ball_pow Metric.nhds_basis_closedBall_pow
 
+/- warning: metric.is_open_iff -> Metric.isOpen_iff is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {s : Set.{u1} α}, Iff (IsOpen.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) s) (forall (x : α), (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) x s) -> (Exists.{1} Real (fun (ε : Real) => Exists.{0} (GT.gt.{0} Real Real.hasLt ε (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero)))) (fun (H : GT.gt.{0} Real Real.hasLt ε (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero)))) => HasSubset.Subset.{u1} (Set.{u1} α) (Set.hasSubset.{u1} α) (Metric.ball.{u1} α _inst_1 x ε) s))))
+but is expected to have type
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {s : Set.{u1} α}, Iff (IsOpen.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) s) (forall (x : α), (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) x s) -> (Exists.{1} Real (fun (ε : Real) => And (GT.gt.{0} Real Real.instLTReal ε (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal))) (HasSubset.Subset.{u1} (Set.{u1} α) (Set.instHasSubsetSet.{u1} α) (Metric.ball.{u1} α _inst_1 x ε) s))))
+Case conversion may be inaccurate. Consider using '#align metric.is_open_iff Metric.isOpen_iffₓ'. -/
 theorem isOpen_iff : IsOpen s ↔ ∀ x ∈ s, ∃ ε > 0, ball x ε ⊆ s := by
   simp only [isOpen_iff_mem_nhds, mem_nhds_iff]
 #align metric.is_open_iff Metric.isOpen_iff
 
+#print Metric.isOpen_ball /-
 theorem isOpen_ball : IsOpen (ball x ε) :=
   isOpen_iff.2 fun y => exists_ball_subset_ball
 #align metric.is_open_ball Metric.isOpen_ball
+-/
 
+/- warning: metric.ball_mem_nhds -> Metric.ball_mem_nhds is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] (x : α) {ε : Real}, (LT.lt.{0} Real Real.hasLt (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero))) ε) -> (Membership.Mem.{u1, u1} (Set.{u1} α) (Filter.{u1} α) (Filter.hasMem.{u1} α) (Metric.ball.{u1} α _inst_1 x ε) (nhds.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) x))
+but is expected to have type
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] (x : α) {ε : Real}, (LT.lt.{0} Real Real.instLTReal (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal)) ε) -> (Membership.mem.{u1, u1} (Set.{u1} α) (Filter.{u1} α) (instMembershipSetFilter.{u1} α) (Metric.ball.{u1} α _inst_1 x ε) (nhds.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) x))
+Case conversion may be inaccurate. Consider using '#align metric.ball_mem_nhds Metric.ball_mem_nhdsₓ'. -/
 theorem ball_mem_nhds (x : α) {ε : ℝ} (ε0 : 0 < ε) : ball x ε ∈ 𝓝 x :=
   isOpen_ball.mem_nhds (mem_ball_self ε0)
 #align metric.ball_mem_nhds Metric.ball_mem_nhds
 
+/- warning: metric.closed_ball_mem_nhds -> Metric.closedBall_mem_nhds is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] (x : α) {ε : Real}, (LT.lt.{0} Real Real.hasLt (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero))) ε) -> (Membership.Mem.{u1, u1} (Set.{u1} α) (Filter.{u1} α) (Filter.hasMem.{u1} α) (Metric.closedBall.{u1} α _inst_1 x ε) (nhds.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) x))
+but is expected to have type
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] (x : α) {ε : Real}, (LT.lt.{0} Real Real.instLTReal (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal)) ε) -> (Membership.mem.{u1, u1} (Set.{u1} α) (Filter.{u1} α) (instMembershipSetFilter.{u1} α) (Metric.closedBall.{u1} α _inst_1 x ε) (nhds.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) x))
+Case conversion may be inaccurate. Consider using '#align metric.closed_ball_mem_nhds Metric.closedBall_mem_nhdsₓ'. -/
 theorem closedBall_mem_nhds (x : α) {ε : ℝ} (ε0 : 0 < ε) : closedBall x ε ∈ 𝓝 x :=
   mem_of_superset (ball_mem_nhds x ε0) ball_subset_closedBall
 #align metric.closed_ball_mem_nhds Metric.closedBall_mem_nhds
 
+#print Metric.closedBall_mem_nhds_of_mem /-
 theorem closedBall_mem_nhds_of_mem {x c : α} {ε : ℝ} (h : x ∈ ball c ε) : closedBall c ε ∈ 𝓝 x :=
   mem_of_superset (isOpen_ball.mem_nhds h) ball_subset_closedBall
 #align metric.closed_ball_mem_nhds_of_mem Metric.closedBall_mem_nhds_of_mem
+-/
 
+/- warning: metric.nhds_within_basis_ball -> Metric.nhdsWithin_basis_ball is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {x : α} {s : Set.{u1} α}, Filter.HasBasis.{u1, 1} α Real (nhdsWithin.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) x s) (fun (ε : Real) => LT.lt.{0} Real Real.hasLt (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero))) ε) (fun (ε : Real) => Inter.inter.{u1} (Set.{u1} α) (Set.hasInter.{u1} α) (Metric.ball.{u1} α _inst_1 x ε) s)
+but is expected to have type
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {x : α} {s : Set.{u1} α}, Filter.HasBasis.{u1, 1} α Real (nhdsWithin.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) x s) (fun (ε : Real) => LT.lt.{0} Real Real.instLTReal (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal)) ε) (fun (ε : Real) => Inter.inter.{u1} (Set.{u1} α) (Set.instInterSet.{u1} α) (Metric.ball.{u1} α _inst_1 x ε) s)
+Case conversion may be inaccurate. Consider using '#align metric.nhds_within_basis_ball Metric.nhdsWithin_basis_ballₓ'. -/
 theorem nhdsWithin_basis_ball {s : Set α} :
     (𝓝[s] x).HasBasis (fun ε : ℝ => 0 < ε) fun ε => ball x ε ∩ s :=
   nhdsWithin_hasBasis nhds_basis_ball s
 #align metric.nhds_within_basis_ball Metric.nhdsWithin_basis_ball
 
+/- warning: metric.mem_nhds_within_iff -> Metric.mem_nhdsWithin_iff is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {x : α} {s : Set.{u1} α} {t : Set.{u1} α}, Iff (Membership.Mem.{u1, u1} (Set.{u1} α) (Filter.{u1} α) (Filter.hasMem.{u1} α) s (nhdsWithin.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) x t)) (Exists.{1} Real (fun (ε : Real) => Exists.{0} (GT.gt.{0} Real Real.hasLt ε (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero)))) (fun (H : GT.gt.{0} Real Real.hasLt ε (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero)))) => HasSubset.Subset.{u1} (Set.{u1} α) (Set.hasSubset.{u1} α) (Inter.inter.{u1} (Set.{u1} α) (Set.hasInter.{u1} α) (Metric.ball.{u1} α _inst_1 x ε) t) s)))
+but is expected to have type
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {x : α} {s : Set.{u1} α} {t : Set.{u1} α}, Iff (Membership.mem.{u1, u1} (Set.{u1} α) (Filter.{u1} α) (instMembershipSetFilter.{u1} α) s (nhdsWithin.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) x t)) (Exists.{1} Real (fun (ε : Real) => And (GT.gt.{0} Real Real.instLTReal ε (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal))) (HasSubset.Subset.{u1} (Set.{u1} α) (Set.instHasSubsetSet.{u1} α) (Inter.inter.{u1} (Set.{u1} α) (Set.instInterSet.{u1} α) (Metric.ball.{u1} α _inst_1 x ε) t) s)))
+Case conversion may be inaccurate. Consider using '#align metric.mem_nhds_within_iff Metric.mem_nhdsWithin_iffₓ'. -/
 theorem mem_nhdsWithin_iff {t : Set α} : s ∈ 𝓝[t] x ↔ ∃ ε > 0, ball x ε ∩ t ⊆ s :=
   nhdsWithin_basis_ball.mem_iff
 #align metric.mem_nhds_within_iff Metric.mem_nhdsWithin_iff
 
+/- warning: metric.tendsto_nhds_within_nhds_within -> Metric.tendsto_nhdsWithin_nhdsWithin is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : PseudoMetricSpace.{u1} α] {s : Set.{u1} α} [_inst_2 : PseudoMetricSpace.{u2} β] {t : Set.{u2} β} {f : α -> β} {a : α} {b : β}, Iff (Filter.Tendsto.{u1, u2} α β f (nhdsWithin.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) a s) (nhdsWithin.{u2} β (UniformSpace.toTopologicalSpace.{u2} β (PseudoMetricSpace.toUniformSpace.{u2} β _inst_2)) b t)) (forall (ε : Real), (GT.gt.{0} Real Real.hasLt ε (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero)))) -> (Exists.{1} Real (fun (δ : Real) => Exists.{0} (GT.gt.{0} Real Real.hasLt δ (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero)))) (fun (H : GT.gt.{0} Real Real.hasLt δ (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero)))) => forall {x : α}, (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) x s) -> (LT.lt.{0} Real Real.hasLt (Dist.dist.{u1} α (PseudoMetricSpace.toHasDist.{u1} α _inst_1) x a) δ) -> (And (Membership.Mem.{u2, u2} β (Set.{u2} β) (Set.hasMem.{u2} β) (f x) t) (LT.lt.{0} Real Real.hasLt (Dist.dist.{u2} β (PseudoMetricSpace.toHasDist.{u2} β _inst_2) (f x) b) ε))))))
+but is expected to have type
+  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : PseudoMetricSpace.{u1} α] {s : Set.{u1} α} [_inst_2 : PseudoMetricSpace.{u2} β] {t : Set.{u2} β} {f : α -> β} {a : α} {b : β}, Iff (Filter.Tendsto.{u1, u2} α β f (nhdsWithin.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) a s) (nhdsWithin.{u2} β (UniformSpace.toTopologicalSpace.{u2} β (PseudoMetricSpace.toUniformSpace.{u2} β _inst_2)) b t)) (forall (ε : Real), (GT.gt.{0} Real Real.instLTReal ε (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal))) -> (Exists.{1} Real (fun (δ : Real) => And (GT.gt.{0} Real Real.instLTReal δ (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal))) (forall {x : α}, (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) x s) -> (LT.lt.{0} Real Real.instLTReal (Dist.dist.{u1} α (PseudoMetricSpace.toDist.{u1} α _inst_1) x a) δ) -> (And (Membership.mem.{u2, u2} β (Set.{u2} β) (Set.instMembershipSet.{u2} β) (f x) t) (LT.lt.{0} Real Real.instLTReal (Dist.dist.{u2} β (PseudoMetricSpace.toDist.{u2} β _inst_2) (f x) b) ε))))))
+Case conversion may be inaccurate. Consider using '#align metric.tendsto_nhds_within_nhds_within Metric.tendsto_nhdsWithin_nhdsWithinₓ'. -/
 theorem tendsto_nhdsWithin_nhdsWithin [PseudoMetricSpace β] {t : Set β} {f : α → β} {a b} :
     Tendsto f (𝓝[s] a) (𝓝[t] b) ↔
       ∀ ε > 0, ∃ δ > 0, ∀ {x : α}, x ∈ s → dist x a < δ → f x ∈ t ∧ dist (f x) b < ε :=
@@ -1136,6 +1928,12 @@ theorem tendsto_nhdsWithin_nhdsWithin [PseudoMetricSpace β] {t : Set β} {f : 
     forall₂_congr fun ε hε => exists₂_congr fun δ hδ => forall_congr' fun x => by simp <;> itauto
 #align metric.tendsto_nhds_within_nhds_within Metric.tendsto_nhdsWithin_nhdsWithin
 
+/- warning: metric.tendsto_nhds_within_nhds -> Metric.tendsto_nhdsWithin_nhds is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : PseudoMetricSpace.{u1} α] {s : Set.{u1} α} [_inst_2 : PseudoMetricSpace.{u2} β] {f : α -> β} {a : α} {b : β}, Iff (Filter.Tendsto.{u1, u2} α β f (nhdsWithin.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) a s) (nhds.{u2} β (UniformSpace.toTopologicalSpace.{u2} β (PseudoMetricSpace.toUniformSpace.{u2} β _inst_2)) b)) (forall (ε : Real), (GT.gt.{0} Real Real.hasLt ε (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero)))) -> (Exists.{1} Real (fun (δ : Real) => Exists.{0} (GT.gt.{0} Real Real.hasLt δ (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero)))) (fun (H : GT.gt.{0} Real Real.hasLt δ (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero)))) => forall {x : α}, (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) x s) -> (LT.lt.{0} Real Real.hasLt (Dist.dist.{u1} α (PseudoMetricSpace.toHasDist.{u1} α _inst_1) x a) δ) -> (LT.lt.{0} Real Real.hasLt (Dist.dist.{u2} β (PseudoMetricSpace.toHasDist.{u2} β _inst_2) (f x) b) ε)))))
+but is expected to have type
+  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : PseudoMetricSpace.{u1} α] {s : Set.{u1} α} [_inst_2 : PseudoMetricSpace.{u2} β] {f : α -> β} {a : α} {b : β}, Iff (Filter.Tendsto.{u1, u2} α β f (nhdsWithin.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) a s) (nhds.{u2} β (UniformSpace.toTopologicalSpace.{u2} β (PseudoMetricSpace.toUniformSpace.{u2} β _inst_2)) b)) (forall (ε : Real), (GT.gt.{0} Real Real.instLTReal ε (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal))) -> (Exists.{1} Real (fun (δ : Real) => And (GT.gt.{0} Real Real.instLTReal δ (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal))) (forall {x : α}, (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) x s) -> (LT.lt.{0} Real Real.instLTReal (Dist.dist.{u1} α (PseudoMetricSpace.toDist.{u1} α _inst_1) x a) δ) -> (LT.lt.{0} Real Real.instLTReal (Dist.dist.{u2} β (PseudoMetricSpace.toDist.{u2} β _inst_2) (f x) b) ε)))))
+Case conversion may be inaccurate. Consider using '#align metric.tendsto_nhds_within_nhds Metric.tendsto_nhdsWithin_nhdsₓ'. -/
 theorem tendsto_nhdsWithin_nhds [PseudoMetricSpace β] {f : α → β} {a b} :
     Tendsto f (𝓝[s] a) (𝓝 b) ↔
       ∀ ε > 0, ∃ δ > 0, ∀ {x : α}, x ∈ s → dist x a < δ → dist (f x) b < ε :=
@@ -1144,57 +1942,123 @@ theorem tendsto_nhdsWithin_nhds [PseudoMetricSpace β] {f : α → β} {a b} :
   simp only [mem_univ, true_and_iff]
 #align metric.tendsto_nhds_within_nhds Metric.tendsto_nhdsWithin_nhds
 
+/- warning: metric.tendsto_nhds_nhds -> Metric.tendsto_nhds_nhds is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : PseudoMetricSpace.{u1} α] [_inst_2 : PseudoMetricSpace.{u2} β] {f : α -> β} {a : α} {b : β}, Iff (Filter.Tendsto.{u1, u2} α β f (nhds.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) a) (nhds.{u2} β (UniformSpace.toTopologicalSpace.{u2} β (PseudoMetricSpace.toUniformSpace.{u2} β _inst_2)) b)) (forall (ε : Real), (GT.gt.{0} Real Real.hasLt ε (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero)))) -> (Exists.{1} Real (fun (δ : Real) => Exists.{0} (GT.gt.{0} Real Real.hasLt δ (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero)))) (fun (H : GT.gt.{0} Real Real.hasLt δ (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero)))) => forall {x : α}, (LT.lt.{0} Real Real.hasLt (Dist.dist.{u1} α (PseudoMetricSpace.toHasDist.{u1} α _inst_1) x a) δ) -> (LT.lt.{0} Real Real.hasLt (Dist.dist.{u2} β (PseudoMetricSpace.toHasDist.{u2} β _inst_2) (f x) b) ε)))))
+but is expected to have type
+  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : PseudoMetricSpace.{u1} α] [_inst_2 : PseudoMetricSpace.{u2} β] {f : α -> β} {a : α} {b : β}, Iff (Filter.Tendsto.{u1, u2} α β f (nhds.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) a) (nhds.{u2} β (UniformSpace.toTopologicalSpace.{u2} β (PseudoMetricSpace.toUniformSpace.{u2} β _inst_2)) b)) (forall (ε : Real), (GT.gt.{0} Real Real.instLTReal ε (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal))) -> (Exists.{1} Real (fun (δ : Real) => And (GT.gt.{0} Real Real.instLTReal δ (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal))) (forall {x : α}, (LT.lt.{0} Real Real.instLTReal (Dist.dist.{u1} α (PseudoMetricSpace.toDist.{u1} α _inst_1) x a) δ) -> (LT.lt.{0} Real Real.instLTReal (Dist.dist.{u2} β (PseudoMetricSpace.toDist.{u2} β _inst_2) (f x) b) ε)))))
+Case conversion may be inaccurate. Consider using '#align metric.tendsto_nhds_nhds Metric.tendsto_nhds_nhdsₓ'. -/
 theorem tendsto_nhds_nhds [PseudoMetricSpace β] {f : α → β} {a b} :
     Tendsto f (𝓝 a) (𝓝 b) ↔ ∀ ε > 0, ∃ δ > 0, ∀ {x : α}, dist x a < δ → dist (f x) b < ε :=
   nhds_basis_ball.tendsto_iffₓ nhds_basis_ball
 #align metric.tendsto_nhds_nhds Metric.tendsto_nhds_nhds
 
+/- warning: metric.continuous_at_iff -> Metric.continuousAt_iff is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : PseudoMetricSpace.{u1} α] [_inst_2 : PseudoMetricSpace.{u2} β] {f : α -> β} {a : α}, Iff (ContinuousAt.{u1, u2} α β (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) (UniformSpace.toTopologicalSpace.{u2} β (PseudoMetricSpace.toUniformSpace.{u2} β _inst_2)) f a) (forall (ε : Real), (GT.gt.{0} Real Real.hasLt ε (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero)))) -> (Exists.{1} Real (fun (δ : Real) => Exists.{0} (GT.gt.{0} Real Real.hasLt δ (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero)))) (fun (H : GT.gt.{0} Real Real.hasLt δ (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero)))) => forall {x : α}, (LT.lt.{0} Real Real.hasLt (Dist.dist.{u1} α (PseudoMetricSpace.toHasDist.{u1} α _inst_1) x a) δ) -> (LT.lt.{0} Real Real.hasLt (Dist.dist.{u2} β (PseudoMetricSpace.toHasDist.{u2} β _inst_2) (f x) (f a)) ε)))))
+but is expected to have type
+  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : PseudoMetricSpace.{u1} α] [_inst_2 : PseudoMetricSpace.{u2} β] {f : α -> β} {a : α}, Iff (ContinuousAt.{u1, u2} α β (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) (UniformSpace.toTopologicalSpace.{u2} β (PseudoMetricSpace.toUniformSpace.{u2} β _inst_2)) f a) (forall (ε : Real), (GT.gt.{0} Real Real.instLTReal ε (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal))) -> (Exists.{1} Real (fun (δ : Real) => And (GT.gt.{0} Real Real.instLTReal δ (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal))) (forall {x : α}, (LT.lt.{0} Real Real.instLTReal (Dist.dist.{u1} α (PseudoMetricSpace.toDist.{u1} α _inst_1) x a) δ) -> (LT.lt.{0} Real Real.instLTReal (Dist.dist.{u2} β (PseudoMetricSpace.toDist.{u2} β _inst_2) (f x) (f a)) ε)))))
+Case conversion may be inaccurate. Consider using '#align metric.continuous_at_iff Metric.continuousAt_iffₓ'. -/
 theorem continuousAt_iff [PseudoMetricSpace β] {f : α → β} {a : α} :
     ContinuousAt f a ↔ ∀ ε > 0, ∃ δ > 0, ∀ {x : α}, dist x a < δ → dist (f x) (f a) < ε := by
   rw [ContinuousAt, tendsto_nhds_nhds]
 #align metric.continuous_at_iff Metric.continuousAt_iff
 
+/- warning: metric.continuous_within_at_iff -> Metric.continuousWithinAt_iff is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : PseudoMetricSpace.{u1} α] [_inst_2 : PseudoMetricSpace.{u2} β] {f : α -> β} {a : α} {s : Set.{u1} α}, Iff (ContinuousWithinAt.{u1, u2} α β (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) (UniformSpace.toTopologicalSpace.{u2} β (PseudoMetricSpace.toUniformSpace.{u2} β _inst_2)) f s a) (forall (ε : Real), (GT.gt.{0} Real Real.hasLt ε (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero)))) -> (Exists.{1} Real (fun (δ : Real) => Exists.{0} (GT.gt.{0} Real Real.hasLt δ (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero)))) (fun (H : GT.gt.{0} Real Real.hasLt δ (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero)))) => forall {x : α}, (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) x s) -> (LT.lt.{0} Real Real.hasLt (Dist.dist.{u1} α (PseudoMetricSpace.toHasDist.{u1} α _inst_1) x a) δ) -> (LT.lt.{0} Real Real.hasLt (Dist.dist.{u2} β (PseudoMetricSpace.toHasDist.{u2} β _inst_2) (f x) (f a)) ε)))))
+but is expected to have type
+  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : PseudoMetricSpace.{u1} α] [_inst_2 : PseudoMetricSpace.{u2} β] {f : α -> β} {a : α} {s : Set.{u1} α}, Iff (ContinuousWithinAt.{u1, u2} α β (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) (UniformSpace.toTopologicalSpace.{u2} β (PseudoMetricSpace.toUniformSpace.{u2} β _inst_2)) f s a) (forall (ε : Real), (GT.gt.{0} Real Real.instLTReal ε (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal))) -> (Exists.{1} Real (fun (δ : Real) => And (GT.gt.{0} Real Real.instLTReal δ (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal))) (forall {x : α}, (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) x s) -> (LT.lt.{0} Real Real.instLTReal (Dist.dist.{u1} α (PseudoMetricSpace.toDist.{u1} α _inst_1) x a) δ) -> (LT.lt.{0} Real Real.instLTReal (Dist.dist.{u2} β (PseudoMetricSpace.toDist.{u2} β _inst_2) (f x) (f a)) ε)))))
+Case conversion may be inaccurate. Consider using '#align metric.continuous_within_at_iff Metric.continuousWithinAt_iffₓ'. -/
 theorem continuousWithinAt_iff [PseudoMetricSpace β] {f : α → β} {a : α} {s : Set α} :
     ContinuousWithinAt f s a ↔
       ∀ ε > 0, ∃ δ > 0, ∀ {x : α}, x ∈ s → dist x a < δ → dist (f x) (f a) < ε :=
   by rw [ContinuousWithinAt, tendsto_nhds_within_nhds]
 #align metric.continuous_within_at_iff Metric.continuousWithinAt_iff
 
+/- warning: metric.continuous_on_iff -> Metric.continuousOn_iff is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : PseudoMetricSpace.{u1} α] [_inst_2 : PseudoMetricSpace.{u2} β] {f : α -> β} {s : Set.{u1} α}, Iff (ContinuousOn.{u1, u2} α β (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) (UniformSpace.toTopologicalSpace.{u2} β (PseudoMetricSpace.toUniformSpace.{u2} β _inst_2)) f s) (forall (b : α), (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) b s) -> (forall (ε : Real), (GT.gt.{0} Real Real.hasLt ε (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero)))) -> (Exists.{1} Real (fun (δ : Real) => Exists.{0} (GT.gt.{0} Real Real.hasLt δ (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero)))) (fun (H : GT.gt.{0} Real Real.hasLt δ (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero)))) => forall (a : α), (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) a s) -> (LT.lt.{0} Real Real.hasLt (Dist.dist.{u1} α (PseudoMetricSpace.toHasDist.{u1} α _inst_1) a b) δ) -> (LT.lt.{0} Real Real.hasLt (Dist.dist.{u2} β (PseudoMetricSpace.toHasDist.{u2} β _inst_2) (f a) (f b)) ε))))))
+but is expected to have type
+  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : PseudoMetricSpace.{u1} α] [_inst_2 : PseudoMetricSpace.{u2} β] {f : α -> β} {s : Set.{u1} α}, Iff (ContinuousOn.{u1, u2} α β (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) (UniformSpace.toTopologicalSpace.{u2} β (PseudoMetricSpace.toUniformSpace.{u2} β _inst_2)) f s) (forall (b : α), (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) b s) -> (forall (ε : Real), (GT.gt.{0} Real Real.instLTReal ε (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal))) -> (Exists.{1} Real (fun (δ : Real) => And (GT.gt.{0} Real Real.instLTReal δ (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal))) (forall (a : α), (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) a s) -> (LT.lt.{0} Real Real.instLTReal (Dist.dist.{u1} α (PseudoMetricSpace.toDist.{u1} α _inst_1) a b) δ) -> (LT.lt.{0} Real Real.instLTReal (Dist.dist.{u2} β (PseudoMetricSpace.toDist.{u2} β _inst_2) (f a) (f b)) ε))))))
+Case conversion may be inaccurate. Consider using '#align metric.continuous_on_iff Metric.continuousOn_iffₓ'. -/
 theorem continuousOn_iff [PseudoMetricSpace β] {f : α → β} {s : Set α} :
     ContinuousOn f s ↔ ∀ b ∈ s, ∀ ε > 0, ∃ δ > 0, ∀ a ∈ s, dist a b < δ → dist (f a) (f b) < ε := by
   simp [ContinuousOn, continuous_within_at_iff]
 #align metric.continuous_on_iff Metric.continuousOn_iff
 
+/- warning: metric.continuous_iff -> Metric.continuous_iff is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : PseudoMetricSpace.{u1} α] [_inst_2 : PseudoMetricSpace.{u2} β] {f : α -> β}, Iff (Continuous.{u1, u2} α β (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) (UniformSpace.toTopologicalSpace.{u2} β (PseudoMetricSpace.toUniformSpace.{u2} β _inst_2)) f) (forall (b : α) (ε : Real), (GT.gt.{0} Real Real.hasLt ε (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero)))) -> (Exists.{1} Real (fun (δ : Real) => Exists.{0} (GT.gt.{0} Real Real.hasLt δ (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero)))) (fun (H : GT.gt.{0} Real Real.hasLt δ (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero)))) => forall (a : α), (LT.lt.{0} Real Real.hasLt (Dist.dist.{u1} α (PseudoMetricSpace.toHasDist.{u1} α _inst_1) a b) δ) -> (LT.lt.{0} Real Real.hasLt (Dist.dist.{u2} β (PseudoMetricSpace.toHasDist.{u2} β _inst_2) (f a) (f b)) ε)))))
+but is expected to have type
+  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : PseudoMetricSpace.{u1} α] [_inst_2 : PseudoMetricSpace.{u2} β] {f : α -> β}, Iff (Continuous.{u1, u2} α β (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) (UniformSpace.toTopologicalSpace.{u2} β (PseudoMetricSpace.toUniformSpace.{u2} β _inst_2)) f) (forall (b : α) (ε : Real), (GT.gt.{0} Real Real.instLTReal ε (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal))) -> (Exists.{1} Real (fun (δ : Real) => And (GT.gt.{0} Real Real.instLTReal δ (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal))) (forall (a : α), (LT.lt.{0} Real Real.instLTReal (Dist.dist.{u1} α (PseudoMetricSpace.toDist.{u1} α _inst_1) a b) δ) -> (LT.lt.{0} Real Real.instLTReal (Dist.dist.{u2} β (PseudoMetricSpace.toDist.{u2} β _inst_2) (f a) (f b)) ε)))))
+Case conversion may be inaccurate. Consider using '#align metric.continuous_iff Metric.continuous_iffₓ'. -/
 theorem continuous_iff [PseudoMetricSpace β] {f : α → β} :
     Continuous f ↔ ∀ (b), ∀ ε > 0, ∃ δ > 0, ∀ a, dist a b < δ → dist (f a) (f b) < ε :=
   continuous_iff_continuousAt.trans <| forall_congr' fun b => tendsto_nhds_nhds
 #align metric.continuous_iff Metric.continuous_iff
 
+/- warning: metric.tendsto_nhds -> Metric.tendsto_nhds is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : PseudoMetricSpace.{u1} α] {f : Filter.{u2} β} {u : β -> α} {a : α}, Iff (Filter.Tendsto.{u2, u1} β α u f (nhds.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) a)) (forall (ε : Real), (GT.gt.{0} Real Real.hasLt ε (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero)))) -> (Filter.Eventually.{u2} β (fun (x : β) => LT.lt.{0} Real Real.hasLt (Dist.dist.{u1} α (PseudoMetricSpace.toHasDist.{u1} α _inst_1) (u x) a) ε) f))
+but is expected to have type
+  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : PseudoMetricSpace.{u1} α] {f : Filter.{u2} β} {u : β -> α} {a : α}, Iff (Filter.Tendsto.{u2, u1} β α u f (nhds.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) a)) (forall (ε : Real), (GT.gt.{0} Real Real.instLTReal ε (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal))) -> (Filter.Eventually.{u2} β (fun (x : β) => LT.lt.{0} Real Real.instLTReal (Dist.dist.{u1} α (PseudoMetricSpace.toDist.{u1} α _inst_1) (u x) a) ε) f))
+Case conversion may be inaccurate. Consider using '#align metric.tendsto_nhds Metric.tendsto_nhdsₓ'. -/
 theorem tendsto_nhds {f : Filter β} {u : β → α} {a : α} :
     Tendsto u f (𝓝 a) ↔ ∀ ε > 0, ∀ᶠ x in f, dist (u x) a < ε :=
   nhds_basis_ball.tendsto_right_iff
 #align metric.tendsto_nhds Metric.tendsto_nhds
 
+/- warning: metric.continuous_at_iff' -> Metric.continuousAt_iff' is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : PseudoMetricSpace.{u1} α] [_inst_2 : TopologicalSpace.{u2} β] {f : β -> α} {b : β}, Iff (ContinuousAt.{u2, u1} β α _inst_2 (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) f b) (forall (ε : Real), (GT.gt.{0} Real Real.hasLt ε (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero)))) -> (Filter.Eventually.{u2} β (fun (x : β) => LT.lt.{0} Real Real.hasLt (Dist.dist.{u1} α (PseudoMetricSpace.toHasDist.{u1} α _inst_1) (f x) (f b)) ε) (nhds.{u2} β _inst_2 b)))
+but is expected to have type
+  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : PseudoMetricSpace.{u1} α] [_inst_2 : TopologicalSpace.{u2} β] {f : β -> α} {b : β}, Iff (ContinuousAt.{u2, u1} β α _inst_2 (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) f b) (forall (ε : Real), (GT.gt.{0} Real Real.instLTReal ε (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal))) -> (Filter.Eventually.{u2} β (fun (x : β) => LT.lt.{0} Real Real.instLTReal (Dist.dist.{u1} α (PseudoMetricSpace.toDist.{u1} α _inst_1) (f x) (f b)) ε) (nhds.{u2} β _inst_2 b)))
+Case conversion may be inaccurate. Consider using '#align metric.continuous_at_iff' Metric.continuousAt_iff'ₓ'. -/
 theorem continuousAt_iff' [TopologicalSpace β] {f : β → α} {b : β} :
     ContinuousAt f b ↔ ∀ ε > 0, ∀ᶠ x in 𝓝 b, dist (f x) (f b) < ε := by
   rw [ContinuousAt, tendsto_nhds]
 #align metric.continuous_at_iff' Metric.continuousAt_iff'
 
+/- warning: metric.continuous_within_at_iff' -> Metric.continuousWithinAt_iff' is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : PseudoMetricSpace.{u1} α] [_inst_2 : TopologicalSpace.{u2} β] {f : β -> α} {b : β} {s : Set.{u2} β}, Iff (ContinuousWithinAt.{u2, u1} β α _inst_2 (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) f s b) (forall (ε : Real), (GT.gt.{0} Real Real.hasLt ε (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero)))) -> (Filter.Eventually.{u2} β (fun (x : β) => LT.lt.{0} Real Real.hasLt (Dist.dist.{u1} α (PseudoMetricSpace.toHasDist.{u1} α _inst_1) (f x) (f b)) ε) (nhdsWithin.{u2} β _inst_2 b s)))
+but is expected to have type
+  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : PseudoMetricSpace.{u1} α] [_inst_2 : TopologicalSpace.{u2} β] {f : β -> α} {b : β} {s : Set.{u2} β}, Iff (ContinuousWithinAt.{u2, u1} β α _inst_2 (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) f s b) (forall (ε : Real), (GT.gt.{0} Real Real.instLTReal ε (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal))) -> (Filter.Eventually.{u2} β (fun (x : β) => LT.lt.{0} Real Real.instLTReal (Dist.dist.{u1} α (PseudoMetricSpace.toDist.{u1} α _inst_1) (f x) (f b)) ε) (nhdsWithin.{u2} β _inst_2 b s)))
+Case conversion may be inaccurate. Consider using '#align metric.continuous_within_at_iff' Metric.continuousWithinAt_iff'ₓ'. -/
 theorem continuousWithinAt_iff' [TopologicalSpace β] {f : β → α} {b : β} {s : Set β} :
     ContinuousWithinAt f s b ↔ ∀ ε > 0, ∀ᶠ x in 𝓝[s] b, dist (f x) (f b) < ε := by
   rw [ContinuousWithinAt, tendsto_nhds]
 #align metric.continuous_within_at_iff' Metric.continuousWithinAt_iff'
 
+/- warning: metric.continuous_on_iff' -> Metric.continuousOn_iff' is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : PseudoMetricSpace.{u1} α] [_inst_2 : TopologicalSpace.{u2} β] {f : β -> α} {s : Set.{u2} β}, Iff (ContinuousOn.{u2, u1} β α _inst_2 (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) f s) (forall (b : β), (Membership.Mem.{u2, u2} β (Set.{u2} β) (Set.hasMem.{u2} β) b s) -> (forall (ε : Real), (GT.gt.{0} Real Real.hasLt ε (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero)))) -> (Filter.Eventually.{u2} β (fun (x : β) => LT.lt.{0} Real Real.hasLt (Dist.dist.{u1} α (PseudoMetricSpace.toHasDist.{u1} α _inst_1) (f x) (f b)) ε) (nhdsWithin.{u2} β _inst_2 b s))))
+but is expected to have type
+  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : PseudoMetricSpace.{u1} α] [_inst_2 : TopologicalSpace.{u2} β] {f : β -> α} {s : Set.{u2} β}, Iff (ContinuousOn.{u2, u1} β α _inst_2 (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) f s) (forall (b : β), (Membership.mem.{u2, u2} β (Set.{u2} β) (Set.instMembershipSet.{u2} β) b s) -> (forall (ε : Real), (GT.gt.{0} Real Real.instLTReal ε (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal))) -> (Filter.Eventually.{u2} β (fun (x : β) => LT.lt.{0} Real Real.instLTReal (Dist.dist.{u1} α (PseudoMetricSpace.toDist.{u1} α _inst_1) (f x) (f b)) ε) (nhdsWithin.{u2} β _inst_2 b s))))
+Case conversion may be inaccurate. Consider using '#align metric.continuous_on_iff' Metric.continuousOn_iff'ₓ'. -/
 theorem continuousOn_iff' [TopologicalSpace β] {f : β → α} {s : Set β} :
     ContinuousOn f s ↔ ∀ b ∈ s, ∀ ε > 0, ∀ᶠ x in 𝓝[s] b, dist (f x) (f b) < ε := by
   simp [ContinuousOn, continuous_within_at_iff']
 #align metric.continuous_on_iff' Metric.continuousOn_iff'
 
+/- warning: metric.continuous_iff' -> Metric.continuous_iff' is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : PseudoMetricSpace.{u1} α] [_inst_2 : TopologicalSpace.{u2} β] {f : β -> α}, Iff (Continuous.{u2, u1} β α _inst_2 (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) f) (forall (a : β) (ε : Real), (GT.gt.{0} Real Real.hasLt ε (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero)))) -> (Filter.Eventually.{u2} β (fun (x : β) => LT.lt.{0} Real Real.hasLt (Dist.dist.{u1} α (PseudoMetricSpace.toHasDist.{u1} α _inst_1) (f x) (f a)) ε) (nhds.{u2} β _inst_2 a)))
+but is expected to have type
+  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : PseudoMetricSpace.{u1} α] [_inst_2 : TopologicalSpace.{u2} β] {f : β -> α}, Iff (Continuous.{u2, u1} β α _inst_2 (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) f) (forall (a : β) (ε : Real), (GT.gt.{0} Real Real.instLTReal ε (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal))) -> (Filter.Eventually.{u2} β (fun (x : β) => LT.lt.{0} Real Real.instLTReal (Dist.dist.{u1} α (PseudoMetricSpace.toDist.{u1} α _inst_1) (f x) (f a)) ε) (nhds.{u2} β _inst_2 a)))
+Case conversion may be inaccurate. Consider using '#align metric.continuous_iff' Metric.continuous_iff'ₓ'. -/
 theorem continuous_iff' [TopologicalSpace β] {f : β → α} :
     Continuous f ↔ ∀ (a), ∀ ε > 0, ∀ᶠ x in 𝓝 a, dist (f x) (f a) < ε :=
   continuous_iff_continuousAt.trans <| forall_congr' fun b => tendsto_nhds
 #align metric.continuous_iff' Metric.continuous_iff'
 
+/- warning: metric.tendsto_at_top -> Metric.tendsto_atTop is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : PseudoMetricSpace.{u1} α] [_inst_2 : Nonempty.{succ u2} β] [_inst_3 : SemilatticeSup.{u2} β] {u : β -> α} {a : α}, Iff (Filter.Tendsto.{u2, u1} β α u (Filter.atTop.{u2} β (PartialOrder.toPreorder.{u2} β (SemilatticeSup.toPartialOrder.{u2} β _inst_3))) (nhds.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) a)) (forall (ε : Real), (GT.gt.{0} Real Real.hasLt ε (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero)))) -> (Exists.{succ u2} β (fun (N : β) => forall (n : β), (GE.ge.{u2} β (Preorder.toLE.{u2} β (PartialOrder.toPreorder.{u2} β (SemilatticeSup.toPartialOrder.{u2} β _inst_3))) n N) -> (LT.lt.{0} Real Real.hasLt (Dist.dist.{u1} α (PseudoMetricSpace.toHasDist.{u1} α _inst_1) (u n) a) ε))))
+but is expected to have type
+  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : PseudoMetricSpace.{u1} α] [_inst_2 : Nonempty.{succ u2} β] [_inst_3 : SemilatticeSup.{u2} β] {u : β -> α} {a : α}, Iff (Filter.Tendsto.{u2, u1} β α u (Filter.atTop.{u2} β (PartialOrder.toPreorder.{u2} β (SemilatticeSup.toPartialOrder.{u2} β _inst_3))) (nhds.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) a)) (forall (ε : Real), (GT.gt.{0} Real Real.instLTReal ε (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal))) -> (Exists.{succ u2} β (fun (N : β) => forall (n : β), (GE.ge.{u2} β (Preorder.toLE.{u2} β (PartialOrder.toPreorder.{u2} β (SemilatticeSup.toPartialOrder.{u2} β _inst_3))) n N) -> (LT.lt.{0} Real Real.instLTReal (Dist.dist.{u1} α (PseudoMetricSpace.toDist.{u1} α _inst_1) (u n) a) ε))))
+Case conversion may be inaccurate. Consider using '#align metric.tendsto_at_top Metric.tendsto_atTopₓ'. -/
 theorem tendsto_atTop [Nonempty β] [SemilatticeSup β] {u : β → α} {a : α} :
     Tendsto u atTop (𝓝 a) ↔ ∀ ε > 0, ∃ N, ∀ n ≥ N, dist (u n) a < ε :=
   (atTop_basis.tendsto_iffₓ nhds_basis_ball).trans <|
@@ -1203,6 +2067,12 @@ theorem tendsto_atTop [Nonempty β] [SemilatticeSup β] {u : β → α} {a : α}
     rfl
 #align metric.tendsto_at_top Metric.tendsto_atTop
 
+/- warning: metric.tendsto_at_top' -> Metric.tendsto_at_top' is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : PseudoMetricSpace.{u1} α] [_inst_2 : Nonempty.{succ u2} β] [_inst_3 : SemilatticeSup.{u2} β] [_inst_4 : NoMaxOrder.{u2} β (Preorder.toLT.{u2} β (PartialOrder.toPreorder.{u2} β (SemilatticeSup.toPartialOrder.{u2} β _inst_3)))] {u : β -> α} {a : α}, Iff (Filter.Tendsto.{u2, u1} β α u (Filter.atTop.{u2} β (PartialOrder.toPreorder.{u2} β (SemilatticeSup.toPartialOrder.{u2} β _inst_3))) (nhds.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) a)) (forall (ε : Real), (GT.gt.{0} Real Real.hasLt ε (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero)))) -> (Exists.{succ u2} β (fun (N : β) => forall (n : β), (GT.gt.{u2} β (Preorder.toLT.{u2} β (PartialOrder.toPreorder.{u2} β (SemilatticeSup.toPartialOrder.{u2} β _inst_3))) n N) -> (LT.lt.{0} Real Real.hasLt (Dist.dist.{u1} α (PseudoMetricSpace.toHasDist.{u1} α _inst_1) (u n) a) ε))))
+but is expected to have type
+  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : PseudoMetricSpace.{u1} α] [_inst_2 : Nonempty.{succ u2} β] [_inst_3 : SemilatticeSup.{u2} β] [_inst_4 : NoMaxOrder.{u2} β (Preorder.toLT.{u2} β (PartialOrder.toPreorder.{u2} β (SemilatticeSup.toPartialOrder.{u2} β _inst_3)))] {u : β -> α} {a : α}, Iff (Filter.Tendsto.{u2, u1} β α u (Filter.atTop.{u2} β (PartialOrder.toPreorder.{u2} β (SemilatticeSup.toPartialOrder.{u2} β _inst_3))) (nhds.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) a)) (forall (ε : Real), (GT.gt.{0} Real Real.instLTReal ε (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal))) -> (Exists.{succ u2} β (fun (N : β) => forall (n : β), (GT.gt.{u2} β (Preorder.toLT.{u2} β (PartialOrder.toPreorder.{u2} β (SemilatticeSup.toPartialOrder.{u2} β _inst_3))) n N) -> (LT.lt.{0} Real Real.instLTReal (Dist.dist.{u1} α (PseudoMetricSpace.toDist.{u1} α _inst_1) (u n) a) ε))))
+Case conversion may be inaccurate. Consider using '#align metric.tendsto_at_top' Metric.tendsto_at_top'ₓ'. -/
 /-- A variant of `tendsto_at_top` that
 uses `∃ N, ∀ n > N, ...` rather than `∃ N, ∀ n ≥ N, ...`
 -/
@@ -1214,11 +2084,23 @@ theorem tendsto_at_top' [Nonempty β] [SemilatticeSup β] [NoMaxOrder β] {u : 
     rfl
 #align metric.tendsto_at_top' Metric.tendsto_at_top'
 
+/- warning: metric.is_open_singleton_iff -> Metric.isOpen_singleton_iff is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} [_inst_2 : PseudoMetricSpace.{u1} α] {x : α}, Iff (IsOpen.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_2)) (Singleton.singleton.{u1, u1} α (Set.{u1} α) (Set.hasSingleton.{u1} α) x)) (Exists.{1} Real (fun (ε : Real) => Exists.{0} (GT.gt.{0} Real Real.hasLt ε (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero)))) (fun (H : GT.gt.{0} Real Real.hasLt ε (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero)))) => forall (y : α), (LT.lt.{0} Real Real.hasLt (Dist.dist.{u1} α (PseudoMetricSpace.toHasDist.{u1} α _inst_2) y x) ε) -> (Eq.{succ u1} α y x))))
+but is expected to have type
+  forall {α : Type.{u1}} [_inst_2 : PseudoMetricSpace.{u1} α] {x : α}, Iff (IsOpen.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_2)) (Singleton.singleton.{u1, u1} α (Set.{u1} α) (Set.instSingletonSet.{u1} α) x)) (Exists.{1} Real (fun (ε : Real) => And (GT.gt.{0} Real Real.instLTReal ε (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal))) (forall (y : α), (LT.lt.{0} Real Real.instLTReal (Dist.dist.{u1} α (PseudoMetricSpace.toDist.{u1} α _inst_2) y x) ε) -> (Eq.{succ u1} α y x))))
+Case conversion may be inaccurate. Consider using '#align metric.is_open_singleton_iff Metric.isOpen_singleton_iffₓ'. -/
 theorem isOpen_singleton_iff {α : Type _} [PseudoMetricSpace α] {x : α} :
     IsOpen ({x} : Set α) ↔ ∃ ε > 0, ∀ y, dist y x < ε → y = x := by
   simp [is_open_iff, subset_singleton_iff, mem_ball]
 #align metric.is_open_singleton_iff Metric.isOpen_singleton_iff
 
+/- warning: metric.exists_ball_inter_eq_singleton_of_mem_discrete -> Metric.exists_ball_inter_eq_singleton_of_mem_discrete is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {s : Set.{u1} α} [_inst_2 : DiscreteTopology.{u1} (coeSort.{succ u1, succ (succ u1)} (Set.{u1} α) Type.{u1} (Set.hasCoeToSort.{u1} α) s) (Subtype.topologicalSpace.{u1} α (fun (x : α) => Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) x s) (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)))] {x : α}, (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) x s) -> (Exists.{1} Real (fun (ε : Real) => Exists.{0} (GT.gt.{0} Real Real.hasLt ε (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero)))) (fun (H : GT.gt.{0} Real Real.hasLt ε (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero)))) => Eq.{succ u1} (Set.{u1} α) (Inter.inter.{u1} (Set.{u1} α) (Set.hasInter.{u1} α) (Metric.ball.{u1} α _inst_1 x ε) s) (Singleton.singleton.{u1, u1} α (Set.{u1} α) (Set.hasSingleton.{u1} α) x))))
+but is expected to have type
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {s : Set.{u1} α} [_inst_2 : DiscreteTopology.{u1} (Set.Elem.{u1} α s) (instTopologicalSpaceSubtype.{u1} α (fun (x : α) => Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) x s) (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)))] {x : α}, (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) x s) -> (Exists.{1} Real (fun (ε : Real) => And (GT.gt.{0} Real Real.instLTReal ε (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal))) (Eq.{succ u1} (Set.{u1} α) (Inter.inter.{u1} (Set.{u1} α) (Set.instInterSet.{u1} α) (Metric.ball.{u1} α _inst_1 x ε) s) (Singleton.singleton.{u1, u1} α (Set.{u1} α) (Set.instSingletonSet.{u1} α) x))))
+Case conversion may be inaccurate. Consider using '#align metric.exists_ball_inter_eq_singleton_of_mem_discrete Metric.exists_ball_inter_eq_singleton_of_mem_discreteₓ'. -/
 /-- Given a point `x` in a discrete subset `s` of a pseudometric space, there is an open ball
 centered at `x` and intersecting `s` only at `x`. -/
 theorem exists_ball_inter_eq_singleton_of_mem_discrete [DiscreteTopology s] {x : α} (hx : x ∈ s) :
@@ -1226,6 +2108,12 @@ theorem exists_ball_inter_eq_singleton_of_mem_discrete [DiscreteTopology s] {x :
   nhds_basis_ball.exists_inter_eq_singleton_of_mem_discrete hx
 #align metric.exists_ball_inter_eq_singleton_of_mem_discrete Metric.exists_ball_inter_eq_singleton_of_mem_discrete
 
+/- warning: metric.exists_closed_ball_inter_eq_singleton_of_discrete -> Metric.exists_closedBall_inter_eq_singleton_of_discrete is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {s : Set.{u1} α} [_inst_2 : DiscreteTopology.{u1} (coeSort.{succ u1, succ (succ u1)} (Set.{u1} α) Type.{u1} (Set.hasCoeToSort.{u1} α) s) (Subtype.topologicalSpace.{u1} α (fun (x : α) => Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) x s) (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)))] {x : α}, (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) x s) -> (Exists.{1} Real (fun (ε : Real) => Exists.{0} (GT.gt.{0} Real Real.hasLt ε (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero)))) (fun (H : GT.gt.{0} Real Real.hasLt ε (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero)))) => Eq.{succ u1} (Set.{u1} α) (Inter.inter.{u1} (Set.{u1} α) (Set.hasInter.{u1} α) (Metric.closedBall.{u1} α _inst_1 x ε) s) (Singleton.singleton.{u1, u1} α (Set.{u1} α) (Set.hasSingleton.{u1} α) x))))
+but is expected to have type
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {s : Set.{u1} α} [_inst_2 : DiscreteTopology.{u1} (Set.Elem.{u1} α s) (instTopologicalSpaceSubtype.{u1} α (fun (x : α) => Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) x s) (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)))] {x : α}, (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) x s) -> (Exists.{1} Real (fun (ε : Real) => And (GT.gt.{0} Real Real.instLTReal ε (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal))) (Eq.{succ u1} (Set.{u1} α) (Inter.inter.{u1} (Set.{u1} α) (Set.instInterSet.{u1} α) (Metric.closedBall.{u1} α _inst_1 x ε) s) (Singleton.singleton.{u1, u1} α (Set.{u1} α) (Set.instSingletonSet.{u1} α) x))))
+Case conversion may be inaccurate. Consider using '#align metric.exists_closed_ball_inter_eq_singleton_of_discrete Metric.exists_closedBall_inter_eq_singleton_of_discreteₓ'. -/
 /-- Given a point `x` in a discrete subset `s` of a pseudometric space, there is a closed ball
 of positive radius centered at `x` and intersecting `s` only at `x`. -/
 theorem exists_closedBall_inter_eq_singleton_of_discrete [DiscreteTopology s] {x : α} (hx : x ∈ s) :
@@ -1233,6 +2121,12 @@ theorem exists_closedBall_inter_eq_singleton_of_discrete [DiscreteTopology s] {x
   nhds_basis_closedBall.exists_inter_eq_singleton_of_mem_discrete hx
 #align metric.exists_closed_ball_inter_eq_singleton_of_discrete Metric.exists_closedBall_inter_eq_singleton_of_discrete
 
+/- warning: dense.exists_dist_lt -> Dense.exists_dist_lt is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {s : Set.{u1} α}, (Dense.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) s) -> (forall (x : α) {ε : Real}, (LT.lt.{0} Real Real.hasLt (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero))) ε) -> (Exists.{succ u1} α (fun (y : α) => Exists.{0} (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) y s) (fun (H : Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) y s) => LT.lt.{0} Real Real.hasLt (Dist.dist.{u1} α (PseudoMetricSpace.toHasDist.{u1} α _inst_1) x y) ε))))
+but is expected to have type
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {s : Set.{u1} α}, (Dense.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) s) -> (forall (x : α) {ε : Real}, (LT.lt.{0} Real Real.instLTReal (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal)) ε) -> (Exists.{succ u1} α (fun (y : α) => And (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) y s) (LT.lt.{0} Real Real.instLTReal (Dist.dist.{u1} α (PseudoMetricSpace.toDist.{u1} α _inst_1) x y) ε))))
+Case conversion may be inaccurate. Consider using '#align dense.exists_dist_lt Dense.exists_dist_ltₓ'. -/
 theorem Dense.exists_dist_lt {s : Set α} (hs : Dense s) (x : α) {ε : ℝ} (hε : 0 < ε) :
     ∃ y ∈ s, dist x y < ε :=
   by
@@ -1240,6 +2134,12 @@ theorem Dense.exists_dist_lt {s : Set α} (hs : Dense s) (x : α) {ε : ℝ} (h
   simpa only [mem_ball'] using hs.exists_mem_open is_open_ball this
 #align dense.exists_dist_lt Dense.exists_dist_lt
 
+/- warning: dense_range.exists_dist_lt -> DenseRange.exists_dist_lt is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {β : Type.{u2}} {f : β -> α}, (DenseRange.{u1, u2} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) β f) -> (forall (x : α) {ε : Real}, (LT.lt.{0} Real Real.hasLt (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero))) ε) -> (Exists.{succ u2} β (fun (y : β) => LT.lt.{0} Real Real.hasLt (Dist.dist.{u1} α (PseudoMetricSpace.toHasDist.{u1} α _inst_1) x (f y)) ε)))
+but is expected to have type
+  forall {α : Type.{u2}} [_inst_1 : PseudoMetricSpace.{u2} α] {β : Type.{u1}} {f : β -> α}, (DenseRange.{u2, u1} α (UniformSpace.toTopologicalSpace.{u2} α (PseudoMetricSpace.toUniformSpace.{u2} α _inst_1)) β f) -> (forall (x : α) {ε : Real}, (LT.lt.{0} Real Real.instLTReal (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal)) ε) -> (Exists.{succ u1} β (fun (y : β) => LT.lt.{0} Real Real.instLTReal (Dist.dist.{u2} α (PseudoMetricSpace.toDist.{u2} α _inst_1) x (f y)) ε)))
+Case conversion may be inaccurate. Consider using '#align dense_range.exists_dist_lt DenseRange.exists_dist_ltₓ'. -/
 theorem DenseRange.exists_dist_lt {β : Type _} {f : β → α} (hf : DenseRange f) (x : α) {ε : ℝ}
     (hε : 0 < ε) : ∃ y, dist x (f y) < ε :=
   exists_range_iff.1 (hf.exists_dist_lt x hε)
@@ -1249,11 +2149,17 @@ end Metric
 
 open Metric
 
+/- warning: pseudo_metric.uniformity_basis_edist -> Metric.uniformity_basis_edist is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α], Filter.HasBasis.{u1, 1} (Prod.{u1, u1} α α) ENNReal (uniformity.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) (fun (ε : ENNReal) => LT.lt.{0} ENNReal (Preorder.toLT.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (CompleteSemilatticeInf.toPartialOrder.{0} ENNReal (CompleteLattice.toCompleteSemilatticeInf.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))))) (OfNat.ofNat.{0} ENNReal 0 (OfNat.mk.{0} ENNReal 0 (Zero.zero.{0} ENNReal ENNReal.hasZero))) ε) (fun (ε : ENNReal) => setOf.{u1} (Prod.{u1, u1} α α) (fun (p : Prod.{u1, u1} α α) => LT.lt.{0} ENNReal (Preorder.toLT.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (CompleteSemilatticeInf.toPartialOrder.{0} ENNReal (CompleteLattice.toCompleteSemilatticeInf.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))))) (EDist.edist.{u1} α (PseudoMetricSpace.toEDist.{u1} α _inst_1) (Prod.fst.{u1, u1} α α p) (Prod.snd.{u1, u1} α α p)) ε))
+but is expected to have type
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α], Filter.HasBasis.{u1, 1} (Prod.{u1, u1} α α) ENNReal (uniformity.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) (fun (ε : ENNReal) => LT.lt.{0} ENNReal (Preorder.toLT.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (CompleteSemilatticeInf.toPartialOrder.{0} ENNReal (CompleteLattice.toCompleteSemilatticeInf.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal))))) (OfNat.ofNat.{0} ENNReal 0 (Zero.toOfNat0.{0} ENNReal instENNRealZero)) ε) (fun (ε : ENNReal) => setOf.{u1} (Prod.{u1, u1} α α) (fun (p : Prod.{u1, u1} α α) => LT.lt.{0} ENNReal (Preorder.toLT.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (CompleteSemilatticeInf.toPartialOrder.{0} ENNReal (CompleteLattice.toCompleteSemilatticeInf.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal))))) (EDist.edist.{u1} α (PseudoEMetricSpace.toEDist.{u1} α (PseudoMetricSpace.toPseudoEMetricSpace.{u1} α _inst_1)) (Prod.fst.{u1, u1} α α p) (Prod.snd.{u1, u1} α α p)) ε))
+Case conversion may be inaccurate. Consider using '#align pseudo_metric.uniformity_basis_edist Metric.uniformity_basis_edistₓ'. -/
 /-Instantiate a pseudometric space as a pseudoemetric space. Before we can state the instance,
 we need to show that the uniform structure coming from the edistance and the
 distance coincide. -/
 /-- Expressing the uniformity in terms of `edist` -/
-protected theorem PseudoMetric.uniformity_basis_edist :
+protected theorem Metric.uniformity_basis_edist :
     (𝓤 α).HasBasis (fun ε : ℝ≥0∞ => 0 < ε) fun ε => { p | edist p.1 p.2 < ε } :=
   ⟨by
     intro t
@@ -1266,15 +2172,22 @@ protected theorem PseudoMetric.uniformity_basis_edist :
       rw [ENNReal.ofReal_pos] at ε0'
       refine' ⟨ε', ε0', fun a b h => Hε (lt_trans _ hε)⟩
       rwa [edist_dist, ENNReal.ofReal_lt_ofReal_iff ε0']⟩
-#align pseudo_metric.uniformity_basis_edist PseudoMetric.uniformity_basis_edist
-
+#align pseudo_metric.uniformity_basis_edist Metric.uniformity_basis_edist
+
+/- warning: metric.uniformity_edist -> Metric.uniformity_edist is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α], Eq.{succ u1} (Filter.{u1} (Prod.{u1, u1} α α)) (uniformity.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) (infᵢ.{u1, 1} (Filter.{u1} (Prod.{u1, u1} α α)) (ConditionallyCompleteLattice.toHasInf.{u1} (Filter.{u1} (Prod.{u1, u1} α α)) (CompleteLattice.toConditionallyCompleteLattice.{u1} (Filter.{u1} (Prod.{u1, u1} α α)) (Filter.completeLattice.{u1} (Prod.{u1, u1} α α)))) ENNReal (fun (ε : ENNReal) => infᵢ.{u1, 0} (Filter.{u1} (Prod.{u1, u1} α α)) (ConditionallyCompleteLattice.toHasInf.{u1} (Filter.{u1} (Prod.{u1, u1} α α)) (CompleteLattice.toConditionallyCompleteLattice.{u1} (Filter.{u1} (Prod.{u1, u1} α α)) (Filter.completeLattice.{u1} (Prod.{u1, u1} α α)))) (GT.gt.{0} ENNReal (Preorder.toLT.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (CompleteSemilatticeInf.toPartialOrder.{0} ENNReal (CompleteLattice.toCompleteSemilatticeInf.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))))) ε (OfNat.ofNat.{0} ENNReal 0 (OfNat.mk.{0} ENNReal 0 (Zero.zero.{0} ENNReal ENNReal.hasZero)))) (fun (H : GT.gt.{0} ENNReal (Preorder.toLT.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (CompleteSemilatticeInf.toPartialOrder.{0} ENNReal (CompleteLattice.toCompleteSemilatticeInf.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))))) ε (OfNat.ofNat.{0} ENNReal 0 (OfNat.mk.{0} ENNReal 0 (Zero.zero.{0} ENNReal ENNReal.hasZero)))) => Filter.principal.{u1} (Prod.{u1, u1} α α) (setOf.{u1} (Prod.{u1, u1} α α) (fun (p : Prod.{u1, u1} α α) => LT.lt.{0} ENNReal (Preorder.toLT.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (CompleteSemilatticeInf.toPartialOrder.{0} ENNReal (CompleteLattice.toCompleteSemilatticeInf.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))))) (EDist.edist.{u1} α (PseudoMetricSpace.toEDist.{u1} α _inst_1) (Prod.fst.{u1, u1} α α p) (Prod.snd.{u1, u1} α α p)) ε)))))
+but is expected to have type
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α], Eq.{succ u1} (Filter.{u1} (Prod.{u1, u1} α α)) (uniformity.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) (infᵢ.{u1, 1} (Filter.{u1} (Prod.{u1, u1} α α)) (ConditionallyCompleteLattice.toInfSet.{u1} (Filter.{u1} (Prod.{u1, u1} α α)) (CompleteLattice.toConditionallyCompleteLattice.{u1} (Filter.{u1} (Prod.{u1, u1} α α)) (Filter.instCompleteLatticeFilter.{u1} (Prod.{u1, u1} α α)))) ENNReal (fun (ε : ENNReal) => infᵢ.{u1, 0} (Filter.{u1} (Prod.{u1, u1} α α)) (ConditionallyCompleteLattice.toInfSet.{u1} (Filter.{u1} (Prod.{u1, u1} α α)) (CompleteLattice.toConditionallyCompleteLattice.{u1} (Filter.{u1} (Prod.{u1, u1} α α)) (Filter.instCompleteLatticeFilter.{u1} (Prod.{u1, u1} α α)))) (GT.gt.{0} ENNReal (Preorder.toLT.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (CompleteSemilatticeInf.toPartialOrder.{0} ENNReal (CompleteLattice.toCompleteSemilatticeInf.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal))))) ε (OfNat.ofNat.{0} ENNReal 0 (Zero.toOfNat0.{0} ENNReal instENNRealZero))) (fun (H : GT.gt.{0} ENNReal (Preorder.toLT.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (CompleteSemilatticeInf.toPartialOrder.{0} ENNReal (CompleteLattice.toCompleteSemilatticeInf.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal))))) ε (OfNat.ofNat.{0} ENNReal 0 (Zero.toOfNat0.{0} ENNReal instENNRealZero))) => Filter.principal.{u1} (Prod.{u1, u1} α α) (setOf.{u1} (Prod.{u1, u1} α α) (fun (p : Prod.{u1, u1} α α) => LT.lt.{0} ENNReal (Preorder.toLT.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (CompleteSemilatticeInf.toPartialOrder.{0} ENNReal (CompleteLattice.toCompleteSemilatticeInf.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal))))) (EDist.edist.{u1} α (PseudoMetricSpace.toEDist.{u1} α _inst_1) (Prod.fst.{u1, u1} α α p) (Prod.snd.{u1, u1} α α p)) ε)))))
+Case conversion may be inaccurate. Consider using '#align metric.uniformity_edist Metric.uniformity_edistₓ'. -/
 theorem Metric.uniformity_edist : 𝓤 α = ⨅ ε > 0, 𝓟 { p : α × α | edist p.1 p.2 < ε } :=
-  PseudoMetric.uniformity_basis_edist.eq_binfᵢ
+  Metric.uniformity_basis_edist.eq_binfᵢ
 #align metric.uniformity_edist Metric.uniformity_edist
 
+#print PseudoMetricSpace.toPseudoEMetricSpace /-
 -- see Note [lower instance priority]
 /-- A pseudometric space induces a pseudoemetric space -/
-instance (priority := 100) PseudoMetricSpace.toPseudoEmetricSpace : PseudoEMetricSpace α :=
+instance (priority := 100) PseudoMetricSpace.toPseudoEMetricSpace : PseudoEMetricSpace α :=
   { ‹PseudoMetricSpace α› with
     edist := edist
     edist_self := by simp [edist_dist]
@@ -1286,13 +2199,21 @@ instance (priority := 100) PseudoMetricSpace.toPseudoEmetricSpace : PseudoEMetri
       · exact dist_triangle _ _ _
       · simpa using add_le_add (dist_nonneg : 0 ≤ dist x y) dist_nonneg
     uniformity_edist := Metric.uniformity_edist }
-#align pseudo_metric_space.to_pseudo_emetric_space PseudoMetricSpace.toPseudoEmetricSpace
+#align pseudo_metric_space.to_pseudo_emetric_space PseudoMetricSpace.toPseudoEMetricSpace
+-/
 
+/- warning: metric.eball_top_eq_univ -> Metric.eball_top_eq_univ is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] (x : α), Eq.{succ u1} (Set.{u1} α) (EMetric.ball.{u1} α (PseudoMetricSpace.toPseudoEMetricSpace.{u1} α _inst_1) x (Top.top.{0} ENNReal (CompleteLattice.toHasTop.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder)))) (Set.univ.{u1} α)
+but is expected to have type
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] (x : α), Eq.{succ u1} (Set.{u1} α) (EMetric.ball.{u1} α (PseudoMetricSpace.toPseudoEMetricSpace.{u1} α _inst_1) x (Top.top.{0} ENNReal (CompleteLattice.toTop.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal)))) (Set.univ.{u1} α)
+Case conversion may be inaccurate. Consider using '#align metric.eball_top_eq_univ Metric.eball_top_eq_univₓ'. -/
 /-- In a pseudometric space, an open ball of infinite radius is the whole space -/
 theorem Metric.eball_top_eq_univ (x : α) : EMetric.ball x ∞ = Set.univ :=
   Set.eq_univ_iff_forall.mpr fun y => edist_lt_top y x
 #align metric.eball_top_eq_univ Metric.eball_top_eq_univ
 
+#print Metric.emetric_ball /-
 /-- Balls defined using the distance or the edistance coincide -/
 @[simp]
 theorem Metric.emetric_ball {x : α} {ε : ℝ} : EMetric.ball x (ENNReal.ofReal ε) = ball x ε :=
@@ -1301,39 +2222,71 @@ theorem Metric.emetric_ball {x : α} {ε : ℝ} : EMetric.ball x (ENNReal.ofReal
   simp only [EMetric.mem_ball, mem_ball, edist_dist]
   exact ENNReal.ofReal_lt_ofReal_iff_of_nonneg dist_nonneg
 #align metric.emetric_ball Metric.emetric_ball
+-/
 
+/- warning: metric.emetric_ball_nnreal -> Metric.emetric_ball_nnreal is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {x : α} {ε : NNReal}, Eq.{succ u1} (Set.{u1} α) (EMetric.ball.{u1} α (PseudoMetricSpace.toPseudoEMetricSpace.{u1} α _inst_1) x ((fun (a : Type) (b : Type) [self : HasLiftT.{1, 1} a b] => self.0) NNReal ENNReal (HasLiftT.mk.{1, 1} NNReal ENNReal (CoeTCₓ.coe.{1, 1} NNReal ENNReal (coeBase.{1, 1} NNReal ENNReal ENNReal.hasCoe))) ε)) (Metric.ball.{u1} α _inst_1 x ((fun (a : Type) (b : Type) [self : HasLiftT.{1, 1} a b] => self.0) NNReal Real (HasLiftT.mk.{1, 1} NNReal Real (CoeTCₓ.coe.{1, 1} NNReal Real (coeBase.{1, 1} NNReal Real NNReal.Real.hasCoe))) ε))
+but is expected to have type
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {x : α} {ε : NNReal}, Eq.{succ u1} (Set.{u1} α) (EMetric.ball.{u1} α (PseudoMetricSpace.toPseudoEMetricSpace.{u1} α _inst_1) x (ENNReal.some ε)) (Metric.ball.{u1} α _inst_1 x (NNReal.toReal ε))
+Case conversion may be inaccurate. Consider using '#align metric.emetric_ball_nnreal Metric.emetric_ball_nnrealₓ'. -/
 /-- Balls defined using the distance or the edistance coincide -/
 @[simp]
-theorem Metric.emetric_ball_nNReal {x : α} {ε : ℝ≥0} : EMetric.ball x ε = ball x ε :=
+theorem Metric.emetric_ball_nnreal {x : α} {ε : ℝ≥0} : EMetric.ball x ε = ball x ε :=
   by
   convert Metric.emetric_ball
   simp
-#align metric.emetric_ball_nnreal Metric.emetric_ball_nNReal
-
+#align metric.emetric_ball_nnreal Metric.emetric_ball_nnreal
+
+/- warning: metric.emetric_closed_ball -> Metric.emetric_closedBall is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {x : α} {ε : Real}, (LE.le.{0} Real Real.hasLe (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero))) ε) -> (Eq.{succ u1} (Set.{u1} α) (EMetric.closedBall.{u1} α (PseudoMetricSpace.toPseudoEMetricSpace.{u1} α _inst_1) x (ENNReal.ofReal ε)) (Metric.closedBall.{u1} α _inst_1 x ε))
+but is expected to have type
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {x : α} {ε : Real}, (LE.le.{0} Real Real.instLEReal (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal)) ε) -> (Eq.{succ u1} (Set.{u1} α) (EMetric.closedBall.{u1} α (PseudoMetricSpace.toPseudoEMetricSpace.{u1} α _inst_1) x (ENNReal.ofReal ε)) (Metric.closedBall.{u1} α _inst_1 x ε))
+Case conversion may be inaccurate. Consider using '#align metric.emetric_closed_ball Metric.emetric_closedBallₓ'. -/
 /-- Closed balls defined using the distance or the edistance coincide -/
 theorem Metric.emetric_closedBall {x : α} {ε : ℝ} (h : 0 ≤ ε) :
     EMetric.closedBall x (ENNReal.ofReal ε) = closedBall x ε := by
   ext y <;> simp [edist_dist] <;> rw [ENNReal.ofReal_le_ofReal_iff h]
 #align metric.emetric_closed_ball Metric.emetric_closedBall
 
+/- warning: metric.emetric_closed_ball_nnreal -> Metric.emetric_closedBall_nnreal is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {x : α} {ε : NNReal}, Eq.{succ u1} (Set.{u1} α) (EMetric.closedBall.{u1} α (PseudoMetricSpace.toPseudoEMetricSpace.{u1} α _inst_1) x ((fun (a : Type) (b : Type) [self : HasLiftT.{1, 1} a b] => self.0) NNReal ENNReal (HasLiftT.mk.{1, 1} NNReal ENNReal (CoeTCₓ.coe.{1, 1} NNReal ENNReal (coeBase.{1, 1} NNReal ENNReal ENNReal.hasCoe))) ε)) (Metric.closedBall.{u1} α _inst_1 x ((fun (a : Type) (b : Type) [self : HasLiftT.{1, 1} a b] => self.0) NNReal Real (HasLiftT.mk.{1, 1} NNReal Real (CoeTCₓ.coe.{1, 1} NNReal Real (coeBase.{1, 1} NNReal Real NNReal.Real.hasCoe))) ε))
+but is expected to have type
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {x : α} {ε : NNReal}, Eq.{succ u1} (Set.{u1} α) (EMetric.closedBall.{u1} α (PseudoMetricSpace.toPseudoEMetricSpace.{u1} α _inst_1) x (ENNReal.some ε)) (Metric.closedBall.{u1} α _inst_1 x (NNReal.toReal ε))
+Case conversion may be inaccurate. Consider using '#align metric.emetric_closed_ball_nnreal Metric.emetric_closedBall_nnrealₓ'. -/
 /-- Closed balls defined using the distance or the edistance coincide -/
 @[simp]
-theorem Metric.emetric_closedBall_nNReal {x : α} {ε : ℝ≥0} :
+theorem Metric.emetric_closedBall_nnreal {x : α} {ε : ℝ≥0} :
     EMetric.closedBall x ε = closedBall x ε :=
   by
   convert Metric.emetric_closedBall ε.2
   simp
-#align metric.emetric_closed_ball_nnreal Metric.emetric_closedBall_nNReal
-
+#align metric.emetric_closed_ball_nnreal Metric.emetric_closedBall_nnreal
+
+/- warning: metric.emetric_ball_top -> Metric.emetric_ball_top is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] (x : α), Eq.{succ u1} (Set.{u1} α) (EMetric.ball.{u1} α (PseudoMetricSpace.toPseudoEMetricSpace.{u1} α _inst_1) x (Top.top.{0} ENNReal (CompleteLattice.toHasTop.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder)))) (Set.univ.{u1} α)
+but is expected to have type
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] (x : α), Eq.{succ u1} (Set.{u1} α) (EMetric.ball.{u1} α (PseudoMetricSpace.toPseudoEMetricSpace.{u1} α _inst_1) x (Top.top.{0} ENNReal (CompleteLattice.toTop.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal)))) (Set.univ.{u1} α)
+Case conversion may be inaccurate. Consider using '#align metric.emetric_ball_top Metric.emetric_ball_topₓ'. -/
 @[simp]
 theorem Metric.emetric_ball_top (x : α) : EMetric.ball x ⊤ = univ :=
   eq_univ_of_forall fun y => edist_lt_top _ _
 #align metric.emetric_ball_top Metric.emetric_ball_top
 
+/- warning: metric.inseparable_iff -> Metric.inseparable_iff is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {x : α} {y : α}, Iff (Inseparable.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) x y) (Eq.{1} Real (Dist.dist.{u1} α (PseudoMetricSpace.toHasDist.{u1} α _inst_1) x y) (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero))))
+but is expected to have type
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {x : α} {y : α}, Iff (Inseparable.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) x y) (Eq.{1} Real (Dist.dist.{u1} α (PseudoMetricSpace.toDist.{u1} α _inst_1) x y) (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal)))
+Case conversion may be inaccurate. Consider using '#align metric.inseparable_iff Metric.inseparable_iffₓ'. -/
 theorem Metric.inseparable_iff {x y : α} : Inseparable x y ↔ dist x y = 0 := by
   rw [EMetric.inseparable_iff, edist_nndist, dist_nndist, ENNReal.coe_eq_zero, NNReal.coe_eq_zero]
 #align metric.inseparable_iff Metric.inseparable_iff
 
+#print PseudoMetricSpace.replaceUniformity /-
 /-- Build a new pseudometric space from an old one where the bundled uniform structure is provably
 (but typically non-definitionaly) equal to some given uniform structure.
 See Note [forgetful inheritance].
@@ -1350,14 +2303,18 @@ def PseudoMetricSpace.replaceUniformity {α} [U : UniformSpace α] (m : PseudoMe
   toUniformSpace := U
   uniformity_dist := H.trans PseudoMetricSpace.uniformity_dist
 #align pseudo_metric_space.replace_uniformity PseudoMetricSpace.replaceUniformity
+-/
 
+#print PseudoMetricSpace.replaceUniformity_eq /-
 theorem PseudoMetricSpace.replaceUniformity_eq {α} [U : UniformSpace α] (m : PseudoMetricSpace α)
     (H : 𝓤[U] = 𝓤[PseudoEMetricSpace.toUniformSpace]) : m.replaceUniformity H = m :=
   by
   ext
   rfl
 #align pseudo_metric_space.replace_uniformity_eq PseudoMetricSpace.replaceUniformity_eq
+-/
 
+#print PseudoMetricSpace.replaceTopology /-
 /-- Build a new pseudo metric space from an old one where the bundled topological structure is
 provably (but typically non-definitionaly) equal to some given topological structure.
 See Note [forgetful inheritance].
@@ -1367,14 +2324,23 @@ def PseudoMetricSpace.replaceTopology {γ} [U : TopologicalSpace γ] (m : Pseudo
     (H : U = m.toUniformSpace.toTopologicalSpace) : PseudoMetricSpace γ :=
   @PseudoMetricSpace.replaceUniformity γ (m.toUniformSpace.replaceTopology H) m rfl
 #align pseudo_metric_space.replace_topology PseudoMetricSpace.replaceTopology
+-/
 
+#print PseudoMetricSpace.replaceTopology_eq /-
 theorem PseudoMetricSpace.replaceTopology_eq {γ} [U : TopologicalSpace γ] (m : PseudoMetricSpace γ)
     (H : U = m.toUniformSpace.toTopologicalSpace) : m.replaceTopology H = m :=
   by
   ext
   rfl
 #align pseudo_metric_space.replace_topology_eq PseudoMetricSpace.replaceTopology_eq
+-/
 
+/- warning: pseudo_emetric_space.to_pseudo_metric_space_of_dist -> PseudoEMetricSpace.toPseudoMetricSpaceOfDist is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} [e : PseudoEMetricSpace.{u1} α] (dist : α -> α -> Real), (forall (x : α) (y : α), Ne.{1} ENNReal (EDist.edist.{u1} α (PseudoEMetricSpace.toHasEdist.{u1} α e) x y) (Top.top.{0} ENNReal (CompleteLattice.toHasTop.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder)))) -> (forall (x : α) (y : α), Eq.{1} Real (dist x y) (ENNReal.toReal (EDist.edist.{u1} α (PseudoEMetricSpace.toHasEdist.{u1} α e) x y))) -> (PseudoMetricSpace.{u1} α)
+but is expected to have type
+  forall {α : Type.{u1}} [e : PseudoEMetricSpace.{u1} α] (dist : α -> α -> Real), (forall (x : α) (y : α), Ne.{1} ENNReal (EDist.edist.{u1} α (PseudoEMetricSpace.toEDist.{u1} α e) x y) (Top.top.{0} ENNReal (CompleteLattice.toTop.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal)))) -> (forall (x : α) (y : α), Eq.{1} Real (dist x y) (ENNReal.toReal (EDist.edist.{u1} α (PseudoEMetricSpace.toEDist.{u1} α e) x y))) -> (PseudoMetricSpace.{u1} α)
+Case conversion may be inaccurate. Consider using '#align pseudo_emetric_space.to_pseudo_metric_space_of_dist PseudoEMetricSpace.toPseudoMetricSpaceOfDistₓ'. -/
 /-- One gets a pseudometric space from an emetric space if the edistance
 is everywhere finite, by pushing the edistance to reals. We set it up so that the edist and the
 uniformity are defeq in the pseudometric space and the pseudoemetric space. In this definition, the
@@ -1401,6 +2367,12 @@ def PseudoEMetricSpace.toPseudoMetricSpaceOfDist {α : Type u} [e : PseudoEMetri
     rfl
 #align pseudo_emetric_space.to_pseudo_metric_space_of_dist PseudoEMetricSpace.toPseudoMetricSpaceOfDist
 
+/- warning: pseudo_emetric_space.to_pseudo_metric_space -> PseudoEMetricSpace.toPseudoMetricSpace is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} [e : PseudoEMetricSpace.{u1} α], (forall (x : α) (y : α), Ne.{1} ENNReal (EDist.edist.{u1} α (PseudoEMetricSpace.toHasEdist.{u1} α e) x y) (Top.top.{0} ENNReal (CompleteLattice.toHasTop.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder)))) -> (PseudoMetricSpace.{u1} α)
+but is expected to have type
+  forall {α : Type.{u1}} [e : PseudoEMetricSpace.{u1} α], (forall (x : α) (y : α), Ne.{1} ENNReal (EDist.edist.{u1} α (PseudoEMetricSpace.toEDist.{u1} α e) x y) (Top.top.{0} ENNReal (CompleteLattice.toTop.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal)))) -> (PseudoMetricSpace.{u1} α)
+Case conversion may be inaccurate. Consider using '#align pseudo_emetric_space.to_pseudo_metric_space PseudoEMetricSpace.toPseudoMetricSpaceₓ'. -/
 /-- One gets a pseudometric space from an emetric space if the edistance
 is everywhere finite, by pushing the edistance to reals. We set it up so that the edist and the
 uniformity are defeq in the pseudometric space and the emetric space. -/
@@ -1410,6 +2382,7 @@ def PseudoEMetricSpace.toPseudoMetricSpace {α : Type u} [e : PseudoEMetricSpace
     rfl
 #align pseudo_emetric_space.to_pseudo_metric_space PseudoEMetricSpace.toPseudoMetricSpace
 
+#print PseudoMetricSpace.replaceBornology /-
 /-- Build a new pseudometric space from an old one where the bundled bornology structure is provably
 (but typically non-definitionaly) equal to some given bornology structure.
 See Note [forgetful inheritance].
@@ -1424,7 +2397,9 @@ def PseudoMetricSpace.replaceBornology {α} [B : Bornology α] (m : PseudoMetric
         compl_surjective.forall.2 fun s =>
           (H s).trans <| by rw [is_bounded_iff, mem_set_of_eq, compl_compl] }
 #align pseudo_metric_space.replace_bornology PseudoMetricSpace.replaceBornology
+-/
 
+#print PseudoMetricSpace.replaceBornology_eq /-
 theorem PseudoMetricSpace.replaceBornology_eq {α} [m : PseudoMetricSpace α] [B : Bornology α]
     (H : ∀ s, @IsBounded _ B s ↔ @IsBounded _ PseudoMetricSpace.toBornology s) :
     PseudoMetricSpace.replaceBornology _ H = m :=
@@ -1432,7 +2407,14 @@ theorem PseudoMetricSpace.replaceBornology_eq {α} [m : PseudoMetricSpace α] [B
   ext
   rfl
 #align pseudo_metric_space.replace_bornology_eq PseudoMetricSpace.replaceBornology_eq
+-/
 
+/- warning: metric.complete_of_convergent_controlled_sequences -> Metric.complete_of_convergent_controlled_sequences is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] (B : Nat -> Real), (forall (n : Nat), LT.lt.{0} Real Real.hasLt (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero))) (B n)) -> (forall (u : Nat -> α), (forall (N : Nat) (n : Nat) (m : Nat), (LE.le.{0} Nat Nat.hasLe N n) -> (LE.le.{0} Nat Nat.hasLe N m) -> (LT.lt.{0} Real Real.hasLt (Dist.dist.{u1} α (PseudoMetricSpace.toHasDist.{u1} α _inst_1) (u n) (u m)) (B N))) -> (Exists.{succ u1} α (fun (x : α) => Filter.Tendsto.{0, u1} Nat α u (Filter.atTop.{0} Nat (PartialOrder.toPreorder.{0} Nat (OrderedCancelAddCommMonoid.toPartialOrder.{0} Nat (StrictOrderedSemiring.toOrderedCancelAddCommMonoid.{0} Nat Nat.strictOrderedSemiring)))) (nhds.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) x)))) -> (CompleteSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1))
+but is expected to have type
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] (B : Nat -> Real), (forall (n : Nat), LT.lt.{0} Real Real.instLTReal (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal)) (B n)) -> (forall (u : Nat -> α), (forall (N : Nat) (n : Nat) (m : Nat), (LE.le.{0} Nat instLENat N n) -> (LE.le.{0} Nat instLENat N m) -> (LT.lt.{0} Real Real.instLTReal (Dist.dist.{u1} α (PseudoMetricSpace.toDist.{u1} α _inst_1) (u n) (u m)) (B N))) -> (Exists.{succ u1} α (fun (x : α) => Filter.Tendsto.{0, u1} Nat α u (Filter.atTop.{0} Nat (PartialOrder.toPreorder.{0} Nat (StrictOrderedSemiring.toPartialOrder.{0} Nat Nat.strictOrderedSemiring))) (nhds.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) x)))) -> (CompleteSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1))
+Case conversion may be inaccurate. Consider using '#align metric.complete_of_convergent_controlled_sequences Metric.complete_of_convergent_controlled_sequencesₓ'. -/
 /-- A very useful criterion to show that a space is complete is to show that all sequences
 which satisfy a bound of the form `dist (u n) (u m) < B N` for all `n m ≥ N` are
 converging. This is often applied for `B N = 2^{-N}`, i.e., with a very fast convergence to
@@ -1447,13 +2429,16 @@ theorem Metric.complete_of_convergent_controlled_sequences (B : ℕ → Real) (h
     (fun n => { p : α × α | dist p.1 p.2 < B n }) (fun n => dist_mem_uniformity <| hB n) H
 #align metric.complete_of_convergent_controlled_sequences Metric.complete_of_convergent_controlled_sequences
 
+#print Metric.complete_of_cauchySeq_tendsto /-
 theorem Metric.complete_of_cauchySeq_tendsto :
     (∀ u : ℕ → α, CauchySeq u → ∃ a, Tendsto u atTop (𝓝 a)) → CompleteSpace α :=
   EMetric.complete_of_cauchySeq_tendsto
 #align metric.complete_of_cauchy_seq_tendsto Metric.complete_of_cauchySeq_tendsto
+-/
 
 section Real
 
+#print Real.pseudoMetricSpace /-
 /-- Instantiate the reals as a pseudometric space. -/
 instance Real.pseudoMetricSpace : PseudoMetricSpace ℝ
     where
@@ -1462,41 +2447,96 @@ instance Real.pseudoMetricSpace : PseudoMetricSpace ℝ
   dist_comm x y := abs_sub_comm _ _
   dist_triangle x y z := abs_sub_le _ _ _
 #align real.pseudo_metric_space Real.pseudoMetricSpace
+-/
 
+/- warning: real.dist_eq -> Real.dist_eq is a dubious translation:
+lean 3 declaration is
+  forall (x : Real) (y : Real), Eq.{1} Real (Dist.dist.{0} Real (PseudoMetricSpace.toHasDist.{0} Real Real.pseudoMetricSpace) x y) (Abs.abs.{0} Real (Neg.toHasAbs.{0} Real Real.hasNeg Real.hasSup) (HSub.hSub.{0, 0, 0} Real Real Real (instHSub.{0} Real Real.hasSub) x y))
+but is expected to have type
+  forall (x : Real) (y : Real), Eq.{1} Real (Dist.dist.{0} Real (PseudoMetricSpace.toDist.{0} Real Real.pseudoMetricSpace) x y) (Abs.abs.{0} Real (Neg.toHasAbs.{0} Real Real.instNegReal Real.instSupReal) (HSub.hSub.{0, 0, 0} Real Real Real (instHSub.{0} Real Real.instSubReal) x y))
+Case conversion may be inaccurate. Consider using '#align real.dist_eq Real.dist_eqₓ'. -/
 theorem Real.dist_eq (x y : ℝ) : dist x y = |x - y| :=
   rfl
 #align real.dist_eq Real.dist_eq
 
+/- warning: real.nndist_eq -> Real.nndist_eq is a dubious translation:
+lean 3 declaration is
+  forall (x : Real) (y : Real), Eq.{1} NNReal (NNDist.nndist.{0} Real (PseudoMetricSpace.toNNDist.{0} Real Real.pseudoMetricSpace) x y) (coeFn.{1, 1} (MonoidWithZeroHom.{0, 0} Real NNReal (NonAssocSemiring.toMulZeroOneClass.{0} Real (NonAssocRing.toNonAssocSemiring.{0} Real (Ring.toNonAssocRing.{0} Real Real.ring))) (NonAssocSemiring.toMulZeroOneClass.{0} NNReal (Semiring.toNonAssocSemiring.{0} NNReal NNReal.semiring))) (fun (_x : MonoidWithZeroHom.{0, 0} Real NNReal (NonAssocSemiring.toMulZeroOneClass.{0} Real (NonAssocRing.toNonAssocSemiring.{0} Real (Ring.toNonAssocRing.{0} Real Real.ring))) (NonAssocSemiring.toMulZeroOneClass.{0} NNReal (Semiring.toNonAssocSemiring.{0} NNReal NNReal.semiring))) => Real -> NNReal) (MonoidWithZeroHom.hasCoeToFun.{0, 0} Real NNReal (NonAssocSemiring.toMulZeroOneClass.{0} Real (NonAssocRing.toNonAssocSemiring.{0} Real (Ring.toNonAssocRing.{0} Real Real.ring))) (NonAssocSemiring.toMulZeroOneClass.{0} NNReal (Semiring.toNonAssocSemiring.{0} NNReal NNReal.semiring))) Real.nnabs (HSub.hSub.{0, 0, 0} Real Real Real (instHSub.{0} Real Real.hasSub) x y))
+but is expected to have type
+  forall (x : Real) (y : Real), Eq.{1} NNReal (NNDist.nndist.{0} Real (PseudoMetricSpace.toNNDist.{0} Real Real.pseudoMetricSpace) x y) (FunLike.coe.{1, 1, 1} (MonoidWithZeroHom.{0, 0} Real NNReal (NonAssocSemiring.toMulZeroOneClass.{0} Real (NonAssocRing.toNonAssocSemiring.{0} Real (Ring.toNonAssocRing.{0} Real Real.instRingReal))) (NonAssocSemiring.toMulZeroOneClass.{0} NNReal (Semiring.toNonAssocSemiring.{0} NNReal instNNRealSemiring))) Real (fun (_x : Real) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2398 : Real) => NNReal) _x) (MulHomClass.toFunLike.{0, 0, 0} (MonoidWithZeroHom.{0, 0} Real NNReal (NonAssocSemiring.toMulZeroOneClass.{0} Real (NonAssocRing.toNonAssocSemiring.{0} Real (Ring.toNonAssocRing.{0} Real Real.instRingReal))) (NonAssocSemiring.toMulZeroOneClass.{0} NNReal (Semiring.toNonAssocSemiring.{0} NNReal instNNRealSemiring))) Real NNReal (MulOneClass.toMul.{0} Real (MulZeroOneClass.toMulOneClass.{0} Real (NonAssocSemiring.toMulZeroOneClass.{0} Real (NonAssocRing.toNonAssocSemiring.{0} Real (Ring.toNonAssocRing.{0} Real Real.instRingReal))))) (MulOneClass.toMul.{0} NNReal (MulZeroOneClass.toMulOneClass.{0} NNReal (NonAssocSemiring.toMulZeroOneClass.{0} NNReal (Semiring.toNonAssocSemiring.{0} NNReal instNNRealSemiring)))) (MonoidHomClass.toMulHomClass.{0, 0, 0} (MonoidWithZeroHom.{0, 0} Real NNReal (NonAssocSemiring.toMulZeroOneClass.{0} Real (NonAssocRing.toNonAssocSemiring.{0} Real (Ring.toNonAssocRing.{0} Real Real.instRingReal))) (NonAssocSemiring.toMulZeroOneClass.{0} NNReal (Semiring.toNonAssocSemiring.{0} NNReal instNNRealSemiring))) Real NNReal (MulZeroOneClass.toMulOneClass.{0} Real (NonAssocSemiring.toMulZeroOneClass.{0} Real (NonAssocRing.toNonAssocSemiring.{0} Real (Ring.toNonAssocRing.{0} Real Real.instRingReal)))) (MulZeroOneClass.toMulOneClass.{0} NNReal (NonAssocSemiring.toMulZeroOneClass.{0} NNReal (Semiring.toNonAssocSemiring.{0} NNReal instNNRealSemiring))) (MonoidWithZeroHomClass.toMonoidHomClass.{0, 0, 0} (MonoidWithZeroHom.{0, 0} Real NNReal (NonAssocSemiring.toMulZeroOneClass.{0} Real (NonAssocRing.toNonAssocSemiring.{0} Real (Ring.toNonAssocRing.{0} Real Real.instRingReal))) (NonAssocSemiring.toMulZeroOneClass.{0} NNReal (Semiring.toNonAssocSemiring.{0} NNReal instNNRealSemiring))) Real NNReal (NonAssocSemiring.toMulZeroOneClass.{0} Real (NonAssocRing.toNonAssocSemiring.{0} Real (Ring.toNonAssocRing.{0} Real Real.instRingReal))) (NonAssocSemiring.toMulZeroOneClass.{0} NNReal (Semiring.toNonAssocSemiring.{0} NNReal instNNRealSemiring)) (MonoidWithZeroHom.monoidWithZeroHomClass.{0, 0} Real NNReal (NonAssocSemiring.toMulZeroOneClass.{0} Real (NonAssocRing.toNonAssocSemiring.{0} Real (Ring.toNonAssocRing.{0} Real Real.instRingReal))) (NonAssocSemiring.toMulZeroOneClass.{0} NNReal (Semiring.toNonAssocSemiring.{0} NNReal instNNRealSemiring)))))) Real.nnabs (HSub.hSub.{0, 0, 0} Real Real Real (instHSub.{0} Real Real.instSubReal) x y))
+Case conversion may be inaccurate. Consider using '#align real.nndist_eq Real.nndist_eqₓ'. -/
 theorem Real.nndist_eq (x y : ℝ) : nndist x y = Real.nnabs (x - y) :=
   rfl
 #align real.nndist_eq Real.nndist_eq
 
+/- warning: real.nndist_eq' -> Real.nndist_eq' is a dubious translation:
+lean 3 declaration is
+  forall (x : Real) (y : Real), Eq.{1} NNReal (NNDist.nndist.{0} Real (PseudoMetricSpace.toNNDist.{0} Real Real.pseudoMetricSpace) x y) (coeFn.{1, 1} (MonoidWithZeroHom.{0, 0} Real NNReal (NonAssocSemiring.toMulZeroOneClass.{0} Real (NonAssocRing.toNonAssocSemiring.{0} Real (Ring.toNonAssocRing.{0} Real Real.ring))) (NonAssocSemiring.toMulZeroOneClass.{0} NNReal (Semiring.toNonAssocSemiring.{0} NNReal NNReal.semiring))) (fun (_x : MonoidWithZeroHom.{0, 0} Real NNReal (NonAssocSemiring.toMulZeroOneClass.{0} Real (NonAssocRing.toNonAssocSemiring.{0} Real (Ring.toNonAssocRing.{0} Real Real.ring))) (NonAssocSemiring.toMulZeroOneClass.{0} NNReal (Semiring.toNonAssocSemiring.{0} NNReal NNReal.semiring))) => Real -> NNReal) (MonoidWithZeroHom.hasCoeToFun.{0, 0} Real NNReal (NonAssocSemiring.toMulZeroOneClass.{0} Real (NonAssocRing.toNonAssocSemiring.{0} Real (Ring.toNonAssocRing.{0} Real Real.ring))) (NonAssocSemiring.toMulZeroOneClass.{0} NNReal (Semiring.toNonAssocSemiring.{0} NNReal NNReal.semiring))) Real.nnabs (HSub.hSub.{0, 0, 0} Real Real Real (instHSub.{0} Real Real.hasSub) y x))
+but is expected to have type
+  forall (x : Real) (y : Real), Eq.{1} NNReal (NNDist.nndist.{0} Real (PseudoMetricSpace.toNNDist.{0} Real Real.pseudoMetricSpace) x y) (FunLike.coe.{1, 1, 1} (MonoidWithZeroHom.{0, 0} Real NNReal (NonAssocSemiring.toMulZeroOneClass.{0} Real (NonAssocRing.toNonAssocSemiring.{0} Real (Ring.toNonAssocRing.{0} Real Real.instRingReal))) (NonAssocSemiring.toMulZeroOneClass.{0} NNReal (Semiring.toNonAssocSemiring.{0} NNReal instNNRealSemiring))) Real (fun (_x : Real) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2398 : Real) => NNReal) _x) (MulHomClass.toFunLike.{0, 0, 0} (MonoidWithZeroHom.{0, 0} Real NNReal (NonAssocSemiring.toMulZeroOneClass.{0} Real (NonAssocRing.toNonAssocSemiring.{0} Real (Ring.toNonAssocRing.{0} Real Real.instRingReal))) (NonAssocSemiring.toMulZeroOneClass.{0} NNReal (Semiring.toNonAssocSemiring.{0} NNReal instNNRealSemiring))) Real NNReal (MulOneClass.toMul.{0} Real (MulZeroOneClass.toMulOneClass.{0} Real (NonAssocSemiring.toMulZeroOneClass.{0} Real (NonAssocRing.toNonAssocSemiring.{0} Real (Ring.toNonAssocRing.{0} Real Real.instRingReal))))) (MulOneClass.toMul.{0} NNReal (MulZeroOneClass.toMulOneClass.{0} NNReal (NonAssocSemiring.toMulZeroOneClass.{0} NNReal (Semiring.toNonAssocSemiring.{0} NNReal instNNRealSemiring)))) (MonoidHomClass.toMulHomClass.{0, 0, 0} (MonoidWithZeroHom.{0, 0} Real NNReal (NonAssocSemiring.toMulZeroOneClass.{0} Real (NonAssocRing.toNonAssocSemiring.{0} Real (Ring.toNonAssocRing.{0} Real Real.instRingReal))) (NonAssocSemiring.toMulZeroOneClass.{0} NNReal (Semiring.toNonAssocSemiring.{0} NNReal instNNRealSemiring))) Real NNReal (MulZeroOneClass.toMulOneClass.{0} Real (NonAssocSemiring.toMulZeroOneClass.{0} Real (NonAssocRing.toNonAssocSemiring.{0} Real (Ring.toNonAssocRing.{0} Real Real.instRingReal)))) (MulZeroOneClass.toMulOneClass.{0} NNReal (NonAssocSemiring.toMulZeroOneClass.{0} NNReal (Semiring.toNonAssocSemiring.{0} NNReal instNNRealSemiring))) (MonoidWithZeroHomClass.toMonoidHomClass.{0, 0, 0} (MonoidWithZeroHom.{0, 0} Real NNReal (NonAssocSemiring.toMulZeroOneClass.{0} Real (NonAssocRing.toNonAssocSemiring.{0} Real (Ring.toNonAssocRing.{0} Real Real.instRingReal))) (NonAssocSemiring.toMulZeroOneClass.{0} NNReal (Semiring.toNonAssocSemiring.{0} NNReal instNNRealSemiring))) Real NNReal (NonAssocSemiring.toMulZeroOneClass.{0} Real (NonAssocRing.toNonAssocSemiring.{0} Real (Ring.toNonAssocRing.{0} Real Real.instRingReal))) (NonAssocSemiring.toMulZeroOneClass.{0} NNReal (Semiring.toNonAssocSemiring.{0} NNReal instNNRealSemiring)) (MonoidWithZeroHom.monoidWithZeroHomClass.{0, 0} Real NNReal (NonAssocSemiring.toMulZeroOneClass.{0} Real (NonAssocRing.toNonAssocSemiring.{0} Real (Ring.toNonAssocRing.{0} Real Real.instRingReal))) (NonAssocSemiring.toMulZeroOneClass.{0} NNReal (Semiring.toNonAssocSemiring.{0} NNReal instNNRealSemiring)))))) Real.nnabs (HSub.hSub.{0, 0, 0} Real Real Real (instHSub.{0} Real Real.instSubReal) y x))
+Case conversion may be inaccurate. Consider using '#align real.nndist_eq' Real.nndist_eq'ₓ'. -/
 theorem Real.nndist_eq' (x y : ℝ) : nndist x y = Real.nnabs (y - x) :=
   nndist_comm _ _
 #align real.nndist_eq' Real.nndist_eq'
 
+/- warning: real.dist_0_eq_abs -> Real.dist_0_eq_abs is a dubious translation:
+lean 3 declaration is
+  forall (x : Real), Eq.{1} Real (Dist.dist.{0} Real (PseudoMetricSpace.toHasDist.{0} Real Real.pseudoMetricSpace) x (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero)))) (Abs.abs.{0} Real (Neg.toHasAbs.{0} Real Real.hasNeg Real.hasSup) x)
+but is expected to have type
+  forall (x : Real), Eq.{1} Real (Dist.dist.{0} Real (PseudoMetricSpace.toDist.{0} Real Real.pseudoMetricSpace) x (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal))) (Abs.abs.{0} Real (Neg.toHasAbs.{0} Real Real.instNegReal Real.instSupReal) x)
+Case conversion may be inaccurate. Consider using '#align real.dist_0_eq_abs Real.dist_0_eq_absₓ'. -/
 theorem Real.dist_0_eq_abs (x : ℝ) : dist x 0 = |x| := by simp [Real.dist_eq]
 #align real.dist_0_eq_abs Real.dist_0_eq_abs
 
+/- warning: real.dist_left_le_of_mem_uIcc -> Real.dist_left_le_of_mem_uIcc is a dubious translation:
+lean 3 declaration is
+  forall {x : Real} {y : Real} {z : Real}, (Membership.Mem.{0, 0} Real (Set.{0} Real) (Set.hasMem.{0} Real) y (Set.uIcc.{0} Real Real.lattice x z)) -> (LE.le.{0} Real Real.hasLe (Dist.dist.{0} Real (PseudoMetricSpace.toHasDist.{0} Real Real.pseudoMetricSpace) x y) (Dist.dist.{0} Real (PseudoMetricSpace.toHasDist.{0} Real Real.pseudoMetricSpace) x z))
+but is expected to have type
+  forall {x : Real} {y : Real} {z : Real}, (Membership.mem.{0, 0} Real (Set.{0} Real) (Set.instMembershipSet.{0} Real) y (Set.uIcc.{0} Real Real.lattice x z)) -> (LE.le.{0} Real Real.instLEReal (Dist.dist.{0} Real (PseudoMetricSpace.toDist.{0} Real Real.pseudoMetricSpace) x y) (Dist.dist.{0} Real (PseudoMetricSpace.toDist.{0} Real Real.pseudoMetricSpace) x z))
+Case conversion may be inaccurate. Consider using '#align real.dist_left_le_of_mem_uIcc Real.dist_left_le_of_mem_uIccₓ'. -/
 theorem Real.dist_left_le_of_mem_uIcc {x y z : ℝ} (h : y ∈ uIcc x z) : dist x y ≤ dist x z := by
   simpa only [dist_comm x] using abs_sub_left_of_mem_uIcc h
 #align real.dist_left_le_of_mem_uIcc Real.dist_left_le_of_mem_uIcc
 
+/- warning: real.dist_right_le_of_mem_uIcc -> Real.dist_right_le_of_mem_uIcc is a dubious translation:
+lean 3 declaration is
+  forall {x : Real} {y : Real} {z : Real}, (Membership.Mem.{0, 0} Real (Set.{0} Real) (Set.hasMem.{0} Real) y (Set.uIcc.{0} Real Real.lattice x z)) -> (LE.le.{0} Real Real.hasLe (Dist.dist.{0} Real (PseudoMetricSpace.toHasDist.{0} Real Real.pseudoMetricSpace) y z) (Dist.dist.{0} Real (PseudoMetricSpace.toHasDist.{0} Real Real.pseudoMetricSpace) x z))
+but is expected to have type
+  forall {x : Real} {y : Real} {z : Real}, (Membership.mem.{0, 0} Real (Set.{0} Real) (Set.instMembershipSet.{0} Real) y (Set.uIcc.{0} Real Real.lattice x z)) -> (LE.le.{0} Real Real.instLEReal (Dist.dist.{0} Real (PseudoMetricSpace.toDist.{0} Real Real.pseudoMetricSpace) y z) (Dist.dist.{0} Real (PseudoMetricSpace.toDist.{0} Real Real.pseudoMetricSpace) x z))
+Case conversion may be inaccurate. Consider using '#align real.dist_right_le_of_mem_uIcc Real.dist_right_le_of_mem_uIccₓ'. -/
 theorem Real.dist_right_le_of_mem_uIcc {x y z : ℝ} (h : y ∈ uIcc x z) : dist y z ≤ dist x z := by
   simpa only [dist_comm _ z] using abs_sub_right_of_mem_uIcc h
 #align real.dist_right_le_of_mem_uIcc Real.dist_right_le_of_mem_uIcc
 
+/- warning: real.dist_le_of_mem_uIcc -> Real.dist_le_of_mem_uIcc is a dubious translation:
+lean 3 declaration is
+  forall {x : Real} {y : Real} {x' : Real} {y' : Real}, (Membership.Mem.{0, 0} Real (Set.{0} Real) (Set.hasMem.{0} Real) x (Set.uIcc.{0} Real Real.lattice x' y')) -> (Membership.Mem.{0, 0} Real (Set.{0} Real) (Set.hasMem.{0} Real) y (Set.uIcc.{0} Real Real.lattice x' y')) -> (LE.le.{0} Real Real.hasLe (Dist.dist.{0} Real (PseudoMetricSpace.toHasDist.{0} Real Real.pseudoMetricSpace) x y) (Dist.dist.{0} Real (PseudoMetricSpace.toHasDist.{0} Real Real.pseudoMetricSpace) x' y'))
+but is expected to have type
+  forall {x : Real} {y : Real} {x' : Real} {y' : Real}, (Membership.mem.{0, 0} Real (Set.{0} Real) (Set.instMembershipSet.{0} Real) x (Set.uIcc.{0} Real Real.lattice x' y')) -> (Membership.mem.{0, 0} Real (Set.{0} Real) (Set.instMembershipSet.{0} Real) y (Set.uIcc.{0} Real Real.lattice x' y')) -> (LE.le.{0} Real Real.instLEReal (Dist.dist.{0} Real (PseudoMetricSpace.toDist.{0} Real Real.pseudoMetricSpace) x y) (Dist.dist.{0} Real (PseudoMetricSpace.toDist.{0} Real Real.pseudoMetricSpace) x' y'))
+Case conversion may be inaccurate. Consider using '#align real.dist_le_of_mem_uIcc Real.dist_le_of_mem_uIccₓ'. -/
 theorem Real.dist_le_of_mem_uIcc {x y x' y' : ℝ} (hx : x ∈ uIcc x' y') (hy : y ∈ uIcc x' y') :
     dist x y ≤ dist x' y' :=
   abs_sub_le_of_uIcc_subset_uIcc <| uIcc_subset_uIcc (by rwa [uIcc_comm]) (by rwa [uIcc_comm])
 #align real.dist_le_of_mem_uIcc Real.dist_le_of_mem_uIcc
 
+/- warning: real.dist_le_of_mem_Icc -> Real.dist_le_of_mem_Icc is a dubious translation:
+lean 3 declaration is
+  forall {x : Real} {y : Real} {x' : Real} {y' : Real}, (Membership.Mem.{0, 0} Real (Set.{0} Real) (Set.hasMem.{0} Real) x (Set.Icc.{0} Real Real.preorder x' y')) -> (Membership.Mem.{0, 0} Real (Set.{0} Real) (Set.hasMem.{0} Real) y (Set.Icc.{0} Real Real.preorder x' y')) -> (LE.le.{0} Real Real.hasLe (Dist.dist.{0} Real (PseudoMetricSpace.toHasDist.{0} Real Real.pseudoMetricSpace) x y) (HSub.hSub.{0, 0, 0} Real Real Real (instHSub.{0} Real Real.hasSub) y' x'))
+but is expected to have type
+  forall {x : Real} {y : Real} {x' : Real} {y' : Real}, (Membership.mem.{0, 0} Real (Set.{0} Real) (Set.instMembershipSet.{0} Real) x (Set.Icc.{0} Real Real.instPreorderReal x' y')) -> (Membership.mem.{0, 0} Real (Set.{0} Real) (Set.instMembershipSet.{0} Real) y (Set.Icc.{0} Real Real.instPreorderReal x' y')) -> (LE.le.{0} Real Real.instLEReal (Dist.dist.{0} Real (PseudoMetricSpace.toDist.{0} Real Real.pseudoMetricSpace) x y) (HSub.hSub.{0, 0, 0} Real Real Real (instHSub.{0} Real Real.instSubReal) y' x'))
+Case conversion may be inaccurate. Consider using '#align real.dist_le_of_mem_Icc Real.dist_le_of_mem_Iccₓ'. -/
 theorem Real.dist_le_of_mem_Icc {x y x' y' : ℝ} (hx : x ∈ Icc x' y') (hy : y ∈ Icc x' y') :
     dist x y ≤ y' - x' := by
   simpa only [Real.dist_eq, abs_of_nonpos (sub_nonpos.2 <| hx.1.trans hx.2), neg_sub] using
     Real.dist_le_of_mem_uIcc (Icc_subset_uIcc hx) (Icc_subset_uIcc hy)
 #align real.dist_le_of_mem_Icc Real.dist_le_of_mem_Icc
 
+/- warning: real.dist_le_of_mem_Icc_01 -> Real.dist_le_of_mem_Icc_01 is a dubious translation:
+lean 3 declaration is
+  forall {x : Real} {y : Real}, (Membership.Mem.{0, 0} Real (Set.{0} Real) (Set.hasMem.{0} Real) x (Set.Icc.{0} Real Real.preorder (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero))) (OfNat.ofNat.{0} Real 1 (OfNat.mk.{0} Real 1 (One.one.{0} Real Real.hasOne))))) -> (Membership.Mem.{0, 0} Real (Set.{0} Real) (Set.hasMem.{0} Real) y (Set.Icc.{0} Real Real.preorder (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero))) (OfNat.ofNat.{0} Real 1 (OfNat.mk.{0} Real 1 (One.one.{0} Real Real.hasOne))))) -> (LE.le.{0} Real Real.hasLe (Dist.dist.{0} Real (PseudoMetricSpace.toHasDist.{0} Real Real.pseudoMetricSpace) x y) (OfNat.ofNat.{0} Real 1 (OfNat.mk.{0} Real 1 (One.one.{0} Real Real.hasOne))))
+but is expected to have type
+  forall {x : Real} {y : Real}, (Membership.mem.{0, 0} Real (Set.{0} Real) (Set.instMembershipSet.{0} Real) x (Set.Icc.{0} Real Real.instPreorderReal (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal)) (OfNat.ofNat.{0} Real 1 (One.toOfNat1.{0} Real Real.instOneReal)))) -> (Membership.mem.{0, 0} Real (Set.{0} Real) (Set.instMembershipSet.{0} Real) y (Set.Icc.{0} Real Real.instPreorderReal (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal)) (OfNat.ofNat.{0} Real 1 (One.toOfNat1.{0} Real Real.instOneReal)))) -> (LE.le.{0} Real Real.instLEReal (Dist.dist.{0} Real (PseudoMetricSpace.toDist.{0} Real Real.pseudoMetricSpace) x y) (OfNat.ofNat.{0} Real 1 (One.toOfNat1.{0} Real Real.instOneReal)))
+Case conversion may be inaccurate. Consider using '#align real.dist_le_of_mem_Icc_01 Real.dist_le_of_mem_Icc_01ₓ'. -/
 theorem Real.dist_le_of_mem_Icc_01 {x y : ℝ} (hx : x ∈ Icc (0 : ℝ) 1) (hy : y ∈ Icc (0 : ℝ) 1) :
     dist x y ≤ 1 := by simpa only [sub_zero] using Real.dist_le_of_mem_Icc hx hy
 #align real.dist_le_of_mem_Icc_01 Real.dist_le_of_mem_Icc_01
@@ -1505,23 +2545,47 @@ instance : OrderTopology ℝ :=
   orderTopology_of_nhds_abs fun x => by
     simp only [nhds_basis_ball.eq_binfi, ball, Real.dist_eq, abs_sub_comm]
 
+/- warning: real.ball_eq_Ioo -> Real.ball_eq_Ioo is a dubious translation:
+lean 3 declaration is
+  forall (x : Real) (r : Real), Eq.{1} (Set.{0} Real) (Metric.ball.{0} Real Real.pseudoMetricSpace x r) (Set.Ioo.{0} Real Real.preorder (HSub.hSub.{0, 0, 0} Real Real Real (instHSub.{0} Real Real.hasSub) x r) (HAdd.hAdd.{0, 0, 0} Real Real Real (instHAdd.{0} Real Real.hasAdd) x r))
+but is expected to have type
+  forall (x : Real) (r : Real), Eq.{1} (Set.{0} Real) (Metric.ball.{0} Real Real.pseudoMetricSpace x r) (Set.Ioo.{0} Real Real.instPreorderReal (HSub.hSub.{0, 0, 0} Real Real Real (instHSub.{0} Real Real.instSubReal) x r) (HAdd.hAdd.{0, 0, 0} Real Real Real (instHAdd.{0} Real Real.instAddReal) x r))
+Case conversion may be inaccurate. Consider using '#align real.ball_eq_Ioo Real.ball_eq_Iooₓ'. -/
 theorem Real.ball_eq_Ioo (x r : ℝ) : ball x r = Ioo (x - r) (x + r) :=
   Set.ext fun y => by
     rw [mem_ball, dist_comm, Real.dist_eq, abs_sub_lt_iff, mem_Ioo, ← sub_lt_iff_lt_add',
       sub_lt_comm]
 #align real.ball_eq_Ioo Real.ball_eq_Ioo
 
+/- warning: real.closed_ball_eq_Icc -> Real.closedBall_eq_Icc is a dubious translation:
+lean 3 declaration is
+  forall {x : Real} {r : Real}, Eq.{1} (Set.{0} Real) (Metric.closedBall.{0} Real Real.pseudoMetricSpace x r) (Set.Icc.{0} Real Real.preorder (HSub.hSub.{0, 0, 0} Real Real Real (instHSub.{0} Real Real.hasSub) x r) (HAdd.hAdd.{0, 0, 0} Real Real Real (instHAdd.{0} Real Real.hasAdd) x r))
+but is expected to have type
+  forall {x : Real} {r : Real}, Eq.{1} (Set.{0} Real) (Metric.closedBall.{0} Real Real.pseudoMetricSpace x r) (Set.Icc.{0} Real Real.instPreorderReal (HSub.hSub.{0, 0, 0} Real Real Real (instHSub.{0} Real Real.instSubReal) x r) (HAdd.hAdd.{0, 0, 0} Real Real Real (instHAdd.{0} Real Real.instAddReal) x r))
+Case conversion may be inaccurate. Consider using '#align real.closed_ball_eq_Icc Real.closedBall_eq_Iccₓ'. -/
 theorem Real.closedBall_eq_Icc {x r : ℝ} : closedBall x r = Icc (x - r) (x + r) := by
   ext y <;>
     rw [mem_closed_ball, dist_comm, Real.dist_eq, abs_sub_le_iff, mem_Icc, ← sub_le_iff_le_add',
       sub_le_comm]
 #align real.closed_ball_eq_Icc Real.closedBall_eq_Icc
 
+/- warning: real.Ioo_eq_ball -> Real.Ioo_eq_ball is a dubious translation:
+lean 3 declaration is
+  forall (x : Real) (y : Real), Eq.{1} (Set.{0} Real) (Set.Ioo.{0} Real Real.preorder x y) (Metric.ball.{0} Real Real.pseudoMetricSpace (HDiv.hDiv.{0, 0, 0} Real Real Real (instHDiv.{0} Real (DivInvMonoid.toHasDiv.{0} Real (DivisionRing.toDivInvMonoid.{0} Real Real.divisionRing))) (HAdd.hAdd.{0, 0, 0} Real Real Real (instHAdd.{0} Real Real.hasAdd) x y) (OfNat.ofNat.{0} Real 2 (OfNat.mk.{0} Real 2 (bit0.{0} Real Real.hasAdd (One.one.{0} Real Real.hasOne))))) (HDiv.hDiv.{0, 0, 0} Real Real Real (instHDiv.{0} Real (DivInvMonoid.toHasDiv.{0} Real (DivisionRing.toDivInvMonoid.{0} Real Real.divisionRing))) (HSub.hSub.{0, 0, 0} Real Real Real (instHSub.{0} Real Real.hasSub) y x) (OfNat.ofNat.{0} Real 2 (OfNat.mk.{0} Real 2 (bit0.{0} Real Real.hasAdd (One.one.{0} Real Real.hasOne))))))
+but is expected to have type
+  forall (x : Real) (y : Real), Eq.{1} (Set.{0} Real) (Set.Ioo.{0} Real Real.instPreorderReal x y) (Metric.ball.{0} Real Real.pseudoMetricSpace (HDiv.hDiv.{0, 0, 0} Real Real Real (instHDiv.{0} Real (LinearOrderedField.toDiv.{0} Real Real.instLinearOrderedFieldReal)) (HAdd.hAdd.{0, 0, 0} Real Real Real (instHAdd.{0} Real Real.instAddReal) x y) (OfNat.ofNat.{0} Real 2 (instOfNat.{0} Real 2 Real.natCast (instAtLeastTwoHAddNatInstHAddInstAddNatOfNat (OfNat.ofNat.{0} Nat 0 (instOfNatNat 0)))))) (HDiv.hDiv.{0, 0, 0} Real Real Real (instHDiv.{0} Real (LinearOrderedField.toDiv.{0} Real Real.instLinearOrderedFieldReal)) (HSub.hSub.{0, 0, 0} Real Real Real (instHSub.{0} Real Real.instSubReal) y x) (OfNat.ofNat.{0} Real 2 (instOfNat.{0} Real 2 Real.natCast (instAtLeastTwoHAddNatInstHAddInstAddNatOfNat (OfNat.ofNat.{0} Nat 0 (instOfNatNat 0)))))))
+Case conversion may be inaccurate. Consider using '#align real.Ioo_eq_ball Real.Ioo_eq_ballₓ'. -/
 theorem Real.Ioo_eq_ball (x y : ℝ) : Ioo x y = ball ((x + y) / 2) ((y - x) / 2) := by
   rw [Real.ball_eq_Ioo, ← sub_div, add_comm, ← sub_add, add_sub_cancel', add_self_div_two, ←
     add_div, add_assoc, add_sub_cancel'_right, add_self_div_two]
 #align real.Ioo_eq_ball Real.Ioo_eq_ball
 
+/- warning: real.Icc_eq_closed_ball -> Real.Icc_eq_closedBall is a dubious translation:
+lean 3 declaration is
+  forall (x : Real) (y : Real), Eq.{1} (Set.{0} Real) (Set.Icc.{0} Real Real.preorder x y) (Metric.closedBall.{0} Real Real.pseudoMetricSpace (HDiv.hDiv.{0, 0, 0} Real Real Real (instHDiv.{0} Real (DivInvMonoid.toHasDiv.{0} Real (DivisionRing.toDivInvMonoid.{0} Real Real.divisionRing))) (HAdd.hAdd.{0, 0, 0} Real Real Real (instHAdd.{0} Real Real.hasAdd) x y) (OfNat.ofNat.{0} Real 2 (OfNat.mk.{0} Real 2 (bit0.{0} Real Real.hasAdd (One.one.{0} Real Real.hasOne))))) (HDiv.hDiv.{0, 0, 0} Real Real Real (instHDiv.{0} Real (DivInvMonoid.toHasDiv.{0} Real (DivisionRing.toDivInvMonoid.{0} Real Real.divisionRing))) (HSub.hSub.{0, 0, 0} Real Real Real (instHSub.{0} Real Real.hasSub) y x) (OfNat.ofNat.{0} Real 2 (OfNat.mk.{0} Real 2 (bit0.{0} Real Real.hasAdd (One.one.{0} Real Real.hasOne))))))
+but is expected to have type
+  forall (x : Real) (y : Real), Eq.{1} (Set.{0} Real) (Set.Icc.{0} Real Real.instPreorderReal x y) (Metric.closedBall.{0} Real Real.pseudoMetricSpace (HDiv.hDiv.{0, 0, 0} Real Real Real (instHDiv.{0} Real (LinearOrderedField.toDiv.{0} Real Real.instLinearOrderedFieldReal)) (HAdd.hAdd.{0, 0, 0} Real Real Real (instHAdd.{0} Real Real.instAddReal) x y) (OfNat.ofNat.{0} Real 2 (instOfNat.{0} Real 2 Real.natCast (instAtLeastTwoHAddNatInstHAddInstAddNatOfNat (OfNat.ofNat.{0} Nat 0 (instOfNatNat 0)))))) (HDiv.hDiv.{0, 0, 0} Real Real Real (instHDiv.{0} Real (LinearOrderedField.toDiv.{0} Real Real.instLinearOrderedFieldReal)) (HSub.hSub.{0, 0, 0} Real Real Real (instHSub.{0} Real Real.instSubReal) y x) (OfNat.ofNat.{0} Real 2 (instOfNat.{0} Real 2 Real.natCast (instAtLeastTwoHAddNatInstHAddInstAddNatOfNat (OfNat.ofNat.{0} Nat 0 (instOfNatNat 0)))))))
+Case conversion may be inaccurate. Consider using '#align real.Icc_eq_closed_ball Real.Icc_eq_closedBallₓ'. -/
 theorem Real.Icc_eq_closedBall (x y : ℝ) : Icc x y = closedBall ((x + y) / 2) ((y - x) / 2) := by
   rw [Real.closedBall_eq_Icc, ← sub_div, add_comm, ← sub_add, add_sub_cancel', add_self_div_two, ←
     add_div, add_assoc, add_sub_cancel'_right, add_self_div_two]
@@ -1531,24 +2595,38 @@ section MetricOrdered
 
 variable [Preorder α] [CompactIccSpace α]
 
+#print totallyBounded_Icc /-
 theorem totallyBounded_Icc (a b : α) : TotallyBounded (Icc a b) :=
   isCompact_Icc.TotallyBounded
 #align totally_bounded_Icc totallyBounded_Icc
+-/
 
+#print totallyBounded_Ico /-
 theorem totallyBounded_Ico (a b : α) : TotallyBounded (Ico a b) :=
   totallyBounded_subset Ico_subset_Icc_self (totallyBounded_Icc a b)
 #align totally_bounded_Ico totallyBounded_Ico
+-/
 
+#print totallyBounded_Ioc /-
 theorem totallyBounded_Ioc (a b : α) : TotallyBounded (Ioc a b) :=
   totallyBounded_subset Ioc_subset_Icc_self (totallyBounded_Icc a b)
 #align totally_bounded_Ioc totallyBounded_Ioc
+-/
 
+#print totallyBounded_Ioo /-
 theorem totallyBounded_Ioo (a b : α) : TotallyBounded (Ioo a b) :=
   totallyBounded_subset Ioo_subset_Icc_self (totallyBounded_Icc a b)
 #align totally_bounded_Ioo totallyBounded_Ioo
+-/
 
 end MetricOrdered
 
+/- warning: squeeze_zero' -> squeeze_zero' is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} {f : α -> Real} {g : α -> Real} {t₀ : Filter.{u1} α}, (Filter.Eventually.{u1} α (fun (t : α) => LE.le.{0} Real Real.hasLe (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero))) (f t)) t₀) -> (Filter.Eventually.{u1} α (fun (t : α) => LE.le.{0} Real Real.hasLe (f t) (g t)) t₀) -> (Filter.Tendsto.{u1, 0} α Real g t₀ (nhds.{0} Real (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero))))) -> (Filter.Tendsto.{u1, 0} α Real f t₀ (nhds.{0} Real (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero)))))
+but is expected to have type
+  forall {α : Type.{u1}} {f : α -> Real} {g : α -> Real} {t₀ : Filter.{u1} α}, (Filter.Eventually.{u1} α (fun (t : α) => LE.le.{0} Real Real.instLEReal (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal)) (f t)) t₀) -> (Filter.Eventually.{u1} α (fun (t : α) => LE.le.{0} Real Real.instLEReal (f t) (g t)) t₀) -> (Filter.Tendsto.{u1, 0} α Real g t₀ (nhds.{0} Real (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal)))) -> (Filter.Tendsto.{u1, 0} α Real f t₀ (nhds.{0} Real (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal))))
+Case conversion may be inaccurate. Consider using '#align squeeze_zero' squeeze_zero'ₓ'. -/
 /-- Special case of the sandwich theorem; see `tendsto_of_tendsto_of_tendsto_of_le_of_le'` for the
 general case. -/
 theorem squeeze_zero' {α} {f g : α → ℝ} {t₀ : Filter α} (hf : ∀ᶠ t in t₀, 0 ≤ f t)
@@ -1556,6 +2634,12 @@ theorem squeeze_zero' {α} {f g : α → ℝ} {t₀ : Filter α} (hf : ∀ᶠ t
   tendsto_of_tendsto_of_tendsto_of_le_of_le' tendsto_const_nhds g0 hf hft
 #align squeeze_zero' squeeze_zero'
 
+/- warning: squeeze_zero -> squeeze_zero is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} {f : α -> Real} {g : α -> Real} {t₀ : Filter.{u1} α}, (forall (t : α), LE.le.{0} Real Real.hasLe (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero))) (f t)) -> (forall (t : α), LE.le.{0} Real Real.hasLe (f t) (g t)) -> (Filter.Tendsto.{u1, 0} α Real g t₀ (nhds.{0} Real (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero))))) -> (Filter.Tendsto.{u1, 0} α Real f t₀ (nhds.{0} Real (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero)))))
+but is expected to have type
+  forall {α : Type.{u1}} {f : α -> Real} {g : α -> Real} {t₀ : Filter.{u1} α}, (forall (t : α), LE.le.{0} Real Real.instLEReal (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal)) (f t)) -> (forall (t : α), LE.le.{0} Real Real.instLEReal (f t) (g t)) -> (Filter.Tendsto.{u1, 0} α Real g t₀ (nhds.{0} Real (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal)))) -> (Filter.Tendsto.{u1, 0} α Real f t₀ (nhds.{0} Real (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal))))
+Case conversion may be inaccurate. Consider using '#align squeeze_zero squeeze_zeroₓ'. -/
 /-- Special case of the sandwich theorem; see `tendsto_of_tendsto_of_tendsto_of_le_of_le`
 and  `tendsto_of_tendsto_of_tendsto_of_le_of_le'` for the general case. -/
 theorem squeeze_zero {α} {f g : α → ℝ} {t₀ : Filter α} (hf : ∀ t, 0 ≤ f t) (hft : ∀ t, f t ≤ g t)
@@ -1563,6 +2647,12 @@ theorem squeeze_zero {α} {f g : α → ℝ} {t₀ : Filter α} (hf : ∀ t, 0 
   squeeze_zero' (eventually_of_forall hf) (eventually_of_forall hft) g0
 #align squeeze_zero squeeze_zero
 
+/- warning: metric.uniformity_eq_comap_nhds_zero -> Metric.uniformity_eq_comap_nhds_zero is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α], Eq.{succ u1} (Filter.{u1} (Prod.{u1, u1} α α)) (uniformity.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) (Filter.comap.{u1, 0} (Prod.{u1, u1} α α) Real (fun (p : Prod.{u1, u1} α α) => Dist.dist.{u1} α (PseudoMetricSpace.toHasDist.{u1} α _inst_1) (Prod.fst.{u1, u1} α α p) (Prod.snd.{u1, u1} α α p)) (nhds.{0} Real (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero)))))
+but is expected to have type
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α], Eq.{succ u1} (Filter.{u1} (Prod.{u1, u1} α α)) (uniformity.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) (Filter.comap.{u1, 0} (Prod.{u1, u1} α α) Real (fun (p : Prod.{u1, u1} α α) => Dist.dist.{u1} α (PseudoMetricSpace.toDist.{u1} α _inst_1) (Prod.fst.{u1, u1} α α p) (Prod.snd.{u1, u1} α α p)) (nhds.{0} Real (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal))))
+Case conversion may be inaccurate. Consider using '#align metric.uniformity_eq_comap_nhds_zero Metric.uniformity_eq_comap_nhds_zeroₓ'. -/
 theorem Metric.uniformity_eq_comap_nhds_zero :
     𝓤 α = comap (fun p : α × α => dist p.1 p.2) (𝓝 (0 : ℝ)) :=
   by
@@ -1570,30 +2660,66 @@ theorem Metric.uniformity_eq_comap_nhds_zero :
   simp [mem_uniformity_dist, (nhds_basis_ball.comap _).mem_iff, subset_def, Real.dist_0_eq_abs]
 #align metric.uniformity_eq_comap_nhds_zero Metric.uniformity_eq_comap_nhds_zero
 
+/- warning: cauchy_seq_iff_tendsto_dist_at_top_0 -> cauchySeq_iff_tendsto_dist_atTop_0 is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : PseudoMetricSpace.{u1} α] [_inst_2 : Nonempty.{succ u2} β] [_inst_3 : SemilatticeSup.{u2} β] {u : β -> α}, Iff (CauchySeq.{u1, u2} α β (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1) _inst_3 u) (Filter.Tendsto.{u2, 0} (Prod.{u2, u2} β β) Real (fun (n : Prod.{u2, u2} β β) => Dist.dist.{u1} α (PseudoMetricSpace.toHasDist.{u1} α _inst_1) (u (Prod.fst.{u2, u2} β β n)) (u (Prod.snd.{u2, u2} β β n))) (Filter.atTop.{u2} (Prod.{u2, u2} β β) (Prod.preorder.{u2, u2} β β (PartialOrder.toPreorder.{u2} β (SemilatticeSup.toPartialOrder.{u2} β _inst_3)) (PartialOrder.toPreorder.{u2} β (SemilatticeSup.toPartialOrder.{u2} β _inst_3)))) (nhds.{0} Real (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero)))))
+but is expected to have type
+  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : PseudoMetricSpace.{u1} α] [_inst_2 : Nonempty.{succ u2} β] [_inst_3 : SemilatticeSup.{u2} β] {u : β -> α}, Iff (CauchySeq.{u1, u2} α β (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1) _inst_3 u) (Filter.Tendsto.{u2, 0} (Prod.{u2, u2} β β) Real (fun (n : Prod.{u2, u2} β β) => Dist.dist.{u1} α (PseudoMetricSpace.toDist.{u1} α _inst_1) (u (Prod.fst.{u2, u2} β β n)) (u (Prod.snd.{u2, u2} β β n))) (Filter.atTop.{u2} (Prod.{u2, u2} β β) (Prod.instPreorderProd.{u2, u2} β β (PartialOrder.toPreorder.{u2} β (SemilatticeSup.toPartialOrder.{u2} β _inst_3)) (PartialOrder.toPreorder.{u2} β (SemilatticeSup.toPartialOrder.{u2} β _inst_3)))) (nhds.{0} Real (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal))))
+Case conversion may be inaccurate. Consider using '#align cauchy_seq_iff_tendsto_dist_at_top_0 cauchySeq_iff_tendsto_dist_atTop_0ₓ'. -/
 theorem cauchySeq_iff_tendsto_dist_atTop_0 [Nonempty β] [SemilatticeSup β] {u : β → α} :
     CauchySeq u ↔ Tendsto (fun n : β × β => dist (u n.1) (u n.2)) atTop (𝓝 0) := by
   rw [cauchySeq_iff_tendsto, Metric.uniformity_eq_comap_nhds_zero, tendsto_comap_iff, Prod.map_def]
 #align cauchy_seq_iff_tendsto_dist_at_top_0 cauchySeq_iff_tendsto_dist_atTop_0
 
+/- warning: tendsto_uniformity_iff_dist_tendsto_zero -> tendsto_uniformity_iff_dist_tendsto_zero is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {ι : Type.{u2}} {f : ι -> (Prod.{u1, u1} α α)} {p : Filter.{u2} ι}, Iff (Filter.Tendsto.{u2, u1} ι (Prod.{u1, u1} α α) f p (uniformity.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1))) (Filter.Tendsto.{u2, 0} ι Real (fun (x : ι) => Dist.dist.{u1} α (PseudoMetricSpace.toHasDist.{u1} α _inst_1) (Prod.fst.{u1, u1} α α (f x)) (Prod.snd.{u1, u1} α α (f x))) p (nhds.{0} Real (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero)))))
+but is expected to have type
+  forall {α : Type.{u2}} [_inst_1 : PseudoMetricSpace.{u2} α] {ι : Type.{u1}} {f : ι -> (Prod.{u2, u2} α α)} {p : Filter.{u1} ι}, Iff (Filter.Tendsto.{u1, u2} ι (Prod.{u2, u2} α α) f p (uniformity.{u2} α (PseudoMetricSpace.toUniformSpace.{u2} α _inst_1))) (Filter.Tendsto.{u1, 0} ι Real (fun (x : ι) => Dist.dist.{u2} α (PseudoMetricSpace.toDist.{u2} α _inst_1) (Prod.fst.{u2, u2} α α (f x)) (Prod.snd.{u2, u2} α α (f x))) p (nhds.{0} Real (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal))))
+Case conversion may be inaccurate. Consider using '#align tendsto_uniformity_iff_dist_tendsto_zero tendsto_uniformity_iff_dist_tendsto_zeroₓ'. -/
 theorem tendsto_uniformity_iff_dist_tendsto_zero {ι : Type _} {f : ι → α × α} {p : Filter ι} :
     Tendsto f p (𝓤 α) ↔ Tendsto (fun x => dist (f x).1 (f x).2) p (𝓝 0) := by
   rw [Metric.uniformity_eq_comap_nhds_zero, tendsto_comap_iff]
 #align tendsto_uniformity_iff_dist_tendsto_zero tendsto_uniformity_iff_dist_tendsto_zero
 
+/- warning: filter.tendsto.congr_dist -> Filter.Tendsto.congr_dist is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {ι : Type.{u2}} {f₁ : ι -> α} {f₂ : ι -> α} {p : Filter.{u2} ι} {a : α}, (Filter.Tendsto.{u2, u1} ι α f₁ p (nhds.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) a)) -> (Filter.Tendsto.{u2, 0} ι Real (fun (x : ι) => Dist.dist.{u1} α (PseudoMetricSpace.toHasDist.{u1} α _inst_1) (f₁ x) (f₂ x)) p (nhds.{0} Real (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero))))) -> (Filter.Tendsto.{u2, u1} ι α f₂ p (nhds.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) a))
+but is expected to have type
+  forall {α : Type.{u2}} [_inst_1 : PseudoMetricSpace.{u2} α] {ι : Type.{u1}} {f₁ : ι -> α} {f₂ : ι -> α} {p : Filter.{u1} ι} {a : α}, (Filter.Tendsto.{u1, u2} ι α f₁ p (nhds.{u2} α (UniformSpace.toTopologicalSpace.{u2} α (PseudoMetricSpace.toUniformSpace.{u2} α _inst_1)) a)) -> (Filter.Tendsto.{u1, 0} ι Real (fun (x : ι) => Dist.dist.{u2} α (PseudoMetricSpace.toDist.{u2} α _inst_1) (f₁ x) (f₂ x)) p (nhds.{0} Real (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal)))) -> (Filter.Tendsto.{u1, u2} ι α f₂ p (nhds.{u2} α (UniformSpace.toTopologicalSpace.{u2} α (PseudoMetricSpace.toUniformSpace.{u2} α _inst_1)) a))
+Case conversion may be inaccurate. Consider using '#align filter.tendsto.congr_dist Filter.Tendsto.congr_distₓ'. -/
 theorem Filter.Tendsto.congr_dist {ι : Type _} {f₁ f₂ : ι → α} {p : Filter ι} {a : α}
     (h₁ : Tendsto f₁ p (𝓝 a)) (h : Tendsto (fun x => dist (f₁ x) (f₂ x)) p (𝓝 0)) :
     Tendsto f₂ p (𝓝 a) :=
   h₁.congr_uniformity <| tendsto_uniformity_iff_dist_tendsto_zero.2 h
 #align filter.tendsto.congr_dist Filter.Tendsto.congr_dist
 
+/- warning: tendsto_of_tendsto_of_dist -> tendsto_of_tendsto_of_dist is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {ι : Type.{u2}} {f₁ : ι -> α} {f₂ : ι -> α} {p : Filter.{u2} ι} {a : α}, (Filter.Tendsto.{u2, u1} ι α f₁ p (nhds.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) a)) -> (Filter.Tendsto.{u2, 0} ι Real (fun (x : ι) => Dist.dist.{u1} α (PseudoMetricSpace.toHasDist.{u1} α _inst_1) (f₁ x) (f₂ x)) p (nhds.{0} Real (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero))))) -> (Filter.Tendsto.{u2, u1} ι α f₂ p (nhds.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) a))
+but is expected to have type
+  forall {α : Type.{u2}} [_inst_1 : PseudoMetricSpace.{u2} α] {ι : Type.{u1}} {f₁ : ι -> α} {f₂ : ι -> α} {p : Filter.{u1} ι} {a : α}, (Filter.Tendsto.{u1, u2} ι α f₁ p (nhds.{u2} α (UniformSpace.toTopologicalSpace.{u2} α (PseudoMetricSpace.toUniformSpace.{u2} α _inst_1)) a)) -> (Filter.Tendsto.{u1, 0} ι Real (fun (x : ι) => Dist.dist.{u2} α (PseudoMetricSpace.toDist.{u2} α _inst_1) (f₁ x) (f₂ x)) p (nhds.{0} Real (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal)))) -> (Filter.Tendsto.{u1, u2} ι α f₂ p (nhds.{u2} α (UniformSpace.toTopologicalSpace.{u2} α (PseudoMetricSpace.toUniformSpace.{u2} α _inst_1)) a))
+Case conversion may be inaccurate. Consider using '#align tendsto_of_tendsto_of_dist tendsto_of_tendsto_of_distₓ'. -/
 alias Filter.Tendsto.congr_dist ← tendsto_of_tendsto_of_dist
 #align tendsto_of_tendsto_of_dist tendsto_of_tendsto_of_dist
 
+/- warning: tendsto_iff_of_dist -> tendsto_iff_of_dist is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {ι : Type.{u2}} {f₁ : ι -> α} {f₂ : ι -> α} {p : Filter.{u2} ι} {a : α}, (Filter.Tendsto.{u2, 0} ι Real (fun (x : ι) => Dist.dist.{u1} α (PseudoMetricSpace.toHasDist.{u1} α _inst_1) (f₁ x) (f₂ x)) p (nhds.{0} Real (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero))))) -> (Iff (Filter.Tendsto.{u2, u1} ι α f₁ p (nhds.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) a)) (Filter.Tendsto.{u2, u1} ι α f₂ p (nhds.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) a)))
+but is expected to have type
+  forall {α : Type.{u2}} [_inst_1 : PseudoMetricSpace.{u2} α] {ι : Type.{u1}} {f₁ : ι -> α} {f₂ : ι -> α} {p : Filter.{u1} ι} {a : α}, (Filter.Tendsto.{u1, 0} ι Real (fun (x : ι) => Dist.dist.{u2} α (PseudoMetricSpace.toDist.{u2} α _inst_1) (f₁ x) (f₂ x)) p (nhds.{0} Real (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal)))) -> (Iff (Filter.Tendsto.{u1, u2} ι α f₁ p (nhds.{u2} α (UniformSpace.toTopologicalSpace.{u2} α (PseudoMetricSpace.toUniformSpace.{u2} α _inst_1)) a)) (Filter.Tendsto.{u1, u2} ι α f₂ p (nhds.{u2} α (UniformSpace.toTopologicalSpace.{u2} α (PseudoMetricSpace.toUniformSpace.{u2} α _inst_1)) a)))
+Case conversion may be inaccurate. Consider using '#align tendsto_iff_of_dist tendsto_iff_of_distₓ'. -/
 theorem tendsto_iff_of_dist {ι : Type _} {f₁ f₂ : ι → α} {p : Filter ι} {a : α}
     (h : Tendsto (fun x => dist (f₁ x) (f₂ x)) p (𝓝 0)) : Tendsto f₁ p (𝓝 a) ↔ Tendsto f₂ p (𝓝 a) :=
   Uniform.tendsto_congr <| tendsto_uniformity_iff_dist_tendsto_zero.2 h
 #align tendsto_iff_of_dist tendsto_iff_of_dist
 
+/- warning: eventually_closed_ball_subset -> eventually_closedBall_subset is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {x : α} {u : Set.{u1} α}, (Membership.Mem.{u1, u1} (Set.{u1} α) (Filter.{u1} α) (Filter.hasMem.{u1} α) u (nhds.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) x)) -> (Filter.Eventually.{0} Real (fun (r : Real) => HasSubset.Subset.{u1} (Set.{u1} α) (Set.hasSubset.{u1} α) (Metric.closedBall.{u1} α _inst_1 x r) u) (nhds.{0} Real (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero)))))
+but is expected to have type
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {x : α} {u : Set.{u1} α}, (Membership.mem.{u1, u1} (Set.{u1} α) (Filter.{u1} α) (instMembershipSetFilter.{u1} α) u (nhds.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) x)) -> (Filter.Eventually.{0} Real (fun (r : Real) => HasSubset.Subset.{u1} (Set.{u1} α) (Set.instHasSubsetSet.{u1} α) (Metric.closedBall.{u1} α _inst_1 x r) u) (nhds.{0} Real (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal))))
+Case conversion may be inaccurate. Consider using '#align eventually_closed_ball_subset eventually_closedBall_subsetₓ'. -/
 /-- If `u` is a neighborhood of `x`, then for small enough `r`, the closed ball
 `closed_ball x r` is contained in `u`. -/
 theorem eventually_closedBall_subset {x : α} {u : Set α} (hu : u ∈ 𝓝 x) :
@@ -1611,6 +2737,12 @@ section CauchySeq
 
 variable [Nonempty β] [SemilatticeSup β]
 
+/- warning: metric.cauchy_seq_iff -> Metric.cauchySeq_iff is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : PseudoMetricSpace.{u1} α] [_inst_2 : Nonempty.{succ u2} β] [_inst_3 : SemilatticeSup.{u2} β] {u : β -> α}, Iff (CauchySeq.{u1, u2} α β (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1) _inst_3 u) (forall (ε : Real), (GT.gt.{0} Real Real.hasLt ε (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero)))) -> (Exists.{succ u2} β (fun (N : β) => forall (m : β), (GE.ge.{u2} β (Preorder.toLE.{u2} β (PartialOrder.toPreorder.{u2} β (SemilatticeSup.toPartialOrder.{u2} β _inst_3))) m N) -> (forall (n : β), (GE.ge.{u2} β (Preorder.toLE.{u2} β (PartialOrder.toPreorder.{u2} β (SemilatticeSup.toPartialOrder.{u2} β _inst_3))) n N) -> (LT.lt.{0} Real Real.hasLt (Dist.dist.{u1} α (PseudoMetricSpace.toHasDist.{u1} α _inst_1) (u m) (u n)) ε)))))
+but is expected to have type
+  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : PseudoMetricSpace.{u1} α] [_inst_2 : Nonempty.{succ u2} β] [_inst_3 : SemilatticeSup.{u2} β] {u : β -> α}, Iff (CauchySeq.{u1, u2} α β (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1) _inst_3 u) (forall (ε : Real), (GT.gt.{0} Real Real.instLTReal ε (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal))) -> (Exists.{succ u2} β (fun (N : β) => forall (m : β), (GE.ge.{u2} β (Preorder.toLE.{u2} β (PartialOrder.toPreorder.{u2} β (SemilatticeSup.toPartialOrder.{u2} β _inst_3))) m N) -> (forall (n : β), (GE.ge.{u2} β (Preorder.toLE.{u2} β (PartialOrder.toPreorder.{u2} β (SemilatticeSup.toPartialOrder.{u2} β _inst_3))) n N) -> (LT.lt.{0} Real Real.instLTReal (Dist.dist.{u1} α (PseudoMetricSpace.toDist.{u1} α _inst_1) (u m) (u n)) ε)))))
+Case conversion may be inaccurate. Consider using '#align metric.cauchy_seq_iff Metric.cauchySeq_iffₓ'. -/
 /- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (m n «expr ≥ » N) -/
 -- see Note [nolint_ge]
 /-- In a pseudometric space, Cauchy sequences are characterized by the fact that, eventually,
@@ -1621,12 +2753,24 @@ theorem Metric.cauchySeq_iff {u : β → α} :
   uniformity_basis_dist.cauchySeq_iff
 #align metric.cauchy_seq_iff Metric.cauchySeq_iff
 
+/- warning: metric.cauchy_seq_iff' -> Metric.cauchySeq_iff' is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : PseudoMetricSpace.{u1} α] [_inst_2 : Nonempty.{succ u2} β] [_inst_3 : SemilatticeSup.{u2} β] {u : β -> α}, Iff (CauchySeq.{u1, u2} α β (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1) _inst_3 u) (forall (ε : Real), (GT.gt.{0} Real Real.hasLt ε (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero)))) -> (Exists.{succ u2} β (fun (N : β) => forall (n : β), (GE.ge.{u2} β (Preorder.toLE.{u2} β (PartialOrder.toPreorder.{u2} β (SemilatticeSup.toPartialOrder.{u2} β _inst_3))) n N) -> (LT.lt.{0} Real Real.hasLt (Dist.dist.{u1} α (PseudoMetricSpace.toHasDist.{u1} α _inst_1) (u n) (u N)) ε))))
+but is expected to have type
+  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : PseudoMetricSpace.{u1} α] [_inst_2 : Nonempty.{succ u2} β] [_inst_3 : SemilatticeSup.{u2} β] {u : β -> α}, Iff (CauchySeq.{u1, u2} α β (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1) _inst_3 u) (forall (ε : Real), (GT.gt.{0} Real Real.instLTReal ε (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal))) -> (Exists.{succ u2} β (fun (N : β) => forall (n : β), (GE.ge.{u2} β (Preorder.toLE.{u2} β (PartialOrder.toPreorder.{u2} β (SemilatticeSup.toPartialOrder.{u2} β _inst_3))) n N) -> (LT.lt.{0} Real Real.instLTReal (Dist.dist.{u1} α (PseudoMetricSpace.toDist.{u1} α _inst_1) (u n) (u N)) ε))))
+Case conversion may be inaccurate. Consider using '#align metric.cauchy_seq_iff' Metric.cauchySeq_iff'ₓ'. -/
 /-- A variation around the pseudometric characterization of Cauchy sequences -/
 theorem Metric.cauchySeq_iff' {u : β → α} :
     CauchySeq u ↔ ∀ ε > 0, ∃ N, ∀ n ≥ N, dist (u n) (u N) < ε :=
   uniformity_basis_dist.cauchySeq_iff'
 #align metric.cauchy_seq_iff' Metric.cauchySeq_iff'
 
+/- warning: metric.uniform_cauchy_seq_on_iff -> Metric.uniformCauchySeqOn_iff is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : PseudoMetricSpace.{u1} α] [_inst_2 : Nonempty.{succ u2} β] [_inst_3 : SemilatticeSup.{u2} β] {γ : Type.{u3}} {F : β -> γ -> α} {s : Set.{u3} γ}, Iff (UniformCauchySeqOn.{u3, u1, u2} γ α β (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1) F (Filter.atTop.{u2} β (PartialOrder.toPreorder.{u2} β (SemilatticeSup.toPartialOrder.{u2} β _inst_3))) s) (forall (ε : Real), (GT.gt.{0} Real Real.hasLt ε (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero)))) -> (Exists.{succ u2} β (fun (N : β) => forall (m : β), (GE.ge.{u2} β (Preorder.toLE.{u2} β (PartialOrder.toPreorder.{u2} β (SemilatticeSup.toPartialOrder.{u2} β _inst_3))) m N) -> (forall (n : β), (GE.ge.{u2} β (Preorder.toLE.{u2} β (PartialOrder.toPreorder.{u2} β (SemilatticeSup.toPartialOrder.{u2} β _inst_3))) n N) -> (forall (x : γ), (Membership.Mem.{u3, u3} γ (Set.{u3} γ) (Set.hasMem.{u3} γ) x s) -> (LT.lt.{0} Real Real.hasLt (Dist.dist.{u1} α (PseudoMetricSpace.toHasDist.{u1} α _inst_1) (F m x) (F n x)) ε))))))
+but is expected to have type
+  forall {α : Type.{u2}} {β : Type.{u3}} [_inst_1 : PseudoMetricSpace.{u2} α] [_inst_2 : Nonempty.{succ u3} β] [_inst_3 : SemilatticeSup.{u3} β] {γ : Type.{u1}} {F : β -> γ -> α} {s : Set.{u1} γ}, Iff (UniformCauchySeqOn.{u1, u2, u3} γ α β (PseudoMetricSpace.toUniformSpace.{u2} α _inst_1) F (Filter.atTop.{u3} β (PartialOrder.toPreorder.{u3} β (SemilatticeSup.toPartialOrder.{u3} β _inst_3))) s) (forall (ε : Real), (GT.gt.{0} Real Real.instLTReal ε (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal))) -> (Exists.{succ u3} β (fun (N : β) => forall (m : β), (GE.ge.{u3} β (Preorder.toLE.{u3} β (PartialOrder.toPreorder.{u3} β (SemilatticeSup.toPartialOrder.{u3} β _inst_3))) m N) -> (forall (n : β), (GE.ge.{u3} β (Preorder.toLE.{u3} β (PartialOrder.toPreorder.{u3} β (SemilatticeSup.toPartialOrder.{u3} β _inst_3))) n N) -> (forall (x : γ), (Membership.mem.{u1, u1} γ (Set.{u1} γ) (Set.instMembershipSet.{u1} γ) x s) -> (LT.lt.{0} Real Real.instLTReal (Dist.dist.{u2} α (PseudoMetricSpace.toDist.{u2} α _inst_1) (F m x) (F n x)) ε))))))
+Case conversion may be inaccurate. Consider using '#align metric.uniform_cauchy_seq_on_iff Metric.uniformCauchySeqOn_iffₓ'. -/
 -- see Note [nolint_ge]
 /-- In a pseudometric space, unifom Cauchy sequences are characterized by the fact that, eventually,
 the distance between all its elements is uniformly, arbitrarily small -/
@@ -1657,6 +2801,12 @@ theorem Metric.uniformCauchySeqOn_iff {γ : Type _} {F : β → γ → α} {s :
     exact hab (hN b.fst hbl.ge b.snd hbr.ge x hx)
 #align metric.uniform_cauchy_seq_on_iff Metric.uniformCauchySeqOn_iff
 
+/- warning: cauchy_seq_of_le_tendsto_0' -> cauchySeq_of_le_tendsto_0' is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : PseudoMetricSpace.{u1} α] [_inst_2 : Nonempty.{succ u2} β] [_inst_3 : SemilatticeSup.{u2} β] {s : β -> α} (b : β -> Real), (forall (n : β) (m : β), (LE.le.{u2} β (Preorder.toLE.{u2} β (PartialOrder.toPreorder.{u2} β (SemilatticeSup.toPartialOrder.{u2} β _inst_3))) n m) -> (LE.le.{0} Real Real.hasLe (Dist.dist.{u1} α (PseudoMetricSpace.toHasDist.{u1} α _inst_1) (s n) (s m)) (b n))) -> (Filter.Tendsto.{u2, 0} β Real b (Filter.atTop.{u2} β (PartialOrder.toPreorder.{u2} β (SemilatticeSup.toPartialOrder.{u2} β _inst_3))) (nhds.{0} Real (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero))))) -> (CauchySeq.{u1, u2} α β (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1) _inst_3 s)
+but is expected to have type
+  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : PseudoMetricSpace.{u1} α] [_inst_2 : Nonempty.{succ u2} β] [_inst_3 : SemilatticeSup.{u2} β] {s : β -> α} (b : β -> Real), (forall (n : β) (m : β), (LE.le.{u2} β (Preorder.toLE.{u2} β (PartialOrder.toPreorder.{u2} β (SemilatticeSup.toPartialOrder.{u2} β _inst_3))) n m) -> (LE.le.{0} Real Real.instLEReal (Dist.dist.{u1} α (PseudoMetricSpace.toDist.{u1} α _inst_1) (s n) (s m)) (b n))) -> (Filter.Tendsto.{u2, 0} β Real b (Filter.atTop.{u2} β (PartialOrder.toPreorder.{u2} β (SemilatticeSup.toPartialOrder.{u2} β _inst_3))) (nhds.{0} Real (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal)))) -> (CauchySeq.{u1, u2} α β (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1) _inst_3 s)
+Case conversion may be inaccurate. Consider using '#align cauchy_seq_of_le_tendsto_0' cauchySeq_of_le_tendsto_0'ₓ'. -/
 /-- If the distance between `s n` and `s m`, `n ≤ m` is bounded above by `b n`
 and `b` converges to zero, then `s` is a Cauchy sequence.  -/
 theorem cauchySeq_of_le_tendsto_0' {s : β → α} (b : β → ℝ)
@@ -1670,6 +2820,12 @@ theorem cauchySeq_of_le_tendsto_0' {s : β → α} (b : β → ℝ)
         
 #align cauchy_seq_of_le_tendsto_0' cauchySeq_of_le_tendsto_0'
 
+/- warning: cauchy_seq_of_le_tendsto_0 -> cauchySeq_of_le_tendsto_0 is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : PseudoMetricSpace.{u1} α] [_inst_2 : Nonempty.{succ u2} β] [_inst_3 : SemilatticeSup.{u2} β] {s : β -> α} (b : β -> Real), (forall (n : β) (m : β) (N : β), (LE.le.{u2} β (Preorder.toLE.{u2} β (PartialOrder.toPreorder.{u2} β (SemilatticeSup.toPartialOrder.{u2} β _inst_3))) N n) -> (LE.le.{u2} β (Preorder.toLE.{u2} β (PartialOrder.toPreorder.{u2} β (SemilatticeSup.toPartialOrder.{u2} β _inst_3))) N m) -> (LE.le.{0} Real Real.hasLe (Dist.dist.{u1} α (PseudoMetricSpace.toHasDist.{u1} α _inst_1) (s n) (s m)) (b N))) -> (Filter.Tendsto.{u2, 0} β Real b (Filter.atTop.{u2} β (PartialOrder.toPreorder.{u2} β (SemilatticeSup.toPartialOrder.{u2} β _inst_3))) (nhds.{0} Real (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero))))) -> (CauchySeq.{u1, u2} α β (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1) _inst_3 s)
+but is expected to have type
+  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : PseudoMetricSpace.{u1} α] [_inst_2 : Nonempty.{succ u2} β] [_inst_3 : SemilatticeSup.{u2} β] {s : β -> α} (b : β -> Real), (forall (n : β) (m : β) (N : β), (LE.le.{u2} β (Preorder.toLE.{u2} β (PartialOrder.toPreorder.{u2} β (SemilatticeSup.toPartialOrder.{u2} β _inst_3))) N n) -> (LE.le.{u2} β (Preorder.toLE.{u2} β (PartialOrder.toPreorder.{u2} β (SemilatticeSup.toPartialOrder.{u2} β _inst_3))) N m) -> (LE.le.{0} Real Real.instLEReal (Dist.dist.{u1} α (PseudoMetricSpace.toDist.{u1} α _inst_1) (s n) (s m)) (b N))) -> (Filter.Tendsto.{u2, 0} β Real b (Filter.atTop.{u2} β (PartialOrder.toPreorder.{u2} β (SemilatticeSup.toPartialOrder.{u2} β _inst_3))) (nhds.{0} Real (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal)))) -> (CauchySeq.{u1, u2} α β (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1) _inst_3 s)
+Case conversion may be inaccurate. Consider using '#align cauchy_seq_of_le_tendsto_0 cauchySeq_of_le_tendsto_0ₓ'. -/
 /-- If the distance between `s n` and `s m`, `n, m ≥ N` is bounded above by `b N`
 and `b` converges to zero, then `s` is a Cauchy sequence.  -/
 theorem cauchySeq_of_le_tendsto_0 {s : β → α} (b : β → ℝ)
@@ -1678,6 +2834,12 @@ theorem cauchySeq_of_le_tendsto_0 {s : β → α} (b : β → ℝ)
   cauchySeq_of_le_tendsto_0' b (fun n m hnm => h _ _ _ le_rfl hnm) h₀
 #align cauchy_seq_of_le_tendsto_0 cauchySeq_of_le_tendsto_0
 
+/- warning: cauchy_seq_bdd -> cauchySeq_bdd is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {u : Nat -> α}, (CauchySeq.{u1, 0} α Nat (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1) (CanonicallyLinearOrderedAddMonoid.semilatticeSup.{0} Nat Nat.canonicallyLinearOrderedAddMonoid) u) -> (Exists.{1} Real (fun (R : Real) => Exists.{0} (GT.gt.{0} Real Real.hasLt R (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero)))) (fun (H : GT.gt.{0} Real Real.hasLt R (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero)))) => forall (m : Nat) (n : Nat), LT.lt.{0} Real Real.hasLt (Dist.dist.{u1} α (PseudoMetricSpace.toHasDist.{u1} α _inst_1) (u m) (u n)) R)))
+but is expected to have type
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {u : Nat -> α}, (CauchySeq.{u1, 0} α Nat (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1) (Lattice.toSemilatticeSup.{0} Nat (DistribLattice.toLattice.{0} Nat instDistribLatticeNat)) u) -> (Exists.{1} Real (fun (R : Real) => And (GT.gt.{0} Real Real.instLTReal R (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal))) (forall (m : Nat) (n : Nat), LT.lt.{0} Real Real.instLTReal (Dist.dist.{u1} α (PseudoMetricSpace.toDist.{u1} α _inst_1) (u m) (u n)) R)))
+Case conversion may be inaccurate. Consider using '#align cauchy_seq_bdd cauchySeq_bddₓ'. -/
 /-- A Cauchy sequence on the natural numbers is bounded. -/
 theorem cauchySeq_bdd {u : ℕ → α} (hu : CauchySeq u) : ∃ R > 0, ∀ m n, dist (u m) (u n) < R :=
   by
@@ -1695,6 +2857,12 @@ theorem cauchySeq_bdd {u : ℕ → α} (hu : CauchySeq u) : ∃ R > 0, ∀ m n,
     exact lt_of_le_of_lt this (lt_add_of_pos_right _ zero_lt_one)
 #align cauchy_seq_bdd cauchySeq_bdd
 
+/- warning: cauchy_seq_iff_le_tendsto_0 -> cauchySeq_iff_le_tendsto_0 is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {s : Nat -> α}, Iff (CauchySeq.{u1, 0} α Nat (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1) (CanonicallyLinearOrderedAddMonoid.semilatticeSup.{0} Nat Nat.canonicallyLinearOrderedAddMonoid) s) (Exists.{1} (Nat -> Real) (fun (b : Nat -> Real) => And (forall (n : Nat), LE.le.{0} Real Real.hasLe (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero))) (b n)) (And (forall (n : Nat) (m : Nat) (N : Nat), (LE.le.{0} Nat Nat.hasLe N n) -> (LE.le.{0} Nat Nat.hasLe N m) -> (LE.le.{0} Real Real.hasLe (Dist.dist.{u1} α (PseudoMetricSpace.toHasDist.{u1} α _inst_1) (s n) (s m)) (b N))) (Filter.Tendsto.{0, 0} Nat Real b (Filter.atTop.{0} Nat (PartialOrder.toPreorder.{0} Nat (OrderedCancelAddCommMonoid.toPartialOrder.{0} Nat (StrictOrderedSemiring.toOrderedCancelAddCommMonoid.{0} Nat Nat.strictOrderedSemiring)))) (nhds.{0} Real (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero))))))))
+but is expected to have type
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {s : Nat -> α}, Iff (CauchySeq.{u1, 0} α Nat (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1) (Lattice.toSemilatticeSup.{0} Nat (DistribLattice.toLattice.{0} Nat instDistribLatticeNat)) s) (Exists.{1} (Nat -> Real) (fun (b : Nat -> Real) => And (forall (n : Nat), LE.le.{0} Real Real.instLEReal (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal)) (b n)) (And (forall (n : Nat) (m : Nat) (N : Nat), (LE.le.{0} Nat instLENat N n) -> (LE.le.{0} Nat instLENat N m) -> (LE.le.{0} Real Real.instLEReal (Dist.dist.{u1} α (PseudoMetricSpace.toDist.{u1} α _inst_1) (s n) (s m)) (b N))) (Filter.Tendsto.{0, 0} Nat Real b (Filter.atTop.{0} Nat (PartialOrder.toPreorder.{0} Nat (StrictOrderedSemiring.toPartialOrder.{0} Nat Nat.strictOrderedSemiring))) (nhds.{0} Real (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal)))))))
+Case conversion may be inaccurate. Consider using '#align cauchy_seq_iff_le_tendsto_0 cauchySeq_iff_le_tendsto_0ₓ'. -/
 /-- Yet another metric characterization of Cauchy sequences on integers. This one is often the
 most efficient. -/
 theorem cauchySeq_iff_le_tendsto_0 {s : ℕ → α} :
@@ -1738,6 +2906,7 @@ theorem cauchySeq_iff_le_tendsto_0 {s : ℕ → α} :
 
 end CauchySeq
 
+#print PseudoMetricSpace.induced /-
 /-- Pseudometric space structure pulled back by a function. -/
 def PseudoMetricSpace.induced {α β} (f : α → β) (m : PseudoMetricSpace β) : PseudoMetricSpace α
     where
@@ -1756,7 +2925,9 @@ def PseudoMetricSpace.induced {α β} (f : α → β) (m : PseudoMetricSpace β)
         simp only [compl_mem_comap, Filter.mem_sets, ← is_bounded_def, mem_set_of_eq, compl_compl,
           is_bounded_iff, ball_image_iff]
 #align pseudo_metric_space.induced PseudoMetricSpace.induced
+-/
 
+#print Inducing.comapPseudoMetricSpace /-
 /-- Pull back a pseudometric space structure by an inducing map. This is a version of
 `pseudo_metric_space.induced` useful in case if the domain already has a `topological_space`
 structure. -/
@@ -1764,7 +2935,9 @@ def Inducing.comapPseudoMetricSpace {α β} [TopologicalSpace α] [PseudoMetricS
     (hf : Inducing f) : PseudoMetricSpace α :=
   (PseudoMetricSpace.induced f ‹_›).replaceTopology hf.induced
 #align inducing.comap_pseudo_metric_space Inducing.comapPseudoMetricSpace
+-/
 
+#print UniformInducing.comapPseudoMetricSpace /-
 /-- Pull back a pseudometric space structure by a uniform inducing map. This is a version of
 `pseudo_metric_space.induced` useful in case if the domain already has a `uniform_space`
 structure. -/
@@ -1772,18 +2945,25 @@ def UniformInducing.comapPseudoMetricSpace {α β} [UniformSpace α] [PseudoMetr
     (h : UniformInducing f) : PseudoMetricSpace α :=
   (PseudoMetricSpace.induced f ‹_›).replaceUniformity h.comap_uniformity.symm
 #align uniform_inducing.comap_pseudo_metric_space UniformInducing.comapPseudoMetricSpace
+-/
 
+#print Subtype.pseudoMetricSpace /-
 instance Subtype.pseudoMetricSpace {p : α → Prop} : PseudoMetricSpace (Subtype p) :=
   PseudoMetricSpace.induced coe ‹_›
 #align subtype.pseudo_metric_space Subtype.pseudoMetricSpace
+-/
 
+#print Subtype.dist_eq /-
 theorem Subtype.dist_eq {p : α → Prop} (x y : Subtype p) : dist x y = dist (x : α) y :=
   rfl
 #align subtype.dist_eq Subtype.dist_eq
+-/
 
+#print Subtype.nndist_eq /-
 theorem Subtype.nndist_eq {p : α → Prop} (x y : Subtype p) : nndist x y = nndist (x : α) y :=
   rfl
 #align subtype.nndist_eq Subtype.nndist_eq
+-/
 
 namespace MulOpposite
 
@@ -1791,29 +2971,37 @@ namespace MulOpposite
 instance : PseudoMetricSpace αᵐᵒᵖ :=
   PseudoMetricSpace.induced MulOpposite.unop ‹_›
 
+#print MulOpposite.dist_unop /-
 @[simp, to_additive]
 theorem dist_unop (x y : αᵐᵒᵖ) : dist (unop x) (unop y) = dist x y :=
   rfl
 #align mul_opposite.dist_unop MulOpposite.dist_unop
 #align add_opposite.dist_unop AddOpposite.dist_unop
+-/
 
+#print MulOpposite.dist_op /-
 @[simp, to_additive]
 theorem dist_op (x y : α) : dist (op x) (op y) = dist x y :=
   rfl
 #align mul_opposite.dist_op MulOpposite.dist_op
 #align add_opposite.dist_op AddOpposite.dist_op
+-/
 
+#print MulOpposite.nndist_unop /-
 @[simp, to_additive]
 theorem nndist_unop (x y : αᵐᵒᵖ) : nndist (unop x) (unop y) = nndist x y :=
   rfl
 #align mul_opposite.nndist_unop MulOpposite.nndist_unop
 #align add_opposite.nndist_unop AddOpposite.nndist_unop
+-/
 
+#print MulOpposite.nndist_op /-
 @[simp, to_additive]
 theorem nndist_op (x y : α) : nndist (op x) (op y) = nndist x y :=
   rfl
 #align mul_opposite.nndist_op MulOpposite.nndist_op
 #align add_opposite.nndist_op AddOpposite.nndist_op
+-/
 
 end MulOpposite
 
@@ -1822,10 +3010,22 @@ section NNReal
 instance : PseudoMetricSpace ℝ≥0 :=
   Subtype.pseudoMetricSpace
 
+/- warning: nnreal.dist_eq -> NNReal.dist_eq is a dubious translation:
+lean 3 declaration is
+  forall (a : NNReal) (b : NNReal), Eq.{1} Real (Dist.dist.{0} NNReal (PseudoMetricSpace.toHasDist.{0} NNReal NNReal.pseudoMetricSpace) a b) (Abs.abs.{0} Real (Neg.toHasAbs.{0} Real Real.hasNeg Real.hasSup) (HSub.hSub.{0, 0, 0} Real Real Real (instHSub.{0} Real Real.hasSub) ((fun (a : Type) (b : Type) [self : HasLiftT.{1, 1} a b] => self.0) NNReal Real (HasLiftT.mk.{1, 1} NNReal Real (CoeTCₓ.coe.{1, 1} NNReal Real (coeBase.{1, 1} NNReal Real NNReal.Real.hasCoe))) a) ((fun (a : Type) (b : Type) [self : HasLiftT.{1, 1} a b] => self.0) NNReal Real (HasLiftT.mk.{1, 1} NNReal Real (CoeTCₓ.coe.{1, 1} NNReal Real (coeBase.{1, 1} NNReal Real NNReal.Real.hasCoe))) b)))
+but is expected to have type
+  forall (a : NNReal) (b : NNReal), Eq.{1} Real (Dist.dist.{0} NNReal (PseudoMetricSpace.toDist.{0} NNReal instPseudoMetricSpaceNNReal) a b) (Abs.abs.{0} Real (Neg.toHasAbs.{0} Real Real.instNegReal Real.instSupReal) (HSub.hSub.{0, 0, 0} Real Real Real (instHSub.{0} Real Real.instSubReal) (NNReal.toReal a) (NNReal.toReal b)))
+Case conversion may be inaccurate. Consider using '#align nnreal.dist_eq NNReal.dist_eqₓ'. -/
 theorem NNReal.dist_eq (a b : ℝ≥0) : dist a b = |(a : ℝ) - b| :=
   rfl
 #align nnreal.dist_eq NNReal.dist_eq
 
+/- warning: nnreal.nndist_eq -> NNReal.nndist_eq is a dubious translation:
+lean 3 declaration is
+  forall (a : NNReal) (b : NNReal), Eq.{1} NNReal (NNDist.nndist.{0} NNReal (PseudoMetricSpace.toNNDist.{0} NNReal NNReal.pseudoMetricSpace) a b) (LinearOrder.max.{0} NNReal (ConditionallyCompleteLinearOrder.toLinearOrder.{0} NNReal (ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder.{0} NNReal NNReal.conditionallyCompleteLinearOrderBot)) (HSub.hSub.{0, 0, 0} NNReal NNReal NNReal (instHSub.{0} NNReal NNReal.hasSub) a b) (HSub.hSub.{0, 0, 0} NNReal NNReal NNReal (instHSub.{0} NNReal NNReal.hasSub) b a))
+but is expected to have type
+  forall (a : NNReal) (b : NNReal), Eq.{1} NNReal (NNDist.nndist.{0} NNReal (PseudoMetricSpace.toNNDist.{0} NNReal instPseudoMetricSpaceNNReal) a b) (Max.max.{0} NNReal (CanonicallyLinearOrderedSemifield.toMax.{0} NNReal NNReal.instCanonicallyLinearOrderedSemifieldNNReal) (HSub.hSub.{0, 0, 0} NNReal NNReal NNReal (instHSub.{0} NNReal NNReal.instSubNNReal) a b) (HSub.hSub.{0, 0, 0} NNReal NNReal NNReal (instHSub.{0} NNReal NNReal.instSubNNReal) b a))
+Case conversion may be inaccurate. Consider using '#align nnreal.nndist_eq NNReal.nndist_eqₓ'. -/
 theorem NNReal.nndist_eq (a b : ℝ≥0) : nndist a b = max (a - b) (b - a) :=
   by
   wlog h : b ≤ a
@@ -1835,11 +3035,23 @@ theorem NNReal.nndist_eq (a b : ℝ≥0) : nndist a b = max (a - b) (b - a) :=
     max_eq_left (zero_le <| a - b), ← NNReal.coe_sub h, abs_of_nonneg (a - b).coe_nonneg]
 #align nnreal.nndist_eq NNReal.nndist_eq
 
+/- warning: nnreal.nndist_zero_eq_val -> NNReal.nndist_zero_eq_val is a dubious translation:
+lean 3 declaration is
+  forall (z : NNReal), Eq.{1} NNReal (NNDist.nndist.{0} NNReal (PseudoMetricSpace.toNNDist.{0} NNReal NNReal.pseudoMetricSpace) (OfNat.ofNat.{0} NNReal 0 (OfNat.mk.{0} NNReal 0 (Zero.zero.{0} NNReal (MulZeroClass.toHasZero.{0} NNReal (NonUnitalNonAssocSemiring.toMulZeroClass.{0} NNReal (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} NNReal (Semiring.toNonAssocSemiring.{0} NNReal NNReal.semiring))))))) z) z
+but is expected to have type
+  forall (z : NNReal), Eq.{1} NNReal (NNDist.nndist.{0} NNReal (PseudoMetricSpace.toNNDist.{0} NNReal instPseudoMetricSpaceNNReal) (OfNat.ofNat.{0} NNReal 0 (Zero.toOfNat0.{0} NNReal instNNRealZero)) z) z
+Case conversion may be inaccurate. Consider using '#align nnreal.nndist_zero_eq_val NNReal.nndist_zero_eq_valₓ'. -/
 @[simp]
 theorem NNReal.nndist_zero_eq_val (z : ℝ≥0) : nndist 0 z = z := by
   simp only [NNReal.nndist_eq, max_eq_right, tsub_zero, zero_tsub, zero_le']
 #align nnreal.nndist_zero_eq_val NNReal.nndist_zero_eq_val
 
+/- warning: nnreal.nndist_zero_eq_val' -> NNReal.nndist_zero_eq_val' is a dubious translation:
+lean 3 declaration is
+  forall (z : NNReal), Eq.{1} NNReal (NNDist.nndist.{0} NNReal (PseudoMetricSpace.toNNDist.{0} NNReal NNReal.pseudoMetricSpace) z (OfNat.ofNat.{0} NNReal 0 (OfNat.mk.{0} NNReal 0 (Zero.zero.{0} NNReal (MulZeroClass.toHasZero.{0} NNReal (NonUnitalNonAssocSemiring.toMulZeroClass.{0} NNReal (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} NNReal (Semiring.toNonAssocSemiring.{0} NNReal NNReal.semiring)))))))) z
+but is expected to have type
+  forall (z : NNReal), Eq.{1} NNReal (NNDist.nndist.{0} NNReal (PseudoMetricSpace.toNNDist.{0} NNReal instPseudoMetricSpaceNNReal) z (OfNat.ofNat.{0} NNReal 0 (Zero.toOfNat0.{0} NNReal instNNRealZero))) z
+Case conversion may be inaccurate. Consider using '#align nnreal.nndist_zero_eq_val' NNReal.nndist_zero_eq_val'ₓ'. -/
 @[simp]
 theorem NNReal.nndist_zero_eq_val' (z : ℝ≥0) : nndist z 0 = z :=
   by
@@ -1847,6 +3059,12 @@ theorem NNReal.nndist_zero_eq_val' (z : ℝ≥0) : nndist z 0 = z :=
   exact NNReal.nndist_zero_eq_val z
 #align nnreal.nndist_zero_eq_val' NNReal.nndist_zero_eq_val'
 
+/- warning: nnreal.le_add_nndist -> NNReal.le_add_nndist is a dubious translation:
+lean 3 declaration is
+  forall (a : NNReal) (b : NNReal), LE.le.{0} NNReal (Preorder.toLE.{0} NNReal (PartialOrder.toPreorder.{0} NNReal (OrderedCancelAddCommMonoid.toPartialOrder.{0} NNReal (StrictOrderedSemiring.toOrderedCancelAddCommMonoid.{0} NNReal NNReal.strictOrderedSemiring)))) a (HAdd.hAdd.{0, 0, 0} NNReal NNReal NNReal (instHAdd.{0} NNReal (Distrib.toHasAdd.{0} NNReal (NonUnitalNonAssocSemiring.toDistrib.{0} NNReal (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} NNReal (Semiring.toNonAssocSemiring.{0} NNReal NNReal.semiring))))) b (NNDist.nndist.{0} NNReal (PseudoMetricSpace.toNNDist.{0} NNReal NNReal.pseudoMetricSpace) a b))
+but is expected to have type
+  forall (a : NNReal) (b : NNReal), LE.le.{0} NNReal (Preorder.toLE.{0} NNReal (PartialOrder.toPreorder.{0} NNReal (StrictOrderedSemiring.toPartialOrder.{0} NNReal instNNRealStrictOrderedSemiring))) a (HAdd.hAdd.{0, 0, 0} NNReal NNReal NNReal (instHAdd.{0} NNReal (Distrib.toAdd.{0} NNReal (NonUnitalNonAssocSemiring.toDistrib.{0} NNReal (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} NNReal (Semiring.toNonAssocSemiring.{0} NNReal instNNRealSemiring))))) b (NNDist.nndist.{0} NNReal (PseudoMetricSpace.toNNDist.{0} NNReal instPseudoMetricSpaceNNReal) a b))
+Case conversion may be inaccurate. Consider using '#align nnreal.le_add_nndist NNReal.le_add_nndistₓ'. -/
 theorem NNReal.le_add_nndist (a b : ℝ≥0) : a ≤ b + nndist a b :=
   by
   suffices (a : ℝ) ≤ (b : ℝ) + dist a b by exact nnreal.coe_le_coe.mp this
@@ -1862,19 +3080,43 @@ variable [PseudoMetricSpace β]
 instance : PseudoMetricSpace (ULift β) :=
   PseudoMetricSpace.induced ULift.down ‹_›
 
+/- warning: ulift.dist_eq -> ULift.dist_eq is a dubious translation:
+lean 3 declaration is
+  forall {β : Type.{u1}} [_inst_2 : PseudoMetricSpace.{u1} β] (x : ULift.{u2, u1} β) (y : ULift.{u2, u1} β), Eq.{1} Real (Dist.dist.{max u1 u2} (ULift.{u2, u1} β) (PseudoMetricSpace.toHasDist.{max u1 u2} (ULift.{u2, u1} β) (ULift.pseudoMetricSpace.{u1, u2} β _inst_2)) x y) (Dist.dist.{u1} β (PseudoMetricSpace.toHasDist.{u1} β _inst_2) (ULift.down.{u2, u1} β x) (ULift.down.{u2, u1} β y))
+but is expected to have type
+  forall {β : Type.{u2}} [_inst_2 : PseudoMetricSpace.{u2} β] (x : ULift.{u1, u2} β) (y : ULift.{u1, u2} β), Eq.{1} Real (Dist.dist.{max u2 u1} (ULift.{u1, u2} β) (PseudoMetricSpace.toDist.{max u2 u1} (ULift.{u1, u2} β) (instPseudoMetricSpaceULift.{u2, u1} β _inst_2)) x y) (Dist.dist.{u2} β (PseudoMetricSpace.toDist.{u2} β _inst_2) (ULift.down.{u1, u2} β x) (ULift.down.{u1, u2} β y))
+Case conversion may be inaccurate. Consider using '#align ulift.dist_eq ULift.dist_eqₓ'. -/
 theorem ULift.dist_eq (x y : ULift β) : dist x y = dist x.down y.down :=
   rfl
 #align ulift.dist_eq ULift.dist_eq
 
+/- warning: ulift.nndist_eq -> ULift.nndist_eq is a dubious translation:
+lean 3 declaration is
+  forall {β : Type.{u1}} [_inst_2 : PseudoMetricSpace.{u1} β] (x : ULift.{u2, u1} β) (y : ULift.{u2, u1} β), Eq.{1} NNReal (NNDist.nndist.{max u1 u2} (ULift.{u2, u1} β) (PseudoMetricSpace.toNNDist.{max u1 u2} (ULift.{u2, u1} β) (ULift.pseudoMetricSpace.{u1, u2} β _inst_2)) x y) (NNDist.nndist.{u1} β (PseudoMetricSpace.toNNDist.{u1} β _inst_2) (ULift.down.{u2, u1} β x) (ULift.down.{u2, u1} β y))
+but is expected to have type
+  forall {β : Type.{u2}} [_inst_2 : PseudoMetricSpace.{u2} β] (x : ULift.{u1, u2} β) (y : ULift.{u1, u2} β), Eq.{1} NNReal (NNDist.nndist.{max u2 u1} (ULift.{u1, u2} β) (PseudoMetricSpace.toNNDist.{max u2 u1} (ULift.{u1, u2} β) (instPseudoMetricSpaceULift.{u2, u1} β _inst_2)) x y) (NNDist.nndist.{u2} β (PseudoMetricSpace.toNNDist.{u2} β _inst_2) (ULift.down.{u1, u2} β x) (ULift.down.{u1, u2} β y))
+Case conversion may be inaccurate. Consider using '#align ulift.nndist_eq ULift.nndist_eqₓ'. -/
 theorem ULift.nndist_eq (x y : ULift β) : nndist x y = nndist x.down y.down :=
   rfl
 #align ulift.nndist_eq ULift.nndist_eq
 
+/- warning: ulift.dist_up_up -> ULift.dist_up_up is a dubious translation:
+lean 3 declaration is
+  forall {β : Type.{u1}} [_inst_2 : PseudoMetricSpace.{u1} β] (x : β) (y : β), Eq.{1} Real (Dist.dist.{max u1 u2} (ULift.{u2, u1} β) (PseudoMetricSpace.toHasDist.{max u1 u2} (ULift.{u2, u1} β) (ULift.pseudoMetricSpace.{u1, u2} β _inst_2)) (ULift.up.{u2, u1} β x) (ULift.up.{u2, u1} β y)) (Dist.dist.{u1} β (PseudoMetricSpace.toHasDist.{u1} β _inst_2) x y)
+but is expected to have type
+  forall {β : Type.{u2}} [_inst_2 : PseudoMetricSpace.{u2} β] (x : β) (y : β), Eq.{1} Real (Dist.dist.{max u2 u1} (ULift.{u1, u2} β) (PseudoMetricSpace.toDist.{max u2 u1} (ULift.{u1, u2} β) (instPseudoMetricSpaceULift.{u2, u1} β _inst_2)) (ULift.up.{u1, u2} β x) (ULift.up.{u1, u2} β y)) (Dist.dist.{u2} β (PseudoMetricSpace.toDist.{u2} β _inst_2) x y)
+Case conversion may be inaccurate. Consider using '#align ulift.dist_up_up ULift.dist_up_upₓ'. -/
 @[simp]
 theorem ULift.dist_up_up (x y : β) : dist (ULift.up x) (ULift.up y) = dist x y :=
   rfl
 #align ulift.dist_up_up ULift.dist_up_up
 
+/- warning: ulift.nndist_up_up -> ULift.nndist_up_up is a dubious translation:
+lean 3 declaration is
+  forall {β : Type.{u1}} [_inst_2 : PseudoMetricSpace.{u1} β] (x : β) (y : β), Eq.{1} NNReal (NNDist.nndist.{max u1 u2} (ULift.{u2, u1} β) (PseudoMetricSpace.toNNDist.{max u1 u2} (ULift.{u2, u1} β) (ULift.pseudoMetricSpace.{u1, u2} β _inst_2)) (ULift.up.{u2, u1} β x) (ULift.up.{u2, u1} β y)) (NNDist.nndist.{u1} β (PseudoMetricSpace.toNNDist.{u1} β _inst_2) x y)
+but is expected to have type
+  forall {β : Type.{u2}} [_inst_2 : PseudoMetricSpace.{u2} β] (x : β) (y : β), Eq.{1} NNReal (NNDist.nndist.{max u2 u1} (ULift.{u1, u2} β) (PseudoMetricSpace.toNNDist.{max u2 u1} (ULift.{u1, u2} β) (instPseudoMetricSpaceULift.{u2, u1} β _inst_2)) (ULift.up.{u1, u2} β x) (ULift.up.{u1, u2} β y)) (NNDist.nndist.{u2} β (PseudoMetricSpace.toNNDist.{u2} β _inst_2) x y)
+Case conversion may be inaccurate. Consider using '#align ulift.nndist_up_up ULift.nndist_up_upₓ'. -/
 @[simp]
 theorem ULift.nndist_up_up (x y : β) : nndist (ULift.up x) (ULift.up y) = nndist x y :=
   rfl
@@ -1886,6 +3128,7 @@ section Prod
 
 variable [PseudoMetricSpace β]
 
+#print Prod.pseudoMetricSpaceMax /-
 instance Prod.pseudoMetricSpaceMax : PseudoMetricSpace (α × β) :=
   (PseudoEMetricSpace.toPseudoMetricSpaceOfDist (fun x y : α × β => dist x.1 y.1 ⊔ dist x.2 y.2)
         (fun x y => (max_lt (edist_lt_top _ _) (edist_lt_top _ _)).Ne) fun x y => by
@@ -1897,34 +3140,55 @@ instance Prod.pseudoMetricSpaceMax : PseudoMetricSpace (α × β) :=
       eventually_and, ← forall_and, ← max_le_iff]
     rfl
 #align prod.pseudo_metric_space_max Prod.pseudoMetricSpaceMax
+-/
 
+/- warning: prod.dist_eq -> Prod.dist_eq is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : PseudoMetricSpace.{u1} α] [_inst_2 : PseudoMetricSpace.{u2} β] {x : Prod.{u1, u2} α β} {y : Prod.{u1, u2} α β}, Eq.{1} Real (Dist.dist.{max u1 u2} (Prod.{u1, u2} α β) (PseudoMetricSpace.toHasDist.{max u1 u2} (Prod.{u1, u2} α β) (Prod.pseudoMetricSpaceMax.{u1, u2} α β _inst_1 _inst_2)) x y) (LinearOrder.max.{0} Real Real.linearOrder (Dist.dist.{u1} α (PseudoMetricSpace.toHasDist.{u1} α _inst_1) (Prod.fst.{u1, u2} α β x) (Prod.fst.{u1, u2} α β y)) (Dist.dist.{u2} β (PseudoMetricSpace.toHasDist.{u2} β _inst_2) (Prod.snd.{u1, u2} α β x) (Prod.snd.{u1, u2} α β y)))
+but is expected to have type
+  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : PseudoMetricSpace.{u1} α] [_inst_2 : PseudoMetricSpace.{u2} β] {x : Prod.{u1, u2} α β} {y : Prod.{u1, u2} α β}, Eq.{1} Real (Dist.dist.{max u1 u2} (Prod.{u1, u2} α β) (PseudoMetricSpace.toDist.{max u1 u2} (Prod.{u1, u2} α β) (Prod.pseudoMetricSpaceMax.{u1, u2} α β _inst_1 _inst_2)) x y) (Max.max.{0} Real (LinearOrderedRing.toMax.{0} Real Real.instLinearOrderedRingReal) (Dist.dist.{u1} α (PseudoMetricSpace.toDist.{u1} α _inst_1) (Prod.fst.{u1, u2} α β x) (Prod.fst.{u1, u2} α β y)) (Dist.dist.{u2} β (PseudoMetricSpace.toDist.{u2} β _inst_2) (Prod.snd.{u1, u2} α β x) (Prod.snd.{u1, u2} α β y)))
+Case conversion may be inaccurate. Consider using '#align prod.dist_eq Prod.dist_eqₓ'. -/
 theorem Prod.dist_eq {x y : α × β} : dist x y = max (dist x.1 y.1) (dist x.2 y.2) :=
   rfl
 #align prod.dist_eq Prod.dist_eq
 
+#print dist_prod_same_left /-
 @[simp]
 theorem dist_prod_same_left {x : α} {y₁ y₂ : β} : dist (x, y₁) (x, y₂) = dist y₁ y₂ := by
   simp [Prod.dist_eq, dist_nonneg]
 #align dist_prod_same_left dist_prod_same_left
+-/
 
+#print dist_prod_same_right /-
 @[simp]
 theorem dist_prod_same_right {x₁ x₂ : α} {y : β} : dist (x₁, y) (x₂, y) = dist x₁ x₂ := by
   simp [Prod.dist_eq, dist_nonneg]
 #align dist_prod_same_right dist_prod_same_right
+-/
 
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
+#print ball_prod_same /-
 theorem ball_prod_same (x : α) (y : β) (r : ℝ) : ball x r ×ˢ ball y r = ball (x, y) r :=
   ext fun z => by simp [Prod.dist_eq]
 #align ball_prod_same ball_prod_same
+-/
 
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
+#print closedBall_prod_same /-
 theorem closedBall_prod_same (x : α) (y : β) (r : ℝ) :
     closedBall x r ×ˢ closedBall y r = closedBall (x, y) r :=
   ext fun z => by simp [Prod.dist_eq]
 #align closed_ball_prod_same closedBall_prod_same
+-/
 
 end Prod
 
+/- warning: uniform_continuous_dist -> uniformContinuous_dist is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α], UniformContinuous.{u1, 0} (Prod.{u1, u1} α α) Real (Prod.uniformSpace.{u1, u1} α α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1) (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace) (fun (p : Prod.{u1, u1} α α) => Dist.dist.{u1} α (PseudoMetricSpace.toHasDist.{u1} α _inst_1) (Prod.fst.{u1, u1} α α p) (Prod.snd.{u1, u1} α α p))
+but is expected to have type
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α], UniformContinuous.{u1, 0} (Prod.{u1, u1} α α) Real (instUniformSpaceProd.{u1, u1} α α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1) (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace) (fun (p : Prod.{u1, u1} α α) => Dist.dist.{u1} α (PseudoMetricSpace.toDist.{u1} α _inst_1) (Prod.fst.{u1, u1} α α p) (Prod.snd.{u1, u1} α α p))
+Case conversion may be inaccurate. Consider using '#align uniform_continuous_dist uniformContinuous_distₓ'. -/
 theorem uniformContinuous_dist : UniformContinuous fun p : α × α => dist p.1 p.2 :=
   Metric.uniformContinuous_iff.2 fun ε ε0 =>
     ⟨ε / 2, half_pos ε0, by
@@ -1947,37 +3211,67 @@ theorem uniformContinuous_dist : UniformContinuous fun p : α × α => dist p.1
       rwa [add_halves, dist_comm p₂, sub_add_sub_cancel, dist_comm q₂] at this⟩
 #align uniform_continuous_dist uniformContinuous_dist
 
+#print UniformContinuous.dist /-
 theorem UniformContinuous.dist [UniformSpace β] {f g : β → α} (hf : UniformContinuous f)
     (hg : UniformContinuous g) : UniformContinuous fun b => dist (f b) (g b) :=
   uniformContinuous_dist.comp (hf.prod_mk hg)
 #align uniform_continuous.dist UniformContinuous.dist
+-/
 
+/- warning: continuous_dist -> continuous_dist is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α], Continuous.{u1, 0} (Prod.{u1, u1} α α) Real (Prod.topologicalSpace.{u1, u1} α α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1))) (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) (fun (p : Prod.{u1, u1} α α) => Dist.dist.{u1} α (PseudoMetricSpace.toHasDist.{u1} α _inst_1) (Prod.fst.{u1, u1} α α p) (Prod.snd.{u1, u1} α α p))
+but is expected to have type
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α], Continuous.{u1, 0} (Prod.{u1, u1} α α) Real (instTopologicalSpaceProd.{u1, u1} α α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1))) (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) (fun (p : Prod.{u1, u1} α α) => Dist.dist.{u1} α (PseudoMetricSpace.toDist.{u1} α _inst_1) (Prod.fst.{u1, u1} α α p) (Prod.snd.{u1, u1} α α p))
+Case conversion may be inaccurate. Consider using '#align continuous_dist continuous_distₓ'. -/
 @[continuity]
 theorem continuous_dist : Continuous fun p : α × α => dist p.1 p.2 :=
   uniformContinuous_dist.Continuous
 #align continuous_dist continuous_dist
 
+#print Continuous.dist /-
 @[continuity]
 theorem Continuous.dist [TopologicalSpace β] {f g : β → α} (hf : Continuous f) (hg : Continuous g) :
     Continuous fun b => dist (f b) (g b) :=
   continuous_dist.comp (hf.prod_mk hg : _)
 #align continuous.dist Continuous.dist
+-/
 
+#print Filter.Tendsto.dist /-
 theorem Filter.Tendsto.dist {f g : β → α} {x : Filter β} {a b : α} (hf : Tendsto f x (𝓝 a))
     (hg : Tendsto g x (𝓝 b)) : Tendsto (fun x => dist (f x) (g x)) x (𝓝 (dist a b)) :=
   (continuous_dist.Tendsto (a, b)).comp (hf.prod_mk_nhds hg)
 #align filter.tendsto.dist Filter.Tendsto.dist
+-/
 
+/- warning: nhds_comap_dist -> nhds_comap_dist is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] (a : α), Eq.{succ u1} (Filter.{u1} α) (Filter.comap.{u1, 0} α Real (fun (a' : α) => Dist.dist.{u1} α (PseudoMetricSpace.toHasDist.{u1} α _inst_1) a' a) (nhds.{0} Real (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero))))) (nhds.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) a)
+but is expected to have type
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] (a : α), Eq.{succ u1} (Filter.{u1} α) (Filter.comap.{u1, 0} α Real (fun (a' : α) => Dist.dist.{u1} α (PseudoMetricSpace.toDist.{u1} α _inst_1) a' a) (nhds.{0} Real (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal)))) (nhds.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) a)
+Case conversion may be inaccurate. Consider using '#align nhds_comap_dist nhds_comap_distₓ'. -/
 theorem nhds_comap_dist (a : α) : ((𝓝 (0 : ℝ)).comap fun a' => dist a' a) = 𝓝 a := by
   simp only [@nhds_eq_comap_uniformity α, Metric.uniformity_eq_comap_nhds_zero, comap_comap,
     (· ∘ ·), dist_comm]
 #align nhds_comap_dist nhds_comap_dist
 
+/- warning: tendsto_iff_dist_tendsto_zero -> tendsto_iff_dist_tendsto_zero is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : PseudoMetricSpace.{u1} α] {f : β -> α} {x : Filter.{u2} β} {a : α}, Iff (Filter.Tendsto.{u2, u1} β α f x (nhds.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) a)) (Filter.Tendsto.{u2, 0} β Real (fun (b : β) => Dist.dist.{u1} α (PseudoMetricSpace.toHasDist.{u1} α _inst_1) (f b) a) x (nhds.{0} Real (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero)))))
+but is expected to have type
+  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : PseudoMetricSpace.{u1} α] {f : β -> α} {x : Filter.{u2} β} {a : α}, Iff (Filter.Tendsto.{u2, u1} β α f x (nhds.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) a)) (Filter.Tendsto.{u2, 0} β Real (fun (b : β) => Dist.dist.{u1} α (PseudoMetricSpace.toDist.{u1} α _inst_1) (f b) a) x (nhds.{0} Real (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal))))
+Case conversion may be inaccurate. Consider using '#align tendsto_iff_dist_tendsto_zero tendsto_iff_dist_tendsto_zeroₓ'. -/
 theorem tendsto_iff_dist_tendsto_zero {f : β → α} {x : Filter β} {a : α} :
     Tendsto f x (𝓝 a) ↔ Tendsto (fun b => dist (f b) a) x (𝓝 0) := by
   rw [← nhds_comap_dist a, tendsto_comap_iff]
 #align tendsto_iff_dist_tendsto_zero tendsto_iff_dist_tendsto_zero
 
+/- warning: continuous_iff_continuous_dist -> continuous_iff_continuous_dist is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : PseudoMetricSpace.{u1} α] [_inst_2 : TopologicalSpace.{u2} β] {f : β -> α}, Iff (Continuous.{u2, u1} β α _inst_2 (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) f) (Continuous.{u2, 0} (Prod.{u2, u2} β β) Real (Prod.topologicalSpace.{u2, u2} β β _inst_2 _inst_2) (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) (fun (x : Prod.{u2, u2} β β) => Dist.dist.{u1} α (PseudoMetricSpace.toHasDist.{u1} α _inst_1) (f (Prod.fst.{u2, u2} β β x)) (f (Prod.snd.{u2, u2} β β x))))
+but is expected to have type
+  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : PseudoMetricSpace.{u1} α] [_inst_2 : TopologicalSpace.{u2} β] {f : β -> α}, Iff (Continuous.{u2, u1} β α _inst_2 (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) f) (Continuous.{u2, 0} (Prod.{u2, u2} β β) Real (instTopologicalSpaceProd.{u2, u2} β β _inst_2 _inst_2) (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) (fun (x : Prod.{u2, u2} β β) => Dist.dist.{u1} α (PseudoMetricSpace.toDist.{u1} α _inst_1) (f (Prod.fst.{u2, u2} β β x)) (f (Prod.snd.{u2, u2} β β x))))
+Case conversion may be inaccurate. Consider using '#align continuous_iff_continuous_dist continuous_iff_continuous_distₓ'. -/
 theorem continuous_iff_continuous_dist [TopologicalSpace β] {f : β → α} :
     Continuous f ↔ Continuous fun x : β × β => dist (f x.1) (f x.2) :=
   ⟨fun h => (h.comp continuous_fst).dist (h.comp continuous_snd), fun h =>
@@ -1986,24 +3280,54 @@ theorem continuous_iff_continuous_dist [TopologicalSpace β] {f : β → α} :
         (h.comp (continuous_id.prod_mk continuous_const)).tendsto' _ _ <| dist_self _⟩
 #align continuous_iff_continuous_dist continuous_iff_continuous_dist
 
+/- warning: uniform_continuous_nndist -> uniformContinuous_nndist is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α], UniformContinuous.{u1, 0} (Prod.{u1, u1} α α) NNReal (Prod.uniformSpace.{u1, u1} α α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1) (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) (PseudoMetricSpace.toUniformSpace.{0} NNReal NNReal.pseudoMetricSpace) (fun (p : Prod.{u1, u1} α α) => NNDist.nndist.{u1} α (PseudoMetricSpace.toNNDist.{u1} α _inst_1) (Prod.fst.{u1, u1} α α p) (Prod.snd.{u1, u1} α α p))
+but is expected to have type
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α], UniformContinuous.{u1, 0} (Prod.{u1, u1} α α) NNReal (instUniformSpaceProd.{u1, u1} α α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1) (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) (PseudoMetricSpace.toUniformSpace.{0} NNReal instPseudoMetricSpaceNNReal) (fun (p : Prod.{u1, u1} α α) => NNDist.nndist.{u1} α (PseudoMetricSpace.toNNDist.{u1} α _inst_1) (Prod.fst.{u1, u1} α α p) (Prod.snd.{u1, u1} α α p))
+Case conversion may be inaccurate. Consider using '#align uniform_continuous_nndist uniformContinuous_nndistₓ'. -/
 theorem uniformContinuous_nndist : UniformContinuous fun p : α × α => nndist p.1 p.2 :=
   uniformContinuous_dist.subtype_mk _
 #align uniform_continuous_nndist uniformContinuous_nndist
 
+/- warning: uniform_continuous.nndist -> UniformContinuous.nndist is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : PseudoMetricSpace.{u1} α] [_inst_2 : UniformSpace.{u2} β] {f : β -> α} {g : β -> α}, (UniformContinuous.{u2, u1} β α _inst_2 (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1) f) -> (UniformContinuous.{u2, u1} β α _inst_2 (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1) g) -> (UniformContinuous.{u2, 0} β NNReal _inst_2 (PseudoMetricSpace.toUniformSpace.{0} NNReal NNReal.pseudoMetricSpace) (fun (b : β) => NNDist.nndist.{u1} α (PseudoMetricSpace.toNNDist.{u1} α _inst_1) (f b) (g b)))
+but is expected to have type
+  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : PseudoMetricSpace.{u1} α] [_inst_2 : UniformSpace.{u2} β] {f : β -> α} {g : β -> α}, (UniformContinuous.{u2, u1} β α _inst_2 (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1) f) -> (UniformContinuous.{u2, u1} β α _inst_2 (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1) g) -> (UniformContinuous.{u2, 0} β NNReal _inst_2 (PseudoMetricSpace.toUniformSpace.{0} NNReal instPseudoMetricSpaceNNReal) (fun (b : β) => NNDist.nndist.{u1} α (PseudoMetricSpace.toNNDist.{u1} α _inst_1) (f b) (g b)))
+Case conversion may be inaccurate. Consider using '#align uniform_continuous.nndist UniformContinuous.nndistₓ'. -/
 theorem UniformContinuous.nndist [UniformSpace β] {f g : β → α} (hf : UniformContinuous f)
     (hg : UniformContinuous g) : UniformContinuous fun b => nndist (f b) (g b) :=
   uniformContinuous_nndist.comp (hf.prod_mk hg)
 #align uniform_continuous.nndist UniformContinuous.nndist
 
+/- warning: continuous_nndist -> continuous_nndist is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α], Continuous.{u1, 0} (Prod.{u1, u1} α α) NNReal (Prod.topologicalSpace.{u1, u1} α α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1))) (UniformSpace.toTopologicalSpace.{0} NNReal (PseudoMetricSpace.toUniformSpace.{0} NNReal NNReal.pseudoMetricSpace)) (fun (p : Prod.{u1, u1} α α) => NNDist.nndist.{u1} α (PseudoMetricSpace.toNNDist.{u1} α _inst_1) (Prod.fst.{u1, u1} α α p) (Prod.snd.{u1, u1} α α p))
+but is expected to have type
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α], Continuous.{u1, 0} (Prod.{u1, u1} α α) NNReal (instTopologicalSpaceProd.{u1, u1} α α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1))) (UniformSpace.toTopologicalSpace.{0} NNReal (PseudoMetricSpace.toUniformSpace.{0} NNReal instPseudoMetricSpaceNNReal)) (fun (p : Prod.{u1, u1} α α) => NNDist.nndist.{u1} α (PseudoMetricSpace.toNNDist.{u1} α _inst_1) (Prod.fst.{u1, u1} α α p) (Prod.snd.{u1, u1} α α p))
+Case conversion may be inaccurate. Consider using '#align continuous_nndist continuous_nndistₓ'. -/
 theorem continuous_nndist : Continuous fun p : α × α => nndist p.1 p.2 :=
   uniformContinuous_nndist.Continuous
 #align continuous_nndist continuous_nndist
 
+/- warning: continuous.nndist -> Continuous.nndist is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : PseudoMetricSpace.{u1} α] [_inst_2 : TopologicalSpace.{u2} β] {f : β -> α} {g : β -> α}, (Continuous.{u2, u1} β α _inst_2 (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) f) -> (Continuous.{u2, u1} β α _inst_2 (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) g) -> (Continuous.{u2, 0} β NNReal _inst_2 (UniformSpace.toTopologicalSpace.{0} NNReal (PseudoMetricSpace.toUniformSpace.{0} NNReal NNReal.pseudoMetricSpace)) (fun (b : β) => NNDist.nndist.{u1} α (PseudoMetricSpace.toNNDist.{u1} α _inst_1) (f b) (g b)))
+but is expected to have type
+  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : PseudoMetricSpace.{u1} α] [_inst_2 : TopologicalSpace.{u2} β] {f : β -> α} {g : β -> α}, (Continuous.{u2, u1} β α _inst_2 (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) f) -> (Continuous.{u2, u1} β α _inst_2 (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) g) -> (Continuous.{u2, 0} β NNReal _inst_2 (UniformSpace.toTopologicalSpace.{0} NNReal (PseudoMetricSpace.toUniformSpace.{0} NNReal instPseudoMetricSpaceNNReal)) (fun (b : β) => NNDist.nndist.{u1} α (PseudoMetricSpace.toNNDist.{u1} α _inst_1) (f b) (g b)))
+Case conversion may be inaccurate. Consider using '#align continuous.nndist Continuous.nndistₓ'. -/
 theorem Continuous.nndist [TopologicalSpace β] {f g : β → α} (hf : Continuous f)
     (hg : Continuous g) : Continuous fun b => nndist (f b) (g b) :=
   continuous_nndist.comp (hf.prod_mk hg : _)
 #align continuous.nndist Continuous.nndist
 
+/- warning: filter.tendsto.nndist -> Filter.Tendsto.nndist is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : PseudoMetricSpace.{u1} α] {f : β -> α} {g : β -> α} {x : Filter.{u2} β} {a : α} {b : α}, (Filter.Tendsto.{u2, u1} β α f x (nhds.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) a)) -> (Filter.Tendsto.{u2, u1} β α g x (nhds.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) b)) -> (Filter.Tendsto.{u2, 0} β NNReal (fun (x : β) => NNDist.nndist.{u1} α (PseudoMetricSpace.toNNDist.{u1} α _inst_1) (f x) (g x)) x (nhds.{0} NNReal (UniformSpace.toTopologicalSpace.{0} NNReal (PseudoMetricSpace.toUniformSpace.{0} NNReal NNReal.pseudoMetricSpace)) (NNDist.nndist.{u1} α (PseudoMetricSpace.toNNDist.{u1} α _inst_1) a b)))
+but is expected to have type
+  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : PseudoMetricSpace.{u1} α] {f : β -> α} {g : β -> α} {x : Filter.{u2} β} {a : α} {b : α}, (Filter.Tendsto.{u2, u1} β α f x (nhds.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) a)) -> (Filter.Tendsto.{u2, u1} β α g x (nhds.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) b)) -> (Filter.Tendsto.{u2, 0} β NNReal (fun (x : β) => NNDist.nndist.{u1} α (PseudoMetricSpace.toNNDist.{u1} α _inst_1) (f x) (g x)) x (nhds.{0} NNReal (UniformSpace.toTopologicalSpace.{0} NNReal (PseudoMetricSpace.toUniformSpace.{0} NNReal instPseudoMetricSpaceNNReal)) (NNDist.nndist.{u1} α (PseudoMetricSpace.toNNDist.{u1} α _inst_1) a b)))
+Case conversion may be inaccurate. Consider using '#align filter.tendsto.nndist Filter.Tendsto.nndistₓ'. -/
 theorem Filter.Tendsto.nndist {f g : β → α} {x : Filter β} {a b : α} (hf : Tendsto f x (𝓝 a))
     (hg : Tendsto g x (𝓝 b)) : Tendsto (fun x => nndist (f x) (g x)) x (𝓝 (nndist a b)) :=
   (continuous_nndist.Tendsto (a, b)).comp (hf.prod_mk_nhds hg)
@@ -2013,56 +3337,100 @@ namespace Metric
 
 variable {x y z : α} {ε ε₁ ε₂ : ℝ} {s : Set α}
 
+#print Metric.isClosed_ball /-
 theorem isClosed_ball : IsClosed (closedBall x ε) :=
   isClosed_le (continuous_id.dist continuous_const) continuous_const
 #align metric.is_closed_ball Metric.isClosed_ball
+-/
 
+#print Metric.isClosed_sphere /-
 theorem isClosed_sphere : IsClosed (sphere x ε) :=
   isClosed_eq (continuous_id.dist continuous_const) continuous_const
 #align metric.is_closed_sphere Metric.isClosed_sphere
+-/
 
+#print Metric.closure_closedBall /-
 @[simp]
 theorem closure_closedBall : closure (closedBall x ε) = closedBall x ε :=
   isClosed_ball.closure_eq
 #align metric.closure_closed_ball Metric.closure_closedBall
+-/
 
+#print Metric.closure_ball_subset_closedBall /-
 theorem closure_ball_subset_closedBall : closure (ball x ε) ⊆ closedBall x ε :=
   closure_minimal ball_subset_closedBall isClosed_ball
 #align metric.closure_ball_subset_closed_ball Metric.closure_ball_subset_closedBall
+-/
 
+#print Metric.frontier_ball_subset_sphere /-
 theorem frontier_ball_subset_sphere : frontier (ball x ε) ⊆ sphere x ε :=
   frontier_lt_subset_eq (continuous_id.dist continuous_const) continuous_const
 #align metric.frontier_ball_subset_sphere Metric.frontier_ball_subset_sphere
+-/
 
+#print Metric.frontier_closedBall_subset_sphere /-
 theorem frontier_closedBall_subset_sphere : frontier (closedBall x ε) ⊆ sphere x ε :=
   frontier_le_subset_eq (continuous_id.dist continuous_const) continuous_const
 #align metric.frontier_closed_ball_subset_sphere Metric.frontier_closedBall_subset_sphere
+-/
 
+#print Metric.ball_subset_interior_closedBall /-
 theorem ball_subset_interior_closedBall : ball x ε ⊆ interior (closedBall x ε) :=
   interior_maximal ball_subset_closedBall isOpen_ball
 #align metric.ball_subset_interior_closed_ball Metric.ball_subset_interior_closedBall
+-/
 
+/- warning: metric.mem_closure_iff -> Metric.mem_closure_iff is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {s : Set.{u1} α} {a : α}, Iff (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) a (closure.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) s)) (forall (ε : Real), (GT.gt.{0} Real Real.hasLt ε (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero)))) -> (Exists.{succ u1} α (fun (b : α) => Exists.{0} (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) b s) (fun (H : Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) b s) => LT.lt.{0} Real Real.hasLt (Dist.dist.{u1} α (PseudoMetricSpace.toHasDist.{u1} α _inst_1) a b) ε))))
+but is expected to have type
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {s : Set.{u1} α} {a : α}, Iff (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) a (closure.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) s)) (forall (ε : Real), (GT.gt.{0} Real Real.instLTReal ε (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal))) -> (Exists.{succ u1} α (fun (b : α) => And (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) b s) (LT.lt.{0} Real Real.instLTReal (Dist.dist.{u1} α (PseudoMetricSpace.toDist.{u1} α _inst_1) a b) ε))))
+Case conversion may be inaccurate. Consider using '#align metric.mem_closure_iff Metric.mem_closure_iffₓ'. -/
 /-- ε-characterization of the closure in pseudometric spaces-/
 theorem mem_closure_iff {s : Set α} {a : α} : a ∈ closure s ↔ ∀ ε > 0, ∃ b ∈ s, dist a b < ε :=
   (mem_closure_iff_nhds_basis nhds_basis_ball).trans <| by simp only [mem_ball, dist_comm]
 #align metric.mem_closure_iff Metric.mem_closure_iff
 
+/- warning: metric.mem_closure_range_iff -> Metric.mem_closure_range_iff is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : PseudoMetricSpace.{u1} α] {e : β -> α} {a : α}, Iff (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) a (closure.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) (Set.range.{u1, succ u2} α β e))) (forall (ε : Real), (GT.gt.{0} Real Real.hasLt ε (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero)))) -> (Exists.{succ u2} β (fun (k : β) => LT.lt.{0} Real Real.hasLt (Dist.dist.{u1} α (PseudoMetricSpace.toHasDist.{u1} α _inst_1) a (e k)) ε)))
+but is expected to have type
+  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : PseudoMetricSpace.{u1} α] {e : β -> α} {a : α}, Iff (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) a (closure.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) (Set.range.{u1, succ u2} α β e))) (forall (ε : Real), (GT.gt.{0} Real Real.instLTReal ε (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal))) -> (Exists.{succ u2} β (fun (k : β) => LT.lt.{0} Real Real.instLTReal (Dist.dist.{u1} α (PseudoMetricSpace.toDist.{u1} α _inst_1) a (e k)) ε)))
+Case conversion may be inaccurate. Consider using '#align metric.mem_closure_range_iff Metric.mem_closure_range_iffₓ'. -/
 theorem mem_closure_range_iff {e : β → α} {a : α} :
     a ∈ closure (range e) ↔ ∀ ε > 0, ∃ k : β, dist a (e k) < ε := by
   simp only [mem_closure_iff, exists_range_iff]
 #align metric.mem_closure_range_iff Metric.mem_closure_range_iff
 
+/- warning: metric.mem_closure_range_iff_nat -> Metric.mem_closure_range_iff_nat is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : PseudoMetricSpace.{u1} α] {e : β -> α} {a : α}, Iff (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) a (closure.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) (Set.range.{u1, succ u2} α β e))) (forall (n : Nat), Exists.{succ u2} β (fun (k : β) => LT.lt.{0} Real Real.hasLt (Dist.dist.{u1} α (PseudoMetricSpace.toHasDist.{u1} α _inst_1) a (e k)) (HDiv.hDiv.{0, 0, 0} Real Real Real (instHDiv.{0} Real (DivInvMonoid.toHasDiv.{0} Real (DivisionRing.toDivInvMonoid.{0} Real Real.divisionRing))) (OfNat.ofNat.{0} Real 1 (OfNat.mk.{0} Real 1 (One.one.{0} Real Real.hasOne))) (HAdd.hAdd.{0, 0, 0} Real Real Real (instHAdd.{0} Real Real.hasAdd) ((fun (a : Type) (b : Type) [self : HasLiftT.{1, 1} a b] => self.0) Nat Real (HasLiftT.mk.{1, 1} Nat Real (CoeTCₓ.coe.{1, 1} Nat Real (Nat.castCoe.{0} Real Real.hasNatCast))) n) (OfNat.ofNat.{0} Real 1 (OfNat.mk.{0} Real 1 (One.one.{0} Real Real.hasOne)))))))
+but is expected to have type
+  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : PseudoMetricSpace.{u1} α] {e : β -> α} {a : α}, Iff (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) a (closure.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) (Set.range.{u1, succ u2} α β e))) (forall (n : Nat), Exists.{succ u2} β (fun (k : β) => LT.lt.{0} Real Real.instLTReal (Dist.dist.{u1} α (PseudoMetricSpace.toDist.{u1} α _inst_1) a (e k)) (HDiv.hDiv.{0, 0, 0} Real Real Real (instHDiv.{0} Real (LinearOrderedField.toDiv.{0} Real Real.instLinearOrderedFieldReal)) (OfNat.ofNat.{0} Real 1 (One.toOfNat1.{0} Real Real.instOneReal)) (HAdd.hAdd.{0, 0, 0} Real Real Real (instHAdd.{0} Real Real.instAddReal) (Nat.cast.{0} Real Real.natCast n) (OfNat.ofNat.{0} Real 1 (One.toOfNat1.{0} Real Real.instOneReal))))))
+Case conversion may be inaccurate. Consider using '#align metric.mem_closure_range_iff_nat Metric.mem_closure_range_iff_natₓ'. -/
 theorem mem_closure_range_iff_nat {e : β → α} {a : α} :
     a ∈ closure (range e) ↔ ∀ n : ℕ, ∃ k : β, dist a (e k) < 1 / ((n : ℝ) + 1) :=
   (mem_closure_iff_nhds_basis nhds_basis_ball_inv_nat_succ).trans <| by
     simp only [mem_ball, dist_comm, exists_range_iff, forall_const]
 #align metric.mem_closure_range_iff_nat Metric.mem_closure_range_iff_nat
 
+/- warning: metric.mem_of_closed' -> Metric.mem_of_closed' is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {s : Set.{u1} α}, (IsClosed.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) s) -> (forall {a : α}, Iff (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) a s) (forall (ε : Real), (GT.gt.{0} Real Real.hasLt ε (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero)))) -> (Exists.{succ u1} α (fun (b : α) => Exists.{0} (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) b s) (fun (H : Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) b s) => LT.lt.{0} Real Real.hasLt (Dist.dist.{u1} α (PseudoMetricSpace.toHasDist.{u1} α _inst_1) a b) ε)))))
+but is expected to have type
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {s : Set.{u1} α}, (IsClosed.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) s) -> (forall {a : α}, Iff (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) a s) (forall (ε : Real), (GT.gt.{0} Real Real.instLTReal ε (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal))) -> (Exists.{succ u1} α (fun (b : α) => And (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) b s) (LT.lt.{0} Real Real.instLTReal (Dist.dist.{u1} α (PseudoMetricSpace.toDist.{u1} α _inst_1) a b) ε)))))
+Case conversion may be inaccurate. Consider using '#align metric.mem_of_closed' Metric.mem_of_closed'ₓ'. -/
 theorem mem_of_closed' {s : Set α} (hs : IsClosed s) {a : α} :
     a ∈ s ↔ ∀ ε > 0, ∃ b ∈ s, dist a b < ε := by
   simpa only [hs.closure_eq] using @mem_closure_iff _ _ s a
 #align metric.mem_of_closed' Metric.mem_of_closed'
 
+/- warning: metric.closed_ball_zero' -> Metric.closedBall_zero' is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] (x : α), Eq.{succ u1} (Set.{u1} α) (Metric.closedBall.{u1} α _inst_1 x (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero)))) (closure.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) (Singleton.singleton.{u1, u1} α (Set.{u1} α) (Set.hasSingleton.{u1} α) x))
+but is expected to have type
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] (x : α), Eq.{succ u1} (Set.{u1} α) (Metric.closedBall.{u1} α _inst_1 x (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal))) (closure.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) (Singleton.singleton.{u1, u1} α (Set.{u1} α) (Set.instSingletonSet.{u1} α) x))
+Case conversion may be inaccurate. Consider using '#align metric.closed_ball_zero' Metric.closedBall_zero'ₓ'. -/
 theorem closedBall_zero' (x : α) : closedBall x 0 = closure {x} :=
   Subset.antisymm
     (fun y hy =>
@@ -2070,15 +3438,33 @@ theorem closedBall_zero' (x : α) : closedBall x 0 = closure {x} :=
     (closure_minimal (singleton_subset_iff.2 (dist_self x).le) isClosed_ball)
 #align metric.closed_ball_zero' Metric.closedBall_zero'
 
+/- warning: metric.dense_iff -> Metric.dense_iff is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {s : Set.{u1} α}, Iff (Dense.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) s) (forall (x : α) (r : Real), (GT.gt.{0} Real Real.hasLt r (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero)))) -> (Set.Nonempty.{u1} α (Inter.inter.{u1} (Set.{u1} α) (Set.hasInter.{u1} α) (Metric.ball.{u1} α _inst_1 x r) s)))
+but is expected to have type
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {s : Set.{u1} α}, Iff (Dense.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) s) (forall (x : α) (r : Real), (GT.gt.{0} Real Real.instLTReal r (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal))) -> (Set.Nonempty.{u1} α (Inter.inter.{u1} (Set.{u1} α) (Set.instInterSet.{u1} α) (Metric.ball.{u1} α _inst_1 x r) s)))
+Case conversion may be inaccurate. Consider using '#align metric.dense_iff Metric.dense_iffₓ'. -/
 theorem dense_iff {s : Set α} : Dense s ↔ ∀ x, ∀ r > 0, (ball x r ∩ s).Nonempty :=
   forall_congr' fun x => by
     simp only [mem_closure_iff, Set.Nonempty, exists_prop, mem_inter_iff, mem_ball', and_comm']
 #align metric.dense_iff Metric.dense_iff
 
+/- warning: metric.dense_range_iff -> Metric.denseRange_iff is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : PseudoMetricSpace.{u1} α] {f : β -> α}, Iff (DenseRange.{u1, u2} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) β f) (forall (x : α) (r : Real), (GT.gt.{0} Real Real.hasLt r (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero)))) -> (Exists.{succ u2} β (fun (y : β) => LT.lt.{0} Real Real.hasLt (Dist.dist.{u1} α (PseudoMetricSpace.toHasDist.{u1} α _inst_1) x (f y)) r)))
+but is expected to have type
+  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : PseudoMetricSpace.{u1} α] {f : β -> α}, Iff (DenseRange.{u1, u2} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) β f) (forall (x : α) (r : Real), (GT.gt.{0} Real Real.instLTReal r (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal))) -> (Exists.{succ u2} β (fun (y : β) => LT.lt.{0} Real Real.instLTReal (Dist.dist.{u1} α (PseudoMetricSpace.toDist.{u1} α _inst_1) x (f y)) r)))
+Case conversion may be inaccurate. Consider using '#align metric.dense_range_iff Metric.denseRange_iffₓ'. -/
 theorem denseRange_iff {f : β → α} : DenseRange f ↔ ∀ x, ∀ r > 0, ∃ y, dist x (f y) < r :=
   forall_congr' fun x => by simp only [mem_closure_iff, exists_range_iff]
 #align metric.dense_range_iff Metric.denseRange_iff
 
+/- warning: topological_space.is_separable.separable_space -> TopologicalSpace.IsSeparable.separableSpace is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {s : Set.{u1} α}, (TopologicalSpace.IsSeparable.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) s) -> (TopologicalSpace.SeparableSpace.{u1} (coeSort.{succ u1, succ (succ u1)} (Set.{u1} α) Type.{u1} (Set.hasCoeToSort.{u1} α) s) (Subtype.topologicalSpace.{u1} α (fun (x : α) => Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) x s) (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1))))
+but is expected to have type
+  forall {α : Type.{u1}} [_inst_1 : PseudoEMetricSpace.{u1} α] {s : Set.{u1} α}, (TopologicalSpace.IsSeparable.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoEMetricSpace.toUniformSpace.{u1} α _inst_1)) s) -> (TopologicalSpace.SeparableSpace.{u1} (Set.Elem.{u1} α s) (instTopologicalSpaceSubtype.{u1} α (fun (x : α) => Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) x s) (UniformSpace.toTopologicalSpace.{u1} α (PseudoEMetricSpace.toUniformSpace.{u1} α _inst_1))))
+Case conversion may be inaccurate. Consider using '#align topological_space.is_separable.separable_space TopologicalSpace.IsSeparable.separableSpaceₓ'. -/
 /-- If a set `s` is separable, then the corresponding subtype is separable in a metric space.
 This is not obvious, as the countable set whose closure covers `s` does not need in general to
 be contained in `s`. -/
@@ -2117,6 +3503,7 @@ theorem TopologicalSpace.IsSeparable.separableSpace {s : Set α} (hs : IsSeparab
       
 #align topological_space.is_separable.separable_space TopologicalSpace.IsSeparable.separableSpace
 
+#print Inducing.isSeparable_preimage /-
 /-- The preimage of a separable set by an inducing map is separable. -/
 protected theorem Inducing.isSeparable_preimage {f : β → α} [TopologicalSpace β] (hf : Inducing f)
     {s : Set α} (hs : IsSeparable s) : IsSeparable (f ⁻¹' s) :=
@@ -2131,12 +3518,16 @@ protected theorem Inducing.isSeparable_preimage {f : β → α} [TopologicalSpac
       simpa only [image_univ, Subtype.range_coe_subtype] ]
   exact (is_separable_of_separable_space _).image continuous_subtype_val
 #align inducing.is_separable_preimage Inducing.isSeparable_preimage
+-/
 
+#print Embedding.isSeparable_preimage /-
 protected theorem Embedding.isSeparable_preimage {f : β → α} [TopologicalSpace β] (hf : Embedding f)
     {s : Set α} (hs : IsSeparable s) : IsSeparable (f ⁻¹' s) :=
   hf.to_inducing.isSeparable_preimage hs
 #align embedding.is_separable_preimage Embedding.isSeparable_preimage
+-/
 
+#print ContinuousOn.isSeparable_image /-
 /-- If a map is continuous on a separable set `s`, then the image of `s` is also separable. -/
 theorem ContinuousOn.isSeparable_image [TopologicalSpace β] {f : α → β} {s : Set α}
     (hf : ContinuousOn f s) (hs : IsSeparable s) : IsSeparable (f '' s) :=
@@ -2145,6 +3536,7 @@ theorem ContinuousOn.isSeparable_image [TopologicalSpace β] {f : α → β} {s
   exact
     (is_separable_univ_iff.2 hs.separable_space).image (continuousOn_iff_continuous_restrict.1 hf)
 #align continuous_on.is_separable_image ContinuousOn.isSeparable_image
+-/
 
 end Metric
 
@@ -2154,6 +3546,7 @@ open Finset
 
 variable {π : β → Type _} [Fintype β] [∀ b, PseudoMetricSpace (π b)]
 
+#print pseudoMetricSpacePi /-
 /-- A finite product of pseudometric spaces is a pseudometric space, with the sup distance. -/
 instance pseudoMetricSpacePi : PseudoMetricSpace (∀ b, π b) :=
   by
@@ -2179,19 +3572,44 @@ instance pseudoMetricSpacePi : PseudoMetricSpace (∀ b, π b) :=
         fun H b x hx y hy => NNReal.coe_le_coe.2 _⟩
     simpa only using Finset.sup_le_iff.1 (NNReal.coe_le_coe.1 <| H hx hy) b (Finset.mem_univ b)
 #align pseudo_metric_space_pi pseudoMetricSpacePi
+-/
 
+/- warning: nndist_pi_def -> nndist_pi_def is a dubious translation:
+lean 3 declaration is
+  forall {β : Type.{u1}} {π : β -> Type.{u2}} [_inst_2 : Fintype.{u1} β] [_inst_3 : forall (b : β), PseudoMetricSpace.{u2} (π b)] (f : forall (b : β), π b) (g : forall (b : β), π b), Eq.{1} NNReal (NNDist.nndist.{max u1 u2} (forall (b : β), π b) (PseudoMetricSpace.toNNDist.{max u1 u2} (forall (b : β), π b) (pseudoMetricSpacePi.{u1, u2} β (fun (b : β) => π b) _inst_2 (fun (b : β) => _inst_3 b))) f g) (Finset.sup.{0, u1} NNReal β NNReal.semilatticeSup NNReal.orderBot (Finset.univ.{u1} β _inst_2) (fun (b : β) => NNDist.nndist.{u2} (π b) (PseudoMetricSpace.toNNDist.{u2} (π b) (_inst_3 b)) (f b) (g b)))
+but is expected to have type
+  forall {β : Type.{u2}} {π : β -> Type.{u1}} [_inst_2 : Fintype.{u2} β] [_inst_3 : forall (b : β), PseudoMetricSpace.{u1} (π b)] (f : forall (b : β), π b) (g : forall (b : β), π b), Eq.{1} NNReal (NNDist.nndist.{max u2 u1} (forall (b : β), π b) (PseudoMetricSpace.toNNDist.{max u2 u1} (forall (b : β), π b) (pseudoMetricSpacePi.{u2, u1} β (fun (b : β) => π b) _inst_2 (fun (b : β) => _inst_3 b))) f g) (Finset.sup.{0, u2} NNReal β instNNRealSemilatticeSup NNReal.instOrderBotNNRealToLEToPreorderToPartialOrderInstNNRealStrictOrderedSemiring (Finset.univ.{u2} β _inst_2) (fun (b : β) => NNDist.nndist.{u1} (π b) (PseudoMetricSpace.toNNDist.{u1} (π b) (_inst_3 b)) (f b) (g b)))
+Case conversion may be inaccurate. Consider using '#align nndist_pi_def nndist_pi_defₓ'. -/
 theorem nndist_pi_def (f g : ∀ b, π b) : nndist f g = sup univ fun b => nndist (f b) (g b) :=
   NNReal.eq rfl
 #align nndist_pi_def nndist_pi_def
 
+/- warning: dist_pi_def -> dist_pi_def is a dubious translation:
+lean 3 declaration is
+  forall {β : Type.{u1}} {π : β -> Type.{u2}} [_inst_2 : Fintype.{u1} β] [_inst_3 : forall (b : β), PseudoMetricSpace.{u2} (π b)] (f : forall (b : β), π b) (g : forall (b : β), π b), Eq.{1} Real (Dist.dist.{max u1 u2} (forall (b : β), π b) (PseudoMetricSpace.toHasDist.{max u1 u2} (forall (b : β), π b) (pseudoMetricSpacePi.{u1, u2} β (fun (b : β) => π b) _inst_2 (fun (b : β) => _inst_3 b))) f g) ((fun (a : Type) (b : Type) [self : HasLiftT.{1, 1} a b] => self.0) NNReal Real (HasLiftT.mk.{1, 1} NNReal Real (CoeTCₓ.coe.{1, 1} NNReal Real (coeBase.{1, 1} NNReal Real NNReal.Real.hasCoe))) (Finset.sup.{0, u1} NNReal β NNReal.semilatticeSup NNReal.orderBot (Finset.univ.{u1} β _inst_2) (fun (b : β) => NNDist.nndist.{u2} (π b) (PseudoMetricSpace.toNNDist.{u2} (π b) (_inst_3 b)) (f b) (g b))))
+but is expected to have type
+  forall {β : Type.{u2}} {π : β -> Type.{u1}} [_inst_2 : Fintype.{u2} β] [_inst_3 : forall (b : β), PseudoMetricSpace.{u1} (π b)] (f : forall (b : β), π b) (g : forall (b : β), π b), Eq.{1} Real (Dist.dist.{max u2 u1} (forall (b : β), π b) (PseudoMetricSpace.toDist.{max u2 u1} (forall (b : β), π b) (pseudoMetricSpacePi.{u2, u1} β (fun (b : β) => π b) _inst_2 (fun (b : β) => _inst_3 b))) f g) (NNReal.toReal (Finset.sup.{0, u2} NNReal β instNNRealSemilatticeSup NNReal.instOrderBotNNRealToLEToPreorderToPartialOrderInstNNRealStrictOrderedSemiring (Finset.univ.{u2} β _inst_2) (fun (b : β) => NNDist.nndist.{u1} (π b) (PseudoMetricSpace.toNNDist.{u1} (π b) (_inst_3 b)) (f b) (g b))))
+Case conversion may be inaccurate. Consider using '#align dist_pi_def dist_pi_defₓ'. -/
 theorem dist_pi_def (f g : ∀ b, π b) : dist f g = (sup univ fun b => nndist (f b) (g b) : ℝ≥0) :=
   rfl
 #align dist_pi_def dist_pi_def
 
+/- warning: nndist_pi_le_iff -> nndist_pi_le_iff is a dubious translation:
+lean 3 declaration is
+  forall {β : Type.{u1}} {π : β -> Type.{u2}} [_inst_2 : Fintype.{u1} β] [_inst_3 : forall (b : β), PseudoMetricSpace.{u2} (π b)] {f : forall (b : β), π b} {g : forall (b : β), π b} {r : NNReal}, Iff (LE.le.{0} NNReal (Preorder.toLE.{0} NNReal (PartialOrder.toPreorder.{0} NNReal (OrderedCancelAddCommMonoid.toPartialOrder.{0} NNReal (StrictOrderedSemiring.toOrderedCancelAddCommMonoid.{0} NNReal NNReal.strictOrderedSemiring)))) (NNDist.nndist.{max u1 u2} (forall (b : β), π b) (PseudoMetricSpace.toNNDist.{max u1 u2} (forall (b : β), π b) (pseudoMetricSpacePi.{u1, u2} β (fun (b : β) => π b) _inst_2 (fun (b : β) => _inst_3 b))) f g) r) (forall (b : β), LE.le.{0} NNReal (Preorder.toLE.{0} NNReal (PartialOrder.toPreorder.{0} NNReal (OrderedCancelAddCommMonoid.toPartialOrder.{0} NNReal (StrictOrderedSemiring.toOrderedCancelAddCommMonoid.{0} NNReal NNReal.strictOrderedSemiring)))) (NNDist.nndist.{u2} (π b) (PseudoMetricSpace.toNNDist.{u2} (π b) (_inst_3 b)) (f b) (g b)) r)
+but is expected to have type
+  forall {β : Type.{u2}} {π : β -> Type.{u1}} [_inst_2 : Fintype.{u2} β] [_inst_3 : forall (b : β), PseudoMetricSpace.{u1} (π b)] {f : forall (b : β), π b} {g : forall (b : β), π b} {r : NNReal}, Iff (LE.le.{0} NNReal (Preorder.toLE.{0} NNReal (PartialOrder.toPreorder.{0} NNReal (StrictOrderedSemiring.toPartialOrder.{0} NNReal instNNRealStrictOrderedSemiring))) (NNDist.nndist.{max u2 u1} (forall (b : β), π b) (PseudoMetricSpace.toNNDist.{max u2 u1} (forall (b : β), π b) (pseudoMetricSpacePi.{u2, u1} β (fun (b : β) => π b) _inst_2 (fun (b : β) => _inst_3 b))) f g) r) (forall (b : β), LE.le.{0} NNReal (Preorder.toLE.{0} NNReal (PartialOrder.toPreorder.{0} NNReal (StrictOrderedSemiring.toPartialOrder.{0} NNReal instNNRealStrictOrderedSemiring))) (NNDist.nndist.{u1} (π b) (PseudoMetricSpace.toNNDist.{u1} (π b) (_inst_3 b)) (f b) (g b)) r)
+Case conversion may be inaccurate. Consider using '#align nndist_pi_le_iff nndist_pi_le_iffₓ'. -/
 theorem nndist_pi_le_iff {f g : ∀ b, π b} {r : ℝ≥0} :
     nndist f g ≤ r ↔ ∀ b, nndist (f b) (g b) ≤ r := by simp [nndist_pi_def]
 #align nndist_pi_le_iff nndist_pi_le_iff
 
+/- warning: dist_pi_lt_iff -> dist_pi_lt_iff is a dubious translation:
+lean 3 declaration is
+  forall {β : Type.{u1}} {π : β -> Type.{u2}} [_inst_2 : Fintype.{u1} β] [_inst_3 : forall (b : β), PseudoMetricSpace.{u2} (π b)] {f : forall (b : β), π b} {g : forall (b : β), π b} {r : Real}, (LT.lt.{0} Real Real.hasLt (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero))) r) -> (Iff (LT.lt.{0} Real Real.hasLt (Dist.dist.{max u1 u2} (forall (b : β), π b) (PseudoMetricSpace.toHasDist.{max u1 u2} (forall (b : β), π b) (pseudoMetricSpacePi.{u1, u2} β (fun (b : β) => π b) _inst_2 (fun (b : β) => _inst_3 b))) f g) r) (forall (b : β), LT.lt.{0} Real Real.hasLt (Dist.dist.{u2} (π b) (PseudoMetricSpace.toHasDist.{u2} (π b) (_inst_3 b)) (f b) (g b)) r))
+but is expected to have type
+  forall {β : Type.{u2}} {π : β -> Type.{u1}} [_inst_2 : Fintype.{u2} β] [_inst_3 : forall (b : β), PseudoMetricSpace.{u1} (π b)] {f : forall (b : β), π b} {g : forall (b : β), π b} {r : Real}, (LT.lt.{0} Real Real.instLTReal (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal)) r) -> (Iff (LT.lt.{0} Real Real.instLTReal (Dist.dist.{max u2 u1} (forall (b : β), π b) (PseudoMetricSpace.toDist.{max u2 u1} (forall (b : β), π b) (pseudoMetricSpacePi.{u2, u1} β (fun (b : β) => π b) _inst_2 (fun (b : β) => _inst_3 b))) f g) r) (forall (b : β), LT.lt.{0} Real Real.instLTReal (Dist.dist.{u1} (π b) (PseudoMetricSpace.toDist.{u1} (π b) (_inst_3 b)) (f b) (g b)) r))
+Case conversion may be inaccurate. Consider using '#align dist_pi_lt_iff dist_pi_lt_iffₓ'. -/
 theorem dist_pi_lt_iff {f g : ∀ b, π b} {r : ℝ} (hr : 0 < r) :
     dist f g < r ↔ ∀ b, dist (f b) (g b) < r :=
   by
@@ -2199,6 +3617,12 @@ theorem dist_pi_lt_iff {f g : ∀ b, π b} {r : ℝ} (hr : 0 < r) :
   simp [dist_pi_def, Finset.sup_lt_iff (show ⊥ < r from hr)]
 #align dist_pi_lt_iff dist_pi_lt_iff
 
+/- warning: dist_pi_le_iff -> dist_pi_le_iff is a dubious translation:
+lean 3 declaration is
+  forall {β : Type.{u1}} {π : β -> Type.{u2}} [_inst_2 : Fintype.{u1} β] [_inst_3 : forall (b : β), PseudoMetricSpace.{u2} (π b)] {f : forall (b : β), π b} {g : forall (b : β), π b} {r : Real}, (LE.le.{0} Real Real.hasLe (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero))) r) -> (Iff (LE.le.{0} Real Real.hasLe (Dist.dist.{max u1 u2} (forall (b : β), π b) (PseudoMetricSpace.toHasDist.{max u1 u2} (forall (b : β), π b) (pseudoMetricSpacePi.{u1, u2} β (fun (b : β) => π b) _inst_2 (fun (b : β) => _inst_3 b))) f g) r) (forall (b : β), LE.le.{0} Real Real.hasLe (Dist.dist.{u2} (π b) (PseudoMetricSpace.toHasDist.{u2} (π b) (_inst_3 b)) (f b) (g b)) r))
+but is expected to have type
+  forall {β : Type.{u2}} {π : β -> Type.{u1}} [_inst_2 : Fintype.{u2} β] [_inst_3 : forall (b : β), PseudoMetricSpace.{u1} (π b)] {f : forall (b : β), π b} {g : forall (b : β), π b} {r : Real}, (LE.le.{0} Real Real.instLEReal (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal)) r) -> (Iff (LE.le.{0} Real Real.instLEReal (Dist.dist.{max u2 u1} (forall (b : β), π b) (PseudoMetricSpace.toDist.{max u2 u1} (forall (b : β), π b) (pseudoMetricSpacePi.{u2, u1} β (fun (b : β) => π b) _inst_2 (fun (b : β) => _inst_3 b))) f g) r) (forall (b : β), LE.le.{0} Real Real.instLEReal (Dist.dist.{u1} (π b) (PseudoMetricSpace.toDist.{u1} (π b) (_inst_3 b)) (f b) (g b)) r))
+Case conversion may be inaccurate. Consider using '#align dist_pi_le_iff dist_pi_le_iffₓ'. -/
 theorem dist_pi_le_iff {f g : ∀ b, π b} {r : ℝ} (hr : 0 ≤ r) :
     dist f g ≤ r ↔ ∀ b, dist (f b) (g b) ≤ r :=
   by
@@ -2206,6 +3630,12 @@ theorem dist_pi_le_iff {f g : ∀ b, π b} {r : ℝ} (hr : 0 ≤ r) :
   exact nndist_pi_le_iff
 #align dist_pi_le_iff dist_pi_le_iff
 
+/- warning: dist_pi_le_iff' -> dist_pi_le_iff' is a dubious translation:
+lean 3 declaration is
+  forall {β : Type.{u1}} {π : β -> Type.{u2}} [_inst_2 : Fintype.{u1} β] [_inst_3 : forall (b : β), PseudoMetricSpace.{u2} (π b)] [_inst_4 : Nonempty.{succ u1} β] {f : forall (b : β), π b} {g : forall (b : β), π b} {r : Real}, Iff (LE.le.{0} Real Real.hasLe (Dist.dist.{max u1 u2} (forall (b : β), π b) (PseudoMetricSpace.toHasDist.{max u1 u2} (forall (b : β), π b) (pseudoMetricSpacePi.{u1, u2} β (fun (b : β) => π b) _inst_2 (fun (b : β) => _inst_3 b))) f g) r) (forall (b : β), LE.le.{0} Real Real.hasLe (Dist.dist.{u2} (π b) (PseudoMetricSpace.toHasDist.{u2} (π b) (_inst_3 b)) (f b) (g b)) r)
+but is expected to have type
+  forall {β : Type.{u2}} {π : β -> Type.{u1}} [_inst_2 : Fintype.{u2} β] [_inst_3 : forall (b : β), PseudoMetricSpace.{u1} (π b)] [_inst_4 : Nonempty.{succ u2} β] {f : forall (b : β), π b} {g : forall (b : β), π b} {r : Real}, Iff (LE.le.{0} Real Real.instLEReal (Dist.dist.{max u2 u1} (forall (b : β), π b) (PseudoMetricSpace.toDist.{max u2 u1} (forall (b : β), π b) (pseudoMetricSpacePi.{u2, u1} β (fun (b : β) => π b) _inst_2 (fun (b : β) => _inst_3 b))) f g) r) (forall (b : β), LE.le.{0} Real Real.instLEReal (Dist.dist.{u1} (π b) (PseudoMetricSpace.toDist.{u1} (π b) (_inst_3 b)) (f b) (g b)) r)
+Case conversion may be inaccurate. Consider using '#align dist_pi_le_iff' dist_pi_le_iff'ₓ'. -/
 theorem dist_pi_le_iff' [Nonempty β] {f g : ∀ b, π b} {r : ℝ} :
     dist f g ≤ r ↔ ∀ b, dist (f b) (g b) ≤ r :=
   by
@@ -2217,35 +3647,65 @@ theorem dist_pi_le_iff' [Nonempty β] {f g : ∀ b, π b} {r : ℝ} :
         hr <| dist_nonneg.trans <| h <| Classical.arbitrary _
 #align dist_pi_le_iff' dist_pi_le_iff'
 
+/- warning: dist_pi_const_le -> dist_pi_const_le is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : PseudoMetricSpace.{u1} α] [_inst_2 : Fintype.{u2} β] (a : α) (b : α), LE.le.{0} Real Real.hasLe (Dist.dist.{max u2 u1} (β -> α) (PseudoMetricSpace.toHasDist.{max u2 u1} (β -> α) (pseudoMetricSpacePi.{u2, u1} β (fun (_x : β) => α) _inst_2 (fun (b : β) => _inst_1))) (fun (_x : β) => a) (fun (_x : β) => b)) (Dist.dist.{u1} α (PseudoMetricSpace.toHasDist.{u1} α _inst_1) a b)
+but is expected to have type
+  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : PseudoMetricSpace.{u1} α] [_inst_2 : Fintype.{u2} β] (a : α) (b : α), LE.le.{0} Real Real.instLEReal (Dist.dist.{max u1 u2} (β -> α) (PseudoMetricSpace.toDist.{max u1 u2} (β -> α) (pseudoMetricSpacePi.{u2, u1} β (fun (_x : β) => α) _inst_2 (fun (b : β) => _inst_1))) (fun (_x : β) => a) (fun (_x : β) => b)) (Dist.dist.{u1} α (PseudoMetricSpace.toDist.{u1} α _inst_1) a b)
+Case conversion may be inaccurate. Consider using '#align dist_pi_const_le dist_pi_const_leₓ'. -/
 theorem dist_pi_const_le (a b : α) : (dist (fun _ : β => a) fun _ => b) ≤ dist a b :=
   (dist_pi_le_iff dist_nonneg).2 fun _ => le_rfl
 #align dist_pi_const_le dist_pi_const_le
 
+#print nndist_pi_const_le /-
 theorem nndist_pi_const_le (a b : α) : (nndist (fun _ : β => a) fun _ => b) ≤ nndist a b :=
   nndist_pi_le_iff.2 fun _ => le_rfl
 #align nndist_pi_const_le nndist_pi_const_le
+-/
 
+#print dist_pi_const /-
 @[simp]
 theorem dist_pi_const [Nonempty β] (a b : α) : (dist (fun x : β => a) fun _ => b) = dist a b := by
   simpa only [dist_edist] using congr_arg ENNReal.toReal (edist_pi_const a b)
 #align dist_pi_const dist_pi_const
+-/
 
+#print nndist_pi_const /-
 @[simp]
 theorem nndist_pi_const [Nonempty β] (a b : α) :
     (nndist (fun x : β => a) fun _ => b) = nndist a b :=
   NNReal.eq <| dist_pi_const a b
 #align nndist_pi_const nndist_pi_const
+-/
 
+/- warning: nndist_le_pi_nndist -> nndist_le_pi_nndist is a dubious translation:
+lean 3 declaration is
+  forall {β : Type.{u1}} {π : β -> Type.{u2}} [_inst_2 : Fintype.{u1} β] [_inst_3 : forall (b : β), PseudoMetricSpace.{u2} (π b)] (f : forall (b : β), π b) (g : forall (b : β), π b) (b : β), LE.le.{0} NNReal (Preorder.toLE.{0} NNReal (PartialOrder.toPreorder.{0} NNReal (OrderedCancelAddCommMonoid.toPartialOrder.{0} NNReal (StrictOrderedSemiring.toOrderedCancelAddCommMonoid.{0} NNReal NNReal.strictOrderedSemiring)))) (NNDist.nndist.{u2} (π b) (PseudoMetricSpace.toNNDist.{u2} (π b) (_inst_3 b)) (f b) (g b)) (NNDist.nndist.{max u1 u2} (forall (b : β), π b) (PseudoMetricSpace.toNNDist.{max u1 u2} (forall (b : β), π b) (pseudoMetricSpacePi.{u1, u2} β (fun (b : β) => π b) _inst_2 (fun (b : β) => _inst_3 b))) f g)
+but is expected to have type
+  forall {β : Type.{u2}} {π : β -> Type.{u1}} [_inst_2 : Fintype.{u2} β] [_inst_3 : forall (b : β), PseudoMetricSpace.{u1} (π b)] (f : forall (b : β), π b) (g : forall (b : β), π b) (b : β), LE.le.{0} NNReal (Preorder.toLE.{0} NNReal (PartialOrder.toPreorder.{0} NNReal (StrictOrderedSemiring.toPartialOrder.{0} NNReal instNNRealStrictOrderedSemiring))) (NNDist.nndist.{u1} (π b) (PseudoMetricSpace.toNNDist.{u1} (π b) (_inst_3 b)) (f b) (g b)) (NNDist.nndist.{max u2 u1} (forall (b : β), π b) (PseudoMetricSpace.toNNDist.{max u2 u1} (forall (b : β), π b) (pseudoMetricSpacePi.{u2, u1} β (fun (b : β) => π b) _inst_2 (fun (b : β) => _inst_3 b))) f g)
+Case conversion may be inaccurate. Consider using '#align nndist_le_pi_nndist nndist_le_pi_nndistₓ'. -/
 theorem nndist_le_pi_nndist (f g : ∀ b, π b) (b : β) : nndist (f b) (g b) ≤ nndist f g :=
   by
   rw [nndist_pi_def]
   exact Finset.le_sup (Finset.mem_univ b)
 #align nndist_le_pi_nndist nndist_le_pi_nndist
 
+/- warning: dist_le_pi_dist -> dist_le_pi_dist is a dubious translation:
+lean 3 declaration is
+  forall {β : Type.{u1}} {π : β -> Type.{u2}} [_inst_2 : Fintype.{u1} β] [_inst_3 : forall (b : β), PseudoMetricSpace.{u2} (π b)] (f : forall (b : β), π b) (g : forall (b : β), π b) (b : β), LE.le.{0} Real Real.hasLe (Dist.dist.{u2} (π b) (PseudoMetricSpace.toHasDist.{u2} (π b) (_inst_3 b)) (f b) (g b)) (Dist.dist.{max u1 u2} (forall (b : β), π b) (PseudoMetricSpace.toHasDist.{max u1 u2} (forall (b : β), π b) (pseudoMetricSpacePi.{u1, u2} β (fun (b : β) => π b) _inst_2 (fun (b : β) => _inst_3 b))) f g)
+but is expected to have type
+  forall {β : Type.{u2}} {π : β -> Type.{u1}} [_inst_2 : Fintype.{u2} β] [_inst_3 : forall (b : β), PseudoMetricSpace.{u1} (π b)] (f : forall (b : β), π b) (g : forall (b : β), π b) (b : β), LE.le.{0} Real Real.instLEReal (Dist.dist.{u1} (π b) (PseudoMetricSpace.toDist.{u1} (π b) (_inst_3 b)) (f b) (g b)) (Dist.dist.{max u2 u1} (forall (b : β), π b) (PseudoMetricSpace.toDist.{max u2 u1} (forall (b : β), π b) (pseudoMetricSpacePi.{u2, u1} β (fun (b : β) => π b) _inst_2 (fun (b : β) => _inst_3 b))) f g)
+Case conversion may be inaccurate. Consider using '#align dist_le_pi_dist dist_le_pi_distₓ'. -/
 theorem dist_le_pi_dist (f g : ∀ b, π b) (b : β) : dist (f b) (g b) ≤ dist f g := by
   simp only [dist_nndist, NNReal.coe_le_coe, nndist_le_pi_nndist f g b]
 #align dist_le_pi_dist dist_le_pi_dist
 
+/- warning: ball_pi -> ball_pi is a dubious translation:
+lean 3 declaration is
+  forall {β : Type.{u1}} {π : β -> Type.{u2}} [_inst_2 : Fintype.{u1} β] [_inst_3 : forall (b : β), PseudoMetricSpace.{u2} (π b)] (x : forall (b : β), π b) {r : Real}, (LT.lt.{0} Real Real.hasLt (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero))) r) -> (Eq.{succ (max u1 u2)} (Set.{max u1 u2} (forall (b : β), π b)) (Metric.ball.{max u1 u2} (forall (b : β), π b) (pseudoMetricSpacePi.{u1, u2} β (fun (b : β) => π b) _inst_2 (fun (b : β) => _inst_3 b)) x r) (Set.pi.{u1, u2} β (fun (b : β) => π b) (Set.univ.{u1} β) (fun (b : β) => Metric.ball.{u2} (π b) (_inst_3 b) (x b) r)))
+but is expected to have type
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+Case conversion may be inaccurate. Consider using '#align ball_pi ball_piₓ'. -/
 /-- An open ball in a product space is a product of open balls. See also `metric.ball_pi'`
 for a version assuming `nonempty β` instead of `0 < r`. -/
 theorem ball_pi (x : ∀ b, π b) {r : ℝ} (hr : 0 < r) :
@@ -2255,6 +3715,12 @@ theorem ball_pi (x : ∀ b, π b) {r : ℝ} (hr : 0 < r) :
   simp [dist_pi_lt_iff hr]
 #align ball_pi ball_pi
 
+/- warning: ball_pi' -> ball_pi' is a dubious translation:
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+Case conversion may be inaccurate. Consider using '#align ball_pi' ball_pi'ₓ'. -/
 /-- An open ball in a product space is a product of open balls. See also `metric.ball_pi`
 for a version assuming `0 < r` instead of `nonempty β`. -/
 theorem ball_pi' [Nonempty β] (x : ∀ b, π b) (r : ℝ) :
@@ -2262,6 +3728,12 @@ theorem ball_pi' [Nonempty β] (x : ∀ b, π b) (r : ℝ) :
   (lt_or_le 0 r).elim (ball_pi x) fun hr => by simp [ball_eq_empty.2 hr]
 #align ball_pi' ball_pi'
 
+/- warning: closed_ball_pi -> closedBall_pi is a dubious translation:
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+Case conversion may be inaccurate. Consider using '#align closed_ball_pi closedBall_piₓ'. -/
 /-- A closed ball in a product space is a product of closed balls. See also `metric.closed_ball_pi'`
 for a version assuming `nonempty β` instead of `0 ≤ r`. -/
 theorem closedBall_pi (x : ∀ b, π b) {r : ℝ} (hr : 0 ≤ r) :
@@ -2271,6 +3743,12 @@ theorem closedBall_pi (x : ∀ b, π b) {r : ℝ} (hr : 0 ≤ r) :
   simp [dist_pi_le_iff hr]
 #align closed_ball_pi closedBall_pi
 
+/- warning: closed_ball_pi' -> closedBall_pi' is a dubious translation:
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+Case conversion may be inaccurate. Consider using '#align closed_ball_pi' closedBall_pi'ₓ'. -/
 /-- A closed ball in a product space is a product of closed balls. See also `metric.closed_ball_pi`
 for a version assuming `0 ≤ r` instead of `nonempty β`. -/
 theorem closedBall_pi' [Nonempty β] (x : ∀ b, π b) (r : ℝ) :
@@ -2278,6 +3756,12 @@ theorem closedBall_pi' [Nonempty β] (x : ∀ b, π b) (r : ℝ) :
   (le_or_lt 0 r).elim (closedBall_pi x) fun hr => by simp [closed_ball_eq_empty.2 hr]
 #align closed_ball_pi' closedBall_pi'
 
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+Case conversion may be inaccurate. Consider using '#align fin.nndist_insert_nth_insert_nth Fin.nndist_insertNth_insertNthₓ'. -/
 @[simp]
 theorem Fin.nndist_insertNth_insertNth {n : ℕ} {α : Fin (n + 1) → Type _}
     [∀ i, PseudoMetricSpace (α i)] (i : Fin (n + 1)) (x y : α i) (f g : ∀ j, α (i.succAbove j)) :
@@ -2285,6 +3769,12 @@ theorem Fin.nndist_insertNth_insertNth {n : ℕ} {α : Fin (n + 1) → Type _}
   eq_of_forall_ge_iff fun c => by simp [nndist_pi_le_iff, i.forall_iff_succ_above]
 #align fin.nndist_insert_nth_insert_nth Fin.nndist_insertNth_insertNth
 
+/- warning: fin.dist_insert_nth_insert_nth -> Fin.dist_insertNth_insertNth is a dubious translation:
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(HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) x._@.Mathlib.Order.Hom.Basic._hyg.695 x._@.Mathlib.Order.Hom.Basic._hyg.697) (Fin.succAbove n i)) b)))) f g))
+Case conversion may be inaccurate. Consider using '#align fin.dist_insert_nth_insert_nth Fin.dist_insertNth_insertNthₓ'. -/
 @[simp]
 theorem Fin.dist_insertNth_insertNth {n : ℕ} {α : Fin (n + 1) → Type _}
     [∀ i, PseudoMetricSpace (α i)] (i : Fin (n + 1)) (x y : α i) (f g : ∀ j, α (i.succAbove j)) :
@@ -2292,6 +3782,12 @@ theorem Fin.dist_insertNth_insertNth {n : ℕ} {α : Fin (n + 1) → Type _}
   simp only [dist_nndist, Fin.nndist_insertNth_insertNth, NNReal.coe_max]
 #align fin.dist_insert_nth_insert_nth Fin.dist_insertNth_insertNth
 
+/- warning: real.dist_le_of_mem_pi_Icc -> Real.dist_le_of_mem_pi_Icc is a dubious translation:
+lean 3 declaration is
+  forall {β : Type.{u1}} [_inst_2 : Fintype.{u1} β] {x : β -> Real} {y : β -> Real} {x' : β -> Real} {y' : β -> Real}, (Membership.Mem.{u1, u1} (β -> Real) (Set.{u1} (β -> Real)) (Set.hasMem.{u1} (β -> Real)) x (Set.Icc.{u1} (β -> Real) (Pi.preorder.{u1, 0} β (fun (ᾰ : β) => Real) (fun (i : β) => Real.preorder)) x' y')) -> (Membership.Mem.{u1, u1} (β -> Real) (Set.{u1} (β -> Real)) (Set.hasMem.{u1} (β -> Real)) y (Set.Icc.{u1} (β -> Real) (Pi.preorder.{u1, 0} β (fun (ᾰ : β) => Real) (fun (i : β) => Real.preorder)) x' y')) -> (LE.le.{0} Real Real.hasLe (Dist.dist.{u1} (β -> Real) (PseudoMetricSpace.toHasDist.{u1} (β -> Real) (pseudoMetricSpacePi.{u1, 0} β (fun (ᾰ : β) => Real) _inst_2 (fun (b : β) => Real.pseudoMetricSpace))) x y) (Dist.dist.{u1} (β -> Real) (PseudoMetricSpace.toHasDist.{u1} (β -> Real) (pseudoMetricSpacePi.{u1, 0} β (fun (ᾰ : β) => Real) _inst_2 (fun (b : β) => Real.pseudoMetricSpace))) x' y'))
+but is expected to have type
+  forall {β : Type.{u1}} [_inst_2 : Fintype.{u1} β] {x : β -> Real} {y : β -> Real} {x' : β -> Real} {y' : β -> Real}, (Membership.mem.{u1, u1} (β -> Real) (Set.{u1} (β -> Real)) (Set.instMembershipSet.{u1} (β -> Real)) x (Set.Icc.{u1} (β -> Real) (Pi.preorder.{u1, 0} β (fun (ᾰ : β) => Real) (fun (i : β) => Real.instPreorderReal)) x' y')) -> (Membership.mem.{u1, u1} (β -> Real) (Set.{u1} (β -> Real)) (Set.instMembershipSet.{u1} (β -> Real)) y (Set.Icc.{u1} (β -> Real) (Pi.preorder.{u1, 0} β (fun (ᾰ : β) => Real) (fun (i : β) => Real.instPreorderReal)) x' y')) -> (LE.le.{0} Real Real.instLEReal (Dist.dist.{u1} (β -> Real) (PseudoMetricSpace.toDist.{u1} (β -> Real) (pseudoMetricSpacePi.{u1, 0} β (fun (ᾰ : β) => Real) _inst_2 (fun (b : β) => Real.pseudoMetricSpace))) x y) (Dist.dist.{u1} (β -> Real) (PseudoMetricSpace.toDist.{u1} (β -> Real) (pseudoMetricSpacePi.{u1, 0} β (fun (ᾰ : β) => Real) _inst_2 (fun (b : β) => Real.pseudoMetricSpace))) x' y'))
+Case conversion may be inaccurate. Consider using '#align real.dist_le_of_mem_pi_Icc Real.dist_le_of_mem_pi_Iccₓ'. -/
 theorem Real.dist_le_of_mem_pi_Icc {x y x' y' : β → ℝ} (hx : x ∈ Icc x' y') (hy : y ∈ Icc x' y') :
     dist x y ≤ dist x' y' :=
   by
@@ -2306,6 +3802,12 @@ end Pi
 
 section Compact
 
+/- warning: finite_cover_balls_of_compact -> finite_cover_balls_of_compact is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} [_inst_2 : PseudoMetricSpace.{u1} α] {s : Set.{u1} α}, (IsCompact.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_2)) s) -> (forall {e : Real}, (LT.lt.{0} Real Real.hasLt (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero))) e) -> (Exists.{succ u1} (Set.{u1} α) (fun (t : Set.{u1} α) => Exists.{0} (HasSubset.Subset.{u1} (Set.{u1} α) (Set.hasSubset.{u1} α) t s) (fun (H : HasSubset.Subset.{u1} (Set.{u1} α) (Set.hasSubset.{u1} α) t s) => And (Set.Finite.{u1} α t) (HasSubset.Subset.{u1} (Set.{u1} α) (Set.hasSubset.{u1} α) s (Set.unionᵢ.{u1, succ u1} α α (fun (x : α) => Set.unionᵢ.{u1, 0} α (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) x t) (fun (H : Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) x t) => Metric.ball.{u1} α _inst_2 x e))))))))
+but is expected to have type
+  forall {α : Type.{u1}} [_inst_2 : PseudoMetricSpace.{u1} α] {s : Set.{u1} α}, (IsCompact.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_2)) s) -> (forall {e : Real}, (LT.lt.{0} Real Real.instLTReal (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal)) e) -> (Exists.{succ u1} (Set.{u1} α) (fun (t : Set.{u1} α) => And (HasSubset.Subset.{u1} (Set.{u1} α) (Set.instHasSubsetSet.{u1} α) t s) (And (Set.Finite.{u1} α t) (HasSubset.Subset.{u1} (Set.{u1} α) (Set.instHasSubsetSet.{u1} α) s (Set.unionᵢ.{u1, succ u1} α α (fun (x : α) => Set.unionᵢ.{u1, 0} α (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) x t) (fun (h._@.Mathlib.Topology.MetricSpace.Basic._hyg.25459 : Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) x t) => Metric.ball.{u1} α _inst_2 x e))))))))
+Case conversion may be inaccurate. Consider using '#align finite_cover_balls_of_compact finite_cover_balls_of_compactₓ'. -/
 /- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (t «expr ⊆ » s) -/
 /-- Any compact set in a pseudometric space can be covered by finitely many balls of a given
 positive radius -/
@@ -2320,6 +3822,12 @@ theorem finite_cover_balls_of_compact {α : Type u} [PseudoMetricSpace α] {s :
     exact ⟨x, ⟨xs, by simpa⟩⟩
 #align finite_cover_balls_of_compact finite_cover_balls_of_compact
 
+/- warning: is_compact.finite_cover_balls -> IsCompact.finite_cover_balls is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} [_inst_2 : PseudoMetricSpace.{u1} α] {s : Set.{u1} α}, (IsCompact.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_2)) s) -> (forall {e : Real}, (LT.lt.{0} Real Real.hasLt (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero))) e) -> (Exists.{succ u1} (Set.{u1} α) (fun (t : Set.{u1} α) => Exists.{0} (HasSubset.Subset.{u1} (Set.{u1} α) (Set.hasSubset.{u1} α) t s) (fun (H : HasSubset.Subset.{u1} (Set.{u1} α) (Set.hasSubset.{u1} α) t s) => And (Set.Finite.{u1} α t) (HasSubset.Subset.{u1} (Set.{u1} α) (Set.hasSubset.{u1} α) s (Set.unionᵢ.{u1, succ u1} α α (fun (x : α) => Set.unionᵢ.{u1, 0} α (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) x t) (fun (H : Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) x t) => Metric.ball.{u1} α _inst_2 x e))))))))
+but is expected to have type
+  forall {α : Type.{u1}} [_inst_2 : PseudoMetricSpace.{u1} α] {s : Set.{u1} α}, (IsCompact.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_2)) s) -> (forall {e : Real}, (LT.lt.{0} Real Real.instLTReal (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal)) e) -> (Exists.{succ u1} (Set.{u1} α) (fun (t : Set.{u1} α) => And (HasSubset.Subset.{u1} (Set.{u1} α) (Set.instHasSubsetSet.{u1} α) t s) (And (Set.Finite.{u1} α t) (HasSubset.Subset.{u1} (Set.{u1} α) (Set.instHasSubsetSet.{u1} α) s (Set.unionᵢ.{u1, succ u1} α α (fun (x : α) => Set.unionᵢ.{u1, 0} α (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) x t) (fun (h._@.Mathlib.Topology.MetricSpace.Basic._hyg.25459 : Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) x t) => Metric.ball.{u1} α _inst_2 x e))))))))
+Case conversion may be inaccurate. Consider using '#align is_compact.finite_cover_balls IsCompact.finite_cover_ballsₓ'. -/
 alias finite_cover_balls_of_compact ← IsCompact.finite_cover_balls
 #align is_compact.finite_cover_balls IsCompact.finite_cover_balls
 
@@ -2329,27 +3837,32 @@ section ProperSpace
 
 open Metric
 
+#print ProperSpace /-
 /-- A pseudometric space is proper if all closed balls are compact. -/
 class ProperSpace (α : Type u) [PseudoMetricSpace α] : Prop where
   isCompact_closedBall : ∀ x : α, ∀ r, IsCompact (closedBall x r)
 #align proper_space ProperSpace
+-/
 
 export ProperSpace (isCompact_closedBall)
 
+#print isCompact_sphere /-
 /-- In a proper pseudometric space, all spheres are compact. -/
 theorem isCompact_sphere {α : Type _} [PseudoMetricSpace α] [ProperSpace α] (x : α) (r : ℝ) :
     IsCompact (sphere x r) :=
   isCompact_of_isClosed_subset (isCompact_closedBall x r) isClosed_sphere sphere_subset_closedBall
 #align is_compact_sphere isCompact_sphere
+-/
 
 /-- In a proper pseudometric space, any sphere is a `compact_space` when considered as a subtype. -/
 instance {α : Type _} [PseudoMetricSpace α] [ProperSpace α] (x : α) (r : ℝ) :
     CompactSpace (sphere x r) :=
   isCompact_iff_compactSpace.mp (isCompact_sphere _ _)
 
+#print secondCountable_of_proper /-
 -- see Note [lower instance priority]
 /-- A proper pseudo metric space is sigma compact, and therefore second countable. -/
-instance (priority := 100) second_countable_of_proper [ProperSpace α] : SecondCountableTopology α :=
+instance (priority := 100) secondCountable_of_proper [ProperSpace α] : SecondCountableTopology α :=
   by
   -- We already have `sigma_compact_space_of_locally_compact_second_countable`, so we don't
   -- add an instance for `sigma_compact_space`.
@@ -2357,19 +3870,30 @@ instance (priority := 100) second_countable_of_proper [ProperSpace α] : SecondC
   rcases em (Nonempty α) with (⟨⟨x⟩⟩ | hn)
   · exact ⟨⟨fun n => closed_ball x n, fun n => is_compact_closed_ball _ _, Union_closed_ball_nat _⟩⟩
   · exact ⟨⟨fun n => ∅, fun n => isCompact_empty, Union_eq_univ_iff.2 fun x => (hn ⟨x⟩).elim⟩⟩
-#align second_countable_of_proper second_countable_of_proper
+#align second_countable_of_proper secondCountable_of_proper
+-/
 
+#print tendsto_dist_right_cocompact_atTop /-
 theorem tendsto_dist_right_cocompact_atTop [ProperSpace α] (x : α) :
     Tendsto (fun y => dist y x) (cocompact α) atTop :=
   (hasBasis_cocompact.tendsto_iffₓ atTop_basis).2 fun r hr =>
     ⟨closedBall x r, isCompact_closedBall x r, fun y hy => (not_le.1 <| mt mem_closedBall.2 hy).le⟩
 #align tendsto_dist_right_cocompact_at_top tendsto_dist_right_cocompact_atTop
+-/
 
+#print tendsto_dist_left_cocompact_atTop /-
 theorem tendsto_dist_left_cocompact_atTop [ProperSpace α] (x : α) :
     Tendsto (dist x) (cocompact α) atTop := by
   simpa only [dist_comm] using tendsto_dist_right_cocompact_atTop x
 #align tendsto_dist_left_cocompact_at_top tendsto_dist_left_cocompact_atTop
+-/
 
+/- warning: proper_space_of_compact_closed_ball_of_le -> properSpace_of_compact_closedBall_of_le is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] (R : Real), (forall (x : α) (r : Real), (LE.le.{0} Real Real.hasLe R r) -> (IsCompact.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) (Metric.closedBall.{u1} α _inst_1 x r))) -> (ProperSpace.{u1} α _inst_1)
+but is expected to have type
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] (R : Real), (forall (x : α) (r : Real), (LE.le.{0} Real Real.instLEReal R r) -> (IsCompact.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) (Metric.closedBall.{u1} α _inst_1 x r))) -> (ProperSpace.{u1} α _inst_1)
+Case conversion may be inaccurate. Consider using '#align proper_space_of_compact_closed_ball_of_le properSpace_of_compact_closedBall_of_leₓ'. -/
 /-- If all closed balls of large enough radius are compact, then the space is proper. Especially
 useful when the lower bound for the radius is 0. -/
 theorem properSpace_of_compact_closedBall_of_le (R : ℝ)
@@ -2387,20 +3911,25 @@ theorem properSpace_of_compact_closedBall_of_le (R : ℝ)
       exact (h x R le_rfl).inter_right is_closed_ball⟩
 #align proper_space_of_compact_closed_ball_of_le properSpace_of_compact_closedBall_of_le
 
+#print proper_of_compact /-
 -- A compact pseudometric space is proper 
 -- see Note [lower instance priority]
 instance (priority := 100) proper_of_compact [CompactSpace α] : ProperSpace α :=
   ⟨fun x r => isClosed_ball.IsCompact⟩
 #align proper_of_compact proper_of_compact
+-/
 
+#print locally_compact_of_proper /-
 -- see Note [lower instance priority]
 /-- A proper space is locally compact -/
 instance (priority := 100) locally_compact_of_proper [ProperSpace α] : LocallyCompactSpace α :=
   locallyCompactSpace_of_hasBasis (fun x => nhds_basis_closedBall) fun x ε ε0 =>
     isCompact_closedBall _ _
 #align locally_compact_of_proper locally_compact_of_proper
+-/
 
 /- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (x y «expr ∈ » t) -/
+#print complete_of_proper /-
 -- see Note [lower instance priority]
 /-- A proper space is complete -/
 instance (priority := 100) complete_of_proper [ProperSpace α] : CompleteSpace α :=
@@ -2417,7 +3946,9 @@ instance (priority := 100) complete_of_proper [ProperSpace α] : CompleteSpace 
       ⟨y, -, hy⟩
     exact ⟨y, hy⟩⟩
 #align complete_of_proper complete_of_proper
+-/
 
+#print prod_properSpace /-
 /-- A binary product of proper spaces is proper. -/
 instance prod_properSpace {α : Type _} {β : Type _} [PseudoMetricSpace α] [PseudoMetricSpace β]
     [ProperSpace α] [ProperSpace β] : ProperSpace (α × β)
@@ -2426,7 +3957,9 @@ instance prod_properSpace {α : Type _} {β : Type _} [PseudoMetricSpace α] [Ps
     rw [← closedBall_prod_same x y]
     apply (is_compact_closed_ball x r).Prod (is_compact_closed_ball y r)
 #align prod_proper_space prod_properSpace
+-/
 
+#print pi_properSpace /-
 /-- A finite product of proper spaces is proper. -/
 instance pi_properSpace {π : β → Type _} [Fintype β] [∀ b, PseudoMetricSpace (π b)]
     [h : ∀ b, ProperSpace (π b)] : ProperSpace (∀ b, π b) :=
@@ -2436,9 +3969,16 @@ instance pi_properSpace {π : β → Type _} [Fintype β] [∀ b, PseudoMetricSp
   apply isCompact_univ_pi fun b => _
   apply (h b).isCompact_closedBall
 #align pi_proper_space pi_properSpace
+-/
 
 variable [ProperSpace α] {x : α} {r : ℝ} {s : Set α}
 
+/- warning: exists_pos_lt_subset_ball -> exists_pos_lt_subset_ball is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] [_inst_2 : ProperSpace.{u1} α _inst_1] {x : α} {r : Real} {s : Set.{u1} α}, (LT.lt.{0} Real Real.hasLt (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero))) r) -> (IsClosed.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) s) -> (HasSubset.Subset.{u1} (Set.{u1} α) (Set.hasSubset.{u1} α) s (Metric.ball.{u1} α _inst_1 x r)) -> (Exists.{1} Real (fun (r' : Real) => Exists.{0} (Membership.Mem.{0, 0} Real (Set.{0} Real) (Set.hasMem.{0} Real) r' (Set.Ioo.{0} Real Real.preorder (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero))) r)) (fun (H : Membership.Mem.{0, 0} Real (Set.{0} Real) (Set.hasMem.{0} Real) r' (Set.Ioo.{0} Real Real.preorder (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero))) r)) => HasSubset.Subset.{u1} (Set.{u1} α) (Set.hasSubset.{u1} α) s (Metric.ball.{u1} α _inst_1 x r'))))
+but is expected to have type
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] [_inst_2 : ProperSpace.{u1} α _inst_1] {x : α} {r : Real} {s : Set.{u1} α}, (LT.lt.{0} Real Real.instLTReal (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal)) r) -> (IsClosed.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) s) -> (HasSubset.Subset.{u1} (Set.{u1} α) (Set.instHasSubsetSet.{u1} α) s (Metric.ball.{u1} α _inst_1 x r)) -> (Exists.{1} Real (fun (r' : Real) => And (Membership.mem.{0, 0} Real (Set.{0} Real) (Set.instMembershipSet.{0} Real) r' (Set.Ioo.{0} Real Real.instPreorderReal (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal)) r)) (HasSubset.Subset.{u1} (Set.{u1} α) (Set.instHasSubsetSet.{u1} α) s (Metric.ball.{u1} α _inst_1 x r'))))
+Case conversion may be inaccurate. Consider using '#align exists_pos_lt_subset_ball exists_pos_lt_subset_ballₓ'. -/
 /-- If a nonempty ball in a proper space includes a closed set `s`, then there exists a nonempty
 ball with the same center and a strictly smaller radius that includes `s`. -/
 theorem exists_pos_lt_subset_ball (hr : 0 < r) (hs : IsClosed s) (h : s ⊆ ball x r) :
@@ -2456,6 +3996,12 @@ theorem exists_pos_lt_subset_ball (hr : 0 < r) (hs : IsClosed s) (h : s ⊆ ball
   exact ⟨r', ⟨dist_nonneg.trans_lt hyr', hrr'⟩, subset.trans hy <| closed_ball_subset_ball hyr'⟩
 #align exists_pos_lt_subset_ball exists_pos_lt_subset_ball
 
+/- warning: exists_lt_subset_ball -> exists_lt_subset_ball is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] [_inst_2 : ProperSpace.{u1} α _inst_1] {x : α} {r : Real} {s : Set.{u1} α}, (IsClosed.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) s) -> (HasSubset.Subset.{u1} (Set.{u1} α) (Set.hasSubset.{u1} α) s (Metric.ball.{u1} α _inst_1 x r)) -> (Exists.{1} Real (fun (r' : Real) => Exists.{0} (LT.lt.{0} Real Real.hasLt r' r) (fun (H : LT.lt.{0} Real Real.hasLt r' r) => HasSubset.Subset.{u1} (Set.{u1} α) (Set.hasSubset.{u1} α) s (Metric.ball.{u1} α _inst_1 x r'))))
+but is expected to have type
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] [_inst_2 : ProperSpace.{u1} α _inst_1] {x : α} {r : Real} {s : Set.{u1} α}, (IsClosed.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) s) -> (HasSubset.Subset.{u1} (Set.{u1} α) (Set.instHasSubsetSet.{u1} α) s (Metric.ball.{u1} α _inst_1 x r)) -> (Exists.{1} Real (fun (r' : Real) => And (LT.lt.{0} Real Real.instLTReal r' r) (HasSubset.Subset.{u1} (Set.{u1} α) (Set.instHasSubsetSet.{u1} α) s (Metric.ball.{u1} α _inst_1 x r'))))
+Case conversion may be inaccurate. Consider using '#align exists_lt_subset_ball exists_lt_subset_ballₓ'. -/
 /-- If a ball in a proper space includes a closed set `s`, then there exists a ball with the same
 center and a strictly smaller radius that includes `s`. -/
 theorem exists_lt_subset_ball (hs : IsClosed s) (h : s ⊆ ball x r) : ∃ r' < r, s ⊆ ball x r' :=
@@ -2469,10 +4015,12 @@ theorem exists_lt_subset_ball (hs : IsClosed s) (h : s ⊆ ball x r) : ∃ r' <
 
 end ProperSpace
 
+#print IsCompact.isSeparable /-
 theorem IsCompact.isSeparable {s : Set α} (hs : IsCompact s) : IsSeparable s :=
   haveI : CompactSpace s := is_compact_iff_compact_space.mp hs
   is_separable_of_separable_space_subtype s
 #align is_compact.is_separable IsCompact.isSeparable
+-/
 
 namespace Metric
 
@@ -2480,9 +4028,15 @@ section SecondCountable
 
 open TopologicalSpace
 
+/- warning: metric.second_countable_of_almost_dense_set -> Metric.secondCountable_of_almost_dense_set is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α], (forall (ε : Real), (GT.gt.{0} Real Real.hasLt ε (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero)))) -> (Exists.{succ u1} (Set.{u1} α) (fun (s : Set.{u1} α) => And (Set.Countable.{u1} α s) (forall (x : α), Exists.{succ u1} α (fun (y : α) => Exists.{0} (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) y s) (fun (H : Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) y s) => LE.le.{0} Real Real.hasLe (Dist.dist.{u1} α (PseudoMetricSpace.toHasDist.{u1} α _inst_1) x y) ε)))))) -> (TopologicalSpace.SecondCountableTopology.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)))
+but is expected to have type
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α], (forall (ε : Real), (GT.gt.{0} Real Real.instLTReal ε (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal))) -> (Exists.{succ u1} (Set.{u1} α) (fun (s : Set.{u1} α) => And (Set.Countable.{u1} α s) (forall (x : α), Exists.{succ u1} α (fun (y : α) => And (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) y s) (LE.le.{0} Real Real.instLEReal (Dist.dist.{u1} α (PseudoMetricSpace.toDist.{u1} α _inst_1) x y) ε)))))) -> (TopologicalSpace.SecondCountableTopology.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)))
+Case conversion may be inaccurate. Consider using '#align metric.second_countable_of_almost_dense_set Metric.secondCountable_of_almost_dense_setₓ'. -/
 /-- A pseudometric space is second countable if, for every `ε > 0`, there is a countable set which
 is `ε`-dense. -/
-theorem second_countable_of_almost_dense_set
+theorem secondCountable_of_almost_dense_set
     (H : ∀ ε > (0 : ℝ), ∃ s : Set α, s.Countable ∧ ∀ x, ∃ y ∈ s, dist x y ≤ ε) :
     SecondCountableTopology α :=
   by
@@ -2491,12 +4045,18 @@ theorem second_countable_of_almost_dense_set
   choose s hsc y hys hyx using H ε' (by exact_mod_cast ε'0)
   refine' ⟨s, hsc, Union₂_eq_univ_iff.2 fun x => ⟨y x, hys _, le_trans _ ε'ε.le⟩⟩
   exact_mod_cast hyx x
-#align metric.second_countable_of_almost_dense_set Metric.second_countable_of_almost_dense_set
+#align metric.second_countable_of_almost_dense_set Metric.secondCountable_of_almost_dense_set
 
 end SecondCountable
 
 end Metric
 
+/- warning: lebesgue_number_lemma_of_metric -> lebesgue_number_lemma_of_metric is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {s : Set.{u1} α} {ι : Sort.{u2}} {c : ι -> (Set.{u1} α)}, (IsCompact.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) s) -> (forall (i : ι), IsOpen.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) (c i)) -> (HasSubset.Subset.{u1} (Set.{u1} α) (Set.hasSubset.{u1} α) s (Set.unionᵢ.{u1, u2} α ι (fun (i : ι) => c i))) -> (Exists.{1} Real (fun (δ : Real) => Exists.{0} (GT.gt.{0} Real Real.hasLt δ (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero)))) (fun (H : GT.gt.{0} Real Real.hasLt δ (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero)))) => forall (x : α), (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) x s) -> (Exists.{u2} ι (fun (i : ι) => HasSubset.Subset.{u1} (Set.{u1} α) (Set.hasSubset.{u1} α) (Metric.ball.{u1} α _inst_1 x δ) (c i))))))
+but is expected to have type
+  forall {α : Type.{u2}} [_inst_1 : PseudoMetricSpace.{u2} α] {s : Set.{u2} α} {ι : Sort.{u1}} {c : ι -> (Set.{u2} α)}, (IsCompact.{u2} α (UniformSpace.toTopologicalSpace.{u2} α (PseudoMetricSpace.toUniformSpace.{u2} α _inst_1)) s) -> (forall (i : ι), IsOpen.{u2} α (UniformSpace.toTopologicalSpace.{u2} α (PseudoMetricSpace.toUniformSpace.{u2} α _inst_1)) (c i)) -> (HasSubset.Subset.{u2} (Set.{u2} α) (Set.instHasSubsetSet.{u2} α) s (Set.unionᵢ.{u2, u1} α ι (fun (i : ι) => c i))) -> (Exists.{1} Real (fun (δ : Real) => And (GT.gt.{0} Real Real.instLTReal δ (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal))) (forall (x : α), (Membership.mem.{u2, u2} α (Set.{u2} α) (Set.instMembershipSet.{u2} α) x s) -> (Exists.{u1} ι (fun (i : ι) => HasSubset.Subset.{u2} (Set.{u2} α) (Set.instHasSubsetSet.{u2} α) (Metric.ball.{u2} α _inst_1 x δ) (c i))))))
+Case conversion may be inaccurate. Consider using '#align lebesgue_number_lemma_of_metric lebesgue_number_lemma_of_metricₓ'. -/
 theorem lebesgue_number_lemma_of_metric {s : Set α} {ι} {c : ι → Set α} (hs : IsCompact s)
     (hc₁ : ∀ i, IsOpen (c i)) (hc₂ : s ⊆ ⋃ i, c i) : ∃ δ > 0, ∀ x ∈ s, ∃ i, ball x δ ⊆ c i :=
   let ⟨n, en, hn⟩ := lebesgue_number_lemma hs hc₁ hc₂
@@ -2506,6 +4066,12 @@ theorem lebesgue_number_lemma_of_metric {s : Set α} {ι} {c : ι → Set α} (h
     ⟨i, fun y hy => hi (hδ (mem_ball'.mp hy))⟩⟩
 #align lebesgue_number_lemma_of_metric lebesgue_number_lemma_of_metric
 
+/- warning: lebesgue_number_lemma_of_metric_sUnion -> lebesgue_number_lemma_of_metric_unionₛ is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {s : Set.{u1} α} {c : Set.{u1} (Set.{u1} α)}, (IsCompact.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) s) -> (forall (t : Set.{u1} α), (Membership.Mem.{u1, u1} (Set.{u1} α) (Set.{u1} (Set.{u1} α)) (Set.hasMem.{u1} (Set.{u1} α)) t c) -> (IsOpen.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) t)) -> (HasSubset.Subset.{u1} (Set.{u1} α) (Set.hasSubset.{u1} α) s (Set.unionₛ.{u1} α c)) -> (Exists.{1} Real (fun (δ : Real) => Exists.{0} (GT.gt.{0} Real Real.hasLt δ (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero)))) (fun (H : GT.gt.{0} Real Real.hasLt δ (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero)))) => forall (x : α), (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) x s) -> (Exists.{succ u1} (Set.{u1} α) (fun (t : Set.{u1} α) => Exists.{0} (Membership.Mem.{u1, u1} (Set.{u1} α) (Set.{u1} (Set.{u1} α)) (Set.hasMem.{u1} (Set.{u1} α)) t c) (fun (H : Membership.Mem.{u1, u1} (Set.{u1} α) (Set.{u1} (Set.{u1} α)) (Set.hasMem.{u1} (Set.{u1} α)) t c) => HasSubset.Subset.{u1} (Set.{u1} α) (Set.hasSubset.{u1} α) (Metric.ball.{u1} α _inst_1 x δ) t))))))
+but is expected to have type
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {s : Set.{u1} α} {c : Set.{u1} (Set.{u1} α)}, (IsCompact.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) s) -> (forall (t : Set.{u1} α), (Membership.mem.{u1, u1} (Set.{u1} α) (Set.{u1} (Set.{u1} α)) (Set.instMembershipSet.{u1} (Set.{u1} α)) t c) -> (IsOpen.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) t)) -> (HasSubset.Subset.{u1} (Set.{u1} α) (Set.instHasSubsetSet.{u1} α) s (Set.unionₛ.{u1} α c)) -> (Exists.{1} Real (fun (δ : Real) => And (GT.gt.{0} Real Real.instLTReal δ (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal))) (forall (x : α), (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) x s) -> (Exists.{succ u1} (Set.{u1} α) (fun (t : Set.{u1} α) => And (Membership.mem.{u1, u1} (Set.{u1} α) (Set.{u1} (Set.{u1} α)) (Set.instMembershipSet.{u1} (Set.{u1} α)) t c) (HasSubset.Subset.{u1} (Set.{u1} α) (Set.instHasSubsetSet.{u1} α) (Metric.ball.{u1} α _inst_1 x δ) t))))))
+Case conversion may be inaccurate. Consider using '#align lebesgue_number_lemma_of_metric_sUnion lebesgue_number_lemma_of_metric_unionₛₓ'. -/
 theorem lebesgue_number_lemma_of_metric_unionₛ {s : Set α} {c : Set (Set α)} (hs : IsCompact s)
     (hc₁ : ∀ t ∈ c, IsOpen t) (hc₂ : s ⊆ ⋃₀ c) : ∃ δ > 0, ∀ x ∈ s, ∃ t ∈ c, ball x δ ⊆ t := by
   rw [sUnion_eq_Union] at hc₂ <;> simpa using lebesgue_number_lemma_of_metric hs (by simpa) hc₂
@@ -2514,37 +4080,48 @@ theorem lebesgue_number_lemma_of_metric_unionₛ {s : Set α} {c : Set (Set α)}
 namespace Metric
 
 /- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (x y «expr ∈ » s) -/
+#print Metric.Bounded /-
 /-- Boundedness of a subset of a pseudometric space. We formulate the definition to work
 even in the empty space. -/
 def Bounded (s : Set α) : Prop :=
   ∃ C, ∀ (x) (_ : x ∈ s) (y) (_ : y ∈ s), dist x y ≤ C
 #align metric.bounded Metric.Bounded
+-/
 
 section Bounded
 
 variable {x : α} {s t : Set α} {r : ℝ}
 
+#print Metric.bounded_iff_isBounded /-
 theorem bounded_iff_isBounded (s : Set α) : Bounded s ↔ IsBounded s :=
   by
   change bounded s ↔ sᶜ ∈ (cobounded α).sets
   simp [PseudoMetricSpace.cobounded_sets, Metric.Bounded]
 #align metric.bounded_iff_is_bounded Metric.bounded_iff_isBounded
+-/
 
+#print Metric.bounded_empty /-
 @[simp]
 theorem bounded_empty : Bounded (∅ : Set α) :=
   ⟨0, by simp⟩
 #align metric.bounded_empty Metric.bounded_empty
+-/
 
+#print Metric.bounded_iff_mem_bounded /-
 theorem bounded_iff_mem_bounded : Bounded s ↔ ∀ x ∈ s, Bounded s :=
   ⟨fun h _ _ => h, fun H =>
     s.eq_empty_or_nonempty.elim (fun hs => hs.symm ▸ bounded_empty) fun ⟨x, hx⟩ => H x hx⟩
 #align metric.bounded_iff_mem_bounded Metric.bounded_iff_mem_bounded
+-/
 
+#print Metric.Bounded.mono /-
 /-- Subsets of a bounded set are also bounded -/
 theorem Bounded.mono (incl : s ⊆ t) : Bounded t → Bounded s :=
   Exists.imp fun C hC x hx y hy => hC x (incl hx) y (incl hy)
 #align metric.bounded.mono Metric.Bounded.mono
+-/
 
+#print Metric.bounded_closedBall /-
 /-- Closed balls are bounded -/
 theorem bounded_closedBall : Bounded (closedBall x r) :=
   ⟨r + r, fun y hy z hz => by
@@ -2554,17 +4131,23 @@ theorem bounded_closedBall : Bounded (closedBall x r) :=
       _ ≤ r + r := add_le_add hy hz
       ⟩
 #align metric.bounded_closed_ball Metric.bounded_closedBall
+-/
 
+#print Metric.bounded_ball /-
 /-- Open balls are bounded -/
 theorem bounded_ball : Bounded (ball x r) :=
   bounded_closedBall.mono ball_subset_closedBall
 #align metric.bounded_ball Metric.bounded_ball
+-/
 
+#print Metric.bounded_sphere /-
 /-- Spheres are bounded -/
 theorem bounded_sphere : Bounded (sphere x r) :=
   bounded_closedBall.mono sphere_subset_closedBall
 #align metric.bounded_sphere Metric.bounded_sphere
+-/
 
+#print Metric.bounded_iff_subset_ball /-
 /-- Given a point, a bounded subset is included in some ball around this point -/
 theorem bounded_iff_subset_ball (c : α) : Bounded s ↔ ∃ r, s ⊆ closedBall c r :=
   by
@@ -2581,11 +4164,20 @@ theorem bounded_iff_subset_ball (c : α) : Bounded s ↔ ∃ r, s ⊆ closedBall
             ⟩
   · exact bounded_closed_ball.mono hC
 #align metric.bounded_iff_subset_ball Metric.bounded_iff_subset_ball
+-/
 
+#print Metric.Bounded.subset_ball /-
 theorem Bounded.subset_ball (h : Bounded s) (c : α) : ∃ r, s ⊆ closedBall c r :=
   (bounded_iff_subset_ball c).1 h
 #align metric.bounded.subset_ball Metric.Bounded.subset_ball
+-/
 
+/- warning: metric.bounded.subset_ball_lt -> Metric.Bounded.subset_ball_lt is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {s : Set.{u1} α}, (Metric.Bounded.{u1} α _inst_1 s) -> (forall (a : Real) (c : α), Exists.{1} Real (fun (r : Real) => And (LT.lt.{0} Real Real.hasLt a r) (HasSubset.Subset.{u1} (Set.{u1} α) (Set.hasSubset.{u1} α) s (Metric.closedBall.{u1} α _inst_1 c r))))
+but is expected to have type
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {s : Set.{u1} α}, (Metric.Bounded.{u1} α _inst_1 s) -> (forall (a : Real) (c : α), Exists.{1} Real (fun (r : Real) => And (LT.lt.{0} Real Real.instLTReal a r) (HasSubset.Subset.{u1} (Set.{u1} α) (Set.instHasSubsetSet.{u1} α) s (Metric.closedBall.{u1} α _inst_1 c r))))
+Case conversion may be inaccurate. Consider using '#align metric.bounded.subset_ball_lt Metric.Bounded.subset_ball_ltₓ'. -/
 theorem Bounded.subset_ball_lt (h : Bounded s) (a : ℝ) (c : α) : ∃ r, a < r ∧ s ⊆ closedBall c r :=
   by
   rcases h.subset_ball c with ⟨r, hr⟩
@@ -2593,19 +4185,29 @@ theorem Bounded.subset_ball_lt (h : Bounded s) (a : ℝ) (c : α) : ∃ r, a < r
   exact subset.trans hr (closed_ball_subset_closed_ball (le_max_left _ _))
 #align metric.bounded.subset_ball_lt Metric.Bounded.subset_ball_lt
 
+#print Metric.bounded_closure_of_bounded /-
 theorem bounded_closure_of_bounded (h : Bounded s) : Bounded (closure s) :=
   let ⟨C, h⟩ := h
   ⟨C, fun a ha b hb => (isClosed_le' C).closure_subset <| map_mem_closure₂ continuous_dist ha hb h⟩
 #align metric.bounded_closure_of_bounded Metric.bounded_closure_of_bounded
+-/
 
 alias bounded_closure_of_bounded ← bounded.closure
 #align metric.bounded.closure Metric.Bounded.closure
 
+#print Metric.bounded_closure_iff /-
 @[simp]
 theorem bounded_closure_iff : Bounded (closure s) ↔ Bounded s :=
   ⟨fun h => h.mono subset_closure, fun h => h.closure⟩
 #align metric.bounded_closure_iff Metric.bounded_closure_iff
+-/
 
+/- warning: metric.bounded.union -> Metric.Bounded.union is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {s : Set.{u1} α} {t : Set.{u1} α}, (Metric.Bounded.{u1} α _inst_1 s) -> (Metric.Bounded.{u1} α _inst_1 t) -> (Metric.Bounded.{u1} α _inst_1 (Union.union.{u1} (Set.{u1} α) (Set.hasUnion.{u1} α) s t))
+but is expected to have type
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {s : Set.{u1} α} {t : Set.{u1} α}, (Metric.Bounded.{u1} α _inst_1 s) -> (Metric.Bounded.{u1} α _inst_1 t) -> (Metric.Bounded.{u1} α _inst_1 (Union.union.{u1} (Set.{u1} α) (Set.instUnionSet.{u1} α) s t))
+Case conversion may be inaccurate. Consider using '#align metric.bounded.union Metric.Bounded.unionₓ'. -/
 /-- The union of two bounded sets is bounded. -/
 theorem Bounded.union (hs : Bounded s) (ht : Bounded t) : Bounded (s ∪ t) :=
   by
@@ -2618,20 +4220,29 @@ theorem Bounded.union (hs : Bounded s) (ht : Bounded t) : Bounded (s ∪ t) :=
         (subset.trans hCt <| closed_ball_subset_closed_ball <| le_max_right _ _)⟩
 #align metric.bounded.union Metric.Bounded.union
 
+/- warning: metric.bounded_union -> Metric.bounded_union is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {s : Set.{u1} α} {t : Set.{u1} α}, Iff (Metric.Bounded.{u1} α _inst_1 (Union.union.{u1} (Set.{u1} α) (Set.hasUnion.{u1} α) s t)) (And (Metric.Bounded.{u1} α _inst_1 s) (Metric.Bounded.{u1} α _inst_1 t))
+but is expected to have type
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {s : Set.{u1} α} {t : Set.{u1} α}, Iff (Metric.Bounded.{u1} α _inst_1 (Union.union.{u1} (Set.{u1} α) (Set.instUnionSet.{u1} α) s t)) (And (Metric.Bounded.{u1} α _inst_1 s) (Metric.Bounded.{u1} α _inst_1 t))
+Case conversion may be inaccurate. Consider using '#align metric.bounded_union Metric.bounded_unionₓ'. -/
 /-- The union of two sets is bounded iff each of the sets is bounded. -/
 @[simp]
 theorem bounded_union : Bounded (s ∪ t) ↔ Bounded s ∧ Bounded t :=
   ⟨fun h => ⟨h.mono (by simp), h.mono (by simp)⟩, fun h => h.1.union h.2⟩
 #align metric.bounded_union Metric.bounded_union
 
+#print Metric.bounded_bunionᵢ /-
 /-- A finite union of bounded sets is bounded -/
-theorem bounded_bUnion {I : Set β} {s : β → Set α} (H : I.Finite) :
+theorem bounded_bunionᵢ {I : Set β} {s : β → Set α} (H : I.Finite) :
     Bounded (⋃ i ∈ I, s i) ↔ ∀ i ∈ I, Bounded (s i) :=
   Finite.induction_on H (by simp) fun x I _ _ IH => by simp [or_imp, forall_and, IH]
-#align metric.bounded_bUnion Metric.bounded_bUnion
+#align metric.bounded_bUnion Metric.bounded_bunionᵢ
+-/
 
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
+#print Metric.Bounded.prod /-
 protected theorem Bounded.prod [PseudoMetricSpace β] {s : Set α} {t : Set β} (hs : Bounded s)
     (ht : Bounded t) : Bounded (s ×ˢ t) :=
   by
@@ -2645,41 +4256,57 @@ protected theorem Bounded.prod [PseudoMetricSpace β] {s : Set α} {t : Set β}
     prod_mono (hrs.trans <| closed_ball_subset_closed_ball <| le_max_left _ _)
       (hrt.trans <| closed_ball_subset_closed_ball <| le_max_right _ _)
 #align metric.bounded.prod Metric.Bounded.prod
+-/
 
+#print TotallyBounded.bounded /-
 /-- A totally bounded set is bounded -/
 theorem TotallyBounded.bounded {s : Set α} (h : TotallyBounded s) : Bounded s :=
   let-- We cover the totally bounded set by finitely many balls of radius 1,
     -- and then argue that a finite union of bounded sets is bounded
     ⟨t, fint, subs⟩ :=
     (totallyBounded_iff.mp h) 1 zero_lt_one
-  Bounded.mono subs <| (bounded_bUnion fint).2 fun i hi => bounded_ball
+  Bounded.mono subs <| (bounded_bunionᵢ fint).2 fun i hi => bounded_ball
 #align totally_bounded.bounded TotallyBounded.bounded
+-/
 
+#print IsCompact.bounded /-
 /-- A compact set is bounded -/
 theorem IsCompact.bounded {s : Set α} (h : IsCompact s) : Bounded s :=
   -- A compact set is totally bounded, thus bounded
       h.TotallyBounded.Bounded
 #align is_compact.bounded IsCompact.bounded
+-/
 
+#print Metric.bounded_of_finite /-
 /-- A finite set is bounded -/
 theorem bounded_of_finite {s : Set α} (h : s.Finite) : Bounded s :=
   h.IsCompact.Bounded
 #align metric.bounded_of_finite Metric.bounded_of_finite
+-/
 
 alias bounded_of_finite ← _root_.set.finite.bounded
 #align set.finite.bounded Set.Finite.bounded
 
+#print Metric.bounded_singleton /-
 /-- A singleton is bounded -/
 theorem bounded_singleton {x : α} : Bounded ({x} : Set α) :=
   bounded_of_finite <| finite_singleton _
 #align metric.bounded_singleton Metric.bounded_singleton
+-/
 
+/- warning: metric.bounded_range_iff -> Metric.bounded_range_iff is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : PseudoMetricSpace.{u1} α] {f : β -> α}, Iff (Metric.Bounded.{u1} α _inst_1 (Set.range.{u1, succ u2} α β f)) (Exists.{1} Real (fun (C : Real) => forall (x : β) (y : β), LE.le.{0} Real Real.hasLe (Dist.dist.{u1} α (PseudoMetricSpace.toHasDist.{u1} α _inst_1) (f x) (f y)) C))
+but is expected to have type
+  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : PseudoMetricSpace.{u1} α] {f : β -> α}, Iff (Metric.Bounded.{u1} α _inst_1 (Set.range.{u1, succ u2} α β f)) (Exists.{1} Real (fun (C : Real) => forall (x : β) (y : β), LE.le.{0} Real Real.instLEReal (Dist.dist.{u1} α (PseudoMetricSpace.toDist.{u1} α _inst_1) (f x) (f y)) C))
+Case conversion may be inaccurate. Consider using '#align metric.bounded_range_iff Metric.bounded_range_iffₓ'. -/
 /-- Characterization of the boundedness of the range of a function -/
 theorem bounded_range_iff {f : β → α} : Bounded (range f) ↔ ∃ C, ∀ x y, dist (f x) (f y) ≤ C :=
   exists_congr fun C =>
     ⟨fun H x y => H _ ⟨x, rfl⟩ _ ⟨y, rfl⟩, by rintro H _ ⟨x, rfl⟩ _ ⟨y, rfl⟩ <;> exact H x y⟩
 #align metric.bounded_range_iff Metric.bounded_range_iff
 
+#print Metric.bounded_range_of_tendsto_cofinite_uniformity /-
 theorem bounded_range_of_tendsto_cofinite_uniformity {f : β → α}
     (hf : Tendsto (Prod.map f f) (cofinite ×ᶠ cofinite) (𝓤 α)) : Bounded (range f) :=
   by
@@ -2690,32 +4317,49 @@ theorem bounded_range_of_tendsto_cofinite_uniformity {f : β → α}
   rintro _ ⟨x, hx, rfl⟩ _ ⟨y, hy, rfl⟩
   exact le_of_lt (hs1 (x, y) ⟨hx, hy⟩)
 #align metric.bounded_range_of_tendsto_cofinite_uniformity Metric.bounded_range_of_tendsto_cofinite_uniformity
+-/
 
+#print Metric.bounded_range_of_cauchy_map_cofinite /-
 theorem bounded_range_of_cauchy_map_cofinite {f : β → α} (hf : Cauchy (map f cofinite)) :
     Bounded (range f) :=
   bounded_range_of_tendsto_cofinite_uniformity <| (cauchy_map_iff.1 hf).2
 #align metric.bounded_range_of_cauchy_map_cofinite Metric.bounded_range_of_cauchy_map_cofinite
+-/
 
+#print CauchySeq.bounded_range /-
 theorem CauchySeq.bounded_range {f : ℕ → α} (hf : CauchySeq f) : Bounded (range f) :=
   bounded_range_of_cauchy_map_cofinite <| by rwa [Nat.cofinite_eq_atTop]
 #align cauchy_seq.bounded_range CauchySeq.bounded_range
+-/
 
+#print Metric.bounded_range_of_tendsto_cofinite /-
 theorem bounded_range_of_tendsto_cofinite {f : β → α} {a : α} (hf : Tendsto f cofinite (𝓝 a)) :
     Bounded (range f) :=
   bounded_range_of_tendsto_cofinite_uniformity <|
     (hf.Prod_map hf).mono_right <| nhds_prod_eq.symm.trans_le (nhds_le_uniformity a)
 #align metric.bounded_range_of_tendsto_cofinite Metric.bounded_range_of_tendsto_cofinite
+-/
 
+#print Metric.bounded_of_compactSpace /-
 /-- In a compact space, all sets are bounded -/
 theorem bounded_of_compactSpace [CompactSpace α] : Bounded s :=
   isCompact_univ.Bounded.mono (subset_univ _)
 #align metric.bounded_of_compact_space Metric.bounded_of_compactSpace
+-/
 
+#print Metric.bounded_range_of_tendsto /-
 theorem bounded_range_of_tendsto (u : ℕ → α) {x : α} (hu : Tendsto u atTop (𝓝 x)) :
     Bounded (range u) :=
   hu.CauchySeq.bounded_range
 #align metric.bounded_range_of_tendsto Metric.bounded_range_of_tendsto
+-/
 
+/- warning: metric.exists_is_open_bounded_image_inter_of_is_compact_of_forall_continuous_within_at -> Metric.exists_isOpen_bounded_image_inter_of_isCompact_of_forall_continuousWithinAt is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : PseudoMetricSpace.{u1} α] [_inst_2 : TopologicalSpace.{u2} β] {k : Set.{u2} β} {s : Set.{u2} β} {f : β -> α}, (IsCompact.{u2} β _inst_2 k) -> (forall (x : β), (Membership.Mem.{u2, u2} β (Set.{u2} β) (Set.hasMem.{u2} β) x k) -> (ContinuousWithinAt.{u2, u1} β α _inst_2 (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) f s x)) -> (Exists.{succ u2} (Set.{u2} β) (fun (t : Set.{u2} β) => And (HasSubset.Subset.{u2} (Set.{u2} β) (Set.hasSubset.{u2} β) k t) (And (IsOpen.{u2} β _inst_2 t) (Metric.Bounded.{u1} α _inst_1 (Set.image.{u2, u1} β α f (Inter.inter.{u2} (Set.{u2} β) (Set.hasInter.{u2} β) t s))))))
+but is expected to have type
+  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : PseudoMetricSpace.{u1} α] [_inst_2 : TopologicalSpace.{u2} β] {k : Set.{u2} β} {s : Set.{u2} β} {f : β -> α}, (IsCompact.{u2} β _inst_2 k) -> (forall (x : β), (Membership.mem.{u2, u2} β (Set.{u2} β) (Set.instMembershipSet.{u2} β) x k) -> (ContinuousWithinAt.{u2, u1} β α _inst_2 (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) f s x)) -> (Exists.{succ u2} (Set.{u2} β) (fun (t : Set.{u2} β) => And (HasSubset.Subset.{u2} (Set.{u2} β) (Set.instHasSubsetSet.{u2} β) k t) (And (IsOpen.{u2} β _inst_2 t) (Metric.Bounded.{u1} α _inst_1 (Set.image.{u2, u1} β α f (Inter.inter.{u2} (Set.{u2} β) (Set.instInterSet.{u2} β) t s))))))
+Case conversion may be inaccurate. Consider using '#align metric.exists_is_open_bounded_image_inter_of_is_compact_of_forall_continuous_within_at Metric.exists_isOpen_bounded_image_inter_of_isCompact_of_forall_continuousWithinAtₓ'. -/
 /-- If a function is continuous within a set `s` at every point of a compact set `k`, then it is
 bounded on some open neighborhood of `k` in `s`. -/
 theorem exists_isOpen_bounded_image_inter_of_isCompact_of_forall_continuousWithinAt
@@ -2742,6 +4386,7 @@ theorem exists_isOpen_bounded_image_inter_of_isCompact_of_forall_continuousWithi
       exact bounded_ball.mono (image_preimage_subset _ _)
 #align metric.exists_is_open_bounded_image_inter_of_is_compact_of_forall_continuous_within_at Metric.exists_isOpen_bounded_image_inter_of_isCompact_of_forall_continuousWithinAt
 
+#print Metric.exists_isOpen_bounded_image_of_isCompact_of_forall_continuousAt /-
 /-- If a function is continuous at every point of a compact set `k`, then it is bounded on
 some open neighborhood of `k`. -/
 theorem exists_isOpen_bounded_image_of_isCompact_of_forall_continuousAt [TopologicalSpace β]
@@ -2752,7 +4397,14 @@ theorem exists_isOpen_bounded_image_of_isCompact_of_forall_continuousAt [Topolog
   simpa only [inter_univ] using
     exists_is_open_bounded_image_inter_of_is_compact_of_forall_continuous_within_at hk hf
 #align metric.exists_is_open_bounded_image_of_is_compact_of_forall_continuous_at Metric.exists_isOpen_bounded_image_of_isCompact_of_forall_continuousAt
+-/
 
+/- warning: metric.exists_is_open_bounded_image_inter_of_is_compact_of_continuous_on -> Metric.exists_isOpen_bounded_image_inter_of_isCompact_of_continuousOn is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : PseudoMetricSpace.{u1} α] [_inst_2 : TopologicalSpace.{u2} β] {k : Set.{u2} β} {s : Set.{u2} β} {f : β -> α}, (IsCompact.{u2} β _inst_2 k) -> (HasSubset.Subset.{u2} (Set.{u2} β) (Set.hasSubset.{u2} β) k s) -> (ContinuousOn.{u2, u1} β α _inst_2 (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) f s) -> (Exists.{succ u2} (Set.{u2} β) (fun (t : Set.{u2} β) => And (HasSubset.Subset.{u2} (Set.{u2} β) (Set.hasSubset.{u2} β) k t) (And (IsOpen.{u2} β _inst_2 t) (Metric.Bounded.{u1} α _inst_1 (Set.image.{u2, u1} β α f (Inter.inter.{u2} (Set.{u2} β) (Set.hasInter.{u2} β) t s))))))
+but is expected to have type
+  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : PseudoMetricSpace.{u1} α] [_inst_2 : TopologicalSpace.{u2} β] {k : Set.{u2} β} {s : Set.{u2} β} {f : β -> α}, (IsCompact.{u2} β _inst_2 k) -> (HasSubset.Subset.{u2} (Set.{u2} β) (Set.instHasSubsetSet.{u2} β) k s) -> (ContinuousOn.{u2, u1} β α _inst_2 (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) f s) -> (Exists.{succ u2} (Set.{u2} β) (fun (t : Set.{u2} β) => And (HasSubset.Subset.{u2} (Set.{u2} β) (Set.instHasSubsetSet.{u2} β) k t) (And (IsOpen.{u2} β _inst_2 t) (Metric.Bounded.{u1} α _inst_1 (Set.image.{u2, u1} β α f (Inter.inter.{u2} (Set.{u2} β) (Set.instInterSet.{u2} β) t s))))))
+Case conversion may be inaccurate. Consider using '#align metric.exists_is_open_bounded_image_inter_of_is_compact_of_continuous_on Metric.exists_isOpen_bounded_image_inter_of_isCompact_of_continuousOnₓ'. -/
 /-- If a function is continuous on a set `s` containing a compact set `k`, then it is bounded on
 some open neighborhood of `k` in `s`. -/
 theorem exists_isOpen_bounded_image_inter_of_isCompact_of_continuousOn [TopologicalSpace β]
@@ -2762,6 +4414,7 @@ theorem exists_isOpen_bounded_image_inter_of_isCompact_of_continuousOn [Topologi
     hf x (hks hx)
 #align metric.exists_is_open_bounded_image_inter_of_is_compact_of_continuous_on Metric.exists_isOpen_bounded_image_inter_of_isCompact_of_continuousOn
 
+#print Metric.exists_isOpen_bounded_image_of_isCompact_of_continuousOn /-
 /-- If a function is continuous on a neighborhood of a compact set `k`, then it is bounded on
 some open neighborhood of `k`. -/
 theorem exists_isOpen_bounded_image_of_isCompact_of_continuousOn [TopologicalSpace β] {k s : Set β}
@@ -2770,7 +4423,9 @@ theorem exists_isOpen_bounded_image_of_isCompact_of_continuousOn [TopologicalSpa
   exists_isOpen_bounded_image_of_isCompact_of_forall_continuousAt hk fun x hx =>
     hf.ContinuousAt (hs.mem_nhds (hks hx))
 #align metric.exists_is_open_bounded_image_of_is_compact_of_continuous_on Metric.exists_isOpen_bounded_image_of_isCompact_of_continuousOn
+-/
 
+#print Metric.isCompact_of_isClosed_bounded /-
 /-- The **Heine–Borel theorem**: In a proper space, a closed bounded set is compact. -/
 theorem isCompact_of_isClosed_bounded [ProperSpace α] (hc : IsClosed s) (hb : Bounded s) :
     IsCompact s := by
@@ -2779,43 +4434,59 @@ theorem isCompact_of_isClosed_bounded [ProperSpace α] (hc : IsClosed s) (hb : B
   · rcases hb.subset_ball x with ⟨r, hr⟩
     exact isCompact_of_isClosed_subset (is_compact_closed_ball x r) hc hr
 #align metric.is_compact_of_is_closed_bounded Metric.isCompact_of_isClosed_bounded
+-/
 
+#print Metric.Bounded.isCompact_closure /-
 /-- The **Heine–Borel theorem**: In a proper space, the closure of a bounded set is compact. -/
 theorem Bounded.isCompact_closure [ProperSpace α] (h : Bounded s) : IsCompact (closure s) :=
   isCompact_of_isClosed_bounded isClosed_closure h.closure
 #align metric.bounded.is_compact_closure Metric.Bounded.isCompact_closure
+-/
 
+#print Metric.isCompact_iff_isClosed_bounded /-
 /-- The **Heine–Borel theorem**:
 In a proper Hausdorff space, a set is compact if and only if it is closed and bounded. -/
 theorem isCompact_iff_isClosed_bounded [T2Space α] [ProperSpace α] :
     IsCompact s ↔ IsClosed s ∧ Bounded s :=
   ⟨fun h => ⟨h.IsClosed, h.Bounded⟩, fun h => isCompact_of_isClosed_bounded h.1 h.2⟩
 #align metric.is_compact_iff_is_closed_bounded Metric.isCompact_iff_isClosed_bounded
+-/
 
+#print Metric.compactSpace_iff_bounded_univ /-
 theorem compactSpace_iff_bounded_univ [ProperSpace α] : CompactSpace α ↔ Bounded (univ : Set α) :=
   ⟨@bounded_of_compactSpace α _ _, fun hb => ⟨isCompact_of_isClosed_bounded isClosed_univ hb⟩⟩
 #align metric.compact_space_iff_bounded_univ Metric.compactSpace_iff_bounded_univ
+-/
 
 section ConditionallyCompleteLinearOrder
 
 variable [Preorder α] [CompactIccSpace α]
 
+#print Metric.bounded_Icc /-
 theorem bounded_Icc (a b : α) : Bounded (Icc a b) :=
   (totallyBounded_Icc a b).Bounded
 #align metric.bounded_Icc Metric.bounded_Icc
+-/
 
+#print Metric.bounded_Ico /-
 theorem bounded_Ico (a b : α) : Bounded (Ico a b) :=
   (totallyBounded_Ico a b).Bounded
 #align metric.bounded_Ico Metric.bounded_Ico
+-/
 
+#print Metric.bounded_Ioc /-
 theorem bounded_Ioc (a b : α) : Bounded (Ioc a b) :=
   (totallyBounded_Ioc a b).Bounded
 #align metric.bounded_Ioc Metric.bounded_Ioc
+-/
 
+#print Metric.bounded_Ioo /-
 theorem bounded_Ioo (a b : α) : Bounded (Ioo a b) :=
   (totallyBounded_Ioo a b).Bounded
 #align metric.bounded_Ioo Metric.bounded_Ioo
+-/
 
+#print Metric.bounded_of_bddAbove_of_bddBelow /-
 /-- In a pseudo metric space with a conditionally complete linear order such that the order and the
     metric structure give the same topology, any order-bounded set is metric-bounded. -/
 theorem bounded_of_bddAbove_of_bddBelow {s : Set α} (h₁ : BddAbove s) (h₂ : BddBelow s) :
@@ -2824,6 +4495,7 @@ theorem bounded_of_bddAbove_of_bddBelow {s : Set α} (h₁ : BddAbove s) (h₂ :
   let ⟨l, hl⟩ := h₂
   Bounded.mono (fun x hx => mem_Icc.mpr ⟨hl hx, hu hx⟩) (bounded_Icc l u)
 #align metric.bounded_of_bdd_above_of_bdd_below Metric.bounded_of_bddAbove_of_bddBelow
+-/
 
 end ConditionallyCompleteLinearOrder
 
@@ -2833,38 +4505,72 @@ section Diam
 
 variable {s : Set α} {x y z : α}
 
+#print Metric.diam /-
 /-- The diameter of a set in a metric space. To get controllable behavior even when the diameter
 should be infinite, we express it in terms of the emetric.diameter -/
 noncomputable def diam (s : Set α) : ℝ :=
   ENNReal.toReal (EMetric.diam s)
 #align metric.diam Metric.diam
+-/
 
+/- warning: metric.diam_nonneg -> Metric.diam_nonneg is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {s : Set.{u1} α}, LE.le.{0} Real Real.hasLe (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero))) (Metric.diam.{u1} α _inst_1 s)
+but is expected to have type
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {s : Set.{u1} α}, LE.le.{0} Real Real.instLEReal (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal)) (Metric.diam.{u1} α _inst_1 s)
+Case conversion may be inaccurate. Consider using '#align metric.diam_nonneg Metric.diam_nonnegₓ'. -/
 /-- The diameter of a set is always nonnegative -/
 theorem diam_nonneg : 0 ≤ diam s :=
   ENNReal.toReal_nonneg
 #align metric.diam_nonneg Metric.diam_nonneg
 
+/- warning: metric.diam_subsingleton -> Metric.diam_subsingleton is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {s : Set.{u1} α}, (Set.Subsingleton.{u1} α s) -> (Eq.{1} Real (Metric.diam.{u1} α _inst_1 s) (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero))))
+but is expected to have type
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {s : Set.{u1} α}, (Set.Subsingleton.{u1} α s) -> (Eq.{1} Real (Metric.diam.{u1} α _inst_1 s) (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal)))
+Case conversion may be inaccurate. Consider using '#align metric.diam_subsingleton Metric.diam_subsingletonₓ'. -/
 theorem diam_subsingleton (hs : s.Subsingleton) : diam s = 0 := by
   simp only [diam, EMetric.diam_subsingleton hs, ENNReal.zero_toReal]
 #align metric.diam_subsingleton Metric.diam_subsingleton
 
+/- warning: metric.diam_empty -> Metric.diam_empty is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α], Eq.{1} Real (Metric.diam.{u1} α _inst_1 (EmptyCollection.emptyCollection.{u1} (Set.{u1} α) (Set.hasEmptyc.{u1} α))) (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero)))
+but is expected to have type
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α], Eq.{1} Real (Metric.diam.{u1} α _inst_1 (EmptyCollection.emptyCollection.{u1} (Set.{u1} α) (Set.instEmptyCollectionSet.{u1} α))) (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal))
+Case conversion may be inaccurate. Consider using '#align metric.diam_empty Metric.diam_emptyₓ'. -/
 /-- The empty set has zero diameter -/
 @[simp]
 theorem diam_empty : diam (∅ : Set α) = 0 :=
   diam_subsingleton subsingleton_empty
 #align metric.diam_empty Metric.diam_empty
 
+/- warning: metric.diam_singleton -> Metric.diam_singleton is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {x : α}, Eq.{1} Real (Metric.diam.{u1} α _inst_1 (Singleton.singleton.{u1, u1} α (Set.{u1} α) (Set.hasSingleton.{u1} α) x)) (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero)))
+but is expected to have type
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {x : α}, Eq.{1} Real (Metric.diam.{u1} α _inst_1 (Singleton.singleton.{u1, u1} α (Set.{u1} α) (Set.instSingletonSet.{u1} α) x)) (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal))
+Case conversion may be inaccurate. Consider using '#align metric.diam_singleton Metric.diam_singletonₓ'. -/
 /-- A singleton has zero diameter -/
 @[simp]
 theorem diam_singleton : diam ({x} : Set α) = 0 :=
   diam_subsingleton subsingleton_singleton
 #align metric.diam_singleton Metric.diam_singleton
 
+#print Metric.diam_pair /-
 -- Does not work as a simp-lemma, since {x, y} reduces to (insert y {x})
 theorem diam_pair : diam ({x, y} : Set α) = dist x y := by
   simp only [diam, EMetric.diam_pair, dist_edist]
 #align metric.diam_pair Metric.diam_pair
+-/
 
+/- warning: metric.diam_triple -> Metric.diam_triple is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {x : α} {y : α} {z : α}, Eq.{1} Real (Metric.diam.{u1} α _inst_1 (Insert.insert.{u1, u1} α (Set.{u1} α) (Set.hasInsert.{u1} α) x (Insert.insert.{u1, u1} α (Set.{u1} α) (Set.hasInsert.{u1} α) y (Singleton.singleton.{u1, u1} α (Set.{u1} α) (Set.hasSingleton.{u1} α) z)))) (LinearOrder.max.{0} Real Real.linearOrder (LinearOrder.max.{0} Real Real.linearOrder (Dist.dist.{u1} α (PseudoMetricSpace.toHasDist.{u1} α _inst_1) x y) (Dist.dist.{u1} α (PseudoMetricSpace.toHasDist.{u1} α _inst_1) x z)) (Dist.dist.{u1} α (PseudoMetricSpace.toHasDist.{u1} α _inst_1) y z))
+but is expected to have type
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {x : α} {y : α} {z : α}, Eq.{1} Real (Metric.diam.{u1} α _inst_1 (Insert.insert.{u1, u1} α (Set.{u1} α) (Set.instInsertSet.{u1} α) x (Insert.insert.{u1, u1} α (Set.{u1} α) (Set.instInsertSet.{u1} α) y (Singleton.singleton.{u1, u1} α (Set.{u1} α) (Set.instSingletonSet.{u1} α) z)))) (Max.max.{0} Real (LinearOrderedRing.toMax.{0} Real Real.instLinearOrderedRingReal) (Max.max.{0} Real (LinearOrderedRing.toMax.{0} Real Real.instLinearOrderedRingReal) (Dist.dist.{u1} α (PseudoMetricSpace.toDist.{u1} α _inst_1) x y) (Dist.dist.{u1} α (PseudoMetricSpace.toDist.{u1} α _inst_1) x z)) (Dist.dist.{u1} α (PseudoMetricSpace.toDist.{u1} α _inst_1) y z))
+Case conversion may be inaccurate. Consider using '#align metric.diam_triple Metric.diam_tripleₓ'. -/
 -- Does not work as a simp-lemma, since {x, y, z} reduces to (insert z (insert y {x}))
 theorem diam_triple :
     Metric.diam ({x, y, z} : Set α) = max (max (dist x y) (dist x z)) (dist y z) :=
@@ -2873,6 +4579,12 @@ theorem diam_triple :
   rw [ENNReal.toReal_max, ENNReal.toReal_max] <;> apply_rules [ne_of_lt, edist_lt_top, max_lt]
 #align metric.diam_triple Metric.diam_triple
 
+/- warning: metric.ediam_le_of_forall_dist_le -> Metric.ediam_le_of_forall_dist_le is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {s : Set.{u1} α} {C : Real}, (forall (x : α), (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) x s) -> (forall (y : α), (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) y s) -> (LE.le.{0} Real Real.hasLe (Dist.dist.{u1} α (PseudoMetricSpace.toHasDist.{u1} α _inst_1) x y) C))) -> (LE.le.{0} ENNReal (Preorder.toLE.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (CompleteSemilatticeInf.toPartialOrder.{0} ENNReal (CompleteLattice.toCompleteSemilatticeInf.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))))) (EMetric.diam.{u1} α (PseudoMetricSpace.toPseudoEMetricSpace.{u1} α _inst_1) s) (ENNReal.ofReal C))
+but is expected to have type
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {s : Set.{u1} α} {C : Real}, (forall (x : α), (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) x s) -> (forall (y : α), (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) y s) -> (LE.le.{0} Real Real.instLEReal (Dist.dist.{u1} α (PseudoMetricSpace.toDist.{u1} α _inst_1) x y) C))) -> (LE.le.{0} ENNReal (Preorder.toLE.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (CompleteSemilatticeInf.toPartialOrder.{0} ENNReal (CompleteLattice.toCompleteSemilatticeInf.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal))))) (EMetric.diam.{u1} α (PseudoMetricSpace.toPseudoEMetricSpace.{u1} α _inst_1) s) (ENNReal.ofReal C))
+Case conversion may be inaccurate. Consider using '#align metric.ediam_le_of_forall_dist_le Metric.ediam_le_of_forall_dist_leₓ'. -/
 /-- If the distance between any two points in a set is bounded by some constant `C`,
 then `ennreal.of_real C`  bounds the emetric diameter of this set. -/
 theorem ediam_le_of_forall_dist_le {C : ℝ} (h : ∀ x ∈ s, ∀ y ∈ s, dist x y ≤ C) :
@@ -2880,6 +4592,12 @@ theorem ediam_le_of_forall_dist_le {C : ℝ} (h : ∀ x ∈ s, ∀ y ∈ s, dist
   EMetric.diam_le fun x hx y hy => (edist_dist x y).symm ▸ ENNReal.ofReal_le_ofReal (h x hx y hy)
 #align metric.ediam_le_of_forall_dist_le Metric.ediam_le_of_forall_dist_le
 
+/- warning: metric.diam_le_of_forall_dist_le -> Metric.diam_le_of_forall_dist_le is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {s : Set.{u1} α} {C : Real}, (LE.le.{0} Real Real.hasLe (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero))) C) -> (forall (x : α), (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) x s) -> (forall (y : α), (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) y s) -> (LE.le.{0} Real Real.hasLe (Dist.dist.{u1} α (PseudoMetricSpace.toHasDist.{u1} α _inst_1) x y) C))) -> (LE.le.{0} Real Real.hasLe (Metric.diam.{u1} α _inst_1 s) C)
+but is expected to have type
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {s : Set.{u1} α} {C : Real}, (LE.le.{0} Real Real.instLEReal (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal)) C) -> (forall (x : α), (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) x s) -> (forall (y : α), (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) y s) -> (LE.le.{0} Real Real.instLEReal (Dist.dist.{u1} α (PseudoMetricSpace.toDist.{u1} α _inst_1) x y) C))) -> (LE.le.{0} Real Real.instLEReal (Metric.diam.{u1} α _inst_1 s) C)
+Case conversion may be inaccurate. Consider using '#align metric.diam_le_of_forall_dist_le Metric.diam_le_of_forall_dist_leₓ'. -/
 /-- If the distance between any two points in a set is bounded by some non-negative constant,
 this constant bounds the diameter. -/
 theorem diam_le_of_forall_dist_le {C : ℝ} (h₀ : 0 ≤ C) (h : ∀ x ∈ s, ∀ y ∈ s, dist x y ≤ C) :
@@ -2887,6 +4605,12 @@ theorem diam_le_of_forall_dist_le {C : ℝ} (h₀ : 0 ≤ C) (h : ∀ x ∈ s, 
   ENNReal.toReal_le_of_le_ofReal h₀ (ediam_le_of_forall_dist_le h)
 #align metric.diam_le_of_forall_dist_le Metric.diam_le_of_forall_dist_le
 
+/- warning: metric.diam_le_of_forall_dist_le_of_nonempty -> Metric.diam_le_of_forall_dist_le_of_nonempty is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {s : Set.{u1} α}, (Set.Nonempty.{u1} α s) -> (forall {C : Real}, (forall (x : α), (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) x s) -> (forall (y : α), (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) y s) -> (LE.le.{0} Real Real.hasLe (Dist.dist.{u1} α (PseudoMetricSpace.toHasDist.{u1} α _inst_1) x y) C))) -> (LE.le.{0} Real Real.hasLe (Metric.diam.{u1} α _inst_1 s) C))
+but is expected to have type
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {s : Set.{u1} α}, (Set.Nonempty.{u1} α s) -> (forall {C : Real}, (forall (x : α), (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) x s) -> (forall (y : α), (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) y s) -> (LE.le.{0} Real Real.instLEReal (Dist.dist.{u1} α (PseudoMetricSpace.toDist.{u1} α _inst_1) x y) C))) -> (LE.le.{0} Real Real.instLEReal (Metric.diam.{u1} α _inst_1 s) C))
+Case conversion may be inaccurate. Consider using '#align metric.diam_le_of_forall_dist_le_of_nonempty Metric.diam_le_of_forall_dist_le_of_nonemptyₓ'. -/
 /-- If the distance between any two points in a nonempty set is bounded by some constant,
 this constant bounds the diameter. -/
 theorem diam_le_of_forall_dist_le_of_nonempty (hs : s.Nonempty) {C : ℝ}
@@ -2897,6 +4621,12 @@ theorem diam_le_of_forall_dist_le_of_nonempty (hs : s.Nonempty) {C : ℝ}
   diam_le_of_forall_dist_le h₀ h
 #align metric.diam_le_of_forall_dist_le_of_nonempty Metric.diam_le_of_forall_dist_le_of_nonempty
 
+/- warning: metric.dist_le_diam_of_mem' -> Metric.dist_le_diam_of_mem' is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {s : Set.{u1} α} {x : α} {y : α}, (Ne.{1} ENNReal (EMetric.diam.{u1} α (PseudoMetricSpace.toPseudoEMetricSpace.{u1} α _inst_1) s) (Top.top.{0} ENNReal (CompleteLattice.toHasTop.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder)))) -> (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) x s) -> (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) y s) -> (LE.le.{0} Real Real.hasLe (Dist.dist.{u1} α (PseudoMetricSpace.toHasDist.{u1} α _inst_1) x y) (Metric.diam.{u1} α _inst_1 s))
+but is expected to have type
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {s : Set.{u1} α} {x : α} {y : α}, (Ne.{1} ENNReal (EMetric.diam.{u1} α (PseudoMetricSpace.toPseudoEMetricSpace.{u1} α _inst_1) s) (Top.top.{0} ENNReal (CompleteLattice.toTop.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal)))) -> (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) x s) -> (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) y s) -> (LE.le.{0} Real Real.instLEReal (Dist.dist.{u1} α (PseudoMetricSpace.toDist.{u1} α _inst_1) x y) (Metric.diam.{u1} α _inst_1 s))
+Case conversion may be inaccurate. Consider using '#align metric.dist_le_diam_of_mem' Metric.dist_le_diam_of_mem'ₓ'. -/
 /-- The distance between two points in a set is controlled by the diameter of the set. -/
 theorem dist_le_diam_of_mem' (h : EMetric.diam s ≠ ⊤) (hx : x ∈ s) (hy : y ∈ s) :
     dist x y ≤ diam s := by
@@ -2905,6 +4635,12 @@ theorem dist_le_diam_of_mem' (h : EMetric.diam s ≠ ⊤) (hx : x ∈ s) (hy : y
   exact EMetric.edist_le_diam_of_mem hx hy
 #align metric.dist_le_diam_of_mem' Metric.dist_le_diam_of_mem'
 
+/- warning: metric.bounded_iff_ediam_ne_top -> Metric.bounded_iff_ediam_ne_top is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {s : Set.{u1} α}, Iff (Metric.Bounded.{u1} α _inst_1 s) (Ne.{1} ENNReal (EMetric.diam.{u1} α (PseudoMetricSpace.toPseudoEMetricSpace.{u1} α _inst_1) s) (Top.top.{0} ENNReal (CompleteLattice.toHasTop.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))))
+but is expected to have type
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {s : Set.{u1} α}, Iff (Metric.Bounded.{u1} α _inst_1 s) (Ne.{1} ENNReal (EMetric.diam.{u1} α (PseudoMetricSpace.toPseudoEMetricSpace.{u1} α _inst_1) s) (Top.top.{0} ENNReal (CompleteLattice.toTop.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal))))
+Case conversion may be inaccurate. Consider using '#align metric.bounded_iff_ediam_ne_top Metric.bounded_iff_ediam_ne_topₓ'. -/
 /-- Characterize the boundedness of a set in terms of the finiteness of its emetric.diameter. -/
 theorem bounded_iff_ediam_ne_top : Bounded s ↔ EMetric.diam s ≠ ⊤ :=
   Iff.intro
@@ -2912,42 +4648,90 @@ theorem bounded_iff_ediam_ne_top : Bounded s ↔ EMetric.diam s ≠ ⊤ :=
     fun h => ⟨diam s, fun x hx y hy => dist_le_diam_of_mem' h hx hy⟩
 #align metric.bounded_iff_ediam_ne_top Metric.bounded_iff_ediam_ne_top
 
+/- warning: metric.bounded.ediam_ne_top -> Metric.Bounded.ediam_ne_top is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {s : Set.{u1} α}, (Metric.Bounded.{u1} α _inst_1 s) -> (Ne.{1} ENNReal (EMetric.diam.{u1} α (PseudoMetricSpace.toPseudoEMetricSpace.{u1} α _inst_1) s) (Top.top.{0} ENNReal (CompleteLattice.toHasTop.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))))
+but is expected to have type
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {s : Set.{u1} α}, (Metric.Bounded.{u1} α _inst_1 s) -> (Ne.{1} ENNReal (EMetric.diam.{u1} α (PseudoMetricSpace.toPseudoEMetricSpace.{u1} α _inst_1) s) (Top.top.{0} ENNReal (CompleteLattice.toTop.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal))))
+Case conversion may be inaccurate. Consider using '#align metric.bounded.ediam_ne_top Metric.Bounded.ediam_ne_topₓ'. -/
 theorem Bounded.ediam_ne_top (h : Bounded s) : EMetric.diam s ≠ ⊤ :=
   bounded_iff_ediam_ne_top.1 h
 #align metric.bounded.ediam_ne_top Metric.Bounded.ediam_ne_top
 
+/- warning: metric.ediam_univ_eq_top_iff_noncompact -> Metric.ediam_univ_eq_top_iff_noncompact is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] [_inst_2 : ProperSpace.{u1} α _inst_1], Iff (Eq.{1} ENNReal (EMetric.diam.{u1} α (PseudoMetricSpace.toPseudoEMetricSpace.{u1} α _inst_1) (Set.univ.{u1} α)) (Top.top.{0} ENNReal (CompleteLattice.toHasTop.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder)))) (NoncompactSpace.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)))
+but is expected to have type
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] [_inst_2 : ProperSpace.{u1} α _inst_1], Iff (Eq.{1} ENNReal (EMetric.diam.{u1} α (PseudoMetricSpace.toPseudoEMetricSpace.{u1} α _inst_1) (Set.univ.{u1} α)) (Top.top.{0} ENNReal (CompleteLattice.toTop.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal)))) (NoncompactSpace.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)))
+Case conversion may be inaccurate. Consider using '#align metric.ediam_univ_eq_top_iff_noncompact Metric.ediam_univ_eq_top_iff_noncompactₓ'. -/
 theorem ediam_univ_eq_top_iff_noncompact [ProperSpace α] :
     EMetric.diam (univ : Set α) = ∞ ↔ NoncompactSpace α := by
   rw [← not_compactSpace_iff, compact_space_iff_bounded_univ, bounded_iff_ediam_ne_top,
     Classical.not_not]
 #align metric.ediam_univ_eq_top_iff_noncompact Metric.ediam_univ_eq_top_iff_noncompact
 
+/- warning: metric.ediam_univ_of_noncompact -> Metric.ediam_univ_of_noncompact is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] [_inst_2 : ProperSpace.{u1} α _inst_1] [_inst_3 : NoncompactSpace.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1))], Eq.{1} ENNReal (EMetric.diam.{u1} α (PseudoMetricSpace.toPseudoEMetricSpace.{u1} α _inst_1) (Set.univ.{u1} α)) (Top.top.{0} ENNReal (CompleteLattice.toHasTop.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder)))
+but is expected to have type
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] [_inst_2 : ProperSpace.{u1} α _inst_1] [_inst_3 : NoncompactSpace.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1))], Eq.{1} ENNReal (EMetric.diam.{u1} α (PseudoMetricSpace.toPseudoEMetricSpace.{u1} α _inst_1) (Set.univ.{u1} α)) (Top.top.{0} ENNReal (CompleteLattice.toTop.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal)))
+Case conversion may be inaccurate. Consider using '#align metric.ediam_univ_of_noncompact Metric.ediam_univ_of_noncompactₓ'. -/
 @[simp]
 theorem ediam_univ_of_noncompact [ProperSpace α] [NoncompactSpace α] :
     EMetric.diam (univ : Set α) = ∞ :=
   ediam_univ_eq_top_iff_noncompact.mpr ‹_›
 #align metric.ediam_univ_of_noncompact Metric.ediam_univ_of_noncompact
 
+/- warning: metric.diam_univ_of_noncompact -> Metric.diam_univ_of_noncompact is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] [_inst_2 : ProperSpace.{u1} α _inst_1] [_inst_3 : NoncompactSpace.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1))], Eq.{1} Real (Metric.diam.{u1} α _inst_1 (Set.univ.{u1} α)) (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero)))
+but is expected to have type
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] [_inst_2 : ProperSpace.{u1} α _inst_1] [_inst_3 : NoncompactSpace.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1))], Eq.{1} Real (Metric.diam.{u1} α _inst_1 (Set.univ.{u1} α)) (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal))
+Case conversion may be inaccurate. Consider using '#align metric.diam_univ_of_noncompact Metric.diam_univ_of_noncompactₓ'. -/
 @[simp]
 theorem diam_univ_of_noncompact [ProperSpace α] [NoncompactSpace α] : diam (univ : Set α) = 0 := by
   simp [diam]
 #align metric.diam_univ_of_noncompact Metric.diam_univ_of_noncompact
 
+/- warning: metric.dist_le_diam_of_mem -> Metric.dist_le_diam_of_mem is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {s : Set.{u1} α} {x : α} {y : α}, (Metric.Bounded.{u1} α _inst_1 s) -> (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) x s) -> (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) y s) -> (LE.le.{0} Real Real.hasLe (Dist.dist.{u1} α (PseudoMetricSpace.toHasDist.{u1} α _inst_1) x y) (Metric.diam.{u1} α _inst_1 s))
+but is expected to have type
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {s : Set.{u1} α} {x : α} {y : α}, (Metric.Bounded.{u1} α _inst_1 s) -> (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) x s) -> (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) y s) -> (LE.le.{0} Real Real.instLEReal (Dist.dist.{u1} α (PseudoMetricSpace.toDist.{u1} α _inst_1) x y) (Metric.diam.{u1} α _inst_1 s))
+Case conversion may be inaccurate. Consider using '#align metric.dist_le_diam_of_mem Metric.dist_le_diam_of_memₓ'. -/
 /-- The distance between two points in a set is controlled by the diameter of the set. -/
 theorem dist_le_diam_of_mem (h : Bounded s) (hx : x ∈ s) (hy : y ∈ s) : dist x y ≤ diam s :=
   dist_le_diam_of_mem' h.ediam_ne_top hx hy
 #align metric.dist_le_diam_of_mem Metric.dist_le_diam_of_mem
 
+/- warning: metric.ediam_of_unbounded -> Metric.ediam_of_unbounded is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {s : Set.{u1} α}, (Not (Metric.Bounded.{u1} α _inst_1 s)) -> (Eq.{1} ENNReal (EMetric.diam.{u1} α (PseudoMetricSpace.toPseudoEMetricSpace.{u1} α _inst_1) s) (Top.top.{0} ENNReal (CompleteLattice.toHasTop.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))))
+but is expected to have type
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {s : Set.{u1} α}, (Not (Metric.Bounded.{u1} α _inst_1 s)) -> (Eq.{1} ENNReal (EMetric.diam.{u1} α (PseudoMetricSpace.toPseudoEMetricSpace.{u1} α _inst_1) s) (Top.top.{0} ENNReal (CompleteLattice.toTop.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal))))
+Case conversion may be inaccurate. Consider using '#align metric.ediam_of_unbounded Metric.ediam_of_unboundedₓ'. -/
 theorem ediam_of_unbounded (h : ¬Bounded s) : EMetric.diam s = ∞ := by
   rwa [bounded_iff_ediam_ne_top, Classical.not_not] at h
 #align metric.ediam_of_unbounded Metric.ediam_of_unbounded
 
+/- warning: metric.diam_eq_zero_of_unbounded -> Metric.diam_eq_zero_of_unbounded is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {s : Set.{u1} α}, (Not (Metric.Bounded.{u1} α _inst_1 s)) -> (Eq.{1} Real (Metric.diam.{u1} α _inst_1 s) (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero))))
+but is expected to have type
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {s : Set.{u1} α}, (Not (Metric.Bounded.{u1} α _inst_1 s)) -> (Eq.{1} Real (Metric.diam.{u1} α _inst_1 s) (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal)))
+Case conversion may be inaccurate. Consider using '#align metric.diam_eq_zero_of_unbounded Metric.diam_eq_zero_of_unboundedₓ'. -/
 /-- An unbounded set has zero diameter. If you would prefer to get the value ∞, use `emetric.diam`.
 This lemma makes it possible to avoid side conditions in some situations -/
 theorem diam_eq_zero_of_unbounded (h : ¬Bounded s) : diam s = 0 := by
   rw [diam, ediam_of_unbounded h, ENNReal.top_toReal]
 #align metric.diam_eq_zero_of_unbounded Metric.diam_eq_zero_of_unbounded
 
+/- warning: metric.diam_mono -> Metric.diam_mono is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {s : Set.{u1} α} {t : Set.{u1} α}, (HasSubset.Subset.{u1} (Set.{u1} α) (Set.hasSubset.{u1} α) s t) -> (Metric.Bounded.{u1} α _inst_1 t) -> (LE.le.{0} Real Real.hasLe (Metric.diam.{u1} α _inst_1 s) (Metric.diam.{u1} α _inst_1 t))
+but is expected to have type
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {s : Set.{u1} α} {t : Set.{u1} α}, (HasSubset.Subset.{u1} (Set.{u1} α) (Set.instHasSubsetSet.{u1} α) s t) -> (Metric.Bounded.{u1} α _inst_1 t) -> (LE.le.{0} Real Real.instLEReal (Metric.diam.{u1} α _inst_1 s) (Metric.diam.{u1} α _inst_1 t))
+Case conversion may be inaccurate. Consider using '#align metric.diam_mono Metric.diam_monoₓ'. -/
 /-- If `s ⊆ t`, then the diameter of `s` is bounded by that of `t`, provided `t` is bounded. -/
 theorem diam_mono {s t : Set α} (h : s ⊆ t) (ht : Bounded t) : diam s ≤ diam t :=
   by
@@ -2956,6 +4740,12 @@ theorem diam_mono {s t : Set α} (h : s ⊆ t) (ht : Bounded t) : diam s ≤ dia
   exact EMetric.diam_mono h
 #align metric.diam_mono Metric.diam_mono
 
+/- warning: metric.diam_union -> Metric.diam_union is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {s : Set.{u1} α} {x : α} {y : α} {t : Set.{u1} α}, (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) x s) -> (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) y t) -> (LE.le.{0} Real Real.hasLe (Metric.diam.{u1} α _inst_1 (Union.union.{u1} (Set.{u1} α) (Set.hasUnion.{u1} α) s t)) (HAdd.hAdd.{0, 0, 0} Real Real Real (instHAdd.{0} Real Real.hasAdd) (HAdd.hAdd.{0, 0, 0} Real Real Real (instHAdd.{0} Real Real.hasAdd) (Metric.diam.{u1} α _inst_1 s) (Dist.dist.{u1} α (PseudoMetricSpace.toHasDist.{u1} α _inst_1) x y)) (Metric.diam.{u1} α _inst_1 t)))
+but is expected to have type
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {s : Set.{u1} α} {x : α} {y : α} {t : Set.{u1} α}, (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) x s) -> (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) y t) -> (LE.le.{0} Real Real.instLEReal (Metric.diam.{u1} α _inst_1 (Union.union.{u1} (Set.{u1} α) (Set.instUnionSet.{u1} α) s t)) (HAdd.hAdd.{0, 0, 0} Real Real Real (instHAdd.{0} Real Real.instAddReal) (HAdd.hAdd.{0, 0, 0} Real Real Real (instHAdd.{0} Real Real.instAddReal) (Metric.diam.{u1} α _inst_1 s) (Dist.dist.{u1} α (PseudoMetricSpace.toDist.{u1} α _inst_1) x y)) (Metric.diam.{u1} α _inst_1 t)))
+Case conversion may be inaccurate. Consider using '#align metric.diam_union Metric.diam_unionₓ'. -/
 /-- The diameter of a union is controlled by the sum of the diameters, and the distance between
 any two points in each of the sets. This lemma is true without any side condition, since it is
 obviously true if `s ∪ t` is unbounded. -/
@@ -2976,6 +4766,12 @@ theorem diam_union {t : Set α} (xs : x ∈ s) (yt : y ∈ t) :
     apply_rules [add_nonneg, diam_nonneg, dist_nonneg]
 #align metric.diam_union Metric.diam_union
 
+/- warning: metric.diam_union' -> Metric.diam_union' is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {s : Set.{u1} α} {t : Set.{u1} α}, (Set.Nonempty.{u1} α (Inter.inter.{u1} (Set.{u1} α) (Set.hasInter.{u1} α) s t)) -> (LE.le.{0} Real Real.hasLe (Metric.diam.{u1} α _inst_1 (Union.union.{u1} (Set.{u1} α) (Set.hasUnion.{u1} α) s t)) (HAdd.hAdd.{0, 0, 0} Real Real Real (instHAdd.{0} Real Real.hasAdd) (Metric.diam.{u1} α _inst_1 s) (Metric.diam.{u1} α _inst_1 t)))
+but is expected to have type
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {s : Set.{u1} α} {t : Set.{u1} α}, (Set.Nonempty.{u1} α (Inter.inter.{u1} (Set.{u1} α) (Set.instInterSet.{u1} α) s t)) -> (LE.le.{0} Real Real.instLEReal (Metric.diam.{u1} α _inst_1 (Union.union.{u1} (Set.{u1} α) (Set.instUnionSet.{u1} α) s t)) (HAdd.hAdd.{0, 0, 0} Real Real Real (instHAdd.{0} Real Real.instAddReal) (Metric.diam.{u1} α _inst_1 s) (Metric.diam.{u1} α _inst_1 t)))
+Case conversion may be inaccurate. Consider using '#align metric.diam_union' Metric.diam_union'ₓ'. -/
 /-- If two sets intersect, the diameter of the union is bounded by the sum of the diameters. -/
 theorem diam_union' {t : Set α} (h : (s ∩ t).Nonempty) : diam (s ∪ t) ≤ diam s + diam t :=
   by
@@ -2983,6 +4779,12 @@ theorem diam_union' {t : Set α} (h : (s ∩ t).Nonempty) : diam (s ∪ t) ≤ d
   simpa using diam_union xs xt
 #align metric.diam_union' Metric.diam_union'
 
+/- warning: metric.diam_le_of_subset_closed_ball -> Metric.diam_le_of_subset_closedBall is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {s : Set.{u1} α} {x : α} {r : Real}, (LE.le.{0} Real Real.hasLe (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero))) r) -> (HasSubset.Subset.{u1} (Set.{u1} α) (Set.hasSubset.{u1} α) s (Metric.closedBall.{u1} α _inst_1 x r)) -> (LE.le.{0} Real Real.hasLe (Metric.diam.{u1} α _inst_1 s) (HMul.hMul.{0, 0, 0} Real Real Real (instHMul.{0} Real Real.hasMul) (OfNat.ofNat.{0} Real 2 (OfNat.mk.{0} Real 2 (bit0.{0} Real Real.hasAdd (One.one.{0} Real Real.hasOne)))) r))
+but is expected to have type
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {s : Set.{u1} α} {x : α} {r : Real}, (LE.le.{0} Real Real.instLEReal (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal)) r) -> (HasSubset.Subset.{u1} (Set.{u1} α) (Set.instHasSubsetSet.{u1} α) s (Metric.closedBall.{u1} α _inst_1 x r)) -> (LE.le.{0} Real Real.instLEReal (Metric.diam.{u1} α _inst_1 s) (HMul.hMul.{0, 0, 0} Real Real Real (instHMul.{0} Real Real.instMulReal) (OfNat.ofNat.{0} Real 2 (instOfNat.{0} Real 2 Real.natCast (instAtLeastTwoHAddNatInstHAddInstAddNatOfNat (OfNat.ofNat.{0} Nat 0 (instOfNatNat 0))))) r))
+Case conversion may be inaccurate. Consider using '#align metric.diam_le_of_subset_closed_ball Metric.diam_le_of_subset_closedBallₓ'. -/
 theorem diam_le_of_subset_closedBall {r : ℝ} (hr : 0 ≤ r) (h : s ⊆ closedBall x r) :
     diam s ≤ 2 * r :=
   diam_le_of_forall_dist_le (mul_nonneg zero_le_two hr) fun a ha b hb =>
@@ -2993,16 +4795,34 @@ theorem diam_le_of_subset_closedBall {r : ℝ} (hr : 0 ≤ r) (h : s ⊆ closedB
       
 #align metric.diam_le_of_subset_closed_ball Metric.diam_le_of_subset_closedBall
 
+/- warning: metric.diam_closed_ball -> Metric.diam_closedBall is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {x : α} {r : Real}, (LE.le.{0} Real Real.hasLe (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero))) r) -> (LE.le.{0} Real Real.hasLe (Metric.diam.{u1} α _inst_1 (Metric.closedBall.{u1} α _inst_1 x r)) (HMul.hMul.{0, 0, 0} Real Real Real (instHMul.{0} Real Real.hasMul) (OfNat.ofNat.{0} Real 2 (OfNat.mk.{0} Real 2 (bit0.{0} Real Real.hasAdd (One.one.{0} Real Real.hasOne)))) r))
+but is expected to have type
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {x : α} {r : Real}, (LE.le.{0} Real Real.instLEReal (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal)) r) -> (LE.le.{0} Real Real.instLEReal (Metric.diam.{u1} α _inst_1 (Metric.closedBall.{u1} α _inst_1 x r)) (HMul.hMul.{0, 0, 0} Real Real Real (instHMul.{0} Real Real.instMulReal) (OfNat.ofNat.{0} Real 2 (instOfNat.{0} Real 2 Real.natCast (instAtLeastTwoHAddNatInstHAddInstAddNatOfNat (OfNat.ofNat.{0} Nat 0 (instOfNatNat 0))))) r))
+Case conversion may be inaccurate. Consider using '#align metric.diam_closed_ball Metric.diam_closedBallₓ'. -/
 /-- The diameter of a closed ball of radius `r` is at most `2 r`. -/
 theorem diam_closedBall {r : ℝ} (h : 0 ≤ r) : diam (closedBall x r) ≤ 2 * r :=
   diam_le_of_subset_closedBall h Subset.rfl
 #align metric.diam_closed_ball Metric.diam_closedBall
 
+/- warning: metric.diam_ball -> Metric.diam_ball is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {x : α} {r : Real}, (LE.le.{0} Real Real.hasLe (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero))) r) -> (LE.le.{0} Real Real.hasLe (Metric.diam.{u1} α _inst_1 (Metric.ball.{u1} α _inst_1 x r)) (HMul.hMul.{0, 0, 0} Real Real Real (instHMul.{0} Real Real.hasMul) (OfNat.ofNat.{0} Real 2 (OfNat.mk.{0} Real 2 (bit0.{0} Real Real.hasAdd (One.one.{0} Real Real.hasOne)))) r))
+but is expected to have type
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {x : α} {r : Real}, (LE.le.{0} Real Real.instLEReal (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal)) r) -> (LE.le.{0} Real Real.instLEReal (Metric.diam.{u1} α _inst_1 (Metric.ball.{u1} α _inst_1 x r)) (HMul.hMul.{0, 0, 0} Real Real Real (instHMul.{0} Real Real.instMulReal) (OfNat.ofNat.{0} Real 2 (instOfNat.{0} Real 2 Real.natCast (instAtLeastTwoHAddNatInstHAddInstAddNatOfNat (OfNat.ofNat.{0} Nat 0 (instOfNatNat 0))))) r))
+Case conversion may be inaccurate. Consider using '#align metric.diam_ball Metric.diam_ballₓ'. -/
 /-- The diameter of a ball of radius `r` is at most `2 r`. -/
 theorem diam_ball {r : ℝ} (h : 0 ≤ r) : diam (ball x r) ≤ 2 * r :=
   diam_le_of_subset_closedBall h ball_subset_closedBall
 #align metric.diam_ball Metric.diam_ball
 
+/- warning: is_complete.nonempty_Inter_of_nonempty_bInter -> IsComplete.nonempty_interᵢ_of_nonempty_bInter is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {s : Nat -> (Set.{u1} α)}, (IsComplete.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1) (s (OfNat.ofNat.{0} Nat 0 (OfNat.mk.{0} Nat 0 (Zero.zero.{0} Nat Nat.hasZero))))) -> (forall (n : Nat), IsClosed.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) (s n)) -> (forall (n : Nat), Metric.Bounded.{u1} α _inst_1 (s n)) -> (forall (N : Nat), Set.Nonempty.{u1} α (Set.interᵢ.{u1, 1} α Nat (fun (n : Nat) => Set.interᵢ.{u1, 0} α (LE.le.{0} Nat Nat.hasLe n N) (fun (H : LE.le.{0} Nat Nat.hasLe n N) => s n)))) -> (Filter.Tendsto.{0, 0} Nat Real (fun (n : Nat) => Metric.diam.{u1} α _inst_1 (s n)) (Filter.atTop.{0} Nat (PartialOrder.toPreorder.{0} Nat (OrderedCancelAddCommMonoid.toPartialOrder.{0} Nat (StrictOrderedSemiring.toOrderedCancelAddCommMonoid.{0} Nat Nat.strictOrderedSemiring)))) (nhds.{0} Real (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero))))) -> (Set.Nonempty.{u1} α (Set.interᵢ.{u1, 1} α Nat (fun (n : Nat) => s n)))
+but is expected to have type
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {s : Nat -> (Set.{u1} α)}, (IsComplete.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1) (s (OfNat.ofNat.{0} Nat 0 (instOfNatNat 0)))) -> (forall (n : Nat), IsClosed.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) (s n)) -> (forall (n : Nat), Metric.Bounded.{u1} α _inst_1 (s n)) -> (forall (N : Nat), Set.Nonempty.{u1} α (Set.interᵢ.{u1, 1} α Nat (fun (n : Nat) => Set.interᵢ.{u1, 0} α (LE.le.{0} Nat instLENat n N) (fun (H : LE.le.{0} Nat instLENat n N) => s n)))) -> (Filter.Tendsto.{0, 0} Nat Real (fun (n : Nat) => Metric.diam.{u1} α _inst_1 (s n)) (Filter.atTop.{0} Nat (PartialOrder.toPreorder.{0} Nat (StrictOrderedSemiring.toPartialOrder.{0} Nat Nat.strictOrderedSemiring))) (nhds.{0} Real (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal)))) -> (Set.Nonempty.{u1} α (Set.interᵢ.{u1, 1} α Nat (fun (n : Nat) => s n)))
+Case conversion may be inaccurate. Consider using '#align is_complete.nonempty_Inter_of_nonempty_bInter IsComplete.nonempty_interᵢ_of_nonempty_bInterₓ'. -/
 /-- If a family of complete sets with diameter tending to `0` is such that each finite intersection
 is nonempty, then the total intersection is also nonempty. -/
 theorem IsComplete.nonempty_interᵢ_of_nonempty_bInter {s : ℕ → Set α} (h0 : IsComplete (s 0))
@@ -3029,6 +4849,12 @@ theorem IsComplete.nonempty_interᵢ_of_nonempty_bInter {s : ℕ → Set α} (h0
   exact I n p hp
 #align is_complete.nonempty_Inter_of_nonempty_bInter IsComplete.nonempty_interᵢ_of_nonempty_bInter
 
+/- warning: metric.nonempty_Inter_of_nonempty_bInter -> Metric.nonempty_interᵢ_of_nonempty_bInter is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] [_inst_2 : CompleteSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)] {s : Nat -> (Set.{u1} α)}, (forall (n : Nat), IsClosed.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) (s n)) -> (forall (n : Nat), Metric.Bounded.{u1} α _inst_1 (s n)) -> (forall (N : Nat), Set.Nonempty.{u1} α (Set.interᵢ.{u1, 1} α Nat (fun (n : Nat) => Set.interᵢ.{u1, 0} α (LE.le.{0} Nat Nat.hasLe n N) (fun (H : LE.le.{0} Nat Nat.hasLe n N) => s n)))) -> (Filter.Tendsto.{0, 0} Nat Real (fun (n : Nat) => Metric.diam.{u1} α _inst_1 (s n)) (Filter.atTop.{0} Nat (PartialOrder.toPreorder.{0} Nat (OrderedCancelAddCommMonoid.toPartialOrder.{0} Nat (StrictOrderedSemiring.toOrderedCancelAddCommMonoid.{0} Nat Nat.strictOrderedSemiring)))) (nhds.{0} Real (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero))))) -> (Set.Nonempty.{u1} α (Set.interᵢ.{u1, 1} α Nat (fun (n : Nat) => s n)))
+but is expected to have type
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] [_inst_2 : CompleteSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)] {s : Nat -> (Set.{u1} α)}, (forall (n : Nat), IsClosed.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) (s n)) -> (forall (n : Nat), Metric.Bounded.{u1} α _inst_1 (s n)) -> (forall (N : Nat), Set.Nonempty.{u1} α (Set.interᵢ.{u1, 1} α Nat (fun (n : Nat) => Set.interᵢ.{u1, 0} α (LE.le.{0} Nat instLENat n N) (fun (H : LE.le.{0} Nat instLENat n N) => s n)))) -> (Filter.Tendsto.{0, 0} Nat Real (fun (n : Nat) => Metric.diam.{u1} α _inst_1 (s n)) (Filter.atTop.{0} Nat (PartialOrder.toPreorder.{0} Nat (StrictOrderedSemiring.toPartialOrder.{0} Nat Nat.strictOrderedSemiring))) (nhds.{0} Real (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal)))) -> (Set.Nonempty.{u1} α (Set.interᵢ.{u1, 1} α Nat (fun (n : Nat) => s n)))
+Case conversion may be inaccurate. Consider using '#align metric.nonempty_Inter_of_nonempty_bInter Metric.nonempty_interᵢ_of_nonempty_bInterₓ'. -/
 /-- In a complete space, if a family of closed sets with diameter tending to `0` is such that each
 finite intersection is nonempty, then the total intersection is also nonempty. -/
 theorem nonempty_interᵢ_of_nonempty_bInter [CompleteSpace α] {s : ℕ → Set α}
@@ -3039,6 +4865,12 @@ theorem nonempty_interᵢ_of_nonempty_bInter [CompleteSpace α] {s : ℕ → Set
 
 end Diam
 
+/- warning: metric.exists_local_min_mem_ball -> Metric.exists_local_min_mem_ball is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : PseudoMetricSpace.{u1} α] [_inst_2 : ProperSpace.{u1} α _inst_1] [_inst_3 : TopologicalSpace.{u2} β] [_inst_4 : ConditionallyCompleteLinearOrder.{u2} β] [_inst_5 : OrderTopology.{u2} β _inst_3 (PartialOrder.toPreorder.{u2} β (SemilatticeInf.toPartialOrder.{u2} β (Lattice.toSemilatticeInf.{u2} β (ConditionallyCompleteLattice.toLattice.{u2} β (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u2} β _inst_4)))))] {f : α -> β} {a : α} {z : α} {r : Real}, (ContinuousOn.{u1, u2} α β (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) _inst_3 f (Metric.closedBall.{u1} α _inst_1 a r)) -> (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) z (Metric.closedBall.{u1} α _inst_1 a r)) -> (forall (z' : α), (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) z' (Metric.sphere.{u1} α _inst_1 a r)) -> (LT.lt.{u2} β (Preorder.toLT.{u2} β (PartialOrder.toPreorder.{u2} β (SemilatticeInf.toPartialOrder.{u2} β (Lattice.toSemilatticeInf.{u2} β (ConditionallyCompleteLattice.toLattice.{u2} β (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u2} β _inst_4)))))) (f z) (f z'))) -> (Exists.{succ u1} α (fun (z : α) => Exists.{0} (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) z (Metric.ball.{u1} α _inst_1 a r)) (fun (H : Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) z (Metric.ball.{u1} α _inst_1 a r)) => IsLocalMin.{u1, u2} α β (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) (PartialOrder.toPreorder.{u2} β (SemilatticeInf.toPartialOrder.{u2} β (Lattice.toSemilatticeInf.{u2} β (ConditionallyCompleteLattice.toLattice.{u2} β (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u2} β _inst_4))))) f z)))
+but is expected to have type
+  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : PseudoMetricSpace.{u1} α] [_inst_2 : ProperSpace.{u1} α _inst_1] [_inst_3 : TopologicalSpace.{u2} β] [_inst_4 : ConditionallyCompleteLinearOrder.{u2} β] [_inst_5 : OrderTopology.{u2} β _inst_3 (PartialOrder.toPreorder.{u2} β (SemilatticeInf.toPartialOrder.{u2} β (Lattice.toSemilatticeInf.{u2} β (ConditionallyCompleteLattice.toLattice.{u2} β (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u2} β _inst_4)))))] {f : α -> β} {a : α} {z : α} {r : Real}, (ContinuousOn.{u1, u2} α β (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) _inst_3 f (Metric.closedBall.{u1} α _inst_1 a r)) -> (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) z (Metric.closedBall.{u1} α _inst_1 a r)) -> (forall (z' : α), (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) z' (Metric.sphere.{u1} α _inst_1 a r)) -> (LT.lt.{u2} β (Preorder.toLT.{u2} β (PartialOrder.toPreorder.{u2} β (SemilatticeInf.toPartialOrder.{u2} β (Lattice.toSemilatticeInf.{u2} β (ConditionallyCompleteLattice.toLattice.{u2} β (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u2} β _inst_4)))))) (f z) (f z'))) -> (Exists.{succ u1} α (fun (z : α) => And (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) z (Metric.ball.{u1} α _inst_1 a r)) (IsLocalMin.{u1, u2} α β (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) (PartialOrder.toPreorder.{u2} β (SemilatticeInf.toPartialOrder.{u2} β (Lattice.toSemilatticeInf.{u2} β (ConditionallyCompleteLattice.toLattice.{u2} β (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u2} β _inst_4))))) f z)))
+Case conversion may be inaccurate. Consider using '#align metric.exists_local_min_mem_ball Metric.exists_local_min_mem_ballₓ'. -/
 theorem exists_local_min_mem_ball [ProperSpace α] [TopologicalSpace β]
     [ConditionallyCompleteLinearOrder β] [OrderTopology β] {f : α → β} {a z : α} {r : ℝ}
     (hf : ContinuousOn f (closedBall a r)) (hz : z ∈ closedBall a r)
@@ -3065,6 +4897,12 @@ unsafe def positivity_diam : expr → tactic strictness
 
 end Tactic
 
+/- warning: comap_dist_right_at_top_le_cocompact -> comap_dist_right_atTop_le_cocompact is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] (x : α), LE.le.{u1} (Filter.{u1} α) (Preorder.toLE.{u1} (Filter.{u1} α) (PartialOrder.toPreorder.{u1} (Filter.{u1} α) (Filter.partialOrder.{u1} α))) (Filter.comap.{u1, 0} α Real (fun (y : α) => Dist.dist.{u1} α (PseudoMetricSpace.toHasDist.{u1} α _inst_1) y x) (Filter.atTop.{0} Real Real.preorder)) (Filter.cocompact.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)))
+but is expected to have type
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] (x : α), LE.le.{u1} (Filter.{u1} α) (Preorder.toLE.{u1} (Filter.{u1} α) (PartialOrder.toPreorder.{u1} (Filter.{u1} α) (Filter.instPartialOrderFilter.{u1} α))) (Filter.comap.{u1, 0} α Real (fun (y : α) => Dist.dist.{u1} α (PseudoMetricSpace.toDist.{u1} α _inst_1) y x) (Filter.atTop.{0} Real Real.instPreorderReal)) (Filter.cocompact.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)))
+Case conversion may be inaccurate. Consider using '#align comap_dist_right_at_top_le_cocompact comap_dist_right_atTop_le_cocompactₓ'. -/
 theorem comap_dist_right_atTop_le_cocompact (x : α) :
     comap (fun y => dist y x) atTop ≤ cocompact α :=
   by
@@ -3073,33 +4911,48 @@ theorem comap_dist_right_atTop_le_cocompact (x : α) :
   exact ⟨Ioi r, Ioi_mem_at_top r, fun y hy hys => (mem_closed_ball.1 <| hr hys).not_lt hy⟩
 #align comap_dist_right_at_top_le_cocompact comap_dist_right_atTop_le_cocompact
 
+/- warning: comap_dist_left_at_top_le_cocompact -> comap_dist_left_atTop_le_cocompact is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] (x : α), LE.le.{u1} (Filter.{u1} α) (Preorder.toLE.{u1} (Filter.{u1} α) (PartialOrder.toPreorder.{u1} (Filter.{u1} α) (Filter.partialOrder.{u1} α))) (Filter.comap.{u1, 0} α Real (Dist.dist.{u1} α (PseudoMetricSpace.toHasDist.{u1} α _inst_1) x) (Filter.atTop.{0} Real Real.preorder)) (Filter.cocompact.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)))
+but is expected to have type
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] (x : α), LE.le.{u1} (Filter.{u1} α) (Preorder.toLE.{u1} (Filter.{u1} α) (PartialOrder.toPreorder.{u1} (Filter.{u1} α) (Filter.instPartialOrderFilter.{u1} α))) (Filter.comap.{u1, 0} α Real (Dist.dist.{u1} α (PseudoMetricSpace.toDist.{u1} α _inst_1) x) (Filter.atTop.{0} Real Real.instPreorderReal)) (Filter.cocompact.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)))
+Case conversion may be inaccurate. Consider using '#align comap_dist_left_at_top_le_cocompact comap_dist_left_atTop_le_cocompactₓ'. -/
 theorem comap_dist_left_atTop_le_cocompact (x : α) : comap (dist x) atTop ≤ cocompact α := by
   simpa only [dist_comm _ x] using comap_dist_right_atTop_le_cocompact x
 #align comap_dist_left_at_top_le_cocompact comap_dist_left_atTop_le_cocompact
 
+#print comap_dist_right_atTop_eq_cocompact /-
 theorem comap_dist_right_atTop_eq_cocompact [ProperSpace α] (x : α) :
     comap (fun y => dist y x) atTop = cocompact α :=
   (comap_dist_right_atTop_le_cocompact x).antisymm <|
     (tendsto_dist_right_cocompact_atTop x).le_comap
 #align comap_dist_right_at_top_eq_cocompact comap_dist_right_atTop_eq_cocompact
+-/
 
+#print comap_dist_left_atTop_eq_cocompact /-
 theorem comap_dist_left_atTop_eq_cocompact [ProperSpace α] (x : α) :
     comap (dist x) atTop = cocompact α :=
   (comap_dist_left_atTop_le_cocompact x).antisymm <| (tendsto_dist_left_cocompact_atTop x).le_comap
 #align comap_dist_left_at_top_eq_cocompact comap_dist_left_atTop_eq_cocompact
+-/
 
+#print tendsto_cocompact_of_tendsto_dist_comp_atTop /-
 theorem tendsto_cocompact_of_tendsto_dist_comp_atTop {f : β → α} {l : Filter β} (x : α)
     (h : Tendsto (fun y => dist (f y) x) l atTop) : Tendsto f l (cocompact α) :=
   by
   refine' tendsto.mono_right _ (comap_dist_right_atTop_le_cocompact x)
   rwa [tendsto_comap_iff]
 #align tendsto_cocompact_of_tendsto_dist_comp_at_top tendsto_cocompact_of_tendsto_dist_comp_atTop
+-/
 
+#print MetricSpace /-
 /-- We now define `metric_space`, extending `pseudo_metric_space`. -/
 class MetricSpace (α : Type u) extends PseudoMetricSpace α : Type u where
   eq_of_dist_eq_zero : ∀ {x y : α}, dist x y = 0 → x = y
 #align metric_space MetricSpace
+-/
 
+#print MetricSpace.ext /-
 /-- Two metric space structures with the same distance coincide. -/
 @[ext]
 theorem MetricSpace.ext {α : Type _} {m m' : MetricSpace α} (h : m.toHasDist = m'.toHasDist) :
@@ -3111,7 +4964,14 @@ theorem MetricSpace.ext {α : Type _} {m m' : MetricSpace α} (h : m.toHasDist =
   dsimp at h'
   subst h'
 #align metric_space.ext MetricSpace.ext
+-/
 
+/- warning: metric_space.of_dist_topology -> MetricSpace.ofDistTopology is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} [_inst_2 : TopologicalSpace.{u1} α] (dist : α -> α -> Real), (forall (x : α), Eq.{1} Real (dist x x) (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero)))) -> (forall (x : α) (y : α), Eq.{1} Real (dist x y) (dist y x)) -> (forall (x : α) (y : α) (z : α), LE.le.{0} Real Real.hasLe (dist x z) (HAdd.hAdd.{0, 0, 0} Real Real Real (instHAdd.{0} Real Real.hasAdd) (dist x y) (dist y z))) -> (forall (s : Set.{u1} α), Iff (IsOpen.{u1} α _inst_2 s) (forall (x : α), (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) x s) -> (Exists.{1} Real (fun (ε : Real) => Exists.{0} (GT.gt.{0} Real Real.hasLt ε (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero)))) (fun (H : GT.gt.{0} Real Real.hasLt ε (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero)))) => forall (y : α), (LT.lt.{0} Real Real.hasLt (dist x y) ε) -> (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) y s)))))) -> (forall (x : α) (y : α), (Eq.{1} Real (dist x y) (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero)))) -> (Eq.{succ u1} α x y)) -> (MetricSpace.{u1} α)
+but is expected to have type
+  forall {α : Type.{u1}} [_inst_2 : TopologicalSpace.{u1} α] (dist : α -> α -> Real), (forall (x : α), Eq.{1} Real (dist x x) (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal))) -> (forall (x : α) (y : α), Eq.{1} Real (dist x y) (dist y x)) -> (forall (x : α) (y : α) (z : α), LE.le.{0} Real Real.instLEReal (dist x z) (HAdd.hAdd.{0, 0, 0} Real Real Real (instHAdd.{0} Real Real.instAddReal) (dist x y) (dist y z))) -> (forall (s : Set.{u1} α), Iff (IsOpen.{u1} α _inst_2 s) (forall (x : α), (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) x s) -> (Exists.{1} Real (fun (ε : Real) => And (GT.gt.{0} Real Real.instLTReal ε (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal))) (forall (y : α), (LT.lt.{0} Real Real.instLTReal (dist x y) ε) -> (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) y s)))))) -> (forall (x : α) (y : α), (Eq.{1} Real (dist x y) (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal))) -> (Eq.{succ u1} α x y)) -> (MetricSpace.{u1} α)
+Case conversion may be inaccurate. Consider using '#align metric_space.of_dist_topology MetricSpace.ofDistTopologyₓ'. -/
 /-- Construct a metric space structure whose underlying topological space structure
 (definitionally) agrees which a pre-existing topology which is compatible with a given distance
 function. -/
@@ -3126,48 +4986,108 @@ def MetricSpace.ofDistTopology {α : Type u} [TopologicalSpace α] (dist : α 
 
 variable {γ : Type w} [MetricSpace γ]
 
+/- warning: eq_of_dist_eq_zero -> eq_of_dist_eq_zero is a dubious translation:
+lean 3 declaration is
+  forall {γ : Type.{u1}} [_inst_2 : MetricSpace.{u1} γ] {x : γ} {y : γ}, (Eq.{1} Real (Dist.dist.{u1} γ (PseudoMetricSpace.toHasDist.{u1} γ (MetricSpace.toPseudoMetricSpace.{u1} γ _inst_2)) x y) (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero)))) -> (Eq.{succ u1} γ x y)
+but is expected to have type
+  forall {γ : Type.{u1}} [_inst_2 : MetricSpace.{u1} γ] {x : γ} {y : γ}, (Eq.{1} Real (Dist.dist.{u1} γ (PseudoMetricSpace.toDist.{u1} γ (MetricSpace.toPseudoMetricSpace.{u1} γ _inst_2)) x y) (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal))) -> (Eq.{succ u1} γ x y)
+Case conversion may be inaccurate. Consider using '#align eq_of_dist_eq_zero eq_of_dist_eq_zeroₓ'. -/
 theorem eq_of_dist_eq_zero {x y : γ} : dist x y = 0 → x = y :=
   MetricSpace.eq_of_dist_eq_zero
 #align eq_of_dist_eq_zero eq_of_dist_eq_zero
 
+/- warning: dist_eq_zero -> dist_eq_zero is a dubious translation:
+lean 3 declaration is
+  forall {γ : Type.{u1}} [_inst_2 : MetricSpace.{u1} γ] {x : γ} {y : γ}, Iff (Eq.{1} Real (Dist.dist.{u1} γ (PseudoMetricSpace.toHasDist.{u1} γ (MetricSpace.toPseudoMetricSpace.{u1} γ _inst_2)) x y) (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero)))) (Eq.{succ u1} γ x y)
+but is expected to have type
+  forall {γ : Type.{u1}} [_inst_2 : MetricSpace.{u1} γ] {x : γ} {y : γ}, Iff (Eq.{1} Real (Dist.dist.{u1} γ (PseudoMetricSpace.toDist.{u1} γ (MetricSpace.toPseudoMetricSpace.{u1} γ _inst_2)) x y) (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal))) (Eq.{succ u1} γ x y)
+Case conversion may be inaccurate. Consider using '#align dist_eq_zero dist_eq_zeroₓ'. -/
 @[simp]
 theorem dist_eq_zero {x y : γ} : dist x y = 0 ↔ x = y :=
   Iff.intro eq_of_dist_eq_zero fun this : x = y => this ▸ dist_self _
 #align dist_eq_zero dist_eq_zero
 
+/- warning: zero_eq_dist -> zero_eq_dist is a dubious translation:
+lean 3 declaration is
+  forall {γ : Type.{u1}} [_inst_2 : MetricSpace.{u1} γ] {x : γ} {y : γ}, Iff (Eq.{1} Real (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero))) (Dist.dist.{u1} γ (PseudoMetricSpace.toHasDist.{u1} γ (MetricSpace.toPseudoMetricSpace.{u1} γ _inst_2)) x y)) (Eq.{succ u1} γ x y)
+but is expected to have type
+  forall {γ : Type.{u1}} [_inst_2 : MetricSpace.{u1} γ] {x : γ} {y : γ}, Iff (Eq.{1} Real (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal)) (Dist.dist.{u1} γ (PseudoMetricSpace.toDist.{u1} γ (MetricSpace.toPseudoMetricSpace.{u1} γ _inst_2)) x y)) (Eq.{succ u1} γ x y)
+Case conversion may be inaccurate. Consider using '#align zero_eq_dist zero_eq_distₓ'. -/
 @[simp]
 theorem zero_eq_dist {x y : γ} : 0 = dist x y ↔ x = y := by rw [eq_comm, dist_eq_zero]
 #align zero_eq_dist zero_eq_dist
 
+/- warning: dist_ne_zero -> dist_ne_zero is a dubious translation:
+lean 3 declaration is
+  forall {γ : Type.{u1}} [_inst_2 : MetricSpace.{u1} γ] {x : γ} {y : γ}, Iff (Ne.{1} Real (Dist.dist.{u1} γ (PseudoMetricSpace.toHasDist.{u1} γ (MetricSpace.toPseudoMetricSpace.{u1} γ _inst_2)) x y) (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero)))) (Ne.{succ u1} γ x y)
+but is expected to have type
+  forall {γ : Type.{u1}} [_inst_2 : MetricSpace.{u1} γ] {x : γ} {y : γ}, Iff (Ne.{1} Real (Dist.dist.{u1} γ (PseudoMetricSpace.toDist.{u1} γ (MetricSpace.toPseudoMetricSpace.{u1} γ _inst_2)) x y) (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal))) (Ne.{succ u1} γ x y)
+Case conversion may be inaccurate. Consider using '#align dist_ne_zero dist_ne_zeroₓ'. -/
 theorem dist_ne_zero {x y : γ} : dist x y ≠ 0 ↔ x ≠ y := by
   simpa only [not_iff_not] using dist_eq_zero
 #align dist_ne_zero dist_ne_zero
 
+/- warning: dist_le_zero -> dist_le_zero is a dubious translation:
+lean 3 declaration is
+  forall {γ : Type.{u1}} [_inst_2 : MetricSpace.{u1} γ] {x : γ} {y : γ}, Iff (LE.le.{0} Real Real.hasLe (Dist.dist.{u1} γ (PseudoMetricSpace.toHasDist.{u1} γ (MetricSpace.toPseudoMetricSpace.{u1} γ _inst_2)) x y) (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero)))) (Eq.{succ u1} γ x y)
+but is expected to have type
+  forall {γ : Type.{u1}} [_inst_2 : MetricSpace.{u1} γ] {x : γ} {y : γ}, Iff (LE.le.{0} Real Real.instLEReal (Dist.dist.{u1} γ (PseudoMetricSpace.toDist.{u1} γ (MetricSpace.toPseudoMetricSpace.{u1} γ _inst_2)) x y) (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal))) (Eq.{succ u1} γ x y)
+Case conversion may be inaccurate. Consider using '#align dist_le_zero dist_le_zeroₓ'. -/
 @[simp]
 theorem dist_le_zero {x y : γ} : dist x y ≤ 0 ↔ x = y := by
   simpa [le_antisymm_iff, dist_nonneg] using @dist_eq_zero _ _ x y
 #align dist_le_zero dist_le_zero
 
+/- warning: dist_pos -> dist_pos is a dubious translation:
+lean 3 declaration is
+  forall {γ : Type.{u1}} [_inst_2 : MetricSpace.{u1} γ] {x : γ} {y : γ}, Iff (LT.lt.{0} Real Real.hasLt (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero))) (Dist.dist.{u1} γ (PseudoMetricSpace.toHasDist.{u1} γ (MetricSpace.toPseudoMetricSpace.{u1} γ _inst_2)) x y)) (Ne.{succ u1} γ x y)
+but is expected to have type
+  forall {γ : Type.{u1}} [_inst_2 : MetricSpace.{u1} γ] {x : γ} {y : γ}, Iff (LT.lt.{0} Real Real.instLTReal (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal)) (Dist.dist.{u1} γ (PseudoMetricSpace.toDist.{u1} γ (MetricSpace.toPseudoMetricSpace.{u1} γ _inst_2)) x y)) (Ne.{succ u1} γ x y)
+Case conversion may be inaccurate. Consider using '#align dist_pos dist_posₓ'. -/
 @[simp]
 theorem dist_pos {x y : γ} : 0 < dist x y ↔ x ≠ y := by
   simpa only [not_le] using not_congr dist_le_zero
 #align dist_pos dist_pos
 
+/- warning: eq_of_forall_dist_le -> eq_of_forall_dist_le is a dubious translation:
+lean 3 declaration is
+  forall {γ : Type.{u1}} [_inst_2 : MetricSpace.{u1} γ] {x : γ} {y : γ}, (forall (ε : Real), (GT.gt.{0} Real Real.hasLt ε (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero)))) -> (LE.le.{0} Real Real.hasLe (Dist.dist.{u1} γ (PseudoMetricSpace.toHasDist.{u1} γ (MetricSpace.toPseudoMetricSpace.{u1} γ _inst_2)) x y) ε)) -> (Eq.{succ u1} γ x y)
+but is expected to have type
+  forall {γ : Type.{u1}} [_inst_2 : MetricSpace.{u1} γ] {x : γ} {y : γ}, (forall (ε : Real), (GT.gt.{0} Real Real.instLTReal ε (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal))) -> (LE.le.{0} Real Real.instLEReal (Dist.dist.{u1} γ (PseudoMetricSpace.toDist.{u1} γ (MetricSpace.toPseudoMetricSpace.{u1} γ _inst_2)) x y) ε)) -> (Eq.{succ u1} γ x y)
+Case conversion may be inaccurate. Consider using '#align eq_of_forall_dist_le eq_of_forall_dist_leₓ'. -/
 theorem eq_of_forall_dist_le {x y : γ} (h : ∀ ε > 0, dist x y ≤ ε) : x = y :=
   eq_of_dist_eq_zero (eq_of_le_of_forall_le_of_dense dist_nonneg h)
 #align eq_of_forall_dist_le eq_of_forall_dist_le
 
+/- warning: eq_of_nndist_eq_zero -> eq_of_nndist_eq_zero is a dubious translation:
+lean 3 declaration is
+  forall {γ : Type.{u1}} [_inst_2 : MetricSpace.{u1} γ] {x : γ} {y : γ}, (Eq.{1} NNReal (NNDist.nndist.{u1} γ (PseudoMetricSpace.toNNDist.{u1} γ (MetricSpace.toPseudoMetricSpace.{u1} γ _inst_2)) x y) (OfNat.ofNat.{0} NNReal 0 (OfNat.mk.{0} NNReal 0 (Zero.zero.{0} NNReal (MulZeroClass.toHasZero.{0} NNReal (NonUnitalNonAssocSemiring.toMulZeroClass.{0} NNReal (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} NNReal (Semiring.toNonAssocSemiring.{0} NNReal NNReal.semiring)))))))) -> (Eq.{succ u1} γ x y)
+but is expected to have type
+  forall {γ : Type.{u1}} [_inst_2 : MetricSpace.{u1} γ] {x : γ} {y : γ}, (Eq.{1} NNReal (NNDist.nndist.{u1} γ (PseudoMetricSpace.toNNDist.{u1} γ (MetricSpace.toPseudoMetricSpace.{u1} γ _inst_2)) x y) (OfNat.ofNat.{0} NNReal 0 (Zero.toOfNat0.{0} NNReal instNNRealZero))) -> (Eq.{succ u1} γ x y)
+Case conversion may be inaccurate. Consider using '#align eq_of_nndist_eq_zero eq_of_nndist_eq_zeroₓ'. -/
 /-- Deduce the equality of points with the vanishing of the nonnegative distance-/
 theorem eq_of_nndist_eq_zero {x y : γ} : nndist x y = 0 → x = y := by
   simp only [← NNReal.eq_iff, ← dist_nndist, imp_self, NNReal.coe_zero, dist_eq_zero]
 #align eq_of_nndist_eq_zero eq_of_nndist_eq_zero
 
+/- warning: nndist_eq_zero -> nndist_eq_zero is a dubious translation:
+lean 3 declaration is
+  forall {γ : Type.{u1}} [_inst_2 : MetricSpace.{u1} γ] {x : γ} {y : γ}, Iff (Eq.{1} NNReal (NNDist.nndist.{u1} γ (PseudoMetricSpace.toNNDist.{u1} γ (MetricSpace.toPseudoMetricSpace.{u1} γ _inst_2)) x y) (OfNat.ofNat.{0} NNReal 0 (OfNat.mk.{0} NNReal 0 (Zero.zero.{0} NNReal (MulZeroClass.toHasZero.{0} NNReal (NonUnitalNonAssocSemiring.toMulZeroClass.{0} NNReal (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} NNReal (Semiring.toNonAssocSemiring.{0} NNReal NNReal.semiring)))))))) (Eq.{succ u1} γ x y)
+but is expected to have type
+  forall {γ : Type.{u1}} [_inst_2 : MetricSpace.{u1} γ] {x : γ} {y : γ}, Iff (Eq.{1} NNReal (NNDist.nndist.{u1} γ (PseudoMetricSpace.toNNDist.{u1} γ (MetricSpace.toPseudoMetricSpace.{u1} γ _inst_2)) x y) (OfNat.ofNat.{0} NNReal 0 (Zero.toOfNat0.{0} NNReal instNNRealZero))) (Eq.{succ u1} γ x y)
+Case conversion may be inaccurate. Consider using '#align nndist_eq_zero nndist_eq_zeroₓ'. -/
 /-- Characterize the equality of points with the vanishing of the nonnegative distance-/
 @[simp]
 theorem nndist_eq_zero {x y : γ} : nndist x y = 0 ↔ x = y := by
   simp only [← NNReal.eq_iff, ← dist_nndist, imp_self, NNReal.coe_zero, dist_eq_zero]
 #align nndist_eq_zero nndist_eq_zero
 
+/- warning: zero_eq_nndist -> zero_eq_nndist is a dubious translation:
+lean 3 declaration is
+  forall {γ : Type.{u1}} [_inst_2 : MetricSpace.{u1} γ] {x : γ} {y : γ}, Iff (Eq.{1} NNReal (OfNat.ofNat.{0} NNReal 0 (OfNat.mk.{0} NNReal 0 (Zero.zero.{0} NNReal (MulZeroClass.toHasZero.{0} NNReal (NonUnitalNonAssocSemiring.toMulZeroClass.{0} NNReal (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} NNReal (Semiring.toNonAssocSemiring.{0} NNReal NNReal.semiring))))))) (NNDist.nndist.{u1} γ (PseudoMetricSpace.toNNDist.{u1} γ (MetricSpace.toPseudoMetricSpace.{u1} γ _inst_2)) x y)) (Eq.{succ u1} γ x y)
+but is expected to have type
+  forall {γ : Type.{u1}} [_inst_2 : MetricSpace.{u1} γ] {x : γ} {y : γ}, Iff (Eq.{1} NNReal (OfNat.ofNat.{0} NNReal 0 (Zero.toOfNat0.{0} NNReal instNNRealZero)) (NNDist.nndist.{u1} γ (PseudoMetricSpace.toNNDist.{u1} γ (MetricSpace.toPseudoMetricSpace.{u1} γ _inst_2)) x y)) (Eq.{succ u1} γ x y)
+Case conversion may be inaccurate. Consider using '#align zero_eq_nndist zero_eq_nndistₓ'. -/
 @[simp]
 theorem zero_eq_nndist {x y : γ} : 0 = nndist x y ↔ x = y := by
   simp only [← NNReal.eq_iff, ← dist_nndist, imp_self, NNReal.coe_zero, zero_eq_dist]
@@ -3177,16 +5097,34 @@ namespace Metric
 
 variable {x : γ} {s : Set γ}
 
+/- warning: metric.closed_ball_zero -> Metric.closedBall_zero is a dubious translation:
+lean 3 declaration is
+  forall {γ : Type.{u1}} [_inst_2 : MetricSpace.{u1} γ] {x : γ}, Eq.{succ u1} (Set.{u1} γ) (Metric.closedBall.{u1} γ (MetricSpace.toPseudoMetricSpace.{u1} γ _inst_2) x (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero)))) (Singleton.singleton.{u1, u1} γ (Set.{u1} γ) (Set.hasSingleton.{u1} γ) x)
+but is expected to have type
+  forall {γ : Type.{u1}} [_inst_2 : MetricSpace.{u1} γ] {x : γ}, Eq.{succ u1} (Set.{u1} γ) (Metric.closedBall.{u1} γ (MetricSpace.toPseudoMetricSpace.{u1} γ _inst_2) x (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal))) (Singleton.singleton.{u1, u1} γ (Set.{u1} γ) (Set.instSingletonSet.{u1} γ) x)
+Case conversion may be inaccurate. Consider using '#align metric.closed_ball_zero Metric.closedBall_zeroₓ'. -/
 @[simp]
 theorem closedBall_zero : closedBall x 0 = {x} :=
   Set.ext fun y => dist_le_zero
 #align metric.closed_ball_zero Metric.closedBall_zero
 
+/- warning: metric.sphere_zero -> Metric.sphere_zero is a dubious translation:
+lean 3 declaration is
+  forall {γ : Type.{u1}} [_inst_2 : MetricSpace.{u1} γ] {x : γ}, Eq.{succ u1} (Set.{u1} γ) (Metric.sphere.{u1} γ (MetricSpace.toPseudoMetricSpace.{u1} γ _inst_2) x (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero)))) (Singleton.singleton.{u1, u1} γ (Set.{u1} γ) (Set.hasSingleton.{u1} γ) x)
+but is expected to have type
+  forall {γ : Type.{u1}} [_inst_2 : MetricSpace.{u1} γ] {x : γ}, Eq.{succ u1} (Set.{u1} γ) (Metric.sphere.{u1} γ (MetricSpace.toPseudoMetricSpace.{u1} γ _inst_2) x (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal))) (Singleton.singleton.{u1, u1} γ (Set.{u1} γ) (Set.instSingletonSet.{u1} γ) x)
+Case conversion may be inaccurate. Consider using '#align metric.sphere_zero Metric.sphere_zeroₓ'. -/
 @[simp]
 theorem sphere_zero : sphere x 0 = {x} :=
   Set.ext fun y => dist_eq_zero
 #align metric.sphere_zero Metric.sphere_zero
 
+/- warning: metric.subsingleton_closed_ball -> Metric.subsingleton_closedBall is a dubious translation:
+lean 3 declaration is
+  forall {γ : Type.{u1}} [_inst_2 : MetricSpace.{u1} γ] (x : γ) {r : Real}, (LE.le.{0} Real Real.hasLe r (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero)))) -> (Set.Subsingleton.{u1} γ (Metric.closedBall.{u1} γ (MetricSpace.toPseudoMetricSpace.{u1} γ _inst_2) x r))
+but is expected to have type
+  forall {γ : Type.{u1}} [_inst_2 : MetricSpace.{u1} γ] (x : γ) {r : Real}, (LE.le.{0} Real Real.instLEReal r (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal))) -> (Set.Subsingleton.{u1} γ (Metric.closedBall.{u1} γ (MetricSpace.toPseudoMetricSpace.{u1} γ _inst_2) x r))
+Case conversion may be inaccurate. Consider using '#align metric.subsingleton_closed_ball Metric.subsingleton_closedBallₓ'. -/
 theorem subsingleton_closedBall (x : γ) {r : ℝ} (hr : r ≤ 0) : (closedBall x r).Subsingleton :=
   by
   rcases hr.lt_or_eq with (hr | rfl)
@@ -3196,16 +5134,30 @@ theorem subsingleton_closedBall (x : γ) {r : ℝ} (hr : r ≤ 0) : (closedBall
     exact subsingleton_singleton
 #align metric.subsingleton_closed_ball Metric.subsingleton_closedBall
 
+/- warning: metric.subsingleton_sphere -> Metric.subsingleton_sphere is a dubious translation:
+lean 3 declaration is
+  forall {γ : Type.{u1}} [_inst_2 : MetricSpace.{u1} γ] (x : γ) {r : Real}, (LE.le.{0} Real Real.hasLe r (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero)))) -> (Set.Subsingleton.{u1} γ (Metric.sphere.{u1} γ (MetricSpace.toPseudoMetricSpace.{u1} γ _inst_2) x r))
+but is expected to have type
+  forall {γ : Type.{u1}} [_inst_2 : MetricSpace.{u1} γ] (x : γ) {r : Real}, (LE.le.{0} Real Real.instLEReal r (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal))) -> (Set.Subsingleton.{u1} γ (Metric.sphere.{u1} γ (MetricSpace.toPseudoMetricSpace.{u1} γ _inst_2) x r))
+Case conversion may be inaccurate. Consider using '#align metric.subsingleton_sphere Metric.subsingleton_sphereₓ'. -/
 theorem subsingleton_sphere (x : γ) {r : ℝ} (hr : r ≤ 0) : (sphere x r).Subsingleton :=
   (subsingleton_closedBall x hr).anti sphere_subset_closedBall
 #align metric.subsingleton_sphere Metric.subsingleton_sphere
 
+#print MetricSpace.to_separated /-
 -- see Note [lower instance priority]
 instance (priority := 100) MetricSpace.to_separated : SeparatedSpace γ :=
   separated_def.2 fun x y h =>
     eq_of_forall_dist_le fun ε ε0 => le_of_lt (h _ (dist_mem_uniformity ε0))
 #align metric_space.to_separated MetricSpace.to_separated
+-/
 
+/- warning: metric.uniform_embedding_iff' -> Metric.uniformEmbedding_iff' is a dubious translation:
+lean 3 declaration is
+  forall {β : Type.{u1}} {γ : Type.{u2}} [_inst_2 : MetricSpace.{u2} γ] [_inst_3 : MetricSpace.{u1} β] {f : γ -> β}, Iff (UniformEmbedding.{u2, u1} γ β (PseudoMetricSpace.toUniformSpace.{u2} γ (MetricSpace.toPseudoMetricSpace.{u2} γ _inst_2)) (PseudoMetricSpace.toUniformSpace.{u1} β (MetricSpace.toPseudoMetricSpace.{u1} β _inst_3)) f) (And (forall (ε : Real), (GT.gt.{0} Real Real.hasLt ε (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero)))) -> (Exists.{1} Real (fun (δ : Real) => Exists.{0} (GT.gt.{0} Real Real.hasLt δ (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero)))) (fun (H : GT.gt.{0} Real Real.hasLt δ (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero)))) => forall {a : γ} {b : γ}, (LT.lt.{0} Real Real.hasLt (Dist.dist.{u2} γ (PseudoMetricSpace.toHasDist.{u2} γ (MetricSpace.toPseudoMetricSpace.{u2} γ _inst_2)) a b) δ) -> (LT.lt.{0} Real Real.hasLt (Dist.dist.{u1} β (PseudoMetricSpace.toHasDist.{u1} β (MetricSpace.toPseudoMetricSpace.{u1} β _inst_3)) (f a) (f b)) ε))))) (forall (δ : Real), (GT.gt.{0} Real Real.hasLt δ (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero)))) -> (Exists.{1} Real (fun (ε : Real) => Exists.{0} (GT.gt.{0} Real Real.hasLt ε (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero)))) (fun (H : GT.gt.{0} Real Real.hasLt ε (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero)))) => forall {a : γ} {b : γ}, (LT.lt.{0} Real Real.hasLt (Dist.dist.{u1} β (PseudoMetricSpace.toHasDist.{u1} β (MetricSpace.toPseudoMetricSpace.{u1} β _inst_3)) (f a) (f b)) ε) -> (LT.lt.{0} Real Real.hasLt (Dist.dist.{u2} γ (PseudoMetricSpace.toHasDist.{u2} γ (MetricSpace.toPseudoMetricSpace.{u2} γ _inst_2)) a b) δ))))))
+but is expected to have type
+  forall {β : Type.{u1}} {γ : Type.{u2}} [_inst_2 : MetricSpace.{u2} γ] [_inst_3 : MetricSpace.{u1} β] {f : γ -> β}, Iff (UniformEmbedding.{u2, u1} γ β (PseudoMetricSpace.toUniformSpace.{u2} γ (MetricSpace.toPseudoMetricSpace.{u2} γ _inst_2)) (PseudoMetricSpace.toUniformSpace.{u1} β (MetricSpace.toPseudoMetricSpace.{u1} β _inst_3)) f) (And (forall (ε : Real), (GT.gt.{0} Real Real.instLTReal ε (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal))) -> (Exists.{1} Real (fun (δ : Real) => And (GT.gt.{0} Real Real.instLTReal δ (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal))) (forall {a : γ} {b : γ}, (LT.lt.{0} Real Real.instLTReal (Dist.dist.{u2} γ (PseudoMetricSpace.toDist.{u2} γ (MetricSpace.toPseudoMetricSpace.{u2} γ _inst_2)) a b) δ) -> (LT.lt.{0} Real Real.instLTReal (Dist.dist.{u1} β (PseudoMetricSpace.toDist.{u1} β (MetricSpace.toPseudoMetricSpace.{u1} β _inst_3)) (f a) (f b)) ε))))) (forall (δ : Real), (GT.gt.{0} Real Real.instLTReal δ (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal))) -> (Exists.{1} Real (fun (ε : Real) => And (GT.gt.{0} Real Real.instLTReal ε (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal))) (forall {a : γ} {b : γ}, (LT.lt.{0} Real Real.instLTReal (Dist.dist.{u1} β (PseudoMetricSpace.toDist.{u1} β (MetricSpace.toPseudoMetricSpace.{u1} β _inst_3)) (f a) (f b)) ε) -> (LT.lt.{0} Real Real.instLTReal (Dist.dist.{u2} γ (PseudoMetricSpace.toDist.{u2} γ (MetricSpace.toPseudoMetricSpace.{u2} γ _inst_2)) a b) δ))))))
+Case conversion may be inaccurate. Consider using '#align metric.uniform_embedding_iff' Metric.uniformEmbedding_iff'ₓ'. -/
 /-- A map between metric spaces is a uniform embedding if and only if the distance between `f x`
 and `f y` is controlled in terms of the distance between `x` and `y` and conversely. -/
 theorem uniformEmbedding_iff' [MetricSpace β] {f : γ → β} :
@@ -3218,30 +5170,52 @@ theorem uniformEmbedding_iff' [MetricSpace β] {f : γ → β} :
   rfl
 #align metric.uniform_embedding_iff' Metric.uniformEmbedding_iff'
 
+#print MetricSpace.ofT0PseudoMetricSpace /-
 /-- If a `pseudo_metric_space` is a T₀ space, then it is a `metric_space`. -/
 def MetricSpace.ofT0PseudoMetricSpace (α : Type _) [PseudoMetricSpace α] [T0Space α] :
     MetricSpace α :=
   { ‹PseudoMetricSpace α› with
     eq_of_dist_eq_zero := fun x y hdist => Inseparable.eq <| Metric.inseparable_iff.2 hdist }
 #align metric_space.of_t0_pseudo_metric_space MetricSpace.ofT0PseudoMetricSpace
+-/
 
+#print MetricSpace.toEMetricSpace /-
 -- see Note [lower instance priority]
 /-- A metric space induces an emetric space -/
-instance (priority := 100) MetricSpace.toEmetricSpace : EMetricSpace γ :=
+instance (priority := 100) MetricSpace.toEMetricSpace : EMetricSpace γ :=
   EMetricSpace.ofT0PseudoEMetricSpace γ
-#align metric_space.to_emetric_space MetricSpace.toEmetricSpace
+#align metric_space.to_emetric_space MetricSpace.toEMetricSpace
+-/
 
+/- warning: metric.is_closed_of_pairwise_le_dist -> Metric.isClosed_of_pairwise_le_dist is a dubious translation:
+lean 3 declaration is
+  forall {γ : Type.{u1}} [_inst_2 : MetricSpace.{u1} γ] {s : Set.{u1} γ} {ε : Real}, (LT.lt.{0} Real Real.hasLt (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero))) ε) -> (Set.Pairwise.{u1} γ s (fun (x : γ) (y : γ) => LE.le.{0} Real Real.hasLe ε (Dist.dist.{u1} γ (PseudoMetricSpace.toHasDist.{u1} γ (MetricSpace.toPseudoMetricSpace.{u1} γ _inst_2)) x y))) -> (IsClosed.{u1} γ (UniformSpace.toTopologicalSpace.{u1} γ (PseudoMetricSpace.toUniformSpace.{u1} γ (MetricSpace.toPseudoMetricSpace.{u1} γ _inst_2))) s)
+but is expected to have type
+  forall {γ : Type.{u1}} [_inst_2 : MetricSpace.{u1} γ] {s : Set.{u1} γ} {ε : Real}, (LT.lt.{0} Real Real.instLTReal (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal)) ε) -> (Set.Pairwise.{u1} γ s (fun (x : γ) (y : γ) => LE.le.{0} Real Real.instLEReal ε (Dist.dist.{u1} γ (PseudoMetricSpace.toDist.{u1} γ (MetricSpace.toPseudoMetricSpace.{u1} γ _inst_2)) x y))) -> (IsClosed.{u1} γ (UniformSpace.toTopologicalSpace.{u1} γ (PseudoMetricSpace.toUniformSpace.{u1} γ (MetricSpace.toPseudoMetricSpace.{u1} γ _inst_2))) s)
+Case conversion may be inaccurate. Consider using '#align metric.is_closed_of_pairwise_le_dist Metric.isClosed_of_pairwise_le_distₓ'. -/
 theorem isClosed_of_pairwise_le_dist {s : Set γ} {ε : ℝ} (hε : 0 < ε)
     (hs : s.Pairwise fun x y => ε ≤ dist x y) : IsClosed s :=
   isClosed_of_spaced_out (dist_mem_uniformity hε) <| by simpa using hs
 #align metric.is_closed_of_pairwise_le_dist Metric.isClosed_of_pairwise_le_dist
 
+/- warning: metric.closed_embedding_of_pairwise_le_dist -> Metric.closedEmbedding_of_pairwise_le_dist is a dubious translation:
+lean 3 declaration is
+  forall {γ : Type.{u1}} [_inst_2 : MetricSpace.{u1} γ] {α : Type.{u2}} [_inst_3 : TopologicalSpace.{u2} α] [_inst_4 : DiscreteTopology.{u2} α _inst_3] {ε : Real}, (LT.lt.{0} Real Real.hasLt (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero))) ε) -> (forall {f : α -> γ}, (Pairwise.{u2} α (fun (x : α) (y : α) => LE.le.{0} Real Real.hasLe ε (Dist.dist.{u1} γ (PseudoMetricSpace.toHasDist.{u1} γ (MetricSpace.toPseudoMetricSpace.{u1} γ _inst_2)) (f x) (f y)))) -> (ClosedEmbedding.{u2, u1} α γ _inst_3 (UniformSpace.toTopologicalSpace.{u1} γ (PseudoMetricSpace.toUniformSpace.{u1} γ (MetricSpace.toPseudoMetricSpace.{u1} γ _inst_2))) f))
+but is expected to have type
+  forall {γ : Type.{u2}} [_inst_2 : MetricSpace.{u2} γ] {α : Type.{u1}} [_inst_3 : TopologicalSpace.{u1} α] [_inst_4 : DiscreteTopology.{u1} α _inst_3] {ε : Real}, (LT.lt.{0} Real Real.instLTReal (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal)) ε) -> (forall {f : α -> γ}, (Pairwise.{u1} α (fun (x : α) (y : α) => LE.le.{0} Real Real.instLEReal ε (Dist.dist.{u2} γ (PseudoMetricSpace.toDist.{u2} γ (MetricSpace.toPseudoMetricSpace.{u2} γ _inst_2)) (f x) (f y)))) -> (ClosedEmbedding.{u1, u2} α γ _inst_3 (UniformSpace.toTopologicalSpace.{u2} γ (PseudoMetricSpace.toUniformSpace.{u2} γ (MetricSpace.toPseudoMetricSpace.{u2} γ _inst_2))) f))
+Case conversion may be inaccurate. Consider using '#align metric.closed_embedding_of_pairwise_le_dist Metric.closedEmbedding_of_pairwise_le_distₓ'. -/
 theorem closedEmbedding_of_pairwise_le_dist {α : Type _} [TopologicalSpace α] [DiscreteTopology α]
     {ε : ℝ} (hε : 0 < ε) {f : α → γ} (hf : Pairwise fun x y => ε ≤ dist (f x) (f y)) :
     ClosedEmbedding f :=
   closedEmbedding_of_spaced_out (dist_mem_uniformity hε) <| by simpa using hf
 #align metric.closed_embedding_of_pairwise_le_dist Metric.closedEmbedding_of_pairwise_le_dist
 
+/- warning: metric.uniform_embedding_bot_of_pairwise_le_dist -> Metric.uniformEmbedding_bot_of_pairwise_le_dist is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {β : Type.{u2}} {ε : Real}, (LT.lt.{0} Real Real.hasLt (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero))) ε) -> (forall {f : β -> α}, (Pairwise.{u2} β (fun (x : β) (y : β) => LE.le.{0} Real Real.hasLe ε (Dist.dist.{u1} α (PseudoMetricSpace.toHasDist.{u1} α _inst_1) (f x) (f y)))) -> (UniformEmbedding.{u2, u1} β α (Bot.bot.{u2} (UniformSpace.{u2} β) (UniformSpace.hasBot.{u2} β)) (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1) f))
+but is expected to have type
+  forall {α : Type.{u2}} [_inst_1 : PseudoMetricSpace.{u2} α] {β : Type.{u1}} {ε : Real}, (LT.lt.{0} Real Real.instLTReal (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal)) ε) -> (forall {f : β -> α}, (Pairwise.{u1} β (fun (x : β) (y : β) => LE.le.{0} Real Real.instLEReal ε (Dist.dist.{u2} α (PseudoMetricSpace.toDist.{u2} α _inst_1) (f x) (f y)))) -> (UniformEmbedding.{u1, u2} β α (Bot.bot.{u1} (UniformSpace.{u1} β) (instBotUniformSpace.{u1} β)) (inferInstance.{succ u2} (UniformSpace.{u2} α) (PseudoMetricSpace.toUniformSpace.{u2} α _inst_1)) f))
+Case conversion may be inaccurate. Consider using '#align metric.uniform_embedding_bot_of_pairwise_le_dist Metric.uniformEmbedding_bot_of_pairwise_le_distₓ'. -/
 /-- If `f : β → α` sends any two distinct points to points at distance at least `ε > 0`, then
 `f` is a uniform embedding with respect to the discrete uniformity on `β`. -/
 theorem uniformEmbedding_bot_of_pairwise_le_dist {β : Type _} {ε : ℝ} (hε : 0 < ε) {f : β → α}
@@ -3252,6 +5226,7 @@ theorem uniformEmbedding_bot_of_pairwise_le_dist {β : Type _} {ε : ℝ} (hε :
 
 end Metric
 
+#print MetricSpace.replaceUniformity /-
 /-- Build a new metric space from an old one where the bundled uniform structure is provably
 (but typically non-definitionaly) equal to some given uniform structure.
 See Note [forgetful inheritance].
@@ -3261,14 +5236,18 @@ def MetricSpace.replaceUniformity {γ} [U : UniformSpace γ] (m : MetricSpace γ
   { PseudoMetricSpace.replaceUniformity m.toPseudoMetricSpace H with
     eq_of_dist_eq_zero := @eq_of_dist_eq_zero _ _ }
 #align metric_space.replace_uniformity MetricSpace.replaceUniformity
+-/
 
+#print MetricSpace.replaceUniformity_eq /-
 theorem MetricSpace.replaceUniformity_eq {γ} [U : UniformSpace γ] (m : MetricSpace γ)
     (H : 𝓤[U] = 𝓤[PseudoEMetricSpace.toUniformSpace]) : m.replaceUniformity H = m :=
   by
   ext
   rfl
 #align metric_space.replace_uniformity_eq MetricSpace.replaceUniformity_eq
+-/
 
+#print MetricSpace.replaceTopology /-
 /-- Build a new metric space from an old one where the bundled topological structure is provably
 (but typically non-definitionaly) equal to some given topological structure.
 See Note [forgetful inheritance].
@@ -3278,14 +5257,23 @@ def MetricSpace.replaceTopology {γ} [U : TopologicalSpace γ] (m : MetricSpace
     (H : U = m.toPseudoMetricSpace.toUniformSpace.toTopologicalSpace) : MetricSpace γ :=
   @MetricSpace.replaceUniformity γ (m.toUniformSpace.replaceTopology H) m rfl
 #align metric_space.replace_topology MetricSpace.replaceTopology
+-/
 
+#print MetricSpace.replaceTopology_eq /-
 theorem MetricSpace.replaceTopology_eq {γ} [U : TopologicalSpace γ] (m : MetricSpace γ)
     (H : U = m.toPseudoMetricSpace.toUniformSpace.toTopologicalSpace) : m.replaceTopology H = m :=
   by
   ext
   rfl
 #align metric_space.replace_topology_eq MetricSpace.replaceTopology_eq
+-/
 
+/- warning: emetric_space.to_metric_space_of_dist -> EMetricSpace.toMetricSpaceOfDist is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} [e : EMetricSpace.{u1} α] (dist : α -> α -> Real), (forall (x : α) (y : α), Ne.{1} ENNReal (EDist.edist.{u1} α (PseudoEMetricSpace.toHasEdist.{u1} α (EMetricSpace.toPseudoEmetricSpace.{u1} α e)) x y) (Top.top.{0} ENNReal (CompleteLattice.toHasTop.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder)))) -> (forall (x : α) (y : α), Eq.{1} Real (dist x y) (ENNReal.toReal (EDist.edist.{u1} α (PseudoEMetricSpace.toHasEdist.{u1} α (EMetricSpace.toPseudoEmetricSpace.{u1} α e)) x y))) -> (MetricSpace.{u1} α)
+but is expected to have type
+  forall {α : Type.{u1}} [e : EMetricSpace.{u1} α] (dist : α -> α -> Real), (forall (x : α) (y : α), Ne.{1} ENNReal (EDist.edist.{u1} α (PseudoEMetricSpace.toEDist.{u1} α (EMetricSpace.toPseudoEMetricSpace.{u1} α e)) x y) (Top.top.{0} ENNReal (CompleteLattice.toTop.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal)))) -> (forall (x : α) (y : α), Eq.{1} Real (dist x y) (ENNReal.toReal (EDist.edist.{u1} α (PseudoEMetricSpace.toEDist.{u1} α (EMetricSpace.toPseudoEMetricSpace.{u1} α e)) x y))) -> (MetricSpace.{u1} α)
+Case conversion may be inaccurate. Consider using '#align emetric_space.to_metric_space_of_dist EMetricSpace.toMetricSpaceOfDistₓ'. -/
 /-- One gets a metric space from an emetric space if the edistance
 is everywhere finite, by pushing the edistance to reals. We set it up so that the edist and the
 uniformity are defeq in the metric space and the emetric space. In this definition, the distance
@@ -3298,6 +5286,12 @@ def EMetricSpace.toMetricSpaceOfDist {α : Type u} [e : EMetricSpace α] (dist :
     (PseudoEMetricSpace.toPseudoMetricSpaceOfDist dist edist_ne_top h) _
 #align emetric_space.to_metric_space_of_dist EMetricSpace.toMetricSpaceOfDist
 
+/- warning: emetric_space.to_metric_space -> EMetricSpace.toMetricSpace is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} [_inst_3 : EMetricSpace.{u1} α], (forall (x : α) (y : α), Ne.{1} ENNReal (EDist.edist.{u1} α (PseudoEMetricSpace.toHasEdist.{u1} α (EMetricSpace.toPseudoEmetricSpace.{u1} α _inst_3)) x y) (Top.top.{0} ENNReal (CompleteLattice.toHasTop.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder)))) -> (MetricSpace.{u1} α)
+but is expected to have type
+  forall {α : Type.{u1}} [_inst_3 : EMetricSpace.{u1} α], (forall (x : α) (y : α), Ne.{1} ENNReal (EDist.edist.{u1} α (PseudoEMetricSpace.toEDist.{u1} α (EMetricSpace.toPseudoEMetricSpace.{u1} α _inst_3)) x y) (Top.top.{0} ENNReal (CompleteLattice.toTop.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal)))) -> (MetricSpace.{u1} α)
+Case conversion may be inaccurate. Consider using '#align emetric_space.to_metric_space EMetricSpace.toMetricSpaceₓ'. -/
 /-- One gets a metric space from an emetric space if the edistance
 is everywhere finite, by pushing the edistance to reals. We set it up so that the edist and the
 uniformity are defeq in the metric space and the emetric space. -/
@@ -3306,6 +5300,7 @@ def EMetricSpace.toMetricSpace {α : Type u} [EMetricSpace α] (h : ∀ x y : α
   EMetricSpace.toMetricSpaceOfDist (fun x y => ENNReal.toReal (edist x y)) h fun x y => rfl
 #align emetric_space.to_metric_space EMetricSpace.toMetricSpace
 
+#print MetricSpace.replaceBornology /-
 /-- Build a new metric space from an old one where the bundled bornology structure is provably
 (but typically non-definitionaly) equal to some given bornology structure.
 See Note [forgetful inheritance].
@@ -3314,14 +5309,18 @@ def MetricSpace.replaceBornology {α} [B : Bornology α] (m : MetricSpace α)
     (H : ∀ s, @IsBounded _ B s ↔ @IsBounded _ PseudoMetricSpace.toBornology s) : MetricSpace α :=
   { PseudoMetricSpace.replaceBornology _ H, m with toBornology := B }
 #align metric_space.replace_bornology MetricSpace.replaceBornology
+-/
 
+#print MetricSpace.replaceBornology_eq /-
 theorem MetricSpace.replaceBornology_eq {α} [m : MetricSpace α] [B : Bornology α]
     (H : ∀ s, @IsBounded _ B s ↔ @IsBounded _ PseudoMetricSpace.toBornology s) :
     MetricSpace.replaceBornology _ H = m := by
   ext
   rfl
 #align metric_space.replace_bornology_eq MetricSpace.replaceBornology_eq
+-/
 
+#print MetricSpace.induced /-
 /-- Metric space structure pulled back by an injective function. Injectivity is necessary to
 ensure that `dist x y = 0` only if `x = y`. -/
 def MetricSpace.induced {γ β} (f : γ → β) (hf : Function.Injective f) (m : MetricSpace β) :
@@ -3329,7 +5328,9 @@ def MetricSpace.induced {γ β} (f : γ → β) (hf : Function.Injective f) (m :
   { PseudoMetricSpace.induced f m.toPseudoMetricSpace with
     eq_of_dist_eq_zero := fun x y h => hf (dist_eq_zero.1 h) }
 #align metric_space.induced MetricSpace.induced
+-/
 
+#print UniformEmbedding.comapMetricSpace /-
 /-- Pull back a metric space structure by a uniform embedding. This is a version of
 `metric_space.induced` useful in case if the domain already has a `uniform_space` structure. -/
 @[reducible]
@@ -3337,7 +5338,9 @@ def UniformEmbedding.comapMetricSpace {α β} [UniformSpace α] [MetricSpace β]
     (h : UniformEmbedding f) : MetricSpace α :=
   (MetricSpace.induced f h.inj ‹_›).replaceUniformity h.comap_uniformity.symm
 #align uniform_embedding.comap_metric_space UniformEmbedding.comapMetricSpace
+-/
 
+#print Embedding.comapMetricSpace /-
 /-- Pull back a metric space structure by an embedding. This is a version of
 `metric_space.induced` useful in case if the domain already has a `topological_space` structure. -/
 @[reducible]
@@ -3346,11 +5349,14 @@ def Embedding.comapMetricSpace {α β} [TopologicalSpace α] [MetricSpace β] (f
   letI : UniformSpace α := Embedding.comapUniformSpace f h
   UniformEmbedding.comapMetricSpace f (h.to_uniform_embedding f)
 #align embedding.comap_metric_space Embedding.comapMetricSpace
+-/
 
+#print Subtype.metricSpace /-
 instance Subtype.metricSpace {α : Type _} {p : α → Prop} [MetricSpace α] :
     MetricSpace (Subtype p) :=
   MetricSpace.induced coe Subtype.coe_injective ‹_›
 #align subtype.metric_space Subtype.metricSpace
+-/
 
 @[to_additive]
 instance {α : Type _} [MetricSpace α] : MetricSpace αᵐᵒᵖ :=
@@ -3378,18 +5384,20 @@ instance : MetricSpace PUnit.{u + 1} where
     simp only
     have : ne_bot (⨅ ε > (0 : ℝ), 𝓟 { p : PUnit.{u + 1} × PUnit.{u + 1} | 0 < ε }) :=
       @uniformity.neBot _
-        (uniformSpaceOfDist (fun _ _ => 0) (fun _ => rfl) (fun _ _ => rfl) fun _ _ _ => by
+        (UniformSpace.ofDist (fun _ _ => 0) (fun _ => rfl) (fun _ _ => rfl) fun _ _ _ => by
           rw [zero_add])
         _
     refine' (eq_top_of_ne_bot _).trans (eq_top_of_ne_bot _).symm
 
 section Real
 
+#print Real.metricSpace /-
 /-- Instantiate the reals as a metric space. -/
 instance Real.metricSpace : MetricSpace ℝ :=
   { Real.pseudoMetricSpace with
     eq_of_dist_eq_zero := fun x y h => by simpa [dist, sub_eq_zero] using h }
 #align real.metric_space Real.metricSpace
+-/
 
 end Real
 
@@ -3405,6 +5413,7 @@ instance [MetricSpace β] : MetricSpace (ULift β) :=
 
 section Prod
 
+#print Prod.metricSpaceMax /-
 instance Prod.metricSpaceMax [MetricSpace β] : MetricSpace (γ × β) :=
   { Prod.pseudoMetricSpaceMax with
     eq_of_dist_eq_zero := fun x y h =>
@@ -3412,6 +5421,7 @@ instance Prod.metricSpaceMax [MetricSpace β] : MetricSpace (γ × β) :=
       cases' max_le_iff.1 (le_of_eq h) with h₁ h₂
       exact Prod.ext_iff.2 ⟨dist_le_zero.1 h₁, dist_le_zero.1 h₂⟩ }
 #align prod.metric_space_max Prod.metricSpaceMax
+-/
 
 end Prod
 
@@ -3421,6 +5431,7 @@ open Finset
 
 variable {π : β → Type _} [Fintype β] [∀ b, MetricSpace (π b)]
 
+#print metricSpacePi /-
 /-- A finite product of metric spaces is a metric space, with the sup distance. -/
 instance metricSpacePi : MetricSpace (∀ b, π b) :=
   {/- we construct the instance from the emetric space instance to avoid checking again that the
@@ -3434,6 +5445,7 @@ instance metricSpacePi : MetricSpace (∀ b, π b) :=
       simp only [Finset.sup_le_iff] at eq2
       exact funext fun b => edist_le_zero.1 <| eq2 b <| mem_univ b }
 #align metric_space_pi metricSpacePi
+-/
 
 end Pi
 
@@ -3443,9 +5455,15 @@ section SecondCountable
 
 open TopologicalSpace
 
+/- warning: metric.second_countable_of_countable_discretization -> Metric.secondCountable_of_countable_discretization is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} [_inst_3 : MetricSpace.{u1} α], (forall (ε : Real), (GT.gt.{0} Real Real.hasLt ε (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero)))) -> (Exists.{succ (succ u2)} Type.{u2} (fun (β : Type.{u2}) => Exists.{succ u2} (Encodable.{u2} β) (fun (_x : Encodable.{u2} β) => Exists.{max (succ u1) (succ u2)} (α -> β) (fun (F : α -> β) => forall (x : α) (y : α), (Eq.{succ u2} β (F x) (F y)) -> (LE.le.{0} Real Real.hasLe (Dist.dist.{u1} α (PseudoMetricSpace.toHasDist.{u1} α (MetricSpace.toPseudoMetricSpace.{u1} α _inst_3)) x y) ε)))))) -> (TopologicalSpace.SecondCountableTopology.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α (MetricSpace.toPseudoMetricSpace.{u1} α _inst_3))))
+but is expected to have type
+  forall {α : Type.{u2}} [_inst_3 : MetricSpace.{u2} α], (forall (ε : Real), (GT.gt.{0} Real Real.instLTReal ε (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal))) -> (Exists.{succ (succ u1)} Type.{u1} (fun (β : Type.{u1}) => Exists.{succ u1} (Encodable.{u1} β) (fun (_x : Encodable.{u1} β) => Exists.{max (succ u2) (succ u1)} (α -> β) (fun (F : α -> β) => forall (x : α) (y : α), (Eq.{succ u1} β (F x) (F y)) -> (LE.le.{0} Real Real.instLEReal (Dist.dist.{u2} α (PseudoMetricSpace.toDist.{u2} α (MetricSpace.toPseudoMetricSpace.{u2} α _inst_3)) x y) ε)))))) -> (TopologicalSpace.SecondCountableTopology.{u2} α (UniformSpace.toTopologicalSpace.{u2} α (PseudoMetricSpace.toUniformSpace.{u2} α (MetricSpace.toPseudoMetricSpace.{u2} α _inst_3))))
+Case conversion may be inaccurate. Consider using '#align metric.second_countable_of_countable_discretization Metric.secondCountable_of_countable_discretizationₓ'. -/
 /-- A metric space is second countable if one can reconstruct up to any `ε>0` any element of the
 space from countably many data. -/
-theorem second_countable_of_countable_discretization {α : Type u} [MetricSpace α]
+theorem secondCountable_of_countable_discretization {α : Type u} [MetricSpace α]
     (H :
       ∀ ε > (0 : ℝ), ∃ (β : Type _)(_ : Encodable β)(F : α → β), ∀ x y, F x = F y → dist x y ≤ ε) :
     SecondCountableTopology α :=
@@ -3463,7 +5481,7 @@ theorem second_countable_of_countable_discretization {α : Type u} [MetricSpace
   let x' := Finv (F x)
   have : F x' = F x := Function.invFun_eq ⟨x, rfl⟩
   exact ⟨x', mem_range_self _, hF _ _ this.symm⟩
-#align metric.second_countable_of_countable_discretization Metric.second_countable_of_countable_discretization
+#align metric.second_countable_of_countable_discretization Metric.secondCountable_of_countable_discretization
 
 end SecondCountable
 
@@ -3471,16 +5489,18 @@ end Metric
 
 section EqRel
 
-instance {α : Type u} [PseudoMetricSpace α] : HasDist (UniformSpace.SeparationQuotient α)
+instance {α : Type u} [PseudoMetricSpace α] : Dist (UniformSpace.SeparationQuotient α)
     where dist p q :=
     Quotient.liftOn₂' p q dist fun x y x' y' hx hy => by
       rw [dist_edist, dist_edist, ← UniformSpace.SeparationQuotient.edist_mk x, ←
         UniformSpace.SeparationQuotient.edist_mk x', Quot.sound hx, Quot.sound hy]
 
+#print UniformSpace.SeparationQuotient.dist_mk /-
 theorem UniformSpace.SeparationQuotient.dist_mk {α : Type u} [PseudoMetricSpace α] (p q : α) :
     @dist (UniformSpace.SeparationQuotient α) _ (Quot.mk _ p) (Quot.mk _ q) = dist p q :=
   rfl
 #align uniform_space.separation_quotient.dist_mk UniformSpace.SeparationQuotient.dist_mk
+-/
 
 instance {α : Type u} [PseudoMetricSpace α] : MetricSpace (UniformSpace.SeparationQuotient α) :=
   EMetricSpace.toMetricSpaceOfDist dist (fun x y => Quotient.inductionOn₂' x y edist_ne_top)
@@ -3499,33 +5519,41 @@ open Additive Multiplicative
 
 section
 
-variable [HasDist X]
+variable [Dist X]
 
-instance : HasDist (Additive X) :=
-  ‹HasDist X›
+instance : Dist (Additive X) :=
+  ‹Dist X›
 
-instance : HasDist (Multiplicative X) :=
-  ‹HasDist X›
+instance : Dist (Multiplicative X) :=
+  ‹Dist X›
 
+#print dist_ofMul /-
 @[simp]
 theorem dist_ofMul (a b : X) : dist (ofMul a) (ofMul b) = dist a b :=
   rfl
 #align dist_of_mul dist_ofMul
+-/
 
+#print dist_ofAdd /-
 @[simp]
 theorem dist_ofAdd (a b : X) : dist (ofAdd a) (ofAdd b) = dist a b :=
   rfl
 #align dist_of_add dist_ofAdd
+-/
 
+#print dist_toMul /-
 @[simp]
 theorem dist_toMul (a b : Additive X) : dist (toMul a) (toMul b) = dist a b :=
   rfl
 #align dist_to_mul dist_toMul
+-/
 
+#print dist_toAdd /-
 @[simp]
 theorem dist_toAdd (a b : Multiplicative X) : dist (toAdd a) (toAdd b) = dist a b :=
   rfl
 #align dist_to_add dist_toAdd
+-/
 
 end
 
@@ -3539,25 +5567,33 @@ instance : PseudoMetricSpace (Additive X) :=
 instance : PseudoMetricSpace (Multiplicative X) :=
   ‹PseudoMetricSpace X›
 
+#print nndist_ofMul /-
 @[simp]
 theorem nndist_ofMul (a b : X) : nndist (ofMul a) (ofMul b) = nndist a b :=
   rfl
 #align nndist_of_mul nndist_ofMul
+-/
 
+#print nndist_ofAdd /-
 @[simp]
 theorem nndist_ofAdd (a b : X) : nndist (ofAdd a) (ofAdd b) = nndist a b :=
   rfl
 #align nndist_of_add nndist_ofAdd
+-/
 
+#print nndist_toMul /-
 @[simp]
 theorem nndist_toMul (a b : Additive X) : nndist (toMul a) (toMul b) = nndist a b :=
   rfl
 #align nndist_to_mul nndist_toMul
+-/
 
+#print nndist_toAdd /-
 @[simp]
 theorem nndist_toAdd (a b : Multiplicative X) : nndist (toAdd a) (toAdd b) = nndist a b :=
   rfl
 #align nndist_to_add nndist_toAdd
+-/
 
 end
 
@@ -3584,20 +5620,24 @@ open OrderDual
 
 section
 
-variable [HasDist X]
+variable [Dist X]
 
-instance : HasDist Xᵒᵈ :=
-  ‹HasDist X›
+instance : Dist Xᵒᵈ :=
+  ‹Dist X›
 
+#print dist_toDual /-
 @[simp]
 theorem dist_toDual (a b : X) : dist (toDual a) (toDual b) = dist a b :=
   rfl
 #align dist_to_dual dist_toDual
+-/
 
+#print dist_ofDual /-
 @[simp]
 theorem dist_ofDual (a b : Xᵒᵈ) : dist (ofDual a) (ofDual b) = dist a b :=
   rfl
 #align dist_of_dual dist_ofDual
+-/
 
 end
 
@@ -3608,15 +5648,19 @@ variable [PseudoMetricSpace X]
 instance : PseudoMetricSpace Xᵒᵈ :=
   ‹PseudoMetricSpace X›
 
+#print nndist_toDual /-
 @[simp]
 theorem nndist_toDual (a b : X) : nndist (toDual a) (toDual b) = nndist a b :=
   rfl
 #align nndist_to_dual nndist_toDual
+-/
 
+#print nndist_ofDual /-
 @[simp]
 theorem nndist_ofDual (a b : Xᵒᵈ) : nndist (ofDual a) (ofDual b) = nndist a b :=
   rfl
 #align nndist_of_dual nndist_ofDual
+-/
 
 end
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/195fcd60ff2bfe392543bceb0ec2adcdb472db4c

@@ -203,7 +203,7 @@ attribute [instance] PseudoMetricSpace.toUniformSpace
 attribute [instance] PseudoMetricSpace.toBornology
 
 -- see Note [lower instance priority]
-instance (priority := 200) PseudoMetricSpace.toHasEdist : HasEdist α :=
+instance (priority := 200) PseudoMetricSpace.toHasEdist : EDist α :=
   ⟨PseudoMetricSpace.edist⟩
 #align pseudo_metric_space.to_has_edist PseudoMetricSpace.toHasEdist
 
@@ -1274,7 +1274,7 @@ theorem Metric.uniformity_edist : 𝓤 α = ⨅ ε > 0, 𝓟 { p : α × α | ed
 
 -- see Note [lower instance priority]
 /-- A pseudometric space induces a pseudoemetric space -/
-instance (priority := 100) PseudoMetricSpace.toPseudoEmetricSpace : PseudoEmetricSpace α :=
+instance (priority := 100) PseudoMetricSpace.toPseudoEmetricSpace : PseudoEMetricSpace α :=
   { ‹PseudoMetricSpace α› with
     edist := edist
     edist_self := by simp [edist_dist]
@@ -1289,22 +1289,22 @@ instance (priority := 100) PseudoMetricSpace.toPseudoEmetricSpace : PseudoEmetri
 #align pseudo_metric_space.to_pseudo_emetric_space PseudoMetricSpace.toPseudoEmetricSpace
 
 /-- In a pseudometric space, an open ball of infinite radius is the whole space -/
-theorem Metric.eball_top_eq_univ (x : α) : Emetric.ball x ∞ = Set.univ :=
+theorem Metric.eball_top_eq_univ (x : α) : EMetric.ball x ∞ = Set.univ :=
   Set.eq_univ_iff_forall.mpr fun y => edist_lt_top y x
 #align metric.eball_top_eq_univ Metric.eball_top_eq_univ
 
 /-- Balls defined using the distance or the edistance coincide -/
 @[simp]
-theorem Metric.emetric_ball {x : α} {ε : ℝ} : Emetric.ball x (ENNReal.ofReal ε) = ball x ε :=
+theorem Metric.emetric_ball {x : α} {ε : ℝ} : EMetric.ball x (ENNReal.ofReal ε) = ball x ε :=
   by
   ext y
-  simp only [Emetric.mem_ball, mem_ball, edist_dist]
+  simp only [EMetric.mem_ball, mem_ball, edist_dist]
   exact ENNReal.ofReal_lt_ofReal_iff_of_nonneg dist_nonneg
 #align metric.emetric_ball Metric.emetric_ball
 
 /-- Balls defined using the distance or the edistance coincide -/
 @[simp]
-theorem Metric.emetric_ball_nNReal {x : α} {ε : ℝ≥0} : Emetric.ball x ε = ball x ε :=
+theorem Metric.emetric_ball_nNReal {x : α} {ε : ℝ≥0} : EMetric.ball x ε = ball x ε :=
   by
   convert Metric.emetric_ball
   simp
@@ -1312,26 +1312,26 @@ theorem Metric.emetric_ball_nNReal {x : α} {ε : ℝ≥0} : Emetric.ball x ε =
 
 /-- Closed balls defined using the distance or the edistance coincide -/
 theorem Metric.emetric_closedBall {x : α} {ε : ℝ} (h : 0 ≤ ε) :
-    Emetric.closedBall x (ENNReal.ofReal ε) = closedBall x ε := by
+    EMetric.closedBall x (ENNReal.ofReal ε) = closedBall x ε := by
   ext y <;> simp [edist_dist] <;> rw [ENNReal.ofReal_le_ofReal_iff h]
 #align metric.emetric_closed_ball Metric.emetric_closedBall
 
 /-- Closed balls defined using the distance or the edistance coincide -/
 @[simp]
 theorem Metric.emetric_closedBall_nNReal {x : α} {ε : ℝ≥0} :
-    Emetric.closedBall x ε = closedBall x ε :=
+    EMetric.closedBall x ε = closedBall x ε :=
   by
   convert Metric.emetric_closedBall ε.2
   simp
 #align metric.emetric_closed_ball_nnreal Metric.emetric_closedBall_nNReal
 
 @[simp]
-theorem Metric.emetric_ball_top (x : α) : Emetric.ball x ⊤ = univ :=
+theorem Metric.emetric_ball_top (x : α) : EMetric.ball x ⊤ = univ :=
   eq_univ_of_forall fun y => edist_lt_top _ _
 #align metric.emetric_ball_top Metric.emetric_ball_top
 
 theorem Metric.inseparable_iff {x y : α} : Inseparable x y ↔ dist x y = 0 := by
-  rw [Emetric.inseparable_iff, edist_nndist, dist_nndist, ENNReal.coe_eq_zero, NNReal.coe_eq_zero]
+  rw [EMetric.inseparable_iff, edist_nndist, dist_nndist, ENNReal.coe_eq_zero, NNReal.coe_eq_zero]
 #align metric.inseparable_iff Metric.inseparable_iff
 
 /-- Build a new pseudometric space from an old one where the bundled uniform structure is provably
@@ -1339,7 +1339,7 @@ theorem Metric.inseparable_iff {x y : α} : Inseparable x y ↔ dist x y = 0 :=
 See Note [forgetful inheritance].
 -/
 def PseudoMetricSpace.replaceUniformity {α} [U : UniformSpace α] (m : PseudoMetricSpace α)
-    (H : 𝓤[U] = 𝓤[PseudoEmetricSpace.toUniformSpace]) : PseudoMetricSpace α
+    (H : 𝓤[U] = 𝓤[PseudoEMetricSpace.toUniformSpace]) : PseudoMetricSpace α
     where
   dist := @dist _ m.toHasDist
   dist_self := dist_self
@@ -1352,7 +1352,7 @@ def PseudoMetricSpace.replaceUniformity {α} [U : UniformSpace α] (m : PseudoMe
 #align pseudo_metric_space.replace_uniformity PseudoMetricSpace.replaceUniformity
 
 theorem PseudoMetricSpace.replaceUniformity_eq {α} [U : UniformSpace α] (m : PseudoMetricSpace α)
-    (H : 𝓤[U] = 𝓤[PseudoEmetricSpace.toUniformSpace]) : m.replaceUniformity H = m :=
+    (H : 𝓤[U] = 𝓤[PseudoEMetricSpace.toUniformSpace]) : m.replaceUniformity H = m :=
   by
   ext
   rfl
@@ -1380,13 +1380,13 @@ is everywhere finite, by pushing the edistance to reals. We set it up so that th
 uniformity are defeq in the pseudometric space and the pseudoemetric space. In this definition, the
 distance is given separately, to be able to prescribe some expression which is not defeq to the
 push-forward of the edistance to reals. -/
-def PseudoEmetricSpace.toPseudoMetricSpaceOfDist {α : Type u} [e : PseudoEmetricSpace α]
+def PseudoEMetricSpace.toPseudoMetricSpaceOfDist {α : Type u} [e : PseudoEMetricSpace α]
     (dist : α → α → ℝ) (edist_ne_top : ∀ x y : α, edist x y ≠ ⊤)
     (h : ∀ x y, dist x y = ENNReal.toReal (edist x y)) : PseudoMetricSpace α :=
   let m : PseudoMetricSpace α :=
     { dist
       dist_self := fun x => by simp [h]
-      dist_comm := fun x y => by simp [h, PseudoEmetricSpace.edist_comm]
+      dist_comm := fun x y => by simp [h, PseudoEMetricSpace.edist_comm]
       dist_triangle := fun x y z => by
         simp only [h]
         rw [← ENNReal.toReal_add (edist_ne_top _ _) (edist_ne_top _ _),
@@ -1399,16 +1399,16 @@ def PseudoEmetricSpace.toPseudoMetricSpaceOfDist {α : Type u} [e : PseudoEmetri
     by
     rw [uniformity_pseudoedist, Metric.uniformity_edist]
     rfl
-#align pseudo_emetric_space.to_pseudo_metric_space_of_dist PseudoEmetricSpace.toPseudoMetricSpaceOfDist
+#align pseudo_emetric_space.to_pseudo_metric_space_of_dist PseudoEMetricSpace.toPseudoMetricSpaceOfDist
 
 /-- One gets a pseudometric space from an emetric space if the edistance
 is everywhere finite, by pushing the edistance to reals. We set it up so that the edist and the
 uniformity are defeq in the pseudometric space and the emetric space. -/
-def PseudoEmetricSpace.toPseudoMetricSpace {α : Type u} [e : PseudoEmetricSpace α]
+def PseudoEMetricSpace.toPseudoMetricSpace {α : Type u} [e : PseudoEMetricSpace α]
     (h : ∀ x y : α, edist x y ≠ ⊤) : PseudoMetricSpace α :=
-  PseudoEmetricSpace.toPseudoMetricSpaceOfDist (fun x y => ENNReal.toReal (edist x y)) h fun x y =>
+  PseudoEMetricSpace.toPseudoMetricSpaceOfDist (fun x y => ENNReal.toReal (edist x y)) h fun x y =>
     rfl
-#align pseudo_emetric_space.to_pseudo_metric_space PseudoEmetricSpace.toPseudoMetricSpace
+#align pseudo_emetric_space.to_pseudo_metric_space PseudoEMetricSpace.toPseudoMetricSpace
 
 /-- Build a new pseudometric space from an old one where the bundled bornology structure is provably
 (but typically non-definitionaly) equal to some given bornology structure.
@@ -1449,7 +1449,7 @@ theorem Metric.complete_of_convergent_controlled_sequences (B : ℕ → Real) (h
 
 theorem Metric.complete_of_cauchySeq_tendsto :
     (∀ u : ℕ → α, CauchySeq u → ∃ a, Tendsto u atTop (𝓝 a)) → CompleteSpace α :=
-  Emetric.complete_of_cauchySeq_tendsto
+  EMetric.complete_of_cauchySeq_tendsto
 #align metric.complete_of_cauchy_seq_tendsto Metric.complete_of_cauchySeq_tendsto
 
 section Real
@@ -1887,7 +1887,7 @@ section Prod
 variable [PseudoMetricSpace β]
 
 instance Prod.pseudoMetricSpaceMax : PseudoMetricSpace (α × β) :=
-  (PseudoEmetricSpace.toPseudoMetricSpaceOfDist (fun x y : α × β => dist x.1 y.1 ⊔ dist x.2 y.2)
+  (PseudoEMetricSpace.toPseudoMetricSpaceOfDist (fun x y : α × β => dist x.1 y.1 ⊔ dist x.2 y.2)
         (fun x y => (max_lt (edist_lt_top _ _) (edist_lt_top _ _)).Ne) fun x y => by
         simp only [sup_eq_max, dist_edist, ←
           ENNReal.toReal_max (edist_ne_top _ _) (edist_ne_top _ _), Prod.edist_eq]).replaceBornology
@@ -2161,7 +2161,7 @@ instance pseudoMetricSpacePi : PseudoMetricSpace (∀ b, π b) :=
     the uniformity is the same as the product uniformity, but we register nevertheless a nice formula
     for the distance -/
   refine'
-    (PseudoEmetricSpace.toPseudoMetricSpaceOfDist
+    (PseudoEMetricSpace.toPseudoMetricSpaceOfDist
           (fun f g : ∀ b, π b => ((sup univ fun b => nndist (f b) (g b) : ℝ≥0) : ℝ)) (fun f g => _)
           fun f g => _).replaceBornology
       fun s => _
@@ -2353,7 +2353,7 @@ instance (priority := 100) second_countable_of_proper [ProperSpace α] : SecondC
   by
   -- We already have `sigma_compact_space_of_locally_compact_second_countable`, so we don't
   -- add an instance for `sigma_compact_space`.
-  suffices SigmaCompactSpace α by exact Emetric.second_countable_of_sigma_compact α
+  suffices SigmaCompactSpace α by exact EMetric.secondCountable_of_sigmaCompact α
   rcases em (Nonempty α) with (⟨⟨x⟩⟩ | hn)
   · exact ⟨⟨fun n => closed_ball x n, fun n => is_compact_closed_ball _ _, Union_closed_ball_nat _⟩⟩
   · exact ⟨⟨fun n => ∅, fun n => isCompact_empty, Union_eq_univ_iff.2 fun x => (hn ⟨x⟩).elim⟩⟩
@@ -2486,7 +2486,7 @@ theorem second_countable_of_almost_dense_set
     (H : ∀ ε > (0 : ℝ), ∃ s : Set α, s.Countable ∧ ∀ x, ∃ y ∈ s, dist x y ≤ ε) :
     SecondCountableTopology α :=
   by
-  refine' Emetric.second_countable_of_almost_dense_set fun ε ε0 => _
+  refine' EMetric.secondCountable_of_almost_dense_set fun ε ε0 => _
   rcases ENNReal.lt_iff_exists_nnreal_btwn.1 ε0 with ⟨ε', ε'0, ε'ε⟩
   choose s hsc y hys hyx using H ε' (by exact_mod_cast ε'0)
   refine' ⟨s, hsc, Union₂_eq_univ_iff.2 fun x => ⟨y x, hys _, le_trans _ ε'ε.le⟩⟩
@@ -2836,7 +2836,7 @@ variable {s : Set α} {x y z : α}
 /-- The diameter of a set in a metric space. To get controllable behavior even when the diameter
 should be infinite, we express it in terms of the emetric.diameter -/
 noncomputable def diam (s : Set α) : ℝ :=
-  ENNReal.toReal (Emetric.diam s)
+  ENNReal.toReal (EMetric.diam s)
 #align metric.diam Metric.diam
 
 /-- The diameter of a set is always nonnegative -/
@@ -2845,7 +2845,7 @@ theorem diam_nonneg : 0 ≤ diam s :=
 #align metric.diam_nonneg Metric.diam_nonneg
 
 theorem diam_subsingleton (hs : s.Subsingleton) : diam s = 0 := by
-  simp only [diam, Emetric.diam_subsingleton hs, ENNReal.zero_toReal]
+  simp only [diam, EMetric.diam_subsingleton hs, ENNReal.zero_toReal]
 #align metric.diam_subsingleton Metric.diam_subsingleton
 
 /-- The empty set has zero diameter -/
@@ -2862,22 +2862,22 @@ theorem diam_singleton : diam ({x} : Set α) = 0 :=
 
 -- Does not work as a simp-lemma, since {x, y} reduces to (insert y {x})
 theorem diam_pair : diam ({x, y} : Set α) = dist x y := by
-  simp only [diam, Emetric.diam_pair, dist_edist]
+  simp only [diam, EMetric.diam_pair, dist_edist]
 #align metric.diam_pair Metric.diam_pair
 
 -- Does not work as a simp-lemma, since {x, y, z} reduces to (insert z (insert y {x}))
 theorem diam_triple :
     Metric.diam ({x, y, z} : Set α) = max (max (dist x y) (dist x z)) (dist y z) :=
   by
-  simp only [Metric.diam, Emetric.diam_triple, dist_edist]
+  simp only [Metric.diam, EMetric.diam_triple, dist_edist]
   rw [ENNReal.toReal_max, ENNReal.toReal_max] <;> apply_rules [ne_of_lt, edist_lt_top, max_lt]
 #align metric.diam_triple Metric.diam_triple
 
 /-- If the distance between any two points in a set is bounded by some constant `C`,
 then `ennreal.of_real C`  bounds the emetric diameter of this set. -/
 theorem ediam_le_of_forall_dist_le {C : ℝ} (h : ∀ x ∈ s, ∀ y ∈ s, dist x y ≤ C) :
-    Emetric.diam s ≤ ENNReal.ofReal C :=
-  Emetric.diam_le fun x hx y hy => (edist_dist x y).symm ▸ ENNReal.ofReal_le_ofReal (h x hx y hy)
+    EMetric.diam s ≤ ENNReal.ofReal C :=
+  EMetric.diam_le fun x hx y hy => (edist_dist x y).symm ▸ ENNReal.ofReal_le_ofReal (h x hx y hy)
 #align metric.ediam_le_of_forall_dist_le Metric.ediam_le_of_forall_dist_le
 
 /-- If the distance between any two points in a set is bounded by some non-negative constant,
@@ -2898,33 +2898,33 @@ theorem diam_le_of_forall_dist_le_of_nonempty (hs : s.Nonempty) {C : ℝ}
 #align metric.diam_le_of_forall_dist_le_of_nonempty Metric.diam_le_of_forall_dist_le_of_nonempty
 
 /-- The distance between two points in a set is controlled by the diameter of the set. -/
-theorem dist_le_diam_of_mem' (h : Emetric.diam s ≠ ⊤) (hx : x ∈ s) (hy : y ∈ s) :
+theorem dist_le_diam_of_mem' (h : EMetric.diam s ≠ ⊤) (hx : x ∈ s) (hy : y ∈ s) :
     dist x y ≤ diam s := by
   rw [diam, dist_edist]
   rw [ENNReal.toReal_le_toReal (edist_ne_top _ _) h]
-  exact Emetric.edist_le_diam_of_mem hx hy
+  exact EMetric.edist_le_diam_of_mem hx hy
 #align metric.dist_le_diam_of_mem' Metric.dist_le_diam_of_mem'
 
 /-- Characterize the boundedness of a set in terms of the finiteness of its emetric.diameter. -/
-theorem bounded_iff_ediam_ne_top : Bounded s ↔ Emetric.diam s ≠ ⊤ :=
+theorem bounded_iff_ediam_ne_top : Bounded s ↔ EMetric.diam s ≠ ⊤ :=
   Iff.intro
     (fun ⟨C, hC⟩ => ne_top_of_le_ne_top ENNReal.ofReal_ne_top <| ediam_le_of_forall_dist_le hC)
     fun h => ⟨diam s, fun x hx y hy => dist_le_diam_of_mem' h hx hy⟩
 #align metric.bounded_iff_ediam_ne_top Metric.bounded_iff_ediam_ne_top
 
-theorem Bounded.ediam_ne_top (h : Bounded s) : Emetric.diam s ≠ ⊤ :=
+theorem Bounded.ediam_ne_top (h : Bounded s) : EMetric.diam s ≠ ⊤ :=
   bounded_iff_ediam_ne_top.1 h
 #align metric.bounded.ediam_ne_top Metric.Bounded.ediam_ne_top
 
 theorem ediam_univ_eq_top_iff_noncompact [ProperSpace α] :
-    Emetric.diam (univ : Set α) = ∞ ↔ NoncompactSpace α := by
+    EMetric.diam (univ : Set α) = ∞ ↔ NoncompactSpace α := by
   rw [← not_compactSpace_iff, compact_space_iff_bounded_univ, bounded_iff_ediam_ne_top,
     Classical.not_not]
 #align metric.ediam_univ_eq_top_iff_noncompact Metric.ediam_univ_eq_top_iff_noncompact
 
 @[simp]
 theorem ediam_univ_of_noncompact [ProperSpace α] [NoncompactSpace α] :
-    Emetric.diam (univ : Set α) = ∞ :=
+    EMetric.diam (univ : Set α) = ∞ :=
   ediam_univ_eq_top_iff_noncompact.mpr ‹_›
 #align metric.ediam_univ_of_noncompact Metric.ediam_univ_of_noncompact
 
@@ -2938,7 +2938,7 @@ theorem dist_le_diam_of_mem (h : Bounded s) (hx : x ∈ s) (hy : y ∈ s) : dist
   dist_le_diam_of_mem' h.ediam_ne_top hx hy
 #align metric.dist_le_diam_of_mem Metric.dist_le_diam_of_mem
 
-theorem ediam_of_unbounded (h : ¬Bounded s) : Emetric.diam s = ∞ := by
+theorem ediam_of_unbounded (h : ¬Bounded s) : EMetric.diam s = ∞ := by
   rwa [bounded_iff_ediam_ne_top, Classical.not_not] at h
 #align metric.ediam_of_unbounded Metric.ediam_of_unbounded
 
@@ -2953,7 +2953,7 @@ theorem diam_mono {s t : Set α} (h : s ⊆ t) (ht : Bounded t) : diam s ≤ dia
   by
   unfold diam
   rw [ENNReal.toReal_le_toReal (bounded.mono h ht).ediam_ne_top ht.ediam_ne_top]
-  exact Emetric.diam_mono h
+  exact EMetric.diam_mono h
 #align metric.diam_mono Metric.diam_mono
 
 /-- The diameter of a union is controlled by the sum of the diameters, and the distance between
@@ -2971,7 +2971,7 @@ theorem diam_union {t : Set α} (xs : x ∈ s) (yt : y ∈ t) :
           repeat' apply ENNReal.add_ne_top.2 <;> constructor <;>
         try assumption <;>
       try apply edist_ne_top
-    exact Emetric.diam_union xs yt
+    exact EMetric.diam_union xs yt
   · rw [diam_eq_zero_of_unbounded H]
     apply_rules [add_nonneg, diam_nonneg, dist_nonneg]
 #align metric.diam_union Metric.diam_union
@@ -3227,8 +3227,8 @@ def MetricSpace.ofT0PseudoMetricSpace (α : Type _) [PseudoMetricSpace α] [T0Sp
 
 -- see Note [lower instance priority]
 /-- A metric space induces an emetric space -/
-instance (priority := 100) MetricSpace.toEmetricSpace : EmetricSpace γ :=
-  EmetricSpace.ofT0PseudoEmetricSpace γ
+instance (priority := 100) MetricSpace.toEmetricSpace : EMetricSpace γ :=
+  EMetricSpace.ofT0PseudoEMetricSpace γ
 #align metric_space.to_emetric_space MetricSpace.toEmetricSpace
 
 theorem isClosed_of_pairwise_le_dist {s : Set γ} {ε : ℝ} (hε : 0 < ε)
@@ -3257,13 +3257,13 @@ end Metric
 See Note [forgetful inheritance].
 -/
 def MetricSpace.replaceUniformity {γ} [U : UniformSpace γ] (m : MetricSpace γ)
-    (H : 𝓤[U] = 𝓤[PseudoEmetricSpace.toUniformSpace]) : MetricSpace γ :=
+    (H : 𝓤[U] = 𝓤[PseudoEMetricSpace.toUniformSpace]) : MetricSpace γ :=
   { PseudoMetricSpace.replaceUniformity m.toPseudoMetricSpace H with
     eq_of_dist_eq_zero := @eq_of_dist_eq_zero _ _ }
 #align metric_space.replace_uniformity MetricSpace.replaceUniformity
 
 theorem MetricSpace.replaceUniformity_eq {γ} [U : UniformSpace γ] (m : MetricSpace γ)
-    (H : 𝓤[U] = 𝓤[PseudoEmetricSpace.toUniformSpace]) : m.replaceUniformity H = m :=
+    (H : 𝓤[U] = 𝓤[PseudoEMetricSpace.toUniformSpace]) : m.replaceUniformity H = m :=
   by
   ext
   rfl
@@ -3291,20 +3291,20 @@ is everywhere finite, by pushing the edistance to reals. We set it up so that th
 uniformity are defeq in the metric space and the emetric space. In this definition, the distance
 is given separately, to be able to prescribe some expression which is not defeq to the push-forward
 of the edistance to reals. -/
-def EmetricSpace.toMetricSpaceOfDist {α : Type u} [e : EmetricSpace α] (dist : α → α → ℝ)
+def EMetricSpace.toMetricSpaceOfDist {α : Type u} [e : EMetricSpace α] (dist : α → α → ℝ)
     (edist_ne_top : ∀ x y : α, edist x y ≠ ⊤) (h : ∀ x y, dist x y = ENNReal.toReal (edist x y)) :
     MetricSpace α :=
   @MetricSpace.ofT0PseudoMetricSpace α
-    (PseudoEmetricSpace.toPseudoMetricSpaceOfDist dist edist_ne_top h) _
-#align emetric_space.to_metric_space_of_dist EmetricSpace.toMetricSpaceOfDist
+    (PseudoEMetricSpace.toPseudoMetricSpaceOfDist dist edist_ne_top h) _
+#align emetric_space.to_metric_space_of_dist EMetricSpace.toMetricSpaceOfDist
 
 /-- One gets a metric space from an emetric space if the edistance
 is everywhere finite, by pushing the edistance to reals. We set it up so that the edist and the
 uniformity are defeq in the metric space and the emetric space. -/
-def EmetricSpace.toMetricSpace {α : Type u} [EmetricSpace α] (h : ∀ x y : α, edist x y ≠ ⊤) :
+def EMetricSpace.toMetricSpace {α : Type u} [EMetricSpace α] (h : ∀ x y : α, edist x y ≠ ⊤) :
     MetricSpace α :=
-  EmetricSpace.toMetricSpaceOfDist (fun x y => ENNReal.toReal (edist x y)) h fun x y => rfl
-#align emetric_space.to_metric_space EmetricSpace.toMetricSpace
+  EMetricSpace.toMetricSpaceOfDist (fun x y => ENNReal.toReal (edist x y)) h fun x y => rfl
+#align emetric_space.to_metric_space EMetricSpace.toMetricSpace
 
 /-- Build a new metric space from an old one where the bundled bornology structure is provably
 (but typically non-definitionaly) equal to some given bornology structure.
@@ -3483,7 +3483,7 @@ theorem UniformSpace.SeparationQuotient.dist_mk {α : Type u} [PseudoMetricSpace
 #align uniform_space.separation_quotient.dist_mk UniformSpace.SeparationQuotient.dist_mk
 
 instance {α : Type u} [PseudoMetricSpace α] : MetricSpace (UniformSpace.SeparationQuotient α) :=
-  EmetricSpace.toMetricSpaceOfDist dist (fun x y => Quotient.inductionOn₂' x y edist_ne_top)
+  EMetricSpace.toMetricSpaceOfDist dist (fun x y => Quotient.inductionOn₂' x y edist_ne_top)
     fun x y => Quotient.inductionOn₂' x y dist_edist
 
 end EqRel

mathlib commit https://github.com/leanprover-community/mathlib/commit/195fcd60ff2bfe392543bceb0ec2adcdb472db4c

@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Jeremy Avigad, Robert Y. Lewis, Johannes Hölzl, Mario Carneiro, Sébastien Gouëzel
 
 ! This file was ported from Lean 3 source module topology.metric_space.basic
-! leanprover-community/mathlib commit e1a7bdeb4fd826b7e71d130d34988f0a2d26a177
+! leanprover-community/mathlib commit 195fcd60ff2bfe392543bceb0ec2adcdb472db4c
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
@@ -916,17 +916,9 @@ theorem uniformEmbedding_iff [PseudoMetricSpace β] {f : α → β} :
     UniformEmbedding f ↔
       Function.Injective f ∧
         UniformContinuous f ∧ ∀ δ > 0, ∃ ε > 0, ∀ {a b : α}, dist (f a) (f b) < ε → dist a b < δ :=
-  uniformEmbedding_def'.trans <|
-    and_congr Iff.rfl <|
-      and_congr Iff.rfl
-        ⟨fun H δ δ0 =>
-          let ⟨t, tu, ht⟩ := H _ (dist_mem_uniformity δ0)
-          let ⟨ε, ε0, hε⟩ := mem_uniformity_dist.1 tu
-          ⟨ε, ε0, fun a b h => ht _ _ (hε h)⟩,
-          fun H s su =>
-          let ⟨δ, δ0, hδ⟩ := mem_uniformity_dist.1 su
-          let ⟨ε, ε0, hε⟩ := H _ δ0
-          ⟨_, dist_mem_uniformity ε0, fun a b h => hδ (hε h)⟩⟩
+  by
+  simp only [uniformity_basis_dist.uniform_embedding_iff uniformity_basis_dist, exists_prop]
+  rfl
 #align metric.uniform_embedding_iff Metric.uniformEmbedding_iff
 
 /-- If a map between pseudometric spaces is a uniform embedding then the distance between `f x`
@@ -937,7 +929,7 @@ theorem controlled_of_uniformEmbedding [PseudoMetricSpace β] {f : α → β} :
         ∀ δ > 0, ∃ ε > 0, ∀ {a b : α}, dist (f a) (f b) < ε → dist a b < δ :=
   by
   intro h
-  exact ⟨uniformContinuous_iff.1 (uniform_embedding_iff.1 h).2.1, (uniform_embedding_iff.1 h).2.2⟩
+  exact ⟨uniformContinuous_iff.1 (uniformEmbedding_iff.1 h).2.1, (uniformEmbedding_iff.1 h).2.2⟩
 #align metric.controlled_of_uniform_embedding Metric.controlled_of_uniformEmbedding
 
 theorem totallyBounded_iff {s : Set α} :
@@ -3208,6 +3200,12 @@ theorem subsingleton_sphere (x : γ) {r : ℝ} (hr : r ≤ 0) : (sphere x r).Sub
   (subsingleton_closedBall x hr).anti sphere_subset_closedBall
 #align metric.subsingleton_sphere Metric.subsingleton_sphere
 
+-- see Note [lower instance priority]
+instance (priority := 100) MetricSpace.to_separated : SeparatedSpace γ :=
+  separated_def.2 fun x y h =>
+    eq_of_forall_dist_le fun ε ε0 => le_of_lt (h _ (dist_mem_uniformity ε0))
+#align metric_space.to_separated MetricSpace.to_separated
+
 /-- A map between metric spaces is a uniform embedding if and only if the distance between `f x`
 and `f y` is controlled in terms of the distance between `x` and `y` and conversely. -/
 theorem uniformEmbedding_iff' [MetricSpace β] {f : γ → β} :
@@ -3215,37 +3213,23 @@ theorem uniformEmbedding_iff' [MetricSpace β] {f : γ → β} :
       (∀ ε > 0, ∃ δ > 0, ∀ {a b : γ}, dist a b < δ → dist (f a) (f b) < ε) ∧
         ∀ δ > 0, ∃ ε > 0, ∀ {a b : γ}, dist (f a) (f b) < ε → dist a b < δ :=
   by
-  constructor
-  · intro h
-    exact ⟨uniformContinuous_iff.1 (uniform_embedding_iff.1 h).2.1, (uniform_embedding_iff.1 h).2.2⟩
-  · rintro ⟨h₁, h₂⟩
-    refine' uniform_embedding_iff.2 ⟨_, uniformContinuous_iff.2 h₁, h₂⟩
-    intro x y hxy
-    have : dist x y ≤ 0 := by
-      refine' le_of_forall_lt' fun δ δpos => _
-      rcases h₂ δ δpos with ⟨ε, εpos, hε⟩
-      have : dist (f x) (f y) < ε := by simpa [hxy]
-      exact hε this
-    simpa using this
+  simp only [uniformEmbedding_iff_uniformInducing,
+    uniformity_basis_dist.uniform_inducing_iff uniformity_basis_dist, exists_prop]
+  rfl
 #align metric.uniform_embedding_iff' Metric.uniformEmbedding_iff'
 
--- see Note [lower instance priority]
-instance (priority := 100) MetricSpace.to_separated : SeparatedSpace γ :=
-  separated_def.2 fun x y h =>
-    eq_of_forall_dist_le fun ε ε0 => le_of_lt (h _ (dist_mem_uniformity ε0))
-#align metric_space.to_separated MetricSpace.to_separated
-
 /-- If a `pseudo_metric_space` is a T₀ space, then it is a `metric_space`. -/
-def ofT0PseudoMetricSpace (α : Type _) [PseudoMetricSpace α] [T0Space α] : MetricSpace α :=
+def MetricSpace.ofT0PseudoMetricSpace (α : Type _) [PseudoMetricSpace α] [T0Space α] :
+    MetricSpace α :=
   { ‹PseudoMetricSpace α› with
     eq_of_dist_eq_zero := fun x y hdist => Inseparable.eq <| Metric.inseparable_iff.2 hdist }
-#align metric.of_t0_pseudo_metric_space Metric.ofT0PseudoMetricSpace
+#align metric_space.of_t0_pseudo_metric_space MetricSpace.ofT0PseudoMetricSpace
 
 -- see Note [lower instance priority]
 /-- A metric space induces an emetric space -/
 instance (priority := 100) MetricSpace.toEmetricSpace : EmetricSpace γ :=
-  Emetric.ofT0PseudoEmetricSpace γ
-#align metric.metric_space.to_emetric_space Metric.MetricSpace.toEmetricSpace
+  EmetricSpace.ofT0PseudoEmetricSpace γ
+#align metric_space.to_emetric_space MetricSpace.toEmetricSpace
 
 theorem isClosed_of_pairwise_le_dist {s : Set γ} {ε : ℝ} (hε : 0 < ε)
     (hs : s.Pairwise fun x y => ε ≤ dist x y) : IsClosed s :=
@@ -3310,18 +3294,14 @@ of the edistance to reals. -/
 def EmetricSpace.toMetricSpaceOfDist {α : Type u} [e : EmetricSpace α] (dist : α → α → ℝ)
     (edist_ne_top : ∀ x y : α, edist x y ≠ ⊤) (h : ∀ x y, dist x y = ENNReal.toReal (edist x y)) :
     MetricSpace α :=
-  {
-    PseudoEmetricSpace.toPseudoMetricSpaceOfDist dist edist_ne_top
-      h with
-    dist
-    eq_of_dist_eq_zero := fun x y hxy => by
-      simpa [h, ENNReal.toReal_eq_zero_iff, edist_ne_top x y] using hxy }
+  @MetricSpace.ofT0PseudoMetricSpace α
+    (PseudoEmetricSpace.toPseudoMetricSpaceOfDist dist edist_ne_top h) _
 #align emetric_space.to_metric_space_of_dist EmetricSpace.toMetricSpaceOfDist
 
 /-- One gets a metric space from an emetric space if the edistance
 is everywhere finite, by pushing the edistance to reals. We set it up so that the edist and the
 uniformity are defeq in the metric space and the emetric space. -/
-def EmetricSpace.toMetricSpace {α : Type u} [e : EmetricSpace α] (h : ∀ x y : α, edist x y ≠ ⊤) :
+def EmetricSpace.toMetricSpace {α : Type u} [EmetricSpace α] (h : ∀ x y : α, edist x y ≠ ⊤) :
     MetricSpace α :=
   EmetricSpace.toMetricSpaceOfDist (fun x y => ENNReal.toReal (edist x y)) h fun x y => rfl
 #align emetric_space.to_metric_space EmetricSpace.toMetricSpace
@@ -3380,6 +3360,7 @@ instance : MetricSpace Empty where
   dist _ _ := 0
   dist_self _ := rfl
   dist_comm _ _ := rfl
+  edist _ _ := 0
   eq_of_dist_eq_zero _ _ _ := Subsingleton.elim _ _
   dist_triangle _ _ _ := show (0 : ℝ) ≤ 0 + 0 by rw [add_zero]
   toUniformSpace := Empty.uniformSpace
@@ -3389,6 +3370,7 @@ instance : MetricSpace PUnit.{u + 1} where
   dist _ _ := 0
   dist_self _ := rfl
   dist_comm _ _ := rfl
+  edist _ _ := 0
   eq_of_dist_eq_zero _ _ _ := Subsingleton.elim _ _
   dist_triangle _ _ _ := show (0 : ℝ) ≤ 0 + 0 by rw [add_zero]
   toUniformSpace := PUnit.uniformSpace

mathlib commit https://github.com/leanprover-community/mathlib/commit/4c586d291f189eecb9d00581aeb3dd998ac34442

@@ -132,14 +132,14 @@ private theorem pseudo_metric_space.dist_nonneg' {α} {x y : α} (dist : α →
   nonneg_of_mul_nonneg_right this zero_lt_two
 #align pseudo_metric_space.dist_nonneg' pseudo_metric_space.dist_nonneg'
 
-/- ./././Mathport/Syntax/Translate/Expr.lean:334:4: warning: unsupported (TODO): `[tacs] -/
+/- ./././Mathport/Syntax/Translate/Expr.lean:330:4: warning: unsupported (TODO): `[tacs] -/
 /-- This tactic is used to populate `pseudo_metric_space.edist_dist` when the default `edist` is
 used. -/
 protected unsafe def pseudo_metric_space.edist_dist_tac : tactic Unit :=
   tactic.intros >> sorry
 #align pseudo_metric_space.edist_dist_tac pseudo_metric_space.edist_dist_tac
 
-/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:72:18: unsupported non-interactive tactic pseudo_metric_space.edist_dist_tac -/
+/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic pseudo_metric_space.edist_dist_tac -/
 /-- Pseudo metric and Metric spaces
 
 A pseudo metric space is endowed with a distance for which the requirement `d(x,y)=0 → x = y` might
@@ -288,7 +288,7 @@ theorem dist_le_Ico_sum_dist (f : ℕ → α) {m n} (h : m ≤ n) :
   · intro n hn hrec
     calc
       dist (f m) (f (n + 1)) ≤ dist (f m) (f n) + dist _ _ := dist_triangle _ _ _
-      _ ≤ (∑ i in Finset.Ico m n, _) + _ := add_le_add hrec le_rfl
+      _ ≤ (∑ i in Finset.Ico m n, _) + _ := (add_le_add hrec le_rfl)
       _ = ∑ i in Finset.Ico m (n + 1), _ := by
         rw [Nat.Ico_succ_right_eq_insert_Ico hn, Finset.sum_insert, add_comm] <;> simp
       
@@ -656,7 +656,7 @@ theorem closedBall_eq_bInter_ball : closedBall x ε = ⋂ δ > ε, ball x δ :=
 theorem ball_subset_ball' (h : ε₁ + dist x y ≤ ε₂) : ball x ε₁ ⊆ ball y ε₂ := fun z hz =>
   calc
     dist z y ≤ dist z x + dist x y := dist_triangle _ _ _
-    _ < ε₁ + dist x y := add_lt_add_right hz _
+    _ < ε₁ + dist x y := (add_lt_add_right hz _)
     _ ≤ ε₂ := h
     
 #align metric.ball_subset_ball' Metric.ball_subset_ball'
@@ -669,7 +669,7 @@ theorem closedBall_subset_closed_ball' (h : ε₁ + dist x y ≤ ε₂) :
     closedBall x ε₁ ⊆ closedBall y ε₂ := fun z hz =>
   calc
     dist z y ≤ dist z x + dist x y := dist_triangle _ _ _
-    _ ≤ ε₁ + dist x y := add_le_add_right hz _
+    _ ≤ ε₁ + dist x y := (add_le_add_right hz _)
     _ ≤ ε₂ := h
     
 #align metric.closed_ball_subset_closed_ball' Metric.closedBall_subset_closed_ball'
@@ -682,7 +682,7 @@ theorem closedBall_subset_ball' (h : ε₁ + dist x y < ε₂) : closedBall x ε
   fun z hz =>
   calc
     dist z y ≤ dist z x + dist x y := dist_triangle _ _ _
-    _ ≤ ε₁ + dist x y := add_le_add_right hz _
+    _ ≤ ε₁ + dist x y := (add_le_add_right hz _)
     _ < ε₂ := h
     
 #align metric.closed_ball_subset_ball' Metric.closedBall_subset_ball'
@@ -898,14 +898,14 @@ theorem uniformContinuous_iff [PseudoMetricSpace β] {f : α → β} :
   uniformity_basis_dist.uniformContinuous_iff uniformity_basis_dist
 #align metric.uniform_continuous_iff Metric.uniformContinuous_iff
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:628:2: warning: expanding binder collection (x y «expr ∈ » s) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (x y «expr ∈ » s) -/
 theorem uniformContinuousOn_iff [PseudoMetricSpace β] {f : α → β} {s : Set α} :
     UniformContinuousOn f s ↔
       ∀ ε > 0, ∃ δ > 0, ∀ (x) (_ : x ∈ s) (y) (_ : y ∈ s), dist x y < δ → dist (f x) (f y) < ε :=
   Metric.uniformity_basis_dist.uniformContinuousOn_iff Metric.uniformity_basis_dist
 #align metric.uniform_continuous_on_iff Metric.uniformContinuousOn_iff
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:628:2: warning: expanding binder collection (x y «expr ∈ » s) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (x y «expr ∈ » s) -/
 theorem uniformContinuousOn_iff_le [PseudoMetricSpace β] {f : α → β} {s : Set α} :
     UniformContinuousOn f s ↔
       ∀ ε > 0, ∃ δ > 0, ∀ (x) (_ : x ∈ s) (y) (_ : y ∈ s), dist x y ≤ δ → dist (f x) (f y) ≤ ε :=
@@ -971,7 +971,7 @@ theorem totallyBounded_of_finite_discretization {s : Set α}
   exact ⟨_, ⟨F ⟨x, xs⟩, rfl⟩, hF _ _ this.symm⟩
 #align metric.totally_bounded_of_finite_discretization Metric.totallyBounded_of_finite_discretization
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:628:2: warning: expanding binder collection (t «expr ⊆ » s) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (t «expr ⊆ » s) -/
 theorem finite_approx_of_totallyBounded {s : Set α} (hs : TotallyBounded s) :
     ∀ ε > 0, ∃ (t : _)(_ : t ⊆ s), Set.Finite t ∧ s ⊆ ⋃ y ∈ t, ball y ε :=
   by
@@ -1030,7 +1030,7 @@ theorem tendstoUniformly_iff {ι : Type _} {F : ι → β → α} {f : β → α
   simp
 #align metric.tendsto_uniformly_iff Metric.tendstoUniformly_iff
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:628:2: warning: expanding binder collection (x y «expr ∈ » t) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (x y «expr ∈ » t) -/
 protected theorem cauchy_iff {f : Filter α} :
     Cauchy f ↔ NeBot f ∧ ∀ ε > 0, ∃ t ∈ f, ∀ (x) (_ : x ∈ t) (y) (_ : y ∈ t), dist x y < ε :=
   uniformity_basis_dist.cauchy_iff
@@ -1619,7 +1619,7 @@ section CauchySeq
 
 variable [Nonempty β] [SemilatticeSup β]
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:628:2: warning: expanding binder collection (m n «expr ≥ » N) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (m n «expr ≥ » N) -/
 -- see Note [nolint_ge]
 /-- In a pseudometric space, Cauchy sequences are characterized by the fact that, eventually,
 the distance between its elements is arbitrarily small -/
@@ -1673,7 +1673,7 @@ theorem cauchySeq_of_le_tendsto_0' {s : β → α} (b : β → ℝ)
     (h₀.Eventually (gt_mem_nhds ε0)).exists.imp fun N hN n hn =>
       calc
         dist (s n) (s N) = dist (s N) (s n) := dist_comm _ _
-        _ ≤ b N := h _ _ hn
+        _ ≤ b N := (h _ _ hn)
         _ < ε := hN
         
 #align cauchy_seq_of_le_tendsto_0' cauchySeq_of_le_tendsto_0'
@@ -2120,7 +2120,7 @@ theorem TopologicalSpace.IsSeparable.separableSpace {s : Set α} (hs : IsSeparab
     simp only [A, dif_pos]
     calc
       dist x A.some ≤ dist x z + dist z A.some := dist_triangle _ _ _
-      _ < r / 2 + r / 2 := add_lt_add (hz.trans hn) ((Metric.mem_ball'.1 A.some_spec.1).trans hn)
+      _ < r / 2 + r / 2 := (add_lt_add (hz.trans hn) ((Metric.mem_ball'.1 A.some_spec.1).trans hn))
       _ = r := add_halves _
       
 #align topological_space.is_separable.separable_space TopologicalSpace.IsSeparable.separableSpace
@@ -2314,7 +2314,7 @@ end Pi
 
 section Compact
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:628:2: warning: expanding binder collection (t «expr ⊆ » s) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (t «expr ⊆ » s) -/
 /-- Any compact set in a pseudometric space can be covered by finitely many balls of a given
 positive radius -/
 theorem finite_cover_balls_of_compact {α : Type u} [PseudoMetricSpace α] {s : Set α}
@@ -2408,7 +2408,7 @@ instance (priority := 100) locally_compact_of_proper [ProperSpace α] : LocallyC
     isCompact_closedBall _ _
 #align locally_compact_of_proper locally_compact_of_proper
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:628:2: warning: expanding binder collection (x y «expr ∈ » t) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (x y «expr ∈ » t) -/
 -- see Note [lower instance priority]
 /-- A proper space is complete -/
 instance (priority := 100) complete_of_proper [ProperSpace α] : CompleteSpace α :=
@@ -2521,7 +2521,7 @@ theorem lebesgue_number_lemma_of_metric_unionₛ {s : Set α} {c : Set (Set α)}
 
 namespace Metric
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:628:2: warning: expanding binder collection (x y «expr ∈ » s) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (x y «expr ∈ » s) -/
 /-- Boundedness of a subset of a pseudometric space. We formulate the definition to work
 even in the empty space. -/
 def Bounded (s : Set α) : Prop :=
@@ -2996,7 +2996,7 @@ theorem diam_le_of_subset_closedBall {r : ℝ} (hr : 0 ≤ r) (h : s ⊆ closedB
   diam_le_of_forall_dist_le (mul_nonneg zero_le_two hr) fun a ha b hb =>
     calc
       dist a b ≤ dist a x + dist b x := dist_triangle_right _ _ _
-      _ ≤ r + r := add_le_add (h ha) (h hb)
+      _ ≤ r + r := (add_le_add (h ha) (h hb))
       _ = 2 * r := by simp [mul_two, mul_comm]
       
 #align metric.diam_le_of_subset_closed_ball Metric.diam_le_of_subset_closedBall

mathlib commit https://github.com/leanprover-community/mathlib/commit/4c586d291f189eecb9d00581aeb3dd998ac34442

@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Jeremy Avigad, Robert Y. Lewis, Johannes Hölzl, Mario Carneiro, Sébastien Gouëzel
 
 ! This file was ported from Lean 3 source module topology.metric_space.basic
-! leanprover-community/mathlib commit 0c1f285a9f6e608ae2bdffa3f993eafb01eba829
+! leanprover-community/mathlib commit e1a7bdeb4fd826b7e71d130d34988f0a2d26a177
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
@@ -60,44 +60,12 @@ universe u v w
 
 variable {α : Type u} {β : Type v} {X ι : Type _}
 
-/-- Construct a uniform structure core from a distance function and metric space axioms.
-This is a technical construction that can be immediately used to construct a uniform structure
-from a distance function and metric space axioms but is also useful when discussing
-metrizable topologies, see `pseudo_metric_space.of_metrizable`. -/
-def UniformSpace.coreOfDist {α : Type _} (dist : α → α → ℝ) (dist_self : ∀ x : α, dist x x = 0)
-    (dist_comm : ∀ x y : α, dist x y = dist y x)
-    (dist_triangle : ∀ x y z : α, dist x z ≤ dist x y + dist y z) : UniformSpace.Core α
-    where
-  uniformity := ⨅ ε > 0, 𝓟 { p : α × α | dist p.1 p.2 < ε }
-  refl :=
-    le_infᵢ fun ε =>
-      le_infᵢ <| by
-        simp (config := { contextual := true }) [Set.subset_def, idRel, dist_self, (· > ·)]
-  comp :=
-    le_infᵢ fun ε =>
-      le_infᵢ fun h =>
-        lift'_le
-            (mem_infᵢ_of_mem (ε / 2) <| mem_infᵢ_of_mem (div_pos h zero_lt_two) (Subset.refl _)) <|
-          by
-          have : ∀ a b c : α, dist a c < ε / 2 → dist c b < ε / 2 → dist a b < ε :=
-            fun a b c hac hcb =>
-            calc
-              dist a b ≤ dist a c + dist c b := dist_triangle _ _ _
-              _ < ε / 2 + ε / 2 := add_lt_add hac hcb
-              _ = ε := by rw [div_add_div_same, add_self_div_two]
-              
-          simpa [compRel]
-  symm :=
-    tendsto_infᵢ.2 fun ε =>
-      tendsto_infᵢ.2 fun h =>
-        tendsto_infᵢ' ε <| tendsto_infᵢ' h <| tendsto_principal_principal.2 <| by simp [dist_comm]
-#align uniform_space.core_of_dist UniformSpace.coreOfDist
-
 /-- Construct a uniform structure from a distance function and metric space axioms -/
 def uniformSpaceOfDist (dist : α → α → ℝ) (dist_self : ∀ x : α, dist x x = 0)
     (dist_comm : ∀ x y : α, dist x y = dist y x)
     (dist_triangle : ∀ x y z : α, dist x z ≤ dist x y + dist y z) : UniformSpace α :=
-  UniformSpace.ofCore (UniformSpace.coreOfDist dist dist_self dist_comm dist_triangle)
+  UniformSpace.ofFun dist dist_self dist_comm dist_triangle fun ε ε0 =>
+    ⟨ε / 2, half_pos ε0, fun x hx y hy => add_halves ε ▸ add_lt_add hx hy⟩
 #align uniform_space_of_dist uniformSpaceOfDist
 
 /-- This is an internal lemma used to construct a bornology from a metric in `bornology.of_dist`. -/
@@ -242,7 +210,7 @@ instance (priority := 200) PseudoMetricSpace.toHasEdist : HasEdist α :=
 /-- Construct a pseudo-metric space structure whose underlying topological space structure
 (definitionally) agrees which a pre-existing topology which is compatible with a given distance
 function. -/
-def PseudoMetricSpace.ofMetrizable {α : Type _} [TopologicalSpace α] (dist : α → α → ℝ)
+def PseudoMetricSpace.ofDistTopology {α : Type u} [TopologicalSpace α] (dist : α → α → ℝ)
     (dist_self : ∀ x : α, dist x x = 0) (dist_comm : ∀ x y : α, dist x y = dist y x)
     (dist_triangle : ∀ x y z : α, dist x z ≤ dist x y + dist y z)
     (H : ∀ s : Set α, IsOpen s ↔ ∀ x ∈ s, ∃ ε > 0, ∀ y, dist x y < ε → y ∈ s) :
@@ -252,31 +220,17 @@ def PseudoMetricSpace.ofMetrizable {α : Type _} [TopologicalSpace α] (dist : 
     dist_comm
     dist_triangle
     toUniformSpace :=
-      { UniformSpace.coreOfDist dist dist_self dist_comm dist_triangle with
-        isOpen_uniformity := by
-          dsimp only [UniformSpace.coreOfDist]
-          intro s
-          change IsOpen s ↔ _
-          rw [H s]
-          refine' forall₂_congr fun x x_in => _
-          erw [(has_basis_binfi_principal _ nonempty_Ioi).mem_iff]
-          · refine' exists₂_congr fun ε ε_pos => _
-            simp only [Prod.forall, set_of_subset_set_of]
-            constructor
-            · rintro h _ y H rfl
-              exact h y H
-            · intro h y hxy
-              exact h _ _ hxy rfl
-          ·
-            exact fun r (hr : 0 < r) p (hp : 0 < p) =>
-              ⟨min r p, lt_min hr hp, fun x (hx : dist _ _ < _) =>
-                lt_of_lt_of_le hx (min_le_left r p), fun x (hx : dist _ _ < _) =>
-                lt_of_lt_of_le hx (min_le_right r p)⟩
-          · infer_instance }
+      { isOpen_uniformity := fun s =>
+          (H s).trans <|
+            forall₂_congr fun x _ =>
+              ((UniformSpace.hasBasis_ofFun (exists_gt (0 : ℝ)) dist _ _ _ _).comap
+                        (Prod.mk x)).mem_iff.symm.trans
+                mem_comap_prod_mk
+        toCore := (uniformSpaceOfDist dist dist_self dist_comm dist_triangle).toCore }
     uniformity_dist := rfl
     toBornology := Bornology.ofDist dist dist_self dist_comm dist_triangle
     cobounded_sets := rfl }
-#align pseudo_metric_space.of_metrizable PseudoMetricSpace.ofMetrizable
+#align pseudo_metric_space.of_dist_topology PseudoMetricSpace.ofDistTopology
 
 @[simp]
 theorem dist_self (x : α) : dist x x = 0 :=
@@ -831,14 +785,17 @@ theorem isBounded_iff_nndist {s : Set α} :
     NNReal.coe_mk, exists_prop]
 #align metric.is_bounded_iff_nndist Metric.isBounded_iff_nndist
 
+theorem toUniformSpace_eq :
+    ‹PseudoMetricSpace α›.toUniformSpace =
+      uniformSpaceOfDist dist dist_self dist_comm dist_triangle :=
+  uniformSpace_eq PseudoMetricSpace.uniformity_dist
+#align metric.to_uniform_space_eq Metric.toUniformSpace_eq
+
 theorem uniformity_basis_dist :
     (𝓤 α).HasBasis (fun ε : ℝ => 0 < ε) fun ε => { p : α × α | dist p.1 p.2 < ε } :=
   by
-  rw [← pseudo_metric_space.uniformity_dist.symm]
-  refine' has_basis_binfi_principal _ nonempty_Ioi
-  exact fun r (hr : 0 < r) p (hp : 0 < p) =>
-    ⟨min r p, lt_min hr hp, fun x (hx : dist _ _ < _) => lt_of_lt_of_le hx (min_le_left r p),
-      fun x (hx : dist _ _ < _) => lt_of_lt_of_le hx (min_le_right r p)⟩
+  rw [to_uniform_space_eq]
+  exact UniformSpace.hasBasis_ofFun (exists_gt _) _ _ _ _ _
 #align metric.uniformity_basis_dist Metric.uniformity_basis_dist
 
 /-- Given `f : β → ℝ`, if `f` sends `{i | p i}` to a set of positive numbers
@@ -859,6 +816,13 @@ protected theorem mk_uniformity_basis {β : Type _} {p : β → Prop} {f : β 
   · exact fun ⟨i, hi, H⟩ => ⟨f i, hf₀ i hi, H⟩
 #align metric.mk_uniformity_basis Metric.mk_uniformity_basis
 
+theorem uniformity_basis_dist_rat :
+    (𝓤 α).HasBasis (fun r : ℚ => 0 < r) fun r => { p : α × α | dist p.1 p.2 < r } :=
+  Metric.mk_uniformity_basis (fun _ => Rat.cast_pos.2) fun ε hε =>
+    let ⟨r, hr0, hrε⟩ := exists_rat_btwn hε
+    ⟨r, Rat.cast_pos.1 hr0, hrε.le⟩
+#align metric.uniformity_basis_dist_rat Metric.uniformity_basis_dist_rat
+
 theorem uniformity_basis_dist_inv_nat_succ :
     (𝓤 α).HasBasis (fun _ => True) fun n : ℕ => { p : α × α | dist p.1 p.2 < 1 / (↑n + 1) } :=
   Metric.mk_uniformity_basis (fun n _ => div_pos zero_lt_one <| Nat.cast_add_one_pos n) fun ε ε0 =>
@@ -1792,11 +1756,7 @@ def PseudoMetricSpace.induced {α β} (f : α → β) (m : PseudoMetricSpace β)
   edist x y := edist (f x) (f y)
   edist_dist x y := edist_dist _ _
   toUniformSpace := UniformSpace.comap f m.toUniformSpace
-  uniformity_dist :=
-    by
-    apply @uniformity_dist_of_mem_uniformity _ _ _ _ _ fun x y => dist (f x) (f y)
-    refine' compl_surjective.forall.2 fun s => compl_mem_comap.trans <| mem_uniformity_dist.trans _
-    simp only [mem_compl_iff, @imp_not_comm _ (_ ∈ _), ← Prod.forall', Prod.mk.eta, ball_image_iff]
+  uniformity_dist := (uniformity_basis_dist.comap _).eq_binfᵢ
   toBornology := Bornology.induced f
   cobounded_sets :=
     Set.ext <|
@@ -3163,14 +3123,14 @@ theorem MetricSpace.ext {α : Type _} {m m' : MetricSpace α} (h : m.toHasDist =
 /-- Construct a metric space structure whose underlying topological space structure
 (definitionally) agrees which a pre-existing topology which is compatible with a given distance
 function. -/
-def MetricSpace.ofMetrizable {α : Type _} [TopologicalSpace α] (dist : α → α → ℝ)
+def MetricSpace.ofDistTopology {α : Type u} [TopologicalSpace α] (dist : α → α → ℝ)
     (dist_self : ∀ x : α, dist x x = 0) (dist_comm : ∀ x y : α, dist x y = dist y x)
     (dist_triangle : ∀ x y z : α, dist x z ≤ dist x y + dist y z)
     (H : ∀ s : Set α, IsOpen s ↔ ∀ x ∈ s, ∃ ε > 0, ∀ y, dist x y < ε → y ∈ s)
     (eq_of_dist_eq_zero : ∀ x y : α, dist x y = 0 → x = y) : MetricSpace α :=
-  { PseudoMetricSpace.ofMetrizable dist dist_self dist_comm dist_triangle H with
+  { PseudoMetricSpace.ofDistTopology dist dist_self dist_comm dist_triangle H with
     eq_of_dist_eq_zero }
-#align metric_space.of_metrizable MetricSpace.ofMetrizable
+#align metric_space.of_dist_topology MetricSpace.ofDistTopology
 
 variable {γ : Type w} [MetricSpace γ]
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/9da1b3534b65d9661eb8f42443598a92bbb49211

@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Jeremy Avigad, Robert Y. Lewis, Johannes Hölzl, Mario Carneiro, Sébastien Gouëzel
 
 ! This file was ported from Lean 3 source module topology.metric_space.basic
-! leanprover-community/mathlib commit 92ca63f0fb391a9ca5f22d2409a6080e786d99f7
+! leanprover-community/mathlib commit 0c1f285a9f6e608ae2bdffa3f993eafb01eba829
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
@@ -3529,71 +3529,20 @@ end Metric
 
 section EqRel
 
-/-- The canonical equivalence relation on a pseudometric space. -/
-def PseudoMetric.distSetoid (α : Type u) [PseudoMetricSpace α] : Setoid α :=
-  Setoid.mk (fun x y => dist x y = 0)
-    (by
-      unfold Equivalence
-      repeat' constructor
-      · exact PseudoMetricSpace.dist_self
-      · intro x y h
-        rwa [PseudoMetricSpace.dist_comm]
-      · intro x y z hxy hyz
-        refine' le_antisymm _ dist_nonneg
-        calc
-          dist x z ≤ dist x y + dist y z := PseudoMetricSpace.dist_triangle _ _ _
-          _ = 0 + 0 := by rw [hxy, hyz]
-          _ = 0 := by simp
-          )
-#align pseudo_metric.dist_setoid PseudoMetric.distSetoid
-
-attribute [local instance] PseudoMetric.distSetoid
-
-/-- The canonical quotient of a pseudometric space, identifying points at distance `0`. -/
-@[reducible]
-def PseudoMetricQuot (α : Type u) [PseudoMetricSpace α] : Type _ :=
-  Quotient (PseudoMetric.distSetoid α)
-#align pseudo_metric_quot PseudoMetricQuot
-
-instance hasDistMetricQuot {α : Type u} [PseudoMetricSpace α] : HasDist (PseudoMetricQuot α)
-    where dist :=
-    Quotient.lift₂ (fun p q : α => dist p q)
-      (by
-        intro x y x' y' hxx' hyy'
-        have Hxx' : dist x x' = 0 := hxx'
-        have Hyy' : dist y y' = 0 := hyy'
-        have A : dist x y ≤ dist x' y' :=
-          calc
-            dist x y ≤ dist x x' + dist x' y := PseudoMetricSpace.dist_triangle _ _ _
-            _ = dist x' y := by simp [Hxx']
-            _ ≤ dist x' y' + dist y' y := PseudoMetricSpace.dist_triangle _ _ _
-            _ = dist x' y' := by simp [PseudoMetricSpace.dist_comm, Hyy']
-            
-        have B : dist x' y' ≤ dist x y :=
-          calc
-            dist x' y' ≤ dist x' x + dist x y' := PseudoMetricSpace.dist_triangle _ _ _
-            _ = dist x y' := by simp [PseudoMetricSpace.dist_comm, Hxx']
-            _ ≤ dist x y + dist y y' := PseudoMetricSpace.dist_triangle _ _ _
-            _ = dist x y := by simp [Hyy']
-            
-        exact le_antisymm A B)
-#align has_dist_metric_quot hasDistMetricQuot
-
-theorem pseudo_metric_quot_dist_eq {α : Type u} [PseudoMetricSpace α] (p q : α) :
-    dist ⟦p⟧ ⟦q⟧ = dist p q :=
+instance {α : Type u} [PseudoMetricSpace α] : HasDist (UniformSpace.SeparationQuotient α)
+    where dist p q :=
+    Quotient.liftOn₂' p q dist fun x y x' y' hx hy => by
+      rw [dist_edist, dist_edist, ← UniformSpace.SeparationQuotient.edist_mk x, ←
+        UniformSpace.SeparationQuotient.edist_mk x', Quot.sound hx, Quot.sound hy]
+
+theorem UniformSpace.SeparationQuotient.dist_mk {α : Type u} [PseudoMetricSpace α] (p q : α) :
+    @dist (UniformSpace.SeparationQuotient α) _ (Quot.mk _ p) (Quot.mk _ q) = dist p q :=
   rfl
-#align pseudo_metric_quot_dist_eq pseudo_metric_quot_dist_eq
+#align uniform_space.separation_quotient.dist_mk UniformSpace.SeparationQuotient.dist_mk
 
-instance metricSpaceQuot {α : Type u} [PseudoMetricSpace α] : MetricSpace (PseudoMetricQuot α)
-    where
-  dist_self := by
-    refine' Quotient.ind fun y => _
-    exact PseudoMetricSpace.dist_self _
-  eq_of_dist_eq_zero xc yc := Quotient.induction_on₂ xc yc fun x y H => Quotient.sound H
-  dist_comm xc yc := Quotient.induction_on₂ xc yc fun x y => PseudoMetricSpace.dist_comm _ _
-  dist_triangle xc yc zc :=
-    Quotient.induction_on₃ xc yc zc fun x y z => PseudoMetricSpace.dist_triangle _ _ _
-#align metric_space_quot metricSpaceQuot
+instance {α : Type u} [PseudoMetricSpace α] : MetricSpace (UniformSpace.SeparationQuotient α) :=
+  EmetricSpace.toMetricSpaceOfDist dist (fun x y => Quotient.inductionOn₂' x y edist_ne_top)
+    fun x y => Quotient.inductionOn₂' x y dist_edist
 
 end EqRel
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/eb0cb4511aaef0da2462207b67358a0e1fe1e2ee

@@ -54,7 +54,7 @@ metric, pseudo_metric, dist
 
 open Set Filter TopologicalSpace Bornology
 
-open uniformity Topology BigOperators Filter NNReal Ennreal
+open uniformity Topology BigOperators Filter NNReal ENNReal
 
 universe u v w
 
@@ -188,7 +188,7 @@ class PseudoMetricSpace (α : Type u) extends HasDist α : Type u where
   dist_triangle : ∀ x y z : α, dist x z ≤ dist x y + dist y z
   edist : α → α → ℝ≥0∞ := fun x y =>
     @coe ℝ≥0 _ _ ⟨dist x y, PseudoMetricSpace.dist_nonneg' _ ‹_› ‹_› ‹_›⟩
-  edist_dist : ∀ x y : α, edist x y = Ennreal.ofReal (dist x y) := by
+  edist_dist : ∀ x y : α, edist x y = ENNReal.ofReal (dist x y) := by
     run_tac
       pseudo_metric_space.edist_dist_tac
   toUniformSpace : UniformSpace α := uniformSpaceOfDist dist dist_self dist_comm dist_triangle
@@ -287,7 +287,7 @@ theorem dist_comm (x y : α) : dist x y = dist y x :=
   PseudoMetricSpace.dist_comm x y
 #align dist_comm dist_comm
 
-theorem edist_dist (x y : α) : edist x y = Ennreal.ofReal (dist x y) :=
+theorem edist_dist (x y : α) : edist x y = ENNReal.ofReal (dist x y) :=
   PseudoMetricSpace.edist_dist x y
 #align edist_dist edist_dist
 
@@ -408,32 +408,32 @@ instance (priority := 100) PseudoMetricSpace.toHasNndist : HasNndist α :=
 
 /-- Express `nndist` in terms of `edist`-/
 theorem nndist_edist (x y : α) : nndist x y = (edist x y).toNNReal := by
-  simp [nndist, edist_dist, Real.toNNReal, max_eq_left dist_nonneg, Ennreal.ofReal]
+  simp [nndist, edist_dist, Real.toNNReal, max_eq_left dist_nonneg, ENNReal.ofReal]
 #align nndist_edist nndist_edist
 
 /-- Express `edist` in terms of `nndist`-/
 theorem edist_nndist (x y : α) : edist x y = ↑(nndist x y) := by
-  simpa only [edist_dist, Ennreal.ofReal_eq_coe_nNReal dist_nonneg]
+  simpa only [edist_dist, ENNReal.ofReal_eq_coe_nnreal dist_nonneg]
 #align edist_nndist edist_nndist
 
 @[simp, norm_cast]
-theorem coe_nNReal_ennreal_nndist (x y : α) : ↑(nndist x y) = edist x y :=
+theorem coe_nNReal_eNNReal_nndist (x y : α) : ↑(nndist x y) = edist x y :=
   (edist_nndist x y).symm
-#align coe_nnreal_ennreal_nndist coe_nNReal_ennreal_nndist
+#align coe_nnreal_ennreal_nndist coe_nNReal_eNNReal_nndist
 
 @[simp, norm_cast]
 theorem edist_lt_coe {x y : α} {c : ℝ≥0} : edist x y < c ↔ nndist x y < c := by
-  rw [edist_nndist, Ennreal.coe_lt_coe]
+  rw [edist_nndist, ENNReal.coe_lt_coe]
 #align edist_lt_coe edist_lt_coe
 
 @[simp, norm_cast]
 theorem edist_le_coe {x y : α} {c : ℝ≥0} : edist x y ≤ c ↔ nndist x y ≤ c := by
-  rw [edist_nndist, Ennreal.coe_le_coe]
+  rw [edist_nndist, ENNReal.coe_le_coe]
 #align edist_le_coe edist_le_coe
 
 /-- In a pseudometric space, the extended distance is always finite-/
 theorem edist_lt_top {α : Type _} [PseudoMetricSpace α] (x y : α) : edist x y < ⊤ :=
-  (edist_dist x y).symm ▸ Ennreal.ofReal_lt_top
+  (edist_dist x y).symm ▸ ENNReal.ofReal_lt_top
 #align edist_lt_top edist_lt_top
 
 /-- In a pseudometric space, the extended distance is always finite-/
@@ -468,14 +468,14 @@ theorem dist_le_coe {x y : α} {c : ℝ≥0} : dist x y ≤ c ↔ nndist x y ≤
 #align dist_le_coe dist_le_coe
 
 @[simp]
-theorem edist_lt_ofReal {x y : α} {r : ℝ} : edist x y < Ennreal.ofReal r ↔ dist x y < r := by
-  rw [edist_dist, Ennreal.ofReal_lt_ofReal_iff_of_nonneg dist_nonneg]
+theorem edist_lt_ofReal {x y : α} {r : ℝ} : edist x y < ENNReal.ofReal r ↔ dist x y < r := by
+  rw [edist_dist, ENNReal.ofReal_lt_ofReal_iff_of_nonneg dist_nonneg]
 #align edist_lt_of_real edist_lt_ofReal
 
 @[simp]
 theorem edist_le_ofReal {x y : α} {r : ℝ} (hr : 0 ≤ r) :
-    edist x y ≤ Ennreal.ofReal r ↔ dist x y ≤ r := by
-  rw [edist_dist, Ennreal.ofReal_le_ofReal_iff hr]
+    edist x y ≤ ENNReal.ofReal r ↔ dist x y ≤ r := by
+  rw [edist_dist, ENNReal.ofReal_le_ofReal_iff hr]
 #align edist_le_of_real edist_le_ofReal
 
 /-- Express `nndist` in terms of `dist`-/
@@ -502,7 +502,7 @@ theorem nndist_triangle_right (x y z : α) : nndist x y ≤ nndist x z + nndist
 
 /-- Express `dist` in terms of `edist`-/
 theorem dist_edist (x y : α) : dist x y = (edist x y).toReal := by
-  rw [edist_dist, Ennreal.toReal_ofReal dist_nonneg]
+  rw [edist_dist, ENNReal.toReal_ofReal dist_nonneg]
 #align dist_edist dist_edist
 
 namespace Metric
@@ -1302,14 +1302,14 @@ protected theorem PseudoMetric.uniformity_basis_edist :
   ⟨by
     intro t
     refine' mem_uniformity_dist.trans ⟨_, _⟩ <;> rintro ⟨ε, ε0, Hε⟩
-    · use Ennreal.ofReal ε, Ennreal.ofReal_pos.2 ε0
+    · use ENNReal.ofReal ε, ENNReal.ofReal_pos.2 ε0
       rintro ⟨a, b⟩
-      simp only [edist_dist, Ennreal.ofReal_lt_ofReal_iff ε0]
+      simp only [edist_dist, ENNReal.ofReal_lt_ofReal_iff ε0]
       exact Hε
-    · rcases Ennreal.lt_iff_exists_real_btwn.1 ε0 with ⟨ε', _, ε0', hε⟩
-      rw [Ennreal.ofReal_pos] at ε0'
+    · rcases ENNReal.lt_iff_exists_real_btwn.1 ε0 with ⟨ε', _, ε0', hε⟩
+      rw [ENNReal.ofReal_pos] at ε0'
       refine' ⟨ε', ε0', fun a b h => Hε (lt_trans _ hε)⟩
-      rwa [edist_dist, Ennreal.ofReal_lt_ofReal_iff ε0']⟩
+      rwa [edist_dist, ENNReal.ofReal_lt_ofReal_iff ε0']⟩
 #align pseudo_metric.uniformity_basis_edist PseudoMetric.uniformity_basis_edist
 
 theorem Metric.uniformity_edist : 𝓤 α = ⨅ ε > 0, 𝓟 { p : α × α | edist p.1 p.2 < ε } :=
@@ -1325,8 +1325,8 @@ instance (priority := 100) PseudoMetricSpace.toPseudoEmetricSpace : PseudoEmetri
     edist_comm := by simp only [edist_dist, dist_comm] <;> simp
     edist_triangle := fun x y z =>
       by
-      simp only [edist_dist, ← Ennreal.ofReal_add, dist_nonneg]
-      rw [Ennreal.ofReal_le_ofReal_iff _]
+      simp only [edist_dist, ← ENNReal.ofReal_add, dist_nonneg]
+      rw [ENNReal.ofReal_le_ofReal_iff _]
       · exact dist_triangle _ _ _
       · simpa using add_le_add (dist_nonneg : 0 ≤ dist x y) dist_nonneg
     uniformity_edist := Metric.uniformity_edist }
@@ -1339,11 +1339,11 @@ theorem Metric.eball_top_eq_univ (x : α) : Emetric.ball x ∞ = Set.univ :=
 
 /-- Balls defined using the distance or the edistance coincide -/
 @[simp]
-theorem Metric.emetric_ball {x : α} {ε : ℝ} : Emetric.ball x (Ennreal.ofReal ε) = ball x ε :=
+theorem Metric.emetric_ball {x : α} {ε : ℝ} : Emetric.ball x (ENNReal.ofReal ε) = ball x ε :=
   by
   ext y
   simp only [Emetric.mem_ball, mem_ball, edist_dist]
-  exact Ennreal.ofReal_lt_ofReal_iff_of_nonneg dist_nonneg
+  exact ENNReal.ofReal_lt_ofReal_iff_of_nonneg dist_nonneg
 #align metric.emetric_ball Metric.emetric_ball
 
 /-- Balls defined using the distance or the edistance coincide -/
@@ -1356,8 +1356,8 @@ theorem Metric.emetric_ball_nNReal {x : α} {ε : ℝ≥0} : Emetric.ball x ε =
 
 /-- Closed balls defined using the distance or the edistance coincide -/
 theorem Metric.emetric_closedBall {x : α} {ε : ℝ} (h : 0 ≤ ε) :
-    Emetric.closedBall x (Ennreal.ofReal ε) = closedBall x ε := by
-  ext y <;> simp [edist_dist] <;> rw [Ennreal.ofReal_le_ofReal_iff h]
+    Emetric.closedBall x (ENNReal.ofReal ε) = closedBall x ε := by
+  ext y <;> simp [edist_dist] <;> rw [ENNReal.ofReal_le_ofReal_iff h]
 #align metric.emetric_closed_ball Metric.emetric_closedBall
 
 /-- Closed balls defined using the distance or the edistance coincide -/
@@ -1375,7 +1375,7 @@ theorem Metric.emetric_ball_top (x : α) : Emetric.ball x ⊤ = univ :=
 #align metric.emetric_ball_top Metric.emetric_ball_top
 
 theorem Metric.inseparable_iff {x y : α} : Inseparable x y ↔ dist x y = 0 := by
-  rw [Emetric.inseparable_iff, edist_nndist, dist_nndist, Ennreal.coe_eq_zero, NNReal.coe_eq_zero]
+  rw [Emetric.inseparable_iff, edist_nndist, dist_nndist, ENNReal.coe_eq_zero, NNReal.coe_eq_zero]
 #align metric.inseparable_iff Metric.inseparable_iff
 
 /-- Build a new pseudometric space from an old one where the bundled uniform structure is provably
@@ -1426,19 +1426,19 @@ distance is given separately, to be able to prescribe some expression which is n
 push-forward of the edistance to reals. -/
 def PseudoEmetricSpace.toPseudoMetricSpaceOfDist {α : Type u} [e : PseudoEmetricSpace α]
     (dist : α → α → ℝ) (edist_ne_top : ∀ x y : α, edist x y ≠ ⊤)
-    (h : ∀ x y, dist x y = Ennreal.toReal (edist x y)) : PseudoMetricSpace α :=
+    (h : ∀ x y, dist x y = ENNReal.toReal (edist x y)) : PseudoMetricSpace α :=
   let m : PseudoMetricSpace α :=
     { dist
       dist_self := fun x => by simp [h]
       dist_comm := fun x y => by simp [h, PseudoEmetricSpace.edist_comm]
       dist_triangle := fun x y z => by
         simp only [h]
-        rw [← Ennreal.toReal_add (edist_ne_top _ _) (edist_ne_top _ _),
-          Ennreal.toReal_le_toReal (edist_ne_top _ _)]
+        rw [← ENNReal.toReal_add (edist_ne_top _ _) (edist_ne_top _ _),
+          ENNReal.toReal_le_toReal (edist_ne_top _ _)]
         · exact edist_triangle _ _ _
-        · simp [Ennreal.add_eq_top, edist_ne_top]
+        · simp [ENNReal.add_eq_top, edist_ne_top]
       edist := edist
-      edist_dist := fun x y => by simp [h, Ennreal.ofReal_toReal, edist_ne_top] }
+      edist_dist := fun x y => by simp [h, ENNReal.ofReal_toReal, edist_ne_top] }
   m.replaceUniformity <|
     by
     rw [uniformity_pseudoedist, Metric.uniformity_edist]
@@ -1450,7 +1450,7 @@ is everywhere finite, by pushing the edistance to reals. We set it up so that th
 uniformity are defeq in the pseudometric space and the emetric space. -/
 def PseudoEmetricSpace.toPseudoMetricSpace {α : Type u} [e : PseudoEmetricSpace α]
     (h : ∀ x y : α, edist x y ≠ ⊤) : PseudoMetricSpace α :=
-  PseudoEmetricSpace.toPseudoMetricSpaceOfDist (fun x y => Ennreal.toReal (edist x y)) h fun x y =>
+  PseudoEmetricSpace.toPseudoMetricSpaceOfDist (fun x y => ENNReal.toReal (edist x y)) h fun x y =>
     rfl
 #align pseudo_emetric_space.to_pseudo_metric_space PseudoEmetricSpace.toPseudoMetricSpace
 
@@ -1938,7 +1938,7 @@ instance Prod.pseudoMetricSpaceMax : PseudoMetricSpace (α × β) :=
   (PseudoEmetricSpace.toPseudoMetricSpaceOfDist (fun x y : α × β => dist x.1 y.1 ⊔ dist x.2 y.2)
         (fun x y => (max_lt (edist_lt_top _ _) (edist_lt_top _ _)).Ne) fun x y => by
         simp only [sup_eq_max, dist_edist, ←
-          Ennreal.toReal_max (edist_ne_top _ _) (edist_ne_top _ _), Prod.edist_eq]).replaceBornology
+          ENNReal.toReal_max (edist_ne_top _ _) (edist_ne_top _ _), Prod.edist_eq]).replaceBornology
     fun s =>
     by
     simp only [← is_bounded_image_fst_and_snd, is_bounded_iff_eventually, ball_image_iff, ←
@@ -2216,7 +2216,7 @@ instance pseudoMetricSpacePi : PseudoMetricSpace (∀ b, π b) :=
   show edist f g ≠ ⊤
   exact ne_of_lt ((Finset.sup_lt_iff bot_lt_top).2 fun b hb => edist_lt_top _ _)
   show ↑(sup univ fun b => nndist (f b) (g b)) = (sup univ fun b => edist (f b) (g b)).toReal
-  · simp only [edist_nndist, ← Ennreal.coe_finset_sup, Ennreal.coe_toReal]
+  · simp only [edist_nndist, ← ENNReal.coe_finset_sup, ENNReal.coe_toReal]
   show @is_bounded _ Pi.bornology s ↔ @is_bounded _ PseudoMetricSpace.toBornology _
   · simp only [← is_bounded_def, is_bounded_iff_eventually, ← forall_is_bounded_image_eval_iff,
       ball_image_iff, ← eventually_all, Function.eval_apply, @dist_nndist (π _)]
@@ -2275,7 +2275,7 @@ theorem nndist_pi_const_le (a b : α) : (nndist (fun _ : β => a) fun _ => b) 
 
 @[simp]
 theorem dist_pi_const [Nonempty β] (a b : α) : (dist (fun x : β => a) fun _ => b) = dist a b := by
-  simpa only [dist_edist] using congr_arg Ennreal.toReal (edist_pi_const a b)
+  simpa only [dist_edist] using congr_arg ENNReal.toReal (edist_pi_const a b)
 #align dist_pi_const dist_pi_const
 
 @[simp]
@@ -2535,7 +2535,7 @@ theorem second_countable_of_almost_dense_set
     SecondCountableTopology α :=
   by
   refine' Emetric.second_countable_of_almost_dense_set fun ε ε0 => _
-  rcases Ennreal.lt_iff_exists_nNReal_btwn.1 ε0 with ⟨ε', ε'0, ε'ε⟩
+  rcases ENNReal.lt_iff_exists_nnreal_btwn.1 ε0 with ⟨ε', ε'0, ε'ε⟩
   choose s hsc y hys hyx using H ε' (by exact_mod_cast ε'0)
   refine' ⟨s, hsc, Union₂_eq_univ_iff.2 fun x => ⟨y x, hys _, le_trans _ ε'ε.le⟩⟩
   exact_mod_cast hyx x
@@ -2884,16 +2884,16 @@ variable {s : Set α} {x y z : α}
 /-- The diameter of a set in a metric space. To get controllable behavior even when the diameter
 should be infinite, we express it in terms of the emetric.diameter -/
 noncomputable def diam (s : Set α) : ℝ :=
-  Ennreal.toReal (Emetric.diam s)
+  ENNReal.toReal (Emetric.diam s)
 #align metric.diam Metric.diam
 
 /-- The diameter of a set is always nonnegative -/
 theorem diam_nonneg : 0 ≤ diam s :=
-  Ennreal.toReal_nonneg
+  ENNReal.toReal_nonneg
 #align metric.diam_nonneg Metric.diam_nonneg
 
 theorem diam_subsingleton (hs : s.Subsingleton) : diam s = 0 := by
-  simp only [diam, Emetric.diam_subsingleton hs, Ennreal.zero_toReal]
+  simp only [diam, Emetric.diam_subsingleton hs, ENNReal.zero_toReal]
 #align metric.diam_subsingleton Metric.diam_subsingleton
 
 /-- The empty set has zero diameter -/
@@ -2918,21 +2918,21 @@ theorem diam_triple :
     Metric.diam ({x, y, z} : Set α) = max (max (dist x y) (dist x z)) (dist y z) :=
   by
   simp only [Metric.diam, Emetric.diam_triple, dist_edist]
-  rw [Ennreal.toReal_max, Ennreal.toReal_max] <;> apply_rules [ne_of_lt, edist_lt_top, max_lt]
+  rw [ENNReal.toReal_max, ENNReal.toReal_max] <;> apply_rules [ne_of_lt, edist_lt_top, max_lt]
 #align metric.diam_triple Metric.diam_triple
 
 /-- If the distance between any two points in a set is bounded by some constant `C`,
 then `ennreal.of_real C`  bounds the emetric diameter of this set. -/
 theorem ediam_le_of_forall_dist_le {C : ℝ} (h : ∀ x ∈ s, ∀ y ∈ s, dist x y ≤ C) :
-    Emetric.diam s ≤ Ennreal.ofReal C :=
-  Emetric.diam_le fun x hx y hy => (edist_dist x y).symm ▸ Ennreal.ofReal_le_ofReal (h x hx y hy)
+    Emetric.diam s ≤ ENNReal.ofReal C :=
+  Emetric.diam_le fun x hx y hy => (edist_dist x y).symm ▸ ENNReal.ofReal_le_ofReal (h x hx y hy)
 #align metric.ediam_le_of_forall_dist_le Metric.ediam_le_of_forall_dist_le
 
 /-- If the distance between any two points in a set is bounded by some non-negative constant,
 this constant bounds the diameter. -/
 theorem diam_le_of_forall_dist_le {C : ℝ} (h₀ : 0 ≤ C) (h : ∀ x ∈ s, ∀ y ∈ s, dist x y ≤ C) :
     diam s ≤ C :=
-  Ennreal.toReal_le_of_le_ofReal h₀ (ediam_le_of_forall_dist_le h)
+  ENNReal.toReal_le_of_le_ofReal h₀ (ediam_le_of_forall_dist_le h)
 #align metric.diam_le_of_forall_dist_le Metric.diam_le_of_forall_dist_le
 
 /-- If the distance between any two points in a nonempty set is bounded by some constant,
@@ -2949,14 +2949,14 @@ theorem diam_le_of_forall_dist_le_of_nonempty (hs : s.Nonempty) {C : ℝ}
 theorem dist_le_diam_of_mem' (h : Emetric.diam s ≠ ⊤) (hx : x ∈ s) (hy : y ∈ s) :
     dist x y ≤ diam s := by
   rw [diam, dist_edist]
-  rw [Ennreal.toReal_le_toReal (edist_ne_top _ _) h]
+  rw [ENNReal.toReal_le_toReal (edist_ne_top _ _) h]
   exact Emetric.edist_le_diam_of_mem hx hy
 #align metric.dist_le_diam_of_mem' Metric.dist_le_diam_of_mem'
 
 /-- Characterize the boundedness of a set in terms of the finiteness of its emetric.diameter. -/
 theorem bounded_iff_ediam_ne_top : Bounded s ↔ Emetric.diam s ≠ ⊤ :=
   Iff.intro
-    (fun ⟨C, hC⟩ => ne_top_of_le_ne_top Ennreal.ofReal_ne_top <| ediam_le_of_forall_dist_le hC)
+    (fun ⟨C, hC⟩ => ne_top_of_le_ne_top ENNReal.ofReal_ne_top <| ediam_le_of_forall_dist_le hC)
     fun h => ⟨diam s, fun x hx y hy => dist_le_diam_of_mem' h hx hy⟩
 #align metric.bounded_iff_ediam_ne_top Metric.bounded_iff_ediam_ne_top
 
@@ -2993,14 +2993,14 @@ theorem ediam_of_unbounded (h : ¬Bounded s) : Emetric.diam s = ∞ := by
 /-- An unbounded set has zero diameter. If you would prefer to get the value ∞, use `emetric.diam`.
 This lemma makes it possible to avoid side conditions in some situations -/
 theorem diam_eq_zero_of_unbounded (h : ¬Bounded s) : diam s = 0 := by
-  rw [diam, ediam_of_unbounded h, Ennreal.top_toReal]
+  rw [diam, ediam_of_unbounded h, ENNReal.top_toReal]
 #align metric.diam_eq_zero_of_unbounded Metric.diam_eq_zero_of_unbounded
 
 /-- If `s ⊆ t`, then the diameter of `s` is bounded by that of `t`, provided `t` is bounded. -/
 theorem diam_mono {s t : Set α} (h : s ⊆ t) (ht : Bounded t) : diam s ≤ diam t :=
   by
   unfold diam
-  rw [Ennreal.toReal_le_toReal (bounded.mono h ht).ediam_ne_top ht.ediam_ne_top]
+  rw [ENNReal.toReal_le_toReal (bounded.mono h ht).ediam_ne_top ht.ediam_ne_top]
   exact Emetric.diam_mono h
 #align metric.diam_mono Metric.diam_mono
 
@@ -3014,9 +3014,9 @@ theorem diam_union {t : Set α} (xs : x ∈ s) (yt : y ∈ t) :
   · have hs : bounded s := H.mono (subset_union_left _ _)
     have ht : bounded t := H.mono (subset_union_right _ _)
     rw [bounded_iff_ediam_ne_top] at H hs ht
-    rw [dist_edist, diam, diam, diam, ← Ennreal.toReal_add, ← Ennreal.toReal_add,
-            Ennreal.toReal_le_toReal] <;>
-          repeat' apply Ennreal.add_ne_top.2 <;> constructor <;>
+    rw [dist_edist, diam, diam, diam, ← ENNReal.toReal_add, ← ENNReal.toReal_add,
+            ENNReal.toReal_le_toReal] <;>
+          repeat' apply ENNReal.add_ne_top.2 <;> constructor <;>
         try assumption <;>
       try apply edist_ne_top
     exact Emetric.diam_union xs yt
@@ -3348,14 +3348,14 @@ uniformity are defeq in the metric space and the emetric space. In this definiti
 is given separately, to be able to prescribe some expression which is not defeq to the push-forward
 of the edistance to reals. -/
 def EmetricSpace.toMetricSpaceOfDist {α : Type u} [e : EmetricSpace α] (dist : α → α → ℝ)
-    (edist_ne_top : ∀ x y : α, edist x y ≠ ⊤) (h : ∀ x y, dist x y = Ennreal.toReal (edist x y)) :
+    (edist_ne_top : ∀ x y : α, edist x y ≠ ⊤) (h : ∀ x y, dist x y = ENNReal.toReal (edist x y)) :
     MetricSpace α :=
   {
     PseudoEmetricSpace.toPseudoMetricSpaceOfDist dist edist_ne_top
       h with
     dist
     eq_of_dist_eq_zero := fun x y hxy => by
-      simpa [h, Ennreal.toReal_eq_zero_iff, edist_ne_top x y] using hxy }
+      simpa [h, ENNReal.toReal_eq_zero_iff, edist_ne_top x y] using hxy }
 #align emetric_space.to_metric_space_of_dist EmetricSpace.toMetricSpaceOfDist
 
 /-- One gets a metric space from an emetric space if the edistance
@@ -3363,7 +3363,7 @@ is everywhere finite, by pushing the edistance to reals. We set it up so that th
 uniformity are defeq in the metric space and the emetric space. -/
 def EmetricSpace.toMetricSpace {α : Type u} [e : EmetricSpace α] (h : ∀ x y : α, edist x y ≠ ⊤) :
     MetricSpace α :=
-  EmetricSpace.toMetricSpaceOfDist (fun x y => Ennreal.toReal (edist x y)) h fun x y => rfl
+  EmetricSpace.toMetricSpaceOfDist (fun x y => ENNReal.toReal (edist x y)) h fun x y => rfl
 #align emetric_space.to_metric_space EmetricSpace.toMetricSpace
 
 /-- Build a new metric space from an old one where the bundled bornology structure is provably
@@ -3487,7 +3487,7 @@ instance metricSpacePi : MetricSpace (∀ b, π b) :=
     pseudoMetricSpacePi with
     eq_of_dist_eq_zero := fun f g eq0 =>
       by
-      have eq1 : edist f g = 0 := by simp only [edist_dist, eq0, Ennreal.ofReal_zero]
+      have eq1 : edist f g = 0 := by simp only [edist_dist, eq0, ENNReal.ofReal_zero]
       have eq2 : (sup univ fun b : β => edist (f b) (g b)) ≤ 0 := le_of_eq eq1
       simp only [Finset.sup_le_iff] at eq2
       exact funext fun b => edist_le_zero.1 <| eq2 b <| mem_univ b }

mathlib commit https://github.com/leanprover-community/mathlib/commit/bd9851ca476957ea4549eb19b40e7b5ade9428cc

by declaring them all in where style with implicit type assumptions and inst prefix

Here to reduce the diff from #11203

@@ -445,6 +445,9 @@ instance [MetricSpace X] : MetricSpace (Multiplicative X) := ‹MetricSpace X›
 instance [PseudoMetricSpace X] [ProperSpace X] : ProperSpace (Additive X) := ‹ProperSpace X›
 instance [PseudoMetricSpace X] [ProperSpace X] : ProperSpace (Multiplicative X) := ‹ProperSpace X›
 
+instance MulOpposite.instMetricSpace [MetricSpace X] : MetricSpace Xᵐᵒᵖ :=
+  MetricSpace.induced unop unop_injective ‹_›
+
 /-!
 ### Order dual
 

We had duplicated API between topological spaces and uniform spaces. In this PR I mostly deduplicate it with some exceptions:

• SeparationQuotient.lift' and SeparationQuotient.map are leftovers from the old version that are designed to work with uniform spaces;
• probably, some theorems/instances still assume UniformSpace when TopologicalSpace would work.

Outside of UniformSpace/Separation, I mostly changed SeparatedSpace to T0Space and separationRel to Inseparable. I also rewrote a few proofs that were broken by the API change.

Fixes #2031

@@ -131,10 +131,9 @@ theorem subsingleton_sphere (x : γ) {r : ℝ} (hr : r ≤ 0) : (sphere x r).Sub
 #align metric.subsingleton_sphere Metric.subsingleton_sphere
 
 -- see Note [lower instance priority]
-instance (priority := 100) _root_.MetricSpace.to_separated : SeparatedSpace γ :=
-  separated_def.2 fun _ _ h =>
-    eq_of_forall_dist_le fun _ ε0 => le_of_lt (h _ (dist_mem_uniformity ε0))
-#align metric_space.to_separated MetricSpace.to_separated
+instance (priority := 100) _root_.MetricSpace.instT0Space : T0Space γ where
+  t0 _ _ h := eq_of_dist_eq_zero <| Metric.inseparable_iff.1 h
+#align metric_space.to_separated MetricSpace.instT0Space
 
 /-- A map between metric spaces is a uniform embedding if and only if the distance between `f x`
 and `f y` is controlled in terms of the distance between `x` and `y` and conversely. -/
@@ -370,19 +369,21 @@ end Metric
 
 section EqRel
 
-instance {α : Type u} [PseudoMetricSpace α] : Dist (UniformSpace.SeparationQuotient α) where
-  dist p q := Quotient.liftOn₂' p q dist fun x y x' y' hx hy => by
-    rw [dist_edist, dist_edist, ← UniformSpace.SeparationQuotient.edist_mk x,
-      ← UniformSpace.SeparationQuotient.edist_mk x', Quot.sound hx, Quot.sound hy]
+-- TODO: add `dist_congr` similar to `edist_congr`?
+instance SeparationQuotient.instDist {α : Type u} [PseudoMetricSpace α] :
+    Dist (SeparationQuotient α) where
+  dist := lift₂ dist fun x y x' y' hx hy ↦ by rw [dist_edist, dist_edist, ← edist_mk x,
+    ← edist_mk x', mk_eq_mk.2 hx, mk_eq_mk.2 hy]
 
-theorem UniformSpace.SeparationQuotient.dist_mk {α : Type u} [PseudoMetricSpace α] (p q : α) :
-    @dist (UniformSpace.SeparationQuotient α) _ (Quot.mk _ p) (Quot.mk _ q) = dist p q :=
+theorem SeparationQuotient.dist_mk {α : Type u} [PseudoMetricSpace α] (p q : α) :
+    dist (mk p) (mk q) = dist p q :=
   rfl
-#align uniform_space.separation_quotient.dist_mk UniformSpace.SeparationQuotient.dist_mk
+#align uniform_space.separation_quotient.dist_mk SeparationQuotient.dist_mk
 
-instance {α : Type u} [PseudoMetricSpace α] : MetricSpace (UniformSpace.SeparationQuotient α) :=
-  EMetricSpace.toMetricSpaceOfDist dist (fun x y => Quotient.inductionOn₂' x y edist_ne_top)
-    fun x y => Quotient.inductionOn₂' x y dist_edist
+instance SeparationQuotient.instMetricSpace {α : Type u} [PseudoMetricSpace α] :
+    MetricSpace (SeparationQuotient α) :=
+  EMetricSpace.toMetricSpaceOfDist dist (surjective_mk.forall₂.2 edist_ne_top) <|
+    surjective_mk.forall₂.2 dist_edist
 
 end EqRel
 

Classifies by adding issue number #11215 to porting notes claiming "TODO".

@@ -348,7 +348,7 @@ section SecondCountable
 
 open TopologicalSpace
 
--- Porting note: todo: use `Countable` instead of `Encodable`
+-- Porting note (#11215): TODO: use `Countable` instead of `Encodable`
 /-- A metric space is second countable if one can reconstruct up to any `ε>0` any element of the
 space from countably many data. -/
 theorem secondCountable_of_countable_discretization {α : Type u} [MetricSpace α]

Homogenises porting notes via capitalisation and addition of whitespace.

It makes the following changes:

• converts "--porting note" into "-- Porting note";
• converts "porting note" into "Porting note".

@@ -287,7 +287,7 @@ instance : MetricSpace Empty where
   dist_self _ := rfl
   dist_comm _ _ := rfl
   edist _ _ := 0
-  edist_dist _ _ := ENNReal.ofReal_zero.symm -- porting note: should not be needed
+  edist_dist _ _ := ENNReal.ofReal_zero.symm -- Porting note: should not be needed
   eq_of_dist_eq_zero _ := Subsingleton.elim _ _
   dist_triangle _ _ _ := show (0 : ℝ) ≤ 0 + 0 by rw [add_zero]
   toUniformSpace := inferInstance
@@ -298,7 +298,7 @@ instance : MetricSpace PUnit.{u + 1} where
   dist_self _ := rfl
   dist_comm _ _ := rfl
   edist _ _ := 0
-  edist_dist _ _ := ENNReal.ofReal_zero.symm -- porting note: should not be needed
+  edist_dist _ _ := ENNReal.ofReal_zero.symm -- Porting note: should not be needed
   eq_of_dist_eq_zero _ := Subsingleton.elim _ _
   dist_triangle _ _ _ := show (0 : ℝ) ≤ 0 + 0 by rw [add_zero]
   toUniformSpace := inferInstance
@@ -348,7 +348,7 @@ section SecondCountable
 
 open TopologicalSpace
 
--- porting note: todo: use `Countable` instead of `Encodable`
+-- Porting note: todo: use `Countable` instead of `Encodable`
 /-- A metric space is second countable if one can reconstruct up to any `ε>0` any element of the
 space from countably many data. -/
 theorem secondCountable_of_countable_discretization {α : Type u} [MetricSpace α]

And move one lemma to a better place.

Shaves off another 130 lines from a 2200 line file; #9815 continues with a more far-reaching split.

Co-authored-by: grunweg <grunweg@posteo.de>

@@ -3,7 +3,7 @@ Copyright (c) 2015, 2017 Jeremy Avigad. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Jeremy Avigad, Robert Y. Lewis, Johannes Hölzl, Mario Carneiro, Sébastien Gouëzel
 -/
-import Mathlib.Topology.MetricSpace.PseudoMetric
+import Mathlib.Topology.MetricSpace.ProperSpace
 
 #align_import topology.metric_space.basic from "leanprover-community/mathlib"@"c8f305514e0d47dfaa710f5a52f0d21b588e6328"
 

Subset of #9319

@@ -371,7 +371,7 @@ end Metric
 section EqRel
 
 instance {α : Type u} [PseudoMetricSpace α] : Dist (UniformSpace.SeparationQuotient α) where
-  dist p q := Quotient.liftOn₂' p q dist <| fun x y x' y' hx hy => by
+  dist p q := Quotient.liftOn₂' p q dist fun x y x' y' hx hy => by
     rw [dist_edist, dist_edist, ← UniformSpace.SeparationQuotient.edist_mk x,
       ← UniformSpace.SeparationQuotient.edist_mk x', Quot.sound hx, Quot.sound hy]
 

@@ -92,12 +92,12 @@ theorem eq_of_forall_dist_le {x y : γ} (h : ∀ ε > 0, dist x y ≤ ε) : x =
   eq_of_dist_eq_zero (eq_of_le_of_forall_le_of_dense dist_nonneg h)
 #align eq_of_forall_dist_le eq_of_forall_dist_le
 
-/-- Deduce the equality of points with the vanishing of the nonnegative distance-/
+/-- Deduce the equality of points from the vanishing of the nonnegative distance-/
 theorem eq_of_nndist_eq_zero {x y : γ} : nndist x y = 0 → x = y := by
   simp only [← NNReal.eq_iff, ← dist_nndist, imp_self, NNReal.coe_zero, dist_eq_zero]
 #align eq_of_nndist_eq_zero eq_of_nndist_eq_zero
 
-/-- Characterize the equality of points with the vanishing of the nonnegative distance-/
+/-- Characterize the equality of points as the vanishing of the nonnegative distance-/
 @[simp]
 theorem nndist_eq_zero {x y : γ} : nndist x y = 0 ↔ x = y := by
   simp only [← NNReal.eq_iff, ← dist_nndist, imp_self, NNReal.coe_zero, dist_eq_zero]

This was "feat(topology/metric_space): diameter of pointwise zero and addition"

@@ -5,7 +5,7 @@ Authors: Jeremy Avigad, Robert Y. Lewis, Johannes Hölzl, Mario Carneiro, Sébas
 -/
 import Mathlib.Topology.MetricSpace.PseudoMetric
 
-#align_import topology.metric_space.basic from "leanprover-community/mathlib"@"8047de4d911cdef39c2d646165eea972f7f9f539"
+#align_import topology.metric_space.basic from "leanprover-community/mathlib"@"c8f305514e0d47dfaa710f5a52f0d21b588e6328"
 
 /-!
 # Metric spaces

This reduces the main file from 3340 to 2220 lines. The remaining file is somewhat entangled, so splitting is less obvious. Help is welcome, though a follow-up PR is probably better :-)

I've kept copyright and authors as they were originally.

@@ -3,2894 +3,38 @@ Copyright (c) 2015, 2017 Jeremy Avigad. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Jeremy Avigad, Robert Y. Lewis, Johannes Hölzl, Mario Carneiro, Sébastien Gouëzel
 -/
-import Mathlib.Tactic.Positivity
-import Mathlib.Topology.Algebra.Order.Compact
-import Mathlib.Topology.EMetricSpace.Basic
-import Mathlib.Topology.Bornology.Constructions
+import Mathlib.Topology.MetricSpace.PseudoMetric
 
 #align_import topology.metric_space.basic from "leanprover-community/mathlib"@"8047de4d911cdef39c2d646165eea972f7f9f539"
 
 /-!
 # Metric spaces
 
-This file defines metric spaces. Many definitions and theorems expected
-on metric spaces are already introduced on uniform spaces and topological spaces.
-For example: open and closed sets, compactness, completeness, continuity and uniform continuity
+This file defines metric spaces and shows some of their basic properties.
 
-## Main definitions
-
-* `Dist α`: Endows a space `α` with a function `dist a b`.
-* `PseudoMetricSpace α`: A space endowed with a distance function, which can
-  be zero even if the two elements are non-equal.
-* `Metric.ball x ε`: The set of all points `y` with `dist y x < ε`.
-* `Metric.Bounded s`: Whether a subset of a `PseudoMetricSpace` is bounded.
-* `MetricSpace α`: A `PseudoMetricSpace` with the guarantee `dist x y = 0 → x = y`.
-
-Additional useful definitions:
-
-* `nndist a b`: `dist` as a function to the non-negative reals.
-* `Metric.closedBall x ε`: The set of all points `y` with `dist y x ≤ ε`.
-* `Metric.sphere x ε`: The set of all points `y` with `dist y x = ε`.
-* `ProperSpace α`: A `PseudoMetricSpace` where all closed balls are compact.
-* `Metric.diam s` : The `iSup` of the distances of members of `s`.
-  Defined in terms of `EMetric.diam`, for better handling of the case when it should be infinite.
+Many definitions and theorems expected on metric spaces are already introduced on uniform spaces and
+topological spaces. This includes open and closed sets, compactness, completeness, continuity
+and uniform continuity.
 
 TODO (anyone): Add "Main results" section.
 
 ## Implementation notes
-
-Since a lot of elementary properties don't require `eq_of_dist_eq_zero` we start setting up the
-theory of `PseudoMetricSpace`, where we don't require `dist x y = 0 → x = y` and we specialize
-to `MetricSpace` at the end.
+A lot of elementary properties don't require `eq_of_dist_eq_zero`, hence are stated and proven
+for `PseudoMetricSpace`s in `PseudoMetric.lean`.
 
 ## Tags
 
 metric, pseudo_metric, dist
 -/
 
-
-open Set Filter TopologicalSpace Bornology
-open scoped BigOperators ENNReal NNReal Uniformity Topology
+open Set Filter Bornology
+open scoped NNReal Uniformity
 
 universe u v w
 
 variable {α : Type u} {β : Type v} {X ι : Type*}
-
-/-- Construct a uniform structure from a distance function and metric space axioms -/
-def UniformSpace.ofDist (dist : α → α → ℝ) (dist_self : ∀ x : α, dist x x = 0)
-    (dist_comm : ∀ x y : α, dist x y = dist y x)
-    (dist_triangle : ∀ x y z : α, dist x z ≤ dist x y + dist y z) : UniformSpace α :=
-  .ofFun dist dist_self dist_comm dist_triangle fun ε ε0 =>
-    ⟨ε / 2, half_pos ε0, fun _x hx _y hy => add_halves ε ▸ add_lt_add hx hy⟩
-#align uniform_space_of_dist UniformSpace.ofDist
-
--- porting note: dropped the `dist_self` argument
-/-- Construct a bornology from a distance function and metric space axioms. -/
-@[reducible]
-def Bornology.ofDist {α : Type*} (dist : α → α → ℝ) (dist_comm : ∀ x y, dist x y = dist y x)
-    (dist_triangle : ∀ x y z, dist x z ≤ dist x y + dist y z) : Bornology α :=
-  Bornology.ofBounded { s : Set α | ∃ C, ∀ ⦃x⦄, x ∈ s → ∀ ⦃y⦄, y ∈ s → dist x y ≤ C }
-    ⟨0, fun x hx y => hx.elim⟩ (fun s ⟨c, hc⟩ t h => ⟨c, fun x hx y hy => hc (h hx) (h hy)⟩)
-    (fun s hs t ht => by
-      rcases s.eq_empty_or_nonempty with rfl | ⟨x, hx⟩
-      · rwa [empty_union]
-      rcases t.eq_empty_or_nonempty with rfl | ⟨y, hy⟩
-      · rwa [union_empty]
-      rsuffices ⟨C, hC⟩ : ∃ C, ∀ z ∈ s ∪ t, dist x z ≤ C
-      · refine ⟨C + C, fun a ha b hb => (dist_triangle a x b).trans ?_⟩
-        simpa only [dist_comm] using add_le_add (hC _ ha) (hC _ hb)
-      rcases hs with ⟨Cs, hs⟩; rcases ht with ⟨Ct, ht⟩
-      refine ⟨max Cs (dist x y + Ct), fun z hz => hz.elim
-        (fun hz => (hs hx hz).trans (le_max_left _ _))
-        (fun hz => (dist_triangle x y z).trans <|
-          (add_le_add le_rfl (ht hy hz)).trans (le_max_right _ _))⟩)
-    fun z => ⟨dist z z, forall_eq.2 <| forall_eq.2 le_rfl⟩
-#align bornology.of_dist Bornology.ofDistₓ
-
-/-- The distance function (given an ambient metric space on `α`), which returns
-  a nonnegative real number `dist x y` given `x y : α`. -/
-@[ext]
-class Dist (α : Type*) where
-  dist : α → α → ℝ
-#align has_dist Dist
-
-export Dist (dist)
-
--- the uniform structure and the emetric space structure are embedded in the metric space structure
--- to avoid instance diamond issues. See Note [forgetful inheritance].
-/-- This is an internal lemma used inside the default of `PseudoMetricSpace.edist`. -/
-private theorem dist_nonneg' {α} {x y : α} (dist : α → α → ℝ)
-    (dist_self : ∀ x : α, dist x x = 0) (dist_comm : ∀ x y : α, dist x y = dist y x)
-    (dist_triangle : ∀ x y z : α, dist x z ≤ dist x y + dist y z) : 0 ≤ dist x y :=
-  have : 0 ≤ 2 * dist x y :=
-    calc 0 = dist x x := (dist_self _).symm
-    _ ≤ dist x y + dist y x := dist_triangle _ _ _
-    _ = 2 * dist x y := by rw [two_mul, dist_comm]
-  nonneg_of_mul_nonneg_right this two_pos
-
-#noalign pseudo_metric_space.edist_dist_tac -- porting note: todo: restore
-
-/-- Pseudo metric and Metric spaces
-
-A pseudo metric space is endowed with a distance for which the requirement `d(x,y)=0 → x = y` might
-not hold. A metric space is a pseudo metric space such that `d(x,y)=0 → x = y`.
-Each pseudo metric space induces a canonical `UniformSpace` and hence a canonical
-`TopologicalSpace` This is enforced in the type class definition, by extending the `UniformSpace`
-structure. When instantiating a `PseudoMetricSpace` structure, the uniformity fields are not
-necessary, they will be filled in by default. In the same way, each (pseudo) metric space induces a
-(pseudo) emetric space structure. It is included in the structure, but filled in by default.
--/
-class PseudoMetricSpace (α : Type u) extends Dist α : Type u where
-  dist_self : ∀ x : α, dist x x = 0
-  dist_comm : ∀ x y : α, dist x y = dist y x
-  dist_triangle : ∀ x y z : α, dist x z ≤ dist x y + dist y z
-  edist : α → α → ℝ≥0∞ := fun x y => ENNReal.some ⟨dist x y, dist_nonneg' _ ‹_› ‹_› ‹_›⟩
-  edist_dist : ∀ x y : α, edist x y = ENNReal.ofReal (dist x y) -- porting note: todo: add := by _
-  toUniformSpace : UniformSpace α := .ofDist dist dist_self dist_comm dist_triangle
-  uniformity_dist : 𝓤 α = ⨅ ε > 0, 𝓟 { p : α × α | dist p.1 p.2 < ε } := by intros; rfl
-  toBornology : Bornology α := Bornology.ofDist dist dist_comm dist_triangle
-  cobounded_sets : (Bornology.cobounded α).sets =
-    { s | ∃ C : ℝ, ∀ x ∈ sᶜ, ∀ y ∈ sᶜ, dist x y ≤ C } := by intros; rfl
-#align pseudo_metric_space PseudoMetricSpace
-
-/-- Two pseudo metric space structures with the same distance function coincide. -/
-@[ext]
-theorem PseudoMetricSpace.ext {α : Type*} {m m' : PseudoMetricSpace α}
-    (h : m.toDist = m'.toDist) : m = m' := by
-  cases' m with d _ _ _ ed hed U hU B hB
-  cases' m' with d' _ _ _ ed' hed' U' hU' B' hB'
-  obtain rfl : d = d' := h
-  congr
-  · ext x y : 2
-    rw [hed, hed']
-  · exact UniformSpace.ext (hU.trans hU'.symm)
-  · ext : 2
-    rw [← Filter.mem_sets, ← Filter.mem_sets, hB, hB']
-#align pseudo_metric_space.ext PseudoMetricSpace.ext
-
 variable [PseudoMetricSpace α]
 
-attribute [instance] PseudoMetricSpace.toUniformSpace PseudoMetricSpace.toBornology
-
--- see Note [lower instance priority]
-instance (priority := 200) PseudoMetricSpace.toEDist : EDist α :=
-  ⟨PseudoMetricSpace.edist⟩
-#align pseudo_metric_space.to_has_edist PseudoMetricSpace.toEDist
-
-/-- Construct a pseudo-metric space structure whose underlying topological space structure
-(definitionally) agrees which a pre-existing topology which is compatible with a given distance
-function. -/
-def PseudoMetricSpace.ofDistTopology {α : Type u} [TopologicalSpace α] (dist : α → α → ℝ)
-    (dist_self : ∀ x : α, dist x x = 0) (dist_comm : ∀ x y : α, dist x y = dist y x)
-    (dist_triangle : ∀ x y z : α, dist x z ≤ dist x y + dist y z)
-    (H : ∀ s : Set α, IsOpen s ↔ ∀ x ∈ s, ∃ ε > 0, ∀ y, dist x y < ε → y ∈ s) :
-    PseudoMetricSpace α :=
-  { dist := dist
-    dist_self := dist_self
-    dist_comm := dist_comm
-    dist_triangle := dist_triangle
-    edist_dist := fun x y => by exact ENNReal.coe_nnreal_eq _
-    toUniformSpace :=
-      { toCore := (UniformSpace.ofDist dist dist_self dist_comm dist_triangle).toCore
-        isOpen_uniformity := fun s => (H s).trans <| forall₂_congr fun x _ =>
-          ((UniformSpace.hasBasis_ofFun (exists_gt (0 : ℝ)) dist _ _ _ _).comap
-            (Prod.mk x)).mem_iff.symm.trans mem_comap_prod_mk }
-    uniformity_dist := rfl
-    toBornology := Bornology.ofDist dist dist_comm dist_triangle
-    cobounded_sets := rfl }
-#align pseudo_metric_space.of_dist_topology PseudoMetricSpace.ofDistTopology
-
-@[simp]
-theorem dist_self (x : α) : dist x x = 0 :=
-  PseudoMetricSpace.dist_self x
-#align dist_self dist_self
-
-theorem dist_comm (x y : α) : dist x y = dist y x :=
-  PseudoMetricSpace.dist_comm x y
-#align dist_comm dist_comm
-
-theorem edist_dist (x y : α) : edist x y = ENNReal.ofReal (dist x y) :=
-  PseudoMetricSpace.edist_dist x y
-#align edist_dist edist_dist
-
-theorem dist_triangle (x y z : α) : dist x z ≤ dist x y + dist y z :=
-  PseudoMetricSpace.dist_triangle x y z
-#align dist_triangle dist_triangle
-
-theorem dist_triangle_left (x y z : α) : dist x y ≤ dist z x + dist z y := by
-  rw [dist_comm z]; apply dist_triangle
-#align dist_triangle_left dist_triangle_left
-
-theorem dist_triangle_right (x y z : α) : dist x y ≤ dist x z + dist y z := by
-  rw [dist_comm y]; apply dist_triangle
-#align dist_triangle_right dist_triangle_right
-
-theorem dist_triangle4 (x y z w : α) : dist x w ≤ dist x y + dist y z + dist z w :=
-  calc
-    dist x w ≤ dist x z + dist z w := dist_triangle x z w
-    _ ≤ dist x y + dist y z + dist z w := add_le_add_right (dist_triangle x y z) _
-#align dist_triangle4 dist_triangle4
-
-theorem dist_triangle4_left (x₁ y₁ x₂ y₂ : α) :
-    dist x₂ y₂ ≤ dist x₁ y₁ + (dist x₁ x₂ + dist y₁ y₂) := by
-  rw [add_left_comm, dist_comm x₁, ← add_assoc]
-  apply dist_triangle4
-#align dist_triangle4_left dist_triangle4_left
-
-theorem dist_triangle4_right (x₁ y₁ x₂ y₂ : α) :
-    dist x₁ y₁ ≤ dist x₁ x₂ + dist y₁ y₂ + dist x₂ y₂ := by
-  rw [add_right_comm, dist_comm y₁]
-  apply dist_triangle4
-#align dist_triangle4_right dist_triangle4_right
-
-/-- The triangle (polygon) inequality for sequences of points; `Finset.Ico` version. -/
-theorem dist_le_Ico_sum_dist (f : ℕ → α) {m n} (h : m ≤ n) :
-    dist (f m) (f n) ≤ ∑ i in Finset.Ico m n, dist (f i) (f (i + 1)) := by
-  induction n, h using Nat.le_induction with
-  | base => rw [Finset.Ico_self, Finset.sum_empty, dist_self]
-  | succ n hle ihn =>
-    calc
-      dist (f m) (f (n + 1)) ≤ dist (f m) (f n) + dist (f n) (f (n + 1)) := dist_triangle _ _ _
-      _ ≤ (∑ i in Finset.Ico m n, _) + _ := add_le_add ihn le_rfl
-      _ = ∑ i in Finset.Ico m (n + 1), _ := by
-      { rw [Nat.Ico_succ_right_eq_insert_Ico hle, Finset.sum_insert, add_comm]; simp }
-#align dist_le_Ico_sum_dist dist_le_Ico_sum_dist
-
-/-- The triangle (polygon) inequality for sequences of points; `Finset.range` version. -/
-theorem dist_le_range_sum_dist (f : ℕ → α) (n : ℕ) :
-    dist (f 0) (f n) ≤ ∑ i in Finset.range n, dist (f i) (f (i + 1)) :=
-  Nat.Ico_zero_eq_range ▸ dist_le_Ico_sum_dist f (Nat.zero_le n)
-#align dist_le_range_sum_dist dist_le_range_sum_dist
-
-/-- A version of `dist_le_Ico_sum_dist` with each intermediate distance replaced
-with an upper estimate. -/
-theorem dist_le_Ico_sum_of_dist_le {f : ℕ → α} {m n} (hmn : m ≤ n) {d : ℕ → ℝ}
-    (hd : ∀ {k}, m ≤ k → k < n → dist (f k) (f (k + 1)) ≤ d k) :
-    dist (f m) (f n) ≤ ∑ i in Finset.Ico m n, d i :=
-  le_trans (dist_le_Ico_sum_dist f hmn) <|
-    Finset.sum_le_sum fun _k hk => hd (Finset.mem_Ico.1 hk).1 (Finset.mem_Ico.1 hk).2
-#align dist_le_Ico_sum_of_dist_le dist_le_Ico_sum_of_dist_le
-
-/-- A version of `dist_le_range_sum_dist` with each intermediate distance replaced
-with an upper estimate. -/
-theorem dist_le_range_sum_of_dist_le {f : ℕ → α} (n : ℕ) {d : ℕ → ℝ}
-    (hd : ∀ {k}, k < n → dist (f k) (f (k + 1)) ≤ d k) :
-    dist (f 0) (f n) ≤ ∑ i in Finset.range n, d i :=
-  Nat.Ico_zero_eq_range ▸ dist_le_Ico_sum_of_dist_le (zero_le n) fun _ => hd
-#align dist_le_range_sum_of_dist_le dist_le_range_sum_of_dist_le
-
-theorem swap_dist : Function.swap (@dist α _) = dist := by funext x y; exact dist_comm _ _
-#align swap_dist swap_dist
-
-theorem abs_dist_sub_le (x y z : α) : |dist x z - dist y z| ≤ dist x y :=
-  abs_sub_le_iff.2
-    ⟨sub_le_iff_le_add.2 (dist_triangle _ _ _), sub_le_iff_le_add.2 (dist_triangle_left _ _ _)⟩
-#align abs_dist_sub_le abs_dist_sub_le
-
-theorem dist_nonneg {x y : α} : 0 ≤ dist x y :=
-  dist_nonneg' dist dist_self dist_comm dist_triangle
-#align dist_nonneg dist_nonneg
-
-namespace Mathlib.Meta.Positivity
-
-open Lean Meta Qq Function
-
-/-- Extension for the `positivity` tactic: distances are nonnegative. -/
-@[positivity Dist.dist _ _]
-def evalDist : PositivityExt where eval {_ _} _zα _pα e := do
-  let .app (.app _ a) b ← whnfR e | throwError "not dist"
-  let p ← mkAppOptM ``dist_nonneg #[none, none, a, b]
-  pure (.nonnegative p)
-
-end Mathlib.Meta.Positivity
-
-example {x y : α} : 0 ≤ dist x y := by positivity
-
-@[simp] theorem abs_dist {a b : α} : |dist a b| = dist a b := abs_of_nonneg dist_nonneg
-#align abs_dist abs_dist
-
-/-- A version of `Dist` that takes value in `ℝ≥0`. -/
-class NNDist (α : Type*) where
-  nndist : α → α → ℝ≥0
-#align has_nndist NNDist
-
-export NNDist (nndist)
-
--- see Note [lower instance priority]
-/-- Distance as a nonnegative real number. -/
-instance (priority := 100) PseudoMetricSpace.toNNDist : NNDist α :=
-  ⟨fun a b => ⟨dist a b, dist_nonneg⟩⟩
-#align pseudo_metric_space.to_has_nndist PseudoMetricSpace.toNNDist
-
-/-- Express `dist` in terms of `nndist`-/
-theorem dist_nndist (x y : α) : dist x y = nndist x y := rfl
-#align dist_nndist dist_nndist
-
-@[simp, norm_cast]
-theorem coe_nndist (x y : α) : ↑(nndist x y) = dist x y := rfl
-#align coe_nndist coe_nndist
-
-/-- Express `edist` in terms of `nndist`-/
-theorem edist_nndist (x y : α) : edist x y = nndist x y := by
-  rw [edist_dist, dist_nndist, ENNReal.ofReal_coe_nnreal]
-#align edist_nndist edist_nndist
-
-/-- Express `nndist` in terms of `edist`-/
-theorem nndist_edist (x y : α) : nndist x y = (edist x y).toNNReal := by
-  simp [edist_nndist]
-#align nndist_edist nndist_edist
-
-@[simp, norm_cast]
-theorem coe_nnreal_ennreal_nndist (x y : α) : ↑(nndist x y) = edist x y :=
-  (edist_nndist x y).symm
-#align coe_nnreal_ennreal_nndist coe_nnreal_ennreal_nndist
-
-@[simp, norm_cast]
-theorem edist_lt_coe {x y : α} {c : ℝ≥0} : edist x y < c ↔ nndist x y < c := by
-  rw [edist_nndist, ENNReal.coe_lt_coe]
-#align edist_lt_coe edist_lt_coe
-
-@[simp, norm_cast]
-theorem edist_le_coe {x y : α} {c : ℝ≥0} : edist x y ≤ c ↔ nndist x y ≤ c := by
-  rw [edist_nndist, ENNReal.coe_le_coe]
-#align edist_le_coe edist_le_coe
-
-/-- In a pseudometric space, the extended distance is always finite-/
-theorem edist_lt_top {α : Type*} [PseudoMetricSpace α] (x y : α) : edist x y < ⊤ :=
-  (edist_dist x y).symm ▸ ENNReal.ofReal_lt_top
-#align edist_lt_top edist_lt_top
-
-/-- In a pseudometric space, the extended distance is always finite-/
-theorem edist_ne_top (x y : α) : edist x y ≠ ⊤ :=
-  (edist_lt_top x y).ne
-#align edist_ne_top edist_ne_top
-
-/-- `nndist x x` vanishes-/
-@[simp]
-theorem nndist_self (a : α) : nndist a a = 0 :=
-  (NNReal.coe_eq_zero _).1 (dist_self a)
-#align nndist_self nndist_self
-
--- porting note: `dist_nndist` and `coe_nndist` moved up
-
-@[simp, norm_cast]
-theorem dist_lt_coe {x y : α} {c : ℝ≥0} : dist x y < c ↔ nndist x y < c :=
-  Iff.rfl
-#align dist_lt_coe dist_lt_coe
-
-@[simp, norm_cast]
-theorem dist_le_coe {x y : α} {c : ℝ≥0} : dist x y ≤ c ↔ nndist x y ≤ c :=
-  Iff.rfl
-#align dist_le_coe dist_le_coe
-
-@[simp]
-theorem edist_lt_ofReal {x y : α} {r : ℝ} : edist x y < ENNReal.ofReal r ↔ dist x y < r := by
-  rw [edist_dist, ENNReal.ofReal_lt_ofReal_iff_of_nonneg dist_nonneg]
-#align edist_lt_of_real edist_lt_ofReal
-
-@[simp]
-theorem edist_le_ofReal {x y : α} {r : ℝ} (hr : 0 ≤ r) :
-    edist x y ≤ ENNReal.ofReal r ↔ dist x y ≤ r := by
-  rw [edist_dist, ENNReal.ofReal_le_ofReal_iff hr]
-#align edist_le_of_real edist_le_ofReal
-
-/-- Express `nndist` in terms of `dist`-/
-theorem nndist_dist (x y : α) : nndist x y = Real.toNNReal (dist x y) := by
-  rw [dist_nndist, Real.toNNReal_coe]
-#align nndist_dist nndist_dist
-
-theorem nndist_comm (x y : α) : nndist x y = nndist y x := NNReal.eq <| dist_comm x y
-#align nndist_comm nndist_comm
-
-/-- Triangle inequality for the nonnegative distance-/
-theorem nndist_triangle (x y z : α) : nndist x z ≤ nndist x y + nndist y z :=
-  dist_triangle _ _ _
-#align nndist_triangle nndist_triangle
-
-theorem nndist_triangle_left (x y z : α) : nndist x y ≤ nndist z x + nndist z y :=
-  dist_triangle_left _ _ _
-#align nndist_triangle_left nndist_triangle_left
-
-theorem nndist_triangle_right (x y z : α) : nndist x y ≤ nndist x z + nndist y z :=
-  dist_triangle_right _ _ _
-#align nndist_triangle_right nndist_triangle_right
-
-/-- Express `dist` in terms of `edist`-/
-theorem dist_edist (x y : α) : dist x y = (edist x y).toReal := by
-  rw [edist_dist, ENNReal.toReal_ofReal dist_nonneg]
-#align dist_edist dist_edist
-
-namespace Metric
-
--- instantiate pseudometric space as a topology
-variable {x y z : α} {δ ε ε₁ ε₂ : ℝ} {s : Set α}
-
-/-- `ball x ε` is the set of all points `y` with `dist y x < ε` -/
-def ball (x : α) (ε : ℝ) : Set α :=
-  { y | dist y x < ε }
-#align metric.ball Metric.ball
-
-@[simp]
-theorem mem_ball : y ∈ ball x ε ↔ dist y x < ε :=
-  Iff.rfl
-#align metric.mem_ball Metric.mem_ball
-
-theorem mem_ball' : y ∈ ball x ε ↔ dist x y < ε := by rw [dist_comm, mem_ball]
-#align metric.mem_ball' Metric.mem_ball'
-
-theorem pos_of_mem_ball (hy : y ∈ ball x ε) : 0 < ε :=
-  dist_nonneg.trans_lt hy
-#align metric.pos_of_mem_ball Metric.pos_of_mem_ball
-
-theorem mem_ball_self (h : 0 < ε) : x ∈ ball x ε := by
-  rwa [mem_ball, dist_self]
-#align metric.mem_ball_self Metric.mem_ball_self
-
-@[simp]
-theorem nonempty_ball : (ball x ε).Nonempty ↔ 0 < ε :=
-  ⟨fun ⟨_x, hx⟩ => pos_of_mem_ball hx, fun h => ⟨x, mem_ball_self h⟩⟩
-#align metric.nonempty_ball Metric.nonempty_ball
-
-@[simp]
-theorem ball_eq_empty : ball x ε = ∅ ↔ ε ≤ 0 := by
-  rw [← not_nonempty_iff_eq_empty, nonempty_ball, not_lt]
-#align metric.ball_eq_empty Metric.ball_eq_empty
-
-@[simp]
-theorem ball_zero : ball x 0 = ∅ := by rw [ball_eq_empty]
-#align metric.ball_zero Metric.ball_zero
-
-/-- If a point belongs to an open ball, then there is a strictly smaller radius whose ball also
-contains it.
-
-See also `exists_lt_subset_ball`. -/
-theorem exists_lt_mem_ball_of_mem_ball (h : x ∈ ball y ε) : ∃ ε' < ε, x ∈ ball y ε' := by
-  simp only [mem_ball] at h ⊢
-  exact ⟨(dist x y + ε) / 2, by linarith, by linarith⟩
-#align metric.exists_lt_mem_ball_of_mem_ball Metric.exists_lt_mem_ball_of_mem_ball
-
-theorem ball_eq_ball (ε : ℝ) (x : α) :
-    UniformSpace.ball x { p | dist p.2 p.1 < ε } = Metric.ball x ε :=
-  rfl
-#align metric.ball_eq_ball Metric.ball_eq_ball
-
-theorem ball_eq_ball' (ε : ℝ) (x : α) :
-    UniformSpace.ball x { p | dist p.1 p.2 < ε } = Metric.ball x ε := by
-  ext
-  simp [dist_comm, UniformSpace.ball]
-#align metric.ball_eq_ball' Metric.ball_eq_ball'
-
-@[simp]
-theorem iUnion_ball_nat (x : α) : ⋃ n : ℕ, ball x n = univ :=
-  iUnion_eq_univ_iff.2 fun y => exists_nat_gt (dist y x)
-#align metric.Union_ball_nat Metric.iUnion_ball_nat
-
-@[simp]
-theorem iUnion_ball_nat_succ (x : α) : ⋃ n : ℕ, ball x (n + 1) = univ :=
-  iUnion_eq_univ_iff.2 fun y => (exists_nat_gt (dist y x)).imp fun _ h => h.trans (lt_add_one _)
-#align metric.Union_ball_nat_succ Metric.iUnion_ball_nat_succ
-
-/-- `closedBall x ε` is the set of all points `y` with `dist y x ≤ ε` -/
-def closedBall (x : α) (ε : ℝ) :=
-  { y | dist y x ≤ ε }
-#align metric.closed_ball Metric.closedBall
-
-@[simp] theorem mem_closedBall : y ∈ closedBall x ε ↔ dist y x ≤ ε := Iff.rfl
-#align metric.mem_closed_ball Metric.mem_closedBall
-
-theorem mem_closedBall' : y ∈ closedBall x ε ↔ dist x y ≤ ε := by rw [dist_comm, mem_closedBall]
-#align metric.mem_closed_ball' Metric.mem_closedBall'
-
-/-- `sphere x ε` is the set of all points `y` with `dist y x = ε` -/
-def sphere (x : α) (ε : ℝ) := { y | dist y x = ε }
-#align metric.sphere Metric.sphere
-
-@[simp] theorem mem_sphere : y ∈ sphere x ε ↔ dist y x = ε := Iff.rfl
-#align metric.mem_sphere Metric.mem_sphere
-
-theorem mem_sphere' : y ∈ sphere x ε ↔ dist x y = ε := by rw [dist_comm, mem_sphere]
-#align metric.mem_sphere' Metric.mem_sphere'
-
-theorem ne_of_mem_sphere (h : y ∈ sphere x ε) (hε : ε ≠ 0) : y ≠ x :=
-  ne_of_mem_of_not_mem h <| by simpa using hε.symm
-#align metric.ne_of_mem_sphere Metric.ne_of_mem_sphere
-
-theorem nonneg_of_mem_sphere (hy : y ∈ sphere x ε) : 0 ≤ ε :=
-  dist_nonneg.trans_eq hy
-#align metric.nonneg_of_mem_sphere Metric.nonneg_of_mem_sphere
-
-@[simp]
-theorem sphere_eq_empty_of_neg (hε : ε < 0) : sphere x ε = ∅ :=
-  Set.eq_empty_iff_forall_not_mem.mpr fun _y hy => (nonneg_of_mem_sphere hy).not_lt hε
-#align metric.sphere_eq_empty_of_neg Metric.sphere_eq_empty_of_neg
-
-theorem sphere_eq_empty_of_subsingleton [Subsingleton α] (hε : ε ≠ 0) : sphere x ε = ∅ :=
-  Set.eq_empty_iff_forall_not_mem.mpr fun _ h => ne_of_mem_sphere h hε (Subsingleton.elim _ _)
-#align metric.sphere_eq_empty_of_subsingleton Metric.sphere_eq_empty_of_subsingleton
-
-theorem sphere_isEmpty_of_subsingleton [Subsingleton α] (hε : ε ≠ 0) : IsEmpty (sphere x ε) := by
-  rw [sphere_eq_empty_of_subsingleton hε]; infer_instance
-#align metric.sphere_is_empty_of_subsingleton Metric.sphere_isEmpty_of_subsingleton
-
-theorem mem_closedBall_self (h : 0 ≤ ε) : x ∈ closedBall x ε := by
-  rwa [mem_closedBall, dist_self]
-#align metric.mem_closed_ball_self Metric.mem_closedBall_self
-
-@[simp]
-theorem nonempty_closedBall : (closedBall x ε).Nonempty ↔ 0 ≤ ε :=
-  ⟨fun ⟨_x, hx⟩ => dist_nonneg.trans hx, fun h => ⟨x, mem_closedBall_self h⟩⟩
-#align metric.nonempty_closed_ball Metric.nonempty_closedBall
-
-@[simp]
-theorem closedBall_eq_empty : closedBall x ε = ∅ ↔ ε < 0 := by
-  rw [← not_nonempty_iff_eq_empty, nonempty_closedBall, not_le]
-#align metric.closed_ball_eq_empty Metric.closedBall_eq_empty
-
-/-- Closed balls and spheres coincide when the radius is non-positive -/
-theorem closedBall_eq_sphere_of_nonpos (hε : ε ≤ 0) : closedBall x ε = sphere x ε :=
-  Set.ext fun _ => (hε.trans dist_nonneg).le_iff_eq
-#align metric.closed_ball_eq_sphere_of_nonpos Metric.closedBall_eq_sphere_of_nonpos
-
-theorem ball_subset_closedBall : ball x ε ⊆ closedBall x ε := fun _y hy =>
-  mem_closedBall.2 (le_of_lt hy)
-#align metric.ball_subset_closed_ball Metric.ball_subset_closedBall
-
-theorem sphere_subset_closedBall : sphere x ε ⊆ closedBall x ε := fun _ => le_of_eq
-#align metric.sphere_subset_closed_ball Metric.sphere_subset_closedBall
-
-lemma sphere_subset_ball {r R : ℝ} (h : r < R) : sphere x r ⊆ ball x R := fun _x hx ↦
-  (mem_sphere.1 hx).trans_lt h
-
-theorem closedBall_disjoint_ball (h : δ + ε ≤ dist x y) : Disjoint (closedBall x δ) (ball y ε) :=
-  Set.disjoint_left.mpr fun _a ha1 ha2 =>
-    (h.trans <| dist_triangle_left _ _ _).not_lt <| add_lt_add_of_le_of_lt ha1 ha2
-#align metric.closed_ball_disjoint_ball Metric.closedBall_disjoint_ball
-
-theorem ball_disjoint_closedBall (h : δ + ε ≤ dist x y) : Disjoint (ball x δ) (closedBall y ε) :=
-  (closedBall_disjoint_ball <| by rwa [add_comm, dist_comm]).symm
-#align metric.ball_disjoint_closed_ball Metric.ball_disjoint_closedBall
-
-theorem ball_disjoint_ball (h : δ + ε ≤ dist x y) : Disjoint (ball x δ) (ball y ε) :=
-  (closedBall_disjoint_ball h).mono_left ball_subset_closedBall
-#align metric.ball_disjoint_ball Metric.ball_disjoint_ball
-
-theorem closedBall_disjoint_closedBall (h : δ + ε < dist x y) :
-    Disjoint (closedBall x δ) (closedBall y ε) :=
-  Set.disjoint_left.mpr fun _a ha1 ha2 =>
-    h.not_le <| (dist_triangle_left _ _ _).trans <| add_le_add ha1 ha2
-#align metric.closed_ball_disjoint_closed_ball Metric.closedBall_disjoint_closedBall
-
-theorem sphere_disjoint_ball : Disjoint (sphere x ε) (ball x ε) :=
-  Set.disjoint_left.mpr fun _y hy₁ hy₂ => absurd hy₁ <| ne_of_lt hy₂
-#align metric.sphere_disjoint_ball Metric.sphere_disjoint_ball
-
-@[simp]
-theorem ball_union_sphere : ball x ε ∪ sphere x ε = closedBall x ε :=
-  Set.ext fun _y => (@le_iff_lt_or_eq ℝ _ _ _).symm
-#align metric.ball_union_sphere Metric.ball_union_sphere
-
-@[simp]
-theorem sphere_union_ball : sphere x ε ∪ ball x ε = closedBall x ε := by
-  rw [union_comm, ball_union_sphere]
-#align metric.sphere_union_ball Metric.sphere_union_ball
-
-@[simp]
-theorem closedBall_diff_sphere : closedBall x ε \ sphere x ε = ball x ε := by
-  rw [← ball_union_sphere, Set.union_diff_cancel_right sphere_disjoint_ball.symm.le_bot]
-#align metric.closed_ball_diff_sphere Metric.closedBall_diff_sphere
-
-@[simp]
-theorem closedBall_diff_ball : closedBall x ε \ ball x ε = sphere x ε := by
-  rw [← ball_union_sphere, Set.union_diff_cancel_left sphere_disjoint_ball.symm.le_bot]
-#align metric.closed_ball_diff_ball Metric.closedBall_diff_ball
-
-theorem mem_ball_comm : x ∈ ball y ε ↔ y ∈ ball x ε := by rw [mem_ball', mem_ball]
-#align metric.mem_ball_comm Metric.mem_ball_comm
-
-theorem mem_closedBall_comm : x ∈ closedBall y ε ↔ y ∈ closedBall x ε := by
-  rw [mem_closedBall', mem_closedBall]
-#align metric.mem_closed_ball_comm Metric.mem_closedBall_comm
-
-theorem mem_sphere_comm : x ∈ sphere y ε ↔ y ∈ sphere x ε := by rw [mem_sphere', mem_sphere]
-#align metric.mem_sphere_comm Metric.mem_sphere_comm
-
-theorem ball_subset_ball (h : ε₁ ≤ ε₂) : ball x ε₁ ⊆ ball x ε₂ := fun _y yx =>
-  lt_of_lt_of_le (mem_ball.1 yx) h
-#align metric.ball_subset_ball Metric.ball_subset_ball
-
-theorem closedBall_eq_bInter_ball : closedBall x ε = ⋂ δ > ε, ball x δ := by
-  ext y; rw [mem_closedBall, ← forall_lt_iff_le', mem_iInter₂]; rfl
-#align metric.closed_ball_eq_bInter_ball Metric.closedBall_eq_bInter_ball
-
-theorem ball_subset_ball' (h : ε₁ + dist x y ≤ ε₂) : ball x ε₁ ⊆ ball y ε₂ := fun z hz =>
-  calc
-    dist z y ≤ dist z x + dist x y := dist_triangle _ _ _
-    _ < ε₁ + dist x y := add_lt_add_right (mem_ball.1 hz) _
-    _ ≤ ε₂ := h
-#align metric.ball_subset_ball' Metric.ball_subset_ball'
-
-theorem closedBall_subset_closedBall (h : ε₁ ≤ ε₂) : closedBall x ε₁ ⊆ closedBall x ε₂ :=
-  fun _y (yx : _ ≤ ε₁) => le_trans yx h
-#align metric.closed_ball_subset_closed_ball Metric.closedBall_subset_closedBall
-
-theorem closedBall_subset_closedBall' (h : ε₁ + dist x y ≤ ε₂) :
-    closedBall x ε₁ ⊆ closedBall y ε₂ := fun z hz =>
-  calc
-    dist z y ≤ dist z x + dist x y := dist_triangle _ _ _
-    _ ≤ ε₁ + dist x y := add_le_add_right (mem_closedBall.1 hz) _
-    _ ≤ ε₂ := h
-#align metric.closed_ball_subset_closed_ball' Metric.closedBall_subset_closedBall'
-
-theorem closedBall_subset_ball (h : ε₁ < ε₂) : closedBall x ε₁ ⊆ ball x ε₂ :=
-  fun y (yh : dist y x ≤ ε₁) => lt_of_le_of_lt yh h
-#align metric.closed_ball_subset_ball Metric.closedBall_subset_ball
-
-theorem closedBall_subset_ball' (h : ε₁ + dist x y < ε₂) :
-    closedBall x ε₁ ⊆ ball y ε₂ := fun z hz =>
-  calc
-    dist z y ≤ dist z x + dist x y := dist_triangle _ _ _
-    _ ≤ ε₁ + dist x y := add_le_add_right (mem_closedBall.1 hz) _
-    _ < ε₂ := h
-#align metric.closed_ball_subset_ball' Metric.closedBall_subset_ball'
-
-theorem dist_le_add_of_nonempty_closedBall_inter_closedBall
-    (h : (closedBall x ε₁ ∩ closedBall y ε₂).Nonempty) : dist x y ≤ ε₁ + ε₂ :=
-  let ⟨z, hz⟩ := h
-  calc
-    dist x y ≤ dist z x + dist z y := dist_triangle_left _ _ _
-    _ ≤ ε₁ + ε₂ := add_le_add hz.1 hz.2
-#align metric.dist_le_add_of_nonempty_closed_ball_inter_closed_ball Metric.dist_le_add_of_nonempty_closedBall_inter_closedBall
-
-theorem dist_lt_add_of_nonempty_closedBall_inter_ball (h : (closedBall x ε₁ ∩ ball y ε₂).Nonempty) :
-    dist x y < ε₁ + ε₂ :=
-  let ⟨z, hz⟩ := h
-  calc
-    dist x y ≤ dist z x + dist z y := dist_triangle_left _ _ _
-    _ < ε₁ + ε₂ := add_lt_add_of_le_of_lt hz.1 hz.2
-#align metric.dist_lt_add_of_nonempty_closed_ball_inter_ball Metric.dist_lt_add_of_nonempty_closedBall_inter_ball
-
-theorem dist_lt_add_of_nonempty_ball_inter_closedBall (h : (ball x ε₁ ∩ closedBall y ε₂).Nonempty) :
-    dist x y < ε₁ + ε₂ := by
-  rw [inter_comm] at h
-  rw [add_comm, dist_comm]
-  exact dist_lt_add_of_nonempty_closedBall_inter_ball h
-#align metric.dist_lt_add_of_nonempty_ball_inter_closed_ball Metric.dist_lt_add_of_nonempty_ball_inter_closedBall
-
-theorem dist_lt_add_of_nonempty_ball_inter_ball (h : (ball x ε₁ ∩ ball y ε₂).Nonempty) :
-    dist x y < ε₁ + ε₂ :=
-  dist_lt_add_of_nonempty_closedBall_inter_ball <|
-    h.mono (inter_subset_inter ball_subset_closedBall Subset.rfl)
-#align metric.dist_lt_add_of_nonempty_ball_inter_ball Metric.dist_lt_add_of_nonempty_ball_inter_ball
-
-@[simp]
-theorem iUnion_closedBall_nat (x : α) : ⋃ n : ℕ, closedBall x n = univ :=
-  iUnion_eq_univ_iff.2 fun y => exists_nat_ge (dist y x)
-#align metric.Union_closed_ball_nat Metric.iUnion_closedBall_nat
-
-theorem iUnion_inter_closedBall_nat (s : Set α) (x : α) : ⋃ n : ℕ, s ∩ closedBall x n = s := by
-  rw [← inter_iUnion, iUnion_closedBall_nat, inter_univ]
-#align metric.Union_inter_closed_ball_nat Metric.iUnion_inter_closedBall_nat
-
-theorem ball_subset (h : dist x y ≤ ε₂ - ε₁) : ball x ε₁ ⊆ ball y ε₂ := fun z zx => by
-  rw [← add_sub_cancel'_right ε₁ ε₂]
-  exact lt_of_le_of_lt (dist_triangle z x y) (add_lt_add_of_lt_of_le zx h)
-#align metric.ball_subset Metric.ball_subset
-
-theorem ball_half_subset (y) (h : y ∈ ball x (ε / 2)) : ball y (ε / 2) ⊆ ball x ε :=
-  ball_subset <| by rw [sub_self_div_two]; exact le_of_lt h
-#align metric.ball_half_subset Metric.ball_half_subset
-
-theorem exists_ball_subset_ball (h : y ∈ ball x ε) : ∃ ε' > 0, ball y ε' ⊆ ball x ε :=
-  ⟨_, sub_pos.2 h, ball_subset <| by rw [sub_sub_self]⟩
-#align metric.exists_ball_subset_ball Metric.exists_ball_subset_ball
-
-/-- If a property holds for all points in closed balls of arbitrarily large radii, then it holds for
-all points. -/
-theorem forall_of_forall_mem_closedBall (p : α → Prop) (x : α)
-    (H : ∃ᶠ R : ℝ in atTop, ∀ y ∈ closedBall x R, p y) (y : α) : p y := by
-  obtain ⟨R, hR, h⟩ : ∃ R ≥ dist y x, ∀ z : α, z ∈ closedBall x R → p z :=
-    frequently_iff.1 H (Ici_mem_atTop (dist y x))
-  exact h _ hR
-#align metric.forall_of_forall_mem_closed_ball Metric.forall_of_forall_mem_closedBall
-
-/-- If a property holds for all points in balls of arbitrarily large radii, then it holds for all
-points. -/
-theorem forall_of_forall_mem_ball (p : α → Prop) (x : α)
-    (H : ∃ᶠ R : ℝ in atTop, ∀ y ∈ ball x R, p y) (y : α) : p y := by
-  obtain ⟨R, hR, h⟩ : ∃ R > dist y x, ∀ z : α, z ∈ ball x R → p z :=
-    frequently_iff.1 H (Ioi_mem_atTop (dist y x))
-  exact h _ hR
-#align metric.forall_of_forall_mem_ball Metric.forall_of_forall_mem_ball
-
-theorem isBounded_iff {s : Set α} :
-    IsBounded s ↔ ∃ C : ℝ, ∀ ⦃x⦄, x ∈ s → ∀ ⦃y⦄, y ∈ s → dist x y ≤ C := by
-  rw [isBounded_def, ← Filter.mem_sets, @PseudoMetricSpace.cobounded_sets α, mem_setOf_eq,
-    compl_compl]
-#align metric.is_bounded_iff Metric.isBounded_iff
-
-theorem isBounded_iff_eventually {s : Set α} :
-    IsBounded s ↔ ∀ᶠ C in atTop, ∀ ⦃x⦄, x ∈ s → ∀ ⦃y⦄, y ∈ s → dist x y ≤ C :=
-  isBounded_iff.trans
-    ⟨fun ⟨C, h⟩ => eventually_atTop.2 ⟨C, fun _C' hC' _x hx _y hy => (h hx hy).trans hC'⟩,
-      Eventually.exists⟩
-#align metric.is_bounded_iff_eventually Metric.isBounded_iff_eventually
-
-theorem isBounded_iff_exists_ge {s : Set α} (c : ℝ) :
-    IsBounded s ↔ ∃ C, c ≤ C ∧ ∀ ⦃x⦄, x ∈ s → ∀ ⦃y⦄, y ∈ s → dist x y ≤ C :=
-  ⟨fun h => ((eventually_ge_atTop c).and (isBounded_iff_eventually.1 h)).exists, fun h =>
-    isBounded_iff.2 <| h.imp fun _ => And.right⟩
-#align metric.is_bounded_iff_exists_ge Metric.isBounded_iff_exists_ge
-
-theorem isBounded_iff_nndist {s : Set α} :
-    IsBounded s ↔ ∃ C : ℝ≥0, ∀ ⦃x⦄, x ∈ s → ∀ ⦃y⦄, y ∈ s → nndist x y ≤ C := by
-  simp only [isBounded_iff_exists_ge 0, NNReal.exists, ← NNReal.coe_le_coe, ← dist_nndist,
-    NNReal.coe_mk, exists_prop]
-#align metric.is_bounded_iff_nndist Metric.isBounded_iff_nndist
-
-theorem toUniformSpace_eq :
-    ‹PseudoMetricSpace α›.toUniformSpace = .ofDist dist dist_self dist_comm dist_triangle :=
-  UniformSpace.ext PseudoMetricSpace.uniformity_dist
-#align metric.to_uniform_space_eq Metric.toUniformSpace_eq
-
-theorem uniformity_basis_dist :
-    (𝓤 α).HasBasis (fun ε : ℝ => 0 < ε) fun ε => { p : α × α | dist p.1 p.2 < ε } := by
-  rw [toUniformSpace_eq]
-  exact UniformSpace.hasBasis_ofFun (exists_gt _) _ _ _ _ _
-#align metric.uniformity_basis_dist Metric.uniformity_basis_dist
-
-/-- Given `f : β → ℝ`, if `f` sends `{i | p i}` to a set of positive numbers
-accumulating to zero, then `f i`-neighborhoods of the diagonal form a basis of `𝓤 α`.
-
-For specific bases see `uniformity_basis_dist`, `uniformity_basis_dist_inv_nat_succ`,
-and `uniformity_basis_dist_inv_nat_pos`. -/
-protected theorem mk_uniformity_basis {β : Type*} {p : β → Prop} {f : β → ℝ}
-    (hf₀ : ∀ i, p i → 0 < f i) (hf : ∀ ⦃ε⦄, 0 < ε → ∃ i, p i ∧ f i ≤ ε) :
-    (𝓤 α).HasBasis p fun i => { p : α × α | dist p.1 p.2 < f i } := by
-  refine' ⟨fun s => uniformity_basis_dist.mem_iff.trans _⟩
-  constructor
-  · rintro ⟨ε, ε₀, hε⟩
-    rcases hf ε₀ with ⟨i, hi, H⟩
-    exact ⟨i, hi, fun x (hx : _ < _) => hε <| lt_of_lt_of_le hx H⟩
-  · exact fun ⟨i, hi, H⟩ => ⟨f i, hf₀ i hi, H⟩
-#align metric.mk_uniformity_basis Metric.mk_uniformity_basis
-
-theorem uniformity_basis_dist_rat :
-    (𝓤 α).HasBasis (fun r : ℚ => 0 < r) fun r => { p : α × α | dist p.1 p.2 < r } :=
-  Metric.mk_uniformity_basis (fun _ => Rat.cast_pos.2) fun _ε hε =>
-    let ⟨r, hr0, hrε⟩ := exists_rat_btwn hε
-    ⟨r, Rat.cast_pos.1 hr0, hrε.le⟩
-#align metric.uniformity_basis_dist_rat Metric.uniformity_basis_dist_rat
-
-theorem uniformity_basis_dist_inv_nat_succ :
-    (𝓤 α).HasBasis (fun _ => True) fun n : ℕ => { p : α × α | dist p.1 p.2 < 1 / (↑n + 1) } :=
-  Metric.mk_uniformity_basis (fun n _ => div_pos zero_lt_one <| Nat.cast_add_one_pos n) fun _ε ε0 =>
-    (exists_nat_one_div_lt ε0).imp fun _n hn => ⟨trivial, le_of_lt hn⟩
-#align metric.uniformity_basis_dist_inv_nat_succ Metric.uniformity_basis_dist_inv_nat_succ
-
-theorem uniformity_basis_dist_inv_nat_pos :
-    (𝓤 α).HasBasis (fun n : ℕ => 0 < n) fun n : ℕ => { p : α × α | dist p.1 p.2 < 1 / ↑n } :=
-  Metric.mk_uniformity_basis (fun n hn => div_pos zero_lt_one <| Nat.cast_pos.2 hn) fun ε ε0 =>
-    let ⟨n, hn⟩ := exists_nat_one_div_lt ε0
-    ⟨n + 1, Nat.succ_pos n, by exact_mod_cast hn.le⟩
-#align metric.uniformity_basis_dist_inv_nat_pos Metric.uniformity_basis_dist_inv_nat_pos
-
-theorem uniformity_basis_dist_pow {r : ℝ} (h0 : 0 < r) (h1 : r < 1) :
-    (𝓤 α).HasBasis (fun _ : ℕ => True) fun n : ℕ => { p : α × α | dist p.1 p.2 < r ^ n } :=
-  Metric.mk_uniformity_basis (fun _ _ => pow_pos h0 _) fun _ε ε0 =>
-    let ⟨n, hn⟩ := exists_pow_lt_of_lt_one ε0 h1
-    ⟨n, trivial, hn.le⟩
-#align metric.uniformity_basis_dist_pow Metric.uniformity_basis_dist_pow
-
-theorem uniformity_basis_dist_lt {R : ℝ} (hR : 0 < R) :
-    (𝓤 α).HasBasis (fun r : ℝ => 0 < r ∧ r < R) fun r => { p : α × α | dist p.1 p.2 < r } :=
-  Metric.mk_uniformity_basis (fun _ => And.left) fun r hr =>
-    ⟨min r (R / 2), ⟨lt_min hr (half_pos hR), min_lt_iff.2 <| Or.inr (half_lt_self hR)⟩,
-      min_le_left _ _⟩
-#align metric.uniformity_basis_dist_lt Metric.uniformity_basis_dist_lt
-
-/-- Given `f : β → ℝ`, if `f` sends `{i | p i}` to a set of positive numbers
-accumulating to zero, then closed neighborhoods of the diagonal of sizes `{f i | p i}`
-form a basis of `𝓤 α`.
-
-Currently we have only one specific basis `uniformity_basis_dist_le` based on this constructor.
-More can be easily added if needed in the future. -/
-protected theorem mk_uniformity_basis_le {β : Type*} {p : β → Prop} {f : β → ℝ}
-    (hf₀ : ∀ x, p x → 0 < f x) (hf : ∀ ε, 0 < ε → ∃ x, p x ∧ f x ≤ ε) :
-    (𝓤 α).HasBasis p fun x => { p : α × α | dist p.1 p.2 ≤ f x } := by
-  refine' ⟨fun s => uniformity_basis_dist.mem_iff.trans _⟩
-  constructor
-  · rintro ⟨ε, ε₀, hε⟩
-    rcases exists_between ε₀ with ⟨ε', hε'⟩
-    rcases hf ε' hε'.1 with ⟨i, hi, H⟩
-    exact ⟨i, hi, fun x (hx : _ ≤ _) => hε <| lt_of_le_of_lt (le_trans hx H) hε'.2⟩
-  · exact fun ⟨i, hi, H⟩ => ⟨f i, hf₀ i hi, fun x (hx : _ < _) => H (mem_setOf.2 hx.le)⟩
-#align metric.mk_uniformity_basis_le Metric.mk_uniformity_basis_le
-
-/-- Constant size closed neighborhoods of the diagonal form a basis
-of the uniformity filter. -/
-theorem uniformity_basis_dist_le :
-    (𝓤 α).HasBasis ((0 : ℝ) < ·) fun ε => { p : α × α | dist p.1 p.2 ≤ ε } :=
-  Metric.mk_uniformity_basis_le (fun _ => id) fun ε ε₀ => ⟨ε, ε₀, le_refl ε⟩
-#align metric.uniformity_basis_dist_le Metric.uniformity_basis_dist_le
-
-theorem uniformity_basis_dist_le_pow {r : ℝ} (h0 : 0 < r) (h1 : r < 1) :
-    (𝓤 α).HasBasis (fun _ : ℕ => True) fun n : ℕ => { p : α × α | dist p.1 p.2 ≤ r ^ n } :=
-  Metric.mk_uniformity_basis_le (fun _ _ => pow_pos h0 _) fun _ε ε0 =>
-    let ⟨n, hn⟩ := exists_pow_lt_of_lt_one ε0 h1
-    ⟨n, trivial, hn.le⟩
-#align metric.uniformity_basis_dist_le_pow Metric.uniformity_basis_dist_le_pow
-
-theorem mem_uniformity_dist {s : Set (α × α)} :
-    s ∈ 𝓤 α ↔ ∃ ε > 0, ∀ {a b : α}, dist a b < ε → (a, b) ∈ s :=
-  uniformity_basis_dist.mem_uniformity_iff
-#align metric.mem_uniformity_dist Metric.mem_uniformity_dist
-
-/-- A constant size neighborhood of the diagonal is an entourage. -/
-theorem dist_mem_uniformity {ε : ℝ} (ε0 : 0 < ε) : { p : α × α | dist p.1 p.2 < ε } ∈ 𝓤 α :=
-  mem_uniformity_dist.2 ⟨ε, ε0, id⟩
-#align metric.dist_mem_uniformity Metric.dist_mem_uniformity
-
-theorem uniformContinuous_iff [PseudoMetricSpace β] {f : α → β} :
-    UniformContinuous f ↔ ∀ ε > 0, ∃ δ > 0, ∀ {a b : α}, dist a b < δ → dist (f a) (f b) < ε :=
-  uniformity_basis_dist.uniformContinuous_iff uniformity_basis_dist
-#align metric.uniform_continuous_iff Metric.uniformContinuous_iff
-
-theorem uniformContinuousOn_iff [PseudoMetricSpace β] {f : α → β} {s : Set α} :
-    UniformContinuousOn f s ↔
-      ∀ ε > 0, ∃ δ > 0, ∀ x ∈ s, ∀ y ∈ s, dist x y < δ → dist (f x) (f y) < ε :=
-  Metric.uniformity_basis_dist.uniformContinuousOn_iff Metric.uniformity_basis_dist
-#align metric.uniform_continuous_on_iff Metric.uniformContinuousOn_iff
-
-theorem uniformContinuousOn_iff_le [PseudoMetricSpace β] {f : α → β} {s : Set α} :
-    UniformContinuousOn f s ↔
-      ∀ ε > 0, ∃ δ > 0, ∀ x ∈ s, ∀ y ∈ s, dist x y ≤ δ → dist (f x) (f y) ≤ ε :=
-  Metric.uniformity_basis_dist_le.uniformContinuousOn_iff Metric.uniformity_basis_dist_le
-#align metric.uniform_continuous_on_iff_le Metric.uniformContinuousOn_iff_le
-
-nonrec theorem uniformInducing_iff [PseudoMetricSpace β] {f : α → β} :
-    UniformInducing f ↔ UniformContinuous f ∧
-      ∀ δ > 0, ∃ ε > 0, ∀ {a b : α}, dist (f a) (f b) < ε → dist a b < δ :=
-  uniformInducing_iff'.trans <| Iff.rfl.and <|
-    ((uniformity_basis_dist.comap _).le_basis_iff uniformity_basis_dist).trans <| by
-      simp only [subset_def, Prod.forall, gt_iff_lt, preimage_setOf_eq, Prod_map, mem_setOf]
-
-nonrec theorem uniformEmbedding_iff [PseudoMetricSpace β] {f : α → β} :
-    UniformEmbedding f ↔ Function.Injective f ∧ UniformContinuous f ∧
-      ∀ δ > 0, ∃ ε > 0, ∀ {a b : α}, dist (f a) (f b) < ε → dist a b < δ := by
-  rw [uniformEmbedding_iff, and_comm, uniformInducing_iff]
-#align metric.uniform_embedding_iff Metric.uniformEmbedding_iff
-
-/-- If a map between pseudometric spaces is a uniform embedding then the distance between `f x`
-and `f y` is controlled in terms of the distance between `x` and `y`. -/
-theorem controlled_of_uniformEmbedding [PseudoMetricSpace β] {f : α → β} (h : UniformEmbedding f) :
-    (∀ ε > 0, ∃ δ > 0, ∀ {a b : α}, dist a b < δ → dist (f a) (f b) < ε) ∧
-      ∀ δ > 0, ∃ ε > 0, ∀ {a b : α}, dist (f a) (f b) < ε → dist a b < δ :=
-  ⟨uniformContinuous_iff.1 h.uniformContinuous, (uniformEmbedding_iff.1 h).2.2⟩
-#align metric.controlled_of_uniform_embedding Metric.controlled_of_uniformEmbedding
-
-theorem totallyBounded_iff {s : Set α} :
-    TotallyBounded s ↔ ∀ ε > 0, ∃ t : Set α, t.Finite ∧ s ⊆ ⋃ y ∈ t, ball y ε :=
-  uniformity_basis_dist.totallyBounded_iff
-#align metric.totally_bounded_iff Metric.totallyBounded_iff
-
-/-- A pseudometric space is totally bounded if one can reconstruct up to any ε>0 any element of the
-space from finitely many data. -/
-theorem totallyBounded_of_finite_discretization {s : Set α}
-    (H : ∀ ε > (0 : ℝ),
-        ∃ (β : Type u) (_ : Fintype β) (F : s → β), ∀ x y, F x = F y → dist (x : α) y < ε) :
-    TotallyBounded s := by
-  cases' s.eq_empty_or_nonempty with hs hs
-  · rw [hs]
-    exact totallyBounded_empty
-  rcases hs with ⟨x0, hx0⟩
-  haveI : Inhabited s := ⟨⟨x0, hx0⟩⟩
-  refine' totallyBounded_iff.2 fun ε ε0 => _
-  rcases H ε ε0 with ⟨β, fβ, F, hF⟩
-  let Finv := Function.invFun F
-  refine' ⟨range (Subtype.val ∘ Finv), finite_range _, fun x xs => _⟩
-  let x' := Finv (F ⟨x, xs⟩)
-  have : F x' = F ⟨x, xs⟩ := Function.invFun_eq ⟨⟨x, xs⟩, rfl⟩
-  simp only [Set.mem_iUnion, Set.mem_range]
-  exact ⟨_, ⟨F ⟨x, xs⟩, rfl⟩, hF _ _ this.symm⟩
-#align metric.totally_bounded_of_finite_discretization Metric.totallyBounded_of_finite_discretization
-
-theorem finite_approx_of_totallyBounded {s : Set α} (hs : TotallyBounded s) :
-    ∀ ε > 0, ∃ t, t ⊆ s ∧ Set.Finite t ∧ s ⊆ ⋃ y ∈ t, ball y ε := by
-  intro ε ε_pos
-  rw [totallyBounded_iff_subset] at hs
-  exact hs _ (dist_mem_uniformity ε_pos)
-#align metric.finite_approx_of_totally_bounded Metric.finite_approx_of_totallyBounded
-
-/-- Expressing uniform convergence using `dist` -/
-theorem tendstoUniformlyOnFilter_iff {F : ι → β → α} {f : β → α} {p : Filter ι} {p' : Filter β} :
-    TendstoUniformlyOnFilter F f p p' ↔
-      ∀ ε > 0, ∀ᶠ n : ι × β in p ×ˢ p', dist (f n.snd) (F n.fst n.snd) < ε := by
-  refine' ⟨fun H ε hε => H _ (dist_mem_uniformity hε), fun H u hu => _⟩
-  rcases mem_uniformity_dist.1 hu with ⟨ε, εpos, hε⟩
-  refine' (H ε εpos).mono fun n hn => hε hn
-#align metric.tendsto_uniformly_on_filter_iff Metric.tendstoUniformlyOnFilter_iff
-
-/-- Expressing locally uniform convergence on a set using `dist`. -/
-theorem tendstoLocallyUniformlyOn_iff [TopologicalSpace β] {F : ι → β → α} {f : β → α}
-    {p : Filter ι} {s : Set β} :
-    TendstoLocallyUniformlyOn F f p s ↔
-      ∀ ε > 0, ∀ x ∈ s, ∃ t ∈ 𝓝[s] x, ∀ᶠ n in p, ∀ y ∈ t, dist (f y) (F n y) < ε := by
-  refine' ⟨fun H ε hε => H _ (dist_mem_uniformity hε), fun H u hu x hx => _⟩
-  rcases mem_uniformity_dist.1 hu with ⟨ε, εpos, hε⟩
-  rcases H ε εpos x hx with ⟨t, ht, Ht⟩
-  exact ⟨t, ht, Ht.mono fun n hs x hx => hε (hs x hx)⟩
-#align metric.tendsto_locally_uniformly_on_iff Metric.tendstoLocallyUniformlyOn_iff
-
-/-- Expressing uniform convergence on a set using `dist`. -/
-theorem tendstoUniformlyOn_iff {F : ι → β → α} {f : β → α} {p : Filter ι} {s : Set β} :
-    TendstoUniformlyOn F f p s ↔ ∀ ε > 0, ∀ᶠ n in p, ∀ x ∈ s, dist (f x) (F n x) < ε := by
-  refine' ⟨fun H ε hε => H _ (dist_mem_uniformity hε), fun H u hu => _⟩
-  rcases mem_uniformity_dist.1 hu with ⟨ε, εpos, hε⟩
-  exact (H ε εpos).mono fun n hs x hx => hε (hs x hx)
-#align metric.tendsto_uniformly_on_iff Metric.tendstoUniformlyOn_iff
-
-/-- Expressing locally uniform convergence using `dist`. -/
-theorem tendstoLocallyUniformly_iff [TopologicalSpace β] {F : ι → β → α} {f : β → α}
-    {p : Filter ι} :
-    TendstoLocallyUniformly F f p ↔
-      ∀ ε > 0, ∀ x : β, ∃ t ∈ 𝓝 x, ∀ᶠ n in p, ∀ y ∈ t, dist (f y) (F n y) < ε := by
-  simp only [← tendstoLocallyUniformlyOn_univ, tendstoLocallyUniformlyOn_iff, nhdsWithin_univ,
-    mem_univ, forall_const, exists_prop]
-#align metric.tendsto_locally_uniformly_iff Metric.tendstoLocallyUniformly_iff
-
-/-- Expressing uniform convergence using `dist`. -/
-theorem tendstoUniformly_iff {F : ι → β → α} {f : β → α} {p : Filter ι} :
-    TendstoUniformly F f p ↔ ∀ ε > 0, ∀ᶠ n in p, ∀ x, dist (f x) (F n x) < ε := by
-  rw [← tendstoUniformlyOn_univ, tendstoUniformlyOn_iff]
-  simp
-#align metric.tendsto_uniformly_iff Metric.tendstoUniformly_iff
-
-protected theorem cauchy_iff {f : Filter α} :
-    Cauchy f ↔ NeBot f ∧ ∀ ε > 0, ∃ t ∈ f, ∀ x ∈ t, ∀ y ∈ t, dist x y < ε :=
-  uniformity_basis_dist.cauchy_iff
-#align metric.cauchy_iff Metric.cauchy_iff
-
-theorem nhds_basis_ball : (𝓝 x).HasBasis (0 < ·) (ball x) :=
-  nhds_basis_uniformity uniformity_basis_dist
-#align metric.nhds_basis_ball Metric.nhds_basis_ball
-
-theorem mem_nhds_iff : s ∈ 𝓝 x ↔ ∃ ε > 0, ball x ε ⊆ s :=
-  nhds_basis_ball.mem_iff
-#align metric.mem_nhds_iff Metric.mem_nhds_iff
-
-theorem eventually_nhds_iff {p : α → Prop} :
-    (∀ᶠ y in 𝓝 x, p y) ↔ ∃ ε > 0, ∀ ⦃y⦄, dist y x < ε → p y :=
-  mem_nhds_iff
-#align metric.eventually_nhds_iff Metric.eventually_nhds_iff
-
-theorem eventually_nhds_iff_ball {p : α → Prop} :
-    (∀ᶠ y in 𝓝 x, p y) ↔ ∃ ε > 0, ∀ y ∈ ball x ε, p y :=
-  mem_nhds_iff
-#align metric.eventually_nhds_iff_ball Metric.eventually_nhds_iff_ball
-
-/-- A version of `Filter.eventually_prod_iff` where the first filter consists of neighborhoods
-in a pseudo-metric space.-/
-theorem eventually_nhds_prod_iff {f : Filter ι} {x₀ : α} {p : α × ι → Prop} :
-    (∀ᶠ x in 𝓝 x₀ ×ˢ f, p x) ↔ ∃ ε > (0 : ℝ), ∃ pa : ι → Prop, (∀ᶠ i in f, pa i) ∧
-      ∀ {x}, dist x x₀ < ε → ∀ {i}, pa i → p (x, i) := by
-  refine (nhds_basis_ball.prod f.basis_sets).eventually_iff.trans ?_
-  simp only [Prod.exists, forall_prod_set, id, mem_ball, and_assoc, exists_and_left, and_imp]
-  rfl
-#align metric.eventually_nhds_prod_iff Metric.eventually_nhds_prod_iff
-
-/-- A version of `Filter.eventually_prod_iff` where the second filter consists of neighborhoods
-in a pseudo-metric space.-/
-theorem eventually_prod_nhds_iff {f : Filter ι} {x₀ : α} {p : ι × α → Prop} :
-    (∀ᶠ x in f ×ˢ 𝓝 x₀, p x) ↔ ∃ pa : ι → Prop, (∀ᶠ i in f, pa i) ∧
-      ∃ ε > 0, ∀ {i}, pa i → ∀ {x}, dist x x₀ < ε → p (i, x) := by
-  rw [eventually_swap_iff, Metric.eventually_nhds_prod_iff]
-  constructor <;>
-    · rintro ⟨a1, a2, a3, a4, a5⟩
-      exact ⟨a3, a4, a1, a2, fun b1 b2 b3 => a5 b3 b1⟩
-#align metric.eventually_prod_nhds_iff Metric.eventually_prod_nhds_iff
-
-theorem nhds_basis_closedBall : (𝓝 x).HasBasis (fun ε : ℝ => 0 < ε) (closedBall x) :=
-  nhds_basis_uniformity uniformity_basis_dist_le
-#align metric.nhds_basis_closed_ball Metric.nhds_basis_closedBall
-
-theorem nhds_basis_ball_inv_nat_succ :
-    (𝓝 x).HasBasis (fun _ => True) fun n : ℕ => ball x (1 / (↑n + 1)) :=
-  nhds_basis_uniformity uniformity_basis_dist_inv_nat_succ
-#align metric.nhds_basis_ball_inv_nat_succ Metric.nhds_basis_ball_inv_nat_succ
-
-theorem nhds_basis_ball_inv_nat_pos :
-    (𝓝 x).HasBasis (fun n => 0 < n) fun n : ℕ => ball x (1 / ↑n) :=
-  nhds_basis_uniformity uniformity_basis_dist_inv_nat_pos
-#align metric.nhds_basis_ball_inv_nat_pos Metric.nhds_basis_ball_inv_nat_pos
-
-theorem nhds_basis_ball_pow {r : ℝ} (h0 : 0 < r) (h1 : r < 1) :
-    (𝓝 x).HasBasis (fun _ => True) fun n : ℕ => ball x (r ^ n) :=
-  nhds_basis_uniformity (uniformity_basis_dist_pow h0 h1)
-#align metric.nhds_basis_ball_pow Metric.nhds_basis_ball_pow
-
-theorem nhds_basis_closedBall_pow {r : ℝ} (h0 : 0 < r) (h1 : r < 1) :
-    (𝓝 x).HasBasis (fun _ => True) fun n : ℕ => closedBall x (r ^ n) :=
-  nhds_basis_uniformity (uniformity_basis_dist_le_pow h0 h1)
-#align metric.nhds_basis_closed_ball_pow Metric.nhds_basis_closedBall_pow
-
-theorem isOpen_iff : IsOpen s ↔ ∀ x ∈ s, ∃ ε > 0, ball x ε ⊆ s := by
-  simp only [isOpen_iff_mem_nhds, mem_nhds_iff]
-#align metric.is_open_iff Metric.isOpen_iff
-
-theorem isOpen_ball : IsOpen (ball x ε) :=
-  isOpen_iff.2 fun _ => exists_ball_subset_ball
-#align metric.is_open_ball Metric.isOpen_ball
-
-theorem ball_mem_nhds (x : α) {ε : ℝ} (ε0 : 0 < ε) : ball x ε ∈ 𝓝 x :=
-  isOpen_ball.mem_nhds (mem_ball_self ε0)
-#align metric.ball_mem_nhds Metric.ball_mem_nhds
-
-theorem closedBall_mem_nhds (x : α) {ε : ℝ} (ε0 : 0 < ε) : closedBall x ε ∈ 𝓝 x :=
-  mem_of_superset (ball_mem_nhds x ε0) ball_subset_closedBall
-#align metric.closed_ball_mem_nhds Metric.closedBall_mem_nhds
-
-theorem closedBall_mem_nhds_of_mem {x c : α} {ε : ℝ} (h : x ∈ ball c ε) : closedBall c ε ∈ 𝓝 x :=
-  mem_of_superset (isOpen_ball.mem_nhds h) ball_subset_closedBall
-#align metric.closed_ball_mem_nhds_of_mem Metric.closedBall_mem_nhds_of_mem
-
-theorem nhdsWithin_basis_ball {s : Set α} :
-    (𝓝[s] x).HasBasis (fun ε : ℝ => 0 < ε) fun ε => ball x ε ∩ s :=
-  nhdsWithin_hasBasis nhds_basis_ball s
-#align metric.nhds_within_basis_ball Metric.nhdsWithin_basis_ball
-
-theorem mem_nhdsWithin_iff {t : Set α} : s ∈ 𝓝[t] x ↔ ∃ ε > 0, ball x ε ∩ t ⊆ s :=
-  nhdsWithin_basis_ball.mem_iff
-#align metric.mem_nhds_within_iff Metric.mem_nhdsWithin_iff
-
-theorem tendsto_nhdsWithin_nhdsWithin [PseudoMetricSpace β] {t : Set β} {f : α → β} {a b} :
-    Tendsto f (𝓝[s] a) (𝓝[t] b) ↔
-      ∀ ε > 0, ∃ δ > 0, ∀ {x : α}, x ∈ s → dist x a < δ → f x ∈ t ∧ dist (f x) b < ε :=
-  (nhdsWithin_basis_ball.tendsto_iff nhdsWithin_basis_ball).trans <| by
-    simp only [inter_comm _ s, inter_comm _ t, mem_inter_iff, and_imp]; rfl
-#align metric.tendsto_nhds_within_nhds_within Metric.tendsto_nhdsWithin_nhdsWithin
-
-theorem tendsto_nhdsWithin_nhds [PseudoMetricSpace β] {f : α → β} {a b} :
-    Tendsto f (𝓝[s] a) (𝓝 b) ↔
-      ∀ ε > 0, ∃ δ > 0, ∀ {x : α}, x ∈ s → dist x a < δ → dist (f x) b < ε := by
-  rw [← nhdsWithin_univ b, tendsto_nhdsWithin_nhdsWithin]
-  simp only [mem_univ, true_and_iff]
-#align metric.tendsto_nhds_within_nhds Metric.tendsto_nhdsWithin_nhds
-
-theorem tendsto_nhds_nhds [PseudoMetricSpace β] {f : α → β} {a b} :
-    Tendsto f (𝓝 a) (𝓝 b) ↔ ∀ ε > 0, ∃ δ > 0, ∀ {x : α}, dist x a < δ → dist (f x) b < ε :=
-  nhds_basis_ball.tendsto_iff nhds_basis_ball
-#align metric.tendsto_nhds_nhds Metric.tendsto_nhds_nhds
-
-theorem continuousAt_iff [PseudoMetricSpace β] {f : α → β} {a : α} :
-    ContinuousAt f a ↔ ∀ ε > 0, ∃ δ > 0, ∀ {x : α}, dist x a < δ → dist (f x) (f a) < ε := by
-  rw [ContinuousAt, tendsto_nhds_nhds]
-#align metric.continuous_at_iff Metric.continuousAt_iff
-
-theorem continuousWithinAt_iff [PseudoMetricSpace β] {f : α → β} {a : α} {s : Set α} :
-    ContinuousWithinAt f s a ↔
-      ∀ ε > 0, ∃ δ > 0, ∀ {x : α}, x ∈ s → dist x a < δ → dist (f x) (f a) < ε :=
-  by rw [ContinuousWithinAt, tendsto_nhdsWithin_nhds]
-#align metric.continuous_within_at_iff Metric.continuousWithinAt_iff
-
-theorem continuousOn_iff [PseudoMetricSpace β] {f : α → β} {s : Set α} :
-    ContinuousOn f s ↔ ∀ b ∈ s, ∀ ε > 0, ∃ δ > 0, ∀ a ∈ s, dist a b < δ → dist (f a) (f b) < ε := by
-  simp [ContinuousOn, continuousWithinAt_iff]
-#align metric.continuous_on_iff Metric.continuousOn_iff
-
-theorem continuous_iff [PseudoMetricSpace β] {f : α → β} :
-    Continuous f ↔ ∀ b, ∀ ε > 0, ∃ δ > 0, ∀ a, dist a b < δ → dist (f a) (f b) < ε :=
-  continuous_iff_continuousAt.trans <| forall_congr' fun _ => tendsto_nhds_nhds
-#align metric.continuous_iff Metric.continuous_iff
-
-theorem tendsto_nhds {f : Filter β} {u : β → α} {a : α} :
-    Tendsto u f (𝓝 a) ↔ ∀ ε > 0, ∀ᶠ x in f, dist (u x) a < ε :=
-  nhds_basis_ball.tendsto_right_iff
-#align metric.tendsto_nhds Metric.tendsto_nhds
-
-theorem continuousAt_iff' [TopologicalSpace β] {f : β → α} {b : β} :
-    ContinuousAt f b ↔ ∀ ε > 0, ∀ᶠ x in 𝓝 b, dist (f x) (f b) < ε := by
-  rw [ContinuousAt, tendsto_nhds]
-#align metric.continuous_at_iff' Metric.continuousAt_iff'
-
-theorem continuousWithinAt_iff' [TopologicalSpace β] {f : β → α} {b : β} {s : Set β} :
-    ContinuousWithinAt f s b ↔ ∀ ε > 0, ∀ᶠ x in 𝓝[s] b, dist (f x) (f b) < ε := by
-  rw [ContinuousWithinAt, tendsto_nhds]
-#align metric.continuous_within_at_iff' Metric.continuousWithinAt_iff'
-
-theorem continuousOn_iff' [TopologicalSpace β] {f : β → α} {s : Set β} :
-    ContinuousOn f s ↔ ∀ b ∈ s, ∀ ε > 0, ∀ᶠ x in 𝓝[s] b, dist (f x) (f b) < ε := by
-  simp [ContinuousOn, continuousWithinAt_iff']
-#align metric.continuous_on_iff' Metric.continuousOn_iff'
-
-theorem continuous_iff' [TopologicalSpace β] {f : β → α} :
-    Continuous f ↔ ∀ (a), ∀ ε > 0, ∀ᶠ x in 𝓝 a, dist (f x) (f a) < ε :=
-  continuous_iff_continuousAt.trans <| forall_congr' fun _ => tendsto_nhds
-#align metric.continuous_iff' Metric.continuous_iff'
-
-theorem tendsto_atTop [Nonempty β] [SemilatticeSup β] {u : β → α} {a : α} :
-    Tendsto u atTop (𝓝 a) ↔ ∀ ε > 0, ∃ N, ∀ n ≥ N, dist (u n) a < ε :=
-  (atTop_basis.tendsto_iff nhds_basis_ball).trans <| by
-    simp only [true_and]; rfl
-#align metric.tendsto_at_top Metric.tendsto_atTop
-
-/-- A variant of `tendsto_atTop` that
-uses `∃ N, ∀ n > N, ...` rather than `∃ N, ∀ n ≥ N, ...`
--/
-theorem tendsto_atTop' [Nonempty β] [SemilatticeSup β] [NoMaxOrder β] {u : β → α} {a : α} :
-    Tendsto u atTop (𝓝 a) ↔ ∀ ε > 0, ∃ N, ∀ n > N, dist (u n) a < ε :=
-  (atTop_basis_Ioi.tendsto_iff nhds_basis_ball).trans <| by
-    simp only [true_and, gt_iff_lt, mem_Ioi, mem_ball]
-#align metric.tendsto_at_top' Metric.tendsto_atTop'
-
-theorem isOpen_singleton_iff {α : Type*} [PseudoMetricSpace α] {x : α} :
-    IsOpen ({x} : Set α) ↔ ∃ ε > 0, ∀ y, dist y x < ε → y = x := by
-  simp [isOpen_iff, subset_singleton_iff, mem_ball]
-#align metric.is_open_singleton_iff Metric.isOpen_singleton_iff
-
-/-- Given a point `x` in a discrete subset `s` of a pseudometric space, there is an open ball
-centered at `x` and intersecting `s` only at `x`. -/
-theorem exists_ball_inter_eq_singleton_of_mem_discrete [DiscreteTopology s] {x : α} (hx : x ∈ s) :
-    ∃ ε > 0, Metric.ball x ε ∩ s = {x} :=
-  nhds_basis_ball.exists_inter_eq_singleton_of_mem_discrete hx
-#align metric.exists_ball_inter_eq_singleton_of_mem_discrete Metric.exists_ball_inter_eq_singleton_of_mem_discrete
-
-/-- Given a point `x` in a discrete subset `s` of a pseudometric space, there is a closed ball
-of positive radius centered at `x` and intersecting `s` only at `x`. -/
-theorem exists_closedBall_inter_eq_singleton_of_discrete [DiscreteTopology s] {x : α} (hx : x ∈ s) :
-    ∃ ε > 0, Metric.closedBall x ε ∩ s = {x} :=
-  nhds_basis_closedBall.exists_inter_eq_singleton_of_mem_discrete hx
-#align metric.exists_closed_ball_inter_eq_singleton_of_discrete Metric.exists_closedBall_inter_eq_singleton_of_discrete
-
-theorem _root_.Dense.exists_dist_lt {s : Set α} (hs : Dense s) (x : α) {ε : ℝ} (hε : 0 < ε) :
-    ∃ y ∈ s, dist x y < ε := by
-  have : (ball x ε).Nonempty := by simp [hε]
-  simpa only [mem_ball'] using hs.exists_mem_open isOpen_ball this
-#align dense.exists_dist_lt Dense.exists_dist_lt
-
-nonrec theorem _root_.DenseRange.exists_dist_lt {β : Type*} {f : β → α} (hf : DenseRange f) (x : α)
-    {ε : ℝ} (hε : 0 < ε) : ∃ y, dist x (f y) < ε :=
-  exists_range_iff.1 (hf.exists_dist_lt x hε)
-#align dense_range.exists_dist_lt DenseRange.exists_dist_lt
-
-end Metric
-
-open Metric
-
-/-Instantiate a pseudometric space as a pseudoemetric space. Before we can state the instance,
-we need to show that the uniform structure coming from the edistance and the
-distance coincide. -/
-
--- porting note: new
-theorem Metric.uniformity_edist_aux {α} (d : α → α → ℝ≥0) :
-    ⨅ ε > (0 : ℝ), 𝓟 { p : α × α | ↑(d p.1 p.2) < ε } =
-      ⨅ ε > (0 : ℝ≥0∞), 𝓟 { p : α × α | ↑(d p.1 p.2) < ε } := by
-  simp only [le_antisymm_iff, le_iInf_iff, le_principal_iff]
-  refine ⟨fun ε hε => ?_, fun ε hε => ?_⟩
-  · rcases ENNReal.lt_iff_exists_nnreal_btwn.1 hε with ⟨ε', ε'0, ε'ε⟩
-    refine mem_iInf_of_mem (ε' : ℝ) (mem_iInf_of_mem (ENNReal.coe_pos.1 ε'0) ?_)
-    exact fun x hx => lt_trans (ENNReal.coe_lt_coe.2 hx) ε'ε
-  · lift ε to ℝ≥0 using le_of_lt hε
-    refine mem_iInf_of_mem (ε : ℝ≥0∞) (mem_iInf_of_mem (ENNReal.coe_pos.2 hε) ?_)
-    exact fun _ => ENNReal.coe_lt_coe.1
-
-theorem Metric.uniformity_edist : 𝓤 α = ⨅ ε > 0, 𝓟 { p : α × α | edist p.1 p.2 < ε } := by
-  simp only [PseudoMetricSpace.uniformity_dist, dist_nndist, edist_nndist,
-    Metric.uniformity_edist_aux]
-#align metric.uniformity_edist Metric.uniformity_edist
-
--- see Note [lower instance priority]
-/-- A pseudometric space induces a pseudoemetric space -/
-instance (priority := 100) PseudoMetricSpace.toPseudoEMetricSpace : PseudoEMetricSpace α :=
-  { ‹PseudoMetricSpace α› with
-    edist_self := by simp [edist_dist]
-    edist_comm := fun _ _ => by simp only [edist_dist, dist_comm]
-    edist_triangle := fun x y z => by
-      simp only [edist_dist, ← ENNReal.ofReal_add, dist_nonneg]
-      rw [ENNReal.ofReal_le_ofReal_iff _]
-      · exact dist_triangle _ _ _
-      · simpa using add_le_add (dist_nonneg : 0 ≤ dist x y) dist_nonneg
-    uniformity_edist := Metric.uniformity_edist }
-#align pseudo_metric_space.to_pseudo_emetric_space PseudoMetricSpace.toPseudoEMetricSpace
-
-/-- Expressing the uniformity in terms of `edist` -/
-@[deprecated _root_.uniformity_basis_edist]
-protected theorem Metric.uniformity_basis_edist :
-    (𝓤 α).HasBasis (fun ε : ℝ≥0∞ => 0 < ε) fun ε => { p | edist p.1 p.2 < ε } :=
-  uniformity_basis_edist
-#align pseudo_metric.uniformity_basis_edist Metric.uniformity_basis_edist
-
-/-- In a pseudometric space, an open ball of infinite radius is the whole space -/
-theorem Metric.eball_top_eq_univ (x : α) : EMetric.ball x ∞ = Set.univ :=
-  Set.eq_univ_iff_forall.mpr fun y => edist_lt_top y x
-#align metric.eball_top_eq_univ Metric.eball_top_eq_univ
-
-/-- Balls defined using the distance or the edistance coincide -/
-@[simp]
-theorem Metric.emetric_ball {x : α} {ε : ℝ} : EMetric.ball x (ENNReal.ofReal ε) = ball x ε := by
-  ext y
-  simp only [EMetric.mem_ball, mem_ball, edist_dist]
-  exact ENNReal.ofReal_lt_ofReal_iff_of_nonneg dist_nonneg
-#align metric.emetric_ball Metric.emetric_ball
-
-/-- Balls defined using the distance or the edistance coincide -/
-@[simp]
-theorem Metric.emetric_ball_nnreal {x : α} {ε : ℝ≥0} : EMetric.ball x ε = ball x ε := by
-  rw [← Metric.emetric_ball]
-  simp
-#align metric.emetric_ball_nnreal Metric.emetric_ball_nnreal
-
-/-- Closed balls defined using the distance or the edistance coincide -/
-theorem Metric.emetric_closedBall {x : α} {ε : ℝ} (h : 0 ≤ ε) :
-    EMetric.closedBall x (ENNReal.ofReal ε) = closedBall x ε := by
-  ext y; simp [edist_le_ofReal h]
-#align metric.emetric_closed_ball Metric.emetric_closedBall
-
-/-- Closed balls defined using the distance or the edistance coincide -/
-@[simp]
-theorem Metric.emetric_closedBall_nnreal {x : α} {ε : ℝ≥0} :
-    EMetric.closedBall x ε = closedBall x ε := by
-  rw [← Metric.emetric_closedBall ε.coe_nonneg, ENNReal.ofReal_coe_nnreal]
-#align metric.emetric_closed_ball_nnreal Metric.emetric_closedBall_nnreal
-
-@[simp]
-theorem Metric.emetric_ball_top (x : α) : EMetric.ball x ⊤ = univ :=
-  eq_univ_of_forall fun _ => edist_lt_top _ _
-#align metric.emetric_ball_top Metric.emetric_ball_top
-
-theorem Metric.inseparable_iff {x y : α} : Inseparable x y ↔ dist x y = 0 := by
-  rw [EMetric.inseparable_iff, edist_nndist, dist_nndist, ENNReal.coe_eq_zero, NNReal.coe_eq_zero]
-#align metric.inseparable_iff Metric.inseparable_iff
-
-/-- Build a new pseudometric space from an old one where the bundled uniform structure is provably
-(but typically non-definitionaly) equal to some given uniform structure.
-See Note [forgetful inheritance].
--/
-@[reducible]
-def PseudoMetricSpace.replaceUniformity {α} [U : UniformSpace α] (m : PseudoMetricSpace α)
-    (H : 𝓤[U] = 𝓤[PseudoEMetricSpace.toUniformSpace]) : PseudoMetricSpace α :=
-  { m with
-    toUniformSpace := U
-    uniformity_dist := H.trans PseudoMetricSpace.uniformity_dist }
-#align pseudo_metric_space.replace_uniformity PseudoMetricSpace.replaceUniformity
-
-theorem PseudoMetricSpace.replaceUniformity_eq {α} [U : UniformSpace α] (m : PseudoMetricSpace α)
-    (H : 𝓤[U] = 𝓤[PseudoEMetricSpace.toUniformSpace]) : m.replaceUniformity H = m := by
-  ext
-  rfl
-#align pseudo_metric_space.replace_uniformity_eq PseudoMetricSpace.replaceUniformity_eq
-
--- ensure that the bornology is unchanged when replacing the uniformity.
-example {α} [U : UniformSpace α] (m : PseudoMetricSpace α)
-    (H : 𝓤[U] = 𝓤[PseudoEMetricSpace.toUniformSpace]) :
-  (PseudoMetricSpace.replaceUniformity m H).toBornology = m.toBornology := rfl
-
-/-- Build a new pseudo metric space from an old one where the bundled topological structure is
-provably (but typically non-definitionaly) equal to some given topological structure.
-See Note [forgetful inheritance].
--/
-@[reducible]
-def PseudoMetricSpace.replaceTopology {γ} [U : TopologicalSpace γ] (m : PseudoMetricSpace γ)
-    (H : U = m.toUniformSpace.toTopologicalSpace) : PseudoMetricSpace γ :=
-  @PseudoMetricSpace.replaceUniformity γ (m.toUniformSpace.replaceTopology H) m rfl
-#align pseudo_metric_space.replace_topology PseudoMetricSpace.replaceTopology
-
-theorem PseudoMetricSpace.replaceTopology_eq {γ} [U : TopologicalSpace γ] (m : PseudoMetricSpace γ)
-    (H : U = m.toUniformSpace.toTopologicalSpace) : m.replaceTopology H = m := by
-  ext
-  rfl
-#align pseudo_metric_space.replace_topology_eq PseudoMetricSpace.replaceTopology_eq
-
-/-- One gets a pseudometric space from an emetric space if the edistance
-is everywhere finite, by pushing the edistance to reals. We set it up so that the edist and the
-uniformity are defeq in the pseudometric space and the pseudoemetric space. In this definition, the
-distance is given separately, to be able to prescribe some expression which is not defeq to the
-push-forward of the edistance to reals. -/
-def PseudoEMetricSpace.toPseudoMetricSpaceOfDist {α : Type u} [e : PseudoEMetricSpace α]
-    (dist : α → α → ℝ) (edist_ne_top : ∀ x y : α, edist x y ≠ ⊤)
-    (h : ∀ x y, dist x y = ENNReal.toReal (edist x y)) : PseudoMetricSpace α where
-  dist := dist
-  dist_self x := by simp [h]
-  dist_comm x y := by simp [h, edist_comm]
-  dist_triangle x y z := by
-    simp only [h]
-    exact ENNReal.toReal_le_add (edist_triangle _ _ _) (edist_ne_top _ _) (edist_ne_top _ _)
-  edist := edist
-  edist_dist _ _ := by simp only [h, ENNReal.ofReal_toReal (edist_ne_top _ _)]
-  toUniformSpace := e.toUniformSpace
-  uniformity_dist := e.uniformity_edist.trans <| by
-    simpa only [ENNReal.coe_toNNReal (edist_ne_top _ _), h]
-      using (Metric.uniformity_edist_aux fun x y : α => (edist x y).toNNReal).symm
-#align pseudo_emetric_space.to_pseudo_metric_space_of_dist PseudoEMetricSpace.toPseudoMetricSpaceOfDist
-
-/-- One gets a pseudometric space from an emetric space if the edistance
-is everywhere finite, by pushing the edistance to reals. We set it up so that the edist and the
-uniformity are defeq in the pseudometric space and the emetric space. -/
-@[reducible]
-def PseudoEMetricSpace.toPseudoMetricSpace {α : Type u} [PseudoEMetricSpace α]
-    (h : ∀ x y : α, edist x y ≠ ⊤) : PseudoMetricSpace α :=
-  PseudoEMetricSpace.toPseudoMetricSpaceOfDist (fun x y => ENNReal.toReal (edist x y)) h fun _ _ =>
-    rfl
-#align pseudo_emetric_space.to_pseudo_metric_space PseudoEMetricSpace.toPseudoMetricSpace
-
-/-- Build a new pseudometric space from an old one where the bundled bornology structure is provably
-(but typically non-definitionaly) equal to some given bornology structure.
-See Note [forgetful inheritance].
--/
-@[reducible]
-def PseudoMetricSpace.replaceBornology {α} [B : Bornology α] (m : PseudoMetricSpace α)
-    (H : ∀ s, @IsBounded _ B s ↔ @IsBounded _ PseudoMetricSpace.toBornology s) :
-    PseudoMetricSpace α :=
-  { m with
-    toBornology := B
-    cobounded_sets := Set.ext <| compl_surjective.forall.2 fun s =>
-        (H s).trans <| by rw [isBounded_iff, mem_setOf_eq, compl_compl] }
-#align pseudo_metric_space.replace_bornology PseudoMetricSpace.replaceBornology
-
-theorem PseudoMetricSpace.replaceBornology_eq {α} [m : PseudoMetricSpace α] [B : Bornology α]
-    (H : ∀ s, @IsBounded _ B s ↔ @IsBounded _ PseudoMetricSpace.toBornology s) :
-    PseudoMetricSpace.replaceBornology _ H = m := by
-  ext
-  rfl
-#align pseudo_metric_space.replace_bornology_eq PseudoMetricSpace.replaceBornology_eq
-
--- ensure that the uniformity is unchanged when replacing the bornology.
-example {α} [B : Bornology α] (m : PseudoMetricSpace α)
-    (H : ∀ s, @IsBounded _ B s ↔ @IsBounded _ PseudoMetricSpace.toBornology s) :
-  (PseudoMetricSpace.replaceBornology m H).toUniformSpace = m.toUniformSpace := rfl
-
-/-- A very useful criterion to show that a space is complete is to show that all sequences
-which satisfy a bound of the form `dist (u n) (u m) < B N` for all `n m ≥ N` are
-converging. This is often applied for `B N = 2^{-N}`, i.e., with a very fast convergence to
-`0`, which makes it possible to use arguments of converging series, while this is impossible
-to do in general for arbitrary Cauchy sequences. -/
-theorem Metric.complete_of_convergent_controlled_sequences (B : ℕ → Real) (hB : ∀ n, 0 < B n)
-    (H : ∀ u : ℕ → α, (∀ N n m : ℕ, N ≤ n → N ≤ m → dist (u n) (u m) < B N) →
-      ∃ x, Tendsto u atTop (𝓝 x)) :
-    CompleteSpace α :=
-  UniformSpace.complete_of_convergent_controlled_sequences
-    (fun n => { p : α × α | dist p.1 p.2 < B n }) (fun n => dist_mem_uniformity <| hB n) H
-#align metric.complete_of_convergent_controlled_sequences Metric.complete_of_convergent_controlled_sequences
-
-theorem Metric.complete_of_cauchySeq_tendsto :
-    (∀ u : ℕ → α, CauchySeq u → ∃ a, Tendsto u atTop (𝓝 a)) → CompleteSpace α :=
-  EMetric.complete_of_cauchySeq_tendsto
-#align metric.complete_of_cauchy_seq_tendsto Metric.complete_of_cauchySeq_tendsto
-
-section Real
-
-/-- Instantiate the reals as a pseudometric space. -/
-instance Real.pseudoMetricSpace : PseudoMetricSpace ℝ where
-  dist x y := |x - y|
-  dist_self := by simp [abs_zero]
-  dist_comm x y := abs_sub_comm _ _
-  dist_triangle x y z := abs_sub_le _ _ _
-  edist_dist := fun x y => by exact ENNReal.coe_nnreal_eq _
-#align real.pseudo_metric_space Real.pseudoMetricSpace
-
-theorem Real.dist_eq (x y : ℝ) : dist x y = |x - y| := rfl
-#align real.dist_eq Real.dist_eq
-
-theorem Real.nndist_eq (x y : ℝ) : nndist x y = Real.nnabs (x - y) := rfl
-#align real.nndist_eq Real.nndist_eq
-
-theorem Real.nndist_eq' (x y : ℝ) : nndist x y = Real.nnabs (y - x) :=
-  nndist_comm _ _
-#align real.nndist_eq' Real.nndist_eq'
-
-theorem Real.dist_0_eq_abs (x : ℝ) : dist x 0 = |x| := by simp [Real.dist_eq]
-#align real.dist_0_eq_abs Real.dist_0_eq_abs
-
-theorem Real.dist_left_le_of_mem_uIcc {x y z : ℝ} (h : y ∈ uIcc x z) : dist x y ≤ dist x z := by
-  simpa only [dist_comm x] using abs_sub_left_of_mem_uIcc h
-#align real.dist_left_le_of_mem_uIcc Real.dist_left_le_of_mem_uIcc
-
-theorem Real.dist_right_le_of_mem_uIcc {x y z : ℝ} (h : y ∈ uIcc x z) : dist y z ≤ dist x z := by
-  simpa only [dist_comm _ z] using abs_sub_right_of_mem_uIcc h
-#align real.dist_right_le_of_mem_uIcc Real.dist_right_le_of_mem_uIcc
-
-theorem Real.dist_le_of_mem_uIcc {x y x' y' : ℝ} (hx : x ∈ uIcc x' y') (hy : y ∈ uIcc x' y') :
-    dist x y ≤ dist x' y' :=
-  abs_sub_le_of_uIcc_subset_uIcc <| uIcc_subset_uIcc (by rwa [uIcc_comm]) (by rwa [uIcc_comm])
-#align real.dist_le_of_mem_uIcc Real.dist_le_of_mem_uIcc
-
-theorem Real.dist_le_of_mem_Icc {x y x' y' : ℝ} (hx : x ∈ Icc x' y') (hy : y ∈ Icc x' y') :
-    dist x y ≤ y' - x' := by
-  simpa only [Real.dist_eq, abs_of_nonpos (sub_nonpos.2 <| hx.1.trans hx.2), neg_sub] using
-    Real.dist_le_of_mem_uIcc (Icc_subset_uIcc hx) (Icc_subset_uIcc hy)
-#align real.dist_le_of_mem_Icc Real.dist_le_of_mem_Icc
-
-theorem Real.dist_le_of_mem_Icc_01 {x y : ℝ} (hx : x ∈ Icc (0 : ℝ) 1) (hy : y ∈ Icc (0 : ℝ) 1) :
-    dist x y ≤ 1 := by simpa only [sub_zero] using Real.dist_le_of_mem_Icc hx hy
-#align real.dist_le_of_mem_Icc_01 Real.dist_le_of_mem_Icc_01
-
-instance : OrderTopology ℝ :=
-  orderTopology_of_nhds_abs fun x => by
-    simp only [nhds_basis_ball.eq_biInf, ball, Real.dist_eq, abs_sub_comm]
-
-theorem Real.ball_eq_Ioo (x r : ℝ) : ball x r = Ioo (x - r) (x + r) :=
-  Set.ext fun y => by
-    rw [mem_ball, dist_comm, Real.dist_eq, abs_sub_lt_iff, mem_Ioo, ← sub_lt_iff_lt_add',
-      sub_lt_comm]
-#align real.ball_eq_Ioo Real.ball_eq_Ioo
-
-theorem Real.closedBall_eq_Icc {x r : ℝ} : closedBall x r = Icc (x - r) (x + r) := by
-  ext y
-  rw [mem_closedBall, dist_comm, Real.dist_eq, abs_sub_le_iff, mem_Icc, ← sub_le_iff_le_add',
-    sub_le_comm]
-#align real.closed_ball_eq_Icc Real.closedBall_eq_Icc
-
-theorem Real.Ioo_eq_ball (x y : ℝ) : Ioo x y = ball ((x + y) / 2) ((y - x) / 2) := by
-  rw [Real.ball_eq_Ioo, ← sub_div, add_comm, ← sub_add, add_sub_cancel', add_self_div_two,
-    ← add_div, add_assoc, add_sub_cancel'_right, add_self_div_two]
-#align real.Ioo_eq_ball Real.Ioo_eq_ball
-
-theorem Real.Icc_eq_closedBall (x y : ℝ) : Icc x y = closedBall ((x + y) / 2) ((y - x) / 2) := by
-  rw [Real.closedBall_eq_Icc, ← sub_div, add_comm, ← sub_add, add_sub_cancel', add_self_div_two, ←
-    add_div, add_assoc, add_sub_cancel'_right, add_self_div_two]
-#align real.Icc_eq_closed_ball Real.Icc_eq_closedBall
-
-section MetricOrdered
-
-variable [Preorder α] [CompactIccSpace α]
-
-theorem totallyBounded_Icc (a b : α) : TotallyBounded (Icc a b) :=
-  isCompact_Icc.totallyBounded
-#align totally_bounded_Icc totallyBounded_Icc
-
-theorem totallyBounded_Ico (a b : α) : TotallyBounded (Ico a b) :=
-  totallyBounded_subset Ico_subset_Icc_self (totallyBounded_Icc a b)
-#align totally_bounded_Ico totallyBounded_Ico
-
-theorem totallyBounded_Ioc (a b : α) : TotallyBounded (Ioc a b) :=
-  totallyBounded_subset Ioc_subset_Icc_self (totallyBounded_Icc a b)
-#align totally_bounded_Ioc totallyBounded_Ioc
-
-theorem totallyBounded_Ioo (a b : α) : TotallyBounded (Ioo a b) :=
-  totallyBounded_subset Ioo_subset_Icc_self (totallyBounded_Icc a b)
-#align totally_bounded_Ioo totallyBounded_Ioo
-
-end MetricOrdered
-
-/-- Special case of the sandwich theorem; see `tendsto_of_tendsto_of_tendsto_of_le_of_le'` for the
-general case. -/
-theorem squeeze_zero' {α} {f g : α → ℝ} {t₀ : Filter α} (hf : ∀ᶠ t in t₀, 0 ≤ f t)
-    (hft : ∀ᶠ t in t₀, f t ≤ g t) (g0 : Tendsto g t₀ (nhds 0)) : Tendsto f t₀ (𝓝 0) :=
-  tendsto_of_tendsto_of_tendsto_of_le_of_le' tendsto_const_nhds g0 hf hft
-#align squeeze_zero' squeeze_zero'
-
-/-- Special case of the sandwich theorem; see `tendsto_of_tendsto_of_tendsto_of_le_of_le`
-and `tendsto_of_tendsto_of_tendsto_of_le_of_le'` for the general case. -/
-theorem squeeze_zero {α} {f g : α → ℝ} {t₀ : Filter α} (hf : ∀ t, 0 ≤ f t) (hft : ∀ t, f t ≤ g t)
-    (g0 : Tendsto g t₀ (𝓝 0)) : Tendsto f t₀ (𝓝 0) :=
-  squeeze_zero' (eventually_of_forall hf) (eventually_of_forall hft) g0
-#align squeeze_zero squeeze_zero
-
-theorem Metric.uniformity_eq_comap_nhds_zero :
-    𝓤 α = comap (fun p : α × α => dist p.1 p.2) (𝓝 (0 : ℝ)) := by
-  ext s
-  simp only [mem_uniformity_dist, (nhds_basis_ball.comap _).mem_iff]
-  simp [subset_def, Real.dist_0_eq_abs]
-#align metric.uniformity_eq_comap_nhds_zero Metric.uniformity_eq_comap_nhds_zero
-
-theorem cauchySeq_iff_tendsto_dist_atTop_0 [Nonempty β] [SemilatticeSup β] {u : β → α} :
-    CauchySeq u ↔ Tendsto (fun n : β × β => dist (u n.1) (u n.2)) atTop (𝓝 0) := by
-  rw [cauchySeq_iff_tendsto, Metric.uniformity_eq_comap_nhds_zero, tendsto_comap_iff]; rfl
-#align cauchy_seq_iff_tendsto_dist_at_top_0 cauchySeq_iff_tendsto_dist_atTop_0
-
-theorem tendsto_uniformity_iff_dist_tendsto_zero {f : ι → α × α} {p : Filter ι} :
-    Tendsto f p (𝓤 α) ↔ Tendsto (fun x => dist (f x).1 (f x).2) p (𝓝 0) := by
-  rw [Metric.uniformity_eq_comap_nhds_zero, tendsto_comap_iff]; rfl
-#align tendsto_uniformity_iff_dist_tendsto_zero tendsto_uniformity_iff_dist_tendsto_zero
-
-theorem Filter.Tendsto.congr_dist {f₁ f₂ : ι → α} {p : Filter ι} {a : α}
-    (h₁ : Tendsto f₁ p (𝓝 a)) (h : Tendsto (fun x => dist (f₁ x) (f₂ x)) p (𝓝 0)) :
-    Tendsto f₂ p (𝓝 a) :=
-  h₁.congr_uniformity <| tendsto_uniformity_iff_dist_tendsto_zero.2 h
-#align filter.tendsto.congr_dist Filter.Tendsto.congr_dist
-
-alias tendsto_of_tendsto_of_dist := Filter.Tendsto.congr_dist
-#align tendsto_of_tendsto_of_dist tendsto_of_tendsto_of_dist
-
-theorem tendsto_iff_of_dist {f₁ f₂ : ι → α} {p : Filter ι} {a : α}
-    (h : Tendsto (fun x => dist (f₁ x) (f₂ x)) p (𝓝 0)) : Tendsto f₁ p (𝓝 a) ↔ Tendsto f₂ p (𝓝 a) :=
-  Uniform.tendsto_congr <| tendsto_uniformity_iff_dist_tendsto_zero.2 h
-#align tendsto_iff_of_dist tendsto_iff_of_dist
-
-/-- If `u` is a neighborhood of `x`, then for small enough `r`, the closed ball
-`Metric.closedBall x r` is contained in `u`. -/
-theorem eventually_closedBall_subset {x : α} {u : Set α} (hu : u ∈ 𝓝 x) :
-    ∀ᶠ r in 𝓝 (0 : ℝ), closedBall x r ⊆ u := by
-  obtain ⟨ε, εpos, hε⟩ : ∃ ε, 0 < ε ∧ closedBall x ε ⊆ u := nhds_basis_closedBall.mem_iff.1 hu
-  have : Iic ε ∈ 𝓝 (0 : ℝ) := Iic_mem_nhds εpos
-  filter_upwards [this] with _ hr using Subset.trans (closedBall_subset_closedBall hr) hε
-#align eventually_closed_ball_subset eventually_closedBall_subset
-
-end Real
-
-section CauchySeq
-
-variable [Nonempty β] [SemilatticeSup β]
-
-/-- In a pseudometric space, Cauchy sequences are characterized by the fact that, eventually,
-the distance between its elements is arbitrarily small -/
--- porting note: @[nolint ge_or_gt] doesn't exist
-theorem Metric.cauchySeq_iff {u : β → α} :
-    CauchySeq u ↔ ∀ ε > 0, ∃ N, ∀ m ≥ N, ∀ n ≥ N, dist (u m) (u n) < ε :=
-  uniformity_basis_dist.cauchySeq_iff
-#align metric.cauchy_seq_iff Metric.cauchySeq_iff
-
-/-- A variation around the pseudometric characterization of Cauchy sequences -/
-theorem Metric.cauchySeq_iff' {u : β → α} :
-    CauchySeq u ↔ ∀ ε > 0, ∃ N, ∀ n ≥ N, dist (u n) (u N) < ε :=
-  uniformity_basis_dist.cauchySeq_iff'
-#align metric.cauchy_seq_iff' Metric.cauchySeq_iff'
-
--- see Note [nolint_ge]
-/-- In a pseudometric space, uniform Cauchy sequences are characterized by the fact that,
-eventually, the distance between all its elements is uniformly, arbitrarily small -/
--- porting note: no attr @[nolint ge_or_gt]
-theorem Metric.uniformCauchySeqOn_iff {γ : Type*} {F : β → γ → α} {s : Set γ} :
-    UniformCauchySeqOn F atTop s ↔ ∀ ε > (0 : ℝ),
-      ∃ N : β, ∀ m ≥ N, ∀ n ≥ N, ∀ x ∈ s, dist (F m x) (F n x) < ε := by
-  constructor
-  · intro h ε hε
-    let u := { a : α × α | dist a.fst a.snd < ε }
-    have hu : u ∈ 𝓤 α := Metric.mem_uniformity_dist.mpr ⟨ε, hε, by simp⟩
-    rw [← @Filter.eventually_atTop_prod_self' _ _ _ fun m =>
-      ∀ x ∈ s, dist (F m.fst x) (F m.snd x) < ε]
-    specialize h u hu
-    rw [prod_atTop_atTop_eq] at h
-    exact h.mono fun n h x hx => h x hx
-  · intro h u hu
-    rcases Metric.mem_uniformity_dist.mp hu with ⟨ε, hε, hab⟩
-    rcases h ε hε with ⟨N, hN⟩
-    rw [prod_atTop_atTop_eq, eventually_atTop]
-    use (N, N)
-    intro b hb x hx
-    rcases hb with ⟨hbl, hbr⟩
-    exact hab (hN b.fst hbl.ge b.snd hbr.ge x hx)
-#align metric.uniform_cauchy_seq_on_iff Metric.uniformCauchySeqOn_iff
-
-/-- If the distance between `s n` and `s m`, `n ≤ m` is bounded above by `b n`
-and `b` converges to zero, then `s` is a Cauchy sequence.  -/
-theorem cauchySeq_of_le_tendsto_0' {s : β → α} (b : β → ℝ)
-    (h : ∀ n m : β, n ≤ m → dist (s n) (s m) ≤ b n) (h₀ : Tendsto b atTop (𝓝 0)) : CauchySeq s :=
-  Metric.cauchySeq_iff'.2 fun ε ε0 => (h₀.eventually (gt_mem_nhds ε0)).exists.imp fun N hN n hn =>
-    calc dist (s n) (s N) = dist (s N) (s n) := dist_comm _ _
-    _ ≤ b N := h _ _ hn
-    _ < ε := hN
-#align cauchy_seq_of_le_tendsto_0' cauchySeq_of_le_tendsto_0'
-
-/-- If the distance between `s n` and `s m`, `n, m ≥ N` is bounded above by `b N`
-and `b` converges to zero, then `s` is a Cauchy sequence.  -/
-theorem cauchySeq_of_le_tendsto_0 {s : β → α} (b : β → ℝ)
-    (h : ∀ n m N : β, N ≤ n → N ≤ m → dist (s n) (s m) ≤ b N) (h₀ : Tendsto b atTop (𝓝 0)) :
-    CauchySeq s :=
-  cauchySeq_of_le_tendsto_0' b (fun _n _m hnm => h _ _ _ le_rfl hnm) h₀
-#align cauchy_seq_of_le_tendsto_0 cauchySeq_of_le_tendsto_0
-
-/-- A Cauchy sequence on the natural numbers is bounded. -/
-theorem cauchySeq_bdd {u : ℕ → α} (hu : CauchySeq u) : ∃ R > 0, ∀ m n, dist (u m) (u n) < R := by
-  rcases Metric.cauchySeq_iff'.1 hu 1 zero_lt_one with ⟨N, hN⟩
-  suffices : ∃ R > 0, ∀ n, dist (u n) (u N) < R
-  · rcases this with ⟨R, R0, H⟩
-    exact ⟨_, add_pos R0 R0, fun m n =>
-      lt_of_le_of_lt (dist_triangle_right _ _ _) (add_lt_add (H m) (H n))⟩
-  let R := Finset.sup (Finset.range N) fun n => nndist (u n) (u N)
-  refine' ⟨↑R + 1, add_pos_of_nonneg_of_pos R.2 zero_lt_one, fun n => _⟩
-  cases' le_or_lt N n with h h
-  · exact lt_of_lt_of_le (hN _ h) (le_add_of_nonneg_left R.2)
-  · have : _ ≤ R := Finset.le_sup (Finset.mem_range.2 h)
-    exact lt_of_le_of_lt this (lt_add_of_pos_right _ zero_lt_one)
-#align cauchy_seq_bdd cauchySeq_bdd
-
-/-- Yet another metric characterization of Cauchy sequences on integers. This one is often the
-most efficient. -/
-theorem cauchySeq_iff_le_tendsto_0 {s : ℕ → α} :
-    CauchySeq s ↔
-      ∃ b : ℕ → ℝ,
-        (∀ n, 0 ≤ b n) ∧
-          (∀ n m N : ℕ, N ≤ n → N ≤ m → dist (s n) (s m) ≤ b N) ∧ Tendsto b atTop (𝓝 0) :=
-  ⟨fun hs => by
-    /- `s` is a Cauchy sequence. The sequence `b` will be constructed by taking
-      the supremum of the distances between `s n` and `s m` for `n m ≥ N`.
-      First, we prove that all these distances are bounded, as otherwise the Sup
-      would not make sense. -/
-    let S N := (fun p : ℕ × ℕ => dist (s p.1) (s p.2)) '' { p | p.1 ≥ N ∧ p.2 ≥ N }
-    have hS : ∀ N, ∃ x, ∀ y ∈ S N, y ≤ x := by
-      rcases cauchySeq_bdd hs with ⟨R, -, hR⟩
-      refine' fun N => ⟨R, _⟩
-      rintro _ ⟨⟨m, n⟩, _, rfl⟩
-      exact le_of_lt (hR m n)
-    -- Prove that it bounds the distances of points in the Cauchy sequence
-    have ub : ∀ m n N, N ≤ m → N ≤ n → dist (s m) (s n) ≤ sSup (S N) := fun m n N hm hn =>
-      le_csSup (hS N) ⟨⟨_, _⟩, ⟨hm, hn⟩, rfl⟩
-    have S0m : ∀ n, (0 : ℝ) ∈ S n := fun n => ⟨⟨n, n⟩, ⟨le_rfl, le_rfl⟩, dist_self _⟩
-    have S0 := fun n => le_csSup (hS n) (S0m n)
-    -- Prove that it tends to `0`, by using the Cauchy property of `s`
-    refine' ⟨fun N => sSup (S N), S0, ub, Metric.tendsto_atTop.2 fun ε ε0 => _⟩
-    refine' (Metric.cauchySeq_iff.1 hs (ε / 2) (half_pos ε0)).imp fun N hN n hn => _
-    rw [Real.dist_0_eq_abs, abs_of_nonneg (S0 n)]
-    refine' lt_of_le_of_lt (csSup_le ⟨_, S0m _⟩ _) (half_lt_self ε0)
-    rintro _ ⟨⟨m', n'⟩, ⟨hm', hn'⟩, rfl⟩
-    exact le_of_lt (hN _ (le_trans hn hm') _ (le_trans hn hn')),
-   fun ⟨b, _, b_bound, b_lim⟩ => cauchySeq_of_le_tendsto_0 b b_bound b_lim⟩
-#align cauchy_seq_iff_le_tendsto_0 cauchySeq_iff_le_tendsto_0
-
-end CauchySeq
-
-/-- Pseudometric space structure pulled back by a function. -/
-@[reducible]
-def PseudoMetricSpace.induced {α β} (f : α → β) (m : PseudoMetricSpace β) :
-    PseudoMetricSpace α where
-  dist x y := dist (f x) (f y)
-  dist_self x := dist_self _
-  dist_comm x y := dist_comm _ _
-  dist_triangle x y z := dist_triangle _ _ _
-  edist x y := edist (f x) (f y)
-  edist_dist x y := edist_dist _ _
-  toUniformSpace := UniformSpace.comap f m.toUniformSpace
-  uniformity_dist := (uniformity_basis_dist.comap _).eq_biInf
-  toBornology := Bornology.induced f
-  cobounded_sets := Set.ext fun s => mem_comap_iff_compl.trans <| by
-    simp only [← isBounded_def, isBounded_iff, ball_image_iff, mem_setOf]
-#align pseudo_metric_space.induced PseudoMetricSpace.induced
-
-/-- Pull back a pseudometric space structure by an inducing map. This is a version of
-`PseudoMetricSpace.induced` useful in case if the domain already has a `TopologicalSpace`
-structure. -/
-def Inducing.comapPseudoMetricSpace {α β} [TopologicalSpace α] [m : PseudoMetricSpace β] {f : α → β}
-    (hf : Inducing f) : PseudoMetricSpace α :=
-  .replaceTopology (.induced f m) hf.induced
-#align inducing.comap_pseudo_metric_space Inducing.comapPseudoMetricSpace
-
-/-- Pull back a pseudometric space structure by a uniform inducing map. This is a version of
-`PseudoMetricSpace.induced` useful in case if the domain already has a `UniformSpace`
-structure. -/
-def UniformInducing.comapPseudoMetricSpace {α β} [UniformSpace α] [m : PseudoMetricSpace β]
-    (f : α → β) (h : UniformInducing f) : PseudoMetricSpace α :=
-  .replaceUniformity (.induced f m) h.comap_uniformity.symm
-#align uniform_inducing.comap_pseudo_metric_space UniformInducing.comapPseudoMetricSpace
-
-instance Subtype.pseudoMetricSpace {p : α → Prop} : PseudoMetricSpace (Subtype p) :=
-  PseudoMetricSpace.induced Subtype.val ‹_›
-#align subtype.pseudo_metric_space Subtype.pseudoMetricSpace
-
-theorem Subtype.dist_eq {p : α → Prop} (x y : Subtype p) : dist x y = dist (x : α) y :=
-  rfl
-#align subtype.dist_eq Subtype.dist_eq
-
-theorem Subtype.nndist_eq {p : α → Prop} (x y : Subtype p) : nndist x y = nndist (x : α) y :=
-  rfl
-#align subtype.nndist_eq Subtype.nndist_eq
-
-namespace MulOpposite
-
-@[to_additive]
-instance instPseudoMetricSpaceMulOpposite : PseudoMetricSpace αᵐᵒᵖ :=
-  PseudoMetricSpace.induced MulOpposite.unop ‹_›
-
-@[to_additive (attr := simp)]
-theorem dist_unop (x y : αᵐᵒᵖ) : dist (unop x) (unop y) = dist x y := rfl
-#align mul_opposite.dist_unop MulOpposite.dist_unop
-#align add_opposite.dist_unop AddOpposite.dist_unop
-
-@[to_additive (attr := simp)]
-theorem dist_op (x y : α) : dist (op x) (op y) = dist x y := rfl
-#align mul_opposite.dist_op MulOpposite.dist_op
-#align add_opposite.dist_op AddOpposite.dist_op
-
-@[to_additive (attr := simp)]
-theorem nndist_unop (x y : αᵐᵒᵖ) : nndist (unop x) (unop y) = nndist x y := rfl
-#align mul_opposite.nndist_unop MulOpposite.nndist_unop
-#align add_opposite.nndist_unop AddOpposite.nndist_unop
-
-@[to_additive (attr := simp)]
-theorem nndist_op (x y : α) : nndist (op x) (op y) = nndist x y := rfl
-#align mul_opposite.nndist_op MulOpposite.nndist_op
-#align add_opposite.nndist_op AddOpposite.nndist_op
-
-end MulOpposite
-
-section NNReal
-
-instance : PseudoMetricSpace ℝ≥0 := Subtype.pseudoMetricSpace
-
-theorem NNReal.dist_eq (a b : ℝ≥0) : dist a b = |(a : ℝ) - b| := rfl
-#align nnreal.dist_eq NNReal.dist_eq
-
-theorem NNReal.nndist_eq (a b : ℝ≥0) : nndist a b = max (a - b) (b - a) :=
-  eq_of_forall_ge_iff fun _ => by
-    simp only [← NNReal.coe_le_coe, coe_nndist, dist_eq, max_le_iff, abs_sub_le_iff,
-      tsub_le_iff_right, NNReal.coe_add]
-#align nnreal.nndist_eq NNReal.nndist_eq
-
-@[simp]
-theorem NNReal.nndist_zero_eq_val (z : ℝ≥0) : nndist 0 z = z := by
-  simp only [NNReal.nndist_eq, max_eq_right, tsub_zero, zero_tsub, zero_le']
-#align nnreal.nndist_zero_eq_val NNReal.nndist_zero_eq_val
-
-@[simp]
-theorem NNReal.nndist_zero_eq_val' (z : ℝ≥0) : nndist z 0 = z := by
-  rw [nndist_comm]
-  exact NNReal.nndist_zero_eq_val z
-#align nnreal.nndist_zero_eq_val' NNReal.nndist_zero_eq_val'
-
-theorem NNReal.le_add_nndist (a b : ℝ≥0) : a ≤ b + nndist a b := by
-  suffices (a : ℝ) ≤ (b : ℝ) + dist a b by
-    rwa [← NNReal.coe_le_coe, NNReal.coe_add, coe_nndist]
-  rw [← sub_le_iff_le_add']
-  exact le_of_abs_le (dist_eq a b).ge
-#align nnreal.le_add_nndist NNReal.le_add_nndist
-
-end NNReal
-
-section ULift
-
-variable [PseudoMetricSpace β]
-
-instance : PseudoMetricSpace (ULift β) :=
-  PseudoMetricSpace.induced ULift.down ‹_›
-
-theorem ULift.dist_eq (x y : ULift β) : dist x y = dist x.down y.down := rfl
-#align ulift.dist_eq ULift.dist_eq
-
-theorem ULift.nndist_eq (x y : ULift β) : nndist x y = nndist x.down y.down := rfl
-#align ulift.nndist_eq ULift.nndist_eq
-
-@[simp]
-theorem ULift.dist_up_up (x y : β) : dist (ULift.up x) (ULift.up y) = dist x y := rfl
-#align ulift.dist_up_up ULift.dist_up_up
-
-@[simp]
-theorem ULift.nndist_up_up (x y : β) : nndist (ULift.up x) (ULift.up y) = nndist x y := rfl
-#align ulift.nndist_up_up ULift.nndist_up_up
-
-end ULift
-
-section Prod
-
-variable [PseudoMetricSpace β]
-
--- porting note: added `let`, otherwise `simp` failed
-instance Prod.pseudoMetricSpaceMax : PseudoMetricSpace (α × β) :=
-  let i := PseudoEMetricSpace.toPseudoMetricSpaceOfDist
-    (fun x y : α × β => dist x.1 y.1 ⊔ dist x.2 y.2)
-    (fun x y => (max_lt (edist_lt_top _ _) (edist_lt_top _ _)).ne) fun x y => by
-      simp only [sup_eq_max, dist_edist, ← ENNReal.toReal_max (edist_ne_top _ _) (edist_ne_top _ _),
-        Prod.edist_eq]
-  i.replaceBornology fun s => by
-    simp only [← isBounded_image_fst_and_snd, isBounded_iff_eventually, ball_image_iff, ←
-      eventually_and, ← forall_and, ← max_le_iff]
-    rfl
-#align prod.pseudo_metric_space_max Prod.pseudoMetricSpaceMax
-
-theorem Prod.dist_eq {x y : α × β} : dist x y = max (dist x.1 y.1) (dist x.2 y.2) := rfl
-#align prod.dist_eq Prod.dist_eq
-
-@[simp]
-theorem dist_prod_same_left {x : α} {y₁ y₂ : β} : dist (x, y₁) (x, y₂) = dist y₁ y₂ := by
-  simp [Prod.dist_eq, dist_nonneg]
-#align dist_prod_same_left dist_prod_same_left
-
-@[simp]
-theorem dist_prod_same_right {x₁ x₂ : α} {y : β} : dist (x₁, y) (x₂, y) = dist x₁ x₂ := by
-  simp [Prod.dist_eq, dist_nonneg]
-#align dist_prod_same_right dist_prod_same_right
-
-theorem ball_prod_same (x : α) (y : β) (r : ℝ) : ball x r ×ˢ ball y r = ball (x, y) r :=
-  ext fun z => by simp [Prod.dist_eq]
-#align ball_prod_same ball_prod_same
-
-theorem closedBall_prod_same (x : α) (y : β) (r : ℝ) :
-    closedBall x r ×ˢ closedBall y r = closedBall (x, y) r :=
-  ext fun z => by simp [Prod.dist_eq]
-#align closed_ball_prod_same closedBall_prod_same
-
-theorem sphere_prod (x : α × β) (r : ℝ) :
-    sphere x r = sphere x.1 r ×ˢ closedBall x.2 r ∪ closedBall x.1 r ×ˢ sphere x.2 r := by
-  obtain hr | rfl | hr := lt_trichotomy r 0
-  · simp [hr]
-  · cases x
-    simp_rw [← closedBall_eq_sphere_of_nonpos le_rfl, union_self, closedBall_prod_same]
-  · ext ⟨x', y'⟩
-    simp_rw [Set.mem_union, Set.mem_prod, Metric.mem_closedBall, Metric.mem_sphere, Prod.dist_eq,
-      max_eq_iff]
-    refine' or_congr (and_congr_right _) (and_comm.trans (and_congr_left _))
-    all_goals rintro rfl; rfl
-#align sphere_prod sphere_prod
-
-end Prod
-
--- porting note: 3 new lemmas
-theorem dist_dist_dist_le_left (x y z : α) : dist (dist x z) (dist y z) ≤ dist x y :=
-  abs_dist_sub_le ..
-
-theorem dist_dist_dist_le_right (x y z : α) : dist (dist x y) (dist x z) ≤ dist y z := by
-  simpa only [dist_comm x] using dist_dist_dist_le_left y z x
-
-theorem dist_dist_dist_le (x y x' y' : α) : dist (dist x y) (dist x' y') ≤ dist x x' + dist y y' :=
-  (dist_triangle _ _ _).trans <|
-    add_le_add (dist_dist_dist_le_left _ _ _) (dist_dist_dist_le_right _ _ _)
-
-theorem uniformContinuous_dist : UniformContinuous fun p : α × α => dist p.1 p.2 :=
-  Metric.uniformContinuous_iff.2 fun ε ε0 =>
-    ⟨ε / 2, half_pos ε0, fun {a b} h =>
-      calc dist (dist a.1 a.2) (dist b.1 b.2) ≤ dist a.1 b.1 + dist a.2 b.2 :=
-        dist_dist_dist_le _ _ _ _
-      _ ≤ dist a b + dist a b := add_le_add (le_max_left _ _) (le_max_right _ _)
-      _ < ε / 2 + ε / 2 := add_lt_add h h
-      _ = ε := add_halves ε⟩
-#align uniform_continuous_dist uniformContinuous_dist
-
-protected theorem UniformContinuous.dist [UniformSpace β] {f g : β → α} (hf : UniformContinuous f)
-    (hg : UniformContinuous g) : UniformContinuous fun b => dist (f b) (g b) :=
-  uniformContinuous_dist.comp (hf.prod_mk hg)
-#align uniform_continuous.dist UniformContinuous.dist
-
-@[continuity]
-theorem continuous_dist : Continuous fun p : α × α => dist p.1 p.2 :=
-  uniformContinuous_dist.continuous
-#align continuous_dist continuous_dist
-
-@[continuity]
-protected theorem Continuous.dist [TopologicalSpace β] {f g : β → α} (hf : Continuous f)
-    (hg : Continuous g) : Continuous fun b => dist (f b) (g b) :=
-  continuous_dist.comp (hf.prod_mk hg : _)
-#align continuous.dist Continuous.dist
-
-protected theorem Filter.Tendsto.dist {f g : β → α} {x : Filter β} {a b : α}
-    (hf : Tendsto f x (𝓝 a)) (hg : Tendsto g x (𝓝 b)) :
-    Tendsto (fun x => dist (f x) (g x)) x (𝓝 (dist a b)) :=
-  (continuous_dist.tendsto (a, b)).comp (hf.prod_mk_nhds hg)
-#align filter.tendsto.dist Filter.Tendsto.dist
-
-theorem nhds_comap_dist (a : α) : ((𝓝 (0 : ℝ)).comap (dist · a)) = 𝓝 a := by
-  simp only [@nhds_eq_comap_uniformity α, Metric.uniformity_eq_comap_nhds_zero, comap_comap,
-    (· ∘ ·), dist_comm]
-#align nhds_comap_dist nhds_comap_dist
-
-theorem tendsto_iff_dist_tendsto_zero {f : β → α} {x : Filter β} {a : α} :
-    Tendsto f x (𝓝 a) ↔ Tendsto (fun b => dist (f b) a) x (𝓝 0) := by
-  rw [← nhds_comap_dist a, tendsto_comap_iff]; rfl
-#align tendsto_iff_dist_tendsto_zero tendsto_iff_dist_tendsto_zero
-
-theorem continuous_iff_continuous_dist [TopologicalSpace β] {f : β → α} :
-    Continuous f ↔ Continuous fun x : β × β => dist (f x.1) (f x.2) :=
-  ⟨fun h => h.fst'.dist h.snd', fun h =>
-    continuous_iff_continuousAt.2 fun _ => tendsto_iff_dist_tendsto_zero.2 <|
-      (h.comp (continuous_id.prod_mk continuous_const)).tendsto' _ _ <| dist_self _⟩
-#align continuous_iff_continuous_dist continuous_iff_continuous_dist
-
-theorem uniformContinuous_nndist : UniformContinuous fun p : α × α => nndist p.1 p.2 :=
-  uniformContinuous_dist.subtype_mk _
-#align uniform_continuous_nndist uniformContinuous_nndist
-
-protected theorem UniformContinuous.nndist [UniformSpace β] {f g : β → α} (hf : UniformContinuous f)
-    (hg : UniformContinuous g) : UniformContinuous fun b => nndist (f b) (g b) :=
-  uniformContinuous_nndist.comp (hf.prod_mk hg)
-#align uniform_continuous.nndist UniformContinuous.nndist
-
-theorem continuous_nndist : Continuous fun p : α × α => nndist p.1 p.2 :=
-  uniformContinuous_nndist.continuous
-#align continuous_nndist continuous_nndist
-
-protected theorem Continuous.nndist [TopologicalSpace β] {f g : β → α} (hf : Continuous f)
-    (hg : Continuous g) : Continuous fun b => nndist (f b) (g b) :=
-  continuous_nndist.comp (hf.prod_mk hg : _)
-#align continuous.nndist Continuous.nndist
-
-protected theorem Filter.Tendsto.nndist {f g : β → α} {x : Filter β} {a b : α}
-    (hf : Tendsto f x (𝓝 a)) (hg : Tendsto g x (𝓝 b)) :
-    Tendsto (fun x => nndist (f x) (g x)) x (𝓝 (nndist a b)) :=
-  (continuous_nndist.tendsto (a, b)).comp (hf.prod_mk_nhds hg)
-#align filter.tendsto.nndist Filter.Tendsto.nndist
-
-namespace Metric
-
-variable {x y z : α} {ε ε₁ ε₂ : ℝ} {s : Set α}
-
-theorem isClosed_ball : IsClosed (closedBall x ε) :=
-  isClosed_le (continuous_id.dist continuous_const) continuous_const
-#align metric.is_closed_ball Metric.isClosed_ball
-
-theorem isClosed_sphere : IsClosed (sphere x ε) :=
-  isClosed_eq (continuous_id.dist continuous_const) continuous_const
-#align metric.is_closed_sphere Metric.isClosed_sphere
-
-@[simp]
-theorem closure_closedBall : closure (closedBall x ε) = closedBall x ε :=
-  isClosed_ball.closure_eq
-#align metric.closure_closed_ball Metric.closure_closedBall
-
-@[simp]
-theorem closure_sphere : closure (sphere x ε) = sphere x ε :=
-  isClosed_sphere.closure_eq
-#align metric.closure_sphere Metric.closure_sphere
-
-theorem closure_ball_subset_closedBall : closure (ball x ε) ⊆ closedBall x ε :=
-  closure_minimal ball_subset_closedBall isClosed_ball
-#align metric.closure_ball_subset_closed_ball Metric.closure_ball_subset_closedBall
-
-theorem frontier_ball_subset_sphere : frontier (ball x ε) ⊆ sphere x ε :=
-  frontier_lt_subset_eq (continuous_id.dist continuous_const) continuous_const
-#align metric.frontier_ball_subset_sphere Metric.frontier_ball_subset_sphere
-
-theorem frontier_closedBall_subset_sphere : frontier (closedBall x ε) ⊆ sphere x ε :=
-  frontier_le_subset_eq (continuous_id.dist continuous_const) continuous_const
-#align metric.frontier_closed_ball_subset_sphere Metric.frontier_closedBall_subset_sphere
-
-theorem ball_subset_interior_closedBall : ball x ε ⊆ interior (closedBall x ε) :=
-  interior_maximal ball_subset_closedBall isOpen_ball
-#align metric.ball_subset_interior_closed_ball Metric.ball_subset_interior_closedBall
-
-/-- ε-characterization of the closure in pseudometric spaces-/
-theorem mem_closure_iff {s : Set α} {a : α} : a ∈ closure s ↔ ∀ ε > 0, ∃ b ∈ s, dist a b < ε :=
-  (mem_closure_iff_nhds_basis nhds_basis_ball).trans <| by simp only [mem_ball, dist_comm]
-#align metric.mem_closure_iff Metric.mem_closure_iff
-
-theorem mem_closure_range_iff {e : β → α} {a : α} :
-    a ∈ closure (range e) ↔ ∀ ε > 0, ∃ k : β, dist a (e k) < ε := by
-  simp only [mem_closure_iff, exists_range_iff]
-#align metric.mem_closure_range_iff Metric.mem_closure_range_iff
-
-theorem mem_closure_range_iff_nat {e : β → α} {a : α} :
-    a ∈ closure (range e) ↔ ∀ n : ℕ, ∃ k : β, dist a (e k) < 1 / ((n : ℝ) + 1) :=
-  (mem_closure_iff_nhds_basis nhds_basis_ball_inv_nat_succ).trans <| by
-    simp only [mem_ball, dist_comm, exists_range_iff, forall_const]
-#align metric.mem_closure_range_iff_nat Metric.mem_closure_range_iff_nat
-
-theorem mem_of_closed' {s : Set α} (hs : IsClosed s) {a : α} :
-    a ∈ s ↔ ∀ ε > 0, ∃ b ∈ s, dist a b < ε := by
-  simpa only [hs.closure_eq] using @mem_closure_iff _ _ s a
-#align metric.mem_of_closed' Metric.mem_of_closed'
-
-theorem closedBall_zero' (x : α) : closedBall x 0 = closure {x} :=
-  Subset.antisymm
-    (fun _y hy =>
-      mem_closure_iff.2 fun _ε ε0 => ⟨x, mem_singleton x, (mem_closedBall.1 hy).trans_lt ε0⟩)
-    (closure_minimal (singleton_subset_iff.2 (dist_self x).le) isClosed_ball)
-#align metric.closed_ball_zero' Metric.closedBall_zero'
-
-lemma eventually_isCompact_closedBall [LocallyCompactSpace α] (x : α) :
-    ∀ᶠ r in 𝓝 (0 : ℝ), IsCompact (closedBall x r) := by
-  rcases local_compact_nhds (x := x) (n := univ) univ_mem with ⟨s, hs, -, s_compact⟩
-  filter_upwards [eventually_closedBall_subset hs] with r hr
-  exact IsCompact.of_isClosed_subset s_compact isClosed_ball hr
-
-lemma exists_isCompact_closedBall [LocallyCompactSpace α] (x : α) :
-    ∃ r, 0 < r ∧ IsCompact (closedBall x r) := by
-  have : ∀ᶠ r in 𝓝[>] 0, IsCompact (closedBall x r) :=
-    eventually_nhdsWithin_of_eventually_nhds (eventually_isCompact_closedBall x)
-  simpa only [and_comm] using (this.and self_mem_nhdsWithin).exists
-
-theorem dense_iff {s : Set α} : Dense s ↔ ∀ x, ∀ r > 0, (ball x r ∩ s).Nonempty :=
-  forall_congr' fun x => by
-    simp only [mem_closure_iff, Set.Nonempty, exists_prop, mem_inter_iff, mem_ball', and_comm]
-#align metric.dense_iff Metric.dense_iff
-
-theorem denseRange_iff {f : β → α} : DenseRange f ↔ ∀ x, ∀ r > 0, ∃ y, dist x (f y) < r :=
-  forall_congr' fun x => by simp only [mem_closure_iff, exists_range_iff]
-#align metric.dense_range_iff Metric.denseRange_iff
-
--- porting note: `TopologicalSpace.IsSeparable.separableSpace` moved to `EMetricSpace`
-
-/-- The preimage of a separable set by an inducing map is separable. -/
-protected theorem _root_.Inducing.isSeparable_preimage {f : β → α} [TopologicalSpace β]
-    (hf : Inducing f) {s : Set α} (hs : IsSeparable s) : IsSeparable (f ⁻¹' s) := by
-  have : SeparableSpace s := hs.separableSpace
-  have : SecondCountableTopology s := UniformSpace.secondCountable_of_separable _
-  have : Inducing ((mapsTo_preimage f s).restrict _ _ _) :=
-    (hf.comp inducing_subtype_val).codRestrict _
-  have := this.secondCountableTopology
-  exact isSeparable_of_separableSpace_subtype _
-#align inducing.is_separable_preimage Inducing.isSeparable_preimage
-
-protected theorem _root_.Embedding.isSeparable_preimage {f : β → α} [TopologicalSpace β]
-    (hf : Embedding f) {s : Set α} (hs : IsSeparable s) : IsSeparable (f ⁻¹' s) :=
-  hf.toInducing.isSeparable_preimage hs
-#align embedding.is_separable_preimage Embedding.isSeparable_preimage
-
-/-- If a map is continuous on a separable set `s`, then the image of `s` is also separable. -/
-theorem _root_.ContinuousOn.isSeparable_image [TopologicalSpace β] {f : α → β} {s : Set α}
-    (hf : ContinuousOn f s) (hs : IsSeparable s) : IsSeparable (f '' s) := by
-  rw [image_eq_range, ← image_univ]
-  exact (isSeparable_univ_iff.2 hs.separableSpace).image hf.restrict
-#align continuous_on.is_separable_image ContinuousOn.isSeparable_image
-
-end Metric
-
-section Pi
-
-open Finset
-
-variable {π : β → Type*} [Fintype β] [∀ b, PseudoMetricSpace (π b)]
-
-/-- A finite product of pseudometric spaces is a pseudometric space, with the sup distance. -/
-instance pseudoMetricSpacePi : PseudoMetricSpace (∀ b, π b) := by
-  /- we construct the instance from the pseudoemetric space instance to avoid checking again that
-    the uniformity is the same as the product uniformity, but we register nevertheless a nice
-    formula for the distance -/
-  let i := PseudoEMetricSpace.toPseudoMetricSpaceOfDist
-    (fun f g : ∀ b, π b => ((sup univ fun b => nndist (f b) (g b) : ℝ≥0) : ℝ))
-    (fun f g => ((Finset.sup_lt_iff bot_lt_top).2 fun b _ => edist_lt_top _ _).ne)
-    (fun f g => by
-      simp only [edist_pi_def, edist_nndist, ← ENNReal.coe_finset_sup, ENNReal.coe_toReal])
-  refine i.replaceBornology fun s => ?_
-  simp only [← isBounded_def, isBounded_iff_eventually, ← forall_isBounded_image_eval_iff,
-    ball_image_iff, ← Filter.eventually_all, Function.eval_apply, @dist_nndist (π _)]
-  refine' eventually_congr ((eventually_ge_atTop 0).mono fun C hC => _)
-  lift C to ℝ≥0 using hC
-  refine' ⟨fun H x hx y hy => NNReal.coe_le_coe.2 <| Finset.sup_le fun b _ => H b x hx y hy,
-    fun H b x hx y hy => NNReal.coe_le_coe.2 _⟩
-  simpa only using Finset.sup_le_iff.1 (NNReal.coe_le_coe.1 <| H hx hy) b (Finset.mem_univ b)
-#align pseudo_metric_space_pi pseudoMetricSpacePi
-
-theorem nndist_pi_def (f g : ∀ b, π b) : nndist f g = sup univ fun b => nndist (f b) (g b) :=
-  NNReal.eq rfl
-#align nndist_pi_def nndist_pi_def
-
-theorem dist_pi_def (f g : ∀ b, π b) : dist f g = (sup univ fun b => nndist (f b) (g b) : ℝ≥0) :=
-  rfl
-#align dist_pi_def dist_pi_def
-
-theorem nndist_pi_le_iff {f g : ∀ b, π b} {r : ℝ≥0} :
-    nndist f g ≤ r ↔ ∀ b, nndist (f b) (g b) ≤ r := by simp [nndist_pi_def]
-#align nndist_pi_le_iff nndist_pi_le_iff
-
-theorem nndist_pi_lt_iff {f g : ∀ b, π b} {r : ℝ≥0} (hr : 0 < r) :
-    nndist f g < r ↔ ∀ b, nndist (f b) (g b) < r := by
-  simp [nndist_pi_def, Finset.sup_lt_iff (show ⊥ < r from hr)]
-#align nndist_pi_lt_iff nndist_pi_lt_iff
-
-theorem nndist_pi_eq_iff {f g : ∀ b, π b} {r : ℝ≥0} (hr : 0 < r) :
-    nndist f g = r ↔ (∃ i, nndist (f i) (g i) = r) ∧ ∀ b, nndist (f b) (g b) ≤ r := by
-  rw [eq_iff_le_not_lt, nndist_pi_lt_iff hr, nndist_pi_le_iff, not_forall, and_comm]
-  simp_rw [not_lt, and_congr_left_iff, le_antisymm_iff]
-  intro h
-  refine' exists_congr fun b => _
-  apply (and_iff_right <| h _).symm
-#align nndist_pi_eq_iff nndist_pi_eq_iff
-
-theorem dist_pi_lt_iff {f g : ∀ b, π b} {r : ℝ} (hr : 0 < r) :
-    dist f g < r ↔ ∀ b, dist (f b) (g b) < r := by
-  lift r to ℝ≥0 using hr.le
-  exact nndist_pi_lt_iff hr
-#align dist_pi_lt_iff dist_pi_lt_iff
-
-theorem dist_pi_le_iff {f g : ∀ b, π b} {r : ℝ} (hr : 0 ≤ r) :
-    dist f g ≤ r ↔ ∀ b, dist (f b) (g b) ≤ r := by
-  lift r to ℝ≥0 using hr
-  exact nndist_pi_le_iff
-#align dist_pi_le_iff dist_pi_le_iff
-
-theorem dist_pi_eq_iff {f g : ∀ b, π b} {r : ℝ} (hr : 0 < r) :
-    dist f g = r ↔ (∃ i, dist (f i) (g i) = r) ∧ ∀ b, dist (f b) (g b) ≤ r := by
-  lift r to ℝ≥0 using hr.le
-  simp_rw [← coe_nndist, NNReal.coe_eq, nndist_pi_eq_iff hr, NNReal.coe_le_coe]
-#align dist_pi_eq_iff dist_pi_eq_iff
-
-theorem dist_pi_le_iff' [Nonempty β] {f g : ∀ b, π b} {r : ℝ} :
-    dist f g ≤ r ↔ ∀ b, dist (f b) (g b) ≤ r := by
-  by_cases hr : 0 ≤ r
-  · exact dist_pi_le_iff hr
-  · exact iff_of_false (fun h => hr <| dist_nonneg.trans h) fun h =>
-      hr <| dist_nonneg.trans <| h <| Classical.arbitrary _
-#align dist_pi_le_iff' dist_pi_le_iff'
-
-theorem dist_pi_const_le (a b : α) : (dist (fun _ : β => a) fun _ => b) ≤ dist a b :=
-  (dist_pi_le_iff dist_nonneg).2 fun _ => le_rfl
-#align dist_pi_const_le dist_pi_const_le
-
-theorem nndist_pi_const_le (a b : α) : (nndist (fun _ : β => a) fun _ => b) ≤ nndist a b :=
-  nndist_pi_le_iff.2 fun _ => le_rfl
-#align nndist_pi_const_le nndist_pi_const_le
-
-@[simp]
-theorem dist_pi_const [Nonempty β] (a b : α) : (dist (fun _ : β => a) fun _ => b) = dist a b := by
-  simpa only [dist_edist] using congr_arg ENNReal.toReal (edist_pi_const a b)
-#align dist_pi_const dist_pi_const
-
-@[simp]
-theorem nndist_pi_const [Nonempty β] (a b : α) :
-    (nndist (fun _ : β => a) fun _ => b) = nndist a b :=
-  NNReal.eq <| dist_pi_const a b
-#align nndist_pi_const nndist_pi_const
-
-theorem nndist_le_pi_nndist (f g : ∀ b, π b) (b : β) : nndist (f b) (g b) ≤ nndist f g := by
-  rw [← ENNReal.coe_le_coe, ← edist_nndist, ← edist_nndist]
-  exact edist_le_pi_edist f g b
-#align nndist_le_pi_nndist nndist_le_pi_nndist
-
-theorem dist_le_pi_dist (f g : ∀ b, π b) (b : β) : dist (f b) (g b) ≤ dist f g := by
-  simp only [dist_nndist, NNReal.coe_le_coe, nndist_le_pi_nndist f g b]
-#align dist_le_pi_dist dist_le_pi_dist
-
-/-- An open ball in a product space is a product of open balls. See also `ball_pi'`
-for a version assuming `Nonempty β` instead of `0 < r`. -/
-theorem ball_pi (x : ∀ b, π b) {r : ℝ} (hr : 0 < r) :
-    ball x r = Set.pi univ fun b => ball (x b) r := by
-  ext p
-  simp [dist_pi_lt_iff hr]
-#align ball_pi ball_pi
-
-/-- An open ball in a product space is a product of open balls. See also `ball_pi`
-for a version assuming `0 < r` instead of `Nonempty β`. -/
-theorem ball_pi' [Nonempty β] (x : ∀ b, π b) (r : ℝ) :
-    ball x r = Set.pi univ fun b => ball (x b) r :=
-  (lt_or_le 0 r).elim (ball_pi x) fun hr => by simp [ball_eq_empty.2 hr]
-#align ball_pi' ball_pi'
-
-/-- A closed ball in a product space is a product of closed balls. See also `closedBall_pi'`
-for a version assuming `Nonempty β` instead of `0 ≤ r`. -/
-theorem closedBall_pi (x : ∀ b, π b) {r : ℝ} (hr : 0 ≤ r) :
-    closedBall x r = Set.pi univ fun b => closedBall (x b) r := by
-  ext p
-  simp [dist_pi_le_iff hr]
-#align closed_ball_pi closedBall_pi
-
-/-- A closed ball in a product space is a product of closed balls. See also `closedBall_pi`
-for a version assuming `0 ≤ r` instead of `Nonempty β`. -/
-theorem closedBall_pi' [Nonempty β] (x : ∀ b, π b) (r : ℝ) :
-    closedBall x r = Set.pi univ fun b => closedBall (x b) r :=
-  (le_or_lt 0 r).elim (closedBall_pi x) fun hr => by simp [closedBall_eq_empty.2 hr]
-#align closed_ball_pi' closedBall_pi'
-
-/-- A sphere in a product space is a union of spheres on each component restricted to the closed
-ball. -/
-theorem sphere_pi (x : ∀ b, π b) {r : ℝ} (h : 0 < r ∨ Nonempty β) :
-    sphere x r = (⋃ i : β, Function.eval i ⁻¹' sphere (x i) r) ∩ closedBall x r := by
-  obtain hr | rfl | hr := lt_trichotomy r 0
-  · simp [hr]
-  · rw [closedBall_eq_sphere_of_nonpos le_rfl, eq_comm, Set.inter_eq_right]
-    letI := h.resolve_left (lt_irrefl _)
-    inhabit β
-    refine' subset_iUnion_of_subset default _
-    intro x hx
-    replace hx := hx.le
-    rw [dist_pi_le_iff le_rfl] at hx
-    exact le_antisymm (hx default) dist_nonneg
-  · ext
-    simp [dist_pi_eq_iff hr, dist_pi_le_iff hr.le]
-#align sphere_pi sphere_pi
-
-@[simp]
-theorem Fin.nndist_insertNth_insertNth {n : ℕ} {α : Fin (n + 1) → Type*}
-    [∀ i, PseudoMetricSpace (α i)] (i : Fin (n + 1)) (x y : α i) (f g : ∀ j, α (i.succAbove j)) :
-    nndist (i.insertNth x f) (i.insertNth y g) = max (nndist x y) (nndist f g) :=
-  eq_of_forall_ge_iff fun c => by simp [nndist_pi_le_iff, i.forall_iff_succAbove]
-#align fin.nndist_insert_nth_insert_nth Fin.nndist_insertNth_insertNth
-
-@[simp]
-theorem Fin.dist_insertNth_insertNth {n : ℕ} {α : Fin (n + 1) → Type*}
-    [∀ i, PseudoMetricSpace (α i)] (i : Fin (n + 1)) (x y : α i) (f g : ∀ j, α (i.succAbove j)) :
-    dist (i.insertNth x f) (i.insertNth y g) = max (dist x y) (dist f g) := by
-  simp only [dist_nndist, Fin.nndist_insertNth_insertNth, NNReal.coe_max]
-#align fin.dist_insert_nth_insert_nth Fin.dist_insertNth_insertNth
-
-theorem Real.dist_le_of_mem_pi_Icc {x y x' y' : β → ℝ} (hx : x ∈ Icc x' y') (hy : y ∈ Icc x' y') :
-    dist x y ≤ dist x' y' := by
-  refine' (dist_pi_le_iff dist_nonneg).2 fun b =>
-    (Real.dist_le_of_mem_uIcc _ _).trans (dist_le_pi_dist x' y' b) <;> refine' Icc_subset_uIcc _
-  exacts [⟨hx.1 _, hx.2 _⟩, ⟨hy.1 _, hy.2 _⟩]
-#align real.dist_le_of_mem_pi_Icc Real.dist_le_of_mem_pi_Icc
-
-end Pi
-
-section Compact
-
-/-- Any compact set in a pseudometric space can be covered by finitely many balls of a given
-positive radius -/
-theorem finite_cover_balls_of_compact {α : Type u} [PseudoMetricSpace α] {s : Set α}
-    (hs : IsCompact s) {e : ℝ} (he : 0 < e) :
-    ∃ t, t ⊆ s ∧ Set.Finite t ∧ s ⊆ ⋃ x ∈ t, ball x e :=
-  let ⟨t, hts, ht⟩ := hs.elim_nhds_subcover _ (fun x _ => ball_mem_nhds x he)
-  ⟨t, hts, t.finite_toSet, ht⟩
-#align finite_cover_balls_of_compact finite_cover_balls_of_compact
-
-alias IsCompact.finite_cover_balls := finite_cover_balls_of_compact
-#align is_compact.finite_cover_balls IsCompact.finite_cover_balls
-
-end Compact
-
-section ProperSpace
-
-open Metric
-
-/-- A pseudometric space is proper if all closed balls are compact. -/
-class ProperSpace (α : Type u) [PseudoMetricSpace α] : Prop where
-  isCompact_closedBall : ∀ x : α, ∀ r, IsCompact (closedBall x r)
-#align proper_space ProperSpace
-
-export ProperSpace (isCompact_closedBall)
-
-/-- In a proper pseudometric space, all spheres are compact. -/
-theorem isCompact_sphere {α : Type*} [PseudoMetricSpace α] [ProperSpace α] (x : α) (r : ℝ) :
-    IsCompact (sphere x r) :=
-  (isCompact_closedBall x r).of_isClosed_subset isClosed_sphere sphere_subset_closedBall
-#align is_compact_sphere isCompact_sphere
-
-/-- In a proper pseudometric space, any sphere is a `CompactSpace` when considered as a subtype. -/
-instance Metric.sphere.compactSpace {α : Type*} [PseudoMetricSpace α] [ProperSpace α]
-    (x : α) (r : ℝ) : CompactSpace (sphere x r) :=
-  isCompact_iff_compactSpace.mp (isCompact_sphere _ _)
-
--- see Note [lower instance priority]
-/-- A proper pseudo metric space is sigma compact, and therefore second countable. -/
-instance (priority := 100) secondCountable_of_proper [ProperSpace α] :
-    SecondCountableTopology α := by
-  -- We already have `sigmaCompactSpace_of_locallyCompact_secondCountable`, so we don't
-  -- add an instance for `SigmaCompactSpace`.
-  suffices SigmaCompactSpace α by exact EMetric.secondCountable_of_sigmaCompact α
-  rcases em (Nonempty α) with (⟨⟨x⟩⟩ | hn)
-  · exact ⟨⟨fun n => closedBall x n, fun n => isCompact_closedBall _ _, iUnion_closedBall_nat _⟩⟩
-  · exact ⟨⟨fun _ => ∅, fun _ => isCompact_empty, iUnion_eq_univ_iff.2 fun x => (hn ⟨x⟩).elim⟩⟩
-#align second_countable_of_proper secondCountable_of_proper
-
-/-- If all closed balls of large enough radius are compact, then the space is proper. Especially
-useful when the lower bound for the radius is 0. -/
-theorem properSpace_of_compact_closedBall_of_le (R : ℝ)
-    (h : ∀ x : α, ∀ r, R ≤ r → IsCompact (closedBall x r)) : ProperSpace α :=
-  ⟨fun x r => IsCompact.of_isClosed_subset (h x (max r R) (le_max_right _ _)) isClosed_ball
-    (closedBall_subset_closedBall <| le_max_left _ _)⟩
-#align proper_space_of_compact_closed_ball_of_le properSpace_of_compact_closedBall_of_le
-
--- A compact pseudometric space is proper
--- see Note [lower instance priority]
-instance (priority := 100) proper_of_compact [CompactSpace α] : ProperSpace α :=
-  ⟨fun _ _ => isClosed_ball.isCompact⟩
-#align proper_of_compact proper_of_compact
-
--- see Note [lower instance priority]
-/-- A proper space is locally compact -/
-instance (priority := 100) locally_compact_of_proper [ProperSpace α] : LocallyCompactSpace α :=
-  locallyCompactSpace_of_hasBasis (fun _ => nhds_basis_closedBall) fun _ _ _ =>
-    isCompact_closedBall _ _
-#align locally_compact_of_proper locally_compact_of_proper
-
--- see Note [lower instance priority]
-/-- A proper space is complete -/
-instance (priority := 100) complete_of_proper [ProperSpace α] : CompleteSpace α :=
-  ⟨fun {f} hf => by
-    /- We want to show that the Cauchy filter `f` is converging. It suffices to find a closed
-      ball (therefore compact by properness) where it is nontrivial. -/
-    obtain ⟨t, t_fset, ht⟩ : ∃ t ∈ f, ∀ x ∈ t, ∀ y ∈ t, dist x y < 1 :=
-      (Metric.cauchy_iff.1 hf).2 1 zero_lt_one
-    rcases hf.1.nonempty_of_mem t_fset with ⟨x, xt⟩
-    have : closedBall x 1 ∈ f := mem_of_superset t_fset fun y yt => (ht y yt x xt).le
-    rcases (isCompact_iff_totallyBounded_isComplete.1 (isCompact_closedBall x 1)).2 f hf
-        (le_principal_iff.2 this) with
-      ⟨y, -, hy⟩
-    exact ⟨y, hy⟩⟩
-#align complete_of_proper complete_of_proper
-
-/-- A binary product of proper spaces is proper. -/
-instance prod_properSpace {α : Type*} {β : Type*} [PseudoMetricSpace α] [PseudoMetricSpace β]
-    [ProperSpace α] [ProperSpace β] : ProperSpace (α × β) where
-  isCompact_closedBall := by
-    rintro ⟨x, y⟩ r
-    rw [← closedBall_prod_same x y]
-    exact (isCompact_closedBall x r).prod (isCompact_closedBall y r)
-#align prod_proper_space prod_properSpace
-
-/-- A finite product of proper spaces is proper. -/
-instance pi_properSpace {π : β → Type*} [Fintype β] [∀ b, PseudoMetricSpace (π b)]
-    [h : ∀ b, ProperSpace (π b)] : ProperSpace (∀ b, π b) := by
-  refine' properSpace_of_compact_closedBall_of_le 0 fun x r hr => _
-  rw [closedBall_pi _ hr]
-  exact isCompact_univ_pi fun _ => isCompact_closedBall _ _
-#align pi_proper_space pi_properSpace
-
-variable [ProperSpace α] {x : α} {r : ℝ} {s : Set α}
-
-/-- If a nonempty ball in a proper space includes a closed set `s`, then there exists a nonempty
-ball with the same center and a strictly smaller radius that includes `s`. -/
-theorem exists_pos_lt_subset_ball (hr : 0 < r) (hs : IsClosed s) (h : s ⊆ ball x r) :
-    ∃ r' ∈ Ioo 0 r, s ⊆ ball x r' := by
-  rcases eq_empty_or_nonempty s with (rfl | hne)
-  · exact ⟨r / 2, ⟨half_pos hr, half_lt_self hr⟩, empty_subset _⟩
-  have : IsCompact s :=
-    (isCompact_closedBall x r).of_isClosed_subset hs (h.trans ball_subset_closedBall)
-  obtain ⟨y, hys, hy⟩ : ∃ y ∈ s, s ⊆ closedBall x (dist y x) :=
-    this.exists_forall_ge hne (continuous_id.dist continuous_const).continuousOn
-  have hyr : dist y x < r := h hys
-  rcases exists_between hyr with ⟨r', hyr', hrr'⟩
-  exact ⟨r', ⟨dist_nonneg.trans_lt hyr', hrr'⟩, hy.trans <| closedBall_subset_ball hyr'⟩
-#align exists_pos_lt_subset_ball exists_pos_lt_subset_ball
-
-/-- If a ball in a proper space includes a closed set `s`, then there exists a ball with the same
-center and a strictly smaller radius that includes `s`. -/
-theorem exists_lt_subset_ball (hs : IsClosed s) (h : s ⊆ ball x r) : ∃ r' < r, s ⊆ ball x r' := by
-  cases' le_or_lt r 0 with hr hr
-  · rw [ball_eq_empty.2 hr, subset_empty_iff] at h
-    subst s
-    exact (exists_lt r).imp fun r' hr' => ⟨hr', empty_subset _⟩
-  · exact (exists_pos_lt_subset_ball hr hs h).imp fun r' hr' => ⟨hr'.1.2, hr'.2⟩
-#align exists_lt_subset_ball exists_lt_subset_ball
-
-end ProperSpace
-
-theorem IsCompact.isSeparable {s : Set α} (hs : IsCompact s) : IsSeparable s :=
-  haveI : CompactSpace s := isCompact_iff_compactSpace.mp hs
-  isSeparable_of_separableSpace_subtype s
-#align is_compact.is_separable IsCompact.isSeparable
-
-namespace Metric
-
-section SecondCountable
-
-open TopologicalSpace
-
-/-- A pseudometric space is second countable if, for every `ε > 0`, there is a countable set which
-is `ε`-dense. -/
-theorem secondCountable_of_almost_dense_set
-    (H : ∀ ε > (0 : ℝ), ∃ s : Set α, s.Countable ∧ ∀ x, ∃ y ∈ s, dist x y ≤ ε) :
-    SecondCountableTopology α := by
-  refine' EMetric.secondCountable_of_almost_dense_set fun ε ε0 => _
-  rcases ENNReal.lt_iff_exists_nnreal_btwn.1 ε0 with ⟨ε', ε'0, ε'ε⟩
-  choose s hsc y hys hyx using H ε' (by exact_mod_cast ε'0)
-  refine' ⟨s, hsc, iUnion₂_eq_univ_iff.2 fun x => ⟨y x, hys _, le_trans _ ε'ε.le⟩⟩
-  exact_mod_cast hyx x
-#align metric.second_countable_of_almost_dense_set Metric.secondCountable_of_almost_dense_set
-
-end SecondCountable
-
-end Metric
-
-theorem lebesgue_number_lemma_of_metric {s : Set α} {ι : Sort*} {c : ι → Set α} (hs : IsCompact s)
-    (hc₁ : ∀ i, IsOpen (c i)) (hc₂ : s ⊆ ⋃ i, c i) : ∃ δ > 0, ∀ x ∈ s, ∃ i, ball x δ ⊆ c i :=
-  let ⟨_n, en, hn⟩ := lebesgue_number_lemma hs hc₁ hc₂
-  let ⟨δ, δ0, hδ⟩ := mem_uniformity_dist.1 en
-  ⟨δ, δ0, fun x hx => let ⟨i, hi⟩ := hn x hx; ⟨i, fun _y hy => hi (hδ (mem_ball'.mp hy))⟩⟩
-#align lebesgue_number_lemma_of_metric lebesgue_number_lemma_of_metric
-
-theorem lebesgue_number_lemma_of_metric_sUnion {s : Set α} {c : Set (Set α)} (hs : IsCompact s)
-    (hc₁ : ∀ t ∈ c, IsOpen t) (hc₂ : s ⊆ ⋃₀ c) : ∃ δ > 0, ∀ x ∈ s, ∃ t ∈ c, ball x δ ⊆ t := by
-  rw [sUnion_eq_iUnion] at hc₂; simpa using lebesgue_number_lemma_of_metric hs (by simpa) hc₂
-#align lebesgue_number_lemma_of_metric_sUnion lebesgue_number_lemma_of_metric_sUnion
-
-namespace Metric
-
-#align metric.bounded Bornology.IsBounded
-
-section Bounded
-
-variable {x : α} {s t : Set α} {r : ℝ}
-
-#noalign metric.bounded_iff_is_bounded
-
-#align metric.bounded_empty Bornology.isBounded_empty
-#align metric.bounded_iff_mem_bounded Bornology.isBounded_iff_forall_mem
-#align metric.bounded.mono Bornology.IsBounded.subset
-
-/-- Closed balls are bounded -/
-theorem isBounded_closedBall : IsBounded (closedBall x r) :=
-  isBounded_iff.2 ⟨r + r, fun y hy z hz =>
-    calc dist y z ≤ dist y x + dist z x := dist_triangle_right _ _ _
-    _ ≤ r + r := add_le_add hy hz⟩
-#align metric.bounded_closed_ball Metric.isBounded_closedBall
-
-/-- Open balls are bounded -/
-theorem isBounded_ball : IsBounded (ball x r) :=
-  isBounded_closedBall.subset ball_subset_closedBall
-#align metric.bounded_ball Metric.isBounded_ball
-
-/-- Spheres are bounded -/
-theorem isBounded_sphere : IsBounded (sphere x r) :=
-  isBounded_closedBall.subset sphere_subset_closedBall
-#align metric.bounded_sphere Metric.isBounded_sphere
-
-/-- Given a point, a bounded subset is included in some ball around this point -/
-theorem isBounded_iff_subset_closedBall (c : α) : IsBounded s ↔ ∃ r, s ⊆ closedBall c r :=
-  ⟨fun h ↦ (isBounded_iff.1 (h.insert c)).imp fun _r hr _x hx ↦ hr (.inr hx) (mem_insert _ _),
-    fun ⟨_r, hr⟩ ↦ isBounded_closedBall.subset hr⟩
-#align metric.bounded_iff_subset_ball Metric.isBounded_iff_subset_closedBall
-
-theorem _root_.Bornology.IsBounded.subset_closedBall (h : IsBounded s) (c : α) :
-    ∃ r, s ⊆ closedBall c r :=
-  (isBounded_iff_subset_closedBall c).1 h
-#align metric.bounded.subset_ball Bornology.IsBounded.subset_closedBall
-
-theorem _root_.Bornology.IsBounded.subset_ball_lt (h : IsBounded s) (a : ℝ) (c : α) :
-    ∃ r, a < r ∧ s ⊆ ball c r :=
-  let ⟨r, hr⟩ := h.subset_closedBall c
-  ⟨max r a + 1, (le_max_right _ _).trans_lt (lt_add_one _), hr.trans <| closedBall_subset_ball <|
-    (le_max_left _ _).trans_lt (lt_add_one _)⟩
-
-theorem _root_.Bornology.IsBounded.subset_ball (h : IsBounded s) (c : α) : ∃ r, s ⊆ ball c r :=
-  (h.subset_ball_lt 0 c).imp fun _ ↦ And.right
-
-theorem isBounded_iff_subset_ball (c : α) : IsBounded s ↔ ∃ r, s ⊆ ball c r :=
-  ⟨(IsBounded.subset_ball · c), fun ⟨_r, hr⟩ ↦ isBounded_ball.subset hr⟩
-
-theorem _root_.Bornology.IsBounded.subset_closedBall_lt (h : IsBounded s) (a : ℝ) (c : α) :
-    ∃ r, a < r ∧ s ⊆ closedBall c r :=
-  let ⟨r, har, hr⟩ := h.subset_ball_lt a c
-  ⟨r, har, hr.trans ball_subset_closedBall⟩
-#align metric.bounded.subset_ball_lt Bornology.IsBounded.subset_closedBall_lt
-
-theorem isBounded_closure_of_isBounded (h : IsBounded s) : IsBounded (closure s) :=
-  let ⟨C, h⟩ := isBounded_iff.1 h
-  isBounded_iff.2 ⟨C, fun _a ha _b hb => (isClosed_le' C).closure_subset <|
-    map_mem_closure₂ continuous_dist ha hb h⟩
-#align metric.bounded_closure_of_bounded Metric.isBounded_closure_of_isBounded
-
-protected theorem _root_.Bornology.IsBounded.closure (h : IsBounded s) : IsBounded (closure s) :=
-  isBounded_closure_of_isBounded h
-#align metric.bounded.closure Bornology.IsBounded.closure
-
-@[simp]
-theorem isBounded_closure_iff : IsBounded (closure s) ↔ IsBounded s :=
-  ⟨fun h => h.subset subset_closure, fun h => h.closure⟩
-#align metric.bounded_closure_iff Metric.isBounded_closure_iff
-
-#align metric.bounded_union Bornology.isBounded_union
-#align metric.bounded.union Bornology.IsBounded.union
-#align metric.bounded_bUnion Bornology.isBounded_biUnion
-#align metric.bounded.prod Bornology.IsBounded.prod
-
-theorem hasBasis_cobounded_compl_closedBall (c : α) :
-    (cobounded α).HasBasis (fun _ ↦ True) (fun r ↦ (closedBall c r)ᶜ) :=
-  ⟨compl_surjective.forall.2 fun _ ↦ (isBounded_iff_subset_closedBall c).trans <| by simp⟩
-
-theorem hasBasis_cobounded_compl_ball (c : α) :
-    (cobounded α).HasBasis (fun _ ↦ True) (fun r ↦ (ball c r)ᶜ) :=
-  ⟨compl_surjective.forall.2 fun _ ↦ (isBounded_iff_subset_ball c).trans <| by simp⟩
-
-@[simp]
-theorem comap_dist_right_atTop (c : α) : comap (dist · c) atTop = cobounded α :=
-  (atTop_basis.comap _).eq_of_same_basis <| by
-    simpa only [compl_def, mem_ball, not_lt] using hasBasis_cobounded_compl_ball c
-
-@[simp]
-theorem comap_dist_left_atTop (c : α) : comap (dist c) atTop = cobounded α := by
-  simpa only [dist_comm _ c] using comap_dist_right_atTop c
-
-@[simp]
-theorem tendsto_dist_right_atTop_iff (c : α) {f : β → α} {l : Filter β} :
-    Tendsto (fun x ↦ dist (f x) c) l atTop ↔ Tendsto f l (cobounded α) := by
-  rw [← comap_dist_right_atTop c, tendsto_comap_iff]; rfl
-
-@[simp]
-theorem tendsto_dist_left_atTop_iff (c : α) {f : β → α} {l : Filter β} :
-    Tendsto (fun x ↦ dist c (f x)) l atTop ↔ Tendsto f l (cobounded α) := by
-  simp only [dist_comm c, tendsto_dist_right_atTop_iff]
-
-theorem tendsto_dist_right_cobounded_atTop (c : α) : Tendsto (dist · c) (cobounded α) atTop :=
-  tendsto_iff_comap.2 (comap_dist_right_atTop c).ge
-
-theorem tendsto_dist_left_cobounded_atTop (c : α) : Tendsto (dist c) (cobounded α) atTop :=
-  tendsto_iff_comap.2 (comap_dist_left_atTop c).ge
-
-/-- A totally bounded set is bounded -/
-theorem _root_.TotallyBounded.isBounded {s : Set α} (h : TotallyBounded s) : IsBounded s :=
-  -- We cover the totally bounded set by finitely many balls of radius 1,
-  -- and then argue that a finite union of bounded sets is bounded
-  let ⟨_t, fint, subs⟩ := (totallyBounded_iff.mp h) 1 zero_lt_one
-  ((isBounded_biUnion fint).2 fun _ _ => isBounded_ball).subset subs
-#align totally_bounded.bounded TotallyBounded.isBounded
-
-/-- A compact set is bounded -/
-theorem _root_.IsCompact.isBounded {s : Set α} (h : IsCompact s) : IsBounded s :=
-  -- A compact set is totally bounded, thus bounded
-  h.totallyBounded.isBounded
-#align is_compact.bounded IsCompact.isBounded
-
-#align metric.bounded_of_finite Set.Finite.isBounded
-#align set.finite.bounded Set.Finite.isBounded
-#align metric.bounded_singleton Bornology.isBounded_singleton
-
-theorem cobounded_le_cocompact : cobounded α ≤ cocompact α :=
-  hasBasis_cocompact.ge_iff.2 fun _s hs ↦ hs.isBounded
-#align comap_dist_right_at_top_le_cocompact Metric.cobounded_le_cocompactₓ
-#align comap_dist_left_at_top_le_cocompact Metric.cobounded_le_cocompactₓ
-
-/-- Characterization of the boundedness of the range of a function -/
-theorem isBounded_range_iff {f : β → α} : IsBounded (range f) ↔ ∃ C, ∀ x y, dist (f x) (f y) ≤ C :=
-  isBounded_iff.trans <| by simp only [forall_range_iff]
-#align metric.bounded_range_iff Metric.isBounded_range_iff
-
-theorem isBounded_image_iff {f : β → α} {s : Set β} :
-    IsBounded (f '' s) ↔ ∃ C, ∀ x ∈ s, ∀ y ∈ s, dist (f x) (f y) ≤ C :=
-  isBounded_iff.trans <| by simp only [ball_image_iff]
-
-theorem isBounded_range_of_tendsto_cofinite_uniformity {f : β → α}
-    (hf : Tendsto (Prod.map f f) (.cofinite ×ˢ .cofinite) (𝓤 α)) : IsBounded (range f) := by
-  rcases (hasBasis_cofinite.prod_self.tendsto_iff uniformity_basis_dist).1 hf 1 zero_lt_one with
-    ⟨s, hsf, hs1⟩
-  rw [← image_union_image_compl_eq_range]
-  refine (hsf.image f).isBounded.union (isBounded_image_iff.2 ⟨1, fun x hx y hy ↦ ?_⟩)
-  exact le_of_lt (hs1 (x, y) ⟨hx, hy⟩)
-#align metric.bounded_range_of_tendsto_cofinite_uniformity Metric.isBounded_range_of_tendsto_cofinite_uniformity
-
-theorem isBounded_range_of_cauchy_map_cofinite {f : β → α} (hf : Cauchy (map f cofinite)) :
-    IsBounded (range f) :=
-  isBounded_range_of_tendsto_cofinite_uniformity <| (cauchy_map_iff.1 hf).2
-#align metric.bounded_range_of_cauchy_map_cofinite Metric.isBounded_range_of_cauchy_map_cofinite
-
-theorem _root_.CauchySeq.isBounded_range {f : ℕ → α} (hf : CauchySeq f) : IsBounded (range f) :=
-  isBounded_range_of_cauchy_map_cofinite <| by rwa [Nat.cofinite_eq_atTop]
-#align cauchy_seq.bounded_range CauchySeq.isBounded_range
-
-theorem isBounded_range_of_tendsto_cofinite {f : β → α} {a : α} (hf : Tendsto f cofinite (𝓝 a)) :
-    IsBounded (range f) :=
-  isBounded_range_of_tendsto_cofinite_uniformity <|
-    (hf.prod_map hf).mono_right <| nhds_prod_eq.symm.trans_le (nhds_le_uniformity a)
-#align metric.bounded_range_of_tendsto_cofinite Metric.isBounded_range_of_tendsto_cofinite
-
-/-- In a compact space, all sets are bounded -/
-theorem isBounded_of_compactSpace [CompactSpace α] : IsBounded s :=
-  isCompact_univ.isBounded.subset (subset_univ _)
-#align metric.bounded_of_compact_space Metric.isBounded_of_compactSpace
-
-theorem isBounded_range_of_tendsto (u : ℕ → α) {x : α} (hu : Tendsto u atTop (𝓝 x)) :
-    IsBounded (range u) :=
-  hu.cauchySeq.isBounded_range
-#align metric.bounded_range_of_tendsto Metric.isBounded_range_of_tendsto
-
-theorem disjoint_nhds_cobounded (x : α) : Disjoint (𝓝 x) (cobounded α) :=
-  disjoint_of_disjoint_of_mem disjoint_compl_right (ball_mem_nhds _ one_pos) isBounded_ball
-
-theorem disjoint_cobounded_nhds (x : α) : Disjoint (cobounded α) (𝓝 x) :=
-  (disjoint_nhds_cobounded x).symm
-
-theorem disjoint_nhdsSet_cobounded {s : Set α} (hs : IsCompact s) : Disjoint (𝓝ˢ s) (cobounded α) :=
-  hs.disjoint_nhdsSet_left.2 fun _ _ ↦ disjoint_nhds_cobounded _
-
-theorem disjoint_cobounded_nhdsSet {s : Set α} (hs : IsCompact s) : Disjoint (cobounded α) (𝓝ˢ s) :=
-  (disjoint_nhdsSet_cobounded hs).symm
-
-/-- If a function is continuous within a set `s` at every point of a compact set `k`, then it is
-bounded on some open neighborhood of `k` in `s`. -/
-theorem exists_isOpen_isBounded_image_inter_of_isCompact_of_forall_continuousWithinAt
-    [TopologicalSpace β] {k s : Set β} {f : β → α} (hk : IsCompact k)
-    (hf : ∀ x ∈ k, ContinuousWithinAt f s x) :
-    ∃ t, k ⊆ t ∧ IsOpen t ∧ IsBounded (f '' (t ∩ s)) := by
-  have : Disjoint (𝓝ˢ k ⊓ 𝓟 s) (comap f (cobounded α)) := by
-    rw [disjoint_assoc, inf_comm, hk.disjoint_nhdsSet_left]
-    exact fun x hx ↦ disjoint_left_comm.2 <|
-      tendsto_comap.disjoint (disjoint_cobounded_nhds _) (hf x hx)
-  rcases ((((hasBasis_nhdsSet _).inf_principal _)).disjoint_iff ((basis_sets _).comap _)).1 this
-    with ⟨U, ⟨hUo, hkU⟩, t, ht, hd⟩
-  refine ⟨U, hkU, hUo, (isBounded_compl_iff.2 ht).subset ?_⟩
-  rwa [image_subset_iff, preimage_compl, subset_compl_iff_disjoint_right]
-#align metric.exists_is_open_bounded_image_inter_of_is_compact_of_forall_continuous_within_at Metric.exists_isOpen_isBounded_image_inter_of_isCompact_of_forall_continuousWithinAt
-
-/-- If a function is continuous at every point of a compact set `k`, then it is bounded on
-some open neighborhood of `k`. -/
-theorem exists_isOpen_isBounded_image_of_isCompact_of_forall_continuousAt [TopologicalSpace β]
-    {k : Set β} {f : β → α} (hk : IsCompact k) (hf : ∀ x ∈ k, ContinuousAt f x) :
-    ∃ t, k ⊆ t ∧ IsOpen t ∧ IsBounded (f '' t) := by
-  simp_rw [← continuousWithinAt_univ] at hf
-  simpa only [inter_univ] using
-    exists_isOpen_isBounded_image_inter_of_isCompact_of_forall_continuousWithinAt hk hf
-#align metric.exists_is_open_bounded_image_of_is_compact_of_forall_continuous_at Metric.exists_isOpen_isBounded_image_of_isCompact_of_forall_continuousAt
-
-/-- If a function is continuous on a set `s` containing a compact set `k`, then it is bounded on
-some open neighborhood of `k` in `s`. -/
-theorem exists_isOpen_isBounded_image_inter_of_isCompact_of_continuousOn [TopologicalSpace β]
-    {k s : Set β} {f : β → α} (hk : IsCompact k) (hks : k ⊆ s) (hf : ContinuousOn f s) :
-    ∃ t, k ⊆ t ∧ IsOpen t ∧ IsBounded (f '' (t ∩ s)) :=
-  exists_isOpen_isBounded_image_inter_of_isCompact_of_forall_continuousWithinAt hk fun x hx =>
-    hf x (hks hx)
-#align metric.exists_is_open_bounded_image_inter_of_is_compact_of_continuous_on Metric.exists_isOpen_isBounded_image_inter_of_isCompact_of_continuousOn
-
-/-- If a function is continuous on a neighborhood of a compact set `k`, then it is bounded on
-some open neighborhood of `k`. -/
-theorem exists_isOpen_isBounded_image_of_isCompact_of_continuousOn [TopologicalSpace β]
-    {k s : Set β} {f : β → α} (hk : IsCompact k) (hs : IsOpen s) (hks : k ⊆ s)
-    (hf : ContinuousOn f s) : ∃ t, k ⊆ t ∧ IsOpen t ∧ IsBounded (f '' t) :=
-  exists_isOpen_isBounded_image_of_isCompact_of_forall_continuousAt hk fun _x hx =>
-    hf.continuousAt (hs.mem_nhds (hks hx))
-#align metric.exists_is_open_bounded_image_of_is_compact_of_continuous_on Metric.exists_isOpen_isBounded_image_of_isCompact_of_continuousOn
-
-/-- The **Heine–Borel theorem**: In a proper space, a closed bounded set is compact. -/
-theorem isCompact_of_isClosed_isBounded [ProperSpace α] (hc : IsClosed s) (hb : IsBounded s) :
-    IsCompact s := by
-  rcases eq_empty_or_nonempty s with (rfl | ⟨x, -⟩)
-  · exact isCompact_empty
-  · rcases hb.subset_closedBall x with ⟨r, hr⟩
-    exact (isCompact_closedBall x r).of_isClosed_subset hc hr
-#align metric.is_compact_of_is_closed_bounded Metric.isCompact_of_isClosed_isBounded
-
-/-- The **Heine–Borel theorem**: In a proper space, the closure of a bounded set is compact. -/
-theorem _root_.Bornology.IsBounded.isCompact_closure [ProperSpace α] (h : IsBounded s) :
-    IsCompact (closure s) :=
-  isCompact_of_isClosed_isBounded isClosed_closure h.closure
-#align metric.bounded.is_compact_closure Bornology.IsBounded.isCompact_closure
-
--- porting note: todo: assume `[MetricSpace α]` instead of `[PseudoMetricSpace α] [T2Space α]`
-/-- The **Heine–Borel theorem**:
-In a proper Hausdorff space, a set is compact if and only if it is closed and bounded. -/
-theorem isCompact_iff_isClosed_bounded [T2Space α] [ProperSpace α] :
-    IsCompact s ↔ IsClosed s ∧ IsBounded s :=
-  ⟨fun h => ⟨h.isClosed, h.isBounded⟩, fun h => isCompact_of_isClosed_isBounded h.1 h.2⟩
-#align metric.is_compact_iff_is_closed_bounded Metric.isCompact_iff_isClosed_bounded
-
-theorem compactSpace_iff_isBounded_univ [ProperSpace α] :
-    CompactSpace α ↔ IsBounded (univ : Set α) :=
-  ⟨@isBounded_of_compactSpace α _ _, fun hb => ⟨isCompact_of_isClosed_isBounded isClosed_univ hb⟩⟩
-#align metric.compact_space_iff_bounded_univ Metric.compactSpace_iff_isBounded_univ
-
-section ConditionallyCompleteLinearOrder
-
-variable [Preorder α] [CompactIccSpace α]
-
-theorem isBounded_Icc (a b : α) : IsBounded (Icc a b) :=
-  (totallyBounded_Icc a b).isBounded
-#align metric.bounded_Icc Metric.isBounded_Icc
-
-theorem isBounded_Ico (a b : α) : IsBounded (Ico a b) :=
-  (totallyBounded_Ico a b).isBounded
-#align metric.bounded_Ico Metric.isBounded_Ico
-
-theorem isBounded_Ioc (a b : α) : IsBounded (Ioc a b) :=
-  (totallyBounded_Ioc a b).isBounded
-#align metric.bounded_Ioc Metric.isBounded_Ioc
-
-theorem isBounded_Ioo (a b : α) : IsBounded (Ioo a b) :=
-  (totallyBounded_Ioo a b).isBounded
-#align metric.bounded_Ioo Metric.isBounded_Ioo
-
-/-- In a pseudo metric space with a conditionally complete linear order such that the order and the
-    metric structure give the same topology, any order-bounded set is metric-bounded. -/
-theorem isBounded_of_bddAbove_of_bddBelow {s : Set α} (h₁ : BddAbove s) (h₂ : BddBelow s) :
-    IsBounded s :=
-  let ⟨u, hu⟩ := h₁
-  let ⟨l, hl⟩ := h₂
-  (isBounded_Icc l u).subset (fun _x hx => mem_Icc.mpr ⟨hl hx, hu hx⟩)
-#align metric.bounded_of_bdd_above_of_bdd_below Metric.isBounded_of_bddAbove_of_bddBelow
-
-end ConditionallyCompleteLinearOrder
-
-end Bounded
-
-section Diam
-
-variable {s : Set α} {x y z : α}
-
-/-- The diameter of a set in a metric space. To get controllable behavior even when the diameter
-should be infinite, we express it in terms of the emetric.diameter -/
-noncomputable def diam (s : Set α) : ℝ :=
-  ENNReal.toReal (EMetric.diam s)
-#align metric.diam Metric.diam
-
-/-- The diameter of a set is always nonnegative -/
-theorem diam_nonneg : 0 ≤ diam s :=
-  ENNReal.toReal_nonneg
-#align metric.diam_nonneg Metric.diam_nonneg
-
-theorem diam_subsingleton (hs : s.Subsingleton) : diam s = 0 := by
-  simp only [diam, EMetric.diam_subsingleton hs, ENNReal.zero_toReal]
-#align metric.diam_subsingleton Metric.diam_subsingleton
-
-/-- The empty set has zero diameter -/
-@[simp]
-theorem diam_empty : diam (∅ : Set α) = 0 :=
-  diam_subsingleton subsingleton_empty
-#align metric.diam_empty Metric.diam_empty
-
-/-- A singleton has zero diameter -/
-@[simp]
-theorem diam_singleton : diam ({x} : Set α) = 0 :=
-  diam_subsingleton subsingleton_singleton
-#align metric.diam_singleton Metric.diam_singleton
-
--- Does not work as a simp-lemma, since {x, y} reduces to (insert y {x})
-theorem diam_pair : diam ({x, y} : Set α) = dist x y := by
-  simp only [diam, EMetric.diam_pair, dist_edist]
-#align metric.diam_pair Metric.diam_pair
-
--- Does not work as a simp-lemma, since {x, y, z} reduces to (insert z (insert y {x}))
-theorem diam_triple :
-    Metric.diam ({x, y, z} : Set α) = max (max (dist x y) (dist x z)) (dist y z) := by
-  simp only [Metric.diam, EMetric.diam_triple, dist_edist]
-  rw [ENNReal.toReal_max, ENNReal.toReal_max] <;> apply_rules [ne_of_lt, edist_lt_top, max_lt]
-#align metric.diam_triple Metric.diam_triple
-
-/-- If the distance between any two points in a set is bounded by some constant `C`,
-then `ENNReal.ofReal C` bounds the emetric diameter of this set. -/
-theorem ediam_le_of_forall_dist_le {C : ℝ} (h : ∀ x ∈ s, ∀ y ∈ s, dist x y ≤ C) :
-    EMetric.diam s ≤ ENNReal.ofReal C :=
-  EMetric.diam_le fun x hx y hy => (edist_dist x y).symm ▸ ENNReal.ofReal_le_ofReal (h x hx y hy)
-#align metric.ediam_le_of_forall_dist_le Metric.ediam_le_of_forall_dist_le
-
-/-- If the distance between any two points in a set is bounded by some non-negative constant,
-this constant bounds the diameter. -/
-theorem diam_le_of_forall_dist_le {C : ℝ} (h₀ : 0 ≤ C) (h : ∀ x ∈ s, ∀ y ∈ s, dist x y ≤ C) :
-    diam s ≤ C :=
-  ENNReal.toReal_le_of_le_ofReal h₀ (ediam_le_of_forall_dist_le h)
-#align metric.diam_le_of_forall_dist_le Metric.diam_le_of_forall_dist_le
-
-/-- If the distance between any two points in a nonempty set is bounded by some constant,
-this constant bounds the diameter. -/
-theorem diam_le_of_forall_dist_le_of_nonempty (hs : s.Nonempty) {C : ℝ}
-    (h : ∀ x ∈ s, ∀ y ∈ s, dist x y ≤ C) : diam s ≤ C :=
-  have h₀ : 0 ≤ C :=
-    let ⟨x, hx⟩ := hs
-    le_trans dist_nonneg (h x hx x hx)
-  diam_le_of_forall_dist_le h₀ h
-#align metric.diam_le_of_forall_dist_le_of_nonempty Metric.diam_le_of_forall_dist_le_of_nonempty
-
-/-- The distance between two points in a set is controlled by the diameter of the set. -/
-theorem dist_le_diam_of_mem' (h : EMetric.diam s ≠ ⊤) (hx : x ∈ s) (hy : y ∈ s) :
-    dist x y ≤ diam s := by
-  rw [diam, dist_edist]
-  rw [ENNReal.toReal_le_toReal (edist_ne_top _ _) h]
-  exact EMetric.edist_le_diam_of_mem hx hy
-#align metric.dist_le_diam_of_mem' Metric.dist_le_diam_of_mem'
-
-/-- Characterize the boundedness of a set in terms of the finiteness of its emetric.diameter. -/
-theorem isBounded_iff_ediam_ne_top : IsBounded s ↔ EMetric.diam s ≠ ⊤ :=
-  isBounded_iff.trans <| Iff.intro
-    (fun ⟨_C, hC⟩ => ne_top_of_le_ne_top ENNReal.ofReal_ne_top <| ediam_le_of_forall_dist_le hC)
-    fun h => ⟨diam s, fun _x hx _y hy => dist_le_diam_of_mem' h hx hy⟩
-#align metric.bounded_iff_ediam_ne_top Metric.isBounded_iff_ediam_ne_top
-
-alias ⟨_root_.Bornology.IsBounded.ediam_ne_top, _⟩ := isBounded_iff_ediam_ne_top
-#align metric.bounded.ediam_ne_top Bornology.IsBounded.ediam_ne_top
-
-theorem ediam_eq_top_iff_unbounded : EMetric.diam s = ⊤ ↔ ¬IsBounded s :=
-  isBounded_iff_ediam_ne_top.not_left.symm
-
-theorem ediam_univ_eq_top_iff_noncompact [ProperSpace α] :
-    EMetric.diam (univ : Set α) = ∞ ↔ NoncompactSpace α := by
-  rw [← not_compactSpace_iff, compactSpace_iff_isBounded_univ, isBounded_iff_ediam_ne_top,
-    Classical.not_not]
-#align metric.ediam_univ_eq_top_iff_noncompact Metric.ediam_univ_eq_top_iff_noncompact
-
-@[simp]
-theorem ediam_univ_of_noncompact [ProperSpace α] [NoncompactSpace α] :
-    EMetric.diam (univ : Set α) = ∞ :=
-  ediam_univ_eq_top_iff_noncompact.mpr ‹_›
-#align metric.ediam_univ_of_noncompact Metric.ediam_univ_of_noncompact
-
-@[simp]
-theorem diam_univ_of_noncompact [ProperSpace α] [NoncompactSpace α] : diam (univ : Set α) = 0 := by
-  simp [diam]
-#align metric.diam_univ_of_noncompact Metric.diam_univ_of_noncompact
-
-/-- The distance between two points in a set is controlled by the diameter of the set. -/
-theorem dist_le_diam_of_mem (h : IsBounded s) (hx : x ∈ s) (hy : y ∈ s) : dist x y ≤ diam s :=
-  dist_le_diam_of_mem' h.ediam_ne_top hx hy
-#align metric.dist_le_diam_of_mem Metric.dist_le_diam_of_mem
-
-theorem ediam_of_unbounded (h : ¬IsBounded s) : EMetric.diam s = ∞ := ediam_eq_top_iff_unbounded.2 h
-#align metric.ediam_of_unbounded Metric.ediam_of_unbounded
-
-/-- An unbounded set has zero diameter. If you would prefer to get the value ∞, use `EMetric.diam`.
-This lemma makes it possible to avoid side conditions in some situations -/
-theorem diam_eq_zero_of_unbounded (h : ¬IsBounded s) : diam s = 0 := by
-  rw [diam, ediam_of_unbounded h, ENNReal.top_toReal]
-#align metric.diam_eq_zero_of_unbounded Metric.diam_eq_zero_of_unbounded
-
-/-- If `s ⊆ t`, then the diameter of `s` is bounded by that of `t`, provided `t` is bounded. -/
-theorem diam_mono {s t : Set α} (h : s ⊆ t) (ht : IsBounded t) : diam s ≤ diam t :=
-  ENNReal.toReal_mono ht.ediam_ne_top <| EMetric.diam_mono h
-#align metric.diam_mono Metric.diam_mono
-
-/-- The diameter of a union is controlled by the sum of the diameters, and the distance between
-any two points in each of the sets. This lemma is true without any side condition, since it is
-obviously true if `s ∪ t` is unbounded. -/
-theorem diam_union {t : Set α} (xs : x ∈ s) (yt : y ∈ t) :
-    diam (s ∪ t) ≤ diam s + dist x y + diam t := by
-  simp only [diam, dist_edist]
-  refine (ENNReal.toReal_le_add' (EMetric.diam_union xs yt) ?_ ?_).trans
-    (add_le_add_right ENNReal.toReal_add_le _)
-  · simp only [ENNReal.add_eq_top, edist_ne_top, or_false]
-    exact fun h ↦ top_unique <| h ▸ EMetric.diam_mono (subset_union_left _ _)
-  · exact fun h ↦ top_unique <| h ▸ EMetric.diam_mono (subset_union_right _ _)
-#align metric.diam_union Metric.diam_union
-
-/-- If two sets intersect, the diameter of the union is bounded by the sum of the diameters. -/
-theorem diam_union' {t : Set α} (h : (s ∩ t).Nonempty) : diam (s ∪ t) ≤ diam s + diam t := by
-  rcases h with ⟨x, ⟨xs, xt⟩⟩
-  simpa using diam_union xs xt
-#align metric.diam_union' Metric.diam_union'
-
-theorem diam_le_of_subset_closedBall {r : ℝ} (hr : 0 ≤ r) (h : s ⊆ closedBall x r) :
-    diam s ≤ 2 * r :=
-  diam_le_of_forall_dist_le (mul_nonneg zero_le_two hr) fun a ha b hb =>
-    calc
-      dist a b ≤ dist a x + dist b x := dist_triangle_right _ _ _
-      _ ≤ r + r := (add_le_add (h ha) (h hb))
-      _ = 2 * r := by simp [mul_two, mul_comm]
-#align metric.diam_le_of_subset_closed_ball Metric.diam_le_of_subset_closedBall
-
-/-- The diameter of a closed ball of radius `r` is at most `2 r`. -/
-theorem diam_closedBall {r : ℝ} (h : 0 ≤ r) : diam (closedBall x r) ≤ 2 * r :=
-  diam_le_of_subset_closedBall h Subset.rfl
-#align metric.diam_closed_ball Metric.diam_closedBall
-
-/-- The diameter of a ball of radius `r` is at most `2 r`. -/
-theorem diam_ball {r : ℝ} (h : 0 ≤ r) : diam (ball x r) ≤ 2 * r :=
-  diam_le_of_subset_closedBall h ball_subset_closedBall
-#align metric.diam_ball Metric.diam_ball
-
-/-- If a family of complete sets with diameter tending to `0` is such that each finite intersection
-is nonempty, then the total intersection is also nonempty. -/
-theorem _root_.IsComplete.nonempty_iInter_of_nonempty_biInter {s : ℕ → Set α}
-    (h0 : IsComplete (s 0)) (hs : ∀ n, IsClosed (s n)) (h's : ∀ n, IsBounded (s n))
-    (h : ∀ N, (⋂ n ≤ N, s n).Nonempty) (h' : Tendsto (fun n => diam (s n)) atTop (𝓝 0)) :
-    (⋂ n, s n).Nonempty := by
-  let u N := (h N).some
-  have I : ∀ n N, n ≤ N → u N ∈ s n := by
-    intro n N hn
-    apply mem_of_subset_of_mem _ (h N).choose_spec
-    intro x hx
-    simp only [mem_iInter] at hx
-    exact hx n hn
-  have : CauchySeq u := by
-    apply cauchySeq_of_le_tendsto_0 _ _ h'
-    intro m n N hm hn
-    exact dist_le_diam_of_mem (h's N) (I _ _ hm) (I _ _ hn)
-  obtain ⟨x, -, xlim⟩ : ∃ x ∈ s 0, Tendsto (fun n : ℕ => u n) atTop (𝓝 x) :=
-    cauchySeq_tendsto_of_isComplete h0 (fun n => I 0 n (zero_le _)) this
-  refine' ⟨x, mem_iInter.2 fun n => _⟩
-  apply (hs n).mem_of_tendsto xlim
-  filter_upwards [Ici_mem_atTop n] with p hp
-  exact I n p hp
-#align is_complete.nonempty_Inter_of_nonempty_bInter IsComplete.nonempty_iInter_of_nonempty_biInter
-
-/-- In a complete space, if a family of closed sets with diameter tending to `0` is such that each
-finite intersection is nonempty, then the total intersection is also nonempty. -/
-theorem nonempty_iInter_of_nonempty_biInter [CompleteSpace α] {s : ℕ → Set α}
-    (hs : ∀ n, IsClosed (s n)) (h's : ∀ n, IsBounded (s n)) (h : ∀ N, (⋂ n ≤ N, s n).Nonempty)
-    (h' : Tendsto (fun n => diam (s n)) atTop (𝓝 0)) : (⋂ n, s n).Nonempty :=
-  (hs 0).isComplete.nonempty_iInter_of_nonempty_biInter hs h's h h'
-#align metric.nonempty_Inter_of_nonempty_bInter Metric.nonempty_iInter_of_nonempty_biInter
-
-end Diam
-
-theorem exists_isLocalMin_mem_ball [ProperSpace α] [TopologicalSpace β]
-    [ConditionallyCompleteLinearOrder β] [OrderTopology β] {f : α → β} {a z : α} {r : ℝ}
-    (hf : ContinuousOn f (closedBall a r)) (hz : z ∈ closedBall a r)
-    (hf1 : ∀ z' ∈ sphere a r, f z < f z') : ∃ z ∈ ball a r, IsLocalMin f z := by
-  simp_rw [← closedBall_diff_ball] at hf1
-  exact (isCompact_closedBall a r).exists_isLocalMin_mem_open ball_subset_closedBall hf hz hf1
-    isOpen_ball
-#align metric.exists_local_min_mem_ball Metric.exists_isLocalMin_mem_ball
-
-end Metric
-
-namespace Mathlib.Meta.Positivity
-
-open Lean Meta Qq Function
-
-/-- Extension for the `positivity` tactic: the diameter of a set is always nonnegative. -/
-@[positivity Metric.diam _]
-def evalDiam : PositivityExt where eval {_ _} _zα _pα e := do
-  let .app _ s ← whnfR e | throwError "not Metric.diam"
-  let p ← mkAppOptM ``Metric.diam_nonneg #[none, none, s]
-  pure (.nonnegative p)
-
-end Mathlib.Meta.Positivity
-
-theorem Metric.cobounded_eq_cocompact [ProperSpace α] : cobounded α = cocompact α := by
-  nontriviality α; inhabit α
-  exact cobounded_le_cocompact.antisymm <| (hasBasis_cobounded_compl_closedBall default).ge_iff.2
-    fun _ _ ↦ (isCompact_closedBall _ _).compl_mem_cocompact
-#align comap_dist_right_at_top_eq_cocompact Metric.cobounded_eq_cocompact
-
-theorem tendsto_dist_right_cocompact_atTop [ProperSpace α] (x : α) :
-    Tendsto (dist · x) (cocompact α) atTop :=
-  (tendsto_dist_right_cobounded_atTop x).mono_left cobounded_eq_cocompact.ge
-#align tendsto_dist_right_cocompact_at_top tendsto_dist_right_cocompact_atTop
-
-theorem tendsto_dist_left_cocompact_atTop [ProperSpace α] (x : α) :
-    Tendsto (dist x) (cocompact α) atTop :=
-  (tendsto_dist_left_cobounded_atTop x).mono_left cobounded_eq_cocompact.ge
-#align tendsto_dist_left_cocompact_at_top tendsto_dist_left_cocompact_atTop
-
-theorem comap_dist_left_atTop_eq_cocompact [ProperSpace α] (x : α) :
-    comap (dist x) atTop = cocompact α := by simp [cobounded_eq_cocompact]
-#align comap_dist_left_at_top_eq_cocompact comap_dist_left_atTop_eq_cocompact
-
-theorem tendsto_cocompact_of_tendsto_dist_comp_atTop {f : β → α} {l : Filter β} (x : α)
-    (h : Tendsto (fun y => dist (f y) x) l atTop) : Tendsto f l (cocompact α) :=
-  ((tendsto_dist_right_atTop_iff _).1 h).mono_right cobounded_le_cocompact
-#align tendsto_cocompact_of_tendsto_dist_comp_at_top tendsto_cocompact_of_tendsto_dist_comp_atTop
-
 /-- We now define `MetricSpace`, extending `PseudoMetricSpace`. -/
 class MetricSpace (α : Type u) extends PseudoMetricSpace α : Type u where
   eq_of_dist_eq_zero : ∀ {x y : α}, dist x y = 0 → x = y
@@ -3034,12 +178,6 @@ theorem uniformEmbedding_bot_of_pairwise_le_dist {β : Type*} {ε : ℝ} (hε :
   uniformEmbedding_of_spaced_out (dist_mem_uniformity hε) <| by simpa using hf
 #align metric.uniform_embedding_bot_of_pairwise_le_dist Metric.uniformEmbedding_bot_of_pairwise_le_dist
 
-theorem finite_isBounded_inter_isClosed [ProperSpace α] {K s : Set α} [DiscreteTopology s]
-    (hK : IsBounded K) (hs : IsClosed s) : Set.Finite (K ∩ s) := by
-  refine Set.Finite.subset (IsCompact.finite ?_ ?_) (Set.inter_subset_inter_left s subset_closure)
-  · exact hK.isCompact_closure.inter_right hs
-  · exact DiscreteTopology.of_subset inferInstance (Set.inter_subset_right _ s)
-
 end Metric
 
 /-- Build a new metric space from an old one where the bundled uniform structure is provably

From the Mandelbrot set boundedness project

Co-Authored-By: @girving

@@ -538,6 +538,9 @@ theorem ball_subset_closedBall : ball x ε ⊆ closedBall x ε := fun _y hy =>
 theorem sphere_subset_closedBall : sphere x ε ⊆ closedBall x ε := fun _ => le_of_eq
 #align metric.sphere_subset_closed_ball Metric.sphere_subset_closedBall
 
+lemma sphere_subset_ball {r R : ℝ} (h : r < R) : sphere x r ⊆ ball x R := fun _x hx ↦
+  (mem_sphere.1 hx).trans_lt h
+
 theorem closedBall_disjoint_ball (h : δ + ε ≤ dist x y) : Disjoint (closedBall x δ) (ball y ε) :=
   Set.disjoint_left.mpr fun _a ha1 ha2 =>
     (h.trans <| dist_triangle_left _ _ _).not_lt <| add_lt_add_of_le_of_lt ha1 ha2

mathport was forgetting a space in filter_upwards [...]with instead of filter_upwards [...] with.

@@ -2825,7 +2825,7 @@ theorem _root_.IsComplete.nonempty_iInter_of_nonempty_biInter {s : ℕ → Set 
     cauchySeq_tendsto_of_isComplete h0 (fun n => I 0 n (zero_le _)) this
   refine' ⟨x, mem_iInter.2 fun n => _⟩
   apply (hs n).mem_of_tendsto xlim
-  filter_upwards [Ici_mem_atTop n]with p hp
+  filter_upwards [Ici_mem_atTop n] with p hp
   exact I n p hp
 #align is_complete.nonempty_Inter_of_nonempty_bInter IsComplete.nonempty_iInter_of_nonempty_biInter
 

@@ -2224,17 +2224,6 @@ instance (priority := 100) secondCountable_of_proper [ProperSpace α] :
   · exact ⟨⟨fun _ => ∅, fun _ => isCompact_empty, iUnion_eq_univ_iff.2 fun x => (hn ⟨x⟩).elim⟩⟩
 #align second_countable_of_proper secondCountable_of_proper
 
-theorem tendsto_dist_right_cocompact_atTop [ProperSpace α] (x : α) :
-    Tendsto (fun y => dist y x) (cocompact α) atTop :=
-  (hasBasis_cocompact.tendsto_iff atTop_basis).2 fun r _ =>
-    ⟨closedBall x r, isCompact_closedBall x r, fun _y hy => (not_le.1 <| mt mem_closedBall.2 hy).le⟩
-#align tendsto_dist_right_cocompact_at_top tendsto_dist_right_cocompact_atTop
-
-theorem tendsto_dist_left_cocompact_atTop [ProperSpace α] (x : α) :
-    Tendsto (dist x) (cocompact α) atTop := by
-  simpa only [dist_comm] using tendsto_dist_right_cocompact_atTop x
-#align tendsto_dist_left_cocompact_at_top tendsto_dist_left_cocompact_atTop
-
 /-- If all closed balls of large enough radius are compact, then the space is proper. Especially
 useful when the lower bound for the radius is 0. -/
 theorem properSpace_of_compact_closedBall_of_le (R : ℝ)
@@ -2408,6 +2397,9 @@ theorem _root_.Bornology.IsBounded.subset_ball_lt (h : IsBounded s) (a : ℝ) (c
 theorem _root_.Bornology.IsBounded.subset_ball (h : IsBounded s) (c : α) : ∃ r, s ⊆ ball c r :=
   (h.subset_ball_lt 0 c).imp fun _ ↦ And.right
 
+theorem isBounded_iff_subset_ball (c : α) : IsBounded s ↔ ∃ r, s ⊆ ball c r :=
+  ⟨(IsBounded.subset_ball · c), fun ⟨_r, hr⟩ ↦ isBounded_ball.subset hr⟩
+
 theorem _root_.Bornology.IsBounded.subset_closedBall_lt (h : IsBounded s) (a : ℝ) (c : α) :
     ∃ r, a < r ∧ s ⊆ closedBall c r :=
   let ⟨r, har, hr⟩ := h.subset_ball_lt a c
@@ -2434,6 +2426,39 @@ theorem isBounded_closure_iff : IsBounded (closure s) ↔ IsBounded s :=
 #align metric.bounded_bUnion Bornology.isBounded_biUnion
 #align metric.bounded.prod Bornology.IsBounded.prod
 
+theorem hasBasis_cobounded_compl_closedBall (c : α) :
+    (cobounded α).HasBasis (fun _ ↦ True) (fun r ↦ (closedBall c r)ᶜ) :=
+  ⟨compl_surjective.forall.2 fun _ ↦ (isBounded_iff_subset_closedBall c).trans <| by simp⟩
+
+theorem hasBasis_cobounded_compl_ball (c : α) :
+    (cobounded α).HasBasis (fun _ ↦ True) (fun r ↦ (ball c r)ᶜ) :=
+  ⟨compl_surjective.forall.2 fun _ ↦ (isBounded_iff_subset_ball c).trans <| by simp⟩
+
+@[simp]
+theorem comap_dist_right_atTop (c : α) : comap (dist · c) atTop = cobounded α :=
+  (atTop_basis.comap _).eq_of_same_basis <| by
+    simpa only [compl_def, mem_ball, not_lt] using hasBasis_cobounded_compl_ball c
+
+@[simp]
+theorem comap_dist_left_atTop (c : α) : comap (dist c) atTop = cobounded α := by
+  simpa only [dist_comm _ c] using comap_dist_right_atTop c
+
+@[simp]
+theorem tendsto_dist_right_atTop_iff (c : α) {f : β → α} {l : Filter β} :
+    Tendsto (fun x ↦ dist (f x) c) l atTop ↔ Tendsto f l (cobounded α) := by
+  rw [← comap_dist_right_atTop c, tendsto_comap_iff]; rfl
+
+@[simp]
+theorem tendsto_dist_left_atTop_iff (c : α) {f : β → α} {l : Filter β} :
+    Tendsto (fun x ↦ dist c (f x)) l atTop ↔ Tendsto f l (cobounded α) := by
+  simp only [dist_comm c, tendsto_dist_right_atTop_iff]
+
+theorem tendsto_dist_right_cobounded_atTop (c : α) : Tendsto (dist · c) (cobounded α) atTop :=
+  tendsto_iff_comap.2 (comap_dist_right_atTop c).ge
+
+theorem tendsto_dist_left_cobounded_atTop (c : α) : Tendsto (dist c) (cobounded α) atTop :=
+  tendsto_iff_comap.2 (comap_dist_left_atTop c).ge
+
 /-- A totally bounded set is bounded -/
 theorem _root_.TotallyBounded.isBounded {s : Set α} (h : TotallyBounded s) : IsBounded s :=
   -- We cover the totally bounded set by finitely many balls of radius 1,
@@ -2452,6 +2477,11 @@ theorem _root_.IsCompact.isBounded {s : Set α} (h : IsCompact s) : IsBounded s
 #align set.finite.bounded Set.Finite.isBounded
 #align metric.bounded_singleton Bornology.isBounded_singleton
 
+theorem cobounded_le_cocompact : cobounded α ≤ cocompact α :=
+  hasBasis_cocompact.ge_iff.2 fun _s hs ↦ hs.isBounded
+#align comap_dist_right_at_top_le_cocompact Metric.cobounded_le_cocompactₓ
+#align comap_dist_left_at_top_le_cocompact Metric.cobounded_le_cocompactₓ
+
 /-- Characterization of the boundedness of the range of a function -/
 theorem isBounded_range_iff {f : β → α} : IsBounded (range f) ↔ ∃ C, ∀ x y, dist (f x) (f y) ≤ C :=
   isBounded_iff.trans <| by simp only [forall_range_iff]
@@ -2833,32 +2863,29 @@ def evalDiam : PositivityExt where eval {_ _} _zα _pα e := do
 
 end Mathlib.Meta.Positivity
 
-theorem comap_dist_right_atTop_le_cocompact (x : α) :
-    comap (fun y => dist y x) atTop ≤ cocompact α := by
-  refine' Filter.hasBasis_cocompact.ge_iff.2 fun s hs => mem_comap.2 _
-  rcases hs.isBounded.subset_closedBall x with ⟨r, hr⟩
-  exact ⟨Ioi r, Ioi_mem_atTop r, fun y hy hys => (mem_closedBall.1 <| hr hys).not_lt hy⟩
-#align comap_dist_right_at_top_le_cocompact comap_dist_right_atTop_le_cocompact
+theorem Metric.cobounded_eq_cocompact [ProperSpace α] : cobounded α = cocompact α := by
+  nontriviality α; inhabit α
+  exact cobounded_le_cocompact.antisymm <| (hasBasis_cobounded_compl_closedBall default).ge_iff.2
+    fun _ _ ↦ (isCompact_closedBall _ _).compl_mem_cocompact
+#align comap_dist_right_at_top_eq_cocompact Metric.cobounded_eq_cocompact
 
-theorem comap_dist_left_atTop_le_cocompact (x : α) : comap (dist x) atTop ≤ cocompact α := by
-  simpa only [dist_comm _ x] using comap_dist_right_atTop_le_cocompact x
-#align comap_dist_left_at_top_le_cocompact comap_dist_left_atTop_le_cocompact
+theorem tendsto_dist_right_cocompact_atTop [ProperSpace α] (x : α) :
+    Tendsto (dist · x) (cocompact α) atTop :=
+  (tendsto_dist_right_cobounded_atTop x).mono_left cobounded_eq_cocompact.ge
+#align tendsto_dist_right_cocompact_at_top tendsto_dist_right_cocompact_atTop
 
-theorem comap_dist_right_atTop_eq_cocompact [ProperSpace α] (x : α) :
-    comap (fun y => dist y x) atTop = cocompact α :=
-  (comap_dist_right_atTop_le_cocompact x).antisymm <|
-    (tendsto_dist_right_cocompact_atTop x).le_comap
-#align comap_dist_right_at_top_eq_cocompact comap_dist_right_atTop_eq_cocompact
+theorem tendsto_dist_left_cocompact_atTop [ProperSpace α] (x : α) :
+    Tendsto (dist x) (cocompact α) atTop :=
+  (tendsto_dist_left_cobounded_atTop x).mono_left cobounded_eq_cocompact.ge
+#align tendsto_dist_left_cocompact_at_top tendsto_dist_left_cocompact_atTop
 
 theorem comap_dist_left_atTop_eq_cocompact [ProperSpace α] (x : α) :
-    comap (dist x) atTop = cocompact α :=
-  (comap_dist_left_atTop_le_cocompact x).antisymm <| (tendsto_dist_left_cocompact_atTop x).le_comap
+    comap (dist x) atTop = cocompact α := by simp [cobounded_eq_cocompact]
 #align comap_dist_left_at_top_eq_cocompact comap_dist_left_atTop_eq_cocompact
 
 theorem tendsto_cocompact_of_tendsto_dist_comp_atTop {f : β → α} {l : Filter β} (x : α)
-    (h : Tendsto (fun y => dist (f y) x) l atTop) : Tendsto f l (cocompact α) := by
-  refine' Tendsto.mono_right _ (comap_dist_right_atTop_le_cocompact x)
-  rwa [tendsto_comap_iff]
+    (h : Tendsto (fun y => dist (f y) x) l atTop) : Tendsto f l (cocompact α) :=
+  ((tendsto_dist_right_atTop_iff _).1 h).mono_right cobounded_le_cocompact
 #align tendsto_cocompact_of_tendsto_dist_comp_at_top tendsto_cocompact_of_tendsto_dist_comp_atTop
 
 /-- We now define `MetricSpace`, extending `PseudoMetricSpace`. -/

Rename union_eq_left_iff_subset to union_eq_left to match sup_eq_left. Similarly for the right and inter versions.

@@ -2139,7 +2139,7 @@ theorem sphere_pi (x : ∀ b, π b) {r : ℝ} (h : 0 < r ∨ Nonempty β) :
     sphere x r = (⋃ i : β, Function.eval i ⁻¹' sphere (x i) r) ∩ closedBall x r := by
   obtain hr | rfl | hr := lt_trichotomy r 0
   · simp [hr]
-  · rw [closedBall_eq_sphere_of_nonpos le_rfl, eq_comm, Set.inter_eq_right_iff_subset]
+  · rw [closedBall_eq_sphere_of_nonpos le_rfl, eq_comm, Set.inter_eq_right]
     letI := h.resolve_left (lt_irrefl _)
     inhabit β
     refine' subset_iUnion_of_subset default _

As discussed on Zulip.

Co-authored-by: grunweg <grunweg@posteo.de>

@@ -1953,7 +1953,7 @@ lemma eventually_isCompact_closedBall [LocallyCompactSpace α] (x : α) :
     ∀ᶠ r in 𝓝 (0 : ℝ), IsCompact (closedBall x r) := by
   rcases local_compact_nhds (x := x) (n := univ) univ_mem with ⟨s, hs, -, s_compact⟩
   filter_upwards [eventually_closedBall_subset hs] with r hr
-  exact isCompact_of_isClosed_subset s_compact isClosed_ball hr
+  exact IsCompact.of_isClosed_subset s_compact isClosed_ball hr
 
 lemma exists_isCompact_closedBall [LocallyCompactSpace α] (x : α) :
     ∃ r, 0 < r ∧ IsCompact (closedBall x r) := by
@@ -2204,7 +2204,7 @@ export ProperSpace (isCompact_closedBall)
 /-- In a proper pseudometric space, all spheres are compact. -/
 theorem isCompact_sphere {α : Type*} [PseudoMetricSpace α] [ProperSpace α] (x : α) (r : ℝ) :
     IsCompact (sphere x r) :=
-  isCompact_of_isClosed_subset (isCompact_closedBall x r) isClosed_sphere sphere_subset_closedBall
+  (isCompact_closedBall x r).of_isClosed_subset isClosed_sphere sphere_subset_closedBall
 #align is_compact_sphere isCompact_sphere
 
 /-- In a proper pseudometric space, any sphere is a `CompactSpace` when considered as a subtype. -/
@@ -2239,7 +2239,7 @@ theorem tendsto_dist_left_cocompact_atTop [ProperSpace α] (x : α) :
 useful when the lower bound for the radius is 0. -/
 theorem properSpace_of_compact_closedBall_of_le (R : ℝ)
     (h : ∀ x : α, ∀ r, R ≤ r → IsCompact (closedBall x r)) : ProperSpace α :=
-  ⟨fun x r => isCompact_of_isClosed_subset (h x (max r R) (le_max_right _ _)) isClosed_ball
+  ⟨fun x r => IsCompact.of_isClosed_subset (h x (max r R) (le_max_right _ _)) isClosed_ball
     (closedBall_subset_closedBall <| le_max_left _ _)⟩
 #align proper_space_of_compact_closed_ball_of_le properSpace_of_compact_closedBall_of_le
 
@@ -2298,8 +2298,7 @@ theorem exists_pos_lt_subset_ball (hr : 0 < r) (hs : IsClosed s) (h : s ⊆ ball
   rcases eq_empty_or_nonempty s with (rfl | hne)
   · exact ⟨r / 2, ⟨half_pos hr, half_lt_self hr⟩, empty_subset _⟩
   have : IsCompact s :=
-    isCompact_of_isClosed_subset (isCompact_closedBall x r) hs
-      (h.trans ball_subset_closedBall)
+    (isCompact_closedBall x r).of_isClosed_subset hs (h.trans ball_subset_closedBall)
   obtain ⟨y, hys, hy⟩ : ∃ y ∈ s, s ⊆ closedBall x (dist y x) :=
     this.exists_forall_ge hne (continuous_id.dist continuous_const).continuousOn
   have hyr : dist y x < r := h hys
@@ -2558,7 +2557,7 @@ theorem isCompact_of_isClosed_isBounded [ProperSpace α] (hc : IsClosed s) (hb :
   rcases eq_empty_or_nonempty s with (rfl | ⟨x, -⟩)
   · exact isCompact_empty
   · rcases hb.subset_closedBall x with ⟨r, hr⟩
-    exact isCompact_of_isClosed_subset (isCompact_closedBall x r) hc hr
+    exact (isCompact_closedBall x r).of_isClosed_subset hc hr
 #align metric.is_compact_of_is_closed_bounded Metric.isCompact_of_isClosed_isBounded
 
 /-- The **Heine–Borel theorem**: In a proper space, the closure of a bounded set is compact. -/

Use Bornology.IsBounded instead.

@@ -899,8 +899,7 @@ theorem finite_approx_of_totallyBounded {s : Set α} (hs : TotallyBounded s) :
 #align metric.finite_approx_of_totally_bounded Metric.finite_approx_of_totallyBounded
 
 /-- Expressing uniform convergence using `dist` -/
-theorem tendstoUniformlyOnFilter_iff {ι : Type*} {F : ι → β → α} {f : β → α} {p : Filter ι}
-    {p' : Filter β} :
+theorem tendstoUniformlyOnFilter_iff {F : ι → β → α} {f : β → α} {p : Filter ι} {p' : Filter β} :
     TendstoUniformlyOnFilter F f p p' ↔
       ∀ ε > 0, ∀ᶠ n : ι × β in p ×ˢ p', dist (f n.snd) (F n.fst n.snd) < ε := by
   refine' ⟨fun H ε hε => H _ (dist_mem_uniformity hε), fun H u hu => _⟩
@@ -909,7 +908,7 @@ theorem tendstoUniformlyOnFilter_iff {ι : Type*} {F : ι → β → α} {f : β
 #align metric.tendsto_uniformly_on_filter_iff Metric.tendstoUniformlyOnFilter_iff
 
 /-- Expressing locally uniform convergence on a set using `dist`. -/
-theorem tendstoLocallyUniformlyOn_iff {ι : Type*} [TopologicalSpace β] {F : ι → β → α} {f : β → α}
+theorem tendstoLocallyUniformlyOn_iff [TopologicalSpace β] {F : ι → β → α} {f : β → α}
     {p : Filter ι} {s : Set β} :
     TendstoLocallyUniformlyOn F f p s ↔
       ∀ ε > 0, ∀ x ∈ s, ∃ t ∈ 𝓝[s] x, ∀ᶠ n in p, ∀ y ∈ t, dist (f y) (F n y) < ε := by
@@ -920,7 +919,7 @@ theorem tendstoLocallyUniformlyOn_iff {ι : Type*} [TopologicalSpace β] {F : ι
 #align metric.tendsto_locally_uniformly_on_iff Metric.tendstoLocallyUniformlyOn_iff
 
 /-- Expressing uniform convergence on a set using `dist`. -/
-theorem tendstoUniformlyOn_iff {ι : Type*} {F : ι → β → α} {f : β → α} {p : Filter ι} {s : Set β} :
+theorem tendstoUniformlyOn_iff {F : ι → β → α} {f : β → α} {p : Filter ι} {s : Set β} :
     TendstoUniformlyOn F f p s ↔ ∀ ε > 0, ∀ᶠ n in p, ∀ x ∈ s, dist (f x) (F n x) < ε := by
   refine' ⟨fun H ε hε => H _ (dist_mem_uniformity hε), fun H u hu => _⟩
   rcases mem_uniformity_dist.1 hu with ⟨ε, εpos, hε⟩
@@ -928,7 +927,7 @@ theorem tendstoUniformlyOn_iff {ι : Type*} {F : ι → β → α} {f : β → 
 #align metric.tendsto_uniformly_on_iff Metric.tendstoUniformlyOn_iff
 
 /-- Expressing locally uniform convergence using `dist`. -/
-theorem tendstoLocallyUniformly_iff {ι : Type*} [TopologicalSpace β] {F : ι → β → α} {f : β → α}
+theorem tendstoLocallyUniformly_iff [TopologicalSpace β] {F : ι → β → α} {f : β → α}
     {p : Filter ι} :
     TendstoLocallyUniformly F f p ↔
       ∀ ε > 0, ∀ x : β, ∃ t ∈ 𝓝 x, ∀ᶠ n in p, ∀ y ∈ t, dist (f y) (F n y) < ε := by
@@ -937,7 +936,7 @@ theorem tendstoLocallyUniformly_iff {ι : Type*} [TopologicalSpace β] {F : ι 
 #align metric.tendsto_locally_uniformly_iff Metric.tendstoLocallyUniformly_iff
 
 /-- Expressing uniform convergence using `dist`. -/
-theorem tendstoUniformly_iff {ι : Type*} {F : ι → β → α} {f : β → α} {p : Filter ι} :
+theorem tendstoUniformly_iff {F : ι → β → α} {f : β → α} {p : Filter ι} :
     TendstoUniformly F f p ↔ ∀ ε > 0, ∀ᶠ n in p, ∀ x, dist (f x) (F n x) < ε := by
   rw [← tendstoUniformlyOn_univ, tendstoUniformlyOn_iff]
   simp
@@ -968,8 +967,7 @@ theorem eventually_nhds_iff_ball {p : α → Prop} :
 
 /-- A version of `Filter.eventually_prod_iff` where the first filter consists of neighborhoods
 in a pseudo-metric space.-/
-theorem eventually_nhds_prod_iff {ι α} [PseudoMetricSpace α] {f : Filter ι} {x₀ : α}
-    {p : α × ι → Prop} :
+theorem eventually_nhds_prod_iff {f : Filter ι} {x₀ : α} {p : α × ι → Prop} :
     (∀ᶠ x in 𝓝 x₀ ×ˢ f, p x) ↔ ∃ ε > (0 : ℝ), ∃ pa : ι → Prop, (∀ᶠ i in f, pa i) ∧
       ∀ {x}, dist x x₀ < ε → ∀ {i}, pa i → p (x, i) := by
   refine (nhds_basis_ball.prod f.basis_sets).eventually_iff.trans ?_
@@ -1476,12 +1474,12 @@ theorem cauchySeq_iff_tendsto_dist_atTop_0 [Nonempty β] [SemilatticeSup β] {u
   rw [cauchySeq_iff_tendsto, Metric.uniformity_eq_comap_nhds_zero, tendsto_comap_iff]; rfl
 #align cauchy_seq_iff_tendsto_dist_at_top_0 cauchySeq_iff_tendsto_dist_atTop_0
 
-theorem tendsto_uniformity_iff_dist_tendsto_zero {ι : Type*} {f : ι → α × α} {p : Filter ι} :
+theorem tendsto_uniformity_iff_dist_tendsto_zero {f : ι → α × α} {p : Filter ι} :
     Tendsto f p (𝓤 α) ↔ Tendsto (fun x => dist (f x).1 (f x).2) p (𝓝 0) := by
   rw [Metric.uniformity_eq_comap_nhds_zero, tendsto_comap_iff]; rfl
 #align tendsto_uniformity_iff_dist_tendsto_zero tendsto_uniformity_iff_dist_tendsto_zero
 
-theorem Filter.Tendsto.congr_dist {ι : Type*} {f₁ f₂ : ι → α} {p : Filter ι} {a : α}
+theorem Filter.Tendsto.congr_dist {f₁ f₂ : ι → α} {p : Filter ι} {a : α}
     (h₁ : Tendsto f₁ p (𝓝 a)) (h : Tendsto (fun x => dist (f₁ x) (f₂ x)) p (𝓝 0)) :
     Tendsto f₂ p (𝓝 a) :=
   h₁.congr_uniformity <| tendsto_uniformity_iff_dist_tendsto_zero.2 h
@@ -1490,7 +1488,7 @@ theorem Filter.Tendsto.congr_dist {ι : Type*} {f₁ f₂ : ι → α} {p : Filt
 alias tendsto_of_tendsto_of_dist := Filter.Tendsto.congr_dist
 #align tendsto_of_tendsto_of_dist tendsto_of_tendsto_of_dist
 
-theorem tendsto_iff_of_dist {ι : Type*} {f₁ f₂ : ι → α} {p : Filter ι} {a : α}
+theorem tendsto_iff_of_dist {f₁ f₂ : ι → α} {p : Filter ι} {a : α}
     (h : Tendsto (fun x => dist (f₁ x) (f₂ x)) p (𝓝 0)) : Tendsto f₁ p (𝓝 a) ↔ Tendsto f₂ p (𝓝 a) :=
   Uniform.tendsto_congr <| tendsto_uniformity_iff_dist_tendsto_zero.2 h
 #align tendsto_iff_of_dist tendsto_iff_of_dist
@@ -2348,13 +2346,11 @@ end SecondCountable
 
 end Metric
 
-theorem lebesgue_number_lemma_of_metric {s : Set α} {ι} {c : ι → Set α} (hs : IsCompact s)
+theorem lebesgue_number_lemma_of_metric {s : Set α} {ι : Sort*} {c : ι → Set α} (hs : IsCompact s)
     (hc₁ : ∀ i, IsOpen (c i)) (hc₂ : s ⊆ ⋃ i, c i) : ∃ δ > 0, ∀ x ∈ s, ∃ i, ball x δ ⊆ c i :=
   let ⟨_n, en, hn⟩ := lebesgue_number_lemma hs hc₁ hc₂
   let ⟨δ, δ0, hδ⟩ := mem_uniformity_dist.1 en
-  ⟨δ, δ0, fun x hx =>
-    let ⟨i, hi⟩ := hn x hx
-    ⟨i, fun _y hy => hi (hδ (mem_ball'.mp hy))⟩⟩
+  ⟨δ, δ0, fun x hx => let ⟨i, hi⟩ := hn x hx; ⟨i, fun _y hy => hi (hδ (mem_ball'.mp hy))⟩⟩
 #align lebesgue_number_lemma_of_metric lebesgue_number_lemma_of_metric
 
 theorem lebesgue_number_lemma_of_metric_sUnion {s : Set α} {c : Set (Set α)} (hs : IsCompact s)
@@ -2364,283 +2360,254 @@ theorem lebesgue_number_lemma_of_metric_sUnion {s : Set α} {c : Set (Set α)} (
 
 namespace Metric
 
-/-- Boundedness of a subset of a pseudometric space. We formulate the definition to work
-even in the empty space. -/
-def Bounded (s : Set α) : Prop :=
-  ∃ C, ∀ (x) (_ : x ∈ s) (y) (_ : y ∈ s), dist x y ≤ C
-#align metric.bounded Metric.Bounded
+#align metric.bounded Bornology.IsBounded
 
 section Bounded
 
 variable {x : α} {s t : Set α} {r : ℝ}
 
-theorem bounded_iff_isBounded (s : Set α) : Bounded s ↔ IsBounded s := by
-  change Bounded s ↔ sᶜ ∈ (cobounded α).sets
-  simp [PseudoMetricSpace.cobounded_sets, Metric.Bounded]
-#align metric.bounded_iff_is_bounded Metric.bounded_iff_isBounded
+#noalign metric.bounded_iff_is_bounded
 
-@[simp] theorem bounded_empty : Bounded (∅ : Set α) := ⟨0, by simp⟩
-#align metric.bounded_empty Metric.bounded_empty
-
-theorem bounded_iff_mem_bounded : Bounded s ↔ ∀ x ∈ s, Bounded s :=
-  ⟨fun h _ _ => h, fun H =>
-    s.eq_empty_or_nonempty.elim (fun hs => hs.symm ▸ bounded_empty) fun ⟨x, hx⟩ => H x hx⟩
-#align metric.bounded_iff_mem_bounded Metric.bounded_iff_mem_bounded
-
-/-- Subsets of a bounded set are also bounded -/
-theorem Bounded.mono (incl : s ⊆ t) : Bounded t → Bounded s :=
-  Exists.imp fun _C hC x hx y hy => hC x (incl hx) y (incl hy)
-#align metric.bounded.mono Metric.Bounded.mono
+#align metric.bounded_empty Bornology.isBounded_empty
+#align metric.bounded_iff_mem_bounded Bornology.isBounded_iff_forall_mem
+#align metric.bounded.mono Bornology.IsBounded.subset
 
 /-- Closed balls are bounded -/
-theorem bounded_closedBall : Bounded (closedBall x r) :=
-  ⟨r + r, fun y hy z hz =>
+theorem isBounded_closedBall : IsBounded (closedBall x r) :=
+  isBounded_iff.2 ⟨r + r, fun y hy z hz =>
     calc dist y z ≤ dist y x + dist z x := dist_triangle_right _ _ _
     _ ≤ r + r := add_le_add hy hz⟩
-#align metric.bounded_closed_ball Metric.bounded_closedBall
+#align metric.bounded_closed_ball Metric.isBounded_closedBall
 
 /-- Open balls are bounded -/
-theorem bounded_ball : Bounded (ball x r) :=
-  bounded_closedBall.mono ball_subset_closedBall
-#align metric.bounded_ball Metric.bounded_ball
+theorem isBounded_ball : IsBounded (ball x r) :=
+  isBounded_closedBall.subset ball_subset_closedBall
+#align metric.bounded_ball Metric.isBounded_ball
 
 /-- Spheres are bounded -/
-theorem bounded_sphere : Bounded (sphere x r) :=
-  bounded_closedBall.mono sphere_subset_closedBall
-#align metric.bounded_sphere Metric.bounded_sphere
+theorem isBounded_sphere : IsBounded (sphere x r) :=
+  isBounded_closedBall.subset sphere_subset_closedBall
+#align metric.bounded_sphere Metric.isBounded_sphere
 
 /-- Given a point, a bounded subset is included in some ball around this point -/
-theorem bounded_iff_subset_ball (c : α) : Bounded s ↔ ∃ r, s ⊆ closedBall c r := by
-  constructor <;> rintro ⟨C, hC⟩
-  · rcases s.eq_empty_or_nonempty with (rfl | ⟨x, hx⟩)
-    · exact ⟨0, by simp⟩
-    · exact ⟨C + dist x c, fun y hy =>
-        calc dist y c ≤ dist y x + dist x c := dist_triangle _ _ _
-        _ ≤ C + dist x c := add_le_add_right (hC y hy x hx) _⟩
-  · exact bounded_closedBall.mono hC
-#align metric.bounded_iff_subset_ball Metric.bounded_iff_subset_ball
-
-theorem Bounded.subset_ball (h : Bounded s) (c : α) : ∃ r, s ⊆ closedBall c r :=
-  (bounded_iff_subset_ball c).1 h
-#align metric.bounded.subset_ball Metric.Bounded.subset_ball
-
-theorem Bounded.subset_ball_lt (h : Bounded s) (a : ℝ) (c : α) :
-    ∃ r, a < r ∧ s ⊆ closedBall c r := by
-  rcases h.subset_ball c with ⟨r, hr⟩
-  refine' ⟨max r (a + 1), lt_of_lt_of_le (by linarith) (le_max_right _ _), _⟩
-  exact hr.trans (closedBall_subset_closedBall (le_max_left _ _))
-#align metric.bounded.subset_ball_lt Metric.Bounded.subset_ball_lt
-
-theorem bounded_closure_of_bounded (h : Bounded s) : Bounded (closure s) :=
-  let ⟨C, h⟩ := h
-  ⟨C, fun _a ha _b hb => (isClosed_le' C).closure_subset <|
+theorem isBounded_iff_subset_closedBall (c : α) : IsBounded s ↔ ∃ r, s ⊆ closedBall c r :=
+  ⟨fun h ↦ (isBounded_iff.1 (h.insert c)).imp fun _r hr _x hx ↦ hr (.inr hx) (mem_insert _ _),
+    fun ⟨_r, hr⟩ ↦ isBounded_closedBall.subset hr⟩
+#align metric.bounded_iff_subset_ball Metric.isBounded_iff_subset_closedBall
+
+theorem _root_.Bornology.IsBounded.subset_closedBall (h : IsBounded s) (c : α) :
+    ∃ r, s ⊆ closedBall c r :=
+  (isBounded_iff_subset_closedBall c).1 h
+#align metric.bounded.subset_ball Bornology.IsBounded.subset_closedBall
+
+theorem _root_.Bornology.IsBounded.subset_ball_lt (h : IsBounded s) (a : ℝ) (c : α) :
+    ∃ r, a < r ∧ s ⊆ ball c r :=
+  let ⟨r, hr⟩ := h.subset_closedBall c
+  ⟨max r a + 1, (le_max_right _ _).trans_lt (lt_add_one _), hr.trans <| closedBall_subset_ball <|
+    (le_max_left _ _).trans_lt (lt_add_one _)⟩
+
+theorem _root_.Bornology.IsBounded.subset_ball (h : IsBounded s) (c : α) : ∃ r, s ⊆ ball c r :=
+  (h.subset_ball_lt 0 c).imp fun _ ↦ And.right
+
+theorem _root_.Bornology.IsBounded.subset_closedBall_lt (h : IsBounded s) (a : ℝ) (c : α) :
+    ∃ r, a < r ∧ s ⊆ closedBall c r :=
+  let ⟨r, har, hr⟩ := h.subset_ball_lt a c
+  ⟨r, har, hr.trans ball_subset_closedBall⟩
+#align metric.bounded.subset_ball_lt Bornology.IsBounded.subset_closedBall_lt
+
+theorem isBounded_closure_of_isBounded (h : IsBounded s) : IsBounded (closure s) :=
+  let ⟨C, h⟩ := isBounded_iff.1 h
+  isBounded_iff.2 ⟨C, fun _a ha _b hb => (isClosed_le' C).closure_subset <|
     map_mem_closure₂ continuous_dist ha hb h⟩
-#align metric.bounded_closure_of_bounded Metric.bounded_closure_of_bounded
+#align metric.bounded_closure_of_bounded Metric.isBounded_closure_of_isBounded
 
-protected theorem Bounded.closure (h : Bounded s) : Bounded (closure s) :=
-  bounded_closure_of_bounded h
-#align metric.bounded.closure Metric.Bounded.closure
+protected theorem _root_.Bornology.IsBounded.closure (h : IsBounded s) : IsBounded (closure s) :=
+  isBounded_closure_of_isBounded h
+#align metric.bounded.closure Bornology.IsBounded.closure
 
 @[simp]
-theorem bounded_closure_iff : Bounded (closure s) ↔ Bounded s :=
-  ⟨fun h => h.mono subset_closure, fun h => h.closure⟩
-#align metric.bounded_closure_iff Metric.bounded_closure_iff
-
-/-- The union of two sets is bounded iff each of the sets is bounded. -/
-@[simp] theorem bounded_union : Bounded (s ∪ t) ↔ Bounded s ∧ Bounded t := by
-  simp only [bounded_iff_isBounded, isBounded_union]
-#align metric.bounded_union Metric.bounded_union
-
-/-- The union of two bounded sets is bounded. -/
-theorem Bounded.union (hs : Bounded s) (ht : Bounded t) : Bounded (s ∪ t) :=
-  bounded_union.2 ⟨hs, ht⟩
-#align metric.bounded.union Metric.Bounded.union
-
-/-- A finite union of bounded sets is bounded -/
-theorem bounded_biUnion {I : Set β} {s : β → Set α} (H : I.Finite) :
-    Bounded (⋃ i ∈ I, s i) ↔ ∀ i ∈ I, Bounded (s i) := by
-  simp only [bounded_iff_isBounded, isBounded_biUnion H]
-#align metric.bounded_bUnion Metric.bounded_biUnion
-
-protected theorem Bounded.prod [PseudoMetricSpace β] {s : Set α} {t : Set β} (hs : Bounded s)
-    (ht : Bounded t) : Bounded (s ×ˢ t) := by
-  rw [bounded_iff_isBounded] at *
-  exact hs.prod ht
-#align metric.bounded.prod Metric.Bounded.prod
+theorem isBounded_closure_iff : IsBounded (closure s) ↔ IsBounded s :=
+  ⟨fun h => h.subset subset_closure, fun h => h.closure⟩
+#align metric.bounded_closure_iff Metric.isBounded_closure_iff
+
+#align metric.bounded_union Bornology.isBounded_union
+#align metric.bounded.union Bornology.IsBounded.union
+#align metric.bounded_bUnion Bornology.isBounded_biUnion
+#align metric.bounded.prod Bornology.IsBounded.prod
 
 /-- A totally bounded set is bounded -/
-theorem _root_.TotallyBounded.bounded {s : Set α} (h : TotallyBounded s) : Bounded s :=
+theorem _root_.TotallyBounded.isBounded {s : Set α} (h : TotallyBounded s) : IsBounded s :=
   -- We cover the totally bounded set by finitely many balls of radius 1,
   -- and then argue that a finite union of bounded sets is bounded
   let ⟨_t, fint, subs⟩ := (totallyBounded_iff.mp h) 1 zero_lt_one
-  Bounded.mono subs <| (bounded_biUnion fint).2 fun _ _ => bounded_ball
-#align totally_bounded.bounded TotallyBounded.bounded
+  ((isBounded_biUnion fint).2 fun _ _ => isBounded_ball).subset subs
+#align totally_bounded.bounded TotallyBounded.isBounded
 
 /-- A compact set is bounded -/
-theorem _root_.IsCompact.bounded {s : Set α} (h : IsCompact s) : Bounded s :=
+theorem _root_.IsCompact.isBounded {s : Set α} (h : IsCompact s) : IsBounded s :=
   -- A compact set is totally bounded, thus bounded
-  h.totallyBounded.bounded
-#align is_compact.bounded IsCompact.bounded
-
-/-- A finite set is bounded -/
-theorem bounded_of_finite {s : Set α} (h : s.Finite) : Bounded s :=
-  h.isCompact.bounded
-#align metric.bounded_of_finite Metric.bounded_of_finite
-
-alias _root_.Set.Finite.bounded := bounded_of_finite
-#align set.finite.bounded Set.Finite.bounded
+  h.totallyBounded.isBounded
+#align is_compact.bounded IsCompact.isBounded
 
-/-- A singleton is bounded -/
-theorem bounded_singleton {x : α} : Bounded ({x} : Set α) := (finite_singleton _).bounded
-#align metric.bounded_singleton Metric.bounded_singleton
+#align metric.bounded_of_finite Set.Finite.isBounded
+#align set.finite.bounded Set.Finite.isBounded
+#align metric.bounded_singleton Bornology.isBounded_singleton
 
 /-- Characterization of the boundedness of the range of a function -/
-theorem bounded_range_iff {f : β → α} : Bounded (range f) ↔ ∃ C, ∀ x y, dist (f x) (f y) ≤ C :=
-  exists_congr fun _ => forall_range_iff.trans <| forall_congr' fun _ => forall_range_iff
-#align metric.bounded_range_iff Metric.bounded_range_iff
+theorem isBounded_range_iff {f : β → α} : IsBounded (range f) ↔ ∃ C, ∀ x y, dist (f x) (f y) ≤ C :=
+  isBounded_iff.trans <| by simp only [forall_range_iff]
+#align metric.bounded_range_iff Metric.isBounded_range_iff
 
-theorem bounded_range_of_tendsto_cofinite_uniformity {f : β → α}
-    (hf : Tendsto (Prod.map f f) (.cofinite ×ˢ .cofinite) (𝓤 α)) : Bounded (range f) := by
+theorem isBounded_image_iff {f : β → α} {s : Set β} :
+    IsBounded (f '' s) ↔ ∃ C, ∀ x ∈ s, ∀ y ∈ s, dist (f x) (f y) ≤ C :=
+  isBounded_iff.trans <| by simp only [ball_image_iff]
+
+theorem isBounded_range_of_tendsto_cofinite_uniformity {f : β → α}
+    (hf : Tendsto (Prod.map f f) (.cofinite ×ˢ .cofinite) (𝓤 α)) : IsBounded (range f) := by
   rcases (hasBasis_cofinite.prod_self.tendsto_iff uniformity_basis_dist).1 hf 1 zero_lt_one with
     ⟨s, hsf, hs1⟩
-  rw [← image_univ, ← union_compl_self s, image_union, bounded_union]
-  use (hsf.image f).bounded, 1
-  rintro _ ⟨x, hx, rfl⟩ _ ⟨y, hy, rfl⟩
+  rw [← image_union_image_compl_eq_range]
+  refine (hsf.image f).isBounded.union (isBounded_image_iff.2 ⟨1, fun x hx y hy ↦ ?_⟩)
   exact le_of_lt (hs1 (x, y) ⟨hx, hy⟩)
-#align metric.bounded_range_of_tendsto_cofinite_uniformity Metric.bounded_range_of_tendsto_cofinite_uniformity
+#align metric.bounded_range_of_tendsto_cofinite_uniformity Metric.isBounded_range_of_tendsto_cofinite_uniformity
 
-theorem bounded_range_of_cauchy_map_cofinite {f : β → α} (hf : Cauchy (map f cofinite)) :
-    Bounded (range f) :=
-  bounded_range_of_tendsto_cofinite_uniformity <| (cauchy_map_iff.1 hf).2
-#align metric.bounded_range_of_cauchy_map_cofinite Metric.bounded_range_of_cauchy_map_cofinite
+theorem isBounded_range_of_cauchy_map_cofinite {f : β → α} (hf : Cauchy (map f cofinite)) :
+    IsBounded (range f) :=
+  isBounded_range_of_tendsto_cofinite_uniformity <| (cauchy_map_iff.1 hf).2
+#align metric.bounded_range_of_cauchy_map_cofinite Metric.isBounded_range_of_cauchy_map_cofinite
 
-theorem _root_.CauchySeq.bounded_range {f : ℕ → α} (hf : CauchySeq f) : Bounded (range f) :=
-  bounded_range_of_cauchy_map_cofinite <| by rwa [Nat.cofinite_eq_atTop]
-#align cauchy_seq.bounded_range CauchySeq.bounded_range
+theorem _root_.CauchySeq.isBounded_range {f : ℕ → α} (hf : CauchySeq f) : IsBounded (range f) :=
+  isBounded_range_of_cauchy_map_cofinite <| by rwa [Nat.cofinite_eq_atTop]
+#align cauchy_seq.bounded_range CauchySeq.isBounded_range
 
-theorem bounded_range_of_tendsto_cofinite {f : β → α} {a : α} (hf : Tendsto f cofinite (𝓝 a)) :
-    Bounded (range f) :=
-  bounded_range_of_tendsto_cofinite_uniformity <|
+theorem isBounded_range_of_tendsto_cofinite {f : β → α} {a : α} (hf : Tendsto f cofinite (𝓝 a)) :
+    IsBounded (range f) :=
+  isBounded_range_of_tendsto_cofinite_uniformity <|
     (hf.prod_map hf).mono_right <| nhds_prod_eq.symm.trans_le (nhds_le_uniformity a)
-#align metric.bounded_range_of_tendsto_cofinite Metric.bounded_range_of_tendsto_cofinite
+#align metric.bounded_range_of_tendsto_cofinite Metric.isBounded_range_of_tendsto_cofinite
 
 /-- In a compact space, all sets are bounded -/
-theorem bounded_of_compactSpace [CompactSpace α] : Bounded s :=
-  isCompact_univ.bounded.mono (subset_univ _)
-#align metric.bounded_of_compact_space Metric.bounded_of_compactSpace
+theorem isBounded_of_compactSpace [CompactSpace α] : IsBounded s :=
+  isCompact_univ.isBounded.subset (subset_univ _)
+#align metric.bounded_of_compact_space Metric.isBounded_of_compactSpace
+
+theorem isBounded_range_of_tendsto (u : ℕ → α) {x : α} (hu : Tendsto u atTop (𝓝 x)) :
+    IsBounded (range u) :=
+  hu.cauchySeq.isBounded_range
+#align metric.bounded_range_of_tendsto Metric.isBounded_range_of_tendsto
+
+theorem disjoint_nhds_cobounded (x : α) : Disjoint (𝓝 x) (cobounded α) :=
+  disjoint_of_disjoint_of_mem disjoint_compl_right (ball_mem_nhds _ one_pos) isBounded_ball
+
+theorem disjoint_cobounded_nhds (x : α) : Disjoint (cobounded α) (𝓝 x) :=
+  (disjoint_nhds_cobounded x).symm
+
+theorem disjoint_nhdsSet_cobounded {s : Set α} (hs : IsCompact s) : Disjoint (𝓝ˢ s) (cobounded α) :=
+  hs.disjoint_nhdsSet_left.2 fun _ _ ↦ disjoint_nhds_cobounded _
 
-theorem bounded_range_of_tendsto (u : ℕ → α) {x : α} (hu : Tendsto u atTop (𝓝 x)) :
-    Bounded (range u) :=
-  hu.cauchySeq.bounded_range
-#align metric.bounded_range_of_tendsto Metric.bounded_range_of_tendsto
+theorem disjoint_cobounded_nhdsSet {s : Set α} (hs : IsCompact s) : Disjoint (cobounded α) (𝓝ˢ s) :=
+  (disjoint_nhdsSet_cobounded hs).symm
 
 /-- If a function is continuous within a set `s` at every point of a compact set `k`, then it is
 bounded on some open neighborhood of `k` in `s`. -/
-theorem exists_isOpen_bounded_image_inter_of_isCompact_of_forall_continuousWithinAt
+theorem exists_isOpen_isBounded_image_inter_of_isCompact_of_forall_continuousWithinAt
     [TopologicalSpace β] {k s : Set β} {f : β → α} (hk : IsCompact k)
-    (hf : ∀ x ∈ k, ContinuousWithinAt f s x) : ∃ t, k ⊆ t ∧ IsOpen t ∧ Bounded (f '' (t ∩ s)) := by
-  refine hk.induction_on ?_ ?_ ?_ ?_
-  · exact ⟨∅, Subset.rfl, isOpen_empty, by simp only [image_empty, bounded_empty, empty_inter]⟩
-  · rintro s s' hss' ⟨t, s't, t_open, t_bounded⟩
-    exact ⟨t, hss'.trans s't, t_open, t_bounded⟩
-  · rintro s s' ⟨t, st, t_open, t_bounded⟩ ⟨t', s't', t'_open, t'_bounded⟩
-    refine' ⟨t ∪ t', union_subset_union st s't', t_open.union t'_open, _⟩
-    rw [union_inter_distrib_right, image_union]
-    exact t_bounded.union t'_bounded
-  · intro x hx
-    have A : ball (f x) 1 ∈ 𝓝 (f x) := ball_mem_nhds _ zero_lt_one
-    have B : f ⁻¹' ball (f x) 1 ∈ 𝓝[s] x := hf x hx A
-    obtain ⟨u, u_open, xu, uf⟩ : ∃ u : Set β, IsOpen u ∧ x ∈ u ∧ u ∩ s ⊆ f ⁻¹' ball (f x) 1
-    exact _root_.mem_nhdsWithin.1 B
-    refine' ⟨u, _, u, Subset.rfl, u_open, _⟩
-    · apply nhdsWithin_le_nhds
-      exact u_open.mem_nhds xu
-    · apply Bounded.mono (image_subset _ uf)
-      exact bounded_ball.mono (image_preimage_subset _ _)
-#align metric.exists_is_open_bounded_image_inter_of_is_compact_of_forall_continuous_within_at Metric.exists_isOpen_bounded_image_inter_of_isCompact_of_forall_continuousWithinAt
+    (hf : ∀ x ∈ k, ContinuousWithinAt f s x) :
+    ∃ t, k ⊆ t ∧ IsOpen t ∧ IsBounded (f '' (t ∩ s)) := by
+  have : Disjoint (𝓝ˢ k ⊓ 𝓟 s) (comap f (cobounded α)) := by
+    rw [disjoint_assoc, inf_comm, hk.disjoint_nhdsSet_left]
+    exact fun x hx ↦ disjoint_left_comm.2 <|
+      tendsto_comap.disjoint (disjoint_cobounded_nhds _) (hf x hx)
+  rcases ((((hasBasis_nhdsSet _).inf_principal _)).disjoint_iff ((basis_sets _).comap _)).1 this
+    with ⟨U, ⟨hUo, hkU⟩, t, ht, hd⟩
+  refine ⟨U, hkU, hUo, (isBounded_compl_iff.2 ht).subset ?_⟩
+  rwa [image_subset_iff, preimage_compl, subset_compl_iff_disjoint_right]
+#align metric.exists_is_open_bounded_image_inter_of_is_compact_of_forall_continuous_within_at Metric.exists_isOpen_isBounded_image_inter_of_isCompact_of_forall_continuousWithinAt
 
 /-- If a function is continuous at every point of a compact set `k`, then it is bounded on
 some open neighborhood of `k`. -/
-theorem exists_isOpen_bounded_image_of_isCompact_of_forall_continuousAt [TopologicalSpace β]
+theorem exists_isOpen_isBounded_image_of_isCompact_of_forall_continuousAt [TopologicalSpace β]
     {k : Set β} {f : β → α} (hk : IsCompact k) (hf : ∀ x ∈ k, ContinuousAt f x) :
-    ∃ t, k ⊆ t ∧ IsOpen t ∧ Bounded (f '' t) := by
+    ∃ t, k ⊆ t ∧ IsOpen t ∧ IsBounded (f '' t) := by
   simp_rw [← continuousWithinAt_univ] at hf
   simpa only [inter_univ] using
-    exists_isOpen_bounded_image_inter_of_isCompact_of_forall_continuousWithinAt hk hf
-#align metric.exists_is_open_bounded_image_of_is_compact_of_forall_continuous_at Metric.exists_isOpen_bounded_image_of_isCompact_of_forall_continuousAt
+    exists_isOpen_isBounded_image_inter_of_isCompact_of_forall_continuousWithinAt hk hf
+#align metric.exists_is_open_bounded_image_of_is_compact_of_forall_continuous_at Metric.exists_isOpen_isBounded_image_of_isCompact_of_forall_continuousAt
 
 /-- If a function is continuous on a set `s` containing a compact set `k`, then it is bounded on
 some open neighborhood of `k` in `s`. -/
-theorem exists_isOpen_bounded_image_inter_of_isCompact_of_continuousOn [TopologicalSpace β]
+theorem exists_isOpen_isBounded_image_inter_of_isCompact_of_continuousOn [TopologicalSpace β]
     {k s : Set β} {f : β → α} (hk : IsCompact k) (hks : k ⊆ s) (hf : ContinuousOn f s) :
-    ∃ t, k ⊆ t ∧ IsOpen t ∧ Bounded (f '' (t ∩ s)) :=
-  exists_isOpen_bounded_image_inter_of_isCompact_of_forall_continuousWithinAt hk fun x hx =>
+    ∃ t, k ⊆ t ∧ IsOpen t ∧ IsBounded (f '' (t ∩ s)) :=
+  exists_isOpen_isBounded_image_inter_of_isCompact_of_forall_continuousWithinAt hk fun x hx =>
     hf x (hks hx)
-#align metric.exists_is_open_bounded_image_inter_of_is_compact_of_continuous_on Metric.exists_isOpen_bounded_image_inter_of_isCompact_of_continuousOn
+#align metric.exists_is_open_bounded_image_inter_of_is_compact_of_continuous_on Metric.exists_isOpen_isBounded_image_inter_of_isCompact_of_continuousOn
 
 /-- If a function is continuous on a neighborhood of a compact set `k`, then it is bounded on
 some open neighborhood of `k`. -/
-theorem exists_isOpen_bounded_image_of_isCompact_of_continuousOn [TopologicalSpace β] {k s : Set β}
-    {f : β → α} (hk : IsCompact k) (hs : IsOpen s) (hks : k ⊆ s) (hf : ContinuousOn f s) :
-    ∃ t, k ⊆ t ∧ IsOpen t ∧ Bounded (f '' t) :=
-  exists_isOpen_bounded_image_of_isCompact_of_forall_continuousAt hk fun _x hx =>
+theorem exists_isOpen_isBounded_image_of_isCompact_of_continuousOn [TopologicalSpace β]
+    {k s : Set β} {f : β → α} (hk : IsCompact k) (hs : IsOpen s) (hks : k ⊆ s)
+    (hf : ContinuousOn f s) : ∃ t, k ⊆ t ∧ IsOpen t ∧ IsBounded (f '' t) :=
+  exists_isOpen_isBounded_image_of_isCompact_of_forall_continuousAt hk fun _x hx =>
     hf.continuousAt (hs.mem_nhds (hks hx))
-#align metric.exists_is_open_bounded_image_of_is_compact_of_continuous_on Metric.exists_isOpen_bounded_image_of_isCompact_of_continuousOn
+#align metric.exists_is_open_bounded_image_of_is_compact_of_continuous_on Metric.exists_isOpen_isBounded_image_of_isCompact_of_continuousOn
 
 /-- The **Heine–Borel theorem**: In a proper space, a closed bounded set is compact. -/
-theorem isCompact_of_isClosed_bounded [ProperSpace α] (hc : IsClosed s) (hb : Bounded s) :
+theorem isCompact_of_isClosed_isBounded [ProperSpace α] (hc : IsClosed s) (hb : IsBounded s) :
     IsCompact s := by
   rcases eq_empty_or_nonempty s with (rfl | ⟨x, -⟩)
   · exact isCompact_empty
-  · rcases hb.subset_ball x with ⟨r, hr⟩
+  · rcases hb.subset_closedBall x with ⟨r, hr⟩
     exact isCompact_of_isClosed_subset (isCompact_closedBall x r) hc hr
-#align metric.is_compact_of_is_closed_bounded Metric.isCompact_of_isClosed_bounded
+#align metric.is_compact_of_is_closed_bounded Metric.isCompact_of_isClosed_isBounded
 
 /-- The **Heine–Borel theorem**: In a proper space, the closure of a bounded set is compact. -/
-theorem Bounded.isCompact_closure [ProperSpace α] (h : Bounded s) : IsCompact (closure s) :=
-  isCompact_of_isClosed_bounded isClosed_closure h.closure
-#align metric.bounded.is_compact_closure Metric.Bounded.isCompact_closure
+theorem _root_.Bornology.IsBounded.isCompact_closure [ProperSpace α] (h : IsBounded s) :
+    IsCompact (closure s) :=
+  isCompact_of_isClosed_isBounded isClosed_closure h.closure
+#align metric.bounded.is_compact_closure Bornology.IsBounded.isCompact_closure
 
 -- porting note: todo: assume `[MetricSpace α]` instead of `[PseudoMetricSpace α] [T2Space α]`
 /-- The **Heine–Borel theorem**:
 In a proper Hausdorff space, a set is compact if and only if it is closed and bounded. -/
 theorem isCompact_iff_isClosed_bounded [T2Space α] [ProperSpace α] :
-    IsCompact s ↔ IsClosed s ∧ Bounded s :=
-  ⟨fun h => ⟨h.isClosed, h.bounded⟩, fun h => isCompact_of_isClosed_bounded h.1 h.2⟩
+    IsCompact s ↔ IsClosed s ∧ IsBounded s :=
+  ⟨fun h => ⟨h.isClosed, h.isBounded⟩, fun h => isCompact_of_isClosed_isBounded h.1 h.2⟩
 #align metric.is_compact_iff_is_closed_bounded Metric.isCompact_iff_isClosed_bounded
 
-theorem compactSpace_iff_bounded_univ [ProperSpace α] : CompactSpace α ↔ Bounded (univ : Set α) :=
-  ⟨@bounded_of_compactSpace α _ _, fun hb => ⟨isCompact_of_isClosed_bounded isClosed_univ hb⟩⟩
-#align metric.compact_space_iff_bounded_univ Metric.compactSpace_iff_bounded_univ
+theorem compactSpace_iff_isBounded_univ [ProperSpace α] :
+    CompactSpace α ↔ IsBounded (univ : Set α) :=
+  ⟨@isBounded_of_compactSpace α _ _, fun hb => ⟨isCompact_of_isClosed_isBounded isClosed_univ hb⟩⟩
+#align metric.compact_space_iff_bounded_univ Metric.compactSpace_iff_isBounded_univ
 
 section ConditionallyCompleteLinearOrder
 
 variable [Preorder α] [CompactIccSpace α]
 
-theorem bounded_Icc (a b : α) : Bounded (Icc a b) :=
-  (totallyBounded_Icc a b).bounded
-#align metric.bounded_Icc Metric.bounded_Icc
+theorem isBounded_Icc (a b : α) : IsBounded (Icc a b) :=
+  (totallyBounded_Icc a b).isBounded
+#align metric.bounded_Icc Metric.isBounded_Icc
 
-theorem bounded_Ico (a b : α) : Bounded (Ico a b) :=
-  (totallyBounded_Ico a b).bounded
-#align metric.bounded_Ico Metric.bounded_Ico
+theorem isBounded_Ico (a b : α) : IsBounded (Ico a b) :=
+  (totallyBounded_Ico a b).isBounded
+#align metric.bounded_Ico Metric.isBounded_Ico
 
-theorem bounded_Ioc (a b : α) : Bounded (Ioc a b) :=
-  (totallyBounded_Ioc a b).bounded
-#align metric.bounded_Ioc Metric.bounded_Ioc
+theorem isBounded_Ioc (a b : α) : IsBounded (Ioc a b) :=
+  (totallyBounded_Ioc a b).isBounded
+#align metric.bounded_Ioc Metric.isBounded_Ioc
 
-theorem bounded_Ioo (a b : α) : Bounded (Ioo a b) :=
-  (totallyBounded_Ioo a b).bounded
-#align metric.bounded_Ioo Metric.bounded_Ioo
+theorem isBounded_Ioo (a b : α) : IsBounded (Ioo a b) :=
+  (totallyBounded_Ioo a b).isBounded
+#align metric.bounded_Ioo Metric.isBounded_Ioo
 
 /-- In a pseudo metric space with a conditionally complete linear order such that the order and the
     metric structure give the same topology, any order-bounded set is metric-bounded. -/
-theorem bounded_of_bddAbove_of_bddBelow {s : Set α} (h₁ : BddAbove s) (h₂ : BddBelow s) :
-    Bounded s :=
+theorem isBounded_of_bddAbove_of_bddBelow {s : Set α} (h₁ : BddAbove s) (h₂ : BddBelow s) :
+    IsBounded s :=
   let ⟨u, hu⟩ := h₁
   let ⟨l, hl⟩ := h₂
-  Bounded.mono (fun _x hx => mem_Icc.mpr ⟨hl hx, hu hx⟩) (bounded_Icc l u)
-#align metric.bounded_of_bdd_above_of_bdd_below Metric.bounded_of_bddAbove_of_bddBelow
+  (isBounded_Icc l u).subset (fun _x hx => mem_Icc.mpr ⟨hl hx, hu hx⟩)
+#align metric.bounded_of_bdd_above_of_bdd_below Metric.isBounded_of_bddAbove_of_bddBelow
 
 end ConditionallyCompleteLinearOrder
 
@@ -2722,22 +2689,21 @@ theorem dist_le_diam_of_mem' (h : EMetric.diam s ≠ ⊤) (hx : x ∈ s) (hy : y
 #align metric.dist_le_diam_of_mem' Metric.dist_le_diam_of_mem'
 
 /-- Characterize the boundedness of a set in terms of the finiteness of its emetric.diameter. -/
-theorem bounded_iff_ediam_ne_top : Bounded s ↔ EMetric.diam s ≠ ⊤ :=
-  Iff.intro
+theorem isBounded_iff_ediam_ne_top : IsBounded s ↔ EMetric.diam s ≠ ⊤ :=
+  isBounded_iff.trans <| Iff.intro
     (fun ⟨_C, hC⟩ => ne_top_of_le_ne_top ENNReal.ofReal_ne_top <| ediam_le_of_forall_dist_le hC)
     fun h => ⟨diam s, fun _x hx _y hy => dist_le_diam_of_mem' h hx hy⟩
-#align metric.bounded_iff_ediam_ne_top Metric.bounded_iff_ediam_ne_top
+#align metric.bounded_iff_ediam_ne_top Metric.isBounded_iff_ediam_ne_top
 
-theorem Bounded.ediam_ne_top (h : Bounded s) : EMetric.diam s ≠ ⊤ :=
-  bounded_iff_ediam_ne_top.1 h
-#align metric.bounded.ediam_ne_top Metric.Bounded.ediam_ne_top
+alias ⟨_root_.Bornology.IsBounded.ediam_ne_top, _⟩ := isBounded_iff_ediam_ne_top
+#align metric.bounded.ediam_ne_top Bornology.IsBounded.ediam_ne_top
 
-theorem ediam_eq_top_iff_unbounded : EMetric.diam s = ⊤ ↔ ¬Bounded s :=
-  bounded_iff_ediam_ne_top.not_left.symm
+theorem ediam_eq_top_iff_unbounded : EMetric.diam s = ⊤ ↔ ¬IsBounded s :=
+  isBounded_iff_ediam_ne_top.not_left.symm
 
 theorem ediam_univ_eq_top_iff_noncompact [ProperSpace α] :
     EMetric.diam (univ : Set α) = ∞ ↔ NoncompactSpace α := by
-  rw [← not_compactSpace_iff, compactSpace_iff_bounded_univ, bounded_iff_ediam_ne_top,
+  rw [← not_compactSpace_iff, compactSpace_iff_isBounded_univ, isBounded_iff_ediam_ne_top,
     Classical.not_not]
 #align metric.ediam_univ_eq_top_iff_noncompact Metric.ediam_univ_eq_top_iff_noncompact
 
@@ -2753,21 +2719,21 @@ theorem diam_univ_of_noncompact [ProperSpace α] [NoncompactSpace α] : diam (un
 #align metric.diam_univ_of_noncompact Metric.diam_univ_of_noncompact
 
 /-- The distance between two points in a set is controlled by the diameter of the set. -/
-theorem dist_le_diam_of_mem (h : Bounded s) (hx : x ∈ s) (hy : y ∈ s) : dist x y ≤ diam s :=
+theorem dist_le_diam_of_mem (h : IsBounded s) (hx : x ∈ s) (hy : y ∈ s) : dist x y ≤ diam s :=
   dist_le_diam_of_mem' h.ediam_ne_top hx hy
 #align metric.dist_le_diam_of_mem Metric.dist_le_diam_of_mem
 
-theorem ediam_of_unbounded (h : ¬Bounded s) : EMetric.diam s = ∞ := ediam_eq_top_iff_unbounded.2 h
+theorem ediam_of_unbounded (h : ¬IsBounded s) : EMetric.diam s = ∞ := ediam_eq_top_iff_unbounded.2 h
 #align metric.ediam_of_unbounded Metric.ediam_of_unbounded
 
 /-- An unbounded set has zero diameter. If you would prefer to get the value ∞, use `EMetric.diam`.
 This lemma makes it possible to avoid side conditions in some situations -/
-theorem diam_eq_zero_of_unbounded (h : ¬Bounded s) : diam s = 0 := by
+theorem diam_eq_zero_of_unbounded (h : ¬IsBounded s) : diam s = 0 := by
   rw [diam, ediam_of_unbounded h, ENNReal.top_toReal]
 #align metric.diam_eq_zero_of_unbounded Metric.diam_eq_zero_of_unbounded
 
 /-- If `s ⊆ t`, then the diameter of `s` is bounded by that of `t`, provided `t` is bounded. -/
-theorem diam_mono {s t : Set α} (h : s ⊆ t) (ht : Bounded t) : diam s ≤ diam t :=
+theorem diam_mono {s t : Set α} (h : s ⊆ t) (ht : IsBounded t) : diam s ≤ diam t :=
   ENNReal.toReal_mono ht.ediam_ne_top <| EMetric.diam_mono h
 #align metric.diam_mono Metric.diam_mono
 
@@ -2812,7 +2778,7 @@ theorem diam_ball {r : ℝ} (h : 0 ≤ r) : diam (ball x r) ≤ 2 * r :=
 /-- If a family of complete sets with diameter tending to `0` is such that each finite intersection
 is nonempty, then the total intersection is also nonempty. -/
 theorem _root_.IsComplete.nonempty_iInter_of_nonempty_biInter {s : ℕ → Set α}
-    (h0 : IsComplete (s 0)) (hs : ∀ n, IsClosed (s n)) (h's : ∀ n, Bounded (s n))
+    (h0 : IsComplete (s 0)) (hs : ∀ n, IsClosed (s n)) (h's : ∀ n, IsBounded (s n))
     (h : ∀ N, (⋂ n ≤ N, s n).Nonempty) (h' : Tendsto (fun n => diam (s n)) atTop (𝓝 0)) :
     (⋂ n, s n).Nonempty := by
   let u N := (h N).some
@@ -2837,7 +2803,7 @@ theorem _root_.IsComplete.nonempty_iInter_of_nonempty_biInter {s : ℕ → Set 
 /-- In a complete space, if a family of closed sets with diameter tending to `0` is such that each
 finite intersection is nonempty, then the total intersection is also nonempty. -/
 theorem nonempty_iInter_of_nonempty_biInter [CompleteSpace α] {s : ℕ → Set α}
-    (hs : ∀ n, IsClosed (s n)) (h's : ∀ n, Bounded (s n)) (h : ∀ N, (⋂ n ≤ N, s n).Nonempty)
+    (hs : ∀ n, IsClosed (s n)) (h's : ∀ n, IsBounded (s n)) (h : ∀ N, (⋂ n ≤ N, s n).Nonempty)
     (h' : Tendsto (fun n => diam (s n)) atTop (𝓝 0)) : (⋂ n, s n).Nonempty :=
   (hs 0).isComplete.nonempty_iInter_of_nonempty_biInter hs h's h h'
 #align metric.nonempty_Inter_of_nonempty_bInter Metric.nonempty_iInter_of_nonempty_biInter
@@ -2871,7 +2837,7 @@ end Mathlib.Meta.Positivity
 theorem comap_dist_right_atTop_le_cocompact (x : α) :
     comap (fun y => dist y x) atTop ≤ cocompact α := by
   refine' Filter.hasBasis_cocompact.ge_iff.2 fun s hs => mem_comap.2 _
-  rcases hs.bounded.subset_ball x with ⟨r, hr⟩
+  rcases hs.isBounded.subset_closedBall x with ⟨r, hr⟩
   exact ⟨Ioi r, Ioi_mem_atTop r, fun y hy hys => (mem_closedBall.1 <| hr hys).not_lt hy⟩
 #align comap_dist_right_at_top_le_cocompact comap_dist_right_atTop_le_cocompact
 
@@ -3039,10 +3005,10 @@ theorem uniformEmbedding_bot_of_pairwise_le_dist {β : Type*} {ε : ℝ} (hε :
   uniformEmbedding_of_spaced_out (dist_mem_uniformity hε) <| by simpa using hf
 #align metric.uniform_embedding_bot_of_pairwise_le_dist Metric.uniformEmbedding_bot_of_pairwise_le_dist
 
-theorem Finite_bounded_inter_isClosed [ProperSpace α] {K s : Set α} [DiscreteTopology s]
-    (hK : Metric.Bounded K) (hs : IsClosed s) : Set.Finite (K ∩ s) := by
+theorem finite_isBounded_inter_isClosed [ProperSpace α] {K s : Set α} [DiscreteTopology s]
+    (hK : IsBounded K) (hs : IsClosed s) : Set.Finite (K ∩ s) := by
   refine Set.Finite.subset (IsCompact.finite ?_ ?_) (Set.inter_subset_inter_left s subset_closure)
-  · exact IsCompact.inter_right (Metric.Bounded.isCompact_closure hK) hs
+  · exact hK.isCompact_closure.inter_right hs
   · exact DiscreteTopology.of_subset inferInstance (Set.inter_subset_right _ s)
 
 end Metric

We reuse the main results that were proved for functions not depending on a parameter (but we need to tweak the definition of the set A to let it cover all cases, replacing a large inequality by a strict one).

@@ -1951,6 +1951,18 @@ theorem closedBall_zero' (x : α) : closedBall x 0 = closure {x} :=
     (closure_minimal (singleton_subset_iff.2 (dist_self x).le) isClosed_ball)
 #align metric.closed_ball_zero' Metric.closedBall_zero'
 
+lemma eventually_isCompact_closedBall [LocallyCompactSpace α] (x : α) :
+    ∀ᶠ r in 𝓝 (0 : ℝ), IsCompact (closedBall x r) := by
+  rcases local_compact_nhds (x := x) (n := univ) univ_mem with ⟨s, hs, -, s_compact⟩
+  filter_upwards [eventually_closedBall_subset hs] with r hr
+  exact isCompact_of_isClosed_subset s_compact isClosed_ball hr
+
+lemma exists_isCompact_closedBall [LocallyCompactSpace α] (x : α) :
+    ∃ r, 0 < r ∧ IsCompact (closedBall x r) := by
+  have : ∀ᶠ r in 𝓝[>] 0, IsCompact (closedBall x r) :=
+    eventually_nhdsWithin_of_eventually_nhds (eventually_isCompact_closedBall x)
+  simpa only [and_comm] using (this.and self_mem_nhdsWithin).exists
+
 theorem dense_iff {s : Set α} : Dense s ↔ ∀ x, ∀ r > 0, (ball x r ∩ s).Nonempty :=
   forall_congr' fun x => by
     simp only [mem_closure_iff, Set.Nonempty, exists_prop, mem_inter_iff, mem_ball', and_comm]

Later I'm going to split files like Lipschitz into 2: one in EMetricSpace/ and one in MetricSpace/.

@@ -5,7 +5,7 @@ Authors: Jeremy Avigad, Robert Y. Lewis, Johannes Hölzl, Mario Carneiro, Sébas
 -/
 import Mathlib.Tactic.Positivity
 import Mathlib.Topology.Algebra.Order.Compact
-import Mathlib.Topology.MetricSpace.EMetricSpace
+import Mathlib.Topology.EMetricSpace.Basic
 import Mathlib.Topology.Bornology.Constructions
 
 #align_import topology.metric_space.basic from "leanprover-community/mathlib"@"8047de4d911cdef39c2d646165eea972f7f9f539"

@@ -1487,7 +1487,7 @@ theorem Filter.Tendsto.congr_dist {ι : Type*} {f₁ f₂ : ι → α} {p : Filt
   h₁.congr_uniformity <| tendsto_uniformity_iff_dist_tendsto_zero.2 h
 #align filter.tendsto.congr_dist Filter.Tendsto.congr_dist
 
-alias Filter.Tendsto.congr_dist ← tendsto_of_tendsto_of_dist
+alias tendsto_of_tendsto_of_dist := Filter.Tendsto.congr_dist
 #align tendsto_of_tendsto_of_dist tendsto_of_tendsto_of_dist
 
 theorem tendsto_iff_of_dist {ι : Type*} {f₁ f₂ : ι → α} {p : Filter ι} {a : α}
@@ -2175,7 +2175,7 @@ theorem finite_cover_balls_of_compact {α : Type u} [PseudoMetricSpace α] {s :
   ⟨t, hts, t.finite_toSet, ht⟩
 #align finite_cover_balls_of_compact finite_cover_balls_of_compact
 
-alias finite_cover_balls_of_compact ← IsCompact.finite_cover_balls
+alias IsCompact.finite_cover_balls := finite_cover_balls_of_compact
 #align is_compact.finite_cover_balls IsCompact.finite_cover_balls
 
 end Compact
@@ -2475,7 +2475,7 @@ theorem bounded_of_finite {s : Set α} (h : s.Finite) : Bounded s :=
   h.isCompact.bounded
 #align metric.bounded_of_finite Metric.bounded_of_finite
 
-alias bounded_of_finite ← _root_.Set.Finite.bounded
+alias _root_.Set.Finite.bounded := bounded_of_finite
 #align set.finite.bounded Set.Finite.bounded
 
 /-- A singleton is bounded -/

@@ -143,7 +143,7 @@ theorem PseudoMetricSpace.ext {α : Type*} {m m' : PseudoMetricSpace α}
   congr
   · ext x y : 2
     rw [hed, hed']
-  · exact uniformSpace_eq (hU.trans hU'.symm)
+  · exact UniformSpace.ext (hU.trans hU'.symm)
   · ext : 2
     rw [← Filter.mem_sets, ← Filter.mem_sets, hB, hB']
 #align pseudo_metric_space.ext PseudoMetricSpace.ext
@@ -726,7 +726,7 @@ theorem isBounded_iff_nndist {s : Set α} :
 
 theorem toUniformSpace_eq :
     ‹PseudoMetricSpace α›.toUniformSpace = .ofDist dist dist_self dist_comm dist_triangle :=
-  uniformSpace_eq PseudoMetricSpace.uniformity_dist
+  UniformSpace.ext PseudoMetricSpace.uniformity_dist
 #align metric.to_uniform_space_eq Metric.toUniformSpace_eq
 
 theorem uniformity_basis_dist :

We prove the main characterisation of $\mathbb{Z}$-lattices: a subgroup $L$ of $\mathbb{R}^n$ that is discrete and that spans $\mathbb{R}^n$ is a free $\mathbb{Z}$-module of rank $n$ (in a slightly more general setting).

@@ -3027,6 +3027,12 @@ theorem uniformEmbedding_bot_of_pairwise_le_dist {β : Type*} {ε : ℝ} (hε :
   uniformEmbedding_of_spaced_out (dist_mem_uniformity hε) <| by simpa using hf
 #align metric.uniform_embedding_bot_of_pairwise_le_dist Metric.uniformEmbedding_bot_of_pairwise_le_dist
 
+theorem Finite_bounded_inter_isClosed [ProperSpace α] {K s : Set α} [DiscreteTopology s]
+    (hK : Metric.Bounded K) (hs : IsClosed s) : Set.Finite (K ∩ s) := by
+  refine Set.Finite.subset (IsCompact.finite ?_ ?_) (Set.inter_subset_inter_left s subset_closure)
+  · exact IsCompact.inter_right (Metric.Bounded.isCompact_closure hK) hs
+  · exact DiscreteTopology.of_subset inferInstance (Set.inter_subset_right _ s)
+
 end Metric
 
 /-- Build a new metric space from an old one where the bundled uniform structure is provably

We remove all possible occurences of Type _ and Sort _ in favor of Type* and Sort*.

This has nice performance benefits.

@@ -54,7 +54,7 @@ open scoped BigOperators ENNReal NNReal Uniformity Topology
 
 universe u v w
 
-variable {α : Type u} {β : Type v} {X ι : Type _}
+variable {α : Type u} {β : Type v} {X ι : Type*}
 
 /-- Construct a uniform structure from a distance function and metric space axioms -/
 def UniformSpace.ofDist (dist : α → α → ℝ) (dist_self : ∀ x : α, dist x x = 0)
@@ -67,7 +67,7 @@ def UniformSpace.ofDist (dist : α → α → ℝ) (dist_self : ∀ x : α, dist
 -- porting note: dropped the `dist_self` argument
 /-- Construct a bornology from a distance function and metric space axioms. -/
 @[reducible]
-def Bornology.ofDist {α : Type _} (dist : α → α → ℝ) (dist_comm : ∀ x y, dist x y = dist y x)
+def Bornology.ofDist {α : Type*} (dist : α → α → ℝ) (dist_comm : ∀ x y, dist x y = dist y x)
     (dist_triangle : ∀ x y z, dist x z ≤ dist x y + dist y z) : Bornology α :=
   Bornology.ofBounded { s : Set α | ∃ C, ∀ ⦃x⦄, x ∈ s → ∀ ⦃y⦄, y ∈ s → dist x y ≤ C }
     ⟨0, fun x hx y => hx.elim⟩ (fun s ⟨c, hc⟩ t h => ⟨c, fun x hx y hy => hc (h hx) (h hy)⟩)
@@ -90,7 +90,7 @@ def Bornology.ofDist {α : Type _} (dist : α → α → ℝ) (dist_comm : ∀ x
 /-- The distance function (given an ambient metric space on `α`), which returns
   a nonnegative real number `dist x y` given `x y : α`. -/
 @[ext]
-class Dist (α : Type _) where
+class Dist (α : Type*) where
   dist : α → α → ℝ
 #align has_dist Dist
 
@@ -135,7 +135,7 @@ class PseudoMetricSpace (α : Type u) extends Dist α : Type u where
 
 /-- Two pseudo metric space structures with the same distance function coincide. -/
 @[ext]
-theorem PseudoMetricSpace.ext {α : Type _} {m m' : PseudoMetricSpace α}
+theorem PseudoMetricSpace.ext {α : Type*} {m m' : PseudoMetricSpace α}
     (h : m.toDist = m'.toDist) : m = m' := by
   cases' m with d _ _ _ ed hed U hU B hB
   cases' m' with d' _ _ _ ed' hed' U' hU' B' hB'
@@ -290,7 +290,7 @@ example {x y : α} : 0 ≤ dist x y := by positivity
 #align abs_dist abs_dist
 
 /-- A version of `Dist` that takes value in `ℝ≥0`. -/
-class NNDist (α : Type _) where
+class NNDist (α : Type*) where
   nndist : α → α → ℝ≥0
 #align has_nndist NNDist
 
@@ -336,7 +336,7 @@ theorem edist_le_coe {x y : α} {c : ℝ≥0} : edist x y ≤ c ↔ nndist x y 
 #align edist_le_coe edist_le_coe
 
 /-- In a pseudometric space, the extended distance is always finite-/
-theorem edist_lt_top {α : Type _} [PseudoMetricSpace α] (x y : α) : edist x y < ⊤ :=
+theorem edist_lt_top {α : Type*} [PseudoMetricSpace α] (x y : α) : edist x y < ⊤ :=
   (edist_dist x y).symm ▸ ENNReal.ofReal_lt_top
 #align edist_lt_top edist_lt_top
 
@@ -740,7 +740,7 @@ accumulating to zero, then `f i`-neighborhoods of the diagonal form a basis of `
 
 For specific bases see `uniformity_basis_dist`, `uniformity_basis_dist_inv_nat_succ`,
 and `uniformity_basis_dist_inv_nat_pos`. -/
-protected theorem mk_uniformity_basis {β : Type _} {p : β → Prop} {f : β → ℝ}
+protected theorem mk_uniformity_basis {β : Type*} {p : β → Prop} {f : β → ℝ}
     (hf₀ : ∀ i, p i → 0 < f i) (hf : ∀ ⦃ε⦄, 0 < ε → ∃ i, p i ∧ f i ≤ ε) :
     (𝓤 α).HasBasis p fun i => { p : α × α | dist p.1 p.2 < f i } := by
   refine' ⟨fun s => uniformity_basis_dist.mem_iff.trans _⟩
@@ -791,7 +791,7 @@ form a basis of `𝓤 α`.
 
 Currently we have only one specific basis `uniformity_basis_dist_le` based on this constructor.
 More can be easily added if needed in the future. -/
-protected theorem mk_uniformity_basis_le {β : Type _} {p : β → Prop} {f : β → ℝ}
+protected theorem mk_uniformity_basis_le {β : Type*} {p : β → Prop} {f : β → ℝ}
     (hf₀ : ∀ x, p x → 0 < f x) (hf : ∀ ε, 0 < ε → ∃ x, p x ∧ f x ≤ ε) :
     (𝓤 α).HasBasis p fun x => { p : α × α | dist p.1 p.2 ≤ f x } := by
   refine' ⟨fun s => uniformity_basis_dist.mem_iff.trans _⟩
@@ -899,7 +899,7 @@ theorem finite_approx_of_totallyBounded {s : Set α} (hs : TotallyBounded s) :
 #align metric.finite_approx_of_totally_bounded Metric.finite_approx_of_totallyBounded
 
 /-- Expressing uniform convergence using `dist` -/
-theorem tendstoUniformlyOnFilter_iff {ι : Type _} {F : ι → β → α} {f : β → α} {p : Filter ι}
+theorem tendstoUniformlyOnFilter_iff {ι : Type*} {F : ι → β → α} {f : β → α} {p : Filter ι}
     {p' : Filter β} :
     TendstoUniformlyOnFilter F f p p' ↔
       ∀ ε > 0, ∀ᶠ n : ι × β in p ×ˢ p', dist (f n.snd) (F n.fst n.snd) < ε := by
@@ -909,7 +909,7 @@ theorem tendstoUniformlyOnFilter_iff {ι : Type _} {F : ι → β → α} {f : 
 #align metric.tendsto_uniformly_on_filter_iff Metric.tendstoUniformlyOnFilter_iff
 
 /-- Expressing locally uniform convergence on a set using `dist`. -/
-theorem tendstoLocallyUniformlyOn_iff {ι : Type _} [TopologicalSpace β] {F : ι → β → α} {f : β → α}
+theorem tendstoLocallyUniformlyOn_iff {ι : Type*} [TopologicalSpace β] {F : ι → β → α} {f : β → α}
     {p : Filter ι} {s : Set β} :
     TendstoLocallyUniformlyOn F f p s ↔
       ∀ ε > 0, ∀ x ∈ s, ∃ t ∈ 𝓝[s] x, ∀ᶠ n in p, ∀ y ∈ t, dist (f y) (F n y) < ε := by
@@ -920,7 +920,7 @@ theorem tendstoLocallyUniformlyOn_iff {ι : Type _} [TopologicalSpace β] {F : 
 #align metric.tendsto_locally_uniformly_on_iff Metric.tendstoLocallyUniformlyOn_iff
 
 /-- Expressing uniform convergence on a set using `dist`. -/
-theorem tendstoUniformlyOn_iff {ι : Type _} {F : ι → β → α} {f : β → α} {p : Filter ι} {s : Set β} :
+theorem tendstoUniformlyOn_iff {ι : Type*} {F : ι → β → α} {f : β → α} {p : Filter ι} {s : Set β} :
     TendstoUniformlyOn F f p s ↔ ∀ ε > 0, ∀ᶠ n in p, ∀ x ∈ s, dist (f x) (F n x) < ε := by
   refine' ⟨fun H ε hε => H _ (dist_mem_uniformity hε), fun H u hu => _⟩
   rcases mem_uniformity_dist.1 hu with ⟨ε, εpos, hε⟩
@@ -928,7 +928,7 @@ theorem tendstoUniformlyOn_iff {ι : Type _} {F : ι → β → α} {f : β →
 #align metric.tendsto_uniformly_on_iff Metric.tendstoUniformlyOn_iff
 
 /-- Expressing locally uniform convergence using `dist`. -/
-theorem tendstoLocallyUniformly_iff {ι : Type _} [TopologicalSpace β] {F : ι → β → α} {f : β → α}
+theorem tendstoLocallyUniformly_iff {ι : Type*} [TopologicalSpace β] {F : ι → β → α} {f : β → α}
     {p : Filter ι} :
     TendstoLocallyUniformly F f p ↔
       ∀ ε > 0, ∀ x : β, ∃ t ∈ 𝓝 x, ∀ᶠ n in p, ∀ y ∈ t, dist (f y) (F n y) < ε := by
@@ -937,7 +937,7 @@ theorem tendstoLocallyUniformly_iff {ι : Type _} [TopologicalSpace β] {F : ι
 #align metric.tendsto_locally_uniformly_iff Metric.tendstoLocallyUniformly_iff
 
 /-- Expressing uniform convergence using `dist`. -/
-theorem tendstoUniformly_iff {ι : Type _} {F : ι → β → α} {f : β → α} {p : Filter ι} :
+theorem tendstoUniformly_iff {ι : Type*} {F : ι → β → α} {f : β → α} {p : Filter ι} :
     TendstoUniformly F f p ↔ ∀ ε > 0, ∀ᶠ n in p, ∀ x, dist (f x) (F n x) < ε := by
   rw [← tendstoUniformlyOn_univ, tendstoUniformlyOn_iff]
   simp
@@ -1121,7 +1121,7 @@ theorem tendsto_atTop' [Nonempty β] [SemilatticeSup β] [NoMaxOrder β] {u : β
     simp only [true_and, gt_iff_lt, mem_Ioi, mem_ball]
 #align metric.tendsto_at_top' Metric.tendsto_atTop'
 
-theorem isOpen_singleton_iff {α : Type _} [PseudoMetricSpace α] {x : α} :
+theorem isOpen_singleton_iff {α : Type*} [PseudoMetricSpace α] {x : α} :
     IsOpen ({x} : Set α) ↔ ∃ ε > 0, ∀ y, dist y x < ε → y = x := by
   simp [isOpen_iff, subset_singleton_iff, mem_ball]
 #align metric.is_open_singleton_iff Metric.isOpen_singleton_iff
@@ -1146,7 +1146,7 @@ theorem _root_.Dense.exists_dist_lt {s : Set α} (hs : Dense s) (x : α) {ε : 
   simpa only [mem_ball'] using hs.exists_mem_open isOpen_ball this
 #align dense.exists_dist_lt Dense.exists_dist_lt
 
-nonrec theorem _root_.DenseRange.exists_dist_lt {β : Type _} {f : β → α} (hf : DenseRange f) (x : α)
+nonrec theorem _root_.DenseRange.exists_dist_lt {β : Type*} {f : β → α} (hf : DenseRange f) (x : α)
     {ε : ℝ} (hε : 0 < ε) : ∃ y, dist x (f y) < ε :=
   exists_range_iff.1 (hf.exists_dist_lt x hε)
 #align dense_range.exists_dist_lt DenseRange.exists_dist_lt
@@ -1476,12 +1476,12 @@ theorem cauchySeq_iff_tendsto_dist_atTop_0 [Nonempty β] [SemilatticeSup β] {u
   rw [cauchySeq_iff_tendsto, Metric.uniformity_eq_comap_nhds_zero, tendsto_comap_iff]; rfl
 #align cauchy_seq_iff_tendsto_dist_at_top_0 cauchySeq_iff_tendsto_dist_atTop_0
 
-theorem tendsto_uniformity_iff_dist_tendsto_zero {ι : Type _} {f : ι → α × α} {p : Filter ι} :
+theorem tendsto_uniformity_iff_dist_tendsto_zero {ι : Type*} {f : ι → α × α} {p : Filter ι} :
     Tendsto f p (𝓤 α) ↔ Tendsto (fun x => dist (f x).1 (f x).2) p (𝓝 0) := by
   rw [Metric.uniformity_eq_comap_nhds_zero, tendsto_comap_iff]; rfl
 #align tendsto_uniformity_iff_dist_tendsto_zero tendsto_uniformity_iff_dist_tendsto_zero
 
-theorem Filter.Tendsto.congr_dist {ι : Type _} {f₁ f₂ : ι → α} {p : Filter ι} {a : α}
+theorem Filter.Tendsto.congr_dist {ι : Type*} {f₁ f₂ : ι → α} {p : Filter ι} {a : α}
     (h₁ : Tendsto f₁ p (𝓝 a)) (h : Tendsto (fun x => dist (f₁ x) (f₂ x)) p (𝓝 0)) :
     Tendsto f₂ p (𝓝 a) :=
   h₁.congr_uniformity <| tendsto_uniformity_iff_dist_tendsto_zero.2 h
@@ -1490,7 +1490,7 @@ theorem Filter.Tendsto.congr_dist {ι : Type _} {f₁ f₂ : ι → α} {p : Fil
 alias Filter.Tendsto.congr_dist ← tendsto_of_tendsto_of_dist
 #align tendsto_of_tendsto_of_dist tendsto_of_tendsto_of_dist
 
-theorem tendsto_iff_of_dist {ι : Type _} {f₁ f₂ : ι → α} {p : Filter ι} {a : α}
+theorem tendsto_iff_of_dist {ι : Type*} {f₁ f₂ : ι → α} {p : Filter ι} {a : α}
     (h : Tendsto (fun x => dist (f₁ x) (f₂ x)) p (𝓝 0)) : Tendsto f₁ p (𝓝 a) ↔ Tendsto f₂ p (𝓝 a) :=
   Uniform.tendsto_congr <| tendsto_uniformity_iff_dist_tendsto_zero.2 h
 #align tendsto_iff_of_dist tendsto_iff_of_dist
@@ -1528,7 +1528,7 @@ theorem Metric.cauchySeq_iff' {u : β → α} :
 /-- In a pseudometric space, uniform Cauchy sequences are characterized by the fact that,
 eventually, the distance between all its elements is uniformly, arbitrarily small -/
 -- porting note: no attr @[nolint ge_or_gt]
-theorem Metric.uniformCauchySeqOn_iff {γ : Type _} {F : β → γ → α} {s : Set γ} :
+theorem Metric.uniformCauchySeqOn_iff {γ : Type*} {F : β → γ → α} {s : Set γ} :
     UniformCauchySeqOn F atTop s ↔ ∀ ε > (0 : ℝ),
       ∃ N : β, ∀ m ≥ N, ∀ n ≥ N, ∀ x ∈ s, dist (F m x) (F n x) < ε := by
   constructor
@@ -1991,7 +1991,7 @@ section Pi
 
 open Finset
 
-variable {π : β → Type _} [Fintype β] [∀ b, PseudoMetricSpace (π b)]
+variable {π : β → Type*} [Fintype β] [∀ b, PseudoMetricSpace (π b)]
 
 /-- A finite product of pseudometric spaces is a pseudometric space, with the sup distance. -/
 instance pseudoMetricSpacePi : PseudoMetricSpace (∀ b, π b) := by
@@ -2142,14 +2142,14 @@ theorem sphere_pi (x : ∀ b, π b) {r : ℝ} (h : 0 < r ∨ Nonempty β) :
 #align sphere_pi sphere_pi
 
 @[simp]
-theorem Fin.nndist_insertNth_insertNth {n : ℕ} {α : Fin (n + 1) → Type _}
+theorem Fin.nndist_insertNth_insertNth {n : ℕ} {α : Fin (n + 1) → Type*}
     [∀ i, PseudoMetricSpace (α i)] (i : Fin (n + 1)) (x y : α i) (f g : ∀ j, α (i.succAbove j)) :
     nndist (i.insertNth x f) (i.insertNth y g) = max (nndist x y) (nndist f g) :=
   eq_of_forall_ge_iff fun c => by simp [nndist_pi_le_iff, i.forall_iff_succAbove]
 #align fin.nndist_insert_nth_insert_nth Fin.nndist_insertNth_insertNth
 
 @[simp]
-theorem Fin.dist_insertNth_insertNth {n : ℕ} {α : Fin (n + 1) → Type _}
+theorem Fin.dist_insertNth_insertNth {n : ℕ} {α : Fin (n + 1) → Type*}
     [∀ i, PseudoMetricSpace (α i)] (i : Fin (n + 1)) (x y : α i) (f g : ∀ j, α (i.succAbove j)) :
     dist (i.insertNth x f) (i.insertNth y g) = max (dist x y) (dist f g) := by
   simp only [dist_nndist, Fin.nndist_insertNth_insertNth, NNReal.coe_max]
@@ -2192,13 +2192,13 @@ class ProperSpace (α : Type u) [PseudoMetricSpace α] : Prop where
 export ProperSpace (isCompact_closedBall)
 
 /-- In a proper pseudometric space, all spheres are compact. -/
-theorem isCompact_sphere {α : Type _} [PseudoMetricSpace α] [ProperSpace α] (x : α) (r : ℝ) :
+theorem isCompact_sphere {α : Type*} [PseudoMetricSpace α] [ProperSpace α] (x : α) (r : ℝ) :
     IsCompact (sphere x r) :=
   isCompact_of_isClosed_subset (isCompact_closedBall x r) isClosed_sphere sphere_subset_closedBall
 #align is_compact_sphere isCompact_sphere
 
 /-- In a proper pseudometric space, any sphere is a `CompactSpace` when considered as a subtype. -/
-instance Metric.sphere.compactSpace {α : Type _} [PseudoMetricSpace α] [ProperSpace α]
+instance Metric.sphere.compactSpace {α : Type*} [PseudoMetricSpace α] [ProperSpace α]
     (x : α) (r : ℝ) : CompactSpace (sphere x r) :=
   isCompact_iff_compactSpace.mp (isCompact_sphere _ _)
 
@@ -2263,7 +2263,7 @@ instance (priority := 100) complete_of_proper [ProperSpace α] : CompleteSpace 
 #align complete_of_proper complete_of_proper
 
 /-- A binary product of proper spaces is proper. -/
-instance prod_properSpace {α : Type _} {β : Type _} [PseudoMetricSpace α] [PseudoMetricSpace β]
+instance prod_properSpace {α : Type*} {β : Type*} [PseudoMetricSpace α] [PseudoMetricSpace β]
     [ProperSpace α] [ProperSpace β] : ProperSpace (α × β) where
   isCompact_closedBall := by
     rintro ⟨x, y⟩ r
@@ -2272,7 +2272,7 @@ instance prod_properSpace {α : Type _} {β : Type _} [PseudoMetricSpace α] [Ps
 #align prod_proper_space prod_properSpace
 
 /-- A finite product of proper spaces is proper. -/
-instance pi_properSpace {π : β → Type _} [Fintype β] [∀ b, PseudoMetricSpace (π b)]
+instance pi_properSpace {π : β → Type*} [Fintype β] [∀ b, PseudoMetricSpace (π b)]
     [h : ∀ b, ProperSpace (π b)] : ProperSpace (∀ b, π b) := by
   refine' properSpace_of_compact_closedBall_of_le 0 fun x r hr => _
   rw [closedBall_pi _ hr]
@@ -2891,7 +2891,7 @@ class MetricSpace (α : Type u) extends PseudoMetricSpace α : Type u where
 
 /-- Two metric space structures with the same distance coincide. -/
 @[ext]
-theorem MetricSpace.ext {α : Type _} {m m' : MetricSpace α} (h : m.toDist = m'.toDist) :
+theorem MetricSpace.ext {α : Type*} {m m' : MetricSpace α} (h : m.toDist = m'.toDist) :
     m = m' := by
   cases m; cases m'; congr; ext1; assumption
 #align metric_space.ext MetricSpace.ext
@@ -2996,7 +2996,7 @@ theorem uniformEmbedding_iff' [MetricSpace β] {f : γ → β} :
 
 /-- If a `PseudoMetricSpace` is a T₀ space, then it is a `MetricSpace`. -/
 @[reducible]
-def _root_.MetricSpace.ofT0PseudoMetricSpace (α : Type _) [PseudoMetricSpace α] [T0Space α] :
+def _root_.MetricSpace.ofT0PseudoMetricSpace (α : Type*) [PseudoMetricSpace α] [T0Space α] :
     MetricSpace α where
   toPseudoMetricSpace := ‹_›
   eq_of_dist_eq_zero hdist := (Metric.inseparable_iff.2 hdist).eq
@@ -3013,7 +3013,7 @@ theorem isClosed_of_pairwise_le_dist {s : Set γ} {ε : ℝ} (hε : 0 < ε)
   isClosed_of_spaced_out (dist_mem_uniformity hε) <| by simpa using hs
 #align metric.is_closed_of_pairwise_le_dist Metric.isClosed_of_pairwise_le_dist
 
-theorem closedEmbedding_of_pairwise_le_dist {α : Type _} [TopologicalSpace α] [DiscreteTopology α]
+theorem closedEmbedding_of_pairwise_le_dist {α : Type*} [TopologicalSpace α] [DiscreteTopology α]
     {ε : ℝ} (hε : 0 < ε) {f : α → γ} (hf : Pairwise fun x y => ε ≤ dist (f x) (f y)) :
     ClosedEmbedding f :=
   closedEmbedding_of_spaced_out (dist_mem_uniformity hε) <| by simpa using hf
@@ -3021,7 +3021,7 @@ theorem closedEmbedding_of_pairwise_le_dist {α : Type _} [TopologicalSpace α]
 
 /-- If `f : β → α` sends any two distinct points to points at distance at least `ε > 0`, then
 `f` is a uniform embedding with respect to the discrete uniformity on `β`. -/
-theorem uniformEmbedding_bot_of_pairwise_le_dist {β : Type _} {ε : ℝ} (hε : 0 < ε) {f : β → α}
+theorem uniformEmbedding_bot_of_pairwise_le_dist {β : Type*} {ε : ℝ} (hε : 0 < ε) {f : β → α}
     (hf : Pairwise fun x y => ε ≤ dist (f x) (f y)) :
     @UniformEmbedding _ _ ⊥ (by infer_instance) f :=
   uniformEmbedding_of_spaced_out (dist_mem_uniformity hε) <| by simpa using hf
@@ -3122,13 +3122,13 @@ def Embedding.comapMetricSpace {α β} [TopologicalSpace α] [m : MetricSpace β
   .replaceTopology (.induced f h.inj m) h.induced
 #align embedding.comap_metric_space Embedding.comapMetricSpace
 
-instance Subtype.metricSpace {α : Type _} {p : α → Prop} [MetricSpace α] :
+instance Subtype.metricSpace {α : Type*} {p : α → Prop} [MetricSpace α] :
     MetricSpace (Subtype p) :=
   .induced Subtype.val Subtype.coe_injective ‹_›
 #align subtype.metric_space Subtype.metricSpace
 
 @[to_additive]
-instance {α : Type _} [MetricSpace α] : MetricSpace αᵐᵒᵖ :=
+instance {α : Type*} [MetricSpace α] : MetricSpace αᵐᵒᵖ :=
   MetricSpace.induced MulOpposite.unop MulOpposite.unop_injective ‹_›
 
 instance : MetricSpace Empty where
@@ -3183,7 +3183,7 @@ section Pi
 
 open Finset
 
-variable {π : β → Type _} [Fintype β] [∀ b, MetricSpace (π b)]
+variable {π : β → Type*} [Fintype β] [∀ b, MetricSpace (π b)]
 
 /-- A finite product of metric spaces is a metric space, with the sup distance. -/
 instance metricSpacePi : MetricSpace (∀ b, π b) := .ofT0PseudoMetricSpace _
@@ -3201,7 +3201,7 @@ open TopologicalSpace
 /-- A metric space is second countable if one can reconstruct up to any `ε>0` any element of the
 space from countably many data. -/
 theorem secondCountable_of_countable_discretization {α : Type u} [MetricSpace α]
-    (H : ∀ ε > (0 : ℝ), ∃ (β : Type _) (_ : Encodable β) (F : α → β),
+    (H : ∀ ε > (0 : ℝ), ∃ (β : Type*) (_ : Encodable β) (F : α → β),
       ∀ x y, F x = F y → dist x y ≤ ε) :
     SecondCountableTopology α := by
   refine secondCountable_of_almost_dense_set fun ε ε0 => ?_

Per https://github.com/leanprover/lean4/issues/2343, we are going to need to change the automatic generation of instance names, as they become too long.

This PR ensures that everywhere in Mathlib that refers to an instance by name, that name is given explicitly, rather than being automatically generated.

There are four exceptions, which are now commented, with links to https://github.com/leanprover/lean4/issues/2343.

This was implemented by running Mathlib against a modified Lean that appended _ᾰ to all automatically generated names, and fixing everything.

Co-authored-by: Scott Morrison <scott.morrison@gmail.com>

@@ -1666,7 +1666,7 @@ theorem Subtype.nndist_eq {p : α → Prop} (x y : Subtype p) : nndist x y = nnd
 namespace MulOpposite
 
 @[to_additive]
-instance : PseudoMetricSpace αᵐᵒᵖ :=
+instance instPseudoMetricSpaceMulOpposite : PseudoMetricSpace αᵐᵒᵖ :=
   PseudoMetricSpace.induced MulOpposite.unop ‹_›
 
 @[to_additive (attr := simp)]

@@ -8,7 +8,7 @@ import Mathlib.Topology.Algebra.Order.Compact
 import Mathlib.Topology.MetricSpace.EMetricSpace
 import Mathlib.Topology.Bornology.Constructions
 
-#align_import topology.metric_space.basic from "leanprover-community/mathlib"@"8000bbbe2e9d39b84edb993d88781f536a8a3fa8"
+#align_import topology.metric_space.basic from "leanprover-community/mathlib"@"8047de4d911cdef39c2d646165eea972f7f9f539"
 
 /-!
 # Metric spaces
@@ -495,6 +495,15 @@ theorem ne_of_mem_sphere (h : y ∈ sphere x ε) (hε : ε ≠ 0) : y ≠ x :=
   ne_of_mem_of_not_mem h <| by simpa using hε.symm
 #align metric.ne_of_mem_sphere Metric.ne_of_mem_sphere
 
+theorem nonneg_of_mem_sphere (hy : y ∈ sphere x ε) : 0 ≤ ε :=
+  dist_nonneg.trans_eq hy
+#align metric.nonneg_of_mem_sphere Metric.nonneg_of_mem_sphere
+
+@[simp]
+theorem sphere_eq_empty_of_neg (hε : ε < 0) : sphere x ε = ∅ :=
+  Set.eq_empty_iff_forall_not_mem.mpr fun _y hy => (nonneg_of_mem_sphere hy).not_lt hε
+#align metric.sphere_eq_empty_of_neg Metric.sphere_eq_empty_of_neg
+
 theorem sphere_eq_empty_of_subsingleton [Subsingleton α] (hε : ε ≠ 0) : sphere x ε = ∅ :=
   Set.eq_empty_iff_forall_not_mem.mpr fun _ h => ne_of_mem_sphere h hε (Subsingleton.elim _ _)
 #align metric.sphere_eq_empty_of_subsingleton Metric.sphere_eq_empty_of_subsingleton
@@ -517,6 +526,11 @@ theorem closedBall_eq_empty : closedBall x ε = ∅ ↔ ε < 0 := by
   rw [← not_nonempty_iff_eq_empty, nonempty_closedBall, not_le]
 #align metric.closed_ball_eq_empty Metric.closedBall_eq_empty
 
+/-- Closed balls and spheres coincide when the radius is non-positive -/
+theorem closedBall_eq_sphere_of_nonpos (hε : ε ≤ 0) : closedBall x ε = sphere x ε :=
+  Set.ext fun _ => (hε.trans dist_nonneg).le_iff_eq
+#align metric.closed_ball_eq_sphere_of_nonpos Metric.closedBall_eq_sphere_of_nonpos
+
 theorem ball_subset_closedBall : ball x ε ⊆ closedBall x ε := fun _y hy =>
   mem_closedBall.2 (le_of_lt hy)
 #align metric.ball_subset_closed_ball Metric.ball_subset_closedBall
@@ -1772,6 +1786,19 @@ theorem closedBall_prod_same (x : α) (y : β) (r : ℝ) :
   ext fun z => by simp [Prod.dist_eq]
 #align closed_ball_prod_same closedBall_prod_same
 
+theorem sphere_prod (x : α × β) (r : ℝ) :
+    sphere x r = sphere x.1 r ×ˢ closedBall x.2 r ∪ closedBall x.1 r ×ˢ sphere x.2 r := by
+  obtain hr | rfl | hr := lt_trichotomy r 0
+  · simp [hr]
+  · cases x
+    simp_rw [← closedBall_eq_sphere_of_nonpos le_rfl, union_self, closedBall_prod_same]
+  · ext ⟨x', y'⟩
+    simp_rw [Set.mem_union, Set.mem_prod, Metric.mem_closedBall, Metric.mem_sphere, Prod.dist_eq,
+      max_eq_iff]
+    refine' or_congr (and_congr_right _) (and_comm.trans (and_congr_left _))
+    all_goals rintro rfl; rfl
+#align sphere_prod sphere_prod
+
 end Prod
 
 -- porting note: 3 new lemmas
@@ -1998,10 +2025,24 @@ theorem nndist_pi_le_iff {f g : ∀ b, π b} {r : ℝ≥0} :
     nndist f g ≤ r ↔ ∀ b, nndist (f b) (g b) ≤ r := by simp [nndist_pi_def]
 #align nndist_pi_le_iff nndist_pi_le_iff
 
+theorem nndist_pi_lt_iff {f g : ∀ b, π b} {r : ℝ≥0} (hr : 0 < r) :
+    nndist f g < r ↔ ∀ b, nndist (f b) (g b) < r := by
+  simp [nndist_pi_def, Finset.sup_lt_iff (show ⊥ < r from hr)]
+#align nndist_pi_lt_iff nndist_pi_lt_iff
+
+theorem nndist_pi_eq_iff {f g : ∀ b, π b} {r : ℝ≥0} (hr : 0 < r) :
+    nndist f g = r ↔ (∃ i, nndist (f i) (g i) = r) ∧ ∀ b, nndist (f b) (g b) ≤ r := by
+  rw [eq_iff_le_not_lt, nndist_pi_lt_iff hr, nndist_pi_le_iff, not_forall, and_comm]
+  simp_rw [not_lt, and_congr_left_iff, le_antisymm_iff]
+  intro h
+  refine' exists_congr fun b => _
+  apply (and_iff_right <| h _).symm
+#align nndist_pi_eq_iff nndist_pi_eq_iff
+
 theorem dist_pi_lt_iff {f g : ∀ b, π b} {r : ℝ} (hr : 0 < r) :
     dist f g < r ↔ ∀ b, dist (f b) (g b) < r := by
   lift r to ℝ≥0 using hr.le
-  simp [dist_pi_def, Finset.sup_lt_iff (show ⊥ < r from hr)]
+  exact nndist_pi_lt_iff hr
 #align dist_pi_lt_iff dist_pi_lt_iff
 
 theorem dist_pi_le_iff {f g : ∀ b, π b} {r : ℝ} (hr : 0 ≤ r) :
@@ -2010,6 +2051,12 @@ theorem dist_pi_le_iff {f g : ∀ b, π b} {r : ℝ} (hr : 0 ≤ r) :
   exact nndist_pi_le_iff
 #align dist_pi_le_iff dist_pi_le_iff
 
+theorem dist_pi_eq_iff {f g : ∀ b, π b} {r : ℝ} (hr : 0 < r) :
+    dist f g = r ↔ (∃ i, dist (f i) (g i) = r) ∧ ∀ b, dist (f b) (g b) ≤ r := by
+  lift r to ℝ≥0 using hr.le
+  simp_rw [← coe_nndist, NNReal.coe_eq, nndist_pi_eq_iff hr, NNReal.coe_le_coe]
+#align dist_pi_eq_iff dist_pi_eq_iff
+
 theorem dist_pi_le_iff' [Nonempty β] {f g : ∀ b, π b} {r : ℝ} :
     dist f g ≤ r ↔ ∀ b, dist (f b) (g b) ≤ r := by
   by_cases hr : 0 ≤ r
@@ -2076,6 +2123,24 @@ theorem closedBall_pi' [Nonempty β] (x : ∀ b, π b) (r : ℝ) :
   (le_or_lt 0 r).elim (closedBall_pi x) fun hr => by simp [closedBall_eq_empty.2 hr]
 #align closed_ball_pi' closedBall_pi'
 
+/-- A sphere in a product space is a union of spheres on each component restricted to the closed
+ball. -/
+theorem sphere_pi (x : ∀ b, π b) {r : ℝ} (h : 0 < r ∨ Nonempty β) :
+    sphere x r = (⋃ i : β, Function.eval i ⁻¹' sphere (x i) r) ∩ closedBall x r := by
+  obtain hr | rfl | hr := lt_trichotomy r 0
+  · simp [hr]
+  · rw [closedBall_eq_sphere_of_nonpos le_rfl, eq_comm, Set.inter_eq_right_iff_subset]
+    letI := h.resolve_left (lt_irrefl _)
+    inhabit β
+    refine' subset_iUnion_of_subset default _
+    intro x hx
+    replace hx := hx.le
+    rw [dist_pi_le_iff le_rfl] at hx
+    exact le_antisymm (hx default) dist_nonneg
+  · ext
+    simp [dist_pi_eq_iff hr, dist_pi_le_iff hr.le]
+#align sphere_pi sphere_pi
+
 @[simp]
 theorem Fin.nndist_insertNth_insertNth {n : ℕ} {α : Fin (n + 1) → Type _}
     [∀ i, PseudoMetricSpace (α i)] (i : Fin (n + 1)) (x y : α i) (f g : ∀ j, α (i.succAbove j)) :

• depends on: #5966

Co-authored-by: Eric Wieser <wieser.eric@gmail.com> Co-authored-by: Scott Morrison <scott.morrison@gmail.com>

@@ -2,17 +2,14 @@
 Copyright (c) 2015, 2017 Jeremy Avigad. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Jeremy Avigad, Robert Y. Lewis, Johannes Hölzl, Mario Carneiro, Sébastien Gouëzel
-
-! This file was ported from Lean 3 source module topology.metric_space.basic
-! leanprover-community/mathlib commit 8000bbbe2e9d39b84edb993d88781f536a8a3fa8
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
 -/
 import Mathlib.Tactic.Positivity
 import Mathlib.Topology.Algebra.Order.Compact
 import Mathlib.Topology.MetricSpace.EMetricSpace
 import Mathlib.Topology.Bornology.Constructions
 
+#align_import topology.metric_space.basic from "leanprover-community/mathlib"@"8000bbbe2e9d39b84edb993d88781f536a8a3fa8"
+
 /-!
 # Metric spaces
 

Grepping for [^ .:{-] [^ :] and reviewing the results. Once I started I couldn't stop. :-)

Co-authored-by: Scott Morrison <scott.morrison@gmail.com>

@@ -1447,7 +1447,7 @@ theorem squeeze_zero' {α} {f g : α → ℝ} {t₀ : Filter α} (hf : ∀ᶠ t
 #align squeeze_zero' squeeze_zero'
 
 /-- Special case of the sandwich theorem; see `tendsto_of_tendsto_of_tendsto_of_le_of_le`
-and  `tendsto_of_tendsto_of_tendsto_of_le_of_le'` for the general case. -/
+and `tendsto_of_tendsto_of_tendsto_of_le_of_le'` for the general case. -/
 theorem squeeze_zero {α} {f g : α → ℝ} {t₀ : Filter α} (hf : ∀ t, 0 ≤ f t) (hft : ∀ t, f t ≤ g t)
     (g0 : Tendsto g t₀ (𝓝 0)) : Tendsto f t₀ (𝓝 0) :=
   squeeze_zero' (eventually_of_forall hf) (eventually_of_forall hft) g0
@@ -2398,7 +2398,7 @@ protected theorem Bounded.prod [PseudoMetricSpace β] {s : Set α} {t : Set β}
 theorem _root_.TotallyBounded.bounded {s : Set α} (h : TotallyBounded s) : Bounded s :=
   -- We cover the totally bounded set by finitely many balls of radius 1,
   -- and then argue that a finite union of bounded sets is bounded
-  let  ⟨_t, fint, subs⟩ := (totallyBounded_iff.mp h) 1 zero_lt_one
+  let ⟨_t, fint, subs⟩ := (totallyBounded_iff.mp h) 1 zero_lt_one
   Bounded.mono subs <| (bounded_biUnion fint).2 fun _ _ => bounded_ball
 #align totally_bounded.bounded TotallyBounded.bounded
 
@@ -2616,7 +2616,7 @@ theorem diam_triple :
 #align metric.diam_triple Metric.diam_triple
 
 /-- If the distance between any two points in a set is bounded by some constant `C`,
-then `ENNReal.ofReal C`  bounds the emetric diameter of this set. -/
+then `ENNReal.ofReal C` bounds the emetric diameter of this set. -/
 theorem ediam_le_of_forall_dist_le {C : ℝ} (h : ∀ x ∈ s, ∀ y ∈ s, dist x y ≤ C) :
     EMetric.diam s ≤ ENNReal.ofReal C :=
   EMetric.diam_le fun x hx y hy => (edist_dist x y).symm ▸ ENNReal.ofReal_le_ofReal (h x hx y hy)

@@ -464,12 +464,12 @@ theorem ball_eq_ball' (ε : ℝ) (x : α) :
 #align metric.ball_eq_ball' Metric.ball_eq_ball'
 
 @[simp]
-theorem iUnion_ball_nat (x : α) : (⋃ n : ℕ, ball x n) = univ :=
+theorem iUnion_ball_nat (x : α) : ⋃ n : ℕ, ball x n = univ :=
   iUnion_eq_univ_iff.2 fun y => exists_nat_gt (dist y x)
 #align metric.Union_ball_nat Metric.iUnion_ball_nat
 
 @[simp]
-theorem iUnion_ball_nat_succ (x : α) : (⋃ n : ℕ, ball x (n + 1)) = univ :=
+theorem iUnion_ball_nat_succ (x : α) : ⋃ n : ℕ, ball x (n + 1) = univ :=
   iUnion_eq_univ_iff.2 fun y => (exists_nat_gt (dist y x)).imp fun _ h => h.trans (lt_add_one _)
 #align metric.Union_ball_nat_succ Metric.iUnion_ball_nat_succ
 
@@ -649,11 +649,11 @@ theorem dist_lt_add_of_nonempty_ball_inter_ball (h : (ball x ε₁ ∩ ball y ε
 #align metric.dist_lt_add_of_nonempty_ball_inter_ball Metric.dist_lt_add_of_nonempty_ball_inter_ball
 
 @[simp]
-theorem iUnion_closedBall_nat (x : α) : (⋃ n : ℕ, closedBall x n) = univ :=
+theorem iUnion_closedBall_nat (x : α) : ⋃ n : ℕ, closedBall x n = univ :=
   iUnion_eq_univ_iff.2 fun y => exists_nat_ge (dist y x)
 #align metric.Union_closed_ball_nat Metric.iUnion_closedBall_nat
 
-theorem iUnion_inter_closedBall_nat (s : Set α) (x : α) : (⋃ n : ℕ, s ∩ closedBall x n) = s := by
+theorem iUnion_inter_closedBall_nat (s : Set α) (x : α) : ⋃ n : ℕ, s ∩ closedBall x n = s := by
   rw [← inter_iUnion, iUnion_closedBall_nat, inter_univ]
 #align metric.Union_inter_closed_ball_nat Metric.iUnion_inter_closedBall_nat
 
@@ -1150,7 +1150,7 @@ distance coincide. -/
 
 -- porting note: new
 theorem Metric.uniformity_edist_aux {α} (d : α → α → ℝ≥0) :
-    (⨅ ε > (0 : ℝ), 𝓟 { p : α × α | ↑(d p.1 p.2) < ε }) =
+    ⨅ ε > (0 : ℝ), 𝓟 { p : α × α | ↑(d p.1 p.2) < ε } =
       ⨅ ε > (0 : ℝ≥0∞), 𝓟 { p : α × α | ↑(d p.1 p.2) < ε } := by
   simp only [le_antisymm_iff, le_iInf_iff, le_principal_iff]
   refine ⟨fun ε hε => ?_, fun ε hε => ?_⟩

This fixes PseudoMetricSpace.replaceUniformity so that it no longer silently changes the bornology. Before the fix, the following example fails, but afterwards, it succeeds.

example {α} [U : UniformSpace α] (m : PseudoMetricSpace α)
    (H : 𝓤[U] = 𝓤[PseudoEMetricSpace.toUniformSpace]) :
  (PseudoMetricSpace.replaceUniformity m H).toBornology = m.toBornology := rfl

@@ -1235,15 +1235,10 @@ See Note [forgetful inheritance].
 -/
 @[reducible]
 def PseudoMetricSpace.replaceUniformity {α} [U : UniformSpace α] (m : PseudoMetricSpace α)
-    (H : 𝓤[U] = 𝓤[PseudoEMetricSpace.toUniformSpace]) : PseudoMetricSpace α where
-  toDist := m.toDist
-  dist_self := dist_self
-  dist_comm := dist_comm
-  dist_triangle := dist_triangle
-  edist := edist
-  edist_dist := edist_dist
-  toUniformSpace := U
-  uniformity_dist := H.trans PseudoMetricSpace.uniformity_dist
+    (H : 𝓤[U] = 𝓤[PseudoEMetricSpace.toUniformSpace]) : PseudoMetricSpace α :=
+  { m with
+    toUniformSpace := U
+    uniformity_dist := H.trans PseudoMetricSpace.uniformity_dist }
 #align pseudo_metric_space.replace_uniformity PseudoMetricSpace.replaceUniformity
 
 theorem PseudoMetricSpace.replaceUniformity_eq {α} [U : UniformSpace α] (m : PseudoMetricSpace α)
@@ -1252,6 +1247,11 @@ theorem PseudoMetricSpace.replaceUniformity_eq {α} [U : UniformSpace α] (m : P
   rfl
 #align pseudo_metric_space.replace_uniformity_eq PseudoMetricSpace.replaceUniformity_eq
 
+-- ensure that the bornology is unchanged when replacing the uniformity.
+example {α} [U : UniformSpace α] (m : PseudoMetricSpace α)
+    (H : 𝓤[U] = 𝓤[PseudoEMetricSpace.toUniformSpace]) :
+  (PseudoMetricSpace.replaceUniformity m H).toBornology = m.toBornology := rfl
+
 /-- Build a new pseudo metric space from an old one where the bundled topological structure is
 provably (but typically non-definitionaly) equal to some given topological structure.
 See Note [forgetful inheritance].
@@ -1321,6 +1321,11 @@ theorem PseudoMetricSpace.replaceBornology_eq {α} [m : PseudoMetricSpace α] [B
   rfl
 #align pseudo_metric_space.replace_bornology_eq PseudoMetricSpace.replaceBornology_eq
 
+-- ensure that the uniformity is unchanged when replacing the bornology.
+example {α} [B : Bornology α] (m : PseudoMetricSpace α)
+    (H : ∀ s, @IsBounded _ B s ↔ @IsBounded _ PseudoMetricSpace.toBornology s) :
+  (PseudoMetricSpace.replaceBornology m H).toUniformSpace = m.toUniformSpace := rfl
+
 /-- A very useful criterion to show that a space is complete is to show that all sequences
 which satisfy a bound of the form `dist (u n) (u m) < B N` for all `n m ≥ N` are
 converging. This is often applied for `B N = 2^{-N}`, i.e., with a very fast convergence to

Also rename 2 lemmas

• IsCompact.exists_local_min_mem_open -> IsCompact.exists_isLocalMin_mem_open;
• Metric.exists_local_min_mem_ball -> Metric.exists_isLocal_min_mem_ball.

@@ -2765,14 +2765,14 @@ theorem nonempty_iInter_of_nonempty_biInter [CompleteSpace α] {s : ℕ → Set
 
 end Diam
 
-theorem exists_local_min_mem_ball [ProperSpace α] [TopologicalSpace β]
+theorem exists_isLocalMin_mem_ball [ProperSpace α] [TopologicalSpace β]
     [ConditionallyCompleteLinearOrder β] [OrderTopology β] {f : α → β} {a z : α} {r : ℝ}
     (hf : ContinuousOn f (closedBall a r)) (hz : z ∈ closedBall a r)
     (hf1 : ∀ z' ∈ sphere a r, f z < f z') : ∃ z ∈ ball a r, IsLocalMin f z := by
   simp_rw [← closedBall_diff_ball] at hf1
-  exact (isCompact_closedBall a r).exists_local_min_mem_open ball_subset_closedBall hf hz hf1
+  exact (isCompact_closedBall a r).exists_isLocalMin_mem_open ball_subset_closedBall hf hz hf1
     isOpen_ball
-#align metric.exists_local_min_mem_ball Metric.exists_local_min_mem_ball
+#align metric.exists_local_min_mem_ball Metric.exists_isLocalMin_mem_ball
 
 end Metric
 

Co-authored-by: Xavier Roblot <46200072+xroblot@users.noreply.github.com> Co-authored-by: Joël Riou <joel.riou@universite-paris-saclay.fr> Co-authored-by: Riccardo Brasca <riccardo.brasca@gmail.com> Co-authored-by: Yury G. Kudryashov <urkud@urkud.name> Co-authored-by: Scott Morrison <scott.morrison@anu.edu.au> Co-authored-by: Scott Morrison <scott.morrison@gmail.com> Co-authored-by: Jeremy Tan Jie Rui <reddeloostw@gmail.com> Co-authored-by: Pol'tta / Miyahara Kō <pol_tta@outlook.jp> Co-authored-by: Jason Yuen <jason_yuen2007@hotmail.com> Co-authored-by: Mario Carneiro <di.gama@gmail.com> Co-authored-by: Jireh Loreaux <loreaujy@gmail.com> Co-authored-by: Ruben Van de Velde <65514131+Ruben-VandeVelde@users.noreply.github.com> Co-authored-by: Kyle Miller <kmill31415@gmail.com> Co-authored-by: Heather Macbeth <25316162+hrmacbeth@users.noreply.github.com> Co-authored-by: Jujian Zhang <jujian.zhang1998@outlook.com> Co-authored-by: Yaël Dillies <yael.dillies@gmail.com>

@@ -144,7 +144,7 @@ theorem PseudoMetricSpace.ext {α : Type _} {m m' : PseudoMetricSpace α}
   cases' m' with d' _ _ _ ed' hed' U' hU' B' hB'
   obtain rfl : d = d' := h
   congr
-  · ext (x y) : 2
+  · ext x y : 2
     rw [hed, hed']
   · exact uniformSpace_eq (hU.trans hU'.symm)
   · ext : 2

Co-authored-by: Scott Morrison <scott.morrison@anu.edu.au> Co-authored-by: Parcly Taxel <reddeloostw@gmail.com>

@@ -863,7 +863,7 @@ theorem totallyBounded_iff {s : Set α} :
 space from finitely many data. -/
 theorem totallyBounded_of_finite_discretization {s : Set α}
     (H : ∀ ε > (0 : ℝ),
-        ∃ (β : Type u)(_ : Fintype β)(F : s → β), ∀ x y, F x = F y → dist (x : α) y < ε) :
+        ∃ (β : Type u) (_ : Fintype β) (F : s → β), ∀ x y, F x = F y → dist (x : α) y < ε) :
     TotallyBounded s := by
   cases' s.eq_empty_or_nonempty with hs hs
   · rw [hs]

These are all doc fixes

@@ -1509,8 +1509,8 @@ theorem Metric.cauchySeq_iff' {u : β → α} :
 #align metric.cauchy_seq_iff' Metric.cauchySeq_iff'
 
 -- see Note [nolint_ge]
-/-- In a pseudometric space, unifom Cauchy sequences are characterized by the fact that, eventually,
-the distance between all its elements is uniformly, arbitrarily small -/
+/-- In a pseudometric space, uniform Cauchy sequences are characterized by the fact that,
+eventually, the distance between all its elements is uniformly, arbitrarily small -/
 -- porting note: no attr @[nolint ge_or_gt]
 theorem Metric.uniformCauchySeqOn_iff {γ : Type _} {F : β → γ → α} {s : Set γ} :
     UniformCauchySeqOn F atTop s ↔ ∀ ε > (0 : ℝ),

Run codespell Mathlib and keep some suggestions.

@@ -792,7 +792,7 @@ protected theorem mk_uniformity_basis_le {β : Type _} {p : β → Prop} {f : β
   · exact fun ⟨i, hi, H⟩ => ⟨f i, hf₀ i hi, fun x (hx : _ < _) => H (mem_setOf.2 hx.le)⟩
 #align metric.mk_uniformity_basis_le Metric.mk_uniformity_basis_le
 
-/-- Contant size closed neighborhoods of the diagonal form a basis
+/-- Constant size closed neighborhoods of the diagonal form a basis
 of the uniformity filter. -/
 theorem uniformity_basis_dist_le :
     (𝓤 α).HasBasis ((0 : ℝ) < ·) fun ε => { p : α × α | dist p.1 p.2 ≤ ε } :=

Currently, the following notations are changed from · ×ˢ · because Lean 4 can't deal with ambiguous notations. | Definition | Notation | | :

Co-authored-by: Jeremy Tan Jie Rui <reddeloostw@gmail.com> Co-authored-by: Kyle Miller <kmill31415@gmail.com> Co-authored-by: Chris Hughes <chrishughes24@gmail.com>

@@ -891,7 +891,7 @@ theorem finite_approx_of_totallyBounded {s : Set α} (hs : TotallyBounded s) :
 theorem tendstoUniformlyOnFilter_iff {ι : Type _} {F : ι → β → α} {f : β → α} {p : Filter ι}
     {p' : Filter β} :
     TendstoUniformlyOnFilter F f p p' ↔
-      ∀ ε > 0, ∀ᶠ n : ι × β in p ×ᶠ p', dist (f n.snd) (F n.fst n.snd) < ε := by
+      ∀ ε > 0, ∀ᶠ n : ι × β in p ×ˢ p', dist (f n.snd) (F n.fst n.snd) < ε := by
   refine' ⟨fun H ε hε => H _ (dist_mem_uniformity hε), fun H u hu => _⟩
   rcases mem_uniformity_dist.1 hu with ⟨ε, εpos, hε⟩
   refine' (H ε εpos).mono fun n hn => hε hn
@@ -959,7 +959,7 @@ theorem eventually_nhds_iff_ball {p : α → Prop} :
 in a pseudo-metric space.-/
 theorem eventually_nhds_prod_iff {ι α} [PseudoMetricSpace α] {f : Filter ι} {x₀ : α}
     {p : α × ι → Prop} :
-    (∀ᶠ x in 𝓝 x₀ ×ᶠ f, p x) ↔ ∃ ε > (0 : ℝ), ∃ pa : ι → Prop, (∀ᶠ i in f, pa i) ∧
+    (∀ᶠ x in 𝓝 x₀ ×ˢ f, p x) ↔ ∃ ε > (0 : ℝ), ∃ pa : ι → Prop, (∀ᶠ i in f, pa i) ∧
       ∀ {x}, dist x x₀ < ε → ∀ {i}, pa i → p (x, i) := by
   refine (nhds_basis_ball.prod f.basis_sets).eventually_iff.trans ?_
   simp only [Prod.exists, forall_prod_set, id, mem_ball, and_assoc, exists_and_left, and_imp]
@@ -969,7 +969,7 @@ theorem eventually_nhds_prod_iff {ι α} [PseudoMetricSpace α] {f : Filter ι}
 /-- A version of `Filter.eventually_prod_iff` where the second filter consists of neighborhoods
 in a pseudo-metric space.-/
 theorem eventually_prod_nhds_iff {f : Filter ι} {x₀ : α} {p : ι × α → Prop} :
-    (∀ᶠ x in f ×ᶠ 𝓝 x₀, p x) ↔ ∃ pa : ι → Prop, (∀ᶠ i in f, pa i) ∧
+    (∀ᶠ x in f ×ˢ 𝓝 x₀, p x) ↔ ∃ pa : ι → Prop, (∀ᶠ i in f, pa i) ∧
       ∃ ε > 0, ∀ {i}, pa i → ∀ {x}, dist x x₀ < ε → p (i, x) := by
   rw [eventually_swap_iff, Metric.eventually_nhds_prod_iff]
   constructor <;>
@@ -2421,7 +2421,7 @@ theorem bounded_range_iff {f : β → α} : Bounded (range f) ↔ ∃ C, ∀ x y
 #align metric.bounded_range_iff Metric.bounded_range_iff
 
 theorem bounded_range_of_tendsto_cofinite_uniformity {f : β → α}
-    (hf : Tendsto (Prod.map f f) (cofinite ×ᶠ cofinite) (𝓤 α)) : Bounded (range f) := by
+    (hf : Tendsto (Prod.map f f) (.cofinite ×ˢ .cofinite) (𝓤 α)) : Bounded (range f) := by
   rcases (hasBasis_cofinite.prod_self.tendsto_iff uniformity_basis_dist).1 hf 1 zero_lt_one with
     ⟨s, hsf, hs1⟩
   rw [← image_univ, ← union_compl_self s, image_union, bounded_union]

Positivity extensions for NonnegHomClass (this includes AbsoluteValue and Seminorm), IsAbsoluteValue, norm, the NNReal-to-Real coercion, factorials, square roots, distance (in a metric space), and diameter.

I tried to do these "properly" using Qq but I hit various errors I couldn't fix -- see https://leanprover.zulipchat.com/#narrow/stream/287929-mathlib4/topic/Qq.20doesn't.20know.20that.20two.20things.20have.20the.20same.20type for some examples.

cc @dwrensha

Co-authored-by: Floris van Doorn <fpvdoorn@gmail.com>

@@ -274,21 +274,20 @@ theorem dist_nonneg {x y : α} : 0 ≤ dist x y :=
   dist_nonneg' dist dist_self dist_comm dist_triangle
 #align dist_nonneg dist_nonneg
 
-/-
-section
+namespace Mathlib.Meta.Positivity
 
-open Tactic Tactic.Positivity
--- porting note: todo: restore `positivity` plugin
+open Lean Meta Qq Function
 
 /-- Extension for the `positivity` tactic: distances are nonnegative. -/
-@[positivity]
-unsafe def _root_.tactic.positivity_dist : expr → tactic strictness
-  | q(dist $(a) $(b)) => nonnegative <$> mk_app `` dist_nonneg [a, b]
-  | _ => failed
-#align tactic.positivity_dist tactic.positivity_dist
+@[positivity Dist.dist _ _]
+def evalDist : PositivityExt where eval {_ _} _zα _pα e := do
+  let .app (.app _ a) b ← whnfR e | throwError "not dist"
+  let p ← mkAppOptM ``dist_nonneg #[none, none, a, b]
+  pure (.nonnegative p)
 
-end
--/
+end Mathlib.Meta.Positivity
+
+example {x y : α} : 0 ≤ dist x y := by positivity
 
 @[simp] theorem abs_dist {a b : α} : |dist a b| = dist a b := abs_of_nonneg dist_nonneg
 #align abs_dist abs_dist
@@ -2777,21 +2776,18 @@ theorem exists_local_min_mem_ball [ProperSpace α] [TopologicalSpace β]
 
 end Metric
 
-/-
-Porting note: restore positivity extension
-namespace Tactic
+namespace Mathlib.Meta.Positivity
 
-open Positivity
+open Lean Meta Qq Function
 
 /-- Extension for the `positivity` tactic: the diameter of a set is always nonnegative. -/
-@[positivity]
-unsafe def positivity_diam : expr → tactic strictness
-  | q(Metric.diam $(s)) => nonnegative <$> mk_app `` Metric.diam_nonneg [s]
-  | e => pp e >>= fail ∘ format.bracket "The expression " " is not of the form `metric.diam s`"
-#align tactic.positivity_diam tactic.positivity_diam
+@[positivity Metric.diam _]
+def evalDiam : PositivityExt where eval {_ _} _zα _pα e := do
+  let .app _ s ← whnfR e | throwError "not Metric.diam"
+  let p ← mkAppOptM ``Metric.diam_nonneg #[none, none, s]
+  pure (.nonnegative p)
 
-end Tactic
--/
+end Mathlib.Meta.Positivity
 
 theorem comap_dist_right_atTop_le_cocompact (x : α) :
     comap (fun y => dist y x) atTop ≤ cocompact α := by

Co-authored-by: Mauricio Collares <mauricio@collares.org>

@@ -2132,8 +2132,8 @@ theorem isCompact_sphere {α : Type _} [PseudoMetricSpace α] [ProperSpace α] (
 #align is_compact_sphere isCompact_sphere
 
 /-- In a proper pseudometric space, any sphere is a `CompactSpace` when considered as a subtype. -/
-instance {α : Type _} [PseudoMetricSpace α] [ProperSpace α] (x : α) (r : ℝ) :
-    CompactSpace (sphere x r) :=
+instance Metric.sphere.compactSpace {α : Type _} [PseudoMetricSpace α] [ProperSpace α]
+    (x : α) (r : ℝ) : CompactSpace (sphere x r) :=
   isCompact_iff_compactSpace.mp (isCompact_sphere _ _)
 
 -- see Note [lower instance priority]

@@ -75,11 +75,12 @@ def Bornology.ofDist {α : Type _} (dist : α → α → ℝ) (dist_comm : ∀ x
   Bornology.ofBounded { s : Set α | ∃ C, ∀ ⦃x⦄, x ∈ s → ∀ ⦃y⦄, y ∈ s → dist x y ≤ C }
     ⟨0, fun x hx y => hx.elim⟩ (fun s ⟨c, hc⟩ t h => ⟨c, fun x hx y hy => hc (h hx) (h hy)⟩)
     (fun s hs t ht => by
-      rcases s.eq_empty_or_nonempty with rfl | ⟨x, hx⟩; · rwa [empty_union]
-      rcases t.eq_empty_or_nonempty with rfl | ⟨y, hy⟩; · rwa [union_empty]
-      suffices : ∃ C, ∀ z ∈ s ∪ t, dist x z ≤ C
-      · rcases this with ⟨C, hC⟩
-        refine ⟨C + C, fun a ha b hb => (dist_triangle a x b).trans ?_⟩
+      rcases s.eq_empty_or_nonempty with rfl | ⟨x, hx⟩
+      · rwa [empty_union]
+      rcases t.eq_empty_or_nonempty with rfl | ⟨y, hy⟩
+      · rwa [union_empty]
+      rsuffices ⟨C, hC⟩ : ∃ C, ∀ z ∈ s ∪ t, dist x z ≤ C
+      · refine ⟨C + C, fun a ha b hb => (dist_triangle a x b).trans ?_⟩
         simpa only [dist_comm] using add_le_add (hC _ ha) (hC _ hb)
       rcases hs with ⟨Cs, hs⟩; rcases ht with ⟨Ct, ht⟩
       refine ⟨max Cs (dist x y + Ct), fun z hz => hz.elim
@@ -100,7 +101,7 @@ export Dist (dist)
 
 -- the uniform structure and the emetric space structure are embedded in the metric space structure
 -- to avoid instance diamond issues. See Note [forgetful inheritance].
-/-- This is an internal lemma used inside the default of `pseudo_metric_space.edist`. -/
+/-- This is an internal lemma used inside the default of `PseudoMetricSpace.edist`. -/
 private theorem dist_nonneg' {α} {x y : α} (dist : α → α → ℝ)
     (dist_self : ∀ x : α, dist x x = 0) (dist_comm : ∀ x y : α, dist x y = dist y x)
     (dist_triangle : ∀ x y z : α, dist x z ≤ dist x y + dist y z) : 0 ≤ dist x y :=
@@ -228,9 +229,9 @@ theorem dist_triangle4_right (x₁ y₁ x₂ y₂ : α) :
 /-- The triangle (polygon) inequality for sequences of points; `Finset.Ico` version. -/
 theorem dist_le_Ico_sum_dist (f : ℕ → α) {m n} (h : m ≤ n) :
     dist (f m) (f n) ≤ ∑ i in Finset.Ico m n, dist (f i) (f (i + 1)) := by
-  induction n, h using Nat.le_induction
-  case base => rw [Finset.Ico_self, Finset.sum_empty, dist_self]
-  case succ n hle ihn =>
+  induction n, h using Nat.le_induction with
+  | base => rw [Finset.Ico_self, Finset.sum_empty, dist_self]
+  | succ n hle ihn =>
     calc
       dist (f m) (f (n + 1)) ≤ dist (f m) (f n) + dist (f n) (f (n + 1)) := dist_triangle _ _ _
       _ ≤ (∑ i in Finset.Ico m n, _) + _ := add_le_add ihn le_rfl
@@ -473,7 +474,7 @@ theorem iUnion_ball_nat_succ (x : α) : (⋃ n : ℕ, ball x (n + 1)) = univ :=
   iUnion_eq_univ_iff.2 fun y => (exists_nat_gt (dist y x)).imp fun _ h => h.trans (lt_add_one _)
 #align metric.Union_ball_nat_succ Metric.iUnion_ball_nat_succ
 
-/-- `closed_ball x ε` is the set of all points `y` with `dist y x ≤ ε` -/
+/-- `closedBall x ε` is the set of all points `y` with `dist y x ≤ ε` -/
 def closedBall (x : α) (ε : ℝ) :=
   { y | dist y x ≤ ε }
 #align metric.closed_ball Metric.closedBall
@@ -838,7 +839,7 @@ nonrec theorem uniformInducing_iff [PseudoMetricSpace β] {f : α → β} :
       ∀ δ > 0, ∃ ε > 0, ∀ {a b : α}, dist (f a) (f b) < ε → dist a b < δ :=
   uniformInducing_iff'.trans <| Iff.rfl.and <|
     ((uniformity_basis_dist.comap _).le_basis_iff uniformity_basis_dist).trans <| by
-      simp only [subset_def, Prod.forall]; rfl
+      simp only [subset_def, Prod.forall, gt_iff_lt, preimage_setOf_eq, Prod_map, mem_setOf]
 
 nonrec theorem uniformEmbedding_iff [PseudoMetricSpace β] {f : α → β} :
     UniformEmbedding f ↔ Function.Injective f ∧ UniformContinuous f ∧
@@ -1101,14 +1102,14 @@ theorem tendsto_atTop [Nonempty β] [SemilatticeSup β] {u : β → α} {a : α}
     simp only [true_and]; rfl
 #align metric.tendsto_at_top Metric.tendsto_atTop
 
-/-- A variant of `tendsto_at_top` that
+/-- A variant of `tendsto_atTop` that
 uses `∃ N, ∀ n > N, ...` rather than `∃ N, ∀ n ≥ N, ...`
 -/
-theorem tendsto_at_top' [Nonempty β] [SemilatticeSup β] [NoMaxOrder β] {u : β → α} {a : α} :
+theorem tendsto_atTop' [Nonempty β] [SemilatticeSup β] [NoMaxOrder β] {u : β → α} {a : α} :
     Tendsto u atTop (𝓝 a) ↔ ∀ ε > 0, ∃ N, ∀ n > N, dist (u n) a < ε :=
   (atTop_basis_Ioi.tendsto_iff nhds_basis_ball).trans <| by
-    simp only [true_and]; rfl
-#align metric.tendsto_at_top' Metric.tendsto_at_top'
+    simp only [true_and, gt_iff_lt, mem_Ioi, mem_ball]
+#align metric.tendsto_at_top' Metric.tendsto_atTop'
 
 theorem isOpen_singleton_iff {α : Type _} [PseudoMetricSpace α] {x : α} :
     IsOpen ({x} : Set α) ↔ ∃ ε > 0, ∀ y, dist y x < ε → y = x := by
@@ -1596,8 +1597,8 @@ theorem cauchySeq_iff_le_tendsto_0 {s : ℕ → α} :
     rw [Real.dist_0_eq_abs, abs_of_nonneg (S0 n)]
     refine' lt_of_le_of_lt (csSup_le ⟨_, S0m _⟩ _) (half_lt_self ε0)
     rintro _ ⟨⟨m', n'⟩, ⟨hm', hn'⟩, rfl⟩
-    exact le_of_lt (hN _ (le_trans hn hm') _ (le_trans hn hn')), fun ⟨b, _, b_bound, b_lim⟩ =>
-    cauchySeq_of_le_tendsto_0 b b_bound b_lim⟩
+    exact le_of_lt (hN _ (le_trans hn hm') _ (le_trans hn hn')),
+   fun ⟨b, _, b_bound, b_lim⟩ => cauchySeq_of_le_tendsto_0 b b_bound b_lim⟩
 #align cauchy_seq_iff_le_tendsto_0 cauchySeq_iff_le_tendsto_0
 
 end CauchySeq
@@ -1700,8 +1701,10 @@ theorem NNReal.nndist_zero_eq_val' (z : ℝ≥0) : nndist z 0 = z := by
 #align nnreal.nndist_zero_eq_val' NNReal.nndist_zero_eq_val'
 
 theorem NNReal.le_add_nndist (a b : ℝ≥0) : a ≤ b + nndist a b := by
-  suffices (a : ℝ) ≤ (b : ℝ) + dist a b by exact NNReal.coe_le_coe.mp this
-  linarith [le_of_abs_le (by rfl : abs (a - b : ℝ) ≤ dist a b)]
+  suffices (a : ℝ) ≤ (b : ℝ) + dist a b by
+    rwa [← NNReal.coe_le_coe, NNReal.coe_add, coe_nndist]
+  rw [← sub_le_iff_le_add']
+  exact le_of_abs_le (dist_eq a b).ge
 #align nnreal.le_add_nndist NNReal.le_add_nndist
 
 end NNReal
@@ -2042,7 +2045,7 @@ theorem dist_le_pi_dist (f g : ∀ b, π b) (b : β) : dist (f b) (g b) ≤ dist
   simp only [dist_nndist, NNReal.coe_le_coe, nndist_le_pi_nndist f g b]
 #align dist_le_pi_dist dist_le_pi_dist
 
-/-- An open ball in a product space is a product of open balls. See also `metric.ball_pi'`
+/-- An open ball in a product space is a product of open balls. See also `ball_pi'`
 for a version assuming `Nonempty β` instead of `0 < r`. -/
 theorem ball_pi (x : ∀ b, π b) {r : ℝ} (hr : 0 < r) :
     ball x r = Set.pi univ fun b => ball (x b) r := by
@@ -2050,14 +2053,14 @@ theorem ball_pi (x : ∀ b, π b) {r : ℝ} (hr : 0 < r) :
   simp [dist_pi_lt_iff hr]
 #align ball_pi ball_pi
 
-/-- An open ball in a product space is a product of open balls. See also `metric.ball_pi`
+/-- An open ball in a product space is a product of open balls. See also `ball_pi`
 for a version assuming `0 < r` instead of `Nonempty β`. -/
 theorem ball_pi' [Nonempty β] (x : ∀ b, π b) (r : ℝ) :
     ball x r = Set.pi univ fun b => ball (x b) r :=
   (lt_or_le 0 r).elim (ball_pi x) fun hr => by simp [ball_eq_empty.2 hr]
 #align ball_pi' ball_pi'
 
-/-- A closed ball in a product space is a product of closed balls. See also `metric.closed_ball_pi'`
+/-- A closed ball in a product space is a product of closed balls. See also `closedBall_pi'`
 for a version assuming `Nonempty β` instead of `0 ≤ r`. -/
 theorem closedBall_pi (x : ∀ b, π b) {r : ℝ} (hr : 0 ≤ r) :
     closedBall x r = Set.pi univ fun b => closedBall (x b) r := by
@@ -2065,7 +2068,7 @@ theorem closedBall_pi (x : ∀ b, π b) {r : ℝ} (hr : 0 ≤ r) :
   simp [dist_pi_le_iff hr]
 #align closed_ball_pi closedBall_pi
 
-/-- A closed ball in a product space is a product of closed balls. See also `metric.closed_ball_pi`
+/-- A closed ball in a product space is a product of closed balls. See also `closedBall_pi`
 for a version assuming `0 ≤ r` instead of `Nonempty β`. -/
 theorem closedBall_pi' [Nonempty β] (x : ∀ b, π b) (r : ℝ) :
     closedBall x r = Set.pi univ fun b => closedBall (x b) r :=
@@ -2933,7 +2936,7 @@ theorem uniformEmbedding_iff' [MetricSpace β] {f : γ → β} :
 def _root_.MetricSpace.ofT0PseudoMetricSpace (α : Type _) [PseudoMetricSpace α] [T0Space α] :
     MetricSpace α where
   toPseudoMetricSpace := ‹_›
-  eq_of_dist_eq_zero := fun hdist => (Metric.inseparable_iff.2 hdist).eq
+  eq_of_dist_eq_zero hdist := (Metric.inseparable_iff.2 hdist).eq
 #align metric_space.of_t0_pseudo_metric_space MetricSpace.ofT0PseudoMetricSpace
 
 -- see Note [lower instance priority]

As discussed on Zulip

Renames

• supₛ → sSup
• infₛ → sInf
• supᵢ → iSup
• infᵢ → iInf
• bsupₛ → bsSup
• binfₛ → bsInf
• bsupᵢ → biSup
• binfᵢ → biInf
• csupₛ → csSup
• cinfₛ → csInf
• csupᵢ → ciSup
• cinfᵢ → ciInf
• unionₛ → sUnion
• interₛ → sInter
• unionᵢ → iUnion
• interᵢ → iInter
• bunionₛ → bsUnion
• binterₛ → bsInter
• bunionᵢ → biUnion
• binterᵢ → biInter

Co-authored-by: Parcly Taxel <reddeloostw@gmail.com>

@@ -35,7 +35,7 @@ Additional useful definitions:
 * `Metric.closedBall x ε`: The set of all points `y` with `dist y x ≤ ε`.
 * `Metric.sphere x ε`: The set of all points `y` with `dist y x = ε`.
 * `ProperSpace α`: A `PseudoMetricSpace` where all closed balls are compact.
-* `Metric.diam s` : The `supᵢ` of the distances of members of `s`.
+* `Metric.diam s` : The `iSup` of the distances of members of `s`.
   Defined in terms of `EMetric.diam`, for better handling of the case when it should be infinite.
 
 TODO (anyone): Add "Main results" section.
@@ -464,14 +464,14 @@ theorem ball_eq_ball' (ε : ℝ) (x : α) :
 #align metric.ball_eq_ball' Metric.ball_eq_ball'
 
 @[simp]
-theorem unionᵢ_ball_nat (x : α) : (⋃ n : ℕ, ball x n) = univ :=
-  unionᵢ_eq_univ_iff.2 fun y => exists_nat_gt (dist y x)
-#align metric.Union_ball_nat Metric.unionᵢ_ball_nat
+theorem iUnion_ball_nat (x : α) : (⋃ n : ℕ, ball x n) = univ :=
+  iUnion_eq_univ_iff.2 fun y => exists_nat_gt (dist y x)
+#align metric.Union_ball_nat Metric.iUnion_ball_nat
 
 @[simp]
-theorem unionᵢ_ball_nat_succ (x : α) : (⋃ n : ℕ, ball x (n + 1)) = univ :=
-  unionᵢ_eq_univ_iff.2 fun y => (exists_nat_gt (dist y x)).imp fun _ h => h.trans (lt_add_one _)
-#align metric.Union_ball_nat_succ Metric.unionᵢ_ball_nat_succ
+theorem iUnion_ball_nat_succ (x : α) : (⋃ n : ℕ, ball x (n + 1)) = univ :=
+  iUnion_eq_univ_iff.2 fun y => (exists_nat_gt (dist y x)).imp fun _ h => h.trans (lt_add_one _)
+#align metric.Union_ball_nat_succ Metric.iUnion_ball_nat_succ
 
 /-- `closed_ball x ε` is the set of all points `y` with `dist y x ≤ ε` -/
 def closedBall (x : α) (ε : ℝ) :=
@@ -585,7 +585,7 @@ theorem ball_subset_ball (h : ε₁ ≤ ε₂) : ball x ε₁ ⊆ ball x ε₂ :
 #align metric.ball_subset_ball Metric.ball_subset_ball
 
 theorem closedBall_eq_bInter_ball : closedBall x ε = ⋂ δ > ε, ball x δ := by
-  ext y; rw [mem_closedBall, ← forall_lt_iff_le', mem_interᵢ₂]; rfl
+  ext y; rw [mem_closedBall, ← forall_lt_iff_le', mem_iInter₂]; rfl
 #align metric.closed_ball_eq_bInter_ball Metric.closedBall_eq_bInter_ball
 
 theorem ball_subset_ball' (h : ε₁ + dist x y ≤ ε₂) : ball x ε₁ ⊆ ball y ε₂ := fun z hz =>
@@ -649,13 +649,13 @@ theorem dist_lt_add_of_nonempty_ball_inter_ball (h : (ball x ε₁ ∩ ball y ε
 #align metric.dist_lt_add_of_nonempty_ball_inter_ball Metric.dist_lt_add_of_nonempty_ball_inter_ball
 
 @[simp]
-theorem unionᵢ_closedBall_nat (x : α) : (⋃ n : ℕ, closedBall x n) = univ :=
-  unionᵢ_eq_univ_iff.2 fun y => exists_nat_ge (dist y x)
-#align metric.Union_closed_ball_nat Metric.unionᵢ_closedBall_nat
+theorem iUnion_closedBall_nat (x : α) : (⋃ n : ℕ, closedBall x n) = univ :=
+  iUnion_eq_univ_iff.2 fun y => exists_nat_ge (dist y x)
+#align metric.Union_closed_ball_nat Metric.iUnion_closedBall_nat
 
-theorem unionᵢ_inter_closedBall_nat (s : Set α) (x : α) : (⋃ n : ℕ, s ∩ closedBall x n) = s := by
-  rw [← inter_unionᵢ, unionᵢ_closedBall_nat, inter_univ]
-#align metric.Union_inter_closed_ball_nat Metric.unionᵢ_inter_closedBall_nat
+theorem iUnion_inter_closedBall_nat (s : Set α) (x : α) : (⋃ n : ℕ, s ∩ closedBall x n) = s := by
+  rw [← inter_iUnion, iUnion_closedBall_nat, inter_univ]
+#align metric.Union_inter_closed_ball_nat Metric.iUnion_inter_closedBall_nat
 
 theorem ball_subset (h : dist x y ≤ ε₂ - ε₁) : ball x ε₁ ⊆ ball y ε₂ := fun z zx => by
   rw [← add_sub_cancel'_right ε₁ ε₂]
@@ -876,7 +876,7 @@ theorem totallyBounded_of_finite_discretization {s : Set α}
   refine' ⟨range (Subtype.val ∘ Finv), finite_range _, fun x xs => _⟩
   let x' := Finv (F ⟨x, xs⟩)
   have : F x' = F ⟨x, xs⟩ := Function.invFun_eq ⟨⟨x, xs⟩, rfl⟩
-  simp only [Set.mem_unionᵢ, Set.mem_range]
+  simp only [Set.mem_iUnion, Set.mem_range]
   exact ⟨_, ⟨F ⟨x, xs⟩, rfl⟩, hF _ _ this.symm⟩
 #align metric.totally_bounded_of_finite_discretization Metric.totallyBounded_of_finite_discretization
 
@@ -1152,13 +1152,13 @@ distance coincide. -/
 theorem Metric.uniformity_edist_aux {α} (d : α → α → ℝ≥0) :
     (⨅ ε > (0 : ℝ), 𝓟 { p : α × α | ↑(d p.1 p.2) < ε }) =
       ⨅ ε > (0 : ℝ≥0∞), 𝓟 { p : α × α | ↑(d p.1 p.2) < ε } := by
-  simp only [le_antisymm_iff, le_infᵢ_iff, le_principal_iff]
+  simp only [le_antisymm_iff, le_iInf_iff, le_principal_iff]
   refine ⟨fun ε hε => ?_, fun ε hε => ?_⟩
   · rcases ENNReal.lt_iff_exists_nnreal_btwn.1 hε with ⟨ε', ε'0, ε'ε⟩
-    refine mem_infᵢ_of_mem (ε' : ℝ) (mem_infᵢ_of_mem (ENNReal.coe_pos.1 ε'0) ?_)
+    refine mem_iInf_of_mem (ε' : ℝ) (mem_iInf_of_mem (ENNReal.coe_pos.1 ε'0) ?_)
     exact fun x hx => lt_trans (ENNReal.coe_lt_coe.2 hx) ε'ε
   · lift ε to ℝ≥0 using le_of_lt hε
-    refine mem_infᵢ_of_mem (ε : ℝ≥0∞) (mem_infᵢ_of_mem (ENNReal.coe_pos.2 hε) ?_)
+    refine mem_iInf_of_mem (ε : ℝ≥0∞) (mem_iInf_of_mem (ENNReal.coe_pos.2 hε) ?_)
     exact fun _ => ENNReal.coe_lt_coe.1
 
 theorem Metric.uniformity_edist : 𝓤 α = ⨅ ε > 0, 𝓟 { p : α × α | edist p.1 p.2 < ε } := by
@@ -1388,7 +1388,7 @@ theorem Real.dist_le_of_mem_Icc_01 {x y : ℝ} (hx : x ∈ Icc (0 : ℝ) 1) (hy
 
 instance : OrderTopology ℝ :=
   orderTopology_of_nhds_abs fun x => by
-    simp only [nhds_basis_ball.eq_binfᵢ, ball, Real.dist_eq, abs_sub_comm]
+    simp only [nhds_basis_ball.eq_biInf, ball, Real.dist_eq, abs_sub_comm]
 
 theorem Real.ball_eq_Ioo (x r : ℝ) : ball x r = Ioo (x - r) (x + r) :=
   Set.ext fun y => by
@@ -1586,15 +1586,15 @@ theorem cauchySeq_iff_le_tendsto_0 {s : ℕ → α} :
       rintro _ ⟨⟨m, n⟩, _, rfl⟩
       exact le_of_lt (hR m n)
     -- Prove that it bounds the distances of points in the Cauchy sequence
-    have ub : ∀ m n N, N ≤ m → N ≤ n → dist (s m) (s n) ≤ supₛ (S N) := fun m n N hm hn =>
-      le_csupₛ (hS N) ⟨⟨_, _⟩, ⟨hm, hn⟩, rfl⟩
+    have ub : ∀ m n N, N ≤ m → N ≤ n → dist (s m) (s n) ≤ sSup (S N) := fun m n N hm hn =>
+      le_csSup (hS N) ⟨⟨_, _⟩, ⟨hm, hn⟩, rfl⟩
     have S0m : ∀ n, (0 : ℝ) ∈ S n := fun n => ⟨⟨n, n⟩, ⟨le_rfl, le_rfl⟩, dist_self _⟩
-    have S0 := fun n => le_csupₛ (hS n) (S0m n)
+    have S0 := fun n => le_csSup (hS n) (S0m n)
     -- Prove that it tends to `0`, by using the Cauchy property of `s`
-    refine' ⟨fun N => supₛ (S N), S0, ub, Metric.tendsto_atTop.2 fun ε ε0 => _⟩
+    refine' ⟨fun N => sSup (S N), S0, ub, Metric.tendsto_atTop.2 fun ε ε0 => _⟩
     refine' (Metric.cauchySeq_iff.1 hs (ε / 2) (half_pos ε0)).imp fun N hN n hn => _
     rw [Real.dist_0_eq_abs, abs_of_nonneg (S0 n)]
-    refine' lt_of_le_of_lt (csupₛ_le ⟨_, S0m _⟩ _) (half_lt_self ε0)
+    refine' lt_of_le_of_lt (csSup_le ⟨_, S0m _⟩ _) (half_lt_self ε0)
     rintro _ ⟨⟨m', n'⟩, ⟨hm', hn'⟩, rfl⟩
     exact le_of_lt (hN _ (le_trans hn hm') _ (le_trans hn hn')), fun ⟨b, _, b_bound, b_lim⟩ =>
     cauchySeq_of_le_tendsto_0 b b_bound b_lim⟩
@@ -1613,7 +1613,7 @@ def PseudoMetricSpace.induced {α β} (f : α → β) (m : PseudoMetricSpace β)
   edist x y := edist (f x) (f y)
   edist_dist x y := edist_dist _ _
   toUniformSpace := UniformSpace.comap f m.toUniformSpace
-  uniformity_dist := (uniformity_basis_dist.comap _).eq_binfᵢ
+  uniformity_dist := (uniformity_basis_dist.comap _).eq_biInf
   toBornology := Bornology.induced f
   cobounded_sets := Set.ext fun s => mem_comap_iff_compl.trans <| by
     simp only [← isBounded_def, isBounded_iff, ball_image_iff, mem_setOf]
@@ -2141,8 +2141,8 @@ instance (priority := 100) secondCountable_of_proper [ProperSpace α] :
   -- add an instance for `SigmaCompactSpace`.
   suffices SigmaCompactSpace α by exact EMetric.secondCountable_of_sigmaCompact α
   rcases em (Nonempty α) with (⟨⟨x⟩⟩ | hn)
-  · exact ⟨⟨fun n => closedBall x n, fun n => isCompact_closedBall _ _, unionᵢ_closedBall_nat _⟩⟩
-  · exact ⟨⟨fun _ => ∅, fun _ => isCompact_empty, unionᵢ_eq_univ_iff.2 fun x => (hn ⟨x⟩).elim⟩⟩
+  · exact ⟨⟨fun n => closedBall x n, fun n => isCompact_closedBall _ _, iUnion_closedBall_nat _⟩⟩
+  · exact ⟨⟨fun _ => ∅, fun _ => isCompact_empty, iUnion_eq_univ_iff.2 fun x => (hn ⟨x⟩).elim⟩⟩
 #align second_countable_of_proper secondCountable_of_proper
 
 theorem tendsto_dist_right_cocompact_atTop [ProperSpace α] (x : α) :
@@ -2259,7 +2259,7 @@ theorem secondCountable_of_almost_dense_set
   refine' EMetric.secondCountable_of_almost_dense_set fun ε ε0 => _
   rcases ENNReal.lt_iff_exists_nnreal_btwn.1 ε0 with ⟨ε', ε'0, ε'ε⟩
   choose s hsc y hys hyx using H ε' (by exact_mod_cast ε'0)
-  refine' ⟨s, hsc, unionᵢ₂_eq_univ_iff.2 fun x => ⟨y x, hys _, le_trans _ ε'ε.le⟩⟩
+  refine' ⟨s, hsc, iUnion₂_eq_univ_iff.2 fun x => ⟨y x, hys _, le_trans _ ε'ε.le⟩⟩
   exact_mod_cast hyx x
 #align metric.second_countable_of_almost_dense_set Metric.secondCountable_of_almost_dense_set
 
@@ -2276,10 +2276,10 @@ theorem lebesgue_number_lemma_of_metric {s : Set α} {ι} {c : ι → Set α} (h
     ⟨i, fun _y hy => hi (hδ (mem_ball'.mp hy))⟩⟩
 #align lebesgue_number_lemma_of_metric lebesgue_number_lemma_of_metric
 
-theorem lebesgue_number_lemma_of_metric_unionₛ {s : Set α} {c : Set (Set α)} (hs : IsCompact s)
+theorem lebesgue_number_lemma_of_metric_sUnion {s : Set α} {c : Set (Set α)} (hs : IsCompact s)
     (hc₁ : ∀ t ∈ c, IsOpen t) (hc₂ : s ⊆ ⋃₀ c) : ∃ δ > 0, ∀ x ∈ s, ∃ t ∈ c, ball x δ ⊆ t := by
-  rw [unionₛ_eq_unionᵢ] at hc₂; simpa using lebesgue_number_lemma_of_metric hs (by simpa) hc₂
-#align lebesgue_number_lemma_of_metric_sUnion lebesgue_number_lemma_of_metric_unionₛ
+  rw [sUnion_eq_iUnion] at hc₂; simpa using lebesgue_number_lemma_of_metric hs (by simpa) hc₂
+#align lebesgue_number_lemma_of_metric_sUnion lebesgue_number_lemma_of_metric_sUnion
 
 namespace Metric
 
@@ -2376,10 +2376,10 @@ theorem Bounded.union (hs : Bounded s) (ht : Bounded t) : Bounded (s ∪ t) :=
 #align metric.bounded.union Metric.Bounded.union
 
 /-- A finite union of bounded sets is bounded -/
-theorem bounded_bunionᵢ {I : Set β} {s : β → Set α} (H : I.Finite) :
+theorem bounded_biUnion {I : Set β} {s : β → Set α} (H : I.Finite) :
     Bounded (⋃ i ∈ I, s i) ↔ ∀ i ∈ I, Bounded (s i) := by
-  simp only [bounded_iff_isBounded, isBounded_bunionᵢ H]
-#align metric.bounded_bUnion Metric.bounded_bunionᵢ
+  simp only [bounded_iff_isBounded, isBounded_biUnion H]
+#align metric.bounded_bUnion Metric.bounded_biUnion
 
 protected theorem Bounded.prod [PseudoMetricSpace β] {s : Set α} {t : Set β} (hs : Bounded s)
     (ht : Bounded t) : Bounded (s ×ˢ t) := by
@@ -2392,7 +2392,7 @@ theorem _root_.TotallyBounded.bounded {s : Set α} (h : TotallyBounded s) : Boun
   -- We cover the totally bounded set by finitely many balls of radius 1,
   -- and then argue that a finite union of bounded sets is bounded
   let  ⟨_t, fint, subs⟩ := (totallyBounded_iff.mp h) 1 zero_lt_one
-  Bounded.mono subs <| (bounded_bunionᵢ fint).2 fun _ _ => bounded_ball
+  Bounded.mono subs <| (bounded_biUnion fint).2 fun _ _ => bounded_ball
 #align totally_bounded.bounded TotallyBounded.bounded
 
 /-- A compact set is bounded -/
@@ -2730,7 +2730,7 @@ theorem diam_ball {r : ℝ} (h : 0 ≤ r) : diam (ball x r) ≤ 2 * r :=
 
 /-- If a family of complete sets with diameter tending to `0` is such that each finite intersection
 is nonempty, then the total intersection is also nonempty. -/
-theorem _root_.IsComplete.nonempty_interᵢ_of_nonempty_binterᵢ {s : ℕ → Set α}
+theorem _root_.IsComplete.nonempty_iInter_of_nonempty_biInter {s : ℕ → Set α}
     (h0 : IsComplete (s 0)) (hs : ∀ n, IsClosed (s n)) (h's : ∀ n, Bounded (s n))
     (h : ∀ N, (⋂ n ≤ N, s n).Nonempty) (h' : Tendsto (fun n => diam (s n)) atTop (𝓝 0)) :
     (⋂ n, s n).Nonempty := by
@@ -2739,7 +2739,7 @@ theorem _root_.IsComplete.nonempty_interᵢ_of_nonempty_binterᵢ {s : ℕ → S
     intro n N hn
     apply mem_of_subset_of_mem _ (h N).choose_spec
     intro x hx
-    simp only [mem_interᵢ] at hx
+    simp only [mem_iInter] at hx
     exact hx n hn
   have : CauchySeq u := by
     apply cauchySeq_of_le_tendsto_0 _ _ h'
@@ -2747,19 +2747,19 @@ theorem _root_.IsComplete.nonempty_interᵢ_of_nonempty_binterᵢ {s : ℕ → S
     exact dist_le_diam_of_mem (h's N) (I _ _ hm) (I _ _ hn)
   obtain ⟨x, -, xlim⟩ : ∃ x ∈ s 0, Tendsto (fun n : ℕ => u n) atTop (𝓝 x) :=
     cauchySeq_tendsto_of_isComplete h0 (fun n => I 0 n (zero_le _)) this
-  refine' ⟨x, mem_interᵢ.2 fun n => _⟩
+  refine' ⟨x, mem_iInter.2 fun n => _⟩
   apply (hs n).mem_of_tendsto xlim
   filter_upwards [Ici_mem_atTop n]with p hp
   exact I n p hp
-#align is_complete.nonempty_Inter_of_nonempty_bInter IsComplete.nonempty_interᵢ_of_nonempty_binterᵢ
+#align is_complete.nonempty_Inter_of_nonempty_bInter IsComplete.nonempty_iInter_of_nonempty_biInter
 
 /-- In a complete space, if a family of closed sets with diameter tending to `0` is such that each
 finite intersection is nonempty, then the total intersection is also nonempty. -/
-theorem nonempty_interᵢ_of_nonempty_binterᵢ [CompleteSpace α] {s : ℕ → Set α}
+theorem nonempty_iInter_of_nonempty_biInter [CompleteSpace α] {s : ℕ → Set α}
     (hs : ∀ n, IsClosed (s n)) (h's : ∀ n, Bounded (s n)) (h : ∀ N, (⋂ n ≤ N, s n).Nonempty)
     (h' : Tendsto (fun n => diam (s n)) atTop (𝓝 0)) : (⋂ n, s n).Nonempty :=
-  (hs 0).isComplete.nonempty_interᵢ_of_nonempty_binterᵢ hs h's h h'
-#align metric.nonempty_Inter_of_nonempty_bInter Metric.nonempty_interᵢ_of_nonempty_binterᵢ
+  (hs 0).isComplete.nonempty_iInter_of_nonempty_biInter hs h's h h'
+#align metric.nonempty_Inter_of_nonempty_bInter Metric.nonempty_iInter_of_nonempty_biInter
 
 end Diam
 

Enable the cancelDenoms preprocessor in linarith. Closes #2714.

• depends on: #3797

Co-authored-by: Kyle Miller <kmill31415@gmail.com> Co-authored-by: Patrick Massot <patrickmassot@free.fr> Co-authored-by: Floris van Doorn <fpvdoorn@gmail.com> Co-authored-by: Scott Morrison <scott.morrison@gmail.com>

@@ -449,8 +449,7 @@ contains it.
 See also `exists_lt_subset_ball`. -/
 theorem exists_lt_mem_ball_of_mem_ball (h : x ∈ ball y ε) : ∃ ε' < ε, x ∈ ball y ε' := by
   simp only [mem_ball] at h ⊢
-  -- porting note: todo: Mathlib 3 used `by linarith` here
-  exact ⟨(dist x y + ε) / 2, add_div_two_lt_right.2 h, left_lt_add_div_two.2 h⟩
+  exact ⟨(dist x y + ε) / 2, by linarith, by linarith⟩
 #align metric.exists_lt_mem_ball_of_mem_ball Metric.exists_lt_mem_ball_of_mem_ball
 
 theorem ball_eq_ball (ε : ℝ) (x : α) :

@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Jeremy Avigad, Robert Y. Lewis, Johannes Hölzl, Mario Carneiro, Sébastien Gouëzel
 
 ! This file was ported from Lean 3 source module topology.metric_space.basic
-! leanprover-community/mathlib commit 195fcd60ff2bfe392543bceb0ec2adcdb472db4c
+! leanprover-community/mathlib commit 8000bbbe2e9d39b84edb993d88781f536a8a3fa8
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
@@ -1872,6 +1872,11 @@ theorem closure_closedBall : closure (closedBall x ε) = closedBall x ε :=
   isClosed_ball.closure_eq
 #align metric.closure_closed_ball Metric.closure_closedBall
 
+@[simp]
+theorem closure_sphere : closure (sphere x ε) = sphere x ε :=
+  isClosed_sphere.closure_eq
+#align metric.closure_sphere Metric.closure_sphere
+
 theorem closure_ball_subset_closedBall : closure (ball x ε) ⊆ closedBall x ε :=
   closure_minimal ball_subset_closedBall isClosed_ball
 #align metric.closure_ball_subset_closed_ball Metric.closure_ball_subset_closedBall

Co-authored-by: Parcly Taxel <reddeloostw@gmail.com>

@@ -2726,9 +2726,10 @@ theorem diam_ball {r : ℝ} (h : 0 ≤ r) : diam (ball x r) ≤ 2 * r :=
 
 /-- If a family of complete sets with diameter tending to `0` is such that each finite intersection
 is nonempty, then the total intersection is also nonempty. -/
-theorem _root_.IsComplete.nonempty_interᵢ_of_nonempty_bInter {s : ℕ → Set α} (h0 : IsComplete (s 0))
-    (hs : ∀ n, IsClosed (s n)) (h's : ∀ n, Bounded (s n)) (h : ∀ N, (⋂ n ≤ N, s n).Nonempty)
-    (h' : Tendsto (fun n => diam (s n)) atTop (𝓝 0)) : (⋂ n, s n).Nonempty := by
+theorem _root_.IsComplete.nonempty_interᵢ_of_nonempty_binterᵢ {s : ℕ → Set α}
+    (h0 : IsComplete (s 0)) (hs : ∀ n, IsClosed (s n)) (h's : ∀ n, Bounded (s n))
+    (h : ∀ N, (⋂ n ≤ N, s n).Nonempty) (h' : Tendsto (fun n => diam (s n)) atTop (𝓝 0)) :
+    (⋂ n, s n).Nonempty := by
   let u N := (h N).some
   have I : ∀ n N, n ≤ N → u N ∈ s n := by
     intro n N hn
@@ -2746,15 +2747,15 @@ theorem _root_.IsComplete.nonempty_interᵢ_of_nonempty_bInter {s : ℕ → Set
   apply (hs n).mem_of_tendsto xlim
   filter_upwards [Ici_mem_atTop n]with p hp
   exact I n p hp
-#align is_complete.nonempty_Inter_of_nonempty_bInter IsComplete.nonempty_interᵢ_of_nonempty_bInter
+#align is_complete.nonempty_Inter_of_nonempty_bInter IsComplete.nonempty_interᵢ_of_nonempty_binterᵢ
 
 /-- In a complete space, if a family of closed sets with diameter tending to `0` is such that each
 finite intersection is nonempty, then the total intersection is also nonempty. -/
-theorem nonempty_interᵢ_of_nonempty_bInter [CompleteSpace α] {s : ℕ → Set α}
+theorem nonempty_interᵢ_of_nonempty_binterᵢ [CompleteSpace α] {s : ℕ → Set α}
     (hs : ∀ n, IsClosed (s n)) (h's : ∀ n, Bounded (s n)) (h : ∀ N, (⋂ n ≤ N, s n).Nonempty)
     (h' : Tendsto (fun n => diam (s n)) atTop (𝓝 0)) : (⋂ n, s n).Nonempty :=
-  (hs 0).isComplete.nonempty_interᵢ_of_nonempty_bInter hs h's h h'
-#align metric.nonempty_Inter_of_nonempty_bInter Metric.nonempty_interᵢ_of_nonempty_bInter
+  (hs 0).isComplete.nonempty_interᵢ_of_nonempty_binterᵢ hs h's h h'
+#align metric.nonempty_Inter_of_nonempty_bInter Metric.nonempty_interᵢ_of_nonempty_binterᵢ
 
 end Diam
 

Co-authored-by: Parcly Taxel <reddeloostw@gmail.com>

@@ -1282,8 +1282,7 @@ def PseudoEMetricSpace.toPseudoMetricSpaceOfDist {α : Type u} [e : PseudoEMetri
   dist_comm x y := by simp [h, edist_comm]
   dist_triangle x y z := by
     simp only [h]
-    exact (ENNReal.toReal_mono (ENNReal.add_ne_top.2 ⟨edist_ne_top _ _, edist_ne_top _ _⟩)
-      (edist_triangle _ _ _)).trans ENNReal.toReal_add_le
+    exact ENNReal.toReal_le_add (edist_triangle _ _ _) (edist_ne_top _ _) (edist_ne_top _ _)
   edist := edist
   edist_dist _ _ := by simp only [h, ENNReal.ofReal_toReal (edist_ne_top _ _)]
   toUniformSpace := e.toUniformSpace
@@ -2648,6 +2647,9 @@ theorem Bounded.ediam_ne_top (h : Bounded s) : EMetric.diam s ≠ ⊤ :=
   bounded_iff_ediam_ne_top.1 h
 #align metric.bounded.ediam_ne_top Metric.Bounded.ediam_ne_top
 
+theorem ediam_eq_top_iff_unbounded : EMetric.diam s = ⊤ ↔ ¬Bounded s :=
+  bounded_iff_ediam_ne_top.not_left.symm
+
 theorem ediam_univ_eq_top_iff_noncompact [ProperSpace α] :
     EMetric.diam (univ : Set α) = ∞ ↔ NoncompactSpace α := by
   rw [← not_compactSpace_iff, compactSpace_iff_bounded_univ, bounded_iff_ediam_ne_top,
@@ -2670,8 +2672,7 @@ theorem dist_le_diam_of_mem (h : Bounded s) (hx : x ∈ s) (hy : y ∈ s) : dist
   dist_le_diam_of_mem' h.ediam_ne_top hx hy
 #align metric.dist_le_diam_of_mem Metric.dist_le_diam_of_mem
 
-theorem ediam_of_unbounded (h : ¬Bounded s) : EMetric.diam s = ∞ := by
-  rwa [bounded_iff_ediam_ne_top, Classical.not_not] at h
+theorem ediam_of_unbounded (h : ¬Bounded s) : EMetric.diam s = ∞ := ediam_eq_top_iff_unbounded.2 h
 #align metric.ediam_of_unbounded Metric.ediam_of_unbounded
 
 /-- An unbounded set has zero diameter. If you would prefer to get the value ∞, use `EMetric.diam`.
@@ -2690,15 +2691,12 @@ any two points in each of the sets. This lemma is true without any side conditio
 obviously true if `s ∪ t` is unbounded. -/
 theorem diam_union {t : Set α} (xs : x ∈ s) (yt : y ∈ t) :
     diam (s ∪ t) ≤ diam s + dist x y + diam t := by
-  by_cases H : Bounded (s ∪ t)
-  · have hs : Bounded s := H.mono (subset_union_left _ _)
-    have ht : Bounded t := H.mono (subset_union_right _ _)
-    rw [bounded_iff_ediam_ne_top] at H hs ht
-    rw [dist_edist, diam, diam, diam, ← ENNReal.toReal_add, ← ENNReal.toReal_add,
-      ENNReal.toReal_le_toReal] <;> apply_rules [ENNReal.add_ne_top.2, And.intro, edist_ne_top]
-    exact EMetric.diam_union xs yt
-  · rw [diam_eq_zero_of_unbounded H]
-    apply_rules [add_nonneg, diam_nonneg, dist_nonneg]
+  simp only [diam, dist_edist]
+  refine (ENNReal.toReal_le_add' (EMetric.diam_union xs yt) ?_ ?_).trans
+    (add_le_add_right ENNReal.toReal_add_le _)
+  · simp only [ENNReal.add_eq_top, edist_ne_top, or_false]
+    exact fun h ↦ top_unique <| h ▸ EMetric.diam_mono (subset_union_left _ _)
+  · exact fun h ↦ top_unique <| h ▸ EMetric.diam_mono (subset_union_right _ _)
 #align metric.diam_union Metric.diam_union
 
 /-- If two sets intersect, the diameter of the union is bounded by the sum of the diameters. -/

vscode is already configured by .vscode/settings.json to trim these on save. It's not clear how they've managed to stick around.

By doing this all in one PR now, it avoids getting random whitespace diffs in PRs later.

This was done with a regex search in vscode,

@@ -2161,7 +2161,7 @@ theorem properSpace_of_compact_closedBall_of_le (R : ℝ)
     (closedBall_subset_closedBall <| le_max_left _ _)⟩
 #align proper_space_of_compact_closed_ball_of_le properSpace_of_compact_closedBall_of_le
 
--- A compact pseudometric space is proper 
+-- A compact pseudometric space is proper
 -- see Note [lower instance priority]
 instance (priority := 100) proper_of_compact [CompactSpace α] : ProperSpace α :=
   ⟨fun _ _ => isClosed_ball.isCompact⟩