topology.metric_space.closeds ⟷ Mathlib.Topology.MetricSpace.Closeds

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

@@ -209,7 +209,7 @@ instance Closeds.completeSpace [CompleteSpace α] : CompleteSpace (Closeds α) :
 #align emetric.closeds.complete_space EMetric.Closeds.completeSpace
 -/
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:641:2: warning: expanding binder collection (v «expr ⊆ » s) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:642:2: warning: expanding binder collection (v «expr ⊆ » s) -/
 #print EMetric.Closeds.compactSpace /-
 /-- In a compact space, the type of closed subsets is compact. -/
 instance Closeds.compactSpace [CompactSpace α] : CompactSpace (Closeds α) :=

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

@@ -90,7 +90,7 @@ theorem isClosed_subsets_of_isClosed (hs : IsClosed s) :
     rcases exists_edist_lt_of_Hausdorff_edist_lt hx Dtu with ⟨y, hy, Dxy⟩
     -- y : α,  hy : y ∈ u, Dxy : edist x y < ε
     exact ⟨y, hu hy, Dxy⟩
-  rwa [hs.closure_eq] at this 
+  rwa [hs.closure_eq] at this
 #align emetric.is_closed_subsets_of_is_closed EMetric.isClosed_subsets_of_isClosed
 -/
 
@@ -181,7 +181,7 @@ instance Closeds.completeSpace [CompleteSpace α] : CompleteSpace (Closeds α) :
     have : x ∈ closure (⋃ m ≥ n, s m : Set α) := by apply mem_Inter.1 xt0 n
     rcases mem_closure_iff.1 this (B n) (B_pos n) with ⟨z, hz, Dxz⟩
     -- z : α,  Dxz : edist x z < B n,
-    simp only [exists_prop, Set.mem_iUnion] at hz 
+    simp only [exists_prop, Set.mem_iUnion] at hz
     rcases hz with ⟨m, ⟨m_ge_n, hm⟩⟩
     -- m : ℕ, m_ge_n : m ≥ n, hm : z ∈ s m
     have : Hausdorff_edist (s m : Set α) (s n) < B n := hs n m n m_ge_n (le_refl n)
@@ -202,7 +202,7 @@ instance Closeds.completeSpace [CompleteSpace α] : CompleteSpace (Closeds α) :
     ENNReal.Tendsto.const_mul
       (ENNReal.tendsto_pow_atTop_nhds_zero_of_lt_one <| by simp [ENNReal.one_lt_two])
       (Or.inr <| by simp)
-  rw [MulZeroClass.mul_zero] at this 
+  rw [MulZeroClass.mul_zero] at this
   obtain ⟨N, hN⟩ : ∃ N, ∀ b ≥ N, ε > 2 * B b
   exact ((tendsto_order.1 this).2 ε εpos).exists_forall_of_atTop
   exact ⟨N, fun n hn => lt_of_le_of_lt (main n) (hN n hn)⟩
@@ -236,7 +236,7 @@ instance Closeds.compactSpace [CompactSpace α] : CompactSpace (Closeds α) :=
       · intro x hx
         have : x ∈ ⋃ y ∈ s, ball y δ := hs (by simp)
         rcases mem_Union₂.1 this with ⟨y, ys, dy⟩
-        have : edist y x < δ := by simp at dy  <;> rwa [edist_comm] at dy 
+        have : edist y x < δ := by simp at dy <;> rwa [edist_comm] at dy
         exact ⟨y, ⟨ys, ⟨x, hx, this⟩⟩, le_of_lt dy⟩
       · rintro x ⟨hx1, ⟨y, yu, hy⟩⟩
         exact ⟨y, yu, le_of_lt hy⟩
@@ -248,7 +248,7 @@ instance Closeds.compactSpace [CompactSpace α] : CompactSpace (Closeds α) :=
       · apply fs.finite_subsets.subset fun b => _
         simp only [and_imp, Set.mem_image, Set.mem_setOf_eq, exists_imp]
         intro x hx hx'
-        rwa [hx'] at hx 
+        rwa [hx'] at hx
       · exact set_like.coe_injective.inj_on F
     -- `F` is ε-dense
     · obtain ⟨t0, t0s, Dut0⟩ := main u
@@ -274,7 +274,7 @@ instance NonemptyCompacts.emetricSpace : EMetricSpace (NonemptyCompacts α)
     NonemptyCompacts.ext <|
       by
       have : closure (s : Set α) = closure t := Hausdorff_edist_zero_iff_closure_eq_closure.1 h
-      rwa [s.is_compact.is_closed.closure_eq, t.is_compact.is_closed.closure_eq] at this 
+      rwa [s.is_compact.is_closed.closure_eq, t.is_compact.is_closed.closure_eq] at this
 #align emetric.nonempty_compacts.emetric_space EMetric.NonemptyCompacts.emetricSpace
 -/
 
@@ -303,7 +303,7 @@ theorem NonemptyCompacts.isClosed_in_closeds [CompleteSpace α] :
   refine' isClosed_of_closure_subset fun s hs => ⟨_, _⟩
   · -- take a set set t which is nonempty and at a finite distance of s
     rcases mem_closure_iff.1 hs ⊤ ENNReal.coe_lt_top with ⟨t, ht, Dst⟩
-    rw [edist_comm] at Dst 
+    rw [edist_comm] at Dst
     -- since `t` is nonempty, so is `s`
     exact nonempty_of_Hausdorff_edist_ne_top ht.1 (ne_of_lt Dst)
   · refine' isCompact_iff_totallyBounded_isComplete.2 ⟨_, s.closed.is_complete⟩

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

@@ -200,7 +200,8 @@ instance Closeds.completeSpace [CompleteSpace α] : CompleteSpace (Closeds α) :
   refine' tendsto_at_top.2 fun ε εpos => _
   have : tendsto (fun n => 2 * B n) at_top (𝓝 (2 * 0)) :=
     ENNReal.Tendsto.const_mul
-      (ENNReal.tendsto_pow_atTop_nhds_0_of_lt_1 <| by simp [ENNReal.one_lt_two]) (Or.inr <| by simp)
+      (ENNReal.tendsto_pow_atTop_nhds_zero_of_lt_one <| by simp [ENNReal.one_lt_two])
+      (Or.inr <| by simp)
   rw [MulZeroClass.mul_zero] at this 
   obtain ⟨N, hN⟩ : ∃ N, ∀ b ≥ N, ε > 2 * B b
   exact ((tendsto_order.1 this).2 ε εpos).exists_forall_of_atTop

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

@@ -3,9 +3,9 @@ Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Sébastien Gouëzel
 -/
-import Mathbin.Analysis.SpecificLimits.Basic
-import Mathbin.Topology.MetricSpace.HausdorffDistance
-import Mathbin.Topology.Sets.Compacts
+import Analysis.SpecificLimits.Basic
+import Topology.MetricSpace.HausdorffDistance
+import Topology.Sets.Compacts
 
 #align_import topology.metric_space.closeds from "leanprover-community/mathlib"@"ce38d86c0b2d427ce208c3cee3159cb421d2b3c4"
 
@@ -208,7 +208,7 @@ instance Closeds.completeSpace [CompleteSpace α] : CompleteSpace (Closeds α) :
 #align emetric.closeds.complete_space EMetric.Closeds.completeSpace
 -/
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (v «expr ⊆ » s) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:641:2: warning: expanding binder collection (v «expr ⊆ » s) -/
 #print EMetric.Closeds.compactSpace /-
 /-- In a compact space, the type of closed subsets is compact. -/
 instance Closeds.compactSpace [CompactSpace α] : CompactSpace (Closeds α) :=

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

@@ -2,16 +2,13 @@
 Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Sébastien Gouëzel
-
-! This file was ported from Lean 3 source module topology.metric_space.closeds
-! leanprover-community/mathlib commit ce38d86c0b2d427ce208c3cee3159cb421d2b3c4
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
 -/
 import Mathbin.Analysis.SpecificLimits.Basic
 import Mathbin.Topology.MetricSpace.HausdorffDistance
 import Mathbin.Topology.Sets.Compacts
 
+#align_import topology.metric_space.closeds from "leanprover-community/mathlib"@"ce38d86c0b2d427ce208c3cee3159cb421d2b3c4"
+
 /-!
 # Closed subsets
 
@@ -211,7 +208,7 @@ instance Closeds.completeSpace [CompleteSpace α] : CompleteSpace (Closeds α) :
 #align emetric.closeds.complete_space EMetric.Closeds.completeSpace
 -/
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:638:2: warning: expanding binder collection (v «expr ⊆ » s) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (v «expr ⊆ » s) -/
 #print EMetric.Closeds.compactSpace /-
 /-- In a compact space, the type of closed subsets is compact. -/
 instance Closeds.compactSpace [CompactSpace α] : CompactSpace (Closeds α) :=

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

@@ -474,15 +474,19 @@ theorem lipschitz_infDist_set (x : α) : LipschitzWith 1 fun s : NonemptyCompact
 #align metric.lipschitz_inf_dist_set Metric.lipschitz_infDist_set
 -/
 
+#print Metric.lipschitz_infDist /-
 theorem lipschitz_infDist : LipschitzWith 2 fun p : α × NonemptyCompacts α => infDist p.1 p.2 :=
   @LipschitzWith.uncurry _ _ _ _ _ _ (fun (x : α) (s : NonemptyCompacts α) => infDist x s) 1 1
     (fun s => lipschitz_infDist_pt s) lipschitz_infDist_set
 #align metric.lipschitz_inf_dist Metric.lipschitz_infDist
+-/
 
+#print Metric.uniformContinuous_infDist_Hausdorff_dist /-
 theorem uniformContinuous_infDist_Hausdorff_dist :
     UniformContinuous fun p : α × NonemptyCompacts α => infDist p.1 p.2 :=
   lipschitz_infDist.UniformContinuous
 #align metric.uniform_continuous_inf_dist_Hausdorff_dist Metric.uniformContinuous_infDist_Hausdorff_dist
+-/
 
 end
 

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

@@ -75,7 +75,6 @@ theorem continuous_infEdist_Hausdorff_edist :
     _ ≤ inf_edist y t + (edist (x, s) (y, t) + edist (x, s) (y, t)) :=
       (add_le_add_left (add_le_add (le_max_left _ _) (le_max_right _ _)) _)
     _ = inf_edist y t + 2 * edist (x, s) (y, t) := by rw [← mul_two, mul_comm]
-    
 #align emetric.continuous_inf_edist_Hausdorff_edist Emetric.continuous_infEdist_Hausdorff_edist
 
 #print EMetric.isClosed_subsets_of_isClosed /-
@@ -196,8 +195,7 @@ instance Closeds.completeSpace [CompleteSpace α] : CompleteSpace (Closeds α) :
         calc
           edist x y ≤ edist x z + edist z y := edist_triangle _ _ _
           _ ≤ B n + B n := (add_le_add (le_of_lt Dxz) (le_of_lt Dzy))
-          _ = 2 * B n := (two_mul _).symm
-          ⟩
+          _ = 2 * B n := (two_mul _).symm⟩
   -- Deduce from the above inequalities that the distance between `s n` and `t0` is at most `2 B n`.
   have main : ∀ n : ℕ, edist (s n) t ≤ 2 * B n := fun n =>
     Hausdorff_edist_le_of_mem_edist (I1 n) (I2 n)
@@ -329,7 +327,6 @@ theorem NonemptyCompacts.isClosed_in_closeds [CompleteSpace α] :
         edist x y ≤ edist x z + edist z y := edist_triangle _ _ _
         _ < ε / 2 + ε / 2 := (ENNReal.add_lt_add Dxz Dzy)
         _ = ε := ENNReal.add_halves _
-        
     exact mem_bUnion hy this
 #align emetric.nonempty_compacts.is_closed_in_closeds EMetric.NonemptyCompacts.isClosed_in_closeds
 -/
@@ -401,7 +398,6 @@ instance NonemptyCompacts.secondCountableTopology [SecondCountableTopology α] :
           edist x (F z) ≤ edist x z + edist z (F z) := edist_triangle _ _ _
           _ < δ / 2 + δ / 2 := (ENNReal.add_lt_add Dxz (Fspec z).2)
           _ = δ := ENNReal.add_halves _
-          
       -- keep only the points in `b` that are close to point in `t`, yielding a new set `c`
       let c := {y ∈ b | ∃ x ∈ t, edist x y < δ}
       have : c.finite := ‹b.finite›.Subset fun x hx => hx.1
@@ -420,7 +416,6 @@ instance NonemptyCompacts.secondCountableTopology [SecondCountableTopology α] :
           calc
             edist y x = edist x y := edist_comm _ _
             _ ≤ δ := le_of_lt Dyx
-            
         exact ⟨x, xt, this⟩
       -- it follows that their Hausdorff distance is small
       have : Hausdorff_edist (t : Set α) c ≤ δ := Hausdorff_edist_le_of_mem_edist tc ct

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

@@ -213,7 +213,7 @@ instance Closeds.completeSpace [CompleteSpace α] : CompleteSpace (Closeds α) :
 #align emetric.closeds.complete_space EMetric.Closeds.completeSpace
 -/
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (v «expr ⊆ » s) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:638:2: warning: expanding binder collection (v «expr ⊆ » s) -/
 #print EMetric.Closeds.compactSpace /-
 /-- In a compact space, the type of closed subsets is compact. -/
 instance Closeds.compactSpace [CompactSpace α] : CompactSpace (Closeds α) :=

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

@@ -81,7 +81,7 @@ theorem continuous_infEdist_Hausdorff_edist :
 #print EMetric.isClosed_subsets_of_isClosed /-
 /-- Subsets of a given closed subset form a closed set -/
 theorem isClosed_subsets_of_isClosed (hs : IsClosed s) :
-    IsClosed { t : Closeds α | (t : Set α) ⊆ s } :=
+    IsClosed {t : Closeds α | (t : Set α) ⊆ s} :=
   by
   refine' isClosed_of_closure_subset fun t ht x hx => _
   -- t : closeds α,  ht : t ∈ closure {t : closeds α | t ⊆ s},
@@ -234,7 +234,7 @@ instance Closeds.compactSpace [CompactSpace α] : CompactSpace (Closeds α) :=
     have main : ∀ u : Set α, ∃ (v : _) (_ : v ⊆ s), Hausdorff_edist u v ≤ δ :=
       by
       intro u
-      let v := { x : α | x ∈ s ∧ ∃ y ∈ u, edist x y < δ }
+      let v := {x : α | x ∈ s ∧ ∃ y ∈ u, edist x y < δ}
       exists v, (fun x hx => hx.1 : v ⊆ s)
       refine' Hausdorff_edist_le_of_mem_edist _ _
       · intro x hx
@@ -245,7 +245,7 @@ instance Closeds.compactSpace [CompactSpace α] : CompactSpace (Closeds α) :=
       · rintro x ⟨hx1, ⟨y, yu, hy⟩⟩
         exact ⟨y, yu, le_of_lt hy⟩
     -- introduce the set F of all subsets of `s` (seen as members of `closeds α`).
-    let F := { f : closeds α | (f : Set α) ⊆ s }
+    let F := {f : closeds α | (f : Set α) ⊆ s}
     refine' ⟨F, _, fun u _ => _⟩
     -- `F` is finite
     · apply @finite.of_finite_image _ _ F coe
@@ -297,7 +297,7 @@ theorem NonemptyCompacts.isClosed_in_closeds [CompleteSpace α] :
   by
   have :
     range nonempty_compacts.to_closeds =
-      { s : closeds α | (s : Set α).Nonempty ∧ IsCompact (s : Set α) } :=
+      {s : closeds α | (s : Set α).Nonempty ∧ IsCompact (s : Set α)} :=
     by
     ext s
     refine' ⟨_, fun h => ⟨⟨⟨s, h.2⟩, h.1⟩, closeds.ext rfl⟩⟩
@@ -368,8 +368,8 @@ instance NonemptyCompacts.secondCountableTopology [SecondCountableTopology α] :
         approximations in `s` of the centers of these balls give the required finite approximation
         of `t`. -/
     rcases exists_countable_dense α with ⟨s, cs, s_dense⟩
-    let v0 := { t : Set α | t.Finite ∧ t ⊆ s }
-    let v : Set (nonempty_compacts α) := { t : nonempty_compacts α | (t : Set α) ∈ v0 }
+    let v0 := {t : Set α | t.Finite ∧ t ⊆ s}
+    let v : Set (nonempty_compacts α) := {t : nonempty_compacts α | (t : Set α) ∈ v0}
     refine' ⟨⟨v, _, _⟩⟩
     · have : v0.countable := countable_set_of_finite_subset cs
       exact this.preimage SetLike.coe_injective
@@ -403,7 +403,7 @@ instance NonemptyCompacts.secondCountableTopology [SecondCountableTopology α] :
           _ = δ := ENNReal.add_halves _
           
       -- keep only the points in `b` that are close to point in `t`, yielding a new set `c`
-      let c := { y ∈ b | ∃ x ∈ t, edist x y < δ }
+      let c := {y ∈ b | ∃ x ∈ t, edist x y < δ}
       have : c.finite := ‹b.finite›.Subset fun x hx => hx.1
       -- points in `t` are well approximated by points in `c`
       have tc : ∀ x ∈ t, ∃ y ∈ c, edist x y ≤ δ :=

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

@@ -94,7 +94,7 @@ theorem isClosed_subsets_of_isClosed (hs : IsClosed s) :
     rcases exists_edist_lt_of_Hausdorff_edist_lt hx Dtu with ⟨y, hy, Dxy⟩
     -- y : α,  hy : y ∈ u, Dxy : edist x y < ε
     exact ⟨y, hu hy, Dxy⟩
-  rwa [hs.closure_eq] at this
+  rwa [hs.closure_eq] at this 
 #align emetric.is_closed_subsets_of_is_closed EMetric.isClosed_subsets_of_isClosed
 -/
 
@@ -185,7 +185,7 @@ instance Closeds.completeSpace [CompleteSpace α] : CompleteSpace (Closeds α) :
     have : x ∈ closure (⋃ m ≥ n, s m : Set α) := by apply mem_Inter.1 xt0 n
     rcases mem_closure_iff.1 this (B n) (B_pos n) with ⟨z, hz, Dxz⟩
     -- z : α,  Dxz : edist x z < B n,
-    simp only [exists_prop, Set.mem_iUnion] at hz
+    simp only [exists_prop, Set.mem_iUnion] at hz 
     rcases hz with ⟨m, ⟨m_ge_n, hm⟩⟩
     -- m : ℕ, m_ge_n : m ≥ n, hm : z ∈ s m
     have : Hausdorff_edist (s m : Set α) (s n) < B n := hs n m n m_ge_n (le_refl n)
@@ -206,7 +206,7 @@ instance Closeds.completeSpace [CompleteSpace α] : CompleteSpace (Closeds α) :
   have : tendsto (fun n => 2 * B n) at_top (𝓝 (2 * 0)) :=
     ENNReal.Tendsto.const_mul
       (ENNReal.tendsto_pow_atTop_nhds_0_of_lt_1 <| by simp [ENNReal.one_lt_two]) (Or.inr <| by simp)
-  rw [MulZeroClass.mul_zero] at this
+  rw [MulZeroClass.mul_zero] at this 
   obtain ⟨N, hN⟩ : ∃ N, ∀ b ≥ N, ε > 2 * B b
   exact ((tendsto_order.1 this).2 ε εpos).exists_forall_of_atTop
   exact ⟨N, fun n hn => lt_of_le_of_lt (main n) (hN n hn)⟩
@@ -231,7 +231,7 @@ instance Closeds.compactSpace [CompactSpace α] : CompactSpace (Closeds α) :=
       ⟨s, fs, hs⟩
     -- s : set α,  fs : s.finite,  hs : univ ⊆ ⋃ (y : α) (H : y ∈ s), eball y δ
     -- we first show that any set is well approximated by a subset of `s`.
-    have main : ∀ u : Set α, ∃ (v : _)(_ : v ⊆ s), Hausdorff_edist u v ≤ δ :=
+    have main : ∀ u : Set α, ∃ (v : _) (_ : v ⊆ s), Hausdorff_edist u v ≤ δ :=
       by
       intro u
       let v := { x : α | x ∈ s ∧ ∃ y ∈ u, edist x y < δ }
@@ -240,7 +240,7 @@ instance Closeds.compactSpace [CompactSpace α] : CompactSpace (Closeds α) :=
       · intro x hx
         have : x ∈ ⋃ y ∈ s, ball y δ := hs (by simp)
         rcases mem_Union₂.1 this with ⟨y, ys, dy⟩
-        have : edist y x < δ := by simp at dy <;> rwa [edist_comm] at dy
+        have : edist y x < δ := by simp at dy  <;> rwa [edist_comm] at dy 
         exact ⟨y, ⟨ys, ⟨x, hx, this⟩⟩, le_of_lt dy⟩
       · rintro x ⟨hx1, ⟨y, yu, hy⟩⟩
         exact ⟨y, yu, le_of_lt hy⟩
@@ -252,7 +252,7 @@ instance Closeds.compactSpace [CompactSpace α] : CompactSpace (Closeds α) :=
       · apply fs.finite_subsets.subset fun b => _
         simp only [and_imp, Set.mem_image, Set.mem_setOf_eq, exists_imp]
         intro x hx hx'
-        rwa [hx'] at hx
+        rwa [hx'] at hx 
       · exact set_like.coe_injective.inj_on F
     -- `F` is ε-dense
     · obtain ⟨t0, t0s, Dut0⟩ := main u
@@ -278,7 +278,7 @@ instance NonemptyCompacts.emetricSpace : EMetricSpace (NonemptyCompacts α)
     NonemptyCompacts.ext <|
       by
       have : closure (s : Set α) = closure t := Hausdorff_edist_zero_iff_closure_eq_closure.1 h
-      rwa [s.is_compact.is_closed.closure_eq, t.is_compact.is_closed.closure_eq] at this
+      rwa [s.is_compact.is_closed.closure_eq, t.is_compact.is_closed.closure_eq] at this 
 #align emetric.nonempty_compacts.emetric_space EMetric.NonemptyCompacts.emetricSpace
 -/
 
@@ -307,7 +307,7 @@ theorem NonemptyCompacts.isClosed_in_closeds [CompleteSpace α] :
   refine' isClosed_of_closure_subset fun s hs => ⟨_, _⟩
   · -- take a set set t which is nonempty and at a finite distance of s
     rcases mem_closure_iff.1 hs ⊤ ENNReal.coe_lt_top with ⟨t, ht, Dst⟩
-    rw [edist_comm] at Dst
+    rw [edist_comm] at Dst 
     -- since `t` is nonempty, so is `s`
     exact nonempty_of_Hausdorff_edist_ne_top ht.1 (ne_of_lt Dst)
   · refine' isCompact_iff_totallyBounded_isComplete.2 ⟨_, s.closed.is_complete⟩

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

@@ -33,7 +33,7 @@ always finite in this context.
 
 noncomputable section
 
-open Classical Topology ENNReal
+open scoped Classical Topology ENNReal
 
 universe u
 

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

@@ -479,23 +479,11 @@ theorem lipschitz_infDist_set (x : α) : LipschitzWith 1 fun s : NonemptyCompact
 #align metric.lipschitz_inf_dist_set Metric.lipschitz_infDist_set
 -/
 
-/- warning: metric.lipschitz_inf_dist -> Metric.lipschitz_infDist is a dubious translation:
-lean 3 declaration is
-  forall {α : Type.{u1}} [_inst_1 : MetricSpace.{u1} α], LipschitzWith.{u1, 0} (Prod.{u1, u1} α (TopologicalSpace.NonemptyCompacts.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α (MetricSpace.toPseudoMetricSpace.{u1} α _inst_1))))) Real (Prod.pseudoEMetricSpaceMax.{u1, u1} α (TopologicalSpace.NonemptyCompacts.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α (MetricSpace.toPseudoMetricSpace.{u1} α _inst_1)))) (PseudoMetricSpace.toPseudoEMetricSpace.{u1} α (MetricSpace.toPseudoMetricSpace.{u1} α _inst_1)) (PseudoMetricSpace.toPseudoEMetricSpace.{u1} (TopologicalSpace.NonemptyCompacts.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α (MetricSpace.toPseudoMetricSpace.{u1} α _inst_1)))) (MetricSpace.toPseudoMetricSpace.{u1} (TopologicalSpace.NonemptyCompacts.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α (MetricSpace.toPseudoMetricSpace.{u1} α _inst_1)))) (Metric.NonemptyCompacts.metricSpace.{u1} α _inst_1)))) (PseudoMetricSpace.toPseudoEMetricSpace.{0} Real Real.pseudoMetricSpace) (OfNat.ofNat.{0} NNReal 2 (OfNat.mk.{0} NNReal 2 (bit0.{0} NNReal (Distrib.toHasAdd.{0} NNReal (NonUnitalNonAssocSemiring.toDistrib.{0} NNReal (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} NNReal (Semiring.toNonAssocSemiring.{0} NNReal NNReal.semiring)))) (One.one.{0} NNReal (AddMonoidWithOne.toOne.{0} NNReal (AddCommMonoidWithOne.toAddMonoidWithOne.{0} NNReal (NonAssocSemiring.toAddCommMonoidWithOne.{0} NNReal (Semiring.toNonAssocSemiring.{0} NNReal NNReal.semiring)))))))) (fun (p : Prod.{u1, u1} α (TopologicalSpace.NonemptyCompacts.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α (MetricSpace.toPseudoMetricSpace.{u1} α _inst_1))))) => Metric.infDist.{u1} α (MetricSpace.toPseudoMetricSpace.{u1} α _inst_1) (Prod.fst.{u1, u1} α (TopologicalSpace.NonemptyCompacts.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α (MetricSpace.toPseudoMetricSpace.{u1} α _inst_1)))) p) ((fun (a : Type.{u1}) (b : Type.{u1}) [self : HasLiftT.{succ u1, succ u1} a b] => self.0) (TopologicalSpace.NonemptyCompacts.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α (MetricSpace.toPseudoMetricSpace.{u1} α _inst_1)))) (Set.{u1} α) (HasLiftT.mk.{succ u1, succ u1} (TopologicalSpace.NonemptyCompacts.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α (MetricSpace.toPseudoMetricSpace.{u1} α _inst_1)))) (Set.{u1} α) (CoeTCₓ.coe.{succ u1, succ u1} (TopologicalSpace.NonemptyCompacts.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α (MetricSpace.toPseudoMetricSpace.{u1} α _inst_1)))) (Set.{u1} α) (SetLike.Set.hasCoeT.{u1, u1} (TopologicalSpace.NonemptyCompacts.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α (MetricSpace.toPseudoMetricSpace.{u1} α _inst_1)))) α (TopologicalSpace.NonemptyCompacts.setLike.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α (MetricSpace.toPseudoMetricSpace.{u1} α _inst_1))))))) (Prod.snd.{u1, u1} α (TopologicalSpace.NonemptyCompacts.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α (MetricSpace.toPseudoMetricSpace.{u1} α _inst_1)))) p)))
-but is expected to have type
-  forall {α : Type.{u1}} [_inst_1 : MetricSpace.{u1} α], LipschitzWith.{u1, 0} (Prod.{u1, u1} α (TopologicalSpace.NonemptyCompacts.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α (MetricSpace.toPseudoMetricSpace.{u1} α _inst_1))))) Real (Prod.pseudoEMetricSpaceMax.{u1, u1} α (TopologicalSpace.NonemptyCompacts.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α (MetricSpace.toPseudoMetricSpace.{u1} α _inst_1)))) (EMetricSpace.toPseudoEMetricSpace.{u1} α (MetricSpace.toEMetricSpace.{u1} α _inst_1)) (EMetricSpace.toPseudoEMetricSpace.{u1} (TopologicalSpace.NonemptyCompacts.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α (MetricSpace.toPseudoMetricSpace.{u1} α _inst_1)))) (EMetric.NonemptyCompacts.emetricSpace.{u1} α (MetricSpace.toEMetricSpace.{u1} α _inst_1)))) (EMetricSpace.toPseudoEMetricSpace.{0} Real (MetricSpace.toEMetricSpace.{0} Real Real.metricSpace)) (OfNat.ofNat.{0} NNReal 2 (instOfNat.{0} NNReal 2 (CanonicallyOrderedCommSemiring.toNatCast.{0} NNReal instNNRealCanonicallyOrderedCommSemiring) (instAtLeastTwoHAddNatInstHAddInstAddNatOfNat (OfNat.ofNat.{0} Nat 0 (instOfNatNat 0))))) (fun (p : Prod.{u1, u1} α (TopologicalSpace.NonemptyCompacts.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α (MetricSpace.toPseudoMetricSpace.{u1} α _inst_1))))) => Metric.infDist.{u1} α (MetricSpace.toPseudoMetricSpace.{u1} α _inst_1) (Prod.fst.{u1, u1} α (TopologicalSpace.NonemptyCompacts.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α (MetricSpace.toPseudoMetricSpace.{u1} α _inst_1)))) p) (SetLike.coe.{u1, u1} (TopologicalSpace.NonemptyCompacts.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α (MetricSpace.toPseudoMetricSpace.{u1} α _inst_1)))) α (TopologicalSpace.NonemptyCompacts.instSetLikeNonemptyCompacts.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α (MetricSpace.toPseudoMetricSpace.{u1} α _inst_1)))) (Prod.snd.{u1, u1} α (TopologicalSpace.NonemptyCompacts.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α (MetricSpace.toPseudoMetricSpace.{u1} α _inst_1)))) p)))
-Case conversion may be inaccurate. Consider using '#align metric.lipschitz_inf_dist Metric.lipschitz_infDistₓ'. -/
 theorem lipschitz_infDist : LipschitzWith 2 fun p : α × NonemptyCompacts α => infDist p.1 p.2 :=
   @LipschitzWith.uncurry _ _ _ _ _ _ (fun (x : α) (s : NonemptyCompacts α) => infDist x s) 1 1
     (fun s => lipschitz_infDist_pt s) lipschitz_infDist_set
 #align metric.lipschitz_inf_dist Metric.lipschitz_infDist
 
-/- warning: metric.uniform_continuous_inf_dist_Hausdorff_dist -> Metric.uniformContinuous_infDist_Hausdorff_dist is a dubious translation:
-lean 3 declaration is
-  forall {α : Type.{u1}} [_inst_1 : MetricSpace.{u1} α], UniformContinuous.{u1, 0} (Prod.{u1, u1} α (TopologicalSpace.NonemptyCompacts.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α (MetricSpace.toPseudoMetricSpace.{u1} α _inst_1))))) Real (Prod.uniformSpace.{u1, u1} α (TopologicalSpace.NonemptyCompacts.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α (MetricSpace.toPseudoMetricSpace.{u1} α _inst_1)))) (PseudoMetricSpace.toUniformSpace.{u1} α (MetricSpace.toPseudoMetricSpace.{u1} α _inst_1)) (PseudoMetricSpace.toUniformSpace.{u1} (TopologicalSpace.NonemptyCompacts.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α (MetricSpace.toPseudoMetricSpace.{u1} α _inst_1)))) (MetricSpace.toPseudoMetricSpace.{u1} (TopologicalSpace.NonemptyCompacts.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α (MetricSpace.toPseudoMetricSpace.{u1} α _inst_1)))) (Metric.NonemptyCompacts.metricSpace.{u1} α _inst_1)))) (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace) (fun (p : Prod.{u1, u1} α (TopologicalSpace.NonemptyCompacts.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α (MetricSpace.toPseudoMetricSpace.{u1} α _inst_1))))) => Metric.infDist.{u1} α (MetricSpace.toPseudoMetricSpace.{u1} α _inst_1) (Prod.fst.{u1, u1} α (TopologicalSpace.NonemptyCompacts.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α (MetricSpace.toPseudoMetricSpace.{u1} α _inst_1)))) p) ((fun (a : Type.{u1}) (b : Type.{u1}) [self : HasLiftT.{succ u1, succ u1} a b] => self.0) (TopologicalSpace.NonemptyCompacts.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α (MetricSpace.toPseudoMetricSpace.{u1} α _inst_1)))) (Set.{u1} α) (HasLiftT.mk.{succ u1, succ u1} (TopologicalSpace.NonemptyCompacts.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α (MetricSpace.toPseudoMetricSpace.{u1} α _inst_1)))) (Set.{u1} α) (CoeTCₓ.coe.{succ u1, succ u1} (TopologicalSpace.NonemptyCompacts.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α (MetricSpace.toPseudoMetricSpace.{u1} α _inst_1)))) (Set.{u1} α) (SetLike.Set.hasCoeT.{u1, u1} (TopologicalSpace.NonemptyCompacts.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α (MetricSpace.toPseudoMetricSpace.{u1} α _inst_1)))) α (TopologicalSpace.NonemptyCompacts.setLike.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α (MetricSpace.toPseudoMetricSpace.{u1} α _inst_1))))))) (Prod.snd.{u1, u1} α (TopologicalSpace.NonemptyCompacts.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α (MetricSpace.toPseudoMetricSpace.{u1} α _inst_1)))) p)))
-but is expected to have type
-  forall {α : Type.{u1}} [_inst_1 : MetricSpace.{u1} α], UniformContinuous.{u1, 0} (Prod.{u1, u1} α (TopologicalSpace.NonemptyCompacts.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α (MetricSpace.toPseudoMetricSpace.{u1} α _inst_1))))) Real (instUniformSpaceProd.{u1, u1} α (TopologicalSpace.NonemptyCompacts.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α (MetricSpace.toPseudoMetricSpace.{u1} α _inst_1)))) (PseudoMetricSpace.toUniformSpace.{u1} α (MetricSpace.toPseudoMetricSpace.{u1} α _inst_1)) (PseudoMetricSpace.toUniformSpace.{u1} (TopologicalSpace.NonemptyCompacts.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α (MetricSpace.toPseudoMetricSpace.{u1} α _inst_1)))) (MetricSpace.toPseudoMetricSpace.{u1} (TopologicalSpace.NonemptyCompacts.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α (MetricSpace.toPseudoMetricSpace.{u1} α _inst_1)))) (Metric.NonemptyCompacts.metricSpace.{u1} α _inst_1)))) (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace) (fun (p : Prod.{u1, u1} α (TopologicalSpace.NonemptyCompacts.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α (MetricSpace.toPseudoMetricSpace.{u1} α _inst_1))))) => Metric.infDist.{u1} α (MetricSpace.toPseudoMetricSpace.{u1} α _inst_1) (Prod.fst.{u1, u1} α (TopologicalSpace.NonemptyCompacts.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α (MetricSpace.toPseudoMetricSpace.{u1} α _inst_1)))) p) (SetLike.coe.{u1, u1} (TopologicalSpace.NonemptyCompacts.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α (MetricSpace.toPseudoMetricSpace.{u1} α _inst_1)))) α (TopologicalSpace.NonemptyCompacts.instSetLikeNonemptyCompacts.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α (MetricSpace.toPseudoMetricSpace.{u1} α _inst_1)))) (Prod.snd.{u1, u1} α (TopologicalSpace.NonemptyCompacts.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α (MetricSpace.toPseudoMetricSpace.{u1} α _inst_1)))) p)))
-Case conversion may be inaccurate. Consider using '#align metric.uniform_continuous_inf_dist_Hausdorff_dist Metric.uniformContinuous_infDist_Hausdorff_distₓ'. -/
 theorem uniformContinuous_infDist_Hausdorff_dist :
     UniformContinuous fun p : α × NonemptyCompacts α => infDist p.1 p.2 :=
   lipschitz_infDist.UniformContinuous

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

@@ -124,7 +124,7 @@ instance Closeds.completeSpace [CompleteSpace α] : CompleteSpace (Closeds α) :
     standard criterion. -/
   refine' complete_of_convergent_controlled_sequences B B_pos fun s hs => _
   let t0 := ⋂ n, closure (⋃ m ≥ n, s m : Set α)
-  let t : closeds α := ⟨t0, isClosed_interᵢ fun _ => isClosed_closure⟩
+  let t : closeds α := ⟨t0, isClosed_iInter fun _ => isClosed_closure⟩
   use t
   -- The inequality is written this way to agree with `edist_le_of_edist_le_geometric_of_tendsto₀`
   have I1 : ∀ n, ∀ x ∈ s n, ∃ y ∈ t0, edist x y ≤ 2 * B n :=
@@ -165,8 +165,8 @@ instance Closeds.completeSpace [CompleteSpace α] : CompleteSpace (Closeds α) :
       mem_Inter.2 fun k =>
         mem_closure_of_tendsto y_lim
           (by
-            simp only [exists_prop, Set.mem_unionᵢ, Filter.eventually_atTop, Set.mem_preimage,
-              Set.preimage_unionᵢ]
+            simp only [exists_prop, Set.mem_iUnion, Filter.eventually_atTop, Set.mem_preimage,
+              Set.preimage_iUnion]
             exact ⟨k, fun m hm => ⟨n + m, zero_add k ▸ add_le_add (zero_le n) hm, (z m).2⟩⟩)
     use this
     -- Then, we check that `y` is close to `x = z n`. This follows from the fact that `y`
@@ -185,7 +185,7 @@ instance Closeds.completeSpace [CompleteSpace α] : CompleteSpace (Closeds α) :
     have : x ∈ closure (⋃ m ≥ n, s m : Set α) := by apply mem_Inter.1 xt0 n
     rcases mem_closure_iff.1 this (B n) (B_pos n) with ⟨z, hz, Dxz⟩
     -- z : α,  Dxz : edist x z < B n,
-    simp only [exists_prop, Set.mem_unionᵢ] at hz
+    simp only [exists_prop, Set.mem_iUnion] at hz
     rcases hz with ⟨m, ⟨m_ge_n, hm⟩⟩
     -- m : ℕ, m_ge_n : m ≥ n, hm : z ∈ s m
     have : Hausdorff_edist (s m : Set α) (s n) < B n := hs n m n m_ge_n (le_refl n)

mathlib commit https://github.com/leanprover-community/mathlib/commit/cd8fafa2fac98e1a67097e8a91ad9901cfde48af

@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Sébastien Gouëzel
 
 ! This file was ported from Lean 3 source module topology.metric_space.closeds
-! leanprover-community/mathlib commit f2ce6086713c78a7f880485f7917ea547a215982
+! leanprover-community/mathlib commit ce38d86c0b2d427ce208c3cee3159cb421d2b3c4
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
@@ -15,6 +15,9 @@ import Mathbin.Topology.Sets.Compacts
 /-!
 # Closed subsets
 
+> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.
+> Any changes to this file require a corresponding PR to mathlib4.
+
 This file defines the metric and emetric space structure on the types of closed subsets and nonempty
 compact subsets of a metric or emetric space.
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/17ad94b4953419f3e3ce3e77da3239c62d1d09f0

@@ -42,6 +42,7 @@ section
 
 variable {α : Type u} [EMetricSpace α] {s : Set α}
 
+#print EMetric.Closeds.emetricSpace /-
 /-- In emetric spaces, the Hausdorff edistance defines an emetric space structure
 on the type of closed subsets -/
 instance Closeds.emetricSpace : EMetricSpace (Closeds α)
@@ -52,7 +53,8 @@ instance Closeds.emetricSpace : EMetricSpace (Closeds α)
   edist_triangle s t u := hausdorffEdist_triangle
   eq_of_edist_eq_zero s t h :=
     Closeds.ext <| (hausdorffEdist_zero_iff_eq_of_closed s.closed t.closed).1 h
-#align emetric.closeds.emetric_space Emetric.Closeds.emetricSpace
+#align emetric.closeds.emetric_space EMetric.Closeds.emetricSpace
+-/
 
 /-- The edistance to a closed set depends continuously on the point and the set -/
 theorem continuous_infEdist_Hausdorff_edist :
@@ -73,6 +75,7 @@ theorem continuous_infEdist_Hausdorff_edist :
     
 #align emetric.continuous_inf_edist_Hausdorff_edist Emetric.continuous_infEdist_Hausdorff_edist
 
+#print EMetric.isClosed_subsets_of_isClosed /-
 /-- Subsets of a given closed subset form a closed set -/
 theorem isClosed_subsets_of_isClosed (hs : IsClosed s) :
     IsClosed { t : Closeds α | (t : Set α) ⊆ s } :=
@@ -89,13 +92,17 @@ theorem isClosed_subsets_of_isClosed (hs : IsClosed s) :
     -- y : α,  hy : y ∈ u, Dxy : edist x y < ε
     exact ⟨y, hu hy, Dxy⟩
   rwa [hs.closure_eq] at this
-#align emetric.is_closed_subsets_of_is_closed Emetric.isClosed_subsets_of_isClosed
+#align emetric.is_closed_subsets_of_is_closed EMetric.isClosed_subsets_of_isClosed
+-/
 
+#print EMetric.Closeds.edist_eq /-
 /-- By definition, the edistance on `closeds α` is given by the Hausdorff edistance -/
 theorem Closeds.edist_eq {s t : Closeds α} : edist s t = hausdorffEdist (s : Set α) t :=
   rfl
-#align emetric.closeds.edist_eq Emetric.Closeds.edist_eq
+#align emetric.closeds.edist_eq EMetric.Closeds.edist_eq
+-/
 
+#print EMetric.Closeds.completeSpace /-
 /-- In a complete space, the type of closed subsets is complete for the
 Hausdorff edistance. -/
 instance Closeds.completeSpace [CompleteSpace α] : CompleteSpace (Closeds α) :=
@@ -200,9 +207,11 @@ instance Closeds.completeSpace [CompleteSpace α] : CompleteSpace (Closeds α) :
   obtain ⟨N, hN⟩ : ∃ N, ∀ b ≥ N, ε > 2 * B b
   exact ((tendsto_order.1 this).2 ε εpos).exists_forall_of_atTop
   exact ⟨N, fun n hn => lt_of_le_of_lt (main n) (hN n hn)⟩
-#align emetric.closeds.complete_space Emetric.Closeds.completeSpace
+#align emetric.closeds.complete_space EMetric.Closeds.completeSpace
+-/
 
 /- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (v «expr ⊆ » s) -/
+#print EMetric.Closeds.compactSpace /-
 /-- In a compact space, the type of closed subsets is compact. -/
 instance Closeds.compactSpace [CompactSpace α] : CompactSpace (Closeds α) :=
   ⟨by
@@ -250,8 +259,10 @@ instance Closeds.compactSpace [CompactSpace α] : CompactSpace (Closeds α) :=
       have : edist u t < ε := lt_of_le_of_lt Dut0 δlt
       apply mem_Union₂.2
       exact ⟨t, ‹t ∈ F›, this⟩⟩
-#align emetric.closeds.compact_space Emetric.Closeds.compactSpace
+#align emetric.closeds.compact_space EMetric.Closeds.compactSpace
+-/
 
+#print EMetric.NonemptyCompacts.emetricSpace /-
 /-- In an emetric space, the type of non-empty compact subsets is an emetric space,
 where the edistance is the Hausdorff edistance -/
 instance NonemptyCompacts.emetricSpace : EMetricSpace (NonemptyCompacts α)
@@ -265,14 +276,18 @@ instance NonemptyCompacts.emetricSpace : EMetricSpace (NonemptyCompacts α)
       by
       have : closure (s : Set α) = closure t := Hausdorff_edist_zero_iff_closure_eq_closure.1 h
       rwa [s.is_compact.is_closed.closure_eq, t.is_compact.is_closed.closure_eq] at this
-#align emetric.nonempty_compacts.emetric_space Emetric.NonemptyCompacts.emetricSpace
+#align emetric.nonempty_compacts.emetric_space EMetric.NonemptyCompacts.emetricSpace
+-/
 
+#print EMetric.NonemptyCompacts.ToCloseds.uniformEmbedding /-
 /-- `nonempty_compacts.to_closeds` is a uniform embedding (as it is an isometry) -/
 theorem NonemptyCompacts.ToCloseds.uniformEmbedding :
     UniformEmbedding (@NonemptyCompacts.toCloseds α _ _) :=
   Isometry.uniformEmbedding fun x y => rfl
-#align emetric.nonempty_compacts.to_closeds.uniform_embedding Emetric.NonemptyCompacts.ToCloseds.uniformEmbedding
+#align emetric.nonempty_compacts.to_closeds.uniform_embedding EMetric.NonemptyCompacts.ToCloseds.uniformEmbedding
+-/
 
+#print EMetric.NonemptyCompacts.isClosed_in_closeds /-
 /-- The range of `nonempty_compacts.to_closeds` is closed in a complete space -/
 theorem NonemptyCompacts.isClosed_in_closeds [CompleteSpace α] :
     IsClosed (range <| @NonemptyCompacts.toCloseds α _ _) :=
@@ -313,16 +328,20 @@ theorem NonemptyCompacts.isClosed_in_closeds [CompleteSpace α] :
         _ = ε := ENNReal.add_halves _
         
     exact mem_bUnion hy this
-#align emetric.nonempty_compacts.is_closed_in_closeds Emetric.NonemptyCompacts.isClosed_in_closeds
+#align emetric.nonempty_compacts.is_closed_in_closeds EMetric.NonemptyCompacts.isClosed_in_closeds
+-/
 
+#print EMetric.NonemptyCompacts.completeSpace /-
 /-- In a complete space, the type of nonempty compact subsets is complete. This follows
 from the same statement for closed subsets -/
 instance NonemptyCompacts.completeSpace [CompleteSpace α] : CompleteSpace (NonemptyCompacts α) :=
   (completeSpace_iff_isComplete_range
         NonemptyCompacts.ToCloseds.uniformEmbedding.to_uniformInducing).2 <|
     NonemptyCompacts.isClosed_in_closeds.IsComplete
-#align emetric.nonempty_compacts.complete_space Emetric.NonemptyCompacts.completeSpace
+#align emetric.nonempty_compacts.complete_space EMetric.NonemptyCompacts.completeSpace
+-/
 
+#print EMetric.NonemptyCompacts.compactSpace /-
 /-- In a compact space, the type of nonempty compact subsets is compact. This follows from
 the same statement for closed subsets -/
 instance NonemptyCompacts.compactSpace [CompactSpace α] : CompactSpace (NonemptyCompacts α) :=
@@ -330,8 +349,10 @@ instance NonemptyCompacts.compactSpace [CompactSpace α] : CompactSpace (Nonempt
     rw [nonempty_compacts.to_closeds.uniform_embedding.embedding.is_compact_iff_is_compact_image]
     rw [image_univ]
     exact nonempty_compacts.is_closed_in_closeds.is_compact⟩
-#align emetric.nonempty_compacts.compact_space Emetric.NonemptyCompacts.compactSpace
+#align emetric.nonempty_compacts.compact_space EMetric.NonemptyCompacts.compactSpace
+-/
 
+#print EMetric.NonemptyCompacts.secondCountableTopology /-
 /-- In a second countable space, the type of nonempty compact subsets is second countable -/
 instance NonemptyCompacts.secondCountableTopology [SecondCountableTopology α] :
     SecondCountableTopology (NonemptyCompacts α) :=
@@ -414,7 +435,8 @@ instance NonemptyCompacts.secondCountableTopology [SecondCountableTopology α] :
       -- we have proved that `d` is a good approximation of `t` as requested
       exact ⟨d, ‹d ∈ v›, Dtc⟩
   UniformSpace.secondCountable_of_separable (nonempty_compacts α)
-#align emetric.nonempty_compacts.second_countable_topology Emetric.NonemptyCompacts.secondCountableTopology
+#align emetric.nonempty_compacts.second_countable_topology EMetric.NonemptyCompacts.secondCountableTopology
+-/
 
 end
 
@@ -428,6 +450,7 @@ section
 
 variable {α : Type u} [MetricSpace α]
 
+#print Metric.NonemptyCompacts.metricSpace /-
 /-- `nonempty_compacts α` inherits a metric space structure, as the Hausdorff
 edistance between two such sets is finite. -/
 instance NonemptyCompacts.metricSpace : MetricSpace (NonemptyCompacts α) :=
@@ -435,24 +458,41 @@ instance NonemptyCompacts.metricSpace : MetricSpace (NonemptyCompacts α) :=
     hausdorffEdist_ne_top_of_nonempty_of_bounded x.Nonempty y.Nonempty x.IsCompact.Bounded
       y.IsCompact.Bounded
 #align metric.nonempty_compacts.metric_space Metric.NonemptyCompacts.metricSpace
+-/
 
+#print Metric.NonemptyCompacts.dist_eq /-
 /-- The distance on `nonempty_compacts α` is the Hausdorff distance, by construction -/
 theorem NonemptyCompacts.dist_eq {x y : NonemptyCompacts α} :
     dist x y = hausdorffDist (x : Set α) y :=
   rfl
 #align metric.nonempty_compacts.dist_eq Metric.NonemptyCompacts.dist_eq
+-/
 
+#print Metric.lipschitz_infDist_set /-
 theorem lipschitz_infDist_set (x : α) : LipschitzWith 1 fun s : NonemptyCompacts α => infDist x s :=
   LipschitzWith.of_le_add fun s t => by
     rw [dist_comm]
     exact inf_dist_le_inf_dist_add_Hausdorff_dist (edist_ne_top t s)
 #align metric.lipschitz_inf_dist_set Metric.lipschitz_infDist_set
+-/
 
+/- warning: metric.lipschitz_inf_dist -> Metric.lipschitz_infDist is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} [_inst_1 : MetricSpace.{u1} α], LipschitzWith.{u1, 0} (Prod.{u1, u1} α (TopologicalSpace.NonemptyCompacts.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α (MetricSpace.toPseudoMetricSpace.{u1} α _inst_1))))) Real (Prod.pseudoEMetricSpaceMax.{u1, u1} α (TopologicalSpace.NonemptyCompacts.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α (MetricSpace.toPseudoMetricSpace.{u1} α _inst_1)))) (PseudoMetricSpace.toPseudoEMetricSpace.{u1} α (MetricSpace.toPseudoMetricSpace.{u1} α _inst_1)) (PseudoMetricSpace.toPseudoEMetricSpace.{u1} (TopologicalSpace.NonemptyCompacts.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α (MetricSpace.toPseudoMetricSpace.{u1} α _inst_1)))) (MetricSpace.toPseudoMetricSpace.{u1} (TopologicalSpace.NonemptyCompacts.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α (MetricSpace.toPseudoMetricSpace.{u1} α _inst_1)))) (Metric.NonemptyCompacts.metricSpace.{u1} α _inst_1)))) (PseudoMetricSpace.toPseudoEMetricSpace.{0} Real Real.pseudoMetricSpace) (OfNat.ofNat.{0} NNReal 2 (OfNat.mk.{0} NNReal 2 (bit0.{0} NNReal (Distrib.toHasAdd.{0} NNReal (NonUnitalNonAssocSemiring.toDistrib.{0} NNReal (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} NNReal (Semiring.toNonAssocSemiring.{0} NNReal NNReal.semiring)))) (One.one.{0} NNReal (AddMonoidWithOne.toOne.{0} NNReal (AddCommMonoidWithOne.toAddMonoidWithOne.{0} NNReal (NonAssocSemiring.toAddCommMonoidWithOne.{0} NNReal (Semiring.toNonAssocSemiring.{0} NNReal NNReal.semiring)))))))) (fun (p : Prod.{u1, u1} α (TopologicalSpace.NonemptyCompacts.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α (MetricSpace.toPseudoMetricSpace.{u1} α _inst_1))))) => Metric.infDist.{u1} α (MetricSpace.toPseudoMetricSpace.{u1} α _inst_1) (Prod.fst.{u1, u1} α (TopologicalSpace.NonemptyCompacts.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α (MetricSpace.toPseudoMetricSpace.{u1} α _inst_1)))) p) ((fun (a : Type.{u1}) (b : Type.{u1}) [self : HasLiftT.{succ u1, succ u1} a b] => self.0) (TopologicalSpace.NonemptyCompacts.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α (MetricSpace.toPseudoMetricSpace.{u1} α _inst_1)))) (Set.{u1} α) (HasLiftT.mk.{succ u1, succ u1} (TopologicalSpace.NonemptyCompacts.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α (MetricSpace.toPseudoMetricSpace.{u1} α _inst_1)))) (Set.{u1} α) (CoeTCₓ.coe.{succ u1, succ u1} (TopologicalSpace.NonemptyCompacts.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α (MetricSpace.toPseudoMetricSpace.{u1} α _inst_1)))) (Set.{u1} α) (SetLike.Set.hasCoeT.{u1, u1} (TopologicalSpace.NonemptyCompacts.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α (MetricSpace.toPseudoMetricSpace.{u1} α _inst_1)))) α (TopologicalSpace.NonemptyCompacts.setLike.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α (MetricSpace.toPseudoMetricSpace.{u1} α _inst_1))))))) (Prod.snd.{u1, u1} α (TopologicalSpace.NonemptyCompacts.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α (MetricSpace.toPseudoMetricSpace.{u1} α _inst_1)))) p)))
+but is expected to have type
+  forall {α : Type.{u1}} [_inst_1 : MetricSpace.{u1} α], LipschitzWith.{u1, 0} (Prod.{u1, u1} α (TopologicalSpace.NonemptyCompacts.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α (MetricSpace.toPseudoMetricSpace.{u1} α _inst_1))))) Real (Prod.pseudoEMetricSpaceMax.{u1, u1} α (TopologicalSpace.NonemptyCompacts.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α (MetricSpace.toPseudoMetricSpace.{u1} α _inst_1)))) (EMetricSpace.toPseudoEMetricSpace.{u1} α (MetricSpace.toEMetricSpace.{u1} α _inst_1)) (EMetricSpace.toPseudoEMetricSpace.{u1} (TopologicalSpace.NonemptyCompacts.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α (MetricSpace.toPseudoMetricSpace.{u1} α _inst_1)))) (EMetric.NonemptyCompacts.emetricSpace.{u1} α (MetricSpace.toEMetricSpace.{u1} α _inst_1)))) (EMetricSpace.toPseudoEMetricSpace.{0} Real (MetricSpace.toEMetricSpace.{0} Real Real.metricSpace)) (OfNat.ofNat.{0} NNReal 2 (instOfNat.{0} NNReal 2 (CanonicallyOrderedCommSemiring.toNatCast.{0} NNReal instNNRealCanonicallyOrderedCommSemiring) (instAtLeastTwoHAddNatInstHAddInstAddNatOfNat (OfNat.ofNat.{0} Nat 0 (instOfNatNat 0))))) (fun (p : Prod.{u1, u1} α (TopologicalSpace.NonemptyCompacts.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α (MetricSpace.toPseudoMetricSpace.{u1} α _inst_1))))) => Metric.infDist.{u1} α (MetricSpace.toPseudoMetricSpace.{u1} α _inst_1) (Prod.fst.{u1, u1} α (TopologicalSpace.NonemptyCompacts.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α (MetricSpace.toPseudoMetricSpace.{u1} α _inst_1)))) p) (SetLike.coe.{u1, u1} (TopologicalSpace.NonemptyCompacts.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α (MetricSpace.toPseudoMetricSpace.{u1} α _inst_1)))) α (TopologicalSpace.NonemptyCompacts.instSetLikeNonemptyCompacts.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α (MetricSpace.toPseudoMetricSpace.{u1} α _inst_1)))) (Prod.snd.{u1, u1} α (TopologicalSpace.NonemptyCompacts.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α (MetricSpace.toPseudoMetricSpace.{u1} α _inst_1)))) p)))
+Case conversion may be inaccurate. Consider using '#align metric.lipschitz_inf_dist Metric.lipschitz_infDistₓ'. -/
 theorem lipschitz_infDist : LipschitzWith 2 fun p : α × NonemptyCompacts α => infDist p.1 p.2 :=
   @LipschitzWith.uncurry _ _ _ _ _ _ (fun (x : α) (s : NonemptyCompacts α) => infDist x s) 1 1
     (fun s => lipschitz_infDist_pt s) lipschitz_infDist_set
 #align metric.lipschitz_inf_dist Metric.lipschitz_infDist
 
+/- warning: metric.uniform_continuous_inf_dist_Hausdorff_dist -> Metric.uniformContinuous_infDist_Hausdorff_dist is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} [_inst_1 : MetricSpace.{u1} α], UniformContinuous.{u1, 0} (Prod.{u1, u1} α (TopologicalSpace.NonemptyCompacts.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α (MetricSpace.toPseudoMetricSpace.{u1} α _inst_1))))) Real (Prod.uniformSpace.{u1, u1} α (TopologicalSpace.NonemptyCompacts.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α (MetricSpace.toPseudoMetricSpace.{u1} α _inst_1)))) (PseudoMetricSpace.toUniformSpace.{u1} α (MetricSpace.toPseudoMetricSpace.{u1} α _inst_1)) (PseudoMetricSpace.toUniformSpace.{u1} (TopologicalSpace.NonemptyCompacts.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α (MetricSpace.toPseudoMetricSpace.{u1} α _inst_1)))) (MetricSpace.toPseudoMetricSpace.{u1} (TopologicalSpace.NonemptyCompacts.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α (MetricSpace.toPseudoMetricSpace.{u1} α _inst_1)))) (Metric.NonemptyCompacts.metricSpace.{u1} α _inst_1)))) (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace) (fun (p : Prod.{u1, u1} α (TopologicalSpace.NonemptyCompacts.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α (MetricSpace.toPseudoMetricSpace.{u1} α _inst_1))))) => Metric.infDist.{u1} α (MetricSpace.toPseudoMetricSpace.{u1} α _inst_1) (Prod.fst.{u1, u1} α (TopologicalSpace.NonemptyCompacts.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α (MetricSpace.toPseudoMetricSpace.{u1} α _inst_1)))) p) ((fun (a : Type.{u1}) (b : Type.{u1}) [self : HasLiftT.{succ u1, succ u1} a b] => self.0) (TopologicalSpace.NonemptyCompacts.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α (MetricSpace.toPseudoMetricSpace.{u1} α _inst_1)))) (Set.{u1} α) (HasLiftT.mk.{succ u1, succ u1} (TopologicalSpace.NonemptyCompacts.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α (MetricSpace.toPseudoMetricSpace.{u1} α _inst_1)))) (Set.{u1} α) (CoeTCₓ.coe.{succ u1, succ u1} (TopologicalSpace.NonemptyCompacts.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α (MetricSpace.toPseudoMetricSpace.{u1} α _inst_1)))) (Set.{u1} α) (SetLike.Set.hasCoeT.{u1, u1} (TopologicalSpace.NonemptyCompacts.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α (MetricSpace.toPseudoMetricSpace.{u1} α _inst_1)))) α (TopologicalSpace.NonemptyCompacts.setLike.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α (MetricSpace.toPseudoMetricSpace.{u1} α _inst_1))))))) (Prod.snd.{u1, u1} α (TopologicalSpace.NonemptyCompacts.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α (MetricSpace.toPseudoMetricSpace.{u1} α _inst_1)))) p)))
+but is expected to have type
+  forall {α : Type.{u1}} [_inst_1 : MetricSpace.{u1} α], UniformContinuous.{u1, 0} (Prod.{u1, u1} α (TopologicalSpace.NonemptyCompacts.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α (MetricSpace.toPseudoMetricSpace.{u1} α _inst_1))))) Real (instUniformSpaceProd.{u1, u1} α (TopologicalSpace.NonemptyCompacts.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α (MetricSpace.toPseudoMetricSpace.{u1} α _inst_1)))) (PseudoMetricSpace.toUniformSpace.{u1} α (MetricSpace.toPseudoMetricSpace.{u1} α _inst_1)) (PseudoMetricSpace.toUniformSpace.{u1} (TopologicalSpace.NonemptyCompacts.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α (MetricSpace.toPseudoMetricSpace.{u1} α _inst_1)))) (MetricSpace.toPseudoMetricSpace.{u1} (TopologicalSpace.NonemptyCompacts.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α (MetricSpace.toPseudoMetricSpace.{u1} α _inst_1)))) (Metric.NonemptyCompacts.metricSpace.{u1} α _inst_1)))) (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace) (fun (p : Prod.{u1, u1} α (TopologicalSpace.NonemptyCompacts.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α (MetricSpace.toPseudoMetricSpace.{u1} α _inst_1))))) => Metric.infDist.{u1} α (MetricSpace.toPseudoMetricSpace.{u1} α _inst_1) (Prod.fst.{u1, u1} α (TopologicalSpace.NonemptyCompacts.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α (MetricSpace.toPseudoMetricSpace.{u1} α _inst_1)))) p) (SetLike.coe.{u1, u1} (TopologicalSpace.NonemptyCompacts.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α (MetricSpace.toPseudoMetricSpace.{u1} α _inst_1)))) α (TopologicalSpace.NonemptyCompacts.instSetLikeNonemptyCompacts.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α (MetricSpace.toPseudoMetricSpace.{u1} α _inst_1)))) (Prod.snd.{u1, u1} α (TopologicalSpace.NonemptyCompacts.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α (MetricSpace.toPseudoMetricSpace.{u1} α _inst_1)))) p)))
+Case conversion may be inaccurate. Consider using '#align metric.uniform_continuous_inf_dist_Hausdorff_dist Metric.uniformContinuous_infDist_Hausdorff_distₓ'. -/
 theorem uniformContinuous_infDist_Hausdorff_dist :
     UniformContinuous fun p : α × NonemptyCompacts α => infDist p.1 p.2 :=
   lipschitz_infDist.UniformContinuous

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

@@ -196,7 +196,7 @@ instance Closeds.completeSpace [CompleteSpace α] : CompleteSpace (Closeds α) :
   have : tendsto (fun n => 2 * B n) at_top (𝓝 (2 * 0)) :=
     ENNReal.Tendsto.const_mul
       (ENNReal.tendsto_pow_atTop_nhds_0_of_lt_1 <| by simp [ENNReal.one_lt_two]) (Or.inr <| by simp)
-  rw [mul_zero] at this
+  rw [MulZeroClass.mul_zero] at this
   obtain ⟨N, hN⟩ : ∃ N, ∀ b ≥ N, ε > 2 * B b
   exact ((tendsto_order.1 this).2 ε εpos).exists_forall_of_atTop
   exact ⟨N, fun n hn => lt_of_le_of_lt (main n) (hN n hn)⟩

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

@@ -40,11 +40,11 @@ namespace Emetric
 
 section
 
-variable {α : Type u} [EmetricSpace α] {s : Set α}
+variable {α : Type u} [EMetricSpace α] {s : Set α}
 
 /-- In emetric spaces, the Hausdorff edistance defines an emetric space structure
 on the type of closed subsets -/
-instance Closeds.emetricSpace : EmetricSpace (Closeds α)
+instance Closeds.emetricSpace : EMetricSpace (Closeds α)
     where
   edist s t := hausdorffEdist (s : Set α) t
   edist_self s := hausdorffEdist_self
@@ -211,10 +211,10 @@ instance Closeds.compactSpace [CompactSpace α] : CompactSpace (Closeds α) :=
         start from a set `s` which is ε-dense in α. Then the subsets of `s`
         are finitely many, and ε-dense for the Hausdorff distance. -/
     refine'
-      isCompact_of_totallyBounded_isClosed (Emetric.totallyBounded_iff.2 fun ε εpos => _)
+      isCompact_of_totallyBounded_isClosed (EMetric.totallyBounded_iff.2 fun ε εpos => _)
         isClosed_univ
     rcases exists_between εpos with ⟨δ, δpos, δlt⟩
-    rcases Emetric.totallyBounded_iff.1
+    rcases EMetric.totallyBounded_iff.1
         (isCompact_iff_totallyBounded_isComplete.1 (@isCompact_univ α _ _)).1 δ δpos with
       ⟨s, fs, hs⟩
     -- s : set α,  fs : s.finite,  hs : univ ⊆ ⋃ (y : α) (H : y ∈ s), eball y δ
@@ -254,7 +254,7 @@ instance Closeds.compactSpace [CompactSpace α] : CompactSpace (Closeds α) :=
 
 /-- In an emetric space, the type of non-empty compact subsets is an emetric space,
 where the edistance is the Hausdorff edistance -/
-instance NonemptyCompacts.emetricSpace : EmetricSpace (NonemptyCompacts α)
+instance NonemptyCompacts.emetricSpace : EMetricSpace (NonemptyCompacts α)
     where
   edist s t := hausdorffEdist (s : Set α) t
   edist_self s := hausdorffEdist_self
@@ -431,7 +431,7 @@ variable {α : Type u} [MetricSpace α]
 /-- `nonempty_compacts α` inherits a metric space structure, as the Hausdorff
 edistance between two such sets is finite. -/
 instance NonemptyCompacts.metricSpace : MetricSpace (NonemptyCompacts α) :=
-  EmetricSpace.toMetricSpace fun x y =>
+  EMetricSpace.toMetricSpace fun x y =>
     hausdorffEdist_ne_top_of_nonempty_of_bounded x.Nonempty y.Nonempty x.IsCompact.Bounded
       y.IsCompact.Bounded
 #align metric.nonempty_compacts.metric_space Metric.NonemptyCompacts.metricSpace

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

@@ -64,11 +64,11 @@ theorem continuous_infEdist_Hausdorff_edist :
     inf_edist x s ≤ inf_edist x t + Hausdorff_edist (t : Set α) s :=
       inf_edist_le_inf_edist_add_Hausdorff_edist
     _ ≤ inf_edist y t + edist x y + Hausdorff_edist (t : Set α) s :=
-      add_le_add_right inf_edist_le_inf_edist_add_edist _
+      (add_le_add_right inf_edist_le_inf_edist_add_edist _)
     _ = inf_edist y t + (edist x y + Hausdorff_edist (s : Set α) t) := by
       rw [add_assoc, Hausdorff_edist_comm]
     _ ≤ inf_edist y t + (edist (x, s) (y, t) + edist (x, s) (y, t)) :=
-      add_le_add_left (add_le_add (le_max_left _ _) (le_max_right _ _)) _
+      (add_le_add_left (add_le_add (le_max_left _ _) (le_max_right _ _)) _)
     _ = inf_edist y t + 2 * edist (x, s) (y, t) := by rw [← mul_two, mul_comm]
     
 #align emetric.continuous_inf_edist_Hausdorff_edist Emetric.continuous_infEdist_Hausdorff_edist
@@ -185,7 +185,7 @@ instance Closeds.completeSpace [CompleteSpace α] : CompleteSpace (Closeds α) :
       ⟨y, hy,
         calc
           edist x y ≤ edist x z + edist z y := edist_triangle _ _ _
-          _ ≤ B n + B n := add_le_add (le_of_lt Dxz) (le_of_lt Dzy)
+          _ ≤ B n + B n := (add_le_add (le_of_lt Dxz) (le_of_lt Dzy))
           _ = 2 * B n := (two_mul _).symm
           ⟩
   -- Deduce from the above inequalities that the distance between `s n` and `t0` is at most `2 B n`.
@@ -202,7 +202,7 @@ instance Closeds.completeSpace [CompleteSpace α] : CompleteSpace (Closeds α) :
   exact ⟨N, fun n hn => lt_of_le_of_lt (main n) (hN n hn)⟩
 #align emetric.closeds.complete_space Emetric.Closeds.completeSpace
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:628:2: warning: expanding binder collection (v «expr ⊆ » s) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (v «expr ⊆ » s) -/
 /-- In a compact space, the type of closed subsets is compact. -/
 instance Closeds.compactSpace [CompactSpace α] : CompactSpace (Closeds α) :=
   ⟨by
@@ -309,7 +309,7 @@ theorem NonemptyCompacts.isClosed_in_closeds [CompleteSpace α] :
     have : edist x y < ε :=
       calc
         edist x y ≤ edist x z + edist z y := edist_triangle _ _ _
-        _ < ε / 2 + ε / 2 := ENNReal.add_lt_add Dxz Dzy
+        _ < ε / 2 + ε / 2 := (ENNReal.add_lt_add Dxz Dzy)
         _ = ε := ENNReal.add_halves _
         
     exact mem_bUnion hy this
@@ -375,7 +375,7 @@ instance NonemptyCompacts.secondCountableTopology [SecondCountableTopology α] :
         exists F z, mem_image_of_mem _ za
         calc
           edist x (F z) ≤ edist x z + edist z (F z) := edist_triangle _ _ _
-          _ < δ / 2 + δ / 2 := ENNReal.add_lt_add Dxz (Fspec z).2
+          _ < δ / 2 + δ / 2 := (ENNReal.add_lt_add Dxz (Fspec z).2)
           _ = δ := ENNReal.add_halves _
           
       -- keep only the points in `b` that are close to point in `t`, yielding a new set `c`

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

@@ -30,7 +30,7 @@ always finite in this context.
 
 noncomputable section
 
-open Classical Topology Ennreal
+open Classical Topology ENNReal
 
 universe u
 
@@ -105,8 +105,8 @@ instance Closeds.completeSpace [CompleteSpace α] : CompleteSpace (Closeds α) :
     completeness, by a standard completeness criterion.
     We use the shorthand `B n = 2^{-n}` in ennreal. -/
   let B : ℕ → ℝ≥0∞ := fun n => 2⁻¹ ^ n
-  have B_pos : ∀ n, (0 : ℝ≥0∞) < B n := by simp [B, Ennreal.pow_pos]
-  have B_ne_top : ∀ n, B n ≠ ⊤ := by simp [B, Ennreal.pow_ne_top]
+  have B_pos : ∀ n, (0 : ℝ≥0∞) < B n := by simp [B, ENNReal.pow_pos]
+  have B_ne_top : ∀ n, B n ≠ ⊤ := by simp [B, ENNReal.pow_ne_top]
   /- Consider a sequence of closed sets `s n` with `edist (s n) (s (n+1)) < B n`.
     We will show that it converges. The limit set is t0 = ⋂n, closure (⋃m≥n, s m).
     We will have to show that a point in `s n` is close to a point in `t0`, and a point
@@ -138,7 +138,7 @@ instance Closeds.completeSpace [CompleteSpace α] : CompleteSpace (Closeds α) :
           refine' exists_edist_lt_of_Hausdorff_edist_lt _ _
           · exact s (n + l)
           · exact z.2
-          simp only [B, Ennreal.inv_pow, div_eq_mul_inv]
+          simp only [B, ENNReal.inv_pow, div_eq_mul_inv]
           rw [← pow_add]
           apply hs <;> simp
         exact ⟨⟨z', z'_mem⟩, le_of_lt hz'⟩
@@ -194,8 +194,8 @@ instance Closeds.completeSpace [CompleteSpace α] : CompleteSpace (Closeds α) :
   -- from this, the convergence of `s n` to `t0` follows.
   refine' tendsto_at_top.2 fun ε εpos => _
   have : tendsto (fun n => 2 * B n) at_top (𝓝 (2 * 0)) :=
-    Ennreal.Tendsto.const_mul
-      (Ennreal.tendsto_pow_atTop_nhds_0_of_lt_1 <| by simp [Ennreal.one_lt_two]) (Or.inr <| by simp)
+    ENNReal.Tendsto.const_mul
+      (ENNReal.tendsto_pow_atTop_nhds_0_of_lt_1 <| by simp [ENNReal.one_lt_two]) (Or.inr <| by simp)
   rw [mul_zero] at this
   obtain ⟨N, hN⟩ : ∃ N, ∀ b ≥ N, ε > 2 * B b
   exact ((tendsto_order.1 this).2 ε εpos).exists_forall_of_atTop
@@ -288,7 +288,7 @@ theorem NonemptyCompacts.isClosed_in_closeds [CompleteSpace α] :
   rw [this]
   refine' isClosed_of_closure_subset fun s hs => ⟨_, _⟩
   · -- take a set set t which is nonempty and at a finite distance of s
-    rcases mem_closure_iff.1 hs ⊤ Ennreal.coe_lt_top with ⟨t, ht, Dst⟩
+    rcases mem_closure_iff.1 hs ⊤ ENNReal.coe_lt_top with ⟨t, ht, Dst⟩
     rw [edist_comm] at Dst
     -- since `t` is nonempty, so is `s`
     exact nonempty_of_Hausdorff_edist_ne_top ht.1 (ne_of_lt Dst)
@@ -296,10 +296,10 @@ theorem NonemptyCompacts.isClosed_in_closeds [CompleteSpace α] :
     refine' totally_bounded_iff.2 fun ε (εpos : 0 < ε) => _
     -- we have to show that s is covered by finitely many eballs of radius ε
     -- pick a nonempty compact set t at distance at most ε/2 of s
-    rcases mem_closure_iff.1 hs (ε / 2) (Ennreal.half_pos εpos.ne') with ⟨t, ht, Dst⟩
+    rcases mem_closure_iff.1 hs (ε / 2) (ENNReal.half_pos εpos.ne') with ⟨t, ht, Dst⟩
     -- cover this space with finitely many balls of radius ε/2
     rcases totally_bounded_iff.1 (isCompact_iff_totallyBounded_isComplete.1 ht.2).1 (ε / 2)
-        (Ennreal.half_pos εpos.ne') with
+        (ENNReal.half_pos εpos.ne') with
       ⟨u, fu, ut⟩
     refine' ⟨u, ⟨fu, fun x hx => _⟩⟩
     -- u : set α,  fu : u.finite,  ut : t ⊆ ⋃ (y : α) (H : y ∈ u), eball y (ε / 2)
@@ -309,8 +309,8 @@ theorem NonemptyCompacts.isClosed_in_closeds [CompleteSpace α] :
     have : edist x y < ε :=
       calc
         edist x y ≤ edist x z + edist z y := edist_triangle _ _ _
-        _ < ε / 2 + ε / 2 := Ennreal.add_lt_add Dxz Dzy
-        _ = ε := Ennreal.add_halves _
+        _ < ε / 2 + ε / 2 := ENNReal.add_lt_add Dxz Dzy
+        _ = ε := ENNReal.add_halves _
         
     exact mem_bUnion hy this
 #align emetric.nonempty_compacts.is_closed_in_closeds Emetric.NonemptyCompacts.isClosed_in_closeds
@@ -352,7 +352,7 @@ instance NonemptyCompacts.secondCountableTopology [SecondCountableTopology α] :
     · refine' fun t => mem_closure_iff.2 fun ε εpos => _
       -- t is a compact nonempty set, that we have to approximate uniformly by a a set in `v`.
       rcases exists_between εpos with ⟨δ, δpos, δlt⟩
-      have δpos' : 0 < δ / 2 := Ennreal.half_pos δpos.ne'
+      have δpos' : 0 < δ / 2 := ENNReal.half_pos δpos.ne'
       -- construct a map F associating to a point in α an approximating point in s, up to δ/2.
       have Exy : ∀ x, ∃ y, y ∈ s ∧ edist x y < δ / 2 :=
         by
@@ -375,8 +375,8 @@ instance NonemptyCompacts.secondCountableTopology [SecondCountableTopology α] :
         exists F z, mem_image_of_mem _ za
         calc
           edist x (F z) ≤ edist x z + edist z (F z) := edist_triangle _ _ _
-          _ < δ / 2 + δ / 2 := Ennreal.add_lt_add Dxz (Fspec z).2
-          _ = δ := Ennreal.add_halves _
+          _ < δ / 2 + δ / 2 := ENNReal.add_lt_add Dxz (Fspec z).2
+          _ = δ := ENNReal.add_halves _
           
       -- keep only the points in `b` that are close to point in `t`, yielding a new set `c`
       let c := { y ∈ b | ∃ x ∈ t, edist x y < δ }

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

Co-authored-by: Moritz Firsching <firsching@google.com>

@@ -171,7 +171,7 @@ instance Closeds.completeSpace [CompleteSpace α] : CompleteSpace (Closeds α) :
       ⟨y, hy,
         calc
           edist x y ≤ edist x z + edist z y := edist_triangle _ _ _
-          _ ≤ B n + B n := (add_le_add (le_of_lt Dxz) (le_of_lt Dzy))
+          _ ≤ B n + B n := add_le_add (le_of_lt Dxz) (le_of_lt Dzy)
           _ = 2 * B n := (two_mul _).symm
           ⟩
   -- Deduce from the above inequalities that the distance between `s n` and `t0` is at most `2 B n`.
@@ -288,7 +288,7 @@ theorem NonemptyCompacts.isClosed_in_closeds [CompleteSpace α] :
     have : edist x y < ε :=
       calc
         edist x y ≤ edist x z + edist z y := edist_triangle _ _ _
-        _ < ε / 2 + ε / 2 := (ENNReal.add_lt_add Dxz Dzy)
+        _ < ε / 2 + ε / 2 := ENNReal.add_lt_add Dxz Dzy
         _ = ε := ENNReal.add_halves _
     exact mem_biUnion hy this
 #align emetric.nonempty_compacts.is_closed_in_closeds EMetric.NonemptyCompacts.isClosed_in_closeds
@@ -349,7 +349,7 @@ instance NonemptyCompacts.secondCountableTopology [SecondCountableTopology α] :
         exists F z, mem_image_of_mem _ za
         calc
           edist x (F z) ≤ edist x z + edist z (F z) := edist_triangle _ _ _
-          _ < δ / 2 + δ / 2 := (ENNReal.add_lt_add Dxz (Fspec z).2)
+          _ < δ / 2 + δ / 2 := ENNReal.add_lt_add Dxz (Fspec z).2
           _ = δ := ENNReal.add_halves _
       -- keep only the points in `b` that are close to point in `t`, yielding a new set `c`
       let c := { y ∈ b | ∃ x ∈ t, edist x y < δ }

@@ -203,7 +203,7 @@ instance Closeds.compactSpace [CompactSpace α] : CompactSpace (Closeds α) :=
       EMetric.totallyBounded_iff.1
         (isCompact_iff_totallyBounded_isComplete.1 (@isCompact_univ α _ _)).1 δ δpos
     -- we first show that any set is well approximated by a subset of `s`.
-    have main : ∀ u : Set α, ∃ (v : _) (_ : v ⊆ s), hausdorffEdist u v ≤ δ := by
+    have main : ∀ u : Set α, ∃ v ⊆ s, hausdorffEdist u v ≤ δ := by
       intro u
       let v := { x : α | x ∈ s ∧ ∃ y ∈ u, edist x y < δ }
       exists v, (fun x hx => hx.1 : v ⊆ s)

@@ -73,16 +73,13 @@ set_option linter.uppercaseLean3 false in
 /-- Subsets of a given closed subset form a closed set -/
 theorem isClosed_subsets_of_isClosed (hs : IsClosed s) :
     IsClosed { t : Closeds α | (t : Set α) ⊆ s } := by
-  refine' isClosed_of_closure_subset fun t ht x hx => _
-  -- t : Closeds α, ht : t ∈ closure {t : Closeds α | t ⊆ s},
-  -- x : α, hx : x ∈ t
-  -- goal : x ∈ s
+  refine isClosed_of_closure_subset fun
+    (t : Closeds α) (ht : t ∈ closure {t : Closeds α | (t : Set α) ⊆ s}) (x : α) (hx : x ∈ t) => ?_
   have : x ∈ closure s := by
     refine' mem_closure_iff.2 fun ε εpos => _
-    rcases mem_closure_iff.1 ht ε εpos with ⟨u, hu, Dtu⟩
-    -- u : Closeds α, hu : u ∈ {t : Closeds α | t ⊆ s}, hu' : edist t u < ε
-    rcases exists_edist_lt_of_hausdorffEdist_lt hx Dtu with ⟨y, hy, Dxy⟩
-    -- y : α, hy : y ∈ u, Dxy : edist x y < ε
+    obtain ⟨u : Closeds α, hu : u ∈ {t : Closeds α | (t : Set α) ⊆ s}, Dtu : edist t u < ε⟩ :=
+      mem_closure_iff.1 ht ε εpos
+    obtain ⟨y : α, hy : y ∈ u, Dxy : edist x y < ε⟩ := exists_edist_lt_of_hausdorffEdist_lt hx Dtu
     exact ⟨y, hu hy, Dxy⟩
   rwa [hs.closure_eq] at this
 #align emetric.is_closed_subsets_of_is_closed EMetric.isClosed_subsets_of_isClosed
@@ -164,14 +161,12 @@ instance Closeds.completeSpace [CompleteSpace α] : CompleteSpace (Closeds α) :
             as required. -/
     intro n x xt0
     have : x ∈ closure (⋃ m ≥ n, s m : Set α) := by apply mem_iInter.1 xt0 n
-    rcases mem_closure_iff.1 this (B n) (B_pos n) with ⟨z, hz, Dxz⟩
-    -- z : α, Dxz : edist x z < B n,
+    obtain ⟨z : α, hz, Dxz : edist x z < B n⟩ := mem_closure_iff.1 this (B n) (B_pos n)
     simp only [exists_prop, Set.mem_iUnion] at hz
-    rcases hz with ⟨m, ⟨m_ge_n, hm⟩⟩
-    -- m : ℕ, m_ge_n : m ≥ n, hm : z ∈ s m
+    obtain ⟨m : ℕ, m_ge_n : m ≥ n, hm : z ∈ (s m : Set α)⟩ := hz
     have : hausdorffEdist (s m : Set α) (s n) < B n := hs n m n m_ge_n (le_refl n)
-    rcases exists_edist_lt_of_hausdorffEdist_lt hm this with ⟨y, hy, Dzy⟩
-    -- y : α, hy : y ∈ s n, Dzy : edist z y < B n
+    obtain ⟨y : α, hy : y ∈ (s n : Set α), Dzy : edist z y < B n⟩ :=
+      exists_edist_lt_of_hausdorffEdist_lt hm this
     exact
       ⟨y, hy,
         calc
@@ -204,10 +199,9 @@ instance Closeds.compactSpace [CompactSpace α] : CompactSpace (Closeds α) :=
       isCompact_of_totallyBounded_isClosed (EMetric.totallyBounded_iff.2 fun ε εpos => _)
         isClosed_univ
     rcases exists_between εpos with ⟨δ, δpos, δlt⟩
-    rcases EMetric.totallyBounded_iff.1
-        (isCompact_iff_totallyBounded_isComplete.1 (@isCompact_univ α _ _)).1 δ δpos with
-      ⟨s, fs, hs⟩
-    -- s : Set α, fs : s.Finite, hs : univ ⊆ ⋃ (y : α) (H : y ∈ s), eball y δ
+    obtain ⟨s : Set α, fs : s.Finite, hs : univ ⊆ ⋃ y ∈ s, ball y δ⟩ :=
+      EMetric.totallyBounded_iff.1
+        (isCompact_iff_totallyBounded_isComplete.1 (@isCompact_univ α _ _)).1 δ δpos
     -- we first show that any set is well approximated by a subset of `s`.
     have main : ∀ u : Set α, ∃ (v : _) (_ : v ⊆ s), hausdorffEdist u v ≤ δ := by
       intro u
@@ -311,8 +305,7 @@ instance NonemptyCompacts.completeSpace [CompleteSpace α] : CompleteSpace (None
 the same statement for closed subsets -/
 instance NonemptyCompacts.compactSpace [CompactSpace α] : CompactSpace (NonemptyCompacts α) :=
   ⟨by
-    rw [NonemptyCompacts.ToCloseds.uniformEmbedding.embedding.isCompact_iff]
-    rw [image_univ]
+    rw [NonemptyCompacts.ToCloseds.uniformEmbedding.embedding.isCompact_iff, image_univ]
     exact NonemptyCompacts.isClosed_in_closeds.isCompact⟩
 #align emetric.nonempty_compacts.compact_space EMetric.NonemptyCompacts.compactSpace
 
@@ -345,8 +338,8 @@ instance NonemptyCompacts.secondCountableTopology [SecondCountableTopology α] :
       have Fspec : ∀ x, F x ∈ s ∧ edist x (F x) < δ / 2 := fun x => (Exy x).choose_spec
       -- cover `t` with finitely many balls. Their centers form a set `a`
       have : TotallyBounded (t : Set α) := t.isCompact.totallyBounded
-      rcases totallyBounded_iff.1 this (δ / 2) δpos' with ⟨a, af, ta⟩
-      -- a : set α, af : a.finite, ta : t ⊆ ⋃ (y : α) (H : y ∈ a), eball y (δ / 2)
+      obtain ⟨a : Set α, af : Set.Finite a, ta : (t : Set α) ⊆ ⋃ y ∈ a, ball y (δ / 2)⟩ :=
+        totallyBounded_iff.1 this (δ / 2) δpos'
       -- replace each center by a nearby approximation in `s`, giving a new set `b`
       let b := F '' a
       have : b.Finite := af.image _

We remove all but one open Classicals, instead preferring to use open scoped Classical. The only real side-effect this led to is moving a couple declarations to use Exists.choose instead of Classical.choose.

The first few commits are explicitly labelled regex replaces for ease of review.

@@ -27,11 +27,13 @@ always finite in this context.
 
 noncomputable section
 
-open Classical Topology ENNReal
+open scoped Classical
+open Topology ENNReal
 
 universe u
 
-open Classical Set Function TopologicalSpace Filter
+open scoped Classical
+open Set Function TopologicalSpace Filter
 
 namespace EMetric
 
@@ -131,9 +133,9 @@ instance Closeds.completeSpace [CompleteSpace α] : CompleteSpace (Closeds α) :
           rw [← pow_add]
           apply hs <;> simp
         exact ⟨⟨z', z'_mem⟩, le_of_lt hz'⟩
-      use fun k => Nat.recOn k ⟨x, hx⟩ fun l z => choose (this l z)
+      use fun k => Nat.recOn k ⟨x, hx⟩ fun l z => (this l z).choose
       simp only [Nat.add_zero, Nat.zero_eq, Nat.rec_zero, Nat.rec_add_one, true_and]
-      exact fun k => choose_spec (this k _)
+      exact fun k => (this k _).choose_spec
     -- it follows from the previous bound that `z` is a Cauchy sequence
     have : CauchySeq fun k => (z k : α) := cauchySeq_of_edist_le_geometric_two (B n) (B_ne_top n) hz
     -- therefore, it converges
@@ -339,8 +341,8 @@ instance NonemptyCompacts.secondCountableTopology [SecondCountableTopology α] :
         intro x
         rcases mem_closure_iff.1 (s_dense x) (δ / 2) δpos' with ⟨y, ys, hy⟩
         exact ⟨y, ⟨ys, hy⟩⟩
-      let F x := choose (Exy x)
-      have Fspec : ∀ x, F x ∈ s ∧ edist x (F x) < δ / 2 := fun x => choose_spec (Exy x)
+      let F x := (Exy x).choose
+      have Fspec : ∀ x, F x ∈ s ∧ edist x (F x) < δ / 2 := fun x => (Exy x).choose_spec
       -- cover `t` with finitely many balls. Their centers form a set `a`
       have : TotallyBounded (t : Set α) := t.isCompact.totallyBounded
       rcases totallyBounded_iff.1 this (δ / 2) δpos' with ⟨a, af, ta⟩

Co-authored-by: Patrick Massot <patrickmassot@free.fr> Co-authored-by: Scott Morrison <scott.morrison@gmail.com>

@@ -98,8 +98,8 @@ instance Closeds.completeSpace [CompleteSpace α] : CompleteSpace (Closeds α) :
     completeness, by a standard completeness criterion.
     We use the shorthand `B n = 2^{-n}` in ennreal. -/
   let B : ℕ → ℝ≥0∞ := fun n => 2⁻¹ ^ n
-  have B_pos : ∀ n, (0 : ℝ≥0∞) < B n := by simp [ENNReal.pow_pos]
-  have B_ne_top : ∀ n, B n ≠ ⊤ := by simp [ENNReal.pow_ne_top]
+  have B_pos : ∀ n, (0 : ℝ≥0∞) < B n := by simp [B, ENNReal.pow_pos]
+  have B_ne_top : ∀ n, B n ≠ ⊤ := by simp [B, ENNReal.pow_ne_top]
   /- Consider a sequence of closed sets `s n` with `edist (s n) (s (n+1)) < B n`.
     We will show that it converges. The limit set is `t0 = ⋂n, closure (⋃m≥n, s m)`.
     We will have to show that a point in `s n` is close to a point in `t0`, and a point
@@ -226,7 +226,7 @@ instance Closeds.compactSpace [CompactSpace α] : CompactSpace (Closeds α) :=
     · apply @Finite.of_finite_image _ _ F _
       · apply fs.finite_subsets.subset fun b => _
         exact fun s => (s : Set α)
-        simp only [and_imp, Set.mem_image, Set.mem_setOf_eq, exists_imp]
+        simp only [F, and_imp, Set.mem_image, Set.mem_setOf_eq, exists_imp]
         intro _ x hx hx'
         rwa [hx'] at hx
       · exact SetLike.coe_injective.injOn F
@@ -363,7 +363,7 @@ instance NonemptyCompacts.secondCountableTopology [SecondCountableTopology α] :
       have tc : ∀ x ∈ t, ∃ y ∈ c, edist x y ≤ δ := by
         intro x hx
         rcases tb x hx with ⟨y, yv, Dxy⟩
-        have : y ∈ c := by simp [-mem_image]; exact ⟨yv, ⟨x, hx, Dxy⟩⟩
+        have : y ∈ c := by simp [c, -mem_image]; exact ⟨yv, ⟨x, hx, Dxy⟩⟩
         exact ⟨y, this, le_of_lt Dxy⟩
       -- points in `c` are well approximated by points in `t`
       have ct : ∀ y ∈ c, ∃ x ∈ t, edist y x ≤ δ := by

See here on Zulip.

This PR changes a bunch of names containing nhds_0 or/and lt_1 to nhds_zero or/and lt_one.

@@ -183,8 +183,8 @@ instance Closeds.completeSpace [CompleteSpace α] : CompleteSpace (Closeds α) :
   -- from this, the convergence of `s n` to `t0` follows.
   refine' tendsto_atTop.2 fun ε εpos => _
   have : Tendsto (fun n => 2 * B n) atTop (𝓝 (2 * 0)) :=
-    ENNReal.Tendsto.const_mul
-      (ENNReal.tendsto_pow_atTop_nhds_0_of_lt_1 <| by simp [ENNReal.one_lt_two]) (Or.inr <| by simp)
+    ENNReal.Tendsto.const_mul (ENNReal.tendsto_pow_atTop_nhds_zero_of_lt_one <|
+      by simp [ENNReal.one_lt_two]) (Or.inr <| by simp)
   rw [mul_zero] at this
   obtain ⟨N, hN⟩ : ∃ N, ∀ b ≥ N, ε > 2 * B b :=
     ((tendsto_order.1 this).2 ε εpos).exists_forall_of_atTop

Replace rcases( with rcases (. Same thing for convert( and congrm(. No other change.

@@ -382,7 +382,7 @@ instance NonemptyCompacts.secondCountableTopology [SecondCountableTopology α] :
       let d : NonemptyCompacts α := ⟨⟨c, ‹c.Finite›.isCompact⟩, hc⟩
       have : c ⊆ s := by
         intro x hx
-        rcases(mem_image _ _ _).1 hx.1 with ⟨y, ⟨_, yx⟩⟩
+        rcases (mem_image _ _ _).1 hx.1 with ⟨y, ⟨_, yx⟩⟩
         rw [← yx]
         exact (Fspec y).1
       have : d ∈ v := ⟨‹c.Finite›, this⟩

Define sigma-compact subsets of a topological space and show their basic properties.

• compact sets are sigma-compact
• countable unions of (sigma-)compact sets are sigma-compact
• closed subsets of sigma-compact sets are sigma-compact.

Relate them to sigma-compact space: a set is sigma-compact iff it is a sigma-compact space (w.r.t. the subspace topology).

In a later PR, we'll show that sigma-compact measure zero sets are nowhere dense.

• depends on: #7528

Co-authored-by: David Loeffler <d.loeffler.01@cantab.net> Co-authored-by: grunweg <grunweg@posteo.de>

@@ -309,7 +309,7 @@ instance NonemptyCompacts.completeSpace [CompleteSpace α] : CompleteSpace (None
 the same statement for closed subsets -/
 instance NonemptyCompacts.compactSpace [CompactSpace α] : CompactSpace (NonemptyCompacts α) :=
   ⟨by
-    rw [NonemptyCompacts.ToCloseds.uniformEmbedding.embedding.isCompact_iff_isCompact_image]
+    rw [NonemptyCompacts.ToCloseds.uniformEmbedding.embedding.isCompact_iff]
     rw [image_univ]
     exact NonemptyCompacts.isClosed_in_closeds.isCompact⟩
 #align emetric.nonempty_compacts.compact_space EMetric.NonemptyCompacts.compactSpace

Use Bornology.IsBounded instead.

@@ -407,8 +407,8 @@ variable {α : Type u} [MetricSpace α]
 edistance between two such sets is finite. -/
 instance NonemptyCompacts.metricSpace : MetricSpace (NonemptyCompacts α) :=
   EMetricSpace.toMetricSpace fun x y =>
-    hausdorffEdist_ne_top_of_nonempty_of_bounded x.nonempty y.nonempty x.isCompact.bounded
-      y.isCompact.bounded
+    hausdorffEdist_ne_top_of_nonempty_of_bounded x.nonempty y.nonempty x.isCompact.isBounded
+      y.isCompact.isBounded
 #align metric.nonempty_compacts.metric_space Metric.NonemptyCompacts.metricSpace
 
 /-- The distance on `NonemptyCompacts α` is the Hausdorff distance, by construction -/

Search&replace MulZeroClass.mul_zero -> mul_zero, MulZeroClass.zero_mul -> zero_mul.

These were introduced by Mathport, as the full name of mul_zero is actually MulZeroClass.mul_zero (it's exported with the short name).

@@ -185,7 +185,7 @@ instance Closeds.completeSpace [CompleteSpace α] : CompleteSpace (Closeds α) :
   have : Tendsto (fun n => 2 * B n) atTop (𝓝 (2 * 0)) :=
     ENNReal.Tendsto.const_mul
       (ENNReal.tendsto_pow_atTop_nhds_0_of_lt_1 <| by simp [ENNReal.one_lt_two]) (Or.inr <| by simp)
-  rw [MulZeroClass.mul_zero] at this
+  rw [mul_zero] at this
   obtain ⟨N, hN⟩ : ∃ N, ∀ b ≥ N, ε > 2 * B b :=
     ((tendsto_order.1 this).2 ε εpos).exists_forall_of_atTop
   exact ⟨N, fun n hn => lt_of_le_of_lt (main n) (hN n hn)⟩

@@ -270,7 +270,7 @@ theorem NonemptyCompacts.isClosed_in_closeds [CompleteSpace α] :
     exact ⟨s.nonempty, s.isCompact⟩
   rw [this]
   refine' isClosed_of_closure_subset fun s hs => ⟨_, _⟩
-  · -- take a set set t which is nonempty and at a finite distance of s
+  · -- take a set t which is nonempty and at a finite distance of s
     rcases mem_closure_iff.1 hs ⊤ ENNReal.coe_lt_top with ⟨t, ht, Dst⟩
     rw [edist_comm] at Dst
     -- since `t` is nonempty, so is `s`

• depends on: #5966

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

@@ -2,16 +2,13 @@
 Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Sébastien Gouëzel
-
-! This file was ported from Lean 3 source module topology.metric_space.closeds
-! leanprover-community/mathlib commit f2ce6086713c78a7f880485f7917ea547a215982
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
 -/
 import Mathlib.Analysis.SpecificLimits.Basic
 import Mathlib.Topology.MetricSpace.HausdorffDistance
 import Mathlib.Topology.Sets.Compacts
 
+#align_import topology.metric_space.closeds from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
+
 /-!
 # Closed subsets
 

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

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

@@ -75,15 +75,15 @@ set_option linter.uppercaseLean3 false in
 theorem isClosed_subsets_of_isClosed (hs : IsClosed s) :
     IsClosed { t : Closeds α | (t : Set α) ⊆ s } := by
   refine' isClosed_of_closure_subset fun t ht x hx => _
-  -- t : Closeds α,  ht : t ∈ closure {t : Closeds α | t ⊆ s},
-  -- x : α,  hx : x ∈ t
+  -- t : Closeds α, ht : t ∈ closure {t : Closeds α | t ⊆ s},
+  -- x : α, hx : x ∈ t
   -- goal : x ∈ s
   have : x ∈ closure s := by
     refine' mem_closure_iff.2 fun ε εpos => _
     rcases mem_closure_iff.1 ht ε εpos with ⟨u, hu, Dtu⟩
-    -- u : Closeds α,  hu : u ∈ {t : Closeds α | t ⊆ s},  hu' : edist t u < ε
+    -- u : Closeds α, hu : u ∈ {t : Closeds α | t ⊆ s}, hu' : edist t u < ε
     rcases exists_edist_lt_of_hausdorffEdist_lt hx Dtu with ⟨y, hy, Dxy⟩
-    -- y : α,  hy : y ∈ u, Dxy : edist x y < ε
+    -- y : α, hy : y ∈ u, Dxy : edist x y < ε
     exact ⟨y, hu hy, Dxy⟩
   rwa [hs.closure_eq] at this
 #align emetric.is_closed_subsets_of_is_closed EMetric.isClosed_subsets_of_isClosed
@@ -166,13 +166,13 @@ instance Closeds.completeSpace [CompleteSpace α] : CompleteSpace (Closeds α) :
     intro n x xt0
     have : x ∈ closure (⋃ m ≥ n, s m : Set α) := by apply mem_iInter.1 xt0 n
     rcases mem_closure_iff.1 this (B n) (B_pos n) with ⟨z, hz, Dxz⟩
-    -- z : α,  Dxz : edist x z < B n,
+    -- z : α, Dxz : edist x z < B n,
     simp only [exists_prop, Set.mem_iUnion] at hz
     rcases hz with ⟨m, ⟨m_ge_n, hm⟩⟩
     -- m : ℕ, m_ge_n : m ≥ n, hm : z ∈ s m
     have : hausdorffEdist (s m : Set α) (s n) < B n := hs n m n m_ge_n (le_refl n)
     rcases exists_edist_lt_of_hausdorffEdist_lt hm this with ⟨y, hy, Dzy⟩
-    -- y : α,  hy : y ∈ s n,  Dzy : edist z y < B n
+    -- y : α, hy : y ∈ s n, Dzy : edist z y < B n
     exact
       ⟨y, hy,
         calc
@@ -208,7 +208,7 @@ instance Closeds.compactSpace [CompactSpace α] : CompactSpace (Closeds α) :=
     rcases EMetric.totallyBounded_iff.1
         (isCompact_iff_totallyBounded_isComplete.1 (@isCompact_univ α _ _)).1 δ δpos with
       ⟨s, fs, hs⟩
-    -- s : Set α,  fs : s.Finite,  hs : univ ⊆ ⋃ (y : α) (H : y ∈ s), eball y δ
+    -- s : Set α, fs : s.Finite, hs : univ ⊆ ⋃ (y : α) (H : y ∈ s), eball y δ
     -- we first show that any set is well approximated by a subset of `s`.
     have main : ∀ u : Set α, ∃ (v : _) (_ : v ⊆ s), hausdorffEdist u v ≤ δ := by
       intro u
@@ -288,7 +288,7 @@ theorem NonemptyCompacts.isClosed_in_closeds [CompleteSpace α] :
         (ENNReal.half_pos εpos.ne') with
       ⟨u, fu, ut⟩
     refine' ⟨u, ⟨fu, fun x hx => _⟩⟩
-    -- u : set α,  fu : u.finite,  ut : t ⊆ ⋃ (y : α) (H : y ∈ u), eball y (ε / 2)
+    -- u : set α, fu : u.finite, ut : t ⊆ ⋃ (y : α) (H : y ∈ u), eball y (ε / 2)
     -- then s is covered by the union of the balls centered at u of radius ε
     rcases exists_edist_lt_of_hausdorffEdist_lt hx Dst with ⟨z, hz, Dxz⟩
     rcases mem_iUnion₂.1 (ut hz) with ⟨y, hy, Dzy⟩
@@ -347,7 +347,7 @@ instance NonemptyCompacts.secondCountableTopology [SecondCountableTopology α] :
       -- cover `t` with finitely many balls. Their centers form a set `a`
       have : TotallyBounded (t : Set α) := t.isCompact.totallyBounded
       rcases totallyBounded_iff.1 this (δ / 2) δpos' with ⟨a, af, ta⟩
-      -- a : set α,  af : a.finite,  ta : t ⊆ ⋃ (y : α) (H : y ∈ a), eball y (δ / 2)
+      -- a : set α, af : a.finite, ta : t ⊆ ⋃ (y : α) (H : y ∈ a), eball y (δ / 2)
       -- replace each center by a nearby approximation in `s`, giving a new set `b`
       let b := F '' a
       have : b.Finite := af.image _

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

@@ -210,7 +210,7 @@ instance Closeds.compactSpace [CompactSpace α] : CompactSpace (Closeds α) :=
       ⟨s, fs, hs⟩
     -- s : Set α,  fs : s.Finite,  hs : univ ⊆ ⋃ (y : α) (H : y ∈ s), eball y δ
     -- we first show that any set is well approximated by a subset of `s`.
-    have main : ∀ u : Set α, ∃ (v : _)(_ : v ⊆ s), hausdorffEdist u v ≤ δ := by
+    have main : ∀ u : Set α, ∃ (v : _) (_ : v ⊆ s), hausdorffEdist u v ≤ δ := by
       intro u
       let v := { x : α | x ∈ s ∧ ∃ y ∈ u, edist x y < δ }
       exists v, (fun x hx => hx.1 : v ⊆ s)

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>

@@ -110,7 +110,7 @@ instance Closeds.completeSpace [CompleteSpace α] : CompleteSpace (Closeds α) :
     standard criterion. -/
   refine' complete_of_convergent_controlled_sequences B B_pos fun s hs => _
   let t0 := ⋂ n, closure (⋃ m ≥ n, s m : Set α)
-  let t : Closeds α := ⟨t0, isClosed_interᵢ fun _ => isClosed_closure⟩
+  let t : Closeds α := ⟨t0, isClosed_iInter fun _ => isClosed_closure⟩
   use t
   -- The inequality is written this way to agree with `edist_le_of_edist_le_geometric_of_tendsto₀`
   have I1 : ∀ n, ∀ x ∈ s n, ∃ y ∈ t0, edist x y ≤ 2 * B n := by
@@ -145,11 +145,11 @@ instance Closeds.completeSpace [CompleteSpace α] : CompleteSpace (Closeds α) :
     -- the limit point `y` will be the desired point, in `t0` and close to our initial point `x`.
     -- First, we check it belongs to `t0`.
     have : y ∈ t0 :=
-      mem_interᵢ.2 fun k =>
+      mem_iInter.2 fun k =>
         mem_closure_of_tendsto y_lim
           (by
-            simp only [exists_prop, Set.mem_unionᵢ, Filter.eventually_atTop, Set.mem_preimage,
-              Set.preimage_unionᵢ]
+            simp only [exists_prop, Set.mem_iUnion, Filter.eventually_atTop, Set.mem_preimage,
+              Set.preimage_iUnion]
             exact ⟨k, fun m hm => ⟨n + m, zero_add k ▸ add_le_add (zero_le n) hm, (z m).2⟩⟩)
     use this
     -- Then, we check that `y` is close to `x = z n`. This follows from the fact that `y`
@@ -164,10 +164,10 @@ instance Closeds.completeSpace [CompleteSpace α] : CompleteSpace (Closeds α) :
             `s n` are close, this point is itself well approximated by a point `y` in `s n`,
             as required. -/
     intro n x xt0
-    have : x ∈ closure (⋃ m ≥ n, s m : Set α) := by apply mem_interᵢ.1 xt0 n
+    have : x ∈ closure (⋃ m ≥ n, s m : Set α) := by apply mem_iInter.1 xt0 n
     rcases mem_closure_iff.1 this (B n) (B_pos n) with ⟨z, hz, Dxz⟩
     -- z : α,  Dxz : edist x z < B n,
-    simp only [exists_prop, Set.mem_unionᵢ] at hz
+    simp only [exists_prop, Set.mem_iUnion] at hz
     rcases hz with ⟨m, ⟨m_ge_n, hm⟩⟩
     -- m : ℕ, m_ge_n : m ≥ n, hm : z ∈ s m
     have : hausdorffEdist (s m : Set α) (s n) < B n := hs n m n m_ge_n (le_refl n)
@@ -217,7 +217,7 @@ instance Closeds.compactSpace [CompactSpace α] : CompactSpace (Closeds α) :=
       refine' hausdorffEdist_le_of_mem_edist _ _
       · intro x hx
         have : x ∈ ⋃ y ∈ s, ball y δ := hs (by simp)
-        rcases mem_unionᵢ₂.1 this with ⟨y, ys, dy⟩
+        rcases mem_iUnion₂.1 this with ⟨y, ys, dy⟩
         have : edist y x < δ := by simp at dy; rwa [edist_comm] at dy
         exact ⟨y, ⟨ys, ⟨x, hx, this⟩⟩, le_of_lt dy⟩
       · rintro x ⟨_, ⟨y, yu, hy⟩⟩
@@ -239,7 +239,7 @@ instance Closeds.compactSpace [CompactSpace α] : CompactSpace (Closeds α) :=
       let t : Closeds α := ⟨t0, this⟩
       have : t ∈ F := t0s
       have : edist u t < ε := lt_of_le_of_lt Dut0 δlt
-      apply mem_unionᵢ₂.2
+      apply mem_iUnion₂.2
       exact ⟨t, ‹t ∈ F›, this⟩⟩
 #align emetric.closeds.compact_space EMetric.Closeds.compactSpace
 
@@ -291,13 +291,13 @@ theorem NonemptyCompacts.isClosed_in_closeds [CompleteSpace α] :
     -- u : set α,  fu : u.finite,  ut : t ⊆ ⋃ (y : α) (H : y ∈ u), eball y (ε / 2)
     -- then s is covered by the union of the balls centered at u of radius ε
     rcases exists_edist_lt_of_hausdorffEdist_lt hx Dst with ⟨z, hz, Dxz⟩
-    rcases mem_unionᵢ₂.1 (ut hz) with ⟨y, hy, Dzy⟩
+    rcases mem_iUnion₂.1 (ut hz) with ⟨y, hy, Dzy⟩
     have : edist x y < ε :=
       calc
         edist x y ≤ edist x z + edist z y := edist_triangle _ _ _
         _ < ε / 2 + ε / 2 := (ENNReal.add_lt_add Dxz Dzy)
         _ = ε := ENNReal.add_halves _
-    exact mem_bunionᵢ hy this
+    exact mem_biUnion hy this
 #align emetric.nonempty_compacts.is_closed_in_closeds EMetric.NonemptyCompacts.isClosed_in_closeds
 
 /-- In a complete space, the type of nonempty compact subsets is complete. This follows
@@ -353,7 +353,7 @@ instance NonemptyCompacts.secondCountableTopology [SecondCountableTopology α] :
       have : b.Finite := af.image _
       have tb : ∀ x ∈ t, ∃ y ∈ b, edist x y < δ := by
         intro x hx
-        rcases mem_unionᵢ₂.1 (ta hx) with ⟨z, za, Dxz⟩
+        rcases mem_iUnion₂.1 (ta hx) with ⟨z, za, Dxz⟩
         exists F z, mem_image_of_mem _ za
         calc
           edist x (F z) ≤ edist x z + edist z (F z) := edist_triangle _ _ _

@@ -75,13 +75,13 @@ set_option linter.uppercaseLean3 false in
 theorem isClosed_subsets_of_isClosed (hs : IsClosed s) :
     IsClosed { t : Closeds α | (t : Set α) ⊆ s } := by
   refine' isClosed_of_closure_subset fun t ht x hx => _
-  -- t : closeds α,  ht : t ∈ closure {t : closeds α | t ⊆ s},
+  -- t : Closeds α,  ht : t ∈ closure {t : Closeds α | t ⊆ s},
   -- x : α,  hx : x ∈ t
   -- goal : x ∈ s
   have : x ∈ closure s := by
     refine' mem_closure_iff.2 fun ε εpos => _
     rcases mem_closure_iff.1 ht ε εpos with ⟨u, hu, Dtu⟩
-    -- u : closeds α,  hu : u ∈ {t : closeds α | t ⊆ s},  hu' : edist t u < ε
+    -- u : Closeds α,  hu : u ∈ {t : Closeds α | t ⊆ s},  hu' : edist t u < ε
     rcases exists_edist_lt_of_hausdorffEdist_lt hx Dtu with ⟨y, hy, Dxy⟩
     -- y : α,  hy : y ∈ u, Dxy : edist x y < ε
     exact ⟨y, hu hy, Dxy⟩
@@ -189,8 +189,8 @@ instance Closeds.completeSpace [CompleteSpace α] : CompleteSpace (Closeds α) :
     ENNReal.Tendsto.const_mul
       (ENNReal.tendsto_pow_atTop_nhds_0_of_lt_1 <| by simp [ENNReal.one_lt_two]) (Or.inr <| by simp)
   rw [MulZeroClass.mul_zero] at this
-  obtain ⟨N, hN⟩ : ∃ N, ∀ b ≥ N, ε > 2 * B b
-  exact ((tendsto_order.1 this).2 ε εpos).exists_forall_of_atTop
+  obtain ⟨N, hN⟩ : ∃ N, ∀ b ≥ N, ε > 2 * B b :=
+    ((tendsto_order.1 this).2 ε εpos).exists_forall_of_atTop
   exact ⟨N, fun n hn => lt_of_le_of_lt (main n) (hN n hn)⟩
 #align emetric.closeds.complete_space EMetric.Closeds.completeSpace
 
@@ -208,7 +208,7 @@ instance Closeds.compactSpace [CompactSpace α] : CompactSpace (Closeds α) :=
     rcases EMetric.totallyBounded_iff.1
         (isCompact_iff_totallyBounded_isComplete.1 (@isCompact_univ α _ _)).1 δ δpos with
       ⟨s, fs, hs⟩
-    -- s : set α,  fs : s.finite,  hs : univ ⊆ ⋃ (y : α) (H : y ∈ s), eball y δ
+    -- s : Set α,  fs : s.Finite,  hs : univ ⊆ ⋃ (y : α) (H : y ∈ s), eball y δ
     -- we first show that any set is well approximated by a subset of `s`.
     have main : ∀ u : Set α, ∃ (v : _)(_ : v ⊆ s), hausdorffEdist u v ≤ δ := by
       intro u
@@ -431,7 +431,7 @@ theorem lipschitz_infDist : LipschitzWith 2 fun p : α × NonemptyCompacts α =>
   convert @LipschitzWith.uncurry α (NonemptyCompacts α) ℝ _ _ _
     (fun (x : α) (s : NonemptyCompacts α) => infDist x s) 1 1
     (fun s => lipschitz_infDist_pt ↑s) lipschitz_infDist_set
-  norm_cast
+  norm_num
 #align metric.lipschitz_inf_dist Metric.lipschitz_infDist
 
 theorem uniformContinuous_infDist_Hausdorff_dist :

Co-authored-by: Moritz Firsching <firsching@google.com> Co-authored-by: int-y1 <jason_yuen2007@hotmail.com>