topology.metric_space.partition_of_unity ⟷ Mathlib.Topology.MetricSpace.PartitionOfUnity

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

@@ -3,7 +3,7 @@ Copyright (c) 2022 Yury Kudryashov. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Yury Kudryashov
 -/
-import Topology.MetricSpace.EmetricParacompact
+import Topology.EMetricSpace.Paracompact
 import Analysis.Convex.PartitionOfUnity
 
 #align_import topology.metric_space.partition_of_unity from "leanprover-community/mathlib"@"50251fd6309cca5ca2e747882ffecd2729f38c5d"

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

@@ -54,7 +54,7 @@ theorem eventually_nhds_zero_forall_closedBall_subset (hK : ∀ i, IsClosed (K i
     filter_upwards [tendsto_snd (hfin.Inter_compl_mem_nhds hK x),
       (eventually_all_finite (hfin.point_finite x)).2 this]
     rintro ⟨r, y⟩ hxy hyU i hi
-    simp only [mem_Inter₂, mem_compl_iff, not_imp_not, mem_preimage] at hxy 
+    simp only [mem_Inter₂, mem_compl_iff, not_imp_not, mem_preimage] at hxy
     exact hyU _ (hxy _ hi)
   intro i hi
   rcases nhds_basis_closed_eball.mem_iff.1 ((hU i).mem_nhds <| hKU i hi) with ⟨R, hR₀, hR⟩

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

@@ -112,12 +112,12 @@ theorem exists_continuous_real_forall_closedBall_subset (hK : ∀ i, IsClosed (K
 #align emetric.exists_continuous_real_forall_closed_ball_subset EMetric.exists_continuous_real_forall_closedBall_subset
 -/
 
-#print EMetric.exists_continuous_nNReal_forall_closedBall_subset /-
+#print EMetric.exists_continuous_nnreal_forall_closedBall_subset /-
 /-- Let `X` be an extended metric space. Let `K : ι → set X` be a locally finite family of closed
 sets, let `U : ι → set X` be a family of open sets such that `K i ⊆ U i` for all `i`. Then there
 exists a positive continuous function `δ : C(X, ℝ≥0)` such that for any `i` and `x ∈ K i`,
 we have `emetric.closed_ball x (δ x) ⊆ U i`. -/
-theorem exists_continuous_nNReal_forall_closedBall_subset (hK : ∀ i, IsClosed (K i))
+theorem exists_continuous_nnreal_forall_closedBall_subset (hK : ∀ i, IsClosed (K i))
     (hU : ∀ i, IsOpen (U i)) (hKU : ∀ i, K i ⊆ U i) (hfin : LocallyFinite K) :
     ∃ δ : C(X, ℝ≥0), (∀ x, 0 < δ x) ∧ ∀ (i), ∀ x ∈ K i, closedBall x (δ x) ⊆ U i :=
   by
@@ -125,7 +125,7 @@ theorem exists_continuous_nNReal_forall_closedBall_subset (hK : ∀ i, IsClosed
   lift δ to C(X, ℝ≥0) using fun x => (hδ₀ x).le
   refine' ⟨δ, hδ₀, fun i x hi => _⟩
   simpa only [← ENNReal.ofReal_coe_nnreal] using hδ i x hi
-#align emetric.exists_continuous_nnreal_forall_closed_ball_subset EMetric.exists_continuous_nNReal_forall_closedBall_subset
+#align emetric.exists_continuous_nnreal_forall_closed_ball_subset EMetric.exists_continuous_nnreal_forall_closedBall_subset
 -/
 
 #print EMetric.exists_continuous_eNNReal_forall_closedBall_subset /-
@@ -136,7 +136,7 @@ we have `emetric.closed_ball x (δ x) ⊆ U i`. -/
 theorem exists_continuous_eNNReal_forall_closedBall_subset (hK : ∀ i, IsClosed (K i))
     (hU : ∀ i, IsOpen (U i)) (hKU : ∀ i, K i ⊆ U i) (hfin : LocallyFinite K) :
     ∃ δ : C(X, ℝ≥0∞), (∀ x, 0 < δ x) ∧ ∀ (i), ∀ x ∈ K i, closedBall x (δ x) ⊆ U i :=
-  let ⟨δ, hδ₀, hδ⟩ := exists_continuous_nNReal_forall_closedBall_subset hK hU hKU hfin
+  let ⟨δ, hδ₀, hδ⟩ := exists_continuous_nnreal_forall_closedBall_subset hK hU hKU hfin
   ⟨ContinuousMap.comp ⟨coe, ENNReal.continuous_coe⟩ δ, fun x => ENNReal.coe_pos.2 (hδ₀ x), hδ⟩
 #align emetric.exists_continuous_ennreal_forall_closed_ball_subset EMetric.exists_continuous_eNNReal_forall_closedBall_subset
 -/
@@ -147,20 +147,20 @@ namespace Metric
 
 variable [MetricSpace X] {K : ι → Set X} {U : ι → Set X}
 
-#print Metric.exists_continuous_nNReal_forall_closedBall_subset /-
+#print Metric.exists_continuous_nnreal_forall_closedBall_subset /-
 /-- Let `X` be a metric space. Let `K : ι → set X` be a locally finite family of closed sets, let
 `U : ι → set X` be a family of open sets such that `K i ⊆ U i` for all `i`. Then there exists a
 positive continuous function `δ : C(X, ℝ≥0)` such that for any `i` and `x ∈ K i`, we have
 `metric.closed_ball x (δ x) ⊆ U i`. -/
-theorem exists_continuous_nNReal_forall_closedBall_subset (hK : ∀ i, IsClosed (K i))
+theorem exists_continuous_nnreal_forall_closedBall_subset (hK : ∀ i, IsClosed (K i))
     (hU : ∀ i, IsOpen (U i)) (hKU : ∀ i, K i ⊆ U i) (hfin : LocallyFinite K) :
     ∃ δ : C(X, ℝ≥0), (∀ x, 0 < δ x) ∧ ∀ (i), ∀ x ∈ K i, closedBall x (δ x) ⊆ U i :=
   by
-  rcases EMetric.exists_continuous_nNReal_forall_closedBall_subset hK hU hKU hfin with ⟨δ, hδ0, hδ⟩
+  rcases EMetric.exists_continuous_nnreal_forall_closedBall_subset hK hU hKU hfin with ⟨δ, hδ0, hδ⟩
   refine' ⟨δ, hδ0, fun i x hx => _⟩
   rw [← emetric_closed_ball_nnreal]
   exact hδ i x hx
-#align metric.exists_continuous_nnreal_forall_closed_ball_subset Metric.exists_continuous_nNReal_forall_closedBall_subset
+#align metric.exists_continuous_nnreal_forall_closed_ball_subset Metric.exists_continuous_nnreal_forall_closedBall_subset
 -/
 
 #print Metric.exists_continuous_real_forall_closedBall_subset /-
@@ -171,7 +171,7 @@ positive continuous function `δ : C(X, ℝ)` such that for any `i` and `x ∈ K
 theorem exists_continuous_real_forall_closedBall_subset (hK : ∀ i, IsClosed (K i))
     (hU : ∀ i, IsOpen (U i)) (hKU : ∀ i, K i ⊆ U i) (hfin : LocallyFinite K) :
     ∃ δ : C(X, ℝ), (∀ x, 0 < δ x) ∧ ∀ (i), ∀ x ∈ K i, closedBall x (δ x) ⊆ U i :=
-  let ⟨δ, hδ₀, hδ⟩ := exists_continuous_nNReal_forall_closedBall_subset hK hU hKU hfin
+  let ⟨δ, hδ₀, hδ⟩ := exists_continuous_nnreal_forall_closedBall_subset hK hU hKU hfin
   ⟨ContinuousMap.comp ⟨coe, NNReal.continuous_coe⟩ δ, hδ₀, hδ⟩
 #align metric.exists_continuous_real_forall_closed_ball_subset Metric.exists_continuous_real_forall_closedBall_subset
 -/

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

@@ -3,8 +3,8 @@ Copyright (c) 2022 Yury Kudryashov. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Yury Kudryashov
 -/
-import Mathbin.Topology.MetricSpace.EmetricParacompact
-import Mathbin.Analysis.Convex.PartitionOfUnity
+import Topology.MetricSpace.EmetricParacompact
+import Analysis.Convex.PartitionOfUnity
 
 #align_import topology.metric_space.partition_of_unity from "leanprover-community/mathlib"@"50251fd6309cca5ca2e747882ffecd2729f38c5d"
 

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

@@ -2,15 +2,12 @@
 Copyright (c) 2022 Yury Kudryashov. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Yury Kudryashov
-
-! This file was ported from Lean 3 source module topology.metric_space.partition_of_unity
-! leanprover-community/mathlib commit 50251fd6309cca5ca2e747882ffecd2729f38c5d
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
 -/
 import Mathbin.Topology.MetricSpace.EmetricParacompact
 import Mathbin.Analysis.Convex.PartitionOfUnity
 
+#align_import topology.metric_space.partition_of_unity from "leanprover-community/mathlib"@"50251fd6309cca5ca2e747882ffecd2729f38c5d"
+
 /-!
 # Lemmas about (e)metric spaces that need partition of unity
 

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

@@ -43,6 +43,7 @@ namespace Emetric
 
 variable [EMetricSpace X] {K : ι → Set X} {U : ι → Set X}
 
+#print EMetric.eventually_nhds_zero_forall_closedBall_subset /-
 /-- Let `K : ι → set X` be a locally finitie family of closed sets in an emetric space. Let
 `U : ι → set X` be a family of open sets such that `K i ⊆ U i` for all `i`. Then for any point
 `x : X`, for sufficiently small `r : ℝ≥0∞` and for `y` sufficiently close to `x`, for all `i`, if
@@ -70,7 +71,9 @@ theorem eventually_nhds_zero_forall_closedBall_subset (hK : ∀ i, IsClosed (K i
     _ ≤ p.1 + (R - p.1) := (add_le_add hz <| le_trans hp.2 <| tsub_le_tsub_left hp.1.out.le _)
     _ = R := add_tsub_cancel_of_le (lt_trans hp.1 hrR).le
 #align emetric.eventually_nhds_zero_forall_closed_ball_subset EMetric.eventually_nhds_zero_forall_closedBall_subset
+-/
 
+#print EMetric.exists_forall_closedBall_subset_aux₁ /-
 theorem exists_forall_closedBall_subset_aux₁ (hK : ∀ i, IsClosed (K i)) (hU : ∀ i, IsOpen (U i))
     (hKU : ∀ i, K i ⊆ U i) (hfin : LocallyFinite K) (x : X) :
     ∃ r : ℝ,
@@ -84,7 +87,9 @@ theorem exists_forall_closedBall_subset_aux₁ (hK : ∀ i, IsClosed (K i)) (hU
   refine' ⟨r, hr.mono fun y hy => ⟨hr0, _⟩⟩
   rwa [mem_preimage, mem_Inter₂]
 #align emetric.exists_forall_closed_ball_subset_aux₁ EMetric.exists_forall_closedBall_subset_aux₁
+-/
 
+#print EMetric.exists_forall_closedBall_subset_aux₂ /-
 theorem exists_forall_closedBall_subset_aux₂ (y : X) :
     Convex ℝ (Ioi (0 : ℝ) ∩ ENNReal.ofReal ⁻¹' ⋂ (i) (hi : y ∈ K i), {r | closedBall y r ⊆ U i}) :=
   (convex_Ioi _).inter <|
@@ -93,7 +98,9 @@ theorem exists_forall_closedBall_subset_aux₂ (y : X) :
         ordConnected_iInter fun i =>
           ordConnected_iInter fun hi => ordConnected_setOf_closedBall_subset y (U i)
 #align emetric.exists_forall_closed_ball_subset_aux₂ EMetric.exists_forall_closedBall_subset_aux₂
+-/
 
+#print EMetric.exists_continuous_real_forall_closedBall_subset /-
 /-- Let `X` be an extended metric space. Let `K : ι → set X` be a locally finite family of closed
 sets, let `U : ι → set X` be a family of open sets such that `K i ⊆ U i` for all `i`. Then there
 exists a positive continuous function `δ : C(X, ℝ)` such that for any `i` and `x ∈ K i`,
@@ -106,7 +113,9 @@ theorem exists_continuous_real_forall_closedBall_subset (hK : ∀ i, IsClosed (K
     exists_continuous_forall_mem_convex_of_local_const exists_forall_closed_ball_subset_aux₂
       (exists_forall_closed_ball_subset_aux₁ hK hU hKU hfin)
 #align emetric.exists_continuous_real_forall_closed_ball_subset EMetric.exists_continuous_real_forall_closedBall_subset
+-/
 
+#print EMetric.exists_continuous_nNReal_forall_closedBall_subset /-
 /-- Let `X` be an extended metric space. Let `K : ι → set X` be a locally finite family of closed
 sets, let `U : ι → set X` be a family of open sets such that `K i ⊆ U i` for all `i`. Then there
 exists a positive continuous function `δ : C(X, ℝ≥0)` such that for any `i` and `x ∈ K i`,
@@ -120,7 +129,9 @@ theorem exists_continuous_nNReal_forall_closedBall_subset (hK : ∀ i, IsClosed
   refine' ⟨δ, hδ₀, fun i x hi => _⟩
   simpa only [← ENNReal.ofReal_coe_nnreal] using hδ i x hi
 #align emetric.exists_continuous_nnreal_forall_closed_ball_subset EMetric.exists_continuous_nNReal_forall_closedBall_subset
+-/
 
+#print EMetric.exists_continuous_eNNReal_forall_closedBall_subset /-
 /-- Let `X` be an extended metric space. Let `K : ι → set X` be a locally finite family of closed
 sets, let `U : ι → set X` be a family of open sets such that `K i ⊆ U i` for all `i`. Then there
 exists a positive continuous function `δ : C(X, ℝ≥0∞)` such that for any `i` and `x ∈ K i`,
@@ -131,6 +142,7 @@ theorem exists_continuous_eNNReal_forall_closedBall_subset (hK : ∀ i, IsClosed
   let ⟨δ, hδ₀, hδ⟩ := exists_continuous_nNReal_forall_closedBall_subset hK hU hKU hfin
   ⟨ContinuousMap.comp ⟨coe, ENNReal.continuous_coe⟩ δ, fun x => ENNReal.coe_pos.2 (hδ₀ x), hδ⟩
 #align emetric.exists_continuous_ennreal_forall_closed_ball_subset EMetric.exists_continuous_eNNReal_forall_closedBall_subset
+-/
 
 end Emetric
 
@@ -138,6 +150,7 @@ namespace Metric
 
 variable [MetricSpace X] {K : ι → Set X} {U : ι → Set X}
 
+#print Metric.exists_continuous_nNReal_forall_closedBall_subset /-
 /-- Let `X` be a metric space. Let `K : ι → set X` be a locally finite family of closed sets, let
 `U : ι → set X` be a family of open sets such that `K i ⊆ U i` for all `i`. Then there exists a
 positive continuous function `δ : C(X, ℝ≥0)` such that for any `i` and `x ∈ K i`, we have
@@ -151,7 +164,9 @@ theorem exists_continuous_nNReal_forall_closedBall_subset (hK : ∀ i, IsClosed
   rw [← emetric_closed_ball_nnreal]
   exact hδ i x hx
 #align metric.exists_continuous_nnreal_forall_closed_ball_subset Metric.exists_continuous_nNReal_forall_closedBall_subset
+-/
 
+#print Metric.exists_continuous_real_forall_closedBall_subset /-
 /-- Let `X` be a metric space. Let `K : ι → set X` be a locally finite family of closed sets, let
 `U : ι → set X` be a family of open sets such that `K i ⊆ U i` for all `i`. Then there exists a
 positive continuous function `δ : C(X, ℝ)` such that for any `i` and `x ∈ K i`, we have
@@ -162,6 +177,7 @@ theorem exists_continuous_real_forall_closedBall_subset (hK : ∀ i, IsClosed (K
   let ⟨δ, hδ₀, hδ⟩ := exists_continuous_nNReal_forall_closedBall_subset hK hU hKU hfin
   ⟨ContinuousMap.comp ⟨coe, NNReal.continuous_coe⟩ δ, hδ₀, hδ⟩
 #align metric.exists_continuous_real_forall_closed_ball_subset Metric.exists_continuous_real_forall_closedBall_subset
+-/
 
 end Metric
 

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

@@ -69,7 +69,6 @@ theorem eventually_nhds_zero_forall_closedBall_subset (hK : ∀ i, IsClosed (K i
     edist z x ≤ edist z p.2 + edist p.2 x := edist_triangle _ _ _
     _ ≤ p.1 + (R - p.1) := (add_le_add hz <| le_trans hp.2 <| tsub_le_tsub_left hp.1.out.le _)
     _ = R := add_tsub_cancel_of_le (lt_trans hp.1 hrR).le
-    
 #align emetric.eventually_nhds_zero_forall_closed_ball_subset EMetric.eventually_nhds_zero_forall_closedBall_subset
 
 theorem exists_forall_closedBall_subset_aux₁ (hK : ∀ i, IsClosed (K i)) (hU : ∀ i, IsOpen (U i))

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

@@ -62,7 +62,8 @@ theorem eventually_nhds_zero_forall_closedBall_subset (hK : ∀ i, IsClosed (K i
   rcases nhds_basis_closed_eball.mem_iff.1 ((hU i).mem_nhds <| hKU i hi) with ⟨R, hR₀, hR⟩
   rcases ennreal.lt_iff_exists_nnreal_btwn.mp hR₀ with ⟨r, hr₀, hrR⟩
   filter_upwards [prod_mem_prod (eventually_lt_nhds hr₀)
-      (closed_ball_mem_nhds x (tsub_pos_iff_lt.2 hrR))]with p hp z hz
+      (closed_ball_mem_nhds x (tsub_pos_iff_lt.2 hrR))] with
+    p hp z hz
   apply hR
   calc
     edist z x ≤ edist z p.2 + edist p.2 x := edist_triangle _ _ _
@@ -75,7 +76,7 @@ theorem exists_forall_closedBall_subset_aux₁ (hK : ∀ i, IsClosed (K i)) (hU
     (hKU : ∀ i, K i ⊆ U i) (hfin : LocallyFinite K) (x : X) :
     ∃ r : ℝ,
       ∀ᶠ y in 𝓝 x,
-        r ∈ Ioi (0 : ℝ) ∩ ENNReal.ofReal ⁻¹' ⋂ (i) (hi : y ∈ K i), { r | closedBall y r ⊆ U i } :=
+        r ∈ Ioi (0 : ℝ) ∩ ENNReal.ofReal ⁻¹' ⋂ (i) (hi : y ∈ K i), {r | closedBall y r ⊆ U i} :=
   by
   have :=
     (ennreal.continuous_of_real.tendsto' 0 0 ENNReal.ofReal_zero).Eventually
@@ -86,8 +87,7 @@ theorem exists_forall_closedBall_subset_aux₁ (hK : ∀ i, IsClosed (K i)) (hU
 #align emetric.exists_forall_closed_ball_subset_aux₁ EMetric.exists_forall_closedBall_subset_aux₁
 
 theorem exists_forall_closedBall_subset_aux₂ (y : X) :
-    Convex ℝ
-      (Ioi (0 : ℝ) ∩ ENNReal.ofReal ⁻¹' ⋂ (i) (hi : y ∈ K i), { r | closedBall y r ⊆ U i }) :=
+    Convex ℝ (Ioi (0 : ℝ) ∩ ENNReal.ofReal ⁻¹' ⋂ (i) (hi : y ∈ K i), {r | closedBall y r ⊆ U i}) :=
   (convex_Ioi _).inter <|
     OrdConnected.convex <|
       OrdConnected.preimage_ennreal_ofReal <|

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

@@ -56,7 +56,7 @@ theorem eventually_nhds_zero_forall_closedBall_subset (hK : ∀ i, IsClosed (K i
     filter_upwards [tendsto_snd (hfin.Inter_compl_mem_nhds hK x),
       (eventually_all_finite (hfin.point_finite x)).2 this]
     rintro ⟨r, y⟩ hxy hyU i hi
-    simp only [mem_Inter₂, mem_compl_iff, not_imp_not, mem_preimage] at hxy
+    simp only [mem_Inter₂, mem_compl_iff, not_imp_not, mem_preimage] at hxy 
     exact hyU _ (hxy _ hi)
   intro i hi
   rcases nhds_basis_closed_eball.mem_iff.1 ((hU i).mem_nhds <| hKU i hi) with ⟨R, hR₀, hR⟩

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

@@ -33,7 +33,7 @@ metric space, partition of unity, locally finite
 -/
 
 
-open Topology ENNReal BigOperators NNReal Filter
+open scoped Topology ENNReal BigOperators NNReal Filter
 
 open Set Function Filter TopologicalSpace
 

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

@@ -43,12 +43,6 @@ namespace Emetric
 
 variable [EMetricSpace X] {K : ι → Set X} {U : ι → Set X}
 
-/- warning: emetric.eventually_nhds_zero_forall_closed_ball_subset -> EMetric.eventually_nhds_zero_forall_closedBall_subset is a dubious translation:
-lean 3 declaration is
-  forall {ι : Type.{u1}} {X : Type.{u2}} [_inst_1 : EMetricSpace.{u2} X] {K : ι -> (Set.{u2} X)} {U : ι -> (Set.{u2} X)}, (forall (i : ι), IsClosed.{u2} X (UniformSpace.toTopologicalSpace.{u2} X (PseudoEMetricSpace.toUniformSpace.{u2} X (EMetricSpace.toPseudoEmetricSpace.{u2} X _inst_1))) (K i)) -> (forall (i : ι), IsOpen.{u2} X (UniformSpace.toTopologicalSpace.{u2} X (PseudoEMetricSpace.toUniformSpace.{u2} X (EMetricSpace.toPseudoEmetricSpace.{u2} X _inst_1))) (U i)) -> (forall (i : ι), HasSubset.Subset.{u2} (Set.{u2} X) (Set.hasSubset.{u2} X) (K i) (U i)) -> (LocallyFinite.{u1, u2} ι X (UniformSpace.toTopologicalSpace.{u2} X (PseudoEMetricSpace.toUniformSpace.{u2} X (EMetricSpace.toPseudoEmetricSpace.{u2} X _inst_1))) K) -> (forall (x : X), Filter.Eventually.{u2} (Prod.{0, u2} ENNReal X) (fun (p : Prod.{0, u2} ENNReal X) => forall (i : ι), (Membership.Mem.{u2, u2} X (Set.{u2} X) (Set.hasMem.{u2} X) (Prod.snd.{0, u2} ENNReal X p) (K i)) -> (HasSubset.Subset.{u2} (Set.{u2} X) (Set.hasSubset.{u2} X) (EMetric.closedBall.{u2} X (EMetricSpace.toPseudoEmetricSpace.{u2} X _inst_1) (Prod.snd.{0, u2} ENNReal X p) (Prod.fst.{0, u2} ENNReal X p)) (U i))) (Filter.prod.{0, u2} ENNReal X (nhds.{0} ENNReal ENNReal.topologicalSpace (OfNat.ofNat.{0} ENNReal 0 (OfNat.mk.{0} ENNReal 0 (Zero.zero.{0} ENNReal ENNReal.hasZero)))) (nhds.{u2} X (UniformSpace.toTopologicalSpace.{u2} X (PseudoEMetricSpace.toUniformSpace.{u2} X (EMetricSpace.toPseudoEmetricSpace.{u2} X _inst_1))) x)))
-but is expected to have type
-  forall {ι : Type.{u1}} {X : Type.{u2}} [_inst_1 : EMetricSpace.{u2} X] {K : ι -> (Set.{u2} X)} {U : ι -> (Set.{u2} X)}, (forall (i : ι), IsClosed.{u2} X (UniformSpace.toTopologicalSpace.{u2} X (PseudoEMetricSpace.toUniformSpace.{u2} X (EMetricSpace.toPseudoEMetricSpace.{u2} X _inst_1))) (K i)) -> (forall (i : ι), IsOpen.{u2} X (UniformSpace.toTopologicalSpace.{u2} X (PseudoEMetricSpace.toUniformSpace.{u2} X (EMetricSpace.toPseudoEMetricSpace.{u2} X _inst_1))) (U i)) -> (forall (i : ι), HasSubset.Subset.{u2} (Set.{u2} X) (Set.instHasSubsetSet.{u2} X) (K i) (U i)) -> (LocallyFinite.{u1, u2} ι X (UniformSpace.toTopologicalSpace.{u2} X (PseudoEMetricSpace.toUniformSpace.{u2} X (EMetricSpace.toPseudoEMetricSpace.{u2} X _inst_1))) K) -> (forall (x : X), Filter.Eventually.{u2} (Prod.{0, u2} ENNReal X) (fun (p : Prod.{0, u2} ENNReal X) => forall (i : ι), (Membership.mem.{u2, u2} X (Set.{u2} X) (Set.instMembershipSet.{u2} X) (Prod.snd.{0, u2} ENNReal X p) (K i)) -> (HasSubset.Subset.{u2} (Set.{u2} X) (Set.instHasSubsetSet.{u2} X) (EMetric.closedBall.{u2} X (EMetricSpace.toPseudoEMetricSpace.{u2} X _inst_1) (Prod.snd.{0, u2} ENNReal X p) (Prod.fst.{0, u2} ENNReal X p)) (U i))) (Filter.prod.{0, u2} ENNReal X (nhds.{0} ENNReal ENNReal.instTopologicalSpaceENNReal (OfNat.ofNat.{0} ENNReal 0 (Zero.toOfNat0.{0} ENNReal instENNRealZero))) (nhds.{u2} X (UniformSpace.toTopologicalSpace.{u2} X (PseudoEMetricSpace.toUniformSpace.{u2} X (EMetricSpace.toPseudoEMetricSpace.{u2} X _inst_1))) x)))
-Case conversion may be inaccurate. Consider using '#align emetric.eventually_nhds_zero_forall_closed_ball_subset EMetric.eventually_nhds_zero_forall_closedBall_subsetₓ'. -/
 /-- Let `K : ι → set X` be a locally finitie family of closed sets in an emetric space. Let
 `U : ι → set X` be a family of open sets such that `K i ⊆ U i` for all `i`. Then for any point
 `x : X`, for sufficiently small `r : ℝ≥0∞` and for `y` sufficiently close to `x`, for all `i`, if
@@ -77,9 +71,6 @@ theorem eventually_nhds_zero_forall_closedBall_subset (hK : ∀ i, IsClosed (K i
     
 #align emetric.eventually_nhds_zero_forall_closed_ball_subset EMetric.eventually_nhds_zero_forall_closedBall_subset
 
-/- warning: emetric.exists_forall_closed_ball_subset_aux₁ -> EMetric.exists_forall_closedBall_subset_aux₁ is a dubious translation:
-<too large>
-Case conversion may be inaccurate. Consider using '#align emetric.exists_forall_closed_ball_subset_aux₁ EMetric.exists_forall_closedBall_subset_aux₁ₓ'. -/
 theorem exists_forall_closedBall_subset_aux₁ (hK : ∀ i, IsClosed (K i)) (hU : ∀ i, IsOpen (U i))
     (hKU : ∀ i, K i ⊆ U i) (hfin : LocallyFinite K) (x : X) :
     ∃ r : ℝ,
@@ -94,12 +85,6 @@ theorem exists_forall_closedBall_subset_aux₁ (hK : ∀ i, IsClosed (K i)) (hU
   rwa [mem_preimage, mem_Inter₂]
 #align emetric.exists_forall_closed_ball_subset_aux₁ EMetric.exists_forall_closedBall_subset_aux₁
 
-/- warning: emetric.exists_forall_closed_ball_subset_aux₂ -> EMetric.exists_forall_closedBall_subset_aux₂ is a dubious translation:
-lean 3 declaration is
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-but is expected to have type
-  forall {ι : Type.{u2}} {X : Type.{u1}} [_inst_1 : EMetricSpace.{u1} X] {K : ι -> (Set.{u1} X)} {U : ι -> (Set.{u1} X)} (y : X), Convex.{0, 0} Real Real Real.orderedSemiring Real.instAddCommMonoidReal (Algebra.toSMul.{0, 0} Real Real Real.instCommSemiringReal Real.semiring (NormedAlgebra.toAlgebra.{0, 0} Real Real Real.normedField (SeminormedCommRing.toSeminormedRing.{0} Real (NormedCommRing.toSeminormedCommRing.{0} Real Real.normedCommRing)) (NormedAlgebra.id.{0} Real Real.normedField))) (Inter.inter.{0} (Set.{0} Real) (Set.instInterSet.{0} Real) (Set.Ioi.{0} Real Real.instPreorderReal (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal))) (Set.preimage.{0, 0} Real ENNReal ENNReal.ofReal (Set.iInter.{0, succ u2} ENNReal ι (fun (i : ι) => Set.iInter.{0, 0} ENNReal (Membership.mem.{u1, u1} X (Set.{u1} X) (Set.instMembershipSet.{u1} X) y (K i)) (fun (hi : Membership.mem.{u1, u1} X (Set.{u1} X) (Set.instMembershipSet.{u1} X) y (K i)) => setOf.{0} ENNReal (fun (r : ENNReal) => HasSubset.Subset.{u1} (Set.{u1} X) (Set.instHasSubsetSet.{u1} X) (EMetric.closedBall.{u1} X (EMetricSpace.toPseudoEMetricSpace.{u1} X _inst_1) y r) (U i)))))))
-Case conversion may be inaccurate. Consider using '#align emetric.exists_forall_closed_ball_subset_aux₂ EMetric.exists_forall_closedBall_subset_aux₂ₓ'. -/
 theorem exists_forall_closedBall_subset_aux₂ (y : X) :
     Convex ℝ
       (Ioi (0 : ℝ) ∩ ENNReal.ofReal ⁻¹' ⋂ (i) (hi : y ∈ K i), { r | closedBall y r ⊆ U i }) :=
@@ -110,9 +95,6 @@ theorem exists_forall_closedBall_subset_aux₂ (y : X) :
           ordConnected_iInter fun hi => ordConnected_setOf_closedBall_subset y (U i)
 #align emetric.exists_forall_closed_ball_subset_aux₂ EMetric.exists_forall_closedBall_subset_aux₂
 
-/- warning: emetric.exists_continuous_real_forall_closed_ball_subset -> EMetric.exists_continuous_real_forall_closedBall_subset is a dubious translation:
-<too large>
-Case conversion may be inaccurate. Consider using '#align emetric.exists_continuous_real_forall_closed_ball_subset EMetric.exists_continuous_real_forall_closedBall_subsetₓ'. -/
 /-- Let `X` be an extended metric space. Let `K : ι → set X` be a locally finite family of closed
 sets, let `U : ι → set X` be a family of open sets such that `K i ⊆ U i` for all `i`. Then there
 exists a positive continuous function `δ : C(X, ℝ)` such that for any `i` and `x ∈ K i`,
@@ -126,9 +108,6 @@ theorem exists_continuous_real_forall_closedBall_subset (hK : ∀ i, IsClosed (K
       (exists_forall_closed_ball_subset_aux₁ hK hU hKU hfin)
 #align emetric.exists_continuous_real_forall_closed_ball_subset EMetric.exists_continuous_real_forall_closedBall_subset
 
-/- warning: emetric.exists_continuous_nnreal_forall_closed_ball_subset -> EMetric.exists_continuous_nNReal_forall_closedBall_subset is a dubious translation:
-<too large>
-Case conversion may be inaccurate. Consider using '#align emetric.exists_continuous_nnreal_forall_closed_ball_subset EMetric.exists_continuous_nNReal_forall_closedBall_subsetₓ'. -/
 /-- Let `X` be an extended metric space. Let `K : ι → set X` be a locally finite family of closed
 sets, let `U : ι → set X` be a family of open sets such that `K i ⊆ U i` for all `i`. Then there
 exists a positive continuous function `δ : C(X, ℝ≥0)` such that for any `i` and `x ∈ K i`,
@@ -143,9 +122,6 @@ theorem exists_continuous_nNReal_forall_closedBall_subset (hK : ∀ i, IsClosed
   simpa only [← ENNReal.ofReal_coe_nnreal] using hδ i x hi
 #align emetric.exists_continuous_nnreal_forall_closed_ball_subset EMetric.exists_continuous_nNReal_forall_closedBall_subset
 
-/- warning: emetric.exists_continuous_ennreal_forall_closed_ball_subset -> EMetric.exists_continuous_eNNReal_forall_closedBall_subset is a dubious translation:
-<too large>
-Case conversion may be inaccurate. Consider using '#align emetric.exists_continuous_ennreal_forall_closed_ball_subset EMetric.exists_continuous_eNNReal_forall_closedBall_subsetₓ'. -/
 /-- Let `X` be an extended metric space. Let `K : ι → set X` be a locally finite family of closed
 sets, let `U : ι → set X` be a family of open sets such that `K i ⊆ U i` for all `i`. Then there
 exists a positive continuous function `δ : C(X, ℝ≥0∞)` such that for any `i` and `x ∈ K i`,
@@ -163,9 +139,6 @@ namespace Metric
 
 variable [MetricSpace X] {K : ι → Set X} {U : ι → Set X}
 
-/- warning: metric.exists_continuous_nnreal_forall_closed_ball_subset -> Metric.exists_continuous_nNReal_forall_closedBall_subset is a dubious translation:
-<too large>
-Case conversion may be inaccurate. Consider using '#align metric.exists_continuous_nnreal_forall_closed_ball_subset Metric.exists_continuous_nNReal_forall_closedBall_subsetₓ'. -/
 /-- Let `X` be a metric space. Let `K : ι → set X` be a locally finite family of closed sets, let
 `U : ι → set X` be a family of open sets such that `K i ⊆ U i` for all `i`. Then there exists a
 positive continuous function `δ : C(X, ℝ≥0)` such that for any `i` and `x ∈ K i`, we have
@@ -180,9 +153,6 @@ theorem exists_continuous_nNReal_forall_closedBall_subset (hK : ∀ i, IsClosed
   exact hδ i x hx
 #align metric.exists_continuous_nnreal_forall_closed_ball_subset Metric.exists_continuous_nNReal_forall_closedBall_subset
 
-/- warning: metric.exists_continuous_real_forall_closed_ball_subset -> Metric.exists_continuous_real_forall_closedBall_subset is a dubious translation:
-<too large>
-Case conversion may be inaccurate. Consider using '#align metric.exists_continuous_real_forall_closed_ball_subset Metric.exists_continuous_real_forall_closedBall_subsetₓ'. -/
 /-- Let `X` be a metric space. Let `K : ι → set X` be a locally finite family of closed sets, let
 `U : ι → set X` be a family of open sets such that `K i ⊆ U i` for all `i`. Then there exists a
 positive continuous function `δ : C(X, ℝ)` such that for any `i` and `x ∈ K i`, we have

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

@@ -78,10 +78,7 @@ theorem eventually_nhds_zero_forall_closedBall_subset (hK : ∀ i, IsClosed (K i
 #align emetric.eventually_nhds_zero_forall_closed_ball_subset EMetric.eventually_nhds_zero_forall_closedBall_subset
 
 /- warning: emetric.exists_forall_closed_ball_subset_aux₁ -> EMetric.exists_forall_closedBall_subset_aux₁ is a dubious translation:
-lean 3 declaration is
-  forall {ι : Type.{u1}} {X : Type.{u2}} [_inst_1 : EMetricSpace.{u2} X] {K : ι -> (Set.{u2} X)} {U : ι -> (Set.{u2} X)}, (forall (i : ι), IsClosed.{u2} X (UniformSpace.toTopologicalSpace.{u2} X (PseudoEMetricSpace.toUniformSpace.{u2} X (EMetricSpace.toPseudoEmetricSpace.{u2} X _inst_1))) (K i)) -> (forall (i : ι), IsOpen.{u2} X (UniformSpace.toTopologicalSpace.{u2} X (PseudoEMetricSpace.toUniformSpace.{u2} X (EMetricSpace.toPseudoEmetricSpace.{u2} X _inst_1))) (U i)) -> (forall (i : ι), HasSubset.Subset.{u2} (Set.{u2} X) (Set.hasSubset.{u2} X) (K i) (U i)) -> (LocallyFinite.{u1, u2} ι X (UniformSpace.toTopologicalSpace.{u2} X (PseudoEMetricSpace.toUniformSpace.{u2} X (EMetricSpace.toPseudoEmetricSpace.{u2} X _inst_1))) K) -> (forall (x : X), Exists.{1} Real (fun (r : Real) => Filter.Eventually.{u2} X (fun (y : X) => Membership.Mem.{0, 0} Real (Set.{0} Real) (Set.hasMem.{0} Real) r (Inter.inter.{0} (Set.{0} Real) (Set.hasInter.{0} Real) (Set.Ioi.{0} Real Real.preorder (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero)))) (Set.preimage.{0, 0} Real ENNReal ENNReal.ofReal (Set.iInter.{0, succ u1} ENNReal ι (fun (i : ι) => Set.iInter.{0, 0} ENNReal (Membership.Mem.{u2, u2} X (Set.{u2} X) (Set.hasMem.{u2} X) y (K i)) (fun (hi : Membership.Mem.{u2, u2} X (Set.{u2} X) (Set.hasMem.{u2} X) y (K i)) => setOf.{0} ENNReal (fun (r : ENNReal) => HasSubset.Subset.{u2} (Set.{u2} X) (Set.hasSubset.{u2} X) (EMetric.closedBall.{u2} X (EMetricSpace.toPseudoEmetricSpace.{u2} X _inst_1) y r) (U i)))))))) (nhds.{u2} X (UniformSpace.toTopologicalSpace.{u2} X (PseudoEMetricSpace.toUniformSpace.{u2} X (EMetricSpace.toPseudoEmetricSpace.{u2} X _inst_1))) x)))
-but is expected to have type
-  forall {ι : Type.{u1}} {X : Type.{u2}} [_inst_1 : EMetricSpace.{u2} X] {K : ι -> (Set.{u2} X)} {U : ι -> (Set.{u2} X)}, (forall (i : ι), IsClosed.{u2} X (UniformSpace.toTopologicalSpace.{u2} X (PseudoEMetricSpace.toUniformSpace.{u2} X (EMetricSpace.toPseudoEMetricSpace.{u2} X _inst_1))) (K i)) -> (forall (i : ι), IsOpen.{u2} X (UniformSpace.toTopologicalSpace.{u2} X (PseudoEMetricSpace.toUniformSpace.{u2} X (EMetricSpace.toPseudoEMetricSpace.{u2} X _inst_1))) (U i)) -> (forall (i : ι), HasSubset.Subset.{u2} (Set.{u2} X) (Set.instHasSubsetSet.{u2} X) (K i) (U i)) -> (LocallyFinite.{u1, u2} ι X (UniformSpace.toTopologicalSpace.{u2} X (PseudoEMetricSpace.toUniformSpace.{u2} X (EMetricSpace.toPseudoEMetricSpace.{u2} X _inst_1))) K) -> (forall (x : X), Exists.{1} Real (fun (r : Real) => Filter.Eventually.{u2} X (fun (y : X) => Membership.mem.{0, 0} Real (Set.{0} Real) (Set.instMembershipSet.{0} Real) r (Inter.inter.{0} (Set.{0} Real) (Set.instInterSet.{0} Real) (Set.Ioi.{0} Real Real.instPreorderReal (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal))) (Set.preimage.{0, 0} Real ENNReal ENNReal.ofReal (Set.iInter.{0, succ u1} ENNReal ι (fun (i : ι) => Set.iInter.{0, 0} ENNReal (Membership.mem.{u2, u2} X (Set.{u2} X) (Set.instMembershipSet.{u2} X) y (K i)) (fun (hi : Membership.mem.{u2, u2} X (Set.{u2} X) (Set.instMembershipSet.{u2} X) y (K i)) => setOf.{0} ENNReal (fun (r : ENNReal) => HasSubset.Subset.{u2} (Set.{u2} X) (Set.instHasSubsetSet.{u2} X) (EMetric.closedBall.{u2} X (EMetricSpace.toPseudoEMetricSpace.{u2} X _inst_1) y r) (U i)))))))) (nhds.{u2} X (UniformSpace.toTopologicalSpace.{u2} X (PseudoEMetricSpace.toUniformSpace.{u2} X (EMetricSpace.toPseudoEMetricSpace.{u2} X _inst_1))) x)))
+<too large>
 Case conversion may be inaccurate. Consider using '#align emetric.exists_forall_closed_ball_subset_aux₁ EMetric.exists_forall_closedBall_subset_aux₁ₓ'. -/
 theorem exists_forall_closedBall_subset_aux₁ (hK : ∀ i, IsClosed (K i)) (hU : ∀ i, IsOpen (U i))
     (hKU : ∀ i, K i ⊆ U i) (hfin : LocallyFinite K) (x : X) :
@@ -114,10 +111,7 @@ theorem exists_forall_closedBall_subset_aux₂ (y : X) :
 #align emetric.exists_forall_closed_ball_subset_aux₂ EMetric.exists_forall_closedBall_subset_aux₂
 
 /- warning: emetric.exists_continuous_real_forall_closed_ball_subset -> EMetric.exists_continuous_real_forall_closedBall_subset is a dubious translation:
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-  forall {ι : Type.{u1}} {X : Type.{u2}} [_inst_1 : EMetricSpace.{u2} X] {K : ι -> (Set.{u2} X)} {U : ι -> (Set.{u2} X)}, (forall (i : ι), IsClosed.{u2} X (UniformSpace.toTopologicalSpace.{u2} X (PseudoEMetricSpace.toUniformSpace.{u2} X (EMetricSpace.toPseudoEmetricSpace.{u2} X _inst_1))) (K i)) -> (forall (i : ι), IsOpen.{u2} X (UniformSpace.toTopologicalSpace.{u2} X (PseudoEMetricSpace.toUniformSpace.{u2} X (EMetricSpace.toPseudoEmetricSpace.{u2} X _inst_1))) (U i)) -> (forall (i : ι), HasSubset.Subset.{u2} (Set.{u2} X) (Set.hasSubset.{u2} X) (K i) (U i)) -> (LocallyFinite.{u1, u2} ι X (UniformSpace.toTopologicalSpace.{u2} X (PseudoEMetricSpace.toUniformSpace.{u2} X (EMetricSpace.toPseudoEmetricSpace.{u2} X _inst_1))) K) -> (Exists.{succ u2} (ContinuousMap.{u2, 0} X Real (UniformSpace.toTopologicalSpace.{u2} X (PseudoEMetricSpace.toUniformSpace.{u2} X (EMetricSpace.toPseudoEmetricSpace.{u2} X _inst_1))) (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) (fun (δ : ContinuousMap.{u2, 0} X Real (UniformSpace.toTopologicalSpace.{u2} X (PseudoEMetricSpace.toUniformSpace.{u2} X (EMetricSpace.toPseudoEmetricSpace.{u2} X _inst_1))) (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) => And (forall (x : X), LT.lt.{0} Real Real.hasLt (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero))) (coeFn.{succ u2, succ u2} (ContinuousMap.{u2, 0} X Real (UniformSpace.toTopologicalSpace.{u2} X (PseudoEMetricSpace.toUniformSpace.{u2} X (EMetricSpace.toPseudoEmetricSpace.{u2} X _inst_1))) (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) (fun (_x : ContinuousMap.{u2, 0} X Real (UniformSpace.toTopologicalSpace.{u2} X (PseudoEMetricSpace.toUniformSpace.{u2} X (EMetricSpace.toPseudoEmetricSpace.{u2} X _inst_1))) (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) => X -> Real) (ContinuousMap.hasCoeToFun.{u2, 0} X Real (UniformSpace.toTopologicalSpace.{u2} X (PseudoEMetricSpace.toUniformSpace.{u2} X (EMetricSpace.toPseudoEmetricSpace.{u2} X _inst_1))) (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) δ x)) (forall (i : ι) (x : X), (Membership.Mem.{u2, u2} X (Set.{u2} X) (Set.hasMem.{u2} X) x (K i)) -> (HasSubset.Subset.{u2} (Set.{u2} X) (Set.hasSubset.{u2} X) (EMetric.closedBall.{u2} X (EMetricSpace.toPseudoEmetricSpace.{u2} X _inst_1) x (ENNReal.ofReal (coeFn.{succ u2, succ u2} (ContinuousMap.{u2, 0} X Real (UniformSpace.toTopologicalSpace.{u2} X (PseudoEMetricSpace.toUniformSpace.{u2} X (EMetricSpace.toPseudoEmetricSpace.{u2} X _inst_1))) (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) (fun (_x : ContinuousMap.{u2, 0} X Real (UniformSpace.toTopologicalSpace.{u2} X (PseudoEMetricSpace.toUniformSpace.{u2} X (EMetricSpace.toPseudoEmetricSpace.{u2} X _inst_1))) (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) => X -> Real) (ContinuousMap.hasCoeToFun.{u2, 0} X Real (UniformSpace.toTopologicalSpace.{u2} X (PseudoEMetricSpace.toUniformSpace.{u2} X (EMetricSpace.toPseudoEmetricSpace.{u2} X _inst_1))) (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) δ x))) (U i)))))
-but is expected to have type
-  forall {ι : Type.{u1}} {X : Type.{u2}} [_inst_1 : EMetricSpace.{u2} X] {K : ι -> (Set.{u2} X)} {U : ι -> (Set.{u2} X)}, (forall (i : ι), IsClosed.{u2} X (UniformSpace.toTopologicalSpace.{u2} X (PseudoEMetricSpace.toUniformSpace.{u2} X (EMetricSpace.toPseudoEMetricSpace.{u2} X _inst_1))) (K i)) -> (forall (i : ι), IsOpen.{u2} X (UniformSpace.toTopologicalSpace.{u2} X (PseudoEMetricSpace.toUniformSpace.{u2} X (EMetricSpace.toPseudoEMetricSpace.{u2} X _inst_1))) (U i)) -> (forall (i : ι), HasSubset.Subset.{u2} (Set.{u2} X) (Set.instHasSubsetSet.{u2} X) (K i) (U i)) -> (LocallyFinite.{u1, u2} ι X (UniformSpace.toTopologicalSpace.{u2} X (PseudoEMetricSpace.toUniformSpace.{u2} X (EMetricSpace.toPseudoEMetricSpace.{u2} X _inst_1))) K) -> (Exists.{succ u2} (ContinuousMap.{u2, 0} X Real (UniformSpace.toTopologicalSpace.{u2} X (PseudoEMetricSpace.toUniformSpace.{u2} X (EMetricSpace.toPseudoEMetricSpace.{u2} X _inst_1))) (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) (fun (δ : ContinuousMap.{u2, 0} X Real (UniformSpace.toTopologicalSpace.{u2} X (PseudoEMetricSpace.toUniformSpace.{u2} X (EMetricSpace.toPseudoEMetricSpace.{u2} X _inst_1))) (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) => And (forall (x : X), LT.lt.{0} ((fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : X) => Real) x) Real.instLTReal (OfNat.ofNat.{0} ((fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : X) => Real) x) 0 (Zero.toOfNat0.{0} ((fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : X) => Real) x) Real.instZeroReal)) (FunLike.coe.{succ u2, succ u2, 1} (ContinuousMap.{u2, 0} X Real (UniformSpace.toTopologicalSpace.{u2} X (PseudoEMetricSpace.toUniformSpace.{u2} X (EMetricSpace.toPseudoEMetricSpace.{u2} X _inst_1))) (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) X (fun (_x : X) => (fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : X) => Real) _x) (ContinuousMapClass.toFunLike.{u2, u2, 0} (ContinuousMap.{u2, 0} X Real (UniformSpace.toTopologicalSpace.{u2} X (PseudoEMetricSpace.toUniformSpace.{u2} X (EMetricSpace.toPseudoEMetricSpace.{u2} X _inst_1))) (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) X Real (UniformSpace.toTopologicalSpace.{u2} X (PseudoEMetricSpace.toUniformSpace.{u2} X (EMetricSpace.toPseudoEMetricSpace.{u2} X _inst_1))) (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) (ContinuousMap.instContinuousMapClassContinuousMap.{u2, 0} X Real (UniformSpace.toTopologicalSpace.{u2} X (PseudoEMetricSpace.toUniformSpace.{u2} X (EMetricSpace.toPseudoEMetricSpace.{u2} X _inst_1))) (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)))) δ x)) (forall (i : ι) (x : X), (Membership.mem.{u2, u2} X (Set.{u2} X) (Set.instMembershipSet.{u2} X) x (K i)) -> (HasSubset.Subset.{u2} (Set.{u2} X) (Set.instHasSubsetSet.{u2} X) (EMetric.closedBall.{u2} X (EMetricSpace.toPseudoEMetricSpace.{u2} X _inst_1) x (ENNReal.ofReal (FunLike.coe.{succ u2, succ u2, 1} (ContinuousMap.{u2, 0} X Real (UniformSpace.toTopologicalSpace.{u2} X (PseudoEMetricSpace.toUniformSpace.{u2} X (EMetricSpace.toPseudoEMetricSpace.{u2} X _inst_1))) (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) X (fun (_x : X) => (fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : X) => Real) _x) (ContinuousMapClass.toFunLike.{u2, u2, 0} (ContinuousMap.{u2, 0} X Real (UniformSpace.toTopologicalSpace.{u2} X (PseudoEMetricSpace.toUniformSpace.{u2} X (EMetricSpace.toPseudoEMetricSpace.{u2} X _inst_1))) (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) X Real (UniformSpace.toTopologicalSpace.{u2} X (PseudoEMetricSpace.toUniformSpace.{u2} X (EMetricSpace.toPseudoEMetricSpace.{u2} X _inst_1))) (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) (ContinuousMap.instContinuousMapClassContinuousMap.{u2, 0} X Real (UniformSpace.toTopologicalSpace.{u2} X (PseudoEMetricSpace.toUniformSpace.{u2} X (EMetricSpace.toPseudoEMetricSpace.{u2} X _inst_1))) (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)))) δ x))) (U i)))))
+<too large>
 Case conversion may be inaccurate. Consider using '#align emetric.exists_continuous_real_forall_closed_ball_subset EMetric.exists_continuous_real_forall_closedBall_subsetₓ'. -/
 /-- Let `X` be an extended metric space. Let `K : ι → set X` be a locally finite family of closed
 sets, let `U : ι → set X` be a family of open sets such that `K i ⊆ U i` for all `i`. Then there
@@ -133,10 +127,7 @@ theorem exists_continuous_real_forall_closedBall_subset (hK : ∀ i, IsClosed (K
 #align emetric.exists_continuous_real_forall_closed_ball_subset EMetric.exists_continuous_real_forall_closedBall_subset
 
 /- warning: emetric.exists_continuous_nnreal_forall_closed_ball_subset -> EMetric.exists_continuous_nNReal_forall_closedBall_subset is a dubious translation:
-lean 3 declaration is
-  forall {ι : Type.{u1}} {X : Type.{u2}} [_inst_1 : EMetricSpace.{u2} X] {K : ι -> (Set.{u2} X)} {U : ι -> (Set.{u2} X)}, (forall (i : ι), IsClosed.{u2} X (UniformSpace.toTopologicalSpace.{u2} X (PseudoEMetricSpace.toUniformSpace.{u2} X (EMetricSpace.toPseudoEmetricSpace.{u2} X _inst_1))) (K i)) -> (forall (i : ι), IsOpen.{u2} X (UniformSpace.toTopologicalSpace.{u2} X (PseudoEMetricSpace.toUniformSpace.{u2} X (EMetricSpace.toPseudoEmetricSpace.{u2} X _inst_1))) (U i)) -> (forall (i : ι), HasSubset.Subset.{u2} (Set.{u2} X) (Set.hasSubset.{u2} X) (K i) (U i)) -> (LocallyFinite.{u1, u2} ι X (UniformSpace.toTopologicalSpace.{u2} X (PseudoEMetricSpace.toUniformSpace.{u2} X (EMetricSpace.toPseudoEmetricSpace.{u2} X _inst_1))) K) -> (Exists.{succ u2} (ContinuousMap.{u2, 0} X NNReal (UniformSpace.toTopologicalSpace.{u2} X (PseudoEMetricSpace.toUniformSpace.{u2} X (EMetricSpace.toPseudoEmetricSpace.{u2} X _inst_1))) NNReal.topologicalSpace) (fun (δ : ContinuousMap.{u2, 0} X NNReal (UniformSpace.toTopologicalSpace.{u2} X (PseudoEMetricSpace.toUniformSpace.{u2} X (EMetricSpace.toPseudoEmetricSpace.{u2} X _inst_1))) NNReal.topologicalSpace) => And (forall (x : X), LT.lt.{0} NNReal (Preorder.toHasLt.{0} NNReal (PartialOrder.toPreorder.{0} NNReal (OrderedCancelAddCommMonoid.toPartialOrder.{0} NNReal (StrictOrderedSemiring.toOrderedCancelAddCommMonoid.{0} NNReal NNReal.strictOrderedSemiring)))) (OfNat.ofNat.{0} NNReal 0 (OfNat.mk.{0} NNReal 0 (Zero.zero.{0} NNReal (MulZeroClass.toHasZero.{0} NNReal (NonUnitalNonAssocSemiring.toMulZeroClass.{0} NNReal (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} NNReal (Semiring.toNonAssocSemiring.{0} NNReal NNReal.semiring))))))) (coeFn.{succ u2, succ u2} (ContinuousMap.{u2, 0} X NNReal (UniformSpace.toTopologicalSpace.{u2} X (PseudoEMetricSpace.toUniformSpace.{u2} X (EMetricSpace.toPseudoEmetricSpace.{u2} X _inst_1))) NNReal.topologicalSpace) (fun (_x : ContinuousMap.{u2, 0} X NNReal (UniformSpace.toTopologicalSpace.{u2} X (PseudoEMetricSpace.toUniformSpace.{u2} X (EMetricSpace.toPseudoEmetricSpace.{u2} X _inst_1))) NNReal.topologicalSpace) => X -> NNReal) (ContinuousMap.hasCoeToFun.{u2, 0} X NNReal (UniformSpace.toTopologicalSpace.{u2} X (PseudoEMetricSpace.toUniformSpace.{u2} X (EMetricSpace.toPseudoEmetricSpace.{u2} X _inst_1))) NNReal.topologicalSpace) δ x)) (forall (i : ι) (x : X), (Membership.Mem.{u2, u2} X (Set.{u2} X) (Set.hasMem.{u2} X) x (K i)) -> (HasSubset.Subset.{u2} (Set.{u2} X) (Set.hasSubset.{u2} X) (EMetric.closedBall.{u2} X (EMetricSpace.toPseudoEmetricSpace.{u2} X _inst_1) x ((fun (a : Type) (b : Type) [self : HasLiftT.{1, 1} a b] => self.0) NNReal ENNReal (HasLiftT.mk.{1, 1} NNReal ENNReal (CoeTCₓ.coe.{1, 1} NNReal ENNReal (coeBase.{1, 1} NNReal ENNReal ENNReal.hasCoe))) (coeFn.{succ u2, succ u2} (ContinuousMap.{u2, 0} X NNReal (UniformSpace.toTopologicalSpace.{u2} X (PseudoEMetricSpace.toUniformSpace.{u2} X (EMetricSpace.toPseudoEmetricSpace.{u2} X _inst_1))) NNReal.topologicalSpace) (fun (_x : ContinuousMap.{u2, 0} X NNReal (UniformSpace.toTopologicalSpace.{u2} X (PseudoEMetricSpace.toUniformSpace.{u2} X (EMetricSpace.toPseudoEmetricSpace.{u2} X _inst_1))) NNReal.topologicalSpace) => X -> NNReal) (ContinuousMap.hasCoeToFun.{u2, 0} X NNReal (UniformSpace.toTopologicalSpace.{u2} X (PseudoEMetricSpace.toUniformSpace.{u2} X (EMetricSpace.toPseudoEmetricSpace.{u2} X _inst_1))) NNReal.topologicalSpace) δ x))) (U i)))))
-but is expected to have type
-  forall {ι : Type.{u1}} {X : Type.{u2}} [_inst_1 : EMetricSpace.{u2} X] {K : ι -> (Set.{u2} X)} {U : ι -> (Set.{u2} X)}, (forall (i : ι), IsClosed.{u2} X (UniformSpace.toTopologicalSpace.{u2} X (PseudoEMetricSpace.toUniformSpace.{u2} X (EMetricSpace.toPseudoEMetricSpace.{u2} X _inst_1))) (K i)) -> (forall (i : ι), IsOpen.{u2} X (UniformSpace.toTopologicalSpace.{u2} X (PseudoEMetricSpace.toUniformSpace.{u2} X (EMetricSpace.toPseudoEMetricSpace.{u2} X _inst_1))) (U i)) -> (forall (i : ι), HasSubset.Subset.{u2} (Set.{u2} X) (Set.instHasSubsetSet.{u2} X) (K i) (U i)) -> (LocallyFinite.{u1, u2} ι X (UniformSpace.toTopologicalSpace.{u2} X (PseudoEMetricSpace.toUniformSpace.{u2} X (EMetricSpace.toPseudoEMetricSpace.{u2} X _inst_1))) K) -> (Exists.{succ u2} (ContinuousMap.{u2, 0} X NNReal (UniformSpace.toTopologicalSpace.{u2} X (PseudoEMetricSpace.toUniformSpace.{u2} X (EMetricSpace.toPseudoEMetricSpace.{u2} X _inst_1))) NNReal.instTopologicalSpaceNNReal) (fun (δ : ContinuousMap.{u2, 0} X NNReal (UniformSpace.toTopologicalSpace.{u2} X (PseudoEMetricSpace.toUniformSpace.{u2} X (EMetricSpace.toPseudoEMetricSpace.{u2} X _inst_1))) NNReal.instTopologicalSpaceNNReal) => And (forall (x : X), LT.lt.{0} ((fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : X) => NNReal) x) (Preorder.toLT.{0} ((fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : X) => NNReal) x) (PartialOrder.toPreorder.{0} ((fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : X) => NNReal) x) (StrictOrderedSemiring.toPartialOrder.{0} ((fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : X) => NNReal) x) instNNRealStrictOrderedSemiring))) (OfNat.ofNat.{0} ((fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : X) => NNReal) x) 0 (Zero.toOfNat0.{0} ((fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : X) => NNReal) x) instNNRealZero)) (FunLike.coe.{succ u2, succ u2, 1} (ContinuousMap.{u2, 0} X NNReal (UniformSpace.toTopologicalSpace.{u2} X (PseudoEMetricSpace.toUniformSpace.{u2} X (EMetricSpace.toPseudoEMetricSpace.{u2} X _inst_1))) NNReal.instTopologicalSpaceNNReal) X (fun (_x : X) => (fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : X) => NNReal) _x) (ContinuousMapClass.toFunLike.{u2, u2, 0} (ContinuousMap.{u2, 0} X NNReal (UniformSpace.toTopologicalSpace.{u2} X (PseudoEMetricSpace.toUniformSpace.{u2} X (EMetricSpace.toPseudoEMetricSpace.{u2} X _inst_1))) NNReal.instTopologicalSpaceNNReal) X NNReal (UniformSpace.toTopologicalSpace.{u2} X (PseudoEMetricSpace.toUniformSpace.{u2} X (EMetricSpace.toPseudoEMetricSpace.{u2} X _inst_1))) NNReal.instTopologicalSpaceNNReal (ContinuousMap.instContinuousMapClassContinuousMap.{u2, 0} X NNReal (UniformSpace.toTopologicalSpace.{u2} X (PseudoEMetricSpace.toUniformSpace.{u2} X (EMetricSpace.toPseudoEMetricSpace.{u2} X _inst_1))) NNReal.instTopologicalSpaceNNReal)) δ x)) (forall (i : ι) (x : X), (Membership.mem.{u2, u2} X (Set.{u2} X) (Set.instMembershipSet.{u2} X) x (K i)) -> (HasSubset.Subset.{u2} (Set.{u2} X) (Set.instHasSubsetSet.{u2} X) (EMetric.closedBall.{u2} X (EMetricSpace.toPseudoEMetricSpace.{u2} X _inst_1) x (ENNReal.some (FunLike.coe.{succ u2, succ u2, 1} (ContinuousMap.{u2, 0} X NNReal (UniformSpace.toTopologicalSpace.{u2} X (PseudoEMetricSpace.toUniformSpace.{u2} X (EMetricSpace.toPseudoEMetricSpace.{u2} X _inst_1))) NNReal.instTopologicalSpaceNNReal) X (fun (_x : X) => (fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : X) => NNReal) _x) (ContinuousMapClass.toFunLike.{u2, u2, 0} (ContinuousMap.{u2, 0} X NNReal (UniformSpace.toTopologicalSpace.{u2} X (PseudoEMetricSpace.toUniformSpace.{u2} X (EMetricSpace.toPseudoEMetricSpace.{u2} X _inst_1))) NNReal.instTopologicalSpaceNNReal) X NNReal (UniformSpace.toTopologicalSpace.{u2} X (PseudoEMetricSpace.toUniformSpace.{u2} X (EMetricSpace.toPseudoEMetricSpace.{u2} X _inst_1))) NNReal.instTopologicalSpaceNNReal (ContinuousMap.instContinuousMapClassContinuousMap.{u2, 0} X NNReal (UniformSpace.toTopologicalSpace.{u2} X (PseudoEMetricSpace.toUniformSpace.{u2} X (EMetricSpace.toPseudoEMetricSpace.{u2} X _inst_1))) NNReal.instTopologicalSpaceNNReal)) δ x))) (U i)))))
+<too large>
 Case conversion may be inaccurate. Consider using '#align emetric.exists_continuous_nnreal_forall_closed_ball_subset EMetric.exists_continuous_nNReal_forall_closedBall_subsetₓ'. -/
 /-- Let `X` be an extended metric space. Let `K : ι → set X` be a locally finite family of closed
 sets, let `U : ι → set X` be a family of open sets such that `K i ⊆ U i` for all `i`. Then there
@@ -153,10 +144,7 @@ theorem exists_continuous_nNReal_forall_closedBall_subset (hK : ∀ i, IsClosed
 #align emetric.exists_continuous_nnreal_forall_closed_ball_subset EMetric.exists_continuous_nNReal_forall_closedBall_subset
 
 /- warning: emetric.exists_continuous_ennreal_forall_closed_ball_subset -> EMetric.exists_continuous_eNNReal_forall_closedBall_subset is a dubious translation:
-lean 3 declaration is
-  forall {ι : Type.{u1}} {X : Type.{u2}} [_inst_1 : EMetricSpace.{u2} X] {K : ι -> (Set.{u2} X)} {U : ι -> (Set.{u2} X)}, (forall (i : ι), IsClosed.{u2} X (UniformSpace.toTopologicalSpace.{u2} X (PseudoEMetricSpace.toUniformSpace.{u2} X (EMetricSpace.toPseudoEmetricSpace.{u2} X _inst_1))) (K i)) -> (forall (i : ι), IsOpen.{u2} X (UniformSpace.toTopologicalSpace.{u2} X (PseudoEMetricSpace.toUniformSpace.{u2} X (EMetricSpace.toPseudoEmetricSpace.{u2} X _inst_1))) (U i)) -> (forall (i : ι), HasSubset.Subset.{u2} (Set.{u2} X) (Set.hasSubset.{u2} X) (K i) (U i)) -> (LocallyFinite.{u1, u2} ι X (UniformSpace.toTopologicalSpace.{u2} X (PseudoEMetricSpace.toUniformSpace.{u2} X (EMetricSpace.toPseudoEmetricSpace.{u2} X _inst_1))) K) -> (Exists.{succ u2} (ContinuousMap.{u2, 0} X ENNReal (UniformSpace.toTopologicalSpace.{u2} X (PseudoEMetricSpace.toUniformSpace.{u2} X (EMetricSpace.toPseudoEmetricSpace.{u2} X _inst_1))) ENNReal.topologicalSpace) (fun (δ : ContinuousMap.{u2, 0} X ENNReal (UniformSpace.toTopologicalSpace.{u2} X (PseudoEMetricSpace.toUniformSpace.{u2} X (EMetricSpace.toPseudoEmetricSpace.{u2} X _inst_1))) ENNReal.topologicalSpace) => And (forall (x : X), LT.lt.{0} ENNReal (Preorder.toHasLt.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (CompleteSemilatticeInf.toPartialOrder.{0} ENNReal (CompleteLattice.toCompleteSemilatticeInf.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))))) (OfNat.ofNat.{0} ENNReal 0 (OfNat.mk.{0} ENNReal 0 (Zero.zero.{0} ENNReal ENNReal.hasZero))) (coeFn.{succ u2, succ u2} (ContinuousMap.{u2, 0} X ENNReal (UniformSpace.toTopologicalSpace.{u2} X (PseudoEMetricSpace.toUniformSpace.{u2} X (EMetricSpace.toPseudoEmetricSpace.{u2} X _inst_1))) ENNReal.topologicalSpace) (fun (_x : ContinuousMap.{u2, 0} X ENNReal (UniformSpace.toTopologicalSpace.{u2} X (PseudoEMetricSpace.toUniformSpace.{u2} X (EMetricSpace.toPseudoEmetricSpace.{u2} X _inst_1))) ENNReal.topologicalSpace) => X -> ENNReal) (ContinuousMap.hasCoeToFun.{u2, 0} X ENNReal (UniformSpace.toTopologicalSpace.{u2} X (PseudoEMetricSpace.toUniformSpace.{u2} X (EMetricSpace.toPseudoEmetricSpace.{u2} X _inst_1))) ENNReal.topologicalSpace) δ x)) (forall (i : ι) (x : X), (Membership.Mem.{u2, u2} X (Set.{u2} X) (Set.hasMem.{u2} X) x (K i)) -> (HasSubset.Subset.{u2} (Set.{u2} X) (Set.hasSubset.{u2} X) (EMetric.closedBall.{u2} X (EMetricSpace.toPseudoEmetricSpace.{u2} X _inst_1) x (coeFn.{succ u2, succ u2} (ContinuousMap.{u2, 0} X ENNReal (UniformSpace.toTopologicalSpace.{u2} X (PseudoEMetricSpace.toUniformSpace.{u2} X (EMetricSpace.toPseudoEmetricSpace.{u2} X _inst_1))) ENNReal.topologicalSpace) (fun (_x : ContinuousMap.{u2, 0} X ENNReal (UniformSpace.toTopologicalSpace.{u2} X (PseudoEMetricSpace.toUniformSpace.{u2} X (EMetricSpace.toPseudoEmetricSpace.{u2} X _inst_1))) ENNReal.topologicalSpace) => X -> ENNReal) (ContinuousMap.hasCoeToFun.{u2, 0} X ENNReal (UniformSpace.toTopologicalSpace.{u2} X (PseudoEMetricSpace.toUniformSpace.{u2} X (EMetricSpace.toPseudoEmetricSpace.{u2} X _inst_1))) ENNReal.topologicalSpace) δ x)) (U i)))))
-but is expected to have type
-  forall {ι : Type.{u1}} {X : Type.{u2}} [_inst_1 : EMetricSpace.{u2} X] {K : ι -> (Set.{u2} X)} {U : ι -> (Set.{u2} X)}, (forall (i : ι), IsClosed.{u2} X (UniformSpace.toTopologicalSpace.{u2} X (PseudoEMetricSpace.toUniformSpace.{u2} X (EMetricSpace.toPseudoEMetricSpace.{u2} X _inst_1))) (K i)) -> (forall (i : ι), IsOpen.{u2} X (UniformSpace.toTopologicalSpace.{u2} X (PseudoEMetricSpace.toUniformSpace.{u2} X (EMetricSpace.toPseudoEMetricSpace.{u2} X _inst_1))) (U i)) -> (forall (i : ι), HasSubset.Subset.{u2} (Set.{u2} X) (Set.instHasSubsetSet.{u2} X) (K i) (U i)) -> (LocallyFinite.{u1, u2} ι X (UniformSpace.toTopologicalSpace.{u2} X (PseudoEMetricSpace.toUniformSpace.{u2} X (EMetricSpace.toPseudoEMetricSpace.{u2} X _inst_1))) K) -> (Exists.{succ u2} (ContinuousMap.{u2, 0} X ENNReal (UniformSpace.toTopologicalSpace.{u2} X (PseudoEMetricSpace.toUniformSpace.{u2} X (EMetricSpace.toPseudoEMetricSpace.{u2} X _inst_1))) ENNReal.instTopologicalSpaceENNReal) (fun (δ : ContinuousMap.{u2, 0} X ENNReal (UniformSpace.toTopologicalSpace.{u2} X (PseudoEMetricSpace.toUniformSpace.{u2} X (EMetricSpace.toPseudoEMetricSpace.{u2} X _inst_1))) ENNReal.instTopologicalSpaceENNReal) => And (forall (x : X), LT.lt.{0} ((fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : X) => ENNReal) x) (Preorder.toLT.{0} ((fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : X) => ENNReal) x) (PartialOrder.toPreorder.{0} ((fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : X) => ENNReal) x) (OmegaCompletePartialOrder.toPartialOrder.{0} ((fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : X) => ENNReal) x) (CompleteLattice.instOmegaCompletePartialOrder.{0} ((fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : X) => ENNReal) x) (CompleteLinearOrder.toCompleteLattice.{0} ((fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : X) => ENNReal) x) ENNReal.instCompleteLinearOrderENNReal))))) (OfNat.ofNat.{0} ((fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : X) => ENNReal) x) 0 (Zero.toOfNat0.{0} ((fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : X) => ENNReal) x) instENNRealZero)) (FunLike.coe.{succ u2, succ u2, 1} (ContinuousMap.{u2, 0} X ENNReal (UniformSpace.toTopologicalSpace.{u2} X (PseudoEMetricSpace.toUniformSpace.{u2} X (EMetricSpace.toPseudoEMetricSpace.{u2} X _inst_1))) ENNReal.instTopologicalSpaceENNReal) X (fun (_x : X) => (fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : X) => ENNReal) _x) (ContinuousMapClass.toFunLike.{u2, u2, 0} (ContinuousMap.{u2, 0} X ENNReal (UniformSpace.toTopologicalSpace.{u2} X (PseudoEMetricSpace.toUniformSpace.{u2} X (EMetricSpace.toPseudoEMetricSpace.{u2} X _inst_1))) ENNReal.instTopologicalSpaceENNReal) X ENNReal (UniformSpace.toTopologicalSpace.{u2} X (PseudoEMetricSpace.toUniformSpace.{u2} X (EMetricSpace.toPseudoEMetricSpace.{u2} X _inst_1))) ENNReal.instTopologicalSpaceENNReal (ContinuousMap.instContinuousMapClassContinuousMap.{u2, 0} X ENNReal (UniformSpace.toTopologicalSpace.{u2} X (PseudoEMetricSpace.toUniformSpace.{u2} X (EMetricSpace.toPseudoEMetricSpace.{u2} X _inst_1))) ENNReal.instTopologicalSpaceENNReal)) δ x)) (forall (i : ι) (x : X), (Membership.mem.{u2, u2} X (Set.{u2} X) (Set.instMembershipSet.{u2} X) x (K i)) -> (HasSubset.Subset.{u2} (Set.{u2} X) (Set.instHasSubsetSet.{u2} X) (EMetric.closedBall.{u2} X (EMetricSpace.toPseudoEMetricSpace.{u2} X _inst_1) x (FunLike.coe.{succ u2, succ u2, 1} (ContinuousMap.{u2, 0} X ENNReal (UniformSpace.toTopologicalSpace.{u2} X (PseudoEMetricSpace.toUniformSpace.{u2} X (EMetricSpace.toPseudoEMetricSpace.{u2} X _inst_1))) ENNReal.instTopologicalSpaceENNReal) X (fun (_x : X) => (fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : X) => ENNReal) _x) (ContinuousMapClass.toFunLike.{u2, u2, 0} (ContinuousMap.{u2, 0} X ENNReal (UniformSpace.toTopologicalSpace.{u2} X (PseudoEMetricSpace.toUniformSpace.{u2} X (EMetricSpace.toPseudoEMetricSpace.{u2} X _inst_1))) ENNReal.instTopologicalSpaceENNReal) X ENNReal (UniformSpace.toTopologicalSpace.{u2} X (PseudoEMetricSpace.toUniformSpace.{u2} X (EMetricSpace.toPseudoEMetricSpace.{u2} X _inst_1))) ENNReal.instTopologicalSpaceENNReal (ContinuousMap.instContinuousMapClassContinuousMap.{u2, 0} X ENNReal (UniformSpace.toTopologicalSpace.{u2} X (PseudoEMetricSpace.toUniformSpace.{u2} X (EMetricSpace.toPseudoEMetricSpace.{u2} X _inst_1))) ENNReal.instTopologicalSpaceENNReal)) δ x)) (U i)))))
+<too large>
 Case conversion may be inaccurate. Consider using '#align emetric.exists_continuous_ennreal_forall_closed_ball_subset EMetric.exists_continuous_eNNReal_forall_closedBall_subsetₓ'. -/
 /-- Let `X` be an extended metric space. Let `K : ι → set X` be a locally finite family of closed
 sets, let `U : ι → set X` be a family of open sets such that `K i ⊆ U i` for all `i`. Then there
@@ -176,10 +164,7 @@ namespace Metric
 variable [MetricSpace X] {K : ι → Set X} {U : ι → Set X}
 
 /- warning: metric.exists_continuous_nnreal_forall_closed_ball_subset -> Metric.exists_continuous_nNReal_forall_closedBall_subset is a dubious translation:
-lean 3 declaration is
-  forall {ι : Type.{u1}} {X : Type.{u2}} [_inst_1 : MetricSpace.{u2} X] {K : ι -> (Set.{u2} X)} {U : ι -> (Set.{u2} X)}, (forall (i : ι), IsClosed.{u2} X (UniformSpace.toTopologicalSpace.{u2} X (PseudoMetricSpace.toUniformSpace.{u2} X (MetricSpace.toPseudoMetricSpace.{u2} X _inst_1))) (K i)) -> (forall (i : ι), IsOpen.{u2} X (UniformSpace.toTopologicalSpace.{u2} X (PseudoMetricSpace.toUniformSpace.{u2} X (MetricSpace.toPseudoMetricSpace.{u2} X _inst_1))) (U i)) -> (forall (i : ι), HasSubset.Subset.{u2} (Set.{u2} X) (Set.hasSubset.{u2} X) (K i) (U i)) -> (LocallyFinite.{u1, u2} ι X (UniformSpace.toTopologicalSpace.{u2} X (PseudoMetricSpace.toUniformSpace.{u2} X (MetricSpace.toPseudoMetricSpace.{u2} X _inst_1))) K) -> (Exists.{succ u2} (ContinuousMap.{u2, 0} X NNReal (UniformSpace.toTopologicalSpace.{u2} X (PseudoMetricSpace.toUniformSpace.{u2} X (MetricSpace.toPseudoMetricSpace.{u2} X _inst_1))) NNReal.topologicalSpace) (fun (δ : ContinuousMap.{u2, 0} X NNReal (UniformSpace.toTopologicalSpace.{u2} X (PseudoMetricSpace.toUniformSpace.{u2} X (MetricSpace.toPseudoMetricSpace.{u2} X _inst_1))) NNReal.topologicalSpace) => And (forall (x : X), LT.lt.{0} NNReal (Preorder.toHasLt.{0} NNReal (PartialOrder.toPreorder.{0} NNReal (OrderedCancelAddCommMonoid.toPartialOrder.{0} NNReal (StrictOrderedSemiring.toOrderedCancelAddCommMonoid.{0} NNReal NNReal.strictOrderedSemiring)))) (OfNat.ofNat.{0} NNReal 0 (OfNat.mk.{0} NNReal 0 (Zero.zero.{0} NNReal (MulZeroClass.toHasZero.{0} NNReal (NonUnitalNonAssocSemiring.toMulZeroClass.{0} NNReal (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} NNReal (Semiring.toNonAssocSemiring.{0} NNReal NNReal.semiring))))))) (coeFn.{succ u2, succ u2} (ContinuousMap.{u2, 0} X NNReal (UniformSpace.toTopologicalSpace.{u2} X (PseudoMetricSpace.toUniformSpace.{u2} X (MetricSpace.toPseudoMetricSpace.{u2} X _inst_1))) NNReal.topologicalSpace) (fun (_x : ContinuousMap.{u2, 0} X NNReal (UniformSpace.toTopologicalSpace.{u2} X (PseudoMetricSpace.toUniformSpace.{u2} X (MetricSpace.toPseudoMetricSpace.{u2} X _inst_1))) NNReal.topologicalSpace) => X -> NNReal) (ContinuousMap.hasCoeToFun.{u2, 0} X NNReal (UniformSpace.toTopologicalSpace.{u2} X (PseudoMetricSpace.toUniformSpace.{u2} X (MetricSpace.toPseudoMetricSpace.{u2} X _inst_1))) NNReal.topologicalSpace) δ x)) (forall (i : ι) (x : X), (Membership.Mem.{u2, u2} X (Set.{u2} X) (Set.hasMem.{u2} X) x (K i)) -> (HasSubset.Subset.{u2} (Set.{u2} X) (Set.hasSubset.{u2} X) (Metric.closedBall.{u2} X (MetricSpace.toPseudoMetricSpace.{u2} X _inst_1) x ((fun (a : Type) (b : Type) [self : HasLiftT.{1, 1} a b] => self.0) NNReal Real (HasLiftT.mk.{1, 1} NNReal Real (CoeTCₓ.coe.{1, 1} NNReal Real (coeBase.{1, 1} NNReal Real NNReal.Real.hasCoe))) (coeFn.{succ u2, succ u2} (ContinuousMap.{u2, 0} X NNReal (UniformSpace.toTopologicalSpace.{u2} X (PseudoMetricSpace.toUniformSpace.{u2} X (MetricSpace.toPseudoMetricSpace.{u2} X _inst_1))) NNReal.topologicalSpace) (fun (_x : ContinuousMap.{u2, 0} X NNReal (UniformSpace.toTopologicalSpace.{u2} X (PseudoMetricSpace.toUniformSpace.{u2} X (MetricSpace.toPseudoMetricSpace.{u2} X _inst_1))) NNReal.topologicalSpace) => X -> NNReal) (ContinuousMap.hasCoeToFun.{u2, 0} X NNReal (UniformSpace.toTopologicalSpace.{u2} X (PseudoMetricSpace.toUniformSpace.{u2} X (MetricSpace.toPseudoMetricSpace.{u2} X _inst_1))) NNReal.topologicalSpace) δ x))) (U i)))))
-but is expected to have type
-  forall {ι : Type.{u1}} {X : Type.{u2}} [_inst_1 : MetricSpace.{u2} X] {K : ι -> (Set.{u2} X)} {U : ι -> (Set.{u2} X)}, (forall (i : ι), IsClosed.{u2} X (UniformSpace.toTopologicalSpace.{u2} X (PseudoMetricSpace.toUniformSpace.{u2} X (MetricSpace.toPseudoMetricSpace.{u2} X _inst_1))) (K i)) -> (forall (i : ι), IsOpen.{u2} X (UniformSpace.toTopologicalSpace.{u2} X (PseudoMetricSpace.toUniformSpace.{u2} X (MetricSpace.toPseudoMetricSpace.{u2} X _inst_1))) (U i)) -> (forall (i : ι), HasSubset.Subset.{u2} (Set.{u2} X) (Set.instHasSubsetSet.{u2} X) (K i) (U i)) -> (LocallyFinite.{u1, u2} ι X (UniformSpace.toTopologicalSpace.{u2} X (PseudoMetricSpace.toUniformSpace.{u2} X (MetricSpace.toPseudoMetricSpace.{u2} X _inst_1))) K) -> (Exists.{succ u2} (ContinuousMap.{u2, 0} X NNReal (UniformSpace.toTopologicalSpace.{u2} X (PseudoMetricSpace.toUniformSpace.{u2} X (MetricSpace.toPseudoMetricSpace.{u2} X _inst_1))) NNReal.instTopologicalSpaceNNReal) (fun (δ : ContinuousMap.{u2, 0} X NNReal (UniformSpace.toTopologicalSpace.{u2} X (PseudoMetricSpace.toUniformSpace.{u2} X (MetricSpace.toPseudoMetricSpace.{u2} X _inst_1))) NNReal.instTopologicalSpaceNNReal) => And (forall (x : X), LT.lt.{0} ((fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : X) => NNReal) x) (Preorder.toLT.{0} ((fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : X) => NNReal) x) (PartialOrder.toPreorder.{0} ((fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : X) => NNReal) x) (StrictOrderedSemiring.toPartialOrder.{0} ((fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : X) => NNReal) x) instNNRealStrictOrderedSemiring))) (OfNat.ofNat.{0} ((fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : X) => NNReal) x) 0 (Zero.toOfNat0.{0} ((fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : X) => NNReal) x) instNNRealZero)) (FunLike.coe.{succ u2, succ u2, 1} (ContinuousMap.{u2, 0} X NNReal (UniformSpace.toTopologicalSpace.{u2} X (PseudoMetricSpace.toUniformSpace.{u2} X (MetricSpace.toPseudoMetricSpace.{u2} X _inst_1))) NNReal.instTopologicalSpaceNNReal) X (fun (_x : X) => (fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : X) => NNReal) _x) (ContinuousMapClass.toFunLike.{u2, u2, 0} (ContinuousMap.{u2, 0} X NNReal (UniformSpace.toTopologicalSpace.{u2} X (PseudoMetricSpace.toUniformSpace.{u2} X (MetricSpace.toPseudoMetricSpace.{u2} X _inst_1))) NNReal.instTopologicalSpaceNNReal) X NNReal (UniformSpace.toTopologicalSpace.{u2} X (PseudoMetricSpace.toUniformSpace.{u2} X (MetricSpace.toPseudoMetricSpace.{u2} X _inst_1))) NNReal.instTopologicalSpaceNNReal (ContinuousMap.instContinuousMapClassContinuousMap.{u2, 0} X NNReal (UniformSpace.toTopologicalSpace.{u2} X (PseudoMetricSpace.toUniformSpace.{u2} X (MetricSpace.toPseudoMetricSpace.{u2} X _inst_1))) NNReal.instTopologicalSpaceNNReal)) δ x)) (forall (i : ι) (x : X), (Membership.mem.{u2, u2} X (Set.{u2} X) (Set.instMembershipSet.{u2} X) x (K i)) -> (HasSubset.Subset.{u2} (Set.{u2} X) (Set.instHasSubsetSet.{u2} X) (Metric.closedBall.{u2} X (MetricSpace.toPseudoMetricSpace.{u2} X _inst_1) x (NNReal.toReal (FunLike.coe.{succ u2, succ u2, 1} (ContinuousMap.{u2, 0} X NNReal (UniformSpace.toTopologicalSpace.{u2} X (PseudoMetricSpace.toUniformSpace.{u2} X (MetricSpace.toPseudoMetricSpace.{u2} X _inst_1))) NNReal.instTopologicalSpaceNNReal) X (fun (_x : X) => (fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : X) => NNReal) _x) (ContinuousMapClass.toFunLike.{u2, u2, 0} (ContinuousMap.{u2, 0} X NNReal (UniformSpace.toTopologicalSpace.{u2} X (PseudoMetricSpace.toUniformSpace.{u2} X (MetricSpace.toPseudoMetricSpace.{u2} X _inst_1))) NNReal.instTopologicalSpaceNNReal) X NNReal (UniformSpace.toTopologicalSpace.{u2} X (PseudoMetricSpace.toUniformSpace.{u2} X (MetricSpace.toPseudoMetricSpace.{u2} X _inst_1))) NNReal.instTopologicalSpaceNNReal (ContinuousMap.instContinuousMapClassContinuousMap.{u2, 0} X NNReal (UniformSpace.toTopologicalSpace.{u2} X (PseudoMetricSpace.toUniformSpace.{u2} X (MetricSpace.toPseudoMetricSpace.{u2} X _inst_1))) NNReal.instTopologicalSpaceNNReal)) δ x))) (U i)))))
+<too large>
 Case conversion may be inaccurate. Consider using '#align metric.exists_continuous_nnreal_forall_closed_ball_subset Metric.exists_continuous_nNReal_forall_closedBall_subsetₓ'. -/
 /-- Let `X` be a metric space. Let `K : ι → set X` be a locally finite family of closed sets, let
 `U : ι → set X` be a family of open sets such that `K i ⊆ U i` for all `i`. Then there exists a
@@ -196,10 +181,7 @@ theorem exists_continuous_nNReal_forall_closedBall_subset (hK : ∀ i, IsClosed
 #align metric.exists_continuous_nnreal_forall_closed_ball_subset Metric.exists_continuous_nNReal_forall_closedBall_subset
 
 /- warning: metric.exists_continuous_real_forall_closed_ball_subset -> Metric.exists_continuous_real_forall_closedBall_subset is a dubious translation:
-lean 3 declaration is
-  forall {ι : Type.{u1}} {X : Type.{u2}} [_inst_1 : MetricSpace.{u2} X] {K : ι -> (Set.{u2} X)} {U : ι -> (Set.{u2} X)}, (forall (i : ι), IsClosed.{u2} X (UniformSpace.toTopologicalSpace.{u2} X (PseudoMetricSpace.toUniformSpace.{u2} X (MetricSpace.toPseudoMetricSpace.{u2} X _inst_1))) (K i)) -> (forall (i : ι), IsOpen.{u2} X (UniformSpace.toTopologicalSpace.{u2} X (PseudoMetricSpace.toUniformSpace.{u2} X (MetricSpace.toPseudoMetricSpace.{u2} X _inst_1))) (U i)) -> (forall (i : ι), HasSubset.Subset.{u2} (Set.{u2} X) (Set.hasSubset.{u2} X) (K i) (U i)) -> (LocallyFinite.{u1, u2} ι X (UniformSpace.toTopologicalSpace.{u2} X (PseudoMetricSpace.toUniformSpace.{u2} X (MetricSpace.toPseudoMetricSpace.{u2} X _inst_1))) K) -> (Exists.{succ u2} (ContinuousMap.{u2, 0} X Real (UniformSpace.toTopologicalSpace.{u2} X (PseudoMetricSpace.toUniformSpace.{u2} X (MetricSpace.toPseudoMetricSpace.{u2} X _inst_1))) (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) (fun (δ : ContinuousMap.{u2, 0} X Real (UniformSpace.toTopologicalSpace.{u2} X (PseudoMetricSpace.toUniformSpace.{u2} X (MetricSpace.toPseudoMetricSpace.{u2} X _inst_1))) (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) => And (forall (x : X), LT.lt.{0} Real Real.hasLt (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero))) (coeFn.{succ u2, succ u2} (ContinuousMap.{u2, 0} X Real (UniformSpace.toTopologicalSpace.{u2} X (PseudoMetricSpace.toUniformSpace.{u2} X (MetricSpace.toPseudoMetricSpace.{u2} X _inst_1))) (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) (fun (_x : ContinuousMap.{u2, 0} X Real (UniformSpace.toTopologicalSpace.{u2} X (PseudoMetricSpace.toUniformSpace.{u2} X (MetricSpace.toPseudoMetricSpace.{u2} X _inst_1))) (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) => X -> Real) (ContinuousMap.hasCoeToFun.{u2, 0} X Real (UniformSpace.toTopologicalSpace.{u2} X (PseudoMetricSpace.toUniformSpace.{u2} X (MetricSpace.toPseudoMetricSpace.{u2} X _inst_1))) (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) δ x)) (forall (i : ι) (x : X), (Membership.Mem.{u2, u2} X (Set.{u2} X) (Set.hasMem.{u2} X) x (K i)) -> (HasSubset.Subset.{u2} (Set.{u2} X) (Set.hasSubset.{u2} X) (Metric.closedBall.{u2} X (MetricSpace.toPseudoMetricSpace.{u2} X _inst_1) x (coeFn.{succ u2, succ u2} (ContinuousMap.{u2, 0} X Real (UniformSpace.toTopologicalSpace.{u2} X (PseudoMetricSpace.toUniformSpace.{u2} X (MetricSpace.toPseudoMetricSpace.{u2} X _inst_1))) (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) (fun (_x : ContinuousMap.{u2, 0} X Real (UniformSpace.toTopologicalSpace.{u2} X (PseudoMetricSpace.toUniformSpace.{u2} X (MetricSpace.toPseudoMetricSpace.{u2} X _inst_1))) (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) => X -> Real) (ContinuousMap.hasCoeToFun.{u2, 0} X Real (UniformSpace.toTopologicalSpace.{u2} X (PseudoMetricSpace.toUniformSpace.{u2} X (MetricSpace.toPseudoMetricSpace.{u2} X _inst_1))) (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) δ x)) (U i)))))
-but is expected to have type
-  forall {ι : Type.{u1}} {X : Type.{u2}} [_inst_1 : MetricSpace.{u2} X] {K : ι -> (Set.{u2} X)} {U : ι -> (Set.{u2} X)}, (forall (i : ι), IsClosed.{u2} X (UniformSpace.toTopologicalSpace.{u2} X (PseudoMetricSpace.toUniformSpace.{u2} X (MetricSpace.toPseudoMetricSpace.{u2} X _inst_1))) (K i)) -> (forall (i : ι), IsOpen.{u2} X (UniformSpace.toTopologicalSpace.{u2} X (PseudoMetricSpace.toUniformSpace.{u2} X (MetricSpace.toPseudoMetricSpace.{u2} X _inst_1))) (U i)) -> (forall (i : ι), HasSubset.Subset.{u2} (Set.{u2} X) (Set.instHasSubsetSet.{u2} X) (K i) (U i)) -> (LocallyFinite.{u1, u2} ι X (UniformSpace.toTopologicalSpace.{u2} X (PseudoMetricSpace.toUniformSpace.{u2} X (MetricSpace.toPseudoMetricSpace.{u2} X _inst_1))) K) -> (Exists.{succ u2} (ContinuousMap.{u2, 0} X Real (UniformSpace.toTopologicalSpace.{u2} X (PseudoMetricSpace.toUniformSpace.{u2} X (MetricSpace.toPseudoMetricSpace.{u2} X _inst_1))) (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) (fun (δ : ContinuousMap.{u2, 0} X Real (UniformSpace.toTopologicalSpace.{u2} X (PseudoMetricSpace.toUniformSpace.{u2} X (MetricSpace.toPseudoMetricSpace.{u2} X _inst_1))) (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) => And (forall (x : X), LT.lt.{0} ((fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : X) => Real) x) Real.instLTReal (OfNat.ofNat.{0} ((fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : X) => Real) x) 0 (Zero.toOfNat0.{0} ((fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : X) => Real) x) Real.instZeroReal)) (FunLike.coe.{succ u2, succ u2, 1} (ContinuousMap.{u2, 0} X Real (UniformSpace.toTopologicalSpace.{u2} X (PseudoMetricSpace.toUniformSpace.{u2} X (MetricSpace.toPseudoMetricSpace.{u2} X _inst_1))) (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) X (fun (_x : X) => (fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : X) => Real) _x) (ContinuousMapClass.toFunLike.{u2, u2, 0} (ContinuousMap.{u2, 0} X Real (UniformSpace.toTopologicalSpace.{u2} X (PseudoMetricSpace.toUniformSpace.{u2} X (MetricSpace.toPseudoMetricSpace.{u2} X _inst_1))) (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) X Real (UniformSpace.toTopologicalSpace.{u2} X (PseudoMetricSpace.toUniformSpace.{u2} X (MetricSpace.toPseudoMetricSpace.{u2} X _inst_1))) (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) (ContinuousMap.instContinuousMapClassContinuousMap.{u2, 0} X Real (UniformSpace.toTopologicalSpace.{u2} X (PseudoMetricSpace.toUniformSpace.{u2} X (MetricSpace.toPseudoMetricSpace.{u2} X _inst_1))) (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)))) δ x)) (forall (i : ι) (x : X), (Membership.mem.{u2, u2} X (Set.{u2} X) (Set.instMembershipSet.{u2} X) x (K i)) -> (HasSubset.Subset.{u2} (Set.{u2} X) (Set.instHasSubsetSet.{u2} X) (Metric.closedBall.{u2} X (MetricSpace.toPseudoMetricSpace.{u2} X _inst_1) x (FunLike.coe.{succ u2, succ u2, 1} (ContinuousMap.{u2, 0} X Real (UniformSpace.toTopologicalSpace.{u2} X (PseudoMetricSpace.toUniformSpace.{u2} X (MetricSpace.toPseudoMetricSpace.{u2} X _inst_1))) (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) X (fun (_x : X) => (fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : X) => Real) _x) (ContinuousMapClass.toFunLike.{u2, u2, 0} (ContinuousMap.{u2, 0} X Real (UniformSpace.toTopologicalSpace.{u2} X (PseudoMetricSpace.toUniformSpace.{u2} X (MetricSpace.toPseudoMetricSpace.{u2} X _inst_1))) (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) X Real (UniformSpace.toTopologicalSpace.{u2} X (PseudoMetricSpace.toUniformSpace.{u2} X (MetricSpace.toPseudoMetricSpace.{u2} X _inst_1))) (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) (ContinuousMap.instContinuousMapClassContinuousMap.{u2, 0} X Real (UniformSpace.toTopologicalSpace.{u2} X (PseudoMetricSpace.toUniformSpace.{u2} X (MetricSpace.toPseudoMetricSpace.{u2} X _inst_1))) (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)))) δ x)) (U i)))))
+<too large>
 Case conversion may be inaccurate. Consider using '#align metric.exists_continuous_real_forall_closed_ball_subset Metric.exists_continuous_real_forall_closedBall_subsetₓ'. -/
 /-- Let `X` be a metric space. Let `K : ι → set X` be a locally finite family of closed sets, let
 `U : ι → set X` be a family of open sets such that `K i ⊆ U i` for all `i`. Then there exists a

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

@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Yury Kudryashov
 
 ! This file was ported from Lean 3 source module topology.metric_space.partition_of_unity
-! leanprover-community/mathlib commit f2ce6086713c78a7f880485f7917ea547a215982
+! leanprover-community/mathlib commit 50251fd6309cca5ca2e747882ffecd2729f38c5d
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
@@ -14,6 +14,9 @@ import Mathbin.Analysis.Convex.PartitionOfUnity
 /-!
 # Lemmas about (e)metric spaces that need partition of unity
 
+> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.
+> Any changes to this file require a corresponding PR to mathlib4.
+
 The main lemma in this file (see `metric.exists_continuous_real_forall_closed_ball_subset`) says the
 following. Let `X` be a metric space. Let `K : ι → set X` be a locally finite family of closed sets,
 let `U : ι → set X` be a family of open sets such that `K i ⊆ U i` for all `i`. Then there exists a

mathlib commit https://github.com/leanprover-community/mathlib/commit/f8c79b0a623404854a2902b836eac32156fd7712

@@ -40,6 +40,12 @@ namespace Emetric
 
 variable [EMetricSpace X] {K : ι → Set X} {U : ι → Set X}
 
+/- warning: emetric.eventually_nhds_zero_forall_closed_ball_subset -> EMetric.eventually_nhds_zero_forall_closedBall_subset is a dubious translation:
+lean 3 declaration is
+  forall {ι : Type.{u1}} {X : Type.{u2}} [_inst_1 : EMetricSpace.{u2} X] {K : ι -> (Set.{u2} X)} {U : ι -> (Set.{u2} X)}, (forall (i : ι), IsClosed.{u2} X (UniformSpace.toTopologicalSpace.{u2} X (PseudoEMetricSpace.toUniformSpace.{u2} X (EMetricSpace.toPseudoEmetricSpace.{u2} X _inst_1))) (K i)) -> (forall (i : ι), IsOpen.{u2} X (UniformSpace.toTopologicalSpace.{u2} X (PseudoEMetricSpace.toUniformSpace.{u2} X (EMetricSpace.toPseudoEmetricSpace.{u2} X _inst_1))) (U i)) -> (forall (i : ι), HasSubset.Subset.{u2} (Set.{u2} X) (Set.hasSubset.{u2} X) (K i) (U i)) -> (LocallyFinite.{u1, u2} ι X (UniformSpace.toTopologicalSpace.{u2} X (PseudoEMetricSpace.toUniformSpace.{u2} X (EMetricSpace.toPseudoEmetricSpace.{u2} X _inst_1))) K) -> (forall (x : X), Filter.Eventually.{u2} (Prod.{0, u2} ENNReal X) (fun (p : Prod.{0, u2} ENNReal X) => forall (i : ι), (Membership.Mem.{u2, u2} X (Set.{u2} X) (Set.hasMem.{u2} X) (Prod.snd.{0, u2} ENNReal X p) (K i)) -> (HasSubset.Subset.{u2} (Set.{u2} X) (Set.hasSubset.{u2} X) (EMetric.closedBall.{u2} X (EMetricSpace.toPseudoEmetricSpace.{u2} X _inst_1) (Prod.snd.{0, u2} ENNReal X p) (Prod.fst.{0, u2} ENNReal X p)) (U i))) (Filter.prod.{0, u2} ENNReal X (nhds.{0} ENNReal ENNReal.topologicalSpace (OfNat.ofNat.{0} ENNReal 0 (OfNat.mk.{0} ENNReal 0 (Zero.zero.{0} ENNReal ENNReal.hasZero)))) (nhds.{u2} X (UniformSpace.toTopologicalSpace.{u2} X (PseudoEMetricSpace.toUniformSpace.{u2} X (EMetricSpace.toPseudoEmetricSpace.{u2} X _inst_1))) x)))
+but is expected to have type
+  forall {ι : Type.{u1}} {X : Type.{u2}} [_inst_1 : EMetricSpace.{u2} X] {K : ι -> (Set.{u2} X)} {U : ι -> (Set.{u2} X)}, (forall (i : ι), IsClosed.{u2} X (UniformSpace.toTopologicalSpace.{u2} X (PseudoEMetricSpace.toUniformSpace.{u2} X (EMetricSpace.toPseudoEMetricSpace.{u2} X _inst_1))) (K i)) -> (forall (i : ι), IsOpen.{u2} X (UniformSpace.toTopologicalSpace.{u2} X (PseudoEMetricSpace.toUniformSpace.{u2} X (EMetricSpace.toPseudoEMetricSpace.{u2} X _inst_1))) (U i)) -> (forall (i : ι), HasSubset.Subset.{u2} (Set.{u2} X) (Set.instHasSubsetSet.{u2} X) (K i) (U i)) -> (LocallyFinite.{u1, u2} ι X (UniformSpace.toTopologicalSpace.{u2} X (PseudoEMetricSpace.toUniformSpace.{u2} X (EMetricSpace.toPseudoEMetricSpace.{u2} X _inst_1))) K) -> (forall (x : X), Filter.Eventually.{u2} (Prod.{0, u2} ENNReal X) (fun (p : Prod.{0, u2} ENNReal X) => forall (i : ι), (Membership.mem.{u2, u2} X (Set.{u2} X) (Set.instMembershipSet.{u2} X) (Prod.snd.{0, u2} ENNReal X p) (K i)) -> (HasSubset.Subset.{u2} (Set.{u2} X) (Set.instHasSubsetSet.{u2} X) (EMetric.closedBall.{u2} X (EMetricSpace.toPseudoEMetricSpace.{u2} X _inst_1) (Prod.snd.{0, u2} ENNReal X p) (Prod.fst.{0, u2} ENNReal X p)) (U i))) (Filter.prod.{0, u2} ENNReal X (nhds.{0} ENNReal ENNReal.instTopologicalSpaceENNReal (OfNat.ofNat.{0} ENNReal 0 (Zero.toOfNat0.{0} ENNReal instENNRealZero))) (nhds.{u2} X (UniformSpace.toTopologicalSpace.{u2} X (PseudoEMetricSpace.toUniformSpace.{u2} X (EMetricSpace.toPseudoEMetricSpace.{u2} X _inst_1))) x)))
+Case conversion may be inaccurate. Consider using '#align emetric.eventually_nhds_zero_forall_closed_ball_subset EMetric.eventually_nhds_zero_forall_closedBall_subsetₓ'. -/
 /-- Let `K : ι → set X` be a locally finitie family of closed sets in an emetric space. Let
 `U : ι → set X` be a family of open sets such that `K i ⊆ U i` for all `i`. Then for any point
 `x : X`, for sufficiently small `r : ℝ≥0∞` and for `y` sufficiently close to `x`, for all `i`, if
@@ -66,8 +72,14 @@ theorem eventually_nhds_zero_forall_closedBall_subset (hK : ∀ i, IsClosed (K i
     _ ≤ p.1 + (R - p.1) := (add_le_add hz <| le_trans hp.2 <| tsub_le_tsub_left hp.1.out.le _)
     _ = R := add_tsub_cancel_of_le (lt_trans hp.1 hrR).le
     
-#align emetric.eventually_nhds_zero_forall_closed_ball_subset Emetric.eventually_nhds_zero_forall_closedBall_subset
-
+#align emetric.eventually_nhds_zero_forall_closed_ball_subset EMetric.eventually_nhds_zero_forall_closedBall_subset
+
+/- warning: emetric.exists_forall_closed_ball_subset_aux₁ -> EMetric.exists_forall_closedBall_subset_aux₁ is a dubious translation:
+lean 3 declaration is
+  forall {ι : Type.{u1}} {X : Type.{u2}} [_inst_1 : EMetricSpace.{u2} X] {K : ι -> (Set.{u2} X)} {U : ι -> (Set.{u2} X)}, (forall (i : ι), IsClosed.{u2} X (UniformSpace.toTopologicalSpace.{u2} X (PseudoEMetricSpace.toUniformSpace.{u2} X (EMetricSpace.toPseudoEmetricSpace.{u2} X _inst_1))) (K i)) -> (forall (i : ι), IsOpen.{u2} X (UniformSpace.toTopologicalSpace.{u2} X (PseudoEMetricSpace.toUniformSpace.{u2} X (EMetricSpace.toPseudoEmetricSpace.{u2} X _inst_1))) (U i)) -> (forall (i : ι), HasSubset.Subset.{u2} (Set.{u2} X) (Set.hasSubset.{u2} X) (K i) (U i)) -> (LocallyFinite.{u1, u2} ι X (UniformSpace.toTopologicalSpace.{u2} X (PseudoEMetricSpace.toUniformSpace.{u2} X (EMetricSpace.toPseudoEmetricSpace.{u2} X _inst_1))) K) -> (forall (x : X), Exists.{1} Real (fun (r : Real) => Filter.Eventually.{u2} X (fun (y : X) => Membership.Mem.{0, 0} Real (Set.{0} Real) (Set.hasMem.{0} Real) r (Inter.inter.{0} (Set.{0} Real) (Set.hasInter.{0} Real) (Set.Ioi.{0} Real Real.preorder (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero)))) (Set.preimage.{0, 0} Real ENNReal ENNReal.ofReal (Set.iInter.{0, succ u1} ENNReal ι (fun (i : ι) => Set.iInter.{0, 0} ENNReal (Membership.Mem.{u2, u2} X (Set.{u2} X) (Set.hasMem.{u2} X) y (K i)) (fun (hi : Membership.Mem.{u2, u2} X (Set.{u2} X) (Set.hasMem.{u2} X) y (K i)) => setOf.{0} ENNReal (fun (r : ENNReal) => HasSubset.Subset.{u2} (Set.{u2} X) (Set.hasSubset.{u2} X) (EMetric.closedBall.{u2} X (EMetricSpace.toPseudoEmetricSpace.{u2} X _inst_1) y r) (U i)))))))) (nhds.{u2} X (UniformSpace.toTopologicalSpace.{u2} X (PseudoEMetricSpace.toUniformSpace.{u2} X (EMetricSpace.toPseudoEmetricSpace.{u2} X _inst_1))) x)))
+but is expected to have type
+  forall {ι : Type.{u1}} {X : Type.{u2}} [_inst_1 : EMetricSpace.{u2} X] {K : ι -> (Set.{u2} X)} {U : ι -> (Set.{u2} X)}, (forall (i : ι), IsClosed.{u2} X (UniformSpace.toTopologicalSpace.{u2} X (PseudoEMetricSpace.toUniformSpace.{u2} X (EMetricSpace.toPseudoEMetricSpace.{u2} X _inst_1))) (K i)) -> (forall (i : ι), IsOpen.{u2} X (UniformSpace.toTopologicalSpace.{u2} X (PseudoEMetricSpace.toUniformSpace.{u2} X (EMetricSpace.toPseudoEMetricSpace.{u2} X _inst_1))) (U i)) -> (forall (i : ι), HasSubset.Subset.{u2} (Set.{u2} X) (Set.instHasSubsetSet.{u2} X) (K i) (U i)) -> (LocallyFinite.{u1, u2} ι X (UniformSpace.toTopologicalSpace.{u2} X (PseudoEMetricSpace.toUniformSpace.{u2} X (EMetricSpace.toPseudoEMetricSpace.{u2} X _inst_1))) K) -> (forall (x : X), Exists.{1} Real (fun (r : Real) => Filter.Eventually.{u2} X (fun (y : X) => Membership.mem.{0, 0} Real (Set.{0} Real) (Set.instMembershipSet.{0} Real) r (Inter.inter.{0} (Set.{0} Real) (Set.instInterSet.{0} Real) (Set.Ioi.{0} Real Real.instPreorderReal (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal))) (Set.preimage.{0, 0} Real ENNReal ENNReal.ofReal (Set.iInter.{0, succ u1} ENNReal ι (fun (i : ι) => Set.iInter.{0, 0} ENNReal (Membership.mem.{u2, u2} X (Set.{u2} X) (Set.instMembershipSet.{u2} X) y (K i)) (fun (hi : Membership.mem.{u2, u2} X (Set.{u2} X) (Set.instMembershipSet.{u2} X) y (K i)) => setOf.{0} ENNReal (fun (r : ENNReal) => HasSubset.Subset.{u2} (Set.{u2} X) (Set.instHasSubsetSet.{u2} X) (EMetric.closedBall.{u2} X (EMetricSpace.toPseudoEMetricSpace.{u2} X _inst_1) y r) (U i)))))))) (nhds.{u2} X (UniformSpace.toTopologicalSpace.{u2} X (PseudoEMetricSpace.toUniformSpace.{u2} X (EMetricSpace.toPseudoEMetricSpace.{u2} X _inst_1))) x)))
+Case conversion may be inaccurate. Consider using '#align emetric.exists_forall_closed_ball_subset_aux₁ EMetric.exists_forall_closedBall_subset_aux₁ₓ'. -/
 theorem exists_forall_closedBall_subset_aux₁ (hK : ∀ i, IsClosed (K i)) (hU : ∀ i, IsOpen (U i))
     (hKU : ∀ i, K i ⊆ U i) (hfin : LocallyFinite K) (x : X) :
     ∃ r : ℝ,
@@ -80,8 +92,14 @@ theorem exists_forall_closedBall_subset_aux₁ (hK : ∀ i, IsClosed (K i)) (hU
   rcases this.exists_gt with ⟨r, hr0, hr⟩
   refine' ⟨r, hr.mono fun y hy => ⟨hr0, _⟩⟩
   rwa [mem_preimage, mem_Inter₂]
-#align emetric.exists_forall_closed_ball_subset_aux₁ Emetric.exists_forall_closedBall_subset_aux₁
-
+#align emetric.exists_forall_closed_ball_subset_aux₁ EMetric.exists_forall_closedBall_subset_aux₁
+
+/- warning: emetric.exists_forall_closed_ball_subset_aux₂ -> EMetric.exists_forall_closedBall_subset_aux₂ is a dubious translation:
+lean 3 declaration is
+  forall {ι : Type.{u1}} {X : Type.{u2}} [_inst_1 : EMetricSpace.{u2} X] {K : ι -> (Set.{u2} X)} {U : ι -> (Set.{u2} X)} (y : X), Convex.{0, 0} Real Real Real.orderedSemiring Real.addCommMonoid (Mul.toSMul.{0} Real Real.hasMul) (Inter.inter.{0} (Set.{0} Real) (Set.hasInter.{0} Real) (Set.Ioi.{0} Real Real.preorder (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero)))) (Set.preimage.{0, 0} Real ENNReal ENNReal.ofReal (Set.iInter.{0, succ u1} ENNReal ι (fun (i : ι) => Set.iInter.{0, 0} ENNReal (Membership.Mem.{u2, u2} X (Set.{u2} X) (Set.hasMem.{u2} X) y (K i)) (fun (hi : Membership.Mem.{u2, u2} X (Set.{u2} X) (Set.hasMem.{u2} X) y (K i)) => setOf.{0} ENNReal (fun (r : ENNReal) => HasSubset.Subset.{u2} (Set.{u2} X) (Set.hasSubset.{u2} X) (EMetric.closedBall.{u2} X (EMetricSpace.toPseudoEmetricSpace.{u2} X _inst_1) y r) (U i)))))))
+but is expected to have type
+  forall {ι : Type.{u2}} {X : Type.{u1}} [_inst_1 : EMetricSpace.{u1} X] {K : ι -> (Set.{u1} X)} {U : ι -> (Set.{u1} X)} (y : X), Convex.{0, 0} Real Real Real.orderedSemiring Real.instAddCommMonoidReal (Algebra.toSMul.{0, 0} Real Real Real.instCommSemiringReal Real.semiring (NormedAlgebra.toAlgebra.{0, 0} Real Real Real.normedField (SeminormedCommRing.toSeminormedRing.{0} Real (NormedCommRing.toSeminormedCommRing.{0} Real Real.normedCommRing)) (NormedAlgebra.id.{0} Real Real.normedField))) (Inter.inter.{0} (Set.{0} Real) (Set.instInterSet.{0} Real) (Set.Ioi.{0} Real Real.instPreorderReal (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal))) (Set.preimage.{0, 0} Real ENNReal ENNReal.ofReal (Set.iInter.{0, succ u2} ENNReal ι (fun (i : ι) => Set.iInter.{0, 0} ENNReal (Membership.mem.{u1, u1} X (Set.{u1} X) (Set.instMembershipSet.{u1} X) y (K i)) (fun (hi : Membership.mem.{u1, u1} X (Set.{u1} X) (Set.instMembershipSet.{u1} X) y (K i)) => setOf.{0} ENNReal (fun (r : ENNReal) => HasSubset.Subset.{u1} (Set.{u1} X) (Set.instHasSubsetSet.{u1} X) (EMetric.closedBall.{u1} X (EMetricSpace.toPseudoEMetricSpace.{u1} X _inst_1) y r) (U i)))))))
+Case conversion may be inaccurate. Consider using '#align emetric.exists_forall_closed_ball_subset_aux₂ EMetric.exists_forall_closedBall_subset_aux₂ₓ'. -/
 theorem exists_forall_closedBall_subset_aux₂ (y : X) :
     Convex ℝ
       (Ioi (0 : ℝ) ∩ ENNReal.ofReal ⁻¹' ⋂ (i) (hi : y ∈ K i), { r | closedBall y r ⊆ U i }) :=
@@ -90,8 +108,14 @@ theorem exists_forall_closedBall_subset_aux₂ (y : X) :
       OrdConnected.preimage_ennreal_ofReal <|
         ordConnected_iInter fun i =>
           ordConnected_iInter fun hi => ordConnected_setOf_closedBall_subset y (U i)
-#align emetric.exists_forall_closed_ball_subset_aux₂ Emetric.exists_forall_closedBall_subset_aux₂
-
+#align emetric.exists_forall_closed_ball_subset_aux₂ EMetric.exists_forall_closedBall_subset_aux₂
+
+/- warning: emetric.exists_continuous_real_forall_closed_ball_subset -> EMetric.exists_continuous_real_forall_closedBall_subset is a dubious translation:
+lean 3 declaration is
+  forall {ι : Type.{u1}} {X : Type.{u2}} [_inst_1 : EMetricSpace.{u2} X] {K : ι -> (Set.{u2} X)} {U : ι -> (Set.{u2} X)}, (forall (i : ι), IsClosed.{u2} X (UniformSpace.toTopologicalSpace.{u2} X (PseudoEMetricSpace.toUniformSpace.{u2} X (EMetricSpace.toPseudoEmetricSpace.{u2} X _inst_1))) (K i)) -> (forall (i : ι), IsOpen.{u2} X (UniformSpace.toTopologicalSpace.{u2} X (PseudoEMetricSpace.toUniformSpace.{u2} X (EMetricSpace.toPseudoEmetricSpace.{u2} X _inst_1))) (U i)) -> (forall (i : ι), HasSubset.Subset.{u2} (Set.{u2} X) (Set.hasSubset.{u2} X) (K i) (U i)) -> (LocallyFinite.{u1, u2} ι X (UniformSpace.toTopologicalSpace.{u2} X (PseudoEMetricSpace.toUniformSpace.{u2} X (EMetricSpace.toPseudoEmetricSpace.{u2} X _inst_1))) K) -> (Exists.{succ u2} (ContinuousMap.{u2, 0} X Real (UniformSpace.toTopologicalSpace.{u2} X (PseudoEMetricSpace.toUniformSpace.{u2} X (EMetricSpace.toPseudoEmetricSpace.{u2} X _inst_1))) (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) (fun (δ : ContinuousMap.{u2, 0} X Real (UniformSpace.toTopologicalSpace.{u2} X (PseudoEMetricSpace.toUniformSpace.{u2} X (EMetricSpace.toPseudoEmetricSpace.{u2} X _inst_1))) (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) => And (forall (x : X), LT.lt.{0} Real Real.hasLt (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero))) (coeFn.{succ u2, succ u2} (ContinuousMap.{u2, 0} X Real (UniformSpace.toTopologicalSpace.{u2} X (PseudoEMetricSpace.toUniformSpace.{u2} X (EMetricSpace.toPseudoEmetricSpace.{u2} X _inst_1))) (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) (fun (_x : ContinuousMap.{u2, 0} X Real (UniformSpace.toTopologicalSpace.{u2} X (PseudoEMetricSpace.toUniformSpace.{u2} X (EMetricSpace.toPseudoEmetricSpace.{u2} X _inst_1))) (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) => X -> Real) (ContinuousMap.hasCoeToFun.{u2, 0} X Real (UniformSpace.toTopologicalSpace.{u2} X (PseudoEMetricSpace.toUniformSpace.{u2} X (EMetricSpace.toPseudoEmetricSpace.{u2} X _inst_1))) (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) δ x)) (forall (i : ι) (x : X), (Membership.Mem.{u2, u2} X (Set.{u2} X) (Set.hasMem.{u2} X) x (K i)) -> (HasSubset.Subset.{u2} (Set.{u2} X) (Set.hasSubset.{u2} X) (EMetric.closedBall.{u2} X (EMetricSpace.toPseudoEmetricSpace.{u2} X _inst_1) x (ENNReal.ofReal (coeFn.{succ u2, succ u2} (ContinuousMap.{u2, 0} X Real (UniformSpace.toTopologicalSpace.{u2} X (PseudoEMetricSpace.toUniformSpace.{u2} X (EMetricSpace.toPseudoEmetricSpace.{u2} X _inst_1))) (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) (fun (_x : ContinuousMap.{u2, 0} X Real (UniformSpace.toTopologicalSpace.{u2} X (PseudoEMetricSpace.toUniformSpace.{u2} X (EMetricSpace.toPseudoEmetricSpace.{u2} X _inst_1))) (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) => X -> Real) (ContinuousMap.hasCoeToFun.{u2, 0} X Real (UniformSpace.toTopologicalSpace.{u2} X (PseudoEMetricSpace.toUniformSpace.{u2} X (EMetricSpace.toPseudoEmetricSpace.{u2} X _inst_1))) (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) δ x))) (U i)))))
+but is expected to have type
+  forall {ι : Type.{u1}} {X : Type.{u2}} [_inst_1 : EMetricSpace.{u2} X] {K : ι -> (Set.{u2} X)} {U : ι -> (Set.{u2} X)}, (forall (i : ι), IsClosed.{u2} X (UniformSpace.toTopologicalSpace.{u2} X (PseudoEMetricSpace.toUniformSpace.{u2} X (EMetricSpace.toPseudoEMetricSpace.{u2} X _inst_1))) (K i)) -> (forall (i : ι), IsOpen.{u2} X (UniformSpace.toTopologicalSpace.{u2} X (PseudoEMetricSpace.toUniformSpace.{u2} X (EMetricSpace.toPseudoEMetricSpace.{u2} X _inst_1))) (U i)) -> (forall (i : ι), HasSubset.Subset.{u2} (Set.{u2} X) (Set.instHasSubsetSet.{u2} X) (K i) (U i)) -> (LocallyFinite.{u1, u2} ι X (UniformSpace.toTopologicalSpace.{u2} X (PseudoEMetricSpace.toUniformSpace.{u2} X (EMetricSpace.toPseudoEMetricSpace.{u2} X _inst_1))) K) -> (Exists.{succ u2} (ContinuousMap.{u2, 0} X Real (UniformSpace.toTopologicalSpace.{u2} X (PseudoEMetricSpace.toUniformSpace.{u2} X (EMetricSpace.toPseudoEMetricSpace.{u2} X _inst_1))) (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) (fun (δ : ContinuousMap.{u2, 0} X Real (UniformSpace.toTopologicalSpace.{u2} X (PseudoEMetricSpace.toUniformSpace.{u2} X (EMetricSpace.toPseudoEMetricSpace.{u2} X _inst_1))) (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) => And (forall (x : X), LT.lt.{0} ((fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : X) => Real) x) Real.instLTReal (OfNat.ofNat.{0} ((fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : X) => Real) x) 0 (Zero.toOfNat0.{0} ((fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : X) => Real) x) Real.instZeroReal)) (FunLike.coe.{succ u2, succ u2, 1} (ContinuousMap.{u2, 0} X Real (UniformSpace.toTopologicalSpace.{u2} X (PseudoEMetricSpace.toUniformSpace.{u2} X (EMetricSpace.toPseudoEMetricSpace.{u2} X _inst_1))) (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) X (fun (_x : X) => (fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : X) => Real) _x) (ContinuousMapClass.toFunLike.{u2, u2, 0} (ContinuousMap.{u2, 0} X Real (UniformSpace.toTopologicalSpace.{u2} X (PseudoEMetricSpace.toUniformSpace.{u2} X (EMetricSpace.toPseudoEMetricSpace.{u2} X _inst_1))) (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) X Real (UniformSpace.toTopologicalSpace.{u2} X (PseudoEMetricSpace.toUniformSpace.{u2} X (EMetricSpace.toPseudoEMetricSpace.{u2} X _inst_1))) (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) (ContinuousMap.instContinuousMapClassContinuousMap.{u2, 0} X Real (UniformSpace.toTopologicalSpace.{u2} X (PseudoEMetricSpace.toUniformSpace.{u2} X (EMetricSpace.toPseudoEMetricSpace.{u2} X _inst_1))) (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)))) δ x)) (forall (i : ι) (x : X), (Membership.mem.{u2, u2} X (Set.{u2} X) (Set.instMembershipSet.{u2} X) x (K i)) -> (HasSubset.Subset.{u2} (Set.{u2} X) (Set.instHasSubsetSet.{u2} X) (EMetric.closedBall.{u2} X (EMetricSpace.toPseudoEMetricSpace.{u2} X _inst_1) x (ENNReal.ofReal (FunLike.coe.{succ u2, succ u2, 1} (ContinuousMap.{u2, 0} X Real (UniformSpace.toTopologicalSpace.{u2} X (PseudoEMetricSpace.toUniformSpace.{u2} X (EMetricSpace.toPseudoEMetricSpace.{u2} X _inst_1))) (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) X (fun (_x : X) => (fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : X) => Real) _x) (ContinuousMapClass.toFunLike.{u2, u2, 0} (ContinuousMap.{u2, 0} X Real (UniformSpace.toTopologicalSpace.{u2} X (PseudoEMetricSpace.toUniformSpace.{u2} X (EMetricSpace.toPseudoEMetricSpace.{u2} X _inst_1))) (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) X Real (UniformSpace.toTopologicalSpace.{u2} X (PseudoEMetricSpace.toUniformSpace.{u2} X (EMetricSpace.toPseudoEMetricSpace.{u2} X _inst_1))) (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) (ContinuousMap.instContinuousMapClassContinuousMap.{u2, 0} X Real (UniformSpace.toTopologicalSpace.{u2} X (PseudoEMetricSpace.toUniformSpace.{u2} X (EMetricSpace.toPseudoEMetricSpace.{u2} X _inst_1))) (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)))) δ x))) (U i)))))
+Case conversion may be inaccurate. Consider using '#align emetric.exists_continuous_real_forall_closed_ball_subset EMetric.exists_continuous_real_forall_closedBall_subsetₓ'. -/
 /-- Let `X` be an extended metric space. Let `K : ι → set X` be a locally finite family of closed
 sets, let `U : ι → set X` be a family of open sets such that `K i ⊆ U i` for all `i`. Then there
 exists a positive continuous function `δ : C(X, ℝ)` such that for any `i` and `x ∈ K i`,
@@ -103,8 +127,14 @@ theorem exists_continuous_real_forall_closedBall_subset (hK : ∀ i, IsClosed (K
   simpa only [mem_inter_iff, forall_and, mem_preimage, mem_Inter, @forall_swap ι X] using
     exists_continuous_forall_mem_convex_of_local_const exists_forall_closed_ball_subset_aux₂
       (exists_forall_closed_ball_subset_aux₁ hK hU hKU hfin)
-#align emetric.exists_continuous_real_forall_closed_ball_subset Emetric.exists_continuous_real_forall_closedBall_subset
-
+#align emetric.exists_continuous_real_forall_closed_ball_subset EMetric.exists_continuous_real_forall_closedBall_subset
+
+/- warning: emetric.exists_continuous_nnreal_forall_closed_ball_subset -> EMetric.exists_continuous_nNReal_forall_closedBall_subset is a dubious translation:
+lean 3 declaration is
+  forall {ι : Type.{u1}} {X : Type.{u2}} [_inst_1 : EMetricSpace.{u2} X] {K : ι -> (Set.{u2} X)} {U : ι -> (Set.{u2} X)}, (forall (i : ι), IsClosed.{u2} X (UniformSpace.toTopologicalSpace.{u2} X (PseudoEMetricSpace.toUniformSpace.{u2} X (EMetricSpace.toPseudoEmetricSpace.{u2} X _inst_1))) (K i)) -> (forall (i : ι), IsOpen.{u2} X (UniformSpace.toTopologicalSpace.{u2} X (PseudoEMetricSpace.toUniformSpace.{u2} X (EMetricSpace.toPseudoEmetricSpace.{u2} X _inst_1))) (U i)) -> (forall (i : ι), HasSubset.Subset.{u2} (Set.{u2} X) (Set.hasSubset.{u2} X) (K i) (U i)) -> (LocallyFinite.{u1, u2} ι X (UniformSpace.toTopologicalSpace.{u2} X (PseudoEMetricSpace.toUniformSpace.{u2} X (EMetricSpace.toPseudoEmetricSpace.{u2} X _inst_1))) K) -> (Exists.{succ u2} (ContinuousMap.{u2, 0} X NNReal (UniformSpace.toTopologicalSpace.{u2} X (PseudoEMetricSpace.toUniformSpace.{u2} X (EMetricSpace.toPseudoEmetricSpace.{u2} X _inst_1))) NNReal.topologicalSpace) (fun (δ : ContinuousMap.{u2, 0} X NNReal (UniformSpace.toTopologicalSpace.{u2} X (PseudoEMetricSpace.toUniformSpace.{u2} X (EMetricSpace.toPseudoEmetricSpace.{u2} X _inst_1))) NNReal.topologicalSpace) => And (forall (x : X), LT.lt.{0} NNReal (Preorder.toHasLt.{0} NNReal (PartialOrder.toPreorder.{0} NNReal (OrderedCancelAddCommMonoid.toPartialOrder.{0} NNReal (StrictOrderedSemiring.toOrderedCancelAddCommMonoid.{0} NNReal NNReal.strictOrderedSemiring)))) (OfNat.ofNat.{0} NNReal 0 (OfNat.mk.{0} NNReal 0 (Zero.zero.{0} NNReal (MulZeroClass.toHasZero.{0} NNReal (NonUnitalNonAssocSemiring.toMulZeroClass.{0} NNReal (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} NNReal (Semiring.toNonAssocSemiring.{0} NNReal NNReal.semiring))))))) (coeFn.{succ u2, succ u2} (ContinuousMap.{u2, 0} X NNReal (UniformSpace.toTopologicalSpace.{u2} X (PseudoEMetricSpace.toUniformSpace.{u2} X (EMetricSpace.toPseudoEmetricSpace.{u2} X _inst_1))) NNReal.topologicalSpace) (fun (_x : ContinuousMap.{u2, 0} X NNReal (UniformSpace.toTopologicalSpace.{u2} X (PseudoEMetricSpace.toUniformSpace.{u2} X (EMetricSpace.toPseudoEmetricSpace.{u2} X _inst_1))) NNReal.topologicalSpace) => X -> NNReal) (ContinuousMap.hasCoeToFun.{u2, 0} X NNReal (UniformSpace.toTopologicalSpace.{u2} X (PseudoEMetricSpace.toUniformSpace.{u2} X (EMetricSpace.toPseudoEmetricSpace.{u2} X _inst_1))) NNReal.topologicalSpace) δ x)) (forall (i : ι) (x : X), (Membership.Mem.{u2, u2} X (Set.{u2} X) (Set.hasMem.{u2} X) x (K i)) -> (HasSubset.Subset.{u2} (Set.{u2} X) (Set.hasSubset.{u2} X) (EMetric.closedBall.{u2} X (EMetricSpace.toPseudoEmetricSpace.{u2} X _inst_1) x ((fun (a : Type) (b : Type) [self : HasLiftT.{1, 1} a b] => self.0) NNReal ENNReal (HasLiftT.mk.{1, 1} NNReal ENNReal (CoeTCₓ.coe.{1, 1} NNReal ENNReal (coeBase.{1, 1} NNReal ENNReal ENNReal.hasCoe))) (coeFn.{succ u2, succ u2} (ContinuousMap.{u2, 0} X NNReal (UniformSpace.toTopologicalSpace.{u2} X (PseudoEMetricSpace.toUniformSpace.{u2} X (EMetricSpace.toPseudoEmetricSpace.{u2} X _inst_1))) NNReal.topologicalSpace) (fun (_x : ContinuousMap.{u2, 0} X NNReal (UniformSpace.toTopologicalSpace.{u2} X (PseudoEMetricSpace.toUniformSpace.{u2} X (EMetricSpace.toPseudoEmetricSpace.{u2} X _inst_1))) NNReal.topologicalSpace) => X -> NNReal) (ContinuousMap.hasCoeToFun.{u2, 0} X NNReal (UniformSpace.toTopologicalSpace.{u2} X (PseudoEMetricSpace.toUniformSpace.{u2} X (EMetricSpace.toPseudoEmetricSpace.{u2} X _inst_1))) NNReal.topologicalSpace) δ x))) (U i)))))
+but is expected to have type
+  forall {ι : Type.{u1}} {X : Type.{u2}} [_inst_1 : EMetricSpace.{u2} X] {K : ι -> (Set.{u2} X)} {U : ι -> (Set.{u2} X)}, (forall (i : ι), IsClosed.{u2} X (UniformSpace.toTopologicalSpace.{u2} X (PseudoEMetricSpace.toUniformSpace.{u2} X (EMetricSpace.toPseudoEMetricSpace.{u2} X _inst_1))) (K i)) -> (forall (i : ι), IsOpen.{u2} X (UniformSpace.toTopologicalSpace.{u2} X (PseudoEMetricSpace.toUniformSpace.{u2} X (EMetricSpace.toPseudoEMetricSpace.{u2} X _inst_1))) (U i)) -> (forall (i : ι), HasSubset.Subset.{u2} (Set.{u2} X) (Set.instHasSubsetSet.{u2} X) (K i) (U i)) -> (LocallyFinite.{u1, u2} ι X (UniformSpace.toTopologicalSpace.{u2} X (PseudoEMetricSpace.toUniformSpace.{u2} X (EMetricSpace.toPseudoEMetricSpace.{u2} X _inst_1))) K) -> (Exists.{succ u2} (ContinuousMap.{u2, 0} X NNReal (UniformSpace.toTopologicalSpace.{u2} X (PseudoEMetricSpace.toUniformSpace.{u2} X (EMetricSpace.toPseudoEMetricSpace.{u2} X _inst_1))) NNReal.instTopologicalSpaceNNReal) (fun (δ : ContinuousMap.{u2, 0} X NNReal (UniformSpace.toTopologicalSpace.{u2} X (PseudoEMetricSpace.toUniformSpace.{u2} X (EMetricSpace.toPseudoEMetricSpace.{u2} X _inst_1))) NNReal.instTopologicalSpaceNNReal) => And (forall (x : X), LT.lt.{0} ((fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : X) => NNReal) x) (Preorder.toLT.{0} ((fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : X) => NNReal) x) (PartialOrder.toPreorder.{0} ((fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : X) => NNReal) x) (StrictOrderedSemiring.toPartialOrder.{0} ((fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : X) => NNReal) x) instNNRealStrictOrderedSemiring))) (OfNat.ofNat.{0} ((fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : X) => NNReal) x) 0 (Zero.toOfNat0.{0} ((fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : X) => NNReal) x) instNNRealZero)) (FunLike.coe.{succ u2, succ u2, 1} (ContinuousMap.{u2, 0} X NNReal (UniformSpace.toTopologicalSpace.{u2} X (PseudoEMetricSpace.toUniformSpace.{u2} X (EMetricSpace.toPseudoEMetricSpace.{u2} X _inst_1))) NNReal.instTopologicalSpaceNNReal) X (fun (_x : X) => (fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : X) => NNReal) _x) (ContinuousMapClass.toFunLike.{u2, u2, 0} (ContinuousMap.{u2, 0} X NNReal (UniformSpace.toTopologicalSpace.{u2} X (PseudoEMetricSpace.toUniformSpace.{u2} X (EMetricSpace.toPseudoEMetricSpace.{u2} X _inst_1))) NNReal.instTopologicalSpaceNNReal) X NNReal (UniformSpace.toTopologicalSpace.{u2} X (PseudoEMetricSpace.toUniformSpace.{u2} X (EMetricSpace.toPseudoEMetricSpace.{u2} X _inst_1))) NNReal.instTopologicalSpaceNNReal (ContinuousMap.instContinuousMapClassContinuousMap.{u2, 0} X NNReal (UniformSpace.toTopologicalSpace.{u2} X (PseudoEMetricSpace.toUniformSpace.{u2} X (EMetricSpace.toPseudoEMetricSpace.{u2} X _inst_1))) NNReal.instTopologicalSpaceNNReal)) δ x)) (forall (i : ι) (x : X), (Membership.mem.{u2, u2} X (Set.{u2} X) (Set.instMembershipSet.{u2} X) x (K i)) -> (HasSubset.Subset.{u2} (Set.{u2} X) (Set.instHasSubsetSet.{u2} X) (EMetric.closedBall.{u2} X (EMetricSpace.toPseudoEMetricSpace.{u2} X _inst_1) x (ENNReal.some (FunLike.coe.{succ u2, succ u2, 1} (ContinuousMap.{u2, 0} X NNReal (UniformSpace.toTopologicalSpace.{u2} X (PseudoEMetricSpace.toUniformSpace.{u2} X (EMetricSpace.toPseudoEMetricSpace.{u2} X _inst_1))) NNReal.instTopologicalSpaceNNReal) X (fun (_x : X) => (fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : X) => NNReal) _x) (ContinuousMapClass.toFunLike.{u2, u2, 0} (ContinuousMap.{u2, 0} X NNReal (UniformSpace.toTopologicalSpace.{u2} X (PseudoEMetricSpace.toUniformSpace.{u2} X (EMetricSpace.toPseudoEMetricSpace.{u2} X _inst_1))) NNReal.instTopologicalSpaceNNReal) X NNReal (UniformSpace.toTopologicalSpace.{u2} X (PseudoEMetricSpace.toUniformSpace.{u2} X (EMetricSpace.toPseudoEMetricSpace.{u2} X _inst_1))) NNReal.instTopologicalSpaceNNReal (ContinuousMap.instContinuousMapClassContinuousMap.{u2, 0} X NNReal (UniformSpace.toTopologicalSpace.{u2} X (PseudoEMetricSpace.toUniformSpace.{u2} X (EMetricSpace.toPseudoEMetricSpace.{u2} X _inst_1))) NNReal.instTopologicalSpaceNNReal)) δ x))) (U i)))))
+Case conversion may be inaccurate. Consider using '#align emetric.exists_continuous_nnreal_forall_closed_ball_subset EMetric.exists_continuous_nNReal_forall_closedBall_subsetₓ'. -/
 /-- Let `X` be an extended metric space. Let `K : ι → set X` be a locally finite family of closed
 sets, let `U : ι → set X` be a family of open sets such that `K i ⊆ U i` for all `i`. Then there
 exists a positive continuous function `δ : C(X, ℝ≥0)` such that for any `i` and `x ∈ K i`,
@@ -117,8 +147,14 @@ theorem exists_continuous_nNReal_forall_closedBall_subset (hK : ∀ i, IsClosed
   lift δ to C(X, ℝ≥0) using fun x => (hδ₀ x).le
   refine' ⟨δ, hδ₀, fun i x hi => _⟩
   simpa only [← ENNReal.ofReal_coe_nnreal] using hδ i x hi
-#align emetric.exists_continuous_nnreal_forall_closed_ball_subset Emetric.exists_continuous_nNReal_forall_closedBall_subset
-
+#align emetric.exists_continuous_nnreal_forall_closed_ball_subset EMetric.exists_continuous_nNReal_forall_closedBall_subset
+
+/- warning: emetric.exists_continuous_ennreal_forall_closed_ball_subset -> EMetric.exists_continuous_eNNReal_forall_closedBall_subset is a dubious translation:
+lean 3 declaration is
+  forall {ι : Type.{u1}} {X : Type.{u2}} [_inst_1 : EMetricSpace.{u2} X] {K : ι -> (Set.{u2} X)} {U : ι -> (Set.{u2} X)}, (forall (i : ι), IsClosed.{u2} X (UniformSpace.toTopologicalSpace.{u2} X (PseudoEMetricSpace.toUniformSpace.{u2} X (EMetricSpace.toPseudoEmetricSpace.{u2} X _inst_1))) (K i)) -> (forall (i : ι), IsOpen.{u2} X (UniformSpace.toTopologicalSpace.{u2} X (PseudoEMetricSpace.toUniformSpace.{u2} X (EMetricSpace.toPseudoEmetricSpace.{u2} X _inst_1))) (U i)) -> (forall (i : ι), HasSubset.Subset.{u2} (Set.{u2} X) (Set.hasSubset.{u2} X) (K i) (U i)) -> (LocallyFinite.{u1, u2} ι X (UniformSpace.toTopologicalSpace.{u2} X (PseudoEMetricSpace.toUniformSpace.{u2} X (EMetricSpace.toPseudoEmetricSpace.{u2} X _inst_1))) K) -> (Exists.{succ u2} (ContinuousMap.{u2, 0} X ENNReal (UniformSpace.toTopologicalSpace.{u2} X (PseudoEMetricSpace.toUniformSpace.{u2} X (EMetricSpace.toPseudoEmetricSpace.{u2} X _inst_1))) ENNReal.topologicalSpace) (fun (δ : ContinuousMap.{u2, 0} X ENNReal (UniformSpace.toTopologicalSpace.{u2} X (PseudoEMetricSpace.toUniformSpace.{u2} X (EMetricSpace.toPseudoEmetricSpace.{u2} X _inst_1))) ENNReal.topologicalSpace) => And (forall (x : X), LT.lt.{0} ENNReal (Preorder.toHasLt.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (CompleteSemilatticeInf.toPartialOrder.{0} ENNReal (CompleteLattice.toCompleteSemilatticeInf.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))))) (OfNat.ofNat.{0} ENNReal 0 (OfNat.mk.{0} ENNReal 0 (Zero.zero.{0} ENNReal ENNReal.hasZero))) (coeFn.{succ u2, succ u2} (ContinuousMap.{u2, 0} X ENNReal (UniformSpace.toTopologicalSpace.{u2} X (PseudoEMetricSpace.toUniformSpace.{u2} X (EMetricSpace.toPseudoEmetricSpace.{u2} X _inst_1))) ENNReal.topologicalSpace) (fun (_x : ContinuousMap.{u2, 0} X ENNReal (UniformSpace.toTopologicalSpace.{u2} X (PseudoEMetricSpace.toUniformSpace.{u2} X (EMetricSpace.toPseudoEmetricSpace.{u2} X _inst_1))) ENNReal.topologicalSpace) => X -> ENNReal) (ContinuousMap.hasCoeToFun.{u2, 0} X ENNReal (UniformSpace.toTopologicalSpace.{u2} X (PseudoEMetricSpace.toUniformSpace.{u2} X (EMetricSpace.toPseudoEmetricSpace.{u2} X _inst_1))) ENNReal.topologicalSpace) δ x)) (forall (i : ι) (x : X), (Membership.Mem.{u2, u2} X (Set.{u2} X) (Set.hasMem.{u2} X) x (K i)) -> (HasSubset.Subset.{u2} (Set.{u2} X) (Set.hasSubset.{u2} X) (EMetric.closedBall.{u2} X (EMetricSpace.toPseudoEmetricSpace.{u2} X _inst_1) x (coeFn.{succ u2, succ u2} (ContinuousMap.{u2, 0} X ENNReal (UniformSpace.toTopologicalSpace.{u2} X (PseudoEMetricSpace.toUniformSpace.{u2} X (EMetricSpace.toPseudoEmetricSpace.{u2} X _inst_1))) ENNReal.topologicalSpace) (fun (_x : ContinuousMap.{u2, 0} X ENNReal (UniformSpace.toTopologicalSpace.{u2} X (PseudoEMetricSpace.toUniformSpace.{u2} X (EMetricSpace.toPseudoEmetricSpace.{u2} X _inst_1))) ENNReal.topologicalSpace) => X -> ENNReal) (ContinuousMap.hasCoeToFun.{u2, 0} X ENNReal (UniformSpace.toTopologicalSpace.{u2} X (PseudoEMetricSpace.toUniformSpace.{u2} X (EMetricSpace.toPseudoEmetricSpace.{u2} X _inst_1))) ENNReal.topologicalSpace) δ x)) (U i)))))
+but is expected to have type
+  forall {ι : Type.{u1}} {X : Type.{u2}} [_inst_1 : EMetricSpace.{u2} X] {K : ι -> (Set.{u2} X)} {U : ι -> (Set.{u2} X)}, (forall (i : ι), IsClosed.{u2} X (UniformSpace.toTopologicalSpace.{u2} X (PseudoEMetricSpace.toUniformSpace.{u2} X (EMetricSpace.toPseudoEMetricSpace.{u2} X _inst_1))) (K i)) -> (forall (i : ι), IsOpen.{u2} X (UniformSpace.toTopologicalSpace.{u2} X (PseudoEMetricSpace.toUniformSpace.{u2} X (EMetricSpace.toPseudoEMetricSpace.{u2} X _inst_1))) (U i)) -> (forall (i : ι), HasSubset.Subset.{u2} (Set.{u2} X) (Set.instHasSubsetSet.{u2} X) (K i) (U i)) -> (LocallyFinite.{u1, u2} ι X (UniformSpace.toTopologicalSpace.{u2} X (PseudoEMetricSpace.toUniformSpace.{u2} X (EMetricSpace.toPseudoEMetricSpace.{u2} X _inst_1))) K) -> (Exists.{succ u2} (ContinuousMap.{u2, 0} X ENNReal (UniformSpace.toTopologicalSpace.{u2} X (PseudoEMetricSpace.toUniformSpace.{u2} X (EMetricSpace.toPseudoEMetricSpace.{u2} X _inst_1))) ENNReal.instTopologicalSpaceENNReal) (fun (δ : ContinuousMap.{u2, 0} X ENNReal (UniformSpace.toTopologicalSpace.{u2} X (PseudoEMetricSpace.toUniformSpace.{u2} X (EMetricSpace.toPseudoEMetricSpace.{u2} X _inst_1))) ENNReal.instTopologicalSpaceENNReal) => And (forall (x : X), LT.lt.{0} ((fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : X) => ENNReal) x) (Preorder.toLT.{0} ((fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : X) => ENNReal) x) (PartialOrder.toPreorder.{0} ((fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : X) => ENNReal) x) (OmegaCompletePartialOrder.toPartialOrder.{0} ((fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : X) => ENNReal) x) (CompleteLattice.instOmegaCompletePartialOrder.{0} ((fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : X) => ENNReal) x) (CompleteLinearOrder.toCompleteLattice.{0} ((fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : X) => ENNReal) x) ENNReal.instCompleteLinearOrderENNReal))))) (OfNat.ofNat.{0} ((fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : X) => ENNReal) x) 0 (Zero.toOfNat0.{0} ((fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : X) => ENNReal) x) instENNRealZero)) (FunLike.coe.{succ u2, succ u2, 1} (ContinuousMap.{u2, 0} X ENNReal (UniformSpace.toTopologicalSpace.{u2} X (PseudoEMetricSpace.toUniformSpace.{u2} X (EMetricSpace.toPseudoEMetricSpace.{u2} X _inst_1))) ENNReal.instTopologicalSpaceENNReal) X (fun (_x : X) => (fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : X) => ENNReal) _x) (ContinuousMapClass.toFunLike.{u2, u2, 0} (ContinuousMap.{u2, 0} X ENNReal (UniformSpace.toTopologicalSpace.{u2} X (PseudoEMetricSpace.toUniformSpace.{u2} X (EMetricSpace.toPseudoEMetricSpace.{u2} X _inst_1))) ENNReal.instTopologicalSpaceENNReal) X ENNReal (UniformSpace.toTopologicalSpace.{u2} X (PseudoEMetricSpace.toUniformSpace.{u2} X (EMetricSpace.toPseudoEMetricSpace.{u2} X _inst_1))) ENNReal.instTopologicalSpaceENNReal (ContinuousMap.instContinuousMapClassContinuousMap.{u2, 0} X ENNReal (UniformSpace.toTopologicalSpace.{u2} X (PseudoEMetricSpace.toUniformSpace.{u2} X (EMetricSpace.toPseudoEMetricSpace.{u2} X _inst_1))) ENNReal.instTopologicalSpaceENNReal)) δ x)) (forall (i : ι) (x : X), (Membership.mem.{u2, u2} X (Set.{u2} X) (Set.instMembershipSet.{u2} X) x (K i)) -> (HasSubset.Subset.{u2} (Set.{u2} X) (Set.instHasSubsetSet.{u2} X) (EMetric.closedBall.{u2} X (EMetricSpace.toPseudoEMetricSpace.{u2} X _inst_1) x (FunLike.coe.{succ u2, succ u2, 1} (ContinuousMap.{u2, 0} X ENNReal (UniformSpace.toTopologicalSpace.{u2} X (PseudoEMetricSpace.toUniformSpace.{u2} X (EMetricSpace.toPseudoEMetricSpace.{u2} X _inst_1))) ENNReal.instTopologicalSpaceENNReal) X (fun (_x : X) => (fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : X) => ENNReal) _x) (ContinuousMapClass.toFunLike.{u2, u2, 0} (ContinuousMap.{u2, 0} X ENNReal (UniformSpace.toTopologicalSpace.{u2} X (PseudoEMetricSpace.toUniformSpace.{u2} X (EMetricSpace.toPseudoEMetricSpace.{u2} X _inst_1))) ENNReal.instTopologicalSpaceENNReal) X ENNReal (UniformSpace.toTopologicalSpace.{u2} X (PseudoEMetricSpace.toUniformSpace.{u2} X (EMetricSpace.toPseudoEMetricSpace.{u2} X _inst_1))) ENNReal.instTopologicalSpaceENNReal (ContinuousMap.instContinuousMapClassContinuousMap.{u2, 0} X ENNReal (UniformSpace.toTopologicalSpace.{u2} X (PseudoEMetricSpace.toUniformSpace.{u2} X (EMetricSpace.toPseudoEMetricSpace.{u2} X _inst_1))) ENNReal.instTopologicalSpaceENNReal)) δ x)) (U i)))))
+Case conversion may be inaccurate. Consider using '#align emetric.exists_continuous_ennreal_forall_closed_ball_subset EMetric.exists_continuous_eNNReal_forall_closedBall_subsetₓ'. -/
 /-- Let `X` be an extended metric space. Let `K : ι → set X` be a locally finite family of closed
 sets, let `U : ι → set X` be a family of open sets such that `K i ⊆ U i` for all `i`. Then there
 exists a positive continuous function `δ : C(X, ℝ≥0∞)` such that for any `i` and `x ∈ K i`,
@@ -128,7 +164,7 @@ theorem exists_continuous_eNNReal_forall_closedBall_subset (hK : ∀ i, IsClosed
     ∃ δ : C(X, ℝ≥0∞), (∀ x, 0 < δ x) ∧ ∀ (i), ∀ x ∈ K i, closedBall x (δ x) ⊆ U i :=
   let ⟨δ, hδ₀, hδ⟩ := exists_continuous_nNReal_forall_closedBall_subset hK hU hKU hfin
   ⟨ContinuousMap.comp ⟨coe, ENNReal.continuous_coe⟩ δ, fun x => ENNReal.coe_pos.2 (hδ₀ x), hδ⟩
-#align emetric.exists_continuous_ennreal_forall_closed_ball_subset Emetric.exists_continuous_eNNReal_forall_closedBall_subset
+#align emetric.exists_continuous_ennreal_forall_closed_ball_subset EMetric.exists_continuous_eNNReal_forall_closedBall_subset
 
 end Emetric
 
@@ -136,6 +172,12 @@ namespace Metric
 
 variable [MetricSpace X] {K : ι → Set X} {U : ι → Set X}
 
+/- warning: metric.exists_continuous_nnreal_forall_closed_ball_subset -> Metric.exists_continuous_nNReal_forall_closedBall_subset is a dubious translation:
+lean 3 declaration is
+  forall {ι : Type.{u1}} {X : Type.{u2}} [_inst_1 : MetricSpace.{u2} X] {K : ι -> (Set.{u2} X)} {U : ι -> (Set.{u2} X)}, (forall (i : ι), IsClosed.{u2} X (UniformSpace.toTopologicalSpace.{u2} X (PseudoMetricSpace.toUniformSpace.{u2} X (MetricSpace.toPseudoMetricSpace.{u2} X _inst_1))) (K i)) -> (forall (i : ι), IsOpen.{u2} X (UniformSpace.toTopologicalSpace.{u2} X (PseudoMetricSpace.toUniformSpace.{u2} X (MetricSpace.toPseudoMetricSpace.{u2} X _inst_1))) (U i)) -> (forall (i : ι), HasSubset.Subset.{u2} (Set.{u2} X) (Set.hasSubset.{u2} X) (K i) (U i)) -> (LocallyFinite.{u1, u2} ι X (UniformSpace.toTopologicalSpace.{u2} X (PseudoMetricSpace.toUniformSpace.{u2} X (MetricSpace.toPseudoMetricSpace.{u2} X _inst_1))) K) -> (Exists.{succ u2} (ContinuousMap.{u2, 0} X NNReal (UniformSpace.toTopologicalSpace.{u2} X (PseudoMetricSpace.toUniformSpace.{u2} X (MetricSpace.toPseudoMetricSpace.{u2} X _inst_1))) NNReal.topologicalSpace) (fun (δ : ContinuousMap.{u2, 0} X NNReal (UniformSpace.toTopologicalSpace.{u2} X (PseudoMetricSpace.toUniformSpace.{u2} X (MetricSpace.toPseudoMetricSpace.{u2} X _inst_1))) NNReal.topologicalSpace) => And (forall (x : X), LT.lt.{0} NNReal (Preorder.toHasLt.{0} NNReal (PartialOrder.toPreorder.{0} NNReal (OrderedCancelAddCommMonoid.toPartialOrder.{0} NNReal (StrictOrderedSemiring.toOrderedCancelAddCommMonoid.{0} NNReal NNReal.strictOrderedSemiring)))) (OfNat.ofNat.{0} NNReal 0 (OfNat.mk.{0} NNReal 0 (Zero.zero.{0} NNReal (MulZeroClass.toHasZero.{0} NNReal (NonUnitalNonAssocSemiring.toMulZeroClass.{0} NNReal (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} NNReal (Semiring.toNonAssocSemiring.{0} NNReal NNReal.semiring))))))) (coeFn.{succ u2, succ u2} (ContinuousMap.{u2, 0} X NNReal (UniformSpace.toTopologicalSpace.{u2} X (PseudoMetricSpace.toUniformSpace.{u2} X (MetricSpace.toPseudoMetricSpace.{u2} X _inst_1))) NNReal.topologicalSpace) (fun (_x : ContinuousMap.{u2, 0} X NNReal (UniformSpace.toTopologicalSpace.{u2} X (PseudoMetricSpace.toUniformSpace.{u2} X (MetricSpace.toPseudoMetricSpace.{u2} X _inst_1))) NNReal.topologicalSpace) => X -> NNReal) (ContinuousMap.hasCoeToFun.{u2, 0} X NNReal (UniformSpace.toTopologicalSpace.{u2} X (PseudoMetricSpace.toUniformSpace.{u2} X (MetricSpace.toPseudoMetricSpace.{u2} X _inst_1))) NNReal.topologicalSpace) δ x)) (forall (i : ι) (x : X), (Membership.Mem.{u2, u2} X (Set.{u2} X) (Set.hasMem.{u2} X) x (K i)) -> (HasSubset.Subset.{u2} (Set.{u2} X) (Set.hasSubset.{u2} X) (Metric.closedBall.{u2} X (MetricSpace.toPseudoMetricSpace.{u2} X _inst_1) x ((fun (a : Type) (b : Type) [self : HasLiftT.{1, 1} a b] => self.0) NNReal Real (HasLiftT.mk.{1, 1} NNReal Real (CoeTCₓ.coe.{1, 1} NNReal Real (coeBase.{1, 1} NNReal Real NNReal.Real.hasCoe))) (coeFn.{succ u2, succ u2} (ContinuousMap.{u2, 0} X NNReal (UniformSpace.toTopologicalSpace.{u2} X (PseudoMetricSpace.toUniformSpace.{u2} X (MetricSpace.toPseudoMetricSpace.{u2} X _inst_1))) NNReal.topologicalSpace) (fun (_x : ContinuousMap.{u2, 0} X NNReal (UniformSpace.toTopologicalSpace.{u2} X (PseudoMetricSpace.toUniformSpace.{u2} X (MetricSpace.toPseudoMetricSpace.{u2} X _inst_1))) NNReal.topologicalSpace) => X -> NNReal) (ContinuousMap.hasCoeToFun.{u2, 0} X NNReal (UniformSpace.toTopologicalSpace.{u2} X (PseudoMetricSpace.toUniformSpace.{u2} X (MetricSpace.toPseudoMetricSpace.{u2} X _inst_1))) NNReal.topologicalSpace) δ x))) (U i)))))
+but is expected to have type
+  forall {ι : Type.{u1}} {X : Type.{u2}} [_inst_1 : MetricSpace.{u2} X] {K : ι -> (Set.{u2} X)} {U : ι -> (Set.{u2} X)}, (forall (i : ι), IsClosed.{u2} X (UniformSpace.toTopologicalSpace.{u2} X (PseudoMetricSpace.toUniformSpace.{u2} X (MetricSpace.toPseudoMetricSpace.{u2} X _inst_1))) (K i)) -> (forall (i : ι), IsOpen.{u2} X (UniformSpace.toTopologicalSpace.{u2} X (PseudoMetricSpace.toUniformSpace.{u2} X (MetricSpace.toPseudoMetricSpace.{u2} X _inst_1))) (U i)) -> (forall (i : ι), HasSubset.Subset.{u2} (Set.{u2} X) (Set.instHasSubsetSet.{u2} X) (K i) (U i)) -> (LocallyFinite.{u1, u2} ι X (UniformSpace.toTopologicalSpace.{u2} X (PseudoMetricSpace.toUniformSpace.{u2} X (MetricSpace.toPseudoMetricSpace.{u2} X _inst_1))) K) -> (Exists.{succ u2} (ContinuousMap.{u2, 0} X NNReal (UniformSpace.toTopologicalSpace.{u2} X (PseudoMetricSpace.toUniformSpace.{u2} X (MetricSpace.toPseudoMetricSpace.{u2} X _inst_1))) NNReal.instTopologicalSpaceNNReal) (fun (δ : ContinuousMap.{u2, 0} X NNReal (UniformSpace.toTopologicalSpace.{u2} X (PseudoMetricSpace.toUniformSpace.{u2} X (MetricSpace.toPseudoMetricSpace.{u2} X _inst_1))) NNReal.instTopologicalSpaceNNReal) => And (forall (x : X), LT.lt.{0} ((fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : X) => NNReal) x) (Preorder.toLT.{0} ((fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : X) => NNReal) x) (PartialOrder.toPreorder.{0} ((fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : X) => NNReal) x) (StrictOrderedSemiring.toPartialOrder.{0} ((fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : X) => NNReal) x) instNNRealStrictOrderedSemiring))) (OfNat.ofNat.{0} ((fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : X) => NNReal) x) 0 (Zero.toOfNat0.{0} ((fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : X) => NNReal) x) instNNRealZero)) (FunLike.coe.{succ u2, succ u2, 1} (ContinuousMap.{u2, 0} X NNReal (UniformSpace.toTopologicalSpace.{u2} X (PseudoMetricSpace.toUniformSpace.{u2} X (MetricSpace.toPseudoMetricSpace.{u2} X _inst_1))) NNReal.instTopologicalSpaceNNReal) X (fun (_x : X) => (fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : X) => NNReal) _x) (ContinuousMapClass.toFunLike.{u2, u2, 0} (ContinuousMap.{u2, 0} X NNReal (UniformSpace.toTopologicalSpace.{u2} X (PseudoMetricSpace.toUniformSpace.{u2} X (MetricSpace.toPseudoMetricSpace.{u2} X _inst_1))) NNReal.instTopologicalSpaceNNReal) X NNReal (UniformSpace.toTopologicalSpace.{u2} X (PseudoMetricSpace.toUniformSpace.{u2} X (MetricSpace.toPseudoMetricSpace.{u2} X _inst_1))) NNReal.instTopologicalSpaceNNReal (ContinuousMap.instContinuousMapClassContinuousMap.{u2, 0} X NNReal (UniformSpace.toTopologicalSpace.{u2} X (PseudoMetricSpace.toUniformSpace.{u2} X (MetricSpace.toPseudoMetricSpace.{u2} X _inst_1))) NNReal.instTopologicalSpaceNNReal)) δ x)) (forall (i : ι) (x : X), (Membership.mem.{u2, u2} X (Set.{u2} X) (Set.instMembershipSet.{u2} X) x (K i)) -> (HasSubset.Subset.{u2} (Set.{u2} X) (Set.instHasSubsetSet.{u2} X) (Metric.closedBall.{u2} X (MetricSpace.toPseudoMetricSpace.{u2} X _inst_1) x (NNReal.toReal (FunLike.coe.{succ u2, succ u2, 1} (ContinuousMap.{u2, 0} X NNReal (UniformSpace.toTopologicalSpace.{u2} X (PseudoMetricSpace.toUniformSpace.{u2} X (MetricSpace.toPseudoMetricSpace.{u2} X _inst_1))) NNReal.instTopologicalSpaceNNReal) X (fun (_x : X) => (fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : X) => NNReal) _x) (ContinuousMapClass.toFunLike.{u2, u2, 0} (ContinuousMap.{u2, 0} X NNReal (UniformSpace.toTopologicalSpace.{u2} X (PseudoMetricSpace.toUniformSpace.{u2} X (MetricSpace.toPseudoMetricSpace.{u2} X _inst_1))) NNReal.instTopologicalSpaceNNReal) X NNReal (UniformSpace.toTopologicalSpace.{u2} X (PseudoMetricSpace.toUniformSpace.{u2} X (MetricSpace.toPseudoMetricSpace.{u2} X _inst_1))) NNReal.instTopologicalSpaceNNReal (ContinuousMap.instContinuousMapClassContinuousMap.{u2, 0} X NNReal (UniformSpace.toTopologicalSpace.{u2} X (PseudoMetricSpace.toUniformSpace.{u2} X (MetricSpace.toPseudoMetricSpace.{u2} X _inst_1))) NNReal.instTopologicalSpaceNNReal)) δ x))) (U i)))))
+Case conversion may be inaccurate. Consider using '#align metric.exists_continuous_nnreal_forall_closed_ball_subset Metric.exists_continuous_nNReal_forall_closedBall_subsetₓ'. -/
 /-- Let `X` be a metric space. Let `K : ι → set X` be a locally finite family of closed sets, let
 `U : ι → set X` be a family of open sets such that `K i ⊆ U i` for all `i`. Then there exists a
 positive continuous function `δ : C(X, ℝ≥0)` such that for any `i` and `x ∈ K i`, we have
@@ -144,12 +186,18 @@ theorem exists_continuous_nNReal_forall_closedBall_subset (hK : ∀ i, IsClosed
     (hU : ∀ i, IsOpen (U i)) (hKU : ∀ i, K i ⊆ U i) (hfin : LocallyFinite K) :
     ∃ δ : C(X, ℝ≥0), (∀ x, 0 < δ x) ∧ ∀ (i), ∀ x ∈ K i, closedBall x (δ x) ⊆ U i :=
   by
-  rcases Emetric.exists_continuous_nNReal_forall_closedBall_subset hK hU hKU hfin with ⟨δ, hδ0, hδ⟩
+  rcases EMetric.exists_continuous_nNReal_forall_closedBall_subset hK hU hKU hfin with ⟨δ, hδ0, hδ⟩
   refine' ⟨δ, hδ0, fun i x hx => _⟩
   rw [← emetric_closed_ball_nnreal]
   exact hδ i x hx
 #align metric.exists_continuous_nnreal_forall_closed_ball_subset Metric.exists_continuous_nNReal_forall_closedBall_subset
 
+/- warning: metric.exists_continuous_real_forall_closed_ball_subset -> Metric.exists_continuous_real_forall_closedBall_subset is a dubious translation:
+lean 3 declaration is
+  forall {ι : Type.{u1}} {X : Type.{u2}} [_inst_1 : MetricSpace.{u2} X] {K : ι -> (Set.{u2} X)} {U : ι -> (Set.{u2} X)}, (forall (i : ι), IsClosed.{u2} X (UniformSpace.toTopologicalSpace.{u2} X (PseudoMetricSpace.toUniformSpace.{u2} X (MetricSpace.toPseudoMetricSpace.{u2} X _inst_1))) (K i)) -> (forall (i : ι), IsOpen.{u2} X (UniformSpace.toTopologicalSpace.{u2} X (PseudoMetricSpace.toUniformSpace.{u2} X (MetricSpace.toPseudoMetricSpace.{u2} X _inst_1))) (U i)) -> (forall (i : ι), HasSubset.Subset.{u2} (Set.{u2} X) (Set.hasSubset.{u2} X) (K i) (U i)) -> (LocallyFinite.{u1, u2} ι X (UniformSpace.toTopologicalSpace.{u2} X (PseudoMetricSpace.toUniformSpace.{u2} X (MetricSpace.toPseudoMetricSpace.{u2} X _inst_1))) K) -> (Exists.{succ u2} (ContinuousMap.{u2, 0} X Real (UniformSpace.toTopologicalSpace.{u2} X (PseudoMetricSpace.toUniformSpace.{u2} X (MetricSpace.toPseudoMetricSpace.{u2} X _inst_1))) (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) (fun (δ : ContinuousMap.{u2, 0} X Real (UniformSpace.toTopologicalSpace.{u2} X (PseudoMetricSpace.toUniformSpace.{u2} X (MetricSpace.toPseudoMetricSpace.{u2} X _inst_1))) (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) => And (forall (x : X), LT.lt.{0} Real Real.hasLt (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero))) (coeFn.{succ u2, succ u2} (ContinuousMap.{u2, 0} X Real (UniformSpace.toTopologicalSpace.{u2} X (PseudoMetricSpace.toUniformSpace.{u2} X (MetricSpace.toPseudoMetricSpace.{u2} X _inst_1))) (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) (fun (_x : ContinuousMap.{u2, 0} X Real (UniformSpace.toTopologicalSpace.{u2} X (PseudoMetricSpace.toUniformSpace.{u2} X (MetricSpace.toPseudoMetricSpace.{u2} X _inst_1))) (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) => X -> Real) (ContinuousMap.hasCoeToFun.{u2, 0} X Real (UniformSpace.toTopologicalSpace.{u2} X (PseudoMetricSpace.toUniformSpace.{u2} X (MetricSpace.toPseudoMetricSpace.{u2} X _inst_1))) (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) δ x)) (forall (i : ι) (x : X), (Membership.Mem.{u2, u2} X (Set.{u2} X) (Set.hasMem.{u2} X) x (K i)) -> (HasSubset.Subset.{u2} (Set.{u2} X) (Set.hasSubset.{u2} X) (Metric.closedBall.{u2} X (MetricSpace.toPseudoMetricSpace.{u2} X _inst_1) x (coeFn.{succ u2, succ u2} (ContinuousMap.{u2, 0} X Real (UniformSpace.toTopologicalSpace.{u2} X (PseudoMetricSpace.toUniformSpace.{u2} X (MetricSpace.toPseudoMetricSpace.{u2} X _inst_1))) (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) (fun (_x : ContinuousMap.{u2, 0} X Real (UniformSpace.toTopologicalSpace.{u2} X (PseudoMetricSpace.toUniformSpace.{u2} X (MetricSpace.toPseudoMetricSpace.{u2} X _inst_1))) (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) => X -> Real) (ContinuousMap.hasCoeToFun.{u2, 0} X Real (UniformSpace.toTopologicalSpace.{u2} X (PseudoMetricSpace.toUniformSpace.{u2} X (MetricSpace.toPseudoMetricSpace.{u2} X _inst_1))) (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) δ x)) (U i)))))
+but is expected to have type
+  forall {ι : Type.{u1}} {X : Type.{u2}} [_inst_1 : MetricSpace.{u2} X] {K : ι -> (Set.{u2} X)} {U : ι -> (Set.{u2} X)}, (forall (i : ι), IsClosed.{u2} X (UniformSpace.toTopologicalSpace.{u2} X (PseudoMetricSpace.toUniformSpace.{u2} X (MetricSpace.toPseudoMetricSpace.{u2} X _inst_1))) (K i)) -> (forall (i : ι), IsOpen.{u2} X (UniformSpace.toTopologicalSpace.{u2} X (PseudoMetricSpace.toUniformSpace.{u2} X (MetricSpace.toPseudoMetricSpace.{u2} X _inst_1))) (U i)) -> (forall (i : ι), HasSubset.Subset.{u2} (Set.{u2} X) (Set.instHasSubsetSet.{u2} X) (K i) (U i)) -> (LocallyFinite.{u1, u2} ι X (UniformSpace.toTopologicalSpace.{u2} X (PseudoMetricSpace.toUniformSpace.{u2} X (MetricSpace.toPseudoMetricSpace.{u2} X _inst_1))) K) -> (Exists.{succ u2} (ContinuousMap.{u2, 0} X Real (UniformSpace.toTopologicalSpace.{u2} X (PseudoMetricSpace.toUniformSpace.{u2} X (MetricSpace.toPseudoMetricSpace.{u2} X _inst_1))) (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) (fun (δ : ContinuousMap.{u2, 0} X Real (UniformSpace.toTopologicalSpace.{u2} X (PseudoMetricSpace.toUniformSpace.{u2} X (MetricSpace.toPseudoMetricSpace.{u2} X _inst_1))) (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) => And (forall (x : X), LT.lt.{0} ((fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : X) => Real) x) Real.instLTReal (OfNat.ofNat.{0} ((fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : X) => Real) x) 0 (Zero.toOfNat0.{0} ((fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : X) => Real) x) Real.instZeroReal)) (FunLike.coe.{succ u2, succ u2, 1} (ContinuousMap.{u2, 0} X Real (UniformSpace.toTopologicalSpace.{u2} X (PseudoMetricSpace.toUniformSpace.{u2} X (MetricSpace.toPseudoMetricSpace.{u2} X _inst_1))) (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) X (fun (_x : X) => (fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : X) => Real) _x) (ContinuousMapClass.toFunLike.{u2, u2, 0} (ContinuousMap.{u2, 0} X Real (UniformSpace.toTopologicalSpace.{u2} X (PseudoMetricSpace.toUniformSpace.{u2} X (MetricSpace.toPseudoMetricSpace.{u2} X _inst_1))) (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) X Real (UniformSpace.toTopologicalSpace.{u2} X (PseudoMetricSpace.toUniformSpace.{u2} X (MetricSpace.toPseudoMetricSpace.{u2} X _inst_1))) (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) (ContinuousMap.instContinuousMapClassContinuousMap.{u2, 0} X Real (UniformSpace.toTopologicalSpace.{u2} X (PseudoMetricSpace.toUniformSpace.{u2} X (MetricSpace.toPseudoMetricSpace.{u2} X _inst_1))) (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)))) δ x)) (forall (i : ι) (x : X), (Membership.mem.{u2, u2} X (Set.{u2} X) (Set.instMembershipSet.{u2} X) x (K i)) -> (HasSubset.Subset.{u2} (Set.{u2} X) (Set.instHasSubsetSet.{u2} X) (Metric.closedBall.{u2} X (MetricSpace.toPseudoMetricSpace.{u2} X _inst_1) x (FunLike.coe.{succ u2, succ u2, 1} (ContinuousMap.{u2, 0} X Real (UniformSpace.toTopologicalSpace.{u2} X (PseudoMetricSpace.toUniformSpace.{u2} X (MetricSpace.toPseudoMetricSpace.{u2} X _inst_1))) (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) X (fun (_x : X) => (fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : X) => Real) _x) (ContinuousMapClass.toFunLike.{u2, u2, 0} (ContinuousMap.{u2, 0} X Real (UniformSpace.toTopologicalSpace.{u2} X (PseudoMetricSpace.toUniformSpace.{u2} X (MetricSpace.toPseudoMetricSpace.{u2} X _inst_1))) (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) X Real (UniformSpace.toTopologicalSpace.{u2} X (PseudoMetricSpace.toUniformSpace.{u2} X (MetricSpace.toPseudoMetricSpace.{u2} X _inst_1))) (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) (ContinuousMap.instContinuousMapClassContinuousMap.{u2, 0} X Real (UniformSpace.toTopologicalSpace.{u2} X (PseudoMetricSpace.toUniformSpace.{u2} X (MetricSpace.toPseudoMetricSpace.{u2} X _inst_1))) (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)))) δ x)) (U i)))))
+Case conversion may be inaccurate. Consider using '#align metric.exists_continuous_real_forall_closed_ball_subset Metric.exists_continuous_real_forall_closedBall_subsetₓ'. -/
 /-- Let `X` be a metric space. Let `K : ι → set X` be a locally finite family of closed sets, let
 `U : ι → set X` be a family of open sets such that `K i ⊆ U i` for all `i`. Then there exists a
 positive continuous function `δ : C(X, ℝ)` such that for any `i` and `x ∈ K i`, we have

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

@@ -88,8 +88,8 @@ theorem exists_forall_closedBall_subset_aux₂ (y : X) :
   (convex_Ioi _).inter <|
     OrdConnected.convex <|
       OrdConnected.preimage_ennreal_ofReal <|
-        ordConnected_interᵢ fun i =>
-          ordConnected_interᵢ fun hi => ordConnected_setOf_closedBall_subset y (U i)
+        ordConnected_iInter fun i =>
+          ordConnected_iInter fun hi => ordConnected_setOf_closedBall_subset y (U i)
 #align emetric.exists_forall_closed_ball_subset_aux₂ Emetric.exists_forall_closedBall_subset_aux₂
 
 /-- Let `X` be an extended metric space. Let `K : ι → set X` be a locally finite family of closed

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

@@ -38,7 +38,7 @@ variable {ι X : Type _}
 
 namespace Emetric
 
-variable [EmetricSpace X] {K : ι → Set X} {U : ι → Set X}
+variable [EMetricSpace X] {K : ι → Set X} {U : ι → Set X}
 
 /-- Let `K : ι → set X` be a locally finitie family of closed sets in an emetric space. Let
 `U : ι → set X` be a family of open sets such that `K i ⊆ U i` for all `i`. Then for any point

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

@@ -63,7 +63,7 @@ theorem eventually_nhds_zero_forall_closedBall_subset (hK : ∀ i, IsClosed (K i
   apply hR
   calc
     edist z x ≤ edist z p.2 + edist p.2 x := edist_triangle _ _ _
-    _ ≤ p.1 + (R - p.1) := add_le_add hz <| le_trans hp.2 <| tsub_le_tsub_left hp.1.out.le _
+    _ ≤ p.1 + (R - p.1) := (add_le_add hz <| le_trans hp.2 <| tsub_le_tsub_left hp.1.out.le _)
     _ = R := add_tsub_cancel_of_le (lt_trans hp.1 hrR).le
     
 #align emetric.eventually_nhds_zero_forall_closed_ball_subset Emetric.eventually_nhds_zero_forall_closedBall_subset

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

@@ -30,7 +30,7 @@ metric space, partition of unity, locally finite
 -/
 
 
-open Topology Ennreal BigOperators NNReal Filter
+open Topology ENNReal BigOperators NNReal Filter
 
 open Set Function Filter TopologicalSpace
 
@@ -72,10 +72,10 @@ theorem exists_forall_closedBall_subset_aux₁ (hK : ∀ i, IsClosed (K i)) (hU
     (hKU : ∀ i, K i ⊆ U i) (hfin : LocallyFinite K) (x : X) :
     ∃ r : ℝ,
       ∀ᶠ y in 𝓝 x,
-        r ∈ Ioi (0 : ℝ) ∩ Ennreal.ofReal ⁻¹' ⋂ (i) (hi : y ∈ K i), { r | closedBall y r ⊆ U i } :=
+        r ∈ Ioi (0 : ℝ) ∩ ENNReal.ofReal ⁻¹' ⋂ (i) (hi : y ∈ K i), { r | closedBall y r ⊆ U i } :=
   by
   have :=
-    (ennreal.continuous_of_real.tendsto' 0 0 Ennreal.ofReal_zero).Eventually
+    (ennreal.continuous_of_real.tendsto' 0 0 ENNReal.ofReal_zero).Eventually
       (eventually_nhds_zero_forall_closed_ball_subset hK hU hKU hfin x).curry
   rcases this.exists_gt with ⟨r, hr0, hr⟩
   refine' ⟨r, hr.mono fun y hy => ⟨hr0, _⟩⟩
@@ -84,7 +84,7 @@ theorem exists_forall_closedBall_subset_aux₁ (hK : ∀ i, IsClosed (K i)) (hU
 
 theorem exists_forall_closedBall_subset_aux₂ (y : X) :
     Convex ℝ
-      (Ioi (0 : ℝ) ∩ Ennreal.ofReal ⁻¹' ⋂ (i) (hi : y ∈ K i), { r | closedBall y r ⊆ U i }) :=
+      (Ioi (0 : ℝ) ∩ ENNReal.ofReal ⁻¹' ⋂ (i) (hi : y ∈ K i), { r | closedBall y r ⊆ U i }) :=
   (convex_Ioi _).inter <|
     OrdConnected.convex <|
       OrdConnected.preimage_ennreal_ofReal <|
@@ -98,7 +98,7 @@ exists a positive continuous function `δ : C(X, ℝ)` such that for any `i` and
 we have `emetric.closed_ball x (ennreal.of_real (δ x)) ⊆ U i`. -/
 theorem exists_continuous_real_forall_closedBall_subset (hK : ∀ i, IsClosed (K i))
     (hU : ∀ i, IsOpen (U i)) (hKU : ∀ i, K i ⊆ U i) (hfin : LocallyFinite K) :
-    ∃ δ : C(X, ℝ), (∀ x, 0 < δ x) ∧ ∀ (i), ∀ x ∈ K i, closedBall x (Ennreal.ofReal <| δ x) ⊆ U i :=
+    ∃ δ : C(X, ℝ), (∀ x, 0 < δ x) ∧ ∀ (i), ∀ x ∈ K i, closedBall x (ENNReal.ofReal <| δ x) ⊆ U i :=
   by
   simpa only [mem_inter_iff, forall_and, mem_preimage, mem_Inter, @forall_swap ι X] using
     exists_continuous_forall_mem_convex_of_local_const exists_forall_closed_ball_subset_aux₂
@@ -116,19 +116,19 @@ theorem exists_continuous_nNReal_forall_closedBall_subset (hK : ∀ i, IsClosed
   rcases exists_continuous_real_forall_closed_ball_subset hK hU hKU hfin with ⟨δ, hδ₀, hδ⟩
   lift δ to C(X, ℝ≥0) using fun x => (hδ₀ x).le
   refine' ⟨δ, hδ₀, fun i x hi => _⟩
-  simpa only [← Ennreal.ofReal_coe_nNReal] using hδ i x hi
+  simpa only [← ENNReal.ofReal_coe_nnreal] using hδ i x hi
 #align emetric.exists_continuous_nnreal_forall_closed_ball_subset Emetric.exists_continuous_nNReal_forall_closedBall_subset
 
 /-- Let `X` be an extended metric space. Let `K : ι → set X` be a locally finite family of closed
 sets, let `U : ι → set X` be a family of open sets such that `K i ⊆ U i` for all `i`. Then there
 exists a positive continuous function `δ : C(X, ℝ≥0∞)` such that for any `i` and `x ∈ K i`,
 we have `emetric.closed_ball x (δ x) ⊆ U i`. -/
-theorem exists_continuous_ennreal_forall_closedBall_subset (hK : ∀ i, IsClosed (K i))
+theorem exists_continuous_eNNReal_forall_closedBall_subset (hK : ∀ i, IsClosed (K i))
     (hU : ∀ i, IsOpen (U i)) (hKU : ∀ i, K i ⊆ U i) (hfin : LocallyFinite K) :
     ∃ δ : C(X, ℝ≥0∞), (∀ x, 0 < δ x) ∧ ∀ (i), ∀ x ∈ K i, closedBall x (δ x) ⊆ U i :=
   let ⟨δ, hδ₀, hδ⟩ := exists_continuous_nNReal_forall_closedBall_subset hK hU hKU hfin
-  ⟨ContinuousMap.comp ⟨coe, Ennreal.continuous_coe⟩ δ, fun x => Ennreal.coe_pos.2 (hδ₀ x), hδ⟩
-#align emetric.exists_continuous_ennreal_forall_closed_ball_subset Emetric.exists_continuous_ennreal_forall_closedBall_subset
+  ⟨ContinuousMap.comp ⟨coe, ENNReal.continuous_coe⟩ δ, fun x => ENNReal.coe_pos.2 (hδ₀ x), hδ⟩
+#align emetric.exists_continuous_ennreal_forall_closed_ball_subset Emetric.exists_continuous_eNNReal_forall_closedBall_subset
 
 end Emetric
 

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

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

@@ -57,7 +57,7 @@ theorem eventually_nhds_zero_forall_closedBall_subset (hK : ∀ i, IsClosed (K i
   apply hR
   calc
     edist z x ≤ edist z p.2 + edist p.2 x := edist_triangle _ _ _
-    _ ≤ p.1 + (R - p.1) := (add_le_add hz <| le_trans hp.2 <| tsub_le_tsub_left hp.1.out.le _)
+    _ ≤ p.1 + (R - p.1) := add_le_add hz <| le_trans hp.2 <| tsub_le_tsub_left hp.1.out.le _
     _ = R := add_tsub_cancel_of_le (lt_trans (by exact hp.1) hrR).le
 #align emetric.eventually_nhds_zero_forall_closed_ball_subset EMetric.eventually_nhds_zero_forall_closedBall_subset
 

@@ -97,14 +97,14 @@ theorem exists_continuous_real_forall_closedBall_subset (hK : ∀ i, IsClosed (K
 sets, let `U : ι → Set X` be a family of open sets such that `K i ⊆ U i` for all `i`. Then there
 exists a positive continuous function `δ : C(X, ℝ≥0)` such that for any `i` and `x ∈ K i`,
 we have `EMetric.closedBall x (δ x) ⊆ U i`. -/
-theorem exists_continuous_nNReal_forall_closedBall_subset (hK : ∀ i, IsClosed (K i))
+theorem exists_continuous_nnreal_forall_closedBall_subset (hK : ∀ i, IsClosed (K i))
     (hU : ∀ i, IsOpen (U i)) (hKU : ∀ i, K i ⊆ U i) (hfin : LocallyFinite K) :
     ∃ δ : C(X, ℝ≥0), (∀ x, 0 < δ x) ∧ ∀ (i), ∀ x ∈ K i, closedBall x (δ x) ⊆ U i := by
   rcases exists_continuous_real_forall_closedBall_subset hK hU hKU hfin with ⟨δ, hδ₀, hδ⟩
   lift δ to C(X, ℝ≥0) using fun x => (hδ₀ x).le
   refine' ⟨δ, hδ₀, fun i x hi => _⟩
   simpa only [← ENNReal.ofReal_coe_nnreal] using hδ i x hi
-#align emetric.exists_continuous_nnreal_forall_closed_ball_subset EMetric.exists_continuous_nNReal_forall_closedBall_subset
+#align emetric.exists_continuous_nnreal_forall_closed_ball_subset EMetric.exists_continuous_nnreal_forall_closedBall_subset
 
 /-- Let `X` be an extended metric space. Let `K : ι → Set X` be a locally finite family of closed
 sets, let `U : ι → Set X` be a family of open sets such that `K i ⊆ U i` for all `i`. Then there
@@ -113,7 +113,7 @@ we have `EMetric.closedBall x (δ x) ⊆ U i`. -/
 theorem exists_continuous_eNNReal_forall_closedBall_subset (hK : ∀ i, IsClosed (K i))
     (hU : ∀ i, IsOpen (U i)) (hKU : ∀ i, K i ⊆ U i) (hfin : LocallyFinite K) :
     ∃ δ : C(X, ℝ≥0∞), (∀ x, 0 < δ x) ∧ ∀ (i), ∀ x ∈ K i, closedBall x (δ x) ⊆ U i :=
-  let ⟨δ, hδ₀, hδ⟩ := exists_continuous_nNReal_forall_closedBall_subset hK hU hKU hfin
+  let ⟨δ, hδ₀, hδ⟩ := exists_continuous_nnreal_forall_closedBall_subset hK hU hKU hfin
   ⟨ContinuousMap.comp ⟨Coe.coe, ENNReal.continuous_coe⟩ δ, fun x => ENNReal.coe_pos.2 (hδ₀ x), hδ⟩
 #align emetric.exists_continuous_ennreal_forall_closed_ball_subset EMetric.exists_continuous_eNNReal_forall_closedBall_subset
 
@@ -127,14 +127,14 @@ variable [MetricSpace X] {K : ι → Set X} {U : ι → Set X}
 `U : ι → Set X` be a family of open sets such that `K i ⊆ U i` for all `i`. Then there exists a
 positive continuous function `δ : C(X, ℝ≥0)` such that for any `i` and `x ∈ K i`, we have
 `Metric.closedBall x (δ x) ⊆ U i`. -/
-theorem exists_continuous_nNReal_forall_closedBall_subset (hK : ∀ i, IsClosed (K i))
+theorem exists_continuous_nnreal_forall_closedBall_subset (hK : ∀ i, IsClosed (K i))
     (hU : ∀ i, IsOpen (U i)) (hKU : ∀ i, K i ⊆ U i) (hfin : LocallyFinite K) :
     ∃ δ : C(X, ℝ≥0), (∀ x, 0 < δ x) ∧ ∀ (i), ∀ x ∈ K i, closedBall x (δ x) ⊆ U i := by
-  rcases EMetric.exists_continuous_nNReal_forall_closedBall_subset hK hU hKU hfin with ⟨δ, hδ0, hδ⟩
+  rcases EMetric.exists_continuous_nnreal_forall_closedBall_subset hK hU hKU hfin with ⟨δ, hδ0, hδ⟩
   refine' ⟨δ, hδ0, fun i x hx => _⟩
   rw [← emetric_closedBall_nnreal]
   exact hδ i x hx
-#align metric.exists_continuous_nnreal_forall_closed_ball_subset Metric.exists_continuous_nNReal_forall_closedBall_subset
+#align metric.exists_continuous_nnreal_forall_closed_ball_subset Metric.exists_continuous_nnreal_forall_closedBall_subset
 
 /-- Let `X` be a metric space. Let `K : ι → Set X` be a locally finite family of closed sets, let
 `U : ι → Set X` be a family of open sets such that `K i ⊆ U i` for all `i`. Then there exists a
@@ -143,7 +143,7 @@ positive continuous function `δ : C(X, ℝ)` such that for any `i` and `x ∈ K
 theorem exists_continuous_real_forall_closedBall_subset (hK : ∀ i, IsClosed (K i))
     (hU : ∀ i, IsOpen (U i)) (hKU : ∀ i, K i ⊆ U i) (hfin : LocallyFinite K) :
     ∃ δ : C(X, ℝ), (∀ x, 0 < δ x) ∧ ∀ (i), ∀ x ∈ K i, closedBall x (δ x) ⊆ U i :=
-  let ⟨δ, hδ₀, hδ⟩ := exists_continuous_nNReal_forall_closedBall_subset hK hU hKU hfin
+  let ⟨δ, hδ₀, hδ⟩ := exists_continuous_nnreal_forall_closedBall_subset hK hU hKU hfin
   ⟨ContinuousMap.comp ⟨Coe.coe, NNReal.continuous_coe⟩ δ, hδ₀, hδ⟩
 #align metric.exists_continuous_real_forall_closed_ball_subset Metric.exists_continuous_real_forall_closedBall_subset
 

This uses the improved shake script from #9772 to reduce imports across mathlib. The corresponding noshake.json file has been added to #9772.

Co-authored-by: Mario Carneiro <di.gama@gmail.com>

@@ -4,6 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Yury Kudryashov
 -/
 import Mathlib.Topology.EMetricSpace.Paracompact
+import Mathlib.Topology.Instances.ENNReal
 import Mathlib.Analysis.Convex.PartitionOfUnity
 
 #align_import topology.metric_space.partition_of_unity from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"

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

@@ -3,7 +3,7 @@ Copyright (c) 2022 Yury Kudryashov. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Yury Kudryashov
 -/
-import Mathlib.Topology.MetricSpace.EMetricParacompact
+import Mathlib.Topology.EMetricSpace.Paracompact
 import Mathlib.Analysis.Convex.PartitionOfUnity
 
 #align_import topology.metric_space.partition_of_unity from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"

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

This has nice performance benefits.

@@ -28,7 +28,7 @@ metric space, partition of unity, locally finite
 
 open Topology ENNReal BigOperators NNReal Filter Set Function TopologicalSpace
 
-variable {ι X : Type _}
+variable {ι X : Type*}
 
 namespace EMetric
 

• depends on: #5966

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

@@ -2,15 +2,12 @@
 Copyright (c) 2022 Yury Kudryashov. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Yury Kudryashov
-
-! This file was ported from Lean 3 source module topology.metric_space.partition_of_unity
-! 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.Topology.MetricSpace.EMetricParacompact
 import Mathlib.Analysis.Convex.PartitionOfUnity
 
+#align_import topology.metric_space.partition_of_unity from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
+
 /-!
 # Lemmas about (e)metric spaces that need partition of unity
 

@@ -66,7 +66,7 @@ theorem eventually_nhds_zero_forall_closedBall_subset (hK : ∀ i, IsClosed (K i
 theorem exists_forall_closedBall_subset_aux₁ (hK : ∀ i, IsClosed (K i)) (hU : ∀ i, IsOpen (U i))
     (hKU : ∀ i, K i ⊆ U i) (hfin : LocallyFinite K) (x : X) :
     ∃ r : ℝ, ∀ᶠ y in 𝓝 x,
-      r ∈ Ioi (0 : ℝ) ∩ ENNReal.ofReal ⁻¹' ⋂ (i) (_hi : y ∈ K i), { r | closedBall y r ⊆ U i } := by
+      r ∈ Ioi (0 : ℝ) ∩ ENNReal.ofReal ⁻¹' ⋂ (i) (_ : y ∈ K i), { r | closedBall y r ⊆ U i } := by
   have := (ENNReal.continuous_ofReal.tendsto' 0 0 ENNReal.ofReal_zero).eventually
     (eventually_nhds_zero_forall_closedBall_subset hK hU hKU hfin x).curry
   rcases this.exists_gt with ⟨r, hr0, hr⟩
@@ -76,7 +76,7 @@ theorem exists_forall_closedBall_subset_aux₁ (hK : ∀ i, IsClosed (K i)) (hU
 
 theorem exists_forall_closedBall_subset_aux₂ (y : X) :
     Convex ℝ
-      (Ioi (0 : ℝ) ∩ ENNReal.ofReal ⁻¹' ⋂ (i) (_hi : y ∈ K i), { r | closedBall y r ⊆ U i }) :=
+      (Ioi (0 : ℝ) ∩ ENNReal.ofReal ⁻¹' ⋂ (i) (_ : y ∈ K i), { r | closedBall y r ⊆ U i }) :=
   (convex_Ioi _).inter <| OrdConnected.convex <| OrdConnected.preimage_ennreal_ofReal <|
     ordConnected_iInter fun i => ordConnected_iInter fun (_ : y ∈ K i) =>
       ordConnected_setOf_closedBall_subset y (U i)

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

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

@@ -43,8 +43,8 @@ variable [EMetricSpace X] {K : ι → Set X} {U : ι → Set X}
 `y ∈ K i`, then `EMetric.closedBall y r ⊆ U i`. -/
 theorem eventually_nhds_zero_forall_closedBall_subset (hK : ∀ i, IsClosed (K i))
     (hU : ∀ i, IsOpen (U i)) (hKU : ∀ i, K i ⊆ U i) (hfin : LocallyFinite K) (x : X) :
-    ∀ᶠ p : ℝ≥0∞ × X in 𝓝 0 ×ᶠ 𝓝 x, ∀ i, p.2 ∈ K i → closedBall p.2 p.1 ⊆ U i := by
-  suffices ∀ i, x ∈ K i → ∀ᶠ p : ℝ≥0∞ × X in 𝓝 0 ×ᶠ 𝓝 x, closedBall p.2 p.1 ⊆ U i by
+    ∀ᶠ p : ℝ≥0∞ × X in 𝓝 0 ×ˢ 𝓝 x, ∀ i, p.2 ∈ K i → closedBall p.2 p.1 ⊆ U i := by
+  suffices ∀ i, x ∈ K i → ∀ᶠ p : ℝ≥0∞ × X in 𝓝 0 ×ˢ 𝓝 x, closedBall p.2 p.1 ⊆ U i by
     apply mp_mem ((eventually_all_finite (hfin.point_finite x)).2 this)
       (mp_mem (@tendsto_snd ℝ≥0∞ _ (𝓝 0) _ _ (hfin.iInter_compl_mem_nhds hK x)) _)
     apply univ_mem'

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

@@ -37,7 +37,7 @@ namespace EMetric
 
 variable [EMetricSpace X] {K : ι → Set X} {U : ι → Set X}
 
-/-- Let `K : ι → Set X` be a locally finitie family of closed sets in an emetric space. Let
+/-- Let `K : ι → Set X` be a locally finite family of closed sets in an emetric space. Let
 `U : ι → Set X` be a family of open sets such that `K i ⊆ U i` for all `i`. Then for any point
 `x : X`, for sufficiently small `r : ℝ≥0∞` and for `y` sufficiently close to `x`, for all `i`, if
 `y ∈ K i`, then `EMetric.closedBall y r ⊆ U i`. -/

Co-authored-by: Ruben Van de Velde <65514131+Ruben-VandeVelde@users.noreply.github.com> Co-authored-by: int-y1 <jason_yuen2007@hotmail.com>