topology.partition_of_unity ⟷ Mathlib.Topology.PartitionOfUnity

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

@@ -6,7 +6,7 @@ Authors: Yury Kudryashov
 import Algebra.BigOperators.Finprod
 import SetTheory.Ordinal.Basic
 import Topology.ContinuousFunction.Algebra
-import Topology.Paracompact
+import Topology.Compactness.Paracompact
 import Topology.ShrinkingLemma
 import Topology.UrysohnsLemma
 

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

@@ -298,7 +298,7 @@ protected def single (i : ι) (s : Set X) : BumpCovering ι X s
     refine' ⟨univ, univ_mem, (finite_singleton i).Subset _⟩
     rintro j ⟨x, hx, -⟩
     contrapose! hx
-    rw [mem_singleton_iff] at hx 
+    rw [mem_singleton_iff] at hx
     simp [hx]
   nonneg' := le_update_iff.2 ⟨fun x => zero_le_one, fun _ _ => le_rfl⟩
   le_one' := update_le_iff.2 ⟨le_rfl, fun _ _ _ => zero_le_one⟩
@@ -467,7 +467,7 @@ theorem toPOUFun_eq_mul_prod (i : ι) (x : X) (t : Finset ι)
     f.toPOUFun i x = f i x * ∏ j in t.filterₓ fun j => WellOrderingRel j i, (1 - f j x) :=
   by
   refine' congr_arg _ (finprod_cond_eq_prod_of_cond_iff _ fun j hj => _)
-  rw [Ne.def, sub_eq_self] at hj 
+  rw [Ne.def, sub_eq_self] at hj
   rw [Finset.mem_filter, Iff.comm, and_iff_right_iff_imp]
   exact flip (ht j) hj
 #align bump_covering.to_pou_fun_eq_mul_prod BumpCovering.toPOUFun_eq_mul_prod

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

@@ -376,7 +376,7 @@ theorem exists_isSubordinate_of_locallyFinite [NormalSpace X] (hs : IsClosed s)
   let ⟨f, _, hfU⟩ :=
     exists_isSubordinate_of_locallyFinite_of_prop (fun _ => True)
       (fun s t hs ht hd =>
-        (exists_continuous_zero_one_of_closed hs ht hd).imp fun f hf => ⟨trivial, hf⟩)
+        (exists_continuous_zero_one_of_isClosed hs ht hd).imp fun f hf => ⟨trivial, hf⟩)
       hs U ho hf hU
   ⟨f, hfU⟩
 #align bump_covering.exists_is_subordinate_of_locally_finite BumpCovering.exists_isSubordinate_of_locallyFinite

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

@@ -3,12 +3,12 @@ Copyright (c) 2021 Yury Kudryashov. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Yury Kudryashov
 -/
-import Mathbin.Algebra.BigOperators.Finprod
-import Mathbin.SetTheory.Ordinal.Basic
-import Mathbin.Topology.ContinuousFunction.Algebra
-import Mathbin.Topology.Paracompact
-import Mathbin.Topology.ShrinkingLemma
-import Mathbin.Topology.UrysohnsLemma
+import Algebra.BigOperators.Finprod
+import SetTheory.Ordinal.Basic
+import Topology.ContinuousFunction.Algebra
+import Topology.Paracompact
+import Topology.ShrinkingLemma
+import Topology.UrysohnsLemma
 
 #align_import topology.partition_of_unity from "leanprover-community/mathlib"@"a2706b55e8d7f7e9b1f93143f0b88f2e34a11eea"
 

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

@@ -2,11 +2,6 @@
 Copyright (c) 2021 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.partition_of_unity
-! leanprover-community/mathlib commit a2706b55e8d7f7e9b1f93143f0b88f2e34a11eea
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
 -/
 import Mathbin.Algebra.BigOperators.Finprod
 import Mathbin.SetTheory.Ordinal.Basic
@@ -15,6 +10,8 @@ import Mathbin.Topology.Paracompact
 import Mathbin.Topology.ShrinkingLemma
 import Mathbin.Topology.UrysohnsLemma
 
+#align_import topology.partition_of_unity from "leanprover-community/mathlib"@"a2706b55e8d7f7e9b1f93143f0b88f2e34a11eea"
+
 /-!
 # Continuous partition of unity
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/8b981918a93bc45a8600de608cde7944a80d92b9

@@ -436,7 +436,7 @@ theorem ind_apply (x : X) (hx : x ∈ s) : f (f.ind x hx) x = 1 :=
 #align bump_covering.ind_apply BumpCovering.ind_apply
 -/
 
-#print BumpCovering.toPouFun /-
+#print BumpCovering.toPOUFun /-
 /-- Partition of unity defined by a `bump_covering`. We use this auxiliary definition to prove some
 properties of the new family of functions before bundling it into a `partition_of_unity`. Do not use
 this definition, use `bump_function.to_partition_of_unity` instead.
@@ -447,37 +447,37 @@ words, `g i x = ∏ᶠ j < i, (1 - f j x) - ∏ᶠ j ≤ i, (1 - f j x)`, so
 of `1 - f j x` vanishes, and `∑ᶠ i, g i x = 1`.
 
 In order to avoid an assumption `linear_order ι`, we use `well_ordering_rel` instead of `(<)`. -/
-def toPouFun (i : ι) (x : X) : ℝ :=
+def toPOUFun (i : ι) (x : X) : ℝ :=
   f i x * ∏ᶠ (j) (hj : WellOrderingRel j i), (1 - f j x)
-#align bump_covering.to_pou_fun BumpCovering.toPouFun
+#align bump_covering.to_pou_fun BumpCovering.toPOUFun
 -/
 
-#print BumpCovering.toPouFun_zero_of_zero /-
-theorem toPouFun_zero_of_zero {i : ι} {x : X} (h : f i x = 0) : f.toPouFun i x = 0 := by
+#print BumpCovering.toPOUFun_zero_of_zero /-
+theorem toPOUFun_zero_of_zero {i : ι} {x : X} (h : f i x = 0) : f.toPOUFun i x = 0 := by
   rw [to_pou_fun, h, MulZeroClass.zero_mul]
-#align bump_covering.to_pou_fun_zero_of_zero BumpCovering.toPouFun_zero_of_zero
+#align bump_covering.to_pou_fun_zero_of_zero BumpCovering.toPOUFun_zero_of_zero
 -/
 
-#print BumpCovering.support_toPouFun_subset /-
-theorem support_toPouFun_subset (i : ι) : support (f.toPouFun i) ⊆ support (f i) := fun x =>
-  mt <| f.toPouFun_zero_of_zero
-#align bump_covering.support_to_pou_fun_subset BumpCovering.support_toPouFun_subset
+#print BumpCovering.support_toPOUFun_subset /-
+theorem support_toPOUFun_subset (i : ι) : support (f.toPOUFun i) ⊆ support (f i) := fun x =>
+  mt <| f.toPOUFun_zero_of_zero
+#align bump_covering.support_to_pou_fun_subset BumpCovering.support_toPOUFun_subset
 -/
 
-#print BumpCovering.toPouFun_eq_mul_prod /-
-theorem toPouFun_eq_mul_prod (i : ι) (x : X) (t : Finset ι)
+#print BumpCovering.toPOUFun_eq_mul_prod /-
+theorem toPOUFun_eq_mul_prod (i : ι) (x : X) (t : Finset ι)
     (ht : ∀ j, WellOrderingRel j i → f j x ≠ 0 → j ∈ t) :
-    f.toPouFun i x = f i x * ∏ j in t.filterₓ fun j => WellOrderingRel j i, (1 - f j x) :=
+    f.toPOUFun i x = f i x * ∏ j in t.filterₓ fun j => WellOrderingRel j i, (1 - f j x) :=
   by
   refine' congr_arg _ (finprod_cond_eq_prod_of_cond_iff _ fun j hj => _)
   rw [Ne.def, sub_eq_self] at hj 
   rw [Finset.mem_filter, Iff.comm, and_iff_right_iff_imp]
   exact flip (ht j) hj
-#align bump_covering.to_pou_fun_eq_mul_prod BumpCovering.toPouFun_eq_mul_prod
+#align bump_covering.to_pou_fun_eq_mul_prod BumpCovering.toPOUFun_eq_mul_prod
 -/
 
-#print BumpCovering.sum_toPouFun_eq /-
-theorem sum_toPouFun_eq (x : X) : ∑ᶠ i, f.toPouFun i x = 1 - ∏ᶠ i, (1 - f i x) :=
+#print BumpCovering.sum_toPOUFun_eq /-
+theorem sum_toPOUFun_eq (x : X) : ∑ᶠ i, f.toPOUFun i x = 1 - ∏ᶠ i, (1 - f i x) :=
   by
   set s := (f.point_finite x).toFinset
   have hs : (s : Set ι) = {i | f i x ≠ 0} := finite.coe_to_finset _
@@ -493,13 +493,13 @@ theorem sum_toPouFun_eq (x : X) : ∑ᶠ i, f.toPouFun i x = 1 - ∏ᶠ i, (1 -
   refine' Finset.sum_congr rfl fun i hi => _
   convert f.to_pou_fun_eq_mul_prod _ _ _ fun j hji hj => _
   rwa [finite.mem_to_finset]
-#align bump_covering.sum_to_pou_fun_eq BumpCovering.sum_toPouFun_eq
+#align bump_covering.sum_to_pou_fun_eq BumpCovering.sum_toPOUFun_eq
 -/
 
-#print BumpCovering.exists_finset_toPouFun_eventuallyEq /-
-theorem exists_finset_toPouFun_eventuallyEq (i : ι) (x : X) :
+#print BumpCovering.exists_finset_toPOUFun_eventuallyEq /-
+theorem exists_finset_toPOUFun_eventuallyEq (i : ι) (x : X) :
     ∃ t : Finset ι,
-      f.toPouFun i =ᶠ[𝓝 x] f i * ∏ j in t.filterₓ fun j => WellOrderingRel j i, (1 - f j) :=
+      f.toPOUFun i =ᶠ[𝓝 x] f i * ∏ j in t.filterₓ fun j => WellOrderingRel j i, (1 - f j) :=
   by
   rcases f.locally_finite x with ⟨U, hU, hf⟩
   use hf.to_finset
@@ -508,18 +508,18 @@ theorem exists_finset_toPouFun_eventuallyEq (i : ι) (x : X) :
   apply to_pou_fun_eq_mul_prod
   intro j hji hj
   exact hf.mem_to_finset.2 ⟨y, ⟨hj, hyU⟩⟩
-#align bump_covering.exists_finset_to_pou_fun_eventually_eq BumpCovering.exists_finset_toPouFun_eventuallyEq
+#align bump_covering.exists_finset_to_pou_fun_eventually_eq BumpCovering.exists_finset_toPOUFun_eventuallyEq
 -/
 
-#print BumpCovering.continuous_toPouFun /-
-theorem continuous_toPouFun (i : ι) : Continuous (f.toPouFun i) :=
+#print BumpCovering.continuous_toPOUFun /-
+theorem continuous_toPOUFun (i : ι) : Continuous (f.toPOUFun i) :=
   by
   refine'
     (f i).Continuous.mul <|
       continuous_finprod_cond (fun j _ => continuous_const.sub (f j).Continuous) _
   simp only [mul_support_one_sub]
   exact f.locally_finite
-#align bump_covering.continuous_to_pou_fun BumpCovering.continuous_toPouFun
+#align bump_covering.continuous_to_pou_fun BumpCovering.continuous_toPOUFun
 -/
 
 #print BumpCovering.toPartitionOfUnity /-
@@ -533,8 +533,8 @@ of `1 - f j x` vanishes, and `∑ᶠ i, g i x = 1`.
 In order to avoid an assumption `linear_order ι`, we use `well_ordering_rel` instead of `(<)`. -/
 def toPartitionOfUnity : PartitionOfUnity ι X s
     where
-  toFun i := ⟨f.toPouFun i, f.continuous_toPouFun i⟩
-  locally_finite' := f.LocallyFinite.Subset f.support_toPouFun_subset
+  toFun i := ⟨f.toPOUFun i, f.continuous_toPOUFun i⟩
+  locally_finite' := f.LocallyFinite.Subset f.support_toPOUFun_subset
   nonneg' i x :=
     mul_nonneg (f.NonNeg i x) (finprod_cond_nonneg fun j hj => sub_nonneg.2 <| f.le_one j x)
   sum_eq_one' x hx :=
@@ -561,7 +561,7 @@ theorem toPartitionOfUnity_apply (i : ι) (x : X) :
 theorem toPartitionOfUnity_eq_mul_prod (i : ι) (x : X) (t : Finset ι)
     (ht : ∀ j, WellOrderingRel j i → f j x ≠ 0 → j ∈ t) :
     f.toPartitionOfUnity i x = f i x * ∏ j in t.filterₓ fun j => WellOrderingRel j i, (1 - f j x) :=
-  f.toPouFun_eq_mul_prod i x t ht
+  f.toPOUFun_eq_mul_prod i x t ht
 #align bump_covering.to_partition_of_unity_eq_mul_prod BumpCovering.toPartitionOfUnity_eq_mul_prod
 -/
 
@@ -570,28 +570,28 @@ theorem exists_finset_toPartitionOfUnity_eventuallyEq (i : ι) (x : X) :
     ∃ t : Finset ι,
       f.toPartitionOfUnity i =ᶠ[𝓝 x]
         f i * ∏ j in t.filterₓ fun j => WellOrderingRel j i, (1 - f j) :=
-  f.exists_finset_toPouFun_eventuallyEq i x
+  f.exists_finset_toPOUFun_eventuallyEq i x
 #align bump_covering.exists_finset_to_partition_of_unity_eventually_eq BumpCovering.exists_finset_toPartitionOfUnity_eventuallyEq
 -/
 
 #print BumpCovering.toPartitionOfUnity_zero_of_zero /-
 theorem toPartitionOfUnity_zero_of_zero {i : ι} {x : X} (h : f i x = 0) :
     f.toPartitionOfUnity i x = 0 :=
-  f.toPouFun_zero_of_zero h
+  f.toPOUFun_zero_of_zero h
 #align bump_covering.to_partition_of_unity_zero_of_zero BumpCovering.toPartitionOfUnity_zero_of_zero
 -/
 
 #print BumpCovering.support_toPartitionOfUnity_subset /-
 theorem support_toPartitionOfUnity_subset (i : ι) :
     support (f.toPartitionOfUnity i) ⊆ support (f i) :=
-  f.support_toPouFun_subset i
+  f.support_toPOUFun_subset i
 #align bump_covering.support_to_partition_of_unity_subset BumpCovering.support_toPartitionOfUnity_subset
 -/
 
 #print BumpCovering.sum_toPartitionOfUnity_eq /-
 theorem sum_toPartitionOfUnity_eq (x : X) :
     ∑ᶠ i, f.toPartitionOfUnity i x = 1 - ∏ᶠ i, (1 - f i x) :=
-  f.sum_toPouFun_eq x
+  f.sum_toPOUFun_eq x
 #align bump_covering.sum_to_partition_of_unity_eq BumpCovering.sum_toPartitionOfUnity_eq
 -/
 

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

@@ -147,22 +147,31 @@ variable {E : Type _} [AddCommMonoid E] [SMulWithZero ℝ E] [TopologicalSpace E
 instance : CoeFun (PartitionOfUnity ι X s) fun _ => ι → C(X, ℝ) :=
   ⟨toFun⟩
 
+#print PartitionOfUnity.locallyFinite /-
 protected theorem locallyFinite : LocallyFinite fun i => support (f i) :=
   f.locally_finite'
 #align partition_of_unity.locally_finite PartitionOfUnity.locallyFinite
+-/
 
+#print PartitionOfUnity.locallyFinite_tsupport /-
 theorem locallyFinite_tsupport : LocallyFinite fun i => tsupport (f i) :=
   f.LocallyFinite.closure
 #align partition_of_unity.locally_finite_tsupport PartitionOfUnity.locallyFinite_tsupport
+-/
 
+#print PartitionOfUnity.nonneg /-
 theorem nonneg (i : ι) (x : X) : 0 ≤ f i x :=
   f.nonneg' i x
 #align partition_of_unity.nonneg PartitionOfUnity.nonneg
+-/
 
+#print PartitionOfUnity.sum_eq_one /-
 theorem sum_eq_one {x : X} (hx : x ∈ s) : ∑ᶠ i, f i x = 1 :=
   f.sum_eq_one' x hx
 #align partition_of_unity.sum_eq_one PartitionOfUnity.sum_eq_one
+-/
 
+#print PartitionOfUnity.exists_pos /-
 /-- If `f` is a partition of unity on `s`, then for every `x ∈ s` there exists an index `i` such
 that `0 < f i x`. -/
 theorem exists_pos {x : X} (hx : x ∈ s) : ∃ i, 0 < f i x :=
@@ -171,19 +180,27 @@ theorem exists_pos {x : X} (hx : x ∈ s) : ∃ i, 0 < f i x :=
   contrapose! H
   simpa only [fun i => (H i).antisymm (f.nonneg i x), finsum_zero] using zero_ne_one
 #align partition_of_unity.exists_pos PartitionOfUnity.exists_pos
+-/
 
+#print PartitionOfUnity.sum_le_one /-
 theorem sum_le_one (x : X) : ∑ᶠ i, f i x ≤ 1 :=
   f.sum_le_one' x
 #align partition_of_unity.sum_le_one PartitionOfUnity.sum_le_one
+-/
 
+#print PartitionOfUnity.sum_nonneg /-
 theorem sum_nonneg (x : X) : 0 ≤ ∑ᶠ i, f i x :=
   finsum_nonneg fun i => f.NonNeg i x
 #align partition_of_unity.sum_nonneg PartitionOfUnity.sum_nonneg
+-/
 
+#print PartitionOfUnity.le_one /-
 theorem le_one (i : ι) (x : X) : f i x ≤ 1 :=
   (single_le_finsum i (f.LocallyFinite.point_finite x) fun j => f.NonNeg j x).trans (f.sum_le_one x)
 #align partition_of_unity.le_one PartitionOfUnity.le_one
+-/
 
+#print PartitionOfUnity.continuous_smul /-
 /-- If `f` is a partition of unity on `s : set X` and `g : X → E` is continuous at every point of
 the topological support of some `f i`, then `λ x, f i x • g x` is continuous on the whole space. -/
 theorem continuous_smul {g : X → E} {i : ι} (hg : ∀ x ∈ tsupport (f i), ContinuousAt g x) :
@@ -191,7 +208,9 @@ theorem continuous_smul {g : X → E} {i : ι} (hg : ∀ x ∈ tsupport (f i), C
   continuous_of_tsupport fun x hx =>
     ((f i).ContinuousAt x).smul <| hg x <| tsupport_smul_subset_left _ _ hx
 #align partition_of_unity.continuous_smul PartitionOfUnity.continuous_smul
+-/
 
+#print PartitionOfUnity.continuous_finsum_smul /-
 /-- If `f` is a partition of unity on a set `s : set X` and `g : ι → X → E` is a family of functions
 such that each `g i` is continuous at every point of the topological support of `f i`, then the sum
 `λ x, ∑ᶠ i, f i x • g i x` is continuous on the whole space. -/
@@ -201,6 +220,7 @@ theorem continuous_finsum_smul [ContinuousAdd E] {g : ι → X → E}
   (continuous_finsum fun i => f.continuous_smul (hg i)) <|
     f.LocallyFinite.Subset fun i => support_smul_subset_left _ _
 #align partition_of_unity.continuous_finsum_smul PartitionOfUnity.continuous_finsum_smul
+-/
 
 #print PartitionOfUnity.IsSubordinate /-
 /-- A partition of unity `f i` is subordinate to a family of sets `U i` indexed by the same type if
@@ -212,13 +232,16 @@ def IsSubordinate (U : ι → Set X) : Prop :=
 
 variable {f}
 
+#print PartitionOfUnity.exists_finset_nhd_support_subset /-
 theorem exists_finset_nhd_support_subset {U : ι → Set X} (hso : f.IsSubordinate U)
     (ho : ∀ i, IsOpen (U i)) (x : X) :
     ∃ (is : Finset ι) (n : Set X) (hn₁ : n ∈ 𝓝 x) (hn₂ : n ⊆ ⋂ i ∈ is, U i),
       ∀ z ∈ n, (support fun i => f i z) ⊆ is :=
   f.LocallyFinite.exists_finset_nhd_support_subset hso ho x
 #align partition_of_unity.exists_finset_nhd_support_subset PartitionOfUnity.exists_finset_nhd_support_subset
+-/
 
+#print PartitionOfUnity.IsSubordinate.continuous_finsum_smul /-
 /-- If `f` is a partition of unity that is subordinate to a family of open sets `U i` and
 `g : ι → X → E` is a family of functions such that each `g i` is continuous on `U i`, then the sum
 `λ x, ∑ᶠ i, f i x • g i x` is a continuous function. -/
@@ -227,6 +250,7 @@ theorem IsSubordinate.continuous_finsum_smul [ContinuousAdd E] {U : ι → Set X
     (hg : ∀ i, ContinuousOn (g i) (U i)) : Continuous fun x => ∑ᶠ i, f i x • g i x :=
   f.continuous_finsum_smul fun i x hx => (hg i).ContinuousAt <| (ho i).mem_nhds <| hf i hx
 #align partition_of_unity.is_subordinate.continuous_finsum_smul PartitionOfUnity.IsSubordinate.continuous_finsum_smul
+-/
 
 end PartitionOfUnity
 
@@ -237,25 +261,35 @@ variable {s : Set X} (f : BumpCovering ι X s)
 instance : CoeFun (BumpCovering ι X s) fun _ => ι → C(X, ℝ) :=
   ⟨toFun⟩
 
+#print BumpCovering.locallyFinite /-
 protected theorem locallyFinite : LocallyFinite fun i => support (f i) :=
   f.locally_finite'
 #align bump_covering.locally_finite BumpCovering.locallyFinite
+-/
 
+#print BumpCovering.locallyFinite_tsupport /-
 theorem locallyFinite_tsupport : LocallyFinite fun i => tsupport (f i) :=
   f.LocallyFinite.closure
 #align bump_covering.locally_finite_tsupport BumpCovering.locallyFinite_tsupport
+-/
 
+#print BumpCovering.point_finite /-
 protected theorem point_finite (x : X) : {i | f i x ≠ 0}.Finite :=
   f.LocallyFinite.point_finite x
 #align bump_covering.point_finite BumpCovering.point_finite
+-/
 
+#print BumpCovering.nonneg /-
 theorem nonneg (i : ι) (x : X) : 0 ≤ f i x :=
   f.nonneg' i x
 #align bump_covering.nonneg BumpCovering.nonneg
+-/
 
+#print BumpCovering.le_one /-
 theorem le_one (i : ι) (x : X) : f i x ≤ 1 :=
   f.le_one' i x
 #align bump_covering.le_one BumpCovering.le_one
+-/
 
 #print BumpCovering.single /-
 /-- A `bump_covering` that consists of a single function, uniformly equal to one, defined as an
@@ -275,10 +309,12 @@ protected def single (i : ι) (s : Set X) : BumpCovering ι X s
 #align bump_covering.single BumpCovering.single
 -/
 
+#print BumpCovering.coe_single /-
 @[simp]
 theorem coe_single (i : ι) (s : Set X) : ⇑(BumpCovering.single i s) = Pi.single i 1 :=
   rfl
 #align bump_covering.coe_single BumpCovering.coe_single
+-/
 
 instance [Inhabited ι] : Inhabited (BumpCovering ι X s) :=
   ⟨BumpCovering.single default s⟩
@@ -297,6 +333,7 @@ theorem IsSubordinate.mono {f : BumpCovering ι X s} {U V : ι → Set X} (hU :
 #align bump_covering.is_subordinate.mono BumpCovering.IsSubordinate.mono
 -/
 
+#print BumpCovering.exists_isSubordinate_of_locallyFinite_of_prop /-
 /-- If `X` is a normal topological space and `U i`, `i : ι`, is a locally finite open covering of a
 closed set `s`, then there exists a `bump_covering ι X s` that is subordinate to `U`. If `X` is a
 paracompact space, then the assumption `hf : locally_finite U` can be omitted, see
@@ -329,6 +366,7 @@ theorem exists_isSubordinate_of_locallyFinite_of_prop [NormalSpace X] (p : (X 
   rcases mem_Union.1 (hsW hx) with ⟨i, hi⟩
   exact ⟨i, ((hf1 i).mono subset_closure).eventuallyEq_of_mem ((hWo i).mem_nhds hi)⟩
 #align bump_covering.exists_is_subordinate_of_locally_finite_of_prop BumpCovering.exists_isSubordinate_of_locallyFinite_of_prop
+-/
 
 #print BumpCovering.exists_isSubordinate_of_locallyFinite /-
 /-- If `X` is a normal topological space and `U i`, `i : ι`, is a locally finite open covering of a
@@ -347,6 +385,7 @@ theorem exists_isSubordinate_of_locallyFinite [NormalSpace X] (hs : IsClosed s)
 #align bump_covering.exists_is_subordinate_of_locally_finite BumpCovering.exists_isSubordinate_of_locallyFinite
 -/
 
+#print BumpCovering.exists_isSubordinate_of_prop /-
 /-- If `X` is a paracompact normal topological space and `U` is an open covering of a closed set
 `s`, then there exists a `bump_covering ι X s` that is subordinate to `U`. This version assumes that
 `p : (X → ℝ) → Prop` is a predicate that satisfies Urysohn's lemma, and provides a
@@ -364,6 +403,7 @@ theorem exists_isSubordinate_of_prop [NormalSpace X] [ParacompactSpace X] (p : (
   rcases exists_is_subordinate_of_locally_finite_of_prop p h01 hs V hVo hVf hsV with ⟨f, hfp, hf⟩
   exact ⟨f, hfp, hf.mono hVU⟩
 #align bump_covering.exists_is_subordinate_of_prop BumpCovering.exists_isSubordinate_of_prop
+-/
 
 #print BumpCovering.exists_isSubordinate /-
 /-- If `X` is a paracompact normal topological space and `U` is an open covering of a closed set
@@ -384,13 +424,17 @@ def ind (x : X) (hx : x ∈ s) : ι :=
 #align bump_covering.ind BumpCovering.ind
 -/
 
+#print BumpCovering.eventuallyEq_one /-
 theorem eventuallyEq_one (x : X) (hx : x ∈ s) : f (f.ind x hx) =ᶠ[𝓝 x] 1 :=
   (f.eventuallyEq_one' x hx).choose_spec
 #align bump_covering.eventually_eq_one BumpCovering.eventuallyEq_one
+-/
 
+#print BumpCovering.ind_apply /-
 theorem ind_apply (x : X) (hx : x ∈ s) : f (f.ind x hx) x = 1 :=
   (f.eventuallyEq_one x hx).eq_of_nhds
 #align bump_covering.ind_apply BumpCovering.ind_apply
+-/
 
 #print BumpCovering.toPouFun /-
 /-- Partition of unity defined by a `bump_covering`. We use this auxiliary definition to prove some
@@ -408,14 +452,19 @@ def toPouFun (i : ι) (x : X) : ℝ :=
 #align bump_covering.to_pou_fun BumpCovering.toPouFun
 -/
 
+#print BumpCovering.toPouFun_zero_of_zero /-
 theorem toPouFun_zero_of_zero {i : ι} {x : X} (h : f i x = 0) : f.toPouFun i x = 0 := by
   rw [to_pou_fun, h, MulZeroClass.zero_mul]
 #align bump_covering.to_pou_fun_zero_of_zero BumpCovering.toPouFun_zero_of_zero
+-/
 
+#print BumpCovering.support_toPouFun_subset /-
 theorem support_toPouFun_subset (i : ι) : support (f.toPouFun i) ⊆ support (f i) := fun x =>
   mt <| f.toPouFun_zero_of_zero
 #align bump_covering.support_to_pou_fun_subset BumpCovering.support_toPouFun_subset
+-/
 
+#print BumpCovering.toPouFun_eq_mul_prod /-
 theorem toPouFun_eq_mul_prod (i : ι) (x : X) (t : Finset ι)
     (ht : ∀ j, WellOrderingRel j i → f j x ≠ 0 → j ∈ t) :
     f.toPouFun i x = f i x * ∏ j in t.filterₓ fun j => WellOrderingRel j i, (1 - f j x) :=
@@ -425,7 +474,9 @@ theorem toPouFun_eq_mul_prod (i : ι) (x : X) (t : Finset ι)
   rw [Finset.mem_filter, Iff.comm, and_iff_right_iff_imp]
   exact flip (ht j) hj
 #align bump_covering.to_pou_fun_eq_mul_prod BumpCovering.toPouFun_eq_mul_prod
+-/
 
+#print BumpCovering.sum_toPouFun_eq /-
 theorem sum_toPouFun_eq (x : X) : ∑ᶠ i, f.toPouFun i x = 1 - ∏ᶠ i, (1 - f i x) :=
   by
   set s := (f.point_finite x).toFinset
@@ -443,7 +494,9 @@ theorem sum_toPouFun_eq (x : X) : ∑ᶠ i, f.toPouFun i x = 1 - ∏ᶠ i, (1 -
   convert f.to_pou_fun_eq_mul_prod _ _ _ fun j hji hj => _
   rwa [finite.mem_to_finset]
 #align bump_covering.sum_to_pou_fun_eq BumpCovering.sum_toPouFun_eq
+-/
 
+#print BumpCovering.exists_finset_toPouFun_eventuallyEq /-
 theorem exists_finset_toPouFun_eventuallyEq (i : ι) (x : X) :
     ∃ t : Finset ι,
       f.toPouFun i =ᶠ[𝓝 x] f i * ∏ j in t.filterₓ fun j => WellOrderingRel j i, (1 - f j) :=
@@ -456,6 +509,7 @@ theorem exists_finset_toPouFun_eventuallyEq (i : ι) (x : X) :
   intro j hji hj
   exact hf.mem_to_finset.2 ⟨y, ⟨hj, hyU⟩⟩
 #align bump_covering.exists_finset_to_pou_fun_eventually_eq BumpCovering.exists_finset_toPouFun_eventuallyEq
+-/
 
 #print BumpCovering.continuous_toPouFun /-
 theorem continuous_toPouFun (i : ι) : Continuous (f.toPouFun i) :=
@@ -496,38 +550,50 @@ def toPartitionOfUnity : PartitionOfUnity ι X s
 #align bump_covering.to_partition_of_unity BumpCovering.toPartitionOfUnity
 -/
 
+#print BumpCovering.toPartitionOfUnity_apply /-
 theorem toPartitionOfUnity_apply (i : ι) (x : X) :
     f.toPartitionOfUnity i x = f i x * ∏ᶠ (j) (hj : WellOrderingRel j i), (1 - f j x) :=
   rfl
 #align bump_covering.to_partition_of_unity_apply BumpCovering.toPartitionOfUnity_apply
+-/
 
+#print BumpCovering.toPartitionOfUnity_eq_mul_prod /-
 theorem toPartitionOfUnity_eq_mul_prod (i : ι) (x : X) (t : Finset ι)
     (ht : ∀ j, WellOrderingRel j i → f j x ≠ 0 → j ∈ t) :
     f.toPartitionOfUnity i x = f i x * ∏ j in t.filterₓ fun j => WellOrderingRel j i, (1 - f j x) :=
   f.toPouFun_eq_mul_prod i x t ht
 #align bump_covering.to_partition_of_unity_eq_mul_prod BumpCovering.toPartitionOfUnity_eq_mul_prod
+-/
 
+#print BumpCovering.exists_finset_toPartitionOfUnity_eventuallyEq /-
 theorem exists_finset_toPartitionOfUnity_eventuallyEq (i : ι) (x : X) :
     ∃ t : Finset ι,
       f.toPartitionOfUnity i =ᶠ[𝓝 x]
         f i * ∏ j in t.filterₓ fun j => WellOrderingRel j i, (1 - f j) :=
   f.exists_finset_toPouFun_eventuallyEq i x
 #align bump_covering.exists_finset_to_partition_of_unity_eventually_eq BumpCovering.exists_finset_toPartitionOfUnity_eventuallyEq
+-/
 
+#print BumpCovering.toPartitionOfUnity_zero_of_zero /-
 theorem toPartitionOfUnity_zero_of_zero {i : ι} {x : X} (h : f i x = 0) :
     f.toPartitionOfUnity i x = 0 :=
   f.toPouFun_zero_of_zero h
 #align bump_covering.to_partition_of_unity_zero_of_zero BumpCovering.toPartitionOfUnity_zero_of_zero
+-/
 
+#print BumpCovering.support_toPartitionOfUnity_subset /-
 theorem support_toPartitionOfUnity_subset (i : ι) :
     support (f.toPartitionOfUnity i) ⊆ support (f i) :=
   f.support_toPouFun_subset i
 #align bump_covering.support_to_partition_of_unity_subset BumpCovering.support_toPartitionOfUnity_subset
+-/
 
+#print BumpCovering.sum_toPartitionOfUnity_eq /-
 theorem sum_toPartitionOfUnity_eq (x : X) :
     ∑ᶠ i, f.toPartitionOfUnity i x = 1 - ∏ᶠ i, (1 - f i x) :=
   f.sum_toPouFun_eq x
 #align bump_covering.sum_to_partition_of_unity_eq BumpCovering.sum_toPartitionOfUnity_eq
+-/
 
 #print BumpCovering.IsSubordinate.toPartitionOfUnity /-
 theorem IsSubordinate.toPartitionOfUnity {f : BumpCovering ι X s} {U : ι → Set X}

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

@@ -107,8 +107,8 @@ structure PartitionOfUnity (ι X : Type _) [TopologicalSpace X] (s : Set X := un
   toFun : ι → C(X, ℝ)
   locally_finite' : LocallyFinite fun i => support (to_fun i)
   nonneg' : 0 ≤ to_fun
-  sum_eq_one' : ∀ x ∈ s, (∑ᶠ i, to_fun i x) = 1
-  sum_le_one' : ∀ x, (∑ᶠ i, to_fun i x) ≤ 1
+  sum_eq_one' : ∀ x ∈ s, ∑ᶠ i, to_fun i x = 1
+  sum_le_one' : ∀ x, ∑ᶠ i, to_fun i x ≤ 1
 #align partition_of_unity PartitionOfUnity
 -/
 
@@ -159,7 +159,7 @@ theorem nonneg (i : ι) (x : X) : 0 ≤ f i x :=
   f.nonneg' i x
 #align partition_of_unity.nonneg PartitionOfUnity.nonneg
 
-theorem sum_eq_one {x : X} (hx : x ∈ s) : (∑ᶠ i, f i x) = 1 :=
+theorem sum_eq_one {x : X} (hx : x ∈ s) : ∑ᶠ i, f i x = 1 :=
   f.sum_eq_one' x hx
 #align partition_of_unity.sum_eq_one PartitionOfUnity.sum_eq_one
 
@@ -172,7 +172,7 @@ theorem exists_pos {x : X} (hx : x ∈ s) : ∃ i, 0 < f i x :=
   simpa only [fun i => (H i).antisymm (f.nonneg i x), finsum_zero] using zero_ne_one
 #align partition_of_unity.exists_pos PartitionOfUnity.exists_pos
 
-theorem sum_le_one (x : X) : (∑ᶠ i, f i x) ≤ 1 :=
+theorem sum_le_one (x : X) : ∑ᶠ i, f i x ≤ 1 :=
   f.sum_le_one' x
 #align partition_of_unity.sum_le_one PartitionOfUnity.sum_le_one
 
@@ -404,7 +404,7 @@ of `1 - f j x` vanishes, and `∑ᶠ i, g i x = 1`.
 
 In order to avoid an assumption `linear_order ι`, we use `well_ordering_rel` instead of `(<)`. -/
 def toPouFun (i : ι) (x : X) : ℝ :=
-  f i x * ∏ᶠ (j) (hj : WellOrderingRel j i), 1 - f j x
+  f i x * ∏ᶠ (j) (hj : WellOrderingRel j i), (1 - f j x)
 #align bump_covering.to_pou_fun BumpCovering.toPouFun
 -/
 
@@ -418,7 +418,7 @@ theorem support_toPouFun_subset (i : ι) : support (f.toPouFun i) ⊆ support (f
 
 theorem toPouFun_eq_mul_prod (i : ι) (x : X) (t : Finset ι)
     (ht : ∀ j, WellOrderingRel j i → f j x ≠ 0 → j ∈ t) :
-    f.toPouFun i x = f i x * ∏ j in t.filterₓ fun j => WellOrderingRel j i, 1 - f j x :=
+    f.toPouFun i x = f i x * ∏ j in t.filterₓ fun j => WellOrderingRel j i, (1 - f j x) :=
   by
   refine' congr_arg _ (finprod_cond_eq_prod_of_cond_iff _ fun j hj => _)
   rw [Ne.def, sub_eq_self] at hj 
@@ -426,7 +426,7 @@ theorem toPouFun_eq_mul_prod (i : ι) (x : X) (t : Finset ι)
   exact flip (ht j) hj
 #align bump_covering.to_pou_fun_eq_mul_prod BumpCovering.toPouFun_eq_mul_prod
 
-theorem sum_toPouFun_eq (x : X) : (∑ᶠ i, f.toPouFun i x) = 1 - ∏ᶠ i, 1 - f i x :=
+theorem sum_toPouFun_eq (x : X) : ∑ᶠ i, f.toPouFun i x = 1 - ∏ᶠ i, (1 - f i x) :=
   by
   set s := (f.point_finite x).toFinset
   have hs : (s : Set ι) = {i | f i x ≠ 0} := finite.coe_to_finset _
@@ -446,7 +446,7 @@ theorem sum_toPouFun_eq (x : X) : (∑ᶠ i, f.toPouFun i x) = 1 - ∏ᶠ i, 1 -
 
 theorem exists_finset_toPouFun_eventuallyEq (i : ι) (x : X) :
     ∃ t : Finset ι,
-      f.toPouFun i =ᶠ[𝓝 x] f i * ∏ j in t.filterₓ fun j => WellOrderingRel j i, 1 - f j :=
+      f.toPouFun i =ᶠ[𝓝 x] f i * ∏ j in t.filterₓ fun j => WellOrderingRel j i, (1 - f j) :=
   by
   rcases f.locally_finite x with ⟨U, hU, hf⟩
   use hf.to_finset
@@ -497,19 +497,20 @@ def toPartitionOfUnity : PartitionOfUnity ι X s
 -/
 
 theorem toPartitionOfUnity_apply (i : ι) (x : X) :
-    f.toPartitionOfUnity i x = f i x * ∏ᶠ (j) (hj : WellOrderingRel j i), 1 - f j x :=
+    f.toPartitionOfUnity i x = f i x * ∏ᶠ (j) (hj : WellOrderingRel j i), (1 - f j x) :=
   rfl
 #align bump_covering.to_partition_of_unity_apply BumpCovering.toPartitionOfUnity_apply
 
 theorem toPartitionOfUnity_eq_mul_prod (i : ι) (x : X) (t : Finset ι)
     (ht : ∀ j, WellOrderingRel j i → f j x ≠ 0 → j ∈ t) :
-    f.toPartitionOfUnity i x = f i x * ∏ j in t.filterₓ fun j => WellOrderingRel j i, 1 - f j x :=
+    f.toPartitionOfUnity i x = f i x * ∏ j in t.filterₓ fun j => WellOrderingRel j i, (1 - f j x) :=
   f.toPouFun_eq_mul_prod i x t ht
 #align bump_covering.to_partition_of_unity_eq_mul_prod BumpCovering.toPartitionOfUnity_eq_mul_prod
 
 theorem exists_finset_toPartitionOfUnity_eventuallyEq (i : ι) (x : X) :
     ∃ t : Finset ι,
-      f.toPartitionOfUnity i =ᶠ[𝓝 x] f i * ∏ j in t.filterₓ fun j => WellOrderingRel j i, 1 - f j :=
+      f.toPartitionOfUnity i =ᶠ[𝓝 x]
+        f i * ∏ j in t.filterₓ fun j => WellOrderingRel j i, (1 - f j) :=
   f.exists_finset_toPouFun_eventuallyEq i x
 #align bump_covering.exists_finset_to_partition_of_unity_eventually_eq BumpCovering.exists_finset_toPartitionOfUnity_eventuallyEq
 
@@ -524,7 +525,7 @@ theorem support_toPartitionOfUnity_subset (i : ι) :
 #align bump_covering.support_to_partition_of_unity_subset BumpCovering.support_toPartitionOfUnity_subset
 
 theorem sum_toPartitionOfUnity_eq (x : X) :
-    (∑ᶠ i, f.toPartitionOfUnity i x) = 1 - ∏ᶠ i, 1 - f i x :=
+    ∑ᶠ i, f.toPartitionOfUnity i x = 1 - ∏ᶠ i, (1 - f i x) :=
   f.sum_toPouFun_eq x
 #align bump_covering.sum_to_partition_of_unity_eq BumpCovering.sum_toPartitionOfUnity_eq
 

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

@@ -245,7 +245,7 @@ theorem locallyFinite_tsupport : LocallyFinite fun i => tsupport (f i) :=
   f.LocallyFinite.closure
 #align bump_covering.locally_finite_tsupport BumpCovering.locallyFinite_tsupport
 
-protected theorem point_finite (x : X) : { i | f i x ≠ 0 }.Finite :=
+protected theorem point_finite (x : X) : {i | f i x ≠ 0}.Finite :=
   f.LocallyFinite.point_finite x
 #align bump_covering.point_finite BumpCovering.point_finite
 
@@ -429,7 +429,7 @@ theorem toPouFun_eq_mul_prod (i : ι) (x : X) (t : Finset ι)
 theorem sum_toPouFun_eq (x : X) : (∑ᶠ i, f.toPouFun i x) = 1 - ∏ᶠ i, 1 - f i x :=
   by
   set s := (f.point_finite x).toFinset
-  have hs : (s : Set ι) = { i | f i x ≠ 0 } := finite.coe_to_finset _
+  have hs : (s : Set ι) = {i | f i x ≠ 0} := finite.coe_to_finset _
   have A : (support fun i => to_pou_fun f i x) ⊆ s :=
     by
     rw [hs]
@@ -450,7 +450,7 @@ theorem exists_finset_toPouFun_eventuallyEq (i : ι) (x : X) :
   by
   rcases f.locally_finite x with ⟨U, hU, hf⟩
   use hf.to_finset
-  filter_upwards [hU]with y hyU
+  filter_upwards [hU] with y hyU
   simp only [Pi.mul_apply, Finset.prod_apply]
   apply to_pou_fun_eq_mul_prod
   intro j hji hj

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

@@ -214,7 +214,7 @@ variable {f}
 
 theorem exists_finset_nhd_support_subset {U : ι → Set X} (hso : f.IsSubordinate U)
     (ho : ∀ i, IsOpen (U i)) (x : X) :
-    ∃ (is : Finset ι)(n : Set X)(hn₁ : n ∈ 𝓝 x)(hn₂ : n ⊆ ⋂ i ∈ is, U i),
+    ∃ (is : Finset ι) (n : Set X) (hn₁ : n ∈ 𝓝 x) (hn₂ : n ⊆ ⋂ i ∈ is, U i),
       ∀ z ∈ n, (support fun i => f i z) ⊆ is :=
   f.LocallyFinite.exists_finset_nhd_support_subset hso ho x
 #align partition_of_unity.exists_finset_nhd_support_subset PartitionOfUnity.exists_finset_nhd_support_subset
@@ -267,7 +267,7 @@ protected def single (i : ι) (s : Set X) : BumpCovering ι X s
     refine' ⟨univ, univ_mem, (finite_singleton i).Subset _⟩
     rintro j ⟨x, hx, -⟩
     contrapose! hx
-    rw [mem_singleton_iff] at hx
+    rw [mem_singleton_iff] at hx 
     simp [hx]
   nonneg' := le_update_iff.2 ⟨fun x => zero_le_one, fun _ _ => le_rfl⟩
   le_one' := update_le_iff.2 ⟨le_rfl, fun _ _ _ => zero_le_one⟩
@@ -421,7 +421,7 @@ theorem toPouFun_eq_mul_prod (i : ι) (x : X) (t : Finset ι)
     f.toPouFun i x = f i x * ∏ j in t.filterₓ fun j => WellOrderingRel j i, 1 - f j x :=
   by
   refine' congr_arg _ (finprod_cond_eq_prod_of_cond_iff _ fun j hj => _)
-  rw [Ne.def, sub_eq_self] at hj
+  rw [Ne.def, sub_eq_self] at hj 
   rw [Finset.mem_filter, Iff.comm, and_iff_right_iff_imp]
   exact flip (ht j) hj
 #align bump_covering.to_pou_fun_eq_mul_prod BumpCovering.toPouFun_eq_mul_prod

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

@@ -85,7 +85,7 @@ universe u v
 
 open Function Set Filter
 
-open BigOperators Topology Classical
+open scoped BigOperators Topology Classical
 
 noncomputable section
 

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

@@ -147,52 +147,22 @@ variable {E : Type _} [AddCommMonoid E] [SMulWithZero ℝ E] [TopologicalSpace E
 instance : CoeFun (PartitionOfUnity ι X s) fun _ => ι → C(X, ℝ) :=
   ⟨toFun⟩
 
-/- warning: partition_of_unity.locally_finite -> PartitionOfUnity.locallyFinite is a dubious translation:
-lean 3 declaration is
-  forall {ι : Type.{u1}} {X : Type.{u2}} [_inst_1 : TopologicalSpace.{u2} X] {s : Set.{u2} X} (f : PartitionOfUnity.{u1, u2} ι X _inst_1 s), LocallyFinite.{u1, u2} ι X _inst_1 (fun (i : ι) => Function.support.{u2, 0} X Real Real.hasZero (coeFn.{succ u2, succ u2} (ContinuousMap.{u2, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) (fun (_x : ContinuousMap.{u2, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) => X -> Real) (ContinuousMap.hasCoeToFun.{u2, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (PartitionOfUnity.{u1, u2} ι X _inst_1 s) (fun (_x : PartitionOfUnity.{u1, u2} ι X _inst_1 s) => ι -> (ContinuousMap.{u2, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)))) (PartitionOfUnity.hasCoeToFun.{u1, u2} ι X _inst_1 s) f i)))
-but is expected to have type
-  forall {ι : Type.{u1}} {X : Type.{u2}} [_inst_1 : TopologicalSpace.{u2} X] {s : Set.{u2} X} (f : PartitionOfUnity.{u1, u2} ι X _inst_1 s), LocallyFinite.{u1, u2} ι X _inst_1 (fun (i : ι) => Function.support.{u2, 0} X Real Real.instZeroReal (FunLike.coe.{succ u2, succ u2, 1} (ContinuousMap.{u2, 0} X Real _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 _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) (ContinuousMap.instContinuousMapClassContinuousMap.{u2, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)))) (PartitionOfUnity.toFun.{u1, u2} ι X _inst_1 s f i)))
-Case conversion may be inaccurate. Consider using '#align partition_of_unity.locally_finite PartitionOfUnity.locallyFiniteₓ'. -/
 protected theorem locallyFinite : LocallyFinite fun i => support (f i) :=
   f.locally_finite'
 #align partition_of_unity.locally_finite PartitionOfUnity.locallyFinite
 
-/- warning: partition_of_unity.locally_finite_tsupport -> PartitionOfUnity.locallyFinite_tsupport is a dubious translation:
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 theorem locallyFinite_tsupport : LocallyFinite fun i => tsupport (f i) :=
   f.LocallyFinite.closure
 #align partition_of_unity.locally_finite_tsupport PartitionOfUnity.locallyFinite_tsupport
 
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 theorem nonneg (i : ι) (x : X) : 0 ≤ f i x :=
   f.nonneg' i x
 #align partition_of_unity.nonneg PartitionOfUnity.nonneg
 
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 theorem sum_eq_one {x : X} (hx : x ∈ s) : (∑ᶠ i, f i x) = 1 :=
   f.sum_eq_one' x hx
 #align partition_of_unity.sum_eq_one PartitionOfUnity.sum_eq_one
 
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 /-- If `f` is a partition of unity on `s`, then for every `x ∈ s` there exists an index `i` such
 that `0 < f i x`. -/
 theorem exists_pos {x : X} (hx : x ∈ s) : ∃ i, 0 < f i x :=
@@ -202,42 +172,18 @@ theorem exists_pos {x : X} (hx : x ∈ s) : ∃ i, 0 < f i x :=
   simpa only [fun i => (H i).antisymm (f.nonneg i x), finsum_zero] using zero_ne_one
 #align partition_of_unity.exists_pos PartitionOfUnity.exists_pos
 
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 theorem sum_le_one (x : X) : (∑ᶠ i, f i x) ≤ 1 :=
   f.sum_le_one' x
 #align partition_of_unity.sum_le_one PartitionOfUnity.sum_le_one
 
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 theorem sum_nonneg (x : X) : 0 ≤ ∑ᶠ i, f i x :=
   finsum_nonneg fun i => f.NonNeg i x
 #align partition_of_unity.sum_nonneg PartitionOfUnity.sum_nonneg
 
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 theorem le_one (i : ι) (x : X) : f i x ≤ 1 :=
   (single_le_finsum i (f.LocallyFinite.point_finite x) fun j => f.NonNeg j x).trans (f.sum_le_one x)
 #align partition_of_unity.le_one PartitionOfUnity.le_one
 
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 /-- If `f` is a partition of unity on `s : set X` and `g : X → E` is continuous at every point of
 the topological support of some `f i`, then `λ x, f i x • g x` is continuous on the whole space. -/
 theorem continuous_smul {g : X → E} {i : ι} (hg : ∀ x ∈ tsupport (f i), ContinuousAt g x) :
@@ -246,9 +192,6 @@ theorem continuous_smul {g : X → E} {i : ι} (hg : ∀ x ∈ tsupport (f i), C
     ((f i).ContinuousAt x).smul <| hg x <| tsupport_smul_subset_left _ _ hx
 #align partition_of_unity.continuous_smul PartitionOfUnity.continuous_smul
 
-/- warning: partition_of_unity.continuous_finsum_smul -> PartitionOfUnity.continuous_finsum_smul is a dubious translation:
-<too large>
-Case conversion may be inaccurate. Consider using '#align partition_of_unity.continuous_finsum_smul PartitionOfUnity.continuous_finsum_smulₓ'. -/
 /-- If `f` is a partition of unity on a set `s : set X` and `g : ι → X → E` is a family of functions
 such that each `g i` is continuous at every point of the topological support of `f i`, then the sum
 `λ x, ∑ᶠ i, f i x • g i x` is continuous on the whole space. -/
@@ -269,9 +212,6 @@ def IsSubordinate (U : ι → Set X) : Prop :=
 
 variable {f}
 
-/- warning: partition_of_unity.exists_finset_nhd_support_subset -> PartitionOfUnity.exists_finset_nhd_support_subset is a dubious translation:
-<too large>
-Case conversion may be inaccurate. Consider using '#align partition_of_unity.exists_finset_nhd_support_subset PartitionOfUnity.exists_finset_nhd_support_subsetₓ'. -/
 theorem exists_finset_nhd_support_subset {U : ι → Set X} (hso : f.IsSubordinate U)
     (ho : ∀ i, IsOpen (U i)) (x : X) :
     ∃ (is : Finset ι)(n : Set X)(hn₁ : n ∈ 𝓝 x)(hn₂ : n ⊆ ⋂ i ∈ is, U i),
@@ -279,9 +219,6 @@ theorem exists_finset_nhd_support_subset {U : ι → Set X} (hso : f.IsSubordina
   f.LocallyFinite.exists_finset_nhd_support_subset hso ho x
 #align partition_of_unity.exists_finset_nhd_support_subset PartitionOfUnity.exists_finset_nhd_support_subset
 
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-<too large>
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 /-- If `f` is a partition of unity that is subordinate to a family of open sets `U i` and
 `g : ι → X → E` is a family of functions such that each `g i` is continuous on `U i`, then the sum
 `λ x, ∑ᶠ i, f i x • g i x` is a continuous function. -/
@@ -300,52 +237,22 @@ variable {s : Set X} (f : BumpCovering ι X s)
 instance : CoeFun (BumpCovering ι X s) fun _ => ι → C(X, ℝ) :=
   ⟨toFun⟩
 
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 protected theorem locallyFinite : LocallyFinite fun i => support (f i) :=
   f.locally_finite'
 #align bump_covering.locally_finite BumpCovering.locallyFinite
 
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 theorem locallyFinite_tsupport : LocallyFinite fun i => tsupport (f i) :=
   f.LocallyFinite.closure
 #align bump_covering.locally_finite_tsupport BumpCovering.locallyFinite_tsupport
 
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 protected theorem point_finite (x : X) : { i | f i x ≠ 0 }.Finite :=
   f.LocallyFinite.point_finite x
 #align bump_covering.point_finite BumpCovering.point_finite
 
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 theorem nonneg (i : ι) (x : X) : 0 ≤ f i x :=
   f.nonneg' i x
 #align bump_covering.nonneg BumpCovering.nonneg
 
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 theorem le_one (i : ι) (x : X) : f i x ≤ 1 :=
   f.le_one' i x
 #align bump_covering.le_one BumpCovering.le_one
@@ -368,12 +275,6 @@ protected def single (i : ι) (s : Set X) : BumpCovering ι X s
 #align bump_covering.single BumpCovering.single
 -/
 
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 @[simp]
 theorem coe_single (i : ι) (s : Set X) : ⇑(BumpCovering.single i s) = Pi.single i 1 :=
   rfl
@@ -396,9 +297,6 @@ theorem IsSubordinate.mono {f : BumpCovering ι X s} {U V : ι → Set X} (hU :
 #align bump_covering.is_subordinate.mono BumpCovering.IsSubordinate.mono
 -/
 
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-<too large>
-Case conversion may be inaccurate. Consider using '#align bump_covering.exists_is_subordinate_of_locally_finite_of_prop BumpCovering.exists_isSubordinate_of_locallyFinite_of_propₓ'. -/
 /-- If `X` is a normal topological space and `U i`, `i : ι`, is a locally finite open covering of a
 closed set `s`, then there exists a `bump_covering ι X s` that is subordinate to `U`. If `X` is a
 paracompact space, then the assumption `hf : locally_finite U` can be omitted, see
@@ -449,9 +347,6 @@ theorem exists_isSubordinate_of_locallyFinite [NormalSpace X] (hs : IsClosed s)
 #align bump_covering.exists_is_subordinate_of_locally_finite BumpCovering.exists_isSubordinate_of_locallyFinite
 -/
 
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-<too large>
-Case conversion may be inaccurate. Consider using '#align bump_covering.exists_is_subordinate_of_prop BumpCovering.exists_isSubordinate_of_propₓ'. -/
 /-- If `X` is a paracompact normal topological space and `U` is an open covering of a closed set
 `s`, then there exists a `bump_covering ι X s` that is subordinate to `U`. This version assumes that
 `p : (X → ℝ) → Prop` is a predicate that satisfies Urysohn's lemma, and provides a
@@ -489,22 +384,10 @@ def ind (x : X) (hx : x ∈ s) : ι :=
 #align bump_covering.ind BumpCovering.ind
 -/
 
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 theorem eventuallyEq_one (x : X) (hx : x ∈ s) : f (f.ind x hx) =ᶠ[𝓝 x] 1 :=
   (f.eventuallyEq_one' x hx).choose_spec
 #align bump_covering.eventually_eq_one BumpCovering.eventuallyEq_one
 
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 theorem ind_apply (x : X) (hx : x ∈ s) : f (f.ind x hx) x = 1 :=
   (f.eventuallyEq_one x hx).eq_of_nhds
 #align bump_covering.ind_apply BumpCovering.ind_apply
@@ -525,32 +408,14 @@ def toPouFun (i : ι) (x : X) : ℝ :=
 #align bump_covering.to_pou_fun BumpCovering.toPouFun
 -/
 
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 theorem toPouFun_zero_of_zero {i : ι} {x : X} (h : f i x = 0) : f.toPouFun i x = 0 := by
   rw [to_pou_fun, h, MulZeroClass.zero_mul]
 #align bump_covering.to_pou_fun_zero_of_zero BumpCovering.toPouFun_zero_of_zero
 
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 theorem support_toPouFun_subset (i : ι) : support (f.toPouFun i) ⊆ support (f i) := fun x =>
   mt <| f.toPouFun_zero_of_zero
 #align bump_covering.support_to_pou_fun_subset BumpCovering.support_toPouFun_subset
 
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 theorem toPouFun_eq_mul_prod (i : ι) (x : X) (t : Finset ι)
     (ht : ∀ j, WellOrderingRel j i → f j x ≠ 0 → j ∈ t) :
     f.toPouFun i x = f i x * ∏ j in t.filterₓ fun j => WellOrderingRel j i, 1 - f j x :=
@@ -561,12 +426,6 @@ theorem toPouFun_eq_mul_prod (i : ι) (x : X) (t : Finset ι)
   exact flip (ht j) hj
 #align bump_covering.to_pou_fun_eq_mul_prod BumpCovering.toPouFun_eq_mul_prod
 
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 theorem sum_toPouFun_eq (x : X) : (∑ᶠ i, f.toPouFun i x) = 1 - ∏ᶠ i, 1 - f i x :=
   by
   set s := (f.point_finite x).toFinset
@@ -585,12 +444,6 @@ theorem sum_toPouFun_eq (x : X) : (∑ᶠ i, f.toPouFun i x) = 1 - ∏ᶠ i, 1 -
   rwa [finite.mem_to_finset]
 #align bump_covering.sum_to_pou_fun_eq BumpCovering.sum_toPouFun_eq
 
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-Case conversion may be inaccurate. Consider using '#align bump_covering.exists_finset_to_pou_fun_eventually_eq BumpCovering.exists_finset_toPouFun_eventuallyEqₓ'. -/
 theorem exists_finset_toPouFun_eventuallyEq (i : ι) (x : X) :
     ∃ t : Finset ι,
       f.toPouFun i =ᶠ[𝓝 x] f i * ∏ j in t.filterₓ fun j => WellOrderingRel j i, 1 - f j :=
@@ -643,66 +496,33 @@ def toPartitionOfUnity : PartitionOfUnity ι X s
 #align bump_covering.to_partition_of_unity BumpCovering.toPartitionOfUnity
 -/
 
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-Case conversion may be inaccurate. Consider using '#align bump_covering.to_partition_of_unity_apply BumpCovering.toPartitionOfUnity_applyₓ'. -/
 theorem toPartitionOfUnity_apply (i : ι) (x : X) :
     f.toPartitionOfUnity i x = f i x * ∏ᶠ (j) (hj : WellOrderingRel j i), 1 - f j x :=
   rfl
 #align bump_covering.to_partition_of_unity_apply BumpCovering.toPartitionOfUnity_apply
 
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 theorem toPartitionOfUnity_eq_mul_prod (i : ι) (x : X) (t : Finset ι)
     (ht : ∀ j, WellOrderingRel j i → f j x ≠ 0 → j ∈ t) :
     f.toPartitionOfUnity i x = f i x * ∏ j in t.filterₓ fun j => WellOrderingRel j i, 1 - f j x :=
   f.toPouFun_eq_mul_prod i x t ht
 #align bump_covering.to_partition_of_unity_eq_mul_prod BumpCovering.toPartitionOfUnity_eq_mul_prod
 
-/- warning: bump_covering.exists_finset_to_partition_of_unity_eventually_eq -> BumpCovering.exists_finset_toPartitionOfUnity_eventuallyEq is a dubious translation:
-<too large>
-Case conversion may be inaccurate. Consider using '#align bump_covering.exists_finset_to_partition_of_unity_eventually_eq BumpCovering.exists_finset_toPartitionOfUnity_eventuallyEqₓ'. -/
 theorem exists_finset_toPartitionOfUnity_eventuallyEq (i : ι) (x : X) :
     ∃ t : Finset ι,
       f.toPartitionOfUnity i =ᶠ[𝓝 x] f i * ∏ j in t.filterₓ fun j => WellOrderingRel j i, 1 - f j :=
   f.exists_finset_toPouFun_eventuallyEq i x
 #align bump_covering.exists_finset_to_partition_of_unity_eventually_eq BumpCovering.exists_finset_toPartitionOfUnity_eventuallyEq
 
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 theorem toPartitionOfUnity_zero_of_zero {i : ι} {x : X} (h : f i x = 0) :
     f.toPartitionOfUnity i x = 0 :=
   f.toPouFun_zero_of_zero h
 #align bump_covering.to_partition_of_unity_zero_of_zero BumpCovering.toPartitionOfUnity_zero_of_zero
 
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-Case conversion may be inaccurate. Consider using '#align bump_covering.support_to_partition_of_unity_subset BumpCovering.support_toPartitionOfUnity_subsetₓ'. -/
 theorem support_toPartitionOfUnity_subset (i : ι) :
     support (f.toPartitionOfUnity i) ⊆ support (f i) :=
   f.support_toPouFun_subset i
 #align bump_covering.support_to_partition_of_unity_subset BumpCovering.support_toPartitionOfUnity_subset
 
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-Case conversion may be inaccurate. Consider using '#align bump_covering.sum_to_partition_of_unity_eq BumpCovering.sum_toPartitionOfUnity_eqₓ'. -/
 theorem sum_toPartitionOfUnity_eq (x : X) :
     (∑ᶠ i, f.toPartitionOfUnity i x) = 1 - ∏ᶠ i, 1 - f i x :=
   f.sum_toPouFun_eq x

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

@@ -575,9 +575,7 @@ theorem sum_toPouFun_eq (x : X) : (∑ᶠ i, f.toPouFun i x) = 1 - ∏ᶠ i, 1 -
     by
     rw [hs]
     exact fun i hi => f.support_to_pou_fun_subset i hi
-  have B : (mul_support fun i => 1 - f i x) ⊆ s :=
-    by
-    rw [hs, mul_support_one_sub]
+  have B : (mul_support fun i => 1 - f i x) ⊆ s := by rw [hs, mul_support_one_sub];
     exact fun i => id
   letI : LinearOrder ι := linearOrderOfSTO WellOrderingRel
   rw [finsum_eq_sum_of_support_subset _ A, finprod_eq_prod_of_mulSupport_subset _ B,
@@ -637,8 +635,7 @@ def toPartitionOfUnity : PartitionOfUnity ι X s
     simp only [ContinuousMap.coe_mk, sum_to_pou_fun_eq, sub_eq_self]
     apply finprod_eq_zero (fun i => 1 - f i x) (f.ind x hx)
     · simp only [f.ind_apply x hx, sub_self]
-    · rw [mul_support_one_sub]
-      exact f.point_finite x
+    · rw [mul_support_one_sub]; exact f.point_finite x
   sum_le_one' x :=
     by
     simp only [ContinuousMap.coe_mk, sum_to_pou_fun_eq, sub_le_self_iff]

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

@@ -247,10 +247,7 @@ theorem continuous_smul {g : X → E} {i : ι} (hg : ∀ x ∈ tsupport (f i), C
 #align partition_of_unity.continuous_smul PartitionOfUnity.continuous_smul
 
 /- warning: partition_of_unity.continuous_finsum_smul -> PartitionOfUnity.continuous_finsum_smul is a dubious translation:
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+<too large>
 Case conversion may be inaccurate. Consider using '#align partition_of_unity.continuous_finsum_smul PartitionOfUnity.continuous_finsum_smulₓ'. -/
 /-- If `f` is a partition of unity on a set `s : set X` and `g : ι → X → E` is a family of functions
 such that each `g i` is continuous at every point of the topological support of `f i`, then the sum
@@ -273,10 +270,7 @@ def IsSubordinate (U : ι → Set X) : Prop :=
 variable {f}
 
 /- warning: partition_of_unity.exists_finset_nhd_support_subset -> PartitionOfUnity.exists_finset_nhd_support_subset is a dubious translation:
-lean 3 declaration is
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+<too large>
 Case conversion may be inaccurate. Consider using '#align partition_of_unity.exists_finset_nhd_support_subset PartitionOfUnity.exists_finset_nhd_support_subsetₓ'. -/
 theorem exists_finset_nhd_support_subset {U : ι → Set X} (hso : f.IsSubordinate U)
     (ho : ∀ i, IsOpen (U i)) (x : X) :
@@ -286,10 +280,7 @@ theorem exists_finset_nhd_support_subset {U : ι → Set X} (hso : f.IsSubordina
 #align partition_of_unity.exists_finset_nhd_support_subset PartitionOfUnity.exists_finset_nhd_support_subset
 
 /- warning: partition_of_unity.is_subordinate.continuous_finsum_smul -> PartitionOfUnity.IsSubordinate.continuous_finsum_smul is a dubious translation:
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+<too large>
 Case conversion may be inaccurate. Consider using '#align partition_of_unity.is_subordinate.continuous_finsum_smul PartitionOfUnity.IsSubordinate.continuous_finsum_smulₓ'. -/
 /-- If `f` is a partition of unity that is subordinate to a family of open sets `U i` and
 `g : ι → X → E` is a family of functions such that each `g i` is continuous on `U i`, then the sum
@@ -406,10 +397,7 @@ theorem IsSubordinate.mono {f : BumpCovering ι X s} {U V : ι → Set X} (hU :
 -/
 
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u2} ι X _inst_1 s f U))))
+<too large>
 Case conversion may be inaccurate. Consider using '#align bump_covering.exists_is_subordinate_of_locally_finite_of_prop BumpCovering.exists_isSubordinate_of_locallyFinite_of_propₓ'. -/
 /-- If `X` is a normal topological space and `U i`, `i : ι`, is a locally finite open covering of a
 closed set `s`, then there exists a `bump_covering ι X s` that is subordinate to `U`. If `X` is a
@@ -462,10 +450,7 @@ theorem exists_isSubordinate_of_locallyFinite [NormalSpace X] (hs : IsClosed s)
 -/
 
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+<too large>
 Case conversion may be inaccurate. Consider using '#align bump_covering.exists_is_subordinate_of_prop BumpCovering.exists_isSubordinate_of_propₓ'. -/
 /-- If `X` is a paracompact normal topological space and `U` is an open covering of a closed set
 `s`, then there exists a `bump_covering ι X s` that is subordinate to `U`. This version assumes that
@@ -685,10 +670,7 @@ theorem toPartitionOfUnity_eq_mul_prod (i : ι) (x : X) (t : Finset ι)
 #align bump_covering.to_partition_of_unity_eq_mul_prod BumpCovering.toPartitionOfUnity_eq_mul_prod
 
 /- warning: bump_covering.exists_finset_to_partition_of_unity_eventually_eq -> BumpCovering.exists_finset_toPartitionOfUnity_eventuallyEq is a dubious translation:
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+<too large>
 Case conversion may be inaccurate. Consider using '#align bump_covering.exists_finset_to_partition_of_unity_eventually_eq BumpCovering.exists_finset_toPartitionOfUnity_eventuallyEqₓ'. -/
 theorem exists_finset_toPartitionOfUnity_eventuallyEq (i : ι) (x : X) :
     ∃ t : Finset ι,

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

@@ -508,7 +508,7 @@ def ind (x : X) (hx : x ∈ s) : ι :=
 lean 3 declaration is
   forall {ι : Type.{u1}} {X : Type.{u2}} [_inst_1 : TopologicalSpace.{u2} X] {s : Set.{u2} X} (f : BumpCovering.{u1, u2} ι X _inst_1 s) (x : X) (hx : Membership.Mem.{u2, u2} X (Set.{u2} X) (Set.hasMem.{u2} X) x s), Filter.EventuallyEq.{u2, 0} X Real (nhds.{u2} X _inst_1 x) (coeFn.{succ u2, succ u2} (ContinuousMap.{u2, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) (fun (_x : ContinuousMap.{u2, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) => X -> Real) (ContinuousMap.hasCoeToFun.{u2, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (BumpCovering.{u1, u2} ι X _inst_1 s) (fun (_x : BumpCovering.{u1, u2} ι X _inst_1 s) => ι -> (ContinuousMap.{u2, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)))) (BumpCovering.hasCoeToFun.{u1, u2} ι X _inst_1 s) f (BumpCovering.ind.{u1, u2} ι X _inst_1 s f x hx))) (OfNat.ofNat.{u2} (X -> Real) 1 (OfNat.mk.{u2} (X -> Real) 1 (One.one.{u2} (X -> Real) (Pi.instOne.{u2, 0} X (fun (ᾰ : X) => Real) (fun (i : X) => Real.hasOne)))))
 but is expected to have type
-  forall {ι : Type.{u1}} {X : Type.{u2}} [_inst_1 : TopologicalSpace.{u2} X] {s : Set.{u2} X} (f : BumpCovering.{u1, u2} ι X _inst_1 s) (x : X) (hx : Membership.mem.{u2, u2} X (Set.{u2} X) (Set.instMembershipSet.{u2} X) x s), Filter.EventuallyEq.{u2, 0} X Real (nhds.{u2} X _inst_1 x) (FunLike.coe.{succ u2, succ u2, 1} (ContinuousMap.{u2, 0} X Real _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 _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) (ContinuousMap.instContinuousMapClassContinuousMap.{u2, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)))) (BumpCovering.toFun.{u1, u2} ι X _inst_1 s f (BumpCovering.ind.{u1, u2} ι X _inst_1 s f x hx))) (OfNat.ofNat.{u2} (X -> Real) 1 (One.toOfNat1.{u2} (X -> Real) (Pi.instOne.{u2, 0} X (fun (a._@.Mathlib.Order.Filter.Basic._hyg.19133 : X) => Real) (fun (i : X) => Real.instOneReal))))
+  forall {ι : Type.{u1}} {X : Type.{u2}} [_inst_1 : TopologicalSpace.{u2} X] {s : Set.{u2} X} (f : BumpCovering.{u1, u2} ι X _inst_1 s) (x : X) (hx : Membership.mem.{u2, u2} X (Set.{u2} X) (Set.instMembershipSet.{u2} X) x s), Filter.EventuallyEq.{u2, 0} X Real (nhds.{u2} X _inst_1 x) (FunLike.coe.{succ u2, succ u2, 1} (ContinuousMap.{u2, 0} X Real _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 _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) (ContinuousMap.instContinuousMapClassContinuousMap.{u2, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)))) (BumpCovering.toFun.{u1, u2} ι X _inst_1 s f (BumpCovering.ind.{u1, u2} ι X _inst_1 s f x hx))) (OfNat.ofNat.{u2} (X -> Real) 1 (One.toOfNat1.{u2} (X -> Real) (Pi.instOne.{u2, 0} X (fun (a._@.Mathlib.Order.Filter.Basic._hyg.19136 : X) => Real) (fun (i : X) => Real.instOneReal))))
 Case conversion may be inaccurate. Consider using '#align bump_covering.eventually_eq_one BumpCovering.eventuallyEq_oneₓ'. -/
 theorem eventuallyEq_one (x : X) (hx : x ∈ s) : f (f.ind x hx) =ᶠ[𝓝 x] 1 :=
   (f.eventuallyEq_one' x hx).choose_spec

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

@@ -274,9 +274,9 @@ variable {f}
 
 /- warning: partition_of_unity.exists_finset_nhd_support_subset -> PartitionOfUnity.exists_finset_nhd_support_subset is a dubious translation:
 lean 3 declaration is
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+  forall {ι : Type.{u1}} {X : Type.{u2}} [_inst_1 : TopologicalSpace.{u2} X] {s : Set.{u2} X} {f : PartitionOfUnity.{u1, u2} ι X _inst_1 s} {U : ι -> (Set.{u2} X)}, (PartitionOfUnity.IsSubordinate.{u1, u2} ι X _inst_1 s f U) -> (forall (i : ι), IsOpen.{u2} X _inst_1 (U i)) -> (forall (x : X), Exists.{succ u1} (Finset.{u1} ι) (fun (is : Finset.{u1} ι) => Exists.{succ u2} (Set.{u2} X) (fun {n : Set.{u2} X} => Exists.{0} (Membership.Mem.{u2, u2} (Set.{u2} X) (Filter.{u2} X) (Filter.hasMem.{u2} X) n (nhds.{u2} X _inst_1 x)) (fun (hn₁ : Membership.Mem.{u2, u2} (Set.{u2} X) (Filter.{u2} X) (Filter.hasMem.{u2} X) n (nhds.{u2} X _inst_1 x)) => Exists.{0} (HasSubset.Subset.{u2} (Set.{u2} X) (Set.hasSubset.{u2} X) n (Set.iInter.{u2, succ u1} X ι (fun (i : ι) => Set.iInter.{u2, 0} X (Membership.Mem.{u1, u1} ι (Finset.{u1} ι) (Finset.hasMem.{u1} ι) i is) (fun (H : Membership.Mem.{u1, u1} ι (Finset.{u1} ι) (Finset.hasMem.{u1} ι) i is) => U i)))) (fun (hn₂ : HasSubset.Subset.{u2} (Set.{u2} X) (Set.hasSubset.{u2} X) n (Set.iInter.{u2, succ u1} X ι (fun (i : ι) => Set.iInter.{u2, 0} X (Membership.Mem.{u1, u1} ι (Finset.{u1} ι) (Finset.hasMem.{u1} ι) i is) (fun (H : Membership.Mem.{u1, u1} ι (Finset.{u1} ι) (Finset.hasMem.{u1} ι) i is) => U i)))) => forall (z : X), (Membership.Mem.{u2, u2} X (Set.{u2} X) (Set.hasMem.{u2} X) z n) -> (HasSubset.Subset.{u1} (Set.{u1} ι) (Set.hasSubset.{u1} ι) (Function.support.{u1, 0} ι Real Real.hasZero (fun (i : ι) => coeFn.{succ u2, succ u2} (ContinuousMap.{u2, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) (fun (_x : ContinuousMap.{u2, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) => X -> Real) (ContinuousMap.hasCoeToFun.{u2, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (PartitionOfUnity.{u1, u2} ι X _inst_1 s) (fun (_x : PartitionOfUnity.{u1, u2} ι X _inst_1 s) => ι -> (ContinuousMap.{u2, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)))) (PartitionOfUnity.hasCoeToFun.{u1, u2} ι X _inst_1 s) f i) z)) ((fun (a : Type.{u1}) (b : Type.{u1}) [self : HasLiftT.{succ u1, succ u1} a b] => self.0) (Finset.{u1} ι) (Set.{u1} ι) (HasLiftT.mk.{succ u1, succ u1} (Finset.{u1} ι) (Set.{u1} ι) (CoeTCₓ.coe.{succ u1, succ u1} (Finset.{u1} ι) (Set.{u1} ι) (Finset.Set.hasCoeT.{u1} ι))) is)))))))
 but is expected to have type
-  forall {ι : Type.{u1}} {X : Type.{u2}} [_inst_1 : TopologicalSpace.{u2} X] {s : Set.{u2} X} {f : PartitionOfUnity.{u1, u2} ι X _inst_1 s} {U : ι -> (Set.{u2} X)}, (PartitionOfUnity.IsSubordinate.{u1, u2} ι X _inst_1 s f U) -> (forall (i : ι), IsOpen.{u2} X _inst_1 (U i)) -> (forall (x : X), Exists.{succ u1} (Finset.{u1} ι) (fun (is : Finset.{u1} ι) => Exists.{succ u2} (Set.{u2} X) (fun (n : Set.{u2} X) => Exists.{0} (Membership.mem.{u2, u2} (Set.{u2} X) (Filter.{u2} X) (instMembershipSetFilter.{u2} X) n (nhds.{u2} X _inst_1 x)) (fun (hn₁ : Membership.mem.{u2, u2} (Set.{u2} X) (Filter.{u2} X) (instMembershipSetFilter.{u2} X) n (nhds.{u2} X _inst_1 x)) => Exists.{0} (HasSubset.Subset.{u2} (Set.{u2} X) (Set.instHasSubsetSet.{u2} X) n (Set.interᵢ.{u2, succ u1} X ι (fun (i : ι) => Set.interᵢ.{u2, 0} X (Membership.mem.{u1, u1} ι (Finset.{u1} ι) (Finset.instMembershipFinset.{u1} ι) i is) (fun (H : Membership.mem.{u1, u1} ι (Finset.{u1} ι) (Finset.instMembershipFinset.{u1} ι) i is) => U i)))) (fun (hn₂ : HasSubset.Subset.{u2} (Set.{u2} X) (Set.instHasSubsetSet.{u2} X) n (Set.interᵢ.{u2, succ u1} X ι (fun (i : ι) => Set.interᵢ.{u2, 0} X (Membership.mem.{u1, u1} ι (Finset.{u1} ι) (Finset.instMembershipFinset.{u1} ι) i is) (fun (H : Membership.mem.{u1, u1} ι (Finset.{u1} ι) (Finset.instMembershipFinset.{u1} ι) i is) => U i)))) => forall (z : X), (Membership.mem.{u2, u2} X (Set.{u2} X) (Set.instMembershipSet.{u2} X) z n) -> (HasSubset.Subset.{u1} (Set.{u1} ι) (Set.instHasSubsetSet.{u1} ι) (Function.support.{u1, 0} ι ((fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : X) => Real) z) Real.instZeroReal (fun (i : ι) => FunLike.coe.{succ u2, succ u2, 1} (ContinuousMap.{u2, 0} X Real _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 _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) (ContinuousMap.instContinuousMapClassContinuousMap.{u2, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)))) (PartitionOfUnity.toFun.{u1, u2} ι X _inst_1 s f i) z)) (Finset.toSet.{u1} ι is)))))))
+  forall {ι : Type.{u1}} {X : Type.{u2}} [_inst_1 : TopologicalSpace.{u2} X] {s : Set.{u2} X} {f : PartitionOfUnity.{u1, u2} ι X _inst_1 s} {U : ι -> (Set.{u2} X)}, (PartitionOfUnity.IsSubordinate.{u1, u2} ι X _inst_1 s f U) -> (forall (i : ι), IsOpen.{u2} X _inst_1 (U i)) -> (forall (x : X), Exists.{succ u1} (Finset.{u1} ι) (fun (is : Finset.{u1} ι) => Exists.{succ u2} (Set.{u2} X) (fun (n : Set.{u2} X) => Exists.{0} (Membership.mem.{u2, u2} (Set.{u2} X) (Filter.{u2} X) (instMembershipSetFilter.{u2} X) n (nhds.{u2} X _inst_1 x)) (fun (hn₁ : Membership.mem.{u2, u2} (Set.{u2} X) (Filter.{u2} X) (instMembershipSetFilter.{u2} X) n (nhds.{u2} X _inst_1 x)) => Exists.{0} (HasSubset.Subset.{u2} (Set.{u2} X) (Set.instHasSubsetSet.{u2} X) n (Set.iInter.{u2, succ u1} X ι (fun (i : ι) => Set.iInter.{u2, 0} X (Membership.mem.{u1, u1} ι (Finset.{u1} ι) (Finset.instMembershipFinset.{u1} ι) i is) (fun (H : Membership.mem.{u1, u1} ι (Finset.{u1} ι) (Finset.instMembershipFinset.{u1} ι) i is) => U i)))) (fun (hn₂ : HasSubset.Subset.{u2} (Set.{u2} X) (Set.instHasSubsetSet.{u2} X) n (Set.iInter.{u2, succ u1} X ι (fun (i : ι) => Set.iInter.{u2, 0} X (Membership.mem.{u1, u1} ι (Finset.{u1} ι) (Finset.instMembershipFinset.{u1} ι) i is) (fun (H : Membership.mem.{u1, u1} ι (Finset.{u1} ι) (Finset.instMembershipFinset.{u1} ι) i is) => U i)))) => forall (z : X), (Membership.mem.{u2, u2} X (Set.{u2} X) (Set.instMembershipSet.{u2} X) z n) -> (HasSubset.Subset.{u1} (Set.{u1} ι) (Set.instHasSubsetSet.{u1} ι) (Function.support.{u1, 0} ι ((fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : X) => Real) z) Real.instZeroReal (fun (i : ι) => FunLike.coe.{succ u2, succ u2, 1} (ContinuousMap.{u2, 0} X Real _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 _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) (ContinuousMap.instContinuousMapClassContinuousMap.{u2, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)))) (PartitionOfUnity.toFun.{u1, u2} ι X _inst_1 s f i) z)) (Finset.toSet.{u1} ι is)))))))
 Case conversion may be inaccurate. Consider using '#align partition_of_unity.exists_finset_nhd_support_subset PartitionOfUnity.exists_finset_nhd_support_subsetₓ'. -/
 theorem exists_finset_nhd_support_subset {U : ι → Set X} (hso : f.IsSubordinate U)
     (ho : ∀ i, IsOpen (U i)) (x : X) :
@@ -407,9 +407,9 @@ theorem IsSubordinate.mono {f : BumpCovering ι X s} {U V : ι → Set X} (hU :
 
 /- warning: bump_covering.exists_is_subordinate_of_locally_finite_of_prop -> BumpCovering.exists_isSubordinate_of_locallyFinite_of_prop is a dubious translation:
 lean 3 declaration is
-  forall {ι : Type.{u1}} {X : Type.{u2}} [_inst_1 : TopologicalSpace.{u2} X] {s : Set.{u2} X} [_inst_2 : NormalSpace.{u2} X _inst_1] (p : (X -> Real) -> Prop), (forall (s : Set.{u2} X) (t : Set.{u2} X), (IsClosed.{u2} X _inst_1 s) -> (IsClosed.{u2} X _inst_1 t) -> (Disjoint.{u2} (Set.{u2} X) (CompleteSemilatticeInf.toPartialOrder.{u2} (Set.{u2} X) (CompleteLattice.toCompleteSemilatticeInf.{u2} (Set.{u2} X) (Order.Coframe.toCompleteLattice.{u2} (Set.{u2} X) (CompleteDistribLattice.toCoframe.{u2} (Set.{u2} X) (CompleteBooleanAlgebra.toCompleteDistribLattice.{u2} (Set.{u2} X) (Set.completeBooleanAlgebra.{u2} X)))))) (GeneralizedBooleanAlgebra.toOrderBot.{u2} (Set.{u2} X) (BooleanAlgebra.toGeneralizedBooleanAlgebra.{u2} (Set.{u2} X) (Set.booleanAlgebra.{u2} X))) s t) -> (Exists.{succ u2} (ContinuousMap.{u2, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) (fun (f : ContinuousMap.{u2, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) => And (p (coeFn.{succ u2, succ u2} (ContinuousMap.{u2, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) (fun (_x : ContinuousMap.{u2, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) => X -> Real) (ContinuousMap.hasCoeToFun.{u2, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) f)) (And (Set.EqOn.{u2, 0} X Real (coeFn.{succ u2, succ u2} (ContinuousMap.{u2, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) (fun (_x : ContinuousMap.{u2, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) => X -> Real) (ContinuousMap.hasCoeToFun.{u2, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) f) (OfNat.ofNat.{u2} (X -> Real) 0 (OfNat.mk.{u2} (X -> Real) 0 (Zero.zero.{u2} (X -> Real) (Pi.instZero.{u2, 0} X (fun (ᾰ : X) => Real) (fun (i : X) => Real.hasZero))))) s) (And (Set.EqOn.{u2, 0} X Real (coeFn.{succ u2, succ u2} (ContinuousMap.{u2, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) (fun (_x : ContinuousMap.{u2, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) => X -> Real) (ContinuousMap.hasCoeToFun.{u2, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) f) (OfNat.ofNat.{u2} (X -> Real) 1 (OfNat.mk.{u2} (X -> Real) 1 (One.one.{u2} (X -> Real) (Pi.instOne.{u2, 0} X (fun (ᾰ : X) => Real) (fun (i : X) => Real.hasOne))))) t) (forall (x : X), Membership.Mem.{0, 0} Real (Set.{0} Real) (Set.hasMem.{0} Real) (coeFn.{succ u2, succ u2} (ContinuousMap.{u2, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) (fun (_x : ContinuousMap.{u2, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) => X -> Real) (ContinuousMap.hasCoeToFun.{u2, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) f x) (Set.Icc.{0} Real Real.preorder (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero))) (OfNat.ofNat.{0} Real 1 (OfNat.mk.{0} Real 1 (One.one.{0} Real Real.hasOne)))))))))) -> (IsClosed.{u2} X _inst_1 s) -> (forall (U : ι -> (Set.{u2} X)), (forall (i : ι), IsOpen.{u2} X _inst_1 (U i)) -> (LocallyFinite.{u1, u2} ι X _inst_1 U) -> (HasSubset.Subset.{u2} (Set.{u2} X) (Set.hasSubset.{u2} X) s (Set.unionᵢ.{u2, succ u1} X ι (fun (i : ι) => U i))) -> (Exists.{max (succ u1) (succ u2)} (BumpCovering.{u1, u2} ι X _inst_1 s) (fun (f : BumpCovering.{u1, u2} ι X _inst_1 s) => And (forall (i : ι), p (coeFn.{succ u2, succ u2} (ContinuousMap.{u2, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) (fun (_x : ContinuousMap.{u2, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) => X -> Real) (ContinuousMap.hasCoeToFun.{u2, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (BumpCovering.{u1, u2} ι X _inst_1 s) (fun (_x : BumpCovering.{u1, u2} ι X _inst_1 s) => ι -> (ContinuousMap.{u2, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)))) (BumpCovering.hasCoeToFun.{u1, u2} ι X _inst_1 s) f i))) (BumpCovering.IsSubordinate.{u1, u2} ι X _inst_1 s f U))))
+  forall {ι : Type.{u1}} {X : Type.{u2}} [_inst_1 : TopologicalSpace.{u2} X] {s : Set.{u2} X} [_inst_2 : NormalSpace.{u2} X _inst_1] (p : (X -> Real) -> Prop), (forall (s : Set.{u2} X) (t : Set.{u2} X), (IsClosed.{u2} X _inst_1 s) -> (IsClosed.{u2} X _inst_1 t) -> (Disjoint.{u2} (Set.{u2} X) (CompleteSemilatticeInf.toPartialOrder.{u2} (Set.{u2} X) (CompleteLattice.toCompleteSemilatticeInf.{u2} (Set.{u2} X) (Order.Coframe.toCompleteLattice.{u2} (Set.{u2} X) (CompleteDistribLattice.toCoframe.{u2} (Set.{u2} X) (CompleteBooleanAlgebra.toCompleteDistribLattice.{u2} (Set.{u2} X) (Set.completeBooleanAlgebra.{u2} X)))))) (GeneralizedBooleanAlgebra.toOrderBot.{u2} (Set.{u2} X) (BooleanAlgebra.toGeneralizedBooleanAlgebra.{u2} (Set.{u2} X) (Set.booleanAlgebra.{u2} X))) s t) -> (Exists.{succ u2} (ContinuousMap.{u2, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) (fun (f : ContinuousMap.{u2, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) => And (p (coeFn.{succ u2, succ u2} (ContinuousMap.{u2, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) (fun (_x : ContinuousMap.{u2, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) => X -> Real) (ContinuousMap.hasCoeToFun.{u2, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) f)) (And (Set.EqOn.{u2, 0} X Real (coeFn.{succ u2, succ u2} (ContinuousMap.{u2, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) (fun (_x : ContinuousMap.{u2, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) => X -> Real) (ContinuousMap.hasCoeToFun.{u2, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) f) (OfNat.ofNat.{u2} (X -> Real) 0 (OfNat.mk.{u2} (X -> Real) 0 (Zero.zero.{u2} (X -> Real) (Pi.instZero.{u2, 0} X (fun (ᾰ : X) => Real) (fun (i : X) => Real.hasZero))))) s) (And (Set.EqOn.{u2, 0} X Real (coeFn.{succ u2, succ u2} (ContinuousMap.{u2, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) (fun (_x : ContinuousMap.{u2, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) => X -> Real) (ContinuousMap.hasCoeToFun.{u2, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) f) (OfNat.ofNat.{u2} (X -> Real) 1 (OfNat.mk.{u2} (X -> Real) 1 (One.one.{u2} (X -> Real) (Pi.instOne.{u2, 0} X (fun (ᾰ : X) => Real) (fun (i : X) => Real.hasOne))))) t) (forall (x : X), Membership.Mem.{0, 0} Real (Set.{0} Real) (Set.hasMem.{0} Real) (coeFn.{succ u2, succ u2} (ContinuousMap.{u2, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) (fun (_x : ContinuousMap.{u2, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) => X -> Real) (ContinuousMap.hasCoeToFun.{u2, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) f x) (Set.Icc.{0} Real Real.preorder (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero))) (OfNat.ofNat.{0} Real 1 (OfNat.mk.{0} Real 1 (One.one.{0} Real Real.hasOne)))))))))) -> (IsClosed.{u2} X _inst_1 s) -> (forall (U : ι -> (Set.{u2} X)), (forall (i : ι), IsOpen.{u2} X _inst_1 (U i)) -> (LocallyFinite.{u1, u2} ι X _inst_1 U) -> (HasSubset.Subset.{u2} (Set.{u2} X) (Set.hasSubset.{u2} X) s (Set.iUnion.{u2, succ u1} X ι (fun (i : ι) => U i))) -> (Exists.{max (succ u1) (succ u2)} (BumpCovering.{u1, u2} ι X _inst_1 s) (fun (f : BumpCovering.{u1, u2} ι X _inst_1 s) => And (forall (i : ι), p (coeFn.{succ u2, succ u2} (ContinuousMap.{u2, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) (fun (_x : ContinuousMap.{u2, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) => X -> Real) (ContinuousMap.hasCoeToFun.{u2, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (BumpCovering.{u1, u2} ι X _inst_1 s) (fun (_x : BumpCovering.{u1, u2} ι X _inst_1 s) => ι -> (ContinuousMap.{u2, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)))) (BumpCovering.hasCoeToFun.{u1, u2} ι X _inst_1 s) f i))) (BumpCovering.IsSubordinate.{u1, u2} ι X _inst_1 s f U))))
 but is expected to have type
-  forall {ι : Type.{u1}} {X : Type.{u2}} [_inst_1 : TopologicalSpace.{u2} X] {s : Set.{u2} X} [_inst_2 : NormalSpace.{u2} X _inst_1] (p : (X -> Real) -> Prop), (forall (s : Set.{u2} X) (t : Set.{u2} X), (IsClosed.{u2} X _inst_1 s) -> (IsClosed.{u2} X _inst_1 t) -> (Disjoint.{u2} (Set.{u2} X) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Set.{u2} X) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Set.{u2} X) (Order.Coframe.toCompleteLattice.{u2} (Set.{u2} X) (CompleteDistribLattice.toCoframe.{u2} (Set.{u2} X) (CompleteBooleanAlgebra.toCompleteDistribLattice.{u2} (Set.{u2} X) (Set.instCompleteBooleanAlgebraSet.{u2} X)))))) (BoundedOrder.toOrderBot.{u2} (Set.{u2} X) (Preorder.toLE.{u2} (Set.{u2} X) (PartialOrder.toPreorder.{u2} (Set.{u2} X) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Set.{u2} X) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Set.{u2} X) (Order.Coframe.toCompleteLattice.{u2} (Set.{u2} X) (CompleteDistribLattice.toCoframe.{u2} (Set.{u2} X) (CompleteBooleanAlgebra.toCompleteDistribLattice.{u2} (Set.{u2} X) (Set.instCompleteBooleanAlgebraSet.{u2} X)))))))) (CompleteLattice.toBoundedOrder.{u2} (Set.{u2} X) (Order.Coframe.toCompleteLattice.{u2} (Set.{u2} X) (CompleteDistribLattice.toCoframe.{u2} (Set.{u2} X) (CompleteBooleanAlgebra.toCompleteDistribLattice.{u2} (Set.{u2} X) (Set.instCompleteBooleanAlgebraSet.{u2} X)))))) s t) -> (Exists.{succ u2} (ContinuousMap.{u2, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) (fun (f : ContinuousMap.{u2, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) => And (p (FunLike.coe.{succ u2, succ u2, 1} (ContinuousMap.{u2, 0} X Real _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 _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) (ContinuousMap.instContinuousMapClassContinuousMap.{u2, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)))) f)) (And (Set.EqOn.{u2, 0} X Real (FunLike.coe.{succ u2, succ u2, 1} (ContinuousMap.{u2, 0} X Real _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 _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) (ContinuousMap.instContinuousMapClassContinuousMap.{u2, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)))) f) (OfNat.ofNat.{u2} (X -> Real) 0 (Zero.toOfNat0.{u2} (X -> Real) (Pi.instZero.{u2, 0} X (fun (a._@.Mathlib.Data.Set.Function._hyg.1349 : X) => Real) (fun (i : X) => Real.instZeroReal)))) s) (And (Set.EqOn.{u2, 0} X Real (FunLike.coe.{succ u2, succ u2, 1} (ContinuousMap.{u2, 0} X Real _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 _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) (ContinuousMap.instContinuousMapClassContinuousMap.{u2, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)))) f) (OfNat.ofNat.{u2} (X -> Real) 1 (One.toOfNat1.{u2} (X -> Real) (Pi.instOne.{u2, 0} X (fun (a._@.Mathlib.Data.Set.Function._hyg.1349 : X) => Real) (fun (i : X) => Real.instOneReal)))) t) (forall (x : X), Membership.mem.{0, 0} ((fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : X) => Real) x) (Set.{0} Real) (Set.instMembershipSet.{0} Real) (FunLike.coe.{succ u2, succ u2, 1} (ContinuousMap.{u2, 0} X Real _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 _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) (ContinuousMap.instContinuousMapClassContinuousMap.{u2, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)))) f x) (Set.Icc.{0} Real Real.instPreorderReal (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal)) (OfNat.ofNat.{0} Real 1 (One.toOfNat1.{0} Real Real.instOneReal))))))))) -> (IsClosed.{u2} X _inst_1 s) -> (forall (U : ι -> (Set.{u2} X)), (forall (i : ι), IsOpen.{u2} X _inst_1 (U i)) -> (LocallyFinite.{u1, u2} ι X _inst_1 U) -> (HasSubset.Subset.{u2} (Set.{u2} X) (Set.instHasSubsetSet.{u2} X) s (Set.unionᵢ.{u2, succ u1} X ι (fun (i : ι) => U i))) -> (Exists.{max (succ u1) (succ u2)} (BumpCovering.{u1, u2} ι X _inst_1 s) (fun (f : BumpCovering.{u1, u2} ι X _inst_1 s) => And (forall (i : ι), p (FunLike.coe.{succ u2, succ u2, 1} (ContinuousMap.{u2, 0} X Real _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 _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) (ContinuousMap.instContinuousMapClassContinuousMap.{u2, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)))) (BumpCovering.toFun.{u1, u2} ι X _inst_1 s f i))) (BumpCovering.IsSubordinate.{u1, u2} ι X _inst_1 s f U))))
+  forall {ι : Type.{u1}} {X : Type.{u2}} [_inst_1 : TopologicalSpace.{u2} X] {s : Set.{u2} X} [_inst_2 : NormalSpace.{u2} X _inst_1] (p : (X -> Real) -> Prop), (forall (s : Set.{u2} X) (t : Set.{u2} X), (IsClosed.{u2} X _inst_1 s) -> (IsClosed.{u2} X _inst_1 t) -> (Disjoint.{u2} (Set.{u2} X) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Set.{u2} X) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Set.{u2} X) (Order.Coframe.toCompleteLattice.{u2} (Set.{u2} X) (CompleteDistribLattice.toCoframe.{u2} (Set.{u2} X) (CompleteBooleanAlgebra.toCompleteDistribLattice.{u2} (Set.{u2} X) (Set.instCompleteBooleanAlgebraSet.{u2} X)))))) (BoundedOrder.toOrderBot.{u2} (Set.{u2} X) (Preorder.toLE.{u2} (Set.{u2} X) (PartialOrder.toPreorder.{u2} (Set.{u2} X) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Set.{u2} X) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Set.{u2} X) (Order.Coframe.toCompleteLattice.{u2} (Set.{u2} X) (CompleteDistribLattice.toCoframe.{u2} (Set.{u2} X) (CompleteBooleanAlgebra.toCompleteDistribLattice.{u2} (Set.{u2} X) (Set.instCompleteBooleanAlgebraSet.{u2} X)))))))) (CompleteLattice.toBoundedOrder.{u2} (Set.{u2} X) (Order.Coframe.toCompleteLattice.{u2} (Set.{u2} X) (CompleteDistribLattice.toCoframe.{u2} (Set.{u2} X) (CompleteBooleanAlgebra.toCompleteDistribLattice.{u2} (Set.{u2} X) (Set.instCompleteBooleanAlgebraSet.{u2} X)))))) s t) -> (Exists.{succ u2} (ContinuousMap.{u2, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) (fun (f : ContinuousMap.{u2, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) => And (p (FunLike.coe.{succ u2, succ u2, 1} (ContinuousMap.{u2, 0} X Real _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 _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) (ContinuousMap.instContinuousMapClassContinuousMap.{u2, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)))) f)) (And (Set.EqOn.{u2, 0} X Real (FunLike.coe.{succ u2, succ u2, 1} (ContinuousMap.{u2, 0} X Real _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 _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) (ContinuousMap.instContinuousMapClassContinuousMap.{u2, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)))) f) (OfNat.ofNat.{u2} (X -> Real) 0 (Zero.toOfNat0.{u2} (X -> Real) (Pi.instZero.{u2, 0} X (fun (a._@.Mathlib.Data.Set.Function._hyg.1349 : X) => Real) (fun (i : X) => Real.instZeroReal)))) s) (And (Set.EqOn.{u2, 0} X Real (FunLike.coe.{succ u2, succ u2, 1} (ContinuousMap.{u2, 0} X Real _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 _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) (ContinuousMap.instContinuousMapClassContinuousMap.{u2, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)))) f) (OfNat.ofNat.{u2} (X -> Real) 1 (One.toOfNat1.{u2} (X -> Real) (Pi.instOne.{u2, 0} X (fun (a._@.Mathlib.Data.Set.Function._hyg.1349 : X) => Real) (fun (i : X) => Real.instOneReal)))) t) (forall (x : X), Membership.mem.{0, 0} ((fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : X) => Real) x) (Set.{0} Real) (Set.instMembershipSet.{0} Real) (FunLike.coe.{succ u2, succ u2, 1} (ContinuousMap.{u2, 0} X Real _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 _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) (ContinuousMap.instContinuousMapClassContinuousMap.{u2, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)))) f x) (Set.Icc.{0} Real Real.instPreorderReal (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal)) (OfNat.ofNat.{0} Real 1 (One.toOfNat1.{0} Real Real.instOneReal))))))))) -> (IsClosed.{u2} X _inst_1 s) -> (forall (U : ι -> (Set.{u2} X)), (forall (i : ι), IsOpen.{u2} X _inst_1 (U i)) -> (LocallyFinite.{u1, u2} ι X _inst_1 U) -> (HasSubset.Subset.{u2} (Set.{u2} X) (Set.instHasSubsetSet.{u2} X) s (Set.iUnion.{u2, succ u1} X ι (fun (i : ι) => U i))) -> (Exists.{max (succ u1) (succ u2)} (BumpCovering.{u1, u2} ι X _inst_1 s) (fun (f : BumpCovering.{u1, u2} ι X _inst_1 s) => And (forall (i : ι), p (FunLike.coe.{succ u2, succ u2, 1} (ContinuousMap.{u2, 0} X Real _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 _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) (ContinuousMap.instContinuousMapClassContinuousMap.{u2, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)))) (BumpCovering.toFun.{u1, u2} ι X _inst_1 s f i))) (BumpCovering.IsSubordinate.{u1, u2} ι X _inst_1 s f U))))
 Case conversion may be inaccurate. Consider using '#align bump_covering.exists_is_subordinate_of_locally_finite_of_prop BumpCovering.exists_isSubordinate_of_locallyFinite_of_propₓ'. -/
 /-- If `X` is a normal topological space and `U i`, `i : ι`, is a locally finite open covering of a
 closed set `s`, then there exists a `bump_covering ι X s` that is subordinate to `U`. If `X` is a
@@ -426,10 +426,10 @@ theorem exists_isSubordinate_of_locallyFinite_of_prop [NormalSpace X] (p : (X 
     (hs : IsClosed s) (U : ι → Set X) (ho : ∀ i, IsOpen (U i)) (hf : LocallyFinite U)
     (hU : s ⊆ ⋃ i, U i) : ∃ f : BumpCovering ι X s, (∀ i, p (f i)) ∧ f.IsSubordinate U :=
   by
-  rcases exists_subset_unionᵢ_closure_subset hs ho (fun x _ => hf.point_finite x) hU with
+  rcases exists_subset_iUnion_closure_subset hs ho (fun x _ => hf.point_finite x) hU with
     ⟨V, hsV, hVo, hVU⟩
   have hVU' : ∀ i, V i ⊆ U i := fun i => subset.trans subset_closure (hVU i)
-  rcases exists_subset_unionᵢ_closure_subset hs hVo (fun x _ => (hf.subset hVU').point_finite x)
+  rcases exists_subset_iUnion_closure_subset hs hVo (fun x _ => (hf.subset hVU').point_finite x)
       hsV with
     ⟨W, hsW, hWo, hWV⟩
   choose f hfp hf0 hf1 hf01 using fun i =>
@@ -463,9 +463,9 @@ theorem exists_isSubordinate_of_locallyFinite [NormalSpace X] (hs : IsClosed s)
 
 /- warning: bump_covering.exists_is_subordinate_of_prop -> BumpCovering.exists_isSubordinate_of_prop is a dubious translation:
 lean 3 declaration is
-  forall {ι : Type.{u1}} {X : Type.{u2}} [_inst_1 : TopologicalSpace.{u2} X] {s : Set.{u2} X} [_inst_2 : NormalSpace.{u2} X _inst_1] [_inst_3 : ParacompactSpace.{u2} X _inst_1] (p : (X -> Real) -> Prop), (forall (s : Set.{u2} X) (t : Set.{u2} X), (IsClosed.{u2} X _inst_1 s) -> (IsClosed.{u2} X _inst_1 t) -> (Disjoint.{u2} (Set.{u2} X) (CompleteSemilatticeInf.toPartialOrder.{u2} (Set.{u2} X) (CompleteLattice.toCompleteSemilatticeInf.{u2} (Set.{u2} X) (Order.Coframe.toCompleteLattice.{u2} (Set.{u2} X) (CompleteDistribLattice.toCoframe.{u2} (Set.{u2} X) (CompleteBooleanAlgebra.toCompleteDistribLattice.{u2} (Set.{u2} X) (Set.completeBooleanAlgebra.{u2} X)))))) (GeneralizedBooleanAlgebra.toOrderBot.{u2} (Set.{u2} X) (BooleanAlgebra.toGeneralizedBooleanAlgebra.{u2} (Set.{u2} X) (Set.booleanAlgebra.{u2} X))) s t) -> (Exists.{succ u2} (ContinuousMap.{u2, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) (fun (f : ContinuousMap.{u2, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) => And (p (coeFn.{succ u2, succ u2} (ContinuousMap.{u2, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) (fun (_x : ContinuousMap.{u2, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) => X -> Real) (ContinuousMap.hasCoeToFun.{u2, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) f)) (And (Set.EqOn.{u2, 0} X Real (coeFn.{succ u2, succ u2} (ContinuousMap.{u2, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) (fun (_x : ContinuousMap.{u2, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) => X -> Real) (ContinuousMap.hasCoeToFun.{u2, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) f) (OfNat.ofNat.{u2} (X -> Real) 0 (OfNat.mk.{u2} (X -> Real) 0 (Zero.zero.{u2} (X -> Real) (Pi.instZero.{u2, 0} X (fun (ᾰ : X) => Real) (fun (i : X) => Real.hasZero))))) s) (And (Set.EqOn.{u2, 0} X Real (coeFn.{succ u2, succ u2} (ContinuousMap.{u2, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) (fun (_x : ContinuousMap.{u2, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) => X -> Real) (ContinuousMap.hasCoeToFun.{u2, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) f) (OfNat.ofNat.{u2} (X -> Real) 1 (OfNat.mk.{u2} (X -> Real) 1 (One.one.{u2} (X -> Real) (Pi.instOne.{u2, 0} X (fun (ᾰ : X) => Real) (fun (i : X) => Real.hasOne))))) t) (forall (x : X), Membership.Mem.{0, 0} Real (Set.{0} Real) (Set.hasMem.{0} Real) (coeFn.{succ u2, succ u2} (ContinuousMap.{u2, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) (fun (_x : ContinuousMap.{u2, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) => X -> Real) (ContinuousMap.hasCoeToFun.{u2, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) f x) (Set.Icc.{0} Real Real.preorder (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero))) (OfNat.ofNat.{0} Real 1 (OfNat.mk.{0} Real 1 (One.one.{0} Real Real.hasOne)))))))))) -> (IsClosed.{u2} X _inst_1 s) -> (forall (U : ι -> (Set.{u2} X)), (forall (i : ι), IsOpen.{u2} X _inst_1 (U i)) -> (HasSubset.Subset.{u2} (Set.{u2} X) (Set.hasSubset.{u2} X) s (Set.unionᵢ.{u2, succ u1} X ι (fun (i : ι) => U i))) -> (Exists.{max (succ u1) (succ u2)} (BumpCovering.{u1, u2} ι X _inst_1 s) (fun (f : BumpCovering.{u1, u2} ι X _inst_1 s) => And (forall (i : ι), p (coeFn.{succ u2, succ u2} (ContinuousMap.{u2, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) (fun (_x : ContinuousMap.{u2, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) => X -> Real) (ContinuousMap.hasCoeToFun.{u2, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (BumpCovering.{u1, u2} ι X _inst_1 s) (fun (_x : BumpCovering.{u1, u2} ι X _inst_1 s) => ι -> (ContinuousMap.{u2, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)))) (BumpCovering.hasCoeToFun.{u1, u2} ι X _inst_1 s) f i))) (BumpCovering.IsSubordinate.{u1, u2} ι X _inst_1 s f U))))
+  forall {ι : Type.{u1}} {X : Type.{u2}} [_inst_1 : TopologicalSpace.{u2} X] {s : Set.{u2} X} [_inst_2 : NormalSpace.{u2} X _inst_1] [_inst_3 : ParacompactSpace.{u2} X _inst_1] (p : (X -> Real) -> Prop), (forall (s : Set.{u2} X) (t : Set.{u2} X), (IsClosed.{u2} X _inst_1 s) -> (IsClosed.{u2} X _inst_1 t) -> (Disjoint.{u2} (Set.{u2} X) (CompleteSemilatticeInf.toPartialOrder.{u2} (Set.{u2} X) (CompleteLattice.toCompleteSemilatticeInf.{u2} (Set.{u2} X) (Order.Coframe.toCompleteLattice.{u2} (Set.{u2} X) (CompleteDistribLattice.toCoframe.{u2} (Set.{u2} X) (CompleteBooleanAlgebra.toCompleteDistribLattice.{u2} (Set.{u2} X) (Set.completeBooleanAlgebra.{u2} X)))))) (GeneralizedBooleanAlgebra.toOrderBot.{u2} (Set.{u2} X) (BooleanAlgebra.toGeneralizedBooleanAlgebra.{u2} (Set.{u2} X) (Set.booleanAlgebra.{u2} X))) s t) -> (Exists.{succ u2} (ContinuousMap.{u2, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) (fun (f : ContinuousMap.{u2, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) => And (p (coeFn.{succ u2, succ u2} (ContinuousMap.{u2, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) (fun (_x : ContinuousMap.{u2, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) => X -> Real) (ContinuousMap.hasCoeToFun.{u2, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) f)) (And (Set.EqOn.{u2, 0} X Real (coeFn.{succ u2, succ u2} (ContinuousMap.{u2, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) (fun (_x : ContinuousMap.{u2, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) => X -> Real) (ContinuousMap.hasCoeToFun.{u2, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) f) (OfNat.ofNat.{u2} (X -> Real) 0 (OfNat.mk.{u2} (X -> Real) 0 (Zero.zero.{u2} (X -> Real) (Pi.instZero.{u2, 0} X (fun (ᾰ : X) => Real) (fun (i : X) => Real.hasZero))))) s) (And (Set.EqOn.{u2, 0} X Real (coeFn.{succ u2, succ u2} (ContinuousMap.{u2, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) (fun (_x : ContinuousMap.{u2, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) => X -> Real) (ContinuousMap.hasCoeToFun.{u2, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) f) (OfNat.ofNat.{u2} (X -> Real) 1 (OfNat.mk.{u2} (X -> Real) 1 (One.one.{u2} (X -> Real) (Pi.instOne.{u2, 0} X (fun (ᾰ : X) => Real) (fun (i : X) => Real.hasOne))))) t) (forall (x : X), Membership.Mem.{0, 0} Real (Set.{0} Real) (Set.hasMem.{0} Real) (coeFn.{succ u2, succ u2} (ContinuousMap.{u2, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) (fun (_x : ContinuousMap.{u2, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) => X -> Real) (ContinuousMap.hasCoeToFun.{u2, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) f x) (Set.Icc.{0} Real Real.preorder (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero))) (OfNat.ofNat.{0} Real 1 (OfNat.mk.{0} Real 1 (One.one.{0} Real Real.hasOne)))))))))) -> (IsClosed.{u2} X _inst_1 s) -> (forall (U : ι -> (Set.{u2} X)), (forall (i : ι), IsOpen.{u2} X _inst_1 (U i)) -> (HasSubset.Subset.{u2} (Set.{u2} X) (Set.hasSubset.{u2} X) s (Set.iUnion.{u2, succ u1} X ι (fun (i : ι) => U i))) -> (Exists.{max (succ u1) (succ u2)} (BumpCovering.{u1, u2} ι X _inst_1 s) (fun (f : BumpCovering.{u1, u2} ι X _inst_1 s) => And (forall (i : ι), p (coeFn.{succ u2, succ u2} (ContinuousMap.{u2, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) (fun (_x : ContinuousMap.{u2, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) => X -> Real) (ContinuousMap.hasCoeToFun.{u2, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (BumpCovering.{u1, u2} ι X _inst_1 s) (fun (_x : BumpCovering.{u1, u2} ι X _inst_1 s) => ι -> (ContinuousMap.{u2, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)))) (BumpCovering.hasCoeToFun.{u1, u2} ι X _inst_1 s) f i))) (BumpCovering.IsSubordinate.{u1, u2} ι X _inst_1 s f U))))
 but is expected to have type
-  forall {ι : Type.{u1}} {X : Type.{u2}} [_inst_1 : TopologicalSpace.{u2} X] {s : Set.{u2} X} [_inst_2 : NormalSpace.{u2} X _inst_1] [_inst_3 : ParacompactSpace.{u2} X _inst_1] (p : (X -> Real) -> Prop), (forall (s : Set.{u2} X) (t : Set.{u2} X), (IsClosed.{u2} X _inst_1 s) -> (IsClosed.{u2} X _inst_1 t) -> (Disjoint.{u2} (Set.{u2} X) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Set.{u2} X) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Set.{u2} X) (Order.Coframe.toCompleteLattice.{u2} (Set.{u2} X) (CompleteDistribLattice.toCoframe.{u2} (Set.{u2} X) (CompleteBooleanAlgebra.toCompleteDistribLattice.{u2} (Set.{u2} X) (Set.instCompleteBooleanAlgebraSet.{u2} X)))))) (BoundedOrder.toOrderBot.{u2} (Set.{u2} X) (Preorder.toLE.{u2} (Set.{u2} X) (PartialOrder.toPreorder.{u2} (Set.{u2} X) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Set.{u2} X) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Set.{u2} X) (Order.Coframe.toCompleteLattice.{u2} (Set.{u2} X) (CompleteDistribLattice.toCoframe.{u2} (Set.{u2} X) (CompleteBooleanAlgebra.toCompleteDistribLattice.{u2} (Set.{u2} X) (Set.instCompleteBooleanAlgebraSet.{u2} X)))))))) (CompleteLattice.toBoundedOrder.{u2} (Set.{u2} X) (Order.Coframe.toCompleteLattice.{u2} (Set.{u2} X) (CompleteDistribLattice.toCoframe.{u2} (Set.{u2} X) (CompleteBooleanAlgebra.toCompleteDistribLattice.{u2} (Set.{u2} X) (Set.instCompleteBooleanAlgebraSet.{u2} X)))))) s t) -> (Exists.{succ u2} (ContinuousMap.{u2, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) (fun (f : ContinuousMap.{u2, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) => And (p (FunLike.coe.{succ u2, succ u2, 1} (ContinuousMap.{u2, 0} X Real _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 _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) (ContinuousMap.instContinuousMapClassContinuousMap.{u2, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)))) f)) (And (Set.EqOn.{u2, 0} X Real (FunLike.coe.{succ u2, succ u2, 1} (ContinuousMap.{u2, 0} X Real _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 _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) (ContinuousMap.instContinuousMapClassContinuousMap.{u2, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)))) f) (OfNat.ofNat.{u2} (X -> Real) 0 (Zero.toOfNat0.{u2} (X -> Real) (Pi.instZero.{u2, 0} X (fun (a._@.Mathlib.Data.Set.Function._hyg.1349 : X) => Real) (fun (i : X) => Real.instZeroReal)))) s) (And (Set.EqOn.{u2, 0} X Real (FunLike.coe.{succ u2, succ u2, 1} (ContinuousMap.{u2, 0} X Real _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 _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) (ContinuousMap.instContinuousMapClassContinuousMap.{u2, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)))) f) (OfNat.ofNat.{u2} (X -> Real) 1 (One.toOfNat1.{u2} (X -> Real) (Pi.instOne.{u2, 0} X (fun (a._@.Mathlib.Data.Set.Function._hyg.1349 : X) => Real) (fun (i : X) => Real.instOneReal)))) t) (forall (x : X), Membership.mem.{0, 0} ((fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : X) => Real) x) (Set.{0} Real) (Set.instMembershipSet.{0} Real) (FunLike.coe.{succ u2, succ u2, 1} (ContinuousMap.{u2, 0} X Real _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 _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) (ContinuousMap.instContinuousMapClassContinuousMap.{u2, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)))) f x) (Set.Icc.{0} Real Real.instPreorderReal (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal)) (OfNat.ofNat.{0} Real 1 (One.toOfNat1.{0} Real Real.instOneReal))))))))) -> (IsClosed.{u2} X _inst_1 s) -> (forall (U : ι -> (Set.{u2} X)), (forall (i : ι), IsOpen.{u2} X _inst_1 (U i)) -> (HasSubset.Subset.{u2} (Set.{u2} X) (Set.instHasSubsetSet.{u2} X) s (Set.unionᵢ.{u2, succ u1} X ι (fun (i : ι) => U i))) -> (Exists.{max (succ u1) (succ u2)} (BumpCovering.{u1, u2} ι X _inst_1 s) (fun (f : BumpCovering.{u1, u2} ι X _inst_1 s) => And (forall (i : ι), p (FunLike.coe.{succ u2, succ u2, 1} (ContinuousMap.{u2, 0} X Real _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 _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) (ContinuousMap.instContinuousMapClassContinuousMap.{u2, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)))) (BumpCovering.toFun.{u1, u2} ι X _inst_1 s f i))) (BumpCovering.IsSubordinate.{u1, u2} ι X _inst_1 s f U))))
+  forall {ι : Type.{u1}} {X : Type.{u2}} [_inst_1 : TopologicalSpace.{u2} X] {s : Set.{u2} X} [_inst_2 : NormalSpace.{u2} X _inst_1] [_inst_3 : ParacompactSpace.{u2} X _inst_1] (p : (X -> Real) -> Prop), (forall (s : Set.{u2} X) (t : Set.{u2} X), (IsClosed.{u2} X _inst_1 s) -> (IsClosed.{u2} X _inst_1 t) -> (Disjoint.{u2} (Set.{u2} X) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Set.{u2} X) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Set.{u2} X) (Order.Coframe.toCompleteLattice.{u2} (Set.{u2} X) (CompleteDistribLattice.toCoframe.{u2} (Set.{u2} X) (CompleteBooleanAlgebra.toCompleteDistribLattice.{u2} (Set.{u2} X) (Set.instCompleteBooleanAlgebraSet.{u2} X)))))) (BoundedOrder.toOrderBot.{u2} (Set.{u2} X) (Preorder.toLE.{u2} (Set.{u2} X) (PartialOrder.toPreorder.{u2} (Set.{u2} X) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Set.{u2} X) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Set.{u2} X) (Order.Coframe.toCompleteLattice.{u2} (Set.{u2} X) (CompleteDistribLattice.toCoframe.{u2} (Set.{u2} X) (CompleteBooleanAlgebra.toCompleteDistribLattice.{u2} (Set.{u2} X) (Set.instCompleteBooleanAlgebraSet.{u2} X)))))))) (CompleteLattice.toBoundedOrder.{u2} (Set.{u2} X) (Order.Coframe.toCompleteLattice.{u2} (Set.{u2} X) (CompleteDistribLattice.toCoframe.{u2} (Set.{u2} X) (CompleteBooleanAlgebra.toCompleteDistribLattice.{u2} (Set.{u2} X) (Set.instCompleteBooleanAlgebraSet.{u2} X)))))) s t) -> (Exists.{succ u2} (ContinuousMap.{u2, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) (fun (f : ContinuousMap.{u2, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) => And (p (FunLike.coe.{succ u2, succ u2, 1} (ContinuousMap.{u2, 0} X Real _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 _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) (ContinuousMap.instContinuousMapClassContinuousMap.{u2, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)))) f)) (And (Set.EqOn.{u2, 0} X Real (FunLike.coe.{succ u2, succ u2, 1} (ContinuousMap.{u2, 0} X Real _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 _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) (ContinuousMap.instContinuousMapClassContinuousMap.{u2, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)))) f) (OfNat.ofNat.{u2} (X -> Real) 0 (Zero.toOfNat0.{u2} (X -> Real) (Pi.instZero.{u2, 0} X (fun (a._@.Mathlib.Data.Set.Function._hyg.1349 : X) => Real) (fun (i : X) => Real.instZeroReal)))) s) (And (Set.EqOn.{u2, 0} X Real (FunLike.coe.{succ u2, succ u2, 1} (ContinuousMap.{u2, 0} X Real _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 _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) (ContinuousMap.instContinuousMapClassContinuousMap.{u2, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)))) f) (OfNat.ofNat.{u2} (X -> Real) 1 (One.toOfNat1.{u2} (X -> Real) (Pi.instOne.{u2, 0} X (fun (a._@.Mathlib.Data.Set.Function._hyg.1349 : X) => Real) (fun (i : X) => Real.instOneReal)))) t) (forall (x : X), Membership.mem.{0, 0} ((fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : X) => Real) x) (Set.{0} Real) (Set.instMembershipSet.{0} Real) (FunLike.coe.{succ u2, succ u2, 1} (ContinuousMap.{u2, 0} X Real _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 _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) (ContinuousMap.instContinuousMapClassContinuousMap.{u2, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)))) f x) (Set.Icc.{0} Real Real.instPreorderReal (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal)) (OfNat.ofNat.{0} Real 1 (One.toOfNat1.{0} Real Real.instOneReal))))))))) -> (IsClosed.{u2} X _inst_1 s) -> (forall (U : ι -> (Set.{u2} X)), (forall (i : ι), IsOpen.{u2} X _inst_1 (U i)) -> (HasSubset.Subset.{u2} (Set.{u2} X) (Set.instHasSubsetSet.{u2} X) s (Set.iUnion.{u2, succ u1} X ι (fun (i : ι) => U i))) -> (Exists.{max (succ u1) (succ u2)} (BumpCovering.{u1, u2} ι X _inst_1 s) (fun (f : BumpCovering.{u1, u2} ι X _inst_1 s) => And (forall (i : ι), p (FunLike.coe.{succ u2, succ u2, 1} (ContinuousMap.{u2, 0} X Real _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 _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) (ContinuousMap.instContinuousMapClassContinuousMap.{u2, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)))) (BumpCovering.toFun.{u1, u2} ι X _inst_1 s f i))) (BumpCovering.IsSubordinate.{u1, u2} ι X _inst_1 s f U))))
 Case conversion may be inaccurate. Consider using '#align bump_covering.exists_is_subordinate_of_prop BumpCovering.exists_isSubordinate_of_propₓ'. -/
 /-- If `X` is a paracompact normal topological space and `U` is an open covering of a closed set
 `s`, then there exists a `bump_covering ι X s` that is subordinate to `U`. This version assumes that

mathlib commit https://github.com/leanprover-community/mathlib/commit/28b2a92f2996d28e580450863c130955de0ed398

@@ -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.partition_of_unity
-! leanprover-community/mathlib commit f2ce6086713c78a7f880485f7917ea547a215982
+! leanprover-community/mathlib commit a2706b55e8d7f7e9b1f93143f0b88f2e34a11eea
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
@@ -18,6 +18,9 @@ import Mathbin.Topology.UrysohnsLemma
 /-!
 # Continuous partition of unity
 
+> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.
+> Any changes to this file require a corresponding PR to mathlib4.
+
 In this file we define `partition_of_unity (ι X : Type*) [topological_space X] (s : set X := univ)`
 to be a continuous partition of unity on `s` indexed by `ι`. More precisely, `f : partition_of_unity
 ι X s` is a collection of continuous functions `f i : C(X, ℝ)`, `i : ι`, such that

mathlib commit https://github.com/leanprover-community/mathlib/commit/d4437c68c8d350fc9d4e95e1e174409db35e30d7

@@ -86,6 +86,7 @@ open BigOperators Topology Classical
 
 noncomputable section
 
+#print PartitionOfUnity /-
 /-- A continuous partition of unity on a set `s : set X` is a collection of continuous functions
 `f i` such that
 
@@ -106,7 +107,9 @@ structure PartitionOfUnity (ι X : Type _) [TopologicalSpace X] (s : Set X := un
   sum_eq_one' : ∀ x ∈ s, (∑ᶠ i, to_fun i x) = 1
   sum_le_one' : ∀ x, (∑ᶠ i, to_fun i x) ≤ 1
 #align partition_of_unity PartitionOfUnity
+-/
 
+#print BumpCovering /-
 /-- A `bump_covering ι X s` is an indexed family of functions `f i`, `i : ι`, such that
 
 * the supports of `f i` form a locally finite family of sets, i.e., for every point `x : X` there
@@ -129,6 +132,7 @@ structure BumpCovering (ι X : Type _) [TopologicalSpace X] (s : Set X := univ)
   le_one' : to_fun ≤ 1
   eventuallyEq_one' : ∀ x ∈ s, ∃ i, to_fun i =ᶠ[𝓝 x] 1
 #align bump_covering BumpCovering
+-/
 
 variable {ι : Type u} {X : Type v} [TopologicalSpace X]
 
@@ -140,22 +144,52 @@ variable {E : Type _} [AddCommMonoid E] [SMulWithZero ℝ E] [TopologicalSpace E
 instance : CoeFun (PartitionOfUnity ι X s) fun _ => ι → C(X, ℝ) :=
   ⟨toFun⟩
 
+/- warning: partition_of_unity.locally_finite -> PartitionOfUnity.locallyFinite is a dubious translation:
+lean 3 declaration is
+  forall {ι : Type.{u1}} {X : Type.{u2}} [_inst_1 : TopologicalSpace.{u2} X] {s : Set.{u2} X} (f : PartitionOfUnity.{u1, u2} ι X _inst_1 s), LocallyFinite.{u1, u2} ι X _inst_1 (fun (i : ι) => Function.support.{u2, 0} X Real Real.hasZero (coeFn.{succ u2, succ u2} (ContinuousMap.{u2, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) (fun (_x : ContinuousMap.{u2, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) => X -> Real) (ContinuousMap.hasCoeToFun.{u2, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (PartitionOfUnity.{u1, u2} ι X _inst_1 s) (fun (_x : PartitionOfUnity.{u1, u2} ι X _inst_1 s) => ι -> (ContinuousMap.{u2, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)))) (PartitionOfUnity.hasCoeToFun.{u1, u2} ι X _inst_1 s) f i)))
+but is expected to have type
+  forall {ι : Type.{u1}} {X : Type.{u2}} [_inst_1 : TopologicalSpace.{u2} X] {s : Set.{u2} X} (f : PartitionOfUnity.{u1, u2} ι X _inst_1 s), LocallyFinite.{u1, u2} ι X _inst_1 (fun (i : ι) => Function.support.{u2, 0} X Real Real.instZeroReal (FunLike.coe.{succ u2, succ u2, 1} (ContinuousMap.{u2, 0} X Real _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 _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) (ContinuousMap.instContinuousMapClassContinuousMap.{u2, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)))) (PartitionOfUnity.toFun.{u1, u2} ι X _inst_1 s f i)))
+Case conversion may be inaccurate. Consider using '#align partition_of_unity.locally_finite PartitionOfUnity.locallyFiniteₓ'. -/
 protected theorem locallyFinite : LocallyFinite fun i => support (f i) :=
   f.locally_finite'
 #align partition_of_unity.locally_finite PartitionOfUnity.locallyFinite
 
+/- warning: partition_of_unity.locally_finite_tsupport -> PartitionOfUnity.locallyFinite_tsupport is a dubious translation:
+lean 3 declaration is
+  forall {ι : Type.{u1}} {X : Type.{u2}} [_inst_1 : TopologicalSpace.{u2} X] {s : Set.{u2} X} (f : PartitionOfUnity.{u1, u2} ι X _inst_1 s), LocallyFinite.{u1, u2} ι X _inst_1 (fun (i : ι) => tsupport.{u2, 0} X Real Real.hasZero _inst_1 (coeFn.{succ u2, succ u2} (ContinuousMap.{u2, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) (fun (_x : ContinuousMap.{u2, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) => X -> Real) (ContinuousMap.hasCoeToFun.{u2, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (PartitionOfUnity.{u1, u2} ι X _inst_1 s) (fun (_x : PartitionOfUnity.{u1, u2} ι X _inst_1 s) => ι -> (ContinuousMap.{u2, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)))) (PartitionOfUnity.hasCoeToFun.{u1, u2} ι X _inst_1 s) f i)))
+but is expected to have type
+  forall {ι : Type.{u1}} {X : Type.{u2}} [_inst_1 : TopologicalSpace.{u2} X] {s : Set.{u2} X} (f : PartitionOfUnity.{u1, u2} ι X _inst_1 s), LocallyFinite.{u1, u2} ι X _inst_1 (fun (i : ι) => tsupport.{u2, 0} X Real Real.instZeroReal _inst_1 (FunLike.coe.{succ u2, succ u2, 1} (ContinuousMap.{u2, 0} X Real _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 _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) (ContinuousMap.instContinuousMapClassContinuousMap.{u2, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)))) (PartitionOfUnity.toFun.{u1, u2} ι X _inst_1 s f i)))
+Case conversion may be inaccurate. Consider using '#align partition_of_unity.locally_finite_tsupport PartitionOfUnity.locallyFinite_tsupportₓ'. -/
 theorem locallyFinite_tsupport : LocallyFinite fun i => tsupport (f i) :=
   f.LocallyFinite.closure
 #align partition_of_unity.locally_finite_tsupport PartitionOfUnity.locallyFinite_tsupport
 
+/- warning: partition_of_unity.nonneg -> PartitionOfUnity.nonneg is a dubious translation:
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+  forall {ι : Type.{u1}} {X : Type.{u2}} [_inst_1 : TopologicalSpace.{u2} X] {s : Set.{u2} X} (f : PartitionOfUnity.{u1, u2} ι X _inst_1 s) (i : ι) (x : X), LE.le.{0} Real Real.hasLe (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 _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) (fun (_x : ContinuousMap.{u2, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) => X -> Real) (ContinuousMap.hasCoeToFun.{u2, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (PartitionOfUnity.{u1, u2} ι X _inst_1 s) (fun (_x : PartitionOfUnity.{u1, u2} ι X _inst_1 s) => ι -> (ContinuousMap.{u2, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)))) (PartitionOfUnity.hasCoeToFun.{u1, u2} ι X _inst_1 s) f i) x)
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+Case conversion may be inaccurate. Consider using '#align partition_of_unity.nonneg PartitionOfUnity.nonnegₓ'. -/
 theorem nonneg (i : ι) (x : X) : 0 ≤ f i x :=
   f.nonneg' i x
 #align partition_of_unity.nonneg PartitionOfUnity.nonneg
 
+/- warning: partition_of_unity.sum_eq_one -> PartitionOfUnity.sum_eq_one is a dubious translation:
+lean 3 declaration is
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+but is expected to have type
+  forall {ι : Type.{u1}} {X : Type.{u2}} [_inst_1 : TopologicalSpace.{u2} X] {s : Set.{u2} X} (f : PartitionOfUnity.{u1, u2} ι X _inst_1 s) {x : X}, (Membership.mem.{u2, u2} X (Set.{u2} X) (Set.instMembershipSet.{u2} X) x s) -> (Eq.{1} ((fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : X) => Real) x) (finsum.{0, succ u1} ((fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : X) => Real) x) ι Real.instAddCommMonoidReal (fun (i : ι) => FunLike.coe.{succ u2, succ u2, 1} (ContinuousMap.{u2, 0} X Real _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 _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) (ContinuousMap.instContinuousMapClassContinuousMap.{u2, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)))) (PartitionOfUnity.toFun.{u1, u2} ι X _inst_1 s f i) x)) (OfNat.ofNat.{0} ((fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : X) => Real) x) 1 (One.toOfNat1.{0} ((fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : X) => Real) x) Real.instOneReal)))
+Case conversion may be inaccurate. Consider using '#align partition_of_unity.sum_eq_one PartitionOfUnity.sum_eq_oneₓ'. -/
 theorem sum_eq_one {x : X} (hx : x ∈ s) : (∑ᶠ i, f i x) = 1 :=
   f.sum_eq_one' x hx
 #align partition_of_unity.sum_eq_one PartitionOfUnity.sum_eq_one
 
+/- warning: partition_of_unity.exists_pos -> PartitionOfUnity.exists_pos is a dubious translation:
+lean 3 declaration is
+  forall {ι : Type.{u1}} {X : Type.{u2}} [_inst_1 : TopologicalSpace.{u2} X] {s : Set.{u2} X} (f : PartitionOfUnity.{u1, u2} ι X _inst_1 s) {x : X}, (Membership.Mem.{u2, u2} X (Set.{u2} X) (Set.hasMem.{u2} X) x s) -> (Exists.{succ u1} ι (fun (i : ι) => 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 _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) (fun (_x : ContinuousMap.{u2, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) => X -> Real) (ContinuousMap.hasCoeToFun.{u2, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (PartitionOfUnity.{u1, u2} ι X _inst_1 s) (fun (_x : PartitionOfUnity.{u1, u2} ι X _inst_1 s) => ι -> (ContinuousMap.{u2, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)))) (PartitionOfUnity.hasCoeToFun.{u1, u2} ι X _inst_1 s) f i) x)))
+but is expected to have type
+  forall {ι : Type.{u1}} {X : Type.{u2}} [_inst_1 : TopologicalSpace.{u2} X] {s : Set.{u2} X} (f : PartitionOfUnity.{u1, u2} ι X _inst_1 s) {x : X}, (Membership.mem.{u2, u2} X (Set.{u2} X) (Set.instMembershipSet.{u2} X) x s) -> (Exists.{succ u1} ι (fun (i : ι) => 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 _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 _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) (ContinuousMap.instContinuousMapClassContinuousMap.{u2, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)))) (PartitionOfUnity.toFun.{u1, u2} ι X _inst_1 s f i) x)))
+Case conversion may be inaccurate. Consider using '#align partition_of_unity.exists_pos PartitionOfUnity.exists_posₓ'. -/
 /-- If `f` is a partition of unity on `s`, then for every `x ∈ s` there exists an index `i` such
 that `0 < f i x`. -/
 theorem exists_pos {x : X} (hx : x ∈ s) : ∃ i, 0 < f i x :=
@@ -165,18 +199,42 @@ theorem exists_pos {x : X} (hx : x ∈ s) : ∃ i, 0 < f i x :=
   simpa only [fun i => (H i).antisymm (f.nonneg i x), finsum_zero] using zero_ne_one
 #align partition_of_unity.exists_pos PartitionOfUnity.exists_pos
 
+/- warning: partition_of_unity.sum_le_one -> PartitionOfUnity.sum_le_one is a dubious translation:
+lean 3 declaration is
+  forall {ι : Type.{u1}} {X : Type.{u2}} [_inst_1 : TopologicalSpace.{u2} X] {s : Set.{u2} X} (f : PartitionOfUnity.{u1, u2} ι X _inst_1 s) (x : X), LE.le.{0} Real Real.hasLe (finsum.{0, succ u1} Real ι Real.addCommMonoid (fun (i : ι) => coeFn.{succ u2, succ u2} (ContinuousMap.{u2, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) (fun (_x : ContinuousMap.{u2, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) => X -> Real) (ContinuousMap.hasCoeToFun.{u2, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (PartitionOfUnity.{u1, u2} ι X _inst_1 s) (fun (_x : PartitionOfUnity.{u1, u2} ι X _inst_1 s) => ι -> (ContinuousMap.{u2, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)))) (PartitionOfUnity.hasCoeToFun.{u1, u2} ι X _inst_1 s) f i) x)) (OfNat.ofNat.{0} Real 1 (OfNat.mk.{0} Real 1 (One.one.{0} Real Real.hasOne)))
+but is expected to have type
+  forall {ι : Type.{u1}} {X : Type.{u2}} [_inst_1 : TopologicalSpace.{u2} X] {s : Set.{u2} X} (f : PartitionOfUnity.{u1, u2} ι X _inst_1 s) (x : X), LE.le.{0} ((fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : X) => Real) x) Real.instLEReal (finsum.{0, succ u1} ((fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : X) => Real) x) ι Real.instAddCommMonoidReal (fun (i : ι) => FunLike.coe.{succ u2, succ u2, 1} (ContinuousMap.{u2, 0} X Real _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 _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) (ContinuousMap.instContinuousMapClassContinuousMap.{u2, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)))) (PartitionOfUnity.toFun.{u1, u2} ι X _inst_1 s f i) x)) (OfNat.ofNat.{0} ((fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : X) => Real) x) 1 (One.toOfNat1.{0} ((fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : X) => Real) x) Real.instOneReal))
+Case conversion may be inaccurate. Consider using '#align partition_of_unity.sum_le_one PartitionOfUnity.sum_le_oneₓ'. -/
 theorem sum_le_one (x : X) : (∑ᶠ i, f i x) ≤ 1 :=
   f.sum_le_one' x
 #align partition_of_unity.sum_le_one PartitionOfUnity.sum_le_one
 
+/- warning: partition_of_unity.sum_nonneg -> PartitionOfUnity.sum_nonneg is a dubious translation:
+lean 3 declaration is
+  forall {ι : Type.{u1}} {X : Type.{u2}} [_inst_1 : TopologicalSpace.{u2} X] {s : Set.{u2} X} (f : PartitionOfUnity.{u1, u2} ι X _inst_1 s) (x : X), LE.le.{0} Real Real.hasLe (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero))) (finsum.{0, succ u1} Real ι Real.addCommMonoid (fun (i : ι) => coeFn.{succ u2, succ u2} (ContinuousMap.{u2, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) (fun (_x : ContinuousMap.{u2, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) => X -> Real) (ContinuousMap.hasCoeToFun.{u2, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (PartitionOfUnity.{u1, u2} ι X _inst_1 s) (fun (_x : PartitionOfUnity.{u1, u2} ι X _inst_1 s) => ι -> (ContinuousMap.{u2, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)))) (PartitionOfUnity.hasCoeToFun.{u1, u2} ι X _inst_1 s) f i) x))
+but is expected to have type
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+Case conversion may be inaccurate. Consider using '#align partition_of_unity.sum_nonneg PartitionOfUnity.sum_nonnegₓ'. -/
 theorem sum_nonneg (x : X) : 0 ≤ ∑ᶠ i, f i x :=
   finsum_nonneg fun i => f.NonNeg i x
 #align partition_of_unity.sum_nonneg PartitionOfUnity.sum_nonneg
 
+/- warning: partition_of_unity.le_one -> PartitionOfUnity.le_one is a dubious translation:
+lean 3 declaration is
+  forall {ι : Type.{u1}} {X : Type.{u2}} [_inst_1 : TopologicalSpace.{u2} X] {s : Set.{u2} X} (f : PartitionOfUnity.{u1, u2} ι X _inst_1 s) (i : ι) (x : X), LE.le.{0} Real Real.hasLe (coeFn.{succ u2, succ u2} (ContinuousMap.{u2, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) (fun (_x : ContinuousMap.{u2, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) => X -> Real) (ContinuousMap.hasCoeToFun.{u2, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (PartitionOfUnity.{u1, u2} ι X _inst_1 s) (fun (_x : PartitionOfUnity.{u1, u2} ι X _inst_1 s) => ι -> (ContinuousMap.{u2, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)))) (PartitionOfUnity.hasCoeToFun.{u1, u2} ι X _inst_1 s) f i) x) (OfNat.ofNat.{0} Real 1 (OfNat.mk.{0} Real 1 (One.one.{0} Real Real.hasOne)))
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+Case conversion may be inaccurate. Consider using '#align partition_of_unity.le_one PartitionOfUnity.le_oneₓ'. -/
 theorem le_one (i : ι) (x : X) : f i x ≤ 1 :=
   (single_le_finsum i (f.LocallyFinite.point_finite x) fun j => f.NonNeg j x).trans (f.sum_le_one x)
 #align partition_of_unity.le_one PartitionOfUnity.le_one
 
+/- warning: partition_of_unity.continuous_smul -> PartitionOfUnity.continuous_smul is a dubious translation:
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+  forall {ι : Type.{u1}} {X : Type.{u2}} [_inst_1 : TopologicalSpace.{u2} X] {E : Type.{u3}} [_inst_2 : AddCommMonoid.{u3} E] [_inst_3 : SMulWithZero.{0, u3} Real E Real.hasZero (AddZeroClass.toHasZero.{u3} E (AddMonoid.toAddZeroClass.{u3} E (AddCommMonoid.toAddMonoid.{u3} E _inst_2)))] [_inst_4 : TopologicalSpace.{u3} E] [_inst_5 : ContinuousSMul.{0, u3} Real E (SMulZeroClass.toHasSmul.{0, u3} Real E (AddZeroClass.toHasZero.{u3} E (AddMonoid.toAddZeroClass.{u3} E (AddCommMonoid.toAddMonoid.{u3} E _inst_2))) (SMulWithZero.toSmulZeroClass.{0, u3} Real E Real.hasZero (AddZeroClass.toHasZero.{u3} E (AddMonoid.toAddZeroClass.{u3} E (AddCommMonoid.toAddMonoid.{u3} E _inst_2))) _inst_3)) (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) _inst_4] {s : Set.{u2} X} (f : PartitionOfUnity.{u1, u2} ι X _inst_1 s) {g : X -> E} {i : ι}, (forall (x : X), (Membership.Mem.{u2, u2} X (Set.{u2} X) (Set.hasMem.{u2} X) x (tsupport.{u2, 0} X Real Real.hasZero _inst_1 (coeFn.{succ u2, succ u2} (ContinuousMap.{u2, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) (fun (_x : ContinuousMap.{u2, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) => X -> Real) (ContinuousMap.hasCoeToFun.{u2, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (PartitionOfUnity.{u1, u2} ι X _inst_1 s) (fun (_x : PartitionOfUnity.{u1, u2} ι X _inst_1 s) => ι -> (ContinuousMap.{u2, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)))) (PartitionOfUnity.hasCoeToFun.{u1, u2} ι X _inst_1 s) f i)))) -> (ContinuousAt.{u2, u3} X E _inst_1 _inst_4 g x)) -> (Continuous.{u2, u3} X E _inst_1 _inst_4 (fun (x : X) => SMul.smul.{0, u3} Real E (SMulZeroClass.toHasSmul.{0, u3} Real E (AddZeroClass.toHasZero.{u3} E (AddMonoid.toAddZeroClass.{u3} E (AddCommMonoid.toAddMonoid.{u3} E _inst_2))) (SMulWithZero.toSmulZeroClass.{0, u3} Real E Real.hasZero (AddZeroClass.toHasZero.{u3} E (AddMonoid.toAddZeroClass.{u3} E (AddCommMonoid.toAddMonoid.{u3} E _inst_2))) _inst_3)) (coeFn.{succ u2, succ u2} (ContinuousMap.{u2, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) (fun (_x : ContinuousMap.{u2, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) => X -> Real) (ContinuousMap.hasCoeToFun.{u2, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (PartitionOfUnity.{u1, u2} ι X _inst_1 s) (fun (_x : PartitionOfUnity.{u1, u2} ι X _inst_1 s) => ι -> (ContinuousMap.{u2, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)))) (PartitionOfUnity.hasCoeToFun.{u1, u2} ι X _inst_1 s) f i) x) (g x)))
+but is expected to have type
+  forall {ι : Type.{u2}} {X : Type.{u3}} [_inst_1 : TopologicalSpace.{u3} X] {E : Type.{u1}} [_inst_2 : AddCommMonoid.{u1} E] [_inst_3 : SMulWithZero.{0, u1} Real E Real.instZeroReal (AddMonoid.toZero.{u1} E (AddCommMonoid.toAddMonoid.{u1} E _inst_2))] [_inst_4 : TopologicalSpace.{u1} E] [_inst_5 : ContinuousSMul.{0, u1} Real E (SMulZeroClass.toSMul.{0, u1} Real E (AddMonoid.toZero.{u1} E (AddCommMonoid.toAddMonoid.{u1} E _inst_2)) (SMulWithZero.toSMulZeroClass.{0, u1} Real E Real.instZeroReal (AddMonoid.toZero.{u1} E (AddCommMonoid.toAddMonoid.{u1} E _inst_2)) _inst_3)) (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) _inst_4] {s : Set.{u3} X} (f : PartitionOfUnity.{u2, u3} ι X _inst_1 s) {g : X -> E} {i : ι}, (forall (x : X), (Membership.mem.{u3, u3} X (Set.{u3} X) (Set.instMembershipSet.{u3} X) x (tsupport.{u3, 0} X Real Real.instZeroReal _inst_1 (FunLike.coe.{succ u3, succ u3, 1} (ContinuousMap.{u3, 0} X Real _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.{u3, u3, 0} (ContinuousMap.{u3, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) (ContinuousMap.instContinuousMapClassContinuousMap.{u3, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)))) (PartitionOfUnity.toFun.{u2, u3} ι X _inst_1 s f i)))) -> (ContinuousAt.{u3, u1} X E _inst_1 _inst_4 g x)) -> (Continuous.{u3, u1} X E _inst_1 _inst_4 (fun (x : X) => HSMul.hSMul.{0, u1, u1} ((fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : X) => Real) x) E E (instHSMul.{0, u1} ((fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : X) => Real) x) E (SMulZeroClass.toSMul.{0, u1} ((fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : X) => Real) x) E (AddMonoid.toZero.{u1} E (AddCommMonoid.toAddMonoid.{u1} E _inst_2)) (SMulWithZero.toSMulZeroClass.{0, u1} ((fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : X) => Real) x) E Real.instZeroReal (AddMonoid.toZero.{u1} E (AddCommMonoid.toAddMonoid.{u1} E _inst_2)) _inst_3))) (FunLike.coe.{succ u3, succ u3, 1} (ContinuousMap.{u3, 0} X Real _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.{u3, u3, 0} (ContinuousMap.{u3, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) (ContinuousMap.instContinuousMapClassContinuousMap.{u3, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)))) (PartitionOfUnity.toFun.{u2, u3} ι X _inst_1 s f i) x) (g x)))
+Case conversion may be inaccurate. Consider using '#align partition_of_unity.continuous_smul PartitionOfUnity.continuous_smulₓ'. -/
 /-- If `f` is a partition of unity on `s : set X` and `g : X → E` is continuous at every point of
 the topological support of some `f i`, then `λ x, f i x • g x` is continuous on the whole space. -/
 theorem continuous_smul {g : X → E} {i : ι} (hg : ∀ x ∈ tsupport (f i), ContinuousAt g x) :
@@ -185,6 +243,12 @@ theorem continuous_smul {g : X → E} {i : ι} (hg : ∀ x ∈ tsupport (f i), C
     ((f i).ContinuousAt x).smul <| hg x <| tsupport_smul_subset_left _ _ hx
 #align partition_of_unity.continuous_smul PartitionOfUnity.continuous_smul
 
+/- warning: partition_of_unity.continuous_finsum_smul -> PartitionOfUnity.continuous_finsum_smul is a dubious translation:
+lean 3 declaration is
+  forall {ι : Type.{u1}} {X : Type.{u2}} [_inst_1 : TopologicalSpace.{u2} X] {E : Type.{u3}} [_inst_2 : AddCommMonoid.{u3} E] [_inst_3 : SMulWithZero.{0, u3} Real E Real.hasZero (AddZeroClass.toHasZero.{u3} E (AddMonoid.toAddZeroClass.{u3} E (AddCommMonoid.toAddMonoid.{u3} E _inst_2)))] [_inst_4 : TopologicalSpace.{u3} E] [_inst_5 : ContinuousSMul.{0, u3} Real E (SMulZeroClass.toHasSmul.{0, u3} Real E (AddZeroClass.toHasZero.{u3} E (AddMonoid.toAddZeroClass.{u3} E (AddCommMonoid.toAddMonoid.{u3} E _inst_2))) (SMulWithZero.toSmulZeroClass.{0, u3} Real E Real.hasZero (AddZeroClass.toHasZero.{u3} E (AddMonoid.toAddZeroClass.{u3} E (AddCommMonoid.toAddMonoid.{u3} E _inst_2))) _inst_3)) (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) _inst_4] {s : Set.{u2} X} (f : PartitionOfUnity.{u1, u2} ι X _inst_1 s) [_inst_6 : ContinuousAdd.{u3} E _inst_4 (AddZeroClass.toHasAdd.{u3} E (AddMonoid.toAddZeroClass.{u3} E (AddCommMonoid.toAddMonoid.{u3} E _inst_2)))] {g : ι -> X -> E}, (forall (i : ι) (x : X), (Membership.Mem.{u2, u2} X (Set.{u2} X) (Set.hasMem.{u2} X) x (tsupport.{u2, 0} X Real Real.hasZero _inst_1 (coeFn.{succ u2, succ u2} (ContinuousMap.{u2, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) (fun (_x : ContinuousMap.{u2, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) => X -> Real) (ContinuousMap.hasCoeToFun.{u2, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (PartitionOfUnity.{u1, u2} ι X _inst_1 s) (fun (_x : PartitionOfUnity.{u1, u2} ι X _inst_1 s) => ι -> (ContinuousMap.{u2, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)))) (PartitionOfUnity.hasCoeToFun.{u1, u2} ι X _inst_1 s) f i)))) -> (ContinuousAt.{u2, u3} X E _inst_1 _inst_4 (g i) x)) -> (Continuous.{u2, u3} X E _inst_1 _inst_4 (fun (x : X) => finsum.{u3, succ u1} E ι _inst_2 (fun (i : ι) => SMul.smul.{0, u3} Real E (SMulZeroClass.toHasSmul.{0, u3} Real E (AddZeroClass.toHasZero.{u3} E (AddMonoid.toAddZeroClass.{u3} E (AddCommMonoid.toAddMonoid.{u3} E _inst_2))) (SMulWithZero.toSmulZeroClass.{0, u3} Real E Real.hasZero (AddZeroClass.toHasZero.{u3} E (AddMonoid.toAddZeroClass.{u3} E (AddCommMonoid.toAddMonoid.{u3} E _inst_2))) _inst_3)) (coeFn.{succ u2, succ u2} (ContinuousMap.{u2, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) (fun (_x : ContinuousMap.{u2, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) => X -> Real) (ContinuousMap.hasCoeToFun.{u2, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (PartitionOfUnity.{u1, u2} ι X _inst_1 s) (fun (_x : PartitionOfUnity.{u1, u2} ι X _inst_1 s) => ι -> (ContinuousMap.{u2, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)))) (PartitionOfUnity.hasCoeToFun.{u1, u2} ι X _inst_1 s) f i) x) (g i x))))
+but is expected to have type
+  forall {ι : Type.{u2}} {X : Type.{u3}} [_inst_1 : TopologicalSpace.{u3} X] {E : Type.{u1}} [_inst_2 : AddCommMonoid.{u1} E] [_inst_3 : SMulWithZero.{0, u1} Real E Real.instZeroReal (AddMonoid.toZero.{u1} E (AddCommMonoid.toAddMonoid.{u1} E _inst_2))] [_inst_4 : TopologicalSpace.{u1} E] [_inst_5 : ContinuousSMul.{0, u1} Real E (SMulZeroClass.toSMul.{0, u1} Real E (AddMonoid.toZero.{u1} E (AddCommMonoid.toAddMonoid.{u1} E _inst_2)) (SMulWithZero.toSMulZeroClass.{0, u1} Real E Real.instZeroReal (AddMonoid.toZero.{u1} E (AddCommMonoid.toAddMonoid.{u1} E _inst_2)) _inst_3)) (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) _inst_4] {s : Set.{u3} X} (f : PartitionOfUnity.{u2, u3} ι X _inst_1 s) [_inst_6 : ContinuousAdd.{u1} E _inst_4 (AddZeroClass.toAdd.{u1} E (AddMonoid.toAddZeroClass.{u1} E (AddCommMonoid.toAddMonoid.{u1} E _inst_2)))] {g : ι -> X -> E}, (forall (i : ι) (x : X), (Membership.mem.{u3, u3} X (Set.{u3} X) (Set.instMembershipSet.{u3} X) x (tsupport.{u3, 0} X Real Real.instZeroReal _inst_1 (FunLike.coe.{succ u3, succ u3, 1} (ContinuousMap.{u3, 0} X Real _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.{u3, u3, 0} (ContinuousMap.{u3, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) (ContinuousMap.instContinuousMapClassContinuousMap.{u3, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)))) (PartitionOfUnity.toFun.{u2, u3} ι X _inst_1 s f i)))) -> (ContinuousAt.{u3, u1} X E _inst_1 _inst_4 (g i) x)) -> (Continuous.{u3, u1} X E _inst_1 _inst_4 (fun (x : X) => finsum.{u1, succ u2} E ι _inst_2 (fun (i : ι) => HSMul.hSMul.{0, u1, u1} ((fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : X) => Real) x) E E (instHSMul.{0, u1} ((fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : X) => Real) x) E (SMulZeroClass.toSMul.{0, u1} ((fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : X) => Real) x) E (AddMonoid.toZero.{u1} E (AddCommMonoid.toAddMonoid.{u1} E _inst_2)) (SMulWithZero.toSMulZeroClass.{0, u1} ((fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : X) => Real) x) E Real.instZeroReal (AddMonoid.toZero.{u1} E (AddCommMonoid.toAddMonoid.{u1} E _inst_2)) _inst_3))) (FunLike.coe.{succ u3, succ u3, 1} (ContinuousMap.{u3, 0} X Real _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.{u3, u3, 0} (ContinuousMap.{u3, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) (ContinuousMap.instContinuousMapClassContinuousMap.{u3, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)))) (PartitionOfUnity.toFun.{u2, u3} ι X _inst_1 s f i) x) (g i x))))
+Case conversion may be inaccurate. Consider using '#align partition_of_unity.continuous_finsum_smul PartitionOfUnity.continuous_finsum_smulₓ'. -/
 /-- If `f` is a partition of unity on a set `s : set X` and `g : ι → X → E` is a family of functions
 such that each `g i` is continuous at every point of the topological support of `f i`, then the sum
 `λ x, ∑ᶠ i, f i x • g i x` is continuous on the whole space. -/
@@ -195,14 +259,22 @@ theorem continuous_finsum_smul [ContinuousAdd E] {g : ι → X → E}
     f.LocallyFinite.Subset fun i => support_smul_subset_left _ _
 #align partition_of_unity.continuous_finsum_smul PartitionOfUnity.continuous_finsum_smul
 
+#print PartitionOfUnity.IsSubordinate /-
 /-- A partition of unity `f i` is subordinate to a family of sets `U i` indexed by the same type if
 for each `i` the closure of the support of `f i` is a subset of `U i`. -/
 def IsSubordinate (U : ι → Set X) : Prop :=
   ∀ i, tsupport (f i) ⊆ U i
 #align partition_of_unity.is_subordinate PartitionOfUnity.IsSubordinate
+-/
 
 variable {f}
 
+/- warning: partition_of_unity.exists_finset_nhd_support_subset -> PartitionOfUnity.exists_finset_nhd_support_subset is a dubious translation:
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+  forall {ι : Type.{u1}} {X : Type.{u2}} [_inst_1 : TopologicalSpace.{u2} X] {s : Set.{u2} X} {f : PartitionOfUnity.{u1, u2} ι X _inst_1 s} {U : ι -> (Set.{u2} X)}, (PartitionOfUnity.IsSubordinate.{u1, u2} ι X _inst_1 s f U) -> (forall (i : ι), IsOpen.{u2} X _inst_1 (U i)) -> (forall (x : X), Exists.{succ u1} (Finset.{u1} ι) (fun (is : Finset.{u1} ι) => Exists.{succ u2} (Set.{u2} X) (fun {n : Set.{u2} X} => Exists.{0} (Membership.Mem.{u2, u2} (Set.{u2} X) (Filter.{u2} X) (Filter.hasMem.{u2} X) n (nhds.{u2} X _inst_1 x)) (fun (hn₁ : Membership.Mem.{u2, u2} (Set.{u2} X) (Filter.{u2} X) (Filter.hasMem.{u2} X) n (nhds.{u2} X _inst_1 x)) => Exists.{0} (HasSubset.Subset.{u2} (Set.{u2} X) (Set.hasSubset.{u2} X) n (Set.interᵢ.{u2, succ u1} X ι (fun (i : ι) => Set.interᵢ.{u2, 0} X (Membership.Mem.{u1, u1} ι (Finset.{u1} ι) (Finset.hasMem.{u1} ι) i is) (fun (H : Membership.Mem.{u1, u1} ι (Finset.{u1} ι) (Finset.hasMem.{u1} ι) i is) => U i)))) (fun (hn₂ : HasSubset.Subset.{u2} (Set.{u2} X) (Set.hasSubset.{u2} X) n (Set.interᵢ.{u2, succ u1} X ι (fun (i : ι) => Set.interᵢ.{u2, 0} X (Membership.Mem.{u1, u1} ι (Finset.{u1} ι) (Finset.hasMem.{u1} ι) i is) (fun (H : Membership.Mem.{u1, u1} ι (Finset.{u1} ι) (Finset.hasMem.{u1} ι) i is) => U i)))) => forall (z : X), (Membership.Mem.{u2, u2} X (Set.{u2} X) (Set.hasMem.{u2} X) z n) -> (HasSubset.Subset.{u1} (Set.{u1} ι) (Set.hasSubset.{u1} ι) (Function.support.{u1, 0} ι Real Real.hasZero (fun (i : ι) => coeFn.{succ u2, succ u2} (ContinuousMap.{u2, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) (fun (_x : ContinuousMap.{u2, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) => X -> Real) (ContinuousMap.hasCoeToFun.{u2, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (PartitionOfUnity.{u1, u2} ι X _inst_1 s) (fun (_x : PartitionOfUnity.{u1, u2} ι X _inst_1 s) => ι -> (ContinuousMap.{u2, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)))) (PartitionOfUnity.hasCoeToFun.{u1, u2} ι X _inst_1 s) f i) z)) ((fun (a : Type.{u1}) (b : Type.{u1}) [self : HasLiftT.{succ u1, succ u1} a b] => self.0) (Finset.{u1} ι) (Set.{u1} ι) (HasLiftT.mk.{succ u1, succ u1} (Finset.{u1} ι) (Set.{u1} ι) (CoeTCₓ.coe.{succ u1, succ u1} (Finset.{u1} ι) (Set.{u1} ι) (Finset.Set.hasCoeT.{u1} ι))) is)))))))
+but is expected to have type
+  forall {ι : Type.{u1}} {X : Type.{u2}} [_inst_1 : TopologicalSpace.{u2} X] {s : Set.{u2} X} {f : PartitionOfUnity.{u1, u2} ι X _inst_1 s} {U : ι -> (Set.{u2} X)}, (PartitionOfUnity.IsSubordinate.{u1, u2} ι X _inst_1 s f U) -> (forall (i : ι), IsOpen.{u2} X _inst_1 (U i)) -> (forall (x : X), Exists.{succ u1} (Finset.{u1} ι) (fun (is : Finset.{u1} ι) => Exists.{succ u2} (Set.{u2} X) (fun (n : Set.{u2} X) => Exists.{0} (Membership.mem.{u2, u2} (Set.{u2} X) (Filter.{u2} X) (instMembershipSetFilter.{u2} X) n (nhds.{u2} X _inst_1 x)) (fun (hn₁ : Membership.mem.{u2, u2} (Set.{u2} X) (Filter.{u2} X) (instMembershipSetFilter.{u2} X) n (nhds.{u2} X _inst_1 x)) => Exists.{0} (HasSubset.Subset.{u2} (Set.{u2} X) (Set.instHasSubsetSet.{u2} X) n (Set.interᵢ.{u2, succ u1} X ι (fun (i : ι) => Set.interᵢ.{u2, 0} X (Membership.mem.{u1, u1} ι (Finset.{u1} ι) (Finset.instMembershipFinset.{u1} ι) i is) (fun (H : Membership.mem.{u1, u1} ι (Finset.{u1} ι) (Finset.instMembershipFinset.{u1} ι) i is) => U i)))) (fun (hn₂ : HasSubset.Subset.{u2} (Set.{u2} X) (Set.instHasSubsetSet.{u2} X) n (Set.interᵢ.{u2, succ u1} X ι (fun (i : ι) => Set.interᵢ.{u2, 0} X (Membership.mem.{u1, u1} ι (Finset.{u1} ι) (Finset.instMembershipFinset.{u1} ι) i is) (fun (H : Membership.mem.{u1, u1} ι (Finset.{u1} ι) (Finset.instMembershipFinset.{u1} ι) i is) => U i)))) => forall (z : X), (Membership.mem.{u2, u2} X (Set.{u2} X) (Set.instMembershipSet.{u2} X) z n) -> (HasSubset.Subset.{u1} (Set.{u1} ι) (Set.instHasSubsetSet.{u1} ι) (Function.support.{u1, 0} ι ((fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : X) => Real) z) Real.instZeroReal (fun (i : ι) => FunLike.coe.{succ u2, succ u2, 1} (ContinuousMap.{u2, 0} X Real _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 _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) (ContinuousMap.instContinuousMapClassContinuousMap.{u2, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)))) (PartitionOfUnity.toFun.{u1, u2} ι X _inst_1 s f i) z)) (Finset.toSet.{u1} ι is)))))))
+Case conversion may be inaccurate. Consider using '#align partition_of_unity.exists_finset_nhd_support_subset PartitionOfUnity.exists_finset_nhd_support_subsetₓ'. -/
 theorem exists_finset_nhd_support_subset {U : ι → Set X} (hso : f.IsSubordinate U)
     (ho : ∀ i, IsOpen (U i)) (x : X) :
     ∃ (is : Finset ι)(n : Set X)(hn₁ : n ∈ 𝓝 x)(hn₂ : n ⊆ ⋂ i ∈ is, U i),
@@ -210,6 +282,12 @@ theorem exists_finset_nhd_support_subset {U : ι → Set X} (hso : f.IsSubordina
   f.LocallyFinite.exists_finset_nhd_support_subset hso ho x
 #align partition_of_unity.exists_finset_nhd_support_subset PartitionOfUnity.exists_finset_nhd_support_subset
 
+/- warning: partition_of_unity.is_subordinate.continuous_finsum_smul -> PartitionOfUnity.IsSubordinate.continuous_finsum_smul is a dubious translation:
+lean 3 declaration is
+  forall {ι : Type.{u1}} {X : Type.{u2}} [_inst_1 : TopologicalSpace.{u2} X] {E : Type.{u3}} [_inst_2 : AddCommMonoid.{u3} E] [_inst_3 : SMulWithZero.{0, u3} Real E Real.hasZero (AddZeroClass.toHasZero.{u3} E (AddMonoid.toAddZeroClass.{u3} E (AddCommMonoid.toAddMonoid.{u3} E _inst_2)))] [_inst_4 : TopologicalSpace.{u3} E] [_inst_5 : ContinuousSMul.{0, u3} Real E (SMulZeroClass.toHasSmul.{0, u3} Real E (AddZeroClass.toHasZero.{u3} E (AddMonoid.toAddZeroClass.{u3} E (AddCommMonoid.toAddMonoid.{u3} E _inst_2))) (SMulWithZero.toSmulZeroClass.{0, u3} Real E Real.hasZero (AddZeroClass.toHasZero.{u3} E (AddMonoid.toAddZeroClass.{u3} E (AddCommMonoid.toAddMonoid.{u3} E _inst_2))) _inst_3)) (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) _inst_4] {s : Set.{u2} X} {f : PartitionOfUnity.{u1, u2} ι X _inst_1 s} [_inst_6 : ContinuousAdd.{u3} E _inst_4 (AddZeroClass.toHasAdd.{u3} E (AddMonoid.toAddZeroClass.{u3} E (AddCommMonoid.toAddMonoid.{u3} E _inst_2)))] {U : ι -> (Set.{u2} X)}, (forall (i : ι), IsOpen.{u2} X _inst_1 (U i)) -> (PartitionOfUnity.IsSubordinate.{u1, u2} ι X _inst_1 s f U) -> (forall {g : ι -> X -> E}, (forall (i : ι), ContinuousOn.{u2, u3} X E _inst_1 _inst_4 (g i) (U i)) -> (Continuous.{u2, u3} X E _inst_1 _inst_4 (fun (x : X) => finsum.{u3, succ u1} E ι _inst_2 (fun (i : ι) => SMul.smul.{0, u3} Real E (SMulZeroClass.toHasSmul.{0, u3} Real E (AddZeroClass.toHasZero.{u3} E (AddMonoid.toAddZeroClass.{u3} E (AddCommMonoid.toAddMonoid.{u3} E _inst_2))) (SMulWithZero.toSmulZeroClass.{0, u3} Real E Real.hasZero (AddZeroClass.toHasZero.{u3} E (AddMonoid.toAddZeroClass.{u3} E (AddCommMonoid.toAddMonoid.{u3} E _inst_2))) _inst_3)) (coeFn.{succ u2, succ u2} (ContinuousMap.{u2, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) (fun (_x : ContinuousMap.{u2, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) => X -> Real) (ContinuousMap.hasCoeToFun.{u2, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (PartitionOfUnity.{u1, u2} ι X _inst_1 s) (fun (_x : PartitionOfUnity.{u1, u2} ι X _inst_1 s) => ι -> (ContinuousMap.{u2, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)))) (PartitionOfUnity.hasCoeToFun.{u1, u2} ι X _inst_1 s) f i) x) (g i x)))))
+but is expected to have type
+  forall {ι : Type.{u2}} {X : Type.{u3}} [_inst_1 : TopologicalSpace.{u3} X] {E : Type.{u1}} [_inst_2 : AddCommMonoid.{u1} E] [_inst_3 : SMulWithZero.{0, u1} Real E Real.instZeroReal (AddMonoid.toZero.{u1} E (AddCommMonoid.toAddMonoid.{u1} E _inst_2))] [_inst_4 : TopologicalSpace.{u1} E] [_inst_5 : ContinuousSMul.{0, u1} Real E (SMulZeroClass.toSMul.{0, u1} Real E (AddMonoid.toZero.{u1} E (AddCommMonoid.toAddMonoid.{u1} E _inst_2)) (SMulWithZero.toSMulZeroClass.{0, u1} Real E Real.instZeroReal (AddMonoid.toZero.{u1} E (AddCommMonoid.toAddMonoid.{u1} E _inst_2)) _inst_3)) (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) _inst_4] {s : Set.{u3} X} {f : PartitionOfUnity.{u2, u3} ι X _inst_1 s} [_inst_6 : ContinuousAdd.{u1} E _inst_4 (AddZeroClass.toAdd.{u1} E (AddMonoid.toAddZeroClass.{u1} E (AddCommMonoid.toAddMonoid.{u1} E _inst_2)))] {U : ι -> (Set.{u3} X)}, (forall (i : ι), IsOpen.{u3} X _inst_1 (U i)) -> (PartitionOfUnity.IsSubordinate.{u2, u3} ι X _inst_1 s f U) -> (forall {g : ι -> X -> E}, (forall (i : ι), ContinuousOn.{u3, u1} X E _inst_1 _inst_4 (g i) (U i)) -> (Continuous.{u3, u1} X E _inst_1 _inst_4 (fun (x : X) => finsum.{u1, succ u2} E ι _inst_2 (fun (i : ι) => HSMul.hSMul.{0, u1, u1} ((fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : X) => Real) x) E E (instHSMul.{0, u1} ((fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : X) => Real) x) E (SMulZeroClass.toSMul.{0, u1} ((fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : X) => Real) x) E (AddMonoid.toZero.{u1} E (AddCommMonoid.toAddMonoid.{u1} E _inst_2)) (SMulWithZero.toSMulZeroClass.{0, u1} ((fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : X) => Real) x) E Real.instZeroReal (AddMonoid.toZero.{u1} E (AddCommMonoid.toAddMonoid.{u1} E _inst_2)) _inst_3))) (FunLike.coe.{succ u3, succ u3, 1} (ContinuousMap.{u3, 0} X Real _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.{u3, u3, 0} (ContinuousMap.{u3, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) (ContinuousMap.instContinuousMapClassContinuousMap.{u3, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)))) (PartitionOfUnity.toFun.{u2, u3} ι X _inst_1 s f i) x) (g i x)))))
+Case conversion may be inaccurate. Consider using '#align partition_of_unity.is_subordinate.continuous_finsum_smul PartitionOfUnity.IsSubordinate.continuous_finsum_smulₓ'. -/
 /-- If `f` is a partition of unity that is subordinate to a family of open sets `U i` and
 `g : ι → X → E` is a family of functions such that each `g i` is continuous on `U i`, then the sum
 `λ x, ∑ᶠ i, f i x • g i x` is a continuous function. -/
@@ -228,26 +306,57 @@ variable {s : Set X} (f : BumpCovering ι X s)
 instance : CoeFun (BumpCovering ι X s) fun _ => ι → C(X, ℝ) :=
   ⟨toFun⟩
 
+/- warning: bump_covering.locally_finite -> BumpCovering.locallyFinite is a dubious translation:
+lean 3 declaration is
+  forall {ι : Type.{u1}} {X : Type.{u2}} [_inst_1 : TopologicalSpace.{u2} X] {s : Set.{u2} X} (f : BumpCovering.{u1, u2} ι X _inst_1 s), LocallyFinite.{u1, u2} ι X _inst_1 (fun (i : ι) => Function.support.{u2, 0} X Real Real.hasZero (coeFn.{succ u2, succ u2} (ContinuousMap.{u2, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) (fun (_x : ContinuousMap.{u2, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) => X -> Real) (ContinuousMap.hasCoeToFun.{u2, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (BumpCovering.{u1, u2} ι X _inst_1 s) (fun (_x : BumpCovering.{u1, u2} ι X _inst_1 s) => ι -> (ContinuousMap.{u2, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)))) (BumpCovering.hasCoeToFun.{u1, u2} ι X _inst_1 s) f i)))
+but is expected to have type
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+Case conversion may be inaccurate. Consider using '#align bump_covering.locally_finite BumpCovering.locallyFiniteₓ'. -/
 protected theorem locallyFinite : LocallyFinite fun i => support (f i) :=
   f.locally_finite'
 #align bump_covering.locally_finite BumpCovering.locallyFinite
 
+/- warning: bump_covering.locally_finite_tsupport -> BumpCovering.locallyFinite_tsupport is a dubious translation:
+lean 3 declaration is
+  forall {ι : Type.{u1}} {X : Type.{u2}} [_inst_1 : TopologicalSpace.{u2} X] {s : Set.{u2} X} (f : BumpCovering.{u1, u2} ι X _inst_1 s), LocallyFinite.{u1, u2} ι X _inst_1 (fun (i : ι) => tsupport.{u2, 0} X Real Real.hasZero _inst_1 (coeFn.{succ u2, succ u2} (ContinuousMap.{u2, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) (fun (_x : ContinuousMap.{u2, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) => X -> Real) (ContinuousMap.hasCoeToFun.{u2, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (BumpCovering.{u1, u2} ι X _inst_1 s) (fun (_x : BumpCovering.{u1, u2} ι X _inst_1 s) => ι -> (ContinuousMap.{u2, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)))) (BumpCovering.hasCoeToFun.{u1, u2} ι X _inst_1 s) f i)))
+but is expected to have type
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+Case conversion may be inaccurate. Consider using '#align bump_covering.locally_finite_tsupport BumpCovering.locallyFinite_tsupportₓ'. -/
 theorem locallyFinite_tsupport : LocallyFinite fun i => tsupport (f i) :=
   f.LocallyFinite.closure
 #align bump_covering.locally_finite_tsupport BumpCovering.locallyFinite_tsupport
 
+/- warning: bump_covering.point_finite -> BumpCovering.point_finite is a dubious translation:
+lean 3 declaration is
+  forall {ι : Type.{u1}} {X : Type.{u2}} [_inst_1 : TopologicalSpace.{u2} X] {s : Set.{u2} X} (f : BumpCovering.{u1, u2} ι X _inst_1 s) (x : X), Set.Finite.{u1} ι (setOf.{u1} ι (fun (i : ι) => Ne.{1} Real (coeFn.{succ u2, succ u2} (ContinuousMap.{u2, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) (fun (_x : ContinuousMap.{u2, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) => X -> Real) (ContinuousMap.hasCoeToFun.{u2, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (BumpCovering.{u1, u2} ι X _inst_1 s) (fun (_x : BumpCovering.{u1, u2} ι X _inst_1 s) => ι -> (ContinuousMap.{u2, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)))) (BumpCovering.hasCoeToFun.{u1, u2} ι X _inst_1 s) f i) x) (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero)))))
+but is expected to have type
+  forall {ι : Type.{u1}} {X : Type.{u2}} [_inst_1 : TopologicalSpace.{u2} X] {s : Set.{u2} X} (f : BumpCovering.{u1, u2} ι X _inst_1 s) (x : X), Set.Finite.{u1} ι (setOf.{u1} ι (fun (i : ι) => Ne.{1} ((fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : X) => Real) x) (FunLike.coe.{succ u2, succ u2, 1} (ContinuousMap.{u2, 0} X Real _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 _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) (ContinuousMap.instContinuousMapClassContinuousMap.{u2, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)))) (BumpCovering.toFun.{u1, u2} ι X _inst_1 s f i) x) (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))))
+Case conversion may be inaccurate. Consider using '#align bump_covering.point_finite BumpCovering.point_finiteₓ'. -/
 protected theorem point_finite (x : X) : { i | f i x ≠ 0 }.Finite :=
   f.LocallyFinite.point_finite x
 #align bump_covering.point_finite BumpCovering.point_finite
 
+/- warning: bump_covering.nonneg -> BumpCovering.nonneg is a dubious translation:
+lean 3 declaration is
+  forall {ι : Type.{u1}} {X : Type.{u2}} [_inst_1 : TopologicalSpace.{u2} X] {s : Set.{u2} X} (f : BumpCovering.{u1, u2} ι X _inst_1 s) (i : ι) (x : X), LE.le.{0} Real Real.hasLe (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 _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) (fun (_x : ContinuousMap.{u2, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) => X -> Real) (ContinuousMap.hasCoeToFun.{u2, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (BumpCovering.{u1, u2} ι X _inst_1 s) (fun (_x : BumpCovering.{u1, u2} ι X _inst_1 s) => ι -> (ContinuousMap.{u2, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)))) (BumpCovering.hasCoeToFun.{u1, u2} ι X _inst_1 s) f i) x)
+but is expected to have type
+  forall {ι : Type.{u1}} {X : Type.{u2}} [_inst_1 : TopologicalSpace.{u2} X] {s : Set.{u2} X} (f : BumpCovering.{u1, u2} ι X _inst_1 s) (i : ι) (x : X), LE.le.{0} ((fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : X) => Real) x) Real.instLEReal (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 _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 _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) (ContinuousMap.instContinuousMapClassContinuousMap.{u2, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)))) (BumpCovering.toFun.{u1, u2} ι X _inst_1 s f i) x)
+Case conversion may be inaccurate. Consider using '#align bump_covering.nonneg BumpCovering.nonnegₓ'. -/
 theorem nonneg (i : ι) (x : X) : 0 ≤ f i x :=
   f.nonneg' i x
 #align bump_covering.nonneg BumpCovering.nonneg
 
+/- warning: bump_covering.le_one -> BumpCovering.le_one is a dubious translation:
+lean 3 declaration is
+  forall {ι : Type.{u1}} {X : Type.{u2}} [_inst_1 : TopologicalSpace.{u2} X] {s : Set.{u2} X} (f : BumpCovering.{u1, u2} ι X _inst_1 s) (i : ι) (x : X), LE.le.{0} Real Real.hasLe (coeFn.{succ u2, succ u2} (ContinuousMap.{u2, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) (fun (_x : ContinuousMap.{u2, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) => X -> Real) (ContinuousMap.hasCoeToFun.{u2, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (BumpCovering.{u1, u2} ι X _inst_1 s) (fun (_x : BumpCovering.{u1, u2} ι X _inst_1 s) => ι -> (ContinuousMap.{u2, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)))) (BumpCovering.hasCoeToFun.{u1, u2} ι X _inst_1 s) f i) x) (OfNat.ofNat.{0} Real 1 (OfNat.mk.{0} Real 1 (One.one.{0} Real Real.hasOne)))
+but is expected to have type
+  forall {ι : Type.{u1}} {X : Type.{u2}} [_inst_1 : TopologicalSpace.{u2} X] {s : Set.{u2} X} (f : BumpCovering.{u1, u2} ι X _inst_1 s) (i : ι) (x : X), LE.le.{0} ((fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : X) => Real) x) Real.instLEReal (FunLike.coe.{succ u2, succ u2, 1} (ContinuousMap.{u2, 0} X Real _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 _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) (ContinuousMap.instContinuousMapClassContinuousMap.{u2, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)))) (BumpCovering.toFun.{u1, u2} ι X _inst_1 s f i) x) (OfNat.ofNat.{0} ((fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : X) => Real) x) 1 (One.toOfNat1.{0} ((fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : X) => Real) x) Real.instOneReal))
+Case conversion may be inaccurate. Consider using '#align bump_covering.le_one BumpCovering.le_oneₓ'. -/
 theorem le_one (i : ι) (x : X) : f i x ≤ 1 :=
   f.le_one' i x
 #align bump_covering.le_one BumpCovering.le_one
 
+#print BumpCovering.single /-
 /-- A `bump_covering` that consists of a single function, uniformly equal to one, defined as an
 example for `inhabited` instance. -/
 protected def single (i : ι) (s : Set X) : BumpCovering ι X s
@@ -263,7 +372,14 @@ protected def single (i : ι) (s : Set X) : BumpCovering ι X s
   le_one' := update_le_iff.2 ⟨le_rfl, fun _ _ _ => zero_le_one⟩
   eventuallyEq_one' x _ := ⟨i, by simp⟩
 #align bump_covering.single BumpCovering.single
+-/
 
+/- warning: bump_covering.coe_single -> BumpCovering.coe_single is a dubious translation:
+lean 3 declaration is
+  forall {ι : Type.{u1}} {X : Type.{u2}} [_inst_1 : TopologicalSpace.{u2} X] (i : ι) (s : Set.{u2} X), Eq.{max (succ u1) (succ u2)} (ι -> (ContinuousMap.{u2, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)))) (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (BumpCovering.{u1, u2} ι X _inst_1 s) (fun (_x : BumpCovering.{u1, u2} ι X _inst_1 s) => ι -> (ContinuousMap.{u2, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)))) (BumpCovering.hasCoeToFun.{u1, u2} ι X _inst_1 s) (BumpCovering.single.{u1, u2} ι X _inst_1 i s)) (Pi.single.{u1, u2} ι (fun (i : ι) => ContinuousMap.{u2, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) (fun (a : ι) (b : ι) => Classical.propDecidable (Eq.{succ u1} ι a b)) (fun (i : ι) => ContinuousMap.hasZero.{u2, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) Real.hasZero) i (OfNat.ofNat.{u2} (ContinuousMap.{u2, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) 1 (OfNat.mk.{u2} (ContinuousMap.{u2, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) 1 (One.one.{u2} (ContinuousMap.{u2, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) (ContinuousMap.hasOne.{u2, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) Real.hasOne)))))
+but is expected to have type
+  forall {ι : Type.{u1}} {X : Type.{u2}} [_inst_1 : TopologicalSpace.{u2} X] (i : ι) (s : Set.{u2} X), Eq.{max (succ u1) (succ u2)} (ι -> (ContinuousMap.{u2, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)))) (BumpCovering.toFun.{u1, u2} ι X _inst_1 s (BumpCovering.single.{u1, u2} ι X _inst_1 i s)) (Pi.single.{u1, u2} ι (fun (i : ι) => ContinuousMap.{u2, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) (fun (a : ι) (b : ι) => Classical.propDecidable (Eq.{succ u1} ι a b)) (fun (i : ι) => ContinuousMap.instZeroContinuousMap.{u2, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) Real.instZeroReal) i (OfNat.ofNat.{u2} (ContinuousMap.{u2, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) 1 (One.toOfNat1.{u2} (ContinuousMap.{u2, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) (ContinuousMap.instOneContinuousMap.{u2, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) Real.instOneReal))))
+Case conversion may be inaccurate. Consider using '#align bump_covering.coe_single BumpCovering.coe_singleₓ'. -/
 @[simp]
 theorem coe_single (i : ι) (s : Set X) : ⇑(BumpCovering.single i s) = Pi.single i 1 :=
   rfl
@@ -272,16 +388,26 @@ theorem coe_single (i : ι) (s : Set X) : ⇑(BumpCovering.single i s) = Pi.sing
 instance [Inhabited ι] : Inhabited (BumpCovering ι X s) :=
   ⟨BumpCovering.single default s⟩
 
+#print BumpCovering.IsSubordinate /-
 /-- A collection of bump functions `f i` is subordinate to a family of sets `U i` indexed by the
 same type if for each `i` the closure of the support of `f i` is a subset of `U i`. -/
 def IsSubordinate (f : BumpCovering ι X s) (U : ι → Set X) : Prop :=
   ∀ i, tsupport (f i) ⊆ U i
 #align bump_covering.is_subordinate BumpCovering.IsSubordinate
+-/
 
+#print BumpCovering.IsSubordinate.mono /-
 theorem IsSubordinate.mono {f : BumpCovering ι X s} {U V : ι → Set X} (hU : f.IsSubordinate U)
     (hV : ∀ i, U i ⊆ V i) : f.IsSubordinate V := fun i => Subset.trans (hU i) (hV i)
 #align bump_covering.is_subordinate.mono BumpCovering.IsSubordinate.mono
+-/
 
+/- warning: bump_covering.exists_is_subordinate_of_locally_finite_of_prop -> BumpCovering.exists_isSubordinate_of_locallyFinite_of_prop is a dubious translation:
+lean 3 declaration is
+  forall {ι : Type.{u1}} {X : Type.{u2}} [_inst_1 : TopologicalSpace.{u2} X] {s : Set.{u2} X} [_inst_2 : NormalSpace.{u2} X _inst_1] (p : (X -> Real) -> Prop), (forall (s : Set.{u2} X) (t : Set.{u2} X), (IsClosed.{u2} X _inst_1 s) -> (IsClosed.{u2} X _inst_1 t) -> (Disjoint.{u2} (Set.{u2} X) (CompleteSemilatticeInf.toPartialOrder.{u2} (Set.{u2} X) (CompleteLattice.toCompleteSemilatticeInf.{u2} (Set.{u2} X) (Order.Coframe.toCompleteLattice.{u2} (Set.{u2} X) (CompleteDistribLattice.toCoframe.{u2} (Set.{u2} X) (CompleteBooleanAlgebra.toCompleteDistribLattice.{u2} (Set.{u2} X) (Set.completeBooleanAlgebra.{u2} X)))))) (GeneralizedBooleanAlgebra.toOrderBot.{u2} (Set.{u2} X) (BooleanAlgebra.toGeneralizedBooleanAlgebra.{u2} (Set.{u2} X) (Set.booleanAlgebra.{u2} X))) s t) -> (Exists.{succ u2} (ContinuousMap.{u2, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) (fun (f : ContinuousMap.{u2, 0} X Real _inst_1 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+but is expected to have type
+  forall {ι : Type.{u1}} {X : Type.{u2}} [_inst_1 : TopologicalSpace.{u2} X] {s : Set.{u2} X} [_inst_2 : NormalSpace.{u2} X _inst_1] (p : (X -> Real) -> Prop), (forall (s : Set.{u2} X) (t : Set.{u2} X), (IsClosed.{u2} X _inst_1 s) -> (IsClosed.{u2} X _inst_1 t) -> (Disjoint.{u2} (Set.{u2} X) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Set.{u2} X) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Set.{u2} X) (Order.Coframe.toCompleteLattice.{u2} (Set.{u2} X) (CompleteDistribLattice.toCoframe.{u2} (Set.{u2} X) (CompleteBooleanAlgebra.toCompleteDistribLattice.{u2} (Set.{u2} X) (Set.instCompleteBooleanAlgebraSet.{u2} X)))))) (BoundedOrder.toOrderBot.{u2} (Set.{u2} X) (Preorder.toLE.{u2} (Set.{u2} X) (PartialOrder.toPreorder.{u2} (Set.{u2} X) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Set.{u2} X) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Set.{u2} X) (Order.Coframe.toCompleteLattice.{u2} (Set.{u2} X) (CompleteDistribLattice.toCoframe.{u2} (Set.{u2} X) (CompleteBooleanAlgebra.toCompleteDistribLattice.{u2} (Set.{u2} X) (Set.instCompleteBooleanAlgebraSet.{u2} X)))))))) (CompleteLattice.toBoundedOrder.{u2} (Set.{u2} X) (Order.Coframe.toCompleteLattice.{u2} (Set.{u2} X) (CompleteDistribLattice.toCoframe.{u2} (Set.{u2} X) (CompleteBooleanAlgebra.toCompleteDistribLattice.{u2} (Set.{u2} X) (Set.instCompleteBooleanAlgebraSet.{u2} X)))))) s t) -> (Exists.{succ u2} (ContinuousMap.{u2, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) (fun (f : ContinuousMap.{u2, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) => And (p (FunLike.coe.{succ u2, succ u2, 1} (ContinuousMap.{u2, 0} X Real _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 _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) (ContinuousMap.instContinuousMapClassContinuousMap.{u2, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)))) f)) (And (Set.EqOn.{u2, 0} X Real (FunLike.coe.{succ u2, succ u2, 1} (ContinuousMap.{u2, 0} X Real _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 _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) (ContinuousMap.instContinuousMapClassContinuousMap.{u2, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)))) f) (OfNat.ofNat.{u2} (X -> Real) 0 (Zero.toOfNat0.{u2} (X -> Real) (Pi.instZero.{u2, 0} X (fun (a._@.Mathlib.Data.Set.Function._hyg.1349 : X) => Real) (fun (i : X) => Real.instZeroReal)))) s) (And (Set.EqOn.{u2, 0} X Real (FunLike.coe.{succ u2, succ u2, 1} (ContinuousMap.{u2, 0} X Real _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 _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) (ContinuousMap.instContinuousMapClassContinuousMap.{u2, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)))) f) (OfNat.ofNat.{u2} (X -> Real) 1 (One.toOfNat1.{u2} (X -> Real) (Pi.instOne.{u2, 0} X (fun (a._@.Mathlib.Data.Set.Function._hyg.1349 : X) => Real) (fun (i : X) => Real.instOneReal)))) t) (forall (x : X), Membership.mem.{0, 0} ((fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : X) => Real) x) (Set.{0} Real) (Set.instMembershipSet.{0} Real) (FunLike.coe.{succ u2, succ u2, 1} (ContinuousMap.{u2, 0} X Real _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 _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) (ContinuousMap.instContinuousMapClassContinuousMap.{u2, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)))) f x) (Set.Icc.{0} Real Real.instPreorderReal (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal)) (OfNat.ofNat.{0} Real 1 (One.toOfNat1.{0} Real Real.instOneReal))))))))) -> (IsClosed.{u2} X _inst_1 s) -> (forall (U : ι -> (Set.{u2} X)), (forall (i : ι), IsOpen.{u2} X _inst_1 (U i)) -> (LocallyFinite.{u1, u2} ι X _inst_1 U) -> (HasSubset.Subset.{u2} (Set.{u2} X) (Set.instHasSubsetSet.{u2} X) s (Set.unionᵢ.{u2, succ u1} X ι (fun (i : ι) => U i))) -> (Exists.{max (succ u1) (succ u2)} (BumpCovering.{u1, u2} ι X _inst_1 s) (fun (f : BumpCovering.{u1, u2} ι X _inst_1 s) => And (forall (i : ι), p (FunLike.coe.{succ u2, succ u2, 1} (ContinuousMap.{u2, 0} X Real _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 _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) (ContinuousMap.instContinuousMapClassContinuousMap.{u2, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)))) (BumpCovering.toFun.{u1, u2} ι X _inst_1 s f i))) (BumpCovering.IsSubordinate.{u1, u2} ι X _inst_1 s f U))))
+Case conversion may be inaccurate. Consider using '#align bump_covering.exists_is_subordinate_of_locally_finite_of_prop BumpCovering.exists_isSubordinate_of_locallyFinite_of_propₓ'. -/
 /-- If `X` is a normal topological space and `U i`, `i : ι`, is a locally finite open covering of a
 closed set `s`, then there exists a `bump_covering ι X s` that is subordinate to `U`. If `X` is a
 paracompact space, then the assumption `hf : locally_finite U` can be omitted, see
@@ -315,6 +441,7 @@ theorem exists_isSubordinate_of_locallyFinite_of_prop [NormalSpace X] (p : (X 
   exact ⟨i, ((hf1 i).mono subset_closure).eventuallyEq_of_mem ((hWo i).mem_nhds hi)⟩
 #align bump_covering.exists_is_subordinate_of_locally_finite_of_prop BumpCovering.exists_isSubordinate_of_locallyFinite_of_prop
 
+#print BumpCovering.exists_isSubordinate_of_locallyFinite /-
 /-- If `X` is a normal topological space and `U i`, `i : ι`, is a locally finite open covering of a
 closed set `s`, then there exists a `bump_covering ι X s` that is subordinate to `U`. If `X` is a
 paracompact space, then the assumption `hf : locally_finite U` can be omitted, see
@@ -329,7 +456,14 @@ theorem exists_isSubordinate_of_locallyFinite [NormalSpace X] (hs : IsClosed s)
       hs U ho hf hU
   ⟨f, hfU⟩
 #align bump_covering.exists_is_subordinate_of_locally_finite BumpCovering.exists_isSubordinate_of_locallyFinite
+-/
 
+/- warning: bump_covering.exists_is_subordinate_of_prop -> BumpCovering.exists_isSubordinate_of_prop is a dubious translation:
+lean 3 declaration is
+  forall {ι : Type.{u1}} {X : Type.{u2}} [_inst_1 : TopologicalSpace.{u2} X] {s : Set.{u2} X} [_inst_2 : NormalSpace.{u2} X _inst_1] [_inst_3 : ParacompactSpace.{u2} X _inst_1] (p : (X -> Real) -> Prop), (forall (s : Set.{u2} X) (t : Set.{u2} X), (IsClosed.{u2} X _inst_1 s) -> (IsClosed.{u2} X _inst_1 t) -> (Disjoint.{u2} (Set.{u2} X) (CompleteSemilatticeInf.toPartialOrder.{u2} (Set.{u2} X) (CompleteLattice.toCompleteSemilatticeInf.{u2} (Set.{u2} X) (Order.Coframe.toCompleteLattice.{u2} (Set.{u2} X) (CompleteDistribLattice.toCoframe.{u2} (Set.{u2} X) (CompleteBooleanAlgebra.toCompleteDistribLattice.{u2} (Set.{u2} X) (Set.completeBooleanAlgebra.{u2} X)))))) (GeneralizedBooleanAlgebra.toOrderBot.{u2} (Set.{u2} X) (BooleanAlgebra.toGeneralizedBooleanAlgebra.{u2} (Set.{u2} X) (Set.booleanAlgebra.{u2} X))) s t) -> (Exists.{succ u2} (ContinuousMap.{u2, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) (fun (f : 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(OfNat.mk.{u2} (X -> Real) 1 (One.one.{u2} (X -> Real) (Pi.instOne.{u2, 0} X (fun (ᾰ : X) => Real) (fun (i : X) => Real.hasOne))))) t) (forall (x : X), Membership.Mem.{0, 0} Real (Set.{0} Real) (Set.hasMem.{0} Real) (coeFn.{succ u2, succ u2} (ContinuousMap.{u2, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) (fun (_x : ContinuousMap.{u2, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) => X -> Real) (ContinuousMap.hasCoeToFun.{u2, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) f x) (Set.Icc.{0} Real Real.preorder (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero))) (OfNat.ofNat.{0} Real 1 (OfNat.mk.{0} Real 1 (One.one.{0} Real Real.hasOne)))))))))) -> (IsClosed.{u2} X _inst_1 s) -> (forall (U : ι -> (Set.{u2} X)), (forall (i : ι), IsOpen.{u2} X _inst_1 (U i)) -> (HasSubset.Subset.{u2} (Set.{u2} X) (Set.hasSubset.{u2} X) s (Set.unionᵢ.{u2, succ u1} X ι (fun (i : ι) => U i))) -> (Exists.{max (succ u1) (succ u2)} (BumpCovering.{u1, u2} ι X _inst_1 s) (fun (f : BumpCovering.{u1, u2} ι X _inst_1 s) => And (forall (i : ι), p (coeFn.{succ u2, succ u2} (ContinuousMap.{u2, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) (fun (_x : ContinuousMap.{u2, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) => X -> Real) (ContinuousMap.hasCoeToFun.{u2, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (BumpCovering.{u1, u2} ι X _inst_1 s) (fun (_x : BumpCovering.{u1, u2} ι X _inst_1 s) => ι -> (ContinuousMap.{u2, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)))) (BumpCovering.hasCoeToFun.{u1, u2} ι X _inst_1 s) f i))) (BumpCovering.IsSubordinate.{u1, u2} ι X _inst_1 s f U))))
+but is expected to have type
+  forall {ι : Type.{u1}} {X : Type.{u2}} [_inst_1 : TopologicalSpace.{u2} X] {s : Set.{u2} X} [_inst_2 : NormalSpace.{u2} X _inst_1] [_inst_3 : ParacompactSpace.{u2} X _inst_1] (p : (X -> Real) -> Prop), (forall (s : Set.{u2} X) (t : Set.{u2} X), (IsClosed.{u2} X _inst_1 s) -> (IsClosed.{u2} X _inst_1 t) -> (Disjoint.{u2} (Set.{u2} X) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Set.{u2} X) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Set.{u2} X) (Order.Coframe.toCompleteLattice.{u2} (Set.{u2} X) (CompleteDistribLattice.toCoframe.{u2} (Set.{u2} X) (CompleteBooleanAlgebra.toCompleteDistribLattice.{u2} (Set.{u2} X) (Set.instCompleteBooleanAlgebraSet.{u2} X)))))) (BoundedOrder.toOrderBot.{u2} (Set.{u2} X) (Preorder.toLE.{u2} (Set.{u2} X) (PartialOrder.toPreorder.{u2} (Set.{u2} X) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Set.{u2} X) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Set.{u2} X) (Order.Coframe.toCompleteLattice.{u2} (Set.{u2} X) (CompleteDistribLattice.toCoframe.{u2} (Set.{u2} X) (CompleteBooleanAlgebra.toCompleteDistribLattice.{u2} (Set.{u2} X) (Set.instCompleteBooleanAlgebraSet.{u2} X)))))))) (CompleteLattice.toBoundedOrder.{u2} (Set.{u2} X) (Order.Coframe.toCompleteLattice.{u2} (Set.{u2} X) (CompleteDistribLattice.toCoframe.{u2} (Set.{u2} X) (CompleteBooleanAlgebra.toCompleteDistribLattice.{u2} (Set.{u2} X) (Set.instCompleteBooleanAlgebraSet.{u2} X)))))) s t) -> (Exists.{succ u2} (ContinuousMap.{u2, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) (fun (f : ContinuousMap.{u2, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) => And (p (FunLike.coe.{succ u2, succ u2, 1} (ContinuousMap.{u2, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) X (fun (_x : X) => (fun 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_inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) (ContinuousMap.instContinuousMapClassContinuousMap.{u2, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)))) f) (OfNat.ofNat.{u2} (X -> Real) 1 (One.toOfNat1.{u2} (X -> Real) (Pi.instOne.{u2, 0} X (fun (a._@.Mathlib.Data.Set.Function._hyg.1349 : X) => Real) (fun (i : X) => Real.instOneReal)))) t) (forall (x : X), Membership.mem.{0, 0} ((fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : X) => Real) x) (Set.{0} Real) (Set.instMembershipSet.{0} Real) (FunLike.coe.{succ u2, succ u2, 1} (ContinuousMap.{u2, 0} X Real _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 _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) (ContinuousMap.instContinuousMapClassContinuousMap.{u2, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)))) f x) (Set.Icc.{0} Real Real.instPreorderReal (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal)) (OfNat.ofNat.{0} Real 1 (One.toOfNat1.{0} Real Real.instOneReal))))))))) -> (IsClosed.{u2} X _inst_1 s) -> (forall (U : ι -> (Set.{u2} X)), (forall (i : ι), IsOpen.{u2} X _inst_1 (U i)) -> (HasSubset.Subset.{u2} (Set.{u2} X) (Set.instHasSubsetSet.{u2} X) s (Set.unionᵢ.{u2, succ u1} X ι (fun (i : ι) => U i))) -> (Exists.{max (succ u1) (succ u2)} (BumpCovering.{u1, u2} ι X _inst_1 s) (fun (f : BumpCovering.{u1, u2} ι X _inst_1 s) => And (forall (i : ι), p (FunLike.coe.{succ u2, succ u2, 1} (ContinuousMap.{u2, 0} X Real _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 _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) (ContinuousMap.instContinuousMapClassContinuousMap.{u2, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)))) (BumpCovering.toFun.{u1, u2} ι X _inst_1 s f i))) (BumpCovering.IsSubordinate.{u1, u2} ι X _inst_1 s f U))))
+Case conversion may be inaccurate. Consider using '#align bump_covering.exists_is_subordinate_of_prop BumpCovering.exists_isSubordinate_of_propₓ'. -/
 /-- If `X` is a paracompact normal topological space and `U` is an open covering of a closed set
 `s`, then there exists a `bump_covering ι X s` that is subordinate to `U`. This version assumes that
 `p : (X → ℝ) → Prop` is a predicate that satisfies Urysohn's lemma, and provides a
@@ -348,6 +482,7 @@ theorem exists_isSubordinate_of_prop [NormalSpace X] [ParacompactSpace X] (p : (
   exact ⟨f, hfp, hf.mono hVU⟩
 #align bump_covering.exists_is_subordinate_of_prop BumpCovering.exists_isSubordinate_of_prop
 
+#print BumpCovering.exists_isSubordinate /-
 /-- If `X` is a paracompact normal topological space and `U` is an open covering of a closed set
 `s`, then there exists a `bump_covering ι X s` that is subordinate to `U`. -/
 theorem exists_isSubordinate [NormalSpace X] [ParacompactSpace X] (hs : IsClosed s) (U : ι → Set X)
@@ -357,20 +492,36 @@ theorem exists_isSubordinate [NormalSpace X] [ParacompactSpace X] (hs : IsClosed
   rcases exists_is_subordinate_of_locally_finite hs V hVo hVf hsV with ⟨f, hf⟩
   exact ⟨f, hf.mono hVU⟩
 #align bump_covering.exists_is_subordinate BumpCovering.exists_isSubordinate
+-/
 
+#print BumpCovering.ind /-
 /-- Index of a bump function such that `fs i =ᶠ[𝓝 x] 1`. -/
 def ind (x : X) (hx : x ∈ s) : ι :=
   (f.eventuallyEq_one' x hx).some
 #align bump_covering.ind BumpCovering.ind
+-/
 
+/- warning: bump_covering.eventually_eq_one -> BumpCovering.eventuallyEq_one is a dubious translation:
+lean 3 declaration is
+  forall {ι : Type.{u1}} {X : Type.{u2}} [_inst_1 : TopologicalSpace.{u2} X] {s : Set.{u2} X} (f : BumpCovering.{u1, u2} ι X _inst_1 s) (x : X) (hx : Membership.Mem.{u2, u2} X (Set.{u2} X) (Set.hasMem.{u2} X) x s), Filter.EventuallyEq.{u2, 0} X Real (nhds.{u2} X _inst_1 x) (coeFn.{succ u2, succ u2} (ContinuousMap.{u2, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) (fun (_x : ContinuousMap.{u2, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) => X -> Real) (ContinuousMap.hasCoeToFun.{u2, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (BumpCovering.{u1, u2} ι X _inst_1 s) (fun (_x : BumpCovering.{u1, u2} ι X _inst_1 s) => ι -> (ContinuousMap.{u2, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)))) (BumpCovering.hasCoeToFun.{u1, u2} ι X _inst_1 s) f (BumpCovering.ind.{u1, u2} ι X _inst_1 s f x hx))) (OfNat.ofNat.{u2} (X -> Real) 1 (OfNat.mk.{u2} (X -> Real) 1 (One.one.{u2} (X -> Real) (Pi.instOne.{u2, 0} X (fun (ᾰ : X) => Real) (fun (i : X) => Real.hasOne)))))
+but is expected to have type
+  forall {ι : Type.{u1}} {X : Type.{u2}} [_inst_1 : TopologicalSpace.{u2} X] {s : Set.{u2} X} (f : BumpCovering.{u1, u2} ι X _inst_1 s) (x : X) (hx : Membership.mem.{u2, u2} X (Set.{u2} X) (Set.instMembershipSet.{u2} X) x s), Filter.EventuallyEq.{u2, 0} X Real (nhds.{u2} X _inst_1 x) (FunLike.coe.{succ u2, succ u2, 1} (ContinuousMap.{u2, 0} X Real _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 _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) (ContinuousMap.instContinuousMapClassContinuousMap.{u2, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)))) (BumpCovering.toFun.{u1, u2} ι X _inst_1 s f (BumpCovering.ind.{u1, u2} ι X _inst_1 s f x hx))) (OfNat.ofNat.{u2} (X -> Real) 1 (One.toOfNat1.{u2} (X -> Real) (Pi.instOne.{u2, 0} X (fun (a._@.Mathlib.Order.Filter.Basic._hyg.19133 : X) => Real) (fun (i : X) => Real.instOneReal))))
+Case conversion may be inaccurate. Consider using '#align bump_covering.eventually_eq_one BumpCovering.eventuallyEq_oneₓ'. -/
 theorem eventuallyEq_one (x : X) (hx : x ∈ s) : f (f.ind x hx) =ᶠ[𝓝 x] 1 :=
   (f.eventuallyEq_one' x hx).choose_spec
 #align bump_covering.eventually_eq_one BumpCovering.eventuallyEq_one
 
+/- warning: bump_covering.ind_apply -> BumpCovering.ind_apply is a dubious translation:
+lean 3 declaration is
+  forall {ι : Type.{u1}} {X : Type.{u2}} [_inst_1 : TopologicalSpace.{u2} X] {s : Set.{u2} X} (f : BumpCovering.{u1, u2} ι X _inst_1 s) (x : X) (hx : Membership.Mem.{u2, u2} X (Set.{u2} X) (Set.hasMem.{u2} X) x s), Eq.{1} Real (coeFn.{succ u2, succ u2} (ContinuousMap.{u2, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) (fun (_x : ContinuousMap.{u2, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) => X -> Real) (ContinuousMap.hasCoeToFun.{u2, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (BumpCovering.{u1, u2} ι X _inst_1 s) (fun (_x : BumpCovering.{u1, u2} ι X _inst_1 s) => ι -> (ContinuousMap.{u2, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)))) (BumpCovering.hasCoeToFun.{u1, u2} ι X _inst_1 s) f (BumpCovering.ind.{u1, u2} ι X _inst_1 s f x hx)) x) (OfNat.ofNat.{0} Real 1 (OfNat.mk.{0} Real 1 (One.one.{0} Real Real.hasOne)))
+but is expected to have type
+  forall {ι : Type.{u1}} {X : Type.{u2}} [_inst_1 : TopologicalSpace.{u2} X] {s : Set.{u2} X} (f : BumpCovering.{u1, u2} ι X _inst_1 s) (x : X) (hx : Membership.mem.{u2, u2} X (Set.{u2} X) (Set.instMembershipSet.{u2} X) x s), Eq.{1} ((fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : X) => Real) x) (FunLike.coe.{succ u2, succ u2, 1} (ContinuousMap.{u2, 0} X Real _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 _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) (ContinuousMap.instContinuousMapClassContinuousMap.{u2, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)))) (BumpCovering.toFun.{u1, u2} ι X _inst_1 s f (BumpCovering.ind.{u1, u2} ι X _inst_1 s f x hx)) x) (OfNat.ofNat.{0} ((fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : X) => Real) x) 1 (One.toOfNat1.{0} ((fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : X) => Real) x) Real.instOneReal))
+Case conversion may be inaccurate. Consider using '#align bump_covering.ind_apply BumpCovering.ind_applyₓ'. -/
 theorem ind_apply (x : X) (hx : x ∈ s) : f (f.ind x hx) x = 1 :=
   (f.eventuallyEq_one x hx).eq_of_nhds
 #align bump_covering.ind_apply BumpCovering.ind_apply
 
+#print BumpCovering.toPouFun /-
 /-- Partition of unity defined by a `bump_covering`. We use this auxiliary definition to prove some
 properties of the new family of functions before bundling it into a `partition_of_unity`. Do not use
 this definition, use `bump_function.to_partition_of_unity` instead.
@@ -384,15 +535,34 @@ In order to avoid an assumption `linear_order ι`, we use `well_ordering_rel` in
 def toPouFun (i : ι) (x : X) : ℝ :=
   f i x * ∏ᶠ (j) (hj : WellOrderingRel j i), 1 - f j x
 #align bump_covering.to_pou_fun BumpCovering.toPouFun
+-/
 
+/- warning: bump_covering.to_pou_fun_zero_of_zero -> BumpCovering.toPouFun_zero_of_zero is a dubious translation:
+lean 3 declaration is
+  forall {ι : Type.{u1}} {X : Type.{u2}} [_inst_1 : TopologicalSpace.{u2} X] {s : Set.{u2} X} (f : BumpCovering.{u1, u2} ι X _inst_1 s) {i : ι} {x : X}, (Eq.{1} Real (coeFn.{succ u2, succ u2} (ContinuousMap.{u2, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) (fun (_x : ContinuousMap.{u2, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) => X -> Real) (ContinuousMap.hasCoeToFun.{u2, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (BumpCovering.{u1, u2} ι X _inst_1 s) (fun (_x : BumpCovering.{u1, u2} ι X _inst_1 s) => ι -> (ContinuousMap.{u2, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)))) (BumpCovering.hasCoeToFun.{u1, u2} ι X _inst_1 s) f i) x) (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero)))) -> (Eq.{1} Real (BumpCovering.toPouFun.{u1, u2} ι X _inst_1 s f i x) (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero))))
+but is expected to have type
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+Case conversion may be inaccurate. Consider using '#align bump_covering.to_pou_fun_zero_of_zero BumpCovering.toPouFun_zero_of_zeroₓ'. -/
 theorem toPouFun_zero_of_zero {i : ι} {x : X} (h : f i x = 0) : f.toPouFun i x = 0 := by
   rw [to_pou_fun, h, MulZeroClass.zero_mul]
 #align bump_covering.to_pou_fun_zero_of_zero BumpCovering.toPouFun_zero_of_zero
 
+/- warning: bump_covering.support_to_pou_fun_subset -> BumpCovering.support_toPouFun_subset is a dubious translation:
+lean 3 declaration is
+  forall {ι : Type.{u1}} {X : Type.{u2}} [_inst_1 : TopologicalSpace.{u2} X] {s : Set.{u2} X} (f : BumpCovering.{u1, u2} ι X _inst_1 s) (i : ι), HasSubset.Subset.{u2} (Set.{u2} X) (Set.hasSubset.{u2} X) (Function.support.{u2, 0} X Real Real.hasZero (BumpCovering.toPouFun.{u1, u2} ι X _inst_1 s f i)) (Function.support.{u2, 0} X Real Real.hasZero (coeFn.{succ u2, succ u2} (ContinuousMap.{u2, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) (fun (_x : ContinuousMap.{u2, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) => X -> Real) (ContinuousMap.hasCoeToFun.{u2, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (BumpCovering.{u1, u2} ι X _inst_1 s) (fun (_x : BumpCovering.{u1, u2} ι X _inst_1 s) => ι -> (ContinuousMap.{u2, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)))) (BumpCovering.hasCoeToFun.{u1, u2} ι X _inst_1 s) f i)))
+but is expected to have type
+  forall {ι : Type.{u1}} {X : Type.{u2}} [_inst_1 : TopologicalSpace.{u2} X] {s : Set.{u2} X} (f : BumpCovering.{u1, u2} ι X _inst_1 s) (i : ι), HasSubset.Subset.{u2} (Set.{u2} X) (Set.instHasSubsetSet.{u2} X) (Function.support.{u2, 0} X Real Real.instZeroReal (BumpCovering.toPouFun.{u1, u2} ι X _inst_1 s f i)) (Function.support.{u2, 0} X Real Real.instZeroReal (FunLike.coe.{succ u2, succ u2, 1} (ContinuousMap.{u2, 0} X Real _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 _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) (ContinuousMap.instContinuousMapClassContinuousMap.{u2, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)))) (BumpCovering.toFun.{u1, u2} ι X _inst_1 s f i)))
+Case conversion may be inaccurate. Consider using '#align bump_covering.support_to_pou_fun_subset BumpCovering.support_toPouFun_subsetₓ'. -/
 theorem support_toPouFun_subset (i : ι) : support (f.toPouFun i) ⊆ support (f i) := fun x =>
   mt <| f.toPouFun_zero_of_zero
 #align bump_covering.support_to_pou_fun_subset BumpCovering.support_toPouFun_subset
 
+/- warning: bump_covering.to_pou_fun_eq_mul_prod -> BumpCovering.toPouFun_eq_mul_prod is a dubious translation:
+lean 3 declaration is
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+Case conversion may be inaccurate. Consider using '#align bump_covering.to_pou_fun_eq_mul_prod BumpCovering.toPouFun_eq_mul_prodₓ'. -/
 theorem toPouFun_eq_mul_prod (i : ι) (x : X) (t : Finset ι)
     (ht : ∀ j, WellOrderingRel j i → f j x ≠ 0 → j ∈ t) :
     f.toPouFun i x = f i x * ∏ j in t.filterₓ fun j => WellOrderingRel j i, 1 - f j x :=
@@ -403,6 +573,12 @@ theorem toPouFun_eq_mul_prod (i : ι) (x : X) (t : Finset ι)
   exact flip (ht j) hj
 #align bump_covering.to_pou_fun_eq_mul_prod BumpCovering.toPouFun_eq_mul_prod
 
+/- warning: bump_covering.sum_to_pou_fun_eq -> BumpCovering.sum_toPouFun_eq is a dubious translation:
+lean 3 declaration is
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+but is expected to have type
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+Case conversion may be inaccurate. Consider using '#align bump_covering.sum_to_pou_fun_eq BumpCovering.sum_toPouFun_eqₓ'. -/
 theorem sum_toPouFun_eq (x : X) : (∑ᶠ i, f.toPouFun i x) = 1 - ∏ᶠ i, 1 - f i x :=
   by
   set s := (f.point_finite x).toFinset
@@ -423,6 +599,12 @@ theorem sum_toPouFun_eq (x : X) : (∑ᶠ i, f.toPouFun i x) = 1 - ∏ᶠ i, 1 -
   rwa [finite.mem_to_finset]
 #align bump_covering.sum_to_pou_fun_eq BumpCovering.sum_toPouFun_eq
 
+/- warning: bump_covering.exists_finset_to_pou_fun_eventually_eq -> BumpCovering.exists_finset_toPouFun_eventuallyEq is a dubious translation:
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(UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) 1 (One.toOfNat1.{u2} (ContinuousMap.{u2, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) (ContinuousMap.instOneContinuousMap.{u2, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) Real.instOneReal))) (BumpCovering.toFun.{u1, u2} ι X _inst_1 s f j))))))
+Case conversion may be inaccurate. Consider using '#align bump_covering.exists_finset_to_pou_fun_eventually_eq BumpCovering.exists_finset_toPouFun_eventuallyEqₓ'. -/
 theorem exists_finset_toPouFun_eventuallyEq (i : ι) (x : X) :
     ∃ t : Finset ι,
       f.toPouFun i =ᶠ[𝓝 x] f i * ∏ j in t.filterₓ fun j => WellOrderingRel j i, 1 - f j :=
@@ -436,6 +618,7 @@ theorem exists_finset_toPouFun_eventuallyEq (i : ι) (x : X) :
   exact hf.mem_to_finset.2 ⟨y, ⟨hj, hyU⟩⟩
 #align bump_covering.exists_finset_to_pou_fun_eventually_eq BumpCovering.exists_finset_toPouFun_eventuallyEq
 
+#print BumpCovering.continuous_toPouFun /-
 theorem continuous_toPouFun (i : ι) : Continuous (f.toPouFun i) :=
   by
   refine'
@@ -444,7 +627,9 @@ theorem continuous_toPouFun (i : ι) : Continuous (f.toPouFun i) :=
   simp only [mul_support_one_sub]
   exact f.locally_finite
 #align bump_covering.continuous_to_pou_fun BumpCovering.continuous_toPouFun
+-/
 
+#print BumpCovering.toPartitionOfUnity /-
 /-- The partition of unity defined by a `bump_covering`.
 
 The partition of unity is given by the formula `g i x = f i x * ∏ᶠ j < i, (1 - f j x)`. In other
@@ -471,43 +656,82 @@ def toPartitionOfUnity : PartitionOfUnity ι X s
     simp only [ContinuousMap.coe_mk, sum_to_pou_fun_eq, sub_le_self_iff]
     exact finprod_nonneg fun i => sub_nonneg.2 <| f.le_one i x
 #align bump_covering.to_partition_of_unity BumpCovering.toPartitionOfUnity
+-/
 
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+Case conversion may be inaccurate. Consider using '#align bump_covering.to_partition_of_unity_apply BumpCovering.toPartitionOfUnity_applyₓ'. -/
 theorem toPartitionOfUnity_apply (i : ι) (x : X) :
     f.toPartitionOfUnity i x = f i x * ∏ᶠ (j) (hj : WellOrderingRel j i), 1 - f j x :=
   rfl
 #align bump_covering.to_partition_of_unity_apply BumpCovering.toPartitionOfUnity_apply
 
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(UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) (ContinuousMap.instContinuousMapClassContinuousMap.{u2, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)))) (BumpCovering.toFun.{u1, u2} ι X _inst_1 s f j) x)))))
+Case conversion may be inaccurate. Consider using '#align bump_covering.to_partition_of_unity_eq_mul_prod BumpCovering.toPartitionOfUnity_eq_mul_prodₓ'. -/
 theorem toPartitionOfUnity_eq_mul_prod (i : ι) (x : X) (t : Finset ι)
     (ht : ∀ j, WellOrderingRel j i → f j x ≠ 0 → j ∈ t) :
     f.toPartitionOfUnity i x = f i x * ∏ j in t.filterₓ fun j => WellOrderingRel j i, 1 - f j x :=
   f.toPouFun_eq_mul_prod i x t ht
 #align bump_covering.to_partition_of_unity_eq_mul_prod BumpCovering.toPartitionOfUnity_eq_mul_prod
 
+/- warning: bump_covering.exists_finset_to_partition_of_unity_eventually_eq -> BumpCovering.exists_finset_toPartitionOfUnity_eventuallyEq is a dubious translation:
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+Case conversion may be inaccurate. Consider using '#align bump_covering.exists_finset_to_partition_of_unity_eventually_eq BumpCovering.exists_finset_toPartitionOfUnity_eventuallyEqₓ'. -/
 theorem exists_finset_toPartitionOfUnity_eventuallyEq (i : ι) (x : X) :
     ∃ t : Finset ι,
       f.toPartitionOfUnity i =ᶠ[𝓝 x] f i * ∏ j in t.filterₓ fun j => WellOrderingRel j i, 1 - f j :=
   f.exists_finset_toPouFun_eventuallyEq i x
 #align bump_covering.exists_finset_to_partition_of_unity_eventually_eq BumpCovering.exists_finset_toPartitionOfUnity_eventuallyEq
 
+/- warning: bump_covering.to_partition_of_unity_zero_of_zero -> BumpCovering.toPartitionOfUnity_zero_of_zero is a dubious translation:
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+Case conversion may be inaccurate. Consider using '#align bump_covering.to_partition_of_unity_zero_of_zero BumpCovering.toPartitionOfUnity_zero_of_zeroₓ'. -/
 theorem toPartitionOfUnity_zero_of_zero {i : ι} {x : X} (h : f i x = 0) :
     f.toPartitionOfUnity i x = 0 :=
   f.toPouFun_zero_of_zero h
 #align bump_covering.to_partition_of_unity_zero_of_zero BumpCovering.toPartitionOfUnity_zero_of_zero
 
+/- warning: bump_covering.support_to_partition_of_unity_subset -> BumpCovering.support_toPartitionOfUnity_subset is a dubious translation:
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+Case conversion may be inaccurate. Consider using '#align bump_covering.support_to_partition_of_unity_subset BumpCovering.support_toPartitionOfUnity_subsetₓ'. -/
 theorem support_toPartitionOfUnity_subset (i : ι) :
     support (f.toPartitionOfUnity i) ⊆ support (f i) :=
   f.support_toPouFun_subset i
 #align bump_covering.support_to_partition_of_unity_subset BumpCovering.support_toPartitionOfUnity_subset
 
+/- warning: bump_covering.sum_to_partition_of_unity_eq -> BumpCovering.sum_toPartitionOfUnity_eq is a dubious translation:
+lean 3 declaration is
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+but is expected to have type
+  forall {ι : Type.{u1}} {X : Type.{u2}} [_inst_1 : TopologicalSpace.{u2} X] {s : Set.{u2} X} (f : BumpCovering.{u1, u2} ι X _inst_1 s) (x : X), Eq.{1} ((fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : X) => Real) x) (finsum.{0, succ u1} ((fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : X) => Real) x) ι Real.instAddCommMonoidReal (fun (i : ι) => FunLike.coe.{succ u2, succ u2, 1} (ContinuousMap.{u2, 0} X Real _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 _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) (ContinuousMap.instContinuousMapClassContinuousMap.{u2, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)))) (PartitionOfUnity.toFun.{u1, u2} ι X _inst_1 s (BumpCovering.toPartitionOfUnity.{u1, u2} ι X _inst_1 s f) i) x)) (HSub.hSub.{0, 0, 0} ((fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : X) => Real) x) ((fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : X) => Real) x) ((fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : X) => Real) x) (instHSub.{0} ((fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : X) => Real) x) Real.instSubReal) (OfNat.ofNat.{0} ((fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : X) => Real) x) 1 (One.toOfNat1.{0} ((fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : X) => Real) x) Real.instOneReal)) (finprod.{0, succ u1} ((fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : X) => Real) x) ι Real.instCommMonoidReal (fun (i : ι) => HSub.hSub.{0, 0, 0} ((fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : X) => Real) x) ((fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : X) => Real) x) ((fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : X) => Real) x) (instHSub.{0} ((fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : X) => Real) x) Real.instSubReal) (OfNat.ofNat.{0} ((fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : X) => Real) x) 1 (One.toOfNat1.{0} ((fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : X) => Real) x) Real.instOneReal)) (FunLike.coe.{succ u2, succ u2, 1} (ContinuousMap.{u2, 0} X Real _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 _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) (ContinuousMap.instContinuousMapClassContinuousMap.{u2, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)))) (BumpCovering.toFun.{u1, u2} ι X _inst_1 s f i) x))))
+Case conversion may be inaccurate. Consider using '#align bump_covering.sum_to_partition_of_unity_eq BumpCovering.sum_toPartitionOfUnity_eqₓ'. -/
 theorem sum_toPartitionOfUnity_eq (x : X) :
     (∑ᶠ i, f.toPartitionOfUnity i x) = 1 - ∏ᶠ i, 1 - f i x :=
   f.sum_toPouFun_eq x
 #align bump_covering.sum_to_partition_of_unity_eq BumpCovering.sum_toPartitionOfUnity_eq
 
+#print BumpCovering.IsSubordinate.toPartitionOfUnity /-
 theorem IsSubordinate.toPartitionOfUnity {f : BumpCovering ι X s} {U : ι → Set X}
     (h : f.IsSubordinate U) : f.toPartitionOfUnity.IsSubordinate U := fun i =>
   Subset.trans (closure_mono <| f.support_toPartitionOfUnity_subset i) (h i)
 #align bump_covering.is_subordinate.to_partition_of_unity BumpCovering.IsSubordinate.toPartitionOfUnity
+-/
 
 end BumpCovering
 
@@ -518,6 +742,7 @@ variable {s : Set X}
 instance [Inhabited ι] : Inhabited (PartitionOfUnity ι X s) :=
   ⟨BumpCovering.toPartitionOfUnity default⟩
 
+#print PartitionOfUnity.exists_isSubordinate_of_locallyFinite /-
 /-- If `X` is a normal topological space and `U` is a locally finite open covering of a closed set
 `s`, then there exists a `partition_of_unity ι X s` that is subordinate to `U`. If `X` is a
 paracompact space, then the assumption `hf : locally_finite U` can be omitted, see
@@ -528,7 +753,9 @@ theorem exists_isSubordinate_of_locallyFinite [NormalSpace X] (hs : IsClosed s)
   let ⟨f, hf⟩ := BumpCovering.exists_isSubordinate_of_locallyFinite hs U ho hf hU
   ⟨f.toPartitionOfUnity, hf.toPartitionOfUnity⟩
 #align partition_of_unity.exists_is_subordinate_of_locally_finite PartitionOfUnity.exists_isSubordinate_of_locallyFinite
+-/
 
+#print PartitionOfUnity.exists_isSubordinate /-
 /-- If `X` is a paracompact normal topological space and `U` is an open covering of a closed set
 `s`, then there exists a `partition_of_unity ι X s` that is subordinate to `U`. -/
 theorem exists_isSubordinate [NormalSpace X] [ParacompactSpace X] (hs : IsClosed s) (U : ι → Set X)
@@ -537,6 +764,7 @@ theorem exists_isSubordinate [NormalSpace X] [ParacompactSpace X] (hs : IsClosed
   let ⟨f, hf⟩ := BumpCovering.exists_isSubordinate hs U ho hU
   ⟨f.toPartitionOfUnity, hf.toPartitionOfUnity⟩
 #align partition_of_unity.exists_is_subordinate PartitionOfUnity.exists_isSubordinate
+-/
 
 end PartitionOfUnity
 

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

@@ -386,7 +386,7 @@ def toPouFun (i : ι) (x : X) : ℝ :=
 #align bump_covering.to_pou_fun BumpCovering.toPouFun
 
 theorem toPouFun_zero_of_zero {i : ι} {x : X} (h : f i x = 0) : f.toPouFun i x = 0 := by
-  rw [to_pou_fun, h, zero_mul]
+  rw [to_pou_fun, h, MulZeroClass.zero_mul]
 #align bump_covering.to_pou_fun_zero_of_zero BumpCovering.toPouFun_zero_of_zero
 
 theorem support_toPouFun_subset (i : ι) : support (f.toPouFun i) ⊆ support (f i) := fun x =>

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

@@ -302,7 +302,7 @@ theorem exists_finset_nhd (ρ : PartitionOfUnity ι X univ) (x₀ : X) :
 
 theorem exists_finset_nhd_support_subset {U : ι → Set X} (hso : f.IsSubordinate U)
     (ho : ∀ i, IsOpen (U i)) (x : X) :
-    ∃ (is : Finset ι),  ∃ n ∈ 𝓝 x, n ⊆ ⋂ i ∈ is, U i ∧ ∀ z ∈ n, (support (f · z)) ⊆ is :=
+    ∃ is : Finset ι, ∃ n ∈ 𝓝 x, n ⊆ ⋂ i ∈ is, U i ∧ ∀ z ∈ n, (support (f · z)) ⊆ is :=
   f.locallyFinite.exists_finset_nhd_support_subset hso ho x
 #align partition_of_unity.exists_finset_nhd_support_subset PartitionOfUnity.exists_finset_nhd_support_subset
 

@@ -483,7 +483,7 @@ theorem toPOUFun_eq_mul_prod (i : ι) (x : X) (t : Finset ι)
     (ht : ∀ j, WellOrderingRel j i → f j x ≠ 0 → j ∈ t) :
     f.toPOUFun i x = f i x * ∏ j in t.filter fun j => WellOrderingRel j i, (1 - f j x) := by
   refine' congr_arg _ (finprod_cond_eq_prod_of_cond_iff _ fun {j} hj => _)
-  rw [Ne.def, sub_eq_self] at hj
+  rw [Ne, sub_eq_self] at hj
   rw [Finset.mem_filter, Iff.comm, and_iff_right_iff_imp]
   exact flip (ht j) hj
 #align bump_covering.to_pou_fun_eq_mul_prod BumpCovering.toPOUFun_eq_mul_prod

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

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

@@ -80,7 +80,8 @@ universe u v
 
 open Function Set Filter
 
-open BigOperators Topology Classical
+open scoped Classical
+open BigOperators Topology
 
 noncomputable section
 

From sphere-eversion: the result is used there for smooth partitions of unity, but also holds in the continuous setting.

In passing,

• use Type* (not Type _) in sum_finsupport_smul_eq_finsum,
• tag SmoothPartitionOfUnity.toPartitionOfUnity with @simps.

@@ -180,7 +180,7 @@ section finsupport
 variable {s : Set X} (ρ : PartitionOfUnity ι X s) (x₀ : X)
 
 /-- The support of a partition of unity at a point `x₀` as a `Finset`.
-  This is the set of `i : ι` such that `f i` doesn't vanish at `x₀`. -/
+  This is the set of `i : ι` such that `x₀ ∈ support f i`, i.e. `f i ≠ x₀`. -/
 def finsupport : Finset ι := (ρ.locallyFinite.point_finite x₀).toFinset
 
 @[simp]
@@ -210,7 +210,7 @@ theorem sum_finsupport' (hx₀ : x₀ ∈ s) {I : Finset ι} (hI : ρ.finsupport
   simp only [Finset.mem_sdiff, ρ.mem_finsupport, mem_support, Classical.not_not] at hx
   exact hx.2
 
-theorem sum_finsupport_smul_eq_finsum {M : Type _} [AddCommGroup M] [Module ℝ M] (φ : ι → X → M) :
+theorem sum_finsupport_smul_eq_finsum {M : Type*} [AddCommGroup M] [Module ℝ M] (φ : ι → X → M) :
     ∑ i in ρ.finsupport x₀, ρ i x₀ • φ i x₀ = ∑ᶠ i, ρ i x₀ • φ i x₀ := by
   apply (finsum_eq_sum_of_support_subset _ _).symm
   have : (fun i ↦ (ρ i) x₀ • φ i x₀) = (fun i ↦ (ρ i) x₀) • (fun i ↦ φ i x₀) :=
@@ -220,6 +220,44 @@ theorem sum_finsupport_smul_eq_finsum {M : Type _} [AddCommGroup M] [Module ℝ
 
 end finsupport
 
+section fintsupport -- partitions of unity have locally finite `tsupport`
+
+variable {s : Set X} (ρ : PartitionOfUnity ι X s) (x₀ : X)
+
+/-- The `tsupport`s of a partition of unity are locally finite. -/
+theorem finite_tsupport : {i | x₀ ∈ tsupport (ρ i)}.Finite := by
+  rcases ρ.locallyFinite x₀ with ⟨t, t_in, ht⟩
+  apply ht.subset
+  rintro i hi
+  simp only [inter_comm]
+  exact mem_closure_iff_nhds.mp hi t t_in
+
+/-- The tsupport of a partition of unity at a point `x₀` as a `Finset`.
+  This is the set of `i : ι` such that `x₀ ∈ tsupport f i`. -/
+def fintsupport (x₀ : X) : Finset ι :=
+  (ρ.finite_tsupport x₀).toFinset
+
+theorem mem_fintsupport_iff (i : ι) : i ∈ ρ.fintsupport x₀ ↔ x₀ ∈ tsupport (ρ i) :=
+  Finite.mem_toFinset _
+
+theorem eventually_fintsupport_subset :
+    ∀ᶠ y in 𝓝 x₀, ρ.fintsupport y ⊆ ρ.fintsupport x₀ := by
+  apply (ρ.locallyFinite.closure.eventually_subset (fun _ ↦ isClosed_closure) x₀).mono
+  intro y hy z hz
+  rw [PartitionOfUnity.mem_fintsupport_iff] at *
+  exact hy hz
+
+theorem finsupport_subset_fintsupport : ρ.finsupport x₀ ⊆ ρ.fintsupport x₀ := fun i hi ↦ by
+  rw [ρ.mem_fintsupport_iff]
+  apply subset_closure
+  exact (ρ.mem_finsupport x₀).mp hi
+
+theorem eventually_finsupport_subset : ∀ᶠ y in 𝓝 x₀, ρ.finsupport y ⊆ ρ.fintsupport x₀ :=
+  (ρ.eventually_fintsupport_subset x₀).mono
+    fun y hy ↦ (ρ.finsupport_subset_fintsupport y).trans hy
+
+end fintsupport
+
 /-- If `f` is a partition of unity on `s : Set X` and `g : X → E` is continuous at every point of
 the topological support of some `f i`, then `fun x ↦ f i x • g x` is continuous on the whole space.
 -/

@@ -183,17 +183,17 @@ variable {s : Set X} (ρ : PartitionOfUnity ι X s) (x₀ : X)
   This is the set of `i : ι` such that `f i` doesn't vanish at `x₀`. -/
 def finsupport : Finset ι := (ρ.locallyFinite.point_finite x₀).toFinset
 
-@[simp]
-theorem coe_finsupport (x₀ : X) :
-    (ρ.finsupport x₀ : Set ι) = support fun i ↦ ρ i x₀ := by
-  rw [finsupport, Finite.coe_toFinset, setOf_set]
-  rfl
-
 @[simp]
 theorem mem_finsupport (x₀ : X) {i} :
     i ∈ ρ.finsupport x₀ ↔ i ∈ support fun i ↦ ρ i x₀ := by
   simp only [finsupport, mem_support, Finite.mem_toFinset, mem_setOf_eq]
 
+@[simp]
+theorem coe_finsupport (x₀ : X) :
+    (ρ.finsupport x₀ : Set ι) = support fun i ↦ ρ i x₀ := by
+  ext
+  rw [Finset.mem_coe, mem_finsupport]
+
 variable {x₀ : X}
 
 theorem sum_finsupport (hx₀ : x₀ ∈ s) : ∑ i in ρ.finsupport x₀, ρ i x₀ = 1 := by
@@ -213,7 +213,8 @@ theorem sum_finsupport' (hx₀ : x₀ ∈ s) {I : Finset ι} (hI : ρ.finsupport
 theorem sum_finsupport_smul_eq_finsum {M : Type _} [AddCommGroup M] [Module ℝ M] (φ : ι → X → M) :
     ∑ i in ρ.finsupport x₀, ρ i x₀ • φ i x₀ = ∑ᶠ i, ρ i x₀ • φ i x₀ := by
   apply (finsum_eq_sum_of_support_subset _ _).symm
-  have : (fun i ↦ (ρ i) x₀ • φ i x₀) = (fun i ↦ (ρ i) x₀) • (fun i ↦ φ i x₀) := rfl
+  have : (fun i ↦ (ρ i) x₀ • φ i x₀) = (fun i ↦ (ρ i) x₀) • (fun i ↦ φ i x₀) :=
+    funext fun _ => (Pi.smul_apply' _ _ _).symm
   rw [ρ.coe_finsupport x₀, this, support_smul]
   exact inter_subset_left _ _
 

This follows up from #9785, which renamed FunLike to DFunLike, by introducing a new abbreviation FunLike F α β := DFunLike F α (fun _ => β), to make the non-dependent use of FunLike easier.

I searched for the pattern DFunLike.*fun and DFunLike.*λ in all files to replace expressions of the form DFunLike F α (fun _ => β) with FunLike F α β. I did this everywhere except for extends clauses for two reasons: it would conflict with #8386, and more importantly extends must directly refer to a structure with no unfolding of defs or abbrevs.

@@ -135,7 +135,7 @@ namespace PartitionOfUnity
 variable {E : Type*} [AddCommMonoid E] [SMulWithZero ℝ E] [TopologicalSpace E] [ContinuousSMul ℝ E]
   {s : Set X} (f : PartitionOfUnity ι X s)
 
-instance : DFunLike (PartitionOfUnity ι X s) ι fun _ ↦ C(X, ℝ) where
+instance : FunLike (PartitionOfUnity ι X s) ι C(X, ℝ) where
   coe := toFun
   coe_injective' := fun f g h ↦ by cases f; cases g; congr
 
@@ -281,7 +281,7 @@ namespace BumpCovering
 
 variable {s : Set X} (f : BumpCovering ι X s)
 
-instance : DFunLike (BumpCovering ι X s) ι fun _ ↦ C(X, ℝ) where
+instance : FunLike (BumpCovering ι X s) ι C(X, ℝ) where
   coe := toFun
   coe_injective' := fun f g h ↦ by cases f; cases g; congr
 

This prepares for the introduction of a non-dependent synonym of FunLike, which helps a lot with keeping #8386 readable.

This is entirely search-and-replace in 680197f combined with manual fixes in 4145626, e900597 and b8428f8. The commands that generated this change:

sed -i 's/\bFunLike\b/DFunLike/g' {Archive,Counterexamples,Mathlib,test}/**/*.lean
sed -i 's/\btoFunLike\b/toDFunLike/g' {Archive,Counterexamples,Mathlib,test}/**/*.lean
sed -i 's/import Mathlib.Data.DFunLike/import Mathlib.Data.FunLike/g' {Archive,Counterexamples,Mathlib,test}/**/*.lean
sed -i 's/\bHom_FunLike\b/Hom_DFunLike/g' {Archive,Counterexamples,Mathlib,test}/**/*.lean     
sed -i 's/\binstFunLike\b/instDFunLike/g' {Archive,Counterexamples,Mathlib,test}/**/*.lean
sed -i 's/\bfunLike\b/instDFunLike/g' {Archive,Counterexamples,Mathlib,test}/**/*.lean
sed -i 's/\btoo many metavariables to apply `fun_like.has_coe_to_fun`/too many metavariables to apply `DFunLike.hasCoeToFun`/g' {Archive,Counterexamples,Mathlib,test}/**/*.lean

Co-authored-by: Anne Baanen <Vierkantor@users.noreply.github.com>

@@ -135,7 +135,7 @@ namespace PartitionOfUnity
 variable {E : Type*} [AddCommMonoid E] [SMulWithZero ℝ E] [TopologicalSpace E] [ContinuousSMul ℝ E]
   {s : Set X} (f : PartitionOfUnity ι X s)
 
-instance : FunLike (PartitionOfUnity ι X s) ι fun _ ↦ C(X, ℝ) where
+instance : DFunLike (PartitionOfUnity ι X s) ι fun _ ↦ C(X, ℝ) where
   coe := toFun
   coe_injective' := fun f g h ↦ by cases f; cases g; congr
 
@@ -281,7 +281,7 @@ namespace BumpCovering
 
 variable {s : Set X} (f : BumpCovering ι X s)
 
-instance : FunLike (BumpCovering ι X s) ι fun _ ↦ C(X, ℝ) where
+instance : DFunLike (BumpCovering ι X s) ι fun _ ↦ C(X, ℝ) where
   coe := toFun
   coe_injective' := fun f g h ↦ by cases f; cases g; congr
 

• add finsupport: the support (the set of indices which functions are non-vanishing) at a point, as a Finset
• every point has a neighbourhood on which only finitely many functions are non-vanishing

From sphere-eversion; I'm just upstreaming results.

@@ -16,8 +16,9 @@ import Mathlib.Topology.UrysohnsLemma
 # Continuous partition of unity
 
 In this file we define `PartitionOfUnity (ι X : Type*) [TopologicalSpace X] (s : Set X := univ)`
-to be a continuous partition of unity on `s` indexed by `ι`. More precisely, `f : PartitionOfUnity
-ι X s` is a collection of continuous functions `f i : C(X, ℝ)`, `i : ι`, such that
+to be a continuous partition of unity on `s` indexed by `ι`. More precisely,
+`f : PartitionOfUnity ι X s` is a collection of continuous functions `f i : C(X, ℝ)`, `i : ι`,
+such that
 
 * the supports of `f i` form a locally finite family of sets;
 * each `f i` is nonnegative;
@@ -174,6 +175,50 @@ theorem le_one (i : ι) (x : X) : f i x ≤ 1 :=
   (single_le_finsum i (f.locallyFinite.point_finite x) fun j => f.nonneg j x).trans (f.sum_le_one x)
 #align partition_of_unity.le_one PartitionOfUnity.le_one
 
+section finsupport
+
+variable {s : Set X} (ρ : PartitionOfUnity ι X s) (x₀ : X)
+
+/-- The support of a partition of unity at a point `x₀` as a `Finset`.
+  This is the set of `i : ι` such that `f i` doesn't vanish at `x₀`. -/
+def finsupport : Finset ι := (ρ.locallyFinite.point_finite x₀).toFinset
+
+@[simp]
+theorem coe_finsupport (x₀ : X) :
+    (ρ.finsupport x₀ : Set ι) = support fun i ↦ ρ i x₀ := by
+  rw [finsupport, Finite.coe_toFinset, setOf_set]
+  rfl
+
+@[simp]
+theorem mem_finsupport (x₀ : X) {i} :
+    i ∈ ρ.finsupport x₀ ↔ i ∈ support fun i ↦ ρ i x₀ := by
+  simp only [finsupport, mem_support, Finite.mem_toFinset, mem_setOf_eq]
+
+variable {x₀ : X}
+
+theorem sum_finsupport (hx₀ : x₀ ∈ s) : ∑ i in ρ.finsupport x₀, ρ i x₀ = 1 := by
+  rw [← ρ.sum_eq_one hx₀, finsum_eq_sum_of_support_subset _ (ρ.coe_finsupport x₀).superset]
+
+theorem sum_finsupport' (hx₀ : x₀ ∈ s) {I : Finset ι} (hI : ρ.finsupport x₀ ⊆ I) :
+    ∑ i in I, ρ i x₀ = 1 := by
+  classical
+  rw [← Finset.sum_sdiff hI, ρ.sum_finsupport hx₀]
+  suffices ∑ i in I \ ρ.finsupport x₀, (ρ i) x₀ = ∑ i in I \ ρ.finsupport x₀, 0 by
+    rw [this, add_left_eq_self, Finset.sum_const_zero]
+  apply Finset.sum_congr rfl
+  rintro x hx
+  simp only [Finset.mem_sdiff, ρ.mem_finsupport, mem_support, Classical.not_not] at hx
+  exact hx.2
+
+theorem sum_finsupport_smul_eq_finsum {M : Type _} [AddCommGroup M] [Module ℝ M] (φ : ι → X → M) :
+    ∑ i in ρ.finsupport x₀, ρ i x₀ • φ i x₀ = ∑ᶠ i, ρ i x₀ • φ i x₀ := by
+  apply (finsum_eq_sum_of_support_subset _ _).symm
+  have : (fun i ↦ (ρ i) x₀ • φ i x₀) = (fun i ↦ (ρ i) x₀) • (fun i ↦ φ i x₀) := rfl
+  rw [ρ.coe_finsupport x₀, this, support_smul]
+  exact inter_subset_left _ _
+
+end finsupport
+
 /-- If `f` is a partition of unity on `s : Set X` and `g : X → E` is continuous at every point of
 the topological support of some `f i`, then `fun x ↦ f i x • g x` is continuous on the whole space.
 -/
@@ -201,6 +246,20 @@ def IsSubordinate (U : ι → Set X) : Prop :=
 
 variable {f}
 
+theorem exists_finset_nhd' {s : Set X} (ρ : PartitionOfUnity ι X s) (x₀ : X) :
+    ∃ I : Finset ι, (∀ᶠ x in 𝓝[s] x₀, ∑ i in I, ρ i x = 1) ∧
+      ∀ᶠ x in 𝓝 x₀, support (ρ · x) ⊆ I := by
+  rcases ρ.locallyFinite.exists_finset_support x₀ with ⟨I, hI⟩
+  refine ⟨I, eventually_nhdsWithin_iff.mpr (hI.mono fun x hx x_in ↦ ?_), hI⟩
+  have : ∑ᶠ i : ι, ρ i x = ∑ i : ι in I, ρ i x := finsum_eq_sum_of_support_subset _ hx
+  rwa [eq_comm, ρ.sum_eq_one x_in] at this
+
+theorem exists_finset_nhd (ρ : PartitionOfUnity ι X univ) (x₀ : X) :
+    ∃ I : Finset ι, ∀ᶠ x in 𝓝 x₀, ∑ i in I, ρ i x = 1 ∧ support (ρ · x) ⊆ I := by
+  rcases ρ.exists_finset_nhd' x₀ with ⟨I, H⟩
+  use I
+  rwa [nhdsWithin_univ, ← eventually_and] at H
+
 theorem exists_finset_nhd_support_subset {U : ι → Set X} (hso : f.IsSubordinate U)
     (ho : ∀ i, IsOpen (U i)) (x : X) :
     ∃ (is : Finset ι),  ∃ n ∈ 𝓝 x, n ⊆ ⋂ i ∈ is, U i ∧ ∀ z ∈ n, (support (f · z)) ⊆ is :=

Follow-up #9184

@@ -203,12 +203,8 @@ variable {f}
 
 theorem exists_finset_nhd_support_subset {U : ι → Set X} (hso : f.IsSubordinate U)
     (ho : ∀ i, IsOpen (U i)) (x : X) :
-    ∃ (is : Finset ι) (n : Set X) (_ : n ∈ 𝓝 x) (_ : n ⊆ ⋂ i ∈ is, U i),
-      ∀ z ∈ n, (support fun i => f i z) ⊆ is :=
-  -- Porting note: Original proof was simply
-  -- `f.locallyFinite.exists_finset_nhd_support_subset hso ho x`
-  let ⟨a, ⟨b, ⟨c, ⟨d, e⟩⟩⟩⟩ := f.locallyFinite.exists_finset_nhd_support_subset hso ho x
-  ⟨a, b, c, d, e⟩
+    ∃ (is : Finset ι),  ∃ n ∈ 𝓝 x, n ⊆ ⋂ i ∈ is, U i ∧ ∀ z ∈ n, (support (f · z)) ⊆ is :=
+  f.locallyFinite.exists_finset_nhd_support_subset hso ho x
 #align partition_of_unity.exists_finset_nhd_support_subset PartitionOfUnity.exists_finset_nhd_support_subset
 
 /-- If `f` is a partition of unity that is subordinate to a family of open sets `U i` and

Three particular examples which caught my eye; not exhaustive.

@@ -321,7 +321,7 @@ theorem exists_isSubordinate_of_locallyFinite [NormalSpace X] (hs : IsClosed s)
   let ⟨f, _, hfU⟩ :=
     exists_isSubordinate_of_locallyFinite_of_prop (fun _ => True)
       (fun _ _ hs ht hd =>
-        (exists_continuous_zero_one_of_closed hs ht hd).imp fun _ hf => ⟨trivial, hf⟩)
+        (exists_continuous_zero_one_of_isClosed hs ht hd).imp fun _ hf => ⟨trivial, hf⟩)
       hs U ho hf hU
   ⟨f, hfU⟩
 #align bump_covering.exists_is_subordinate_of_locally_finite BumpCovering.exists_isSubordinate_of_locallyFinite

Split up the 2000-line Topology/SubsetProperties.lean into several smaller files. Not only is it too huge, but the name is very unhelpful, since actually about 90% of the file is about compactness; I've moved this material into various files inside a new subdirectory Topology/Compactness/.

@@ -6,7 +6,7 @@ Authors: Yury Kudryashov
 import Mathlib.Algebra.BigOperators.Finprod
 import Mathlib.SetTheory.Ordinal.Basic
 import Mathlib.Topology.ContinuousFunction.Algebra
-import Mathlib.Topology.Paracompact
+import Mathlib.Topology.Compactness.Paracompact
 import Mathlib.Topology.ShrinkingLemma
 import Mathlib.Topology.UrysohnsLemma
 

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

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

@@ -377,7 +377,7 @@ def toPOUFun (i : ι) (x : X) : ℝ :=
 #align bump_covering.to_pou_fun BumpCovering.toPOUFun
 
 theorem toPOUFun_zero_of_zero {i : ι} {x : X} (h : f i x = 0) : f.toPOUFun i x = 0 := by
-  rw [toPOUFun, h, MulZeroClass.zero_mul]
+  rw [toPOUFun, h, zero_mul]
 #align bump_covering.to_pou_fun_zero_of_zero BumpCovering.toPOUFun_zero_of_zero
 
 theorem support_toPOUFun_subset (i : ι) : support (f.toPOUFun i) ⊆ support (f i) :=

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

This has nice performance benefits.

@@ -15,7 +15,7 @@ import Mathlib.Topology.UrysohnsLemma
 /-!
 # Continuous partition of unity
 
-In this file we define `PartitionOfUnity (ι X : Type _) [TopologicalSpace X] (s : Set X := univ)`
+In this file we define `PartitionOfUnity (ι X : Type*) [TopologicalSpace X] (s : Set X := univ)`
 to be a continuous partition of unity on `s` indexed by `ι`. More precisely, `f : PartitionOfUnity
 ι X s` is a collection of continuous functions `f i : C(X, ℝ)`, `i : ι`, such that
 
@@ -28,7 +28,7 @@ In the case `s = univ` the last assumption follows from the previous one but it
 have this assumption in the case `s ≠ univ`.
 
 We also define a bump function covering,
-`BumpCovering (ι X : Type _) [TopologicalSpace X] (s : Set X := univ)`, to be a collection of
+`BumpCovering (ι X : Type*) [TopologicalSpace X] (s : Set X := univ)`, to be a collection of
 functions `f i : C(X, ℝ)`, `i : ι`, such that
 
 * the supports of `f i` form a locally finite family of sets;
@@ -96,7 +96,7 @@ If `X` is a normal paracompact space, then `PartitionOfUnity.exists_isSubordinat
 that for every open covering `U : Set (Set X)` of `s` there exists a partition of unity that is
 subordinate to `U`.
 -/
-structure PartitionOfUnity (ι X : Type _) [TopologicalSpace X] (s : Set X := univ) where
+structure PartitionOfUnity (ι X : Type*) [TopologicalSpace X] (s : Set X := univ) where
   toFun : ι → C(X, ℝ)
   locallyFinite' : LocallyFinite fun i => support (toFun i)
   nonneg' : 0 ≤ toFun
@@ -119,7 +119,7 @@ If `X` is a normal paracompact space, then `BumpCovering.exists_isSubordinate` g
 every open covering `U : Set (Set X)` of `s` there exists a `BumpCovering` of `s` that is
 subordinate to `U`.
 -/
-structure BumpCovering (ι X : Type _) [TopologicalSpace X] (s : Set X := univ) where
+structure BumpCovering (ι X : Type*) [TopologicalSpace X] (s : Set X := univ) where
   toFun : ι → C(X, ℝ)
   locallyFinite' : LocallyFinite fun i => support (toFun i)
   nonneg' : 0 ≤ toFun
@@ -131,7 +131,7 @@ variable {ι : Type u} {X : Type v} [TopologicalSpace X]
 
 namespace PartitionOfUnity
 
-variable {E : Type _} [AddCommMonoid E] [SMulWithZero ℝ E] [TopologicalSpace E] [ContinuousSMul ℝ E]
+variable {E : Type*} [AddCommMonoid E] [SMulWithZero ℝ E] [TopologicalSpace E] [ContinuousSMul ℝ E]
   {s : Set X} (f : PartitionOfUnity ι X s)
 
 instance : FunLike (PartitionOfUnity ι X s) ι fun _ ↦ C(X, ℝ) where

• depends on: #5966

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

@@ -2,11 +2,6 @@
 Copyright (c) 2021 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.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.Algebra.BigOperators.Finprod
 import Mathlib.SetTheory.Ordinal.Basic
@@ -15,6 +10,8 @@ import Mathlib.Topology.Paracompact
 import Mathlib.Topology.ShrinkingLemma
 import Mathlib.Topology.UrysohnsLemma
 
+#align_import topology.partition_of_unity from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
+
 /-!
 # Continuous partition of unity
 

• Also remove most superfluous parentheses around big operators (∑, ∏ and variants).
• roughly the used regex: ([^a-zA-Zα-ωΑ-Ω'𝓝ℳ₀𝕂ₛ)]) \(([∑∏][^()∑∏]*,[^()∑∏:]*)\) ([⊂⊆=<≤]) replaced by $1 $2 $3

@@ -103,8 +103,8 @@ structure PartitionOfUnity (ι X : Type _) [TopologicalSpace X] (s : Set X := un
   toFun : ι → C(X, ℝ)
   locallyFinite' : LocallyFinite fun i => support (toFun i)
   nonneg' : 0 ≤ toFun
-  sum_eq_one' : ∀ x ∈ s, (∑ᶠ i, toFun i x) = 1
-  sum_le_one' : ∀ x, (∑ᶠ i, toFun i x) ≤ 1
+  sum_eq_one' : ∀ x ∈ s, ∑ᶠ i, toFun i x = 1
+  sum_le_one' : ∀ x, ∑ᶠ i, toFun i x ≤ 1
 #align partition_of_unity PartitionOfUnity
 
 /-- A `BumpCovering ι X s` is an indexed family of functions `f i`, `i : ι`, such that
@@ -153,7 +153,7 @@ theorem nonneg (i : ι) (x : X) : 0 ≤ f i x :=
   f.nonneg' i x
 #align partition_of_unity.nonneg PartitionOfUnity.nonneg
 
-theorem sum_eq_one {x : X} (hx : x ∈ s) : (∑ᶠ i, f i x) = 1 :=
+theorem sum_eq_one {x : X} (hx : x ∈ s) : ∑ᶠ i, f i x = 1 :=
   f.sum_eq_one' x hx
 #align partition_of_unity.sum_eq_one PartitionOfUnity.sum_eq_one
 
@@ -165,7 +165,7 @@ theorem exists_pos {x : X} (hx : x ∈ s) : ∃ i, 0 < f i x := by
   simpa only [fun i => (H i).antisymm (f.nonneg i x), finsum_zero] using zero_ne_one
 #align partition_of_unity.exists_pos PartitionOfUnity.exists_pos
 
-theorem sum_le_one (x : X) : (∑ᶠ i, f i x) ≤ 1 :=
+theorem sum_le_one (x : X) : ∑ᶠ i, f i x ≤ 1 :=
   f.sum_le_one' x
 #align partition_of_unity.sum_le_one PartitionOfUnity.sum_le_one
 
@@ -396,7 +396,7 @@ theorem toPOUFun_eq_mul_prod (i : ι) (x : X) (t : Finset ι)
   exact flip (ht j) hj
 #align bump_covering.to_pou_fun_eq_mul_prod BumpCovering.toPOUFun_eq_mul_prod
 
-theorem sum_toPOUFun_eq (x : X) : (∑ᶠ i, f.toPOUFun i x) = 1 - ∏ᶠ i, (1 - f i x) := by
+theorem sum_toPOUFun_eq (x : X) : ∑ᶠ i, f.toPOUFun i x = 1 - ∏ᶠ i, (1 - f i x) := by
   set s := (f.point_finite x).toFinset
   have hs : (s : Set ι) = { i | f i x ≠ 0 } := Finite.coe_toFinset _
   have A : (support fun i => toPOUFun f i x) ⊆ s := by
@@ -481,7 +481,7 @@ theorem support_toPartitionOfUnity_subset (i : ι) :
 #align bump_covering.support_to_partition_of_unity_subset BumpCovering.support_toPartitionOfUnity_subset
 
 theorem sum_toPartitionOfUnity_eq (x : X) :
-    (∑ᶠ i, f.toPartitionOfUnity i x) = 1 - ∏ᶠ i, (1 - f i x) :=
+    ∑ᶠ i, f.toPartitionOfUnity i x = 1 - ∏ᶠ i, (1 - f i x) :=
   f.sum_toPOUFun_eq x
 #align bump_covering.sum_to_partition_of_unity_eq BumpCovering.sum_toPartitionOfUnity_eq
 

• Replace notation3 lines with the latest versions in mathport.
• Fix Topology.PartitionOfUnity.
• Fix names in Topology.PartitionOfUnity (locally_finite' -> locallyFinite').
• Use FunLike for PartitionOfUnity and BumpCovering

@@ -101,7 +101,7 @@ subordinate to `U`.
 -/
 structure PartitionOfUnity (ι X : Type _) [TopologicalSpace X] (s : Set X := univ) where
   toFun : ι → C(X, ℝ)
-  locally_finite' : LocallyFinite fun i => support (toFun i)
+  locallyFinite' : LocallyFinite fun i => support (toFun i)
   nonneg' : 0 ≤ toFun
   sum_eq_one' : ∀ x ∈ s, (∑ᶠ i, toFun i x) = 1
   sum_le_one' : ∀ x, (∑ᶠ i, toFun i x) ≤ 1
@@ -124,7 +124,7 @@ subordinate to `U`.
 -/
 structure BumpCovering (ι X : Type _) [TopologicalSpace X] (s : Set X := univ) where
   toFun : ι → C(X, ℝ)
-  locally_finite' : LocallyFinite fun i => support (toFun i)
+  locallyFinite' : LocallyFinite fun i => support (toFun i)
   nonneg' : 0 ≤ toFun
   le_one' : toFun ≤ 1
   eventuallyEq_one' : ∀ x ∈ s, ∃ i, toFun i =ᶠ[𝓝 x] 1
@@ -137,11 +137,12 @@ namespace PartitionOfUnity
 variable {E : Type _} [AddCommMonoid E] [SMulWithZero ℝ E] [TopologicalSpace E] [ContinuousSMul ℝ E]
   {s : Set X} (f : PartitionOfUnity ι X s)
 
-instance : CoeFun (PartitionOfUnity ι X s) fun _ => ι → C(X, ℝ) :=
-  ⟨toFun⟩
+instance : FunLike (PartitionOfUnity ι X s) ι fun _ ↦ C(X, ℝ) where
+  coe := toFun
+  coe_injective' := fun f g h ↦ by cases f; cases g; congr
 
 protected theorem locallyFinite : LocallyFinite fun i => support (f i) :=
-  f.locally_finite'
+  f.locallyFinite'
 #align partition_of_unity.locally_finite PartitionOfUnity.locallyFinite
 
 theorem locallyFinite_tsupport : LocallyFinite fun i => tsupport (f i) :=
@@ -228,11 +229,12 @@ namespace BumpCovering
 
 variable {s : Set X} (f : BumpCovering ι X s)
 
-instance : CoeFun (BumpCovering ι X s) fun _ => ι → C(X, ℝ) :=
-  ⟨toFun⟩
+instance : FunLike (BumpCovering ι X s) ι fun _ ↦ C(X, ℝ) where
+  coe := toFun
+  coe_injective' := fun f g h ↦ by cases f; cases g; congr
 
 protected theorem locallyFinite : LocallyFinite fun i => support (f i) :=
-  f.locally_finite'
+  f.locallyFinite'
 #align bump_covering.locally_finite BumpCovering.locallyFinite
 
 theorem locallyFinite_tsupport : LocallyFinite fun i => tsupport (f i) :=
@@ -255,7 +257,7 @@ theorem le_one (i : ι) (x : X) : f i x ≤ 1 :=
 example for `Inhabited` instance. -/
 protected def single (i : ι) (s : Set X) : BumpCovering ι X s where
   toFun := Pi.single i 1
-  locally_finite' x := by
+  locallyFinite' x := by
     refine' ⟨univ, univ_mem, (finite_singleton i).subset _⟩
     rintro j ⟨x, hx, -⟩
     contrapose! hx
@@ -373,33 +375,33 @@ words, `g i x = ∏ᶠ j < i, (1 - f j x) - ∏ᶠ j ≤ i, (1 - f j x)`, so
 of `1 - f j x` vanishes, and `∑ᶠ i, g i x = 1`.
 
 In order to avoid an assumption `LinearOrder ι`, we use `WellOrderingRel` instead of `(<)`. -/
-def toPouFun (i : ι) (x : X) : ℝ :=
-  f i x * ∏ᶠ (j) (_ : WellOrderingRel j i), 1 - f j x
-#align bump_covering.to_pou_fun BumpCovering.toPouFun
+def toPOUFun (i : ι) (x : X) : ℝ :=
+  f i x * ∏ᶠ (j) (_ : WellOrderingRel j i), (1 - f j x)
+#align bump_covering.to_pou_fun BumpCovering.toPOUFun
 
-theorem toPouFun_zero_of_zero {i : ι} {x : X} (h : f i x = 0) : f.toPouFun i x = 0 := by
-  rw [toPouFun, h, MulZeroClass.zero_mul]
-#align bump_covering.to_pou_fun_zero_of_zero BumpCovering.toPouFun_zero_of_zero
+theorem toPOUFun_zero_of_zero {i : ι} {x : X} (h : f i x = 0) : f.toPOUFun i x = 0 := by
+  rw [toPOUFun, h, MulZeroClass.zero_mul]
+#align bump_covering.to_pou_fun_zero_of_zero BumpCovering.toPOUFun_zero_of_zero
 
-theorem support_toPouFun_subset (i : ι) : support (f.toPouFun i) ⊆ support (f i) :=
-  fun _ => mt <| f.toPouFun_zero_of_zero
-#align bump_covering.support_to_pou_fun_subset BumpCovering.support_toPouFun_subset
+theorem support_toPOUFun_subset (i : ι) : support (f.toPOUFun i) ⊆ support (f i) :=
+  fun _ => mt <| f.toPOUFun_zero_of_zero
+#align bump_covering.support_to_pou_fun_subset BumpCovering.support_toPOUFun_subset
 
-theorem toPouFun_eq_mul_prod (i : ι) (x : X) (t : Finset ι)
+theorem toPOUFun_eq_mul_prod (i : ι) (x : X) (t : Finset ι)
     (ht : ∀ j, WellOrderingRel j i → f j x ≠ 0 → j ∈ t) :
-    f.toPouFun i x = f i x * ∏ j in t.filter fun j => WellOrderingRel j i, (1 - f j x) := by
+    f.toPOUFun i x = f i x * ∏ j in t.filter fun j => WellOrderingRel j i, (1 - f j x) := by
   refine' congr_arg _ (finprod_cond_eq_prod_of_cond_iff _ fun {j} hj => _)
   rw [Ne.def, sub_eq_self] at hj
   rw [Finset.mem_filter, Iff.comm, and_iff_right_iff_imp]
   exact flip (ht j) hj
-#align bump_covering.to_pou_fun_eq_mul_prod BumpCovering.toPouFun_eq_mul_prod
+#align bump_covering.to_pou_fun_eq_mul_prod BumpCovering.toPOUFun_eq_mul_prod
 
-theorem sum_toPouFun_eq (x : X) : (∑ᶠ i, f.toPouFun i x) = 1 - ∏ᶠ i, 1 - f i x := by
+theorem sum_toPOUFun_eq (x : X) : (∑ᶠ i, f.toPOUFun i x) = 1 - ∏ᶠ i, (1 - f i x) := by
   set s := (f.point_finite x).toFinset
   have hs : (s : Set ι) = { i | f i x ≠ 0 } := Finite.coe_toFinset _
-  have A : (support fun i => toPouFun f i x) ⊆ s := by
+  have A : (support fun i => toPOUFun f i x) ⊆ s := by
     rw [hs]
-    exact fun i hi => f.support_toPouFun_subset i hi
+    exact fun i hi => f.support_toPOUFun_subset i hi
   have B : (mulSupport fun i => 1 - f i x) ⊆ s := by
     rw [hs, mulSupport_one_sub]
     exact fun i => id
@@ -407,27 +409,27 @@ theorem sum_toPouFun_eq (x : X) : (∑ᶠ i, f.toPouFun i x) = 1 - ∏ᶠ i, 1 -
   rw [finsum_eq_sum_of_support_subset _ A, finprod_eq_prod_of_mulSupport_subset _ B,
     Finset.prod_one_sub_ordered, sub_sub_cancel]
   refine' Finset.sum_congr rfl fun i _ => _
-  convert f.toPouFun_eq_mul_prod _ _ _ fun j _ hj => _
+  convert f.toPOUFun_eq_mul_prod _ _ _ fun j _ hj => _
   rwa [Finite.mem_toFinset]
-#align bump_covering.sum_to_pou_fun_eq BumpCovering.sum_toPouFun_eq
+#align bump_covering.sum_to_pou_fun_eq BumpCovering.sum_toPOUFun_eq
 
-theorem exists_finset_toPouFun_eventuallyEq (i : ι) (x : X) : ∃ t : Finset ι,
-    f.toPouFun i =ᶠ[𝓝 x] f i * ∏ j in t.filter fun j => WellOrderingRel j i, (1 - f j) := by
+theorem exists_finset_toPOUFun_eventuallyEq (i : ι) (x : X) : ∃ t : Finset ι,
+    f.toPOUFun i =ᶠ[𝓝 x] f i * ∏ j in t.filter fun j => WellOrderingRel j i, (1 - f j) := by
   rcases f.locallyFinite x with ⟨U, hU, hf⟩
   use hf.toFinset
-  filter_upwards [hU]with y hyU
+  filter_upwards [hU] with y hyU
   simp only [ContinuousMap.coe_prod, Pi.mul_apply, Finset.prod_apply]
-  apply toPouFun_eq_mul_prod
+  apply toPOUFun_eq_mul_prod
   intro j _ hj
   exact hf.mem_toFinset.2 ⟨y, ⟨hj, hyU⟩⟩
-#align bump_covering.exists_finset_to_pou_fun_eventually_eq BumpCovering.exists_finset_toPouFun_eventuallyEq
+#align bump_covering.exists_finset_to_pou_fun_eventually_eq BumpCovering.exists_finset_toPOUFun_eventuallyEq
 
-theorem continuous_toPouFun (i : ι) : Continuous (f.toPouFun i) := by
+theorem continuous_toPOUFun (i : ι) : Continuous (f.toPOUFun i) := by
   refine' (f i).continuous.mul <|
     continuous_finprod_cond (fun j _ => continuous_const.sub (f j).continuous) _
   simp only [mulSupport_one_sub]
   exact f.locallyFinite
-#align bump_covering.continuous_to_pou_fun BumpCovering.continuous_toPouFun
+#align bump_covering.continuous_to_pou_fun BumpCovering.continuous_toPOUFun
 
 /-- The partition of unity defined by a `BumpCovering`.
 
@@ -438,49 +440,49 @@ of `1 - f j x` vanishes, and `∑ᶠ i, g i x = 1`.
 
 In order to avoid an assumption `LinearOrder ι`, we use `WellOrderingRel` instead of `(<)`. -/
 def toPartitionOfUnity : PartitionOfUnity ι X s where
-  toFun i := ⟨f.toPouFun i, f.continuous_toPouFun i⟩
-  locally_finite' := f.locallyFinite.subset f.support_toPouFun_subset
+  toFun i := ⟨f.toPOUFun i, f.continuous_toPOUFun i⟩
+  locallyFinite' := f.locallyFinite.subset f.support_toPOUFun_subset
   nonneg' i x :=
     mul_nonneg (f.nonneg i x) (finprod_cond_nonneg fun j hj => sub_nonneg.2 <| f.le_one j x)
   sum_eq_one' x hx := by
-    simp only [ContinuousMap.coe_mk, sum_toPouFun_eq, sub_eq_self]
+    simp only [ContinuousMap.coe_mk, sum_toPOUFun_eq, sub_eq_self]
     apply finprod_eq_zero (fun i => 1 - f i x) (f.ind x hx)
     · simp only [f.ind_apply x hx, sub_self]
     · rw [mulSupport_one_sub]
       exact f.point_finite x
   sum_le_one' x := by
-    simp only [ContinuousMap.coe_mk, sum_toPouFun_eq, sub_le_self_iff]
+    simp only [ContinuousMap.coe_mk, sum_toPOUFun_eq, sub_le_self_iff]
     exact finprod_nonneg fun i => sub_nonneg.2 <| f.le_one i x
 #align bump_covering.to_partition_of_unity BumpCovering.toPartitionOfUnity
 
 theorem toPartitionOfUnity_apply (i : ι) (x : X) :
-    f.toPartitionOfUnity i x = f i x * ∏ᶠ (j) (_ : WellOrderingRel j i), 1 - f j x := rfl
+    f.toPartitionOfUnity i x = f i x * ∏ᶠ (j) (_ : WellOrderingRel j i), (1 - f j x) := rfl
 #align bump_covering.to_partition_of_unity_apply BumpCovering.toPartitionOfUnity_apply
 
 theorem toPartitionOfUnity_eq_mul_prod (i : ι) (x : X) (t : Finset ι)
     (ht : ∀ j, WellOrderingRel j i → f j x ≠ 0 → j ∈ t) :
     f.toPartitionOfUnity i x = f i x * ∏ j in t.filter fun j => WellOrderingRel j i, (1 - f j x) :=
-  f.toPouFun_eq_mul_prod i x t ht
+  f.toPOUFun_eq_mul_prod i x t ht
 #align bump_covering.to_partition_of_unity_eq_mul_prod BumpCovering.toPartitionOfUnity_eq_mul_prod
 
 theorem exists_finset_toPartitionOfUnity_eventuallyEq (i : ι) (x : X) : ∃ t : Finset ι,
     f.toPartitionOfUnity i =ᶠ[𝓝 x] f i * ∏ j in t.filter fun j => WellOrderingRel j i, (1 - f j) :=
-  f.exists_finset_toPouFun_eventuallyEq i x
+  f.exists_finset_toPOUFun_eventuallyEq i x
 #align bump_covering.exists_finset_to_partition_of_unity_eventually_eq BumpCovering.exists_finset_toPartitionOfUnity_eventuallyEq
 
 theorem toPartitionOfUnity_zero_of_zero {i : ι} {x : X} (h : f i x = 0) :
     f.toPartitionOfUnity i x = 0 :=
-  f.toPouFun_zero_of_zero h
+  f.toPOUFun_zero_of_zero h
 #align bump_covering.to_partition_of_unity_zero_of_zero BumpCovering.toPartitionOfUnity_zero_of_zero
 
 theorem support_toPartitionOfUnity_subset (i : ι) :
     support (f.toPartitionOfUnity i) ⊆ support (f i) :=
-  f.support_toPouFun_subset i
+  f.support_toPOUFun_subset i
 #align bump_covering.support_to_partition_of_unity_subset BumpCovering.support_toPartitionOfUnity_subset
 
 theorem sum_toPartitionOfUnity_eq (x : X) :
-    (∑ᶠ i, f.toPartitionOfUnity i x) = 1 - ∏ᶠ i, 1 - f i x :=
-  f.sum_toPouFun_eq x
+    (∑ᶠ i, f.toPartitionOfUnity i x) = 1 - ∏ᶠ i, (1 - f i x) :=
+  f.sum_toPOUFun_eq x
 #align bump_covering.sum_to_partition_of_unity_eq BumpCovering.sum_toPartitionOfUnity_eq
 
 theorem IsSubordinate.toPartitionOfUnity {f : BumpCovering ι X s} {U : ι → Set X}

These are just three cases where a mathported proof had to be changed, due to unwanted behaviour with push_neg/contradiction. Since #5082 is supposed to fix some issues with these tactics, some of these workarounds can be removed.

Note: this was not in any way systematic, I simply grepped the line after a contra, then grepped for a not and then selected 3 likely candidates. There are potentially more situations like these.

@@ -161,7 +161,6 @@ that `0 < f i x`. -/
 theorem exists_pos {x : X} (hx : x ∈ s) : ∃ i, 0 < f i x := by
   have H := f.sum_eq_one hx
   contrapose! H
-  simp_rw [not_exists, not_lt] at H
   simpa only [fun i => (H i).antisymm (f.nonneg i x), finsum_zero] using zero_ne_one
 #align partition_of_unity.exists_pos PartitionOfUnity.exists_pos
 

@@ -375,7 +375,7 @@ of `1 - f j x` vanishes, and `∑ᶠ i, g i x = 1`.
 
 In order to avoid an assumption `LinearOrder ι`, we use `WellOrderingRel` instead of `(<)`. -/
 def toPouFun (i : ι) (x : X) : ℝ :=
-  f i x * ∏ᶠ (j) (_hj : WellOrderingRel j i), 1 - f j x
+  f i x * ∏ᶠ (j) (_ : WellOrderingRel j i), 1 - f j x
 #align bump_covering.to_pou_fun BumpCovering.toPouFun
 
 theorem toPouFun_zero_of_zero {i : ι} {x : X} (h : f i x = 0) : f.toPouFun i x = 0 := by
@@ -455,7 +455,7 @@ def toPartitionOfUnity : PartitionOfUnity ι X s where
 #align bump_covering.to_partition_of_unity BumpCovering.toPartitionOfUnity
 
 theorem toPartitionOfUnity_apply (i : ι) (x : X) :
-    f.toPartitionOfUnity i x = f i x * ∏ᶠ (j) (_hj : WellOrderingRel j i), 1 - f j x := rfl
+    f.toPartitionOfUnity i x = f i x * ∏ᶠ (j) (_ : WellOrderingRel j i), 1 - f j x := rfl
 #align bump_covering.to_partition_of_unity_apply BumpCovering.toPartitionOfUnity_apply
 
 theorem toPartitionOfUnity_eq_mul_prod (i : ι) (x : X) (t : Finset ι)

As discussed on Zulip

Renames

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

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

@@ -296,10 +296,10 @@ theorem exists_isSubordinate_of_locallyFinite_of_prop [NormalSpace X] (p : (X 
       ∃ f : C(X, ℝ), p f ∧ EqOn f 0 s ∧ EqOn f 1 t ∧ ∀ x, f x ∈ Icc (0 : ℝ) 1)
     (hs : IsClosed s) (U : ι → Set X) (ho : ∀ i, IsOpen (U i)) (hf : LocallyFinite U)
     (hU : s ⊆ ⋃ i, U i) : ∃ f : BumpCovering ι X s, (∀ i, p (f i)) ∧ f.IsSubordinate U := by
-  rcases exists_subset_unionᵢ_closure_subset hs ho (fun x _ => hf.point_finite x) hU with
+  rcases exists_subset_iUnion_closure_subset hs ho (fun x _ => hf.point_finite x) hU with
     ⟨V, hsV, hVo, hVU⟩
   have hVU' : ∀ i, V i ⊆ U i := fun i => Subset.trans subset_closure (hVU i)
-  rcases exists_subset_unionᵢ_closure_subset hs hVo (fun x _ => (hf.subset hVU').point_finite x)
+  rcases exists_subset_iUnion_closure_subset hs hVo (fun x _ => (hf.subset hVU').point_finite x)
       hsV with
     ⟨W, hsW, hWo, hWV⟩
   choose f hfp hf0 hf1 hf01 using fun i =>
@@ -309,7 +309,7 @@ theorem exists_isSubordinate_of_locallyFinite_of_prop [NormalSpace X] (p : (X 
   refine' ⟨⟨f, hf.subset fun i => Subset.trans (hsupp i) (hVU' i), fun i x => (hf01 i x).1,
       fun i x => (hf01 i x).2, fun x hx => _⟩,
     hfp, fun i => Subset.trans (closure_mono (hsupp i)) (hVU i)⟩
-  rcases mem_unionᵢ.1 (hsW hx) with ⟨i, hi⟩
+  rcases mem_iUnion.1 (hsW hx) with ⟨i, hi⟩
   exact ⟨i, ((hf1 i).mono subset_closure).eventuallyEq_of_mem ((hWo i).mem_nhds hi)⟩
 #align bump_covering.exists_is_subordinate_of_locally_finite_of_prop BumpCovering.exists_isSubordinate_of_locallyFinite_of_prop