geometry.manifold.partition_of_unity ⟷ Mathlib.Geometry.Manifold.PartitionOfUnity

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

@@ -567,11 +567,11 @@ end SmoothBumpCovering
 
 variable (I)
 
-#print exists_smooth_zero_one_of_closed /-
+#print exists_smooth_zero_one_of_isClosed /-
 /-- Given two disjoint closed sets in a Hausdorff σ-compact finite dimensional manifold, there
 exists an infinitely smooth function that is equal to `0` on one of them and is equal to one on the
 other. -/
-theorem exists_smooth_zero_one_of_closed [T2Space M] [SigmaCompactSpace M] {s t : Set M}
+theorem exists_smooth_zero_one_of_isClosed [T2Space M] [SigmaCompactSpace M] {s t : Set M}
     (hs : IsClosed s) (ht : IsClosed t) (hd : Disjoint s t) :
     ∃ f : C^∞⟮I, M; 𝓘(ℝ), ℝ⟯, EqOn f 0 s ∧ EqOn f 1 t ∧ ∀ x, f x ∈ Icc (0 : ℝ) 1 :=
   by
@@ -584,7 +584,7 @@ theorem exists_smooth_zero_one_of_closed [T2Space M] [SigmaCompactSpace M] {s t
   suffices ∀ i, g i x = 0 by simp only [this, ContMDiffMap.coeFn_mk, finsum_zero, Pi.zero_apply]
   refine' fun i => f.to_smooth_partition_of_unity_zero_of_zero _
   exact nmem_support.1 (subset_compl_comm.1 (hf.support_subset i) hx)
-#align exists_smooth_zero_one_of_closed exists_smooth_zero_one_of_closed
+#align exists_smooth_zero_one_of_closed exists_smooth_zero_one_of_isClosed
 -/
 
 namespace SmoothPartitionOfUnity
@@ -618,7 +618,7 @@ theorem exists_isSubordinate {s : Set M} (hs : IsClosed s) (U : ι → Set M) (h
   rcases BumpCovering.exists_isSubordinate_of_prop (Smooth I 𝓘(ℝ)) _ hs U ho hU with ⟨f, hf, hfU⟩
   · exact ⟨f.to_smooth_partition_of_unity hf, hfU.to_smooth_partition_of_unity hf⟩
   · intro s t hs ht hd
-    rcases exists_smooth_zero_one_of_closed I hs ht hd with ⟨f, hf⟩
+    rcases exists_smooth_zero_one_of_isClosed I hs ht hd with ⟨f, hf⟩
     exact ⟨f, f.smooth, hf⟩
 #align smooth_partition_of_unity.exists_is_subordinate SmoothPartitionOfUnity.exists_isSubordinate
 -/

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

@@ -192,7 +192,7 @@ theorem sum_nonneg (x : M) : 0 ≤ ∑ᶠ i, f i x :=
 #print SmoothPartitionOfUnity.contMDiff_smul /-
 theorem contMDiff_smul {g : M → F} {i} (hg : ∀ x ∈ tsupport (f i), ContMDiffAt I 𝓘(ℝ, F) n g x) :
     ContMDiff I 𝓘(ℝ, F) n fun x => f i x • g x :=
-  contMDiff_of_support fun x hx =>
+  contMDiff_of_tsupport fun x hx =>
     ((f i).ContMDiff.ContMDiffAt.of_le le_top).smul <| hg x <| tsupport_smul_subset_left _ _ hx
 #align smooth_partition_of_unity.cont_mdiff_smul SmoothPartitionOfUnity.contMDiff_smul
 -/

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

@@ -3,10 +3,10 @@ Copyright (c) 2021 Yury G. Kudryashov. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Yury G. Kudryashov
 -/
-import Mathbin.Geometry.Manifold.Algebra.Structures
-import Mathbin.Geometry.Manifold.BumpFunction
-import Mathbin.Topology.MetricSpace.PartitionOfUnity
-import Mathbin.Topology.ShrinkingLemma
+import Geometry.Manifold.Algebra.Structures
+import Geometry.Manifold.BumpFunction
+import Topology.MetricSpace.PartitionOfUnity
+import Topology.ShrinkingLemma
 
 #align_import geometry.manifold.partition_of_unity from "leanprover-community/mathlib"@"9240e8be927a0955b9a82c6c85ef499ee3a626b8"
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/001ffdc42920050657fd45bd2b8bfbec8eaaeb29

@@ -383,7 +383,7 @@ theorem exists_isSubordinate [T2Space M] [SigmaCompactSpace M] (hs : IsClosed s)
   -- First we deduce some missing instances
   haveI : LocallyCompactSpace H := I.locally_compact
   haveI : LocallyCompactSpace M := ChartedSpace.locallyCompactSpace H M
-  haveI : NormalSpace M := normal_of_paracompact_t2
+  haveI : NormalSpace M := T4Space.of_paracompactSpace_t2Space
   -- Next we choose a covering by supports of smooth bump functions
   have hB := fun x hx => SmoothBumpFunction.nhds_basis_support I (hU x hx)
   rcases refinement_of_locallyCompact_sigmaCompact_of_nhds_basis_set hs hB with
@@ -614,7 +614,7 @@ theorem exists_isSubordinate {s : Set M} (hs : IsClosed s) (U : ι → Set M) (h
   by
   haveI : LocallyCompactSpace H := I.locally_compact
   haveI : LocallyCompactSpace M := ChartedSpace.locallyCompactSpace H M
-  haveI : NormalSpace M := normal_of_paracompact_t2
+  haveI : NormalSpace M := T4Space.of_paracompactSpace_t2Space
   rcases BumpCovering.exists_isSubordinate_of_prop (Smooth I 𝓘(ℝ)) _ hs U ho hU with ⟨f, hf, hfU⟩
   · exact ⟨f.to_smooth_partition_of_unity hf, hfU.to_smooth_partition_of_unity hf⟩
   · intro s t hs ht hd

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

@@ -382,7 +382,7 @@ theorem exists_isSubordinate [T2Space M] [SigmaCompactSpace M] (hs : IsClosed s)
   by
   -- First we deduce some missing instances
   haveI : LocallyCompactSpace H := I.locally_compact
-  haveI : LocallyCompactSpace M := ChartedSpace.locallyCompact H M
+  haveI : LocallyCompactSpace M := ChartedSpace.locallyCompactSpace H M
   haveI : NormalSpace M := normal_of_paracompact_t2
   -- Next we choose a covering by supports of smooth bump functions
   have hB := fun x hx => SmoothBumpFunction.nhds_basis_support I (hU x hx)
@@ -613,7 +613,7 @@ theorem exists_isSubordinate {s : Set M} (hs : IsClosed s) (U : ι → Set M) (h
     (hU : s ⊆ ⋃ i, U i) : ∃ f : SmoothPartitionOfUnity ι I M s, f.IsSubordinate U :=
   by
   haveI : LocallyCompactSpace H := I.locally_compact
-  haveI : LocallyCompactSpace M := ChartedSpace.locallyCompact H M
+  haveI : LocallyCompactSpace M := ChartedSpace.locallyCompactSpace H M
   haveI : NormalSpace M := normal_of_paracompact_t2
   rcases BumpCovering.exists_isSubordinate_of_prop (Smooth I 𝓘(ℝ)) _ hs U ho hU with ⟨f, hf, hfU⟩
   · exact ⟨f.to_smooth_partition_of_unity hf, hfU.to_smooth_partition_of_unity hf⟩

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

@@ -252,7 +252,7 @@ theorem isSubordinate_toPartitionOfUnity :
 #align smooth_partition_of_unity.is_subordinate_to_partition_of_unity SmoothPartitionOfUnity.isSubordinate_toPartitionOfUnity
 -/
 
-alias is_subordinate_to_partition_of_unity ↔ _ is_subordinate.to_partition_of_unity
+alias ⟨_, is_subordinate.to_partition_of_unity⟩ := is_subordinate_to_partition_of_unity
 #align smooth_partition_of_unity.is_subordinate.to_partition_of_unity SmoothPartitionOfUnity.IsSubordinate.toPartitionOfUnity
 
 #print SmoothPartitionOfUnity.IsSubordinate.contMDiff_finsum_smul /-
@@ -500,7 +500,7 @@ theorem isSubordinate_toBumpCovering {f : SmoothBumpCovering ι I M s} {U : M 
 #align smooth_bump_covering.is_subordinate_to_bump_covering SmoothBumpCovering.isSubordinate_toBumpCovering
 -/
 
-alias is_subordinate_to_bump_covering ↔ _ is_subordinate.to_bump_covering
+alias ⟨_, is_subordinate.to_bump_covering⟩ := is_subordinate_to_bump_covering
 #align smooth_bump_covering.is_subordinate.to_bump_covering SmoothBumpCovering.IsSubordinate.toBumpCovering
 
 #print SmoothBumpCovering.toSmoothPartitionOfUnity /-

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

@@ -627,14 +627,14 @@ end SmoothPartitionOfUnity
 
 variable [SigmaCompactSpace M] [T2Space M] {t : M → Set F} {n : ℕ∞}
 
-#print exists_cont_mdiff_forall_mem_convex_of_local /-
+#print exists_contMDiffOn_forall_mem_convex_of_local /-
 /-- Let `M` be a σ-compact Hausdorff finite dimensional topological manifold. Let `t : M → set F`
 be a family of convex sets. Suppose that for each point `x : M` there exists a neighborhood
 `U ∈ 𝓝 x` and a function `g : M → F` such that `g` is $C^n$ smooth on `U` and `g y ∈ t y` for all
 `y ∈ U`. Then there exists a $C^n$ smooth function `g : C^∞⟮I, M; 𝓘(ℝ, F), F⟯` such that `g x ∈ t x`
 for all `x`. See also `exists_smooth_forall_mem_convex_of_local` and
 `exists_smooth_forall_mem_convex_of_local_const`. -/
-theorem exists_cont_mdiff_forall_mem_convex_of_local (ht : ∀ x, Convex ℝ (t x))
+theorem exists_contMDiffOn_forall_mem_convex_of_local (ht : ∀ x, Convex ℝ (t x))
     (Hloc : ∀ x : M, ∃ U ∈ 𝓝 x, ∃ g : M → F, ContMDiffOn I 𝓘(ℝ, F) n g U ∧ ∀ y ∈ U, g y ∈ t y) :
     ∃ g : C^n⟮I, M; 𝓘(ℝ, F), F⟯, ∀ x, g x ∈ t x :=
   by
@@ -647,7 +647,7 @@ theorem exists_cont_mdiff_forall_mem_convex_of_local (ht : ∀ x, Convex ℝ (t
         hf.cont_mdiff_finsum_smul (fun i => isOpen_interior) fun i => (hgs i).mono interior_subset⟩,
       fun x => f.finsum_smul_mem_convex (mem_univ x) (fun i hi => hgt _ _ _) (ht _)⟩
   exact interior_subset (hf _ <| subset_closure hi)
-#align exists_cont_mdiff_forall_mem_convex_of_local exists_cont_mdiff_forall_mem_convex_of_local
+#align exists_cont_mdiff_forall_mem_convex_of_local exists_contMDiffOn_forall_mem_convex_of_local
 -/
 
 #print exists_smooth_forall_mem_convex_of_local /-
@@ -660,7 +660,7 @@ See also `exists_cont_mdiff_forall_mem_convex_of_local` and
 theorem exists_smooth_forall_mem_convex_of_local (ht : ∀ x, Convex ℝ (t x))
     (Hloc : ∀ x : M, ∃ U ∈ 𝓝 x, ∃ g : M → F, SmoothOn I 𝓘(ℝ, F) g U ∧ ∀ y ∈ U, g y ∈ t y) :
     ∃ g : C^∞⟮I, M; 𝓘(ℝ, F), F⟯, ∀ x, g x ∈ t x :=
-  exists_cont_mdiff_forall_mem_convex_of_local I ht Hloc
+  exists_contMDiffOn_forall_mem_convex_of_local I ht Hloc
 #align exists_smooth_forall_mem_convex_of_local exists_smooth_forall_mem_convex_of_local
 -/
 

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

@@ -2,17 +2,14 @@
 Copyright (c) 2021 Yury G. Kudryashov. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Yury G. Kudryashov
-
-! This file was ported from Lean 3 source module geometry.manifold.partition_of_unity
-! leanprover-community/mathlib commit 9240e8be927a0955b9a82c6c85ef499ee3a626b8
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
 -/
 import Mathbin.Geometry.Manifold.Algebra.Structures
 import Mathbin.Geometry.Manifold.BumpFunction
 import Mathbin.Topology.MetricSpace.PartitionOfUnity
 import Mathbin.Topology.ShrinkingLemma
 
+#align_import geometry.manifold.partition_of_unity from "leanprover-community/mathlib"@"9240e8be927a0955b9a82c6c85ef499ee3a626b8"
+
 /-!
 # Smooth partition of unity
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/9240e8be927a0955b9a82c6c85ef499ee3a626b8

@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Yury G. Kudryashov
 
 ! This file was ported from Lean 3 source module geometry.manifold.partition_of_unity
-! leanprover-community/mathlib commit f2ce6086713c78a7f880485f7917ea547a215982
+! leanprover-community/mathlib commit 9240e8be927a0955b9a82c6c85ef499ee3a626b8
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
@@ -16,6 +16,9 @@ import Mathbin.Topology.ShrinkingLemma
 /-!
 # Smooth 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 two structures, `smooth_bump_covering` and `smooth_partition_of_unity`. Both
 structures describe coverings of a set by a locally finite family of supports of smooth functions
 with some additional properties. The former structure is mostly useful as an intermediate step in

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

@@ -91,6 +91,7 @@ compact real manifold can be embedded into `ℝ^n` for large enough `n`.  -/
 
 variable (ι M)
 
+#print SmoothBumpCovering /-
 /-- We say that a collection of `smooth_bump_function`s is a `smooth_bump_covering` of a set `s` if
 
 * `(f i).c ∈ s` for all `i`;
@@ -112,7 +113,9 @@ structure SmoothBumpCovering (s : Set M := univ) where
   locally_finite' : LocallyFinite fun i => support (to_fun i)
   eventuallyEq_one' : ∀ x ∈ s, ∃ i, to_fun i =ᶠ[𝓝 x] 1
 #align smooth_bump_covering SmoothBumpCovering
+-/
 
+#print SmoothPartitionOfUnity /-
 /-- We say that that a collection of functions form a smooth partition of unity on a set `s` if
 
 * all functions are infinitely smooth and nonnegative;
@@ -126,6 +129,7 @@ structure SmoothPartitionOfUnity (s : Set M := univ) where
   sum_eq_one' : ∀ x ∈ s, ∑ᶠ i, to_fun i x = 1
   sum_le_one' : ∀ x, ∑ᶠ i, to_fun i x ≤ 1
 #align smooth_partition_of_unity SmoothPartitionOfUnity
+-/
 
 variable {ι I M}
 
@@ -136,50 +140,71 @@ variable {s : Set M} (f : SmoothPartitionOfUnity ι I M s) {n : ℕ∞}
 instance {s : Set M} : CoeFun (SmoothPartitionOfUnity ι I M s) fun _ => ι → C^∞⟮I, M; 𝓘(ℝ), ℝ⟯ :=
   ⟨SmoothPartitionOfUnity.toFun⟩
 
+#print SmoothPartitionOfUnity.locallyFinite /-
 protected theorem locallyFinite : LocallyFinite fun i => support (f i) :=
   f.locally_finite'
 #align smooth_partition_of_unity.locally_finite SmoothPartitionOfUnity.locallyFinite
+-/
 
+#print SmoothPartitionOfUnity.nonneg /-
 theorem nonneg (i : ι) (x : M) : 0 ≤ f i x :=
   f.nonneg' i x
 #align smooth_partition_of_unity.nonneg SmoothPartitionOfUnity.nonneg
+-/
 
+#print SmoothPartitionOfUnity.sum_eq_one /-
 theorem sum_eq_one {x} (hx : x ∈ s) : ∑ᶠ i, f i x = 1 :=
   f.sum_eq_one' x hx
 #align smooth_partition_of_unity.sum_eq_one SmoothPartitionOfUnity.sum_eq_one
+-/
 
+#print SmoothPartitionOfUnity.sum_le_one /-
 theorem sum_le_one (x : M) : ∑ᶠ i, f i x ≤ 1 :=
   f.sum_le_one' x
 #align smooth_partition_of_unity.sum_le_one SmoothPartitionOfUnity.sum_le_one
+-/
 
+#print SmoothPartitionOfUnity.toPartitionOfUnity /-
 /-- Reinterpret a smooth partition of unity as a continuous partition of unity. -/
 def toPartitionOfUnity : PartitionOfUnity ι M s :=
   { f with toFun := fun i => f i }
 #align smooth_partition_of_unity.to_partition_of_unity SmoothPartitionOfUnity.toPartitionOfUnity
+-/
 
+#print SmoothPartitionOfUnity.smooth_sum /-
 theorem smooth_sum : Smooth I 𝓘(ℝ) fun x => ∑ᶠ i, f i x :=
   smooth_finsum (fun i => (f i).Smooth) f.LocallyFinite
 #align smooth_partition_of_unity.smooth_sum SmoothPartitionOfUnity.smooth_sum
+-/
 
+#print SmoothPartitionOfUnity.le_one /-
 theorem le_one (i : ι) (x : M) : f i x ≤ 1 :=
   f.toPartitionOfUnity.le_one i x
 #align smooth_partition_of_unity.le_one SmoothPartitionOfUnity.le_one
+-/
 
+#print SmoothPartitionOfUnity.sum_nonneg /-
 theorem sum_nonneg (x : M) : 0 ≤ ∑ᶠ i, f i x :=
   f.toPartitionOfUnity.sum_nonneg x
 #align smooth_partition_of_unity.sum_nonneg SmoothPartitionOfUnity.sum_nonneg
+-/
 
+#print SmoothPartitionOfUnity.contMDiff_smul /-
 theorem contMDiff_smul {g : M → F} {i} (hg : ∀ x ∈ tsupport (f i), ContMDiffAt I 𝓘(ℝ, F) n g x) :
     ContMDiff I 𝓘(ℝ, F) n fun x => f i x • g x :=
   contMDiff_of_support fun x hx =>
     ((f i).ContMDiff.ContMDiffAt.of_le le_top).smul <| hg x <| tsupport_smul_subset_left _ _ hx
 #align smooth_partition_of_unity.cont_mdiff_smul SmoothPartitionOfUnity.contMDiff_smul
+-/
 
+#print SmoothPartitionOfUnity.smooth_smul /-
 theorem smooth_smul {g : M → F} {i} (hg : ∀ x ∈ tsupport (f i), SmoothAt I 𝓘(ℝ, F) g x) :
     Smooth I 𝓘(ℝ, F) fun x => f i x • g x :=
   f.contMDiff_smul hg
 #align smooth_partition_of_unity.smooth_smul SmoothPartitionOfUnity.smooth_smul
+-/
 
+#print SmoothPartitionOfUnity.contMDiff_finsum_smul /-
 /-- If `f` is a smooth partition of unity on a set `s : set M` and `g : ι → M → F` is a family of
 functions such that `g i` is $C^n$ smooth at every point of the topological support of `f i`, then
 the sum `λ x, ∑ᶠ i, f i x • g i x` is smooth on the whole manifold. -/
@@ -189,7 +214,9 @@ theorem contMDiff_finsum_smul {g : ι → M → F}
   (contMDiff_finsum fun i => f.contMDiff_smul (hg i)) <|
     f.LocallyFinite.Subset fun i => support_smul_subset_left _ _
 #align smooth_partition_of_unity.cont_mdiff_finsum_smul SmoothPartitionOfUnity.contMDiff_finsum_smul
+-/
 
+#print SmoothPartitionOfUnity.smooth_finsum_smul /-
 /-- If `f` is a smooth partition of unity on a set `s : set M` and `g : ι → M → F` is a family of
 functions such that `g i` is smooth at every point of the topological support of `f i`, then the sum
 `λ x, ∑ᶠ i, f i x • g i x` is smooth on the whole manifold. -/
@@ -198,29 +225,37 @@ theorem smooth_finsum_smul {g : ι → M → F}
     Smooth I 𝓘(ℝ, F) fun x => ∑ᶠ i, f i x • g i x :=
   f.contMDiff_finsum_smul hg
 #align smooth_partition_of_unity.smooth_finsum_smul SmoothPartitionOfUnity.smooth_finsum_smul
+-/
 
+#print SmoothPartitionOfUnity.finsum_smul_mem_convex /-
 theorem finsum_smul_mem_convex {g : ι → M → F} {t : Set F} {x : M} (hx : x ∈ s)
     (hg : ∀ i, f i x ≠ 0 → g i x ∈ t) (ht : Convex ℝ t) : ∑ᶠ i, f i x • g i x ∈ t :=
   ht.finsum_mem (fun i => f.NonNeg _ _) (f.sum_eq_one hx) hg
 #align smooth_partition_of_unity.finsum_smul_mem_convex SmoothPartitionOfUnity.finsum_smul_mem_convex
+-/
 
+#print SmoothPartitionOfUnity.IsSubordinate /-
 /-- A smooth 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 (f : SmoothPartitionOfUnity ι I M s) (U : ι → Set M) :=
   ∀ i, tsupport (f i) ⊆ U i
 #align smooth_partition_of_unity.is_subordinate SmoothPartitionOfUnity.IsSubordinate
+-/
 
 variable {f} {U : ι → Set M}
 
+#print SmoothPartitionOfUnity.isSubordinate_toPartitionOfUnity /-
 @[simp]
 theorem isSubordinate_toPartitionOfUnity :
     f.toPartitionOfUnity.IsSubordinate U ↔ f.IsSubordinate U :=
   Iff.rfl
 #align smooth_partition_of_unity.is_subordinate_to_partition_of_unity SmoothPartitionOfUnity.isSubordinate_toPartitionOfUnity
+-/
 
 alias is_subordinate_to_partition_of_unity ↔ _ is_subordinate.to_partition_of_unity
 #align smooth_partition_of_unity.is_subordinate.to_partition_of_unity SmoothPartitionOfUnity.IsSubordinate.toPartitionOfUnity
 
+#print SmoothPartitionOfUnity.IsSubordinate.contMDiff_finsum_smul /-
 /-- If `f` is a smooth partition of unity on a set `s : set M` subordinate to a family of open sets
 `U : ι → set M` and `g : ι → M → F` is a family of functions such that `g i` is $C^n$ smooth on
 `U i`, then the sum `λ x, ∑ᶠ i, f i x • g i x` is $C^n$ smooth on the whole manifold. -/
@@ -229,7 +264,9 @@ theorem IsSubordinate.contMDiff_finsum_smul {g : ι → M → F} (hf : f.IsSubor
     ContMDiff I 𝓘(ℝ, F) n fun x => ∑ᶠ i, f i x • g i x :=
   f.contMDiff_finsum_smul fun i x hx => (hg i).ContMDiffAt <| (ho i).mem_nhds (hf i hx)
 #align smooth_partition_of_unity.is_subordinate.cont_mdiff_finsum_smul SmoothPartitionOfUnity.IsSubordinate.contMDiff_finsum_smul
+-/
 
+#print SmoothPartitionOfUnity.IsSubordinate.smooth_finsum_smul /-
 /-- If `f` is a smooth partition of unity on a set `s : set M` subordinate to a family of open sets
 `U : ι → set M` and `g : ι → M → F` is a family of functions such that `g i` is smooth on `U i`,
 then the sum `λ x, ∑ᶠ i, f i x • g i x` is smooth on the whole manifold. -/
@@ -238,11 +275,13 @@ theorem IsSubordinate.smooth_finsum_smul {g : ι → M → F} (hf : f.IsSubordin
     Smooth I 𝓘(ℝ, F) fun x => ∑ᶠ i, f i x • g i x :=
   hf.contMDiff_finsum_smul ho hg
 #align smooth_partition_of_unity.is_subordinate.smooth_finsum_smul SmoothPartitionOfUnity.IsSubordinate.smooth_finsum_smul
+-/
 
 end SmoothPartitionOfUnity
 
 namespace BumpCovering
 
+#print BumpCovering.smooth_toPartitionOfUnity /-
 -- Repeat variables to drop [finite_dimensional ℝ E] and [smooth_manifold_with_corners I M]
 theorem smooth_toPartitionOfUnity {E : Type uE} [NormedAddCommGroup E] [NormedSpace ℝ E]
     {H : Type uH} [TopologicalSpace H] {I : ModelWithCorners ℝ E H} {M : Type uM}
@@ -252,9 +291,11 @@ theorem smooth_toPartitionOfUnity {E : Type uE} [NormedAddCommGroup E] [NormedSp
     (smooth_finprod_cond fun j _ => smooth_const.sub (hf j)) <| by simp only [mul_support_one_sub];
       exact f.locally_finite
 #align bump_covering.smooth_to_partition_of_unity BumpCovering.smooth_toPartitionOfUnity
+-/
 
 variable {s : Set M}
 
+#print BumpCovering.toSmoothPartitionOfUnity /-
 /-- A `bump_covering` such that all functions in this covering are smooth generates a smooth
 partition of unity.
 
@@ -267,25 +308,32 @@ def toSmoothPartitionOfUnity (f : BumpCovering ι M s) (hf : ∀ i, Smooth I 
   { f.toPartitionOfUnity with
     toFun := fun i => ⟨f.toPartitionOfUnity i, f.smooth_toPartitionOfUnity hf i⟩ }
 #align bump_covering.to_smooth_partition_of_unity BumpCovering.toSmoothPartitionOfUnity
+-/
 
+#print BumpCovering.toSmoothPartitionOfUnity_toPartitionOfUnity /-
 @[simp]
 theorem toSmoothPartitionOfUnity_toPartitionOfUnity (f : BumpCovering ι M s)
     (hf : ∀ i, Smooth I 𝓘(ℝ) (f i)) :
     (f.toSmoothPartitionOfUnity hf).toPartitionOfUnity = f.toPartitionOfUnity :=
   rfl
 #align bump_covering.to_smooth_partition_of_unity_to_partition_of_unity BumpCovering.toSmoothPartitionOfUnity_toPartitionOfUnity
+-/
 
+#print BumpCovering.coe_toSmoothPartitionOfUnity /-
 @[simp]
 theorem coe_toSmoothPartitionOfUnity (f : BumpCovering ι M s) (hf : ∀ i, Smooth I 𝓘(ℝ) (f i))
     (i : ι) : ⇑(f.toSmoothPartitionOfUnity hf i) = f.toPartitionOfUnity i :=
   rfl
 #align bump_covering.coe_to_smooth_partition_of_unity BumpCovering.coe_toSmoothPartitionOfUnity
+-/
 
+#print BumpCovering.IsSubordinate.toSmoothPartitionOfUnity /-
 theorem IsSubordinate.toSmoothPartitionOfUnity {f : BumpCovering ι M s} {U : ι → Set M}
     (h : f.IsSubordinate U) (hf : ∀ i, Smooth I 𝓘(ℝ) (f i)) :
     (f.toSmoothPartitionOfUnity hf).IsSubordinate U :=
   h.toPartitionOfUnity
 #align bump_covering.is_subordinate.to_smooth_partition_of_unity BumpCovering.IsSubordinate.toSmoothPartitionOfUnity
+-/
 
 end BumpCovering
 
@@ -302,6 +350,7 @@ theorem coe_mk (c : ι → M) (to_fun : ∀ i, SmoothBumpFunction I (c i)) (h₁
   rfl
 #align smooth_bump_covering.coe_mk SmoothBumpCovering.coe_mk
 
+#print SmoothBumpCovering.IsSubordinate /-
 /--
 We say that `f : smooth_bump_covering ι I M s` is *subordinate* to a map `U : M → set M` if for each
 index `i`, we have `tsupport (f i) ⊆ U (f i).c`. This notion is a bit more general than
@@ -311,14 +360,18 @@ depends on `x`.
 def IsSubordinate {s : Set M} (f : SmoothBumpCovering ι I M s) (U : M → Set M) :=
   ∀ i, tsupport (f i) ⊆ U (f.c i)
 #align smooth_bump_covering.is_subordinate SmoothBumpCovering.IsSubordinate
+-/
 
+#print SmoothBumpCovering.IsSubordinate.support_subset /-
 theorem IsSubordinate.support_subset {fs : SmoothBumpCovering ι I M s} {U : M → Set M}
     (h : fs.IsSubordinate U) (i : ι) : support (fs i) ⊆ U (fs.c i) :=
   Subset.trans subset_closure (h i)
 #align smooth_bump_covering.is_subordinate.support_subset SmoothBumpCovering.IsSubordinate.support_subset
+-/
 
 variable (I)
 
+#print SmoothBumpCovering.exists_isSubordinate /-
 /-- Let `M` be a smooth manifold with corners modelled on a finite dimensional real vector space.
 Suppose also that `M` is a Hausdorff `σ`-compact topological space. Let `s` be a closed set
 in `M` and `U : M → set M` be a collection of sets such that `U x ∈ 𝓝 x` for every `x ∈ s`.
@@ -349,60 +402,84 @@ theorem exists_isSubordinate [T2Space M] [SigmaCompactSpace M] (hs : IsClosed s)
         (hr i hi).2
   · simpa only [coe_mk, SmoothBumpFunction.support_updateRIn, tsupport] using hfU i
 #align smooth_bump_covering.exists_is_subordinate SmoothBumpCovering.exists_isSubordinate
+-/
 
 variable {I M}
 
+#print SmoothBumpCovering.locallyFinite /-
 protected theorem locallyFinite : LocallyFinite fun i => support (fs i) :=
   fs.locally_finite'
 #align smooth_bump_covering.locally_finite SmoothBumpCovering.locallyFinite
+-/
 
+#print SmoothBumpCovering.point_finite /-
 protected theorem point_finite (x : M) : {i | fs i x ≠ 0}.Finite :=
   fs.LocallyFinite.point_finite x
 #align smooth_bump_covering.point_finite SmoothBumpCovering.point_finite
+-/
 
+#print SmoothBumpCovering.mem_chartAt_source_of_eq_one /-
 theorem mem_chartAt_source_of_eq_one {i : ι} {x : M} (h : fs i x = 1) :
     x ∈ (chartAt H (fs.c i)).source :=
   (fs i).support_subset_source <| by simp [h]
 #align smooth_bump_covering.mem_chart_at_source_of_eq_one SmoothBumpCovering.mem_chartAt_source_of_eq_one
+-/
 
+#print SmoothBumpCovering.mem_extChartAt_source_of_eq_one /-
 theorem mem_extChartAt_source_of_eq_one {i : ι} {x : M} (h : fs i x = 1) :
     x ∈ (extChartAt I (fs.c i)).source := by rw [extChartAt_source];
   exact fs.mem_chart_at_source_of_eq_one h
 #align smooth_bump_covering.mem_ext_chart_at_source_of_eq_one SmoothBumpCovering.mem_extChartAt_source_of_eq_one
+-/
 
+#print SmoothBumpCovering.ind /-
 /-- Index of a bump function such that `fs i =ᶠ[𝓝 x] 1`. -/
 def ind (x : M) (hx : x ∈ s) : ι :=
   (fs.eventuallyEq_one' x hx).some
 #align smooth_bump_covering.ind SmoothBumpCovering.ind
+-/
 
+#print SmoothBumpCovering.eventuallyEq_one /-
 theorem eventuallyEq_one (x : M) (hx : x ∈ s) : fs (fs.ind x hx) =ᶠ[𝓝 x] 1 :=
   (fs.eventuallyEq_one' x hx).choose_spec
 #align smooth_bump_covering.eventually_eq_one SmoothBumpCovering.eventuallyEq_one
+-/
 
+#print SmoothBumpCovering.apply_ind /-
 theorem apply_ind (x : M) (hx : x ∈ s) : fs (fs.ind x hx) x = 1 :=
   (fs.eventuallyEq_one x hx).eq_of_nhds
 #align smooth_bump_covering.apply_ind SmoothBumpCovering.apply_ind
+-/
 
+#print SmoothBumpCovering.mem_support_ind /-
 theorem mem_support_ind (x : M) (hx : x ∈ s) : x ∈ support (fs <| fs.ind x hx) := by
   simp [fs.apply_ind x hx]
 #align smooth_bump_covering.mem_support_ind SmoothBumpCovering.mem_support_ind
+-/
 
+#print SmoothBumpCovering.mem_chartAt_ind_source /-
 theorem mem_chartAt_ind_source (x : M) (hx : x ∈ s) : x ∈ (chartAt H (fs.c (fs.ind x hx))).source :=
   fs.mem_chartAt_source_of_eq_one (fs.apply_ind x hx)
 #align smooth_bump_covering.mem_chart_at_ind_source SmoothBumpCovering.mem_chartAt_ind_source
+-/
 
+#print SmoothBumpCovering.mem_extChartAt_ind_source /-
 theorem mem_extChartAt_ind_source (x : M) (hx : x ∈ s) :
     x ∈ (extChartAt I (fs.c (fs.ind x hx))).source :=
   fs.mem_extChartAt_source_of_eq_one (fs.apply_ind x hx)
 #align smooth_bump_covering.mem_ext_chart_at_ind_source SmoothBumpCovering.mem_extChartAt_ind_source
+-/
 
+#print SmoothBumpCovering.fintype /-
 /-- The index type of a `smooth_bump_covering` of a compact manifold is finite. -/
 protected def fintype [CompactSpace M] : Fintype ι :=
   fs.LocallyFinite.fintypeOfCompact fun i => (fs i).nonempty_support
 #align smooth_bump_covering.fintype SmoothBumpCovering.fintype
+-/
 
 variable [T2Space M]
 
+#print SmoothBumpCovering.toBumpCovering /-
 /-- Reinterpret a `smooth_bump_covering` as a continuous `bump_covering`. Note that not every
 `f : bump_covering ι M s` with smooth functions `f i` is a `smooth_bump_covering`. -/
 def toBumpCovering : BumpCovering ι M s
@@ -413,64 +490,84 @@ def toBumpCovering : BumpCovering ι M s
   le_one' i x := (fs i).le_one
   eventuallyEq_one' := fs.eventuallyEq_one'
 #align smooth_bump_covering.to_bump_covering SmoothBumpCovering.toBumpCovering
+-/
 
+#print SmoothBumpCovering.isSubordinate_toBumpCovering /-
 @[simp]
 theorem isSubordinate_toBumpCovering {f : SmoothBumpCovering ι I M s} {U : M → Set M} :
     (f.toBumpCovering.IsSubordinate fun i => U (f.c i)) ↔ f.IsSubordinate U :=
   Iff.rfl
 #align smooth_bump_covering.is_subordinate_to_bump_covering SmoothBumpCovering.isSubordinate_toBumpCovering
+-/
 
 alias is_subordinate_to_bump_covering ↔ _ is_subordinate.to_bump_covering
 #align smooth_bump_covering.is_subordinate.to_bump_covering SmoothBumpCovering.IsSubordinate.toBumpCovering
 
+#print SmoothBumpCovering.toSmoothPartitionOfUnity /-
 /-- Every `smooth_bump_covering` defines a smooth partition of unity. -/
 def toSmoothPartitionOfUnity : SmoothPartitionOfUnity ι I M s :=
   fs.toBumpCovering.toSmoothPartitionOfUnity fun i => (fs i).Smooth
 #align smooth_bump_covering.to_smooth_partition_of_unity SmoothBumpCovering.toSmoothPartitionOfUnity
+-/
 
+#print SmoothBumpCovering.toSmoothPartitionOfUnity_apply /-
 theorem toSmoothPartitionOfUnity_apply (i : ι) (x : M) :
     fs.toSmoothPartitionOfUnity i x = fs i x * ∏ᶠ (j) (hj : WellOrderingRel j i), (1 - fs j x) :=
   rfl
 #align smooth_bump_covering.to_smooth_partition_of_unity_apply SmoothBumpCovering.toSmoothPartitionOfUnity_apply
+-/
 
+#print SmoothBumpCovering.toSmoothPartitionOfUnity_eq_mul_prod /-
 theorem toSmoothPartitionOfUnity_eq_mul_prod (i : ι) (x : M) (t : Finset ι)
     (ht : ∀ j, WellOrderingRel j i → fs j x ≠ 0 → j ∈ t) :
     fs.toSmoothPartitionOfUnity i x =
       fs i x * ∏ j in t.filterₓ fun j => WellOrderingRel j i, (1 - fs j x) :=
   fs.toBumpCovering.toPartitionOfUnity_eq_mul_prod i x t ht
 #align smooth_bump_covering.to_smooth_partition_of_unity_eq_mul_prod SmoothBumpCovering.toSmoothPartitionOfUnity_eq_mul_prod
+-/
 
+#print SmoothBumpCovering.exists_finset_toSmoothPartitionOfUnity_eventuallyEq /-
 theorem exists_finset_toSmoothPartitionOfUnity_eventuallyEq (i : ι) (x : M) :
     ∃ t : Finset ι,
       fs.toSmoothPartitionOfUnity i =ᶠ[𝓝 x]
         fs i * ∏ j in t.filterₓ fun j => WellOrderingRel j i, (1 - fs j) :=
   fs.toBumpCovering.exists_finset_toPartitionOfUnity_eventuallyEq i x
 #align smooth_bump_covering.exists_finset_to_smooth_partition_of_unity_eventually_eq SmoothBumpCovering.exists_finset_toSmoothPartitionOfUnity_eventuallyEq
+-/
 
+#print SmoothBumpCovering.toSmoothPartitionOfUnity_zero_of_zero /-
 theorem toSmoothPartitionOfUnity_zero_of_zero {i : ι} {x : M} (h : fs i x = 0) :
     fs.toSmoothPartitionOfUnity i x = 0 :=
   fs.toBumpCovering.toPartitionOfUnity_zero_of_zero h
 #align smooth_bump_covering.to_smooth_partition_of_unity_zero_of_zero SmoothBumpCovering.toSmoothPartitionOfUnity_zero_of_zero
+-/
 
+#print SmoothBumpCovering.support_toSmoothPartitionOfUnity_subset /-
 theorem support_toSmoothPartitionOfUnity_subset (i : ι) :
     support (fs.toSmoothPartitionOfUnity i) ⊆ support (fs i) :=
   fs.toBumpCovering.support_toPartitionOfUnity_subset i
 #align smooth_bump_covering.support_to_smooth_partition_of_unity_subset SmoothBumpCovering.support_toSmoothPartitionOfUnity_subset
+-/
 
+#print SmoothBumpCovering.IsSubordinate.toSmoothPartitionOfUnity /-
 theorem IsSubordinate.toSmoothPartitionOfUnity {f : SmoothBumpCovering ι I M s} {U : M → Set M}
     (h : f.IsSubordinate U) : f.toSmoothPartitionOfUnity.IsSubordinate fun i => U (f.c i) :=
   h.toBumpCovering.toPartitionOfUnity
 #align smooth_bump_covering.is_subordinate.to_smooth_partition_of_unity SmoothBumpCovering.IsSubordinate.toSmoothPartitionOfUnity
+-/
 
+#print SmoothBumpCovering.sum_toSmoothPartitionOfUnity_eq /-
 theorem sum_toSmoothPartitionOfUnity_eq (x : M) :
     ∑ᶠ i, fs.toSmoothPartitionOfUnity i x = 1 - ∏ᶠ i, (1 - fs i x) :=
   fs.toBumpCovering.sum_toPartitionOfUnity_eq x
 #align smooth_bump_covering.sum_to_smooth_partition_of_unity_eq SmoothBumpCovering.sum_toSmoothPartitionOfUnity_eq
+-/
 
 end SmoothBumpCovering
 
 variable (I)
 
+#print exists_smooth_zero_one_of_closed /-
 /-- Given two disjoint closed sets in a Hausdorff σ-compact finite dimensional manifold, there
 exists an infinitely smooth function that is equal to `0` on one of them and is equal to one on the
 other. -/
@@ -488,9 +585,11 @@ theorem exists_smooth_zero_one_of_closed [T2Space M] [SigmaCompactSpace M] {s t
   refine' fun i => f.to_smooth_partition_of_unity_zero_of_zero _
   exact nmem_support.1 (subset_compl_comm.1 (hf.support_subset i) hx)
 #align exists_smooth_zero_one_of_closed exists_smooth_zero_one_of_closed
+-/
 
 namespace SmoothPartitionOfUnity
 
+#print SmoothPartitionOfUnity.single /-
 /-- A `smooth_partition_of_unity` that consists of a single function, uniformly equal to one,
 defined as an example for `inhabited` instance. -/
 def single (i : ι) (s : Set M) : SmoothPartitionOfUnity ι I M s :=
@@ -500,12 +599,14 @@ def single (i : ι) (s : Set M) : SmoothPartitionOfUnity ι I M s :=
     · simp only [smooth_one, ContinuousMap.coe_one, BumpCovering.coe_single, Pi.single_eq_same]
     · simp only [smooth_zero, BumpCovering.coe_single, Pi.single_eq_of_ne h, ContinuousMap.coe_zero]
 #align smooth_partition_of_unity.single SmoothPartitionOfUnity.single
+-/
 
 instance [Inhabited ι] (s : Set M) : Inhabited (SmoothPartitionOfUnity ι I M s) :=
   ⟨single I default s⟩
 
 variable [T2Space M] [SigmaCompactSpace M]
 
+#print SmoothPartitionOfUnity.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 {s : Set M} (hs : IsClosed s) (U : ι → Set M) (ho : ∀ i, IsOpen (U i))
@@ -520,11 +621,13 @@ theorem exists_isSubordinate {s : Set M} (hs : IsClosed s) (U : ι → Set M) (h
     rcases exists_smooth_zero_one_of_closed I hs ht hd with ⟨f, hf⟩
     exact ⟨f, f.smooth, hf⟩
 #align smooth_partition_of_unity.exists_is_subordinate SmoothPartitionOfUnity.exists_isSubordinate
+-/
 
 end SmoothPartitionOfUnity
 
 variable [SigmaCompactSpace M] [T2Space M] {t : M → Set F} {n : ℕ∞}
 
+#print exists_cont_mdiff_forall_mem_convex_of_local /-
 /-- Let `M` be a σ-compact Hausdorff finite dimensional topological manifold. Let `t : M → set F`
 be a family of convex sets. Suppose that for each point `x : M` there exists a neighborhood
 `U ∈ 𝓝 x` and a function `g : M → F` such that `g` is $C^n$ smooth on `U` and `g y ∈ t y` for all
@@ -545,7 +648,9 @@ theorem exists_cont_mdiff_forall_mem_convex_of_local (ht : ∀ x, Convex ℝ (t
       fun x => f.finsum_smul_mem_convex (mem_univ x) (fun i hi => hgt _ _ _) (ht _)⟩
   exact interior_subset (hf _ <| subset_closure hi)
 #align exists_cont_mdiff_forall_mem_convex_of_local exists_cont_mdiff_forall_mem_convex_of_local
+-/
 
+#print exists_smooth_forall_mem_convex_of_local /-
 /-- Let `M` be a σ-compact Hausdorff finite dimensional topological manifold. Let `t : M → set F`
 be a family of convex sets. Suppose that for each point `x : M` there exists a neighborhood
 `U ∈ 𝓝 x` and a function `g : M → F` such that `g` is smooth on `U` and `g y ∈ t y` for all `y ∈ U`.
@@ -557,7 +662,9 @@ theorem exists_smooth_forall_mem_convex_of_local (ht : ∀ x, Convex ℝ (t x))
     ∃ g : C^∞⟮I, M; 𝓘(ℝ, F), F⟯, ∀ x, g x ∈ t x :=
   exists_cont_mdiff_forall_mem_convex_of_local I ht Hloc
 #align exists_smooth_forall_mem_convex_of_local exists_smooth_forall_mem_convex_of_local
+-/
 
+#print exists_smooth_forall_mem_convex_of_local_const /-
 /-- Let `M` be a σ-compact Hausdorff finite dimensional topological manifold. Let `t : M → set F` be
 a family of convex sets. Suppose that for each point `x : M` there exists a vector `c : F` such that
 for all `y` in a neighborhood of `x` we have `c ∈ t y`. Then there exists a smooth function
@@ -569,7 +676,9 @@ theorem exists_smooth_forall_mem_convex_of_local_const (ht : ∀ x, Convex ℝ (
     let ⟨c, hc⟩ := Hloc x
     ⟨_, hc, fun _ => c, smoothOn_const, fun y => id⟩
 #align exists_smooth_forall_mem_convex_of_local_const exists_smooth_forall_mem_convex_of_local_const
+-/
 
+#print Emetric.exists_smooth_forall_closedBall_subset /-
 /-- Let `M` be a smooth σ-compact manifold with extended distance. Let `K : ι → set M` be a locally
 finite family of closed sets, let `U : ι → set M` be a family of open sets such that `K i ⊆ U i` for
 all `i`. Then there exists a positive smooth function `δ : M → ℝ≥0` such that for any `i` and
@@ -585,7 +694,9 @@ theorem Emetric.exists_smooth_forall_closedBall_subset {M} [EMetricSpace M] [Cha
     exists_smooth_forall_mem_convex_of_local_const I EMetric.exists_forall_closedBall_subset_aux₂
       (EMetric.exists_forall_closedBall_subset_aux₁ hK hU hKU hfin)
 #align emetric.exists_smooth_forall_closed_ball_subset Emetric.exists_smooth_forall_closedBall_subset
+-/
 
+#print Metric.exists_smooth_forall_closedBall_subset /-
 /-- Let `M` be a smooth σ-compact manifold with a metric. Let `K : ι → set M` be a locally finite
 family of closed sets, let `U : ι → set M` be a family of open sets such that `K i ⊆ U i` for all
 `i`. Then there exists a positive smooth function `δ : M → ℝ≥0` such that for any `i` and `x ∈ K i`,
@@ -602,4 +713,5 @@ theorem Metric.exists_smooth_forall_closedBall_subset {M} [MetricSpace M] [Chart
   rw [← Metric.emetric_closedBall (hδ0 _).le]
   exact hδ i x hx
 #align metric.exists_smooth_forall_closed_ball_subset Metric.exists_smooth_forall_closedBall_subset
+-/
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/93f880918cb51905fd51b76add8273cbc27718ab

@@ -186,7 +186,7 @@ the sum `λ x, ∑ᶠ i, f i x • g i x` is smooth on the whole manifold. -/
 theorem contMDiff_finsum_smul {g : ι → M → F}
     (hg : ∀ (i), ∀ x ∈ tsupport (f i), ContMDiffAt I 𝓘(ℝ, F) n (g i) x) :
     ContMDiff I 𝓘(ℝ, F) n fun x => ∑ᶠ i, f i x • g i x :=
-  (cont_mdiff_finsum fun i => f.contMDiff_smul (hg i)) <|
+  (contMDiff_finsum fun i => f.contMDiff_smul (hg i)) <|
     f.LocallyFinite.Subset fun i => support_smul_subset_left _ _
 #align smooth_partition_of_unity.cont_mdiff_finsum_smul SmoothPartitionOfUnity.contMDiff_finsum_smul
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/93f880918cb51905fd51b76add8273cbc27718ab

@@ -341,13 +341,13 @@ theorem exists_isSubordinate [T2Space M] [SigmaCompactSpace M] (hs : IsClosed s)
       (fun x hx => hfin.point_finite x) hsub' with
     ⟨V, hsV, hVc, hVf⟩
   choose r hrR hr using fun i => (f i).exists_r_pos_lt_subset_ball (hVc i) (hVf i)
-  refine' ⟨ι, ⟨c, fun i => (f i).updateR (r i) (hrR i), hcs, _, fun x hx => _⟩, fun i => _⟩
-  · simpa only [SmoothBumpFunction.support_updateR]
+  refine' ⟨ι, ⟨c, fun i => (f i).updateRIn (r i) (hrR i), hcs, _, fun x hx => _⟩, fun i => _⟩
+  · simpa only [SmoothBumpFunction.support_updateRIn]
   · refine' (mem_Union.1 <| hsV hx).imp fun i hi => _
     exact
-      ((f i).updateR _ _).eventuallyEq_one_of_dist_lt ((f i).support_subset_source <| hVf _ hi)
+      ((f i).updateRIn _ _).eventuallyEq_one_of_dist_lt ((f i).support_subset_source <| hVf _ hi)
         (hr i hi).2
-  · simpa only [coe_mk, SmoothBumpFunction.support_updateR, tsupport] using hfU i
+  · simpa only [coe_mk, SmoothBumpFunction.support_updateRIn, tsupport] using hfU i
 #align smooth_bump_covering.exists_is_subordinate SmoothBumpCovering.exists_isSubordinate
 
 variable {I M}
@@ -484,7 +484,7 @@ theorem exists_smooth_zero_one_of_closed [T2Space M] [SigmaCompactSpace M] {s t
   refine'
     ⟨⟨_, g.smooth_sum⟩, fun x hx => _, fun x => g.sum_eq_one, fun x =>
       ⟨g.sum_nonneg x, g.sum_le_one x⟩⟩
-  suffices ∀ i, g i x = 0 by simp only [this, ContMdiffMap.coeFn_mk, finsum_zero, Pi.zero_apply]
+  suffices ∀ i, g i x = 0 by simp only [this, ContMDiffMap.coeFn_mk, finsum_zero, Pi.zero_apply]
   refine' fun i => f.to_smooth_partition_of_unity_zero_of_zero _
   exact nmem_support.1 (subset_compl_comm.1 (hf.support_subset i) hx)
 #align exists_smooth_zero_one_of_closed exists_smooth_zero_one_of_closed

mathlib commit https://github.com/leanprover-community/mathlib/commit/93f880918cb51905fd51b76add8273cbc27718ab

@@ -169,26 +169,26 @@ theorem sum_nonneg (x : M) : 0 ≤ ∑ᶠ i, f i x :=
   f.toPartitionOfUnity.sum_nonneg x
 #align smooth_partition_of_unity.sum_nonneg SmoothPartitionOfUnity.sum_nonneg
 
-theorem contMdiff_smul {g : M → F} {i} (hg : ∀ x ∈ tsupport (f i), ContMdiffAt I 𝓘(ℝ, F) n g x) :
-    ContMdiff I 𝓘(ℝ, F) n fun x => f i x • g x :=
-  contMdiff_of_support fun x hx =>
-    ((f i).ContMdiff.ContMdiffAt.of_le le_top).smul <| hg x <| tsupport_smul_subset_left _ _ hx
-#align smooth_partition_of_unity.cont_mdiff_smul SmoothPartitionOfUnity.contMdiff_smul
+theorem contMDiff_smul {g : M → F} {i} (hg : ∀ x ∈ tsupport (f i), ContMDiffAt I 𝓘(ℝ, F) n g x) :
+    ContMDiff I 𝓘(ℝ, F) n fun x => f i x • g x :=
+  contMDiff_of_support fun x hx =>
+    ((f i).ContMDiff.ContMDiffAt.of_le le_top).smul <| hg x <| tsupport_smul_subset_left _ _ hx
+#align smooth_partition_of_unity.cont_mdiff_smul SmoothPartitionOfUnity.contMDiff_smul
 
 theorem smooth_smul {g : M → F} {i} (hg : ∀ x ∈ tsupport (f i), SmoothAt I 𝓘(ℝ, F) g x) :
     Smooth I 𝓘(ℝ, F) fun x => f i x • g x :=
-  f.contMdiff_smul hg
+  f.contMDiff_smul hg
 #align smooth_partition_of_unity.smooth_smul SmoothPartitionOfUnity.smooth_smul
 
 /-- If `f` is a smooth partition of unity on a set `s : set M` and `g : ι → M → F` is a family of
 functions such that `g i` is $C^n$ smooth at every point of the topological support of `f i`, then
 the sum `λ x, ∑ᶠ i, f i x • g i x` is smooth on the whole manifold. -/
-theorem contMdiff_finsum_smul {g : ι → M → F}
-    (hg : ∀ (i), ∀ x ∈ tsupport (f i), ContMdiffAt I 𝓘(ℝ, F) n (g i) x) :
-    ContMdiff I 𝓘(ℝ, F) n fun x => ∑ᶠ i, f i x • g i x :=
-  (cont_mdiff_finsum fun i => f.contMdiff_smul (hg i)) <|
+theorem contMDiff_finsum_smul {g : ι → M → F}
+    (hg : ∀ (i), ∀ x ∈ tsupport (f i), ContMDiffAt I 𝓘(ℝ, F) n (g i) x) :
+    ContMDiff I 𝓘(ℝ, F) n fun x => ∑ᶠ i, f i x • g i x :=
+  (cont_mdiff_finsum fun i => f.contMDiff_smul (hg i)) <|
     f.LocallyFinite.Subset fun i => support_smul_subset_left _ _
-#align smooth_partition_of_unity.cont_mdiff_finsum_smul SmoothPartitionOfUnity.contMdiff_finsum_smul
+#align smooth_partition_of_unity.cont_mdiff_finsum_smul SmoothPartitionOfUnity.contMDiff_finsum_smul
 
 /-- If `f` is a smooth partition of unity on a set `s : set M` and `g : ι → M → F` is a family of
 functions such that `g i` is smooth at every point of the topological support of `f i`, then the sum
@@ -196,7 +196,7 @@ functions such that `g i` is smooth at every point of the topological support of
 theorem smooth_finsum_smul {g : ι → M → F}
     (hg : ∀ (i), ∀ x ∈ tsupport (f i), SmoothAt I 𝓘(ℝ, F) (g i) x) :
     Smooth I 𝓘(ℝ, F) fun x => ∑ᶠ i, f i x • g i x :=
-  f.contMdiff_finsum_smul hg
+  f.contMDiff_finsum_smul hg
 #align smooth_partition_of_unity.smooth_finsum_smul SmoothPartitionOfUnity.smooth_finsum_smul
 
 theorem finsum_smul_mem_convex {g : ι → M → F} {t : Set F} {x : M} (hx : x ∈ s)
@@ -224,11 +224,11 @@ alias is_subordinate_to_partition_of_unity ↔ _ is_subordinate.to_partition_of_
 /-- If `f` is a smooth partition of unity on a set `s : set M` subordinate to a family of open sets
 `U : ι → set M` and `g : ι → M → F` is a family of functions such that `g i` is $C^n$ smooth on
 `U i`, then the sum `λ x, ∑ᶠ i, f i x • g i x` is $C^n$ smooth on the whole manifold. -/
-theorem IsSubordinate.contMdiff_finsum_smul {g : ι → M → F} (hf : f.IsSubordinate U)
-    (ho : ∀ i, IsOpen (U i)) (hg : ∀ i, ContMdiffOn I 𝓘(ℝ, F) n (g i) (U i)) :
-    ContMdiff I 𝓘(ℝ, F) n fun x => ∑ᶠ i, f i x • g i x :=
-  f.contMdiff_finsum_smul fun i x hx => (hg i).ContMdiffAt <| (ho i).mem_nhds (hf i hx)
-#align smooth_partition_of_unity.is_subordinate.cont_mdiff_finsum_smul SmoothPartitionOfUnity.IsSubordinate.contMdiff_finsum_smul
+theorem IsSubordinate.contMDiff_finsum_smul {g : ι → M → F} (hf : f.IsSubordinate U)
+    (ho : ∀ i, IsOpen (U i)) (hg : ∀ i, ContMDiffOn I 𝓘(ℝ, F) n (g i) (U i)) :
+    ContMDiff I 𝓘(ℝ, F) n fun x => ∑ᶠ i, f i x • g i x :=
+  f.contMDiff_finsum_smul fun i x hx => (hg i).ContMDiffAt <| (ho i).mem_nhds (hf i hx)
+#align smooth_partition_of_unity.is_subordinate.cont_mdiff_finsum_smul SmoothPartitionOfUnity.IsSubordinate.contMDiff_finsum_smul
 
 /-- If `f` is a smooth partition of unity on a set `s : set M` subordinate to a family of open sets
 `U : ι → set M` and `g : ι → M → F` is a family of functions such that `g i` is smooth on `U i`,
@@ -236,7 +236,7 @@ then the sum `λ x, ∑ᶠ i, f i x • g i x` is smooth on the whole manifold.
 theorem IsSubordinate.smooth_finsum_smul {g : ι → M → F} (hf : f.IsSubordinate U)
     (ho : ∀ i, IsOpen (U i)) (hg : ∀ i, SmoothOn I 𝓘(ℝ, F) (g i) (U i)) :
     Smooth I 𝓘(ℝ, F) fun x => ∑ᶠ i, f i x • g i x :=
-  hf.contMdiff_finsum_smul ho hg
+  hf.contMDiff_finsum_smul ho hg
 #align smooth_partition_of_unity.is_subordinate.smooth_finsum_smul SmoothPartitionOfUnity.IsSubordinate.smooth_finsum_smul
 
 end SmoothPartitionOfUnity
@@ -532,7 +532,7 @@ be a family of convex sets. Suppose that for each point `x : M` there exists a n
 for all `x`. See also `exists_smooth_forall_mem_convex_of_local` and
 `exists_smooth_forall_mem_convex_of_local_const`. -/
 theorem exists_cont_mdiff_forall_mem_convex_of_local (ht : ∀ x, Convex ℝ (t x))
-    (Hloc : ∀ x : M, ∃ U ∈ 𝓝 x, ∃ g : M → F, ContMdiffOn I 𝓘(ℝ, F) n g U ∧ ∀ y ∈ U, g y ∈ t y) :
+    (Hloc : ∀ x : M, ∃ U ∈ 𝓝 x, ∃ g : M → F, ContMDiffOn I 𝓘(ℝ, F) n g U ∧ ∀ y ∈ U, g y ∈ t y) :
     ∃ g : C^n⟮I, M; 𝓘(ℝ, F), F⟯, ∀ x, g x ∈ t x :=
   by
   choose U hU g hgs hgt using Hloc

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

@@ -123,8 +123,8 @@ structure SmoothPartitionOfUnity (s : Set M := univ) where
   toFun : ι → C^∞⟮I, M; 𝓘(ℝ), ℝ⟯
   locally_finite' : LocallyFinite fun i => support (to_fun i)
   nonneg' : ∀ i x, 0 ≤ to_fun i x
-  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 smooth_partition_of_unity SmoothPartitionOfUnity
 
 variable {ι I M}
@@ -144,11 +144,11 @@ theorem nonneg (i : ι) (x : M) : 0 ≤ f i x :=
   f.nonneg' i x
 #align smooth_partition_of_unity.nonneg SmoothPartitionOfUnity.nonneg
 
-theorem sum_eq_one {x} (hx : x ∈ s) : (∑ᶠ i, f i x) = 1 :=
+theorem sum_eq_one {x} (hx : x ∈ s) : ∑ᶠ i, f i x = 1 :=
   f.sum_eq_one' x hx
 #align smooth_partition_of_unity.sum_eq_one SmoothPartitionOfUnity.sum_eq_one
 
-theorem sum_le_one (x : M) : (∑ᶠ i, f i x) ≤ 1 :=
+theorem sum_le_one (x : M) : ∑ᶠ i, f i x ≤ 1 :=
   f.sum_le_one' x
 #align smooth_partition_of_unity.sum_le_one SmoothPartitionOfUnity.sum_le_one
 
@@ -200,7 +200,7 @@ theorem smooth_finsum_smul {g : ι → M → F}
 #align smooth_partition_of_unity.smooth_finsum_smul SmoothPartitionOfUnity.smooth_finsum_smul
 
 theorem finsum_smul_mem_convex {g : ι → M → F} {t : Set F} {x : M} (hx : x ∈ s)
-    (hg : ∀ i, f i x ≠ 0 → g i x ∈ t) (ht : Convex ℝ t) : (∑ᶠ i, f i x • g i x) ∈ t :=
+    (hg : ∀ i, f i x ≠ 0 → g i x ∈ t) (ht : Convex ℝ t) : ∑ᶠ i, f i x • g i x ∈ t :=
   ht.finsum_mem (fun i => f.NonNeg _ _) (f.sum_eq_one hx) hg
 #align smooth_partition_of_unity.finsum_smul_mem_convex SmoothPartitionOfUnity.finsum_smul_mem_convex
 
@@ -429,21 +429,21 @@ def toSmoothPartitionOfUnity : SmoothPartitionOfUnity ι I M s :=
 #align smooth_bump_covering.to_smooth_partition_of_unity SmoothBumpCovering.toSmoothPartitionOfUnity
 
 theorem toSmoothPartitionOfUnity_apply (i : ι) (x : M) :
-    fs.toSmoothPartitionOfUnity i x = fs i x * ∏ᶠ (j) (hj : WellOrderingRel j i), 1 - fs j x :=
+    fs.toSmoothPartitionOfUnity i x = fs i x * ∏ᶠ (j) (hj : WellOrderingRel j i), (1 - fs j x) :=
   rfl
 #align smooth_bump_covering.to_smooth_partition_of_unity_apply SmoothBumpCovering.toSmoothPartitionOfUnity_apply
 
 theorem toSmoothPartitionOfUnity_eq_mul_prod (i : ι) (x : M) (t : Finset ι)
     (ht : ∀ j, WellOrderingRel j i → fs j x ≠ 0 → j ∈ t) :
     fs.toSmoothPartitionOfUnity i x =
-      fs i x * ∏ j in t.filterₓ fun j => WellOrderingRel j i, 1 - fs j x :=
+      fs i x * ∏ j in t.filterₓ fun j => WellOrderingRel j i, (1 - fs j x) :=
   fs.toBumpCovering.toPartitionOfUnity_eq_mul_prod i x t ht
 #align smooth_bump_covering.to_smooth_partition_of_unity_eq_mul_prod SmoothBumpCovering.toSmoothPartitionOfUnity_eq_mul_prod
 
 theorem exists_finset_toSmoothPartitionOfUnity_eventuallyEq (i : ι) (x : M) :
     ∃ t : Finset ι,
       fs.toSmoothPartitionOfUnity i =ᶠ[𝓝 x]
-        fs i * ∏ j in t.filterₓ fun j => WellOrderingRel j i, 1 - fs j :=
+        fs i * ∏ j in t.filterₓ fun j => WellOrderingRel j i, (1 - fs j) :=
   fs.toBumpCovering.exists_finset_toPartitionOfUnity_eventuallyEq i x
 #align smooth_bump_covering.exists_finset_to_smooth_partition_of_unity_eventually_eq SmoothBumpCovering.exists_finset_toSmoothPartitionOfUnity_eventuallyEq
 
@@ -463,7 +463,7 @@ theorem IsSubordinate.toSmoothPartitionOfUnity {f : SmoothBumpCovering ι I M s}
 #align smooth_bump_covering.is_subordinate.to_smooth_partition_of_unity SmoothBumpCovering.IsSubordinate.toSmoothPartitionOfUnity
 
 theorem sum_toSmoothPartitionOfUnity_eq (x : M) :
-    (∑ᶠ i, fs.toSmoothPartitionOfUnity i x) = 1 - ∏ᶠ i, 1 - fs i x :=
+    ∑ᶠ i, fs.toSmoothPartitionOfUnity i x = 1 - ∏ᶠ i, (1 - fs i x) :=
   fs.toBumpCovering.sum_toPartitionOfUnity_eq x
 #align smooth_bump_covering.sum_to_smooth_partition_of_unity_eq SmoothBumpCovering.sum_toSmoothPartitionOfUnity_eq
 

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

@@ -356,7 +356,7 @@ protected theorem locallyFinite : LocallyFinite fun i => support (fs i) :=
   fs.locally_finite'
 #align smooth_bump_covering.locally_finite SmoothBumpCovering.locallyFinite
 
-protected theorem point_finite (x : M) : { i | fs i x ≠ 0 }.Finite :=
+protected theorem point_finite (x : M) : {i | fs i x ≠ 0}.Finite :=
   fs.LocallyFinite.point_finite x
 #align smooth_bump_covering.point_finite SmoothBumpCovering.point_finite
 

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

@@ -325,7 +325,7 @@ in `M` and `U : M → set M` be a collection of sets such that `U x ∈ 𝓝 x`
 Then there exists a smooth bump covering of `s` that is subordinate to `U`. -/
 theorem exists_isSubordinate [T2Space M] [SigmaCompactSpace M] (hs : IsClosed s)
     (hU : ∀ x ∈ s, U x ∈ 𝓝 x) :
-    ∃ (ι : Type uM)(f : SmoothBumpCovering ι I M s), f.IsSubordinate U :=
+    ∃ (ι : Type uM) (f : SmoothBumpCovering ι I M s), f.IsSubordinate U :=
   by
   -- First we deduce some missing instances
   haveI : LocallyCompactSpace H := I.locally_compact

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

@@ -65,7 +65,7 @@ universe uι uE uH uM uF
 
 open Function Filter FiniteDimensional Set
 
-open Topology Manifold Classical Filter BigOperators
+open scoped Topology Manifold Classical Filter BigOperators
 
 noncomputable section
 

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

@@ -249,9 +249,7 @@ theorem smooth_toPartitionOfUnity {E : Type uE} [NormedAddCommGroup E] [NormedSp
     [TopologicalSpace M] [ChartedSpace H M] {s : Set M} (f : BumpCovering ι M s)
     (hf : ∀ i, Smooth I 𝓘(ℝ) (f i)) (i : ι) : Smooth I 𝓘(ℝ) (f.toPartitionOfUnity i) :=
   (hf i).mul <|
-    (smooth_finprod_cond fun j _ => smooth_const.sub (hf j)) <|
-      by
-      simp only [mul_support_one_sub]
+    (smooth_finprod_cond fun j _ => smooth_const.sub (hf j)) <| by simp only [mul_support_one_sub];
       exact f.locally_finite
 #align bump_covering.smooth_to_partition_of_unity BumpCovering.smooth_toPartitionOfUnity
 
@@ -368,9 +366,7 @@ theorem mem_chartAt_source_of_eq_one {i : ι} {x : M} (h : fs i x = 1) :
 #align smooth_bump_covering.mem_chart_at_source_of_eq_one SmoothBumpCovering.mem_chartAt_source_of_eq_one
 
 theorem mem_extChartAt_source_of_eq_one {i : ι} {x : M} (h : fs i x = 1) :
-    x ∈ (extChartAt I (fs.c i)).source :=
-  by
-  rw [extChartAt_source]
+    x ∈ (extChartAt I (fs.c i)).source := by rw [extChartAt_source];
   exact fs.mem_chart_at_source_of_eq_one h
 #align smooth_bump_covering.mem_ext_chart_at_source_of_eq_one SmoothBumpCovering.mem_extChartAt_source_of_eq_one
 

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

@@ -586,8 +586,8 @@ theorem Emetric.exists_smooth_forall_closedBall_subset {M} [EMetricSpace M] [Cha
       (∀ x, 0 < δ x) ∧ ∀ (i), ∀ x ∈ K i, EMetric.closedBall x (ENNReal.ofReal (δ x)) ⊆ U i :=
   by
   simpa only [mem_inter_iff, forall_and, mem_preimage, mem_Inter, @forall_swap ι M] using
-    exists_smooth_forall_mem_convex_of_local_const I Emetric.exists_forall_closedBall_subset_aux₂
-      (Emetric.exists_forall_closedBall_subset_aux₁ hK hU hKU hfin)
+    exists_smooth_forall_mem_convex_of_local_const I EMetric.exists_forall_closedBall_subset_aux₂
+      (EMetric.exists_forall_closedBall_subset_aux₁ hK hU hKU hfin)
 #align emetric.exists_smooth_forall_closed_ball_subset Emetric.exists_smooth_forall_closedBall_subset
 
 /-- Let `M` be a smooth σ-compact manifold with a metric. Let `K : ι → set M` be a locally finite

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

@@ -339,7 +339,7 @@ theorem exists_isSubordinate [T2Space M] [SigmaCompactSpace M] (hs : IsClosed s)
     ⟨ι, c, f, hf, hsub', hfin⟩
   choose hcs hfU using hf
   -- Then we use the shrinking lemma to get a covering by smaller open
-  rcases exists_subset_unionᵢ_closed_subset hs (fun i => (f i).isOpen_support)
+  rcases exists_subset_iUnion_closed_subset hs (fun i => (f i).isOpen_support)
       (fun x hx => hfin.point_finite x) hsub' with
     ⟨V, hsV, hVc, hVf⟩
   choose r hrR hr using fun i => (f i).exists_r_pos_lt_subset_ball (hVc i) (hVf i)

mathlib commit https://github.com/leanprover-community/mathlib/commit/c9236f47f5b9df573443aa499c0d3968769628b7

@@ -186,7 +186,7 @@ the sum `λ x, ∑ᶠ i, f i x • g i x` is smooth on the whole manifold. -/
 theorem contMdiff_finsum_smul {g : ι → M → F}
     (hg : ∀ (i), ∀ x ∈ tsupport (f i), ContMdiffAt I 𝓘(ℝ, F) n (g i) x) :
     ContMdiff I 𝓘(ℝ, F) n fun x => ∑ᶠ i, f i x • g i x :=
-  (contMdiff_finsum fun i => f.contMdiff_smul (hg i)) <|
+  (cont_mdiff_finsum fun i => f.contMdiff_smul (hg i)) <|
     f.LocallyFinite.Subset fun i => support_smul_subset_left _ _
 #align smooth_partition_of_unity.cont_mdiff_finsum_smul SmoothPartitionOfUnity.contMdiff_finsum_smul
 
@@ -331,7 +331,7 @@ theorem exists_isSubordinate [T2Space M] [SigmaCompactSpace M] (hs : IsClosed s)
   by
   -- First we deduce some missing instances
   haveI : LocallyCompactSpace H := I.locally_compact
-  haveI : LocallyCompactSpace M := ChartedSpace.locally_compact H M
+  haveI : LocallyCompactSpace M := ChartedSpace.locallyCompact H M
   haveI : NormalSpace M := normal_of_paracompact_t2
   -- Next we choose a covering by supports of smooth bump functions
   have hB := fun x hx => SmoothBumpFunction.nhds_basis_support I (hU x hx)
@@ -516,7 +516,7 @@ theorem exists_isSubordinate {s : Set M} (hs : IsClosed s) (U : ι → Set M) (h
     (hU : s ⊆ ⋃ i, U i) : ∃ f : SmoothPartitionOfUnity ι I M s, f.IsSubordinate U :=
   by
   haveI : LocallyCompactSpace H := I.locally_compact
-  haveI : LocallyCompactSpace M := ChartedSpace.locally_compact H M
+  haveI : LocallyCompactSpace M := ChartedSpace.locallyCompact H M
   haveI : NormalSpace M := normal_of_paracompact_t2
   rcases BumpCovering.exists_isSubordinate_of_prop (Smooth I 𝓘(ℝ)) _ hs U ho hU with ⟨f, hf, hfU⟩
   · exact ⟨f.to_smooth_partition_of_unity hf, hfU.to_smooth_partition_of_unity hf⟩

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

@@ -186,7 +186,7 @@ the sum `λ x, ∑ᶠ i, f i x • g i x` is smooth on the whole manifold. -/
 theorem contMdiff_finsum_smul {g : ι → M → F}
     (hg : ∀ (i), ∀ x ∈ tsupport (f i), ContMdiffAt I 𝓘(ℝ, F) n (g i) x) :
     ContMdiff I 𝓘(ℝ, F) n fun x => ∑ᶠ i, f i x • g i x :=
-  (cont_mdiff_finsum fun i => f.contMdiff_smul (hg i)) <|
+  (contMdiff_finsum fun i => f.contMdiff_smul (hg i)) <|
     f.LocallyFinite.Subset fun i => support_smul_subset_left _ _
 #align smooth_partition_of_unity.cont_mdiff_finsum_smul SmoothPartitionOfUnity.contMdiff_finsum_smul
 

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

@@ -578,12 +578,12 @@ theorem exists_smooth_forall_mem_convex_of_local_const (ht : ∀ x, Convex ℝ (
 finite family of closed sets, let `U : ι → set M` be a family of open sets such that `K i ⊆ U i` for
 all `i`. Then there exists a positive smooth function `δ : M → ℝ≥0` such that for any `i` and
 `x ∈ K i`, we have `emetric.closed_ball x (δ x) ⊆ U i`. -/
-theorem Emetric.exists_smooth_forall_closedBall_subset {M} [EmetricSpace M] [ChartedSpace H M]
+theorem Emetric.exists_smooth_forall_closedBall_subset {M} [EMetricSpace M] [ChartedSpace H M]
     [SmoothManifoldWithCorners I M] [SigmaCompactSpace M] {K : ι → Set M} {U : ι → Set M}
     (hK : ∀ i, IsClosed (K i)) (hU : ∀ i, IsOpen (U i)) (hKU : ∀ i, K i ⊆ U i)
     (hfin : LocallyFinite K) :
     ∃ δ : C^∞⟮I, M; 𝓘(ℝ, ℝ), ℝ⟯,
-      (∀ x, 0 < δ x) ∧ ∀ (i), ∀ x ∈ K i, Emetric.closedBall x (ENNReal.ofReal (δ x)) ⊆ U i :=
+      (∀ x, 0 < δ x) ∧ ∀ (i), ∀ x ∈ K i, EMetric.closedBall x (ENNReal.ofReal (δ x)) ⊆ U i :=
   by
   simpa only [mem_inter_iff, forall_and, mem_preimage, mem_Inter, @forall_swap ι M] using
     exists_smooth_forall_mem_convex_of_local_const I Emetric.exists_forall_closedBall_subset_aux₂

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

@@ -583,7 +583,7 @@ theorem Emetric.exists_smooth_forall_closedBall_subset {M} [EmetricSpace M] [Cha
     (hK : ∀ i, IsClosed (K i)) (hU : ∀ i, IsOpen (U i)) (hKU : ∀ i, K i ⊆ U i)
     (hfin : LocallyFinite K) :
     ∃ δ : C^∞⟮I, M; 𝓘(ℝ, ℝ), ℝ⟯,
-      (∀ x, 0 < δ x) ∧ ∀ (i), ∀ x ∈ K i, Emetric.closedBall x (Ennreal.ofReal (δ x)) ⊆ U i :=
+      (∀ x, 0 < δ x) ∧ ∀ (i), ∀ x ∈ K i, Emetric.closedBall x (ENNReal.ofReal (δ x)) ⊆ U i :=
   by
   simpa only [mem_inter_iff, forall_and, mem_preimage, mem_Inter, @forall_swap ι M] using
     exists_smooth_forall_mem_convex_of_local_const I Emetric.exists_forall_closedBall_subset_aux₂

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

Reference the newly created issues #12094 and #12096, as well as the pre-existing #5171. Change all references to #10927 to #5171. Some of these changes were not labelled as "porting note"; change this for good measure.

@@ -99,7 +99,7 @@ subordinate to `U`, see `SmoothBumpCovering.exists_isSubordinate`.
 
 This covering can be used, e.g., to construct a partition of unity and to prove the weak
 Whitney embedding theorem. -/
--- Porting note: was @[nolint has_nonempty_instance]
+-- Porting note(#5171): was @[nolint has_nonempty_instance]
 structure SmoothBumpCovering (s : Set M := univ) where
   /-- The center point of each bump in the smooth covering. -/
   c : ι → M

@@ -271,14 +271,13 @@ theorem finite_tsupport : {i | x₀ ∈ tsupport (ρ i)}.Finite :=
 
 /-- 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 (x : M ): Finset ι :=
+def fintsupport (x : M) : Finset ι :=
   (ρ.finite_tsupport x).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 :=
+theorem eventually_fintsupport_subset : ∀ᶠ y in 𝓝 x₀, ρ.fintsupport y ⊆ ρ.fintsupport x₀ :=
   ρ.toPartitionOfUnity.eventually_fintsupport_subset _
 
 theorem finsupport_subset_fintsupport : ρ.finsupport x₀ ⊆ ρ.fintsupport x₀ :=

Homogenises porting notes via capitalisation and addition of whitespace.

It makes the following changes:

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

@@ -99,7 +99,7 @@ subordinate to `U`, see `SmoothBumpCovering.exists_isSubordinate`.
 
 This covering can be used, e.g., to construct a partition of unity and to prove the weak
 Whitney embedding theorem. -/
--- porting note: was @[nolint has_nonempty_instance]
+-- Porting note: was @[nolint has_nonempty_instance]
 structure SmoothBumpCovering (s : Set M := univ) where
   /-- The center point of each bump in the smooth covering. -/
   c : ι → M
@@ -525,7 +525,7 @@ theorem exists_finset_toSmoothPartitionOfUnity_eventuallyEq (i : ι) (x : M) :
     ∃ t : Finset ι,
       fs.toSmoothPartitionOfUnity i =ᶠ[𝓝 x]
         fs i * ∏ j in t.filter fun j => WellOrderingRel j i, ((1 : M → ℝ) - fs j) := by
-  -- porting note: was defeq, now the continuous lemma uses bundled homs
+  -- Porting note: was defeq, now the continuous lemma uses bundled homs
   simpa using fs.toBumpCovering.exists_finset_toPartitionOfUnity_eventuallyEq i x
 #align smooth_bump_covering.exists_finset_to_smooth_partition_of_unity_eventually_eq SmoothBumpCovering.exists_finset_toSmoothPartitionOfUnity_eventuallyEq
 

Use Lean 4 syntax fun x ↦ instead, matching the style guide. This is close to exhaustive for doc comments; mathlib has about 460 remaining uses of λ (not all in Lean 3 syntax).

@@ -89,7 +89,7 @@ variable (ι M)
 /-- We say that a collection of `SmoothBumpFunction`s is a `SmoothBumpCovering` of a set `s` if
 
 * `(f i).c ∈ s` for all `i`;
-* the family `λ i, support (f i)` is locally finite;
+* the family `fun i ↦ support (f i)` is locally finite;
 * for each point `x ∈ s` there exists `i` such that `f i =ᶠ[𝓝 x] 1`;
   in other words, `x` belongs to the interior of `{y | f i y = 1}`;
 
@@ -116,7 +116,7 @@ structure SmoothBumpCovering (s : Set M := univ) where
 /-- We say that a collection of functions form a smooth partition of unity on a set `s` if
 
 * all functions are infinitely smooth and nonnegative;
-* the family `λ i, support (f i)` is locally finite;
+* the family `fun i ↦ support (f i)` is locally finite;
 * for all `x ∈ s` the sum `∑ᶠ i, f i x` equals one;
 * for all `x`, the sum `∑ᶠ i, f i x` is less than or equal to one. -/
 structure SmoothPartitionOfUnity (s : Set M := univ) where
@@ -196,7 +196,7 @@ theorem smooth_smul {g : M → F} {i} (hg : ∀ x ∈ tsupport (f i), SmoothAt I
 
 /-- If `f` is a smooth partition of unity on a set `s : Set M` and `g : ι → M → F` is a family of
 functions such that `g i` is $C^n$ smooth at every point of the topological support of `f i`, then
-the sum `λ x, ∑ᶠ i, f i x • g i x` is smooth on the whole manifold. -/
+the sum `fun x ↦ ∑ᶠ i, f i x • g i x` is smooth on the whole manifold. -/
 theorem contMDiff_finsum_smul {g : ι → M → F}
     (hg : ∀ (i), ∀ x ∈ tsupport (f i), ContMDiffAt I 𝓘(ℝ, F) n (g i) x) :
     ContMDiff I 𝓘(ℝ, F) n fun x => ∑ᶠ i, f i x • g i x :=
@@ -206,7 +206,7 @@ theorem contMDiff_finsum_smul {g : ι → M → F}
 
 /-- If `f` is a smooth partition of unity on a set `s : Set M` and `g : ι → M → F` is a family of
 functions such that `g i` is smooth at every point of the topological support of `f i`, then the sum
-`λ x, ∑ᶠ i, f i x • g i x` is smooth on the whole manifold. -/
+`fun x ↦ ∑ᶠ i, f i x • g i x` is smooth on the whole manifold. -/
 theorem smooth_finsum_smul {g : ι → M → F}
     (hg : ∀ (i), ∀ x ∈ tsupport (f i), SmoothAt I 𝓘(ℝ, F) (g i) x) :
     Smooth I 𝓘(ℝ, F) fun x => ∑ᶠ i, f i x • g i x :=
@@ -310,7 +310,7 @@ alias ⟨_, IsSubordinate.toPartitionOfUnity⟩ := isSubordinate_toPartitionOfUn
 
 /-- If `f` is a smooth partition of unity on a set `s : Set M` subordinate to a family of open sets
 `U : ι → Set M` and `g : ι → M → F` is a family of functions such that `g i` is $C^n$ smooth on
-`U i`, then the sum `λ x, ∑ᶠ i, f i x • g i x` is $C^n$ smooth on the whole manifold. -/
+`U i`, then the sum `fun x ↦ ∑ᶠ i, f i x • g i x` is $C^n$ smooth on the whole manifold. -/
 theorem IsSubordinate.contMDiff_finsum_smul {g : ι → M → F} (hf : f.IsSubordinate U)
     (ho : ∀ i, IsOpen (U i)) (hg : ∀ i, ContMDiffOn I 𝓘(ℝ, F) n (g i) (U i)) :
     ContMDiff I 𝓘(ℝ, F) n fun x => ∑ᶠ i, f i x • g i x :=
@@ -319,7 +319,7 @@ theorem IsSubordinate.contMDiff_finsum_smul {g : ι → M → F} (hf : f.IsSubor
 
 /-- If `f` is a smooth partition of unity on a set `s : Set M` subordinate to a family of open sets
 `U : ι → Set M` and `g : ι → M → F` is a family of functions such that `g i` is smooth on `U i`,
-then the sum `λ x, ∑ᶠ i, f i x • g i x` is smooth on the whole manifold. -/
+then the sum `fun x ↦ ∑ᶠ i, f i x • g i x` is smooth on the whole manifold. -/
 theorem IsSubordinate.smooth_finsum_smul {g : ι → M → F} (hf : f.IsSubordinate U)
     (ho : ∀ i, IsOpen (U i)) (hg : ∀ i, SmoothOn I 𝓘(ℝ, F) (g i) (U i)) :
     Smooth I 𝓘(ℝ, F) fun x => ∑ᶠ i, f i x • g i x :=

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.

@@ -166,6 +166,7 @@ theorem sum_le_one (x : M) : ∑ᶠ i, f i x ≤ 1 :=
 #align smooth_partition_of_unity.sum_le_one SmoothPartitionOfUnity.sum_le_one
 
 /-- Reinterpret a smooth partition of unity as a continuous partition of unity. -/
+@[simps]
 def toPartitionOfUnity : PartitionOfUnity ι M s :=
   { f with toFun := fun i => f i }
 #align smooth_partition_of_unity.to_partition_of_unity SmoothPartitionOfUnity.toPartitionOfUnity
@@ -232,6 +233,64 @@ theorem finsum_smul_mem_convex {g : ι → M → F} {t : Set F} {x : M} (hx : x
   ht.finsum_mem (fun _ => f.nonneg _ _) (f.sum_eq_one hx) hg
 #align smooth_partition_of_unity.finsum_smul_mem_convex SmoothPartitionOfUnity.finsum_smul_mem_convex
 
+section finsupport
+
+variable {s : Set M} (ρ : SmoothPartitionOfUnity ι I M s) (x₀ : M)
+
+/-- The support of a smooth partition of unity at a point `x₀` as a `Finset`.
+  This is the set of `i : ι` such that `x₀ ∈ support f i`, i.e. `f i ≠ x₀`. -/
+def finsupport : Finset ι := ρ.toPartitionOfUnity.finsupport x₀
+
+@[simp]
+theorem mem_finsupport {i : ι} : i ∈ ρ.finsupport x₀ ↔ i ∈ support fun i ↦ ρ i x₀ :=
+  ρ.toPartitionOfUnity.mem_finsupport x₀
+
+@[simp]
+theorem coe_finsupport : (ρ.finsupport x₀ : Set ι) = support fun i ↦ ρ i x₀ :=
+  ρ.toPartitionOfUnity.coe_finsupport x₀
+
+theorem sum_finsupport (hx₀ : x₀ ∈ s) : ∑ i in ρ.finsupport x₀, ρ i x₀ = 1 :=
+  ρ.toPartitionOfUnity.sum_finsupport hx₀
+
+theorem sum_finsupport' (hx₀ : x₀ ∈ s) {I : Finset ι} (hI : ρ.finsupport x₀ ⊆ I) :
+    ∑ i in I, ρ i x₀ = 1 :=
+  ρ.toPartitionOfUnity.sum_finsupport' hx₀ hI
+
+theorem sum_finsupport_smul_eq_finsum {A : Type*} [AddCommGroup A] [Module ℝ A] (φ : ι → M → A) :
+    ∑ i in ρ.finsupport x₀, ρ i x₀ • φ i x₀ = ∑ᶠ i, ρ i x₀ • φ i x₀ :=
+  ρ.toPartitionOfUnity.sum_finsupport_smul_eq_finsum φ
+
+end finsupport
+
+section fintsupport -- smooth partitions of unity have locally finite `tsupport`
+variable {s : Set M} (ρ : SmoothPartitionOfUnity ι I M s) (x₀ : M)
+
+/-- The `tsupport`s of a smooth partition of unity are locally finite. -/
+theorem finite_tsupport : {i | x₀ ∈ tsupport (ρ i)}.Finite :=
+  ρ.toPartitionOfUnity.finite_tsupport _
+
+/-- 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 (x : M ): Finset ι :=
+  (ρ.finite_tsupport x).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 :=
+  ρ.toPartitionOfUnity.eventually_fintsupport_subset _
+
+theorem finsupport_subset_fintsupport : ρ.finsupport x₀ ⊆ ρ.fintsupport x₀ :=
+  ρ.toPartitionOfUnity.finsupport_subset_fintsupport x₀
+
+theorem eventually_finsupport_subset : ∀ᶠ y in 𝓝 x₀, ρ.finsupport y ⊆ ρ.fintsupport x₀ :=
+  ρ.toPartitionOfUnity.eventually_finsupport_subset x₀
+
+end fintsupport
+
+section IsSubordinate
+
 /-- A smooth 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 (f : SmoothPartitionOfUnity ι I M s) (U : ι → Set M) :=
@@ -267,6 +326,8 @@ theorem IsSubordinate.smooth_finsum_smul {g : ι → M → F} (hf : f.IsSubordin
   hf.contMDiff_finsum_smul ho hg
 #align smooth_partition_of_unity.is_subordinate.smooth_finsum_smul SmoothPartitionOfUnity.IsSubordinate.smooth_finsum_smul
 
+end IsSubordinate
+
 end SmoothPartitionOfUnity
 
 namespace BumpCovering

A point-wise version of contMDiff_finsum_smul. From sphere-eversion.

@@ -212,6 +212,21 @@ theorem smooth_finsum_smul {g : ι → M → F}
   f.contMDiff_finsum_smul hg
 #align smooth_partition_of_unity.smooth_finsum_smul SmoothPartitionOfUnity.smooth_finsum_smul
 
+theorem contMDiffAt_finsum {x₀ : M} {g : ι → M → F}
+    (hφ : ∀ i, x₀ ∈ tsupport (f i) → ContMDiffAt I 𝓘(ℝ, F) n (g i) x₀) :
+    ContMDiffAt I 𝓘(ℝ, F) n (fun x ↦ ∑ᶠ i, f i x • g i x) x₀ := by
+  refine _root_.contMDiffAt_finsum (f.locallyFinite.smul_left _) fun i ↦ ?_
+  by_cases hx : x₀ ∈ tsupport (f i)
+  · exact ContMDiffAt.smul ((f i).smooth.of_le le_top).contMDiffAt (hφ i hx)
+  · exact contMDiffAt_of_not_mem (compl_subset_compl.mpr
+      (tsupport_smul_subset_left (f i) (g i)) hx) n
+
+theorem contDiffAt_finsum {s : Set E} (f : SmoothPartitionOfUnity ι 𝓘(ℝ, E) E s) {x₀ : E}
+    {g : ι → E → F} (hφ : ∀ i, x₀ ∈ tsupport (f i) → ContDiffAt ℝ n (g i) x₀) :
+    ContDiffAt ℝ n (fun x ↦ ∑ᶠ i, f i x • g i x) x₀ := by
+  simp only [← contMDiffAt_iff_contDiffAt] at *
+  exact f.contMDiffAt_finsum hφ
+
 theorem finsum_smul_mem_convex {g : ι → M → F} {t : Set F} {x : M} (hx : x ∈ s)
     (hg : ∀ i, f i x ≠ 0 → g i x ∈ t) (ht : Convex ℝ t) : ∑ᶠ i, f i x • g i x ∈ t :=
   ht.finsum_mem (fun _ => f.nonneg _ _) (f.sum_eq_one hx) hg

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

The version in sphere-eversion uses an unbundled design; we provide a bundled version (for now) to match the remaining file.

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

@@ -483,7 +483,7 @@ See also `exists_msmooth_zero_iff_one_iff_of_isClosed`, which ensures additional
 `f` is equal to `0` exactly on `s` and to `1` exactly on `t`. -/
 theorem exists_smooth_zero_one_of_isClosed [T2Space M] [SigmaCompactSpace M] {s t : Set M}
     (hs : IsClosed s) (ht : IsClosed t) (hd : Disjoint s t) :
-    ∃ f : C^∞⟮I, M; 𝓘(ℝ), ℝ⟯, EqOn f 0 s ∧ EqOn f 1 t ∧ ∀ x, f x ∈ Icc (0 : ℝ) 1 := by
+    ∃ f : C^∞⟮I, M; 𝓘(ℝ), ℝ⟯, EqOn f 0 s ∧ EqOn f 1 t ∧ ∀ x, f x ∈ Icc 0 1 := by
   have : ∀ x ∈ t, sᶜ ∈ 𝓝 x := fun x hx => hs.isOpen_compl.mem_nhds (disjoint_right.1 hd hx)
   rcases SmoothBumpCovering.exists_isSubordinate I ht this with ⟨ι, f, hf⟩
   set g := f.toSmoothPartitionOfUnity
@@ -495,6 +495,35 @@ theorem exists_smooth_zero_one_of_isClosed [T2Space M] [SigmaCompactSpace M] {s
   exact nmem_support.1 (subset_compl_comm.1 (hf.support_subset i) hx)
 #align exists_smooth_zero_one_of_closed exists_smooth_zero_one_of_isClosed
 
+/-- Given two disjoint closed sets `s, t` in a Hausdorff normal σ-compact finite dimensional
+manifold `M`, there exists a smooth function `f : M → [0,1]` that vanishes in a neighbourhood of `s`
+and is equal to `1` in a neighbourhood of `t`. -/
+theorem exists_smooth_zero_one_nhds_of_isClosed [T2Space M] [NormalSpace M] [SigmaCompactSpace M]
+    {s t : Set M} (hs : IsClosed s) (ht : IsClosed t) (hd : Disjoint s t) :
+    ∃ f : C^∞⟮I, M; 𝓘(ℝ), ℝ⟯, (∀ᶠ x in 𝓝ˢ s, f x = 0) ∧ (∀ᶠ x in 𝓝ˢ t, f x = 1) ∧
+      ∀ x, f x ∈ Icc 0 1 := by
+  obtain ⟨u, u_op, hsu, hut⟩ := normal_exists_closure_subset hs ht.isOpen_compl
+    (subset_compl_iff_disjoint_left.mpr hd.symm)
+  obtain ⟨v, v_op, htv, hvu⟩ := normal_exists_closure_subset ht isClosed_closure.isOpen_compl
+    (subset_compl_comm.mp hut)
+  obtain ⟨f, hfu, hfv, hf⟩ := exists_smooth_zero_one_of_isClosed I isClosed_closure isClosed_closure
+    (subset_compl_iff_disjoint_left.mp hvu)
+  refine ⟨f, ?_, ?_, hf⟩
+  · exact eventually_of_mem (mem_of_superset (u_op.mem_nhdsSet.mpr hsu) subset_closure) hfu
+  · exact eventually_of_mem (mem_of_superset (v_op.mem_nhdsSet.mpr htv) subset_closure) hfv
+
+/-- Given two sets `s, t` in a Hausdorff normal σ-compact finite-dimensional manifold `M`
+with `s` open and `s ⊆ interior t`, there is a smooth function `f : M → [0,1]` which is equal to `s`
+in a neighbourhood of `s` and has support contained in `t`. -/
+theorem exists_smooth_one_nhds_of_subset_interior [T2Space M] [NormalSpace M] [SigmaCompactSpace M]
+    {s t : Set M} (hs : IsClosed s) (hd : s ⊆ interior t) :
+    ∃ f : C^∞⟮I, M; 𝓘(ℝ), ℝ⟯, (∀ᶠ x in 𝓝ˢ s, f x = 1) ∧ (∀ x ∉ t, f x = 0) ∧
+      ∀ x, f x ∈ Icc 0 1 := by
+  rcases exists_smooth_zero_one_nhds_of_isClosed I isOpen_interior.isClosed_compl hs
+    (by rwa [← subset_compl_iff_disjoint_left, compl_compl]) with ⟨f, h0, h1, hf⟩
+  refine ⟨f, h1, fun x hx ↦ ?_, hf⟩
+  exact h0.self_of_nhdsSet _ fun hx' ↦ hx <| interior_subset hx'
+
 namespace SmoothPartitionOfUnity
 
 /-- A `SmoothPartitionOfUnity` that consists of a single function, uniformly equal to one,

And fix a typo-ed lemma name in a doc comment.

@@ -479,9 +479,9 @@ variable (I)
 
 /-- Given two disjoint closed sets `s, t` in a Hausdorff σ-compact finite dimensional manifold,
 there exists an infinitely smooth function that is equal to `0` on `s` and to `1` on `t`.
-See also `exists_smooth_zero_iff_one_iff_of_closed`, which ensures additionally that
+See also `exists_msmooth_zero_iff_one_iff_of_isClosed`, which ensures additionally that
 `f` is equal to `0` exactly on `s` and to `1` exactly on `t`. -/
-theorem exists_smooth_zero_one_of_closed [T2Space M] [SigmaCompactSpace M] {s t : Set M}
+theorem exists_smooth_zero_one_of_isClosed [T2Space M] [SigmaCompactSpace M] {s t : Set M}
     (hs : IsClosed s) (ht : IsClosed t) (hd : Disjoint s t) :
     ∃ f : C^∞⟮I, M; 𝓘(ℝ), ℝ⟯, EqOn f 0 s ∧ EqOn f 1 t ∧ ∀ x, f x ∈ Icc (0 : ℝ) 1 := by
   have : ∀ x ∈ t, sᶜ ∈ 𝓝 x := fun x hx => hs.isOpen_compl.mem_nhds (disjoint_right.1 hd hx)
@@ -493,7 +493,7 @@ theorem exists_smooth_zero_one_of_closed [T2Space M] [SigmaCompactSpace M] {s t
   suffices ∀ i, g i x = 0 by simp only [this, ContMDiffMap.coeFn_mk, finsum_zero, Pi.zero_apply]
   refine' fun i => f.toSmoothPartitionOfUnity_zero_of_zero _
   exact nmem_support.1 (subset_compl_comm.1 (hf.support_subset i) hx)
-#align exists_smooth_zero_one_of_closed exists_smooth_zero_one_of_closed
+#align exists_smooth_zero_one_of_closed exists_smooth_zero_one_of_isClosed
 
 namespace SmoothPartitionOfUnity
 
@@ -523,7 +523,7 @@ theorem exists_isSubordinate {s : Set M} (hs : IsClosed s) (U : ι → Set M) (h
   · rcases this with ⟨f, hf, hfU⟩
     exact ⟨f.toSmoothPartitionOfUnity hf, hfU.toSmoothPartitionOfUnity hf⟩
   · intro s t hs ht hd
-    rcases exists_smooth_zero_one_of_closed I hs ht hd with ⟨f, hf⟩
+    rcases exists_smooth_zero_one_of_isClosed I hs ht hd with ⟨f, hf⟩
     exact ⟨f, f.smooth, hf⟩
 #align smooth_partition_of_unity.exists_is_subordinate SmoothPartitionOfUnity.exists_isSubordinate
 
@@ -716,8 +716,8 @@ theorem exists_msmooth_support_eq_eq_one_iff
 
 /-- Given two disjoint closed sets `s, t` in a Hausdorff σ-compact finite dimensional manifold,
 there exists an infinitely smooth function that is equal to `0` exactly on `s` and to `1`
-exactly on `t`. See also `exists_smooth_zero_one_of_closed` for a slightly weaker version. -/
-theorem exists_msmooth_zero_iff_one_iff_of_closed {s t : Set M}
+exactly on `t`. See also `exists_smooth_zero_one_of_isClosed` for a slightly weaker version. -/
+theorem exists_msmooth_zero_iff_one_iff_of_isClosed {s t : Set M}
     (hs : IsClosed s) (ht : IsClosed t) (hd : Disjoint s t) :
     ∃ f : M → ℝ, Smooth I 𝓘(ℝ) f ∧ range f ⊆ Icc 0 1 ∧ (∀ x, x ∈ s ↔ f x = 0)
       ∧ (∀ x, x ∈ t ↔ f x = 1) := by

This comment was written by @urkud back then - but presumably forgotten to adjust then the proof of Whitney embedding moved into its own file.

@@ -82,8 +82,7 @@ partition of unity in some proofs.
 
 We prove that on a smooth finite dimensional real manifold with `σ`-compact Hausdorff topology, for
 any `U : M → Set M` such that `∀ x ∈ s, U x ∈ 𝓝 x` there exists a `SmoothBumpCovering ι I M s`
-subordinate to `U`. Then we use this fact to prove a version of the Whitney embedding theorem: any
-compact real manifold can be embedded into `ℝ^n` for large enough `n`.  -/
+subordinate to `U`. -/
 
 variable (ι M)
 

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.

@@ -139,7 +139,7 @@ namespace SmoothPartitionOfUnity
 
 variable {s : Set M} (f : SmoothPartitionOfUnity ι I M s) {n : ℕ∞}
 
-instance {s : Set M} : DFunLike (SmoothPartitionOfUnity ι I M s) ι fun _ => C^∞⟮I, M; 𝓘(ℝ), ℝ⟯ where
+instance {s : Set M} : FunLike (SmoothPartitionOfUnity ι I M s) ι C^∞⟮I, M; 𝓘(ℝ), ℝ⟯ where
   coe := toFun
   coe_injective' 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>

@@ -139,7 +139,7 @@ namespace SmoothPartitionOfUnity
 
 variable {s : Set M} (f : SmoothPartitionOfUnity ι I M s) {n : ℕ∞}
 
-instance {s : Set M} : FunLike (SmoothPartitionOfUnity ι I M s) ι fun _ => C^∞⟮I, M; 𝓘(ℝ), ℝ⟯ where
+instance {s : Set M} : DFunLike (SmoothPartitionOfUnity ι I M s) ι fun _ => C^∞⟮I, M; 𝓘(ℝ), ℝ⟯ where
   coe := toFun
   coe_injective' f g h := by cases f; cases g; congr
 

• multiplicativise contMDiff_of_support, contMDiffWithinAt_of_not_mem and contMDiffAt_of_not_mem and use to_additive
• golf extend_one with the multiplicative version
• generalise these lemmas to manifolds with zero/one
• slight clean-up: remove unused variables and opens
• slight drive-by golfing of one proof
• deprecate eventuallyEq_zero_nhds in favor of not_mem_tsupport_iff_eventuallyEq

Addresses the post-merge review comments in #9669.

Co-authored-by: ADedecker <anatolededecker@gmail.com> Co-authored-by: grunweg <grunweg@posteo.de>

@@ -185,7 +185,7 @@ theorem sum_nonneg (x : M) : 0 ≤ ∑ᶠ i, f i x :=
 
 theorem contMDiff_smul {g : M → F} {i} (hg : ∀ x ∈ tsupport (f i), ContMDiffAt I 𝓘(ℝ, F) n g x) :
     ContMDiff I 𝓘(ℝ, F) n fun x => f i x • g x :=
-  contMDiff_of_support fun x hx =>
+  contMDiff_of_tsupport fun x hx =>
     ((f i).contMDiff.contMDiffAt.of_le le_top).smul <| hg x <| tsupport_smul_subset_left _ _ hx
 #align smooth_partition_of_unity.cont_mdiff_smul SmoothPartitionOfUnity.contMDiff_smul
 

It's just used once for one variable. That's not pulling its weight.

@@ -85,9 +85,6 @@ any `U : M → Set M` such that `∀ x ∈ s, U x ∈ 𝓝 x` there exists a `Sm
 subordinate to `U`. Then we use this fact to prove a version of the Whitney embedding theorem: any
 compact real manifold can be embedded into `ℝ^n` for large enough `n`.  -/
 
-set_option autoImplicit true
-
-
 variable (ι M)
 
 /-- We say that a collection of `SmoothBumpFunction`s is a `SmoothBumpCovering` of a set `s` if
@@ -634,7 +631,7 @@ lemma IsOpen.exists_msmooth_support_eq_aux {s : Set H} (hs : IsOpen s) :
 
 /-- Given an open set in a finite-dimensional real manifold, there exists a nonnegative smooth
 function with support equal to `s`. -/
-theorem IsOpen.exists_msmooth_support_eq (hs : IsOpen s) :
+theorem IsOpen.exists_msmooth_support_eq {s : Set M} (hs : IsOpen s) :
     ∃ f : M → ℝ, f.support = s ∧ Smooth I 𝓘(ℝ) f ∧ ∀ x, 0 ≤ f x := by
   rcases SmoothPartitionOfUnity.exists_isSubordinate_chartAt_source I M with ⟨f, hf⟩
   have A : ∀ (c : M), ∃ g : H → ℝ,

Removes nonterminal uses of simp at. Replaces most of these with instances of simp? ... says.

Co-authored-by: Scott Morrison <scott.morrison@gmail.com> Co-authored-by: Mario Carneiro <di.gama@gmail.com>

@@ -701,7 +701,7 @@ theorem exists_msmooth_support_eq_eq_one_iff
         apply lt_of_le_of_ne (g_pos x) (Ne.symm ?_)
         rw [← mem_support, g_supp]
         contrapose! xs
-        simp at xs
+        simp? at xs says simp only [mem_compl_iff, not_not] at xs
         exact h.trans f_supp.symm.subset xs
       linarith [f_pos x]
   refine ⟨fun x ↦ f x / (f x + g x), ?_, ?_, ?_, ?_⟩

LocalHomeomorph evokes a "local homeomorphism": this is not what this means. Instead, this is a homeomorphism on an open set of the domain (extended to the whole space, by the junk value pattern). Hence, partial homeomorphism is more appropriate, and avoids confusion with IsLocallyHomeomorph.

A future PR will rename LocalEquiv to PartialEquiv.

Zulip discussion

@@ -642,7 +642,7 @@ theorem IsOpen.exists_msmooth_support_eq (hs : IsOpen s) :
       Smooth I 𝓘(ℝ) g ∧ Set.range g ⊆ Set.Icc 0 1 := by
     intro i
     apply IsOpen.exists_msmooth_support_eq_aux
-    exact LocalHomeomorph.isOpen_inter_preimage_symm _ hs
+    exact PartialHomeomorph.isOpen_inter_preimage_symm _ hs
   choose g g_supp g_diff hg using A
   have h'g : ∀ c x, 0 ≤ g c x := fun c x ↦ (hg c (mem_range_self (f := g c) x)).1
   have h''g : ∀ c x, 0 ≤ f c x * g c (chartAt H c x) :=

To fit with the "please try harder" convention of ! tactics.

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

@@ -159,7 +159,7 @@ theorem sum_eq_one {x} (hx : x ∈ s) : ∑ᶠ i, f i x = 1 :=
 #align smooth_partition_of_unity.sum_eq_one SmoothPartitionOfUnity.sum_eq_one
 
 theorem exists_pos_of_mem {x} (hx : x ∈ s) : ∃ i, 0 < f i x := by
-  by_contra' h
+  by_contra! h
   have H : ∀ i, f i x = 0 := fun i ↦ le_antisymm (h i) (f.nonneg i x)
   have := f.sum_eq_one hx
   simp_rw [H] at this

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

@@ -159,8 +159,7 @@ theorem sum_eq_one {x} (hx : x ∈ s) : ∑ᶠ i, f i x = 1 :=
 #align smooth_partition_of_unity.sum_eq_one SmoothPartitionOfUnity.sum_eq_one
 
 theorem exists_pos_of_mem {x} (hx : x ∈ s) : ∃ i, 0 < f i x := by
-  by_contra h
-  push_neg at h
+  by_contra' h
   have H : ∀ i, f i x = 0 := fun i ↦ le_antisymm (h i) (f.nonneg i x)
   have := f.sum_eq_one hx
   simp_rw [H] at this

Mostly, this means replacing "of_open" by "of_isOpen". A few lemmas names were misleading and are corrected differently. Zulip discussion.

@@ -643,7 +643,7 @@ theorem IsOpen.exists_msmooth_support_eq (hs : IsOpen s) :
       Smooth I 𝓘(ℝ) g ∧ Set.range g ⊆ Set.Icc 0 1 := by
     intro i
     apply IsOpen.exists_msmooth_support_eq_aux
-    exact LocalHomeomorph.preimage_open_of_open_symm _ hs
+    exact LocalHomeomorph.isOpen_inter_preimage_symm _ hs
   choose g g_supp g_diff hg using A
   have h'g : ∀ c x, 0 ≤ g c x := fun c x ↦ (hg c (mem_range_self (f := g c) x)).1
   have h''g : ∀ c x, 0 ≤ f c x * g c (chartAt H c x) :=

• Rename NormalSpace to T4Space.
• Add NormalSpace, a version without the T1Space assumption.
• Adjust some theorems.
• Supersedes thus closes #6892.
• Add some instance cycles, see #2030

@@ -343,7 +343,6 @@ theorem exists_isSubordinate [T2Space M] [SigmaCompactSpace M] (hs : IsClosed s)
   -- First we deduce some missing instances
   haveI : LocallyCompactSpace H := I.locallyCompactSpace
   haveI : LocallyCompactSpace M := ChartedSpace.locallyCompactSpace H M
-  haveI : NormalSpace M := normal_of_paracompact_t2
   -- Next we choose a covering by supports of smooth bump functions
   have hB := fun x hx => SmoothBumpFunction.nhds_basis_support I (hU x hx)
   rcases refinement_of_locallyCompact_sigmaCompact_of_nhds_basis_set hs hB with
@@ -523,7 +522,6 @@ theorem exists_isSubordinate {s : Set M} (hs : IsClosed s) (U : ι → Set M) (h
     (hU : s ⊆ ⋃ i, U i) : ∃ f : SmoothPartitionOfUnity ι I M s, f.IsSubordinate U := by
   haveI : LocallyCompactSpace H := I.locallyCompactSpace
   haveI : LocallyCompactSpace M := ChartedSpace.locallyCompactSpace H M
-  haveI : NormalSpace M := normal_of_paracompact_t2
   -- porting note(https://github.com/leanprover/std4/issues/116):
   -- split `rcases` into `have` + `rcases`
   have := BumpCovering.exists_isSubordinate_of_prop (Smooth I 𝓘(ℝ)) ?_ hs U ho hU

• ChartedSpace.locallyCompact → ChartedSpace.locallyCompactSpace
• ModelWithCorners.locally_compact → ModelWithCorners.locallyCompactSpace

@@ -341,8 +341,8 @@ theorem exists_isSubordinate [T2Space M] [SigmaCompactSpace M] (hs : IsClosed s)
     (hU : ∀ x ∈ s, U x ∈ 𝓝 x) :
     ∃ (ι : Type uM) (f : SmoothBumpCovering ι I M s), f.IsSubordinate U := by
   -- First we deduce some missing instances
-  haveI : LocallyCompactSpace H := I.locally_compact
-  haveI : LocallyCompactSpace M := ChartedSpace.locallyCompact H M
+  haveI : LocallyCompactSpace H := I.locallyCompactSpace
+  haveI : LocallyCompactSpace M := ChartedSpace.locallyCompactSpace H M
   haveI : NormalSpace M := normal_of_paracompact_t2
   -- Next we choose a covering by supports of smooth bump functions
   have hB := fun x hx => SmoothBumpFunction.nhds_basis_support I (hU x hx)
@@ -521,8 +521,8 @@ variable [T2Space M] [SigmaCompactSpace M]
 `s`, then there exists a `SmoothPartitionOfUnity ι M s` that is subordinate to `U`. -/
 theorem exists_isSubordinate {s : Set M} (hs : IsClosed s) (U : ι → Set M) (ho : ∀ i, IsOpen (U i))
     (hU : s ⊆ ⋃ i, U i) : ∃ f : SmoothPartitionOfUnity ι I M s, f.IsSubordinate U := by
-  haveI : LocallyCompactSpace H := I.locally_compact
-  haveI : LocallyCompactSpace M := ChartedSpace.locallyCompact H M
+  haveI : LocallyCompactSpace H := I.locallyCompactSpace
+  haveI : LocallyCompactSpace M := ChartedSpace.locallyCompactSpace H M
   haveI : NormalSpace M := normal_of_paracompact_t2
   -- porting note(https://github.com/leanprover/std4/issues/116):
   -- split `rcases` into `have` + `rcases`

@@ -236,7 +236,7 @@ theorem isSubordinate_toPartitionOfUnity :
   Iff.rfl
 #align smooth_partition_of_unity.is_subordinate_to_partition_of_unity SmoothPartitionOfUnity.isSubordinate_toPartitionOfUnity
 
-alias isSubordinate_toPartitionOfUnity ↔ _ IsSubordinate.toPartitionOfUnity
+alias ⟨_, IsSubordinate.toPartitionOfUnity⟩ := isSubordinate_toPartitionOfUnity
 #align smooth_partition_of_unity.is_subordinate.to_partition_of_unity SmoothPartitionOfUnity.IsSubordinate.toPartitionOfUnity
 
 /-- If `f` is a smooth partition of unity on a set `s : Set M` subordinate to a family of open sets
@@ -431,7 +431,7 @@ theorem isSubordinate_toBumpCovering {f : SmoothBumpCovering ι I M s} {U : M 
   Iff.rfl
 #align smooth_bump_covering.is_subordinate_to_bump_covering SmoothBumpCovering.isSubordinate_toBumpCovering
 
-alias isSubordinate_toBumpCovering ↔ _ IsSubordinate.toBumpCovering
+alias ⟨_, IsSubordinate.toBumpCovering⟩ := isSubordinate_toBumpCovering
 #align smooth_bump_covering.is_subordinate.to_bump_covering SmoothBumpCovering.IsSubordinate.toBumpCovering
 
 /-- Every `SmoothBumpCovering` defines a smooth partition of unity. -/

Autoimplicits are highly controversial and also defeat the performance-improving work in #6474.

The intent of this PR is to make autoImplicit opt-in on a per-file basis, by disabling it in the lakefile and enabling it again with set_option autoImplicit true in the few files that rely on it.

That also keeps this PR small, as opposed to attempting to "fix" files to not need it any more.

I claim that many of the uses of autoImplicit in these files are accidental; situations such as:

• Assuming variables are in scope, but pasting the lemma in the wrong section
• Pasting in a lemma from a scratch file without checking to see if the variable names are consistent with the rest of the file
• Making a copy-paste error between lemmas and forgetting to add an explicit arguments.

Having set_option autoImplicit false as the default prevents these types of mistake being made in the 90% of files where autoImplicits are not used at all, and causes them to be caught by CI during review.

I think there were various points during the port where we encouraged porters to delete the universes u v lines; I think having autoparams for universe variables only would cover a lot of the cases we actually use them, while avoiding any real shortcomings.

A Zulip poll (after combining overlapping votes accordingly) was in favor of this change with 5:5:18 as the no:dontcare:yes vote ratio.

While this PR was being reviewed, a handful of files gained some more likely-accidental autoImplicits. In these places, set_option autoImplicit true has been placed locally within a section, rather than at the top of the file.

@@ -85,6 +85,8 @@ any `U : M → Set M` such that `∀ x ∈ s, U x ∈ 𝓝 x` there exists a `Sm
 subordinate to `U`. Then we use this fact to prove a version of the Whitney embedding theorem: any
 compact real manifold can be embedded into `ℝ^n` for large enough `n`.  -/
 
+set_option autoImplicit true
+
 
 variable (ι M)
 

@@ -259,7 +259,7 @@ end SmoothPartitionOfUnity
 
 namespace BumpCovering
 
--- Repeat variables to drop [finite_dimensional ℝ E] and [smooth_manifold_with_corners I M]
+-- Repeat variables to drop `[FiniteDimensional ℝ E]` and `[SmoothManifoldWithCorners I M]`
 theorem smooth_toPartitionOfUnity {E : Type uE} [NormedAddCommGroup E] [NormedSpace ℝ E]
     {H : Type uH} [TopologicalSpace H] {I : ModelWithCorners ℝ E H} {M : Type uM}
     [TopologicalSpace M] [ChartedSpace H M] {s : Set M} (f : BumpCovering ι M s)
@@ -522,7 +522,8 @@ theorem exists_isSubordinate {s : Set M} (hs : IsClosed s) (U : ι → Set M) (h
   haveI : LocallyCompactSpace H := I.locally_compact
   haveI : LocallyCompactSpace M := ChartedSpace.locallyCompact H M
   haveI : NormalSpace M := normal_of_paracompact_t2
-  -- porting note: split `rcases` into `have` + `rcases`
+  -- porting note(https://github.com/leanprover/std4/issues/116):
+  -- split `rcases` into `have` + `rcases`
   have := BumpCovering.exists_isSubordinate_of_prop (Smooth I 𝓘(ℝ)) ?_ hs U ho hU
   · rcases this with ⟨f, hf, hfU⟩
     exact ⟨f.toSmoothPartitionOfUnity hf, hfU.toSmoothPartitionOfUnity hf⟩
@@ -553,7 +554,7 @@ be a family of convex sets. Suppose that for each point `x : M` there exists a n
 `y ∈ U`. Then there exists a $C^n$ smooth function `g : C^∞⟮I, M; 𝓘(ℝ, F), F⟯` such that `g x ∈ t x`
 for all `x`. See also `exists_smooth_forall_mem_convex_of_local` and
 `exists_smooth_forall_mem_convex_of_local_const`. -/
-theorem exists_cont_mdiff_forall_mem_convex_of_local (ht : ∀ x, Convex ℝ (t x))
+theorem exists_contMDiffOn_forall_mem_convex_of_local (ht : ∀ x, Convex ℝ (t x))
     (Hloc : ∀ x : M, ∃ U ∈ 𝓝 x, ∃ g : M → F, ContMDiffOn I 𝓘(ℝ, F) n g U ∧ ∀ y ∈ U, g y ∈ t y) :
     ∃ g : C^n⟮I, M; 𝓘(ℝ, F), F⟯, ∀ x, g x ∈ t x := by
   choose U hU g hgs hgt using Hloc
@@ -564,25 +565,25 @@ theorem exists_cont_mdiff_forall_mem_convex_of_local (ht : ∀ x, Convex ℝ (t
       hf.contMDiff_finsum_smul (fun i => isOpen_interior) fun i => (hgs i).mono interior_subset⟩,
     fun x => f.finsum_smul_mem_convex (mem_univ x) (fun i hi => hgt _ _ _) (ht _)⟩
   exact interior_subset (hf _ <| subset_closure hi)
-#align exists_cont_mdiff_forall_mem_convex_of_local exists_cont_mdiff_forall_mem_convex_of_local
+#align exists_cont_mdiff_forall_mem_convex_of_local exists_contMDiffOn_forall_mem_convex_of_local
 
 /-- Let `M` be a σ-compact Hausdorff finite dimensional topological manifold. Let `t : M → Set F`
 be a family of convex sets. Suppose that for each point `x : M` there exists a neighborhood
 `U ∈ 𝓝 x` and a function `g : M → F` such that `g` is smooth on `U` and `g y ∈ t y` for all `y ∈ U`.
 Then there exists a smooth function `g : C^∞⟮I, M; 𝓘(ℝ, F), F⟯` such that `g x ∈ t x` for all `x`.
-See also `exists_cont_mdiff_forall_mem_convex_of_local` and
+See also `exists_contMDiffOn_forall_mem_convex_of_local` and
 `exists_smooth_forall_mem_convex_of_local_const`. -/
 theorem exists_smooth_forall_mem_convex_of_local (ht : ∀ x, Convex ℝ (t x))
     (Hloc : ∀ x : M, ∃ U ∈ 𝓝 x, ∃ g : M → F, SmoothOn I 𝓘(ℝ, F) g U ∧ ∀ y ∈ U, g y ∈ t y) :
     ∃ g : C^∞⟮I, M; 𝓘(ℝ, F), F⟯, ∀ x, g x ∈ t x :=
-  exists_cont_mdiff_forall_mem_convex_of_local I ht Hloc
+  exists_contMDiffOn_forall_mem_convex_of_local I ht Hloc
 #align exists_smooth_forall_mem_convex_of_local exists_smooth_forall_mem_convex_of_local
 
 /-- Let `M` be a σ-compact Hausdorff finite dimensional topological manifold. Let `t : M → Set F` be
 a family of convex sets. Suppose that for each point `x : M` there exists a vector `c : F` such that
 for all `y` in a neighborhood of `x` we have `c ∈ t y`. Then there exists a smooth function
 `g : C^∞⟮I, M; 𝓘(ℝ, F), F⟯` such that `g x ∈ t x` for all `x`.  See also
-`exists_cont_mdiff_forall_mem_convex_of_local` and `exists_smooth_forall_mem_convex_of_local`. -/
+`exists_contMDiffOn_forall_mem_convex_of_local` and `exists_smooth_forall_mem_convex_of_local`. -/
 theorem exists_smooth_forall_mem_convex_of_local_const (ht : ∀ x, Convex ℝ (t x))
     (Hloc : ∀ x : M, ∃ c : F, ∀ᶠ y in 𝓝 x, c ∈ t y) : ∃ g : C^∞⟮I, M; 𝓘(ℝ, F), F⟯, ∀ x, g x ∈ t x :=
   exists_smooth_forall_mem_convex_of_local I ht fun x =>

@@ -75,10 +75,10 @@ variable {ι : Type uι} {E : Type uE} [NormedAddCommGroup E] [NormedSpace ℝ E
 ### Covering by supports of smooth bump functions
 
 In this section we define `SmoothBumpCovering ι I M s` to be a collection of
-`SmoothBumpFunction`s such that their supports is a locally finite family of sets and for each `x
-∈ s` some function `f i` from the collection is equal to `1` in a neighborhood of `x`. A covering of
-this type is useful to construct a smooth partition of unity and can be used instead of a partition
-of unity in some proofs.
+`SmoothBumpFunction`s such that their supports is a locally finite family of sets and for each
+`x ∈ s` some function `f i` from the collection is equal to `1` in a neighborhood of `x`. A covering
+of this type is useful to construct a smooth partition of unity and can be used instead of a
+partition of unity in some proofs.
 
 We prove that on a smooth finite dimensional real manifold with `σ`-compact Hausdorff topology, for
 any `U : M → Set M` such that `∀ x ∈ s, U x ∈ 𝓝 x` there exists a `SmoothBumpCovering ι I M s`
@@ -103,10 +103,15 @@ This covering can be used, e.g., to construct a partition of unity and to prove
 Whitney embedding theorem. -/
 -- porting note: was @[nolint has_nonempty_instance]
 structure SmoothBumpCovering (s : Set M := univ) where
+  /-- The center point of each bump in the smooth covering. -/
   c : ι → M
+  /-- A smooth bump function around `c i`. -/
   toFun : ∀ i, SmoothBumpFunction I (c i)
+  /-- All the bump functions in the covering are centered at points in `s`. -/
   c_mem' : ∀ i, c i ∈ s
+  /-- Around each point, there are only finitely many nonzero bump functions in the family. -/
   locallyFinite' : LocallyFinite fun i => support (toFun i)
+  /-- Around each point in `s`, one of the bump functions is equal to `1`. -/
   eventuallyEq_one' : ∀ x ∈ s, ∃ i, toFun i =ᶠ[𝓝 x] 1
 #align smooth_bump_covering SmoothBumpCovering
 
@@ -117,10 +122,15 @@ structure SmoothBumpCovering (s : Set M := univ) where
 * for all `x ∈ s` the sum `∑ᶠ i, f i x` equals one;
 * for all `x`, the sum `∑ᶠ i, f i x` is less than or equal to one. -/
 structure SmoothPartitionOfUnity (s : Set M := univ) where
+  /-- The family of functions forming the partition of unity. -/
   toFun : ι → C^∞⟮I, M; 𝓘(ℝ), ℝ⟯
+  /-- Around each point, there are only finitely many nonzero functions in the family. -/
   locallyFinite' : LocallyFinite fun i => support (toFun i)
+  /-- All the functions in the partition of unity are nonnegative. -/
   nonneg' : ∀ i x, 0 ≤ toFun i x
+  /-- The functions in the partition of unity add up to `1` at any point of `s`. -/
   sum_eq_one' : ∀ x ∈ s, ∑ᶠ i, toFun i x = 1
+  /-- The functions in the partition of unity add up to at most `1` everywhere. -/
   sum_le_one' : ∀ x, ∑ᶠ i, toFun i x ≤ 1
 #align smooth_partition_of_unity SmoothPartitionOfUnity
 
@@ -146,6 +156,14 @@ theorem sum_eq_one {x} (hx : x ∈ s) : ∑ᶠ i, f i x = 1 :=
   f.sum_eq_one' x hx
 #align smooth_partition_of_unity.sum_eq_one SmoothPartitionOfUnity.sum_eq_one
 
+theorem exists_pos_of_mem {x} (hx : x ∈ s) : ∃ i, 0 < f i x := by
+  by_contra h
+  push_neg at h
+  have H : ∀ i, f i x = 0 := fun i ↦ le_antisymm (h i) (f.nonneg i x)
+  have := f.sum_eq_one hx
+  simp_rw [H] at this
+  simpa
+
 theorem sum_le_one (x : M) : ∑ᶠ i, f i x ≤ 1 :=
   f.sum_le_one' x
 #align smooth_partition_of_unity.sum_le_one SmoothPartitionOfUnity.sum_le_one
@@ -463,9 +481,10 @@ end SmoothBumpCovering
 
 variable (I)
 
-/-- Given two disjoint closed sets in a Hausdorff σ-compact finite dimensional manifold, there
-exists an infinitely smooth function that is equal to `0` on one of them and is equal to one on the
-other. -/
+/-- Given two disjoint closed sets `s, t` in a Hausdorff σ-compact finite dimensional manifold,
+there exists an infinitely smooth function that is equal to `0` on `s` and to `1` on `t`.
+See also `exists_smooth_zero_iff_one_iff_of_closed`, which ensures additionally that
+`f` is equal to `0` exactly on `s` and to `1` exactly on `t`. -/
 theorem exists_smooth_zero_one_of_closed [T2Space M] [SigmaCompactSpace M] {s t : Set M}
     (hs : IsClosed s) (ht : IsClosed t) (hd : Disjoint s t) :
     ∃ f : C^∞⟮I, M; 𝓘(ℝ), ℝ⟯, EqOn f 0 s ∧ EqOn f 1 t ∧ ∀ x, f x ∈ Icc (0 : ℝ) 1 := by
@@ -497,7 +516,7 @@ instance [Inhabited ι] (s : Set M) : Inhabited (SmoothPartitionOfUnity ι I M s
 variable [T2Space M] [SigmaCompactSpace M]
 
 /-- If `X` is a paracompact normal topological space and `U` is an open covering of a closed set
-`s`, then there exists a `BumpCovering ι X s` that is subordinate to `U`. -/
+`s`, then there exists a `SmoothPartitionOfUnity ι M s` that is subordinate to `U`. -/
 theorem exists_isSubordinate {s : Set M} (hs : IsClosed s) (U : ι → Set M) (ho : ∀ i, IsOpen (U i))
     (hU : s ⊆ ⋃ i, U i) : ∃ f : SmoothPartitionOfUnity ι I M s, f.IsSubordinate U := by
   haveI : LocallyCompactSpace H := I.locally_compact
@@ -512,6 +531,18 @@ theorem exists_isSubordinate {s : Set M} (hs : IsClosed s) (U : ι → Set M) (h
     exact ⟨f, f.smooth, hf⟩
 #align smooth_partition_of_unity.exists_is_subordinate SmoothPartitionOfUnity.exists_isSubordinate
 
+theorem exists_isSubordinate_chartAt_source_of_isClosed {s : Set M} (hs : IsClosed s) :
+    ∃ f : SmoothPartitionOfUnity s I M s,
+      f.IsSubordinate (fun x ↦ (chartAt H (x : M)).source) := by
+  apply exists_isSubordinate _ hs _ (fun i ↦ (chartAt H _).open_source) (fun x hx ↦ ?_)
+  exact mem_iUnion_of_mem ⟨x, hx⟩ (mem_chart_source H x)
+
+variable (M)
+theorem exists_isSubordinate_chartAt_source :
+    ∃ f : SmoothPartitionOfUnity M I M univ, f.IsSubordinate (fun x ↦ (chartAt H x).source) := by
+  apply exists_isSubordinate _ isClosed_univ _ (fun i ↦ (chartAt H _).open_source) (fun x _ ↦ ?_)
+  exact mem_iUnion_of_mem x (mem_chart_source H x)
+
 end SmoothPartitionOfUnity
 
 variable [SigmaCompactSpace M] [T2Space M] {t : M → Set F} {n : ℕ∞}
@@ -590,3 +621,111 @@ theorem Metric.exists_smooth_forall_closedBall_subset {M} [MetricSpace M] [Chart
   rw [← Metric.emetric_closedBall (hδ0 _).le]
   exact hδ i x hx
 #align metric.exists_smooth_forall_closed_ball_subset Metric.exists_smooth_forall_closedBall_subset
+
+lemma IsOpen.exists_msmooth_support_eq_aux {s : Set H} (hs : IsOpen s) :
+    ∃ f : H → ℝ, f.support = s ∧ Smooth I 𝓘(ℝ) f ∧ Set.range f ⊆ Set.Icc 0 1 := by
+  have h's : IsOpen (I.symm ⁻¹' s) := I.continuous_symm.isOpen_preimage _ hs
+  rcases h's.exists_smooth_support_eq with ⟨f, f_supp, f_diff, f_range⟩
+  refine ⟨f ∘ I, ?_, ?_, ?_⟩
+  · rw [support_comp_eq_preimage, f_supp, ← preimage_comp]
+    simp only [ModelWithCorners.symm_comp_self, preimage_id_eq, id_eq]
+  · exact f_diff.comp_contMDiff contMDiff_model
+  · exact Subset.trans (range_comp_subset_range _ _) f_range
+
+/-- Given an open set in a finite-dimensional real manifold, there exists a nonnegative smooth
+function with support equal to `s`. -/
+theorem IsOpen.exists_msmooth_support_eq (hs : IsOpen s) :
+    ∃ f : M → ℝ, f.support = s ∧ Smooth I 𝓘(ℝ) f ∧ ∀ x, 0 ≤ f x := by
+  rcases SmoothPartitionOfUnity.exists_isSubordinate_chartAt_source I M with ⟨f, hf⟩
+  have A : ∀ (c : M), ∃ g : H → ℝ,
+      g.support = (chartAt H c).target ∩ (chartAt H c).symm ⁻¹' s ∧
+      Smooth I 𝓘(ℝ) g ∧ Set.range g ⊆ Set.Icc 0 1 := by
+    intro i
+    apply IsOpen.exists_msmooth_support_eq_aux
+    exact LocalHomeomorph.preimage_open_of_open_symm _ hs
+  choose g g_supp g_diff hg using A
+  have h'g : ∀ c x, 0 ≤ g c x := fun c x ↦ (hg c (mem_range_self (f := g c) x)).1
+  have h''g : ∀ c x, 0 ≤ f c x * g c (chartAt H c x) :=
+    fun c x ↦ mul_nonneg (f.nonneg c x) (h'g c _)
+  refine ⟨fun x ↦ ∑ᶠ c, f c x * g c (chartAt H c x), ?_, ?_, ?_⟩
+  · refine support_eq_iff.2 ⟨fun x hx ↦ ?_, fun x hx ↦ ?_⟩
+    · apply ne_of_gt
+      have B : ∃ c, 0 < f c x * g c (chartAt H c x) := by
+        obtain ⟨c, hc⟩ : ∃ c, 0 < f c x := f.exists_pos_of_mem (mem_univ x)
+        refine ⟨c, mul_pos hc ?_⟩
+        apply lt_of_le_of_ne (h'g _ _) (Ne.symm _)
+        rw [← mem_support, g_supp, ← mem_preimage, preimage_inter]
+        have Hx : x ∈ tsupport (f c) := subset_tsupport _ (ne_of_gt hc)
+        simp [(chartAt H c).left_inv (hf c Hx), hx, (chartAt H c).map_source (hf c Hx)]
+      apply finsum_pos' (fun c ↦ h''g c x) B
+      apply (f.locallyFinite.point_finite x).subset
+      apply compl_subset_compl.2
+      rintro c (hc : f c x = 0)
+      simpa only [mul_eq_zero] using Or.inl hc
+    · apply finsum_eq_zero_of_forall_eq_zero
+      intro c
+      by_cases Hx : x ∈ tsupport (f c)
+      · suffices g c (chartAt H c x) = 0 by simp only [this, mul_zero]
+        rw [← nmem_support, g_supp, ← mem_preimage, preimage_inter]
+        contrapose! hx
+        simp only [mem_inter_iff, mem_preimage, (chartAt H c).left_inv (hf c Hx)] at hx
+        exact hx.2
+      · have : x ∉ support (f c) := by contrapose! Hx; exact subset_tsupport _ Hx
+        rw [nmem_support] at this
+        simp [this]
+  · apply SmoothPartitionOfUnity.smooth_finsum_smul
+    intro c x hx
+    apply (g_diff c (chartAt H c x)).comp
+    exact contMDiffAt_of_mem_maximalAtlas (SmoothManifoldWithCorners.chart_mem_maximalAtlas I _)
+      (hf c hx)
+  · intro x
+    apply finsum_nonneg (fun c ↦ h''g c x)
+
+/-- Given an open set `s` containing a closed set `t` in a finite-dimensional real manifold, there
+exists a smooth function with support equal to `s`, taking values in `[0,1]`, and equal to `1`
+exactly on `t`. -/
+theorem exists_msmooth_support_eq_eq_one_iff
+    {s t : Set M} (hs : IsOpen s) (ht : IsClosed t) (h : t ⊆ s) :
+    ∃ f : M → ℝ, Smooth I 𝓘(ℝ) f ∧ range f ⊆ Icc 0 1 ∧ support f = s
+      ∧ (∀ x, x ∈ t ↔ f x = 1) := by
+  /- Take `f` with support equal to `s`, and `g` with support equal to `tᶜ`. Then `f / (f + g)`
+  satisfies the conclusion of the theorem. -/
+  rcases hs.exists_msmooth_support_eq I with ⟨f, f_supp, f_diff, f_pos⟩
+  rcases ht.isOpen_compl.exists_msmooth_support_eq I with ⟨g, g_supp, g_diff, g_pos⟩
+  have A : ∀ x, 0 < f x + g x := by
+    intro x
+    by_cases xs : x ∈ support f
+    · have : 0 < f x := lt_of_le_of_ne (f_pos x) (Ne.symm xs)
+      linarith [g_pos x]
+    · have : 0 < g x := by
+        apply lt_of_le_of_ne (g_pos x) (Ne.symm ?_)
+        rw [← mem_support, g_supp]
+        contrapose! xs
+        simp at xs
+        exact h.trans f_supp.symm.subset xs
+      linarith [f_pos x]
+  refine ⟨fun x ↦ f x / (f x + g x), ?_, ?_, ?_, ?_⟩
+  -- show that `f / (f + g)` is smooth
+  · exact f_diff.div₀ (f_diff.add g_diff) (fun x ↦ ne_of_gt (A x))
+  -- show that the range is included in `[0, 1]`
+  · refine range_subset_iff.2 (fun x ↦ ⟨div_nonneg (f_pos x) (A x).le, ?_⟩)
+    apply div_le_one_of_le _ (A x).le
+    simpa only [le_add_iff_nonneg_right] using g_pos x
+  -- show that the support is `s`
+  · have B : support (fun x ↦ f x + g x) = univ := eq_univ_of_forall (fun x ↦ (A x).ne')
+    simp only [support_div, f_supp, B, inter_univ]
+  -- show that the function equals one exactly on `t`
+  · intro x
+    simp [div_eq_one_iff_eq (A x).ne', self_eq_add_right, ← nmem_support, g_supp]
+
+/-- Given two disjoint closed sets `s, t` in a Hausdorff σ-compact finite dimensional manifold,
+there exists an infinitely smooth function that is equal to `0` exactly on `s` and to `1`
+exactly on `t`. See also `exists_smooth_zero_one_of_closed` for a slightly weaker version. -/
+theorem exists_msmooth_zero_iff_one_iff_of_closed {s t : Set M}
+    (hs : IsClosed s) (ht : IsClosed t) (hd : Disjoint s t) :
+    ∃ f : M → ℝ, Smooth I 𝓘(ℝ) f ∧ range f ⊆ Icc 0 1 ∧ (∀ x, x ∈ s ↔ f x = 0)
+      ∧ (∀ x, x ∈ t ↔ f x = 1) := by
+  rcases exists_msmooth_support_eq_eq_one_iff I hs.isOpen_compl ht hd.subset_compl_left with
+    ⟨f, f_diff, f_range, fs, ft⟩
+  refine ⟨f, f_diff, f_range, ?_, ft⟩
+  simp [← nmem_support, fs]

@@ -110,7 +110,7 @@ structure SmoothBumpCovering (s : Set M := univ) where
   eventuallyEq_one' : ∀ x ∈ s, ∃ i, toFun i =ᶠ[𝓝 x] 1
 #align smooth_bump_covering SmoothBumpCovering
 
-/-- We say that that a collection of functions form a smooth partition of unity on a set `s` if
+/-- We say that a collection of functions form a smooth partition of unity on a set `s` if
 
 * all functions are infinitely smooth and nonnegative;
 * the family `λ i, support (f i)` is locally finite;
@@ -590,4 +590,3 @@ theorem Metric.exists_smooth_forall_closedBall_subset {M} [MetricSpace M] [Chart
   rw [← Metric.emetric_closedBall (hδ0 _).le]
   exact hδ i x hx
 #align metric.exists_smooth_forall_closed_ball_subset Metric.exists_smooth_forall_closedBall_subset
-

• depends on: #5966

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

@@ -2,17 +2,14 @@
 Copyright (c) 2021 Yury G. Kudryashov. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Yury G. Kudryashov
-
-! This file was ported from Lean 3 source module geometry.manifold.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.Geometry.Manifold.Algebra.Structures
 import Mathlib.Geometry.Manifold.BumpFunction
 import Mathlib.Topology.MetricSpace.PartitionOfUnity
 import Mathlib.Topology.ShrinkingLemma
 
+#align_import geometry.manifold.partition_of_unity from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
+
 /-!
 # Smooth partition of unity