analysis.convex.partition_of_unity ⟷ Mathlib.Analysis.Convex.PartitionOfUnity

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

@@ -3,8 +3,8 @@ Copyright (c) 2022 Yury Kudryashov. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Yury Kudryashov
 -/
-import Mathbin.Topology.PartitionOfUnity
-import Mathbin.Analysis.Convex.Combination
+import Topology.PartitionOfUnity
+import Analysis.Convex.Combination
 
 #align_import analysis.convex.partition_of_unity from "leanprover-community/mathlib"@"781cb2eed038c4caf53bdbd8d20a95e5822d77df"
 

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

@@ -2,15 +2,12 @@
 Copyright (c) 2022 Yury Kudryashov. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Yury Kudryashov
-
-! This file was ported from Lean 3 source module analysis.convex.partition_of_unity
-! leanprover-community/mathlib commit 781cb2eed038c4caf53bdbd8d20a95e5822d77df
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
 -/
 import Mathbin.Topology.PartitionOfUnity
 import Mathbin.Analysis.Convex.Combination
 
+#align_import analysis.convex.partition_of_unity from "leanprover-community/mathlib"@"781cb2eed038c4caf53bdbd8d20a95e5822d77df"
+
 /-!
 # Partition of unity and convex sets
 

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

@@ -39,15 +39,18 @@ open scoped BigOperators Topology
 
 variable {ι X E : Type _} [TopologicalSpace X] [AddCommGroup E] [Module ℝ E]
 
+#print PartitionOfUnity.finsum_smul_mem_convex /-
 theorem PartitionOfUnity.finsum_smul_mem_convex {s : Set X} (f : PartitionOfUnity ι X s)
     {g : ι → X → E} {t : Set E} {x : X} (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 partition_of_unity.finsum_smul_mem_convex PartitionOfUnity.finsum_smul_mem_convex
+-/
 
 variable [NormalSpace X] [ParacompactSpace X] [TopologicalSpace E] [ContinuousAdd E]
   [ContinuousSMul ℝ E] {t : X → Set E}
 
+#print exists_continuous_forall_mem_convex_of_local /-
 /-- Let `X` be a normal paracompact topological space (e.g., any extended metric space). Let `E` be
 a topological real vector space. Let `t : X → set E` be a family of convex sets. Suppose that for
 each point `x : X`, there exists a neighborhood `U ∈ 𝓝 X` and a function `g : X → E` that is
@@ -68,7 +71,9 @@ theorem exists_continuous_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_continuous_forall_mem_convex_of_local exists_continuous_forall_mem_convex_of_local
+-/
 
+#print exists_continuous_forall_mem_convex_of_local_const /-
 /-- Let `X` be a normal paracompact topological space (e.g., any extended metric space). Let `E` be
 a topological real vector space. Let `t : X → set E` be a family of convex sets. Suppose that for
 each point `x : X`, there exists a vector `c : E` that belongs to `t y` for all `y` in a
@@ -80,4 +85,5 @@ theorem exists_continuous_forall_mem_convex_of_local_const (ht : ∀ x, Convex 
     let ⟨c, hc⟩ := H x
     ⟨_, hc, fun _ => c, continuousOn_const, fun y => id⟩
 #align exists_continuous_forall_mem_convex_of_local_const exists_continuous_forall_mem_convex_of_local_const
+-/
 

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

@@ -41,7 +41,7 @@ variable {ι X E : Type _} [TopologicalSpace X] [AddCommGroup E] [Module ℝ E]
 
 theorem PartitionOfUnity.finsum_smul_mem_convex {s : Set X} (f : PartitionOfUnity ι X s)
     {g : ι → X → E} {t : Set E} {x : X} (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 : 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 partition_of_unity.finsum_smul_mem_convex PartitionOfUnity.finsum_smul_mem_convex
 

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

@@ -35,7 +35,7 @@ partition of unity
 
 open Set Function
 
-open BigOperators Topology
+open scoped BigOperators Topology
 
 variable {ι X E : Type _} [TopologicalSpace X] [AddCommGroup E] [Module ℝ E]
 

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

@@ -39,9 +39,6 @@ open BigOperators Topology
 
 variable {ι X E : Type _} [TopologicalSpace X] [AddCommGroup E] [Module ℝ E]
 
-/- warning: partition_of_unity.finsum_smul_mem_convex -> PartitionOfUnity.finsum_smul_mem_convex is a dubious translation:
-<too large>
-Case conversion may be inaccurate. Consider using '#align partition_of_unity.finsum_smul_mem_convex PartitionOfUnity.finsum_smul_mem_convexₓ'. -/
 theorem PartitionOfUnity.finsum_smul_mem_convex {s : Set X} (f : PartitionOfUnity ι X s)
     {g : ι → X → E} {t : Set E} {x : X} (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 :=
@@ -51,9 +48,6 @@ theorem PartitionOfUnity.finsum_smul_mem_convex {s : Set X} (f : PartitionOfUnit
 variable [NormalSpace X] [ParacompactSpace X] [TopologicalSpace E] [ContinuousAdd E]
   [ContinuousSMul ℝ E] {t : X → Set E}
 
-/- warning: exists_continuous_forall_mem_convex_of_local -> exists_continuous_forall_mem_convex_of_local is a dubious translation:
-<too large>
-Case conversion may be inaccurate. Consider using '#align exists_continuous_forall_mem_convex_of_local exists_continuous_forall_mem_convex_of_localₓ'. -/
 /-- Let `X` be a normal paracompact topological space (e.g., any extended metric space). Let `E` be
 a topological real vector space. Let `t : X → set E` be a family of convex sets. Suppose that for
 each point `x : X`, there exists a neighborhood `U ∈ 𝓝 X` and a function `g : X → E` that is
@@ -75,9 +69,6 @@ theorem exists_continuous_forall_mem_convex_of_local (ht : ∀ x, Convex ℝ (t
   exact interior_subset (hf _ <| subset_closure hi)
 #align exists_continuous_forall_mem_convex_of_local exists_continuous_forall_mem_convex_of_local
 
-/- warning: exists_continuous_forall_mem_convex_of_local_const -> exists_continuous_forall_mem_convex_of_local_const is a dubious translation:
-<too large>
-Case conversion may be inaccurate. Consider using '#align exists_continuous_forall_mem_convex_of_local_const exists_continuous_forall_mem_convex_of_local_constₓ'. -/
 /-- Let `X` be a normal paracompact topological space (e.g., any extended metric space). Let `E` be
 a topological real vector space. Let `t : X → set E` be a family of convex sets. Suppose that for
 each point `x : X`, there exists a vector `c : E` that belongs to `t y` for all `y` in a

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

@@ -40,10 +40,7 @@ open BigOperators Topology
 variable {ι X E : Type _} [TopologicalSpace X] [AddCommGroup E] [Module ℝ E]
 
 /- warning: partition_of_unity.finsum_smul_mem_convex -> PartitionOfUnity.finsum_smul_mem_convex is a dubious translation:
-lean 3 declaration is
-  forall {ι : Type.{u1}} {X : Type.{u2}} {E : Type.{u3}} [_inst_1 : TopologicalSpace.{u2} X] [_inst_2 : AddCommGroup.{u3} E] [_inst_3 : Module.{0, u3} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u3} E _inst_2)] {s : Set.{u2} X} (f : PartitionOfUnity.{u1, u2} ι X _inst_1 s) {g : ι -> X -> E} {t : Set.{u3} E} {x : X}, (Membership.Mem.{u2, u2} X (Set.{u2} X) (Set.hasMem.{u2} X) x s) -> (forall (i : ι), (Ne.{1} Real (coeFn.{succ u2, succ u2} (ContinuousMap.{u2, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) (fun (_x : ContinuousMap.{u2, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) => X -> Real) (ContinuousMap.hasCoeToFun.{u2, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (PartitionOfUnity.{u1, u2} ι X _inst_1 s) (fun (_x : PartitionOfUnity.{u1, u2} ι X _inst_1 s) => ι -> (ContinuousMap.{u2, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)))) (PartitionOfUnity.hasCoeToFun.{u1, u2} ι X _inst_1 s) f i) x) (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero)))) -> (Membership.Mem.{u3, u3} E (Set.{u3} E) (Set.hasMem.{u3} E) (g i x) t)) -> (Convex.{0, u3} Real E Real.orderedSemiring (AddCommGroup.toAddCommMonoid.{u3} E _inst_2) (SMulZeroClass.toHasSmul.{0, u3} Real E (AddZeroClass.toHasZero.{u3} E (AddMonoid.toAddZeroClass.{u3} E (AddCommMonoid.toAddMonoid.{u3} E (AddCommGroup.toAddCommMonoid.{u3} E _inst_2)))) (SMulWithZero.toSmulZeroClass.{0, u3} Real E (MulZeroClass.toHasZero.{0} Real (MulZeroOneClass.toMulZeroClass.{0} Real (MonoidWithZero.toMulZeroOneClass.{0} Real (Semiring.toMonoidWithZero.{0} Real Real.semiring)))) (AddZeroClass.toHasZero.{u3} E (AddMonoid.toAddZeroClass.{u3} E (AddCommMonoid.toAddMonoid.{u3} E (AddCommGroup.toAddCommMonoid.{u3} E _inst_2)))) (MulActionWithZero.toSMulWithZero.{0, u3} Real E (Semiring.toMonoidWithZero.{0} Real Real.semiring) (AddZeroClass.toHasZero.{u3} E (AddMonoid.toAddZeroClass.{u3} E (AddCommMonoid.toAddMonoid.{u3} E (AddCommGroup.toAddCommMonoid.{u3} E _inst_2)))) (Module.toMulActionWithZero.{0, u3} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u3} E _inst_2) _inst_3)))) t) -> (Membership.Mem.{u3, u3} E (Set.{u3} E) (Set.hasMem.{u3} E) (finsum.{u3, succ u1} E ι (AddCommGroup.toAddCommMonoid.{u3} E _inst_2) (fun (i : ι) => SMul.smul.{0, u3} Real E (SMulZeroClass.toHasSmul.{0, u3} Real E (AddZeroClass.toHasZero.{u3} E (AddMonoid.toAddZeroClass.{u3} E (AddCommMonoid.toAddMonoid.{u3} E (AddCommGroup.toAddCommMonoid.{u3} E _inst_2)))) (SMulWithZero.toSmulZeroClass.{0, u3} Real E (MulZeroClass.toHasZero.{0} Real (MulZeroOneClass.toMulZeroClass.{0} Real (MonoidWithZero.toMulZeroOneClass.{0} Real (Semiring.toMonoidWithZero.{0} Real Real.semiring)))) (AddZeroClass.toHasZero.{u3} E (AddMonoid.toAddZeroClass.{u3} E (AddCommMonoid.toAddMonoid.{u3} E (AddCommGroup.toAddCommMonoid.{u3} E _inst_2)))) (MulActionWithZero.toSMulWithZero.{0, u3} Real E (Semiring.toMonoidWithZero.{0} Real Real.semiring) (AddZeroClass.toHasZero.{u3} E (AddMonoid.toAddZeroClass.{u3} E (AddCommMonoid.toAddMonoid.{u3} E (AddCommGroup.toAddCommMonoid.{u3} E _inst_2)))) (Module.toMulActionWithZero.{0, u3} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u3} E _inst_2) _inst_3)))) (coeFn.{succ u2, succ u2} (ContinuousMap.{u2, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) (fun (_x : ContinuousMap.{u2, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) => X -> Real) (ContinuousMap.hasCoeToFun.{u2, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (PartitionOfUnity.{u1, u2} ι X _inst_1 s) (fun (_x : PartitionOfUnity.{u1, u2} ι X _inst_1 s) => ι -> (ContinuousMap.{u2, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)))) (PartitionOfUnity.hasCoeToFun.{u1, u2} ι X _inst_1 s) f i) x) (g i x))) t)
-but is expected to have type
-  forall {ι : Type.{u2}} {X : Type.{u3}} {E : Type.{u1}} [_inst_1 : TopologicalSpace.{u3} X] [_inst_2 : AddCommGroup.{u1} E] [_inst_3 : Module.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_2)] {s : Set.{u3} X} (f : PartitionOfUnity.{u2, u3} ι X _inst_1 s) {g : ι -> X -> E} {t : Set.{u1} E} {x : X}, (Membership.mem.{u3, u3} X (Set.{u3} X) (Set.instMembershipSet.{u3} X) x s) -> (forall (i : ι), (Ne.{1} ((fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : X) => Real) x) (FunLike.coe.{succ u3, succ u3, 1} (ContinuousMap.{u3, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) X (fun (_x : X) => (fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : X) => Real) _x) (ContinuousMapClass.toFunLike.{u3, u3, 0} (ContinuousMap.{u3, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) (ContinuousMap.instContinuousMapClassContinuousMap.{u3, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)))) (PartitionOfUnity.toFun.{u2, u3} ι X _inst_1 s f i) x) (OfNat.ofNat.{0} ((fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : X) => Real) x) 0 (Zero.toOfNat0.{0} ((fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : X) => Real) x) Real.instZeroReal))) -> (Membership.mem.{u1, u1} E (Set.{u1} E) (Set.instMembershipSet.{u1} E) (g i x) t)) -> (Convex.{0, u1} Real E Real.orderedSemiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_2) (SMulZeroClass.toSMul.{0, u1} Real E (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_2))))) (SMulWithZero.toSMulZeroClass.{0, u1} Real E Real.instZeroReal (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_2))))) (MulActionWithZero.toSMulWithZero.{0, u1} Real E Real.instMonoidWithZeroReal (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_2))))) (Module.toMulActionWithZero.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_2) _inst_3)))) t) -> (Membership.mem.{u1, u1} E (Set.{u1} E) (Set.instMembershipSet.{u1} E) (finsum.{u1, succ u2} E ι (AddCommGroup.toAddCommMonoid.{u1} E _inst_2) (fun (i : ι) => HSMul.hSMul.{0, u1, u1} ((fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : X) => Real) x) E E (instHSMul.{0, u1} ((fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : X) => Real) x) E (SMulZeroClass.toSMul.{0, u1} ((fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : X) => Real) x) E (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_2))))) (SMulWithZero.toSMulZeroClass.{0, u1} ((fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : X) => Real) x) E Real.instZeroReal (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_2))))) (MulActionWithZero.toSMulWithZero.{0, u1} ((fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : X) => Real) x) E Real.instMonoidWithZeroReal (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_2))))) (Module.toMulActionWithZero.{0, u1} ((fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : X) => Real) x) E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_2) _inst_3))))) (FunLike.coe.{succ u3, succ u3, 1} (ContinuousMap.{u3, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) X (fun (_x : X) => (fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : X) => Real) _x) (ContinuousMapClass.toFunLike.{u3, u3, 0} (ContinuousMap.{u3, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) (ContinuousMap.instContinuousMapClassContinuousMap.{u3, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)))) (PartitionOfUnity.toFun.{u2, u3} ι X _inst_1 s f i) x) (g i x))) t)
+<too large>
 Case conversion may be inaccurate. Consider using '#align partition_of_unity.finsum_smul_mem_convex PartitionOfUnity.finsum_smul_mem_convexₓ'. -/
 theorem PartitionOfUnity.finsum_smul_mem_convex {s : Set X} (f : PartitionOfUnity ι X s)
     {g : ι → X → E} {t : Set E} {x : X} (hx : x ∈ s) (hg : ∀ i, f i x ≠ 0 → g i x ∈ t)
@@ -55,10 +52,7 @@ variable [NormalSpace X] [ParacompactSpace X] [TopologicalSpace E] [ContinuousAd
   [ContinuousSMul ℝ E] {t : X → Set E}
 
 /- warning: exists_continuous_forall_mem_convex_of_local -> exists_continuous_forall_mem_convex_of_local is a dubious translation:
-lean 3 declaration is
-  forall {X : Type.{u1}} {E : Type.{u2}} [_inst_1 : TopologicalSpace.{u1} X] [_inst_2 : AddCommGroup.{u2} E] [_inst_3 : Module.{0, u2} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u2} E _inst_2)] [_inst_4 : NormalSpace.{u1} X _inst_1] [_inst_5 : ParacompactSpace.{u1} X _inst_1] [_inst_6 : TopologicalSpace.{u2} E] [_inst_7 : ContinuousAdd.{u2} E _inst_6 (AddZeroClass.toHasAdd.{u2} E (AddMonoid.toAddZeroClass.{u2} E (SubNegMonoid.toAddMonoid.{u2} E (AddGroup.toSubNegMonoid.{u2} E (AddCommGroup.toAddGroup.{u2} E _inst_2)))))] [_inst_8 : ContinuousSMul.{0, u2} Real E (SMulZeroClass.toHasSmul.{0, u2} Real E (AddZeroClass.toHasZero.{u2} E (AddMonoid.toAddZeroClass.{u2} E (AddCommMonoid.toAddMonoid.{u2} E (AddCommGroup.toAddCommMonoid.{u2} E _inst_2)))) (SMulWithZero.toSmulZeroClass.{0, u2} Real E (MulZeroClass.toHasZero.{0} Real (MulZeroOneClass.toMulZeroClass.{0} Real (MonoidWithZero.toMulZeroOneClass.{0} Real (Semiring.toMonoidWithZero.{0} Real Real.semiring)))) (AddZeroClass.toHasZero.{u2} E (AddMonoid.toAddZeroClass.{u2} E (AddCommMonoid.toAddMonoid.{u2} E (AddCommGroup.toAddCommMonoid.{u2} E _inst_2)))) (MulActionWithZero.toSMulWithZero.{0, u2} Real E (Semiring.toMonoidWithZero.{0} Real Real.semiring) (AddZeroClass.toHasZero.{u2} E (AddMonoid.toAddZeroClass.{u2} E (AddCommMonoid.toAddMonoid.{u2} E (AddCommGroup.toAddCommMonoid.{u2} E _inst_2)))) (Module.toMulActionWithZero.{0, u2} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u2} E _inst_2) _inst_3)))) (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) _inst_6] {t : X -> (Set.{u2} E)}, (forall (x : X), Convex.{0, u2} Real E Real.orderedSemiring (AddCommGroup.toAddCommMonoid.{u2} E _inst_2) (SMulZeroClass.toHasSmul.{0, u2} Real E (AddZeroClass.toHasZero.{u2} E (AddMonoid.toAddZeroClass.{u2} E (AddCommMonoid.toAddMonoid.{u2} E (AddCommGroup.toAddCommMonoid.{u2} E _inst_2)))) (SMulWithZero.toSmulZeroClass.{0, u2} Real E (MulZeroClass.toHasZero.{0} Real (MulZeroOneClass.toMulZeroClass.{0} Real (MonoidWithZero.toMulZeroOneClass.{0} Real (Semiring.toMonoidWithZero.{0} Real Real.semiring)))) (AddZeroClass.toHasZero.{u2} E (AddMonoid.toAddZeroClass.{u2} E (AddCommMonoid.toAddMonoid.{u2} E (AddCommGroup.toAddCommMonoid.{u2} E _inst_2)))) (MulActionWithZero.toSMulWithZero.{0, u2} Real E (Semiring.toMonoidWithZero.{0} Real Real.semiring) (AddZeroClass.toHasZero.{u2} E (AddMonoid.toAddZeroClass.{u2} E (AddCommMonoid.toAddMonoid.{u2} E (AddCommGroup.toAddCommMonoid.{u2} E _inst_2)))) (Module.toMulActionWithZero.{0, u2} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u2} E _inst_2) _inst_3)))) (t x)) -> (forall (x : X), Exists.{succ u1} (Set.{u1} X) (fun (U : Set.{u1} X) => Exists.{0} (Membership.Mem.{u1, u1} (Set.{u1} X) (Filter.{u1} X) (Filter.hasMem.{u1} X) U (nhds.{u1} X _inst_1 x)) (fun (H : Membership.Mem.{u1, u1} (Set.{u1} X) (Filter.{u1} X) (Filter.hasMem.{u1} X) U (nhds.{u1} X _inst_1 x)) => Exists.{max (succ u1) (succ u2)} (X -> E) (fun (g : X -> E) => And (ContinuousOn.{u1, u2} X E _inst_1 _inst_6 g U) (forall (y : X), (Membership.Mem.{u1, u1} X (Set.{u1} X) (Set.hasMem.{u1} X) y U) -> (Membership.Mem.{u2, u2} E (Set.{u2} E) (Set.hasMem.{u2} E) (g y) (t y))))))) -> (Exists.{max (succ u1) (succ u2)} (ContinuousMap.{u1, u2} X E _inst_1 _inst_6) (fun (g : ContinuousMap.{u1, u2} X E _inst_1 _inst_6) => forall (x : X), Membership.Mem.{u2, u2} E (Set.{u2} E) (Set.hasMem.{u2} E) (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (ContinuousMap.{u1, u2} X E _inst_1 _inst_6) (fun (_x : ContinuousMap.{u1, u2} X E _inst_1 _inst_6) => X -> E) (ContinuousMap.hasCoeToFun.{u1, u2} X E _inst_1 _inst_6) g x) (t x)))
-but is expected to have type
-  forall {X : Type.{u1}} {E : Type.{u2}} [_inst_1 : TopologicalSpace.{u1} X] [_inst_2 : AddCommGroup.{u2} E] [_inst_3 : Module.{0, u2} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u2} E _inst_2)] [_inst_4 : NormalSpace.{u1} X _inst_1] [_inst_5 : ParacompactSpace.{u1} X _inst_1] [_inst_6 : TopologicalSpace.{u2} E] [_inst_7 : ContinuousAdd.{u2} E _inst_6 (AddZeroClass.toAdd.{u2} E (AddMonoid.toAddZeroClass.{u2} E (SubNegMonoid.toAddMonoid.{u2} E (AddGroup.toSubNegMonoid.{u2} E (AddCommGroup.toAddGroup.{u2} E _inst_2)))))] [_inst_8 : ContinuousSMul.{0, u2} Real E (SMulZeroClass.toSMul.{0, u2} Real E (NegZeroClass.toZero.{u2} E (SubNegZeroMonoid.toNegZeroClass.{u2} E (SubtractionMonoid.toSubNegZeroMonoid.{u2} E (SubtractionCommMonoid.toSubtractionMonoid.{u2} E (AddCommGroup.toDivisionAddCommMonoid.{u2} E _inst_2))))) (SMulWithZero.toSMulZeroClass.{0, u2} Real E Real.instZeroReal (NegZeroClass.toZero.{u2} E (SubNegZeroMonoid.toNegZeroClass.{u2} E (SubtractionMonoid.toSubNegZeroMonoid.{u2} E (SubtractionCommMonoid.toSubtractionMonoid.{u2} E (AddCommGroup.toDivisionAddCommMonoid.{u2} E _inst_2))))) (MulActionWithZero.toSMulWithZero.{0, u2} Real E Real.instMonoidWithZeroReal (NegZeroClass.toZero.{u2} E (SubNegZeroMonoid.toNegZeroClass.{u2} E (SubtractionMonoid.toSubNegZeroMonoid.{u2} E (SubtractionCommMonoid.toSubtractionMonoid.{u2} E (AddCommGroup.toDivisionAddCommMonoid.{u2} E _inst_2))))) (Module.toMulActionWithZero.{0, u2} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u2} E _inst_2) _inst_3)))) (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) _inst_6] {t : X -> (Set.{u2} E)}, (forall (x : X), Convex.{0, u2} Real E Real.orderedSemiring (AddCommGroup.toAddCommMonoid.{u2} E _inst_2) (SMulZeroClass.toSMul.{0, u2} Real E (NegZeroClass.toZero.{u2} E (SubNegZeroMonoid.toNegZeroClass.{u2} E (SubtractionMonoid.toSubNegZeroMonoid.{u2} E (SubtractionCommMonoid.toSubtractionMonoid.{u2} E (AddCommGroup.toDivisionAddCommMonoid.{u2} E _inst_2))))) (SMulWithZero.toSMulZeroClass.{0, u2} Real E Real.instZeroReal (NegZeroClass.toZero.{u2} E (SubNegZeroMonoid.toNegZeroClass.{u2} E (SubtractionMonoid.toSubNegZeroMonoid.{u2} E (SubtractionCommMonoid.toSubtractionMonoid.{u2} E (AddCommGroup.toDivisionAddCommMonoid.{u2} E _inst_2))))) (MulActionWithZero.toSMulWithZero.{0, u2} Real E Real.instMonoidWithZeroReal (NegZeroClass.toZero.{u2} E (SubNegZeroMonoid.toNegZeroClass.{u2} E (SubtractionMonoid.toSubNegZeroMonoid.{u2} E (SubtractionCommMonoid.toSubtractionMonoid.{u2} E (AddCommGroup.toDivisionAddCommMonoid.{u2} E _inst_2))))) (Module.toMulActionWithZero.{0, u2} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u2} E _inst_2) _inst_3)))) (t x)) -> (forall (x : X), Exists.{succ u1} (Set.{u1} X) (fun (U : Set.{u1} X) => And (Membership.mem.{u1, u1} (Set.{u1} X) (Filter.{u1} X) (instMembershipSetFilter.{u1} X) U (nhds.{u1} X _inst_1 x)) (Exists.{max (succ u1) (succ u2)} (X -> E) (fun (g : X -> E) => And (ContinuousOn.{u1, u2} X E _inst_1 _inst_6 g U) (forall (y : X), (Membership.mem.{u1, u1} X (Set.{u1} X) (Set.instMembershipSet.{u1} X) y U) -> (Membership.mem.{u2, u2} E (Set.{u2} E) (Set.instMembershipSet.{u2} E) (g y) (t y))))))) -> (Exists.{max (succ u1) (succ u2)} (ContinuousMap.{u1, u2} X E _inst_1 _inst_6) (fun (g : ContinuousMap.{u1, u2} X E _inst_1 _inst_6) => forall (x : X), Membership.mem.{u2, u2} ((fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : X) => E) x) (Set.{u2} E) (Set.instMembershipSet.{u2} E) (FunLike.coe.{max (succ u1) (succ u2), succ u1, succ u2} (ContinuousMap.{u1, u2} X E _inst_1 _inst_6) X (fun (_x : X) => (fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : X) => E) _x) (ContinuousMapClass.toFunLike.{max u1 u2, u1, u2} (ContinuousMap.{u1, u2} X E _inst_1 _inst_6) X E _inst_1 _inst_6 (ContinuousMap.instContinuousMapClassContinuousMap.{u1, u2} X E _inst_1 _inst_6)) g x) (t x)))
+<too large>
 Case conversion may be inaccurate. Consider using '#align exists_continuous_forall_mem_convex_of_local exists_continuous_forall_mem_convex_of_localₓ'. -/
 /-- Let `X` be a normal paracompact topological space (e.g., any extended metric space). Let `E` be
 a topological real vector space. Let `t : X → set E` be a family of convex sets. Suppose that for
@@ -82,10 +76,7 @@ theorem exists_continuous_forall_mem_convex_of_local (ht : ∀ x, Convex ℝ (t
 #align exists_continuous_forall_mem_convex_of_local exists_continuous_forall_mem_convex_of_local
 
 /- warning: exists_continuous_forall_mem_convex_of_local_const -> exists_continuous_forall_mem_convex_of_local_const is a dubious translation:
-lean 3 declaration is
-  forall {X : Type.{u1}} {E : Type.{u2}} [_inst_1 : TopologicalSpace.{u1} X] [_inst_2 : AddCommGroup.{u2} E] [_inst_3 : Module.{0, u2} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u2} E _inst_2)] [_inst_4 : NormalSpace.{u1} X _inst_1] [_inst_5 : ParacompactSpace.{u1} X _inst_1] [_inst_6 : TopologicalSpace.{u2} E] [_inst_7 : ContinuousAdd.{u2} E _inst_6 (AddZeroClass.toHasAdd.{u2} E (AddMonoid.toAddZeroClass.{u2} E (SubNegMonoid.toAddMonoid.{u2} E (AddGroup.toSubNegMonoid.{u2} E (AddCommGroup.toAddGroup.{u2} E _inst_2)))))] [_inst_8 : ContinuousSMul.{0, u2} Real E (SMulZeroClass.toHasSmul.{0, u2} Real E (AddZeroClass.toHasZero.{u2} E (AddMonoid.toAddZeroClass.{u2} E (AddCommMonoid.toAddMonoid.{u2} E (AddCommGroup.toAddCommMonoid.{u2} E _inst_2)))) (SMulWithZero.toSmulZeroClass.{0, u2} Real E (MulZeroClass.toHasZero.{0} Real (MulZeroOneClass.toMulZeroClass.{0} Real (MonoidWithZero.toMulZeroOneClass.{0} Real (Semiring.toMonoidWithZero.{0} Real Real.semiring)))) (AddZeroClass.toHasZero.{u2} E (AddMonoid.toAddZeroClass.{u2} E (AddCommMonoid.toAddMonoid.{u2} E (AddCommGroup.toAddCommMonoid.{u2} E _inst_2)))) (MulActionWithZero.toSMulWithZero.{0, u2} Real E (Semiring.toMonoidWithZero.{0} Real Real.semiring) (AddZeroClass.toHasZero.{u2} E (AddMonoid.toAddZeroClass.{u2} E (AddCommMonoid.toAddMonoid.{u2} E (AddCommGroup.toAddCommMonoid.{u2} E _inst_2)))) (Module.toMulActionWithZero.{0, u2} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u2} E _inst_2) _inst_3)))) (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) _inst_6] {t : X -> (Set.{u2} E)}, (forall (x : X), Convex.{0, u2} Real E Real.orderedSemiring (AddCommGroup.toAddCommMonoid.{u2} E _inst_2) (SMulZeroClass.toHasSmul.{0, u2} Real E (AddZeroClass.toHasZero.{u2} E (AddMonoid.toAddZeroClass.{u2} E (AddCommMonoid.toAddMonoid.{u2} E (AddCommGroup.toAddCommMonoid.{u2} E _inst_2)))) (SMulWithZero.toSmulZeroClass.{0, u2} Real E (MulZeroClass.toHasZero.{0} Real (MulZeroOneClass.toMulZeroClass.{0} Real (MonoidWithZero.toMulZeroOneClass.{0} Real (Semiring.toMonoidWithZero.{0} Real Real.semiring)))) (AddZeroClass.toHasZero.{u2} E (AddMonoid.toAddZeroClass.{u2} E (AddCommMonoid.toAddMonoid.{u2} E (AddCommGroup.toAddCommMonoid.{u2} E _inst_2)))) (MulActionWithZero.toSMulWithZero.{0, u2} Real E (Semiring.toMonoidWithZero.{0} Real Real.semiring) (AddZeroClass.toHasZero.{u2} E (AddMonoid.toAddZeroClass.{u2} E (AddCommMonoid.toAddMonoid.{u2} E (AddCommGroup.toAddCommMonoid.{u2} E _inst_2)))) (Module.toMulActionWithZero.{0, u2} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u2} E _inst_2) _inst_3)))) (t x)) -> (forall (x : X), Exists.{succ u2} E (fun (c : E) => Filter.Eventually.{u1} X (fun (y : X) => Membership.Mem.{u2, u2} E (Set.{u2} E) (Set.hasMem.{u2} E) c (t y)) (nhds.{u1} X _inst_1 x))) -> (Exists.{max (succ u1) (succ u2)} (ContinuousMap.{u1, u2} X E _inst_1 _inst_6) (fun (g : ContinuousMap.{u1, u2} X E _inst_1 _inst_6) => forall (x : X), Membership.Mem.{u2, u2} E (Set.{u2} E) (Set.hasMem.{u2} E) (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (ContinuousMap.{u1, u2} X E _inst_1 _inst_6) (fun (_x : ContinuousMap.{u1, u2} X E _inst_1 _inst_6) => X -> E) (ContinuousMap.hasCoeToFun.{u1, u2} X E _inst_1 _inst_6) g x) (t x)))
-but is expected to have type
-  forall {X : Type.{u1}} {E : Type.{u2}} [_inst_1 : TopologicalSpace.{u1} X] [_inst_2 : AddCommGroup.{u2} E] [_inst_3 : Module.{0, u2} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u2} E _inst_2)] [_inst_4 : NormalSpace.{u1} X _inst_1] [_inst_5 : ParacompactSpace.{u1} X _inst_1] [_inst_6 : TopologicalSpace.{u2} E] [_inst_7 : ContinuousAdd.{u2} E _inst_6 (AddZeroClass.toAdd.{u2} E (AddMonoid.toAddZeroClass.{u2} E (SubNegMonoid.toAddMonoid.{u2} E (AddGroup.toSubNegMonoid.{u2} E (AddCommGroup.toAddGroup.{u2} E _inst_2)))))] [_inst_8 : ContinuousSMul.{0, u2} Real E (SMulZeroClass.toSMul.{0, u2} Real E (NegZeroClass.toZero.{u2} E (SubNegZeroMonoid.toNegZeroClass.{u2} E (SubtractionMonoid.toSubNegZeroMonoid.{u2} E (SubtractionCommMonoid.toSubtractionMonoid.{u2} E (AddCommGroup.toDivisionAddCommMonoid.{u2} E _inst_2))))) (SMulWithZero.toSMulZeroClass.{0, u2} Real E Real.instZeroReal (NegZeroClass.toZero.{u2} E (SubNegZeroMonoid.toNegZeroClass.{u2} E (SubtractionMonoid.toSubNegZeroMonoid.{u2} E (SubtractionCommMonoid.toSubtractionMonoid.{u2} E (AddCommGroup.toDivisionAddCommMonoid.{u2} E _inst_2))))) (MulActionWithZero.toSMulWithZero.{0, u2} Real E Real.instMonoidWithZeroReal (NegZeroClass.toZero.{u2} E (SubNegZeroMonoid.toNegZeroClass.{u2} E (SubtractionMonoid.toSubNegZeroMonoid.{u2} E (SubtractionCommMonoid.toSubtractionMonoid.{u2} E (AddCommGroup.toDivisionAddCommMonoid.{u2} E _inst_2))))) (Module.toMulActionWithZero.{0, u2} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u2} E _inst_2) _inst_3)))) (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) _inst_6] {t : X -> (Set.{u2} E)}, (forall (x : X), Convex.{0, u2} Real E Real.orderedSemiring (AddCommGroup.toAddCommMonoid.{u2} E _inst_2) (SMulZeroClass.toSMul.{0, u2} Real E (NegZeroClass.toZero.{u2} E (SubNegZeroMonoid.toNegZeroClass.{u2} E (SubtractionMonoid.toSubNegZeroMonoid.{u2} E (SubtractionCommMonoid.toSubtractionMonoid.{u2} E (AddCommGroup.toDivisionAddCommMonoid.{u2} E _inst_2))))) (SMulWithZero.toSMulZeroClass.{0, u2} Real E Real.instZeroReal (NegZeroClass.toZero.{u2} E (SubNegZeroMonoid.toNegZeroClass.{u2} E (SubtractionMonoid.toSubNegZeroMonoid.{u2} E (SubtractionCommMonoid.toSubtractionMonoid.{u2} E (AddCommGroup.toDivisionAddCommMonoid.{u2} E _inst_2))))) (MulActionWithZero.toSMulWithZero.{0, u2} Real E Real.instMonoidWithZeroReal (NegZeroClass.toZero.{u2} E (SubNegZeroMonoid.toNegZeroClass.{u2} E (SubtractionMonoid.toSubNegZeroMonoid.{u2} E (SubtractionCommMonoid.toSubtractionMonoid.{u2} E (AddCommGroup.toDivisionAddCommMonoid.{u2} E _inst_2))))) (Module.toMulActionWithZero.{0, u2} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u2} E _inst_2) _inst_3)))) (t x)) -> (forall (x : X), Exists.{succ u2} E (fun (c : E) => Filter.Eventually.{u1} X (fun (y : X) => Membership.mem.{u2, u2} E (Set.{u2} E) (Set.instMembershipSet.{u2} E) c (t y)) (nhds.{u1} X _inst_1 x))) -> (Exists.{max (succ u1) (succ u2)} (ContinuousMap.{u1, u2} X E _inst_1 _inst_6) (fun (g : ContinuousMap.{u1, u2} X E _inst_1 _inst_6) => forall (x : X), Membership.mem.{u2, u2} ((fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : X) => E) x) (Set.{u2} E) (Set.instMembershipSet.{u2} E) (FunLike.coe.{max (succ u1) (succ u2), succ u1, succ u2} (ContinuousMap.{u1, u2} X E _inst_1 _inst_6) X (fun (_x : X) => (fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : X) => E) _x) (ContinuousMapClass.toFunLike.{max u1 u2, u1, u2} (ContinuousMap.{u1, u2} X E _inst_1 _inst_6) X E _inst_1 _inst_6 (ContinuousMap.instContinuousMapClassContinuousMap.{u1, u2} X E _inst_1 _inst_6)) g x) (t x)))
+<too large>
 Case conversion may be inaccurate. Consider using '#align exists_continuous_forall_mem_convex_of_local_const exists_continuous_forall_mem_convex_of_local_constₓ'. -/
 /-- Let `X` be a normal paracompact topological space (e.g., any extended metric space). Let `E` be
 a topological real vector space. Let `t : X → set E` be a family of convex sets. Suppose that for

mathlib commit https://github.com/leanprover-community/mathlib/commit/2f8347015b12b0864dfaf366ec4909eb70c78740

@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Yury Kudryashov
 
 ! This file was ported from Lean 3 source module analysis.convex.partition_of_unity
-! leanprover-community/mathlib commit f2ce6086713c78a7f880485f7917ea547a215982
+! leanprover-community/mathlib commit 781cb2eed038c4caf53bdbd8d20a95e5822d77df
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
@@ -14,6 +14,9 @@ import Mathbin.Analysis.Convex.Combination
 /-!
 # Partition of unity and convex sets
 
+> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.
+> Any changes to this file require a corresponding PR to mathlib4.
+
 In this file we prove the following lemma, see `exists_continuous_forall_mem_convex_of_local`. Let
 `X` be a normal paracompact topological space (e.g., any extended metric space). Let `E` be a
 topological real vector space. Let `t : X → set E` be a family of convex sets. Suppose that for each

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

@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Yury Kudryashov
 
 ! This file was ported from Lean 3 source module analysis.convex.partition_of_unity
-! leanprover-community/mathlib commit 781cb2eed038c4caf53bdbd8d20a95e5822d77df
+! leanprover-community/mathlib commit f2ce6086713c78a7f880485f7917ea547a215982
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
@@ -14,9 +14,6 @@ import Mathbin.Analysis.Convex.Combination
 /-!
 # Partition of unity and convex sets
 
-> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.
-> Any changes to this file require a corresponding PR to mathlib4.
-
 In this file we prove the following lemma, see `exists_continuous_forall_mem_convex_of_local`. Let
 `X` be a normal paracompact topological space (e.g., any extended metric space). Let `E` be a
 topological real vector space. Let `t : X → set E` be a family of convex sets. Suppose that for each

mathlib commit https://github.com/leanprover-community/mathlib/commit/2f8347015b12b0864dfaf366ec4909eb70c78740

@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Yury Kudryashov
 
 ! This file was ported from Lean 3 source module analysis.convex.partition_of_unity
-! leanprover-community/mathlib commit f2ce6086713c78a7f880485f7917ea547a215982
+! leanprover-community/mathlib commit 781cb2eed038c4caf53bdbd8d20a95e5822d77df
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
@@ -14,6 +14,9 @@ import Mathbin.Analysis.Convex.Combination
 /-!
 # Partition of unity and convex sets
 
+> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.
+> Any changes to this file require a corresponding PR to mathlib4.
+
 In this file we prove the following lemma, see `exists_continuous_forall_mem_convex_of_local`. Let
 `X` be a normal paracompact topological space (e.g., any extended metric space). Let `E` be a
 topological real vector space. Let `t : X → set E` be a family of convex sets. Suppose that for each

mathlib commit https://github.com/leanprover-community/mathlib/commit/cc5dd6244981976cc9da7afc4eee5682b037a013

@@ -36,6 +36,12 @@ open BigOperators Topology
 
 variable {ι X E : Type _} [TopologicalSpace X] [AddCommGroup E] [Module ℝ E]
 
+/- warning: partition_of_unity.finsum_smul_mem_convex -> PartitionOfUnity.finsum_smul_mem_convex is a dubious translation:
+lean 3 declaration is
+  forall {ι : Type.{u1}} {X : Type.{u2}} {E : Type.{u3}} [_inst_1 : TopologicalSpace.{u2} X] [_inst_2 : AddCommGroup.{u3} E] [_inst_3 : Module.{0, u3} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u3} E _inst_2)] {s : Set.{u2} X} (f : PartitionOfUnity.{u1, u2} ι X _inst_1 s) {g : ι -> X -> E} {t : Set.{u3} E} {x : X}, (Membership.Mem.{u2, u2} X (Set.{u2} X) (Set.hasMem.{u2} X) x s) -> (forall (i : ι), (Ne.{1} Real (coeFn.{succ u2, succ u2} (ContinuousMap.{u2, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) (fun (_x : ContinuousMap.{u2, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) => X -> Real) (ContinuousMap.hasCoeToFun.{u2, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (PartitionOfUnity.{u1, u2} ι X _inst_1 s) (fun (_x : PartitionOfUnity.{u1, u2} ι X _inst_1 s) => ι -> (ContinuousMap.{u2, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)))) (PartitionOfUnity.hasCoeToFun.{u1, u2} ι X _inst_1 s) f i) x) (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero)))) -> (Membership.Mem.{u3, u3} E (Set.{u3} E) (Set.hasMem.{u3} E) (g i x) t)) -> (Convex.{0, u3} Real E Real.orderedSemiring (AddCommGroup.toAddCommMonoid.{u3} E _inst_2) (SMulZeroClass.toHasSmul.{0, u3} Real E (AddZeroClass.toHasZero.{u3} E (AddMonoid.toAddZeroClass.{u3} E (AddCommMonoid.toAddMonoid.{u3} E (AddCommGroup.toAddCommMonoid.{u3} E _inst_2)))) (SMulWithZero.toSmulZeroClass.{0, u3} Real E (MulZeroClass.toHasZero.{0} Real (MulZeroOneClass.toMulZeroClass.{0} Real (MonoidWithZero.toMulZeroOneClass.{0} Real (Semiring.toMonoidWithZero.{0} Real Real.semiring)))) (AddZeroClass.toHasZero.{u3} E (AddMonoid.toAddZeroClass.{u3} E (AddCommMonoid.toAddMonoid.{u3} E (AddCommGroup.toAddCommMonoid.{u3} E _inst_2)))) (MulActionWithZero.toSMulWithZero.{0, u3} Real E (Semiring.toMonoidWithZero.{0} Real Real.semiring) (AddZeroClass.toHasZero.{u3} E (AddMonoid.toAddZeroClass.{u3} E (AddCommMonoid.toAddMonoid.{u3} E (AddCommGroup.toAddCommMonoid.{u3} E _inst_2)))) (Module.toMulActionWithZero.{0, u3} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u3} E _inst_2) _inst_3)))) t) -> (Membership.Mem.{u3, u3} E (Set.{u3} E) (Set.hasMem.{u3} E) (finsum.{u3, succ u1} E ι (AddCommGroup.toAddCommMonoid.{u3} E _inst_2) (fun (i : ι) => SMul.smul.{0, u3} Real E (SMulZeroClass.toHasSmul.{0, u3} Real E (AddZeroClass.toHasZero.{u3} E (AddMonoid.toAddZeroClass.{u3} E (AddCommMonoid.toAddMonoid.{u3} E (AddCommGroup.toAddCommMonoid.{u3} E _inst_2)))) (SMulWithZero.toSmulZeroClass.{0, u3} Real E (MulZeroClass.toHasZero.{0} Real (MulZeroOneClass.toMulZeroClass.{0} Real (MonoidWithZero.toMulZeroOneClass.{0} Real (Semiring.toMonoidWithZero.{0} Real Real.semiring)))) (AddZeroClass.toHasZero.{u3} E (AddMonoid.toAddZeroClass.{u3} E (AddCommMonoid.toAddMonoid.{u3} E (AddCommGroup.toAddCommMonoid.{u3} E _inst_2)))) (MulActionWithZero.toSMulWithZero.{0, u3} Real E (Semiring.toMonoidWithZero.{0} Real Real.semiring) (AddZeroClass.toHasZero.{u3} E (AddMonoid.toAddZeroClass.{u3} E (AddCommMonoid.toAddMonoid.{u3} E (AddCommGroup.toAddCommMonoid.{u3} E _inst_2)))) (Module.toMulActionWithZero.{0, u3} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u3} E _inst_2) _inst_3)))) (coeFn.{succ u2, succ u2} (ContinuousMap.{u2, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) (fun (_x : ContinuousMap.{u2, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) => X -> Real) (ContinuousMap.hasCoeToFun.{u2, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (PartitionOfUnity.{u1, u2} ι X _inst_1 s) (fun (_x : PartitionOfUnity.{u1, u2} ι X _inst_1 s) => ι -> (ContinuousMap.{u2, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)))) (PartitionOfUnity.hasCoeToFun.{u1, u2} ι X _inst_1 s) f i) x) (g i x))) t)
+but is expected to have type
+  forall {ι : Type.{u2}} {X : Type.{u3}} {E : Type.{u1}} [_inst_1 : TopologicalSpace.{u3} X] [_inst_2 : AddCommGroup.{u1} E] [_inst_3 : Module.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_2)] {s : Set.{u3} X} (f : PartitionOfUnity.{u2, u3} ι X _inst_1 s) {g : ι -> X -> E} {t : Set.{u1} E} {x : X}, (Membership.mem.{u3, u3} X (Set.{u3} X) (Set.instMembershipSet.{u3} X) x s) -> (forall (i : ι), (Ne.{1} ((fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : X) => Real) x) (FunLike.coe.{succ u3, succ u3, 1} (ContinuousMap.{u3, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) X (fun (_x : X) => (fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : X) => Real) _x) (ContinuousMapClass.toFunLike.{u3, u3, 0} (ContinuousMap.{u3, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) (ContinuousMap.instContinuousMapClassContinuousMap.{u3, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)))) (PartitionOfUnity.toFun.{u2, u3} ι X _inst_1 s f i) x) (OfNat.ofNat.{0} ((fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : X) => Real) x) 0 (Zero.toOfNat0.{0} ((fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : X) => Real) x) Real.instZeroReal))) -> (Membership.mem.{u1, u1} E (Set.{u1} E) (Set.instMembershipSet.{u1} E) (g i x) t)) -> (Convex.{0, u1} Real E Real.orderedSemiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_2) (SMulZeroClass.toSMul.{0, u1} Real E (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_2))))) (SMulWithZero.toSMulZeroClass.{0, u1} Real E Real.instZeroReal (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_2))))) (MulActionWithZero.toSMulWithZero.{0, u1} Real E Real.instMonoidWithZeroReal (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_2))))) (Module.toMulActionWithZero.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_2) _inst_3)))) t) -> (Membership.mem.{u1, u1} E (Set.{u1} E) (Set.instMembershipSet.{u1} E) (finsum.{u1, succ u2} E ι (AddCommGroup.toAddCommMonoid.{u1} E _inst_2) (fun (i : ι) => HSMul.hSMul.{0, u1, u1} ((fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : X) => Real) x) E E (instHSMul.{0, u1} ((fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : X) => Real) x) E (SMulZeroClass.toSMul.{0, u1} ((fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : X) => Real) x) E (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_2))))) (SMulWithZero.toSMulZeroClass.{0, u1} ((fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : X) => Real) x) E Real.instZeroReal (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_2))))) (MulActionWithZero.toSMulWithZero.{0, u1} ((fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : X) => Real) x) E Real.instMonoidWithZeroReal (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_2))))) (Module.toMulActionWithZero.{0, u1} ((fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : X) => Real) x) E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_2) _inst_3))))) (FunLike.coe.{succ u3, succ u3, 1} (ContinuousMap.{u3, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) X (fun (_x : X) => (fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : X) => Real) _x) (ContinuousMapClass.toFunLike.{u3, u3, 0} (ContinuousMap.{u3, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) (ContinuousMap.instContinuousMapClassContinuousMap.{u3, 0} X Real _inst_1 (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)))) (PartitionOfUnity.toFun.{u2, u3} ι X _inst_1 s f i) x) (g i x))) t)
+Case conversion may be inaccurate. Consider using '#align partition_of_unity.finsum_smul_mem_convex PartitionOfUnity.finsum_smul_mem_convexₓ'. -/
 theorem PartitionOfUnity.finsum_smul_mem_convex {s : Set X} (f : PartitionOfUnity ι X s)
     {g : ι → X → E} {t : Set E} {x : X} (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 :=
@@ -45,6 +51,12 @@ theorem PartitionOfUnity.finsum_smul_mem_convex {s : Set X} (f : PartitionOfUnit
 variable [NormalSpace X] [ParacompactSpace X] [TopologicalSpace E] [ContinuousAdd E]
   [ContinuousSMul ℝ E] {t : X → Set E}
 
+/- warning: exists_continuous_forall_mem_convex_of_local -> exists_continuous_forall_mem_convex_of_local is a dubious translation:
+lean 3 declaration is
+  forall {X : Type.{u1}} {E : Type.{u2}} [_inst_1 : TopologicalSpace.{u1} X] [_inst_2 : AddCommGroup.{u2} E] [_inst_3 : Module.{0, u2} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u2} E _inst_2)] [_inst_4 : NormalSpace.{u1} X _inst_1] [_inst_5 : ParacompactSpace.{u1} X _inst_1] [_inst_6 : TopologicalSpace.{u2} E] [_inst_7 : ContinuousAdd.{u2} E _inst_6 (AddZeroClass.toHasAdd.{u2} E (AddMonoid.toAddZeroClass.{u2} E (SubNegMonoid.toAddMonoid.{u2} E (AddGroup.toSubNegMonoid.{u2} E (AddCommGroup.toAddGroup.{u2} E _inst_2)))))] [_inst_8 : ContinuousSMul.{0, u2} Real E (SMulZeroClass.toHasSmul.{0, u2} Real E (AddZeroClass.toHasZero.{u2} E (AddMonoid.toAddZeroClass.{u2} E (AddCommMonoid.toAddMonoid.{u2} E (AddCommGroup.toAddCommMonoid.{u2} E _inst_2)))) (SMulWithZero.toSmulZeroClass.{0, u2} Real E (MulZeroClass.toHasZero.{0} Real (MulZeroOneClass.toMulZeroClass.{0} Real (MonoidWithZero.toMulZeroOneClass.{0} Real (Semiring.toMonoidWithZero.{0} Real Real.semiring)))) (AddZeroClass.toHasZero.{u2} E (AddMonoid.toAddZeroClass.{u2} E (AddCommMonoid.toAddMonoid.{u2} E (AddCommGroup.toAddCommMonoid.{u2} E _inst_2)))) (MulActionWithZero.toSMulWithZero.{0, u2} Real E (Semiring.toMonoidWithZero.{0} Real Real.semiring) (AddZeroClass.toHasZero.{u2} E (AddMonoid.toAddZeroClass.{u2} E (AddCommMonoid.toAddMonoid.{u2} E (AddCommGroup.toAddCommMonoid.{u2} E _inst_2)))) (Module.toMulActionWithZero.{0, u2} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u2} E _inst_2) _inst_3)))) (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) _inst_6] {t : X -> (Set.{u2} E)}, (forall (x : X), Convex.{0, u2} Real E Real.orderedSemiring (AddCommGroup.toAddCommMonoid.{u2} E _inst_2) (SMulZeroClass.toHasSmul.{0, u2} Real E (AddZeroClass.toHasZero.{u2} E (AddMonoid.toAddZeroClass.{u2} E (AddCommMonoid.toAddMonoid.{u2} E (AddCommGroup.toAddCommMonoid.{u2} E _inst_2)))) (SMulWithZero.toSmulZeroClass.{0, u2} Real E (MulZeroClass.toHasZero.{0} Real (MulZeroOneClass.toMulZeroClass.{0} Real (MonoidWithZero.toMulZeroOneClass.{0} Real (Semiring.toMonoidWithZero.{0} Real Real.semiring)))) (AddZeroClass.toHasZero.{u2} E (AddMonoid.toAddZeroClass.{u2} E (AddCommMonoid.toAddMonoid.{u2} E (AddCommGroup.toAddCommMonoid.{u2} E _inst_2)))) (MulActionWithZero.toSMulWithZero.{0, u2} Real E (Semiring.toMonoidWithZero.{0} Real Real.semiring) (AddZeroClass.toHasZero.{u2} E (AddMonoid.toAddZeroClass.{u2} E (AddCommMonoid.toAddMonoid.{u2} E (AddCommGroup.toAddCommMonoid.{u2} E _inst_2)))) (Module.toMulActionWithZero.{0, u2} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u2} E _inst_2) _inst_3)))) (t x)) -> (forall (x : X), Exists.{succ u1} (Set.{u1} X) (fun (U : Set.{u1} X) => Exists.{0} (Membership.Mem.{u1, u1} (Set.{u1} X) (Filter.{u1} X) (Filter.hasMem.{u1} X) U (nhds.{u1} X _inst_1 x)) (fun (H : Membership.Mem.{u1, u1} (Set.{u1} X) (Filter.{u1} X) (Filter.hasMem.{u1} X) U (nhds.{u1} X _inst_1 x)) => Exists.{max (succ u1) (succ u2)} (X -> E) (fun (g : X -> E) => And (ContinuousOn.{u1, u2} X E _inst_1 _inst_6 g U) (forall (y : X), (Membership.Mem.{u1, u1} X (Set.{u1} X) (Set.hasMem.{u1} X) y U) -> (Membership.Mem.{u2, u2} E (Set.{u2} E) (Set.hasMem.{u2} E) (g y) (t y))))))) -> (Exists.{max (succ u1) (succ u2)} (ContinuousMap.{u1, u2} X E _inst_1 _inst_6) (fun (g : ContinuousMap.{u1, u2} X E _inst_1 _inst_6) => forall (x : X), Membership.Mem.{u2, u2} E (Set.{u2} E) (Set.hasMem.{u2} E) (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (ContinuousMap.{u1, u2} X E _inst_1 _inst_6) (fun (_x : ContinuousMap.{u1, u2} X E _inst_1 _inst_6) => X -> E) (ContinuousMap.hasCoeToFun.{u1, u2} X E _inst_1 _inst_6) g x) (t x)))
+but is expected to have type
+  forall {X : Type.{u1}} {E : Type.{u2}} [_inst_1 : TopologicalSpace.{u1} X] [_inst_2 : AddCommGroup.{u2} E] [_inst_3 : Module.{0, u2} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u2} E _inst_2)] [_inst_4 : NormalSpace.{u1} X _inst_1] [_inst_5 : ParacompactSpace.{u1} X _inst_1] [_inst_6 : TopologicalSpace.{u2} E] [_inst_7 : ContinuousAdd.{u2} E _inst_6 (AddZeroClass.toAdd.{u2} E (AddMonoid.toAddZeroClass.{u2} E (SubNegMonoid.toAddMonoid.{u2} E (AddGroup.toSubNegMonoid.{u2} E (AddCommGroup.toAddGroup.{u2} E _inst_2)))))] [_inst_8 : ContinuousSMul.{0, u2} Real E (SMulZeroClass.toSMul.{0, u2} Real E (NegZeroClass.toZero.{u2} E (SubNegZeroMonoid.toNegZeroClass.{u2} E (SubtractionMonoid.toSubNegZeroMonoid.{u2} E (SubtractionCommMonoid.toSubtractionMonoid.{u2} E (AddCommGroup.toDivisionAddCommMonoid.{u2} E _inst_2))))) (SMulWithZero.toSMulZeroClass.{0, u2} Real E Real.instZeroReal (NegZeroClass.toZero.{u2} E (SubNegZeroMonoid.toNegZeroClass.{u2} E (SubtractionMonoid.toSubNegZeroMonoid.{u2} E (SubtractionCommMonoid.toSubtractionMonoid.{u2} E (AddCommGroup.toDivisionAddCommMonoid.{u2} E _inst_2))))) (MulActionWithZero.toSMulWithZero.{0, u2} Real E Real.instMonoidWithZeroReal (NegZeroClass.toZero.{u2} E (SubNegZeroMonoid.toNegZeroClass.{u2} E (SubtractionMonoid.toSubNegZeroMonoid.{u2} E (SubtractionCommMonoid.toSubtractionMonoid.{u2} E (AddCommGroup.toDivisionAddCommMonoid.{u2} E _inst_2))))) (Module.toMulActionWithZero.{0, u2} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u2} E _inst_2) _inst_3)))) (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) _inst_6] {t : X -> (Set.{u2} E)}, (forall (x : X), Convex.{0, u2} Real E Real.orderedSemiring (AddCommGroup.toAddCommMonoid.{u2} E _inst_2) (SMulZeroClass.toSMul.{0, u2} Real E (NegZeroClass.toZero.{u2} E (SubNegZeroMonoid.toNegZeroClass.{u2} E (SubtractionMonoid.toSubNegZeroMonoid.{u2} E (SubtractionCommMonoid.toSubtractionMonoid.{u2} E (AddCommGroup.toDivisionAddCommMonoid.{u2} E _inst_2))))) (SMulWithZero.toSMulZeroClass.{0, u2} Real E Real.instZeroReal (NegZeroClass.toZero.{u2} E (SubNegZeroMonoid.toNegZeroClass.{u2} E (SubtractionMonoid.toSubNegZeroMonoid.{u2} E (SubtractionCommMonoid.toSubtractionMonoid.{u2} E (AddCommGroup.toDivisionAddCommMonoid.{u2} E _inst_2))))) (MulActionWithZero.toSMulWithZero.{0, u2} Real E Real.instMonoidWithZeroReal (NegZeroClass.toZero.{u2} E (SubNegZeroMonoid.toNegZeroClass.{u2} E (SubtractionMonoid.toSubNegZeroMonoid.{u2} E (SubtractionCommMonoid.toSubtractionMonoid.{u2} E (AddCommGroup.toDivisionAddCommMonoid.{u2} E _inst_2))))) (Module.toMulActionWithZero.{0, u2} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u2} E _inst_2) _inst_3)))) (t x)) -> (forall (x : X), Exists.{succ u1} (Set.{u1} X) (fun (U : Set.{u1} X) => And (Membership.mem.{u1, u1} (Set.{u1} X) (Filter.{u1} X) (instMembershipSetFilter.{u1} X) U (nhds.{u1} X _inst_1 x)) (Exists.{max (succ u1) (succ u2)} (X -> E) (fun (g : X -> E) => And (ContinuousOn.{u1, u2} X E _inst_1 _inst_6 g U) (forall (y : X), (Membership.mem.{u1, u1} X (Set.{u1} X) (Set.instMembershipSet.{u1} X) y U) -> (Membership.mem.{u2, u2} E (Set.{u2} E) (Set.instMembershipSet.{u2} E) (g y) (t y))))))) -> (Exists.{max (succ u1) (succ u2)} (ContinuousMap.{u1, u2} X E _inst_1 _inst_6) (fun (g : ContinuousMap.{u1, u2} X E _inst_1 _inst_6) => forall (x : X), Membership.mem.{u2, u2} ((fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : X) => E) x) (Set.{u2} E) (Set.instMembershipSet.{u2} E) (FunLike.coe.{max (succ u1) (succ u2), succ u1, succ u2} (ContinuousMap.{u1, u2} X E _inst_1 _inst_6) X (fun (_x : X) => (fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : X) => E) _x) (ContinuousMapClass.toFunLike.{max u1 u2, u1, u2} (ContinuousMap.{u1, u2} X E _inst_1 _inst_6) X E _inst_1 _inst_6 (ContinuousMap.instContinuousMapClassContinuousMap.{u1, u2} X E _inst_1 _inst_6)) g x) (t x)))
+Case conversion may be inaccurate. Consider using '#align exists_continuous_forall_mem_convex_of_local exists_continuous_forall_mem_convex_of_localₓ'. -/
 /-- Let `X` be a normal paracompact topological space (e.g., any extended metric space). Let `E` be
 a topological real vector space. Let `t : X → set E` be a family of convex sets. Suppose that for
 each point `x : X`, there exists a neighborhood `U ∈ 𝓝 X` and a function `g : X → E` that is
@@ -66,6 +78,12 @@ theorem exists_continuous_forall_mem_convex_of_local (ht : ∀ x, Convex ℝ (t
   exact interior_subset (hf _ <| subset_closure hi)
 #align exists_continuous_forall_mem_convex_of_local exists_continuous_forall_mem_convex_of_local
 
+/- warning: exists_continuous_forall_mem_convex_of_local_const -> exists_continuous_forall_mem_convex_of_local_const is a dubious translation:
+lean 3 declaration is
+  forall {X : Type.{u1}} {E : Type.{u2}} [_inst_1 : TopologicalSpace.{u1} X] [_inst_2 : AddCommGroup.{u2} E] [_inst_3 : Module.{0, u2} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u2} E _inst_2)] [_inst_4 : NormalSpace.{u1} X _inst_1] [_inst_5 : ParacompactSpace.{u1} X _inst_1] [_inst_6 : TopologicalSpace.{u2} E] [_inst_7 : ContinuousAdd.{u2} E _inst_6 (AddZeroClass.toHasAdd.{u2} E (AddMonoid.toAddZeroClass.{u2} E (SubNegMonoid.toAddMonoid.{u2} E (AddGroup.toSubNegMonoid.{u2} E (AddCommGroup.toAddGroup.{u2} E _inst_2)))))] [_inst_8 : ContinuousSMul.{0, u2} Real E (SMulZeroClass.toHasSmul.{0, u2} Real E (AddZeroClass.toHasZero.{u2} E (AddMonoid.toAddZeroClass.{u2} E (AddCommMonoid.toAddMonoid.{u2} E (AddCommGroup.toAddCommMonoid.{u2} E _inst_2)))) (SMulWithZero.toSmulZeroClass.{0, u2} Real E (MulZeroClass.toHasZero.{0} Real (MulZeroOneClass.toMulZeroClass.{0} Real (MonoidWithZero.toMulZeroOneClass.{0} Real (Semiring.toMonoidWithZero.{0} Real Real.semiring)))) (AddZeroClass.toHasZero.{u2} E (AddMonoid.toAddZeroClass.{u2} E (AddCommMonoid.toAddMonoid.{u2} E (AddCommGroup.toAddCommMonoid.{u2} E _inst_2)))) (MulActionWithZero.toSMulWithZero.{0, u2} Real E (Semiring.toMonoidWithZero.{0} Real Real.semiring) (AddZeroClass.toHasZero.{u2} E (AddMonoid.toAddZeroClass.{u2} E (AddCommMonoid.toAddMonoid.{u2} E (AddCommGroup.toAddCommMonoid.{u2} E _inst_2)))) (Module.toMulActionWithZero.{0, u2} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u2} E _inst_2) _inst_3)))) (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) _inst_6] {t : X -> (Set.{u2} E)}, (forall (x : X), Convex.{0, u2} Real E Real.orderedSemiring (AddCommGroup.toAddCommMonoid.{u2} E _inst_2) (SMulZeroClass.toHasSmul.{0, u2} Real E (AddZeroClass.toHasZero.{u2} E (AddMonoid.toAddZeroClass.{u2} E (AddCommMonoid.toAddMonoid.{u2} E (AddCommGroup.toAddCommMonoid.{u2} E _inst_2)))) (SMulWithZero.toSmulZeroClass.{0, u2} Real E (MulZeroClass.toHasZero.{0} Real (MulZeroOneClass.toMulZeroClass.{0} Real (MonoidWithZero.toMulZeroOneClass.{0} Real (Semiring.toMonoidWithZero.{0} Real Real.semiring)))) (AddZeroClass.toHasZero.{u2} E (AddMonoid.toAddZeroClass.{u2} E (AddCommMonoid.toAddMonoid.{u2} E (AddCommGroup.toAddCommMonoid.{u2} E _inst_2)))) (MulActionWithZero.toSMulWithZero.{0, u2} Real E (Semiring.toMonoidWithZero.{0} Real Real.semiring) (AddZeroClass.toHasZero.{u2} E (AddMonoid.toAddZeroClass.{u2} E (AddCommMonoid.toAddMonoid.{u2} E (AddCommGroup.toAddCommMonoid.{u2} E _inst_2)))) (Module.toMulActionWithZero.{0, u2} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u2} E _inst_2) _inst_3)))) (t x)) -> (forall (x : X), Exists.{succ u2} E (fun (c : E) => Filter.Eventually.{u1} X (fun (y : X) => Membership.Mem.{u2, u2} E (Set.{u2} E) (Set.hasMem.{u2} E) c (t y)) (nhds.{u1} X _inst_1 x))) -> (Exists.{max (succ u1) (succ u2)} (ContinuousMap.{u1, u2} X E _inst_1 _inst_6) (fun (g : ContinuousMap.{u1, u2} X E _inst_1 _inst_6) => forall (x : X), Membership.Mem.{u2, u2} E (Set.{u2} E) (Set.hasMem.{u2} E) (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (ContinuousMap.{u1, u2} X E _inst_1 _inst_6) (fun (_x : ContinuousMap.{u1, u2} X E _inst_1 _inst_6) => X -> E) (ContinuousMap.hasCoeToFun.{u1, u2} X E _inst_1 _inst_6) g x) (t x)))
+but is expected to have type
+  forall {X : Type.{u1}} {E : Type.{u2}} [_inst_1 : TopologicalSpace.{u1} X] [_inst_2 : AddCommGroup.{u2} E] [_inst_3 : Module.{0, u2} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u2} E _inst_2)] [_inst_4 : NormalSpace.{u1} X _inst_1] [_inst_5 : ParacompactSpace.{u1} X _inst_1] [_inst_6 : TopologicalSpace.{u2} E] [_inst_7 : ContinuousAdd.{u2} E _inst_6 (AddZeroClass.toAdd.{u2} E (AddMonoid.toAddZeroClass.{u2} E (SubNegMonoid.toAddMonoid.{u2} E (AddGroup.toSubNegMonoid.{u2} E (AddCommGroup.toAddGroup.{u2} E _inst_2)))))] [_inst_8 : ContinuousSMul.{0, u2} Real E (SMulZeroClass.toSMul.{0, u2} Real E (NegZeroClass.toZero.{u2} E (SubNegZeroMonoid.toNegZeroClass.{u2} E (SubtractionMonoid.toSubNegZeroMonoid.{u2} E (SubtractionCommMonoid.toSubtractionMonoid.{u2} E (AddCommGroup.toDivisionAddCommMonoid.{u2} E _inst_2))))) (SMulWithZero.toSMulZeroClass.{0, u2} Real E Real.instZeroReal (NegZeroClass.toZero.{u2} E (SubNegZeroMonoid.toNegZeroClass.{u2} E (SubtractionMonoid.toSubNegZeroMonoid.{u2} E (SubtractionCommMonoid.toSubtractionMonoid.{u2} E (AddCommGroup.toDivisionAddCommMonoid.{u2} E _inst_2))))) (MulActionWithZero.toSMulWithZero.{0, u2} Real E Real.instMonoidWithZeroReal (NegZeroClass.toZero.{u2} E (SubNegZeroMonoid.toNegZeroClass.{u2} E (SubtractionMonoid.toSubNegZeroMonoid.{u2} E (SubtractionCommMonoid.toSubtractionMonoid.{u2} E (AddCommGroup.toDivisionAddCommMonoid.{u2} E _inst_2))))) (Module.toMulActionWithZero.{0, u2} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u2} E _inst_2) _inst_3)))) (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) _inst_6] {t : X -> (Set.{u2} E)}, (forall (x : X), Convex.{0, u2} Real E Real.orderedSemiring (AddCommGroup.toAddCommMonoid.{u2} E _inst_2) (SMulZeroClass.toSMul.{0, u2} Real E (NegZeroClass.toZero.{u2} E (SubNegZeroMonoid.toNegZeroClass.{u2} E (SubtractionMonoid.toSubNegZeroMonoid.{u2} E (SubtractionCommMonoid.toSubtractionMonoid.{u2} E (AddCommGroup.toDivisionAddCommMonoid.{u2} E _inst_2))))) (SMulWithZero.toSMulZeroClass.{0, u2} Real E Real.instZeroReal (NegZeroClass.toZero.{u2} E (SubNegZeroMonoid.toNegZeroClass.{u2} E (SubtractionMonoid.toSubNegZeroMonoid.{u2} E (SubtractionCommMonoid.toSubtractionMonoid.{u2} E (AddCommGroup.toDivisionAddCommMonoid.{u2} E _inst_2))))) (MulActionWithZero.toSMulWithZero.{0, u2} Real E Real.instMonoidWithZeroReal (NegZeroClass.toZero.{u2} E (SubNegZeroMonoid.toNegZeroClass.{u2} E (SubtractionMonoid.toSubNegZeroMonoid.{u2} E (SubtractionCommMonoid.toSubtractionMonoid.{u2} E (AddCommGroup.toDivisionAddCommMonoid.{u2} E _inst_2))))) (Module.toMulActionWithZero.{0, u2} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u2} E _inst_2) _inst_3)))) (t x)) -> (forall (x : X), Exists.{succ u2} E (fun (c : E) => Filter.Eventually.{u1} X (fun (y : X) => Membership.mem.{u2, u2} E (Set.{u2} E) (Set.instMembershipSet.{u2} E) c (t y)) (nhds.{u1} X _inst_1 x))) -> (Exists.{max (succ u1) (succ u2)} (ContinuousMap.{u1, u2} X E _inst_1 _inst_6) (fun (g : ContinuousMap.{u1, u2} X E _inst_1 _inst_6) => forall (x : X), Membership.mem.{u2, u2} ((fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : X) => E) x) (Set.{u2} E) (Set.instMembershipSet.{u2} E) (FunLike.coe.{max (succ u1) (succ u2), succ u1, succ u2} (ContinuousMap.{u1, u2} X E _inst_1 _inst_6) X (fun (_x : X) => (fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : X) => E) _x) (ContinuousMapClass.toFunLike.{max u1 u2, u1, u2} (ContinuousMap.{u1, u2} X E _inst_1 _inst_6) X E _inst_1 _inst_6 (ContinuousMap.instContinuousMapClassContinuousMap.{u1, u2} X E _inst_1 _inst_6)) g x) (t x)))
+Case conversion may be inaccurate. Consider using '#align exists_continuous_forall_mem_convex_of_local_const exists_continuous_forall_mem_convex_of_local_constₓ'. -/
 /-- Let `X` be a normal paracompact topological space (e.g., any extended metric space). Let `E` be
 a topological real vector space. Let `t : X → set E` be a family of convex sets. Suppose that for
 each point `x : X`, there exists a vector `c : E` that belongs to `t y` for all `y` in a

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

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

This has nice performance benefits.

@@ -31,7 +31,7 @@ open Set Function
 
 open BigOperators Topology
 
-variable {ι X E : Type _} [TopologicalSpace X] [AddCommGroup E] [Module ℝ E]
+variable {ι X E : Type*} [TopologicalSpace X] [AddCommGroup E] [Module ℝ E]
 
 theorem PartitionOfUnity.finsum_smul_mem_convex {s : Set X} (f : PartitionOfUnity ι X s)
     {g : ι → X → E} {t : Set E} {x : X} (hx : x ∈ s) (hg : ∀ i, f i x ≠ 0 → g i x ∈ t)

• depends on: #5966

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

@@ -2,15 +2,12 @@
 Copyright (c) 2022 Yury Kudryashov. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Yury Kudryashov
-
-! This file was ported from Lean 3 source module analysis.convex.partition_of_unity
-! leanprover-community/mathlib commit f2ce6086713c78a7f880485f7917ea547a215982
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
 -/
 import Mathlib.Topology.PartitionOfUnity
 import Mathlib.Analysis.Convex.Combination
 
+#align_import analysis.convex.partition_of_unity from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
+
 /-!
 # Partition of unity and convex sets
 

As discussed on Zulip

Renames

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

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

@@ -56,7 +56,7 @@ theorem exists_continuous_forall_mem_convex_of_local (ht : ∀ x, Convex ℝ (t
     ∃ g : C(X, E), ∀ x, g x ∈ t x := by
   choose U hU g hgc hgt using H
   obtain ⟨f, hf⟩ := PartitionOfUnity.exists_isSubordinate isClosed_univ (fun x => interior (U x))
-    (fun x => isOpen_interior) fun x _ => mem_unionᵢ.2 ⟨x, mem_interior_iff_mem_nhds.2 (hU x)⟩
+    (fun x => isOpen_interior) fun x _ => mem_iUnion.2 ⟨x, mem_interior_iff_mem_nhds.2 (hU x)⟩
   refine' ⟨⟨fun x => ∑ᶠ i, f i x • g i x,
     hf.continuous_finsum_smul (fun i => isOpen_interior) fun i => (hgc i).mono interior_subset⟩,
     fun x => f.finsum_smul_mem_convex (mem_univ x) (fun i hi => hgt _ _ _) (ht _)⟩