topology.continuous_function.zero_at_infty ⟷ Mathlib.Topology.ContinuousFunction.ZeroAtInfty

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

@@ -247,18 +247,20 @@ section AddMonoid
 
 variable [AddMonoid β] [ContinuousAdd β] (f g : C₀(α, β))
 
-#print ZeroAtInftyContinuousMap.coe_nsmulRec /-
+/- warning: zero_at_infty_continuous_map.coe_nsmul_rec clashes with zero_at_infty_continuous_map.coe_smul -> ZeroAtInftyContinuousMap.coe_smul
+Case conversion may be inaccurate. Consider using '#align zero_at_infty_continuous_map.coe_nsmul_rec ZeroAtInftyContinuousMap.coe_smulₓ'. -/
+#print ZeroAtInftyContinuousMap.coe_smul /-
 @[simp]
-theorem coe_nsmulRec : ∀ n, ⇑(nsmulRec n f) = n • f
+theorem coe_smul : ∀ n, ⇑(nsmulRec n f) = n • f
   | 0 => by rw [nsmulRec, zero_smul, coe_zero]
   | n + 1 => by rw [nsmulRec, succ_nsmul', coe_add, coe_nsmul_rec]
-#align zero_at_infty_continuous_map.coe_nsmul_rec ZeroAtInftyContinuousMap.coe_nsmulRec
+#align zero_at_infty_continuous_map.coe_nsmul_rec ZeroAtInftyContinuousMap.coe_smul
 -/
 
-#print ZeroAtInftyContinuousMap.instNatSMul /-
-instance instNatSMul : SMul ℕ C₀(α, β) :=
+#print ZeroAtInftyContinuousMap.instSMul /-
+instance instSMul : SMul ℕ C₀(α, β) :=
   ⟨fun n f => ⟨n • f, by simpa [coe_nsmul_rec] using zero_at_infty (nsmulRec n f)⟩⟩
-#align zero_at_infty_continuous_map.has_nat_scalar ZeroAtInftyContinuousMap.instNatSMul
+#align zero_at_infty_continuous_map.has_nat_scalar ZeroAtInftyContinuousMap.instSMul
 -/
 
 instance : AddMonoid C₀(α, β) :=
@@ -305,18 +307,22 @@ theorem sub_apply : (f - g) x = f x - g x :=
 #align zero_at_infty_continuous_map.sub_apply ZeroAtInftyContinuousMap.sub_apply
 -/
 
-#print ZeroAtInftyContinuousMap.coe_zsmulRec /-
+/- warning: zero_at_infty_continuous_map.coe_zsmul_rec clashes with zero_at_infty_continuous_map.coe_smul -> ZeroAtInftyContinuousMap.coe_smul
+Case conversion may be inaccurate. Consider using '#align zero_at_infty_continuous_map.coe_zsmul_rec ZeroAtInftyContinuousMap.coe_smulₓ'. -/
+#print ZeroAtInftyContinuousMap.coe_smul /-
 @[simp]
-theorem coe_zsmulRec : ∀ z, ⇑(zsmulRec z f) = z • f
+theorem coe_smul : ∀ z, ⇑(zsmulRec z f) = z • f
   | Int.ofNat n => by rw [zsmulRec, Int.ofNat_eq_coe, coe_nsmul_rec, natCast_zsmul]
   | -[n+1] => by rw [zsmulRec, negSucc_zsmul, coe_neg, coe_nsmul_rec]
-#align zero_at_infty_continuous_map.coe_zsmul_rec ZeroAtInftyContinuousMap.coe_zsmulRec
+#align zero_at_infty_continuous_map.coe_zsmul_rec ZeroAtInftyContinuousMap.coe_smul
 -/
 
-#print ZeroAtInftyContinuousMap.instIntSMul /-
-instance instIntSMul : SMul ℤ C₀(α, β) :=
+/- warning: zero_at_infty_continuous_map.has_int_scalar clashes with zero_at_infty_continuous_map.has_nat_scalar -> ZeroAtInftyContinuousMap.instSMul
+Case conversion may be inaccurate. Consider using '#align zero_at_infty_continuous_map.has_int_scalar ZeroAtInftyContinuousMap.instSMulₓ'. -/
+#print ZeroAtInftyContinuousMap.instSMul /-
+instance instSMul : SMul ℤ C₀(α, β) :=
   ⟨fun n f => ⟨n • f, by simpa using zero_at_infty (zsmulRec n f)⟩⟩
-#align zero_at_infty_continuous_map.has_int_scalar ZeroAtInftyContinuousMap.instIntSMul
+#align zero_at_infty_continuous_map.has_int_scalar ZeroAtInftyContinuousMap.instSMul
 -/
 
 instance : AddGroup C₀(α, β) :=

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

@@ -251,7 +251,7 @@ variable [AddMonoid β] [ContinuousAdd β] (f g : C₀(α, β))
 @[simp]
 theorem coe_nsmulRec : ∀ n, ⇑(nsmulRec n f) = n • f
   | 0 => by rw [nsmulRec, zero_smul, coe_zero]
-  | n + 1 => by rw [nsmulRec, succ_nsmul, coe_add, coe_nsmul_rec]
+  | n + 1 => by rw [nsmulRec, succ_nsmul', coe_add, coe_nsmul_rec]
 #align zero_at_infty_continuous_map.coe_nsmul_rec ZeroAtInftyContinuousMap.coe_nsmulRec
 -/
 
@@ -308,7 +308,7 @@ theorem sub_apply : (f - g) x = f x - g x :=
 #print ZeroAtInftyContinuousMap.coe_zsmulRec /-
 @[simp]
 theorem coe_zsmulRec : ∀ z, ⇑(zsmulRec z f) = z • f
-  | Int.ofNat n => by rw [zsmulRec, Int.ofNat_eq_coe, coe_nsmul_rec, coe_nat_zsmul]
+  | Int.ofNat n => by rw [zsmulRec, Int.ofNat_eq_coe, coe_nsmul_rec, natCast_zsmul]
   | -[n+1] => by rw [zsmulRec, negSucc_zsmul, coe_neg, coe_nsmul_rec]
 #align zero_at_infty_continuous_map.coe_zsmul_rec ZeroAtInftyContinuousMap.coe_zsmulRec
 -/

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

@@ -515,8 +515,8 @@ theorem isometry_toBCF : Isometry (toBCF : C₀(α, β) → α →ᵇ β) := by
 #align zero_at_infty_continuous_map.isometry_to_bcf ZeroAtInftyContinuousMap.isometry_toBCF
 -/
 
-#print ZeroAtInftyContinuousMap.closed_range_toBCF /-
-theorem closed_range_toBCF : IsClosed (range (toBCF : C₀(α, β) → α →ᵇ β)) :=
+#print ZeroAtInftyContinuousMap.isClosed_range_toBCF /-
+theorem isClosed_range_toBCF : IsClosed (range (toBCF : C₀(α, β) → α →ᵇ β)) :=
   by
   refine' is_closed_iff_cluster_pt.mpr fun f hf => _
   rw [clusterPt_principal_iff] at hf
@@ -532,14 +532,14 @@ theorem closed_range_toBCF : IsClosed (range (toBCF : C₀(α, β) → α →ᵇ
       _ < dist g.to_bcf f + ε / 2 := (add_lt_add_of_le_of_lt (dist_coe_le_dist x) hx)
       _ < ε := by simpa [add_halves ε] using add_lt_add_right hg (ε / 2)
   exact ⟨⟨f.to_continuous_map, this⟩, by ext; rfl⟩
-#align zero_at_infty_continuous_map.closed_range_to_bcf ZeroAtInftyContinuousMap.closed_range_toBCF
+#align zero_at_infty_continuous_map.closed_range_to_bcf ZeroAtInftyContinuousMap.isClosed_range_toBCF
 -/
 
 /-- Continuous functions vanishing at infinity taking values in a complete space form a
 complete space. -/
 instance [CompleteSpace β] : CompleteSpace C₀(α, β) :=
   (completeSpace_iff_isComplete_range isometry_toBCF.UniformInducing).mpr
-    closed_range_toBCF.IsComplete
+    isClosed_range_toBCF.IsComplete
 
 end Metric
 

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

@@ -519,7 +519,7 @@ theorem isometry_toBCF : Isometry (toBCF : C₀(α, β) → α →ᵇ β) := by
 theorem closed_range_toBCF : IsClosed (range (toBCF : C₀(α, β) → α →ᵇ β)) :=
   by
   refine' is_closed_iff_cluster_pt.mpr fun f hf => _
-  rw [clusterPt_principal_iff] at hf 
+  rw [clusterPt_principal_iff] at hf
   have : tendsto f (cocompact α) (𝓝 0) :=
     by
     refine' metric.tendsto_nhds.mpr fun ε hε => _

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

@@ -86,7 +86,7 @@ instance : ZeroAtInftyContinuousMapClass C₀(α, β) α β
 /-- Helper instance for when there's too many metavariables to apply `fun_like.has_coe_to_fun`
 directly. -/
 instance : CoeFun C₀(α, β) fun _ => α → β :=
-  FunLike.hasCoeToFun
+  DFunLike.hasCoeToFun
 
 instance : CoeTC F C₀(α, β) :=
   ⟨fun f =>
@@ -104,7 +104,7 @@ theorem coe_toContinuousMap (f : C₀(α, β)) : (f.toContinuousMap : α → β)
 #print ZeroAtInftyContinuousMap.ext /-
 @[ext]
 theorem ext {f g : C₀(α, β)} (h : ∀ x, f x = g x) : f = g :=
-  FunLike.ext _ _ h
+  DFunLike.ext _ _ h
 #align zero_at_infty_continuous_map.ext ZeroAtInftyContinuousMap.ext
 -/
 
@@ -128,7 +128,7 @@ theorem coe_copy (f : C₀(α, β)) (f' : α → β) (h : f' = f) : ⇑(f.copy f
 
 #print ZeroAtInftyContinuousMap.copy_eq /-
 theorem copy_eq (f : C₀(α, β)) (f' : α → β) (h : f' = f) : f.copy f' h = f :=
-  FunLike.ext' h
+  DFunLike.ext' h
 #align zero_at_infty_continuous_map.copy_eq ZeroAtInftyContinuousMap.copy_eq
 -/
 
@@ -162,7 +162,7 @@ def zeroAtInftyContinuousMapClass.ofCompact {G : Type _} [ContinuousMapClass G 
     [CompactSpace α] : ZeroAtInftyContinuousMapClass G α β
     where
   coe g := g
-  coe_injective' f g h := FunLike.coe_fn_eq.mp h
+  coe_injective' f g h := DFunLike.coe_fn_eq.mp h
   map_continuous := map_continuous
   zero_at_infty := by simp
 #align zero_at_infty_continuous_map.zero_at_infty_continuous_map_class.of_compact ZeroAtInftyContinuousMap.zeroAtInftyContinuousMapClass.ofCompact
@@ -219,10 +219,10 @@ theorem mul_apply [MulZeroClass β] [ContinuousMul β] (f g : C₀(α, β)) : (f
 -/
 
 instance [MulZeroClass β] [ContinuousMul β] : MulZeroClass C₀(α, β) :=
-  FunLike.coe_injective.MulZeroClass _ coe_zero coe_mul
+  DFunLike.coe_injective.MulZeroClass _ coe_zero coe_mul
 
 instance [SemigroupWithZero β] [ContinuousMul β] : SemigroupWithZero C₀(α, β) :=
-  FunLike.coe_injective.SemigroupWithZero _ coe_zero coe_mul
+  DFunLike.coe_injective.SemigroupWithZero _ coe_zero coe_mul
 
 instance [AddZeroClass β] [ContinuousAdd β] : Add C₀(α, β) :=
   ⟨fun f g => ⟨f + g, by simpa only [add_zero] using (zero_at_infty f).add (zero_at_infty g)⟩⟩
@@ -241,7 +241,7 @@ theorem add_apply [AddZeroClass β] [ContinuousAdd β] (f g : C₀(α, β)) : (f
 -/
 
 instance [AddZeroClass β] [ContinuousAdd β] : AddZeroClass C₀(α, β) :=
-  FunLike.coe_injective.AddZeroClass _ coe_zero coe_add
+  DFunLike.coe_injective.AddZeroClass _ coe_zero coe_add
 
 section AddMonoid
 
@@ -262,12 +262,12 @@ instance instNatSMul : SMul ℕ C₀(α, β) :=
 -/
 
 instance : AddMonoid C₀(α, β) :=
-  FunLike.coe_injective.AddMonoid _ coe_zero coe_add fun _ _ => rfl
+  DFunLike.coe_injective.AddMonoid _ coe_zero coe_add fun _ _ => rfl
 
 end AddMonoid
 
 instance [AddCommMonoid β] [ContinuousAdd β] : AddCommMonoid C₀(α, β) :=
-  FunLike.coe_injective.AddCommMonoid _ coe_zero coe_add fun _ _ => rfl
+  DFunLike.coe_injective.AddCommMonoid _ coe_zero coe_add fun _ _ => rfl
 
 section AddGroup
 
@@ -320,12 +320,12 @@ instance instIntSMul : SMul ℤ C₀(α, β) :=
 -/
 
 instance : AddGroup C₀(α, β) :=
-  FunLike.coe_injective.AddGroup _ coe_zero coe_add coe_neg coe_sub (fun _ _ => rfl) fun _ _ => rfl
+  DFunLike.coe_injective.AddGroup _ coe_zero coe_add coe_neg coe_sub (fun _ _ => rfl) fun _ _ => rfl
 
 end AddGroup
 
 instance [AddCommGroup β] [TopologicalAddGroup β] : AddCommGroup C₀(α, β) :=
-  FunLike.coe_injective.AddCommGroup _ coe_zero coe_add coe_neg coe_sub (fun _ _ => rfl) fun _ _ =>
+  DFunLike.coe_injective.AddCommGroup _ coe_zero coe_add coe_neg coe_sub (fun _ _ => rfl) fun _ _ =>
     rfl
 
 instance [Zero β] {R : Type _} [Zero R] [SMulWithZero R β] [ContinuousConstSMul R β] :
@@ -353,36 +353,36 @@ instance [Zero β] {R : Type _} [Zero R] [SMulWithZero R β] [SMulWithZero Rᵐ
 
 instance [Zero β] {R : Type _} [Zero R] [SMulWithZero R β] [ContinuousConstSMul R β] :
     SMulWithZero R C₀(α, β) :=
-  Function.Injective.smulWithZero ⟨_, coe_zero⟩ FunLike.coe_injective coe_smul
+  Function.Injective.smulWithZero ⟨_, coe_zero⟩ DFunLike.coe_injective coe_smul
 
 instance [Zero β] {R : Type _} [MonoidWithZero R] [MulActionWithZero R β]
     [ContinuousConstSMul R β] : MulActionWithZero R C₀(α, β) :=
-  Function.Injective.mulActionWithZero ⟨_, coe_zero⟩ FunLike.coe_injective coe_smul
+  Function.Injective.mulActionWithZero ⟨_, coe_zero⟩ DFunLike.coe_injective coe_smul
 
 instance [AddCommMonoid β] [ContinuousAdd β] {R : Type _} [Semiring R] [Module R β]
     [ContinuousConstSMul R β] : Module R C₀(α, β) :=
-  Function.Injective.module R ⟨_, coe_zero, coe_add⟩ FunLike.coe_injective coe_smul
+  Function.Injective.module R ⟨_, coe_zero, coe_add⟩ DFunLike.coe_injective coe_smul
 
 instance [NonUnitalNonAssocSemiring β] [TopologicalSemiring β] :
     NonUnitalNonAssocSemiring C₀(α, β) :=
-  FunLike.coe_injective.NonUnitalNonAssocSemiring _ coe_zero coe_add coe_mul fun _ _ => rfl
+  DFunLike.coe_injective.NonUnitalNonAssocSemiring _ coe_zero coe_add coe_mul fun _ _ => rfl
 
 instance [NonUnitalSemiring β] [TopologicalSemiring β] : NonUnitalSemiring C₀(α, β) :=
-  FunLike.coe_injective.NonUnitalSemiring _ coe_zero coe_add coe_mul fun _ _ => rfl
+  DFunLike.coe_injective.NonUnitalSemiring _ coe_zero coe_add coe_mul fun _ _ => rfl
 
 instance [NonUnitalCommSemiring β] [TopologicalSemiring β] : NonUnitalCommSemiring C₀(α, β) :=
-  FunLike.coe_injective.NonUnitalCommSemiring _ coe_zero coe_add coe_mul fun _ _ => rfl
+  DFunLike.coe_injective.NonUnitalCommSemiring _ coe_zero coe_add coe_mul fun _ _ => rfl
 
 instance [NonUnitalNonAssocRing β] [TopologicalRing β] : NonUnitalNonAssocRing C₀(α, β) :=
-  FunLike.coe_injective.NonUnitalNonAssocRing _ coe_zero coe_add coe_mul coe_neg coe_sub
+  DFunLike.coe_injective.NonUnitalNonAssocRing _ coe_zero coe_add coe_mul coe_neg coe_sub
     (fun _ _ => rfl) fun _ _ => rfl
 
 instance [NonUnitalRing β] [TopologicalRing β] : NonUnitalRing C₀(α, β) :=
-  FunLike.coe_injective.NonUnitalRing _ coe_zero coe_add coe_mul coe_neg coe_sub (fun _ _ => rfl)
+  DFunLike.coe_injective.NonUnitalRing _ coe_zero coe_add coe_mul coe_neg coe_sub (fun _ _ => rfl)
     fun _ _ => rfl
 
 instance [NonUnitalCommRing β] [TopologicalRing β] : NonUnitalCommRing C₀(α, β) :=
-  FunLike.coe_injective.NonUnitalCommRing _ coe_zero coe_add coe_mul coe_neg coe_sub
+  DFunLike.coe_injective.NonUnitalCommRing _ coe_zero coe_add coe_mul coe_neg coe_sub
     (fun _ _ => rfl) fun _ _ => rfl
 
 instance {R : Type _} [Semiring R] [NonUnitalNonAssocSemiring β] [TopologicalSemiring β]
@@ -478,7 +478,7 @@ variable (α) (β)
 
 #print ZeroAtInftyContinuousMap.toBCF_injective /-
 theorem toBCF_injective : Function.Injective (toBCF : C₀(α, β) → α →ᵇ β) := fun f g h => by ext;
-  simpa only using FunLike.congr_fun h x
+  simpa only using DFunLike.congr_fun h x
 #align zero_at_infty_continuous_map.to_bcf_injective ZeroAtInftyContinuousMap.toBCF_injective
 -/
 

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

@@ -464,22 +464,22 @@ instance (priority := 100) : BoundedContinuousMapClass F α β :=
   { ‹ZeroAtInftyContinuousMapClass F α β› with
     map_bounded := fun f => ZeroAtInftyContinuousMap.bounded f }
 
-#print ZeroAtInftyContinuousMap.toBcf /-
+#print ZeroAtInftyContinuousMap.toBCF /-
 /-- Construct a bounded continuous function from a continuous function vanishing at infinity. -/
 @[simps]
-def toBcf (f : C₀(α, β)) : α →ᵇ β :=
+def toBCF (f : C₀(α, β)) : α →ᵇ β :=
   ⟨f, map_bounded f⟩
-#align zero_at_infty_continuous_map.to_bcf ZeroAtInftyContinuousMap.toBcf
+#align zero_at_infty_continuous_map.to_bcf ZeroAtInftyContinuousMap.toBCF
 -/
 
 section
 
 variable (α) (β)
 
-#print ZeroAtInftyContinuousMap.toBcf_injective /-
-theorem toBcf_injective : Function.Injective (toBcf : C₀(α, β) → α →ᵇ β) := fun f g h => by ext;
+#print ZeroAtInftyContinuousMap.toBCF_injective /-
+theorem toBCF_injective : Function.Injective (toBCF : C₀(α, β) → α →ᵇ β) := fun f g h => by ext;
   simpa only using FunLike.congr_fun h x
-#align zero_at_infty_continuous_map.to_bcf_injective ZeroAtInftyContinuousMap.toBcf_injective
+#align zero_at_infty_continuous_map.to_bcf_injective ZeroAtInftyContinuousMap.toBCF_injective
 -/
 
 end
@@ -489,13 +489,13 @@ variable {C : ℝ} {f g : C₀(α, β)}
 /-- The type of continuous functions vanishing at infinity, with the uniform distance induced by the
 inclusion `zero_at_infinity_continuous_map.to_bcf`, is a metric space. -/
 noncomputable instance : MetricSpace C₀(α, β) :=
-  MetricSpace.induced _ (toBcf_injective α β) (by infer_instance)
+  MetricSpace.induced _ (toBCF_injective α β) (by infer_instance)
 
-#print ZeroAtInftyContinuousMap.dist_toBcf_eq_dist /-
+#print ZeroAtInftyContinuousMap.dist_toBCF_eq_dist /-
 @[simp]
-theorem dist_toBcf_eq_dist {f g : C₀(α, β)} : dist f.toBcf g.toBcf = dist f g :=
+theorem dist_toBCF_eq_dist {f g : C₀(α, β)} : dist f.toBCF g.toBCF = dist f g :=
   rfl
-#align zero_at_infty_continuous_map.dist_to_bcf_eq_dist ZeroAtInftyContinuousMap.dist_toBcf_eq_dist
+#align zero_at_infty_continuous_map.dist_to_bcf_eq_dist ZeroAtInftyContinuousMap.dist_toBCF_eq_dist
 -/
 
 open BoundedContinuousFunction
@@ -505,18 +505,18 @@ open BoundedContinuousFunction
 theorem tendsto_iff_tendstoUniformly {ι : Type _} {F : ι → C₀(α, β)} {f : C₀(α, β)} {l : Filter ι} :
     Tendsto F l (𝓝 f) ↔ TendstoUniformly (fun i => F i) f l := by
   simpa only [Metric.tendsto_nhds] using
-    @BoundedContinuousFunction.tendsto_iff_tendstoUniformly _ _ _ _ _ (fun i => (F i).toBcf)
+    @BoundedContinuousFunction.tendsto_iff_tendstoUniformly _ _ _ _ _ (fun i => (F i).toBCF)
       f.to_bcf l
 #align zero_at_infty_continuous_map.tendsto_iff_tendsto_uniformly ZeroAtInftyContinuousMap.tendsto_iff_tendstoUniformly
 -/
 
-#print ZeroAtInftyContinuousMap.isometry_toBcf /-
-theorem isometry_toBcf : Isometry (toBcf : C₀(α, β) → α →ᵇ β) := by tauto
-#align zero_at_infty_continuous_map.isometry_to_bcf ZeroAtInftyContinuousMap.isometry_toBcf
+#print ZeroAtInftyContinuousMap.isometry_toBCF /-
+theorem isometry_toBCF : Isometry (toBCF : C₀(α, β) → α →ᵇ β) := by tauto
+#align zero_at_infty_continuous_map.isometry_to_bcf ZeroAtInftyContinuousMap.isometry_toBCF
 -/
 
-#print ZeroAtInftyContinuousMap.closed_range_toBcf /-
-theorem closed_range_toBcf : IsClosed (range (toBcf : C₀(α, β) → α →ᵇ β)) :=
+#print ZeroAtInftyContinuousMap.closed_range_toBCF /-
+theorem closed_range_toBCF : IsClosed (range (toBCF : C₀(α, β) → α →ᵇ β)) :=
   by
   refine' is_closed_iff_cluster_pt.mpr fun f hf => _
   rw [clusterPt_principal_iff] at hf 
@@ -532,14 +532,14 @@ theorem closed_range_toBcf : IsClosed (range (toBcf : C₀(α, β) → α →ᵇ
       _ < dist g.to_bcf f + ε / 2 := (add_lt_add_of_le_of_lt (dist_coe_le_dist x) hx)
       _ < ε := by simpa [add_halves ε] using add_lt_add_right hg (ε / 2)
   exact ⟨⟨f.to_continuous_map, this⟩, by ext; rfl⟩
-#align zero_at_infty_continuous_map.closed_range_to_bcf ZeroAtInftyContinuousMap.closed_range_toBcf
+#align zero_at_infty_continuous_map.closed_range_to_bcf ZeroAtInftyContinuousMap.closed_range_toBCF
 -/
 
 /-- Continuous functions vanishing at infinity taking values in a complete space form a
 complete space. -/
 instance [CompleteSpace β] : CompleteSpace C₀(α, β) :=
-  (completeSpace_iff_isComplete_range isometry_toBcf.UniformInducing).mpr
-    closed_range_toBcf.IsComplete
+  (completeSpace_iff_isComplete_range isometry_toBCF.UniformInducing).mpr
+    closed_range_toBCF.IsComplete
 
 end Metric
 
@@ -558,17 +558,17 @@ section NormedSpace
 variable [NormedAddCommGroup β] {𝕜 : Type _} [NormedField 𝕜] [NormedSpace 𝕜 β]
 
 noncomputable instance : NormedAddCommGroup C₀(α, β) :=
-  NormedAddCommGroup.induced C₀(α, β) (α →ᵇ β) (⟨toBcf, rfl, fun x y => rfl⟩ : C₀(α, β) →+ α →ᵇ β)
-    (toBcf_injective α β)
+  NormedAddCommGroup.induced C₀(α, β) (α →ᵇ β) (⟨toBCF, rfl, fun x y => rfl⟩ : C₀(α, β) →+ α →ᵇ β)
+    (toBCF_injective α β)
 
-#print ZeroAtInftyContinuousMap.norm_toBcf_eq_norm /-
+#print ZeroAtInftyContinuousMap.norm_toBCF_eq_norm /-
 @[simp]
-theorem norm_toBcf_eq_norm {f : C₀(α, β)} : ‖f.toBcf‖ = ‖f‖ :=
+theorem norm_toBCF_eq_norm {f : C₀(α, β)} : ‖f.toBCF‖ = ‖f‖ :=
   rfl
-#align zero_at_infty_continuous_map.norm_to_bcf_eq_norm ZeroAtInftyContinuousMap.norm_toBcf_eq_norm
+#align zero_at_infty_continuous_map.norm_to_bcf_eq_norm ZeroAtInftyContinuousMap.norm_toBCF_eq_norm
 -/
 
-instance : NormedSpace 𝕜 C₀(α, β) where norm_smul_le k f := (norm_smul_le k f.toBcf : _)
+instance : NormedSpace 𝕜 C₀(α, β) where norm_smul_le k f := (norm_smul_le k f.toBCF : _)
 
 end NormedSpace
 
@@ -578,7 +578,7 @@ variable [NonUnitalNormedRing β]
 
 noncomputable instance : NonUnitalNormedRing C₀(α, β) :=
   { ZeroAtInftyContinuousMap.nonUnitalRing, ZeroAtInftyContinuousMap.normedAddCommGroup with
-    norm_hMul := fun f g => norm_mul_le f.toBcf g.toBcf }
+    norm_hMul := fun f g => norm_mul_le f.toBCF g.toBCF }
 
 end NormedRing
 
@@ -630,7 +630,7 @@ section NormedStar
 
 variable [NormedAddCommGroup β] [StarAddMonoid β] [NormedStarGroup β]
 
-instance : NormedStarGroup C₀(α, β) where norm_star f := (norm_star f.toBcf : _)
+instance : NormedStarGroup C₀(α, β) where norm_star f := (norm_star f.toBCF : _)
 
 end NormedStar
 
@@ -657,7 +657,7 @@ end StarRing
 section CstarRing
 
 instance [NonUnitalNormedRing β] [StarRing β] [CstarRing β] : CstarRing C₀(α, β)
-    where norm_star_hMul_self f := @CstarRing.norm_star_hMul_self _ _ _ _ f.toBcf
+    where norm_star_hMul_self f := @CstarRing.norm_star_hMul_self _ _ _ _ f.toBCF
 
 end CstarRing
 

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

@@ -3,8 +3,8 @@ Copyright (c) 2022 Jireh Loreaux. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Jireh Loreaux
 -/
-import Mathbin.Topology.ContinuousFunction.Bounded
-import Mathbin.Topology.ContinuousFunction.CocompactMap
+import Topology.ContinuousFunction.Bounded
+import Topology.ContinuousFunction.CocompactMap
 
 #align_import topology.continuous_function.zero_at_infty from "leanprover-community/mathlib"@"2a0ce625dbb0ffbc7d1316597de0b25c1ec75303"
 

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

@@ -434,7 +434,7 @@ protected theorem bounded (f : F) : ∃ C, ∀ x y : α, dist ((f : α → β) x
   obtain ⟨K : Set α, hK₁, hK₂⟩ :=
     mem_cocompact.mp
       (tendsto_def.mp (zero_at_infty (f : F)) _ (closed_ball_mem_nhds (0 : β) zero_lt_one))
-  obtain ⟨C, hC⟩ := (hK₁.image (map_continuous f)).Bounded.subset_ball (0 : β)
+  obtain ⟨C, hC⟩ := (hK₁.image (map_continuous f)).Bounded.subset_closedBall (0 : β)
   refine' ⟨max C 1 + max C 1, fun x y => _⟩
   have : ∀ x, f x ∈ closed_ball (0 : β) (max C 1) :=
     by
@@ -448,16 +448,16 @@ protected theorem bounded (f : F) : ∃ C, ∀ x y : α, dist ((f : α → β) x
 #align zero_at_infty_continuous_map.bounded ZeroAtInftyContinuousMap.bounded
 -/
 
-#print ZeroAtInftyContinuousMap.bounded_range /-
-theorem bounded_range (f : C₀(α, β)) : Bounded (range f) :=
-  bounded_range_iff.2 f.Bounded
-#align zero_at_infty_continuous_map.bounded_range ZeroAtInftyContinuousMap.bounded_range
+#print ZeroAtInftyContinuousMap.isBounded_range /-
+theorem isBounded_range (f : C₀(α, β)) : Bounded (range f) :=
+  isBounded_range_iff.2 f.Bounded
+#align zero_at_infty_continuous_map.bounded_range ZeroAtInftyContinuousMap.isBounded_range
 -/
 
-#print ZeroAtInftyContinuousMap.bounded_image /-
-theorem bounded_image (f : C₀(α, β)) (s : Set α) : Bounded (f '' s) :=
-  f.bounded_range.mono <| image_subset_range _ _
-#align zero_at_infty_continuous_map.bounded_image ZeroAtInftyContinuousMap.bounded_image
+#print ZeroAtInftyContinuousMap.isBounded_image /-
+theorem isBounded_image (f : C₀(α, β)) (s : Set α) : Bounded (f '' s) :=
+  f.isBounded_range.mono <| image_subset_range _ _
+#align zero_at_infty_continuous_map.bounded_image ZeroAtInftyContinuousMap.isBounded_image
 -/
 
 instance (priority := 100) : BoundedContinuousMapClass F α β :=

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

@@ -578,7 +578,7 @@ variable [NonUnitalNormedRing β]
 
 noncomputable instance : NonUnitalNormedRing C₀(α, β) :=
   { ZeroAtInftyContinuousMap.nonUnitalRing, ZeroAtInftyContinuousMap.normedAddCommGroup with
-    norm_mul := fun f g => norm_mul_le f.toBcf g.toBcf }
+    norm_hMul := fun f g => norm_mul_le f.toBcf g.toBcf }
 
 end NormedRing
 
@@ -650,14 +650,14 @@ variable [NonUnitalSemiring β] [StarRing β] [TopologicalSpace β] [ContinuousS
 
 instance : StarRing C₀(α, β) :=
   { ZeroAtInftyContinuousMap.starAddMonoid with
-    star_mul := fun f g => ext fun x => star_mul (f x) (g x) }
+    star_hMul := fun f g => ext fun x => star_hMul (f x) (g x) }
 
 end StarRing
 
 section CstarRing
 
 instance [NonUnitalNormedRing β] [StarRing β] [CstarRing β] : CstarRing C₀(α, β)
-    where norm_star_mul_self f := @CstarRing.norm_star_mul_self _ _ _ _ f.toBcf
+    where norm_star_hMul_self f := @CstarRing.norm_star_hMul_self _ _ _ _ f.toBcf
 
 end CstarRing
 

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

@@ -2,15 +2,12 @@
 Copyright (c) 2022 Jireh Loreaux. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Jireh Loreaux
-
-! This file was ported from Lean 3 source module topology.continuous_function.zero_at_infty
-! leanprover-community/mathlib commit 2a0ce625dbb0ffbc7d1316597de0b25c1ec75303
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
 -/
 import Mathbin.Topology.ContinuousFunction.Bounded
 import Mathbin.Topology.ContinuousFunction.CocompactMap
 
+#align_import topology.continuous_function.zero_at_infty from "leanprover-community/mathlib"@"2a0ce625dbb0ffbc7d1316597de0b25c1ec75303"
+
 /-!
 # Continuous functions vanishing at infinity
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/2a0ce625dbb0ffbc7d1316597de0b25c1ec75303

@@ -38,6 +38,7 @@ open scoped BoundedContinuousFunction Topology
 
 open Filter Metric
 
+#print ZeroAtInftyContinuousMap /-
 /-- `C₀(α, β)` is the type of continuous functions `α → β` which vanish at infinity from a
 topological space to a metric space with a zero element.
 
@@ -49,6 +50,7 @@ structure ZeroAtInftyContinuousMap (α : Type u) (β : Type v) [TopologicalSpace
     [TopologicalSpace β] extends ContinuousMap α β : Type max u v where
   zero_at_infty' : Tendsto to_fun (cocompact α) (𝓝 0)
 #align zero_at_infty_continuous_map ZeroAtInftyContinuousMap
+-/
 
 scoped[ZeroAtInfty] notation (priority := 2000) "C₀(" α ", " β ")" => ZeroAtInftyContinuousMap α β
 
@@ -56,6 +58,7 @@ scoped[ZeroAtInfty] notation α " →C₀ " β => ZeroAtInftyContinuousMap α β
 
 section
 
+#print ZeroAtInftyContinuousMapClass /-
 /-- `zero_at_infty_continuous_map_class F α β` states that `F` is a type of continuous maps which
 vanish at infinity.
 
@@ -64,6 +67,7 @@ class ZeroAtInftyContinuousMapClass (F : Type _) (α β : outParam <| Type _) [T
     [Zero β] [TopologicalSpace β] extends ContinuousMapClass F α β where
   zero_at_infty (f : F) : Tendsto f (cocompact α) (𝓝 0)
 #align zero_at_infty_continuous_map_class ZeroAtInftyContinuousMapClass
+-/
 
 end
 
@@ -93,16 +97,21 @@ instance : CoeTC F C₀(α, β) :=
       continuous_toFun := map_continuous f
       zero_at_infty' := zero_at_infty f }⟩
 
+#print ZeroAtInftyContinuousMap.coe_toContinuousMap /-
 @[simp]
-theorem coe_to_continuous_fun (f : C₀(α, β)) : (f.toContinuousMap : α → β) = f :=
+theorem coe_toContinuousMap (f : C₀(α, β)) : (f.toContinuousMap : α → β) = f :=
   rfl
-#align zero_at_infty_continuous_map.coe_to_continuous_fun ZeroAtInftyContinuousMap.coe_to_continuous_fun
+#align zero_at_infty_continuous_map.coe_to_continuous_fun ZeroAtInftyContinuousMap.coe_toContinuousMap
+-/
 
+#print ZeroAtInftyContinuousMap.ext /-
 @[ext]
 theorem ext {f g : C₀(α, β)} (h : ∀ x, f x = g x) : f = g :=
   FunLike.ext _ _ h
 #align zero_at_infty_continuous_map.ext ZeroAtInftyContinuousMap.ext
+-/
 
+#print ZeroAtInftyContinuousMap.copy /-
 /-- Copy of a `zero_at_infinity_continuous_map` with a new `to_fun` equal to the old one. Useful
 to fix definitional equalities. -/
 protected def copy (f : C₀(α, β)) (f' : α → β) (h : f' = f) : C₀(α, β)
@@ -111,20 +120,28 @@ protected def copy (f : C₀(α, β)) (f' : α → β) (h : f' = f) : C₀(α, 
   continuous_toFun := by rw [h]; exact f.continuous_to_fun
   zero_at_infty' := by simp_rw [h]; exact f.zero_at_infty'
 #align zero_at_infty_continuous_map.copy ZeroAtInftyContinuousMap.copy
+-/
 
+#print ZeroAtInftyContinuousMap.coe_copy /-
 @[simp]
 theorem coe_copy (f : C₀(α, β)) (f' : α → β) (h : f' = f) : ⇑(f.copy f' h) = f' :=
   rfl
 #align zero_at_infty_continuous_map.coe_copy ZeroAtInftyContinuousMap.coe_copy
+-/
 
+#print ZeroAtInftyContinuousMap.copy_eq /-
 theorem copy_eq (f : C₀(α, β)) (f' : α → β) (h : f' = f) : f.copy f' h = f :=
   FunLike.ext' h
 #align zero_at_infty_continuous_map.copy_eq ZeroAtInftyContinuousMap.copy_eq
+-/
 
+#print ZeroAtInftyContinuousMap.eq_of_empty /-
 theorem eq_of_empty [IsEmpty α] (f g : C₀(α, β)) : f = g :=
   ext <| IsEmpty.elim ‹_›
 #align zero_at_infty_continuous_map.eq_of_empty ZeroAtInftyContinuousMap.eq_of_empty
+-/
 
+#print ZeroAtInftyContinuousMap.ContinuousMap.liftZeroAtInfty /-
 /-- A continuous function on a compact space is automatically a continuous function vanishing at
 infinity. -/
 @[simps]
@@ -138,7 +155,9 @@ def ContinuousMap.liftZeroAtInfty [CompactSpace α] : C(α, β) ≃ C₀(α, β)
   left_inv f := by ext; rfl
   right_inv f := by ext; rfl
 #align zero_at_infty_continuous_map.continuous_map.lift_zero_at_infty ZeroAtInftyContinuousMap.ContinuousMap.liftZeroAtInfty
+-/
 
+#print ZeroAtInftyContinuousMap.zeroAtInftyContinuousMapClass.ofCompact /-
 /-- A continuous function on a compact space is automatically a continuous function vanishing at
 infinity. This is not an instance to avoid type class loops. -/
 @[simps]
@@ -150,6 +169,7 @@ def zeroAtInftyContinuousMapClass.ofCompact {G : Type _} [ContinuousMapClass G 
   map_continuous := map_continuous
   zero_at_infty := by simp
 #align zero_at_infty_continuous_map.zero_at_infty_continuous_map_class.of_compact ZeroAtInftyContinuousMap.zeroAtInftyContinuousMapClass.ofCompact
+-/
 
 end Basics
 
@@ -171,27 +191,35 @@ instance [Zero β] : Zero C₀(α, β) :=
 instance [Zero β] : Inhabited C₀(α, β) :=
   ⟨0⟩
 
+#print ZeroAtInftyContinuousMap.coe_zero /-
 @[simp]
 theorem coe_zero [Zero β] : ⇑(0 : C₀(α, β)) = 0 :=
   rfl
 #align zero_at_infty_continuous_map.coe_zero ZeroAtInftyContinuousMap.coe_zero
+-/
 
+#print ZeroAtInftyContinuousMap.zero_apply /-
 theorem zero_apply [Zero β] : (0 : C₀(α, β)) x = 0 :=
   rfl
 #align zero_at_infty_continuous_map.zero_apply ZeroAtInftyContinuousMap.zero_apply
+-/
 
 instance [MulZeroClass β] [ContinuousMul β] : Mul C₀(α, β) :=
   ⟨fun f g =>
     ⟨f * g, by simpa only [MulZeroClass.mul_zero] using (zero_at_infty f).mul (zero_at_infty g)⟩⟩
 
+#print ZeroAtInftyContinuousMap.coe_mul /-
 @[simp]
 theorem coe_mul [MulZeroClass β] [ContinuousMul β] (f g : C₀(α, β)) : ⇑(f * g) = f * g :=
   rfl
 #align zero_at_infty_continuous_map.coe_mul ZeroAtInftyContinuousMap.coe_mul
+-/
 
+#print ZeroAtInftyContinuousMap.mul_apply /-
 theorem mul_apply [MulZeroClass β] [ContinuousMul β] (f g : C₀(α, β)) : (f * g) x = f x * g x :=
   rfl
 #align zero_at_infty_continuous_map.mul_apply ZeroAtInftyContinuousMap.mul_apply
+-/
 
 instance [MulZeroClass β] [ContinuousMul β] : MulZeroClass C₀(α, β) :=
   FunLike.coe_injective.MulZeroClass _ coe_zero coe_mul
@@ -202,14 +230,18 @@ instance [SemigroupWithZero β] [ContinuousMul β] : SemigroupWithZero C₀(α,
 instance [AddZeroClass β] [ContinuousAdd β] : Add C₀(α, β) :=
   ⟨fun f g => ⟨f + g, by simpa only [add_zero] using (zero_at_infty f).add (zero_at_infty g)⟩⟩
 
+#print ZeroAtInftyContinuousMap.coe_add /-
 @[simp]
 theorem coe_add [AddZeroClass β] [ContinuousAdd β] (f g : C₀(α, β)) : ⇑(f + g) = f + g :=
   rfl
 #align zero_at_infty_continuous_map.coe_add ZeroAtInftyContinuousMap.coe_add
+-/
 
+#print ZeroAtInftyContinuousMap.add_apply /-
 theorem add_apply [AddZeroClass β] [ContinuousAdd β] (f g : C₀(α, β)) : (f + g) x = f x + g x :=
   rfl
 #align zero_at_infty_continuous_map.add_apply ZeroAtInftyContinuousMap.add_apply
+-/
 
 instance [AddZeroClass β] [ContinuousAdd β] : AddZeroClass C₀(α, β) :=
   FunLike.coe_injective.AddZeroClass _ coe_zero coe_add
@@ -218,15 +250,19 @@ section AddMonoid
 
 variable [AddMonoid β] [ContinuousAdd β] (f g : C₀(α, β))
 
+#print ZeroAtInftyContinuousMap.coe_nsmulRec /-
 @[simp]
 theorem coe_nsmulRec : ∀ n, ⇑(nsmulRec n f) = n • f
   | 0 => by rw [nsmulRec, zero_smul, coe_zero]
   | n + 1 => by rw [nsmulRec, succ_nsmul, coe_add, coe_nsmul_rec]
 #align zero_at_infty_continuous_map.coe_nsmul_rec ZeroAtInftyContinuousMap.coe_nsmulRec
+-/
 
-instance hasNatScalar : SMul ℕ C₀(α, β) :=
+#print ZeroAtInftyContinuousMap.instNatSMul /-
+instance instNatSMul : SMul ℕ C₀(α, β) :=
   ⟨fun n f => ⟨n • f, by simpa [coe_nsmul_rec] using zero_at_infty (nsmulRec n f)⟩⟩
-#align zero_at_infty_continuous_map.has_nat_scalar ZeroAtInftyContinuousMap.hasNatScalar
+#align zero_at_infty_continuous_map.has_nat_scalar ZeroAtInftyContinuousMap.instNatSMul
+-/
 
 instance : AddMonoid C₀(α, β) :=
   FunLike.coe_injective.AddMonoid _ coe_zero coe_add fun _ _ => rfl
@@ -243,36 +279,48 @@ variable [AddGroup β] [TopologicalAddGroup β] (f g : C₀(α, β))
 instance : Neg C₀(α, β) :=
   ⟨fun f => ⟨-f, by simpa only [neg_zero] using (zero_at_infty f).neg⟩⟩
 
+#print ZeroAtInftyContinuousMap.coe_neg /-
 @[simp]
 theorem coe_neg : ⇑(-f) = -f :=
   rfl
 #align zero_at_infty_continuous_map.coe_neg ZeroAtInftyContinuousMap.coe_neg
+-/
 
+#print ZeroAtInftyContinuousMap.neg_apply /-
 theorem neg_apply : (-f) x = -f x :=
   rfl
 #align zero_at_infty_continuous_map.neg_apply ZeroAtInftyContinuousMap.neg_apply
+-/
 
 instance : Sub C₀(α, β) :=
   ⟨fun f g => ⟨f - g, by simpa only [sub_zero] using (zero_at_infty f).sub (zero_at_infty g)⟩⟩
 
+#print ZeroAtInftyContinuousMap.coe_sub /-
 @[simp]
 theorem coe_sub : ⇑(f - g) = f - g :=
   rfl
 #align zero_at_infty_continuous_map.coe_sub ZeroAtInftyContinuousMap.coe_sub
+-/
 
+#print ZeroAtInftyContinuousMap.sub_apply /-
 theorem sub_apply : (f - g) x = f x - g x :=
   rfl
 #align zero_at_infty_continuous_map.sub_apply ZeroAtInftyContinuousMap.sub_apply
+-/
 
+#print ZeroAtInftyContinuousMap.coe_zsmulRec /-
 @[simp]
 theorem coe_zsmulRec : ∀ z, ⇑(zsmulRec z f) = z • f
   | Int.ofNat n => by rw [zsmulRec, Int.ofNat_eq_coe, coe_nsmul_rec, coe_nat_zsmul]
   | -[n+1] => by rw [zsmulRec, negSucc_zsmul, coe_neg, coe_nsmul_rec]
 #align zero_at_infty_continuous_map.coe_zsmul_rec ZeroAtInftyContinuousMap.coe_zsmulRec
+-/
 
-instance hasIntScalar : SMul ℤ C₀(α, β) :=
+#print ZeroAtInftyContinuousMap.instIntSMul /-
+instance instIntSMul : SMul ℤ C₀(α, β) :=
   ⟨fun n f => ⟨n • f, by simpa using zero_at_infty (zsmulRec n f)⟩⟩
-#align zero_at_infty_continuous_map.has_int_scalar ZeroAtInftyContinuousMap.hasIntScalar
+#align zero_at_infty_continuous_map.has_int_scalar ZeroAtInftyContinuousMap.instIntSMul
+-/
 
 instance : AddGroup C₀(α, β) :=
   FunLike.coe_injective.AddGroup _ coe_zero coe_add coe_neg coe_sub (fun _ _ => rfl) fun _ _ => rfl
@@ -287,16 +335,20 @@ instance [Zero β] {R : Type _} [Zero R] [SMulWithZero R β] [ContinuousConstSMu
     SMul R C₀(α, β) :=
   ⟨fun r f => ⟨r • f, by simpa [smul_zero] using (zero_at_infty f).const_smul r⟩⟩
 
+#print ZeroAtInftyContinuousMap.coe_smul /-
 @[simp]
 theorem coe_smul [Zero β] {R : Type _} [Zero R] [SMulWithZero R β] [ContinuousConstSMul R β] (r : R)
     (f : C₀(α, β)) : ⇑(r • f) = r • f :=
   rfl
 #align zero_at_infty_continuous_map.coe_smul ZeroAtInftyContinuousMap.coe_smul
+-/
 
+#print ZeroAtInftyContinuousMap.smul_apply /-
 theorem smul_apply [Zero β] {R : Type _} [Zero R] [SMulWithZero R β] [ContinuousConstSMul R β]
     (r : R) (f : C₀(α, β)) (x : α) : (r • f) x = r • f x :=
   rfl
 #align zero_at_infty_continuous_map.smul_apply ZeroAtInftyContinuousMap.smul_apply
+-/
 
 instance [Zero β] {R : Type _} [Zero R] [SMulWithZero R β] [SMulWithZero Rᵐᵒᵖ β]
     [ContinuousConstSMul R β] [IsCentralScalar R β] : IsCentralScalar R C₀(α, β) :=
@@ -356,9 +408,11 @@ section Uniform
 
 variable [UniformSpace β] [UniformSpace γ] [Zero γ] [ZeroAtInftyContinuousMapClass F β γ]
 
+#print ZeroAtInftyContinuousMap.uniformContinuous /-
 theorem uniformContinuous (f : F) : UniformContinuous (f : β → γ) :=
   (map_continuous f).uniformContinuous_of_tendsto_cocompact (zero_at_infty f)
 #align zero_at_infty_continuous_map.uniform_continuous ZeroAtInftyContinuousMap.uniformContinuous
+-/
 
 end Uniform
 
@@ -377,6 +431,7 @@ open Metric Set
 
 variable [MetricSpace β] [Zero β] [ZeroAtInftyContinuousMapClass F α β]
 
+#print ZeroAtInftyContinuousMap.bounded /-
 protected theorem bounded (f : F) : ∃ C, ∀ x y : α, dist ((f : α → β) x) (f y) ≤ C :=
   by
   obtain ⟨K : Set α, hK₁, hK₂⟩ :=
@@ -394,32 +449,41 @@ protected theorem bounded (f : F) : ∃ C, ∀ x y : α, dist ((f : α → β) x
     (dist_triangle (f x) 0 (f y)).trans
       (add_le_add (mem_closed_ball.mp <| this x) (mem_closed_ball'.mp <| this y))
 #align zero_at_infty_continuous_map.bounded ZeroAtInftyContinuousMap.bounded
+-/
 
+#print ZeroAtInftyContinuousMap.bounded_range /-
 theorem bounded_range (f : C₀(α, β)) : Bounded (range f) :=
   bounded_range_iff.2 f.Bounded
 #align zero_at_infty_continuous_map.bounded_range ZeroAtInftyContinuousMap.bounded_range
+-/
 
+#print ZeroAtInftyContinuousMap.bounded_image /-
 theorem bounded_image (f : C₀(α, β)) (s : Set α) : Bounded (f '' s) :=
   f.bounded_range.mono <| image_subset_range _ _
 #align zero_at_infty_continuous_map.bounded_image ZeroAtInftyContinuousMap.bounded_image
+-/
 
 instance (priority := 100) : BoundedContinuousMapClass F α β :=
   { ‹ZeroAtInftyContinuousMapClass F α β› with
     map_bounded := fun f => ZeroAtInftyContinuousMap.bounded f }
 
+#print ZeroAtInftyContinuousMap.toBcf /-
 /-- Construct a bounded continuous function from a continuous function vanishing at infinity. -/
 @[simps]
 def toBcf (f : C₀(α, β)) : α →ᵇ β :=
   ⟨f, map_bounded f⟩
 #align zero_at_infty_continuous_map.to_bcf ZeroAtInftyContinuousMap.toBcf
+-/
 
 section
 
 variable (α) (β)
 
+#print ZeroAtInftyContinuousMap.toBcf_injective /-
 theorem toBcf_injective : Function.Injective (toBcf : C₀(α, β) → α →ᵇ β) := fun f g h => by ext;
   simpa only using FunLike.congr_fun h x
 #align zero_at_infty_continuous_map.to_bcf_injective ZeroAtInftyContinuousMap.toBcf_injective
+-/
 
 end
 
@@ -430,13 +494,16 @@ inclusion `zero_at_infinity_continuous_map.to_bcf`, is a metric space. -/
 noncomputable instance : MetricSpace C₀(α, β) :=
   MetricSpace.induced _ (toBcf_injective α β) (by infer_instance)
 
+#print ZeroAtInftyContinuousMap.dist_toBcf_eq_dist /-
 @[simp]
 theorem dist_toBcf_eq_dist {f g : C₀(α, β)} : dist f.toBcf g.toBcf = dist f g :=
   rfl
 #align zero_at_infty_continuous_map.dist_to_bcf_eq_dist ZeroAtInftyContinuousMap.dist_toBcf_eq_dist
+-/
 
 open BoundedContinuousFunction
 
+#print ZeroAtInftyContinuousMap.tendsto_iff_tendstoUniformly /-
 /-- Convergence in the metric on `C₀(α, β)` is uniform convergence. -/
 theorem tendsto_iff_tendstoUniformly {ι : Type _} {F : ι → C₀(α, β)} {f : C₀(α, β)} {l : Filter ι} :
     Tendsto F l (𝓝 f) ↔ TendstoUniformly (fun i => F i) f l := by
@@ -444,10 +511,14 @@ theorem tendsto_iff_tendstoUniformly {ι : Type _} {F : ι → C₀(α, β)} {f
     @BoundedContinuousFunction.tendsto_iff_tendstoUniformly _ _ _ _ _ (fun i => (F i).toBcf)
       f.to_bcf l
 #align zero_at_infty_continuous_map.tendsto_iff_tendsto_uniformly ZeroAtInftyContinuousMap.tendsto_iff_tendstoUniformly
+-/
 
+#print ZeroAtInftyContinuousMap.isometry_toBcf /-
 theorem isometry_toBcf : Isometry (toBcf : C₀(α, β) → α →ᵇ β) := by tauto
 #align zero_at_infty_continuous_map.isometry_to_bcf ZeroAtInftyContinuousMap.isometry_toBcf
+-/
 
+#print ZeroAtInftyContinuousMap.closed_range_toBcf /-
 theorem closed_range_toBcf : IsClosed (range (toBcf : C₀(α, β) → α →ᵇ β)) :=
   by
   refine' is_closed_iff_cluster_pt.mpr fun f hf => _
@@ -465,6 +536,7 @@ theorem closed_range_toBcf : IsClosed (range (toBcf : C₀(α, β) → α →ᵇ
       _ < ε := by simpa [add_halves ε] using add_lt_add_right hg (ε / 2)
   exact ⟨⟨f.to_continuous_map, this⟩, by ext; rfl⟩
 #align zero_at_infty_continuous_map.closed_range_to_bcf ZeroAtInftyContinuousMap.closed_range_toBcf
+-/
 
 /-- Continuous functions vanishing at infinity taking values in a complete space form a
 complete space. -/
@@ -492,10 +564,12 @@ noncomputable instance : NormedAddCommGroup C₀(α, β) :=
   NormedAddCommGroup.induced C₀(α, β) (α →ᵇ β) (⟨toBcf, rfl, fun x y => rfl⟩ : C₀(α, β) →+ α →ᵇ β)
     (toBcf_injective α β)
 
+#print ZeroAtInftyContinuousMap.norm_toBcf_eq_norm /-
 @[simp]
 theorem norm_toBcf_eq_norm {f : C₀(α, β)} : ‖f.toBcf‖ = ‖f‖ :=
   rfl
 #align zero_at_infty_continuous_map.norm_to_bcf_eq_norm ZeroAtInftyContinuousMap.norm_toBcf_eq_norm
+-/
 
 instance : NormedSpace 𝕜 C₀(α, β) where norm_smul_le k f := (norm_smul_le k f.toBcf : _)
 
@@ -535,14 +609,18 @@ instance : Star C₀(α, β)
       zero_at_infty' := by
         simpa only [star_zero] using (continuous_star.tendsto (0 : β)).comp (zero_at_infty f) }
 
+#print ZeroAtInftyContinuousMap.coe_star /-
 @[simp]
 theorem coe_star (f : C₀(α, β)) : ⇑(star f) = star f :=
   rfl
 #align zero_at_infty_continuous_map.coe_star ZeroAtInftyContinuousMap.coe_star
+-/
 
+#print ZeroAtInftyContinuousMap.star_apply /-
 theorem star_apply (f : C₀(α, β)) (x : α) : (star f) x = star (f x) :=
   rfl
 #align zero_at_infty_continuous_map.star_apply ZeroAtInftyContinuousMap.star_apply
+-/
 
 instance [ContinuousAdd β] : StarAddMonoid C₀(α, β)
     where
@@ -601,6 +679,7 @@ section
 
 variable [Zero δ]
 
+#print ZeroAtInftyContinuousMap.comp /-
 /-- Composition of a continuous function vanishing at infinity with a cocompact map yields another
 continuous function vanishing at infinity. -/
 def comp (f : C₀(γ, δ)) (g : β →co γ) : C₀(β, δ)
@@ -608,31 +687,41 @@ def comp (f : C₀(γ, δ)) (g : β →co γ) : C₀(β, δ)
   toContinuousMap := (f : C(γ, δ)).comp g
   zero_at_infty' := (zero_at_infty f).comp (cocompact_tendsto g)
 #align zero_at_infty_continuous_map.comp ZeroAtInftyContinuousMap.comp
+-/
 
+#print ZeroAtInftyContinuousMap.coe_comp_to_continuous_fun /-
 @[simp]
 theorem coe_comp_to_continuous_fun (f : C₀(γ, δ)) (g : β →co γ) :
     ((f.comp g).toContinuousMap : β → δ) = f ∘ g :=
   rfl
 #align zero_at_infty_continuous_map.coe_comp_to_continuous_fun ZeroAtInftyContinuousMap.coe_comp_to_continuous_fun
+-/
 
+#print ZeroAtInftyContinuousMap.comp_id /-
 @[simp]
 theorem comp_id (f : C₀(γ, δ)) : f.comp (CocompactMap.id γ) = f :=
   ext fun x => rfl
 #align zero_at_infty_continuous_map.comp_id ZeroAtInftyContinuousMap.comp_id
+-/
 
+#print ZeroAtInftyContinuousMap.comp_assoc /-
 @[simp]
 theorem comp_assoc (f : C₀(γ, δ)) (g : β →co γ) (h : α →co β) :
     (f.comp g).comp h = f.comp (g.comp h) :=
   rfl
 #align zero_at_infty_continuous_map.comp_assoc ZeroAtInftyContinuousMap.comp_assoc
+-/
 
+#print ZeroAtInftyContinuousMap.zero_comp /-
 @[simp]
 theorem zero_comp (g : β →co γ) : (0 : C₀(γ, δ)).comp g = 0 :=
   rfl
 #align zero_at_infty_continuous_map.zero_comp ZeroAtInftyContinuousMap.zero_comp
+-/
 
 end
 
+#print ZeroAtInftyContinuousMap.compAddMonoidHom /-
 /-- Composition as an additive monoid homomorphism. -/
 def compAddMonoidHom [AddMonoid δ] [ContinuousAdd δ] (g : β →co γ) : C₀(γ, δ) →+ C₀(β, δ)
     where
@@ -640,14 +729,18 @@ def compAddMonoidHom [AddMonoid δ] [ContinuousAdd δ] (g : β →co γ) : C₀(
   map_zero' := zero_comp g
   map_add' f₁ f₂ := rfl
 #align zero_at_infty_continuous_map.comp_add_monoid_hom ZeroAtInftyContinuousMap.compAddMonoidHom
+-/
 
+#print ZeroAtInftyContinuousMap.compMulHom /-
 /-- Composition as a semigroup homomorphism. -/
 def compMulHom [MulZeroClass δ] [ContinuousMul δ] (g : β →co γ) : C₀(γ, δ) →ₙ* C₀(β, δ)
     where
   toFun f := f.comp g
   map_mul' f₁ f₂ := rfl
 #align zero_at_infty_continuous_map.comp_mul_hom ZeroAtInftyContinuousMap.compMulHom
+-/
 
+#print ZeroAtInftyContinuousMap.compLinearMap /-
 /-- Composition as a linear map. -/
 def compLinearMap [AddCommMonoid δ] [ContinuousAdd δ] {R : Type _} [Semiring R] [Module R δ]
     [ContinuousConstSMul R δ] (g : β →co γ) : C₀(γ, δ) →ₗ[R] C₀(β, δ)
@@ -656,7 +749,9 @@ def compLinearMap [AddCommMonoid δ] [ContinuousAdd δ] {R : Type _} [Semiring R
   map_add' f₁ f₂ := rfl
   map_smul' r f := rfl
 #align zero_at_infty_continuous_map.comp_linear_map ZeroAtInftyContinuousMap.compLinearMap
+-/
 
+#print ZeroAtInftyContinuousMap.compNonUnitalAlgHom /-
 /-- Composition as a non-unital algebra homomorphism. -/
 def compNonUnitalAlgHom {R : Type _} [Semiring R] [NonUnitalNonAssocSemiring δ]
     [TopologicalSemiring δ] [Module R δ] [ContinuousConstSMul R δ] (g : β →co γ) :
@@ -667,6 +762,7 @@ def compNonUnitalAlgHom {R : Type _} [Semiring R] [NonUnitalNonAssocSemiring δ]
   map_add' f₁ f₂ := rfl
   map_mul' f₁ f₂ := rfl
 #align zero_at_infty_continuous_map.comp_non_unital_alg_hom ZeroAtInftyContinuousMap.compNonUnitalAlgHom
+-/
 
 end ZeroAtInftyContinuousMap
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/2a0ce625dbb0ffbc7d1316597de0b25c1ec75303

@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Jireh Loreaux
 
 ! This file was ported from Lean 3 source module topology.continuous_function.zero_at_infty
-! leanprover-community/mathlib commit ba5ff5ad5d120fb0ef094ad2994967e9bfaf5112
+! leanprover-community/mathlib commit 2a0ce625dbb0ffbc7d1316597de0b25c1ec75303
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
@@ -14,6 +14,9 @@ import Mathbin.Topology.ContinuousFunction.CocompactMap
 /-!
 # Continuous functions vanishing at infinity
 
+> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.
+> Any changes to this file require a corresponding PR to mathlib4.
+
 The type of continuous functions vanishing at infinity. When the domain is compact
 `C(α, β) ≃ C₀(α, β)` via the identity map. When the codomain is a metric space, every continuous
 map which vanishes at infinity is a bounded continuous function. When the domain is a locally

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

@@ -47,10 +47,8 @@ structure ZeroAtInftyContinuousMap (α : Type u) (β : Type v) [TopologicalSpace
   zero_at_infty' : Tendsto to_fun (cocompact α) (𝓝 0)
 #align zero_at_infty_continuous_map ZeroAtInftyContinuousMap
 
--- mathport name: zero_at_infty_continuous_map
 scoped[ZeroAtInfty] notation (priority := 2000) "C₀(" α ", " β ")" => ZeroAtInftyContinuousMap α β
 
--- mathport name: zero_at_infty_continuous_map.arrow
 scoped[ZeroAtInfty] notation α " →C₀ " β => ZeroAtInftyContinuousMap α β
 
 section
@@ -594,7 +592,6 @@ category of topological spaces with morphisms given by `cocompact_map`s.
 
 variable {δ : Type _} [TopologicalSpace β] [TopologicalSpace γ] [TopologicalSpace δ]
 
--- mathport name: «expr →co »
 local notation α " →co " β => CocompactMap α β
 
 section

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

@@ -462,7 +462,6 @@ theorem closed_range_toBcf : IsClosed (range (toBcf : C₀(α, β) → α →ᵇ
       dist (f x) 0 ≤ dist (g.to_bcf x) (f x) + dist (g x) 0 := dist_triangle_left _ _ _
       _ < dist g.to_bcf f + ε / 2 := (add_lt_add_of_le_of_lt (dist_coe_le_dist x) hx)
       _ < ε := by simpa [add_halves ε] using add_lt_add_right hg (ε / 2)
-      
   exact ⟨⟨f.to_continuous_map, this⟩, by ext; rfl⟩
 #align zero_at_infty_continuous_map.closed_range_to_bcf ZeroAtInftyContinuousMap.closed_range_toBcf
 

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

@@ -43,7 +43,7 @@ you should parametrize over `(F : Type*) [zero_at_infty_continuous_map_class F 
 
 When you extend this structure, make sure to extend `zero_at_infty_continuous_map_class`. -/
 structure ZeroAtInftyContinuousMap (α : Type u) (β : Type v) [TopologicalSpace α] [Zero β]
-  [TopologicalSpace β] extends ContinuousMap α β : Type max u v where
+    [TopologicalSpace β] extends ContinuousMap α β : Type max u v where
   zero_at_infty' : Tendsto to_fun (cocompact α) (𝓝 0)
 #align zero_at_infty_continuous_map ZeroAtInftyContinuousMap
 
@@ -60,7 +60,7 @@ vanish at infinity.
 
 You should also extend this typeclass when you extend `zero_at_infty_continuous_map`. -/
 class ZeroAtInftyContinuousMapClass (F : Type _) (α β : outParam <| Type _) [TopologicalSpace α]
-  [Zero β] [TopologicalSpace β] extends ContinuousMapClass F α β where
+    [Zero β] [TopologicalSpace β] extends ContinuousMapClass F α β where
   zero_at_infty (f : F) : Tendsto f (cocompact α) (𝓝 0)
 #align zero_at_infty_continuous_map_class ZeroAtInftyContinuousMapClass
 
@@ -450,7 +450,7 @@ theorem isometry_toBcf : Isometry (toBcf : C₀(α, β) → α →ᵇ β) := by
 theorem closed_range_toBcf : IsClosed (range (toBcf : C₀(α, β) → α →ᵇ β)) :=
   by
   refine' is_closed_iff_cluster_pt.mpr fun f hf => _
-  rw [clusterPt_principal_iff] at hf
+  rw [clusterPt_principal_iff] at hf 
   have : tendsto f (cocompact α) (𝓝 0) :=
     by
     refine' metric.tendsto_nhds.mpr fun ε hε => _

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

@@ -31,7 +31,7 @@ universe u v w
 
 variable {F : Type _} {α : Type u} {β : Type v} {γ : Type w} [TopologicalSpace α]
 
-open BoundedContinuousFunction Topology
+open scoped BoundedContinuousFunction Topology
 
 open Filter Metric
 

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

@@ -77,10 +77,7 @@ variable [TopologicalSpace β] [Zero β] [ZeroAtInftyContinuousMapClass F α β]
 instance : ZeroAtInftyContinuousMapClass C₀(α, β) α β
     where
   coe f := f.toFun
-  coe_injective' f g h := by
-    obtain ⟨⟨_, _⟩, _⟩ := f
-    obtain ⟨⟨_, _⟩, _⟩ := g
-    congr
+  coe_injective' f g h := by obtain ⟨⟨_, _⟩, _⟩ := f; obtain ⟨⟨_, _⟩, _⟩ := g; congr
   map_continuous f := f.continuous_toFun
   zero_at_infty f := f.zero_at_infty'
 
@@ -110,12 +107,8 @@ to fix definitional equalities. -/
 protected def copy (f : C₀(α, β)) (f' : α → β) (h : f' = f) : C₀(α, β)
     where
   toFun := f'
-  continuous_toFun := by
-    rw [h]
-    exact f.continuous_to_fun
-  zero_at_infty' := by
-    simp_rw [h]
-    exact f.zero_at_infty'
+  continuous_toFun := by rw [h]; exact f.continuous_to_fun
+  zero_at_infty' := by simp_rw [h]; exact f.zero_at_infty'
 #align zero_at_infty_continuous_map.copy ZeroAtInftyContinuousMap.copy
 
 @[simp]
@@ -141,12 +134,8 @@ def ContinuousMap.liftZeroAtInfty [CompactSpace α] : C(α, β) ≃ C₀(α, β)
       continuous_toFun := f.Continuous
       zero_at_infty' := by simp }
   invFun f := f
-  left_inv f := by
-    ext
-    rfl
-  right_inv f := by
-    ext
-    rfl
+  left_inv f := by ext; rfl
+  right_inv f := by ext; rfl
 #align zero_at_infty_continuous_map.continuous_map.lift_zero_at_infty ZeroAtInftyContinuousMap.ContinuousMap.liftZeroAtInfty
 
 /-- A continuous function on a compact space is automatically a continuous function vanishing at
@@ -427,9 +416,7 @@ section
 
 variable (α) (β)
 
-theorem toBcf_injective : Function.Injective (toBcf : C₀(α, β) → α →ᵇ β) := fun f g h =>
-  by
-  ext
+theorem toBcf_injective : Function.Injective (toBcf : C₀(α, β) → α →ᵇ β) := fun f g h => by ext;
   simpa only using FunLike.congr_fun h x
 #align zero_at_infty_continuous_map.to_bcf_injective ZeroAtInftyContinuousMap.toBcf_injective
 
@@ -476,10 +463,7 @@ theorem closed_range_toBcf : IsClosed (range (toBcf : C₀(α, β) → α →ᵇ
       _ < dist g.to_bcf f + ε / 2 := (add_lt_add_of_le_of_lt (dist_coe_le_dist x) hx)
       _ < ε := by simpa [add_halves ε] using add_lt_add_right hg (ε / 2)
       
-  exact
-    ⟨⟨f.to_continuous_map, this⟩, by
-      ext
-      rfl⟩
+  exact ⟨⟨f.to_continuous_map, this⟩, by ext; rfl⟩
 #align zero_at_infty_continuous_map.closed_range_to_bcf ZeroAtInftyContinuousMap.closed_range_toBcf
 
 /-- Continuous functions vanishing at infinity taking values in a complete space form a

mathlib commit https://github.com/leanprover-community/mathlib/commit/8d33f09cd7089ecf074b4791907588245aec5d1b

@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Jireh Loreaux
 
 ! This file was ported from Lean 3 source module topology.continuous_function.zero_at_infty
-! leanprover-community/mathlib commit d3af0609f6db8691dffdc3e1fb7feb7da72698f2
+! leanprover-community/mathlib commit ba5ff5ad5d120fb0ef094ad2994967e9bfaf5112
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
@@ -513,7 +513,7 @@ theorem norm_toBcf_eq_norm {f : C₀(α, β)} : ‖f.toBcf‖ = ‖f‖ :=
   rfl
 #align zero_at_infty_continuous_map.norm_to_bcf_eq_norm ZeroAtInftyContinuousMap.norm_toBcf_eq_norm
 
-instance : NormedSpace 𝕜 C₀(α, β) where norm_smul_le k f := norm_smul_le k f.toBcf
+instance : NormedSpace 𝕜 C₀(α, β) where norm_smul_le k f := (norm_smul_le k f.toBcf : _)
 
 end NormedSpace
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/55d771df074d0dd020139ee1cd4b95521422df9f

@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Jireh Loreaux
 
 ! This file was ported from Lean 3 source module topology.continuous_function.zero_at_infty
-! leanprover-community/mathlib commit 735b22f8f9ff9792cf4212d7cb051c4c994bc685
+! leanprover-community/mathlib commit d3af0609f6db8691dffdc3e1fb7feb7da72698f2
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
@@ -513,7 +513,7 @@ theorem norm_toBcf_eq_norm {f : C₀(α, β)} : ‖f.toBcf‖ = ‖f‖ :=
   rfl
 #align zero_at_infty_continuous_map.norm_to_bcf_eq_norm ZeroAtInftyContinuousMap.norm_toBcf_eq_norm
 
-instance : NormedSpace 𝕜 C₀(α, β) where norm_smul_le k f := (norm_smul k f.toBcf).le
+instance : NormedSpace 𝕜 C₀(α, β) where norm_smul_le k f := norm_smul_le k f.toBcf
 
 end NormedSpace
 

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

@@ -191,7 +191,8 @@ theorem zero_apply [Zero β] : (0 : C₀(α, β)) x = 0 :=
 #align zero_at_infty_continuous_map.zero_apply ZeroAtInftyContinuousMap.zero_apply
 
 instance [MulZeroClass β] [ContinuousMul β] : Mul C₀(α, β) :=
-  ⟨fun f g => ⟨f * g, by simpa only [mul_zero] using (zero_at_infty f).mul (zero_at_infty g)⟩⟩
+  ⟨fun f g =>
+    ⟨f * g, by simpa only [MulZeroClass.mul_zero] using (zero_at_infty f).mul (zero_at_infty g)⟩⟩
 
 @[simp]
 theorem coe_mul [MulZeroClass β] [ContinuousMul β] (f g : C₀(α, β)) : ⇑(f * g) = f * g :=

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

@@ -472,7 +472,7 @@ theorem closed_range_toBcf : IsClosed (range (toBcf : C₀(α, β) → α →ᵇ
         (eventually_of_forall fun x hx => _)
     calc
       dist (f x) 0 ≤ dist (g.to_bcf x) (f x) + dist (g x) 0 := dist_triangle_left _ _ _
-      _ < dist g.to_bcf f + ε / 2 := add_lt_add_of_le_of_lt (dist_coe_le_dist x) hx
+      _ < dist g.to_bcf f + ε / 2 := (add_lt_add_of_le_of_lt (dist_coe_le_dist x) hx)
       _ < ε := by simpa [add_halves ε] using add_lt_add_right hg (ε / 2)
       
   exact

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

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

@@ -456,7 +456,7 @@ theorem isClosed_range_toBCF : IsClosed (range (toBCF : C₀(α, β) → α →
       (eventually_of_forall fun x hx => _)
     calc
       dist (f x) 0 ≤ dist (g.toBCF x) (f x) + dist (g x) 0 := dist_triangle_left _ _ _
-      _ < dist g.toBCF f + ε / 2 := (add_lt_add_of_le_of_lt (dist_coe_le_dist x) hx)
+      _ < dist g.toBCF f + ε / 2 := add_lt_add_of_le_of_lt (dist_coe_le_dist x) hx
       _ < ε := by simpa [add_halves ε] using add_lt_add_right (mem_ball.1 hg) (ε / 2)
   exact ⟨⟨f.toContinuousMap, this⟩, rfl⟩
 #align zero_at_infty_continuous_map.closed_range_to_bcf ZeroAtInftyContinuousMap.isClosed_range_toBCF

It's annoying that zsmulRec uses nsmulRec to define zsmul even when the user already set nsmul explicitly. This PR changes zsmulRec to take nsmul as an argument.

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

@@ -216,20 +216,30 @@ theorem add_apply [AddZeroClass β] [ContinuousAdd β] (f g : C₀(α, β)) : (f
 instance instAddZeroClass [AddZeroClass β] [ContinuousAdd β] : AddZeroClass C₀(α, β) :=
   DFunLike.coe_injective.addZeroClass _ coe_zero coe_add
 
-section AddMonoid
+instance instSMul [Zero β] {R : Type*} [Zero R] [SMulWithZero R β] [ContinuousConstSMul R β] :
+    SMul R C₀(α, β) :=
+  -- Porting note: Original version didn't have `Continuous.const_smul f.continuous r`
+  ⟨fun r f => ⟨⟨r • ⇑f, Continuous.const_smul f.continuous r⟩,
+    by simpa [smul_zero] using (zero_at_infty f).const_smul r⟩⟩
+#align zero_at_infty_continuous_map.has_nat_scalar ZeroAtInftyContinuousMap.instSMul
+#align zero_at_infty_continuous_map.has_int_scalar ZeroAtInftyContinuousMap.instSMul
 
-variable [AddMonoid β] [ContinuousAdd β] (f g : C₀(α, β))
+@[simp, norm_cast]
+theorem coe_smul [Zero β] {R : Type*} [Zero R] [SMulWithZero R β] [ContinuousConstSMul R β] (r : R)
+    (f : C₀(α, β)) : ⇑(r • f) = r • ⇑f :=
+  rfl
+#align zero_at_infty_continuous_map.coe_smul ZeroAtInftyContinuousMap.coe_smul
+#align zero_at_infty_continuous_map.coe_nsmul_rec ZeroAtInftyContinuousMap.coe_smul
+#align zero_at_infty_continuous_map.coe_zsmul_rec ZeroAtInftyContinuousMap.coe_smul
 
-@[simp]
-theorem coe_nsmulRec : ∀ n, ⇑(nsmulRec n f) = n • ⇑f
-  | 0 => by rw [nsmulRec, zero_smul, coe_zero]
-  | n + 1 => by rw [nsmulRec, succ_nsmul, coe_add, coe_nsmulRec n]
-#align zero_at_infty_continuous_map.coe_nsmul_rec ZeroAtInftyContinuousMap.coe_nsmulRec
+theorem smul_apply [Zero β] {R : Type*} [Zero R] [SMulWithZero R β] [ContinuousConstSMul R β]
+    (r : R) (f : C₀(α, β)) (x : α) : (r • f) x = r • f x :=
+  rfl
+#align zero_at_infty_continuous_map.smul_apply ZeroAtInftyContinuousMap.smul_apply
 
-instance instNatSMul : SMul ℕ C₀(α, β) :=
-  ⟨fun n f => ⟨n • (f : C(α, β)),
-    by simpa [coe_nsmulRec] using zero_at_infty (nsmulRec n f)⟩⟩
-#align zero_at_infty_continuous_map.has_nat_scalar ZeroAtInftyContinuousMap.instNatSMul
+section AddMonoid
+
+variable [AddMonoid β] [ContinuousAdd β] (f g : C₀(α, β))
 
 instance instAddMonoid : AddMonoid C₀(α, β) :=
   DFunLike.coe_injective.addMonoid _ coe_zero coe_add fun _ _ => rfl
@@ -267,18 +277,6 @@ theorem sub_apply : (f - g) x = f x - g x :=
   rfl
 #align zero_at_infty_continuous_map.sub_apply ZeroAtInftyContinuousMap.sub_apply
 
-@[simp]
-theorem coe_zsmulRec : ∀ z, ⇑(zsmulRec z f) = z • ⇑f
-  | Int.ofNat n => by rw [zsmulRec, Int.ofNat_eq_coe, coe_nsmulRec, natCast_zsmul]
-  | Int.negSucc n => by rw [zsmulRec, negSucc_zsmul, coe_neg, coe_nsmulRec]
-#align zero_at_infty_continuous_map.coe_zsmul_rec ZeroAtInftyContinuousMap.coe_zsmulRec
-
-instance instIntSMul : SMul ℤ C₀(α, β) :=
-  -- Porting note: Original version didn't have `Continuous.const_smul f.continuous n`
-  ⟨fun n f => ⟨⟨n • ⇑f, Continuous.const_smul f.continuous n⟩,
-    by simpa using zero_at_infty (zsmulRec n f)⟩⟩
-#align zero_at_infty_continuous_map.has_int_scalar ZeroAtInftyContinuousMap.instIntSMul
-
 instance instAddGroup : AddGroup C₀(α, β) :=
   DFunLike.coe_injective.addGroup _ coe_zero coe_add coe_neg coe_sub (fun _ _ => rfl) fun _ _ => rfl
 
@@ -288,23 +286,6 @@ instance instAddCommGroup [AddCommGroup β] [TopologicalAddGroup β] : AddCommGr
   DFunLike.coe_injective.addCommGroup _ coe_zero coe_add coe_neg coe_sub (fun _ _ => rfl) fun _ _ =>
     rfl
 
-instance instSMul [Zero β] {R : Type*} [Zero R] [SMulWithZero R β] [ContinuousConstSMul R β] :
-    SMul R C₀(α, β) :=
-  -- Porting note: Original version didn't have `Continuous.const_smul f.continuous r`
-  ⟨fun r f => ⟨⟨r • ⇑f, Continuous.const_smul f.continuous r⟩,
-    by simpa [smul_zero] using (zero_at_infty f).const_smul r⟩⟩
-
-@[simp]
-theorem coe_smul [Zero β] {R : Type*} [Zero R] [SMulWithZero R β] [ContinuousConstSMul R β] (r : R)
-    (f : C₀(α, β)) : ⇑(r • f) = r • ⇑f :=
-  rfl
-#align zero_at_infty_continuous_map.coe_smul ZeroAtInftyContinuousMap.coe_smul
-
-theorem smul_apply [Zero β] {R : Type*} [Zero R] [SMulWithZero R β] [ContinuousConstSMul R β]
-    (r : R) (f : C₀(α, β)) (x : α) : (r • f) x = r • f x :=
-  rfl
-#align zero_at_infty_continuous_map.smul_apply ZeroAtInftyContinuousMap.smul_apply
-
 instance instIsCentralScalar [Zero β] {R : Type*} [Zero R] [SMulWithZero R β] [SMulWithZero Rᵐᵒᵖ β]
     [ContinuousConstSMul R β] [IsCentralScalar R β] : IsCentralScalar R C₀(α, β) :=
   ⟨fun _ _ => ext fun _ => op_smul_eq_smul _ _⟩

... and add a deprecated alias for the old name. This is mostly just me discovering the power of F2

@@ -269,7 +269,7 @@ theorem sub_apply : (f - g) x = f x - g x :=
 
 @[simp]
 theorem coe_zsmulRec : ∀ z, ⇑(zsmulRec z f) = z • ⇑f
-  | Int.ofNat n => by rw [zsmulRec, Int.ofNat_eq_coe, coe_nsmulRec, coe_nat_zsmul]
+  | Int.ofNat n => by rw [zsmulRec, Int.ofNat_eq_coe, coe_nsmulRec, natCast_zsmul]
   | Int.negSucc n => by rw [zsmulRec, negSucc_zsmul, coe_neg, coe_nsmulRec]
 #align zero_at_infty_continuous_map.coe_zsmul_rec ZeroAtInftyContinuousMap.coe_zsmulRec
 

All these lemmas refer to the range of some function being open/range (i.e. isOpen or isClosed).

@@ -465,7 +465,7 @@ theorem tendsto_iff_tendstoUniformly {ι : Type*} {F : ι → C₀(α, β)} {f :
 theorem isometry_toBCF : Isometry (toBCF : C₀(α, β) → α →ᵇ β) := by tauto
 #align zero_at_infty_continuous_map.isometry_to_bcf ZeroAtInftyContinuousMap.isometry_toBCF
 
-theorem closed_range_toBCF : IsClosed (range (toBCF : C₀(α, β) → α →ᵇ β)) := by
+theorem isClosed_range_toBCF : IsClosed (range (toBCF : C₀(α, β) → α →ᵇ β)) := by
   refine' isClosed_iff_clusterPt.mpr fun f hf => _
   rw [clusterPt_principal_iff] at hf
   have : Tendsto f (cocompact α) (𝓝 0) := by
@@ -478,13 +478,15 @@ theorem closed_range_toBCF : IsClosed (range (toBCF : C₀(α, β) → α →ᵇ
       _ < dist g.toBCF f + ε / 2 := (add_lt_add_of_le_of_lt (dist_coe_le_dist x) hx)
       _ < ε := by simpa [add_halves ε] using add_lt_add_right (mem_ball.1 hg) (ε / 2)
   exact ⟨⟨f.toContinuousMap, this⟩, rfl⟩
-#align zero_at_infty_continuous_map.closed_range_to_bcf ZeroAtInftyContinuousMap.closed_range_toBCF
+#align zero_at_infty_continuous_map.closed_range_to_bcf ZeroAtInftyContinuousMap.isClosed_range_toBCF
+
+@[deprecated] alias closed_range_toBCF := isClosed_range_toBCF -- 2024-03-17
 
 /-- Continuous functions vanishing at infinity taking values in a complete space form a
 complete space. -/
 instance instCompleteSpace [CompleteSpace β] : CompleteSpace C₀(α, β) :=
   (completeSpace_iff_isComplete_range isometry_toBCF.uniformInducing).mpr
-    closed_range_toBCF.isComplete
+    isClosed_range_toBCF.isComplete
 
 end Metric
 

The FunLike hierarchy is very big and gets scanned through each time we need a coercion (via the CoeFun instance). It looks like unbundled inheritance suits Lean 4 better here. The only class that still extends FunLike is EquivLike, since that has a custom coe_injective' field that is easier to implement. All other classes should take FunLike or EquivLike as a parameter.

Zulip thread

Important changes

Previously, morphism classes would be Type-valued and extend FunLike:

/-- `MyHomClass F A B` states that `F` is a type of `MyClass.op`-preserving morphisms.
You should extend this class when you extend `MyHom`. -/
class MyHomClass (F : Type*) (A B : outParam <| Type*) [MyClass A] [MyClass B]
  extends FunLike F A B :=
(map_op : ∀ (f : F) (x y : A), f (MyClass.op x y) = MyClass.op (f x) (f y))

After this PR, they should be Prop-valued and take FunLike as a parameter:

/-- `MyHomClass F A B` states that `F` is a type of `MyClass.op`-preserving morphisms.
You should extend this class when you extend `MyHom`. -/
class MyHomClass (F : Type*) (A B : outParam <| Type*) [MyClass A] [MyClass B]
  [FunLike F A B] : Prop :=
(map_op : ∀ (f : F) (x y : A), f (MyClass.op x y) = MyClass.op (f x) (f y))

(Note that A B stay marked as outParam even though they are not purely required to be so due to the FunLike parameter already filling them in. This is required to see through type synonyms, which is important in the category theory library. Also, I think keeping them as outParam is slightly faster.)

Similarly, MyEquivClass should take EquivLike as a parameter.

As a result, every mention of [MyHomClass F A B] should become [FunLike F A B] [MyHomClass F A B].

Remaining issues

Slower (failing) search

While overall this gives some great speedups, there are some cases that are noticeably slower. In particular, a failing application of a lemma such as map_mul is more expensive. This is due to suboptimal processing of arguments. For example:

variable [FunLike F M N] [Mul M] [Mul N] (f : F) (x : M) (y : M)

theorem map_mul [MulHomClass F M N] : f (x * y) = f x * f y

example [AddHomClass F A B] : f (x * y) = f x * f y := map_mul f _ _

Before this PR, applying map_mul f gives the goals [Mul ?M] [Mul ?N] [MulHomClass F ?M ?N]. Since M and N are out_params, [MulHomClass F ?M ?N] is synthesized first, supplies values for ?M and ?N and then the Mul M and Mul N instances can be found.

After this PR, the goals become [FunLike F ?M ?N] [Mul ?M] [Mul ?N] [MulHomClass F ?M ?N]. Now [FunLike F ?M ?N] is synthesized first, supplies values for ?M and ?N and then the Mul M and Mul N instances can be found, before trying MulHomClass F M N which fails. Since the Mul hierarchy is very big, this can be slow to fail, especially when there is no such Mul instance.

A long-term but harder to achieve solution would be to specify the order in which instance goals get solved. For example, we'd like to change the arguments to map_mul to look like [FunLike F M N] [Mul M] [Mul N] [highPriority <| MulHomClass F M N] because MulHomClass fails or succeeds much faster than the others.

As a consequence, the simpNF linter is much slower since by design it tries and fails to apply many map_ lemmas. The same issue occurs a few times in existing calls to simp [map_mul], where map_mul is tried "too soon" and fails. Thanks to the speedup of leanprover/lean4#2478 the impact is very limited, only in files that already were close to the timeout.

simp not firing sometimes

This affects map_smulₛₗ and related definitions. For simp lemmas Lean apparently uses a slightly different mechanism to find instances, so that rw can find every argument to map_smulₛₗ successfully but simp can't: leanprover/lean4#3701.

Missing instances due to unification failing

Especially in the category theory library, we might sometimes have a type A which is also accessible as a synonym (Bundled A hA).1. Instance synthesis doesn't always work if we have f : A →* B but x * y : (Bundled A hA).1 or vice versa. This seems to be mostly fixed by keeping A B as outParams in MulHomClass F A B. (Presumably because Lean will do a definitional check A =?= (Bundled A hA).1 instead of using the syntax in the discrimination tree.)

Workaround for issues

The timeouts can be worked around for now by specifying which map_mul we mean, either as map_mul f for some explicit f, or as e.g. MonoidHomClass.map_mul.

map_smulₛₗ not firing as simp lemma can be worked around by going back to the pre-FunLike situation and making LinearMap.map_smulₛₗ a simp lemma instead of the generic map_smulₛₗ. Writing simp [map_smulₛₗ _] also works.

Co-authored-by: Matthew Ballard <matt@mrb.email> Co-authored-by: Scott Morrison <scott.morrison@gmail.com> Co-authored-by: Scott Morrison <scott@tqft.net> Co-authored-by: Anne Baanen <Vierkantor@users.noreply.github.com>

@@ -60,7 +60,7 @@ vanish at infinity.
 
 You should also extend this typeclass when you extend `ZeroAtInftyContinuousMap`. -/
 class ZeroAtInftyContinuousMapClass (F : Type*) (α β : outParam <| Type*) [TopologicalSpace α]
-    [Zero β] [TopologicalSpace β] extends ContinuousMapClass F α β where
+    [Zero β] [TopologicalSpace β] [FunLike F α β] extends ContinuousMapClass F α β : Prop where
   /-- Each member of the class tends to zero along the `cocompact` filter. -/
   zero_at_infty (f : F) : Tendsto f (cocompact α) (𝓝 0)
 #align zero_at_infty_continuous_map_class ZeroAtInftyContinuousMapClass
@@ -73,22 +73,19 @@ namespace ZeroAtInftyContinuousMap
 
 section Basics
 
-variable [TopologicalSpace β] [Zero β] [ZeroAtInftyContinuousMapClass F α β]
+variable [TopologicalSpace β] [Zero β] [FunLike F α β] [ZeroAtInftyContinuousMapClass F α β]
 
-instance instZeroAtInftyContinuousMapClass : ZeroAtInftyContinuousMapClass C₀(α, β) α β where
+instance instFunLike : FunLike C₀(α, β) α β where
   coe f := f.toFun
   coe_injective' f g h := by
     obtain ⟨⟨_, _⟩, _⟩ := f
     obtain ⟨⟨_, _⟩, _⟩ := g
     congr
+
+instance instZeroAtInftyContinuousMapClass : ZeroAtInftyContinuousMapClass C₀(α, β) α β where
   map_continuous f := f.continuous_toFun
   zero_at_infty f := f.zero_at_infty'
 
-/-- Helper instance for when there's too many metavariables to apply `DFunLike.hasCoeToFun`
-directly. -/
-instance instCoeFun : CoeFun C₀(α, β) fun _ => α → β :=
-  DFunLike.hasCoeToFun
-
 instance instCoeTC : CoeTC F C₀(α, β) :=
   ⟨fun f =>
     { toFun := f
@@ -149,10 +146,8 @@ def ContinuousMap.liftZeroAtInfty [CompactSpace α] : C(α, β) ≃ C₀(α, β)
 
 /-- A continuous function on a compact space is automatically a continuous function vanishing at
 infinity. This is not an instance to avoid type class loops. -/
-def zeroAtInftyContinuousMapClass.ofCompact {G : Type*} [ContinuousMapClass G α β]
-    [CompactSpace α] : ZeroAtInftyContinuousMapClass G α β where
-  coe g := g
-  coe_injective' f g h := DFunLike.coe_fn_eq.mp h
+lemma zeroAtInftyContinuousMapClass.ofCompact {G : Type*} [FunLike G α β]
+    [ContinuousMapClass G α β] [CompactSpace α] : ZeroAtInftyContinuousMapClass G α β where
   map_continuous := map_continuous
   zero_at_infty := by simp
 #align zero_at_infty_continuous_map.zero_at_infty_continuous_map_class.of_compact ZeroAtInftyContinuousMap.zeroAtInftyContinuousMapClass.ofCompact
@@ -372,7 +367,8 @@ end AlgebraicStructure
 
 section Uniform
 
-variable [UniformSpace β] [UniformSpace γ] [Zero γ] [ZeroAtInftyContinuousMapClass F β γ]
+variable [UniformSpace β] [UniformSpace γ] [Zero γ]
+variable [FunLike F β γ] [ZeroAtInftyContinuousMapClass F β γ]
 
 theorem uniformContinuous (f : F) : UniformContinuous (f : β → γ) :=
   (map_continuous f).uniformContinuous_of_tendsto_cocompact (zero_at_infty f)
@@ -393,7 +389,7 @@ section Metric
 
 open Metric Set
 
-variable [PseudoMetricSpace β] [Zero β] [ZeroAtInftyContinuousMapClass F α β]
+variable [PseudoMetricSpace β] [Zero β] [FunLike F α β] [ZeroAtInftyContinuousMapClass F α β]
 
 protected theorem bounded (f : F) : ∃ C, ∀ x y : α, dist ((f : α → β) x) (f y) ≤ C := by
   obtain ⟨K : Set α, hK₁, hK₂⟩ := mem_cocompact.mp

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>

@@ -84,10 +84,10 @@ instance instZeroAtInftyContinuousMapClass : ZeroAtInftyContinuousMapClass C₀(
   map_continuous f := f.continuous_toFun
   zero_at_infty f := f.zero_at_infty'
 
-/-- Helper instance for when there's too many metavariables to apply `FunLike.hasCoeToFun`
+/-- Helper instance for when there's too many metavariables to apply `DFunLike.hasCoeToFun`
 directly. -/
 instance instCoeFun : CoeFun C₀(α, β) fun _ => α → β :=
-  FunLike.hasCoeToFun
+  DFunLike.hasCoeToFun
 
 instance instCoeTC : CoeTC F C₀(α, β) :=
   ⟨fun f =>
@@ -102,7 +102,7 @@ theorem coe_toContinuousMap (f : C₀(α, β)) : (f.toContinuousMap : α → β)
 
 @[ext]
 theorem ext {f g : C₀(α, β)} (h : ∀ x, f x = g x) : f = g :=
-  FunLike.ext _ _ h
+  DFunLike.ext _ _ h
 #align zero_at_infty_continuous_map.ext ZeroAtInftyContinuousMap.ext
 
 /-- Copy of a `ZeroAtInftyContinuousMap` with a new `toFun` equal to the old one. Useful
@@ -123,7 +123,7 @@ theorem coe_copy (f : C₀(α, β)) (f' : α → β) (h : f' = f) : ⇑(f.copy f
 #align zero_at_infty_continuous_map.coe_copy ZeroAtInftyContinuousMap.coe_copy
 
 theorem copy_eq (f : C₀(α, β)) (f' : α → β) (h : f' = f) : f.copy f' h = f :=
-  FunLike.ext' h
+  DFunLike.ext' h
 #align zero_at_infty_continuous_map.copy_eq ZeroAtInftyContinuousMap.copy_eq
 
 theorem eq_of_empty [IsEmpty α] (f g : C₀(α, β)) : f = g :=
@@ -152,7 +152,7 @@ infinity. This is not an instance to avoid type class loops. -/
 def zeroAtInftyContinuousMapClass.ofCompact {G : Type*} [ContinuousMapClass G α β]
     [CompactSpace α] : ZeroAtInftyContinuousMapClass G α β where
   coe g := g
-  coe_injective' f g h := FunLike.coe_fn_eq.mp h
+  coe_injective' f g h := DFunLike.coe_fn_eq.mp h
   map_continuous := map_continuous
   zero_at_infty := by simp
 #align zero_at_infty_continuous_map.zero_at_infty_continuous_map_class.of_compact ZeroAtInftyContinuousMap.zeroAtInftyContinuousMapClass.ofCompact
@@ -200,11 +200,11 @@ theorem mul_apply [MulZeroClass β] [ContinuousMul β] (f g : C₀(α, β)) : (f
 #align zero_at_infty_continuous_map.mul_apply ZeroAtInftyContinuousMap.mul_apply
 
 instance instMulZeroClass [MulZeroClass β] [ContinuousMul β] : MulZeroClass C₀(α, β) :=
-  FunLike.coe_injective.mulZeroClass _ coe_zero coe_mul
+  DFunLike.coe_injective.mulZeroClass _ coe_zero coe_mul
 
 instance instSemigroupWithZero [SemigroupWithZero β] [ContinuousMul β] :
     SemigroupWithZero C₀(α, β) :=
-  FunLike.coe_injective.semigroupWithZero _ coe_zero coe_mul
+  DFunLike.coe_injective.semigroupWithZero _ coe_zero coe_mul
 
 instance instAdd [AddZeroClass β] [ContinuousAdd β] : Add C₀(α, β) :=
   ⟨fun f g => ⟨f + g, by simpa only [add_zero] using (zero_at_infty f).add (zero_at_infty g)⟩⟩
@@ -219,7 +219,7 @@ theorem add_apply [AddZeroClass β] [ContinuousAdd β] (f g : C₀(α, β)) : (f
 #align zero_at_infty_continuous_map.add_apply ZeroAtInftyContinuousMap.add_apply
 
 instance instAddZeroClass [AddZeroClass β] [ContinuousAdd β] : AddZeroClass C₀(α, β) :=
-  FunLike.coe_injective.addZeroClass _ coe_zero coe_add
+  DFunLike.coe_injective.addZeroClass _ coe_zero coe_add
 
 section AddMonoid
 
@@ -237,12 +237,12 @@ instance instNatSMul : SMul ℕ C₀(α, β) :=
 #align zero_at_infty_continuous_map.has_nat_scalar ZeroAtInftyContinuousMap.instNatSMul
 
 instance instAddMonoid : AddMonoid C₀(α, β) :=
-  FunLike.coe_injective.addMonoid _ coe_zero coe_add fun _ _ => rfl
+  DFunLike.coe_injective.addMonoid _ coe_zero coe_add fun _ _ => rfl
 
 end AddMonoid
 
 instance instAddCommMonoid [AddCommMonoid β] [ContinuousAdd β] : AddCommMonoid C₀(α, β) :=
-  FunLike.coe_injective.addCommMonoid _ coe_zero coe_add fun _ _ => rfl
+  DFunLike.coe_injective.addCommMonoid _ coe_zero coe_add fun _ _ => rfl
 
 section AddGroup
 
@@ -285,12 +285,12 @@ instance instIntSMul : SMul ℤ C₀(α, β) :=
 #align zero_at_infty_continuous_map.has_int_scalar ZeroAtInftyContinuousMap.instIntSMul
 
 instance instAddGroup : AddGroup C₀(α, β) :=
-  FunLike.coe_injective.addGroup _ coe_zero coe_add coe_neg coe_sub (fun _ _ => rfl) fun _ _ => rfl
+  DFunLike.coe_injective.addGroup _ coe_zero coe_add coe_neg coe_sub (fun _ _ => rfl) fun _ _ => rfl
 
 end AddGroup
 
 instance instAddCommGroup [AddCommGroup β] [TopologicalAddGroup β] : AddCommGroup C₀(α, β) :=
-  FunLike.coe_injective.addCommGroup _ coe_zero coe_add coe_neg coe_sub (fun _ _ => rfl) fun _ _ =>
+  DFunLike.coe_injective.addCommGroup _ coe_zero coe_add coe_neg coe_sub (fun _ _ => rfl) fun _ _ =>
     rfl
 
 instance instSMul [Zero β] {R : Type*} [Zero R] [SMulWithZero R β] [ContinuousConstSMul R β] :
@@ -316,40 +316,40 @@ instance instIsCentralScalar [Zero β] {R : Type*} [Zero R] [SMulWithZero R β]
 
 instance instSMulWithZero [Zero β] {R : Type*} [Zero R] [SMulWithZero R β]
     [ContinuousConstSMul R β] : SMulWithZero R C₀(α, β) :=
-  Function.Injective.smulWithZero ⟨_, coe_zero⟩ FunLike.coe_injective coe_smul
+  Function.Injective.smulWithZero ⟨_, coe_zero⟩ DFunLike.coe_injective coe_smul
 
 instance instMulActionWithZero [Zero β] {R : Type*} [MonoidWithZero R] [MulActionWithZero R β]
     [ContinuousConstSMul R β] : MulActionWithZero R C₀(α, β) :=
-  Function.Injective.mulActionWithZero ⟨_, coe_zero⟩ FunLike.coe_injective coe_smul
+  Function.Injective.mulActionWithZero ⟨_, coe_zero⟩ DFunLike.coe_injective coe_smul
 
 instance instModule [AddCommMonoid β] [ContinuousAdd β] {R : Type*} [Semiring R] [Module R β]
     [ContinuousConstSMul R β] : Module R C₀(α, β) :=
-  Function.Injective.module R ⟨⟨_, coe_zero⟩, coe_add⟩ FunLike.coe_injective coe_smul
+  Function.Injective.module R ⟨⟨_, coe_zero⟩, coe_add⟩ DFunLike.coe_injective coe_smul
 
 instance instNonUnitalNonAssocSemiring [NonUnitalNonAssocSemiring β] [TopologicalSemiring β] :
     NonUnitalNonAssocSemiring C₀(α, β) :=
-  FunLike.coe_injective.nonUnitalNonAssocSemiring _ coe_zero coe_add coe_mul fun _ _ => rfl
+  DFunLike.coe_injective.nonUnitalNonAssocSemiring _ coe_zero coe_add coe_mul fun _ _ => rfl
 
 instance instNonUnitalSemiring [NonUnitalSemiring β] [TopologicalSemiring β] :
     NonUnitalSemiring C₀(α, β) :=
-  FunLike.coe_injective.nonUnitalSemiring _ coe_zero coe_add coe_mul fun _ _ => rfl
+  DFunLike.coe_injective.nonUnitalSemiring _ coe_zero coe_add coe_mul fun _ _ => rfl
 
 instance instNonUnitalCommSemiring [NonUnitalCommSemiring β] [TopologicalSemiring β] :
     NonUnitalCommSemiring C₀(α, β) :=
-  FunLike.coe_injective.nonUnitalCommSemiring _ coe_zero coe_add coe_mul fun _ _ => rfl
+  DFunLike.coe_injective.nonUnitalCommSemiring _ coe_zero coe_add coe_mul fun _ _ => rfl
 
 instance instNonUnitalNonAssocRing [NonUnitalNonAssocRing β] [TopologicalRing β] :
     NonUnitalNonAssocRing C₀(α, β) :=
-  FunLike.coe_injective.nonUnitalNonAssocRing _ coe_zero coe_add coe_mul coe_neg coe_sub
+  DFunLike.coe_injective.nonUnitalNonAssocRing _ coe_zero coe_add coe_mul coe_neg coe_sub
     (fun _ _ => rfl) fun _ _ => rfl
 
 instance instNonUnitalRing [NonUnitalRing β] [TopologicalRing β] : NonUnitalRing C₀(α, β) :=
-  FunLike.coe_injective.nonUnitalRing _ coe_zero coe_add coe_mul coe_neg coe_sub (fun _ _ => rfl)
+  DFunLike.coe_injective.nonUnitalRing _ coe_zero coe_add coe_mul coe_neg coe_sub (fun _ _ => rfl)
     fun _ _ => rfl
 
 instance instNonUnitalCommRing [NonUnitalCommRing β] [TopologicalRing β] :
     NonUnitalCommRing C₀(α, β) :=
-  FunLike.coe_injective.nonUnitalCommRing _ coe_zero coe_add coe_mul coe_neg coe_sub
+  DFunLike.coe_injective.nonUnitalCommRing _ coe_zero coe_add coe_mul coe_neg coe_sub
     (fun _ _ => rfl) fun _ _ => rfl
 
 instance instIsScalarTower {R : Type*} [Semiring R] [NonUnitalNonAssocSemiring β]
@@ -433,7 +433,7 @@ variable (α) (β)
 
 theorem toBCF_injective : Function.Injective (toBCF : C₀(α, β) → α →ᵇ β) := fun f g h => by
   ext x
-  simpa only using FunLike.congr_fun h x
+  simpa only using DFunLike.congr_fun h x
 #align zero_at_infty_continuous_map.to_bcf_injective ZeroAtInftyContinuousMap.toBCF_injective
 
 end

adds instances for the new NonUnital(Semi)NormedCommRing classes, and also adds some missing instances to Analysis.Normed.Field.Basic.

This gives us our first example of an actually non-unital commutative C⋆-algebra: C₀(X, ℂ) where X is a locally compact, noncompact Hausdorff space.

• depends on: #8664

@@ -393,7 +393,7 @@ section Metric
 
 open Metric Set
 
-variable [MetricSpace β] [Zero β] [ZeroAtInftyContinuousMapClass F α β]
+variable [PseudoMetricSpace β] [Zero β] [ZeroAtInftyContinuousMapClass F α β]
 
 protected theorem bounded (f : F) : ∃ C, ∀ x y : α, dist ((f : α → β) x) (f y) ≤ C := by
   obtain ⟨K : Set α, hK₁, hK₂⟩ := mem_cocompact.mp
@@ -440,9 +440,15 @@ end
 
 variable {C : ℝ} {f g : C₀(α, β)}
 
+/-- The type of continuous functions vanishing at infinity, with the uniform distance induced by the
+inclusion `ZeroAtInftyContinuousMap.toBCF`, is a pseudo-metric space. -/
+noncomputable instance instPseudoMetricSpace : PseudoMetricSpace C₀(α, β) :=
+  PseudoMetricSpace.induced toBCF inferInstance
+
 /-- The type of continuous functions vanishing at infinity, with the uniform distance induced by the
 inclusion `ZeroAtInftyContinuousMap.toBCF`, is a metric space. -/
-noncomputable instance instMetricSpace : MetricSpace C₀(α, β) :=
+noncomputable instance instMetricSpace {β : Type*} [MetricSpace β] [Zero β] :
+    MetricSpace C₀(α, β) :=
   MetricSpace.induced _ (toBCF_injective α β) inferInstance
 
 @[simp]
@@ -498,12 +504,17 @@ field `𝕜` whenever `β` is as well.
 
 section NormedSpace
 
-variable [NormedAddCommGroup β] {𝕜 : Type*} [NormedField 𝕜] [NormedSpace 𝕜 β]
+noncomputable instance instSeminormedAddCommGroup [SeminormedAddCommGroup β] :
+    SeminormedAddCommGroup C₀(α, β) :=
+  SeminormedAddCommGroup.induced _ _ (⟨⟨toBCF, rfl⟩, fun _ _ => rfl⟩ : C₀(α, β) →+ α →ᵇ β)
 
-noncomputable instance instNormedAddCommGroup : NormedAddCommGroup C₀(α, β) :=
-  NormedAddCommGroup.induced C₀(α, β) (α →ᵇ β) (⟨⟨toBCF, rfl⟩, fun _ _ => rfl⟩ : C₀(α, β) →+ α →ᵇ β)
+noncomputable instance instNormedAddCommGroup [NormedAddCommGroup β] :
+    NormedAddCommGroup C₀(α, β) :=
+  NormedAddCommGroup.induced _ _ (⟨⟨toBCF, rfl⟩, fun _ _ => rfl⟩ : C₀(α, β) →+ α →ᵇ β)
     (toBCF_injective α β)
 
+variable [SeminormedAddCommGroup β] {𝕜 : Type*} [NormedField 𝕜] [NormedSpace 𝕜 β]
+
 @[simp]
 theorem norm_toBCF_eq_norm {f : C₀(α, β)} : ‖f.toBCF‖ = ‖f‖ :=
   rfl
@@ -515,12 +526,24 @@ end NormedSpace
 
 section NormedRing
 
-variable [NonUnitalNormedRing β]
+noncomputable instance instNonUnitalSeminormedRing [NonUnitalSeminormedRing β] :
+    NonUnitalSeminormedRing C₀(α, β) :=
+  { instNonUnitalRing, instSeminormedAddCommGroup with
+    norm_mul := fun f g => norm_mul_le f.toBCF g.toBCF }
 
-noncomputable instance instNonUnitalNormedRing : NonUnitalNormedRing C₀(α, β) :=
-  { ZeroAtInftyContinuousMap.instNonUnitalRing, ZeroAtInftyContinuousMap.instNormedAddCommGroup with
+noncomputable instance instNonUnitalNormedRing [NonUnitalNormedRing β] :
+    NonUnitalNormedRing C₀(α, β) :=
+  { instNonUnitalRing, instNormedAddCommGroup with
     norm_mul := fun f g => norm_mul_le f.toBCF g.toBCF }
 
+noncomputable instance instNonUnitalSeminormedCommRing [NonUnitalSeminormedCommRing β] :
+    NonUnitalSeminormedCommRing C₀(α, β) :=
+  { instNonUnitalSeminormedRing, instNonUnitalCommRing with }
+
+noncomputable instance instNonUnitalNormedCommRing [NonUnitalNormedCommRing β] :
+    NonUnitalNormedCommRing C₀(α, β) :=
+  { instNonUnitalNormedRing, instNonUnitalCommRing with }
+
 end NormedRing
 
 end Norm

@@ -383,7 +383,7 @@ end Uniform
 /-! ### Metric structure
 
 When `β` is a metric space, then every element of `C₀(α, β)` is bounded, and so there is a natural
-inclusion map `ZeroAtInftyContinuousMap.toBcf : C₀(α, β) → (α →ᵇ β)`. Via this map `C₀(α, β)`
+inclusion map `ZeroAtInftyContinuousMap.toBCF : C₀(α, β) → (α →ᵇ β)`. Via this map `C₀(α, β)`
 inherits a metric as the pullback of the metric on `α →ᵇ β`. Moreover, this map has closed range
 in `α →ᵇ β` and consequently `C₀(α, β)` is a complete space whenever `β` is complete.
 -/
@@ -423,32 +423,32 @@ instance (priority := 100) instBoundedContinuousMapClass : BoundedContinuousMapC
 
 /-- Construct a bounded continuous function from a continuous function vanishing at infinity. -/
 @[simps!]
-def toBcf (f : C₀(α, β)) : α →ᵇ β :=
+def toBCF (f : C₀(α, β)) : α →ᵇ β :=
   ⟨f, map_bounded f⟩
-#align zero_at_infty_continuous_map.to_bcf ZeroAtInftyContinuousMap.toBcf
+#align zero_at_infty_continuous_map.to_bcf ZeroAtInftyContinuousMap.toBCF
 
 section
 
 variable (α) (β)
 
-theorem toBcf_injective : Function.Injective (toBcf : C₀(α, β) → α →ᵇ β) := fun f g h => by
+theorem toBCF_injective : Function.Injective (toBCF : C₀(α, β) → α →ᵇ β) := fun f g h => by
   ext x
   simpa only using FunLike.congr_fun h x
-#align zero_at_infty_continuous_map.to_bcf_injective ZeroAtInftyContinuousMap.toBcf_injective
+#align zero_at_infty_continuous_map.to_bcf_injective ZeroAtInftyContinuousMap.toBCF_injective
 
 end
 
 variable {C : ℝ} {f g : C₀(α, β)}
 
 /-- The type of continuous functions vanishing at infinity, with the uniform distance induced by the
-inclusion `ZeroAtInftyContinuousMap.toBcf`, is a metric space. -/
+inclusion `ZeroAtInftyContinuousMap.toBCF`, is a metric space. -/
 noncomputable instance instMetricSpace : MetricSpace C₀(α, β) :=
-  MetricSpace.induced _ (toBcf_injective α β) inferInstance
+  MetricSpace.induced _ (toBCF_injective α β) inferInstance
 
 @[simp]
-theorem dist_toBcf_eq_dist {f g : C₀(α, β)} : dist f.toBcf g.toBcf = dist f g :=
+theorem dist_toBCF_eq_dist {f g : C₀(α, β)} : dist f.toBCF g.toBCF = dist f g :=
   rfl
-#align zero_at_infty_continuous_map.dist_to_bcf_eq_dist ZeroAtInftyContinuousMap.dist_toBcf_eq_dist
+#align zero_at_infty_continuous_map.dist_to_bcf_eq_dist ZeroAtInftyContinuousMap.dist_toBCF_eq_dist
 
 open BoundedContinuousFunction
 
@@ -456,14 +456,14 @@ open BoundedContinuousFunction
 theorem tendsto_iff_tendstoUniformly {ι : Type*} {F : ι → C₀(α, β)} {f : C₀(α, β)} {l : Filter ι} :
     Tendsto F l (𝓝 f) ↔ TendstoUniformly (fun i => F i) f l := by
   simpa only [Metric.tendsto_nhds] using
-    @BoundedContinuousFunction.tendsto_iff_tendstoUniformly _ _ _ _ _ (fun i => (F i).toBcf)
-      f.toBcf l
+    @BoundedContinuousFunction.tendsto_iff_tendstoUniformly _ _ _ _ _ (fun i => (F i).toBCF)
+      f.toBCF l
 #align zero_at_infty_continuous_map.tendsto_iff_tendsto_uniformly ZeroAtInftyContinuousMap.tendsto_iff_tendstoUniformly
 
-theorem isometry_toBcf : Isometry (toBcf : C₀(α, β) → α →ᵇ β) := by tauto
-#align zero_at_infty_continuous_map.isometry_to_bcf ZeroAtInftyContinuousMap.isometry_toBcf
+theorem isometry_toBCF : Isometry (toBCF : C₀(α, β) → α →ᵇ β) := by tauto
+#align zero_at_infty_continuous_map.isometry_to_bcf ZeroAtInftyContinuousMap.isometry_toBCF
 
-theorem closed_range_toBcf : IsClosed (range (toBcf : C₀(α, β) → α →ᵇ β)) := by
+theorem closed_range_toBCF : IsClosed (range (toBCF : C₀(α, β) → α →ᵇ β)) := by
   refine' isClosed_iff_clusterPt.mpr fun f hf => _
   rw [clusterPt_principal_iff] at hf
   have : Tendsto f (cocompact α) (𝓝 0) := by
@@ -472,17 +472,17 @@ theorem closed_range_toBcf : IsClosed (range (toBcf : C₀(α, β) → α →ᵇ
     refine' (Metric.tendsto_nhds.mp (zero_at_infty g) (ε / 2) (half_pos hε)).mp
       (eventually_of_forall fun x hx => _)
     calc
-      dist (f x) 0 ≤ dist (g.toBcf x) (f x) + dist (g x) 0 := dist_triangle_left _ _ _
-      _ < dist g.toBcf f + ε / 2 := (add_lt_add_of_le_of_lt (dist_coe_le_dist x) hx)
+      dist (f x) 0 ≤ dist (g.toBCF x) (f x) + dist (g x) 0 := dist_triangle_left _ _ _
+      _ < dist g.toBCF f + ε / 2 := (add_lt_add_of_le_of_lt (dist_coe_le_dist x) hx)
       _ < ε := by simpa [add_halves ε] using add_lt_add_right (mem_ball.1 hg) (ε / 2)
   exact ⟨⟨f.toContinuousMap, this⟩, rfl⟩
-#align zero_at_infty_continuous_map.closed_range_to_bcf ZeroAtInftyContinuousMap.closed_range_toBcf
+#align zero_at_infty_continuous_map.closed_range_to_bcf ZeroAtInftyContinuousMap.closed_range_toBCF
 
 /-- Continuous functions vanishing at infinity taking values in a complete space form a
 complete space. -/
 instance instCompleteSpace [CompleteSpace β] : CompleteSpace C₀(α, β) :=
-  (completeSpace_iff_isComplete_range isometry_toBcf.uniformInducing).mpr
-    closed_range_toBcf.isComplete
+  (completeSpace_iff_isComplete_range isometry_toBCF.uniformInducing).mpr
+    closed_range_toBCF.isComplete
 
 end Metric
 
@@ -490,7 +490,7 @@ section Norm
 
 /-! ### Normed space
 
-The norm structure on `C₀(α, β)` is the one induced by the inclusion `toBcf : C₀(α, β) → (α →ᵇ b)`,
+The norm structure on `C₀(α, β)` is the one induced by the inclusion `toBCF : C₀(α, β) → (α →ᵇ b)`,
 viewed as an additive monoid homomorphism. Then `C₀(α, β)` is naturally a normed space over a normed
 field `𝕜` whenever `β` is as well.
 -/
@@ -501,15 +501,15 @@ section NormedSpace
 variable [NormedAddCommGroup β] {𝕜 : Type*} [NormedField 𝕜] [NormedSpace 𝕜 β]
 
 noncomputable instance instNormedAddCommGroup : NormedAddCommGroup C₀(α, β) :=
-  NormedAddCommGroup.induced C₀(α, β) (α →ᵇ β) (⟨⟨toBcf, rfl⟩, fun _ _ => rfl⟩ : C₀(α, β) →+ α →ᵇ β)
-    (toBcf_injective α β)
+  NormedAddCommGroup.induced C₀(α, β) (α →ᵇ β) (⟨⟨toBCF, rfl⟩, fun _ _ => rfl⟩ : C₀(α, β) →+ α →ᵇ β)
+    (toBCF_injective α β)
 
 @[simp]
-theorem norm_toBcf_eq_norm {f : C₀(α, β)} : ‖f.toBcf‖ = ‖f‖ :=
+theorem norm_toBCF_eq_norm {f : C₀(α, β)} : ‖f.toBCF‖ = ‖f‖ :=
   rfl
-#align zero_at_infty_continuous_map.norm_to_bcf_eq_norm ZeroAtInftyContinuousMap.norm_toBcf_eq_norm
+#align zero_at_infty_continuous_map.norm_to_bcf_eq_norm ZeroAtInftyContinuousMap.norm_toBCF_eq_norm
 
-instance : NormedSpace 𝕜 C₀(α, β) where norm_smul_le k f := (norm_smul_le k f.toBcf : _)
+instance : NormedSpace 𝕜 C₀(α, β) where norm_smul_le k f := (norm_smul_le k f.toBCF : _)
 
 end NormedSpace
 
@@ -519,7 +519,7 @@ variable [NonUnitalNormedRing β]
 
 noncomputable instance instNonUnitalNormedRing : NonUnitalNormedRing C₀(α, β) :=
   { ZeroAtInftyContinuousMap.instNonUnitalRing, ZeroAtInftyContinuousMap.instNormedAddCommGroup with
-    norm_mul := fun f g => norm_mul_le f.toBcf g.toBcf }
+    norm_mul := fun f g => norm_mul_le f.toBCF g.toBCF }
 
 end NormedRing
 
@@ -567,7 +567,7 @@ section NormedStar
 variable [NormedAddCommGroup β] [StarAddMonoid β] [NormedStarGroup β]
 
 instance instNormedStarGroup : NormedStarGroup C₀(α, β) where
-  norm_star f := (norm_star f.toBcf : _)
+  norm_star f := (norm_star f.toBCF : _)
 
 end NormedStar
 
@@ -595,7 +595,7 @@ end StarRing
 section CstarRing
 
 instance instCstarRing [NonUnitalNormedRing β] [StarRing β] [CstarRing β] : CstarRing C₀(α, β) where
-  norm_star_mul_self {f} := CstarRing.norm_star_mul_self (x := f.toBcf)
+  norm_star_mul_self {f} := CstarRing.norm_star_mul_self (x := f.toBCF)
 
 end CstarRing
 

Use Bornology.IsBounded instead.

@@ -28,7 +28,7 @@ universe u v w
 
 variable {F : Type*} {α : Type u} {β : Type v} {γ : Type w} [TopologicalSpace α]
 
-open BoundedContinuousFunction Topology
+open BoundedContinuousFunction Topology Bornology
 
 open Filter Metric
 
@@ -398,7 +398,7 @@ variable [MetricSpace β] [Zero β] [ZeroAtInftyContinuousMapClass F α β]
 protected theorem bounded (f : F) : ∃ C, ∀ x y : α, dist ((f : α → β) x) (f y) ≤ C := by
   obtain ⟨K : Set α, hK₁, hK₂⟩ := mem_cocompact.mp
     (tendsto_def.mp (zero_at_infty (f : F)) _ (closedBall_mem_nhds (0 : β) zero_lt_one))
-  obtain ⟨C, hC⟩ := (hK₁.image (map_continuous f)).bounded.subset_ball (0 : β)
+  obtain ⟨C, hC⟩ := (hK₁.image (map_continuous f)).isBounded.subset_closedBall (0 : β)
   refine' ⟨max C 1 + max C 1, fun x y => _⟩
   have : ∀ x, f x ∈ closedBall (0 : β) (max C 1) := by
     intro x
@@ -409,13 +409,13 @@ protected theorem bounded (f : F) : ∃ C, ∀ x y : α, dist ((f : α → β) x
     (add_le_add (mem_closedBall.mp <| this x) (mem_closedBall'.mp <| this y))
 #align zero_at_infty_continuous_map.bounded ZeroAtInftyContinuousMap.bounded
 
-theorem bounded_range (f : C₀(α, β)) : Bounded (range f) :=
-  bounded_range_iff.2 (ZeroAtInftyContinuousMap.bounded f)
-#align zero_at_infty_continuous_map.bounded_range ZeroAtInftyContinuousMap.bounded_range
+theorem isBounded_range (f : C₀(α, β)) : IsBounded (range f) :=
+  isBounded_range_iff.2 (ZeroAtInftyContinuousMap.bounded f)
+#align zero_at_infty_continuous_map.bounded_range ZeroAtInftyContinuousMap.isBounded_range
 
-theorem bounded_image (f : C₀(α, β)) (s : Set α) : Bounded (f '' s) :=
-  f.bounded_range.mono <| image_subset_range _ _
-#align zero_at_infty_continuous_map.bounded_image ZeroAtInftyContinuousMap.bounded_image
+theorem isBounded_image (f : C₀(α, β)) (s : Set α) : IsBounded (f '' s) :=
+  f.isBounded_range.subset <| image_subset_range _ _
+#align zero_at_infty_continuous_map.bounded_image ZeroAtInftyContinuousMap.isBounded_image
 
 instance (priority := 100) instBoundedContinuousMapClass : BoundedContinuousMapClass F α β :=
   { ‹ZeroAtInftyContinuousMapClass F α β› with

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

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

@@ -188,7 +188,7 @@ theorem zero_apply [Zero β] : (0 : C₀(α, β)) x = 0 :=
 
 instance instMul [MulZeroClass β] [ContinuousMul β] : Mul C₀(α, β) :=
   ⟨fun f g =>
-    ⟨f * g, by simpa only [MulZeroClass.mul_zero] using (zero_at_infty f).mul (zero_at_infty g)⟩⟩
+    ⟨f * g, by simpa only [mul_zero] using (zero_at_infty f).mul (zero_at_infty g)⟩⟩
 
 @[simp]
 theorem coe_mul [MulZeroClass β] [ContinuousMul β] (f g : C₀(α, β)) : ⇑(f * g) = f * g :=

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

This has nice performance benefits.

@@ -26,7 +26,7 @@ compact space, this type has nice properties.
 
 universe u v w
 
-variable {F : Type _} {α : Type u} {β : Type v} {γ : Type w} [TopologicalSpace α]
+variable {F : Type*} {α : Type u} {β : Type v} {γ : Type w} [TopologicalSpace α]
 
 open BoundedContinuousFunction Topology
 
@@ -36,7 +36,7 @@ open Filter Metric
 topological space to a metric space with a zero element.
 
 When possible, instead of parametrizing results over `(f : C₀(α, β))`,
-you should parametrize over `(F : Type _) [ZeroAtInftyContinuousMapClass F α β] (f : F)`.
+you should parametrize over `(F : Type*) [ZeroAtInftyContinuousMapClass F α β] (f : F)`.
 
 When you extend this structure, make sure to extend `ZeroAtInftyContinuousMapClass`. -/
 structure ZeroAtInftyContinuousMap (α : Type u) (β : Type v) [TopologicalSpace α] [Zero β]
@@ -59,7 +59,7 @@ section
 vanish at infinity.
 
 You should also extend this typeclass when you extend `ZeroAtInftyContinuousMap`. -/
-class ZeroAtInftyContinuousMapClass (F : Type _) (α β : outParam <| Type _) [TopologicalSpace α]
+class ZeroAtInftyContinuousMapClass (F : Type*) (α β : outParam <| Type*) [TopologicalSpace α]
     [Zero β] [TopologicalSpace β] extends ContinuousMapClass F α β where
   /-- Each member of the class tends to zero along the `cocompact` filter. -/
   zero_at_infty (f : F) : Tendsto f (cocompact α) (𝓝 0)
@@ -149,7 +149,7 @@ def ContinuousMap.liftZeroAtInfty [CompactSpace α] : C(α, β) ≃ C₀(α, β)
 
 /-- A continuous function on a compact space is automatically a continuous function vanishing at
 infinity. This is not an instance to avoid type class loops. -/
-def zeroAtInftyContinuousMapClass.ofCompact {G : Type _} [ContinuousMapClass G α β]
+def zeroAtInftyContinuousMapClass.ofCompact {G : Type*} [ContinuousMapClass G α β]
     [CompactSpace α] : ZeroAtInftyContinuousMapClass G α β where
   coe g := g
   coe_injective' f g h := FunLike.coe_fn_eq.mp h
@@ -293,36 +293,36 @@ instance instAddCommGroup [AddCommGroup β] [TopologicalAddGroup β] : AddCommGr
   FunLike.coe_injective.addCommGroup _ coe_zero coe_add coe_neg coe_sub (fun _ _ => rfl) fun _ _ =>
     rfl
 
-instance instSMul [Zero β] {R : Type _} [Zero R] [SMulWithZero R β] [ContinuousConstSMul R β] :
+instance instSMul [Zero β] {R : Type*} [Zero R] [SMulWithZero R β] [ContinuousConstSMul R β] :
     SMul R C₀(α, β) :=
   -- Porting note: Original version didn't have `Continuous.const_smul f.continuous r`
   ⟨fun r f => ⟨⟨r • ⇑f, Continuous.const_smul f.continuous r⟩,
     by simpa [smul_zero] using (zero_at_infty f).const_smul r⟩⟩
 
 @[simp]
-theorem coe_smul [Zero β] {R : Type _} [Zero R] [SMulWithZero R β] [ContinuousConstSMul R β] (r : R)
+theorem coe_smul [Zero β] {R : Type*} [Zero R] [SMulWithZero R β] [ContinuousConstSMul R β] (r : R)
     (f : C₀(α, β)) : ⇑(r • f) = r • ⇑f :=
   rfl
 #align zero_at_infty_continuous_map.coe_smul ZeroAtInftyContinuousMap.coe_smul
 
-theorem smul_apply [Zero β] {R : Type _} [Zero R] [SMulWithZero R β] [ContinuousConstSMul R β]
+theorem smul_apply [Zero β] {R : Type*} [Zero R] [SMulWithZero R β] [ContinuousConstSMul R β]
     (r : R) (f : C₀(α, β)) (x : α) : (r • f) x = r • f x :=
   rfl
 #align zero_at_infty_continuous_map.smul_apply ZeroAtInftyContinuousMap.smul_apply
 
-instance instIsCentralScalar [Zero β] {R : Type _} [Zero R] [SMulWithZero R β] [SMulWithZero Rᵐᵒᵖ β]
+instance instIsCentralScalar [Zero β] {R : Type*} [Zero R] [SMulWithZero R β] [SMulWithZero Rᵐᵒᵖ β]
     [ContinuousConstSMul R β] [IsCentralScalar R β] : IsCentralScalar R C₀(α, β) :=
   ⟨fun _ _ => ext fun _ => op_smul_eq_smul _ _⟩
 
-instance instSMulWithZero [Zero β] {R : Type _} [Zero R] [SMulWithZero R β]
+instance instSMulWithZero [Zero β] {R : Type*} [Zero R] [SMulWithZero R β]
     [ContinuousConstSMul R β] : SMulWithZero R C₀(α, β) :=
   Function.Injective.smulWithZero ⟨_, coe_zero⟩ FunLike.coe_injective coe_smul
 
-instance instMulActionWithZero [Zero β] {R : Type _} [MonoidWithZero R] [MulActionWithZero R β]
+instance instMulActionWithZero [Zero β] {R : Type*} [MonoidWithZero R] [MulActionWithZero R β]
     [ContinuousConstSMul R β] : MulActionWithZero R C₀(α, β) :=
   Function.Injective.mulActionWithZero ⟨_, coe_zero⟩ FunLike.coe_injective coe_smul
 
-instance instModule [AddCommMonoid β] [ContinuousAdd β] {R : Type _} [Semiring R] [Module R β]
+instance instModule [AddCommMonoid β] [ContinuousAdd β] {R : Type*} [Semiring R] [Module R β]
     [ContinuousConstSMul R β] : Module R C₀(α, β) :=
   Function.Injective.module R ⟨⟨_, coe_zero⟩, coe_add⟩ FunLike.coe_injective coe_smul
 
@@ -352,7 +352,7 @@ instance instNonUnitalCommRing [NonUnitalCommRing β] [TopologicalRing β] :
   FunLike.coe_injective.nonUnitalCommRing _ coe_zero coe_add coe_mul coe_neg coe_sub
     (fun _ _ => rfl) fun _ _ => rfl
 
-instance instIsScalarTower {R : Type _} [Semiring R] [NonUnitalNonAssocSemiring β]
+instance instIsScalarTower {R : Type*} [Semiring R] [NonUnitalNonAssocSemiring β]
     [TopologicalSemiring β] [Module R β] [ContinuousConstSMul R β] [IsScalarTower R β β] :
     IsScalarTower R C₀(α, β) C₀(α, β) where
   smul_assoc r f g := by
@@ -360,7 +360,7 @@ instance instIsScalarTower {R : Type _} [Semiring R] [NonUnitalNonAssocSemiring
     simp only [smul_eq_mul, coe_mul, coe_smul, Pi.mul_apply, Pi.smul_apply]
     rw [← smul_eq_mul, ← smul_eq_mul, smul_assoc]
 
-instance instSMulCommClass {R : Type _} [Semiring R] [NonUnitalNonAssocSemiring β]
+instance instSMulCommClass {R : Type*} [Semiring R] [NonUnitalNonAssocSemiring β]
     [TopologicalSemiring β] [Module R β] [ContinuousConstSMul R β] [SMulCommClass R β β] :
     SMulCommClass R C₀(α, β) C₀(α, β) where
   smul_comm r f g := by
@@ -453,7 +453,7 @@ theorem dist_toBcf_eq_dist {f g : C₀(α, β)} : dist f.toBcf g.toBcf = dist f
 open BoundedContinuousFunction
 
 /-- Convergence in the metric on `C₀(α, β)` is uniform convergence. -/
-theorem tendsto_iff_tendstoUniformly {ι : Type _} {F : ι → C₀(α, β)} {f : C₀(α, β)} {l : Filter ι} :
+theorem tendsto_iff_tendstoUniformly {ι : Type*} {F : ι → C₀(α, β)} {f : C₀(α, β)} {l : Filter ι} :
     Tendsto F l (𝓝 f) ↔ TendstoUniformly (fun i => F i) f l := by
   simpa only [Metric.tendsto_nhds] using
     @BoundedContinuousFunction.tendsto_iff_tendstoUniformly _ _ _ _ _ (fun i => (F i).toBcf)
@@ -498,7 +498,7 @@ field `𝕜` whenever `β` is as well.
 
 section NormedSpace
 
-variable [NormedAddCommGroup β] {𝕜 : Type _} [NormedField 𝕜] [NormedSpace 𝕜 β]
+variable [NormedAddCommGroup β] {𝕜 : Type*} [NormedField 𝕜] [NormedSpace 𝕜 β]
 
 noncomputable instance instNormedAddCommGroup : NormedAddCommGroup C₀(α, β) :=
   NormedAddCommGroup.induced C₀(α, β) (α →ᵇ β) (⟨⟨toBcf, rfl⟩, fun _ _ => rfl⟩ : C₀(α, β) →+ α →ᵇ β)
@@ -573,7 +573,7 @@ end NormedStar
 
 section StarModule
 
-variable {𝕜 : Type _} [Zero 𝕜] [Star 𝕜] [AddMonoid β] [StarAddMonoid β] [TopologicalSpace β]
+variable {𝕜 : Type*} [Zero 𝕜] [Star 𝕜] [AddMonoid β] [StarAddMonoid β] [TopologicalSpace β]
   [ContinuousStar β] [SMulWithZero 𝕜 β] [ContinuousConstSMul 𝕜 β] [StarModule 𝕜 β]
 
 instance instStarModule : StarModule 𝕜 C₀(α, β) where
@@ -606,7 +606,7 @@ category of topological spaces with morphisms given by `CocompactMap`s.
 -/
 
 
-variable {δ : Type _} [TopologicalSpace β] [TopologicalSpace γ] [TopologicalSpace δ]
+variable {δ : Type*} [TopologicalSpace β] [TopologicalSpace γ] [TopologicalSpace δ]
 
 local notation α " →co " β => CocompactMap α β
 
@@ -658,7 +658,7 @@ def compMulHom [MulZeroClass δ] [ContinuousMul δ] (g : β →co γ) : C₀(γ,
 #align zero_at_infty_continuous_map.comp_mul_hom ZeroAtInftyContinuousMap.compMulHom
 
 /-- Composition as a linear map. -/
-def compLinearMap [AddCommMonoid δ] [ContinuousAdd δ] {R : Type _} [Semiring R] [Module R δ]
+def compLinearMap [AddCommMonoid δ] [ContinuousAdd δ] {R : Type*} [Semiring R] [Module R δ]
     [ContinuousConstSMul R δ] (g : β →co γ) : C₀(γ, δ) →ₗ[R] C₀(β, δ) where
   toFun f := f.comp g
   map_add' _ _ := rfl
@@ -666,7 +666,7 @@ def compLinearMap [AddCommMonoid δ] [ContinuousAdd δ] {R : Type _} [Semiring R
 #align zero_at_infty_continuous_map.comp_linear_map ZeroAtInftyContinuousMap.compLinearMap
 
 /-- Composition as a non-unital algebra homomorphism. -/
-def compNonUnitalAlgHom {R : Type _} [Semiring R] [NonUnitalNonAssocSemiring δ]
+def compNonUnitalAlgHom {R : Type*} [Semiring R] [NonUnitalNonAssocSemiring δ]
     [TopologicalSemiring δ] [Module R δ] [ContinuousConstSMul R δ] (g : β →co γ) :
     C₀(γ, δ) →ₙₐ[R] C₀(β, δ) where
   toFun f := f.comp g

• 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 Jireh Loreaux. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Jireh Loreaux
-
-! This file was ported from Lean 3 source module topology.continuous_function.zero_at_infty
-! leanprover-community/mathlib commit ba5ff5ad5d120fb0ef094ad2994967e9bfaf5112
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
 -/
 import Mathlib.Topology.ContinuousFunction.Bounded
 import Mathlib.Topology.ContinuousFunction.CocompactMap
 
+#align_import topology.continuous_function.zero_at_infty from "leanprover-community/mathlib"@"ba5ff5ad5d120fb0ef094ad2994967e9bfaf5112"
+
 /-!
 # Continuous functions vanishing at infinity
 

@@ -425,7 +425,7 @@ instance (priority := 100) instBoundedContinuousMapClass : BoundedContinuousMapC
     map_bounded := fun f => ZeroAtInftyContinuousMap.bounded f }
 
 /-- Construct a bounded continuous function from a continuous function vanishing at infinity. -/
-@[simps]
+@[simps!]
 def toBcf (f : C₀(α, β)) : α →ᵇ β :=
   ⟨f, map_bounded f⟩
 #align zero_at_infty_continuous_map.to_bcf ZeroAtInftyContinuousMap.toBcf

Co-authored-by: int-y1 <jason_yuen2007@hotmail.com> Co-authored-by: Jireh Loreaux <loreaujy@gmail.com> Co-authored-by: Chris Hughes <chrishughes24@gmail.com>