topology.uniform_space.equicontinuity ⟷ Mathlib.Topology.UniformSpace.Equicontinuity

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

@@ -143,7 +143,7 @@ protected abbrev Set.UniformEquicontinuous (H : Set <| β → α) : Prop :=
 #align set.uniform_equicontinuous Set.UniformEquicontinuous
 -/
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:641:2: warning: expanding binder collection (x y «expr ∈ » V) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:642:2: warning: expanding binder collection (x y «expr ∈ » V) -/
 #print equicontinuousAt_iff_pair /-
 /-- Reformulation of equicontinuity at `x₀` comparing two variables near `x₀` instead of comparing
 only one with `x₀`. -/

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

@@ -173,7 +173,7 @@ theorem UniformEquicontinuous.equicontinuous {F : ι → β → α} (h : Uniform
 theorem EquicontinuousAt.continuousAt {F : ι → X → α} {x₀ : X} (h : EquicontinuousAt F x₀) (i : ι) :
     ContinuousAt (F i) x₀ := by
   intro U hU
-  rw [UniformSpace.mem_nhds_iff] at hU 
+  rw [UniformSpace.mem_nhds_iff] at hU
   rcases hU with ⟨V, hV₁, hV₂⟩
   exact mem_map.mpr (mem_of_superset (h V hV₁) fun x hx => hV₂ (hx i))
 #align equicontinuous_at.continuous_at EquicontinuousAt.continuousAt
@@ -450,7 +450,7 @@ theorem EquicontinuousAt.closure' {A : Set Y} {u : Y → X → α} {x₀ : X}
   rcases mem_uniformity_isClosed hU with ⟨V, hV, hVclosed, hVU⟩
   filter_upwards [hA V hV] with x hx
   rw [SetCoe.forall] at *
-  change A ⊆ (fun f => (u f x₀, u f x)) ⁻¹' V at hx 
+  change A ⊆ (fun f => (u f x₀, u f x)) ⁻¹' V at hx
   refine' (closure_minimal hx <| hVclosed.preimage <| _).trans (preimage_mono hVU)
   exact Continuous.prod_mk ((continuous_apply x₀).comp hu) ((continuous_apply x).comp hu)
 #align equicontinuous_at.closure' EquicontinuousAt.closure'
@@ -518,7 +518,7 @@ theorem UniformEquicontinuous.closure' {A : Set Y} {u : Y → β → α}
   filter_upwards [hA V hV]
   rintro ⟨x, y⟩ hxy
   rw [SetCoe.forall] at *
-  change A ⊆ (fun f => (u f x, u f y)) ⁻¹' V at hxy 
+  change A ⊆ (fun f => (u f x, u f y)) ⁻¹' V at hxy
   refine' (closure_minimal hxy <| hVclosed.preimage <| _).trans (preimage_mono hVU)
   exact Continuous.prod_mk ((continuous_apply x).comp hu) ((continuous_apply y).comp hu)
 #align uniform_equicontinuous.closure' UniformEquicontinuous.closure'

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

@@ -277,14 +277,14 @@ theorem equicontinuous_iff_range {F : ι → X → α} :
 #align equicontinuous_iff_range equicontinuous_iff_range
 -/
 
-#print uniformEquicontinuous_at_iff_range /-
+#print uniformEquicontinuous_iff_range /-
 /-- A family `𝓕 : ι → β → α` is uniformly equicontinuous iff `range 𝓕` is uniformly equicontinuous,
 i.e the family `coe : range F → β → α` is uniformly equicontinuous. -/
-theorem uniformEquicontinuous_at_iff_range {F : ι → β → α} :
+theorem uniformEquicontinuous_iff_range {F : ι → β → α} :
     UniformEquicontinuous F ↔ UniformEquicontinuous (coe : range F → β → α) :=
   ⟨fun h => by rw [← comp_range_splitting F] <;> exact h.comp _, fun h =>
     h.comp (rangeFactorization F)⟩
-#align uniform_equicontinuous_at_iff_range uniformEquicontinuous_at_iff_range
+#align uniform_equicontinuous_at_iff_range uniformEquicontinuous_iff_range
 -/
 
 section
@@ -456,13 +456,13 @@ theorem EquicontinuousAt.closure' {A : Set Y} {u : Y → X → α} {x₀ : X}
 #align equicontinuous_at.closure' EquicontinuousAt.closure'
 -/
 
-#print EquicontinuousAt.closure /-
+#print Set.EquicontinuousAt.closure /-
 /-- If a set of functions is equicontinuous at some `x₀`, its closure for the product topology is
 also equicontinuous at `x₀`. -/
-theorem EquicontinuousAt.closure {A : Set <| X → α} {x₀ : X} (hA : A.EquicontinuousAt x₀) :
+theorem Set.EquicontinuousAt.closure {A : Set <| X → α} {x₀ : X} (hA : A.EquicontinuousAt x₀) :
     (closure A).EquicontinuousAt x₀ :=
   @EquicontinuousAt.closure' _ _ _ _ _ _ _ id _ hA continuous_id
-#align equicontinuous_at.closure EquicontinuousAt.closure
+#align equicontinuous_at.closure Set.EquicontinuousAt.closure
 -/
 
 #print Filter.Tendsto.continuousAt_of_equicontinuousAt /-
@@ -487,21 +487,21 @@ theorem Equicontinuous.closure' {A : Set Y} {u : Y → X → α}
 #align equicontinuous.closure' Equicontinuous.closure'
 -/
 
-#print Equicontinuous.closure /-
+#print Set.Equicontinuous.closure /-
 /-- If a set of functions is equicontinuous, its closure for the product topology is also
 equicontinuous. -/
-theorem Equicontinuous.closure {A : Set <| X → α} (hA : A.Equicontinuous) :
+theorem Set.Equicontinuous.closure {A : Set <| X → α} (hA : A.Equicontinuous) :
     (closure A).Equicontinuous := fun x => (hA x).closure
-#align equicontinuous.closure Equicontinuous.closure
+#align equicontinuous.closure Set.Equicontinuous.closure
 -/
 
-#print Filter.Tendsto.continuous_of_equicontinuousAt /-
+#print Filter.Tendsto.continuous_of_equicontinuous /-
 /-- If `𝓕 : ι → X → α` tends to `f : X → α` *pointwise* along some nontrivial filter, and if the
 family `𝓕` is equicontinuous, then the limit is continuous. -/
-theorem Filter.Tendsto.continuous_of_equicontinuousAt {l : Filter ι} [l.ne_bot] {F : ι → X → α}
+theorem Filter.Tendsto.continuous_of_equicontinuous {l : Filter ι} [l.ne_bot] {F : ι → X → α}
     {f : X → α} (h₁ : Tendsto F l (𝓝 f)) (h₂ : Equicontinuous F) : Continuous f :=
   continuous_iff_continuousAt.mpr fun x => h₁.continuousAt_of_equicontinuousAt (h₂ x)
-#align filter.tendsto.continuous_of_equicontinuous_at Filter.Tendsto.continuous_of_equicontinuousAt
+#align filter.tendsto.continuous_of_equicontinuous_at Filter.Tendsto.continuous_of_equicontinuous
 -/
 
 #print UniformEquicontinuous.closure' /-
@@ -524,13 +524,13 @@ theorem UniformEquicontinuous.closure' {A : Set Y} {u : Y → β → α}
 #align uniform_equicontinuous.closure' UniformEquicontinuous.closure'
 -/
 
-#print UniformEquicontinuous.closure /-
+#print Set.UniformEquicontinuous.closure /-
 /-- If a set of functions is uniformly equicontinuous, its closure for the product topology is also
 uniformly equicontinuous. -/
-theorem UniformEquicontinuous.closure {A : Set <| β → α} (hA : A.UniformEquicontinuous) :
+theorem Set.UniformEquicontinuous.closure {A : Set <| β → α} (hA : A.UniformEquicontinuous) :
     (closure A).UniformEquicontinuous :=
   @UniformEquicontinuous.closure' _ _ _ _ _ _ _ id hA continuous_id
-#align uniform_equicontinuous.closure UniformEquicontinuous.closure
+#align uniform_equicontinuous.closure Set.UniformEquicontinuous.closure
 -/
 
 #print Filter.Tendsto.uniformContinuous_of_uniformEquicontinuous /-
@@ -539,7 +539,7 @@ family `𝓕` is uniformly equicontinuous, then the limit is uniformly continuou
 theorem Filter.Tendsto.uniformContinuous_of_uniformEquicontinuous {l : Filter ι} [l.ne_bot]
     {F : ι → β → α} {f : β → α} (h₁ : Tendsto F l (𝓝 f)) (h₂ : UniformEquicontinuous F) :
     UniformContinuous f :=
-  (uniformEquicontinuous_at_iff_range.mp h₂).closure.UniformContinuous
+  (uniformEquicontinuous_iff_range.mp h₂).closure.UniformContinuous
     ⟨f, mem_closure_of_tendsto h₁ <| eventually_of_forall mem_range_self⟩
 #align filter.tendsto.uniform_continuous_of_uniform_equicontinuous Filter.Tendsto.uniformContinuous_of_uniformEquicontinuous
 -/

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

@@ -3,7 +3,7 @@ Copyright (c) 2022 Anatole Dedecker. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Anatole Dedecker
 -/
-import Mathbin.Topology.UniformSpace.UniformConvergenceTopology
+import Topology.UniformSpace.UniformConvergenceTopology
 
 #align_import topology.uniform_space.equicontinuity from "leanprover-community/mathlib"@"ee05e9ce1322178f0c12004eb93c00d2c8c00ed2"
 
@@ -143,7 +143,7 @@ protected abbrev Set.UniformEquicontinuous (H : Set <| β → α) : Prop :=
 #align set.uniform_equicontinuous Set.UniformEquicontinuous
 -/
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (x y «expr ∈ » V) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:641:2: warning: expanding binder collection (x y «expr ∈ » V) -/
 #print equicontinuousAt_iff_pair /-
 /-- Reformulation of equicontinuity at `x₀` comparing two variables near `x₀` instead of comparing
 only one with `x₀`. -/

mathlib commit https://github.com/leanprover-community/mathlib/commit/442a83d738cb208d3600056c489be16900ba701d

@@ -495,13 +495,13 @@ theorem Equicontinuous.closure {A : Set <| X → α} (hA : A.Equicontinuous) :
 #align equicontinuous.closure Equicontinuous.closure
 -/
 
-#print Filter.Tendsto.continuous_of_equicontinuous_at /-
+#print Filter.Tendsto.continuous_of_equicontinuousAt /-
 /-- If `𝓕 : ι → X → α` tends to `f : X → α` *pointwise* along some nontrivial filter, and if the
 family `𝓕` is equicontinuous, then the limit is continuous. -/
-theorem Filter.Tendsto.continuous_of_equicontinuous_at {l : Filter ι} [l.ne_bot] {F : ι → X → α}
+theorem Filter.Tendsto.continuous_of_equicontinuousAt {l : Filter ι} [l.ne_bot] {F : ι → X → α}
     {f : X → α} (h₁ : Tendsto F l (𝓝 f)) (h₂ : Equicontinuous F) : Continuous f :=
   continuous_iff_continuousAt.mpr fun x => h₁.continuousAt_of_equicontinuousAt (h₂ x)
-#align filter.tendsto.continuous_of_equicontinuous_at Filter.Tendsto.continuous_of_equicontinuous_at
+#align filter.tendsto.continuous_of_equicontinuous_at Filter.Tendsto.continuous_of_equicontinuousAt
 -/
 
 #print UniformEquicontinuous.closure' /-

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

@@ -2,14 +2,11 @@
 Copyright (c) 2022 Anatole Dedecker. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Anatole Dedecker
-
-! This file was ported from Lean 3 source module topology.uniform_space.equicontinuity
-! leanprover-community/mathlib commit ee05e9ce1322178f0c12004eb93c00d2c8c00ed2
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
 -/
 import Mathbin.Topology.UniformSpace.UniformConvergenceTopology
 
+#align_import topology.uniform_space.equicontinuity from "leanprover-community/mathlib"@"ee05e9ce1322178f0c12004eb93c00d2c8c00ed2"
+
 /-!
 # Equicontinuity of a family of functions
 
@@ -146,7 +143,7 @@ protected abbrev Set.UniformEquicontinuous (H : Set <| β → α) : Prop :=
 #align set.uniform_equicontinuous Set.UniformEquicontinuous
 -/
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:638:2: warning: expanding binder collection (x y «expr ∈ » V) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (x y «expr ∈ » V) -/
 #print equicontinuousAt_iff_pair /-
 /-- Reformulation of equicontinuity at `x₀` comparing two variables near `x₀` instead of comparing
 only one with `x₀`. -/

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

@@ -147,6 +147,7 @@ protected abbrev Set.UniformEquicontinuous (H : Set <| β → α) : Prop :=
 -/
 
 /- ./././Mathport/Syntax/Translate/Basic.lean:638:2: warning: expanding binder collection (x y «expr ∈ » V) -/
+#print equicontinuousAt_iff_pair /-
 /-- Reformulation of equicontinuity at `x₀` comparing two variables near `x₀` instead of comparing
 only one with `x₀`. -/
 theorem equicontinuousAt_iff_pair {F : ι → X → α} {x₀ : X} :
@@ -160,13 +161,17 @@ theorem equicontinuousAt_iff_pair {F : ι → X → α} {x₀ : X} :
   · rcases H U hU with ⟨V, hV, hVU⟩
     filter_upwards [hV] using fun x hx i => hVU x₀ (mem_of_mem_nhds hV) x hx i
 #align equicontinuous_at_iff_pair equicontinuousAt_iff_pair
+-/
 
+#print UniformEquicontinuous.equicontinuous /-
 /-- Uniform equicontinuity implies equicontinuity. -/
 theorem UniformEquicontinuous.equicontinuous {F : ι → β → α} (h : UniformEquicontinuous F) :
     Equicontinuous F := fun x₀ U hU =>
   mem_of_superset (ball_mem_nhds x₀ (h U hU)) fun x hx i => hx i
 #align uniform_equicontinuous.equicontinuous UniformEquicontinuous.equicontinuous
+-/
 
+#print EquicontinuousAt.continuousAt /-
 /-- Each function of a family equicontinuous at `x₀` is continuous at `x₀`. -/
 theorem EquicontinuousAt.continuousAt {F : ι → X → α} {x₀ : X} (h : EquicontinuousAt F x₀) (i : ι) :
     ContinuousAt (F i) x₀ := by
@@ -175,64 +180,88 @@ theorem EquicontinuousAt.continuousAt {F : ι → X → α} {x₀ : X} (h : Equi
   rcases hU with ⟨V, hV₁, hV₂⟩
   exact mem_map.mpr (mem_of_superset (h V hV₁) fun x hx => hV₂ (hx i))
 #align equicontinuous_at.continuous_at EquicontinuousAt.continuousAt
+-/
 
+#print Set.EquicontinuousAt.continuousAt_of_mem /-
 protected theorem Set.EquicontinuousAt.continuousAt_of_mem {H : Set <| X → α} {x₀ : X}
     (h : H.EquicontinuousAt x₀) {f : X → α} (hf : f ∈ H) : ContinuousAt f x₀ :=
   h.ContinuousAt ⟨f, hf⟩
 #align set.equicontinuous_at.continuous_at_of_mem Set.EquicontinuousAt.continuousAt_of_mem
+-/
 
+#print Equicontinuous.continuous /-
 /-- Each function of an equicontinuous family is continuous. -/
 theorem Equicontinuous.continuous {F : ι → X → α} (h : Equicontinuous F) (i : ι) :
     Continuous (F i) :=
   continuous_iff_continuousAt.mpr fun x => (h x).ContinuousAt i
 #align equicontinuous.continuous Equicontinuous.continuous
+-/
 
+#print Set.Equicontinuous.continuous_of_mem /-
 protected theorem Set.Equicontinuous.continuous_of_mem {H : Set <| X → α} (h : H.Equicontinuous)
     {f : X → α} (hf : f ∈ H) : Continuous f :=
   h.Continuous ⟨f, hf⟩
 #align set.equicontinuous.continuous_of_mem Set.Equicontinuous.continuous_of_mem
+-/
 
+#print UniformEquicontinuous.uniformContinuous /-
 /-- Each function of a uniformly equicontinuous family is uniformly continuous. -/
 theorem UniformEquicontinuous.uniformContinuous {F : ι → β → α} (h : UniformEquicontinuous F)
     (i : ι) : UniformContinuous (F i) := fun U hU =>
   mem_map.mpr (mem_of_superset (h U hU) fun xy hxy => hxy i)
 #align uniform_equicontinuous.uniform_continuous UniformEquicontinuous.uniformContinuous
+-/
 
+#print Set.UniformEquicontinuous.uniformContinuous_of_mem /-
 protected theorem Set.UniformEquicontinuous.uniformContinuous_of_mem {H : Set <| β → α}
     (h : H.UniformEquicontinuous) {f : β → α} (hf : f ∈ H) : UniformContinuous f :=
   h.UniformContinuous ⟨f, hf⟩
 #align set.uniform_equicontinuous.uniform_continuous_of_mem Set.UniformEquicontinuous.uniformContinuous_of_mem
+-/
 
+#print EquicontinuousAt.comp /-
 /-- Taking sub-families preserves equicontinuity at a point. -/
 theorem EquicontinuousAt.comp {F : ι → X → α} {x₀ : X} (h : EquicontinuousAt F x₀) (u : κ → ι) :
     EquicontinuousAt (F ∘ u) x₀ := fun U hU => (h U hU).mono fun x H k => H (u k)
 #align equicontinuous_at.comp EquicontinuousAt.comp
+-/
 
+#print Set.EquicontinuousAt.mono /-
 protected theorem Set.EquicontinuousAt.mono {H H' : Set <| X → α} {x₀ : X}
     (h : H.EquicontinuousAt x₀) (hH : H' ⊆ H) : H'.EquicontinuousAt x₀ :=
   h.comp (inclusion hH)
 #align set.equicontinuous_at.mono Set.EquicontinuousAt.mono
+-/
 
+#print Equicontinuous.comp /-
 /-- Taking sub-families preserves equicontinuity. -/
 theorem Equicontinuous.comp {F : ι → X → α} (h : Equicontinuous F) (u : κ → ι) :
     Equicontinuous (F ∘ u) := fun x => (h x).comp u
 #align equicontinuous.comp Equicontinuous.comp
+-/
 
+#print Set.Equicontinuous.mono /-
 protected theorem Set.Equicontinuous.mono {H H' : Set <| X → α} (h : H.Equicontinuous)
     (hH : H' ⊆ H) : H'.Equicontinuous :=
   h.comp (inclusion hH)
 #align set.equicontinuous.mono Set.Equicontinuous.mono
+-/
 
+#print UniformEquicontinuous.comp /-
 /-- Taking sub-families preserves uniform equicontinuity. -/
 theorem UniformEquicontinuous.comp {F : ι → β → α} (h : UniformEquicontinuous F) (u : κ → ι) :
     UniformEquicontinuous (F ∘ u) := fun U hU => (h U hU).mono fun x H k => H (u k)
 #align uniform_equicontinuous.comp UniformEquicontinuous.comp
+-/
 
+#print Set.UniformEquicontinuous.mono /-
 protected theorem Set.UniformEquicontinuous.mono {H H' : Set <| β → α} (h : H.UniformEquicontinuous)
     (hH : H' ⊆ H) : H'.UniformEquicontinuous :=
   h.comp (inclusion hH)
 #align set.uniform_equicontinuous.mono Set.UniformEquicontinuous.mono
+-/
 
+#print equicontinuousAt_iff_range /-
 /-- A family `𝓕 : ι → X → α` is equicontinuous at `x₀` iff `range 𝓕` is equicontinuous at `x₀`,
 i.e the family `coe : range F → X → α` is equicontinuous at `x₀`. -/
 theorem equicontinuousAt_iff_range {F : ι → X → α} {x₀ : X} :
@@ -240,14 +269,18 @@ theorem equicontinuousAt_iff_range {F : ι → X → α} {x₀ : X} :
   ⟨fun h => by rw [← comp_range_splitting F] <;> exact h.comp _, fun h =>
     h.comp (rangeFactorization F)⟩
 #align equicontinuous_at_iff_range equicontinuousAt_iff_range
+-/
 
+#print equicontinuous_iff_range /-
 /-- A family `𝓕 : ι → X → α` is equicontinuous iff `range 𝓕` is equicontinuous,
 i.e the family `coe : range F → X → α` is equicontinuous. -/
 theorem equicontinuous_iff_range {F : ι → X → α} :
     Equicontinuous F ↔ Equicontinuous (coe : range F → X → α) :=
   forall_congr' fun x₀ => equicontinuousAt_iff_range
 #align equicontinuous_iff_range equicontinuous_iff_range
+-/
 
+#print uniformEquicontinuous_at_iff_range /-
 /-- A family `𝓕 : ι → β → α` is uniformly equicontinuous iff `range 𝓕` is uniformly equicontinuous,
 i.e the family `coe : range F → β → α` is uniformly equicontinuous. -/
 theorem uniformEquicontinuous_at_iff_range {F : ι → β → α} :
@@ -255,11 +288,13 @@ theorem uniformEquicontinuous_at_iff_range {F : ι → β → α} :
   ⟨fun h => by rw [← comp_range_splitting F] <;> exact h.comp _, fun h =>
     h.comp (rangeFactorization F)⟩
 #align uniform_equicontinuous_at_iff_range uniformEquicontinuous_at_iff_range
+-/
 
 section
 
 open UniformFun
 
+#print equicontinuousAt_iff_continuousAt /-
 /-- A family `𝓕 : ι → X → α` is equicontinuous at `x₀` iff the function `swap 𝓕 : X → ι → α` is
 continuous at `x₀` *when `ι → α` is equipped with the topology of uniform convergence*. This is
 very useful for developping the equicontinuity API, but it should not be used directly for other
@@ -268,7 +303,9 @@ theorem equicontinuousAt_iff_continuousAt {F : ι → X → α} {x₀ : X} :
     EquicontinuousAt F x₀ ↔ ContinuousAt (ofFun ∘ Function.swap F : X → ι →ᵤ α) x₀ := by
   rw [ContinuousAt, (UniformFun.hasBasis_nhds ι α _).tendsto_right_iff] <;> rfl
 #align equicontinuous_at_iff_continuous_at equicontinuousAt_iff_continuousAt
+-/
 
+#print equicontinuous_iff_continuous /-
 /-- A family `𝓕 : ι → X → α` is equicontinuous iff the function `swap 𝓕 : X → ι → α` is
 continuous *when `ι → α` is equipped with the topology of uniform convergence*. This is
 very useful for developping the equicontinuity API, but it should not be used directly for other
@@ -277,7 +314,9 @@ theorem equicontinuous_iff_continuous {F : ι → X → α} :
     Equicontinuous F ↔ Continuous (ofFun ∘ Function.swap F : X → ι →ᵤ α) := by
   simp_rw [Equicontinuous, continuous_iff_continuousAt, equicontinuousAt_iff_continuousAt]
 #align equicontinuous_iff_continuous equicontinuous_iff_continuous
+-/
 
+#print uniformEquicontinuous_iff_uniformContinuous /-
 /-- A family `𝓕 : ι → β → α` is uniformly equicontinuous iff the function `swap 𝓕 : β → ι → α` is
 uniformly continuous *when `ι → α` is equipped with the uniform structure of uniform convergence*.
 This is very useful for developping the equicontinuity API, but it should not be used directly
@@ -286,7 +325,9 @@ theorem uniformEquicontinuous_iff_uniformContinuous {F : ι → β → α} :
     UniformEquicontinuous F ↔ UniformContinuous (ofFun ∘ Function.swap F : β → ι →ᵤ α) := by
   rw [UniformContinuous, (UniformFun.hasBasis_uniformity ι α).tendsto_right_iff] <;> rfl
 #align uniform_equicontinuous_iff_uniform_continuous uniformEquicontinuous_iff_uniformContinuous
+-/
 
+#print Filter.HasBasis.equicontinuousAt_iff_left /-
 theorem Filter.HasBasis.equicontinuousAt_iff_left {κ : Type _} {p : κ → Prop} {s : κ → Set X}
     {F : ι → X → α} {x₀ : X} (hX : (𝓝 x₀).HasBasis p s) :
     EquicontinuousAt F x₀ ↔ ∀ U ∈ 𝓤 α, ∃ (k : _) (_ : p k), ∀ x ∈ s k, ∀ i, (F i x₀, F i x) ∈ U :=
@@ -295,7 +336,9 @@ theorem Filter.HasBasis.equicontinuousAt_iff_left {κ : Type _} {p : κ → Prop
     hX.tendsto_iff (UniformFun.hasBasis_nhds ι α _)]
   rfl
 #align filter.has_basis.equicontinuous_at_iff_left Filter.HasBasis.equicontinuousAt_iff_left
+-/
 
+#print Filter.HasBasis.equicontinuousAt_iff_right /-
 theorem Filter.HasBasis.equicontinuousAt_iff_right {κ : Type _} {p : κ → Prop} {s : κ → Set (α × α)}
     {F : ι → X → α} {x₀ : X} (hα : (𝓤 α).HasBasis p s) :
     EquicontinuousAt F x₀ ↔ ∀ k, p k → ∀ᶠ x in 𝓝 x₀, ∀ i, (F i x₀, F i x) ∈ s k :=
@@ -304,7 +347,9 @@ theorem Filter.HasBasis.equicontinuousAt_iff_right {κ : Type _} {p : κ → Pro
     (UniformFun.hasBasis_nhds_of_basis ι α _ hα).tendsto_right_iff]
   rfl
 #align filter.has_basis.equicontinuous_at_iff_right Filter.HasBasis.equicontinuousAt_iff_right
+-/
 
+#print Filter.HasBasis.equicontinuousAt_iff /-
 theorem Filter.HasBasis.equicontinuousAt_iff {κ₁ κ₂ : Type _} {p₁ : κ₁ → Prop} {s₁ : κ₁ → Set X}
     {p₂ : κ₂ → Prop} {s₂ : κ₂ → Set (α × α)} {F : ι → X → α} {x₀ : X} (hX : (𝓝 x₀).HasBasis p₁ s₁)
     (hα : (𝓤 α).HasBasis p₂ s₂) :
@@ -315,7 +360,9 @@ theorem Filter.HasBasis.equicontinuousAt_iff {κ₁ κ₂ : Type _} {p₁ : κ
     hX.tendsto_iff (UniformFun.hasBasis_nhds_of_basis ι α _ hα)]
   rfl
 #align filter.has_basis.equicontinuous_at_iff Filter.HasBasis.equicontinuousAt_iff
+-/
 
+#print Filter.HasBasis.uniformEquicontinuous_iff_left /-
 theorem Filter.HasBasis.uniformEquicontinuous_iff_left {κ : Type _} {p : κ → Prop}
     {s : κ → Set (β × β)} {F : ι → β → α} (hβ : (𝓤 β).HasBasis p s) :
     UniformEquicontinuous F ↔
@@ -326,7 +373,9 @@ theorem Filter.HasBasis.uniformEquicontinuous_iff_left {κ : Type _} {p : κ →
   simp_rw [Prod.forall]
   rfl
 #align filter.has_basis.uniform_equicontinuous_iff_left Filter.HasBasis.uniformEquicontinuous_iff_left
+-/
 
+#print Filter.HasBasis.uniformEquicontinuous_iff_right /-
 theorem Filter.HasBasis.uniformEquicontinuous_iff_right {κ : Type _} {p : κ → Prop}
     {s : κ → Set (α × α)} {F : ι → β → α} (hα : (𝓤 α).HasBasis p s) :
     UniformEquicontinuous F ↔ ∀ k, p k → ∀ᶠ xy : β × β in 𝓤 β, ∀ i, (F i xy.1, F i xy.2) ∈ s k :=
@@ -335,7 +384,9 @@ theorem Filter.HasBasis.uniformEquicontinuous_iff_right {κ : Type _} {p : κ 
     (UniformFun.hasBasis_uniformity_of_basis ι α hα).tendsto_right_iff]
   rfl
 #align filter.has_basis.uniform_equicontinuous_iff_right Filter.HasBasis.uniformEquicontinuous_iff_right
+-/
 
+#print Filter.HasBasis.uniformEquicontinuous_iff /-
 theorem Filter.HasBasis.uniformEquicontinuous_iff {κ₁ κ₂ : Type _} {p₁ : κ₁ → Prop}
     {s₁ : κ₁ → Set (β × β)} {p₂ : κ₂ → Prop} {s₂ : κ₂ → Set (α × α)} {F : ι → β → α}
     (hβ : (𝓤 β).HasBasis p₁ s₁) (hα : (𝓤 α).HasBasis p₂ s₂) :
@@ -347,7 +398,9 @@ theorem Filter.HasBasis.uniformEquicontinuous_iff {κ₁ κ₂ : Type _} {p₁ :
   simp_rw [Prod.forall]
   rfl
 #align filter.has_basis.uniform_equicontinuous_iff Filter.HasBasis.uniformEquicontinuous_iff
+-/
 
+#print UniformInducing.equicontinuousAt_iff /-
 /-- Given `u : α → β` a uniform inducing map, a family `𝓕 : ι → X → α` is equicontinuous at a point
 `x₀ : X` iff the family `𝓕'`, obtained by precomposing each function of `𝓕` by `u`, is
 equicontinuous at `x₀`. -/
@@ -358,8 +411,10 @@ theorem UniformInducing.equicontinuousAt_iff {F : ι → X → α} {x₀ : X} {u
   rw [equicontinuousAt_iff_continuousAt, equicontinuousAt_iff_continuousAt, this.continuous_at_iff]
   rfl
 #align uniform_inducing.equicontinuous_at_iff UniformInducing.equicontinuousAt_iff
+-/
 
 /- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:73:14: unsupported tactic `congrm #[[expr ∀ x, (_ : exprProp())]] -/
+#print UniformInducing.equicontinuous_iff /-
 /-- Given `u : α → β` a uniform inducing map, a family `𝓕 : ι → X → α` is equicontinuous iff the
 family `𝓕'`, obtained by precomposing each function of `𝓕` by `u`, is equicontinuous. -/
 theorem UniformInducing.equicontinuous_iff {F : ι → X → α} {u : α → β} (hu : UniformInducing u) :
@@ -369,7 +424,9 @@ theorem UniformInducing.equicontinuous_iff {F : ι → X → α} {u : α → β}
     "./././Mathport/Syntax/Translate/Tactic/Builtin.lean:73:14: unsupported tactic `congrm #[[expr ∀ x, (_ : exprProp())]]"
   rw [hu.equicontinuous_at_iff]
 #align uniform_inducing.equicontinuous_iff UniformInducing.equicontinuous_iff
+-/
 
+#print UniformInducing.uniformEquicontinuous_iff /-
 /-- Given `u : α → γ` a uniform inducing map, a family `𝓕 : ι → β → α` is uniformly equicontinuous
 iff the family `𝓕'`, obtained by precomposing each function of `𝓕` by `u`, is uniformly
 equicontinuous. -/
@@ -381,7 +438,9 @@ theorem UniformInducing.uniformEquicontinuous_iff {F : ι → β → α} {u : α
     this.uniform_continuous_iff]
   rfl
 #align uniform_inducing.uniform_equicontinuous_iff UniformInducing.uniformEquicontinuous_iff
+-/
 
+#print EquicontinuousAt.closure' /-
 /-- A version of `equicontinuous_at.closure` applicable to subsets of types which embed continuously
 into `X → α` with the product topology. It turns out we don't need any other condition on the
 embedding than continuity, but in practice this will mostly be applied to `fun_like` types where
@@ -398,14 +457,18 @@ theorem EquicontinuousAt.closure' {A : Set Y} {u : Y → X → α} {x₀ : X}
   refine' (closure_minimal hx <| hVclosed.preimage <| _).trans (preimage_mono hVU)
   exact Continuous.prod_mk ((continuous_apply x₀).comp hu) ((continuous_apply x).comp hu)
 #align equicontinuous_at.closure' EquicontinuousAt.closure'
+-/
 
+#print EquicontinuousAt.closure /-
 /-- If a set of functions is equicontinuous at some `x₀`, its closure for the product topology is
 also equicontinuous at `x₀`. -/
 theorem EquicontinuousAt.closure {A : Set <| X → α} {x₀ : X} (hA : A.EquicontinuousAt x₀) :
     (closure A).EquicontinuousAt x₀ :=
   @EquicontinuousAt.closure' _ _ _ _ _ _ _ id _ hA continuous_id
 #align equicontinuous_at.closure EquicontinuousAt.closure
+-/
 
+#print Filter.Tendsto.continuousAt_of_equicontinuousAt /-
 /-- If `𝓕 : ι → X → α` tends to `f : X → α` *pointwise* along some nontrivial filter, and if the
 family `𝓕` is equicontinuous at some `x₀ : X`, then the limit is continuous at `x₀`. -/
 theorem Filter.Tendsto.continuousAt_of_equicontinuousAt {l : Filter ι} [l.ne_bot] {F : ι → X → α}
@@ -414,7 +477,9 @@ theorem Filter.Tendsto.continuousAt_of_equicontinuousAt {l : Filter ι} [l.ne_bo
   (equicontinuousAt_iff_range.mp h₂).closure.ContinuousAt
     ⟨f, mem_closure_of_tendsto h₁ <| eventually_of_forall mem_range_self⟩
 #align filter.tendsto.continuous_at_of_equicontinuous_at Filter.Tendsto.continuousAt_of_equicontinuousAt
+-/
 
+#print Equicontinuous.closure' /-
 /-- A version of `equicontinuous.closure` applicable to subsets of types which embed continuously
 into `X → α` with the product topology. It turns out we don't need any other condition on the
 embedding than continuity, but in practice this will mostly be applied to `fun_like` types where
@@ -423,20 +488,26 @@ theorem Equicontinuous.closure' {A : Set Y} {u : Y → X → α}
     (hA : Equicontinuous (u ∘ coe : A → X → α)) (hu : Continuous u) :
     Equicontinuous (u ∘ coe : closure A → X → α) := fun x => (hA x).closure' hu
 #align equicontinuous.closure' Equicontinuous.closure'
+-/
 
+#print Equicontinuous.closure /-
 /-- If a set of functions is equicontinuous, its closure for the product topology is also
 equicontinuous. -/
 theorem Equicontinuous.closure {A : Set <| X → α} (hA : A.Equicontinuous) :
     (closure A).Equicontinuous := fun x => (hA x).closure
 #align equicontinuous.closure Equicontinuous.closure
+-/
 
+#print Filter.Tendsto.continuous_of_equicontinuous_at /-
 /-- If `𝓕 : ι → X → α` tends to `f : X → α` *pointwise* along some nontrivial filter, and if the
 family `𝓕` is equicontinuous, then the limit is continuous. -/
 theorem Filter.Tendsto.continuous_of_equicontinuous_at {l : Filter ι} [l.ne_bot] {F : ι → X → α}
     {f : X → α} (h₁ : Tendsto F l (𝓝 f)) (h₂ : Equicontinuous F) : Continuous f :=
   continuous_iff_continuousAt.mpr fun x => h₁.continuousAt_of_equicontinuousAt (h₂ x)
 #align filter.tendsto.continuous_of_equicontinuous_at Filter.Tendsto.continuous_of_equicontinuous_at
+-/
 
+#print UniformEquicontinuous.closure' /-
 /-- A version of `uniform_equicontinuous.closure` applicable to subsets of types which embed
 continuously into `β → α` with the product topology. It turns out we don't need any other condition
 on the embedding than continuity, but in practice this will mostly be applied to `fun_like` types
@@ -454,14 +525,18 @@ theorem UniformEquicontinuous.closure' {A : Set Y} {u : Y → β → α}
   refine' (closure_minimal hxy <| hVclosed.preimage <| _).trans (preimage_mono hVU)
   exact Continuous.prod_mk ((continuous_apply x).comp hu) ((continuous_apply y).comp hu)
 #align uniform_equicontinuous.closure' UniformEquicontinuous.closure'
+-/
 
+#print UniformEquicontinuous.closure /-
 /-- If a set of functions is uniformly equicontinuous, its closure for the product topology is also
 uniformly equicontinuous. -/
 theorem UniformEquicontinuous.closure {A : Set <| β → α} (hA : A.UniformEquicontinuous) :
     (closure A).UniformEquicontinuous :=
   @UniformEquicontinuous.closure' _ _ _ _ _ _ _ id hA continuous_id
 #align uniform_equicontinuous.closure UniformEquicontinuous.closure
+-/
 
+#print Filter.Tendsto.uniformContinuous_of_uniformEquicontinuous /-
 /-- If `𝓕 : ι → β → α` tends to `f : β → α` *pointwise* along some nontrivial filter, and if the
 family `𝓕` is uniformly equicontinuous, then the limit is uniformly continuous. -/
 theorem Filter.Tendsto.uniformContinuous_of_uniformEquicontinuous {l : Filter ι} [l.ne_bot]
@@ -470,6 +545,7 @@ theorem Filter.Tendsto.uniformContinuous_of_uniformEquicontinuous {l : Filter ι
   (uniformEquicontinuous_at_iff_range.mp h₂).closure.UniformContinuous
     ⟨f, mem_closure_of_tendsto h₁ <| eventually_of_forall mem_range_self⟩
 #align filter.tendsto.uniform_continuous_of_uniform_equicontinuous Filter.Tendsto.uniformContinuous_of_uniformEquicontinuous
+-/
 
 end
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/31c24aa72e7b3e5ed97a8412470e904f82b81004

@@ -146,7 +146,7 @@ protected abbrev Set.UniformEquicontinuous (H : Set <| β → α) : Prop :=
 #align set.uniform_equicontinuous Set.UniformEquicontinuous
 -/
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (x y «expr ∈ » V) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:638:2: warning: expanding binder collection (x y «expr ∈ » V) -/
 /-- Reformulation of equicontinuity at `x₀` comparing two variables near `x₀` instead of comparing
 only one with `x₀`. -/
 theorem equicontinuousAt_iff_pair {F : ι → X → α} {x₀ : X} :

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

@@ -158,7 +158,7 @@ theorem equicontinuousAt_iff_pair {F : ι → X → α} {x₀ : X} :
     refine' ⟨_, H V hV, fun x hx y hy i => hVU (prod_mk_mem_compRel _ (hy i))⟩
     exact hVsymm.mk_mem_comm.mp (hx i)
   · rcases H U hU with ⟨V, hV, hVU⟩
-    filter_upwards [hV]using fun x hx i => hVU x₀ (mem_of_mem_nhds hV) x hx i
+    filter_upwards [hV] using fun x hx i => hVU x₀ (mem_of_mem_nhds hV) x hx i
 #align equicontinuous_at_iff_pair equicontinuousAt_iff_pair
 
 /-- Uniform equicontinuity implies equicontinuity. -/
@@ -392,7 +392,7 @@ theorem EquicontinuousAt.closure' {A : Set Y} {u : Y → X → α} {x₀ : X}
   by
   intro U hU
   rcases mem_uniformity_isClosed hU with ⟨V, hV, hVclosed, hVU⟩
-  filter_upwards [hA V hV]with x hx
+  filter_upwards [hA V hV] with x hx
   rw [SetCoe.forall] at *
   change A ⊆ (fun f => (u f x₀, u f x)) ⁻¹' V at hx 
   refine' (closure_minimal hx <| hVclosed.preimage <| _).trans (preimage_mono hVU)

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

@@ -171,7 +171,7 @@ theorem UniformEquicontinuous.equicontinuous {F : ι → β → α} (h : Uniform
 theorem EquicontinuousAt.continuousAt {F : ι → X → α} {x₀ : X} (h : EquicontinuousAt F x₀) (i : ι) :
     ContinuousAt (F i) x₀ := by
   intro U hU
-  rw [UniformSpace.mem_nhds_iff] at hU
+  rw [UniformSpace.mem_nhds_iff] at hU 
   rcases hU with ⟨V, hV₁, hV₂⟩
   exact mem_map.mpr (mem_of_superset (h V hV₁) fun x hx => hV₂ (hx i))
 #align equicontinuous_at.continuous_at EquicontinuousAt.continuousAt
@@ -289,7 +289,7 @@ theorem uniformEquicontinuous_iff_uniformContinuous {F : ι → β → α} :
 
 theorem Filter.HasBasis.equicontinuousAt_iff_left {κ : Type _} {p : κ → Prop} {s : κ → Set X}
     {F : ι → X → α} {x₀ : X} (hX : (𝓝 x₀).HasBasis p s) :
-    EquicontinuousAt F x₀ ↔ ∀ U ∈ 𝓤 α, ∃ (k : _)(_ : p k), ∀ x ∈ s k, ∀ i, (F i x₀, F i x) ∈ U :=
+    EquicontinuousAt F x₀ ↔ ∀ U ∈ 𝓤 α, ∃ (k : _) (_ : p k), ∀ x ∈ s k, ∀ i, (F i x₀, F i x) ∈ U :=
   by
   rw [equicontinuousAt_iff_continuousAt, ContinuousAt,
     hX.tendsto_iff (UniformFun.hasBasis_nhds ι α _)]
@@ -309,7 +309,7 @@ theorem Filter.HasBasis.equicontinuousAt_iff {κ₁ κ₂ : Type _} {p₁ : κ
     {p₂ : κ₂ → Prop} {s₂ : κ₂ → Set (α × α)} {F : ι → X → α} {x₀ : X} (hX : (𝓝 x₀).HasBasis p₁ s₁)
     (hα : (𝓤 α).HasBasis p₂ s₂) :
     EquicontinuousAt F x₀ ↔
-      ∀ k₂, p₂ k₂ → ∃ (k₁ : _)(_ : p₁ k₁), ∀ x ∈ s₁ k₁, ∀ i, (F i x₀, F i x) ∈ s₂ k₂ :=
+      ∀ k₂, p₂ k₂ → ∃ (k₁ : _) (_ : p₁ k₁), ∀ x ∈ s₁ k₁, ∀ i, (F i x₀, F i x) ∈ s₂ k₂ :=
   by
   rw [equicontinuousAt_iff_continuousAt, ContinuousAt,
     hX.tendsto_iff (UniformFun.hasBasis_nhds_of_basis ι α _ hα)]
@@ -319,7 +319,7 @@ theorem Filter.HasBasis.equicontinuousAt_iff {κ₁ κ₂ : Type _} {p₁ : κ
 theorem Filter.HasBasis.uniformEquicontinuous_iff_left {κ : Type _} {p : κ → Prop}
     {s : κ → Set (β × β)} {F : ι → β → α} (hβ : (𝓤 β).HasBasis p s) :
     UniformEquicontinuous F ↔
-      ∀ U ∈ 𝓤 α, ∃ (k : _)(_ : p k), ∀ x y, (x, y) ∈ s k → ∀ i, (F i x, F i y) ∈ U :=
+      ∀ U ∈ 𝓤 α, ∃ (k : _) (_ : p k), ∀ x y, (x, y) ∈ s k → ∀ i, (F i x, F i y) ∈ U :=
   by
   rw [uniformEquicontinuous_iff_uniformContinuous, UniformContinuous,
     hβ.tendsto_iff (UniformFun.hasBasis_uniformity ι α)]
@@ -340,7 +340,7 @@ theorem Filter.HasBasis.uniformEquicontinuous_iff {κ₁ κ₂ : Type _} {p₁ :
     {s₁ : κ₁ → Set (β × β)} {p₂ : κ₂ → Prop} {s₂ : κ₂ → Set (α × α)} {F : ι → β → α}
     (hβ : (𝓤 β).HasBasis p₁ s₁) (hα : (𝓤 α).HasBasis p₂ s₂) :
     UniformEquicontinuous F ↔
-      ∀ k₂, p₂ k₂ → ∃ (k₁ : _)(_ : p₁ k₁), ∀ x y, (x, y) ∈ s₁ k₁ → ∀ i, (F i x, F i y) ∈ s₂ k₂ :=
+      ∀ k₂, p₂ k₂ → ∃ (k₁ : _) (_ : p₁ k₁), ∀ x y, (x, y) ∈ s₁ k₁ → ∀ i, (F i x, F i y) ∈ s₂ k₂ :=
   by
   rw [uniformEquicontinuous_iff_uniformContinuous, UniformContinuous,
     hβ.tendsto_iff (UniformFun.hasBasis_uniformity_of_basis ι α hα)]
@@ -394,7 +394,7 @@ theorem EquicontinuousAt.closure' {A : Set Y} {u : Y → X → α} {x₀ : X}
   rcases mem_uniformity_isClosed hU with ⟨V, hV, hVclosed, hVU⟩
   filter_upwards [hA V hV]with x hx
   rw [SetCoe.forall] at *
-  change A ⊆ (fun f => (u f x₀, u f x)) ⁻¹' V at hx
+  change A ⊆ (fun f => (u f x₀, u f x)) ⁻¹' V at hx 
   refine' (closure_minimal hx <| hVclosed.preimage <| _).trans (preimage_mono hVU)
   exact Continuous.prod_mk ((continuous_apply x₀).comp hu) ((continuous_apply x).comp hu)
 #align equicontinuous_at.closure' EquicontinuousAt.closure'
@@ -450,7 +450,7 @@ theorem UniformEquicontinuous.closure' {A : Set Y} {u : Y → β → α}
   filter_upwards [hA V hV]
   rintro ⟨x, y⟩ hxy
   rw [SetCoe.forall] at *
-  change A ⊆ (fun f => (u f x, u f y)) ⁻¹' V at hxy
+  change A ⊆ (fun f => (u f x, u f y)) ⁻¹' V at hxy 
   refine' (closure_minimal hxy <| hVclosed.preimage <| _).trans (preimage_mono hVU)
   exact Continuous.prod_mk ((continuous_apply x).comp hu) ((continuous_apply y).comp hu)
 #align uniform_equicontinuous.closure' UniformEquicontinuous.closure'

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

@@ -91,7 +91,7 @@ section
 
 open UniformSpace Filter Set
 
-open uniformity Topology UniformConvergence
+open scoped uniformity Topology UniformConvergence
 
 variable {ι κ X Y Z α β γ 𝓕 : Type _} [TopologicalSpace X] [TopologicalSpace Y] [TopologicalSpace Z]
   [UniformSpace α] [UniformSpace β] [UniformSpace γ]

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

@@ -146,12 +146,6 @@ protected abbrev Set.UniformEquicontinuous (H : Set <| β → α) : Prop :=
 #align set.uniform_equicontinuous Set.UniformEquicontinuous
 -/
 
-/- warning: equicontinuous_at_iff_pair -> equicontinuousAt_iff_pair is a dubious translation:
-lean 3 declaration is
-  forall {ι : Type.{u1}} {X : Type.{u2}} {α : Type.{u3}} [_inst_1 : TopologicalSpace.{u2} X] [_inst_4 : UniformSpace.{u3} α] {F : ι -> X -> α} {x₀ : X}, Iff (EquicontinuousAt.{u1, u2, u3} ι X α _inst_1 _inst_4 F x₀) (forall (U : Set.{u3} (Prod.{u3, u3} α α)), (Membership.Mem.{u3, u3} (Set.{u3} (Prod.{u3, u3} α α)) (Filter.{u3} (Prod.{u3, u3} α α)) (Filter.hasMem.{u3} (Prod.{u3, u3} α α)) U (uniformity.{u3} α _inst_4)) -> (Exists.{succ u2} (Set.{u2} X) (fun (V : Set.{u2} X) => Exists.{0} (Membership.Mem.{u2, u2} (Set.{u2} X) (Filter.{u2} X) (Filter.hasMem.{u2} X) V (nhds.{u2} X _inst_1 x₀)) (fun (H : Membership.Mem.{u2, u2} (Set.{u2} X) (Filter.{u2} X) (Filter.hasMem.{u2} X) V (nhds.{u2} X _inst_1 x₀)) => forall (x : X), (Membership.Mem.{u2, u2} X (Set.{u2} X) (Set.hasMem.{u2} X) x V) -> (forall (y : X), (Membership.Mem.{u2, u2} X (Set.{u2} X) (Set.hasMem.{u2} X) y V) -> (forall (i : ι), Membership.Mem.{u3, u3} (Prod.{u3, u3} α α) (Set.{u3} (Prod.{u3, u3} α α)) (Set.hasMem.{u3} (Prod.{u3, u3} α α)) (Prod.mk.{u3, u3} α α (F i x) (F i y)) U))))))
-but is expected to have type
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 /- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (x y «expr ∈ » V) -/
 /-- Reformulation of equicontinuity at `x₀` comparing two variables near `x₀` instead of comparing
 only one with `x₀`. -/
@@ -167,24 +161,12 @@ theorem equicontinuousAt_iff_pair {F : ι → X → α} {x₀ : X} :
     filter_upwards [hV]using fun x hx i => hVU x₀ (mem_of_mem_nhds hV) x hx i
 #align equicontinuous_at_iff_pair equicontinuousAt_iff_pair
 
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 /-- Uniform equicontinuity implies equicontinuity. -/
 theorem UniformEquicontinuous.equicontinuous {F : ι → β → α} (h : UniformEquicontinuous F) :
     Equicontinuous F := fun x₀ U hU =>
   mem_of_superset (ball_mem_nhds x₀ (h U hU)) fun x hx i => hx i
 #align uniform_equicontinuous.equicontinuous UniformEquicontinuous.equicontinuous
 
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 /-- Each function of a family equicontinuous at `x₀` is continuous at `x₀`. -/
 theorem EquicontinuousAt.continuousAt {F : ι → X → α} {x₀ : X} (h : EquicontinuousAt F x₀) (i : ι) :
     ContinuousAt (F i) x₀ := by
@@ -194,135 +176,63 @@ theorem EquicontinuousAt.continuousAt {F : ι → X → α} {x₀ : X} (h : Equi
   exact mem_map.mpr (mem_of_superset (h V hV₁) fun x hx => hV₂ (hx i))
 #align equicontinuous_at.continuous_at EquicontinuousAt.continuousAt
 
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 protected theorem Set.EquicontinuousAt.continuousAt_of_mem {H : Set <| X → α} {x₀ : X}
     (h : H.EquicontinuousAt x₀) {f : X → α} (hf : f ∈ H) : ContinuousAt f x₀ :=
   h.ContinuousAt ⟨f, hf⟩
 #align set.equicontinuous_at.continuous_at_of_mem Set.EquicontinuousAt.continuousAt_of_mem
 
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 /-- Each function of an equicontinuous family is continuous. -/
 theorem Equicontinuous.continuous {F : ι → X → α} (h : Equicontinuous F) (i : ι) :
     Continuous (F i) :=
   continuous_iff_continuousAt.mpr fun x => (h x).ContinuousAt i
 #align equicontinuous.continuous Equicontinuous.continuous
 
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 protected theorem Set.Equicontinuous.continuous_of_mem {H : Set <| X → α} (h : H.Equicontinuous)
     {f : X → α} (hf : f ∈ H) : Continuous f :=
   h.Continuous ⟨f, hf⟩
 #align set.equicontinuous.continuous_of_mem Set.Equicontinuous.continuous_of_mem
 
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 /-- Each function of a uniformly equicontinuous family is uniformly continuous. -/
 theorem UniformEquicontinuous.uniformContinuous {F : ι → β → α} (h : UniformEquicontinuous F)
     (i : ι) : UniformContinuous (F i) := fun U hU =>
   mem_map.mpr (mem_of_superset (h U hU) fun xy hxy => hxy i)
 #align uniform_equicontinuous.uniform_continuous UniformEquicontinuous.uniformContinuous
 
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 protected theorem Set.UniformEquicontinuous.uniformContinuous_of_mem {H : Set <| β → α}
     (h : H.UniformEquicontinuous) {f : β → α} (hf : f ∈ H) : UniformContinuous f :=
   h.UniformContinuous ⟨f, hf⟩
 #align set.uniform_equicontinuous.uniform_continuous_of_mem Set.UniformEquicontinuous.uniformContinuous_of_mem
 
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 /-- Taking sub-families preserves equicontinuity at a point. -/
 theorem EquicontinuousAt.comp {F : ι → X → α} {x₀ : X} (h : EquicontinuousAt F x₀) (u : κ → ι) :
     EquicontinuousAt (F ∘ u) x₀ := fun U hU => (h U hU).mono fun x H k => H (u k)
 #align equicontinuous_at.comp EquicontinuousAt.comp
 
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 protected theorem Set.EquicontinuousAt.mono {H H' : Set <| X → α} {x₀ : X}
     (h : H.EquicontinuousAt x₀) (hH : H' ⊆ H) : H'.EquicontinuousAt x₀ :=
   h.comp (inclusion hH)
 #align set.equicontinuous_at.mono Set.EquicontinuousAt.mono
 
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 /-- Taking sub-families preserves equicontinuity. -/
 theorem Equicontinuous.comp {F : ι → X → α} (h : Equicontinuous F) (u : κ → ι) :
     Equicontinuous (F ∘ u) := fun x => (h x).comp u
 #align equicontinuous.comp Equicontinuous.comp
 
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 protected theorem Set.Equicontinuous.mono {H H' : Set <| X → α} (h : H.Equicontinuous)
     (hH : H' ⊆ H) : H'.Equicontinuous :=
   h.comp (inclusion hH)
 #align set.equicontinuous.mono Set.Equicontinuous.mono
 
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 /-- Taking sub-families preserves uniform equicontinuity. -/
 theorem UniformEquicontinuous.comp {F : ι → β → α} (h : UniformEquicontinuous F) (u : κ → ι) :
     UniformEquicontinuous (F ∘ u) := fun U hU => (h U hU).mono fun x H k => H (u k)
 #align uniform_equicontinuous.comp UniformEquicontinuous.comp
 
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 protected theorem Set.UniformEquicontinuous.mono {H H' : Set <| β → α} (h : H.UniformEquicontinuous)
     (hH : H' ⊆ H) : H'.UniformEquicontinuous :=
   h.comp (inclusion hH)
 #align set.uniform_equicontinuous.mono Set.UniformEquicontinuous.mono
 
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 /-- A family `𝓕 : ι → X → α` is equicontinuous at `x₀` iff `range 𝓕` is equicontinuous at `x₀`,
 i.e the family `coe : range F → X → α` is equicontinuous at `x₀`. -/
 theorem equicontinuousAt_iff_range {F : ι → X → α} {x₀ : X} :
@@ -331,12 +241,6 @@ theorem equicontinuousAt_iff_range {F : ι → X → α} {x₀ : X} :
     h.comp (rangeFactorization F)⟩
 #align equicontinuous_at_iff_range equicontinuousAt_iff_range
 
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 /-- A family `𝓕 : ι → X → α` is equicontinuous iff `range 𝓕` is equicontinuous,
 i.e the family `coe : range F → X → α` is equicontinuous. -/
 theorem equicontinuous_iff_range {F : ι → X → α} :
@@ -344,12 +248,6 @@ theorem equicontinuous_iff_range {F : ι → X → α} :
   forall_congr' fun x₀ => equicontinuousAt_iff_range
 #align equicontinuous_iff_range equicontinuous_iff_range
 
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 /-- A family `𝓕 : ι → β → α` is uniformly equicontinuous iff `range 𝓕` is uniformly equicontinuous,
 i.e the family `coe : range F → β → α` is uniformly equicontinuous. -/
 theorem uniformEquicontinuous_at_iff_range {F : ι → β → α} :
@@ -362,12 +260,6 @@ section
 
 open UniformFun
 
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 /-- A family `𝓕 : ι → X → α` is equicontinuous at `x₀` iff the function `swap 𝓕 : X → ι → α` is
 continuous at `x₀` *when `ι → α` is equipped with the topology of uniform convergence*. This is
 very useful for developping the equicontinuity API, but it should not be used directly for other
@@ -377,12 +269,6 @@ theorem equicontinuousAt_iff_continuousAt {F : ι → X → α} {x₀ : X} :
   rw [ContinuousAt, (UniformFun.hasBasis_nhds ι α _).tendsto_right_iff] <;> rfl
 #align equicontinuous_at_iff_continuous_at equicontinuousAt_iff_continuousAt
 
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 /-- A family `𝓕 : ι → X → α` is equicontinuous iff the function `swap 𝓕 : X → ι → α` is
 continuous *when `ι → α` is equipped with the topology of uniform convergence*. This is
 very useful for developping the equicontinuity API, but it should not be used directly for other
@@ -392,12 +278,6 @@ theorem equicontinuous_iff_continuous {F : ι → X → α} :
   simp_rw [Equicontinuous, continuous_iff_continuousAt, equicontinuousAt_iff_continuousAt]
 #align equicontinuous_iff_continuous equicontinuous_iff_continuous
 
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 /-- A family `𝓕 : ι → β → α` is uniformly equicontinuous iff the function `swap 𝓕 : β → ι → α` is
 uniformly continuous *when `ι → α` is equipped with the uniform structure of uniform convergence*.
 This is very useful for developping the equicontinuity API, but it should not be used directly
@@ -407,12 +287,6 @@ theorem uniformEquicontinuous_iff_uniformContinuous {F : ι → β → α} :
   rw [UniformContinuous, (UniformFun.hasBasis_uniformity ι α).tendsto_right_iff] <;> rfl
 #align uniform_equicontinuous_iff_uniform_continuous uniformEquicontinuous_iff_uniformContinuous
 
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 theorem Filter.HasBasis.equicontinuousAt_iff_left {κ : Type _} {p : κ → Prop} {s : κ → Set X}
     {F : ι → X → α} {x₀ : X} (hX : (𝓝 x₀).HasBasis p s) :
     EquicontinuousAt F x₀ ↔ ∀ U ∈ 𝓤 α, ∃ (k : _)(_ : p k), ∀ x ∈ s k, ∀ i, (F i x₀, F i x) ∈ U :=
@@ -422,12 +296,6 @@ theorem Filter.HasBasis.equicontinuousAt_iff_left {κ : Type _} {p : κ → Prop
   rfl
 #align filter.has_basis.equicontinuous_at_iff_left Filter.HasBasis.equicontinuousAt_iff_left
 
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 theorem Filter.HasBasis.equicontinuousAt_iff_right {κ : Type _} {p : κ → Prop} {s : κ → Set (α × α)}
     {F : ι → X → α} {x₀ : X} (hα : (𝓤 α).HasBasis p s) :
     EquicontinuousAt F x₀ ↔ ∀ k, p k → ∀ᶠ x in 𝓝 x₀, ∀ i, (F i x₀, F i x) ∈ s k :=
@@ -437,9 +305,6 @@ theorem Filter.HasBasis.equicontinuousAt_iff_right {κ : Type _} {p : κ → Pro
   rfl
 #align filter.has_basis.equicontinuous_at_iff_right Filter.HasBasis.equicontinuousAt_iff_right
 
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 theorem Filter.HasBasis.equicontinuousAt_iff {κ₁ κ₂ : Type _} {p₁ : κ₁ → Prop} {s₁ : κ₁ → Set X}
     {p₂ : κ₂ → Prop} {s₂ : κ₂ → Set (α × α)} {F : ι → X → α} {x₀ : X} (hX : (𝓝 x₀).HasBasis p₁ s₁)
     (hα : (𝓤 α).HasBasis p₂ s₂) :
@@ -451,12 +316,6 @@ theorem Filter.HasBasis.equicontinuousAt_iff {κ₁ κ₂ : Type _} {p₁ : κ
   rfl
 #align filter.has_basis.equicontinuous_at_iff Filter.HasBasis.equicontinuousAt_iff
 
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 theorem Filter.HasBasis.uniformEquicontinuous_iff_left {κ : Type _} {p : κ → Prop}
     {s : κ → Set (β × β)} {F : ι → β → α} (hβ : (𝓤 β).HasBasis p s) :
     UniformEquicontinuous F ↔
@@ -468,12 +327,6 @@ theorem Filter.HasBasis.uniformEquicontinuous_iff_left {κ : Type _} {p : κ →
   rfl
 #align filter.has_basis.uniform_equicontinuous_iff_left Filter.HasBasis.uniformEquicontinuous_iff_left
 
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 theorem Filter.HasBasis.uniformEquicontinuous_iff_right {κ : Type _} {p : κ → Prop}
     {s : κ → Set (α × α)} {F : ι → β → α} (hα : (𝓤 α).HasBasis p s) :
     UniformEquicontinuous F ↔ ∀ k, p k → ∀ᶠ xy : β × β in 𝓤 β, ∀ i, (F i xy.1, F i xy.2) ∈ s k :=
@@ -483,9 +336,6 @@ theorem Filter.HasBasis.uniformEquicontinuous_iff_right {κ : Type _} {p : κ 
   rfl
 #align filter.has_basis.uniform_equicontinuous_iff_right Filter.HasBasis.uniformEquicontinuous_iff_right
 
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 theorem Filter.HasBasis.uniformEquicontinuous_iff {κ₁ κ₂ : Type _} {p₁ : κ₁ → Prop}
     {s₁ : κ₁ → Set (β × β)} {p₂ : κ₂ → Prop} {s₂ : κ₂ → Set (α × α)} {F : ι → β → α}
     (hβ : (𝓤 β).HasBasis p₁ s₁) (hα : (𝓤 α).HasBasis p₂ s₂) :
@@ -498,12 +348,6 @@ theorem Filter.HasBasis.uniformEquicontinuous_iff {κ₁ κ₂ : Type _} {p₁ :
   rfl
 #align filter.has_basis.uniform_equicontinuous_iff Filter.HasBasis.uniformEquicontinuous_iff
 
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 /-- Given `u : α → β` a uniform inducing map, a family `𝓕 : ι → X → α` is equicontinuous at a point
 `x₀ : X` iff the family `𝓕'`, obtained by precomposing each function of `𝓕` by `u`, is
 equicontinuous at `x₀`. -/
@@ -515,12 +359,6 @@ theorem UniformInducing.equicontinuousAt_iff {F : ι → X → α} {x₀ : X} {u
   rfl
 #align uniform_inducing.equicontinuous_at_iff UniformInducing.equicontinuousAt_iff
 
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 /- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:73:14: unsupported tactic `congrm #[[expr ∀ x, (_ : exprProp())]] -/
 /-- Given `u : α → β` a uniform inducing map, a family `𝓕 : ι → X → α` is equicontinuous iff the
 family `𝓕'`, obtained by precomposing each function of `𝓕` by `u`, is equicontinuous. -/
@@ -532,12 +370,6 @@ theorem UniformInducing.equicontinuous_iff {F : ι → X → α} {u : α → β}
   rw [hu.equicontinuous_at_iff]
 #align uniform_inducing.equicontinuous_iff UniformInducing.equicontinuous_iff
 
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 /-- Given `u : α → γ` a uniform inducing map, a family `𝓕 : ι → β → α` is uniformly equicontinuous
 iff the family `𝓕'`, obtained by precomposing each function of `𝓕` by `u`, is uniformly
 equicontinuous. -/
@@ -550,12 +382,6 @@ theorem UniformInducing.uniformEquicontinuous_iff {F : ι → β → α} {u : α
   rfl
 #align uniform_inducing.uniform_equicontinuous_iff UniformInducing.uniformEquicontinuous_iff
 
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 /-- A version of `equicontinuous_at.closure` applicable to subsets of types which embed continuously
 into `X → α` with the product topology. It turns out we don't need any other condition on the
 embedding than continuity, but in practice this will mostly be applied to `fun_like` types where
@@ -573,12 +399,6 @@ theorem EquicontinuousAt.closure' {A : Set Y} {u : Y → X → α} {x₀ : X}
   exact Continuous.prod_mk ((continuous_apply x₀).comp hu) ((continuous_apply x).comp hu)
 #align equicontinuous_at.closure' EquicontinuousAt.closure'
 
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 /-- If a set of functions is equicontinuous at some `x₀`, its closure for the product topology is
 also equicontinuous at `x₀`. -/
 theorem EquicontinuousAt.closure {A : Set <| X → α} {x₀ : X} (hA : A.EquicontinuousAt x₀) :
@@ -586,12 +406,6 @@ theorem EquicontinuousAt.closure {A : Set <| X → α} {x₀ : X} (hA : A.Equico
   @EquicontinuousAt.closure' _ _ _ _ _ _ _ id _ hA continuous_id
 #align equicontinuous_at.closure EquicontinuousAt.closure
 
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 /-- If `𝓕 : ι → X → α` tends to `f : X → α` *pointwise* along some nontrivial filter, and if the
 family `𝓕` is equicontinuous at some `x₀ : X`, then the limit is continuous at `x₀`. -/
 theorem Filter.Tendsto.continuousAt_of_equicontinuousAt {l : Filter ι} [l.ne_bot] {F : ι → X → α}
@@ -601,12 +415,6 @@ theorem Filter.Tendsto.continuousAt_of_equicontinuousAt {l : Filter ι} [l.ne_bo
     ⟨f, mem_closure_of_tendsto h₁ <| eventually_of_forall mem_range_self⟩
 #align filter.tendsto.continuous_at_of_equicontinuous_at Filter.Tendsto.continuousAt_of_equicontinuousAt
 
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 /-- A version of `equicontinuous.closure` applicable to subsets of types which embed continuously
 into `X → α` with the product topology. It turns out we don't need any other condition on the
 embedding than continuity, but in practice this will mostly be applied to `fun_like` types where
@@ -616,24 +424,12 @@ theorem Equicontinuous.closure' {A : Set Y} {u : Y → X → α}
     Equicontinuous (u ∘ coe : closure A → X → α) := fun x => (hA x).closure' hu
 #align equicontinuous.closure' Equicontinuous.closure'
 
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 /-- If a set of functions is equicontinuous, its closure for the product topology is also
 equicontinuous. -/
 theorem Equicontinuous.closure {A : Set <| X → α} (hA : A.Equicontinuous) :
     (closure A).Equicontinuous := fun x => (hA x).closure
 #align equicontinuous.closure Equicontinuous.closure
 
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 /-- If `𝓕 : ι → X → α` tends to `f : X → α` *pointwise* along some nontrivial filter, and if the
 family `𝓕` is equicontinuous, then the limit is continuous. -/
 theorem Filter.Tendsto.continuous_of_equicontinuous_at {l : Filter ι} [l.ne_bot] {F : ι → X → α}
@@ -641,12 +437,6 @@ theorem Filter.Tendsto.continuous_of_equicontinuous_at {l : Filter ι} [l.ne_bot
   continuous_iff_continuousAt.mpr fun x => h₁.continuousAt_of_equicontinuousAt (h₂ x)
 #align filter.tendsto.continuous_of_equicontinuous_at Filter.Tendsto.continuous_of_equicontinuous_at
 
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 /-- A version of `uniform_equicontinuous.closure` applicable to subsets of types which embed
 continuously into `β → α` with the product topology. It turns out we don't need any other condition
 on the embedding than continuity, but in practice this will mostly be applied to `fun_like` types
@@ -665,12 +455,6 @@ theorem UniformEquicontinuous.closure' {A : Set Y} {u : Y → β → α}
   exact Continuous.prod_mk ((continuous_apply x).comp hu) ((continuous_apply y).comp hu)
 #align uniform_equicontinuous.closure' UniformEquicontinuous.closure'
 
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 /-- If a set of functions is uniformly equicontinuous, its closure for the product topology is also
 uniformly equicontinuous. -/
 theorem UniformEquicontinuous.closure {A : Set <| β → α} (hA : A.UniformEquicontinuous) :
@@ -678,12 +462,6 @@ theorem UniformEquicontinuous.closure {A : Set <| β → α} (hA : A.UniformEqui
   @UniformEquicontinuous.closure' _ _ _ _ _ _ _ id hA continuous_id
 #align uniform_equicontinuous.closure UniformEquicontinuous.closure
 
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 /-- If `𝓕 : ι → β → α` tends to `f : β → α` *pointwise* along some nontrivial filter, and if the
 family `𝓕` is uniformly equicontinuous, then the limit is uniformly continuous. -/
 theorem Filter.Tendsto.uniformContinuous_of_uniformEquicontinuous {l : Filter ι} [l.ne_bot]

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

@@ -438,10 +438,7 @@ theorem Filter.HasBasis.equicontinuousAt_iff_right {κ : Type _} {p : κ → Pro
 #align filter.has_basis.equicontinuous_at_iff_right Filter.HasBasis.equicontinuousAt_iff_right
 
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+<too large>
 Case conversion may be inaccurate. Consider using '#align filter.has_basis.equicontinuous_at_iff Filter.HasBasis.equicontinuousAt_iffₓ'. -/
 theorem Filter.HasBasis.equicontinuousAt_iff {κ₁ κ₂ : Type _} {p₁ : κ₁ → Prop} {s₁ : κ₁ → Set X}
     {p₂ : κ₂ → Prop} {s₂ : κ₂ → Set (α × α)} {F : ι → X → α} {x₀ : X} (hX : (𝓝 x₀).HasBasis p₁ s₁)
@@ -487,10 +484,7 @@ theorem Filter.HasBasis.uniformEquicontinuous_iff_right {κ : Type _} {p : κ 
 #align filter.has_basis.uniform_equicontinuous_iff_right Filter.HasBasis.uniformEquicontinuous_iff_right
 
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+<too large>
 Case conversion may be inaccurate. Consider using '#align filter.has_basis.uniform_equicontinuous_iff Filter.HasBasis.uniformEquicontinuous_iffₓ'. -/
 theorem Filter.HasBasis.uniformEquicontinuous_iff {κ₁ κ₂ : Type _} {p₁ : κ₁ → Prop}
     {s₁ : κ₁ → Set (β × β)} {p₂ : κ₂ → Prop} {s₂ : κ₂ → Set (α × α)} {F : ι → β → α}

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

@@ -366,7 +366,7 @@ open UniformFun
 lean 3 declaration is
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 but is expected to have type
-  forall {ι : Type.{u3}} {X : Type.{u2}} {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u2} X] [_inst_4 : UniformSpace.{u1} α] {F : ι -> X -> α} {x₀ : X}, Iff (EquicontinuousAt.{u3, u2, u1} ι X α _inst_1 _inst_4 F x₀) (ContinuousAt.{u2, max u3 u1} X (UniformFun.{u3, u1} ι α) _inst_1 (UniformFun.topologicalSpace.{u3, u1} ι α _inst_4) (Function.comp.{succ u2, max (succ u1) (succ u3), max (succ u3) (succ u1)} X (ι -> α) (UniformFun.{u3, u1} ι α) (FunLike.coe.{max (succ u1) (succ u3), max (succ u1) (succ u3), max (succ u1) (succ u3)} (Equiv.{max (succ u3) (succ u1), max (succ u1) (succ u3)} (ι -> α) (UniformFun.{u3, u1} ι α)) (ι -> α) (fun (_x : ι -> α) => (fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : ι -> α) => UniformFun.{u3, u1} ι α) _x) (Equiv.instFunLikeEquiv.{max (succ u1) (succ u3), max (succ u1) (succ u3)} (ι -> α) (UniformFun.{u3, u1} ι α)) (UniformFun.ofFun.{u3, u1} ι α)) (Function.swap.{succ u3, succ u2, succ u1} ι X (fun (ᾰ : ι) (ᾰ : X) => α) F)) x₀)
+  forall {ι : Type.{u3}} {X : Type.{u2}} {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u2} X] [_inst_4 : UniformSpace.{u1} α] {F : ι -> X -> α} {x₀ : X}, Iff (EquicontinuousAt.{u3, u2, u1} ι X α _inst_1 _inst_4 F x₀) (ContinuousAt.{u2, max u3 u1} X (UniformFun.{u3, u1} ι α) _inst_1 (UniformFun.topologicalSpace.{u3, u1} ι α _inst_4) (Function.comp.{succ u2, max (succ u1) (succ u3), max (succ u3) (succ u1)} X (ι -> α) (UniformFun.{u3, u1} ι α) (FunLike.coe.{max (succ u1) (succ u3), max (succ u1) (succ u3), max (succ u1) (succ u3)} (Equiv.{max (succ u3) (succ u1), max (succ u1) (succ u3)} (ι -> α) (UniformFun.{u3, u1} ι α)) (ι -> α) (fun (_x : ι -> α) => (fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.812 : ι -> α) => UniformFun.{u3, u1} ι α) _x) (Equiv.instFunLikeEquiv.{max (succ u1) (succ u3), max (succ u1) (succ u3)} (ι -> α) (UniformFun.{u3, u1} ι α)) (UniformFun.ofFun.{u3, u1} ι α)) (Function.swap.{succ u3, succ u2, succ u1} ι X (fun (ᾰ : ι) (ᾰ : X) => α) F)) x₀)
 Case conversion may be inaccurate. Consider using '#align equicontinuous_at_iff_continuous_at equicontinuousAt_iff_continuousAtₓ'. -/
 /-- A family `𝓕 : ι → X → α` is equicontinuous at `x₀` iff the function `swap 𝓕 : X → ι → α` is
 continuous at `x₀` *when `ι → α` is equipped with the topology of uniform convergence*. This is
@@ -381,7 +381,7 @@ theorem equicontinuousAt_iff_continuousAt {F : ι → X → α} {x₀ : X} :
 lean 3 declaration is
   forall {ι : Type.{u1}} {X : Type.{u2}} {α : Type.{u3}} [_inst_1 : TopologicalSpace.{u2} X] [_inst_4 : UniformSpace.{u3} α] {F : ι -> X -> α}, Iff (Equicontinuous.{u1, u2, u3} ι X α _inst_1 _inst_4 F) (Continuous.{u2, max u1 u3} X (UniformFun.{u1, u3} ι α) _inst_1 (UniformFun.topologicalSpace.{u1, u3} ι α _inst_4) (Function.comp.{succ u2, max (succ u1) (succ u3), max (succ u1) (succ u3)} X (ι -> α) (UniformFun.{u1, u3} ι α) (coeFn.{max 1 (succ u1) (succ u3), max (succ u1) (succ u3)} (Equiv.{max (succ u1) (succ u3), max (succ u1) (succ u3)} (ι -> α) (UniformFun.{u1, u3} ι α)) (fun (_x : Equiv.{max (succ u1) (succ u3), max (succ u1) (succ u3)} (ι -> α) (UniformFun.{u1, u3} ι α)) => (ι -> α) -> (UniformFun.{u1, u3} ι α)) (Equiv.hasCoeToFun.{max (succ u1) (succ u3), max (succ u1) (succ u3)} (ι -> α) (UniformFun.{u1, u3} ι α)) (UniformFun.ofFun.{u1, u3} ι α)) (Function.swap.{succ u1, succ u2, succ u3} ι X (fun (ᾰ : ι) (ᾰ : X) => α) F)))
 but is expected to have type
-  forall {ι : Type.{u3}} {X : Type.{u2}} {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u2} X] [_inst_4 : UniformSpace.{u1} α] {F : ι -> X -> α}, Iff (Equicontinuous.{u3, u2, u1} ι X α _inst_1 _inst_4 F) (Continuous.{u2, max u3 u1} X (UniformFun.{u3, u1} ι α) _inst_1 (UniformFun.topologicalSpace.{u3, u1} ι α _inst_4) (Function.comp.{succ u2, max (succ u1) (succ u3), max (succ u3) (succ u1)} X (ι -> α) (UniformFun.{u3, u1} ι α) (FunLike.coe.{max (succ u1) (succ u3), max (succ u1) (succ u3), max (succ u1) (succ u3)} (Equiv.{max (succ u3) (succ u1), max (succ u1) (succ u3)} (ι -> α) (UniformFun.{u3, u1} ι α)) (ι -> α) (fun (_x : ι -> α) => (fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : ι -> α) => UniformFun.{u3, u1} ι α) _x) (Equiv.instFunLikeEquiv.{max (succ u1) (succ u3), max (succ u1) (succ u3)} (ι -> α) (UniformFun.{u3, u1} ι α)) (UniformFun.ofFun.{u3, u1} ι α)) (Function.swap.{succ u3, succ u2, succ u1} ι X (fun (ᾰ : ι) (ᾰ : X) => α) F)))
+  forall {ι : Type.{u3}} {X : Type.{u2}} {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u2} X] [_inst_4 : UniformSpace.{u1} α] {F : ι -> X -> α}, Iff (Equicontinuous.{u3, u2, u1} ι X α _inst_1 _inst_4 F) (Continuous.{u2, max u3 u1} X (UniformFun.{u3, u1} ι α) _inst_1 (UniformFun.topologicalSpace.{u3, u1} ι α _inst_4) (Function.comp.{succ u2, max (succ u1) (succ u3), max (succ u3) (succ u1)} X (ι -> α) (UniformFun.{u3, u1} ι α) (FunLike.coe.{max (succ u1) (succ u3), max (succ u1) (succ u3), max (succ u1) (succ u3)} (Equiv.{max (succ u3) (succ u1), max (succ u1) (succ u3)} (ι -> α) (UniformFun.{u3, u1} ι α)) (ι -> α) (fun (_x : ι -> α) => (fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.812 : ι -> α) => UniformFun.{u3, u1} ι α) _x) (Equiv.instFunLikeEquiv.{max (succ u1) (succ u3), max (succ u1) (succ u3)} (ι -> α) (UniformFun.{u3, u1} ι α)) (UniformFun.ofFun.{u3, u1} ι α)) (Function.swap.{succ u3, succ u2, succ u1} ι X (fun (ᾰ : ι) (ᾰ : X) => α) F)))
 Case conversion may be inaccurate. Consider using '#align equicontinuous_iff_continuous equicontinuous_iff_continuousₓ'. -/
 /-- A family `𝓕 : ι → X → α` is equicontinuous iff the function `swap 𝓕 : X → ι → α` is
 continuous *when `ι → α` is equipped with the topology of uniform convergence*. This is
@@ -396,7 +396,7 @@ theorem equicontinuous_iff_continuous {F : ι → X → α} :
 lean 3 declaration is
   forall {ι : Type.{u1}} {α : Type.{u2}} {β : Type.{u3}} [_inst_4 : UniformSpace.{u2} α] [_inst_5 : UniformSpace.{u3} β] {F : ι -> β -> α}, Iff (UniformEquicontinuous.{u1, u2, u3} ι α β _inst_4 _inst_5 F) (UniformContinuous.{u3, max u1 u2} β (UniformFun.{u1, u2} ι α) _inst_5 (UniformFun.uniformSpace.{u1, u2} ι α _inst_4) (Function.comp.{succ u3, max (succ u1) (succ u2), max (succ u1) (succ u2)} β (ι -> α) (UniformFun.{u1, u2} ι α) (coeFn.{max 1 (succ u1) (succ u2), max (succ u1) (succ u2)} (Equiv.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (ι -> α) (UniformFun.{u1, u2} ι α)) (fun (_x : Equiv.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (ι -> α) (UniformFun.{u1, u2} ι α)) => (ι -> α) -> (UniformFun.{u1, u2} ι α)) (Equiv.hasCoeToFun.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (ι -> α) (UniformFun.{u1, u2} ι α)) (UniformFun.ofFun.{u1, u2} ι α)) (Function.swap.{succ u1, succ u3, succ u2} ι β (fun (ᾰ : ι) (ᾰ : β) => α) F)))
 but is expected to have type
-  forall {ι : Type.{u3}} {α : Type.{u2}} {β : Type.{u1}} [_inst_4 : UniformSpace.{u2} α] [_inst_5 : UniformSpace.{u1} β] {F : ι -> β -> α}, Iff (UniformEquicontinuous.{u3, u2, u1} ι α β _inst_4 _inst_5 F) (UniformContinuous.{u1, max u3 u2} β (UniformFun.{u3, u2} ι α) _inst_5 (UniformFun.uniformSpace.{u3, u2} ι α _inst_4) (Function.comp.{succ u1, max (succ u2) (succ u3), max (succ u3) (succ u2)} β (ι -> α) (UniformFun.{u3, u2} ι α) (FunLike.coe.{max (succ u2) (succ u3), max (succ u2) (succ u3), max (succ u2) (succ u3)} (Equiv.{max (succ u3) (succ u2), max (succ u2) (succ u3)} (ι -> α) (UniformFun.{u3, u2} ι α)) (ι -> α) (fun (_x : ι -> α) => (fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : ι -> α) => UniformFun.{u3, u2} ι α) _x) (Equiv.instFunLikeEquiv.{max (succ u2) (succ u3), max (succ u2) (succ u3)} (ι -> α) (UniformFun.{u3, u2} ι α)) (UniformFun.ofFun.{u3, u2} ι α)) (Function.swap.{succ u3, succ u1, succ u2} ι β (fun (ᾰ : ι) (ᾰ : β) => α) F)))
+  forall {ι : Type.{u3}} {α : Type.{u2}} {β : Type.{u1}} [_inst_4 : UniformSpace.{u2} α] [_inst_5 : UniformSpace.{u1} β] {F : ι -> β -> α}, Iff (UniformEquicontinuous.{u3, u2, u1} ι α β _inst_4 _inst_5 F) (UniformContinuous.{u1, max u3 u2} β (UniformFun.{u3, u2} ι α) _inst_5 (UniformFun.uniformSpace.{u3, u2} ι α _inst_4) (Function.comp.{succ u1, max (succ u2) (succ u3), max (succ u3) (succ u2)} β (ι -> α) (UniformFun.{u3, u2} ι α) (FunLike.coe.{max (succ u2) (succ u3), max (succ u2) (succ u3), max (succ u2) (succ u3)} (Equiv.{max (succ u3) (succ u2), max (succ u2) (succ u3)} (ι -> α) (UniformFun.{u3, u2} ι α)) (ι -> α) (fun (_x : ι -> α) => (fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.812 : ι -> α) => UniformFun.{u3, u2} ι α) _x) (Equiv.instFunLikeEquiv.{max (succ u2) (succ u3), max (succ u2) (succ u3)} (ι -> α) (UniformFun.{u3, u2} ι α)) (UniformFun.ofFun.{u3, u2} ι α)) (Function.swap.{succ u3, succ u1, succ u2} ι β (fun (ᾰ : ι) (ᾰ : β) => α) F)))
 Case conversion may be inaccurate. Consider using '#align uniform_equicontinuous_iff_uniform_continuous uniformEquicontinuous_iff_uniformContinuousₓ'. -/
 /-- A family `𝓕 : ι → β → α` is uniformly equicontinuous iff the function `swap 𝓕 : β → ι → α` is
 uniformly continuous *when `ι → α` is equipped with the uniform structure of uniform convergence*.

mathlib commit https://github.com/leanprover-community/mathlib/commit/36b8aa61ea7c05727161f96a0532897bd72aedab

@@ -508,7 +508,7 @@ theorem Filter.HasBasis.uniformEquicontinuous_iff {κ₁ κ₂ : Type _} {p₁ :
 lean 3 declaration is
   forall {ι : Type.{u1}} {X : Type.{u2}} {α : Type.{u3}} {β : Type.{u4}} [_inst_1 : TopologicalSpace.{u2} X] [_inst_4 : UniformSpace.{u3} α] [_inst_5 : UniformSpace.{u4} β] {F : ι -> X -> α} {x₀ : X} {u : α -> β}, (UniformInducing.{u3, u4} α β _inst_4 _inst_5 u) -> (Iff (EquicontinuousAt.{u1, u2, u3} ι X α _inst_1 _inst_4 F x₀) (EquicontinuousAt.{u1, u2, u4} ι X β _inst_1 _inst_5 (Function.comp.{succ u1, max (succ u2) (succ u3), max (succ u2) (succ u4)} ι (X -> α) (X -> β) (Function.comp.{succ u2, succ u3, succ u4} X α β u) F) x₀))
 but is expected to have type
-  forall {ι : Type.{u2}} {X : Type.{u1}} {α : Type.{u4}} {β : Type.{u3}} [_inst_1 : TopologicalSpace.{u1} X] [_inst_4 : UniformSpace.{u4} α] [_inst_5 : UniformSpace.{u3} β] {F : ι -> X -> α} {x₀ : X} {u : α -> β}, (UniformInducing.{u4, u3} α β _inst_4 _inst_5 u) -> (Iff (EquicontinuousAt.{u2, u1, u4} ι X α _inst_1 _inst_4 F x₀) (EquicontinuousAt.{u2, u1, u3} ι X β _inst_1 _inst_5 (Function.comp.{succ u2, max (succ u4) (succ u1), max (succ u3) (succ u1)} ι (X -> α) (X -> β) ((fun (x._@.Mathlib.Topology.UniformSpace.Equicontinuity._hyg.3102 : α -> β) (x._@.Mathlib.Topology.UniformSpace.Equicontinuity._hyg.3104 : X -> α) => Function.comp.{succ u1, succ u4, succ u3} X α β x._@.Mathlib.Topology.UniformSpace.Equicontinuity._hyg.3102 x._@.Mathlib.Topology.UniformSpace.Equicontinuity._hyg.3104) u) F) x₀))
+  forall {ι : Type.{u2}} {X : Type.{u1}} {α : Type.{u4}} {β : Type.{u3}} [_inst_1 : TopologicalSpace.{u1} X] [_inst_4 : UniformSpace.{u4} α] [_inst_5 : UniformSpace.{u3} β] {F : ι -> X -> α} {x₀ : X} {u : α -> β}, (UniformInducing.{u4, u3} α β _inst_4 _inst_5 u) -> (Iff (EquicontinuousAt.{u2, u1, u4} ι X α _inst_1 _inst_4 F x₀) (EquicontinuousAt.{u2, u1, u3} ι X β _inst_1 _inst_5 (Function.comp.{succ u2, max (succ u4) (succ u1), max (succ u3) (succ u1)} ι (X -> α) (X -> β) ((fun (x._@.Mathlib.Topology.UniformSpace.Equicontinuity._hyg.3100 : α -> β) (x._@.Mathlib.Topology.UniformSpace.Equicontinuity._hyg.3102 : X -> α) => Function.comp.{succ u1, succ u4, succ u3} X α β x._@.Mathlib.Topology.UniformSpace.Equicontinuity._hyg.3100 x._@.Mathlib.Topology.UniformSpace.Equicontinuity._hyg.3102) u) F) x₀))
 Case conversion may be inaccurate. Consider using '#align uniform_inducing.equicontinuous_at_iff UniformInducing.equicontinuousAt_iffₓ'. -/
 /-- Given `u : α → β` a uniform inducing map, a family `𝓕 : ι → X → α` is equicontinuous at a point
 `x₀ : X` iff the family `𝓕'`, obtained by precomposing each function of `𝓕` by `u`, is
@@ -525,7 +525,7 @@ theorem UniformInducing.equicontinuousAt_iff {F : ι → X → α} {x₀ : X} {u
 lean 3 declaration is
   forall {ι : Type.{u1}} {X : Type.{u2}} {α : Type.{u3}} {β : Type.{u4}} [_inst_1 : TopologicalSpace.{u2} X] [_inst_4 : UniformSpace.{u3} α] [_inst_5 : UniformSpace.{u4} β] {F : ι -> X -> α} {u : α -> β}, (UniformInducing.{u3, u4} α β _inst_4 _inst_5 u) -> (Iff (Equicontinuous.{u1, u2, u3} ι X α _inst_1 _inst_4 F) (Equicontinuous.{u1, u2, u4} ι X β _inst_1 _inst_5 (Function.comp.{succ u1, max (succ u2) (succ u3), max (succ u2) (succ u4)} ι (X -> α) (X -> β) (Function.comp.{succ u2, succ u3, succ u4} X α β u) F)))
 but is expected to have type
-  forall {ι : Type.{u2}} {X : Type.{u1}} {α : Type.{u4}} {β : Type.{u3}} [_inst_1 : TopologicalSpace.{u1} X] [_inst_4 : UniformSpace.{u4} α] [_inst_5 : UniformSpace.{u3} β] {F : ι -> X -> α} {u : α -> β}, (UniformInducing.{u4, u3} α β _inst_4 _inst_5 u) -> (Iff (Equicontinuous.{u2, u1, u4} ι X α _inst_1 _inst_4 F) (Equicontinuous.{u2, u1, u3} ι X β _inst_1 _inst_5 (Function.comp.{succ u2, max (succ u4) (succ u1), max (succ u3) (succ u1)} ι (X -> α) (X -> β) ((fun (x._@.Mathlib.Topology.UniformSpace.Equicontinuity._hyg.3238 : α -> β) (x._@.Mathlib.Topology.UniformSpace.Equicontinuity._hyg.3240 : X -> α) => Function.comp.{succ u1, succ u4, succ u3} X α β x._@.Mathlib.Topology.UniformSpace.Equicontinuity._hyg.3238 x._@.Mathlib.Topology.UniformSpace.Equicontinuity._hyg.3240) u) F)))
+  forall {ι : Type.{u2}} {X : Type.{u1}} {α : Type.{u4}} {β : Type.{u3}} [_inst_1 : TopologicalSpace.{u1} X] [_inst_4 : UniformSpace.{u4} α] [_inst_5 : UniformSpace.{u3} β] {F : ι -> X -> α} {u : α -> β}, (UniformInducing.{u4, u3} α β _inst_4 _inst_5 u) -> (Iff (Equicontinuous.{u2, u1, u4} ι X α _inst_1 _inst_4 F) (Equicontinuous.{u2, u1, u3} ι X β _inst_1 _inst_5 (Function.comp.{succ u2, max (succ u4) (succ u1), max (succ u3) (succ u1)} ι (X -> α) (X -> β) ((fun (x._@.Mathlib.Topology.UniformSpace.Equicontinuity._hyg.3236 : α -> β) (x._@.Mathlib.Topology.UniformSpace.Equicontinuity._hyg.3238 : X -> α) => Function.comp.{succ u1, succ u4, succ u3} X α β x._@.Mathlib.Topology.UniformSpace.Equicontinuity._hyg.3236 x._@.Mathlib.Topology.UniformSpace.Equicontinuity._hyg.3238) u) F)))
 Case conversion may be inaccurate. Consider using '#align uniform_inducing.equicontinuous_iff UniformInducing.equicontinuous_iffₓ'. -/
 /- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:73:14: unsupported tactic `congrm #[[expr ∀ x, (_ : exprProp())]] -/
 /-- Given `u : α → β` a uniform inducing map, a family `𝓕 : ι → X → α` is equicontinuous iff the
@@ -542,7 +542,7 @@ theorem UniformInducing.equicontinuous_iff {F : ι → X → α} {u : α → β}
 lean 3 declaration is
   forall {ι : Type.{u1}} {α : Type.{u2}} {β : Type.{u3}} {γ : Type.{u4}} [_inst_4 : UniformSpace.{u2} α] [_inst_5 : UniformSpace.{u3} β] [_inst_6 : UniformSpace.{u4} γ] {F : ι -> β -> α} {u : α -> γ}, (UniformInducing.{u2, u4} α γ _inst_4 _inst_6 u) -> (Iff (UniformEquicontinuous.{u1, u2, u3} ι α β _inst_4 _inst_5 F) (UniformEquicontinuous.{u1, u4, u3} ι γ β _inst_6 _inst_5 (Function.comp.{succ u1, max (succ u3) (succ u2), max (succ u3) (succ u4)} ι (β -> α) (β -> γ) (Function.comp.{succ u3, succ u2, succ u4} β α γ u) F)))
 but is expected to have type
-  forall {ι : Type.{u2}} {α : Type.{u4}} {β : Type.{u1}} {γ : Type.{u3}} [_inst_4 : UniformSpace.{u4} α] [_inst_5 : UniformSpace.{u1} β] [_inst_6 : UniformSpace.{u3} γ] {F : ι -> β -> α} {u : α -> γ}, (UniformInducing.{u4, u3} α γ _inst_4 _inst_6 u) -> (Iff (UniformEquicontinuous.{u2, u4, u1} ι α β _inst_4 _inst_5 F) (UniformEquicontinuous.{u2, u3, u1} ι γ β _inst_6 _inst_5 (Function.comp.{succ u2, max (succ u4) (succ u1), max (succ u1) (succ u3)} ι (β -> α) (β -> γ) ((fun (x._@.Mathlib.Topology.UniformSpace.Equicontinuity._hyg.3425 : α -> γ) (x._@.Mathlib.Topology.UniformSpace.Equicontinuity._hyg.3427 : β -> α) => Function.comp.{succ u1, succ u4, succ u3} β α γ x._@.Mathlib.Topology.UniformSpace.Equicontinuity._hyg.3425 x._@.Mathlib.Topology.UniformSpace.Equicontinuity._hyg.3427) u) F)))
+  forall {ι : Type.{u2}} {α : Type.{u4}} {β : Type.{u1}} {γ : Type.{u3}} [_inst_4 : UniformSpace.{u4} α] [_inst_5 : UniformSpace.{u1} β] [_inst_6 : UniformSpace.{u3} γ] {F : ι -> β -> α} {u : α -> γ}, (UniformInducing.{u4, u3} α γ _inst_4 _inst_6 u) -> (Iff (UniformEquicontinuous.{u2, u4, u1} ι α β _inst_4 _inst_5 F) (UniformEquicontinuous.{u2, u3, u1} ι γ β _inst_6 _inst_5 (Function.comp.{succ u2, max (succ u4) (succ u1), max (succ u1) (succ u3)} ι (β -> α) (β -> γ) ((fun (x._@.Mathlib.Topology.UniformSpace.Equicontinuity._hyg.3423 : α -> γ) (x._@.Mathlib.Topology.UniformSpace.Equicontinuity._hyg.3425 : β -> α) => Function.comp.{succ u1, succ u4, succ u3} β α γ x._@.Mathlib.Topology.UniformSpace.Equicontinuity._hyg.3423 x._@.Mathlib.Topology.UniformSpace.Equicontinuity._hyg.3425) u) F)))
 Case conversion may be inaccurate. Consider using '#align uniform_inducing.uniform_equicontinuous_iff UniformInducing.uniformEquicontinuous_iffₓ'. -/
 /-- Given `u : α → γ` a uniform inducing map, a family `𝓕 : ι → β → α` is uniformly equicontinuous
 iff the family `𝓕'`, obtained by precomposing each function of `𝓕` by `u`, is uniformly

mathlib commit https://github.com/leanprover-community/mathlib/commit/730c6d4cab72b9d84fcfb9e95e8796e9cd8f40ba

@@ -508,7 +508,7 @@ theorem Filter.HasBasis.uniformEquicontinuous_iff {κ₁ κ₂ : Type _} {p₁ :
 lean 3 declaration is
   forall {ι : Type.{u1}} {X : Type.{u2}} {α : Type.{u3}} {β : Type.{u4}} [_inst_1 : TopologicalSpace.{u2} X] [_inst_4 : UniformSpace.{u3} α] [_inst_5 : UniformSpace.{u4} β] {F : ι -> X -> α} {x₀ : X} {u : α -> β}, (UniformInducing.{u3, u4} α β _inst_4 _inst_5 u) -> (Iff (EquicontinuousAt.{u1, u2, u3} ι X α _inst_1 _inst_4 F x₀) (EquicontinuousAt.{u1, u2, u4} ι X β _inst_1 _inst_5 (Function.comp.{succ u1, max (succ u2) (succ u3), max (succ u2) (succ u4)} ι (X -> α) (X -> β) (Function.comp.{succ u2, succ u3, succ u4} X α β u) F) x₀))
 but is expected to have type
-  forall {ι : Type.{u2}} {X : Type.{u1}} {α : Type.{u4}} {β : Type.{u3}} [_inst_1 : TopologicalSpace.{u1} X] [_inst_4 : UniformSpace.{u4} α] [_inst_5 : UniformSpace.{u3} β] {F : ι -> X -> α} {x₀ : X} {u : α -> β}, (UniformInducing.{u4, u3} α β _inst_4 _inst_5 u) -> (Iff (EquicontinuousAt.{u2, u1, u4} ι X α _inst_1 _inst_4 F x₀) (EquicontinuousAt.{u2, u1, u3} ι X β _inst_1 _inst_5 (Function.comp.{succ u2, max (succ u4) (succ u1), max (succ u3) (succ u1)} ι (X -> α) (X -> β) ((fun (x._@.Mathlib.Topology.UniformSpace.Equicontinuity._hyg.3103 : α -> β) (x._@.Mathlib.Topology.UniformSpace.Equicontinuity._hyg.3105 : X -> α) => Function.comp.{succ u1, succ u4, succ u3} X α β x._@.Mathlib.Topology.UniformSpace.Equicontinuity._hyg.3103 x._@.Mathlib.Topology.UniformSpace.Equicontinuity._hyg.3105) u) F) x₀))
+  forall {ι : Type.{u2}} {X : Type.{u1}} {α : Type.{u4}} {β : Type.{u3}} [_inst_1 : TopologicalSpace.{u1} X] [_inst_4 : UniformSpace.{u4} α] [_inst_5 : UniformSpace.{u3} β] {F : ι -> X -> α} {x₀ : X} {u : α -> β}, (UniformInducing.{u4, u3} α β _inst_4 _inst_5 u) -> (Iff (EquicontinuousAt.{u2, u1, u4} ι X α _inst_1 _inst_4 F x₀) (EquicontinuousAt.{u2, u1, u3} ι X β _inst_1 _inst_5 (Function.comp.{succ u2, max (succ u4) (succ u1), max (succ u3) (succ u1)} ι (X -> α) (X -> β) ((fun (x._@.Mathlib.Topology.UniformSpace.Equicontinuity._hyg.3102 : α -> β) (x._@.Mathlib.Topology.UniformSpace.Equicontinuity._hyg.3104 : X -> α) => Function.comp.{succ u1, succ u4, succ u3} X α β x._@.Mathlib.Topology.UniformSpace.Equicontinuity._hyg.3102 x._@.Mathlib.Topology.UniformSpace.Equicontinuity._hyg.3104) u) F) x₀))
 Case conversion may be inaccurate. Consider using '#align uniform_inducing.equicontinuous_at_iff UniformInducing.equicontinuousAt_iffₓ'. -/
 /-- Given `u : α → β` a uniform inducing map, a family `𝓕 : ι → X → α` is equicontinuous at a point
 `x₀ : X` iff the family `𝓕'`, obtained by precomposing each function of `𝓕` by `u`, is
@@ -525,7 +525,7 @@ theorem UniformInducing.equicontinuousAt_iff {F : ι → X → α} {x₀ : X} {u
 lean 3 declaration is
   forall {ι : Type.{u1}} {X : Type.{u2}} {α : Type.{u3}} {β : Type.{u4}} [_inst_1 : TopologicalSpace.{u2} X] [_inst_4 : UniformSpace.{u3} α] [_inst_5 : UniformSpace.{u4} β] {F : ι -> X -> α} {u : α -> β}, (UniformInducing.{u3, u4} α β _inst_4 _inst_5 u) -> (Iff (Equicontinuous.{u1, u2, u3} ι X α _inst_1 _inst_4 F) (Equicontinuous.{u1, u2, u4} ι X β _inst_1 _inst_5 (Function.comp.{succ u1, max (succ u2) (succ u3), max (succ u2) (succ u4)} ι (X -> α) (X -> β) (Function.comp.{succ u2, succ u3, succ u4} X α β u) F)))
 but is expected to have type
-  forall {ι : Type.{u2}} {X : Type.{u1}} {α : Type.{u4}} {β : Type.{u3}} [_inst_1 : TopologicalSpace.{u1} X] [_inst_4 : UniformSpace.{u4} α] [_inst_5 : UniformSpace.{u3} β] {F : ι -> X -> α} {u : α -> β}, (UniformInducing.{u4, u3} α β _inst_4 _inst_5 u) -> (Iff (Equicontinuous.{u2, u1, u4} ι X α _inst_1 _inst_4 F) (Equicontinuous.{u2, u1, u3} ι X β _inst_1 _inst_5 (Function.comp.{succ u2, max (succ u4) (succ u1), max (succ u3) (succ u1)} ι (X -> α) (X -> β) ((fun (x._@.Mathlib.Topology.UniformSpace.Equicontinuity._hyg.3239 : α -> β) (x._@.Mathlib.Topology.UniformSpace.Equicontinuity._hyg.3241 : X -> α) => Function.comp.{succ u1, succ u4, succ u3} X α β x._@.Mathlib.Topology.UniformSpace.Equicontinuity._hyg.3239 x._@.Mathlib.Topology.UniformSpace.Equicontinuity._hyg.3241) u) F)))
+  forall {ι : Type.{u2}} {X : Type.{u1}} {α : Type.{u4}} {β : Type.{u3}} [_inst_1 : TopologicalSpace.{u1} X] [_inst_4 : UniformSpace.{u4} α] [_inst_5 : UniformSpace.{u3} β] {F : ι -> X -> α} {u : α -> β}, (UniformInducing.{u4, u3} α β _inst_4 _inst_5 u) -> (Iff (Equicontinuous.{u2, u1, u4} ι X α _inst_1 _inst_4 F) (Equicontinuous.{u2, u1, u3} ι X β _inst_1 _inst_5 (Function.comp.{succ u2, max (succ u4) (succ u1), max (succ u3) (succ u1)} ι (X -> α) (X -> β) ((fun (x._@.Mathlib.Topology.UniformSpace.Equicontinuity._hyg.3238 : α -> β) (x._@.Mathlib.Topology.UniformSpace.Equicontinuity._hyg.3240 : X -> α) => Function.comp.{succ u1, succ u4, succ u3} X α β x._@.Mathlib.Topology.UniformSpace.Equicontinuity._hyg.3238 x._@.Mathlib.Topology.UniformSpace.Equicontinuity._hyg.3240) u) F)))
 Case conversion may be inaccurate. Consider using '#align uniform_inducing.equicontinuous_iff UniformInducing.equicontinuous_iffₓ'. -/
 /- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:73:14: unsupported tactic `congrm #[[expr ∀ x, (_ : exprProp())]] -/
 /-- Given `u : α → β` a uniform inducing map, a family `𝓕 : ι → X → α` is equicontinuous iff the
@@ -542,7 +542,7 @@ theorem UniformInducing.equicontinuous_iff {F : ι → X → α} {u : α → β}
 lean 3 declaration is
   forall {ι : Type.{u1}} {α : Type.{u2}} {β : Type.{u3}} {γ : Type.{u4}} [_inst_4 : UniformSpace.{u2} α] [_inst_5 : UniformSpace.{u3} β] [_inst_6 : UniformSpace.{u4} γ] {F : ι -> β -> α} {u : α -> γ}, (UniformInducing.{u2, u4} α γ _inst_4 _inst_6 u) -> (Iff (UniformEquicontinuous.{u1, u2, u3} ι α β _inst_4 _inst_5 F) (UniformEquicontinuous.{u1, u4, u3} ι γ β _inst_6 _inst_5 (Function.comp.{succ u1, max (succ u3) (succ u2), max (succ u3) (succ u4)} ι (β -> α) (β -> γ) (Function.comp.{succ u3, succ u2, succ u4} β α γ u) F)))
 but is expected to have type
-  forall {ι : Type.{u2}} {α : Type.{u4}} {β : Type.{u1}} {γ : Type.{u3}} [_inst_4 : UniformSpace.{u4} α] [_inst_5 : UniformSpace.{u1} β] [_inst_6 : UniformSpace.{u3} γ] {F : ι -> β -> α} {u : α -> γ}, (UniformInducing.{u4, u3} α γ _inst_4 _inst_6 u) -> (Iff (UniformEquicontinuous.{u2, u4, u1} ι α β _inst_4 _inst_5 F) (UniformEquicontinuous.{u2, u3, u1} ι γ β _inst_6 _inst_5 (Function.comp.{succ u2, max (succ u4) (succ u1), max (succ u1) (succ u3)} ι (β -> α) (β -> γ) ((fun (x._@.Mathlib.Topology.UniformSpace.Equicontinuity._hyg.3426 : α -> γ) (x._@.Mathlib.Topology.UniformSpace.Equicontinuity._hyg.3428 : β -> α) => Function.comp.{succ u1, succ u4, succ u3} β α γ x._@.Mathlib.Topology.UniformSpace.Equicontinuity._hyg.3426 x._@.Mathlib.Topology.UniformSpace.Equicontinuity._hyg.3428) u) F)))
+  forall {ι : Type.{u2}} {α : Type.{u4}} {β : Type.{u1}} {γ : Type.{u3}} [_inst_4 : UniformSpace.{u4} α] [_inst_5 : UniformSpace.{u1} β] [_inst_6 : UniformSpace.{u3} γ] {F : ι -> β -> α} {u : α -> γ}, (UniformInducing.{u4, u3} α γ _inst_4 _inst_6 u) -> (Iff (UniformEquicontinuous.{u2, u4, u1} ι α β _inst_4 _inst_5 F) (UniformEquicontinuous.{u2, u3, u1} ι γ β _inst_6 _inst_5 (Function.comp.{succ u2, max (succ u4) (succ u1), max (succ u1) (succ u3)} ι (β -> α) (β -> γ) ((fun (x._@.Mathlib.Topology.UniformSpace.Equicontinuity._hyg.3425 : α -> γ) (x._@.Mathlib.Topology.UniformSpace.Equicontinuity._hyg.3427 : β -> α) => Function.comp.{succ u1, succ u4, succ u3} β α γ x._@.Mathlib.Topology.UniformSpace.Equicontinuity._hyg.3425 x._@.Mathlib.Topology.UniformSpace.Equicontinuity._hyg.3427) u) F)))
 Case conversion may be inaccurate. Consider using '#align uniform_inducing.uniform_equicontinuous_iff UniformInducing.uniformEquicontinuous_iffₓ'. -/
 /-- Given `u : α → γ` a uniform inducing map, a family `𝓕 : ι → β → α` is uniformly equicontinuous
 iff the family `𝓕'`, obtained by precomposing each function of `𝓕` by `u`, is uniformly

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

@@ -366,7 +366,7 @@ open UniformFun
 lean 3 declaration is
   forall {ι : Type.{u1}} {X : Type.{u2}} {α : Type.{u3}} [_inst_1 : TopologicalSpace.{u2} X] [_inst_4 : UniformSpace.{u3} α] {F : ι -> X -> α} {x₀ : X}, Iff (EquicontinuousAt.{u1, u2, u3} ι X α _inst_1 _inst_4 F x₀) (ContinuousAt.{u2, max u1 u3} X (UniformFun.{u1, u3} ι α) _inst_1 (UniformFun.topologicalSpace.{u1, u3} ι α _inst_4) (Function.comp.{succ u2, max (succ u1) (succ u3), max (succ u1) (succ u3)} X (ι -> α) (UniformFun.{u1, u3} ι α) (coeFn.{max 1 (succ u1) (succ u3), max (succ u1) (succ u3)} (Equiv.{max (succ u1) (succ u3), max (succ u1) (succ u3)} (ι -> α) (UniformFun.{u1, u3} ι α)) (fun (_x : Equiv.{max (succ u1) (succ u3), max (succ u1) (succ u3)} (ι -> α) (UniformFun.{u1, u3} ι α)) => (ι -> α) -> (UniformFun.{u1, u3} ι α)) (Equiv.hasCoeToFun.{max (succ u1) (succ u3), max (succ u1) (succ u3)} (ι -> α) (UniformFun.{u1, u3} ι α)) (UniformFun.ofFun.{u1, u3} ι α)) (Function.swap.{succ u1, succ u2, succ u3} ι X (fun (ᾰ : ι) (ᾰ : X) => α) F)) x₀)
 but is expected to have type
-  forall {ι : Type.{u3}} {X : Type.{u2}} {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u2} X] [_inst_4 : UniformSpace.{u1} α] {F : ι -> X -> α} {x₀ : X}, Iff (EquicontinuousAt.{u3, u2, u1} ι X α _inst_1 _inst_4 F x₀) (ContinuousAt.{u2, max u3 u1} X (UniformFun.{u3, u1} ι α) _inst_1 (UniformFun.topologicalSpace.{u3, u1} ι α _inst_4) (Function.comp.{succ u2, max (succ u1) (succ u3), max (succ u3) (succ u1)} X (ι -> α) (UniformFun.{u3, u1} ι α) (FunLike.coe.{max (succ u1) (succ u3), max (succ u1) (succ u3), max (succ u1) (succ u3)} (Equiv.{max (succ u3) (succ u1), max (succ u1) (succ u3)} (ι -> α) (UniformFun.{u3, u1} ι α)) (ι -> α) (fun (_x : ι -> α) => (fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.805 : ι -> α) => UniformFun.{u3, u1} ι α) _x) (Equiv.instFunLikeEquiv.{max (succ u1) (succ u3), max (succ u1) (succ u3)} (ι -> α) (UniformFun.{u3, u1} ι α)) (UniformFun.ofFun.{u3, u1} ι α)) (Function.swap.{succ u3, succ u2, succ u1} ι X (fun (ᾰ : ι) (ᾰ : X) => α) F)) x₀)
+  forall {ι : Type.{u3}} {X : Type.{u2}} {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u2} X] [_inst_4 : UniformSpace.{u1} α] {F : ι -> X -> α} {x₀ : X}, Iff (EquicontinuousAt.{u3, u2, u1} ι X α _inst_1 _inst_4 F x₀) (ContinuousAt.{u2, max u3 u1} X (UniformFun.{u3, u1} ι α) _inst_1 (UniformFun.topologicalSpace.{u3, u1} ι α _inst_4) (Function.comp.{succ u2, max (succ u1) (succ u3), max (succ u3) (succ u1)} X (ι -> α) (UniformFun.{u3, u1} ι α) (FunLike.coe.{max (succ u1) (succ u3), max (succ u1) (succ u3), max (succ u1) (succ u3)} (Equiv.{max (succ u3) (succ u1), max (succ u1) (succ u3)} (ι -> α) (UniformFun.{u3, u1} ι α)) (ι -> α) (fun (_x : ι -> α) => (fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : ι -> α) => UniformFun.{u3, u1} ι α) _x) (Equiv.instFunLikeEquiv.{max (succ u1) (succ u3), max (succ u1) (succ u3)} (ι -> α) (UniformFun.{u3, u1} ι α)) (UniformFun.ofFun.{u3, u1} ι α)) (Function.swap.{succ u3, succ u2, succ u1} ι X (fun (ᾰ : ι) (ᾰ : X) => α) F)) x₀)
 Case conversion may be inaccurate. Consider using '#align equicontinuous_at_iff_continuous_at equicontinuousAt_iff_continuousAtₓ'. -/
 /-- A family `𝓕 : ι → X → α` is equicontinuous at `x₀` iff the function `swap 𝓕 : X → ι → α` is
 continuous at `x₀` *when `ι → α` is equipped with the topology of uniform convergence*. This is
@@ -381,7 +381,7 @@ theorem equicontinuousAt_iff_continuousAt {F : ι → X → α} {x₀ : X} :
 lean 3 declaration is
   forall {ι : Type.{u1}} {X : Type.{u2}} {α : Type.{u3}} [_inst_1 : TopologicalSpace.{u2} X] [_inst_4 : UniformSpace.{u3} α] {F : ι -> X -> α}, Iff (Equicontinuous.{u1, u2, u3} ι X α _inst_1 _inst_4 F) (Continuous.{u2, max u1 u3} X (UniformFun.{u1, u3} ι α) _inst_1 (UniformFun.topologicalSpace.{u1, u3} ι α _inst_4) (Function.comp.{succ u2, max (succ u1) (succ u3), max (succ u1) (succ u3)} X (ι -> α) (UniformFun.{u1, u3} ι α) (coeFn.{max 1 (succ u1) (succ u3), max (succ u1) (succ u3)} (Equiv.{max (succ u1) (succ u3), max (succ u1) (succ u3)} (ι -> α) (UniformFun.{u1, u3} ι α)) (fun (_x : Equiv.{max (succ u1) (succ u3), max (succ u1) (succ u3)} (ι -> α) (UniformFun.{u1, u3} ι α)) => (ι -> α) -> (UniformFun.{u1, u3} ι α)) (Equiv.hasCoeToFun.{max (succ u1) (succ u3), max (succ u1) (succ u3)} (ι -> α) (UniformFun.{u1, u3} ι α)) (UniformFun.ofFun.{u1, u3} ι α)) (Function.swap.{succ u1, succ u2, succ u3} ι X (fun (ᾰ : ι) (ᾰ : X) => α) F)))
 but is expected to have type
-  forall {ι : Type.{u3}} {X : Type.{u2}} {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u2} X] [_inst_4 : UniformSpace.{u1} α] {F : ι -> X -> α}, Iff (Equicontinuous.{u3, u2, u1} ι X α _inst_1 _inst_4 F) (Continuous.{u2, max u3 u1} X (UniformFun.{u3, u1} ι α) _inst_1 (UniformFun.topologicalSpace.{u3, u1} ι α _inst_4) (Function.comp.{succ u2, max (succ u1) (succ u3), max (succ u3) (succ u1)} X (ι -> α) (UniformFun.{u3, u1} ι α) (FunLike.coe.{max (succ u1) (succ u3), max (succ u1) (succ u3), max (succ u1) (succ u3)} (Equiv.{max (succ u3) (succ u1), max (succ u1) (succ u3)} (ι -> α) (UniformFun.{u3, u1} ι α)) (ι -> α) (fun (_x : ι -> α) => (fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.805 : ι -> α) => UniformFun.{u3, u1} ι α) _x) (Equiv.instFunLikeEquiv.{max (succ u1) (succ u3), max (succ u1) (succ u3)} (ι -> α) (UniformFun.{u3, u1} ι α)) (UniformFun.ofFun.{u3, u1} ι α)) (Function.swap.{succ u3, succ u2, succ u1} ι X (fun (ᾰ : ι) (ᾰ : X) => α) F)))
+  forall {ι : Type.{u3}} {X : Type.{u2}} {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u2} X] [_inst_4 : UniformSpace.{u1} α] {F : ι -> X -> α}, Iff (Equicontinuous.{u3, u2, u1} ι X α _inst_1 _inst_4 F) (Continuous.{u2, max u3 u1} X (UniformFun.{u3, u1} ι α) _inst_1 (UniformFun.topologicalSpace.{u3, u1} ι α _inst_4) (Function.comp.{succ u2, max (succ u1) (succ u3), max (succ u3) (succ u1)} X (ι -> α) (UniformFun.{u3, u1} ι α) (FunLike.coe.{max (succ u1) (succ u3), max (succ u1) (succ u3), max (succ u1) (succ u3)} (Equiv.{max (succ u3) (succ u1), max (succ u1) (succ u3)} (ι -> α) (UniformFun.{u3, u1} ι α)) (ι -> α) (fun (_x : ι -> α) => (fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : ι -> α) => UniformFun.{u3, u1} ι α) _x) (Equiv.instFunLikeEquiv.{max (succ u1) (succ u3), max (succ u1) (succ u3)} (ι -> α) (UniformFun.{u3, u1} ι α)) (UniformFun.ofFun.{u3, u1} ι α)) (Function.swap.{succ u3, succ u2, succ u1} ι X (fun (ᾰ : ι) (ᾰ : X) => α) F)))
 Case conversion may be inaccurate. Consider using '#align equicontinuous_iff_continuous equicontinuous_iff_continuousₓ'. -/
 /-- A family `𝓕 : ι → X → α` is equicontinuous iff the function `swap 𝓕 : X → ι → α` is
 continuous *when `ι → α` is equipped with the topology of uniform convergence*. This is
@@ -396,7 +396,7 @@ theorem equicontinuous_iff_continuous {F : ι → X → α} :
 lean 3 declaration is
   forall {ι : Type.{u1}} {α : Type.{u2}} {β : Type.{u3}} [_inst_4 : UniformSpace.{u2} α] [_inst_5 : UniformSpace.{u3} β] {F : ι -> β -> α}, Iff (UniformEquicontinuous.{u1, u2, u3} ι α β _inst_4 _inst_5 F) (UniformContinuous.{u3, max u1 u2} β (UniformFun.{u1, u2} ι α) _inst_5 (UniformFun.uniformSpace.{u1, u2} ι α _inst_4) (Function.comp.{succ u3, max (succ u1) (succ u2), max (succ u1) (succ u2)} β (ι -> α) (UniformFun.{u1, u2} ι α) (coeFn.{max 1 (succ u1) (succ u2), max (succ u1) (succ u2)} (Equiv.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (ι -> α) (UniformFun.{u1, u2} ι α)) (fun (_x : Equiv.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (ι -> α) (UniformFun.{u1, u2} ι α)) => (ι -> α) -> (UniformFun.{u1, u2} ι α)) (Equiv.hasCoeToFun.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (ι -> α) (UniformFun.{u1, u2} ι α)) (UniformFun.ofFun.{u1, u2} ι α)) (Function.swap.{succ u1, succ u3, succ u2} ι β (fun (ᾰ : ι) (ᾰ : β) => α) F)))
 but is expected to have type
-  forall {ι : Type.{u3}} {α : Type.{u2}} {β : Type.{u1}} [_inst_4 : UniformSpace.{u2} α] [_inst_5 : UniformSpace.{u1} β] {F : ι -> β -> α}, Iff (UniformEquicontinuous.{u3, u2, u1} ι α β _inst_4 _inst_5 F) (UniformContinuous.{u1, max u3 u2} β (UniformFun.{u3, u2} ι α) _inst_5 (UniformFun.uniformSpace.{u3, u2} ι α _inst_4) (Function.comp.{succ u1, max (succ u2) (succ u3), max (succ u3) (succ u2)} β (ι -> α) (UniformFun.{u3, u2} ι α) (FunLike.coe.{max (succ u2) (succ u3), max (succ u2) (succ u3), max (succ u2) (succ u3)} (Equiv.{max (succ u3) (succ u2), max (succ u2) (succ u3)} (ι -> α) (UniformFun.{u3, u2} ι α)) (ι -> α) (fun (_x : ι -> α) => (fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.805 : ι -> α) => UniformFun.{u3, u2} ι α) _x) (Equiv.instFunLikeEquiv.{max (succ u2) (succ u3), max (succ u2) (succ u3)} (ι -> α) (UniformFun.{u3, u2} ι α)) (UniformFun.ofFun.{u3, u2} ι α)) (Function.swap.{succ u3, succ u1, succ u2} ι β (fun (ᾰ : ι) (ᾰ : β) => α) F)))
+  forall {ι : Type.{u3}} {α : Type.{u2}} {β : Type.{u1}} [_inst_4 : UniformSpace.{u2} α] [_inst_5 : UniformSpace.{u1} β] {F : ι -> β -> α}, Iff (UniformEquicontinuous.{u3, u2, u1} ι α β _inst_4 _inst_5 F) (UniformContinuous.{u1, max u3 u2} β (UniformFun.{u3, u2} ι α) _inst_5 (UniformFun.uniformSpace.{u3, u2} ι α _inst_4) (Function.comp.{succ u1, max (succ u2) (succ u3), max (succ u3) (succ u2)} β (ι -> α) (UniformFun.{u3, u2} ι α) (FunLike.coe.{max (succ u2) (succ u3), max (succ u2) (succ u3), max (succ u2) (succ u3)} (Equiv.{max (succ u3) (succ u2), max (succ u2) (succ u3)} (ι -> α) (UniformFun.{u3, u2} ι α)) (ι -> α) (fun (_x : ι -> α) => (fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : ι -> α) => UniformFun.{u3, u2} ι α) _x) (Equiv.instFunLikeEquiv.{max (succ u2) (succ u3), max (succ u2) (succ u3)} (ι -> α) (UniformFun.{u3, u2} ι α)) (UniformFun.ofFun.{u3, u2} ι α)) (Function.swap.{succ u3, succ u1, succ u2} ι β (fun (ᾰ : ι) (ᾰ : β) => α) F)))
 Case conversion may be inaccurate. Consider using '#align uniform_equicontinuous_iff_uniform_continuous uniformEquicontinuous_iff_uniformContinuousₓ'. -/
 /-- A family `𝓕 : ι → β → α` is uniformly equicontinuous iff the function `swap 𝓕 : β → ι → α` is
 uniformly continuous *when `ι → α` is equipped with the uniform structure of uniform convergence*.
@@ -508,7 +508,7 @@ theorem Filter.HasBasis.uniformEquicontinuous_iff {κ₁ κ₂ : Type _} {p₁ :
 lean 3 declaration is
   forall {ι : Type.{u1}} {X : Type.{u2}} {α : Type.{u3}} {β : Type.{u4}} [_inst_1 : TopologicalSpace.{u2} X] [_inst_4 : UniformSpace.{u3} α] [_inst_5 : UniformSpace.{u4} β] {F : ι -> X -> α} {x₀ : X} {u : α -> β}, (UniformInducing.{u3, u4} α β _inst_4 _inst_5 u) -> (Iff (EquicontinuousAt.{u1, u2, u3} ι X α _inst_1 _inst_4 F x₀) (EquicontinuousAt.{u1, u2, u4} ι X β _inst_1 _inst_5 (Function.comp.{succ u1, max (succ u2) (succ u3), max (succ u2) (succ u4)} ι (X -> α) (X -> β) (Function.comp.{succ u2, succ u3, succ u4} X α β u) F) x₀))
 but is expected to have type
-  forall {ι : Type.{u2}} {X : Type.{u1}} {α : Type.{u4}} {β : Type.{u3}} [_inst_1 : TopologicalSpace.{u1} X] [_inst_4 : UniformSpace.{u4} α] [_inst_5 : UniformSpace.{u3} β] {F : ι -> X -> α} {x₀ : X} {u : α -> β}, (UniformInducing.{u4, u3} α β _inst_4 _inst_5 u) -> (Iff (EquicontinuousAt.{u2, u1, u4} ι X α _inst_1 _inst_4 F x₀) (EquicontinuousAt.{u2, u1, u3} ι X β _inst_1 _inst_5 (Function.comp.{succ u2, max (succ u4) (succ u1), max (succ u3) (succ u1)} ι (X -> α) (X -> β) ((fun (x._@.Mathlib.Topology.UniformSpace.Equicontinuity._hyg.3088 : α -> β) (x._@.Mathlib.Topology.UniformSpace.Equicontinuity._hyg.3090 : X -> α) => Function.comp.{succ u1, succ u4, succ u3} X α β x._@.Mathlib.Topology.UniformSpace.Equicontinuity._hyg.3088 x._@.Mathlib.Topology.UniformSpace.Equicontinuity._hyg.3090) u) F) x₀))
+  forall {ι : Type.{u2}} {X : Type.{u1}} {α : Type.{u4}} {β : Type.{u3}} [_inst_1 : TopologicalSpace.{u1} X] [_inst_4 : UniformSpace.{u4} α] [_inst_5 : UniformSpace.{u3} β] {F : ι -> X -> α} {x₀ : X} {u : α -> β}, (UniformInducing.{u4, u3} α β _inst_4 _inst_5 u) -> (Iff (EquicontinuousAt.{u2, u1, u4} ι X α _inst_1 _inst_4 F x₀) (EquicontinuousAt.{u2, u1, u3} ι X β _inst_1 _inst_5 (Function.comp.{succ u2, max (succ u4) (succ u1), max (succ u3) (succ u1)} ι (X -> α) (X -> β) ((fun (x._@.Mathlib.Topology.UniformSpace.Equicontinuity._hyg.3103 : α -> β) (x._@.Mathlib.Topology.UniformSpace.Equicontinuity._hyg.3105 : X -> α) => Function.comp.{succ u1, succ u4, succ u3} X α β x._@.Mathlib.Topology.UniformSpace.Equicontinuity._hyg.3103 x._@.Mathlib.Topology.UniformSpace.Equicontinuity._hyg.3105) u) F) x₀))
 Case conversion may be inaccurate. Consider using '#align uniform_inducing.equicontinuous_at_iff UniformInducing.equicontinuousAt_iffₓ'. -/
 /-- Given `u : α → β` a uniform inducing map, a family `𝓕 : ι → X → α` is equicontinuous at a point
 `x₀ : X` iff the family `𝓕'`, obtained by precomposing each function of `𝓕` by `u`, is
@@ -525,7 +525,7 @@ theorem UniformInducing.equicontinuousAt_iff {F : ι → X → α} {x₀ : X} {u
 lean 3 declaration is
   forall {ι : Type.{u1}} {X : Type.{u2}} {α : Type.{u3}} {β : Type.{u4}} [_inst_1 : TopologicalSpace.{u2} X] [_inst_4 : UniformSpace.{u3} α] [_inst_5 : UniformSpace.{u4} β] {F : ι -> X -> α} {u : α -> β}, (UniformInducing.{u3, u4} α β _inst_4 _inst_5 u) -> (Iff (Equicontinuous.{u1, u2, u3} ι X α _inst_1 _inst_4 F) (Equicontinuous.{u1, u2, u4} ι X β _inst_1 _inst_5 (Function.comp.{succ u1, max (succ u2) (succ u3), max (succ u2) (succ u4)} ι (X -> α) (X -> β) (Function.comp.{succ u2, succ u3, succ u4} X α β u) F)))
 but is expected to have type
-  forall {ι : Type.{u2}} {X : Type.{u1}} {α : Type.{u4}} {β : Type.{u3}} [_inst_1 : TopologicalSpace.{u1} X] [_inst_4 : UniformSpace.{u4} α] [_inst_5 : UniformSpace.{u3} β] {F : ι -> X -> α} {u : α -> β}, (UniformInducing.{u4, u3} α β _inst_4 _inst_5 u) -> (Iff (Equicontinuous.{u2, u1, u4} ι X α _inst_1 _inst_4 F) (Equicontinuous.{u2, u1, u3} ι X β _inst_1 _inst_5 (Function.comp.{succ u2, max (succ u4) (succ u1), max (succ u3) (succ u1)} ι (X -> α) (X -> β) ((fun (x._@.Mathlib.Topology.UniformSpace.Equicontinuity._hyg.3224 : α -> β) (x._@.Mathlib.Topology.UniformSpace.Equicontinuity._hyg.3226 : X -> α) => Function.comp.{succ u1, succ u4, succ u3} X α β x._@.Mathlib.Topology.UniformSpace.Equicontinuity._hyg.3224 x._@.Mathlib.Topology.UniformSpace.Equicontinuity._hyg.3226) u) F)))
+  forall {ι : Type.{u2}} {X : Type.{u1}} {α : Type.{u4}} {β : Type.{u3}} [_inst_1 : TopologicalSpace.{u1} X] [_inst_4 : UniformSpace.{u4} α] [_inst_5 : UniformSpace.{u3} β] {F : ι -> X -> α} {u : α -> β}, (UniformInducing.{u4, u3} α β _inst_4 _inst_5 u) -> (Iff (Equicontinuous.{u2, u1, u4} ι X α _inst_1 _inst_4 F) (Equicontinuous.{u2, u1, u3} ι X β _inst_1 _inst_5 (Function.comp.{succ u2, max (succ u4) (succ u1), max (succ u3) (succ u1)} ι (X -> α) (X -> β) ((fun (x._@.Mathlib.Topology.UniformSpace.Equicontinuity._hyg.3239 : α -> β) (x._@.Mathlib.Topology.UniformSpace.Equicontinuity._hyg.3241 : X -> α) => Function.comp.{succ u1, succ u4, succ u3} X α β x._@.Mathlib.Topology.UniformSpace.Equicontinuity._hyg.3239 x._@.Mathlib.Topology.UniformSpace.Equicontinuity._hyg.3241) u) F)))
 Case conversion may be inaccurate. Consider using '#align uniform_inducing.equicontinuous_iff UniformInducing.equicontinuous_iffₓ'. -/
 /- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:73:14: unsupported tactic `congrm #[[expr ∀ x, (_ : exprProp())]] -/
 /-- Given `u : α → β` a uniform inducing map, a family `𝓕 : ι → X → α` is equicontinuous iff the
@@ -542,7 +542,7 @@ theorem UniformInducing.equicontinuous_iff {F : ι → X → α} {u : α → β}
 lean 3 declaration is
   forall {ι : Type.{u1}} {α : Type.{u2}} {β : Type.{u3}} {γ : Type.{u4}} [_inst_4 : UniformSpace.{u2} α] [_inst_5 : UniformSpace.{u3} β] [_inst_6 : UniformSpace.{u4} γ] {F : ι -> β -> α} {u : α -> γ}, (UniformInducing.{u2, u4} α γ _inst_4 _inst_6 u) -> (Iff (UniformEquicontinuous.{u1, u2, u3} ι α β _inst_4 _inst_5 F) (UniformEquicontinuous.{u1, u4, u3} ι γ β _inst_6 _inst_5 (Function.comp.{succ u1, max (succ u3) (succ u2), max (succ u3) (succ u4)} ι (β -> α) (β -> γ) (Function.comp.{succ u3, succ u2, succ u4} β α γ u) F)))
 but is expected to have type
-  forall {ι : Type.{u2}} {α : Type.{u4}} {β : Type.{u1}} {γ : Type.{u3}} [_inst_4 : UniformSpace.{u4} α] [_inst_5 : UniformSpace.{u1} β] [_inst_6 : UniformSpace.{u3} γ] {F : ι -> β -> α} {u : α -> γ}, (UniformInducing.{u4, u3} α γ _inst_4 _inst_6 u) -> (Iff (UniformEquicontinuous.{u2, u4, u1} ι α β _inst_4 _inst_5 F) (UniformEquicontinuous.{u2, u3, u1} ι γ β _inst_6 _inst_5 (Function.comp.{succ u2, max (succ u4) (succ u1), max (succ u1) (succ u3)} ι (β -> α) (β -> γ) ((fun (x._@.Mathlib.Topology.UniformSpace.Equicontinuity._hyg.3411 : α -> γ) (x._@.Mathlib.Topology.UniformSpace.Equicontinuity._hyg.3413 : β -> α) => Function.comp.{succ u1, succ u4, succ u3} β α γ x._@.Mathlib.Topology.UniformSpace.Equicontinuity._hyg.3411 x._@.Mathlib.Topology.UniformSpace.Equicontinuity._hyg.3413) u) F)))
+  forall {ι : Type.{u2}} {α : Type.{u4}} {β : Type.{u1}} {γ : Type.{u3}} [_inst_4 : UniformSpace.{u4} α] [_inst_5 : UniformSpace.{u1} β] [_inst_6 : UniformSpace.{u3} γ] {F : ι -> β -> α} {u : α -> γ}, (UniformInducing.{u4, u3} α γ _inst_4 _inst_6 u) -> (Iff (UniformEquicontinuous.{u2, u4, u1} ι α β _inst_4 _inst_5 F) (UniformEquicontinuous.{u2, u3, u1} ι γ β _inst_6 _inst_5 (Function.comp.{succ u2, max (succ u4) (succ u1), max (succ u1) (succ u3)} ι (β -> α) (β -> γ) ((fun (x._@.Mathlib.Topology.UniformSpace.Equicontinuity._hyg.3426 : α -> γ) (x._@.Mathlib.Topology.UniformSpace.Equicontinuity._hyg.3428 : β -> α) => Function.comp.{succ u1, succ u4, succ u3} β α γ x._@.Mathlib.Topology.UniformSpace.Equicontinuity._hyg.3426 x._@.Mathlib.Topology.UniformSpace.Equicontinuity._hyg.3428) u) F)))
 Case conversion may be inaccurate. Consider using '#align uniform_inducing.uniform_equicontinuous_iff UniformInducing.uniformEquicontinuous_iffₓ'. -/
 /-- Given `u : α → γ` a uniform inducing map, a family `𝓕 : ι → β → α` is uniformly equicontinuous
 iff the family `𝓕'`, obtained by precomposing each function of `𝓕` by `u`, is uniformly

mathlib commit https://github.com/leanprover-community/mathlib/commit/641b6a82006416ec431b2987b354af9311fed4f2

@@ -411,7 +411,7 @@ theorem uniformEquicontinuous_iff_uniformContinuous {F : ι → β → α} :
 lean 3 declaration is
   forall {ι : Type.{u1}} {X : Type.{u2}} {α : Type.{u3}} [_inst_1 : TopologicalSpace.{u2} X] [_inst_4 : UniformSpace.{u3} α] {κ : Type.{u4}} {p : κ -> Prop} {s : κ -> (Set.{u2} X)} {F : ι -> X -> α} {x₀ : X}, (Filter.HasBasis.{u2, succ u4} X κ (nhds.{u2} X _inst_1 x₀) p s) -> (Iff (EquicontinuousAt.{u1, u2, u3} ι X α _inst_1 _inst_4 F x₀) (forall (U : Set.{u3} (Prod.{u3, u3} α α)), (Membership.Mem.{u3, u3} (Set.{u3} (Prod.{u3, u3} α α)) (Filter.{u3} (Prod.{u3, u3} α α)) (Filter.hasMem.{u3} (Prod.{u3, u3} α α)) U (uniformity.{u3} α _inst_4)) -> (Exists.{succ u4} κ (fun (k : κ) => Exists.{0} (p k) (fun (_x : p k) => forall (x : X), (Membership.Mem.{u2, u2} X (Set.{u2} X) (Set.hasMem.{u2} X) x (s k)) -> (forall (i : ι), Membership.Mem.{u3, u3} (Prod.{u3, u3} α α) (Set.{u3} (Prod.{u3, u3} α α)) (Set.hasMem.{u3} (Prod.{u3, u3} α α)) (Prod.mk.{u3, u3} α α (F i x₀) (F i x)) U))))))
 but is expected to have type
-  forall {ι : Type.{u2}} {X : Type.{u3}} {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u3} X] [_inst_4 : UniformSpace.{u1} α] {κ : Type.{u4}} {p : κ -> Prop} {s : κ -> (Set.{u3} X)} {F : ι -> X -> α} {x₀ : X}, (Filter.HasBasis.{u3, succ u4} X κ (nhds.{u3} X _inst_1 x₀) p s) -> (Iff (EquicontinuousAt.{u2, u3, u1} ι X α _inst_1 _inst_4 F x₀) (forall (U : Set.{u1} (Prod.{u1, u1} α α)), (Membership.mem.{u1, u1} (Set.{u1} (Prod.{u1, u1} α α)) (Filter.{u1} (Prod.{u1, u1} α α)) (instMembershipSetFilter.{u1} (Prod.{u1, u1} α α)) U (uniformity.{u1} α _inst_4)) -> (Exists.{succ u4} κ (fun (k : κ) => Exists.{0} (p k) (fun (_x : p k) => forall (x : X), (Membership.mem.{u3, u3} X (Set.{u3} X) (Set.instMembershipSet.{u3} X) x (s k)) -> (forall (i : ι), Membership.mem.{u1, u1} (Prod.{u1, u1} α α) (Set.{u1} (Prod.{u1, u1} α α)) (Set.instMembershipSet.{u1} (Prod.{u1, u1} α α)) (Prod.mk.{u1, u1} α α (F i x₀) (F i x)) U))))))
+  forall {ι : Type.{u2}} {X : Type.{u3}} {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u3} X] [_inst_4 : UniformSpace.{u1} α] {κ : Type.{u4}} {p : κ -> Prop} {s : κ -> (Set.{u3} X)} {F : ι -> X -> α} {x₀ : X}, (Filter.HasBasis.{u3, succ u4} X κ (nhds.{u3} X _inst_1 x₀) p s) -> (Iff (EquicontinuousAt.{u2, u3, u1} ι X α _inst_1 _inst_4 F x₀) (forall (U : Set.{u1} (Prod.{u1, u1} α α)), (Membership.mem.{u1, u1} (Set.{u1} (Prod.{u1, u1} α α)) (Filter.{u1} (Prod.{u1, u1} α α)) (instMembershipSetFilter.{u1} (Prod.{u1, u1} α α)) U (uniformity.{u1} α _inst_4)) -> (Exists.{succ u4} κ (fun (k : κ) => And (p k) (forall (x : X), (Membership.mem.{u3, u3} X (Set.{u3} X) (Set.instMembershipSet.{u3} X) x (s k)) -> (forall (i : ι), Membership.mem.{u1, u1} (Prod.{u1, u1} α α) (Set.{u1} (Prod.{u1, u1} α α)) (Set.instMembershipSet.{u1} (Prod.{u1, u1} α α)) (Prod.mk.{u1, u1} α α (F i x₀) (F i x)) U))))))
 Case conversion may be inaccurate. Consider using '#align filter.has_basis.equicontinuous_at_iff_left Filter.HasBasis.equicontinuousAt_iff_leftₓ'. -/
 theorem Filter.HasBasis.equicontinuousAt_iff_left {κ : Type _} {p : κ → Prop} {s : κ → Set X}
     {F : ι → X → α} {x₀ : X} (hX : (𝓝 x₀).HasBasis p s) :
@@ -441,7 +441,7 @@ theorem Filter.HasBasis.equicontinuousAt_iff_right {κ : Type _} {p : κ → Pro
 lean 3 declaration is
   forall {ι : Type.{u1}} {X : Type.{u2}} {α : Type.{u3}} [_inst_1 : TopologicalSpace.{u2} X] [_inst_4 : UniformSpace.{u3} α] {κ₁ : Type.{u4}} {κ₂ : Type.{u5}} {p₁ : κ₁ -> Prop} {s₁ : κ₁ -> (Set.{u2} X)} {p₂ : κ₂ -> Prop} {s₂ : κ₂ -> (Set.{u3} (Prod.{u3, u3} α α))} {F : ι -> X -> α} {x₀ : X}, (Filter.HasBasis.{u2, succ u4} X κ₁ (nhds.{u2} X _inst_1 x₀) p₁ s₁) -> (Filter.HasBasis.{u3, succ u5} (Prod.{u3, u3} α α) κ₂ (uniformity.{u3} α _inst_4) p₂ s₂) -> (Iff (EquicontinuousAt.{u1, u2, u3} ι X α _inst_1 _inst_4 F x₀) (forall (k₂ : κ₂), (p₂ k₂) -> (Exists.{succ u4} κ₁ (fun (k₁ : κ₁) => Exists.{0} (p₁ k₁) (fun (_x : p₁ k₁) => forall (x : X), (Membership.Mem.{u2, u2} X (Set.{u2} X) (Set.hasMem.{u2} X) x (s₁ k₁)) -> (forall (i : ι), Membership.Mem.{u3, u3} (Prod.{u3, u3} α α) (Set.{u3} (Prod.{u3, u3} α α)) (Set.hasMem.{u3} (Prod.{u3, u3} α α)) (Prod.mk.{u3, u3} α α (F i x₀) (F i x)) (s₂ k₂)))))))
 but is expected to have type
-  forall {ι : Type.{u1}} {X : Type.{u3}} {α : Type.{u2}} [_inst_1 : TopologicalSpace.{u3} X] [_inst_4 : UniformSpace.{u2} α] {κ₁ : Type.{u5}} {κ₂ : Type.{u4}} {p₁ : κ₁ -> Prop} {s₁ : κ₁ -> (Set.{u3} X)} {p₂ : κ₂ -> Prop} {s₂ : κ₂ -> (Set.{u2} (Prod.{u2, u2} α α))} {F : ι -> X -> α} {x₀ : X}, (Filter.HasBasis.{u3, succ u5} X κ₁ (nhds.{u3} X _inst_1 x₀) p₁ s₁) -> (Filter.HasBasis.{u2, succ u4} (Prod.{u2, u2} α α) κ₂ (uniformity.{u2} α _inst_4) p₂ s₂) -> (Iff (EquicontinuousAt.{u1, u3, u2} ι X α _inst_1 _inst_4 F x₀) (forall (k₂ : κ₂), (p₂ k₂) -> (Exists.{succ u5} κ₁ (fun (k₁ : κ₁) => Exists.{0} (p₁ k₁) (fun (_x : p₁ k₁) => forall (x : X), (Membership.mem.{u3, u3} X (Set.{u3} X) (Set.instMembershipSet.{u3} X) x (s₁ k₁)) -> (forall (i : ι), Membership.mem.{u2, u2} (Prod.{u2, u2} α α) (Set.{u2} (Prod.{u2, u2} α α)) (Set.instMembershipSet.{u2} (Prod.{u2, u2} α α)) (Prod.mk.{u2, u2} α α (F i x₀) (F i x)) (s₂ k₂)))))))
+  forall {ι : Type.{u1}} {X : Type.{u3}} {α : Type.{u2}} [_inst_1 : TopologicalSpace.{u3} X] [_inst_4 : UniformSpace.{u2} α] {κ₁ : Type.{u5}} {κ₂ : Type.{u4}} {p₁ : κ₁ -> Prop} {s₁ : κ₁ -> (Set.{u3} X)} {p₂ : κ₂ -> Prop} {s₂ : κ₂ -> (Set.{u2} (Prod.{u2, u2} α α))} {F : ι -> X -> α} {x₀ : X}, (Filter.HasBasis.{u3, succ u5} X κ₁ (nhds.{u3} X _inst_1 x₀) p₁ s₁) -> (Filter.HasBasis.{u2, succ u4} (Prod.{u2, u2} α α) κ₂ (uniformity.{u2} α _inst_4) p₂ s₂) -> (Iff (EquicontinuousAt.{u1, u3, u2} ι X α _inst_1 _inst_4 F x₀) (forall (k₂ : κ₂), (p₂ k₂) -> (Exists.{succ u5} κ₁ (fun (k₁ : κ₁) => And (p₁ k₁) (forall (x : X), (Membership.mem.{u3, u3} X (Set.{u3} X) (Set.instMembershipSet.{u3} X) x (s₁ k₁)) -> (forall (i : ι), Membership.mem.{u2, u2} (Prod.{u2, u2} α α) (Set.{u2} (Prod.{u2, u2} α α)) (Set.instMembershipSet.{u2} (Prod.{u2, u2} α α)) (Prod.mk.{u2, u2} α α (F i x₀) (F i x)) (s₂ k₂)))))))
 Case conversion may be inaccurate. Consider using '#align filter.has_basis.equicontinuous_at_iff Filter.HasBasis.equicontinuousAt_iffₓ'. -/
 theorem Filter.HasBasis.equicontinuousAt_iff {κ₁ κ₂ : Type _} {p₁ : κ₁ → Prop} {s₁ : κ₁ → Set X}
     {p₂ : κ₂ → Prop} {s₂ : κ₂ → Set (α × α)} {F : ι → X → α} {x₀ : X} (hX : (𝓝 x₀).HasBasis p₁ s₁)
@@ -458,7 +458,7 @@ theorem Filter.HasBasis.equicontinuousAt_iff {κ₁ κ₂ : Type _} {p₁ : κ
 lean 3 declaration is
   forall {ι : Type.{u1}} {α : Type.{u2}} {β : Type.{u3}} [_inst_4 : UniformSpace.{u2} α] [_inst_5 : UniformSpace.{u3} β] {κ : Type.{u4}} {p : κ -> Prop} {s : κ -> (Set.{u3} (Prod.{u3, u3} β β))} {F : ι -> β -> α}, (Filter.HasBasis.{u3, succ u4} (Prod.{u3, u3} β β) κ (uniformity.{u3} β _inst_5) p s) -> (Iff (UniformEquicontinuous.{u1, u2, u3} ι α β _inst_4 _inst_5 F) (forall (U : Set.{u2} (Prod.{u2, u2} α α)), (Membership.Mem.{u2, u2} (Set.{u2} (Prod.{u2, u2} α α)) (Filter.{u2} (Prod.{u2, u2} α α)) (Filter.hasMem.{u2} (Prod.{u2, u2} α α)) U (uniformity.{u2} α _inst_4)) -> (Exists.{succ u4} κ (fun (k : κ) => Exists.{0} (p k) (fun (_x : p k) => forall (x : β) (y : β), (Membership.Mem.{u3, u3} (Prod.{u3, u3} β β) (Set.{u3} (Prod.{u3, u3} β β)) (Set.hasMem.{u3} (Prod.{u3, u3} β β)) (Prod.mk.{u3, u3} β β x y) (s k)) -> (forall (i : ι), Membership.Mem.{u2, u2} (Prod.{u2, u2} α α) (Set.{u2} (Prod.{u2, u2} α α)) (Set.hasMem.{u2} (Prod.{u2, u2} α α)) (Prod.mk.{u2, u2} α α (F i x) (F i y)) U))))))
 but is expected to have type
-  forall {ι : Type.{u2}} {α : Type.{u1}} {β : Type.{u3}} [_inst_4 : UniformSpace.{u1} α] [_inst_5 : UniformSpace.{u3} β] {κ : Type.{u4}} {p : κ -> Prop} {s : κ -> (Set.{u3} (Prod.{u3, u3} β β))} {F : ι -> β -> α}, (Filter.HasBasis.{u3, succ u4} (Prod.{u3, u3} β β) κ (uniformity.{u3} β _inst_5) p s) -> (Iff (UniformEquicontinuous.{u2, u1, u3} ι α β _inst_4 _inst_5 F) (forall (U : Set.{u1} (Prod.{u1, u1} α α)), (Membership.mem.{u1, u1} (Set.{u1} (Prod.{u1, u1} α α)) (Filter.{u1} (Prod.{u1, u1} α α)) (instMembershipSetFilter.{u1} (Prod.{u1, u1} α α)) U (uniformity.{u1} α _inst_4)) -> (Exists.{succ u4} κ (fun (k : κ) => Exists.{0} (p k) (fun (_x : p k) => forall (x : β) (y : β), (Membership.mem.{u3, u3} (Prod.{u3, u3} β β) (Set.{u3} (Prod.{u3, u3} β β)) (Set.instMembershipSet.{u3} (Prod.{u3, u3} β β)) (Prod.mk.{u3, u3} β β x y) (s k)) -> (forall (i : ι), Membership.mem.{u1, u1} (Prod.{u1, u1} α α) (Set.{u1} (Prod.{u1, u1} α α)) (Set.instMembershipSet.{u1} (Prod.{u1, u1} α α)) (Prod.mk.{u1, u1} α α (F i x) (F i y)) U))))))
+  forall {ι : Type.{u2}} {α : Type.{u1}} {β : Type.{u3}} [_inst_4 : UniformSpace.{u1} α] [_inst_5 : UniformSpace.{u3} β] {κ : Type.{u4}} {p : κ -> Prop} {s : κ -> (Set.{u3} (Prod.{u3, u3} β β))} {F : ι -> β -> α}, (Filter.HasBasis.{u3, succ u4} (Prod.{u3, u3} β β) κ (uniformity.{u3} β _inst_5) p s) -> (Iff (UniformEquicontinuous.{u2, u1, u3} ι α β _inst_4 _inst_5 F) (forall (U : Set.{u1} (Prod.{u1, u1} α α)), (Membership.mem.{u1, u1} (Set.{u1} (Prod.{u1, u1} α α)) (Filter.{u1} (Prod.{u1, u1} α α)) (instMembershipSetFilter.{u1} (Prod.{u1, u1} α α)) U (uniformity.{u1} α _inst_4)) -> (Exists.{succ u4} κ (fun (k : κ) => And (p k) (forall (x : β) (y : β), (Membership.mem.{u3, u3} (Prod.{u3, u3} β β) (Set.{u3} (Prod.{u3, u3} β β)) (Set.instMembershipSet.{u3} (Prod.{u3, u3} β β)) (Prod.mk.{u3, u3} β β x y) (s k)) -> (forall (i : ι), Membership.mem.{u1, u1} (Prod.{u1, u1} α α) (Set.{u1} (Prod.{u1, u1} α α)) (Set.instMembershipSet.{u1} (Prod.{u1, u1} α α)) (Prod.mk.{u1, u1} α α (F i x) (F i y)) U))))))
 Case conversion may be inaccurate. Consider using '#align filter.has_basis.uniform_equicontinuous_iff_left Filter.HasBasis.uniformEquicontinuous_iff_leftₓ'. -/
 theorem Filter.HasBasis.uniformEquicontinuous_iff_left {κ : Type _} {p : κ → Prop}
     {s : κ → Set (β × β)} {F : ι → β → α} (hβ : (𝓤 β).HasBasis p s) :
@@ -490,7 +490,7 @@ theorem Filter.HasBasis.uniformEquicontinuous_iff_right {κ : Type _} {p : κ 
 lean 3 declaration is
   forall {ι : Type.{u1}} {α : Type.{u2}} {β : Type.{u3}} [_inst_4 : UniformSpace.{u2} α] [_inst_5 : UniformSpace.{u3} β] {κ₁ : Type.{u4}} {κ₂ : Type.{u5}} {p₁ : κ₁ -> Prop} {s₁ : κ₁ -> (Set.{u3} (Prod.{u3, u3} β β))} {p₂ : κ₂ -> Prop} {s₂ : κ₂ -> (Set.{u2} (Prod.{u2, u2} α α))} {F : ι -> β -> α}, (Filter.HasBasis.{u3, succ u4} (Prod.{u3, u3} β β) κ₁ (uniformity.{u3} β _inst_5) p₁ s₁) -> (Filter.HasBasis.{u2, succ u5} (Prod.{u2, u2} α α) κ₂ (uniformity.{u2} α _inst_4) p₂ s₂) -> (Iff (UniformEquicontinuous.{u1, u2, u3} ι α β _inst_4 _inst_5 F) (forall (k₂ : κ₂), (p₂ k₂) -> (Exists.{succ u4} κ₁ (fun (k₁ : κ₁) => Exists.{0} (p₁ k₁) (fun (_x : p₁ k₁) => forall (x : β) (y : β), (Membership.Mem.{u3, u3} (Prod.{u3, u3} β β) (Set.{u3} (Prod.{u3, u3} β β)) (Set.hasMem.{u3} (Prod.{u3, u3} β β)) (Prod.mk.{u3, u3} β β x y) (s₁ k₁)) -> (forall (i : ι), Membership.Mem.{u2, u2} (Prod.{u2, u2} α α) (Set.{u2} (Prod.{u2, u2} α α)) (Set.hasMem.{u2} (Prod.{u2, u2} α α)) (Prod.mk.{u2, u2} α α (F i x) (F i y)) (s₂ k₂)))))))
 but is expected to have type
-  forall {ι : Type.{u1}} {α : Type.{u2}} {β : Type.{u3}} [_inst_4 : UniformSpace.{u2} α] [_inst_5 : UniformSpace.{u3} β] {κ₁ : Type.{u5}} {κ₂ : Type.{u4}} {p₁ : κ₁ -> Prop} {s₁ : κ₁ -> (Set.{u3} (Prod.{u3, u3} β β))} {p₂ : κ₂ -> Prop} {s₂ : κ₂ -> (Set.{u2} (Prod.{u2, u2} α α))} {F : ι -> β -> α}, (Filter.HasBasis.{u3, succ u5} (Prod.{u3, u3} β β) κ₁ (uniformity.{u3} β _inst_5) p₁ s₁) -> (Filter.HasBasis.{u2, succ u4} (Prod.{u2, u2} α α) κ₂ (uniformity.{u2} α _inst_4) p₂ s₂) -> (Iff (UniformEquicontinuous.{u1, u2, u3} ι α β _inst_4 _inst_5 F) (forall (k₂ : κ₂), (p₂ k₂) -> (Exists.{succ u5} κ₁ (fun (k₁ : κ₁) => Exists.{0} (p₁ k₁) (fun (_x : p₁ k₁) => forall (x : β) (y : β), (Membership.mem.{u3, u3} (Prod.{u3, u3} β β) (Set.{u3} (Prod.{u3, u3} β β)) (Set.instMembershipSet.{u3} (Prod.{u3, u3} β β)) (Prod.mk.{u3, u3} β β x y) (s₁ k₁)) -> (forall (i : ι), Membership.mem.{u2, u2} (Prod.{u2, u2} α α) (Set.{u2} (Prod.{u2, u2} α α)) (Set.instMembershipSet.{u2} (Prod.{u2, u2} α α)) (Prod.mk.{u2, u2} α α (F i x) (F i y)) (s₂ k₂)))))))
+  forall {ι : Type.{u1}} {α : Type.{u2}} {β : Type.{u3}} [_inst_4 : UniformSpace.{u2} α] [_inst_5 : UniformSpace.{u3} β] {κ₁ : Type.{u5}} {κ₂ : Type.{u4}} {p₁ : κ₁ -> Prop} {s₁ : κ₁ -> (Set.{u3} (Prod.{u3, u3} β β))} {p₂ : κ₂ -> Prop} {s₂ : κ₂ -> (Set.{u2} (Prod.{u2, u2} α α))} {F : ι -> β -> α}, (Filter.HasBasis.{u3, succ u5} (Prod.{u3, u3} β β) κ₁ (uniformity.{u3} β _inst_5) p₁ s₁) -> (Filter.HasBasis.{u2, succ u4} (Prod.{u2, u2} α α) κ₂ (uniformity.{u2} α _inst_4) p₂ s₂) -> (Iff (UniformEquicontinuous.{u1, u2, u3} ι α β _inst_4 _inst_5 F) (forall (k₂ : κ₂), (p₂ k₂) -> (Exists.{succ u5} κ₁ (fun (k₁ : κ₁) => And (p₁ k₁) (forall (x : β) (y : β), (Membership.mem.{u3, u3} (Prod.{u3, u3} β β) (Set.{u3} (Prod.{u3, u3} β β)) (Set.instMembershipSet.{u3} (Prod.{u3, u3} β β)) (Prod.mk.{u3, u3} β β x y) (s₁ k₁)) -> (forall (i : ι), Membership.mem.{u2, u2} (Prod.{u2, u2} α α) (Set.{u2} (Prod.{u2, u2} α α)) (Set.instMembershipSet.{u2} (Prod.{u2, u2} α α)) (Prod.mk.{u2, u2} α α (F i x) (F i y)) (s₂ k₂)))))))
 Case conversion may be inaccurate. Consider using '#align filter.has_basis.uniform_equicontinuous_iff Filter.HasBasis.uniformEquicontinuous_iffₓ'. -/
 theorem Filter.HasBasis.uniformEquicontinuous_iff {κ₁ κ₂ : Type _} {p₁ : κ₁ → Prop}
     {s₁ : κ₁ → Set (β × β)} {p₂ : κ₂ → Prop} {s₂ : κ₂ → Set (α × α)} {F : ι → β → α}
@@ -508,7 +508,7 @@ theorem Filter.HasBasis.uniformEquicontinuous_iff {κ₁ κ₂ : Type _} {p₁ :
 lean 3 declaration is
   forall {ι : Type.{u1}} {X : Type.{u2}} {α : Type.{u3}} {β : Type.{u4}} [_inst_1 : TopologicalSpace.{u2} X] [_inst_4 : UniformSpace.{u3} α] [_inst_5 : UniformSpace.{u4} β] {F : ι -> X -> α} {x₀ : X} {u : α -> β}, (UniformInducing.{u3, u4} α β _inst_4 _inst_5 u) -> (Iff (EquicontinuousAt.{u1, u2, u3} ι X α _inst_1 _inst_4 F x₀) (EquicontinuousAt.{u1, u2, u4} ι X β _inst_1 _inst_5 (Function.comp.{succ u1, max (succ u2) (succ u3), max (succ u2) (succ u4)} ι (X -> α) (X -> β) (Function.comp.{succ u2, succ u3, succ u4} X α β u) F) x₀))
 but is expected to have type
-  forall {ι : Type.{u2}} {X : Type.{u1}} {α : Type.{u4}} {β : Type.{u3}} [_inst_1 : TopologicalSpace.{u1} X] [_inst_4 : UniformSpace.{u4} α] [_inst_5 : UniformSpace.{u3} β] {F : ι -> X -> α} {x₀ : X} {u : α -> β}, (UniformInducing.{u4, u3} α β _inst_4 _inst_5 u) -> (Iff (EquicontinuousAt.{u2, u1, u4} ι X α _inst_1 _inst_4 F x₀) (EquicontinuousAt.{u2, u1, u3} ι X β _inst_1 _inst_5 (Function.comp.{succ u2, max (succ u4) (succ u1), max (succ u3) (succ u1)} ι (X -> α) (X -> β) ((fun (x._@.Mathlib.Topology.UniformSpace.Equicontinuity._hyg.3096 : α -> β) (x._@.Mathlib.Topology.UniformSpace.Equicontinuity._hyg.3098 : X -> α) => Function.comp.{succ u1, succ u4, succ u3} X α β x._@.Mathlib.Topology.UniformSpace.Equicontinuity._hyg.3096 x._@.Mathlib.Topology.UniformSpace.Equicontinuity._hyg.3098) u) F) x₀))
+  forall {ι : Type.{u2}} {X : Type.{u1}} {α : Type.{u4}} {β : Type.{u3}} [_inst_1 : TopologicalSpace.{u1} X] [_inst_4 : UniformSpace.{u4} α] [_inst_5 : UniformSpace.{u3} β] {F : ι -> X -> α} {x₀ : X} {u : α -> β}, (UniformInducing.{u4, u3} α β _inst_4 _inst_5 u) -> (Iff (EquicontinuousAt.{u2, u1, u4} ι X α _inst_1 _inst_4 F x₀) (EquicontinuousAt.{u2, u1, u3} ι X β _inst_1 _inst_5 (Function.comp.{succ u2, max (succ u4) (succ u1), max (succ u3) (succ u1)} ι (X -> α) (X -> β) ((fun (x._@.Mathlib.Topology.UniformSpace.Equicontinuity._hyg.3088 : α -> β) (x._@.Mathlib.Topology.UniformSpace.Equicontinuity._hyg.3090 : X -> α) => Function.comp.{succ u1, succ u4, succ u3} X α β x._@.Mathlib.Topology.UniformSpace.Equicontinuity._hyg.3088 x._@.Mathlib.Topology.UniformSpace.Equicontinuity._hyg.3090) u) F) x₀))
 Case conversion may be inaccurate. Consider using '#align uniform_inducing.equicontinuous_at_iff UniformInducing.equicontinuousAt_iffₓ'. -/
 /-- Given `u : α → β` a uniform inducing map, a family `𝓕 : ι → X → α` is equicontinuous at a point
 `x₀ : X` iff the family `𝓕'`, obtained by precomposing each function of `𝓕` by `u`, is
@@ -525,7 +525,7 @@ theorem UniformInducing.equicontinuousAt_iff {F : ι → X → α} {x₀ : X} {u
 lean 3 declaration is
   forall {ι : Type.{u1}} {X : Type.{u2}} {α : Type.{u3}} {β : Type.{u4}} [_inst_1 : TopologicalSpace.{u2} X] [_inst_4 : UniformSpace.{u3} α] [_inst_5 : UniformSpace.{u4} β] {F : ι -> X -> α} {u : α -> β}, (UniformInducing.{u3, u4} α β _inst_4 _inst_5 u) -> (Iff (Equicontinuous.{u1, u2, u3} ι X α _inst_1 _inst_4 F) (Equicontinuous.{u1, u2, u4} ι X β _inst_1 _inst_5 (Function.comp.{succ u1, max (succ u2) (succ u3), max (succ u2) (succ u4)} ι (X -> α) (X -> β) (Function.comp.{succ u2, succ u3, succ u4} X α β u) F)))
 but is expected to have type
-  forall {ι : Type.{u2}} {X : Type.{u1}} {α : Type.{u4}} {β : Type.{u3}} [_inst_1 : TopologicalSpace.{u1} X] [_inst_4 : UniformSpace.{u4} α] [_inst_5 : UniformSpace.{u3} β] {F : ι -> X -> α} {u : α -> β}, (UniformInducing.{u4, u3} α β _inst_4 _inst_5 u) -> (Iff (Equicontinuous.{u2, u1, u4} ι X α _inst_1 _inst_4 F) (Equicontinuous.{u2, u1, u3} ι X β _inst_1 _inst_5 (Function.comp.{succ u2, max (succ u4) (succ u1), max (succ u3) (succ u1)} ι (X -> α) (X -> β) ((fun (x._@.Mathlib.Topology.UniformSpace.Equicontinuity._hyg.3235 : α -> β) (x._@.Mathlib.Topology.UniformSpace.Equicontinuity._hyg.3237 : X -> α) => Function.comp.{succ u1, succ u4, succ u3} X α β x._@.Mathlib.Topology.UniformSpace.Equicontinuity._hyg.3235 x._@.Mathlib.Topology.UniformSpace.Equicontinuity._hyg.3237) u) F)))
+  forall {ι : Type.{u2}} {X : Type.{u1}} {α : Type.{u4}} {β : Type.{u3}} [_inst_1 : TopologicalSpace.{u1} X] [_inst_4 : UniformSpace.{u4} α] [_inst_5 : UniformSpace.{u3} β] {F : ι -> X -> α} {u : α -> β}, (UniformInducing.{u4, u3} α β _inst_4 _inst_5 u) -> (Iff (Equicontinuous.{u2, u1, u4} ι X α _inst_1 _inst_4 F) (Equicontinuous.{u2, u1, u3} ι X β _inst_1 _inst_5 (Function.comp.{succ u2, max (succ u4) (succ u1), max (succ u3) (succ u1)} ι (X -> α) (X -> β) ((fun (x._@.Mathlib.Topology.UniformSpace.Equicontinuity._hyg.3224 : α -> β) (x._@.Mathlib.Topology.UniformSpace.Equicontinuity._hyg.3226 : X -> α) => Function.comp.{succ u1, succ u4, succ u3} X α β x._@.Mathlib.Topology.UniformSpace.Equicontinuity._hyg.3224 x._@.Mathlib.Topology.UniformSpace.Equicontinuity._hyg.3226) u) F)))
 Case conversion may be inaccurate. Consider using '#align uniform_inducing.equicontinuous_iff UniformInducing.equicontinuous_iffₓ'. -/
 /- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:73:14: unsupported tactic `congrm #[[expr ∀ x, (_ : exprProp())]] -/
 /-- Given `u : α → β` a uniform inducing map, a family `𝓕 : ι → X → α` is equicontinuous iff the
@@ -542,7 +542,7 @@ theorem UniformInducing.equicontinuous_iff {F : ι → X → α} {u : α → β}
 lean 3 declaration is
   forall {ι : Type.{u1}} {α : Type.{u2}} {β : Type.{u3}} {γ : Type.{u4}} [_inst_4 : UniformSpace.{u2} α] [_inst_5 : UniformSpace.{u3} β] [_inst_6 : UniformSpace.{u4} γ] {F : ι -> β -> α} {u : α -> γ}, (UniformInducing.{u2, u4} α γ _inst_4 _inst_6 u) -> (Iff (UniformEquicontinuous.{u1, u2, u3} ι α β _inst_4 _inst_5 F) (UniformEquicontinuous.{u1, u4, u3} ι γ β _inst_6 _inst_5 (Function.comp.{succ u1, max (succ u3) (succ u2), max (succ u3) (succ u4)} ι (β -> α) (β -> γ) (Function.comp.{succ u3, succ u2, succ u4} β α γ u) F)))
 but is expected to have type
-  forall {ι : Type.{u2}} {α : Type.{u4}} {β : Type.{u1}} {γ : Type.{u3}} [_inst_4 : UniformSpace.{u4} α] [_inst_5 : UniformSpace.{u1} β] [_inst_6 : UniformSpace.{u3} γ] {F : ι -> β -> α} {u : α -> γ}, (UniformInducing.{u4, u3} α γ _inst_4 _inst_6 u) -> (Iff (UniformEquicontinuous.{u2, u4, u1} ι α β _inst_4 _inst_5 F) (UniformEquicontinuous.{u2, u3, u1} ι γ β _inst_6 _inst_5 (Function.comp.{succ u2, max (succ u4) (succ u1), max (succ u1) (succ u3)} ι (β -> α) (β -> γ) ((fun (x._@.Mathlib.Topology.UniformSpace.Equicontinuity._hyg.3422 : α -> γ) (x._@.Mathlib.Topology.UniformSpace.Equicontinuity._hyg.3424 : β -> α) => Function.comp.{succ u1, succ u4, succ u3} β α γ x._@.Mathlib.Topology.UniformSpace.Equicontinuity._hyg.3422 x._@.Mathlib.Topology.UniformSpace.Equicontinuity._hyg.3424) u) F)))
+  forall {ι : Type.{u2}} {α : Type.{u4}} {β : Type.{u1}} {γ : Type.{u3}} [_inst_4 : UniformSpace.{u4} α] [_inst_5 : UniformSpace.{u1} β] [_inst_6 : UniformSpace.{u3} γ] {F : ι -> β -> α} {u : α -> γ}, (UniformInducing.{u4, u3} α γ _inst_4 _inst_6 u) -> (Iff (UniformEquicontinuous.{u2, u4, u1} ι α β _inst_4 _inst_5 F) (UniformEquicontinuous.{u2, u3, u1} ι γ β _inst_6 _inst_5 (Function.comp.{succ u2, max (succ u4) (succ u1), max (succ u1) (succ u3)} ι (β -> α) (β -> γ) ((fun (x._@.Mathlib.Topology.UniformSpace.Equicontinuity._hyg.3411 : α -> γ) (x._@.Mathlib.Topology.UniformSpace.Equicontinuity._hyg.3413 : β -> α) => Function.comp.{succ u1, succ u4, succ u3} β α γ x._@.Mathlib.Topology.UniformSpace.Equicontinuity._hyg.3411 x._@.Mathlib.Topology.UniformSpace.Equicontinuity._hyg.3413) u) F)))
 Case conversion may be inaccurate. Consider using '#align uniform_inducing.uniform_equicontinuous_iff UniformInducing.uniformEquicontinuous_iffₓ'. -/
 /-- Given `u : α → γ` a uniform inducing map, a family `𝓕 : ι → β → α` is uniformly equicontinuous
 iff the family `𝓕'`, obtained by precomposing each function of `𝓕` by `u`, is uniformly

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

@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Anatole Dedecker
 
 ! This file was ported from Lean 3 source module topology.uniform_space.equicontinuity
-! leanprover-community/mathlib commit f2ce6086713c78a7f880485f7917ea547a215982
+! leanprover-community/mathlib commit ee05e9ce1322178f0c12004eb93c00d2c8c00ed2
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
@@ -13,6 +13,9 @@ import Mathbin.Topology.UniformSpace.UniformConvergenceTopology
 /-!
 # Equicontinuity of a family of functions
 
+> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.
+> Any changes to this file require a corresponding PR to mathlib4.
+
 Let `X` be a topological space and `α` a `uniform_space`. A family of functions `F : ι → X → α`
 is said to be *equicontinuous at a point `x₀ : X`* when, for any entourage `U` in `α`, there is a
 neighborhood `V` of `x₀` such that, for all `x ∈ V`, and *for all `i`*, `F i x` is `U`-close to

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

@@ -149,7 +149,7 @@ lean 3 declaration is
 but is expected to have type
   forall {ι : Type.{u3}} {X : Type.{u2}} {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u2} X] [_inst_4 : UniformSpace.{u1} α] {F : ι -> X -> α} {x₀ : X}, Iff (EquicontinuousAt.{u3, u2, u1} ι X α _inst_1 _inst_4 F x₀) (forall (U : Set.{u1} (Prod.{u1, u1} α α)), (Membership.mem.{u1, u1} (Set.{u1} (Prod.{u1, u1} α α)) (Filter.{u1} (Prod.{u1, u1} α α)) (instMembershipSetFilter.{u1} (Prod.{u1, u1} α α)) U (uniformity.{u1} α _inst_4)) -> (Exists.{succ u2} (Set.{u2} X) (fun (V : Set.{u2} X) => And (Membership.mem.{u2, u2} (Set.{u2} X) (Filter.{u2} X) (instMembershipSetFilter.{u2} X) V (nhds.{u2} X _inst_1 x₀)) (forall (x : X), (Membership.mem.{u2, u2} X (Set.{u2} X) (Set.instMembershipSet.{u2} X) x V) -> (forall (y : X), (Membership.mem.{u2, u2} X (Set.{u2} X) (Set.instMembershipSet.{u2} X) y V) -> (forall (i : ι), Membership.mem.{u1, u1} (Prod.{u1, u1} α α) (Set.{u1} (Prod.{u1, u1} α α)) (Set.instMembershipSet.{u1} (Prod.{u1, u1} α α)) (Prod.mk.{u1, u1} α α (F i x) (F i y)) U))))))
 Case conversion may be inaccurate. Consider using '#align equicontinuous_at_iff_pair equicontinuousAt_iff_pairₓ'. -/
-/- ./././Mathport/Syntax/Translate/Basic.lean:628:2: warning: expanding binder collection (x y «expr ∈ » V) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (x y «expr ∈ » V) -/
 /-- Reformulation of equicontinuity at `x₀` comparing two variables near `x₀` instead of comparing
 only one with `x₀`. -/
 theorem equicontinuousAt_iff_pair {F : ι → X → α} {x₀ : X} :
@@ -524,14 +524,14 @@ lean 3 declaration is
 but is expected to have type
   forall {ι : Type.{u2}} {X : Type.{u1}} {α : Type.{u4}} {β : Type.{u3}} [_inst_1 : TopologicalSpace.{u1} X] [_inst_4 : UniformSpace.{u4} α] [_inst_5 : UniformSpace.{u3} β] {F : ι -> X -> α} {u : α -> β}, (UniformInducing.{u4, u3} α β _inst_4 _inst_5 u) -> (Iff (Equicontinuous.{u2, u1, u4} ι X α _inst_1 _inst_4 F) (Equicontinuous.{u2, u1, u3} ι X β _inst_1 _inst_5 (Function.comp.{succ u2, max (succ u4) (succ u1), max (succ u3) (succ u1)} ι (X -> α) (X -> β) ((fun (x._@.Mathlib.Topology.UniformSpace.Equicontinuity._hyg.3235 : α -> β) (x._@.Mathlib.Topology.UniformSpace.Equicontinuity._hyg.3237 : X -> α) => Function.comp.{succ u1, succ u4, succ u3} X α β x._@.Mathlib.Topology.UniformSpace.Equicontinuity._hyg.3235 x._@.Mathlib.Topology.UniformSpace.Equicontinuity._hyg.3237) u) F)))
 Case conversion may be inaccurate. Consider using '#align uniform_inducing.equicontinuous_iff UniformInducing.equicontinuous_iffₓ'. -/
-/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:76:14: unsupported tactic `congrm #[[expr ∀ x, (_ : exprProp())]] -/
+/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:73:14: unsupported tactic `congrm #[[expr ∀ x, (_ : exprProp())]] -/
 /-- Given `u : α → β` a uniform inducing map, a family `𝓕 : ι → X → α` is equicontinuous iff the
 family `𝓕'`, obtained by precomposing each function of `𝓕` by `u`, is equicontinuous. -/
 theorem UniformInducing.equicontinuous_iff {F : ι → X → α} {u : α → β} (hu : UniformInducing u) :
     Equicontinuous F ↔ Equicontinuous ((· ∘ ·) u ∘ F) :=
   by
   trace
-    "./././Mathport/Syntax/Translate/Tactic/Builtin.lean:76:14: unsupported tactic `congrm #[[expr ∀ x, (_ : exprProp())]]"
+    "./././Mathport/Syntax/Translate/Tactic/Builtin.lean:73:14: unsupported tactic `congrm #[[expr ∀ x, (_ : exprProp())]]"
   rw [hu.equicontinuous_at_iff]
 #align uniform_inducing.equicontinuous_iff UniformInducing.equicontinuous_iff
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/22131150f88a2d125713ffa0f4693e3355b1eb49

@@ -93,44 +93,62 @@ open uniformity Topology UniformConvergence
 variable {ι κ X Y Z α β γ 𝓕 : Type _} [TopologicalSpace X] [TopologicalSpace Y] [TopologicalSpace Z]
   [UniformSpace α] [UniformSpace β] [UniformSpace γ]
 
+#print EquicontinuousAt /-
 /-- A family `F : ι → X → α` of functions from a topological space to a uniform space is
 *equicontinuous at `x₀ : X`* if, for all entourage `U ∈ 𝓤 α`, there is a neighborhood `V` of `x₀`
 such that, for all `x ∈ V` and for all `i : ι`, `F i x` is `U`-close to `F i x₀`. -/
 def EquicontinuousAt (F : ι → X → α) (x₀ : X) : Prop :=
   ∀ U ∈ 𝓤 α, ∀ᶠ x in 𝓝 x₀, ∀ i, (F i x₀, F i x) ∈ U
 #align equicontinuous_at EquicontinuousAt
+-/
 
+#print Set.EquicontinuousAt /-
 /-- We say that a set `H : set (X → α)` of functions is equicontinuous at a point if the family
 `coe : ↥H → (X → α)` is equicontinuous at that point. -/
 protected abbrev Set.EquicontinuousAt (H : Set <| X → α) (x₀ : X) : Prop :=
   EquicontinuousAt (coe : H → X → α) x₀
 #align set.equicontinuous_at Set.EquicontinuousAt
+-/
 
+#print Equicontinuous /-
 /-- A family `F : ι → X → α` of functions from a topological space to a uniform space is
 *equicontinuous* on all of `X` if it is equicontinuous at each point of `X`. -/
 def Equicontinuous (F : ι → X → α) : Prop :=
   ∀ x₀, EquicontinuousAt F x₀
 #align equicontinuous Equicontinuous
+-/
 
+#print Set.Equicontinuous /-
 /-- We say that a set `H : set (X → α)` of functions is equicontinuous if the family
 `coe : ↥H → (X → α)` is equicontinuous. -/
 protected abbrev Set.Equicontinuous (H : Set <| X → α) : Prop :=
   Equicontinuous (coe : H → X → α)
 #align set.equicontinuous Set.Equicontinuous
+-/
 
+#print UniformEquicontinuous /-
 /-- A family `F : ι → β → α` of functions between uniform spaces is *uniformly equicontinuous* if,
 for all entourage `U ∈ 𝓤 α`, there is an entourage `V ∈ 𝓤 β` such that, whenever `x` and `y` are
 `V`-close, we have that, *for all `i : ι`*, `F i x` is `U`-close to `F i x₀`. -/
 def UniformEquicontinuous (F : ι → β → α) : Prop :=
   ∀ U ∈ 𝓤 α, ∀ᶠ xy : β × β in 𝓤 β, ∀ i, (F i xy.1, F i xy.2) ∈ U
 #align uniform_equicontinuous UniformEquicontinuous
+-/
 
+#print Set.UniformEquicontinuous /-
 /-- We say that a set `H : set (X → α)` of functions is uniformly equicontinuous if the family
 `coe : ↥H → (X → α)` is uniformly equicontinuous. -/
 protected abbrev Set.UniformEquicontinuous (H : Set <| β → α) : Prop :=
   UniformEquicontinuous (coe : H → β → α)
 #align set.uniform_equicontinuous Set.UniformEquicontinuous
+-/
 
+/- warning: equicontinuous_at_iff_pair -> equicontinuousAt_iff_pair is a dubious translation:
+lean 3 declaration is
+  forall {ι : Type.{u1}} {X : Type.{u2}} {α : Type.{u3}} [_inst_1 : TopologicalSpace.{u2} X] [_inst_4 : UniformSpace.{u3} α] {F : ι -> X -> α} {x₀ : X}, Iff (EquicontinuousAt.{u1, u2, u3} ι X α _inst_1 _inst_4 F x₀) (forall (U : Set.{u3} (Prod.{u3, u3} α α)), (Membership.Mem.{u3, u3} (Set.{u3} (Prod.{u3, u3} α α)) (Filter.{u3} (Prod.{u3, u3} α α)) (Filter.hasMem.{u3} (Prod.{u3, u3} α α)) U (uniformity.{u3} α _inst_4)) -> (Exists.{succ u2} (Set.{u2} X) (fun (V : Set.{u2} X) => Exists.{0} (Membership.Mem.{u2, u2} (Set.{u2} X) (Filter.{u2} X) (Filter.hasMem.{u2} X) V (nhds.{u2} X _inst_1 x₀)) (fun (H : Membership.Mem.{u2, u2} (Set.{u2} X) (Filter.{u2} X) (Filter.hasMem.{u2} X) V (nhds.{u2} X _inst_1 x₀)) => forall (x : X), (Membership.Mem.{u2, u2} X (Set.{u2} X) (Set.hasMem.{u2} X) x V) -> (forall (y : X), (Membership.Mem.{u2, u2} X (Set.{u2} X) (Set.hasMem.{u2} X) y V) -> (forall (i : ι), Membership.Mem.{u3, u3} (Prod.{u3, u3} α α) (Set.{u3} (Prod.{u3, u3} α α)) (Set.hasMem.{u3} (Prod.{u3, u3} α α)) (Prod.mk.{u3, u3} α α (F i x) (F i y)) U))))))
+but is expected to have type
+  forall {ι : Type.{u3}} {X : Type.{u2}} {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u2} X] [_inst_4 : UniformSpace.{u1} α] {F : ι -> X -> α} {x₀ : X}, Iff (EquicontinuousAt.{u3, u2, u1} ι X α _inst_1 _inst_4 F x₀) (forall (U : Set.{u1} (Prod.{u1, u1} α α)), (Membership.mem.{u1, u1} (Set.{u1} (Prod.{u1, u1} α α)) (Filter.{u1} (Prod.{u1, u1} α α)) (instMembershipSetFilter.{u1} (Prod.{u1, u1} α α)) U (uniformity.{u1} α _inst_4)) -> (Exists.{succ u2} (Set.{u2} X) (fun (V : Set.{u2} X) => And (Membership.mem.{u2, u2} (Set.{u2} X) (Filter.{u2} X) (instMembershipSetFilter.{u2} X) V (nhds.{u2} X _inst_1 x₀)) (forall (x : X), (Membership.mem.{u2, u2} X (Set.{u2} X) (Set.instMembershipSet.{u2} X) x V) -> (forall (y : X), (Membership.mem.{u2, u2} X (Set.{u2} X) (Set.instMembershipSet.{u2} X) y V) -> (forall (i : ι), Membership.mem.{u1, u1} (Prod.{u1, u1} α α) (Set.{u1} (Prod.{u1, u1} α α)) (Set.instMembershipSet.{u1} (Prod.{u1, u1} α α)) (Prod.mk.{u1, u1} α α (F i x) (F i y)) U))))))
+Case conversion may be inaccurate. Consider using '#align equicontinuous_at_iff_pair equicontinuousAt_iff_pairₓ'. -/
 /- ./././Mathport/Syntax/Translate/Basic.lean:628:2: warning: expanding binder collection (x y «expr ∈ » V) -/
 /-- Reformulation of equicontinuity at `x₀` comparing two variables near `x₀` instead of comparing
 only one with `x₀`. -/
@@ -146,12 +164,24 @@ theorem equicontinuousAt_iff_pair {F : ι → X → α} {x₀ : X} :
     filter_upwards [hV]using fun x hx i => hVU x₀ (mem_of_mem_nhds hV) x hx i
 #align equicontinuous_at_iff_pair equicontinuousAt_iff_pair
 
+/- warning: uniform_equicontinuous.equicontinuous -> UniformEquicontinuous.equicontinuous is a dubious translation:
+lean 3 declaration is
+  forall {ι : Type.{u1}} {α : Type.{u2}} {β : Type.{u3}} [_inst_4 : UniformSpace.{u2} α] [_inst_5 : UniformSpace.{u3} β] {F : ι -> β -> α}, (UniformEquicontinuous.{u1, u2, u3} ι α β _inst_4 _inst_5 F) -> (Equicontinuous.{u1, u3, u2} ι β α (UniformSpace.toTopologicalSpace.{u3} β _inst_5) _inst_4 F)
+but is expected to have type
+  forall {ι : Type.{u3}} {α : Type.{u2}} {β : Type.{u1}} [_inst_4 : UniformSpace.{u2} α] [_inst_5 : UniformSpace.{u1} β] {F : ι -> β -> α}, (UniformEquicontinuous.{u3, u2, u1} ι α β _inst_4 _inst_5 F) -> (Equicontinuous.{u3, u1, u2} ι β α (UniformSpace.toTopologicalSpace.{u1} β _inst_5) _inst_4 F)
+Case conversion may be inaccurate. Consider using '#align uniform_equicontinuous.equicontinuous UniformEquicontinuous.equicontinuousₓ'. -/
 /-- Uniform equicontinuity implies equicontinuity. -/
 theorem UniformEquicontinuous.equicontinuous {F : ι → β → α} (h : UniformEquicontinuous F) :
     Equicontinuous F := fun x₀ U hU =>
   mem_of_superset (ball_mem_nhds x₀ (h U hU)) fun x hx i => hx i
 #align uniform_equicontinuous.equicontinuous UniformEquicontinuous.equicontinuous
 
+/- warning: equicontinuous_at.continuous_at -> EquicontinuousAt.continuousAt is a dubious translation:
+lean 3 declaration is
+  forall {ι : Type.{u1}} {X : Type.{u2}} {α : Type.{u3}} [_inst_1 : TopologicalSpace.{u2} X] [_inst_4 : UniformSpace.{u3} α] {F : ι -> X -> α} {x₀ : X}, (EquicontinuousAt.{u1, u2, u3} ι X α _inst_1 _inst_4 F x₀) -> (forall (i : ι), ContinuousAt.{u2, u3} X α _inst_1 (UniformSpace.toTopologicalSpace.{u3} α _inst_4) (F i) x₀)
+but is expected to have type
+  forall {ι : Type.{u3}} {X : Type.{u2}} {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u2} X] [_inst_4 : UniformSpace.{u1} α] {F : ι -> X -> α} {x₀ : X}, (EquicontinuousAt.{u3, u2, u1} ι X α _inst_1 _inst_4 F x₀) -> (forall (i : ι), ContinuousAt.{u2, u1} X α _inst_1 (UniformSpace.toTopologicalSpace.{u1} α _inst_4) (F i) x₀)
+Case conversion may be inaccurate. Consider using '#align equicontinuous_at.continuous_at EquicontinuousAt.continuousAtₓ'. -/
 /-- Each function of a family equicontinuous at `x₀` is continuous at `x₀`. -/
 theorem EquicontinuousAt.continuousAt {F : ι → X → α} {x₀ : X} (h : EquicontinuousAt F x₀) (i : ι) :
     ContinuousAt (F i) x₀ := by
@@ -161,63 +191,135 @@ theorem EquicontinuousAt.continuousAt {F : ι → X → α} {x₀ : X} (h : Equi
   exact mem_map.mpr (mem_of_superset (h V hV₁) fun x hx => hV₂ (hx i))
 #align equicontinuous_at.continuous_at EquicontinuousAt.continuousAt
 
+/- warning: set.equicontinuous_at.continuous_at_of_mem -> Set.EquicontinuousAt.continuousAt_of_mem is a dubious translation:
+lean 3 declaration is
+  forall {X : Type.{u1}} {α : Type.{u2}} [_inst_1 : TopologicalSpace.{u1} X] [_inst_4 : UniformSpace.{u2} α] {H : Set.{max u1 u2} (X -> α)} {x₀ : X}, (Set.EquicontinuousAt.{u1, u2} X α _inst_1 _inst_4 H x₀) -> (forall {f : X -> α}, (Membership.Mem.{max u1 u2, max u1 u2} (X -> α) (Set.{max u1 u2} (X -> α)) (Set.hasMem.{max u1 u2} (X -> α)) f H) -> (ContinuousAt.{u1, u2} X α _inst_1 (UniformSpace.toTopologicalSpace.{u2} α _inst_4) f x₀))
+but is expected to have type
+  forall {X : Type.{u2}} {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u2} X] [_inst_4 : UniformSpace.{u1} α] {H : Set.{max u2 u1} (X -> α)} {x₀ : X}, (Set.EquicontinuousAt.{u2, u1} X α _inst_1 _inst_4 H x₀) -> (forall {f : X -> α}, (Membership.mem.{max u2 u1, max u2 u1} (X -> α) (Set.{max u2 u1} (X -> α)) (Set.instMembershipSet.{max u2 u1} (X -> α)) f H) -> (ContinuousAt.{u2, u1} X α _inst_1 (UniformSpace.toTopologicalSpace.{u1} α _inst_4) f x₀))
+Case conversion may be inaccurate. Consider using '#align set.equicontinuous_at.continuous_at_of_mem Set.EquicontinuousAt.continuousAt_of_memₓ'. -/
 protected theorem Set.EquicontinuousAt.continuousAt_of_mem {H : Set <| X → α} {x₀ : X}
     (h : H.EquicontinuousAt x₀) {f : X → α} (hf : f ∈ H) : ContinuousAt f x₀ :=
   h.ContinuousAt ⟨f, hf⟩
 #align set.equicontinuous_at.continuous_at_of_mem Set.EquicontinuousAt.continuousAt_of_mem
 
+/- warning: equicontinuous.continuous -> Equicontinuous.continuous is a dubious translation:
+lean 3 declaration is
+  forall {ι : Type.{u1}} {X : Type.{u2}} {α : Type.{u3}} [_inst_1 : TopologicalSpace.{u2} X] [_inst_4 : UniformSpace.{u3} α] {F : ι -> X -> α}, (Equicontinuous.{u1, u2, u3} ι X α _inst_1 _inst_4 F) -> (forall (i : ι), Continuous.{u2, u3} X α _inst_1 (UniformSpace.toTopologicalSpace.{u3} α _inst_4) (F i))
+but is expected to have type
+  forall {ι : Type.{u3}} {X : Type.{u2}} {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u2} X] [_inst_4 : UniformSpace.{u1} α] {F : ι -> X -> α}, (Equicontinuous.{u3, u2, u1} ι X α _inst_1 _inst_4 F) -> (forall (i : ι), Continuous.{u2, u1} X α _inst_1 (UniformSpace.toTopologicalSpace.{u1} α _inst_4) (F i))
+Case conversion may be inaccurate. Consider using '#align equicontinuous.continuous Equicontinuous.continuousₓ'. -/
 /-- Each function of an equicontinuous family is continuous. -/
 theorem Equicontinuous.continuous {F : ι → X → α} (h : Equicontinuous F) (i : ι) :
     Continuous (F i) :=
   continuous_iff_continuousAt.mpr fun x => (h x).ContinuousAt i
 #align equicontinuous.continuous Equicontinuous.continuous
 
+/- warning: set.equicontinuous.continuous_of_mem -> Set.Equicontinuous.continuous_of_mem is a dubious translation:
+lean 3 declaration is
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+Case conversion may be inaccurate. Consider using '#align set.equicontinuous.continuous_of_mem Set.Equicontinuous.continuous_of_memₓ'. -/
 protected theorem Set.Equicontinuous.continuous_of_mem {H : Set <| X → α} (h : H.Equicontinuous)
     {f : X → α} (hf : f ∈ H) : Continuous f :=
   h.Continuous ⟨f, hf⟩
 #align set.equicontinuous.continuous_of_mem Set.Equicontinuous.continuous_of_mem
 
+/- warning: uniform_equicontinuous.uniform_continuous -> UniformEquicontinuous.uniformContinuous is a dubious translation:
+lean 3 declaration is
+  forall {ι : Type.{u1}} {α : Type.{u2}} {β : Type.{u3}} [_inst_4 : UniformSpace.{u2} α] [_inst_5 : UniformSpace.{u3} β] {F : ι -> β -> α}, (UniformEquicontinuous.{u1, u2, u3} ι α β _inst_4 _inst_5 F) -> (forall (i : ι), UniformContinuous.{u3, u2} β α _inst_5 _inst_4 (F i))
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+Case conversion may be inaccurate. Consider using '#align uniform_equicontinuous.uniform_continuous UniformEquicontinuous.uniformContinuousₓ'. -/
 /-- Each function of a uniformly equicontinuous family is uniformly continuous. -/
 theorem UniformEquicontinuous.uniformContinuous {F : ι → β → α} (h : UniformEquicontinuous F)
     (i : ι) : UniformContinuous (F i) := fun U hU =>
   mem_map.mpr (mem_of_superset (h U hU) fun xy hxy => hxy i)
 #align uniform_equicontinuous.uniform_continuous UniformEquicontinuous.uniformContinuous
 
+/- warning: set.uniform_equicontinuous.uniform_continuous_of_mem -> Set.UniformEquicontinuous.uniformContinuous_of_mem is a dubious translation:
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+Case conversion may be inaccurate. Consider using '#align set.uniform_equicontinuous.uniform_continuous_of_mem Set.UniformEquicontinuous.uniformContinuous_of_memₓ'. -/
 protected theorem Set.UniformEquicontinuous.uniformContinuous_of_mem {H : Set <| β → α}
     (h : H.UniformEquicontinuous) {f : β → α} (hf : f ∈ H) : UniformContinuous f :=
   h.UniformContinuous ⟨f, hf⟩
 #align set.uniform_equicontinuous.uniform_continuous_of_mem Set.UniformEquicontinuous.uniformContinuous_of_mem
 
+/- warning: equicontinuous_at.comp -> EquicontinuousAt.comp is a dubious translation:
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+Case conversion may be inaccurate. Consider using '#align equicontinuous_at.comp EquicontinuousAt.compₓ'. -/
 /-- Taking sub-families preserves equicontinuity at a point. -/
 theorem EquicontinuousAt.comp {F : ι → X → α} {x₀ : X} (h : EquicontinuousAt F x₀) (u : κ → ι) :
     EquicontinuousAt (F ∘ u) x₀ := fun U hU => (h U hU).mono fun x H k => H (u k)
 #align equicontinuous_at.comp EquicontinuousAt.comp
 
+/- warning: set.equicontinuous_at.mono -> Set.EquicontinuousAt.mono is a dubious translation:
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+but is expected to have type
+  forall {X : Type.{u2}} {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u2} X] [_inst_4 : UniformSpace.{u1} α] {H : Set.{max u2 u1} (X -> α)} {H' : Set.{max u2 u1} (X -> α)} {x₀ : X}, (Set.EquicontinuousAt.{u2, u1} X α _inst_1 _inst_4 H x₀) -> (HasSubset.Subset.{max u2 u1} (Set.{max u2 u1} (X -> α)) (Set.instHasSubsetSet.{max u2 u1} (X -> α)) H' H) -> (Set.EquicontinuousAt.{u2, u1} X α _inst_1 _inst_4 H' x₀)
+Case conversion may be inaccurate. Consider using '#align set.equicontinuous_at.mono Set.EquicontinuousAt.monoₓ'. -/
 protected theorem Set.EquicontinuousAt.mono {H H' : Set <| X → α} {x₀ : X}
     (h : H.EquicontinuousAt x₀) (hH : H' ⊆ H) : H'.EquicontinuousAt x₀ :=
   h.comp (inclusion hH)
 #align set.equicontinuous_at.mono Set.EquicontinuousAt.mono
 
+/- warning: equicontinuous.comp -> Equicontinuous.comp is a dubious translation:
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+Case conversion may be inaccurate. Consider using '#align equicontinuous.comp Equicontinuous.compₓ'. -/
 /-- Taking sub-families preserves equicontinuity. -/
 theorem Equicontinuous.comp {F : ι → X → α} (h : Equicontinuous F) (u : κ → ι) :
     Equicontinuous (F ∘ u) := fun x => (h x).comp u
 #align equicontinuous.comp Equicontinuous.comp
 
+/- warning: set.equicontinuous.mono -> Set.Equicontinuous.mono is a dubious translation:
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+but is expected to have type
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 protected theorem Set.Equicontinuous.mono {H H' : Set <| X → α} (h : H.Equicontinuous)
     (hH : H' ⊆ H) : H'.Equicontinuous :=
   h.comp (inclusion hH)
 #align set.equicontinuous.mono Set.Equicontinuous.mono
 
+/- warning: uniform_equicontinuous.comp -> UniformEquicontinuous.comp is a dubious translation:
+lean 3 declaration is
+  forall {ι : Type.{u1}} {κ : Type.{u2}} {α : Type.{u3}} {β : Type.{u4}} [_inst_4 : UniformSpace.{u3} α] [_inst_5 : UniformSpace.{u4} β] {F : ι -> β -> α}, (UniformEquicontinuous.{u1, u3, u4} ι α β _inst_4 _inst_5 F) -> (forall (u : κ -> ι), UniformEquicontinuous.{u2, u3, u4} κ α β _inst_4 _inst_5 (Function.comp.{succ u2, succ u1, max (succ u4) (succ u3)} κ ι (β -> α) F u))
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+Case conversion may be inaccurate. Consider using '#align uniform_equicontinuous.comp UniformEquicontinuous.compₓ'. -/
 /-- Taking sub-families preserves uniform equicontinuity. -/
 theorem UniformEquicontinuous.comp {F : ι → β → α} (h : UniformEquicontinuous F) (u : κ → ι) :
     UniformEquicontinuous (F ∘ u) := fun U hU => (h U hU).mono fun x H k => H (u k)
 #align uniform_equicontinuous.comp UniformEquicontinuous.comp
 
+/- warning: set.uniform_equicontinuous.mono -> Set.UniformEquicontinuous.mono is a dubious translation:
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+  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_4 : UniformSpace.{u1} α] [_inst_5 : UniformSpace.{u2} β] {H : Set.{max u2 u1} (β -> α)} {H' : Set.{max u2 u1} (β -> α)}, (Set.UniformEquicontinuous.{u1, u2} α β _inst_4 _inst_5 H) -> (HasSubset.Subset.{max u2 u1} (Set.{max u2 u1} (β -> α)) (Set.hasSubset.{max u2 u1} (β -> α)) H' H) -> (Set.UniformEquicontinuous.{u1, u2} α β _inst_4 _inst_5 H')
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+Case conversion may be inaccurate. Consider using '#align set.uniform_equicontinuous.mono Set.UniformEquicontinuous.monoₓ'. -/
 protected theorem Set.UniformEquicontinuous.mono {H H' : Set <| β → α} (h : H.UniformEquicontinuous)
     (hH : H' ⊆ H) : H'.UniformEquicontinuous :=
   h.comp (inclusion hH)
 #align set.uniform_equicontinuous.mono Set.UniformEquicontinuous.mono
 
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 /-- A family `𝓕 : ι → X → α` is equicontinuous at `x₀` iff `range 𝓕` is equicontinuous at `x₀`,
 i.e the family `coe : range F → X → α` is equicontinuous at `x₀`. -/
 theorem equicontinuousAt_iff_range {F : ι → X → α} {x₀ : X} :
@@ -226,6 +328,12 @@ theorem equicontinuousAt_iff_range {F : ι → X → α} {x₀ : X} :
     h.comp (rangeFactorization F)⟩
 #align equicontinuous_at_iff_range equicontinuousAt_iff_range
 
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 /-- A family `𝓕 : ι → X → α` is equicontinuous iff `range 𝓕` is equicontinuous,
 i.e the family `coe : range F → X → α` is equicontinuous. -/
 theorem equicontinuous_iff_range {F : ι → X → α} :
@@ -233,6 +341,12 @@ theorem equicontinuous_iff_range {F : ι → X → α} :
   forall_congr' fun x₀ => equicontinuousAt_iff_range
 #align equicontinuous_iff_range equicontinuous_iff_range
 
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+Case conversion may be inaccurate. Consider using '#align uniform_equicontinuous_at_iff_range uniformEquicontinuous_at_iff_rangeₓ'. -/
 /-- A family `𝓕 : ι → β → α` is uniformly equicontinuous iff `range 𝓕` is uniformly equicontinuous,
 i.e the family `coe : range F → β → α` is uniformly equicontinuous. -/
 theorem uniformEquicontinuous_at_iff_range {F : ι → β → α} :
@@ -245,6 +359,12 @@ section
 
 open UniformFun
 
+/- warning: equicontinuous_at_iff_continuous_at -> equicontinuousAt_iff_continuousAt is a dubious translation:
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 /-- A family `𝓕 : ι → X → α` is equicontinuous at `x₀` iff the function `swap 𝓕 : X → ι → α` is
 continuous at `x₀` *when `ι → α` is equipped with the topology of uniform convergence*. This is
 very useful for developping the equicontinuity API, but it should not be used directly for other
@@ -254,6 +374,12 @@ theorem equicontinuousAt_iff_continuousAt {F : ι → X → α} {x₀ : X} :
   rw [ContinuousAt, (UniformFun.hasBasis_nhds ι α _).tendsto_right_iff] <;> rfl
 #align equicontinuous_at_iff_continuous_at equicontinuousAt_iff_continuousAt
 
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 /-- A family `𝓕 : ι → X → α` is equicontinuous iff the function `swap 𝓕 : X → ι → α` is
 continuous *when `ι → α` is equipped with the topology of uniform convergence*. This is
 very useful for developping the equicontinuity API, but it should not be used directly for other
@@ -263,6 +389,12 @@ theorem equicontinuous_iff_continuous {F : ι → X → α} :
   simp_rw [Equicontinuous, continuous_iff_continuousAt, equicontinuousAt_iff_continuousAt]
 #align equicontinuous_iff_continuous equicontinuous_iff_continuous
 
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+Case conversion may be inaccurate. Consider using '#align uniform_equicontinuous_iff_uniform_continuous uniformEquicontinuous_iff_uniformContinuousₓ'. -/
 /-- A family `𝓕 : ι → β → α` is uniformly equicontinuous iff the function `swap 𝓕 : β → ι → α` is
 uniformly continuous *when `ι → α` is equipped with the uniform structure of uniform convergence*.
 This is very useful for developping the equicontinuity API, but it should not be used directly
@@ -272,6 +404,12 @@ theorem uniformEquicontinuous_iff_uniformContinuous {F : ι → β → α} :
   rw [UniformContinuous, (UniformFun.hasBasis_uniformity ι α).tendsto_right_iff] <;> rfl
 #align uniform_equicontinuous_iff_uniform_continuous uniformEquicontinuous_iff_uniformContinuous
 
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+Case conversion may be inaccurate. Consider using '#align filter.has_basis.equicontinuous_at_iff_left Filter.HasBasis.equicontinuousAt_iff_leftₓ'. -/
 theorem Filter.HasBasis.equicontinuousAt_iff_left {κ : Type _} {p : κ → Prop} {s : κ → Set X}
     {F : ι → X → α} {x₀ : X} (hX : (𝓝 x₀).HasBasis p s) :
     EquicontinuousAt F x₀ ↔ ∀ U ∈ 𝓤 α, ∃ (k : _)(_ : p k), ∀ x ∈ s k, ∀ i, (F i x₀, F i x) ∈ U :=
@@ -281,6 +419,12 @@ theorem Filter.HasBasis.equicontinuousAt_iff_left {κ : Type _} {p : κ → Prop
   rfl
 #align filter.has_basis.equicontinuous_at_iff_left Filter.HasBasis.equicontinuousAt_iff_left
 
+/- warning: filter.has_basis.equicontinuous_at_iff_right -> Filter.HasBasis.equicontinuousAt_iff_right is a dubious translation:
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+Case conversion may be inaccurate. Consider using '#align filter.has_basis.equicontinuous_at_iff_right Filter.HasBasis.equicontinuousAt_iff_rightₓ'. -/
 theorem Filter.HasBasis.equicontinuousAt_iff_right {κ : Type _} {p : κ → Prop} {s : κ → Set (α × α)}
     {F : ι → X → α} {x₀ : X} (hα : (𝓤 α).HasBasis p s) :
     EquicontinuousAt F x₀ ↔ ∀ k, p k → ∀ᶠ x in 𝓝 x₀, ∀ i, (F i x₀, F i x) ∈ s k :=
@@ -290,6 +434,12 @@ theorem Filter.HasBasis.equicontinuousAt_iff_right {κ : Type _} {p : κ → Pro
   rfl
 #align filter.has_basis.equicontinuous_at_iff_right Filter.HasBasis.equicontinuousAt_iff_right
 
+/- warning: filter.has_basis.equicontinuous_at_iff -> Filter.HasBasis.equicontinuousAt_iff is a dubious translation:
+lean 3 declaration is
+  forall {ι : Type.{u1}} {X : Type.{u2}} {α : Type.{u3}} [_inst_1 : TopologicalSpace.{u2} X] [_inst_4 : UniformSpace.{u3} α] {κ₁ : Type.{u4}} {κ₂ : Type.{u5}} {p₁ : κ₁ -> Prop} {s₁ : κ₁ -> (Set.{u2} X)} {p₂ : κ₂ -> Prop} {s₂ : κ₂ -> (Set.{u3} (Prod.{u3, u3} α α))} {F : ι -> X -> α} {x₀ : X}, (Filter.HasBasis.{u2, succ u4} X κ₁ (nhds.{u2} X _inst_1 x₀) p₁ s₁) -> (Filter.HasBasis.{u3, succ u5} (Prod.{u3, u3} α α) κ₂ (uniformity.{u3} α _inst_4) p₂ s₂) -> (Iff (EquicontinuousAt.{u1, u2, u3} ι X α _inst_1 _inst_4 F x₀) (forall (k₂ : κ₂), (p₂ k₂) -> (Exists.{succ u4} κ₁ (fun (k₁ : κ₁) => Exists.{0} (p₁ k₁) (fun (_x : p₁ k₁) => forall (x : X), (Membership.Mem.{u2, u2} X (Set.{u2} X) (Set.hasMem.{u2} X) x (s₁ k₁)) -> (forall (i : ι), Membership.Mem.{u3, u3} (Prod.{u3, u3} α α) (Set.{u3} (Prod.{u3, u3} α α)) (Set.hasMem.{u3} (Prod.{u3, u3} α α)) (Prod.mk.{u3, u3} α α (F i x₀) (F i x)) (s₂ k₂)))))))
+but is expected to have type
+  forall {ι : Type.{u1}} {X : Type.{u3}} {α : Type.{u2}} [_inst_1 : TopologicalSpace.{u3} X] [_inst_4 : UniformSpace.{u2} α] {κ₁ : Type.{u5}} {κ₂ : Type.{u4}} {p₁ : κ₁ -> Prop} {s₁ : κ₁ -> (Set.{u3} X)} {p₂ : κ₂ -> Prop} {s₂ : κ₂ -> (Set.{u2} (Prod.{u2, u2} α α))} {F : ι -> X -> α} {x₀ : X}, (Filter.HasBasis.{u3, succ u5} X κ₁ (nhds.{u3} X _inst_1 x₀) p₁ s₁) -> (Filter.HasBasis.{u2, succ u4} (Prod.{u2, u2} α α) κ₂ (uniformity.{u2} α _inst_4) p₂ s₂) -> (Iff (EquicontinuousAt.{u1, u3, u2} ι X α _inst_1 _inst_4 F x₀) (forall (k₂ : κ₂), (p₂ k₂) -> (Exists.{succ u5} κ₁ (fun (k₁ : κ₁) => Exists.{0} (p₁ k₁) (fun (_x : p₁ k₁) => forall (x : X), (Membership.mem.{u3, u3} X (Set.{u3} X) (Set.instMembershipSet.{u3} X) x (s₁ k₁)) -> (forall (i : ι), Membership.mem.{u2, u2} (Prod.{u2, u2} α α) (Set.{u2} (Prod.{u2, u2} α α)) (Set.instMembershipSet.{u2} (Prod.{u2, u2} α α)) (Prod.mk.{u2, u2} α α (F i x₀) (F i x)) (s₂ k₂)))))))
+Case conversion may be inaccurate. Consider using '#align filter.has_basis.equicontinuous_at_iff Filter.HasBasis.equicontinuousAt_iffₓ'. -/
 theorem Filter.HasBasis.equicontinuousAt_iff {κ₁ κ₂ : Type _} {p₁ : κ₁ → Prop} {s₁ : κ₁ → Set X}
     {p₂ : κ₂ → Prop} {s₂ : κ₂ → Set (α × α)} {F : ι → X → α} {x₀ : X} (hX : (𝓝 x₀).HasBasis p₁ s₁)
     (hα : (𝓤 α).HasBasis p₂ s₂) :
@@ -301,6 +451,12 @@ theorem Filter.HasBasis.equicontinuousAt_iff {κ₁ κ₂ : Type _} {p₁ : κ
   rfl
 #align filter.has_basis.equicontinuous_at_iff Filter.HasBasis.equicontinuousAt_iff
 
+/- warning: filter.has_basis.uniform_equicontinuous_iff_left -> Filter.HasBasis.uniformEquicontinuous_iff_left is a dubious translation:
+lean 3 declaration is
+  forall {ι : Type.{u1}} {α : Type.{u2}} {β : Type.{u3}} [_inst_4 : UniformSpace.{u2} α] [_inst_5 : UniformSpace.{u3} β] {κ : Type.{u4}} {p : κ -> Prop} {s : κ -> (Set.{u3} (Prod.{u3, u3} β β))} {F : ι -> β -> α}, (Filter.HasBasis.{u3, succ u4} (Prod.{u3, u3} β β) κ (uniformity.{u3} β _inst_5) p s) -> (Iff (UniformEquicontinuous.{u1, u2, u3} ι α β _inst_4 _inst_5 F) (forall (U : Set.{u2} (Prod.{u2, u2} α α)), (Membership.Mem.{u2, u2} (Set.{u2} (Prod.{u2, u2} α α)) (Filter.{u2} (Prod.{u2, u2} α α)) (Filter.hasMem.{u2} (Prod.{u2, u2} α α)) U (uniformity.{u2} α _inst_4)) -> (Exists.{succ u4} κ (fun (k : κ) => Exists.{0} (p k) (fun (_x : p k) => forall (x : β) (y : β), (Membership.Mem.{u3, u3} (Prod.{u3, u3} β β) (Set.{u3} (Prod.{u3, u3} β β)) (Set.hasMem.{u3} (Prod.{u3, u3} β β)) (Prod.mk.{u3, u3} β β x y) (s k)) -> (forall (i : ι), Membership.Mem.{u2, u2} (Prod.{u2, u2} α α) (Set.{u2} (Prod.{u2, u2} α α)) (Set.hasMem.{u2} (Prod.{u2, u2} α α)) (Prod.mk.{u2, u2} α α (F i x) (F i y)) U))))))
+but is expected to have type
+  forall {ι : Type.{u2}} {α : Type.{u1}} {β : Type.{u3}} [_inst_4 : UniformSpace.{u1} α] [_inst_5 : UniformSpace.{u3} β] {κ : Type.{u4}} {p : κ -> Prop} {s : κ -> (Set.{u3} (Prod.{u3, u3} β β))} {F : ι -> β -> α}, (Filter.HasBasis.{u3, succ u4} (Prod.{u3, u3} β β) κ (uniformity.{u3} β _inst_5) p s) -> (Iff (UniformEquicontinuous.{u2, u1, u3} ι α β _inst_4 _inst_5 F) (forall (U : Set.{u1} (Prod.{u1, u1} α α)), (Membership.mem.{u1, u1} (Set.{u1} (Prod.{u1, u1} α α)) (Filter.{u1} (Prod.{u1, u1} α α)) (instMembershipSetFilter.{u1} (Prod.{u1, u1} α α)) U (uniformity.{u1} α _inst_4)) -> (Exists.{succ u4} κ (fun (k : κ) => Exists.{0} (p k) (fun (_x : p k) => forall (x : β) (y : β), (Membership.mem.{u3, u3} (Prod.{u3, u3} β β) (Set.{u3} (Prod.{u3, u3} β β)) (Set.instMembershipSet.{u3} (Prod.{u3, u3} β β)) (Prod.mk.{u3, u3} β β x y) (s k)) -> (forall (i : ι), Membership.mem.{u1, u1} (Prod.{u1, u1} α α) (Set.{u1} (Prod.{u1, u1} α α)) (Set.instMembershipSet.{u1} (Prod.{u1, u1} α α)) (Prod.mk.{u1, u1} α α (F i x) (F i y)) U))))))
+Case conversion may be inaccurate. Consider using '#align filter.has_basis.uniform_equicontinuous_iff_left Filter.HasBasis.uniformEquicontinuous_iff_leftₓ'. -/
 theorem Filter.HasBasis.uniformEquicontinuous_iff_left {κ : Type _} {p : κ → Prop}
     {s : κ → Set (β × β)} {F : ι → β → α} (hβ : (𝓤 β).HasBasis p s) :
     UniformEquicontinuous F ↔
@@ -312,6 +468,12 @@ theorem Filter.HasBasis.uniformEquicontinuous_iff_left {κ : Type _} {p : κ →
   rfl
 #align filter.has_basis.uniform_equicontinuous_iff_left Filter.HasBasis.uniformEquicontinuous_iff_left
 
+/- warning: filter.has_basis.uniform_equicontinuous_iff_right -> Filter.HasBasis.uniformEquicontinuous_iff_right is a dubious translation:
+lean 3 declaration is
+  forall {ι : Type.{u1}} {α : Type.{u2}} {β : Type.{u3}} [_inst_4 : UniformSpace.{u2} α] [_inst_5 : UniformSpace.{u3} β] {κ : Type.{u4}} {p : κ -> Prop} {s : κ -> (Set.{u2} (Prod.{u2, u2} α α))} {F : ι -> β -> α}, (Filter.HasBasis.{u2, succ u4} (Prod.{u2, u2} α α) κ (uniformity.{u2} α _inst_4) p s) -> (Iff (UniformEquicontinuous.{u1, u2, u3} ι α β _inst_4 _inst_5 F) (forall (k : κ), (p k) -> (Filter.Eventually.{u3} (Prod.{u3, u3} β β) (fun (xy : Prod.{u3, u3} β β) => forall (i : ι), Membership.Mem.{u2, u2} (Prod.{u2, u2} α α) (Set.{u2} (Prod.{u2, u2} α α)) (Set.hasMem.{u2} (Prod.{u2, u2} α α)) (Prod.mk.{u2, u2} α α (F i (Prod.fst.{u3, u3} β β xy)) (F i (Prod.snd.{u3, u3} β β xy))) (s k)) (uniformity.{u3} β _inst_5))))
+but is expected to have type
+  forall {ι : Type.{u2}} {α : Type.{u3}} {β : Type.{u1}} [_inst_4 : UniformSpace.{u3} α] [_inst_5 : UniformSpace.{u1} β] {κ : Type.{u4}} {p : κ -> Prop} {s : κ -> (Set.{u3} (Prod.{u3, u3} α α))} {F : ι -> β -> α}, (Filter.HasBasis.{u3, succ u4} (Prod.{u3, u3} α α) κ (uniformity.{u3} α _inst_4) p s) -> (Iff (UniformEquicontinuous.{u2, u3, u1} ι α β _inst_4 _inst_5 F) (forall (k : κ), (p k) -> (Filter.Eventually.{u1} (Prod.{u1, u1} β β) (fun (xy : Prod.{u1, u1} β β) => forall (i : ι), Membership.mem.{u3, u3} (Prod.{u3, u3} α α) (Set.{u3} (Prod.{u3, u3} α α)) (Set.instMembershipSet.{u3} (Prod.{u3, u3} α α)) (Prod.mk.{u3, u3} α α (F i (Prod.fst.{u1, u1} β β xy)) (F i (Prod.snd.{u1, u1} β β xy))) (s k)) (uniformity.{u1} β _inst_5))))
+Case conversion may be inaccurate. Consider using '#align filter.has_basis.uniform_equicontinuous_iff_right Filter.HasBasis.uniformEquicontinuous_iff_rightₓ'. -/
 theorem Filter.HasBasis.uniformEquicontinuous_iff_right {κ : Type _} {p : κ → Prop}
     {s : κ → Set (α × α)} {F : ι → β → α} (hα : (𝓤 α).HasBasis p s) :
     UniformEquicontinuous F ↔ ∀ k, p k → ∀ᶠ xy : β × β in 𝓤 β, ∀ i, (F i xy.1, F i xy.2) ∈ s k :=
@@ -321,6 +483,12 @@ theorem Filter.HasBasis.uniformEquicontinuous_iff_right {κ : Type _} {p : κ 
   rfl
 #align filter.has_basis.uniform_equicontinuous_iff_right Filter.HasBasis.uniformEquicontinuous_iff_right
 
+/- warning: filter.has_basis.uniform_equicontinuous_iff -> Filter.HasBasis.uniformEquicontinuous_iff is a dubious translation:
+lean 3 declaration is
+  forall {ι : Type.{u1}} {α : Type.{u2}} {β : Type.{u3}} [_inst_4 : UniformSpace.{u2} α] [_inst_5 : UniformSpace.{u3} β] {κ₁ : Type.{u4}} {κ₂ : Type.{u5}} {p₁ : κ₁ -> Prop} {s₁ : κ₁ -> (Set.{u3} (Prod.{u3, u3} β β))} {p₂ : κ₂ -> Prop} {s₂ : κ₂ -> (Set.{u2} (Prod.{u2, u2} α α))} {F : ι -> β -> α}, (Filter.HasBasis.{u3, succ u4} (Prod.{u3, u3} β β) κ₁ (uniformity.{u3} β _inst_5) p₁ s₁) -> (Filter.HasBasis.{u2, succ u5} (Prod.{u2, u2} α α) κ₂ (uniformity.{u2} α _inst_4) p₂ s₂) -> (Iff (UniformEquicontinuous.{u1, u2, u3} ι α β _inst_4 _inst_5 F) (forall (k₂ : κ₂), (p₂ k₂) -> (Exists.{succ u4} κ₁ (fun (k₁ : κ₁) => Exists.{0} (p₁ k₁) (fun (_x : p₁ k₁) => forall (x : β) (y : β), (Membership.Mem.{u3, u3} (Prod.{u3, u3} β β) (Set.{u3} (Prod.{u3, u3} β β)) (Set.hasMem.{u3} (Prod.{u3, u3} β β)) (Prod.mk.{u3, u3} β β x y) (s₁ k₁)) -> (forall (i : ι), Membership.Mem.{u2, u2} (Prod.{u2, u2} α α) (Set.{u2} (Prod.{u2, u2} α α)) (Set.hasMem.{u2} (Prod.{u2, u2} α α)) (Prod.mk.{u2, u2} α α (F i x) (F i y)) (s₂ k₂)))))))
+but is expected to have type
+  forall {ι : Type.{u1}} {α : Type.{u2}} {β : Type.{u3}} [_inst_4 : UniformSpace.{u2} α] [_inst_5 : UniformSpace.{u3} β] {κ₁ : Type.{u5}} {κ₂ : Type.{u4}} {p₁ : κ₁ -> Prop} {s₁ : κ₁ -> (Set.{u3} (Prod.{u3, u3} β β))} {p₂ : κ₂ -> Prop} {s₂ : κ₂ -> (Set.{u2} (Prod.{u2, u2} α α))} {F : ι -> β -> α}, (Filter.HasBasis.{u3, succ u5} (Prod.{u3, u3} β β) κ₁ (uniformity.{u3} β _inst_5) p₁ s₁) -> (Filter.HasBasis.{u2, succ u4} (Prod.{u2, u2} α α) κ₂ (uniformity.{u2} α _inst_4) p₂ s₂) -> (Iff (UniformEquicontinuous.{u1, u2, u3} ι α β _inst_4 _inst_5 F) (forall (k₂ : κ₂), (p₂ k₂) -> (Exists.{succ u5} κ₁ (fun (k₁ : κ₁) => Exists.{0} (p₁ k₁) (fun (_x : p₁ k₁) => forall (x : β) (y : β), (Membership.mem.{u3, u3} (Prod.{u3, u3} β β) (Set.{u3} (Prod.{u3, u3} β β)) (Set.instMembershipSet.{u3} (Prod.{u3, u3} β β)) (Prod.mk.{u3, u3} β β x y) (s₁ k₁)) -> (forall (i : ι), Membership.mem.{u2, u2} (Prod.{u2, u2} α α) (Set.{u2} (Prod.{u2, u2} α α)) (Set.instMembershipSet.{u2} (Prod.{u2, u2} α α)) (Prod.mk.{u2, u2} α α (F i x) (F i y)) (s₂ k₂)))))))
+Case conversion may be inaccurate. Consider using '#align filter.has_basis.uniform_equicontinuous_iff Filter.HasBasis.uniformEquicontinuous_iffₓ'. -/
 theorem Filter.HasBasis.uniformEquicontinuous_iff {κ₁ κ₂ : Type _} {p₁ : κ₁ → Prop}
     {s₁ : κ₁ → Set (β × β)} {p₂ : κ₂ → Prop} {s₂ : κ₂ → Set (α × α)} {F : ι → β → α}
     (hβ : (𝓤 β).HasBasis p₁ s₁) (hα : (𝓤 α).HasBasis p₂ s₂) :
@@ -333,6 +501,12 @@ theorem Filter.HasBasis.uniformEquicontinuous_iff {κ₁ κ₂ : Type _} {p₁ :
   rfl
 #align filter.has_basis.uniform_equicontinuous_iff Filter.HasBasis.uniformEquicontinuous_iff
 
+/- warning: uniform_inducing.equicontinuous_at_iff -> UniformInducing.equicontinuousAt_iff is a dubious translation:
+lean 3 declaration is
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+but is expected to have type
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+Case conversion may be inaccurate. Consider using '#align uniform_inducing.equicontinuous_at_iff UniformInducing.equicontinuousAt_iffₓ'. -/
 /-- Given `u : α → β` a uniform inducing map, a family `𝓕 : ι → X → α` is equicontinuous at a point
 `x₀ : X` iff the family `𝓕'`, obtained by precomposing each function of `𝓕` by `u`, is
 equicontinuous at `x₀`. -/
@@ -344,6 +518,12 @@ theorem UniformInducing.equicontinuousAt_iff {F : ι → X → α} {x₀ : X} {u
   rfl
 #align uniform_inducing.equicontinuous_at_iff UniformInducing.equicontinuousAt_iff
 
+/- warning: uniform_inducing.equicontinuous_iff -> UniformInducing.equicontinuous_iff is a dubious translation:
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+Case conversion may be inaccurate. Consider using '#align uniform_inducing.equicontinuous_iff UniformInducing.equicontinuous_iffₓ'. -/
 /- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:76:14: unsupported tactic `congrm #[[expr ∀ x, (_ : exprProp())]] -/
 /-- Given `u : α → β` a uniform inducing map, a family `𝓕 : ι → X → α` is equicontinuous iff the
 family `𝓕'`, obtained by precomposing each function of `𝓕` by `u`, is equicontinuous. -/
@@ -355,6 +535,12 @@ theorem UniformInducing.equicontinuous_iff {F : ι → X → α} {u : α → β}
   rw [hu.equicontinuous_at_iff]
 #align uniform_inducing.equicontinuous_iff UniformInducing.equicontinuous_iff
 
+/- warning: uniform_inducing.uniform_equicontinuous_iff -> UniformInducing.uniformEquicontinuous_iff is a dubious translation:
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+Case conversion may be inaccurate. Consider using '#align uniform_inducing.uniform_equicontinuous_iff UniformInducing.uniformEquicontinuous_iffₓ'. -/
 /-- Given `u : α → γ` a uniform inducing map, a family `𝓕 : ι → β → α` is uniformly equicontinuous
 iff the family `𝓕'`, obtained by precomposing each function of `𝓕` by `u`, is uniformly
 equicontinuous. -/
@@ -367,6 +553,12 @@ theorem UniformInducing.uniformEquicontinuous_iff {F : ι → β → α} {u : α
   rfl
 #align uniform_inducing.uniform_equicontinuous_iff UniformInducing.uniformEquicontinuous_iff
 
+/- warning: equicontinuous_at.closure' -> EquicontinuousAt.closure' is a dubious translation:
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+but is expected to have type
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+Case conversion may be inaccurate. Consider using '#align equicontinuous_at.closure' EquicontinuousAt.closure'ₓ'. -/
 /-- A version of `equicontinuous_at.closure` applicable to subsets of types which embed continuously
 into `X → α` with the product topology. It turns out we don't need any other condition on the
 embedding than continuity, but in practice this will mostly be applied to `fun_like` types where
@@ -384,6 +576,12 @@ theorem EquicontinuousAt.closure' {A : Set Y} {u : Y → X → α} {x₀ : X}
   exact Continuous.prod_mk ((continuous_apply x₀).comp hu) ((continuous_apply x).comp hu)
 #align equicontinuous_at.closure' EquicontinuousAt.closure'
 
+/- warning: equicontinuous_at.closure -> EquicontinuousAt.closure is a dubious translation:
+lean 3 declaration is
+  forall {X : Type.{u1}} {α : Type.{u2}} [_inst_1 : TopologicalSpace.{u1} X] [_inst_4 : UniformSpace.{u2} α] {A : Set.{max u1 u2} (X -> α)} {x₀ : X}, (Set.EquicontinuousAt.{u1, u2} X α _inst_1 _inst_4 A x₀) -> (Set.EquicontinuousAt.{u1, u2} X α _inst_1 _inst_4 (closure.{max u1 u2} (X -> α) (Pi.topologicalSpace.{u1, u2} X (fun (ᾰ : X) => α) (fun (a : X) => UniformSpace.toTopologicalSpace.{u2} α _inst_4)) A) x₀)
+but is expected to have type
+  forall {X : Type.{u2}} {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u2} X] [_inst_4 : UniformSpace.{u1} α] {A : Set.{max u2 u1} (X -> α)} {x₀ : X}, (Set.EquicontinuousAt.{u2, u1} X α _inst_1 _inst_4 A x₀) -> (Set.EquicontinuousAt.{u2, u1} X α _inst_1 _inst_4 (closure.{max u2 u1} (X -> α) (Pi.topologicalSpace.{u2, u1} X (fun (ᾰ : X) => α) (fun (a : X) => UniformSpace.toTopologicalSpace.{u1} α _inst_4)) A) x₀)
+Case conversion may be inaccurate. Consider using '#align equicontinuous_at.closure EquicontinuousAt.closureₓ'. -/
 /-- If a set of functions is equicontinuous at some `x₀`, its closure for the product topology is
 also equicontinuous at `x₀`. -/
 theorem EquicontinuousAt.closure {A : Set <| X → α} {x₀ : X} (hA : A.EquicontinuousAt x₀) :
@@ -391,6 +589,12 @@ theorem EquicontinuousAt.closure {A : Set <| X → α} {x₀ : X} (hA : A.Equico
   @EquicontinuousAt.closure' _ _ _ _ _ _ _ id _ hA continuous_id
 #align equicontinuous_at.closure EquicontinuousAt.closure
 
+/- warning: filter.tendsto.continuous_at_of_equicontinuous_at -> Filter.Tendsto.continuousAt_of_equicontinuousAt is a dubious translation:
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+  forall {ι : Type.{u1}} {X : Type.{u2}} {α : Type.{u3}} [_inst_1 : TopologicalSpace.{u2} X] [_inst_4 : UniformSpace.{u3} α] {l : Filter.{u1} ι} [_inst_7 : Filter.NeBot.{u1} ι l] {F : ι -> X -> α} {f : X -> α} {x₀ : X}, (Filter.Tendsto.{u1, max u2 u3} ι (X -> α) F l (nhds.{max u2 u3} (X -> α) (Pi.topologicalSpace.{u2, u3} X (fun (ᾰ : X) => α) (fun (a : X) => UniformSpace.toTopologicalSpace.{u3} α _inst_4)) f)) -> (EquicontinuousAt.{u1, u2, u3} ι X α _inst_1 _inst_4 F x₀) -> (ContinuousAt.{u2, u3} X α _inst_1 (UniformSpace.toTopologicalSpace.{u3} α _inst_4) f x₀)
+but is expected to have type
+  forall {ι : Type.{u3}} {X : Type.{u2}} {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u2} X] [_inst_4 : UniformSpace.{u1} α] {l : Filter.{u3} ι} [_inst_7 : Filter.NeBot.{u3} ι l] {F : ι -> X -> α} {f : X -> α} {x₀ : X}, (Filter.Tendsto.{u3, max u2 u1} ι (X -> α) F l (nhds.{max u2 u1} (X -> α) (Pi.topologicalSpace.{u2, u1} X (fun (ᾰ : X) => α) (fun (a : X) => UniformSpace.toTopologicalSpace.{u1} α _inst_4)) f)) -> (EquicontinuousAt.{u3, u2, u1} ι X α _inst_1 _inst_4 F x₀) -> (ContinuousAt.{u2, u1} X α _inst_1 (UniformSpace.toTopologicalSpace.{u1} α _inst_4) f x₀)
+Case conversion may be inaccurate. Consider using '#align filter.tendsto.continuous_at_of_equicontinuous_at Filter.Tendsto.continuousAt_of_equicontinuousAtₓ'. -/
 /-- If `𝓕 : ι → X → α` tends to `f : X → α` *pointwise* along some nontrivial filter, and if the
 family `𝓕` is equicontinuous at some `x₀ : X`, then the limit is continuous at `x₀`. -/
 theorem Filter.Tendsto.continuousAt_of_equicontinuousAt {l : Filter ι} [l.ne_bot] {F : ι → X → α}
@@ -400,6 +604,12 @@ theorem Filter.Tendsto.continuousAt_of_equicontinuousAt {l : Filter ι} [l.ne_bo
     ⟨f, mem_closure_of_tendsto h₁ <| eventually_of_forall mem_range_self⟩
 #align filter.tendsto.continuous_at_of_equicontinuous_at Filter.Tendsto.continuousAt_of_equicontinuousAt
 
+/- warning: equicontinuous.closure' -> Equicontinuous.closure' is a dubious translation:
+lean 3 declaration is
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+Case conversion may be inaccurate. Consider using '#align equicontinuous.closure' Equicontinuous.closure'ₓ'. -/
 /-- A version of `equicontinuous.closure` applicable to subsets of types which embed continuously
 into `X → α` with the product topology. It turns out we don't need any other condition on the
 embedding than continuity, but in practice this will mostly be applied to `fun_like` types where
@@ -409,12 +619,24 @@ theorem Equicontinuous.closure' {A : Set Y} {u : Y → X → α}
     Equicontinuous (u ∘ coe : closure A → X → α) := fun x => (hA x).closure' hu
 #align equicontinuous.closure' Equicontinuous.closure'
 
+/- warning: equicontinuous.closure -> Equicontinuous.closure is a dubious translation:
+lean 3 declaration is
+  forall {X : Type.{u1}} {α : Type.{u2}} [_inst_1 : TopologicalSpace.{u1} X] [_inst_4 : UniformSpace.{u2} α] {A : Set.{max u1 u2} (X -> α)}, (Set.Equicontinuous.{u1, u2} X α _inst_1 _inst_4 A) -> (Set.Equicontinuous.{u1, u2} X α _inst_1 _inst_4 (closure.{max u1 u2} (X -> α) (Pi.topologicalSpace.{u1, u2} X (fun (ᾰ : X) => α) (fun (a : X) => UniformSpace.toTopologicalSpace.{u2} α _inst_4)) A))
+but is expected to have type
+  forall {X : Type.{u2}} {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u2} X] [_inst_4 : UniformSpace.{u1} α] {A : Set.{max u2 u1} (X -> α)}, (Set.Equicontinuous.{u2, u1} X α _inst_1 _inst_4 A) -> (Set.Equicontinuous.{u2, u1} X α _inst_1 _inst_4 (closure.{max u2 u1} (X -> α) (Pi.topologicalSpace.{u2, u1} X (fun (ᾰ : X) => α) (fun (a : X) => UniformSpace.toTopologicalSpace.{u1} α _inst_4)) A))
+Case conversion may be inaccurate. Consider using '#align equicontinuous.closure Equicontinuous.closureₓ'. -/
 /-- If a set of functions is equicontinuous, its closure for the product topology is also
 equicontinuous. -/
 theorem Equicontinuous.closure {A : Set <| X → α} (hA : A.Equicontinuous) :
     (closure A).Equicontinuous := fun x => (hA x).closure
 #align equicontinuous.closure Equicontinuous.closure
 
+/- warning: filter.tendsto.continuous_of_equicontinuous_at -> Filter.Tendsto.continuous_of_equicontinuous_at is a dubious translation:
+lean 3 declaration is
+  forall {ι : Type.{u1}} {X : Type.{u2}} {α : Type.{u3}} [_inst_1 : TopologicalSpace.{u2} X] [_inst_4 : UniformSpace.{u3} α] {l : Filter.{u1} ι} [_inst_7 : Filter.NeBot.{u1} ι l] {F : ι -> X -> α} {f : X -> α}, (Filter.Tendsto.{u1, max u2 u3} ι (X -> α) F l (nhds.{max u2 u3} (X -> α) (Pi.topologicalSpace.{u2, u3} X (fun (ᾰ : X) => α) (fun (a : X) => UniformSpace.toTopologicalSpace.{u3} α _inst_4)) f)) -> (Equicontinuous.{u1, u2, u3} ι X α _inst_1 _inst_4 F) -> (Continuous.{u2, u3} X α _inst_1 (UniformSpace.toTopologicalSpace.{u3} α _inst_4) f)
+but is expected to have type
+  forall {ι : Type.{u3}} {X : Type.{u2}} {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u2} X] [_inst_4 : UniformSpace.{u1} α] {l : Filter.{u3} ι} [_inst_7 : Filter.NeBot.{u3} ι l] {F : ι -> X -> α} {f : X -> α}, (Filter.Tendsto.{u3, max u2 u1} ι (X -> α) F l (nhds.{max u2 u1} (X -> α) (Pi.topologicalSpace.{u2, u1} X (fun (ᾰ : X) => α) (fun (a : X) => UniformSpace.toTopologicalSpace.{u1} α _inst_4)) f)) -> (Equicontinuous.{u3, u2, u1} ι X α _inst_1 _inst_4 F) -> (Continuous.{u2, u1} X α _inst_1 (UniformSpace.toTopologicalSpace.{u1} α _inst_4) f)
+Case conversion may be inaccurate. Consider using '#align filter.tendsto.continuous_of_equicontinuous_at Filter.Tendsto.continuous_of_equicontinuous_atₓ'. -/
 /-- If `𝓕 : ι → X → α` tends to `f : X → α` *pointwise* along some nontrivial filter, and if the
 family `𝓕` is equicontinuous, then the limit is continuous. -/
 theorem Filter.Tendsto.continuous_of_equicontinuous_at {l : Filter ι} [l.ne_bot] {F : ι → X → α}
@@ -422,6 +644,12 @@ theorem Filter.Tendsto.continuous_of_equicontinuous_at {l : Filter ι} [l.ne_bot
   continuous_iff_continuousAt.mpr fun x => h₁.continuousAt_of_equicontinuousAt (h₂ x)
 #align filter.tendsto.continuous_of_equicontinuous_at Filter.Tendsto.continuous_of_equicontinuous_at
 
+/- warning: uniform_equicontinuous.closure' -> UniformEquicontinuous.closure' is a dubious translation:
+lean 3 declaration is
+  forall {Y : Type.{u1}} {α : Type.{u2}} {β : Type.{u3}} [_inst_2 : TopologicalSpace.{u1} Y] [_inst_4 : UniformSpace.{u2} α] [_inst_5 : UniformSpace.{u3} β] {A : Set.{u1} Y} {u : Y -> β -> α}, (UniformEquicontinuous.{u1, u2, u3} (coeSort.{succ u1, succ (succ u1)} (Set.{u1} Y) Type.{u1} (Set.hasCoeToSort.{u1} Y) A) α β _inst_4 _inst_5 (Function.comp.{succ u1, succ u1, max (succ u3) (succ u2)} (coeSort.{succ u1, succ (succ u1)} (Set.{u1} Y) Type.{u1} (Set.hasCoeToSort.{u1} Y) A) Y (β -> α) u ((fun (a : Type.{u1}) (b : Type.{u1}) [self : HasLiftT.{succ u1, succ u1} a b] => self.0) (coeSort.{succ u1, succ (succ u1)} (Set.{u1} Y) Type.{u1} (Set.hasCoeToSort.{u1} Y) A) Y (HasLiftT.mk.{succ u1, succ u1} (coeSort.{succ u1, succ (succ u1)} (Set.{u1} Y) Type.{u1} (Set.hasCoeToSort.{u1} Y) A) Y (CoeTCₓ.coe.{succ u1, succ u1} (coeSort.{succ u1, succ (succ u1)} (Set.{u1} Y) Type.{u1} (Set.hasCoeToSort.{u1} Y) A) Y (coeBase.{succ u1, succ u1} (coeSort.{succ u1, succ (succ u1)} (Set.{u1} Y) Type.{u1} (Set.hasCoeToSort.{u1} Y) A) Y (coeSubtype.{succ u1} Y (fun (x : Y) => Membership.Mem.{u1, u1} Y (Set.{u1} Y) (Set.hasMem.{u1} Y) x A)))))))) -> (Continuous.{u1, max u3 u2} Y (β -> α) _inst_2 (Pi.topologicalSpace.{u3, u2} β (fun (ᾰ : β) => α) (fun (a : β) => UniformSpace.toTopologicalSpace.{u2} α _inst_4)) u) -> (UniformEquicontinuous.{u1, u2, u3} (coeSort.{succ u1, succ (succ u1)} (Set.{u1} Y) Type.{u1} (Set.hasCoeToSort.{u1} Y) (closure.{u1} Y _inst_2 A)) α β _inst_4 _inst_5 (Function.comp.{succ u1, succ u1, max (succ u3) (succ u2)} (coeSort.{succ u1, succ (succ u1)} (Set.{u1} Y) Type.{u1} (Set.hasCoeToSort.{u1} Y) (closure.{u1} Y _inst_2 A)) Y (β -> α) u ((fun (a : Type.{u1}) (b : Type.{u1}) [self : HasLiftT.{succ u1, succ u1} a b] => self.0) (coeSort.{succ u1, succ (succ u1)} (Set.{u1} Y) Type.{u1} (Set.hasCoeToSort.{u1} Y) (closure.{u1} Y _inst_2 A)) Y (HasLiftT.mk.{succ u1, succ u1} (coeSort.{succ u1, succ (succ u1)} (Set.{u1} Y) Type.{u1} (Set.hasCoeToSort.{u1} Y) (closure.{u1} Y _inst_2 A)) Y (CoeTCₓ.coe.{succ u1, succ u1} (coeSort.{succ u1, succ (succ u1)} (Set.{u1} Y) Type.{u1} (Set.hasCoeToSort.{u1} Y) (closure.{u1} Y _inst_2 A)) Y (coeBase.{succ u1, succ u1} (coeSort.{succ u1, succ (succ u1)} (Set.{u1} Y) Type.{u1} (Set.hasCoeToSort.{u1} Y) (closure.{u1} Y _inst_2 A)) Y (coeSubtype.{succ u1} Y (fun (x : Y) => Membership.Mem.{u1, u1} Y (Set.{u1} Y) (Set.hasMem.{u1} Y) x (closure.{u1} Y _inst_2 A)))))))))
+but is expected to have type
+  forall {Y : Type.{u3}} {α : Type.{u2}} {β : Type.{u1}} [_inst_2 : TopologicalSpace.{u3} Y] [_inst_4 : UniformSpace.{u2} α] [_inst_5 : UniformSpace.{u1} β] {A : Set.{u3} Y} {u : Y -> β -> α}, (UniformEquicontinuous.{u3, u2, u1} (Set.Elem.{u3} Y A) α β _inst_4 _inst_5 (Function.comp.{succ u3, succ u3, max (succ u2) (succ u1)} (Set.Elem.{u3} Y A) Y (β -> α) u (Subtype.val.{succ u3} Y (fun (x : Y) => Membership.mem.{u3, u3} Y (Set.{u3} Y) (Set.instMembershipSet.{u3} Y) x A)))) -> (Continuous.{u3, max u2 u1} Y (β -> α) _inst_2 (Pi.topologicalSpace.{u1, u2} β (fun (ᾰ : β) => α) (fun (a : β) => UniformSpace.toTopologicalSpace.{u2} α _inst_4)) u) -> (UniformEquicontinuous.{u3, u2, u1} (Set.Elem.{u3} Y (closure.{u3} Y _inst_2 A)) α β _inst_4 _inst_5 (Function.comp.{succ u3, succ u3, max (succ u2) (succ u1)} (Set.Elem.{u3} Y (closure.{u3} Y _inst_2 A)) Y (β -> α) u (Subtype.val.{succ u3} Y (fun (x : Y) => Membership.mem.{u3, u3} Y (Set.{u3} Y) (Set.instMembershipSet.{u3} Y) x (closure.{u3} Y _inst_2 A)))))
+Case conversion may be inaccurate. Consider using '#align uniform_equicontinuous.closure' UniformEquicontinuous.closure'ₓ'. -/
 /-- A version of `uniform_equicontinuous.closure` applicable to subsets of types which embed
 continuously into `β → α` with the product topology. It turns out we don't need any other condition
 on the embedding than continuity, but in practice this will mostly be applied to `fun_like` types
@@ -440,6 +668,12 @@ theorem UniformEquicontinuous.closure' {A : Set Y} {u : Y → β → α}
   exact Continuous.prod_mk ((continuous_apply x).comp hu) ((continuous_apply y).comp hu)
 #align uniform_equicontinuous.closure' UniformEquicontinuous.closure'
 
+/- warning: uniform_equicontinuous.closure -> UniformEquicontinuous.closure is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_4 : UniformSpace.{u1} α] [_inst_5 : UniformSpace.{u2} β] {A : Set.{max u2 u1} (β -> α)}, (Set.UniformEquicontinuous.{u1, u2} α β _inst_4 _inst_5 A) -> (Set.UniformEquicontinuous.{u1, u2} α β _inst_4 _inst_5 (closure.{max u2 u1} (β -> α) (Pi.topologicalSpace.{u2, u1} β (fun (ᾰ : β) => α) (fun (a : β) => UniformSpace.toTopologicalSpace.{u1} α _inst_4)) A))
+but is expected to have type
+  forall {α : Type.{u2}} {β : Type.{u1}} [_inst_4 : UniformSpace.{u2} α] [_inst_5 : UniformSpace.{u1} β] {A : Set.{max u2 u1} (β -> α)}, (Set.UniformEquicontinuous.{u2, u1} α β _inst_4 _inst_5 A) -> (Set.UniformEquicontinuous.{u2, u1} α β _inst_4 _inst_5 (closure.{max u2 u1} (β -> α) (Pi.topologicalSpace.{u1, u2} β (fun (ᾰ : β) => α) (fun (a : β) => UniformSpace.toTopologicalSpace.{u2} α _inst_4)) A))
+Case conversion may be inaccurate. Consider using '#align uniform_equicontinuous.closure UniformEquicontinuous.closureₓ'. -/
 /-- If a set of functions is uniformly equicontinuous, its closure for the product topology is also
 uniformly equicontinuous. -/
 theorem UniformEquicontinuous.closure {A : Set <| β → α} (hA : A.UniformEquicontinuous) :
@@ -447,6 +681,12 @@ theorem UniformEquicontinuous.closure {A : Set <| β → α} (hA : A.UniformEqui
   @UniformEquicontinuous.closure' _ _ _ _ _ _ _ id hA continuous_id
 #align uniform_equicontinuous.closure UniformEquicontinuous.closure
 
+/- warning: filter.tendsto.uniform_continuous_of_uniform_equicontinuous -> Filter.Tendsto.uniformContinuous_of_uniformEquicontinuous is a dubious translation:
+lean 3 declaration is
+  forall {ι : Type.{u1}} {α : Type.{u2}} {β : Type.{u3}} [_inst_4 : UniformSpace.{u2} α] [_inst_5 : UniformSpace.{u3} β] {l : Filter.{u1} ι} [_inst_7 : Filter.NeBot.{u1} ι l] {F : ι -> β -> α} {f : β -> α}, (Filter.Tendsto.{u1, max u3 u2} ι (β -> α) F l (nhds.{max u3 u2} (β -> α) (Pi.topologicalSpace.{u3, u2} β (fun (ᾰ : β) => α) (fun (a : β) => UniformSpace.toTopologicalSpace.{u2} α _inst_4)) f)) -> (UniformEquicontinuous.{u1, u2, u3} ι α β _inst_4 _inst_5 F) -> (UniformContinuous.{u3, u2} β α _inst_5 _inst_4 f)
+but is expected to have type
+  forall {ι : Type.{u3}} {α : Type.{u2}} {β : Type.{u1}} [_inst_4 : UniformSpace.{u2} α] [_inst_5 : UniformSpace.{u1} β] {l : Filter.{u3} ι} [_inst_7 : Filter.NeBot.{u3} ι l] {F : ι -> β -> α} {f : β -> α}, (Filter.Tendsto.{u3, max u2 u1} ι (β -> α) F l (nhds.{max u2 u1} (β -> α) (Pi.topologicalSpace.{u1, u2} β (fun (ᾰ : β) => α) (fun (a : β) => UniformSpace.toTopologicalSpace.{u2} α _inst_4)) f)) -> (UniformEquicontinuous.{u3, u2, u1} ι α β _inst_4 _inst_5 F) -> (UniformContinuous.{u1, u2} β α _inst_5 _inst_4 f)
+Case conversion may be inaccurate. Consider using '#align filter.tendsto.uniform_continuous_of_uniform_equicontinuous Filter.Tendsto.uniformContinuous_of_uniformEquicontinuousₓ'. -/
 /-- If `𝓕 : ι → β → α` tends to `f : β → α` *pointwise* along some nontrivial filter, and if the
 family `𝓕` is uniformly equicontinuous, then the limit is uniformly continuous. -/
 theorem Filter.Tendsto.uniformContinuous_of_uniformEquicontinuous {l : Filter ι} [l.ne_bot]

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

Purely automatic replacement. If this is in any way controversial; I'm happy to just close this PR.

@@ -909,7 +909,7 @@ protected theorem Set.UniformEquicontinuous.closure {A : Set <| β → α}
 
 /-- If a set of functions is uniformly equicontinuous on a set `S`, its closure for the product
 topology is also uniformly equicontinuous. This would also be true for the coarser topology of
-pointwise convergence on `S`, see `UniformEquicontinuousOn.closure'`.-/
+pointwise convergence on `S`, see `UniformEquicontinuousOn.closure'`. -/
 protected theorem Set.UniformEquicontinuousOn.closure {A : Set <| β → α} {S : Set β}
     (hA : A.UniformEquicontinuousOn S) : (closure A).UniformEquicontinuousOn S :=
   UniformEquicontinuousOn.closure' (u := id) hA (Pi.continuous_restrict _)

@@ -88,7 +88,7 @@ variable {ι κ X X' Y Z α α' β β' γ 𝓕 : Type*} [tX : TopologicalSpace X
   [tZ : TopologicalSpace Z] [uα : UniformSpace α] [uβ : UniformSpace β] [uγ : UniformSpace γ]
 
 /-- A family `F : ι → X → α` of functions from a topological space to a uniform space is
-*equicontinuous at `x₀ : X`* if, for all entourage `U ∈ 𝓤 α`, there is a neighborhood `V` of `x₀`
+*equicontinuous at `x₀ : X`* if, for all entourages `U ∈ 𝓤 α`, there is a neighborhood `V` of `x₀`
 such that, for all `x ∈ V` and for all `i : ι`, `F i x` is `U`-close to `F i x₀`. -/
 def EquicontinuousAt (F : ι → X → α) (x₀ : X) : Prop :=
   ∀ U ∈ 𝓤 α, ∀ᶠ x in 𝓝 x₀, ∀ i, (F i x₀, F i x) ∈ U
@@ -101,7 +101,7 @@ protected abbrev Set.EquicontinuousAt (H : Set <| X → α) (x₀ : X) : Prop :=
 #align set.equicontinuous_at Set.EquicontinuousAt
 
 /-- A family `F : ι → X → α` of functions from a topological space to a uniform space is
-*equicontinuous at `x₀ : X` within `S : Set X`* if, for all entourage `U ∈ 𝓤 α`, there is a
+*equicontinuous at `x₀ : X` within `S : Set X`* if, for all entourages `U ∈ 𝓤 α`, there is a
 neighborhood `V` of `x₀` within `S` such that, for all `x ∈ V` and for all `i : ι`, `F i x` is
 `U`-close to `F i x₀`. -/
 def EquicontinuousWithinAt (F : ι → X → α) (S : Set X) (x₀ : X) : Prop :=
@@ -135,7 +135,7 @@ protected abbrev Set.EquicontinuousOn (H : Set <| X → α) (S : Set X) : Prop :
   EquicontinuousOn ((↑) : H → X → α) S
 
 /-- A family `F : ι → β → α` of functions between uniform spaces is *uniformly equicontinuous* if,
-for all entourage `U ∈ 𝓤 α`, there is an entourage `V ∈ 𝓤 β` such that, whenever `x` and `y` are
+for all entourages `U ∈ 𝓤 α`, there is an entourage `V ∈ 𝓤 β` such that, whenever `x` and `y` are
 `V`-close, we have that, *for all `i : ι`*, `F i x` is `U`-close to `F i y`. -/
 def UniformEquicontinuous (F : ι → β → α) : Prop :=
   ∀ U ∈ 𝓤 α, ∀ᶠ xy : β × β in 𝓤 β, ∀ i, (F i xy.1, F i xy.2) ∈ U
@@ -148,7 +148,7 @@ protected abbrev Set.UniformEquicontinuous (H : Set <| β → α) : Prop :=
 #align set.uniform_equicontinuous Set.UniformEquicontinuous
 
 /-- A family `F : ι → β → α` of functions between uniform spaces is
-*uniformly equicontinuous on `S : Set β`* if, for all entourage `U ∈ 𝓤 α`, there is a relative
+*uniformly equicontinuous on `S : Set β`* if, for all entourages `U ∈ 𝓤 α`, there is a relative
 entourage `V ∈ 𝓤 β ⊓ 𝓟 (S ×ˢ S)` such that, whenever `x` and `y` are `V`-close, we have that,
 *for all `i : ι`*, `F i x` is `U`-close to `F i y`. -/
 def UniformEquicontinuousOn (F : ι → β → α) (S : Set β) : Prop :=

• depends on: #7568

Co-authored-by: Heather Macbeth <25316162+hrmacbeth@users.noreply.github.com>

@@ -33,12 +33,16 @@ For maps between metric spaces, this corresponds to
 * `Equicontinuous`: equicontinuity of a family of functions on the whole domain
 * `UniformEquicontinuous`: uniform equicontinuity of a family of functions on the whole domain
 
+We also introduce relative versions, namely `EquicontinuousWithinAt`, `EquicontinuousOn` and
+`UniformEquicontinuousOn`, akin to `ContinuousWithinAt`, `ContinuousOn` and `UniformContinuousOn`
+respectively.
+
 ## Main statements
 
 * `equicontinuous_iff_continuous`: equicontinuity can be expressed as a simple continuity
   condition between well-chosen function spaces. This is really useful for building up the theory.
 * `Equicontinuous.closure`: if a set of functions is equicontinuous, its closure
-  *for the topology of uniform convergence* is also equicontinuous.
+  *for the topology of pointwise convergence* is also equicontinuous.
 
 ## Notations
 
@@ -66,11 +70,6 @@ and `Set.UniformEquicontinuous` asserting the corresponding fact about the famil
 `(↑) : ↥H → (X → α)` where `H : Set (X → α)`. Note however that these won't work for sets of hom
 types, and in that case one should go back to the family definition rather than using `Set.image`.
 
-Since we have no use case for it yet, we don't introduce any relative version
-(i.e no `EquicontinuousWithinAt` or `EquicontinuousOn`), but this is more of a conservative
-position than a design decision, so anyone needing relative versions should feel free to add them,
-and that should hopefully be a straightforward task.
-
 ## References
 
 * [N. Bourbaki, *General Topology, Chapter X*][bourbaki1966]
@@ -83,10 +82,10 @@ equicontinuity, uniform convergence, ascoli
 
 section
 
-open UniformSpace Filter Set Uniformity Topology UniformConvergence
+open UniformSpace Filter Set Uniformity Topology UniformConvergence Function
 
-variable {ι κ X Y Z α β γ 𝓕 : Type*} [TopologicalSpace X] [TopologicalSpace Y] [TopologicalSpace Z]
-  [UniformSpace α] [UniformSpace β] [UniformSpace γ]
+variable {ι κ X X' Y Z α α' β β' γ 𝓕 : Type*} [tX : TopologicalSpace X] [tY : TopologicalSpace Y]
+  [tZ : TopologicalSpace Z] [uα : UniformSpace α] [uβ : UniformSpace β] [uγ : UniformSpace γ]
 
 /-- A family `F : ι → X → α` of functions from a topological space to a uniform space is
 *equicontinuous at `x₀ : X`* if, for all entourage `U ∈ 𝓤 α`, there is a neighborhood `V` of `x₀`
@@ -101,6 +100,18 @@ protected abbrev Set.EquicontinuousAt (H : Set <| X → α) (x₀ : X) : Prop :=
   EquicontinuousAt ((↑) : H → X → α) x₀
 #align set.equicontinuous_at Set.EquicontinuousAt
 
+/-- A family `F : ι → X → α` of functions from a topological space to a uniform space is
+*equicontinuous at `x₀ : X` within `S : Set X`* if, for all entourage `U ∈ 𝓤 α`, there is a
+neighborhood `V` of `x₀` within `S` such that, for all `x ∈ V` and for all `i : ι`, `F i x` is
+`U`-close to `F i x₀`. -/
+def EquicontinuousWithinAt (F : ι → X → α) (S : Set X) (x₀ : X) : Prop :=
+  ∀ U ∈ 𝓤 α, ∀ᶠ x in 𝓝[S] x₀, ∀ i, (F i x₀, F i x) ∈ U
+
+/-- We say that a set `H : Set (X → α)` of functions is equicontinuous at a point within a subset
+if the family `(↑) : ↥H → (X → α)` is equicontinuous at that point within that same subset. -/
+protected abbrev Set.EquicontinuousWithinAt (H : Set <| X → α) (S : Set X) (x₀ : X) : Prop :=
+  EquicontinuousWithinAt ((↑) : H → X → α) S x₀
+
 /-- A family `F : ι → X → α` of functions from a topological space to a uniform space is
 *equicontinuous* on all of `X` if it is equicontinuous at each point of `X`. -/
 def Equicontinuous (F : ι → X → α) : Prop :=
@@ -113,9 +124,19 @@ protected abbrev Set.Equicontinuous (H : Set <| X → α) : Prop :=
   Equicontinuous ((↑) : H → X → α)
 #align set.equicontinuous Set.Equicontinuous
 
+/-- A family `F : ι → X → α` of functions from a topological space to a uniform space is
+*equicontinuous on `S : Set X`* if it is equicontinuous *within `S`* at each point of `S`. -/
+def EquicontinuousOn (F : ι → X → α) (S : Set X) : Prop :=
+  ∀ x₀ ∈ S, EquicontinuousWithinAt F S x₀
+
+/-- We say that a set `H : Set (X → α)` of functions is equicontinuous on a subset if the family
+`(↑) : ↥H → (X → α)` is equicontinuous on that subset. -/
+protected abbrev Set.EquicontinuousOn (H : Set <| X → α) (S : Set X) : Prop :=
+  EquicontinuousOn ((↑) : H → X → α) S
+
 /-- A family `F : ι → β → α` of functions between uniform spaces is *uniformly equicontinuous* if,
 for all entourage `U ∈ 𝓤 α`, there is an entourage `V ∈ 𝓤 β` such that, whenever `x` and `y` are
-`V`-close, we have that, *for all `i : ι`*, `F i x` is `U`-close to `F i x₀`. -/
+`V`-close, we have that, *for all `i : ι`*, `F i x` is `U`-close to `F i y`. -/
 def UniformEquicontinuous (F : ι → β → α) : Prop :=
   ∀ U ∈ 𝓤 α, ∀ᶠ xy : β × β in 𝓤 β, ∀ i, (F i xy.1, F i xy.2) ∈ U
 #align uniform_equicontinuous UniformEquicontinuous
@@ -126,6 +147,68 @@ protected abbrev Set.UniformEquicontinuous (H : Set <| β → α) : Prop :=
   UniformEquicontinuous ((↑) : H → β → α)
 #align set.uniform_equicontinuous Set.UniformEquicontinuous
 
+/-- A family `F : ι → β → α` of functions between uniform spaces is
+*uniformly equicontinuous on `S : Set β`* if, for all entourage `U ∈ 𝓤 α`, there is a relative
+entourage `V ∈ 𝓤 β ⊓ 𝓟 (S ×ˢ S)` such that, whenever `x` and `y` are `V`-close, we have that,
+*for all `i : ι`*, `F i x` is `U`-close to `F i y`. -/
+def UniformEquicontinuousOn (F : ι → β → α) (S : Set β) : Prop :=
+  ∀ U ∈ 𝓤 α, ∀ᶠ xy : β × β in 𝓤 β ⊓ 𝓟 (S ×ˢ S), ∀ i, (F i xy.1, F i xy.2) ∈ U
+
+/-- We say that a set `H : Set (X → α)` of functions is uniformly equicontinuous on a subset if the
+family `(↑) : ↥H → (X → α)` is uniformly equicontinuous on that subset. -/
+protected abbrev Set.UniformEquicontinuousOn (H : Set <| β → α) (S : Set β) : Prop :=
+  UniformEquicontinuousOn ((↑) : H → β → α) S
+
+lemma EquicontinuousAt.equicontinuousWithinAt {F : ι → X → α} {x₀ : X} (H : EquicontinuousAt F x₀)
+    (S : Set X) : EquicontinuousWithinAt F S x₀ :=
+  fun U hU ↦ (H U hU).filter_mono inf_le_left
+
+lemma EquicontinuousWithinAt.mono {F : ι → X → α} {x₀ : X} {S T : Set X}
+    (H : EquicontinuousWithinAt F T x₀) (hST : S ⊆ T) : EquicontinuousWithinAt F S x₀ :=
+  fun U hU ↦ (H U hU).filter_mono <| nhdsWithin_mono x₀ hST
+
+@[simp] lemma equicontinuousWithinAt_univ (F : ι → X → α) (x₀ : X) :
+    EquicontinuousWithinAt F univ x₀ ↔ EquicontinuousAt F x₀ := by
+  rw [EquicontinuousWithinAt, EquicontinuousAt, nhdsWithin_univ]
+
+lemma equicontinuousAt_restrict_iff (F : ι → X → α) {S : Set X} (x₀ : S) :
+    EquicontinuousAt (S.restrict ∘ F) x₀ ↔ EquicontinuousWithinAt F S x₀ := by
+  simp [EquicontinuousWithinAt, EquicontinuousAt,
+    ← eventually_nhds_subtype_iff]
+
+lemma Equicontinuous.equicontinuousOn {F : ι → X → α} (H : Equicontinuous F)
+    (S : Set X) : EquicontinuousOn F S :=
+  fun x _ ↦ (H x).equicontinuousWithinAt S
+
+lemma EquicontinuousOn.mono {F : ι → X → α} {S T : Set X}
+    (H : EquicontinuousOn F T) (hST : S ⊆ T) : EquicontinuousOn F S :=
+  fun x hx ↦ (H x (hST hx)).mono hST
+
+lemma equicontinuousOn_univ (F : ι → X → α) :
+    EquicontinuousOn F univ ↔ Equicontinuous F := by
+  simp [EquicontinuousOn, Equicontinuous]
+
+lemma equicontinuous_restrict_iff (F : ι → X → α) {S : Set X} :
+    Equicontinuous (S.restrict ∘ F) ↔ EquicontinuousOn F S := by
+  simp [Equicontinuous, EquicontinuousOn, equicontinuousAt_restrict_iff]
+
+lemma UniformEquicontinuous.uniformEquicontinuousOn {F : ι → β → α} (H : UniformEquicontinuous F)
+    (S : Set β) : UniformEquicontinuousOn F S :=
+  fun U hU ↦ (H U hU).filter_mono inf_le_left
+
+lemma UniformEquicontinuousOn.mono {F : ι → β → α} {S T : Set β}
+    (H : UniformEquicontinuousOn F T) (hST : S ⊆ T) : UniformEquicontinuousOn F S :=
+  fun U hU ↦ (H U hU).filter_mono <| inf_le_inf_left _ <| principal_mono.mpr <| prod_mono hST hST
+
+lemma uniformEquicontinuousOn_univ (F : ι → β → α) :
+    UniformEquicontinuousOn F univ ↔ UniformEquicontinuous F := by
+  simp [UniformEquicontinuousOn, UniformEquicontinuous]
+
+lemma uniformEquicontinuous_restrict_iff (F : ι → β → α) {S : Set β} :
+    UniformEquicontinuous (S.restrict ∘ F) ↔ UniformEquicontinuousOn F S := by
+  rw [UniformEquicontinuous, UniformEquicontinuousOn]
+  conv in _ ⊓ _ => rw [← Subtype.range_val (s := S), ← range_prod_map, ← map_comap]
+
 /-!
 ### Empty index type
 -/
@@ -135,16 +218,31 @@ lemma equicontinuousAt_empty [h : IsEmpty ι] (F : ι → X → α) (x₀ : X) :
     EquicontinuousAt F x₀ :=
   fun _ _ ↦ eventually_of_forall (fun _ ↦ h.elim)
 
+@[simp]
+lemma equicontinuousWithinAt_empty [h : IsEmpty ι] (F : ι → X → α) (S : Set X) (x₀ : X) :
+    EquicontinuousWithinAt F S x₀ :=
+  fun _ _ ↦ eventually_of_forall (fun _ ↦ h.elim)
+
 @[simp]
 lemma equicontinuous_empty [IsEmpty ι] (F : ι → X → α) :
     Equicontinuous F :=
   equicontinuousAt_empty F
 
+@[simp]
+lemma equicontinuousOn_empty [IsEmpty ι] (F : ι → X → α) (S : Set X) :
+    EquicontinuousOn F S :=
+  fun x₀ _ ↦ equicontinuousWithinAt_empty F S x₀
+
 @[simp]
 lemma uniformEquicontinuous_empty [h : IsEmpty ι] (F : ι → β → α) :
     UniformEquicontinuous F :=
   fun _ _ ↦ eventually_of_forall (fun _ ↦ h.elim)
 
+@[simp]
+lemma uniformEquicontinuousOn_empty [h : IsEmpty ι] (F : ι → β → α) (S : Set β) :
+    UniformEquicontinuousOn F S :=
+  fun _ _ ↦ eventually_of_forall (fun _ ↦ h.elim)
+
 /-!
 ### Finite index type
 -/
@@ -154,14 +252,28 @@ theorem equicontinuousAt_finite [Finite ι] {F : ι → X → α} {x₀ : X} :
   simp [EquicontinuousAt, ContinuousAt, (nhds_basis_uniformity' (𝓤 α).basis_sets).tendsto_right_iff,
     UniformSpace.ball, @forall_swap _ ι]
 
+theorem equicontinuousWithinAt_finite [Finite ι] {F : ι → X → α} {S : Set X} {x₀ : X} :
+    EquicontinuousWithinAt F S x₀ ↔ ∀ i, ContinuousWithinAt (F i) S x₀ := by
+  simp [EquicontinuousWithinAt, ContinuousWithinAt,
+    (nhds_basis_uniformity' (𝓤 α).basis_sets).tendsto_right_iff, UniformSpace.ball,
+    @forall_swap _ ι]
+
 theorem equicontinuous_finite [Finite ι] {F : ι → X → α} :
     Equicontinuous F ↔ ∀ i, Continuous (F i) := by
   simp only [Equicontinuous, equicontinuousAt_finite, continuous_iff_continuousAt, @forall_swap ι]
 
+theorem equicontinuousOn_finite [Finite ι] {F : ι → X → α} {S : Set X} :
+    EquicontinuousOn F S ↔ ∀ i, ContinuousOn (F i) S := by
+  simp only [EquicontinuousOn, equicontinuousWithinAt_finite, ContinuousOn, @forall_swap ι]
+
 theorem uniformEquicontinuous_finite [Finite ι] {F : ι → β → α} :
     UniformEquicontinuous F ↔ ∀ i, UniformContinuous (F i) := by
   simp only [UniformEquicontinuous, eventually_all, @forall_swap _ ι]; rfl
 
+theorem uniformEquicontinuousOn_finite [Finite ι] {F : ι → β → α} {S : Set β} :
+    UniformEquicontinuousOn F S ↔ ∀ i, UniformContinuousOn (F i) S := by
+  simp only [UniformEquicontinuousOn, eventually_all, @forall_swap _ ι]; rfl
+
 /-!
 ### Index type with a unique element
 -/
@@ -170,96 +282,180 @@ theorem equicontinuousAt_unique [Unique ι] {F : ι → X → α} {x : X} :
     EquicontinuousAt F x ↔ ContinuousAt (F default) x :=
   equicontinuousAt_finite.trans Unique.forall_iff
 
+theorem equicontinuousWithinAt_unique [Unique ι] {F : ι → X → α} {S : Set X} {x : X} :
+    EquicontinuousWithinAt F S x ↔ ContinuousWithinAt (F default) S x :=
+  equicontinuousWithinAt_finite.trans Unique.forall_iff
+
 theorem equicontinuous_unique [Unique ι] {F : ι → X → α} :
     Equicontinuous F ↔ Continuous (F default) :=
   equicontinuous_finite.trans Unique.forall_iff
 
+theorem equicontinuousOn_unique [Unique ι] {F : ι → X → α} {S : Set X} :
+    EquicontinuousOn F S ↔ ContinuousOn (F default) S :=
+  equicontinuousOn_finite.trans Unique.forall_iff
+
 theorem uniformEquicontinuous_unique [Unique ι] {F : ι → β → α} :
     UniformEquicontinuous F ↔ UniformContinuous (F default) :=
   uniformEquicontinuous_finite.trans Unique.forall_iff
 
-/-- Reformulation of equicontinuity at `x₀` comparing two variables near `x₀` instead of comparing
-only one with `x₀`. -/
-theorem equicontinuousAt_iff_pair {F : ι → X → α} {x₀ : X} :
-    EquicontinuousAt F x₀ ↔
-      ∀ U ∈ 𝓤 α, ∃ V ∈ 𝓝 x₀, ∀ x ∈ V, ∀ y ∈ V, ∀ i, (F i x, F i y) ∈ U := by
+theorem uniformEquicontinuousOn_unique [Unique ι] {F : ι → β → α} {S : Set β} :
+    UniformEquicontinuousOn F S ↔ UniformContinuousOn (F default) S :=
+  uniformEquicontinuousOn_finite.trans Unique.forall_iff
+
+/-- Reformulation of equicontinuity at `x₀` within a set `S`, comparing two variables near `x₀`
+instead of comparing only one with `x₀`. -/
+theorem equicontinuousWithinAt_iff_pair {F : ι → X → α} {S : Set X} {x₀ : X} (hx₀ : x₀ ∈ S) :
+    EquicontinuousWithinAt F S x₀ ↔
+      ∀ U ∈ 𝓤 α, ∃ V ∈ 𝓝[S] x₀, ∀ x ∈ V, ∀ y ∈ V, ∀ i, (F i x, F i y) ∈ U := by
   constructor <;> intro H U hU
   · rcases comp_symm_mem_uniformity_sets hU with ⟨V, hV, hVsymm, hVU⟩
     refine' ⟨_, H V hV, fun x hx y hy i => hVU (prod_mk_mem_compRel _ (hy i))⟩
     exact hVsymm.mk_mem_comm.mp (hx i)
   · rcases H U hU with ⟨V, hV, hVU⟩
-    filter_upwards [hV] using fun x hx i => hVU x₀ (mem_of_mem_nhds hV) x hx i
+    filter_upwards [hV] using fun x hx i => hVU x₀ (mem_of_mem_nhdsWithin hx₀ hV) x hx i
+
+/-- Reformulation of equicontinuity at `x₀` comparing two variables near `x₀` instead of comparing
+only one with `x₀`. -/
+theorem equicontinuousAt_iff_pair {F : ι → X → α} {x₀ : X} :
+    EquicontinuousAt F x₀ ↔
+      ∀ U ∈ 𝓤 α, ∃ V ∈ 𝓝 x₀, ∀ x ∈ V, ∀ y ∈ V, ∀ i, (F i x, F i y) ∈ U := by
+  simp_rw [← equicontinuousWithinAt_univ, equicontinuousWithinAt_iff_pair (mem_univ x₀),
+    nhdsWithin_univ]
 #align equicontinuous_at_iff_pair equicontinuousAt_iff_pair
 
 /-- Uniform equicontinuity implies equicontinuity. -/
 theorem UniformEquicontinuous.equicontinuous {F : ι → β → α} (h : UniformEquicontinuous F) :
-    Equicontinuous F := fun x₀ U hU =>
-  mem_of_superset (ball_mem_nhds x₀ (h U hU)) fun _ hx i => hx i
+    Equicontinuous F := fun x₀ U hU ↦
+  mem_of_superset (ball_mem_nhds x₀ (h U hU)) fun _ hx i ↦ hx i
 #align uniform_equicontinuous.equicontinuous UniformEquicontinuous.equicontinuous
 
+/-- Uniform equicontinuity on a subset implies equicontinuity on that subset. -/
+theorem UniformEquicontinuousOn.equicontinuousOn {F : ι → β → α} {S : Set β}
+    (h : UniformEquicontinuousOn F S) :
+    EquicontinuousOn F S := fun _ hx₀ U hU ↦
+  mem_of_superset (ball_mem_nhdsWithin hx₀ (h U hU)) fun _ hx i ↦ hx i
+
 /-- Each function of a family equicontinuous at `x₀` is continuous at `x₀`. -/
 theorem EquicontinuousAt.continuousAt {F : ι → X → α} {x₀ : X} (h : EquicontinuousAt F x₀) (i : ι) :
     ContinuousAt (F i) x₀ :=
   (UniformSpace.hasBasis_nhds _).tendsto_right_iff.2 fun U ⟨hU, _⟩ ↦ (h U hU).mono fun _x hx ↦ hx i
 #align equicontinuous_at.continuous_at EquicontinuousAt.continuousAt
 
+/-- Each function of a family equicontinuous at `x₀` within `S` is continuous at `x₀` within `S`. -/
+theorem EquicontinuousWithinAt.continuousWithinAt {F : ι → X → α} {S : Set X} {x₀ : X}
+    (h : EquicontinuousWithinAt F S x₀) (i : ι) :
+    ContinuousWithinAt (F i) S x₀ :=
+  (UniformSpace.hasBasis_nhds _).tendsto_right_iff.2 fun U ⟨hU, _⟩ ↦ (h U hU).mono fun _x hx ↦ hx i
+
 protected theorem Set.EquicontinuousAt.continuousAt_of_mem {H : Set <| X → α} {x₀ : X}
     (h : H.EquicontinuousAt x₀) {f : X → α} (hf : f ∈ H) : ContinuousAt f x₀ :=
   h.continuousAt ⟨f, hf⟩
 #align set.equicontinuous_at.continuous_at_of_mem Set.EquicontinuousAt.continuousAt_of_mem
 
+protected theorem Set.EquicontinuousWithinAt.continuousWithinAt_of_mem {H : Set <| X → α}
+    {S : Set X} {x₀ : X} (h : H.EquicontinuousWithinAt S x₀) {f : X → α} (hf : f ∈ H) :
+    ContinuousWithinAt f S x₀ :=
+  h.continuousWithinAt ⟨f, hf⟩
+
 /-- Each function of an equicontinuous family is continuous. -/
 theorem Equicontinuous.continuous {F : ι → X → α} (h : Equicontinuous F) (i : ι) :
     Continuous (F i) :=
   continuous_iff_continuousAt.mpr fun x => (h x).continuousAt i
 #align equicontinuous.continuous Equicontinuous.continuous
 
+/-- Each function of a family equicontinuous on `S` is continuous on `S`. -/
+theorem EquicontinuousOn.continuousOn {F : ι → X → α} {S : Set X} (h : EquicontinuousOn F S)
+    (i : ι) : ContinuousOn (F i) S :=
+  fun x hx ↦ (h x hx).continuousWithinAt i
+
 protected theorem Set.Equicontinuous.continuous_of_mem {H : Set <| X → α} (h : H.Equicontinuous)
     {f : X → α} (hf : f ∈ H) : Continuous f :=
   h.continuous ⟨f, hf⟩
 #align set.equicontinuous.continuous_of_mem Set.Equicontinuous.continuous_of_mem
 
+protected theorem Set.EquicontinuousOn.continuousOn_of_mem {H : Set <| X → α} {S : Set X}
+    (h : H.EquicontinuousOn S) {f : X → α} (hf : f ∈ H) : ContinuousOn f S :=
+  h.continuousOn ⟨f, hf⟩
+
 /-- Each function of a uniformly equicontinuous family is uniformly continuous. -/
 theorem UniformEquicontinuous.uniformContinuous {F : ι → β → α} (h : UniformEquicontinuous F)
     (i : ι) : UniformContinuous (F i) := fun U hU =>
   mem_map.mpr (mem_of_superset (h U hU) fun _ hxy => hxy i)
 #align uniform_equicontinuous.uniform_continuous UniformEquicontinuous.uniformContinuous
 
+/-- Each function of a family uniformly equicontinuous on `S` is uniformly continuous on `S`. -/
+theorem UniformEquicontinuousOn.uniformContinuousOn {F : ι → β → α} {S : Set β}
+    (h : UniformEquicontinuousOn F S) (i : ι) :
+    UniformContinuousOn (F i) S := fun U hU =>
+  mem_map.mpr (mem_of_superset (h U hU) fun _ hxy => hxy i)
+
 protected theorem Set.UniformEquicontinuous.uniformContinuous_of_mem {H : Set <| β → α}
     (h : H.UniformEquicontinuous) {f : β → α} (hf : f ∈ H) : UniformContinuous f :=
   h.uniformContinuous ⟨f, hf⟩
 #align set.uniform_equicontinuous.uniform_continuous_of_mem Set.UniformEquicontinuous.uniformContinuous_of_mem
 
+protected theorem Set.UniformEquicontinuousOn.uniformContinuousOn_of_mem {H : Set <| β → α}
+    {S : Set β} (h : H.UniformEquicontinuousOn S) {f : β → α} (hf : f ∈ H) :
+    UniformContinuousOn f S :=
+  h.uniformContinuousOn ⟨f, hf⟩
+
 /-- Taking sub-families preserves equicontinuity at a point. -/
 theorem EquicontinuousAt.comp {F : ι → X → α} {x₀ : X} (h : EquicontinuousAt F x₀) (u : κ → ι) :
     EquicontinuousAt (F ∘ u) x₀ := fun U hU => (h U hU).mono fun _ H k => H (u k)
 #align equicontinuous_at.comp EquicontinuousAt.comp
 
+/-- Taking sub-families preserves equicontinuity at a point within a subset. -/
+theorem EquicontinuousWithinAt.comp {F : ι → X → α} {S : Set X} {x₀ : X}
+    (h : EquicontinuousWithinAt F S x₀) (u : κ → ι) :
+    EquicontinuousWithinAt (F ∘ u) S x₀ :=
+  fun U hU ↦ (h U hU).mono fun _ H k => H (u k)
+
 protected theorem Set.EquicontinuousAt.mono {H H' : Set <| X → α} {x₀ : X}
     (h : H.EquicontinuousAt x₀) (hH : H' ⊆ H) : H'.EquicontinuousAt x₀ :=
   h.comp (inclusion hH)
 #align set.equicontinuous_at.mono Set.EquicontinuousAt.mono
 
+protected theorem Set.EquicontinuousWithinAt.mono {H H' : Set <| X → α} {S : Set X} {x₀ : X}
+    (h : H.EquicontinuousWithinAt S x₀) (hH : H' ⊆ H) : H'.EquicontinuousWithinAt S x₀ :=
+  h.comp (inclusion hH)
+
 /-- Taking sub-families preserves equicontinuity. -/
 theorem Equicontinuous.comp {F : ι → X → α} (h : Equicontinuous F) (u : κ → ι) :
     Equicontinuous (F ∘ u) := fun x => (h x).comp u
 #align equicontinuous.comp Equicontinuous.comp
 
+/-- Taking sub-families preserves equicontinuity on a subset. -/
+theorem EquicontinuousOn.comp {F : ι → X → α} {S : Set X} (h : EquicontinuousOn F S) (u : κ → ι) :
+    EquicontinuousOn (F ∘ u) S := fun x hx ↦ (h x hx).comp u
+
 protected theorem Set.Equicontinuous.mono {H H' : Set <| X → α} (h : H.Equicontinuous)
     (hH : H' ⊆ H) : H'.Equicontinuous :=
   h.comp (inclusion hH)
 #align set.equicontinuous.mono Set.Equicontinuous.mono
 
+protected theorem Set.EquicontinuousOn.mono {H H' : Set <| X → α} {S : Set X}
+    (h : H.EquicontinuousOn S) (hH : H' ⊆ H) : H'.EquicontinuousOn S :=
+  h.comp (inclusion hH)
+
 /-- Taking sub-families preserves uniform equicontinuity. -/
 theorem UniformEquicontinuous.comp {F : ι → β → α} (h : UniformEquicontinuous F) (u : κ → ι) :
     UniformEquicontinuous (F ∘ u) := fun U hU => (h U hU).mono fun _ H k => H (u k)
 #align uniform_equicontinuous.comp UniformEquicontinuous.comp
 
+/-- Taking sub-families preserves uniform equicontinuity on a subset. -/
+theorem UniformEquicontinuousOn.comp {F : ι → β → α} {S : Set β} (h : UniformEquicontinuousOn F S)
+    (u : κ → ι) : UniformEquicontinuousOn (F ∘ u) S :=
+  fun U hU ↦ (h U hU).mono fun _ H k => H (u k)
+
 protected theorem Set.UniformEquicontinuous.mono {H H' : Set <| β → α} (h : H.UniformEquicontinuous)
     (hH : H' ⊆ H) : H'.UniformEquicontinuous :=
   h.comp (inclusion hH)
 #align set.uniform_equicontinuous.mono Set.UniformEquicontinuous.mono
 
+protected theorem Set.UniformEquicontinuousOn.mono {H H' : Set <| β → α} {S : Set β}
+    (h : H.UniformEquicontinuousOn S) (hH : H' ⊆ H) : H'.UniformEquicontinuousOn S :=
+  h.comp (inclusion hH)
+
 /-- A family `𝓕 : ι → X → α` is equicontinuous at `x₀` iff `range 𝓕` is equicontinuous at `x₀`,
 i.e the family `(↑) : range F → X → α` is equicontinuous at `x₀`. -/
 theorem equicontinuousAt_iff_range {F : ι → X → α} {x₀ : X} :
@@ -267,6 +463,12 @@ theorem equicontinuousAt_iff_range {F : ι → X → α} {x₀ : X} :
   simp only [EquicontinuousAt, forall_subtype_range_iff]
 #align equicontinuous_at_iff_range equicontinuousAt_iff_range
 
+/-- A family `𝓕 : ι → X → α` is equicontinuous at `x₀` within `S` iff `range 𝓕` is equicontinuous
+at `x₀` within `S`, i.e the family `(↑) : range F → X → α` is equicontinuous at `x₀` within `S`. -/
+theorem equicontinuousWithinAt_iff_range {F : ι → X → α} {S : Set X} {x₀ : X} :
+    EquicontinuousWithinAt F S x₀ ↔ EquicontinuousWithinAt ((↑) : range F → X → α) S x₀ := by
+  simp only [EquicontinuousWithinAt, forall_subtype_range_iff]
+
 /-- A family `𝓕 : ι → X → α` is equicontinuous iff `range 𝓕` is equicontinuous,
 i.e the family `(↑) : range F → X → α` is equicontinuous. -/
 theorem equicontinuous_iff_range {F : ι → X → α} :
@@ -274,13 +476,26 @@ theorem equicontinuous_iff_range {F : ι → X → α} :
   forall_congr' fun _ => equicontinuousAt_iff_range
 #align equicontinuous_iff_range equicontinuous_iff_range
 
+/-- A family `𝓕 : ι → X → α` is equicontinuous on `S` iff `range 𝓕` is equicontinuous on `S`,
+i.e the family `(↑) : range F → X → α` is equicontinuous on `S`. -/
+theorem equicontinuousOn_iff_range {F : ι → X → α} {S : Set X} :
+    EquicontinuousOn F S ↔ EquicontinuousOn ((↑) : range F → X → α) S :=
+  forall_congr' fun _ ↦ forall_congr' fun _ ↦ equicontinuousWithinAt_iff_range
+
 /-- A family `𝓕 : ι → β → α` is uniformly equicontinuous iff `range 𝓕` is uniformly equicontinuous,
 i.e the family `(↑) : range F → β → α` is uniformly equicontinuous. -/
-theorem uniformEquicontinuous_at_iff_range {F : ι → β → α} :
+theorem uniformEquicontinuous_iff_range {F : ι → β → α} :
     UniformEquicontinuous F ↔ UniformEquicontinuous ((↑) : range F → β → α) :=
   ⟨fun h => by rw [← comp_rangeSplitting F]; exact h.comp _, fun h =>
     h.comp (rangeFactorization F)⟩
-#align uniform_equicontinuous_at_iff_range uniformEquicontinuous_at_iff_range
+#align uniform_equicontinuous_at_iff_range uniformEquicontinuous_iff_range
+
+/-- A family `𝓕 : ι → β → α` is uniformly equicontinuous on `S` iff `range 𝓕` is uniformly
+equicontinuous on `S`, i.e the family `(↑) : range F → β → α` is uniformly equicontinuous on `S`. -/
+theorem uniformEquicontinuousOn_iff_range {F : ι → β → α} {S : Set β} :
+    UniformEquicontinuousOn F S ↔ UniformEquicontinuousOn ((↑) : range F → β → α) S :=
+  ⟨fun h => by rw [← comp_rangeSplitting F]; exact h.comp _, fun h =>
+    h.comp (rangeFactorization F)⟩
 
 section
 
@@ -296,6 +511,16 @@ theorem equicontinuousAt_iff_continuousAt {F : ι → X → α} {x₀ : X} :
   rfl
 #align equicontinuous_at_iff_continuous_at equicontinuousAt_iff_continuousAt
 
+/-- A family `𝓕 : ι → X → α` is equicontinuous at `x₀` within `S` iff the function
+`swap 𝓕 : X → ι → α` is continuous at `x₀` within `S`
+*when `ι → α` is equipped with the topology of uniform convergence*. This is very useful for
+developping the equicontinuity API, but it should not be used directly for other purposes. -/
+theorem equicontinuousWithinAt_iff_continuousWithinAt {F : ι → X → α} {S : Set X} {x₀ : X} :
+    EquicontinuousWithinAt F S x₀ ↔
+    ContinuousWithinAt (ofFun ∘ Function.swap F : X → ι →ᵤ α) S x₀ := by
+  rw [ContinuousWithinAt, (UniformFun.hasBasis_nhds ι α _).tendsto_right_iff]
+  rfl
+
 /-- A family `𝓕 : ι → X → α` is equicontinuous iff the function `swap 𝓕 : X → ι → α` is
 continuous *when `ι → α` is equipped with the topology of uniform convergence*. This is
 very useful for developping the equicontinuity API, but it should not be used directly for other
@@ -305,6 +530,14 @@ theorem equicontinuous_iff_continuous {F : ι → X → α} :
   simp_rw [Equicontinuous, continuous_iff_continuousAt, equicontinuousAt_iff_continuousAt]
 #align equicontinuous_iff_continuous equicontinuous_iff_continuous
 
+/-- A family `𝓕 : ι → X → α` is equicontinuous on `S` iff the function `swap 𝓕 : X → ι → α` is
+continuous on `S` *when `ι → α` is equipped with the topology of uniform convergence*. This is
+very useful for developping the equicontinuity API, but it should not be used directly for other
+purposes. -/
+theorem equicontinuousOn_iff_continuousOn {F : ι → X → α} {S : Set X} :
+    EquicontinuousOn F S ↔ ContinuousOn (ofFun ∘ Function.swap F : X → ι →ᵤ α) S := by
+  simp_rw [EquicontinuousOn, ContinuousOn, equicontinuousWithinAt_iff_continuousWithinAt]
+
 /-- A family `𝓕 : ι → β → α` is uniformly equicontinuous iff the function `swap 𝓕 : β → ι → α` is
 uniformly continuous *when `ι → α` is equipped with the uniform structure of uniform convergence*.
 This is very useful for developping the equicontinuity API, but it should not be used directly
@@ -315,54 +548,103 @@ theorem uniformEquicontinuous_iff_uniformContinuous {F : ι → β → α} :
   rfl
 #align uniform_equicontinuous_iff_uniform_continuous uniformEquicontinuous_iff_uniformContinuous
 
-theorem equicontinuousAt_iInf_rng {α' : Type*} {u : κ → UniformSpace α'} {F : ι → X → α'}
-    {x₀ : X} :
-    @EquicontinuousAt _ _ _ _ (⨅ k, u k) F x₀ ↔ ∀ k, @EquicontinuousAt _ _ _ _ (u k) F x₀ := by
-  simp only [@equicontinuousAt_iff_continuousAt _ _ _ _ _, topologicalSpace]
-  unfold ContinuousAt
+/-- A family `𝓕 : ι → β → α` is uniformly equicontinuous on `S` iff the function
+`swap 𝓕 : β → ι → α` is uniformly continuous on `S`
+*when `ι → α` is equipped with the uniform structure of uniform convergence*. This is very useful
+for developping the equicontinuity API, but it should not be used directly for other purposes. -/
+theorem uniformEquicontinuousOn_iff_uniformContinuousOn {F : ι → β → α} {S : Set β} :
+    UniformEquicontinuousOn F S ↔ UniformContinuousOn (ofFun ∘ Function.swap F : β → ι →ᵤ α) S := by
+  rw [UniformContinuousOn, (UniformFun.hasBasis_uniformity ι α).tendsto_right_iff]
+  rfl
+
+theorem equicontinuousWithinAt_iInf_rng {u : κ → UniformSpace α'} {F : ι → X → α'}
+    {S : Set X} {x₀ : X} : EquicontinuousWithinAt (uα :=  ⨅ k, u k) F S x₀ ↔
+      ∀ k, EquicontinuousWithinAt (uα :=  u k) F S x₀ := by
+  simp only [equicontinuousWithinAt_iff_continuousWithinAt (uα := _), topologicalSpace]
+  unfold ContinuousWithinAt
   rw [UniformFun.iInf_eq, toTopologicalSpace_iInf, nhds_iInf, tendsto_iInf]
 
-theorem equicontinuous_iInf_rng {α' : Type*} {u : κ → UniformSpace α'} {F : ι → X → α'} :
-    @Equicontinuous _ _ _ _ (⨅ k, u k) F ↔ ∀ k, @Equicontinuous _ _ _ _ (u k) F := by
-  simp_rw [@equicontinuous_iff_continuous _ _ _ _ _, UniformFun.topologicalSpace]
+theorem equicontinuousAt_iInf_rng {u : κ → UniformSpace α'} {F : ι → X → α'}
+    {x₀ : X} :
+    EquicontinuousAt (uα := ⨅ k, u k) F x₀ ↔ ∀ k, EquicontinuousAt (uα := u k) F x₀ := by
+  simp only [← equicontinuousWithinAt_univ (uα := _), equicontinuousWithinAt_iInf_rng]
+
+theorem equicontinuous_iInf_rng {u : κ → UniformSpace α'} {F : ι → X → α'} :
+    Equicontinuous (uα := ⨅ k, u k) F ↔ ∀ k, Equicontinuous (uα := u k) F := by
+  simp_rw [equicontinuous_iff_continuous (uα := _), UniformFun.topologicalSpace]
   rw [UniformFun.iInf_eq, toTopologicalSpace_iInf, continuous_iInf_rng]
 
-theorem uniformEquicontinuous_iInf_rng {α' : Type*} {u : κ → UniformSpace α'} {F : ι → β → α'} :
-    @UniformEquicontinuous _ _ _ (⨅ k, u k) _ F ↔ ∀ k, @UniformEquicontinuous _ _ _ (u k) _ F := by
-  simp_rw [@uniformEquicontinuous_iff_uniformContinuous _ _ _ _]
+theorem equicontinuousOn_iInf_rng {u : κ → UniformSpace α'} {F : ι → X → α'}
+    {S : Set X} :
+    EquicontinuousOn (uα := ⨅ k, u k) F S ↔ ∀ k, EquicontinuousOn (uα := u k) F S := by
+  simp_rw [EquicontinuousOn, equicontinuousWithinAt_iInf_rng, @forall_swap _ κ]
+
+theorem uniformEquicontinuous_iInf_rng {u : κ → UniformSpace α'} {F : ι → β → α'} :
+    UniformEquicontinuous (uα := ⨅ k, u k) F ↔ ∀ k, UniformEquicontinuous (uα := u k) F := by
+  simp_rw [uniformEquicontinuous_iff_uniformContinuous (uα := _)]
   rw [UniformFun.iInf_eq, uniformContinuous_iInf_rng]
 
-theorem equicontinuousAt_iInf_dom {X' : Type*} {t : κ → TopologicalSpace X'} {F : ι → X' → α}
-    {x₀ : X'} {k : κ} (hk : @EquicontinuousAt _ _ _ (t k) _ F x₀) :
-    @EquicontinuousAt _ _ _ (⨅ k, t k) _ F x₀ := by
-  simp? [@equicontinuousAt_iff_continuousAt _ _ _ _] at hk ⊢ says
-    simp only [@equicontinuousAt_iff_continuousAt _ _ _ _] at hk ⊢
-  unfold ContinuousAt at hk ⊢
+theorem uniformEquicontinuousOn_iInf_rng {u : κ → UniformSpace α'} {F : ι → β → α'}
+    {S : Set β} : UniformEquicontinuousOn (uα := ⨅ k, u k) F S ↔
+      ∀ k, UniformEquicontinuousOn (uα := u k) F S := by
+  simp_rw [uniformEquicontinuousOn_iff_uniformContinuousOn (uα := _)]
+  unfold UniformContinuousOn
+  rw [UniformFun.iInf_eq, iInf_uniformity, tendsto_iInf]
+
+theorem equicontinuousWithinAt_iInf_dom {t : κ → TopologicalSpace X'} {F : ι → X' → α}
+    {S : Set X'} {x₀ : X'} {k : κ} (hk : EquicontinuousWithinAt (tX := t k) F S x₀) :
+    EquicontinuousWithinAt (tX := ⨅ k, t k) F S x₀ := by
+  simp [equicontinuousWithinAt_iff_continuousWithinAt (tX := _)] at hk ⊢
+  unfold ContinuousWithinAt nhdsWithin at hk ⊢
   rw [nhds_iInf]
-  exact tendsto_iInf' k hk
-
-theorem equicontinuous_iInf_dom {X' : Type*} {t : κ → TopologicalSpace X'} {F : ι → X' → α}
-    {k : κ} (hk : @Equicontinuous _ _ _ (t k) _ F) :
-    @Equicontinuous _ _ _ (⨅ k, t k) _ F := by
-  simp_rw [@equicontinuous_iff_continuous _ _ _ _] at hk ⊢
-  exact continuous_iInf_dom hk
-
-theorem uniform_equicontinuous_infi_dom {β' : Type*} {u : κ → UniformSpace β'} {F : ι → β' → α}
-    {k : κ} (hk : @UniformEquicontinuous _ _ _ _ (u k) F) :
-    @UniformEquicontinuous _ _ _ _ (⨅ k, u k) F := by
-  simp_rw [@uniformEquicontinuous_iff_uniformContinuous _ _ _ _ _] at hk ⊢
+  exact hk.mono_left <| inf_le_inf_right _ <| iInf_le _ k
+
+theorem equicontinuousAt_iInf_dom {t : κ → TopologicalSpace X'} {F : ι → X' → α}
+    {x₀ : X'} {k : κ} (hk : EquicontinuousAt (tX := t k) F x₀) :
+    EquicontinuousAt (tX := ⨅ k, t k) F x₀ := by
+  rw [← equicontinuousWithinAt_univ (tX := _)] at hk ⊢
+  exact equicontinuousWithinAt_iInf_dom hk
+
+theorem equicontinuous_iInf_dom {t : κ → TopologicalSpace X'} {F : ι → X' → α}
+    {k : κ} (hk : Equicontinuous (tX := t k) F) :
+    Equicontinuous (tX := ⨅ k, t k) F :=
+  fun x ↦ equicontinuousAt_iInf_dom (hk x)
+
+theorem equicontinuousOn_iInf_dom {t : κ → TopologicalSpace X'} {F : ι → X' → α}
+    {S : Set X'} {k : κ} (hk : EquicontinuousOn (tX := t k) F S) :
+    EquicontinuousOn (tX := ⨅ k, t k) F S :=
+  fun x hx ↦ equicontinuousWithinAt_iInf_dom (hk x hx)
+
+theorem uniformEquicontinuous_iInf_dom {u : κ → UniformSpace β'} {F : ι → β' → α}
+    {k : κ} (hk : UniformEquicontinuous (uβ := u k) F) :
+    UniformEquicontinuous (uβ := ⨅ k, u k) F := by
+  simp_rw [uniformEquicontinuous_iff_uniformContinuous (uβ := _)] at hk ⊢
   exact uniformContinuous_iInf_dom hk
 
-theorem Filter.HasBasis.equicontinuousAt_iff_left {κ : Type*} {p : κ → Prop} {s : κ → Set X}
+theorem uniformEquicontinuousOn_iInf_dom {u : κ → UniformSpace β'} {F : ι → β' → α}
+    {S : Set β'} {k : κ} (hk : UniformEquicontinuousOn (uβ := u k) F S) :
+    UniformEquicontinuousOn (uβ := ⨅ k, u k) F S := by
+  simp_rw [uniformEquicontinuousOn_iff_uniformContinuousOn (uβ := _)] at hk ⊢
+  unfold UniformContinuousOn
+  rw [iInf_uniformity]
+  exact hk.mono_left <| inf_le_inf_right _ <| iInf_le _ k
+
+theorem Filter.HasBasis.equicontinuousAt_iff_left {p : κ → Prop} {s : κ → Set X}
     {F : ι → X → α} {x₀ : X} (hX : (𝓝 x₀).HasBasis p s) :
     EquicontinuousAt F x₀ ↔ ∀ U ∈ 𝓤 α, ∃ k, p k ∧ ∀ x ∈ s k, ∀ i, (F i x₀, F i x) ∈ U := by
   rw [equicontinuousAt_iff_continuousAt, ContinuousAt,
     hX.tendsto_iff (UniformFun.hasBasis_nhds ι α _)]
-  simp only [Function.comp_apply, mem_setOf_eq, exists_prop]
   rfl
 #align filter.has_basis.equicontinuous_at_iff_left Filter.HasBasis.equicontinuousAt_iff_left
 
-theorem Filter.HasBasis.equicontinuousAt_iff_right {κ : Type*} {p : κ → Prop} {s : κ → Set (α × α)}
+theorem Filter.HasBasis.equicontinuousWithinAt_iff_left {p : κ → Prop} {s : κ → Set X}
+    {F : ι → X → α} {S : Set X} {x₀ : X} (hX : (𝓝[S] x₀).HasBasis p s) :
+    EquicontinuousWithinAt F S x₀ ↔ ∀ U ∈ 𝓤 α, ∃ k, p k ∧ ∀ x ∈ s k, ∀ i, (F i x₀, F i x) ∈ U := by
+  rw [equicontinuousWithinAt_iff_continuousWithinAt, ContinuousWithinAt,
+    hX.tendsto_iff (UniformFun.hasBasis_nhds ι α _)]
+  rfl
+
+theorem Filter.HasBasis.equicontinuousAt_iff_right {p : κ → Prop} {s : κ → Set (α × α)}
     {F : ι → X → α} {x₀ : X} (hα : (𝓤 α).HasBasis p s) :
     EquicontinuousAt F x₀ ↔ ∀ k, p k → ∀ᶠ x in 𝓝 x₀, ∀ i, (F i x₀, F i x) ∈ s k := by
   rw [equicontinuousAt_iff_continuousAt, ContinuousAt,
@@ -370,6 +652,13 @@ theorem Filter.HasBasis.equicontinuousAt_iff_right {κ : Type*} {p : κ → Prop
   rfl
 #align filter.has_basis.equicontinuous_at_iff_right Filter.HasBasis.equicontinuousAt_iff_right
 
+theorem Filter.HasBasis.equicontinuousWithinAt_iff_right {p : κ → Prop}
+    {s : κ → Set (α × α)} {F : ι → X → α} {S : Set X} {x₀ : X} (hα : (𝓤 α).HasBasis p s) :
+    EquicontinuousWithinAt F S x₀ ↔ ∀ k, p k → ∀ᶠ x in 𝓝[S] x₀, ∀ i, (F i x₀, F i x) ∈ s k := by
+  rw [equicontinuousWithinAt_iff_continuousWithinAt, ContinuousWithinAt,
+    (UniformFun.hasBasis_nhds_of_basis ι α _ hα).tendsto_right_iff]
+  rfl
+
 theorem Filter.HasBasis.equicontinuousAt_iff {κ₁ κ₂ : Type*} {p₁ : κ₁ → Prop} {s₁ : κ₁ → Set X}
     {p₂ : κ₂ → Prop} {s₂ : κ₂ → Set (α × α)} {F : ι → X → α} {x₀ : X} (hX : (𝓝 x₀).HasBasis p₁ s₁)
     (hα : (𝓤 α).HasBasis p₂ s₂) :
@@ -377,21 +666,38 @@ theorem Filter.HasBasis.equicontinuousAt_iff {κ₁ κ₂ : Type*} {p₁ : κ₁
       ∀ k₂, p₂ k₂ → ∃ k₁, p₁ k₁ ∧ ∀ x ∈ s₁ k₁, ∀ i, (F i x₀, F i x) ∈ s₂ k₂ := by
   rw [equicontinuousAt_iff_continuousAt, ContinuousAt,
     hX.tendsto_iff (UniformFun.hasBasis_nhds_of_basis ι α _ hα)]
-  simp only [Function.comp_apply, mem_setOf_eq, exists_prop]
   rfl
 #align filter.has_basis.equicontinuous_at_iff Filter.HasBasis.equicontinuousAt_iff
 
-theorem Filter.HasBasis.uniformEquicontinuous_iff_left {κ : Type*} {p : κ → Prop}
+theorem Filter.HasBasis.equicontinuousWithinAt_iff {κ₁ κ₂ : Type*} {p₁ : κ₁ → Prop}
+    {s₁ : κ₁ → Set X} {p₂ : κ₂ → Prop} {s₂ : κ₂ → Set (α × α)} {F : ι → X → α} {S : Set X} {x₀ : X}
+    (hX : (𝓝[S] x₀).HasBasis p₁ s₁) (hα : (𝓤 α).HasBasis p₂ s₂) :
+    EquicontinuousWithinAt F S x₀ ↔
+      ∀ k₂, p₂ k₂ → ∃ k₁, p₁ k₁ ∧ ∀ x ∈ s₁ k₁, ∀ i, (F i x₀, F i x) ∈ s₂ k₂ := by
+  rw [equicontinuousWithinAt_iff_continuousWithinAt, ContinuousWithinAt,
+    hX.tendsto_iff (UniformFun.hasBasis_nhds_of_basis ι α _ hα)]
+  rfl
+
+theorem Filter.HasBasis.uniformEquicontinuous_iff_left {p : κ → Prop}
     {s : κ → Set (β × β)} {F : ι → β → α} (hβ : (𝓤 β).HasBasis p s) :
     UniformEquicontinuous F ↔
       ∀ U ∈ 𝓤 α, ∃ k, p k ∧ ∀ x y, (x, y) ∈ s k → ∀ i, (F i x, F i y) ∈ U := by
   rw [uniformEquicontinuous_iff_uniformContinuous, UniformContinuous,
     hβ.tendsto_iff (UniformFun.hasBasis_uniformity ι α)]
-  simp only [Prod.forall, Function.comp_apply, mem_setOf_eq, exists_prop]
+  simp only [Prod.forall]
   rfl
 #align filter.has_basis.uniform_equicontinuous_iff_left Filter.HasBasis.uniformEquicontinuous_iff_left
 
-theorem Filter.HasBasis.uniformEquicontinuous_iff_right {κ : Type*} {p : κ → Prop}
+theorem Filter.HasBasis.uniformEquicontinuousOn_iff_left {p : κ → Prop}
+    {s : κ → Set (β × β)} {F : ι → β → α} {S : Set β} (hβ : (𝓤 β ⊓ 𝓟 (S ×ˢ S)).HasBasis p s) :
+    UniformEquicontinuousOn F S ↔
+      ∀ U ∈ 𝓤 α, ∃ k, p k ∧ ∀ x y, (x, y) ∈ s k → ∀ i, (F i x, F i y) ∈ U := by
+  rw [uniformEquicontinuousOn_iff_uniformContinuousOn, UniformContinuousOn,
+    hβ.tendsto_iff (UniformFun.hasBasis_uniformity ι α)]
+  simp only [Prod.forall]
+  rfl
+
+theorem Filter.HasBasis.uniformEquicontinuous_iff_right {p : κ → Prop}
     {s : κ → Set (α × α)} {F : ι → β → α} (hα : (𝓤 α).HasBasis p s) :
     UniformEquicontinuous F ↔ ∀ k, p k → ∀ᶠ xy : β × β in 𝓤 β, ∀ i, (F i xy.1, F i xy.2) ∈ s k := by
   rw [uniformEquicontinuous_iff_uniformContinuous, UniformContinuous,
@@ -399,6 +705,14 @@ theorem Filter.HasBasis.uniformEquicontinuous_iff_right {κ : Type*} {p : κ →
   rfl
 #align filter.has_basis.uniform_equicontinuous_iff_right Filter.HasBasis.uniformEquicontinuous_iff_right
 
+theorem Filter.HasBasis.uniformEquicontinuousOn_iff_right {p : κ → Prop}
+    {s : κ → Set (α × α)} {F : ι → β → α} {S : Set β} (hα : (𝓤 α).HasBasis p s) :
+    UniformEquicontinuousOn F S ↔
+      ∀ k, p k → ∀ᶠ xy : β × β in 𝓤 β ⊓ 𝓟 (S ×ˢ S), ∀ i, (F i xy.1, F i xy.2) ∈ s k := by
+  rw [uniformEquicontinuousOn_iff_uniformContinuousOn, UniformContinuousOn,
+    (UniformFun.hasBasis_uniformity_of_basis ι α hα).tendsto_right_iff]
+  rfl
+
 theorem Filter.HasBasis.uniformEquicontinuous_iff {κ₁ κ₂ : Type*} {p₁ : κ₁ → Prop}
     {s₁ : κ₁ → Set (β × β)} {p₂ : κ₂ → Prop} {s₂ : κ₂ → Set (α × α)} {F : ι → β → α}
     (hβ : (𝓤 β).HasBasis p₁ s₁) (hα : (𝓤 α).HasBasis p₂ s₂) :
@@ -406,12 +720,22 @@ theorem Filter.HasBasis.uniformEquicontinuous_iff {κ₁ κ₂ : Type*} {p₁ :
       ∀ k₂, p₂ k₂ → ∃ k₁, p₁ k₁ ∧ ∀ x y, (x, y) ∈ s₁ k₁ → ∀ i, (F i x, F i y) ∈ s₂ k₂ := by
   rw [uniformEquicontinuous_iff_uniformContinuous, UniformContinuous,
     hβ.tendsto_iff (UniformFun.hasBasis_uniformity_of_basis ι α hα)]
-  simp only [Prod.forall, Function.comp_apply, mem_setOf_eq, exists_prop]
+  simp only [Prod.forall]
   rfl
 #align filter.has_basis.uniform_equicontinuous_iff Filter.HasBasis.uniformEquicontinuous_iff
 
+theorem Filter.HasBasis.uniformEquicontinuousOn_iff {κ₁ κ₂ : Type*} {p₁ : κ₁ → Prop}
+    {s₁ : κ₁ → Set (β × β)} {p₂ : κ₂ → Prop} {s₂ : κ₂ → Set (α × α)} {F : ι → β → α}
+    {S : Set β} (hβ : (𝓤 β ⊓ 𝓟 (S ×ˢ S)).HasBasis p₁ s₁) (hα : (𝓤 α).HasBasis p₂ s₂) :
+    UniformEquicontinuousOn F S ↔
+      ∀ k₂, p₂ k₂ → ∃ k₁, p₁ k₁ ∧ ∀ x y, (x, y) ∈ s₁ k₁ → ∀ i, (F i x, F i y) ∈ s₂ k₂ := by
+  rw [uniformEquicontinuousOn_iff_uniformContinuousOn, UniformContinuousOn,
+    hβ.tendsto_iff (UniformFun.hasBasis_uniformity_of_basis ι α hα)]
+  simp only [Prod.forall]
+  rfl
+
 /-- Given `u : α → β` a uniform inducing map, a family `𝓕 : ι → X → α` is equicontinuous at a point
-`x₀ : X` iff the family `𝓕'`, obtained by precomposing each function of `𝓕` by `u`, is
+`x₀ : X` iff the family `𝓕'`, obtained by composing each function of `𝓕` by `u`, is
 equicontinuous at `x₀`. -/
 theorem UniformInducing.equicontinuousAt_iff {F : ι → X → α} {x₀ : X} {u : α → β}
     (hu : UniformInducing u) : EquicontinuousAt F x₀ ↔ EquicontinuousAt ((u ∘ ·) ∘ F) x₀ := by
@@ -420,110 +744,255 @@ theorem UniformInducing.equicontinuousAt_iff {F : ι → X → α} {x₀ : X} {u
   rfl
 #align uniform_inducing.equicontinuous_at_iff UniformInducing.equicontinuousAt_iff
 
+/-- Given `u : α → β` a uniform inducing map, a family `𝓕 : ι → X → α` is equicontinuous at a point
+`x₀ : X` within a subset `S : Set X` iff the family `𝓕'`, obtained by composing each function
+of `𝓕` by `u`, is equicontinuous at `x₀` within `S`. -/
+theorem UniformInducing.equicontinuousWithinAt_iff {F : ι → X → α} {S : Set X} {x₀ : X} {u : α → β}
+    (hu : UniformInducing u) : EquicontinuousWithinAt F S x₀ ↔
+      EquicontinuousWithinAt ((u ∘ ·) ∘ F) S x₀ := by
+  have := (UniformFun.postcomp_uniformInducing (α := ι) hu).inducing
+  simp only [equicontinuousWithinAt_iff_continuousWithinAt, this.continuousWithinAt_iff]
+  rfl
+
 /-- Given `u : α → β` a uniform inducing map, a family `𝓕 : ι → X → α` is equicontinuous iff the
-family `𝓕'`, obtained by precomposing each function of `𝓕` by `u`, is equicontinuous. -/
+family `𝓕'`, obtained by composing each function of `𝓕` by `u`, is equicontinuous. -/
 theorem UniformInducing.equicontinuous_iff {F : ι → X → α} {u : α → β} (hu : UniformInducing u) :
     Equicontinuous F ↔ Equicontinuous ((u ∘ ·) ∘ F) := by
   congrm ∀ x, ?_
   rw [hu.equicontinuousAt_iff]
 #align uniform_inducing.equicontinuous_iff UniformInducing.equicontinuous_iff
 
+/-- Given `u : α → β` a uniform inducing map, a family `𝓕 : ι → X → α` is equicontinuous on a
+subset `S : Set X` iff the family `𝓕'`, obtained by composing each function of `𝓕` by `u`, is
+equicontinuous on `S`. -/
+theorem UniformInducing.equicontinuousOn_iff {F : ι → X → α} {S : Set X} {u : α → β}
+    (hu : UniformInducing u) : EquicontinuousOn F S ↔ EquicontinuousOn ((u ∘ ·) ∘ F) S := by
+  congrm ∀ x ∈ S, ?_
+  rw [hu.equicontinuousWithinAt_iff]
+
 /-- Given `u : α → γ` a uniform inducing map, a family `𝓕 : ι → β → α` is uniformly equicontinuous
-iff the family `𝓕'`, obtained by precomposing each function of `𝓕` by `u`, is uniformly
+iff the family `𝓕'`, obtained by composing each function of `𝓕` by `u`, is uniformly
 equicontinuous. -/
 theorem UniformInducing.uniformEquicontinuous_iff {F : ι → β → α} {u : α → γ}
     (hu : UniformInducing u) : UniformEquicontinuous F ↔ UniformEquicontinuous ((u ∘ ·) ∘ F) := by
   have := UniformFun.postcomp_uniformInducing (α := ι) hu
-  rw [uniformEquicontinuous_iff_uniformContinuous, uniformEquicontinuous_iff_uniformContinuous,
-    this.uniformContinuous_iff]
+  simp only [uniformEquicontinuous_iff_uniformContinuous, this.uniformContinuous_iff]
   rfl
 #align uniform_inducing.uniform_equicontinuous_iff UniformInducing.uniformEquicontinuous_iff
 
-/-- A version of `EquicontinuousAt.closure` applicable to subsets of types which embed continuously
-into `X → α` with the product topology. It turns out we don't need any other condition on the
-embedding than continuity, but in practice this will mostly be applied to `DFunLike` types where
-the coercion is injective. -/
-theorem EquicontinuousAt.closure' {A : Set Y} {u : Y → X → α} {x₀ : X}
-    (hA : EquicontinuousAt (u ∘ (↑) : A → X → α) x₀) (hu : Continuous u) :
-    EquicontinuousAt (u ∘ (↑) : closure A → X → α) x₀ := by
+/-- Given `u : α → γ` a uniform inducing map, a family `𝓕 : ι → β → α` is uniformly equicontinuous
+on a subset `S : Set β` iff the family `𝓕'`, obtained by composing each function of `𝓕` by `u`,
+is uniformly equicontinuous on `S`. -/
+theorem UniformInducing.uniformEquicontinuousOn_iff {F : ι → β → α} {S : Set β} {u : α → γ}
+    (hu : UniformInducing u) :
+    UniformEquicontinuousOn F S ↔ UniformEquicontinuousOn ((u ∘ ·) ∘ F) S := by
+  have := UniformFun.postcomp_uniformInducing (α := ι) hu
+  simp only [uniformEquicontinuousOn_iff_uniformContinuousOn, this.uniformContinuousOn_iff]
+  rfl
+
+/-- If a set of functions is equicontinuous at some `x₀` within a set `S`, the same is true for its
+closure in *any* topology for which evaluation at any `x ∈ S ∪ {x₀}` is continuous. Since
+this will be applied to `DFunLike` types, we state it for any topological space whith a map
+to `X → α` satisfying the right continuity conditions. See also `Set.EquicontinuousWithinAt.closure`
+for a more familiar (but weaker) statement.
+
+Note: This could *technically* be called `EquicontinuousWithinAt.closure` without name clashes
+with `Set.EquicontinuousWithinAt.closure`, but we don't do it because, even with a `protected`
+marker, it would introduce ambiguities while working in namespace `Set` (e.g, in the proof of
+any theorem called `Set.something`). -/
+theorem EquicontinuousWithinAt.closure' {A : Set Y} {u : Y → X → α} {S : Set X} {x₀ : X}
+    (hA : EquicontinuousWithinAt (u ∘ (↑) : A → X → α) S x₀) (hu₁ : Continuous (S.restrict ∘ u))
+    (hu₂ : Continuous (eval x₀ ∘ u)) :
+    EquicontinuousWithinAt (u ∘ (↑) : closure A → X → α) S x₀ := by
   intro U hU
   rcases mem_uniformity_isClosed hU with ⟨V, hV, hVclosed, hVU⟩
-  filter_upwards [hA V hV] with x hx
+  filter_upwards [hA V hV, eventually_mem_nhdsWithin] with x hx hxS
   rw [SetCoe.forall] at *
   change A ⊆ (fun f => (u f x₀, u f x)) ⁻¹' V at hx
-  refine' (closure_minimal hx <| hVclosed.preimage <| _).trans (preimage_mono hVU)
-  exact Continuous.prod_mk ((continuous_apply x₀).comp hu) ((continuous_apply x).comp hu)
+  refine (closure_minimal hx <| hVclosed.preimage <| hu₂.prod_mk ?_).trans (preimage_mono hVU)
+  exact (continuous_apply ⟨x, hxS⟩).comp hu₁
+
+/-- If a set of functions is equicontinuous at some `x₀`, the same is true for its closure in *any*
+topology for which evaluation at any point is continuous. Since this will be applied to
+`DFunLike` types, we state it for any topological space whith a map to `X → α` satisfying the right
+continuity conditions. See also `Set.EquicontinuousAt.closure` for a more familiar statement. -/
+theorem EquicontinuousAt.closure' {A : Set Y} {u : Y → X → α} {x₀ : X}
+    (hA : EquicontinuousAt (u ∘ (↑) : A → X → α) x₀) (hu : Continuous u) :
+    EquicontinuousAt (u ∘ (↑) : closure A → X → α) x₀ := by
+  rw [← equicontinuousWithinAt_univ] at hA ⊢
+  exact hA.closure' (Pi.continuous_restrict _ |>.comp hu) (continuous_apply x₀ |>.comp hu)
 #align equicontinuous_at.closure' EquicontinuousAt.closure'
 
 /-- If a set of functions is equicontinuous at some `x₀`, its closure for the product topology is
 also equicontinuous at `x₀`. -/
-protected theorem EquicontinuousAt.closure {A : Set (X → α)} {x₀ : X} (hA : A.EquicontinuousAt x₀) :
-    (closure A).EquicontinuousAt x₀ :=
-  EquicontinuousAt.closure' (u := id) hA continuous_id
-#align equicontinuous_at.closure EquicontinuousAt.closure
-
-/-- If `𝓕 : ι → X → α` tends to `f : X → α` *pointwise* along some nontrivial filter, and if the
-family `𝓕` is equicontinuous at some `x₀ : X`, then the limit is continuous at `x₀`. -/
-theorem Filter.Tendsto.continuousAt_of_equicontinuousAt {l : Filter ι} [l.NeBot] {F : ι → X → α}
-    {f : X → α} {x₀ : X} (h₁ : Tendsto F l (𝓝 f)) (h₂ : EquicontinuousAt F x₀) :
-    ContinuousAt f x₀ :=
-  (equicontinuousAt_iff_range.mp h₂).closure.continuousAt
-    ⟨f, mem_closure_of_tendsto h₁ <| eventually_of_forall mem_range_self⟩
-#align filter.tendsto.continuous_at_of_equicontinuous_at Filter.Tendsto.continuousAt_of_equicontinuousAt
-
-/-- A version of `Equicontinuous.closure` applicable to subsets of types which embed continuously
-into `X → α` with the product topology. It turns out we don't need any other condition on the
-embedding than continuity, but in practice this will mostly be applied to `DFunLike` types where
-the coercion is injective. -/
+protected theorem Set.EquicontinuousAt.closure {A : Set (X → α)} {x₀ : X}
+    (hA : A.EquicontinuousAt x₀) : (closure A).EquicontinuousAt x₀ :=
+  hA.closure' (u := id) continuous_id
+#align equicontinuous_at.closure Set.EquicontinuousAt.closure
+
+/-- If a set of functions is equicontinuous at some `x₀` within a set `S`, its closure for the
+product topology is also equicontinuous at `x₀` within `S`. This would also be true for the coarser
+topology of pointwise convergence on `S ∪ {x₀}`, see `Set.EquicontinuousWithinAt.closure'`. -/
+protected theorem Set.EquicontinuousWithinAt.closure {A : Set (X → α)} {S : Set X} {x₀ : X}
+    (hA : A.EquicontinuousWithinAt S x₀) :
+    (closure A).EquicontinuousWithinAt S x₀ :=
+  hA.closure' (u := id) (Pi.continuous_restrict _) (continuous_apply _)
+
+/-- If a set of functions is equicontinuous, the same is true for its closure in *any*
+topology for which evaluation at any point is continuous. Since this will be applied to
+`DFunLike` types, we state it for any topological space whith a map to `X → α` satisfying the right
+continuity conditions. See also `Set.Equicontinuous.closure` for a more familiar statement. -/
 theorem Equicontinuous.closure' {A : Set Y} {u : Y → X → α}
     (hA : Equicontinuous (u ∘ (↑) : A → X → α)) (hu : Continuous u) :
-    Equicontinuous (u ∘ (↑) : closure A → X → α) := fun x => (hA x).closure' hu
+    Equicontinuous (u ∘ (↑) : closure A → X → α) := fun x ↦ (hA x).closure' hu
 #align equicontinuous.closure' Equicontinuous.closure'
 
+/-- If a set of functions is equicontinuous on a set `S`, the same is true for its closure in *any*
+topology for which evaluation at any `x ∈ S` is continuous. Since this will be applied to
+`DFunLike` types, we state it for any topological space whith a map to `X → α` satisfying the right
+continuity conditions. See also `Set.EquicontinuousOn.closure` for a more familiar
+(but weaker) statement. -/
+theorem EquicontinuousOn.closure' {A : Set Y} {u : Y → X → α} {S : Set X}
+    (hA : EquicontinuousOn (u ∘ (↑) : A → X → α) S) (hu : Continuous (S.restrict ∘ u)) :
+    EquicontinuousOn (u ∘ (↑) : closure A → X → α) S :=
+  fun x hx ↦ (hA x hx).closure' hu <| by exact continuous_apply ⟨x, hx⟩ |>.comp hu
+
 /-- If a set of functions is equicontinuous, its closure for the product topology is also
 equicontinuous. -/
-theorem Equicontinuous.closure {A : Set <| X → α} (hA : A.Equicontinuous) :
-    (closure A).Equicontinuous := fun x => (hA x).closure
-#align equicontinuous.closure Equicontinuous.closure
+protected theorem Set.Equicontinuous.closure {A : Set <| X → α} (hA : A.Equicontinuous) :
+    (closure A).Equicontinuous := fun x ↦ Set.EquicontinuousAt.closure (hA x)
+#align equicontinuous.closure Set.Equicontinuous.closure
 
-/-- If `𝓕 : ι → X → α` tends to `f : X → α` *pointwise* along some nontrivial filter, and if the
-family `𝓕` is equicontinuous, then the limit is continuous. -/
-theorem Filter.Tendsto.continuous_of_equicontinuousAt {l : Filter ι} [l.NeBot] {F : ι → X → α}
-    {f : X → α} (h₁ : Tendsto F l (𝓝 f)) (h₂ : Equicontinuous F) : Continuous f :=
-  continuous_iff_continuousAt.mpr fun x => h₁.continuousAt_of_equicontinuousAt (h₂ x)
-#align filter.tendsto.continuous_of_equicontinuous_at Filter.Tendsto.continuous_of_equicontinuousAt
-
-/-- A version of `UniformEquicontinuous.closure` applicable to subsets of types which embed
-continuously into `β → α` with the product topology. It turns out we don't need any other condition
-on the embedding than continuity, but in practice this will mostly be applied to `DFunLike` types
-where the coercion is injective. -/
-theorem UniformEquicontinuous.closure' {A : Set Y} {u : Y → β → α}
-    (hA : UniformEquicontinuous (u ∘ (↑) : A → β → α)) (hu : Continuous u) :
-    UniformEquicontinuous (u ∘ (↑) : closure A → β → α) := by
+/-- If a set of functions is equicontinuous, its closure for the product topology is also
+equicontinuous. This would also be true for the coarser topology of pointwise convergence on `S`,
+see `EquicontinuousOn.closure'`. -/
+protected theorem Set.EquicontinuousOn.closure {A : Set <| X → α} {S : Set X}
+    (hA : A.EquicontinuousOn S) : (closure A).EquicontinuousOn S :=
+  fun x hx ↦ Set.EquicontinuousWithinAt.closure (hA x hx)
+
+/-- If a set of functions is uniformly equicontinuous on a set `S`, the same is true for its
+closure in *any* topology for which evaluation at any `x ∈ S` i continuous. Since this will be
+applied to `DFunLike` types, we state it for any topological space whith a map to `β → α` satisfying
+the right continuity conditions. See also `Set.UniformEquicontinuousOn.closure` for a more familiar
+(but weaker) statement. -/
+theorem UniformEquicontinuousOn.closure' {A : Set Y} {u : Y → β → α} {S : Set β}
+    (hA : UniformEquicontinuousOn (u ∘ (↑) : A → β → α) S) (hu : Continuous (S.restrict ∘ u)) :
+    UniformEquicontinuousOn (u ∘ (↑) : closure A → β → α) S := by
   intro U hU
   rcases mem_uniformity_isClosed hU with ⟨V, hV, hVclosed, hVU⟩
-  filter_upwards [hA V hV]
-  rintro ⟨x, y⟩ hxy
+  filter_upwards [hA V hV, mem_inf_of_right (mem_principal_self _)]
+  rintro ⟨x, y⟩ hxy ⟨hxS, hyS⟩
   rw [SetCoe.forall] at *
   change A ⊆ (fun f => (u f x, u f y)) ⁻¹' V at hxy
-  refine' (closure_minimal hxy <| hVclosed.preimage <| _).trans (preimage_mono hVU)
-  exact Continuous.prod_mk ((continuous_apply x).comp hu) ((continuous_apply y).comp hu)
+  refine (closure_minimal hxy <| hVclosed.preimage <| .prod_mk ?_ ?_).trans (preimage_mono hVU)
+  · exact (continuous_apply ⟨x, hxS⟩).comp hu
+  · exact (continuous_apply ⟨y, hyS⟩).comp hu
+
+/-- If a set of functions is uniformly equicontinuous, the same is true for its closure in *any*
+topology for which evaluation at any point is continuous. Since this will be applied to
+`DFunLike` types, we state it for any topological space whith a map to `β → α` satisfying the right
+continuity conditions. See also `Set.UniformEquicontinuous.closure` for a more familiar statement.
+-/
+theorem UniformEquicontinuous.closure' {A : Set Y} {u : Y → β → α}
+    (hA : UniformEquicontinuous (u ∘ (↑) : A → β → α)) (hu : Continuous u) :
+    UniformEquicontinuous (u ∘ (↑) : closure A → β → α) := by
+  rw [← uniformEquicontinuousOn_univ] at hA ⊢
+  exact hA.closure' (Pi.continuous_restrict _ |>.comp hu)
 #align uniform_equicontinuous.closure' UniformEquicontinuous.closure'
 
 /-- If a set of functions is uniformly equicontinuous, its closure for the product topology is also
 uniformly equicontinuous. -/
-theorem UniformEquicontinuous.closure {A : Set <| β → α} (hA : A.UniformEquicontinuous) :
-    (closure A).UniformEquicontinuous :=
+protected theorem Set.UniformEquicontinuous.closure {A : Set <| β → α}
+    (hA : A.UniformEquicontinuous) : (closure A).UniformEquicontinuous :=
   UniformEquicontinuous.closure' (u := id) hA continuous_id
-#align uniform_equicontinuous.closure UniformEquicontinuous.closure
+#align uniform_equicontinuous.closure Set.UniformEquicontinuous.closure
+
+/-- If a set of functions is uniformly equicontinuous on a set `S`, its closure for the product
+topology is also uniformly equicontinuous. This would also be true for the coarser topology of
+pointwise convergence on `S`, see `UniformEquicontinuousOn.closure'`.-/
+protected theorem Set.UniformEquicontinuousOn.closure {A : Set <| β → α} {S : Set β}
+    (hA : A.UniformEquicontinuousOn S) : (closure A).UniformEquicontinuousOn S :=
+  UniformEquicontinuousOn.closure' (u := id) hA (Pi.continuous_restrict _)
+
+/-
+Implementation note: The following lemma (as well as all the following variations) could
+theoretically be deduced from the "closure" statements above. For example, we could do:
+```lean
+theorem Filter.Tendsto.continuousAt_of_equicontinuousAt {l : Filter ι} [l.NeBot] {F : ι → X → α}
+    {f : X → α} {x₀ : X} (h₁ : Tendsto F l (𝓝 f)) (h₂ : EquicontinuousAt F x₀) :
+    ContinuousAt f x₀ :=
+  (equicontinuousAt_iff_range.mp h₂).closure.continuousAt
+    ⟨f, mem_closure_of_tendsto h₁ <| eventually_of_forall mem_range_self⟩
 
-/-- If `𝓕 : ι → β → α` tends to `f : β → α` *pointwise* along some nontrivial filter, and if the
-family `𝓕` is uniformly equicontinuous, then the limit is uniformly continuous. -/
 theorem Filter.Tendsto.uniformContinuous_of_uniformEquicontinuous {l : Filter ι} [l.NeBot]
     {F : ι → β → α} {f : β → α} (h₁ : Tendsto F l (𝓝 f)) (h₂ : UniformEquicontinuous F) :
     UniformContinuous f :=
-  (uniformEquicontinuous_at_iff_range.mp h₂).closure.uniformContinuous
+  (uniformEquicontinuous_iff_range.mp h₂).closure.uniformContinuous
     ⟨f, mem_closure_of_tendsto h₁ <| eventually_of_forall mem_range_self⟩
+```
+
+Unfortunately, the proofs get painful when dealing with the relative case as one needs to change
+the ambient topology. So it turns out to be easier to re-do the proof by hand.
+-/
+
+/-- If `𝓕 : ι → X → α` tends to `f : X → α` *pointwise on `S ∪ {x₀} : Set X`* along some nontrivial
+filter, and if the family `𝓕` is equicontinuous at `x₀ : X` within `S`, then the limit is
+continuous at `x₀` within `S`. -/
+theorem Filter.Tendsto.continuousWithinAt_of_equicontinuousWithinAt {l : Filter ι} [l.NeBot]
+    {F : ι → X → α} {f : X → α} {S : Set X} {x₀ : X} (h₁ : ∀ x ∈ S, Tendsto (F · x) l (𝓝 (f x)))
+    (h₂ : Tendsto (F · x₀) l (𝓝 (f x₀))) (h₃ : EquicontinuousWithinAt F S x₀) :
+    ContinuousWithinAt f S x₀ := by
+  intro U hU; rw [mem_map]
+  rcases UniformSpace.mem_nhds_iff.mp hU with ⟨V, hV, hVU⟩
+  rcases mem_uniformity_isClosed hV with ⟨W, hW, hWclosed, hWV⟩
+  filter_upwards [h₃ W hW, eventually_mem_nhdsWithin] with x hx hxS using
+    hVU <| ball_mono hWV (f x₀) <| hWclosed.mem_of_tendsto (h₂.prod_mk_nhds (h₁ x hxS)) <|
+    eventually_of_forall hx
+
+/-- If `𝓕 : ι → X → α` tends to `f : X → α` *pointwise* along some nontrivial filter, and if the
+family `𝓕` is equicontinuous at some `x₀ : X`, then the limit is continuous at `x₀`. -/
+theorem Filter.Tendsto.continuousAt_of_equicontinuousAt {l : Filter ι} [l.NeBot] {F : ι → X → α}
+    {f : X → α} {x₀ : X} (h₁ : Tendsto F l (𝓝 f)) (h₂ : EquicontinuousAt F x₀) :
+    ContinuousAt f x₀ := by
+  rw [← continuousWithinAt_univ, ← equicontinuousWithinAt_univ, tendsto_pi_nhds] at *
+  exact continuousWithinAt_of_equicontinuousWithinAt (fun x _ ↦ h₁ x) (h₁ x₀) h₂
+#align filter.tendsto.continuous_at_of_equicontinuous_at Filter.Tendsto.continuousAt_of_equicontinuousAt
+
+/-- If `𝓕 : ι → X → α` tends to `f : X → α` *pointwise* along some nontrivial filter, and if the
+family `𝓕` is equicontinuous, then the limit is continuous. -/
+theorem Filter.Tendsto.continuous_of_equicontinuous {l : Filter ι} [l.NeBot] {F : ι → X → α}
+    {f : X → α} (h₁ : Tendsto F l (𝓝 f)) (h₂ : Equicontinuous F) : Continuous f :=
+  continuous_iff_continuousAt.mpr fun x => h₁.continuousAt_of_equicontinuousAt (h₂ x)
+#align filter.tendsto.continuous_of_equicontinuous_at Filter.Tendsto.continuous_of_equicontinuous
+
+/-- If `𝓕 : ι → X → α` tends to `f : X → α` *pointwise on `S : Set X`* along some nontrivial
+filter, and if the family `𝓕` is equicontinuous, then the limit is continuous on `S`. -/
+theorem Filter.Tendsto.continuousOn_of_equicontinuousOn {l : Filter ι} [l.NeBot] {F : ι → X → α}
+    {f : X → α} {S : Set X} (h₁ : ∀ x ∈ S, Tendsto (F · x) l (𝓝 (f x)))
+    (h₂ : EquicontinuousOn F S) : ContinuousOn f S :=
+  fun x hx ↦ Filter.Tendsto.continuousWithinAt_of_equicontinuousWithinAt h₁ (h₁ x hx) (h₂ x hx)
+
+/-- If `𝓕 : ι → β → α` tends to `f : β → α` *pointwise on `S : Set β`* along some nontrivial
+filter, and if the family `𝓕` is uniformly equicontinuous on `S`, then the limit is uniformly
+continuous on `S`. -/
+theorem Filter.Tendsto.uniformContinuousOn_of_uniformEquicontinuousOn {l : Filter ι} [l.NeBot]
+    {F : ι → β → α} {f : β → α} {S : Set β} (h₁ : ∀ x ∈ S, Tendsto (F · x) l (𝓝 (f x)))
+    (h₂ : UniformEquicontinuousOn F S) :
+    UniformContinuousOn f S := by
+  intro U hU; rw [mem_map]
+  rcases mem_uniformity_isClosed hU with ⟨V, hV, hVclosed, hVU⟩
+  filter_upwards [h₂ V hV, mem_inf_of_right (mem_principal_self _)]
+  rintro ⟨x, y⟩ hxy ⟨hxS, hyS⟩
+  exact hVU <| hVclosed.mem_of_tendsto ((h₁ x hxS).prod_mk_nhds (h₁ y hyS)) <|
+    eventually_of_forall hxy
+
+/-- If `𝓕 : ι → β → α` tends to `f : β → α` *pointwise* along some nontrivial filter, and if the
+family `𝓕` is uniformly equicontinuous, then the limit is uniformly continuous. -/
+theorem Filter.Tendsto.uniformContinuous_of_uniformEquicontinuous {l : Filter ι} [l.NeBot]
+    {F : ι → β → α} {f : β → α} (h₁ : Tendsto F l (𝓝 f)) (h₂ : UniformEquicontinuous F) :
+    UniformContinuous f := by
+  rw [← uniformContinuousOn_univ, ← uniformEquicontinuousOn_univ, tendsto_pi_nhds] at *
+  exact uniformContinuousOn_of_uniformEquicontinuousOn (fun x _ ↦ h₁ x) h₂
 #align filter.tendsto.uniform_continuous_of_uniform_equicontinuous Filter.Tendsto.uniformContinuous_of_uniformEquicontinuous
 
 /-- If `F : ι → X → α` is a family of functions equicontinuous at `x`,

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>

@@ -441,7 +441,7 @@ theorem UniformInducing.uniformEquicontinuous_iff {F : ι → β → α} {u : α
 
 /-- A version of `EquicontinuousAt.closure` applicable to subsets of types which embed continuously
 into `X → α` with the product topology. It turns out we don't need any other condition on the
-embedding than continuity, but in practice this will mostly be applied to `FunLike` types where
+embedding than continuity, but in practice this will mostly be applied to `DFunLike` types where
 the coercion is injective. -/
 theorem EquicontinuousAt.closure' {A : Set Y} {u : Y → X → α} {x₀ : X}
     (hA : EquicontinuousAt (u ∘ (↑) : A → X → α) x₀) (hu : Continuous u) :
@@ -473,7 +473,7 @@ theorem Filter.Tendsto.continuousAt_of_equicontinuousAt {l : Filter ι} [l.NeBot
 
 /-- A version of `Equicontinuous.closure` applicable to subsets of types which embed continuously
 into `X → α` with the product topology. It turns out we don't need any other condition on the
-embedding than continuity, but in practice this will mostly be applied to `FunLike` types where
+embedding than continuity, but in practice this will mostly be applied to `DFunLike` types where
 the coercion is injective. -/
 theorem Equicontinuous.closure' {A : Set Y} {u : Y → X → α}
     (hA : Equicontinuous (u ∘ (↑) : A → X → α)) (hu : Continuous u) :
@@ -495,7 +495,7 @@ theorem Filter.Tendsto.continuous_of_equicontinuousAt {l : Filter ι} [l.NeBot]
 
 /-- A version of `UniformEquicontinuous.closure` applicable to subsets of types which embed
 continuously into `β → α` with the product topology. It turns out we don't need any other condition
-on the embedding than continuity, but in practice this will mostly be applied to `FunLike` types
+on the embedding than continuity, but in practice this will mostly be applied to `DFunLike` types
 where the coercion is injective. -/
 theorem UniformEquicontinuous.closure' {A : Set Y} {u : Y → β → α}
     (hA : UniformEquicontinuous (u ∘ (↑) : A → β → α)) (hu : Continuous u) :

Search for [∀∃].*(_ and manually replace some occurrences with more readable versions. In case of ∀, the new expressions are defeq to the old ones. In case of ∃, they differ by exists_prop.

In some rare cases, golf proofs that needed fixing.

@@ -353,8 +353,6 @@ theorem uniform_equicontinuous_infi_dom {β' : Type*} {u : κ → UniformSpace 
   simp_rw [@uniformEquicontinuous_iff_uniformContinuous _ _ _ _ _] at hk ⊢
   exact uniformContinuous_iInf_dom hk
 
--- Porting note: changed from `∃ k (_ : p k), _` to `∃ k, p k ∧ _` since Lean 4 generates the
--- second one when parsing expressions like `∃ δ > 0, _`.
 theorem Filter.HasBasis.equicontinuousAt_iff_left {κ : Type*} {p : κ → Prop} {s : κ → Set X}
     {F : ι → X → α} {x₀ : X} (hX : (𝓝 x₀).HasBasis p s) :
     EquicontinuousAt F x₀ ↔ ∀ U ∈ 𝓤 α, ∃ k, p k ∧ ∀ x ∈ s k, ∀ i, (F i x₀, F i x) ∈ U := by
@@ -372,8 +370,6 @@ theorem Filter.HasBasis.equicontinuousAt_iff_right {κ : Type*} {p : κ → Prop
   rfl
 #align filter.has_basis.equicontinuous_at_iff_right Filter.HasBasis.equicontinuousAt_iff_right
 
--- Porting note: changed from `∃ k (_ : p k), _` to `∃ k, p k ∧ _` since Lean 4 generates the
--- second one when parsing expressions like `∃ δ > 0, _`.
 theorem Filter.HasBasis.equicontinuousAt_iff {κ₁ κ₂ : Type*} {p₁ : κ₁ → Prop} {s₁ : κ₁ → Set X}
     {p₂ : κ₂ → Prop} {s₂ : κ₂ → Set (α × α)} {F : ι → X → α} {x₀ : X} (hX : (𝓝 x₀).HasBasis p₁ s₁)
     (hα : (𝓤 α).HasBasis p₂ s₂) :
@@ -385,8 +381,6 @@ theorem Filter.HasBasis.equicontinuousAt_iff {κ₁ κ₂ : Type*} {p₁ : κ₁
   rfl
 #align filter.has_basis.equicontinuous_at_iff Filter.HasBasis.equicontinuousAt_iff
 
--- Porting note: changed from `∃ k (_ : p k), _` to `∃ k, p k ∧ _` since Lean 4 generates the
--- second one when parsing expressions like `∃ δ > 0, _`.
 theorem Filter.HasBasis.uniformEquicontinuous_iff_left {κ : Type*} {p : κ → Prop}
     {s : κ → Set (β × β)} {F : ι → β → α} (hβ : (𝓤 β).HasBasis p s) :
     UniformEquicontinuous F ↔
@@ -405,8 +399,6 @@ theorem Filter.HasBasis.uniformEquicontinuous_iff_right {κ : Type*} {p : κ →
   rfl
 #align filter.has_basis.uniform_equicontinuous_iff_right Filter.HasBasis.uniformEquicontinuous_iff_right
 
--- Porting note: changed from `∃ k (_ : p k), _` to `∃ k, p k ∧ _` since Lean 4 generates the
--- second one when parsing expressions like `∃ δ > 0, _`.
 theorem Filter.HasBasis.uniformEquicontinuous_iff {κ₁ κ₂ : Type*} {p₁ : κ₁ → Prop}
     {s₁ : κ₁ → Set (β × β)} {p₂ : κ₂ → Prop} {s₂ : κ₂ → Set (α × α)} {F : ι → β → α}
     (hβ : (𝓤 β).HasBasis p₁ s₁) (hα : (𝓤 α).HasBasis p₂ s₂) :

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

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

@@ -335,7 +335,8 @@ theorem uniformEquicontinuous_iInf_rng {α' : Type*} {u : κ → UniformSpace α
 theorem equicontinuousAt_iInf_dom {X' : Type*} {t : κ → TopologicalSpace X'} {F : ι → X' → α}
     {x₀ : X'} {k : κ} (hk : @EquicontinuousAt _ _ _ (t k) _ F x₀) :
     @EquicontinuousAt _ _ _ (⨅ k, t k) _ F x₀ := by
-  simp [@equicontinuousAt_iff_continuousAt _ _ _ _] at hk ⊢
+  simp? [@equicontinuousAt_iff_continuousAt _ _ _ _] at hk ⊢ says
+    simp only [@equicontinuousAt_iff_continuousAt _ _ _ _] at hk ⊢
   unfold ContinuousAt at hk ⊢
   rw [nhds_iInf]
   exact tendsto_iInf' k hk

I used the regex \(\(· (.) ·\) (.)\), replacing with ($2 $1 ·).

@@ -421,7 +421,7 @@ theorem Filter.HasBasis.uniformEquicontinuous_iff {κ₁ κ₂ : Type*} {p₁ :
 `x₀ : X` iff the family `𝓕'`, obtained by precomposing each function of `𝓕` by `u`, is
 equicontinuous at `x₀`. -/
 theorem UniformInducing.equicontinuousAt_iff {F : ι → X → α} {x₀ : X} {u : α → β}
-    (hu : UniformInducing u) : EquicontinuousAt F x₀ ↔ EquicontinuousAt ((· ∘ ·) u ∘ F) x₀ := by
+    (hu : UniformInducing u) : EquicontinuousAt F x₀ ↔ EquicontinuousAt ((u ∘ ·) ∘ F) x₀ := by
   have := (UniformFun.postcomp_uniformInducing (α := ι) hu).inducing
   rw [equicontinuousAt_iff_continuousAt, equicontinuousAt_iff_continuousAt, this.continuousAt_iff]
   rfl
@@ -430,7 +430,7 @@ theorem UniformInducing.equicontinuousAt_iff {F : ι → X → α} {x₀ : X} {u
 /-- Given `u : α → β` a uniform inducing map, a family `𝓕 : ι → X → α` is equicontinuous iff the
 family `𝓕'`, obtained by precomposing each function of `𝓕` by `u`, is equicontinuous. -/
 theorem UniformInducing.equicontinuous_iff {F : ι → X → α} {u : α → β} (hu : UniformInducing u) :
-    Equicontinuous F ↔ Equicontinuous ((· ∘ ·) u ∘ F) := by
+    Equicontinuous F ↔ Equicontinuous ((u ∘ ·) ∘ F) := by
   congrm ∀ x, ?_
   rw [hu.equicontinuousAt_iff]
 #align uniform_inducing.equicontinuous_iff UniformInducing.equicontinuous_iff
@@ -439,7 +439,7 @@ theorem UniformInducing.equicontinuous_iff {F : ι → X → α} {u : α → β}
 iff the family `𝓕'`, obtained by precomposing each function of `𝓕` by `u`, is uniformly
 equicontinuous. -/
 theorem UniformInducing.uniformEquicontinuous_iff {F : ι → β → α} {u : α → γ}
-    (hu : UniformInducing u) : UniformEquicontinuous F ↔ UniformEquicontinuous ((· ∘ ·) u ∘ F) := by
+    (hu : UniformInducing u) : UniformEquicontinuous F ↔ UniformEquicontinuous ((u ∘ ·) ∘ F) := by
   have := UniformFun.postcomp_uniformInducing (α := ι) hu
   rw [uniformEquicontinuous_iff_uniformContinuous, uniformEquicontinuous_iff_uniformContinuous,
     this.uniformContinuous_iff]

Removes nonterminal simps on lines looking like simp [...]

@@ -318,7 +318,7 @@ theorem uniformEquicontinuous_iff_uniformContinuous {F : ι → β → α} :
 theorem equicontinuousAt_iInf_rng {α' : Type*} {u : κ → UniformSpace α'} {F : ι → X → α'}
     {x₀ : X} :
     @EquicontinuousAt _ _ _ _ (⨅ k, u k) F x₀ ↔ ∀ k, @EquicontinuousAt _ _ _ _ (u k) F x₀ := by
-  simp [@equicontinuousAt_iff_continuousAt _ _ _ _ _, UniformFun.topologicalSpace]
+  simp only [@equicontinuousAt_iff_continuousAt _ _ _ _ _, topologicalSpace]
   unfold ContinuousAt
   rw [UniformFun.iInf_eq, toTopologicalSpace_iInf, nhds_iInf, tendsto_iInf]
 

If F : ι → X → α is an equicontinuous family of functions and f : X → α is a continuous function, then {x | Tendsto (F · x) l (𝓝 (f x))} is a closed set.

@@ -464,7 +464,7 @@ theorem EquicontinuousAt.closure' {A : Set Y} {u : Y → X → α} {x₀ : X}
 
 /-- If a set of functions is equicontinuous at some `x₀`, its closure for the product topology is
 also equicontinuous at `x₀`. -/
-theorem EquicontinuousAt.closure {A : Set <| X → α} {x₀ : X} (hA : A.EquicontinuousAt x₀) :
+protected theorem EquicontinuousAt.closure {A : Set (X → α)} {x₀ : X} (hA : A.EquicontinuousAt x₀) :
     (closure A).EquicontinuousAt x₀ :=
   EquicontinuousAt.closure' (u := id) hA continuous_id
 #align equicontinuous_at.closure EquicontinuousAt.closure
@@ -533,6 +533,35 @@ theorem Filter.Tendsto.uniformContinuous_of_uniformEquicontinuous {l : Filter ι
     ⟨f, mem_closure_of_tendsto h₁ <| eventually_of_forall mem_range_self⟩
 #align filter.tendsto.uniform_continuous_of_uniform_equicontinuous Filter.Tendsto.uniformContinuous_of_uniformEquicontinuous
 
+/-- If `F : ι → X → α` is a family of functions equicontinuous at `x`,
+it tends to `f y` along a filter `l` for any `y ∈ s`,
+the limit function `f` tends to `z` along `𝓝[s] x`, and `x ∈ closure s`,
+then `(F · x)` tends to `z` along `l`.
+
+In some sense, this is a converse of `EquicontinuousAt.closure`. -/
+theorem EquicontinuousAt.tendsto_of_mem_closure {l : Filter ι} {F : ι → X → α} {f : X → α}
+    {s : Set X} {x : X} {z : α} (hF : EquicontinuousAt F x) (hf : Tendsto f (𝓝[s] x) (𝓝 z))
+    (hs : ∀ y ∈ s, Tendsto (F · y) l (𝓝 (f y))) (hx : x ∈ closure s) :
+    Tendsto (F · x) l (𝓝 z) := by
+  rw [(nhds_basis_uniformity (𝓤 α).basis_sets).tendsto_right_iff] at hf ⊢
+  intro U hU
+  rcases comp_comp_symm_mem_uniformity_sets hU with ⟨V, hV, hVs, hVU⟩
+  rw [mem_closure_iff_nhdsWithin_neBot] at hx
+  have : ∀ᶠ y in 𝓝[s] x, y ∈ s ∧ (∀ i, (F i x, F i y) ∈ V) ∧ (f y, z) ∈ V :=
+    eventually_mem_nhdsWithin.and <| ((hF V hV).filter_mono nhdsWithin_le_nhds).and (hf V hV)
+  rcases this.exists with ⟨y, hys, hFy, hfy⟩
+  filter_upwards [hs y hys (ball_mem_nhds _ hV)] with i hi
+  exact hVU ⟨_, ⟨_, hFy i, (mem_ball_symmetry hVs).2 hi⟩, hfy⟩
+
+/-- If `F : ι → X → α` is an equicontinuous family of functions,
+`f : X → α` is a continuous function, and `l` is a filter on `ι`,
+then `{x | Filter.Tendsto (F · x) l (𝓝 (f x))}` is a closed set. -/
+theorem Equicontinuous.isClosed_setOf_tendsto {l : Filter ι} {F : ι → X → α} {f : X → α}
+    (hF : Equicontinuous F) (hf : Continuous f) :
+    IsClosed {x | Tendsto (F · x) l (𝓝 (f x))} :=
+  closure_subset_iff_isClosed.mp fun x hx ↦
+    (hF x).tendsto_of_mem_closure (hf.continuousAt.mono_left inf_le_left) (fun _ ↦ id) hx
+
 end
 
 end

Rename:

• tendsto_iff_norm_tendsto_one → tendsto_iff_norm_div_tendsto_zero;
• tendsto_iff_norm_tendsto_zero → tendsto_iff_norm_sub_tendsto_zero;
• tendsto_one_iff_norm_tendsto_one → tendsto_one_iff_norm_tendsto_zero;
• Filter.Tendsto.continuous_of_equicontinuous_at → Filter.Tendsto.continuous_of_equicontinuousAt.

@@ -495,10 +495,10 @@ theorem Equicontinuous.closure {A : Set <| X → α} (hA : A.Equicontinuous) :
 
 /-- If `𝓕 : ι → X → α` tends to `f : X → α` *pointwise* along some nontrivial filter, and if the
 family `𝓕` is equicontinuous, then the limit is continuous. -/
-theorem Filter.Tendsto.continuous_of_equicontinuous_at {l : Filter ι} [l.NeBot] {F : ι → X → α}
+theorem Filter.Tendsto.continuous_of_equicontinuousAt {l : Filter ι} [l.NeBot] {F : ι → X → α}
     {f : X → α} (h₁ : Tendsto F l (𝓝 f)) (h₂ : Equicontinuous F) : Continuous f :=
   continuous_iff_continuousAt.mpr fun x => h₁.continuousAt_of_equicontinuousAt (h₂ x)
-#align filter.tendsto.continuous_of_equicontinuous_at Filter.Tendsto.continuous_of_equicontinuous_at
+#align filter.tendsto.continuous_of_equicontinuous_at Filter.Tendsto.continuous_of_equicontinuousAt
 
 /-- A version of `UniformEquicontinuous.closure` applicable to subsets of types which embed
 continuously into `β → α` with the product topology. It turns out we don't need any other condition

@@ -126,18 +126,58 @@ protected abbrev Set.UniformEquicontinuous (H : Set <| β → α) : Prop :=
   UniformEquicontinuous ((↑) : H → β → α)
 #align set.uniform_equicontinuous Set.UniformEquicontinuous
 
+/-!
+### Empty index type
+-/
+
+@[simp]
 lemma equicontinuousAt_empty [h : IsEmpty ι] (F : ι → X → α) (x₀ : X) :
     EquicontinuousAt F x₀ :=
   fun _ _ ↦ eventually_of_forall (fun _ ↦ h.elim)
 
+@[simp]
 lemma equicontinuous_empty [IsEmpty ι] (F : ι → X → α) :
     Equicontinuous F :=
   equicontinuousAt_empty F
 
+@[simp]
 lemma uniformEquicontinuous_empty [h : IsEmpty ι] (F : ι → β → α) :
     UniformEquicontinuous F :=
   fun _ _ ↦ eventually_of_forall (fun _ ↦ h.elim)
 
+/-!
+### Finite index type
+-/
+
+theorem equicontinuousAt_finite [Finite ι] {F : ι → X → α} {x₀ : X} :
+    EquicontinuousAt F x₀ ↔ ∀ i, ContinuousAt (F i) x₀ := by
+  simp [EquicontinuousAt, ContinuousAt, (nhds_basis_uniformity' (𝓤 α).basis_sets).tendsto_right_iff,
+    UniformSpace.ball, @forall_swap _ ι]
+
+theorem equicontinuous_finite [Finite ι] {F : ι → X → α} :
+    Equicontinuous F ↔ ∀ i, Continuous (F i) := by
+  simp only [Equicontinuous, equicontinuousAt_finite, continuous_iff_continuousAt, @forall_swap ι]
+
+theorem uniformEquicontinuous_finite [Finite ι] {F : ι → β → α} :
+    UniformEquicontinuous F ↔ ∀ i, UniformContinuous (F i) := by
+  simp only [UniformEquicontinuous, eventually_all, @forall_swap _ ι]; rfl
+
+/-!
+### Index type with a unique element
+-/
+
+theorem equicontinuousAt_unique [Unique ι] {F : ι → X → α} {x : X} :
+    EquicontinuousAt F x ↔ ContinuousAt (F default) x :=
+  equicontinuousAt_finite.trans Unique.forall_iff
+
+theorem equicontinuous_unique [Unique ι] {F : ι → X → α} :
+    Equicontinuous F ↔ Continuous (F default) :=
+  equicontinuous_finite.trans Unique.forall_iff
+
+theorem uniformEquicontinuous_unique [Unique ι] {F : ι → β → α} :
+    UniformEquicontinuous F ↔ UniformContinuous (F default) :=
+  uniformEquicontinuous_finite.trans Unique.forall_iff
+
 /-- Reformulation of equicontinuity at `x₀` comparing two variables near `x₀` instead of comparing
 only one with `x₀`. -/
 theorem equicontinuousAt_iff_pair {F : ι → X → α} {x₀ : X} :
@@ -148,7 +188,7 @@ theorem equicontinuousAt_iff_pair {F : ι → X → α} {x₀ : X} :
     refine' ⟨_, H V hV, fun x hx y hy i => hVU (prod_mk_mem_compRel _ (hy i))⟩
     exact hVsymm.mk_mem_comm.mp (hx i)
   · rcases H U hU with ⟨V, hV, hVU⟩
-    filter_upwards [hV]using fun x hx i => hVU x₀ (mem_of_mem_nhds hV) x hx i
+    filter_upwards [hV] using fun x hx i => hVU x₀ (mem_of_mem_nhds hV) x hx i
 #align equicontinuous_at_iff_pair equicontinuousAt_iff_pair
 
 /-- Uniform equicontinuity implies equicontinuity. -/
@@ -159,11 +199,8 @@ theorem UniformEquicontinuous.equicontinuous {F : ι → β → α} (h : Uniform
 
 /-- Each function of a family equicontinuous at `x₀` is continuous at `x₀`. -/
 theorem EquicontinuousAt.continuousAt {F : ι → X → α} {x₀ : X} (h : EquicontinuousAt F x₀) (i : ι) :
-    ContinuousAt (F i) x₀ := by
-  intro U hU
-  rw [UniformSpace.mem_nhds_iff] at hU
-  rcases hU with ⟨V, hV₁, hV₂⟩
-  exact mem_map.mpr (mem_of_superset (h V hV₁) fun x hx => hV₂ (hx i))
+    ContinuousAt (F i) x₀ :=
+  (UniformSpace.hasBasis_nhds _).tendsto_right_iff.2 fun U ⟨hU, _⟩ ↦ (h U hU).mono fun _x hx ↦ hx i
 #align equicontinuous_at.continuous_at EquicontinuousAt.continuousAt
 
 protected theorem Set.EquicontinuousAt.continuousAt_of_mem {H : Set <| X → α} {x₀ : X}
@@ -226,9 +263,8 @@ protected theorem Set.UniformEquicontinuous.mono {H H' : Set <| β → α} (h :
 /-- A family `𝓕 : ι → X → α` is equicontinuous at `x₀` iff `range 𝓕` is equicontinuous at `x₀`,
 i.e the family `(↑) : range F → X → α` is equicontinuous at `x₀`. -/
 theorem equicontinuousAt_iff_range {F : ι → X → α} {x₀ : X} :
-    EquicontinuousAt F x₀ ↔ EquicontinuousAt ((↑) : range F → X → α) x₀ :=
-  ⟨fun h => by rw [← comp_rangeSplitting F]; exact h.comp _,
-   fun h => h.comp (rangeFactorization F)⟩
+    EquicontinuousAt F x₀ ↔ EquicontinuousAt ((↑) : range F → X → α) x₀ := by
+  simp only [EquicontinuousAt, forall_subtype_range_iff]
 #align equicontinuous_at_iff_range equicontinuousAt_iff_range
 
 /-- A family `𝓕 : ι → X → α` is equicontinuous iff `range 𝓕` is equicontinuous,
@@ -419,7 +455,7 @@ theorem EquicontinuousAt.closure' {A : Set Y} {u : Y → X → α} {x₀ : X}
     EquicontinuousAt (u ∘ (↑) : closure A → X → α) x₀ := by
   intro U hU
   rcases mem_uniformity_isClosed hU with ⟨V, hV, hVclosed, hVU⟩
-  filter_upwards [hA V hV]with x hx
+  filter_upwards [hA V hV] with x hx
   rw [SetCoe.forall] at *
   change A ⊆ (fun f => (u f x₀, u f x)) ⁻¹' V at hx
   refine' (closure_minimal hx <| hVclosed.preimage <| _).trans (preimage_mono hVU)

Adds a term elaborator for congr(...) "congruence quotations". For example, if hf : f = f' and hx : x = x', then we have congr($hf $x) : f x = f' x'. This supports the functions having implicit arguments, and it has support for subsingleton instance arguments. So for example, if s t : Set X are sets with Fintype instances and h : s = t then congr(Fintype.card $h) : Fintype.card s = Fintype.card t works.

Ports the congrm tactic as a convenient frontend for applying a congruence quotation to the goal. Holes are turned into congruence holes. For example, congrm 1 + ?_ uses congr(1 + $(?_)). Placeholders (_) do not turn into congruence holes; that's not to say they have to be identical on the LHS and RHS, but congrm itself is responsible for finding a congruence lemma for such arguments.

Co-authored-by: Scott Morrison <scott.morrison@gmail.com> Co-authored-by: Moritz Doll <moritz.doll@googlemail.com>

@@ -395,13 +395,8 @@ theorem UniformInducing.equicontinuousAt_iff {F : ι → X → α} {x₀ : X} {u
 family `𝓕'`, obtained by precomposing each function of `𝓕` by `u`, is equicontinuous. -/
 theorem UniformInducing.equicontinuous_iff {F : ι → X → α} {u : α → β} (hu : UniformInducing u) :
     Equicontinuous F ↔ Equicontinuous ((· ∘ ·) u ∘ F) := by
-  have : ∀ x, EquicontinuousAt F x ↔ EquicontinuousAt ((fun x x_1 => x ∘ x_1) u ∘ F) x := by
-    intro
-    rw [hu.equicontinuousAt_iff]
-  exact ⟨fun h x => (this x).mp (h x), fun h x => (this x).mpr (h x)⟩
-  -- Porting note: proof was:
-  -- congrm (∀ x, _ : Prop)
-  -- rw [hu.equicontinuousAt_iff]
+  congrm ∀ x, ?_
+  rw [hu.equicontinuousAt_iff]
 #align uniform_inducing.equicontinuous_iff UniformInducing.equicontinuous_iff
 
 /-- Given `u : α → γ` a uniform inducing map, a family `𝓕 : ι → β → α` is uniformly equicontinuous

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

This has nice performance benefits.

@@ -85,7 +85,7 @@ section
 
 open UniformSpace Filter Set Uniformity Topology UniformConvergence
 
-variable {ι κ X Y Z α β γ 𝓕 : Type _} [TopologicalSpace X] [TopologicalSpace Y] [TopologicalSpace Z]
+variable {ι κ X Y Z α β γ 𝓕 : Type*} [TopologicalSpace X] [TopologicalSpace Y] [TopologicalSpace Z]
   [UniformSpace α] [UniformSpace β] [UniformSpace γ]
 
 /-- A family `F : ι → X → α` of functions from a topological space to a uniform space is
@@ -279,24 +279,24 @@ theorem uniformEquicontinuous_iff_uniformContinuous {F : ι → β → α} :
   rfl
 #align uniform_equicontinuous_iff_uniform_continuous uniformEquicontinuous_iff_uniformContinuous
 
-theorem equicontinuousAt_iInf_rng {α' : Type _} {u : κ → UniformSpace α'} {F : ι → X → α'}
+theorem equicontinuousAt_iInf_rng {α' : Type*} {u : κ → UniformSpace α'} {F : ι → X → α'}
     {x₀ : X} :
     @EquicontinuousAt _ _ _ _ (⨅ k, u k) F x₀ ↔ ∀ k, @EquicontinuousAt _ _ _ _ (u k) F x₀ := by
   simp [@equicontinuousAt_iff_continuousAt _ _ _ _ _, UniformFun.topologicalSpace]
   unfold ContinuousAt
   rw [UniformFun.iInf_eq, toTopologicalSpace_iInf, nhds_iInf, tendsto_iInf]
 
-theorem equicontinuous_iInf_rng {α' : Type _} {u : κ → UniformSpace α'} {F : ι → X → α'} :
+theorem equicontinuous_iInf_rng {α' : Type*} {u : κ → UniformSpace α'} {F : ι → X → α'} :
     @Equicontinuous _ _ _ _ (⨅ k, u k) F ↔ ∀ k, @Equicontinuous _ _ _ _ (u k) F := by
   simp_rw [@equicontinuous_iff_continuous _ _ _ _ _, UniformFun.topologicalSpace]
   rw [UniformFun.iInf_eq, toTopologicalSpace_iInf, continuous_iInf_rng]
 
-theorem uniformEquicontinuous_iInf_rng {α' : Type _} {u : κ → UniformSpace α'} {F : ι → β → α'} :
+theorem uniformEquicontinuous_iInf_rng {α' : Type*} {u : κ → UniformSpace α'} {F : ι → β → α'} :
     @UniformEquicontinuous _ _ _ (⨅ k, u k) _ F ↔ ∀ k, @UniformEquicontinuous _ _ _ (u k) _ F := by
   simp_rw [@uniformEquicontinuous_iff_uniformContinuous _ _ _ _]
   rw [UniformFun.iInf_eq, uniformContinuous_iInf_rng]
 
-theorem equicontinuousAt_iInf_dom {X' : Type _} {t : κ → TopologicalSpace X'} {F : ι → X' → α}
+theorem equicontinuousAt_iInf_dom {X' : Type*} {t : κ → TopologicalSpace X'} {F : ι → X' → α}
     {x₀ : X'} {k : κ} (hk : @EquicontinuousAt _ _ _ (t k) _ F x₀) :
     @EquicontinuousAt _ _ _ (⨅ k, t k) _ F x₀ := by
   simp [@equicontinuousAt_iff_continuousAt _ _ _ _] at hk ⊢
@@ -304,13 +304,13 @@ theorem equicontinuousAt_iInf_dom {X' : Type _} {t : κ → TopologicalSpace X'}
   rw [nhds_iInf]
   exact tendsto_iInf' k hk
 
-theorem equicontinuous_iInf_dom {X' : Type _} {t : κ → TopologicalSpace X'} {F : ι → X' → α}
+theorem equicontinuous_iInf_dom {X' : Type*} {t : κ → TopologicalSpace X'} {F : ι → X' → α}
     {k : κ} (hk : @Equicontinuous _ _ _ (t k) _ F) :
     @Equicontinuous _ _ _ (⨅ k, t k) _ F := by
   simp_rw [@equicontinuous_iff_continuous _ _ _ _] at hk ⊢
   exact continuous_iInf_dom hk
 
-theorem uniform_equicontinuous_infi_dom {β' : Type _} {u : κ → UniformSpace β'} {F : ι → β' → α}
+theorem uniform_equicontinuous_infi_dom {β' : Type*} {u : κ → UniformSpace β'} {F : ι → β' → α}
     {k : κ} (hk : @UniformEquicontinuous _ _ _ _ (u k) F) :
     @UniformEquicontinuous _ _ _ _ (⨅ k, u k) F := by
   simp_rw [@uniformEquicontinuous_iff_uniformContinuous _ _ _ _ _] at hk ⊢
@@ -318,7 +318,7 @@ theorem uniform_equicontinuous_infi_dom {β' : Type _} {u : κ → UniformSpace
 
 -- Porting note: changed from `∃ k (_ : p k), _` to `∃ k, p k ∧ _` since Lean 4 generates the
 -- second one when parsing expressions like `∃ δ > 0, _`.
-theorem Filter.HasBasis.equicontinuousAt_iff_left {κ : Type _} {p : κ → Prop} {s : κ → Set X}
+theorem Filter.HasBasis.equicontinuousAt_iff_left {κ : Type*} {p : κ → Prop} {s : κ → Set X}
     {F : ι → X → α} {x₀ : X} (hX : (𝓝 x₀).HasBasis p s) :
     EquicontinuousAt F x₀ ↔ ∀ U ∈ 𝓤 α, ∃ k, p k ∧ ∀ x ∈ s k, ∀ i, (F i x₀, F i x) ∈ U := by
   rw [equicontinuousAt_iff_continuousAt, ContinuousAt,
@@ -327,7 +327,7 @@ theorem Filter.HasBasis.equicontinuousAt_iff_left {κ : Type _} {p : κ → Prop
   rfl
 #align filter.has_basis.equicontinuous_at_iff_left Filter.HasBasis.equicontinuousAt_iff_left
 
-theorem Filter.HasBasis.equicontinuousAt_iff_right {κ : Type _} {p : κ → Prop} {s : κ → Set (α × α)}
+theorem Filter.HasBasis.equicontinuousAt_iff_right {κ : Type*} {p : κ → Prop} {s : κ → Set (α × α)}
     {F : ι → X → α} {x₀ : X} (hα : (𝓤 α).HasBasis p s) :
     EquicontinuousAt F x₀ ↔ ∀ k, p k → ∀ᶠ x in 𝓝 x₀, ∀ i, (F i x₀, F i x) ∈ s k := by
   rw [equicontinuousAt_iff_continuousAt, ContinuousAt,
@@ -337,7 +337,7 @@ theorem Filter.HasBasis.equicontinuousAt_iff_right {κ : Type _} {p : κ → Pro
 
 -- Porting note: changed from `∃ k (_ : p k), _` to `∃ k, p k ∧ _` since Lean 4 generates the
 -- second one when parsing expressions like `∃ δ > 0, _`.
-theorem Filter.HasBasis.equicontinuousAt_iff {κ₁ κ₂ : Type _} {p₁ : κ₁ → Prop} {s₁ : κ₁ → Set X}
+theorem Filter.HasBasis.equicontinuousAt_iff {κ₁ κ₂ : Type*} {p₁ : κ₁ → Prop} {s₁ : κ₁ → Set X}
     {p₂ : κ₂ → Prop} {s₂ : κ₂ → Set (α × α)} {F : ι → X → α} {x₀ : X} (hX : (𝓝 x₀).HasBasis p₁ s₁)
     (hα : (𝓤 α).HasBasis p₂ s₂) :
     EquicontinuousAt F x₀ ↔
@@ -350,7 +350,7 @@ theorem Filter.HasBasis.equicontinuousAt_iff {κ₁ κ₂ : Type _} {p₁ : κ
 
 -- Porting note: changed from `∃ k (_ : p k), _` to `∃ k, p k ∧ _` since Lean 4 generates the
 -- second one when parsing expressions like `∃ δ > 0, _`.
-theorem Filter.HasBasis.uniformEquicontinuous_iff_left {κ : Type _} {p : κ → Prop}
+theorem Filter.HasBasis.uniformEquicontinuous_iff_left {κ : Type*} {p : κ → Prop}
     {s : κ → Set (β × β)} {F : ι → β → α} (hβ : (𝓤 β).HasBasis p s) :
     UniformEquicontinuous F ↔
       ∀ U ∈ 𝓤 α, ∃ k, p k ∧ ∀ x y, (x, y) ∈ s k → ∀ i, (F i x, F i y) ∈ U := by
@@ -360,7 +360,7 @@ theorem Filter.HasBasis.uniformEquicontinuous_iff_left {κ : Type _} {p : κ →
   rfl
 #align filter.has_basis.uniform_equicontinuous_iff_left Filter.HasBasis.uniformEquicontinuous_iff_left
 
-theorem Filter.HasBasis.uniformEquicontinuous_iff_right {κ : Type _} {p : κ → Prop}
+theorem Filter.HasBasis.uniformEquicontinuous_iff_right {κ : Type*} {p : κ → Prop}
     {s : κ → Set (α × α)} {F : ι → β → α} (hα : (𝓤 α).HasBasis p s) :
     UniformEquicontinuous F ↔ ∀ k, p k → ∀ᶠ xy : β × β in 𝓤 β, ∀ i, (F i xy.1, F i xy.2) ∈ s k := by
   rw [uniformEquicontinuous_iff_uniformContinuous, UniformContinuous,
@@ -370,7 +370,7 @@ theorem Filter.HasBasis.uniformEquicontinuous_iff_right {κ : Type _} {p : κ 
 
 -- Porting note: changed from `∃ k (_ : p k), _` to `∃ k, p k ∧ _` since Lean 4 generates the
 -- second one when parsing expressions like `∃ δ > 0, _`.
-theorem Filter.HasBasis.uniformEquicontinuous_iff {κ₁ κ₂ : Type _} {p₁ : κ₁ → Prop}
+theorem Filter.HasBasis.uniformEquicontinuous_iff {κ₁ κ₂ : Type*} {p₁ : κ₁ → Prop}
     {s₁ : κ₁ → Set (β × β)} {p₂ : κ₂ → Prop} {s₂ : κ₂ → Set (α × α)} {F : ι → β → α}
     (hβ : (𝓤 β).HasBasis p₁ s₁) (hα : (𝓤 α).HasBasis p₂ s₂) :
     UniformEquicontinuous F ↔

@@ -126,6 +126,18 @@ protected abbrev Set.UniformEquicontinuous (H : Set <| β → α) : Prop :=
   UniformEquicontinuous ((↑) : H → β → α)
 #align set.uniform_equicontinuous Set.UniformEquicontinuous
 
+lemma equicontinuousAt_empty [h : IsEmpty ι] (F : ι → X → α) (x₀ : X) :
+    EquicontinuousAt F x₀ :=
+  fun _ _ ↦ eventually_of_forall (fun _ ↦ h.elim)
+
+lemma equicontinuous_empty [IsEmpty ι] (F : ι → X → α) :
+    Equicontinuous F :=
+  equicontinuousAt_empty F
+
+lemma uniformEquicontinuous_empty [h : IsEmpty ι] (F : ι → β → α) :
+    UniformEquicontinuous F :=
+  fun _ _ ↦ eventually_of_forall (fun _ ↦ h.elim)
+
 /-- Reformulation of equicontinuity at `x₀` comparing two variables near `x₀` instead of comparing
 only one with `x₀`. -/
 theorem equicontinuousAt_iff_pair {F : ι → X → α} {x₀ : X} :
@@ -267,6 +279,43 @@ theorem uniformEquicontinuous_iff_uniformContinuous {F : ι → β → α} :
   rfl
 #align uniform_equicontinuous_iff_uniform_continuous uniformEquicontinuous_iff_uniformContinuous
 
+theorem equicontinuousAt_iInf_rng {α' : Type _} {u : κ → UniformSpace α'} {F : ι → X → α'}
+    {x₀ : X} :
+    @EquicontinuousAt _ _ _ _ (⨅ k, u k) F x₀ ↔ ∀ k, @EquicontinuousAt _ _ _ _ (u k) F x₀ := by
+  simp [@equicontinuousAt_iff_continuousAt _ _ _ _ _, UniformFun.topologicalSpace]
+  unfold ContinuousAt
+  rw [UniformFun.iInf_eq, toTopologicalSpace_iInf, nhds_iInf, tendsto_iInf]
+
+theorem equicontinuous_iInf_rng {α' : Type _} {u : κ → UniformSpace α'} {F : ι → X → α'} :
+    @Equicontinuous _ _ _ _ (⨅ k, u k) F ↔ ∀ k, @Equicontinuous _ _ _ _ (u k) F := by
+  simp_rw [@equicontinuous_iff_continuous _ _ _ _ _, UniformFun.topologicalSpace]
+  rw [UniformFun.iInf_eq, toTopologicalSpace_iInf, continuous_iInf_rng]
+
+theorem uniformEquicontinuous_iInf_rng {α' : Type _} {u : κ → UniformSpace α'} {F : ι → β → α'} :
+    @UniformEquicontinuous _ _ _ (⨅ k, u k) _ F ↔ ∀ k, @UniformEquicontinuous _ _ _ (u k) _ F := by
+  simp_rw [@uniformEquicontinuous_iff_uniformContinuous _ _ _ _]
+  rw [UniformFun.iInf_eq, uniformContinuous_iInf_rng]
+
+theorem equicontinuousAt_iInf_dom {X' : Type _} {t : κ → TopologicalSpace X'} {F : ι → X' → α}
+    {x₀ : X'} {k : κ} (hk : @EquicontinuousAt _ _ _ (t k) _ F x₀) :
+    @EquicontinuousAt _ _ _ (⨅ k, t k) _ F x₀ := by
+  simp [@equicontinuousAt_iff_continuousAt _ _ _ _] at hk ⊢
+  unfold ContinuousAt at hk ⊢
+  rw [nhds_iInf]
+  exact tendsto_iInf' k hk
+
+theorem equicontinuous_iInf_dom {X' : Type _} {t : κ → TopologicalSpace X'} {F : ι → X' → α}
+    {k : κ} (hk : @Equicontinuous _ _ _ (t k) _ F) :
+    @Equicontinuous _ _ _ (⨅ k, t k) _ F := by
+  simp_rw [@equicontinuous_iff_continuous _ _ _ _] at hk ⊢
+  exact continuous_iInf_dom hk
+
+theorem uniform_equicontinuous_infi_dom {β' : Type _} {u : κ → UniformSpace β'} {F : ι → β' → α}
+    {k : κ} (hk : @UniformEquicontinuous _ _ _ _ (u k) F) :
+    @UniformEquicontinuous _ _ _ _ (⨅ k, u k) F := by
+  simp_rw [@uniformEquicontinuous_iff_uniformContinuous _ _ _ _ _] at hk ⊢
+  exact uniformContinuous_iInf_dom hk
+
 -- Porting note: changed from `∃ k (_ : p k), _` to `∃ k, p k ∧ _` since Lean 4 generates the
 -- second one when parsing expressions like `∃ δ > 0, _`.
 theorem Filter.HasBasis.equicontinuousAt_iff_left {κ : Type _} {p : κ → Prop} {s : κ → Set X}

• depends on: #5966

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

@@ -2,14 +2,11 @@
 Copyright (c) 2022 Anatole Dedecker. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Anatole Dedecker
-
-! This file was ported from Lean 3 source module topology.uniform_space.equicontinuity
-! leanprover-community/mathlib commit f2ce6086713c78a7f880485f7917ea547a215982
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
 -/
 import Mathlib.Topology.UniformSpace.UniformConvergenceTopology
 
+#align_import topology.uniform_space.equicontinuity from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
+
 /-!
 # Equicontinuity of a family of functions
 

• Run a non-interactive version of fix-comments.py on all files.
• Go through the diff and manually add/discard/edit chunks.

@@ -371,7 +371,7 @@ theorem UniformInducing.uniformEquicontinuous_iff {F : ι → β → α} {u : α
 
 /-- A version of `EquicontinuousAt.closure` applicable to subsets of types which embed continuously
 into `X → α` with the product topology. It turns out we don't need any other condition on the
-embedding than continuity, but in practice this will mostly be applied to `fun_like` types where
+embedding than continuity, but in practice this will mostly be applied to `FunLike` types where
 the coercion is injective. -/
 theorem EquicontinuousAt.closure' {A : Set Y} {u : Y → X → α} {x₀ : X}
     (hA : EquicontinuousAt (u ∘ (↑) : A → X → α) x₀) (hu : Continuous u) :

@@ -54,12 +54,12 @@ Throughout this file, we use :
 
 We choose to express equicontinuity as a properties of indexed families of functions rather
 than sets of functions for the following reasons:
-- it is really easy to express equicontinuity of `H : set (X → α)` using our setup: it is just
+- it is really easy to express equicontinuity of `H : Set (X → α)` using our setup: it is just
   equicontinuity of the family `(↑) : ↥H → (X → α)`. On the other hand, going the other way around
   would require working with the range of the family, which is always annoying because it
   introduces useless existentials.
 - in most applications, one doesn't work with bare functions but with a more specific hom type
-  `hom`. Equicontinuity of a set `H : set hom` would then have to be expressed as equicontinuity
+  `hom`. Equicontinuity of a set `H : Set hom` would then have to be expressed as equicontinuity
   of `coe_fn '' H`, which is super annoying to work with. This is much simpler with families,
   because equicontinuity of a family `𝓕 : ι → hom` would simply be expressed as equicontinuity
   of `coe_fn ∘ 𝓕`, which doesn't introduce any nasty existentials.
@@ -86,9 +86,7 @@ equicontinuity, uniform convergence, ascoli
 
 section
 
-open UniformSpace Filter Set
-
-open Uniformity Topology UniformConvergence
+open UniformSpace Filter Set Uniformity Topology UniformConvergence
 
 variable {ι κ X Y Z α β γ 𝓕 : Type _} [TopologicalSpace X] [TopologicalSpace Y] [TopologicalSpace Z]
   [UniformSpace α] [UniformSpace β] [UniformSpace γ]
@@ -100,7 +98,7 @@ def EquicontinuousAt (F : ι → X → α) (x₀ : X) : Prop :=
   ∀ U ∈ 𝓤 α, ∀ᶠ x in 𝓝 x₀, ∀ i, (F i x₀, F i x) ∈ U
 #align equicontinuous_at EquicontinuousAt
 
-/-- We say that a set `H : set (X → α)` of functions is equicontinuous at a point if the family
+/-- We say that a set `H : Set (X → α)` of functions is equicontinuous at a point if the family
 `(↑) : ↥H → (X → α)` is equicontinuous at that point. -/
 protected abbrev Set.EquicontinuousAt (H : Set <| X → α) (x₀ : X) : Prop :=
   EquicontinuousAt ((↑) : H → X → α) x₀
@@ -112,7 +110,7 @@ def Equicontinuous (F : ι → X → α) : Prop :=
   ∀ x₀, EquicontinuousAt F x₀
 #align equicontinuous Equicontinuous
 
-/-- We say that a set `H : set (X → α)` of functions is equicontinuous if the family
+/-- We say that a set `H : Set (X → α)` of functions is equicontinuous if the family
 `(↑) : ↥H → (X → α)` is equicontinuous. -/
 protected abbrev Set.Equicontinuous (H : Set <| X → α) : Prop :=
   Equicontinuous ((↑) : H → X → α)
@@ -125,7 +123,7 @@ def UniformEquicontinuous (F : ι → β → α) : Prop :=
   ∀ U ∈ 𝓤 α, ∀ᶠ xy : β × β in 𝓤 β, ∀ i, (F i xy.1, F i xy.2) ∈ U
 #align uniform_equicontinuous UniformEquicontinuous
 
-/-- We say that a set `H : set (X → α)` of functions is uniformly equicontinuous if the family
+/-- We say that a set `H : Set (X → α)` of functions is uniformly equicontinuous if the family
 `(↑) : ↥H → (X → α)` is uniformly equicontinuous. -/
 protected abbrev Set.UniformEquicontinuous (H : Set <| β → α) : Prop :=
   UniformEquicontinuous ((↑) : H → β → α)
@@ -220,8 +218,8 @@ protected theorem Set.UniformEquicontinuous.mono {H H' : Set <| β → α} (h :
 i.e the family `(↑) : range F → X → α` is equicontinuous at `x₀`. -/
 theorem equicontinuousAt_iff_range {F : ι → X → α} {x₀ : X} :
     EquicontinuousAt F x₀ ↔ EquicontinuousAt ((↑) : range F → X → α) x₀ :=
-  ⟨fun h => by rw [← comp_rangeSplitting F]; exact h.comp _, fun h =>
-    h.comp (rangeFactorization F)⟩
+  ⟨fun h => by rw [← comp_rangeSplitting F]; exact h.comp _,
+   fun h => h.comp (rangeFactorization F)⟩
 #align equicontinuous_at_iff_range equicontinuousAt_iff_range
 
 /-- A family `𝓕 : ι → X → α` is equicontinuous iff `range 𝓕` is equicontinuous,
@@ -382,9 +380,7 @@ theorem EquicontinuousAt.closure' {A : Set Y} {u : Y → X → α} {x₀ : X}
   rcases mem_uniformity_isClosed hU with ⟨V, hV, hVclosed, hVU⟩
   filter_upwards [hA V hV]with x hx
   rw [SetCoe.forall] at *
-  have hx : A ⊆ (fun f => (u f x₀, u f x)) ⁻¹' V := hx
-  -- Porting note: was
-  -- change A ⊆ (fun f => (u f x₀, u f x)) ⁻¹' V at hx
+  change A ⊆ (fun f => (u f x₀, u f x)) ⁻¹' V at hx
   refine' (closure_minimal hx <| hVclosed.preimage <| _).trans (preimage_mono hVU)
   exact Continuous.prod_mk ((continuous_apply x₀).comp hu) ((continuous_apply x).comp hu)
 #align equicontinuous_at.closure' EquicontinuousAt.closure'
@@ -407,7 +403,7 @@ theorem Filter.Tendsto.continuousAt_of_equicontinuousAt {l : Filter ι} [l.NeBot
 
 /-- A version of `Equicontinuous.closure` applicable to subsets of types which embed continuously
 into `X → α` with the product topology. It turns out we don't need any other condition on the
-embedding than continuity, but in practice this will mostly be applied to `fun_like` types where
+embedding than continuity, but in practice this will mostly be applied to `FunLike` types where
 the coercion is injective. -/
 theorem Equicontinuous.closure' {A : Set Y} {u : Y → X → α}
     (hA : Equicontinuous (u ∘ (↑) : A → X → α)) (hu : Continuous u) :
@@ -429,7 +425,7 @@ theorem Filter.Tendsto.continuous_of_equicontinuous_at {l : Filter ι} [l.NeBot]
 
 /-- A version of `UniformEquicontinuous.closure` applicable to subsets of types which embed
 continuously into `β → α` with the product topology. It turns out we don't need any other condition
-on the embedding than continuity, but in practice this will mostly be applied to `fun_like` types
+on the embedding than continuity, but in practice this will mostly be applied to `FunLike` types
 where the coercion is injective. -/
 theorem UniformEquicontinuous.closure' {A : Set Y} {u : Y → β → α}
     (hA : UniformEquicontinuous (u ∘ (↑) : A → β → α)) (hu : Continuous u) :
@@ -439,9 +435,7 @@ theorem UniformEquicontinuous.closure' {A : Set Y} {u : Y → β → α}
   filter_upwards [hA V hV]
   rintro ⟨x, y⟩ hxy
   rw [SetCoe.forall] at *
-  have hxy : A ⊆ (fun f => (u f x, u f y)) ⁻¹' V := hxy
-  -- Porting note: was
-  -- change A ⊆ (fun f => (u f x, u f y)) ⁻¹' V at hxy
+  change A ⊆ (fun f => (u f x, u f y)) ⁻¹' V at hxy
   refine' (closure_minimal hxy <| hVclosed.preimage <| _).trans (preimage_mono hVU)
   exact Continuous.prod_mk ((continuous_apply x).comp hu) ((continuous_apply y).comp hu)
 #align uniform_equicontinuous.closure' UniformEquicontinuous.closure'

Also changed a few small things in Topology/UniformSpace/Equicontinuity

@@ -135,7 +135,7 @@ protected abbrev Set.UniformEquicontinuous (H : Set <| β → α) : Prop :=
 only one with `x₀`. -/
 theorem equicontinuousAt_iff_pair {F : ι → X → α} {x₀ : X} :
     EquicontinuousAt F x₀ ↔
-      ∀ U ∈ 𝓤 α, ∃ V ∈ 𝓝 x₀, ∀ (x) (_ : x ∈ V) (y) (_ : y ∈ V) (i), (F i x, F i y) ∈ U := by
+      ∀ U ∈ 𝓤 α, ∃ V ∈ 𝓝 x₀, ∀ x ∈ V, ∀ y ∈ V, ∀ i, (F i x, F i y) ∈ U := by
   constructor <;> intro H U hU
   · rcases comp_symm_mem_uniformity_sets hU with ⟨V, hV, hVsymm, hVU⟩
     refine' ⟨_, H V hV, fun x hx y hy i => hVU (prod_mk_mem_compRel _ (hy i))⟩
@@ -272,9 +272,11 @@ theorem uniformEquicontinuous_iff_uniformContinuous {F : ι → β → α} :
   rfl
 #align uniform_equicontinuous_iff_uniform_continuous uniformEquicontinuous_iff_uniformContinuous
 
+-- Porting note: changed from `∃ k (_ : p k), _` to `∃ k, p k ∧ _` since Lean 4 generates the
+-- second one when parsing expressions like `∃ δ > 0, _`.
 theorem Filter.HasBasis.equicontinuousAt_iff_left {κ : Type _} {p : κ → Prop} {s : κ → Set X}
     {F : ι → X → α} {x₀ : X} (hX : (𝓝 x₀).HasBasis p s) :
-    EquicontinuousAt F x₀ ↔ ∀ U ∈ 𝓤 α, ∃ (k : _)(_ : p k), ∀ x ∈ s k, ∀ i, (F i x₀, F i x) ∈ U := by
+    EquicontinuousAt F x₀ ↔ ∀ U ∈ 𝓤 α, ∃ k, p k ∧ ∀ x ∈ s k, ∀ i, (F i x₀, F i x) ∈ U := by
   rw [equicontinuousAt_iff_continuousAt, ContinuousAt,
     hX.tendsto_iff (UniformFun.hasBasis_nhds ι α _)]
   simp only [Function.comp_apply, mem_setOf_eq, exists_prop]
@@ -289,21 +291,25 @@ theorem Filter.HasBasis.equicontinuousAt_iff_right {κ : Type _} {p : κ → Pro
   rfl
 #align filter.has_basis.equicontinuous_at_iff_right Filter.HasBasis.equicontinuousAt_iff_right
 
+-- Porting note: changed from `∃ k (_ : p k), _` to `∃ k, p k ∧ _` since Lean 4 generates the
+-- second one when parsing expressions like `∃ δ > 0, _`.
 theorem Filter.HasBasis.equicontinuousAt_iff {κ₁ κ₂ : Type _} {p₁ : κ₁ → Prop} {s₁ : κ₁ → Set X}
     {p₂ : κ₂ → Prop} {s₂ : κ₂ → Set (α × α)} {F : ι → X → α} {x₀ : X} (hX : (𝓝 x₀).HasBasis p₁ s₁)
     (hα : (𝓤 α).HasBasis p₂ s₂) :
     EquicontinuousAt F x₀ ↔
-      ∀ k₂, p₂ k₂ → ∃ (k₁ : _)(_ : p₁ k₁), ∀ x ∈ s₁ k₁, ∀ i, (F i x₀, F i x) ∈ s₂ k₂ := by
+      ∀ k₂, p₂ k₂ → ∃ k₁, p₁ k₁ ∧ ∀ x ∈ s₁ k₁, ∀ i, (F i x₀, F i x) ∈ s₂ k₂ := by
   rw [equicontinuousAt_iff_continuousAt, ContinuousAt,
     hX.tendsto_iff (UniformFun.hasBasis_nhds_of_basis ι α _ hα)]
   simp only [Function.comp_apply, mem_setOf_eq, exists_prop]
   rfl
 #align filter.has_basis.equicontinuous_at_iff Filter.HasBasis.equicontinuousAt_iff
 
+-- Porting note: changed from `∃ k (_ : p k), _` to `∃ k, p k ∧ _` since Lean 4 generates the
+-- second one when parsing expressions like `∃ δ > 0, _`.
 theorem Filter.HasBasis.uniformEquicontinuous_iff_left {κ : Type _} {p : κ → Prop}
     {s : κ → Set (β × β)} {F : ι → β → α} (hβ : (𝓤 β).HasBasis p s) :
     UniformEquicontinuous F ↔
-      ∀ U ∈ 𝓤 α, ∃ (k : _)(_ : p k), ∀ x y, (x, y) ∈ s k → ∀ i, (F i x, F i y) ∈ U := by
+      ∀ U ∈ 𝓤 α, ∃ k, p k ∧ ∀ x y, (x, y) ∈ s k → ∀ i, (F i x, F i y) ∈ U := by
   rw [uniformEquicontinuous_iff_uniformContinuous, UniformContinuous,
     hβ.tendsto_iff (UniformFun.hasBasis_uniformity ι α)]
   simp only [Prod.forall, Function.comp_apply, mem_setOf_eq, exists_prop]
@@ -318,11 +324,13 @@ theorem Filter.HasBasis.uniformEquicontinuous_iff_right {κ : Type _} {p : κ 
   rfl
 #align filter.has_basis.uniform_equicontinuous_iff_right Filter.HasBasis.uniformEquicontinuous_iff_right
 
+-- Porting note: changed from `∃ k (_ : p k), _` to `∃ k, p k ∧ _` since Lean 4 generates the
+-- second one when parsing expressions like `∃ δ > 0, _`.
 theorem Filter.HasBasis.uniformEquicontinuous_iff {κ₁ κ₂ : Type _} {p₁ : κ₁ → Prop}
     {s₁ : κ₁ → Set (β × β)} {p₂ : κ₂ → Prop} {s₂ : κ₂ → Set (α × α)} {F : ι → β → α}
     (hβ : (𝓤 β).HasBasis p₁ s₁) (hα : (𝓤 α).HasBasis p₂ s₂) :
     UniformEquicontinuous F ↔
-      ∀ k₂, p₂ k₂ → ∃ (k₁ : _)(_ : p₁ k₁), ∀ x y, (x, y) ∈ s₁ k₁ → ∀ i, (F i x, F i y) ∈ s₂ k₂ := by
+      ∀ k₂, p₂ k₂ → ∃ k₁, p₁ k₁ ∧ ∀ x y, (x, y) ∈ s₁ k₁ → ∀ i, (F i x, F i y) ∈ s₂ k₂ := by
   rw [uniformEquicontinuous_iff_uniformContinuous, UniformContinuous,
     hβ.tendsto_iff (UniformFun.hasBasis_uniformity_of_basis ι α hα)]
   simp only [Prod.forall, Function.comp_apply, mem_setOf_eq, exists_prop]
@@ -334,7 +342,7 @@ theorem Filter.HasBasis.uniformEquicontinuous_iff {κ₁ κ₂ : Type _} {p₁ :
 equicontinuous at `x₀`. -/
 theorem UniformInducing.equicontinuousAt_iff {F : ι → X → α} {x₀ : X} {u : α → β}
     (hu : UniformInducing u) : EquicontinuousAt F x₀ ↔ EquicontinuousAt ((· ∘ ·) u ∘ F) x₀ := by
-  have := (@UniformFun.postcomp_uniformInducing ι _ _ _ _ _ hu).inducing
+  have := (UniformFun.postcomp_uniformInducing (α := ι) hu).inducing
   rw [equicontinuousAt_iff_continuousAt, equicontinuousAt_iff_continuousAt, this.continuousAt_iff]
   rfl
 #align uniform_inducing.equicontinuous_at_iff UniformInducing.equicontinuousAt_iff
@@ -357,7 +365,7 @@ iff the family `𝓕'`, obtained by precomposing each function of `𝓕` by `u`,
 equicontinuous. -/
 theorem UniformInducing.uniformEquicontinuous_iff {F : ι → β → α} {u : α → γ}
     (hu : UniformInducing u) : UniformEquicontinuous F ↔ UniformEquicontinuous ((· ∘ ·) u ∘ F) := by
-  have := @UniformFun.postcomp_uniformInducing ι _ _ _ _ _ hu
+  have := UniformFun.postcomp_uniformInducing (α := ι) hu
   rw [uniformEquicontinuous_iff_uniformContinuous, uniformEquicontinuous_iff_uniformContinuous,
     this.uniformContinuous_iff]
   rfl
@@ -385,7 +393,7 @@ theorem EquicontinuousAt.closure' {A : Set Y} {u : Y → X → α} {x₀ : X}
 also equicontinuous at `x₀`. -/
 theorem EquicontinuousAt.closure {A : Set <| X → α} {x₀ : X} (hA : A.EquicontinuousAt x₀) :
     (closure A).EquicontinuousAt x₀ :=
-  @EquicontinuousAt.closure' _ _ _ _ _ _ _ id _ hA continuous_id
+  EquicontinuousAt.closure' (u := id) hA continuous_id
 #align equicontinuous_at.closure EquicontinuousAt.closure
 
 /-- If `𝓕 : ι → X → α` tends to `f : X → α` *pointwise* along some nontrivial filter, and if the
@@ -442,7 +450,7 @@ theorem UniformEquicontinuous.closure' {A : Set Y} {u : Y → β → α}
 uniformly equicontinuous. -/
 theorem UniformEquicontinuous.closure {A : Set <| β → α} (hA : A.UniformEquicontinuous) :
     (closure A).UniformEquicontinuous :=
-  @UniformEquicontinuous.closure' _ _ _ _ _ _ _ id hA continuous_id
+  UniformEquicontinuous.closure' (u := id) hA continuous_id
 #align uniform_equicontinuous.closure UniformEquicontinuous.closure
 
 /-- If `𝓕 : ι → β → α` tends to `f : β → α` *pointwise* along some nontrivial filter, and if the

Co-authored-by: RemyDegenne <Remydegenne@gmail.com>