topology.continuous_on ⟷ Mathlib.Topology.ContinuousOn

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

@@ -117,7 +117,7 @@ theorem nhdsWithin_basis_open (a : α) (t : Set α) :
 #print mem_nhdsWithin /-
 theorem mem_nhdsWithin {t : Set α} {a : α} {s : Set α} :
     t ∈ 𝓝[s] a ↔ ∃ u, IsOpen u ∧ a ∈ u ∧ u ∩ s ⊆ t := by
-  simpa only [exists_prop, and_assoc', and_comm'] using (nhdsWithin_basis_open a s).mem_iff
+  simpa only [exists_prop, and_assoc, and_comm] using (nhdsWithin_basis_open a s).mem_iff
 #align mem_nhds_within mem_nhdsWithin
 -/
 

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

@@ -1172,7 +1172,7 @@ theorem ContinuousWithinAt.continuousAt {f : α → β} {s : Set α} {x : α}
 #print IsOpen.continuousOn_iff /-
 theorem IsOpen.continuousOn_iff {f : α → β} {s : Set α} (hs : IsOpen s) :
     ContinuousOn f s ↔ ∀ ⦃a⦄, a ∈ s → ContinuousAt f a :=
-  ball_congr fun _ => continuousWithinAt_iff_continuousAt ∘ hs.mem_nhds
+  forall₂_congr fun _ => continuousWithinAt_iff_continuousAt ∘ hs.mem_nhds
 #align is_open.continuous_on_iff IsOpen.continuousOn_iff
 -/
 

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

@@ -443,7 +443,7 @@ theorem nhdsWithin_pi_eq' {ι : Type _} {α : ι → Type _} [∀ i, Topological
 #align nhds_within_pi_eq' nhdsWithin_pi_eq'
 -/
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:641:2: warning: expanding binder collection (i «expr ∉ » I) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:642:2: warning: expanding binder collection (i «expr ∉ » I) -/
 #print nhdsWithin_pi_eq /-
 theorem nhdsWithin_pi_eq {ι : Type _} {α : ι → Type _} [∀ i, TopologicalSpace (α i)] {I : Set ι}
     (hI : I.Finite) (s : ∀ i, Set (α i)) (x : ∀ i, α i) :

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

@@ -192,7 +192,7 @@ theorem preimage_nhdsWithin_coinduced' {π : α → β} {s : Set β} {t : Set α
       ⟨π ⁻¹' V, mem_nhds_iff.mpr ⟨t ∩ π ⁻¹' V, inter_subset_right t (π ⁻¹' V), _, mem_sep h mem_V⟩,
         subset.trans (inter_subset_left _ _) (preimage_mono hVs)⟩
   obtain ⟨u, hu1, hu2⟩ := is_open_induced_iff.mp (isOpen_coinduced.1 V_op)
-  rw [preimage_comp] at hu2 
+  rw [preimage_comp] at hu2
   rw [Set.inter_comm, ← subtype.preimage_coe_eq_preimage_coe_iff.mp hu2]
   exact hu1.inter ht
 #align preimage_nhds_within_coinduced' preimage_nhdsWithin_coinduced'ₓ
@@ -529,7 +529,7 @@ theorem eventually_mem_of_tendsto_nhdsWithin {f : β → α} {a : α} {s : Set 
     (h : Tendsto f l (𝓝[s] a)) : ∀ᶠ i in l, f i ∈ s :=
   by
   simp_rw [nhdsWithin_eq, tendsto_infi, mem_set_of_eq, tendsto_principal, mem_inter_iff,
-    eventually_and] at h 
+    eventually_and] at h
   exact (h univ ⟨mem_univ a, isOpen_univ⟩).2
 #align eventually_mem_of_tendsto_nhds_within eventually_mem_of_tendsto_nhdsWithin
 -/
@@ -1128,8 +1128,8 @@ theorem ContinuousOn.congr_mono {f g : α → β} {s s₁ : Set α} (h : Continu
   intro x hx
   unfold ContinuousWithinAt
   have A := (h x (h₁ hx)).mono h₁
-  unfold ContinuousWithinAt at A 
-  rw [← h' hx] at A 
+  unfold ContinuousWithinAt at A
+  rw [← h' hx] at A
   exact A.congr' h'.eventually_eq_nhds_within.symm
 #align continuous_on.congr_mono ContinuousOn.congr_mono
 -/
@@ -1241,7 +1241,7 @@ theorem ContinuousOn.comp' {g : β → γ} {f : α → β} {s : Set α} {t : Set
 #print Continuous.continuousOn /-
 theorem Continuous.continuousOn {f : α → β} {s : Set α} (h : Continuous f) : ContinuousOn f s :=
   by
-  rw [continuous_iff_continuousOn_univ] at h 
+  rw [continuous_iff_continuousOn_univ] at h
   exact h.mono (subset_univ _)
 #align continuous.continuous_on Continuous.continuousOn
 -/
@@ -1421,7 +1421,7 @@ theorem continuousOn_of_locally_continuousOn {f : α → β} {s : Set α}
   intro x xs
   rcases h x xs with ⟨t, open_t, xt, ct⟩
   have := ct x ⟨xs, xt⟩
-  rwa [ContinuousWithinAt, ← nhdsWithin_restrict _ xt open_t] at this 
+  rwa [ContinuousWithinAt, ← nhdsWithin_restrict _ xt open_t] at this
 #align continuous_on_of_locally_continuous_on continuousOn_of_locally_continuousOn
 -/
 
@@ -1502,7 +1502,7 @@ theorem continuousWithinAt_of_not_mem_closure {f : α → β} {s : Set α} {x :
     x ∉ closure s → ContinuousWithinAt f s x :=
   by
   intro hx
-  rw [mem_closure_iff_nhdsWithin_neBot, ne_bot_iff, Classical.not_not] at hx 
+  rw [mem_closure_iff_nhdsWithin_neBot, ne_bot_iff, Classical.not_not] at hx
   rw [ContinuousWithinAt, hx]
   exact tendsto_bot
 #align continuous_within_at_of_not_mem_closure continuousWithinAt_of_not_mem_closure
@@ -1524,7 +1524,7 @@ theorem ContinuousOn.piecewise' {s t : Set α} {f g : α → β} [∀ a, Decidab
         exact
           (hf x hx).congr (fun y hy => piecewise_eq_of_mem _ _ _ hy.2)
             (piecewise_eq_of_mem _ _ _ hx.2)
-      · have : x ∉ closure (tᶜ) := fun h => hx' ⟨subset_closure hx.2, by rwa [closure_compl] at h ⟩
+      · have : x ∉ closure (tᶜ) := fun h => hx' ⟨subset_closure hx.2, by rwa [closure_compl] at h⟩
         exact
           continuousWithinAt_of_not_mem_closure fun h =>
             this (closure_inter_subset_inter_closure _ _ h).2
@@ -1568,7 +1568,7 @@ theorem ContinuousOn.if {α β : Type _} [TopologicalSpace α] [TopologicalSpace
     simp only [hp a ha, if_t_t]
     apply tendsto_nhdsWithin_mono_left (inter_subset_inter_right s subset_closure)
     rcases ha with ⟨has, ⟨_, ha⟩⟩
-    rw [← mem_compl_iff, ← closure_compl] at ha 
+    rw [← mem_compl_iff, ← closure_compl] at ha
     apply hg a ⟨has, ha⟩
   · exact hf.mono (inter_subset_inter_right s subset_closure)
   · exact hg.mono (inter_subset_inter_right s subset_closure)

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

@@ -1645,7 +1645,12 @@ theorem Continuous.piecewise {s : Set α} {f g : α → β} [∀ a, Decidable (a
 
 #print IsOpen.ite' /-
 theorem IsOpen.ite' {s s' t : Set α} (hs : IsOpen s) (hs' : IsOpen s')
-    (ht : ∀ x ∈ frontier t, x ∈ s ↔ x ∈ s') : IsOpen (t.ite s s') := by classical
+    (ht : ∀ x ∈ frontier t, x ∈ s ↔ x ∈ s') : IsOpen (t.ite s s') := by
+  classical
+  simp only [isOpen_iff_continuous_mem, Set.ite] at *
+  convert continuous_piecewise (fun x hx => propext (ht x hx)) hs.continuous_on hs'.continuous_on
+  ext x
+  by_cases hx : x ∈ t <;> simp [hx]
 #align is_open.ite' IsOpen.ite'
 -/
 

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

@@ -1645,12 +1645,7 @@ theorem Continuous.piecewise {s : Set α} {f g : α → β} [∀ a, Decidable (a
 
 #print IsOpen.ite' /-
 theorem IsOpen.ite' {s s' t : Set α} (hs : IsOpen s) (hs' : IsOpen s')
-    (ht : ∀ x ∈ frontier t, x ∈ s ↔ x ∈ s') : IsOpen (t.ite s s') := by
-  classical
-  simp only [isOpen_iff_continuous_mem, Set.ite] at *
-  convert continuous_piecewise (fun x hx => propext (ht x hx)) hs.continuous_on hs'.continuous_on
-  ext x
-  by_cases hx : x ∈ t <;> simp [hx]
+    (ht : ∀ x ∈ frontier t, x ∈ s ↔ x ∈ s') : IsOpen (t.ite s s') := by classical
 #align is_open.ite' IsOpen.ite'
 -/
 

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

@@ -1377,11 +1377,11 @@ theorem continuousOn_open_iff {f : α → β} {s : Set α} (hs : IsOpen s) :
 #align continuous_on_open_iff continuousOn_open_iff
 -/
 
-#print ContinuousOn.preimage_open_of_open /-
-theorem ContinuousOn.preimage_open_of_open {f : α → β} {s : Set α} {t : Set β}
+#print ContinuousOn.isOpen_inter_preimage /-
+theorem ContinuousOn.isOpen_inter_preimage {f : α → β} {s : Set α} {t : Set β}
     (hf : ContinuousOn f s) (hs : IsOpen s) (ht : IsOpen t) : IsOpen (s ∩ f ⁻¹' t) :=
   (continuousOn_open_iff hs).1 hf t ht
-#align continuous_on.preimage_open_of_open ContinuousOn.preimage_open_of_open
+#align continuous_on.preimage_open_of_open ContinuousOn.isOpen_inter_preimage
 -/
 
 #print ContinuousOn.isOpen_preimage /-
@@ -1409,7 +1409,7 @@ theorem ContinuousOn.preimage_interior_subset_interior_preimage {f : α → β}
   calc
     s ∩ f ⁻¹' interior t ⊆ interior (s ∩ f ⁻¹' t) :=
       interior_maximal (inter_subset_inter (Subset.refl _) (preimage_mono interior_subset))
-        (hf.preimage_open_of_open hs isOpen_interior)
+        (hf.isOpen_inter_preimage hs isOpen_interior)
     _ = s ∩ interior (f ⁻¹' t) := by rw [interior_inter, hs.interior_eq]
 #align continuous_on.preimage_interior_subset_interior_preimage ContinuousOn.preimage_interior_subset_interior_preimage
 -/
@@ -1425,7 +1425,7 @@ theorem continuousOn_of_locally_continuousOn {f : α → β} {s : Set α}
 #align continuous_on_of_locally_continuous_on continuousOn_of_locally_continuousOn
 -/
 
-theorem continuousOn_open_of_generateFrom {β : Type _} {s : Set α} {T : Set (Set β)} {f : α → β}
+theorem continuousOn_isOpen_of_generateFrom {β : Type _} {s : Set α} {T : Set (Set β)} {f : α → β}
     (hs : IsOpen s) (h : ∀ t ∈ T, IsOpen (s ∩ f ⁻¹' t)) :
     @ContinuousOn α β _ (TopologicalSpace.generateFrom T) f s :=
   by
@@ -1441,7 +1441,7 @@ theorem continuousOn_open_of_generateFrom {β : Type _} {s : Set α} {T : Set (S
   · rw [preimage_sUnion, inter_Union₂]
     exact isOpen_biUnion hU'
   · exact hs
-#align continuous_on_open_of_generate_from continuousOn_open_of_generateFromₓ
+#align continuous_on_open_of_generate_from continuousOn_isOpen_of_generateFromₓ
 
 #print ContinuousWithinAt.prod /-
 theorem ContinuousWithinAt.prod {f : α → β} {g : α → γ} {s : Set α} {x : α}

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

@@ -1393,14 +1393,14 @@ theorem ContinuousOn.isOpen_preimage {f : α → β} {s : Set α} {t : Set β} (
 #align continuous_on.is_open_preimage ContinuousOn.isOpen_preimage
 -/
 
-#print ContinuousOn.preimage_closed_of_closed /-
-theorem ContinuousOn.preimage_closed_of_closed {f : α → β} {s : Set α} {t : Set β}
+#print ContinuousOn.preimage_isClosed_of_isClosed /-
+theorem ContinuousOn.preimage_isClosed_of_isClosed {f : α → β} {s : Set α} {t : Set β}
     (hf : ContinuousOn f s) (hs : IsClosed s) (ht : IsClosed t) : IsClosed (s ∩ f ⁻¹' t) :=
   by
   rcases continuousOn_iff_isClosed.1 hf t ht with ⟨u, hu⟩
   rw [inter_comm, hu.2]
   apply IsClosed.inter hu.1 hs
-#align continuous_on.preimage_closed_of_closed ContinuousOn.preimage_closed_of_closed
+#align continuous_on.preimage_closed_of_closed ContinuousOn.preimage_isClosed_of_isClosed
 -/
 
 #print ContinuousOn.preimage_interior_subset_interior_preimage /-

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

@@ -405,8 +405,8 @@ theorem insert_mem_nhdsWithin_insert {a : α} {s t : Set α} (h : t ∈ 𝓝[s]
 
 #print insert_mem_nhds_iff /-
 theorem insert_mem_nhds_iff {a : α} {s : Set α} : insert a s ∈ 𝓝 a ↔ s ∈ 𝓝[≠] a := by
-  simp only [nhdsWithin, mem_inf_principal, mem_compl_iff, mem_singleton_iff, or_iff_not_imp_left,
-    insert_def]
+  simp only [nhdsWithin, mem_inf_principal, mem_compl_iff, mem_singleton_iff,
+    Classical.or_iff_not_imp_left, insert_def]
 #align insert_mem_nhds_iff insert_mem_nhds_iff
 -/
 

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

@@ -3,7 +3,7 @@ Copyright (c) 2019 Reid Barton. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Sébastien Gouëzel
 -/
-import Mathbin.Topology.Constructions
+import Topology.Constructions
 
 #align_import topology.continuous_on from "leanprover-community/mathlib"@"d4f691b9e5f94cfc64639973f3544c95f8d5d494"
 
@@ -443,7 +443,7 @@ theorem nhdsWithin_pi_eq' {ι : Type _} {α : ι → Type _} [∀ i, Topological
 #align nhds_within_pi_eq' nhdsWithin_pi_eq'
 -/
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (i «expr ∉ » I) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:641:2: warning: expanding binder collection (i «expr ∉ » I) -/
 #print nhdsWithin_pi_eq /-
 theorem nhdsWithin_pi_eq {ι : Type _} {α : ι → Type _} [∀ i, TopologicalSpace (α i)] {I : Set ι}
     (hI : I.Finite) (s : ∀ i, Set (α i)) (x : ∀ i, α i) :

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

@@ -1053,7 +1053,7 @@ theorem continuousWithinAt_insert_self {f : α → β} {x : α} {s : Set α} :
 #align continuous_within_at_insert_self continuousWithinAt_insert_self
 -/
 
-alias continuousWithinAt_insert_self ↔ _ ContinuousWithinAt.insert_self
+alias ⟨_, ContinuousWithinAt.insert_self⟩ := continuousWithinAt_insert_self
 #align continuous_within_at.insert_self ContinuousWithinAt.insert_self
 
 #print ContinuousWithinAt.diff_iff /-

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

@@ -2,14 +2,11 @@
 Copyright (c) 2019 Reid Barton. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Sébastien Gouëzel
-
-! This file was ported from Lean 3 source module topology.continuous_on
-! leanprover-community/mathlib commit d4f691b9e5f94cfc64639973f3544c95f8d5d494
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
 -/
 import Mathbin.Topology.Constructions
 
+#align_import topology.continuous_on from "leanprover-community/mathlib"@"d4f691b9e5f94cfc64639973f3544c95f8d5d494"
+
 /-!
 # Neighborhoods and continuity relative to a subset
 
@@ -446,7 +443,7 @@ theorem nhdsWithin_pi_eq' {ι : Type _} {α : ι → Type _} [∀ i, Topological
 #align nhds_within_pi_eq' nhdsWithin_pi_eq'
 -/
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:638:2: warning: expanding binder collection (i «expr ∉ » I) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (i «expr ∉ » I) -/
 #print nhdsWithin_pi_eq /-
 theorem nhdsWithin_pi_eq {ι : Type _} {α : ι → Type _} [∀ i, TopologicalSpace (α i)] {I : Set ι}
     (hI : I.Finite) (s : ∀ i, Set (α i)) (x : ∀ i, α i) :

mathlib commit https://github.com/leanprover-community/mathlib/commit/2fe465deb81bcd7ccafa065bb686888a82f15372

@@ -807,7 +807,7 @@ theorem continuousOn_pi {ι : Type _} {π : ι → Type _} [∀ i, TopologicalSp
 #print ContinuousWithinAt.fin_insertNth /-
 theorem ContinuousWithinAt.fin_insertNth {n} {π : Fin (n + 1) → Type _}
     [∀ i, TopologicalSpace (π i)] (i : Fin (n + 1)) {f : α → π i} {a : α} {s : Set α}
-    (hf : ContinuousWithinAt f s a) {g : α → ∀ j : Fin n, π (i.succAbove j)}
+    (hf : ContinuousWithinAt f s a) {g : α → ∀ j : Fin n, π (i.succAboveEmb j)}
     (hg : ContinuousWithinAt g s a) : ContinuousWithinAt (fun a => i.insertNth (f a) (g a)) s a :=
   hf.fin_insertNth i hg
 #align continuous_within_at.fin_insert_nth ContinuousWithinAt.fin_insertNth
@@ -816,7 +816,7 @@ theorem ContinuousWithinAt.fin_insertNth {n} {π : Fin (n + 1) → Type _}
 #print ContinuousOn.fin_insertNth /-
 theorem ContinuousOn.fin_insertNth {n} {π : Fin (n + 1) → Type _} [∀ i, TopologicalSpace (π i)]
     (i : Fin (n + 1)) {f : α → π i} {s : Set α} (hf : ContinuousOn f s)
-    {g : α → ∀ j : Fin n, π (i.succAbove j)} (hg : ContinuousOn g s) :
+    {g : α → ∀ j : Fin n, π (i.succAboveEmb j)} (hg : ContinuousOn g s) :
     ContinuousOn (fun a => i.insertNth (f a) (g a)) s := fun a ha =>
   (hf a ha).fin_insertNth i (hg a ha)
 #align continuous_on.fin_insert_nth ContinuousOn.fin_insertNth

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

@@ -72,10 +72,12 @@ theorem frequently_nhdsWithin_iff {z : α} {s : Set α} {p : α → Prop} :
 #align frequently_nhds_within_iff frequently_nhdsWithin_iff
 -/
 
+#print mem_closure_ne_iff_frequently_within /-
 theorem mem_closure_ne_iff_frequently_within {z : α} {s : Set α} :
     z ∈ closure (s \ {z}) ↔ ∃ᶠ x in 𝓝[≠] z, x ∈ s := by
   simp [mem_closure_iff_frequently, frequently_nhdsWithin_iff]
 #align mem_closure_ne_iff_frequently_within mem_closure_ne_iff_frequently_within
+-/
 
 #print eventually_nhdsWithin_nhdsWithin /-
 @[simp]
@@ -88,10 +90,12 @@ theorem eventually_nhdsWithin_nhdsWithin {a : α} {s : Set α} {p : α → Prop}
 #align eventually_nhds_within_nhds_within eventually_nhdsWithin_nhdsWithin
 -/
 
+#print nhdsWithin_eq /-
 theorem nhdsWithin_eq (a : α) (s : Set α) :
     𝓝[s] a = ⨅ t ∈ {t : Set α | a ∈ t ∧ IsOpen t}, 𝓟 (t ∩ s) :=
   ((nhds_basis_opens a).inf_principal s).eq_biInf
 #align nhds_within_eq nhdsWithin_eq
+-/
 
 #print nhdsWithin_univ /-
 theorem nhdsWithin_univ (a : α) : 𝓝[Set.univ] a = 𝓝 a := by
@@ -99,37 +103,49 @@ theorem nhdsWithin_univ (a : α) : 𝓝[Set.univ] a = 𝓝 a := by
 #align nhds_within_univ nhdsWithin_univ
 -/
 
+#print nhdsWithin_hasBasis /-
 theorem nhdsWithin_hasBasis {p : β → Prop} {s : β → Set α} {a : α} (h : (𝓝 a).HasBasis p s)
     (t : Set α) : (𝓝[t] a).HasBasis p fun i => s i ∩ t :=
   h.inf_principal t
 #align nhds_within_has_basis nhdsWithin_hasBasis
+-/
 
+#print nhdsWithin_basis_open /-
 theorem nhdsWithin_basis_open (a : α) (t : Set α) :
     (𝓝[t] a).HasBasis (fun u => a ∈ u ∧ IsOpen u) fun u => u ∩ t :=
   nhdsWithin_hasBasis (nhds_basis_opens a) t
 #align nhds_within_basis_open nhdsWithin_basis_open
+-/
 
+#print mem_nhdsWithin /-
 theorem mem_nhdsWithin {t : Set α} {a : α} {s : Set α} :
     t ∈ 𝓝[s] a ↔ ∃ u, IsOpen u ∧ a ∈ u ∧ u ∩ s ⊆ t := by
   simpa only [exists_prop, and_assoc', and_comm'] using (nhdsWithin_basis_open a s).mem_iff
 #align mem_nhds_within mem_nhdsWithin
+-/
 
+#print mem_nhdsWithin_iff_exists_mem_nhds_inter /-
 theorem mem_nhdsWithin_iff_exists_mem_nhds_inter {t : Set α} {a : α} {s : Set α} :
     t ∈ 𝓝[s] a ↔ ∃ u ∈ 𝓝 a, u ∩ s ⊆ t :=
   (nhdsWithin_hasBasis (𝓝 a).basis_sets s).mem_iff
 #align mem_nhds_within_iff_exists_mem_nhds_inter mem_nhdsWithin_iff_exists_mem_nhds_inter
+-/
 
+#print diff_mem_nhdsWithin_compl /-
 theorem diff_mem_nhdsWithin_compl {x : α} {s : Set α} (hs : s ∈ 𝓝 x) (t : Set α) :
     s \ t ∈ 𝓝[tᶜ] x :=
   diff_mem_inf_principal_compl hs t
 #align diff_mem_nhds_within_compl diff_mem_nhdsWithin_compl
+-/
 
+#print diff_mem_nhdsWithin_diff /-
 theorem diff_mem_nhdsWithin_diff {x : α} {s t : Set α} (hs : s ∈ 𝓝[t] x) (t' : Set α) :
     s \ t' ∈ 𝓝[t \ t'] x :=
   by
   rw [nhdsWithin, diff_eq, diff_eq, ← inf_principal, ← inf_assoc]
   exact inter_mem_inf hs (mem_principal_self _)
 #align diff_mem_nhds_within_diff diff_mem_nhdsWithin_diff
+-/
 
 #print nhds_of_nhdsWithin_of_nhds /-
 theorem nhds_of_nhdsWithin_of_nhds {s t : Set α} {a : α} (h1 : s ∈ 𝓝 a) (h2 : t ∈ 𝓝[s] a) :
@@ -147,10 +163,12 @@ theorem mem_nhdsWithin_iff_eventually {s t : Set α} {x : α} :
 #align mem_nhds_within_iff_eventually mem_nhdsWithin_iff_eventually
 -/
 
+#print mem_nhdsWithin_iff_eventuallyEq /-
 theorem mem_nhdsWithin_iff_eventuallyEq {s t : Set α} {x : α} :
     t ∈ 𝓝[s] x ↔ s =ᶠ[𝓝 x] (s ∩ t : Set α) := by
   simp_rw [mem_nhdsWithin_iff_eventually, eventually_eq_set, mem_inter_iff, iff_self_and]
 #align mem_nhds_within_iff_eventually_eq mem_nhdsWithin_iff_eventuallyEq
+-/
 
 #print nhdsWithin_eq_iff_eventuallyEq /-
 theorem nhdsWithin_eq_iff_eventuallyEq {s t : Set α} {x : α} : 𝓝[s] x = 𝓝[t] x ↔ s =ᶠ[𝓝 x] t :=
@@ -158,9 +176,11 @@ theorem nhdsWithin_eq_iff_eventuallyEq {s t : Set α} {x : α} : 𝓝[s] x = 
 #align nhds_within_eq_iff_eventually_eq nhdsWithin_eq_iff_eventuallyEq
 -/
 
+#print nhdsWithin_le_iff /-
 theorem nhdsWithin_le_iff {s t : Set α} {x : α} : 𝓝[s] x ≤ 𝓝[t] x ↔ t ∈ 𝓝[s] x :=
   set_eventuallyLE_iff_inf_principal_le.symm.trans set_eventuallyLE_iff_mem_inf_principal
 #align nhds_within_le_iff nhdsWithin_le_iff
+-/
 
 theorem preimage_nhdsWithin_coinduced' {π : α → β} {s : Set β} {t : Set α} {a : α} (h : a ∈ t)
     (ht : IsOpen t)
@@ -198,17 +218,23 @@ theorem eventually_mem_nhdsWithin {a : α} {s : Set α} : ∀ᶠ x in 𝓝[s] a,
 #align eventually_mem_nhds_within eventually_mem_nhdsWithin
 -/
 
+#print inter_mem_nhdsWithin /-
 theorem inter_mem_nhdsWithin (s : Set α) {t : Set α} {a : α} (h : t ∈ 𝓝 a) : s ∩ t ∈ 𝓝[s] a :=
   inter_mem self_mem_nhdsWithin (mem_inf_of_left h)
 #align inter_mem_nhds_within inter_mem_nhdsWithin
+-/
 
+#print nhdsWithin_mono /-
 theorem nhdsWithin_mono (a : α) {s t : Set α} (h : s ⊆ t) : 𝓝[s] a ≤ 𝓝[t] a :=
   inf_le_inf_left _ (principal_mono.mpr h)
 #align nhds_within_mono nhdsWithin_mono
+-/
 
+#print pure_le_nhdsWithin /-
 theorem pure_le_nhdsWithin {a : α} {s : Set α} (ha : a ∈ s) : pure a ≤ 𝓝[s] a :=
   le_inf (pure_le_nhds a) (le_principal_iff.2 ha)
 #align pure_le_nhds_within pure_le_nhdsWithin
+-/
 
 #print mem_of_mem_nhdsWithin /-
 theorem mem_of_mem_nhdsWithin {a : α} {s t : Set α} (ha : a ∈ s) (ht : t ∈ 𝓝[s] a) : a ∈ t :=
@@ -230,37 +256,51 @@ theorem tendsto_const_nhdsWithin {l : Filter β} {s : Set α} {a : α} (ha : a 
 #align tendsto_const_nhds_within tendsto_const_nhdsWithin
 -/
 
+#print nhdsWithin_restrict'' /-
 theorem nhdsWithin_restrict'' {a : α} (s : Set α) {t : Set α} (h : t ∈ 𝓝[s] a) :
     𝓝[s] a = 𝓝[s ∩ t] a :=
   le_antisymm (le_inf inf_le_left (le_principal_iff.mpr (inter_mem self_mem_nhdsWithin h)))
     (inf_le_inf_left _ (principal_mono.mpr (Set.inter_subset_left _ _)))
 #align nhds_within_restrict'' nhdsWithin_restrict''
+-/
 
+#print nhdsWithin_restrict' /-
 theorem nhdsWithin_restrict' {a : α} (s : Set α) {t : Set α} (h : t ∈ 𝓝 a) : 𝓝[s] a = 𝓝[s ∩ t] a :=
   nhdsWithin_restrict'' s <| mem_inf_of_left h
 #align nhds_within_restrict' nhdsWithin_restrict'
+-/
 
+#print nhdsWithin_restrict /-
 theorem nhdsWithin_restrict {a : α} (s : Set α) {t : Set α} (h₀ : a ∈ t) (h₁ : IsOpen t) :
     𝓝[s] a = 𝓝[s ∩ t] a :=
   nhdsWithin_restrict' s (IsOpen.mem_nhds h₁ h₀)
 #align nhds_within_restrict nhdsWithin_restrict
+-/
 
+#print nhdsWithin_le_of_mem /-
 theorem nhdsWithin_le_of_mem {a : α} {s t : Set α} (h : s ∈ 𝓝[t] a) : 𝓝[t] a ≤ 𝓝[s] a :=
   nhdsWithin_le_iff.mpr h
 #align nhds_within_le_of_mem nhdsWithin_le_of_mem
+-/
 
+#print nhdsWithin_le_nhds /-
 theorem nhdsWithin_le_nhds {a : α} {s : Set α} : 𝓝[s] a ≤ 𝓝 a := by rw [← nhdsWithin_univ];
   apply nhdsWithin_le_of_mem; exact univ_mem
 #align nhds_within_le_nhds nhdsWithin_le_nhds
+-/
 
+#print nhdsWithin_eq_nhdsWithin' /-
 theorem nhdsWithin_eq_nhdsWithin' {a : α} {s t u : Set α} (hs : s ∈ 𝓝 a) (h₂ : t ∩ s = u ∩ s) :
     𝓝[t] a = 𝓝[u] a := by rw [nhdsWithin_restrict' t hs, nhdsWithin_restrict' u hs, h₂]
 #align nhds_within_eq_nhds_within' nhdsWithin_eq_nhdsWithin'
+-/
 
+#print nhdsWithin_eq_nhdsWithin /-
 theorem nhdsWithin_eq_nhdsWithin {a : α} {s t u : Set α} (h₀ : a ∈ s) (h₁ : IsOpen s)
     (h₂ : t ∩ s = u ∩ s) : 𝓝[t] a = 𝓝[u] a := by
   rw [nhdsWithin_restrict t h₀ h₁, nhdsWithin_restrict u h₀ h₁, h₂]
 #align nhds_within_eq_nhds_within nhdsWithin_eq_nhdsWithin
+-/
 
 #print nhdsWithin_eq_nhds /-
 @[simp]
@@ -284,43 +324,61 @@ theorem preimage_nhds_within_coinduced {π : α → β} {s : Set β} {t : Set α
 #align preimage_nhds_within_coinduced preimage_nhds_within_coinduced
 -/
 
+#print nhdsWithin_empty /-
 @[simp]
 theorem nhdsWithin_empty (a : α) : 𝓝[∅] a = ⊥ := by rw [nhdsWithin, principal_empty, inf_bot_eq]
 #align nhds_within_empty nhdsWithin_empty
+-/
 
+#print nhdsWithin_union /-
 theorem nhdsWithin_union (a : α) (s t : Set α) : 𝓝[s ∪ t] a = 𝓝[s] a ⊔ 𝓝[t] a := by
   delta nhdsWithin; rw [← inf_sup_left, sup_principal]
 #align nhds_within_union nhdsWithin_union
+-/
 
+#print nhdsWithin_biUnion /-
 theorem nhdsWithin_biUnion {ι} {I : Set ι} (hI : I.Finite) (s : ι → Set α) (a : α) :
     𝓝[⋃ i ∈ I, s i] a = ⨆ i ∈ I, 𝓝[s i] a :=
   Set.Finite.induction_on hI (by simp) fun t T _ _ hT => by
     simp only [hT, nhdsWithin_union, iSup_insert, bUnion_insert]
 #align nhds_within_bUnion nhdsWithin_biUnion
+-/
 
+#print nhdsWithin_sUnion /-
 theorem nhdsWithin_sUnion {S : Set (Set α)} (hS : S.Finite) (a : α) : 𝓝[⋃₀ S] a = ⨆ s ∈ S, 𝓝[s] a :=
   by rw [sUnion_eq_bUnion, nhdsWithin_biUnion hS]
 #align nhds_within_sUnion nhdsWithin_sUnion
+-/
 
+#print nhdsWithin_iUnion /-
 theorem nhdsWithin_iUnion {ι} [Finite ι] (s : ι → Set α) (a : α) : 𝓝[⋃ i, s i] a = ⨆ i, 𝓝[s i] a :=
   by rw [← sUnion_range, nhdsWithin_sUnion (finite_range s), iSup_range]
 #align nhds_within_Union nhdsWithin_iUnion
+-/
 
+#print nhdsWithin_inter /-
 theorem nhdsWithin_inter (a : α) (s t : Set α) : 𝓝[s ∩ t] a = 𝓝[s] a ⊓ 𝓝[t] a := by
   delta nhdsWithin; rw [inf_left_comm, inf_assoc, inf_principal, ← inf_assoc, inf_idem]
 #align nhds_within_inter nhdsWithin_inter
+-/
 
+#print nhdsWithin_inter' /-
 theorem nhdsWithin_inter' (a : α) (s t : Set α) : 𝓝[s ∩ t] a = 𝓝[s] a ⊓ 𝓟 t := by delta nhdsWithin;
   rw [← inf_principal, inf_assoc]
 #align nhds_within_inter' nhdsWithin_inter'
+-/
 
+#print nhdsWithin_inter_of_mem /-
 theorem nhdsWithin_inter_of_mem {a : α} {s t : Set α} (h : s ∈ 𝓝[t] a) : 𝓝[s ∩ t] a = 𝓝[t] a := by
   rw [nhdsWithin_inter, inf_eq_right]; exact nhdsWithin_le_of_mem h
 #align nhds_within_inter_of_mem nhdsWithin_inter_of_mem
+-/
 
+#print nhdsWithin_inter_of_mem' /-
 theorem nhdsWithin_inter_of_mem' {a : α} {s t : Set α} (h : s ∈ 𝓝[t] a) : 𝓝[t ∩ s] a = 𝓝[t] a := by
   rw [inter_comm, nhdsWithin_inter_of_mem h]
 #align nhds_within_inter_of_mem' nhdsWithin_inter_of_mem'
+-/
 
 #print nhdsWithin_singleton /-
 @[simp]
@@ -329,10 +387,12 @@ theorem nhdsWithin_singleton (a : α) : 𝓝[{a}] a = pure a := by
 #align nhds_within_singleton nhdsWithin_singleton
 -/
 
+#print nhdsWithin_insert /-
 @[simp]
 theorem nhdsWithin_insert (a : α) (s : Set α) : 𝓝[insert a s] a = pure a ⊔ 𝓝[s] a := by
   rw [← singleton_union, nhdsWithin_union, nhdsWithin_singleton]
 #align nhds_within_insert nhdsWithin_insert
+-/
 
 #print mem_nhdsWithin_insert /-
 theorem mem_nhdsWithin_insert {a : α} {s t : Set α} : t ∈ 𝓝[insert a s] a ↔ a ∈ t ∧ t ∈ 𝓝[s] a := by
@@ -346,37 +406,48 @@ theorem insert_mem_nhdsWithin_insert {a : α} {s t : Set α} (h : t ∈ 𝓝[s]
 #align insert_mem_nhds_within_insert insert_mem_nhdsWithin_insert
 -/
 
+#print insert_mem_nhds_iff /-
 theorem insert_mem_nhds_iff {a : α} {s : Set α} : insert a s ∈ 𝓝 a ↔ s ∈ 𝓝[≠] a := by
   simp only [nhdsWithin, mem_inf_principal, mem_compl_iff, mem_singleton_iff, or_iff_not_imp_left,
     insert_def]
 #align insert_mem_nhds_iff insert_mem_nhds_iff
+-/
 
+#print nhdsWithin_compl_singleton_sup_pure /-
 @[simp]
 theorem nhdsWithin_compl_singleton_sup_pure (a : α) : 𝓝[≠] a ⊔ pure a = 𝓝 a := by
   rw [← nhdsWithin_singleton, ← nhdsWithin_union, compl_union_self, nhdsWithin_univ]
 #align nhds_within_compl_singleton_sup_pure nhdsWithin_compl_singleton_sup_pure
+-/
 
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
+#print nhdsWithin_prod_eq /-
 theorem nhdsWithin_prod_eq {α : Type _} [TopologicalSpace α] {β : Type _} [TopologicalSpace β]
     (a : α) (b : β) (s : Set α) (t : Set β) : 𝓝[s ×ˢ t] (a, b) = 𝓝[s] a ×ᶠ 𝓝[t] b := by
   delta nhdsWithin; rw [nhds_prod_eq, ← Filter.prod_inf_prod, Filter.prod_principal_principal]
 #align nhds_within_prod_eq nhdsWithin_prod_eq
+-/
 
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
+#print nhdsWithin_prod /-
 theorem nhdsWithin_prod {α : Type _} [TopologicalSpace α] {β : Type _} [TopologicalSpace β]
     {s u : Set α} {t v : Set β} {a : α} {b : β} (hu : u ∈ 𝓝[s] a) (hv : v ∈ 𝓝[t] b) :
     u ×ˢ v ∈ 𝓝[s ×ˢ t] (a, b) := by rw [nhdsWithin_prod_eq]; exact prod_mem_prod hu hv
 #align nhds_within_prod nhdsWithin_prod
+-/
 
+#print nhdsWithin_pi_eq' /-
 theorem nhdsWithin_pi_eq' {ι : Type _} {α : ι → Type _} [∀ i, TopologicalSpace (α i)] {I : Set ι}
     (hI : I.Finite) (s : ∀ i, Set (α i)) (x : ∀ i, α i) :
     𝓝[pi I s] x = ⨅ i, comap (fun x => x i) (𝓝 (x i) ⊓ ⨅ hi : i ∈ I, 𝓟 (s i)) := by
   simp only [nhdsWithin, nhds_pi, Filter.pi, comap_inf, comap_infi, pi_def, comap_principal, ←
     infi_principal_finite hI, ← iInf_inf_eq]
 #align nhds_within_pi_eq' nhdsWithin_pi_eq'
+-/
 
 /- ./././Mathport/Syntax/Translate/Basic.lean:638:2: warning: expanding binder collection (i «expr ∉ » I) -/
+#print nhdsWithin_pi_eq /-
 theorem nhdsWithin_pi_eq {ι : Type _} {α : ι → Type _} [∀ i, TopologicalSpace (α i)] {I : Set ι}
     (hI : I.Finite) (s : ∀ i, Set (α i)) (x : ∀ i, α i) :
     𝓝[pi I s] x =
@@ -388,44 +459,59 @@ theorem nhdsWithin_pi_eq {ι : Type _} {α : ι → Type _} [∀ i, TopologicalS
   rw [iInf_split _ fun i => i ∈ I, inf_right_comm]
   simp only [iInf_inf_eq]
 #align nhds_within_pi_eq nhdsWithin_pi_eq
+-/
 
+#print nhdsWithin_pi_univ_eq /-
 theorem nhdsWithin_pi_univ_eq {ι : Type _} {α : ι → Type _} [Finite ι] [∀ i, TopologicalSpace (α i)]
     (s : ∀ i, Set (α i)) (x : ∀ i, α i) : 𝓝[pi univ s] x = ⨅ i, comap (fun x => x i) (𝓝[s i] x i) :=
   by simpa [nhdsWithin] using nhdsWithin_pi_eq finite_univ s x
 #align nhds_within_pi_univ_eq nhdsWithin_pi_univ_eq
+-/
 
+#print nhdsWithin_pi_eq_bot /-
 theorem nhdsWithin_pi_eq_bot {ι : Type _} {α : ι → Type _} [∀ i, TopologicalSpace (α i)] {I : Set ι}
     {s : ∀ i, Set (α i)} {x : ∀ i, α i} : 𝓝[pi I s] x = ⊥ ↔ ∃ i ∈ I, 𝓝[s i] x i = ⊥ := by
   simp only [nhdsWithin, nhds_pi, pi_inf_principal_pi_eq_bot]
 #align nhds_within_pi_eq_bot nhdsWithin_pi_eq_bot
+-/
 
+#print nhdsWithin_pi_neBot /-
 theorem nhdsWithin_pi_neBot {ι : Type _} {α : ι → Type _} [∀ i, TopologicalSpace (α i)] {I : Set ι}
     {s : ∀ i, Set (α i)} {x : ∀ i, α i} : (𝓝[pi I s] x).ne_bot ↔ ∀ i ∈ I, (𝓝[s i] x i).ne_bot := by
   simp [ne_bot_iff, nhdsWithin_pi_eq_bot]
 #align nhds_within_pi_ne_bot nhdsWithin_pi_neBot
+-/
 
+#print Filter.Tendsto.piecewise_nhdsWithin /-
 theorem Filter.Tendsto.piecewise_nhdsWithin {f g : α → β} {t : Set α} [∀ x, Decidable (x ∈ t)]
     {a : α} {s : Set α} {l : Filter β} (h₀ : Tendsto f (𝓝[s ∩ t] a) l)
     (h₁ : Tendsto g (𝓝[s ∩ tᶜ] a) l) : Tendsto (piecewise t f g) (𝓝[s] a) l := by
   apply tendsto.piecewise <;> rwa [← nhdsWithin_inter']
 #align filter.tendsto.piecewise_nhds_within Filter.Tendsto.piecewise_nhdsWithin
+-/
 
+#print Filter.Tendsto.if_nhdsWithin /-
 theorem Filter.Tendsto.if_nhdsWithin {f g : α → β} {p : α → Prop} [DecidablePred p] {a : α}
     {s : Set α} {l : Filter β} (h₀ : Tendsto f (𝓝[s ∩ {x | p x}] a) l)
     (h₁ : Tendsto g (𝓝[s ∩ {x | ¬p x}] a) l) :
     Tendsto (fun x => if p x then f x else g x) (𝓝[s] a) l :=
   h₀.piecewise_nhdsWithin h₁
 #align filter.tendsto.if_nhds_within Filter.Tendsto.if_nhdsWithin
+-/
 
+#print map_nhdsWithin /-
 theorem map_nhdsWithin (f : α → β) (a : α) (s : Set α) :
     map f (𝓝[s] a) = ⨅ t ∈ {t : Set α | a ∈ t ∧ IsOpen t}, 𝓟 (f '' (t ∩ s)) :=
   ((nhdsWithin_basis_open a s).map f).eq_biInf
 #align map_nhds_within map_nhdsWithin
+-/
 
+#print tendsto_nhdsWithin_mono_left /-
 theorem tendsto_nhdsWithin_mono_left {f : α → β} {a : α} {s t : Set α} {l : Filter β} (hst : s ⊆ t)
     (h : Tendsto f (𝓝[t] a) l) : Tendsto f (𝓝[s] a) l :=
   h.mono_left <| nhdsWithin_mono a hst
 #align tendsto_nhds_within_mono_left tendsto_nhdsWithin_mono_left
+-/
 
 #print tendsto_nhdsWithin_mono_right /-
 theorem tendsto_nhdsWithin_mono_right {f : β → α} {l : Filter β} {a : α} {s t : Set α} (hst : s ⊆ t)
@@ -434,11 +520,14 @@ theorem tendsto_nhdsWithin_mono_right {f : β → α} {l : Filter β} {a : α} {
 #align tendsto_nhds_within_mono_right tendsto_nhdsWithin_mono_right
 -/
 
+#print tendsto_nhdsWithin_of_tendsto_nhds /-
 theorem tendsto_nhdsWithin_of_tendsto_nhds {f : α → β} {a : α} {s : Set α} {l : Filter β}
     (h : Tendsto f (𝓝 a) l) : Tendsto f (𝓝[s] a) l :=
   h.mono_left inf_le_left
 #align tendsto_nhds_within_of_tendsto_nhds tendsto_nhdsWithin_of_tendsto_nhds
+-/
 
+#print eventually_mem_of_tendsto_nhdsWithin /-
 theorem eventually_mem_of_tendsto_nhdsWithin {f : β → α} {a : α} {s : Set α} {l : Filter β}
     (h : Tendsto f l (𝓝[s] a)) : ∀ᶠ i in l, f i ∈ s :=
   by
@@ -446,11 +535,14 @@ theorem eventually_mem_of_tendsto_nhdsWithin {f : β → α} {a : α} {s : Set 
     eventually_and] at h 
   exact (h univ ⟨mem_univ a, isOpen_univ⟩).2
 #align eventually_mem_of_tendsto_nhds_within eventually_mem_of_tendsto_nhdsWithin
+-/
 
+#print tendsto_nhds_of_tendsto_nhdsWithin /-
 theorem tendsto_nhds_of_tendsto_nhdsWithin {f : β → α} {a : α} {s : Set α} {l : Filter β}
     (h : Tendsto f l (𝓝[s] a)) : Tendsto f l (𝓝 a) :=
   h.mono_right nhdsWithin_le_nhds
 #align tendsto_nhds_of_tendsto_nhds_within tendsto_nhds_of_tendsto_nhdsWithin
+-/
 
 #print principal_subtype /-
 theorem principal_subtype {α : Type _} (s : Set α) (t : Set { x // x ∈ s }) :
@@ -479,41 +571,55 @@ theorem DenseRange.nhdsWithin_neBot {ι : Type _} {f : ι → α} (h : DenseRang
 #align dense_range.nhds_within_ne_bot DenseRange.nhdsWithin_neBot
 -/
 
+#print mem_closure_pi /-
 theorem mem_closure_pi {ι : Type _} {α : ι → Type _} [∀ i, TopologicalSpace (α i)] {I : Set ι}
     {s : ∀ i, Set (α i)} {x : ∀ i, α i} : x ∈ closure (pi I s) ↔ ∀ i ∈ I, x i ∈ closure (s i) := by
   simp only [mem_closure_iff_nhdsWithin_neBot, nhdsWithin_pi_neBot]
 #align mem_closure_pi mem_closure_pi
+-/
 
+#print closure_pi_set /-
 theorem closure_pi_set {ι : Type _} {α : ι → Type _} [∀ i, TopologicalSpace (α i)] (I : Set ι)
     (s : ∀ i, Set (α i)) : closure (pi I s) = pi I fun i => closure (s i) :=
   Set.ext fun x => mem_closure_pi
 #align closure_pi_set closure_pi_set
+-/
 
+#print dense_pi /-
 theorem dense_pi {ι : Type _} {α : ι → Type _} [∀ i, TopologicalSpace (α i)] {s : ∀ i, Set (α i)}
     (I : Set ι) (hs : ∀ i ∈ I, Dense (s i)) : Dense (pi I s) := by
   simp only [dense_iff_closure_eq, closure_pi_set, pi_congr rfl fun i hi => (hs i hi).closure_eq,
     pi_univ]
 #align dense_pi dense_pi
+-/
 
+#print eventuallyEq_nhdsWithin_iff /-
 theorem eventuallyEq_nhdsWithin_iff {f g : α → β} {s : Set α} {a : α} :
     f =ᶠ[𝓝[s] a] g ↔ ∀ᶠ x in 𝓝 a, x ∈ s → f x = g x :=
   mem_inf_principal
 #align eventually_eq_nhds_within_iff eventuallyEq_nhdsWithin_iff
+-/
 
+#print eventuallyEq_nhdsWithin_of_eqOn /-
 theorem eventuallyEq_nhdsWithin_of_eqOn {f g : α → β} {s : Set α} {a : α} (h : EqOn f g s) :
     f =ᶠ[𝓝[s] a] g :=
   mem_inf_of_right h
 #align eventually_eq_nhds_within_of_eq_on eventuallyEq_nhdsWithin_of_eqOn
+-/
 
+#print Set.EqOn.eventuallyEq_nhdsWithin /-
 theorem Set.EqOn.eventuallyEq_nhdsWithin {f g : α → β} {s : Set α} {a : α} (h : EqOn f g s) :
     f =ᶠ[𝓝[s] a] g :=
   eventuallyEq_nhdsWithin_of_eqOn h
 #align set.eq_on.eventually_eq_nhds_within Set.EqOn.eventuallyEq_nhdsWithin
+-/
 
+#print tendsto_nhdsWithin_congr /-
 theorem tendsto_nhdsWithin_congr {f g : α → β} {s : Set α} {a : α} {l : Filter β}
     (hfg : ∀ x ∈ s, f x = g x) (hf : Tendsto f (𝓝[s] a) l) : Tendsto g (𝓝[s] a) l :=
   (tendsto_congr' <| eventuallyEq_nhdsWithin_of_eqOn hfg).1 hf
 #align tendsto_nhds_within_congr tendsto_nhdsWithin_congr
+-/
 
 #print eventually_nhdsWithin_of_forall /-
 theorem eventually_nhdsWithin_of_forall {s : Set α} {a : α} {p : α → Prop} (h : ∀ x ∈ s, p x) :
@@ -546,10 +652,12 @@ theorem tendsto_nhdsWithin_range {a : α} {l : Filter β} {f : β → α} :
 #align tendsto_nhds_within_range tendsto_nhdsWithin_range
 -/
 
+#print Filter.EventuallyEq.eq_of_nhdsWithin /-
 theorem Filter.EventuallyEq.eq_of_nhdsWithin {s : Set α} {f g : α → β} {a : α} (h : f =ᶠ[𝓝[s] a] g)
     (hmem : a ∈ s) : f a = g a :=
   h.self_of_nhdsWithin hmem
 #align filter.eventually_eq.eq_of_nhds_within Filter.EventuallyEq.eq_of_nhdsWithin
+-/
 
 #print eventually_nhdsWithin_of_eventually_nhds /-
 theorem eventually_nhdsWithin_of_eventually_nhds {α : Type _} [TopologicalSpace α] {s : Set α}
@@ -599,10 +707,12 @@ theorem preimage_coe_mem_nhds_subtype {s t : Set α} {a : s} : coe ⁻¹' t ∈
 #align preimage_coe_mem_nhds_subtype preimage_coe_mem_nhds_subtype
 -/
 
+#print tendsto_nhdsWithin_iff_subtype /-
 theorem tendsto_nhdsWithin_iff_subtype {s : Set α} {a : α} (h : a ∈ s) (f : α → β) (l : Filter β) :
     Tendsto f (𝓝[s] a) l ↔ Tendsto (s.restrict f) (𝓝 ⟨a, h⟩) l := by
   simp only [tendsto, nhdsWithin_eq_map_subtype_coe h, Filter.map_map, restrict]
 #align tendsto_nhds_within_iff_subtype tendsto_nhdsWithin_iff_subtype
+-/
 
 variable [TopologicalSpace β] [TopologicalSpace γ] [TopologicalSpace δ]
 
@@ -614,6 +724,7 @@ def ContinuousWithinAt (f : α → β) (s : Set α) (x : α) : Prop :=
 #align continuous_within_at ContinuousWithinAt
 -/
 
+#print ContinuousWithinAt.tendsto /-
 /-- If a function is continuous within `s` at `x`, then it tends to `f x` within `s` by definition.
 We register this fact for use with the dot notation, especially to use `tendsto.comp` as
 `continuous_within_at.comp` will have a different meaning. -/
@@ -621,6 +732,7 @@ theorem ContinuousWithinAt.tendsto {f : α → β} {s : Set α} {x : α} (h : Co
     Tendsto f (𝓝[s] x) (𝓝 (f x)) :=
   h
 #align continuous_within_at.tendsto ContinuousWithinAt.tendsto
+-/
 
 #print ContinuousOn /-
 /-- A function between topological spaces is continuous on a subset `s`
@@ -630,32 +742,43 @@ def ContinuousOn (f : α → β) (s : Set α) : Prop :=
 #align continuous_on ContinuousOn
 -/
 
+#print ContinuousOn.continuousWithinAt /-
 theorem ContinuousOn.continuousWithinAt {f : α → β} {s : Set α} {x : α} (hf : ContinuousOn f s)
     (hx : x ∈ s) : ContinuousWithinAt f s x :=
   hf x hx
 #align continuous_on.continuous_within_at ContinuousOn.continuousWithinAt
+-/
 
+#print continuousWithinAt_univ /-
 theorem continuousWithinAt_univ (f : α → β) (x : α) :
     ContinuousWithinAt f Set.univ x ↔ ContinuousAt f x := by
   rw [ContinuousAt, ContinuousWithinAt, nhdsWithin_univ]
 #align continuous_within_at_univ continuousWithinAt_univ
+-/
 
+#print continuousWithinAt_iff_continuousAt_restrict /-
 theorem continuousWithinAt_iff_continuousAt_restrict (f : α → β) {x : α} {s : Set α} (h : x ∈ s) :
     ContinuousWithinAt f s x ↔ ContinuousAt (s.restrict f) ⟨x, h⟩ :=
   tendsto_nhdsWithin_iff_subtype h f _
 #align continuous_within_at_iff_continuous_at_restrict continuousWithinAt_iff_continuousAt_restrict
+-/
 
+#print ContinuousWithinAt.tendsto_nhdsWithin /-
 theorem ContinuousWithinAt.tendsto_nhdsWithin {f : α → β} {x : α} {s : Set α} {t : Set β}
     (h : ContinuousWithinAt f s x) (ht : MapsTo f s t) : Tendsto f (𝓝[s] x) (𝓝[t] f x) :=
   tendsto_inf.2 ⟨h, tendsto_principal.2 <| mem_inf_of_right <| mem_principal.2 <| ht⟩
 #align continuous_within_at.tendsto_nhds_within ContinuousWithinAt.tendsto_nhdsWithin
+-/
 
+#print ContinuousWithinAt.tendsto_nhdsWithin_image /-
 theorem ContinuousWithinAt.tendsto_nhdsWithin_image {f : α → β} {x : α} {s : Set α}
     (h : ContinuousWithinAt f s x) : Tendsto f (𝓝[s] x) (𝓝[f '' s] f x) :=
   h.tendsto_nhdsWithin (mapsTo_image _ _)
 #align continuous_within_at.tendsto_nhds_within_image ContinuousWithinAt.tendsto_nhdsWithin_image
+-/
 
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
+#print ContinuousWithinAt.prod_map /-
 theorem ContinuousWithinAt.prod_map {f : α → γ} {g : β → δ} {s : Set α} {t : Set β} {x : α} {y : β}
     (hf : ContinuousWithinAt f s x) (hg : ContinuousWithinAt g t y) :
     ContinuousWithinAt (Prod.map f g) (s ×ˢ t) (x, y) :=
@@ -664,38 +787,50 @@ theorem ContinuousWithinAt.prod_map {f : α → γ} {g : β → δ} {s : Set α}
   rw [nhdsWithin_prod_eq, Prod.map, nhds_prod_eq]
   exact hf.prod_map hg
 #align continuous_within_at.prod_map ContinuousWithinAt.prod_map
+-/
 
+#print continuousWithinAt_pi /-
 theorem continuousWithinAt_pi {ι : Type _} {π : ι → Type _} [∀ i, TopologicalSpace (π i)]
     {f : α → ∀ i, π i} {s : Set α} {x : α} :
     ContinuousWithinAt f s x ↔ ∀ i, ContinuousWithinAt (fun y => f y i) s x :=
   tendsto_pi_nhds
 #align continuous_within_at_pi continuousWithinAt_pi
+-/
 
+#print continuousOn_pi /-
 theorem continuousOn_pi {ι : Type _} {π : ι → Type _} [∀ i, TopologicalSpace (π i)]
     {f : α → ∀ i, π i} {s : Set α} : ContinuousOn f s ↔ ∀ i, ContinuousOn (fun y => f y i) s :=
   ⟨fun h i x hx => tendsto_pi_nhds.1 (h x hx) i, fun h x hx => tendsto_pi_nhds.2 fun i => h i x hx⟩
 #align continuous_on_pi continuousOn_pi
+-/
 
+#print ContinuousWithinAt.fin_insertNth /-
 theorem ContinuousWithinAt.fin_insertNth {n} {π : Fin (n + 1) → Type _}
     [∀ i, TopologicalSpace (π i)] (i : Fin (n + 1)) {f : α → π i} {a : α} {s : Set α}
     (hf : ContinuousWithinAt f s a) {g : α → ∀ j : Fin n, π (i.succAbove j)}
     (hg : ContinuousWithinAt g s a) : ContinuousWithinAt (fun a => i.insertNth (f a) (g a)) s a :=
   hf.fin_insertNth i hg
 #align continuous_within_at.fin_insert_nth ContinuousWithinAt.fin_insertNth
+-/
 
+#print ContinuousOn.fin_insertNth /-
 theorem ContinuousOn.fin_insertNth {n} {π : Fin (n + 1) → Type _} [∀ i, TopologicalSpace (π i)]
     (i : Fin (n + 1)) {f : α → π i} {s : Set α} (hf : ContinuousOn f s)
     {g : α → ∀ j : Fin n, π (i.succAbove j)} (hg : ContinuousOn g s) :
     ContinuousOn (fun a => i.insertNth (f a) (g a)) s := fun a ha =>
   (hf a ha).fin_insertNth i (hg a ha)
 #align continuous_on.fin_insert_nth ContinuousOn.fin_insertNth
+-/
 
+#print continuousOn_iff /-
 theorem continuousOn_iff {f : α → β} {s : Set α} :
     ContinuousOn f s ↔
       ∀ x ∈ s, ∀ t : Set β, IsOpen t → f x ∈ t → ∃ u, IsOpen u ∧ x ∈ u ∧ u ∩ s ⊆ f ⁻¹' t :=
   by simp only [ContinuousOn, ContinuousWithinAt, tendsto_nhds, mem_nhdsWithin]
 #align continuous_on_iff continuousOn_iff
+-/
 
+#print continuousOn_iff_continuous_restrict /-
 theorem continuousOn_iff_continuous_restrict {f : α → β} {s : Set α} :
     ContinuousOn f s ↔ Continuous (s.restrict f) :=
   by
@@ -705,7 +840,9 @@ theorem continuousOn_iff_continuous_restrict {f : α → β} {s : Set α} :
   intro h x xs
   exact (continuousWithinAt_iff_continuousAt_restrict f xs).mpr (h ⟨x, xs⟩)
 #align continuous_on_iff_continuous_restrict continuousOn_iff_continuous_restrict
+-/
 
+#print continuousOn_iff' /-
 theorem continuousOn_iff' {f : α → β} {s : Set α} :
     ContinuousOn f s ↔ ∀ t : Set β, IsOpen t → ∃ u, IsOpen u ∧ f ⁻¹' t ∩ s = u ∩ s :=
   by
@@ -717,7 +854,9 @@ theorem continuousOn_iff' {f : α → β} {s : Set α} :
     constructor <;> · rintro ⟨u, ou, useq⟩; exact ⟨u, ou, useq.symm⟩
   rw [continuousOn_iff_continuous_restrict, continuous_def] <;> simp only [this]
 #align continuous_on_iff' continuousOn_iff'
+-/
 
+#print ContinuousOn.mono_dom /-
 /-- If a function is continuous on a set for some topologies, then it is
 continuous on the same set with respect to any finer topology on the source space. -/
 theorem ContinuousOn.mono_dom {α β : Type _} {t₁ t₂ : TopologicalSpace α} {t₃ : TopologicalSpace β}
@@ -729,7 +868,9 @@ theorem ContinuousOn.mono_dom {α β : Type _} {t₁ t₂ : TopologicalSpace α}
   rcases h₂ t ht with ⟨u, hu, h'u⟩
   exact ⟨u, h₁ u hu, h'u⟩
 #align continuous_on.mono_dom ContinuousOn.mono_dom
+-/
 
+#print ContinuousOn.mono_rng /-
 /-- If a function is continuous on a set for some topologies, then it is
 continuous on the same set with respect to any coarser topology on the target space. -/
 theorem ContinuousOn.mono_rng {α β : Type _} {t₁ : TopologicalSpace α} {t₂ t₃ : TopologicalSpace β}
@@ -740,7 +881,9 @@ theorem ContinuousOn.mono_rng {α β : Type _} {t₁ : TopologicalSpace α} {t
   intro t ht
   exact h₂ t (h₁ t ht)
 #align continuous_on.mono_rng ContinuousOn.mono_rng
+-/
 
+#print continuousOn_iff_isClosed /-
 theorem continuousOn_iff_isClosed {f : α → β} {s : Set α} :
     ContinuousOn f s ↔ ∀ t : Set β, IsClosed t → ∃ u, IsClosed u ∧ f ⁻¹' t ∩ s = u ∩ s :=
   by
@@ -751,13 +894,17 @@ theorem continuousOn_iff_isClosed {f : α → β} {s : Set α} :
     simp only [Subtype.preimage_coe_eq_preimage_coe_iff, eq_comm]
   rw [continuousOn_iff_continuous_restrict, continuous_iff_isClosed] <;> simp only [this]
 #align continuous_on_iff_is_closed continuousOn_iff_isClosed
+-/
 
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
+#print ContinuousOn.prod_map /-
 theorem ContinuousOn.prod_map {f : α → γ} {g : β → δ} {s : Set α} {t : Set β}
     (hf : ContinuousOn f s) (hg : ContinuousOn g t) : ContinuousOn (Prod.map f g) (s ×ˢ t) :=
   fun ⟨x, y⟩ ⟨hx, hy⟩ => ContinuousWithinAt.prod_map (hf x hx) (hg y hy)
 #align continuous_on.prod_map ContinuousOn.prod_map
+-/
 
+#print continuous_of_cover_nhds /-
 theorem continuous_of_cover_nhds {ι : Sort _} {f : α → β} {s : ι → Set α}
     (hs : ∀ x : α, ∃ i, s i ∈ 𝓝 x) (hf : ∀ i, ContinuousOn f (s i)) : Continuous f :=
   continuous_iff_continuousAt.mpr fun x =>
@@ -765,25 +912,34 @@ theorem continuous_of_cover_nhds {ι : Sort _} {f : α → β} {s : ι → Set 
     let ⟨i, hi⟩ := hs x
     rw [ContinuousAt, ← nhdsWithin_eq_nhds.2 hi]; exact hf _ _ (mem_of_mem_nhds hi)
 #align continuous_of_cover_nhds continuous_of_cover_nhds
+-/
 
+#print continuousOn_empty /-
 theorem continuousOn_empty (f : α → β) : ContinuousOn f ∅ := fun x => False.elim
 #align continuous_on_empty continuousOn_empty
+-/
 
+#print continuousOn_singleton /-
 theorem continuousOn_singleton (f : α → β) (a : α) : ContinuousOn f {a} :=
   forall_eq.2 <| by
     simpa only [ContinuousWithinAt, nhdsWithin_singleton, tendsto_pure_left] using fun s =>
       mem_of_mem_nhds
 #align continuous_on_singleton continuousOn_singleton
+-/
 
+#print Set.Subsingleton.continuousOn /-
 theorem Set.Subsingleton.continuousOn {s : Set α} (hs : s.Subsingleton) (f : α → β) :
     ContinuousOn f s :=
   hs.inductionOn (continuousOn_empty f) (continuousOn_singleton f)
 #align set.subsingleton.continuous_on Set.Subsingleton.continuousOn
+-/
 
+#print nhdsWithin_le_comap /-
 theorem nhdsWithin_le_comap {x : α} {s : Set α} {f : α → β} (ctsf : ContinuousWithinAt f s x) :
     𝓝[s] x ≤ comap f (𝓝[f '' s] f x) :=
   ctsf.tendsto_nhdsWithin_image.le_comap
 #align nhds_within_le_comap nhdsWithin_le_comap
+-/
 
 #print comap_nhdsWithin_range /-
 @[simp]
@@ -792,105 +948,142 @@ theorem comap_nhdsWithin_range {α} (f : α → β) (y : β) : comap f (𝓝[ran
 #align comap_nhds_within_range comap_nhdsWithin_range
 -/
 
+#print continuous_iff_continuousOn_univ /-
 theorem continuous_iff_continuousOn_univ {f : α → β} : Continuous f ↔ ContinuousOn f univ := by
   simp [continuous_iff_continuousAt, ContinuousOn, ContinuousAt, ContinuousWithinAt,
     nhdsWithin_univ]
 #align continuous_iff_continuous_on_univ continuous_iff_continuousOn_univ
+-/
 
+#print ContinuousWithinAt.mono /-
 theorem ContinuousWithinAt.mono {f : α → β} {s t : Set α} {x : α} (h : ContinuousWithinAt f t x)
     (hs : s ⊆ t) : ContinuousWithinAt f s x :=
   h.mono_left (nhdsWithin_mono x hs)
 #align continuous_within_at.mono ContinuousWithinAt.mono
+-/
 
+#print ContinuousWithinAt.mono_of_mem /-
 theorem ContinuousWithinAt.mono_of_mem {f : α → β} {s t : Set α} {x : α}
     (h : ContinuousWithinAt f t x) (hs : t ∈ 𝓝[s] x) : ContinuousWithinAt f s x :=
   h.mono_left (nhdsWithin_le_of_mem hs)
 #align continuous_within_at.mono_of_mem ContinuousWithinAt.mono_of_mem
+-/
 
+#print continuousWithinAt_inter' /-
 theorem continuousWithinAt_inter' {f : α → β} {s t : Set α} {x : α} (h : t ∈ 𝓝[s] x) :
     ContinuousWithinAt f (s ∩ t) x ↔ ContinuousWithinAt f s x := by
   simp [ContinuousWithinAt, nhdsWithin_restrict'' s h]
 #align continuous_within_at_inter' continuousWithinAt_inter'
+-/
 
+#print continuousWithinAt_inter /-
 theorem continuousWithinAt_inter {f : α → β} {s t : Set α} {x : α} (h : t ∈ 𝓝 x) :
     ContinuousWithinAt f (s ∩ t) x ↔ ContinuousWithinAt f s x := by
   simp [ContinuousWithinAt, nhdsWithin_restrict' s h]
 #align continuous_within_at_inter continuousWithinAt_inter
+-/
 
+#print continuousWithinAt_union /-
 theorem continuousWithinAt_union {f : α → β} {s t : Set α} {x : α} :
     ContinuousWithinAt f (s ∪ t) x ↔ ContinuousWithinAt f s x ∧ ContinuousWithinAt f t x := by
   simp only [ContinuousWithinAt, nhdsWithin_union, tendsto_sup]
 #align continuous_within_at_union continuousWithinAt_union
+-/
 
+#print ContinuousWithinAt.union /-
 theorem ContinuousWithinAt.union {f : α → β} {s t : Set α} {x : α} (hs : ContinuousWithinAt f s x)
     (ht : ContinuousWithinAt f t x) : ContinuousWithinAt f (s ∪ t) x :=
   continuousWithinAt_union.2 ⟨hs, ht⟩
 #align continuous_within_at.union ContinuousWithinAt.union
+-/
 
+#print ContinuousWithinAt.mem_closure_image /-
 theorem ContinuousWithinAt.mem_closure_image {f : α → β} {s : Set α} {x : α}
     (h : ContinuousWithinAt f s x) (hx : x ∈ closure s) : f x ∈ closure (f '' s) :=
   haveI := mem_closure_iff_nhdsWithin_neBot.1 hx
   mem_closure_of_tendsto h <| mem_of_superset self_mem_nhdsWithin (subset_preimage_image f s)
 #align continuous_within_at.mem_closure_image ContinuousWithinAt.mem_closure_image
+-/
 
+#print ContinuousWithinAt.mem_closure /-
 theorem ContinuousWithinAt.mem_closure {f : α → β} {s : Set α} {x : α} {A : Set β}
     (h : ContinuousWithinAt f s x) (hx : x ∈ closure s) (hA : MapsTo f s A) : f x ∈ closure A :=
   closure_mono (image_subset_iff.2 hA) (h.mem_closure_image hx)
 #align continuous_within_at.mem_closure ContinuousWithinAt.mem_closure
+-/
 
+#print Set.MapsTo.closure_of_continuousWithinAt /-
 theorem Set.MapsTo.closure_of_continuousWithinAt {f : α → β} {s : Set α} {t : Set β}
     (h : MapsTo f s t) (hc : ∀ x ∈ closure s, ContinuousWithinAt f s x) :
     MapsTo f (closure s) (closure t) := fun x hx => (hc x hx).mem_closure hx h
 #align set.maps_to.closure_of_continuous_within_at Set.MapsTo.closure_of_continuousWithinAt
+-/
 
+#print Set.MapsTo.closure_of_continuousOn /-
 theorem Set.MapsTo.closure_of_continuousOn {f : α → β} {s : Set α} {t : Set β} (h : MapsTo f s t)
     (hc : ContinuousOn f (closure s)) : MapsTo f (closure s) (closure t) :=
   h.closure_of_continuousWithinAt fun x hx => (hc x hx).mono subset_closure
 #align set.maps_to.closure_of_continuous_on Set.MapsTo.closure_of_continuousOn
+-/
 
+#print ContinuousWithinAt.image_closure /-
 theorem ContinuousWithinAt.image_closure {f : α → β} {s : Set α}
     (hf : ∀ x ∈ closure s, ContinuousWithinAt f s x) : f '' closure s ⊆ closure (f '' s) :=
   mapsTo'.1 <| (mapsTo_image f s).closure_of_continuousWithinAt hf
 #align continuous_within_at.image_closure ContinuousWithinAt.image_closure
+-/
 
+#print ContinuousOn.image_closure /-
 theorem ContinuousOn.image_closure {f : α → β} {s : Set α} (hf : ContinuousOn f (closure s)) :
     f '' closure s ⊆ closure (f '' s) :=
   ContinuousWithinAt.image_closure fun x hx => (hf x hx).mono subset_closure
 #align continuous_on.image_closure ContinuousOn.image_closure
+-/
 
+#print continuousWithinAt_singleton /-
 @[simp]
 theorem continuousWithinAt_singleton {f : α → β} {x : α} : ContinuousWithinAt f {x} x := by
   simp only [ContinuousWithinAt, nhdsWithin_singleton, tendsto_pure_nhds]
 #align continuous_within_at_singleton continuousWithinAt_singleton
+-/
 
+#print continuousWithinAt_insert_self /-
 @[simp]
 theorem continuousWithinAt_insert_self {f : α → β} {x : α} {s : Set α} :
     ContinuousWithinAt f (insert x s) x ↔ ContinuousWithinAt f s x := by
   simp only [← singleton_union, continuousWithinAt_union, continuousWithinAt_singleton,
     true_and_iff]
 #align continuous_within_at_insert_self continuousWithinAt_insert_self
+-/
 
 alias continuousWithinAt_insert_self ↔ _ ContinuousWithinAt.insert_self
 #align continuous_within_at.insert_self ContinuousWithinAt.insert_self
 
+#print ContinuousWithinAt.diff_iff /-
 theorem ContinuousWithinAt.diff_iff {f : α → β} {s t : Set α} {x : α}
     (ht : ContinuousWithinAt f t x) : ContinuousWithinAt f (s \ t) x ↔ ContinuousWithinAt f s x :=
   ⟨fun h => (h.union ht).mono <| by simp only [diff_union_self, subset_union_left], fun h =>
     h.mono (diff_subset _ _)⟩
 #align continuous_within_at.diff_iff ContinuousWithinAt.diff_iff
+-/
 
+#print continuousWithinAt_diff_self /-
 @[simp]
 theorem continuousWithinAt_diff_self {f : α → β} {s : Set α} {x : α} :
     ContinuousWithinAt f (s \ {x}) x ↔ ContinuousWithinAt f s x :=
   continuousWithinAt_singleton.diff_iff
 #align continuous_within_at_diff_self continuousWithinAt_diff_self
+-/
 
+#print continuousWithinAt_compl_self /-
 @[simp]
 theorem continuousWithinAt_compl_self {f : α → β} {a : α} :
     ContinuousWithinAt f ({a}ᶜ) a ↔ ContinuousAt f a := by
   rw [compl_eq_univ_diff, continuousWithinAt_diff_self, continuousWithinAt_univ]
 #align continuous_within_at_compl_self continuousWithinAt_compl_self
+-/
 
+#print continuousWithinAt_update_same /-
 @[simp]
 theorem continuousWithinAt_update_same [DecidableEq α] {f : α → β} {s : Set α} {x : α} {y : β} :
     ContinuousWithinAt (update f x y) s x ↔ Tendsto f (𝓝[s \ {x}] x) (𝓝 y) :=
@@ -901,13 +1094,17 @@ theorem continuousWithinAt_update_same [DecidableEq α] {f : α → β} {s : Set
       tendsto_congr' <|
         eventually_nhdsWithin_iff.2 <| eventually_of_forall fun z hz => update_noteq hz.2 _ _
 #align continuous_within_at_update_same continuousWithinAt_update_same
+-/
 
+#print continuousAt_update_same /-
 @[simp]
 theorem continuousAt_update_same [DecidableEq α] {f : α → β} {x : α} {y : β} :
     ContinuousAt (Function.update f x y) x ↔ Tendsto f (𝓝[≠] x) (𝓝 y) := by
   rw [← continuousWithinAt_univ, continuousWithinAt_update_same, compl_eq_univ_diff]
 #align continuous_at_update_same continuousAt_update_same
+-/
 
+#print IsOpenMap.continuousOn_image_of_leftInvOn /-
 theorem IsOpenMap.continuousOn_image_of_leftInvOn {f : α → β} {s : Set α}
     (h : IsOpenMap (s.restrict f)) {finv : β → α} (hleft : LeftInvOn finv f s) :
     ContinuousOn finv (f '' s) :=
@@ -916,14 +1113,18 @@ theorem IsOpenMap.continuousOn_image_of_leftInvOn {f : α → β} {s : Set α}
   · rw [← image_restrict]; exact h _ (ht.preimage continuous_subtype_val)
   · rw [inter_eq_self_of_subset_left (image_subset f (inter_subset_right t s)), hleft.image_inter']
 #align is_open_map.continuous_on_image_of_left_inv_on IsOpenMap.continuousOn_image_of_leftInvOn
+-/
 
+#print IsOpenMap.continuousOn_range_of_leftInverse /-
 theorem IsOpenMap.continuousOn_range_of_leftInverse {f : α → β} (hf : IsOpenMap f) {finv : β → α}
     (hleft : Function.LeftInverse finv f) : ContinuousOn finv (range f) :=
   by
   rw [← image_univ]
   exact (hf.restrict isOpen_univ).continuousOn_image_of_leftInvOn fun x _ => hleft x
 #align is_open_map.continuous_on_range_of_left_inverse IsOpenMap.continuousOn_range_of_leftInverse
+-/
 
+#print ContinuousOn.congr_mono /-
 theorem ContinuousOn.congr_mono {f g : α → β} {s s₁ : Set α} (h : ContinuousOn f s)
     (h' : EqOn g f s₁) (h₁ : s₁ ⊆ s) : ContinuousOn g s₁ :=
   by
@@ -934,108 +1135,149 @@ theorem ContinuousOn.congr_mono {f g : α → β} {s s₁ : Set α} (h : Continu
   rw [← h' hx] at A 
   exact A.congr' h'.eventually_eq_nhds_within.symm
 #align continuous_on.congr_mono ContinuousOn.congr_mono
+-/
 
+#print ContinuousOn.congr /-
 theorem ContinuousOn.congr {f g : α → β} {s : Set α} (h : ContinuousOn f s) (h' : EqOn g f s) :
     ContinuousOn g s :=
   h.congr_mono h' (Subset.refl _)
 #align continuous_on.congr ContinuousOn.congr
+-/
 
+#print continuousOn_congr /-
 theorem continuousOn_congr {f g : α → β} {s : Set α} (h' : EqOn g f s) :
     ContinuousOn g s ↔ ContinuousOn f s :=
   ⟨fun h => ContinuousOn.congr h h'.symm, fun h => h.congr h'⟩
 #align continuous_on_congr continuousOn_congr
+-/
 
+#print ContinuousAt.continuousWithinAt /-
 theorem ContinuousAt.continuousWithinAt {f : α → β} {s : Set α} {x : α} (h : ContinuousAt f x) :
     ContinuousWithinAt f s x :=
   ContinuousWithinAt.mono ((continuousWithinAt_univ f x).2 h) (subset_univ _)
 #align continuous_at.continuous_within_at ContinuousAt.continuousWithinAt
+-/
 
+#print continuousWithinAt_iff_continuousAt /-
 theorem continuousWithinAt_iff_continuousAt {f : α → β} {s : Set α} {x : α} (h : s ∈ 𝓝 x) :
     ContinuousWithinAt f s x ↔ ContinuousAt f x := by
   rw [← univ_inter s, continuousWithinAt_inter h, continuousWithinAt_univ]
 #align continuous_within_at_iff_continuous_at continuousWithinAt_iff_continuousAt
+-/
 
+#print ContinuousWithinAt.continuousAt /-
 theorem ContinuousWithinAt.continuousAt {f : α → β} {s : Set α} {x : α}
     (h : ContinuousWithinAt f s x) (hs : s ∈ 𝓝 x) : ContinuousAt f x :=
   (continuousWithinAt_iff_continuousAt hs).mp h
 #align continuous_within_at.continuous_at ContinuousWithinAt.continuousAt
+-/
 
+#print IsOpen.continuousOn_iff /-
 theorem IsOpen.continuousOn_iff {f : α → β} {s : Set α} (hs : IsOpen s) :
     ContinuousOn f s ↔ ∀ ⦃a⦄, a ∈ s → ContinuousAt f a :=
   ball_congr fun _ => continuousWithinAt_iff_continuousAt ∘ hs.mem_nhds
 #align is_open.continuous_on_iff IsOpen.continuousOn_iff
+-/
 
+#print ContinuousOn.continuousAt /-
 theorem ContinuousOn.continuousAt {f : α → β} {s : Set α} {x : α} (h : ContinuousOn f s)
     (hx : s ∈ 𝓝 x) : ContinuousAt f x :=
   (h x (mem_of_mem_nhds hx)).ContinuousAt hx
 #align continuous_on.continuous_at ContinuousOn.continuousAt
+-/
 
+#print ContinuousAt.continuousOn /-
 theorem ContinuousAt.continuousOn {f : α → β} {s : Set α} (hcont : ∀ x ∈ s, ContinuousAt f x) :
     ContinuousOn f s := fun x hx => (hcont x hx).ContinuousWithinAt
 #align continuous_at.continuous_on ContinuousAt.continuousOn
+-/
 
+#print ContinuousWithinAt.comp /-
 theorem ContinuousWithinAt.comp {g : β → γ} {f : α → β} {s : Set α} {t : Set β} {x : α}
     (hg : ContinuousWithinAt g t (f x)) (hf : ContinuousWithinAt f s x) (h : MapsTo f s t) :
     ContinuousWithinAt (g ∘ f) s x :=
   hg.Tendsto.comp (hf.tendsto_nhdsWithin h)
 #align continuous_within_at.comp ContinuousWithinAt.comp
+-/
 
+#print ContinuousWithinAt.comp' /-
 theorem ContinuousWithinAt.comp' {g : β → γ} {f : α → β} {s : Set α} {t : Set β} {x : α}
     (hg : ContinuousWithinAt g t (f x)) (hf : ContinuousWithinAt f s x) :
     ContinuousWithinAt (g ∘ f) (s ∩ f ⁻¹' t) x :=
   hg.comp (hf.mono (inter_subset_left _ _)) (inter_subset_right _ _)
 #align continuous_within_at.comp' ContinuousWithinAt.comp'
+-/
 
+#print ContinuousAt.comp_continuousWithinAt /-
 theorem ContinuousAt.comp_continuousWithinAt {g : β → γ} {f : α → β} {s : Set α} {x : α}
     (hg : ContinuousAt g (f x)) (hf : ContinuousWithinAt f s x) : ContinuousWithinAt (g ∘ f) s x :=
   hg.ContinuousWithinAt.comp hf (mapsTo_univ _ _)
 #align continuous_at.comp_continuous_within_at ContinuousAt.comp_continuousWithinAt
+-/
 
+#print ContinuousOn.comp /-
 theorem ContinuousOn.comp {g : β → γ} {f : α → β} {s : Set α} {t : Set β} (hg : ContinuousOn g t)
     (hf : ContinuousOn f s) (h : MapsTo f s t) : ContinuousOn (g ∘ f) s := fun x hx =>
   ContinuousWithinAt.comp (hg _ (h hx)) (hf x hx) h
 #align continuous_on.comp ContinuousOn.comp
+-/
 
+#print ContinuousOn.mono /-
 theorem ContinuousOn.mono {f : α → β} {s t : Set α} (hf : ContinuousOn f s) (h : t ⊆ s) :
     ContinuousOn f t := fun x hx => (hf x (h hx)).mono_left (nhdsWithin_mono _ h)
 #align continuous_on.mono ContinuousOn.mono
+-/
 
+#print antitone_continuousOn /-
 theorem antitone_continuousOn {f : α → β} : Antitone (ContinuousOn f) := fun s t hst hf =>
   hf.mono hst
 #align antitone_continuous_on antitone_continuousOn
+-/
 
+#print ContinuousOn.comp' /-
 theorem ContinuousOn.comp' {g : β → γ} {f : α → β} {s : Set α} {t : Set β} (hg : ContinuousOn g t)
     (hf : ContinuousOn f s) : ContinuousOn (g ∘ f) (s ∩ f ⁻¹' t) :=
   hg.comp (hf.mono (inter_subset_left _ _)) (inter_subset_right _ _)
 #align continuous_on.comp' ContinuousOn.comp'
+-/
 
+#print Continuous.continuousOn /-
 theorem Continuous.continuousOn {f : α → β} {s : Set α} (h : Continuous f) : ContinuousOn f s :=
   by
   rw [continuous_iff_continuousOn_univ] at h 
   exact h.mono (subset_univ _)
 #align continuous.continuous_on Continuous.continuousOn
+-/
 
+#print Continuous.continuousWithinAt /-
 theorem Continuous.continuousWithinAt {f : α → β} {s : Set α} {x : α} (h : Continuous f) :
     ContinuousWithinAt f s x :=
   h.ContinuousAt.ContinuousWithinAt
 #align continuous.continuous_within_at Continuous.continuousWithinAt
+-/
 
+#print Continuous.comp_continuousOn /-
 theorem Continuous.comp_continuousOn {g : β → γ} {f : α → β} {s : Set α} (hg : Continuous g)
     (hf : ContinuousOn f s) : ContinuousOn (g ∘ f) s :=
   hg.ContinuousOn.comp hf (mapsTo_univ _ _)
 #align continuous.comp_continuous_on Continuous.comp_continuousOn
+-/
 
+#print ContinuousOn.comp_continuous /-
 theorem ContinuousOn.comp_continuous {g : β → γ} {f : α → β} {s : Set β} (hg : ContinuousOn g s)
     (hf : Continuous f) (hs : ∀ x, f x ∈ s) : Continuous (g ∘ f) :=
   by
   rw [continuous_iff_continuousOn_univ] at *
   exact hg.comp hf fun x _ => hs x
 #align continuous_on.comp_continuous ContinuousOn.comp_continuous
+-/
 
+#print ContinuousWithinAt.preimage_mem_nhdsWithin /-
 theorem ContinuousWithinAt.preimage_mem_nhdsWithin {f : α → β} {x : α} {s : Set α} {t : Set β}
     (h : ContinuousWithinAt f s x) (ht : t ∈ 𝓝 (f x)) : f ⁻¹' t ∈ 𝓝[s] x :=
   h ht
 #align continuous_within_at.preimage_mem_nhds_within ContinuousWithinAt.preimage_mem_nhdsWithin
+-/
 
 #print Set.LeftInvOn.map_nhdsWithin_eq /-
 theorem Set.LeftInvOn.map_nhdsWithin_eq {f : α → β} {g : β → α} {x : β} {s : Set β}
@@ -1060,41 +1302,55 @@ theorem Function.LeftInverse.map_nhds_eq {f : α → β} {g : β → α} {x : β
 #align function.left_inverse.map_nhds_eq Function.LeftInverse.map_nhds_eq
 -/
 
+#print ContinuousWithinAt.preimage_mem_nhdsWithin' /-
 theorem ContinuousWithinAt.preimage_mem_nhdsWithin' {f : α → β} {x : α} {s : Set α} {t : Set β}
     (h : ContinuousWithinAt f s x) (ht : t ∈ 𝓝[f '' s] f x) : f ⁻¹' t ∈ 𝓝[s] x :=
   h.tendsto_nhdsWithin (mapsTo_image _ _) ht
 #align continuous_within_at.preimage_mem_nhds_within' ContinuousWithinAt.preimage_mem_nhdsWithin'
+-/
 
+#print Filter.EventuallyEq.congr_continuousWithinAt /-
 theorem Filter.EventuallyEq.congr_continuousWithinAt {f g : α → β} {s : Set α} {x : α}
     (h : f =ᶠ[𝓝[s] x] g) (hx : f x = g x) : ContinuousWithinAt f s x ↔ ContinuousWithinAt g s x :=
   by rw [ContinuousWithinAt, hx, tendsto_congr' h, ContinuousWithinAt]
 #align filter.eventually_eq.congr_continuous_within_at Filter.EventuallyEq.congr_continuousWithinAt
+-/
 
+#print ContinuousWithinAt.congr_of_eventuallyEq /-
 theorem ContinuousWithinAt.congr_of_eventuallyEq {f f₁ : α → β} {s : Set α} {x : α}
     (h : ContinuousWithinAt f s x) (h₁ : f₁ =ᶠ[𝓝[s] x] f) (hx : f₁ x = f x) :
     ContinuousWithinAt f₁ s x :=
   (h₁.congr_continuousWithinAt hx).2 h
 #align continuous_within_at.congr_of_eventually_eq ContinuousWithinAt.congr_of_eventuallyEq
+-/
 
+#print ContinuousWithinAt.congr /-
 theorem ContinuousWithinAt.congr {f f₁ : α → β} {s : Set α} {x : α} (h : ContinuousWithinAt f s x)
     (h₁ : ∀ y ∈ s, f₁ y = f y) (hx : f₁ x = f x) : ContinuousWithinAt f₁ s x :=
   h.congr_of_eventuallyEq (mem_of_superset self_mem_nhdsWithin h₁) hx
 #align continuous_within_at.congr ContinuousWithinAt.congr
+-/
 
+#print ContinuousWithinAt.congr_mono /-
 theorem ContinuousWithinAt.congr_mono {f g : α → β} {s s₁ : Set α} {x : α}
     (h : ContinuousWithinAt f s x) (h' : EqOn g f s₁) (h₁ : s₁ ⊆ s) (hx : g x = f x) :
     ContinuousWithinAt g s₁ x :=
   (h.mono h₁).congr h' hx
 #align continuous_within_at.congr_mono ContinuousWithinAt.congr_mono
+-/
 
+#print continuousOn_const /-
 theorem continuousOn_const {s : Set α} {c : β} : ContinuousOn (fun x => c) s :=
   continuous_const.ContinuousOn
 #align continuous_on_const continuousOn_const
+-/
 
+#print continuousWithinAt_const /-
 theorem continuousWithinAt_const {b : β} {s : Set α} {x : α} :
     ContinuousWithinAt (fun _ : α => b) s x :=
   continuous_const.ContinuousWithinAt
 #align continuous_within_at_const continuousWithinAt_const
+-/
 
 #print continuousOn_id /-
 theorem continuousOn_id {s : Set α} : ContinuousOn id s :=
@@ -1108,6 +1364,7 @@ theorem continuousWithinAt_id {s : Set α} {x : α} : ContinuousWithinAt id s x
 #align continuous_within_at_id continuousWithinAt_id
 -/
 
+#print continuousOn_open_iff /-
 theorem continuousOn_open_iff {f : α → β} {s : Set α} (hs : IsOpen s) :
     ContinuousOn f s ↔ ∀ t, IsOpen t → IsOpen (s ∩ f ⁻¹' t) :=
   by
@@ -1121,19 +1378,25 @@ theorem continuousOn_open_iff {f : α → β} {s : Set α} (hs : IsOpen s) :
     refine' ⟨s ∩ f ⁻¹' t, h t ht, _⟩
     rw [@inter_comm _ s (f ⁻¹' t), inter_assoc, inter_self]
 #align continuous_on_open_iff continuousOn_open_iff
+-/
 
+#print ContinuousOn.preimage_open_of_open /-
 theorem ContinuousOn.preimage_open_of_open {f : α → β} {s : Set α} {t : Set β}
     (hf : ContinuousOn f s) (hs : IsOpen s) (ht : IsOpen t) : IsOpen (s ∩ f ⁻¹' t) :=
   (continuousOn_open_iff hs).1 hf t ht
 #align continuous_on.preimage_open_of_open ContinuousOn.preimage_open_of_open
+-/
 
+#print ContinuousOn.isOpen_preimage /-
 theorem ContinuousOn.isOpen_preimage {f : α → β} {s : Set α} {t : Set β} (h : ContinuousOn f s)
     (hs : IsOpen s) (hp : f ⁻¹' t ⊆ s) (ht : IsOpen t) : IsOpen (f ⁻¹' t) :=
   by
   convert (continuousOn_open_iff hs).mp h t ht
   rw [inter_comm, inter_eq_self_of_subset_left hp]
 #align continuous_on.is_open_preimage ContinuousOn.isOpen_preimage
+-/
 
+#print ContinuousOn.preimage_closed_of_closed /-
 theorem ContinuousOn.preimage_closed_of_closed {f : α → β} {s : Set α} {t : Set β}
     (hf : ContinuousOn f s) (hs : IsClosed s) (ht : IsClosed t) : IsClosed (s ∩ f ⁻¹' t) :=
   by
@@ -1141,7 +1404,9 @@ theorem ContinuousOn.preimage_closed_of_closed {f : α → β} {s : Set α} {t :
   rw [inter_comm, hu.2]
   apply IsClosed.inter hu.1 hs
 #align continuous_on.preimage_closed_of_closed ContinuousOn.preimage_closed_of_closed
+-/
 
+#print ContinuousOn.preimage_interior_subset_interior_preimage /-
 theorem ContinuousOn.preimage_interior_subset_interior_preimage {f : α → β} {s : Set α} {t : Set β}
     (hf : ContinuousOn f s) (hs : IsOpen s) : s ∩ f ⁻¹' interior t ⊆ s ∩ interior (f ⁻¹' t) :=
   calc
@@ -1150,7 +1415,9 @@ theorem ContinuousOn.preimage_interior_subset_interior_preimage {f : α → β}
         (hf.preimage_open_of_open hs isOpen_interior)
     _ = s ∩ interior (f ⁻¹' t) := by rw [interior_inter, hs.interior_eq]
 #align continuous_on.preimage_interior_subset_interior_preimage ContinuousOn.preimage_interior_subset_interior_preimage
+-/
 
+#print continuousOn_of_locally_continuousOn /-
 theorem continuousOn_of_locally_continuousOn {f : α → β} {s : Set α}
     (h : ∀ x ∈ s, ∃ t, IsOpen t ∧ x ∈ t ∧ ContinuousOn f (s ∩ t)) : ContinuousOn f s :=
   by
@@ -1159,6 +1426,7 @@ theorem continuousOn_of_locally_continuousOn {f : α → β} {s : Set α}
   have := ct x ⟨xs, xt⟩
   rwa [ContinuousWithinAt, ← nhdsWithin_restrict _ xt open_t] at this 
 #align continuous_on_of_locally_continuous_on continuousOn_of_locally_continuousOn
+-/
 
 theorem continuousOn_open_of_generateFrom {β : Type _} {s : Set α} {T : Set (Set β)} {f : α → β}
     (hs : IsOpen s) (h : ∀ t ∈ T, IsOpen (s ∩ f ⁻¹' t)) :
@@ -1178,38 +1446,51 @@ theorem continuousOn_open_of_generateFrom {β : Type _} {s : Set α} {T : Set (S
   · exact hs
 #align continuous_on_open_of_generate_from continuousOn_open_of_generateFromₓ
 
+#print ContinuousWithinAt.prod /-
 theorem ContinuousWithinAt.prod {f : α → β} {g : α → γ} {s : Set α} {x : α}
     (hf : ContinuousWithinAt f s x) (hg : ContinuousWithinAt g s x) :
     ContinuousWithinAt (fun x => (f x, g x)) s x :=
   hf.prod_mk_nhds hg
 #align continuous_within_at.prod ContinuousWithinAt.prod
+-/
 
+#print ContinuousOn.prod /-
 theorem ContinuousOn.prod {f : α → β} {g : α → γ} {s : Set α} (hf : ContinuousOn f s)
     (hg : ContinuousOn g s) : ContinuousOn (fun x => (f x, g x)) s := fun x hx =>
   ContinuousWithinAt.prod (hf x hx) (hg x hx)
 #align continuous_on.prod ContinuousOn.prod
+-/
 
+#print Inducing.continuousWithinAt_iff /-
 theorem Inducing.continuousWithinAt_iff {f : α → β} {g : β → γ} (hg : Inducing g) {s : Set α}
     {x : α} : ContinuousWithinAt f s x ↔ ContinuousWithinAt (g ∘ f) s x := by
   simp_rw [ContinuousWithinAt, Inducing.tendsto_nhds_iff hg]
 #align inducing.continuous_within_at_iff Inducing.continuousWithinAt_iff
+-/
 
+#print Inducing.continuousOn_iff /-
 theorem Inducing.continuousOn_iff {f : α → β} {g : β → γ} (hg : Inducing g) {s : Set α} :
     ContinuousOn f s ↔ ContinuousOn (g ∘ f) s := by
   simp_rw [ContinuousOn, hg.continuous_within_at_iff]
 #align inducing.continuous_on_iff Inducing.continuousOn_iff
+-/
 
+#print Embedding.continuousOn_iff /-
 theorem Embedding.continuousOn_iff {f : α → β} {g : β → γ} (hg : Embedding g) {s : Set α} :
     ContinuousOn f s ↔ ContinuousOn (g ∘ f) s :=
   Inducing.continuousOn_iff hg.1
 #align embedding.continuous_on_iff Embedding.continuousOn_iff
+-/
 
+#print Embedding.map_nhdsWithin_eq /-
 theorem Embedding.map_nhdsWithin_eq {f : α → β} (hf : Embedding f) (s : Set α) (x : α) :
     map f (𝓝[s] x) = 𝓝[f '' s] f x := by
   rw [nhdsWithin, map_inf hf.inj, hf.map_nhds_eq, map_principal, ← nhdsWithin_inter',
     inter_eq_self_of_subset_right (image_subset_range _ _)]
 #align embedding.map_nhds_within_eq Embedding.map_nhdsWithin_eq
+-/
 
+#print OpenEmbedding.map_nhdsWithin_preimage_eq /-
 theorem OpenEmbedding.map_nhdsWithin_preimage_eq {f : α → β} (hf : OpenEmbedding f) (s : Set β)
     (x : α) : map f (𝓝[f ⁻¹' s] x) = 𝓝[s] f x :=
   by
@@ -1217,7 +1498,9 @@ theorem OpenEmbedding.map_nhdsWithin_preimage_eq {f : α → β} (hf : OpenEmbed
   apply nhdsWithin_eq_nhdsWithin (mem_range_self _) hf.open_range
   rw [inter_assoc, inter_self]
 #align open_embedding.map_nhds_within_preimage_eq OpenEmbedding.map_nhdsWithin_preimage_eq
+-/
 
+#print continuousWithinAt_of_not_mem_closure /-
 theorem continuousWithinAt_of_not_mem_closure {f : α → β} {s : Set α} {x : α} :
     x ∉ closure s → ContinuousWithinAt f s x :=
   by
@@ -1226,7 +1509,9 @@ theorem continuousWithinAt_of_not_mem_closure {f : α → β} {s : Set α} {x :
   rw [ContinuousWithinAt, hx]
   exact tendsto_bot
 #align continuous_within_at_of_not_mem_closure continuousWithinAt_of_not_mem_closure
+-/
 
+#print ContinuousOn.piecewise' /-
 theorem ContinuousOn.piecewise' {s t : Set α} {f g : α → β} [∀ a, Decidable (a ∈ t)]
     (hpf : ∀ a ∈ s ∩ frontier t, Tendsto f (𝓝[s ∩ t] a) (𝓝 (piecewise t f g a)))
     (hpg : ∀ a ∈ s ∩ frontier t, Tendsto g (𝓝[s ∩ tᶜ] a) (𝓝 (piecewise t f g a)))
@@ -1257,7 +1542,9 @@ theorem ContinuousOn.piecewise' {s t : Set α} {f g : α → β} [∀ a, Decidab
           (hg x hx).congr (fun y hy => piecewise_eq_of_not_mem _ _ _ hy.2)
             (piecewise_eq_of_not_mem _ _ _ hx.2)
 #align continuous_on.piecewise' ContinuousOn.piecewise'
+-/
 
+#print ContinuousOn.if' /-
 theorem ContinuousOn.if' {s : Set α} {p : α → Prop} {f g : α → β} [∀ a, Decidable (p a)]
     (hpf :
       ∀ a ∈ s ∩ frontier {a | p a}, Tendsto f (𝓝[s ∩ {a | p a}] a) (𝓝 <| if p a then f a else g a))
@@ -1267,7 +1554,9 @@ theorem ContinuousOn.if' {s : Set α} {p : α → Prop} {f g : α → β} [∀ a
     ContinuousOn (fun a => if p a then f a else g a) s :=
   hf.piecewise' hpf hpg hg
 #align continuous_on.if' ContinuousOn.if'
+-/
 
+#print ContinuousOn.if /-
 theorem ContinuousOn.if {α β : Type _} [TopologicalSpace α] [TopologicalSpace β] {p : α → Prop}
     [∀ a, Decidable (p a)] {s : Set α} {f g : α → β} (hp : ∀ a ∈ s ∩ frontier {a | p a}, f a = g a)
     (hf : ContinuousOn f <| s ∩ closure {a | p a}) (hg : ContinuousOn g <| s ∩ closure {a | ¬p a}) :
@@ -1287,13 +1576,17 @@ theorem ContinuousOn.if {α β : Type _} [TopologicalSpace α] [TopologicalSpace
   · exact hf.mono (inter_subset_inter_right s subset_closure)
   · exact hg.mono (inter_subset_inter_right s subset_closure)
 #align continuous_on.if ContinuousOn.if
+-/
 
+#print ContinuousOn.piecewise /-
 theorem ContinuousOn.piecewise {s t : Set α} {f g : α → β} [∀ a, Decidable (a ∈ t)]
     (ht : ∀ a ∈ s ∩ frontier t, f a = g a) (hf : ContinuousOn f <| s ∩ closure t)
     (hg : ContinuousOn g <| s ∩ closure (tᶜ)) : ContinuousOn (piecewise t f g) s :=
   hf.if ht hg
 #align continuous_on.piecewise ContinuousOn.piecewise
+-/
 
+#print continuous_if' /-
 theorem continuous_if' {p : α → Prop} {f g : α → β} [∀ a, Decidable (p a)]
     (hpf : ∀ a ∈ frontier {x | p x}, Tendsto f (𝓝[{x | p x}] a) (𝓝 <| ite (p a) (f a) (g a)))
     (hpg : ∀ a ∈ frontier {x | p x}, Tendsto g (𝓝[{x | ¬p x}] a) (𝓝 <| ite (p a) (f a) (g a)))
@@ -1303,7 +1596,9 @@ theorem continuous_if' {p : α → Prop} {f g : α → β} [∀ a, Decidable (p
   rw [continuous_iff_continuousOn_univ]
   apply ContinuousOn.if' <;> simp [*] <;> assumption
 #align continuous_if' continuous_if'
+-/
 
+#print continuous_if /-
 theorem continuous_if {p : α → Prop} {f g : α → β} [∀ a, Decidable (p a)]
     (hp : ∀ a ∈ frontier {x | p x}, f a = g a) (hf : ContinuousOn f (closure {x | p x}))
     (hg : ContinuousOn g (closure {x | ¬p x})) : Continuous fun a => if p a then f a else g a :=
@@ -1311,34 +1606,45 @@ theorem continuous_if {p : α → Prop} {f g : α → β} [∀ a, Decidable (p a
   rw [continuous_iff_continuousOn_univ]
   apply ContinuousOn.if <;> simp <;> assumption
 #align continuous_if continuous_if
+-/
 
+#print Continuous.if /-
 theorem Continuous.if {p : α → Prop} {f g : α → β} [∀ a, Decidable (p a)]
     (hp : ∀ a ∈ frontier {x | p x}, f a = g a) (hf : Continuous f) (hg : Continuous g) :
     Continuous fun a => if p a then f a else g a :=
   continuous_if hp hf.ContinuousOn hg.ContinuousOn
 #align continuous.if Continuous.if
+-/
 
+#print continuous_if_const /-
 theorem continuous_if_const (p : Prop) {f g : α → β} [Decidable p] (hf : p → Continuous f)
     (hg : ¬p → Continuous g) : Continuous fun a => if p then f a else g a := by split_ifs;
   exact hf h; exact hg h
 #align continuous_if_const continuous_if_const
+-/
 
+#print Continuous.if_const /-
 theorem Continuous.if_const (p : Prop) {f g : α → β} [Decidable p] (hf : Continuous f)
     (hg : Continuous g) : Continuous fun a => if p then f a else g a :=
   continuous_if_const p (fun _ => hf) fun _ => hg
 #align continuous.if_const Continuous.if_const
+-/
 
+#print continuous_piecewise /-
 theorem continuous_piecewise {s : Set α} {f g : α → β} [∀ a, Decidable (a ∈ s)]
     (hs : ∀ a ∈ frontier s, f a = g a) (hf : ContinuousOn f (closure s))
     (hg : ContinuousOn g (closure (sᶜ))) : Continuous (piecewise s f g) :=
   continuous_if hs hf hg
 #align continuous_piecewise continuous_piecewise
+-/
 
+#print Continuous.piecewise /-
 theorem Continuous.piecewise {s : Set α} {f g : α → β} [∀ a, Decidable (a ∈ s)]
     (hs : ∀ a ∈ frontier s, f a = g a) (hf : Continuous f) (hg : Continuous g) :
     Continuous (piecewise s f g) :=
   hf.if hs hg
 #align continuous.piecewise Continuous.piecewise
+-/
 
 #print IsOpen.ite' /-
 theorem IsOpen.ite' {s s' t : Set α} (hs : IsOpen s) (hs' : IsOpen s')
@@ -1351,22 +1657,29 @@ theorem IsOpen.ite' {s s' t : Set α} (hs : IsOpen s) (hs' : IsOpen s')
 #align is_open.ite' IsOpen.ite'
 -/
 
+#print IsOpen.ite /-
 theorem IsOpen.ite {s s' t : Set α} (hs : IsOpen s) (hs' : IsOpen s')
     (ht : s ∩ frontier t = s' ∩ frontier t) : IsOpen (t.ite s s') :=
   hs.ite' hs' fun x hx => by simpa [hx] using ext_iff.1 ht x
 #align is_open.ite IsOpen.ite
+-/
 
+#print ite_inter_closure_eq_of_inter_frontier_eq /-
 theorem ite_inter_closure_eq_of_inter_frontier_eq {s s' t : Set α}
     (ht : s ∩ frontier t = s' ∩ frontier t) : t.ite s s' ∩ closure t = s ∩ closure t := by
   rw [closure_eq_self_union_frontier, inter_union_distrib_left, inter_union_distrib_left,
     ite_inter_self, ite_inter_of_inter_eq _ ht]
 #align ite_inter_closure_eq_of_inter_frontier_eq ite_inter_closure_eq_of_inter_frontier_eq
+-/
 
+#print ite_inter_closure_compl_eq_of_inter_frontier_eq /-
 theorem ite_inter_closure_compl_eq_of_inter_frontier_eq {s s' t : Set α}
     (ht : s ∩ frontier t = s' ∩ frontier t) : t.ite s s' ∩ closure (tᶜ) = s' ∩ closure (tᶜ) := by
   rw [← ite_compl, ite_inter_closure_eq_of_inter_frontier_eq]; rwa [frontier_compl, eq_comm]
 #align ite_inter_closure_compl_eq_of_inter_frontier_eq ite_inter_closure_compl_eq_of_inter_frontier_eq
+-/
 
+#print continuousOn_piecewise_ite' /-
 theorem continuousOn_piecewise_ite' {s s' t : Set α} {f f' : α → β} [∀ x, Decidable (x ∈ t)]
     (h : ContinuousOn f (s ∩ closure t)) (h' : ContinuousOn f' (s' ∩ closure (tᶜ)))
     (H : s ∩ frontier t = s' ∩ frontier t) (Heq : EqOn f f' (s ∩ frontier t)) :
@@ -1377,60 +1690,83 @@ theorem continuousOn_piecewise_ite' {s s' t : Set α} {f f' : α → β} [∀ x,
   · rwa [ite_inter_closure_eq_of_inter_frontier_eq H]
   · rwa [ite_inter_closure_compl_eq_of_inter_frontier_eq H]
 #align continuous_on_piecewise_ite' continuousOn_piecewise_ite'
+-/
 
+#print continuousOn_piecewise_ite /-
 theorem continuousOn_piecewise_ite {s s' t : Set α} {f f' : α → β} [∀ x, Decidable (x ∈ t)]
     (h : ContinuousOn f s) (h' : ContinuousOn f' s') (H : s ∩ frontier t = s' ∩ frontier t)
     (Heq : EqOn f f' (s ∩ frontier t)) : ContinuousOn (t.piecewise f f') (t.ite s s') :=
   continuousOn_piecewise_ite' (h.mono (inter_subset_left _ _)) (h'.mono (inter_subset_left _ _)) H
     Heq
 #align continuous_on_piecewise_ite continuousOn_piecewise_ite
+-/
 
+#print frontier_inter_open_inter /-
 theorem frontier_inter_open_inter {s t : Set α} (ht : IsOpen t) :
     frontier (s ∩ t) ∩ t = frontier s ∩ t := by
   simp only [← Subtype.preimage_coe_eq_preimage_coe_iff,
     ht.is_open_map_subtype_coe.preimage_frontier_eq_frontier_preimage continuous_subtype_val,
     Subtype.preimage_coe_inter_self]
 #align frontier_inter_open_inter frontier_inter_open_inter
+-/
 
+#print continuousOn_fst /-
 theorem continuousOn_fst {s : Set (α × β)} : ContinuousOn Prod.fst s :=
   continuous_fst.ContinuousOn
 #align continuous_on_fst continuousOn_fst
+-/
 
+#print continuousWithinAt_fst /-
 theorem continuousWithinAt_fst {s : Set (α × β)} {p : α × β} : ContinuousWithinAt Prod.fst s p :=
   continuous_fst.ContinuousWithinAt
 #align continuous_within_at_fst continuousWithinAt_fst
+-/
 
+#print ContinuousOn.fst /-
 theorem ContinuousOn.fst {f : α → β × γ} {s : Set α} (hf : ContinuousOn f s) :
     ContinuousOn (fun x => (f x).1) s :=
   continuous_fst.comp_continuousOn hf
 #align continuous_on.fst ContinuousOn.fst
+-/
 
+#print ContinuousWithinAt.fst /-
 theorem ContinuousWithinAt.fst {f : α → β × γ} {s : Set α} {a : α} (h : ContinuousWithinAt f s a) :
     ContinuousWithinAt (fun x => (f x).fst) s a :=
   continuousAt_fst.comp_continuousWithinAt h
 #align continuous_within_at.fst ContinuousWithinAt.fst
+-/
 
+#print continuousOn_snd /-
 theorem continuousOn_snd {s : Set (α × β)} : ContinuousOn Prod.snd s :=
   continuous_snd.ContinuousOn
 #align continuous_on_snd continuousOn_snd
+-/
 
+#print continuousWithinAt_snd /-
 theorem continuousWithinAt_snd {s : Set (α × β)} {p : α × β} : ContinuousWithinAt Prod.snd s p :=
   continuous_snd.ContinuousWithinAt
 #align continuous_within_at_snd continuousWithinAt_snd
+-/
 
+#print ContinuousOn.snd /-
 theorem ContinuousOn.snd {f : α → β × γ} {s : Set α} (hf : ContinuousOn f s) :
     ContinuousOn (fun x => (f x).2) s :=
   continuous_snd.comp_continuousOn hf
 #align continuous_on.snd ContinuousOn.snd
+-/
 
+#print ContinuousWithinAt.snd /-
 theorem ContinuousWithinAt.snd {f : α → β × γ} {s : Set α} {a : α} (h : ContinuousWithinAt f s a) :
     ContinuousWithinAt (fun x => (f x).snd) s a :=
   continuousAt_snd.comp_continuousWithinAt h
 #align continuous_within_at.snd ContinuousWithinAt.snd
+-/
 
+#print continuousWithinAt_prod_iff /-
 theorem continuousWithinAt_prod_iff {f : α → β × γ} {s : Set α} {x : α} :
     ContinuousWithinAt f s x ↔
       ContinuousWithinAt (Prod.fst ∘ f) s x ∧ ContinuousWithinAt (Prod.snd ∘ f) s x :=
   ⟨fun h => ⟨h.fst, h.snd⟩, by rintro ⟨h1, h2⟩; convert h1.prod h2; ext; rfl; rfl⟩
 #align continuous_within_at_prod_iff continuousWithinAt_prod_iff
+-/
 

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

@@ -900,7 +900,6 @@ theorem continuousWithinAt_update_same [DecidableEq α] {f : α → β} {s : Set
     _ ↔ Tendsto f (𝓝[s \ {x}] x) (𝓝 y) :=
       tendsto_congr' <|
         eventually_nhdsWithin_iff.2 <| eventually_of_forall fun z hz => update_noteq hz.2 _ _
-    
 #align continuous_within_at_update_same continuousWithinAt_update_same
 
 @[simp]
@@ -1150,7 +1149,6 @@ theorem ContinuousOn.preimage_interior_subset_interior_preimage {f : α → β}
       interior_maximal (inter_subset_inter (Subset.refl _) (preimage_mono interior_subset))
         (hf.preimage_open_of_open hs isOpen_interior)
     _ = s ∩ interior (f ⁻¹' t) := by rw [interior_inter, hs.interior_eq]
-    
 #align continuous_on.preimage_interior_subset_interior_preimage ContinuousOn.preimage_interior_subset_interior_preimage
 
 theorem continuousOn_of_locally_continuousOn {f : α → β} {s : Set α}

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

@@ -376,7 +376,7 @@ theorem nhdsWithin_pi_eq' {ι : Type _} {α : ι → Type _} [∀ i, Topological
     infi_principal_finite hI, ← iInf_inf_eq]
 #align nhds_within_pi_eq' nhdsWithin_pi_eq'
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (i «expr ∉ » I) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:638:2: warning: expanding binder collection (i «expr ∉ » I) -/
 theorem nhdsWithin_pi_eq {ι : Type _} {α : ι → Type _} [∀ i, TopologicalSpace (α i)] {I : Set ι}
     (hI : I.Finite) (s : ∀ i, Set (α i)) (x : ∀ i, α i) :
     𝓝[pi I s] x =

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

@@ -54,7 +54,7 @@ theorem nhds_bind_nhdsWithin {a : α} {s : Set α} : ((𝓝 a).bind fun x => 
 @[simp]
 theorem eventually_nhds_nhdsWithin {a : α} {s : Set α} {p : α → Prop} :
     (∀ᶠ y in 𝓝 a, ∀ᶠ x in 𝓝[s] y, p x) ↔ ∀ᶠ x in 𝓝[s] a, p x :=
-  Filter.ext_iff.1 nhds_bind_nhdsWithin { x | p x }
+  Filter.ext_iff.1 nhds_bind_nhdsWithin {x | p x}
 #align eventually_nhds_nhds_within eventually_nhds_nhdsWithin
 -/
 
@@ -89,7 +89,7 @@ theorem eventually_nhdsWithin_nhdsWithin {a : α} {s : Set α} {p : α → Prop}
 -/
 
 theorem nhdsWithin_eq (a : α) (s : Set α) :
-    𝓝[s] a = ⨅ t ∈ { t : Set α | a ∈ t ∧ IsOpen t }, 𝓟 (t ∩ s) :=
+    𝓝[s] a = ⨅ t ∈ {t : Set α | a ∈ t ∧ IsOpen t}, 𝓟 (t ∩ s) :=
   ((nhds_basis_opens a).inf_principal s).eq_biInf
 #align nhds_within_eq nhdsWithin_eq
 
@@ -253,9 +253,9 @@ theorem nhdsWithin_le_nhds {a : α} {s : Set α} : 𝓝[s] a ≤ 𝓝 a := by rw
   apply nhdsWithin_le_of_mem; exact univ_mem
 #align nhds_within_le_nhds nhdsWithin_le_nhds
 
-theorem nhdsWithin_eq_nhds_within' {a : α} {s t u : Set α} (hs : s ∈ 𝓝 a) (h₂ : t ∩ s = u ∩ s) :
+theorem nhdsWithin_eq_nhdsWithin' {a : α} {s t u : Set α} (hs : s ∈ 𝓝 a) (h₂ : t ∩ s = u ∩ s) :
     𝓝[t] a = 𝓝[u] a := by rw [nhdsWithin_restrict' t hs, nhdsWithin_restrict' u hs, h₂]
-#align nhds_within_eq_nhds_within' nhdsWithin_eq_nhds_within'
+#align nhds_within_eq_nhds_within' nhdsWithin_eq_nhdsWithin'
 
 theorem nhdsWithin_eq_nhdsWithin {a : α} {s t u : Set α} (h₀ : a ∈ s) (h₁ : IsOpen s)
     (h₂ : t ∩ s = u ∩ s) : 𝓝[t] a = 𝓝[u] a := by
@@ -411,14 +411,14 @@ theorem Filter.Tendsto.piecewise_nhdsWithin {f g : α → β} {t : Set α} [∀
 #align filter.tendsto.piecewise_nhds_within Filter.Tendsto.piecewise_nhdsWithin
 
 theorem Filter.Tendsto.if_nhdsWithin {f g : α → β} {p : α → Prop} [DecidablePred p] {a : α}
-    {s : Set α} {l : Filter β} (h₀ : Tendsto f (𝓝[s ∩ { x | p x }] a) l)
-    (h₁ : Tendsto g (𝓝[s ∩ { x | ¬p x }] a) l) :
+    {s : Set α} {l : Filter β} (h₀ : Tendsto f (𝓝[s ∩ {x | p x}] a) l)
+    (h₁ : Tendsto g (𝓝[s ∩ {x | ¬p x}] a) l) :
     Tendsto (fun x => if p x then f x else g x) (𝓝[s] a) l :=
   h₀.piecewise_nhdsWithin h₁
 #align filter.tendsto.if_nhds_within Filter.Tendsto.if_nhdsWithin
 
 theorem map_nhdsWithin (f : α → β) (a : α) (s : Set α) :
-    map f (𝓝[s] a) = ⨅ t ∈ { t : Set α | a ∈ t ∧ IsOpen t }, 𝓟 (f '' (t ∩ s)) :=
+    map f (𝓝[s] a) = ⨅ t ∈ {t : Set α | a ∈ t ∧ IsOpen t}, 𝓟 (f '' (t ∩ s)) :=
   ((nhdsWithin_basis_open a s).map f).eq_biInf
 #align map_nhds_within map_nhdsWithin
 
@@ -1061,10 +1061,10 @@ theorem Function.LeftInverse.map_nhds_eq {f : α → β} {g : β → α} {x : β
 #align function.left_inverse.map_nhds_eq Function.LeftInverse.map_nhds_eq
 -/
 
-theorem ContinuousWithinAt.preimage_mem_nhds_within' {f : α → β} {x : α} {s : Set α} {t : Set β}
+theorem ContinuousWithinAt.preimage_mem_nhdsWithin' {f : α → β} {x : α} {s : Set α} {t : Set β}
     (h : ContinuousWithinAt f s x) (ht : t ∈ 𝓝[f '' s] f x) : f ⁻¹' t ∈ 𝓝[s] x :=
   h.tendsto_nhdsWithin (mapsTo_image _ _) ht
-#align continuous_within_at.preimage_mem_nhds_within' ContinuousWithinAt.preimage_mem_nhds_within'
+#align continuous_within_at.preimage_mem_nhds_within' ContinuousWithinAt.preimage_mem_nhdsWithin'
 
 theorem Filter.EventuallyEq.congr_continuousWithinAt {f g : α → β} {s : Set α} {x : α}
     (h : f =ᶠ[𝓝[s] x] g) (hx : f x = g x) : ContinuousWithinAt f s x ↔ ContinuousWithinAt g s x :=
@@ -1131,7 +1131,7 @@ theorem ContinuousOn.preimage_open_of_open {f : α → β} {s : Set α} {t : Set
 theorem ContinuousOn.isOpen_preimage {f : α → β} {s : Set α} {t : Set β} (h : ContinuousOn f s)
     (hs : IsOpen s) (hp : f ⁻¹' t ⊆ s) (ht : IsOpen t) : IsOpen (f ⁻¹' t) :=
   by
-  convert(continuousOn_open_iff hs).mp h t ht
+  convert (continuousOn_open_iff hs).mp h t ht
   rw [inter_comm, inter_eq_self_of_subset_left hp]
 #align continuous_on.is_open_preimage ContinuousOn.isOpen_preimage
 
@@ -1262,21 +1262,17 @@ theorem ContinuousOn.piecewise' {s t : Set α} {f g : α → β} [∀ a, Decidab
 
 theorem ContinuousOn.if' {s : Set α} {p : α → Prop} {f g : α → β} [∀ a, Decidable (p a)]
     (hpf :
-      ∀ a ∈ s ∩ frontier { a | p a },
-        Tendsto f (𝓝[s ∩ { a | p a }] a) (𝓝 <| if p a then f a else g a))
+      ∀ a ∈ s ∩ frontier {a | p a}, Tendsto f (𝓝[s ∩ {a | p a}] a) (𝓝 <| if p a then f a else g a))
     (hpg :
-      ∀ a ∈ s ∩ frontier { a | p a },
-        Tendsto g (𝓝[s ∩ { a | ¬p a }] a) (𝓝 <| if p a then f a else g a))
-    (hf : ContinuousOn f <| s ∩ { a | p a }) (hg : ContinuousOn g <| s ∩ { a | ¬p a }) :
+      ∀ a ∈ s ∩ frontier {a | p a}, Tendsto g (𝓝[s ∩ {a | ¬p a}] a) (𝓝 <| if p a then f a else g a))
+    (hf : ContinuousOn f <| s ∩ {a | p a}) (hg : ContinuousOn g <| s ∩ {a | ¬p a}) :
     ContinuousOn (fun a => if p a then f a else g a) s :=
   hf.piecewise' hpf hpg hg
 #align continuous_on.if' ContinuousOn.if'
 
 theorem ContinuousOn.if {α β : Type _} [TopologicalSpace α] [TopologicalSpace β] {p : α → Prop}
-    [∀ a, Decidable (p a)] {s : Set α} {f g : α → β}
-    (hp : ∀ a ∈ s ∩ frontier { a | p a }, f a = g a)
-    (hf : ContinuousOn f <| s ∩ closure { a | p a })
-    (hg : ContinuousOn g <| s ∩ closure { a | ¬p a }) :
+    [∀ a, Decidable (p a)] {s : Set α} {f g : α → β} (hp : ∀ a ∈ s ∩ frontier {a | p a}, f a = g a)
+    (hf : ContinuousOn f <| s ∩ closure {a | p a}) (hg : ContinuousOn g <| s ∩ closure {a | ¬p a}) :
     ContinuousOn (fun a => if p a then f a else g a) s :=
   by
   apply ContinuousOn.if'
@@ -1301,9 +1297,9 @@ theorem ContinuousOn.piecewise {s t : Set α} {f g : α → β} [∀ a, Decidabl
 #align continuous_on.piecewise ContinuousOn.piecewise
 
 theorem continuous_if' {p : α → Prop} {f g : α → β} [∀ a, Decidable (p a)]
-    (hpf : ∀ a ∈ frontier { x | p x }, Tendsto f (𝓝[{ x | p x }] a) (𝓝 <| ite (p a) (f a) (g a)))
-    (hpg : ∀ a ∈ frontier { x | p x }, Tendsto g (𝓝[{ x | ¬p x }] a) (𝓝 <| ite (p a) (f a) (g a)))
-    (hf : ContinuousOn f { x | p x }) (hg : ContinuousOn g { x | ¬p x }) :
+    (hpf : ∀ a ∈ frontier {x | p x}, Tendsto f (𝓝[{x | p x}] a) (𝓝 <| ite (p a) (f a) (g a)))
+    (hpg : ∀ a ∈ frontier {x | p x}, Tendsto g (𝓝[{x | ¬p x}] a) (𝓝 <| ite (p a) (f a) (g a)))
+    (hf : ContinuousOn f {x | p x}) (hg : ContinuousOn g {x | ¬p x}) :
     Continuous fun a => ite (p a) (f a) (g a) :=
   by
   rw [continuous_iff_continuousOn_univ]
@@ -1311,15 +1307,15 @@ theorem continuous_if' {p : α → Prop} {f g : α → β} [∀ a, Decidable (p
 #align continuous_if' continuous_if'
 
 theorem continuous_if {p : α → Prop} {f g : α → β} [∀ a, Decidable (p a)]
-    (hp : ∀ a ∈ frontier { x | p x }, f a = g a) (hf : ContinuousOn f (closure { x | p x }))
-    (hg : ContinuousOn g (closure { x | ¬p x })) : Continuous fun a => if p a then f a else g a :=
+    (hp : ∀ a ∈ frontier {x | p x}, f a = g a) (hf : ContinuousOn f (closure {x | p x}))
+    (hg : ContinuousOn g (closure {x | ¬p x})) : Continuous fun a => if p a then f a else g a :=
   by
   rw [continuous_iff_continuousOn_univ]
   apply ContinuousOn.if <;> simp <;> assumption
 #align continuous_if continuous_if
 
 theorem Continuous.if {p : α → Prop} {f g : α → β} [∀ a, Decidable (p a)]
-    (hp : ∀ a ∈ frontier { x | p x }, f a = g a) (hf : Continuous f) (hg : Continuous g) :
+    (hp : ∀ a ∈ frontier {x | p x}, f a = g a) (hf : Continuous f) (hg : Continuous g) :
     Continuous fun a => if p a then f a else g a :=
   continuous_if hp hf.ContinuousOn hg.ContinuousOn
 #align continuous.if Continuous.if
@@ -1350,10 +1346,10 @@ theorem Continuous.piecewise {s : Set α} {f g : α → β} [∀ a, Decidable (a
 theorem IsOpen.ite' {s s' t : Set α} (hs : IsOpen s) (hs' : IsOpen s')
     (ht : ∀ x ∈ frontier t, x ∈ s ↔ x ∈ s') : IsOpen (t.ite s s') := by
   classical
-    simp only [isOpen_iff_continuous_mem, Set.ite] at *
-    convert continuous_piecewise (fun x hx => propext (ht x hx)) hs.continuous_on hs'.continuous_on
-    ext x
-    by_cases hx : x ∈ t <;> simp [hx]
+  simp only [isOpen_iff_continuous_mem, Set.ite] at *
+  convert continuous_piecewise (fun x hx => propext (ht x hx)) hs.continuous_on hs'.continuous_on
+  ext x
+  by_cases hx : x ∈ t <;> simp [hx]
 #align is_open.ite' IsOpen.ite'
 -/
 

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

@@ -83,7 +83,7 @@ theorem eventually_nhdsWithin_nhdsWithin {a : α} {s : Set α} {p : α → Prop}
     (∀ᶠ y in 𝓝[s] a, ∀ᶠ x in 𝓝[s] y, p x) ↔ ∀ᶠ x in 𝓝[s] a, p x :=
   by
   refine' ⟨fun h => _, fun h => (eventually_nhds_nhdsWithin.2 h).filter_mono inf_le_left⟩
-  simp only [eventually_nhdsWithin_iff] at h⊢
+  simp only [eventually_nhdsWithin_iff] at h ⊢
   exact h.mono fun x hx hxs => (hx hxs).self_of_nhds hxs
 #align eventually_nhds_within_nhds_within eventually_nhdsWithin_nhdsWithin
 -/
@@ -175,7 +175,7 @@ theorem preimage_nhdsWithin_coinduced' {π : α → β} {s : Set β} {t : Set α
       ⟨π ⁻¹' V, mem_nhds_iff.mpr ⟨t ∩ π ⁻¹' V, inter_subset_right t (π ⁻¹' V), _, mem_sep h mem_V⟩,
         subset.trans (inter_subset_left _ _) (preimage_mono hVs)⟩
   obtain ⟨u, hu1, hu2⟩ := is_open_induced_iff.mp (isOpen_coinduced.1 V_op)
-  rw [preimage_comp] at hu2
+  rw [preimage_comp] at hu2 
   rw [Set.inter_comm, ← subtype.preimage_coe_eq_preimage_coe_iff.mp hu2]
   exact hu1.inter ht
 #align preimage_nhds_within_coinduced' preimage_nhdsWithin_coinduced'ₓ
@@ -443,7 +443,7 @@ theorem eventually_mem_of_tendsto_nhdsWithin {f : β → α} {a : α} {s : Set 
     (h : Tendsto f l (𝓝[s] a)) : ∀ᶠ i in l, f i ∈ s :=
   by
   simp_rw [nhdsWithin_eq, tendsto_infi, mem_set_of_eq, tendsto_principal, mem_inter_iff,
-    eventually_and] at h
+    eventually_and] at h 
   exact (h univ ⟨mem_univ a, isOpen_univ⟩).2
 #align eventually_mem_of_tendsto_nhds_within eventually_mem_of_tendsto_nhdsWithin
 
@@ -722,8 +722,9 @@ theorem continuousOn_iff' {f : α → β} {s : Set α} :
 continuous on the same set with respect to any finer topology on the source space. -/
 theorem ContinuousOn.mono_dom {α β : Type _} {t₁ t₂ : TopologicalSpace α} {t₃ : TopologicalSpace β}
     (h₁ : t₂ ≤ t₁) {s : Set α} {f : α → β} (h₂ : @ContinuousOn α β t₁ t₃ f s) :
-    @ContinuousOn α β t₂ t₃ f s := by
-  rw [continuousOn_iff'] at h₂⊢
+    @ContinuousOn α β t₂ t₃ f s :=
+  by
+  rw [continuousOn_iff'] at h₂ ⊢
   intro t ht
   rcases h₂ t ht with ⟨u, hu, h'u⟩
   exact ⟨u, h₁ u hu, h'u⟩
@@ -733,8 +734,9 @@ theorem ContinuousOn.mono_dom {α β : Type _} {t₁ t₂ : TopologicalSpace α}
 continuous on the same set with respect to any coarser topology on the target space. -/
 theorem ContinuousOn.mono_rng {α β : Type _} {t₁ : TopologicalSpace α} {t₂ t₃ : TopologicalSpace β}
     (h₁ : t₂ ≤ t₃) {s : Set α} {f : α → β} (h₂ : @ContinuousOn α β t₁ t₂ f s) :
-    @ContinuousOn α β t₁ t₃ f s := by
-  rw [continuousOn_iff'] at h₂⊢
+    @ContinuousOn α β t₁ t₃ f s :=
+  by
+  rw [continuousOn_iff'] at h₂ ⊢
   intro t ht
   exact h₂ t (h₁ t ht)
 #align continuous_on.mono_rng ContinuousOn.mono_rng
@@ -929,8 +931,8 @@ theorem ContinuousOn.congr_mono {f g : α → β} {s s₁ : Set α} (h : Continu
   intro x hx
   unfold ContinuousWithinAt
   have A := (h x (h₁ hx)).mono h₁
-  unfold ContinuousWithinAt at A
-  rw [← h' hx] at A
+  unfold ContinuousWithinAt at A 
+  rw [← h' hx] at A 
   exact A.congr' h'.eventually_eq_nhds_within.symm
 #align continuous_on.congr_mono ContinuousOn.congr_mono
 
@@ -1010,7 +1012,7 @@ theorem ContinuousOn.comp' {g : β → γ} {f : α → β} {s : Set α} {t : Set
 
 theorem Continuous.continuousOn {f : α → β} {s : Set α} (h : Continuous f) : ContinuousOn f s :=
   by
-  rw [continuous_iff_continuousOn_univ] at h
+  rw [continuous_iff_continuousOn_univ] at h 
   exact h.mono (subset_univ _)
 #align continuous.continuous_on Continuous.continuousOn
 
@@ -1157,7 +1159,7 @@ theorem continuousOn_of_locally_continuousOn {f : α → β} {s : Set α}
   intro x xs
   rcases h x xs with ⟨t, open_t, xt, ct⟩
   have := ct x ⟨xs, xt⟩
-  rwa [ContinuousWithinAt, ← nhdsWithin_restrict _ xt open_t] at this
+  rwa [ContinuousWithinAt, ← nhdsWithin_restrict _ xt open_t] at this 
 #align continuous_on_of_locally_continuous_on continuousOn_of_locally_continuousOn
 
 theorem continuousOn_open_of_generateFrom {β : Type _} {s : Set α} {T : Set (Set β)} {f : α → β}
@@ -1222,7 +1224,7 @@ theorem continuousWithinAt_of_not_mem_closure {f : α → β} {s : Set α} {x :
     x ∉ closure s → ContinuousWithinAt f s x :=
   by
   intro hx
-  rw [mem_closure_iff_nhdsWithin_neBot, ne_bot_iff, Classical.not_not] at hx
+  rw [mem_closure_iff_nhdsWithin_neBot, ne_bot_iff, Classical.not_not] at hx 
   rw [ContinuousWithinAt, hx]
   exact tendsto_bot
 #align continuous_within_at_of_not_mem_closure continuousWithinAt_of_not_mem_closure
@@ -1235,14 +1237,14 @@ theorem ContinuousOn.piecewise' {s t : Set α} {f g : α → β} [∀ a, Decidab
   intro x hx
   by_cases hx' : x ∈ frontier t
   · exact (hpf x ⟨hx, hx'⟩).piecewise_nhdsWithin (hpg x ⟨hx, hx'⟩)
-  · rw [← inter_univ s, ← union_compl_self t, inter_union_distrib_left] at hx⊢
+  · rw [← inter_univ s, ← union_compl_self t, inter_union_distrib_left] at hx ⊢
     cases hx
     · apply ContinuousWithinAt.union
       ·
         exact
           (hf x hx).congr (fun y hy => piecewise_eq_of_mem _ _ _ hy.2)
             (piecewise_eq_of_mem _ _ _ hx.2)
-      · have : x ∉ closure (tᶜ) := fun h => hx' ⟨subset_closure hx.2, by rwa [closure_compl] at h⟩
+      · have : x ∉ closure (tᶜ) := fun h => hx' ⟨subset_closure hx.2, by rwa [closure_compl] at h ⟩
         exact
           continuousWithinAt_of_not_mem_closure fun h =>
             this (closure_inter_subset_inter_closure _ _ h).2
@@ -1286,7 +1288,7 @@ theorem ContinuousOn.if {α β : Type _} [TopologicalSpace α] [TopologicalSpace
     simp only [hp a ha, if_t_t]
     apply tendsto_nhdsWithin_mono_left (inter_subset_inter_right s subset_closure)
     rcases ha with ⟨has, ⟨_, ha⟩⟩
-    rw [← mem_compl_iff, ← closure_compl] at ha
+    rw [← mem_compl_iff, ← closure_compl] at ha 
     apply hg a ⟨has, ha⟩
   · exact hf.mono (inter_subset_inter_right s subset_closure)
   · exact hg.mono (inter_subset_inter_right s subset_closure)

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

@@ -37,7 +37,7 @@ equipped with the subspace topology.
 
 open Set Filter Function
 
-open Topology Filter
+open scoped Topology Filter
 
 variable {α : Type _} {β : Type _} {γ : Type _} {δ : Type _}
 

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

@@ -72,12 +72,6 @@ theorem frequently_nhdsWithin_iff {z : α} {s : Set α} {p : α → Prop} :
 #align frequently_nhds_within_iff frequently_nhdsWithin_iff
 -/
 
-/- warning: mem_closure_ne_iff_frequently_within -> mem_closure_ne_iff_frequently_within is a dubious translation:
-lean 3 declaration is
-  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] {z : α} {s : Set.{u1} α}, Iff (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) z (closure.{u1} α _inst_1 (SDiff.sdiff.{u1} (Set.{u1} α) (BooleanAlgebra.toHasSdiff.{u1} (Set.{u1} α) (Set.booleanAlgebra.{u1} α)) s (Singleton.singleton.{u1, u1} α (Set.{u1} α) (Set.hasSingleton.{u1} α) z)))) (Filter.Frequently.{u1} α (fun (x : α) => Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) x s) (nhdsWithin.{u1} α _inst_1 z (HasCompl.compl.{u1} (Set.{u1} α) (BooleanAlgebra.toHasCompl.{u1} (Set.{u1} α) (Set.booleanAlgebra.{u1} α)) (Singleton.singleton.{u1, u1} α (Set.{u1} α) (Set.hasSingleton.{u1} α) z))))
-but is expected to have type
-  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] {z : α} {s : Set.{u1} α}, Iff (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) z (closure.{u1} α _inst_1 (SDiff.sdiff.{u1} (Set.{u1} α) (Set.instSDiffSet.{u1} α) s (Singleton.singleton.{u1, u1} α (Set.{u1} α) (Set.instSingletonSet.{u1} α) z)))) (Filter.Frequently.{u1} α (fun (x : α) => Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) x s) (nhdsWithin.{u1} α _inst_1 z (HasCompl.compl.{u1} (Set.{u1} α) (BooleanAlgebra.toHasCompl.{u1} (Set.{u1} α) (Set.instBooleanAlgebraSet.{u1} α)) (Singleton.singleton.{u1, u1} α (Set.{u1} α) (Set.instSingletonSet.{u1} α) z))))
-Case conversion may be inaccurate. Consider using '#align mem_closure_ne_iff_frequently_within mem_closure_ne_iff_frequently_withinₓ'. -/
 theorem mem_closure_ne_iff_frequently_within {z : α} {s : Set α} :
     z ∈ closure (s \ {z}) ↔ ∃ᶠ x in 𝓝[≠] z, x ∈ s := by
   simp [mem_closure_iff_frequently, frequently_nhdsWithin_iff]
@@ -94,12 +88,6 @@ theorem eventually_nhdsWithin_nhdsWithin {a : α} {s : Set α} {p : α → Prop}
 #align eventually_nhds_within_nhds_within eventually_nhdsWithin_nhdsWithin
 -/
 
-/- warning: nhds_within_eq -> nhdsWithin_eq is a dubious translation:
-lean 3 declaration is
-  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] (a : α) (s : Set.{u1} α), Eq.{succ u1} (Filter.{u1} α) (nhdsWithin.{u1} α _inst_1 a s) (iInf.{u1, succ u1} (Filter.{u1} α) (ConditionallyCompleteLattice.toHasInf.{u1} (Filter.{u1} α) (CompleteLattice.toConditionallyCompleteLattice.{u1} (Filter.{u1} α) (Filter.completeLattice.{u1} α))) (Set.{u1} α) (fun (t : Set.{u1} α) => iInf.{u1, 0} (Filter.{u1} α) (ConditionallyCompleteLattice.toHasInf.{u1} (Filter.{u1} α) (CompleteLattice.toConditionallyCompleteLattice.{u1} (Filter.{u1} α) (Filter.completeLattice.{u1} α))) (Membership.Mem.{u1, u1} (Set.{u1} α) (Set.{u1} (Set.{u1} α)) (Set.hasMem.{u1} (Set.{u1} α)) t (setOf.{u1} (Set.{u1} α) (fun (t : Set.{u1} α) => And (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) a t) (IsOpen.{u1} α _inst_1 t)))) (fun (H : Membership.Mem.{u1, u1} (Set.{u1} α) (Set.{u1} (Set.{u1} α)) (Set.hasMem.{u1} (Set.{u1} α)) t (setOf.{u1} (Set.{u1} α) (fun (t : Set.{u1} α) => And (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) a t) (IsOpen.{u1} α _inst_1 t)))) => Filter.principal.{u1} α (Inter.inter.{u1} (Set.{u1} α) (Set.hasInter.{u1} α) t s))))
-but is expected to have type
-  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] (a : α) (s : Set.{u1} α), Eq.{succ u1} (Filter.{u1} α) (nhdsWithin.{u1} α _inst_1 a s) (iInf.{u1, succ u1} (Filter.{u1} α) (ConditionallyCompleteLattice.toInfSet.{u1} (Filter.{u1} α) (CompleteLattice.toConditionallyCompleteLattice.{u1} (Filter.{u1} α) (Filter.instCompleteLatticeFilter.{u1} α))) (Set.{u1} α) (fun (t : Set.{u1} α) => iInf.{u1, 0} (Filter.{u1} α) (ConditionallyCompleteLattice.toInfSet.{u1} (Filter.{u1} α) (CompleteLattice.toConditionallyCompleteLattice.{u1} (Filter.{u1} α) (Filter.instCompleteLatticeFilter.{u1} α))) (Membership.mem.{u1, u1} (Set.{u1} α) (Set.{u1} (Set.{u1} α)) (Set.instMembershipSet.{u1} (Set.{u1} α)) t (setOf.{u1} (Set.{u1} α) (fun (t : Set.{u1} α) => And (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) a t) (IsOpen.{u1} α _inst_1 t)))) (fun (H : Membership.mem.{u1, u1} (Set.{u1} α) (Set.{u1} (Set.{u1} α)) (Set.instMembershipSet.{u1} (Set.{u1} α)) t (setOf.{u1} (Set.{u1} α) (fun (t : Set.{u1} α) => And (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) a t) (IsOpen.{u1} α _inst_1 t)))) => Filter.principal.{u1} α (Inter.inter.{u1} (Set.{u1} α) (Set.instInterSet.{u1} α) t s))))
-Case conversion may be inaccurate. Consider using '#align nhds_within_eq nhdsWithin_eqₓ'. -/
 theorem nhdsWithin_eq (a : α) (s : Set α) :
     𝓝[s] a = ⨅ t ∈ { t : Set α | a ∈ t ∧ IsOpen t }, 𝓟 (t ∩ s) :=
   ((nhds_basis_opens a).inf_principal s).eq_biInf
@@ -111,67 +99,31 @@ theorem nhdsWithin_univ (a : α) : 𝓝[Set.univ] a = 𝓝 a := by
 #align nhds_within_univ nhdsWithin_univ
 -/
 
-/- warning: nhds_within_has_basis -> nhdsWithin_hasBasis is a dubious translation:
-lean 3 declaration is
-  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : TopologicalSpace.{u1} α] {p : β -> Prop} {s : β -> (Set.{u1} α)} {a : α}, (Filter.HasBasis.{u1, succ u2} α β (nhds.{u1} α _inst_1 a) p s) -> (forall (t : Set.{u1} α), Filter.HasBasis.{u1, succ u2} α β (nhdsWithin.{u1} α _inst_1 a t) p (fun (i : β) => Inter.inter.{u1} (Set.{u1} α) (Set.hasInter.{u1} α) (s i) t))
-but is expected to have type
-  forall {α : Type.{u2}} {β : Type.{u1}} [_inst_1 : TopologicalSpace.{u2} α] {p : β -> Prop} {s : β -> (Set.{u2} α)} {a : α}, (Filter.HasBasis.{u2, succ u1} α β (nhds.{u2} α _inst_1 a) p s) -> (forall (t : Set.{u2} α), Filter.HasBasis.{u2, succ u1} α β (nhdsWithin.{u2} α _inst_1 a t) p (fun (i : β) => Inter.inter.{u2} (Set.{u2} α) (Set.instInterSet.{u2} α) (s i) t))
-Case conversion may be inaccurate. Consider using '#align nhds_within_has_basis nhdsWithin_hasBasisₓ'. -/
 theorem nhdsWithin_hasBasis {p : β → Prop} {s : β → Set α} {a : α} (h : (𝓝 a).HasBasis p s)
     (t : Set α) : (𝓝[t] a).HasBasis p fun i => s i ∩ t :=
   h.inf_principal t
 #align nhds_within_has_basis nhdsWithin_hasBasis
 
-/- warning: nhds_within_basis_open -> nhdsWithin_basis_open is a dubious translation:
-lean 3 declaration is
-  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] (a : α) (t : Set.{u1} α), Filter.HasBasis.{u1, succ u1} α (Set.{u1} α) (nhdsWithin.{u1} α _inst_1 a t) (fun (u : Set.{u1} α) => And (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) a u) (IsOpen.{u1} α _inst_1 u)) (fun (u : Set.{u1} α) => Inter.inter.{u1} (Set.{u1} α) (Set.hasInter.{u1} α) u t)
-but is expected to have type
-  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] (a : α) (t : Set.{u1} α), Filter.HasBasis.{u1, succ u1} α (Set.{u1} α) (nhdsWithin.{u1} α _inst_1 a t) (fun (u : Set.{u1} α) => And (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) a u) (IsOpen.{u1} α _inst_1 u)) (fun (u : Set.{u1} α) => Inter.inter.{u1} (Set.{u1} α) (Set.instInterSet.{u1} α) u t)
-Case conversion may be inaccurate. Consider using '#align nhds_within_basis_open nhdsWithin_basis_openₓ'. -/
 theorem nhdsWithin_basis_open (a : α) (t : Set α) :
     (𝓝[t] a).HasBasis (fun u => a ∈ u ∧ IsOpen u) fun u => u ∩ t :=
   nhdsWithin_hasBasis (nhds_basis_opens a) t
 #align nhds_within_basis_open nhdsWithin_basis_open
 
-/- warning: mem_nhds_within -> mem_nhdsWithin is a dubious translation:
-lean 3 declaration is
-  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] {t : Set.{u1} α} {a : α} {s : Set.{u1} α}, Iff (Membership.Mem.{u1, u1} (Set.{u1} α) (Filter.{u1} α) (Filter.hasMem.{u1} α) t (nhdsWithin.{u1} α _inst_1 a s)) (Exists.{succ u1} (Set.{u1} α) (fun (u : Set.{u1} α) => And (IsOpen.{u1} α _inst_1 u) (And (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) a u) (HasSubset.Subset.{u1} (Set.{u1} α) (Set.hasSubset.{u1} α) (Inter.inter.{u1} (Set.{u1} α) (Set.hasInter.{u1} α) u s) t))))
-but is expected to have type
-  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] {t : Set.{u1} α} {a : α} {s : Set.{u1} α}, Iff (Membership.mem.{u1, u1} (Set.{u1} α) (Filter.{u1} α) (instMembershipSetFilter.{u1} α) t (nhdsWithin.{u1} α _inst_1 a s)) (Exists.{succ u1} (Set.{u1} α) (fun (u : Set.{u1} α) => And (IsOpen.{u1} α _inst_1 u) (And (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) a u) (HasSubset.Subset.{u1} (Set.{u1} α) (Set.instHasSubsetSet.{u1} α) (Inter.inter.{u1} (Set.{u1} α) (Set.instInterSet.{u1} α) u s) t))))
-Case conversion may be inaccurate. Consider using '#align mem_nhds_within mem_nhdsWithinₓ'. -/
 theorem mem_nhdsWithin {t : Set α} {a : α} {s : Set α} :
     t ∈ 𝓝[s] a ↔ ∃ u, IsOpen u ∧ a ∈ u ∧ u ∩ s ⊆ t := by
   simpa only [exists_prop, and_assoc', and_comm'] using (nhdsWithin_basis_open a s).mem_iff
 #align mem_nhds_within mem_nhdsWithin
 
-/- warning: mem_nhds_within_iff_exists_mem_nhds_inter -> mem_nhdsWithin_iff_exists_mem_nhds_inter is a dubious translation:
-lean 3 declaration is
-  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] {t : Set.{u1} α} {a : α} {s : Set.{u1} α}, Iff (Membership.Mem.{u1, u1} (Set.{u1} α) (Filter.{u1} α) (Filter.hasMem.{u1} α) t (nhdsWithin.{u1} α _inst_1 a s)) (Exists.{succ u1} (Set.{u1} α) (fun (u : Set.{u1} α) => Exists.{0} (Membership.Mem.{u1, u1} (Set.{u1} α) (Filter.{u1} α) (Filter.hasMem.{u1} α) u (nhds.{u1} α _inst_1 a)) (fun (H : Membership.Mem.{u1, u1} (Set.{u1} α) (Filter.{u1} α) (Filter.hasMem.{u1} α) u (nhds.{u1} α _inst_1 a)) => HasSubset.Subset.{u1} (Set.{u1} α) (Set.hasSubset.{u1} α) (Inter.inter.{u1} (Set.{u1} α) (Set.hasInter.{u1} α) u s) t)))
-but is expected to have type
-  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] {t : Set.{u1} α} {a : α} {s : Set.{u1} α}, Iff (Membership.mem.{u1, u1} (Set.{u1} α) (Filter.{u1} α) (instMembershipSetFilter.{u1} α) t (nhdsWithin.{u1} α _inst_1 a s)) (Exists.{succ u1} (Set.{u1} α) (fun (u : Set.{u1} α) => And (Membership.mem.{u1, u1} (Set.{u1} α) (Filter.{u1} α) (instMembershipSetFilter.{u1} α) u (nhds.{u1} α _inst_1 a)) (HasSubset.Subset.{u1} (Set.{u1} α) (Set.instHasSubsetSet.{u1} α) (Inter.inter.{u1} (Set.{u1} α) (Set.instInterSet.{u1} α) u s) t)))
-Case conversion may be inaccurate. Consider using '#align mem_nhds_within_iff_exists_mem_nhds_inter mem_nhdsWithin_iff_exists_mem_nhds_interₓ'. -/
 theorem mem_nhdsWithin_iff_exists_mem_nhds_inter {t : Set α} {a : α} {s : Set α} :
     t ∈ 𝓝[s] a ↔ ∃ u ∈ 𝓝 a, u ∩ s ⊆ t :=
   (nhdsWithin_hasBasis (𝓝 a).basis_sets s).mem_iff
 #align mem_nhds_within_iff_exists_mem_nhds_inter mem_nhdsWithin_iff_exists_mem_nhds_inter
 
-/- warning: diff_mem_nhds_within_compl -> diff_mem_nhdsWithin_compl is a dubious translation:
-lean 3 declaration is
-  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] {x : α} {s : Set.{u1} α}, (Membership.Mem.{u1, u1} (Set.{u1} α) (Filter.{u1} α) (Filter.hasMem.{u1} α) s (nhds.{u1} α _inst_1 x)) -> (forall (t : Set.{u1} α), Membership.Mem.{u1, u1} (Set.{u1} α) (Filter.{u1} α) (Filter.hasMem.{u1} α) (SDiff.sdiff.{u1} (Set.{u1} α) (BooleanAlgebra.toHasSdiff.{u1} (Set.{u1} α) (Set.booleanAlgebra.{u1} α)) s t) (nhdsWithin.{u1} α _inst_1 x (HasCompl.compl.{u1} (Set.{u1} α) (BooleanAlgebra.toHasCompl.{u1} (Set.{u1} α) (Set.booleanAlgebra.{u1} α)) t)))
-but is expected to have type
-  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] {x : α} {s : Set.{u1} α}, (Membership.mem.{u1, u1} (Set.{u1} α) (Filter.{u1} α) (instMembershipSetFilter.{u1} α) s (nhds.{u1} α _inst_1 x)) -> (forall (t : Set.{u1} α), Membership.mem.{u1, u1} (Set.{u1} α) (Filter.{u1} α) (instMembershipSetFilter.{u1} α) (SDiff.sdiff.{u1} (Set.{u1} α) (Set.instSDiffSet.{u1} α) s t) (nhdsWithin.{u1} α _inst_1 x (HasCompl.compl.{u1} (Set.{u1} α) (BooleanAlgebra.toHasCompl.{u1} (Set.{u1} α) (Set.instBooleanAlgebraSet.{u1} α)) t)))
-Case conversion may be inaccurate. Consider using '#align diff_mem_nhds_within_compl diff_mem_nhdsWithin_complₓ'. -/
 theorem diff_mem_nhdsWithin_compl {x : α} {s : Set α} (hs : s ∈ 𝓝 x) (t : Set α) :
     s \ t ∈ 𝓝[tᶜ] x :=
   diff_mem_inf_principal_compl hs t
 #align diff_mem_nhds_within_compl diff_mem_nhdsWithin_compl
 
-/- warning: diff_mem_nhds_within_diff -> diff_mem_nhdsWithin_diff is a dubious translation:
-lean 3 declaration is
-  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] {x : α} {s : Set.{u1} α} {t : Set.{u1} α}, (Membership.Mem.{u1, u1} (Set.{u1} α) (Filter.{u1} α) (Filter.hasMem.{u1} α) s (nhdsWithin.{u1} α _inst_1 x t)) -> (forall (t' : Set.{u1} α), Membership.Mem.{u1, u1} (Set.{u1} α) (Filter.{u1} α) (Filter.hasMem.{u1} α) (SDiff.sdiff.{u1} (Set.{u1} α) (BooleanAlgebra.toHasSdiff.{u1} (Set.{u1} α) (Set.booleanAlgebra.{u1} α)) s t') (nhdsWithin.{u1} α _inst_1 x (SDiff.sdiff.{u1} (Set.{u1} α) (BooleanAlgebra.toHasSdiff.{u1} (Set.{u1} α) (Set.booleanAlgebra.{u1} α)) t t')))
-but is expected to have type
-  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] {x : α} {s : Set.{u1} α} {t : Set.{u1} α}, (Membership.mem.{u1, u1} (Set.{u1} α) (Filter.{u1} α) (instMembershipSetFilter.{u1} α) s (nhdsWithin.{u1} α _inst_1 x t)) -> (forall (t' : Set.{u1} α), Membership.mem.{u1, u1} (Set.{u1} α) (Filter.{u1} α) (instMembershipSetFilter.{u1} α) (SDiff.sdiff.{u1} (Set.{u1} α) (Set.instSDiffSet.{u1} α) s t') (nhdsWithin.{u1} α _inst_1 x (SDiff.sdiff.{u1} (Set.{u1} α) (Set.instSDiffSet.{u1} α) t t')))
-Case conversion may be inaccurate. Consider using '#align diff_mem_nhds_within_diff diff_mem_nhdsWithin_diffₓ'. -/
 theorem diff_mem_nhdsWithin_diff {x : α} {s t : Set α} (hs : s ∈ 𝓝[t] x) (t' : Set α) :
     s \ t' ∈ 𝓝[t \ t'] x :=
   by
@@ -195,12 +147,6 @@ theorem mem_nhdsWithin_iff_eventually {s t : Set α} {x : α} :
 #align mem_nhds_within_iff_eventually mem_nhdsWithin_iff_eventually
 -/
 
-/- warning: mem_nhds_within_iff_eventually_eq -> mem_nhdsWithin_iff_eventuallyEq is a dubious translation:
-lean 3 declaration is
-  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] {s : Set.{u1} α} {t : Set.{u1} α} {x : α}, Iff (Membership.Mem.{u1, u1} (Set.{u1} α) (Filter.{u1} α) (Filter.hasMem.{u1} α) t (nhdsWithin.{u1} α _inst_1 x s)) (Filter.EventuallyEq.{u1, 0} α Prop (nhds.{u1} α _inst_1 x) s (Inter.inter.{u1} (Set.{u1} α) (Set.hasInter.{u1} α) s t))
-but is expected to have type
-  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] {s : Set.{u1} α} {t : Set.{u1} α} {x : α}, Iff (Membership.mem.{u1, u1} (Set.{u1} α) (Filter.{u1} α) (instMembershipSetFilter.{u1} α) t (nhdsWithin.{u1} α _inst_1 x s)) (Filter.EventuallyEq.{u1, 0} α Prop (nhds.{u1} α _inst_1 x) s (Inter.inter.{u1} (Set.{u1} α) (Set.instInterSet.{u1} α) s t))
-Case conversion may be inaccurate. Consider using '#align mem_nhds_within_iff_eventually_eq mem_nhdsWithin_iff_eventuallyEqₓ'. -/
 theorem mem_nhdsWithin_iff_eventuallyEq {s t : Set α} {x : α} :
     t ∈ 𝓝[s] x ↔ s =ᶠ[𝓝 x] (s ∩ t : Set α) := by
   simp_rw [mem_nhdsWithin_iff_eventually, eventually_eq_set, mem_inter_iff, iff_self_and]
@@ -212,12 +158,6 @@ theorem nhdsWithin_eq_iff_eventuallyEq {s t : Set α} {x : α} : 𝓝[s] x = 
 #align nhds_within_eq_iff_eventually_eq nhdsWithin_eq_iff_eventuallyEq
 -/
 
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-lean 3 declaration is
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-Case conversion may be inaccurate. Consider using '#align nhds_within_le_iff nhdsWithin_le_iffₓ'. -/
 theorem nhdsWithin_le_iff {s t : Set α} {x : α} : 𝓝[s] x ≤ 𝓝[t] x ↔ t ∈ 𝓝[s] x :=
   set_eventuallyLE_iff_inf_principal_le.symm.trans set_eventuallyLE_iff_mem_inf_principal
 #align nhds_within_le_iff nhdsWithin_le_iff
@@ -258,32 +198,14 @@ theorem eventually_mem_nhdsWithin {a : α} {s : Set α} : ∀ᶠ x in 𝓝[s] a,
 #align eventually_mem_nhds_within eventually_mem_nhdsWithin
 -/
 
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-Case conversion may be inaccurate. Consider using '#align inter_mem_nhds_within inter_mem_nhdsWithinₓ'. -/
 theorem inter_mem_nhdsWithin (s : Set α) {t : Set α} {a : α} (h : t ∈ 𝓝 a) : s ∩ t ∈ 𝓝[s] a :=
   inter_mem self_mem_nhdsWithin (mem_inf_of_left h)
 #align inter_mem_nhds_within inter_mem_nhdsWithin
 
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-Case conversion may be inaccurate. Consider using '#align nhds_within_mono nhdsWithin_monoₓ'. -/
 theorem nhdsWithin_mono (a : α) {s t : Set α} (h : s ⊆ t) : 𝓝[s] a ≤ 𝓝[t] a :=
   inf_le_inf_left _ (principal_mono.mpr h)
 #align nhds_within_mono nhdsWithin_mono
 
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-Case conversion may be inaccurate. Consider using '#align pure_le_nhds_within pure_le_nhdsWithinₓ'. -/
 theorem pure_le_nhdsWithin {a : α} {s : Set α} (ha : a ∈ s) : pure a ≤ 𝓝[s] a :=
   le_inf (pure_le_nhds a) (le_principal_iff.2 ha)
 #align pure_le_nhds_within pure_le_nhdsWithin
@@ -308,75 +230,33 @@ theorem tendsto_const_nhdsWithin {l : Filter β} {s : Set α} {a : α} (ha : a 
 #align tendsto_const_nhds_within tendsto_const_nhdsWithin
 -/
 
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 theorem nhdsWithin_restrict'' {a : α} (s : Set α) {t : Set α} (h : t ∈ 𝓝[s] a) :
     𝓝[s] a = 𝓝[s ∩ t] a :=
   le_antisymm (le_inf inf_le_left (le_principal_iff.mpr (inter_mem self_mem_nhdsWithin h)))
     (inf_le_inf_left _ (principal_mono.mpr (Set.inter_subset_left _ _)))
 #align nhds_within_restrict'' nhdsWithin_restrict''
 
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 theorem nhdsWithin_restrict' {a : α} (s : Set α) {t : Set α} (h : t ∈ 𝓝 a) : 𝓝[s] a = 𝓝[s ∩ t] a :=
   nhdsWithin_restrict'' s <| mem_inf_of_left h
 #align nhds_within_restrict' nhdsWithin_restrict'
 
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-Case conversion may be inaccurate. Consider using '#align nhds_within_restrict nhdsWithin_restrictₓ'. -/
 theorem nhdsWithin_restrict {a : α} (s : Set α) {t : Set α} (h₀ : a ∈ t) (h₁ : IsOpen t) :
     𝓝[s] a = 𝓝[s ∩ t] a :=
   nhdsWithin_restrict' s (IsOpen.mem_nhds h₁ h₀)
 #align nhds_within_restrict nhdsWithin_restrict
 
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-Case conversion may be inaccurate. Consider using '#align nhds_within_le_of_mem nhdsWithin_le_of_memₓ'. -/
 theorem nhdsWithin_le_of_mem {a : α} {s t : Set α} (h : s ∈ 𝓝[t] a) : 𝓝[t] a ≤ 𝓝[s] a :=
   nhdsWithin_le_iff.mpr h
 #align nhds_within_le_of_mem nhdsWithin_le_of_mem
 
-/- warning: nhds_within_le_nhds -> nhdsWithin_le_nhds is a dubious translation:
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-Case conversion may be inaccurate. Consider using '#align nhds_within_le_nhds nhdsWithin_le_nhdsₓ'. -/
 theorem nhdsWithin_le_nhds {a : α} {s : Set α} : 𝓝[s] a ≤ 𝓝 a := by rw [← nhdsWithin_univ];
   apply nhdsWithin_le_of_mem; exact univ_mem
 #align nhds_within_le_nhds nhdsWithin_le_nhds
 
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-Case conversion may be inaccurate. Consider using '#align nhds_within_eq_nhds_within' nhdsWithin_eq_nhds_within'ₓ'. -/
 theorem nhdsWithin_eq_nhds_within' {a : α} {s t u : Set α} (hs : s ∈ 𝓝 a) (h₂ : t ∩ s = u ∩ s) :
     𝓝[t] a = 𝓝[u] a := by rw [nhdsWithin_restrict' t hs, nhdsWithin_restrict' u hs, h₂]
 #align nhds_within_eq_nhds_within' nhdsWithin_eq_nhds_within'
 
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 theorem nhdsWithin_eq_nhdsWithin {a : α} {s t u : Set α} (h₀ : a ∈ s) (h₁ : IsOpen s)
     (h₂ : t ∩ s = u ∩ s) : 𝓝[t] a = 𝓝[u] a := by
   rw [nhdsWithin_restrict t h₀ h₁, nhdsWithin_restrict u h₀ h₁, h₂]
@@ -404,94 +284,40 @@ theorem preimage_nhds_within_coinduced {π : α → β} {s : Set β} {t : Set α
 #align preimage_nhds_within_coinduced preimage_nhds_within_coinduced
 -/
 
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 @[simp]
 theorem nhdsWithin_empty (a : α) : 𝓝[∅] a = ⊥ := by rw [nhdsWithin, principal_empty, inf_bot_eq]
 #align nhds_within_empty nhdsWithin_empty
 
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 theorem nhdsWithin_union (a : α) (s t : Set α) : 𝓝[s ∪ t] a = 𝓝[s] a ⊔ 𝓝[t] a := by
   delta nhdsWithin; rw [← inf_sup_left, sup_principal]
 #align nhds_within_union nhdsWithin_union
 
-/- warning: nhds_within_bUnion -> nhdsWithin_biUnion is a dubious translation:
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-Case conversion may be inaccurate. Consider using '#align nhds_within_bUnion nhdsWithin_biUnionₓ'. -/
 theorem nhdsWithin_biUnion {ι} {I : Set ι} (hI : I.Finite) (s : ι → Set α) (a : α) :
     𝓝[⋃ i ∈ I, s i] a = ⨆ i ∈ I, 𝓝[s i] a :=
   Set.Finite.induction_on hI (by simp) fun t T _ _ hT => by
     simp only [hT, nhdsWithin_union, iSup_insert, bUnion_insert]
 #align nhds_within_bUnion nhdsWithin_biUnion
 
-/- warning: nhds_within_sUnion -> nhdsWithin_sUnion is a dubious translation:
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-Case conversion may be inaccurate. Consider using '#align nhds_within_sUnion nhdsWithin_sUnionₓ'. -/
 theorem nhdsWithin_sUnion {S : Set (Set α)} (hS : S.Finite) (a : α) : 𝓝[⋃₀ S] a = ⨆ s ∈ S, 𝓝[s] a :=
   by rw [sUnion_eq_bUnion, nhdsWithin_biUnion hS]
 #align nhds_within_sUnion nhdsWithin_sUnion
 
-/- warning: nhds_within_Union -> nhdsWithin_iUnion is a dubious translation:
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-Case conversion may be inaccurate. Consider using '#align nhds_within_Union nhdsWithin_iUnionₓ'. -/
 theorem nhdsWithin_iUnion {ι} [Finite ι] (s : ι → Set α) (a : α) : 𝓝[⋃ i, s i] a = ⨆ i, 𝓝[s i] a :=
   by rw [← sUnion_range, nhdsWithin_sUnion (finite_range s), iSup_range]
 #align nhds_within_Union nhdsWithin_iUnion
 
-/- warning: nhds_within_inter -> nhdsWithin_inter is a dubious translation:
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-Case conversion may be inaccurate. Consider using '#align nhds_within_inter nhdsWithin_interₓ'. -/
 theorem nhdsWithin_inter (a : α) (s t : Set α) : 𝓝[s ∩ t] a = 𝓝[s] a ⊓ 𝓝[t] a := by
   delta nhdsWithin; rw [inf_left_comm, inf_assoc, inf_principal, ← inf_assoc, inf_idem]
 #align nhds_within_inter nhdsWithin_inter
 
-/- warning: nhds_within_inter' -> nhdsWithin_inter' is a dubious translation:
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-Case conversion may be inaccurate. Consider using '#align nhds_within_inter' nhdsWithin_inter'ₓ'. -/
 theorem nhdsWithin_inter' (a : α) (s t : Set α) : 𝓝[s ∩ t] a = 𝓝[s] a ⊓ 𝓟 t := by delta nhdsWithin;
   rw [← inf_principal, inf_assoc]
 #align nhds_within_inter' nhdsWithin_inter'
 
-/- warning: nhds_within_inter_of_mem -> nhdsWithin_inter_of_mem is a dubious translation:
-lean 3 declaration is
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-Case conversion may be inaccurate. Consider using '#align nhds_within_inter_of_mem nhdsWithin_inter_of_memₓ'. -/
 theorem nhdsWithin_inter_of_mem {a : α} {s t : Set α} (h : s ∈ 𝓝[t] a) : 𝓝[s ∩ t] a = 𝓝[t] a := by
   rw [nhdsWithin_inter, inf_eq_right]; exact nhdsWithin_le_of_mem h
 #align nhds_within_inter_of_mem nhdsWithin_inter_of_mem
 
-/- warning: nhds_within_inter_of_mem' -> nhdsWithin_inter_of_mem' is a dubious translation:
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-Case conversion may be inaccurate. Consider using '#align nhds_within_inter_of_mem' nhdsWithin_inter_of_mem'ₓ'. -/
 theorem nhdsWithin_inter_of_mem' {a : α} {s t : Set α} (h : s ∈ 𝓝[t] a) : 𝓝[t ∩ s] a = 𝓝[t] a := by
   rw [inter_comm, nhdsWithin_inter_of_mem h]
 #align nhds_within_inter_of_mem' nhdsWithin_inter_of_mem'
@@ -503,12 +329,6 @@ theorem nhdsWithin_singleton (a : α) : 𝓝[{a}] a = pure a := by
 #align nhds_within_singleton nhdsWithin_singleton
 -/
 
-/- warning: nhds_within_insert -> nhdsWithin_insert is a dubious translation:
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-Case conversion may be inaccurate. Consider using '#align nhds_within_insert nhdsWithin_insertₓ'. -/
 @[simp]
 theorem nhdsWithin_insert (a : α) (s : Set α) : 𝓝[insert a s] a = pure a ⊔ 𝓝[s] a := by
   rw [← singleton_union, nhdsWithin_union, nhdsWithin_singleton]
@@ -526,46 +346,22 @@ theorem insert_mem_nhdsWithin_insert {a : α} {s t : Set α} (h : t ∈ 𝓝[s]
 #align insert_mem_nhds_within_insert insert_mem_nhdsWithin_insert
 -/
 
-/- warning: insert_mem_nhds_iff -> insert_mem_nhds_iff is a dubious translation:
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-Case conversion may be inaccurate. Consider using '#align insert_mem_nhds_iff insert_mem_nhds_iffₓ'. -/
 theorem insert_mem_nhds_iff {a : α} {s : Set α} : insert a s ∈ 𝓝 a ↔ s ∈ 𝓝[≠] a := by
   simp only [nhdsWithin, mem_inf_principal, mem_compl_iff, mem_singleton_iff, or_iff_not_imp_left,
     insert_def]
 #align insert_mem_nhds_iff insert_mem_nhds_iff
 
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-Case conversion may be inaccurate. Consider using '#align nhds_within_compl_singleton_sup_pure nhdsWithin_compl_singleton_sup_pureₓ'. -/
 @[simp]
 theorem nhdsWithin_compl_singleton_sup_pure (a : α) : 𝓝[≠] a ⊔ pure a = 𝓝 a := by
   rw [← nhdsWithin_singleton, ← nhdsWithin_union, compl_union_self, nhdsWithin_univ]
 #align nhds_within_compl_singleton_sup_pure nhdsWithin_compl_singleton_sup_pure
 
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 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
 theorem nhdsWithin_prod_eq {α : Type _} [TopologicalSpace α] {β : Type _} [TopologicalSpace β]
     (a : α) (b : β) (s : Set α) (t : Set β) : 𝓝[s ×ˢ t] (a, b) = 𝓝[s] a ×ᶠ 𝓝[t] b := by
   delta nhdsWithin; rw [nhds_prod_eq, ← Filter.prod_inf_prod, Filter.prod_principal_principal]
 #align nhds_within_prod_eq nhdsWithin_prod_eq
 
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 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
 theorem nhdsWithin_prod {α : Type _} [TopologicalSpace α] {β : Type _} [TopologicalSpace β]
@@ -573,12 +369,6 @@ theorem nhdsWithin_prod {α : Type _} [TopologicalSpace α] {β : Type _} [Topol
     u ×ˢ v ∈ 𝓝[s ×ˢ t] (a, b) := by rw [nhdsWithin_prod_eq]; exact prod_mem_prod hu hv
 #align nhds_within_prod nhdsWithin_prod
 
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 theorem nhdsWithin_pi_eq' {ι : Type _} {α : ι → Type _} [∀ i, TopologicalSpace (α i)] {I : Set ι}
     (hI : I.Finite) (s : ∀ i, Set (α i)) (x : ∀ i, α i) :
     𝓝[pi I s] x = ⨅ i, comap (fun x => x i) (𝓝 (x i) ⊓ ⨅ hi : i ∈ I, 𝓟 (s i)) := by
@@ -586,12 +376,6 @@ theorem nhdsWithin_pi_eq' {ι : Type _} {α : ι → Type _} [∀ i, Topological
     infi_principal_finite hI, ← iInf_inf_eq]
 #align nhds_within_pi_eq' nhdsWithin_pi_eq'
 
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 /- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (i «expr ∉ » I) -/
 theorem nhdsWithin_pi_eq {ι : Type _} {α : ι → Type _} [∀ i, TopologicalSpace (α i)] {I : Set ι}
     (hI : I.Finite) (s : ∀ i, Set (α i)) (x : ∀ i, α i) :
@@ -605,57 +389,27 @@ theorem nhdsWithin_pi_eq {ι : Type _} {α : ι → Type _} [∀ i, TopologicalS
   simp only [iInf_inf_eq]
 #align nhds_within_pi_eq nhdsWithin_pi_eq
 
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 theorem nhdsWithin_pi_univ_eq {ι : Type _} {α : ι → Type _} [Finite ι] [∀ i, TopologicalSpace (α i)]
     (s : ∀ i, Set (α i)) (x : ∀ i, α i) : 𝓝[pi univ s] x = ⨅ i, comap (fun x => x i) (𝓝[s i] x i) :=
   by simpa [nhdsWithin] using nhdsWithin_pi_eq finite_univ s x
 #align nhds_within_pi_univ_eq nhdsWithin_pi_univ_eq
 
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-Case conversion may be inaccurate. Consider using '#align nhds_within_pi_eq_bot nhdsWithin_pi_eq_botₓ'. -/
 theorem nhdsWithin_pi_eq_bot {ι : Type _} {α : ι → Type _} [∀ i, TopologicalSpace (α i)] {I : Set ι}
     {s : ∀ i, Set (α i)} {x : ∀ i, α i} : 𝓝[pi I s] x = ⊥ ↔ ∃ i ∈ I, 𝓝[s i] x i = ⊥ := by
   simp only [nhdsWithin, nhds_pi, pi_inf_principal_pi_eq_bot]
 #align nhds_within_pi_eq_bot nhdsWithin_pi_eq_bot
 
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 theorem nhdsWithin_pi_neBot {ι : Type _} {α : ι → Type _} [∀ i, TopologicalSpace (α i)] {I : Set ι}
     {s : ∀ i, Set (α i)} {x : ∀ i, α i} : (𝓝[pi I s] x).ne_bot ↔ ∀ i ∈ I, (𝓝[s i] x i).ne_bot := by
   simp [ne_bot_iff, nhdsWithin_pi_eq_bot]
 #align nhds_within_pi_ne_bot nhdsWithin_pi_neBot
 
-/- warning: filter.tendsto.piecewise_nhds_within -> Filter.Tendsto.piecewise_nhdsWithin is a dubious translation:
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 theorem Filter.Tendsto.piecewise_nhdsWithin {f g : α → β} {t : Set α} [∀ x, Decidable (x ∈ t)]
     {a : α} {s : Set α} {l : Filter β} (h₀ : Tendsto f (𝓝[s ∩ t] a) l)
     (h₁ : Tendsto g (𝓝[s ∩ tᶜ] a) l) : Tendsto (piecewise t f g) (𝓝[s] a) l := by
   apply tendsto.piecewise <;> rwa [← nhdsWithin_inter']
 #align filter.tendsto.piecewise_nhds_within Filter.Tendsto.piecewise_nhdsWithin
 
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-Case conversion may be inaccurate. Consider using '#align filter.tendsto.if_nhds_within Filter.Tendsto.if_nhdsWithinₓ'. -/
 theorem Filter.Tendsto.if_nhdsWithin {f g : α → β} {p : α → Prop} [DecidablePred p] {a : α}
     {s : Set α} {l : Filter β} (h₀ : Tendsto f (𝓝[s ∩ { x | p x }] a) l)
     (h₁ : Tendsto g (𝓝[s ∩ { x | ¬p x }] a) l) :
@@ -663,23 +417,11 @@ theorem Filter.Tendsto.if_nhdsWithin {f g : α → β} {p : α → Prop} [Decida
   h₀.piecewise_nhdsWithin h₁
 #align filter.tendsto.if_nhds_within Filter.Tendsto.if_nhdsWithin
 
-/- warning: map_nhds_within -> map_nhdsWithin is a dubious translation:
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-Case conversion may be inaccurate. Consider using '#align map_nhds_within map_nhdsWithinₓ'. -/
 theorem map_nhdsWithin (f : α → β) (a : α) (s : Set α) :
     map f (𝓝[s] a) = ⨅ t ∈ { t : Set α | a ∈ t ∧ IsOpen t }, 𝓟 (f '' (t ∩ s)) :=
   ((nhdsWithin_basis_open a s).map f).eq_biInf
 #align map_nhds_within map_nhdsWithin
 
-/- warning: tendsto_nhds_within_mono_left -> tendsto_nhdsWithin_mono_left is a dubious translation:
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-but is expected to have type
-  forall {α : Type.{u2}} {β : Type.{u1}} [_inst_1 : TopologicalSpace.{u2} α] {f : α -> β} {a : α} {s : Set.{u2} α} {t : Set.{u2} α} {l : Filter.{u1} β}, (HasSubset.Subset.{u2} (Set.{u2} α) (Set.instHasSubsetSet.{u2} α) s t) -> (Filter.Tendsto.{u2, u1} α β f (nhdsWithin.{u2} α _inst_1 a t) l) -> (Filter.Tendsto.{u2, u1} α β f (nhdsWithin.{u2} α _inst_1 a s) l)
-Case conversion may be inaccurate. Consider using '#align tendsto_nhds_within_mono_left tendsto_nhdsWithin_mono_leftₓ'. -/
 theorem tendsto_nhdsWithin_mono_left {f : α → β} {a : α} {s t : Set α} {l : Filter β} (hst : s ⊆ t)
     (h : Tendsto f (𝓝[t] a) l) : Tendsto f (𝓝[s] a) l :=
   h.mono_left <| nhdsWithin_mono a hst
@@ -692,23 +434,11 @@ theorem tendsto_nhdsWithin_mono_right {f : β → α} {l : Filter β} {a : α} {
 #align tendsto_nhds_within_mono_right tendsto_nhdsWithin_mono_right
 -/
 
-/- warning: tendsto_nhds_within_of_tendsto_nhds -> tendsto_nhdsWithin_of_tendsto_nhds is a dubious translation:
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-  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : TopologicalSpace.{u1} α] {f : α -> β} {a : α} {s : Set.{u1} α} {l : Filter.{u2} β}, (Filter.Tendsto.{u1, u2} α β f (nhds.{u1} α _inst_1 a) l) -> (Filter.Tendsto.{u1, u2} α β f (nhdsWithin.{u1} α _inst_1 a s) l)
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-Case conversion may be inaccurate. Consider using '#align tendsto_nhds_within_of_tendsto_nhds tendsto_nhdsWithin_of_tendsto_nhdsₓ'. -/
 theorem tendsto_nhdsWithin_of_tendsto_nhds {f : α → β} {a : α} {s : Set α} {l : Filter β}
     (h : Tendsto f (𝓝 a) l) : Tendsto f (𝓝[s] a) l :=
   h.mono_left inf_le_left
 #align tendsto_nhds_within_of_tendsto_nhds tendsto_nhdsWithin_of_tendsto_nhds
 
-/- warning: eventually_mem_of_tendsto_nhds_within -> eventually_mem_of_tendsto_nhdsWithin is a dubious translation:
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-  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : TopologicalSpace.{u1} α] {f : β -> α} {a : α} {s : Set.{u1} α} {l : Filter.{u2} β}, (Filter.Tendsto.{u2, u1} β α f l (nhdsWithin.{u1} α _inst_1 a s)) -> (Filter.Eventually.{u2} β (fun (i : β) => Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) (f i) s) l)
-but is expected to have type
-  forall {α : Type.{u2}} {β : Type.{u1}} [_inst_1 : TopologicalSpace.{u2} α] {f : β -> α} {a : α} {s : Set.{u2} α} {l : Filter.{u1} β}, (Filter.Tendsto.{u1, u2} β α f l (nhdsWithin.{u2} α _inst_1 a s)) -> (Filter.Eventually.{u1} β (fun (i : β) => Membership.mem.{u2, u2} α (Set.{u2} α) (Set.instMembershipSet.{u2} α) (f i) s) l)
-Case conversion may be inaccurate. Consider using '#align eventually_mem_of_tendsto_nhds_within eventually_mem_of_tendsto_nhdsWithinₓ'. -/
 theorem eventually_mem_of_tendsto_nhdsWithin {f : β → α} {a : α} {s : Set α} {l : Filter β}
     (h : Tendsto f l (𝓝[s] a)) : ∀ᶠ i in l, f i ∈ s :=
   by
@@ -717,12 +447,6 @@ theorem eventually_mem_of_tendsto_nhdsWithin {f : β → α} {a : α} {s : Set 
   exact (h univ ⟨mem_univ a, isOpen_univ⟩).2
 #align eventually_mem_of_tendsto_nhds_within eventually_mem_of_tendsto_nhdsWithin
 
-/- warning: tendsto_nhds_of_tendsto_nhds_within -> tendsto_nhds_of_tendsto_nhdsWithin is a dubious translation:
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-  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : TopologicalSpace.{u1} α] {f : β -> α} {a : α} {s : Set.{u1} α} {l : Filter.{u2} β}, (Filter.Tendsto.{u2, u1} β α f l (nhdsWithin.{u1} α _inst_1 a s)) -> (Filter.Tendsto.{u2, u1} β α f l (nhds.{u1} α _inst_1 a))
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-Case conversion may be inaccurate. Consider using '#align tendsto_nhds_of_tendsto_nhds_within tendsto_nhds_of_tendsto_nhdsWithinₓ'. -/
 theorem tendsto_nhds_of_tendsto_nhdsWithin {f : β → α} {a : α} {s : Set α} {l : Filter β}
     (h : Tendsto f l (𝓝[s] a)) : Tendsto f l (𝓝 a) :=
   h.mono_right nhdsWithin_le_nhds
@@ -755,79 +479,37 @@ theorem DenseRange.nhdsWithin_neBot {ι : Type _} {f : ι → α} (h : DenseRang
 #align dense_range.nhds_within_ne_bot DenseRange.nhdsWithin_neBot
 -/
 
-/- warning: mem_closure_pi -> mem_closure_pi is a dubious translation:
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 theorem mem_closure_pi {ι : Type _} {α : ι → Type _} [∀ i, TopologicalSpace (α i)] {I : Set ι}
     {s : ∀ i, Set (α i)} {x : ∀ i, α i} : x ∈ closure (pi I s) ↔ ∀ i ∈ I, x i ∈ closure (s i) := by
   simp only [mem_closure_iff_nhdsWithin_neBot, nhdsWithin_pi_neBot]
 #align mem_closure_pi mem_closure_pi
 
-/- warning: closure_pi_set -> closure_pi_set is a dubious translation:
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 theorem closure_pi_set {ι : Type _} {α : ι → Type _} [∀ i, TopologicalSpace (α i)] (I : Set ι)
     (s : ∀ i, Set (α i)) : closure (pi I s) = pi I fun i => closure (s i) :=
   Set.ext fun x => mem_closure_pi
 #align closure_pi_set closure_pi_set
 
-/- warning: dense_pi -> dense_pi is a dubious translation:
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-Case conversion may be inaccurate. Consider using '#align dense_pi dense_piₓ'. -/
 theorem dense_pi {ι : Type _} {α : ι → Type _} [∀ i, TopologicalSpace (α i)] {s : ∀ i, Set (α i)}
     (I : Set ι) (hs : ∀ i ∈ I, Dense (s i)) : Dense (pi I s) := by
   simp only [dense_iff_closure_eq, closure_pi_set, pi_congr rfl fun i hi => (hs i hi).closure_eq,
     pi_univ]
 #align dense_pi dense_pi
 
-/- warning: eventually_eq_nhds_within_iff -> eventuallyEq_nhdsWithin_iff is a dubious translation:
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 theorem eventuallyEq_nhdsWithin_iff {f g : α → β} {s : Set α} {a : α} :
     f =ᶠ[𝓝[s] a] g ↔ ∀ᶠ x in 𝓝 a, x ∈ s → f x = g x :=
   mem_inf_principal
 #align eventually_eq_nhds_within_iff eventuallyEq_nhdsWithin_iff
 
-/- warning: eventually_eq_nhds_within_of_eq_on -> eventuallyEq_nhdsWithin_of_eqOn is a dubious translation:
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 theorem eventuallyEq_nhdsWithin_of_eqOn {f g : α → β} {s : Set α} {a : α} (h : EqOn f g s) :
     f =ᶠ[𝓝[s] a] g :=
   mem_inf_of_right h
 #align eventually_eq_nhds_within_of_eq_on eventuallyEq_nhdsWithin_of_eqOn
 
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 theorem Set.EqOn.eventuallyEq_nhdsWithin {f g : α → β} {s : Set α} {a : α} (h : EqOn f g s) :
     f =ᶠ[𝓝[s] a] g :=
   eventuallyEq_nhdsWithin_of_eqOn h
 #align set.eq_on.eventually_eq_nhds_within Set.EqOn.eventuallyEq_nhdsWithin
 
-/- warning: tendsto_nhds_within_congr -> tendsto_nhdsWithin_congr is a dubious translation:
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 theorem tendsto_nhdsWithin_congr {f g : α → β} {s : Set α} {a : α} {l : Filter β}
     (hfg : ∀ x ∈ s, f x = g x) (hf : Tendsto f (𝓝[s] a) l) : Tendsto g (𝓝[s] a) l :=
   (tendsto_congr' <| eventuallyEq_nhdsWithin_of_eqOn hfg).1 hf
@@ -864,12 +546,6 @@ theorem tendsto_nhdsWithin_range {a : α} {l : Filter β} {f : β → α} :
 #align tendsto_nhds_within_range tendsto_nhdsWithin_range
 -/
 
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 theorem Filter.EventuallyEq.eq_of_nhdsWithin {s : Set α} {f g : α → β} {a : α} (h : f =ᶠ[𝓝[s] a] g)
     (hmem : a ∈ s) : f a = g a :=
   h.self_of_nhdsWithin hmem
@@ -923,12 +599,6 @@ theorem preimage_coe_mem_nhds_subtype {s t : Set α} {a : s} : coe ⁻¹' t ∈
 #align preimage_coe_mem_nhds_subtype preimage_coe_mem_nhds_subtype
 -/
 
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 theorem tendsto_nhdsWithin_iff_subtype {s : Set α} {a : α} (h : a ∈ s) (f : α → β) (l : Filter β) :
     Tendsto f (𝓝[s] a) l ↔ Tendsto (s.restrict f) (𝓝 ⟨a, h⟩) l := by
   simp only [tendsto, nhdsWithin_eq_map_subtype_coe h, Filter.map_map, restrict]
@@ -944,12 +614,6 @@ def ContinuousWithinAt (f : α → β) (s : Set α) (x : α) : Prop :=
 #align continuous_within_at ContinuousWithinAt
 -/
 
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 /-- If a function is continuous within `s` at `x`, then it tends to `f x` within `s` by definition.
 We register this fact for use with the dot notation, especially to use `tendsto.comp` as
 `continuous_within_at.comp` will have a different meaning. -/
@@ -966,67 +630,31 @@ def ContinuousOn (f : α → β) (s : Set α) : Prop :=
 #align continuous_on ContinuousOn
 -/
 
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 theorem ContinuousOn.continuousWithinAt {f : α → β} {s : Set α} {x : α} (hf : ContinuousOn f s)
     (hx : x ∈ s) : ContinuousWithinAt f s x :=
   hf x hx
 #align continuous_on.continuous_within_at ContinuousOn.continuousWithinAt
 
-/- warning: continuous_within_at_univ -> continuousWithinAt_univ is a dubious translation:
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 theorem continuousWithinAt_univ (f : α → β) (x : α) :
     ContinuousWithinAt f Set.univ x ↔ ContinuousAt f x := by
   rw [ContinuousAt, ContinuousWithinAt, nhdsWithin_univ]
 #align continuous_within_at_univ continuousWithinAt_univ
 
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 theorem continuousWithinAt_iff_continuousAt_restrict (f : α → β) {x : α} {s : Set α} (h : x ∈ s) :
     ContinuousWithinAt f s x ↔ ContinuousAt (s.restrict f) ⟨x, h⟩ :=
   tendsto_nhdsWithin_iff_subtype h f _
 #align continuous_within_at_iff_continuous_at_restrict continuousWithinAt_iff_continuousAt_restrict
 
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 theorem ContinuousWithinAt.tendsto_nhdsWithin {f : α → β} {x : α} {s : Set α} {t : Set β}
     (h : ContinuousWithinAt f s x) (ht : MapsTo f s t) : Tendsto f (𝓝[s] x) (𝓝[t] f x) :=
   tendsto_inf.2 ⟨h, tendsto_principal.2 <| mem_inf_of_right <| mem_principal.2 <| ht⟩
 #align continuous_within_at.tendsto_nhds_within ContinuousWithinAt.tendsto_nhdsWithin
 
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 theorem ContinuousWithinAt.tendsto_nhdsWithin_image {f : α → β} {x : α} {s : Set α}
     (h : ContinuousWithinAt f s x) : Tendsto f (𝓝[s] x) (𝓝[f '' s] f x) :=
   h.tendsto_nhdsWithin (mapsTo_image _ _)
 #align continuous_within_at.tendsto_nhds_within_image ContinuousWithinAt.tendsto_nhdsWithin_image
 
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 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
 theorem ContinuousWithinAt.prod_map {f : α → γ} {g : β → δ} {s : Set α} {t : Set β} {x : α} {y : β}
     (hf : ContinuousWithinAt f s x) (hg : ContinuousWithinAt g t y) :
@@ -1037,32 +665,17 @@ theorem ContinuousWithinAt.prod_map {f : α → γ} {g : β → δ} {s : Set α}
   exact hf.prod_map hg
 #align continuous_within_at.prod_map ContinuousWithinAt.prod_map
 
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 theorem continuousWithinAt_pi {ι : Type _} {π : ι → Type _} [∀ i, TopologicalSpace (π i)]
     {f : α → ∀ i, π i} {s : Set α} {x : α} :
     ContinuousWithinAt f s x ↔ ∀ i, ContinuousWithinAt (fun y => f y i) s x :=
   tendsto_pi_nhds
 #align continuous_within_at_pi continuousWithinAt_pi
 
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 theorem continuousOn_pi {ι : Type _} {π : ι → Type _} [∀ i, TopologicalSpace (π i)]
     {f : α → ∀ i, π i} {s : Set α} : ContinuousOn f s ↔ ∀ i, ContinuousOn (fun y => f y i) s :=
   ⟨fun h i x hx => tendsto_pi_nhds.1 (h x hx) i, fun h x hx => tendsto_pi_nhds.2 fun i => h i x hx⟩
 #align continuous_on_pi continuousOn_pi
 
-/- warning: continuous_within_at.fin_insert_nth -> ContinuousWithinAt.fin_insertNth is a dubious translation:
-<too large>
-Case conversion may be inaccurate. Consider using '#align continuous_within_at.fin_insert_nth ContinuousWithinAt.fin_insertNthₓ'. -/
 theorem ContinuousWithinAt.fin_insertNth {n} {π : Fin (n + 1) → Type _}
     [∀ i, TopologicalSpace (π i)] (i : Fin (n + 1)) {f : α → π i} {a : α} {s : Set α}
     (hf : ContinuousWithinAt f s a) {g : α → ∀ j : Fin n, π (i.succAbove j)}
@@ -1070,9 +683,6 @@ theorem ContinuousWithinAt.fin_insertNth {n} {π : Fin (n + 1) → Type _}
   hf.fin_insertNth i hg
 #align continuous_within_at.fin_insert_nth ContinuousWithinAt.fin_insertNth
 
-/- warning: continuous_on.fin_insert_nth -> ContinuousOn.fin_insertNth is a dubious translation:
-<too large>
-Case conversion may be inaccurate. Consider using '#align continuous_on.fin_insert_nth ContinuousOn.fin_insertNthₓ'. -/
 theorem ContinuousOn.fin_insertNth {n} {π : Fin (n + 1) → Type _} [∀ i, TopologicalSpace (π i)]
     (i : Fin (n + 1)) {f : α → π i} {s : Set α} (hf : ContinuousOn f s)
     {g : α → ∀ j : Fin n, π (i.succAbove j)} (hg : ContinuousOn g s) :
@@ -1080,24 +690,12 @@ theorem ContinuousOn.fin_insertNth {n} {π : Fin (n + 1) → Type _} [∀ i, Top
   (hf a ha).fin_insertNth i (hg a ha)
 #align continuous_on.fin_insert_nth ContinuousOn.fin_insertNth
 
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 theorem continuousOn_iff {f : α → β} {s : Set α} :
     ContinuousOn f s ↔
       ∀ x ∈ s, ∀ t : Set β, IsOpen t → f x ∈ t → ∃ u, IsOpen u ∧ x ∈ u ∧ u ∩ s ⊆ f ⁻¹' t :=
   by simp only [ContinuousOn, ContinuousWithinAt, tendsto_nhds, mem_nhdsWithin]
 #align continuous_on_iff continuousOn_iff
 
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 theorem continuousOn_iff_continuous_restrict {f : α → β} {s : Set α} :
     ContinuousOn f s ↔ Continuous (s.restrict f) :=
   by
@@ -1108,12 +706,6 @@ theorem continuousOn_iff_continuous_restrict {f : α → β} {s : Set α} :
   exact (continuousWithinAt_iff_continuousAt_restrict f xs).mpr (h ⟨x, xs⟩)
 #align continuous_on_iff_continuous_restrict continuousOn_iff_continuous_restrict
 
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 theorem continuousOn_iff' {f : α → β} {s : Set α} :
     ContinuousOn f s ↔ ∀ t : Set β, IsOpen t → ∃ u, IsOpen u ∧ f ⁻¹' t ∩ s = u ∩ s :=
   by
@@ -1126,12 +718,6 @@ theorem continuousOn_iff' {f : α → β} {s : Set α} :
   rw [continuousOn_iff_continuous_restrict, continuous_def] <;> simp only [this]
 #align continuous_on_iff' continuousOn_iff'
 
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 /-- If a function is continuous on a set for some topologies, then it is
 continuous on the same set with respect to any finer topology on the source space. -/
 theorem ContinuousOn.mono_dom {α β : Type _} {t₁ t₂ : TopologicalSpace α} {t₃ : TopologicalSpace β}
@@ -1143,12 +729,6 @@ theorem ContinuousOn.mono_dom {α β : Type _} {t₁ t₂ : TopologicalSpace α}
   exact ⟨u, h₁ u hu, h'u⟩
 #align continuous_on.mono_dom ContinuousOn.mono_dom
 
-/- warning: continuous_on.mono_rng -> ContinuousOn.mono_rng is a dubious translation:
-lean 3 declaration is
-  forall {α : Type.{u1}} {β : Type.{u2}} {t₁ : TopologicalSpace.{u1} α} {t₂ : TopologicalSpace.{u2} β} {t₃ : TopologicalSpace.{u2} β}, (LE.le.{u2} (TopologicalSpace.{u2} β) (Preorder.toHasLe.{u2} (TopologicalSpace.{u2} β) (PartialOrder.toPreorder.{u2} (TopologicalSpace.{u2} β) (TopologicalSpace.partialOrder.{u2} β))) t₂ t₃) -> (forall {s : Set.{u1} α} {f : α -> β}, (ContinuousOn.{u1, u2} α β t₁ t₂ f s) -> (ContinuousOn.{u1, u2} α β t₁ t₃ f s))
-but is expected to have type
-  forall {α : Type.{u2}} {β : Type.{u1}} {t₁ : TopologicalSpace.{u2} α} {t₂ : TopologicalSpace.{u1} β} {t₃ : TopologicalSpace.{u1} β}, (LE.le.{u1} (TopologicalSpace.{u1} β) (Preorder.toLE.{u1} (TopologicalSpace.{u1} β) (PartialOrder.toPreorder.{u1} (TopologicalSpace.{u1} β) (TopologicalSpace.instPartialOrderTopologicalSpace.{u1} β))) t₂ t₃) -> (forall {s : Set.{u2} α} {f : α -> β}, (ContinuousOn.{u2, u1} α β t₁ t₂ f s) -> (ContinuousOn.{u2, u1} α β t₁ t₃ f s))
-Case conversion may be inaccurate. Consider using '#align continuous_on.mono_rng ContinuousOn.mono_rngₓ'. -/
 /-- If a function is continuous on a set for some topologies, then it is
 continuous on the same set with respect to any coarser topology on the target space. -/
 theorem ContinuousOn.mono_rng {α β : Type _} {t₁ : TopologicalSpace α} {t₂ t₃ : TopologicalSpace β}
@@ -1159,12 +739,6 @@ theorem ContinuousOn.mono_rng {α β : Type _} {t₁ : TopologicalSpace α} {t
   exact h₂ t (h₁ t ht)
 #align continuous_on.mono_rng ContinuousOn.mono_rng
 
-/- warning: continuous_on_iff_is_closed -> continuousOn_iff_isClosed is a dubious translation:
-lean 3 declaration is
-  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : TopologicalSpace.{u1} α] [_inst_2 : TopologicalSpace.{u2} β] {f : α -> β} {s : Set.{u1} α}, Iff (ContinuousOn.{u1, u2} α β _inst_1 _inst_2 f s) (forall (t : Set.{u2} β), (IsClosed.{u2} β _inst_2 t) -> (Exists.{succ u1} (Set.{u1} α) (fun (u : Set.{u1} α) => And (IsClosed.{u1} α _inst_1 u) (Eq.{succ u1} (Set.{u1} α) (Inter.inter.{u1} (Set.{u1} α) (Set.hasInter.{u1} α) (Set.preimage.{u1, u2} α β f t) s) (Inter.inter.{u1} (Set.{u1} α) (Set.hasInter.{u1} α) u s)))))
-but is expected to have type
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-Case conversion may be inaccurate. Consider using '#align continuous_on_iff_is_closed continuousOn_iff_isClosedₓ'. -/
 theorem continuousOn_iff_isClosed {f : α → β} {s : Set α} :
     ContinuousOn f s ↔ ∀ t : Set β, IsClosed t → ∃ u, IsClosed u ∧ f ⁻¹' t ∩ s = u ∩ s :=
   by
@@ -1176,24 +750,12 @@ theorem continuousOn_iff_isClosed {f : α → β} {s : Set α} :
   rw [continuousOn_iff_continuous_restrict, continuous_iff_isClosed] <;> simp only [this]
 #align continuous_on_iff_is_closed continuousOn_iff_isClosed
 
-/- warning: continuous_on.prod_map -> ContinuousOn.prod_map 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 continuous_on.prod_map ContinuousOn.prod_mapₓ'. -/
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
 theorem ContinuousOn.prod_map {f : α → γ} {g : β → δ} {s : Set α} {t : Set β}
     (hf : ContinuousOn f s) (hg : ContinuousOn g t) : ContinuousOn (Prod.map f g) (s ×ˢ t) :=
   fun ⟨x, y⟩ ⟨hx, hy⟩ => ContinuousWithinAt.prod_map (hf x hx) (hg y hy)
 #align continuous_on.prod_map ContinuousOn.prod_map
 
-/- warning: continuous_of_cover_nhds -> continuous_of_cover_nhds is a dubious translation:
-lean 3 declaration is
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-but is expected to have type
-  forall {α : Type.{u2}} {β : Type.{u1}} [_inst_1 : TopologicalSpace.{u2} α] [_inst_2 : TopologicalSpace.{u1} β] {ι : Sort.{u3}} {f : α -> β} {s : ι -> (Set.{u2} α)}, (forall (x : α), Exists.{u3} ι (fun (i : ι) => Membership.mem.{u2, u2} (Set.{u2} α) (Filter.{u2} α) (instMembershipSetFilter.{u2} α) (s i) (nhds.{u2} α _inst_1 x))) -> (forall (i : ι), ContinuousOn.{u2, u1} α β _inst_1 _inst_2 f (s i)) -> (Continuous.{u2, u1} α β _inst_1 _inst_2 f)
-Case conversion may be inaccurate. Consider using '#align continuous_of_cover_nhds continuous_of_cover_nhdsₓ'. -/
 theorem continuous_of_cover_nhds {ι : Sort _} {f : α → β} {s : ι → Set α}
     (hs : ∀ x : α, ∃ i, s i ∈ 𝓝 x) (hf : ∀ i, ContinuousOn f (s i)) : Continuous f :=
   continuous_iff_continuousAt.mpr fun x =>
@@ -1202,44 +764,20 @@ theorem continuous_of_cover_nhds {ι : Sort _} {f : α → β} {s : ι → Set 
     rw [ContinuousAt, ← nhdsWithin_eq_nhds.2 hi]; exact hf _ _ (mem_of_mem_nhds hi)
 #align continuous_of_cover_nhds continuous_of_cover_nhds
 
-/- warning: continuous_on_empty -> continuousOn_empty is a dubious translation:
-lean 3 declaration is
-  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : TopologicalSpace.{u1} α] [_inst_2 : TopologicalSpace.{u2} β] (f : α -> β), ContinuousOn.{u1, u2} α β _inst_1 _inst_2 f (EmptyCollection.emptyCollection.{u1} (Set.{u1} α) (Set.hasEmptyc.{u1} α))
-but is expected to have type
-  forall {α : Type.{u2}} {β : Type.{u1}} [_inst_1 : TopologicalSpace.{u2} α] [_inst_2 : TopologicalSpace.{u1} β] (f : α -> β), ContinuousOn.{u2, u1} α β _inst_1 _inst_2 f (EmptyCollection.emptyCollection.{u2} (Set.{u2} α) (Set.instEmptyCollectionSet.{u2} α))
-Case conversion may be inaccurate. Consider using '#align continuous_on_empty continuousOn_emptyₓ'. -/
 theorem continuousOn_empty (f : α → β) : ContinuousOn f ∅ := fun x => False.elim
 #align continuous_on_empty continuousOn_empty
 
-/- warning: continuous_on_singleton -> continuousOn_singleton is a dubious translation:
-lean 3 declaration is
-  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : TopologicalSpace.{u1} α] [_inst_2 : TopologicalSpace.{u2} β] (f : α -> β) (a : α), ContinuousOn.{u1, u2} α β _inst_1 _inst_2 f (Singleton.singleton.{u1, u1} α (Set.{u1} α) (Set.hasSingleton.{u1} α) a)
-but is expected to have type
-  forall {α : Type.{u2}} {β : Type.{u1}} [_inst_1 : TopologicalSpace.{u2} α] [_inst_2 : TopologicalSpace.{u1} β] (f : α -> β) (a : α), ContinuousOn.{u2, u1} α β _inst_1 _inst_2 f (Singleton.singleton.{u2, u2} α (Set.{u2} α) (Set.instSingletonSet.{u2} α) a)
-Case conversion may be inaccurate. Consider using '#align continuous_on_singleton continuousOn_singletonₓ'. -/
 theorem continuousOn_singleton (f : α → β) (a : α) : ContinuousOn f {a} :=
   forall_eq.2 <| by
     simpa only [ContinuousWithinAt, nhdsWithin_singleton, tendsto_pure_left] using fun s =>
       mem_of_mem_nhds
 #align continuous_on_singleton continuousOn_singleton
 
-/- warning: set.subsingleton.continuous_on -> Set.Subsingleton.continuousOn is a dubious translation:
-lean 3 declaration is
-  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : TopologicalSpace.{u1} α] [_inst_2 : TopologicalSpace.{u2} β] {s : Set.{u1} α}, (Set.Subsingleton.{u1} α s) -> (forall (f : α -> β), ContinuousOn.{u1, u2} α β _inst_1 _inst_2 f s)
-but is expected to have type
-  forall {α : Type.{u2}} {β : Type.{u1}} [_inst_1 : TopologicalSpace.{u2} α] [_inst_2 : TopologicalSpace.{u1} β] {s : Set.{u2} α}, (Set.Subsingleton.{u2} α s) -> (forall (f : α -> β), ContinuousOn.{u2, u1} α β _inst_1 _inst_2 f s)
-Case conversion may be inaccurate. Consider using '#align set.subsingleton.continuous_on Set.Subsingleton.continuousOnₓ'. -/
 theorem Set.Subsingleton.continuousOn {s : Set α} (hs : s.Subsingleton) (f : α → β) :
     ContinuousOn f s :=
   hs.inductionOn (continuousOn_empty f) (continuousOn_singleton f)
 #align set.subsingleton.continuous_on Set.Subsingleton.continuousOn
 
-/- warning: nhds_within_le_comap -> nhdsWithin_le_comap is a dubious translation:
-lean 3 declaration is
-  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : TopologicalSpace.{u1} α] [_inst_2 : TopologicalSpace.{u2} β] {x : α} {s : Set.{u1} α} {f : α -> β}, (ContinuousWithinAt.{u1, u2} α β _inst_1 _inst_2 f s x) -> (LE.le.{u1} (Filter.{u1} α) (Preorder.toHasLe.{u1} (Filter.{u1} α) (PartialOrder.toPreorder.{u1} (Filter.{u1} α) (Filter.partialOrder.{u1} α))) (nhdsWithin.{u1} α _inst_1 x s) (Filter.comap.{u1, u2} α β f (nhdsWithin.{u2} β _inst_2 (f x) (Set.image.{u1, u2} α β f s))))
-but is expected to have type
-  forall {α : Type.{u2}} {β : Type.{u1}} [_inst_1 : TopologicalSpace.{u2} α] [_inst_2 : TopologicalSpace.{u1} β] {x : α} {s : Set.{u2} α} {f : α -> β}, (ContinuousWithinAt.{u2, u1} α β _inst_1 _inst_2 f s x) -> (LE.le.{u2} (Filter.{u2} α) (Preorder.toLE.{u2} (Filter.{u2} α) (PartialOrder.toPreorder.{u2} (Filter.{u2} α) (Filter.instPartialOrderFilter.{u2} α))) (nhdsWithin.{u2} α _inst_1 x s) (Filter.comap.{u2, u1} α β f (nhdsWithin.{u1} β _inst_2 (f x) (Set.image.{u2, u1} α β f s))))
-Case conversion may be inaccurate. Consider using '#align nhds_within_le_comap nhdsWithin_le_comapₓ'. -/
 theorem nhdsWithin_le_comap {x : α} {s : Set α} {f : α → β} (ctsf : ContinuousWithinAt f s x) :
     𝓝[s] x ≤ comap f (𝓝[f '' s] f x) :=
   ctsf.tendsto_nhdsWithin_image.le_comap
@@ -1252,167 +790,77 @@ theorem comap_nhdsWithin_range {α} (f : α → β) (y : β) : comap f (𝓝[ran
 #align comap_nhds_within_range comap_nhdsWithin_range
 -/
 
-/- warning: continuous_iff_continuous_on_univ -> continuous_iff_continuousOn_univ is a dubious translation:
-lean 3 declaration is
-  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : TopologicalSpace.{u1} α] [_inst_2 : TopologicalSpace.{u2} β] {f : α -> β}, Iff (Continuous.{u1, u2} α β _inst_1 _inst_2 f) (ContinuousOn.{u1, u2} α β _inst_1 _inst_2 f (Set.univ.{u1} α))
-but is expected to have type
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-Case conversion may be inaccurate. Consider using '#align continuous_iff_continuous_on_univ continuous_iff_continuousOn_univₓ'. -/
 theorem continuous_iff_continuousOn_univ {f : α → β} : Continuous f ↔ ContinuousOn f univ := by
   simp [continuous_iff_continuousAt, ContinuousOn, ContinuousAt, ContinuousWithinAt,
     nhdsWithin_univ]
 #align continuous_iff_continuous_on_univ continuous_iff_continuousOn_univ
 
-/- warning: continuous_within_at.mono -> ContinuousWithinAt.mono 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 continuous_within_at.mono ContinuousWithinAt.monoₓ'. -/
 theorem ContinuousWithinAt.mono {f : α → β} {s t : Set α} {x : α} (h : ContinuousWithinAt f t x)
     (hs : s ⊆ t) : ContinuousWithinAt f s x :=
   h.mono_left (nhdsWithin_mono x hs)
 #align continuous_within_at.mono ContinuousWithinAt.mono
 
-/- warning: continuous_within_at.mono_of_mem -> ContinuousWithinAt.mono_of_mem 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 continuous_within_at.mono_of_mem ContinuousWithinAt.mono_of_memₓ'. -/
 theorem ContinuousWithinAt.mono_of_mem {f : α → β} {s t : Set α} {x : α}
     (h : ContinuousWithinAt f t x) (hs : t ∈ 𝓝[s] x) : ContinuousWithinAt f s x :=
   h.mono_left (nhdsWithin_le_of_mem hs)
 #align continuous_within_at.mono_of_mem ContinuousWithinAt.mono_of_mem
 
-/- warning: continuous_within_at_inter' -> continuousWithinAt_inter' is a dubious translation:
-lean 3 declaration is
-  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : TopologicalSpace.{u1} α] [_inst_2 : TopologicalSpace.{u2} β] {f : α -> β} {s : Set.{u1} α} {t : Set.{u1} α} {x : α}, (Membership.Mem.{u1, u1} (Set.{u1} α) (Filter.{u1} α) (Filter.hasMem.{u1} α) t (nhdsWithin.{u1} α _inst_1 x s)) -> (Iff (ContinuousWithinAt.{u1, u2} α β _inst_1 _inst_2 f (Inter.inter.{u1} (Set.{u1} α) (Set.hasInter.{u1} α) s t) x) (ContinuousWithinAt.{u1, u2} α β _inst_1 _inst_2 f s x))
-but is expected to have type
-  forall {α : Type.{u2}} {β : Type.{u1}} [_inst_1 : TopologicalSpace.{u2} α] [_inst_2 : TopologicalSpace.{u1} β] {f : α -> β} {s : Set.{u2} α} {t : Set.{u2} α} {x : α}, (Membership.mem.{u2, u2} (Set.{u2} α) (Filter.{u2} α) (instMembershipSetFilter.{u2} α) t (nhdsWithin.{u2} α _inst_1 x s)) -> (Iff (ContinuousWithinAt.{u2, u1} α β _inst_1 _inst_2 f (Inter.inter.{u2} (Set.{u2} α) (Set.instInterSet.{u2} α) s t) x) (ContinuousWithinAt.{u2, u1} α β _inst_1 _inst_2 f s x))
-Case conversion may be inaccurate. Consider using '#align continuous_within_at_inter' continuousWithinAt_inter'ₓ'. -/
 theorem continuousWithinAt_inter' {f : α → β} {s t : Set α} {x : α} (h : t ∈ 𝓝[s] x) :
     ContinuousWithinAt f (s ∩ t) x ↔ ContinuousWithinAt f s x := by
   simp [ContinuousWithinAt, nhdsWithin_restrict'' s h]
 #align continuous_within_at_inter' continuousWithinAt_inter'
 
-/- warning: continuous_within_at_inter -> continuousWithinAt_inter 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 continuous_within_at_inter continuousWithinAt_interₓ'. -/
 theorem continuousWithinAt_inter {f : α → β} {s t : Set α} {x : α} (h : t ∈ 𝓝 x) :
     ContinuousWithinAt f (s ∩ t) x ↔ ContinuousWithinAt f s x := by
   simp [ContinuousWithinAt, nhdsWithin_restrict' s h]
 #align continuous_within_at_inter continuousWithinAt_inter
 
-/- warning: continuous_within_at_union -> continuousWithinAt_union 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 continuous_within_at_union continuousWithinAt_unionₓ'. -/
 theorem continuousWithinAt_union {f : α → β} {s t : Set α} {x : α} :
     ContinuousWithinAt f (s ∪ t) x ↔ ContinuousWithinAt f s x ∧ ContinuousWithinAt f t x := by
   simp only [ContinuousWithinAt, nhdsWithin_union, tendsto_sup]
 #align continuous_within_at_union continuousWithinAt_union
 
-/- warning: continuous_within_at.union -> ContinuousWithinAt.union 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 continuous_within_at.union ContinuousWithinAt.unionₓ'. -/
 theorem ContinuousWithinAt.union {f : α → β} {s t : Set α} {x : α} (hs : ContinuousWithinAt f s x)
     (ht : ContinuousWithinAt f t x) : ContinuousWithinAt f (s ∪ t) x :=
   continuousWithinAt_union.2 ⟨hs, ht⟩
 #align continuous_within_at.union ContinuousWithinAt.union
 
-/- warning: continuous_within_at.mem_closure_image -> ContinuousWithinAt.mem_closure_image 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 continuous_within_at.mem_closure_image ContinuousWithinAt.mem_closure_imageₓ'. -/
 theorem ContinuousWithinAt.mem_closure_image {f : α → β} {s : Set α} {x : α}
     (h : ContinuousWithinAt f s x) (hx : x ∈ closure s) : f x ∈ closure (f '' s) :=
   haveI := mem_closure_iff_nhdsWithin_neBot.1 hx
   mem_closure_of_tendsto h <| mem_of_superset self_mem_nhdsWithin (subset_preimage_image f s)
 #align continuous_within_at.mem_closure_image ContinuousWithinAt.mem_closure_image
 
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-lean 3 declaration is
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-Case conversion may be inaccurate. Consider using '#align continuous_within_at.mem_closure ContinuousWithinAt.mem_closureₓ'. -/
 theorem ContinuousWithinAt.mem_closure {f : α → β} {s : Set α} {x : α} {A : Set β}
     (h : ContinuousWithinAt f s x) (hx : x ∈ closure s) (hA : MapsTo f s A) : f x ∈ closure A :=
   closure_mono (image_subset_iff.2 hA) (h.mem_closure_image hx)
 #align continuous_within_at.mem_closure ContinuousWithinAt.mem_closure
 
-/- warning: set.maps_to.closure_of_continuous_within_at -> Set.MapsTo.closure_of_continuousWithinAt 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 set.maps_to.closure_of_continuous_within_at Set.MapsTo.closure_of_continuousWithinAtₓ'. -/
 theorem Set.MapsTo.closure_of_continuousWithinAt {f : α → β} {s : Set α} {t : Set β}
     (h : MapsTo f s t) (hc : ∀ x ∈ closure s, ContinuousWithinAt f s x) :
     MapsTo f (closure s) (closure t) := fun x hx => (hc x hx).mem_closure hx h
 #align set.maps_to.closure_of_continuous_within_at Set.MapsTo.closure_of_continuousWithinAt
 
-/- warning: set.maps_to.closure_of_continuous_on -> Set.MapsTo.closure_of_continuousOn 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 set.maps_to.closure_of_continuous_on Set.MapsTo.closure_of_continuousOnₓ'. -/
 theorem Set.MapsTo.closure_of_continuousOn {f : α → β} {s : Set α} {t : Set β} (h : MapsTo f s t)
     (hc : ContinuousOn f (closure s)) : MapsTo f (closure s) (closure t) :=
   h.closure_of_continuousWithinAt fun x hx => (hc x hx).mono subset_closure
 #align set.maps_to.closure_of_continuous_on Set.MapsTo.closure_of_continuousOn
 
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-lean 3 declaration is
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-Case conversion may be inaccurate. Consider using '#align continuous_within_at.image_closure ContinuousWithinAt.image_closureₓ'. -/
 theorem ContinuousWithinAt.image_closure {f : α → β} {s : Set α}
     (hf : ∀ x ∈ closure s, ContinuousWithinAt f s x) : f '' closure s ⊆ closure (f '' s) :=
   mapsTo'.1 <| (mapsTo_image f s).closure_of_continuousWithinAt hf
 #align continuous_within_at.image_closure ContinuousWithinAt.image_closure
 
-/- warning: continuous_on.image_closure -> ContinuousOn.image_closure 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 continuous_on.image_closure ContinuousOn.image_closureₓ'. -/
 theorem ContinuousOn.image_closure {f : α → β} {s : Set α} (hf : ContinuousOn f (closure s)) :
     f '' closure s ⊆ closure (f '' s) :=
   ContinuousWithinAt.image_closure fun x hx => (hf x hx).mono subset_closure
 #align continuous_on.image_closure ContinuousOn.image_closure
 
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-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 continuous_within_at_singleton continuousWithinAt_singletonₓ'. -/
 @[simp]
 theorem continuousWithinAt_singleton {f : α → β} {x : α} : ContinuousWithinAt f {x} x := by
   simp only [ContinuousWithinAt, nhdsWithin_singleton, tendsto_pure_nhds]
 #align continuous_within_at_singleton continuousWithinAt_singleton
 
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-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 continuous_within_at_insert_self continuousWithinAt_insert_selfₓ'. -/
 @[simp]
 theorem continuousWithinAt_insert_self {f : α → β} {x : α} {s : Set α} :
     ContinuousWithinAt f (insert x s) x ↔ ContinuousWithinAt f s x := by
@@ -1420,57 +868,27 @@ theorem continuousWithinAt_insert_self {f : α → β} {x : α} {s : Set α} :
     true_and_iff]
 #align continuous_within_at_insert_self continuousWithinAt_insert_self
 
-/- warning: continuous_within_at.insert_self -> ContinuousWithinAt.insert_self 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 continuous_within_at.insert_self ContinuousWithinAt.insert_selfₓ'. -/
 alias continuousWithinAt_insert_self ↔ _ ContinuousWithinAt.insert_self
 #align continuous_within_at.insert_self ContinuousWithinAt.insert_self
 
-/- warning: continuous_within_at.diff_iff -> ContinuousWithinAt.diff_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 continuous_within_at.diff_iff ContinuousWithinAt.diff_iffₓ'. -/
 theorem ContinuousWithinAt.diff_iff {f : α → β} {s t : Set α} {x : α}
     (ht : ContinuousWithinAt f t x) : ContinuousWithinAt f (s \ t) x ↔ ContinuousWithinAt f s x :=
   ⟨fun h => (h.union ht).mono <| by simp only [diff_union_self, subset_union_left], fun h =>
     h.mono (diff_subset _ _)⟩
 #align continuous_within_at.diff_iff ContinuousWithinAt.diff_iff
 
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-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 continuous_within_at_diff_self continuousWithinAt_diff_selfₓ'. -/
 @[simp]
 theorem continuousWithinAt_diff_self {f : α → β} {s : Set α} {x : α} :
     ContinuousWithinAt f (s \ {x}) x ↔ ContinuousWithinAt f s x :=
   continuousWithinAt_singleton.diff_iff
 #align continuous_within_at_diff_self continuousWithinAt_diff_self
 
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-lean 3 declaration is
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 @[simp]
 theorem continuousWithinAt_compl_self {f : α → β} {a : α} :
     ContinuousWithinAt f ({a}ᶜ) a ↔ ContinuousAt f a := by
   rw [compl_eq_univ_diff, continuousWithinAt_diff_self, continuousWithinAt_univ]
 #align continuous_within_at_compl_self continuousWithinAt_compl_self
 
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 @[simp]
 theorem continuousWithinAt_update_same [DecidableEq α] {f : α → β} {s : Set α} {x : α} {y : β} :
     ContinuousWithinAt (update f x y) s x ↔ Tendsto f (𝓝[s \ {x}] x) (𝓝 y) :=
@@ -1483,24 +901,12 @@ theorem continuousWithinAt_update_same [DecidableEq α] {f : α → β} {s : Set
     
 #align continuous_within_at_update_same continuousWithinAt_update_same
 
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 @[simp]
 theorem continuousAt_update_same [DecidableEq α] {f : α → β} {x : α} {y : β} :
     ContinuousAt (Function.update f x y) x ↔ Tendsto f (𝓝[≠] x) (𝓝 y) := by
   rw [← continuousWithinAt_univ, continuousWithinAt_update_same, compl_eq_univ_diff]
 #align continuous_at_update_same continuousAt_update_same
 
-/- warning: is_open_map.continuous_on_image_of_left_inv_on -> IsOpenMap.continuousOn_image_of_leftInvOn is a dubious translation:
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 theorem IsOpenMap.continuousOn_image_of_leftInvOn {f : α → β} {s : Set α}
     (h : IsOpenMap (s.restrict f)) {finv : β → α} (hleft : LeftInvOn finv f s) :
     ContinuousOn finv (f '' s) :=
@@ -1510,12 +916,6 @@ theorem IsOpenMap.continuousOn_image_of_leftInvOn {f : α → β} {s : Set α}
   · rw [inter_eq_self_of_subset_left (image_subset f (inter_subset_right t s)), hleft.image_inter']
 #align is_open_map.continuous_on_image_of_left_inv_on IsOpenMap.continuousOn_image_of_leftInvOn
 
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 theorem IsOpenMap.continuousOn_range_of_leftInverse {f : α → β} (hf : IsOpenMap f) {finv : β → α}
     (hleft : Function.LeftInverse finv f) : ContinuousOn finv (range f) :=
   by
@@ -1523,12 +923,6 @@ theorem IsOpenMap.continuousOn_range_of_leftInverse {f : α → β} (hf : IsOpen
   exact (hf.restrict isOpen_univ).continuousOn_image_of_leftInvOn fun x _ => hleft x
 #align is_open_map.continuous_on_range_of_left_inverse IsOpenMap.continuousOn_range_of_leftInverse
 
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 theorem ContinuousOn.congr_mono {f g : α → β} {s s₁ : Set α} (h : ContinuousOn f s)
     (h' : EqOn g f s₁) (h₁ : s₁ ⊆ s) : ContinuousOn g s₁ :=
   by
@@ -1540,210 +934,96 @@ theorem ContinuousOn.congr_mono {f g : α → β} {s s₁ : Set α} (h : Continu
   exact A.congr' h'.eventually_eq_nhds_within.symm
 #align continuous_on.congr_mono ContinuousOn.congr_mono
 
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 theorem ContinuousOn.congr {f g : α → β} {s : Set α} (h : ContinuousOn f s) (h' : EqOn g f s) :
     ContinuousOn g s :=
   h.congr_mono h' (Subset.refl _)
 #align continuous_on.congr ContinuousOn.congr
 
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 theorem continuousOn_congr {f g : α → β} {s : Set α} (h' : EqOn g f s) :
     ContinuousOn g s ↔ ContinuousOn f s :=
   ⟨fun h => ContinuousOn.congr h h'.symm, fun h => h.congr h'⟩
 #align continuous_on_congr continuousOn_congr
 
-/- warning: continuous_at.continuous_within_at -> ContinuousAt.continuousWithinAt is a dubious translation:
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 theorem ContinuousAt.continuousWithinAt {f : α → β} {s : Set α} {x : α} (h : ContinuousAt f x) :
     ContinuousWithinAt f s x :=
   ContinuousWithinAt.mono ((continuousWithinAt_univ f x).2 h) (subset_univ _)
 #align continuous_at.continuous_within_at ContinuousAt.continuousWithinAt
 
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 theorem continuousWithinAt_iff_continuousAt {f : α → β} {s : Set α} {x : α} (h : s ∈ 𝓝 x) :
     ContinuousWithinAt f s x ↔ ContinuousAt f x := by
   rw [← univ_inter s, continuousWithinAt_inter h, continuousWithinAt_univ]
 #align continuous_within_at_iff_continuous_at continuousWithinAt_iff_continuousAt
 
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 theorem ContinuousWithinAt.continuousAt {f : α → β} {s : Set α} {x : α}
     (h : ContinuousWithinAt f s x) (hs : s ∈ 𝓝 x) : ContinuousAt f x :=
   (continuousWithinAt_iff_continuousAt hs).mp h
 #align continuous_within_at.continuous_at ContinuousWithinAt.continuousAt
 
-/- warning: is_open.continuous_on_iff -> IsOpen.continuousOn_iff is a dubious translation:
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 theorem IsOpen.continuousOn_iff {f : α → β} {s : Set α} (hs : IsOpen s) :
     ContinuousOn f s ↔ ∀ ⦃a⦄, a ∈ s → ContinuousAt f a :=
   ball_congr fun _ => continuousWithinAt_iff_continuousAt ∘ hs.mem_nhds
 #align is_open.continuous_on_iff IsOpen.continuousOn_iff
 
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-Case conversion may be inaccurate. Consider using '#align continuous_on.continuous_at ContinuousOn.continuousAtₓ'. -/
 theorem ContinuousOn.continuousAt {f : α → β} {s : Set α} {x : α} (h : ContinuousOn f s)
     (hx : s ∈ 𝓝 x) : ContinuousAt f x :=
   (h x (mem_of_mem_nhds hx)).ContinuousAt hx
 #align continuous_on.continuous_at ContinuousOn.continuousAt
 
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-Case conversion may be inaccurate. Consider using '#align continuous_at.continuous_on ContinuousAt.continuousOnₓ'. -/
 theorem ContinuousAt.continuousOn {f : α → β} {s : Set α} (hcont : ∀ x ∈ s, ContinuousAt f x) :
     ContinuousOn f s := fun x hx => (hcont x hx).ContinuousWithinAt
 #align continuous_at.continuous_on ContinuousAt.continuousOn
 
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-Case conversion may be inaccurate. Consider using '#align continuous_within_at.comp ContinuousWithinAt.compₓ'. -/
 theorem ContinuousWithinAt.comp {g : β → γ} {f : α → β} {s : Set α} {t : Set β} {x : α}
     (hg : ContinuousWithinAt g t (f x)) (hf : ContinuousWithinAt f s x) (h : MapsTo f s t) :
     ContinuousWithinAt (g ∘ f) s x :=
   hg.Tendsto.comp (hf.tendsto_nhdsWithin h)
 #align continuous_within_at.comp ContinuousWithinAt.comp
 
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-Case conversion may be inaccurate. Consider using '#align continuous_within_at.comp' ContinuousWithinAt.comp'ₓ'. -/
 theorem ContinuousWithinAt.comp' {g : β → γ} {f : α → β} {s : Set α} {t : Set β} {x : α}
     (hg : ContinuousWithinAt g t (f x)) (hf : ContinuousWithinAt f s x) :
     ContinuousWithinAt (g ∘ f) (s ∩ f ⁻¹' t) x :=
   hg.comp (hf.mono (inter_subset_left _ _)) (inter_subset_right _ _)
 #align continuous_within_at.comp' ContinuousWithinAt.comp'
 
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-Case conversion may be inaccurate. Consider using '#align continuous_at.comp_continuous_within_at ContinuousAt.comp_continuousWithinAtₓ'. -/
 theorem ContinuousAt.comp_continuousWithinAt {g : β → γ} {f : α → β} {s : Set α} {x : α}
     (hg : ContinuousAt g (f x)) (hf : ContinuousWithinAt f s x) : ContinuousWithinAt (g ∘ f) s x :=
   hg.ContinuousWithinAt.comp hf (mapsTo_univ _ _)
 #align continuous_at.comp_continuous_within_at ContinuousAt.comp_continuousWithinAt
 
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-Case conversion may be inaccurate. Consider using '#align continuous_on.comp ContinuousOn.compₓ'. -/
 theorem ContinuousOn.comp {g : β → γ} {f : α → β} {s : Set α} {t : Set β} (hg : ContinuousOn g t)
     (hf : ContinuousOn f s) (h : MapsTo f s t) : ContinuousOn (g ∘ f) s := fun x hx =>
   ContinuousWithinAt.comp (hg _ (h hx)) (hf x hx) h
 #align continuous_on.comp ContinuousOn.comp
 
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-Case conversion may be inaccurate. Consider using '#align continuous_on.mono ContinuousOn.monoₓ'. -/
 theorem ContinuousOn.mono {f : α → β} {s t : Set α} (hf : ContinuousOn f s) (h : t ⊆ s) :
     ContinuousOn f t := fun x hx => (hf x (h hx)).mono_left (nhdsWithin_mono _ h)
 #align continuous_on.mono ContinuousOn.mono
 
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-Case conversion may be inaccurate. Consider using '#align antitone_continuous_on antitone_continuousOnₓ'. -/
 theorem antitone_continuousOn {f : α → β} : Antitone (ContinuousOn f) := fun s t hst hf =>
   hf.mono hst
 #align antitone_continuous_on antitone_continuousOn
 
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-Case conversion may be inaccurate. Consider using '#align continuous_on.comp' ContinuousOn.comp'ₓ'. -/
 theorem ContinuousOn.comp' {g : β → γ} {f : α → β} {s : Set α} {t : Set β} (hg : ContinuousOn g t)
     (hf : ContinuousOn f s) : ContinuousOn (g ∘ f) (s ∩ f ⁻¹' t) :=
   hg.comp (hf.mono (inter_subset_left _ _)) (inter_subset_right _ _)
 #align continuous_on.comp' ContinuousOn.comp'
 
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-lean 3 declaration is
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-Case conversion may be inaccurate. Consider using '#align continuous.continuous_on Continuous.continuousOnₓ'. -/
 theorem Continuous.continuousOn {f : α → β} {s : Set α} (h : Continuous f) : ContinuousOn f s :=
   by
   rw [continuous_iff_continuousOn_univ] at h
   exact h.mono (subset_univ _)
 #align continuous.continuous_on Continuous.continuousOn
 
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-lean 3 declaration is
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-Case conversion may be inaccurate. Consider using '#align continuous.continuous_within_at Continuous.continuousWithinAtₓ'. -/
 theorem Continuous.continuousWithinAt {f : α → β} {s : Set α} {x : α} (h : Continuous f) :
     ContinuousWithinAt f s x :=
   h.ContinuousAt.ContinuousWithinAt
 #align continuous.continuous_within_at Continuous.continuousWithinAt
 
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-Case conversion may be inaccurate. Consider using '#align continuous.comp_continuous_on Continuous.comp_continuousOnₓ'. -/
 theorem Continuous.comp_continuousOn {g : β → γ} {f : α → β} {s : Set α} (hg : Continuous g)
     (hf : ContinuousOn f s) : ContinuousOn (g ∘ f) s :=
   hg.ContinuousOn.comp hf (mapsTo_univ _ _)
 #align continuous.comp_continuous_on Continuous.comp_continuousOn
 
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-lean 3 declaration is
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-Case conversion may be inaccurate. Consider using '#align continuous_on.comp_continuous ContinuousOn.comp_continuousₓ'. -/
 theorem ContinuousOn.comp_continuous {g : β → γ} {f : α → β} {s : Set β} (hg : ContinuousOn g s)
     (hf : Continuous f) (hs : ∀ x, f x ∈ s) : Continuous (g ∘ f) :=
   by
@@ -1751,12 +1031,6 @@ theorem ContinuousOn.comp_continuous {g : β → γ} {f : α → β} {s : Set β
   exact hg.comp hf fun x _ => hs x
 #align continuous_on.comp_continuous ContinuousOn.comp_continuous
 
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-Case conversion may be inaccurate. Consider using '#align continuous_within_at.preimage_mem_nhds_within ContinuousWithinAt.preimage_mem_nhdsWithinₓ'. -/
 theorem ContinuousWithinAt.preimage_mem_nhdsWithin {f : α → β} {x : α} {s : Set α} {t : Set β}
     (h : ContinuousWithinAt f s x) (ht : t ∈ 𝓝 (f x)) : f ⁻¹' t ∈ 𝓝[s] x :=
   h ht
@@ -1785,79 +1059,37 @@ theorem Function.LeftInverse.map_nhds_eq {f : α → β} {g : β → α} {x : β
 #align function.left_inverse.map_nhds_eq Function.LeftInverse.map_nhds_eq
 -/
 
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-lean 3 declaration is
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-Case conversion may be inaccurate. Consider using '#align continuous_within_at.preimage_mem_nhds_within' ContinuousWithinAt.preimage_mem_nhds_within'ₓ'. -/
 theorem ContinuousWithinAt.preimage_mem_nhds_within' {f : α → β} {x : α} {s : Set α} {t : Set β}
     (h : ContinuousWithinAt f s x) (ht : t ∈ 𝓝[f '' s] f x) : f ⁻¹' t ∈ 𝓝[s] x :=
   h.tendsto_nhdsWithin (mapsTo_image _ _) ht
 #align continuous_within_at.preimage_mem_nhds_within' ContinuousWithinAt.preimage_mem_nhds_within'
 
-/- warning: filter.eventually_eq.congr_continuous_within_at -> Filter.EventuallyEq.congr_continuousWithinAt 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 filter.eventually_eq.congr_continuous_within_at Filter.EventuallyEq.congr_continuousWithinAtₓ'. -/
 theorem Filter.EventuallyEq.congr_continuousWithinAt {f g : α → β} {s : Set α} {x : α}
     (h : f =ᶠ[𝓝[s] x] g) (hx : f x = g x) : ContinuousWithinAt f s x ↔ ContinuousWithinAt g s x :=
   by rw [ContinuousWithinAt, hx, tendsto_congr' h, ContinuousWithinAt]
 #align filter.eventually_eq.congr_continuous_within_at Filter.EventuallyEq.congr_continuousWithinAt
 
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-Case conversion may be inaccurate. Consider using '#align continuous_within_at.congr_of_eventually_eq ContinuousWithinAt.congr_of_eventuallyEqₓ'. -/
 theorem ContinuousWithinAt.congr_of_eventuallyEq {f f₁ : α → β} {s : Set α} {x : α}
     (h : ContinuousWithinAt f s x) (h₁ : f₁ =ᶠ[𝓝[s] x] f) (hx : f₁ x = f x) :
     ContinuousWithinAt f₁ s x :=
   (h₁.congr_continuousWithinAt hx).2 h
 #align continuous_within_at.congr_of_eventually_eq ContinuousWithinAt.congr_of_eventuallyEq
 
-/- warning: continuous_within_at.congr -> ContinuousWithinAt.congr 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 continuous_within_at.congr ContinuousWithinAt.congrₓ'. -/
 theorem ContinuousWithinAt.congr {f f₁ : α → β} {s : Set α} {x : α} (h : ContinuousWithinAt f s x)
     (h₁ : ∀ y ∈ s, f₁ y = f y) (hx : f₁ x = f x) : ContinuousWithinAt f₁ s x :=
   h.congr_of_eventuallyEq (mem_of_superset self_mem_nhdsWithin h₁) hx
 #align continuous_within_at.congr ContinuousWithinAt.congr
 
-/- warning: continuous_within_at.congr_mono -> ContinuousWithinAt.congr_mono is a dubious translation:
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-Case conversion may be inaccurate. Consider using '#align continuous_within_at.congr_mono ContinuousWithinAt.congr_monoₓ'. -/
 theorem ContinuousWithinAt.congr_mono {f g : α → β} {s s₁ : Set α} {x : α}
     (h : ContinuousWithinAt f s x) (h' : EqOn g f s₁) (h₁ : s₁ ⊆ s) (hx : g x = f x) :
     ContinuousWithinAt g s₁ x :=
   (h.mono h₁).congr h' hx
 #align continuous_within_at.congr_mono ContinuousWithinAt.congr_mono
 
-/- warning: continuous_on_const -> continuousOn_const 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 continuous_on_const continuousOn_constₓ'. -/
 theorem continuousOn_const {s : Set α} {c : β} : ContinuousOn (fun x => c) s :=
   continuous_const.ContinuousOn
 #align continuous_on_const continuousOn_const
 
-/- warning: continuous_within_at_const -> continuousWithinAt_const 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 continuous_within_at_const continuousWithinAt_constₓ'. -/
 theorem continuousWithinAt_const {b : β} {s : Set α} {x : α} :
     ContinuousWithinAt (fun _ : α => b) s x :=
   continuous_const.ContinuousWithinAt
@@ -1875,12 +1107,6 @@ theorem continuousWithinAt_id {s : Set α} {x : α} : ContinuousWithinAt id s x
 #align continuous_within_at_id continuousWithinAt_id
 -/
 
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-but is expected to have type
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-Case conversion may be inaccurate. Consider using '#align continuous_on_open_iff continuousOn_open_iffₓ'. -/
 theorem continuousOn_open_iff {f : α → β} {s : Set α} (hs : IsOpen s) :
     ContinuousOn f s ↔ ∀ t, IsOpen t → IsOpen (s ∩ f ⁻¹' t) :=
   by
@@ -1895,23 +1121,11 @@ theorem continuousOn_open_iff {f : α → β} {s : Set α} (hs : IsOpen s) :
     rw [@inter_comm _ s (f ⁻¹' t), inter_assoc, inter_self]
 #align continuous_on_open_iff continuousOn_open_iff
 
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-but is expected to have type
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-Case conversion may be inaccurate. Consider using '#align continuous_on.preimage_open_of_open ContinuousOn.preimage_open_of_openₓ'. -/
 theorem ContinuousOn.preimage_open_of_open {f : α → β} {s : Set α} {t : Set β}
     (hf : ContinuousOn f s) (hs : IsOpen s) (ht : IsOpen t) : IsOpen (s ∩ f ⁻¹' t) :=
   (continuousOn_open_iff hs).1 hf t ht
 #align continuous_on.preimage_open_of_open ContinuousOn.preimage_open_of_open
 
-/- warning: continuous_on.is_open_preimage -> ContinuousOn.isOpen_preimage 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 continuous_on.is_open_preimage ContinuousOn.isOpen_preimageₓ'. -/
 theorem ContinuousOn.isOpen_preimage {f : α → β} {s : Set α} {t : Set β} (h : ContinuousOn f s)
     (hs : IsOpen s) (hp : f ⁻¹' t ⊆ s) (ht : IsOpen t) : IsOpen (f ⁻¹' t) :=
   by
@@ -1919,12 +1133,6 @@ theorem ContinuousOn.isOpen_preimage {f : α → β} {s : Set α} {t : Set β} (
   rw [inter_comm, inter_eq_self_of_subset_left hp]
 #align continuous_on.is_open_preimage ContinuousOn.isOpen_preimage
 
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-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 continuous_on.preimage_closed_of_closed ContinuousOn.preimage_closed_of_closedₓ'. -/
 theorem ContinuousOn.preimage_closed_of_closed {f : α → β} {s : Set α} {t : Set β}
     (hf : ContinuousOn f s) (hs : IsClosed s) (ht : IsClosed t) : IsClosed (s ∩ f ⁻¹' t) :=
   by
@@ -1933,12 +1141,6 @@ theorem ContinuousOn.preimage_closed_of_closed {f : α → β} {s : Set α} {t :
   apply IsClosed.inter hu.1 hs
 #align continuous_on.preimage_closed_of_closed ContinuousOn.preimage_closed_of_closed
 
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-but is expected to have type
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-Case conversion may be inaccurate. Consider using '#align continuous_on.preimage_interior_subset_interior_preimage ContinuousOn.preimage_interior_subset_interior_preimageₓ'. -/
 theorem ContinuousOn.preimage_interior_subset_interior_preimage {f : α → β} {s : Set α} {t : Set β}
     (hf : ContinuousOn f s) (hs : IsOpen s) : s ∩ f ⁻¹' interior t ⊆ s ∩ interior (f ⁻¹' t) :=
   calc
@@ -1949,12 +1151,6 @@ theorem ContinuousOn.preimage_interior_subset_interior_preimage {f : α → β}
     
 #align continuous_on.preimage_interior_subset_interior_preimage ContinuousOn.preimage_interior_subset_interior_preimage
 
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-Case conversion may be inaccurate. Consider using '#align continuous_on_of_locally_continuous_on continuousOn_of_locally_continuousOnₓ'. -/
 theorem continuousOn_of_locally_continuousOn {f : α → β} {s : Set α}
     (h : ∀ x ∈ s, ∃ t, IsOpen t ∧ x ∈ t ∧ ContinuousOn f (s ∩ t)) : ContinuousOn f s :=
   by
@@ -1982,80 +1178,38 @@ theorem continuousOn_open_of_generateFrom {β : Type _} {s : Set α} {T : Set (S
   · exact hs
 #align continuous_on_open_of_generate_from continuousOn_open_of_generateFromₓ
 
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-Case conversion may be inaccurate. Consider using '#align continuous_within_at.prod ContinuousWithinAt.prodₓ'. -/
 theorem ContinuousWithinAt.prod {f : α → β} {g : α → γ} {s : Set α} {x : α}
     (hf : ContinuousWithinAt f s x) (hg : ContinuousWithinAt g s x) :
     ContinuousWithinAt (fun x => (f x, g x)) s x :=
   hf.prod_mk_nhds hg
 #align continuous_within_at.prod ContinuousWithinAt.prod
 
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-Case conversion may be inaccurate. Consider using '#align continuous_on.prod ContinuousOn.prodₓ'. -/
 theorem ContinuousOn.prod {f : α → β} {g : α → γ} {s : Set α} (hf : ContinuousOn f s)
     (hg : ContinuousOn g s) : ContinuousOn (fun x => (f x, g x)) s := fun x hx =>
   ContinuousWithinAt.prod (hf x hx) (hg x hx)
 #align continuous_on.prod ContinuousOn.prod
 
-/- warning: inducing.continuous_within_at_iff -> Inducing.continuousWithinAt_iff is a dubious translation:
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-Case conversion may be inaccurate. Consider using '#align inducing.continuous_within_at_iff Inducing.continuousWithinAt_iffₓ'. -/
 theorem Inducing.continuousWithinAt_iff {f : α → β} {g : β → γ} (hg : Inducing g) {s : Set α}
     {x : α} : ContinuousWithinAt f s x ↔ ContinuousWithinAt (g ∘ f) s x := by
   simp_rw [ContinuousWithinAt, Inducing.tendsto_nhds_iff hg]
 #align inducing.continuous_within_at_iff Inducing.continuousWithinAt_iff
 
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-but is expected to have type
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-Case conversion may be inaccurate. Consider using '#align inducing.continuous_on_iff Inducing.continuousOn_iffₓ'. -/
 theorem Inducing.continuousOn_iff {f : α → β} {g : β → γ} (hg : Inducing g) {s : Set α} :
     ContinuousOn f s ↔ ContinuousOn (g ∘ f) s := by
   simp_rw [ContinuousOn, hg.continuous_within_at_iff]
 #align inducing.continuous_on_iff Inducing.continuousOn_iff
 
-/- warning: embedding.continuous_on_iff -> Embedding.continuousOn_iff 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 embedding.continuous_on_iff Embedding.continuousOn_iffₓ'. -/
 theorem Embedding.continuousOn_iff {f : α → β} {g : β → γ} (hg : Embedding g) {s : Set α} :
     ContinuousOn f s ↔ ContinuousOn (g ∘ f) s :=
   Inducing.continuousOn_iff hg.1
 #align embedding.continuous_on_iff Embedding.continuousOn_iff
 
-/- warning: embedding.map_nhds_within_eq -> Embedding.map_nhdsWithin_eq is a dubious translation:
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-Case conversion may be inaccurate. Consider using '#align embedding.map_nhds_within_eq Embedding.map_nhdsWithin_eqₓ'. -/
 theorem Embedding.map_nhdsWithin_eq {f : α → β} (hf : Embedding f) (s : Set α) (x : α) :
     map f (𝓝[s] x) = 𝓝[f '' s] f x := by
   rw [nhdsWithin, map_inf hf.inj, hf.map_nhds_eq, map_principal, ← nhdsWithin_inter',
     inter_eq_self_of_subset_right (image_subset_range _ _)]
 #align embedding.map_nhds_within_eq Embedding.map_nhdsWithin_eq
 
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-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 open_embedding.map_nhds_within_preimage_eq OpenEmbedding.map_nhdsWithin_preimage_eqₓ'. -/
 theorem OpenEmbedding.map_nhdsWithin_preimage_eq {f : α → β} (hf : OpenEmbedding f) (s : Set β)
     (x : α) : map f (𝓝[f ⁻¹' s] x) = 𝓝[s] f x :=
   by
@@ -2064,12 +1218,6 @@ theorem OpenEmbedding.map_nhdsWithin_preimage_eq {f : α → β} (hf : OpenEmbed
   rw [inter_assoc, inter_self]
 #align open_embedding.map_nhds_within_preimage_eq OpenEmbedding.map_nhdsWithin_preimage_eq
 
-/- warning: continuous_within_at_of_not_mem_closure -> continuousWithinAt_of_not_mem_closure 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 continuous_within_at_of_not_mem_closure continuousWithinAt_of_not_mem_closureₓ'. -/
 theorem continuousWithinAt_of_not_mem_closure {f : α → β} {s : Set α} {x : α} :
     x ∉ closure s → ContinuousWithinAt f s x :=
   by
@@ -2079,12 +1227,6 @@ theorem continuousWithinAt_of_not_mem_closure {f : α → β} {s : Set α} {x :
   exact tendsto_bot
 #align continuous_within_at_of_not_mem_closure continuousWithinAt_of_not_mem_closure
 
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-lean 3 declaration is
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-Case conversion may be inaccurate. Consider using '#align continuous_on.piecewise' ContinuousOn.piecewise'ₓ'. -/
 theorem ContinuousOn.piecewise' {s t : Set α} {f g : α → β} [∀ a, Decidable (a ∈ t)]
     (hpf : ∀ a ∈ s ∩ frontier t, Tendsto f (𝓝[s ∩ t] a) (𝓝 (piecewise t f g a)))
     (hpg : ∀ a ∈ s ∩ frontier t, Tendsto g (𝓝[s ∩ tᶜ] a) (𝓝 (piecewise t f g a)))
@@ -2116,12 +1258,6 @@ theorem ContinuousOn.piecewise' {s t : Set α} {f g : α → β} [∀ a, Decidab
             (piecewise_eq_of_not_mem _ _ _ hx.2)
 #align continuous_on.piecewise' ContinuousOn.piecewise'
 
-/- warning: continuous_on.if' -> ContinuousOn.if' is a dubious translation:
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-Case conversion may be inaccurate. Consider using '#align continuous_on.if' ContinuousOn.if'ₓ'. -/
 theorem ContinuousOn.if' {s : Set α} {p : α → Prop} {f g : α → β} [∀ a, Decidable (p a)]
     (hpf :
       ∀ a ∈ s ∩ frontier { a | p a },
@@ -2134,12 +1270,6 @@ theorem ContinuousOn.if' {s : Set α} {p : α → Prop} {f g : α → β} [∀ a
   hf.piecewise' hpf hpg hg
 #align continuous_on.if' ContinuousOn.if'
 
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-Case conversion may be inaccurate. Consider using '#align continuous_on.if ContinuousOn.ifₓ'. -/
 theorem ContinuousOn.if {α β : Type _} [TopologicalSpace α] [TopologicalSpace β] {p : α → Prop}
     [∀ a, Decidable (p a)] {s : Set α} {f g : α → β}
     (hp : ∀ a ∈ s ∩ frontier { a | p a }, f a = g a)
@@ -2162,24 +1292,12 @@ theorem ContinuousOn.if {α β : Type _} [TopologicalSpace α] [TopologicalSpace
   · exact hg.mono (inter_subset_inter_right s subset_closure)
 #align continuous_on.if ContinuousOn.if
 
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-Case conversion may be inaccurate. Consider using '#align continuous_on.piecewise ContinuousOn.piecewiseₓ'. -/
 theorem ContinuousOn.piecewise {s t : Set α} {f g : α → β} [∀ a, Decidable (a ∈ t)]
     (ht : ∀ a ∈ s ∩ frontier t, f a = g a) (hf : ContinuousOn f <| s ∩ closure t)
     (hg : ContinuousOn g <| s ∩ closure (tᶜ)) : ContinuousOn (piecewise t f g) s :=
   hf.if ht hg
 #align continuous_on.piecewise ContinuousOn.piecewise
 
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-Case conversion may be inaccurate. Consider using '#align continuous_if' continuous_if'ₓ'. -/
 theorem continuous_if' {p : α → Prop} {f g : α → β} [∀ a, Decidable (p a)]
     (hpf : ∀ a ∈ frontier { x | p x }, Tendsto f (𝓝[{ x | p x }] a) (𝓝 <| ite (p a) (f a) (g a)))
     (hpg : ∀ a ∈ frontier { x | p x }, Tendsto g (𝓝[{ x | ¬p x }] a) (𝓝 <| ite (p a) (f a) (g a)))
@@ -2190,12 +1308,6 @@ theorem continuous_if' {p : α → Prop} {f g : α → β} [∀ a, Decidable (p
   apply ContinuousOn.if' <;> simp [*] <;> assumption
 #align continuous_if' continuous_if'
 
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-Case conversion may be inaccurate. Consider using '#align continuous_if continuous_ifₓ'. -/
 theorem continuous_if {p : α → Prop} {f g : α → β} [∀ a, Decidable (p a)]
     (hp : ∀ a ∈ frontier { x | p x }, f a = g a) (hf : ContinuousOn f (closure { x | p x }))
     (hg : ContinuousOn g (closure { x | ¬p x })) : Continuous fun a => if p a then f a else g a :=
@@ -2204,58 +1316,28 @@ theorem continuous_if {p : α → Prop} {f g : α → β} [∀ a, Decidable (p a
   apply ContinuousOn.if <;> simp <;> assumption
 #align continuous_if continuous_if
 
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-Case conversion may be inaccurate. Consider using '#align continuous.if Continuous.ifₓ'. -/
 theorem Continuous.if {p : α → Prop} {f g : α → β} [∀ a, Decidable (p a)]
     (hp : ∀ a ∈ frontier { x | p x }, f a = g a) (hf : Continuous f) (hg : Continuous g) :
     Continuous fun a => if p a then f a else g a :=
   continuous_if hp hf.ContinuousOn hg.ContinuousOn
 #align continuous.if Continuous.if
 
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-Case conversion may be inaccurate. Consider using '#align continuous_if_const continuous_if_constₓ'. -/
 theorem continuous_if_const (p : Prop) {f g : α → β} [Decidable p] (hf : p → Continuous f)
     (hg : ¬p → Continuous g) : Continuous fun a => if p then f a else g a := by split_ifs;
   exact hf h; exact hg h
 #align continuous_if_const continuous_if_const
 
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 theorem Continuous.if_const (p : Prop) {f g : α → β} [Decidable p] (hf : Continuous f)
     (hg : Continuous g) : Continuous fun a => if p then f a else g a :=
   continuous_if_const p (fun _ => hf) fun _ => hg
 #align continuous.if_const Continuous.if_const
 
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 theorem continuous_piecewise {s : Set α} {f g : α → β} [∀ a, Decidable (a ∈ s)]
     (hs : ∀ a ∈ frontier s, f a = g a) (hf : ContinuousOn f (closure s))
     (hg : ContinuousOn g (closure (sᶜ))) : Continuous (piecewise s f g) :=
   continuous_if hs hf hg
 #align continuous_piecewise continuous_piecewise
 
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 theorem Continuous.piecewise {s : Set α} {f g : α → β} [∀ a, Decidable (a ∈ s)]
     (hs : ∀ a ∈ frontier s, f a = g a) (hf : Continuous f) (hg : Continuous g) :
     Continuous (piecewise s f g) :=
@@ -2273,46 +1355,22 @@ theorem IsOpen.ite' {s s' t : Set α} (hs : IsOpen s) (hs' : IsOpen s')
 #align is_open.ite' IsOpen.ite'
 -/
 
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 theorem IsOpen.ite {s s' t : Set α} (hs : IsOpen s) (hs' : IsOpen s')
     (ht : s ∩ frontier t = s' ∩ frontier t) : IsOpen (t.ite s s') :=
   hs.ite' hs' fun x hx => by simpa [hx] using ext_iff.1 ht x
 #align is_open.ite IsOpen.ite
 
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 theorem ite_inter_closure_eq_of_inter_frontier_eq {s s' t : Set α}
     (ht : s ∩ frontier t = s' ∩ frontier t) : t.ite s s' ∩ closure t = s ∩ closure t := by
   rw [closure_eq_self_union_frontier, inter_union_distrib_left, inter_union_distrib_left,
     ite_inter_self, ite_inter_of_inter_eq _ ht]
 #align ite_inter_closure_eq_of_inter_frontier_eq ite_inter_closure_eq_of_inter_frontier_eq
 
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 theorem ite_inter_closure_compl_eq_of_inter_frontier_eq {s s' t : Set α}
     (ht : s ∩ frontier t = s' ∩ frontier t) : t.ite s s' ∩ closure (tᶜ) = s' ∩ closure (tᶜ) := by
   rw [← ite_compl, ite_inter_closure_eq_of_inter_frontier_eq]; rwa [frontier_compl, eq_comm]
 #align ite_inter_closure_compl_eq_of_inter_frontier_eq ite_inter_closure_compl_eq_of_inter_frontier_eq
 
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 theorem continuousOn_piecewise_ite' {s s' t : Set α} {f f' : α → β} [∀ x, Decidable (x ∈ t)]
     (h : ContinuousOn f (s ∩ closure t)) (h' : ContinuousOn f' (s' ∩ closure (tᶜ)))
     (H : s ∩ frontier t = s' ∩ frontier t) (Heq : EqOn f f' (s ∩ frontier t)) :
@@ -2324,12 +1382,6 @@ theorem continuousOn_piecewise_ite' {s s' t : Set α} {f f' : α → β} [∀ x,
   · rwa [ite_inter_closure_compl_eq_of_inter_frontier_eq H]
 #align continuous_on_piecewise_ite' continuousOn_piecewise_ite'
 
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 theorem continuousOn_piecewise_ite {s s' t : Set α} {f f' : α → β} [∀ x, Decidable (x ∈ t)]
     (h : ContinuousOn f s) (h' : ContinuousOn f' s') (H : s ∩ frontier t = s' ∩ frontier t)
     (Heq : EqOn f f' (s ∩ frontier t)) : ContinuousOn (t.piecewise f f') (t.ite s s') :=
@@ -2337,12 +1389,6 @@ theorem continuousOn_piecewise_ite {s s' t : Set α} {f f' : α → β} [∀ x,
     Heq
 #align continuous_on_piecewise_ite continuousOn_piecewise_ite
 
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 theorem frontier_inter_open_inter {s t : Set α} (ht : IsOpen t) :
     frontier (s ∩ t) ∩ t = frontier s ∩ t := by
   simp only [← Subtype.preimage_coe_eq_preimage_coe_iff,
@@ -2350,96 +1396,42 @@ theorem frontier_inter_open_inter {s t : Set α} (ht : IsOpen t) :
     Subtype.preimage_coe_inter_self]
 #align frontier_inter_open_inter frontier_inter_open_inter
 
-/- warning: continuous_on_fst -> continuousOn_fst is a dubious translation:
-lean 3 declaration is
-  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : TopologicalSpace.{u1} α] [_inst_2 : TopologicalSpace.{u2} β] {s : Set.{max u1 u2} (Prod.{u1, u2} α β)}, ContinuousOn.{max u1 u2, u1} (Prod.{u1, u2} α β) α (Prod.topologicalSpace.{u1, u2} α β _inst_1 _inst_2) _inst_1 (Prod.fst.{u1, u2} α β) s
-but is expected to have type
-  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : TopologicalSpace.{u1} α] [_inst_2 : TopologicalSpace.{u2} β] {s : Set.{max u2 u1} (Prod.{u1, u2} α β)}, ContinuousOn.{max u2 u1, u1} (Prod.{u1, u2} α β) α (instTopologicalSpaceProd.{u1, u2} α β _inst_1 _inst_2) _inst_1 (Prod.fst.{u1, u2} α β) s
-Case conversion may be inaccurate. Consider using '#align continuous_on_fst continuousOn_fstₓ'. -/
 theorem continuousOn_fst {s : Set (α × β)} : ContinuousOn Prod.fst s :=
   continuous_fst.ContinuousOn
 #align continuous_on_fst continuousOn_fst
 
-/- warning: continuous_within_at_fst -> continuousWithinAt_fst is a dubious translation:
-lean 3 declaration is
-  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : TopologicalSpace.{u1} α] [_inst_2 : TopologicalSpace.{u2} β] {s : Set.{max u1 u2} (Prod.{u1, u2} α β)} {p : Prod.{u1, u2} α β}, ContinuousWithinAt.{max u1 u2, u1} (Prod.{u1, u2} α β) α (Prod.topologicalSpace.{u1, u2} α β _inst_1 _inst_2) _inst_1 (Prod.fst.{u1, u2} α β) s p
-but is expected to have type
-  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : TopologicalSpace.{u1} α] [_inst_2 : TopologicalSpace.{u2} β] {s : Set.{max u2 u1} (Prod.{u1, u2} α β)} {p : Prod.{u1, u2} α β}, ContinuousWithinAt.{max u2 u1, u1} (Prod.{u1, u2} α β) α (instTopologicalSpaceProd.{u1, u2} α β _inst_1 _inst_2) _inst_1 (Prod.fst.{u1, u2} α β) s p
-Case conversion may be inaccurate. Consider using '#align continuous_within_at_fst continuousWithinAt_fstₓ'. -/
 theorem continuousWithinAt_fst {s : Set (α × β)} {p : α × β} : ContinuousWithinAt Prod.fst s p :=
   continuous_fst.ContinuousWithinAt
 #align continuous_within_at_fst continuousWithinAt_fst
 
-/- warning: continuous_on.fst -> ContinuousOn.fst is a dubious translation:
-lean 3 declaration is
-  forall {α : Type.{u1}} {β : Type.{u2}} {γ : Type.{u3}} [_inst_1 : TopologicalSpace.{u1} α] [_inst_2 : TopologicalSpace.{u2} β] [_inst_3 : TopologicalSpace.{u3} γ] {f : α -> (Prod.{u2, u3} β γ)} {s : Set.{u1} α}, (ContinuousOn.{u1, max u2 u3} α (Prod.{u2, u3} β γ) _inst_1 (Prod.topologicalSpace.{u2, u3} β γ _inst_2 _inst_3) f s) -> (ContinuousOn.{u1, u2} α β _inst_1 _inst_2 (fun (x : α) => Prod.fst.{u2, u3} β γ (f x)) s)
-but is expected to have type
-  forall {α : Type.{u1}} {β : Type.{u3}} {γ : Type.{u2}} [_inst_1 : TopologicalSpace.{u1} α] [_inst_2 : TopologicalSpace.{u3} β] [_inst_3 : TopologicalSpace.{u2} γ] {f : α -> (Prod.{u3, u2} β γ)} {s : Set.{u1} α}, (ContinuousOn.{u1, max u3 u2} α (Prod.{u3, u2} β γ) _inst_1 (instTopologicalSpaceProd.{u3, u2} β γ _inst_2 _inst_3) f s) -> (ContinuousOn.{u1, u3} α β _inst_1 _inst_2 (fun (x : α) => Prod.fst.{u3, u2} β γ (f x)) s)
-Case conversion may be inaccurate. Consider using '#align continuous_on.fst ContinuousOn.fstₓ'. -/
 theorem ContinuousOn.fst {f : α → β × γ} {s : Set α} (hf : ContinuousOn f s) :
     ContinuousOn (fun x => (f x).1) s :=
   continuous_fst.comp_continuousOn hf
 #align continuous_on.fst ContinuousOn.fst
 
-/- warning: continuous_within_at.fst -> ContinuousWithinAt.fst is a dubious translation:
-lean 3 declaration is
-  forall {α : Type.{u1}} {β : Type.{u2}} {γ : Type.{u3}} [_inst_1 : TopologicalSpace.{u1} α] [_inst_2 : TopologicalSpace.{u2} β] [_inst_3 : TopologicalSpace.{u3} γ] {f : α -> (Prod.{u2, u3} β γ)} {s : Set.{u1} α} {a : α}, (ContinuousWithinAt.{u1, max u2 u3} α (Prod.{u2, u3} β γ) _inst_1 (Prod.topologicalSpace.{u2, u3} β γ _inst_2 _inst_3) f s a) -> (ContinuousWithinAt.{u1, u2} α β _inst_1 _inst_2 (fun (x : α) => Prod.fst.{u2, u3} β γ (f x)) s a)
-but is expected to have type
-  forall {α : Type.{u1}} {β : Type.{u3}} {γ : Type.{u2}} [_inst_1 : TopologicalSpace.{u1} α] [_inst_2 : TopologicalSpace.{u3} β] [_inst_3 : TopologicalSpace.{u2} γ] {f : α -> (Prod.{u3, u2} β γ)} {s : Set.{u1} α} {a : α}, (ContinuousWithinAt.{u1, max u3 u2} α (Prod.{u3, u2} β γ) _inst_1 (instTopologicalSpaceProd.{u3, u2} β γ _inst_2 _inst_3) f s a) -> (ContinuousWithinAt.{u1, u3} α β _inst_1 _inst_2 (fun (x : α) => Prod.fst.{u3, u2} β γ (f x)) s a)
-Case conversion may be inaccurate. Consider using '#align continuous_within_at.fst ContinuousWithinAt.fstₓ'. -/
 theorem ContinuousWithinAt.fst {f : α → β × γ} {s : Set α} {a : α} (h : ContinuousWithinAt f s a) :
     ContinuousWithinAt (fun x => (f x).fst) s a :=
   continuousAt_fst.comp_continuousWithinAt h
 #align continuous_within_at.fst ContinuousWithinAt.fst
 
-/- warning: continuous_on_snd -> continuousOn_snd is a dubious translation:
-lean 3 declaration is
-  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : TopologicalSpace.{u1} α] [_inst_2 : TopologicalSpace.{u2} β] {s : Set.{max u1 u2} (Prod.{u1, u2} α β)}, ContinuousOn.{max u1 u2, u2} (Prod.{u1, u2} α β) β (Prod.topologicalSpace.{u1, u2} α β _inst_1 _inst_2) _inst_2 (Prod.snd.{u1, u2} α β) s
-but is expected to have type
-  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : TopologicalSpace.{u1} α] [_inst_2 : TopologicalSpace.{u2} β] {s : Set.{max u2 u1} (Prod.{u1, u2} α β)}, ContinuousOn.{max u2 u1, u2} (Prod.{u1, u2} α β) β (instTopologicalSpaceProd.{u1, u2} α β _inst_1 _inst_2) _inst_2 (Prod.snd.{u1, u2} α β) s
-Case conversion may be inaccurate. Consider using '#align continuous_on_snd continuousOn_sndₓ'. -/
 theorem continuousOn_snd {s : Set (α × β)} : ContinuousOn Prod.snd s :=
   continuous_snd.ContinuousOn
 #align continuous_on_snd continuousOn_snd
 
-/- warning: continuous_within_at_snd -> continuousWithinAt_snd is a dubious translation:
-lean 3 declaration is
-  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : TopologicalSpace.{u1} α] [_inst_2 : TopologicalSpace.{u2} β] {s : Set.{max u1 u2} (Prod.{u1, u2} α β)} {p : Prod.{u1, u2} α β}, ContinuousWithinAt.{max u1 u2, u2} (Prod.{u1, u2} α β) β (Prod.topologicalSpace.{u1, u2} α β _inst_1 _inst_2) _inst_2 (Prod.snd.{u1, u2} α β) s p
-but is expected to have type
-  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : TopologicalSpace.{u1} α] [_inst_2 : TopologicalSpace.{u2} β] {s : Set.{max u2 u1} (Prod.{u1, u2} α β)} {p : Prod.{u1, u2} α β}, ContinuousWithinAt.{max u2 u1, u2} (Prod.{u1, u2} α β) β (instTopologicalSpaceProd.{u1, u2} α β _inst_1 _inst_2) _inst_2 (Prod.snd.{u1, u2} α β) s p
-Case conversion may be inaccurate. Consider using '#align continuous_within_at_snd continuousWithinAt_sndₓ'. -/
 theorem continuousWithinAt_snd {s : Set (α × β)} {p : α × β} : ContinuousWithinAt Prod.snd s p :=
   continuous_snd.ContinuousWithinAt
 #align continuous_within_at_snd continuousWithinAt_snd
 
-/- warning: continuous_on.snd -> ContinuousOn.snd is a dubious translation:
-lean 3 declaration is
-  forall {α : Type.{u1}} {β : Type.{u2}} {γ : Type.{u3}} [_inst_1 : TopologicalSpace.{u1} α] [_inst_2 : TopologicalSpace.{u2} β] [_inst_3 : TopologicalSpace.{u3} γ] {f : α -> (Prod.{u2, u3} β γ)} {s : Set.{u1} α}, (ContinuousOn.{u1, max u2 u3} α (Prod.{u2, u3} β γ) _inst_1 (Prod.topologicalSpace.{u2, u3} β γ _inst_2 _inst_3) f s) -> (ContinuousOn.{u1, u3} α γ _inst_1 _inst_3 (fun (x : α) => Prod.snd.{u2, u3} β γ (f x)) s)
-but is expected to have type
-  forall {α : Type.{u1}} {β : Type.{u3}} {γ : Type.{u2}} [_inst_1 : TopologicalSpace.{u1} α] [_inst_2 : TopologicalSpace.{u3} β] [_inst_3 : TopologicalSpace.{u2} γ] {f : α -> (Prod.{u3, u2} β γ)} {s : Set.{u1} α}, (ContinuousOn.{u1, max u3 u2} α (Prod.{u3, u2} β γ) _inst_1 (instTopologicalSpaceProd.{u3, u2} β γ _inst_2 _inst_3) f s) -> (ContinuousOn.{u1, u2} α γ _inst_1 _inst_3 (fun (x : α) => Prod.snd.{u3, u2} β γ (f x)) s)
-Case conversion may be inaccurate. Consider using '#align continuous_on.snd ContinuousOn.sndₓ'. -/
 theorem ContinuousOn.snd {f : α → β × γ} {s : Set α} (hf : ContinuousOn f s) :
     ContinuousOn (fun x => (f x).2) s :=
   continuous_snd.comp_continuousOn hf
 #align continuous_on.snd ContinuousOn.snd
 
-/- warning: continuous_within_at.snd -> ContinuousWithinAt.snd is a dubious translation:
-lean 3 declaration is
-  forall {α : Type.{u1}} {β : Type.{u2}} {γ : Type.{u3}} [_inst_1 : TopologicalSpace.{u1} α] [_inst_2 : TopologicalSpace.{u2} β] [_inst_3 : TopologicalSpace.{u3} γ] {f : α -> (Prod.{u2, u3} β γ)} {s : Set.{u1} α} {a : α}, (ContinuousWithinAt.{u1, max u2 u3} α (Prod.{u2, u3} β γ) _inst_1 (Prod.topologicalSpace.{u2, u3} β γ _inst_2 _inst_3) f s a) -> (ContinuousWithinAt.{u1, u3} α γ _inst_1 _inst_3 (fun (x : α) => Prod.snd.{u2, u3} β γ (f x)) s a)
-but is expected to have type
-  forall {α : Type.{u1}} {β : Type.{u3}} {γ : Type.{u2}} [_inst_1 : TopologicalSpace.{u1} α] [_inst_2 : TopologicalSpace.{u3} β] [_inst_3 : TopologicalSpace.{u2} γ] {f : α -> (Prod.{u3, u2} β γ)} {s : Set.{u1} α} {a : α}, (ContinuousWithinAt.{u1, max u3 u2} α (Prod.{u3, u2} β γ) _inst_1 (instTopologicalSpaceProd.{u3, u2} β γ _inst_2 _inst_3) f s a) -> (ContinuousWithinAt.{u1, u2} α γ _inst_1 _inst_3 (fun (x : α) => Prod.snd.{u3, u2} β γ (f x)) s a)
-Case conversion may be inaccurate. Consider using '#align continuous_within_at.snd ContinuousWithinAt.sndₓ'. -/
 theorem ContinuousWithinAt.snd {f : α → β × γ} {s : Set α} {a : α} (h : ContinuousWithinAt f s a) :
     ContinuousWithinAt (fun x => (f x).snd) s a :=
   continuousAt_snd.comp_continuousWithinAt h
 #align continuous_within_at.snd ContinuousWithinAt.snd
 
-/- warning: continuous_within_at_prod_iff -> continuousWithinAt_prod_iff is a dubious translation:
-lean 3 declaration is
-  forall {α : Type.{u1}} {β : Type.{u2}} {γ : Type.{u3}} [_inst_1 : TopologicalSpace.{u1} α] [_inst_2 : TopologicalSpace.{u2} β] [_inst_3 : TopologicalSpace.{u3} γ] {f : α -> (Prod.{u2, u3} β γ)} {s : Set.{u1} α} {x : α}, Iff (ContinuousWithinAt.{u1, max u2 u3} α (Prod.{u2, u3} β γ) _inst_1 (Prod.topologicalSpace.{u2, u3} β γ _inst_2 _inst_3) f s x) (And (ContinuousWithinAt.{u1, u2} α β _inst_1 _inst_2 (Function.comp.{succ u1, max (succ u2) (succ u3), succ u2} α (Prod.{u2, u3} β γ) β (Prod.fst.{u2, u3} β γ) f) s x) (ContinuousWithinAt.{u1, u3} α γ _inst_1 _inst_3 (Function.comp.{succ u1, max (succ u2) (succ u3), succ u3} α (Prod.{u2, u3} β γ) γ (Prod.snd.{u2, u3} β γ) f) s x))
-but is expected to have type
-  forall {α : Type.{u1}} {β : Type.{u3}} {γ : Type.{u2}} [_inst_1 : TopologicalSpace.{u1} α] [_inst_2 : TopologicalSpace.{u3} β] [_inst_3 : TopologicalSpace.{u2} γ] {f : α -> (Prod.{u3, u2} β γ)} {s : Set.{u1} α} {x : α}, Iff (ContinuousWithinAt.{u1, max u3 u2} α (Prod.{u3, u2} β γ) _inst_1 (instTopologicalSpaceProd.{u3, u2} β γ _inst_2 _inst_3) f s x) (And (ContinuousWithinAt.{u1, u3} α β _inst_1 _inst_2 (Function.comp.{succ u1, max (succ u2) (succ u3), succ u3} α (Prod.{u3, u2} β γ) β (Prod.fst.{u3, u2} β γ) f) s x) (ContinuousWithinAt.{u1, u2} α γ _inst_1 _inst_3 (Function.comp.{succ u1, max (succ u2) (succ u3), succ u2} α (Prod.{u3, u2} β γ) γ (Prod.snd.{u3, u2} β γ) f) s x))
-Case conversion may be inaccurate. Consider using '#align continuous_within_at_prod_iff continuousWithinAt_prod_iffₓ'. -/
 theorem continuousWithinAt_prod_iff {f : α → β × γ} {s : Set α} {x : α} :
     ContinuousWithinAt f s x ↔
       ContinuousWithinAt (Prod.fst ∘ f) s x ∧ ContinuousWithinAt (Prod.snd ∘ f) s x :=

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

@@ -357,11 +357,8 @@ lean 3 declaration is
 but is expected to have type
   forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] {a : α} {s : Set.{u1} α}, LE.le.{u1} (Filter.{u1} α) (Preorder.toLE.{u1} (Filter.{u1} α) (PartialOrder.toPreorder.{u1} (Filter.{u1} α) (Filter.instPartialOrderFilter.{u1} α))) (nhdsWithin.{u1} α _inst_1 a s) (nhds.{u1} α _inst_1 a)
 Case conversion may be inaccurate. Consider using '#align nhds_within_le_nhds nhdsWithin_le_nhdsₓ'. -/
-theorem nhdsWithin_le_nhds {a : α} {s : Set α} : 𝓝[s] a ≤ 𝓝 a :=
-  by
-  rw [← nhdsWithin_univ]
-  apply nhdsWithin_le_of_mem
-  exact univ_mem
+theorem nhdsWithin_le_nhds {a : α} {s : Set α} : 𝓝[s] a ≤ 𝓝 a := by rw [← nhdsWithin_univ];
+  apply nhdsWithin_le_of_mem; exact univ_mem
 #align nhds_within_le_nhds nhdsWithin_le_nhds
 
 /- warning: nhds_within_eq_nhds_within' -> nhdsWithin_eq_nhds_within' is a dubious translation:
@@ -403,9 +400,7 @@ theorem preimage_nhds_within_coinduced {π : α → β} {s : Set β} {t : Set α
     (ht : IsOpen t)
     (hs :
       s ∈ @nhds β (TopologicalSpace.coinduced (fun x : t => π x) Subtype.topologicalSpace) (π a)) :
-    π ⁻¹' s ∈ 𝓝 a := by
-  rw [← ht.nhds_within_eq h]
-  exact preimage_nhdsWithin_coinduced' h ht hs
+    π ⁻¹' s ∈ 𝓝 a := by rw [← ht.nhds_within_eq h]; exact preimage_nhdsWithin_coinduced' h ht hs
 #align preimage_nhds_within_coinduced preimage_nhds_within_coinduced
 -/
 
@@ -425,10 +420,8 @@ lean 3 declaration is
 but is expected to have type
   forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] (a : α) (s : Set.{u1} α) (t : Set.{u1} α), Eq.{succ u1} (Filter.{u1} α) (nhdsWithin.{u1} α _inst_1 a (Union.union.{u1} (Set.{u1} α) (Set.instUnionSet.{u1} α) s t)) (Sup.sup.{u1} (Filter.{u1} α) (SemilatticeSup.toSup.{u1} (Filter.{u1} α) (Lattice.toSemilatticeSup.{u1} (Filter.{u1} α) (ConditionallyCompleteLattice.toLattice.{u1} (Filter.{u1} α) (CompleteLattice.toConditionallyCompleteLattice.{u1} (Filter.{u1} α) (Filter.instCompleteLatticeFilter.{u1} α))))) (nhdsWithin.{u1} α _inst_1 a s) (nhdsWithin.{u1} α _inst_1 a t))
 Case conversion may be inaccurate. Consider using '#align nhds_within_union nhdsWithin_unionₓ'. -/
-theorem nhdsWithin_union (a : α) (s t : Set α) : 𝓝[s ∪ t] a = 𝓝[s] a ⊔ 𝓝[t] a :=
-  by
-  delta nhdsWithin
-  rw [← inf_sup_left, sup_principal]
+theorem nhdsWithin_union (a : α) (s t : Set α) : 𝓝[s ∪ t] a = 𝓝[s] a ⊔ 𝓝[t] a := by
+  delta nhdsWithin; rw [← inf_sup_left, sup_principal]
 #align nhds_within_union nhdsWithin_union
 
 /- warning: nhds_within_bUnion -> nhdsWithin_biUnion is a dubious translation:
@@ -469,10 +462,8 @@ lean 3 declaration is
 but is expected to have type
   forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] (a : α) (s : Set.{u1} α) (t : Set.{u1} α), Eq.{succ u1} (Filter.{u1} α) (nhdsWithin.{u1} α _inst_1 a (Inter.inter.{u1} (Set.{u1} α) (Set.instInterSet.{u1} α) s t)) (Inf.inf.{u1} (Filter.{u1} α) (Filter.instInfFilter.{u1} α) (nhdsWithin.{u1} α _inst_1 a s) (nhdsWithin.{u1} α _inst_1 a t))
 Case conversion may be inaccurate. Consider using '#align nhds_within_inter nhdsWithin_interₓ'. -/
-theorem nhdsWithin_inter (a : α) (s t : Set α) : 𝓝[s ∩ t] a = 𝓝[s] a ⊓ 𝓝[t] a :=
-  by
-  delta nhdsWithin
-  rw [inf_left_comm, inf_assoc, inf_principal, ← inf_assoc, inf_idem]
+theorem nhdsWithin_inter (a : α) (s t : Set α) : 𝓝[s ∩ t] a = 𝓝[s] a ⊓ 𝓝[t] a := by
+  delta nhdsWithin; rw [inf_left_comm, inf_assoc, inf_principal, ← inf_assoc, inf_idem]
 #align nhds_within_inter nhdsWithin_inter
 
 /- warning: nhds_within_inter' -> nhdsWithin_inter' is a dubious translation:
@@ -481,9 +472,7 @@ lean 3 declaration is
 but is expected to have type
   forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] (a : α) (s : Set.{u1} α) (t : Set.{u1} α), Eq.{succ u1} (Filter.{u1} α) (nhdsWithin.{u1} α _inst_1 a (Inter.inter.{u1} (Set.{u1} α) (Set.instInterSet.{u1} α) s t)) (Inf.inf.{u1} (Filter.{u1} α) (Filter.instInfFilter.{u1} α) (nhdsWithin.{u1} α _inst_1 a s) (Filter.principal.{u1} α t))
 Case conversion may be inaccurate. Consider using '#align nhds_within_inter' nhdsWithin_inter'ₓ'. -/
-theorem nhdsWithin_inter' (a : α) (s t : Set α) : 𝓝[s ∩ t] a = 𝓝[s] a ⊓ 𝓟 t :=
-  by
-  delta nhdsWithin
+theorem nhdsWithin_inter' (a : α) (s t : Set α) : 𝓝[s ∩ t] a = 𝓝[s] a ⊓ 𝓟 t := by delta nhdsWithin;
   rw [← inf_principal, inf_assoc]
 #align nhds_within_inter' nhdsWithin_inter'
 
@@ -493,10 +482,8 @@ lean 3 declaration is
 but is expected to have type
   forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] {a : α} {s : Set.{u1} α} {t : Set.{u1} α}, (Membership.mem.{u1, u1} (Set.{u1} α) (Filter.{u1} α) (instMembershipSetFilter.{u1} α) s (nhdsWithin.{u1} α _inst_1 a t)) -> (Eq.{succ u1} (Filter.{u1} α) (nhdsWithin.{u1} α _inst_1 a (Inter.inter.{u1} (Set.{u1} α) (Set.instInterSet.{u1} α) s t)) (nhdsWithin.{u1} α _inst_1 a t))
 Case conversion may be inaccurate. Consider using '#align nhds_within_inter_of_mem nhdsWithin_inter_of_memₓ'. -/
-theorem nhdsWithin_inter_of_mem {a : α} {s t : Set α} (h : s ∈ 𝓝[t] a) : 𝓝[s ∩ t] a = 𝓝[t] a :=
-  by
-  rw [nhdsWithin_inter, inf_eq_right]
-  exact nhdsWithin_le_of_mem h
+theorem nhdsWithin_inter_of_mem {a : α} {s t : Set α} (h : s ∈ 𝓝[t] a) : 𝓝[s ∩ t] a = 𝓝[t] a := by
+  rw [nhdsWithin_inter, inf_eq_right]; exact nhdsWithin_le_of_mem h
 #align nhds_within_inter_of_mem nhdsWithin_inter_of_mem
 
 /- warning: nhds_within_inter_of_mem' -> nhdsWithin_inter_of_mem' is a dubious translation:
@@ -569,10 +556,8 @@ but is expected to have type
 Case conversion may be inaccurate. Consider using '#align nhds_within_prod_eq nhdsWithin_prod_eqₓ'. -/
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
 theorem nhdsWithin_prod_eq {α : Type _} [TopologicalSpace α] {β : Type _} [TopologicalSpace β]
-    (a : α) (b : β) (s : Set α) (t : Set β) : 𝓝[s ×ˢ t] (a, b) = 𝓝[s] a ×ᶠ 𝓝[t] b :=
-  by
-  delta nhdsWithin
-  rw [nhds_prod_eq, ← Filter.prod_inf_prod, Filter.prod_principal_principal]
+    (a : α) (b : β) (s : Set α) (t : Set β) : 𝓝[s ×ˢ t] (a, b) = 𝓝[s] a ×ᶠ 𝓝[t] b := by
+  delta nhdsWithin; rw [nhds_prod_eq, ← Filter.prod_inf_prod, Filter.prod_principal_principal]
 #align nhds_within_prod_eq nhdsWithin_prod_eq
 
 /- warning: nhds_within_prod -> nhdsWithin_prod is a dubious translation:
@@ -585,9 +570,7 @@ Case conversion may be inaccurate. Consider using '#align nhds_within_prod nhdsW
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
 theorem nhdsWithin_prod {α : Type _} [TopologicalSpace α] {β : Type _} [TopologicalSpace β]
     {s u : Set α} {t v : Set β} {a : α} {b : β} (hu : u ∈ 𝓝[s] a) (hv : v ∈ 𝓝[t] b) :
-    u ×ˢ v ∈ 𝓝[s ×ˢ t] (a, b) := by
-  rw [nhdsWithin_prod_eq]
-  exact prod_mem_prod hu hv
+    u ×ˢ v ∈ 𝓝[s ×ˢ t] (a, b) := by rw [nhdsWithin_prod_eq]; exact prod_mem_prod hu hv
 #align nhds_within_prod nhdsWithin_prod
 
 /- warning: nhds_within_pi_eq' -> nhdsWithin_pi_eq' is a dubious translation:
@@ -1139,9 +1122,7 @@ theorem continuousOn_iff' {f : α → β} {s : Set α} :
     intro t
     rw [isOpen_induced_iff, Set.restrict_eq, Set.preimage_comp]
     simp only [Subtype.preimage_coe_eq_preimage_coe_iff]
-    constructor <;>
-      · rintro ⟨u, ou, useq⟩
-        exact ⟨u, ou, useq.symm⟩
+    constructor <;> · rintro ⟨u, ou, useq⟩; exact ⟨u, ou, useq.symm⟩
   rw [continuousOn_iff_continuous_restrict, continuous_def] <;> simp only [this]
 #align continuous_on_iff' continuousOn_iff'
 
@@ -1218,8 +1199,7 @@ theorem continuous_of_cover_nhds {ι : Sort _} {f : α → β} {s : ι → Set 
   continuous_iff_continuousAt.mpr fun x =>
     by
     let ⟨i, hi⟩ := hs x
-    rw [ContinuousAt, ← nhdsWithin_eq_nhds.2 hi]
-    exact hf _ _ (mem_of_mem_nhds hi)
+    rw [ContinuousAt, ← nhdsWithin_eq_nhds.2 hi]; exact hf _ _ (mem_of_mem_nhds hi)
 #align continuous_of_cover_nhds continuous_of_cover_nhds
 
 /- warning: continuous_on_empty -> continuousOn_empty is a dubious translation:
@@ -1526,8 +1506,7 @@ theorem IsOpenMap.continuousOn_image_of_leftInvOn {f : α → β} {s : Set α}
     ContinuousOn finv (f '' s) :=
   by
   refine' continuousOn_iff'.2 fun t ht => ⟨f '' (t ∩ s), _, _⟩
-  · rw [← image_restrict]
-    exact h _ (ht.preimage continuous_subtype_val)
+  · rw [← image_restrict]; exact h _ (ht.preimage continuous_subtype_val)
   · rw [inter_eq_self_of_subset_left (image_subset f (inter_subset_right t s)), hleft.image_inter']
 #align is_open_map.continuous_on_image_of_left_inv_on IsOpenMap.continuousOn_image_of_leftInvOn
 
@@ -1993,8 +1972,7 @@ theorem continuousOn_open_of_generateFrom {β : Type _} {s : Set α} {T : Set (S
   intro t ht
   induction' ht with u hu u v Tu Tv hu hv U hU hU'
   · exact h u hu
-  · simp only [preimage_univ, inter_univ]
-    exact hs
+  · simp only [preimage_univ, inter_univ]; exact hs
   · have : s ∩ f ⁻¹' (u ∩ v) = s ∩ f ⁻¹' u ∩ (s ∩ f ⁻¹' v) := by
       rw [preimage_inter, inter_assoc, inter_left_comm _ s, ← inter_assoc s s, inter_self]
     rw [this]
@@ -2245,11 +2223,8 @@ but is expected to have type
   forall {α : Type.{u2}} {β : Type.{u1}} [_inst_1 : TopologicalSpace.{u2} α] [_inst_2 : TopologicalSpace.{u1} β] (p : Prop) {f : α -> β} {g : α -> β} [_inst_5 : Decidable p], (p -> (Continuous.{u2, u1} α β _inst_1 _inst_2 f)) -> ((Not p) -> (Continuous.{u2, u1} α β _inst_1 _inst_2 g)) -> (Continuous.{u2, u1} α β _inst_1 _inst_2 (fun (a : α) => ite.{succ u1} β p _inst_5 (f a) (g a)))
 Case conversion may be inaccurate. Consider using '#align continuous_if_const continuous_if_constₓ'. -/
 theorem continuous_if_const (p : Prop) {f g : α → β} [Decidable p] (hf : p → Continuous f)
-    (hg : ¬p → Continuous g) : Continuous fun a => if p then f a else g a :=
-  by
-  split_ifs
-  exact hf h
-  exact hg h
+    (hg : ¬p → Continuous g) : Continuous fun a => if p then f a else g a := by split_ifs;
+  exact hf h; exact hg h
 #align continuous_if_const continuous_if_const
 
 /- warning: continuous.if_const -> Continuous.if_const is a dubious translation:
@@ -2328,10 +2303,8 @@ but is expected to have type
   forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] {s : Set.{u1} α} {s' : Set.{u1} α} {t : Set.{u1} α}, (Eq.{succ u1} (Set.{u1} α) (Inter.inter.{u1} (Set.{u1} α) (Set.instInterSet.{u1} α) s (frontier.{u1} α _inst_1 t)) (Inter.inter.{u1} (Set.{u1} α) (Set.instInterSet.{u1} α) s' (frontier.{u1} α _inst_1 t))) -> (Eq.{succ u1} (Set.{u1} α) (Inter.inter.{u1} (Set.{u1} α) (Set.instInterSet.{u1} α) (Set.ite.{u1} α t s s') (closure.{u1} α _inst_1 (HasCompl.compl.{u1} (Set.{u1} α) (BooleanAlgebra.toHasCompl.{u1} (Set.{u1} α) (Set.instBooleanAlgebraSet.{u1} α)) t))) (Inter.inter.{u1} (Set.{u1} α) (Set.instInterSet.{u1} α) s' (closure.{u1} α _inst_1 (HasCompl.compl.{u1} (Set.{u1} α) (BooleanAlgebra.toHasCompl.{u1} (Set.{u1} α) (Set.instBooleanAlgebraSet.{u1} α)) t))))
 Case conversion may be inaccurate. Consider using '#align ite_inter_closure_compl_eq_of_inter_frontier_eq ite_inter_closure_compl_eq_of_inter_frontier_eqₓ'. -/
 theorem ite_inter_closure_compl_eq_of_inter_frontier_eq {s s' t : Set α}
-    (ht : s ∩ frontier t = s' ∩ frontier t) : t.ite s s' ∩ closure (tᶜ) = s' ∩ closure (tᶜ) :=
-  by
-  rw [← ite_compl, ite_inter_closure_eq_of_inter_frontier_eq]
-  rwa [frontier_compl, eq_comm]
+    (ht : s ∩ frontier t = s' ∩ frontier t) : t.ite s s' ∩ closure (tᶜ) = s' ∩ closure (tᶜ) := by
+  rw [← ite_compl, ite_inter_closure_eq_of_inter_frontier_eq]; rwa [frontier_compl, eq_comm]
 #align ite_inter_closure_compl_eq_of_inter_frontier_eq ite_inter_closure_compl_eq_of_inter_frontier_eq
 
 /- warning: continuous_on_piecewise_ite' -> continuousOn_piecewise_ite' is a dubious translation:
@@ -2470,11 +2443,6 @@ Case conversion may be inaccurate. Consider using '#align continuous_within_at_p
 theorem continuousWithinAt_prod_iff {f : α → β × γ} {s : Set α} {x : α} :
     ContinuousWithinAt f s x ↔
       ContinuousWithinAt (Prod.fst ∘ f) s x ∧ ContinuousWithinAt (Prod.snd ∘ f) s x :=
-  ⟨fun h => ⟨h.fst, h.snd⟩, by
-    rintro ⟨h1, h2⟩
-    convert h1.prod h2
-    ext
-    rfl
-    rfl⟩
+  ⟨fun h => ⟨h.fst, h.snd⟩, by rintro ⟨h1, h2⟩; convert h1.prod h2; ext; rfl; rfl⟩
 #align continuous_within_at_prod_iff continuousWithinAt_prod_iff
 

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

@@ -1078,10 +1078,7 @@ theorem continuousOn_pi {ι : Type _} {π : ι → Type _} [∀ i, TopologicalSp
 #align continuous_on_pi continuousOn_pi
 
 /- warning: continuous_within_at.fin_insert_nth -> ContinuousWithinAt.fin_insertNth is a dubious translation:
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+<too large>
 Case conversion may be inaccurate. Consider using '#align continuous_within_at.fin_insert_nth ContinuousWithinAt.fin_insertNthₓ'. -/
 theorem ContinuousWithinAt.fin_insertNth {n} {π : Fin (n + 1) → Type _}
     [∀ i, TopologicalSpace (π i)] (i : Fin (n + 1)) {f : α → π i} {a : α} {s : Set α}
@@ -1091,10 +1088,7 @@ theorem ContinuousWithinAt.fin_insertNth {n} {π : Fin (n + 1) → Type _}
 #align continuous_within_at.fin_insert_nth ContinuousWithinAt.fin_insertNth
 
 /- warning: continuous_on.fin_insert_nth -> ContinuousOn.fin_insertNth is a dubious translation:
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+<too large>
 Case conversion may be inaccurate. Consider using '#align continuous_on.fin_insert_nth ContinuousOn.fin_insertNthₓ'. -/
 theorem ContinuousOn.fin_insertNth {n} {π : Fin (n + 1) → Type _} [∀ i, TopologicalSpace (π i)]
     (i : Fin (n + 1)) {f : α → π i} {s : Set α} (hf : ContinuousOn f s)

mathlib commit https://github.com/leanprover-community/mathlib/commit/ef95945cd48c932c9e034872bd25c3c220d9c946

@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Sébastien Gouëzel
 
 ! This file was ported from Lean 3 source module topology.continuous_on
-! leanprover-community/mathlib commit 55d771df074d0dd020139ee1cd4b95521422df9f
+! leanprover-community/mathlib commit d4f691b9e5f94cfc64639973f3544c95f8d5d494
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
@@ -191,12 +191,7 @@ theorem nhds_of_nhdsWithin_of_nhds {s t : Set α} {a : α} (h1 : s ∈ 𝓝 a) (
 #print mem_nhdsWithin_iff_eventually /-
 theorem mem_nhdsWithin_iff_eventually {s t : Set α} {x : α} :
     t ∈ 𝓝[s] x ↔ ∀ᶠ y in 𝓝 x, y ∈ s → y ∈ t :=
-  by
-  rw [mem_nhdsWithin_iff_exists_mem_nhds_inter]
-  constructor
-  · rintro ⟨u, hu, hut⟩
-    exact eventually_of_mem hu fun x hxu hxs => hut ⟨hxu, hxs⟩
-  · refine' fun h => ⟨_, h, fun y hy => hy.1 hy.2⟩
+  set_eventuallyLE_iff_mem_inf_principal.symm
 #align mem_nhds_within_iff_eventually mem_nhdsWithin_iff_eventually
 -/
 
@@ -213,15 +208,7 @@ theorem mem_nhdsWithin_iff_eventuallyEq {s t : Set α} {x : α} :
 
 #print nhdsWithin_eq_iff_eventuallyEq /-
 theorem nhdsWithin_eq_iff_eventuallyEq {s t : Set α} {x : α} : 𝓝[s] x = 𝓝[t] x ↔ s =ᶠ[𝓝 x] t :=
-  by
-  simp_rw [Filter.ext_iff, mem_nhdsWithin_iff_eventually, eventually_eq_set]
-  constructor
-  · intro h
-    filter_upwards [(h t).mpr (eventually_of_forall fun x => id),
-      (h s).mp (eventually_of_forall fun x => id)]
-    exact fun x => Iff.intro
-  · refine' fun h u => eventually_congr (h.mono fun x h => _)
-    rw [h]
+  set_eventuallyEq_iff_inf_principal.symm
 #align nhds_within_eq_iff_eventually_eq nhdsWithin_eq_iff_eventuallyEq
 -/
 
@@ -232,11 +219,7 @@ but is expected to have type
   forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] {s : Set.{u1} α} {t : Set.{u1} α} {x : α}, Iff (LE.le.{u1} (Filter.{u1} α) (Preorder.toLE.{u1} (Filter.{u1} α) (PartialOrder.toPreorder.{u1} (Filter.{u1} α) (Filter.instPartialOrderFilter.{u1} α))) (nhdsWithin.{u1} α _inst_1 x s) (nhdsWithin.{u1} α _inst_1 x t)) (Membership.mem.{u1, u1} (Set.{u1} α) (Filter.{u1} α) (instMembershipSetFilter.{u1} α) t (nhdsWithin.{u1} α _inst_1 x s))
 Case conversion may be inaccurate. Consider using '#align nhds_within_le_iff nhdsWithin_le_iffₓ'. -/
 theorem nhdsWithin_le_iff {s t : Set α} {x : α} : 𝓝[s] x ≤ 𝓝[t] x ↔ t ∈ 𝓝[s] x :=
-  by
-  simp_rw [Filter.le_def, mem_nhdsWithin_iff_eventually]
-  constructor
-  · exact fun h => (h t <| eventually_of_forall fun x => id).mono fun x => id
-  · exact fun h u hu => (h.And hu).mono fun x hx h => hx.2 <| hx.1 h
+  set_eventuallyLE_iff_inf_principal_le.symm.trans set_eventuallyLE_iff_mem_inf_principal
 #align nhds_within_le_iff nhdsWithin_le_iff
 
 theorem preimage_nhdsWithin_coinduced' {π : α → β} {s : Set β} {t : Set α} {a : α} (h : a ∈ t)

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

@@ -1098,7 +1098,7 @@ theorem continuousOn_pi {ι : Type _} {π : ι → Type _} [∀ i, TopologicalSp
 lean 3 declaration is
   forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] {n : Nat} {π : (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) -> Type.{u2}} [_inst_5 : forall (i : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))), TopologicalSpace.{u2} (π i)] (i : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) {f : α -> (π i)} {a : α} {s : Set.{u1} α}, (ContinuousWithinAt.{u1, u2} α (π i) _inst_1 (_inst_5 i) f s a) -> (forall {g : α -> (forall (j : Fin n), π (coeFn.{1, 1} (OrderEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (Fin.hasLe n) (Preorder.toHasLe.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (PartialOrder.toPreorder.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (Fin.partialOrder (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne)))))))) (fun (_x : RelEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (LE.le.{0} (Fin n) (Fin.hasLe n)) (LE.le.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (Preorder.toHasLe.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (PartialOrder.toPreorder.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (Fin.partialOrder (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))))))) => (Fin n) -> (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne)))))) (RelEmbedding.hasCoeToFun.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (LE.le.{0} (Fin n) (Fin.hasLe n)) (LE.le.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (Preorder.toHasLe.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 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Nat Nat.hasOne))))) (Fin.partialOrder (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne)))))))) (fun (_x : RelEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (LE.le.{0} (Fin n) (Fin.hasLe n)) (LE.le.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (Preorder.toHasLe.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (PartialOrder.toPreorder.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (Fin.partialOrder (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} 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(instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))))))) (Fin.succAbove n i) j)) _inst_1 (Pi.topologicalSpace.{0, u2} (Fin n) (fun (j : Fin n) => π (coeFn.{1, 1} (OrderEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (Fin.hasLe n) (Preorder.toHasLe.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (PartialOrder.toPreorder.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (Fin.partialOrder (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne)))))))) (fun (_x : RelEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat 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(Preorder.toHasLe.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (PartialOrder.toPreorder.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (Fin.partialOrder (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))))))) => (Fin n) -> (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne)))))) (RelEmbedding.hasCoeToFun.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (LE.le.{0} (Fin n) (Fin.hasLe n)) (LE.le.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (Preorder.toHasLe.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (PartialOrder.toPreorder.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (Fin.partialOrder (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))))))) (Fin.succAbove n i) a))) g s a) -> (ContinuousWithinAt.{u1, u2} α (forall (j : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))), π j) _inst_1 (Pi.topologicalSpace.{0, u2} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (fun (j : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) => π j) (fun (a : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) => _inst_5 a)) (fun (a : α) => Fin.insertNth.{u2} n (fun (i : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) => π i) i (f a) (g a)) s a))
 but is expected to have type
-  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] {n : Nat} {π : (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) -> Type.{u2}} [_inst_5 : forall (i : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))), TopologicalSpace.{u2} (π i)] (i : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) {f : α -> (π i)} {a : α} {s : Set.{u1} α}, (ContinuousWithinAt.{u1, u2} α (π i) _inst_1 (_inst_5 i) f s a) -> (forall {g : α -> (forall (j : Fin n), π (FunLike.coe.{1, 1, 1} (OrderEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin n) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))))) (Fin n) (fun (_x : Fin n) => (fun (x._@.Mathlib.Order.Hom.Lattice._hyg.494 : Fin n) => Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) _x) (InfHomClass.toFunLike.{0, 0, 0} (OrderEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin n) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))))) (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (Lattice.toInf.{0} (Fin n) (DistribLattice.toLattice.{0} (Fin n) (instDistribLattice.{0} (Fin n) (Fin.instLinearOrderFin n)))) (Lattice.toInf.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (Fin.instLatticeFinHAddNatInstHAddInstAddNatOfNat n)) (LatticeHomClass.toInfHomClass.{0, 0, 0} (OrderEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin n) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))))) (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (DistribLattice.toLattice.{0} (Fin n) (instDistribLattice.{0} (Fin n) (Fin.instLinearOrderFin n))) (Fin.instLatticeFinHAddNatInstHAddInstAddNatOfNat n) (OrderHomClass.toLatticeHomClass.{0, 0, 0} (OrderEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin n) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))))) (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (Fin.instLinearOrderFin n) (Fin.instLatticeFinHAddNatInstHAddInstAddNatOfNat n) (RelEmbedding.instRelHomClassRelEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.680 : Fin n) (x._@.Mathlib.Order.Hom.Basic._hyg.682 : Fin n) => LE.le.{0} (Fin n) (instLEFin n) x._@.Mathlib.Order.Hom.Basic._hyg.680 x._@.Mathlib.Order.Hom.Basic._hyg.682) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.695 : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (x._@.Mathlib.Order.Hom.Basic._hyg.697 : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) => LE.le.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) x._@.Mathlib.Order.Hom.Basic._hyg.695 x._@.Mathlib.Order.Hom.Basic._hyg.697))))) (Fin.succAbove n i) j))}, (ContinuousWithinAt.{u1, u2} α (forall (j : Fin n), π (FunLike.coe.{1, 1, 1} (OrderEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin n) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))))) (Fin n) (fun (_x : Fin n) => (fun (x._@.Mathlib.Order.Hom.Lattice._hyg.494 : Fin n) => Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) _x) (InfHomClass.toFunLike.{0, 0, 0} (OrderEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin n) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))))) (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (Lattice.toInf.{0} (Fin n) (DistribLattice.toLattice.{0} (Fin n) (instDistribLattice.{0} (Fin n) (Fin.instLinearOrderFin n)))) (Lattice.toInf.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (Fin.instLatticeFinHAddNatInstHAddInstAddNatOfNat n)) (LatticeHomClass.toInfHomClass.{0, 0, 0} (OrderEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin n) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))))) (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (DistribLattice.toLattice.{0} (Fin n) (instDistribLattice.{0} (Fin n) (Fin.instLinearOrderFin n))) (Fin.instLatticeFinHAddNatInstHAddInstAddNatOfNat n) (OrderHomClass.toLatticeHomClass.{0, 0, 0} (OrderEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin n) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))))) (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (Fin.instLinearOrderFin n) (Fin.instLatticeFinHAddNatInstHAddInstAddNatOfNat n) (RelEmbedding.instRelHomClassRelEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.680 : Fin n) (x._@.Mathlib.Order.Hom.Basic._hyg.682 : Fin n) => LE.le.{0} (Fin n) (instLEFin n) x._@.Mathlib.Order.Hom.Basic._hyg.680 x._@.Mathlib.Order.Hom.Basic._hyg.682) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.695 : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (x._@.Mathlib.Order.Hom.Basic._hyg.697 : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) => LE.le.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) x._@.Mathlib.Order.Hom.Basic._hyg.695 x._@.Mathlib.Order.Hom.Basic._hyg.697))))) (Fin.succAbove n i) j)) _inst_1 (Pi.topologicalSpace.{0, u2} (Fin n) (fun (j : Fin n) => π (FunLike.coe.{1, 1, 1} (OrderEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin n) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))))) (Fin n) (fun (_x : Fin n) => (fun (x._@.Mathlib.Order.Hom.Lattice._hyg.494 : Fin n) => Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) _x) (InfHomClass.toFunLike.{0, 0, 0} (OrderEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin n) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))))) (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (Lattice.toInf.{0} (Fin n) (DistribLattice.toLattice.{0} (Fin n) (instDistribLattice.{0} (Fin n) (Fin.instLinearOrderFin n)))) (Lattice.toInf.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (Fin.instLatticeFinHAddNatInstHAddInstAddNatOfNat n)) (LatticeHomClass.toInfHomClass.{0, 0, 0} (OrderEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin n) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))))) (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (DistribLattice.toLattice.{0} (Fin n) (instDistribLattice.{0} (Fin n) (Fin.instLinearOrderFin n))) (Fin.instLatticeFinHAddNatInstHAddInstAddNatOfNat n) (OrderHomClass.toLatticeHomClass.{0, 0, 0} (OrderEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin n) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))))) (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (Fin.instLinearOrderFin n) (Fin.instLatticeFinHAddNatInstHAddInstAddNatOfNat n) (RelEmbedding.instRelHomClassRelEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.680 : Fin n) (x._@.Mathlib.Order.Hom.Basic._hyg.682 : Fin n) => LE.le.{0} (Fin n) (instLEFin n) x._@.Mathlib.Order.Hom.Basic._hyg.680 x._@.Mathlib.Order.Hom.Basic._hyg.682) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.695 : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (x._@.Mathlib.Order.Hom.Basic._hyg.697 : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) => LE.le.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) x._@.Mathlib.Order.Hom.Basic._hyg.695 x._@.Mathlib.Order.Hom.Basic._hyg.697))))) (Fin.succAbove n i) j)) (fun (a : Fin n) => _inst_5 (FunLike.coe.{1, 1, 1} (OrderEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin n) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))))) (Fin n) (fun (_x : Fin n) => (fun (x._@.Mathlib.Order.Hom.Lattice._hyg.494 : Fin n) => Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) _x) (InfHomClass.toFunLike.{0, 0, 0} (OrderEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin n) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))))) (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (Lattice.toInf.{0} (Fin n) (DistribLattice.toLattice.{0} (Fin n) (instDistribLattice.{0} (Fin n) (Fin.instLinearOrderFin n)))) (Lattice.toInf.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (Fin.instLatticeFinHAddNatInstHAddInstAddNatOfNat n)) (LatticeHomClass.toInfHomClass.{0, 0, 0} (OrderEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin n) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))))) (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (DistribLattice.toLattice.{0} (Fin n) (instDistribLattice.{0} (Fin n) (Fin.instLinearOrderFin n))) (Fin.instLatticeFinHAddNatInstHAddInstAddNatOfNat n) (OrderHomClass.toLatticeHomClass.{0, 0, 0} (OrderEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin n) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))))) (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (Fin.instLinearOrderFin n) (Fin.instLatticeFinHAddNatInstHAddInstAddNatOfNat n) (RelEmbedding.instRelHomClassRelEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.680 : Fin n) (x._@.Mathlib.Order.Hom.Basic._hyg.682 : Fin n) => LE.le.{0} (Fin n) (instLEFin n) x._@.Mathlib.Order.Hom.Basic._hyg.680 x._@.Mathlib.Order.Hom.Basic._hyg.682) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.695 : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (x._@.Mathlib.Order.Hom.Basic._hyg.697 : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) => LE.le.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) x._@.Mathlib.Order.Hom.Basic._hyg.695 x._@.Mathlib.Order.Hom.Basic._hyg.697))))) (Fin.succAbove n i) a))) g s a) -> (ContinuousWithinAt.{u1, u2} α (forall (j : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))), π j) _inst_1 (Pi.topologicalSpace.{0, u2} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (fun (j : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) => π j) (fun (a : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) => _inst_5 a)) (fun (a : α) => Fin.insertNth.{u2} n π i (f a) (g a)) s a))
+  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] {n : Nat} {π : (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) -> Type.{u2}} [_inst_5 : forall (i : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))), TopologicalSpace.{u2} (π i)] (i : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) {f : α -> (π i)} {a : α} {s : Set.{u1} α}, (ContinuousWithinAt.{u1, u2} α (π i) _inst_1 (_inst_5 i) f s a) -> (forall {g : α -> (forall (j : Fin n), π (FunLike.coe.{1, 1, 1} (OrderEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin n) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))))) (Fin n) (fun (_x : Fin n) => (fun (x._@.Mathlib.Order.Hom.Lattice._hyg.494 : Fin n) => Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) _x) (InfHomClass.toFunLike.{0, 0, 0} (OrderEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin n) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))))) (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (Lattice.toInf.{0} (Fin n) (DistribLattice.toLattice.{0} (Fin n) (instDistribLattice.{0} (Fin n) (Fin.instLinearOrderFin n)))) (Lattice.toInf.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (Fin.instLatticeFinHAddNatInstHAddInstAddNatOfNat n)) (LatticeHomClass.toInfHomClass.{0, 0, 0} (OrderEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin n) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))))) (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (DistribLattice.toLattice.{0} (Fin n) (instDistribLattice.{0} (Fin n) (Fin.instLinearOrderFin n))) (Fin.instLatticeFinHAddNatInstHAddInstAddNatOfNat n) (OrderHomClass.toLatticeHomClass.{0, 0, 0} (OrderEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin n) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))))) (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (Fin.instLinearOrderFin n) (Fin.instLatticeFinHAddNatInstHAddInstAddNatOfNat n) (RelEmbedding.instRelHomClassRelEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.682 : Fin n) (x._@.Mathlib.Order.Hom.Basic._hyg.684 : Fin n) => LE.le.{0} (Fin n) (instLEFin n) x._@.Mathlib.Order.Hom.Basic._hyg.682 x._@.Mathlib.Order.Hom.Basic._hyg.684) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.697 : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (x._@.Mathlib.Order.Hom.Basic._hyg.699 : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) => LE.le.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) x._@.Mathlib.Order.Hom.Basic._hyg.697 x._@.Mathlib.Order.Hom.Basic._hyg.699))))) (Fin.succAbove n i) j))}, (ContinuousWithinAt.{u1, u2} α (forall (j : Fin n), π (FunLike.coe.{1, 1, 1} (OrderEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin n) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))))) (Fin n) (fun (_x : Fin n) => (fun (x._@.Mathlib.Order.Hom.Lattice._hyg.494 : Fin n) => Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) _x) (InfHomClass.toFunLike.{0, 0, 0} (OrderEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin n) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))))) (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (Lattice.toInf.{0} (Fin n) (DistribLattice.toLattice.{0} (Fin n) (instDistribLattice.{0} (Fin n) (Fin.instLinearOrderFin n)))) (Lattice.toInf.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (Fin.instLatticeFinHAddNatInstHAddInstAddNatOfNat n)) (LatticeHomClass.toInfHomClass.{0, 0, 0} (OrderEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin n) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))))) (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (DistribLattice.toLattice.{0} (Fin n) (instDistribLattice.{0} (Fin n) (Fin.instLinearOrderFin n))) (Fin.instLatticeFinHAddNatInstHAddInstAddNatOfNat n) (OrderHomClass.toLatticeHomClass.{0, 0, 0} (OrderEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin n) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))))) (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (Fin.instLinearOrderFin n) (Fin.instLatticeFinHAddNatInstHAddInstAddNatOfNat n) (RelEmbedding.instRelHomClassRelEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.682 : Fin n) (x._@.Mathlib.Order.Hom.Basic._hyg.684 : Fin n) => LE.le.{0} (Fin n) (instLEFin n) x._@.Mathlib.Order.Hom.Basic._hyg.682 x._@.Mathlib.Order.Hom.Basic._hyg.684) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.697 : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (x._@.Mathlib.Order.Hom.Basic._hyg.699 : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) => LE.le.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) x._@.Mathlib.Order.Hom.Basic._hyg.697 x._@.Mathlib.Order.Hom.Basic._hyg.699))))) (Fin.succAbove n i) j)) _inst_1 (Pi.topologicalSpace.{0, u2} (Fin n) (fun (j : Fin n) => π (FunLike.coe.{1, 1, 1} (OrderEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin n) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))))) (Fin n) (fun (_x : Fin n) => (fun (x._@.Mathlib.Order.Hom.Lattice._hyg.494 : Fin n) => Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) _x) (InfHomClass.toFunLike.{0, 0, 0} (OrderEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin n) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))))) (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (Lattice.toInf.{0} (Fin n) (DistribLattice.toLattice.{0} (Fin n) (instDistribLattice.{0} (Fin n) (Fin.instLinearOrderFin n)))) (Lattice.toInf.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (Fin.instLatticeFinHAddNatInstHAddInstAddNatOfNat n)) (LatticeHomClass.toInfHomClass.{0, 0, 0} (OrderEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin n) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))))) (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (DistribLattice.toLattice.{0} (Fin n) (instDistribLattice.{0} (Fin n) (Fin.instLinearOrderFin n))) (Fin.instLatticeFinHAddNatInstHAddInstAddNatOfNat n) (OrderHomClass.toLatticeHomClass.{0, 0, 0} (OrderEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin n) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))))) (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (Fin.instLinearOrderFin n) (Fin.instLatticeFinHAddNatInstHAddInstAddNatOfNat n) (RelEmbedding.instRelHomClassRelEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.682 : Fin n) (x._@.Mathlib.Order.Hom.Basic._hyg.684 : Fin n) => LE.le.{0} (Fin n) (instLEFin n) x._@.Mathlib.Order.Hom.Basic._hyg.682 x._@.Mathlib.Order.Hom.Basic._hyg.684) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.697 : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (x._@.Mathlib.Order.Hom.Basic._hyg.699 : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) => LE.le.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) x._@.Mathlib.Order.Hom.Basic._hyg.697 x._@.Mathlib.Order.Hom.Basic._hyg.699))))) (Fin.succAbove n i) j)) (fun (a : Fin n) => _inst_5 (FunLike.coe.{1, 1, 1} (OrderEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin n) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))))) (Fin n) (fun (_x : Fin n) => (fun (x._@.Mathlib.Order.Hom.Lattice._hyg.494 : Fin n) => Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) _x) (InfHomClass.toFunLike.{0, 0, 0} (OrderEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin n) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))))) (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (Lattice.toInf.{0} (Fin n) (DistribLattice.toLattice.{0} (Fin n) (instDistribLattice.{0} (Fin n) (Fin.instLinearOrderFin n)))) (Lattice.toInf.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (Fin.instLatticeFinHAddNatInstHAddInstAddNatOfNat n)) (LatticeHomClass.toInfHomClass.{0, 0, 0} (OrderEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin n) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))))) (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (DistribLattice.toLattice.{0} (Fin n) (instDistribLattice.{0} (Fin n) (Fin.instLinearOrderFin n))) (Fin.instLatticeFinHAddNatInstHAddInstAddNatOfNat n) (OrderHomClass.toLatticeHomClass.{0, 0, 0} (OrderEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin n) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))))) (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (Fin.instLinearOrderFin n) (Fin.instLatticeFinHAddNatInstHAddInstAddNatOfNat n) (RelEmbedding.instRelHomClassRelEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.682 : Fin n) (x._@.Mathlib.Order.Hom.Basic._hyg.684 : Fin n) => LE.le.{0} (Fin n) (instLEFin n) x._@.Mathlib.Order.Hom.Basic._hyg.682 x._@.Mathlib.Order.Hom.Basic._hyg.684) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.697 : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (x._@.Mathlib.Order.Hom.Basic._hyg.699 : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) => LE.le.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) x._@.Mathlib.Order.Hom.Basic._hyg.697 x._@.Mathlib.Order.Hom.Basic._hyg.699))))) (Fin.succAbove n i) a))) g s a) -> (ContinuousWithinAt.{u1, u2} α (forall (j : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))), π j) _inst_1 (Pi.topologicalSpace.{0, u2} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (fun (j : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) => π j) (fun (a : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) => _inst_5 a)) (fun (a : α) => Fin.insertNth.{u2} n π i (f a) (g a)) s a))
 Case conversion may be inaccurate. Consider using '#align continuous_within_at.fin_insert_nth ContinuousWithinAt.fin_insertNthₓ'. -/
 theorem ContinuousWithinAt.fin_insertNth {n} {π : Fin (n + 1) → Type _}
     [∀ i, TopologicalSpace (π i)] (i : Fin (n + 1)) {f : α → π i} {a : α} {s : Set α}
@@ -1111,7 +1111,7 @@ theorem ContinuousWithinAt.fin_insertNth {n} {π : Fin (n + 1) → Type _}
 lean 3 declaration is
   forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] {n : Nat} {π : (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) -> Type.{u2}} [_inst_5 : forall (i : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))), TopologicalSpace.{u2} (π i)] (i : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) {f : α -> (π i)} {s : Set.{u1} α}, (ContinuousOn.{u1, u2} α (π i) _inst_1 (_inst_5 i) f s) -> (forall {g : α -> (forall (j : Fin n), π (coeFn.{1, 1} (OrderEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (Fin.hasLe n) (Preorder.toHasLe.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} 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 but is expected to have type
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instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (Lattice.toInf.{0} (Fin n) (DistribLattice.toLattice.{0} (Fin n) (instDistribLattice.{0} (Fin n) (Fin.instLinearOrderFin n)))) (Lattice.toInf.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (Fin.instLatticeFinHAddNatInstHAddInstAddNatOfNat n)) (LatticeHomClass.toInfHomClass.{0, 0, 0} (OrderEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin n) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))))) (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (DistribLattice.toLattice.{0} (Fin n) (instDistribLattice.{0} (Fin n) (Fin.instLinearOrderFin n))) (Fin.instLatticeFinHAddNatInstHAddInstAddNatOfNat n) (OrderHomClass.toLatticeHomClass.{0, 0, 0} (OrderEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin n) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))))) (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (Fin.instLinearOrderFin n) (Fin.instLatticeFinHAddNatInstHAddInstAddNatOfNat n) (RelEmbedding.instRelHomClassRelEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.680 : Fin n) (x._@.Mathlib.Order.Hom.Basic._hyg.682 : Fin n) => LE.le.{0} (Fin n) (instLEFin n) x._@.Mathlib.Order.Hom.Basic._hyg.680 x._@.Mathlib.Order.Hom.Basic._hyg.682) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.695 : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (x._@.Mathlib.Order.Hom.Basic._hyg.697 : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) => LE.le.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) x._@.Mathlib.Order.Hom.Basic._hyg.695 x._@.Mathlib.Order.Hom.Basic._hyg.697))))) (Fin.succAbove n i) j)) _inst_1 (Pi.topologicalSpace.{0, u2} (Fin n) (fun (j : Fin n) => π (FunLike.coe.{1, 1, 1} (OrderEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin n) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))))) (Fin n) (fun (_x : Fin n) => (fun (x._@.Mathlib.Order.Hom.Lattice._hyg.494 : Fin n) => Fin 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(RelEmbedding.instRelHomClassRelEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.680 : Fin n) (x._@.Mathlib.Order.Hom.Basic._hyg.682 : Fin n) => LE.le.{0} (Fin n) (instLEFin n) x._@.Mathlib.Order.Hom.Basic._hyg.680 x._@.Mathlib.Order.Hom.Basic._hyg.682) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.695 : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (x._@.Mathlib.Order.Hom.Basic._hyg.697 : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) => LE.le.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) x._@.Mathlib.Order.Hom.Basic._hyg.695 x._@.Mathlib.Order.Hom.Basic._hyg.697))))) (Fin.succAbove n i) j)) (fun (a : Fin n) => _inst_5 (FunLike.coe.{1, 1, 1} (OrderEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin n) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))))) (Fin n) (fun (_x : Fin n) => (fun (x._@.Mathlib.Order.Hom.Lattice._hyg.494 : Fin n) => Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) _x) (InfHomClass.toFunLike.{0, 0, 0} (OrderEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin n) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))))) (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (Lattice.toInf.{0} (Fin n) (DistribLattice.toLattice.{0} (Fin n) (instDistribLattice.{0} (Fin n) (Fin.instLinearOrderFin n)))) (Lattice.toInf.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (Fin.instLatticeFinHAddNatInstHAddInstAddNatOfNat n)) (LatticeHomClass.toInfHomClass.{0, 0, 0} (OrderEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin n) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))))) (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (DistribLattice.toLattice.{0} (Fin n) (instDistribLattice.{0} (Fin n) (Fin.instLinearOrderFin n))) (Fin.instLatticeFinHAddNatInstHAddInstAddNatOfNat n) (OrderHomClass.toLatticeHomClass.{0, 0, 0} (OrderEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin n) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))))) (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (Fin.instLinearOrderFin n) (Fin.instLatticeFinHAddNatInstHAddInstAddNatOfNat n) (RelEmbedding.instRelHomClassRelEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.680 : Fin n) (x._@.Mathlib.Order.Hom.Basic._hyg.682 : Fin n) => LE.le.{0} (Fin n) (instLEFin n) x._@.Mathlib.Order.Hom.Basic._hyg.680 x._@.Mathlib.Order.Hom.Basic._hyg.682) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.695 : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (x._@.Mathlib.Order.Hom.Basic._hyg.697 : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) => LE.le.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) x._@.Mathlib.Order.Hom.Basic._hyg.695 x._@.Mathlib.Order.Hom.Basic._hyg.697))))) (Fin.succAbove n i) a))) g s) -> (ContinuousOn.{u1, u2} α (forall (j : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))), π j) _inst_1 (Pi.topologicalSpace.{0, u2} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (fun (j : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) => π j) (fun (a : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) => _inst_5 a)) (fun (a : α) => Fin.insertNth.{u2} n π i (f a) (g a)) s))
+  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] {n : Nat} {π : (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) -> Type.{u2}} [_inst_5 : forall (i : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))), TopologicalSpace.{u2} (π i)] (i : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) {f : α -> (π i)} {s : Set.{u1} α}, (ContinuousOn.{u1, u2} α (π i) _inst_1 (_inst_5 i) f s) -> (forall {g : α -> (forall (j : Fin n), π (FunLike.coe.{1, 1, 1} (OrderEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin n) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))))) (Fin n) (fun (_x : Fin n) => (fun (x._@.Mathlib.Order.Hom.Lattice._hyg.494 : 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(RelEmbedding.instRelHomClassRelEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.682 : Fin n) (x._@.Mathlib.Order.Hom.Basic._hyg.684 : Fin n) => LE.le.{0} (Fin n) (instLEFin n) x._@.Mathlib.Order.Hom.Basic._hyg.682 x._@.Mathlib.Order.Hom.Basic._hyg.684) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.697 : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (x._@.Mathlib.Order.Hom.Basic._hyg.699 : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) => LE.le.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) x._@.Mathlib.Order.Hom.Basic._hyg.697 x._@.Mathlib.Order.Hom.Basic._hyg.699))))) (Fin.succAbove n i) j))}, (ContinuousOn.{u1, u2} α (forall (j : Fin n), π (FunLike.coe.{1, 1, 1} (OrderEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin n) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))))) (Fin n) (fun (_x : Fin n) => (fun (x._@.Mathlib.Order.Hom.Lattice._hyg.494 : Fin n) => Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) _x) (InfHomClass.toFunLike.{0, 0, 0} (OrderEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin n) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))))) (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (Lattice.toInf.{0} (Fin n) (DistribLattice.toLattice.{0} (Fin n) (instDistribLattice.{0} (Fin n) (Fin.instLinearOrderFin n)))) (Lattice.toInf.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (Fin.instLatticeFinHAddNatInstHAddInstAddNatOfNat n)) (LatticeHomClass.toInfHomClass.{0, 0, 0} (OrderEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin n) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))))) (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (DistribLattice.toLattice.{0} (Fin n) (instDistribLattice.{0} (Fin n) (Fin.instLinearOrderFin n))) (Fin.instLatticeFinHAddNatInstHAddInstAddNatOfNat n) (OrderHomClass.toLatticeHomClass.{0, 0, 0} (OrderEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin n) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))))) (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (Fin.instLinearOrderFin n) (Fin.instLatticeFinHAddNatInstHAddInstAddNatOfNat n) (RelEmbedding.instRelHomClassRelEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.682 : Fin n) (x._@.Mathlib.Order.Hom.Basic._hyg.684 : Fin n) => LE.le.{0} (Fin n) (instLEFin n) x._@.Mathlib.Order.Hom.Basic._hyg.682 x._@.Mathlib.Order.Hom.Basic._hyg.684) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.697 : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (x._@.Mathlib.Order.Hom.Basic._hyg.699 : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) => LE.le.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) x._@.Mathlib.Order.Hom.Basic._hyg.697 x._@.Mathlib.Order.Hom.Basic._hyg.699))))) (Fin.succAbove n i) j)) _inst_1 (Pi.topologicalSpace.{0, u2} (Fin n) (fun (j : Fin n) => π (FunLike.coe.{1, 1, 1} (OrderEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin n) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))))) (Fin n) (fun (_x : Fin n) => (fun (x._@.Mathlib.Order.Hom.Lattice._hyg.494 : Fin n) => Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) _x) (InfHomClass.toFunLike.{0, 0, 0} (OrderEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin n) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))))) (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (Lattice.toInf.{0} (Fin n) (DistribLattice.toLattice.{0} (Fin n) (instDistribLattice.{0} (Fin n) (Fin.instLinearOrderFin n)))) (Lattice.toInf.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (Fin.instLatticeFinHAddNatInstHAddInstAddNatOfNat n)) (LatticeHomClass.toInfHomClass.{0, 0, 0} (OrderEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin n) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))))) (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (DistribLattice.toLattice.{0} (Fin n) (instDistribLattice.{0} (Fin n) (Fin.instLinearOrderFin n))) (Fin.instLatticeFinHAddNatInstHAddInstAddNatOfNat n) (OrderHomClass.toLatticeHomClass.{0, 0, 0} (OrderEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin n) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))))) (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (Fin.instLinearOrderFin n) (Fin.instLatticeFinHAddNatInstHAddInstAddNatOfNat n) (RelEmbedding.instRelHomClassRelEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.682 : Fin n) (x._@.Mathlib.Order.Hom.Basic._hyg.684 : Fin n) => LE.le.{0} (Fin n) (instLEFin n) x._@.Mathlib.Order.Hom.Basic._hyg.682 x._@.Mathlib.Order.Hom.Basic._hyg.684) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.697 : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (x._@.Mathlib.Order.Hom.Basic._hyg.699 : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) => LE.le.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) x._@.Mathlib.Order.Hom.Basic._hyg.697 x._@.Mathlib.Order.Hom.Basic._hyg.699))))) (Fin.succAbove n i) j)) (fun (a : Fin n) => _inst_5 (FunLike.coe.{1, 1, 1} (OrderEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin n) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))))) (Fin n) (fun (_x : Fin n) => (fun (x._@.Mathlib.Order.Hom.Lattice._hyg.494 : Fin n) => Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) _x) (InfHomClass.toFunLike.{0, 0, 0} (OrderEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin n) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))))) (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (Lattice.toInf.{0} (Fin n) (DistribLattice.toLattice.{0} (Fin n) (instDistribLattice.{0} (Fin n) (Fin.instLinearOrderFin n)))) (Lattice.toInf.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (Fin.instLatticeFinHAddNatInstHAddInstAddNatOfNat n)) (LatticeHomClass.toInfHomClass.{0, 0, 0} (OrderEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin n) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))))) (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (DistribLattice.toLattice.{0} (Fin n) (instDistribLattice.{0} (Fin n) (Fin.instLinearOrderFin n))) (Fin.instLatticeFinHAddNatInstHAddInstAddNatOfNat n) (OrderHomClass.toLatticeHomClass.{0, 0, 0} (OrderEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin n) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))))) (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (Fin.instLinearOrderFin n) (Fin.instLatticeFinHAddNatInstHAddInstAddNatOfNat n) (RelEmbedding.instRelHomClassRelEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.682 : Fin n) (x._@.Mathlib.Order.Hom.Basic._hyg.684 : Fin n) => LE.le.{0} (Fin n) (instLEFin n) x._@.Mathlib.Order.Hom.Basic._hyg.682 x._@.Mathlib.Order.Hom.Basic._hyg.684) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.697 : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (x._@.Mathlib.Order.Hom.Basic._hyg.699 : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) => LE.le.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) x._@.Mathlib.Order.Hom.Basic._hyg.697 x._@.Mathlib.Order.Hom.Basic._hyg.699))))) (Fin.succAbove n i) a))) g s) -> (ContinuousOn.{u1, u2} α (forall (j : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))), π j) _inst_1 (Pi.topologicalSpace.{0, u2} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (fun (j : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) => π j) (fun (a : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) => _inst_5 a)) (fun (a : α) => Fin.insertNth.{u2} n π i (f a) (g a)) s))
 Case conversion may be inaccurate. Consider using '#align continuous_on.fin_insert_nth ContinuousOn.fin_insertNthₓ'. -/
 theorem ContinuousOn.fin_insertNth {n} {π : Fin (n + 1) → Type _} [∀ i, TopologicalSpace (π i)]
     (i : Fin (n + 1)) {f : α → π i} {s : Set α} (hf : ContinuousOn f s)

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

@@ -227,7 +227,7 @@ theorem nhdsWithin_eq_iff_eventuallyEq {s t : Set α} {x : α} : 𝓝[s] x = 
 
 /- warning: nhds_within_le_iff -> nhdsWithin_le_iff is a dubious translation:
 lean 3 declaration is
-  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] {s : Set.{u1} α} {t : Set.{u1} α} {x : α}, Iff (LE.le.{u1} (Filter.{u1} α) (Preorder.toLE.{u1} (Filter.{u1} α) (PartialOrder.toPreorder.{u1} (Filter.{u1} α) (Filter.partialOrder.{u1} α))) (nhdsWithin.{u1} α _inst_1 x s) (nhdsWithin.{u1} α _inst_1 x t)) (Membership.Mem.{u1, u1} (Set.{u1} α) (Filter.{u1} α) (Filter.hasMem.{u1} α) t (nhdsWithin.{u1} α _inst_1 x s))
+  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] {s : Set.{u1} α} {t : Set.{u1} α} {x : α}, Iff (LE.le.{u1} (Filter.{u1} α) (Preorder.toHasLe.{u1} (Filter.{u1} α) (PartialOrder.toPreorder.{u1} (Filter.{u1} α) (Filter.partialOrder.{u1} α))) (nhdsWithin.{u1} α _inst_1 x s) (nhdsWithin.{u1} α _inst_1 x t)) (Membership.Mem.{u1, u1} (Set.{u1} α) (Filter.{u1} α) (Filter.hasMem.{u1} α) t (nhdsWithin.{u1} α _inst_1 x s))
 but is expected to have type
   forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] {s : Set.{u1} α} {t : Set.{u1} α} {x : α}, Iff (LE.le.{u1} (Filter.{u1} α) (Preorder.toLE.{u1} (Filter.{u1} α) (PartialOrder.toPreorder.{u1} (Filter.{u1} α) (Filter.instPartialOrderFilter.{u1} α))) (nhdsWithin.{u1} α _inst_1 x s) (nhdsWithin.{u1} α _inst_1 x t)) (Membership.mem.{u1, u1} (Set.{u1} α) (Filter.{u1} α) (instMembershipSetFilter.{u1} α) t (nhdsWithin.{u1} α _inst_1 x s))
 Case conversion may be inaccurate. Consider using '#align nhds_within_le_iff nhdsWithin_le_iffₓ'. -/
@@ -287,7 +287,7 @@ theorem inter_mem_nhdsWithin (s : Set α) {t : Set α} {a : α} (h : t ∈ 𝓝
 
 /- warning: nhds_within_mono -> nhdsWithin_mono is a dubious translation:
 lean 3 declaration is
-  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] (a : α) {s : Set.{u1} α} {t : Set.{u1} α}, (HasSubset.Subset.{u1} (Set.{u1} α) (Set.hasSubset.{u1} α) s t) -> (LE.le.{u1} (Filter.{u1} α) (Preorder.toLE.{u1} (Filter.{u1} α) (PartialOrder.toPreorder.{u1} (Filter.{u1} α) (Filter.partialOrder.{u1} α))) (nhdsWithin.{u1} α _inst_1 a s) (nhdsWithin.{u1} α _inst_1 a t))
+  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] (a : α) {s : Set.{u1} α} {t : Set.{u1} α}, (HasSubset.Subset.{u1} (Set.{u1} α) (Set.hasSubset.{u1} α) s t) -> (LE.le.{u1} (Filter.{u1} α) (Preorder.toHasLe.{u1} (Filter.{u1} α) (PartialOrder.toPreorder.{u1} (Filter.{u1} α) (Filter.partialOrder.{u1} α))) (nhdsWithin.{u1} α _inst_1 a s) (nhdsWithin.{u1} α _inst_1 a t))
 but is expected to have type
   forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] (a : α) {s : Set.{u1} α} {t : Set.{u1} α}, (HasSubset.Subset.{u1} (Set.{u1} α) (Set.instHasSubsetSet.{u1} α) s t) -> (LE.le.{u1} (Filter.{u1} α) (Preorder.toLE.{u1} (Filter.{u1} α) (PartialOrder.toPreorder.{u1} (Filter.{u1} α) (Filter.instPartialOrderFilter.{u1} α))) (nhdsWithin.{u1} α _inst_1 a s) (nhdsWithin.{u1} α _inst_1 a t))
 Case conversion may be inaccurate. Consider using '#align nhds_within_mono nhdsWithin_monoₓ'. -/
@@ -297,7 +297,7 @@ theorem nhdsWithin_mono (a : α) {s t : Set α} (h : s ⊆ t) : 𝓝[s] a ≤ 
 
 /- warning: pure_le_nhds_within -> pure_le_nhdsWithin is a dubious translation:
 lean 3 declaration is
-  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] {a : α} {s : Set.{u1} α}, (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) a s) -> (LE.le.{u1} (Filter.{u1} α) (Preorder.toLE.{u1} (Filter.{u1} α) (PartialOrder.toPreorder.{u1} (Filter.{u1} α) (Filter.partialOrder.{u1} α))) (Pure.pure.{u1, u1} Filter.{u1} Filter.hasPure.{u1} α a) (nhdsWithin.{u1} α _inst_1 a s))
+  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] {a : α} {s : Set.{u1} α}, (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) a s) -> (LE.le.{u1} (Filter.{u1} α) (Preorder.toHasLe.{u1} (Filter.{u1} α) (PartialOrder.toPreorder.{u1} (Filter.{u1} α) (Filter.partialOrder.{u1} α))) (Pure.pure.{u1, u1} Filter.{u1} Filter.hasPure.{u1} α a) (nhdsWithin.{u1} α _inst_1 a s))
 but is expected to have type
   forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] {a : α} {s : Set.{u1} α}, (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) a s) -> (LE.le.{u1} (Filter.{u1} α) (Preorder.toLE.{u1} (Filter.{u1} α) (PartialOrder.toPreorder.{u1} (Filter.{u1} α) (Filter.instPartialOrderFilter.{u1} α))) (Pure.pure.{u1, u1} Filter.{u1} Filter.instPureFilter.{u1} α a) (nhdsWithin.{u1} α _inst_1 a s))
 Case conversion may be inaccurate. Consider using '#align pure_le_nhds_within pure_le_nhdsWithinₓ'. -/
@@ -360,7 +360,7 @@ theorem nhdsWithin_restrict {a : α} (s : Set α) {t : Set α} (h₀ : a ∈ t)
 
 /- warning: nhds_within_le_of_mem -> nhdsWithin_le_of_mem is a dubious translation:
 lean 3 declaration is
-  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] {a : α} {s : Set.{u1} α} {t : Set.{u1} α}, (Membership.Mem.{u1, u1} (Set.{u1} α) (Filter.{u1} α) (Filter.hasMem.{u1} α) s (nhdsWithin.{u1} α _inst_1 a t)) -> (LE.le.{u1} (Filter.{u1} α) (Preorder.toLE.{u1} (Filter.{u1} α) (PartialOrder.toPreorder.{u1} (Filter.{u1} α) (Filter.partialOrder.{u1} α))) (nhdsWithin.{u1} α _inst_1 a t) (nhdsWithin.{u1} α _inst_1 a s))
+  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] {a : α} {s : Set.{u1} α} {t : Set.{u1} α}, (Membership.Mem.{u1, u1} (Set.{u1} α) (Filter.{u1} α) (Filter.hasMem.{u1} α) s (nhdsWithin.{u1} α _inst_1 a t)) -> (LE.le.{u1} (Filter.{u1} α) (Preorder.toHasLe.{u1} (Filter.{u1} α) (PartialOrder.toPreorder.{u1} (Filter.{u1} α) (Filter.partialOrder.{u1} α))) (nhdsWithin.{u1} α _inst_1 a t) (nhdsWithin.{u1} α _inst_1 a s))
 but is expected to have type
   forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] {a : α} {s : Set.{u1} α} {t : Set.{u1} α}, (Membership.mem.{u1, u1} (Set.{u1} α) (Filter.{u1} α) (instMembershipSetFilter.{u1} α) s (nhdsWithin.{u1} α _inst_1 a t)) -> (LE.le.{u1} (Filter.{u1} α) (Preorder.toLE.{u1} (Filter.{u1} α) (PartialOrder.toPreorder.{u1} (Filter.{u1} α) (Filter.instPartialOrderFilter.{u1} α))) (nhdsWithin.{u1} α _inst_1 a t) (nhdsWithin.{u1} α _inst_1 a s))
 Case conversion may be inaccurate. Consider using '#align nhds_within_le_of_mem nhdsWithin_le_of_memₓ'. -/
@@ -370,7 +370,7 @@ theorem nhdsWithin_le_of_mem {a : α} {s t : Set α} (h : s ∈ 𝓝[t] a) : 
 
 /- warning: nhds_within_le_nhds -> nhdsWithin_le_nhds is a dubious translation:
 lean 3 declaration is
-  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] {a : α} {s : Set.{u1} α}, LE.le.{u1} (Filter.{u1} α) (Preorder.toLE.{u1} (Filter.{u1} α) (PartialOrder.toPreorder.{u1} (Filter.{u1} α) (Filter.partialOrder.{u1} α))) (nhdsWithin.{u1} α _inst_1 a s) (nhds.{u1} α _inst_1 a)
+  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] {a : α} {s : Set.{u1} α}, LE.le.{u1} (Filter.{u1} α) (Preorder.toHasLe.{u1} (Filter.{u1} α) (PartialOrder.toPreorder.{u1} (Filter.{u1} α) (Filter.partialOrder.{u1} α))) (nhdsWithin.{u1} α _inst_1 a s) (nhds.{u1} α _inst_1 a)
 but is expected to have type
   forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] {a : α} {s : Set.{u1} α}, LE.le.{u1} (Filter.{u1} α) (Preorder.toLE.{u1} (Filter.{u1} α) (PartialOrder.toPreorder.{u1} (Filter.{u1} α) (Filter.instPartialOrderFilter.{u1} α))) (nhdsWithin.{u1} α _inst_1 a s) (nhds.{u1} α _inst_1 a)
 Case conversion may be inaccurate. Consider using '#align nhds_within_le_nhds nhdsWithin_le_nhdsₓ'. -/
@@ -1096,7 +1096,7 @@ theorem continuousOn_pi {ι : Type _} {π : ι → Type _} [∀ i, TopologicalSp
 
 /- warning: continuous_within_at.fin_insert_nth -> ContinuousWithinAt.fin_insertNth is a dubious translation:
 lean 3 declaration is
-  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] {n : Nat} {π : (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) -> Type.{u2}} [_inst_5 : forall (i : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))), TopologicalSpace.{u2} (π i)] (i : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) {f : α -> (π i)} {a : α} {s : Set.{u1} α}, (ContinuousWithinAt.{u1, u2} α (π i) _inst_1 (_inst_5 i) f s a) -> (forall {g : α -> (forall (j : Fin n), π (coeFn.{1, 1} (OrderEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (Fin.hasLe n) (Preorder.toLE.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (PartialOrder.toPreorder.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (Fin.partialOrder (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne)))))))) (fun (_x : RelEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (LE.le.{0} (Fin n) (Fin.hasLe n)) (LE.le.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (Preorder.toLE.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (PartialOrder.toPreorder.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (Fin.partialOrder (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))))))) => (Fin n) -> (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne)))))) (RelEmbedding.hasCoeToFun.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (LE.le.{0} (Fin n) (Fin.hasLe n)) (LE.le.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (Preorder.toLE.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 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Nat.hasOne))))) (Fin.partialOrder (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne)))))))) (fun (_x : RelEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (LE.le.{0} (Fin n) (Fin.hasLe n)) (LE.le.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (Preorder.toLE.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (PartialOrder.toPreorder.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (Fin.partialOrder (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))))))) => (Fin n) -> (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne)))))) (RelEmbedding.hasCoeToFun.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (LE.le.{0} (Fin n) (Fin.hasLe n)) (LE.le.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (Preorder.toLE.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (PartialOrder.toPreorder.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (Fin.partialOrder (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))))))) (Fin.succAbove n i) j)) _inst_1 (Pi.topologicalSpace.{0, u2} (Fin n) (fun (j : Fin n) => π (coeFn.{1, 1} (OrderEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (Fin.hasLe n) (Preorder.toLE.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (PartialOrder.toPreorder.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (Fin.partialOrder (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne)))))))) (fun (_x : RelEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (LE.le.{0} (Fin n) (Fin.hasLe n)) (LE.le.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (Preorder.toLE.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (PartialOrder.toPreorder.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (Fin.partialOrder (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))))))) => (Fin n) -> (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne)))))) (RelEmbedding.hasCoeToFun.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (LE.le.{0} (Fin n) (Fin.hasLe n)) (LE.le.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (Preorder.toLE.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (PartialOrder.toPreorder.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (Fin.partialOrder (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))))))) (Fin.succAbove n i) j)) (fun (a : Fin n) => _inst_5 (coeFn.{1, 1} (OrderEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (Fin.hasLe n) (Preorder.toLE.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (PartialOrder.toPreorder.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (Fin.partialOrder (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne)))))))) (fun (_x : RelEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (LE.le.{0} (Fin n) (Fin.hasLe n)) (LE.le.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (Preorder.toLE.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (PartialOrder.toPreorder.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (Fin.partialOrder (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))))))) => (Fin n) -> (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne)))))) (RelEmbedding.hasCoeToFun.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (LE.le.{0} (Fin n) (Fin.hasLe n)) (LE.le.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (Preorder.toLE.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (PartialOrder.toPreorder.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (Fin.partialOrder (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))))))) (Fin.succAbove n i) a))) g s a) -> (ContinuousWithinAt.{u1, u2} α (forall (j : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))), π j) _inst_1 (Pi.topologicalSpace.{0, u2} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (fun (j : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) => π j) (fun (a : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) => _inst_5 a)) (fun (a : α) => Fin.insertNth.{u2} n (fun (i : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) => π i) i (f a) (g a)) s a))
+  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] {n : Nat} {π : (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) -> Type.{u2}} [_inst_5 : forall (i : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))), TopologicalSpace.{u2} (π i)] (i : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) {f : α -> (π i)} {a : α} {s : Set.{u1} α}, (ContinuousWithinAt.{u1, u2} α (π i) _inst_1 (_inst_5 i) f s a) -> (forall {g : α -> (forall (j : Fin n), π (coeFn.{1, 1} (OrderEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (Fin.hasLe n) (Preorder.toHasLe.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (PartialOrder.toPreorder.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (Fin.partialOrder (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne)))))))) (fun (_x : RelEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (LE.le.{0} (Fin n) (Fin.hasLe n)) (LE.le.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (Preorder.toHasLe.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (PartialOrder.toPreorder.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (Fin.partialOrder (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))))))) => (Fin n) -> (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne)))))) (RelEmbedding.hasCoeToFun.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (LE.le.{0} (Fin n) (Fin.hasLe n)) (LE.le.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (Preorder.toHasLe.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (PartialOrder.toPreorder.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (Fin.partialOrder (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))))))) (Fin.succAbove n i) j))}, (ContinuousWithinAt.{u1, u2} α (forall (j : Fin n), π (coeFn.{1, 1} (OrderEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (Fin.hasLe n) (Preorder.toHasLe.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (PartialOrder.toPreorder.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (Fin.partialOrder (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne)))))))) (fun (_x : RelEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (LE.le.{0} (Fin n) (Fin.hasLe n)) (LE.le.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (Preorder.toHasLe.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (PartialOrder.toPreorder.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (Fin.partialOrder (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))))))) => (Fin n) -> (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne)))))) (RelEmbedding.hasCoeToFun.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (LE.le.{0} (Fin n) (Fin.hasLe n)) (LE.le.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (Preorder.toHasLe.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (PartialOrder.toPreorder.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (Fin.partialOrder (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))))))) (Fin.succAbove n i) j)) _inst_1 (Pi.topologicalSpace.{0, u2} (Fin n) (fun (j : Fin n) => π (coeFn.{1, 1} (OrderEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (Fin.hasLe n) (Preorder.toHasLe.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (PartialOrder.toPreorder.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (Fin.partialOrder (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne)))))))) (fun (_x : RelEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (LE.le.{0} (Fin n) (Fin.hasLe n)) (LE.le.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (Preorder.toHasLe.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (PartialOrder.toPreorder.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (Fin.partialOrder (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))))))) => (Fin n) -> (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne)))))) (RelEmbedding.hasCoeToFun.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (LE.le.{0} (Fin n) (Fin.hasLe n)) (LE.le.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (Preorder.toHasLe.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (PartialOrder.toPreorder.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (Fin.partialOrder (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))))))) (Fin.succAbove n i) j)) (fun (a : Fin n) => _inst_5 (coeFn.{1, 1} (OrderEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (Fin.hasLe n) (Preorder.toHasLe.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (PartialOrder.toPreorder.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (Fin.partialOrder (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne)))))))) (fun (_x : RelEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (LE.le.{0} (Fin n) (Fin.hasLe n)) (LE.le.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (Preorder.toHasLe.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (PartialOrder.toPreorder.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (Fin.partialOrder (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))))))) => (Fin n) -> (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne)))))) (RelEmbedding.hasCoeToFun.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (LE.le.{0} (Fin n) (Fin.hasLe n)) (LE.le.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (Preorder.toHasLe.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (PartialOrder.toPreorder.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (Fin.partialOrder (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))))))) (Fin.succAbove n i) a))) g s a) -> (ContinuousWithinAt.{u1, u2} α (forall (j : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))), π j) _inst_1 (Pi.topologicalSpace.{0, u2} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (fun (j : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) => π j) (fun (a : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) => _inst_5 a)) (fun (a : α) => Fin.insertNth.{u2} n (fun (i : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) => π i) i (f a) (g a)) s a))
 but is expected to have type
   forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] {n : Nat} {π : (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) -> Type.{u2}} [_inst_5 : forall (i : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))), TopologicalSpace.{u2} (π i)] (i : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) {f : α -> (π i)} {a : α} {s : Set.{u1} α}, (ContinuousWithinAt.{u1, u2} α (π i) _inst_1 (_inst_5 i) f s a) -> (forall {g : α -> (forall (j : Fin n), π (FunLike.coe.{1, 1, 1} (OrderEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin n) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))))) (Fin n) (fun (_x : Fin n) => (fun (x._@.Mathlib.Order.Hom.Lattice._hyg.494 : Fin n) => Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) _x) (InfHomClass.toFunLike.{0, 0, 0} (OrderEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin n) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))))) (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (Lattice.toInf.{0} (Fin n) (DistribLattice.toLattice.{0} (Fin n) (instDistribLattice.{0} (Fin n) (Fin.instLinearOrderFin n)))) (Lattice.toInf.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (Fin.instLatticeFinHAddNatInstHAddInstAddNatOfNat n)) (LatticeHomClass.toInfHomClass.{0, 0, 0} (OrderEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin n) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))))) (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (DistribLattice.toLattice.{0} (Fin n) (instDistribLattice.{0} (Fin n) (Fin.instLinearOrderFin n))) (Fin.instLatticeFinHAddNatInstHAddInstAddNatOfNat n) (OrderHomClass.toLatticeHomClass.{0, 0, 0} (OrderEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin n) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))))) (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (Fin.instLinearOrderFin n) (Fin.instLatticeFinHAddNatInstHAddInstAddNatOfNat n) (RelEmbedding.instRelHomClassRelEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.680 : Fin n) (x._@.Mathlib.Order.Hom.Basic._hyg.682 : Fin n) => LE.le.{0} (Fin n) (instLEFin n) x._@.Mathlib.Order.Hom.Basic._hyg.680 x._@.Mathlib.Order.Hom.Basic._hyg.682) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.695 : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (x._@.Mathlib.Order.Hom.Basic._hyg.697 : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) => LE.le.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) x._@.Mathlib.Order.Hom.Basic._hyg.695 x._@.Mathlib.Order.Hom.Basic._hyg.697))))) (Fin.succAbove n i) j))}, (ContinuousWithinAt.{u1, u2} α (forall (j : Fin n), π (FunLike.coe.{1, 1, 1} (OrderEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin n) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))))) (Fin n) (fun (_x : Fin n) => (fun (x._@.Mathlib.Order.Hom.Lattice._hyg.494 : Fin n) => Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) _x) (InfHomClass.toFunLike.{0, 0, 0} (OrderEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin n) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))))) (Fin n) (Fin 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(x._@.Mathlib.Order.Hom.Basic._hyg.695 : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (x._@.Mathlib.Order.Hom.Basic._hyg.697 : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) => LE.le.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) x._@.Mathlib.Order.Hom.Basic._hyg.695 x._@.Mathlib.Order.Hom.Basic._hyg.697))))) (Fin.succAbove n i) j)) _inst_1 (Pi.topologicalSpace.{0, u2} (Fin n) (fun (j : Fin n) => π (FunLike.coe.{1, 1, 1} (OrderEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin n) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 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Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) x._@.Mathlib.Order.Hom.Basic._hyg.695 x._@.Mathlib.Order.Hom.Basic._hyg.697))))) (Fin.succAbove n i) j)) (fun (a : Fin n) => _inst_5 (FunLike.coe.{1, 1, 1} (OrderEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin n) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))))) (Fin n) (fun (_x : Fin n) => (fun (x._@.Mathlib.Order.Hom.Lattice._hyg.494 : Fin n) => Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) _x) (InfHomClass.toFunLike.{0, 0, 0} (OrderEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin n) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 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(x._@.Mathlib.Order.Hom.Basic._hyg.695 : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (x._@.Mathlib.Order.Hom.Basic._hyg.697 : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) => LE.le.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) x._@.Mathlib.Order.Hom.Basic._hyg.695 x._@.Mathlib.Order.Hom.Basic._hyg.697))))) (Fin.succAbove n i) a))) g s a) -> (ContinuousWithinAt.{u1, u2} α (forall (j : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))), π j) _inst_1 (Pi.topologicalSpace.{0, u2} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (fun (j : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) => π j) (fun (a : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) => _inst_5 a)) (fun (a : α) => Fin.insertNth.{u2} n π i (f a) (g a)) s a))
 Case conversion may be inaccurate. Consider using '#align continuous_within_at.fin_insert_nth ContinuousWithinAt.fin_insertNthₓ'. -/
@@ -1109,7 +1109,7 @@ theorem ContinuousWithinAt.fin_insertNth {n} {π : Fin (n + 1) → Type _}
 
 /- warning: continuous_on.fin_insert_nth -> ContinuousOn.fin_insertNth is a dubious translation:
 lean 3 declaration is
-  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] {n : Nat} {π : (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) -> Type.{u2}} [_inst_5 : forall (i : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))), TopologicalSpace.{u2} (π i)] (i : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) {f : α -> (π i)} {s : Set.{u1} α}, (ContinuousOn.{u1, u2} α (π i) _inst_1 (_inst_5 i) f s) -> (forall {g : α -> (forall (j : Fin n), π (coeFn.{1, 1} (OrderEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (Fin.hasLe n) (Preorder.toLE.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat 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(Preorder.toHasLe.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (PartialOrder.toPreorder.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (Fin.partialOrder (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))))))) => (Fin n) -> (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne)))))) (RelEmbedding.hasCoeToFun.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (LE.le.{0} (Fin n) (Fin.hasLe n)) (LE.le.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (Preorder.toHasLe.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (PartialOrder.toPreorder.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (Fin.partialOrder (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))))))) (Fin.succAbove n i) a))) g s) -> (ContinuousOn.{u1, u2} α (forall (j : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))), π j) _inst_1 (Pi.topologicalSpace.{0, u2} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (fun (j : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) => π j) (fun (a : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) => _inst_5 a)) (fun (a : α) => Fin.insertNth.{u2} n (fun (i : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) => π i) i (f a) (g a)) s))
 but is expected to have type
   forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] {n : Nat} {π : (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) -> Type.{u2}} [_inst_5 : forall (i : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))), TopologicalSpace.{u2} (π i)] (i : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) {f : α -> (π i)} {s : Set.{u1} α}, (ContinuousOn.{u1, u2} α (π i) _inst_1 (_inst_5 i) f s) -> (forall {g : α -> (forall (j : Fin n), π (FunLike.coe.{1, 1, 1} (OrderEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin n) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))))) (Fin n) (fun (_x : Fin n) => (fun (x._@.Mathlib.Order.Hom.Lattice._hyg.494 : 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(RelEmbedding.instRelHomClassRelEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.680 : Fin n) (x._@.Mathlib.Order.Hom.Basic._hyg.682 : Fin n) => LE.le.{0} (Fin n) (instLEFin n) x._@.Mathlib.Order.Hom.Basic._hyg.680 x._@.Mathlib.Order.Hom.Basic._hyg.682) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.695 : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (x._@.Mathlib.Order.Hom.Basic._hyg.697 : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) => LE.le.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) x._@.Mathlib.Order.Hom.Basic._hyg.695 x._@.Mathlib.Order.Hom.Basic._hyg.697))))) (Fin.succAbove n i) j))}, (ContinuousOn.{u1, u2} α (forall (j : Fin n), π (FunLike.coe.{1, 1, 1} (OrderEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin n) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))))) (Fin n) (fun (_x : Fin n) => (fun (x._@.Mathlib.Order.Hom.Lattice._hyg.494 : Fin n) => Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) _x) (InfHomClass.toFunLike.{0, 0, 0} (OrderEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin n) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))))) (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat 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0, 0} (OrderEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin n) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))))) (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (Fin.instLinearOrderFin n) (Fin.instLatticeFinHAddNatInstHAddInstAddNatOfNat n) (RelEmbedding.instRelHomClassRelEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.680 : Fin n) (x._@.Mathlib.Order.Hom.Basic._hyg.682 : Fin n) => LE.le.{0} (Fin n) (instLEFin n) x._@.Mathlib.Order.Hom.Basic._hyg.680 x._@.Mathlib.Order.Hom.Basic._hyg.682) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.695 : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (x._@.Mathlib.Order.Hom.Basic._hyg.697 : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) => LE.le.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) x._@.Mathlib.Order.Hom.Basic._hyg.695 x._@.Mathlib.Order.Hom.Basic._hyg.697))))) (Fin.succAbove n i) j)) _inst_1 (Pi.topologicalSpace.{0, u2} (Fin n) (fun (j : Fin n) => π (FunLike.coe.{1, 1, 1} (OrderEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin n) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))))) (Fin n) (fun (_x : Fin n) => (fun (x._@.Mathlib.Order.Hom.Lattice._hyg.494 : Fin n) => Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) _x) (InfHomClass.toFunLike.{0, 0, 0} (OrderEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin n) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))))) (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (Lattice.toInf.{0} (Fin n) (DistribLattice.toLattice.{0} (Fin n) (instDistribLattice.{0} (Fin n) (Fin.instLinearOrderFin n)))) (Lattice.toInf.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (Fin.instLatticeFinHAddNatInstHAddInstAddNatOfNat n)) (LatticeHomClass.toInfHomClass.{0, 0, 0} (OrderEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin n) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))))) (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (DistribLattice.toLattice.{0} (Fin n) (instDistribLattice.{0} (Fin n) (Fin.instLinearOrderFin n))) (Fin.instLatticeFinHAddNatInstHAddInstAddNatOfNat n) (OrderHomClass.toLatticeHomClass.{0, 0, 0} (OrderEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin n) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))))) (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (Fin.instLinearOrderFin n) (Fin.instLatticeFinHAddNatInstHAddInstAddNatOfNat n) (RelEmbedding.instRelHomClassRelEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.680 : Fin n) (x._@.Mathlib.Order.Hom.Basic._hyg.682 : Fin n) => LE.le.{0} (Fin n) (instLEFin n) x._@.Mathlib.Order.Hom.Basic._hyg.680 x._@.Mathlib.Order.Hom.Basic._hyg.682) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.695 : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (x._@.Mathlib.Order.Hom.Basic._hyg.697 : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) => LE.le.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) x._@.Mathlib.Order.Hom.Basic._hyg.695 x._@.Mathlib.Order.Hom.Basic._hyg.697))))) (Fin.succAbove n i) j)) (fun (a : Fin n) => _inst_5 (FunLike.coe.{1, 1, 1} (OrderEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin n) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))))) (Fin n) (fun (_x : Fin n) => (fun (x._@.Mathlib.Order.Hom.Lattice._hyg.494 : Fin n) => Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) _x) (InfHomClass.toFunLike.{0, 0, 0} (OrderEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin n) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))))) (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (Lattice.toInf.{0} (Fin n) (DistribLattice.toLattice.{0} (Fin n) (instDistribLattice.{0} (Fin n) (Fin.instLinearOrderFin n)))) (Lattice.toInf.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (Fin.instLatticeFinHAddNatInstHAddInstAddNatOfNat n)) (LatticeHomClass.toInfHomClass.{0, 0, 0} (OrderEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin n) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))))) (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (DistribLattice.toLattice.{0} (Fin n) (instDistribLattice.{0} (Fin n) (Fin.instLinearOrderFin n))) (Fin.instLatticeFinHAddNatInstHAddInstAddNatOfNat n) (OrderHomClass.toLatticeHomClass.{0, 0, 0} (OrderEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin n) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))))) (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (Fin.instLinearOrderFin n) (Fin.instLatticeFinHAddNatInstHAddInstAddNatOfNat n) (RelEmbedding.instRelHomClassRelEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.680 : Fin n) (x._@.Mathlib.Order.Hom.Basic._hyg.682 : Fin n) => LE.le.{0} (Fin n) (instLEFin n) x._@.Mathlib.Order.Hom.Basic._hyg.680 x._@.Mathlib.Order.Hom.Basic._hyg.682) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.695 : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (x._@.Mathlib.Order.Hom.Basic._hyg.697 : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) => LE.le.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) x._@.Mathlib.Order.Hom.Basic._hyg.695 x._@.Mathlib.Order.Hom.Basic._hyg.697))))) (Fin.succAbove n i) a))) g s) -> (ContinuousOn.{u1, u2} α (forall (j : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))), π j) _inst_1 (Pi.topologicalSpace.{0, u2} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (fun (j : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) => π j) (fun (a : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) => _inst_5 a)) (fun (a : α) => Fin.insertNth.{u2} n π i (f a) (g a)) s))
 Case conversion may be inaccurate. Consider using '#align continuous_on.fin_insert_nth ContinuousOn.fin_insertNthₓ'. -/
@@ -1170,7 +1170,7 @@ theorem continuousOn_iff' {f : α → β} {s : Set α} :
 
 /- warning: continuous_on.mono_dom -> ContinuousOn.mono_dom is a dubious translation:
 lean 3 declaration is
-  forall {α : Type.{u1}} {β : Type.{u2}} {t₁ : TopologicalSpace.{u1} α} {t₂ : TopologicalSpace.{u1} α} {t₃ : TopologicalSpace.{u2} β}, (LE.le.{u1} (TopologicalSpace.{u1} α) (Preorder.toLE.{u1} (TopologicalSpace.{u1} α) (PartialOrder.toPreorder.{u1} (TopologicalSpace.{u1} α) (TopologicalSpace.partialOrder.{u1} α))) t₂ t₁) -> (forall {s : Set.{u1} α} {f : α -> β}, (ContinuousOn.{u1, u2} α β t₁ t₃ f s) -> (ContinuousOn.{u1, u2} α β t₂ t₃ f s))
+  forall {α : Type.{u1}} {β : Type.{u2}} {t₁ : TopologicalSpace.{u1} α} {t₂ : TopologicalSpace.{u1} α} {t₃ : TopologicalSpace.{u2} β}, (LE.le.{u1} (TopologicalSpace.{u1} α) (Preorder.toHasLe.{u1} (TopologicalSpace.{u1} α) (PartialOrder.toPreorder.{u1} (TopologicalSpace.{u1} α) (TopologicalSpace.partialOrder.{u1} α))) t₂ t₁) -> (forall {s : Set.{u1} α} {f : α -> β}, (ContinuousOn.{u1, u2} α β t₁ t₃ f s) -> (ContinuousOn.{u1, u2} α β t₂ t₃ f s))
 but is expected to have type
   forall {α : Type.{u2}} {β : Type.{u1}} {t₁ : TopologicalSpace.{u2} α} {t₂ : TopologicalSpace.{u2} α} {t₃ : TopologicalSpace.{u1} β}, (LE.le.{u2} (TopologicalSpace.{u2} α) (Preorder.toLE.{u2} (TopologicalSpace.{u2} α) (PartialOrder.toPreorder.{u2} (TopologicalSpace.{u2} α) (TopologicalSpace.instPartialOrderTopologicalSpace.{u2} α))) t₂ t₁) -> (forall {s : Set.{u2} α} {f : α -> β}, (ContinuousOn.{u2, u1} α β t₁ t₃ f s) -> (ContinuousOn.{u2, u1} α β t₂ t₃ f s))
 Case conversion may be inaccurate. Consider using '#align continuous_on.mono_dom ContinuousOn.mono_domₓ'. -/
@@ -1187,7 +1187,7 @@ theorem ContinuousOn.mono_dom {α β : Type _} {t₁ t₂ : TopologicalSpace α}
 
 /- warning: continuous_on.mono_rng -> ContinuousOn.mono_rng is a dubious translation:
 lean 3 declaration is
-  forall {α : Type.{u1}} {β : Type.{u2}} {t₁ : TopologicalSpace.{u1} α} {t₂ : TopologicalSpace.{u2} β} {t₃ : TopologicalSpace.{u2} β}, (LE.le.{u2} (TopologicalSpace.{u2} β) (Preorder.toLE.{u2} (TopologicalSpace.{u2} β) (PartialOrder.toPreorder.{u2} (TopologicalSpace.{u2} β) (TopologicalSpace.partialOrder.{u2} β))) t₂ t₃) -> (forall {s : Set.{u1} α} {f : α -> β}, (ContinuousOn.{u1, u2} α β t₁ t₂ f s) -> (ContinuousOn.{u1, u2} α β t₁ t₃ f s))
+  forall {α : Type.{u1}} {β : Type.{u2}} {t₁ : TopologicalSpace.{u1} α} {t₂ : TopologicalSpace.{u2} β} {t₃ : TopologicalSpace.{u2} β}, (LE.le.{u2} (TopologicalSpace.{u2} β) (Preorder.toHasLe.{u2} (TopologicalSpace.{u2} β) (PartialOrder.toPreorder.{u2} (TopologicalSpace.{u2} β) (TopologicalSpace.partialOrder.{u2} β))) t₂ t₃) -> (forall {s : Set.{u1} α} {f : α -> β}, (ContinuousOn.{u1, u2} α β t₁ t₂ f s) -> (ContinuousOn.{u1, u2} α β t₁ t₃ f s))
 but is expected to have type
   forall {α : Type.{u2}} {β : Type.{u1}} {t₁ : TopologicalSpace.{u2} α} {t₂ : TopologicalSpace.{u1} β} {t₃ : TopologicalSpace.{u1} β}, (LE.le.{u1} (TopologicalSpace.{u1} β) (Preorder.toLE.{u1} (TopologicalSpace.{u1} β) (PartialOrder.toPreorder.{u1} (TopologicalSpace.{u1} β) (TopologicalSpace.instPartialOrderTopologicalSpace.{u1} β))) t₂ t₃) -> (forall {s : Set.{u2} α} {f : α -> β}, (ContinuousOn.{u2, u1} α β t₁ t₂ f s) -> (ContinuousOn.{u2, u1} α β t₁ t₃ f s))
 Case conversion may be inaccurate. Consider using '#align continuous_on.mono_rng ContinuousOn.mono_rngₓ'. -/
@@ -1279,7 +1279,7 @@ theorem Set.Subsingleton.continuousOn {s : Set α} (hs : s.Subsingleton) (f : α
 
 /- warning: nhds_within_le_comap -> nhdsWithin_le_comap is a dubious translation:
 lean 3 declaration is
-  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : TopologicalSpace.{u1} α] [_inst_2 : TopologicalSpace.{u2} β] {x : α} {s : Set.{u1} α} {f : α -> β}, (ContinuousWithinAt.{u1, u2} α β _inst_1 _inst_2 f s x) -> (LE.le.{u1} (Filter.{u1} α) (Preorder.toLE.{u1} (Filter.{u1} α) (PartialOrder.toPreorder.{u1} (Filter.{u1} α) (Filter.partialOrder.{u1} α))) (nhdsWithin.{u1} α _inst_1 x s) (Filter.comap.{u1, u2} α β f (nhdsWithin.{u2} β _inst_2 (f x) (Set.image.{u1, u2} α β f s))))
+  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : TopologicalSpace.{u1} α] [_inst_2 : TopologicalSpace.{u2} β] {x : α} {s : Set.{u1} α} {f : α -> β}, (ContinuousWithinAt.{u1, u2} α β _inst_1 _inst_2 f s x) -> (LE.le.{u1} (Filter.{u1} α) (Preorder.toHasLe.{u1} (Filter.{u1} α) (PartialOrder.toPreorder.{u1} (Filter.{u1} α) (Filter.partialOrder.{u1} α))) (nhdsWithin.{u1} α _inst_1 x s) (Filter.comap.{u1, u2} α β f (nhdsWithin.{u2} β _inst_2 (f x) (Set.image.{u1, u2} α β f s))))
 but is expected to have type
   forall {α : Type.{u2}} {β : Type.{u1}} [_inst_1 : TopologicalSpace.{u2} α] [_inst_2 : TopologicalSpace.{u1} β] {x : α} {s : Set.{u2} α} {f : α -> β}, (ContinuousWithinAt.{u2, u1} α β _inst_1 _inst_2 f s x) -> (LE.le.{u2} (Filter.{u2} α) (Preorder.toLE.{u2} (Filter.{u2} α) (PartialOrder.toPreorder.{u2} (Filter.{u2} α) (Filter.instPartialOrderFilter.{u2} α))) (nhdsWithin.{u2} α _inst_1 x s) (Filter.comap.{u2, u1} α β f (nhdsWithin.{u1} β _inst_2 (f x) (Set.image.{u2, u1} α β f s))))
 Case conversion may be inaccurate. Consider using '#align nhds_within_le_comap nhdsWithin_le_comapₓ'. -/

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

@@ -96,13 +96,13 @@ theorem eventually_nhdsWithin_nhdsWithin {a : α} {s : Set α} {p : α → Prop}
 
 /- warning: nhds_within_eq -> nhdsWithin_eq is a dubious translation:
 lean 3 declaration is
-  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] (a : α) (s : Set.{u1} α), Eq.{succ u1} (Filter.{u1} α) (nhdsWithin.{u1} α _inst_1 a s) (infᵢ.{u1, succ u1} (Filter.{u1} α) (ConditionallyCompleteLattice.toHasInf.{u1} (Filter.{u1} α) (CompleteLattice.toConditionallyCompleteLattice.{u1} (Filter.{u1} α) (Filter.completeLattice.{u1} α))) (Set.{u1} α) (fun (t : Set.{u1} α) => infᵢ.{u1, 0} (Filter.{u1} α) (ConditionallyCompleteLattice.toHasInf.{u1} (Filter.{u1} α) (CompleteLattice.toConditionallyCompleteLattice.{u1} (Filter.{u1} α) (Filter.completeLattice.{u1} α))) (Membership.Mem.{u1, u1} (Set.{u1} α) (Set.{u1} (Set.{u1} α)) (Set.hasMem.{u1} (Set.{u1} α)) t (setOf.{u1} (Set.{u1} α) (fun (t : Set.{u1} α) => And (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) a t) (IsOpen.{u1} α _inst_1 t)))) (fun (H : Membership.Mem.{u1, u1} (Set.{u1} α) (Set.{u1} (Set.{u1} α)) (Set.hasMem.{u1} (Set.{u1} α)) t (setOf.{u1} (Set.{u1} α) (fun (t : Set.{u1} α) => And (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) a t) (IsOpen.{u1} α _inst_1 t)))) => Filter.principal.{u1} α (Inter.inter.{u1} (Set.{u1} α) (Set.hasInter.{u1} α) t s))))
+  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] (a : α) (s : Set.{u1} α), Eq.{succ u1} (Filter.{u1} α) (nhdsWithin.{u1} α _inst_1 a s) (iInf.{u1, succ u1} (Filter.{u1} α) (ConditionallyCompleteLattice.toHasInf.{u1} (Filter.{u1} α) (CompleteLattice.toConditionallyCompleteLattice.{u1} (Filter.{u1} α) (Filter.completeLattice.{u1} α))) (Set.{u1} α) (fun (t : Set.{u1} α) => iInf.{u1, 0} (Filter.{u1} α) (ConditionallyCompleteLattice.toHasInf.{u1} (Filter.{u1} α) (CompleteLattice.toConditionallyCompleteLattice.{u1} (Filter.{u1} α) (Filter.completeLattice.{u1} α))) (Membership.Mem.{u1, u1} (Set.{u1} α) (Set.{u1} (Set.{u1} α)) (Set.hasMem.{u1} (Set.{u1} α)) t (setOf.{u1} (Set.{u1} α) (fun (t : Set.{u1} α) => And (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) a t) (IsOpen.{u1} α _inst_1 t)))) (fun (H : Membership.Mem.{u1, u1} (Set.{u1} α) (Set.{u1} (Set.{u1} α)) (Set.hasMem.{u1} (Set.{u1} α)) t (setOf.{u1} (Set.{u1} α) (fun (t : Set.{u1} α) => And (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) a t) (IsOpen.{u1} α _inst_1 t)))) => Filter.principal.{u1} α (Inter.inter.{u1} (Set.{u1} α) (Set.hasInter.{u1} α) t s))))
 but is expected to have type
-  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] (a : α) (s : Set.{u1} α), Eq.{succ u1} (Filter.{u1} α) (nhdsWithin.{u1} α _inst_1 a s) (infᵢ.{u1, succ u1} (Filter.{u1} α) (ConditionallyCompleteLattice.toInfSet.{u1} (Filter.{u1} α) (CompleteLattice.toConditionallyCompleteLattice.{u1} (Filter.{u1} α) (Filter.instCompleteLatticeFilter.{u1} α))) (Set.{u1} α) (fun (t : Set.{u1} α) => infᵢ.{u1, 0} (Filter.{u1} α) (ConditionallyCompleteLattice.toInfSet.{u1} (Filter.{u1} α) (CompleteLattice.toConditionallyCompleteLattice.{u1} (Filter.{u1} α) (Filter.instCompleteLatticeFilter.{u1} α))) (Membership.mem.{u1, u1} (Set.{u1} α) (Set.{u1} (Set.{u1} α)) (Set.instMembershipSet.{u1} (Set.{u1} α)) t (setOf.{u1} (Set.{u1} α) (fun (t : Set.{u1} α) => And (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) a t) (IsOpen.{u1} α _inst_1 t)))) (fun (H : Membership.mem.{u1, u1} (Set.{u1} α) (Set.{u1} (Set.{u1} α)) (Set.instMembershipSet.{u1} (Set.{u1} α)) t (setOf.{u1} (Set.{u1} α) (fun (t : Set.{u1} α) => And (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) a t) (IsOpen.{u1} α _inst_1 t)))) => Filter.principal.{u1} α (Inter.inter.{u1} (Set.{u1} α) (Set.instInterSet.{u1} α) t s))))
+  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] (a : α) (s : Set.{u1} α), Eq.{succ u1} (Filter.{u1} α) (nhdsWithin.{u1} α _inst_1 a s) (iInf.{u1, succ u1} (Filter.{u1} α) (ConditionallyCompleteLattice.toInfSet.{u1} (Filter.{u1} α) (CompleteLattice.toConditionallyCompleteLattice.{u1} (Filter.{u1} α) (Filter.instCompleteLatticeFilter.{u1} α))) (Set.{u1} α) (fun (t : Set.{u1} α) => iInf.{u1, 0} (Filter.{u1} α) (ConditionallyCompleteLattice.toInfSet.{u1} (Filter.{u1} α) (CompleteLattice.toConditionallyCompleteLattice.{u1} (Filter.{u1} α) (Filter.instCompleteLatticeFilter.{u1} α))) (Membership.mem.{u1, u1} (Set.{u1} α) (Set.{u1} (Set.{u1} α)) (Set.instMembershipSet.{u1} (Set.{u1} α)) t (setOf.{u1} (Set.{u1} α) (fun (t : Set.{u1} α) => And (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) a t) (IsOpen.{u1} α _inst_1 t)))) (fun (H : Membership.mem.{u1, u1} (Set.{u1} α) (Set.{u1} (Set.{u1} α)) (Set.instMembershipSet.{u1} (Set.{u1} α)) t (setOf.{u1} (Set.{u1} α) (fun (t : Set.{u1} α) => And (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) a t) (IsOpen.{u1} α _inst_1 t)))) => Filter.principal.{u1} α (Inter.inter.{u1} (Set.{u1} α) (Set.instInterSet.{u1} α) t s))))
 Case conversion may be inaccurate. Consider using '#align nhds_within_eq nhdsWithin_eqₓ'. -/
 theorem nhdsWithin_eq (a : α) (s : Set α) :
     𝓝[s] a = ⨅ t ∈ { t : Set α | a ∈ t ∧ IsOpen t }, 𝓟 (t ∩ s) :=
-  ((nhds_basis_opens a).inf_principal s).eq_binfᵢ
+  ((nhds_basis_opens a).inf_principal s).eq_biInf
 #align nhds_within_eq nhdsWithin_eq
 
 #print nhdsWithin_univ /-
@@ -448,37 +448,37 @@ theorem nhdsWithin_union (a : α) (s t : Set α) : 𝓝[s ∪ t] a = 𝓝[s] a 
   rw [← inf_sup_left, sup_principal]
 #align nhds_within_union nhdsWithin_union
 
-/- warning: nhds_within_bUnion -> nhdsWithin_bunionᵢ is a dubious translation:
+/- warning: nhds_within_bUnion -> nhdsWithin_biUnion is a dubious translation:
 lean 3 declaration is
-  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] {ι : Type.{u2}} {I : Set.{u2} ι}, (Set.Finite.{u2} ι I) -> (forall (s : ι -> (Set.{u1} α)) (a : α), Eq.{succ u1} (Filter.{u1} α) (nhdsWithin.{u1} α _inst_1 a (Set.unionᵢ.{u1, succ u2} α ι (fun (i : ι) => Set.unionᵢ.{u1, 0} α (Membership.Mem.{u2, u2} ι (Set.{u2} ι) (Set.hasMem.{u2} ι) i I) (fun (H : Membership.Mem.{u2, u2} ι (Set.{u2} ι) (Set.hasMem.{u2} ι) i I) => s i)))) (supᵢ.{u1, succ u2} (Filter.{u1} α) (ConditionallyCompleteLattice.toHasSup.{u1} (Filter.{u1} α) (CompleteLattice.toConditionallyCompleteLattice.{u1} (Filter.{u1} α) (Filter.completeLattice.{u1} α))) ι (fun (i : ι) => supᵢ.{u1, 0} (Filter.{u1} α) (ConditionallyCompleteLattice.toHasSup.{u1} (Filter.{u1} α) (CompleteLattice.toConditionallyCompleteLattice.{u1} (Filter.{u1} α) (Filter.completeLattice.{u1} α))) (Membership.Mem.{u2, u2} ι (Set.{u2} ι) (Set.hasMem.{u2} ι) i I) (fun (H : Membership.Mem.{u2, u2} ι (Set.{u2} ι) (Set.hasMem.{u2} ι) i I) => nhdsWithin.{u1} α _inst_1 a (s i)))))
+  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] {ι : Type.{u2}} {I : Set.{u2} ι}, (Set.Finite.{u2} ι I) -> (forall (s : ι -> (Set.{u1} α)) (a : α), Eq.{succ u1} (Filter.{u1} α) (nhdsWithin.{u1} α _inst_1 a (Set.iUnion.{u1, succ u2} α ι (fun (i : ι) => Set.iUnion.{u1, 0} α (Membership.Mem.{u2, u2} ι (Set.{u2} ι) (Set.hasMem.{u2} ι) i I) (fun (H : Membership.Mem.{u2, u2} ι (Set.{u2} ι) (Set.hasMem.{u2} ι) i I) => s i)))) (iSup.{u1, succ u2} (Filter.{u1} α) (ConditionallyCompleteLattice.toHasSup.{u1} (Filter.{u1} α) (CompleteLattice.toConditionallyCompleteLattice.{u1} (Filter.{u1} α) (Filter.completeLattice.{u1} α))) ι (fun (i : ι) => iSup.{u1, 0} (Filter.{u1} α) (ConditionallyCompleteLattice.toHasSup.{u1} (Filter.{u1} α) (CompleteLattice.toConditionallyCompleteLattice.{u1} (Filter.{u1} α) (Filter.completeLattice.{u1} α))) (Membership.Mem.{u2, u2} ι (Set.{u2} ι) (Set.hasMem.{u2} ι) i I) (fun (H : Membership.Mem.{u2, u2} ι (Set.{u2} ι) (Set.hasMem.{u2} ι) i I) => nhdsWithin.{u1} α _inst_1 a (s i)))))
 but is expected to have type
-  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] {ι : Type.{u2}} {I : Set.{u2} ι}, (Set.Finite.{u2} ι I) -> (forall (s : ι -> (Set.{u1} α)) (a : α), Eq.{succ u1} (Filter.{u1} α) (nhdsWithin.{u1} α _inst_1 a (Set.unionᵢ.{u1, succ u2} α ι (fun (i : ι) => Set.unionᵢ.{u1, 0} α (Membership.mem.{u2, u2} ι (Set.{u2} ι) (Set.instMembershipSet.{u2} ι) i I) (fun (H : Membership.mem.{u2, u2} ι (Set.{u2} ι) (Set.instMembershipSet.{u2} ι) i I) => s i)))) (supᵢ.{u1, succ u2} (Filter.{u1} α) (ConditionallyCompleteLattice.toSupSet.{u1} (Filter.{u1} α) (CompleteLattice.toConditionallyCompleteLattice.{u1} (Filter.{u1} α) (Filter.instCompleteLatticeFilter.{u1} α))) ι (fun (i : ι) => supᵢ.{u1, 0} (Filter.{u1} α) (ConditionallyCompleteLattice.toSupSet.{u1} (Filter.{u1} α) (CompleteLattice.toConditionallyCompleteLattice.{u1} (Filter.{u1} α) (Filter.instCompleteLatticeFilter.{u1} α))) (Membership.mem.{u2, u2} ι (Set.{u2} ι) (Set.instMembershipSet.{u2} ι) i I) (fun (H : Membership.mem.{u2, u2} ι (Set.{u2} ι) (Set.instMembershipSet.{u2} ι) i I) => nhdsWithin.{u1} α _inst_1 a (s i)))))
-Case conversion may be inaccurate. Consider using '#align nhds_within_bUnion nhdsWithin_bunionᵢₓ'. -/
-theorem nhdsWithin_bunionᵢ {ι} {I : Set ι} (hI : I.Finite) (s : ι → Set α) (a : α) :
+  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] {ι : Type.{u2}} {I : Set.{u2} ι}, (Set.Finite.{u2} ι I) -> (forall (s : ι -> (Set.{u1} α)) (a : α), Eq.{succ u1} (Filter.{u1} α) (nhdsWithin.{u1} α _inst_1 a (Set.iUnion.{u1, succ u2} α ι (fun (i : ι) => Set.iUnion.{u1, 0} α (Membership.mem.{u2, u2} ι (Set.{u2} ι) (Set.instMembershipSet.{u2} ι) i I) (fun (H : Membership.mem.{u2, u2} ι (Set.{u2} ι) (Set.instMembershipSet.{u2} ι) i I) => s i)))) (iSup.{u1, succ u2} (Filter.{u1} α) (ConditionallyCompleteLattice.toSupSet.{u1} (Filter.{u1} α) (CompleteLattice.toConditionallyCompleteLattice.{u1} (Filter.{u1} α) (Filter.instCompleteLatticeFilter.{u1} α))) ι (fun (i : ι) => iSup.{u1, 0} (Filter.{u1} α) (ConditionallyCompleteLattice.toSupSet.{u1} (Filter.{u1} α) (CompleteLattice.toConditionallyCompleteLattice.{u1} (Filter.{u1} α) (Filter.instCompleteLatticeFilter.{u1} α))) (Membership.mem.{u2, u2} ι (Set.{u2} ι) (Set.instMembershipSet.{u2} ι) i I) (fun (H : Membership.mem.{u2, u2} ι (Set.{u2} ι) (Set.instMembershipSet.{u2} ι) i I) => nhdsWithin.{u1} α _inst_1 a (s i)))))
+Case conversion may be inaccurate. Consider using '#align nhds_within_bUnion nhdsWithin_biUnionₓ'. -/
+theorem nhdsWithin_biUnion {ι} {I : Set ι} (hI : I.Finite) (s : ι → Set α) (a : α) :
     𝓝[⋃ i ∈ I, s i] a = ⨆ i ∈ I, 𝓝[s i] a :=
   Set.Finite.induction_on hI (by simp) fun t T _ _ hT => by
-    simp only [hT, nhdsWithin_union, supᵢ_insert, bUnion_insert]
-#align nhds_within_bUnion nhdsWithin_bunionᵢ
+    simp only [hT, nhdsWithin_union, iSup_insert, bUnion_insert]
+#align nhds_within_bUnion nhdsWithin_biUnion
 
-/- warning: nhds_within_sUnion -> nhdsWithin_unionₛ is a dubious translation:
+/- warning: nhds_within_sUnion -> nhdsWithin_sUnion is a dubious translation:
 lean 3 declaration is
-  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] {S : Set.{u1} (Set.{u1} α)}, (Set.Finite.{u1} (Set.{u1} α) S) -> (forall (a : α), Eq.{succ u1} (Filter.{u1} α) (nhdsWithin.{u1} α _inst_1 a (Set.unionₛ.{u1} α S)) (supᵢ.{u1, succ u1} (Filter.{u1} α) (ConditionallyCompleteLattice.toHasSup.{u1} (Filter.{u1} α) (CompleteLattice.toConditionallyCompleteLattice.{u1} (Filter.{u1} α) (Filter.completeLattice.{u1} α))) (Set.{u1} α) (fun (s : Set.{u1} α) => supᵢ.{u1, 0} (Filter.{u1} α) (ConditionallyCompleteLattice.toHasSup.{u1} (Filter.{u1} α) (CompleteLattice.toConditionallyCompleteLattice.{u1} (Filter.{u1} α) (Filter.completeLattice.{u1} α))) (Membership.Mem.{u1, u1} (Set.{u1} α) (Set.{u1} (Set.{u1} α)) (Set.hasMem.{u1} (Set.{u1} α)) s S) (fun (H : Membership.Mem.{u1, u1} (Set.{u1} α) (Set.{u1} (Set.{u1} α)) (Set.hasMem.{u1} (Set.{u1} α)) s S) => nhdsWithin.{u1} α _inst_1 a s))))
+  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] {S : Set.{u1} (Set.{u1} α)}, (Set.Finite.{u1} (Set.{u1} α) S) -> (forall (a : α), Eq.{succ u1} (Filter.{u1} α) (nhdsWithin.{u1} α _inst_1 a (Set.sUnion.{u1} α S)) (iSup.{u1, succ u1} (Filter.{u1} α) (ConditionallyCompleteLattice.toHasSup.{u1} (Filter.{u1} α) (CompleteLattice.toConditionallyCompleteLattice.{u1} (Filter.{u1} α) (Filter.completeLattice.{u1} α))) (Set.{u1} α) (fun (s : Set.{u1} α) => iSup.{u1, 0} (Filter.{u1} α) (ConditionallyCompleteLattice.toHasSup.{u1} (Filter.{u1} α) (CompleteLattice.toConditionallyCompleteLattice.{u1} (Filter.{u1} α) (Filter.completeLattice.{u1} α))) (Membership.Mem.{u1, u1} (Set.{u1} α) (Set.{u1} (Set.{u1} α)) (Set.hasMem.{u1} (Set.{u1} α)) s S) (fun (H : Membership.Mem.{u1, u1} (Set.{u1} α) (Set.{u1} (Set.{u1} α)) (Set.hasMem.{u1} (Set.{u1} α)) s S) => nhdsWithin.{u1} α _inst_1 a s))))
 but is expected to have type
-  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] {S : Set.{u1} (Set.{u1} α)}, (Set.Finite.{u1} (Set.{u1} α) S) -> (forall (a : α), Eq.{succ u1} (Filter.{u1} α) (nhdsWithin.{u1} α _inst_1 a (Set.unionₛ.{u1} α S)) (supᵢ.{u1, succ u1} (Filter.{u1} α) (ConditionallyCompleteLattice.toSupSet.{u1} (Filter.{u1} α) (CompleteLattice.toConditionallyCompleteLattice.{u1} (Filter.{u1} α) (Filter.instCompleteLatticeFilter.{u1} α))) (Set.{u1} α) (fun (s : Set.{u1} α) => supᵢ.{u1, 0} (Filter.{u1} α) (ConditionallyCompleteLattice.toSupSet.{u1} (Filter.{u1} α) (CompleteLattice.toConditionallyCompleteLattice.{u1} (Filter.{u1} α) (Filter.instCompleteLatticeFilter.{u1} α))) (Membership.mem.{u1, u1} (Set.{u1} α) (Set.{u1} (Set.{u1} α)) (Set.instMembershipSet.{u1} (Set.{u1} α)) s S) (fun (H : Membership.mem.{u1, u1} (Set.{u1} α) (Set.{u1} (Set.{u1} α)) (Set.instMembershipSet.{u1} (Set.{u1} α)) s S) => nhdsWithin.{u1} α _inst_1 a s))))
-Case conversion may be inaccurate. Consider using '#align nhds_within_sUnion nhdsWithin_unionₛₓ'. -/
-theorem nhdsWithin_unionₛ {S : Set (Set α)} (hS : S.Finite) (a : α) : 𝓝[⋃₀ S] a = ⨆ s ∈ S, 𝓝[s] a :=
-  by rw [sUnion_eq_bUnion, nhdsWithin_bunionᵢ hS]
-#align nhds_within_sUnion nhdsWithin_unionₛ
+  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] {S : Set.{u1} (Set.{u1} α)}, (Set.Finite.{u1} (Set.{u1} α) S) -> (forall (a : α), Eq.{succ u1} (Filter.{u1} α) (nhdsWithin.{u1} α _inst_1 a (Set.sUnion.{u1} α S)) (iSup.{u1, succ u1} (Filter.{u1} α) (ConditionallyCompleteLattice.toSupSet.{u1} (Filter.{u1} α) (CompleteLattice.toConditionallyCompleteLattice.{u1} (Filter.{u1} α) (Filter.instCompleteLatticeFilter.{u1} α))) (Set.{u1} α) (fun (s : Set.{u1} α) => iSup.{u1, 0} (Filter.{u1} α) (ConditionallyCompleteLattice.toSupSet.{u1} (Filter.{u1} α) (CompleteLattice.toConditionallyCompleteLattice.{u1} (Filter.{u1} α) (Filter.instCompleteLatticeFilter.{u1} α))) (Membership.mem.{u1, u1} (Set.{u1} α) (Set.{u1} (Set.{u1} α)) (Set.instMembershipSet.{u1} (Set.{u1} α)) s S) (fun (H : Membership.mem.{u1, u1} (Set.{u1} α) (Set.{u1} (Set.{u1} α)) (Set.instMembershipSet.{u1} (Set.{u1} α)) s S) => nhdsWithin.{u1} α _inst_1 a s))))
+Case conversion may be inaccurate. Consider using '#align nhds_within_sUnion nhdsWithin_sUnionₓ'. -/
+theorem nhdsWithin_sUnion {S : Set (Set α)} (hS : S.Finite) (a : α) : 𝓝[⋃₀ S] a = ⨆ s ∈ S, 𝓝[s] a :=
+  by rw [sUnion_eq_bUnion, nhdsWithin_biUnion hS]
+#align nhds_within_sUnion nhdsWithin_sUnion
 
-/- warning: nhds_within_Union -> nhdsWithin_unionᵢ is a dubious translation:
+/- warning: nhds_within_Union -> nhdsWithin_iUnion is a dubious translation:
 lean 3 declaration is
-  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] {ι : Sort.{u2}} [_inst_2 : Finite.{u2} ι] (s : ι -> (Set.{u1} α)) (a : α), Eq.{succ u1} (Filter.{u1} α) (nhdsWithin.{u1} α _inst_1 a (Set.unionᵢ.{u1, u2} α ι (fun (i : ι) => s i))) (supᵢ.{u1, u2} (Filter.{u1} α) (ConditionallyCompleteLattice.toHasSup.{u1} (Filter.{u1} α) (CompleteLattice.toConditionallyCompleteLattice.{u1} (Filter.{u1} α) (Filter.completeLattice.{u1} α))) ι (fun (i : ι) => nhdsWithin.{u1} α _inst_1 a (s i)))
+  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] {ι : Sort.{u2}} [_inst_2 : Finite.{u2} ι] (s : ι -> (Set.{u1} α)) (a : α), Eq.{succ u1} (Filter.{u1} α) (nhdsWithin.{u1} α _inst_1 a (Set.iUnion.{u1, u2} α ι (fun (i : ι) => s i))) (iSup.{u1, u2} (Filter.{u1} α) (ConditionallyCompleteLattice.toHasSup.{u1} (Filter.{u1} α) (CompleteLattice.toConditionallyCompleteLattice.{u1} (Filter.{u1} α) (Filter.completeLattice.{u1} α))) ι (fun (i : ι) => nhdsWithin.{u1} α _inst_1 a (s i)))
 but is expected to have type
-  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] {ι : Sort.{u2}} [_inst_2 : Finite.{u2} ι] (s : ι -> (Set.{u1} α)) (a : α), Eq.{succ u1} (Filter.{u1} α) (nhdsWithin.{u1} α _inst_1 a (Set.unionᵢ.{u1, u2} α ι (fun (i : ι) => s i))) (supᵢ.{u1, u2} (Filter.{u1} α) (ConditionallyCompleteLattice.toSupSet.{u1} (Filter.{u1} α) (CompleteLattice.toConditionallyCompleteLattice.{u1} (Filter.{u1} α) (Filter.instCompleteLatticeFilter.{u1} α))) ι (fun (i : ι) => nhdsWithin.{u1} α _inst_1 a (s i)))
-Case conversion may be inaccurate. Consider using '#align nhds_within_Union nhdsWithin_unionᵢₓ'. -/
-theorem nhdsWithin_unionᵢ {ι} [Finite ι] (s : ι → Set α) (a : α) : 𝓝[⋃ i, s i] a = ⨆ i, 𝓝[s i] a :=
-  by rw [← sUnion_range, nhdsWithin_unionₛ (finite_range s), supᵢ_range]
-#align nhds_within_Union nhdsWithin_unionᵢ
+  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] {ι : Sort.{u2}} [_inst_2 : Finite.{u2} ι] (s : ι -> (Set.{u1} α)) (a : α), Eq.{succ u1} (Filter.{u1} α) (nhdsWithin.{u1} α _inst_1 a (Set.iUnion.{u1, u2} α ι (fun (i : ι) => s i))) (iSup.{u1, u2} (Filter.{u1} α) (ConditionallyCompleteLattice.toSupSet.{u1} (Filter.{u1} α) (CompleteLattice.toConditionallyCompleteLattice.{u1} (Filter.{u1} α) (Filter.instCompleteLatticeFilter.{u1} α))) ι (fun (i : ι) => nhdsWithin.{u1} α _inst_1 a (s i)))
+Case conversion may be inaccurate. Consider using '#align nhds_within_Union nhdsWithin_iUnionₓ'. -/
+theorem nhdsWithin_iUnion {ι} [Finite ι] (s : ι → Set α) (a : α) : 𝓝[⋃ i, s i] a = ⨆ i, 𝓝[s i] a :=
+  by rw [← sUnion_range, nhdsWithin_sUnion (finite_range s), iSup_range]
+#align nhds_within_Union nhdsWithin_iUnion
 
 /- warning: nhds_within_inter -> nhdsWithin_inter is a dubious translation:
 lean 3 declaration is
@@ -609,22 +609,22 @@ theorem nhdsWithin_prod {α : Type _} [TopologicalSpace α] {β : Type _} [Topol
 
 /- warning: nhds_within_pi_eq' -> nhdsWithin_pi_eq' is a dubious translation:
 lean 3 declaration is
-  forall {ι : Type.{u1}} {α : ι -> Type.{u2}} [_inst_2 : forall (i : ι), TopologicalSpace.{u2} (α i)] {I : Set.{u1} ι}, (Set.Finite.{u1} ι I) -> (forall (s : forall (i : ι), Set.{u2} (α i)) (x : forall (i : ι), α i), Eq.{succ (max u1 u2)} (Filter.{max u1 u2} (forall (i : ι), α i)) (nhdsWithin.{max u1 u2} (forall (i : ι), α i) (Pi.topologicalSpace.{u1, u2} ι (fun (i : ι) => α i) (fun (a : ι) => _inst_2 a)) x (Set.pi.{u1, u2} ι (fun (i : ι) => α i) I s)) (infᵢ.{max u1 u2, succ u1} (Filter.{max u1 u2} (forall (i : ι), α i)) (ConditionallyCompleteLattice.toHasInf.{max u1 u2} (Filter.{max u1 u2} (forall (i : ι), α i)) (CompleteLattice.toConditionallyCompleteLattice.{max u1 u2} (Filter.{max u1 u2} (forall (i : ι), α i)) (Filter.completeLattice.{max u1 u2} (forall (i : ι), α i)))) ι (fun (i : ι) => Filter.comap.{max u1 u2, u2} (forall (i : ι), α i) (α i) (fun (x : forall (i : ι), α i) => x i) (Inf.inf.{u2} (Filter.{u2} (α i)) (Filter.hasInf.{u2} (α i)) (nhds.{u2} (α i) (_inst_2 i) (x i)) (infᵢ.{u2, 0} (Filter.{u2} (α i)) (ConditionallyCompleteLattice.toHasInf.{u2} (Filter.{u2} (α i)) (CompleteLattice.toConditionallyCompleteLattice.{u2} (Filter.{u2} (α i)) (Filter.completeLattice.{u2} (α i)))) (Membership.Mem.{u1, u1} ι (Set.{u1} ι) (Set.hasMem.{u1} ι) i I) (fun (hi : Membership.Mem.{u1, u1} ι (Set.{u1} ι) (Set.hasMem.{u1} ι) i I) => Filter.principal.{u2} (α i) (s i)))))))
+  forall {ι : Type.{u1}} {α : ι -> Type.{u2}} [_inst_2 : forall (i : ι), TopologicalSpace.{u2} (α i)] {I : Set.{u1} ι}, (Set.Finite.{u1} ι I) -> (forall (s : forall (i : ι), Set.{u2} (α i)) (x : forall (i : ι), α i), Eq.{succ (max u1 u2)} (Filter.{max u1 u2} (forall (i : ι), α i)) (nhdsWithin.{max u1 u2} (forall (i : ι), α i) (Pi.topologicalSpace.{u1, u2} ι (fun (i : ι) => α i) (fun (a : ι) => _inst_2 a)) x (Set.pi.{u1, u2} ι (fun (i : ι) => α i) I s)) (iInf.{max u1 u2, succ u1} (Filter.{max u1 u2} (forall (i : ι), α i)) (ConditionallyCompleteLattice.toHasInf.{max u1 u2} (Filter.{max u1 u2} (forall (i : ι), α i)) (CompleteLattice.toConditionallyCompleteLattice.{max u1 u2} (Filter.{max u1 u2} (forall (i : ι), α i)) (Filter.completeLattice.{max u1 u2} (forall (i : ι), α i)))) ι (fun (i : ι) => Filter.comap.{max u1 u2, u2} (forall (i : ι), α i) (α i) (fun (x : forall (i : ι), α i) => x i) (Inf.inf.{u2} (Filter.{u2} (α i)) (Filter.hasInf.{u2} (α i)) (nhds.{u2} (α i) (_inst_2 i) (x i)) (iInf.{u2, 0} (Filter.{u2} (α i)) (ConditionallyCompleteLattice.toHasInf.{u2} (Filter.{u2} (α i)) (CompleteLattice.toConditionallyCompleteLattice.{u2} (Filter.{u2} (α i)) (Filter.completeLattice.{u2} (α i)))) (Membership.Mem.{u1, u1} ι (Set.{u1} ι) (Set.hasMem.{u1} ι) i I) (fun (hi : Membership.Mem.{u1, u1} ι (Set.{u1} ι) (Set.hasMem.{u1} ι) i I) => Filter.principal.{u2} (α i) (s i)))))))
 but is expected to have type
-  forall {ι : Type.{u2}} {α : ι -> Type.{u1}} [_inst_2 : forall (i : ι), TopologicalSpace.{u1} (α i)] {I : Set.{u2} ι}, (Set.Finite.{u2} ι I) -> (forall (s : forall (i : ι), Set.{u1} (α i)) (x : forall (i : ι), α i), Eq.{max (succ u2) (succ u1)} (Filter.{max u2 u1} (forall (i : ι), α i)) (nhdsWithin.{max u2 u1} (forall (i : ι), α i) (Pi.topologicalSpace.{u2, u1} ι (fun (i : ι) => α i) (fun (a : ι) => _inst_2 a)) x (Set.pi.{u2, u1} ι (fun (i : ι) => α i) I s)) (infᵢ.{max u2 u1, succ u2} (Filter.{max u2 u1} (forall (i : ι), α i)) (ConditionallyCompleteLattice.toInfSet.{max u2 u1} (Filter.{max u2 u1} (forall (i : ι), α i)) (CompleteLattice.toConditionallyCompleteLattice.{max u2 u1} (Filter.{max u2 u1} (forall (i : ι), α i)) (Filter.instCompleteLatticeFilter.{max u2 u1} (forall (i : ι), α i)))) ι (fun (i : ι) => Filter.comap.{max u2 u1, u1} (forall (i : ι), α i) (α i) (fun (x : forall (i : ι), α i) => x i) (Inf.inf.{u1} (Filter.{u1} (α i)) (Filter.instInfFilter.{u1} (α i)) (nhds.{u1} (α i) (_inst_2 i) (x i)) (infᵢ.{u1, 0} (Filter.{u1} (α i)) (ConditionallyCompleteLattice.toInfSet.{u1} (Filter.{u1} (α i)) (CompleteLattice.toConditionallyCompleteLattice.{u1} (Filter.{u1} (α i)) (Filter.instCompleteLatticeFilter.{u1} (α i)))) (Membership.mem.{u2, u2} ι (Set.{u2} ι) (Set.instMembershipSet.{u2} ι) i I) (fun (hi : Membership.mem.{u2, u2} ι (Set.{u2} ι) (Set.instMembershipSet.{u2} ι) i I) => Filter.principal.{u1} (α i) (s i)))))))
+  forall {ι : Type.{u2}} {α : ι -> Type.{u1}} [_inst_2 : forall (i : ι), TopologicalSpace.{u1} (α i)] {I : Set.{u2} ι}, (Set.Finite.{u2} ι I) -> (forall (s : forall (i : ι), Set.{u1} (α i)) (x : forall (i : ι), α i), Eq.{max (succ u2) (succ u1)} (Filter.{max u2 u1} (forall (i : ι), α i)) (nhdsWithin.{max u2 u1} (forall (i : ι), α i) (Pi.topologicalSpace.{u2, u1} ι (fun (i : ι) => α i) (fun (a : ι) => _inst_2 a)) x (Set.pi.{u2, u1} ι (fun (i : ι) => α i) I s)) (iInf.{max u2 u1, succ u2} (Filter.{max u2 u1} (forall (i : ι), α i)) (ConditionallyCompleteLattice.toInfSet.{max u2 u1} (Filter.{max u2 u1} (forall (i : ι), α i)) (CompleteLattice.toConditionallyCompleteLattice.{max u2 u1} (Filter.{max u2 u1} (forall (i : ι), α i)) (Filter.instCompleteLatticeFilter.{max u2 u1} (forall (i : ι), α i)))) ι (fun (i : ι) => Filter.comap.{max u2 u1, u1} (forall (i : ι), α i) (α i) (fun (x : forall (i : ι), α i) => x i) (Inf.inf.{u1} (Filter.{u1} (α i)) (Filter.instInfFilter.{u1} (α i)) (nhds.{u1} (α i) (_inst_2 i) (x i)) (iInf.{u1, 0} (Filter.{u1} (α i)) (ConditionallyCompleteLattice.toInfSet.{u1} (Filter.{u1} (α i)) (CompleteLattice.toConditionallyCompleteLattice.{u1} (Filter.{u1} (α i)) (Filter.instCompleteLatticeFilter.{u1} (α i)))) (Membership.mem.{u2, u2} ι (Set.{u2} ι) (Set.instMembershipSet.{u2} ι) i I) (fun (hi : Membership.mem.{u2, u2} ι (Set.{u2} ι) (Set.instMembershipSet.{u2} ι) i I) => Filter.principal.{u1} (α i) (s i)))))))
 Case conversion may be inaccurate. Consider using '#align nhds_within_pi_eq' nhdsWithin_pi_eq'ₓ'. -/
 theorem nhdsWithin_pi_eq' {ι : Type _} {α : ι → Type _} [∀ i, TopologicalSpace (α i)] {I : Set ι}
     (hI : I.Finite) (s : ∀ i, Set (α i)) (x : ∀ i, α i) :
     𝓝[pi I s] x = ⨅ i, comap (fun x => x i) (𝓝 (x i) ⊓ ⨅ hi : i ∈ I, 𝓟 (s i)) := by
   simp only [nhdsWithin, nhds_pi, Filter.pi, comap_inf, comap_infi, pi_def, comap_principal, ←
-    infi_principal_finite hI, ← infᵢ_inf_eq]
+    infi_principal_finite hI, ← iInf_inf_eq]
 #align nhds_within_pi_eq' nhdsWithin_pi_eq'
 
 /- warning: nhds_within_pi_eq -> nhdsWithin_pi_eq is a dubious translation:
 lean 3 declaration is
-  forall {ι : Type.{u1}} {α : ι -> Type.{u2}} [_inst_2 : forall (i : ι), TopologicalSpace.{u2} (α i)] {I : Set.{u1} ι}, (Set.Finite.{u1} ι I) -> (forall (s : forall (i : ι), Set.{u2} (α i)) (x : forall (i : ι), α i), Eq.{succ (max u1 u2)} (Filter.{max u1 u2} (forall (i : ι), α i)) (nhdsWithin.{max u1 u2} (forall (i : ι), α i) (Pi.topologicalSpace.{u1, u2} ι (fun (i : ι) => α i) (fun (a : ι) => _inst_2 a)) x (Set.pi.{u1, u2} ι (fun (i : ι) => α i) I s)) (Inf.inf.{max u1 u2} (Filter.{max u1 u2} (forall (i : ι), α i)) (Filter.hasInf.{max u1 u2} (forall (i : ι), α i)) (infᵢ.{max u1 u2, succ u1} (Filter.{max u1 u2} (forall (i : ι), α i)) (ConditionallyCompleteLattice.toHasInf.{max u1 u2} (Filter.{max u1 u2} (forall (i : ι), α i)) (CompleteLattice.toConditionallyCompleteLattice.{max u1 u2} (Filter.{max u1 u2} (forall (i : ι), α i)) (Filter.completeLattice.{max u1 u2} (forall (i : ι), α i)))) ι (fun (i : ι) => infᵢ.{max u1 u2, 0} (Filter.{max u1 u2} (forall (i : ι), α i)) (ConditionallyCompleteLattice.toHasInf.{max u1 u2} (Filter.{max u1 u2} (forall (i : ι), α i)) (CompleteLattice.toConditionallyCompleteLattice.{max u1 u2} (Filter.{max u1 u2} (forall (i : ι), α i)) (Filter.completeLattice.{max u1 u2} (forall (i : ι), α i)))) (Membership.Mem.{u1, u1} ι (Set.{u1} ι) (Set.hasMem.{u1} ι) i I) (fun (H : Membership.Mem.{u1, u1} ι (Set.{u1} ι) (Set.hasMem.{u1} ι) i I) => Filter.comap.{max u1 u2, u2} (forall (i : ι), α i) (α i) (fun (x : forall (i : ι), α i) => x i) (nhdsWithin.{u2} (α i) (_inst_2 i) (x i) (s i))))) (infᵢ.{max u1 u2, succ u1} (Filter.{max u1 u2} (forall (i : ι), α i)) (ConditionallyCompleteLattice.toHasInf.{max u1 u2} (Filter.{max u1 u2} (forall (i : ι), α i)) (CompleteLattice.toConditionallyCompleteLattice.{max u1 u2} (Filter.{max u1 u2} (forall (i : ι), α i)) (Filter.completeLattice.{max u1 u2} (forall (i : ι), α i)))) ι (fun (i : ι) => infᵢ.{max u1 u2, 0} (Filter.{max u1 u2} (forall (i : ι), α i)) (ConditionallyCompleteLattice.toHasInf.{max u1 u2} (Filter.{max u1 u2} (forall (i : ι), α i)) (CompleteLattice.toConditionallyCompleteLattice.{max u1 u2} (Filter.{max u1 u2} (forall (i : ι), α i)) (Filter.completeLattice.{max u1 u2} (forall (i : ι), α i)))) (Not (Membership.Mem.{u1, u1} ι (Set.{u1} ι) (Set.hasMem.{u1} ι) i I)) (fun (H : Not (Membership.Mem.{u1, u1} ι (Set.{u1} ι) (Set.hasMem.{u1} ι) i I)) => Filter.comap.{max u1 u2, u2} (forall (i : ι), α i) (α i) (fun (x : forall (i : ι), α i) => x i) (nhds.{u2} (α i) (_inst_2 i) (x i)))))))
+  forall {ι : Type.{u1}} {α : ι -> Type.{u2}} [_inst_2 : forall (i : ι), TopologicalSpace.{u2} (α i)] {I : Set.{u1} ι}, (Set.Finite.{u1} ι I) -> (forall (s : forall (i : ι), Set.{u2} (α i)) (x : forall (i : ι), α i), Eq.{succ (max u1 u2)} (Filter.{max u1 u2} (forall (i : ι), α i)) (nhdsWithin.{max u1 u2} (forall (i : ι), α i) (Pi.topologicalSpace.{u1, u2} ι (fun (i : ι) => α i) (fun (a : ι) => _inst_2 a)) x (Set.pi.{u1, u2} ι (fun (i : ι) => α i) I s)) (Inf.inf.{max u1 u2} (Filter.{max u1 u2} (forall (i : ι), α i)) (Filter.hasInf.{max u1 u2} (forall (i : ι), α i)) (iInf.{max u1 u2, succ u1} (Filter.{max u1 u2} (forall (i : ι), α i)) (ConditionallyCompleteLattice.toHasInf.{max u1 u2} (Filter.{max u1 u2} (forall (i : ι), α i)) (CompleteLattice.toConditionallyCompleteLattice.{max u1 u2} (Filter.{max u1 u2} (forall (i : ι), α i)) (Filter.completeLattice.{max u1 u2} (forall (i : ι), α i)))) ι (fun (i : ι) => iInf.{max u1 u2, 0} (Filter.{max u1 u2} (forall (i : ι), α i)) (ConditionallyCompleteLattice.toHasInf.{max u1 u2} (Filter.{max u1 u2} (forall (i : ι), α i)) (CompleteLattice.toConditionallyCompleteLattice.{max u1 u2} (Filter.{max u1 u2} (forall (i : ι), α i)) (Filter.completeLattice.{max u1 u2} (forall (i : ι), α i)))) (Membership.Mem.{u1, u1} ι (Set.{u1} ι) (Set.hasMem.{u1} ι) i I) (fun (H : Membership.Mem.{u1, u1} ι (Set.{u1} ι) (Set.hasMem.{u1} ι) i I) => Filter.comap.{max u1 u2, u2} (forall (i : ι), α i) (α i) (fun (x : forall (i : ι), α i) => x i) (nhdsWithin.{u2} (α i) (_inst_2 i) (x i) (s i))))) (iInf.{max u1 u2, succ u1} (Filter.{max u1 u2} (forall (i : ι), α i)) (ConditionallyCompleteLattice.toHasInf.{max u1 u2} (Filter.{max u1 u2} (forall (i : ι), α i)) (CompleteLattice.toConditionallyCompleteLattice.{max u1 u2} (Filter.{max u1 u2} (forall (i : ι), α i)) (Filter.completeLattice.{max u1 u2} (forall (i : ι), α i)))) ι (fun (i : ι) => iInf.{max u1 u2, 0} (Filter.{max u1 u2} (forall (i : ι), α i)) (ConditionallyCompleteLattice.toHasInf.{max u1 u2} (Filter.{max u1 u2} (forall (i : ι), α i)) (CompleteLattice.toConditionallyCompleteLattice.{max u1 u2} (Filter.{max u1 u2} (forall (i : ι), α i)) (Filter.completeLattice.{max u1 u2} (forall (i : ι), α i)))) (Not (Membership.Mem.{u1, u1} ι (Set.{u1} ι) (Set.hasMem.{u1} ι) i I)) (fun (H : Not (Membership.Mem.{u1, u1} ι (Set.{u1} ι) (Set.hasMem.{u1} ι) i I)) => Filter.comap.{max u1 u2, u2} (forall (i : ι), α i) (α i) (fun (x : forall (i : ι), α i) => x i) (nhds.{u2} (α i) (_inst_2 i) (x i)))))))
 but is expected to have type
-  forall {ι : Type.{u2}} {α : ι -> Type.{u1}} [_inst_2 : forall (i : ι), TopologicalSpace.{u1} (α i)] {I : Set.{u2} ι}, (Set.Finite.{u2} ι I) -> (forall (s : forall (i : ι), Set.{u1} (α i)) (x : forall (i : ι), α i), Eq.{max (succ u2) (succ u1)} (Filter.{max u2 u1} (forall (i : ι), α i)) (nhdsWithin.{max u2 u1} (forall (i : ι), α i) (Pi.topologicalSpace.{u2, u1} ι (fun (i : ι) => α i) (fun (a : ι) => _inst_2 a)) x (Set.pi.{u2, u1} ι (fun (i : ι) => α i) I s)) (Inf.inf.{max u2 u1} (Filter.{max u2 u1} (forall (i : ι), α i)) (Filter.instInfFilter.{max u2 u1} (forall (i : ι), α i)) (infᵢ.{max u2 u1, succ u2} (Filter.{max u2 u1} (forall (i : ι), α i)) (ConditionallyCompleteLattice.toInfSet.{max u2 u1} (Filter.{max u2 u1} (forall (i : ι), α i)) (CompleteLattice.toConditionallyCompleteLattice.{max u2 u1} (Filter.{max u2 u1} (forall (i : ι), α i)) (Filter.instCompleteLatticeFilter.{max u2 u1} (forall (i : ι), α i)))) ι (fun (i : ι) => infᵢ.{max u2 u1, 0} (Filter.{max u2 u1} (forall (i : ι), α i)) (ConditionallyCompleteLattice.toInfSet.{max u2 u1} (Filter.{max u2 u1} (forall (i : ι), α i)) (CompleteLattice.toConditionallyCompleteLattice.{max u2 u1} (Filter.{max u2 u1} (forall (i : ι), α i)) (Filter.instCompleteLatticeFilter.{max u2 u1} (forall (i : ι), α i)))) (Membership.mem.{u2, u2} ι (Set.{u2} ι) (Set.instMembershipSet.{u2} ι) i I) (fun (H : Membership.mem.{u2, u2} ι (Set.{u2} ι) (Set.instMembershipSet.{u2} ι) i I) => Filter.comap.{max u2 u1, u1} (forall (i : ι), α i) (α i) (fun (x : forall (i : ι), α i) => x i) (nhdsWithin.{u1} (α i) (_inst_2 i) (x i) (s i))))) (infᵢ.{max u2 u1, succ u2} (Filter.{max u2 u1} (forall (i : ι), α i)) (ConditionallyCompleteLattice.toInfSet.{max u2 u1} (Filter.{max u2 u1} (forall (i : ι), α i)) (CompleteLattice.toConditionallyCompleteLattice.{max u2 u1} (Filter.{max u2 u1} (forall (i : ι), α i)) (Filter.instCompleteLatticeFilter.{max u2 u1} (forall (i : ι), α i)))) ι (fun (i : ι) => infᵢ.{max u2 u1, 0} (Filter.{max u2 u1} (forall (i : ι), α i)) (ConditionallyCompleteLattice.toInfSet.{max u2 u1} (Filter.{max u2 u1} (forall (i : ι), α i)) (CompleteLattice.toConditionallyCompleteLattice.{max u2 u1} (Filter.{max u2 u1} (forall (i : ι), α i)) (Filter.instCompleteLatticeFilter.{max u2 u1} (forall (i : ι), α i)))) (Not (Membership.mem.{u2, u2} ι (Set.{u2} ι) (Set.instMembershipSet.{u2} ι) i I)) (fun (H : Not (Membership.mem.{u2, u2} ι (Set.{u2} ι) (Set.instMembershipSet.{u2} ι) i I)) => Filter.comap.{max u2 u1, u1} (forall (i : ι), α i) (α i) (fun (x : forall (i : ι), α i) => x i) (nhds.{u1} (α i) (_inst_2 i) (x i)))))))
+  forall {ι : Type.{u2}} {α : ι -> Type.{u1}} [_inst_2 : forall (i : ι), TopologicalSpace.{u1} (α i)] {I : Set.{u2} ι}, (Set.Finite.{u2} ι I) -> (forall (s : forall (i : ι), Set.{u1} (α i)) (x : forall (i : ι), α i), Eq.{max (succ u2) (succ u1)} (Filter.{max u2 u1} (forall (i : ι), α i)) (nhdsWithin.{max u2 u1} (forall (i : ι), α i) (Pi.topologicalSpace.{u2, u1} ι (fun (i : ι) => α i) (fun (a : ι) => _inst_2 a)) x (Set.pi.{u2, u1} ι (fun (i : ι) => α i) I s)) (Inf.inf.{max u2 u1} (Filter.{max u2 u1} (forall (i : ι), α i)) (Filter.instInfFilter.{max u2 u1} (forall (i : ι), α i)) (iInf.{max u2 u1, succ u2} (Filter.{max u2 u1} (forall (i : ι), α i)) (ConditionallyCompleteLattice.toInfSet.{max u2 u1} (Filter.{max u2 u1} (forall (i : ι), α i)) (CompleteLattice.toConditionallyCompleteLattice.{max u2 u1} (Filter.{max u2 u1} (forall (i : ι), α i)) (Filter.instCompleteLatticeFilter.{max u2 u1} (forall (i : ι), α i)))) ι (fun (i : ι) => iInf.{max u2 u1, 0} (Filter.{max u2 u1} (forall (i : ι), α i)) (ConditionallyCompleteLattice.toInfSet.{max u2 u1} (Filter.{max u2 u1} (forall (i : ι), α i)) (CompleteLattice.toConditionallyCompleteLattice.{max u2 u1} (Filter.{max u2 u1} (forall (i : ι), α i)) (Filter.instCompleteLatticeFilter.{max u2 u1} (forall (i : ι), α i)))) (Membership.mem.{u2, u2} ι (Set.{u2} ι) (Set.instMembershipSet.{u2} ι) i I) (fun (H : Membership.mem.{u2, u2} ι (Set.{u2} ι) (Set.instMembershipSet.{u2} ι) i I) => Filter.comap.{max u2 u1, u1} (forall (i : ι), α i) (α i) (fun (x : forall (i : ι), α i) => x i) (nhdsWithin.{u1} (α i) (_inst_2 i) (x i) (s i))))) (iInf.{max u2 u1, succ u2} (Filter.{max u2 u1} (forall (i : ι), α i)) (ConditionallyCompleteLattice.toInfSet.{max u2 u1} (Filter.{max u2 u1} (forall (i : ι), α i)) (CompleteLattice.toConditionallyCompleteLattice.{max u2 u1} (Filter.{max u2 u1} (forall (i : ι), α i)) (Filter.instCompleteLatticeFilter.{max u2 u1} (forall (i : ι), α i)))) ι (fun (i : ι) => iInf.{max u2 u1, 0} (Filter.{max u2 u1} (forall (i : ι), α i)) (ConditionallyCompleteLattice.toInfSet.{max u2 u1} (Filter.{max u2 u1} (forall (i : ι), α i)) (CompleteLattice.toConditionallyCompleteLattice.{max u2 u1} (Filter.{max u2 u1} (forall (i : ι), α i)) (Filter.instCompleteLatticeFilter.{max u2 u1} (forall (i : ι), α i)))) (Not (Membership.mem.{u2, u2} ι (Set.{u2} ι) (Set.instMembershipSet.{u2} ι) i I)) (fun (H : Not (Membership.mem.{u2, u2} ι (Set.{u2} ι) (Set.instMembershipSet.{u2} ι) i I)) => Filter.comap.{max u2 u1, u1} (forall (i : ι), α i) (α i) (fun (x : forall (i : ι), α i) => x i) (nhds.{u1} (α i) (_inst_2 i) (x i)))))))
 Case conversion may be inaccurate. Consider using '#align nhds_within_pi_eq nhdsWithin_pi_eqₓ'. -/
 /- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (i «expr ∉ » I) -/
 theorem nhdsWithin_pi_eq {ι : Type _} {α : ι → Type _} [∀ i, TopologicalSpace (α i)] {I : Set ι}
@@ -635,15 +635,15 @@ theorem nhdsWithin_pi_eq {ι : Type _} {α : ι → Type _} [∀ i, TopologicalS
   by
   simp only [nhdsWithin, nhds_pi, Filter.pi, pi_def, ← infi_principal_finite hI, comap_inf,
     comap_principal, eval]
-  rw [infᵢ_split _ fun i => i ∈ I, inf_right_comm]
-  simp only [infᵢ_inf_eq]
+  rw [iInf_split _ fun i => i ∈ I, inf_right_comm]
+  simp only [iInf_inf_eq]
 #align nhds_within_pi_eq nhdsWithin_pi_eq
 
 /- warning: nhds_within_pi_univ_eq -> nhdsWithin_pi_univ_eq is a dubious translation:
 lean 3 declaration is
-  forall {ι : Type.{u1}} {α : ι -> Type.{u2}} [_inst_2 : Finite.{succ u1} ι] [_inst_3 : forall (i : ι), TopologicalSpace.{u2} (α i)] (s : forall (i : ι), Set.{u2} (α i)) (x : forall (i : ι), α i), Eq.{succ (max u1 u2)} (Filter.{max u1 u2} (forall (i : ι), α i)) (nhdsWithin.{max u1 u2} (forall (i : ι), α i) (Pi.topologicalSpace.{u1, u2} ι (fun (i : ι) => α i) (fun (a : ι) => _inst_3 a)) x (Set.pi.{u1, u2} ι (fun (i : ι) => α i) (Set.univ.{u1} ι) s)) (infᵢ.{max u1 u2, succ u1} (Filter.{max u1 u2} (forall (i : ι), α i)) (ConditionallyCompleteLattice.toHasInf.{max u1 u2} (Filter.{max u1 u2} (forall (i : ι), α i)) (CompleteLattice.toConditionallyCompleteLattice.{max u1 u2} (Filter.{max u1 u2} (forall (i : ι), α i)) (Filter.completeLattice.{max u1 u2} (forall (i : ι), α i)))) ι (fun (i : ι) => Filter.comap.{max u1 u2, u2} (forall (i : ι), α i) (α i) (fun (x : forall (i : ι), α i) => x i) (nhdsWithin.{u2} (α i) (_inst_3 i) (x i) (s i))))
+  forall {ι : Type.{u1}} {α : ι -> Type.{u2}} [_inst_2 : Finite.{succ u1} ι] [_inst_3 : forall (i : ι), TopologicalSpace.{u2} (α i)] (s : forall (i : ι), Set.{u2} (α i)) (x : forall (i : ι), α i), Eq.{succ (max u1 u2)} (Filter.{max u1 u2} (forall (i : ι), α i)) (nhdsWithin.{max u1 u2} (forall (i : ι), α i) (Pi.topologicalSpace.{u1, u2} ι (fun (i : ι) => α i) (fun (a : ι) => _inst_3 a)) x (Set.pi.{u1, u2} ι (fun (i : ι) => α i) (Set.univ.{u1} ι) s)) (iInf.{max u1 u2, succ u1} (Filter.{max u1 u2} (forall (i : ι), α i)) (ConditionallyCompleteLattice.toHasInf.{max u1 u2} (Filter.{max u1 u2} (forall (i : ι), α i)) (CompleteLattice.toConditionallyCompleteLattice.{max u1 u2} (Filter.{max u1 u2} (forall (i : ι), α i)) (Filter.completeLattice.{max u1 u2} (forall (i : ι), α i)))) ι (fun (i : ι) => Filter.comap.{max u1 u2, u2} (forall (i : ι), α i) (α i) (fun (x : forall (i : ι), α i) => x i) (nhdsWithin.{u2} (α i) (_inst_3 i) (x i) (s i))))
 but is expected to have type
-  forall {ι : Type.{u2}} {α : ι -> Type.{u1}} [_inst_2 : Finite.{succ u2} ι] [_inst_3 : forall (i : ι), TopologicalSpace.{u1} (α i)] (s : forall (i : ι), Set.{u1} (α i)) (x : forall (i : ι), α i), Eq.{max (succ u2) (succ u1)} (Filter.{max u2 u1} (forall (i : ι), α i)) (nhdsWithin.{max u2 u1} (forall (i : ι), α i) (Pi.topologicalSpace.{u2, u1} ι (fun (i : ι) => α i) (fun (a : ι) => _inst_3 a)) x (Set.pi.{u2, u1} ι (fun (i : ι) => α i) (Set.univ.{u2} ι) s)) (infᵢ.{max u2 u1, succ u2} (Filter.{max u2 u1} (forall (i : ι), α i)) (ConditionallyCompleteLattice.toInfSet.{max u2 u1} (Filter.{max u2 u1} (forall (i : ι), α i)) (CompleteLattice.toConditionallyCompleteLattice.{max u2 u1} (Filter.{max u2 u1} (forall (i : ι), α i)) (Filter.instCompleteLatticeFilter.{max u2 u1} (forall (i : ι), α i)))) ι (fun (i : ι) => Filter.comap.{max u2 u1, u1} (forall (i : ι), α i) (α i) (fun (x : forall (i : ι), α i) => x i) (nhdsWithin.{u1} (α i) (_inst_3 i) (x i) (s i))))
+  forall {ι : Type.{u2}} {α : ι -> Type.{u1}} [_inst_2 : Finite.{succ u2} ι] [_inst_3 : forall (i : ι), TopologicalSpace.{u1} (α i)] (s : forall (i : ι), Set.{u1} (α i)) (x : forall (i : ι), α i), Eq.{max (succ u2) (succ u1)} (Filter.{max u2 u1} (forall (i : ι), α i)) (nhdsWithin.{max u2 u1} (forall (i : ι), α i) (Pi.topologicalSpace.{u2, u1} ι (fun (i : ι) => α i) (fun (a : ι) => _inst_3 a)) x (Set.pi.{u2, u1} ι (fun (i : ι) => α i) (Set.univ.{u2} ι) s)) (iInf.{max u2 u1, succ u2} (Filter.{max u2 u1} (forall (i : ι), α i)) (ConditionallyCompleteLattice.toInfSet.{max u2 u1} (Filter.{max u2 u1} (forall (i : ι), α i)) (CompleteLattice.toConditionallyCompleteLattice.{max u2 u1} (Filter.{max u2 u1} (forall (i : ι), α i)) (Filter.instCompleteLatticeFilter.{max u2 u1} (forall (i : ι), α i)))) ι (fun (i : ι) => Filter.comap.{max u2 u1, u1} (forall (i : ι), α i) (α i) (fun (x : forall (i : ι), α i) => x i) (nhdsWithin.{u1} (α i) (_inst_3 i) (x i) (s i))))
 Case conversion may be inaccurate. Consider using '#align nhds_within_pi_univ_eq nhdsWithin_pi_univ_eqₓ'. -/
 theorem nhdsWithin_pi_univ_eq {ι : Type _} {α : ι → Type _} [Finite ι] [∀ i, TopologicalSpace (α i)]
     (s : ∀ i, Set (α i)) (x : ∀ i, α i) : 𝓝[pi univ s] x = ⨅ i, comap (fun x => x i) (𝓝[s i] x i) :=
@@ -699,13 +699,13 @@ theorem Filter.Tendsto.if_nhdsWithin {f g : α → β} {p : α → Prop} [Decida
 
 /- warning: map_nhds_within -> map_nhdsWithin is a dubious translation:
 lean 3 declaration is
-  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : TopologicalSpace.{u1} α] (f : α -> β) (a : α) (s : Set.{u1} α), Eq.{succ u2} (Filter.{u2} β) (Filter.map.{u1, u2} α β f (nhdsWithin.{u1} α _inst_1 a s)) (infᵢ.{u2, succ u1} (Filter.{u2} β) (ConditionallyCompleteLattice.toHasInf.{u2} (Filter.{u2} β) (CompleteLattice.toConditionallyCompleteLattice.{u2} (Filter.{u2} β) (Filter.completeLattice.{u2} β))) (Set.{u1} α) (fun (t : Set.{u1} α) => infᵢ.{u2, 0} (Filter.{u2} β) (ConditionallyCompleteLattice.toHasInf.{u2} (Filter.{u2} β) (CompleteLattice.toConditionallyCompleteLattice.{u2} (Filter.{u2} β) (Filter.completeLattice.{u2} β))) (Membership.Mem.{u1, u1} (Set.{u1} α) (Set.{u1} (Set.{u1} α)) (Set.hasMem.{u1} (Set.{u1} α)) t (setOf.{u1} (Set.{u1} α) (fun (t : Set.{u1} α) => And (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) a t) (IsOpen.{u1} α _inst_1 t)))) (fun (H : Membership.Mem.{u1, u1} (Set.{u1} α) (Set.{u1} (Set.{u1} α)) (Set.hasMem.{u1} (Set.{u1} α)) t (setOf.{u1} (Set.{u1} α) (fun (t : Set.{u1} α) => And (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) a t) (IsOpen.{u1} α _inst_1 t)))) => Filter.principal.{u2} β (Set.image.{u1, u2} α β f (Inter.inter.{u1} (Set.{u1} α) (Set.hasInter.{u1} α) t s)))))
+  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : TopologicalSpace.{u1} α] (f : α -> β) (a : α) (s : Set.{u1} α), Eq.{succ u2} (Filter.{u2} β) (Filter.map.{u1, u2} α β f (nhdsWithin.{u1} α _inst_1 a s)) (iInf.{u2, succ u1} (Filter.{u2} β) (ConditionallyCompleteLattice.toHasInf.{u2} (Filter.{u2} β) (CompleteLattice.toConditionallyCompleteLattice.{u2} (Filter.{u2} β) (Filter.completeLattice.{u2} β))) (Set.{u1} α) (fun (t : Set.{u1} α) => iInf.{u2, 0} (Filter.{u2} β) (ConditionallyCompleteLattice.toHasInf.{u2} (Filter.{u2} β) (CompleteLattice.toConditionallyCompleteLattice.{u2} (Filter.{u2} β) (Filter.completeLattice.{u2} β))) (Membership.Mem.{u1, u1} (Set.{u1} α) (Set.{u1} (Set.{u1} α)) (Set.hasMem.{u1} (Set.{u1} α)) t (setOf.{u1} (Set.{u1} α) (fun (t : Set.{u1} α) => And (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) a t) (IsOpen.{u1} α _inst_1 t)))) (fun (H : Membership.Mem.{u1, u1} (Set.{u1} α) (Set.{u1} (Set.{u1} α)) (Set.hasMem.{u1} (Set.{u1} α)) t (setOf.{u1} (Set.{u1} α) (fun (t : Set.{u1} α) => And (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) a t) (IsOpen.{u1} α _inst_1 t)))) => Filter.principal.{u2} β (Set.image.{u1, u2} α β f (Inter.inter.{u1} (Set.{u1} α) (Set.hasInter.{u1} α) t s)))))
 but is expected to have type
-  forall {α : Type.{u2}} {β : Type.{u1}} [_inst_1 : TopologicalSpace.{u2} α] (f : α -> β) (a : α) (s : Set.{u2} α), Eq.{succ u1} (Filter.{u1} β) (Filter.map.{u2, u1} α β f (nhdsWithin.{u2} α _inst_1 a s)) (infᵢ.{u1, succ u2} (Filter.{u1} β) (ConditionallyCompleteLattice.toInfSet.{u1} (Filter.{u1} β) (CompleteLattice.toConditionallyCompleteLattice.{u1} (Filter.{u1} β) (Filter.instCompleteLatticeFilter.{u1} β))) (Set.{u2} α) (fun (t : Set.{u2} α) => infᵢ.{u1, 0} (Filter.{u1} β) (ConditionallyCompleteLattice.toInfSet.{u1} (Filter.{u1} β) (CompleteLattice.toConditionallyCompleteLattice.{u1} (Filter.{u1} β) (Filter.instCompleteLatticeFilter.{u1} β))) (Membership.mem.{u2, u2} (Set.{u2} α) (Set.{u2} (Set.{u2} α)) (Set.instMembershipSet.{u2} (Set.{u2} α)) t (setOf.{u2} (Set.{u2} α) (fun (t : Set.{u2} α) => And (Membership.mem.{u2, u2} α (Set.{u2} α) (Set.instMembershipSet.{u2} α) a t) (IsOpen.{u2} α _inst_1 t)))) (fun (H : Membership.mem.{u2, u2} (Set.{u2} α) (Set.{u2} (Set.{u2} α)) (Set.instMembershipSet.{u2} (Set.{u2} α)) t (setOf.{u2} (Set.{u2} α) (fun (t : Set.{u2} α) => And (Membership.mem.{u2, u2} α (Set.{u2} α) (Set.instMembershipSet.{u2} α) a t) (IsOpen.{u2} α _inst_1 t)))) => Filter.principal.{u1} β (Set.image.{u2, u1} α β f (Inter.inter.{u2} (Set.{u2} α) (Set.instInterSet.{u2} α) t s)))))
+  forall {α : Type.{u2}} {β : Type.{u1}} [_inst_1 : TopologicalSpace.{u2} α] (f : α -> β) (a : α) (s : Set.{u2} α), Eq.{succ u1} (Filter.{u1} β) (Filter.map.{u2, u1} α β f (nhdsWithin.{u2} α _inst_1 a s)) (iInf.{u1, succ u2} (Filter.{u1} β) (ConditionallyCompleteLattice.toInfSet.{u1} (Filter.{u1} β) (CompleteLattice.toConditionallyCompleteLattice.{u1} (Filter.{u1} β) (Filter.instCompleteLatticeFilter.{u1} β))) (Set.{u2} α) (fun (t : Set.{u2} α) => iInf.{u1, 0} (Filter.{u1} β) (ConditionallyCompleteLattice.toInfSet.{u1} (Filter.{u1} β) (CompleteLattice.toConditionallyCompleteLattice.{u1} (Filter.{u1} β) (Filter.instCompleteLatticeFilter.{u1} β))) (Membership.mem.{u2, u2} (Set.{u2} α) (Set.{u2} (Set.{u2} α)) (Set.instMembershipSet.{u2} (Set.{u2} α)) t (setOf.{u2} (Set.{u2} α) (fun (t : Set.{u2} α) => And (Membership.mem.{u2, u2} α (Set.{u2} α) (Set.instMembershipSet.{u2} α) a t) (IsOpen.{u2} α _inst_1 t)))) (fun (H : Membership.mem.{u2, u2} (Set.{u2} α) (Set.{u2} (Set.{u2} α)) (Set.instMembershipSet.{u2} (Set.{u2} α)) t (setOf.{u2} (Set.{u2} α) (fun (t : Set.{u2} α) => And (Membership.mem.{u2, u2} α (Set.{u2} α) (Set.instMembershipSet.{u2} α) a t) (IsOpen.{u2} α _inst_1 t)))) => Filter.principal.{u1} β (Set.image.{u2, u1} α β f (Inter.inter.{u2} (Set.{u2} α) (Set.instInterSet.{u2} α) t s)))))
 Case conversion may be inaccurate. Consider using '#align map_nhds_within map_nhdsWithinₓ'. -/
 theorem map_nhdsWithin (f : α → β) (a : α) (s : Set α) :
     map f (𝓝[s] a) = ⨅ t ∈ { t : Set α | a ∈ t ∧ IsOpen t }, 𝓟 (f '' (t ∩ s)) :=
-  ((nhdsWithin_basis_open a s).map f).eq_binfᵢ
+  ((nhdsWithin_basis_open a s).map f).eq_biInf
 #align map_nhds_within map_nhdsWithin
 
 /- warning: tendsto_nhds_within_mono_left -> tendsto_nhdsWithin_mono_left is a dubious translation:
@@ -2023,7 +2023,7 @@ theorem continuousOn_open_of_generateFrom {β : Type _} {s : Set α} {T : Set (S
     rw [this]
     exact hu.inter hv
   · rw [preimage_sUnion, inter_Union₂]
-    exact isOpen_bunionᵢ hU'
+    exact isOpen_biUnion hU'
   · exact hs
 #align continuous_on_open_of_generate_from continuousOn_open_of_generateFromₓ
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/8b8ba04e2f326f3f7cf24ad129beda58531ada61

@@ -1098,7 +1098,7 @@ theorem continuousOn_pi {ι : Type _} {π : ι → Type _} [∀ i, TopologicalSp
 lean 3 declaration is
   forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] {n : Nat} {π : (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) -> Type.{u2}} [_inst_5 : forall (i : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))), TopologicalSpace.{u2} (π i)] (i : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) {f : α -> (π i)} {a : α} {s : Set.{u1} α}, (ContinuousWithinAt.{u1, u2} α (π i) _inst_1 (_inst_5 i) f s a) -> (forall {g : α -> (forall (j : Fin n), π (coeFn.{1, 1} (OrderEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (Fin.hasLe n) (Preorder.toLE.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (PartialOrder.toPreorder.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (Fin.partialOrder (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne)))))))) (fun (_x : RelEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (LE.le.{0} (Fin n) (Fin.hasLe n)) (LE.le.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (Preorder.toLE.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (PartialOrder.toPreorder.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (Fin.partialOrder (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))))))) => (Fin n) -> (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne)))))) (RelEmbedding.hasCoeToFun.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (LE.le.{0} (Fin n) (Fin.hasLe n)) (LE.le.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (Preorder.toLE.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 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 but is expected to have type
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(x._@.Mathlib.Order.Hom.Basic._hyg.697 : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) => LE.le.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) x._@.Mathlib.Order.Hom.Basic._hyg.695 x._@.Mathlib.Order.Hom.Basic._hyg.697) (RelEmbedding.instRelHomClassRelEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.680 : Fin n) (x._@.Mathlib.Order.Hom.Basic._hyg.682 : Fin n) => LE.le.{0} (Fin n) (instLEFin n) x._@.Mathlib.Order.Hom.Basic._hyg.680 x._@.Mathlib.Order.Hom.Basic._hyg.682) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.695 : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 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(OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) => LE.le.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) x._@.Mathlib.Order.Hom.Basic._hyg.695 x._@.Mathlib.Order.Hom.Basic._hyg.697) (RelEmbedding.instRelHomClassRelEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.680 : Fin n) (x._@.Mathlib.Order.Hom.Basic._hyg.682 : Fin n) => LE.le.{0} (Fin n) (instLEFin n) x._@.Mathlib.Order.Hom.Basic._hyg.680 x._@.Mathlib.Order.Hom.Basic._hyg.682) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.695 : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (x._@.Mathlib.Order.Hom.Basic._hyg.697 : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} 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(RelHomClass.toFunLike.{0, 0, 0} (OrderEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin n) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))))) (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.680 : Fin n) (x._@.Mathlib.Order.Hom.Basic._hyg.682 : Fin n) => LE.le.{0} (Fin n) (instLEFin n) x._@.Mathlib.Order.Hom.Basic._hyg.680 x._@.Mathlib.Order.Hom.Basic._hyg.682) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.695 : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (x._@.Mathlib.Order.Hom.Basic._hyg.697 : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) => LE.le.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) x._@.Mathlib.Order.Hom.Basic._hyg.695 x._@.Mathlib.Order.Hom.Basic._hyg.697) (RelEmbedding.instRelHomClassRelEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.680 : Fin n) (x._@.Mathlib.Order.Hom.Basic._hyg.682 : Fin n) => LE.le.{0} (Fin n) (instLEFin n) x._@.Mathlib.Order.Hom.Basic._hyg.680 x._@.Mathlib.Order.Hom.Basic._hyg.682) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.695 : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (x._@.Mathlib.Order.Hom.Basic._hyg.697 : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) => LE.le.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) x._@.Mathlib.Order.Hom.Basic._hyg.695 x._@.Mathlib.Order.Hom.Basic._hyg.697))) (Fin.succAbove n i) j)) (fun (a : Fin n) => _inst_5 (FunLike.coe.{1, 1, 1} (OrderEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin n) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))))) (Fin n) (fun (_x : Fin n) => (fun (x._@.Mathlib.Order.RelIso.Basic._hyg.867 : Fin n) => Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) _x) (RelHomClass.toFunLike.{0, 0, 0} (OrderEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin n) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))))) (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.680 : Fin n) (x._@.Mathlib.Order.Hom.Basic._hyg.682 : Fin n) => LE.le.{0} (Fin n) (instLEFin n) x._@.Mathlib.Order.Hom.Basic._hyg.680 x._@.Mathlib.Order.Hom.Basic._hyg.682) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.695 : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (x._@.Mathlib.Order.Hom.Basic._hyg.697 : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) => LE.le.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) x._@.Mathlib.Order.Hom.Basic._hyg.695 x._@.Mathlib.Order.Hom.Basic._hyg.697) (RelEmbedding.instRelHomClassRelEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.680 : Fin n) (x._@.Mathlib.Order.Hom.Basic._hyg.682 : Fin n) => LE.le.{0} (Fin n) (instLEFin n) x._@.Mathlib.Order.Hom.Basic._hyg.680 x._@.Mathlib.Order.Hom.Basic._hyg.682) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.695 : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (x._@.Mathlib.Order.Hom.Basic._hyg.697 : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) => LE.le.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) x._@.Mathlib.Order.Hom.Basic._hyg.695 x._@.Mathlib.Order.Hom.Basic._hyg.697))) (Fin.succAbove n i) a))) g s a) -> (ContinuousWithinAt.{u1, u2} α (forall (j : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))), π j) _inst_1 (Pi.topologicalSpace.{0, u2} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (fun (j : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) => π j) (fun (a : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) => _inst_5 a)) (fun (a : α) => Fin.insertNth.{u2} n π i (f a) (g a)) s a))
+  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] {n : Nat} {π : (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) -> Type.{u2}} [_inst_5 : forall (i : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))), TopologicalSpace.{u2} (π i)] (i : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) {f : α -> (π i)} {a : α} {s : Set.{u1} α}, (ContinuousWithinAt.{u1, u2} α (π i) _inst_1 (_inst_5 i) f s a) -> (forall {g : α -> (forall (j : Fin n), π (FunLike.coe.{1, 1, 1} (OrderEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin n) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))))) (Fin n) (fun (_x : Fin n) => (fun (x._@.Mathlib.Order.Hom.Lattice._hyg.494 : Fin n) => Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) _x) (InfHomClass.toFunLike.{0, 0, 0} (OrderEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin n) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))))) (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (Lattice.toInf.{0} (Fin n) (DistribLattice.toLattice.{0} (Fin n) (instDistribLattice.{0} (Fin n) (Fin.instLinearOrderFin n)))) (Lattice.toInf.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (Fin.instLatticeFinHAddNatInstHAddInstAddNatOfNat n)) (LatticeHomClass.toInfHomClass.{0, 0, 0} (OrderEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin n) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))))) (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (DistribLattice.toLattice.{0} (Fin n) (instDistribLattice.{0} (Fin n) (Fin.instLinearOrderFin n))) (Fin.instLatticeFinHAddNatInstHAddInstAddNatOfNat n) (OrderHomClass.toLatticeHomClass.{0, 0, 0} (OrderEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin n) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))))) (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (Fin.instLinearOrderFin n) (Fin.instLatticeFinHAddNatInstHAddInstAddNatOfNat n) (RelEmbedding.instRelHomClassRelEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.680 : Fin n) (x._@.Mathlib.Order.Hom.Basic._hyg.682 : Fin n) => LE.le.{0} (Fin n) (instLEFin n) x._@.Mathlib.Order.Hom.Basic._hyg.680 x._@.Mathlib.Order.Hom.Basic._hyg.682) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.695 : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (x._@.Mathlib.Order.Hom.Basic._hyg.697 : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) => LE.le.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) x._@.Mathlib.Order.Hom.Basic._hyg.695 x._@.Mathlib.Order.Hom.Basic._hyg.697))))) (Fin.succAbove n i) j))}, (ContinuousWithinAt.{u1, u2} α (forall (j : Fin n), π (FunLike.coe.{1, 1, 1} (OrderEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin n) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))))) (Fin n) (fun (_x : Fin n) => (fun (x._@.Mathlib.Order.Hom.Lattice._hyg.494 : Fin n) => Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) _x) (InfHomClass.toFunLike.{0, 0, 0} (OrderEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin n) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))))) (Fin n) (Fin 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(Fin.instLatticeFinHAddNatInstHAddInstAddNatOfNat n) (OrderHomClass.toLatticeHomClass.{0, 0, 0} (OrderEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin n) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))))) (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (Fin.instLinearOrderFin n) (Fin.instLatticeFinHAddNatInstHAddInstAddNatOfNat n) (RelEmbedding.instRelHomClassRelEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.680 : Fin n) (x._@.Mathlib.Order.Hom.Basic._hyg.682 : Fin n) => LE.le.{0} (Fin n) (instLEFin n) x._@.Mathlib.Order.Hom.Basic._hyg.680 x._@.Mathlib.Order.Hom.Basic._hyg.682) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.695 : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (x._@.Mathlib.Order.Hom.Basic._hyg.697 : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) => LE.le.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) x._@.Mathlib.Order.Hom.Basic._hyg.695 x._@.Mathlib.Order.Hom.Basic._hyg.697))))) (Fin.succAbove n i) j)) _inst_1 (Pi.topologicalSpace.{0, u2} (Fin n) (fun (j : Fin n) => π (FunLike.coe.{1, 1, 1} (OrderEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin n) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))))) (Fin n) (fun (_x : Fin n) => (fun (x._@.Mathlib.Order.Hom.Lattice._hyg.494 : Fin n) => Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) _x) (InfHomClass.toFunLike.{0, 0, 0} (OrderEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin n) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))))) (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (Lattice.toInf.{0} (Fin n) (DistribLattice.toLattice.{0} (Fin n) (instDistribLattice.{0} (Fin n) (Fin.instLinearOrderFin n)))) (Lattice.toInf.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (Fin.instLatticeFinHAddNatInstHAddInstAddNatOfNat n)) (LatticeHomClass.toInfHomClass.{0, 0, 0} (OrderEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin n) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))))) (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (DistribLattice.toLattice.{0} (Fin n) (instDistribLattice.{0} (Fin n) (Fin.instLinearOrderFin n))) (Fin.instLatticeFinHAddNatInstHAddInstAddNatOfNat n) (OrderHomClass.toLatticeHomClass.{0, 0, 0} (OrderEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin n) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))))) (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (Fin.instLinearOrderFin n) (Fin.instLatticeFinHAddNatInstHAddInstAddNatOfNat n) (RelEmbedding.instRelHomClassRelEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.680 : Fin n) (x._@.Mathlib.Order.Hom.Basic._hyg.682 : Fin n) => LE.le.{0} (Fin n) (instLEFin n) x._@.Mathlib.Order.Hom.Basic._hyg.680 x._@.Mathlib.Order.Hom.Basic._hyg.682) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.695 : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (x._@.Mathlib.Order.Hom.Basic._hyg.697 : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) => LE.le.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) x._@.Mathlib.Order.Hom.Basic._hyg.695 x._@.Mathlib.Order.Hom.Basic._hyg.697))))) (Fin.succAbove n i) j)) (fun (a : Fin n) => _inst_5 (FunLike.coe.{1, 1, 1} (OrderEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin n) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))))) (Fin n) (fun (_x : Fin n) => (fun (x._@.Mathlib.Order.Hom.Lattice._hyg.494 : Fin n) => Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) _x) (InfHomClass.toFunLike.{0, 0, 0} (OrderEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin n) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))))) (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (Lattice.toInf.{0} (Fin n) (DistribLattice.toLattice.{0} (Fin n) (instDistribLattice.{0} (Fin n) (Fin.instLinearOrderFin n)))) (Lattice.toInf.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (Fin.instLatticeFinHAddNatInstHAddInstAddNatOfNat n)) (LatticeHomClass.toInfHomClass.{0, 0, 0} (OrderEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin n) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))))) (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (DistribLattice.toLattice.{0} (Fin n) (instDistribLattice.{0} (Fin n) (Fin.instLinearOrderFin n))) (Fin.instLatticeFinHAddNatInstHAddInstAddNatOfNat n) (OrderHomClass.toLatticeHomClass.{0, 0, 0} (OrderEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin n) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))))) (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (Fin.instLinearOrderFin n) (Fin.instLatticeFinHAddNatInstHAddInstAddNatOfNat n) (RelEmbedding.instRelHomClassRelEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.680 : Fin n) (x._@.Mathlib.Order.Hom.Basic._hyg.682 : Fin n) => LE.le.{0} (Fin n) (instLEFin n) x._@.Mathlib.Order.Hom.Basic._hyg.680 x._@.Mathlib.Order.Hom.Basic._hyg.682) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.695 : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (x._@.Mathlib.Order.Hom.Basic._hyg.697 : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) => LE.le.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) x._@.Mathlib.Order.Hom.Basic._hyg.695 x._@.Mathlib.Order.Hom.Basic._hyg.697))))) (Fin.succAbove n i) a))) g s a) -> (ContinuousWithinAt.{u1, u2} α (forall (j : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))), π j) _inst_1 (Pi.topologicalSpace.{0, u2} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (fun (j : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) => π j) (fun (a : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) => _inst_5 a)) (fun (a : α) => Fin.insertNth.{u2} n π i (f a) (g a)) s a))
 Case conversion may be inaccurate. Consider using '#align continuous_within_at.fin_insert_nth ContinuousWithinAt.fin_insertNthₓ'. -/
 theorem ContinuousWithinAt.fin_insertNth {n} {π : Fin (n + 1) → Type _}
     [∀ i, TopologicalSpace (π i)] (i : Fin (n + 1)) {f : α → π i} {a : α} {s : Set α}
@@ -1111,7 +1111,7 @@ theorem ContinuousWithinAt.fin_insertNth {n} {π : Fin (n + 1) → Type _}
 lean 3 declaration is
   forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] {n : Nat} {π : (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) -> Type.{u2}} [_inst_5 : forall (i : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))), TopologicalSpace.{u2} (π i)] (i : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) {f : α -> (π i)} {s : Set.{u1} α}, (ContinuousOn.{u1, u2} α (π i) _inst_1 (_inst_5 i) f s) -> (forall {g : α -> (forall (j : Fin n), π (coeFn.{1, 1} (OrderEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (Fin.hasLe n) (Preorder.toLE.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (PartialOrder.toPreorder.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (Fin.partialOrder (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne)))))))) (fun (_x : RelEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (LE.le.{0} (Fin n) (Fin.hasLe n)) (LE.le.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (Preorder.toLE.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (PartialOrder.toPreorder.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (Fin.partialOrder (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))))))) => (Fin n) -> (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne)))))) (RelEmbedding.hasCoeToFun.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (LE.le.{0} (Fin n) (Fin.hasLe n)) (LE.le.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (Preorder.toLE.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat 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(HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne)))))))) (fun (_x : RelEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (LE.le.{0} (Fin n) (Fin.hasLe n)) (LE.le.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (Preorder.toLE.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (PartialOrder.toPreorder.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (Fin.partialOrder (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} 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(OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))))))) (Fin.succAbove n i) j)) _inst_1 (Pi.topologicalSpace.{0, u2} (Fin n) (fun (j : Fin n) => π (coeFn.{1, 1} (OrderEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (Fin.hasLe n) (Preorder.toLE.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (PartialOrder.toPreorder.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (Fin.partialOrder (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne)))))))) (fun (_x : RelEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n 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 but is expected to have type
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1)))) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) x._@.Mathlib.Order.Hom.Basic._hyg.695 x._@.Mathlib.Order.Hom.Basic._hyg.697))) (Fin.succAbove n i) j)) (fun (a : Fin n) => _inst_5 (FunLike.coe.{1, 1, 1} (OrderEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin n) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))))) (Fin n) (fun (_x : Fin n) => (fun (x._@.Mathlib.Order.RelIso.Basic._hyg.867 : Fin n) => Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) _x) (RelHomClass.toFunLike.{0, 0, 0} (OrderEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin n) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))))) (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.680 : Fin n) (x._@.Mathlib.Order.Hom.Basic._hyg.682 : Fin n) => LE.le.{0} (Fin n) (instLEFin n) x._@.Mathlib.Order.Hom.Basic._hyg.680 x._@.Mathlib.Order.Hom.Basic._hyg.682) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.695 : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (x._@.Mathlib.Order.Hom.Basic._hyg.697 : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) => LE.le.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) x._@.Mathlib.Order.Hom.Basic._hyg.695 x._@.Mathlib.Order.Hom.Basic._hyg.697) (RelEmbedding.instRelHomClassRelEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.680 : Fin n) (x._@.Mathlib.Order.Hom.Basic._hyg.682 : Fin n) => LE.le.{0} (Fin n) (instLEFin n) x._@.Mathlib.Order.Hom.Basic._hyg.680 x._@.Mathlib.Order.Hom.Basic._hyg.682) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.695 : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (x._@.Mathlib.Order.Hom.Basic._hyg.697 : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) => LE.le.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) x._@.Mathlib.Order.Hom.Basic._hyg.695 x._@.Mathlib.Order.Hom.Basic._hyg.697))) (Fin.succAbove n i) a))) g s) -> (ContinuousOn.{u1, u2} α (forall (j : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))), π j) _inst_1 (Pi.topologicalSpace.{0, u2} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (fun (j : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) => π j) (fun (a : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) => _inst_5 a)) (fun (a : α) => Fin.insertNth.{u2} n π i (f a) (g a)) s))
+  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] {n : Nat} {π : (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) -> Type.{u2}} [_inst_5 : forall (i : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))), TopologicalSpace.{u2} (π i)] (i : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) {f : α -> (π i)} {s : Set.{u1} α}, (ContinuousOn.{u1, u2} α (π i) _inst_1 (_inst_5 i) f s) -> (forall {g : α -> (forall (j : Fin n), π (FunLike.coe.{1, 1, 1} (OrderEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin n) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))))) (Fin n) (fun (_x : Fin n) => (fun (x._@.Mathlib.Order.Hom.Lattice._hyg.494 : 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(RelEmbedding.instRelHomClassRelEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.680 : Fin n) (x._@.Mathlib.Order.Hom.Basic._hyg.682 : Fin n) => LE.le.{0} (Fin n) (instLEFin n) x._@.Mathlib.Order.Hom.Basic._hyg.680 x._@.Mathlib.Order.Hom.Basic._hyg.682) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.695 : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (x._@.Mathlib.Order.Hom.Basic._hyg.697 : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) => LE.le.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) x._@.Mathlib.Order.Hom.Basic._hyg.695 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(OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (Lattice.toInf.{0} (Fin n) (DistribLattice.toLattice.{0} (Fin n) (instDistribLattice.{0} (Fin n) (Fin.instLinearOrderFin n)))) (Lattice.toInf.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (Fin.instLatticeFinHAddNatInstHAddInstAddNatOfNat n)) (LatticeHomClass.toInfHomClass.{0, 0, 0} (OrderEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin n) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))))) (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (DistribLattice.toLattice.{0} (Fin n) (instDistribLattice.{0} (Fin n) (Fin.instLinearOrderFin n))) (Fin.instLatticeFinHAddNatInstHAddInstAddNatOfNat n) (OrderHomClass.toLatticeHomClass.{0, 0, 0} (OrderEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin n) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))))) (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (Fin.instLinearOrderFin n) (Fin.instLatticeFinHAddNatInstHAddInstAddNatOfNat n) (RelEmbedding.instRelHomClassRelEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.680 : Fin n) (x._@.Mathlib.Order.Hom.Basic._hyg.682 : Fin n) => LE.le.{0} (Fin n) (instLEFin n) x._@.Mathlib.Order.Hom.Basic._hyg.680 x._@.Mathlib.Order.Hom.Basic._hyg.682) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.695 : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (x._@.Mathlib.Order.Hom.Basic._hyg.697 : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) => LE.le.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) x._@.Mathlib.Order.Hom.Basic._hyg.695 x._@.Mathlib.Order.Hom.Basic._hyg.697))))) (Fin.succAbove n i) a))) g s) -> (ContinuousOn.{u1, u2} α (forall (j : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))), π j) _inst_1 (Pi.topologicalSpace.{0, u2} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (fun (j : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) => π j) (fun (a : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) => _inst_5 a)) (fun (a : α) => Fin.insertNth.{u2} n π i (f a) (g a)) s))
 Case conversion may be inaccurate. Consider using '#align continuous_on.fin_insert_nth ContinuousOn.fin_insertNthₓ'. -/
 theorem ContinuousOn.fin_insertNth {n} {π : Fin (n + 1) → Type _} [∀ i, TopologicalSpace (π i)]
     (i : Fin (n + 1)) {f : α → π i} {s : Set α} (hf : ContinuousOn f s)

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

@@ -1098,7 +1098,7 @@ theorem continuousOn_pi {ι : Type _} {π : ι → Type _} [∀ i, TopologicalSp
 lean 3 declaration is
   forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] {n : Nat} {π : (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) -> Type.{u2}} [_inst_5 : forall (i : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))), TopologicalSpace.{u2} (π i)] (i : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) {f : α -> (π i)} {a : α} {s : Set.{u1} α}, (ContinuousWithinAt.{u1, u2} α (π i) _inst_1 (_inst_5 i) f s a) -> (forall {g : α -> (forall (j : Fin n), π (coeFn.{1, 1} (OrderEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (Fin.hasLe n) (Preorder.toLE.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (PartialOrder.toPreorder.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (Fin.partialOrder (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne)))))))) (fun (_x : RelEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (LE.le.{0} (Fin n) (Fin.hasLe n)) (LE.le.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (Preorder.toLE.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (PartialOrder.toPreorder.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (Fin.partialOrder (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))))))) => (Fin n) -> (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne)))))) (RelEmbedding.hasCoeToFun.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (LE.le.{0} (Fin n) (Fin.hasLe n)) (LE.le.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (Preorder.toLE.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (PartialOrder.toPreorder.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (Fin.partialOrder (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))))))) (Fin.succAbove n i) j))}, (ContinuousWithinAt.{u1, u2} α (forall (j : Fin n), π (coeFn.{1, 1} (OrderEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (Fin.hasLe n) (Preorder.toLE.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (PartialOrder.toPreorder.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (Fin.partialOrder (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne)))))))) (fun (_x : RelEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (LE.le.{0} (Fin n) (Fin.hasLe n)) (LE.le.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (Preorder.toLE.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (PartialOrder.toPreorder.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (Fin.partialOrder (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))))))) => (Fin n) -> (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne)))))) (RelEmbedding.hasCoeToFun.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (LE.le.{0} (Fin n) (Fin.hasLe n)) (LE.le.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (Preorder.toLE.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (PartialOrder.toPreorder.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (Fin.partialOrder (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))))))) (Fin.succAbove n i) j)) _inst_1 (Pi.topologicalSpace.{0, u2} (Fin n) (fun (j : Fin n) => π (coeFn.{1, 1} (OrderEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (Fin.hasLe n) (Preorder.toLE.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (PartialOrder.toPreorder.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (Fin.partialOrder (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne)))))))) (fun (_x : RelEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (LE.le.{0} (Fin n) (Fin.hasLe n)) (LE.le.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (Preorder.toLE.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (PartialOrder.toPreorder.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (Fin.partialOrder (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))))))) => (Fin n) -> (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne)))))) (RelEmbedding.hasCoeToFun.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (LE.le.{0} (Fin n) (Fin.hasLe n)) (LE.le.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (Preorder.toLE.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (PartialOrder.toPreorder.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (Fin.partialOrder (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))))))) (Fin.succAbove n i) j)) (fun (a : Fin n) => _inst_5 (coeFn.{1, 1} (OrderEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (Fin.hasLe n) (Preorder.toLE.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (PartialOrder.toPreorder.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (Fin.partialOrder (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne)))))))) (fun (_x : RelEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (LE.le.{0} (Fin n) (Fin.hasLe n)) (LE.le.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (Preorder.toLE.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (PartialOrder.toPreorder.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (Fin.partialOrder (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))))))) => (Fin n) -> (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne)))))) (RelEmbedding.hasCoeToFun.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (LE.le.{0} (Fin n) (Fin.hasLe n)) (LE.le.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (Preorder.toLE.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (PartialOrder.toPreorder.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (Fin.partialOrder (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))))))) (Fin.succAbove n i) a))) g s a) -> (ContinuousWithinAt.{u1, u2} α (forall (j : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))), π j) _inst_1 (Pi.topologicalSpace.{0, u2} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (fun (j : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) => π j) (fun (a : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) => _inst_5 a)) (fun (a : α) => Fin.insertNth.{u2} n (fun (i : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) => π i) i (f a) (g a)) s a))
 but is expected to have type
-  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] {n : Nat} {π : (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) -> Type.{u2}} [_inst_5 : forall (i : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))), TopologicalSpace.{u2} (π i)] (i : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) {f : α -> (π i)} {a : α} {s : Set.{u1} α}, (ContinuousWithinAt.{u1, u2} α (π i) _inst_1 (_inst_5 i) f s a) -> (forall {g : α -> (forall (j : Fin n), π (FunLike.coe.{1, 1, 1} (Function.Embedding.{1, 1} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))))) (Fin n) (fun (_x : Fin n) => (fun (x._@.Mathlib.Data.FunLike.Embedding._hyg.19 : Fin n) => Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) _x) (EmbeddingLike.toFunLike.{1, 1, 1} (Function.Embedding.{1, 1} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))))) (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (Function.instEmbeddingLikeEmbedding.{1, 1} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))))) (RelEmbedding.toEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.680 : Fin n) (x._@.Mathlib.Order.Hom.Basic._hyg.682 : Fin n) => LE.le.{0} (Fin n) (instLEFin n) x._@.Mathlib.Order.Hom.Basic._hyg.680 x._@.Mathlib.Order.Hom.Basic._hyg.682) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.695 : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (x._@.Mathlib.Order.Hom.Basic._hyg.697 : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) => LE.le.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) x._@.Mathlib.Order.Hom.Basic._hyg.695 x._@.Mathlib.Order.Hom.Basic._hyg.697) (Fin.succAbove n i)) j))}, (ContinuousWithinAt.{u1, u2} α (forall (j : Fin n), π (FunLike.coe.{1, 1, 1} (Function.Embedding.{1, 1} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))))) (Fin n) (fun (_x : Fin n) => (fun (x._@.Mathlib.Data.FunLike.Embedding._hyg.19 : Fin n) => Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) _x) (EmbeddingLike.toFunLike.{1, 1, 1} (Function.Embedding.{1, 1} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))))) (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (Function.instEmbeddingLikeEmbedding.{1, 1} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))))) (RelEmbedding.toEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.680 : Fin n) (x._@.Mathlib.Order.Hom.Basic._hyg.682 : Fin n) => LE.le.{0} (Fin n) (instLEFin n) x._@.Mathlib.Order.Hom.Basic._hyg.680 x._@.Mathlib.Order.Hom.Basic._hyg.682) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.695 : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (x._@.Mathlib.Order.Hom.Basic._hyg.697 : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) => LE.le.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) x._@.Mathlib.Order.Hom.Basic._hyg.695 x._@.Mathlib.Order.Hom.Basic._hyg.697) (Fin.succAbove n i)) j)) _inst_1 (Pi.topologicalSpace.{0, u2} (Fin n) (fun (j : Fin n) => π (FunLike.coe.{1, 1, 1} (Function.Embedding.{1, 1} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))))) (Fin n) (fun (_x : Fin n) => (fun (x._@.Mathlib.Data.FunLike.Embedding._hyg.19 : Fin n) => Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) _x) (EmbeddingLike.toFunLike.{1, 1, 1} (Function.Embedding.{1, 1} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))))) (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (Function.instEmbeddingLikeEmbedding.{1, 1} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))))) (RelEmbedding.toEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.680 : Fin n) (x._@.Mathlib.Order.Hom.Basic._hyg.682 : Fin n) => LE.le.{0} (Fin n) (instLEFin n) x._@.Mathlib.Order.Hom.Basic._hyg.680 x._@.Mathlib.Order.Hom.Basic._hyg.682) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.695 : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (x._@.Mathlib.Order.Hom.Basic._hyg.697 : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) => LE.le.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) x._@.Mathlib.Order.Hom.Basic._hyg.695 x._@.Mathlib.Order.Hom.Basic._hyg.697) (Fin.succAbove n i)) j)) (fun (a : Fin n) => _inst_5 (FunLike.coe.{1, 1, 1} (Function.Embedding.{1, 1} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))))) (Fin n) (fun (_x : Fin n) => (fun (x._@.Mathlib.Data.FunLike.Embedding._hyg.19 : Fin n) => Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) _x) (EmbeddingLike.toFunLike.{1, 1, 1} (Function.Embedding.{1, 1} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))))) (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (Function.instEmbeddingLikeEmbedding.{1, 1} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))))) (RelEmbedding.toEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.680 : Fin n) (x._@.Mathlib.Order.Hom.Basic._hyg.682 : Fin n) => LE.le.{0} (Fin n) (instLEFin n) x._@.Mathlib.Order.Hom.Basic._hyg.680 x._@.Mathlib.Order.Hom.Basic._hyg.682) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.695 : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (x._@.Mathlib.Order.Hom.Basic._hyg.697 : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) => LE.le.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) x._@.Mathlib.Order.Hom.Basic._hyg.695 x._@.Mathlib.Order.Hom.Basic._hyg.697) (Fin.succAbove n i)) a))) g s a) -> (ContinuousWithinAt.{u1, u2} α (forall (j : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))), π j) _inst_1 (Pi.topologicalSpace.{0, u2} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (fun (j : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) => π j) (fun (a : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) => _inst_5 a)) (fun (a : α) => Fin.insertNth.{u2} n π i (f a) (g a)) s a))
+  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] {n : Nat} {π : (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) -> Type.{u2}} [_inst_5 : forall (i : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))), TopologicalSpace.{u2} (π i)] (i : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) {f : α -> (π i)} {a : α} {s : Set.{u1} α}, (ContinuousWithinAt.{u1, u2} α (π i) _inst_1 (_inst_5 i) f s a) -> (forall {g : α -> (forall (j : Fin n), π (FunLike.coe.{1, 1, 1} (OrderEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin n) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))))) (Fin n) (fun (_x : Fin n) => (fun (x._@.Mathlib.Order.RelIso.Basic._hyg.867 : Fin n) => Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) _x) (RelHomClass.toFunLike.{0, 0, 0} (OrderEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin n) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))))) (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.680 : Fin n) (x._@.Mathlib.Order.Hom.Basic._hyg.682 : Fin n) => LE.le.{0} (Fin n) (instLEFin n) x._@.Mathlib.Order.Hom.Basic._hyg.680 x._@.Mathlib.Order.Hom.Basic._hyg.682) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.695 : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (x._@.Mathlib.Order.Hom.Basic._hyg.697 : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) => LE.le.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) x._@.Mathlib.Order.Hom.Basic._hyg.695 x._@.Mathlib.Order.Hom.Basic._hyg.697) (RelEmbedding.instRelHomClassRelEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.680 : Fin n) (x._@.Mathlib.Order.Hom.Basic._hyg.682 : Fin n) => LE.le.{0} (Fin n) (instLEFin n) x._@.Mathlib.Order.Hom.Basic._hyg.680 x._@.Mathlib.Order.Hom.Basic._hyg.682) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.695 : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (x._@.Mathlib.Order.Hom.Basic._hyg.697 : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) => LE.le.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) x._@.Mathlib.Order.Hom.Basic._hyg.695 x._@.Mathlib.Order.Hom.Basic._hyg.697))) (Fin.succAbove n i) j))}, (ContinuousWithinAt.{u1, u2} α (forall (j : Fin n), π (FunLike.coe.{1, 1, 1} (OrderEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin n) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))))) (Fin n) (fun (_x : Fin n) => (fun (x._@.Mathlib.Order.RelIso.Basic._hyg.867 : Fin n) => Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) _x) (RelHomClass.toFunLike.{0, 0, 0} (OrderEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin n) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))))) (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.680 : Fin n) (x._@.Mathlib.Order.Hom.Basic._hyg.682 : Fin n) => LE.le.{0} (Fin n) (instLEFin n) x._@.Mathlib.Order.Hom.Basic._hyg.680 x._@.Mathlib.Order.Hom.Basic._hyg.682) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.695 : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (x._@.Mathlib.Order.Hom.Basic._hyg.697 : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) => LE.le.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) x._@.Mathlib.Order.Hom.Basic._hyg.695 x._@.Mathlib.Order.Hom.Basic._hyg.697) (RelEmbedding.instRelHomClassRelEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.680 : Fin n) (x._@.Mathlib.Order.Hom.Basic._hyg.682 : Fin n) => LE.le.{0} (Fin n) (instLEFin n) x._@.Mathlib.Order.Hom.Basic._hyg.680 x._@.Mathlib.Order.Hom.Basic._hyg.682) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.695 : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (x._@.Mathlib.Order.Hom.Basic._hyg.697 : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) => LE.le.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) x._@.Mathlib.Order.Hom.Basic._hyg.695 x._@.Mathlib.Order.Hom.Basic._hyg.697))) (Fin.succAbove n i) j)) _inst_1 (Pi.topologicalSpace.{0, u2} (Fin n) (fun (j : Fin n) => π (FunLike.coe.{1, 1, 1} (OrderEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin n) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))))) (Fin n) (fun (_x : Fin n) => (fun (x._@.Mathlib.Order.RelIso.Basic._hyg.867 : Fin n) => Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) _x) (RelHomClass.toFunLike.{0, 0, 0} (OrderEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin n) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))))) (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.680 : Fin n) (x._@.Mathlib.Order.Hom.Basic._hyg.682 : Fin n) => LE.le.{0} (Fin n) (instLEFin n) x._@.Mathlib.Order.Hom.Basic._hyg.680 x._@.Mathlib.Order.Hom.Basic._hyg.682) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.695 : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (x._@.Mathlib.Order.Hom.Basic._hyg.697 : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) => LE.le.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) x._@.Mathlib.Order.Hom.Basic._hyg.695 x._@.Mathlib.Order.Hom.Basic._hyg.697) (RelEmbedding.instRelHomClassRelEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.680 : Fin n) (x._@.Mathlib.Order.Hom.Basic._hyg.682 : Fin n) => LE.le.{0} (Fin n) (instLEFin n) x._@.Mathlib.Order.Hom.Basic._hyg.680 x._@.Mathlib.Order.Hom.Basic._hyg.682) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.695 : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (x._@.Mathlib.Order.Hom.Basic._hyg.697 : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) => LE.le.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) x._@.Mathlib.Order.Hom.Basic._hyg.695 x._@.Mathlib.Order.Hom.Basic._hyg.697))) (Fin.succAbove n i) j)) (fun (a : Fin n) => _inst_5 (FunLike.coe.{1, 1, 1} (OrderEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin n) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))))) (Fin n) (fun (_x : Fin n) => (fun (x._@.Mathlib.Order.RelIso.Basic._hyg.867 : Fin n) => Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) _x) (RelHomClass.toFunLike.{0, 0, 0} (OrderEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin n) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))))) (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.680 : Fin n) (x._@.Mathlib.Order.Hom.Basic._hyg.682 : Fin n) => LE.le.{0} (Fin n) (instLEFin n) x._@.Mathlib.Order.Hom.Basic._hyg.680 x._@.Mathlib.Order.Hom.Basic._hyg.682) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.695 : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (x._@.Mathlib.Order.Hom.Basic._hyg.697 : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) => LE.le.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) x._@.Mathlib.Order.Hom.Basic._hyg.695 x._@.Mathlib.Order.Hom.Basic._hyg.697) (RelEmbedding.instRelHomClassRelEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.680 : Fin n) (x._@.Mathlib.Order.Hom.Basic._hyg.682 : Fin n) => LE.le.{0} (Fin n) (instLEFin n) x._@.Mathlib.Order.Hom.Basic._hyg.680 x._@.Mathlib.Order.Hom.Basic._hyg.682) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.695 : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (x._@.Mathlib.Order.Hom.Basic._hyg.697 : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) => LE.le.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) x._@.Mathlib.Order.Hom.Basic._hyg.695 x._@.Mathlib.Order.Hom.Basic._hyg.697))) (Fin.succAbove n i) a))) g s a) -> (ContinuousWithinAt.{u1, u2} α (forall (j : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))), π j) _inst_1 (Pi.topologicalSpace.{0, u2} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (fun (j : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) => π j) (fun (a : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) => _inst_5 a)) (fun (a : α) => Fin.insertNth.{u2} n π i (f a) (g a)) s a))
 Case conversion may be inaccurate. Consider using '#align continuous_within_at.fin_insert_nth ContinuousWithinAt.fin_insertNthₓ'. -/
 theorem ContinuousWithinAt.fin_insertNth {n} {π : Fin (n + 1) → Type _}
     [∀ i, TopologicalSpace (π i)] (i : Fin (n + 1)) {f : α → π i} {a : α} {s : Set α}
@@ -1111,7 +1111,7 @@ theorem ContinuousWithinAt.fin_insertNth {n} {π : Fin (n + 1) → Type _}
 lean 3 declaration is
   forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] {n : Nat} {π : (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) -> Type.{u2}} [_inst_5 : forall (i : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))), TopologicalSpace.{u2} (π i)] (i : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) {f : α -> (π i)} {s : Set.{u1} α}, (ContinuousOn.{u1, u2} α (π i) _inst_1 (_inst_5 i) f s) -> (forall {g : α -> (forall (j : Fin n), π (coeFn.{1, 1} (OrderEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (Fin.hasLe n) (Preorder.toLE.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat 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(HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne)))))))) (fun (_x : RelEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (LE.le.{0} (Fin n) (Fin.hasLe n)) (LE.le.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (Preorder.toLE.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (PartialOrder.toPreorder.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (Fin.partialOrder (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} 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Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (PartialOrder.toPreorder.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (Fin.partialOrder (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))))))) => (Fin n) -> (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne)))))) (RelEmbedding.hasCoeToFun.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (LE.le.{0} (Fin n) (Fin.hasLe n)) (LE.le.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (Preorder.toLE.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (PartialOrder.toPreorder.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (Fin.partialOrder (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))))))) (Fin.succAbove n i) a))) g s) -> (ContinuousOn.{u1, u2} α (forall (j : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))), π j) _inst_1 (Pi.topologicalSpace.{0, u2} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (fun (j : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) => π j) (fun (a : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) => _inst_5 a)) (fun (a : α) => Fin.insertNth.{u2} n (fun (i : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) => π i) i (f a) (g a)) s))
 but is expected to have type
-  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] {n : Nat} {π : (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) -> Type.{u2}} [_inst_5 : forall (i : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))), TopologicalSpace.{u2} (π i)] (i : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) {f : α -> (π i)} {s : Set.{u1} α}, (ContinuousOn.{u1, u2} α (π i) _inst_1 (_inst_5 i) f s) -> (forall {g : α -> (forall (j : Fin n), π (FunLike.coe.{1, 1, 1} (Function.Embedding.{1, 1} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))))) (Fin n) (fun (_x : Fin n) => (fun (x._@.Mathlib.Data.FunLike.Embedding._hyg.19 : Fin n) => Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) _x) (EmbeddingLike.toFunLike.{1, 1, 1} (Function.Embedding.{1, 1} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))))) (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (Function.instEmbeddingLikeEmbedding.{1, 1} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))))) (RelEmbedding.toEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.680 : Fin n) (x._@.Mathlib.Order.Hom.Basic._hyg.682 : Fin n) => LE.le.{0} (Fin n) (instLEFin n) x._@.Mathlib.Order.Hom.Basic._hyg.680 x._@.Mathlib.Order.Hom.Basic._hyg.682) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.695 : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (x._@.Mathlib.Order.Hom.Basic._hyg.697 : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) => LE.le.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) x._@.Mathlib.Order.Hom.Basic._hyg.695 x._@.Mathlib.Order.Hom.Basic._hyg.697) (Fin.succAbove n i)) j))}, (ContinuousOn.{u1, u2} α (forall (j : Fin n), π (FunLike.coe.{1, 1, 1} (Function.Embedding.{1, 1} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))))) (Fin n) (fun (_x : Fin n) => (fun (x._@.Mathlib.Data.FunLike.Embedding._hyg.19 : Fin n) => Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) _x) (EmbeddingLike.toFunLike.{1, 1, 1} (Function.Embedding.{1, 1} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))))) (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (Function.instEmbeddingLikeEmbedding.{1, 1} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))))) (RelEmbedding.toEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.680 : Fin n) (x._@.Mathlib.Order.Hom.Basic._hyg.682 : Fin n) => LE.le.{0} (Fin n) (instLEFin n) x._@.Mathlib.Order.Hom.Basic._hyg.680 x._@.Mathlib.Order.Hom.Basic._hyg.682) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.695 : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (x._@.Mathlib.Order.Hom.Basic._hyg.697 : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) => LE.le.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) x._@.Mathlib.Order.Hom.Basic._hyg.695 x._@.Mathlib.Order.Hom.Basic._hyg.697) (Fin.succAbove n i)) j)) _inst_1 (Pi.topologicalSpace.{0, u2} (Fin n) (fun (j : Fin n) => π (FunLike.coe.{1, 1, 1} (Function.Embedding.{1, 1} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))))) (Fin n) (fun (_x : Fin n) => (fun (x._@.Mathlib.Data.FunLike.Embedding._hyg.19 : Fin n) => Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) _x) (EmbeddingLike.toFunLike.{1, 1, 1} (Function.Embedding.{1, 1} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))))) (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (Function.instEmbeddingLikeEmbedding.{1, 1} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))))) (RelEmbedding.toEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.680 : Fin n) (x._@.Mathlib.Order.Hom.Basic._hyg.682 : Fin n) => LE.le.{0} (Fin n) (instLEFin n) x._@.Mathlib.Order.Hom.Basic._hyg.680 x._@.Mathlib.Order.Hom.Basic._hyg.682) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.695 : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (x._@.Mathlib.Order.Hom.Basic._hyg.697 : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) => LE.le.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) x._@.Mathlib.Order.Hom.Basic._hyg.695 x._@.Mathlib.Order.Hom.Basic._hyg.697) (Fin.succAbove n i)) j)) (fun (a : Fin n) => _inst_5 (FunLike.coe.{1, 1, 1} (Function.Embedding.{1, 1} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))))) (Fin n) (fun (_x : Fin n) => (fun (x._@.Mathlib.Data.FunLike.Embedding._hyg.19 : Fin n) => Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) _x) (EmbeddingLike.toFunLike.{1, 1, 1} (Function.Embedding.{1, 1} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))))) (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (Function.instEmbeddingLikeEmbedding.{1, 1} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))))) (RelEmbedding.toEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.680 : Fin n) (x._@.Mathlib.Order.Hom.Basic._hyg.682 : Fin n) => LE.le.{0} (Fin n) (instLEFin n) x._@.Mathlib.Order.Hom.Basic._hyg.680 x._@.Mathlib.Order.Hom.Basic._hyg.682) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.695 : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (x._@.Mathlib.Order.Hom.Basic._hyg.697 : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) => LE.le.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) x._@.Mathlib.Order.Hom.Basic._hyg.695 x._@.Mathlib.Order.Hom.Basic._hyg.697) (Fin.succAbove n i)) a))) g s) -> (ContinuousOn.{u1, u2} α (forall (j : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))), π j) _inst_1 (Pi.topologicalSpace.{0, u2} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (fun (j : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) => π j) (fun (a : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) => _inst_5 a)) (fun (a : α) => Fin.insertNth.{u2} n π i (f a) (g a)) s))
+  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] {n : Nat} {π : (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) -> Type.{u2}} [_inst_5 : forall (i : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))), TopologicalSpace.{u2} (π i)] (i : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) {f : α -> (π i)} {s : Set.{u1} α}, (ContinuousOn.{u1, u2} α (π i) _inst_1 (_inst_5 i) f s) -> (forall {g : α -> (forall (j : Fin n), π (FunLike.coe.{1, 1, 1} (OrderEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin n) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))))) (Fin n) (fun (_x : Fin n) => (fun (x._@.Mathlib.Order.RelIso.Basic._hyg.867 : Fin n) => Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) _x) (RelHomClass.toFunLike.{0, 0, 0} (OrderEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin n) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))))) (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.680 : Fin n) (x._@.Mathlib.Order.Hom.Basic._hyg.682 : Fin n) => LE.le.{0} (Fin n) (instLEFin n) x._@.Mathlib.Order.Hom.Basic._hyg.680 x._@.Mathlib.Order.Hom.Basic._hyg.682) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.695 : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (x._@.Mathlib.Order.Hom.Basic._hyg.697 : Fin (HAdd.hAdd.{0, 0, 0} Nat 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(HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) => LE.le.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) x._@.Mathlib.Order.Hom.Basic._hyg.695 x._@.Mathlib.Order.Hom.Basic._hyg.697))) (Fin.succAbove n i) j))}, (ContinuousOn.{u1, u2} α (forall (j : Fin n), π (FunLike.coe.{1, 1, 1} (OrderEmbedding.{0, 0} (Fin n) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin n) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))))) (Fin n) (fun (_x : Fin n) => (fun (x._@.Mathlib.Order.RelIso.Basic._hyg.867 : Fin n) => Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 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(instOfNatNat 1)))) x._@.Mathlib.Order.Hom.Basic._hyg.695 x._@.Mathlib.Order.Hom.Basic._hyg.697))) (Fin.succAbove n i) a))) g s) -> (ContinuousOn.{u1, u2} α (forall (j : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))), π j) _inst_1 (Pi.topologicalSpace.{0, u2} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (fun (j : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) => π j) (fun (a : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) => _inst_5 a)) (fun (a : α) => Fin.insertNth.{u2} n π i (f a) (g a)) s))
 Case conversion may be inaccurate. Consider using '#align continuous_on.fin_insert_nth ContinuousOn.fin_insertNthₓ'. -/
 theorem ContinuousOn.fin_insertNth {n} {π : Fin (n + 1) → Type _} [∀ i, TopologicalSpace (π i)]
     (i : Fin (n + 1)) {f : α → π i} {s : Set α} (hf : ContinuousOn f s)

mathlib commit https://github.com/leanprover-community/mathlib/commit/0148d455199ed64bf8eb2f493a1e7eb9211ce170

@@ -448,16 +448,34 @@ theorem nhdsWithin_union (a : α) (s t : Set α) : 𝓝[s ∪ t] a = 𝓝[s] a 
   rw [← inf_sup_left, sup_principal]
 #align nhds_within_union nhdsWithin_union
 
-theorem nhdsWithin_bUnion {ι} {I : Set ι} (hI : I.Finite) (s : ι → Set α) (a : α) :
+/- warning: nhds_within_bUnion -> nhdsWithin_bunionᵢ is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] {ι : Type.{u2}} {I : Set.{u2} ι}, (Set.Finite.{u2} ι I) -> (forall (s : ι -> (Set.{u1} α)) (a : α), Eq.{succ u1} (Filter.{u1} α) (nhdsWithin.{u1} α _inst_1 a (Set.unionᵢ.{u1, succ u2} α ι (fun (i : ι) => Set.unionᵢ.{u1, 0} α (Membership.Mem.{u2, u2} ι (Set.{u2} ι) (Set.hasMem.{u2} ι) i I) (fun (H : Membership.Mem.{u2, u2} ι (Set.{u2} ι) (Set.hasMem.{u2} ι) i I) => s i)))) (supᵢ.{u1, succ u2} (Filter.{u1} α) (ConditionallyCompleteLattice.toHasSup.{u1} (Filter.{u1} α) (CompleteLattice.toConditionallyCompleteLattice.{u1} (Filter.{u1} α) (Filter.completeLattice.{u1} α))) ι (fun (i : ι) => supᵢ.{u1, 0} (Filter.{u1} α) (ConditionallyCompleteLattice.toHasSup.{u1} (Filter.{u1} α) (CompleteLattice.toConditionallyCompleteLattice.{u1} (Filter.{u1} α) (Filter.completeLattice.{u1} α))) (Membership.Mem.{u2, u2} ι (Set.{u2} ι) (Set.hasMem.{u2} ι) i I) (fun (H : Membership.Mem.{u2, u2} ι (Set.{u2} ι) (Set.hasMem.{u2} ι) i I) => nhdsWithin.{u1} α _inst_1 a (s i)))))
+but is expected to have type
+  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] {ι : Type.{u2}} {I : Set.{u2} ι}, (Set.Finite.{u2} ι I) -> (forall (s : ι -> (Set.{u1} α)) (a : α), Eq.{succ u1} (Filter.{u1} α) (nhdsWithin.{u1} α _inst_1 a (Set.unionᵢ.{u1, succ u2} α ι (fun (i : ι) => Set.unionᵢ.{u1, 0} α (Membership.mem.{u2, u2} ι (Set.{u2} ι) (Set.instMembershipSet.{u2} ι) i I) (fun (H : Membership.mem.{u2, u2} ι (Set.{u2} ι) (Set.instMembershipSet.{u2} ι) i I) => s i)))) (supᵢ.{u1, succ u2} (Filter.{u1} α) (ConditionallyCompleteLattice.toSupSet.{u1} (Filter.{u1} α) (CompleteLattice.toConditionallyCompleteLattice.{u1} (Filter.{u1} α) (Filter.instCompleteLatticeFilter.{u1} α))) ι (fun (i : ι) => supᵢ.{u1, 0} (Filter.{u1} α) (ConditionallyCompleteLattice.toSupSet.{u1} (Filter.{u1} α) (CompleteLattice.toConditionallyCompleteLattice.{u1} (Filter.{u1} α) (Filter.instCompleteLatticeFilter.{u1} α))) (Membership.mem.{u2, u2} ι (Set.{u2} ι) (Set.instMembershipSet.{u2} ι) i I) (fun (H : Membership.mem.{u2, u2} ι (Set.{u2} ι) (Set.instMembershipSet.{u2} ι) i I) => nhdsWithin.{u1} α _inst_1 a (s i)))))
+Case conversion may be inaccurate. Consider using '#align nhds_within_bUnion nhdsWithin_bunionᵢₓ'. -/
+theorem nhdsWithin_bunionᵢ {ι} {I : Set ι} (hI : I.Finite) (s : ι → Set α) (a : α) :
     𝓝[⋃ i ∈ I, s i] a = ⨆ i ∈ I, 𝓝[s i] a :=
   Set.Finite.induction_on hI (by simp) fun t T _ _ hT => by
     simp only [hT, nhdsWithin_union, supᵢ_insert, bUnion_insert]
-#align nhds_within_bUnion nhdsWithin_bUnion
+#align nhds_within_bUnion nhdsWithin_bunionᵢ
 
+/- warning: nhds_within_sUnion -> nhdsWithin_unionₛ is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] {S : Set.{u1} (Set.{u1} α)}, (Set.Finite.{u1} (Set.{u1} α) S) -> (forall (a : α), Eq.{succ u1} (Filter.{u1} α) (nhdsWithin.{u1} α _inst_1 a (Set.unionₛ.{u1} α S)) (supᵢ.{u1, succ u1} (Filter.{u1} α) (ConditionallyCompleteLattice.toHasSup.{u1} (Filter.{u1} α) (CompleteLattice.toConditionallyCompleteLattice.{u1} (Filter.{u1} α) (Filter.completeLattice.{u1} α))) (Set.{u1} α) (fun (s : Set.{u1} α) => supᵢ.{u1, 0} (Filter.{u1} α) (ConditionallyCompleteLattice.toHasSup.{u1} (Filter.{u1} α) (CompleteLattice.toConditionallyCompleteLattice.{u1} (Filter.{u1} α) (Filter.completeLattice.{u1} α))) (Membership.Mem.{u1, u1} (Set.{u1} α) (Set.{u1} (Set.{u1} α)) (Set.hasMem.{u1} (Set.{u1} α)) s S) (fun (H : Membership.Mem.{u1, u1} (Set.{u1} α) (Set.{u1} (Set.{u1} α)) (Set.hasMem.{u1} (Set.{u1} α)) s S) => nhdsWithin.{u1} α _inst_1 a s))))
+but is expected to have type
+  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] {S : Set.{u1} (Set.{u1} α)}, (Set.Finite.{u1} (Set.{u1} α) S) -> (forall (a : α), Eq.{succ u1} (Filter.{u1} α) (nhdsWithin.{u1} α _inst_1 a (Set.unionₛ.{u1} α S)) (supᵢ.{u1, succ u1} (Filter.{u1} α) (ConditionallyCompleteLattice.toSupSet.{u1} (Filter.{u1} α) (CompleteLattice.toConditionallyCompleteLattice.{u1} (Filter.{u1} α) (Filter.instCompleteLatticeFilter.{u1} α))) (Set.{u1} α) (fun (s : Set.{u1} α) => supᵢ.{u1, 0} (Filter.{u1} α) (ConditionallyCompleteLattice.toSupSet.{u1} (Filter.{u1} α) (CompleteLattice.toConditionallyCompleteLattice.{u1} (Filter.{u1} α) (Filter.instCompleteLatticeFilter.{u1} α))) (Membership.mem.{u1, u1} (Set.{u1} α) (Set.{u1} (Set.{u1} α)) (Set.instMembershipSet.{u1} (Set.{u1} α)) s S) (fun (H : Membership.mem.{u1, u1} (Set.{u1} α) (Set.{u1} (Set.{u1} α)) (Set.instMembershipSet.{u1} (Set.{u1} α)) s S) => nhdsWithin.{u1} α _inst_1 a s))))
+Case conversion may be inaccurate. Consider using '#align nhds_within_sUnion nhdsWithin_unionₛₓ'. -/
 theorem nhdsWithin_unionₛ {S : Set (Set α)} (hS : S.Finite) (a : α) : 𝓝[⋃₀ S] a = ⨆ s ∈ S, 𝓝[s] a :=
-  by rw [sUnion_eq_bUnion, nhdsWithin_bUnion hS]
+  by rw [sUnion_eq_bUnion, nhdsWithin_bunionᵢ hS]
 #align nhds_within_sUnion nhdsWithin_unionₛ
 
+/- warning: nhds_within_Union -> nhdsWithin_unionᵢ is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] {ι : Sort.{u2}} [_inst_2 : Finite.{u2} ι] (s : ι -> (Set.{u1} α)) (a : α), Eq.{succ u1} (Filter.{u1} α) (nhdsWithin.{u1} α _inst_1 a (Set.unionᵢ.{u1, u2} α ι (fun (i : ι) => s i))) (supᵢ.{u1, u2} (Filter.{u1} α) (ConditionallyCompleteLattice.toHasSup.{u1} (Filter.{u1} α) (CompleteLattice.toConditionallyCompleteLattice.{u1} (Filter.{u1} α) (Filter.completeLattice.{u1} α))) ι (fun (i : ι) => nhdsWithin.{u1} α _inst_1 a (s i)))
+but is expected to have type
+  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] {ι : Sort.{u2}} [_inst_2 : Finite.{u2} ι] (s : ι -> (Set.{u1} α)) (a : α), Eq.{succ u1} (Filter.{u1} α) (nhdsWithin.{u1} α _inst_1 a (Set.unionᵢ.{u1, u2} α ι (fun (i : ι) => s i))) (supᵢ.{u1, u2} (Filter.{u1} α) (ConditionallyCompleteLattice.toSupSet.{u1} (Filter.{u1} α) (CompleteLattice.toConditionallyCompleteLattice.{u1} (Filter.{u1} α) (Filter.instCompleteLatticeFilter.{u1} α))) ι (fun (i : ι) => nhdsWithin.{u1} α _inst_1 a (s i)))
+Case conversion may be inaccurate. Consider using '#align nhds_within_Union nhdsWithin_unionᵢₓ'. -/
 theorem nhdsWithin_unionᵢ {ι} [Finite ι] (s : ι → Set α) (a : α) : 𝓝[⋃ i, s i] a = ⨆ i, 𝓝[s i] a :=
   by rw [← sUnion_range, nhdsWithin_unionₛ (finite_range s), supᵢ_range]
 #align nhds_within_Union nhdsWithin_unionᵢ
@@ -1212,6 +1230,12 @@ theorem ContinuousOn.prod_map {f : α → γ} {g : β → δ} {s : Set α} {t :
   fun ⟨x, y⟩ ⟨hx, hy⟩ => ContinuousWithinAt.prod_map (hf x hx) (hg y hy)
 #align continuous_on.prod_map ContinuousOn.prod_map
 
+/- warning: continuous_of_cover_nhds -> continuous_of_cover_nhds is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : TopologicalSpace.{u1} α] [_inst_2 : TopologicalSpace.{u2} β] {ι : Sort.{u3}} {f : α -> β} {s : ι -> (Set.{u1} α)}, (forall (x : α), Exists.{u3} ι (fun (i : ι) => Membership.Mem.{u1, u1} (Set.{u1} α) (Filter.{u1} α) (Filter.hasMem.{u1} α) (s i) (nhds.{u1} α _inst_1 x))) -> (forall (i : ι), ContinuousOn.{u1, u2} α β _inst_1 _inst_2 f (s i)) -> (Continuous.{u1, u2} α β _inst_1 _inst_2 f)
+but is expected to have type
+  forall {α : Type.{u2}} {β : Type.{u1}} [_inst_1 : TopologicalSpace.{u2} α] [_inst_2 : TopologicalSpace.{u1} β] {ι : Sort.{u3}} {f : α -> β} {s : ι -> (Set.{u2} α)}, (forall (x : α), Exists.{u3} ι (fun (i : ι) => Membership.mem.{u2, u2} (Set.{u2} α) (Filter.{u2} α) (instMembershipSetFilter.{u2} α) (s i) (nhds.{u2} α _inst_1 x))) -> (forall (i : ι), ContinuousOn.{u2, u1} α β _inst_1 _inst_2 f (s i)) -> (Continuous.{u2, u1} α β _inst_1 _inst_2 f)
+Case conversion may be inaccurate. Consider using '#align continuous_of_cover_nhds continuous_of_cover_nhdsₓ'. -/
 theorem continuous_of_cover_nhds {ι : Sort _} {f : α → β} {s : ι → Set α}
     (hs : ∀ x : α, ∃ i, s i ∈ 𝓝 x) (hf : ∀ i, ContinuousOn f (s i)) : Continuous f :=
   continuous_iff_continuousAt.mpr fun x =>

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

@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Sébastien Gouëzel
 
 ! This file was ported from Lean 3 source module topology.continuous_on
-! leanprover-community/mathlib commit e46da4e335b8671848ac711ccb34b42538c0d800
+! leanprover-community/mathlib commit 55d771df074d0dd020139ee1cd4b95521422df9f
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
@@ -402,9 +402,16 @@ theorem nhdsWithin_eq_nhdsWithin {a : α} {s t u : Set α} (h₀ : a ∈ s) (h
   rw [nhdsWithin_restrict t h₀ h₁, nhdsWithin_restrict u h₀ h₁, h₂]
 #align nhds_within_eq_nhds_within nhdsWithin_eq_nhdsWithin
 
+#print nhdsWithin_eq_nhds /-
+@[simp]
+theorem nhdsWithin_eq_nhds {a : α} {s : Set α} : 𝓝[s] a = 𝓝 a ↔ s ∈ 𝓝 a := by
+  rw [nhdsWithin, inf_eq_left, le_principal_iff]
+#align nhds_within_eq_nhds nhdsWithin_eq_nhds
+-/
+
 #print IsOpen.nhdsWithin_eq /-
 theorem IsOpen.nhdsWithin_eq {a : α} {s : Set α} (h : IsOpen s) (ha : a ∈ s) : 𝓝[s] a = 𝓝 a :=
-  inf_eq_left.2 <| le_principal_iff.2 <| IsOpen.mem_nhds h ha
+  nhdsWithin_eq_nhds.2 <| IsOpen.mem_nhds h ha
 #align is_open.nhds_within_eq IsOpen.nhdsWithin_eq
 -/
 
@@ -441,6 +448,20 @@ theorem nhdsWithin_union (a : α) (s t : Set α) : 𝓝[s ∪ t] a = 𝓝[s] a 
   rw [← inf_sup_left, sup_principal]
 #align nhds_within_union nhdsWithin_union
 
+theorem nhdsWithin_bUnion {ι} {I : Set ι} (hI : I.Finite) (s : ι → Set α) (a : α) :
+    𝓝[⋃ i ∈ I, s i] a = ⨆ i ∈ I, 𝓝[s i] a :=
+  Set.Finite.induction_on hI (by simp) fun t T _ _ hT => by
+    simp only [hT, nhdsWithin_union, supᵢ_insert, bUnion_insert]
+#align nhds_within_bUnion nhdsWithin_bUnion
+
+theorem nhdsWithin_unionₛ {S : Set (Set α)} (hS : S.Finite) (a : α) : 𝓝[⋃₀ S] a = ⨆ s ∈ S, 𝓝[s] a :=
+  by rw [sUnion_eq_bUnion, nhdsWithin_bUnion hS]
+#align nhds_within_sUnion nhdsWithin_unionₛ
+
+theorem nhdsWithin_unionᵢ {ι} [Finite ι] (s : ι → Set α) (a : α) : 𝓝[⋃ i, s i] a = ⨆ i, 𝓝[s i] a :=
+  by rw [← sUnion_range, nhdsWithin_unionₛ (finite_range s), supᵢ_range]
+#align nhds_within_Union nhdsWithin_unionᵢ
+
 /- warning: nhds_within_inter -> nhdsWithin_inter is a dubious translation:
 lean 3 declaration is
   forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] (a : α) (s : Set.{u1} α) (t : Set.{u1} α), Eq.{succ u1} (Filter.{u1} α) (nhdsWithin.{u1} α _inst_1 a (Inter.inter.{u1} (Set.{u1} α) (Set.hasInter.{u1} α) s t)) (Inf.inf.{u1} (Filter.{u1} α) (Filter.hasInf.{u1} α) (nhdsWithin.{u1} α _inst_1 a s) (nhdsWithin.{u1} α _inst_1 a t))
@@ -477,6 +498,16 @@ theorem nhdsWithin_inter_of_mem {a : α} {s t : Set α} (h : s ∈ 𝓝[t] a) :
   exact nhdsWithin_le_of_mem h
 #align nhds_within_inter_of_mem nhdsWithin_inter_of_mem
 
+/- warning: nhds_within_inter_of_mem' -> nhdsWithin_inter_of_mem' is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] {a : α} {s : Set.{u1} α} {t : Set.{u1} α}, (Membership.Mem.{u1, u1} (Set.{u1} α) (Filter.{u1} α) (Filter.hasMem.{u1} α) s (nhdsWithin.{u1} α _inst_1 a t)) -> (Eq.{succ u1} (Filter.{u1} α) (nhdsWithin.{u1} α _inst_1 a (Inter.inter.{u1} (Set.{u1} α) (Set.hasInter.{u1} α) t s)) (nhdsWithin.{u1} α _inst_1 a t))
+but is expected to have type
+  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] {a : α} {s : Set.{u1} α} {t : Set.{u1} α}, (Membership.mem.{u1, u1} (Set.{u1} α) (Filter.{u1} α) (instMembershipSetFilter.{u1} α) t (nhdsWithin.{u1} α _inst_1 a s)) -> (Eq.{succ u1} (Filter.{u1} α) (nhdsWithin.{u1} α _inst_1 a (Inter.inter.{u1} (Set.{u1} α) (Set.instInterSet.{u1} α) s t)) (nhdsWithin.{u1} α _inst_1 a s))
+Case conversion may be inaccurate. Consider using '#align nhds_within_inter_of_mem' nhdsWithin_inter_of_mem'ₓ'. -/
+theorem nhdsWithin_inter_of_mem' {a : α} {s t : Set α} (h : s ∈ 𝓝[t] a) : 𝓝[t ∩ s] a = 𝓝[t] a := by
+  rw [inter_comm, nhdsWithin_inter_of_mem h]
+#align nhds_within_inter_of_mem' nhdsWithin_inter_of_mem'
+
 #print nhdsWithin_singleton /-
 @[simp]
 theorem nhdsWithin_singleton (a : α) : 𝓝[{a}] a = pure a := by
@@ -1181,6 +1212,15 @@ theorem ContinuousOn.prod_map {f : α → γ} {g : β → δ} {s : Set α} {t :
   fun ⟨x, y⟩ ⟨hx, hy⟩ => ContinuousWithinAt.prod_map (hf x hx) (hg y hy)
 #align continuous_on.prod_map ContinuousOn.prod_map
 
+theorem continuous_of_cover_nhds {ι : Sort _} {f : α → β} {s : ι → Set α}
+    (hs : ∀ x : α, ∃ i, s i ∈ 𝓝 x) (hf : ∀ i, ContinuousOn f (s i)) : Continuous f :=
+  continuous_iff_continuousAt.mpr fun x =>
+    by
+    let ⟨i, hi⟩ := hs x
+    rw [ContinuousAt, ← nhdsWithin_eq_nhds.2 hi]
+    exact hf _ _ (mem_of_mem_nhds hi)
+#align continuous_of_cover_nhds continuous_of_cover_nhds
+
 /- warning: continuous_on_empty -> continuousOn_empty is a dubious translation:
 lean 3 declaration is
   forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : TopologicalSpace.{u1} α] [_inst_2 : TopologicalSpace.{u2} β] (f : α -> β), ContinuousOn.{u1, u2} α β _inst_1 _inst_2 f (EmptyCollection.emptyCollection.{u1} (Set.{u1} α) (Set.hasEmptyc.{u1} α))

mathlib commit https://github.com/leanprover-community/mathlib/commit/ce7e9d53d4bbc38065db3b595cd5bd73c323bc1d

@@ -1895,7 +1895,7 @@ Case conversion may be inaccurate. Consider using '#align continuous_on.is_open_
 theorem ContinuousOn.isOpen_preimage {f : α → β} {s : Set α} {t : Set β} (h : ContinuousOn f s)
     (hs : IsOpen s) (hp : f ⁻¹' t ⊆ s) (ht : IsOpen t) : IsOpen (f ⁻¹' t) :=
   by
-  convert (continuousOn_open_iff hs).mp h t ht
+  convert(continuousOn_open_iff hs).mp h t ht
   rw [inter_comm, inter_eq_self_of_subset_left hp]
 #align continuous_on.is_open_preimage ContinuousOn.isOpen_preimage
 

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

@@ -577,7 +577,7 @@ lean 3 declaration is
 but is expected to have type
   forall {ι : Type.{u2}} {α : ι -> Type.{u1}} [_inst_2 : forall (i : ι), TopologicalSpace.{u1} (α i)] {I : Set.{u2} ι}, (Set.Finite.{u2} ι I) -> (forall (s : forall (i : ι), Set.{u1} (α i)) (x : forall (i : ι), α i), Eq.{max (succ u2) (succ u1)} (Filter.{max u2 u1} (forall (i : ι), α i)) (nhdsWithin.{max u2 u1} (forall (i : ι), α i) (Pi.topologicalSpace.{u2, u1} ι (fun (i : ι) => α i) (fun (a : ι) => _inst_2 a)) x (Set.pi.{u2, u1} ι (fun (i : ι) => α i) I s)) (Inf.inf.{max u2 u1} (Filter.{max u2 u1} (forall (i : ι), α i)) (Filter.instInfFilter.{max u2 u1} (forall (i : ι), α i)) (infᵢ.{max u2 u1, succ u2} (Filter.{max u2 u1} (forall (i : ι), α i)) (ConditionallyCompleteLattice.toInfSet.{max u2 u1} (Filter.{max u2 u1} (forall (i : ι), α i)) (CompleteLattice.toConditionallyCompleteLattice.{max u2 u1} (Filter.{max u2 u1} (forall (i : ι), α i)) (Filter.instCompleteLatticeFilter.{max u2 u1} (forall (i : ι), α i)))) ι (fun (i : ι) => infᵢ.{max u2 u1, 0} (Filter.{max u2 u1} (forall (i : ι), α i)) (ConditionallyCompleteLattice.toInfSet.{max u2 u1} (Filter.{max u2 u1} (forall (i : ι), α i)) (CompleteLattice.toConditionallyCompleteLattice.{max u2 u1} (Filter.{max u2 u1} (forall (i : ι), α i)) (Filter.instCompleteLatticeFilter.{max u2 u1} (forall (i : ι), α i)))) (Membership.mem.{u2, u2} ι (Set.{u2} ι) (Set.instMembershipSet.{u2} ι) i I) (fun (H : Membership.mem.{u2, u2} ι (Set.{u2} ι) (Set.instMembershipSet.{u2} ι) i I) => Filter.comap.{max u2 u1, u1} (forall (i : ι), α i) (α i) (fun (x : forall (i : ι), α i) => x i) (nhdsWithin.{u1} (α i) (_inst_2 i) (x i) (s i))))) (infᵢ.{max u2 u1, succ u2} (Filter.{max u2 u1} (forall (i : ι), α i)) (ConditionallyCompleteLattice.toInfSet.{max u2 u1} (Filter.{max u2 u1} (forall (i : ι), α i)) (CompleteLattice.toConditionallyCompleteLattice.{max u2 u1} (Filter.{max u2 u1} (forall (i : ι), α i)) (Filter.instCompleteLatticeFilter.{max u2 u1} (forall (i : ι), α i)))) ι (fun (i : ι) => infᵢ.{max u2 u1, 0} (Filter.{max u2 u1} (forall (i : ι), α i)) (ConditionallyCompleteLattice.toInfSet.{max u2 u1} (Filter.{max u2 u1} (forall (i : ι), α i)) (CompleteLattice.toConditionallyCompleteLattice.{max u2 u1} (Filter.{max u2 u1} (forall (i : ι), α i)) (Filter.instCompleteLatticeFilter.{max u2 u1} (forall (i : ι), α i)))) (Not (Membership.mem.{u2, u2} ι (Set.{u2} ι) (Set.instMembershipSet.{u2} ι) i I)) (fun (H : Not (Membership.mem.{u2, u2} ι (Set.{u2} ι) (Set.instMembershipSet.{u2} ι) i I)) => Filter.comap.{max u2 u1, u1} (forall (i : ι), α i) (α i) (fun (x : forall (i : ι), α i) => x i) (nhds.{u1} (α i) (_inst_2 i) (x i)))))))
 Case conversion may be inaccurate. Consider using '#align nhds_within_pi_eq nhdsWithin_pi_eqₓ'. -/
-/- ./././Mathport/Syntax/Translate/Basic.lean:628:2: warning: expanding binder collection (i «expr ∉ » I) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (i «expr ∉ » I) -/
 theorem nhdsWithin_pi_eq {ι : Type _} {α : ι → Type _} [∀ i, TopologicalSpace (α i)] {I : Set ι}
     (hI : I.Finite) (s : ∀ i, Set (α i)) (x : ∀ i, α i) :
     𝓝[pi I s] x =

mathlib commit https://github.com/leanprover-community/mathlib/commit/9da1b3534b65d9661eb8f42443598a92bbb49211

@@ -431,9 +431,9 @@ theorem nhdsWithin_empty (a : α) : 𝓝[∅] a = ⊥ := by rw [nhdsWithin, prin
 
 /- warning: nhds_within_union -> nhdsWithin_union is a dubious translation:
 lean 3 declaration is
-  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] (a : α) (s : Set.{u1} α) (t : Set.{u1} α), Eq.{succ u1} (Filter.{u1} α) (nhdsWithin.{u1} α _inst_1 a (Union.union.{u1} (Set.{u1} α) (Set.hasUnion.{u1} α) s t)) (HasSup.sup.{u1} (Filter.{u1} α) (SemilatticeSup.toHasSup.{u1} (Filter.{u1} α) (Lattice.toSemilatticeSup.{u1} (Filter.{u1} α) (ConditionallyCompleteLattice.toLattice.{u1} (Filter.{u1} α) (CompleteLattice.toConditionallyCompleteLattice.{u1} (Filter.{u1} α) (Filter.completeLattice.{u1} α))))) (nhdsWithin.{u1} α _inst_1 a s) (nhdsWithin.{u1} α _inst_1 a t))
+  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] (a : α) (s : Set.{u1} α) (t : Set.{u1} α), Eq.{succ u1} (Filter.{u1} α) (nhdsWithin.{u1} α _inst_1 a (Union.union.{u1} (Set.{u1} α) (Set.hasUnion.{u1} α) s t)) (Sup.sup.{u1} (Filter.{u1} α) (SemilatticeSup.toHasSup.{u1} (Filter.{u1} α) (Lattice.toSemilatticeSup.{u1} (Filter.{u1} α) (ConditionallyCompleteLattice.toLattice.{u1} (Filter.{u1} α) (CompleteLattice.toConditionallyCompleteLattice.{u1} (Filter.{u1} α) (Filter.completeLattice.{u1} α))))) (nhdsWithin.{u1} α _inst_1 a s) (nhdsWithin.{u1} α _inst_1 a t))
 but is expected to have type
-  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] (a : α) (s : Set.{u1} α) (t : Set.{u1} α), Eq.{succ u1} (Filter.{u1} α) (nhdsWithin.{u1} α _inst_1 a (Union.union.{u1} (Set.{u1} α) (Set.instUnionSet.{u1} α) s t)) (HasSup.sup.{u1} (Filter.{u1} α) (SemilatticeSup.toHasSup.{u1} (Filter.{u1} α) (Lattice.toSemilatticeSup.{u1} (Filter.{u1} α) (ConditionallyCompleteLattice.toLattice.{u1} (Filter.{u1} α) (CompleteLattice.toConditionallyCompleteLattice.{u1} (Filter.{u1} α) (Filter.instCompleteLatticeFilter.{u1} α))))) (nhdsWithin.{u1} α _inst_1 a s) (nhdsWithin.{u1} α _inst_1 a t))
+  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] (a : α) (s : Set.{u1} α) (t : Set.{u1} α), Eq.{succ u1} (Filter.{u1} α) (nhdsWithin.{u1} α _inst_1 a (Union.union.{u1} (Set.{u1} α) (Set.instUnionSet.{u1} α) s t)) (Sup.sup.{u1} (Filter.{u1} α) (SemilatticeSup.toSup.{u1} (Filter.{u1} α) (Lattice.toSemilatticeSup.{u1} (Filter.{u1} α) (ConditionallyCompleteLattice.toLattice.{u1} (Filter.{u1} α) (CompleteLattice.toConditionallyCompleteLattice.{u1} (Filter.{u1} α) (Filter.instCompleteLatticeFilter.{u1} α))))) (nhdsWithin.{u1} α _inst_1 a s) (nhdsWithin.{u1} α _inst_1 a t))
 Case conversion may be inaccurate. Consider using '#align nhds_within_union nhdsWithin_unionₓ'. -/
 theorem nhdsWithin_union (a : α) (s t : Set α) : 𝓝[s ∪ t] a = 𝓝[s] a ⊔ 𝓝[t] a :=
   by
@@ -443,9 +443,9 @@ theorem nhdsWithin_union (a : α) (s t : Set α) : 𝓝[s ∪ t] a = 𝓝[s] a 
 
 /- warning: nhds_within_inter -> nhdsWithin_inter is a dubious translation:
 lean 3 declaration is
-  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] (a : α) (s : Set.{u1} α) (t : Set.{u1} α), Eq.{succ u1} (Filter.{u1} α) (nhdsWithin.{u1} α _inst_1 a (Inter.inter.{u1} (Set.{u1} α) (Set.hasInter.{u1} α) s t)) (HasInf.inf.{u1} (Filter.{u1} α) (Filter.hasInf.{u1} α) (nhdsWithin.{u1} α _inst_1 a s) (nhdsWithin.{u1} α _inst_1 a t))
+  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] (a : α) (s : Set.{u1} α) (t : Set.{u1} α), Eq.{succ u1} (Filter.{u1} α) (nhdsWithin.{u1} α _inst_1 a (Inter.inter.{u1} (Set.{u1} α) (Set.hasInter.{u1} α) s t)) (Inf.inf.{u1} (Filter.{u1} α) (Filter.hasInf.{u1} α) (nhdsWithin.{u1} α _inst_1 a s) (nhdsWithin.{u1} α _inst_1 a t))
 but is expected to have type
-  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] (a : α) (s : Set.{u1} α) (t : Set.{u1} α), Eq.{succ u1} (Filter.{u1} α) (nhdsWithin.{u1} α _inst_1 a (Inter.inter.{u1} (Set.{u1} α) (Set.instInterSet.{u1} α) s t)) (HasInf.inf.{u1} (Filter.{u1} α) (Filter.instHasInfFilter.{u1} α) (nhdsWithin.{u1} α _inst_1 a s) (nhdsWithin.{u1} α _inst_1 a t))
+  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] (a : α) (s : Set.{u1} α) (t : Set.{u1} α), Eq.{succ u1} (Filter.{u1} α) (nhdsWithin.{u1} α _inst_1 a (Inter.inter.{u1} (Set.{u1} α) (Set.instInterSet.{u1} α) s t)) (Inf.inf.{u1} (Filter.{u1} α) (Filter.instInfFilter.{u1} α) (nhdsWithin.{u1} α _inst_1 a s) (nhdsWithin.{u1} α _inst_1 a t))
 Case conversion may be inaccurate. Consider using '#align nhds_within_inter nhdsWithin_interₓ'. -/
 theorem nhdsWithin_inter (a : α) (s t : Set α) : 𝓝[s ∩ t] a = 𝓝[s] a ⊓ 𝓝[t] a :=
   by
@@ -455,9 +455,9 @@ theorem nhdsWithin_inter (a : α) (s t : Set α) : 𝓝[s ∩ t] a = 𝓝[s] a 
 
 /- warning: nhds_within_inter' -> nhdsWithin_inter' is a dubious translation:
 lean 3 declaration is
-  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] (a : α) (s : Set.{u1} α) (t : Set.{u1} α), Eq.{succ u1} (Filter.{u1} α) (nhdsWithin.{u1} α _inst_1 a (Inter.inter.{u1} (Set.{u1} α) (Set.hasInter.{u1} α) s t)) (HasInf.inf.{u1} (Filter.{u1} α) (Filter.hasInf.{u1} α) (nhdsWithin.{u1} α _inst_1 a s) (Filter.principal.{u1} α t))
+  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] (a : α) (s : Set.{u1} α) (t : Set.{u1} α), Eq.{succ u1} (Filter.{u1} α) (nhdsWithin.{u1} α _inst_1 a (Inter.inter.{u1} (Set.{u1} α) (Set.hasInter.{u1} α) s t)) (Inf.inf.{u1} (Filter.{u1} α) (Filter.hasInf.{u1} α) (nhdsWithin.{u1} α _inst_1 a s) (Filter.principal.{u1} α t))
 but is expected to have type
-  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] (a : α) (s : Set.{u1} α) (t : Set.{u1} α), Eq.{succ u1} (Filter.{u1} α) (nhdsWithin.{u1} α _inst_1 a (Inter.inter.{u1} (Set.{u1} α) (Set.instInterSet.{u1} α) s t)) (HasInf.inf.{u1} (Filter.{u1} α) (Filter.instHasInfFilter.{u1} α) (nhdsWithin.{u1} α _inst_1 a s) (Filter.principal.{u1} α t))
+  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] (a : α) (s : Set.{u1} α) (t : Set.{u1} α), Eq.{succ u1} (Filter.{u1} α) (nhdsWithin.{u1} α _inst_1 a (Inter.inter.{u1} (Set.{u1} α) (Set.instInterSet.{u1} α) s t)) (Inf.inf.{u1} (Filter.{u1} α) (Filter.instInfFilter.{u1} α) (nhdsWithin.{u1} α _inst_1 a s) (Filter.principal.{u1} α t))
 Case conversion may be inaccurate. Consider using '#align nhds_within_inter' nhdsWithin_inter'ₓ'. -/
 theorem nhdsWithin_inter' (a : α) (s t : Set α) : 𝓝[s ∩ t] a = 𝓝[s] a ⊓ 𝓟 t :=
   by
@@ -486,9 +486,9 @@ theorem nhdsWithin_singleton (a : α) : 𝓝[{a}] a = pure a := by
 
 /- warning: nhds_within_insert -> nhdsWithin_insert is a dubious translation:
 lean 3 declaration is
-  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] (a : α) (s : Set.{u1} α), Eq.{succ u1} (Filter.{u1} α) (nhdsWithin.{u1} α _inst_1 a (Insert.insert.{u1, u1} α (Set.{u1} α) (Set.hasInsert.{u1} α) a s)) (HasSup.sup.{u1} (Filter.{u1} α) (SemilatticeSup.toHasSup.{u1} (Filter.{u1} α) (Lattice.toSemilatticeSup.{u1} (Filter.{u1} α) (ConditionallyCompleteLattice.toLattice.{u1} (Filter.{u1} α) (CompleteLattice.toConditionallyCompleteLattice.{u1} (Filter.{u1} α) (Filter.completeLattice.{u1} α))))) (Pure.pure.{u1, u1} Filter.{u1} Filter.hasPure.{u1} α a) (nhdsWithin.{u1} α _inst_1 a s))
+  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] (a : α) (s : Set.{u1} α), Eq.{succ u1} (Filter.{u1} α) (nhdsWithin.{u1} α _inst_1 a (Insert.insert.{u1, u1} α (Set.{u1} α) (Set.hasInsert.{u1} α) a s)) (Sup.sup.{u1} (Filter.{u1} α) (SemilatticeSup.toHasSup.{u1} (Filter.{u1} α) (Lattice.toSemilatticeSup.{u1} (Filter.{u1} α) (ConditionallyCompleteLattice.toLattice.{u1} (Filter.{u1} α) (CompleteLattice.toConditionallyCompleteLattice.{u1} (Filter.{u1} α) (Filter.completeLattice.{u1} α))))) (Pure.pure.{u1, u1} Filter.{u1} Filter.hasPure.{u1} α a) (nhdsWithin.{u1} α _inst_1 a s))
 but is expected to have type
-  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] (a : α) (s : Set.{u1} α), Eq.{succ u1} (Filter.{u1} α) (nhdsWithin.{u1} α _inst_1 a (Insert.insert.{u1, u1} α (Set.{u1} α) (Set.instInsertSet.{u1} α) a s)) (HasSup.sup.{u1} (Filter.{u1} α) (SemilatticeSup.toHasSup.{u1} (Filter.{u1} α) (Lattice.toSemilatticeSup.{u1} (Filter.{u1} α) (ConditionallyCompleteLattice.toLattice.{u1} (Filter.{u1} α) (CompleteLattice.toConditionallyCompleteLattice.{u1} (Filter.{u1} α) (Filter.instCompleteLatticeFilter.{u1} α))))) (Pure.pure.{u1, u1} Filter.{u1} Filter.instPureFilter.{u1} α a) (nhdsWithin.{u1} α _inst_1 a s))
+  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] (a : α) (s : Set.{u1} α), Eq.{succ u1} (Filter.{u1} α) (nhdsWithin.{u1} α _inst_1 a (Insert.insert.{u1, u1} α (Set.{u1} α) (Set.instInsertSet.{u1} α) a s)) (Sup.sup.{u1} (Filter.{u1} α) (SemilatticeSup.toSup.{u1} (Filter.{u1} α) (Lattice.toSemilatticeSup.{u1} (Filter.{u1} α) (ConditionallyCompleteLattice.toLattice.{u1} (Filter.{u1} α) (CompleteLattice.toConditionallyCompleteLattice.{u1} (Filter.{u1} α) (Filter.instCompleteLatticeFilter.{u1} α))))) (Pure.pure.{u1, u1} Filter.{u1} Filter.instPureFilter.{u1} α a) (nhdsWithin.{u1} α _inst_1 a s))
 Case conversion may be inaccurate. Consider using '#align nhds_within_insert nhdsWithin_insertₓ'. -/
 @[simp]
 theorem nhdsWithin_insert (a : α) (s : Set α) : 𝓝[insert a s] a = pure a ⊔ 𝓝[s] a := by
@@ -520,9 +520,9 @@ theorem insert_mem_nhds_iff {a : α} {s : Set α} : insert a s ∈ 𝓝 a ↔ s
 
 /- warning: nhds_within_compl_singleton_sup_pure -> nhdsWithin_compl_singleton_sup_pure is a dubious translation:
 lean 3 declaration is
-  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] (a : α), Eq.{succ u1} (Filter.{u1} α) (HasSup.sup.{u1} (Filter.{u1} α) (SemilatticeSup.toHasSup.{u1} (Filter.{u1} α) (Lattice.toSemilatticeSup.{u1} (Filter.{u1} α) (ConditionallyCompleteLattice.toLattice.{u1} (Filter.{u1} α) (CompleteLattice.toConditionallyCompleteLattice.{u1} (Filter.{u1} α) (Filter.completeLattice.{u1} α))))) (nhdsWithin.{u1} α _inst_1 a (HasCompl.compl.{u1} (Set.{u1} α) (BooleanAlgebra.toHasCompl.{u1} (Set.{u1} α) (Set.booleanAlgebra.{u1} α)) (Singleton.singleton.{u1, u1} α (Set.{u1} α) (Set.hasSingleton.{u1} α) a))) (Pure.pure.{u1, u1} Filter.{u1} Filter.hasPure.{u1} α a)) (nhds.{u1} α _inst_1 a)
+  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] (a : α), Eq.{succ u1} (Filter.{u1} α) (Sup.sup.{u1} (Filter.{u1} α) (SemilatticeSup.toHasSup.{u1} (Filter.{u1} α) (Lattice.toSemilatticeSup.{u1} (Filter.{u1} α) (ConditionallyCompleteLattice.toLattice.{u1} (Filter.{u1} α) (CompleteLattice.toConditionallyCompleteLattice.{u1} (Filter.{u1} α) (Filter.completeLattice.{u1} α))))) (nhdsWithin.{u1} α _inst_1 a (HasCompl.compl.{u1} (Set.{u1} α) (BooleanAlgebra.toHasCompl.{u1} (Set.{u1} α) (Set.booleanAlgebra.{u1} α)) (Singleton.singleton.{u1, u1} α (Set.{u1} α) (Set.hasSingleton.{u1} α) a))) (Pure.pure.{u1, u1} Filter.{u1} Filter.hasPure.{u1} α a)) (nhds.{u1} α _inst_1 a)
 but is expected to have type
-  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] (a : α), Eq.{succ u1} (Filter.{u1} α) (HasSup.sup.{u1} (Filter.{u1} α) (SemilatticeSup.toHasSup.{u1} (Filter.{u1} α) (Lattice.toSemilatticeSup.{u1} (Filter.{u1} α) (ConditionallyCompleteLattice.toLattice.{u1} (Filter.{u1} α) (CompleteLattice.toConditionallyCompleteLattice.{u1} (Filter.{u1} α) (Filter.instCompleteLatticeFilter.{u1} α))))) (nhdsWithin.{u1} α _inst_1 a (HasCompl.compl.{u1} (Set.{u1} α) (BooleanAlgebra.toHasCompl.{u1} (Set.{u1} α) (Set.instBooleanAlgebraSet.{u1} α)) (Singleton.singleton.{u1, u1} α (Set.{u1} α) (Set.instSingletonSet.{u1} α) a))) (Pure.pure.{u1, u1} Filter.{u1} Filter.instPureFilter.{u1} α a)) (nhds.{u1} α _inst_1 a)
+  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] (a : α), Eq.{succ u1} (Filter.{u1} α) (Sup.sup.{u1} (Filter.{u1} α) (SemilatticeSup.toSup.{u1} (Filter.{u1} α) (Lattice.toSemilatticeSup.{u1} (Filter.{u1} α) (ConditionallyCompleteLattice.toLattice.{u1} (Filter.{u1} α) (CompleteLattice.toConditionallyCompleteLattice.{u1} (Filter.{u1} α) (Filter.instCompleteLatticeFilter.{u1} α))))) (nhdsWithin.{u1} α _inst_1 a (HasCompl.compl.{u1} (Set.{u1} α) (BooleanAlgebra.toHasCompl.{u1} (Set.{u1} α) (Set.instBooleanAlgebraSet.{u1} α)) (Singleton.singleton.{u1, u1} α (Set.{u1} α) (Set.instSingletonSet.{u1} α) a))) (Pure.pure.{u1, u1} Filter.{u1} Filter.instPureFilter.{u1} α a)) (nhds.{u1} α _inst_1 a)
 Case conversion may be inaccurate. Consider using '#align nhds_within_compl_singleton_sup_pure nhdsWithin_compl_singleton_sup_pureₓ'. -/
 @[simp]
 theorem nhdsWithin_compl_singleton_sup_pure (a : α) : 𝓝[≠] a ⊔ pure a = 𝓝 a := by
@@ -560,9 +560,9 @@ theorem nhdsWithin_prod {α : Type _} [TopologicalSpace α] {β : Type _} [Topol
 
 /- warning: nhds_within_pi_eq' -> nhdsWithin_pi_eq' is a dubious translation:
 lean 3 declaration is
-  forall {ι : Type.{u1}} {α : ι -> Type.{u2}} [_inst_2 : forall (i : ι), TopologicalSpace.{u2} (α i)] {I : Set.{u1} ι}, (Set.Finite.{u1} ι I) -> (forall (s : forall (i : ι), Set.{u2} (α i)) (x : forall (i : ι), α i), Eq.{succ (max u1 u2)} (Filter.{max u1 u2} (forall (i : ι), α i)) (nhdsWithin.{max u1 u2} (forall (i : ι), α i) (Pi.topologicalSpace.{u1, u2} ι (fun (i : ι) => α i) (fun (a : ι) => _inst_2 a)) x (Set.pi.{u1, u2} ι (fun (i : ι) => α i) I s)) (infᵢ.{max u1 u2, succ u1} (Filter.{max u1 u2} (forall (i : ι), α i)) (ConditionallyCompleteLattice.toHasInf.{max u1 u2} (Filter.{max u1 u2} (forall (i : ι), α i)) (CompleteLattice.toConditionallyCompleteLattice.{max u1 u2} (Filter.{max u1 u2} (forall (i : ι), α i)) (Filter.completeLattice.{max u1 u2} (forall (i : ι), α i)))) ι (fun (i : ι) => Filter.comap.{max u1 u2, u2} (forall (i : ι), α i) (α i) (fun (x : forall (i : ι), α i) => x i) (HasInf.inf.{u2} (Filter.{u2} (α i)) (Filter.hasInf.{u2} (α i)) (nhds.{u2} (α i) (_inst_2 i) (x i)) (infᵢ.{u2, 0} (Filter.{u2} (α i)) (ConditionallyCompleteLattice.toHasInf.{u2} (Filter.{u2} (α i)) (CompleteLattice.toConditionallyCompleteLattice.{u2} (Filter.{u2} (α i)) (Filter.completeLattice.{u2} (α i)))) (Membership.Mem.{u1, u1} ι (Set.{u1} ι) (Set.hasMem.{u1} ι) i I) (fun (hi : Membership.Mem.{u1, u1} ι (Set.{u1} ι) (Set.hasMem.{u1} ι) i I) => Filter.principal.{u2} (α i) (s i)))))))
+  forall {ι : Type.{u1}} {α : ι -> Type.{u2}} [_inst_2 : forall (i : ι), TopologicalSpace.{u2} (α i)] {I : Set.{u1} ι}, (Set.Finite.{u1} ι I) -> (forall (s : forall (i : ι), Set.{u2} (α i)) (x : forall (i : ι), α i), Eq.{succ (max u1 u2)} (Filter.{max u1 u2} (forall (i : ι), α i)) (nhdsWithin.{max u1 u2} (forall (i : ι), α i) (Pi.topologicalSpace.{u1, u2} ι (fun (i : ι) => α i) (fun (a : ι) => _inst_2 a)) x (Set.pi.{u1, u2} ι (fun (i : ι) => α i) I s)) (infᵢ.{max u1 u2, succ u1} (Filter.{max u1 u2} (forall (i : ι), α i)) (ConditionallyCompleteLattice.toHasInf.{max u1 u2} (Filter.{max u1 u2} (forall (i : ι), α i)) (CompleteLattice.toConditionallyCompleteLattice.{max u1 u2} (Filter.{max u1 u2} (forall (i : ι), α i)) (Filter.completeLattice.{max u1 u2} (forall (i : ι), α i)))) ι (fun (i : ι) => Filter.comap.{max u1 u2, u2} (forall (i : ι), α i) (α i) (fun (x : forall (i : ι), α i) => x i) (Inf.inf.{u2} (Filter.{u2} (α i)) (Filter.hasInf.{u2} (α i)) (nhds.{u2} (α i) (_inst_2 i) (x i)) (infᵢ.{u2, 0} (Filter.{u2} (α i)) (ConditionallyCompleteLattice.toHasInf.{u2} (Filter.{u2} (α i)) (CompleteLattice.toConditionallyCompleteLattice.{u2} (Filter.{u2} (α i)) (Filter.completeLattice.{u2} (α i)))) (Membership.Mem.{u1, u1} ι (Set.{u1} ι) (Set.hasMem.{u1} ι) i I) (fun (hi : Membership.Mem.{u1, u1} ι (Set.{u1} ι) (Set.hasMem.{u1} ι) i I) => Filter.principal.{u2} (α i) (s i)))))))
 but is expected to have type
-  forall {ι : Type.{u2}} {α : ι -> Type.{u1}} [_inst_2 : forall (i : ι), TopologicalSpace.{u1} (α i)] {I : Set.{u2} ι}, (Set.Finite.{u2} ι I) -> (forall (s : forall (i : ι), Set.{u1} (α i)) (x : forall (i : ι), α i), Eq.{max (succ u2) (succ u1)} (Filter.{max u2 u1} (forall (i : ι), α i)) (nhdsWithin.{max u2 u1} (forall (i : ι), α i) (Pi.topologicalSpace.{u2, u1} ι (fun (i : ι) => α i) (fun (a : ι) => _inst_2 a)) x (Set.pi.{u2, u1} ι (fun (i : ι) => α i) I s)) (infᵢ.{max u2 u1, succ u2} (Filter.{max u2 u1} (forall (i : ι), α i)) (ConditionallyCompleteLattice.toInfSet.{max u2 u1} (Filter.{max u2 u1} (forall (i : ι), α i)) (CompleteLattice.toConditionallyCompleteLattice.{max u2 u1} (Filter.{max u2 u1} (forall (i : ι), α i)) (Filter.instCompleteLatticeFilter.{max u2 u1} (forall (i : ι), α i)))) ι (fun (i : ι) => Filter.comap.{max u2 u1, u1} (forall (i : ι), α i) (α i) (fun (x : forall (i : ι), α i) => x i) (HasInf.inf.{u1} (Filter.{u1} (α i)) (Filter.instHasInfFilter.{u1} (α i)) (nhds.{u1} (α i) (_inst_2 i) (x i)) (infᵢ.{u1, 0} (Filter.{u1} (α i)) (ConditionallyCompleteLattice.toInfSet.{u1} (Filter.{u1} (α i)) (CompleteLattice.toConditionallyCompleteLattice.{u1} (Filter.{u1} (α i)) (Filter.instCompleteLatticeFilter.{u1} (α i)))) (Membership.mem.{u2, u2} ι (Set.{u2} ι) (Set.instMembershipSet.{u2} ι) i I) (fun (hi : Membership.mem.{u2, u2} ι (Set.{u2} ι) (Set.instMembershipSet.{u2} ι) i I) => Filter.principal.{u1} (α i) (s i)))))))
+  forall {ι : Type.{u2}} {α : ι -> Type.{u1}} [_inst_2 : forall (i : ι), TopologicalSpace.{u1} (α i)] {I : Set.{u2} ι}, (Set.Finite.{u2} ι I) -> (forall (s : forall (i : ι), Set.{u1} (α i)) (x : forall (i : ι), α i), Eq.{max (succ u2) (succ u1)} (Filter.{max u2 u1} (forall (i : ι), α i)) (nhdsWithin.{max u2 u1} (forall (i : ι), α i) (Pi.topologicalSpace.{u2, u1} ι (fun (i : ι) => α i) (fun (a : ι) => _inst_2 a)) x (Set.pi.{u2, u1} ι (fun (i : ι) => α i) I s)) (infᵢ.{max u2 u1, succ u2} (Filter.{max u2 u1} (forall (i : ι), α i)) (ConditionallyCompleteLattice.toInfSet.{max u2 u1} (Filter.{max u2 u1} (forall (i : ι), α i)) (CompleteLattice.toConditionallyCompleteLattice.{max u2 u1} (Filter.{max u2 u1} (forall (i : ι), α i)) (Filter.instCompleteLatticeFilter.{max u2 u1} (forall (i : ι), α i)))) ι (fun (i : ι) => Filter.comap.{max u2 u1, u1} (forall (i : ι), α i) (α i) (fun (x : forall (i : ι), α i) => x i) (Inf.inf.{u1} (Filter.{u1} (α i)) (Filter.instInfFilter.{u1} (α i)) (nhds.{u1} (α i) (_inst_2 i) (x i)) (infᵢ.{u1, 0} (Filter.{u1} (α i)) (ConditionallyCompleteLattice.toInfSet.{u1} (Filter.{u1} (α i)) (CompleteLattice.toConditionallyCompleteLattice.{u1} (Filter.{u1} (α i)) (Filter.instCompleteLatticeFilter.{u1} (α i)))) (Membership.mem.{u2, u2} ι (Set.{u2} ι) (Set.instMembershipSet.{u2} ι) i I) (fun (hi : Membership.mem.{u2, u2} ι (Set.{u2} ι) (Set.instMembershipSet.{u2} ι) i I) => Filter.principal.{u1} (α i) (s i)))))))
 Case conversion may be inaccurate. Consider using '#align nhds_within_pi_eq' nhdsWithin_pi_eq'ₓ'. -/
 theorem nhdsWithin_pi_eq' {ι : Type _} {α : ι → Type _} [∀ i, TopologicalSpace (α i)] {I : Set ι}
     (hI : I.Finite) (s : ∀ i, Set (α i)) (x : ∀ i, α i) :
@@ -573,9 +573,9 @@ theorem nhdsWithin_pi_eq' {ι : Type _} {α : ι → Type _} [∀ i, Topological
 
 /- warning: nhds_within_pi_eq -> nhdsWithin_pi_eq is a dubious translation:
 lean 3 declaration is
-  forall {ι : Type.{u1}} {α : ι -> Type.{u2}} [_inst_2 : forall (i : ι), TopologicalSpace.{u2} (α i)] {I : Set.{u1} ι}, (Set.Finite.{u1} ι I) -> (forall (s : forall (i : ι), Set.{u2} (α i)) (x : forall (i : ι), α i), Eq.{succ (max u1 u2)} (Filter.{max u1 u2} (forall (i : ι), α i)) (nhdsWithin.{max u1 u2} (forall (i : ι), α i) (Pi.topologicalSpace.{u1, u2} ι (fun (i : ι) => α i) (fun (a : ι) => _inst_2 a)) x (Set.pi.{u1, u2} ι (fun (i : ι) => α i) I s)) (HasInf.inf.{max u1 u2} (Filter.{max u1 u2} (forall (i : ι), α i)) (Filter.hasInf.{max u1 u2} (forall (i : ι), α i)) (infᵢ.{max u1 u2, succ u1} (Filter.{max u1 u2} (forall (i : ι), α i)) (ConditionallyCompleteLattice.toHasInf.{max u1 u2} (Filter.{max u1 u2} (forall (i : ι), α i)) (CompleteLattice.toConditionallyCompleteLattice.{max u1 u2} (Filter.{max u1 u2} (forall (i : ι), α i)) (Filter.completeLattice.{max u1 u2} (forall (i : ι), α i)))) ι (fun (i : ι) => infᵢ.{max u1 u2, 0} (Filter.{max u1 u2} (forall (i : ι), α i)) (ConditionallyCompleteLattice.toHasInf.{max u1 u2} (Filter.{max u1 u2} (forall (i : ι), α i)) (CompleteLattice.toConditionallyCompleteLattice.{max u1 u2} (Filter.{max u1 u2} (forall (i : ι), α i)) (Filter.completeLattice.{max u1 u2} (forall (i : ι), α i)))) (Membership.Mem.{u1, u1} ι (Set.{u1} ι) (Set.hasMem.{u1} ι) i I) (fun (H : Membership.Mem.{u1, u1} ι (Set.{u1} ι) (Set.hasMem.{u1} ι) i I) => Filter.comap.{max u1 u2, u2} (forall (i : ι), α i) (α i) (fun (x : forall (i : ι), α i) => x i) (nhdsWithin.{u2} (α i) (_inst_2 i) (x i) (s i))))) (infᵢ.{max u1 u2, succ u1} (Filter.{max u1 u2} (forall (i : ι), α i)) (ConditionallyCompleteLattice.toHasInf.{max u1 u2} (Filter.{max u1 u2} (forall (i : ι), α i)) (CompleteLattice.toConditionallyCompleteLattice.{max u1 u2} (Filter.{max u1 u2} (forall (i : ι), α i)) (Filter.completeLattice.{max u1 u2} (forall (i : ι), α i)))) ι (fun (i : ι) => infᵢ.{max u1 u2, 0} (Filter.{max u1 u2} (forall (i : ι), α i)) (ConditionallyCompleteLattice.toHasInf.{max u1 u2} (Filter.{max u1 u2} (forall (i : ι), α i)) (CompleteLattice.toConditionallyCompleteLattice.{max u1 u2} (Filter.{max u1 u2} (forall (i : ι), α i)) (Filter.completeLattice.{max u1 u2} (forall (i : ι), α i)))) (Not (Membership.Mem.{u1, u1} ι (Set.{u1} ι) (Set.hasMem.{u1} ι) i I)) (fun (H : Not (Membership.Mem.{u1, u1} ι (Set.{u1} ι) (Set.hasMem.{u1} ι) i I)) => Filter.comap.{max u1 u2, u2} (forall (i : ι), α i) (α i) (fun (x : forall (i : ι), α i) => x i) (nhds.{u2} (α i) (_inst_2 i) (x i)))))))
+  forall {ι : Type.{u1}} {α : ι -> Type.{u2}} [_inst_2 : forall (i : ι), TopologicalSpace.{u2} (α i)] {I : Set.{u1} ι}, (Set.Finite.{u1} ι I) -> (forall (s : forall (i : ι), Set.{u2} (α i)) (x : forall (i : ι), α i), Eq.{succ (max u1 u2)} (Filter.{max u1 u2} (forall (i : ι), α i)) (nhdsWithin.{max u1 u2} (forall (i : ι), α i) (Pi.topologicalSpace.{u1, u2} ι (fun (i : ι) => α i) (fun (a : ι) => _inst_2 a)) x (Set.pi.{u1, u2} ι (fun (i : ι) => α i) I s)) (Inf.inf.{max u1 u2} (Filter.{max u1 u2} (forall (i : ι), α i)) (Filter.hasInf.{max u1 u2} (forall (i : ι), α i)) (infᵢ.{max u1 u2, succ u1} (Filter.{max u1 u2} (forall (i : ι), α i)) (ConditionallyCompleteLattice.toHasInf.{max u1 u2} (Filter.{max u1 u2} (forall (i : ι), α i)) (CompleteLattice.toConditionallyCompleteLattice.{max u1 u2} (Filter.{max u1 u2} (forall (i : ι), α i)) (Filter.completeLattice.{max u1 u2} (forall (i : ι), α i)))) ι (fun (i : ι) => infᵢ.{max u1 u2, 0} (Filter.{max u1 u2} (forall (i : ι), α i)) (ConditionallyCompleteLattice.toHasInf.{max u1 u2} (Filter.{max u1 u2} (forall (i : ι), α i)) (CompleteLattice.toConditionallyCompleteLattice.{max u1 u2} (Filter.{max u1 u2} (forall (i : ι), α i)) (Filter.completeLattice.{max u1 u2} (forall (i : ι), α i)))) (Membership.Mem.{u1, u1} ι (Set.{u1} ι) (Set.hasMem.{u1} ι) i I) (fun (H : Membership.Mem.{u1, u1} ι (Set.{u1} ι) (Set.hasMem.{u1} ι) i I) => Filter.comap.{max u1 u2, u2} (forall (i : ι), α i) (α i) (fun (x : forall (i : ι), α i) => x i) (nhdsWithin.{u2} (α i) (_inst_2 i) (x i) (s i))))) (infᵢ.{max u1 u2, succ u1} (Filter.{max u1 u2} (forall (i : ι), α i)) (ConditionallyCompleteLattice.toHasInf.{max u1 u2} (Filter.{max u1 u2} (forall (i : ι), α i)) (CompleteLattice.toConditionallyCompleteLattice.{max u1 u2} (Filter.{max u1 u2} (forall (i : ι), α i)) (Filter.completeLattice.{max u1 u2} (forall (i : ι), α i)))) ι (fun (i : ι) => infᵢ.{max u1 u2, 0} (Filter.{max u1 u2} (forall (i : ι), α i)) (ConditionallyCompleteLattice.toHasInf.{max u1 u2} (Filter.{max u1 u2} (forall (i : ι), α i)) (CompleteLattice.toConditionallyCompleteLattice.{max u1 u2} (Filter.{max u1 u2} (forall (i : ι), α i)) (Filter.completeLattice.{max u1 u2} (forall (i : ι), α i)))) (Not (Membership.Mem.{u1, u1} ι (Set.{u1} ι) (Set.hasMem.{u1} ι) i I)) (fun (H : Not (Membership.Mem.{u1, u1} ι (Set.{u1} ι) (Set.hasMem.{u1} ι) i I)) => Filter.comap.{max u1 u2, u2} (forall (i : ι), α i) (α i) (fun (x : forall (i : ι), α i) => x i) (nhds.{u2} (α i) (_inst_2 i) (x i)))))))
 but is expected to have type
-  forall {ι : Type.{u2}} {α : ι -> Type.{u1}} [_inst_2 : forall (i : ι), TopologicalSpace.{u1} (α i)] {I : Set.{u2} ι}, (Set.Finite.{u2} ι I) -> (forall (s : forall (i : ι), Set.{u1} (α i)) (x : forall (i : ι), α i), Eq.{max (succ u2) (succ u1)} (Filter.{max u2 u1} (forall (i : ι), α i)) (nhdsWithin.{max u2 u1} (forall (i : ι), α i) (Pi.topologicalSpace.{u2, u1} ι (fun (i : ι) => α i) (fun (a : ι) => _inst_2 a)) x (Set.pi.{u2, u1} ι (fun (i : ι) => α i) I s)) (HasInf.inf.{max u2 u1} (Filter.{max u2 u1} (forall (i : ι), α i)) (Filter.instHasInfFilter.{max u2 u1} (forall (i : ι), α i)) (infᵢ.{max u2 u1, succ u2} (Filter.{max u2 u1} (forall (i : ι), α i)) (ConditionallyCompleteLattice.toInfSet.{max u2 u1} (Filter.{max u2 u1} (forall (i : ι), α i)) (CompleteLattice.toConditionallyCompleteLattice.{max u2 u1} (Filter.{max u2 u1} (forall (i : ι), α i)) (Filter.instCompleteLatticeFilter.{max u2 u1} (forall (i : ι), α i)))) ι (fun (i : ι) => infᵢ.{max u2 u1, 0} (Filter.{max u2 u1} (forall (i : ι), α i)) (ConditionallyCompleteLattice.toInfSet.{max u2 u1} (Filter.{max u2 u1} (forall (i : ι), α i)) (CompleteLattice.toConditionallyCompleteLattice.{max u2 u1} (Filter.{max u2 u1} (forall (i : ι), α i)) (Filter.instCompleteLatticeFilter.{max u2 u1} (forall (i : ι), α i)))) (Membership.mem.{u2, u2} ι (Set.{u2} ι) (Set.instMembershipSet.{u2} ι) i I) (fun (H : Membership.mem.{u2, u2} ι (Set.{u2} ι) (Set.instMembershipSet.{u2} ι) i I) => Filter.comap.{max u2 u1, u1} (forall (i : ι), α i) (α i) (fun (x : forall (i : ι), α i) => x i) (nhdsWithin.{u1} (α i) (_inst_2 i) (x i) (s i))))) (infᵢ.{max u2 u1, succ u2} (Filter.{max u2 u1} (forall (i : ι), α i)) (ConditionallyCompleteLattice.toInfSet.{max u2 u1} (Filter.{max u2 u1} (forall (i : ι), α i)) (CompleteLattice.toConditionallyCompleteLattice.{max u2 u1} (Filter.{max u2 u1} (forall (i : ι), α i)) (Filter.instCompleteLatticeFilter.{max u2 u1} (forall (i : ι), α i)))) ι (fun (i : ι) => infᵢ.{max u2 u1, 0} (Filter.{max u2 u1} (forall (i : ι), α i)) (ConditionallyCompleteLattice.toInfSet.{max u2 u1} (Filter.{max u2 u1} (forall (i : ι), α i)) (CompleteLattice.toConditionallyCompleteLattice.{max u2 u1} (Filter.{max u2 u1} (forall (i : ι), α i)) (Filter.instCompleteLatticeFilter.{max u2 u1} (forall (i : ι), α i)))) (Not (Membership.mem.{u2, u2} ι (Set.{u2} ι) (Set.instMembershipSet.{u2} ι) i I)) (fun (H : Not (Membership.mem.{u2, u2} ι (Set.{u2} ι) (Set.instMembershipSet.{u2} ι) i I)) => Filter.comap.{max u2 u1, u1} (forall (i : ι), α i) (α i) (fun (x : forall (i : ι), α i) => x i) (nhds.{u1} (α i) (_inst_2 i) (x i)))))))
+  forall {ι : Type.{u2}} {α : ι -> Type.{u1}} [_inst_2 : forall (i : ι), TopologicalSpace.{u1} (α i)] {I : Set.{u2} ι}, (Set.Finite.{u2} ι I) -> (forall (s : forall (i : ι), Set.{u1} (α i)) (x : forall (i : ι), α i), Eq.{max (succ u2) (succ u1)} (Filter.{max u2 u1} (forall (i : ι), α i)) (nhdsWithin.{max u2 u1} (forall (i : ι), α i) (Pi.topologicalSpace.{u2, u1} ι (fun (i : ι) => α i) (fun (a : ι) => _inst_2 a)) x (Set.pi.{u2, u1} ι (fun (i : ι) => α i) I s)) (Inf.inf.{max u2 u1} (Filter.{max u2 u1} (forall (i : ι), α i)) (Filter.instInfFilter.{max u2 u1} (forall (i : ι), α i)) (infᵢ.{max u2 u1, succ u2} (Filter.{max u2 u1} (forall (i : ι), α i)) (ConditionallyCompleteLattice.toInfSet.{max u2 u1} (Filter.{max u2 u1} (forall (i : ι), α i)) (CompleteLattice.toConditionallyCompleteLattice.{max u2 u1} (Filter.{max u2 u1} (forall (i : ι), α i)) (Filter.instCompleteLatticeFilter.{max u2 u1} (forall (i : ι), α i)))) ι (fun (i : ι) => infᵢ.{max u2 u1, 0} (Filter.{max u2 u1} (forall (i : ι), α i)) (ConditionallyCompleteLattice.toInfSet.{max u2 u1} (Filter.{max u2 u1} (forall (i : ι), α i)) (CompleteLattice.toConditionallyCompleteLattice.{max u2 u1} (Filter.{max u2 u1} (forall (i : ι), α i)) (Filter.instCompleteLatticeFilter.{max u2 u1} (forall (i : ι), α i)))) (Membership.mem.{u2, u2} ι (Set.{u2} ι) (Set.instMembershipSet.{u2} ι) i I) (fun (H : Membership.mem.{u2, u2} ι (Set.{u2} ι) (Set.instMembershipSet.{u2} ι) i I) => Filter.comap.{max u2 u1, u1} (forall (i : ι), α i) (α i) (fun (x : forall (i : ι), α i) => x i) (nhdsWithin.{u1} (α i) (_inst_2 i) (x i) (s i))))) (infᵢ.{max u2 u1, succ u2} (Filter.{max u2 u1} (forall (i : ι), α i)) (ConditionallyCompleteLattice.toInfSet.{max u2 u1} (Filter.{max u2 u1} (forall (i : ι), α i)) (CompleteLattice.toConditionallyCompleteLattice.{max u2 u1} (Filter.{max u2 u1} (forall (i : ι), α i)) (Filter.instCompleteLatticeFilter.{max u2 u1} (forall (i : ι), α i)))) ι (fun (i : ι) => infᵢ.{max u2 u1, 0} (Filter.{max u2 u1} (forall (i : ι), α i)) (ConditionallyCompleteLattice.toInfSet.{max u2 u1} (Filter.{max u2 u1} (forall (i : ι), α i)) (CompleteLattice.toConditionallyCompleteLattice.{max u2 u1} (Filter.{max u2 u1} (forall (i : ι), α i)) (Filter.instCompleteLatticeFilter.{max u2 u1} (forall (i : ι), α i)))) (Not (Membership.mem.{u2, u2} ι (Set.{u2} ι) (Set.instMembershipSet.{u2} ι) i I)) (fun (H : Not (Membership.mem.{u2, u2} ι (Set.{u2} ι) (Set.instMembershipSet.{u2} ι) i I)) => Filter.comap.{max u2 u1, u1} (forall (i : ι), α i) (α i) (fun (x : forall (i : ι), α i) => x i) (nhds.{u1} (α i) (_inst_2 i) (x i)))))))
 Case conversion may be inaccurate. Consider using '#align nhds_within_pi_eq nhdsWithin_pi_eqₓ'. -/
 /- ./././Mathport/Syntax/Translate/Basic.lean:628:2: warning: expanding binder collection (i «expr ∉ » I) -/
 theorem nhdsWithin_pi_eq {ι : Type _} {α : ι → Type _} [∀ i, TopologicalSpace (α i)] {I : Set ι}

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

Follow-up to #10816.

Remaining places containing such lemmas are

• Option.bex_ne_none and Option.ball_ne_none: defined in Lean core
• Nat.decidableBallLT and Nat.decidableBallLE: defined in Lean core
• bef_def is still used in a number of places and could be renamed
• BAll.imp_{left,right}, BEx.imp_{left,right}, BEx.intro and BEx.elim

I only audited the first ~150 lemmas mentioning "ball"; too many lemmas named after Metric.ball/openBall/closedBall.

Co-authored-by: Yaël Dillies <yael.dillies@gmail.com>

@@ -908,7 +908,7 @@ theorem ContinuousWithinAt.continuousAt {f : α → β} {s : Set α} {x : α}
 
 theorem IsOpen.continuousOn_iff {f : α → β} {s : Set α} (hs : IsOpen s) :
     ContinuousOn f s ↔ ∀ ⦃a⦄, a ∈ s → ContinuousAt f a :=
-  ball_congr fun _ => continuousWithinAt_iff_continuousAt ∘ hs.mem_nhds
+  forall₂_congr fun _ => continuousWithinAt_iff_continuousAt ∘ hs.mem_nhds
 #align is_open.continuous_on_iff IsOpen.continuousOn_iff
 
 theorem ContinuousOn.continuousAt {f : α → β} {s : Set α} {x : α} (h : ContinuousOn f s)

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

@@ -1188,7 +1188,7 @@ theorem Embedding.map_nhdsWithin_eq {f : α → β} (hf : Embedding f) (s : Set
 theorem OpenEmbedding.map_nhdsWithin_preimage_eq {f : α → β} (hf : OpenEmbedding f) (s : Set β)
     (x : α) : map f (𝓝[f ⁻¹' s] x) = 𝓝[s] f x := by
   rw [hf.toEmbedding.map_nhdsWithin_eq, image_preimage_eq_inter_range]
-  apply nhdsWithin_eq_nhdsWithin (mem_range_self _) hf.open_range
+  apply nhdsWithin_eq_nhdsWithin (mem_range_self _) hf.isOpen_range
   rw [inter_assoc, inter_self]
 #align open_embedding.map_nhds_within_preimage_eq OpenEmbedding.map_nhdsWithin_preimage_eq
 

Empty lines were removed by executing the following Python script twice

import os
import re


# Loop through each file in the repository
for dir_path, dirs, files in os.walk('.'):
  for filename in files:
    if filename.endswith('.lean'):
      file_path = os.path.join(dir_path, filename)

      # Open the file and read its contents
      with open(file_path, 'r') as file:
        content = file.read()

      # Use a regular expression to replace sequences of "variable" lines separated by empty lines
      # with sequences without empty lines
      modified_content = re.sub(r'(variable.*\n)\n(variable(?! .* in))', r'\1\2', content)

      # Write the modified content back to the file
      with open(file_path, 'w') as file:
        file.write(modified_content)

@@ -32,7 +32,6 @@ equipped with the subspace topology.
 open Set Filter Function Topology Filter
 
 variable {α : Type*} {β : Type*} {γ : Type*} {δ : Type*}
-
 variable [TopologicalSpace α]
 
 @[simp]

1. ContinuousAt.comp₂, ContinuousAt.comp₂_continuousWithinAt, and _of_eq versions.
2. ContinuousAt.along_{fst,snd}: Continuous functions are continuous in their first and second arguments.

@@ -1152,6 +1152,19 @@ theorem ContinuousOn.prod {f : α → β} {g : α → γ} {s : Set α} (hf : Con
   ContinuousWithinAt.prod (hf x hx) (hg x hx)
 #align continuous_on.prod ContinuousOn.prod
 
+theorem ContinuousAt.comp₂_continuousWithinAt {f : β × γ → δ} {g : α → β} {h : α → γ} {x : α}
+    {s : Set α} (hf : ContinuousAt f (g x, h x)) (hg : ContinuousWithinAt g s x)
+    (hh : ContinuousWithinAt h s x) :
+    ContinuousWithinAt (fun x ↦ f (g x, h x)) s x :=
+  ContinuousAt.comp_continuousWithinAt hf (hg.prod hh)
+
+theorem ContinuousAt.comp₂_continuousWithinAt_of_eq {f : β × γ → δ} {g : α → β}
+    {h : α → γ} {x : α} {s : Set α} {y : β × γ} (hf : ContinuousAt f y)
+    (hg : ContinuousWithinAt g s x) (hh : ContinuousWithinAt h s x) (e : (g x, h x) = y) :
+    ContinuousWithinAt (fun x ↦ f (g x, h x)) s x := by
+  rw [← e] at hf
+  exact hf.comp₂_continuousWithinAt hg hh
+
 theorem Inducing.continuousWithinAt_iff {f : α → β} {g : β → γ} (hg : Inducing g) {s : Set α}
     {x : α} : ContinuousWithinAt f s x ↔ ContinuousWithinAt (g ∘ f) s x := by
   simp_rw [ContinuousWithinAt, Inducing.tendsto_nhds_iff hg]; rfl

Classifies by adding issue number #10756 to porting notes claiming anything semantically equivalent to:

• "new lemma"
• "added lemma"

@@ -1123,7 +1123,7 @@ theorem continuousOn_of_locally_continuousOn {f : α → β} {s : Set α}
   rwa [ContinuousWithinAt, ← nhdsWithin_restrict _ xt open_t] at this
 #align continuous_on_of_locally_continuous_on continuousOn_of_locally_continuousOn
 
--- Porting note: new lemma
+-- Porting note (#10756): new lemma
 theorem continuousOn_to_generateFrom_iff {s : Set α} {T : Set (Set β)} {f : α → β} :
     @ContinuousOn α β _ (.generateFrom T) f s ↔ ∀ x ∈ s, ∀ t ∈ T, f x ∈ t → f ⁻¹' t ∈ 𝓝[s] x :=
   forall₂_congr fun x _ => by

Homogenises porting notes via capitalisation and addition of whitespace.

It makes the following changes:

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

@@ -133,7 +133,7 @@ theorem nhdsWithin_le_iff {s t : Set α} {x : α} : 𝓝[s] x ≤ 𝓝[t] x ↔
   set_eventuallyLE_iff_inf_principal_le.symm.trans set_eventuallyLE_iff_mem_inf_principal
 #align nhds_within_le_iff nhdsWithin_le_iff
 
--- porting note: golfed, dropped an unneeded assumption
+-- Porting note: golfed, dropped an unneeded assumption
 theorem preimage_nhdsWithin_coinduced' {π : α → β} {s : Set β} {t : Set α} {a : α} (h : a ∈ t)
     (hs : s ∈ @nhds β (.coinduced (fun x : t => π x) inferInstance) (π a)) :
     π ⁻¹' s ∈ 𝓝[t] a := by
@@ -660,7 +660,7 @@ theorem continuousOn_iff_continuous_restrict {f : α → β} {s : Set α} :
   exact (continuousWithinAt_iff_continuousAt_restrict f xs).mpr (h ⟨x, xs⟩)
 #align continuous_on_iff_continuous_restrict continuousOn_iff_continuous_restrict
 
--- porting note: 2 new lemmas
+-- Porting note: 2 new lemmas
 alias ⟨ContinuousOn.restrict, _⟩ := continuousOn_iff_continuous_restrict
 
 theorem ContinuousOn.restrict_mapsTo {f : α → β} {s : Set α} {t : Set β} (hf : ContinuousOn f s)
@@ -1123,7 +1123,7 @@ theorem continuousOn_of_locally_continuousOn {f : α → β} {s : Set α}
   rwa [ContinuousWithinAt, ← nhdsWithin_restrict _ xt open_t] at this
 #align continuous_on_of_locally_continuous_on continuousOn_of_locally_continuousOn
 
--- porting note: new lemma
+-- Porting note: new lemma
 theorem continuousOn_to_generateFrom_iff {s : Set α} {T : Set (Set β)} {f : α → β} :
     @ContinuousOn α β _ (.generateFrom T) f s ↔ ∀ x ∈ s, ∀ t ∈ T, f x ∈ t → f ⁻¹' t ∈ 𝓝[s] x :=
   forall₂_congr fun x _ => by
@@ -1132,7 +1132,7 @@ theorem continuousOn_to_generateFrom_iff {s : Set α} {T : Set (Set β)} {f : α
       and_imp]
     exact forall_congr' fun t => forall_swap
 
--- porting note: dropped an unneeded assumption
+-- Porting note: dropped an unneeded assumption
 theorem continuousOn_isOpen_of_generateFrom {β : Type*} {s : Set α} {T : Set (Set β)} {f : α → β}
     (h : ∀ t ∈ T, IsOpen (s ∩ f ⁻¹' t)) :
     @ContinuousOn α β _ (.generateFrom T) f s :=

@@ -978,6 +978,13 @@ theorem Continuous.comp_continuousOn {g : β → γ} {f : α → β} {s : Set α
   hg.continuousOn.comp hf (mapsTo_univ _ _)
 #align continuous.comp_continuous_on Continuous.comp_continuousOn
 
+@[fun_prop]
+theorem Continuous.comp_continuousOn'
+    {α β γ : Type*} [TopologicalSpace α] [TopologicalSpace β] [TopologicalSpace γ] {g : β → γ}
+    {f : α → β} {s : Set α} (hg : Continuous g) (hf : ContinuousOn f s) :
+    ContinuousOn (fun x ↦ g (f x)) s :=
+  hg.comp_continuousOn hf
+
 theorem ContinuousOn.comp_continuous {g : β → γ} {f : α → β} {s : Set β} (hg : ContinuousOn g s)
     (hf : Continuous f) (hs : ∀ x, f x ∈ s) : Continuous (g ∘ f) := by
   rw [continuous_iff_continuousOn_univ] at *

This is partial work to make s ∩ . be consistently used for passing to a subset of s. This is sort of an adjoint to (Subtype.val : s -> _) '' ., except for the fact that it does not produce a Set s.

The main API changes are to Subtype.image_preimage_val and Subtype.preimage_val_eq_preimage_val_iff in Mathlib.Data.Set.Image. Changes in other modules are all proof fixups.

Zulip discussion

@@ -675,7 +675,7 @@ theorem continuousOn_iff' {f : α → β} {s : Set α} :
     simp only [Subtype.preimage_coe_eq_preimage_coe_iff]
     constructor <;>
       · rintro ⟨u, ou, useq⟩
-        exact ⟨u, ou, useq.symm⟩
+        exact ⟨u, ou, by simpa only [Set.inter_comm, eq_comm] using useq⟩
   rw [continuousOn_iff_continuous_restrict, continuous_def]; simp only [this]
 #align continuous_on_iff' continuousOn_iff'
 
@@ -700,7 +700,7 @@ theorem continuousOn_iff_isClosed {f : α → β} {s : Set α} :
   have : ∀ t, IsClosed (s.restrict f ⁻¹' t) ↔ ∃ u : Set α, IsClosed u ∧ f ⁻¹' t ∩ s = u ∩ s := by
     intro t
     rw [isClosed_induced_iff, Set.restrict_eq, Set.preimage_comp]
-    simp only [Subtype.preimage_coe_eq_preimage_coe_iff, eq_comm]
+    simp only [Subtype.preimage_coe_eq_preimage_coe_iff, eq_comm, Set.inter_comm s]
   rw [continuousOn_iff_continuous_restrict, continuous_iff_isClosed]; simp only [this]
 #align continuous_on_iff_is_closed continuousOn_iff_isClosed
 
@@ -1348,9 +1348,9 @@ theorem continuousOn_piecewise_ite {s s' t : Set α} {f f' : α → β} [∀ x,
 
 theorem frontier_inter_open_inter {s t : Set α} (ht : IsOpen t) :
     frontier (s ∩ t) ∩ t = frontier s ∩ t := by
-  simp only [← Subtype.preimage_coe_eq_preimage_coe_iff,
+  simp only [Set.inter_comm _ t, ← Subtype.preimage_coe_eq_preimage_coe_iff,
     ht.isOpenMap_subtype_val.preimage_frontier_eq_frontier_preimage continuous_subtype_val,
-    Subtype.preimage_coe_inter_self]
+    Subtype.preimage_coe_self_inter]
 #align frontier_inter_open_inter frontier_inter_open_inter
 
 theorem continuousOn_fst {s : Set (α × β)} : ContinuousOn Prod.fst s :=

Co-authored-by: Yury G. Kudryashov <urkud@urkud.name>

@@ -625,6 +625,12 @@ theorem continuousOn_pi {ι : Type*} {π : ι → Type*} [∀ i, TopologicalSpac
   ⟨fun h i x hx => tendsto_pi_nhds.1 (h x hx) i, fun h x hx => tendsto_pi_nhds.2 fun i => h i x hx⟩
 #align continuous_on_pi continuousOn_pi
 
+@[fun_prop]
+theorem continuousOn_pi' {ι : Type*} {π : ι → Type*} [∀ i, TopologicalSpace (π i)]
+    {f : α → ∀ i, π i} {s : Set α} (hf : ∀ i, ContinuousOn (fun y => f y i) s) :
+    ContinuousOn f s :=
+  continuousOn_pi.2 hf
+
 theorem ContinuousWithinAt.fin_insertNth {n} {π : Fin (n + 1) → Type*}
     [∀ i, TopologicalSpace (π i)] (i : Fin (n + 1)) {f : α → π i} {a : α} {s : Set α}
     (hf : ContinuousWithinAt f s a) {g : α → ∀ j : Fin n, π (i.succAbove j)}
@@ -937,6 +943,11 @@ theorem ContinuousOn.comp {g : β → γ} {f : α → β} {s : Set α} {t : Set
   ContinuousWithinAt.comp (hg _ (h hx)) (hf x hx) h
 #align continuous_on.comp ContinuousOn.comp
 
+@[fun_prop]
+theorem ContinuousOn.comp'' {g : β → γ} {f : α → β} {s : Set α} {t : Set β} (hg : ContinuousOn g t)
+    (hf : ContinuousOn f s) (h : Set.MapsTo f s t) : ContinuousOn (fun x => g (f x)) s :=
+  ContinuousOn.comp hg hf h
+
 theorem ContinuousOn.mono {f : α → β} {s t : Set α} (hf : ContinuousOn f s) (h : t ⊆ s) :
     ContinuousOn f t := fun x hx => (hf x (h hx)).mono_left (nhdsWithin_mono _ h)
 #align continuous_on.mono ContinuousOn.mono
@@ -945,11 +956,13 @@ theorem antitone_continuousOn {f : α → β} : Antitone (ContinuousOn f) := fun
   hf.mono hst
 #align antitone_continuous_on antitone_continuousOn
 
+@[fun_prop]
 theorem ContinuousOn.comp' {g : β → γ} {f : α → β} {s : Set α} {t : Set β} (hg : ContinuousOn g t)
     (hf : ContinuousOn f s) : ContinuousOn (g ∘ f) (s ∩ f ⁻¹' t) :=
   hg.comp (hf.mono (inter_subset_left _ _)) (inter_subset_right _ _)
 #align continuous_on.comp' ContinuousOn.comp'
 
+@[fun_prop]
 theorem Continuous.continuousOn {f : α → β} {s : Set α} (h : Continuous f) : ContinuousOn f s := by
   rw [continuous_iff_continuousOn_univ] at h
   exact h.mono (subset_univ _)
@@ -971,6 +984,11 @@ theorem ContinuousOn.comp_continuous {g : β → γ} {f : α → β} {s : Set β
   exact hg.comp hf fun x _ => hs x
 #align continuous_on.comp_continuous ContinuousOn.comp_continuous
 
+@[fun_prop]
+theorem continuousOn_apply {ι : Type*} {π : ι → Type*} [∀ i, TopologicalSpace (π i)]
+    (i : ι) (s) : ContinuousOn (fun p : ∀ i, π i => p i) s :=
+  Continuous.continuousOn (continuous_apply i)
+
 theorem ContinuousWithinAt.preimage_mem_nhdsWithin {f : α → β} {x : α} {s : Set α} {t : Set β}
     (h : ContinuousWithinAt f s x) (ht : t ∈ 𝓝 (f x)) : f ⁻¹' t ∈ 𝓝[s] x :=
   h ht
@@ -1029,6 +1047,7 @@ theorem ContinuousWithinAt.congr_mono {f g : α → β} {s s₁ : Set α} {x : 
   (h.mono h₁).congr h' hx
 #align continuous_within_at.congr_mono ContinuousWithinAt.congr_mono
 
+@[fun_prop]
 theorem continuousOn_const {s : Set α} {c : β} : ContinuousOn (fun _ => c) s :=
   continuous_const.continuousOn
 #align continuous_on_const continuousOn_const
@@ -1042,6 +1061,9 @@ theorem continuousOn_id {s : Set α} : ContinuousOn id s :=
   continuous_id.continuousOn
 #align continuous_on_id continuousOn_id
 
+@[fun_prop]
+theorem continuousOn_id' (s : Set α) : ContinuousOn (fun x : α => x) s := continuousOn_id
+
 theorem continuousWithinAt_id {s : Set α} {x : α} : ContinuousWithinAt id s x :=
   continuous_id.continuousWithinAt
 #align continuous_within_at_id continuousWithinAt_id
@@ -1117,6 +1139,7 @@ theorem ContinuousWithinAt.prod {f : α → β} {g : α → γ} {s : Set α} {x
   hf.prod_mk_nhds hg
 #align continuous_within_at.prod ContinuousWithinAt.prod
 
+@[fun_prop]
 theorem ContinuousOn.prod {f : α → β} {g : α → γ} {s : Set α} (hf : ContinuousOn f s)
     (hg : ContinuousOn g s) : ContinuousOn (fun x => (f x, g x)) s := fun x hx =>
   ContinuousWithinAt.prod (hf x hx) (hg x hx)
@@ -1338,6 +1361,7 @@ theorem continuousWithinAt_fst {s : Set (α × β)} {p : α × β} : ContinuousW
   continuous_fst.continuousWithinAt
 #align continuous_within_at_fst continuousWithinAt_fst
 
+@[fun_prop]
 theorem ContinuousOn.fst {f : α → β × γ} {s : Set α} (hf : ContinuousOn f s) :
     ContinuousOn (fun x => (f x).1) s :=
   continuous_fst.comp_continuousOn hf
@@ -1356,6 +1380,7 @@ theorem continuousWithinAt_snd {s : Set (α × β)} {p : α × β} : ContinuousW
   continuous_snd.continuousWithinAt
 #align continuous_within_at_snd continuousWithinAt_snd
 
+@[fun_prop]
 theorem ContinuousOn.snd {f : α → β × γ} {s : Set α} (hf : ContinuousOn f s) :
     ContinuousOn (fun x => (f x).2) s :=
   continuous_snd.comp_continuousOn hf

Also fix GeneralizedContinuedFraction.of_convergence: it worked for the Preorder.topology only.

@@ -112,7 +112,7 @@ theorem diff_mem_nhdsWithin_diff {x : α} {s t : Set α} (hs : s ∈ 𝓝[t] x)
 theorem nhds_of_nhdsWithin_of_nhds {s t : Set α} {a : α} (h1 : s ∈ 𝓝 a) (h2 : t ∈ 𝓝[s] a) :
     t ∈ 𝓝 a := by
   rcases mem_nhdsWithin_iff_exists_mem_nhds_inter.mp h2 with ⟨_, Hw, hw⟩
-  exact (nhds a).sets_of_superset ((nhds a).inter_sets Hw h1) hw
+  exact (𝓝 a).sets_of_superset ((𝓝 a).inter_sets Hw h1) hw
 #align nhds_of_nhds_within_of_nhds nhds_of_nhdsWithin_of_nhds
 
 theorem mem_nhdsWithin_iff_eventually {s t : Set α} {x : α} :

In some cases, the order of implicit arguments changed because now they appear in a different order in variables.

Also, some definitions used greek letters for topological spaces, changed to X/Y.

@@ -520,12 +520,6 @@ theorem tendsto_nhdsWithin_iff_subtype {s : Set α} {a : α} (h : a ∈ s) (f :
 
 variable [TopologicalSpace β] [TopologicalSpace γ] [TopologicalSpace δ]
 
-/-- A function between topological spaces is continuous at a point `x₀` within a subset `s`
-if `f x` tends to `f x₀` when `x` tends to `x₀` while staying within `s`. -/
-def ContinuousWithinAt (f : α → β) (s : Set α) (x : α) : Prop :=
-  Tendsto f (𝓝[s] x) (𝓝 (f x))
-#align continuous_within_at ContinuousWithinAt
-
 /-- If a function is continuous within `s` at `x`, then it tends to `f x` within `s` by definition.
 We register this fact for use with the dot notation, especially to use `Filter.Tendsto.comp` as
 `ContinuousWithinAt.comp` will have a different meaning. -/
@@ -534,12 +528,6 @@ theorem ContinuousWithinAt.tendsto {f : α → β} {s : Set α} {x : α} (h : Co
   h
 #align continuous_within_at.tendsto ContinuousWithinAt.tendsto
 
-/-- A function between topological spaces is continuous on a subset `s`
-when it's continuous at every point of `s` within `s`. -/
-def ContinuousOn (f : α → β) (s : Set α) : Prop :=
-  ∀ x ∈ s, ContinuousWithinAt f s x
-#align continuous_on ContinuousOn
-
 theorem ContinuousOn.continuousWithinAt {f : α → β} {s : Set α} {x : α} (hf : ContinuousOn f s)
     (hx : x ∈ s) : ContinuousWithinAt f s x :=
   hf x hx

This was unnoticed during the review of #7568

@@ -505,13 +505,13 @@ theorem preimage_coe_mem_nhds_subtype {s t : Set α} {a : s} : (↑) ⁻¹' t 
   rw [← map_nhds_subtype_val, mem_map]
 #align preimage_coe_mem_nhds_subtype preimage_coe_mem_nhds_subtype
 
-theorem eventually_nhds_subtype_if (s : Set α) (a : s) (P : α → Prop) :
+theorem eventually_nhds_subtype_iff (s : Set α) (a : s) (P : α → Prop) :
     (∀ᶠ x : s in 𝓝 a, P x) ↔ ∀ᶠ x in 𝓝[s] a, P x :=
   preimage_coe_mem_nhds_subtype
 
 theorem frequently_nhds_subtype_iff (s : Set α) (a : s) (P : α → Prop) :
     (∃ᶠ x : s in 𝓝 a, P x) ↔ ∃ᶠ x in 𝓝[s] a, P x :=
-  eventually_nhds_subtype_if s a (¬ P ·) |>.not
+  eventually_nhds_subtype_iff s a (¬ P ·) |>.not
 
 theorem tendsto_nhdsWithin_iff_subtype {s : Set α} {a : α} (h : a ∈ s) (f : α → β) (l : Filter β) :
     Tendsto f (𝓝[s] a) l ↔ Tendsto (s.restrict f) (𝓝 ⟨a, h⟩) l := by

Co-authored-by: Anatole Dedecker <anatolededecker@gmail.com> Co-authored-by: ADedecker <anatolededecker@gmail.com>

@@ -505,6 +505,14 @@ theorem preimage_coe_mem_nhds_subtype {s t : Set α} {a : s} : (↑) ⁻¹' t 
   rw [← map_nhds_subtype_val, mem_map]
 #align preimage_coe_mem_nhds_subtype preimage_coe_mem_nhds_subtype
 
+theorem eventually_nhds_subtype_if (s : Set α) (a : s) (P : α → Prop) :
+    (∀ᶠ x : s in 𝓝 a, P x) ↔ ∀ᶠ x in 𝓝[s] a, P x :=
+  preimage_coe_mem_nhds_subtype
+
+theorem frequently_nhds_subtype_iff (s : Set α) (a : s) (P : α → Prop) :
+    (∃ᶠ x : s in 𝓝 a, P x) ↔ ∃ᶠ x in 𝓝[s] a, P x :=
+  eventually_nhds_subtype_if s a (¬ P ·) |>.not
+
 theorem tendsto_nhdsWithin_iff_subtype {s : Set α} {a : α} (h : a ∈ s) (f : α → β) (l : Filter β) :
     Tendsto f (𝓝[s] a) l ↔ Tendsto (s.restrict f) (𝓝 ⟨a, h⟩) l := by
   rw [nhdsWithin_eq_map_subtype_coe h, tendsto_map'_iff]; rfl

@@ -29,9 +29,6 @@ equipped with the subspace topology.
 
 -/
 
-set_option autoImplicit true
-
-
 open Set Filter Function Topology Filter
 
 variable {α : Type*} {β : Type*} {γ : Type*} {δ : Type*}
@@ -753,7 +750,7 @@ theorem ContinuousWithinAt.mono_of_mem {f : α → β} {s t : Set α} {x : α}
   h.mono_left (nhdsWithin_le_of_mem hs)
 #align continuous_within_at.mono_of_mem ContinuousWithinAt.mono_of_mem
 
-theorem continuousWithinAt_congr_nhds {f : α → β} (h : 𝓝[s] x = 𝓝[t] x) :
+theorem continuousWithinAt_congr_nhds {f : α → β} {s t : Set α} {x : α} (h : 𝓝[s] x = 𝓝[t] x) :
     ContinuousWithinAt f s x ↔ ContinuousWithinAt f t x := by
   simp only [ContinuousWithinAt, h]
 

Subset of #9319

@@ -1104,7 +1104,7 @@ theorem continuousOn_of_locally_continuousOn {f : α → β} {s : Set α}
 -- porting note: new lemma
 theorem continuousOn_to_generateFrom_iff {s : Set α} {T : Set (Set β)} {f : α → β} :
     @ContinuousOn α β _ (.generateFrom T) f s ↔ ∀ x ∈ s, ∀ t ∈ T, f x ∈ t → f ⁻¹' t ∈ 𝓝[s] x :=
-  forall₂_congr <| fun x _ => by
+  forall₂_congr fun x _ => by
     delta ContinuousWithinAt
     simp only [TopologicalSpace.nhds_generateFrom, tendsto_iInf, tendsto_principal, mem_setOf_eq,
       and_imp]

Function.support is a very basic definition. Nevertheless, it is a pretty heavy import because it imports most objects a support lemma can be written about.

This PR reverses the dependencies between those objects and Function.support, so that the latter can become a much more lightweight import.

Only two import could not easily be reversed, namely the ones to Data.Set.Finite and Order.ConditionallyCompleteLattice.Basic, so I created two new files instead.

I credit:

• Yury for https://github.com/leanprover-community/mathlib/pull/6791
• Oliver for https://github.com/leanprover-community/mathlib/pull/7439

@@ -3,7 +3,7 @@ Copyright (c) 2019 Reid Barton. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Sébastien Gouëzel
 -/
-import Mathlib.Algebra.IndicatorFunction
+import Mathlib.Algebra.Function.Indicator
 import Mathlib.Topology.Constructions
 
 #align_import topology.continuous_on from "leanprover-community/mathlib"@"d4f691b9e5f94cfc64639973f3544c95f8d5d494"

Assorted golf I did while working on a refactor. Submitting as a separate PR.

• Move not_differentiableAt_abs_zero to Calculus.Deriv.Add, golf.
• Rename HasFDerivWithinAt_of_nhdsWithin_eq_bot to HasFDerivWithinAt.of_nhdsWithin_eq_bot, golf.
• Protect Filter.EventuallyEq.rfl.
• Golf here and there.

@@ -1157,10 +1157,9 @@ theorem OpenEmbedding.map_nhdsWithin_preimage_eq {f : α → β} (hf : OpenEmbed
   rw [inter_assoc, inter_self]
 #align open_embedding.map_nhds_within_preimage_eq OpenEmbedding.map_nhdsWithin_preimage_eq
 
-theorem continuousWithinAt_of_not_mem_closure {f : α → β} {s : Set α} {x : α} :
-    x ∉ closure s → ContinuousWithinAt f s x := by
-  intro hx
-  rw [mem_closure_iff_nhdsWithin_neBot, neBot_iff, not_not] at hx
+theorem continuousWithinAt_of_not_mem_closure {f : α → β} {s : Set α} {x : α} (hx : x ∉ closure s) :
+    ContinuousWithinAt f s x := by
+  rw [mem_closure_iff_nhdsWithin_neBot, not_neBot] at hx
   rw [ContinuousWithinAt, hx]
   exact tendsto_bot
 #align continuous_within_at_of_not_mem_closure continuousWithinAt_of_not_mem_closure

From PFR

@@ -3,6 +3,7 @@ Copyright (c) 2019 Reid Barton. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Sébastien Gouëzel
 -/
+import Mathlib.Algebra.IndicatorFunction
 import Mathlib.Topology.Constructions
 
 #align_import topology.continuous_on from "leanprover-community/mathlib"@"d4f691b9e5f94cfc64639973f3544c95f8d5d494"
@@ -720,6 +721,7 @@ theorem continuous_of_cover_nhds {ι : Sort*} {f : α → β} {s : ι → Set α
 theorem continuousOn_empty (f : α → β) : ContinuousOn f ∅ := fun _ => False.elim
 #align continuous_on_empty continuousOn_empty
 
+@[simp]
 theorem continuousOn_singleton (f : α → β) (a : α) : ContinuousOn f {a} :=
   forall_eq.2 <| by
     simpa only [ContinuousWithinAt, nhdsWithin_singleton, tendsto_pure_left] using fun s =>
@@ -1271,6 +1273,21 @@ theorem Continuous.piecewise {s : Set α} {f g : α → β} [∀ a, Decidable (a
   hf.if hs hg
 #align continuous.piecewise Continuous.piecewise
 
+section Indicator
+variable [One β] {f : α → β} {s : Set α}
+
+@[to_additive]
+lemma continuous_mulIndicator (hs : ∀ a ∈ frontier s, f a = 1) (hf : ContinuousOn f (closure s)) :
+    Continuous (mulIndicator s f) := by
+  classical exact continuous_piecewise hs hf continuousOn_const
+
+@[to_additive]
+protected lemma Continuous.mulIndicator (hs : ∀ a ∈ frontier s, f a = 1) (hf : Continuous f) :
+    Continuous (mulIndicator s f) := by
+  classical exact hf.piecewise hs continuous_const
+
+end Indicator
+
 theorem IsOpen.ite' {s s' t : Set α} (hs : IsOpen s) (hs' : IsOpen s')
     (ht : ∀ x ∈ frontier t, x ∈ s ↔ x ∈ s') : IsOpen (t.ite s s') := by
   classical

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

@@ -1064,10 +1064,10 @@ theorem continuousOn_open_iff {f : α → β} {s : Set α} (hs : IsOpen s) :
     rw [@inter_comm _ s (f ⁻¹' t), inter_assoc, inter_self]
 #align continuous_on_open_iff continuousOn_open_iff
 
-theorem ContinuousOn.preimage_open_of_open {f : α → β} {s : Set α} {t : Set β}
+theorem ContinuousOn.isOpen_inter_preimage {f : α → β} {s : Set α} {t : Set β}
     (hf : ContinuousOn f s) (hs : IsOpen s) (ht : IsOpen t) : IsOpen (s ∩ f ⁻¹' t) :=
   (continuousOn_open_iff hs).1 hf t ht
-#align continuous_on.preimage_open_of_open ContinuousOn.preimage_open_of_open
+#align continuous_on.preimage_open_of_open ContinuousOn.isOpen_inter_preimage
 
 theorem ContinuousOn.isOpen_preimage {f : α → β} {s : Set α} {t : Set β} (h : ContinuousOn f s)
     (hs : IsOpen s) (hp : f ⁻¹' t ⊆ s) (ht : IsOpen t) : IsOpen (f ⁻¹' t) := by
@@ -1087,7 +1087,7 @@ theorem ContinuousOn.preimage_interior_subset_interior_preimage {f : α → β}
   calc
     s ∩ f ⁻¹' interior t ⊆ interior (s ∩ f ⁻¹' t) :=
       interior_maximal (inter_subset_inter (Subset.refl _) (preimage_mono interior_subset))
-        (hf.preimage_open_of_open hs isOpen_interior)
+        (hf.isOpen_inter_preimage hs isOpen_interior)
     _ = s ∩ interior (f ⁻¹' t) := by rw [interior_inter, hs.interior_eq]
 #align continuous_on.preimage_interior_subset_interior_preimage ContinuousOn.preimage_interior_subset_interior_preimage
 
@@ -1109,12 +1109,12 @@ theorem continuousOn_to_generateFrom_iff {s : Set α} {T : Set (Set β)} {f : α
     exact forall_congr' fun t => forall_swap
 
 -- porting note: dropped an unneeded assumption
-theorem continuousOn_open_of_generateFrom {β : Type*} {s : Set α} {T : Set (Set β)} {f : α → β}
+theorem continuousOn_isOpen_of_generateFrom {β : Type*} {s : Set α} {T : Set (Set β)} {f : α → β}
     (h : ∀ t ∈ T, IsOpen (s ∩ f ⁻¹' t)) :
     @ContinuousOn α β _ (.generateFrom T) f s :=
   continuousOn_to_generateFrom_iff.2 fun _x hx t ht hxt => mem_nhdsWithin.2
     ⟨_, h t ht, ⟨hx, hxt⟩, fun _y hy => hy.1.2⟩
-#align continuous_on_open_of_generate_from continuousOn_open_of_generateFromₓ
+#align continuous_on_open_of_generate_from continuousOn_isOpen_of_generateFromₓ
 
 theorem ContinuousWithinAt.prod {f : α → β} {g : α → γ} {s : Set α} {x : α}
     (hf : ContinuousWithinAt f s x) (hg : ContinuousWithinAt g s x) :

This PR renames the field Clopens.clopen' -> Clopens.isClopen', and the lemmas

• preimage_closed_of_closed -> ContinuousOn.preimage_isClosed_of_isClosed

as well as: ClopenUpperSet.clopen -> ClopenUpperSet.isClopen connectedComponent_eq_iInter_clopen -> connectedComponent_eq_iInter_isClopen connectedComponent_subset_iInter_clopen -> connectedComponent_subset_iInter_isClopen continuous_boolIndicator_iff_clopen -> continuous_boolIndicator_iff_isClopen continuousOn_boolIndicator_iff_clopen -> continuousOn_boolIndicator_iff_isClopen DiscreteQuotient.ofClopen -> DiscreteQuotient.ofIsClopen disjoint_or_subset_of_clopen -> disjoint_or_subset_of_isClopen exists_clopen_{lower,upper}of_not_le -> exists_isClopen{lower,upper}_of_not_le exists_clopen_of_cofiltered -> exists_isClopen_of_cofiltered exists_clopen_of_totally_separated -> exists_isClopen_of_totally_separated exists_clopen_upper_or_lower_of_ne -> exists_isClopen_upper_or_lower_of_ne IsPreconnected.subset_clopen -> IsPreconnected.subset_isClopen isTotallyDisconnected_of_clopen_set -> isTotallyDisconnected_of_isClopen_set LocallyConstant.ofClopen_fiber_one -> LocallyConstant.ofIsClopen_fiber_one LocallyConstant.ofClopen_fiber_zero -> LocallyConstant.ofIsClopen_fiber_zero LocallyConstant.ofClopen -> LocallyConstant.ofIsClopen preimage_clopen_of_clopen -> preimage_isClopen_of_isClopen TopologicalSpace.Clopens.clopen -> TopologicalSpace.Clopens.isClopen

@@ -1075,12 +1075,12 @@ theorem ContinuousOn.isOpen_preimage {f : α → β} {s : Set α} {t : Set β} (
   rw [inter_comm, inter_eq_self_of_subset_left hp]
 #align continuous_on.is_open_preimage ContinuousOn.isOpen_preimage
 
-theorem ContinuousOn.preimage_closed_of_closed {f : α → β} {s : Set α} {t : Set β}
+theorem ContinuousOn.preimage_isClosed_of_isClosed {f : α → β} {s : Set α} {t : Set β}
     (hf : ContinuousOn f s) (hs : IsClosed s) (ht : IsClosed t) : IsClosed (s ∩ f ⁻¹' t) := by
   rcases continuousOn_iff_isClosed.1 hf t ht with ⟨u, hu⟩
   rw [inter_comm, hu.2]
   apply IsClosed.inter hu.1 hs
-#align continuous_on.preimage_closed_of_closed ContinuousOn.preimage_closed_of_closed
+#align continuous_on.preimage_closed_of_closed ContinuousOn.preimage_isClosed_of_isClosed
 
 theorem ContinuousOn.preimage_interior_subset_interior_preimage {f : α → β} {s : Set α} {t : Set β}
     (hf : ContinuousOn f s) (hs : IsOpen s) : s ∩ f ⁻¹' interior t ⊆ s ∩ interior (f ⁻¹' t) :=

Co-authored-by: Eric Wieser <wieser.eric@gmail.com>

@@ -330,8 +330,9 @@ theorem nhdsWithin_pi_eq {ι : Type*} {α : ι → Type*} [∀ i, TopologicalSpa
 #align nhds_within_pi_eq nhdsWithin_pi_eq
 
 theorem nhdsWithin_pi_univ_eq {ι : Type*} {α : ι → Type*} [Finite ι] [∀ i, TopologicalSpace (α i)]
-    (s : ∀ i, Set (α i)) (x : ∀ i, α i) : 𝓝[pi univ s] x = ⨅ i, comap (fun x => x i) (𝓝[s i] x i) :=
-  by simpa [nhdsWithin] using nhdsWithin_pi_eq finite_univ s x
+    (s : ∀ i, Set (α i)) (x : ∀ i, α i) :
+    𝓝[pi univ s] x = ⨅ i, comap (fun x => x i) (𝓝[s i] x i) := by
+  simpa [nhdsWithin] using nhdsWithin_pi_eq finite_univ s x
 #align nhds_within_pi_univ_eq nhdsWithin_pi_univ_eq
 
 theorem nhdsWithin_pi_eq_bot {ι : Type*} {α : ι → Type*} [∀ i, TopologicalSpace (α i)] {I : Set ι}
@@ -646,8 +647,8 @@ nonrec theorem ContinuousOn.fin_insertNth {n} {π : Fin (n + 1) → Type*}
 
 theorem continuousOn_iff {f : α → β} {s : Set α} :
     ContinuousOn f s ↔
-      ∀ x ∈ s, ∀ t : Set β, IsOpen t → f x ∈ t → ∃ u, IsOpen u ∧ x ∈ u ∧ u ∩ s ⊆ f ⁻¹' t :=
-  by simp only [ContinuousOn, ContinuousWithinAt, tendsto_nhds, mem_nhdsWithin]
+      ∀ x ∈ s, ∀ t : Set β, IsOpen t → f x ∈ t → ∃ u, IsOpen u ∧ x ∈ u ∧ u ∩ s ⊆ f ⁻¹' t := by
+  simp only [ContinuousOn, ContinuousWithinAt, tendsto_nhds, mem_nhdsWithin]
 #align continuous_on_iff continuousOn_iff
 
 theorem continuousOn_iff_continuous_restrict {f : α → β} {s : Set α} :
@@ -1011,8 +1012,9 @@ theorem ContinuousWithinAt.preimage_mem_nhdsWithin''
   exact h.preimage_mem_nhdsWithin' (nhdsWithin_mono _ (image_preimage_subset f s) ht)
 
 theorem Filter.EventuallyEq.congr_continuousWithinAt {f g : α → β} {s : Set α} {x : α}
-    (h : f =ᶠ[𝓝[s] x] g) (hx : f x = g x) : ContinuousWithinAt f s x ↔ ContinuousWithinAt g s x :=
-  by rw [ContinuousWithinAt, hx, tendsto_congr' h, ContinuousWithinAt]
+    (h : f =ᶠ[𝓝[s] x] g) (hx : f x = g x) :
+    ContinuousWithinAt f s x ↔ ContinuousWithinAt g s x := by
+  rw [ContinuousWithinAt, hx, tendsto_congr' h, ContinuousWithinAt]
 #align filter.eventually_eq.congr_continuous_within_at Filter.EventuallyEq.congr_continuousWithinAt
 
 theorem ContinuousWithinAt.congr_of_eventuallyEq {f f₁ : α → β} {s : Set α} {x : α}

Co-authored-by: Mario Carneiro <di.gama@gmail.com>

@@ -609,6 +609,16 @@ theorem continuous_prod_of_discrete_right [DiscreteTopology β] {f : α × β 
     Continuous f ↔ ∀ b, Continuous (f ⟨·, b⟩) := by
   simp_rw [continuous_iff_continuousOn_univ]; exact continuousOn_prod_of_discrete_right
 
+theorem isOpenMap_prod_of_discrete_left [DiscreteTopology α] {f : α × β → γ} :
+    IsOpenMap f ↔ ∀ a, IsOpenMap (f ⟨a, ·⟩) := by
+  simp_rw [isOpenMap_iff_nhds_le, Prod.forall, nhds_prod_eq, nhds_discrete, pure_prod, map_map]
+  rfl
+
+theorem isOpenMap_prod_of_discrete_right [DiscreteTopology β] {f : α × β → γ} :
+    IsOpenMap f ↔ ∀ b, IsOpenMap (f ⟨·, b⟩) := by
+  simp_rw [isOpenMap_iff_nhds_le, Prod.forall, forall_swap (α := α) (β := β), nhds_prod_eq,
+    nhds_discrete, prod_pure, map_map]; rfl
+
 theorem continuousWithinAt_pi {ι : Type*} {π : ι → Type*} [∀ i, TopologicalSpace (π i)]
     {f : α → ∀ i, π i} {s : Set α} {x : α} :
     ContinuousWithinAt f s x ↔ ∀ i, ContinuousWithinAt (fun y => f y i) s x :=

• Add four pairs of lemmas continuous((Within)At/On)_prod_of_discrete_left/right in ContinuousOn.lean: to check continuity of a function from X × Y to Z with X discrete, it suffices to check continuity of every slice of it with x : X fixed.

• Remove duplicate lemmas continuous_uncurry_of_discreteTopology(_left) from Constructions.lean in favor of the more general (iff) version.

• Move the lemma continuous_iff_continuousOn_univ up.

Co-authored-by: Junyan Xu <junyanxu.math@gmail.com>

@@ -543,6 +543,11 @@ theorem continuousWithinAt_univ (f : α → β) (x : α) :
   rw [ContinuousAt, ContinuousWithinAt, nhdsWithin_univ]
 #align continuous_within_at_univ continuousWithinAt_univ
 
+theorem continuous_iff_continuousOn_univ {f : α → β} : Continuous f ↔ ContinuousOn f univ := by
+  simp [continuous_iff_continuousAt, ContinuousOn, ContinuousAt, ContinuousWithinAt,
+    nhdsWithin_univ]
+#align continuous_iff_continuous_on_univ continuous_iff_continuousOn_univ
+
 theorem continuousWithinAt_iff_continuousAt_restrict (f : α → β) {x : α} {s : Set α} (h : x ∈ s) :
     ContinuousWithinAt f s x ↔ ContinuousAt (s.restrict f) ⟨x, h⟩ :=
   tendsto_nhdsWithin_iff_subtype h f _
@@ -566,6 +571,44 @@ theorem ContinuousWithinAt.prod_map {f : α → γ} {g : β → δ} {s : Set α}
   exact hf.prod_map hg
 #align continuous_within_at.prod_map ContinuousWithinAt.prod_map
 
+theorem continuousWithinAt_prod_of_discrete_left [DiscreteTopology α]
+    {f : α × β → γ} {s : Set (α × β)} {x : α × β} :
+    ContinuousWithinAt f s x ↔ ContinuousWithinAt (f ⟨x.1, ·⟩) {b | (x.1, b) ∈ s} x.2 := by
+  rw [← x.eta]; simp_rw [ContinuousWithinAt, nhdsWithin, nhds_prod_eq, nhds_discrete, pure_prod,
+    ← map_inf_principal_preimage]; rfl
+
+theorem continuousWithinAt_prod_of_discrete_right [DiscreteTopology β]
+    {f : α × β → γ} {s : Set (α × β)} {x : α × β} :
+    ContinuousWithinAt f s x ↔ ContinuousWithinAt (f ⟨·, x.2⟩) {a | (a, x.2) ∈ s} x.1 := by
+  rw [← x.eta]; simp_rw [ContinuousWithinAt, nhdsWithin, nhds_prod_eq, nhds_discrete, prod_pure,
+    ← map_inf_principal_preimage]; rfl
+
+theorem continuousAt_prod_of_discrete_left [DiscreteTopology α] {f : α × β → γ} {x : α × β} :
+    ContinuousAt f x ↔ ContinuousAt (f ⟨x.1, ·⟩) x.2 := by
+  simp_rw [← continuousWithinAt_univ]; exact continuousWithinAt_prod_of_discrete_left
+
+theorem continuousAt_prod_of_discrete_right [DiscreteTopology β] {f : α × β → γ} {x : α × β} :
+    ContinuousAt f x ↔ ContinuousAt (f ⟨·, x.2⟩) x.1 := by
+  simp_rw [← continuousWithinAt_univ]; exact continuousWithinAt_prod_of_discrete_right
+
+theorem continuousOn_prod_of_discrete_left [DiscreteTopology α] {f : α × β → γ} {s : Set (α × β)} :
+    ContinuousOn f s ↔ ∀ a, ContinuousOn (f ⟨a, ·⟩) {b | (a, b) ∈ s} := by
+  simp_rw [ContinuousOn, Prod.forall, continuousWithinAt_prod_of_discrete_left]; rfl
+
+theorem continuousOn_prod_of_discrete_right [DiscreteTopology β] {f : α × β → γ} {s : Set (α × β)} :
+    ContinuousOn f s ↔ ∀ b, ContinuousOn (f ⟨·, b⟩) {a | (a, b) ∈ s} := by
+  simp_rw [ContinuousOn, Prod.forall, continuousWithinAt_prod_of_discrete_right]; apply forall_swap
+
+/-- If a function `f a b` is such that `y ↦ f a b` is continuous for all `a`, and `a` lives in a
+discrete space, then `f` is continuous, and vice versa. -/
+theorem continuous_prod_of_discrete_left [DiscreteTopology α] {f : α × β → γ} :
+    Continuous f ↔ ∀ a, Continuous (f ⟨a, ·⟩) := by
+  simp_rw [continuous_iff_continuousOn_univ]; exact continuousOn_prod_of_discrete_left
+
+theorem continuous_prod_of_discrete_right [DiscreteTopology β] {f : α × β → γ} :
+    Continuous f ↔ ∀ b, Continuous (f ⟨·, b⟩) := by
+  simp_rw [continuous_iff_continuousOn_univ]; exact continuousOn_prod_of_discrete_right
+
 theorem continuousWithinAt_pi {ι : Type*} {π : ι → Type*} [∀ i, TopologicalSpace (π i)]
     {f : α → ∀ i, π i} {s : Set α} {x : α} :
     ContinuousWithinAt f s x ↔ ∀ i, ContinuousWithinAt (fun y => f y i) s x :=
@@ -687,11 +730,6 @@ theorem comap_nhdsWithin_range {α} (f : α → β) (y : β) : comap f (𝓝[ran
   comap_inf_principal_range
 #align comap_nhds_within_range comap_nhdsWithin_range
 
-theorem continuous_iff_continuousOn_univ {f : α → β} : Continuous f ↔ ContinuousOn f univ := by
-  simp [continuous_iff_continuousAt, ContinuousOn, ContinuousAt, ContinuousWithinAt,
-    nhdsWithin_univ]
-#align continuous_iff_continuous_on_univ continuous_iff_continuousOn_univ
-
 theorem ContinuousWithinAt.mono {f : α → β} {s t : Set α} {x : α} (h : ContinuousWithinAt f t x)
     (hs : s ⊆ t) : ContinuousWithinAt f s x :=
   h.mono_left (nhdsWithin_mono x hs)

Topological prerequisites for Rademacher theorem in #7003.

@@ -955,6 +955,13 @@ theorem ContinuousWithinAt.preimage_mem_nhdsWithin' {f : α → β} {x : α} {s
   h.tendsto_nhdsWithin (mapsTo_image _ _) ht
 #align continuous_within_at.preimage_mem_nhds_within' ContinuousWithinAt.preimage_mem_nhdsWithin'
 
+theorem ContinuousWithinAt.preimage_mem_nhdsWithin''
+    {f : α → β} {x : α} {y : β} {s t : Set β}
+    (h : ContinuousWithinAt f (f ⁻¹' s) x) (ht : t ∈ 𝓝[s] y) (hxy : y = f x) :
+    f ⁻¹' t ∈ 𝓝[f ⁻¹' s] x := by
+  rw [hxy] at ht
+  exact h.preimage_mem_nhdsWithin' (nhdsWithin_mono _ (image_preimage_subset f s) ht)
+
 theorem Filter.EventuallyEq.congr_continuousWithinAt {f g : α → β} {s : Set α} {x : α}
     (h : f =ᶠ[𝓝[s] x] g) (hx : f x = g x) : ContinuousWithinAt f s x ↔ ContinuousWithinAt g s x :=
   by rw [ContinuousWithinAt, hx, tendsto_congr' h, ContinuousWithinAt]

@@ -607,7 +607,7 @@ theorem continuousOn_iff_continuous_restrict {f : α → β} {s : Set α} :
 #align continuous_on_iff_continuous_restrict continuousOn_iff_continuous_restrict
 
 -- porting note: 2 new lemmas
-alias continuousOn_iff_continuous_restrict ↔ ContinuousOn.restrict _
+alias ⟨ContinuousOn.restrict, _⟩ := continuousOn_iff_continuous_restrict
 
 theorem ContinuousOn.restrict_mapsTo {f : α → β} {s : Set α} {t : Set β} (hf : ContinuousOn f s)
     (ht : MapsTo f s t) : Continuous (ht.restrict f s t) :=
@@ -769,7 +769,7 @@ theorem continuousWithinAt_insert_self {f : α → β} {x : α} {s : Set α} :
     true_and_iff]
 #align continuous_within_at_insert_self continuousWithinAt_insert_self
 
-alias continuousWithinAt_insert_self ↔ _ ContinuousWithinAt.insert_self
+alias ⟨_, ContinuousWithinAt.insert_self⟩ := continuousWithinAt_insert_self
 #align continuous_within_at.insert_self ContinuousWithinAt.insert_self
 
 theorem ContinuousWithinAt.diff_iff {f : α → β} {s t : Set α} {x : α}

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

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

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

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

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

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

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

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

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

@@ -28,6 +28,8 @@ equipped with the subspace topology.
 
 -/
 
+set_option autoImplicit true
+
 
 open Set Filter Function Topology Filter
 

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

This has nice performance benefits.

@@ -31,7 +31,7 @@ equipped with the subspace topology.
 
 open Set Filter Function Topology Filter
 
-variable {α : Type _} {β : Type _} {γ : Type _} {δ : Type _}
+variable {α : Type*} {β : Type*} {γ : Type*} {δ : Type*}
 
 variable [TopologicalSpace α]
 
@@ -302,21 +302,21 @@ theorem nhdsWithin_compl_singleton_sup_pure (a : α) : 𝓝[≠] a ⊔ pure a =
   rw [← nhdsWithin_singleton, ← nhdsWithin_union, compl_union_self, nhdsWithin_univ]
 #align nhds_within_compl_singleton_sup_pure nhdsWithin_compl_singleton_sup_pure
 
-theorem nhdsWithin_prod {α : Type _} [TopologicalSpace α] {β : Type _} [TopologicalSpace β]
+theorem nhdsWithin_prod {α : Type*} [TopologicalSpace α] {β : Type*} [TopologicalSpace β]
     {s u : Set α} {t v : Set β} {a : α} {b : β} (hu : u ∈ 𝓝[s] a) (hv : v ∈ 𝓝[t] b) :
     u ×ˢ v ∈ 𝓝[s ×ˢ t] (a, b) := by
   rw [nhdsWithin_prod_eq]
   exact prod_mem_prod hu hv
 #align nhds_within_prod nhdsWithin_prod
 
-theorem nhdsWithin_pi_eq' {ι : Type _} {α : ι → Type _} [∀ i, TopologicalSpace (α i)] {I : Set ι}
+theorem nhdsWithin_pi_eq' {ι : Type*} {α : ι → Type*} [∀ i, TopologicalSpace (α i)] {I : Set ι}
     (hI : I.Finite) (s : ∀ i, Set (α i)) (x : ∀ i, α i) :
     𝓝[pi I s] x = ⨅ i, comap (fun x => x i) (𝓝 (x i) ⊓ ⨅ (_ : i ∈ I), 𝓟 (s i)) := by
   simp only [nhdsWithin, nhds_pi, Filter.pi, comap_inf, comap_iInf, pi_def, comap_principal, ←
     iInf_principal_finite hI, ← iInf_inf_eq]
 #align nhds_within_pi_eq' nhdsWithin_pi_eq'
 
-theorem nhdsWithin_pi_eq {ι : Type _} {α : ι → Type _} [∀ i, TopologicalSpace (α i)] {I : Set ι}
+theorem nhdsWithin_pi_eq {ι : Type*} {α : ι → Type*} [∀ i, TopologicalSpace (α i)] {I : Set ι}
     (hI : I.Finite) (s : ∀ i, Set (α i)) (x : ∀ i, α i) :
     𝓝[pi I s] x =
       (⨅ i ∈ I, comap (fun x => x i) (𝓝[s i] x i)) ⊓
@@ -327,17 +327,17 @@ theorem nhdsWithin_pi_eq {ι : Type _} {α : ι → Type _} [∀ i, TopologicalS
   simp only [iInf_inf_eq]
 #align nhds_within_pi_eq nhdsWithin_pi_eq
 
-theorem nhdsWithin_pi_univ_eq {ι : Type _} {α : ι → Type _} [Finite ι] [∀ i, TopologicalSpace (α i)]
+theorem nhdsWithin_pi_univ_eq {ι : Type*} {α : ι → Type*} [Finite ι] [∀ i, TopologicalSpace (α i)]
     (s : ∀ i, Set (α i)) (x : ∀ i, α i) : 𝓝[pi univ s] x = ⨅ i, comap (fun x => x i) (𝓝[s i] x i) :=
   by simpa [nhdsWithin] using nhdsWithin_pi_eq finite_univ s x
 #align nhds_within_pi_univ_eq nhdsWithin_pi_univ_eq
 
-theorem nhdsWithin_pi_eq_bot {ι : Type _} {α : ι → Type _} [∀ i, TopologicalSpace (α i)] {I : Set ι}
+theorem nhdsWithin_pi_eq_bot {ι : Type*} {α : ι → Type*} [∀ i, TopologicalSpace (α i)] {I : Set ι}
     {s : ∀ i, Set (α i)} {x : ∀ i, α i} : 𝓝[pi I s] x = ⊥ ↔ ∃ i ∈ I, 𝓝[s i] x i = ⊥ := by
   simp only [nhdsWithin, nhds_pi, pi_inf_principal_pi_eq_bot]
 #align nhds_within_pi_eq_bot nhdsWithin_pi_eq_bot
 
-theorem nhdsWithin_pi_neBot {ι : Type _} {α : ι → Type _} [∀ i, TopologicalSpace (α i)] {I : Set ι}
+theorem nhdsWithin_pi_neBot {ι : Type*} {α : ι → Type*} [∀ i, TopologicalSpace (α i)] {I : Set ι}
     {s : ∀ i, Set (α i)} {x : ∀ i, α i} : (𝓝[pi I s] x).NeBot ↔ ∀ i ∈ I, (𝓝[s i] x i).NeBot := by
   simp [neBot_iff, nhdsWithin_pi_eq_bot]
 #align nhds_within_pi_ne_bot nhdsWithin_pi_neBot
@@ -388,7 +388,7 @@ theorem tendsto_nhds_of_tendsto_nhdsWithin {f : β → α} {a : α} {s : Set α}
 #align tendsto_nhds_of_tendsto_nhds_within tendsto_nhds_of_tendsto_nhdsWithin
 
 -- todo: move to `Mathlib.Filter.Order.Basic` or drop
-theorem principal_subtype {α : Type _} (s : Set α) (t : Set s) :
+theorem principal_subtype {α : Type*} (s : Set α) (t : Set s) :
     𝓟 t = comap (↑) (𝓟 (((↑) : s → α) '' t)) := by
   rw [comap_principal, preimage_image_eq _ Subtype.coe_injective]
 #align principal_subtype principal_subtype
@@ -402,22 +402,22 @@ theorem IsClosed.mem_of_nhdsWithin_neBot {s : Set α} (hs : IsClosed s) {x : α}
   hs.closure_eq ▸ mem_closure_iff_nhdsWithin_neBot.2 hx
 #align is_closed.mem_of_nhds_within_ne_bot IsClosed.mem_of_nhdsWithin_neBot
 
-theorem DenseRange.nhdsWithin_neBot {ι : Type _} {f : ι → α} (h : DenseRange f) (x : α) :
+theorem DenseRange.nhdsWithin_neBot {ι : Type*} {f : ι → α} (h : DenseRange f) (x : α) :
     NeBot (𝓝[range f] x) :=
   mem_closure_iff_clusterPt.1 (h x)
 #align dense_range.nhds_within_ne_bot DenseRange.nhdsWithin_neBot
 
-theorem mem_closure_pi {ι : Type _} {α : ι → Type _} [∀ i, TopologicalSpace (α i)] {I : Set ι}
+theorem mem_closure_pi {ι : Type*} {α : ι → Type*} [∀ i, TopologicalSpace (α i)] {I : Set ι}
     {s : ∀ i, Set (α i)} {x : ∀ i, α i} : x ∈ closure (pi I s) ↔ ∀ i ∈ I, x i ∈ closure (s i) := by
   simp only [mem_closure_iff_nhdsWithin_neBot, nhdsWithin_pi_neBot]
 #align mem_closure_pi mem_closure_pi
 
-theorem closure_pi_set {ι : Type _} {α : ι → Type _} [∀ i, TopologicalSpace (α i)] (I : Set ι)
+theorem closure_pi_set {ι : Type*} {α : ι → Type*} [∀ i, TopologicalSpace (α i)] (I : Set ι)
     (s : ∀ i, Set (α i)) : closure (pi I s) = pi I fun i => closure (s i) :=
   Set.ext fun _ => mem_closure_pi
 #align closure_pi_set closure_pi_set
 
-theorem dense_pi {ι : Type _} {α : ι → Type _} [∀ i, TopologicalSpace (α i)] {s : ∀ i, Set (α i)}
+theorem dense_pi {ι : Type*} {α : ι → Type*} [∀ i, TopologicalSpace (α i)] {s : ∀ i, Set (α i)}
     (I : Set ι) (hs : ∀ i ∈ I, Dense (s i)) : Dense (pi I s) := by
   simp only [dense_iff_closure_eq, closure_pi_set, pi_congr rfl fun i hi => (hs i hi).closure_eq,
     pi_univ]
@@ -471,7 +471,7 @@ theorem Filter.EventuallyEq.eq_of_nhdsWithin {s : Set α} {f g : α → β} {a :
   h.self_of_nhdsWithin hmem
 #align filter.eventually_eq.eq_of_nhds_within Filter.EventuallyEq.eq_of_nhdsWithin
 
-theorem eventually_nhdsWithin_of_eventually_nhds {α : Type _} [TopologicalSpace α] {s : Set α}
+theorem eventually_nhdsWithin_of_eventually_nhds {α : Type*} [TopologicalSpace α] {s : Set α}
     {a : α} {p : α → Prop} (h : ∀ᶠ x in 𝓝 a, p x) : ∀ᶠ x in 𝓝[s] a, p x :=
   mem_nhdsWithin_of_mem_nhds h
 #align eventually_nhds_within_of_eventually_nhds eventually_nhdsWithin_of_eventually_nhds
@@ -564,25 +564,25 @@ theorem ContinuousWithinAt.prod_map {f : α → γ} {g : β → δ} {s : Set α}
   exact hf.prod_map hg
 #align continuous_within_at.prod_map ContinuousWithinAt.prod_map
 
-theorem continuousWithinAt_pi {ι : Type _} {π : ι → Type _} [∀ i, TopologicalSpace (π i)]
+theorem continuousWithinAt_pi {ι : Type*} {π : ι → Type*} [∀ i, TopologicalSpace (π i)]
     {f : α → ∀ i, π i} {s : Set α} {x : α} :
     ContinuousWithinAt f s x ↔ ∀ i, ContinuousWithinAt (fun y => f y i) s x :=
   tendsto_pi_nhds
 #align continuous_within_at_pi continuousWithinAt_pi
 
-theorem continuousOn_pi {ι : Type _} {π : ι → Type _} [∀ i, TopologicalSpace (π i)]
+theorem continuousOn_pi {ι : Type*} {π : ι → Type*} [∀ i, TopologicalSpace (π i)]
     {f : α → ∀ i, π i} {s : Set α} : ContinuousOn f s ↔ ∀ i, ContinuousOn (fun y => f y i) s :=
   ⟨fun h i x hx => tendsto_pi_nhds.1 (h x hx) i, fun h x hx => tendsto_pi_nhds.2 fun i => h i x hx⟩
 #align continuous_on_pi continuousOn_pi
 
-theorem ContinuousWithinAt.fin_insertNth {n} {π : Fin (n + 1) → Type _}
+theorem ContinuousWithinAt.fin_insertNth {n} {π : Fin (n + 1) → Type*}
     [∀ i, TopologicalSpace (π i)] (i : Fin (n + 1)) {f : α → π i} {a : α} {s : Set α}
     (hf : ContinuousWithinAt f s a) {g : α → ∀ j : Fin n, π (i.succAbove j)}
     (hg : ContinuousWithinAt g s a) : ContinuousWithinAt (fun a => i.insertNth (f a) (g a)) s a :=
   hf.tendsto.fin_insertNth i hg
 #align continuous_within_at.fin_insert_nth ContinuousWithinAt.fin_insertNth
 
-nonrec theorem ContinuousOn.fin_insertNth {n} {π : Fin (n + 1) → Type _}
+nonrec theorem ContinuousOn.fin_insertNth {n} {π : Fin (n + 1) → Type*}
     [∀ i, TopologicalSpace (π i)] (i : Fin (n + 1)) {f : α → π i} {s : Set α}
     (hf : ContinuousOn f s) {g : α → ∀ j : Fin n, π (i.succAbove j)} (hg : ContinuousOn g s) :
     ContinuousOn (fun a => i.insertNth (f a) (g a)) s := fun a ha =>
@@ -625,7 +625,7 @@ theorem continuousOn_iff' {f : α → β} {s : Set α} :
 
 /-- If a function is continuous on a set for some topologies, then it is
 continuous on the same set with respect to any finer topology on the source space. -/
-theorem ContinuousOn.mono_dom {α β : Type _} {t₁ t₂ : TopologicalSpace α} {t₃ : TopologicalSpace β}
+theorem ContinuousOn.mono_dom {α β : Type*} {t₁ t₂ : TopologicalSpace α} {t₃ : TopologicalSpace β}
     (h₁ : t₂ ≤ t₁) {s : Set α} {f : α → β} (h₂ : @ContinuousOn α β t₁ t₃ f s) :
     @ContinuousOn α β t₂ t₃ f s := fun x hx _u hu =>
   map_mono (inf_le_inf_right _ <| nhds_mono h₁) (h₂ x hx hu)
@@ -633,7 +633,7 @@ theorem ContinuousOn.mono_dom {α β : Type _} {t₁ t₂ : TopologicalSpace α}
 
 /-- If a function is continuous on a set for some topologies, then it is
 continuous on the same set with respect to any coarser topology on the target space. -/
-theorem ContinuousOn.mono_rng {α β : Type _} {t₁ : TopologicalSpace α} {t₂ t₃ : TopologicalSpace β}
+theorem ContinuousOn.mono_rng {α β : Type*} {t₁ : TopologicalSpace α} {t₂ t₃ : TopologicalSpace β}
     (h₁ : t₂ ≤ t₃) {s : Set α} {f : α → β} (h₂ : @ContinuousOn α β t₁ t₂ f s) :
     @ContinuousOn α β t₁ t₃ f s := fun x hx _u hu =>
   h₂ x hx <| nhds_mono h₁ hu
@@ -653,7 +653,7 @@ theorem ContinuousOn.prod_map {f : α → γ} {g : β → δ} {s : Set α} {t :
   fun ⟨x, y⟩ ⟨hx, hy⟩ => ContinuousWithinAt.prod_map (hf x hx) (hg y hy)
 #align continuous_on.prod_map ContinuousOn.prod_map
 
-theorem continuous_of_cover_nhds {ι : Sort _} {f : α → β} {s : ι → Set α}
+theorem continuous_of_cover_nhds {ι : Sort*} {f : α → β} {s : ι → Set α}
     (hs : ∀ x : α, ∃ i, s i ∈ 𝓝 x) (hf : ∀ i, ContinuousOn f (s i)) :
     Continuous f :=
   continuous_iff_continuousAt.mpr fun x ↦ let ⟨i, hi⟩ := hs x; by
@@ -1050,7 +1050,7 @@ theorem continuousOn_to_generateFrom_iff {s : Set α} {T : Set (Set β)} {f : α
     exact forall_congr' fun t => forall_swap
 
 -- porting note: dropped an unneeded assumption
-theorem continuousOn_open_of_generateFrom {β : Type _} {s : Set α} {T : Set (Set β)} {f : α → β}
+theorem continuousOn_open_of_generateFrom {β : Type*} {s : Set α} {T : Set (Set β)} {f : α → β}
     (h : ∀ t ∈ T, IsOpen (s ∩ f ⁻¹' t)) :
     @ContinuousOn α β _ (.generateFrom T) f s :=
   continuousOn_to_generateFrom_iff.2 fun _x hx t ht hxt => mem_nhdsWithin.2
@@ -1139,7 +1139,7 @@ theorem ContinuousOn.piecewise' {s t : Set α} {f g : α → β} [∀ a, Decidab
   hf.if' hpf hpg hg
 #align continuous_on.piecewise' ContinuousOn.piecewise'
 
-theorem ContinuousOn.if {α β : Type _} [TopologicalSpace α] [TopologicalSpace β] {p : α → Prop}
+theorem ContinuousOn.if {α β : Type*} [TopologicalSpace α] [TopologicalSpace β] {p : α → Prop}
     [∀ a, Decidable (p a)] {s : Set α} {f g : α → β}
     (hp : ∀ a ∈ s ∩ frontier { a | p a }, f a = g a)
     (hf : ContinuousOn f <| s ∩ closure { a | p a })

• 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) 2019 Reid Barton. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Sébastien Gouëzel
-
-! This file was ported from Lean 3 source module topology.continuous_on
-! leanprover-community/mathlib commit d4f691b9e5f94cfc64639973f3544c95f8d5d494
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
 -/
 import Mathlib.Topology.Constructions
 
+#align_import topology.continuous_on from "leanprover-community/mathlib"@"d4f691b9e5f94cfc64639973f3544c95f8d5d494"
+
 /-!
 # Neighborhoods and continuity relative to a subset
 

Co-authored-by: Yury G. Kudryashov <urkud@urkud.name>

@@ -787,7 +787,7 @@ theorem continuousWithinAt_diff_self {f : α → β} {s : Set α} {x : α} :
 
 @[simp]
 theorem continuousWithinAt_compl_self {f : α → β} {a : α} :
-    ContinuousWithinAt f ({a}ᶜ) a ↔ ContinuousAt f a := by
+    ContinuousWithinAt f {a}ᶜ a ↔ ContinuousAt f a := by
   rw [compl_eq_univ_diff, continuousWithinAt_diff_self, continuousWithinAt_univ]
 #align continuous_within_at_compl_self continuousWithinAt_compl_self
 
@@ -1122,7 +1122,7 @@ theorem ContinuousOn.if' {s : Set α} {p : α → Prop} {f g : α → β} [∀ a
     cases' hx with hx hx
     · apply ContinuousWithinAt.union
       · exact (hf x hx).congr (fun y hy => if_pos hy.2) (if_pos hx.2)
-      · have : x ∉ closure ({ a | p a }ᶜ) := fun h => hx' ⟨subset_closure hx.2, by
+      · have : x ∉ closure { a | p a }ᶜ := fun h => hx' ⟨subset_closure hx.2, by
           rwa [closure_compl] at h⟩
         exact continuousWithinAt_of_not_mem_closure fun h =>
           this (closure_inter_subset_inter_closure _ _ h).2
@@ -1165,7 +1165,7 @@ theorem ContinuousOn.if {α β : Type _} [TopologicalSpace α] [TopologicalSpace
 
 theorem ContinuousOn.piecewise {s t : Set α} {f g : α → β} [∀ a, Decidable (a ∈ t)]
     (ht : ∀ a ∈ s ∩ frontier t, f a = g a) (hf : ContinuousOn f <| s ∩ closure t)
-    (hg : ContinuousOn g <| s ∩ closure (tᶜ)) : ContinuousOn (piecewise t f g) s :=
+    (hg : ContinuousOn g <| s ∩ closure tᶜ) : ContinuousOn (piecewise t f g) s :=
   hf.if ht hg
 #align continuous_on.piecewise ContinuousOn.piecewise
 
@@ -1205,7 +1205,7 @@ theorem Continuous.if_const (p : Prop) {f g : α → β} [Decidable p] (hf : Con
 
 theorem continuous_piecewise {s : Set α} {f g : α → β} [∀ a, Decidable (a ∈ s)]
     (hs : ∀ a ∈ frontier s, f a = g a) (hf : ContinuousOn f (closure s))
-    (hg : ContinuousOn g (closure (sᶜ))) : Continuous (piecewise s f g) :=
+    (hg : ContinuousOn g (closure sᶜ)) : Continuous (piecewise s f g) :=
   continuous_if hs hf hg
 #align continuous_piecewise continuous_piecewise
 
@@ -1236,13 +1236,13 @@ theorem ite_inter_closure_eq_of_inter_frontier_eq {s s' t : Set α}
 #align ite_inter_closure_eq_of_inter_frontier_eq ite_inter_closure_eq_of_inter_frontier_eq
 
 theorem ite_inter_closure_compl_eq_of_inter_frontier_eq {s s' t : Set α}
-    (ht : s ∩ frontier t = s' ∩ frontier t) : t.ite s s' ∩ closure (tᶜ) = s' ∩ closure (tᶜ) := by
+    (ht : s ∩ frontier t = s' ∩ frontier t) : t.ite s s' ∩ closure tᶜ = s' ∩ closure tᶜ := by
   rw [← ite_compl, ite_inter_closure_eq_of_inter_frontier_eq]
   rwa [frontier_compl, eq_comm]
 #align ite_inter_closure_compl_eq_of_inter_frontier_eq ite_inter_closure_compl_eq_of_inter_frontier_eq
 
 theorem continuousOn_piecewise_ite' {s s' t : Set α} {f f' : α → β} [∀ x, Decidable (x ∈ t)]
-    (h : ContinuousOn f (s ∩ closure t)) (h' : ContinuousOn f' (s' ∩ closure (tᶜ)))
+    (h : ContinuousOn f (s ∩ closure t)) (h' : ContinuousOn f' (s' ∩ closure tᶜ))
     (H : s ∩ frontier t = s' ∩ frontier t) (Heq : EqOn f f' (s ∩ frontier t)) :
     ContinuousOn (t.piecewise f f') (t.ite s s') := by
   apply ContinuousOn.piecewise

Instead of fixing a proof "as is", I'm golfing it and moving parts of it to lemmas.

@@ -703,6 +703,10 @@ theorem ContinuousWithinAt.mono_of_mem {f : α → β} {s t : Set α} {x : α}
   h.mono_left (nhdsWithin_le_of_mem hs)
 #align continuous_within_at.mono_of_mem ContinuousWithinAt.mono_of_mem
 
+theorem continuousWithinAt_congr_nhds {f : α → β} (h : 𝓝[s] x = 𝓝[t] x) :
+    ContinuousWithinAt f s x ↔ ContinuousWithinAt f t x := by
+  simp only [ContinuousWithinAt, h]
+
 theorem continuousWithinAt_inter' {f : α → β} {s t : Set α} {x : α} (h : t ∈ 𝓝[s] x) :
     ContinuousWithinAt f (s ∩ t) x ↔ ContinuousWithinAt f s x := by
   simp [ContinuousWithinAt, nhdsWithin_restrict'' s h]

Changes are of the form

• some_tactic at h⊢ -> some_tactic at h ⊢
• some_tactic at h -> some_tactic at h

@@ -68,7 +68,7 @@ theorem mem_closure_ne_iff_frequently_within {z : α} {s : Set α} :
 theorem eventually_nhdsWithin_nhdsWithin {a : α} {s : Set α} {p : α → Prop} :
     (∀ᶠ y in 𝓝[s] a, ∀ᶠ x in 𝓝[s] y, p x) ↔ ∀ᶠ x in 𝓝[s] a, p x := by
   refine' ⟨fun h => _, fun h => (eventually_nhds_nhdsWithin.2 h).filter_mono inf_le_left⟩
-  simp only [eventually_nhdsWithin_iff] at h⊢
+  simp only [eventually_nhdsWithin_iff] at h ⊢
   exact h.mono fun x hx hxs => (hx hxs).self_of_nhds hxs
 #align eventually_nhds_within_nhds_within eventually_nhdsWithin_nhdsWithin
 

@@ -314,7 +314,7 @@ theorem nhdsWithin_prod {α : Type _} [TopologicalSpace α] {β : Type _} [Topol
 
 theorem nhdsWithin_pi_eq' {ι : Type _} {α : ι → Type _} [∀ i, TopologicalSpace (α i)] {I : Set ι}
     (hI : I.Finite) (s : ∀ i, Set (α i)) (x : ∀ i, α i) :
-    𝓝[pi I s] x = ⨅ i, comap (fun x => x i) (𝓝 (x i) ⊓ ⨅ (_hi : i ∈ I), 𝓟 (s i)) := by
+    𝓝[pi I s] x = ⨅ i, comap (fun x => x i) (𝓝 (x i) ⊓ ⨅ (_ : i ∈ I), 𝓟 (s i)) := by
   simp only [nhdsWithin, nhds_pi, Filter.pi, comap_inf, comap_iInf, pi_def, comap_principal, ←
     iInf_principal_finite hI, ← iInf_inf_eq]
 #align nhds_within_pi_eq' nhdsWithin_pi_eq'
@@ -323,7 +323,7 @@ theorem nhdsWithin_pi_eq {ι : Type _} {α : ι → Type _} [∀ i, TopologicalS
     (hI : I.Finite) (s : ∀ i, Set (α i)) (x : ∀ i, α i) :
     𝓝[pi I s] x =
       (⨅ i ∈ I, comap (fun x => x i) (𝓝[s i] x i)) ⊓
-        ⨅ (i) (_hi : i ∉ I), comap (fun x => x i) (𝓝 (x i)) := by
+        ⨅ (i) (_ : i ∉ I), comap (fun x => x i) (𝓝 (x i)) := by
   simp only [nhdsWithin, nhds_pi, Filter.pi, pi_def, ← iInf_principal_finite hI, comap_inf,
     comap_principal, eval]
   rw [iInf_split _ fun i => i ∈ I, inf_right_comm]

• Protect HasFTaylorSeriesUpToOn.fderivWithin, IsBoundedBilinearMap.fderiv, and IsBoundedBilinearMap.fderivWithin.

• Rename

  • isBoundedBilinearMapSmul -> isBoundedBilinearMap_smul;
  • isBoundedBilinearMapMul -> isBoundedBilinearMap_mul;
  • isBoundedBilinearMapComp -> isBoundedBilinearMap_comp;
  • isBoundedBilinearMapSmulRight -> isBoundedBilinearMap_smulRight;
  • isBoundedBilinearMapCompMultilinear -> isBoundedBilinearMap_compMultilinear;
  • ContinuousLinearMap.mulLeftRightIsBoundedBilinear -> ContinuousLinearMap.mulLeftRight_isBoundedBilinear;
  • nhdsWithin_eq_nhds_within' -> nhdsWithin_eq_nhdsWithin';
  • ContinuousWithinAt.preimage_mem_nhds_within' -> ContinuousWithinAt.preimage_mem_nhdsWithin'.

Co-authored-by: Floris van Doorn <fpvdoorn@gmail.com> Co-authored-by: Parcly Taxel <reddeloostw@gmail.com>

@@ -208,9 +208,9 @@ theorem nhdsWithin_le_nhds {a : α} {s : Set α} : 𝓝[s] a ≤ 𝓝 a := by
   exact univ_mem
 #align nhds_within_le_nhds nhdsWithin_le_nhds
 
-theorem nhdsWithin_eq_nhds_within' {a : α} {s t u : Set α} (hs : s ∈ 𝓝 a) (h₂ : t ∩ s = u ∩ s) :
+theorem nhdsWithin_eq_nhdsWithin' {a : α} {s t u : Set α} (hs : s ∈ 𝓝 a) (h₂ : t ∩ s = u ∩ s) :
     𝓝[t] a = 𝓝[u] a := by rw [nhdsWithin_restrict' t hs, nhdsWithin_restrict' u hs, h₂]
-#align nhds_within_eq_nhds_within' nhdsWithin_eq_nhds_within'
+#align nhds_within_eq_nhds_within' nhdsWithin_eq_nhdsWithin'
 
 theorem nhdsWithin_eq_nhdsWithin {a : α} {s t u : Set α} (h₀ : a ∈ s) (h₁ : IsOpen s)
     (h₂ : t ∩ s = u ∩ s) : 𝓝[t] a = 𝓝[u] a := by
@@ -947,10 +947,10 @@ theorem Function.LeftInverse.map_nhds_eq {f : α → β} {g : β → α} {x : β
     (h.leftInvOn univ).map_nhdsWithin_eq (h x) (by rwa [image_univ]) hg.continuousWithinAt
 #align function.left_inverse.map_nhds_eq Function.LeftInverse.map_nhds_eq
 
-theorem ContinuousWithinAt.preimage_mem_nhds_within' {f : α → β} {x : α} {s : Set α} {t : Set β}
+theorem ContinuousWithinAt.preimage_mem_nhdsWithin' {f : α → β} {x : α} {s : Set α} {t : Set β}
     (h : ContinuousWithinAt f s x) (ht : t ∈ 𝓝[f '' s] f x) : f ⁻¹' t ∈ 𝓝[s] x :=
   h.tendsto_nhdsWithin (mapsTo_image _ _) ht
-#align continuous_within_at.preimage_mem_nhds_within' ContinuousWithinAt.preimage_mem_nhds_within'
+#align continuous_within_at.preimage_mem_nhds_within' ContinuousWithinAt.preimage_mem_nhdsWithin'
 
 theorem Filter.EventuallyEq.congr_continuousWithinAt {f g : α → β} {s : Set α} {x : α}
     (h : f =ᶠ[𝓝[s] x] g) (hx : f x = g x) : ContinuousWithinAt f s x ↔ ContinuousWithinAt g s x :=

Run codespell Mathlib and keep some suggestions.

@@ -136,7 +136,7 @@ theorem nhdsWithin_le_iff {s t : Set α} {x : α} : 𝓝[s] x ≤ 𝓝[t] x ↔
   set_eventuallyLE_iff_inf_principal_le.symm.trans set_eventuallyLE_iff_mem_inf_principal
 #align nhds_within_le_iff nhdsWithin_le_iff
 
--- porting note: golfed, droped an unneeded assumption
+-- porting note: golfed, dropped an unneeded assumption
 theorem preimage_nhdsWithin_coinduced' {π : α → β} {s : Set β} {t : Set α} {a : α} (h : a ∈ t)
     (hs : s ∈ @nhds β (.coinduced (fun x : t => π x) inferInstance) (π a)) :
     π ⁻¹' s ∈ 𝓝[t] a := by
@@ -1048,7 +1048,7 @@ theorem continuousOn_to_generateFrom_iff {s : Set α} {T : Set (Set β)} {f : α
       and_imp]
     exact forall_congr' fun t => forall_swap
 
--- porting note: droped an unneeded assumption
+-- porting note: dropped an unneeded assumption
 theorem continuousOn_open_of_generateFrom {β : Type _} {s : Set α} {T : Set (Set β)} {f : α → β}
     (h : ∀ t ∈ T, IsOpen (s ∩ f ⁻¹' t)) :
     @ContinuousOn α β _ (.generateFrom T) f s :=

The actual forward-porting was done in #4327 and #4328

@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Sébastien Gouëzel
 
 ! This file was ported from Lean 3 source module topology.continuous_on
-! leanprover-community/mathlib commit 55d771df074d0dd020139ee1cd4b95521422df9f
+! leanprover-community/mathlib commit d4f691b9e5f94cfc64639973f3544c95f8d5d494
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/

Forward-port leanprover-community/mathlib#19070:

• add Filter.set_eventuallyLE_iff_mem_inf_principal, Filter.set_eventuallyLE_iff_inf_principal_le, and Filter.set_eventuallyEq_iff_inf_principal;
• golf nhdsWithin_eq_iff_eventuallyEq and nhdsWithin_le_iff.

The original PR golfs one more lemma which was already golfed to 1 line in Mathlib 4.

@@ -128,22 +128,12 @@ theorem mem_nhdsWithin_iff_eventuallyEq {s t : Set α} {x : α} :
   simp_rw [mem_nhdsWithin_iff_eventually, eventuallyEq_set, mem_inter_iff, iff_self_and]
 #align mem_nhds_within_iff_eventually_eq mem_nhdsWithin_iff_eventuallyEq
 
-theorem nhdsWithin_eq_iff_eventuallyEq {s t : Set α} {x : α} : 𝓝[s] x = 𝓝[t] x ↔ s =ᶠ[𝓝 x] t := by
-  simp_rw [Filter.ext_iff, mem_nhdsWithin_iff_eventually, eventuallyEq_set]
-  constructor
-  · intro h
-    filter_upwards [(h t).mpr (eventually_of_forall fun x => id),
-      (h s).mp (eventually_of_forall fun x => id)]
-    exact fun x => Iff.intro
-  · refine' fun h u => eventually_congr (h.mono fun x h => _)
-    rw [h]
+theorem nhdsWithin_eq_iff_eventuallyEq {s t : Set α} {x : α} : 𝓝[s] x = 𝓝[t] x ↔ s =ᶠ[𝓝 x] t :=
+  set_eventuallyEq_iff_inf_principal.symm
 #align nhds_within_eq_iff_eventually_eq nhdsWithin_eq_iff_eventuallyEq
 
-theorem nhdsWithin_le_iff {s t : Set α} {x : α} : 𝓝[s] x ≤ 𝓝[t] x ↔ t ∈ 𝓝[s] x := by
-  simp_rw [Filter.le_def, mem_nhdsWithin_iff_eventually]
-  constructor
-  · exact fun h => (h t <| eventually_of_forall fun x => id).mono fun x => id
-  · exact fun h u hu => (h.and hu).mono fun x hx h => hx.2 <| hx.1 h
+theorem nhdsWithin_le_iff {s t : Set α} {x : α} : 𝓝[s] x ≤ 𝓝[t] x ↔ t ∈ 𝓝[s] x :=
+  set_eventuallyLE_iff_inf_principal_le.symm.trans set_eventuallyLE_iff_mem_inf_principal
 #align nhds_within_le_iff nhdsWithin_le_iff
 
 -- porting note: golfed, droped an unneeded assumption

As discussed on Zulip

Renames

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

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

@@ -74,7 +74,7 @@ theorem eventually_nhdsWithin_nhdsWithin {a : α} {s : Set α} {p : α → Prop}
 
 theorem nhdsWithin_eq (a : α) (s : Set α) :
     𝓝[s] a = ⨅ t ∈ { t : Set α | a ∈ t ∧ IsOpen t }, 𝓟 (t ∩ s) :=
-  ((nhds_basis_opens a).inf_principal s).eq_binfᵢ
+  ((nhds_basis_opens a).inf_principal s).eq_biInf
 #align nhds_within_eq nhdsWithin_eq
 
 theorem nhdsWithin_univ (a : α) : 𝓝[Set.univ] a = 𝓝 a := by
@@ -252,21 +252,21 @@ theorem nhdsWithin_union (a : α) (s t : Set α) : 𝓝[s ∪ t] a = 𝓝[s] a 
   rw [← inf_sup_left, sup_principal]
 #align nhds_within_union nhdsWithin_union
 
-theorem nhdsWithin_bunionᵢ {ι} {I : Set ι} (hI : I.Finite) (s : ι → Set α) (a : α) :
+theorem nhdsWithin_biUnion {ι} {I : Set ι} (hI : I.Finite) (s : ι → Set α) (a : α) :
     𝓝[⋃ i ∈ I, s i] a = ⨆ i ∈ I, 𝓝[s i] a :=
   Set.Finite.induction_on hI (by simp) fun _ _ hT ↦ by
-    simp only [hT, nhdsWithin_union, supᵢ_insert, bunionᵢ_insert]
-#align nhds_within_bUnion nhdsWithin_bunionᵢ
+    simp only [hT, nhdsWithin_union, iSup_insert, biUnion_insert]
+#align nhds_within_bUnion nhdsWithin_biUnion
 
-theorem nhdsWithin_unionₛ {S : Set (Set α)} (hS : S.Finite) (a : α) :
+theorem nhdsWithin_sUnion {S : Set (Set α)} (hS : S.Finite) (a : α) :
     𝓝[⋃₀ S] a = ⨆ s ∈ S, 𝓝[s] a := by
-  rw [unionₛ_eq_bunionᵢ, nhdsWithin_bunionᵢ hS]
-#align nhds_within_sUnion nhdsWithin_unionₛ
+  rw [sUnion_eq_biUnion, nhdsWithin_biUnion hS]
+#align nhds_within_sUnion nhdsWithin_sUnion
 
-theorem nhdsWithin_unionᵢ {ι} [Finite ι] (s : ι → Set α) (a : α) :
+theorem nhdsWithin_iUnion {ι} [Finite ι] (s : ι → Set α) (a : α) :
     𝓝[⋃ i, s i] a = ⨆ i, 𝓝[s i] a := by
-  rw [← unionₛ_range, nhdsWithin_unionₛ (finite_range s), supᵢ_range]
-#align nhds_within_Union nhdsWithin_unionᵢ
+  rw [← sUnion_range, nhdsWithin_sUnion (finite_range s), iSup_range]
+#align nhds_within_Union nhdsWithin_iUnion
 
 theorem nhdsWithin_inter (a : α) (s t : Set α) : 𝓝[s ∩ t] a = 𝓝[s] a ⊓ 𝓝[t] a := by
   delta nhdsWithin
@@ -325,8 +325,8 @@ theorem nhdsWithin_prod {α : Type _} [TopologicalSpace α] {β : Type _} [Topol
 theorem nhdsWithin_pi_eq' {ι : Type _} {α : ι → Type _} [∀ i, TopologicalSpace (α i)] {I : Set ι}
     (hI : I.Finite) (s : ∀ i, Set (α i)) (x : ∀ i, α i) :
     𝓝[pi I s] x = ⨅ i, comap (fun x => x i) (𝓝 (x i) ⊓ ⨅ (_hi : i ∈ I), 𝓟 (s i)) := by
-  simp only [nhdsWithin, nhds_pi, Filter.pi, comap_inf, comap_infᵢ, pi_def, comap_principal, ←
-    infᵢ_principal_finite hI, ← infᵢ_inf_eq]
+  simp only [nhdsWithin, nhds_pi, Filter.pi, comap_inf, comap_iInf, pi_def, comap_principal, ←
+    iInf_principal_finite hI, ← iInf_inf_eq]
 #align nhds_within_pi_eq' nhdsWithin_pi_eq'
 
 theorem nhdsWithin_pi_eq {ι : Type _} {α : ι → Type _} [∀ i, TopologicalSpace (α i)] {I : Set ι}
@@ -334,10 +334,10 @@ theorem nhdsWithin_pi_eq {ι : Type _} {α : ι → Type _} [∀ i, TopologicalS
     𝓝[pi I s] x =
       (⨅ i ∈ I, comap (fun x => x i) (𝓝[s i] x i)) ⊓
         ⨅ (i) (_hi : i ∉ I), comap (fun x => x i) (𝓝 (x i)) := by
-  simp only [nhdsWithin, nhds_pi, Filter.pi, pi_def, ← infᵢ_principal_finite hI, comap_inf,
+  simp only [nhdsWithin, nhds_pi, Filter.pi, pi_def, ← iInf_principal_finite hI, comap_inf,
     comap_principal, eval]
-  rw [infᵢ_split _ fun i => i ∈ I, inf_right_comm]
-  simp only [infᵢ_inf_eq]
+  rw [iInf_split _ fun i => i ∈ I, inf_right_comm]
+  simp only [iInf_inf_eq]
 #align nhds_within_pi_eq nhdsWithin_pi_eq
 
 theorem nhdsWithin_pi_univ_eq {ι : Type _} {α : ι → Type _} [Finite ι] [∀ i, TopologicalSpace (α i)]
@@ -370,7 +370,7 @@ theorem Filter.Tendsto.if_nhdsWithin {f g : α → β} {p : α → Prop} [Decida
 
 theorem map_nhdsWithin (f : α → β) (a : α) (s : Set α) :
     map f (𝓝[s] a) = ⨅ t ∈ { t : Set α | a ∈ t ∧ IsOpen t }, 𝓟 (f '' (t ∩ s)) :=
-  ((nhdsWithin_basis_open a s).map f).eq_binfᵢ
+  ((nhdsWithin_basis_open a s).map f).eq_biInf
 #align map_nhds_within map_nhdsWithin
 
 theorem tendsto_nhdsWithin_mono_left {f : α → β} {a : α} {s t : Set α} {l : Filter β} (hst : s ⊆ t)
@@ -390,7 +390,7 @@ theorem tendsto_nhdsWithin_of_tendsto_nhds {f : α → β} {a : α} {s : Set α}
 
 theorem eventually_mem_of_tendsto_nhdsWithin {f : β → α} {a : α} {s : Set α} {l : Filter β}
     (h : Tendsto f l (𝓝[s] a)) : ∀ᶠ i in l, f i ∈ s := by
-  simp_rw [nhdsWithin_eq, tendsto_infᵢ, mem_setOf_eq, tendsto_principal, mem_inter_iff,
+  simp_rw [nhdsWithin_eq, tendsto_iInf, mem_setOf_eq, tendsto_principal, mem_inter_iff,
     eventually_and] at h
   exact (h univ ⟨mem_univ a, isOpen_univ⟩).2
 #align eventually_mem_of_tendsto_nhds_within eventually_mem_of_tendsto_nhdsWithin
@@ -1054,7 +1054,7 @@ theorem continuousOn_to_generateFrom_iff {s : Set α} {T : Set (Set β)} {f : α
     @ContinuousOn α β _ (.generateFrom T) f s ↔ ∀ x ∈ s, ∀ t ∈ T, f x ∈ t → f ⁻¹' t ∈ 𝓝[s] x :=
   forall₂_congr <| fun x _ => by
     delta ContinuousWithinAt
-    simp only [TopologicalSpace.nhds_generateFrom, tendsto_infᵢ, tendsto_principal, mem_setOf_eq,
+    simp only [TopologicalSpace.nhds_generateFrom, tendsto_iInf, tendsto_principal, mem_setOf_eq,
       and_imp]
     exact forall_congr' fun t => forall_swap
 

This PR puts, with one exception, every single remaining by that lies all by itself on its own line to the previous line, thus matching the current behaviour of start-port.sh. The exception is when the by begins the second or later argument to a tuple or anonymous constructor; see https://github.com/leanprover-community/mathlib4/pull/3825#discussion_r1186702599.

Essentially this is s/\n *by$/ by/g, but with manual editing to satisfy the linter's max-100-char-line requirement. The Python style linter is also modified to catch these "isolated bys".

@@ -626,8 +626,7 @@ theorem ContinuousOn.restrict_mapsTo {f : α → β} {s : Set α} {t : Set β} (
 
 theorem continuousOn_iff' {f : α → β} {s : Set α} :
     ContinuousOn f s ↔ ∀ t : Set β, IsOpen t → ∃ u, IsOpen u ∧ f ⁻¹' t ∩ s = u ∩ s := by
-  have : ∀ t, IsOpen (s.restrict f ⁻¹' t) ↔ ∃ u : Set α, IsOpen u ∧ f ⁻¹' t ∩ s = u ∩ s :=
-    by
+  have : ∀ t, IsOpen (s.restrict f ⁻¹' t) ↔ ∃ u : Set α, IsOpen u ∧ f ⁻¹' t ∩ s = u ∩ s := by
     intro t
     rw [isOpen_induced_iff, Set.restrict_eq, Set.preimage_comp]
     simp only [Subtype.preimage_coe_eq_preimage_coe_iff]
@@ -655,8 +654,7 @@ theorem ContinuousOn.mono_rng {α β : Type _} {t₁ : TopologicalSpace α} {t
 
 theorem continuousOn_iff_isClosed {f : α → β} {s : Set α} :
     ContinuousOn f s ↔ ∀ t : Set β, IsClosed t → ∃ u, IsClosed u ∧ f ⁻¹' t ∩ s = u ∩ s := by
-  have : ∀ t, IsClosed (s.restrict f ⁻¹' t) ↔ ∃ u : Set α, IsClosed u ∧ f ⁻¹' t ∩ s = u ∩ s :=
-    by
+  have : ∀ t, IsClosed (s.restrict f ⁻¹' t) ↔ ∃ u : Set α, IsClosed u ∧ f ⁻¹' t ∩ s = u ∩ s := by
     intro t
     rw [isClosed_induced_iff, Set.restrict_eq, Set.preimage_comp]
     simp only [Subtype.preimage_coe_eq_preimage_coe_iff, eq_comm]

Match https://github.com/leanprover-community/mathlib/pull/18900

Co-authored-by: Eric Wieser <wieser.eric@gmail.com>

@@ -1096,7 +1096,7 @@ theorem Embedding.continuousOn_iff {f : α → β} {g : β → γ} (hg : Embeddi
 
 theorem Embedding.map_nhdsWithin_eq {f : α → β} (hf : Embedding f) (s : Set α) (x : α) :
     map f (𝓝[s] x) = 𝓝[f '' s] f x := by
-  rw [nhdsWithin, map_inf hf.inj, hf.map_nhds_eq, map_principal, ← nhdsWithin_inter',
+  rw [nhdsWithin, Filter.map_inf hf.inj, hf.map_nhds_eq, map_principal, ← nhdsWithin_inter',
     inter_eq_self_of_subset_right (image_subset_range _ _)]
 #align embedding.map_nhds_within_eq Embedding.map_nhdsWithin_eq
 

Co-authored-by: Eric Wieser <wieser.eric@gmail.com>

@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Sébastien Gouëzel
 
 ! This file was ported from Lean 3 source module topology.continuous_on
-! leanprover-community/mathlib commit bcfa726826abd57587355b4b5b7e78ad6527b7e4
+! leanprover-community/mathlib commit 55d771df074d0dd020139ee1cd4b95521422df9f
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
@@ -227,9 +227,9 @@ theorem nhdsWithin_eq_nhdsWithin {a : α} {s t u : Set α} (h₀ : a ∈ s) (h
   rw [nhdsWithin_restrict t h₀ h₁, nhdsWithin_restrict u h₀ h₁, h₂]
 #align nhds_within_eq_nhds_within nhdsWithin_eq_nhdsWithin
 
--- porting note: new lemma; todo: make it `@[simp]`
-theorem nhdsWithin_eq_nhds {a : α} {s : Set α} : 𝓝[s] a = 𝓝 a ↔ s ∈ 𝓝 a :=
+@[simp] theorem nhdsWithin_eq_nhds {a : α} {s : Set α} : 𝓝[s] a = 𝓝 a ↔ s ∈ 𝓝 a :=
   inf_eq_left.trans le_principal_iff
+#align nhds_within_eq_nhds nhdsWithin_eq_nhds
 
 theorem IsOpen.nhdsWithin_eq {a : α} {s : Set α} (h : IsOpen s) (ha : a ∈ s) : 𝓝[s] a = 𝓝 a :=
   nhdsWithin_eq_nhds.2 <| h.mem_nhds ha
@@ -252,6 +252,22 @@ theorem nhdsWithin_union (a : α) (s t : Set α) : 𝓝[s ∪ t] a = 𝓝[s] a 
   rw [← inf_sup_left, sup_principal]
 #align nhds_within_union nhdsWithin_union
 
+theorem nhdsWithin_bunionᵢ {ι} {I : Set ι} (hI : I.Finite) (s : ι → Set α) (a : α) :
+    𝓝[⋃ i ∈ I, s i] a = ⨆ i ∈ I, 𝓝[s i] a :=
+  Set.Finite.induction_on hI (by simp) fun _ _ hT ↦ by
+    simp only [hT, nhdsWithin_union, supᵢ_insert, bunionᵢ_insert]
+#align nhds_within_bUnion nhdsWithin_bunionᵢ
+
+theorem nhdsWithin_unionₛ {S : Set (Set α)} (hS : S.Finite) (a : α) :
+    𝓝[⋃₀ S] a = ⨆ s ∈ S, 𝓝[s] a := by
+  rw [unionₛ_eq_bunionᵢ, nhdsWithin_bunionᵢ hS]
+#align nhds_within_sUnion nhdsWithin_unionₛ
+
+theorem nhdsWithin_unionᵢ {ι} [Finite ι] (s : ι → Set α) (a : α) :
+    𝓝[⋃ i, s i] a = ⨆ i, 𝓝[s i] a := by
+  rw [← unionₛ_range, nhdsWithin_unionₛ (finite_range s), supᵢ_range]
+#align nhds_within_Union nhdsWithin_unionᵢ
+
 theorem nhdsWithin_inter (a : α) (s t : Set α) : 𝓝[s ∩ t] a = 𝓝[s] a ⊓ 𝓝[t] a := by
   delta nhdsWithin
   rw [inf_left_comm, inf_assoc, inf_principal, ← inf_assoc, inf_idem]
@@ -267,9 +283,9 @@ theorem nhdsWithin_inter_of_mem {a : α} {s t : Set α} (h : s ∈ 𝓝[t] a) :
   exact nhdsWithin_le_of_mem h
 #align nhds_within_inter_of_mem nhdsWithin_inter_of_mem
 
--- porting note: new lemma
 theorem nhdsWithin_inter_of_mem' {a : α} {s t : Set α} (h : t ∈ 𝓝[s] a) : 𝓝[s ∩ t] a = 𝓝[s] a := by
   rw [inter_comm, nhdsWithin_inter_of_mem h]
+#align nhds_within_inter_of_mem' nhdsWithin_inter_of_mem'
 
 @[simp]
 theorem nhdsWithin_singleton (a : α) : 𝓝[{a}] a = pure a := by
@@ -652,6 +668,14 @@ theorem ContinuousOn.prod_map {f : α → γ} {g : β → δ} {s : Set α} {t :
   fun ⟨x, y⟩ ⟨hx, hy⟩ => ContinuousWithinAt.prod_map (hf x hx) (hg y hy)
 #align continuous_on.prod_map ContinuousOn.prod_map
 
+theorem continuous_of_cover_nhds {ι : Sort _} {f : α → β} {s : ι → Set α}
+    (hs : ∀ x : α, ∃ i, s i ∈ 𝓝 x) (hf : ∀ i, ContinuousOn f (s i)) :
+    Continuous f :=
+  continuous_iff_continuousAt.mpr fun x ↦ let ⟨i, hi⟩ := hs x; by
+    rw [ContinuousAt, ← nhdsWithin_eq_nhds.2 hi]
+    exact hf _ _ (mem_of_mem_nhds hi)
+#align continuous_of_cover_nhds continuous_of_cover_nhds
+
 theorem continuousOn_empty (f : α → β) : ContinuousOn f ∅ := fun _ => False.elim
 #align continuous_on_empty continuousOn_empty
 

This introduces a tactic congr! that is an analogue to mathlib 3's congr'. It is a more insistent version of congr that makes use of more congruence lemmas (including user congruence lemmas), propext, funext, and Subsingleton instances. It also has a feature to lift reflexive relations to equalities. Along with funext, the tactic does intros, allowing congr! to get access to function bodies; the introduced variables can be named using rename_i if needed.

This also modifies convert to use congr! rather than congr, which makes it work more like the mathlib3 version of the tactic.

@@ -1204,7 +1204,7 @@ theorem IsOpen.ite' {s s' t : Set α} (hs : IsOpen s) (hs' : IsOpen s')
   classical
     simp only [isOpen_iff_continuous_mem, Set.ite] at *
     convert continuous_piecewise (fun x hx => propext (ht x hx)) hs.continuousOn hs'.continuousOn
-    ext x
+    rename_i x
     by_cases hx : x ∈ t <;> simp [hx]
 #align is_open.ite' IsOpen.ite'
 
@@ -1290,4 +1290,3 @@ theorem continuousWithinAt_prod_iff {f : α → β × γ} {s : Set α} {x : α}
       ContinuousWithinAt (Prod.fst ∘ f) s x ∧ ContinuousWithinAt (Prod.snd ∘ f) s x :=
   ⟨fun h => ⟨h.fst, h.snd⟩, fun ⟨h1, h2⟩ => h1.prod h2⟩
 #align continuous_within_at_prod_iff continuousWithinAt_prod_iff
-

@@ -119,12 +119,8 @@ theorem nhds_of_nhdsWithin_of_nhds {s t : Set α} {a : α} (h1 : s ∈ 𝓝 a) (
 #align nhds_of_nhds_within_of_nhds nhds_of_nhdsWithin_of_nhds
 
 theorem mem_nhdsWithin_iff_eventually {s t : Set α} {x : α} :
-    t ∈ 𝓝[s] x ↔ ∀ᶠ y in 𝓝 x, y ∈ s → y ∈ t := by
-  rw [mem_nhdsWithin_iff_exists_mem_nhds_inter]
-  constructor
-  · rintro ⟨u, hu, hut⟩
-    exact eventually_of_mem hu fun x hxu hxs => hut ⟨hxu, hxs⟩
-  · refine' fun h => ⟨_, h, fun y hy => hy.1 hy.2⟩
+    t ∈ 𝓝[s] x ↔ ∀ᶠ y in 𝓝 x, y ∈ s → y ∈ t :=
+  eventually_inf_principal
 #align mem_nhds_within_iff_eventually mem_nhdsWithin_iff_eventually
 
 theorem mem_nhdsWithin_iff_eventuallyEq {s t : Set α} {x : α} :
@@ -271,6 +267,10 @@ theorem nhdsWithin_inter_of_mem {a : α} {s t : Set α} (h : s ∈ 𝓝[t] a) :
   exact nhdsWithin_le_of_mem h
 #align nhds_within_inter_of_mem nhdsWithin_inter_of_mem
 
+-- porting note: new lemma
+theorem nhdsWithin_inter_of_mem' {a : α} {s t : Set α} (h : t ∈ 𝓝[s] a) : 𝓝[s ∩ t] a = 𝓝[s] a := by
+  rw [inter_comm, nhdsWithin_inter_of_mem h]
+
 @[simp]
 theorem nhdsWithin_singleton (a : α) : 𝓝[{a}] a = pure a := by
   rw [nhdsWithin, principal_singleton, inf_eq_right.2 (pure_le_nhds a)]

Co-authored-by: Anatole Dedecker <anatolededecker@gmail.com> Co-authored-by: Johan Commelin <johan@commelin.net>

@@ -231,8 +231,12 @@ theorem nhdsWithin_eq_nhdsWithin {a : α} {s t u : Set α} (h₀ : a ∈ s) (h
   rw [nhdsWithin_restrict t h₀ h₁, nhdsWithin_restrict u h₀ h₁, h₂]
 #align nhds_within_eq_nhds_within nhdsWithin_eq_nhdsWithin
 
+-- porting note: new lemma; todo: make it `@[simp]`
+theorem nhdsWithin_eq_nhds {a : α} {s : Set α} : 𝓝[s] a = 𝓝 a ↔ s ∈ 𝓝 a :=
+  inf_eq_left.trans le_principal_iff
+
 theorem IsOpen.nhdsWithin_eq {a : α} {s : Set α} (h : IsOpen s) (ha : a ∈ s) : 𝓝[s] a = 𝓝 a :=
-  inf_eq_left.2 <| le_principal_iff.2 <| IsOpen.mem_nhds h ha
+  nhdsWithin_eq_nhds.2 <| h.mem_nhds ha
 #align is_open.nhds_within_eq IsOpen.nhdsWithin_eq
 
 theorem preimage_nhds_within_coinduced {π : α → β} {s : Set β} {t : Set α} {a : α} (h : a ∈ t)

@@ -597,6 +597,13 @@ theorem continuousOn_iff_continuous_restrict {f : α → β} {s : Set α} :
   exact (continuousWithinAt_iff_continuousAt_restrict f xs).mpr (h ⟨x, xs⟩)
 #align continuous_on_iff_continuous_restrict continuousOn_iff_continuous_restrict
 
+-- porting note: 2 new lemmas
+alias continuousOn_iff_continuous_restrict ↔ ContinuousOn.restrict _
+
+theorem ContinuousOn.restrict_mapsTo {f : α → β} {s : Set α} {t : Set β} (hf : ContinuousOn f s)
+    (ht : MapsTo f s t) : Continuous (ht.restrict f s t) :=
+  hf.restrict.codRestrict _
+
 theorem continuousOn_iff' {f : α → β} {s : Set α} :
     ContinuousOn f s ↔ ∀ t : Set β, IsOpen t → ∃ u, IsOpen u ∧ f ⁻¹' t ∩ s = u ∩ s := by
   have : ∀ t, IsOpen (s.restrict f ⁻¹' t) ↔ ∃ u : Set α, IsOpen u ∧ f ⁻¹' t ∩ s = u ∩ s :=