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

@@ -892,7 +892,7 @@ theorem isOpen_prod_iff {s : Set (α × β)} :
   rw [isOpen_iff_nhds]
   simp_rw [le_principal_iff, Prod.forall,
     ((nhds_basis_opens _).prod_nhds (nhds_basis_opens _)).mem_iff, Prod.exists, exists_prop]
-  simp only [and_assoc', and_left_comm]
+  simp only [and_assoc, and_left_comm]
 #align is_open_prod_iff isOpen_prod_iff
 -/
 

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

@@ -1744,7 +1744,7 @@ theorem isOpen_pi_iff {s : Set (∀ a, π a)} :
   by
   rw [isOpen_iff_nhds]
   simp_rw [le_principal_iff, nhds_pi, Filter.mem_pi', mem_nhds_iff, exists_prop]
-  refine' ball_congr fun a h => ⟨_, _⟩
+  refine' forall₂_congr fun a h => ⟨_, _⟩
   · rintro ⟨I, t, ⟨h1, h2⟩⟩
     refine' ⟨I, fun a => eval a '' (I : Set ι).pi fun a => (h1 a).some, fun i hi => _, _⟩
     · simp_rw [Set.eval_image_pi (finset.mem_coe.mpr hi)
@@ -1776,7 +1776,7 @@ theorem isOpen_pi_iff' [Finite ι] {s : Set (∀ a, π a)} :
   cases nonempty_fintype ι
   rw [isOpen_iff_nhds]
   simp_rw [le_principal_iff, nhds_pi, Filter.mem_pi', mem_nhds_iff, exists_prop]
-  refine' ball_congr fun a h => ⟨_, _⟩
+  refine' forall₂_congr fun a h => ⟨_, _⟩
   · rintro ⟨I, t, ⟨h1, h2⟩⟩
     refine'
       ⟨fun i => (h1 i).some,

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

@@ -1631,11 +1631,11 @@ theorem continuousAt_apply (i : ι) (x : ∀ i, π i) : ContinuousAt (fun p : 
 #align continuous_at_apply continuousAt_apply
 -/
 
-#print Filter.Tendsto.apply /-
-theorem Filter.Tendsto.apply {l : Filter β} {f : β → ∀ i, π i} {x : ∀ i, π i}
+#print Filter.Tendsto.apply_nhds /-
+theorem Filter.Tendsto.apply_nhds {l : Filter β} {f : β → ∀ i, π i} {x : ∀ i, π i}
     (h : Tendsto f l (𝓝 x)) (i : ι) : Tendsto (fun a => f a i) l (𝓝 <| x i) :=
   (continuousAt_apply i _).Tendsto.comp h
-#align filter.tendsto.apply Filter.Tendsto.apply
+#align filter.tendsto.apply Filter.Tendsto.apply_nhds
 -/
 
 #print nhds_pi /-

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

@@ -1401,7 +1401,7 @@ theorem IsClosed.closedEmbedding_subtype_val {s : Set α} (hs : IsClosed s) :
     ClosedEmbedding (coe : { x // x ∈ s } → α) :=
   { induced := rfl
     inj := Subtype.coe_injective
-    closed_range := (Subtype.range_coe : range coe = s).symm ▸ hs }
+    isClosed_range := (Subtype.range_coe : range coe = s).symm ▸ hs }
 #align is_closed.closed_embedding_subtype_coe IsClosed.closedEmbedding_subtype_val
 -/
 
@@ -1985,7 +1985,7 @@ theorem isClosedMap_sigmaMk {i : ι} : IsClosedMap (@Sigma.mk ι σ i) :=
 
 #print isClosed_range_sigmaMk /-
 theorem isClosed_range_sigmaMk {i : ι} : IsClosed (Set.range (@Sigma.mk ι σ i)) :=
-  isClosedMap_sigmaMk.closed_range
+  isClosedMap_sigmaMk.isClosed_range
 #align is_closed_range_sigma_mk isClosed_range_sigmaMk
 -/
 

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

@@ -669,18 +669,18 @@ theorem continuous_swap : Continuous (Prod.swap : α × β → β × α) :=
 #align continuous_swap continuous_swap
 -/
 
-#print continuous_uncurry_left /-
-theorem continuous_uncurry_left {f : α → β → γ} (a : α) (h : Continuous (uncurry f)) :
+#print Continuous.uncurry_left /-
+theorem Continuous.uncurry_left {f : α → β → γ} (a : α) (h : Continuous (uncurry f)) :
     Continuous (f a) :=
   show Continuous (uncurry f ∘ fun b => (a, b)) from h.comp (by continuity)
-#align continuous_uncurry_left continuous_uncurry_left
+#align continuous_uncurry_left Continuous.uncurry_left
 -/
 
-#print continuous_uncurry_right /-
-theorem continuous_uncurry_right {f : α → β → γ} (b : β) (h : Continuous (uncurry f)) :
+#print Continuous.uncurry_right /-
+theorem Continuous.uncurry_right {f : α → β → γ} (b : β) (h : Continuous (uncurry f)) :
     Continuous fun a => f a b :=
   show Continuous (uncurry f ∘ fun a => (a, b)) from h.comp (by continuity)
-#align continuous_uncurry_right continuous_uncurry_right
+#align continuous_uncurry_right Continuous.uncurry_right
 -/
 
 #print continuous_curry /-
@@ -808,7 +808,7 @@ theorem Filter.Tendsto.prod_mk_nhds {γ} {a : α} {b : β} {f : Filter γ} {ma :
 #print Filter.Eventually.curry_nhds /-
 theorem Filter.Eventually.curry_nhds {p : α × β → Prop} {x : α} {y : β}
     (h : ∀ᶠ x in 𝓝 (x, y), p x) : ∀ᶠ x' in 𝓝 x, ∀ᶠ y' in 𝓝 y, p (x', y') := by
-  rw [nhds_prod_eq] at h ; exact h.curry
+  rw [nhds_prod_eq] at h; exact h.curry
 #align filter.eventually.curry_nhds Filter.Eventually.curry_nhds
 -/
 
@@ -929,9 +929,9 @@ theorem map_fst_nhdsWithin (x : α × β) : map Prod.fst (𝓝[Prod.snd ⁻¹' {
   by
   refine' le_antisymm (continuous_at_fst.mono_left inf_le_left) fun s hs => _
   rcases x with ⟨x, y⟩
-  rw [mem_map, nhdsWithin, mem_inf_principal, mem_nhds_prod_iff] at hs 
+  rw [mem_map, nhdsWithin, mem_inf_principal, mem_nhds_prod_iff] at hs
   rcases hs with ⟨u, hu, v, hv, H⟩
-  simp only [prod_subset_iff, mem_singleton_iff, mem_set_of_eq, mem_preimage] at H 
+  simp only [prod_subset_iff, mem_singleton_iff, mem_set_of_eq, mem_preimage] at H
   exact mem_of_superset hu fun z hz => H _ hz _ (mem_of_mem_nhds hv) rfl
 #align map_fst_nhds_within map_fst_nhdsWithin
 -/
@@ -957,9 +957,9 @@ theorem map_snd_nhdsWithin (x : α × β) : map Prod.snd (𝓝[Prod.fst ⁻¹' {
   by
   refine' le_antisymm (continuous_at_snd.mono_left inf_le_left) fun s hs => _
   rcases x with ⟨x, y⟩
-  rw [mem_map, nhdsWithin, mem_inf_principal, mem_nhds_prod_iff] at hs 
+  rw [mem_map, nhdsWithin, mem_inf_principal, mem_nhds_prod_iff] at hs
   rcases hs with ⟨u, hu, v, hv, H⟩
-  simp only [prod_subset_iff, mem_singleton_iff, mem_set_of_eq, mem_preimage] at H 
+  simp only [prod_subset_iff, mem_singleton_iff, mem_set_of_eq, mem_preimage] at H
   exact mem_of_superset hv fun z hz => H _ (mem_of_mem_nhds hu) _ hz rfl
 #align map_snd_nhds_within map_snd_nhdsWithin
 -/
@@ -1000,7 +1000,7 @@ theorem isOpen_prod_iff' {s : Set α} {t : Set β} :
       · rw [← snd_image_prod st.1 t]
         exact isOpenMap_snd _ H
     · intro H
-      simp only [st.1.ne_empty, st.2.ne_empty, not_false_iff, or_false_iff] at H 
+      simp only [st.1.ne_empty, st.2.ne_empty, not_false_iff, or_false_iff] at H
       exact H.1.Prod H.2
 #align is_open_prod_iff' isOpen_prod_iff'
 -/
@@ -1800,7 +1800,7 @@ theorem isClosed_set_pi {i : Set ι} {s : ∀ a, Set (π a)} (hs : ∀ a ∈ i,
 
 #print mem_nhds_of_pi_mem_nhds /-
 theorem mem_nhds_of_pi_mem_nhds {I : Set ι} {s : ∀ i, Set (π i)} (a : ∀ i, π i) (hs : I.pi s ∈ 𝓝 a)
-    {i : ι} (hi : i ∈ I) : s i ∈ 𝓝 (a i) := by rw [nhds_pi] at hs ; exact mem_of_pi_mem_pi hs hi
+    {i : ι} (hi : i ∈ I) : s i ∈ 𝓝 (a i) := by rw [nhds_pi] at hs; exact mem_of_pi_mem_pi hs hi
 #align mem_nhds_of_pi_mem_nhds mem_nhds_of_pi_mem_nhds
 -/
 
@@ -1829,7 +1829,7 @@ theorem interior_pi_set {I : Set ι} (hI : I.Finite) {s : ∀ i, Set (π i)} :
 theorem exists_finset_piecewise_mem_of_mem_nhds [DecidableEq ι] {s : Set (∀ a, π a)} {x : ∀ a, π a}
     (hs : s ∈ 𝓝 x) (y : ∀ a, π a) : ∃ I : Finset ι, I.piecewise x y ∈ s :=
   by
-  simp only [nhds_pi, Filter.mem_pi'] at hs 
+  simp only [nhds_pi, Filter.mem_pi'] at hs
   rcases hs with ⟨I, t, htx, hts⟩
   refine' ⟨I, hts fun i hi => _⟩
   simpa [Finset.mem_coe.1 hi] using mem_of_mem_nhds (htx i)

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

@@ -397,7 +397,7 @@ theorem isOpen_iff {s : Set (CofiniteTopology α)} : IsOpen s ↔ s.Nonempty →
 
 #print CofiniteTopology.isOpen_iff' /-
 theorem isOpen_iff' {s : Set (CofiniteTopology α)} : IsOpen s ↔ s = ∅ ∨ sᶜ.Finite := by
-  simp only [is_open_iff, nonempty_iff_ne_empty, or_iff_not_imp_left]
+  simp only [is_open_iff, nonempty_iff_ne_empty, Classical.or_iff_not_imp_left]
 #align cofinite_topology.is_open_iff' CofiniteTopology.isOpen_iff'
 -/
 

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

@@ -702,7 +702,6 @@ theorem nhds_prod_eq {a : α} {b : β} : 𝓝 (a, b) = 𝓝 a ×ᶠ 𝓝 b := by
 #align nhds_prod_eq nhds_prod_eq
 -/
 
-#print continuous_uncurry_of_discreteTopology /-
 /-- If a function `f x y` is such that `y ↦ f x y` is continuous for all `x`, and `x` lives in a
 discrete space, then `f` is continuous. -/
 theorem continuous_uncurry_of_discreteTopology [DiscreteTopology α] {f : α → β → γ}
@@ -714,7 +713,6 @@ theorem continuous_uncurry_of_discreteTopology [DiscreteTopology α] {f : α →
   rw [nhds_prod_eq, nhds_discrete, Filter.map_pure_prod]
   exact (hf a).ContinuousAt
 #align continuous_uncurry_of_discrete_topology continuous_uncurry_of_discreteTopology
--/
 
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
 #print mem_nhds_prod_iff /-
@@ -907,14 +905,12 @@ theorem prod_induced_induced {α γ : Type _} (f : α → β) (g : γ → δ) :
 #align prod_induced_induced prod_induced_induced
 -/
 
-#print continuous_uncurry_of_discreteTopology_left /-
 theorem continuous_uncurry_of_discreteTopology_left [DiscreteTopology α] {f : α → β → γ}
     (h : ∀ a, Continuous (f a)) : Continuous (uncurry f) :=
   continuous_iff_continuousAt.2 fun ⟨a, b⟩ => by
     simp only [ContinuousAt, nhds_prod_eq, nhds_discrete α, pure_prod, tendsto_map'_iff, (· ∘ ·),
       uncurry, (h a).Tendsto]
 #align continuous_uncurry_of_discrete_topology_left continuous_uncurry_of_discreteTopology_left
--/
 
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
 #print exists_nhds_square /-

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

@@ -3,8 +3,8 @@ Copyright (c) 2017 Johannes Hölzl. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Johannes Hölzl, Mario Carneiro, Patrick Massot
 -/
-import Mathbin.Topology.Maps
-import Mathbin.Order.Filter.Pi
+import Topology.Maps
+import Order.Filter.Pi
 
 #align_import topology.constructions from "leanprover-community/mathlib"@"f7ebde7ee0d1505dfccac8644ae12371aa3c1c9f"
 

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

@@ -276,7 +276,7 @@ theorem Quotient.preimage_mem_nhds [TopologicalSpace α] [s : Setoid α] {V : Se
 /-- The image of a dense set under `quotient.mk` is a dense set. -/
 theorem Dense.quotient [Setoid α] [TopologicalSpace α] {s : Set α} (H : Dense s) :
     Dense (Quotient.mk' '' s) :=
-  (surjective_quotient_mk α).DenseRange.dense_image continuous_coinduced_rng H
+  (surjective_quotient_mk' α).DenseRange.dense_image continuous_coinduced_rng H
 #align dense.quotient Dense.quotient
 -/
 
@@ -284,7 +284,7 @@ theorem Dense.quotient [Setoid α] [TopologicalSpace α] {s : Set α} (H : Dense
 /-- The composition of `quotient.mk` and a function with dense range has dense range. -/
 theorem DenseRange.quotient [Setoid α] [TopologicalSpace α] {f : β → α} (hf : DenseRange f) :
     DenseRange (Quotient.mk' ∘ f) :=
-  (surjective_quotient_mk α).DenseRange.comp hf continuous_coinduced_rng
+  (surjective_quotient_mk' α).DenseRange.comp hf continuous_coinduced_rng
 #align dense_range.quotient DenseRange.quotient
 -/
 

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

@@ -1733,7 +1733,8 @@ theorem Continuous.fin_insertNth {n} {π : Fin (n + 1) → Type _} [∀ i, Topol
 #print isOpen_set_pi /-
 theorem isOpen_set_pi {i : Set ι} {s : ∀ a, Set (π a)} (hi : i.Finite)
     (hs : ∀ a ∈ i, IsOpen (s a)) : IsOpen (pi i s) := by
-  rw [pi_def] <;> exact isOpen_biInter hi fun a ha => (hs _ ha).Preimage (continuous_apply _)
+  rw [pi_def] <;>
+    exact Set.Finite.isOpen_biInter hi fun a ha => (hs _ ha).Preimage (continuous_apply _)
 #align is_open_set_pi isOpen_set_pi
 -/
 
@@ -1864,7 +1865,7 @@ theorem pi_generateFrom_eq {π : ι → Type _} {g : ∀ a, Set (Set (π a))} :
   exact fun s ⟨t, i, ht, Eq⟩ => ⟨t, i, fun a ha => generate_open.basic _ (ht a ha), Eq⟩
   · rintro s ⟨t, i, hi, rfl⟩
     rw [pi_def]
-    apply isOpen_biInter (Finset.finite_toSet _)
+    apply Set.Finite.isOpen_biInter (Finset.finite_toSet _)
     intro a ha; show ((generate_from G).coinduced fun f : ∀ a, π a => f a).IsOpen (t a)
     refine' le_generateFrom _ _ (hi a ha)
     exact fun s hs => generate_open.basic _ ⟨update (fun a => univ) a s, {a}, by simp [hs]⟩
@@ -1922,7 +1923,9 @@ instance Pi.discreteTopology : DiscreteTopology (∀ i, π i) :=
     by
     rw [show {x} = ⋂ i, {y : ∀ i, π i | y i = x i} by ext;
         simp only [funext_iff, Set.mem_singleton_iff, Set.mem_iInter, Set.mem_setOf_eq]]
-    exact isOpen_iInter fun i => (continuous_apply i).isOpen_preimage {x i} (isOpen_discrete {x i})
+    exact
+      isOpen_iInter_of_finite fun i =>
+        (continuous_apply i).isOpen_preimage {x i} (isOpen_discrete {x i})
 #align Pi.discrete_topology Pi.discreteTopology
 -/
 

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

@@ -1477,7 +1477,7 @@ theorem continuousAt_codRestrict_iff {f : α → β} {t : Set β} (h1 : ∀ x, f
 #align continuous_at_cod_restrict_iff continuousAt_codRestrict_iff
 -/
 
-alias continuousAt_codRestrict_iff ↔ _ ContinuousAt.codRestrict
+alias ⟨_, ContinuousAt.codRestrict⟩ := continuousAt_codRestrict_iff
 #align continuous_at.cod_restrict ContinuousAt.codRestrict
 
 #print ContinuousAt.restrict /-

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

@@ -2,15 +2,12 @@
 Copyright (c) 2017 Johannes Hölzl. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Johannes Hölzl, Mario Carneiro, Patrick Massot
-
-! This file was ported from Lean 3 source module topology.constructions
-! leanprover-community/mathlib commit f7ebde7ee0d1505dfccac8644ae12371aa3c1c9f
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
 -/
 import Mathbin.Topology.Maps
 import Mathbin.Order.Filter.Pi
 
+#align_import topology.constructions from "leanprover-community/mathlib"@"f7ebde7ee0d1505dfccac8644ae12371aa3c1c9f"
+
 /-!
 # Constructions of new topological spaces from old ones
 

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

@@ -1709,7 +1709,8 @@ theorem continuous_mulSingle [∀ i, One (π i)] [DecidableEq ι] (i : ι) :
 #print Filter.Tendsto.fin_insertNth /-
 theorem Filter.Tendsto.fin_insertNth {n} {π : Fin (n + 1) → Type _} [∀ i, TopologicalSpace (π i)]
     (i : Fin (n + 1)) {f : β → π i} {l : Filter β} {x : π i} (hf : Tendsto f l (𝓝 x))
-    {g : β → ∀ j : Fin n, π (i.succAbove j)} {y : ∀ j, π (i.succAbove j)} (hg : Tendsto g l (𝓝 y)) :
+    {g : β → ∀ j : Fin n, π (i.succAboveEmb j)} {y : ∀ j, π (i.succAboveEmb j)}
+    (hg : Tendsto g l (𝓝 y)) :
     Tendsto (fun a => i.insertNth (f a) (g a)) l (𝓝 <| i.insertNth x y) :=
   tendsto_pi_nhds.2 fun j => Fin.succAboveCases i (by simpa) (by simpa using tendsto_pi_nhds.1 hg) j
 #align filter.tendsto.fin_insert_nth Filter.Tendsto.fin_insertNth
@@ -1718,7 +1719,7 @@ theorem Filter.Tendsto.fin_insertNth {n} {π : Fin (n + 1) → Type _} [∀ i, T
 #print ContinuousAt.fin_insertNth /-
 theorem ContinuousAt.fin_insertNth {n} {π : Fin (n + 1) → Type _} [∀ i, TopologicalSpace (π i)]
     (i : Fin (n + 1)) {f : α → π i} {a : α} (hf : ContinuousAt f a)
-    {g : α → ∀ j : Fin n, π (i.succAbove j)} (hg : ContinuousAt g a) :
+    {g : α → ∀ j : Fin n, π (i.succAboveEmb j)} (hg : ContinuousAt g a) :
     ContinuousAt (fun a => i.insertNth (f a) (g a)) a :=
   hf.fin_insertNth i hg
 #align continuous_at.fin_insert_nth ContinuousAt.fin_insertNth
@@ -1726,7 +1727,7 @@ theorem ContinuousAt.fin_insertNth {n} {π : Fin (n + 1) → Type _} [∀ i, Top
 
 #print Continuous.fin_insertNth /-
 theorem Continuous.fin_insertNth {n} {π : Fin (n + 1) → Type _} [∀ i, TopologicalSpace (π i)]
-    (i : Fin (n + 1)) {f : α → π i} (hf : Continuous f) {g : α → ∀ j : Fin n, π (i.succAbove j)}
+    (i : Fin (n + 1)) {f : α → π i} (hf : Continuous f) {g : α → ∀ j : Fin n, π (i.succAboveEmb j)}
     (hg : Continuous g) : Continuous fun a => i.insertNth (f a) (g a) :=
   continuous_iff_continuousAt.2 fun a => hf.ContinuousAt.fin_insertNth i hg.ContinuousAt
 #align continuous.fin_insert_nth Continuous.fin_insertNth

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

@@ -295,20 +295,25 @@ instance {p : α → Prop} [TopologicalSpace α] [DiscreteTopology α] : Discret
   ⟨bot_unique fun s hs =>
       ⟨coe '' s, isOpen_discrete _, Set.preimage_image_eq _ Subtype.coe_injective⟩⟩
 
+#print Sum.discreteTopology /-
 instance Sum.discreteTopology [TopologicalSpace α] [TopologicalSpace β] [hα : DiscreteTopology α]
     [hβ : DiscreteTopology β] : DiscreteTopology (Sum α β) :=
   ⟨by unfold Sum.topologicalSpace <;> simp [hα.eq_bot, hβ.eq_bot]⟩
 #align sum.discrete_topology Sum.discreteTopology
+-/
 
+#print Sigma.discreteTopology /-
 instance Sigma.discreteTopology {β : α → Type v} [∀ a, TopologicalSpace (β a)]
     [h : ∀ a, DiscreteTopology (β a)] : DiscreteTopology (Sigma β) :=
   ⟨by unfold Sigma.topologicalSpace; simp [fun a => (h a).eq_bot]⟩
 #align sigma.discrete_topology Sigma.discreteTopology
+-/
 
 section Topα
 
 variable [TopologicalSpace α]
 
+#print mem_nhds_subtype /-
 /-
 The 𝓝 filter and the subspace topology.
 -/
@@ -316,6 +321,7 @@ theorem mem_nhds_subtype (s : Set α) (a : { x // x ∈ s }) (t : Set { x // x 
     t ∈ 𝓝 a ↔ ∃ u ∈ 𝓝 (a : α), coe ⁻¹' u ⊆ t :=
   mem_nhds_induced coe a t
 #align mem_nhds_subtype mem_nhds_subtype
+-/
 
 #print nhds_subtype /-
 theorem nhds_subtype (s : Set α) (a : { x // x ∈ s }) : 𝓝 a = comap coe (𝓝 (a : α)) :=
@@ -323,26 +329,34 @@ theorem nhds_subtype (s : Set α) (a : { x // x ∈ s }) : 𝓝 a = comap coe (
 #align nhds_subtype nhds_subtype
 -/
 
+#print nhdsWithin_subtype_eq_bot_iff /-
 theorem nhdsWithin_subtype_eq_bot_iff {s t : Set α} {x : s} :
     𝓝[(coe : s → α) ⁻¹' t] x = ⊥ ↔ 𝓝[t] (x : α) ⊓ 𝓟 s = ⊥ := by
   rw [inf_principal_eq_bot_iff_comap, nhdsWithin, nhdsWithin, comap_inf, comap_principal,
     nhds_induced]
 #align nhds_within_subtype_eq_bot_iff nhdsWithin_subtype_eq_bot_iff
+-/
 
+#print nhds_ne_subtype_eq_bot_iff /-
 theorem nhds_ne_subtype_eq_bot_iff {S : Set α} {x : S} :
     𝓝[{x}ᶜ] x = ⊥ ↔ 𝓝[{x}ᶜ] (x : α) ⊓ 𝓟 S = ⊥ := by
   rw [← nhdsWithin_subtype_eq_bot_iff, preimage_compl, ← image_singleton,
     subtype.coe_injective.preimage_image]
 #align nhds_ne_subtype_eq_bot_iff nhds_ne_subtype_eq_bot_iff
+-/
 
+#print nhds_ne_subtype_neBot_iff /-
 theorem nhds_ne_subtype_neBot_iff {S : Set α} {x : S} :
     (𝓝[{x}ᶜ] x).ne_bot ↔ (𝓝[{x}ᶜ] (x : α) ⊓ 𝓟 S).ne_bot := by
   rw [ne_bot_iff, ne_bot_iff, not_iff_not, nhds_ne_subtype_eq_bot_iff]
 #align nhds_ne_subtype_ne_bot_iff nhds_ne_subtype_neBot_iff
+-/
 
+#print discreteTopology_subtype_iff /-
 theorem discreteTopology_subtype_iff {S : Set α} : DiscreteTopology S ↔ ∀ x ∈ S, 𝓝[≠] x ⊓ 𝓟 S = ⊥ :=
   by simp_rw [discreteTopology_iff_nhds_ne, SetCoe.forall', nhds_ne_subtype_eq_bot_iff]
 #align discrete_topology_subtype_iff discreteTopology_subtype_iff
+-/
 
 end Topα
 
@@ -378,18 +392,25 @@ instance : TopologicalSpace (CofiniteTopology α)
     rw [Set.compl_sUnion]
     exact Set.Finite.sInter (mem_image_of_mem _ hts) (h t hts ⟨x, hzt⟩)
 
+#print CofiniteTopology.isOpen_iff /-
 theorem isOpen_iff {s : Set (CofiniteTopology α)} : IsOpen s ↔ s.Nonempty → sᶜ.Finite :=
   Iff.rfl
 #align cofinite_topology.is_open_iff CofiniteTopology.isOpen_iff
+-/
 
+#print CofiniteTopology.isOpen_iff' /-
 theorem isOpen_iff' {s : Set (CofiniteTopology α)} : IsOpen s ↔ s = ∅ ∨ sᶜ.Finite := by
   simp only [is_open_iff, nonempty_iff_ne_empty, or_iff_not_imp_left]
 #align cofinite_topology.is_open_iff' CofiniteTopology.isOpen_iff'
+-/
 
+#print CofiniteTopology.isClosed_iff /-
 theorem isClosed_iff {s : Set (CofiniteTopology α)} : IsClosed s ↔ s = univ ∨ s.Finite := by
   simp [← isOpen_compl_iff, is_open_iff']
 #align cofinite_topology.is_closed_iff CofiniteTopology.isClosed_iff
+-/
 
+#print CofiniteTopology.nhds_eq /-
 theorem nhds_eq (a : CofiniteTopology α) : 𝓝 a = pure a ⊔ cofinite :=
   by
   ext U
@@ -400,10 +421,13 @@ theorem nhds_eq (a : CofiniteTopology α) : 𝓝 a = pure a ⊔ cofinite :=
   · rintro ⟨hU : a ∈ U, hU' : Uᶜ.Finite⟩
     exact ⟨U, subset.rfl, fun h => hU', hU⟩
 #align cofinite_topology.nhds_eq CofiniteTopology.nhds_eq
+-/
 
+#print CofiniteTopology.mem_nhds_iff /-
 theorem mem_nhds_iff {a : CofiniteTopology α} {s : Set (CofiniteTopology α)} :
     s ∈ 𝓝 a ↔ a ∈ s ∧ sᶜ.Finite := by simp [nhds_eq]
 #align cofinite_topology.mem_nhds_iff CofiniteTopology.mem_nhds_iff
+-/
 
 end CofiniteTopology
 
@@ -414,124 +438,169 @@ section Prod
 variable [TopologicalSpace α] [TopologicalSpace β] [TopologicalSpace γ] [TopologicalSpace δ]
   [TopologicalSpace ε] [TopologicalSpace ζ]
 
+#print continuous_fst /-
 @[continuity]
 theorem continuous_fst : Continuous (@Prod.fst α β) :=
   continuous_inf_dom_left continuous_induced_dom
 #align continuous_fst continuous_fst
+-/
 
+#print Continuous.fst /-
 /-- Postcomposing `f` with `prod.fst` is continuous -/
 theorem Continuous.fst {f : α → β × γ} (hf : Continuous f) : Continuous fun a : α => (f a).1 :=
   continuous_fst.comp hf
 #align continuous.fst Continuous.fst
+-/
 
+#print Continuous.fst' /-
 /-- Precomposing `f` with `prod.fst` is continuous -/
 theorem Continuous.fst' {f : α → γ} (hf : Continuous f) : Continuous fun x : α × β => f x.fst :=
   hf.comp continuous_fst
 #align continuous.fst' Continuous.fst'
+-/
 
+#print continuousAt_fst /-
 theorem continuousAt_fst {p : α × β} : ContinuousAt Prod.fst p :=
   continuous_fst.ContinuousAt
 #align continuous_at_fst continuousAt_fst
+-/
 
+#print ContinuousAt.fst /-
 /-- Postcomposing `f` with `prod.fst` is continuous at `x` -/
 theorem ContinuousAt.fst {f : α → β × γ} {x : α} (hf : ContinuousAt f x) :
     ContinuousAt (fun a : α => (f a).1) x :=
   continuousAt_fst.comp hf
 #align continuous_at.fst ContinuousAt.fst
+-/
 
+#print ContinuousAt.fst' /-
 /-- Precomposing `f` with `prod.fst` is continuous at `(x, y)` -/
 theorem ContinuousAt.fst' {f : α → γ} {x : α} {y : β} (hf : ContinuousAt f x) :
     ContinuousAt (fun x : α × β => f x.fst) (x, y) :=
   ContinuousAt.comp hf continuousAt_fst
 #align continuous_at.fst' ContinuousAt.fst'
+-/
 
+#print ContinuousAt.fst'' /-
 /-- Precomposing `f` with `prod.fst` is continuous at `x : α × β` -/
 theorem ContinuousAt.fst'' {f : α → γ} {x : α × β} (hf : ContinuousAt f x.fst) :
     ContinuousAt (fun x : α × β => f x.fst) x :=
   hf.comp continuousAt_fst
 #align continuous_at.fst'' ContinuousAt.fst''
+-/
 
+#print continuous_snd /-
 @[continuity]
 theorem continuous_snd : Continuous (@Prod.snd α β) :=
   continuous_inf_dom_right continuous_induced_dom
 #align continuous_snd continuous_snd
+-/
 
+#print Continuous.snd /-
 /-- Postcomposing `f` with `prod.snd` is continuous -/
 theorem Continuous.snd {f : α → β × γ} (hf : Continuous f) : Continuous fun a : α => (f a).2 :=
   continuous_snd.comp hf
 #align continuous.snd Continuous.snd
+-/
 
+#print Continuous.snd' /-
 /-- Precomposing `f` with `prod.snd` is continuous -/
 theorem Continuous.snd' {f : β → γ} (hf : Continuous f) : Continuous fun x : α × β => f x.snd :=
   hf.comp continuous_snd
 #align continuous.snd' Continuous.snd'
+-/
 
+#print continuousAt_snd /-
 theorem continuousAt_snd {p : α × β} : ContinuousAt Prod.snd p :=
   continuous_snd.ContinuousAt
 #align continuous_at_snd continuousAt_snd
+-/
 
+#print ContinuousAt.snd /-
 /-- Postcomposing `f` with `prod.snd` is continuous at `x` -/
 theorem ContinuousAt.snd {f : α → β × γ} {x : α} (hf : ContinuousAt f x) :
     ContinuousAt (fun a : α => (f a).2) x :=
   continuousAt_snd.comp hf
 #align continuous_at.snd ContinuousAt.snd
+-/
 
+#print ContinuousAt.snd' /-
 /-- Precomposing `f` with `prod.snd` is continuous at `(x, y)` -/
 theorem ContinuousAt.snd' {f : β → γ} {x : α} {y : β} (hf : ContinuousAt f y) :
     ContinuousAt (fun x : α × β => f x.snd) (x, y) :=
   ContinuousAt.comp hf continuousAt_snd
 #align continuous_at.snd' ContinuousAt.snd'
+-/
 
+#print ContinuousAt.snd'' /-
 /-- Precomposing `f` with `prod.snd` is continuous at `x : α × β` -/
 theorem ContinuousAt.snd'' {f : β → γ} {x : α × β} (hf : ContinuousAt f x.snd) :
     ContinuousAt (fun x : α × β => f x.snd) x :=
   hf.comp continuousAt_snd
 #align continuous_at.snd'' ContinuousAt.snd''
+-/
 
+#print Continuous.prod_mk /-
 @[continuity]
 theorem Continuous.prod_mk {f : γ → α} {g : γ → β} (hf : Continuous f) (hg : Continuous g) :
     Continuous fun x => (f x, g x) :=
   continuous_inf_rng.2 ⟨continuous_induced_rng.2 hf, continuous_induced_rng.2 hg⟩
 #align continuous.prod_mk Continuous.prod_mk
+-/
 
+#print continuous_prod_mk /-
 @[simp]
 theorem continuous_prod_mk {f : α → β} {g : α → γ} :
     (Continuous fun x => (f x, g x)) ↔ Continuous f ∧ Continuous g :=
   ⟨fun h => ⟨h.fst, h.snd⟩, fun h => h.1.prod_mk h.2⟩
 #align continuous_prod_mk continuous_prod_mk
+-/
 
+#print Continuous.Prod.mk /-
 @[continuity]
 theorem Continuous.Prod.mk (a : α) : Continuous fun b : β => (a, b) :=
   continuous_const.prod_mk continuous_id'
 #align continuous.prod.mk Continuous.Prod.mk
+-/
 
+#print Continuous.Prod.mk_left /-
 @[continuity]
 theorem Continuous.Prod.mk_left (b : β) : Continuous fun a : α => (a, b) :=
   continuous_id'.prod_mk continuous_const
 #align continuous.prod.mk_left Continuous.Prod.mk_left
+-/
 
+#print Continuous.comp₂ /-
 theorem Continuous.comp₂ {g : α × β → γ} (hg : Continuous g) {e : δ → α} (he : Continuous e)
     {f : δ → β} (hf : Continuous f) : Continuous fun x => g (e x, f x) :=
   hg.comp <| he.prod_mk hf
 #align continuous.comp₂ Continuous.comp₂
+-/
 
+#print Continuous.comp₃ /-
 theorem Continuous.comp₃ {g : α × β × γ → ε} (hg : Continuous g) {e : δ → α} (he : Continuous e)
     {f : δ → β} (hf : Continuous f) {k : δ → γ} (hk : Continuous k) :
     Continuous fun x => g (e x, f x, k x) :=
   hg.comp₂ he <| hf.prod_mk hk
 #align continuous.comp₃ Continuous.comp₃
+-/
 
+#print Continuous.comp₄ /-
 theorem Continuous.comp₄ {g : α × β × γ × ζ → ε} (hg : Continuous g) {e : δ → α} (he : Continuous e)
     {f : δ → β} (hf : Continuous f) {k : δ → γ} (hk : Continuous k) {l : δ → ζ}
     (hl : Continuous l) : Continuous fun x => g (e x, f x, k x, l x) :=
   hg.comp₃ he hf <| hk.prod_mk hl
 #align continuous.comp₄ Continuous.comp₄
+-/
 
+#print Continuous.prod_map /-
 theorem Continuous.prod_map {f : γ → α} {g : δ → β} (hf : Continuous f) (hg : Continuous g) :
     Continuous fun x : γ × δ => (f x.1, g x.2) :=
   hf.fst'.prod_mk hg.snd'
 #align continuous.prod_map Continuous.prod_map
+-/
 
+#print continuous_inf_dom_left₂ /-
 /-- A version of `continuous_inf_dom_left` for binary functions -/
 theorem continuous_inf_dom_left₂ {α β γ} {f : α → β → γ} {ta1 ta2 : TopologicalSpace α}
     {tb1 tb2 : TopologicalSpace β} {tc1 : TopologicalSpace γ}
@@ -543,7 +612,9 @@ theorem continuous_inf_dom_left₂ {α β γ} {f : α → β → γ} {ta1 ta2 :
   have h_continuous_id := @Continuous.prod_map _ _ _ _ ta1 tb1 (ta1 ⊓ ta2) (tb1 ⊓ tb2) _ _ ha hb
   exact @Continuous.comp _ _ _ (id _) (id _) _ _ _ h h_continuous_id
 #align continuous_inf_dom_left₂ continuous_inf_dom_left₂
+-/
 
+#print continuous_inf_dom_right₂ /-
 /-- A version of `continuous_inf_dom_right` for binary functions -/
 theorem continuous_inf_dom_right₂ {α β γ} {f : α → β → γ} {ta1 ta2 : TopologicalSpace α}
     {tb1 tb2 : TopologicalSpace β} {tc1 : TopologicalSpace γ}
@@ -555,7 +626,9 @@ theorem continuous_inf_dom_right₂ {α β γ} {f : α → β → γ} {ta1 ta2 :
   have h_continuous_id := @Continuous.prod_map _ _ _ _ ta2 tb2 (ta1 ⊓ ta2) (tb1 ⊓ tb2) _ _ ha hb
   exact @Continuous.comp _ _ _ (id _) (id _) _ _ _ h h_continuous_id
 #align continuous_inf_dom_right₂ continuous_inf_dom_right₂
+-/
 
+#print continuous_sInf_dom₂ /-
 /-- A version of `continuous_Inf_dom` for binary functions -/
 theorem continuous_sInf_dom₂ {α β γ} {f : α → β → γ} {tas : Set (TopologicalSpace α)}
     {tbs : Set (TopologicalSpace β)} {ta : TopologicalSpace α} {tb : TopologicalSpace β}
@@ -570,49 +643,69 @@ theorem continuous_sInf_dom₂ {α β γ} {f : α → β → γ} {tas : Set (Top
   have h_continuous_id := @Continuous.prod_map _ _ _ _ ta tb (Inf tas) (Inf tbs) _ _ ha hb
   exact @Continuous.comp _ _ _ (id _) (id _) _ _ _ hf h_continuous_id
 #align continuous_Inf_dom₂ continuous_sInf_dom₂
+-/
 
+#print Filter.Eventually.prod_inl_nhds /-
 theorem Filter.Eventually.prod_inl_nhds {p : α → Prop} {a : α} (h : ∀ᶠ x in 𝓝 a, p x) (b : β) :
     ∀ᶠ x in 𝓝 (a, b), p (x : α × β).1 :=
   continuousAt_fst h
 #align filter.eventually.prod_inl_nhds Filter.Eventually.prod_inl_nhds
+-/
 
+#print Filter.Eventually.prod_inr_nhds /-
 theorem Filter.Eventually.prod_inr_nhds {p : β → Prop} {b : β} (h : ∀ᶠ x in 𝓝 b, p x) (a : α) :
     ∀ᶠ x in 𝓝 (a, b), p (x : α × β).2 :=
   continuousAt_snd h
 #align filter.eventually.prod_inr_nhds Filter.Eventually.prod_inr_nhds
+-/
 
+#print Filter.Eventually.prod_mk_nhds /-
 theorem Filter.Eventually.prod_mk_nhds {pa : α → Prop} {a} (ha : ∀ᶠ x in 𝓝 a, pa x) {pb : β → Prop}
     {b} (hb : ∀ᶠ y in 𝓝 b, pb y) : ∀ᶠ p in 𝓝 (a, b), pa (p : α × β).1 ∧ pb p.2 :=
   (ha.prod_inl_nhds b).And (hb.prod_inr_nhds a)
 #align filter.eventually.prod_mk_nhds Filter.Eventually.prod_mk_nhds
+-/
 
+#print continuous_swap /-
 theorem continuous_swap : Continuous (Prod.swap : α × β → β × α) :=
   continuous_snd.prod_mk continuous_fst
 #align continuous_swap continuous_swap
+-/
 
+#print continuous_uncurry_left /-
 theorem continuous_uncurry_left {f : α → β → γ} (a : α) (h : Continuous (uncurry f)) :
     Continuous (f a) :=
   show Continuous (uncurry f ∘ fun b => (a, b)) from h.comp (by continuity)
 #align continuous_uncurry_left continuous_uncurry_left
+-/
 
+#print continuous_uncurry_right /-
 theorem continuous_uncurry_right {f : α → β → γ} (b : β) (h : Continuous (uncurry f)) :
     Continuous fun a => f a b :=
   show Continuous (uncurry f ∘ fun a => (a, b)) from h.comp (by continuity)
 #align continuous_uncurry_right continuous_uncurry_right
+-/
 
+#print continuous_curry /-
 theorem continuous_curry {g : α × β → γ} (a : α) (h : Continuous g) : Continuous (curry g a) :=
   show Continuous (g ∘ fun b => (a, b)) from h.comp (by continuity)
 #align continuous_curry continuous_curry
+-/
 
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
+#print IsOpen.prod /-
 theorem IsOpen.prod {s : Set α} {t : Set β} (hs : IsOpen s) (ht : IsOpen t) : IsOpen (s ×ˢ t) :=
   (hs.Preimage continuous_fst).inter (ht.Preimage continuous_snd)
 #align is_open.prod IsOpen.prod
+-/
 
+#print nhds_prod_eq /-
 theorem nhds_prod_eq {a : α} {b : β} : 𝓝 (a, b) = 𝓝 a ×ᶠ 𝓝 b := by
   rw [Filter.prod, Prod.topologicalSpace, nhds_inf, nhds_induced, nhds_induced]
 #align nhds_prod_eq nhds_prod_eq
+-/
 
+#print continuous_uncurry_of_discreteTopology /-
 /-- If a function `f x y` is such that `y ↦ f x y` is continuous for all `x`, and `x` lives in a
 discrete space, then `f` is continuous. -/
 theorem continuous_uncurry_of_discreteTopology [DiscreteTopology α] {f : α → β → γ}
@@ -624,13 +717,17 @@ theorem continuous_uncurry_of_discreteTopology [DiscreteTopology α] {f : α →
   rw [nhds_prod_eq, nhds_discrete, Filter.map_pure_prod]
   exact (hf a).ContinuousAt
 #align continuous_uncurry_of_discrete_topology continuous_uncurry_of_discreteTopology
+-/
 
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
+#print mem_nhds_prod_iff /-
 theorem mem_nhds_prod_iff {a : α} {b : β} {s : Set (α × β)} :
     s ∈ 𝓝 (a, b) ↔ ∃ u ∈ 𝓝 a, ∃ v ∈ 𝓝 b, u ×ˢ v ⊆ s := by rw [nhds_prod_eq, mem_prod_iff]
 #align mem_nhds_prod_iff mem_nhds_prod_iff
+-/
 
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
+#print mem_nhds_prod_iff' /-
 theorem mem_nhds_prod_iff' {a : α} {b : β} {s : Set (α × β)} :
     s ∈ 𝓝 (a, b) ↔ ∃ (u : Set α) (v : Set β), IsOpen u ∧ a ∈ u ∧ IsOpen v ∧ b ∈ v ∧ u ×ˢ v ⊆ s :=
   by
@@ -643,83 +740,109 @@ theorem mem_nhds_prod_iff' {a : α} {b : β} {s : Set (α × β)} :
   · rintro ⟨u, v, u_open, au, v_open, bv, huv⟩
     exact ⟨u, u_open.mem_nhds au, v, v_open.mem_nhds bv, huv⟩
 #align mem_nhds_prod_iff' mem_nhds_prod_iff'
+-/
 
+#print Prod.tendsto_iff /-
 theorem Prod.tendsto_iff {α} (seq : α → β × γ) {f : Filter α} (x : β × γ) :
     Tendsto seq f (𝓝 x) ↔
       Tendsto (fun n => (seq n).fst) f (𝓝 x.fst) ∧ Tendsto (fun n => (seq n).snd) f (𝓝 x.snd) :=
   by cases x; rw [nhds_prod_eq, Filter.tendsto_prod_iff']
 #align prod.tendsto_iff Prod.tendsto_iff
+-/
 
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
+#print Filter.HasBasis.prod_nhds /-
 theorem Filter.HasBasis.prod_nhds {ιa ιb : Type _} {pa : ιa → Prop} {pb : ιb → Prop}
     {sa : ιa → Set α} {sb : ιb → Set β} {a : α} {b : β} (ha : (𝓝 a).HasBasis pa sa)
     (hb : (𝓝 b).HasBasis pb sb) :
     (𝓝 (a, b)).HasBasis (fun i : ιa × ιb => pa i.1 ∧ pb i.2) fun i => sa i.1 ×ˢ sb i.2 := by
   rw [nhds_prod_eq]; exact ha.prod hb
 #align filter.has_basis.prod_nhds Filter.HasBasis.prod_nhds
+-/
 
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
+#print Filter.HasBasis.prod_nhds' /-
 theorem Filter.HasBasis.prod_nhds' {ιa ιb : Type _} {pa : ιa → Prop} {pb : ιb → Prop}
     {sa : ιa → Set α} {sb : ιb → Set β} {ab : α × β} (ha : (𝓝 ab.1).HasBasis pa sa)
     (hb : (𝓝 ab.2).HasBasis pb sb) :
     (𝓝 ab).HasBasis (fun i : ιa × ιb => pa i.1 ∧ pb i.2) fun i => sa i.1 ×ˢ sb i.2 := by cases ab;
   exact ha.prod_nhds hb
 #align filter.has_basis.prod_nhds' Filter.HasBasis.prod_nhds'
+-/
 
 instance [DiscreteTopology α] [DiscreteTopology β] : DiscreteTopology (α × β) :=
   discreteTopology_iff_nhds.2 fun ⟨a, b⟩ => by
     rw [nhds_prod_eq, nhds_discrete α, nhds_discrete β, Filter.prod_pure_pure]
 
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
+#print prod_mem_nhds_iff /-
 theorem prod_mem_nhds_iff {s : Set α} {t : Set β} {a : α} {b : β} :
     s ×ˢ t ∈ 𝓝 (a, b) ↔ s ∈ 𝓝 a ∧ t ∈ 𝓝 b := by rw [nhds_prod_eq, prod_mem_prod_iff]
 #align prod_mem_nhds_iff prod_mem_nhds_iff
+-/
 
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
+#print prod_mem_nhds /-
 theorem prod_mem_nhds {s : Set α} {t : Set β} {a : α} {b : β} (ha : s ∈ 𝓝 a) (hb : t ∈ 𝓝 b) :
     s ×ˢ t ∈ 𝓝 (a, b) :=
   prod_mem_nhds_iff.2 ⟨ha, hb⟩
 #align prod_mem_nhds prod_mem_nhds
+-/
 
+#print Filter.Eventually.prod_nhds /-
 theorem Filter.Eventually.prod_nhds {p : α → Prop} {q : β → Prop} {a : α} {b : β}
     (ha : ∀ᶠ x in 𝓝 a, p x) (hb : ∀ᶠ y in 𝓝 b, q y) : ∀ᶠ z : α × β in 𝓝 (a, b), p z.1 ∧ q z.2 :=
   prod_mem_nhds ha hb
 #align filter.eventually.prod_nhds Filter.Eventually.prod_nhds
+-/
 
+#print nhds_swap /-
 theorem nhds_swap (a : α) (b : β) : 𝓝 (a, b) = (𝓝 (b, a)).map Prod.swap := by
   rw [nhds_prod_eq, Filter.prod_comm, nhds_prod_eq] <;> rfl
 #align nhds_swap nhds_swap
+-/
 
+#print Filter.Tendsto.prod_mk_nhds /-
 theorem Filter.Tendsto.prod_mk_nhds {γ} {a : α} {b : β} {f : Filter γ} {ma : γ → α} {mb : γ → β}
     (ha : Tendsto ma f (𝓝 a)) (hb : Tendsto mb f (𝓝 b)) :
     Tendsto (fun c => (ma c, mb c)) f (𝓝 (a, b)) := by
   rw [nhds_prod_eq] <;> exact Filter.Tendsto.prod_mk ha hb
 #align filter.tendsto.prod_mk_nhds Filter.Tendsto.prod_mk_nhds
+-/
 
+#print Filter.Eventually.curry_nhds /-
 theorem Filter.Eventually.curry_nhds {p : α × β → Prop} {x : α} {y : β}
     (h : ∀ᶠ x in 𝓝 (x, y), p x) : ∀ᶠ x' in 𝓝 x, ∀ᶠ y' in 𝓝 y, p (x', y') := by
   rw [nhds_prod_eq] at h ; exact h.curry
 #align filter.eventually.curry_nhds Filter.Eventually.curry_nhds
+-/
 
+#print ContinuousAt.prod /-
 theorem ContinuousAt.prod {f : α → β} {g : α → γ} {x : α} (hf : ContinuousAt f x)
     (hg : ContinuousAt g x) : ContinuousAt (fun x => (f x, g x)) x :=
   hf.prod_mk_nhds hg
 #align continuous_at.prod ContinuousAt.prod
+-/
 
+#print ContinuousAt.prod_map /-
 theorem ContinuousAt.prod_map {f : α → γ} {g : β → δ} {p : α × β} (hf : ContinuousAt f p.fst)
     (hg : ContinuousAt g p.snd) : ContinuousAt (fun p : α × β => (f p.1, g p.2)) p :=
   hf.fst''.Prod hg.snd''
 #align continuous_at.prod_map ContinuousAt.prod_map
+-/
 
+#print ContinuousAt.prod_map' /-
 theorem ContinuousAt.prod_map' {f : α → γ} {g : β → δ} {x : α} {y : β} (hf : ContinuousAt f x)
     (hg : ContinuousAt g y) : ContinuousAt (fun p : α × β => (f p.1, g p.2)) (x, y) :=
   hf.fst'.Prod hg.snd'
 #align continuous_at.prod_map' ContinuousAt.prod_map'
+-/
 
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
+#print prod_generateFrom_generateFrom_eq /-
 theorem prod_generateFrom_generateFrom_eq {α β : Type _} {s : Set (Set α)} {t : Set (Set β)}
     (hs : ⋃₀ s = univ) (ht : ⋃₀ t = univ) :
     @Prod.topologicalSpace α β (generateFrom s) (generateFrom t) =
@@ -748,8 +871,10 @@ theorem prod_generateFrom_generateFrom_eq {α β : Type _} {s : Set (Set α)} {t
               isOpen_iUnion fun u =>
                 isOpen_iUnion fun hu => generate_open.basic _ ⟨_, hu, _, hv, rfl⟩))
 #align prod_generate_from_generate_from_eq prod_generateFrom_generateFrom_eq
+-/
 
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
+#print prod_eq_generateFrom /-
 theorem prod_eq_generateFrom :
     Prod.topologicalSpace =
       generateFrom {g | ∃ (s : Set α) (t : Set β), IsOpen s ∧ IsOpen t ∧ g = s ×ˢ t} :=
@@ -760,8 +885,10 @@ theorem prod_eq_generateFrom :
       (ball_image_of_ball fun t ht =>
         GenerateOpen.basic _ ⟨univ, t, by simpa [Set.prod_eq] using ht⟩))
 #align prod_eq_generate_from prod_eq_generateFrom
+-/
 
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
+#print isOpen_prod_iff /-
 theorem isOpen_prod_iff {s : Set (α × β)} :
     IsOpen s ↔
       ∀ a b,
@@ -772,29 +899,37 @@ theorem isOpen_prod_iff {s : Set (α × β)} :
     ((nhds_basis_opens _).prod_nhds (nhds_basis_opens _)).mem_iff, Prod.exists, exists_prop]
   simp only [and_assoc', and_left_comm]
 #align is_open_prod_iff isOpen_prod_iff
+-/
 
+#print prod_induced_induced /-
 /-- A product of induced topologies is induced by the product map -/
 theorem prod_induced_induced {α γ : Type _} (f : α → β) (g : γ → δ) :
     @Prod.topologicalSpace α γ (induced f ‹_›) (induced g ‹_›) =
       induced (fun p => (f p.1, g p.2)) Prod.topologicalSpace :=
   by simp_rw [Prod.topologicalSpace, induced_inf, induced_compose]
 #align prod_induced_induced prod_induced_induced
+-/
 
+#print continuous_uncurry_of_discreteTopology_left /-
 theorem continuous_uncurry_of_discreteTopology_left [DiscreteTopology α] {f : α → β → γ}
     (h : ∀ a, Continuous (f a)) : Continuous (uncurry f) :=
   continuous_iff_continuousAt.2 fun ⟨a, b⟩ => by
     simp only [ContinuousAt, nhds_prod_eq, nhds_discrete α, pure_prod, tendsto_map'_iff, (· ∘ ·),
       uncurry, (h a).Tendsto]
 #align continuous_uncurry_of_discrete_topology_left continuous_uncurry_of_discreteTopology_left
+-/
 
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
+#print exists_nhds_square /-
 /-- Given a neighborhood `s` of `(x, x)`, then `(x, x)` has a square open neighborhood
   that is a subset of `s`. -/
 theorem exists_nhds_square {s : Set (α × α)} {x : α} (hx : s ∈ 𝓝 (x, x)) :
     ∃ U : Set α, IsOpen U ∧ x ∈ U ∧ U ×ˢ U ⊆ s := by
   simpa [nhds_prod_eq, (nhds_basis_opens x).prod_self.mem_iff, and_assoc, and_left_comm] using hx
 #align exists_nhds_square exists_nhds_square
+-/
 
+#print map_fst_nhdsWithin /-
 /-- `prod.fst` maps neighborhood of `x : α × β` within the section `prod.snd ⁻¹' {x.2}`
 to `𝓝 x.1`. -/
 theorem map_fst_nhdsWithin (x : α × β) : map Prod.fst (𝓝[Prod.snd ⁻¹' {x.2}] x) = 𝓝 x.1 :=
@@ -806,17 +941,23 @@ theorem map_fst_nhdsWithin (x : α × β) : map Prod.fst (𝓝[Prod.snd ⁻¹' {
   simp only [prod_subset_iff, mem_singleton_iff, mem_set_of_eq, mem_preimage] at H 
   exact mem_of_superset hu fun z hz => H _ hz _ (mem_of_mem_nhds hv) rfl
 #align map_fst_nhds_within map_fst_nhdsWithin
+-/
 
+#print map_fst_nhds /-
 @[simp]
 theorem map_fst_nhds (x : α × β) : map Prod.fst (𝓝 x) = 𝓝 x.1 :=
   le_antisymm continuousAt_fst <| (map_fst_nhdsWithin x).symm.trans_le (map_mono inf_le_left)
 #align map_fst_nhds map_fst_nhds
+-/
 
+#print isOpenMap_fst /-
 /-- The first projection in a product of topological spaces sends open sets to open sets. -/
 theorem isOpenMap_fst : IsOpenMap (@Prod.fst α β) :=
   isOpenMap_iff_nhds_le.2 fun x => (map_fst_nhds x).ge
 #align is_open_map_fst isOpenMap_fst
+-/
 
+#print map_snd_nhdsWithin /-
 /-- `prod.snd` maps neighborhood of `x : α × β` within the section `prod.fst ⁻¹' {x.1}`
 to `𝓝 x.2`. -/
 theorem map_snd_nhdsWithin (x : α × β) : map Prod.snd (𝓝[Prod.fst ⁻¹' {x.1}] x) = 𝓝 x.2 :=
@@ -828,20 +969,26 @@ theorem map_snd_nhdsWithin (x : α × β) : map Prod.snd (𝓝[Prod.fst ⁻¹' {
   simp only [prod_subset_iff, mem_singleton_iff, mem_set_of_eq, mem_preimage] at H 
   exact mem_of_superset hv fun z hz => H _ (mem_of_mem_nhds hu) _ hz rfl
 #align map_snd_nhds_within map_snd_nhdsWithin
+-/
 
+#print map_snd_nhds /-
 @[simp]
 theorem map_snd_nhds (x : α × β) : map Prod.snd (𝓝 x) = 𝓝 x.2 :=
   le_antisymm continuousAt_snd <| (map_snd_nhdsWithin x).symm.trans_le (map_mono inf_le_left)
 #align map_snd_nhds map_snd_nhds
+-/
 
+#print isOpenMap_snd /-
 /-- The second projection in a product of topological spaces sends open sets to open sets. -/
 theorem isOpenMap_snd : IsOpenMap (@Prod.snd α β) :=
   isOpenMap_iff_nhds_le.2 fun x => (map_snd_nhds x).ge
 #align is_open_map_snd isOpenMap_snd
+-/
 
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
+#print isOpen_prod_iff' /-
 /-- A product set is open in a product space if and only if each factor is open, or one of them is
 empty -/
 theorem isOpen_prod_iff' {s : Set α} {t : Set β} :
@@ -863,10 +1010,12 @@ theorem isOpen_prod_iff' {s : Set α} {t : Set β} :
       simp only [st.1.ne_empty, st.2.ne_empty, not_false_iff, or_false_iff] at H 
       exact H.1.Prod H.2
 #align is_open_prod_iff' isOpen_prod_iff'
+-/
 
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
+#print closure_prod_eq /-
 theorem closure_prod_eq {s : Set α} {t : Set β} : closure (s ×ˢ t) = closure s ×ˢ closure t :=
   Set.ext fun ⟨a, b⟩ =>
     by
@@ -874,37 +1023,47 @@ theorem closure_prod_eq {s : Set α} {t : Set β} : closure (s ×ˢ t) = closure
       rw [← prod_inf_prod, prod_principal_principal]
     simp [closure_eq_cluster_pts, ClusterPt, nhds_prod_eq, this] <;> exact prod_ne_bot
 #align closure_prod_eq closure_prod_eq
+-/
 
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
+#print interior_prod_eq /-
 theorem interior_prod_eq (s : Set α) (t : Set β) : interior (s ×ˢ t) = interior s ×ˢ interior t :=
   Set.ext fun ⟨a, b⟩ => by simp only [mem_interior_iff_mem_nhds, mem_prod, prod_mem_nhds_iff]
 #align interior_prod_eq interior_prod_eq
+-/
 
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
+#print frontier_prod_eq /-
 theorem frontier_prod_eq (s : Set α) (t : Set β) :
     frontier (s ×ˢ t) = closure s ×ˢ frontier t ∪ frontier s ×ˢ closure t := by
   simp only [frontier, closure_prod_eq, interior_prod_eq, prod_diff_prod]
 #align frontier_prod_eq frontier_prod_eq
+-/
 
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
+#print frontier_prod_univ_eq /-
 @[simp]
 theorem frontier_prod_univ_eq (s : Set α) : frontier (s ×ˢ (univ : Set β)) = frontier s ×ˢ univ :=
   by simp [frontier_prod_eq]
 #align frontier_prod_univ_eq frontier_prod_univ_eq
+-/
 
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
+#print frontier_univ_prod_eq /-
 @[simp]
 theorem frontier_univ_prod_eq (s : Set β) : frontier ((univ : Set α) ×ˢ s) = univ ×ˢ frontier s :=
   by simp [frontier_prod_eq]
 #align frontier_univ_prod_eq frontier_univ_prod_eq
+-/
 
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
+#print map_mem_closure₂ /-
 theorem map_mem_closure₂ {f : α → β → γ} {a : α} {b : β} {s : Set α} {t : Set β} {u : Set γ}
     (hf : Continuous (uncurry f)) (ha : a ∈ closure s) (hb : b ∈ closure t)
     (h : ∀ a ∈ s, ∀ b ∈ t, f a b ∈ u) : f a b ∈ closure u :=
@@ -912,50 +1071,66 @@ theorem map_mem_closure₂ {f : α → β → γ} {a : α} {b : β} {s : Set α}
   have H₂ : MapsTo (uncurry f) (s ×ˢ t) u := forall_prod_set.2 h
   H₂.closure hf H₁
 #align map_mem_closure₂ map_mem_closure₂
+-/
 
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
+#print IsClosed.prod /-
 theorem IsClosed.prod {s₁ : Set α} {s₂ : Set β} (h₁ : IsClosed s₁) (h₂ : IsClosed s₂) :
     IsClosed (s₁ ×ˢ s₂) :=
   closure_eq_iff_isClosed.mp <| by simp only [h₁.closure_eq, h₂.closure_eq, closure_prod_eq]
 #align is_closed.prod IsClosed.prod
+-/
 
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
+#print Dense.prod /-
 /-- The product of two dense sets is a dense set. -/
 theorem Dense.prod {s : Set α} {t : Set β} (hs : Dense s) (ht : Dense t) : Dense (s ×ˢ t) :=
   fun x => by rw [closure_prod_eq]; exact ⟨hs x.1, ht x.2⟩
 #align dense.prod Dense.prod
+-/
 
+#print DenseRange.prod_map /-
 /-- If `f` and `g` are maps with dense range, then `prod.map f g` has dense range. -/
 theorem DenseRange.prod_map {ι : Type _} {κ : Type _} {f : ι → β} {g : κ → γ} (hf : DenseRange f)
     (hg : DenseRange g) : DenseRange (Prod.map f g) := by
   simpa only [DenseRange, prod_range_range_eq] using hf.prod hg
 #align dense_range.prod_map DenseRange.prod_map
+-/
 
+#print Inducing.prod_map /-
 theorem Inducing.prod_map {f : α → β} {g : γ → δ} (hf : Inducing f) (hg : Inducing g) :
     Inducing fun x : α × γ => (f x.1, g x.2) :=
   ⟨by
     rw [Prod.topologicalSpace, Prod.topologicalSpace, hf.induced, hg.induced, induced_compose,
       induced_compose, induced_inf, induced_compose, induced_compose]⟩
 #align inducing.prod_mk Inducing.prod_map
+-/
 
+#print inducing_const_prod /-
 @[simp]
 theorem inducing_const_prod {a : α} {f : β → γ} : (Inducing fun x => (a, f x)) ↔ Inducing f := by
   simp_rw [inducing_iff, Prod.topologicalSpace, induced_inf, induced_compose, Function.comp,
     induced_const, top_inf_eq]
 #align inducing_const_prod inducing_const_prod
+-/
 
+#print inducing_prod_const /-
 @[simp]
 theorem inducing_prod_const {b : β} {f : α → γ} : (Inducing fun x => (f x, b)) ↔ Inducing f := by
   simp_rw [inducing_iff, Prod.topologicalSpace, induced_inf, induced_compose, Function.comp,
     induced_const, inf_top_eq]
 #align inducing_prod_const inducing_prod_const
+-/
 
+#print Embedding.prod_map /-
 theorem Embedding.prod_map {f : α → β} {g : γ → δ} (hf : Embedding f) (hg : Embedding g) :
     Embedding fun x : α × γ => (f x.1, g x.2) :=
   { hf.to_inducing.prod_mk hg.to_inducing with
     inj := fun ⟨x₁, x₂⟩ ⟨y₁, y₂⟩ => by simp <;> exact fun h₁ h₂ => ⟨hf.inj h₁, hg.inj h₂⟩ }
 #align embedding.prod_mk Embedding.prod_map
+-/
 
+#print IsOpenMap.prod /-
 protected theorem IsOpenMap.prod {f : α → β} {g : γ → δ} (hf : IsOpenMap f) (hg : IsOpenMap g) :
     IsOpenMap fun p : α × γ => (f p.1, g p.2) :=
   by
@@ -964,15 +1139,20 @@ protected theorem IsOpenMap.prod {f : α → β} {g : γ → δ} (hf : IsOpenMap
   rw [nhds_prod_eq, nhds_prod_eq, ← Filter.prod_map_map_eq]
   exact Filter.prod_mono (hf.nhds_le a) (hg.nhds_le b)
 #align is_open_map.prod IsOpenMap.prod
+-/
 
+#print OpenEmbedding.prod /-
 protected theorem OpenEmbedding.prod {f : α → β} {g : γ → δ} (hf : OpenEmbedding f)
     (hg : OpenEmbedding g) : OpenEmbedding fun x : α × γ => (f x.1, g x.2) :=
   openEmbedding_of_embedding_open (hf.1.prod_mk hg.1) (hf.IsOpenMap.Prod hg.IsOpenMap)
 #align open_embedding.prod OpenEmbedding.prod
+-/
 
+#print embedding_graph /-
 theorem embedding_graph {f : α → β} (hf : Continuous f) : Embedding fun x => (x, f x) :=
   embedding_of_embedding_compose (continuous_id.prod_mk hf) continuous_fst embedding_id
 #align embedding_graph embedding_graph
+-/
 
 end Prod
 
@@ -982,120 +1162,170 @@ open Sum
 
 variable [TopologicalSpace α] [TopologicalSpace β] [TopologicalSpace γ] [TopologicalSpace δ]
 
+#print continuous_inl /-
 @[continuity]
 theorem continuous_inl : Continuous (@inl α β) :=
   continuous_sup_rng_left continuous_coinduced_rng
 #align continuous_inl continuous_inl
+-/
 
+#print continuous_inr /-
 @[continuity]
 theorem continuous_inr : Continuous (@inr α β) :=
   continuous_sup_rng_right continuous_coinduced_rng
 #align continuous_inr continuous_inr
+-/
 
+#print isOpen_sum_iff /-
 theorem isOpen_sum_iff {s : Set (Sum α β)} : IsOpen s ↔ IsOpen (inl ⁻¹' s) ∧ IsOpen (inr ⁻¹' s) :=
   Iff.rfl
 #align is_open_sum_iff isOpen_sum_iff
+-/
 
+#print isOpenMap_inl /-
 theorem isOpenMap_inl : IsOpenMap (@inl α β) := fun u hu => by
   simpa [isOpen_sum_iff, preimage_image_eq u Sum.inl_injective]
 #align is_open_map_inl isOpenMap_inl
+-/
 
+#print isOpenMap_inr /-
 theorem isOpenMap_inr : IsOpenMap (@inr α β) := fun u hu => by
   simpa [isOpen_sum_iff, preimage_image_eq u Sum.inr_injective]
 #align is_open_map_inr isOpenMap_inr
+-/
 
+#print openEmbedding_inl /-
 theorem openEmbedding_inl : OpenEmbedding (@inl α β) :=
   openEmbedding_of_continuous_injective_open continuous_inl inl_injective isOpenMap_inl
 #align open_embedding_inl openEmbedding_inl
+-/
 
+#print openEmbedding_inr /-
 theorem openEmbedding_inr : OpenEmbedding (@inr α β) :=
   openEmbedding_of_continuous_injective_open continuous_inr inr_injective isOpenMap_inr
 #align open_embedding_inr openEmbedding_inr
+-/
 
+#print embedding_inl /-
 theorem embedding_inl : Embedding (@inl α β) :=
   openEmbedding_inl.1
 #align embedding_inl embedding_inl
+-/
 
+#print embedding_inr /-
 theorem embedding_inr : Embedding (@inr α β) :=
   openEmbedding_inr.1
 #align embedding_inr embedding_inr
+-/
 
+#print isOpen_range_inl /-
 theorem isOpen_range_inl : IsOpen (range (inl : α → Sum α β)) :=
   openEmbedding_inl.2
 #align is_open_range_inl isOpen_range_inl
+-/
 
+#print isOpen_range_inr /-
 theorem isOpen_range_inr : IsOpen (range (inr : β → Sum α β)) :=
   openEmbedding_inr.2
 #align is_open_range_inr isOpen_range_inr
+-/
 
+#print isClosed_range_inl /-
 theorem isClosed_range_inl : IsClosed (range (inl : α → Sum α β)) := by
   rw [← isOpen_compl_iff, compl_range_inl]; exact isOpen_range_inr
 #align is_closed_range_inl isClosed_range_inl
+-/
 
+#print isClosed_range_inr /-
 theorem isClosed_range_inr : IsClosed (range (inr : β → Sum α β)) := by
   rw [← isOpen_compl_iff, compl_range_inr]; exact isOpen_range_inl
 #align is_closed_range_inr isClosed_range_inr
+-/
 
+#print closedEmbedding_inl /-
 theorem closedEmbedding_inl : ClosedEmbedding (inl : α → Sum α β) :=
   ⟨embedding_inl, isClosed_range_inl⟩
 #align closed_embedding_inl closedEmbedding_inl
+-/
 
+#print closedEmbedding_inr /-
 theorem closedEmbedding_inr : ClosedEmbedding (inr : β → Sum α β) :=
   ⟨embedding_inr, isClosed_range_inr⟩
 #align closed_embedding_inr closedEmbedding_inr
+-/
 
+#print nhds_inl /-
 theorem nhds_inl (x : α) : 𝓝 (inl x : Sum α β) = map inl (𝓝 x) :=
   (openEmbedding_inl.map_nhds_eq _).symm
 #align nhds_inl nhds_inl
+-/
 
+#print nhds_inr /-
 theorem nhds_inr (x : β) : 𝓝 (inr x : Sum α β) = map inr (𝓝 x) :=
   (openEmbedding_inr.map_nhds_eq _).symm
 #align nhds_inr nhds_inr
+-/
 
+#print continuous_sum_dom /-
 theorem continuous_sum_dom {f : Sum α β → γ} :
     Continuous f ↔ Continuous (f ∘ Sum.inl) ∧ Continuous (f ∘ Sum.inr) := by
   simp only [continuous_sup_dom, continuous_coinduced_dom]
 #align continuous_sum_dom continuous_sum_dom
+-/
 
+#print continuous_sum_elim /-
 theorem continuous_sum_elim {f : α → γ} {g : β → γ} :
     Continuous (Sum.elim f g) ↔ Continuous f ∧ Continuous g :=
   continuous_sum_dom
 #align continuous_sum_elim continuous_sum_elim
+-/
 
+#print Continuous.sum_elim /-
 @[continuity]
 theorem Continuous.sum_elim {f : α → γ} {g : β → γ} (hf : Continuous f) (hg : Continuous g) :
     Continuous (Sum.elim f g) :=
   continuous_sum_elim.2 ⟨hf, hg⟩
 #align continuous.sum_elim Continuous.sum_elim
+-/
 
+#print continuous_sum_map /-
 @[simp]
 theorem continuous_sum_map {f : α → β} {g : γ → δ} :
     Continuous (Sum.map f g) ↔ Continuous f ∧ Continuous g :=
   continuous_sum_elim.trans <|
     embedding_inl.continuous_iff.symm.And embedding_inr.continuous_iff.symm
 #align continuous_sum_map continuous_sum_map
+-/
 
+#print Continuous.sum_map /-
 @[continuity]
 theorem Continuous.sum_map {f : α → β} {g : γ → δ} (hf : Continuous f) (hg : Continuous g) :
     Continuous (Sum.map f g) :=
   continuous_sum_map.2 ⟨hf, hg⟩
 #align continuous.sum_map Continuous.sum_map
+-/
 
+#print isOpenMap_sum /-
 theorem isOpenMap_sum {f : Sum α β → γ} :
     IsOpenMap f ↔ (IsOpenMap fun a => f (inl a)) ∧ IsOpenMap fun b => f (inr b) := by
   simp only [isOpenMap_iff_nhds_le, Sum.forall, nhds_inl, nhds_inr, Filter.map_map]
 #align is_open_map_sum isOpenMap_sum
+-/
 
+#print isOpenMap_sum_elim /-
 @[simp]
 theorem isOpenMap_sum_elim {f : α → γ} {g : β → γ} :
     IsOpenMap (Sum.elim f g) ↔ IsOpenMap f ∧ IsOpenMap g := by
   simp only [isOpenMap_sum, elim_inl, elim_inr]
 #align is_open_map_sum_elim isOpenMap_sum_elim
+-/
 
+#print IsOpenMap.sum_elim /-
 theorem IsOpenMap.sum_elim {f : α → γ} {g : β → γ} (hf : IsOpenMap f) (hg : IsOpenMap g) :
     IsOpenMap (Sum.elim f g) :=
   isOpenMap_sum_elim.2 ⟨hf, hg⟩
 #align is_open_map.sum_elim IsOpenMap.sum_elim
+-/
 
 end Sum
 
@@ -1229,10 +1459,12 @@ theorem nhds_subtype_eq_comap {a : α} {h : p a} : 𝓝 (⟨a, h⟩ : Subtype p)
 #align nhds_subtype_eq_comap nhds_subtype_eq_comap
 -/
 
+#print tendsto_subtype_rng /-
 theorem tendsto_subtype_rng {β : Type _} {p : α → Prop} {b : Filter β} {f : β → Subtype p} :
     ∀ {a : Subtype p}, Tendsto f b (𝓝 a) ↔ Tendsto (fun x => (f x : α)) b (𝓝 (a : α))
   | ⟨a, ha⟩ => by rw [nhds_subtype_eq_comap, tendsto_comap_iff, Subtype.coe_mk]
 #align tendsto_subtype_rng tendsto_subtype_rng
+-/
 
 #print closure_subtype /-
 theorem closure_subtype {x : { a // p a }} {s : Set { a // p a }} :
@@ -1372,71 +1604,97 @@ section Pi
 variable {ι : Type _} {π : ι → Type _} {κ : Type _} [TopologicalSpace α]
   [∀ i, TopologicalSpace (π i)] {f : α → ∀ i : ι, π i}
 
+#print continuous_pi_iff /-
 theorem continuous_pi_iff : Continuous f ↔ ∀ i, Continuous fun a => f a i := by
   simp only [continuous_iInf_rng, continuous_induced_rng]
 #align continuous_pi_iff continuous_pi_iff
+-/
 
+#print continuous_pi /-
 @[continuity]
 theorem continuous_pi (h : ∀ i, Continuous fun a => f a i) : Continuous f :=
   continuous_pi_iff.2 h
 #align continuous_pi continuous_pi
+-/
 
+#print continuous_apply /-
 @[continuity]
 theorem continuous_apply (i : ι) : Continuous fun p : ∀ i, π i => p i :=
   continuous_iInf_dom continuous_induced_dom
 #align continuous_apply continuous_apply
+-/
 
+#print continuous_apply_apply /-
 @[continuity]
 theorem continuous_apply_apply {ρ : κ → ι → Type _} [∀ j i, TopologicalSpace (ρ j i)] (j : κ)
     (i : ι) : Continuous fun p : ∀ j, ∀ i, ρ j i => p j i :=
   (continuous_apply i).comp (continuous_apply j)
 #align continuous_apply_apply continuous_apply_apply
+-/
 
+#print continuousAt_apply /-
 theorem continuousAt_apply (i : ι) (x : ∀ i, π i) : ContinuousAt (fun p : ∀ i, π i => p i) x :=
   (continuous_apply i).ContinuousAt
 #align continuous_at_apply continuousAt_apply
+-/
 
+#print Filter.Tendsto.apply /-
 theorem Filter.Tendsto.apply {l : Filter β} {f : β → ∀ i, π i} {x : ∀ i, π i}
     (h : Tendsto f l (𝓝 x)) (i : ι) : Tendsto (fun a => f a i) l (𝓝 <| x i) :=
   (continuousAt_apply i _).Tendsto.comp h
 #align filter.tendsto.apply Filter.Tendsto.apply
+-/
 
+#print nhds_pi /-
 theorem nhds_pi {a : ∀ i, π i} : 𝓝 a = pi fun i => 𝓝 (a i) := by
   simp only [nhds_iInf, nhds_induced, Filter.pi]
 #align nhds_pi nhds_pi
+-/
 
+#print tendsto_pi_nhds /-
 theorem tendsto_pi_nhds {f : β → ∀ i, π i} {g : ∀ i, π i} {u : Filter β} :
     Tendsto f u (𝓝 g) ↔ ∀ x, Tendsto (fun i => f i x) u (𝓝 (g x)) := by
   rw [nhds_pi, Filter.tendsto_pi]
 #align tendsto_pi_nhds tendsto_pi_nhds
+-/
 
+#print continuousAt_pi /-
 theorem continuousAt_pi {f : α → ∀ i, π i} {x : α} :
     ContinuousAt f x ↔ ∀ i, ContinuousAt (fun y => f y i) x :=
   tendsto_pi_nhds
 #align continuous_at_pi continuousAt_pi
+-/
 
+#print Filter.Tendsto.update /-
 theorem Filter.Tendsto.update [DecidableEq ι] {l : Filter β} {f : β → ∀ i, π i} {x : ∀ i, π i}
     (hf : Tendsto f l (𝓝 x)) (i : ι) {g : β → π i} {xi : π i} (hg : Tendsto g l (𝓝 xi)) :
     Tendsto (fun a => update (f a) i (g a)) l (𝓝 <| update x i xi) :=
   tendsto_pi_nhds.2 fun j => by rcases em (j = i) with (rfl | hj) <;> simp [*, hf.apply]
 #align filter.tendsto.update Filter.Tendsto.update
+-/
 
+#print ContinuousAt.update /-
 theorem ContinuousAt.update [DecidableEq ι] {a : α} (hf : ContinuousAt f a) (i : ι) {g : α → π i}
     (hg : ContinuousAt g a) : ContinuousAt (fun a => update (f a) i (g a)) a :=
   hf.update i hg
 #align continuous_at.update ContinuousAt.update
+-/
 
+#print Continuous.update /-
 theorem Continuous.update [DecidableEq ι] (hf : Continuous f) (i : ι) {g : α → π i}
     (hg : Continuous g) : Continuous fun a => update (f a) i (g a) :=
   continuous_iff_continuousAt.2 fun x => hf.ContinuousAt.update i hg.ContinuousAt
 #align continuous.update Continuous.update
+-/
 
+#print continuous_update /-
 /-- `function.update f i x` is continuous in `(f, x)`. -/
 @[continuity]
 theorem continuous_update [DecidableEq ι] (i : ι) :
     Continuous fun f : (∀ j, π j) × π i => update f.1 i f.2 :=
   continuous_fst.update i continuous_snd
 #align continuous_update continuous_update
+-/
 
 #print continuous_mulSingle /-
 /-- `pi.mul_single i x` is continuous in `x`. -/
@@ -1448,31 +1706,40 @@ theorem continuous_mulSingle [∀ i, One (π i)] [DecidableEq ι] (i : ι) :
 #align continuous_single continuous_single
 -/
 
+#print Filter.Tendsto.fin_insertNth /-
 theorem Filter.Tendsto.fin_insertNth {n} {π : Fin (n + 1) → Type _} [∀ i, TopologicalSpace (π i)]
     (i : Fin (n + 1)) {f : β → π i} {l : Filter β} {x : π i} (hf : Tendsto f l (𝓝 x))
     {g : β → ∀ j : Fin n, π (i.succAbove j)} {y : ∀ j, π (i.succAbove j)} (hg : Tendsto g l (𝓝 y)) :
     Tendsto (fun a => i.insertNth (f a) (g a)) l (𝓝 <| i.insertNth x y) :=
   tendsto_pi_nhds.2 fun j => Fin.succAboveCases i (by simpa) (by simpa using tendsto_pi_nhds.1 hg) j
 #align filter.tendsto.fin_insert_nth Filter.Tendsto.fin_insertNth
+-/
 
+#print ContinuousAt.fin_insertNth /-
 theorem ContinuousAt.fin_insertNth {n} {π : Fin (n + 1) → Type _} [∀ i, TopologicalSpace (π i)]
     (i : Fin (n + 1)) {f : α → π i} {a : α} (hf : ContinuousAt f a)
     {g : α → ∀ j : Fin n, π (i.succAbove j)} (hg : ContinuousAt g a) :
     ContinuousAt (fun a => i.insertNth (f a) (g a)) a :=
   hf.fin_insertNth i hg
 #align continuous_at.fin_insert_nth ContinuousAt.fin_insertNth
+-/
 
+#print Continuous.fin_insertNth /-
 theorem Continuous.fin_insertNth {n} {π : Fin (n + 1) → Type _} [∀ i, TopologicalSpace (π i)]
     (i : Fin (n + 1)) {f : α → π i} (hf : Continuous f) {g : α → ∀ j : Fin n, π (i.succAbove j)}
     (hg : Continuous g) : Continuous fun a => i.insertNth (f a) (g a) :=
   continuous_iff_continuousAt.2 fun a => hf.ContinuousAt.fin_insertNth i hg.ContinuousAt
 #align continuous.fin_insert_nth Continuous.fin_insertNth
+-/
 
+#print isOpen_set_pi /-
 theorem isOpen_set_pi {i : Set ι} {s : ∀ a, Set (π a)} (hi : i.Finite)
     (hs : ∀ a ∈ i, IsOpen (s a)) : IsOpen (pi i s) := by
   rw [pi_def] <;> exact isOpen_biInter hi fun a ha => (hs _ ha).Preimage (continuous_apply _)
 #align is_open_set_pi isOpen_set_pi
+-/
 
+#print isOpen_pi_iff /-
 theorem isOpen_pi_iff {s : Set (∀ a, π a)} :
     IsOpen s ↔
       ∀ f,
@@ -1504,7 +1771,9 @@ theorem isOpen_pi_iff {s : Set (∀ a, π a)} :
     · rw [← Set.univ_pi_ite]
       simp only [← ite_and, ← Finset.mem_coe, and_self_iff, Set.univ_pi_ite, h2]
 #align is_open_pi_iff isOpen_pi_iff
+-/
 
+#print isOpen_pi_iff' /-
 theorem isOpen_pi_iff' [Finite ι] {s : Set (∀ a, π a)} :
     IsOpen s ↔
       ∀ f, f ∈ s → ∃ u : ∀ a, Set (π a), (∀ a, IsOpen (u a) ∧ f a ∈ u a) ∧ Set.univ.pi u ⊆ s :=
@@ -1524,32 +1793,44 @@ theorem isOpen_pi_iff' [Finite ι] {s : Set (∀ a, π a)} :
     exact fun ⟨u, ⟨h1, _⟩⟩ =>
       ⟨Finset.univ, u, ⟨fun i => ⟨u i, ⟨rfl.subset, h1 i⟩⟩, by rwa [Finset.coe_univ]⟩⟩
 #align is_open_pi_iff' isOpen_pi_iff'
+-/
 
+#print isClosed_set_pi /-
 theorem isClosed_set_pi {i : Set ι} {s : ∀ a, Set (π a)} (hs : ∀ a ∈ i, IsClosed (s a)) :
     IsClosed (pi i s) := by
   rw [pi_def] <;>
     exact isClosed_iInter fun a => isClosed_iInter fun ha => (hs _ ha).Preimage (continuous_apply _)
 #align is_closed_set_pi isClosed_set_pi
+-/
 
+#print mem_nhds_of_pi_mem_nhds /-
 theorem mem_nhds_of_pi_mem_nhds {I : Set ι} {s : ∀ i, Set (π i)} (a : ∀ i, π i) (hs : I.pi s ∈ 𝓝 a)
     {i : ι} (hi : i ∈ I) : s i ∈ 𝓝 (a i) := by rw [nhds_pi] at hs ; exact mem_of_pi_mem_pi hs hi
 #align mem_nhds_of_pi_mem_nhds mem_nhds_of_pi_mem_nhds
+-/
 
+#print set_pi_mem_nhds /-
 theorem set_pi_mem_nhds {i : Set ι} {s : ∀ a, Set (π a)} {x : ∀ a, π a} (hi : i.Finite)
     (hs : ∀ a ∈ i, s a ∈ 𝓝 (x a)) : pi i s ∈ 𝓝 x := by rw [pi_def, bInter_mem hi];
   exact fun a ha => (continuous_apply a).ContinuousAt (hs a ha)
 #align set_pi_mem_nhds set_pi_mem_nhds
+-/
 
+#print set_pi_mem_nhds_iff /-
 theorem set_pi_mem_nhds_iff {I : Set ι} (hI : I.Finite) {s : ∀ i, Set (π i)} (a : ∀ i, π i) :
     I.pi s ∈ 𝓝 a ↔ ∀ i : ι, i ∈ I → s i ∈ 𝓝 (a i) := by rw [nhds_pi, pi_mem_pi_iff hI];
   infer_instance
 #align set_pi_mem_nhds_iff set_pi_mem_nhds_iff
+-/
 
+#print interior_pi_set /-
 theorem interior_pi_set {I : Set ι} (hI : I.Finite) {s : ∀ i, Set (π i)} :
     interior (pi I s) = I.pi fun i => interior (s i) := by ext a;
   simp only [Set.mem_pi, mem_interior_iff_mem_nhds, set_pi_mem_nhds_iff hI]
 #align interior_pi_set interior_pi_set
+-/
 
+#print exists_finset_piecewise_mem_of_mem_nhds /-
 theorem exists_finset_piecewise_mem_of_mem_nhds [DecidableEq ι] {s : Set (∀ a, π a)} {x : ∀ a, π a}
     (hs : s ∈ 𝓝 x) (y : ∀ a, π a) : ∃ I : Finset ι, I.piecewise x y ∈ s :=
   by
@@ -1558,6 +1839,7 @@ theorem exists_finset_piecewise_mem_of_mem_nhds [DecidableEq ι] {s : Set (∀ a
   refine' ⟨I, hts fun i hi => _⟩
   simpa [Finset.mem_coe.1 hi] using mem_of_mem_nhds (htx i)
 #align exists_finset_piecewise_mem_of_mem_nhds exists_finset_piecewise_mem_of_mem_nhds
+-/
 
 #print pi_eq_generateFrom /-
 theorem pi_eq_generateFrom :
@@ -1616,6 +1898,7 @@ theorem pi_generateFrom_eq_finite {π : ι → Type _} {g : ∀ a, Set (Set (π
 #align pi_generate_from_eq_finite pi_generateFrom_eq_finite
 -/
 
+#print inducing_iInf_to_pi /-
 /-- Suppose `π i` is a family of topological spaces indexed by `i : ι`, and `X` is a type
 endowed with a family of maps `f i : X → π i` for every `i : ι`, hence inducing a
 map `g : X → Π i, π i`. This lemma shows that infimum of the topologies on `X` induced by
@@ -1630,6 +1913,7 @@ theorem inducing_iInf_to_pi {X : Type _} (f : ∀ i, X → π i) :
   funext
   erw [induced_compose]
 #align inducing_infi_to_pi inducing_iInf_to_pi
+-/
 
 variable [Finite ι] [∀ i, DiscreteTopology (π i)]
 
@@ -1651,19 +1935,26 @@ section Sigma
 variable {ι κ : Type _} {σ : ι → Type _} {τ : κ → Type _} [∀ i, TopologicalSpace (σ i)]
   [∀ k, TopologicalSpace (τ k)] [TopologicalSpace α]
 
+#print continuous_sigmaMk /-
 @[continuity]
 theorem continuous_sigmaMk {i : ι} : Continuous (@Sigma.mk ι σ i) :=
   continuous_iSup_rng continuous_coinduced_rng
 #align continuous_sigma_mk continuous_sigmaMk
+-/
 
+#print isOpen_sigma_iff /-
 theorem isOpen_sigma_iff {s : Set (Sigma σ)} : IsOpen s ↔ ∀ i, IsOpen (Sigma.mk i ⁻¹' s) := by
   simp only [isOpen_iSup_iff, isOpen_coinduced]
 #align is_open_sigma_iff isOpen_sigma_iff
+-/
 
+#print isClosed_sigma_iff /-
 theorem isClosed_sigma_iff {s : Set (Sigma σ)} : IsClosed s ↔ ∀ i, IsClosed (Sigma.mk i ⁻¹' s) := by
   simp only [← isOpen_compl_iff, isOpen_sigma_iff, preimage_compl]
 #align is_closed_sigma_iff isClosed_sigma_iff
+-/
 
+#print isOpenMap_sigmaMk /-
 theorem isOpenMap_sigmaMk {i : ι} : IsOpenMap (@Sigma.mk ι σ i) :=
   by
   intro s hs
@@ -1674,11 +1965,15 @@ theorem isOpenMap_sigmaMk {i : ι} : IsOpenMap (@Sigma.mk ι σ i) :=
   · rw [preimage_image_sigma_mk_of_ne hne]
     exact isOpen_empty
 #align is_open_map_sigma_mk isOpenMap_sigmaMk
+-/
 
+#print isOpen_range_sigmaMk /-
 theorem isOpen_range_sigmaMk {i : ι} : IsOpen (Set.range (@Sigma.mk ι σ i)) :=
   isOpenMap_sigmaMk.isOpen_range
 #align is_open_range_sigma_mk isOpen_range_sigmaMk
+-/
 
+#print isClosedMap_sigmaMk /-
 theorem isClosedMap_sigmaMk {i : ι} : IsClosedMap (@Sigma.mk ι σ i) :=
   by
   intro s hs
@@ -1689,94 +1984,129 @@ theorem isClosedMap_sigmaMk {i : ι} : IsClosedMap (@Sigma.mk ι σ i) :=
   · rw [preimage_image_sigma_mk_of_ne hne]
     exact isClosed_empty
 #align is_closed_map_sigma_mk isClosedMap_sigmaMk
+-/
 
+#print isClosed_range_sigmaMk /-
 theorem isClosed_range_sigmaMk {i : ι} : IsClosed (Set.range (@Sigma.mk ι σ i)) :=
   isClosedMap_sigmaMk.closed_range
 #align is_closed_range_sigma_mk isClosed_range_sigmaMk
+-/
 
+#print openEmbedding_sigmaMk /-
 theorem openEmbedding_sigmaMk {i : ι} : OpenEmbedding (@Sigma.mk ι σ i) :=
   openEmbedding_of_continuous_injective_open continuous_sigmaMk sigma_mk_injective isOpenMap_sigmaMk
 #align open_embedding_sigma_mk openEmbedding_sigmaMk
+-/
 
+#print closedEmbedding_sigmaMk /-
 theorem closedEmbedding_sigmaMk {i : ι} : ClosedEmbedding (@Sigma.mk ι σ i) :=
   closedEmbedding_of_continuous_injective_closed continuous_sigmaMk sigma_mk_injective
     isClosedMap_sigmaMk
 #align closed_embedding_sigma_mk closedEmbedding_sigmaMk
+-/
 
+#print embedding_sigmaMk /-
 theorem embedding_sigmaMk {i : ι} : Embedding (@Sigma.mk ι σ i) :=
   closedEmbedding_sigmaMk.1
 #align embedding_sigma_mk embedding_sigmaMk
+-/
 
+#print Sigma.nhds_mk /-
 theorem Sigma.nhds_mk (i : ι) (x : σ i) : 𝓝 (⟨i, x⟩ : Sigma σ) = map (Sigma.mk i) (𝓝 x) :=
   (openEmbedding_sigmaMk.map_nhds_eq x).symm
 #align sigma.nhds_mk Sigma.nhds_mk
+-/
 
+#print Sigma.nhds_eq /-
 theorem Sigma.nhds_eq (x : Sigma σ) : 𝓝 x = map (Sigma.mk x.1) (𝓝 x.2) := by cases x;
   apply Sigma.nhds_mk
 #align sigma.nhds_eq Sigma.nhds_eq
+-/
 
+#print comap_sigmaMk_nhds /-
 theorem comap_sigmaMk_nhds (i : ι) (x : σ i) : comap (Sigma.mk i) (𝓝 ⟨i, x⟩) = 𝓝 x :=
   (embedding_sigmaMk.to_inducing.nhds_eq_comap _).symm
 #align comap_sigma_mk_nhds comap_sigmaMk_nhds
+-/
 
+#print isOpen_sigma_fst_preimage /-
 theorem isOpen_sigma_fst_preimage (s : Set ι) : IsOpen (Sigma.fst ⁻¹' s : Set (Σ a, σ a)) :=
   by
   rw [← bUnion_of_singleton s, preimage_Union₂]
   simp only [← range_sigma_mk]
   exact isOpen_biUnion fun _ _ => isOpen_range_sigmaMk
 #align is_open_sigma_fst_preimage isOpen_sigma_fst_preimage
+-/
 
+#print continuous_sigma_iff /-
 /-- A map out of a sum type is continuous iff its restriction to each summand is. -/
 @[simp]
 theorem continuous_sigma_iff {f : Sigma σ → α} : Continuous f ↔ ∀ i, Continuous fun a => f ⟨i, a⟩ :=
   by simp only [continuous_iSup_dom, continuous_coinduced_dom]
 #align continuous_sigma_iff continuous_sigma_iff
+-/
 
+#print continuous_sigma /-
 /-- A map out of a sum type is continuous if its restriction to each summand is. -/
 @[continuity]
 theorem continuous_sigma {f : Sigma σ → α} (hf : ∀ i, Continuous fun a => f ⟨i, a⟩) :
     Continuous f :=
   continuous_sigma_iff.2 hf
 #align continuous_sigma continuous_sigma
+-/
 
+#print continuous_sigma_map /-
 @[simp]
 theorem continuous_sigma_map {f₁ : ι → κ} {f₂ : ∀ i, σ i → τ (f₁ i)} :
     Continuous (Sigma.map f₁ f₂) ↔ ∀ i, Continuous (f₂ i) :=
   continuous_sigma_iff.trans <| by simp only [Sigma.map, embedding_sigma_mk.continuous_iff]
 #align continuous_sigma_map continuous_sigma_map
+-/
 
+#print Continuous.sigma_map /-
 @[continuity]
 theorem Continuous.sigma_map {f₁ : ι → κ} {f₂ : ∀ i, σ i → τ (f₁ i)} (hf : ∀ i, Continuous (f₂ i)) :
     Continuous (Sigma.map f₁ f₂) :=
   continuous_sigma_map.2 hf
 #align continuous.sigma_map Continuous.sigma_map
+-/
 
+#print isOpenMap_sigma /-
 theorem isOpenMap_sigma {f : Sigma σ → α} : IsOpenMap f ↔ ∀ i, IsOpenMap fun a => f ⟨i, a⟩ := by
   simp only [isOpenMap_iff_nhds_le, Sigma.forall, Sigma.nhds_eq, map_map]
 #align is_open_map_sigma isOpenMap_sigma
+-/
 
+#print isOpenMap_sigma_map /-
 theorem isOpenMap_sigma_map {f₁ : ι → κ} {f₂ : ∀ i, σ i → τ (f₁ i)} :
     IsOpenMap (Sigma.map f₁ f₂) ↔ ∀ i, IsOpenMap (f₂ i) :=
   isOpenMap_sigma.trans <|
     forall_congr' fun i => (@openEmbedding_sigmaMk _ _ _ (f₁ i)).isOpenMap_iff.symm
 #align is_open_map_sigma_map isOpenMap_sigma_map
+-/
 
+#print inducing_sigma_map /-
 theorem inducing_sigma_map {f₁ : ι → κ} {f₂ : ∀ i, σ i → τ (f₁ i)} (h₁ : Injective f₁) :
     Inducing (Sigma.map f₁ f₂) ↔ ∀ i, Inducing (f₂ i) := by
   simp only [inducing_iff_nhds, Sigma.forall, Sigma.nhds_mk, Sigma.map, ← map_sigma_mk_comap h₁,
     map_inj sigma_mk_injective]
 #align inducing_sigma_map inducing_sigma_map
+-/
 
+#print embedding_sigma_map /-
 theorem embedding_sigma_map {f₁ : ι → κ} {f₂ : ∀ i, σ i → τ (f₁ i)} (h : Injective f₁) :
     Embedding (Sigma.map f₁ f₂) ↔ ∀ i, Embedding (f₂ i) := by
   simp only [embedding_iff, injective.sigma_map, inducing_sigma_map h, forall_and, h.sigma_map_iff]
 #align embedding_sigma_map embedding_sigma_map
+-/
 
+#print openEmbedding_sigma_map /-
 theorem openEmbedding_sigma_map {f₁ : ι → κ} {f₂ : ∀ i, σ i → τ (f₁ i)} (h : Injective f₁) :
     OpenEmbedding (Sigma.map f₁ f₂) ↔ ∀ i, OpenEmbedding (f₂ i) := by
   simp only [openEmbedding_iff_embedding_open, isOpenMap_sigma_map, embedding_sigma_map h,
     forall_and]
 #align open_embedding_sigma_map openEmbedding_sigma_map
+-/
 
 end Sigma
 

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

@@ -723,8 +723,8 @@ theorem ContinuousAt.prod_map' {f : α → γ} {g : β → δ} {x : α} {y : β}
 theorem prod_generateFrom_generateFrom_eq {α β : Type _} {s : Set (Set α)} {t : Set (Set β)}
     (hs : ⋃₀ s = univ) (ht : ⋃₀ t = univ) :
     @Prod.topologicalSpace α β (generateFrom s) (generateFrom t) =
-      generateFrom { g | ∃ u ∈ s, ∃ v ∈ t, g = u ×ˢ v } :=
-  let G := generateFrom { g | ∃ u ∈ s, ∃ v ∈ t, g = u ×ˢ v }
+      generateFrom {g | ∃ u ∈ s, ∃ v ∈ t, g = u ×ˢ v} :=
+  let G := generateFrom {g | ∃ u ∈ s, ∃ v ∈ t, g = u ×ˢ v}
   le_antisymm
     (le_generateFrom fun g ⟨u, hu, v, hv, g_eq⟩ =>
       g_eq.symm ▸
@@ -752,7 +752,7 @@ theorem prod_generateFrom_generateFrom_eq {α β : Type _} {s : Set (Set α)} {t
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
 theorem prod_eq_generateFrom :
     Prod.topologicalSpace =
-      generateFrom { g | ∃ (s : Set α) (t : Set β), IsOpen s ∧ IsOpen t ∧ g = s ×ˢ t } :=
+      generateFrom {g | ∃ (s : Set α) (t : Set β), IsOpen s ∧ IsOpen t ∧ g = s ×ˢ t} :=
   le_antisymm (le_generateFrom fun g ⟨s, t, hs, ht, g_eq⟩ => g_eq.symm ▸ hs.Prod ht)
     (le_inf
       (ball_image_of_ball fun t ht =>
@@ -1123,7 +1123,7 @@ theorem embedding_subtype_val : Embedding (coe : Subtype p → α) :=
 -/
 
 #print closedEmbedding_subtype_val /-
-theorem closedEmbedding_subtype_val (h : IsClosed { a | p a }) :
+theorem closedEmbedding_subtype_val (h : IsClosed {a | p a}) :
     ClosedEmbedding (coe : Subtype p → α) :=
   ⟨embedding_subtype_val, by rwa [Subtype.range_coe_subtype]⟩
 #align closed_embedding_subtype_coe closedEmbedding_subtype_val
@@ -1217,7 +1217,7 @@ theorem Subtype.dense_iff {s : Set α} {t : Set s} : Dense t ↔ s ⊆ closure (
 -/
 
 #print map_nhds_subtype_coe_eq_nhds /-
-theorem map_nhds_subtype_coe_eq_nhds {a : α} (ha : p a) (h : { a | p a } ∈ 𝓝 a) :
+theorem map_nhds_subtype_coe_eq_nhds {a : α} (ha : p a) (h : {a | p a} ∈ 𝓝 a) :
     map (coe : Subtype p → α) (𝓝 ⟨a, ha⟩) = 𝓝 a :=
   map_nhds_induced_of_mem <| by simpa only [Subtype.coe_mk, Subtype.range_coe] using h
 #align map_nhds_subtype_coe_eq map_nhds_subtype_coe_eq_nhds
@@ -1563,7 +1563,7 @@ theorem exists_finset_piecewise_mem_of_mem_nhds [DecidableEq ι] {s : Set (∀ a
 theorem pi_eq_generateFrom :
     Pi.topologicalSpace =
       generateFrom
-        { g | ∃ (s : ∀ a, Set (π a)) (i : Finset ι), (∀ a ∈ i, IsOpen (s a)) ∧ g = pi (↑i) s } :=
+        {g | ∃ (s : ∀ a, Set (π a)) (i : Finset ι), (∀ a ∈ i, IsOpen (s a)) ∧ g = pi (↑i) s} :=
   le_antisymm
     (le_generateFrom fun g ⟨s, i, hi, Eq⟩ => Eq.symm ▸ isOpen_set_pi (Finset.finite_toSet _) hi)
     (le_iInf fun a s ⟨t, ht, s_eq⟩ =>
@@ -1576,9 +1576,9 @@ theorem pi_eq_generateFrom :
 theorem pi_generateFrom_eq {π : ι → Type _} {g : ∀ a, Set (Set (π a))} :
     (@Pi.topologicalSpace ι π fun a => generateFrom (g a)) =
       generateFrom
-        { t | ∃ (s : ∀ a, Set (π a)) (i : Finset ι), (∀ a ∈ i, s a ∈ g a) ∧ t = pi (↑i) s } :=
+        {t | ∃ (s : ∀ a, Set (π a)) (i : Finset ι), (∀ a ∈ i, s a ∈ g a) ∧ t = pi (↑i) s} :=
   by
-  let G := { t | ∃ (s : ∀ a, Set (π a)) (i : Finset ι), (∀ a ∈ i, s a ∈ g a) ∧ t = pi (↑i) s }
+  let G := {t | ∃ (s : ∀ a, Set (π a)) (i : Finset ι), (∀ a ∈ i, s a ∈ g a) ∧ t = pi (↑i) s}
   rw [pi_eq_generateFrom]
   refine' le_antisymm (generate_from_anti _) (le_generateFrom _)
   exact fun s ⟨t, i, ht, Eq⟩ => ⟨t, i, fun a ha => generate_open.basic _ (ht a ha), Eq⟩
@@ -1595,7 +1595,7 @@ theorem pi_generateFrom_eq {π : ι → Type _} {g : ∀ a, Set (Set (π a))} :
 theorem pi_generateFrom_eq_finite {π : ι → Type _} {g : ∀ a, Set (Set (π a))} [Finite ι]
     (hg : ∀ a, ⋃₀ g a = univ) :
     (@Pi.topologicalSpace ι π fun a => generateFrom (g a)) =
-      generateFrom { t | ∃ s : ∀ a, Set (π a), (∀ a, s a ∈ g a) ∧ t = pi univ s } :=
+      generateFrom {t | ∃ s : ∀ a, Set (π a), (∀ a, s a ∈ g a) ∧ t = pi univ s} :=
   by
   cases nonempty_fintype ι
   rw [pi_generateFrom_eq]
@@ -1611,7 +1611,7 @@ theorem pi_generateFrom_eq_finite {π : ι → Type _} {g : ∀ a, Set (Set (π
     · simp [pi_if]
     · refine' generate_open.basic _ ⟨_, fun a => _, rfl⟩
       by_cases a ∈ i <;> simp_all [Set.pi]
-    · have : f ∈ pi { a | a ∉ i } c := by simp_all [Set.pi]
+    · have : f ∈ pi {a | a ∉ i} c := by simp_all [Set.pi]
       simpa [pi_if, hf]
 #align pi_generate_from_eq_finite pi_generateFrom_eq_finite
 -/
@@ -1638,7 +1638,7 @@ variable [Finite ι] [∀ i, DiscreteTopology (π i)]
 instance Pi.discreteTopology : DiscreteTopology (∀ i, π i) :=
   singletons_open_iff_discrete.mp fun x =>
     by
-    rw [show {x} = ⋂ i, { y : ∀ i, π i | y i = x i } by ext;
+    rw [show {x} = ⋂ i, {y : ∀ i, π i | y i = x i} by ext;
         simp only [funext_iff, Set.mem_singleton_iff, Set.mem_iInter, Set.mem_setOf_eq]]
     exact isOpen_iInter fun i => (continuous_apply i).isOpen_preimage {x i} (isOpen_discrete {x i})
 #align Pi.discrete_topology Pi.discreteTopology

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

@@ -632,7 +632,7 @@ theorem mem_nhds_prod_iff {a : α} {b : β} {s : Set (α × β)} :
 
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
 theorem mem_nhds_prod_iff' {a : α} {b : β} {s : Set (α × β)} :
-    s ∈ 𝓝 (a, b) ↔ ∃ (u : Set α)(v : Set β), IsOpen u ∧ a ∈ u ∧ IsOpen v ∧ b ∈ v ∧ u ×ˢ v ⊆ s :=
+    s ∈ 𝓝 (a, b) ↔ ∃ (u : Set α) (v : Set β), IsOpen u ∧ a ∈ u ∧ IsOpen v ∧ b ∈ v ∧ u ×ˢ v ⊆ s :=
   by
   rw [mem_nhds_prod_iff]
   constructor
@@ -698,7 +698,7 @@ theorem Filter.Tendsto.prod_mk_nhds {γ} {a : α} {b : β} {f : Filter γ} {ma :
 
 theorem Filter.Eventually.curry_nhds {p : α × β → Prop} {x : α} {y : β}
     (h : ∀ᶠ x in 𝓝 (x, y), p x) : ∀ᶠ x' in 𝓝 x, ∀ᶠ y' in 𝓝 y, p (x', y') := by
-  rw [nhds_prod_eq] at h; exact h.curry
+  rw [nhds_prod_eq] at h ; exact h.curry
 #align filter.eventually.curry_nhds Filter.Eventually.curry_nhds
 
 theorem ContinuousAt.prod {f : α → β} {g : α → γ} {x : α} (hf : ContinuousAt f x)
@@ -752,7 +752,7 @@ theorem prod_generateFrom_generateFrom_eq {α β : Type _} {s : Set (Set α)} {t
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
 theorem prod_eq_generateFrom :
     Prod.topologicalSpace =
-      generateFrom { g | ∃ (s : Set α)(t : Set β), IsOpen s ∧ IsOpen t ∧ g = s ×ˢ t } :=
+      generateFrom { g | ∃ (s : Set α) (t : Set β), IsOpen s ∧ IsOpen t ∧ g = s ×ˢ t } :=
   le_antisymm (le_generateFrom fun g ⟨s, t, hs, ht, g_eq⟩ => g_eq.symm ▸ hs.Prod ht)
     (le_inf
       (ball_image_of_ball fun t ht =>
@@ -765,7 +765,7 @@ theorem prod_eq_generateFrom :
 theorem isOpen_prod_iff {s : Set (α × β)} :
     IsOpen s ↔
       ∀ a b,
-        (a, b) ∈ s → ∃ (u : Set α)(v : Set β), IsOpen u ∧ IsOpen v ∧ a ∈ u ∧ b ∈ v ∧ u ×ˢ v ⊆ s :=
+        (a, b) ∈ s → ∃ (u : Set α) (v : Set β), IsOpen u ∧ IsOpen v ∧ a ∈ u ∧ b ∈ v ∧ u ×ˢ v ⊆ s :=
   by
   rw [isOpen_iff_nhds]
   simp_rw [le_principal_iff, Prod.forall,
@@ -801,9 +801,9 @@ theorem map_fst_nhdsWithin (x : α × β) : map Prod.fst (𝓝[Prod.snd ⁻¹' {
   by
   refine' le_antisymm (continuous_at_fst.mono_left inf_le_left) fun s hs => _
   rcases x with ⟨x, y⟩
-  rw [mem_map, nhdsWithin, mem_inf_principal, mem_nhds_prod_iff] at hs
+  rw [mem_map, nhdsWithin, mem_inf_principal, mem_nhds_prod_iff] at hs 
   rcases hs with ⟨u, hu, v, hv, H⟩
-  simp only [prod_subset_iff, mem_singleton_iff, mem_set_of_eq, mem_preimage] at H
+  simp only [prod_subset_iff, mem_singleton_iff, mem_set_of_eq, mem_preimage] at H 
   exact mem_of_superset hu fun z hz => H _ hz _ (mem_of_mem_nhds hv) rfl
 #align map_fst_nhds_within map_fst_nhdsWithin
 
@@ -823,9 +823,9 @@ theorem map_snd_nhdsWithin (x : α × β) : map Prod.snd (𝓝[Prod.fst ⁻¹' {
   by
   refine' le_antisymm (continuous_at_snd.mono_left inf_le_left) fun s hs => _
   rcases x with ⟨x, y⟩
-  rw [mem_map, nhdsWithin, mem_inf_principal, mem_nhds_prod_iff] at hs
+  rw [mem_map, nhdsWithin, mem_inf_principal, mem_nhds_prod_iff] at hs 
   rcases hs with ⟨u, hu, v, hv, H⟩
-  simp only [prod_subset_iff, mem_singleton_iff, mem_set_of_eq, mem_preimage] at H
+  simp only [prod_subset_iff, mem_singleton_iff, mem_set_of_eq, mem_preimage] at H 
   exact mem_of_superset hv fun z hz => H _ (mem_of_mem_nhds hu) _ hz rfl
 #align map_snd_nhds_within map_snd_nhdsWithin
 
@@ -860,7 +860,7 @@ theorem isOpen_prod_iff' {s : Set α} {t : Set β} :
       · rw [← snd_image_prod st.1 t]
         exact isOpenMap_snd _ H
     · intro H
-      simp only [st.1.ne_empty, st.2.ne_empty, not_false_iff, or_false_iff] at H
+      simp only [st.1.ne_empty, st.2.ne_empty, not_false_iff, or_false_iff] at H 
       exact H.1.Prod H.2
 #align is_open_prod_iff' isOpen_prod_iff'
 
@@ -1477,7 +1477,7 @@ theorem isOpen_pi_iff {s : Set (∀ a, π a)} :
     IsOpen s ↔
       ∀ f,
         f ∈ s →
-          ∃ (I : Finset ι)(u : ∀ a, Set (π a)),
+          ∃ (I : Finset ι) (u : ∀ a, Set (π a)),
             (∀ a, a ∈ I → IsOpen (u a) ∧ f a ∈ u a) ∧ (I : Set ι).pi u ⊆ s :=
   by
   rw [isOpen_iff_nhds]
@@ -1532,7 +1532,7 @@ theorem isClosed_set_pi {i : Set ι} {s : ∀ a, Set (π a)} (hs : ∀ a ∈ i,
 #align is_closed_set_pi isClosed_set_pi
 
 theorem mem_nhds_of_pi_mem_nhds {I : Set ι} {s : ∀ i, Set (π i)} (a : ∀ i, π i) (hs : I.pi s ∈ 𝓝 a)
-    {i : ι} (hi : i ∈ I) : s i ∈ 𝓝 (a i) := by rw [nhds_pi] at hs; exact mem_of_pi_mem_pi hs hi
+    {i : ι} (hi : i ∈ I) : s i ∈ 𝓝 (a i) := by rw [nhds_pi] at hs ; exact mem_of_pi_mem_pi hs hi
 #align mem_nhds_of_pi_mem_nhds mem_nhds_of_pi_mem_nhds
 
 theorem set_pi_mem_nhds {i : Set ι} {s : ∀ a, Set (π a)} {x : ∀ a, π a} (hi : i.Finite)
@@ -1553,7 +1553,7 @@ theorem interior_pi_set {I : Set ι} (hI : I.Finite) {s : ∀ i, Set (π i)} :
 theorem exists_finset_piecewise_mem_of_mem_nhds [DecidableEq ι] {s : Set (∀ a, π a)} {x : ∀ a, π a}
     (hs : s ∈ 𝓝 x) (y : ∀ a, π a) : ∃ I : Finset ι, I.piecewise x y ∈ s :=
   by
-  simp only [nhds_pi, Filter.mem_pi'] at hs
+  simp only [nhds_pi, Filter.mem_pi'] at hs 
   rcases hs with ⟨I, t, htx, hts⟩
   refine' ⟨I, hts fun i hi => _⟩
   simpa [Finset.mem_coe.1 hi] using mem_of_mem_nhds (htx i)
@@ -1563,7 +1563,7 @@ theorem exists_finset_piecewise_mem_of_mem_nhds [DecidableEq ι] {s : Set (∀ a
 theorem pi_eq_generateFrom :
     Pi.topologicalSpace =
       generateFrom
-        { g | ∃ (s : ∀ a, Set (π a))(i : Finset ι), (∀ a ∈ i, IsOpen (s a)) ∧ g = pi (↑i) s } :=
+        { g | ∃ (s : ∀ a, Set (π a)) (i : Finset ι), (∀ a ∈ i, IsOpen (s a)) ∧ g = pi (↑i) s } :=
   le_antisymm
     (le_generateFrom fun g ⟨s, i, hi, Eq⟩ => Eq.symm ▸ isOpen_set_pi (Finset.finite_toSet _) hi)
     (le_iInf fun a s ⟨t, ht, s_eq⟩ =>
@@ -1576,9 +1576,9 @@ theorem pi_eq_generateFrom :
 theorem pi_generateFrom_eq {π : ι → Type _} {g : ∀ a, Set (Set (π a))} :
     (@Pi.topologicalSpace ι π fun a => generateFrom (g a)) =
       generateFrom
-        { t | ∃ (s : ∀ a, Set (π a))(i : Finset ι), (∀ a ∈ i, s a ∈ g a) ∧ t = pi (↑i) s } :=
+        { t | ∃ (s : ∀ a, Set (π a)) (i : Finset ι), (∀ a ∈ i, s a ∈ g a) ∧ t = pi (↑i) s } :=
   by
-  let G := { t | ∃ (s : ∀ a, Set (π a))(i : Finset ι), (∀ a ∈ i, s a ∈ g a) ∧ t = pi (↑i) s }
+  let G := { t | ∃ (s : ∀ a, Set (π a)) (i : Finset ι), (∀ a ∈ i, s a ∈ g a) ∧ t = pi (↑i) s }
   rw [pi_eq_generateFrom]
   refine' le_antisymm (generate_from_anti _) (le_generateFrom _)
   exact fun s ⟨t, i, ht, Eq⟩ => ⟨t, i, fun a ha => generate_open.basic _ (ht a ha), Eq⟩
@@ -1719,7 +1719,7 @@ theorem comap_sigmaMk_nhds (i : ι) (x : σ i) : comap (Sigma.mk i) (𝓝 ⟨i,
   (embedding_sigmaMk.to_inducing.nhds_eq_comap _).symm
 #align comap_sigma_mk_nhds comap_sigmaMk_nhds
 
-theorem isOpen_sigma_fst_preimage (s : Set ι) : IsOpen (Sigma.fst ⁻¹' s : Set (Σa, σ a)) :=
+theorem isOpen_sigma_fst_preimage (s : Set ι) : IsOpen (Sigma.fst ⁻¹' s : Set (Σ a, σ a)) :=
   by
   rw [← bUnion_of_singleton s, preimage_Union₂]
   simp only [← range_sigma_mk]

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

@@ -1802,10 +1802,12 @@ theorem embedding_uLift_down [TopologicalSpace α] : Embedding (ULift.down : ULi
 #align embedding_ulift_down embedding_uLift_down
 -/
 
+#print ULift.closedEmbedding_down /-
 theorem ULift.closedEmbedding_down [TopologicalSpace α] :
     ClosedEmbedding (ULift.down : ULift.{v, u} α → α) :=
   ⟨embedding_uLift_down, by simp only [ulift.down_surjective.range_eq, isClosed_univ]⟩
 #align ulift.closed_embedding_down ULift.closedEmbedding_down
+-/
 
 instance [TopologicalSpace α] [DiscreteTopology α] : DiscreteTopology (ULift α) :=
   embedding_uLift_down.DiscreteTopology

mathlib commit https://github.com/leanprover-community/mathlib/commit/88a563b158f59f2983cfad685664da95502e8cdd

@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Johannes Hölzl, Mario Carneiro, Patrick Massot
 
 ! This file was ported from Lean 3 source module topology.constructions
-! leanprover-community/mathlib commit 76f9c990d4b7c3dd26b87c4c4b51759e249d9e66
+! leanprover-community/mathlib commit f7ebde7ee0d1505dfccac8644ae12371aa3c1c9f
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
@@ -938,6 +938,18 @@ theorem Inducing.prod_map {f : α → β} {g : γ → δ} (hf : Inducing f) (hg
       induced_compose, induced_inf, induced_compose, induced_compose]⟩
 #align inducing.prod_mk Inducing.prod_map
 
+@[simp]
+theorem inducing_const_prod {a : α} {f : β → γ} : (Inducing fun x => (a, f x)) ↔ Inducing f := by
+  simp_rw [inducing_iff, Prod.topologicalSpace, induced_inf, induced_compose, Function.comp,
+    induced_const, top_inf_eq]
+#align inducing_const_prod inducing_const_prod
+
+@[simp]
+theorem inducing_prod_const {b : β} {f : α → γ} : (Inducing fun x => (f x, b)) ↔ Inducing f := by
+  simp_rw [inducing_iff, Prod.topologicalSpace, induced_inf, induced_compose, Function.comp,
+    induced_const, inf_top_eq]
+#align inducing_prod_const inducing_prod_const
+
 theorem Embedding.prod_map {f : α → β} {g : γ → δ} (hf : Embedding f) (hg : Embedding g) :
     Embedding fun x : α × γ => (f x.1, g x.2) :=
   { hf.to_inducing.prod_mk hg.to_inducing with

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

@@ -43,7 +43,7 @@ noncomputable section
 
 open TopologicalSpace Set Filter Function
 
-open Classical Topology Filter
+open scoped Classical Topology Filter
 
 universe u v
 

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

@@ -295,23 +295,11 @@ instance {p : α → Prop} [TopologicalSpace α] [DiscreteTopology α] : Discret
   ⟨bot_unique fun s hs =>
       ⟨coe '' s, isOpen_discrete _, Set.preimage_image_eq _ Subtype.coe_injective⟩⟩
 
-/- warning: sum.discrete_topology -> Sum.discreteTopology 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] [hα : DiscreteTopology.{u1} α _inst_1] [hβ : DiscreteTopology.{u2} β _inst_2], DiscreteTopology.{max u1 u2} (Sum.{u1, u2} α β) (Sum.topologicalSpace.{u1, u2} α β _inst_1 _inst_2)
-but is expected to have type
-  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : TopologicalSpace.{u1} α] [_inst_2 : TopologicalSpace.{u2} β] [hα : DiscreteTopology.{u1} α _inst_1] [hβ : DiscreteTopology.{u2} β _inst_2], DiscreteTopology.{max u2 u1} (Sum.{u1, u2} α β) (instTopologicalSpaceSum.{u1, u2} α β _inst_1 _inst_2)
-Case conversion may be inaccurate. Consider using '#align sum.discrete_topology Sum.discreteTopologyₓ'. -/
 instance Sum.discreteTopology [TopologicalSpace α] [TopologicalSpace β] [hα : DiscreteTopology α]
     [hβ : DiscreteTopology β] : DiscreteTopology (Sum α β) :=
   ⟨by unfold Sum.topologicalSpace <;> simp [hα.eq_bot, hβ.eq_bot]⟩
 #align sum.discrete_topology Sum.discreteTopology
 
-/- warning: sigma.discrete_topology -> Sigma.discreteTopology is a dubious translation:
-lean 3 declaration is
-  forall {α : Type.{u1}} {β : α -> Type.{u2}} [_inst_1 : forall (a : α), TopologicalSpace.{u2} (β a)] [h : forall (a : α), DiscreteTopology.{u2} (β a) (_inst_1 a)], DiscreteTopology.{max u1 u2} (Sigma.{u1, u2} α β) (Sigma.topologicalSpace.{u1, u2} α β (fun (a : α) => _inst_1 a))
-but is expected to have type
-  forall {α : Type.{u1}} {β : α -> Type.{u2}} [_inst_1 : forall (a : α), TopologicalSpace.{u2} (β a)] [h : forall (a : α), DiscreteTopology.{u2} (β a) (_inst_1 a)], DiscreteTopology.{max u2 u1} (Sigma.{u1, u2} α β) (instTopologicalSpaceSigma.{u1, u2} α β (fun (a : α) => _inst_1 a))
-Case conversion may be inaccurate. Consider using '#align sigma.discrete_topology Sigma.discreteTopologyₓ'. -/
 instance Sigma.discreteTopology {β : α → Type v} [∀ a, TopologicalSpace (β a)]
     [h : ∀ a, DiscreteTopology (β a)] : DiscreteTopology (Sigma β) :=
   ⟨by unfold Sigma.topologicalSpace; simp [fun a => (h a).eq_bot]⟩
@@ -321,12 +309,6 @@ section Topα
 
 variable [TopologicalSpace α]
 
-/- warning: mem_nhds_subtype -> mem_nhds_subtype is a dubious translation:
-lean 3 declaration is
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 /-
 The 𝓝 filter and the subspace topology.
 -/
@@ -341,47 +323,23 @@ theorem nhds_subtype (s : Set α) (a : { x // x ∈ s }) : 𝓝 a = comap coe (
 #align nhds_subtype nhds_subtype
 -/
 
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 theorem nhdsWithin_subtype_eq_bot_iff {s t : Set α} {x : s} :
     𝓝[(coe : s → α) ⁻¹' t] x = ⊥ ↔ 𝓝[t] (x : α) ⊓ 𝓟 s = ⊥ := by
   rw [inf_principal_eq_bot_iff_comap, nhdsWithin, nhdsWithin, comap_inf, comap_principal,
     nhds_induced]
 #align nhds_within_subtype_eq_bot_iff nhdsWithin_subtype_eq_bot_iff
 
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 theorem nhds_ne_subtype_eq_bot_iff {S : Set α} {x : S} :
     𝓝[{x}ᶜ] x = ⊥ ↔ 𝓝[{x}ᶜ] (x : α) ⊓ 𝓟 S = ⊥ := by
   rw [← nhdsWithin_subtype_eq_bot_iff, preimage_compl, ← image_singleton,
     subtype.coe_injective.preimage_image]
 #align nhds_ne_subtype_eq_bot_iff nhds_ne_subtype_eq_bot_iff
 
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 theorem nhds_ne_subtype_neBot_iff {S : Set α} {x : S} :
     (𝓝[{x}ᶜ] x).ne_bot ↔ (𝓝[{x}ᶜ] (x : α) ⊓ 𝓟 S).ne_bot := by
   rw [ne_bot_iff, ne_bot_iff, not_iff_not, nhds_ne_subtype_eq_bot_iff]
 #align nhds_ne_subtype_ne_bot_iff nhds_ne_subtype_neBot_iff
 
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 theorem discreteTopology_subtype_iff {S : Set α} : DiscreteTopology S ↔ ∀ x ∈ S, 𝓝[≠] x ⊓ 𝓟 S = ⊥ :=
   by simp_rw [discreteTopology_iff_nhds_ne, SetCoe.forall', nhds_ne_subtype_eq_bot_iff]
 #align discrete_topology_subtype_iff discreteTopology_subtype_iff
@@ -420,42 +378,18 @@ instance : TopologicalSpace (CofiniteTopology α)
     rw [Set.compl_sUnion]
     exact Set.Finite.sInter (mem_image_of_mem _ hts) (h t hts ⟨x, hzt⟩)
 
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 theorem isOpen_iff {s : Set (CofiniteTopology α)} : IsOpen s ↔ s.Nonempty → sᶜ.Finite :=
   Iff.rfl
 #align cofinite_topology.is_open_iff CofiniteTopology.isOpen_iff
 
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 theorem isOpen_iff' {s : Set (CofiniteTopology α)} : IsOpen s ↔ s = ∅ ∨ sᶜ.Finite := by
   simp only [is_open_iff, nonempty_iff_ne_empty, or_iff_not_imp_left]
 #align cofinite_topology.is_open_iff' CofiniteTopology.isOpen_iff'
 
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 theorem isClosed_iff {s : Set (CofiniteTopology α)} : IsClosed s ↔ s = univ ∨ s.Finite := by
   simp [← isOpen_compl_iff, is_open_iff']
 #align cofinite_topology.is_closed_iff CofiniteTopology.isClosed_iff
 
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 theorem nhds_eq (a : CofiniteTopology α) : 𝓝 a = pure a ⊔ cofinite :=
   by
   ext U
@@ -467,12 +401,6 @@ theorem nhds_eq (a : CofiniteTopology α) : 𝓝 a = pure a ⊔ cofinite :=
     exact ⟨U, subset.rfl, fun h => hU', hU⟩
 #align cofinite_topology.nhds_eq CofiniteTopology.nhds_eq
 
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 theorem mem_nhds_iff {a : CofiniteTopology α} {s : Set (CofiniteTopology α)} :
     s ∈ 𝓝 a ↔ a ∈ s ∧ sᶜ.Finite := by simp [nhds_eq]
 #align cofinite_topology.mem_nhds_iff CofiniteTopology.mem_nhds_iff
@@ -486,259 +414,124 @@ section Prod
 variable [TopologicalSpace α] [TopologicalSpace β] [TopologicalSpace γ] [TopologicalSpace δ]
   [TopologicalSpace ε] [TopologicalSpace ζ]
 
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 @[continuity]
 theorem continuous_fst : Continuous (@Prod.fst α β) :=
   continuous_inf_dom_left continuous_induced_dom
 #align continuous_fst continuous_fst
 
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 /-- Postcomposing `f` with `prod.fst` is continuous -/
 theorem Continuous.fst {f : α → β × γ} (hf : Continuous f) : Continuous fun a : α => (f a).1 :=
   continuous_fst.comp hf
 #align continuous.fst Continuous.fst
 
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 /-- Precomposing `f` with `prod.fst` is continuous -/
 theorem Continuous.fst' {f : α → γ} (hf : Continuous f) : Continuous fun x : α × β => f x.fst :=
   hf.comp continuous_fst
 #align continuous.fst' Continuous.fst'
 
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 theorem continuousAt_fst {p : α × β} : ContinuousAt Prod.fst p :=
   continuous_fst.ContinuousAt
 #align continuous_at_fst continuousAt_fst
 
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 /-- Postcomposing `f` with `prod.fst` is continuous at `x` -/
 theorem ContinuousAt.fst {f : α → β × γ} {x : α} (hf : ContinuousAt f x) :
     ContinuousAt (fun a : α => (f a).1) x :=
   continuousAt_fst.comp hf
 #align continuous_at.fst ContinuousAt.fst
 
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 /-- Precomposing `f` with `prod.fst` is continuous at `(x, y)` -/
 theorem ContinuousAt.fst' {f : α → γ} {x : α} {y : β} (hf : ContinuousAt f x) :
     ContinuousAt (fun x : α × β => f x.fst) (x, y) :=
   ContinuousAt.comp hf continuousAt_fst
 #align continuous_at.fst' ContinuousAt.fst'
 
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 /-- Precomposing `f` with `prod.fst` is continuous at `x : α × β` -/
 theorem ContinuousAt.fst'' {f : α → γ} {x : α × β} (hf : ContinuousAt f x.fst) :
     ContinuousAt (fun x : α × β => f x.fst) x :=
   hf.comp continuousAt_fst
 #align continuous_at.fst'' ContinuousAt.fst''
 
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 @[continuity]
 theorem continuous_snd : Continuous (@Prod.snd α β) :=
   continuous_inf_dom_right continuous_induced_dom
 #align continuous_snd continuous_snd
 
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 /-- Postcomposing `f` with `prod.snd` is continuous -/
 theorem Continuous.snd {f : α → β × γ} (hf : Continuous f) : Continuous fun a : α => (f a).2 :=
   continuous_snd.comp hf
 #align continuous.snd Continuous.snd
 
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 /-- Precomposing `f` with `prod.snd` is continuous -/
 theorem Continuous.snd' {f : β → γ} (hf : Continuous f) : Continuous fun x : α × β => f x.snd :=
   hf.comp continuous_snd
 #align continuous.snd' Continuous.snd'
 
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 theorem continuousAt_snd {p : α × β} : ContinuousAt Prod.snd p :=
   continuous_snd.ContinuousAt
 #align continuous_at_snd continuousAt_snd
 
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 /-- Postcomposing `f` with `prod.snd` is continuous at `x` -/
 theorem ContinuousAt.snd {f : α → β × γ} {x : α} (hf : ContinuousAt f x) :
     ContinuousAt (fun a : α => (f a).2) x :=
   continuousAt_snd.comp hf
 #align continuous_at.snd ContinuousAt.snd
 
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 /-- Precomposing `f` with `prod.snd` is continuous at `(x, y)` -/
 theorem ContinuousAt.snd' {f : β → γ} {x : α} {y : β} (hf : ContinuousAt f y) :
     ContinuousAt (fun x : α × β => f x.snd) (x, y) :=
   ContinuousAt.comp hf continuousAt_snd
 #align continuous_at.snd' ContinuousAt.snd'
 
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 /-- Precomposing `f` with `prod.snd` is continuous at `x : α × β` -/
 theorem ContinuousAt.snd'' {f : β → γ} {x : α × β} (hf : ContinuousAt f x.snd) :
     ContinuousAt (fun x : α × β => f x.snd) x :=
   hf.comp continuousAt_snd
 #align continuous_at.snd'' ContinuousAt.snd''
 
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 @[continuity]
 theorem Continuous.prod_mk {f : γ → α} {g : γ → β} (hf : Continuous f) (hg : Continuous g) :
     Continuous fun x => (f x, g x) :=
   continuous_inf_rng.2 ⟨continuous_induced_rng.2 hf, continuous_induced_rng.2 hg⟩
 #align continuous.prod_mk Continuous.prod_mk
 
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 @[simp]
 theorem continuous_prod_mk {f : α → β} {g : α → γ} :
     (Continuous fun x => (f x, g x)) ↔ Continuous f ∧ Continuous g :=
   ⟨fun h => ⟨h.fst, h.snd⟩, fun h => h.1.prod_mk h.2⟩
 #align continuous_prod_mk continuous_prod_mk
 
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-Case conversion may be inaccurate. Consider using '#align continuous.prod.mk Continuous.Prod.mkₓ'. -/
 @[continuity]
 theorem Continuous.Prod.mk (a : α) : Continuous fun b : β => (a, b) :=
   continuous_const.prod_mk continuous_id'
 #align continuous.prod.mk Continuous.Prod.mk
 
-/- warning: continuous.prod.mk_left -> Continuous.Prod.mk_left 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.prod.mk_left Continuous.Prod.mk_leftₓ'. -/
 @[continuity]
 theorem Continuous.Prod.mk_left (b : β) : Continuous fun a : α => (a, b) :=
   continuous_id'.prod_mk continuous_const
 #align continuous.prod.mk_left Continuous.Prod.mk_left
 
-/- warning: continuous.comp₂ -> Continuous.comp₂ 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.comp₂ Continuous.comp₂ₓ'. -/
 theorem Continuous.comp₂ {g : α × β → γ} (hg : Continuous g) {e : δ → α} (he : Continuous e)
     {f : δ → β} (hf : Continuous f) : Continuous fun x => g (e x, f x) :=
   hg.comp <| he.prod_mk hf
 #align continuous.comp₂ Continuous.comp₂
 
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-lean 3 declaration is
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-Case conversion may be inaccurate. Consider using '#align continuous.comp₃ Continuous.comp₃ₓ'. -/
 theorem Continuous.comp₃ {g : α × β × γ → ε} (hg : Continuous g) {e : δ → α} (he : Continuous e)
     {f : δ → β} (hf : Continuous f) {k : δ → γ} (hk : Continuous k) :
     Continuous fun x => g (e x, f x, k x) :=
   hg.comp₂ he <| hf.prod_mk hk
 #align continuous.comp₃ Continuous.comp₃
 
-/- warning: continuous.comp₄ -> Continuous.comp₄ is a dubious translation:
-<too large>
-Case conversion may be inaccurate. Consider using '#align continuous.comp₄ Continuous.comp₄ₓ'. -/
 theorem Continuous.comp₄ {g : α × β × γ × ζ → ε} (hg : Continuous g) {e : δ → α} (he : Continuous e)
     {f : δ → β} (hf : Continuous f) {k : δ → γ} (hk : Continuous k) {l : δ → ζ}
     (hl : Continuous l) : Continuous fun x => g (e x, f x, k x, l x) :=
   hg.comp₃ he hf <| hk.prod_mk hl
 #align continuous.comp₄ Continuous.comp₄
 
-/- warning: continuous.prod_map -> Continuous.prod_map 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.prod_map Continuous.prod_mapₓ'. -/
 theorem Continuous.prod_map {f : γ → α} {g : δ → β} (hf : Continuous f) (hg : Continuous g) :
     Continuous fun x : γ × δ => (f x.1, g x.2) :=
   hf.fst'.prod_mk hg.snd'
 #align continuous.prod_map Continuous.prod_map
 
-/- warning: continuous_inf_dom_left₂ -> continuous_inf_dom_left₂ is a dubious translation:
-lean 3 declaration is
-  forall {α : Type.{u1}} {β : Type.{u2}} {γ : Type.{u3}} {f : α -> β -> γ} {ta1 : TopologicalSpace.{u1} α} {ta2 : TopologicalSpace.{u1} α} {tb1 : TopologicalSpace.{u2} β} {tb2 : TopologicalSpace.{u2} β} {tc1 : TopologicalSpace.{u3} γ}, (Continuous.{max u1 u2, u3} (Prod.{u1, u2} α β) γ (Prod.topologicalSpace.{u1, u2} α β ta1 tb1) tc1 (fun (p : Prod.{u1, u2} α β) => f (Prod.fst.{u1, u2} α β p) (Prod.snd.{u1, u2} α β p))) -> (Continuous.{max u1 u2, u3} (Prod.{u1, u2} α β) γ (Prod.topologicalSpace.{u1, u2} α β (Inf.inf.{u1} (TopologicalSpace.{u1} α) (SemilatticeInf.toHasInf.{u1} (TopologicalSpace.{u1} α) (Lattice.toSemilatticeInf.{u1} (TopologicalSpace.{u1} α) (ConditionallyCompleteLattice.toLattice.{u1} (TopologicalSpace.{u1} α) (CompleteLattice.toConditionallyCompleteLattice.{u1} (TopologicalSpace.{u1} α) (TopologicalSpace.completeLattice.{u1} α))))) ta1 ta2) (Inf.inf.{u2} (TopologicalSpace.{u2} β) (SemilatticeInf.toHasInf.{u2} (TopologicalSpace.{u2} β) (Lattice.toSemilatticeInf.{u2} (TopologicalSpace.{u2} β) (ConditionallyCompleteLattice.toLattice.{u2} (TopologicalSpace.{u2} β) (CompleteLattice.toConditionallyCompleteLattice.{u2} (TopologicalSpace.{u2} β) (TopologicalSpace.completeLattice.{u2} β))))) tb1 tb2)) tc1 (fun (p : Prod.{u1, u2} α β) => f (Prod.fst.{u1, u2} α β p) (Prod.snd.{u1, u2} α β p)))
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-Case conversion may be inaccurate. Consider using '#align continuous_inf_dom_left₂ continuous_inf_dom_left₂ₓ'. -/
 /-- A version of `continuous_inf_dom_left` for binary functions -/
 theorem continuous_inf_dom_left₂ {α β γ} {f : α → β → γ} {ta1 ta2 : TopologicalSpace α}
     {tb1 tb2 : TopologicalSpace β} {tc1 : TopologicalSpace γ}
@@ -751,12 +544,6 @@ theorem continuous_inf_dom_left₂ {α β γ} {f : α → β → γ} {ta1 ta2 :
   exact @Continuous.comp _ _ _ (id _) (id _) _ _ _ h h_continuous_id
 #align continuous_inf_dom_left₂ continuous_inf_dom_left₂
 
-/- warning: continuous_inf_dom_right₂ -> continuous_inf_dom_right₂ is a dubious translation:
-lean 3 declaration is
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-Case conversion may be inaccurate. Consider using '#align continuous_inf_dom_right₂ continuous_inf_dom_right₂ₓ'. -/
 /-- A version of `continuous_inf_dom_right` for binary functions -/
 theorem continuous_inf_dom_right₂ {α β γ} {f : α → β → γ} {ta1 ta2 : TopologicalSpace α}
     {tb1 tb2 : TopologicalSpace β} {tc1 : TopologicalSpace γ}
@@ -769,12 +556,6 @@ theorem continuous_inf_dom_right₂ {α β γ} {f : α → β → γ} {ta1 ta2 :
   exact @Continuous.comp _ _ _ (id _) (id _) _ _ _ h h_continuous_id
 #align continuous_inf_dom_right₂ continuous_inf_dom_right₂
 
-/- warning: continuous_Inf_dom₂ -> continuous_sInf_dom₂ is a dubious translation:
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-Case conversion may be inaccurate. Consider using '#align continuous_Inf_dom₂ continuous_sInf_dom₂ₓ'. -/
 /-- A version of `continuous_Inf_dom` for binary functions -/
 theorem continuous_sInf_dom₂ {α β γ} {f : α → β → γ} {tas : Set (TopologicalSpace α)}
     {tbs : Set (TopologicalSpace β)} {ta : TopologicalSpace α} {tb : TopologicalSpace β}
@@ -790,108 +571,48 @@ theorem continuous_sInf_dom₂ {α β γ} {f : α → β → γ} {tas : Set (Top
   exact @Continuous.comp _ _ _ (id _) (id _) _ _ _ hf h_continuous_id
 #align continuous_Inf_dom₂ continuous_sInf_dom₂
 
-/- warning: filter.eventually.prod_inl_nhds -> Filter.Eventually.prod_inl_nhds is a dubious translation:
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 theorem Filter.Eventually.prod_inl_nhds {p : α → Prop} {a : α} (h : ∀ᶠ x in 𝓝 a, p x) (b : β) :
     ∀ᶠ x in 𝓝 (a, b), p (x : α × β).1 :=
   continuousAt_fst h
 #align filter.eventually.prod_inl_nhds Filter.Eventually.prod_inl_nhds
 
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 theorem Filter.Eventually.prod_inr_nhds {p : β → Prop} {b : β} (h : ∀ᶠ x in 𝓝 b, p x) (a : α) :
     ∀ᶠ x in 𝓝 (a, b), p (x : α × β).2 :=
   continuousAt_snd h
 #align filter.eventually.prod_inr_nhds Filter.Eventually.prod_inr_nhds
 
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 theorem Filter.Eventually.prod_mk_nhds {pa : α → Prop} {a} (ha : ∀ᶠ x in 𝓝 a, pa x) {pb : β → Prop}
     {b} (hb : ∀ᶠ y in 𝓝 b, pb y) : ∀ᶠ p in 𝓝 (a, b), pa (p : α × β).1 ∧ pb p.2 :=
   (ha.prod_inl_nhds b).And (hb.prod_inr_nhds a)
 #align filter.eventually.prod_mk_nhds Filter.Eventually.prod_mk_nhds
 
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 theorem continuous_swap : Continuous (Prod.swap : α × β → β × α) :=
   continuous_snd.prod_mk continuous_fst
 #align continuous_swap continuous_swap
 
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 theorem continuous_uncurry_left {f : α → β → γ} (a : α) (h : Continuous (uncurry f)) :
     Continuous (f a) :=
   show Continuous (uncurry f ∘ fun b => (a, b)) from h.comp (by continuity)
 #align continuous_uncurry_left continuous_uncurry_left
 
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 theorem continuous_uncurry_right {f : α → β → γ} (b : β) (h : Continuous (uncurry f)) :
     Continuous fun a => f a b :=
   show Continuous (uncurry f ∘ fun a => (a, b)) from h.comp (by continuity)
 #align continuous_uncurry_right continuous_uncurry_right
 
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 theorem continuous_curry {g : α × β → γ} (a : α) (h : Continuous g) : Continuous (curry g a) :=
   show Continuous (g ∘ fun b => (a, b)) from h.comp (by continuity)
 #align continuous_curry continuous_curry
 
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 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
 theorem IsOpen.prod {s : Set α} {t : Set β} (hs : IsOpen s) (ht : IsOpen t) : IsOpen (s ×ˢ t) :=
   (hs.Preimage continuous_fst).inter (ht.Preimage continuous_snd)
 #align is_open.prod IsOpen.prod
 
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 theorem nhds_prod_eq {a : α} {b : β} : 𝓝 (a, b) = 𝓝 a ×ᶠ 𝓝 b := by
   rw [Filter.prod, Prod.topologicalSpace, nhds_inf, nhds_induced, nhds_induced]
 #align nhds_prod_eq nhds_prod_eq
 
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 /-- If a function `f x y` is such that `y ↦ f x y` is continuous for all `x`, and `x` lives in a
 discrete space, then `f` is continuous. -/
 theorem continuous_uncurry_of_discreteTopology [DiscreteTopology α] {f : α → β → γ}
@@ -904,23 +625,11 @@ theorem continuous_uncurry_of_discreteTopology [DiscreteTopology α] {f : α →
   exact (hf a).ContinuousAt
 #align continuous_uncurry_of_discrete_topology continuous_uncurry_of_discreteTopology
 
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 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
 theorem mem_nhds_prod_iff {a : α} {b : β} {s : Set (α × β)} :
     s ∈ 𝓝 (a, b) ↔ ∃ u ∈ 𝓝 a, ∃ v ∈ 𝓝 b, u ×ˢ v ⊆ s := by rw [nhds_prod_eq, mem_prod_iff]
 #align mem_nhds_prod_iff mem_nhds_prod_iff
 
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 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
 theorem mem_nhds_prod_iff' {a : α} {b : β} {s : Set (α × β)} :
     s ∈ 𝓝 (a, b) ↔ ∃ (u : Set α)(v : Set β), IsOpen u ∧ a ∈ u ∧ IsOpen v ∧ b ∈ v ∧ u ×ˢ v ⊆ s :=
@@ -935,24 +644,12 @@ theorem mem_nhds_prod_iff' {a : α} {b : β} {s : Set (α × β)} :
     exact ⟨u, u_open.mem_nhds au, v, v_open.mem_nhds bv, huv⟩
 #align mem_nhds_prod_iff' mem_nhds_prod_iff'
 
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 theorem Prod.tendsto_iff {α} (seq : α → β × γ) {f : Filter α} (x : β × γ) :
     Tendsto seq f (𝓝 x) ↔
       Tendsto (fun n => (seq n).fst) f (𝓝 x.fst) ∧ Tendsto (fun n => (seq n).snd) f (𝓝 x.snd) :=
   by cases x; rw [nhds_prod_eq, Filter.tendsto_prod_iff']
 #align prod.tendsto_iff Prod.tendsto_iff
 
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 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
 theorem Filter.HasBasis.prod_nhds {ιa ιb : Type _} {pa : ιa → Prop} {pb : ιb → Prop}
     {sa : ιa → Set α} {sb : ιb → Set β} {a : α} {b : β} (ha : (𝓝 a).HasBasis pa sa)
@@ -961,12 +658,6 @@ theorem Filter.HasBasis.prod_nhds {ιa ιb : Type _} {pa : ιa → Prop} {pb : 
   rw [nhds_prod_eq]; exact ha.prod hb
 #align filter.has_basis.prod_nhds Filter.HasBasis.prod_nhds
 
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 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
 theorem Filter.HasBasis.prod_nhds' {ιa ιb : Type _} {pa : ιa → Prop} {pb : ιb → Prop}
     {sa : ιa → Set α} {sb : ιb → Set β} {ab : α × β} (ha : (𝓝 ab.1).HasBasis pa sa)
@@ -979,112 +670,52 @@ instance [DiscreteTopology α] [DiscreteTopology β] : DiscreteTopology (α × 
   discreteTopology_iff_nhds.2 fun ⟨a, b⟩ => by
     rw [nhds_prod_eq, nhds_discrete α, nhds_discrete β, Filter.prod_pure_pure]
 
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 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
 theorem prod_mem_nhds_iff {s : Set α} {t : Set β} {a : α} {b : β} :
     s ×ˢ t ∈ 𝓝 (a, b) ↔ s ∈ 𝓝 a ∧ t ∈ 𝓝 b := by rw [nhds_prod_eq, prod_mem_prod_iff]
 #align prod_mem_nhds_iff prod_mem_nhds_iff
 
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 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
 theorem prod_mem_nhds {s : Set α} {t : Set β} {a : α} {b : β} (ha : s ∈ 𝓝 a) (hb : t ∈ 𝓝 b) :
     s ×ˢ t ∈ 𝓝 (a, b) :=
   prod_mem_nhds_iff.2 ⟨ha, hb⟩
 #align prod_mem_nhds prod_mem_nhds
 
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 theorem Filter.Eventually.prod_nhds {p : α → Prop} {q : β → Prop} {a : α} {b : β}
     (ha : ∀ᶠ x in 𝓝 a, p x) (hb : ∀ᶠ y in 𝓝 b, q y) : ∀ᶠ z : α × β in 𝓝 (a, b), p z.1 ∧ q z.2 :=
   prod_mem_nhds ha hb
 #align filter.eventually.prod_nhds Filter.Eventually.prod_nhds
 
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 theorem nhds_swap (a : α) (b : β) : 𝓝 (a, b) = (𝓝 (b, a)).map Prod.swap := by
   rw [nhds_prod_eq, Filter.prod_comm, nhds_prod_eq] <;> rfl
 #align nhds_swap nhds_swap
 
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 theorem Filter.Tendsto.prod_mk_nhds {γ} {a : α} {b : β} {f : Filter γ} {ma : γ → α} {mb : γ → β}
     (ha : Tendsto ma f (𝓝 a)) (hb : Tendsto mb f (𝓝 b)) :
     Tendsto (fun c => (ma c, mb c)) f (𝓝 (a, b)) := by
   rw [nhds_prod_eq] <;> exact Filter.Tendsto.prod_mk ha hb
 #align filter.tendsto.prod_mk_nhds Filter.Tendsto.prod_mk_nhds
 
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 theorem Filter.Eventually.curry_nhds {p : α × β → Prop} {x : α} {y : β}
     (h : ∀ᶠ x in 𝓝 (x, y), p x) : ∀ᶠ x' in 𝓝 x, ∀ᶠ y' in 𝓝 y, p (x', y') := by
   rw [nhds_prod_eq] at h; exact h.curry
 #align filter.eventually.curry_nhds Filter.Eventually.curry_nhds
 
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 theorem ContinuousAt.prod {f : α → β} {g : α → γ} {x : α} (hf : ContinuousAt f x)
     (hg : ContinuousAt g x) : ContinuousAt (fun x => (f x, g x)) x :=
   hf.prod_mk_nhds hg
 #align continuous_at.prod ContinuousAt.prod
 
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 theorem ContinuousAt.prod_map {f : α → γ} {g : β → δ} {p : α × β} (hf : ContinuousAt f p.fst)
     (hg : ContinuousAt g p.snd) : ContinuousAt (fun p : α × β => (f p.1, g p.2)) p :=
   hf.fst''.Prod hg.snd''
 #align continuous_at.prod_map ContinuousAt.prod_map
 
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 theorem ContinuousAt.prod_map' {f : α → γ} {g : β → δ} {x : α} {y : β} (hf : ContinuousAt f x)
     (hg : ContinuousAt g y) : ContinuousAt (fun p : α × β => (f p.1, g p.2)) (x, y) :=
   hf.fst'.Prod hg.snd'
 #align continuous_at.prod_map' ContinuousAt.prod_map'
 
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 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
@@ -1118,12 +749,6 @@ theorem prod_generateFrom_generateFrom_eq {α β : Type _} {s : Set (Set α)} {t
                 isOpen_iUnion fun hu => generate_open.basic _ ⟨_, hu, _, hv, rfl⟩))
 #align prod_generate_from_generate_from_eq prod_generateFrom_generateFrom_eq
 
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 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
 theorem prod_eq_generateFrom :
     Prod.topologicalSpace =
@@ -1136,12 +761,6 @@ theorem prod_eq_generateFrom :
         GenerateOpen.basic _ ⟨univ, t, by simpa [Set.prod_eq] using ht⟩))
 #align prod_eq_generate_from prod_eq_generateFrom
 
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 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
 theorem isOpen_prod_iff {s : Set (α × β)} :
     IsOpen s ↔
@@ -1154,12 +773,6 @@ theorem isOpen_prod_iff {s : Set (α × β)} :
   simp only [and_assoc', and_left_comm]
 #align is_open_prod_iff isOpen_prod_iff
 
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 /-- A product of induced topologies is induced by the product map -/
 theorem prod_induced_induced {α γ : Type _} (f : α → β) (g : γ → δ) :
     @Prod.topologicalSpace α γ (induced f ‹_›) (induced g ‹_›) =
@@ -1167,12 +780,6 @@ theorem prod_induced_induced {α γ : Type _} (f : α → β) (g : γ → δ) :
   by simp_rw [Prod.topologicalSpace, induced_inf, induced_compose]
 #align prod_induced_induced prod_induced_induced
 
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 theorem continuous_uncurry_of_discreteTopology_left [DiscreteTopology α] {f : α → β → γ}
     (h : ∀ a, Continuous (f a)) : Continuous (uncurry f) :=
   continuous_iff_continuousAt.2 fun ⟨a, b⟩ => by
@@ -1180,12 +787,6 @@ theorem continuous_uncurry_of_discreteTopology_left [DiscreteTopology α] {f : 
       uncurry, (h a).Tendsto]
 #align continuous_uncurry_of_discrete_topology_left continuous_uncurry_of_discreteTopology_left
 
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 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
 /-- Given a neighborhood `s` of `(x, x)`, then `(x, x)` has a square open neighborhood
   that is a subset of `s`. -/
@@ -1194,12 +795,6 @@ theorem exists_nhds_square {s : Set (α × α)} {x : α} (hx : s ∈ 𝓝 (x, x)
   simpa [nhds_prod_eq, (nhds_basis_opens x).prod_self.mem_iff, and_assoc, and_left_comm] using hx
 #align exists_nhds_square exists_nhds_square
 
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 /-- `prod.fst` maps neighborhood of `x : α × β` within the section `prod.snd ⁻¹' {x.2}`
 to `𝓝 x.1`. -/
 theorem map_fst_nhdsWithin (x : α × β) : map Prod.fst (𝓝[Prod.snd ⁻¹' {x.2}] x) = 𝓝 x.1 :=
@@ -1212,34 +807,16 @@ theorem map_fst_nhdsWithin (x : α × β) : map Prod.fst (𝓝[Prod.snd ⁻¹' {
   exact mem_of_superset hu fun z hz => H _ hz _ (mem_of_mem_nhds hv) rfl
 #align map_fst_nhds_within map_fst_nhdsWithin
 
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 @[simp]
 theorem map_fst_nhds (x : α × β) : map Prod.fst (𝓝 x) = 𝓝 x.1 :=
   le_antisymm continuousAt_fst <| (map_fst_nhdsWithin x).symm.trans_le (map_mono inf_le_left)
 #align map_fst_nhds map_fst_nhds
 
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 /-- The first projection in a product of topological spaces sends open sets to open sets. -/
 theorem isOpenMap_fst : IsOpenMap (@Prod.fst α β) :=
   isOpenMap_iff_nhds_le.2 fun x => (map_fst_nhds x).ge
 #align is_open_map_fst isOpenMap_fst
 
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 /-- `prod.snd` maps neighborhood of `x : α × β` within the section `prod.fst ⁻¹' {x.1}`
 to `𝓝 x.2`. -/
 theorem map_snd_nhdsWithin (x : α × β) : map Prod.snd (𝓝[Prod.fst ⁻¹' {x.1}] x) = 𝓝 x.2 :=
@@ -1252,34 +829,16 @@ theorem map_snd_nhdsWithin (x : α × β) : map Prod.snd (𝓝[Prod.fst ⁻¹' {
   exact mem_of_superset hv fun z hz => H _ (mem_of_mem_nhds hu) _ hz rfl
 #align map_snd_nhds_within map_snd_nhdsWithin
 
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 @[simp]
 theorem map_snd_nhds (x : α × β) : map Prod.snd (𝓝 x) = 𝓝 x.2 :=
   le_antisymm continuousAt_snd <| (map_snd_nhdsWithin x).symm.trans_le (map_mono inf_le_left)
 #align map_snd_nhds map_snd_nhds
 
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 /-- The second projection in a product of topological spaces sends open sets to open sets. -/
 theorem isOpenMap_snd : IsOpenMap (@Prod.snd α β) :=
   isOpenMap_iff_nhds_le.2 fun x => (map_snd_nhds x).ge
 #align is_open_map_snd isOpenMap_snd
 
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 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
@@ -1305,12 +864,6 @@ theorem isOpen_prod_iff' {s : Set α} {t : Set β} :
       exact H.1.Prod H.2
 #align is_open_prod_iff' isOpen_prod_iff'
 
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 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
@@ -1322,24 +875,12 @@ theorem closure_prod_eq {s : Set α} {t : Set β} : closure (s ×ˢ t) = closure
     simp [closure_eq_cluster_pts, ClusterPt, nhds_prod_eq, this] <;> exact prod_ne_bot
 #align closure_prod_eq closure_prod_eq
 
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 theorem interior_prod_eq (s : Set α) (t : Set β) : interior (s ×ˢ t) = interior s ×ˢ interior t :=
   Set.ext fun ⟨a, b⟩ => by simp only [mem_interior_iff_mem_nhds, mem_prod, prod_mem_nhds_iff]
 #align interior_prod_eq interior_prod_eq
 
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 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
@@ -1348,12 +889,6 @@ theorem frontier_prod_eq (s : Set α) (t : Set β) :
   simp only [frontier, closure_prod_eq, interior_prod_eq, prod_diff_prod]
 #align frontier_prod_eq frontier_prod_eq
 
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 @[simp]
@@ -1361,12 +896,6 @@ theorem frontier_prod_univ_eq (s : Set α) : frontier (s ×ˢ (univ : Set β)) =
   by simp [frontier_prod_eq]
 #align frontier_prod_univ_eq frontier_prod_univ_eq
 
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 @[simp]
@@ -1374,12 +903,6 @@ theorem frontier_univ_prod_eq (s : Set β) : frontier ((univ : Set α) ×ˢ s) =
   by simp [frontier_prod_eq]
 #align frontier_univ_prod_eq frontier_univ_prod_eq
 
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 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
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 theorem map_mem_closure₂ {f : α → β → γ} {a : α} {b : β} {s : Set α} {t : Set β} {u : Set γ}
@@ -1390,48 +913,24 @@ theorem map_mem_closure₂ {f : α → β → γ} {a : α} {b : β} {s : Set α}
   H₂.closure hf H₁
 #align map_mem_closure₂ map_mem_closure₂
 
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 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
 theorem IsClosed.prod {s₁ : Set α} {s₂ : Set β} (h₁ : IsClosed s₁) (h₂ : IsClosed s₂) :
     IsClosed (s₁ ×ˢ s₂) :=
   closure_eq_iff_isClosed.mp <| by simp only [h₁.closure_eq, h₂.closure_eq, closure_prod_eq]
 #align is_closed.prod IsClosed.prod
 
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 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
 /-- The product of two dense sets is a dense set. -/
 theorem Dense.prod {s : Set α} {t : Set β} (hs : Dense s) (ht : Dense t) : Dense (s ×ˢ t) :=
   fun x => by rw [closure_prod_eq]; exact ⟨hs x.1, ht x.2⟩
 #align dense.prod Dense.prod
 
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 /-- If `f` and `g` are maps with dense range, then `prod.map f g` has dense range. -/
 theorem DenseRange.prod_map {ι : Type _} {κ : Type _} {f : ι → β} {g : κ → γ} (hf : DenseRange f)
     (hg : DenseRange g) : DenseRange (Prod.map f g) := by
   simpa only [DenseRange, prod_range_range_eq] using hf.prod hg
 #align dense_range.prod_map DenseRange.prod_map
 
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-Case conversion may be inaccurate. Consider using '#align inducing.prod_mk Inducing.prod_mapₓ'. -/
 theorem Inducing.prod_map {f : α → β} {g : γ → δ} (hf : Inducing f) (hg : Inducing g) :
     Inducing fun x : α × γ => (f x.1, g x.2) :=
   ⟨by
@@ -1439,24 +938,12 @@ theorem Inducing.prod_map {f : α → β} {g : γ → δ} (hf : Inducing f) (hg
       induced_compose, induced_inf, induced_compose, induced_compose]⟩
 #align inducing.prod_mk Inducing.prod_map
 
-/- warning: embedding.prod_mk -> Embedding.prod_map is a dubious translation:
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-Case conversion may be inaccurate. Consider using '#align embedding.prod_mk Embedding.prod_mapₓ'. -/
 theorem Embedding.prod_map {f : α → β} {g : γ → δ} (hf : Embedding f) (hg : Embedding g) :
     Embedding fun x : α × γ => (f x.1, g x.2) :=
   { hf.to_inducing.prod_mk hg.to_inducing with
     inj := fun ⟨x₁, x₂⟩ ⟨y₁, y₂⟩ => by simp <;> exact fun h₁ h₂ => ⟨hf.inj h₁, hg.inj h₂⟩ }
 #align embedding.prod_mk Embedding.prod_map
 
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-Case conversion may be inaccurate. Consider using '#align is_open_map.prod IsOpenMap.prodₓ'. -/
 protected theorem IsOpenMap.prod {f : α → β} {g : γ → δ} (hf : IsOpenMap f) (hg : IsOpenMap g) :
     IsOpenMap fun p : α × γ => (f p.1, g p.2) :=
   by
@@ -1466,23 +953,11 @@ protected theorem IsOpenMap.prod {f : α → β} {g : γ → δ} (hf : IsOpenMap
   exact Filter.prod_mono (hf.nhds_le a) (hg.nhds_le b)
 #align is_open_map.prod IsOpenMap.prod
 
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-Case conversion may be inaccurate. Consider using '#align open_embedding.prod OpenEmbedding.prodₓ'. -/
 protected theorem OpenEmbedding.prod {f : α → β} {g : γ → δ} (hf : OpenEmbedding f)
     (hg : OpenEmbedding g) : OpenEmbedding fun x : α × γ => (f x.1, g x.2) :=
   openEmbedding_of_embedding_open (hf.1.prod_mk hg.1) (hf.IsOpenMap.Prod hg.IsOpenMap)
 #align open_embedding.prod OpenEmbedding.prod
 
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-Case conversion may be inaccurate. Consider using '#align embedding_graph embedding_graphₓ'. -/
 theorem embedding_graph {f : α → β} (hf : Continuous f) : Embedding fun x => (x, f x) :=
   embedding_of_embedding_compose (continuous_id.prod_mk hf) continuous_fst embedding_id
 #align embedding_graph embedding_graph
@@ -1495,218 +970,92 @@ open Sum
 
 variable [TopologicalSpace α] [TopologicalSpace β] [TopologicalSpace γ] [TopologicalSpace δ]
 
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-but is expected to have type
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-Case conversion may be inaccurate. Consider using '#align continuous_inl continuous_inlₓ'. -/
 @[continuity]
 theorem continuous_inl : Continuous (@inl α β) :=
   continuous_sup_rng_left continuous_coinduced_rng
 #align continuous_inl continuous_inl
 
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-Case conversion may be inaccurate. Consider using '#align continuous_inr continuous_inrₓ'. -/
 @[continuity]
 theorem continuous_inr : Continuous (@inr α β) :=
   continuous_sup_rng_right continuous_coinduced_rng
 #align continuous_inr continuous_inr
 
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-Case conversion may be inaccurate. Consider using '#align is_open_sum_iff isOpen_sum_iffₓ'. -/
 theorem isOpen_sum_iff {s : Set (Sum α β)} : IsOpen s ↔ IsOpen (inl ⁻¹' s) ∧ IsOpen (inr ⁻¹' s) :=
   Iff.rfl
 #align is_open_sum_iff isOpen_sum_iff
 
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-Case conversion may be inaccurate. Consider using '#align is_open_map_inl isOpenMap_inlₓ'. -/
 theorem isOpenMap_inl : IsOpenMap (@inl α β) := fun u hu => by
   simpa [isOpen_sum_iff, preimage_image_eq u Sum.inl_injective]
 #align is_open_map_inl isOpenMap_inl
 
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-Case conversion may be inaccurate. Consider using '#align is_open_map_inr isOpenMap_inrₓ'. -/
 theorem isOpenMap_inr : IsOpenMap (@inr α β) := fun u hu => by
   simpa [isOpen_sum_iff, preimage_image_eq u Sum.inr_injective]
 #align is_open_map_inr isOpenMap_inr
 
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-Case conversion may be inaccurate. Consider using '#align open_embedding_inl openEmbedding_inlₓ'. -/
 theorem openEmbedding_inl : OpenEmbedding (@inl α β) :=
   openEmbedding_of_continuous_injective_open continuous_inl inl_injective isOpenMap_inl
 #align open_embedding_inl openEmbedding_inl
 
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-but is expected to have type
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-Case conversion may be inaccurate. Consider using '#align open_embedding_inr openEmbedding_inrₓ'. -/
 theorem openEmbedding_inr : OpenEmbedding (@inr α β) :=
   openEmbedding_of_continuous_injective_open continuous_inr inr_injective isOpenMap_inr
 #align open_embedding_inr openEmbedding_inr
 
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-but is expected to have type
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-Case conversion may be inaccurate. Consider using '#align embedding_inl embedding_inlₓ'. -/
 theorem embedding_inl : Embedding (@inl α β) :=
   openEmbedding_inl.1
 #align embedding_inl embedding_inl
 
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-but is expected to have type
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-Case conversion may be inaccurate. Consider using '#align embedding_inr embedding_inrₓ'. -/
 theorem embedding_inr : Embedding (@inr α β) :=
   openEmbedding_inr.1
 #align embedding_inr embedding_inr
 
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-but is expected to have type
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-Case conversion may be inaccurate. Consider using '#align is_open_range_inl isOpen_range_inlₓ'. -/
 theorem isOpen_range_inl : IsOpen (range (inl : α → Sum α β)) :=
   openEmbedding_inl.2
 #align is_open_range_inl isOpen_range_inl
 
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-Case conversion may be inaccurate. Consider using '#align is_open_range_inr isOpen_range_inrₓ'. -/
 theorem isOpen_range_inr : IsOpen (range (inr : β → Sum α β)) :=
   openEmbedding_inr.2
 #align is_open_range_inr isOpen_range_inr
 
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 theorem isClosed_range_inl : IsClosed (range (inl : α → Sum α β)) := by
   rw [← isOpen_compl_iff, compl_range_inl]; exact isOpen_range_inr
 #align is_closed_range_inl isClosed_range_inl
 
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 theorem isClosed_range_inr : IsClosed (range (inr : β → Sum α β)) := by
   rw [← isOpen_compl_iff, compl_range_inr]; exact isOpen_range_inl
 #align is_closed_range_inr isClosed_range_inr
 
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 theorem closedEmbedding_inl : ClosedEmbedding (inl : α → Sum α β) :=
   ⟨embedding_inl, isClosed_range_inl⟩
 #align closed_embedding_inl closedEmbedding_inl
 
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 theorem closedEmbedding_inr : ClosedEmbedding (inr : β → Sum α β) :=
   ⟨embedding_inr, isClosed_range_inr⟩
 #align closed_embedding_inr closedEmbedding_inr
 
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 theorem nhds_inl (x : α) : 𝓝 (inl x : Sum α β) = map inl (𝓝 x) :=
   (openEmbedding_inl.map_nhds_eq _).symm
 #align nhds_inl nhds_inl
 
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 theorem nhds_inr (x : β) : 𝓝 (inr x : Sum α β) = map inr (𝓝 x) :=
   (openEmbedding_inr.map_nhds_eq _).symm
 #align nhds_inr nhds_inr
 
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 theorem continuous_sum_dom {f : Sum α β → γ} :
     Continuous f ↔ Continuous (f ∘ Sum.inl) ∧ Continuous (f ∘ Sum.inr) := by
   simp only [continuous_sup_dom, continuous_coinduced_dom]
 #align continuous_sum_dom continuous_sum_dom
 
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 theorem continuous_sum_elim {f : α → γ} {g : β → γ} :
     Continuous (Sum.elim f g) ↔ Continuous f ∧ Continuous g :=
   continuous_sum_dom
 #align continuous_sum_elim continuous_sum_elim
 
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 @[continuity]
 theorem Continuous.sum_elim {f : α → γ} {g : β → γ} (hf : Continuous f) (hg : Continuous g) :
     Continuous (Sum.elim f g) :=
   continuous_sum_elim.2 ⟨hf, hg⟩
 #align continuous.sum_elim Continuous.sum_elim
 
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 @[simp]
 theorem continuous_sum_map {f : α → β} {g : γ → δ} :
     Continuous (Sum.map f g) ↔ Continuous f ∧ Continuous g :=
@@ -1714,47 +1063,23 @@ theorem continuous_sum_map {f : α → β} {g : γ → δ} :
     embedding_inl.continuous_iff.symm.And embedding_inr.continuous_iff.symm
 #align continuous_sum_map continuous_sum_map
 
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 @[continuity]
 theorem Continuous.sum_map {f : α → β} {g : γ → δ} (hf : Continuous f) (hg : Continuous g) :
     Continuous (Sum.map f g) :=
   continuous_sum_map.2 ⟨hf, hg⟩
 #align continuous.sum_map Continuous.sum_map
 
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 theorem isOpenMap_sum {f : Sum α β → γ} :
     IsOpenMap f ↔ (IsOpenMap fun a => f (inl a)) ∧ IsOpenMap fun b => f (inr b) := by
   simp only [isOpenMap_iff_nhds_le, Sum.forall, nhds_inl, nhds_inr, Filter.map_map]
 #align is_open_map_sum isOpenMap_sum
 
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 @[simp]
 theorem isOpenMap_sum_elim {f : α → γ} {g : β → γ} :
     IsOpenMap (Sum.elim f g) ↔ IsOpenMap f ∧ IsOpenMap g := by
   simp only [isOpenMap_sum, elim_inl, elim_inr]
 #align is_open_map_sum_elim isOpenMap_sum_elim
 
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 theorem IsOpenMap.sum_elim {f : α → γ} {g : β → γ} (hf : IsOpenMap f) (hg : IsOpenMap g) :
     IsOpenMap (Sum.elim f g) :=
   isOpenMap_sum_elim.2 ⟨hf, hg⟩
@@ -1892,12 +1217,6 @@ theorem nhds_subtype_eq_comap {a : α} {h : p a} : 𝓝 (⟨a, h⟩ : Subtype p)
 #align nhds_subtype_eq_comap nhds_subtype_eq_comap
 -/
 
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 theorem tendsto_subtype_rng {β : Type _} {p : α → Prop} {b : Filter β} {f : β → Subtype p} :
     ∀ {a : Subtype p}, Tendsto f b (𝓝 a) ↔ Tendsto (fun x => (f x : α)) b (𝓝 (a : α))
   | ⟨a, ha⟩ => by rw [nhds_subtype_eq_comap, tendsto_comap_iff, Subtype.coe_mk]
@@ -2041,143 +1360,65 @@ section Pi
 variable {ι : Type _} {π : ι → Type _} {κ : Type _} [TopologicalSpace α]
   [∀ i, TopologicalSpace (π i)] {f : α → ∀ i : ι, π i}
 
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 theorem continuous_pi_iff : Continuous f ↔ ∀ i, Continuous fun a => f a i := by
   simp only [continuous_iInf_rng, continuous_induced_rng]
 #align continuous_pi_iff continuous_pi_iff
 
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 @[continuity]
 theorem continuous_pi (h : ∀ i, Continuous fun a => f a i) : Continuous f :=
   continuous_pi_iff.2 h
 #align continuous_pi continuous_pi
 
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 @[continuity]
 theorem continuous_apply (i : ι) : Continuous fun p : ∀ i, π i => p i :=
   continuous_iInf_dom continuous_induced_dom
 #align continuous_apply continuous_apply
 
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 @[continuity]
 theorem continuous_apply_apply {ρ : κ → ι → Type _} [∀ j i, TopologicalSpace (ρ j i)] (j : κ)
     (i : ι) : Continuous fun p : ∀ j, ∀ i, ρ j i => p j i :=
   (continuous_apply i).comp (continuous_apply j)
 #align continuous_apply_apply continuous_apply_apply
 
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 theorem continuousAt_apply (i : ι) (x : ∀ i, π i) : ContinuousAt (fun p : ∀ i, π i => p i) x :=
   (continuous_apply i).ContinuousAt
 #align continuous_at_apply continuousAt_apply
 
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 theorem Filter.Tendsto.apply {l : Filter β} {f : β → ∀ i, π i} {x : ∀ i, π i}
     (h : Tendsto f l (𝓝 x)) (i : ι) : Tendsto (fun a => f a i) l (𝓝 <| x i) :=
   (continuousAt_apply i _).Tendsto.comp h
 #align filter.tendsto.apply Filter.Tendsto.apply
 
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 theorem nhds_pi {a : ∀ i, π i} : 𝓝 a = pi fun i => 𝓝 (a i) := by
   simp only [nhds_iInf, nhds_induced, Filter.pi]
 #align nhds_pi nhds_pi
 
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 theorem tendsto_pi_nhds {f : β → ∀ i, π i} {g : ∀ i, π i} {u : Filter β} :
     Tendsto f u (𝓝 g) ↔ ∀ x, Tendsto (fun i => f i x) u (𝓝 (g x)) := by
   rw [nhds_pi, Filter.tendsto_pi]
 #align tendsto_pi_nhds tendsto_pi_nhds
 
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 theorem continuousAt_pi {f : α → ∀ i, π i} {x : α} :
     ContinuousAt f x ↔ ∀ i, ContinuousAt (fun y => f y i) x :=
   tendsto_pi_nhds
 #align continuous_at_pi continuousAt_pi
 
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 theorem Filter.Tendsto.update [DecidableEq ι] {l : Filter β} {f : β → ∀ i, π i} {x : ∀ i, π i}
     (hf : Tendsto f l (𝓝 x)) (i : ι) {g : β → π i} {xi : π i} (hg : Tendsto g l (𝓝 xi)) :
     Tendsto (fun a => update (f a) i (g a)) l (𝓝 <| update x i xi) :=
   tendsto_pi_nhds.2 fun j => by rcases em (j = i) with (rfl | hj) <;> simp [*, hf.apply]
 #align filter.tendsto.update Filter.Tendsto.update
 
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 theorem ContinuousAt.update [DecidableEq ι] {a : α} (hf : ContinuousAt f a) (i : ι) {g : α → π i}
     (hg : ContinuousAt g a) : ContinuousAt (fun a => update (f a) i (g a)) a :=
   hf.update i hg
 #align continuous_at.update ContinuousAt.update
 
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 theorem Continuous.update [DecidableEq ι] (hf : Continuous f) (i : ι) {g : α → π i}
     (hg : Continuous g) : Continuous fun a => update (f a) i (g a) :=
   continuous_iff_continuousAt.2 fun x => hf.ContinuousAt.update i hg.ContinuousAt
 #align continuous.update Continuous.update
 
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 /-- `function.update f i x` is continuous in `(f, x)`. -/
 @[continuity]
 theorem continuous_update [DecidableEq ι] (i : ι) :
@@ -2195,9 +1436,6 @@ theorem continuous_mulSingle [∀ i, One (π i)] [DecidableEq ι] (i : ι) :
 #align continuous_single continuous_single
 -/
 
-/- warning: filter.tendsto.fin_insert_nth -> Filter.Tendsto.fin_insertNth is a dubious translation:
-<too large>
-Case conversion may be inaccurate. Consider using '#align filter.tendsto.fin_insert_nth Filter.Tendsto.fin_insertNthₓ'. -/
 theorem Filter.Tendsto.fin_insertNth {n} {π : Fin (n + 1) → Type _} [∀ i, TopologicalSpace (π i)]
     (i : Fin (n + 1)) {f : β → π i} {l : Filter β} {x : π i} (hf : Tendsto f l (𝓝 x))
     {g : β → ∀ j : Fin n, π (i.succAbove j)} {y : ∀ j, π (i.succAbove j)} (hg : Tendsto g l (𝓝 y)) :
@@ -2205,9 +1443,6 @@ theorem Filter.Tendsto.fin_insertNth {n} {π : Fin (n + 1) → Type _} [∀ i, T
   tendsto_pi_nhds.2 fun j => Fin.succAboveCases i (by simpa) (by simpa using tendsto_pi_nhds.1 hg) j
 #align filter.tendsto.fin_insert_nth Filter.Tendsto.fin_insertNth
 
-/- warning: continuous_at.fin_insert_nth -> ContinuousAt.fin_insertNth is a dubious translation:
-<too large>
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 theorem ContinuousAt.fin_insertNth {n} {π : Fin (n + 1) → Type _} [∀ i, TopologicalSpace (π i)]
     (i : Fin (n + 1)) {f : α → π i} {a : α} (hf : ContinuousAt f a)
     {g : α → ∀ j : Fin n, π (i.succAbove j)} (hg : ContinuousAt g a) :
@@ -2215,32 +1450,17 @@ theorem ContinuousAt.fin_insertNth {n} {π : Fin (n + 1) → Type _} [∀ i, Top
   hf.fin_insertNth i hg
 #align continuous_at.fin_insert_nth ContinuousAt.fin_insertNth
 
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-<too large>
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 theorem Continuous.fin_insertNth {n} {π : Fin (n + 1) → Type _} [∀ i, TopologicalSpace (π i)]
     (i : Fin (n + 1)) {f : α → π i} (hf : Continuous f) {g : α → ∀ j : Fin n, π (i.succAbove j)}
     (hg : Continuous g) : Continuous fun a => i.insertNth (f a) (g a) :=
   continuous_iff_continuousAt.2 fun a => hf.ContinuousAt.fin_insertNth i hg.ContinuousAt
 #align continuous.fin_insert_nth Continuous.fin_insertNth
 
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 theorem isOpen_set_pi {i : Set ι} {s : ∀ a, Set (π a)} (hi : i.Finite)
     (hs : ∀ a ∈ i, IsOpen (s a)) : IsOpen (pi i s) := by
   rw [pi_def] <;> exact isOpen_biInter hi fun a ha => (hs _ ha).Preimage (continuous_apply _)
 #align is_open_set_pi isOpen_set_pi
 
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 theorem isOpen_pi_iff {s : Set (∀ a, π a)} :
     IsOpen s ↔
       ∀ f,
@@ -2273,12 +1493,6 @@ theorem isOpen_pi_iff {s : Set (∀ a, π a)} :
       simp only [← ite_and, ← Finset.mem_coe, and_self_iff, Set.univ_pi_ite, h2]
 #align is_open_pi_iff isOpen_pi_iff
 
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 theorem isOpen_pi_iff' [Finite ι] {s : Set (∀ a, π a)} :
     IsOpen s ↔
       ∀ f, f ∈ s → ∃ u : ∀ a, Set (π a), (∀ a, IsOpen (u a) ∧ f a ∈ u a) ∧ Set.univ.pi u ⊆ s :=
@@ -2299,67 +1513,31 @@ theorem isOpen_pi_iff' [Finite ι] {s : Set (∀ a, π a)} :
       ⟨Finset.univ, u, ⟨fun i => ⟨u i, ⟨rfl.subset, h1 i⟩⟩, by rwa [Finset.coe_univ]⟩⟩
 #align is_open_pi_iff' isOpen_pi_iff'
 
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 theorem isClosed_set_pi {i : Set ι} {s : ∀ a, Set (π a)} (hs : ∀ a ∈ i, IsClosed (s a)) :
     IsClosed (pi i s) := by
   rw [pi_def] <;>
     exact isClosed_iInter fun a => isClosed_iInter fun ha => (hs _ ha).Preimage (continuous_apply _)
 #align is_closed_set_pi isClosed_set_pi
 
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-  forall {ι : Type.{u1}} {π : ι -> Type.{u2}} [_inst_2 : forall (i : ι), TopologicalSpace.{u2} (π i)] {I : Set.{u1} ι} {s : forall (i : ι), Set.{u2} (π i)} (a : forall (i : ι), π i), (Membership.Mem.{max u1 u2, max u1 u2} (Set.{max u1 u2} (forall (i : ι), π i)) (Filter.{max u1 u2} (forall (i : ι), π i)) (Filter.hasMem.{max u1 u2} (forall (i : ι), π i)) (Set.pi.{u1, u2} ι (fun (i : ι) => π i) I s) (nhds.{max u1 u2} (forall (i : ι), π i) (Pi.topologicalSpace.{u1, u2} ι (fun (i : ι) => π i) (fun (a : ι) => _inst_2 a)) a)) -> (forall {i : ι}, (Membership.Mem.{u1, u1} ι (Set.{u1} ι) (Set.hasMem.{u1} ι) i I) -> (Membership.Mem.{u2, u2} (Set.{u2} (π i)) (Filter.{u2} (π i)) (Filter.hasMem.{u2} (π i)) (s i) (nhds.{u2} (π i) (_inst_2 i) (a i))))
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-Case conversion may be inaccurate. Consider using '#align mem_nhds_of_pi_mem_nhds mem_nhds_of_pi_mem_nhdsₓ'. -/
 theorem mem_nhds_of_pi_mem_nhds {I : Set ι} {s : ∀ i, Set (π i)} (a : ∀ i, π i) (hs : I.pi s ∈ 𝓝 a)
     {i : ι} (hi : i ∈ I) : s i ∈ 𝓝 (a i) := by rw [nhds_pi] at hs; exact mem_of_pi_mem_pi hs hi
 #align mem_nhds_of_pi_mem_nhds mem_nhds_of_pi_mem_nhds
 
-/- warning: set_pi_mem_nhds -> set_pi_mem_nhds is a dubious translation:
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-Case conversion may be inaccurate. Consider using '#align set_pi_mem_nhds set_pi_mem_nhdsₓ'. -/
 theorem set_pi_mem_nhds {i : Set ι} {s : ∀ a, Set (π a)} {x : ∀ a, π a} (hi : i.Finite)
     (hs : ∀ a ∈ i, s a ∈ 𝓝 (x a)) : pi i s ∈ 𝓝 x := by rw [pi_def, bInter_mem hi];
   exact fun a ha => (continuous_apply a).ContinuousAt (hs a ha)
 #align set_pi_mem_nhds set_pi_mem_nhds
 
-/- warning: set_pi_mem_nhds_iff -> set_pi_mem_nhds_iff is a dubious translation:
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-Case conversion may be inaccurate. Consider using '#align set_pi_mem_nhds_iff set_pi_mem_nhds_iffₓ'. -/
 theorem set_pi_mem_nhds_iff {I : Set ι} (hI : I.Finite) {s : ∀ i, Set (π i)} (a : ∀ i, π i) :
     I.pi s ∈ 𝓝 a ↔ ∀ i : ι, i ∈ I → s i ∈ 𝓝 (a i) := by rw [nhds_pi, pi_mem_pi_iff hI];
   infer_instance
 #align set_pi_mem_nhds_iff set_pi_mem_nhds_iff
 
-/- warning: interior_pi_set -> interior_pi_set is a dubious translation:
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-  forall {ι : Type.{u1}} {π : ι -> Type.{u2}} [_inst_2 : forall (i : ι), TopologicalSpace.{u2} (π i)] {I : Set.{u1} ι}, (Set.Finite.{u1} ι I) -> (forall {s : forall (i : ι), Set.{u2} (π i)}, Eq.{succ (max u1 u2)} (Set.{max u1 u2} (forall (i : ι), π i)) (interior.{max u1 u2} (forall (i : ι), π i) (Pi.topologicalSpace.{u1, u2} ι (fun (i : ι) => π i) (fun (a : ι) => _inst_2 a)) (Set.pi.{u1, u2} ι (fun (i : ι) => π i) I s)) (Set.pi.{u1, u2} ι (fun (i : ι) => π i) I (fun (i : ι) => interior.{u2} (π i) (_inst_2 i) (s i))))
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-Case conversion may be inaccurate. Consider using '#align interior_pi_set interior_pi_setₓ'. -/
 theorem interior_pi_set {I : Set ι} (hI : I.Finite) {s : ∀ i, Set (π i)} :
     interior (pi I s) = I.pi fun i => interior (s i) := by ext a;
   simp only [Set.mem_pi, mem_interior_iff_mem_nhds, set_pi_mem_nhds_iff hI]
 #align interior_pi_set interior_pi_set
 
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 theorem exists_finset_piecewise_mem_of_mem_nhds [DecidableEq ι] {s : Set (∀ a, π a)} {x : ∀ a, π a}
     (hs : s ∈ 𝓝 x) (y : ∀ a, π a) : ∃ I : Finset ι, I.piecewise x y ∈ s :=
   by
@@ -2426,12 +1604,6 @@ theorem pi_generateFrom_eq_finite {π : ι → Type _} {g : ∀ a, Set (Set (π
 #align pi_generate_from_eq_finite pi_generateFrom_eq_finite
 -/
 
-/- warning: inducing_infi_to_pi -> inducing_iInf_to_pi is a dubious translation:
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-  forall {ι : Type.{u1}} {π : ι -> Type.{u2}} [_inst_2 : forall (i : ι), TopologicalSpace.{u2} (π i)] {X : Type.{u3}} (f : forall (i : ι), X -> (π i)), Inducing.{u3, max u1 u2} X (forall (i : ι), π i) (iInf.{u3, succ u1} (TopologicalSpace.{u3} X) (ConditionallyCompleteLattice.toHasInf.{u3} (TopologicalSpace.{u3} X) (CompleteLattice.toConditionallyCompleteLattice.{u3} (TopologicalSpace.{u3} X) (TopologicalSpace.completeLattice.{u3} X))) ι (fun (i : ι) => TopologicalSpace.induced.{u3, u2} X (π i) (f i) (inferInstance.{succ u2} (TopologicalSpace.{u2} (π i)) (_inst_2 i)))) (Pi.topologicalSpace.{u1, u2} ι (fun (i : ι) => π i) (fun (a : ι) => _inst_2 a)) (fun (x : X) (i : ι) => f i x)
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-Case conversion may be inaccurate. Consider using '#align inducing_infi_to_pi inducing_iInf_to_piₓ'. -/
 /-- Suppose `π i` is a family of topological spaces indexed by `i : ι`, and `X` is a type
 endowed with a family of maps `f i : X → π i` for every `i : ι`, hence inducing a
 map `g : X → Π i, π i`. This lemma shows that infimum of the topologies on `X` induced by
@@ -2467,43 +1639,19 @@ section Sigma
 variable {ι κ : Type _} {σ : ι → Type _} {τ : κ → Type _} [∀ i, TopologicalSpace (σ i)]
   [∀ k, TopologicalSpace (τ k)] [TopologicalSpace α]
 
-/- warning: continuous_sigma_mk -> continuous_sigmaMk is a dubious translation:
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-Case conversion may be inaccurate. Consider using '#align continuous_sigma_mk continuous_sigmaMkₓ'. -/
 @[continuity]
 theorem continuous_sigmaMk {i : ι} : Continuous (@Sigma.mk ι σ i) :=
   continuous_iSup_rng continuous_coinduced_rng
 #align continuous_sigma_mk continuous_sigmaMk
 
-/- warning: is_open_sigma_iff -> isOpen_sigma_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 is_open_sigma_iff isOpen_sigma_iffₓ'. -/
 theorem isOpen_sigma_iff {s : Set (Sigma σ)} : IsOpen s ↔ ∀ i, IsOpen (Sigma.mk i ⁻¹' s) := by
   simp only [isOpen_iSup_iff, isOpen_coinduced]
 #align is_open_sigma_iff isOpen_sigma_iff
 
-/- warning: is_closed_sigma_iff -> isClosed_sigma_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 is_closed_sigma_iff isClosed_sigma_iffₓ'. -/
 theorem isClosed_sigma_iff {s : Set (Sigma σ)} : IsClosed s ↔ ∀ i, IsClosed (Sigma.mk i ⁻¹' s) := by
   simp only [← isOpen_compl_iff, isOpen_sigma_iff, preimage_compl]
 #align is_closed_sigma_iff isClosed_sigma_iff
 
-/- warning: is_open_map_sigma_mk -> isOpenMap_sigmaMk is a dubious translation:
-lean 3 declaration is
-  forall {ι : Type.{u1}} {σ : ι -> Type.{u2}} [_inst_1 : forall (i : ι), TopologicalSpace.{u2} (σ i)] {i : ι}, IsOpenMap.{u2, max u1 u2} (σ i) (Sigma.{u1, u2} ι σ) (_inst_1 i) (Sigma.topologicalSpace.{u1, u2} ι σ (fun (a : ι) => _inst_1 a)) (Sigma.mk.{u1, u2} ι σ i)
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-Case conversion may be inaccurate. Consider using '#align is_open_map_sigma_mk isOpenMap_sigmaMkₓ'. -/
 theorem isOpenMap_sigmaMk {i : ι} : IsOpenMap (@Sigma.mk ι σ i) :=
   by
   intro s hs
@@ -2515,22 +1663,10 @@ theorem isOpenMap_sigmaMk {i : ι} : IsOpenMap (@Sigma.mk ι σ i) :=
     exact isOpen_empty
 #align is_open_map_sigma_mk isOpenMap_sigmaMk
 
-/- warning: is_open_range_sigma_mk -> isOpen_range_sigmaMk is a dubious translation:
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-Case conversion may be inaccurate. Consider using '#align is_open_range_sigma_mk isOpen_range_sigmaMkₓ'. -/
 theorem isOpen_range_sigmaMk {i : ι} : IsOpen (Set.range (@Sigma.mk ι σ i)) :=
   isOpenMap_sigmaMk.isOpen_range
 #align is_open_range_sigma_mk isOpen_range_sigmaMk
 
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 theorem isClosedMap_sigmaMk {i : ι} : IsClosedMap (@Sigma.mk ι σ i) :=
   by
   intro s hs
@@ -2542,83 +1678,35 @@ theorem isClosedMap_sigmaMk {i : ι} : IsClosedMap (@Sigma.mk ι σ i) :=
     exact isClosed_empty
 #align is_closed_map_sigma_mk isClosedMap_sigmaMk
 
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-Case conversion may be inaccurate. Consider using '#align is_closed_range_sigma_mk isClosed_range_sigmaMkₓ'. -/
 theorem isClosed_range_sigmaMk {i : ι} : IsClosed (Set.range (@Sigma.mk ι σ i)) :=
   isClosedMap_sigmaMk.closed_range
 #align is_closed_range_sigma_mk isClosed_range_sigmaMk
 
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-Case conversion may be inaccurate. Consider using '#align open_embedding_sigma_mk openEmbedding_sigmaMkₓ'. -/
 theorem openEmbedding_sigmaMk {i : ι} : OpenEmbedding (@Sigma.mk ι σ i) :=
   openEmbedding_of_continuous_injective_open continuous_sigmaMk sigma_mk_injective isOpenMap_sigmaMk
 #align open_embedding_sigma_mk openEmbedding_sigmaMk
 
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 theorem closedEmbedding_sigmaMk {i : ι} : ClosedEmbedding (@Sigma.mk ι σ i) :=
   closedEmbedding_of_continuous_injective_closed continuous_sigmaMk sigma_mk_injective
     isClosedMap_sigmaMk
 #align closed_embedding_sigma_mk closedEmbedding_sigmaMk
 
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 theorem embedding_sigmaMk {i : ι} : Embedding (@Sigma.mk ι σ i) :=
   closedEmbedding_sigmaMk.1
 #align embedding_sigma_mk embedding_sigmaMk
 
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 theorem Sigma.nhds_mk (i : ι) (x : σ i) : 𝓝 (⟨i, x⟩ : Sigma σ) = map (Sigma.mk i) (𝓝 x) :=
   (openEmbedding_sigmaMk.map_nhds_eq x).symm
 #align sigma.nhds_mk Sigma.nhds_mk
 
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 theorem Sigma.nhds_eq (x : Sigma σ) : 𝓝 x = map (Sigma.mk x.1) (𝓝 x.2) := by cases x;
   apply Sigma.nhds_mk
 #align sigma.nhds_eq Sigma.nhds_eq
 
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 theorem comap_sigmaMk_nhds (i : ι) (x : σ i) : comap (Sigma.mk i) (𝓝 ⟨i, x⟩) = 𝓝 x :=
   (embedding_sigmaMk.to_inducing.nhds_eq_comap _).symm
 #align comap_sigma_mk_nhds comap_sigmaMk_nhds
 
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 theorem isOpen_sigma_fst_preimage (s : Set ι) : IsOpen (Sigma.fst ⁻¹' s : Set (Σa, σ a)) :=
   by
   rw [← bUnion_of_singleton s, preimage_Union₂]
@@ -2626,24 +1714,12 @@ theorem isOpen_sigma_fst_preimage (s : Set ι) : IsOpen (Sigma.fst ⁻¹' s : Se
   exact isOpen_biUnion fun _ _ => isOpen_range_sigmaMk
 #align is_open_sigma_fst_preimage isOpen_sigma_fst_preimage
 
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 /-- A map out of a sum type is continuous iff its restriction to each summand is. -/
 @[simp]
 theorem continuous_sigma_iff {f : Sigma σ → α} : Continuous f ↔ ∀ i, Continuous fun a => f ⟨i, a⟩ :=
   by simp only [continuous_iSup_dom, continuous_coinduced_dom]
 #align continuous_sigma_iff continuous_sigma_iff
 
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 /-- A map out of a sum type is continuous if its restriction to each summand is. -/
 @[continuity]
 theorem continuous_sigma {f : Sigma σ → α} (hf : ∀ i, Continuous fun a => f ⟨i, a⟩) :
@@ -2651,81 +1727,39 @@ theorem continuous_sigma {f : Sigma σ → α} (hf : ∀ i, Continuous fun a =>
   continuous_sigma_iff.2 hf
 #align continuous_sigma continuous_sigma
 
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 @[simp]
 theorem continuous_sigma_map {f₁ : ι → κ} {f₂ : ∀ i, σ i → τ (f₁ i)} :
     Continuous (Sigma.map f₁ f₂) ↔ ∀ i, Continuous (f₂ i) :=
   continuous_sigma_iff.trans <| by simp only [Sigma.map, embedding_sigma_mk.continuous_iff]
 #align continuous_sigma_map continuous_sigma_map
 
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-  forall {ι : Type.{u2}} {κ : Type.{u1}} {σ : ι -> Type.{u4}} {τ : κ -> Type.{u3}} [_inst_1 : forall (i : ι), TopologicalSpace.{u4} (σ i)] [_inst_2 : forall (k : κ), TopologicalSpace.{u3} (τ k)] {f₁ : ι -> κ} {f₂ : forall (i : ι), (σ i) -> (τ (f₁ i))}, (forall (i : ι), Continuous.{u4, u3} (σ i) (τ (f₁ i)) (_inst_1 i) (_inst_2 (f₁ i)) (f₂ i)) -> (Continuous.{max u4 u2, max u3 u1} (Sigma.{u2, u4} ι (fun (i : ι) => σ i)) (Sigma.{u1, u3} κ τ) (instTopologicalSpaceSigma.{u2, u4} ι (fun (i : ι) => σ i) (fun (a : ι) => _inst_1 a)) (instTopologicalSpaceSigma.{u1, u3} κ τ (fun (a : κ) => _inst_2 a)) (Sigma.map.{u2, u1, u4, u3} ι κ (fun (i : ι) => σ i) τ f₁ f₂))
-Case conversion may be inaccurate. Consider using '#align continuous.sigma_map Continuous.sigma_mapₓ'. -/
 @[continuity]
 theorem Continuous.sigma_map {f₁ : ι → κ} {f₂ : ∀ i, σ i → τ (f₁ i)} (hf : ∀ i, Continuous (f₂ i)) :
     Continuous (Sigma.map f₁ f₂) :=
   continuous_sigma_map.2 hf
 #align continuous.sigma_map Continuous.sigma_map
 
-/- warning: is_open_map_sigma -> isOpenMap_sigma is a dubious translation:
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-but is expected to have type
-  forall {α : Type.{u3}} {ι : Type.{u2}} {σ : ι -> Type.{u1}} [_inst_1 : forall (i : ι), TopologicalSpace.{u1} (σ i)] [_inst_3 : TopologicalSpace.{u3} α] {f : (Sigma.{u2, u1} ι σ) -> α}, Iff (IsOpenMap.{max u2 u1, u3} (Sigma.{u2, u1} ι σ) α (instTopologicalSpaceSigma.{u2, u1} ι σ (fun (a : ι) => _inst_1 a)) _inst_3 f) (forall (i : ι), IsOpenMap.{u1, u3} (σ i) α (_inst_1 i) _inst_3 (fun (a : σ i) => f (Sigma.mk.{u2, u1} ι σ i a)))
-Case conversion may be inaccurate. Consider using '#align is_open_map_sigma isOpenMap_sigmaₓ'. -/
 theorem isOpenMap_sigma {f : Sigma σ → α} : IsOpenMap f ↔ ∀ i, IsOpenMap fun a => f ⟨i, a⟩ := by
   simp only [isOpenMap_iff_nhds_le, Sigma.forall, Sigma.nhds_eq, map_map]
 #align is_open_map_sigma isOpenMap_sigma
 
-/- warning: is_open_map_sigma_map -> isOpenMap_sigma_map is a dubious translation:
-lean 3 declaration is
-  forall {ι : Type.{u1}} {κ : Type.{u2}} {σ : ι -> Type.{u3}} {τ : κ -> Type.{u4}} [_inst_1 : forall (i : ι), TopologicalSpace.{u3} (σ i)] [_inst_2 : forall (k : κ), TopologicalSpace.{u4} (τ k)] {f₁ : ι -> κ} {f₂ : forall (i : ι), (σ i) -> (τ (f₁ i))}, Iff (IsOpenMap.{max u1 u3, max u2 u4} (Sigma.{u1, u3} ι (fun (i : ι) => σ i)) (Sigma.{u2, u4} κ τ) (Sigma.topologicalSpace.{u1, u3} ι (fun (i : ι) => σ i) (fun (a : ι) => _inst_1 a)) (Sigma.topologicalSpace.{u2, u4} κ τ (fun (a : κ) => _inst_2 a)) (Sigma.map.{u1, u2, u3, u4} ι κ (fun (i : ι) => σ i) τ f₁ f₂)) (forall (i : ι), IsOpenMap.{u3, u4} (σ i) (τ (f₁ i)) (_inst_1 i) (_inst_2 (f₁ i)) (f₂ i))
-but is expected to have type
-  forall {ι : Type.{u3}} {κ : Type.{u1}} {σ : ι -> Type.{u4}} {τ : κ -> Type.{u2}} [_inst_1 : forall (i : ι), TopologicalSpace.{u4} (σ i)] [_inst_2 : forall (k : κ), TopologicalSpace.{u2} (τ k)] {f₁ : ι -> κ} {f₂ : forall (i : ι), (σ i) -> (τ (f₁ i))}, Iff (IsOpenMap.{max u4 u3, max u2 u1} (Sigma.{u3, u4} ι (fun (i : ι) => σ i)) (Sigma.{u1, u2} κ τ) (instTopologicalSpaceSigma.{u3, u4} ι (fun (i : ι) => σ i) (fun (a : ι) => _inst_1 a)) (instTopologicalSpaceSigma.{u1, u2} κ τ (fun (a : κ) => _inst_2 a)) (Sigma.map.{u3, u1, u4, u2} ι κ (fun (i : ι) => σ i) τ f₁ f₂)) (forall (i : ι), IsOpenMap.{u4, u2} (σ i) (τ (f₁ i)) (_inst_1 i) (_inst_2 (f₁ i)) (f₂ i))
-Case conversion may be inaccurate. Consider using '#align is_open_map_sigma_map isOpenMap_sigma_mapₓ'. -/
 theorem isOpenMap_sigma_map {f₁ : ι → κ} {f₂ : ∀ i, σ i → τ (f₁ i)} :
     IsOpenMap (Sigma.map f₁ f₂) ↔ ∀ i, IsOpenMap (f₂ i) :=
   isOpenMap_sigma.trans <|
     forall_congr' fun i => (@openEmbedding_sigmaMk _ _ _ (f₁ i)).isOpenMap_iff.symm
 #align is_open_map_sigma_map isOpenMap_sigma_map
 
-/- warning: inducing_sigma_map -> inducing_sigma_map is a dubious translation:
-lean 3 declaration is
-  forall {ι : Type.{u1}} {κ : Type.{u2}} {σ : ι -> Type.{u3}} {τ : κ -> Type.{u4}} [_inst_1 : forall (i : ι), TopologicalSpace.{u3} (σ i)] [_inst_2 : forall (k : κ), TopologicalSpace.{u4} (τ k)] {f₁ : ι -> κ} {f₂ : forall (i : ι), (σ i) -> (τ (f₁ i))}, (Function.Injective.{succ u1, succ u2} ι κ f₁) -> (Iff (Inducing.{max u1 u3, max u2 u4} (Sigma.{u1, u3} ι (fun (i : ι) => σ i)) (Sigma.{u2, u4} κ τ) (Sigma.topologicalSpace.{u1, u3} ι (fun (i : ι) => σ i) (fun (a : ι) => _inst_1 a)) (Sigma.topologicalSpace.{u2, u4} κ τ (fun (a : κ) => _inst_2 a)) (Sigma.map.{u1, u2, u3, u4} ι κ (fun (i : ι) => σ i) τ f₁ f₂)) (forall (i : ι), Inducing.{u3, u4} (σ i) (τ (f₁ i)) (_inst_1 i) (_inst_2 (f₁ i)) (f₂ i)))
-but is expected to have type
-  forall {ι : Type.{u4}} {κ : Type.{u3}} {σ : ι -> Type.{u2}} {τ : κ -> Type.{u1}} [_inst_1 : forall (i : ι), TopologicalSpace.{u2} (σ i)] [_inst_2 : forall (k : κ), TopologicalSpace.{u1} (τ k)] {f₁ : ι -> κ} {f₂ : forall (i : ι), (σ i) -> (τ (f₁ i))}, (Function.Injective.{succ u4, succ u3} ι κ f₁) -> (Iff (Inducing.{max u2 u4, max u1 u3} (Sigma.{u4, u2} ι (fun (i : ι) => σ i)) (Sigma.{u3, u1} κ τ) (instTopologicalSpaceSigma.{u4, u2} ι (fun (i : ι) => σ i) (fun (a : ι) => _inst_1 a)) (instTopologicalSpaceSigma.{u3, u1} κ τ (fun (a : κ) => _inst_2 a)) (Sigma.map.{u4, u3, u2, u1} ι κ (fun (i : ι) => σ i) τ f₁ f₂)) (forall (i : ι), Inducing.{u2, u1} (σ i) (τ (f₁ i)) (_inst_1 i) (_inst_2 (f₁ i)) (f₂ i)))
-Case conversion may be inaccurate. Consider using '#align inducing_sigma_map inducing_sigma_mapₓ'. -/
 theorem inducing_sigma_map {f₁ : ι → κ} {f₂ : ∀ i, σ i → τ (f₁ i)} (h₁ : Injective f₁) :
     Inducing (Sigma.map f₁ f₂) ↔ ∀ i, Inducing (f₂ i) := by
   simp only [inducing_iff_nhds, Sigma.forall, Sigma.nhds_mk, Sigma.map, ← map_sigma_mk_comap h₁,
     map_inj sigma_mk_injective]
 #align inducing_sigma_map inducing_sigma_map
 
-/- warning: embedding_sigma_map -> embedding_sigma_map is a dubious translation:
-lean 3 declaration is
-  forall {ι : Type.{u1}} {κ : Type.{u2}} {σ : ι -> Type.{u3}} {τ : κ -> Type.{u4}} [_inst_1 : forall (i : ι), TopologicalSpace.{u3} (σ i)] [_inst_2 : forall (k : κ), TopologicalSpace.{u4} (τ k)] {f₁ : ι -> κ} {f₂ : forall (i : ι), (σ i) -> (τ (f₁ i))}, (Function.Injective.{succ u1, succ u2} ι κ f₁) -> (Iff (Embedding.{max u1 u3, max u2 u4} (Sigma.{u1, u3} ι (fun (i : ι) => σ i)) (Sigma.{u2, u4} κ τ) (Sigma.topologicalSpace.{u1, u3} ι (fun (i : ι) => σ i) (fun (a : ι) => _inst_1 a)) (Sigma.topologicalSpace.{u2, u4} κ τ (fun (a : κ) => _inst_2 a)) (Sigma.map.{u1, u2, u3, u4} ι κ (fun (i : ι) => σ i) τ f₁ f₂)) (forall (i : ι), Embedding.{u3, u4} (σ i) (τ (f₁ i)) (_inst_1 i) (_inst_2 (f₁ i)) (f₂ i)))
-but is expected to have type
-  forall {ι : Type.{u4}} {κ : Type.{u3}} {σ : ι -> Type.{u2}} {τ : κ -> Type.{u1}} [_inst_1 : forall (i : ι), TopologicalSpace.{u2} (σ i)] [_inst_2 : forall (k : κ), TopologicalSpace.{u1} (τ k)] {f₁ : ι -> κ} {f₂ : forall (i : ι), (σ i) -> (τ (f₁ i))}, (Function.Injective.{succ u4, succ u3} ι κ f₁) -> (Iff (Embedding.{max u2 u4, max u1 u3} (Sigma.{u4, u2} ι (fun (i : ι) => σ i)) (Sigma.{u3, u1} κ τ) (instTopologicalSpaceSigma.{u4, u2} ι (fun (i : ι) => σ i) (fun (a : ι) => _inst_1 a)) (instTopologicalSpaceSigma.{u3, u1} κ τ (fun (a : κ) => _inst_2 a)) (Sigma.map.{u4, u3, u2, u1} ι κ (fun (i : ι) => σ i) τ f₁ f₂)) (forall (i : ι), Embedding.{u2, u1} (σ i) (τ (f₁ i)) (_inst_1 i) (_inst_2 (f₁ i)) (f₂ i)))
-Case conversion may be inaccurate. Consider using '#align embedding_sigma_map embedding_sigma_mapₓ'. -/
 theorem embedding_sigma_map {f₁ : ι → κ} {f₂ : ∀ i, σ i → τ (f₁ i)} (h : Injective f₁) :
     Embedding (Sigma.map f₁ f₂) ↔ ∀ i, Embedding (f₂ i) := by
   simp only [embedding_iff, injective.sigma_map, inducing_sigma_map h, forall_and, h.sigma_map_iff]
 #align embedding_sigma_map embedding_sigma_map
 
-/- warning: open_embedding_sigma_map -> openEmbedding_sigma_map is a dubious translation:
-lean 3 declaration is
-  forall {ι : Type.{u1}} {κ : Type.{u2}} {σ : ι -> Type.{u3}} {τ : κ -> Type.{u4}} [_inst_1 : forall (i : ι), TopologicalSpace.{u3} (σ i)] [_inst_2 : forall (k : κ), TopologicalSpace.{u4} (τ k)] {f₁ : ι -> κ} {f₂ : forall (i : ι), (σ i) -> (τ (f₁ i))}, (Function.Injective.{succ u1, succ u2} ι κ f₁) -> (Iff (OpenEmbedding.{max u1 u3, max u2 u4} (Sigma.{u1, u3} ι (fun (i : ι) => σ i)) (Sigma.{u2, u4} κ τ) (Sigma.topologicalSpace.{u1, u3} ι (fun (i : ι) => σ i) (fun (a : ι) => _inst_1 a)) (Sigma.topologicalSpace.{u2, u4} κ τ (fun (a : κ) => _inst_2 a)) (Sigma.map.{u1, u2, u3, u4} ι κ (fun (i : ι) => σ i) τ f₁ f₂)) (forall (i : ι), OpenEmbedding.{u3, u4} (σ i) (τ (f₁ i)) (_inst_1 i) (_inst_2 (f₁ i)) (f₂ i)))
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-Case conversion may be inaccurate. Consider using '#align open_embedding_sigma_map openEmbedding_sigma_mapₓ'. -/
 theorem openEmbedding_sigma_map {f₁ : ι → κ} {f₂ : ∀ i, σ i → τ (f₁ i)} (h : Injective f₁) :
     OpenEmbedding (Sigma.map f₁ f₂) ↔ ∀ i, OpenEmbedding (f₂ i) := by
   simp only [openEmbedding_iff_embedding_open, isOpenMap_sigma_map, embedding_sigma_map h,

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

@@ -180,34 +180,22 @@ theorem isClosedMap_toAdd : IsClosedMap (toAdd : Multiplicative α → α) :=
 -/
 
 #print nhds_ofMul /-
-theorem nhds_ofMul (a : α) : 𝓝 (ofMul a) = map ofMul (𝓝 a) :=
-  by
-  unfold nhds
-  rfl
+theorem nhds_ofMul (a : α) : 𝓝 (ofMul a) = map ofMul (𝓝 a) := by unfold nhds; rfl
 #align nhds_of_mul nhds_ofMul
 -/
 
 #print nhds_ofAdd /-
-theorem nhds_ofAdd (a : α) : 𝓝 (ofAdd a) = map ofAdd (𝓝 a) :=
-  by
-  unfold nhds
-  rfl
+theorem nhds_ofAdd (a : α) : 𝓝 (ofAdd a) = map ofAdd (𝓝 a) := by unfold nhds; rfl
 #align nhds_of_add nhds_ofAdd
 -/
 
 #print nhds_toMul /-
-theorem nhds_toMul (a : Additive α) : 𝓝 (toMul a) = map toMul (𝓝 a) :=
-  by
-  unfold nhds
-  rfl
+theorem nhds_toMul (a : Additive α) : 𝓝 (toMul a) = map toMul (𝓝 a) := by unfold nhds; rfl
 #align nhds_to_mul nhds_toMul
 -/
 
 #print nhds_toAdd /-
-theorem nhds_toAdd (a : Multiplicative α) : 𝓝 (toAdd a) = map toAdd (𝓝 a) :=
-  by
-  unfold nhds
-  rfl
+theorem nhds_toAdd (a : Multiplicative α) : 𝓝 (toAdd a) = map toAdd (𝓝 a) := by unfold nhds; rfl
 #align nhds_to_add nhds_toAdd
 -/
 
@@ -269,18 +257,12 @@ theorem isClosedMap_ofDual : IsClosedMap (ofDual : αᵒᵈ → α) :=
 -/
 
 #print nhds_toDual /-
-theorem nhds_toDual (a : α) : 𝓝 (toDual a) = map toDual (𝓝 a) :=
-  by
-  unfold nhds
-  rfl
+theorem nhds_toDual (a : α) : 𝓝 (toDual a) = map toDual (𝓝 a) := by unfold nhds; rfl
 #align nhds_to_dual nhds_toDual
 -/
 
 #print nhds_ofDual /-
-theorem nhds_ofDual (a : α) : 𝓝 (ofDual a) = map ofDual (𝓝 a) :=
-  by
-  unfold nhds
-  rfl
+theorem nhds_ofDual (a : α) : 𝓝 (ofDual a) = map ofDual (𝓝 a) := by unfold nhds; rfl
 #align nhds_of_dual nhds_ofDual
 -/
 
@@ -332,9 +314,7 @@ but is expected to have type
 Case conversion may be inaccurate. Consider using '#align sigma.discrete_topology Sigma.discreteTopologyₓ'. -/
 instance Sigma.discreteTopology {β : α → Type v} [∀ a, TopologicalSpace (β a)]
     [h : ∀ a, DiscreteTopology (β a)] : DiscreteTopology (Sigma β) :=
-  ⟨by
-    unfold Sigma.topologicalSpace
-    simp [fun a => (h a).eq_bot]⟩
+  ⟨by unfold Sigma.topologicalSpace; simp [fun a => (h a).eq_bot]⟩
 #align sigma.discrete_topology Sigma.discreteTopology
 
 section Topα
@@ -964,9 +944,7 @@ Case conversion may be inaccurate. Consider using '#align prod.tendsto_iff Prod.
 theorem Prod.tendsto_iff {α} (seq : α → β × γ) {f : Filter α} (x : β × γ) :
     Tendsto seq f (𝓝 x) ↔
       Tendsto (fun n => (seq n).fst) f (𝓝 x.fst) ∧ Tendsto (fun n => (seq n).snd) f (𝓝 x.snd) :=
-  by
-  cases x
-  rw [nhds_prod_eq, Filter.tendsto_prod_iff']
+  by cases x; rw [nhds_prod_eq, Filter.tendsto_prod_iff']
 #align prod.tendsto_iff Prod.tendsto_iff
 
 /- warning: filter.has_basis.prod_nhds -> Filter.HasBasis.prod_nhds is a dubious translation:
@@ -979,10 +957,8 @@ Case conversion may be inaccurate. Consider using '#align filter.has_basis.prod_
 theorem Filter.HasBasis.prod_nhds {ιa ιb : Type _} {pa : ιa → Prop} {pb : ιb → Prop}
     {sa : ιa → Set α} {sb : ιb → Set β} {a : α} {b : β} (ha : (𝓝 a).HasBasis pa sa)
     (hb : (𝓝 b).HasBasis pb sb) :
-    (𝓝 (a, b)).HasBasis (fun i : ιa × ιb => pa i.1 ∧ pb i.2) fun i => sa i.1 ×ˢ sb i.2 :=
-  by
-  rw [nhds_prod_eq]
-  exact ha.prod hb
+    (𝓝 (a, b)).HasBasis (fun i : ιa × ιb => pa i.1 ∧ pb i.2) fun i => sa i.1 ×ˢ sb i.2 := by
+  rw [nhds_prod_eq]; exact ha.prod hb
 #align filter.has_basis.prod_nhds Filter.HasBasis.prod_nhds
 
 /- warning: filter.has_basis.prod_nhds' -> Filter.HasBasis.prod_nhds' is a dubious translation:
@@ -995,9 +971,7 @@ Case conversion may be inaccurate. Consider using '#align filter.has_basis.prod_
 theorem Filter.HasBasis.prod_nhds' {ιa ιb : Type _} {pa : ιa → Prop} {pb : ιb → Prop}
     {sa : ιa → Set α} {sb : ιb → Set β} {ab : α × β} (ha : (𝓝 ab.1).HasBasis pa sa)
     (hb : (𝓝 ab.2).HasBasis pb sb) :
-    (𝓝 ab).HasBasis (fun i : ιa × ιb => pa i.1 ∧ pb i.2) fun i => sa i.1 ×ˢ sb i.2 :=
-  by
-  cases ab
+    (𝓝 ab).HasBasis (fun i : ιa × ιb => pa i.1 ∧ pb i.2) fun i => sa i.1 ×ˢ sb i.2 := by cases ab;
   exact ha.prod_nhds hb
 #align filter.has_basis.prod_nhds' Filter.HasBasis.prod_nhds'
 
@@ -1068,10 +1042,8 @@ but is expected to have type
   forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : TopologicalSpace.{u1} α] [_inst_2 : TopologicalSpace.{u2} β] {p : (Prod.{u1, u2} α β) -> Prop} {x : α} {y : β}, (Filter.Eventually.{max u1 u2} (Prod.{u1, u2} α β) (fun (x : Prod.{u1, u2} α β) => p x) (nhds.{max u1 u2} (Prod.{u1, u2} α β) (instTopologicalSpaceProd.{u1, u2} α β _inst_1 _inst_2) (Prod.mk.{u1, u2} α β x y))) -> (Filter.Eventually.{u1} α (fun (x' : α) => Filter.Eventually.{u2} β (fun (y' : β) => p (Prod.mk.{u1, u2} α β x' y')) (nhds.{u2} β _inst_2 y)) (nhds.{u1} α _inst_1 x))
 Case conversion may be inaccurate. Consider using '#align filter.eventually.curry_nhds Filter.Eventually.curry_nhdsₓ'. -/
 theorem Filter.Eventually.curry_nhds {p : α × β → Prop} {x : α} {y : β}
-    (h : ∀ᶠ x in 𝓝 (x, y), p x) : ∀ᶠ x' in 𝓝 x, ∀ᶠ y' in 𝓝 y, p (x', y') :=
-  by
-  rw [nhds_prod_eq] at h
-  exact h.curry
+    (h : ∀ᶠ x in 𝓝 (x, y), p x) : ∀ᶠ x' in 𝓝 x, ∀ᶠ y' in 𝓝 y, p (x', y') := by
+  rw [nhds_prod_eq] at h; exact h.curry
 #align filter.eventually.curry_nhds Filter.Eventually.curry_nhds
 
 /- warning: continuous_at.prod -> ContinuousAt.prod is a dubious translation:
@@ -1132,8 +1104,7 @@ theorem prod_generateFrom_generateFrom_eq {α β : Type _} {s : Set (Set α)} {t
         le_generateFrom fun u hu =>
           have : (⋃ v ∈ t, u ×ˢ v) = Prod.fst ⁻¹' u := by
             simp_rw [← prod_Union, ← sUnion_eq_bUnion, ht, prod_univ]
-          show G.IsOpen (Prod.fst ⁻¹' u) by
-            rw [← this]
+          show G.IsOpen (Prod.fst ⁻¹' u) by rw [← this];
             exact
               isOpen_iUnion fun v =>
                 isOpen_iUnion fun hv => generate_open.basic _ ⟨_, hu, _, hv, rfl⟩)
@@ -1141,8 +1112,7 @@ theorem prod_generateFrom_generateFrom_eq {α β : Type _} {s : Set (Set α)} {t
         le_generateFrom fun v hv =>
           have : (⋃ u ∈ s, u ×ˢ v) = Prod.snd ⁻¹' v := by
             simp_rw [← Union_prod_const, ← sUnion_eq_bUnion, hs, univ_prod]
-          show G.IsOpen (Prod.snd ⁻¹' v) by
-            rw [← this]
+          show G.IsOpen (Prod.snd ⁻¹' v) by rw [← this];
             exact
               isOpen_iUnion fun u =>
                 isOpen_iUnion fun hu => generate_open.basic _ ⟨_, hu, _, hv, rfl⟩))
@@ -1441,9 +1411,7 @@ Case conversion may be inaccurate. Consider using '#align dense.prod Dense.prod
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
 /-- The product of two dense sets is a dense set. -/
 theorem Dense.prod {s : Set α} {t : Set β} (hs : Dense s) (ht : Dense t) : Dense (s ×ˢ t) :=
-  fun x => by
-  rw [closure_prod_eq]
-  exact ⟨hs x.1, ht x.2⟩
+  fun x => by rw [closure_prod_eq]; exact ⟨hs x.1, ht x.2⟩
 #align dense.prod Dense.prod
 
 /- warning: dense_range.prod_map -> DenseRange.prod_map is a dubious translation:
@@ -1645,10 +1613,8 @@ lean 3 declaration is
 but is expected to have type
   forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : TopologicalSpace.{u1} α] [_inst_2 : TopologicalSpace.{u2} β], IsClosed.{max u1 u2} (Sum.{u1, u2} α β) (instTopologicalSpaceSum.{u1, u2} α β _inst_1 _inst_2) (Set.range.{max u1 u2, succ u1} (Sum.{u1, u2} α β) α (Sum.inl.{u1, u2} α β))
 Case conversion may be inaccurate. Consider using '#align is_closed_range_inl isClosed_range_inlₓ'. -/
-theorem isClosed_range_inl : IsClosed (range (inl : α → Sum α β)) :=
-  by
-  rw [← isOpen_compl_iff, compl_range_inl]
-  exact isOpen_range_inr
+theorem isClosed_range_inl : IsClosed (range (inl : α → Sum α β)) := by
+  rw [← isOpen_compl_iff, compl_range_inl]; exact isOpen_range_inr
 #align is_closed_range_inl isClosed_range_inl
 
 /- warning: is_closed_range_inr -> isClosed_range_inr is a dubious translation:
@@ -1657,10 +1623,8 @@ lean 3 declaration is
 but is expected to have type
   forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : TopologicalSpace.{u1} α] [_inst_2 : TopologicalSpace.{u2} β], IsClosed.{max u1 u2} (Sum.{u1, u2} α β) (instTopologicalSpaceSum.{u1, u2} α β _inst_1 _inst_2) (Set.range.{max u1 u2, succ u2} (Sum.{u1, u2} α β) β (Sum.inr.{u1, u2} α β))
 Case conversion may be inaccurate. Consider using '#align is_closed_range_inr isClosed_range_inrₓ'. -/
-theorem isClosed_range_inr : IsClosed (range (inr : β → Sum α β)) :=
-  by
-  rw [← isOpen_compl_iff, compl_range_inr]
-  exact isOpen_range_inl
+theorem isClosed_range_inr : IsClosed (range (inr : β → Sum α β)) := by
+  rw [← isOpen_compl_iff, compl_range_inr]; exact isOpen_range_inl
 #align is_closed_range_inr isClosed_range_inr
 
 /- warning: closed_embedding_inl -> closedEmbedding_inl is a dubious translation:
@@ -1910,10 +1874,8 @@ theorem continuousAt_subtype_val {p : α → Prop} {a : Subtype p} :
 -/
 
 #print Subtype.dense_iff /-
-theorem Subtype.dense_iff {s : Set α} {t : Set s} : Dense t ↔ s ⊆ closure (coe '' t) :=
-  by
-  rw [inducing_coe.dense_iff, SetCoe.forall]
-  rfl
+theorem Subtype.dense_iff {s : Set α} {t : Set s} : Dense t ↔ s ⊆ closure (coe '' t) := by
+  rw [inducing_coe.dense_iff, SetCoe.forall]; rfl
 #align subtype.dense_iff Subtype.dense_iff
 -/
 
@@ -2356,10 +2318,7 @@ but is expected to have type
   forall {ι : Type.{u2}} {π : ι -> Type.{u1}} [_inst_2 : forall (i : ι), TopologicalSpace.{u1} (π i)] {I : Set.{u2} ι} {s : forall (i : ι), Set.{u1} (π i)} (a : forall (i : ι), π i), (Membership.mem.{max u2 u1, max u2 u1} (Set.{max u2 u1} (forall (i : ι), π i)) (Filter.{max u2 u1} (forall (i : ι), π i)) (instMembershipSetFilter.{max u2 u1} (forall (i : ι), π i)) (Set.pi.{u2, u1} ι (fun (i : ι) => π i) I s) (nhds.{max u2 u1} (forall (i : ι), π i) (Pi.topologicalSpace.{u2, u1} ι (fun (i : ι) => π i) (fun (a : ι) => _inst_2 a)) a)) -> (forall {i : ι}, (Membership.mem.{u2, u2} ι (Set.{u2} ι) (Set.instMembershipSet.{u2} ι) i I) -> (Membership.mem.{u1, u1} (Set.{u1} (π i)) (Filter.{u1} (π i)) (instMembershipSetFilter.{u1} (π i)) (s i) (nhds.{u1} (π i) (_inst_2 i) (a i))))
 Case conversion may be inaccurate. Consider using '#align mem_nhds_of_pi_mem_nhds mem_nhds_of_pi_mem_nhdsₓ'. -/
 theorem mem_nhds_of_pi_mem_nhds {I : Set ι} {s : ∀ i, Set (π i)} (a : ∀ i, π i) (hs : I.pi s ∈ 𝓝 a)
-    {i : ι} (hi : i ∈ I) : s i ∈ 𝓝 (a i) :=
-  by
-  rw [nhds_pi] at hs
-  exact mem_of_pi_mem_pi hs hi
+    {i : ι} (hi : i ∈ I) : s i ∈ 𝓝 (a i) := by rw [nhds_pi] at hs; exact mem_of_pi_mem_pi hs hi
 #align mem_nhds_of_pi_mem_nhds mem_nhds_of_pi_mem_nhds
 
 /- warning: set_pi_mem_nhds -> set_pi_mem_nhds is a dubious translation:
@@ -2369,9 +2328,7 @@ but is expected to have type
   forall {ι : Type.{u2}} {π : ι -> Type.{u1}} [_inst_2 : forall (i : ι), TopologicalSpace.{u1} (π i)] {i : Set.{u2} ι} {s : forall (a : ι), Set.{u1} (π a)} {x : forall (a : ι), π a}, (Set.Finite.{u2} ι i) -> (forall (a : ι), (Membership.mem.{u2, u2} ι (Set.{u2} ι) (Set.instMembershipSet.{u2} ι) a i) -> (Membership.mem.{u1, u1} (Set.{u1} (π a)) (Filter.{u1} (π a)) (instMembershipSetFilter.{u1} (π a)) (s a) (nhds.{u1} (π a) (_inst_2 a) (x a)))) -> (Membership.mem.{max u1 u2, max u2 u1} (Set.{max u2 u1} (forall (i : ι), π i)) (Filter.{max u2 u1} (forall (a : ι), π a)) (instMembershipSetFilter.{max u2 u1} (forall (a : ι), π a)) (Set.pi.{u2, u1} ι (fun (a : ι) => π a) i s) (nhds.{max u2 u1} (forall (a : ι), π a) (Pi.topologicalSpace.{u2, u1} ι (fun (a : ι) => π a) (fun (a : ι) => _inst_2 a)) x))
 Case conversion may be inaccurate. Consider using '#align set_pi_mem_nhds set_pi_mem_nhdsₓ'. -/
 theorem set_pi_mem_nhds {i : Set ι} {s : ∀ a, Set (π a)} {x : ∀ a, π a} (hi : i.Finite)
-    (hs : ∀ a ∈ i, s a ∈ 𝓝 (x a)) : pi i s ∈ 𝓝 x :=
-  by
-  rw [pi_def, bInter_mem hi]
+    (hs : ∀ a ∈ i, s a ∈ 𝓝 (x a)) : pi i s ∈ 𝓝 x := by rw [pi_def, bInter_mem hi];
   exact fun a ha => (continuous_apply a).ContinuousAt (hs a ha)
 #align set_pi_mem_nhds set_pi_mem_nhds
 
@@ -2382,9 +2339,7 @@ but is expected to have type
   forall {ι : Type.{u2}} {π : ι -> Type.{u1}} [_inst_2 : forall (i : ι), TopologicalSpace.{u1} (π i)] {I : Set.{u2} ι}, (Set.Finite.{u2} ι I) -> (forall {s : forall (i : ι), Set.{u1} (π i)} (a : forall (i : ι), π i), Iff (Membership.mem.{max u2 u1, max u2 u1} (Set.{max u2 u1} (forall (i : ι), π i)) (Filter.{max u2 u1} (forall (i : ι), π i)) (instMembershipSetFilter.{max u2 u1} (forall (i : ι), π i)) (Set.pi.{u2, u1} ι (fun (i : ι) => π i) I s) (nhds.{max u2 u1} (forall (i : ι), π i) (Pi.topologicalSpace.{u2, u1} ι (fun (i : ι) => π i) (fun (a : ι) => _inst_2 a)) a)) (forall (i : ι), (Membership.mem.{u2, u2} ι (Set.{u2} ι) (Set.instMembershipSet.{u2} ι) i I) -> (Membership.mem.{u1, u1} (Set.{u1} (π i)) (Filter.{u1} (π i)) (instMembershipSetFilter.{u1} (π i)) (s i) (nhds.{u1} (π i) (_inst_2 i) (a i)))))
 Case conversion may be inaccurate. Consider using '#align set_pi_mem_nhds_iff set_pi_mem_nhds_iffₓ'. -/
 theorem set_pi_mem_nhds_iff {I : Set ι} (hI : I.Finite) {s : ∀ i, Set (π i)} (a : ∀ i, π i) :
-    I.pi s ∈ 𝓝 a ↔ ∀ i : ι, i ∈ I → s i ∈ 𝓝 (a i) :=
-  by
-  rw [nhds_pi, pi_mem_pi_iff hI]
+    I.pi s ∈ 𝓝 a ↔ ∀ i : ι, i ∈ I → s i ∈ 𝓝 (a i) := by rw [nhds_pi, pi_mem_pi_iff hI];
   infer_instance
 #align set_pi_mem_nhds_iff set_pi_mem_nhds_iff
 
@@ -2395,9 +2350,7 @@ but is expected to have type
   forall {ι : Type.{u2}} {π : ι -> Type.{u1}} [_inst_2 : forall (i : ι), TopologicalSpace.{u1} (π i)] {I : Set.{u2} ι}, (Set.Finite.{u2} ι I) -> (forall {s : forall (i : ι), Set.{u1} (π i)}, Eq.{max (succ u2) (succ u1)} (Set.{max u1 u2} (forall (i : ι), π i)) (interior.{max u1 u2} (forall (i : ι), π i) (Pi.topologicalSpace.{u2, u1} ι (fun (i : ι) => π i) (fun (a : ι) => _inst_2 a)) (Set.pi.{u2, u1} ι (fun (i : ι) => π i) I s)) (Set.pi.{u2, u1} ι (fun (i : ι) => π i) I (fun (i : ι) => interior.{u1} (π i) (_inst_2 i) (s i))))
 Case conversion may be inaccurate. Consider using '#align interior_pi_set interior_pi_setₓ'. -/
 theorem interior_pi_set {I : Set ι} (hI : I.Finite) {s : ∀ i, Set (π i)} :
-    interior (pi I s) = I.pi fun i => interior (s i) :=
-  by
-  ext a
+    interior (pi I s) = I.pi fun i => interior (s i) := by ext a;
   simp only [Set.mem_pi, mem_interior_iff_mem_nhds, set_pi_mem_nhds_iff hI]
 #align interior_pi_set interior_pi_set
 
@@ -2442,8 +2395,7 @@ theorem pi_generateFrom_eq {π : ι → Type _} {g : ∀ a, Set (Set (π a))} :
   · rintro s ⟨t, i, hi, rfl⟩
     rw [pi_def]
     apply isOpen_biInter (Finset.finite_toSet _)
-    intro a ha
-    show ((generate_from G).coinduced fun f : ∀ a, π a => f a).IsOpen (t a)
+    intro a ha; show ((generate_from G).coinduced fun f : ∀ a, π a => f a).IsOpen (t a)
     refine' le_generateFrom _ _ (hi a ha)
     exact fun s hs => generate_open.basic _ ⟨update (fun a => univ) a s, {a}, by simp [hs]⟩
 #align pi_generate_from_eq pi_generateFrom_eq
@@ -2458,18 +2410,13 @@ theorem pi_generateFrom_eq_finite {π : ι → Type _} {g : ∀ a, Set (Set (π
   cases nonempty_fintype ι
   rw [pi_generateFrom_eq]
   refine' le_antisymm (generate_from_anti _) (le_generateFrom _)
-  · rintro s ⟨t, ht, rfl⟩
-    exact ⟨t, Finset.univ, by simp [ht]⟩
+  · rintro s ⟨t, ht, rfl⟩; exact ⟨t, Finset.univ, by simp [ht]⟩
   · rintro s ⟨t, i, ht, rfl⟩
     apply isOpen_iff_forall_mem_open.2 _
     intro f hf
     choose c hc using
-      show ∀ a, ∃ s, s ∈ g a ∧ f a ∈ s by
-        intro a
-        have : f a ∈ ⋃₀ g a := by
-          rw [hg]
-          apply mem_univ
-        simpa
+      show ∀ a, ∃ s, s ∈ g a ∧ f a ∈ s by intro a;
+        have : f a ∈ ⋃₀ g a := by rw [hg]; apply mem_univ; simpa
     refine' ⟨pi univ fun a => if a ∈ i then t a else (c : ∀ a, Set (π a)) a, _, _, _⟩
     · simp [pi_if]
     · refine' generate_open.basic _ ⟨_, fun a => _, rfl⟩
@@ -2652,9 +2599,7 @@ lean 3 declaration is
 but is expected to have type
   forall {ι : Type.{u2}} {σ : ι -> Type.{u1}} [_inst_1 : forall (i : ι), TopologicalSpace.{u1} (σ i)] (x : Sigma.{u2, u1} ι σ), Eq.{max (succ u2) (succ u1)} (Filter.{max u2 u1} (Sigma.{u2, u1} ι σ)) (nhds.{max u2 u1} (Sigma.{u2, u1} ι σ) (instTopologicalSpaceSigma.{u2, u1} ι σ (fun (a : ι) => _inst_1 a)) x) (Filter.map.{u1, max u2 u1} (σ (Sigma.fst.{u2, u1} ι σ x)) (Sigma.{u2, u1} ι σ) (Sigma.mk.{u2, u1} ι σ (Sigma.fst.{u2, u1} ι σ x)) (nhds.{u1} (σ (Sigma.fst.{u2, u1} ι σ x)) (_inst_1 (Sigma.fst.{u2, u1} ι σ x)) (Sigma.snd.{u2, u1} ι σ x)))
 Case conversion may be inaccurate. Consider using '#align sigma.nhds_eq Sigma.nhds_eqₓ'. -/
-theorem Sigma.nhds_eq (x : Sigma σ) : 𝓝 x = map (Sigma.mk x.1) (𝓝 x.2) :=
-  by
-  cases x
+theorem Sigma.nhds_eq (x : Sigma σ) : 𝓝 x = map (Sigma.mk x.1) (𝓝 x.2) := by cases x;
   apply Sigma.nhds_mk
 #align sigma.nhds_eq Sigma.nhds_eq
 

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

@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Johannes Hölzl, Mario Carneiro, Patrick Massot
 
 ! This file was ported from Lean 3 source module topology.constructions
-! leanprover-community/mathlib commit 55d771df074d0dd020139ee1cd4b95521422df9f
+! leanprover-community/mathlib commit 76f9c990d4b7c3dd26b87c4c4b51759e249d9e66
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
@@ -734,10 +734,7 @@ theorem Continuous.comp₃ {g : α × β × γ → ε} (hg : Continuous g) {e :
 #align continuous.comp₃ Continuous.comp₃
 
 /- warning: continuous.comp₄ -> Continuous.comp₄ is a dubious translation:
-lean 3 declaration is
-  forall {α : Type.{u1}} {β : Type.{u2}} {γ : Type.{u3}} {δ : Type.{u4}} {ε : Type.{u5}} {ζ : Type.{u6}} [_inst_1 : TopologicalSpace.{u1} α] [_inst_2 : TopologicalSpace.{u2} β] [_inst_3 : TopologicalSpace.{u3} γ] [_inst_4 : TopologicalSpace.{u4} δ] [_inst_5 : TopologicalSpace.{u5} ε] [_inst_6 : TopologicalSpace.{u6} ζ] {g : (Prod.{u1, max u2 u3 u6} α (Prod.{u2, max u3 u6} β (Prod.{u3, u6} γ ζ))) -> ε}, (Continuous.{max u1 u2 u3 u6, u5} (Prod.{u1, max u2 u3 u6} α (Prod.{u2, max u3 u6} β (Prod.{u3, u6} γ ζ))) ε (Prod.topologicalSpace.{u1, max u2 u3 u6} α (Prod.{u2, max u3 u6} β (Prod.{u3, u6} γ ζ)) _inst_1 (Prod.topologicalSpace.{u2, max u3 u6} β (Prod.{u3, u6} γ ζ) _inst_2 (Prod.topologicalSpace.{u3, u6} γ ζ _inst_3 _inst_6))) _inst_5 g) -> (forall {e : δ -> α}, (Continuous.{u4, u1} δ α _inst_4 _inst_1 e) -> (forall {f : δ -> β}, (Continuous.{u4, u2} δ β _inst_4 _inst_2 f) -> (forall {k : δ -> γ}, (Continuous.{u4, u3} δ γ _inst_4 _inst_3 k) -> (forall {l : δ -> ζ}, (Continuous.{u4, u6} δ ζ _inst_4 _inst_6 l) -> (Continuous.{u4, u5} δ ε _inst_4 _inst_5 (fun (x : δ) => g (Prod.mk.{u1, max u2 u3 u6} α (Prod.{u2, max u3 u6} β (Prod.{u3, u6} γ ζ)) (e x) (Prod.mk.{u2, max u3 u6} β (Prod.{u3, u6} γ ζ) (f x) (Prod.mk.{u3, u6} γ ζ (k x) (l x))))))))))
-but is expected to have type
-  forall {α : Type.{u5}} {β : Type.{u6}} {γ : Type.{u3}} {δ : Type.{u1}} {ε : Type.{u2}} {ζ : Type.{u4}} [_inst_1 : TopologicalSpace.{u5} α] [_inst_2 : TopologicalSpace.{u6} β] [_inst_3 : TopologicalSpace.{u3} γ] [_inst_4 : TopologicalSpace.{u1} δ] [_inst_5 : TopologicalSpace.{u2} ε] [_inst_6 : TopologicalSpace.{u4} ζ] {g : (Prod.{u5, max (max u4 u3) u6} α (Prod.{u6, max u4 u3} β (Prod.{u3, u4} γ ζ))) -> ε}, (Continuous.{max (max (max u5 u6) u3) u4, u2} (Prod.{u5, max (max u4 u3) u6} α (Prod.{u6, max u4 u3} β (Prod.{u3, u4} γ ζ))) ε (instTopologicalSpaceProd.{u5, max (max u6 u3) u4} α (Prod.{u6, max u4 u3} β (Prod.{u3, u4} γ ζ)) _inst_1 (instTopologicalSpaceProd.{u6, max u3 u4} β (Prod.{u3, u4} γ ζ) _inst_2 (instTopologicalSpaceProd.{u3, u4} γ ζ _inst_3 _inst_6))) _inst_5 g) -> (forall {e : δ -> α}, (Continuous.{u1, u5} δ α _inst_4 _inst_1 e) -> (forall {f : δ -> β}, (Continuous.{u1, u6} δ β _inst_4 _inst_2 f) -> (forall {k : δ -> γ}, (Continuous.{u1, u3} δ γ _inst_4 _inst_3 k) -> (forall {l : δ -> ζ}, (Continuous.{u1, u4} δ ζ _inst_4 _inst_6 l) -> (Continuous.{u1, u2} δ ε _inst_4 _inst_5 (fun (x : δ) => g (Prod.mk.{u5, max (max u6 u3) u4} α (Prod.{u6, max u4 u3} β (Prod.{u3, u4} γ ζ)) (e x) (Prod.mk.{u6, max u3 u4} β (Prod.{u3, u4} γ ζ) (f x) (Prod.mk.{u3, u4} γ ζ (k x) (l x))))))))))
+<too large>
 Case conversion may be inaccurate. Consider using '#align continuous.comp₄ Continuous.comp₄ₓ'. -/
 theorem Continuous.comp₄ {g : α × β × γ × ζ → ε} (hg : Continuous g) {e : δ → α} (he : Continuous e)
     {f : δ → β} (hf : Continuous f) {k : δ → γ} (hk : Continuous k) {l : δ → ζ}
@@ -2237,10 +2234,7 @@ theorem continuous_mulSingle [∀ i, One (π i)] [DecidableEq ι] (i : ι) :
 -/
 
 /- warning: filter.tendsto.fin_insert_nth -> Filter.Tendsto.fin_insertNth is a dubious translation:
-lean 3 declaration is
-  forall {β : Type.{u1}} {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_3 : 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 : β -> (π i)} {l : Filter.{u1} β} {x : π i}, (Filter.Tendsto.{u1, u2} β (π i) f l (nhds.{u2} (π i) (_inst_3 i) x)) -> (forall {g : β -> (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} 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+<too large>
 Case conversion may be inaccurate. Consider using '#align filter.tendsto.fin_insert_nth Filter.Tendsto.fin_insertNthₓ'. -/
 theorem Filter.Tendsto.fin_insertNth {n} {π : Fin (n + 1) → Type _} [∀ i, TopologicalSpace (π i)]
     (i : Fin (n + 1)) {f : β → π i} {l : Filter β} {x : π i} (hf : Tendsto f l (𝓝 x))
@@ -2250,10 +2244,7 @@ theorem Filter.Tendsto.fin_insertNth {n} {π : Fin (n + 1) → Type _} [∀ i, T
 #align filter.tendsto.fin_insert_nth Filter.Tendsto.fin_insertNth
 
 /- warning: continuous_at.fin_insert_nth -> ContinuousAt.fin_insertNth is a dubious translation:
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+<too large>
 Case conversion may be inaccurate. Consider using '#align continuous_at.fin_insert_nth ContinuousAt.fin_insertNthₓ'. -/
 theorem ContinuousAt.fin_insertNth {n} {π : Fin (n + 1) → Type _} [∀ i, TopologicalSpace (π i)]
     (i : Fin (n + 1)) {f : α → π i} {a : α} (hf : ContinuousAt f a)
@@ -2263,10 +2254,7 @@ theorem ContinuousAt.fin_insertNth {n} {π : Fin (n + 1) → Type _} [∀ i, Top
 #align continuous_at.fin_insert_nth ContinuousAt.fin_insertNth
 
 /- warning: continuous.fin_insert_nth -> Continuous.fin_insertNth is a dubious translation:
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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)) (fun (a : Fin n) => _inst_3 (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 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(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) -> (Continuous.{u2, u1} 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, u1} (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_3 a)) (fun (a : α) => Fin.insertNth.{u1} n π i (f a) (g a))))
+<too large>
 Case conversion may be inaccurate. Consider using '#align continuous.fin_insert_nth Continuous.fin_insertNthₓ'. -/
 theorem Continuous.fin_insertNth {n} {π : Fin (n + 1) → Type _} [∀ i, TopologicalSpace (π i)]
     (i : Fin (n + 1)) {f : α → π i} (hf : Continuous f) {g : α → ∀ j : Fin n, π (i.succAbove j)}
@@ -2823,6 +2811,11 @@ theorem embedding_uLift_down [TopologicalSpace α] : Embedding (ULift.down : ULi
 #align embedding_ulift_down embedding_uLift_down
 -/
 
+theorem ULift.closedEmbedding_down [TopologicalSpace α] :
+    ClosedEmbedding (ULift.down : ULift.{v, u} α → α) :=
+  ⟨embedding_uLift_down, by simp only [ulift.down_surjective.range_eq, isClosed_univ]⟩
+#align ulift.closed_embedding_down ULift.closedEmbedding_down
+
 instance [TopologicalSpace α] [DiscreteTopology α] : DiscreteTopology (ULift α) :=
   embedding_uLift_down.DiscreteTopology
 

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

@@ -2240,7 +2240,7 @@ theorem continuous_mulSingle [∀ i, One (π i)] [DecidableEq ι] (i : ι) :
 lean 3 declaration is
   forall {β : Type.{u1}} {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_3 : 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 : β -> (π i)} {l : Filter.{u1} β} {x : π i}, (Filter.Tendsto.{u1, u2} β (π i) f l (nhds.{u2} (π i) (_inst_3 i) x)) -> (forall {g : β -> (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))} {y : 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} 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 but is expected to have type
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+  forall {β : Type.{u2}} {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.{u1}} [_inst_3 : forall (i : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))), TopologicalSpace.{u1} (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 : β -> (π i)} {l : Filter.{u2} β} {x : π i}, (Filter.Tendsto.{u2, u1} β (π i) f l (nhds.{u1} (π i) (_inst_3 i) x)) -> (forall {g : β -> (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) => 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(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} 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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_3 (FunLike.coe.{1, 1, 1} 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(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} 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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))) y)) -> (Filter.Tendsto.{u2, u1} β (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) (fun (a : β) => Fin.insertNth.{u1} n π i (f a) (g a)) l (nhds.{u1} (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) (Pi.topologicalSpace.{0, u1} (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_3 a)) (Fin.insertNth.{u1} n (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) i x y))))
 Case conversion may be inaccurate. Consider using '#align filter.tendsto.fin_insert_nth Filter.Tendsto.fin_insertNthₓ'. -/
 theorem Filter.Tendsto.fin_insertNth {n} {π : Fin (n + 1) → Type _} [∀ i, TopologicalSpace (π i)]
     (i : Fin (n + 1)) {f : β → π i} {l : Filter β} {x : π i} (hf : Tendsto f l (𝓝 x))
@@ -2253,7 +2253,7 @@ theorem Filter.Tendsto.fin_insertNth {n} {π : Fin (n + 1) → Type _} [∀ i, T
 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_3 : 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}, (ContinuousAt.{u1, u2} α (π i) _inst_1 (_inst_3 i) f 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))}, (ContinuousAt.{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) 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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.succAbove n i) a))) g a) -> (ContinuousAt.{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 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 but is expected to have type
-  forall {α : Type.{u2}} [_inst_1 : TopologicalSpace.{u2} 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.{u1}} [_inst_3 : forall (i : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))), TopologicalSpace.{u1} (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}, (ContinuousAt.{u2, u1} α (π i) _inst_1 (_inst_3 i) f 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) => 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(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_3 (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 a) -> (ContinuousAt.{u2, u1} 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, u1} (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_3 a)) (fun (a : α) => Fin.insertNth.{u1} n π i (f a) (g a)) a))
+  forall {α : Type.{u2}} [_inst_1 : TopologicalSpace.{u2} 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.{u1}} [_inst_3 : forall (i : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))), TopologicalSpace.{u1} (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}, (ContinuousAt.{u2, u1} α (π i) _inst_1 (_inst_3 i) f 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) => 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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))}, (ContinuousAt.{u2, u1} 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, 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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, u1} (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.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_3 (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 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(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 a) -> (ContinuousAt.{u2, u1} 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, u1} (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_3 a)) (fun (a : α) => Fin.insertNth.{u1} n π i (f a) (g a)) a))
 Case conversion may be inaccurate. Consider using '#align continuous_at.fin_insert_nth ContinuousAt.fin_insertNthₓ'. -/
 theorem ContinuousAt.fin_insertNth {n} {π : Fin (n + 1) → Type _} [∀ i, TopologicalSpace (π i)]
     (i : Fin (n + 1)) {f : α → π i} {a : α} (hf : ContinuousAt f a)
@@ -2266,7 +2266,7 @@ theorem ContinuousAt.fin_insertNth {n} {π : Fin (n + 1) → Type _} [∀ i, Top
 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_3 : 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)}, (Continuous.{u1, u2} α (π i) _inst_1 (_inst_3 i) f) -> (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))}, (Continuous.{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 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0} Nat 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) -> (Continuous.{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_3 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))))
 but is expected to have type
-  forall {α : Type.{u2}} [_inst_1 : TopologicalSpace.{u2} 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.{u1}} [_inst_3 : forall (i : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))), TopologicalSpace.{u1} (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)}, (Continuous.{u2, u1} α (π i) _inst_1 (_inst_3 i) f) -> (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 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(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))}, (Continuous.{u2, u1} 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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x._@.Mathlib.Order.Hom.Basic._hyg.697))))) (Fin.succAbove n i) j)) (fun (a : Fin n) => _inst_3 (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) -> (Continuous.{u2, u1} 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, u1} (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_3 a)) (fun (a : α) => Fin.insertNth.{u1} n π i (f a) (g a))))
+  forall {α : Type.{u2}} [_inst_1 : TopologicalSpace.{u2} 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.{u1}} [_inst_3 : forall (i : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))), TopologicalSpace.{u1} (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)}, (Continuous.{u2, u1} α (π i) _inst_1 (_inst_3 i) f) -> (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 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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))}, (Continuous.{u2, u1} 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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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, u1} (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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x._@.Mathlib.Order.Hom.Basic._hyg.699))))) (Fin.succAbove n i) j)) (fun (a : Fin n) => _inst_3 (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 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Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) => _inst_3 a)) (fun (a : α) => Fin.insertNth.{u1} n π i (f a) (g a))))
 Case conversion may be inaccurate. Consider using '#align continuous.fin_insert_nth Continuous.fin_insertNthₓ'. -/
 theorem Continuous.fin_insertNth {n} {π : Fin (n + 1) → Type _} [∀ i, TopologicalSpace (π i)]
     (i : Fin (n + 1)) {f : α → π i} (hf : Continuous f) {g : α → ∀ j : Fin n, π (i.succAbove j)}

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

@@ -2238,7 +2238,7 @@ theorem continuous_mulSingle [∀ i, One (π i)] [DecidableEq ι] (i : ι) :
 
 /- warning: filter.tendsto.fin_insert_nth -> Filter.Tendsto.fin_insertNth is a dubious translation:
 lean 3 declaration is
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+  forall {β : Type.{u1}} {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_3 : 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 : β -> (π i)} {l : Filter.{u1} β} {x : π i}, (Filter.Tendsto.{u1, u2} β (π i) f l (nhds.{u2} (π i) (_inst_3 i) x)) -> (forall {g : β -> (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 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 but is expected to have type
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(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_3 a)) (Fin.insertNth.{u1} n (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) i x y))))
 Case conversion may be inaccurate. Consider using '#align filter.tendsto.fin_insert_nth Filter.Tendsto.fin_insertNthₓ'. -/
@@ -2251,7 +2251,7 @@ theorem Filter.Tendsto.fin_insertNth {n} {π : Fin (n + 1) → Type _} [∀ i, T
 
 /- warning: continuous_at.fin_insert_nth -> ContinuousAt.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_3 : 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}, (ContinuousAt.{u1, u2} α (π i) _inst_1 (_inst_3 i) f 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} 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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} 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} 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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 a) -> (ContinuousAt.{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_3 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)) 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_3 : 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}, (ContinuousAt.{u1, u2} α (π i) _inst_1 (_inst_3 i) f 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))}, (ContinuousAt.{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_3 (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 a) -> (ContinuousAt.{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_3 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)) a))
 but is expected to have type
   forall {α : Type.{u2}} [_inst_1 : TopologicalSpace.{u2} 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.{u1}} [_inst_3 : forall (i : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))), TopologicalSpace.{u1} (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}, (ContinuousAt.{u2, u1} α (π i) _inst_1 (_inst_3 i) f 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))}, (ContinuousAt.{u2, u1} 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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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, u1} (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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x._@.Mathlib.Order.Hom.Basic._hyg.697))))) (Fin.succAbove n i) j)) (fun (a : Fin n) => _inst_3 (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 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(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 a) -> (ContinuousAt.{u2, u1} 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, u1} (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_3 a)) (fun (a : α) => Fin.insertNth.{u1} n π i (f a) (g a)) a))
 Case conversion may be inaccurate. Consider using '#align continuous_at.fin_insert_nth ContinuousAt.fin_insertNthₓ'. -/
@@ -2264,7 +2264,7 @@ theorem ContinuousAt.fin_insertNth {n} {π : Fin (n + 1) → Type _} [∀ i, Top
 
 /- warning: continuous.fin_insert_nth -> Continuous.fin_insertNth is a dubious translation:
 lean 3 declaration is
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+  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_3 : 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)}, (Continuous.{u1, u2} α (π i) _inst_1 (_inst_3 i) f) -> (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 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0} Nat 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) -> (Continuous.{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 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 but is expected to have type
   forall {α : Type.{u2}} [_inst_1 : TopologicalSpace.{u2} 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.{u1}} [_inst_3 : forall (i : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))), TopologicalSpace.{u1} (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)}, (Continuous.{u2, u1} α (π i) _inst_1 (_inst_3 i) f) -> (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 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(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))}, (Continuous.{u2, u1} 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, u1} (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_3 (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) -> (Continuous.{u2, u1} 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, u1} (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_3 a)) (fun (a : α) => Fin.insertNth.{u1} n π i (f a) (g a))))
 Case conversion may be inaccurate. Consider using '#align continuous.fin_insert_nth Continuous.fin_insertNthₓ'. -/

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

@@ -435,10 +435,10 @@ instance : TopologicalSpace (CofiniteTopology α)
     rintro hs ht ⟨x, hxs, hxt⟩
     rw [compl_inter]
     exact (hs ⟨x, hxs⟩).union (ht ⟨x, hxt⟩)
-  isOpen_unionₛ := by
+  isOpen_sUnion := by
     rintro s h ⟨x, t, hts, hzt⟩
-    rw [Set.compl_unionₛ]
-    exact Set.Finite.interₛ (mem_image_of_mem _ hts) (h t hts ⟨x, hzt⟩)
+    rw [Set.compl_sUnion]
+    exact Set.Finite.sInter (mem_image_of_mem _ hts) (h t hts ⟨x, hzt⟩)
 
 /- warning: cofinite_topology.is_open_iff -> CofiniteTopology.isOpen_iff is a dubious translation:
 lean 3 declaration is
@@ -792,14 +792,14 @@ theorem continuous_inf_dom_right₂ {α β γ} {f : α → β → γ} {ta1 ta2 :
   exact @Continuous.comp _ _ _ (id _) (id _) _ _ _ h h_continuous_id
 #align continuous_inf_dom_right₂ continuous_inf_dom_right₂
 
-/- warning: continuous_Inf_dom₂ -> continuous_infₛ_dom₂ is a dubious translation:
+/- warning: continuous_Inf_dom₂ -> continuous_sInf_dom₂ is a dubious translation:
 lean 3 declaration is
-  forall {α : Type.{u1}} {β : Type.{u2}} {γ : Type.{u3}} {f : α -> β -> γ} {tas : Set.{u1} (TopologicalSpace.{u1} α)} {tbs : Set.{u2} (TopologicalSpace.{u2} β)} {ta : TopologicalSpace.{u1} α} {tb : TopologicalSpace.{u2} β} {tc : TopologicalSpace.{u3} γ}, (Membership.Mem.{u1, u1} (TopologicalSpace.{u1} α) (Set.{u1} (TopologicalSpace.{u1} α)) (Set.hasMem.{u1} (TopologicalSpace.{u1} α)) ta tas) -> (Membership.Mem.{u2, u2} (TopologicalSpace.{u2} β) (Set.{u2} (TopologicalSpace.{u2} β)) (Set.hasMem.{u2} (TopologicalSpace.{u2} β)) tb tbs) -> (Continuous.{max u1 u2, u3} (Prod.{u1, u2} α β) γ (Prod.topologicalSpace.{u1, u2} α β ta tb) tc (fun (p : Prod.{u1, u2} α β) => f (Prod.fst.{u1, u2} α β p) (Prod.snd.{u1, u2} α β p))) -> (Continuous.{max u1 u2, u3} (Prod.{u1, u2} α β) γ (Prod.topologicalSpace.{u1, u2} α β (InfSet.infₛ.{u1} (TopologicalSpace.{u1} α) (ConditionallyCompleteLattice.toHasInf.{u1} (TopologicalSpace.{u1} α) (CompleteLattice.toConditionallyCompleteLattice.{u1} (TopologicalSpace.{u1} α) (TopologicalSpace.completeLattice.{u1} α))) tas) (InfSet.infₛ.{u2} (TopologicalSpace.{u2} β) (ConditionallyCompleteLattice.toHasInf.{u2} (TopologicalSpace.{u2} β) (CompleteLattice.toConditionallyCompleteLattice.{u2} (TopologicalSpace.{u2} β) (TopologicalSpace.completeLattice.{u2} β))) tbs)) tc (fun (p : Prod.{u1, u2} α β) => f (Prod.fst.{u1, u2} α β p) (Prod.snd.{u1, u2} α β p)))
+  forall {α : Type.{u1}} {β : Type.{u2}} {γ : Type.{u3}} {f : α -> β -> γ} {tas : Set.{u1} (TopologicalSpace.{u1} α)} {tbs : Set.{u2} (TopologicalSpace.{u2} β)} {ta : TopologicalSpace.{u1} α} {tb : TopologicalSpace.{u2} β} {tc : TopologicalSpace.{u3} γ}, (Membership.Mem.{u1, u1} (TopologicalSpace.{u1} α) (Set.{u1} (TopologicalSpace.{u1} α)) (Set.hasMem.{u1} (TopologicalSpace.{u1} α)) ta tas) -> (Membership.Mem.{u2, u2} (TopologicalSpace.{u2} β) (Set.{u2} (TopologicalSpace.{u2} β)) (Set.hasMem.{u2} (TopologicalSpace.{u2} β)) tb tbs) -> (Continuous.{max u1 u2, u3} (Prod.{u1, u2} α β) γ (Prod.topologicalSpace.{u1, u2} α β ta tb) tc (fun (p : Prod.{u1, u2} α β) => f (Prod.fst.{u1, u2} α β p) (Prod.snd.{u1, u2} α β p))) -> (Continuous.{max u1 u2, u3} (Prod.{u1, u2} α β) γ (Prod.topologicalSpace.{u1, u2} α β (InfSet.sInf.{u1} (TopologicalSpace.{u1} α) (ConditionallyCompleteLattice.toHasInf.{u1} (TopologicalSpace.{u1} α) (CompleteLattice.toConditionallyCompleteLattice.{u1} (TopologicalSpace.{u1} α) (TopologicalSpace.completeLattice.{u1} α))) tas) (InfSet.sInf.{u2} (TopologicalSpace.{u2} β) (ConditionallyCompleteLattice.toHasInf.{u2} (TopologicalSpace.{u2} β) (CompleteLattice.toConditionallyCompleteLattice.{u2} (TopologicalSpace.{u2} β) (TopologicalSpace.completeLattice.{u2} β))) tbs)) tc (fun (p : Prod.{u1, u2} α β) => f (Prod.fst.{u1, u2} α β p) (Prod.snd.{u1, u2} α β p)))
 but is expected to have type
-  forall {α : Type.{u3}} {β : Type.{u2}} {γ : Type.{u1}} {f : α -> β -> γ} {tas : Set.{u3} (TopologicalSpace.{u3} α)} {tbs : Set.{u2} (TopologicalSpace.{u2} β)} {ta : TopologicalSpace.{u3} α} {tb : TopologicalSpace.{u2} β} {tc : TopologicalSpace.{u1} γ}, (Membership.mem.{u3, u3} (TopologicalSpace.{u3} α) (Set.{u3} (TopologicalSpace.{u3} α)) (Set.instMembershipSet.{u3} (TopologicalSpace.{u3} α)) ta tas) -> (Membership.mem.{u2, u2} (TopologicalSpace.{u2} β) (Set.{u2} (TopologicalSpace.{u2} β)) (Set.instMembershipSet.{u2} (TopologicalSpace.{u2} β)) tb tbs) -> (Continuous.{max u3 u2, u1} (Prod.{u3, u2} α β) γ (instTopologicalSpaceProd.{u3, u2} α β ta tb) tc (fun (p : Prod.{u3, u2} α β) => f (Prod.fst.{u3, u2} α β p) (Prod.snd.{u3, u2} α β p))) -> (Continuous.{max u3 u2, u1} (Prod.{u3, u2} α β) γ (instTopologicalSpaceProd.{u3, u2} α β (InfSet.infₛ.{u3} (TopologicalSpace.{u3} α) (ConditionallyCompleteLattice.toInfSet.{u3} (TopologicalSpace.{u3} α) (CompleteLattice.toConditionallyCompleteLattice.{u3} (TopologicalSpace.{u3} α) (TopologicalSpace.instCompleteLatticeTopologicalSpace.{u3} α))) tas) (InfSet.infₛ.{u2} (TopologicalSpace.{u2} β) (ConditionallyCompleteLattice.toInfSet.{u2} (TopologicalSpace.{u2} β) (CompleteLattice.toConditionallyCompleteLattice.{u2} (TopologicalSpace.{u2} β) (TopologicalSpace.instCompleteLatticeTopologicalSpace.{u2} β))) tbs)) tc (fun (p : Prod.{u3, u2} α β) => f (Prod.fst.{u3, u2} α β p) (Prod.snd.{u3, u2} α β p)))
-Case conversion may be inaccurate. Consider using '#align continuous_Inf_dom₂ continuous_infₛ_dom₂ₓ'. -/
+  forall {α : Type.{u3}} {β : Type.{u2}} {γ : Type.{u1}} {f : α -> β -> γ} {tas : Set.{u3} (TopologicalSpace.{u3} α)} {tbs : Set.{u2} (TopologicalSpace.{u2} β)} {ta : TopologicalSpace.{u3} α} {tb : TopologicalSpace.{u2} β} {tc : TopologicalSpace.{u1} γ}, (Membership.mem.{u3, u3} (TopologicalSpace.{u3} α) (Set.{u3} (TopologicalSpace.{u3} α)) (Set.instMembershipSet.{u3} (TopologicalSpace.{u3} α)) ta tas) -> (Membership.mem.{u2, u2} (TopologicalSpace.{u2} β) (Set.{u2} (TopologicalSpace.{u2} β)) (Set.instMembershipSet.{u2} (TopologicalSpace.{u2} β)) tb tbs) -> (Continuous.{max u3 u2, u1} (Prod.{u3, u2} α β) γ (instTopologicalSpaceProd.{u3, u2} α β ta tb) tc (fun (p : Prod.{u3, u2} α β) => f (Prod.fst.{u3, u2} α β p) (Prod.snd.{u3, u2} α β p))) -> (Continuous.{max u3 u2, u1} (Prod.{u3, u2} α β) γ (instTopologicalSpaceProd.{u3, u2} α β (InfSet.sInf.{u3} (TopologicalSpace.{u3} α) (ConditionallyCompleteLattice.toInfSet.{u3} (TopologicalSpace.{u3} α) (CompleteLattice.toConditionallyCompleteLattice.{u3} (TopologicalSpace.{u3} α) (TopologicalSpace.instCompleteLatticeTopologicalSpace.{u3} α))) tas) (InfSet.sInf.{u2} (TopologicalSpace.{u2} β) (ConditionallyCompleteLattice.toInfSet.{u2} (TopologicalSpace.{u2} β) (CompleteLattice.toConditionallyCompleteLattice.{u2} (TopologicalSpace.{u2} β) (TopologicalSpace.instCompleteLatticeTopologicalSpace.{u2} β))) tbs)) tc (fun (p : Prod.{u3, u2} α β) => f (Prod.fst.{u3, u2} α β p) (Prod.snd.{u3, u2} α β p)))
+Case conversion may be inaccurate. Consider using '#align continuous_Inf_dom₂ continuous_sInf_dom₂ₓ'. -/
 /-- A version of `continuous_Inf_dom` for binary functions -/
-theorem continuous_infₛ_dom₂ {α β γ} {f : α → β → γ} {tas : Set (TopologicalSpace α)}
+theorem continuous_sInf_dom₂ {α β γ} {f : α → β → γ} {tas : Set (TopologicalSpace α)}
     {tbs : Set (TopologicalSpace β)} {ta : TopologicalSpace α} {tb : TopologicalSpace β}
     {tc : TopologicalSpace γ} (ha : ta ∈ tas) (hb : tb ∈ tbs)
     (hf : Continuous fun p : α × β => f p.1 p.2) : by
@@ -807,11 +807,11 @@ theorem continuous_infₛ_dom₂ {α β γ} {f : α → β → γ} {tas : Set (T
       exact @Continuous _ _ _ tc fun p : α × β => f p.1 p.2 :=
   by
   let t : TopologicalSpace (α × β) := Prod.topologicalSpace
-  have ha := continuous_infₛ_dom ha continuous_id
-  have hb := continuous_infₛ_dom hb continuous_id
+  have ha := continuous_sInf_dom ha continuous_id
+  have hb := continuous_sInf_dom hb continuous_id
   have h_continuous_id := @Continuous.prod_map _ _ _ _ ta tb (Inf tas) (Inf tbs) _ _ ha hb
   exact @Continuous.comp _ _ _ (id _) (id _) _ _ _ hf h_continuous_id
-#align continuous_Inf_dom₂ continuous_infₛ_dom₂
+#align continuous_Inf_dom₂ continuous_sInf_dom₂
 
 /- warning: filter.eventually.prod_inl_nhds -> Filter.Eventually.prod_inl_nhds is a dubious translation:
 lean 3 declaration is
@@ -1112,9 +1112,9 @@ theorem ContinuousAt.prod_map' {f : α → γ} {g : β → δ} {x : α} {y : β}
 
 /- warning: prod_generate_from_generate_from_eq -> prod_generateFrom_generateFrom_eq is a dubious translation:
 lean 3 declaration is
-  forall {α : Type.{u1}} {β : Type.{u2}} {s : Set.{u1} (Set.{u1} α)} {t : Set.{u2} (Set.{u2} β)}, (Eq.{succ u1} (Set.{u1} α) (Set.unionₛ.{u1} α s) (Set.univ.{u1} α)) -> (Eq.{succ u2} (Set.{u2} β) (Set.unionₛ.{u2} β t) (Set.univ.{u2} β)) -> (Eq.{succ (max u1 u2)} (TopologicalSpace.{max u1 u2} (Prod.{u1, u2} α β)) (Prod.topologicalSpace.{u1, u2} α β (TopologicalSpace.generateFrom.{u1} α s) (TopologicalSpace.generateFrom.{u2} β t)) (TopologicalSpace.generateFrom.{max u1 u2} (Prod.{u1, u2} α β) (setOf.{max u1 u2} (Set.{max u1 u2} (Prod.{u1, u2} α β)) (fun (g : Set.{max u1 u2} (Prod.{u1, u2} α β)) => Exists.{succ u1} (Set.{u1} α) (fun (u : Set.{u1} α) => Exists.{0} (Membership.Mem.{u1, u1} (Set.{u1} α) (Set.{u1} (Set.{u1} α)) (Set.hasMem.{u1} (Set.{u1} α)) u s) (fun (H : Membership.Mem.{u1, u1} (Set.{u1} α) (Set.{u1} (Set.{u1} α)) (Set.hasMem.{u1} (Set.{u1} α)) u s) => Exists.{succ u2} (Set.{u2} β) (fun (v : Set.{u2} β) => Exists.{0} (Membership.Mem.{u2, u2} (Set.{u2} β) (Set.{u2} (Set.{u2} β)) (Set.hasMem.{u2} (Set.{u2} β)) v t) (fun (H : Membership.Mem.{u2, u2} (Set.{u2} β) (Set.{u2} (Set.{u2} β)) (Set.hasMem.{u2} (Set.{u2} β)) v t) => Eq.{succ (max u1 u2)} (Set.{max u1 u2} (Prod.{u1, u2} α β)) g (Set.prod.{u1, u2} α β u v)))))))))
+  forall {α : Type.{u1}} {β : Type.{u2}} {s : Set.{u1} (Set.{u1} α)} {t : Set.{u2} (Set.{u2} β)}, (Eq.{succ u1} (Set.{u1} α) (Set.sUnion.{u1} α s) (Set.univ.{u1} α)) -> (Eq.{succ u2} (Set.{u2} β) (Set.sUnion.{u2} β t) (Set.univ.{u2} β)) -> (Eq.{succ (max u1 u2)} (TopologicalSpace.{max u1 u2} (Prod.{u1, u2} α β)) (Prod.topologicalSpace.{u1, u2} α β (TopologicalSpace.generateFrom.{u1} α s) (TopologicalSpace.generateFrom.{u2} β t)) (TopologicalSpace.generateFrom.{max u1 u2} (Prod.{u1, u2} α β) (setOf.{max u1 u2} (Set.{max u1 u2} (Prod.{u1, u2} α β)) (fun (g : Set.{max u1 u2} (Prod.{u1, u2} α β)) => Exists.{succ u1} (Set.{u1} α) (fun (u : Set.{u1} α) => Exists.{0} (Membership.Mem.{u1, u1} (Set.{u1} α) (Set.{u1} (Set.{u1} α)) (Set.hasMem.{u1} (Set.{u1} α)) u s) (fun (H : Membership.Mem.{u1, u1} (Set.{u1} α) (Set.{u1} (Set.{u1} α)) (Set.hasMem.{u1} (Set.{u1} α)) u s) => Exists.{succ u2} (Set.{u2} β) (fun (v : Set.{u2} β) => Exists.{0} (Membership.Mem.{u2, u2} (Set.{u2} β) (Set.{u2} (Set.{u2} β)) (Set.hasMem.{u2} (Set.{u2} β)) v t) (fun (H : Membership.Mem.{u2, u2} (Set.{u2} β) (Set.{u2} (Set.{u2} β)) (Set.hasMem.{u2} (Set.{u2} β)) v t) => Eq.{succ (max u1 u2)} (Set.{max u1 u2} (Prod.{u1, u2} α β)) g (Set.prod.{u1, u2} α β u v)))))))))
 but is expected to have type
-  forall {α : Type.{u2}} {β : Type.{u1}} {s : Set.{u2} (Set.{u2} α)} {t : Set.{u1} (Set.{u1} β)}, (Eq.{succ u2} (Set.{u2} α) (Set.unionₛ.{u2} α s) (Set.univ.{u2} α)) -> (Eq.{succ u1} (Set.{u1} β) (Set.unionₛ.{u1} β t) (Set.univ.{u1} β)) -> (Eq.{max (succ u2) (succ u1)} (TopologicalSpace.{max u1 u2} (Prod.{u2, u1} α β)) (instTopologicalSpaceProd.{u2, u1} α β (TopologicalSpace.generateFrom.{u2} α s) (TopologicalSpace.generateFrom.{u1} β t)) (TopologicalSpace.generateFrom.{max u2 u1} (Prod.{u2, u1} α β) (setOf.{max u2 u1} (Set.{max u2 u1} (Prod.{u2, u1} α β)) (fun (g : Set.{max u2 u1} (Prod.{u2, u1} α β)) => Exists.{succ u2} (Set.{u2} α) (fun (u : Set.{u2} α) => And (Membership.mem.{u2, u2} (Set.{u2} α) (Set.{u2} (Set.{u2} α)) (Set.instMembershipSet.{u2} (Set.{u2} α)) u s) (Exists.{succ u1} (Set.{u1} β) (fun (v : Set.{u1} β) => And (Membership.mem.{u1, u1} (Set.{u1} β) (Set.{u1} (Set.{u1} β)) (Set.instMembershipSet.{u1} (Set.{u1} β)) v t) (Eq.{max (succ u2) (succ u1)} (Set.{max u2 u1} (Prod.{u2, u1} α β)) g (Set.prod.{u2, u1} α β u v)))))))))
+  forall {α : Type.{u2}} {β : Type.{u1}} {s : Set.{u2} (Set.{u2} α)} {t : Set.{u1} (Set.{u1} β)}, (Eq.{succ u2} (Set.{u2} α) (Set.sUnion.{u2} α s) (Set.univ.{u2} α)) -> (Eq.{succ u1} (Set.{u1} β) (Set.sUnion.{u1} β t) (Set.univ.{u1} β)) -> (Eq.{max (succ u2) (succ u1)} (TopologicalSpace.{max u1 u2} (Prod.{u2, u1} α β)) (instTopologicalSpaceProd.{u2, u1} α β (TopologicalSpace.generateFrom.{u2} α s) (TopologicalSpace.generateFrom.{u1} β t)) (TopologicalSpace.generateFrom.{max u2 u1} (Prod.{u2, u1} α β) (setOf.{max u2 u1} (Set.{max u2 u1} (Prod.{u2, u1} α β)) (fun (g : Set.{max u2 u1} (Prod.{u2, u1} α β)) => Exists.{succ u2} (Set.{u2} α) (fun (u : Set.{u2} α) => And (Membership.mem.{u2, u2} (Set.{u2} α) (Set.{u2} (Set.{u2} α)) (Set.instMembershipSet.{u2} (Set.{u2} α)) u s) (Exists.{succ u1} (Set.{u1} β) (fun (v : Set.{u1} β) => And (Membership.mem.{u1, u1} (Set.{u1} β) (Set.{u1} (Set.{u1} β)) (Set.instMembershipSet.{u1} (Set.{u1} β)) v t) (Eq.{max (succ u2) (succ u1)} (Set.{max u2 u1} (Prod.{u2, u1} α β)) g (Set.prod.{u2, u1} α β u v)))))))))
 Case conversion may be inaccurate. Consider using '#align prod_generate_from_generate_from_eq prod_generateFrom_generateFrom_eqₓ'. -/
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
@@ -1138,8 +1138,8 @@ theorem prod_generateFrom_generateFrom_eq {α β : Type _} {s : Set (Set α)} {t
           show G.IsOpen (Prod.fst ⁻¹' u) by
             rw [← this]
             exact
-              isOpen_unionᵢ fun v =>
-                isOpen_unionᵢ fun hv => generate_open.basic _ ⟨_, hu, _, hv, rfl⟩)
+              isOpen_iUnion fun v =>
+                isOpen_iUnion fun hv => generate_open.basic _ ⟨_, hu, _, hv, rfl⟩)
       (coinduced_le_iff_le_induced.mp <|
         le_generateFrom fun v hv =>
           have : (⋃ u ∈ s, u ×ˢ v) = Prod.snd ⁻¹' v := by
@@ -1147,8 +1147,8 @@ theorem prod_generateFrom_generateFrom_eq {α β : Type _} {s : Set (Set α)} {t
           show G.IsOpen (Prod.snd ⁻¹' v) by
             rw [← this]
             exact
-              isOpen_unionᵢ fun u =>
-                isOpen_unionᵢ fun hu => generate_open.basic _ ⟨_, hu, _, hv, rfl⟩))
+              isOpen_iUnion fun u =>
+                isOpen_iUnion fun hu => generate_open.basic _ ⟨_, hu, _, hv, rfl⟩))
 #align prod_generate_from_generate_from_eq prod_generateFrom_generateFrom_eq
 
 /- warning: prod_eq_generate_from -> prod_eq_generateFrom is a dubious translation:
@@ -2089,7 +2089,7 @@ but is expected to have type
   forall {α : Type.{u3}} {ι : Type.{u2}} {π : ι -> Type.{u1}} [_inst_1 : TopologicalSpace.{u3} α] [_inst_2 : forall (i : ι), TopologicalSpace.{u1} (π i)] {f : α -> (forall (i : ι), π i)}, Iff (Continuous.{u3, max u2 u1} α (forall (i : ι), π i) _inst_1 (Pi.topologicalSpace.{u2, u1} ι (fun (i : ι) => π i) (fun (a : ι) => _inst_2 a)) f) (forall (i : ι), Continuous.{u3, u1} α (π i) _inst_1 (_inst_2 i) (fun (a : α) => f a i))
 Case conversion may be inaccurate. Consider using '#align continuous_pi_iff continuous_pi_iffₓ'. -/
 theorem continuous_pi_iff : Continuous f ↔ ∀ i, Continuous fun a => f a i := by
-  simp only [continuous_infᵢ_rng, continuous_induced_rng]
+  simp only [continuous_iInf_rng, continuous_induced_rng]
 #align continuous_pi_iff continuous_pi_iff
 
 /- warning: continuous_pi -> continuous_pi is a dubious translation:
@@ -2111,7 +2111,7 @@ but is expected to have type
 Case conversion may be inaccurate. Consider using '#align continuous_apply continuous_applyₓ'. -/
 @[continuity]
 theorem continuous_apply (i : ι) : Continuous fun p : ∀ i, π i => p i :=
-  continuous_infᵢ_dom continuous_induced_dom
+  continuous_iInf_dom continuous_induced_dom
 #align continuous_apply continuous_apply
 
 /- warning: continuous_apply_apply -> continuous_apply_apply is a dubious translation:
@@ -2154,7 +2154,7 @@ but is expected to have type
   forall {ι : Type.{u2}} {π : ι -> Type.{u1}} [_inst_2 : forall (i : ι), TopologicalSpace.{u1} (π i)] {a : forall (i : ι), π i}, Eq.{max (succ u2) (succ u1)} (Filter.{max u2 u1} (forall (i : ι), π i)) (nhds.{max u2 u1} (forall (i : ι), π i) (Pi.topologicalSpace.{u2, u1} ι (fun (i : ι) => π i) (fun (a : ι) => _inst_2 a)) a) (Filter.pi.{u2, u1} ι (fun (i : ι) => π i) (fun (i : ι) => nhds.{u1} (π i) (_inst_2 i) (a i)))
 Case conversion may be inaccurate. Consider using '#align nhds_pi nhds_piₓ'. -/
 theorem nhds_pi {a : ∀ i, π i} : 𝓝 a = pi fun i => 𝓝 (a i) := by
-  simp only [nhds_infᵢ, nhds_induced, Filter.pi]
+  simp only [nhds_iInf, nhds_induced, Filter.pi]
 #align nhds_pi nhds_pi
 
 /- warning: tendsto_pi_nhds -> tendsto_pi_nhds is a dubious translation:
@@ -2282,7 +2282,7 @@ but is expected to have type
 Case conversion may be inaccurate. Consider using '#align is_open_set_pi isOpen_set_piₓ'. -/
 theorem isOpen_set_pi {i : Set ι} {s : ∀ a, Set (π a)} (hi : i.Finite)
     (hs : ∀ a ∈ i, IsOpen (s a)) : IsOpen (pi i s) := by
-  rw [pi_def] <;> exact isOpen_binterᵢ hi fun a ha => (hs _ ha).Preimage (continuous_apply _)
+  rw [pi_def] <;> exact isOpen_biInter hi fun a ha => (hs _ ha).Preimage (continuous_apply _)
 #align is_open_set_pi isOpen_set_pi
 
 /- warning: is_open_pi_iff -> isOpen_pi_iff is a dubious translation:
@@ -2358,7 +2358,7 @@ Case conversion may be inaccurate. Consider using '#align is_closed_set_pi isClo
 theorem isClosed_set_pi {i : Set ι} {s : ∀ a, Set (π a)} (hs : ∀ a ∈ i, IsClosed (s a)) :
     IsClosed (pi i s) := by
   rw [pi_def] <;>
-    exact isClosed_interᵢ fun a => isClosed_interᵢ fun ha => (hs _ ha).Preimage (continuous_apply _)
+    exact isClosed_iInter fun a => isClosed_iInter fun ha => (hs _ ha).Preimage (continuous_apply _)
 #align is_closed_set_pi isClosed_set_pi
 
 /- warning: mem_nhds_of_pi_mem_nhds -> mem_nhds_of_pi_mem_nhds is a dubious translation:
@@ -2435,7 +2435,7 @@ theorem pi_eq_generateFrom :
         { g | ∃ (s : ∀ a, Set (π a))(i : Finset ι), (∀ a ∈ i, IsOpen (s a)) ∧ g = pi (↑i) s } :=
   le_antisymm
     (le_generateFrom fun g ⟨s, i, hi, Eq⟩ => Eq.symm ▸ isOpen_set_pi (Finset.finite_toSet _) hi)
-    (le_infᵢ fun a s ⟨t, ht, s_eq⟩ =>
+    (le_iInf fun a s ⟨t, ht, s_eq⟩ =>
       GenerateOpen.basic _ <|
         ⟨update (fun a => univ) a t, {a}, by simpa using ht, s_eq ▸ by ext f <;> simp [Set.pi]⟩)
 #align pi_eq_generate_from pi_eq_generateFrom
@@ -2453,7 +2453,7 @@ theorem pi_generateFrom_eq {π : ι → Type _} {g : ∀ a, Set (Set (π a))} :
   exact fun s ⟨t, i, ht, Eq⟩ => ⟨t, i, fun a ha => generate_open.basic _ (ht a ha), Eq⟩
   · rintro s ⟨t, i, hi, rfl⟩
     rw [pi_def]
-    apply isOpen_binterᵢ (Finset.finite_toSet _)
+    apply isOpen_biInter (Finset.finite_toSet _)
     intro a ha
     show ((generate_from G).coinduced fun f : ∀ a, π a => f a).IsOpen (t a)
     refine' le_generateFrom _ _ (hi a ha)
@@ -2491,26 +2491,26 @@ theorem pi_generateFrom_eq_finite {π : ι → Type _} {g : ∀ a, Set (Set (π
 #align pi_generate_from_eq_finite pi_generateFrom_eq_finite
 -/
 
-/- warning: inducing_infi_to_pi -> inducing_infᵢ_to_pi is a dubious translation:
+/- warning: inducing_infi_to_pi -> inducing_iInf_to_pi is a dubious translation:
 lean 3 declaration is
-  forall {ι : Type.{u1}} {π : ι -> Type.{u2}} [_inst_2 : forall (i : ι), TopologicalSpace.{u2} (π i)] {X : Type.{u3}} (f : forall (i : ι), X -> (π i)), Inducing.{u3, max u1 u2} X (forall (i : ι), π i) (infᵢ.{u3, succ u1} (TopologicalSpace.{u3} X) (ConditionallyCompleteLattice.toHasInf.{u3} (TopologicalSpace.{u3} X) (CompleteLattice.toConditionallyCompleteLattice.{u3} (TopologicalSpace.{u3} X) (TopologicalSpace.completeLattice.{u3} X))) ι (fun (i : ι) => TopologicalSpace.induced.{u3, u2} X (π i) (f i) (inferInstance.{succ u2} (TopologicalSpace.{u2} (π i)) (_inst_2 i)))) (Pi.topologicalSpace.{u1, u2} ι (fun (i : ι) => π i) (fun (a : ι) => _inst_2 a)) (fun (x : X) (i : ι) => f i x)
+  forall {ι : Type.{u1}} {π : ι -> Type.{u2}} [_inst_2 : forall (i : ι), TopologicalSpace.{u2} (π i)] {X : Type.{u3}} (f : forall (i : ι), X -> (π i)), Inducing.{u3, max u1 u2} X (forall (i : ι), π i) (iInf.{u3, succ u1} (TopologicalSpace.{u3} X) (ConditionallyCompleteLattice.toHasInf.{u3} (TopologicalSpace.{u3} X) (CompleteLattice.toConditionallyCompleteLattice.{u3} (TopologicalSpace.{u3} X) (TopologicalSpace.completeLattice.{u3} X))) ι (fun (i : ι) => TopologicalSpace.induced.{u3, u2} X (π i) (f i) (inferInstance.{succ u2} (TopologicalSpace.{u2} (π i)) (_inst_2 i)))) (Pi.topologicalSpace.{u1, u2} ι (fun (i : ι) => π i) (fun (a : ι) => _inst_2 a)) (fun (x : X) (i : ι) => f i x)
 but is expected to have type
-  forall {ι : Type.{u2}} {π : ι -> Type.{u1}} [_inst_2 : forall (i : ι), TopologicalSpace.{u1} (π i)] {X : Type.{u3}} (f : forall (i : ι), X -> (π i)), Inducing.{u3, max u2 u1} X (forall (i : ι), π i) (infᵢ.{u3, succ u2} (TopologicalSpace.{u3} X) (ConditionallyCompleteLattice.toInfSet.{u3} (TopologicalSpace.{u3} X) (CompleteLattice.toConditionallyCompleteLattice.{u3} (TopologicalSpace.{u3} X) (TopologicalSpace.instCompleteLatticeTopologicalSpace.{u3} X))) ι (fun (i : ι) => TopologicalSpace.induced.{u3, u1} X (π i) (f i) (inferInstance.{succ u1} (TopologicalSpace.{u1} (π i)) (_inst_2 i)))) (Pi.topologicalSpace.{u2, u1} ι (fun (i : ι) => π i) (fun (a : ι) => _inst_2 a)) (fun (x : X) (i : ι) => f i x)
-Case conversion may be inaccurate. Consider using '#align inducing_infi_to_pi inducing_infᵢ_to_piₓ'. -/
+  forall {ι : Type.{u2}} {π : ι -> Type.{u1}} [_inst_2 : forall (i : ι), TopologicalSpace.{u1} (π i)] {X : Type.{u3}} (f : forall (i : ι), X -> (π i)), Inducing.{u3, max u2 u1} X (forall (i : ι), π i) (iInf.{u3, succ u2} (TopologicalSpace.{u3} X) (ConditionallyCompleteLattice.toInfSet.{u3} (TopologicalSpace.{u3} X) (CompleteLattice.toConditionallyCompleteLattice.{u3} (TopologicalSpace.{u3} X) (TopologicalSpace.instCompleteLatticeTopologicalSpace.{u3} X))) ι (fun (i : ι) => TopologicalSpace.induced.{u3, u1} X (π i) (f i) (inferInstance.{succ u1} (TopologicalSpace.{u1} (π i)) (_inst_2 i)))) (Pi.topologicalSpace.{u2, u1} ι (fun (i : ι) => π i) (fun (a : ι) => _inst_2 a)) (fun (x : X) (i : ι) => f i x)
+Case conversion may be inaccurate. Consider using '#align inducing_infi_to_pi inducing_iInf_to_piₓ'. -/
 /-- Suppose `π i` is a family of topological spaces indexed by `i : ι`, and `X` is a type
 endowed with a family of maps `f i : X → π i` for every `i : ι`, hence inducing a
 map `g : X → Π i, π i`. This lemma shows that infimum of the topologies on `X` induced by
 the `f i` as `i : ι` varies is simply the topology on `X` induced by `g : X → Π i, π i`
 where `Π i, π i` is endowed with the usual product topology. -/
-theorem inducing_infᵢ_to_pi {X : Type _} (f : ∀ i, X → π i) :
+theorem inducing_iInf_to_pi {X : Type _} (f : ∀ i, X → π i) :
     @Inducing X (∀ i, π i) (⨅ i, induced (f i) inferInstance) _ fun x i => f i x :=
   by
   constructor
-  erw [induced_infᵢ]
+  erw [induced_iInf]
   congr 1
   funext
   erw [induced_compose]
-#align inducing_infi_to_pi inducing_infᵢ_to_pi
+#align inducing_infi_to_pi inducing_iInf_to_pi
 
 variable [Finite ι] [∀ i, DiscreteTopology (π i)]
 
@@ -2520,8 +2520,8 @@ instance Pi.discreteTopology : DiscreteTopology (∀ i, π i) :=
   singletons_open_iff_discrete.mp fun x =>
     by
     rw [show {x} = ⋂ i, { y : ∀ i, π i | y i = x i } by ext;
-        simp only [funext_iff, Set.mem_singleton_iff, Set.mem_interᵢ, Set.mem_setOf_eq]]
-    exact isOpen_interᵢ fun i => (continuous_apply i).isOpen_preimage {x i} (isOpen_discrete {x i})
+        simp only [funext_iff, Set.mem_singleton_iff, Set.mem_iInter, Set.mem_setOf_eq]]
+    exact isOpen_iInter fun i => (continuous_apply i).isOpen_preimage {x i} (isOpen_discrete {x i})
 #align Pi.discrete_topology Pi.discreteTopology
 -/
 
@@ -2540,7 +2540,7 @@ but is expected to have type
 Case conversion may be inaccurate. Consider using '#align continuous_sigma_mk continuous_sigmaMkₓ'. -/
 @[continuity]
 theorem continuous_sigmaMk {i : ι} : Continuous (@Sigma.mk ι σ i) :=
-  continuous_supᵢ_rng continuous_coinduced_rng
+  continuous_iSup_rng continuous_coinduced_rng
 #align continuous_sigma_mk continuous_sigmaMk
 
 /- warning: is_open_sigma_iff -> isOpen_sigma_iff is a dubious translation:
@@ -2550,7 +2550,7 @@ but is expected to have type
   forall {ι : Type.{u1}} {σ : ι -> Type.{u2}} [_inst_1 : forall (i : ι), TopologicalSpace.{u2} (σ i)] {s : Set.{max u2 u1} (Sigma.{u1, u2} ι σ)}, Iff (IsOpen.{max u1 u2} (Sigma.{u1, u2} ι σ) (instTopologicalSpaceSigma.{u1, u2} ι σ (fun (a : ι) => _inst_1 a)) s) (forall (i : ι), IsOpen.{u2} (σ i) (_inst_1 i) (Set.preimage.{u2, max u2 u1} (σ i) (Sigma.{u1, u2} ι σ) (Sigma.mk.{u1, u2} ι σ i) s))
 Case conversion may be inaccurate. Consider using '#align is_open_sigma_iff isOpen_sigma_iffₓ'. -/
 theorem isOpen_sigma_iff {s : Set (Sigma σ)} : IsOpen s ↔ ∀ i, IsOpen (Sigma.mk i ⁻¹' s) := by
-  simp only [isOpen_supᵢ_iff, isOpen_coinduced]
+  simp only [isOpen_iSup_iff, isOpen_coinduced]
 #align is_open_sigma_iff isOpen_sigma_iff
 
 /- warning: is_closed_sigma_iff -> isClosed_sigma_iff is a dubious translation:
@@ -2690,7 +2690,7 @@ theorem isOpen_sigma_fst_preimage (s : Set ι) : IsOpen (Sigma.fst ⁻¹' s : Se
   by
   rw [← bUnion_of_singleton s, preimage_Union₂]
   simp only [← range_sigma_mk]
-  exact isOpen_bunionᵢ fun _ _ => isOpen_range_sigmaMk
+  exact isOpen_biUnion fun _ _ => isOpen_range_sigmaMk
 #align is_open_sigma_fst_preimage isOpen_sigma_fst_preimage
 
 /- warning: continuous_sigma_iff -> continuous_sigma_iff is a dubious translation:
@@ -2702,7 +2702,7 @@ Case conversion may be inaccurate. Consider using '#align continuous_sigma_iff c
 /-- A map out of a sum type is continuous iff its restriction to each summand is. -/
 @[simp]
 theorem continuous_sigma_iff {f : Sigma σ → α} : Continuous f ↔ ∀ i, Continuous fun a => f ⟨i, a⟩ :=
-  by simp only [continuous_supᵢ_dom, continuous_coinduced_dom]
+  by simp only [continuous_iSup_dom, continuous_coinduced_dom]
 #align continuous_sigma_iff continuous_sigma_iff
 
 /- warning: continuous_sigma -> continuous_sigma is a dubious translation:

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

@@ -2240,7 +2240,7 @@ theorem continuous_mulSingle [∀ i, One (π i)] [DecidableEq ι] (i : ι) :
 lean 3 declaration is
   forall {β : Type.{u1}} {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_3 : 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 : β -> (π i)} {l : Filter.{u1} β} {x : π i}, (Filter.Tendsto.{u1, u2} β (π i) f l (nhds.{u2} (π i) (_inst_3 i) x)) -> (forall {g : β -> (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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=> _inst_3 a)) (Fin.insertNth.{u2} n (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) i x y))))
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(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))) y)) -> (Filter.Tendsto.{u2, u1} β (forall (j : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat 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(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)) g l (nhds.{u1} (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) 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(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)) (Pi.topologicalSpace.{0, u1} (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} 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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_3 (FunLike.coe.{1, 1, 1} 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(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} 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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))) y)) -> (Filter.Tendsto.{u2, u1} β (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) (fun (a : β) => Fin.insertNth.{u1} n π i (f a) (g a)) l (nhds.{u1} (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) (Pi.topologicalSpace.{0, u1} (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_3 a)) (Fin.insertNth.{u1} n (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) i x y))))
 Case conversion may be inaccurate. Consider using '#align filter.tendsto.fin_insert_nth Filter.Tendsto.fin_insertNthₓ'. -/
 theorem Filter.Tendsto.fin_insertNth {n} {π : Fin (n + 1) → Type _} [∀ i, TopologicalSpace (π i)]
     (i : Fin (n + 1)) {f : β → π i} {l : Filter β} {x : π i} (hf : Tendsto f l (𝓝 x))
@@ -2253,7 +2253,7 @@ theorem Filter.Tendsto.fin_insertNth {n} {π : Fin (n + 1) → Type _} [∀ i, T
 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_3 : 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}, (ContinuousAt.{u1, u2} α (π i) _inst_1 (_inst_3 i) f 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 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 but is expected to have type
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(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, u1} (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 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(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 a) -> (ContinuousAt.{u2, u1} 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, u1} (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_3 a)) (fun (a : α) => Fin.insertNth.{u1} n π i (f a) (g a)) a))
+  forall {α : Type.{u2}} [_inst_1 : TopologicalSpace.{u2} 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.{u1}} [_inst_3 : forall (i : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))), TopologicalSpace.{u1} (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}, (ContinuousAt.{u2, u1} α (π i) _inst_1 (_inst_3 i) f 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) => 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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, u1} (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_3 (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 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(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 a) -> (ContinuousAt.{u2, u1} 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, u1} (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_3 a)) (fun (a : α) => Fin.insertNth.{u1} n π i (f a) (g a)) a))
 Case conversion may be inaccurate. Consider using '#align continuous_at.fin_insert_nth ContinuousAt.fin_insertNthₓ'. -/
 theorem ContinuousAt.fin_insertNth {n} {π : Fin (n + 1) → Type _} [∀ i, TopologicalSpace (π i)]
     (i : Fin (n + 1)) {f : α → π i} {a : α} (hf : ContinuousAt f a)
@@ -2266,7 +2266,7 @@ theorem ContinuousAt.fin_insertNth {n} {π : Fin (n + 1) → Type _} [∀ i, Top
 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_3 : 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)}, (Continuous.{u1, u2} α (π i) _inst_1 (_inst_3 i) f) -> (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 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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))))) (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))}, (Continuous.{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} 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(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) -> (Continuous.{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} 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 but is expected to have type
-  forall {α : Type.{u2}} [_inst_1 : TopologicalSpace.{u2} 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.{u1}} [_inst_3 : forall (i : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))), TopologicalSpace.{u1} (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)}, (Continuous.{u2, u1} α (π i) _inst_1 (_inst_3 i) f) -> (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 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(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, 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(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_3 (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) -> (Continuous.{u2, u1} 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, u1} (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_3 a)) (fun (a : α) => Fin.insertNth.{u1} n π i (f a) (g a))))
+  forall {α : Type.{u2}} [_inst_1 : TopologicalSpace.{u2} 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.{u1}} [_inst_3 : forall (i : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))), TopologicalSpace.{u1} (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)}, (Continuous.{u2, u1} α (π i) _inst_1 (_inst_3 i) f) -> (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 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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))}, (Continuous.{u2, u1} 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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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, u1} (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_3 (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) -> (Continuous.{u2, u1} 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, u1} (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_3 a)) (fun (a : α) => Fin.insertNth.{u1} n π i (f a) (g a))))
 Case conversion may be inaccurate. Consider using '#align continuous.fin_insert_nth Continuous.fin_insertNthₓ'. -/
 theorem Continuous.fin_insertNth {n} {π : Fin (n + 1) → Type _} [∀ i, TopologicalSpace (π i)]
     (i : Fin (n + 1)) {f : α → π i} (hf : Continuous f) {g : α → ∀ j : Fin n, π (i.succAbove j)}

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

@@ -2240,7 +2240,7 @@ theorem continuous_mulSingle [∀ i, One (π i)] [DecidableEq ι] (i : ι) :
 lean 3 declaration is
   forall {β : Type.{u1}} {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_3 : 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 : β -> (π i)} {l : Filter.{u1} β} {x : π i}, (Filter.Tendsto.{u1, u2} β (π i) f l (nhds.{u2} (π i) (_inst_3 i) x)) -> (forall {g : β -> (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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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))} {y : 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))))))))) => 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=> _inst_3 a)) (Fin.insertNth.{u2} n (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) i x y))))
 but is expected to have type
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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 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(OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) => π j) i x y))))
+  forall {β : Type.{u2}} {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.{u1}} [_inst_3 : forall (i : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))), TopologicalSpace.{u1} (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 : β -> (π i)} {l : Filter.{u2} β} {x : π i}, (Filter.Tendsto.{u2, u1} β (π i) f l (nhds.{u1} (π i) (_inst_3 i) x)) -> (forall {g : β -> (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) => 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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)))) 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(a : Fin n) => _inst_3 (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 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(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))) y)) -> (Filter.Tendsto.{u2, u1} β (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) (fun (a : β) => Fin.insertNth.{u1} n π i (f a) (g a)) l (nhds.{u1} (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) (Pi.topologicalSpace.{0, u1} (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_3 a)) (Fin.insertNth.{u1} n (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) i x y))))
 Case conversion may be inaccurate. Consider using '#align filter.tendsto.fin_insert_nth Filter.Tendsto.fin_insertNthₓ'. -/
 theorem Filter.Tendsto.fin_insertNth {n} {π : Fin (n + 1) → Type _} [∀ i, TopologicalSpace (π i)]
     (i : Fin (n + 1)) {f : β → π i} {l : Filter β} {x : π i} (hf : Tendsto f l (𝓝 x))
@@ -2253,7 +2253,7 @@ theorem Filter.Tendsto.fin_insertNth {n} {π : Fin (n + 1) → Type _} [∀ i, T
 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_3 : 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}, (ContinuousAt.{u1, u2} α (π i) _inst_1 (_inst_3 i) f 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 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(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 a) -> (ContinuousAt.{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_3 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)) a))
 but is expected to have type
-  forall {α : Type.{u2}} [_inst_1 : TopologicalSpace.{u2} 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.{u1}} [_inst_3 : forall (i : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))), TopologicalSpace.{u1} (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}, (ContinuousAt.{u2, u1} α (π i) _inst_1 (_inst_3 i) f 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) 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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) (Fin.succAbove n i)) j))}, (ContinuousAt.{u2, u1} 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, u1} (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 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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_3 (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 a) -> (ContinuousAt.{u2, u1} 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, u1} (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_3 a)) (fun (a : α) => Fin.insertNth.{u1} n π i (f a) (g a)) a))
+  forall {α : Type.{u2}} [_inst_1 : TopologicalSpace.{u2} 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.{u1}} [_inst_3 : forall (i : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))), TopologicalSpace.{u1} (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}, (ContinuousAt.{u2, u1} α (π i) _inst_1 (_inst_3 i) f 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))}, (ContinuousAt.{u2, u1} 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, u1} (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_3 (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 a) -> (ContinuousAt.{u2, u1} 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, u1} (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_3 a)) (fun (a : α) => Fin.insertNth.{u1} n π i (f a) (g a)) a))
 Case conversion may be inaccurate. Consider using '#align continuous_at.fin_insert_nth ContinuousAt.fin_insertNthₓ'. -/
 theorem ContinuousAt.fin_insertNth {n} {π : Fin (n + 1) → Type _} [∀ i, TopologicalSpace (π i)]
     (i : Fin (n + 1)) {f : α → π i} {a : α} (hf : ContinuousAt f a)
@@ -2266,7 +2266,7 @@ theorem ContinuousAt.fin_insertNth {n} {π : Fin (n + 1) → Type _} [∀ i, Top
 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_3 : 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)}, (Continuous.{u1, u2} α (π i) _inst_1 (_inst_3 i) f) -> (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 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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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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) -> (Continuous.{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_3 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))))
 but is expected to have type
-  forall {α : Type.{u2}} [_inst_1 : TopologicalSpace.{u2} 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.{u1}} [_inst_3 : forall (i : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))), TopologicalSpace.{u1} (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)}, (Continuous.{u2, u1} α (π i) _inst_1 (_inst_3 i) f) -> (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))}, (Continuous.{u2, u1} 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, u1} (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_3 (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) -> (Continuous.{u2, u1} 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, u1} (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_3 a)) (fun (a : α) => Fin.insertNth.{u1} n π i (f a) (g a))))
+  forall {α : Type.{u2}} [_inst_1 : TopologicalSpace.{u2} 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.{u1}} [_inst_3 : forall (i : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))), TopologicalSpace.{u1} (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)}, (Continuous.{u2, u1} α (π i) _inst_1 (_inst_3 i) f) -> (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))}, (Continuous.{u2, u1} 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) 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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, u1} (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 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(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 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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) -> (Continuous.{u2, u1} 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, u1} (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_3 a)) (fun (a : α) => Fin.insertNth.{u1} n π i (f a) (g a))))
 Case conversion may be inaccurate. Consider using '#align continuous.fin_insert_nth Continuous.fin_insertNthₓ'. -/
 theorem Continuous.fin_insertNth {n} {π : Fin (n + 1) → Type _} [∀ i, TopologicalSpace (π i)]
     (i : Fin (n + 1)) {f : α → π i} (hf : Continuous f) {g : α → ∀ j : Fin n, π (i.succAbove j)}

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

@@ -4,12 +4,11 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Johannes Hölzl, Mario Carneiro, Patrick Massot
 
 ! This file was ported from Lean 3 source module topology.constructions
-! leanprover-community/mathlib commit 0c1f285a9f6e608ae2bdffa3f993eafb01eba829
+! leanprover-community/mathlib commit 55d771df074d0dd020139ee1cd4b95521422df9f
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
 import Mathbin.Topology.Maps
-import Mathbin.Topology.LocallyFinite
 import Mathbin.Order.Filter.Pi
 
 /-!
@@ -1945,55 +1944,6 @@ theorem tendsto_subtype_rng {β : Type _} {p : α → Prop} {b : Filter β} {f :
   | ⟨a, ha⟩ => by rw [nhds_subtype_eq_comap, tendsto_comap_iff, Subtype.coe_mk]
 #align tendsto_subtype_rng tendsto_subtype_rng
 
-/- warning: continuous_subtype_nhds_cover -> continuous_subtype_nhds_cover 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} {c : ι -> α -> Prop}, (forall (x : α), Exists.{u3} ι (fun (i : ι) => Membership.Mem.{u1, u1} (Set.{u1} α) (Filter.{u1} α) (Filter.hasMem.{u1} α) (setOf.{u1} α (fun (x : α) => c i x)) (nhds.{u1} α _inst_1 x))) -> (forall (i : ι), Continuous.{u1, u2} (Subtype.{succ u1} α (c i)) β (Subtype.topologicalSpace.{u1} α (c i) _inst_1) _inst_2 (fun (x : Subtype.{succ u1} α (c i)) => f ((fun (a : Type.{u1}) (b : Type.{u1}) [self : HasLiftT.{succ u1, succ u1} a b] => self.0) (Subtype.{succ u1} α (c i)) α (HasLiftT.mk.{succ u1, succ u1} (Subtype.{succ u1} α (c i)) α (CoeTCₓ.coe.{succ u1, succ u1} (Subtype.{succ u1} α (c i)) α (coeBase.{succ u1, succ u1} (Subtype.{succ u1} α (c i)) α (coeSubtype.{succ u1} α (fun (x : α) => c i x))))) x))) -> (Continuous.{u1, u2} α β _inst_1 _inst_2 f)
-but is expected to have type
-  forall {α : Type.{u2}} {β : Type.{u3}} [_inst_1 : TopologicalSpace.{u2} α] [_inst_2 : TopologicalSpace.{u3} β] {ι : Sort.{u1}} {f : α -> β} {c : ι -> α -> Prop}, (forall (x : α), Exists.{u1} ι (fun (i : ι) => Membership.mem.{u2, u2} (Set.{u2} α) (Filter.{u2} α) (instMembershipSetFilter.{u2} α) (setOf.{u2} α (fun (x : α) => c i x)) (nhds.{u2} α _inst_1 x))) -> (forall (i : ι), Continuous.{u2, u3} (Subtype.{succ u2} α (c i)) β (instTopologicalSpaceSubtype.{u2} α (c i) _inst_1) _inst_2 (fun (x : Subtype.{succ u2} α (c i)) => f (Subtype.val.{succ u2} α (c i) x))) -> (Continuous.{u2, u3} α β _inst_1 _inst_2 f)
-Case conversion may be inaccurate. Consider using '#align continuous_subtype_nhds_cover continuous_subtype_nhds_coverₓ'. -/
-theorem continuous_subtype_nhds_cover {ι : Sort _} {f : α → β} {c : ι → α → Prop}
-    (c_cover : ∀ x : α, ∃ i, { x | c i x } ∈ 𝓝 x)
-    (f_cont : ∀ i, Continuous fun x : Subtype (c i) => f x) : Continuous f :=
-  continuous_iff_continuousAt.mpr fun x =>
-    let ⟨i, (c_sets : { x | c i x } ∈ 𝓝 x)⟩ := c_cover x
-    let x' : Subtype (c i) := ⟨x, mem_of_mem_nhds c_sets⟩
-    calc
-      map f (𝓝 x) = map f (map coe (𝓝 x')) :=
-        congr_arg (map f) (map_nhds_subtype_coe_eq_nhds _ <| c_sets).symm
-      _ = map (fun x : Subtype (c i) => f x) (𝓝 x') := rfl
-      _ ≤ 𝓝 (f x) := continuous_iff_continuousAt.mp (f_cont i) x'
-      
-#align continuous_subtype_nhds_cover continuous_subtype_nhds_cover
-
-/- warning: continuous_subtype_is_closed_cover clashes with [anonymous] -> [anonymous]
-warning: continuous_subtype_is_closed_cover -> [anonymous] is a dubious translation:
-lean 3 declaration is
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-but is expected to have type
-  forall {α : Type.{u}} {β : Type.{v}}, (Nat -> α -> β) -> Nat -> (List.{u} α) -> (List.{v} β)
-Case conversion may be inaccurate. Consider using '#align continuous_subtype_is_closed_cover [anonymous]ₓ'. -/
-theorem [anonymous] {ι : Sort _} {f : α → β} (c : ι → α → Prop)
-    (h_lf : LocallyFinite fun i => { x | c i x }) (h_is_closed : ∀ i, IsClosed { x | c i x })
-    (h_cover : ∀ x, ∃ i, c i x) (f_cont : ∀ i, Continuous fun x : Subtype (c i) => f x) :
-    Continuous f :=
-  continuous_iff_isClosed.mpr fun s hs =>
-    by
-    have : ∀ i, IsClosed ((coe : { x | c i x } → α) '' (f ∘ coe ⁻¹' s)) := fun i =>
-      (closedEmbedding_subtype_val (h_is_closed _)).IsClosedMap _ (hs.Preimage (f_cont i))
-    have : IsClosed (⋃ i, (coe : { x | c i x } → α) '' (f ∘ coe ⁻¹' s)) :=
-      LocallyFinite.isClosed_unionᵢ (h_lf.Subset fun i x ⟨⟨x', hx'⟩, _, HEq⟩ => HEq ▸ hx') this
-    have : f ⁻¹' s = ⋃ i, (coe : { x | c i x } → α) '' (f ∘ coe ⁻¹' s) :=
-      by
-      apply Set.ext
-      have : ∀ x : α, f x ∈ s ↔ ∃ i : ι, c i x ∧ f x ∈ s := fun x =>
-        ⟨fun hx =>
-          let ⟨i, hi⟩ := h_cover x
-          ⟨i, hi, hx⟩,
-          fun ⟨i, hi, hx⟩ => hx⟩
-      simpa [and_comm, @and_left_comm (c _ _), ← exists_and_right]
-    rwa [this]
-#align continuous_subtype_is_closed_cover [anonymous]
-
 #print closure_subtype /-
 theorem closure_subtype {x : { a // p a }} {s : Set { a // p a }} :
     x ∈ closure s ↔ (x : α) ∈ closure ((coe : _ → α) '' s) :=

mathlib commit https://github.com/leanprover-community/mathlib/commit/38f16f960f5006c6c0c2bac7b0aba5273188f4e5

@@ -2276,6 +2276,7 @@ theorem continuous_update [DecidableEq ι] (i : ι) :
   continuous_fst.update i continuous_snd
 #align continuous_update continuous_update
 
+#print continuous_mulSingle /-
 /-- `pi.mul_single i x` is continuous in `x`. -/
 @[continuity, to_additive "`pi.single i x` is continuous in `x`."]
 theorem continuous_mulSingle [∀ i, One (π i)] [DecidableEq ι] (i : ι) :
@@ -2283,6 +2284,7 @@ theorem continuous_mulSingle [∀ i, One (π i)] [DecidableEq ι] (i : ι) :
   continuous_const.update _ continuous_id
 #align continuous_mul_single continuous_mulSingle
 #align continuous_single continuous_single
+-/
 
 /- warning: filter.tendsto.fin_insert_nth -> Filter.Tendsto.fin_insertNth is a dubious translation:
 lean 3 declaration is

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

@@ -364,9 +364,9 @@ theorem nhds_subtype (s : Set α) (a : { x // x ∈ s }) : 𝓝 a = comap coe (
 
 /- warning: nhds_within_subtype_eq_bot_iff -> nhdsWithin_subtype_eq_bot_iff is a dubious translation:
 lean 3 declaration is
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+  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] {s : Set.{u1} α} {t : Set.{u1} α} {x : coeSort.{succ u1, succ (succ u1)} (Set.{u1} α) Type.{u1} (Set.hasCoeToSort.{u1} α) s}, Iff (Eq.{succ u1} (Filter.{u1} (coeSort.{succ u1, succ (succ u1)} (Set.{u1} α) Type.{u1} (Set.hasCoeToSort.{u1} α) s)) (nhdsWithin.{u1} (coeSort.{succ u1, succ (succ u1)} (Set.{u1} α) Type.{u1} (Set.hasCoeToSort.{u1} α) s) (Subtype.topologicalSpace.{u1} α (fun (x : α) => Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) x s) _inst_1) x (Set.preimage.{u1, u1} (coeSort.{succ u1, succ (succ u1)} (Set.{u1} α) Type.{u1} (Set.hasCoeToSort.{u1} α) s) α ((fun (a : Type.{u1}) (b : Type.{u1}) [self : HasLiftT.{succ u1, succ u1} a b] => self.0) (coeSort.{succ u1, succ (succ u1)} (Set.{u1} α) Type.{u1} (Set.hasCoeToSort.{u1} α) s) α (HasLiftT.mk.{succ u1, succ u1} (coeSort.{succ u1, succ (succ u1)} (Set.{u1} α) Type.{u1} (Set.hasCoeToSort.{u1} α) s) α (CoeTCₓ.coe.{succ u1, succ u1} (coeSort.{succ u1, succ (succ u1)} (Set.{u1} α) Type.{u1} (Set.hasCoeToSort.{u1} α) s) α (coeBase.{succ u1, succ u1} (coeSort.{succ u1, succ (succ u1)} (Set.{u1} α) Type.{u1} (Set.hasCoeToSort.{u1} α) s) α (coeSubtype.{succ u1} α (fun (x : α) => Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) x s)))))) t)) (Bot.bot.{u1} (Filter.{u1} (coeSort.{succ u1, succ (succ u1)} (Set.{u1} α) Type.{u1} (Set.hasCoeToSort.{u1} α) s)) (CompleteLattice.toHasBot.{u1} (Filter.{u1} (coeSort.{succ u1, succ (succ u1)} (Set.{u1} α) Type.{u1} (Set.hasCoeToSort.{u1} α) s)) (Filter.completeLattice.{u1} (coeSort.{succ u1, succ (succ u1)} (Set.{u1} α) Type.{u1} (Set.hasCoeToSort.{u1} α) s))))) (Eq.{succ u1} (Filter.{u1} α) (Inf.inf.{u1} (Filter.{u1} α) (Filter.hasInf.{u1} α) (nhdsWithin.{u1} α _inst_1 ((fun (a : Type.{u1}) (b : Type.{u1}) [self : HasLiftT.{succ u1, succ u1} a b] => self.0) (coeSort.{succ u1, succ (succ u1)} (Set.{u1} α) Type.{u1} (Set.hasCoeToSort.{u1} α) s) α (HasLiftT.mk.{succ u1, succ u1} (coeSort.{succ u1, succ (succ u1)} (Set.{u1} α) Type.{u1} (Set.hasCoeToSort.{u1} α) s) α (CoeTCₓ.coe.{succ u1, succ u1} (coeSort.{succ u1, succ (succ u1)} (Set.{u1} α) Type.{u1} (Set.hasCoeToSort.{u1} α) s) α (coeBase.{succ u1, succ u1} (coeSort.{succ u1, succ (succ u1)} (Set.{u1} α) Type.{u1} (Set.hasCoeToSort.{u1} α) s) α (coeSubtype.{succ u1} α (fun (x : α) => Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) x s))))) x) t) (Filter.principal.{u1} α s)) (Bot.bot.{u1} (Filter.{u1} α) (CompleteLattice.toHasBot.{u1} (Filter.{u1} α) (Filter.completeLattice.{u1} α))))
 but is expected to have type
-  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] {s : Set.{u1} α} {t : Set.{u1} α} {x : Set.Elem.{u1} α s}, Iff (Eq.{succ u1} (Filter.{u1} (Set.Elem.{u1} α s)) (nhdsWithin.{u1} (Set.Elem.{u1} α s) (instTopologicalSpaceSubtype.{u1} α (fun (x : α) => Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) x s) _inst_1) x (Set.preimage.{u1, u1} (Set.Elem.{u1} α s) α (Subtype.val.{succ u1} α (fun (x : α) => Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) x s)) t)) (Bot.bot.{u1} (Filter.{u1} (Set.Elem.{u1} α s)) (CompleteLattice.toBot.{u1} (Filter.{u1} (Set.Elem.{u1} α s)) (Filter.instCompleteLatticeFilter.{u1} (Set.Elem.{u1} α s))))) (Eq.{succ u1} (Filter.{u1} α) (HasInf.inf.{u1} (Filter.{u1} α) (Filter.instHasInfFilter.{u1} α) (nhdsWithin.{u1} α _inst_1 (Subtype.val.{succ u1} α (fun (x : α) => Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) x s) x) t) (Filter.principal.{u1} α s)) (Bot.bot.{u1} (Filter.{u1} α) (CompleteLattice.toBot.{u1} (Filter.{u1} α) (Filter.instCompleteLatticeFilter.{u1} α))))
+  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] {s : Set.{u1} α} {t : Set.{u1} α} {x : Set.Elem.{u1} α s}, Iff (Eq.{succ u1} (Filter.{u1} (Set.Elem.{u1} α s)) (nhdsWithin.{u1} (Set.Elem.{u1} α s) (instTopologicalSpaceSubtype.{u1} α (fun (x : α) => Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) x s) _inst_1) x (Set.preimage.{u1, u1} (Set.Elem.{u1} α s) α (Subtype.val.{succ u1} α (fun (x : α) => Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) x s)) t)) (Bot.bot.{u1} (Filter.{u1} (Set.Elem.{u1} α s)) (CompleteLattice.toBot.{u1} (Filter.{u1} (Set.Elem.{u1} α s)) (Filter.instCompleteLatticeFilter.{u1} (Set.Elem.{u1} α s))))) (Eq.{succ u1} (Filter.{u1} α) (Inf.inf.{u1} (Filter.{u1} α) (Filter.instInfFilter.{u1} α) (nhdsWithin.{u1} α _inst_1 (Subtype.val.{succ u1} α (fun (x : α) => Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) x s) x) t) (Filter.principal.{u1} α s)) (Bot.bot.{u1} (Filter.{u1} α) (CompleteLattice.toBot.{u1} (Filter.{u1} α) (Filter.instCompleteLatticeFilter.{u1} α))))
 Case conversion may be inaccurate. Consider using '#align nhds_within_subtype_eq_bot_iff nhdsWithin_subtype_eq_bot_iffₓ'. -/
 theorem nhdsWithin_subtype_eq_bot_iff {s t : Set α} {x : s} :
     𝓝[(coe : s → α) ⁻¹' t] x = ⊥ ↔ 𝓝[t] (x : α) ⊓ 𝓟 s = ⊥ := by
@@ -376,9 +376,9 @@ theorem nhdsWithin_subtype_eq_bot_iff {s t : Set α} {x : s} :
 
 /- warning: nhds_ne_subtype_eq_bot_iff -> nhds_ne_subtype_eq_bot_iff is a dubious translation:
 lean 3 declaration is
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 but is expected to have type
-  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] {S : Set.{u1} α} {x : Set.Elem.{u1} α S}, Iff (Eq.{succ u1} (Filter.{u1} (Set.Elem.{u1} α S)) (nhdsWithin.{u1} (Set.Elem.{u1} α S) (instTopologicalSpaceSubtype.{u1} α (fun (x : α) => Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) x S) _inst_1) x (HasCompl.compl.{u1} (Set.{u1} (Set.Elem.{u1} α S)) (BooleanAlgebra.toHasCompl.{u1} (Set.{u1} (Set.Elem.{u1} α S)) (Set.instBooleanAlgebraSet.{u1} (Set.Elem.{u1} α S))) (Singleton.singleton.{u1, u1} (Set.Elem.{u1} α S) (Set.{u1} (Set.Elem.{u1} α S)) (Set.instSingletonSet.{u1} (Set.Elem.{u1} α S)) x))) (Bot.bot.{u1} (Filter.{u1} (Set.Elem.{u1} α S)) (CompleteLattice.toBot.{u1} (Filter.{u1} (Set.Elem.{u1} α S)) (Filter.instCompleteLatticeFilter.{u1} (Set.Elem.{u1} α S))))) (Eq.{succ u1} (Filter.{u1} α) (HasInf.inf.{u1} (Filter.{u1} α) (Filter.instHasInfFilter.{u1} α) (nhdsWithin.{u1} α _inst_1 (Subtype.val.{succ u1} α (fun (x : α) => Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) x S) x) (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) (Subtype.val.{succ u1} α (fun (x : α) => Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) x S) x)))) (Filter.principal.{u1} α S)) (Bot.bot.{u1} (Filter.{u1} α) (CompleteLattice.toBot.{u1} (Filter.{u1} α) (Filter.instCompleteLatticeFilter.{u1} α))))
+  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] {S : Set.{u1} α} {x : Set.Elem.{u1} α S}, Iff (Eq.{succ u1} (Filter.{u1} (Set.Elem.{u1} α S)) (nhdsWithin.{u1} (Set.Elem.{u1} α S) (instTopologicalSpaceSubtype.{u1} α (fun (x : α) => Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) x S) _inst_1) x (HasCompl.compl.{u1} (Set.{u1} (Set.Elem.{u1} α S)) (BooleanAlgebra.toHasCompl.{u1} (Set.{u1} (Set.Elem.{u1} α S)) (Set.instBooleanAlgebraSet.{u1} (Set.Elem.{u1} α S))) (Singleton.singleton.{u1, u1} (Set.Elem.{u1} α S) (Set.{u1} (Set.Elem.{u1} α S)) (Set.instSingletonSet.{u1} (Set.Elem.{u1} α S)) x))) (Bot.bot.{u1} (Filter.{u1} (Set.Elem.{u1} α S)) (CompleteLattice.toBot.{u1} (Filter.{u1} (Set.Elem.{u1} α S)) (Filter.instCompleteLatticeFilter.{u1} (Set.Elem.{u1} α S))))) (Eq.{succ u1} (Filter.{u1} α) (Inf.inf.{u1} (Filter.{u1} α) (Filter.instInfFilter.{u1} α) (nhdsWithin.{u1} α _inst_1 (Subtype.val.{succ u1} α (fun (x : α) => Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) x S) x) (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) (Subtype.val.{succ u1} α (fun (x : α) => Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) x S) x)))) (Filter.principal.{u1} α S)) (Bot.bot.{u1} (Filter.{u1} α) (CompleteLattice.toBot.{u1} (Filter.{u1} α) (Filter.instCompleteLatticeFilter.{u1} α))))
 Case conversion may be inaccurate. Consider using '#align nhds_ne_subtype_eq_bot_iff nhds_ne_subtype_eq_bot_iffₓ'. -/
 theorem nhds_ne_subtype_eq_bot_iff {S : Set α} {x : S} :
     𝓝[{x}ᶜ] x = ⊥ ↔ 𝓝[{x}ᶜ] (x : α) ⊓ 𝓟 S = ⊥ := by
@@ -388,9 +388,9 @@ theorem nhds_ne_subtype_eq_bot_iff {S : Set α} {x : S} :
 
 /- warning: nhds_ne_subtype_ne_bot_iff -> nhds_ne_subtype_neBot_iff is a dubious translation:
 lean 3 declaration is
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+  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] {S : Set.{u1} α} {x : coeSort.{succ u1, succ (succ u1)} (Set.{u1} α) Type.{u1} (Set.hasCoeToSort.{u1} α) S}, Iff (Filter.NeBot.{u1} (coeSort.{succ u1, succ (succ u1)} (Set.{u1} α) Type.{u1} (Set.hasCoeToSort.{u1} α) S) (nhdsWithin.{u1} (coeSort.{succ u1, succ (succ u1)} (Set.{u1} α) Type.{u1} (Set.hasCoeToSort.{u1} α) S) (Subtype.topologicalSpace.{u1} α (fun (x : α) => Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) x S) _inst_1) x (HasCompl.compl.{u1} (Set.{u1} (coeSort.{succ u1, succ (succ u1)} (Set.{u1} α) Type.{u1} (Set.hasCoeToSort.{u1} α) S)) (BooleanAlgebra.toHasCompl.{u1} (Set.{u1} (coeSort.{succ u1, succ (succ u1)} (Set.{u1} α) Type.{u1} (Set.hasCoeToSort.{u1} α) S)) (Set.booleanAlgebra.{u1} (coeSort.{succ u1, succ (succ u1)} (Set.{u1} α) Type.{u1} (Set.hasCoeToSort.{u1} α) S))) (Singleton.singleton.{u1, u1} (coeSort.{succ u1, succ (succ u1)} (Set.{u1} α) Type.{u1} (Set.hasCoeToSort.{u1} α) S) (Set.{u1} (coeSort.{succ u1, succ (succ u1)} (Set.{u1} α) Type.{u1} (Set.hasCoeToSort.{u1} α) S)) (Set.hasSingleton.{u1} (coeSort.{succ u1, succ (succ u1)} (Set.{u1} α) Type.{u1} (Set.hasCoeToSort.{u1} α) S)) x)))) (Filter.NeBot.{u1} α (Inf.inf.{u1} (Filter.{u1} α) (Filter.hasInf.{u1} α) (nhdsWithin.{u1} α _inst_1 ((fun (a : Type.{u1}) (b : Type.{u1}) [self : HasLiftT.{succ u1, succ u1} a b] => self.0) (coeSort.{succ u1, succ (succ u1)} (Set.{u1} α) Type.{u1} (Set.hasCoeToSort.{u1} α) S) α (HasLiftT.mk.{succ u1, succ u1} (coeSort.{succ u1, succ (succ u1)} (Set.{u1} α) Type.{u1} (Set.hasCoeToSort.{u1} α) S) α (CoeTCₓ.coe.{succ u1, succ u1} (coeSort.{succ u1, succ (succ u1)} (Set.{u1} α) Type.{u1} (Set.hasCoeToSort.{u1} α) S) α (coeBase.{succ u1, succ u1} (coeSort.{succ u1, succ (succ u1)} (Set.{u1} α) Type.{u1} (Set.hasCoeToSort.{u1} α) S) α (coeSubtype.{succ u1} α (fun (x : α) => Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) x S))))) x) (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) ((fun (a : Type.{u1}) (b : Type.{u1}) [self : HasLiftT.{succ u1, succ u1} a b] => self.0) (coeSort.{succ u1, succ (succ u1)} (Set.{u1} α) Type.{u1} (Set.hasCoeToSort.{u1} α) S) α (HasLiftT.mk.{succ u1, succ u1} (coeSort.{succ u1, succ (succ u1)} (Set.{u1} α) Type.{u1} (Set.hasCoeToSort.{u1} α) S) α (CoeTCₓ.coe.{succ u1, succ u1} (coeSort.{succ u1, succ (succ u1)} (Set.{u1} α) Type.{u1} (Set.hasCoeToSort.{u1} α) S) α (coeBase.{succ u1, succ u1} (coeSort.{succ u1, succ (succ u1)} (Set.{u1} α) Type.{u1} (Set.hasCoeToSort.{u1} α) S) α (coeSubtype.{succ u1} α (fun (x : α) => Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) x S))))) x)))) (Filter.principal.{u1} α S)))
 but is expected to have type
-  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] {S : Set.{u1} α} {x : Set.Elem.{u1} α S}, Iff (Filter.NeBot.{u1} (Set.Elem.{u1} α S) (nhdsWithin.{u1} (Set.Elem.{u1} α S) (instTopologicalSpaceSubtype.{u1} α (fun (x : α) => Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) x S) _inst_1) x (HasCompl.compl.{u1} (Set.{u1} (Set.Elem.{u1} α S)) (BooleanAlgebra.toHasCompl.{u1} (Set.{u1} (Set.Elem.{u1} α S)) (Set.instBooleanAlgebraSet.{u1} (Set.Elem.{u1} α S))) (Singleton.singleton.{u1, u1} (Set.Elem.{u1} α S) (Set.{u1} (Set.Elem.{u1} α S)) (Set.instSingletonSet.{u1} (Set.Elem.{u1} α S)) x)))) (Filter.NeBot.{u1} α (HasInf.inf.{u1} (Filter.{u1} α) (Filter.instHasInfFilter.{u1} α) (nhdsWithin.{u1} α _inst_1 (Subtype.val.{succ u1} α (fun (x : α) => Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) x S) x) (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) (Subtype.val.{succ u1} α (fun (x : α) => Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) x S) x)))) (Filter.principal.{u1} α S)))
+  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] {S : Set.{u1} α} {x : Set.Elem.{u1} α S}, Iff (Filter.NeBot.{u1} (Set.Elem.{u1} α S) (nhdsWithin.{u1} (Set.Elem.{u1} α S) (instTopologicalSpaceSubtype.{u1} α (fun (x : α) => Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) x S) _inst_1) x (HasCompl.compl.{u1} (Set.{u1} (Set.Elem.{u1} α S)) (BooleanAlgebra.toHasCompl.{u1} (Set.{u1} (Set.Elem.{u1} α S)) (Set.instBooleanAlgebraSet.{u1} (Set.Elem.{u1} α S))) (Singleton.singleton.{u1, u1} (Set.Elem.{u1} α S) (Set.{u1} (Set.Elem.{u1} α S)) (Set.instSingletonSet.{u1} (Set.Elem.{u1} α S)) x)))) (Filter.NeBot.{u1} α (Inf.inf.{u1} (Filter.{u1} α) (Filter.instInfFilter.{u1} α) (nhdsWithin.{u1} α _inst_1 (Subtype.val.{succ u1} α (fun (x : α) => Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) x S) x) (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) (Subtype.val.{succ u1} α (fun (x : α) => Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) x S) x)))) (Filter.principal.{u1} α S)))
 Case conversion may be inaccurate. Consider using '#align nhds_ne_subtype_ne_bot_iff nhds_ne_subtype_neBot_iffₓ'. -/
 theorem nhds_ne_subtype_neBot_iff {S : Set α} {x : S} :
     (𝓝[{x}ᶜ] x).ne_bot ↔ (𝓝[{x}ᶜ] (x : α) ⊓ 𝓟 S).ne_bot := by
@@ -399,9 +399,9 @@ theorem nhds_ne_subtype_neBot_iff {S : Set α} {x : S} :
 
 /- warning: discrete_topology_subtype_iff -> discreteTopology_subtype_iff is a dubious translation:
 lean 3 declaration is
-  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] {S : Set.{u1} α}, Iff (DiscreteTopology.{u1} (coeSort.{succ u1, succ (succ u1)} (Set.{u1} α) Type.{u1} (Set.hasCoeToSort.{u1} α) S) (Subtype.topologicalSpace.{u1} α (fun (x : α) => Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) x S) _inst_1)) (forall (x : α), (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) x S) -> (Eq.{succ u1} (Filter.{u1} α) (HasInf.inf.{u1} (Filter.{u1} α) (Filter.hasInf.{u1} α) (nhdsWithin.{u1} α _inst_1 x (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) x))) (Filter.principal.{u1} α S)) (Bot.bot.{u1} (Filter.{u1} α) (CompleteLattice.toHasBot.{u1} (Filter.{u1} α) (Filter.completeLattice.{u1} α)))))
+  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] {S : Set.{u1} α}, Iff (DiscreteTopology.{u1} (coeSort.{succ u1, succ (succ u1)} (Set.{u1} α) Type.{u1} (Set.hasCoeToSort.{u1} α) S) (Subtype.topologicalSpace.{u1} α (fun (x : α) => Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) x S) _inst_1)) (forall (x : α), (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) x S) -> (Eq.{succ u1} (Filter.{u1} α) (Inf.inf.{u1} (Filter.{u1} α) (Filter.hasInf.{u1} α) (nhdsWithin.{u1} α _inst_1 x (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) x))) (Filter.principal.{u1} α S)) (Bot.bot.{u1} (Filter.{u1} α) (CompleteLattice.toHasBot.{u1} (Filter.{u1} α) (Filter.completeLattice.{u1} α)))))
 but is expected to have type
-  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] {S : Set.{u1} α}, Iff (DiscreteTopology.{u1} (Set.Elem.{u1} α S) (instTopologicalSpaceSubtype.{u1} α (fun (x : α) => Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) x S) _inst_1)) (forall (x : α), (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) x S) -> (Eq.{succ u1} (Filter.{u1} α) (HasInf.inf.{u1} (Filter.{u1} α) (Filter.instHasInfFilter.{u1} α) (nhdsWithin.{u1} α _inst_1 x (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) x))) (Filter.principal.{u1} α S)) (Bot.bot.{u1} (Filter.{u1} α) (CompleteLattice.toBot.{u1} (Filter.{u1} α) (Filter.instCompleteLatticeFilter.{u1} α)))))
+  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] {S : Set.{u1} α}, Iff (DiscreteTopology.{u1} (Set.Elem.{u1} α S) (instTopologicalSpaceSubtype.{u1} α (fun (x : α) => Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) x S) _inst_1)) (forall (x : α), (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) x S) -> (Eq.{succ u1} (Filter.{u1} α) (Inf.inf.{u1} (Filter.{u1} α) (Filter.instInfFilter.{u1} α) (nhdsWithin.{u1} α _inst_1 x (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) x))) (Filter.principal.{u1} α S)) (Bot.bot.{u1} (Filter.{u1} α) (CompleteLattice.toBot.{u1} (Filter.{u1} α) (Filter.instCompleteLatticeFilter.{u1} α)))))
 Case conversion may be inaccurate. Consider using '#align discrete_topology_subtype_iff discreteTopology_subtype_iffₓ'. -/
 theorem discreteTopology_subtype_iff {S : Set α} : DiscreteTopology S ↔ ∀ x ∈ S, 𝓝[≠] x ⊓ 𝓟 S = ⊥ :=
   by simp_rw [discreteTopology_iff_nhds_ne, SetCoe.forall', nhds_ne_subtype_eq_bot_iff]
@@ -473,9 +473,9 @@ theorem isClosed_iff {s : Set (CofiniteTopology α)} : IsClosed s ↔ s = univ 
 
 /- warning: cofinite_topology.nhds_eq -> CofiniteTopology.nhds_eq is a dubious translation:
 lean 3 declaration is
-  forall {α : Type.{u1}} (a : CofiniteTopology.{u1} α), Eq.{succ u1} (Filter.{u1} (CofiniteTopology.{u1} α)) (nhds.{u1} (CofiniteTopology.{u1} α) (CofiniteTopology.topologicalSpace.{u1} α) a) (HasSup.sup.{u1} (Filter.{u1} (CofiniteTopology.{u1} α)) (SemilatticeSup.toHasSup.{u1} (Filter.{u1} (CofiniteTopology.{u1} α)) (Lattice.toSemilatticeSup.{u1} (Filter.{u1} (CofiniteTopology.{u1} α)) (ConditionallyCompleteLattice.toLattice.{u1} (Filter.{u1} (CofiniteTopology.{u1} α)) (CompleteLattice.toConditionallyCompleteLattice.{u1} (Filter.{u1} (CofiniteTopology.{u1} α)) (Filter.completeLattice.{u1} (CofiniteTopology.{u1} α)))))) (Pure.pure.{u1, u1} Filter.{u1} Filter.hasPure.{u1} (CofiniteTopology.{u1} α) a) (Filter.cofinite.{u1} (CofiniteTopology.{u1} α)))
+  forall {α : Type.{u1}} (a : CofiniteTopology.{u1} α), Eq.{succ u1} (Filter.{u1} (CofiniteTopology.{u1} α)) (nhds.{u1} (CofiniteTopology.{u1} α) (CofiniteTopology.topologicalSpace.{u1} α) a) (Sup.sup.{u1} (Filter.{u1} (CofiniteTopology.{u1} α)) (SemilatticeSup.toHasSup.{u1} (Filter.{u1} (CofiniteTopology.{u1} α)) (Lattice.toSemilatticeSup.{u1} (Filter.{u1} (CofiniteTopology.{u1} α)) (ConditionallyCompleteLattice.toLattice.{u1} (Filter.{u1} (CofiniteTopology.{u1} α)) (CompleteLattice.toConditionallyCompleteLattice.{u1} (Filter.{u1} (CofiniteTopology.{u1} α)) (Filter.completeLattice.{u1} (CofiniteTopology.{u1} α)))))) (Pure.pure.{u1, u1} Filter.{u1} Filter.hasPure.{u1} (CofiniteTopology.{u1} α) a) (Filter.cofinite.{u1} (CofiniteTopology.{u1} α)))
 but is expected to have type
-  forall {α : Type.{u1}} (a : CofiniteTopology.{u1} α), Eq.{succ u1} (Filter.{u1} (CofiniteTopology.{u1} α)) (nhds.{u1} (CofiniteTopology.{u1} α) (CofiniteTopology.instTopologicalSpaceCofiniteTopology.{u1} α) a) (HasSup.sup.{u1} (Filter.{u1} (CofiniteTopology.{u1} α)) (SemilatticeSup.toHasSup.{u1} (Filter.{u1} (CofiniteTopology.{u1} α)) (Lattice.toSemilatticeSup.{u1} (Filter.{u1} (CofiniteTopology.{u1} α)) (ConditionallyCompleteLattice.toLattice.{u1} (Filter.{u1} (CofiniteTopology.{u1} α)) (CompleteLattice.toConditionallyCompleteLattice.{u1} (Filter.{u1} (CofiniteTopology.{u1} α)) (Filter.instCompleteLatticeFilter.{u1} (CofiniteTopology.{u1} α)))))) (Pure.pure.{u1, u1} Filter.{u1} Filter.instPureFilter.{u1} (CofiniteTopology.{u1} α) a) (Filter.cofinite.{u1} (CofiniteTopology.{u1} α)))
+  forall {α : Type.{u1}} (a : CofiniteTopology.{u1} α), Eq.{succ u1} (Filter.{u1} (CofiniteTopology.{u1} α)) (nhds.{u1} (CofiniteTopology.{u1} α) (CofiniteTopology.instTopologicalSpaceCofiniteTopology.{u1} α) a) (Sup.sup.{u1} (Filter.{u1} (CofiniteTopology.{u1} α)) (SemilatticeSup.toSup.{u1} (Filter.{u1} (CofiniteTopology.{u1} α)) (Lattice.toSemilatticeSup.{u1} (Filter.{u1} (CofiniteTopology.{u1} α)) (ConditionallyCompleteLattice.toLattice.{u1} (Filter.{u1} (CofiniteTopology.{u1} α)) (CompleteLattice.toConditionallyCompleteLattice.{u1} (Filter.{u1} (CofiniteTopology.{u1} α)) (Filter.instCompleteLatticeFilter.{u1} (CofiniteTopology.{u1} α)))))) (Pure.pure.{u1, u1} Filter.{u1} Filter.instPureFilter.{u1} (CofiniteTopology.{u1} α) a) (Filter.cofinite.{u1} (CofiniteTopology.{u1} α)))
 Case conversion may be inaccurate. Consider using '#align cofinite_topology.nhds_eq CofiniteTopology.nhds_eqₓ'. -/
 theorem nhds_eq (a : CofiniteTopology α) : 𝓝 a = pure a ⊔ cofinite :=
   by
@@ -759,9 +759,9 @@ theorem Continuous.prod_map {f : γ → α} {g : δ → β} (hf : Continuous f)
 
 /- warning: continuous_inf_dom_left₂ -> continuous_inf_dom_left₂ is a dubious translation:
 lean 3 declaration is
-  forall {α : Type.{u1}} {β : Type.{u2}} {γ : Type.{u3}} {f : α -> β -> γ} {ta1 : TopologicalSpace.{u1} α} {ta2 : TopologicalSpace.{u1} α} {tb1 : TopologicalSpace.{u2} β} {tb2 : TopologicalSpace.{u2} β} {tc1 : TopologicalSpace.{u3} γ}, (Continuous.{max u1 u2, u3} (Prod.{u1, u2} α β) γ (Prod.topologicalSpace.{u1, u2} α β ta1 tb1) tc1 (fun (p : Prod.{u1, u2} α β) => f (Prod.fst.{u1, u2} α β p) (Prod.snd.{u1, u2} α β p))) -> (Continuous.{max u1 u2, u3} (Prod.{u1, u2} α β) γ (Prod.topologicalSpace.{u1, u2} α β (HasInf.inf.{u1} (TopologicalSpace.{u1} α) (SemilatticeInf.toHasInf.{u1} (TopologicalSpace.{u1} α) (Lattice.toSemilatticeInf.{u1} (TopologicalSpace.{u1} α) (ConditionallyCompleteLattice.toLattice.{u1} (TopologicalSpace.{u1} α) (CompleteLattice.toConditionallyCompleteLattice.{u1} (TopologicalSpace.{u1} α) (TopologicalSpace.completeLattice.{u1} α))))) ta1 ta2) (HasInf.inf.{u2} (TopologicalSpace.{u2} β) (SemilatticeInf.toHasInf.{u2} (TopologicalSpace.{u2} β) (Lattice.toSemilatticeInf.{u2} (TopologicalSpace.{u2} β) (ConditionallyCompleteLattice.toLattice.{u2} (TopologicalSpace.{u2} β) (CompleteLattice.toConditionallyCompleteLattice.{u2} (TopologicalSpace.{u2} β) (TopologicalSpace.completeLattice.{u2} β))))) tb1 tb2)) tc1 (fun (p : Prod.{u1, u2} α β) => f (Prod.fst.{u1, u2} α β p) (Prod.snd.{u1, u2} α β p)))
+  forall {α : Type.{u1}} {β : Type.{u2}} {γ : Type.{u3}} {f : α -> β -> γ} {ta1 : TopologicalSpace.{u1} α} {ta2 : TopologicalSpace.{u1} α} {tb1 : TopologicalSpace.{u2} β} {tb2 : TopologicalSpace.{u2} β} {tc1 : TopologicalSpace.{u3} γ}, (Continuous.{max u1 u2, u3} (Prod.{u1, u2} α β) γ (Prod.topologicalSpace.{u1, u2} α β ta1 tb1) tc1 (fun (p : Prod.{u1, u2} α β) => f (Prod.fst.{u1, u2} α β p) (Prod.snd.{u1, u2} α β p))) -> (Continuous.{max u1 u2, u3} (Prod.{u1, u2} α β) γ (Prod.topologicalSpace.{u1, u2} α β (Inf.inf.{u1} (TopologicalSpace.{u1} α) (SemilatticeInf.toHasInf.{u1} (TopologicalSpace.{u1} α) (Lattice.toSemilatticeInf.{u1} (TopologicalSpace.{u1} α) (ConditionallyCompleteLattice.toLattice.{u1} (TopologicalSpace.{u1} α) (CompleteLattice.toConditionallyCompleteLattice.{u1} (TopologicalSpace.{u1} α) (TopologicalSpace.completeLattice.{u1} α))))) ta1 ta2) (Inf.inf.{u2} (TopologicalSpace.{u2} β) (SemilatticeInf.toHasInf.{u2} (TopologicalSpace.{u2} β) (Lattice.toSemilatticeInf.{u2} (TopologicalSpace.{u2} β) (ConditionallyCompleteLattice.toLattice.{u2} (TopologicalSpace.{u2} β) (CompleteLattice.toConditionallyCompleteLattice.{u2} (TopologicalSpace.{u2} β) (TopologicalSpace.completeLattice.{u2} β))))) tb1 tb2)) tc1 (fun (p : Prod.{u1, u2} α β) => f (Prod.fst.{u1, u2} α β p) (Prod.snd.{u1, u2} α β p)))
 but is expected to have type
-  forall {α : Type.{u3}} {β : Type.{u2}} {γ : Type.{u1}} {f : α -> β -> γ} {ta1 : TopologicalSpace.{u3} α} {ta2 : TopologicalSpace.{u3} α} {tb1 : TopologicalSpace.{u2} β} {tb2 : TopologicalSpace.{u2} β} {tc1 : TopologicalSpace.{u1} γ}, (Continuous.{max u3 u2, u1} (Prod.{u3, u2} α β) γ (instTopologicalSpaceProd.{u3, u2} α β ta1 tb1) tc1 (fun (p : Prod.{u3, u2} α β) => f (Prod.fst.{u3, u2} α β p) (Prod.snd.{u3, u2} α β p))) -> (Continuous.{max u3 u2, u1} (Prod.{u3, u2} α β) γ (instTopologicalSpaceProd.{u3, u2} α β (HasInf.inf.{u3} (TopologicalSpace.{u3} α) (Lattice.toHasInf.{u3} (TopologicalSpace.{u3} α) (ConditionallyCompleteLattice.toLattice.{u3} (TopologicalSpace.{u3} α) (CompleteLattice.toConditionallyCompleteLattice.{u3} (TopologicalSpace.{u3} α) (TopologicalSpace.instCompleteLatticeTopologicalSpace.{u3} α)))) ta1 ta2) (HasInf.inf.{u2} (TopologicalSpace.{u2} β) (Lattice.toHasInf.{u2} (TopologicalSpace.{u2} β) (ConditionallyCompleteLattice.toLattice.{u2} (TopologicalSpace.{u2} β) (CompleteLattice.toConditionallyCompleteLattice.{u2} (TopologicalSpace.{u2} β) (TopologicalSpace.instCompleteLatticeTopologicalSpace.{u2} β)))) tb1 tb2)) tc1 (fun (p : Prod.{u3, u2} α β) => f (Prod.fst.{u3, u2} α β p) (Prod.snd.{u3, u2} α β p)))
+  forall {α : Type.{u3}} {β : Type.{u2}} {γ : Type.{u1}} {f : α -> β -> γ} {ta1 : TopologicalSpace.{u3} α} {ta2 : TopologicalSpace.{u3} α} {tb1 : TopologicalSpace.{u2} β} {tb2 : TopologicalSpace.{u2} β} {tc1 : TopologicalSpace.{u1} γ}, (Continuous.{max u3 u2, u1} (Prod.{u3, u2} α β) γ (instTopologicalSpaceProd.{u3, u2} α β ta1 tb1) tc1 (fun (p : Prod.{u3, u2} α β) => f (Prod.fst.{u3, u2} α β p) (Prod.snd.{u3, u2} α β p))) -> (Continuous.{max u3 u2, u1} (Prod.{u3, u2} α β) γ (instTopologicalSpaceProd.{u3, u2} α β (Inf.inf.{u3} (TopologicalSpace.{u3} α) (Lattice.toInf.{u3} (TopologicalSpace.{u3} α) (ConditionallyCompleteLattice.toLattice.{u3} (TopologicalSpace.{u3} α) (CompleteLattice.toConditionallyCompleteLattice.{u3} (TopologicalSpace.{u3} α) (TopologicalSpace.instCompleteLatticeTopologicalSpace.{u3} α)))) ta1 ta2) (Inf.inf.{u2} (TopologicalSpace.{u2} β) (Lattice.toInf.{u2} (TopologicalSpace.{u2} β) (ConditionallyCompleteLattice.toLattice.{u2} (TopologicalSpace.{u2} β) (CompleteLattice.toConditionallyCompleteLattice.{u2} (TopologicalSpace.{u2} β) (TopologicalSpace.instCompleteLatticeTopologicalSpace.{u2} β)))) tb1 tb2)) tc1 (fun (p : Prod.{u3, u2} α β) => f (Prod.fst.{u3, u2} α β p) (Prod.snd.{u3, u2} α β p)))
 Case conversion may be inaccurate. Consider using '#align continuous_inf_dom_left₂ continuous_inf_dom_left₂ₓ'. -/
 /-- A version of `continuous_inf_dom_left` for binary functions -/
 theorem continuous_inf_dom_left₂ {α β γ} {f : α → β → γ} {ta1 ta2 : TopologicalSpace α}
@@ -777,9 +777,9 @@ theorem continuous_inf_dom_left₂ {α β γ} {f : α → β → γ} {ta1 ta2 :
 
 /- warning: continuous_inf_dom_right₂ -> continuous_inf_dom_right₂ is a dubious translation:
 lean 3 declaration is
-  forall {α : Type.{u1}} {β : Type.{u2}} {γ : Type.{u3}} {f : α -> β -> γ} {ta1 : TopologicalSpace.{u1} α} {ta2 : TopologicalSpace.{u1} α} {tb1 : TopologicalSpace.{u2} β} {tb2 : TopologicalSpace.{u2} β} {tc1 : TopologicalSpace.{u3} γ}, (Continuous.{max u1 u2, u3} (Prod.{u1, u2} α β) γ (Prod.topologicalSpace.{u1, u2} α β ta2 tb2) tc1 (fun (p : Prod.{u1, u2} α β) => f (Prod.fst.{u1, u2} α β p) (Prod.snd.{u1, u2} α β p))) -> (Continuous.{max u1 u2, u3} (Prod.{u1, u2} α β) γ (Prod.topologicalSpace.{u1, u2} α β (HasInf.inf.{u1} (TopologicalSpace.{u1} α) (SemilatticeInf.toHasInf.{u1} (TopologicalSpace.{u1} α) (Lattice.toSemilatticeInf.{u1} (TopologicalSpace.{u1} α) (ConditionallyCompleteLattice.toLattice.{u1} (TopologicalSpace.{u1} α) (CompleteLattice.toConditionallyCompleteLattice.{u1} (TopologicalSpace.{u1} α) (TopologicalSpace.completeLattice.{u1} α))))) ta1 ta2) (HasInf.inf.{u2} (TopologicalSpace.{u2} β) (SemilatticeInf.toHasInf.{u2} (TopologicalSpace.{u2} β) (Lattice.toSemilatticeInf.{u2} (TopologicalSpace.{u2} β) (ConditionallyCompleteLattice.toLattice.{u2} (TopologicalSpace.{u2} β) (CompleteLattice.toConditionallyCompleteLattice.{u2} (TopologicalSpace.{u2} β) (TopologicalSpace.completeLattice.{u2} β))))) tb1 tb2)) tc1 (fun (p : Prod.{u1, u2} α β) => f (Prod.fst.{u1, u2} α β p) (Prod.snd.{u1, u2} α β p)))
+  forall {α : Type.{u1}} {β : Type.{u2}} {γ : Type.{u3}} {f : α -> β -> γ} {ta1 : TopologicalSpace.{u1} α} {ta2 : TopologicalSpace.{u1} α} {tb1 : TopologicalSpace.{u2} β} {tb2 : TopologicalSpace.{u2} β} {tc1 : TopologicalSpace.{u3} γ}, (Continuous.{max u1 u2, u3} (Prod.{u1, u2} α β) γ (Prod.topologicalSpace.{u1, u2} α β ta2 tb2) tc1 (fun (p : Prod.{u1, u2} α β) => f (Prod.fst.{u1, u2} α β p) (Prod.snd.{u1, u2} α β p))) -> (Continuous.{max u1 u2, u3} (Prod.{u1, u2} α β) γ (Prod.topologicalSpace.{u1, u2} α β (Inf.inf.{u1} (TopologicalSpace.{u1} α) (SemilatticeInf.toHasInf.{u1} (TopologicalSpace.{u1} α) (Lattice.toSemilatticeInf.{u1} (TopologicalSpace.{u1} α) (ConditionallyCompleteLattice.toLattice.{u1} (TopologicalSpace.{u1} α) (CompleteLattice.toConditionallyCompleteLattice.{u1} (TopologicalSpace.{u1} α) (TopologicalSpace.completeLattice.{u1} α))))) ta1 ta2) (Inf.inf.{u2} (TopologicalSpace.{u2} β) (SemilatticeInf.toHasInf.{u2} (TopologicalSpace.{u2} β) (Lattice.toSemilatticeInf.{u2} (TopologicalSpace.{u2} β) (ConditionallyCompleteLattice.toLattice.{u2} (TopologicalSpace.{u2} β) (CompleteLattice.toConditionallyCompleteLattice.{u2} (TopologicalSpace.{u2} β) (TopologicalSpace.completeLattice.{u2} β))))) tb1 tb2)) tc1 (fun (p : Prod.{u1, u2} α β) => f (Prod.fst.{u1, u2} α β p) (Prod.snd.{u1, u2} α β p)))
 but is expected to have type
-  forall {α : Type.{u3}} {β : Type.{u2}} {γ : Type.{u1}} {f : α -> β -> γ} {ta1 : TopologicalSpace.{u3} α} {ta2 : TopologicalSpace.{u3} α} {tb1 : TopologicalSpace.{u2} β} {tb2 : TopologicalSpace.{u2} β} {tc1 : TopologicalSpace.{u1} γ}, (Continuous.{max u3 u2, u1} (Prod.{u3, u2} α β) γ (instTopologicalSpaceProd.{u3, u2} α β ta2 tb2) tc1 (fun (p : Prod.{u3, u2} α β) => f (Prod.fst.{u3, u2} α β p) (Prod.snd.{u3, u2} α β p))) -> (Continuous.{max u3 u2, u1} (Prod.{u3, u2} α β) γ (instTopologicalSpaceProd.{u3, u2} α β (HasInf.inf.{u3} (TopologicalSpace.{u3} α) (Lattice.toHasInf.{u3} (TopologicalSpace.{u3} α) (ConditionallyCompleteLattice.toLattice.{u3} (TopologicalSpace.{u3} α) (CompleteLattice.toConditionallyCompleteLattice.{u3} (TopologicalSpace.{u3} α) (TopologicalSpace.instCompleteLatticeTopologicalSpace.{u3} α)))) ta1 ta2) (HasInf.inf.{u2} (TopologicalSpace.{u2} β) (Lattice.toHasInf.{u2} (TopologicalSpace.{u2} β) (ConditionallyCompleteLattice.toLattice.{u2} (TopologicalSpace.{u2} β) (CompleteLattice.toConditionallyCompleteLattice.{u2} (TopologicalSpace.{u2} β) (TopologicalSpace.instCompleteLatticeTopologicalSpace.{u2} β)))) tb1 tb2)) tc1 (fun (p : Prod.{u3, u2} α β) => f (Prod.fst.{u3, u2} α β p) (Prod.snd.{u3, u2} α β p)))
+  forall {α : Type.{u3}} {β : Type.{u2}} {γ : Type.{u1}} {f : α -> β -> γ} {ta1 : TopologicalSpace.{u3} α} {ta2 : TopologicalSpace.{u3} α} {tb1 : TopologicalSpace.{u2} β} {tb2 : TopologicalSpace.{u2} β} {tc1 : TopologicalSpace.{u1} γ}, (Continuous.{max u3 u2, u1} (Prod.{u3, u2} α β) γ (instTopologicalSpaceProd.{u3, u2} α β ta2 tb2) tc1 (fun (p : Prod.{u3, u2} α β) => f (Prod.fst.{u3, u2} α β p) (Prod.snd.{u3, u2} α β p))) -> (Continuous.{max u3 u2, u1} (Prod.{u3, u2} α β) γ (instTopologicalSpaceProd.{u3, u2} α β (Inf.inf.{u3} (TopologicalSpace.{u3} α) (Lattice.toInf.{u3} (TopologicalSpace.{u3} α) (ConditionallyCompleteLattice.toLattice.{u3} (TopologicalSpace.{u3} α) (CompleteLattice.toConditionallyCompleteLattice.{u3} (TopologicalSpace.{u3} α) (TopologicalSpace.instCompleteLatticeTopologicalSpace.{u3} α)))) ta1 ta2) (Inf.inf.{u2} (TopologicalSpace.{u2} β) (Lattice.toInf.{u2} (TopologicalSpace.{u2} β) (ConditionallyCompleteLattice.toLattice.{u2} (TopologicalSpace.{u2} β) (CompleteLattice.toConditionallyCompleteLattice.{u2} (TopologicalSpace.{u2} β) (TopologicalSpace.instCompleteLatticeTopologicalSpace.{u2} β)))) tb1 tb2)) tc1 (fun (p : Prod.{u3, u2} α β) => f (Prod.fst.{u3, u2} α β p) (Prod.snd.{u3, u2} α β p)))
 Case conversion may be inaccurate. Consider using '#align continuous_inf_dom_right₂ continuous_inf_dom_right₂ₓ'. -/
 /-- A version of `continuous_inf_dom_right` for binary functions -/
 theorem continuous_inf_dom_right₂ {α β γ} {f : α → β → γ} {ta1 ta2 : TopologicalSpace α}

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

@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Johannes Hölzl, Mario Carneiro, Patrick Massot
 
 ! This file was ported from Lean 3 source module topology.constructions
-! leanprover-community/mathlib commit 4b83fd2855cbf5e9da1fcf0fc1eba89a48c21551
+! leanprover-community/mathlib commit 0c1f285a9f6e608ae2bdffa3f993eafb01eba829
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
@@ -1707,6 +1707,17 @@ theorem nhds_inr (x : β) : 𝓝 (inr x : Sum α β) = map inr (𝓝 x) :=
   (openEmbedding_inr.map_nhds_eq _).symm
 #align nhds_inr nhds_inr
 
+/- warning: continuous_sum_dom -> continuous_sum_dom 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 : (Sum.{u1, u2} α β) -> γ}, Iff (Continuous.{max u1 u2, u3} (Sum.{u1, u2} α β) γ (Sum.topologicalSpace.{u1, u2} α β _inst_1 _inst_2) _inst_3 f) (And (Continuous.{u1, u3} α γ _inst_1 _inst_3 (Function.comp.{succ u1, max (succ u1) (succ u2), succ u3} α (Sum.{u1, u2} α β) γ f (Sum.inl.{u1, u2} α β))) (Continuous.{u2, u3} β γ _inst_2 _inst_3 (Function.comp.{succ u2, max (succ u1) (succ u2), succ u3} β (Sum.{u1, u2} α β) γ f (Sum.inr.{u1, u2} α β))))
+but is expected to have type
+  forall {α : Type.{u2}} {β : Type.{u3}} {γ : Type.{u1}} [_inst_1 : TopologicalSpace.{u2} α] [_inst_2 : TopologicalSpace.{u3} β] [_inst_3 : TopologicalSpace.{u1} γ] {f : (Sum.{u2, u3} α β) -> γ}, Iff (Continuous.{max u2 u3, u1} (Sum.{u2, u3} α β) γ (instTopologicalSpaceSum.{u2, u3} α β _inst_1 _inst_2) _inst_3 f) (And (Continuous.{u2, u1} α γ _inst_1 _inst_3 (Function.comp.{succ u2, max (succ u2) (succ u3), succ u1} α (Sum.{u2, u3} α β) γ f (Sum.inl.{u2, u3} α β))) (Continuous.{u3, u1} β γ _inst_2 _inst_3 (Function.comp.{succ u3, max (succ u2) (succ u3), succ u1} β (Sum.{u2, u3} α β) γ f (Sum.inr.{u2, u3} α β))))
+Case conversion may be inaccurate. Consider using '#align continuous_sum_dom continuous_sum_domₓ'. -/
+theorem continuous_sum_dom {f : Sum α β → γ} :
+    Continuous f ↔ Continuous (f ∘ Sum.inl) ∧ Continuous (f ∘ Sum.inr) := by
+  simp only [continuous_sup_dom, continuous_coinduced_dom]
+#align continuous_sum_dom continuous_sum_dom
+
 /- warning: continuous_sum_elim -> continuous_sum_elim 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 -> γ} {g : β -> γ}, Iff (Continuous.{max u1 u2, u3} (Sum.{u1, u2} α β) γ (Sum.topologicalSpace.{u1, u2} α β _inst_1 _inst_2) _inst_3 (Sum.elim.{u1, u2, succ u3} α β γ f g)) (And (Continuous.{u1, u3} α γ _inst_1 _inst_3 f) (Continuous.{u2, u3} β γ _inst_2 _inst_3 g))
@@ -1714,8 +1725,8 @@ but is expected to have type
   forall {α : Type.{u2}} {β : Type.{u3}} {γ : Type.{u1}} [_inst_1 : TopologicalSpace.{u2} α] [_inst_2 : TopologicalSpace.{u3} β] [_inst_3 : TopologicalSpace.{u1} γ] {f : α -> γ} {g : β -> γ}, Iff (Continuous.{max u3 u2, u1} (Sum.{u2, u3} α β) γ (instTopologicalSpaceSum.{u2, u3} α β _inst_1 _inst_2) _inst_3 (Sum.elim.{u2, u3, succ u1} α β γ f g)) (And (Continuous.{u2, u1} α γ _inst_1 _inst_3 f) (Continuous.{u3, u1} β γ _inst_2 _inst_3 g))
 Case conversion may be inaccurate. Consider using '#align continuous_sum_elim continuous_sum_elimₓ'. -/
 theorem continuous_sum_elim {f : α → γ} {g : β → γ} :
-    Continuous (Sum.elim f g) ↔ Continuous f ∧ Continuous g := by
-  simp only [continuous_sup_dom, continuous_coinduced_dom, Sum.elim_comp_inl, Sum.elim_comp_inr]
+    Continuous (Sum.elim f g) ↔ Continuous f ∧ Continuous g :=
+  continuous_sum_dom
 #align continuous_sum_elim continuous_sum_elim
 
 /- warning: continuous.sum_elim -> Continuous.sum_elim is a dubious translation:

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

These · are scoping when there is a single active goal.

These were found using a modification of the linter at #12339.

@@ -796,8 +796,8 @@ theorem isOpen_prod_iff' {s : Set X} {t : Set Y} :
       show IsOpen s
       · rw [← fst_image_prod s st.2]
         exact isOpenMap_fst _ H
-      show IsOpen t
-      · rw [← snd_image_prod st.1 t]
+      · show IsOpen t
+        rw [← snd_image_prod st.1 t]
         exact isOpenMap_snd _ H
     · intro H
       simp only [st.1.ne_empty, st.2.ne_empty, not_false_iff, or_false_iff] at H

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>

@@ -1420,7 +1420,7 @@ theorem isOpen_pi_iff {s : Set (∀ a, π a)} :
         (∀ a, a ∈ I → IsOpen (u a) ∧ f a ∈ u a) ∧ (I : Set ι).pi u ⊆ s := by
   rw [isOpen_iff_nhds]
   simp_rw [le_principal_iff, nhds_pi, Filter.mem_pi', mem_nhds_iff]
-  refine ball_congr fun a _ => ⟨?_, ?_⟩
+  refine forall₂_congr fun a _ => ⟨?_, ?_⟩
   · rintro ⟨I, t, ⟨h1, h2⟩⟩
     refine ⟨I, fun a => eval a '' (I : Set ι).pi fun a => (h1 a).choose, fun i hi => ?_, ?_⟩
     · simp_rw [Set.eval_image_pi (Finset.mem_coe.mpr hi)
@@ -1447,7 +1447,7 @@ theorem isOpen_pi_iff' [Finite ι] {s : Set (∀ a, π a)} :
   cases nonempty_fintype ι
   rw [isOpen_iff_nhds]
   simp_rw [le_principal_iff, nhds_pi, Filter.mem_pi', mem_nhds_iff]
-  refine ball_congr fun a _ => ⟨?_, ?_⟩
+  refine forall₂_congr fun a _ => ⟨?_, ?_⟩
   · rintro ⟨I, t, ⟨h1, h2⟩⟩
     refine
       ⟨fun i => (h1 i).choose,

A mix of various changes; generated with a script and manually tweaked.

@@ -545,7 +545,7 @@ theorem nhds_prod_eq {x : X} {y : Y} : 𝓝 (x, y) = 𝓝 x ×ˢ 𝓝 y := by
     (t₂ := TopologicalSpace.induced Prod.snd _), nhds_induced, nhds_induced]
 #align nhds_prod_eq nhds_prod_eq
 
--- Porting note: moved from `topology.continuous_on`
+-- Porting note: moved from `Topology.ContinuousOn`
 theorem nhdsWithin_prod_eq (x : X) (y : Y) (s : Set X) (t : Set Y) :
     𝓝[s ×ˢ t] (x, y) = 𝓝[s] x ×ˢ 𝓝[t] y := by
   simp only [nhdsWithin, nhds_prod_eq, ← prod_inf_prod, prod_principal_principal]

Co-authored-by: @vbeffara (port to Lean4), @tb65536 (suggested to skip the totally bounded case and go straight to compactness using Tykhonov)

This was discussed on Zulip recently and a while ago.

@@ -1352,6 +1352,11 @@ lemma Pi.induced_restrict (S : Set ι) :
   simp (config := { unfoldPartialApp := true }) [← iInf_subtype'', ← induced_precomp' ((↑) : S → ι),
     Set.restrict]
 
+lemma Pi.induced_restrict_sUnion (𝔖 : Set (Set ι)) :
+    induced (⋃₀ 𝔖).restrict (Pi.topologicalSpace (Y := fun i : (⋃₀ 𝔖) ↦ π i)) =
+    ⨅ S ∈ 𝔖, induced S.restrict Pi.topologicalSpace := by
+  simp_rw [Pi.induced_restrict, iInf_sUnion]
+
 theorem Filter.Tendsto.update [DecidableEq ι] {l : Filter Y} {f : Y → ∀ i, π i} {x : ∀ i, π i}
     (hf : Tendsto f l (𝓝 x)) (i : ι) {g : Y → π i} {xi : π i} (hg : Tendsto g l (𝓝 xi)) :
     Tendsto (fun a => update (f a) i (g a)) l (𝓝 <| update x i xi) :=

• add the theorem image_restrictPreimage; use it to simplify the proof of Set.restrictPreimage_isClosedMap
• add the equivalent for open maps Set.restrictPreimage_isOpenMap.
• delete IsClosedMap.restrictPreimage, which duplicates Set.restrictPreimage_isClosedMap

Co-authored-by: BGuillemet <115193742+BGuillemet@users.noreply.github.com>

@@ -1097,17 +1097,6 @@ theorem IsOpenMap.restrict {f : X → Y} (hf : IsOpenMap f) {s : Set X} (hs : Is
   hf.comp hs.isOpenMap_subtype_val
 #align is_open_map.restrict IsOpenMap.restrict
 
-lemma IsClosedMap.restrictPreimage {f : X → Y} (hcl : IsClosedMap f) (T : Set Y) :
-    IsClosedMap (T.restrictPreimage f) := by
-  rw [isClosedMap_iff_clusterPt] at hcl ⊢
-  intro A ⟨y, hyT⟩ hy
-  rw [Set.restrictPreimage, MapClusterPt, ← inducing_subtype_val.mapClusterPt_iff, MapClusterPt,
-      map_map, MapsTo.restrict_commutes, ← map_map, ← MapClusterPt, map_principal] at hy
-  rcases hcl _ y hy with ⟨x, hxy, hx⟩
-  have hxT : f x ∈ T := hxy ▸ hyT
-  refine ⟨⟨x, hxT⟩, Subtype.ext hxy, ?_⟩
-  rwa [← inducing_subtype_val.mapClusterPt_iff, MapClusterPt, map_principal]
-
 nonrec theorem IsClosed.closedEmbedding_subtype_val {s : Set X} (hs : IsClosed s) :
     ClosedEmbedding ((↑) : s → X) :=
   closedEmbedding_subtype_val hs

• Add Tendsto.fst, Tendsto.snd and Tendsto.apply (these are about product of filters)
• Move the current Tendsto.apply to Tendsto.apply_nhds, and add Tendsto.fst_nhds and Tendsto.snd_nhds (these are about neighborhoods in a product space)

@@ -366,6 +366,10 @@ theorem ContinuousAt.fst'' {f : X → Z} {x : X × Y} (hf : ContinuousAt f x.fst
   hf.comp continuousAt_fst
 #align continuous_at.fst'' ContinuousAt.fst''
 
+theorem Filter.Tendsto.fst_nhds {l : Filter X} {f : X → Y × Z} {p : Y × Z}
+    (h : Tendsto f l (𝓝 p)) : Tendsto (fun a ↦ (f a).1) l (𝓝 <| p.1) :=
+  continuousAt_fst.tendsto.comp h
+
 @[continuity]
 theorem continuous_snd : Continuous (@Prod.snd X Y) :=
   (continuous_prod_mk.1 continuous_id).2
@@ -405,6 +409,10 @@ theorem ContinuousAt.snd'' {f : Y → Z} {x : X × Y} (hf : ContinuousAt f x.snd
   hf.comp continuousAt_snd
 #align continuous_at.snd'' ContinuousAt.snd''
 
+theorem Filter.Tendsto.snd_nhds {l : Filter X} {f : X → Y × Z} {p : Y × Z}
+    (h : Tendsto f l (𝓝 p)) : Tendsto (fun a ↦ (f a).2) l (𝓝 <| p.2) :=
+  continuousAt_snd.tendsto.comp h
+
 @[continuity, fun_prop]
 theorem Continuous.prod_mk {f : Z → X} {g : Z → Y} (hf : Continuous f) (hg : Continuous g) :
     Continuous fun x => (f x, g x) :=
@@ -1294,10 +1302,10 @@ theorem continuousAt_apply (i : ι) (x : ∀ i, π i) : ContinuousAt (fun p : 
   (continuous_apply i).continuousAt
 #align continuous_at_apply continuousAt_apply
 
-theorem Filter.Tendsto.apply {l : Filter Y} {f : Y → ∀ i, π i} {x : ∀ i, π i}
+theorem Filter.Tendsto.apply_nhds {l : Filter Y} {f : Y → ∀ i, π i} {x : ∀ i, π i}
     (h : Tendsto f l (𝓝 x)) (i : ι) : Tendsto (fun a => f a i) l (𝓝 <| x i) :=
   (continuousAt_apply i _).tendsto.comp h
-#align filter.tendsto.apply Filter.Tendsto.apply
+#align filter.tendsto.apply Filter.Tendsto.apply_nhds
 
 theorem nhds_pi {a : ∀ i, π i} : 𝓝 a = pi fun i => 𝓝 (a i) := by
   simp only [nhds_iInf, nhds_induced, Filter.pi]
@@ -1358,7 +1366,7 @@ lemma Pi.induced_restrict (S : Set ι) :
 theorem Filter.Tendsto.update [DecidableEq ι] {l : Filter Y} {f : Y → ∀ i, π i} {x : ∀ i, π i}
     (hf : Tendsto f l (𝓝 x)) (i : ι) {g : Y → π i} {xi : π i} (hg : Tendsto g l (𝓝 xi)) :
     Tendsto (fun a => update (f a) i (g a)) l (𝓝 <| update x i xi) :=
-  tendsto_pi_nhds.2 fun j => by rcases eq_or_ne j i with (rfl | hj) <;> simp [*, hf.apply]
+  tendsto_pi_nhds.2 fun j => by rcases eq_or_ne j i with (rfl | hj) <;> simp [*, hf.apply_nhds]
 #align filter.tendsto.update Filter.Tendsto.update
 
 theorem ContinuousAt.update [DecidableEq ι] {x : X} (hf : ContinuousAt f x) (i : ι) {g : X → π i}

@@ -1534,12 +1534,9 @@ theorem pi_generateFrom_eq_finite {π : ι → Type*} {g : ∀ a, Set (Set (π a
     by_cases a ∈ i <;> simp [*]
 #align pi_generate_from_eq_finite pi_generateFrom_eq_finite
 
--- Porting note (#10756): new lemma
 theorem induced_to_pi {X : Type*} (f : X → ∀ i, π i) :
     induced f Pi.topologicalSpace = ⨅ i, induced (f · i) inferInstance := by
-  erw [induced_iInf]
-  simp only [induced_compose]
-  rfl
+  simp_rw [Pi.topologicalSpace, induced_iInf, induced_compose, Function.comp]
 
 /-- Suppose `π i` is a family of topological spaces indexed by `i : ι`, and `X` is a type
 endowed with a family of maps `f i : X → π i` for every `i : ι`, hence inducing a
@@ -1721,9 +1718,7 @@ section ULift
 
 theorem ULift.isOpen_iff [TopologicalSpace X] {s : Set (ULift.{v} X)} :
     IsOpen s ↔ IsOpen (ULift.up ⁻¹' s) := by
-  unfold ULift.topologicalSpace
-  erw [← Equiv.ulift.coinduced_symm]
-  rfl
+  rw [ULift.topologicalSpace, ← Equiv.ulift_apply, ← Equiv.ulift.coinduced_symm, ← isOpen_coinduced]
 
 theorem ULift.isClosed_iff [TopologicalSpace X] {s : Set (ULift.{v} X)} :
     IsClosed s ↔ IsClosed (ULift.up ⁻¹' s) := by

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

@@ -1608,7 +1608,7 @@ theorem isClosedMap_sigmaMk {i : ι} : IsClosedMap (@Sigma.mk ι σ i) := by
 #align is_closed_map_sigma_mk isClosedMap_sigmaMk
 
 theorem isClosed_range_sigmaMk {i : ι} : IsClosed (Set.range (@Sigma.mk ι σ i)) :=
-  isClosedMap_sigmaMk.closed_range
+  isClosedMap_sigmaMk.isClosed_range
 #align is_closed_range_sigma_mk isClosed_range_sigmaMk
 
 theorem openEmbedding_sigmaMk {i : ι} : OpenEmbedding (@Sigma.mk ι σ i) :=

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)

@@ -1222,7 +1222,6 @@ end Subtype
 section Quotient
 
 variable [TopologicalSpace X] [TopologicalSpace Y] [TopologicalSpace Z]
-
 variable {r : X → X → Prop} {s : Setoid X}
 
 theorem quotientMap_quot_mk : QuotientMap (@Quot.mk X r) :=

See the Zulip thread

@@ -1209,6 +1209,14 @@ theorem DiscreteTopology.of_subset {X : Type*} [TopologicalSpace X] {s t : Set X
   (embedding_inclusion ts).discreteTopology
 #align discrete_topology.of_subset DiscreteTopology.of_subset
 
+/-- Let `s` be a discrete subset of a topological space. Then the preimage of `s` by
+a continuous injective map is also discrete. -/
+theorem DiscreteTopology.preimage_of_continuous_injective {X Y : Type*} [TopologicalSpace X]
+    [TopologicalSpace Y] (s : Set Y) [DiscreteTopology s] {f : X → Y} (hc : Continuous f)
+    (hinj : Function.Injective f) : DiscreteTopology (f ⁻¹' s) :=
+  DiscreteTopology.of_continuous_injective (β := s) (Continuous.restrict
+    (by exact fun _ x ↦ x) hc) ((Set.MapsTo.restrict_inj _).mpr <| Set.injOn_of_injective hinj _)
+
 end Subtype
 
 section Quotient

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

• "added theorem"
• "added theorems"
• "new theorem"
• "new theorems"
• "added lemma"
• "new lemma"
• "new lemmas"

@@ -949,7 +949,7 @@ theorem isOpen_sum_iff {s : Set (X ⊕ Y)} : IsOpen s ↔ IsOpen (inl ⁻¹' s)
   Iff.rfl
 #align is_open_sum_iff isOpen_sum_iff
 
--- Porting note: new theorem
+-- Porting note (#10756): new theorem
 theorem isClosed_sum_iff {s : Set (X ⊕ Y)} :
     IsClosed s ↔ IsClosed (inl ⁻¹' s) ∧ IsClosed (inr ⁻¹' s) := by
   simp only [← isOpen_compl_iff, isOpen_sum_iff, preimage_compl]

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.

@@ -508,18 +508,22 @@ lemma isClosedMap_swap : IsClosedMap (Prod.swap : X × Y → Y × X) := fun s hs
   rw [image_swap_eq_preimage_swap]
   exact hs.preimage continuous_swap
 
-theorem continuous_uncurry_left {f : X → Y → Z} (x : X) (h : Continuous (uncurry f)) :
+theorem Continuous.uncurry_left {f : X → Y → Z} (x : X) (h : Continuous (uncurry f)) :
     Continuous (f x) :=
   h.comp (Continuous.Prod.mk _)
-#align continuous_uncurry_left continuous_uncurry_left
+#align continuous_uncurry_left Continuous.uncurry_left
 
-theorem continuous_uncurry_right {f : X → Y → Z} (y : Y) (h : Continuous (uncurry f)) :
+theorem Continuous.uncurry_right {f : X → Y → Z} (y : Y) (h : Continuous (uncurry f)) :
     Continuous fun a => f a y :=
   h.comp (Continuous.Prod.mk_left _)
-#align continuous_uncurry_right continuous_uncurry_right
+#align continuous_uncurry_right Continuous.uncurry_right
+
+-- 2024-03-09
+@[deprecated] alias continuous_uncurry_left := Continuous.uncurry_left
+@[deprecated] alias continuous_uncurry_right := Continuous.uncurry_right
 
 theorem continuous_curry {g : X × Y → Z} (x : X) (h : Continuous g) : Continuous (curry g x) :=
-  continuous_uncurry_left x h
+  Continuous.uncurry_left x h
 #align continuous_curry continuous_curry
 
 theorem IsOpen.prod {s : Set X} {t : Set Y} (hs : IsOpen s) (ht : IsOpen t) : IsOpen (s ×ˢ t) :=
@@ -636,6 +640,29 @@ theorem ContinuousAt.prod_map' {f : X → Z} {g : Y → W} {x : X} {y : Y} (hf :
   hf.fst'.prod hg.snd'
 #align continuous_at.prod_map' ContinuousAt.prod_map'
 
+theorem ContinuousAt.comp₂ {f : Y × Z → W} {g : X → Y} {h : X → Z} {x : X}
+    (hf : ContinuousAt f (g x, h x)) (hg : ContinuousAt g x) (hh : ContinuousAt h x) :
+    ContinuousAt (fun x ↦ f (g x, h x)) x :=
+  ContinuousAt.comp hf (hg.prod hh)
+
+theorem ContinuousAt.comp₂_of_eq {f : Y × Z → W} {g : X → Y} {h : X → Z} {x : X} {y : Y × Z}
+    (hf : ContinuousAt f y) (hg : ContinuousAt g x) (hh : ContinuousAt h x) (e : (g x, h x) = y) :
+    ContinuousAt (fun x ↦ f (g x, h x)) x := by
+  rw [← e] at hf
+  exact hf.comp₂ hg hh
+
+/-- Continuous functions on products are continuous in their first argument -/
+theorem Continuous.curry_left {f : X × Y → Z} (hf : Continuous f) {y : Y} :
+    Continuous fun x ↦ f (x, y) :=
+  hf.comp (continuous_id.prod_mk continuous_const)
+alias Continuous.along_fst := Continuous.curry_left
+
+/-- Continuous functions on products are continuous in their second argument -/
+theorem Continuous.curry_right {f : X × Y → Z} (hf : Continuous f) {x : X} :
+    Continuous fun y ↦ f (x, y) :=
+  hf.comp (continuous_const.prod_mk continuous_id)
+alias Continuous.along_snd := Continuous.curry_right
+
 -- todo: reformulate using `Set.image2`
 -- todo: prove a version of `generateFrom_union` with `image2 (∩) s t` in the LHS and use it here
 theorem prod_generateFrom_generateFrom_eq {X Y : Type*} {s : Set (Set X)} {t : Set (Set Y)}

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

• "new lemma"
• "added lemma"

@@ -1103,7 +1103,7 @@ theorem Subtype.dense_iff {s : Set X} {t : Set s} : Dense t ↔ s ⊆ closure ((
   rfl
 #align subtype.dense_iff Subtype.dense_iff
 
--- Porting note: new lemma
+-- Porting note (#10756): new lemma
 theorem map_nhds_subtype_val {s : Set X} (x : s) : map ((↑) : s → X) (𝓝 x) = 𝓝[s] ↑x := by
   rw [inducing_subtype_val.map_nhds_eq, Subtype.range_val]
 
@@ -1500,7 +1500,7 @@ theorem pi_generateFrom_eq_finite {π : ι → Type*} {g : ∀ a, Set (Set (π a
     by_cases a ∈ i <;> simp [*]
 #align pi_generate_from_eq_finite pi_generateFrom_eq_finite
 
--- Porting note: new lemma
+-- Porting note (#10756): new lemma
 theorem induced_to_pi {X : Type*} (f : X → ∀ i, π i) :
     induced f Pi.topologicalSpace = ⨅ i, induced (f · i) inferInstance := by
   erw [induced_iInf]

Classifies by adding issue number #11215 to porting notes claiming "TODO".

@@ -319,7 +319,7 @@ section Prod
 variable [TopologicalSpace X] [TopologicalSpace Y] [TopologicalSpace Z] [TopologicalSpace W]
   [TopologicalSpace ε] [TopologicalSpace ζ]
 
--- Porting note: todo: Lean 4 fails to deduce implicit args
+-- Porting note (#11215): TODO: Lean 4 fails to deduce implicit args
 @[simp] theorem continuous_prod_mk {f : X → Y} {g : X → Z} :
     (Continuous fun x => (f x, g x)) ↔ Continuous f ∧ Continuous g :=
   (@continuous_inf_rng X (Y × Z) _ _ (TopologicalSpace.induced Prod.fst _)
@@ -526,7 +526,7 @@ theorem IsOpen.prod {s : Set X} {t : Set Y} (hs : IsOpen s) (ht : IsOpen t) : Is
   (hs.preimage continuous_fst).inter (ht.preimage continuous_snd)
 #align is_open.prod IsOpen.prod
 
--- Porting note: todo: Lean fails to find `t₁` and `t₂` by unification
+-- Porting note (#11215): TODO: Lean fails to find `t₁` and `t₂` by unification
 theorem nhds_prod_eq {x : X} {y : Y} : 𝓝 (x, y) = 𝓝 x ×ˢ 𝓝 y := by
   dsimp only [SProd.sprod]
   rw [Filter.prod, instTopologicalSpaceProd, nhds_inf (t₁ := TopologicalSpace.induced Prod.fst _)
@@ -681,7 +681,7 @@ theorem prod_eq_generateFrom :
         GenerateOpen.basic _ ⟨univ, t, by simpa [Set.prod_eq] using ht⟩))
 #align prod_eq_generate_from prod_eq_generateFrom
 
--- Porting note: todo: align with `mem_nhds_prod_iff'`
+-- Porting note (#11215): TODO: align with `mem_nhds_prod_iff'`
 theorem isOpen_prod_iff {s : Set (X × Y)} :
     IsOpen s ↔ ∀ a b, (a, b) ∈ s →
       ∃ u v, IsOpen u ∧ IsOpen v ∧ a ∈ u ∧ b ∈ v ∧ u ×ˢ v ⊆ s :=
@@ -1345,7 +1345,7 @@ theorem continuous_update [DecidableEq ι] (i : ι) :
 #align continuous_update continuous_update
 
 /-- `Pi.mulSingle i x` is continuous in `x`. -/
--- Porting note: todo: restore @[continuity]
+-- Porting note (#11215): TODO: restore @[continuity]
 @[to_additive "`Pi.single i x` is continuous in `x`."]
 theorem continuous_mulSingle [∀ i, One (π i)] [DecidableEq ι] (i : ι) :
     Continuous fun x => (Pi.mulSingle i x : ∀ i, π i) :=

ball for "bounded forall" and bex for "bounded exists" are from experience very confusing abbreviations. This PR renames them to forall_mem and exists_mem in the few Set lemma names that mention them.

Also deprecate ball_image_of_ball, mem_image_elim, mem_image_elim_on since those lemmas are duplicates of the renamed lemmas (apart from argument order and implicitness, which I am also fixing by making the binder in the RHS of forall_mem_image semi-implicit), have obscure names and are completely unused.

@@ -675,9 +675,9 @@ theorem prod_eq_generateFrom :
       generateFrom { g | ∃ (s : Set X) (t : Set Y), IsOpen s ∧ IsOpen t ∧ g = s ×ˢ t } :=
   le_antisymm (le_generateFrom fun g ⟨s, t, hs, ht, g_eq⟩ => g_eq.symm ▸ hs.prod ht)
     (le_inf
-      (ball_image_of_ball fun t ht =>
+      (forall_mem_image.2 fun t ht =>
         GenerateOpen.basic _ ⟨t, univ, by simpa [Set.prod_eq] using ht⟩)
-      (ball_image_of_ball fun t ht =>
+      (forall_mem_image.2 fun t ht =>
         GenerateOpen.basic _ ⟨univ, t, by simpa [Set.prod_eq] using ht⟩))
 #align prod_eq_generate_from prod_eq_generateFrom
 

We remove all but one open Classicals, instead preferring to use open scoped Classical. The only real side-effect this led to is moving a couple declarations to use Exists.choose instead of Classical.choose.

The first few commits are explicitly labelled regex replaces for ease of review.

@@ -36,7 +36,8 @@ product, sum, disjoint union, subspace, quotient space
 
 noncomputable section
 
-open Topology TopologicalSpace Set Filter Function Classical
+open scoped Classical
+open Topology TopologicalSpace Set Filter Function
 
 universe u v
 

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".

@@ -318,7 +318,7 @@ section Prod
 variable [TopologicalSpace X] [TopologicalSpace Y] [TopologicalSpace Z] [TopologicalSpace W]
   [TopologicalSpace ε] [TopologicalSpace ζ]
 
--- porting note: todo: Lean 4 fails to deduce implicit args
+-- Porting note: todo: Lean 4 fails to deduce implicit args
 @[simp] theorem continuous_prod_mk {f : X → Y} {g : X → Z} :
     (Continuous fun x => (f x, g x)) ↔ Continuous f ∧ Continuous g :=
   (@continuous_inf_rng X (Y × Z) _ _ (TopologicalSpace.induced Prod.fst _)
@@ -525,14 +525,14 @@ theorem IsOpen.prod {s : Set X} {t : Set Y} (hs : IsOpen s) (ht : IsOpen t) : Is
   (hs.preimage continuous_fst).inter (ht.preimage continuous_snd)
 #align is_open.prod IsOpen.prod
 
--- porting note: todo: Lean fails to find `t₁` and `t₂` by unification
+-- Porting note: todo: Lean fails to find `t₁` and `t₂` by unification
 theorem nhds_prod_eq {x : X} {y : Y} : 𝓝 (x, y) = 𝓝 x ×ˢ 𝓝 y := by
   dsimp only [SProd.sprod]
   rw [Filter.prod, instTopologicalSpaceProd, nhds_inf (t₁ := TopologicalSpace.induced Prod.fst _)
     (t₂ := TopologicalSpace.induced Prod.snd _), nhds_induced, nhds_induced]
 #align nhds_prod_eq nhds_prod_eq
 
--- porting note: moved from `topology.continuous_on`
+-- Porting note: moved from `topology.continuous_on`
 theorem nhdsWithin_prod_eq (x : X) (y : Y) (s : Set X) (t : Set Y) :
     𝓝[s ×ˢ t] (x, y) = 𝓝[s] x ×ˢ 𝓝[t] y := by
   simp only [nhdsWithin, nhds_prod_eq, ← prod_inf_prod, prod_principal_principal]
@@ -548,7 +548,7 @@ theorem mem_nhdsWithin_prod_iff {x : X} {y : Y} {s : Set (X × Y)} {tx : Set X}
     s ∈ 𝓝[tx ×ˢ ty] (x, y) ↔ ∃ u ∈ 𝓝[tx] x, ∃ v ∈ 𝓝[ty] y, u ×ˢ v ⊆ s := by
   rw [nhdsWithin_prod_eq, mem_prod_iff]
 
--- porting note: moved up
+-- Porting note: moved up
 theorem Filter.HasBasis.prod_nhds {ιX ιY : Type*} {px : ιX → Prop} {py : ιY → Prop}
     {sx : ιX → Set X} {sy : ιY → Set Y} {x : X} {y : Y} (hx : (𝓝 x).HasBasis px sx)
     (hy : (𝓝 y).HasBasis py sy) :
@@ -557,7 +557,7 @@ theorem Filter.HasBasis.prod_nhds {ιX ιY : Type*} {px : ιX → Prop} {py : ι
   exact hx.prod hy
 #align filter.has_basis.prod_nhds Filter.HasBasis.prod_nhds
 
--- porting note: moved up
+-- Porting note: moved up
 theorem Filter.HasBasis.prod_nhds' {ιX ιY : Type*} {pX : ιX → Prop} {pY : ιY → Prop}
     {sx : ιX → Set X} {sy : ιY → Set Y} {p : X × Y} (hx : (𝓝 p.1).HasBasis pX sx)
     (hy : (𝓝 p.2).HasBasis pY sy) :
@@ -680,7 +680,7 @@ theorem prod_eq_generateFrom :
         GenerateOpen.basic _ ⟨univ, t, by simpa [Set.prod_eq] using ht⟩))
 #align prod_eq_generate_from prod_eq_generateFrom
 
--- porting note: todo: align with `mem_nhds_prod_iff'`
+-- Porting note: todo: align with `mem_nhds_prod_iff'`
 theorem isOpen_prod_iff {s : Set (X × Y)} :
     IsOpen s ↔ ∀ a b, (a, b) ∈ s →
       ∃ u v, IsOpen u ∧ IsOpen v ∧ a ∈ u ∧ b ∈ v ∧ u ×ˢ v ⊆ s :=
@@ -908,12 +908,12 @@ theorem continuous_isRight : Continuous (isRight : X ⊕ Y → Bool) :=
   continuous_sum_dom.2 ⟨continuous_const, continuous_const⟩
 
 @[continuity]
--- porting note: the proof was `continuous_sup_rng_left continuous_coinduced_rng`
+-- Porting note: the proof was `continuous_sup_rng_left continuous_coinduced_rng`
 theorem continuous_inl : Continuous (@inl X Y) := ⟨fun _ => And.left⟩
 #align continuous_inl continuous_inl
 
 @[continuity]
--- porting note: the proof was `continuous_sup_rng_right continuous_coinduced_rng`
+-- Porting note: the proof was `continuous_sup_rng_right continuous_coinduced_rng`
 theorem continuous_inr : Continuous (@inr X Y) := ⟨fun _ => And.right⟩
 #align continuous_inr continuous_inr
 
@@ -921,7 +921,7 @@ theorem isOpen_sum_iff {s : Set (X ⊕ Y)} : IsOpen s ↔ IsOpen (inl ⁻¹' s)
   Iff.rfl
 #align is_open_sum_iff isOpen_sum_iff
 
--- porting note: new theorem
+-- Porting note: new theorem
 theorem isClosed_sum_iff {s : Set (X ⊕ Y)} :
     IsClosed s ↔ IsClosed (inl ⁻¹' s) ∧ IsClosed (inr ⁻¹' s) := by
   simp only [← isOpen_compl_iff, isOpen_sum_iff, preimage_compl]
@@ -1102,7 +1102,7 @@ theorem Subtype.dense_iff {s : Set X} {t : Set s} : Dense t ↔ s ⊆ closure ((
   rfl
 #align subtype.dense_iff Subtype.dense_iff
 
--- porting note: new lemma
+-- Porting note: new lemma
 theorem map_nhds_subtype_val {s : Set X} (x : s) : map ((↑) : s → X) (𝓝 x) = 𝓝[s] ↑x := by
   rw [inducing_subtype_val.map_nhds_eq, Subtype.range_val]
 
@@ -1344,7 +1344,7 @@ theorem continuous_update [DecidableEq ι] (i : ι) :
 #align continuous_update continuous_update
 
 /-- `Pi.mulSingle i x` is continuous in `x`. -/
--- porting note: todo: restore @[continuity]
+-- Porting note: todo: restore @[continuity]
 @[to_additive "`Pi.single i x` is continuous in `x`."]
 theorem continuous_mulSingle [∀ i, One (π i)] [DecidableEq ι] (i : ι) :
     Continuous fun x => (Pi.mulSingle i x : ∀ i, π i) :=
@@ -1499,7 +1499,7 @@ theorem pi_generateFrom_eq_finite {π : ι → Type*} {g : ∀ a, Set (Set (π a
     by_cases a ∈ i <;> simp [*]
 #align pi_generate_from_eq_finite pi_generateFrom_eq_finite
 
--- porting note: new lemma
+-- Porting note: new lemma
 theorem induced_to_pi {X : Type*} (f : X → ∀ i, π i) :
     induced f Pi.topologicalSpace = ⨅ i, induced (f · i) inferInstance := by
   erw [induced_iInf]
@@ -1537,7 +1537,7 @@ theorem continuous_sigmaMk {i : ι} : Continuous (@Sigma.mk ι σ i) :=
   continuous_iSup_rng continuous_coinduced_rng
 #align continuous_sigma_mk continuous_sigmaMk
 
--- porting note: the proof was `by simp only [isOpen_iSup_iff, isOpen_coinduced]`
+-- Porting note: the proof was `by simp only [isOpen_iSup_iff, isOpen_coinduced]`
 theorem isOpen_sigma_iff {s : Set (Sigma σ)} : IsOpen s ↔ ∀ i, IsOpen (Sigma.mk i ⁻¹' s) := by
   delta instTopologicalSpaceSigma
   rw [isOpen_iSup_iff]

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

@@ -1725,12 +1725,12 @@ variable [TopologicalSpace X] {s : Set X} {t : Set s}
 theorem IsOpen.trans (ht : IsOpen t) (hs : IsOpen s) : IsOpen (t : Set X) := by
   rcases isOpen_induced_iff.mp ht with ⟨s', hs', rfl⟩
   rw [Subtype.image_preimage_coe]
-  exact hs'.inter hs
+  exact hs.inter hs'
 
 theorem IsClosed.trans (ht : IsClosed t) (hs : IsClosed s) : IsClosed (t : Set X) := by
   rcases isClosed_induced_iff.mp ht with ⟨s', hs', rfl⟩
   rw [Subtype.image_preimage_coe]
-  convert hs'.inter hs
+  exact hs.inter hs'
 
 end Monad
 

Provide continuity criteria for functions X → Bool and, more generally, X → Y, where Y has discrete topology.

@@ -866,6 +866,15 @@ theorem embedding_prod_mk (x : X) : Embedding (Prod.mk x : Y → X × Y) :=
 
 end Prod
 
+section Bool
+
+lemma continuous_bool_rng [TopologicalSpace X] {f : X → Bool} (b : Bool) :
+    Continuous f ↔ IsClopen (f ⁻¹' {b}) := by
+  rw [continuous_discrete_rng, Bool.forall_bool' b, IsClopen, ← isOpen_compl_iff, ← preimage_compl,
+    Bool.compl_singleton, and_comm]
+
+end Bool
+
 section Sum
 
 open Sum

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

@@ -332,6 +332,7 @@ theorem continuous_fst : Continuous (@Prod.fst X Y) :=
 #align continuous_fst continuous_fst
 
 /-- Postcomposing `f` with `Prod.fst` is continuous -/
+@[fun_prop]
 theorem Continuous.fst {f : X → Y × Z} (hf : Continuous f) : Continuous fun x : X => (f x).1 :=
   continuous_fst.comp hf
 #align continuous.fst Continuous.fst
@@ -346,6 +347,7 @@ theorem continuousAt_fst {p : X × Y} : ContinuousAt Prod.fst p :=
 #align continuous_at_fst continuousAt_fst
 
 /-- Postcomposing `f` with `Prod.fst` is continuous at `x` -/
+@[fun_prop]
 theorem ContinuousAt.fst {f : X → Y × Z} {x : X} (hf : ContinuousAt f x) :
     ContinuousAt (fun x : X => (f x).1) x :=
   continuousAt_fst.comp hf
@@ -369,6 +371,7 @@ theorem continuous_snd : Continuous (@Prod.snd X Y) :=
 #align continuous_snd continuous_snd
 
 /-- Postcomposing `f` with `Prod.snd` is continuous -/
+@[fun_prop]
 theorem Continuous.snd {f : X → Y × Z} (hf : Continuous f) : Continuous fun x : X => (f x).2 :=
   continuous_snd.comp hf
 #align continuous.snd Continuous.snd
@@ -383,6 +386,7 @@ theorem continuousAt_snd {p : X × Y} : ContinuousAt Prod.snd p :=
 #align continuous_at_snd continuousAt_snd
 
 /-- Postcomposing `f` with `Prod.snd` is continuous at `x` -/
+@[fun_prop]
 theorem ContinuousAt.snd {f : X → Y × Z} {x : X} (hf : ContinuousAt f x) :
     ContinuousAt (fun x : X => (f x).2) x :=
   continuousAt_snd.comp hf
@@ -400,7 +404,7 @@ theorem ContinuousAt.snd'' {f : Y → Z} {x : X × Y} (hf : ContinuousAt f x.snd
   hf.comp continuousAt_snd
 #align continuous_at.snd'' ContinuousAt.snd''
 
-@[continuity]
+@[continuity, fun_prop]
 theorem Continuous.prod_mk {f : Z → X} {g : Z → Y} (hf : Continuous f) (hg : Continuous g) :
     Continuous fun x => (f x, g x) :=
   continuous_prod_mk.2 ⟨hf, hg⟩
@@ -615,6 +619,7 @@ theorem Filter.Eventually.curry_nhds {p : X × Y → Prop} {x : X} {y : Y}
   exact h.curry
 #align filter.eventually.curry_nhds Filter.Eventually.curry_nhds
 
+@[fun_prop]
 theorem ContinuousAt.prod {f : X → Y} {g : X → Z} {x : X} (hf : ContinuousAt f x)
     (hg : ContinuousAt g x) : ContinuousAt (fun x => (f x, g x)) x :=
   hf.prod_mk_nhds hg
@@ -1225,12 +1230,12 @@ theorem continuous_pi_iff : Continuous f ↔ ∀ i, Continuous fun a => f a i :=
   simp only [continuous_iInf_rng, continuous_induced_rng, comp]
 #align continuous_pi_iff continuous_pi_iff
 
-@[continuity]
+@[continuity, fun_prop]
 theorem continuous_pi (h : ∀ i, Continuous fun a => f a i) : Continuous f :=
   continuous_pi_iff.2 h
 #align continuous_pi continuous_pi
 
-@[continuity]
+@[continuity, fun_prop]
 theorem continuous_apply (i : ι) : Continuous fun p : ∀ i, π i => p i :=
   continuous_iInf_dom continuous_induced_dom
 #align continuous_apply continuous_apply
@@ -1264,6 +1269,11 @@ theorem continuousAt_pi {f : X → ∀ i, π i} {x : X} :
   tendsto_pi_nhds
 #align continuous_at_pi continuousAt_pi
 
+@[fun_prop]
+theorem continuousAt_pi' {f : X → ∀ i, π i} {x : X} (hf : ∀ i, ContinuousAt (fun y => f y i) x) :
+    ContinuousAt f x :=
+  continuousAt_pi.2 hf
+
 theorem Pi.continuous_precomp' {ι' : Type*} (φ : ι' → ι) :
     Continuous (fun (f : (∀ i, π i)) (j : ι') ↦ f (φ j)) :=
   continuous_pi fun j ↦ continuous_apply (φ j)

@@ -1729,8 +1729,8 @@ theorem Filter.eventually_nhdsSet_prod_iff {p : X × Y → Prop} :
     (∀ᶠ q in 𝓝ˢ (s ×ˢ t), p q) ↔
       ∀ x ∈ s, ∀ y ∈ t,
           ∃ px : X → Prop, (∀ᶠ x' in 𝓝 x, px x') ∧ ∃ py : Y → Prop, (∀ᶠ y' in 𝓝 y, py y') ∧
-            ∀ {x : X}, px x → ∀ {y : Y}, py y → p (x, y) :=
-  by simp_rw [eventually_nhdsSet_iff_forall, Set.forall_prod_set, nhds_prod_eq, eventually_prod_iff]
+            ∀ {x : X}, px x → ∀ {y : Y}, py y → p (x, y) := by
+  simp_rw [eventually_nhdsSet_iff_forall, Set.forall_prod_set, nhds_prod_eq, eventually_prod_iff]
 
 theorem Filter.Eventually.prod_nhdsSet {p : X × Y → Prop} {px : X → Prop} {py : Y → Prop}
     (hp : ∀ {x : X}, px x → ∀ {y : Y}, py y → p (x, y)) (hs : ∀ᶠ x in 𝓝ˢ s, px x)

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.

@@ -5,6 +5,7 @@ Authors: Johannes Hölzl, Mario Carneiro, Patrick Massot
 -/
 import Mathlib.Topology.Maps
 import Mathlib.Order.Filter.Pi
+import Mathlib.Topology.NhdsSet
 
 #align_import topology.constructions from "leanprover-community/mathlib"@"f7ebde7ee0d1505dfccac8644ae12371aa3c1c9f"
 

Main API changes

• Define R1Space, a.k.a. preregular space.
• Drop T2OrLocallyCompactRegularSpace.
• Generalize all existing theorems about T2OrLocallyCompactRegularSpace to R1Space.
• Drop the [T2OrLocallyCompactRegularSpace _] assumption if the space is known to be regular for other reason (e.g., because it's a topological group).

New theorems

• Specializes.not_disjoint: if x ⤳ y, then 𝓝 x and 𝓝 y aren't disjoint;
• specializes_iff_not_disjoint, Specializes.inseparable, disjoint_nhds_nhds_iff_not_inseparable, r1Space_iff_inseparable_or_disjoint_nhds: basic API about R1Spaces;
• Inducing.r1Space, R1Space.induced, R1Space.sInf, R1Space.iInf, R1Space.inf, instances for Subtype _, X × Y, and ∀ i, X i: basic instances for R1Space;
• IsCompact.mem_closure_iff_exists_inseparable, IsCompact.closure_eq_biUnion_inseparable: characterizations of the closure of a compact set in a preregular space;
• Inseparable.mem_measurableSet_iff: topologically inseparable points can't be separated by a Borel measurable set;
• IsCompact.closure_subset_measurableSet, IsCompact.measure_closure: in a preregular space, a measurable superset of a compact set includes its closure as well; as a corollary, closure K has the same measure as K.
• exists_mem_nhds_isCompact_mapsTo_of_isCompact_mem_nhds: an auxiliary lemma extracted from a LocallyCompactPair instance;
• IsCompact.isCompact_isClosed_basis_nhds: if x admits a compact neighborhood, then it admits a basis of compact closed neighborhoods; in particular, a weakly locally compact preregular space is a locally compact regular space;
• isCompact_isClosed_basis_nhds: a version of the previous theorem for weakly locally compact spaces;
• exists_mem_nhds_isCompact_isClosed: in a locally compact regular space, each point admits a compact closed neighborhood.

Deprecated theorems

Some theorems about topological groups are true for any (pre)regular space, so we deprecate the special cases.

• exists_isCompact_isClosed_subset_isCompact_nhds_one: use new IsCompact.isCompact_isClosed_basis_nhds instead;
• instLocallyCompactSpaceOfWeaklyOfGroup, instLocallyCompactSpaceOfWeaklyOfAddGroup: are now implied by WeaklyLocallyCompactSpace.locallyCompactSpace;
• local_isCompact_isClosed_nhds_of_group, local_isCompact_isClosed_nhds_of_addGroup: use isCompact_isClosed_basis_nhds instead;
• exists_isCompact_isClosed_nhds_one, exists_isCompact_isClosed_nhds_zero: use exists_mem_nhds_isCompact_isClosed instead.

Renamed/moved theorems

For each renamed theorem, the old theorem is redefined as a deprecated alias.

• isOpen_setOf_disjoint_nhds_nhds: moved to Constructions;
• isCompact_closure_of_subset_compact -> IsCompact.closure_of_subset;
• IsCompact.measure_eq_infi_isOpen -> IsCompact.measure_eq_iInf_isOpen;
• exists_compact_superset_iff -> exists_isCompact_superset_iff;
• separatedNhds_of_isCompact_isCompact_isClosed -> SeparatedNhds.of_isCompact_isCompact_isClosed;
• separatedNhds_of_isCompact_isCompact -> SeparatedNhds.of_isCompact_isCompact;
• separatedNhds_of_finset_finset -> SeparatedNhds.of_finset_finset;
• point_disjoint_finset_opens_of_t2 -> SeparatedNhds.of_singleton_finset;
• separatedNhds_of_isCompact_isClosed -> SeparatedNhds.of_isCompact_isClosed;
• exists_open_superset_and_isCompact_closure -> exists_isOpen_superset_and_isCompact_closure;
• exists_open_with_compact_closure -> exists_isOpen_mem_isCompact_closure;

@@ -585,6 +585,14 @@ theorem prod_mem_nhds {s : Set X} {t : Set Y} {x : X} {y : Y} (hx : s ∈ 𝓝 x
   prod_mem_nhds_iff.2 ⟨hx, hy⟩
 #align prod_mem_nhds prod_mem_nhds
 
+theorem isOpen_setOf_disjoint_nhds_nhds : IsOpen { p : X × X | Disjoint (𝓝 p.1) (𝓝 p.2) } := by
+  simp only [isOpen_iff_mem_nhds, Prod.forall, mem_setOf_eq]
+  intro x y h
+  obtain ⟨U, hU, V, hV, hd⟩ := ((nhds_basis_opens x).disjoint_iff (nhds_basis_opens y)).mp h
+  exact mem_nhds_prod_iff'.mpr ⟨U, V, hU.2, hU.1, hV.2, hV.1, fun ⟨x', y'⟩ ⟨hx', hy'⟩ =>
+    disjoint_of_disjoint_of_mem hd (hU.2.mem_nhds hx') (hV.2.mem_nhds hy')⟩
+#align is_open_set_of_disjoint_nhds_nhds isOpen_setOf_disjoint_nhds_nhds
+
 theorem Filter.Eventually.prod_nhds {p : X → Prop} {q : Y → Prop} {x : X} {y : Y}
     (hx : ∀ᶠ x in 𝓝 x, p x) (hy : ∀ᶠ y in 𝓝 y, q y) : ∀ᶠ z : X × Y in 𝓝 (x, y), p z.1 ∧ q z.2 :=
   prod_mem_nhds hx hy

Now we use letters X and Y for topological spaces, not Greek letters.

I didn't replace all letters; some lemmas need a large number of different letters. Volunteers for the last instances welcome.

@@ -39,39 +39,39 @@ open Topology TopologicalSpace Set Filter Function Classical
 
 universe u v
 
-variable {α : Type u} {β : Type v} {γ δ ε ζ : Type*}
+variable {X : Type u} {Y : Type v} {Z W ε ζ : Type*}
 
 section Constructions
 
-instance instTopologicalSpaceSubtype {p : α → Prop} [t : TopologicalSpace α] :
+instance instTopologicalSpaceSubtype {p : X → Prop} [t : TopologicalSpace X] :
     TopologicalSpace (Subtype p) :=
   induced (↑) t
 
-instance {r : α → α → Prop} [t : TopologicalSpace α] : TopologicalSpace (Quot r) :=
+instance {r : X → X → Prop} [t : TopologicalSpace X] : TopologicalSpace (Quot r) :=
   coinduced (Quot.mk r) t
 
-instance instTopologicalSpaceQuotient {s : Setoid α} [t : TopologicalSpace α] :
+instance instTopologicalSpaceQuotient {s : Setoid X} [t : TopologicalSpace X] :
     TopologicalSpace (Quotient s) :=
   coinduced Quotient.mk' t
 
-instance instTopologicalSpaceProd [t₁ : TopologicalSpace α] [t₂ : TopologicalSpace β] :
-    TopologicalSpace (α × β) :=
+instance instTopologicalSpaceProd [t₁ : TopologicalSpace X] [t₂ : TopologicalSpace Y] :
+    TopologicalSpace (X × Y) :=
   induced Prod.fst t₁ ⊓ induced Prod.snd t₂
 
-instance instTopologicalSpaceSum [t₁ : TopologicalSpace α] [t₂ : TopologicalSpace β] :
-    TopologicalSpace (α ⊕ β) :=
+instance instTopologicalSpaceSum [t₁ : TopologicalSpace X] [t₂ : TopologicalSpace Y] :
+    TopologicalSpace (X ⊕ Y) :=
   coinduced Sum.inl t₁ ⊔ coinduced Sum.inr t₂
 
-instance instTopologicalSpaceSigma {β : α → Type v} [t₂ : ∀ a, TopologicalSpace (β a)] :
-    TopologicalSpace (Sigma β) :=
-  ⨆ a, coinduced (Sigma.mk a) (t₂ a)
+instance instTopologicalSpaceSigma {ι : Type*} {X : ι → Type v} [t₂ : ∀ i, TopologicalSpace (X i)] :
+    TopologicalSpace (Sigma X) :=
+  ⨆ i, coinduced (Sigma.mk i) (t₂ i)
 
-instance Pi.topologicalSpace {β : α → Type v} [t₂ : (a : α) → TopologicalSpace (β a)] :
-    TopologicalSpace ((a : α) → β a) :=
-  ⨅ a, induced (fun f => f a) (t₂ a)
+instance Pi.topologicalSpace {ι : Type*} {Y : ι → Type v} [t₂ : (i : ι) → TopologicalSpace (Y i)] :
+    TopologicalSpace ((i : ι) → Y i) :=
+  ⨅ i, induced (fun f => f i) (t₂ i)
 #align Pi.topological_space Pi.topologicalSpace
 
-instance ULift.topologicalSpace [t : TopologicalSpace α] : TopologicalSpace (ULift.{v, u} α) :=
+instance ULift.topologicalSpace [t : TopologicalSpace X] : TopologicalSpace (ULift.{v, u} X) :=
   t.induced ULift.down
 #align ulift.topological_space ULift.topologicalSpace
 
@@ -83,62 +83,62 @@ The topology on those type synonyms is inherited without change.
 
 section
 
-variable [TopologicalSpace α]
+variable [TopologicalSpace X]
 
 open Additive Multiplicative
 
-instance : TopologicalSpace (Additive α) := ‹TopologicalSpace α›
-instance : TopologicalSpace (Multiplicative α) := ‹TopologicalSpace α›
+instance : TopologicalSpace (Additive X) := ‹TopologicalSpace X›
+instance : TopologicalSpace (Multiplicative X) := ‹TopologicalSpace X›
 
-instance [DiscreteTopology α] : DiscreteTopology (Additive α) := ‹DiscreteTopology α›
-instance [DiscreteTopology α] : DiscreteTopology (Multiplicative α) := ‹DiscreteTopology α›
+instance [DiscreteTopology X] : DiscreteTopology (Additive X) := ‹DiscreteTopology X›
+instance [DiscreteTopology X] : DiscreteTopology (Multiplicative X) := ‹DiscreteTopology X›
 
-theorem continuous_ofMul : Continuous (ofMul : α → Additive α) := continuous_id
+theorem continuous_ofMul : Continuous (ofMul : X → Additive X) := continuous_id
 #align continuous_of_mul continuous_ofMul
 
-theorem continuous_toMul : Continuous (toMul : Additive α → α) := continuous_id
+theorem continuous_toMul : Continuous (toMul : Additive X → X) := continuous_id
 #align continuous_to_mul continuous_toMul
 
-theorem continuous_ofAdd : Continuous (ofAdd : α → Multiplicative α) := continuous_id
+theorem continuous_ofAdd : Continuous (ofAdd : X → Multiplicative X) := continuous_id
 #align continuous_of_add continuous_ofAdd
 
-theorem continuous_toAdd : Continuous (toAdd : Multiplicative α → α) := continuous_id
+theorem continuous_toAdd : Continuous (toAdd : Multiplicative X → X) := continuous_id
 #align continuous_to_add continuous_toAdd
 
-theorem isOpenMap_ofMul : IsOpenMap (ofMul : α → Additive α) := IsOpenMap.id
+theorem isOpenMap_ofMul : IsOpenMap (ofMul : X → Additive X) := IsOpenMap.id
 #align is_open_map_of_mul isOpenMap_ofMul
 
-theorem isOpenMap_toMul : IsOpenMap (toMul : Additive α → α) := IsOpenMap.id
+theorem isOpenMap_toMul : IsOpenMap (toMul : Additive X → X) := IsOpenMap.id
 #align is_open_map_to_mul isOpenMap_toMul
 
-theorem isOpenMap_ofAdd : IsOpenMap (ofAdd : α → Multiplicative α) := IsOpenMap.id
+theorem isOpenMap_ofAdd : IsOpenMap (ofAdd : X → Multiplicative X) := IsOpenMap.id
 #align is_open_map_of_add isOpenMap_ofAdd
 
-theorem isOpenMap_toAdd : IsOpenMap (toAdd : Multiplicative α → α) := IsOpenMap.id
+theorem isOpenMap_toAdd : IsOpenMap (toAdd : Multiplicative X → X) := IsOpenMap.id
 #align is_open_map_to_add isOpenMap_toAdd
 
-theorem isClosedMap_ofMul : IsClosedMap (ofMul : α → Additive α) := IsClosedMap.id
+theorem isClosedMap_ofMul : IsClosedMap (ofMul : X → Additive X) := IsClosedMap.id
 #align is_closed_map_of_mul isClosedMap_ofMul
 
-theorem isClosedMap_toMul : IsClosedMap (toMul : Additive α → α) := IsClosedMap.id
+theorem isClosedMap_toMul : IsClosedMap (toMul : Additive X → X) := IsClosedMap.id
 #align is_closed_map_to_mul isClosedMap_toMul
 
-theorem isClosedMap_ofAdd : IsClosedMap (ofAdd : α → Multiplicative α) := IsClosedMap.id
+theorem isClosedMap_ofAdd : IsClosedMap (ofAdd : X → Multiplicative X) := IsClosedMap.id
 #align is_closed_map_of_add isClosedMap_ofAdd
 
-theorem isClosedMap_toAdd : IsClosedMap (toAdd : Multiplicative α → α) := IsClosedMap.id
+theorem isClosedMap_toAdd : IsClosedMap (toAdd : Multiplicative X → X) := IsClosedMap.id
 #align is_closed_map_to_add isClosedMap_toAdd
 
-theorem nhds_ofMul (a : α) : 𝓝 (ofMul a) = map ofMul (𝓝 a) := rfl
+theorem nhds_ofMul (x : X) : 𝓝 (ofMul x) = map ofMul (𝓝 x) := rfl
 #align nhds_of_mul nhds_ofMul
 
-theorem nhds_ofAdd (a : α) : 𝓝 (ofAdd a) = map ofAdd (𝓝 a) := rfl
+theorem nhds_ofAdd (x : X) : 𝓝 (ofAdd x) = map ofAdd (𝓝 x) := rfl
 #align nhds_of_add nhds_ofAdd
 
-theorem nhds_toMul (a : Additive α) : 𝓝 (toMul a) = map toMul (𝓝 a) := rfl
+theorem nhds_toMul (x : Additive X) : 𝓝 (toMul x) = map toMul (𝓝 x) := rfl
 #align nhds_to_mul nhds_toMul
 
-theorem nhds_toAdd (a : Multiplicative α) : 𝓝 (toAdd a) = map toAdd (𝓝 a) := rfl
+theorem nhds_toAdd (x : Multiplicative X) : 𝓝 (toAdd x) = map toAdd (𝓝 x) := rfl
 #align nhds_to_add nhds_toAdd
 
 end
@@ -152,125 +152,125 @@ The topology on this type synonym is inherited without change.
 
 section
 
-variable [TopologicalSpace α]
+variable [TopologicalSpace X]
 
 open OrderDual
 
-instance : TopologicalSpace αᵒᵈ := ‹TopologicalSpace α›
-instance [DiscreteTopology α] : DiscreteTopology αᵒᵈ := ‹DiscreteTopology α›
+instance : TopologicalSpace Xᵒᵈ := ‹TopologicalSpace X›
+instance [DiscreteTopology X] : DiscreteTopology Xᵒᵈ := ‹DiscreteTopology X›
 
-theorem continuous_toDual : Continuous (toDual : α → αᵒᵈ) := continuous_id
+theorem continuous_toDual : Continuous (toDual : X → Xᵒᵈ) := continuous_id
 #align continuous_to_dual continuous_toDual
 
-theorem continuous_ofDual : Continuous (ofDual : αᵒᵈ → α) := continuous_id
+theorem continuous_ofDual : Continuous (ofDual : Xᵒᵈ → X) := continuous_id
 #align continuous_of_dual continuous_ofDual
 
-theorem isOpenMap_toDual : IsOpenMap (toDual : α → αᵒᵈ) := IsOpenMap.id
+theorem isOpenMap_toDual : IsOpenMap (toDual : X → Xᵒᵈ) := IsOpenMap.id
 #align is_open_map_to_dual isOpenMap_toDual
 
-theorem isOpenMap_ofDual : IsOpenMap (ofDual : αᵒᵈ → α) := IsOpenMap.id
+theorem isOpenMap_ofDual : IsOpenMap (ofDual : Xᵒᵈ → X) := IsOpenMap.id
 #align is_open_map_of_dual isOpenMap_ofDual
 
-theorem isClosedMap_toDual : IsClosedMap (toDual : α → αᵒᵈ) := IsClosedMap.id
+theorem isClosedMap_toDual : IsClosedMap (toDual : X → Xᵒᵈ) := IsClosedMap.id
 #align is_closed_map_to_dual isClosedMap_toDual
 
-theorem isClosedMap_ofDual : IsClosedMap (ofDual : αᵒᵈ → α) := IsClosedMap.id
+theorem isClosedMap_ofDual : IsClosedMap (ofDual : Xᵒᵈ → X) := IsClosedMap.id
 #align is_closed_map_of_dual isClosedMap_ofDual
 
-theorem nhds_toDual (a : α) : 𝓝 (toDual a) = map toDual (𝓝 a) := rfl
+theorem nhds_toDual (x : X) : 𝓝 (toDual x) = map toDual (𝓝 x) := rfl
 #align nhds_to_dual nhds_toDual
 
-theorem nhds_ofDual (a : α) : 𝓝 (ofDual a) = map ofDual (𝓝 a) := rfl
+theorem nhds_ofDual (x : X) : 𝓝 (ofDual x) = map ofDual (𝓝 x) := rfl
 #align nhds_of_dual nhds_ofDual
 
 end
 
-theorem Quotient.preimage_mem_nhds [TopologicalSpace α] [s : Setoid α] {V : Set <| Quotient s}
-    {a : α} (hs : V ∈ 𝓝 (Quotient.mk' a)) : Quotient.mk' ⁻¹' V ∈ 𝓝 a :=
+theorem Quotient.preimage_mem_nhds [TopologicalSpace X] [s : Setoid X] {V : Set <| Quotient s}
+    {x : X} (hs : V ∈ 𝓝 (Quotient.mk' x)) : Quotient.mk' ⁻¹' V ∈ 𝓝 x :=
   preimage_nhds_coinduced hs
 #align quotient.preimage_mem_nhds Quotient.preimage_mem_nhds
 
 /-- The image of a dense set under `Quotient.mk'` is a dense set. -/
-theorem Dense.quotient [Setoid α] [TopologicalSpace α] {s : Set α} (H : Dense s) :
+theorem Dense.quotient [Setoid X] [TopologicalSpace X] {s : Set X} (H : Dense s) :
     Dense (Quotient.mk' '' s) :=
   Quotient.surjective_Quotient_mk''.denseRange.dense_image continuous_coinduced_rng H
 #align dense.quotient Dense.quotient
 
 /-- The composition of `Quotient.mk'` and a function with dense range has dense range. -/
-theorem DenseRange.quotient [Setoid α] [TopologicalSpace α] {f : β → α} (hf : DenseRange f) :
+theorem DenseRange.quotient [Setoid X] [TopologicalSpace X] {f : Y → X} (hf : DenseRange f) :
     DenseRange (Quotient.mk' ∘ f) :=
   Quotient.surjective_Quotient_mk''.denseRange.comp hf continuous_coinduced_rng
 #align dense_range.quotient DenseRange.quotient
 
-instance {p : α → Prop} [TopologicalSpace α] [DiscreteTopology α] : DiscreteTopology (Subtype p) :=
+instance {p : X → Prop} [TopologicalSpace X] [DiscreteTopology X] : DiscreteTopology (Subtype p) :=
   ⟨bot_unique fun s _ => ⟨(↑) '' s, isOpen_discrete _, preimage_image_eq _ Subtype.val_injective⟩⟩
 
-instance Sum.discreteTopology [TopologicalSpace α] [TopologicalSpace β] [hα : DiscreteTopology α]
-    [hβ : DiscreteTopology β] : DiscreteTopology (Sum α β) :=
-  ⟨sup_eq_bot_iff.2 <| by simp [hα.eq_bot, hβ.eq_bot]⟩
+instance Sum.discreteTopology [TopologicalSpace X] [TopologicalSpace Y] [h : DiscreteTopology X]
+    [hY : DiscreteTopology Y] : DiscreteTopology (X ⊕ Y) :=
+  ⟨sup_eq_bot_iff.2 <| by simp [h.eq_bot, hY.eq_bot]⟩
 #align sum.discrete_topology Sum.discreteTopology
 
-instance Sigma.discreteTopology {β : α → Type v} [∀ a, TopologicalSpace (β a)]
-    [h : ∀ a, DiscreteTopology (β a)] : DiscreteTopology (Sigma β) :=
+instance Sigma.discreteTopology {ι : Type*} {Y : ι → Type v} [∀ i, TopologicalSpace (Y i)]
+    [h : ∀ i, DiscreteTopology (Y i)] : DiscreteTopology (Sigma Y) :=
   ⟨iSup_eq_bot.2 fun _ => by simp only [(h _).eq_bot, coinduced_bot]⟩
 #align sigma.discrete_topology Sigma.discreteTopology
 
-section Topα
+section Top
 
-variable [TopologicalSpace α]
+variable [TopologicalSpace X]
 
 /-
 The 𝓝 filter and the subspace topology.
 -/
-theorem mem_nhds_subtype (s : Set α) (a : { x // x ∈ s }) (t : Set { x // x ∈ s }) :
-    t ∈ 𝓝 a ↔ ∃ u ∈ 𝓝 (a : α), Subtype.val ⁻¹' u ⊆ t :=
-  mem_nhds_induced _ a t
+theorem mem_nhds_subtype (s : Set X) (x : { x // x ∈ s }) (t : Set { x // x ∈ s }) :
+    t ∈ 𝓝 x ↔ ∃ u ∈ 𝓝 (x : X), Subtype.val ⁻¹' u ⊆ t :=
+  mem_nhds_induced _ x t
 #align mem_nhds_subtype mem_nhds_subtype
 
-theorem nhds_subtype (s : Set α) (a : { x // x ∈ s }) : 𝓝 a = comap (↑) (𝓝 (a : α)) :=
-  nhds_induced _ a
+theorem nhds_subtype (s : Set X) (x : { x // x ∈ s }) : 𝓝 x = comap (↑) (𝓝 (x : X)) :=
+  nhds_induced _ x
 #align nhds_subtype nhds_subtype
 
-theorem nhdsWithin_subtype_eq_bot_iff {s t : Set α} {x : s} :
-    𝓝[((↑) : s → α) ⁻¹' t] x = ⊥ ↔ 𝓝[t] (x : α) ⊓ 𝓟 s = ⊥ := by
+theorem nhdsWithin_subtype_eq_bot_iff {s t : Set X} {x : s} :
+    𝓝[((↑) : s → X) ⁻¹' t] x = ⊥ ↔ 𝓝[t] (x : X) ⊓ 𝓟 s = ⊥ := by
   rw [inf_principal_eq_bot_iff_comap, nhdsWithin, nhdsWithin, comap_inf, comap_principal,
     nhds_induced]
 #align nhds_within_subtype_eq_bot_iff nhdsWithin_subtype_eq_bot_iff
 
-theorem nhds_ne_subtype_eq_bot_iff {S : Set α} {x : S} :
-    𝓝[≠] x = ⊥ ↔ 𝓝[≠] (x : α) ⊓ 𝓟 S = ⊥ := by
+theorem nhds_ne_subtype_eq_bot_iff {S : Set X} {x : S} :
+    𝓝[≠] x = ⊥ ↔ 𝓝[≠] (x : X) ⊓ 𝓟 S = ⊥ := by
   rw [← nhdsWithin_subtype_eq_bot_iff, preimage_compl, ← image_singleton,
     Subtype.coe_injective.preimage_image]
 #align nhds_ne_subtype_eq_bot_iff nhds_ne_subtype_eq_bot_iff
 
-theorem nhds_ne_subtype_neBot_iff {S : Set α} {x : S} :
-    (𝓝[≠] x).NeBot ↔ (𝓝[≠] (x : α) ⊓ 𝓟 S).NeBot := by
+theorem nhds_ne_subtype_neBot_iff {S : Set X} {x : S} :
+    (𝓝[≠] x).NeBot ↔ (𝓝[≠] (x : X) ⊓ 𝓟 S).NeBot := by
   rw [neBot_iff, neBot_iff, not_iff_not, nhds_ne_subtype_eq_bot_iff]
 #align nhds_ne_subtype_ne_bot_iff nhds_ne_subtype_neBot_iff
 
-theorem discreteTopology_subtype_iff {S : Set α} :
+theorem discreteTopology_subtype_iff {S : Set X} :
     DiscreteTopology S ↔ ∀ x ∈ S, 𝓝[≠] x ⊓ 𝓟 S = ⊥ := by
   simp_rw [discreteTopology_iff_nhds_ne, SetCoe.forall', nhds_ne_subtype_eq_bot_iff]
 #align discrete_topology_subtype_iff discreteTopology_subtype_iff
 
-end Topα
+end Top
 
 /-- A type synonym equipped with the topology whose open sets are the empty set and the sets with
 finite complements. -/
-def CofiniteTopology (α : Type*) :=
-  α
+def CofiniteTopology (X : Type*) := X
+
 #align cofinite_topology CofiniteTopology
 
 namespace CofiniteTopology
 
-/-- The identity equivalence between `α` and `CofiniteTopology α`. -/
-def of : α ≃ CofiniteTopology α :=
-  Equiv.refl α
+/-- The identity equivalence between `` and `CofiniteTopology `. -/
+def of : X ≃ CofiniteTopology X :=
+  Equiv.refl X
 #align cofinite_topology.of CofiniteTopology.of
 
-instance [Inhabited α] : Inhabited (CofiniteTopology α) where default := of default
+instance [Inhabited X] : Inhabited (CofiniteTopology X) where default := of default
 
-instance : TopologicalSpace (CofiniteTopology α) where
+instance : TopologicalSpace (CofiniteTopology X) where
   IsOpen s := s.Nonempty → Set.Finite sᶜ
   isOpen_univ := by simp
   isOpen_inter s t := by
@@ -282,30 +282,30 @@ instance : TopologicalSpace (CofiniteTopology α) where
     rw [Set.compl_sUnion]
     exact Set.Finite.sInter (mem_image_of_mem _ hts) (h t hts ⟨x, hzt⟩)
 
-theorem isOpen_iff {s : Set (CofiniteTopology α)} : IsOpen s ↔ s.Nonempty → sᶜ.Finite :=
+theorem isOpen_iff {s : Set (CofiniteTopology X)} : IsOpen s ↔ s.Nonempty → sᶜ.Finite :=
   Iff.rfl
 #align cofinite_topology.is_open_iff CofiniteTopology.isOpen_iff
 
-theorem isOpen_iff' {s : Set (CofiniteTopology α)} : IsOpen s ↔ s = ∅ ∨ sᶜ.Finite := by
+theorem isOpen_iff' {s : Set (CofiniteTopology X)} : IsOpen s ↔ s = ∅ ∨ sᶜ.Finite := by
   simp only [isOpen_iff, nonempty_iff_ne_empty, or_iff_not_imp_left]
 #align cofinite_topology.is_open_iff' CofiniteTopology.isOpen_iff'
 
-theorem isClosed_iff {s : Set (CofiniteTopology α)} : IsClosed s ↔ s = univ ∨ s.Finite := by
+theorem isClosed_iff {s : Set (CofiniteTopology X)} : IsClosed s ↔ s = univ ∨ s.Finite := by
   simp only [← isOpen_compl_iff, isOpen_iff', compl_compl, compl_empty_iff]
 #align cofinite_topology.is_closed_iff CofiniteTopology.isClosed_iff
 
-theorem nhds_eq (a : CofiniteTopology α) : 𝓝 a = pure a ⊔ cofinite := by
+theorem nhds_eq (x : CofiniteTopology X) : 𝓝 x = pure x ⊔ cofinite := by
   ext U
   rw [mem_nhds_iff]
   constructor
   · rintro ⟨V, hVU, V_op, haV⟩
     exact mem_sup.mpr ⟨hVU haV, mem_of_superset (V_op ⟨_, haV⟩) hVU⟩
-  · rintro ⟨hU : a ∈ U, hU' : Uᶜ.Finite⟩
+  · rintro ⟨hU : x ∈ U, hU' : Uᶜ.Finite⟩
     exact ⟨U, Subset.rfl, fun _ => hU', hU⟩
 #align cofinite_topology.nhds_eq CofiniteTopology.nhds_eq
 
-theorem mem_nhds_iff {a : CofiniteTopology α} {s : Set (CofiniteTopology α)} :
-    s ∈ 𝓝 a ↔ a ∈ s ∧ sᶜ.Finite := by simp [nhds_eq]
+theorem mem_nhds_iff {x : CofiniteTopology X} {s : Set (CofiniteTopology X)} :
+    s ∈ 𝓝 x ↔ x ∈ s ∧ sᶜ.Finite := by simp [nhds_eq]
 #align cofinite_topology.mem_nhds_iff CofiniteTopology.mem_nhds_iff
 
 end CofiniteTopology
@@ -314,141 +314,141 @@ end Constructions
 
 section Prod
 
-variable [TopologicalSpace α] [TopologicalSpace β] [TopologicalSpace γ] [TopologicalSpace δ]
+variable [TopologicalSpace X] [TopologicalSpace Y] [TopologicalSpace Z] [TopologicalSpace W]
   [TopologicalSpace ε] [TopologicalSpace ζ]
 
 -- porting note: todo: Lean 4 fails to deduce implicit args
-@[simp] theorem continuous_prod_mk {f : α → β} {g : α → γ} :
+@[simp] theorem continuous_prod_mk {f : X → Y} {g : X → Z} :
     (Continuous fun x => (f x, g x)) ↔ Continuous f ∧ Continuous g :=
-  (@continuous_inf_rng α (β × γ) _ _ (TopologicalSpace.induced Prod.fst _)
+  (@continuous_inf_rng X (Y × Z) _ _ (TopologicalSpace.induced Prod.fst _)
     (TopologicalSpace.induced Prod.snd _)).trans <|
     continuous_induced_rng.and continuous_induced_rng
 #align continuous_prod_mk continuous_prod_mk
 
 @[continuity]
-theorem continuous_fst : Continuous (@Prod.fst α β) :=
+theorem continuous_fst : Continuous (@Prod.fst X Y) :=
   (continuous_prod_mk.1 continuous_id).1
 #align continuous_fst continuous_fst
 
 /-- Postcomposing `f` with `Prod.fst` is continuous -/
-theorem Continuous.fst {f : α → β × γ} (hf : Continuous f) : Continuous fun a : α => (f a).1 :=
+theorem Continuous.fst {f : X → Y × Z} (hf : Continuous f) : Continuous fun x : X => (f x).1 :=
   continuous_fst.comp hf
 #align continuous.fst Continuous.fst
 
 /-- Precomposing `f` with `Prod.fst` is continuous -/
-theorem Continuous.fst' {f : α → γ} (hf : Continuous f) : Continuous fun x : α × β => f x.fst :=
+theorem Continuous.fst' {f : X → Z} (hf : Continuous f) : Continuous fun x : X × Y => f x.fst :=
   hf.comp continuous_fst
 #align continuous.fst' Continuous.fst'
 
-theorem continuousAt_fst {p : α × β} : ContinuousAt Prod.fst p :=
+theorem continuousAt_fst {p : X × Y} : ContinuousAt Prod.fst p :=
   continuous_fst.continuousAt
 #align continuous_at_fst continuousAt_fst
 
 /-- Postcomposing `f` with `Prod.fst` is continuous at `x` -/
-theorem ContinuousAt.fst {f : α → β × γ} {x : α} (hf : ContinuousAt f x) :
-    ContinuousAt (fun a : α => (f a).1) x :=
+theorem ContinuousAt.fst {f : X → Y × Z} {x : X} (hf : ContinuousAt f x) :
+    ContinuousAt (fun x : X => (f x).1) x :=
   continuousAt_fst.comp hf
 #align continuous_at.fst ContinuousAt.fst
 
 /-- Precomposing `f` with `Prod.fst` is continuous at `(x, y)` -/
-theorem ContinuousAt.fst' {f : α → γ} {x : α} {y : β} (hf : ContinuousAt f x) :
-    ContinuousAt (fun x : α × β => f x.fst) (x, y) :=
+theorem ContinuousAt.fst' {f : X → Z} {x : X} {y : Y} (hf : ContinuousAt f x) :
+    ContinuousAt (fun x : X × Y => f x.fst) (x, y) :=
   ContinuousAt.comp hf continuousAt_fst
 #align continuous_at.fst' ContinuousAt.fst'
 
-/-- Precomposing `f` with `Prod.fst` is continuous at `x : α × β` -/
-theorem ContinuousAt.fst'' {f : α → γ} {x : α × β} (hf : ContinuousAt f x.fst) :
-    ContinuousAt (fun x : α × β => f x.fst) x :=
+/-- Precomposing `f` with `Prod.fst` is continuous at `x : X × Y` -/
+theorem ContinuousAt.fst'' {f : X → Z} {x : X × Y} (hf : ContinuousAt f x.fst) :
+    ContinuousAt (fun x : X × Y => f x.fst) x :=
   hf.comp continuousAt_fst
 #align continuous_at.fst'' ContinuousAt.fst''
 
 @[continuity]
-theorem continuous_snd : Continuous (@Prod.snd α β) :=
+theorem continuous_snd : Continuous (@Prod.snd X Y) :=
   (continuous_prod_mk.1 continuous_id).2
 #align continuous_snd continuous_snd
 
 /-- Postcomposing `f` with `Prod.snd` is continuous -/
-theorem Continuous.snd {f : α → β × γ} (hf : Continuous f) : Continuous fun a : α => (f a).2 :=
+theorem Continuous.snd {f : X → Y × Z} (hf : Continuous f) : Continuous fun x : X => (f x).2 :=
   continuous_snd.comp hf
 #align continuous.snd Continuous.snd
 
 /-- Precomposing `f` with `Prod.snd` is continuous -/
-theorem Continuous.snd' {f : β → γ} (hf : Continuous f) : Continuous fun x : α × β => f x.snd :=
+theorem Continuous.snd' {f : Y → Z} (hf : Continuous f) : Continuous fun x : X × Y => f x.snd :=
   hf.comp continuous_snd
 #align continuous.snd' Continuous.snd'
 
-theorem continuousAt_snd {p : α × β} : ContinuousAt Prod.snd p :=
+theorem continuousAt_snd {p : X × Y} : ContinuousAt Prod.snd p :=
   continuous_snd.continuousAt
 #align continuous_at_snd continuousAt_snd
 
 /-- Postcomposing `f` with `Prod.snd` is continuous at `x` -/
-theorem ContinuousAt.snd {f : α → β × γ} {x : α} (hf : ContinuousAt f x) :
-    ContinuousAt (fun a : α => (f a).2) x :=
+theorem ContinuousAt.snd {f : X → Y × Z} {x : X} (hf : ContinuousAt f x) :
+    ContinuousAt (fun x : X => (f x).2) x :=
   continuousAt_snd.comp hf
 #align continuous_at.snd ContinuousAt.snd
 
 /-- Precomposing `f` with `Prod.snd` is continuous at `(x, y)` -/
-theorem ContinuousAt.snd' {f : β → γ} {x : α} {y : β} (hf : ContinuousAt f y) :
-    ContinuousAt (fun x : α × β => f x.snd) (x, y) :=
+theorem ContinuousAt.snd' {f : Y → Z} {x : X} {y : Y} (hf : ContinuousAt f y) :
+    ContinuousAt (fun x : X × Y => f x.snd) (x, y) :=
   ContinuousAt.comp hf continuousAt_snd
 #align continuous_at.snd' ContinuousAt.snd'
 
-/-- Precomposing `f` with `Prod.snd` is continuous at `x : α × β` -/
-theorem ContinuousAt.snd'' {f : β → γ} {x : α × β} (hf : ContinuousAt f x.snd) :
-    ContinuousAt (fun x : α × β => f x.snd) x :=
+/-- Precomposing `f` with `Prod.snd` is continuous at `x : X × Y` -/
+theorem ContinuousAt.snd'' {f : Y → Z} {x : X × Y} (hf : ContinuousAt f x.snd) :
+    ContinuousAt (fun x : X × Y => f x.snd) x :=
   hf.comp continuousAt_snd
 #align continuous_at.snd'' ContinuousAt.snd''
 
 @[continuity]
-theorem Continuous.prod_mk {f : γ → α} {g : γ → β} (hf : Continuous f) (hg : Continuous g) :
+theorem Continuous.prod_mk {f : Z → X} {g : Z → Y} (hf : Continuous f) (hg : Continuous g) :
     Continuous fun x => (f x, g x) :=
   continuous_prod_mk.2 ⟨hf, hg⟩
 #align continuous.prod_mk Continuous.prod_mk
 
 @[continuity]
-theorem Continuous.Prod.mk (a : α) : Continuous fun b : β => (a, b) :=
+theorem Continuous.Prod.mk (x : X) : Continuous fun y : Y => (x, y) :=
   continuous_const.prod_mk continuous_id
 #align continuous.prod.mk Continuous.Prod.mk
 
 @[continuity]
-theorem Continuous.Prod.mk_left (b : β) : Continuous fun a : α => (a, b) :=
+theorem Continuous.Prod.mk_left (y : Y) : Continuous fun x : X => (x, y) :=
   continuous_id.prod_mk continuous_const
 #align continuous.prod.mk_left Continuous.Prod.mk_left
 
 /-- If `f x y` is continuous in `x` for all `y ∈ s`,
 then the set of `x` such that `f x` maps `s` to `t` is closed. -/
-lemma IsClosed.setOf_mapsTo {f : α → β → γ} {s : Set β} {t : Set γ} (ht : IsClosed t)
-    (hf : ∀ y ∈ s, Continuous (f · y)) : IsClosed {x | MapsTo (f x) s t} := by
+lemma IsClosed.setOf_mapsTo {α : Type*} {f : X → α → Z} {s : Set α} {t : Set Z} (ht : IsClosed t)
+    (hf : ∀ a ∈ s, Continuous (f · a)) : IsClosed {x | MapsTo (f x) s t} := by
   simpa only [MapsTo, setOf_forall] using isClosed_biInter fun y hy ↦ ht.preimage (hf y hy)
 
-theorem Continuous.comp₂ {g : α × β → γ} (hg : Continuous g) {e : δ → α} (he : Continuous e)
-    {f : δ → β} (hf : Continuous f) : Continuous fun x => g (e x, f x) :=
+theorem Continuous.comp₂ {g : X × Y → Z} (hg : Continuous g) {e : W → X} (he : Continuous e)
+    {f : W → Y} (hf : Continuous f) : Continuous fun w => g (e w, f w) :=
   hg.comp <| he.prod_mk hf
 #align continuous.comp₂ Continuous.comp₂
 
-theorem Continuous.comp₃ {g : α × β × γ → ε} (hg : Continuous g) {e : δ → α} (he : Continuous e)
-    {f : δ → β} (hf : Continuous f) {k : δ → γ} (hk : Continuous k) :
-    Continuous fun x => g (e x, f x, k x) :=
+theorem Continuous.comp₃ {g : X × Y × Z → ε} (hg : Continuous g) {e : W → X} (he : Continuous e)
+    {f : W → Y} (hf : Continuous f) {k : W → Z} (hk : Continuous k) :
+    Continuous fun w => g (e w, f w, k w) :=
   hg.comp₂ he <| hf.prod_mk hk
 #align continuous.comp₃ Continuous.comp₃
 
-theorem Continuous.comp₄ {g : α × β × γ × ζ → ε} (hg : Continuous g) {e : δ → α} (he : Continuous e)
-    {f : δ → β} (hf : Continuous f) {k : δ → γ} (hk : Continuous k) {l : δ → ζ}
-    (hl : Continuous l) : Continuous fun x => g (e x, f x, k x, l x) :=
+theorem Continuous.comp₄ {g : X × Y × Z × ζ → ε} (hg : Continuous g) {e : W → X} (he : Continuous e)
+    {f : W → Y} (hf : Continuous f) {k : W → Z} (hk : Continuous k) {l : W → ζ}
+    (hl : Continuous l) : Continuous fun w => g (e w, f w, k w, l w) :=
   hg.comp₃ he hf <| hk.prod_mk hl
 #align continuous.comp₄ Continuous.comp₄
 
 @[continuity]
-theorem Continuous.prod_map {f : γ → α} {g : δ → β} (hf : Continuous f) (hg : Continuous g) :
-    Continuous fun x : γ × δ => (f x.1, g x.2) :=
+theorem Continuous.prod_map {f : Z → X} {g : W → Y} (hf : Continuous f) (hg : Continuous g) :
+    Continuous fun p : Z × W => (f p.1, g p.2) :=
   hf.fst'.prod_mk hg.snd'
 #align continuous.prod_map Continuous.prod_map
 
 /-- A version of `continuous_inf_dom_left` for binary functions -/
-theorem continuous_inf_dom_left₂ {α β γ} {f : α → β → γ} {ta1 ta2 : TopologicalSpace α}
-    {tb1 tb2 : TopologicalSpace β} {tc1 : TopologicalSpace γ}
-    (h : by haveI := ta1; haveI := tb1; exact Continuous fun p : α × β => f p.1 p.2) : by
-    haveI := ta1 ⊓ ta2; haveI := tb1 ⊓ tb2; exact Continuous fun p : α × β => f p.1 p.2 := by
+theorem continuous_inf_dom_left₂ {X Y Z} {f : X → Y → Z} {ta1 ta2 : TopologicalSpace X}
+    {tb1 tb2 : TopologicalSpace Y} {tc1 : TopologicalSpace Z}
+    (h : by haveI := ta1; haveI := tb1; exact Continuous fun p : X × Y => f p.1 p.2) : by
+    haveI := ta1 ⊓ ta2; haveI := tb1 ⊓ tb2; exact Continuous fun p : X × Y => f p.1 p.2 := by
   have ha := @continuous_inf_dom_left _ _ id ta1 ta2 ta1 (@continuous_id _ (id _))
   have hb := @continuous_inf_dom_left _ _ id tb1 tb2 tb1 (@continuous_id _ (id _))
   have h_continuous_id := @Continuous.prod_map _ _ _ _ ta1 tb1 (ta1 ⊓ ta2) (tb1 ⊓ tb2) _ _ ha hb
@@ -456,10 +456,10 @@ theorem continuous_inf_dom_left₂ {α β γ} {f : α → β → γ} {ta1 ta2 :
 #align continuous_inf_dom_left₂ continuous_inf_dom_left₂
 
 /-- A version of `continuous_inf_dom_right` for binary functions -/
-theorem continuous_inf_dom_right₂ {α β γ} {f : α → β → γ} {ta1 ta2 : TopologicalSpace α}
-    {tb1 tb2 : TopologicalSpace β} {tc1 : TopologicalSpace γ}
-    (h : by haveI := ta2; haveI := tb2; exact Continuous fun p : α × β => f p.1 p.2) : by
-    haveI := ta1 ⊓ ta2; haveI := tb1 ⊓ tb2; exact Continuous fun p : α × β => f p.1 p.2 := by
+theorem continuous_inf_dom_right₂ {X Y Z} {f : X → Y → Z} {ta1 ta2 : TopologicalSpace X}
+    {tb1 tb2 : TopologicalSpace Y} {tc1 : TopologicalSpace Z}
+    (h : by haveI := ta2; haveI := tb2; exact Continuous fun p : X × Y => f p.1 p.2) : by
+    haveI := ta1 ⊓ ta2; haveI := tb1 ⊓ tb2; exact Continuous fun p : X × Y => f p.1 p.2 := by
   have ha := @continuous_inf_dom_right _ _ id ta1 ta2 ta2 (@continuous_id _ (id _))
   have hb := @continuous_inf_dom_right _ _ id tb1 tb2 tb2 (@continuous_id _ (id _))
   have h_continuous_id := @Continuous.prod_map _ _ _ _ ta2 tb2 (ta1 ⊓ ta2) (tb1 ⊓ tb2) _ _ ha hb
@@ -467,165 +467,165 @@ theorem continuous_inf_dom_right₂ {α β γ} {f : α → β → γ} {ta1 ta2 :
 #align continuous_inf_dom_right₂ continuous_inf_dom_right₂
 
 /-- A version of `continuous_sInf_dom` for binary functions -/
-theorem continuous_sInf_dom₂ {α β γ} {f : α → β → γ} {tas : Set (TopologicalSpace α)}
-    {tbs : Set (TopologicalSpace β)} {ta : TopologicalSpace α} {tb : TopologicalSpace β}
-    {tc : TopologicalSpace γ} (ha : ta ∈ tas) (hb : tb ∈ tbs)
-    (hf : Continuous fun p : α × β => f p.1 p.2) : by
+theorem continuous_sInf_dom₂ {X Y Z} {f : X → Y → Z} {tas : Set (TopologicalSpace X)}
+    {tbs : Set (TopologicalSpace Y)} {tX : TopologicalSpace X} {tY : TopologicalSpace Y}
+    {tc : TopologicalSpace Z} (hX : tX ∈ tas) (hY : tY ∈ tbs)
+    (hf : Continuous fun p : X × Y => f p.1 p.2) : by
     haveI := sInf tas; haveI := sInf tbs;
-      exact @Continuous _ _ _ tc fun p : α × β => f p.1 p.2 := by
-  have ha := continuous_sInf_dom ha continuous_id
-  have hb := continuous_sInf_dom hb continuous_id
-  have h_continuous_id := @Continuous.prod_map _ _ _ _ ta tb (sInf tas) (sInf tbs) _ _ ha hb
+      exact @Continuous _ _ _ tc fun p : X × Y => f p.1 p.2 := by
+  have hX := continuous_sInf_dom hX continuous_id
+  have hY := continuous_sInf_dom hY continuous_id
+  have h_continuous_id := @Continuous.prod_map _ _ _ _ tX tY (sInf tas) (sInf tbs) _ _ hX hY
   exact @Continuous.comp _ _ _ (id _) (id _) _ _ _ hf h_continuous_id
 #align continuous_Inf_dom₂ continuous_sInf_dom₂
 
-theorem Filter.Eventually.prod_inl_nhds {p : α → Prop} {a : α} (h : ∀ᶠ x in 𝓝 a, p x) (b : β) :
-    ∀ᶠ x in 𝓝 (a, b), p (x : α × β).1 :=
+theorem Filter.Eventually.prod_inl_nhds {p : X → Prop} {x : X} (h : ∀ᶠ x in 𝓝 x, p x) (y : Y) :
+    ∀ᶠ x in 𝓝 (x, y), p (x : X × Y).1 :=
   continuousAt_fst h
 #align filter.eventually.prod_inl_nhds Filter.Eventually.prod_inl_nhds
 
-theorem Filter.Eventually.prod_inr_nhds {p : β → Prop} {b : β} (h : ∀ᶠ x in 𝓝 b, p x) (a : α) :
-    ∀ᶠ x in 𝓝 (a, b), p (x : α × β).2 :=
+theorem Filter.Eventually.prod_inr_nhds {p : Y → Prop} {y : Y} (h : ∀ᶠ x in 𝓝 y, p x) (x : X) :
+    ∀ᶠ x in 𝓝 (x, y), p (x : X × Y).2 :=
   continuousAt_snd h
 #align filter.eventually.prod_inr_nhds Filter.Eventually.prod_inr_nhds
 
-theorem Filter.Eventually.prod_mk_nhds {pa : α → Prop} {a} (ha : ∀ᶠ x in 𝓝 a, pa x) {pb : β → Prop}
-    {b} (hb : ∀ᶠ y in 𝓝 b, pb y) : ∀ᶠ p in 𝓝 (a, b), pa (p : α × β).1 ∧ pb p.2 :=
-  (ha.prod_inl_nhds b).and (hb.prod_inr_nhds a)
+theorem Filter.Eventually.prod_mk_nhds {px : X → Prop} {x} (hx : ∀ᶠ x in 𝓝 x, px x) {py : Y → Prop}
+    {y} (hy : ∀ᶠ y in 𝓝 y, py y) : ∀ᶠ p in 𝓝 (x, y), px (p : X × Y).1 ∧ py p.2 :=
+  (hx.prod_inl_nhds y).and (hy.prod_inr_nhds x)
 #align filter.eventually.prod_mk_nhds Filter.Eventually.prod_mk_nhds
 
-theorem continuous_swap : Continuous (Prod.swap : α × β → β × α) :=
+theorem continuous_swap : Continuous (Prod.swap : X × Y → Y × X) :=
   continuous_snd.prod_mk continuous_fst
 #align continuous_swap continuous_swap
 
-lemma isClosedMap_swap : IsClosedMap (Prod.swap : α × β → β × α) := fun s hs ↦ by
+lemma isClosedMap_swap : IsClosedMap (Prod.swap : X × Y → Y × X) := fun s hs ↦ by
   rw [image_swap_eq_preimage_swap]
   exact hs.preimage continuous_swap
 
-theorem continuous_uncurry_left {f : α → β → γ} (a : α) (h : Continuous (uncurry f)) :
-    Continuous (f a) :=
+theorem continuous_uncurry_left {f : X → Y → Z} (x : X) (h : Continuous (uncurry f)) :
+    Continuous (f x) :=
   h.comp (Continuous.Prod.mk _)
 #align continuous_uncurry_left continuous_uncurry_left
 
-theorem continuous_uncurry_right {f : α → β → γ} (b : β) (h : Continuous (uncurry f)) :
-    Continuous fun a => f a b :=
+theorem continuous_uncurry_right {f : X → Y → Z} (y : Y) (h : Continuous (uncurry f)) :
+    Continuous fun a => f a y :=
   h.comp (Continuous.Prod.mk_left _)
 #align continuous_uncurry_right continuous_uncurry_right
 
-theorem continuous_curry {g : α × β → γ} (a : α) (h : Continuous g) : Continuous (curry g a) :=
-  continuous_uncurry_left a h
+theorem continuous_curry {g : X × Y → Z} (x : X) (h : Continuous g) : Continuous (curry g x) :=
+  continuous_uncurry_left x h
 #align continuous_curry continuous_curry
 
-theorem IsOpen.prod {s : Set α} {t : Set β} (hs : IsOpen s) (ht : IsOpen t) : IsOpen (s ×ˢ t) :=
+theorem IsOpen.prod {s : Set X} {t : Set Y} (hs : IsOpen s) (ht : IsOpen t) : IsOpen (s ×ˢ t) :=
   (hs.preimage continuous_fst).inter (ht.preimage continuous_snd)
 #align is_open.prod IsOpen.prod
 
 -- porting note: todo: Lean fails to find `t₁` and `t₂` by unification
-theorem nhds_prod_eq {a : α} {b : β} : 𝓝 (a, b) = 𝓝 a ×ˢ 𝓝 b := by
+theorem nhds_prod_eq {x : X} {y : Y} : 𝓝 (x, y) = 𝓝 x ×ˢ 𝓝 y := by
   dsimp only [SProd.sprod]
   rw [Filter.prod, instTopologicalSpaceProd, nhds_inf (t₁ := TopologicalSpace.induced Prod.fst _)
     (t₂ := TopologicalSpace.induced Prod.snd _), nhds_induced, nhds_induced]
 #align nhds_prod_eq nhds_prod_eq
 
 -- porting note: moved from `topology.continuous_on`
-theorem nhdsWithin_prod_eq (a : α) (b : β) (s : Set α) (t : Set β) :
-    𝓝[s ×ˢ t] (a, b) = 𝓝[s] a ×ˢ 𝓝[t] b := by
+theorem nhdsWithin_prod_eq (x : X) (y : Y) (s : Set X) (t : Set Y) :
+    𝓝[s ×ˢ t] (x, y) = 𝓝[s] x ×ˢ 𝓝[t] y := by
   simp only [nhdsWithin, nhds_prod_eq, ← prod_inf_prod, prod_principal_principal]
 #align nhds_within_prod_eq nhdsWithin_prod_eq
 
 #noalign continuous_uncurry_of_discrete_topology
 
-theorem mem_nhds_prod_iff {a : α} {b : β} {s : Set (α × β)} :
-    s ∈ 𝓝 (a, b) ↔ ∃ u ∈ 𝓝 a, ∃ v ∈ 𝓝 b, u ×ˢ v ⊆ s := by rw [nhds_prod_eq, mem_prod_iff]
+theorem mem_nhds_prod_iff {x : X} {y : Y} {s : Set (X × Y)} :
+    s ∈ 𝓝 (x, y) ↔ ∃ u ∈ 𝓝 x, ∃ v ∈ 𝓝 y, u ×ˢ v ⊆ s := by rw [nhds_prod_eq, mem_prod_iff]
 #align mem_nhds_prod_iff mem_nhds_prod_iff
 
-theorem mem_nhdsWithin_prod_iff {a : α} {b : β} {s : Set (α × β)} {ta : Set α} {tb : Set β} :
-    s ∈ 𝓝[ta ×ˢ tb] (a, b) ↔ ∃ u ∈ 𝓝[ta] a, ∃ v ∈ 𝓝[tb] b, u ×ˢ v ⊆ s := by
+theorem mem_nhdsWithin_prod_iff {x : X} {y : Y} {s : Set (X × Y)} {tx : Set X} {ty : Set Y} :
+    s ∈ 𝓝[tx ×ˢ ty] (x, y) ↔ ∃ u ∈ 𝓝[tx] x, ∃ v ∈ 𝓝[ty] y, u ×ˢ v ⊆ s := by
   rw [nhdsWithin_prod_eq, mem_prod_iff]
 
 -- porting note: moved up
-theorem Filter.HasBasis.prod_nhds {ιa ιb : Type*} {pa : ιa → Prop} {pb : ιb → Prop}
-    {sa : ιa → Set α} {sb : ιb → Set β} {a : α} {b : β} (ha : (𝓝 a).HasBasis pa sa)
-    (hb : (𝓝 b).HasBasis pb sb) :
-    (𝓝 (a, b)).HasBasis (fun i : ιa × ιb => pa i.1 ∧ pb i.2) fun i => sa i.1 ×ˢ sb i.2 := by
+theorem Filter.HasBasis.prod_nhds {ιX ιY : Type*} {px : ιX → Prop} {py : ιY → Prop}
+    {sx : ιX → Set X} {sy : ιY → Set Y} {x : X} {y : Y} (hx : (𝓝 x).HasBasis px sx)
+    (hy : (𝓝 y).HasBasis py sy) :
+    (𝓝 (x, y)).HasBasis (fun i : ιX × ιY => px i.1 ∧ py i.2) fun i => sx i.1 ×ˢ sy i.2 := by
   rw [nhds_prod_eq]
-  exact ha.prod hb
+  exact hx.prod hy
 #align filter.has_basis.prod_nhds Filter.HasBasis.prod_nhds
 
 -- porting note: moved up
-theorem Filter.HasBasis.prod_nhds' {ιa ιb : Type*} {pa : ιa → Prop} {pb : ιb → Prop}
-    {sa : ιa → Set α} {sb : ιb → Set β} {ab : α × β} (ha : (𝓝 ab.1).HasBasis pa sa)
-    (hb : (𝓝 ab.2).HasBasis pb sb) :
-    (𝓝 ab).HasBasis (fun i : ιa × ιb => pa i.1 ∧ pb i.2) fun i => sa i.1 ×ˢ sb i.2 :=
-  ha.prod_nhds hb
+theorem Filter.HasBasis.prod_nhds' {ιX ιY : Type*} {pX : ιX → Prop} {pY : ιY → Prop}
+    {sx : ιX → Set X} {sy : ιY → Set Y} {p : X × Y} (hx : (𝓝 p.1).HasBasis pX sx)
+    (hy : (𝓝 p.2).HasBasis pY sy) :
+    (𝓝 p).HasBasis (fun i : ιX × ιY => pX i.1 ∧ pY i.2) fun i => sx i.1 ×ˢ sy i.2 :=
+  hx.prod_nhds hy
 #align filter.has_basis.prod_nhds' Filter.HasBasis.prod_nhds'
 
-theorem mem_nhds_prod_iff' {a : α} {b : β} {s : Set (α × β)} :
-    s ∈ 𝓝 (a, b) ↔ ∃ u v, IsOpen u ∧ a ∈ u ∧ IsOpen v ∧ b ∈ v ∧ u ×ˢ v ⊆ s :=
-  ((nhds_basis_opens a).prod_nhds (nhds_basis_opens b)).mem_iff.trans <| by
+theorem mem_nhds_prod_iff' {x : X} {y : Y} {s : Set (X × Y)} :
+    s ∈ 𝓝 (x, y) ↔ ∃ u v, IsOpen u ∧ x ∈ u ∧ IsOpen v ∧ y ∈ v ∧ u ×ˢ v ⊆ s :=
+  ((nhds_basis_opens x).prod_nhds (nhds_basis_opens y)).mem_iff.trans <| by
     simp only [Prod.exists, and_comm, and_assoc, and_left_comm]
 #align mem_nhds_prod_iff' mem_nhds_prod_iff'
 
-theorem Prod.tendsto_iff {α} (seq : α → β × γ) {f : Filter α} (x : β × γ) :
-    Tendsto seq f (𝓝 x) ↔
-      Tendsto (fun n => (seq n).fst) f (𝓝 x.fst) ∧ Tendsto (fun n => (seq n).snd) f (𝓝 x.snd) := by
+theorem Prod.tendsto_iff {X} (seq : X → Y × Z) {f : Filter X} (p : Y × Z) :
+    Tendsto seq f (𝓝 p) ↔
+      Tendsto (fun n => (seq n).fst) f (𝓝 p.fst) ∧ Tendsto (fun n => (seq n).snd) f (𝓝 p.snd) := by
   rw [nhds_prod_eq, Filter.tendsto_prod_iff']
 #align prod.tendsto_iff Prod.tendsto_iff
 
-instance [DiscreteTopology α] [DiscreteTopology β] : DiscreteTopology (α × β) :=
+instance [DiscreteTopology X] [DiscreteTopology Y] : DiscreteTopology (X × Y) :=
   discreteTopology_iff_nhds.2 fun (a, b) => by
-    rw [nhds_prod_eq, nhds_discrete α, nhds_discrete β, prod_pure_pure]
+    rw [nhds_prod_eq, nhds_discrete X, nhds_discrete Y, prod_pure_pure]
 
-theorem prod_mem_nhds_iff {s : Set α} {t : Set β} {a : α} {b : β} :
-    s ×ˢ t ∈ 𝓝 (a, b) ↔ s ∈ 𝓝 a ∧ t ∈ 𝓝 b := by rw [nhds_prod_eq, prod_mem_prod_iff]
+theorem prod_mem_nhds_iff {s : Set X} {t : Set Y} {x : X} {y : Y} :
+    s ×ˢ t ∈ 𝓝 (x, y) ↔ s ∈ 𝓝 x ∧ t ∈ 𝓝 y := by rw [nhds_prod_eq, prod_mem_prod_iff]
 #align prod_mem_nhds_iff prod_mem_nhds_iff
 
-theorem prod_mem_nhds {s : Set α} {t : Set β} {a : α} {b : β} (ha : s ∈ 𝓝 a) (hb : t ∈ 𝓝 b) :
-    s ×ˢ t ∈ 𝓝 (a, b) :=
-  prod_mem_nhds_iff.2 ⟨ha, hb⟩
+theorem prod_mem_nhds {s : Set X} {t : Set Y} {x : X} {y : Y} (hx : s ∈ 𝓝 x) (hy : t ∈ 𝓝 y) :
+    s ×ˢ t ∈ 𝓝 (x, y) :=
+  prod_mem_nhds_iff.2 ⟨hx, hy⟩
 #align prod_mem_nhds prod_mem_nhds
 
-theorem Filter.Eventually.prod_nhds {p : α → Prop} {q : β → Prop} {a : α} {b : β}
-    (ha : ∀ᶠ x in 𝓝 a, p x) (hb : ∀ᶠ y in 𝓝 b, q y) : ∀ᶠ z : α × β in 𝓝 (a, b), p z.1 ∧ q z.2 :=
-  prod_mem_nhds ha hb
+theorem Filter.Eventually.prod_nhds {p : X → Prop} {q : Y → Prop} {x : X} {y : Y}
+    (hx : ∀ᶠ x in 𝓝 x, p x) (hy : ∀ᶠ y in 𝓝 y, q y) : ∀ᶠ z : X × Y in 𝓝 (x, y), p z.1 ∧ q z.2 :=
+  prod_mem_nhds hx hy
 #align filter.eventually.prod_nhds Filter.Eventually.prod_nhds
 
-theorem nhds_swap (a : α) (b : β) : 𝓝 (a, b) = (𝓝 (b, a)).map Prod.swap := by
+theorem nhds_swap (x : X) (y : Y) : 𝓝 (x, y) = (𝓝 (y, x)).map Prod.swap := by
   rw [nhds_prod_eq, Filter.prod_comm, nhds_prod_eq]; rfl
 #align nhds_swap nhds_swap
 
-theorem Filter.Tendsto.prod_mk_nhds {γ} {a : α} {b : β} {f : Filter γ} {ma : γ → α} {mb : γ → β}
-    (ha : Tendsto ma f (𝓝 a)) (hb : Tendsto mb f (𝓝 b)) :
-    Tendsto (fun c => (ma c, mb c)) f (𝓝 (a, b)) := by
-  rw [nhds_prod_eq]; exact Filter.Tendsto.prod_mk ha hb
+theorem Filter.Tendsto.prod_mk_nhds {γ} {x : X} {y : Y} {f : Filter γ} {mx : γ → X} {my : γ → Y}
+    (hx : Tendsto mx f (𝓝 x)) (hy : Tendsto my f (𝓝 y)) :
+    Tendsto (fun c => (mx c, my c)) f (𝓝 (x, y)) := by
+  rw [nhds_prod_eq]; exact Filter.Tendsto.prod_mk hx hy
 #align filter.tendsto.prod_mk_nhds Filter.Tendsto.prod_mk_nhds
 
-theorem Filter.Eventually.curry_nhds {p : α × β → Prop} {x : α} {y : β}
+theorem Filter.Eventually.curry_nhds {p : X × Y → Prop} {x : X} {y : Y}
     (h : ∀ᶠ x in 𝓝 (x, y), p x) : ∀ᶠ x' in 𝓝 x, ∀ᶠ y' in 𝓝 y, p (x', y') := by
   rw [nhds_prod_eq] at h
   exact h.curry
 #align filter.eventually.curry_nhds Filter.Eventually.curry_nhds
 
-theorem ContinuousAt.prod {f : α → β} {g : α → γ} {x : α} (hf : ContinuousAt f x)
+theorem ContinuousAt.prod {f : X → Y} {g : X → Z} {x : X} (hf : ContinuousAt f x)
     (hg : ContinuousAt g x) : ContinuousAt (fun x => (f x, g x)) x :=
   hf.prod_mk_nhds hg
 #align continuous_at.prod ContinuousAt.prod
 
-theorem ContinuousAt.prod_map {f : α → γ} {g : β → δ} {p : α × β} (hf : ContinuousAt f p.fst)
-    (hg : ContinuousAt g p.snd) : ContinuousAt (fun p : α × β => (f p.1, g p.2)) p :=
+theorem ContinuousAt.prod_map {f : X → Z} {g : Y → W} {p : X × Y} (hf : ContinuousAt f p.fst)
+    (hg : ContinuousAt g p.snd) : ContinuousAt (fun p : X × Y => (f p.1, g p.2)) p :=
   hf.fst''.prod hg.snd''
 #align continuous_at.prod_map ContinuousAt.prod_map
 
-theorem ContinuousAt.prod_map' {f : α → γ} {g : β → δ} {x : α} {y : β} (hf : ContinuousAt f x)
-    (hg : ContinuousAt g y) : ContinuousAt (fun p : α × β => (f p.1, g p.2)) (x, y) :=
+theorem ContinuousAt.prod_map' {f : X → Z} {g : Y → W} {x : X} {y : Y} (hf : ContinuousAt f x)
+    (hg : ContinuousAt g y) : ContinuousAt (fun p : X × Y => (f p.1, g p.2)) (x, y) :=
   hf.fst'.prod hg.snd'
 #align continuous_at.prod_map' ContinuousAt.prod_map'
 
 -- todo: reformulate using `Set.image2`
 -- todo: prove a version of `generateFrom_union` with `image2 (∩) s t` in the LHS and use it here
-theorem prod_generateFrom_generateFrom_eq {α β : Type*} {s : Set (Set α)} {t : Set (Set β)}
+theorem prod_generateFrom_generateFrom_eq {X Y : Type*} {s : Set (Set X)} {t : Set (Set Y)}
     (hs : ⋃₀ s = univ) (ht : ⋃₀ t = univ) :
-    @instTopologicalSpaceProd α β (generateFrom s) (generateFrom t) =
+    @instTopologicalSpaceProd X Y (generateFrom s) (generateFrom t) =
       generateFrom { g | ∃ u ∈ s, ∃ v ∈ t, g = u ×ˢ v } :=
   let G := generateFrom { g | ∃ u ∈ s, ∃ v ∈ t, g = u ×ˢ v }
   le_antisymm
@@ -657,7 +657,7 @@ theorem prod_generateFrom_generateFrom_eq {α β : Type*} {s : Set (Set α)} {t
 -- todo: use the previous lemma?
 theorem prod_eq_generateFrom :
     instTopologicalSpaceProd =
-      generateFrom { g | ∃ (s : Set α) (t : Set β), IsOpen s ∧ IsOpen t ∧ g = s ×ˢ t } :=
+      generateFrom { g | ∃ (s : Set X) (t : Set Y), IsOpen s ∧ IsOpen t ∧ g = s ×ˢ t } :=
   le_antisymm (le_generateFrom fun g ⟨s, t, hs, ht, g_eq⟩ => g_eq.symm ▸ hs.prod ht)
     (le_inf
       (ball_image_of_ball fun t ht =>
@@ -667,15 +667,15 @@ theorem prod_eq_generateFrom :
 #align prod_eq_generate_from prod_eq_generateFrom
 
 -- porting note: todo: align with `mem_nhds_prod_iff'`
-theorem isOpen_prod_iff {s : Set (α × β)} :
+theorem isOpen_prod_iff {s : Set (X × Y)} :
     IsOpen s ↔ ∀ a b, (a, b) ∈ s →
       ∃ u v, IsOpen u ∧ IsOpen v ∧ a ∈ u ∧ b ∈ v ∧ u ×ˢ v ⊆ s :=
   isOpen_iff_mem_nhds.trans <| by simp_rw [Prod.forall, mem_nhds_prod_iff', and_left_comm]
 #align is_open_prod_iff isOpen_prod_iff
 
 /-- A product of induced topologies is induced by the product map -/
-theorem prod_induced_induced (f : α → β) (g : γ → δ) :
-    @instTopologicalSpaceProd α γ (induced f ‹_›) (induced g ‹_›) =
+theorem prod_induced_induced (f : X → Y) (g : Z → W) :
+    @instTopologicalSpaceProd X Z (induced f ‹_›) (induced g ‹_›) =
       induced (fun p => (f p.1, g p.2)) instTopologicalSpaceProd := by
   delta instTopologicalSpaceProd
   simp_rw [induced_inf, induced_compose]
@@ -686,14 +686,14 @@ theorem prod_induced_induced (f : α → β) (g : γ → δ) :
 
 /-- Given a neighborhood `s` of `(x, x)`, then `(x, x)` has a square open neighborhood
   that is a subset of `s`. -/
-theorem exists_nhds_square {s : Set (α × α)} {x : α} (hx : s ∈ 𝓝 (x, x)) :
-    ∃ U : Set α, IsOpen U ∧ x ∈ U ∧ U ×ˢ U ⊆ s := by
+theorem exists_nhds_square {s : Set (X × X)} {x : X} (hx : s ∈ 𝓝 (x, x)) :
+    ∃ U : Set X, IsOpen U ∧ x ∈ U ∧ U ×ˢ U ⊆ s := by
   simpa [nhds_prod_eq, (nhds_basis_opens x).prod_self.mem_iff, and_assoc, and_left_comm] using hx
 #align exists_nhds_square exists_nhds_square
 
-/-- `Prod.fst` maps neighborhood of `x : α × β` within the section `Prod.snd ⁻¹' {x.2}`
+/-- `Prod.fst` maps neighborhood of `x : X × Y` within the section `Prod.snd ⁻¹' {x.2}`
 to `𝓝 x.1`. -/
-theorem map_fst_nhdsWithin (x : α × β) : map Prod.fst (𝓝[Prod.snd ⁻¹' {x.2}] x) = 𝓝 x.1 := by
+theorem map_fst_nhdsWithin (x : X × Y) : map Prod.fst (𝓝[Prod.snd ⁻¹' {x.2}] x) = 𝓝 x.1 := by
   refine' le_antisymm (continuousAt_fst.mono_left inf_le_left) fun s hs => _
   rcases x with ⟨x, y⟩
   rw [mem_map, nhdsWithin, mem_inf_principal, mem_nhds_prod_iff] at hs
@@ -703,18 +703,18 @@ theorem map_fst_nhdsWithin (x : α × β) : map Prod.fst (𝓝[Prod.snd ⁻¹' {
 #align map_fst_nhds_within map_fst_nhdsWithin
 
 @[simp]
-theorem map_fst_nhds (x : α × β) : map Prod.fst (𝓝 x) = 𝓝 x.1 :=
+theorem map_fst_nhds (x : X × Y) : map Prod.fst (𝓝 x) = 𝓝 x.1 :=
   le_antisymm continuousAt_fst <| (map_fst_nhdsWithin x).symm.trans_le (map_mono inf_le_left)
 #align map_fst_nhds map_fst_nhds
 
 /-- The first projection in a product of topological spaces sends open sets to open sets. -/
-theorem isOpenMap_fst : IsOpenMap (@Prod.fst α β) :=
+theorem isOpenMap_fst : IsOpenMap (@Prod.fst X Y) :=
   isOpenMap_iff_nhds_le.2 fun x => (map_fst_nhds x).ge
 #align is_open_map_fst isOpenMap_fst
 
-/-- `Prod.snd` maps neighborhood of `x : α × β` within the section `Prod.fst ⁻¹' {x.1}`
+/-- `Prod.snd` maps neighborhood of `x : X × Y` within the section `Prod.fst ⁻¹' {x.1}`
 to `𝓝 x.2`. -/
-theorem map_snd_nhdsWithin (x : α × β) : map Prod.snd (𝓝[Prod.fst ⁻¹' {x.1}] x) = 𝓝 x.2 := by
+theorem map_snd_nhdsWithin (x : X × Y) : map Prod.snd (𝓝[Prod.fst ⁻¹' {x.1}] x) = 𝓝 x.2 := by
   refine' le_antisymm (continuousAt_snd.mono_left inf_le_left) fun s hs => _
   rcases x with ⟨x, y⟩
   rw [mem_map, nhdsWithin, mem_inf_principal, mem_nhds_prod_iff] at hs
@@ -724,18 +724,18 @@ theorem map_snd_nhdsWithin (x : α × β) : map Prod.snd (𝓝[Prod.fst ⁻¹' {
 #align map_snd_nhds_within map_snd_nhdsWithin
 
 @[simp]
-theorem map_snd_nhds (x : α × β) : map Prod.snd (𝓝 x) = 𝓝 x.2 :=
+theorem map_snd_nhds (x : X × Y) : map Prod.snd (𝓝 x) = 𝓝 x.2 :=
   le_antisymm continuousAt_snd <| (map_snd_nhdsWithin x).symm.trans_le (map_mono inf_le_left)
 #align map_snd_nhds map_snd_nhds
 
 /-- The second projection in a product of topological spaces sends open sets to open sets. -/
-theorem isOpenMap_snd : IsOpenMap (@Prod.snd α β) :=
+theorem isOpenMap_snd : IsOpenMap (@Prod.snd X Y) :=
   isOpenMap_iff_nhds_le.2 fun x => (map_snd_nhds x).ge
 #align is_open_map_snd isOpenMap_snd
 
 /-- A product set is open in a product space if and only if each factor is open, or one of them is
 empty -/
-theorem isOpen_prod_iff' {s : Set α} {t : Set β} :
+theorem isOpen_prod_iff' {s : Set X} {t : Set Y} :
     IsOpen (s ×ˢ t) ↔ IsOpen s ∧ IsOpen t ∨ s = ∅ ∨ t = ∅ := by
   rcases (s ×ˢ t).eq_empty_or_nonempty with h | h
   · simp [h, prod_eq_empty_iff.1 h]
@@ -754,100 +754,100 @@ theorem isOpen_prod_iff' {s : Set α} {t : Set β} :
       exact H.1.prod H.2
 #align is_open_prod_iff' isOpen_prod_iff'
 
-theorem closure_prod_eq {s : Set α} {t : Set β} : closure (s ×ˢ t) = closure s ×ˢ closure t :=
+theorem closure_prod_eq {s : Set X} {t : Set Y} : closure (s ×ˢ t) = closure s ×ˢ closure t :=
   Set.ext fun ⟨a, b⟩ => by
     simp_rw [mem_prod, mem_closure_iff_nhdsWithin_neBot, nhdsWithin_prod_eq, prod_neBot]
 #align closure_prod_eq closure_prod_eq
 
-theorem interior_prod_eq (s : Set α) (t : Set β) : interior (s ×ˢ t) = interior s ×ˢ interior t :=
+theorem interior_prod_eq (s : Set X) (t : Set Y) : interior (s ×ˢ t) = interior s ×ˢ interior t :=
   Set.ext fun ⟨a, b⟩ => by simp only [mem_interior_iff_mem_nhds, mem_prod, prod_mem_nhds_iff]
 #align interior_prod_eq interior_prod_eq
 
-theorem frontier_prod_eq (s : Set α) (t : Set β) :
+theorem frontier_prod_eq (s : Set X) (t : Set Y) :
     frontier (s ×ˢ t) = closure s ×ˢ frontier t ∪ frontier s ×ˢ closure t := by
   simp only [frontier, closure_prod_eq, interior_prod_eq, prod_diff_prod]
 #align frontier_prod_eq frontier_prod_eq
 
 @[simp]
-theorem frontier_prod_univ_eq (s : Set α) :
-    frontier (s ×ˢ (univ : Set β)) = frontier s ×ˢ univ := by
+theorem frontier_prod_univ_eq (s : Set X) :
+    frontier (s ×ˢ (univ : Set Y)) = frontier s ×ˢ univ := by
   simp [frontier_prod_eq]
 #align frontier_prod_univ_eq frontier_prod_univ_eq
 
 @[simp]
-theorem frontier_univ_prod_eq (s : Set β) :
-    frontier ((univ : Set α) ×ˢ s) = univ ×ˢ frontier s := by
+theorem frontier_univ_prod_eq (s : Set Y) :
+    frontier ((univ : Set X) ×ˢ s) = univ ×ˢ frontier s := by
   simp [frontier_prod_eq]
 #align frontier_univ_prod_eq frontier_univ_prod_eq
 
-theorem map_mem_closure₂ {f : α → β → γ} {a : α} {b : β} {s : Set α} {t : Set β} {u : Set γ}
-    (hf : Continuous (uncurry f)) (ha : a ∈ closure s) (hb : b ∈ closure t)
-    (h : ∀ a ∈ s, ∀ b ∈ t, f a b ∈ u) : f a b ∈ closure u :=
-  have H₁ : (a, b) ∈ closure (s ×ˢ t) := by simpa only [closure_prod_eq] using mk_mem_prod ha hb
+theorem map_mem_closure₂ {f : X → Y → Z} {x : X} {y : Y} {s : Set X} {t : Set Y} {u : Set Z}
+    (hf : Continuous (uncurry f)) (hx : x ∈ closure s) (hy : y ∈ closure t)
+    (h : ∀ a ∈ s, ∀ b ∈ t, f a b ∈ u) : f x y ∈ closure u :=
+  have H₁ : (x, y) ∈ closure (s ×ˢ t) := by simpa only [closure_prod_eq] using mk_mem_prod hx hy
   have H₂ : MapsTo (uncurry f) (s ×ˢ t) u := forall_prod_set.2 h
   H₂.closure hf H₁
 #align map_mem_closure₂ map_mem_closure₂
 
-theorem IsClosed.prod {s₁ : Set α} {s₂ : Set β} (h₁ : IsClosed s₁) (h₂ : IsClosed s₂) :
+theorem IsClosed.prod {s₁ : Set X} {s₂ : Set Y} (h₁ : IsClosed s₁) (h₂ : IsClosed s₂) :
     IsClosed (s₁ ×ˢ s₂) :=
   closure_eq_iff_isClosed.mp <| by simp only [h₁.closure_eq, h₂.closure_eq, closure_prod_eq]
 #align is_closed.prod IsClosed.prod
 
 /-- The product of two dense sets is a dense set. -/
-theorem Dense.prod {s : Set α} {t : Set β} (hs : Dense s) (ht : Dense t) : Dense (s ×ˢ t) :=
+theorem Dense.prod {s : Set X} {t : Set Y} (hs : Dense s) (ht : Dense t) : Dense (s ×ˢ t) :=
   fun x => by
   rw [closure_prod_eq]
   exact ⟨hs x.1, ht x.2⟩
 #align dense.prod Dense.prod
 
 /-- If `f` and `g` are maps with dense range, then `Prod.map f g` has dense range. -/
-theorem DenseRange.prod_map {ι : Type*} {κ : Type*} {f : ι → β} {g : κ → γ} (hf : DenseRange f)
+theorem DenseRange.prod_map {ι : Type*} {κ : Type*} {f : ι → Y} {g : κ → Z} (hf : DenseRange f)
     (hg : DenseRange g) : DenseRange (Prod.map f g) := by
   simpa only [DenseRange, prod_range_range_eq] using hf.prod hg
 #align dense_range.prod_map DenseRange.prod_map
 
-theorem Inducing.prod_map {f : α → β} {g : γ → δ} (hf : Inducing f) (hg : Inducing g) :
+theorem Inducing.prod_map {f : X → Y} {g : Z → W} (hf : Inducing f) (hg : Inducing g) :
     Inducing (Prod.map f g) :=
-  inducing_iff_nhds.2 fun (a, b) => by simp_rw [Prod.map_def, nhds_prod_eq, hf.nhds_eq_comap,
+  inducing_iff_nhds.2 fun (x, z) => by simp_rw [Prod.map_def, nhds_prod_eq, hf.nhds_eq_comap,
     hg.nhds_eq_comap, prod_comap_comap_eq]
 #align inducing.prod_mk Inducing.prod_map
 
 @[simp]
-theorem inducing_const_prod {a : α} {f : β → γ} : (Inducing fun x => (a, f x)) ↔ Inducing f := by
+theorem inducing_const_prod {x : X} {f : Y → Z} : (Inducing fun x' => (x, f x')) ↔ Inducing f := by
   simp_rw [inducing_iff, instTopologicalSpaceProd, induced_inf, induced_compose, Function.comp,
     induced_const, top_inf_eq]
 #align inducing_const_prod inducing_const_prod
 
 @[simp]
-theorem inducing_prod_const {b : β} {f : α → γ} : (Inducing fun x => (f x, b)) ↔ Inducing f := by
+theorem inducing_prod_const {y : Y} {f : X → Z} : (Inducing fun x => (f x, y)) ↔ Inducing f := by
   simp_rw [inducing_iff, instTopologicalSpaceProd, induced_inf, induced_compose, Function.comp,
     induced_const, inf_top_eq]
 #align inducing_prod_const inducing_prod_const
 
-theorem Embedding.prod_map {f : α → β} {g : γ → δ} (hf : Embedding f) (hg : Embedding g) :
+theorem Embedding.prod_map {f : X → Y} {g : Z → W} (hf : Embedding f) (hg : Embedding g) :
     Embedding (Prod.map f g) :=
   { hf.toInducing.prod_map hg.toInducing with
-    inj := fun ⟨x₁, x₂⟩ ⟨y₁, y₂⟩ => by simp [hf.inj.eq_iff, hg.inj.eq_iff] }
+    inj := fun ⟨x₁, z₁⟩ ⟨x₂, z₂⟩ => by simp [hf.inj.eq_iff, hg.inj.eq_iff] }
 #align embedding.prod_mk Embedding.prod_map
 
-protected theorem IsOpenMap.prod {f : α → β} {g : γ → δ} (hf : IsOpenMap f) (hg : IsOpenMap g) :
-    IsOpenMap fun p : α × γ => (f p.1, g p.2) := by
+protected theorem IsOpenMap.prod {f : X → Y} {g : Z → W} (hf : IsOpenMap f) (hg : IsOpenMap g) :
+    IsOpenMap fun p : X × Z => (f p.1, g p.2) := by
   rw [isOpenMap_iff_nhds_le]
   rintro ⟨a, b⟩
   rw [nhds_prod_eq, nhds_prod_eq, ← Filter.prod_map_map_eq]
   exact Filter.prod_mono (hf.nhds_le a) (hg.nhds_le b)
 #align is_open_map.prod IsOpenMap.prod
 
-protected theorem OpenEmbedding.prod {f : α → β} {g : γ → δ} (hf : OpenEmbedding f)
-    (hg : OpenEmbedding g) : OpenEmbedding fun x : α × γ => (f x.1, g x.2) :=
+protected theorem OpenEmbedding.prod {f : X → Y} {g : Z → W} (hf : OpenEmbedding f)
+    (hg : OpenEmbedding g) : OpenEmbedding fun x : X × Z => (f x.1, g x.2) :=
   openEmbedding_of_embedding_open (hf.1.prod_map hg.1) (hf.isOpenMap.prod hg.isOpenMap)
 #align open_embedding.prod OpenEmbedding.prod
 
-theorem embedding_graph {f : α → β} (hf : Continuous f) : Embedding fun x => (x, f x) :=
+theorem embedding_graph {f : X → Y} (hf : Continuous f) : Embedding fun x => (x, f x) :=
   embedding_of_embedding_compose (continuous_id.prod_mk hf) continuous_fst embedding_id
 #align embedding_graph embedding_graph
 
-theorem embedding_prod_mk (x : α) : Embedding (Prod.mk x : β → α × β) :=
+theorem embedding_prod_mk (x : X) : Embedding (Prod.mk x : Y → X × Y) :=
   embedding_of_embedding_compose (Continuous.Prod.mk x) continuous_snd embedding_id
 
 end Prod
@@ -856,136 +856,136 @@ section Sum
 
 open Sum
 
-variable [TopologicalSpace α] [TopologicalSpace β] [TopologicalSpace γ] [TopologicalSpace δ]
+variable [TopologicalSpace X] [TopologicalSpace Y] [TopologicalSpace Z] [TopologicalSpace W]
 
-theorem continuous_sum_dom {f : α ⊕ β → γ} :
+theorem continuous_sum_dom {f : X ⊕ Y → Z} :
     Continuous f ↔ Continuous (f ∘ Sum.inl) ∧ Continuous (f ∘ Sum.inr) :=
   (continuous_sup_dom (t₁ := TopologicalSpace.coinduced Sum.inl _)
     (t₂ := TopologicalSpace.coinduced Sum.inr _)).trans <|
     continuous_coinduced_dom.and continuous_coinduced_dom
 #align continuous_sum_dom continuous_sum_dom
 
-theorem continuous_sum_elim {f : α → γ} {g : β → γ} :
+theorem continuous_sum_elim {f : X → Z} {g : Y → Z} :
     Continuous (Sum.elim f g) ↔ Continuous f ∧ Continuous g :=
   continuous_sum_dom
 #align continuous_sum_elim continuous_sum_elim
 
 @[continuity]
-theorem Continuous.sum_elim {f : α → γ} {g : β → γ} (hf : Continuous f) (hg : Continuous g) :
+theorem Continuous.sum_elim {f : X → Z} {g : Y → Z} (hf : Continuous f) (hg : Continuous g) :
     Continuous (Sum.elim f g) :=
   continuous_sum_elim.2 ⟨hf, hg⟩
 #align continuous.sum_elim Continuous.sum_elim
 
 @[continuity]
-theorem continuous_isLeft : Continuous (isLeft : α ⊕ β → Bool) :=
+theorem continuous_isLeft : Continuous (isLeft : X ⊕ Y → Bool) :=
   continuous_sum_dom.2 ⟨continuous_const, continuous_const⟩
 
 @[continuity]
-theorem continuous_isRight : Continuous (isRight : α ⊕ β → Bool) :=
+theorem continuous_isRight : Continuous (isRight : X ⊕ Y → Bool) :=
   continuous_sum_dom.2 ⟨continuous_const, continuous_const⟩
 
 @[continuity]
 -- porting note: the proof was `continuous_sup_rng_left continuous_coinduced_rng`
-theorem continuous_inl : Continuous (@inl α β) := ⟨fun _ => And.left⟩
+theorem continuous_inl : Continuous (@inl X Y) := ⟨fun _ => And.left⟩
 #align continuous_inl continuous_inl
 
 @[continuity]
 -- porting note: the proof was `continuous_sup_rng_right continuous_coinduced_rng`
-theorem continuous_inr : Continuous (@inr α β) := ⟨fun _ => And.right⟩
+theorem continuous_inr : Continuous (@inr X Y) := ⟨fun _ => And.right⟩
 #align continuous_inr continuous_inr
 
-theorem isOpen_sum_iff {s : Set (Sum α β)} : IsOpen s ↔ IsOpen (inl ⁻¹' s) ∧ IsOpen (inr ⁻¹' s) :=
+theorem isOpen_sum_iff {s : Set (X ⊕ Y)} : IsOpen s ↔ IsOpen (inl ⁻¹' s) ∧ IsOpen (inr ⁻¹' s) :=
   Iff.rfl
 #align is_open_sum_iff isOpen_sum_iff
 
 -- porting note: new theorem
-theorem isClosed_sum_iff {s : Set (α ⊕ β)} :
+theorem isClosed_sum_iff {s : Set (X ⊕ Y)} :
     IsClosed s ↔ IsClosed (inl ⁻¹' s) ∧ IsClosed (inr ⁻¹' s) := by
   simp only [← isOpen_compl_iff, isOpen_sum_iff, preimage_compl]
 
-theorem isOpenMap_inl : IsOpenMap (@inl α β) := fun u hu => by
+theorem isOpenMap_inl : IsOpenMap (@inl X Y) := fun u hu => by
   simpa [isOpen_sum_iff, preimage_image_eq u Sum.inl_injective]
 #align is_open_map_inl isOpenMap_inl
 
-theorem isOpenMap_inr : IsOpenMap (@inr α β) := fun u hu => by
+theorem isOpenMap_inr : IsOpenMap (@inr X Y) := fun u hu => by
   simpa [isOpen_sum_iff, preimage_image_eq u Sum.inr_injective]
 #align is_open_map_inr isOpenMap_inr
 
-theorem openEmbedding_inl : OpenEmbedding (@inl α β) :=
+theorem openEmbedding_inl : OpenEmbedding (@inl X Y) :=
   openEmbedding_of_continuous_injective_open continuous_inl inl_injective isOpenMap_inl
 #align open_embedding_inl openEmbedding_inl
 
-theorem openEmbedding_inr : OpenEmbedding (@inr α β) :=
+theorem openEmbedding_inr : OpenEmbedding (@inr X Y) :=
   openEmbedding_of_continuous_injective_open continuous_inr inr_injective isOpenMap_inr
 #align open_embedding_inr openEmbedding_inr
 
-theorem embedding_inl : Embedding (@inl α β) :=
+theorem embedding_inl : Embedding (@inl X Y) :=
   openEmbedding_inl.1
 #align embedding_inl embedding_inl
 
-theorem embedding_inr : Embedding (@inr α β) :=
+theorem embedding_inr : Embedding (@inr X Y) :=
   openEmbedding_inr.1
 #align embedding_inr embedding_inr
 
-theorem isOpen_range_inl : IsOpen (range (inl : α → Sum α β)) :=
+theorem isOpen_range_inl : IsOpen (range (inl : X → X ⊕ Y)) :=
   openEmbedding_inl.2
 #align is_open_range_inl isOpen_range_inl
 
-theorem isOpen_range_inr : IsOpen (range (inr : β → Sum α β)) :=
+theorem isOpen_range_inr : IsOpen (range (inr : Y → X ⊕ Y)) :=
   openEmbedding_inr.2
 #align is_open_range_inr isOpen_range_inr
 
-theorem isClosed_range_inl : IsClosed (range (inl : α → Sum α β)) := by
+theorem isClosed_range_inl : IsClosed (range (inl : X → X ⊕ Y)) := by
   rw [← isOpen_compl_iff, compl_range_inl]
   exact isOpen_range_inr
 #align is_closed_range_inl isClosed_range_inl
 
-theorem isClosed_range_inr : IsClosed (range (inr : β → Sum α β)) := by
+theorem isClosed_range_inr : IsClosed (range (inr : Y → X ⊕ Y)) := by
   rw [← isOpen_compl_iff, compl_range_inr]
   exact isOpen_range_inl
 #align is_closed_range_inr isClosed_range_inr
 
-theorem closedEmbedding_inl : ClosedEmbedding (inl : α → Sum α β) :=
+theorem closedEmbedding_inl : ClosedEmbedding (inl : X → X ⊕ Y) :=
   ⟨embedding_inl, isClosed_range_inl⟩
 #align closed_embedding_inl closedEmbedding_inl
 
-theorem closedEmbedding_inr : ClosedEmbedding (inr : β → Sum α β) :=
+theorem closedEmbedding_inr : ClosedEmbedding (inr : Y → X ⊕ Y) :=
   ⟨embedding_inr, isClosed_range_inr⟩
 #align closed_embedding_inr closedEmbedding_inr
 
-theorem nhds_inl (x : α) : 𝓝 (inl x : Sum α β) = map inl (𝓝 x) :=
+theorem nhds_inl (x : X) : 𝓝 (inl x : X ⊕ Y) = map inl (𝓝 x) :=
   (openEmbedding_inl.map_nhds_eq _).symm
 #align nhds_inl nhds_inl
 
-theorem nhds_inr (x : β) : 𝓝 (inr x : Sum α β) = map inr (𝓝 x) :=
+theorem nhds_inr (y : Y) : 𝓝 (inr y : X ⊕ Y) = map inr (𝓝 y) :=
   (openEmbedding_inr.map_nhds_eq _).symm
 #align nhds_inr nhds_inr
 
 @[simp]
-theorem continuous_sum_map {f : α → β} {g : γ → δ} :
+theorem continuous_sum_map {f : X → Y} {g : Z → W} :
     Continuous (Sum.map f g) ↔ Continuous f ∧ Continuous g :=
   continuous_sum_elim.trans <|
     embedding_inl.continuous_iff.symm.and embedding_inr.continuous_iff.symm
 #align continuous_sum_map continuous_sum_map
 
 @[continuity]
-theorem Continuous.sum_map {f : α → β} {g : γ → δ} (hf : Continuous f) (hg : Continuous g) :
+theorem Continuous.sum_map {f : X → Y} {g : Z → W} (hf : Continuous f) (hg : Continuous g) :
     Continuous (Sum.map f g) :=
   continuous_sum_map.2 ⟨hf, hg⟩
 #align continuous.sum_map Continuous.sum_map
 
-theorem isOpenMap_sum {f : Sum α β → γ} :
+theorem isOpenMap_sum {f : X ⊕ Y → Z} :
     IsOpenMap f ↔ (IsOpenMap fun a => f (inl a)) ∧ IsOpenMap fun b => f (inr b) := by
   simp only [isOpenMap_iff_nhds_le, Sum.forall, nhds_inl, nhds_inr, Filter.map_map, comp]
 #align is_open_map_sum isOpenMap_sum
 
 @[simp]
-theorem isOpenMap_sum_elim {f : α → γ} {g : β → γ} :
+theorem isOpenMap_sum_elim {f : X → Z} {g : Y → Z} :
     IsOpenMap (Sum.elim f g) ↔ IsOpenMap f ∧ IsOpenMap g := by
   simp only [isOpenMap_sum, elim_inl, elim_inr]
 #align is_open_map_sum_elim isOpenMap_sum_elim
 
-theorem IsOpenMap.sum_elim {f : α → γ} {g : β → γ} (hf : IsOpenMap f) (hg : IsOpenMap g) :
+theorem IsOpenMap.sum_elim {f : X → Z} {g : Y → Z} (hf : IsOpenMap f) (hg : IsOpenMap g) :
     IsOpenMap (Sum.elim f g) :=
   isOpenMap_sum_elim.2 ⟨hf, hg⟩
 #align is_open_map.sum_elim IsOpenMap.sum_elim
@@ -994,51 +994,51 @@ end Sum
 
 section Subtype
 
-variable [TopologicalSpace α] [TopologicalSpace β] [TopologicalSpace γ] {p : α → Prop}
+variable [TopologicalSpace X] [TopologicalSpace Y] [TopologicalSpace Z] {p : X → Prop}
 
-theorem inducing_subtype_val {b : Set β} : Inducing ((↑) : b → β) := ⟨rfl⟩
+theorem inducing_subtype_val {t : Set Y} : Inducing ((↑) : t → Y) := ⟨rfl⟩
 #align inducing_coe inducing_subtype_val
 
-theorem Inducing.of_codRestrict {f : α → β} {b : Set β} (hb : ∀ a, f a ∈ b)
-    (h : Inducing (b.codRestrict f hb)) : Inducing f :=
+theorem Inducing.of_codRestrict {f : X → Y} {t : Set Y} (ht : ∀ x, f x ∈ t)
+    (h : Inducing (t.codRestrict f ht)) : Inducing f :=
   inducing_subtype_val.comp h
 #align inducing.of_cod_restrict Inducing.of_codRestrict
 
-theorem embedding_subtype_val : Embedding ((↑) : Subtype p → α) :=
+theorem embedding_subtype_val : Embedding ((↑) : Subtype p → X) :=
   ⟨inducing_subtype_val, Subtype.coe_injective⟩
 #align embedding_subtype_coe embedding_subtype_val
 
 theorem closedEmbedding_subtype_val (h : IsClosed { a | p a }) :
-    ClosedEmbedding ((↑) : Subtype p → α) :=
+    ClosedEmbedding ((↑) : Subtype p → X) :=
   ⟨embedding_subtype_val, by rwa [Subtype.range_coe_subtype]⟩
 #align closed_embedding_subtype_coe closedEmbedding_subtype_val
 
 @[continuity]
-theorem continuous_subtype_val : Continuous (@Subtype.val α p) :=
+theorem continuous_subtype_val : Continuous (@Subtype.val X p) :=
   continuous_induced_dom
 #align continuous_subtype_val continuous_subtype_val
 #align continuous_subtype_coe continuous_subtype_val
 
-theorem Continuous.subtype_val {f : β → Subtype p} (hf : Continuous f) :
-    Continuous fun x => (f x : α) :=
+theorem Continuous.subtype_val {f : Y → Subtype p} (hf : Continuous f) :
+    Continuous fun x => (f x : X) :=
   continuous_subtype_val.comp hf
 #align continuous.subtype_coe Continuous.subtype_val
 
-theorem IsOpen.openEmbedding_subtype_val {s : Set α} (hs : IsOpen s) :
-    OpenEmbedding ((↑) : s → α) :=
+theorem IsOpen.openEmbedding_subtype_val {s : Set X} (hs : IsOpen s) :
+    OpenEmbedding ((↑) : s → X) :=
   ⟨embedding_subtype_val, (@Subtype.range_coe _ s).symm ▸ hs⟩
 #align is_open.open_embedding_subtype_coe IsOpen.openEmbedding_subtype_val
 
-theorem IsOpen.isOpenMap_subtype_val {s : Set α} (hs : IsOpen s) : IsOpenMap ((↑) : s → α) :=
+theorem IsOpen.isOpenMap_subtype_val {s : Set X} (hs : IsOpen s) : IsOpenMap ((↑) : s → X) :=
   hs.openEmbedding_subtype_val.isOpenMap
 #align is_open.is_open_map_subtype_coe IsOpen.isOpenMap_subtype_val
 
-theorem IsOpenMap.restrict {f : α → β} (hf : IsOpenMap f) {s : Set α} (hs : IsOpen s) :
+theorem IsOpenMap.restrict {f : X → Y} (hf : IsOpenMap f) {s : Set X} (hs : IsOpen s) :
     IsOpenMap (s.restrict f) :=
   hf.comp hs.isOpenMap_subtype_val
 #align is_open_map.restrict IsOpenMap.restrict
 
-lemma IsClosedMap.restrictPreimage {f : α → β} (hcl : IsClosedMap f) (T : Set β) :
+lemma IsClosedMap.restrictPreimage {f : X → Y} (hcl : IsClosedMap f) (T : Set Y) :
     IsClosedMap (T.restrictPreimage f) := by
   rw [isClosedMap_iff_clusterPt] at hcl ⊢
   intro A ⟨y, hyT⟩ hy
@@ -1049,61 +1049,61 @@ lemma IsClosedMap.restrictPreimage {f : α → β} (hcl : IsClosedMap f) (T : Se
   refine ⟨⟨x, hxT⟩, Subtype.ext hxy, ?_⟩
   rwa [← inducing_subtype_val.mapClusterPt_iff, MapClusterPt, map_principal]
 
-nonrec theorem IsClosed.closedEmbedding_subtype_val {s : Set α} (hs : IsClosed s) :
-    ClosedEmbedding ((↑) : s → α) :=
+nonrec theorem IsClosed.closedEmbedding_subtype_val {s : Set X} (hs : IsClosed s) :
+    ClosedEmbedding ((↑) : s → X) :=
   closedEmbedding_subtype_val hs
 #align is_closed.closed_embedding_subtype_coe IsClosed.closedEmbedding_subtype_val
 
 @[continuity]
-theorem Continuous.subtype_mk {f : β → α} (h : Continuous f) (hp : ∀ x, p (f x)) :
+theorem Continuous.subtype_mk {f : Y → X} (h : Continuous f) (hp : ∀ x, p (f x)) :
     Continuous fun x => (⟨f x, hp x⟩ : Subtype p) :=
   continuous_induced_rng.2 h
 #align continuous.subtype_mk Continuous.subtype_mk
 
-theorem Continuous.subtype_map {f : α → β} (h : Continuous f) {q : β → Prop}
+theorem Continuous.subtype_map {f : X → Y} (h : Continuous f) {q : Y → Prop}
     (hpq : ∀ x, p x → q (f x)) : Continuous (Subtype.map f hpq) :=
   (h.comp continuous_subtype_val).subtype_mk _
 #align continuous.subtype_map Continuous.subtype_map
 
-theorem continuous_inclusion {s t : Set α} (h : s ⊆ t) : Continuous (inclusion h) :=
+theorem continuous_inclusion {s t : Set X} (h : s ⊆ t) : Continuous (inclusion h) :=
   continuous_id.subtype_map h
 #align continuous_inclusion continuous_inclusion
 
-theorem continuousAt_subtype_val {p : α → Prop} {a : Subtype p} :
-    ContinuousAt ((↑) : Subtype p → α) a :=
+theorem continuousAt_subtype_val {p : X → Prop} {x : Subtype p} :
+    ContinuousAt ((↑) : Subtype p → X) x :=
   continuous_subtype_val.continuousAt
 #align continuous_at_subtype_coe continuousAt_subtype_val
 
-theorem Subtype.dense_iff {s : Set α} {t : Set s} : Dense t ↔ s ⊆ closure ((↑) '' t) := by
+theorem Subtype.dense_iff {s : Set X} {t : Set s} : Dense t ↔ s ⊆ closure ((↑) '' t) := by
   rw [inducing_subtype_val.dense_iff, SetCoe.forall]
   rfl
 #align subtype.dense_iff Subtype.dense_iff
 
 -- porting note: new lemma
-theorem map_nhds_subtype_val {s : Set α} (a : s) : map ((↑) : s → α) (𝓝 a) = 𝓝[s] ↑a := by
+theorem map_nhds_subtype_val {s : Set X} (x : s) : map ((↑) : s → X) (𝓝 x) = 𝓝[s] ↑x := by
   rw [inducing_subtype_val.map_nhds_eq, Subtype.range_val]
 
-theorem map_nhds_subtype_coe_eq_nhds {a : α} (ha : p a) (h : ∀ᶠ x in 𝓝 a, p x) :
-    map ((↑) : Subtype p → α) (𝓝 ⟨a, ha⟩) = 𝓝 a :=
+theorem map_nhds_subtype_coe_eq_nhds {x : X} (hx : p x) (h : ∀ᶠ x in 𝓝 x, p x) :
+    map ((↑) : Subtype p → X) (𝓝 ⟨x, hx⟩) = 𝓝 x :=
   map_nhds_induced_of_mem <| by rw [Subtype.range_val]; exact h
 #align map_nhds_subtype_coe_eq map_nhds_subtype_coe_eq_nhds
 
-theorem nhds_subtype_eq_comap {a : α} {h : p a} : 𝓝 (⟨a, h⟩ : Subtype p) = comap (↑) (𝓝 a) :=
+theorem nhds_subtype_eq_comap {x : X} {h : p x} : 𝓝 (⟨x, h⟩ : Subtype p) = comap (↑) (𝓝 x) :=
   nhds_induced _ _
 #align nhds_subtype_eq_comap nhds_subtype_eq_comap
 
-theorem tendsto_subtype_rng {β : Type*} {p : α → Prop} {b : Filter β} {f : β → Subtype p} :
-    ∀ {a : Subtype p}, Tendsto f b (𝓝 a) ↔ Tendsto (fun x => (f x : α)) b (𝓝 (a : α))
+theorem tendsto_subtype_rng {Y : Type*} {p : X → Prop} {l : Filter Y} {f : Y → Subtype p} :
+    ∀ {x : Subtype p}, Tendsto f l (𝓝 x) ↔ Tendsto (fun x => (f x : X)) l (𝓝 (x : X))
   | ⟨a, ha⟩ => by rw [nhds_subtype_eq_comap, tendsto_comap_iff]; rfl
 #align tendsto_subtype_rng tendsto_subtype_rng
 
 theorem closure_subtype {x : { a // p a }} {s : Set { a // p a }} :
-    x ∈ closure s ↔ (x : α) ∈ closure (((↑) : _ → α) '' s) :=
+    x ∈ closure s ↔ (x : X) ∈ closure (((↑) : _ → X) '' s) :=
   closure_induced
 #align closure_subtype closure_subtype
 
 @[simp]
-theorem continuousAt_codRestrict_iff {f : α → β} {t : Set β} (h1 : ∀ x, f x ∈ t) {x : α} :
+theorem continuousAt_codRestrict_iff {f : X → Y} {t : Set Y} (h1 : ∀ x, f x ∈ t) {x : X} :
     ContinuousAt (codRestrict f t h1) x ↔ ContinuousAt f x :=
   inducing_subtype_val.continuousAt_iff
 #align continuous_at_cod_restrict_iff continuousAt_codRestrict_iff
@@ -1111,43 +1111,43 @@ theorem continuousAt_codRestrict_iff {f : α → β} {t : Set β} (h1 : ∀ x, f
 alias ⟨_, ContinuousAt.codRestrict⟩ := continuousAt_codRestrict_iff
 #align continuous_at.cod_restrict ContinuousAt.codRestrict
 
-theorem ContinuousAt.restrict {f : α → β} {s : Set α} {t : Set β} (h1 : MapsTo f s t) {x : s}
+theorem ContinuousAt.restrict {f : X → Y} {s : Set X} {t : Set Y} (h1 : MapsTo f s t) {x : s}
     (h2 : ContinuousAt f x) : ContinuousAt (h1.restrict f s t) x :=
   (h2.comp continuousAt_subtype_val).codRestrict _
 #align continuous_at.restrict ContinuousAt.restrict
 
-theorem ContinuousAt.restrictPreimage {f : α → β} {s : Set β} {x : f ⁻¹' s} (h : ContinuousAt f x) :
+theorem ContinuousAt.restrictPreimage {f : X → Y} {s : Set Y} {x : f ⁻¹' s} (h : ContinuousAt f x) :
     ContinuousAt (s.restrictPreimage f) x :=
   h.restrict _
 #align continuous_at.restrict_preimage ContinuousAt.restrictPreimage
 
 @[continuity]
-theorem Continuous.codRestrict {f : α → β} {s : Set β} (hf : Continuous f) (hs : ∀ a, f a ∈ s) :
+theorem Continuous.codRestrict {f : X → Y} {s : Set Y} (hf : Continuous f) (hs : ∀ a, f a ∈ s) :
     Continuous (s.codRestrict f hs) :=
   hf.subtype_mk hs
 #align continuous.cod_restrict Continuous.codRestrict
 
 @[continuity]
-theorem Continuous.restrict {f : α → β} {s : Set α} {t : Set β} (h1 : MapsTo f s t)
+theorem Continuous.restrict {f : X → Y} {s : Set X} {t : Set Y} (h1 : MapsTo f s t)
     (h2 : Continuous f) : Continuous (h1.restrict f s t) :=
   (h2.comp continuous_subtype_val).codRestrict _
 
 @[continuity]
-theorem Continuous.restrictPreimage {f : α → β} {s : Set β} (h : Continuous f) :
+theorem Continuous.restrictPreimage {f : X → Y} {s : Set Y} (h : Continuous f) :
     Continuous (s.restrictPreimage f) :=
   h.restrict _
 
-theorem Inducing.codRestrict {e : α → β} (he : Inducing e) {s : Set β} (hs : ∀ x, e x ∈ s) :
+theorem Inducing.codRestrict {e : X → Y} (he : Inducing e) {s : Set Y} (hs : ∀ x, e x ∈ s) :
     Inducing (codRestrict e s hs) :=
   inducing_of_inducing_compose (he.continuous.codRestrict hs) continuous_subtype_val he
 #align inducing.cod_restrict Inducing.codRestrict
 
-theorem Embedding.codRestrict {e : α → β} (he : Embedding e) (s : Set β) (hs : ∀ x, e x ∈ s) :
+theorem Embedding.codRestrict {e : X → Y} (he : Embedding e) (s : Set Y) (hs : ∀ x, e x ∈ s) :
     Embedding (codRestrict e s hs) :=
   embedding_of_embedding_compose (he.continuous.codRestrict hs) continuous_subtype_val he
 #align embedding.cod_restrict Embedding.codRestrict
 
-theorem embedding_inclusion {s t : Set α} (h : s ⊆ t) : Embedding (Set.inclusion h) :=
+theorem embedding_inclusion {s t : Set X} (h : s ⊆ t) : Embedding (Set.inclusion h) :=
   embedding_subtype_val.codRestrict _ _
 #align embedding_inclusion embedding_inclusion
 
@@ -1162,45 +1162,45 @@ end Subtype
 
 section Quotient
 
-variable [TopologicalSpace α] [TopologicalSpace β] [TopologicalSpace γ]
+variable [TopologicalSpace X] [TopologicalSpace Y] [TopologicalSpace Z]
 
-variable {r : α → α → Prop} {s : Setoid α}
+variable {r : X → X → Prop} {s : Setoid X}
 
-theorem quotientMap_quot_mk : QuotientMap (@Quot.mk α r) :=
+theorem quotientMap_quot_mk : QuotientMap (@Quot.mk X r) :=
   ⟨Quot.exists_rep, rfl⟩
 #align quotient_map_quot_mk quotientMap_quot_mk
 
 @[continuity]
-theorem continuous_quot_mk : Continuous (@Quot.mk α r) :=
+theorem continuous_quot_mk : Continuous (@Quot.mk X r) :=
   continuous_coinduced_rng
 #align continuous_quot_mk continuous_quot_mk
 
 @[continuity]
-theorem continuous_quot_lift {f : α → β} (hr : ∀ a b, r a b → f a = f b) (h : Continuous f) :
-    Continuous (Quot.lift f hr : Quot r → β) :=
+theorem continuous_quot_lift {f : X → Y} (hr : ∀ a b, r a b → f a = f b) (h : Continuous f) :
+    Continuous (Quot.lift f hr : Quot r → Y) :=
   continuous_coinduced_dom.2 h
 #align continuous_quot_lift continuous_quot_lift
 
-theorem quotientMap_quotient_mk' : QuotientMap (@Quotient.mk' α s) :=
+theorem quotientMap_quotient_mk' : QuotientMap (@Quotient.mk' X s) :=
   quotientMap_quot_mk
 #align quotient_map_quotient_mk quotientMap_quotient_mk'
 
-theorem continuous_quotient_mk' : Continuous (@Quotient.mk' α s) :=
+theorem continuous_quotient_mk' : Continuous (@Quotient.mk' X s) :=
   continuous_coinduced_rng
 #align continuous_quotient_mk continuous_quotient_mk'
 
-theorem Continuous.quotient_lift {f : α → β} (h : Continuous f) (hs : ∀ a b, a ≈ b → f a = f b) :
-    Continuous (Quotient.lift f hs : Quotient s → β) :=
+theorem Continuous.quotient_lift {f : X → Y} (h : Continuous f) (hs : ∀ a b, a ≈ b → f a = f b) :
+    Continuous (Quotient.lift f hs : Quotient s → Y) :=
   continuous_coinduced_dom.2 h
 #align continuous.quotient_lift Continuous.quotient_lift
 
-theorem Continuous.quotient_liftOn' {f : α → β} (h : Continuous f)
+theorem Continuous.quotient_liftOn' {f : X → Y} (h : Continuous f)
     (hs : ∀ a b, @Setoid.r _ s a b → f a = f b) :
-    Continuous (fun x => Quotient.liftOn' x f hs : Quotient s → β) :=
+    Continuous (fun x => Quotient.liftOn' x f hs : Quotient s → Y) :=
   h.quotient_lift hs
 #align continuous.quotient_lift_on' Continuous.quotient_liftOn'
 
-@[continuity] theorem Continuous.quotient_map' {t : Setoid β} {f : α → β} (hf : Continuous f)
+@[continuity] theorem Continuous.quotient_map' {t : Setoid Y} {f : X → Y} (hf : Continuous f)
     (H : (s.r ⇒ t.r) f f) : Continuous (Quotient.map' f H) :=
   (continuous_quotient_mk'.comp hf).quotient_lift _
 #align continuous.quotient_map' Continuous.quotient_map'
@@ -1209,8 +1209,8 @@ end Quotient
 
 section Pi
 
-variable {ι : Type*} {π : ι → Type*} {κ : Type*} [TopologicalSpace α]
-  [T : ∀ i, TopologicalSpace (π i)] {f : α → ∀ i : ι, π i}
+variable {ι : Type*} {π : ι → Type*} {κ : Type*} [TopologicalSpace X]
+  [T : ∀ i, TopologicalSpace (π i)] {f : X → ∀ i : ι, π i}
 
 theorem continuous_pi_iff : Continuous f ↔ ∀ i, Continuous fun a => f a i := by
   simp only [continuous_iInf_rng, continuous_induced_rng, comp]
@@ -1236,7 +1236,7 @@ theorem continuousAt_apply (i : ι) (x : ∀ i, π i) : ContinuousAt (fun p : 
   (continuous_apply i).continuousAt
 #align continuous_at_apply continuousAt_apply
 
-theorem Filter.Tendsto.apply {l : Filter β} {f : β → ∀ i, π i} {x : ∀ i, π i}
+theorem Filter.Tendsto.apply {l : Filter Y} {f : Y → ∀ i, π i} {x : ∀ i, π i}
     (h : Tendsto f l (𝓝 x)) (i : ι) : Tendsto (fun a => f a i) l (𝓝 <| x i) :=
   (continuousAt_apply i _).tendsto.comp h
 #align filter.tendsto.apply Filter.Tendsto.apply
@@ -1245,12 +1245,12 @@ theorem nhds_pi {a : ∀ i, π i} : 𝓝 a = pi fun i => 𝓝 (a i) := by
   simp only [nhds_iInf, nhds_induced, Filter.pi]
 #align nhds_pi nhds_pi
 
-theorem tendsto_pi_nhds {f : β → ∀ i, π i} {g : ∀ i, π i} {u : Filter β} :
+theorem tendsto_pi_nhds {f : Y → ∀ i, π i} {g : ∀ i, π i} {u : Filter Y} :
     Tendsto f u (𝓝 g) ↔ ∀ x, Tendsto (fun i => f i x) u (𝓝 (g x)) := by
   rw [nhds_pi, Filter.tendsto_pi]
 #align tendsto_pi_nhds tendsto_pi_nhds
 
-theorem continuousAt_pi {f : α → ∀ i, π i} {x : α} :
+theorem continuousAt_pi {f : X → ∀ i, π i} {x : X} :
     ContinuousAt f x ↔ ∀ i, ContinuousAt (fun y => f y i) x :=
   tendsto_pi_nhds
 #align continuous_at_pi continuousAt_pi
@@ -1260,16 +1260,16 @@ theorem Pi.continuous_precomp' {ι' : Type*} (φ : ι' → ι) :
   continuous_pi fun j ↦ continuous_apply (φ j)
 
 theorem Pi.continuous_precomp {ι' : Type*} (φ : ι' → ι) :
-    Continuous (· ∘ φ : (ι → α) → (ι' → α)) :=
+    Continuous (· ∘ φ : (ι → X) → (ι' → X)) :=
   Pi.continuous_precomp' φ
 
-theorem Pi.continuous_postcomp' {ρ : ι → Type*} [∀ i, TopologicalSpace (ρ i)]
-    {g : ∀ i, π i → ρ i} (hg : ∀ i, Continuous (g i)) :
+theorem Pi.continuous_postcomp' {X : ι → Type*} [∀ i, TopologicalSpace (X i)]
+    {g : ∀ i, π i → X i} (hg : ∀ i, Continuous (g i)) :
     Continuous (fun (f : (∀ i, π i)) (i : ι) ↦ g i (f i)) :=
   continuous_pi fun i ↦ (hg i).comp <| continuous_apply i
 
-theorem Pi.continuous_postcomp [TopologicalSpace β] {g : α → β} (hg : Continuous g) :
-    Continuous (g ∘ · : (ι → α) → (ι → β)) :=
+theorem Pi.continuous_postcomp [TopologicalSpace Y] {g : X → Y} (hg : Continuous g) :
+    Continuous (g ∘ · : (ι → X) → (ι → Y)) :=
   Pi.continuous_postcomp' fun _ ↦ hg
 
 lemma Pi.induced_precomp' {ι' : Type*} (φ : ι' → ι) :
@@ -1277,9 +1277,9 @@ lemma Pi.induced_precomp' {ι' : Type*} (φ : ι' → ι) :
     ⨅ i', induced (eval (φ i')) (T (φ i')) := by
   simp [Pi.topologicalSpace, induced_iInf, induced_compose, comp]
 
-lemma Pi.induced_precomp [TopologicalSpace β] {ι' : Type*} (φ : ι' → ι) :
+lemma Pi.induced_precomp [TopologicalSpace Y] {ι' : Type*} (φ : ι' → ι) :
     induced (· ∘ φ) Pi.topologicalSpace =
-    ⨅ i', induced (eval (φ i')) ‹TopologicalSpace β› :=
+    ⨅ i', induced (eval (φ i')) ‹TopologicalSpace Y› :=
   induced_precomp' φ
 
 lemma Pi.continuous_restrict (S : Set ι) :
@@ -1292,18 +1292,18 @@ lemma Pi.induced_restrict (S : Set ι) :
   simp (config := { unfoldPartialApp := true }) [← iInf_subtype'', ← induced_precomp' ((↑) : S → ι),
     Set.restrict]
 
-theorem Filter.Tendsto.update [DecidableEq ι] {l : Filter β} {f : β → ∀ i, π i} {x : ∀ i, π i}
-    (hf : Tendsto f l (𝓝 x)) (i : ι) {g : β → π i} {xi : π i} (hg : Tendsto g l (𝓝 xi)) :
+theorem Filter.Tendsto.update [DecidableEq ι] {l : Filter Y} {f : Y → ∀ i, π i} {x : ∀ i, π i}
+    (hf : Tendsto f l (𝓝 x)) (i : ι) {g : Y → π i} {xi : π i} (hg : Tendsto g l (𝓝 xi)) :
     Tendsto (fun a => update (f a) i (g a)) l (𝓝 <| update x i xi) :=
   tendsto_pi_nhds.2 fun j => by rcases eq_or_ne j i with (rfl | hj) <;> simp [*, hf.apply]
 #align filter.tendsto.update Filter.Tendsto.update
 
-theorem ContinuousAt.update [DecidableEq ι] {a : α} (hf : ContinuousAt f a) (i : ι) {g : α → π i}
-    (hg : ContinuousAt g a) : ContinuousAt (fun a => update (f a) i (g a)) a :=
+theorem ContinuousAt.update [DecidableEq ι] {x : X} (hf : ContinuousAt f x) (i : ι) {g : X → π i}
+    (hg : ContinuousAt g x) : ContinuousAt (fun a => update (f a) i (g a)) x :=
   hf.tendsto.update i hg
 #align continuous_at.update ContinuousAt.update
 
-theorem Continuous.update [DecidableEq ι] (hf : Continuous f) (i : ι) {g : α → π i}
+theorem Continuous.update [DecidableEq ι] (hf : Continuous f) (i : ι) {g : X → π i}
     (hg : Continuous g) : Continuous fun a => update (f a) i (g a) :=
   continuous_iff_continuousAt.2 fun _ => hf.continuousAt.update i hg.continuousAt
 #align continuous.update Continuous.update
@@ -1325,21 +1325,21 @@ theorem continuous_mulSingle [∀ i, One (π i)] [DecidableEq ι] (i : ι) :
 #align continuous_single continuous_single
 
 theorem Filter.Tendsto.fin_insertNth {n} {π : Fin (n + 1) → Type*} [∀ i, TopologicalSpace (π i)]
-    (i : Fin (n + 1)) {f : β → π i} {l : Filter β} {x : π i} (hf : Tendsto f l (𝓝 x))
-    {g : β → ∀ j : Fin n, π (i.succAbove j)} {y : ∀ j, π (i.succAbove j)} (hg : Tendsto g l (𝓝 y)) :
+    (i : Fin (n + 1)) {f : Y → π i} {l : Filter Y} {x : π i} (hf : Tendsto f l (𝓝 x))
+    {g : Y → ∀ j : Fin n, π (i.succAbove j)} {y : ∀ j, π (i.succAbove j)} (hg : Tendsto g l (𝓝 y)) :
     Tendsto (fun a => i.insertNth (f a) (g a)) l (𝓝 <| i.insertNth x y) :=
   tendsto_pi_nhds.2 fun j => Fin.succAboveCases i (by simpa) (by simpa using tendsto_pi_nhds.1 hg) j
 #align filter.tendsto.fin_insert_nth Filter.Tendsto.fin_insertNth
 
 theorem ContinuousAt.fin_insertNth {n} {π : Fin (n + 1) → Type*} [∀ i, TopologicalSpace (π i)]
-    (i : Fin (n + 1)) {f : α → π i} {a : α} (hf : ContinuousAt f a)
-    {g : α → ∀ j : Fin n, π (i.succAbove j)} (hg : ContinuousAt g a) :
-    ContinuousAt (fun a => i.insertNth (f a) (g a)) a :=
+    (i : Fin (n + 1)) {f : X → π i} {x : X} (hf : ContinuousAt f x)
+    {g : X → ∀ j : Fin n, π (i.succAbove j)} (hg : ContinuousAt g x) :
+    ContinuousAt (fun a => i.insertNth (f a) (g a)) x :=
   hf.tendsto.fin_insertNth i hg
 #align continuous_at.fin_insert_nth ContinuousAt.fin_insertNth
 
 theorem Continuous.fin_insertNth {n} {π : Fin (n + 1) → Type*} [∀ i, TopologicalSpace (π i)]
-    (i : Fin (n + 1)) {f : α → π i} (hf : Continuous f) {g : α → ∀ j : Fin n, π (i.succAbove j)}
+    (i : Fin (n + 1)) {f : X → π i} (hf : Continuous f) {g : X → ∀ j : Fin n, π (i.succAbove j)}
     (hg : Continuous g) : Continuous fun a => i.insertNth (f a) (g a) :=
   continuous_iff_continuousAt.2 fun _ => hf.continuousAt.fin_insertNth i hg.continuousAt
 #align continuous.fin_insert_nth Continuous.fin_insertNth
@@ -1438,8 +1438,8 @@ theorem pi_generateFrom_eq {π : ι → Type*} {g : ∀ a, Set (Set (π a))} :
     rintro _ ⟨s, i, hi, rfl⟩
     letI := fun a => generateFrom (g a)
     exact isOpen_set_pi i.finite_toSet (fun a ha => GenerateOpen.basic _ (hi a ha))
-  · refine le_iInf fun a => coinduced_le_iff_le_induced.1 <| le_generateFrom fun s hs => ?_
-    refine GenerateOpen.basic _ ⟨update (fun a => univ) a s, {a}, ?_⟩
+  · refine le_iInf fun i => coinduced_le_iff_le_induced.1 <| le_generateFrom fun s hs => ?_
+    refine GenerateOpen.basic _ ⟨update (fun i => univ) i s, {i}, ?_⟩
     simp [hs]
 #align pi_generate_from_eq pi_generateFrom_eq
 
@@ -1502,7 +1502,7 @@ end Pi
 section Sigma
 
 variable {ι κ : Type*} {σ : ι → Type*} {τ : κ → Type*} [∀ i, TopologicalSpace (σ i)]
-  [∀ k, TopologicalSpace (τ k)] [TopologicalSpace α]
+  [∀ k, TopologicalSpace (τ k)] [TopologicalSpace X]
 
 @[continuity]
 theorem continuous_sigmaMk {i : ι} : Continuous (@Sigma.mk ι σ i) :=
@@ -1583,7 +1583,7 @@ theorem isOpen_sigma_fst_preimage (s : Set ι) : IsOpen (Sigma.fst ⁻¹' s : Se
 
 /-- A map out of a sum type is continuous iff its restriction to each summand is. -/
 @[simp]
-theorem continuous_sigma_iff {f : Sigma σ → α} :
+theorem continuous_sigma_iff {f : Sigma σ → X} :
     Continuous f ↔ ∀ i, Continuous fun a => f ⟨i, a⟩ := by
   delta instTopologicalSpaceSigma
   rw [continuous_iSup_dom]
@@ -1592,7 +1592,7 @@ theorem continuous_sigma_iff {f : Sigma σ → α} :
 
 /-- A map out of a sum type is continuous if its restriction to each summand is. -/
 @[continuity]
-theorem continuous_sigma {f : Sigma σ → α} (hf : ∀ i, Continuous fun a => f ⟨i, a⟩) :
+theorem continuous_sigma {f : Sigma σ → X} (hf : ∀ i, Continuous fun a => f ⟨i, a⟩) :
     Continuous f :=
   continuous_sigma_iff.2 hf
 #align continuous_sigma continuous_sigma
@@ -1600,7 +1600,7 @@ theorem continuous_sigma {f : Sigma σ → α} (hf : ∀ i, Continuous fun a =>
 /-- A map defined on a sigma type (a.k.a. the disjoint union of an indexed family of topological
 spaces) is inducing iff its restriction to each component is inducing and each the image of each
 component under `f` can be separated from the images of all other components by an open set. -/
-theorem inducing_sigma {f : Sigma σ → α} :
+theorem inducing_sigma {f : Sigma σ → X} :
     Inducing f ↔ (∀ i, Inducing (f ∘ Sigma.mk i)) ∧
       (∀ i, ∃ U, IsOpen U ∧ ∀ x, f x ∈ U ↔ x.1 = i) := by
   refine ⟨fun h ↦ ⟨fun i ↦ h.comp embedding_sigmaMk.1, fun i ↦ ?_⟩, ?_⟩
@@ -1625,7 +1625,7 @@ theorem Continuous.sigma_map {f₁ : ι → κ} {f₂ : ∀ i, σ i → τ (f₁
   continuous_sigma_map.2 hf
 #align continuous.sigma_map Continuous.sigma_map
 
-theorem isOpenMap_sigma {f : Sigma σ → α} : IsOpenMap f ↔ ∀ i, IsOpenMap fun a => f ⟨i, a⟩ := by
+theorem isOpenMap_sigma {f : Sigma σ → X} : IsOpenMap f ↔ ∀ i, IsOpenMap fun a => f ⟨i, a⟩ := by
   simp only [isOpenMap_iff_nhds_le, Sigma.forall, Sigma.nhds_eq, map_map, comp]
 #align is_open_map_sigma isOpenMap_sigma
 
@@ -1656,53 +1656,53 @@ end Sigma
 
 section ULift
 
-theorem ULift.isOpen_iff [TopologicalSpace α] {s : Set (ULift.{v} α)} :
+theorem ULift.isOpen_iff [TopologicalSpace X] {s : Set (ULift.{v} X)} :
     IsOpen s ↔ IsOpen (ULift.up ⁻¹' s) := by
   unfold ULift.topologicalSpace
   erw [← Equiv.ulift.coinduced_symm]
   rfl
 
-theorem ULift.isClosed_iff [TopologicalSpace α] {s : Set (ULift.{v} α)} :
+theorem ULift.isClosed_iff [TopologicalSpace X] {s : Set (ULift.{v} X)} :
     IsClosed s ↔ IsClosed (ULift.up ⁻¹' s) := by
   rw [← isOpen_compl_iff, ← isOpen_compl_iff, isOpen_iff, preimage_compl]
 
 @[continuity]
-theorem continuous_uLift_down [TopologicalSpace α] : Continuous (ULift.down : ULift.{v, u} α → α) :=
+theorem continuous_uLift_down [TopologicalSpace X] : Continuous (ULift.down : ULift.{v, u} X → X) :=
   continuous_induced_dom
 #align continuous_ulift_down continuous_uLift_down
 
 @[continuity]
-theorem continuous_uLift_up [TopologicalSpace α] : Continuous (ULift.up : α → ULift.{v, u} α) :=
+theorem continuous_uLift_up [TopologicalSpace X] : Continuous (ULift.up : X → ULift.{v, u} X) :=
   continuous_induced_rng.2 continuous_id
 #align continuous_ulift_up continuous_uLift_up
 
-theorem embedding_uLift_down [TopologicalSpace α] : Embedding (ULift.down : ULift.{v, u} α → α) :=
+theorem embedding_uLift_down [TopologicalSpace X] : Embedding (ULift.down : ULift.{v, u} X → X) :=
   ⟨⟨rfl⟩, ULift.down_injective⟩
 #align embedding_ulift_down embedding_uLift_down
 
-theorem ULift.closedEmbedding_down [TopologicalSpace α] :
-    ClosedEmbedding (ULift.down : ULift.{v, u} α → α) :=
+theorem ULift.closedEmbedding_down [TopologicalSpace X] :
+    ClosedEmbedding (ULift.down : ULift.{v, u} X → X) :=
   ⟨embedding_uLift_down, by simp only [ULift.down_surjective.range_eq, isClosed_univ]⟩
 #align ulift.closed_embedding_down ULift.closedEmbedding_down
 
-instance [TopologicalSpace α] [DiscreteTopology α] : DiscreteTopology (ULift α) :=
+instance [TopologicalSpace X] [DiscreteTopology X] : DiscreteTopology (ULift X) :=
   embedding_uLift_down.discreteTopology
 
 end ULift
 
 section Monad
 
-variable [TopologicalSpace α] {β : Set α} {γ : Set β}
+variable [TopologicalSpace X] {s : Set X} {t : Set s}
 
-theorem IsOpen.trans (hγ : IsOpen γ) (hβ : IsOpen β) : IsOpen (γ : Set α) := by
-  rcases isOpen_induced_iff.mp hγ with ⟨δ, hδ, rfl⟩
+theorem IsOpen.trans (ht : IsOpen t) (hs : IsOpen s) : IsOpen (t : Set X) := by
+  rcases isOpen_induced_iff.mp ht with ⟨s', hs', rfl⟩
   rw [Subtype.image_preimage_coe]
-  exact IsOpen.inter hδ hβ
+  exact hs'.inter hs
 
-theorem IsClosed.trans (hγ : IsClosed γ) (hβ : IsClosed β) : IsClosed (γ : Set α) := by
-  rcases isClosed_induced_iff.mp hγ with ⟨δ, hδ, rfl⟩
+theorem IsClosed.trans (ht : IsClosed t) (hs : IsClosed s) : IsClosed (t : Set X) := by
+  rcases isClosed_induced_iff.mp ht with ⟨s', hs', rfl⟩
   rw [Subtype.image_preimage_coe]
-  convert IsClosed.inter hδ hβ
+  convert hs'.inter hs
 
 end Monad
 
@@ -1719,13 +1719,13 @@ theorem nhdsSet_prod_le (s : Set X) (t : Set Y) : 𝓝ˢ (s ×ˢ t) ≤ 𝓝ˢ s
 theorem Filter.eventually_nhdsSet_prod_iff {p : X × Y → Prop} :
     (∀ᶠ q in 𝓝ˢ (s ×ˢ t), p q) ↔
       ∀ x ∈ s, ∀ y ∈ t,
-          ∃ pa : X → Prop, (∀ᶠ x' in 𝓝 x, pa x') ∧ ∃ pb : Y → Prop, (∀ᶠ y' in 𝓝 y, pb y') ∧
-            ∀ {x : X}, pa x → ∀ {y : Y}, pb y → p (x, y) :=
+          ∃ px : X → Prop, (∀ᶠ x' in 𝓝 x, px x') ∧ ∃ py : Y → Prop, (∀ᶠ y' in 𝓝 y, py y') ∧
+            ∀ {x : X}, px x → ∀ {y : Y}, py y → p (x, y) :=
   by simp_rw [eventually_nhdsSet_iff_forall, Set.forall_prod_set, nhds_prod_eq, eventually_prod_iff]
 
-theorem Filter.Eventually.prod_nhdsSet {p : X × Y → Prop} {pa : X → Prop} {pb : Y → Prop}
-    (hp : ∀ {x : X}, pa x → ∀ {y : Y}, pb y → p (x, y)) (hs : ∀ᶠ x in 𝓝ˢ s, pa x)
-    (ht : ∀ᶠ y in 𝓝ˢ t, pb y) : ∀ᶠ q in 𝓝ˢ (s ×ˢ t), p q :=
+theorem Filter.Eventually.prod_nhdsSet {p : X × Y → Prop} {px : X → Prop} {py : Y → Prop}
+    (hp : ∀ {x : X}, px x → ∀ {y : Y}, py y → p (x, y)) (hs : ∀ᶠ x in 𝓝ˢ s, px x)
+    (ht : ∀ᶠ y in 𝓝ˢ t, py y) : ∀ᶠ q in 𝓝ˢ (s ×ˢ t), p q :=
   nhdsSet_prod_le _ _ (mem_of_superset (prod_mem_prod hs ht) fun _ ⟨hx, hy⟩ ↦ hp hx hy)
 
 end NhdsSet

From sphere-eversion; I'm just submitting these. Thanks to @urkud for some review suggestions.

@@ -1705,3 +1705,27 @@ theorem IsClosed.trans (hγ : IsClosed γ) (hβ : IsClosed β) : IsClosed (γ :
   convert IsClosed.inter hδ hβ
 
 end Monad
+
+section NhdsSet
+variable {X Y : Type*} [TopologicalSpace X] [TopologicalSpace Y] {f : Filter X}
+  {s : Set X} {t : Set Y} {x : X}
+
+/-- The product of a neighborhood of `s` and a neighborhood of `t` is a neighborhood of `s ×ˢ t`,
+formulated in terms of a filter inequality. -/
+theorem nhdsSet_prod_le (s : Set X) (t : Set Y) : 𝓝ˢ (s ×ˢ t) ≤ 𝓝ˢ s ×ˢ 𝓝ˢ t :=
+  ((hasBasis_nhdsSet _).prod (hasBasis_nhdsSet _)).ge_iff.2 fun (_u, _v) ⟨⟨huo, hsu⟩, hvo, htv⟩ ↦
+    (huo.prod hvo).mem_nhdsSet.2 <| prod_mono hsu htv
+
+theorem Filter.eventually_nhdsSet_prod_iff {p : X × Y → Prop} :
+    (∀ᶠ q in 𝓝ˢ (s ×ˢ t), p q) ↔
+      ∀ x ∈ s, ∀ y ∈ t,
+          ∃ pa : X → Prop, (∀ᶠ x' in 𝓝 x, pa x') ∧ ∃ pb : Y → Prop, (∀ᶠ y' in 𝓝 y, pb y') ∧
+            ∀ {x : X}, pa x → ∀ {y : Y}, pb y → p (x, y) :=
+  by simp_rw [eventually_nhdsSet_iff_forall, Set.forall_prod_set, nhds_prod_eq, eventually_prod_iff]
+
+theorem Filter.Eventually.prod_nhdsSet {p : X × Y → Prop} {pa : X → Prop} {pb : Y → Prop}
+    (hp : ∀ {x : X}, pa x → ∀ {y : Y}, pb y → p (x, y)) (hs : ∀ᶠ x in 𝓝ˢ s, pa x)
+    (ht : ∀ᶠ y in 𝓝ˢ t, pb y) : ∀ᶠ q in 𝓝ˢ (s ×ˢ t), p q :=
+  nhdsSet_prod_le _ _ (mem_of_superset (prod_mem_prod hs ht) fun _ ⟨hx, hy⟩ ↦ hp hx hy)
+
+end NhdsSet

Subset of #9319

@@ -212,7 +212,7 @@ instance Sum.discreteTopology [TopologicalSpace α] [TopologicalSpace β] [hα :
 
 instance Sigma.discreteTopology {β : α → Type v} [∀ a, TopologicalSpace (β a)]
     [h : ∀ a, DiscreteTopology (β a)] : DiscreteTopology (Sigma β) :=
-  ⟨iSup_eq_bot.2 <| fun _ => by simp only [(h _).eq_bot, coinduced_bot]⟩
+  ⟨iSup_eq_bot.2 fun _ => by simp only [(h _).eq_bot, coinduced_bot]⟩
 #align sigma.discrete_topology Sigma.discreteTopology
 
 section Topα

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

@@ -415,6 +415,12 @@ theorem Continuous.Prod.mk_left (b : β) : Continuous fun a : α => (a, b) :=
   continuous_id.prod_mk continuous_const
 #align continuous.prod.mk_left Continuous.Prod.mk_left
 
+/-- If `f x y` is continuous in `x` for all `y ∈ s`,
+then the set of `x` such that `f x` maps `s` to `t` is closed. -/
+lemma IsClosed.setOf_mapsTo {f : α → β → γ} {s : Set β} {t : Set γ} (ht : IsClosed t)
+    (hf : ∀ y ∈ s, Continuous (f · y)) : IsClosed {x | MapsTo (f x) s t} := by
+  simpa only [MapsTo, setOf_forall] using isClosed_biInter fun y hy ↦ ht.preimage (hf y hy)
+
 theorem Continuous.comp₂ {g : α × β → γ} (hg : Continuous g) {e : δ → α} (he : Continuous e)
     {f : δ → β} (hf : Continuous f) : Continuous fun x => g (e x, f x) :=
   hg.comp <| he.prod_mk hf

I literally went through and regex'd some uses of cases', replacing them with rcases; this is meant to be a low effort PR as I hope that tools can do this in the future.

rcases is an easier replacement than cases, though with better tools we could in future do a second pass converting simple rcases added here (and existing ones) to cases.

@@ -731,7 +731,7 @@ theorem isOpenMap_snd : IsOpenMap (@Prod.snd α β) :=
 empty -/
 theorem isOpen_prod_iff' {s : Set α} {t : Set β} :
     IsOpen (s ×ˢ t) ↔ IsOpen s ∧ IsOpen t ∨ s = ∅ ∨ t = ∅ := by
-  cases' (s ×ˢ t).eq_empty_or_nonempty with h h
+  rcases (s ×ˢ t).eq_empty_or_nonempty with h | h
   · simp [h, prod_eq_empty_iff.1 h]
   · have st : s.Nonempty ∧ t.Nonempty := prod_nonempty_iff.1 h
     constructor

This also adds a handful of easy instances

Arguably the topological space instance should be defined via coinduced to make these true by Iff.rfl, but that creates a headache with constructors for normed spaces, so is not in this PR.

@@ -1650,6 +1650,16 @@ end Sigma
 
 section ULift
 
+theorem ULift.isOpen_iff [TopologicalSpace α] {s : Set (ULift.{v} α)} :
+    IsOpen s ↔ IsOpen (ULift.up ⁻¹' s) := by
+  unfold ULift.topologicalSpace
+  erw [← Equiv.ulift.coinduced_symm]
+  rfl
+
+theorem ULift.isClosed_iff [TopologicalSpace α] {s : Set (ULift.{v} α)} :
+    IsClosed s ↔ IsClosed (ULift.up ⁻¹' s) := by
+  rw [← isOpen_compl_iff, ← isOpen_compl_iff, isOpen_iff, preimage_compl]
+
 @[continuity]
 theorem continuous_uLift_down [TopologicalSpace α] : Continuous (ULift.down : ULift.{v, u} α → α) :=
   continuous_induced_dom

PR contents

This is the supremum of

• #8284
• #8056
• #8023
• #8332
• #8226 (already approved)
• #7834 (already approved)

along with some minor fixes from failures on nightly-testing as Mathlib master is merged into it.

Note that some PRs for changes that are already compatible with the current toolchain and will be necessary have already been split out: #8380.

I am hopeful that in future we will be able to progressively merge adaptation PRs into a bump/v4.X.0 branch, so we never end up with a "big merge" like this. However one of these adaptation PRs (#8056) predates my new scheme for combined CI, and it wasn't possible to keep that PR viable in the meantime.

Lean PRs involved in this bump

In particular this includes adjustments for the Lean PRs

• leanprover/lean4#2778
• leanprover/lean4#2790
• leanprover/lean4#2783
• leanprover/lean4#2825
• leanprover/lean4#2722

leanprover/lean4#2778

We can get rid of all the

local macro_rules | `($x ^ $y) => `(HPow.hPow $x $y) -- Porting note: See issue [lean4#2220](https://github.com/leanprover/lean4/pull/2220)

macros across Mathlib (and in any projects that want to write natural number powers of reals).

leanprover/lean4#2722

Changes the default behaviour of simp to (config := {decide := false}). This makes simp (and consequentially norm_num) less powerful, but also more consistent, and less likely to blow up in long failures. This requires a variety of changes: changing some previously by simp or norm_num to decide or rfl, or adding (config := {decide := true}).

leanprover/lean4#2783

This changed the behaviour of simp so that simp [f] will only unfold "fully applied" occurrences of f. The old behaviour can be recovered with simp (config := { unfoldPartialApp := true }). We may in future add a syntax for this, e.g. simp [!f]; please provide feedback! In the meantime, we have made the following changes:

• switching to using explicit lemmas that have the intended level of application
• (config := { unfoldPartialApp := true }) in some places, to recover the old behaviour
• Using @[eqns] to manually adjust the equation lemmas for a particular definition, recovering the old behaviour just for that definition. See #8371, where we do this for Function.comp and Function.flip.

This change in Lean may require further changes down the line (e.g. adding the !f syntax, and/or upstreaming the special treatment for Function.comp and Function.flip, and/or removing this special treatment). Please keep an open and skeptical mind about these changes!

Co-authored-by: leanprover-community-mathlib4-bot <leanprover-community-mathlib4-bot@users.noreply.github.com> Co-authored-by: Scott Morrison <scott.morrison@gmail.com> Co-authored-by: Eric Wieser <wieser.eric@gmail.com> Co-authored-by: Mauricio Collares <mauricio@collares.org>

@@ -802,7 +802,7 @@ theorem DenseRange.prod_map {ι : Type*} {κ : Type*} {f : ι → β} {g : κ 
 
 theorem Inducing.prod_map {f : α → β} {g : γ → δ} (hf : Inducing f) (hg : Inducing g) :
     Inducing (Prod.map f g) :=
-  inducing_iff_nhds.2 fun (a, b) => by simp_rw [Prod.map, nhds_prod_eq, hf.nhds_eq_comap,
+  inducing_iff_nhds.2 fun (a, b) => by simp_rw [Prod.map_def, nhds_prod_eq, hf.nhds_eq_comap,
     hg.nhds_eq_comap, prod_comap_comap_eq]
 #align inducing.prod_mk Inducing.prod_map
 
@@ -1283,7 +1283,8 @@ lemma Pi.continuous_restrict (S : Set ι) :
 lemma Pi.induced_restrict (S : Set ι) :
     induced (S.restrict) Pi.topologicalSpace =
     ⨅ i ∈ S, induced (eval i) (T i) := by
-  simp [← iInf_subtype'', ← induced_precomp' ((↑) : S → ι), Set.restrict]
+  simp (config := { unfoldPartialApp := true }) [← iInf_subtype'', ← induced_precomp' ((↑) : S → ι),
+    Set.restrict]
 
 theorem Filter.Tendsto.update [DecidableEq ι] {l : Filter β} {f : β → ∀ i, π i} {x : ∀ i, π i}
     (hf : Tendsto f l (𝓝 x)) (i : ι) {g : β → π i} {xi : π i} (hg : Tendsto g l (𝓝 xi)) :

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

@@ -248,8 +248,9 @@ theorem nhds_ne_subtype_neBot_iff {S : Set α} {x : S} :
   rw [neBot_iff, neBot_iff, not_iff_not, nhds_ne_subtype_eq_bot_iff]
 #align nhds_ne_subtype_ne_bot_iff nhds_ne_subtype_neBot_iff
 
-theorem discreteTopology_subtype_iff {S : Set α} : DiscreteTopology S ↔ ∀ x ∈ S, 𝓝[≠] x ⊓ 𝓟 S = ⊥ :=
-  by simp_rw [discreteTopology_iff_nhds_ne, SetCoe.forall', nhds_ne_subtype_eq_bot_iff]
+theorem discreteTopology_subtype_iff {S : Set α} :
+    DiscreteTopology S ↔ ∀ x ∈ S, 𝓝[≠] x ⊓ 𝓟 S = ⊥ := by
+  simp_rw [discreteTopology_iff_nhds_ne, SetCoe.forall', nhds_ne_subtype_eq_bot_iff]
 #align discrete_topology_subtype_iff discreteTopology_subtype_iff
 
 end Topα
@@ -533,8 +534,8 @@ theorem mem_nhds_prod_iff {a : α} {b : β} {s : Set (α × β)} :
 #align mem_nhds_prod_iff mem_nhds_prod_iff
 
 theorem mem_nhdsWithin_prod_iff {a : α} {b : β} {s : Set (α × β)} {ta : Set α} {tb : Set β} :
-    s ∈ 𝓝[ta ×ˢ tb] (a, b) ↔ ∃ u ∈ 𝓝[ta] a, ∃ v ∈ 𝓝[tb] b, u ×ˢ v ⊆ s :=
-  by rw [nhdsWithin_prod_eq, mem_prod_iff]
+    s ∈ 𝓝[ta ×ˢ tb] (a, b) ↔ ∃ u ∈ 𝓝[ta] a, ∃ v ∈ 𝓝[tb] b, u ×ˢ v ⊆ s := by
+  rw [nhdsWithin_prod_eq, mem_prod_iff]
 
 -- porting note: moved up
 theorem Filter.HasBasis.prod_nhds {ιa ιb : Type*} {pa : ιa → Prop} {pb : ιb → Prop}
@@ -762,13 +763,15 @@ theorem frontier_prod_eq (s : Set α) (t : Set β) :
 #align frontier_prod_eq frontier_prod_eq
 
 @[simp]
-theorem frontier_prod_univ_eq (s : Set α) : frontier (s ×ˢ (univ : Set β)) = frontier s ×ˢ univ :=
-  by simp [frontier_prod_eq]
+theorem frontier_prod_univ_eq (s : Set α) :
+    frontier (s ×ˢ (univ : Set β)) = frontier s ×ˢ univ := by
+  simp [frontier_prod_eq]
 #align frontier_prod_univ_eq frontier_prod_univ_eq
 
 @[simp]
-theorem frontier_univ_prod_eq (s : Set β) : frontier ((univ : Set α) ×ˢ s) = univ ×ˢ frontier s :=
-  by simp [frontier_prod_eq]
+theorem frontier_univ_prod_eq (s : Set β) :
+    frontier ((univ : Set α) ×ˢ s) = univ ×ˢ frontier s := by
+  simp [frontier_prod_eq]
 #align frontier_univ_prod_eq frontier_univ_prod_eq
 
 theorem map_mem_closure₂ {f : α → β → γ} {a : α} {b : β} {s : Set α} {t : Set β} {u : Set γ}
@@ -1438,8 +1441,8 @@ theorem pi_eq_generateFrom :
       generateFrom
         { g | ∃ (s : ∀ a, Set (π a)) (i : Finset ι), (∀ a ∈ i, IsOpen (s a)) ∧ g = pi (↑i) s } :=
   calc Pi.topologicalSpace
-  _ = @Pi.topologicalSpace ι π fun a => generateFrom { s | IsOpen s } :=
-    by simp only [generateFrom_setOf_isOpen]
+  _ = @Pi.topologicalSpace ι π fun a => generateFrom { s | IsOpen s } := by
+    simp only [generateFrom_setOf_isOpen]
   _ = _ := pi_generateFrom_eq
 #align pi_eq_generate_from pi_eq_generateFrom
 

The monad instance on Set isn't computationally relevant, and it causes Lean's monad lifting coercion logic to activate. We introduce a coercion instance for the case that's actually used in practice: when s : Set X and t : Set s then (t : Set X) ought to be Subtype.val '' t. This way we do not see Lean.Internal.coeM terms.

If the monad is still wanted, it can be activated using a local attribute or by using the SetM.run function.

@@ -1676,14 +1676,12 @@ variable [TopologicalSpace α] {β : Set α} {γ : Set β}
 
 theorem IsOpen.trans (hγ : IsOpen γ) (hβ : IsOpen β) : IsOpen (γ : Set α) := by
   rcases isOpen_induced_iff.mp hγ with ⟨δ, hδ, rfl⟩
-  convert IsOpen.inter hβ hδ
-  ext
-  exact ⟨fun h => ⟨coe_subset h, mem_of_mem_coe h⟩, fun ⟨hβ, hδ⟩ => mem_coe_of_mem hβ hδ⟩
+  rw [Subtype.image_preimage_coe]
+  exact IsOpen.inter hδ hβ
 
 theorem IsClosed.trans (hγ : IsClosed γ) (hβ : IsClosed β) : IsClosed (γ : Set α) := by
   rcases isClosed_induced_iff.mp hγ with ⟨δ, hδ, rfl⟩
-  convert IsClosed.inter hβ hδ
-  ext
-  exact ⟨fun h => ⟨coe_subset h, mem_of_mem_coe h⟩, fun ⟨hβ, hδ⟩ => mem_coe_of_mem hβ hδ⟩
+  rw [Subtype.image_preimage_coe]
+  convert IsClosed.inter hδ hβ
 
 end Monad

Useful to uniformize proofs for T2 spaces and regular spaces, notably to discuss regular measures in topological groups.

@@ -532,6 +532,10 @@ theorem mem_nhds_prod_iff {a : α} {b : β} {s : Set (α × β)} :
     s ∈ 𝓝 (a, b) ↔ ∃ u ∈ 𝓝 a, ∃ v ∈ 𝓝 b, u ×ˢ v ⊆ s := by rw [nhds_prod_eq, mem_prod_iff]
 #align mem_nhds_prod_iff mem_nhds_prod_iff
 
+theorem mem_nhdsWithin_prod_iff {a : α} {b : β} {s : Set (α × β)} {ta : Set α} {tb : Set β} :
+    s ∈ 𝓝[ta ×ˢ tb] (a, b) ↔ ∃ u ∈ 𝓝[ta] a, ∃ v ∈ 𝓝[tb] b, u ×ˢ v ⊆ s :=
+  by rw [nhdsWithin_prod_eq, mem_prod_iff]
+
 -- porting note: moved up
 theorem Filter.HasBasis.prod_nhds {ιa ιb : Type*} {pa : ιa → Prop} {pb : ιb → Prop}
     {sa : ιa → Set α} {sb : ιb → Set β} {a : α} {b : β} (ha : (𝓝 a).HasBasis pa sa)

This incorporates changes from

• #7845
• #7847
• #7853
• #7872 (was never actually made to work, but the diffs in nightly-testing are unexciting: we need to fully qualify a few names)

They can all be closed when this is merged.

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

@@ -1029,7 +1029,7 @@ lemma IsClosedMap.restrictPreimage {f : α → β} (hcl : IsClosedMap f) (T : Se
     IsClosedMap (T.restrictPreimage f) := by
   rw [isClosedMap_iff_clusterPt] at hcl ⊢
   intro A ⟨y, hyT⟩ hy
-  rw [restrictPreimage, MapClusterPt, ← inducing_subtype_val.mapClusterPt_iff, MapClusterPt,
+  rw [Set.restrictPreimage, MapClusterPt, ← inducing_subtype_val.mapClusterPt_iff, MapClusterPt,
       map_map, MapsTo.restrict_commutes, ← map_map, ← MapClusterPt, map_principal] at hy
   rcases hcl _ y hy with ⟨x, hxy, hx⟩
   have hxT : f x ∈ T := hxy ▸ hyT

From the Mandelbrot set connectedness project

Co-Authored-By: @girving

@@ -491,6 +491,10 @@ theorem continuous_swap : Continuous (Prod.swap : α × β → β × α) :=
   continuous_snd.prod_mk continuous_fst
 #align continuous_swap continuous_swap
 
+lemma isClosedMap_swap : IsClosedMap (Prod.swap : α × β → β × α) := fun s hs ↦ by
+  rw [image_swap_eq_preimage_swap]
+  exact hs.preimage continuous_swap
+
 theorem continuous_uncurry_left {f : α → β → γ} (a : α) (h : Continuous (uncurry f)) :
     Continuous (f a) :=
   h.comp (Continuous.Prod.mk _)

• 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>

@@ -522,16 +522,7 @@ theorem nhdsWithin_prod_eq (a : α) (b : β) (s : Set α) (t : Set β) :
   simp only [nhdsWithin, nhds_prod_eq, ← prod_inf_prod, prod_principal_principal]
 #align nhds_within_prod_eq nhdsWithin_prod_eq
 
-/-- If a function `f x y` is such that `y ↦ f x y` is continuous for all `x`, and `x` lives in a
-discrete space, then `f` is continuous. -/
-theorem continuous_uncurry_of_discreteTopology [DiscreteTopology α] {f : α → β → γ}
-    (hf : ∀ a, Continuous (f a)) : Continuous (uncurry f) := by
-  apply continuous_iff_continuousAt.2
-  rintro ⟨a, x⟩
-  change map _ _ ≤ _
-  rw [nhds_prod_eq, nhds_discrete, Filter.map_pure_prod]
-  exact (hf a).continuousAt
-#align continuous_uncurry_of_discrete_topology continuous_uncurry_of_discreteTopology
+#noalign continuous_uncurry_of_discrete_topology
 
 theorem mem_nhds_prod_iff {a : α} {b : β} {s : Set (α × β)} :
     s ∈ 𝓝 (a, b) ↔ ∃ u ∈ 𝓝 a, ∃ v ∈ 𝓝 b, u ×ˢ v ⊆ s := by rw [nhds_prod_eq, mem_prod_iff]
@@ -676,12 +667,7 @@ theorem prod_induced_induced (f : α → β) (g : γ → δ) :
   rfl
 #align prod_induced_induced prod_induced_induced
 
-theorem continuous_uncurry_of_discreteTopology_left [DiscreteTopology α] {f : α → β → γ}
-    (h : ∀ a, Continuous (f a)) : Continuous (uncurry f) :=
-  continuous_iff_continuousAt.2 fun ⟨a, b⟩ => by
-    simp only [ContinuousAt, nhds_prod_eq, nhds_discrete α, pure_prod, tendsto_map'_iff, (· ∘ ·),
-      uncurry, (h a).tendsto]
-#align continuous_uncurry_of_discrete_topology_left continuous_uncurry_of_discreteTopology_left
+#noalign continuous_uncurry_of_discrete_topology_left
 
 /-- Given a neighborhood `s` of `(x, x)`, then `(x, x)` has a square open neighborhood
   that is a subset of `s`. -/

@@ -1207,7 +1207,7 @@ end Quotient
 section Pi
 
 variable {ι : Type*} {π : ι → Type*} {κ : Type*} [TopologicalSpace α]
-  [∀ i, TopologicalSpace (π i)] {f : α → ∀ i : ι, π i}
+  [T : ∀ i, TopologicalSpace (π i)] {f : α → ∀ i : ι, π i}
 
 theorem continuous_pi_iff : Continuous f ↔ ∀ i, Continuous fun a => f a i := by
   simp only [continuous_iInf_rng, continuous_induced_rng, comp]
@@ -1252,6 +1252,42 @@ theorem continuousAt_pi {f : α → ∀ i, π i} {x : α} :
   tendsto_pi_nhds
 #align continuous_at_pi continuousAt_pi
 
+theorem Pi.continuous_precomp' {ι' : Type*} (φ : ι' → ι) :
+    Continuous (fun (f : (∀ i, π i)) (j : ι') ↦ f (φ j)) :=
+  continuous_pi fun j ↦ continuous_apply (φ j)
+
+theorem Pi.continuous_precomp {ι' : Type*} (φ : ι' → ι) :
+    Continuous (· ∘ φ : (ι → α) → (ι' → α)) :=
+  Pi.continuous_precomp' φ
+
+theorem Pi.continuous_postcomp' {ρ : ι → Type*} [∀ i, TopologicalSpace (ρ i)]
+    {g : ∀ i, π i → ρ i} (hg : ∀ i, Continuous (g i)) :
+    Continuous (fun (f : (∀ i, π i)) (i : ι) ↦ g i (f i)) :=
+  continuous_pi fun i ↦ (hg i).comp <| continuous_apply i
+
+theorem Pi.continuous_postcomp [TopologicalSpace β] {g : α → β} (hg : Continuous g) :
+    Continuous (g ∘ · : (ι → α) → (ι → β)) :=
+  Pi.continuous_postcomp' fun _ ↦ hg
+
+lemma Pi.induced_precomp' {ι' : Type*} (φ : ι' → ι) :
+    induced (fun (f : (∀ i, π i)) (j : ι') ↦ f (φ j)) Pi.topologicalSpace =
+    ⨅ i', induced (eval (φ i')) (T (φ i')) := by
+  simp [Pi.topologicalSpace, induced_iInf, induced_compose, comp]
+
+lemma Pi.induced_precomp [TopologicalSpace β] {ι' : Type*} (φ : ι' → ι) :
+    induced (· ∘ φ) Pi.topologicalSpace =
+    ⨅ i', induced (eval (φ i')) ‹TopologicalSpace β› :=
+  induced_precomp' φ
+
+lemma Pi.continuous_restrict (S : Set ι) :
+    Continuous (S.restrict : (∀ i : ι, π i) → (∀ i : S, π i)) :=
+  Pi.continuous_precomp' ((↑) : S → ι)
+
+lemma Pi.induced_restrict (S : Set ι) :
+    induced (S.restrict) Pi.topologicalSpace =
+    ⨅ i ∈ S, induced (eval i) (T i) := by
+  simp [← iInf_subtype'', ← induced_precomp' ((↑) : S → ι), Set.restrict]
+
 theorem Filter.Tendsto.update [DecidableEq ι] {l : Filter β} {f : β → ∀ i, π i} {x : ∀ i, π i}
     (hf : Tendsto f l (𝓝 x)) (i : ι) {g : β → π i} {xi : π i} (hg : Tendsto g l (𝓝 xi)) :
     Tendsto (fun a => update (f a) i (g a)) l (𝓝 <| update x i xi) :=

We define Alexandrov-discrete spaces as topological spaces where the intersection of a family of open sets is open.

This PR only gives a minimal API because the goal is to ensure that lemma names like isOpen_sInter are free to use for AlexandrovDiscrete. The existing lemmas are getting prefixed by Set.Finite or suffixed by _of_finite.

@@ -1306,7 +1306,7 @@ theorem Continuous.fin_insertNth {n} {π : Fin (n + 1) → Type*} [∀ i, Topolo
 
 theorem isOpen_set_pi {i : Set ι} {s : ∀ a, Set (π a)} (hi : i.Finite)
     (hs : ∀ a ∈ i, IsOpen (s a)) : IsOpen (pi i s) := by
-  rw [pi_def]; exact isOpen_biInter hi fun a ha => (hs _ ha).preimage (continuous_apply _)
+  rw [pi_def]; exact hi.isOpen_biInter fun a ha => (hs _ ha).preimage (continuous_apply _)
 #align is_open_set_pi isOpen_set_pi
 
 theorem isOpen_pi_iff {s : Set (∀ a, π a)} :

@@ -1105,7 +1105,7 @@ theorem continuousAt_codRestrict_iff {f : α → β} {t : Set β} (h1 : ∀ x, f
   inducing_subtype_val.continuousAt_iff
 #align continuous_at_cod_restrict_iff continuousAt_codRestrict_iff
 
-alias continuousAt_codRestrict_iff ↔ _ ContinuousAt.codRestrict
+alias ⟨_, ContinuousAt.codRestrict⟩ := continuousAt_codRestrict_iff
 #align continuous_at.cod_restrict ContinuousAt.codRestrict
 
 theorem ContinuousAt.restrict {f : α → β} {s : Set α} {t : Set β} (h1 : MapsTo f s t) {x : s}

@@ -1099,9 +1099,10 @@ theorem closure_subtype {x : { a // p a }} {s : Set { a // p a }} :
   closure_induced
 #align closure_subtype closure_subtype
 
+@[simp]
 theorem continuousAt_codRestrict_iff {f : α → β} {t : Set β} (h1 : ∀ x, f x ∈ t) {x : α} :
-    ContinuousAt (codRestrict f t h1) x ↔ ContinuousAt f x := by
-  simp_rw [inducing_subtype_val.continuousAt_iff, Function.comp, val_codRestrict_apply]
+    ContinuousAt (codRestrict f t h1) x ↔ ContinuousAt f x :=
+  inducing_subtype_val.continuousAt_iff
 #align continuous_at_cod_restrict_iff continuousAt_codRestrict_iff
 
 alias continuousAt_codRestrict_iff ↔ _ ContinuousAt.codRestrict
@@ -1123,6 +1124,16 @@ theorem Continuous.codRestrict {f : α → β} {s : Set β} (hf : Continuous f)
   hf.subtype_mk hs
 #align continuous.cod_restrict Continuous.codRestrict
 
+@[continuity]
+theorem Continuous.restrict {f : α → β} {s : Set α} {t : Set β} (h1 : MapsTo f s t)
+    (h2 : Continuous f) : Continuous (h1.restrict f s t) :=
+  (h2.comp continuous_subtype_val).codRestrict _
+
+@[continuity]
+theorem Continuous.restrictPreimage {f : α → β} {s : Set β} (h : Continuous f) :
+    Continuous (s.restrictPreimage f) :=
+  h.restrict _
+
 theorem Inducing.codRestrict {e : α → β} (he : Inducing e) {s : Set β} (hs : ∀ x, e x ∈ s) :
     Inducing (codRestrict e s hs) :=
   inducing_of_inducing_compose (he.continuous.codRestrict hs) continuous_subtype_val he

This reverts commit c978143b.

@@ -1100,8 +1100,8 @@ theorem closure_subtype {x : { a // p a }} {s : Set { a // p a }} :
 #align closure_subtype closure_subtype
 
 theorem continuousAt_codRestrict_iff {f : α → β} {t : Set β} (h1 : ∀ x, f x ∈ t) {x : α} :
-    ContinuousAt (codRestrict f t h1) x ↔ ContinuousAt f x :=
-  inducing_subtype_val.continuousAt_iff
+    ContinuousAt (codRestrict f t h1) x ↔ ContinuousAt f x := by
+  simp_rw [inducing_subtype_val.continuousAt_iff, Function.comp, val_codRestrict_apply]
 #align continuous_at_cod_restrict_iff continuousAt_codRestrict_iff
 
 alias continuousAt_codRestrict_iff ↔ _ ContinuousAt.codRestrict
@@ -1123,14 +1123,6 @@ theorem Continuous.codRestrict {f : α → β} {s : Set β} (hf : Continuous f)
   hf.subtype_mk hs
 #align continuous.cod_restrict Continuous.codRestrict
 
-theorem Continuous.restrict {f : α → β} {s : Set α} {t : Set β} (h1 : MapsTo f s t)
-    (h2 : Continuous f) : Continuous (h1.restrict f s t) :=
-  (h2.comp continuous_subtype_val).codRestrict _
-
-theorem Continuous.restrictPreimage {f : α → β} {s : Set β} (h : Continuous f) :
-    Continuous (s.restrictPreimage f) :=
-  h.restrict _
-
 theorem Inducing.codRestrict {e : α → β} (he : Inducing e) {s : Set β} (hs : ∀ x, e x ∈ s) :
     Inducing (codRestrict e s hs) :=
   inducing_of_inducing_compose (he.continuous.codRestrict hs) continuous_subtype_val he

@@ -1100,8 +1100,8 @@ theorem closure_subtype {x : { a // p a }} {s : Set { a // p a }} :
 #align closure_subtype closure_subtype
 
 theorem continuousAt_codRestrict_iff {f : α → β} {t : Set β} (h1 : ∀ x, f x ∈ t) {x : α} :
-    ContinuousAt (codRestrict f t h1) x ↔ ContinuousAt f x := by
-  simp_rw [inducing_subtype_val.continuousAt_iff, Function.comp, val_codRestrict_apply]
+    ContinuousAt (codRestrict f t h1) x ↔ ContinuousAt f x :=
+  inducing_subtype_val.continuousAt_iff
 #align continuous_at_cod_restrict_iff continuousAt_codRestrict_iff
 
 alias continuousAt_codRestrict_iff ↔ _ ContinuousAt.codRestrict
@@ -1123,6 +1123,14 @@ theorem Continuous.codRestrict {f : α → β} {s : Set β} (hf : Continuous f)
   hf.subtype_mk hs
 #align continuous.cod_restrict Continuous.codRestrict
 
+theorem Continuous.restrict {f : α → β} {s : Set α} {t : Set β} (h1 : MapsTo f s t)
+    (h2 : Continuous f) : Continuous (h1.restrict f s t) :=
+  (h2.comp continuous_subtype_val).codRestrict _
+
+theorem Continuous.restrictPreimage {f : α → β} {s : Set β} (h : Continuous f) :
+    Continuous (s.restrictPreimage f) :=
+  h.restrict _
+
 theorem Inducing.codRestrict {e : α → β} (he : Inducing e) {s : Set β} (hs : ∀ x, e x ∈ s) :
     Inducing (codRestrict e s hs) :=
   inducing_of_inducing_compose (he.continuous.codRestrict hs) continuous_subtype_val he

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

This has nice performance benefits.

@@ -39,7 +39,7 @@ open Topology TopologicalSpace Set Filter Function Classical
 
 universe u v
 
-variable {α : Type u} {β : Type v} {γ δ ε ζ : Type _}
+variable {α : Type u} {β : Type v} {γ δ ε ζ : Type*}
 
 section Constructions
 
@@ -256,7 +256,7 @@ end Topα
 
 /-- A type synonym equipped with the topology whose open sets are the empty set and the sets with
 finite complements. -/
-def CofiniteTopology (α : Type _) :=
+def CofiniteTopology (α : Type*) :=
   α
 #align cofinite_topology CofiniteTopology
 
@@ -538,7 +538,7 @@ theorem mem_nhds_prod_iff {a : α} {b : β} {s : Set (α × β)} :
 #align mem_nhds_prod_iff mem_nhds_prod_iff
 
 -- porting note: moved up
-theorem Filter.HasBasis.prod_nhds {ιa ιb : Type _} {pa : ιa → Prop} {pb : ιb → Prop}
+theorem Filter.HasBasis.prod_nhds {ιa ιb : Type*} {pa : ιa → Prop} {pb : ιb → Prop}
     {sa : ιa → Set α} {sb : ιb → Set β} {a : α} {b : β} (ha : (𝓝 a).HasBasis pa sa)
     (hb : (𝓝 b).HasBasis pb sb) :
     (𝓝 (a, b)).HasBasis (fun i : ιa × ιb => pa i.1 ∧ pb i.2) fun i => sa i.1 ×ˢ sb i.2 := by
@@ -547,7 +547,7 @@ theorem Filter.HasBasis.prod_nhds {ιa ιb : Type _} {pa : ιa → Prop} {pb : 
 #align filter.has_basis.prod_nhds Filter.HasBasis.prod_nhds
 
 -- porting note: moved up
-theorem Filter.HasBasis.prod_nhds' {ιa ιb : Type _} {pa : ιa → Prop} {pb : ιb → Prop}
+theorem Filter.HasBasis.prod_nhds' {ιa ιb : Type*} {pa : ιa → Prop} {pb : ιb → Prop}
     {sa : ιa → Set α} {sb : ιb → Set β} {ab : α × β} (ha : (𝓝 ab.1).HasBasis pa sa)
     (hb : (𝓝 ab.2).HasBasis pb sb) :
     (𝓝 ab).HasBasis (fun i : ιa × ιb => pa i.1 ∧ pb i.2) fun i => sa i.1 ×ˢ sb i.2 :=
@@ -617,7 +617,7 @@ theorem ContinuousAt.prod_map' {f : α → γ} {g : β → δ} {x : α} {y : β}
 
 -- todo: reformulate using `Set.image2`
 -- todo: prove a version of `generateFrom_union` with `image2 (∩) s t` in the LHS and use it here
-theorem prod_generateFrom_generateFrom_eq {α β : Type _} {s : Set (Set α)} {t : Set (Set β)}
+theorem prod_generateFrom_generateFrom_eq {α β : Type*} {s : Set (Set α)} {t : Set (Set β)}
     (hs : ⋃₀ s = univ) (ht : ⋃₀ t = univ) :
     @instTopologicalSpaceProd α β (generateFrom s) (generateFrom t) =
       generateFrom { g | ∃ u ∈ s, ∃ v ∈ t, g = u ×ˢ v } :=
@@ -798,7 +798,7 @@ theorem Dense.prod {s : Set α} {t : Set β} (hs : Dense s) (ht : Dense t) : Den
 #align dense.prod Dense.prod
 
 /-- If `f` and `g` are maps with dense range, then `Prod.map f g` has dense range. -/
-theorem DenseRange.prod_map {ι : Type _} {κ : Type _} {f : ι → β} {g : κ → γ} (hf : DenseRange f)
+theorem DenseRange.prod_map {ι : Type*} {κ : Type*} {f : ι → β} {g : κ → γ} (hf : DenseRange f)
     (hg : DenseRange g) : DenseRange (Prod.map f g) := by
   simpa only [DenseRange, prod_range_range_eq] using hf.prod hg
 #align dense_range.prod_map DenseRange.prod_map
@@ -1089,7 +1089,7 @@ theorem nhds_subtype_eq_comap {a : α} {h : p a} : 𝓝 (⟨a, h⟩ : Subtype p)
   nhds_induced _ _
 #align nhds_subtype_eq_comap nhds_subtype_eq_comap
 
-theorem tendsto_subtype_rng {β : Type _} {p : α → Prop} {b : Filter β} {f : β → Subtype p} :
+theorem tendsto_subtype_rng {β : Type*} {p : α → Prop} {b : Filter β} {f : β → Subtype p} :
     ∀ {a : Subtype p}, Tendsto f b (𝓝 a) ↔ Tendsto (fun x => (f x : α)) b (𝓝 (a : α))
   | ⟨a, ha⟩ => by rw [nhds_subtype_eq_comap, tendsto_comap_iff]; rfl
 #align tendsto_subtype_rng tendsto_subtype_rng
@@ -1139,7 +1139,7 @@ theorem embedding_inclusion {s t : Set α} (h : s ⊆ t) : Embedding (Set.inclus
 
 /-- Let `s, t ⊆ X` be two subsets of a topological space `X`.  If `t ⊆ s` and the topology induced
 by `X`on `s` is discrete, then also the topology induces on `t` is discrete.  -/
-theorem DiscreteTopology.of_subset {X : Type _} [TopologicalSpace X] {s t : Set X}
+theorem DiscreteTopology.of_subset {X : Type*} [TopologicalSpace X] {s t : Set X}
     (_ : DiscreteTopology s) (ts : t ⊆ s) : DiscreteTopology t :=
   (embedding_inclusion ts).discreteTopology
 #align discrete_topology.of_subset DiscreteTopology.of_subset
@@ -1195,7 +1195,7 @@ end Quotient
 
 section Pi
 
-variable {ι : Type _} {π : ι → Type _} {κ : Type _} [TopologicalSpace α]
+variable {ι : Type*} {π : ι → Type*} {κ : Type*} [TopologicalSpace α]
   [∀ i, TopologicalSpace (π i)] {f : α → ∀ i : ι, π i}
 
 theorem continuous_pi_iff : Continuous f ↔ ∀ i, Continuous fun a => f a i := by
@@ -1213,7 +1213,7 @@ theorem continuous_apply (i : ι) : Continuous fun p : ∀ i, π i => p i :=
 #align continuous_apply continuous_apply
 
 @[continuity]
-theorem continuous_apply_apply {ρ : κ → ι → Type _} [∀ j i, TopologicalSpace (ρ j i)] (j : κ)
+theorem continuous_apply_apply {ρ : κ → ι → Type*} [∀ j i, TopologicalSpace (ρ j i)] (j : κ)
     (i : ι) : Continuous fun p : ∀ j, ∀ i, ρ j i => p j i :=
   (continuous_apply i).comp (continuous_apply j)
 #align continuous_apply_apply continuous_apply_apply
@@ -1273,21 +1273,21 @@ theorem continuous_mulSingle [∀ i, One (π i)] [DecidableEq ι] (i : ι) :
 #align continuous_mul_single continuous_mulSingle
 #align continuous_single continuous_single
 
-theorem Filter.Tendsto.fin_insertNth {n} {π : Fin (n + 1) → Type _} [∀ i, TopologicalSpace (π i)]
+theorem Filter.Tendsto.fin_insertNth {n} {π : Fin (n + 1) → Type*} [∀ i, TopologicalSpace (π i)]
     (i : Fin (n + 1)) {f : β → π i} {l : Filter β} {x : π i} (hf : Tendsto f l (𝓝 x))
     {g : β → ∀ j : Fin n, π (i.succAbove j)} {y : ∀ j, π (i.succAbove j)} (hg : Tendsto g l (𝓝 y)) :
     Tendsto (fun a => i.insertNth (f a) (g a)) l (𝓝 <| i.insertNth x y) :=
   tendsto_pi_nhds.2 fun j => Fin.succAboveCases i (by simpa) (by simpa using tendsto_pi_nhds.1 hg) j
 #align filter.tendsto.fin_insert_nth Filter.Tendsto.fin_insertNth
 
-theorem ContinuousAt.fin_insertNth {n} {π : Fin (n + 1) → Type _} [∀ i, TopologicalSpace (π i)]
+theorem ContinuousAt.fin_insertNth {n} {π : Fin (n + 1) → Type*} [∀ i, TopologicalSpace (π i)]
     (i : Fin (n + 1)) {f : α → π i} {a : α} (hf : ContinuousAt f a)
     {g : α → ∀ j : Fin n, π (i.succAbove j)} (hg : ContinuousAt g a) :
     ContinuousAt (fun a => i.insertNth (f a) (g a)) a :=
   hf.tendsto.fin_insertNth i hg
 #align continuous_at.fin_insert_nth ContinuousAt.fin_insertNth
 
-theorem Continuous.fin_insertNth {n} {π : Fin (n + 1) → Type _} [∀ i, TopologicalSpace (π i)]
+theorem Continuous.fin_insertNth {n} {π : Fin (n + 1) → Type*} [∀ i, TopologicalSpace (π i)]
     (i : Fin (n + 1)) {f : α → π i} (hf : Continuous f) {g : α → ∀ j : Fin n, π (i.succAbove j)}
     (hg : Continuous g) : Continuous fun a => i.insertNth (f a) (g a) :=
   continuous_iff_continuousAt.2 fun _ => hf.continuousAt.fin_insertNth i hg.continuousAt
@@ -1378,7 +1378,7 @@ theorem exists_finset_piecewise_mem_of_mem_nhds [DecidableEq ι] {s : Set (∀ a
   simpa [Finset.mem_coe.1 hi] using mem_of_mem_nhds (htx i)
 #align exists_finset_piecewise_mem_of_mem_nhds exists_finset_piecewise_mem_of_mem_nhds
 
-theorem pi_generateFrom_eq {π : ι → Type _} {g : ∀ a, Set (Set (π a))} :
+theorem pi_generateFrom_eq {π : ι → Type*} {g : ∀ a, Set (Set (π a))} :
     (@Pi.topologicalSpace ι π fun a => generateFrom (g a)) =
       generateFrom
         { t | ∃ (s : ∀ a, Set (π a)) (i : Finset ι), (∀ a ∈ i, s a ∈ g a) ∧ t = pi (↑i) s } := by
@@ -1402,7 +1402,7 @@ theorem pi_eq_generateFrom :
   _ = _ := pi_generateFrom_eq
 #align pi_eq_generate_from pi_eq_generateFrom
 
-theorem pi_generateFrom_eq_finite {π : ι → Type _} {g : ∀ a, Set (Set (π a))} [Finite ι]
+theorem pi_generateFrom_eq_finite {π : ι → Type*} {g : ∀ a, Set (Set (π a))} [Finite ι]
     (hg : ∀ a, ⋃₀ g a = univ) :
     (@Pi.topologicalSpace ι π fun a => generateFrom (g a)) =
       generateFrom { t | ∃ s : ∀ a, Set (π a), (∀ a, s a ∈ g a) ∧ t = pi univ s } := by
@@ -1421,7 +1421,7 @@ theorem pi_generateFrom_eq_finite {π : ι → Type _} {g : ∀ a, Set (Set (π
 #align pi_generate_from_eq_finite pi_generateFrom_eq_finite
 
 -- porting note: new lemma
-theorem induced_to_pi {X : Type _} (f : X → ∀ i, π i) :
+theorem induced_to_pi {X : Type*} (f : X → ∀ i, π i) :
     induced f Pi.topologicalSpace = ⨅ i, induced (f · i) inferInstance := by
   erw [induced_iInf]
   simp only [induced_compose]
@@ -1432,7 +1432,7 @@ endowed with a family of maps `f i : X → π i` for every `i : ι`, hence induc
 map `g : X → Π i, π i`. This lemma shows that infimum of the topologies on `X` induced by
 the `f i` as `i : ι` varies is simply the topology on `X` induced by `g : X → Π i, π i`
 where `Π i, π i` is endowed with the usual product topology. -/
-theorem inducing_iInf_to_pi {X : Type _} (f : ∀ i, X → π i) :
+theorem inducing_iInf_to_pi {X : Type*} (f : ∀ i, X → π i) :
     @Inducing X (∀ i, π i) (⨅ i, induced (f i) inferInstance) _ fun x i => f i x :=
   letI := ⨅ i, induced (f i) inferInstance; ⟨(induced_to_pi _).symm⟩
 #align inducing_infi_to_pi inducing_iInf_to_pi
@@ -1450,7 +1450,7 @@ end Pi
 
 section Sigma
 
-variable {ι κ : Type _} {σ : ι → Type _} {τ : κ → Type _} [∀ i, TopologicalSpace (σ i)]
+variable {ι κ : Type*} {σ : ι → Type*} {τ : κ → Type*} [∀ i, TopologicalSpace (σ i)]
   [∀ k, TopologicalSpace (τ k)] [TopologicalSpace α]
 
 @[continuity]

Per https://github.com/leanprover/lean4/issues/2343, we are going to need to change the automatic generation of instance names, as they become too long.

This PR ensures that everywhere in Mathlib that refers to an instance by name, that name is given explicitly, rather than being automatically generated.

There are four exceptions, which are now commented, with links to https://github.com/leanprover/lean4/issues/2343.

This was implemented by running Mathlib against a modified Lean that appended _ᾰ to all automatically generated names, and fixing everything.

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

@@ -43,22 +43,27 @@ variable {α : Type u} {β : Type v} {γ δ ε ζ : Type _}
 
 section Constructions
 
-instance {p : α → Prop} [t : TopologicalSpace α] : TopologicalSpace (Subtype p) :=
+instance instTopologicalSpaceSubtype {p : α → Prop} [t : TopologicalSpace α] :
+    TopologicalSpace (Subtype p) :=
   induced (↑) t
 
 instance {r : α → α → Prop} [t : TopologicalSpace α] : TopologicalSpace (Quot r) :=
   coinduced (Quot.mk r) t
 
-instance {s : Setoid α} [t : TopologicalSpace α] : TopologicalSpace (Quotient s) :=
+instance instTopologicalSpaceQuotient {s : Setoid α} [t : TopologicalSpace α] :
+    TopologicalSpace (Quotient s) :=
   coinduced Quotient.mk' t
 
-instance [t₁ : TopologicalSpace α] [t₂ : TopologicalSpace β] : TopologicalSpace (α × β) :=
+instance instTopologicalSpaceProd [t₁ : TopologicalSpace α] [t₂ : TopologicalSpace β] :
+    TopologicalSpace (α × β) :=
   induced Prod.fst t₁ ⊓ induced Prod.snd t₂
 
-instance [t₁ : TopologicalSpace α] [t₂ : TopologicalSpace β] : TopologicalSpace (α ⊕ β) :=
+instance instTopologicalSpaceSum [t₁ : TopologicalSpace α] [t₂ : TopologicalSpace β] :
+    TopologicalSpace (α ⊕ β) :=
   coinduced Sum.inl t₁ ⊔ coinduced Sum.inr t₂
 
-instance {β : α → Type v} [t₂ : ∀ a, TopologicalSpace (β a)] : TopologicalSpace (Sigma β) :=
+instance instTopologicalSpaceSigma {β : α → Type v} [t₂ : ∀ a, TopologicalSpace (β a)] :
+    TopologicalSpace (Sigma β) :=
   ⨆ a, coinduced (Sigma.mk a) (t₂ a)
 
 instance Pi.topologicalSpace {β : α → Type v} [t₂ : (a : α) → TopologicalSpace (β a)] :

This work was done during the 2023 Copenhagen masterclass on formalisation of condensed mathematics. Numerous participants contributed.

Co-authored-by: Filippo A E Nuccio <filippo.nuccio@univ-st-etienne.fr> Co-authored-by: Dagur Ásgeirsson <dagurtomas@gmail.com> Co-authored-by: Nikolas Kuhn <nikolaskuhn@gmx.de>

@@ -1623,3 +1623,21 @@ instance [TopologicalSpace α] [DiscreteTopology α] : DiscreteTopology (ULift 
   embedding_uLift_down.discreteTopology
 
 end ULift
+
+section Monad
+
+variable [TopologicalSpace α] {β : Set α} {γ : Set β}
+
+theorem IsOpen.trans (hγ : IsOpen γ) (hβ : IsOpen β) : IsOpen (γ : Set α) := by
+  rcases isOpen_induced_iff.mp hγ with ⟨δ, hδ, rfl⟩
+  convert IsOpen.inter hβ hδ
+  ext
+  exact ⟨fun h => ⟨coe_subset h, mem_of_mem_coe h⟩, fun ⟨hβ, hδ⟩ => mem_coe_of_mem hβ hδ⟩
+
+theorem IsClosed.trans (hγ : IsClosed γ) (hβ : IsClosed β) : IsClosed (γ : Set α) := by
+  rcases isClosed_induced_iff.mp hγ with ⟨δ, hδ, rfl⟩
+  convert IsClosed.inter hβ hδ
+  ext
+  exact ⟨fun h => ⟨coe_subset h, mem_of_mem_coe h⟩, fun ⟨hβ, hδ⟩ => mem_coe_of_mem hβ hδ⟩
+
+end Monad

@@ -1030,6 +1030,17 @@ theorem IsOpenMap.restrict {f : α → β} (hf : IsOpenMap f) {s : Set α} (hs :
   hf.comp hs.isOpenMap_subtype_val
 #align is_open_map.restrict IsOpenMap.restrict
 
+lemma IsClosedMap.restrictPreimage {f : α → β} (hcl : IsClosedMap f) (T : Set β) :
+    IsClosedMap (T.restrictPreimage f) := by
+  rw [isClosedMap_iff_clusterPt] at hcl ⊢
+  intro A ⟨y, hyT⟩ hy
+  rw [restrictPreimage, MapClusterPt, ← inducing_subtype_val.mapClusterPt_iff, MapClusterPt,
+      map_map, MapsTo.restrict_commutes, ← map_map, ← MapClusterPt, map_principal] at hy
+  rcases hcl _ y hy with ⟨x, hxy, hx⟩
+  have hxT : f x ∈ T := hxy ▸ hyT
+  refine ⟨⟨x, hxT⟩, Subtype.ext hxy, ?_⟩
+  rwa [← inducing_subtype_val.mapClusterPt_iff, MapClusterPt, map_principal]
+
 nonrec theorem IsClosed.closedEmbedding_subtype_val {s : Set α} (hs : IsClosed s) :
     ClosedEmbedding ((↑) : s → α) :=
   closedEmbedding_subtype_val hs

• depends on: #5966

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

@@ -2,15 +2,12 @@
 Copyright (c) 2017 Johannes Hölzl. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Johannes Hölzl, Mario Carneiro, Patrick Massot
-
-! This file was ported from Lean 3 source module topology.constructions
-! leanprover-community/mathlib commit f7ebde7ee0d1505dfccac8644ae12371aa3c1c9f
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
 -/
 import Mathlib.Topology.Maps
 import Mathlib.Order.Filter.Pi
 
+#align_import topology.constructions from "leanprover-community/mathlib"@"f7ebde7ee0d1505dfccac8644ae12371aa3c1c9f"
+
 /-!
 # Constructions of new topological spaces from old ones
 

This is a follow-up to #4325.

@@ -1533,6 +1533,22 @@ theorem continuous_sigma {f : Sigma σ → α} (hf : ∀ i, Continuous fun a =>
   continuous_sigma_iff.2 hf
 #align continuous_sigma continuous_sigma
 
+/-- A map defined on a sigma type (a.k.a. the disjoint union of an indexed family of topological
+spaces) is inducing iff its restriction to each component is inducing and each the image of each
+component under `f` can be separated from the images of all other components by an open set. -/
+theorem inducing_sigma {f : Sigma σ → α} :
+    Inducing f ↔ (∀ i, Inducing (f ∘ Sigma.mk i)) ∧
+      (∀ i, ∃ U, IsOpen U ∧ ∀ x, f x ∈ U ↔ x.1 = i) := by
+  refine ⟨fun h ↦ ⟨fun i ↦ h.comp embedding_sigmaMk.1, fun i ↦ ?_⟩, ?_⟩
+  · rcases h.isOpen_iff.1 (isOpen_range_sigmaMk (i := i)) with ⟨U, hUo, hU⟩
+    refine ⟨U, hUo, ?_⟩
+    simpa [Set.ext_iff] using hU
+  · refine fun ⟨h₁, h₂⟩ ↦ inducing_iff_nhds.2 fun ⟨i, x⟩ ↦ ?_
+    rw [Sigma.nhds_mk, (h₁ i).nhds_eq_comap, comp_apply, ← comap_comap, map_comap_of_mem]
+    rcases h₂ i with ⟨U, hUo, hU⟩
+    filter_upwards [preimage_mem_comap <| hUo.mem_nhds <| (hU _).2 rfl] with y hy
+    simpa [hU] using hy
+
 @[simp 1100]
 theorem continuous_sigma_map {f₁ : ι → κ} {f₂ : ∀ i, σ i → τ (f₁ i)} :
     Continuous (Sigma.map f₁ f₂) ↔ ∀ i, Continuous (f₂ i) :=

@@ -628,7 +628,7 @@ theorem prod_generateFrom_generateFrom_eq {α β : Type _} {s : Set (Set α)} {t
     (le_inf
       (coinduced_le_iff_le_induced.mp <|
         le_generateFrom fun u hu =>
-          have : (⋃ v ∈ t, u ×ˢ v) = Prod.fst ⁻¹' u := by
+          have : ⋃ v ∈ t, u ×ˢ v = Prod.fst ⁻¹' u := by
             simp_rw [← prod_iUnion, ← sUnion_eq_biUnion, ht, prod_univ]
           show G.IsOpen (Prod.fst ⁻¹' u) by
             rw [← this]
@@ -637,7 +637,7 @@ theorem prod_generateFrom_generateFrom_eq {α β : Type _} {s : Set (Set α)} {t
                 isOpen_iUnion fun hv => GenerateOpen.basic _ ⟨_, hu, _, hv, rfl⟩)
       (coinduced_le_iff_le_induced.mp <|
         le_generateFrom fun v hv =>
-          have : (⋃ u ∈ s, u ×ˢ v) = Prod.snd ⁻¹' v := by
+          have : ⋃ u ∈ s, u ×ˢ v = Prod.snd ⁻¹' v := by
             simp_rw [← iUnion_prod_const, ← sUnion_eq_biUnion, hs, univ_prod]
           show G.IsOpen (Prod.snd ⁻¹' v) by
             rw [← this]

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

@@ -268,7 +268,7 @@ def of : α ≃ CofiniteTopology α :=
 instance [Inhabited α] : Inhabited (CofiniteTopology α) where default := of default
 
 instance : TopologicalSpace (CofiniteTopology α) where
-  IsOpen s := s.Nonempty → Set.Finite (sᶜ)
+  IsOpen s := s.Nonempty → Set.Finite sᶜ
   isOpen_univ := by simp
   isOpen_inter s t := by
     rintro hs ht ⟨x, hxs, hxt⟩
@@ -1401,7 +1401,7 @@ theorem pi_generateFrom_eq_finite {π : ι → Type _} {g : ∀ a, Set (Set (π
     letI := generateFrom { t | ∃ s : ∀ a, Set (π a), (∀ a, s a ∈ g a) ∧ t = pi univ s }
     refine isOpen_iff_forall_mem_open.2 fun f hf => ?_
     choose c hcg hfc using fun a => sUnion_eq_univ_iff.1 (hg a) (f a)
-    refine ⟨pi i t ∩ pi (↑iᶜ : Set ι) c, inter_subset_left _ _, ?_, ⟨hf, fun a _ => hfc a⟩⟩
+    refine ⟨pi i t ∩ pi ((↑i)ᶜ : Set ι) c, inter_subset_left _ _, ?_, ⟨hf, fun a _ => hfc a⟩⟩
     rw [← univ_pi_piecewise]
     refine GenerateOpen.basic _ ⟨_, fun a => ?_, rfl⟩
     by_cases a ∈ i <;> simp [*]

Co-authored-by: Scott Morrison <scott.morrison@anu.edu.au> Co-authored-by: Parcly Taxel <reddeloostw@gmail.com>

@@ -1287,7 +1287,7 @@ theorem isOpen_set_pi {i : Set ι} {s : ∀ a, Set (π a)} (hi : i.Finite)
 
 theorem isOpen_pi_iff {s : Set (∀ a, π a)} :
     IsOpen s ↔
-      ∀ f, f ∈ s → ∃ (I : Finset ι)(u : ∀ a, Set (π a)),
+      ∀ f, f ∈ s → ∃ (I : Finset ι) (u : ∀ a, Set (π a)),
         (∀ a, a ∈ I → IsOpen (u a) ∧ f a ∈ u a) ∧ (I : Set ι).pi u ⊆ s := by
   rw [isOpen_iff_nhds]
   simp_rw [le_principal_iff, nhds_pi, Filter.mem_pi', mem_nhds_iff]

Run codespell Mathlib and keep some suggestions.

@@ -252,7 +252,7 @@ theorem discreteTopology_subtype_iff {S : Set α} : DiscreteTopology S ↔ ∀ x
 
 end Topα
 
-/-- A type synonym equiped with the topology whose open sets are the empty set and the sets with
+/-- A type synonym equipped with the topology whose open sets are the empty set and the sets with
 finite complements. -/
 def CofiniteTopology (α : Type _) :=
   α

Forward-port leanprover-community/mathlib#19107

@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Johannes Hölzl, Mario Carneiro, Patrick Massot
 
 ! This file was ported from Lean 3 source module topology.constructions
-! leanprover-community/mathlib commit 76f9c990d4b7c3dd26b87c4c4b51759e249d9e66
+! leanprover-community/mathlib commit f7ebde7ee0d1505dfccac8644ae12371aa3c1c9f
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
@@ -807,6 +807,18 @@ theorem Inducing.prod_map {f : α → β} {g : γ → δ} (hf : Inducing f) (hg
     hg.nhds_eq_comap, prod_comap_comap_eq]
 #align inducing.prod_mk Inducing.prod_map
 
+@[simp]
+theorem inducing_const_prod {a : α} {f : β → γ} : (Inducing fun x => (a, f x)) ↔ Inducing f := by
+  simp_rw [inducing_iff, instTopologicalSpaceProd, induced_inf, induced_compose, Function.comp,
+    induced_const, top_inf_eq]
+#align inducing_const_prod inducing_const_prod
+
+@[simp]
+theorem inducing_prod_const {b : β} {f : α → γ} : (Inducing fun x => (f x, b)) ↔ Inducing f := by
+  simp_rw [inducing_iff, instTopologicalSpaceProd, induced_inf, induced_compose, Function.comp,
+    induced_const, inf_top_eq]
+#align inducing_prod_const inducing_prod_const
+
 theorem Embedding.prod_map {f : α → β} {g : γ → δ} (hf : Embedding f) (hg : Embedding g) :
     Embedding (Prod.map f g) :=
   { hf.toInducing.prod_map hg.toInducing with

forward-port of leanprover-community/mathlib#19100. Also adds a new lemma IsClosed.sigmaCompactSpace.

@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Johannes Hölzl, Mario Carneiro, Patrick Massot
 
 ! This file was ported from Lean 3 source module topology.constructions
-! leanprover-community/mathlib commit 55d771df074d0dd020139ee1cd4b95521422df9f
+! leanprover-community/mathlib commit 76f9c990d4b7c3dd26b87c4c4b51759e249d9e66
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
@@ -1578,6 +1578,11 @@ theorem embedding_uLift_down [TopologicalSpace α] : Embedding (ULift.down : ULi
   ⟨⟨rfl⟩, ULift.down_injective⟩
 #align embedding_ulift_down embedding_uLift_down
 
+theorem ULift.closedEmbedding_down [TopologicalSpace α] :
+    ClosedEmbedding (ULift.down : ULift.{v, u} α → α) :=
+  ⟨embedding_uLift_down, by simp only [ULift.down_surjective.range_eq, isClosed_univ]⟩
+#align ulift.closed_embedding_down ULift.closedEmbedding_down
+
 instance [TopologicalSpace α] [DiscreteTopology α] : DiscreteTopology (ULift α) :=
   embedding_uLift_down.discreteTopology
 

Currently, the following notations are changed from · ×ˢ · because Lean 4 can't deal with ambiguous notations. | Definition | Notation | | :

Co-authored-by: Jeremy Tan Jie Rui <reddeloostw@gmail.com> Co-authored-by: Kyle Miller <kmill31415@gmail.com> Co-authored-by: Chris Hughes <chrishughes24@gmail.com>

@@ -508,14 +508,15 @@ theorem IsOpen.prod {s : Set α} {t : Set β} (hs : IsOpen s) (ht : IsOpen t) :
 #align is_open.prod IsOpen.prod
 
 -- porting note: todo: Lean fails to find `t₁` and `t₂` by unification
-theorem nhds_prod_eq {a : α} {b : β} : 𝓝 (a, b) = 𝓝 a ×ᶠ 𝓝 b := by
+theorem nhds_prod_eq {a : α} {b : β} : 𝓝 (a, b) = 𝓝 a ×ˢ 𝓝 b := by
+  dsimp only [SProd.sprod]
   rw [Filter.prod, instTopologicalSpaceProd, nhds_inf (t₁ := TopologicalSpace.induced Prod.fst _)
     (t₂ := TopologicalSpace.induced Prod.snd _), nhds_induced, nhds_induced]
 #align nhds_prod_eq nhds_prod_eq
 
 -- porting note: moved from `topology.continuous_on`
 theorem nhdsWithin_prod_eq (a : α) (b : β) (s : Set α) (t : Set β) :
-    𝓝[s ×ˢ t] (a, b) = 𝓝[s] a ×ᶠ 𝓝[t] b := by
+    𝓝[s ×ˢ t] (a, b) = 𝓝[s] a ×ˢ 𝓝[t] b := by
   simp only [nhdsWithin, nhds_prod_eq, ← prod_inf_prod, prod_principal_principal]
 #align nhds_within_prod_eq nhdsWithin_prod_eq
 

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>

@@ -210,7 +210,7 @@ instance Sum.discreteTopology [TopologicalSpace α] [TopologicalSpace β] [hα :
 
 instance Sigma.discreteTopology {β : α → Type v} [∀ a, TopologicalSpace (β a)]
     [h : ∀ a, DiscreteTopology (β a)] : DiscreteTopology (Sigma β) :=
-  ⟨supᵢ_eq_bot.2 <| fun _ => by simp only [(h _).eq_bot, coinduced_bot]⟩
+  ⟨iSup_eq_bot.2 <| fun _ => by simp only [(h _).eq_bot, coinduced_bot]⟩
 #align sigma.discrete_topology Sigma.discreteTopology
 
 section Topα
@@ -274,10 +274,10 @@ instance : TopologicalSpace (CofiniteTopology α) where
     rintro hs ht ⟨x, hxs, hxt⟩
     rw [compl_inter]
     exact (hs ⟨x, hxs⟩).union (ht ⟨x, hxt⟩)
-  isOpen_unionₛ := by
+  isOpen_sUnion := by
     rintro s h ⟨x, t, hts, hzt⟩
-    rw [Set.compl_unionₛ]
-    exact Set.Finite.interₛ (mem_image_of_mem _ hts) (h t hts ⟨x, hzt⟩)
+    rw [Set.compl_sUnion]
+    exact Set.Finite.sInter (mem_image_of_mem _ hts) (h t hts ⟨x, hzt⟩)
 
 theorem isOpen_iff {s : Set (CofiniteTopology α)} : IsOpen s ↔ s.Nonempty → sᶜ.Finite :=
   Iff.rfl
@@ -457,18 +457,18 @@ theorem continuous_inf_dom_right₂ {α β γ} {f : α → β → γ} {ta1 ta2 :
   exact @Continuous.comp _ _ _ (id _) (id _) _ _ _ h h_continuous_id
 #align continuous_inf_dom_right₂ continuous_inf_dom_right₂
 
-/-- A version of `continuous_infₛ_dom` for binary functions -/
-theorem continuous_infₛ_dom₂ {α β γ} {f : α → β → γ} {tas : Set (TopologicalSpace α)}
+/-- A version of `continuous_sInf_dom` for binary functions -/
+theorem continuous_sInf_dom₂ {α β γ} {f : α → β → γ} {tas : Set (TopologicalSpace α)}
     {tbs : Set (TopologicalSpace β)} {ta : TopologicalSpace α} {tb : TopologicalSpace β}
     {tc : TopologicalSpace γ} (ha : ta ∈ tas) (hb : tb ∈ tbs)
     (hf : Continuous fun p : α × β => f p.1 p.2) : by
-    haveI := infₛ tas; haveI := infₛ tbs;
+    haveI := sInf tas; haveI := sInf tbs;
       exact @Continuous _ _ _ tc fun p : α × β => f p.1 p.2 := by
-  have ha := continuous_infₛ_dom ha continuous_id
-  have hb := continuous_infₛ_dom hb continuous_id
-  have h_continuous_id := @Continuous.prod_map _ _ _ _ ta tb (infₛ tas) (infₛ tbs) _ _ ha hb
+  have ha := continuous_sInf_dom ha continuous_id
+  have hb := continuous_sInf_dom hb continuous_id
+  have h_continuous_id := @Continuous.prod_map _ _ _ _ ta tb (sInf tas) (sInf tbs) _ _ ha hb
   exact @Continuous.comp _ _ _ (id _) (id _) _ _ _ hf h_continuous_id
-#align continuous_Inf_dom₂ continuous_infₛ_dom₂
+#align continuous_Inf_dom₂ continuous_sInf_dom₂
 
 theorem Filter.Eventually.prod_inl_nhds {p : α → Prop} {a : α} (h : ∀ᶠ x in 𝓝 a, p x) (b : β) :
     ∀ᶠ x in 𝓝 (a, b), p (x : α × β).1 :=
@@ -628,21 +628,21 @@ theorem prod_generateFrom_generateFrom_eq {α β : Type _} {s : Set (Set α)} {t
       (coinduced_le_iff_le_induced.mp <|
         le_generateFrom fun u hu =>
           have : (⋃ v ∈ t, u ×ˢ v) = Prod.fst ⁻¹' u := by
-            simp_rw [← prod_unionᵢ, ← unionₛ_eq_bunionᵢ, ht, prod_univ]
+            simp_rw [← prod_iUnion, ← sUnion_eq_biUnion, ht, prod_univ]
           show G.IsOpen (Prod.fst ⁻¹' u) by
             rw [← this]
             exact
-              isOpen_unionᵢ fun v =>
-                isOpen_unionᵢ fun hv => GenerateOpen.basic _ ⟨_, hu, _, hv, rfl⟩)
+              isOpen_iUnion fun v =>
+                isOpen_iUnion fun hv => GenerateOpen.basic _ ⟨_, hu, _, hv, rfl⟩)
       (coinduced_le_iff_le_induced.mp <|
         le_generateFrom fun v hv =>
           have : (⋃ u ∈ s, u ×ˢ v) = Prod.snd ⁻¹' v := by
-            simp_rw [← unionᵢ_prod_const, ← unionₛ_eq_bunionᵢ, hs, univ_prod]
+            simp_rw [← iUnion_prod_const, ← sUnion_eq_biUnion, hs, univ_prod]
           show G.IsOpen (Prod.snd ⁻¹' v) by
             rw [← this]
             exact
-              isOpen_unionᵢ fun u =>
-                isOpen_unionᵢ fun hu => GenerateOpen.basic _ ⟨_, hu, _, hv, rfl⟩))
+              isOpen_iUnion fun u =>
+                isOpen_iUnion fun hu => GenerateOpen.basic _ ⟨_, hu, _, hv, rfl⟩))
 #align prod_generate_from_generate_from_eq prod_generateFrom_generateFrom_eq
 
 -- todo: use the previous lemma?
@@ -1173,7 +1173,7 @@ variable {ι : Type _} {π : ι → Type _} {κ : Type _} [TopologicalSpace α]
   [∀ i, TopologicalSpace (π i)] {f : α → ∀ i : ι, π i}
 
 theorem continuous_pi_iff : Continuous f ↔ ∀ i, Continuous fun a => f a i := by
-  simp only [continuous_infᵢ_rng, continuous_induced_rng, comp]
+  simp only [continuous_iInf_rng, continuous_induced_rng, comp]
 #align continuous_pi_iff continuous_pi_iff
 
 @[continuity]
@@ -1183,7 +1183,7 @@ theorem continuous_pi (h : ∀ i, Continuous fun a => f a i) : Continuous f :=
 
 @[continuity]
 theorem continuous_apply (i : ι) : Continuous fun p : ∀ i, π i => p i :=
-  continuous_infᵢ_dom continuous_induced_dom
+  continuous_iInf_dom continuous_induced_dom
 #align continuous_apply continuous_apply
 
 @[continuity]
@@ -1202,7 +1202,7 @@ theorem Filter.Tendsto.apply {l : Filter β} {f : β → ∀ i, π i} {x : ∀ i
 #align filter.tendsto.apply Filter.Tendsto.apply
 
 theorem nhds_pi {a : ∀ i, π i} : 𝓝 a = pi fun i => 𝓝 (a i) := by
-  simp only [nhds_infᵢ, nhds_induced, Filter.pi]
+  simp only [nhds_iInf, nhds_induced, Filter.pi]
 #align nhds_pi nhds_pi
 
 theorem tendsto_pi_nhds {f : β → ∀ i, π i} {g : ∀ i, π i} {u : Filter β} :
@@ -1269,7 +1269,7 @@ theorem Continuous.fin_insertNth {n} {π : Fin (n + 1) → Type _} [∀ i, Topol
 
 theorem isOpen_set_pi {i : Set ι} {s : ∀ a, Set (π a)} (hi : i.Finite)
     (hs : ∀ a ∈ i, IsOpen (s a)) : IsOpen (pi i s) := by
-  rw [pi_def]; exact isOpen_binterᵢ hi fun a ha => (hs _ ha).preimage (continuous_apply _)
+  rw [pi_def]; exact isOpen_biInter hi fun a ha => (hs _ ha).preimage (continuous_apply _)
 #align is_open_set_pi isOpen_set_pi
 
 theorem isOpen_pi_iff {s : Set (∀ a, π a)} :
@@ -1319,7 +1319,7 @@ theorem isOpen_pi_iff' [Finite ι] {s : Set (∀ a, π a)} :
 
 theorem isClosed_set_pi {i : Set ι} {s : ∀ a, Set (π a)} (hs : ∀ a ∈ i, IsClosed (s a)) :
     IsClosed (pi i s) := by
-  rw [pi_def]; exact isClosed_binterᵢ fun a ha => (hs _ ha).preimage (continuous_apply _)
+  rw [pi_def]; exact isClosed_biInter fun a ha => (hs _ ha).preimage (continuous_apply _)
 #align is_closed_set_pi isClosed_set_pi
 
 theorem mem_nhds_of_pi_mem_nhds {I : Set ι} {s : ∀ i, Set (π i)} (a : ∀ i, π i) (hs : I.pi s ∈ 𝓝 a)
@@ -1329,7 +1329,7 @@ theorem mem_nhds_of_pi_mem_nhds {I : Set ι} {s : ∀ i, Set (π i)} (a : ∀ i,
 
 theorem set_pi_mem_nhds {i : Set ι} {s : ∀ a, Set (π a)} {x : ∀ a, π a} (hi : i.Finite)
     (hs : ∀ a ∈ i, s a ∈ 𝓝 (x a)) : pi i s ∈ 𝓝 x := by
-  rw [pi_def, binterᵢ_mem hi]
+  rw [pi_def, biInter_mem hi]
   exact fun a ha => (continuous_apply a).continuousAt (hs a ha)
 #align set_pi_mem_nhds set_pi_mem_nhds
 
@@ -1361,7 +1361,7 @@ theorem pi_generateFrom_eq {π : ι → Type _} {g : ∀ a, Set (Set (π a))} :
     rintro _ ⟨s, i, hi, rfl⟩
     letI := fun a => generateFrom (g a)
     exact isOpen_set_pi i.finite_toSet (fun a ha => GenerateOpen.basic _ (hi a ha))
-  · refine le_infᵢ fun a => coinduced_le_iff_le_induced.1 <| le_generateFrom fun s hs => ?_
+  · refine le_iInf fun a => coinduced_le_iff_le_induced.1 <| le_generateFrom fun s hs => ?_
     refine GenerateOpen.basic _ ⟨update (fun a => univ) a s, {a}, ?_⟩
     simp [hs]
 #align pi_generate_from_eq pi_generateFrom_eq
@@ -1387,7 +1387,7 @@ theorem pi_generateFrom_eq_finite {π : ι → Type _} {g : ∀ a, Set (Set (π
   · rintro s ⟨t, i, ht, rfl⟩
     letI := generateFrom { t | ∃ s : ∀ a, Set (π a), (∀ a, s a ∈ g a) ∧ t = pi univ s }
     refine isOpen_iff_forall_mem_open.2 fun f hf => ?_
-    choose c hcg hfc using fun a => unionₛ_eq_univ_iff.1 (hg a) (f a)
+    choose c hcg hfc using fun a => sUnion_eq_univ_iff.1 (hg a) (f a)
     refine ⟨pi i t ∩ pi (↑iᶜ : Set ι) c, inter_subset_left _ _, ?_, ⟨hf, fun a _ => hfc a⟩⟩
     rw [← univ_pi_piecewise]
     refine GenerateOpen.basic _ ⟨_, fun a => ?_, rfl⟩
@@ -1397,7 +1397,7 @@ theorem pi_generateFrom_eq_finite {π : ι → Type _} {g : ∀ a, Set (Set (π
 -- porting note: new lemma
 theorem induced_to_pi {X : Type _} (f : X → ∀ i, π i) :
     induced f Pi.topologicalSpace = ⨅ i, induced (f · i) inferInstance := by
-  erw [induced_infᵢ]
+  erw [induced_iInf]
   simp only [induced_compose]
   rfl
 
@@ -1406,10 +1406,10 @@ endowed with a family of maps `f i : X → π i` for every `i : ι`, hence induc
 map `g : X → Π i, π i`. This lemma shows that infimum of the topologies on `X` induced by
 the `f i` as `i : ι` varies is simply the topology on `X` induced by `g : X → Π i, π i`
 where `Π i, π i` is endowed with the usual product topology. -/
-theorem inducing_infᵢ_to_pi {X : Type _} (f : ∀ i, X → π i) :
+theorem inducing_iInf_to_pi {X : Type _} (f : ∀ i, X → π i) :
     @Inducing X (∀ i, π i) (⨅ i, induced (f i) inferInstance) _ fun x i => f i x :=
   letI := ⨅ i, induced (f i) inferInstance; ⟨(induced_to_pi _).symm⟩
-#align inducing_infi_to_pi inducing_infᵢ_to_pi
+#align inducing_infi_to_pi inducing_iInf_to_pi
 
 variable [Finite ι] [∀ i, DiscreteTopology (π i)]
 
@@ -1429,13 +1429,13 @@ variable {ι κ : Type _} {σ : ι → Type _} {τ : κ → Type _} [∀ i, Topo
 
 @[continuity]
 theorem continuous_sigmaMk {i : ι} : Continuous (@Sigma.mk ι σ i) :=
-  continuous_supᵢ_rng continuous_coinduced_rng
+  continuous_iSup_rng continuous_coinduced_rng
 #align continuous_sigma_mk continuous_sigmaMk
 
--- porting note: the proof was `by simp only [isOpen_supᵢ_iff, isOpen_coinduced]`
+-- porting note: the proof was `by simp only [isOpen_iSup_iff, isOpen_coinduced]`
 theorem isOpen_sigma_iff {s : Set (Sigma σ)} : IsOpen s ↔ ∀ i, IsOpen (Sigma.mk i ⁻¹' s) := by
   delta instTopologicalSpaceSigma
-  rw [isOpen_supᵢ_iff]
+  rw [isOpen_iSup_iff]
   rfl
 #align is_open_sigma_iff isOpen_sigma_iff
 
@@ -1499,9 +1499,9 @@ theorem comap_sigmaMk_nhds (i : ι) (x : σ i) : comap (Sigma.mk i) (𝓝 ⟨i,
 #align comap_sigma_mk_nhds comap_sigmaMk_nhds
 
 theorem isOpen_sigma_fst_preimage (s : Set ι) : IsOpen (Sigma.fst ⁻¹' s : Set (Σ a, σ a)) := by
-  rw [← bunionᵢ_of_singleton s, preimage_unionᵢ₂]
+  rw [← biUnion_of_singleton s, preimage_iUnion₂]
   simp only [← range_sigmaMk]
-  exact isOpen_bunionᵢ fun _ _ => isOpen_range_sigmaMk
+  exact isOpen_biUnion fun _ _ => isOpen_range_sigmaMk
 #align is_open_sigma_fst_preimage isOpen_sigma_fst_preimage
 
 /-- A map out of a sum type is continuous iff its restriction to each summand is. -/
@@ -1509,7 +1509,7 @@ theorem isOpen_sigma_fst_preimage (s : Set ι) : IsOpen (Sigma.fst ⁻¹' s : Se
 theorem continuous_sigma_iff {f : Sigma σ → α} :
     Continuous f ↔ ∀ i, Continuous fun a => f ⟨i, a⟩ := by
   delta instTopologicalSpaceSigma
-  rw [continuous_supᵢ_dom]
+  rw [continuous_iSup_dom]
   exact forall_congr' fun _ => continuous_coinduced_dom
 #align continuous_sigma_iff continuous_sigma_iff
 

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".

@@ -1301,8 +1301,7 @@ theorem isOpen_pi_iff {s : Set (∀ a, π a)} :
 
 theorem isOpen_pi_iff' [Finite ι] {s : Set (∀ a, π a)} :
     IsOpen s ↔
-      ∀ f, f ∈ s → ∃ u : ∀ a, Set (π a), (∀ a, IsOpen (u a) ∧ f a ∈ u a) ∧ Set.univ.pi u ⊆ s :=
-  by
+      ∀ f, f ∈ s → ∃ u : ∀ a, Set (π a), (∀ a, IsOpen (u a) ∧ f a ∈ u a) ∧ Set.univ.pi u ⊆ s := by
   cases nonempty_fintype ι
   rw [isOpen_iff_nhds]
   simp_rw [le_principal_iff, nhds_pi, Filter.mem_pi', mem_nhds_iff]

closes #3680, see https://leanprover.zulipchat.com/#narrow/stream/287929-mathlib4/topic/Stepping.20through.20simp_rw/near/326712986

@@ -1277,7 +1277,7 @@ theorem isOpen_pi_iff {s : Set (∀ a, π a)} :
       ∀ f, f ∈ s → ∃ (I : Finset ι)(u : ∀ a, Set (π a)),
         (∀ a, a ∈ I → IsOpen (u a) ∧ f a ∈ u a) ∧ (I : Set ι).pi u ⊆ s := by
   rw [isOpen_iff_nhds]
-  simp_rw [le_principal_iff, nhds_pi, Filter.mem_pi', mem_nhds_iff, exists_prop]
+  simp_rw [le_principal_iff, nhds_pi, Filter.mem_pi', mem_nhds_iff]
   refine ball_congr fun a _ => ⟨?_, ?_⟩
   · rintro ⟨I, t, ⟨h1, h2⟩⟩
     refine ⟨I, fun a => eval a '' (I : Set ι).pi fun a => (h1 a).choose, fun i hi => ?_, ?_⟩
@@ -1305,7 +1305,7 @@ theorem isOpen_pi_iff' [Finite ι] {s : Set (∀ a, π a)} :
   by
   cases nonempty_fintype ι
   rw [isOpen_iff_nhds]
-  simp_rw [le_principal_iff, nhds_pi, Filter.mem_pi', mem_nhds_iff, exists_prop]
+  simp_rw [le_principal_iff, nhds_pi, Filter.mem_pi', mem_nhds_iff]
   refine ball_congr fun a _ => ⟨?_, ?_⟩
   · rintro ⟨I, t, ⟨h1, h2⟩⟩
     refine

Co-authored-by: Scott Morrison <scott@tqft.net>

@@ -1395,19 +1395,21 @@ theorem pi_generateFrom_eq_finite {π : ι → Type _} {g : ∀ a, Set (Set (π
     by_cases a ∈ i <;> simp [*]
 #align pi_generate_from_eq_finite pi_generateFrom_eq_finite
 
+-- porting note: new lemma
+theorem induced_to_pi {X : Type _} (f : X → ∀ i, π i) :
+    induced f Pi.topologicalSpace = ⨅ i, induced (f · i) inferInstance := by
+  erw [induced_infᵢ]
+  simp only [induced_compose]
+  rfl
+
 /-- Suppose `π i` is a family of topological spaces indexed by `i : ι`, and `X` is a type
 endowed with a family of maps `f i : X → π i` for every `i : ι`, hence inducing a
 map `g : X → Π i, π i`. This lemma shows that infimum of the topologies on `X` induced by
 the `f i` as `i : ι` varies is simply the topology on `X` induced by `g : X → Π i, π i`
 where `Π i, π i` is endowed with the usual product topology. -/
 theorem inducing_infᵢ_to_pi {X : Type _} (f : ∀ i, X → π i) :
-    @Inducing X (∀ i, π i) (⨅ i, induced (f i) inferInstance) _ fun x i => f i x := by
-  letI := ⨅ i, induced (f i) inferInstance
-  constructor
-  erw [induced_infᵢ]
-  congr 1
-  funext
-  erw [induced_compose]; rfl
+    @Inducing X (∀ i, π i) (⨅ i, induced (f i) inferInstance) _ fun x i => f i x :=
+  letI := ⨅ i, induced (f i) inferInstance; ⟨(induced_to_pi _).symm⟩
 #align inducing_infi_to_pi inducing_infᵢ_to_pi
 
 variable [Finite ι] [∀ i, DiscreteTopology (π i)]

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

@@ -4,12 +4,11 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Johannes Hölzl, Mario Carneiro, Patrick Massot
 
 ! This file was ported from Lean 3 source module topology.constructions
-! leanprover-community/mathlib commit 0c1f285a9f6e608ae2bdffa3f993eafb01eba829
+! leanprover-community/mathlib commit 55d771df074d0dd020139ee1cd4b95521422df9f
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
 import Mathlib.Topology.Maps
-import Mathlib.Topology.LocallyFinite
 import Mathlib.Order.Filter.Pi
 
 /-!
@@ -1069,24 +1068,6 @@ theorem tendsto_subtype_rng {β : Type _} {p : α → Prop} {b : Filter β} {f :
   | ⟨a, ha⟩ => by rw [nhds_subtype_eq_comap, tendsto_comap_iff]; rfl
 #align tendsto_subtype_rng tendsto_subtype_rng
 
--- porting note: todo: see https://github.com/leanprover-community/mathlib/pull/18321
-theorem continuous_subtype_nhds_cover {ι : Sort _} {f : α → β} {c : ι → α → Prop}
-    (c_cover : ∀ x : α, ∃ i, { x | c i x } ∈ 𝓝 x)
-    (f_cont : ∀ i, Continuous fun x : Subtype (c i) => f x) : Continuous f :=
-  continuous_iff_continuousAt.mpr fun x => by
-    rcases c_cover x with ⟨i, c_sets⟩
-    lift x to Subtype (c i) using mem_of_mem_nhds c_sets
-    refine' (inducing_subtype_val.continuousAt_iff' _).1 (f_cont i).continuousAt
-    rwa [Subtype.range_coe]
-#align continuous_subtype_nhds_cover continuous_subtype_nhds_cover
-
-/- porting note: todo: see https://github.com/leanprover-community/mathlib/pull/18321
-
-I failed to quickly fix the proof. This is a leaf lemma, and it is going to be replaced by a lemma
-formulated using `ContinuousOn`.
--/
-#noalign continuous_subtype_is_closed_cover
-
 theorem closure_subtype {x : { a // p a }} {s : Set { a // p a }} :
     x ∈ closure s ↔ (x : α) ∈ closure (((↑) : _ → α) '' s) :=
   closure_induced

@@ -830,6 +830,9 @@ theorem embedding_graph {f : α → β} (hf : Continuous f) : Embedding fun x =>
   embedding_of_embedding_compose (continuous_id.prod_mk hf) continuous_fst embedding_id
 #align embedding_graph embedding_graph
 
+theorem embedding_prod_mk (x : α) : Embedding (Prod.mk x : β → α × β) :=
+  embedding_of_embedding_compose (Continuous.Prod.mk x) continuous_snd embedding_id
+
 end Prod
 
 section Sum

This was incorrectly marked as forward-ported in #2110. You can see that by looking at the status page and noting that no commits in mathlib4 contain the diff from the highlighted commits on the left.

@@ -1254,6 +1254,15 @@ theorem continuous_update [DecidableEq ι] (i : ι) :
   continuous_fst.update i continuous_snd
 #align continuous_update continuous_update
 
+/-- `Pi.mulSingle i x` is continuous in `x`. -/
+-- porting note: todo: restore @[continuity]
+@[to_additive "`Pi.single i x` is continuous in `x`."]
+theorem continuous_mulSingle [∀ i, One (π i)] [DecidableEq ι] (i : ι) :
+    Continuous fun x => (Pi.mulSingle i x : ∀ i, π i) :=
+  continuous_const.update _ continuous_id
+#align continuous_mul_single continuous_mulSingle
+#align continuous_single continuous_single
+
 theorem Filter.Tendsto.fin_insertNth {n} {π : Fin (n + 1) → Type _} [∀ i, TopologicalSpace (π i)]
     (i : Fin (n + 1)) {f : β → π i} {l : Filter β} {x : π i} (hf : Tendsto f l (𝓝 x))
     {g : β → ∀ j : Fin n, π (i.succAbove j)} {y : ∀ j, π (i.succAbove j)} (hg : Tendsto g l (𝓝 y)) :

@@ -1378,7 +1378,8 @@ theorem pi_eq_generateFrom :
     Pi.topologicalSpace =
       generateFrom
         { g | ∃ (s : ∀ a, Set (π a)) (i : Finset ι), (∀ a ∈ i, IsOpen (s a)) ∧ g = pi (↑i) s } :=
-  calc Pi.topologicalSpace = @Pi.topologicalSpace ι π fun a => generateFrom { s | IsOpen s } :=
+  calc Pi.topologicalSpace
+  _ = @Pi.topologicalSpace ι π fun a => generateFrom { s | IsOpen s } :=
     by simp only [generateFrom_setOf_isOpen]
   _ = _ := pi_generateFrom_eq
 #align pi_eq_generate_from pi_eq_generateFrom

Some Mathlib 4 lemmas were backported to Mathlib 3 in leanprover-community/mathlib#18502, sync SHAs

• topology.uniform_space.separation@d90e4e186f1d18e375dcd4e5b5f6364b01cb3e46..0c1f285a9f6e608ae2bdffa3f993eafb01eba829
• topology.constructions@dc6c365e751e34d100e80fe6e314c3c3e0fd2988..0c1f285a9f6e608ae2bdffa3f993eafb01eba829

@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Johannes Hölzl, Mario Carneiro, Patrick Massot
 
 ! This file was ported from Lean 3 source module topology.constructions
-! leanprover-community/mathlib commit dc6c365e751e34d100e80fe6e314c3c3e0fd2988
+! leanprover-community/mathlib commit 0c1f285a9f6e608ae2bdffa3f993eafb01eba829
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
@@ -843,6 +843,7 @@ theorem continuous_sum_dom {f : α ⊕ β → γ} :
   (continuous_sup_dom (t₁ := TopologicalSpace.coinduced Sum.inl _)
     (t₂ := TopologicalSpace.coinduced Sum.inr _)).trans <|
     continuous_coinduced_dom.and continuous_coinduced_dom
+#align continuous_sum_dom continuous_sum_dom
 
 theorem continuous_sum_elim {f : α → γ} {g : β → γ} :
     Continuous (Sum.elim f g) ↔ Continuous f ∧ Continuous g :=

Co-authored-by: qawbecrdtey <qawbecrdtey@naver.com> Co-authored-by: Johan Commelin <johan@commelin.net>

@@ -1175,7 +1175,7 @@ theorem Continuous.quotient_liftOn' {f : α → β} (h : Continuous f)
   h.quotient_lift hs
 #align continuous.quotient_lift_on' Continuous.quotient_liftOn'
 
-theorem Continuous.quotient_map' {t : Setoid β} {f : α → β} (hf : Continuous f)
+@[continuity] theorem Continuous.quotient_map' {t : Setoid β} {f : α → β} (hf : Continuous f)
     (H : (s.r ⇒ t.r) f f) : Continuous (Quotient.map' f H) :=
   (continuous_quotient_mk'.comp hf).quotient_lift _
 #align continuous.quotient_map' Continuous.quotient_map'

We implement the continuity tactic using aesop, this makes it more robust and reduces the code to trivial macros.

@@ -323,7 +323,7 @@ variable [TopologicalSpace α] [TopologicalSpace β] [TopologicalSpace γ] [Topo
     continuous_induced_rng.and continuous_induced_rng
 #align continuous_prod_mk continuous_prod_mk
 
--- porting note: todo: restore @[continuity]
+@[continuity]
 theorem continuous_fst : Continuous (@Prod.fst α β) :=
   (continuous_prod_mk.1 continuous_id).1
 #align continuous_fst continuous_fst
@@ -360,7 +360,7 @@ theorem ContinuousAt.fst'' {f : α → γ} {x : α × β} (hf : ContinuousAt f x
   hf.comp continuousAt_fst
 #align continuous_at.fst'' ContinuousAt.fst''
 
--- porting note: todo: restore @[continuity]
+@[continuity]
 theorem continuous_snd : Continuous (@Prod.snd α β) :=
   (continuous_prod_mk.1 continuous_id).2
 #align continuous_snd continuous_snd
@@ -397,18 +397,18 @@ theorem ContinuousAt.snd'' {f : β → γ} {x : α × β} (hf : ContinuousAt f x
   hf.comp continuousAt_snd
 #align continuous_at.snd'' ContinuousAt.snd''
 
--- porting note: todo: restore @[continuity]
+@[continuity]
 theorem Continuous.prod_mk {f : γ → α} {g : γ → β} (hf : Continuous f) (hg : Continuous g) :
     Continuous fun x => (f x, g x) :=
   continuous_prod_mk.2 ⟨hf, hg⟩
 #align continuous.prod_mk Continuous.prod_mk
 
--- porting note: todo: restore @[continuity]
+@[continuity]
 theorem Continuous.Prod.mk (a : α) : Continuous fun b : β => (a, b) :=
   continuous_const.prod_mk continuous_id
 #align continuous.prod.mk Continuous.Prod.mk
 
--- porting note: todo: restore @[continuity]
+@[continuity]
 theorem Continuous.Prod.mk_left (b : β) : Continuous fun a : α => (a, b) :=
   continuous_id.prod_mk continuous_const
 #align continuous.prod.mk_left Continuous.Prod.mk_left
@@ -430,6 +430,7 @@ theorem Continuous.comp₄ {g : α × β × γ × ζ → ε} (hg : Continuous g)
   hg.comp₃ he hf <| hk.prod_mk hl
 #align continuous.comp₄ Continuous.comp₄
 
+@[continuity]
 theorem Continuous.prod_map {f : γ → α} {g : δ → β} (hf : Continuous f) (hg : Continuous g) :
     Continuous fun x : γ × δ => (f x.1, g x.2) :=
   hf.fst'.prod_mk hg.snd'
@@ -848,26 +849,26 @@ theorem continuous_sum_elim {f : α → γ} {g : β → γ} :
   continuous_sum_dom
 #align continuous_sum_elim continuous_sum_elim
 
--- porting note: todo: restore @[continuity]
+@[continuity]
 theorem Continuous.sum_elim {f : α → γ} {g : β → γ} (hf : Continuous f) (hg : Continuous g) :
     Continuous (Sum.elim f g) :=
   continuous_sum_elim.2 ⟨hf, hg⟩
 #align continuous.sum_elim Continuous.sum_elim
 
--- porting note: todo: add @[continuity]
+@[continuity]
 theorem continuous_isLeft : Continuous (isLeft : α ⊕ β → Bool) :=
   continuous_sum_dom.2 ⟨continuous_const, continuous_const⟩
 
--- porting note: todo: add @[continuity]
+@[continuity]
 theorem continuous_isRight : Continuous (isRight : α ⊕ β → Bool) :=
   continuous_sum_dom.2 ⟨continuous_const, continuous_const⟩
 
--- porting note: todo: restore @[continuity]
+@[continuity]
 -- porting note: the proof was `continuous_sup_rng_left continuous_coinduced_rng`
 theorem continuous_inl : Continuous (@inl α β) := ⟨fun _ => And.left⟩
 #align continuous_inl continuous_inl
 
--- porting note: todo: restore @[continuity]
+@[continuity]
 -- porting note: the proof was `continuous_sup_rng_right continuous_coinduced_rng`
 theorem continuous_inr : Continuous (@inr α β) := ⟨fun _ => And.right⟩
 #align continuous_inr continuous_inr
@@ -946,7 +947,7 @@ theorem continuous_sum_map {f : α → β} {g : γ → δ} :
     embedding_inl.continuous_iff.symm.and embedding_inr.continuous_iff.symm
 #align continuous_sum_map continuous_sum_map
 
--- porting note: todo: restore @[continuity]
+@[continuity]
 theorem Continuous.sum_map {f : α → β} {g : γ → δ} (hf : Continuous f) (hg : Continuous g) :
     Continuous (Sum.map f g) :=
   continuous_sum_map.2 ⟨hf, hg⟩
@@ -991,7 +992,7 @@ theorem closedEmbedding_subtype_val (h : IsClosed { a | p a }) :
   ⟨embedding_subtype_val, by rwa [Subtype.range_coe_subtype]⟩
 #align closed_embedding_subtype_coe closedEmbedding_subtype_val
 
--- porting note: todo: restore @[continuity]
+@[continuity]
 theorem continuous_subtype_val : Continuous (@Subtype.val α p) :=
   continuous_induced_dom
 #align continuous_subtype_val continuous_subtype_val
@@ -1021,7 +1022,7 @@ nonrec theorem IsClosed.closedEmbedding_subtype_val {s : Set α} (hs : IsClosed
   closedEmbedding_subtype_val hs
 #align is_closed.closed_embedding_subtype_coe IsClosed.closedEmbedding_subtype_val
 
--- porting note: todo: restore @[continuity]
+@[continuity]
 theorem Continuous.subtype_mk {f : β → α} (h : Continuous f) (hp : ∀ x, p (f x)) :
     Continuous fun x => (⟨f x, hp x⟩ : Subtype p) :=
   continuous_induced_rng.2 h
@@ -1105,7 +1106,7 @@ theorem ContinuousAt.restrictPreimage {f : α → β} {s : Set β} {x : f ⁻¹'
   h.restrict _
 #align continuous_at.restrict_preimage ContinuousAt.restrictPreimage
 
--- porting note: todo: restore @[continuity]
+@[continuity]
 theorem Continuous.codRestrict {f : α → β} {s : Set β} (hf : Continuous f) (hs : ∀ a, f a ∈ s) :
     Continuous (s.codRestrict f hs) :=
   hf.subtype_mk hs
@@ -1144,12 +1145,12 @@ theorem quotientMap_quot_mk : QuotientMap (@Quot.mk α r) :=
   ⟨Quot.exists_rep, rfl⟩
 #align quotient_map_quot_mk quotientMap_quot_mk
 
--- porting note: todo: restore @[continuity]
+@[continuity]
 theorem continuous_quot_mk : Continuous (@Quot.mk α r) :=
   continuous_coinduced_rng
 #align continuous_quot_mk continuous_quot_mk
 
--- porting note: todo: restore @[continuity]
+@[continuity]
 theorem continuous_quot_lift {f : α → β} (hr : ∀ a b, r a b → f a = f b) (h : Continuous f) :
     Continuous (Quot.lift f hr : Quot r → β) :=
   continuous_coinduced_dom.2 h
@@ -1190,17 +1191,17 @@ theorem continuous_pi_iff : Continuous f ↔ ∀ i, Continuous fun a => f a i :=
   simp only [continuous_infᵢ_rng, continuous_induced_rng, comp]
 #align continuous_pi_iff continuous_pi_iff
 
--- porting note: todo: restore @[continuity]
+@[continuity]
 theorem continuous_pi (h : ∀ i, Continuous fun a => f a i) : Continuous f :=
   continuous_pi_iff.2 h
 #align continuous_pi continuous_pi
 
--- porting note: todo: restore @[continuity]
+@[continuity]
 theorem continuous_apply (i : ι) : Continuous fun p : ∀ i, π i => p i :=
   continuous_infᵢ_dom continuous_induced_dom
 #align continuous_apply continuous_apply
 
--- porting note: todo: restore @[continuity]
+@[continuity]
 theorem continuous_apply_apply {ρ : κ → ι → Type _} [∀ j i, TopologicalSpace (ρ j i)] (j : κ)
     (i : ι) : Continuous fun p : ∀ j, ∀ i, ρ j i => p j i :=
   (continuous_apply i).comp (continuous_apply j)
@@ -1246,7 +1247,7 @@ theorem Continuous.update [DecidableEq ι] (hf : Continuous f) (i : ι) {g : α
 #align continuous.update Continuous.update
 
 /-- `Function.update f i x` is continuous in `(f, x)`. -/
--- porting note: todo: restore @[continuity]
+@[continuity]
 theorem continuous_update [DecidableEq ι] (i : ι) :
     Continuous fun f : (∀ j, π j) × π i => update f.1 i f.2 :=
   continuous_fst.update i continuous_snd
@@ -1430,7 +1431,7 @@ section Sigma
 variable {ι κ : Type _} {σ : ι → Type _} {τ : κ → Type _} [∀ i, TopologicalSpace (σ i)]
   [∀ k, TopologicalSpace (τ k)] [TopologicalSpace α]
 
--- porting note: todo: restore @[continuity]
+@[continuity]
 theorem continuous_sigmaMk {i : ι} : Continuous (@Sigma.mk ι σ i) :=
   continuous_supᵢ_rng continuous_coinduced_rng
 #align continuous_sigma_mk continuous_sigmaMk
@@ -1517,7 +1518,7 @@ theorem continuous_sigma_iff {f : Sigma σ → α} :
 #align continuous_sigma_iff continuous_sigma_iff
 
 /-- A map out of a sum type is continuous if its restriction to each summand is. -/
--- porting note: todo: restore @[continuity]
+@[continuity]
 theorem continuous_sigma {f : Sigma σ → α} (hf : ∀ i, Continuous fun a => f ⟨i, a⟩) :
     Continuous f :=
   continuous_sigma_iff.2 hf
@@ -1529,7 +1530,7 @@ theorem continuous_sigma_map {f₁ : ι → κ} {f₂ : ∀ i, σ i → τ (f₁
   continuous_sigma_iff.trans <| by simp only [Sigma.map, embedding_sigmaMk.continuous_iff, comp]
 #align continuous_sigma_map continuous_sigma_map
 
--- porting note: todo: restore @[continuity]
+@[continuity]
 theorem Continuous.sigma_map {f₁ : ι → κ} {f₂ : ∀ i, σ i → τ (f₁ i)} (hf : ∀ i, Continuous (f₂ i)) :
     Continuous (Sigma.map f₁ f₂) :=
   continuous_sigma_map.2 hf
@@ -1566,12 +1567,12 @@ end Sigma
 
 section ULift
 
--- porting note: todo: restore @[continuity]
+@[continuity]
 theorem continuous_uLift_down [TopologicalSpace α] : Continuous (ULift.down : ULift.{v, u} α → α) :=
   continuous_induced_dom
 #align continuous_ulift_down continuous_uLift_down
 
--- porting note: todo: restore @[continuity]
+@[continuity]
 theorem continuous_uLift_up [TopologicalSpace α] : Continuous (ULift.up : α → ULift.{v, u} α) :=
   continuous_induced_rng.2 continuous_id
 #align continuous_ulift_up continuous_uLift_up

This is the mathlib4 pendant of https://github.com/leanprover-community/mathlib/pull/18390

@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Johannes Hölzl, Mario Carneiro, Patrick Massot
 
 ! This file was ported from Lean 3 source module topology.constructions
-! leanprover-community/mathlib commit bcfa726826abd57587355b4b5b7e78ad6527b7e4
+! leanprover-community/mathlib commit dc6c365e751e34d100e80fe6e314c3c3e0fd2988
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
@@ -1277,6 +1277,52 @@ theorem isOpen_set_pi {i : Set ι} {s : ∀ a, Set (π a)} (hi : i.Finite)
   rw [pi_def]; exact isOpen_binterᵢ hi fun a ha => (hs _ ha).preimage (continuous_apply _)
 #align is_open_set_pi isOpen_set_pi
 
+theorem isOpen_pi_iff {s : Set (∀ a, π a)} :
+    IsOpen s ↔
+      ∀ f, f ∈ s → ∃ (I : Finset ι)(u : ∀ a, Set (π a)),
+        (∀ a, a ∈ I → IsOpen (u a) ∧ f a ∈ u a) ∧ (I : Set ι).pi u ⊆ s := by
+  rw [isOpen_iff_nhds]
+  simp_rw [le_principal_iff, nhds_pi, Filter.mem_pi', mem_nhds_iff, exists_prop]
+  refine ball_congr fun a _ => ⟨?_, ?_⟩
+  · rintro ⟨I, t, ⟨h1, h2⟩⟩
+    refine ⟨I, fun a => eval a '' (I : Set ι).pi fun a => (h1 a).choose, fun i hi => ?_, ?_⟩
+    · simp_rw [Set.eval_image_pi (Finset.mem_coe.mpr hi)
+          (pi_nonempty_iff.mpr fun i => ⟨_, fun _ => (h1 i).choose_spec.2.2⟩)]
+      exact (h1 i).choose_spec.2
+    · exact Subset.trans
+        (Set.pi_mono fun i hi => (Set.eval_image_pi_subset hi).trans (h1 i).choose_spec.1) h2
+  · rintro ⟨I, t, ⟨h1, h2⟩⟩
+    refine ⟨I, fun a => ite (a ∈ I) (t a) Set.univ, fun i => ?_, ?_⟩
+    · by_cases hi : i ∈ I
+      · use t i
+        simp_rw [if_pos hi]
+        exact ⟨Subset.rfl, (h1 i) hi⟩
+      · use Set.univ
+        simp_rw [if_neg hi]
+        exact ⟨Subset.rfl, isOpen_univ, mem_univ _⟩
+    · rw [← Set.univ_pi_ite]
+      simp only [← ite_and, ← Finset.mem_coe, and_self_iff, Set.univ_pi_ite, h2]
+#align is_open_pi_iff isOpen_pi_iff
+
+theorem isOpen_pi_iff' [Finite ι] {s : Set (∀ a, π a)} :
+    IsOpen s ↔
+      ∀ f, f ∈ s → ∃ u : ∀ a, Set (π a), (∀ a, IsOpen (u a) ∧ f a ∈ u a) ∧ Set.univ.pi u ⊆ s :=
+  by
+  cases nonempty_fintype ι
+  rw [isOpen_iff_nhds]
+  simp_rw [le_principal_iff, nhds_pi, Filter.mem_pi', mem_nhds_iff, exists_prop]
+  refine ball_congr fun a _ => ⟨?_, ?_⟩
+  · rintro ⟨I, t, ⟨h1, h2⟩⟩
+    refine
+      ⟨fun i => (h1 i).choose,
+        ⟨fun i => (h1 i).choose_spec.2,
+          (Set.pi_mono fun i _ => (h1 i).choose_spec.1).trans (Subset.trans ?_ h2)⟩⟩
+    rw [← Set.pi_inter_compl (I : Set ι)]
+    exact inter_subset_left _ _
+  · exact fun ⟨u, ⟨h1, _⟩⟩ =>
+      ⟨Finset.univ, u, ⟨fun i => ⟨u i, ⟨rfl.subset, h1 i⟩⟩, by rwa [Finset.coe_univ]⟩⟩
+#align is_open_pi_iff' isOpen_pi_iff'
+
 theorem isClosed_set_pi {i : Set ι} {s : ∀ a, Set (π a)} (hs : ∀ a ∈ i, IsClosed (s a)) :
     IsClosed (pi i s) := by
   rw [pi_def]; exact isClosed_binterᵢ fun a ha => (hs _ ha).preimage (continuous_apply _)

Co-authored-by: Johan Commelin <johan@commelin.net>

@@ -80,7 +80,6 @@ instance ULift.topologicalSpace [t : TopologicalSpace α] : TopologicalSpace (UL
 The topology on those type synonyms is inherited without change.
 -/
 
-
 section
 
 variable [TopologicalSpace α]

@@ -838,11 +838,15 @@ open Sum
 
 variable [TopologicalSpace α] [TopologicalSpace β] [TopologicalSpace γ] [TopologicalSpace δ]
 
-theorem continuous_sum_elim {f : α → γ} {g : β → γ} :
-    Continuous (Sum.elim f g) ↔ Continuous f ∧ Continuous g :=
+theorem continuous_sum_dom {f : α ⊕ β → γ} :
+    Continuous f ↔ Continuous (f ∘ Sum.inl) ∧ Continuous (f ∘ Sum.inr) :=
   (continuous_sup_dom (t₁ := TopologicalSpace.coinduced Sum.inl _)
     (t₂ := TopologicalSpace.coinduced Sum.inr _)).trans <|
     continuous_coinduced_dom.and continuous_coinduced_dom
+
+theorem continuous_sum_elim {f : α → γ} {g : β → γ} :
+    Continuous (Sum.elim f g) ↔ Continuous f ∧ Continuous g :=
+  continuous_sum_dom
 #align continuous_sum_elim continuous_sum_elim
 
 -- porting note: todo: restore @[continuity]
@@ -851,6 +855,14 @@ theorem Continuous.sum_elim {f : α → γ} {g : β → γ} (hf : Continuous f)
   continuous_sum_elim.2 ⟨hf, hg⟩
 #align continuous.sum_elim Continuous.sum_elim
 
+-- porting note: todo: add @[continuity]
+theorem continuous_isLeft : Continuous (isLeft : α ⊕ β → Bool) :=
+  continuous_sum_dom.2 ⟨continuous_const, continuous_const⟩
+
+-- porting note: todo: add @[continuity]
+theorem continuous_isRight : Continuous (isRight : α ⊕ β → Bool) :=
+  continuous_sum_dom.2 ⟨continuous_const, continuous_const⟩
+
 -- porting note: todo: restore @[continuity]
 -- porting note: the proof was `continuous_sup_rng_left continuous_coinduced_rng`
 theorem continuous_inl : Continuous (@inl α β) := ⟨fun _ => And.left⟩

@@ -1035,10 +1035,14 @@ theorem Subtype.dense_iff {s : Set α} {t : Set s} : Dense t ↔ s ⊆ closure (
   rfl
 #align subtype.dense_iff Subtype.dense_iff
 
-theorem map_nhds_subtype_coe_eq {a : α} (ha : p a) (h : ∀ᶠ x in 𝓝 a, p x) :
+-- porting note: new lemma
+theorem map_nhds_subtype_val {s : Set α} (a : s) : map ((↑) : s → α) (𝓝 a) = 𝓝[s] ↑a := by
+  rw [inducing_subtype_val.map_nhds_eq, Subtype.range_val]
+
+theorem map_nhds_subtype_coe_eq_nhds {a : α} (ha : p a) (h : ∀ᶠ x in 𝓝 a, p x) :
     map ((↑) : Subtype p → α) (𝓝 ⟨a, ha⟩) = 𝓝 a :=
   map_nhds_induced_of_mem <| by rw [Subtype.range_val]; exact h
-#align map_nhds_subtype_coe_eq map_nhds_subtype_coe_eq
+#align map_nhds_subtype_coe_eq map_nhds_subtype_coe_eq_nhds
 
 theorem nhds_subtype_eq_comap {a : α} {h : p a} : 𝓝 (⟨a, h⟩ : Subtype p) = comap (↑) (𝓝 a) :=
   nhds_induced _ _