topology.metric_space.isometry ⟷ Mathlib.Topology.MetricSpace.Isometry

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

@@ -465,7 +465,7 @@ theorem ediam_image (h : α ≃ᵢ β) (s : Set α) : EMetric.diam (h '' s) = EM
 
 #print IsometryEquiv.toEquiv_injective /-
 theorem toEquiv_injective : ∀ ⦃h₁ h₂ : α ≃ᵢ β⦄, h₁.toEquiv = h₂.toEquiv → h₁ = h₂
-  | ⟨e₁, h₁⟩, ⟨e₂, h₂⟩, H => by dsimp at H ; subst e₁
+  | ⟨e₁, h₁⟩, ⟨e₂, h₂⟩, H => by dsimp at H; subst e₁
 #align isometry_equiv.to_equiv_inj IsometryEquiv.toEquiv_injective
 -/
 

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

@@ -54,7 +54,7 @@ theorem isometry_iff_nndist_eq [PseudoMetricSpace α] [PseudoMetricSpace β] {f
 /-- On pseudometric spaces, a map is an isometry if and only if it preserves distances. -/
 theorem isometry_iff_dist_eq [PseudoMetricSpace α] [PseudoMetricSpace β] {f : α → β} :
     Isometry f ↔ ∀ x y, dist (f x) (f y) = dist x y := by
-  simp only [isometry_iff_nndist_eq, ← coe_nndist, NNReal.coe_eq]
+  simp only [isometry_iff_nndist_eq, ← coe_nndist, NNReal.coe_inj]
 #align isometry_iff_dist_eq isometry_iff_dist_eq
 -/
 

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

@@ -46,7 +46,7 @@ def Isometry [PseudoEMetricSpace α] [PseudoEMetricSpace β] (f : α → β) : P
 distances. -/
 theorem isometry_iff_nndist_eq [PseudoMetricSpace α] [PseudoMetricSpace β] {f : α → β} :
     Isometry f ↔ ∀ x y, nndist (f x) (f y) = nndist x y := by
-  simp only [Isometry, edist_nndist, ENNReal.coe_eq_coe]
+  simp only [Isometry, edist_nndist, ENNReal.coe_inj]
 #align isometry_iff_nndist_eq isometry_iff_nndist_eq
 -/
 

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

@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Isometries of emetric and metric spaces
 Authors: Sébastien Gouëzel
 -/
-import Mathbin.Topology.MetricSpace.Antilipschitz
+import Topology.MetricSpace.Antilipschitz
 
 #align_import topology.metric_space.isometry from "leanprover-community/mathlib"@"b1859b6d4636fdbb78c5d5cefd24530653cfd3eb"
 

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

@@ -59,19 +59,19 @@ theorem isometry_iff_dist_eq [PseudoMetricSpace α] [PseudoMetricSpace β] {f :
 -/
 
 /-- An isometry preserves distances. -/
-alias isometry_iff_dist_eq ↔ Isometry.dist_eq _
+alias ⟨Isometry.dist_eq, _⟩ := isometry_iff_dist_eq
 #align isometry.dist_eq Isometry.dist_eq
 
 /-- A map that preserves distances is an isometry -/
-alias isometry_iff_dist_eq ↔ _ Isometry.of_dist_eq
+alias ⟨_, Isometry.of_dist_eq⟩ := isometry_iff_dist_eq
 #align isometry.of_dist_eq Isometry.of_dist_eq
 
 /-- An isometry preserves non-negative distances. -/
-alias isometry_iff_nndist_eq ↔ Isometry.nndist_eq _
+alias ⟨Isometry.nndist_eq, _⟩ := isometry_iff_nndist_eq
 #align isometry.nndist_eq Isometry.nndist_eq
 
 /-- A map that preserves non-negative distances is an isometry. -/
-alias isometry_iff_nndist_eq ↔ _ Isometry.of_nndist_eq
+alias ⟨_, Isometry.of_nndist_eq⟩ := isometry_iff_nndist_eq
 #align isometry.of_nndist_eq Isometry.of_nndist_eq
 
 namespace Isometry

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

@@ -711,7 +711,7 @@ instance : Group (α ≃ᵢ α) where
   mul_assoc e₁ e₂ e₃ := rfl
   one_mul e := ext fun _ => rfl
   mul_one e := ext fun _ => rfl
-  mul_left_inv e := ext e.symm_apply_apply
+  hMul_left_inv e := ext e.symm_apply_apply
 
 #print IsometryEquiv.coe_one /-
 @[simp]

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

@@ -3,14 +3,11 @@ Copyright (c) 2018 Sébastien Gouëzel. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Isometries of emetric and metric spaces
 Authors: Sébastien Gouëzel
-
-! This file was ported from Lean 3 source module topology.metric_space.isometry
-! leanprover-community/mathlib commit b1859b6d4636fdbb78c5d5cefd24530653cfd3eb
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
 -/
 import Mathbin.Topology.MetricSpace.Antilipschitz
 
+#align_import topology.metric_space.isometry from "leanprover-community/mathlib"@"b1859b6d4636fdbb78c5d5cefd24530653cfd3eb"
+
 /-!
 # Isometries
 

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

@@ -118,15 +118,19 @@ theorem isometry_id : Isometry (id : α → α) := fun x y => rfl
 #align isometry_id isometry_id
 -/
 
+#print Isometry.prod_map /-
 theorem prod_map {δ} [PseudoEMetricSpace δ] {f : α → β} {g : γ → δ} (hf : Isometry f)
     (hg : Isometry g) : Isometry (Prod.map f g) := fun x y => by
   simp only [Prod.edist_eq, hf.edist_eq, hg.edist_eq, Prod_map]
 #align isometry.prod_map Isometry.prod_map
+-/
 
+#print isometry_dcomp /-
 theorem isometry_dcomp {ι} [Fintype ι] {α β : ι → Type _} [∀ i, PseudoEMetricSpace (α i)]
     [∀ i, PseudoEMetricSpace (β i)] (f : ∀ i, α i → β i) (hf : ∀ i, Isometry (f i)) :
     Isometry (dcomp f) := fun x y => by simp only [edist_pi_def, (hf _).edist_eq]
 #align isometry_dcomp isometry_dcomp
+-/
 
 #print Isometry.comp /-
 /-- The composition of isometries is an isometry. -/
@@ -149,10 +153,12 @@ protected theorem uniformInducing (hf : Isometry f) : UniformInducing f :=
 #align isometry.uniform_inducing Isometry.uniformInducing
 -/
 
+#print Isometry.tendsto_nhds_iff /-
 theorem tendsto_nhds_iff {ι : Type _} {f : α → β} {g : ι → α} {a : Filter ι} {b : α}
     (hf : Isometry f) : Filter.Tendsto g a (𝓝 b) ↔ Filter.Tendsto (f ∘ g) a (𝓝 (f b)) :=
   hf.UniformInducing.Inducing.tendsto_nhds_iff
 #align isometry.tendsto_nhds_iff Isometry.tendsto_nhds_iff
+-/
 
 #print Isometry.continuous /-
 /-- An isometry is continuous. -/
@@ -213,15 +219,19 @@ theorem isometry_subtype_coe {s : Set α} : Isometry (coe : s → α) := fun x y
 #align isometry_subtype_coe isometry_subtype_coe
 -/
 
+#print Isometry.comp_continuousOn_iff /-
 theorem comp_continuousOn_iff {γ} [TopologicalSpace γ] (hf : Isometry f) {g : γ → α} {s : Set γ} :
     ContinuousOn (f ∘ g) s ↔ ContinuousOn g s :=
   hf.UniformInducing.Inducing.continuousOn_iff.symm
 #align isometry.comp_continuous_on_iff Isometry.comp_continuousOn_iff
+-/
 
+#print Isometry.comp_continuous_iff /-
 theorem comp_continuous_iff {γ} [TopologicalSpace γ] (hf : Isometry f) {g : γ → α} :
     Continuous (f ∘ g) ↔ Continuous g :=
   hf.UniformInducing.Inducing.continuous_iff.symm
 #align isometry.comp_continuous_iff Isometry.comp_continuous_iff
+-/
 
 end PseudoEmetricIsometry
 
@@ -332,6 +342,7 @@ end PseudoMetricIsometry
 -- section
 end Isometry
 
+#print UniformEmbedding.to_isometry /-
 -- namespace
 /-- A uniform embedding from a uniform space to a metric space is an isometry with respect to the
 induced metric space structure on the source space. -/
@@ -346,7 +357,9 @@ theorem UniformEmbedding.to_isometry {α β} [UniformSpace α] [MetricSpace β]
   intro x y
   rfl
 #align uniform_embedding.to_isometry UniformEmbedding.to_isometry
+-/
 
+#print Embedding.to_isometry /-
 /-- An embedding from a topological space to a metric space is an isometry with respect to the
 induced metric space structure on the source space. -/
 theorem Embedding.to_isometry {α β} [TopologicalSpace α] [MetricSpace β] {f : α → β}
@@ -360,6 +373,7 @@ theorem Embedding.to_isometry {α β} [TopologicalSpace α] [MetricSpace β] {f
   intro x y
   rfl
 #align embedding.to_isometry Embedding.to_isometry
+-/
 
 #print IsometryEquiv /-
 -- such a bijection need not exist
@@ -371,7 +385,6 @@ structure IsometryEquiv (α β : Type _) [PseudoEMetricSpace α] [PseudoEMetricS
 #align isometry_equiv IsometryEquiv
 -/
 
--- mathport name: «expr ≃ᵢ »
 infixl:25 " ≃ᵢ " => IsometryEquiv
 
 namespace IsometryEquiv
@@ -383,9 +396,11 @@ variable [PseudoEMetricSpace α] [PseudoEMetricSpace β] [PseudoEMetricSpace γ]
 instance : CoeFun (α ≃ᵢ β) fun _ => α → β :=
   ⟨fun e => e.toEquiv⟩
 
+#print IsometryEquiv.coe_eq_toEquiv /-
 theorem coe_eq_toEquiv (h : α ≃ᵢ β) (a : α) : h a = h.toEquiv a :=
   rfl
 #align isometry_equiv.coe_eq_to_equiv IsometryEquiv.coe_eq_toEquiv
+-/
 
 #print IsometryEquiv.coe_toEquiv /-
 @[simp]
@@ -394,39 +409,55 @@ theorem coe_toEquiv (h : α ≃ᵢ β) : ⇑h.toEquiv = h :=
 #align isometry_equiv.coe_to_equiv IsometryEquiv.coe_toEquiv
 -/
 
+#print IsometryEquiv.isometry /-
 protected theorem isometry (h : α ≃ᵢ β) : Isometry h :=
   h.isometry_toFun
 #align isometry_equiv.isometry IsometryEquiv.isometry
+-/
 
+#print IsometryEquiv.bijective /-
 protected theorem bijective (h : α ≃ᵢ β) : Bijective h :=
   h.toEquiv.Bijective
 #align isometry_equiv.bijective IsometryEquiv.bijective
+-/
 
+#print IsometryEquiv.injective /-
 protected theorem injective (h : α ≃ᵢ β) : Injective h :=
   h.toEquiv.Injective
 #align isometry_equiv.injective IsometryEquiv.injective
+-/
 
+#print IsometryEquiv.surjective /-
 protected theorem surjective (h : α ≃ᵢ β) : Surjective h :=
   h.toEquiv.Surjective
 #align isometry_equiv.surjective IsometryEquiv.surjective
+-/
 
+#print IsometryEquiv.edist_eq /-
 protected theorem edist_eq (h : α ≃ᵢ β) (x y : α) : edist (h x) (h y) = edist x y :=
   h.Isometry.edist_eq x y
 #align isometry_equiv.edist_eq IsometryEquiv.edist_eq
+-/
 
+#print IsometryEquiv.dist_eq /-
 protected theorem dist_eq {α β : Type _} [PseudoMetricSpace α] [PseudoMetricSpace β] (h : α ≃ᵢ β)
     (x y : α) : dist (h x) (h y) = dist x y :=
   h.Isometry.dist_eq x y
 #align isometry_equiv.dist_eq IsometryEquiv.dist_eq
+-/
 
+#print IsometryEquiv.nndist_eq /-
 protected theorem nndist_eq {α β : Type _} [PseudoMetricSpace α] [PseudoMetricSpace β] (h : α ≃ᵢ β)
     (x y : α) : nndist (h x) (h y) = nndist x y :=
   h.Isometry.nndist_eq x y
 #align isometry_equiv.nndist_eq IsometryEquiv.nndist_eq
+-/
 
+#print IsometryEquiv.continuous /-
 protected theorem continuous (h : α ≃ᵢ β) : Continuous h :=
   h.Isometry.Continuous
 #align isometry_equiv.continuous IsometryEquiv.continuous
+-/
 
 #print IsometryEquiv.ediam_image /-
 @[simp]
@@ -441,10 +472,12 @@ theorem toEquiv_injective : ∀ ⦃h₁ h₂ : α ≃ᵢ β⦄, h₁.toEquiv = h
 #align isometry_equiv.to_equiv_inj IsometryEquiv.toEquiv_injective
 -/
 
+#print IsometryEquiv.ext /-
 @[ext]
 theorem ext ⦃h₁ h₂ : α ≃ᵢ β⦄ (H : ∀ x, h₁ x = h₂ x) : h₁ = h₂ :=
   toEquiv_injective <| Equiv.ext H
 #align isometry_equiv.ext IsometryEquiv.ext
+-/
 
 #print IsometryEquiv.mk' /-
 /-- Alternative constructor for isometric bijections,
@@ -474,10 +507,12 @@ protected def trans (h₁ : α ≃ᵢ β) (h₂ : β ≃ᵢ γ) : α ≃ᵢ γ :
 #align isometry_equiv.trans IsometryEquiv.trans
 -/
 
+#print IsometryEquiv.trans_apply /-
 @[simp]
 theorem trans_apply (h₁ : α ≃ᵢ β) (h₂ : β ≃ᵢ γ) (x : α) : h₁.trans h₂ x = h₂ (h₁ x) :=
   rfl
 #align isometry_equiv.trans_apply IsometryEquiv.trans_apply
+-/
 
 #print IsometryEquiv.symm /-
 /-- The inverse of an isometric isomorphism, as an isometric isomorphism. -/
@@ -512,50 +547,70 @@ theorem symm_symm (h : α ≃ᵢ β) : h.symm.symm = h :=
 #align isometry_equiv.symm_symm IsometryEquiv.symm_symm
 -/
 
+#print IsometryEquiv.apply_symm_apply /-
 @[simp]
 theorem apply_symm_apply (h : α ≃ᵢ β) (y : β) : h (h.symm y) = y :=
   h.toEquiv.apply_symm_apply y
 #align isometry_equiv.apply_symm_apply IsometryEquiv.apply_symm_apply
+-/
 
+#print IsometryEquiv.symm_apply_apply /-
 @[simp]
 theorem symm_apply_apply (h : α ≃ᵢ β) (x : α) : h.symm (h x) = x :=
   h.toEquiv.symm_apply_apply x
 #align isometry_equiv.symm_apply_apply IsometryEquiv.symm_apply_apply
+-/
 
+#print IsometryEquiv.symm_apply_eq /-
 theorem symm_apply_eq (h : α ≃ᵢ β) {x : α} {y : β} : h.symm y = x ↔ y = h x :=
   h.toEquiv.symm_apply_eq
 #align isometry_equiv.symm_apply_eq IsometryEquiv.symm_apply_eq
+-/
 
+#print IsometryEquiv.eq_symm_apply /-
 theorem eq_symm_apply (h : α ≃ᵢ β) {x : α} {y : β} : x = h.symm y ↔ h x = y :=
   h.toEquiv.eq_symm_apply
 #align isometry_equiv.eq_symm_apply IsometryEquiv.eq_symm_apply
+-/
 
+#print IsometryEquiv.symm_comp_self /-
 theorem symm_comp_self (h : α ≃ᵢ β) : ⇑h.symm ∘ ⇑h = id :=
   funext fun a => h.toEquiv.left_inv a
 #align isometry_equiv.symm_comp_self IsometryEquiv.symm_comp_self
+-/
 
+#print IsometryEquiv.self_comp_symm /-
 theorem self_comp_symm (h : α ≃ᵢ β) : ⇑h ∘ ⇑h.symm = id :=
   funext fun a => h.toEquiv.right_inv a
 #align isometry_equiv.self_comp_symm IsometryEquiv.self_comp_symm
+-/
 
+#print IsometryEquiv.range_eq_univ /-
 @[simp]
 theorem range_eq_univ (h : α ≃ᵢ β) : range h = univ :=
   h.toEquiv.range_eq_univ
 #align isometry_equiv.range_eq_univ IsometryEquiv.range_eq_univ
+-/
 
+#print IsometryEquiv.image_symm /-
 theorem image_symm (h : α ≃ᵢ β) : image h.symm = preimage h :=
   image_eq_preimage_of_inverse h.symm.toEquiv.left_inv h.symm.toEquiv.right_inv
 #align isometry_equiv.image_symm IsometryEquiv.image_symm
+-/
 
+#print IsometryEquiv.preimage_symm /-
 theorem preimage_symm (h : α ≃ᵢ β) : preimage h.symm = image h :=
   (image_eq_preimage_of_inverse h.toEquiv.left_inv h.toEquiv.right_inv).symm
 #align isometry_equiv.preimage_symm IsometryEquiv.preimage_symm
+-/
 
+#print IsometryEquiv.symm_trans_apply /-
 @[simp]
 theorem symm_trans_apply (h₁ : α ≃ᵢ β) (h₂ : β ≃ᵢ γ) (x : γ) :
     (h₁.trans h₂).symm x = h₁.symm (h₂.symm x) :=
   rfl
 #align isometry_equiv.symm_trans_apply IsometryEquiv.symm_trans_apply
+-/
 
 #print IsometryEquiv.ediam_univ /-
 theorem ediam_univ (h : α ≃ᵢ β) : EMetric.diam (univ : Set α) = EMetric.diam (univ : Set β) := by
@@ -570,29 +625,37 @@ theorem ediam_preimage (h : α ≃ᵢ β) (s : Set β) : EMetric.diam (h ⁻¹'
 #align isometry_equiv.ediam_preimage IsometryEquiv.ediam_preimage
 -/
 
+#print IsometryEquiv.preimage_emetric_ball /-
 @[simp]
 theorem preimage_emetric_ball (h : α ≃ᵢ β) (x : β) (r : ℝ≥0∞) :
     h ⁻¹' EMetric.ball x r = EMetric.ball (h.symm x) r := by
   rw [← h.isometry.preimage_emetric_ball (h.symm x) r, h.apply_symm_apply]
 #align isometry_equiv.preimage_emetric_ball IsometryEquiv.preimage_emetric_ball
+-/
 
+#print IsometryEquiv.preimage_emetric_closedBall /-
 @[simp]
 theorem preimage_emetric_closedBall (h : α ≃ᵢ β) (x : β) (r : ℝ≥0∞) :
     h ⁻¹' EMetric.closedBall x r = EMetric.closedBall (h.symm x) r := by
   rw [← h.isometry.preimage_emetric_closed_ball (h.symm x) r, h.apply_symm_apply]
 #align isometry_equiv.preimage_emetric_closed_ball IsometryEquiv.preimage_emetric_closedBall
+-/
 
+#print IsometryEquiv.image_emetric_ball /-
 @[simp]
 theorem image_emetric_ball (h : α ≃ᵢ β) (x : α) (r : ℝ≥0∞) :
     h '' EMetric.ball x r = EMetric.ball (h x) r := by
   rw [← h.preimage_symm, h.symm.preimage_emetric_ball, symm_symm]
 #align isometry_equiv.image_emetric_ball IsometryEquiv.image_emetric_ball
+-/
 
+#print IsometryEquiv.image_emetric_closedBall /-
 @[simp]
 theorem image_emetric_closedBall (h : α ≃ᵢ β) (x : α) (r : ℝ≥0∞) :
     h '' EMetric.closedBall x r = EMetric.closedBall (h x) r := by
   rw [← h.preimage_symm, h.symm.preimage_emetric_closed_ball, symm_symm]
 #align isometry_equiv.image_emetric_closed_ball IsometryEquiv.image_emetric_closedBall
+-/
 
 #print IsometryEquiv.toHomeomorph /-
 /-- The (bundled) homeomorphism associated to an isometric isomorphism. -/
@@ -605,33 +668,43 @@ protected def toHomeomorph (h : α ≃ᵢ β) : α ≃ₜ β
 #align isometry_equiv.to_homeomorph IsometryEquiv.toHomeomorph
 -/
 
+#print IsometryEquiv.coe_toHomeomorph /-
 @[simp]
 theorem coe_toHomeomorph (h : α ≃ᵢ β) : ⇑h.toHomeomorph = h :=
   rfl
 #align isometry_equiv.coe_to_homeomorph IsometryEquiv.coe_toHomeomorph
+-/
 
+#print IsometryEquiv.coe_toHomeomorph_symm /-
 @[simp]
 theorem coe_toHomeomorph_symm (h : α ≃ᵢ β) : ⇑h.toHomeomorph.symm = h.symm :=
   rfl
 #align isometry_equiv.coe_to_homeomorph_symm IsometryEquiv.coe_toHomeomorph_symm
+-/
 
+#print IsometryEquiv.comp_continuousOn_iff /-
 @[simp]
 theorem comp_continuousOn_iff {γ} [TopologicalSpace γ] (h : α ≃ᵢ β) {f : γ → α} {s : Set γ} :
     ContinuousOn (h ∘ f) s ↔ ContinuousOn f s :=
   h.toHomeomorph.comp_continuousOn_iff _ _
 #align isometry_equiv.comp_continuous_on_iff IsometryEquiv.comp_continuousOn_iff
+-/
 
+#print IsometryEquiv.comp_continuous_iff /-
 @[simp]
 theorem comp_continuous_iff {γ} [TopologicalSpace γ] (h : α ≃ᵢ β) {f : γ → α} :
     Continuous (h ∘ f) ↔ Continuous f :=
   h.toHomeomorph.comp_continuous_iff
 #align isometry_equiv.comp_continuous_iff IsometryEquiv.comp_continuous_iff
+-/
 
+#print IsometryEquiv.comp_continuous_iff' /-
 @[simp]
 theorem comp_continuous_iff' {γ} [TopologicalSpace γ] (h : α ≃ᵢ β) {f : β → γ} :
     Continuous (f ∘ h) ↔ Continuous f :=
   h.toHomeomorph.comp_continuous_iff'
 #align isometry_equiv.comp_continuous_iff' IsometryEquiv.comp_continuous_iff'
+-/
 
 /-- The group of isometries. -/
 instance : Group (α ≃ᵢ α) where
@@ -643,29 +716,39 @@ instance : Group (α ≃ᵢ α) where
   mul_one e := ext fun _ => rfl
   mul_left_inv e := ext e.symm_apply_apply
 
+#print IsometryEquiv.coe_one /-
 @[simp]
 theorem coe_one : ⇑(1 : α ≃ᵢ α) = id :=
   rfl
 #align isometry_equiv.coe_one IsometryEquiv.coe_one
+-/
 
+#print IsometryEquiv.coe_mul /-
 @[simp]
 theorem coe_mul (e₁ e₂ : α ≃ᵢ α) : ⇑(e₁ * e₂) = e₁ ∘ e₂ :=
   rfl
 #align isometry_equiv.coe_mul IsometryEquiv.coe_mul
+-/
 
+#print IsometryEquiv.mul_apply /-
 theorem mul_apply (e₁ e₂ : α ≃ᵢ α) (x : α) : (e₁ * e₂) x = e₁ (e₂ x) :=
   rfl
 #align isometry_equiv.mul_apply IsometryEquiv.mul_apply
+-/
 
+#print IsometryEquiv.inv_apply_self /-
 @[simp]
 theorem inv_apply_self (e : α ≃ᵢ α) (x : α) : e⁻¹ (e x) = x :=
   e.symm_apply_apply x
 #align isometry_equiv.inv_apply_self IsometryEquiv.inv_apply_self
+-/
 
+#print IsometryEquiv.apply_inv_self /-
 @[simp]
 theorem apply_inv_self (e : α ≃ᵢ α) (x : α) : e (e⁻¹ x) = x :=
   e.apply_symm_apply x
 #align isometry_equiv.apply_inv_self IsometryEquiv.apply_inv_self
+-/
 
 #print IsometryEquiv.completeSpace /-
 protected theorem completeSpace [CompleteSpace β] (e : α ≃ᵢ β) : CompleteSpace α :=
@@ -693,6 +776,7 @@ def funUnique [Unique ι] [Fintype ι] : (ι → α) ≃ᵢ α
 #align isometry_equiv.fun_unique IsometryEquiv.funUnique
 -/
 
+#print IsometryEquiv.piFinTwo /-
 /-- `pi_fin_two_equiv` as an `isometry_equiv`. -/
 @[simps]
 def piFinTwo (α : Fin 2 → Type _) [∀ i, PseudoEMetricSpace (α i)] : (∀ i, α i) ≃ᵢ α 0 × α 1
@@ -700,6 +784,7 @@ def piFinTwo (α : Fin 2 → Type _) [∀ i, PseudoEMetricSpace (α i)] : (∀ i
   toEquiv := piFinTwoEquiv α
   isometry_toFun x hx := by simp [edist_pi_def, Fin.univ_succ, Prod.edist_eq]
 #align isometry_equiv.pi_fin_two IsometryEquiv.piFinTwo
+-/
 
 end PseudoEMetricSpace
 
@@ -727,40 +812,52 @@ theorem diam_univ : Metric.diam (univ : Set α) = Metric.diam (univ : Set β) :=
 #align isometry_equiv.diam_univ IsometryEquiv.diam_univ
 -/
 
+#print IsometryEquiv.preimage_ball /-
 @[simp]
 theorem preimage_ball (h : α ≃ᵢ β) (x : β) (r : ℝ) :
     h ⁻¹' Metric.ball x r = Metric.ball (h.symm x) r := by
   rw [← h.isometry.preimage_ball (h.symm x) r, h.apply_symm_apply]
 #align isometry_equiv.preimage_ball IsometryEquiv.preimage_ball
+-/
 
+#print IsometryEquiv.preimage_sphere /-
 @[simp]
 theorem preimage_sphere (h : α ≃ᵢ β) (x : β) (r : ℝ) :
     h ⁻¹' Metric.sphere x r = Metric.sphere (h.symm x) r := by
   rw [← h.isometry.preimage_sphere (h.symm x) r, h.apply_symm_apply]
 #align isometry_equiv.preimage_sphere IsometryEquiv.preimage_sphere
+-/
 
+#print IsometryEquiv.preimage_closedBall /-
 @[simp]
 theorem preimage_closedBall (h : α ≃ᵢ β) (x : β) (r : ℝ) :
     h ⁻¹' Metric.closedBall x r = Metric.closedBall (h.symm x) r := by
   rw [← h.isometry.preimage_closed_ball (h.symm x) r, h.apply_symm_apply]
 #align isometry_equiv.preimage_closed_ball IsometryEquiv.preimage_closedBall
+-/
 
+#print IsometryEquiv.image_ball /-
 @[simp]
 theorem image_ball (h : α ≃ᵢ β) (x : α) (r : ℝ) : h '' Metric.ball x r = Metric.ball (h x) r := by
   rw [← h.preimage_symm, h.symm.preimage_ball, symm_symm]
 #align isometry_equiv.image_ball IsometryEquiv.image_ball
+-/
 
+#print IsometryEquiv.image_sphere /-
 @[simp]
 theorem image_sphere (h : α ≃ᵢ β) (x : α) (r : ℝ) :
     h '' Metric.sphere x r = Metric.sphere (h x) r := by
   rw [← h.preimage_symm, h.symm.preimage_sphere, symm_symm]
 #align isometry_equiv.image_sphere IsometryEquiv.image_sphere
+-/
 
+#print IsometryEquiv.image_closedBall /-
 @[simp]
 theorem image_closedBall (h : α ≃ᵢ β) (x : α) (r : ℝ) :
     h '' Metric.closedBall x r = Metric.closedBall (h x) r := by
   rw [← h.preimage_symm, h.symm.preimage_closed_ball, symm_symm]
 #align isometry_equiv.image_closed_ball IsometryEquiv.image_closedBall
+-/
 
 end PseudoMetricSpace
 

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

@@ -281,7 +281,7 @@ theorem diam_range (hf : Isometry f) : Metric.diam (range f) = Metric.diam (univ
 
 #print Isometry.preimage_setOf_dist /-
 theorem preimage_setOf_dist (hf : Isometry f) (x : α) (p : ℝ → Prop) :
-    f ⁻¹' { y | p (dist y (f x)) } = { y | p (dist y x) } := by ext y; simp [hf.dist_eq]
+    f ⁻¹' {y | p (dist y (f x))} = {y | p (dist y x)} := by ext y; simp [hf.dist_eq]
 #align isometry.preimage_set_of_dist Isometry.preimage_setOf_dist
 -/
 

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

@@ -366,7 +366,7 @@ theorem Embedding.to_isometry {α β} [TopologicalSpace α] [MetricSpace β] {f
 /-- `α` and `β` are isometric if there is an isometric bijection between them. -/
 @[nolint has_nonempty_instance]
 structure IsometryEquiv (α β : Type _) [PseudoEMetricSpace α] [PseudoEMetricSpace β] extends
-  α ≃ β where
+    α ≃ β where
   isometry_toFun : Isometry to_fun
 #align isometry_equiv IsometryEquiv
 -/
@@ -437,7 +437,7 @@ theorem ediam_image (h : α ≃ᵢ β) (s : Set α) : EMetric.diam (h '' s) = EM
 
 #print IsometryEquiv.toEquiv_injective /-
 theorem toEquiv_injective : ∀ ⦃h₁ h₂ : α ≃ᵢ β⦄, h₁.toEquiv = h₂.toEquiv → h₁ = h₂
-  | ⟨e₁, h₁⟩, ⟨e₂, h₂⟩, H => by dsimp at H; subst e₁
+  | ⟨e₁, h₁⟩, ⟨e₂, h₂⟩, H => by dsimp at H ; subst e₁
 #align isometry_equiv.to_equiv_inj IsometryEquiv.toEquiv_injective
 -/
 
@@ -677,7 +677,7 @@ protected theorem completeSpace [CompleteSpace β] (e : α ≃ᵢ β) : Complete
 
 #print IsometryEquiv.completeSpace_iff /-
 theorem completeSpace_iff (e : α ≃ᵢ β) : CompleteSpace α ↔ CompleteSpace β := by
-  constructor <;> intro H; exacts[e.symm.complete_space, e.complete_space]
+  constructor <;> intro H; exacts [e.symm.complete_space, e.complete_space]
 #align isometry_equiv.complete_space_iff IsometryEquiv.completeSpace_iff
 -/
 

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

@@ -34,7 +34,7 @@ variable {ι : Type _} {α : Type u} {β : Type v} {γ : Type w}
 
 open Function Set
 
-open Topology ENNReal
+open scoped Topology ENNReal
 
 #print Isometry /-
 /-- An isometry (also known as isometric embedding) is a map preserving the edistance

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

@@ -118,23 +118,11 @@ theorem isometry_id : Isometry (id : α → α) := fun x y => rfl
 #align isometry_id isometry_id
 -/
 
-/- warning: isometry.prod_map -> Isometry.prod_map is a dubious translation:
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-but is expected to have type
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-Case conversion may be inaccurate. Consider using '#align isometry.prod_map Isometry.prod_mapₓ'. -/
 theorem prod_map {δ} [PseudoEMetricSpace δ] {f : α → β} {g : γ → δ} (hf : Isometry f)
     (hg : Isometry g) : Isometry (Prod.map f g) := fun x y => by
   simp only [Prod.edist_eq, hf.edist_eq, hg.edist_eq, Prod_map]
 #align isometry.prod_map Isometry.prod_map
 
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-but is expected to have type
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-Case conversion may be inaccurate. Consider using '#align isometry_dcomp isometry_dcompₓ'. -/
 theorem isometry_dcomp {ι} [Fintype ι] {α β : ι → Type _} [∀ i, PseudoEMetricSpace (α i)]
     [∀ i, PseudoEMetricSpace (β i)] (f : ∀ i, α i → β i) (hf : ∀ i, Isometry (f i)) :
     Isometry (dcomp f) := fun x y => by simp only [edist_pi_def, (hf _).edist_eq]
@@ -161,12 +149,6 @@ protected theorem uniformInducing (hf : Isometry f) : UniformInducing f :=
 #align isometry.uniform_inducing Isometry.uniformInducing
 -/
 
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-lean 3 declaration is
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-but is expected to have type
-  forall {α : Type.{u2}} {β : Type.{u3}} [_inst_1 : PseudoEMetricSpace.{u2} α] [_inst_2 : PseudoEMetricSpace.{u3} β] {ι : Type.{u1}} {f : α -> β} {g : ι -> α} {a : Filter.{u1} ι} {b : α}, (Isometry.{u2, u3} α β _inst_1 _inst_2 f) -> (Iff (Filter.Tendsto.{u1, u2} ι α g a (nhds.{u2} α (UniformSpace.toTopologicalSpace.{u2} α (PseudoEMetricSpace.toUniformSpace.{u2} α _inst_1)) b)) (Filter.Tendsto.{u1, u3} ι β (Function.comp.{succ u1, succ u2, succ u3} ι α β f g) a (nhds.{u3} β (UniformSpace.toTopologicalSpace.{u3} β (PseudoEMetricSpace.toUniformSpace.{u3} β _inst_2)) (f b))))
-Case conversion may be inaccurate. Consider using '#align isometry.tendsto_nhds_iff Isometry.tendsto_nhds_iffₓ'. -/
 theorem tendsto_nhds_iff {ι : Type _} {f : α → β} {g : ι → α} {a : Filter ι} {b : α}
     (hf : Isometry f) : Filter.Tendsto g a (𝓝 b) ↔ Filter.Tendsto (f ∘ g) a (𝓝 (f b)) :=
   hf.UniformInducing.Inducing.tendsto_nhds_iff
@@ -231,23 +213,11 @@ theorem isometry_subtype_coe {s : Set α} : Isometry (coe : s → α) := fun x y
 #align isometry_subtype_coe isometry_subtype_coe
 -/
 
-/- warning: isometry.comp_continuous_on_iff -> Isometry.comp_continuousOn_iff is a dubious translation:
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-but is expected to have type
-  forall {α : Type.{u2}} {β : Type.{u3}} [_inst_1 : PseudoEMetricSpace.{u2} α] [_inst_2 : PseudoEMetricSpace.{u3} β] {f : α -> β} {γ : Type.{u1}} [_inst_4 : TopologicalSpace.{u1} γ], (Isometry.{u2, u3} α β _inst_1 _inst_2 f) -> (forall {g : γ -> α} {s : Set.{u1} γ}, Iff (ContinuousOn.{u1, u3} γ β _inst_4 (UniformSpace.toTopologicalSpace.{u3} β (PseudoEMetricSpace.toUniformSpace.{u3} β _inst_2)) (Function.comp.{succ u1, succ u2, succ u3} γ α β f g) s) (ContinuousOn.{u1, u2} γ α _inst_4 (UniformSpace.toTopologicalSpace.{u2} α (PseudoEMetricSpace.toUniformSpace.{u2} α _inst_1)) g s))
-Case conversion may be inaccurate. Consider using '#align isometry.comp_continuous_on_iff Isometry.comp_continuousOn_iffₓ'. -/
 theorem comp_continuousOn_iff {γ} [TopologicalSpace γ] (hf : Isometry f) {g : γ → α} {s : Set γ} :
     ContinuousOn (f ∘ g) s ↔ ContinuousOn g s :=
   hf.UniformInducing.Inducing.continuousOn_iff.symm
 #align isometry.comp_continuous_on_iff Isometry.comp_continuousOn_iff
 
-/- warning: isometry.comp_continuous_iff -> Isometry.comp_continuous_iff is a dubious translation:
-lean 3 declaration is
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-but is expected to have type
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-Case conversion may be inaccurate. Consider using '#align isometry.comp_continuous_iff Isometry.comp_continuous_iffₓ'. -/
 theorem comp_continuous_iff {γ} [TopologicalSpace γ] (hf : Isometry f) {g : γ → α} :
     Continuous (f ∘ g) ↔ Continuous g :=
   hf.UniformInducing.Inducing.continuous_iff.symm
@@ -362,12 +332,6 @@ end PseudoMetricIsometry
 -- section
 end Isometry
 
-/- warning: uniform_embedding.to_isometry -> UniformEmbedding.to_isometry is a dubious translation:
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-Case conversion may be inaccurate. Consider using '#align uniform_embedding.to_isometry UniformEmbedding.to_isometryₓ'. -/
 -- namespace
 /-- A uniform embedding from a uniform space to a metric space is an isometry with respect to the
 induced metric space structure on the source space. -/
@@ -383,12 +347,6 @@ theorem UniformEmbedding.to_isometry {α β} [UniformSpace α] [MetricSpace β]
   rfl
 #align uniform_embedding.to_isometry UniformEmbedding.to_isometry
 
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-Case conversion may be inaccurate. Consider using '#align embedding.to_isometry Embedding.to_isometryₓ'. -/
 /-- An embedding from a topological space to a metric space is an isometry with respect to the
 induced metric space structure on the source space. -/
 theorem Embedding.to_isometry {α β} [TopologicalSpace α] [MetricSpace β] {f : α → β}
@@ -425,12 +383,6 @@ variable [PseudoEMetricSpace α] [PseudoEMetricSpace β] [PseudoEMetricSpace γ]
 instance : CoeFun (α ≃ᵢ β) fun _ => α → β :=
   ⟨fun e => e.toEquiv⟩
 
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 theorem coe_eq_toEquiv (h : α ≃ᵢ β) (a : α) : h a = h.toEquiv a :=
   rfl
 #align isometry_equiv.coe_eq_to_equiv IsometryEquiv.coe_eq_toEquiv
@@ -442,84 +394,36 @@ theorem coe_toEquiv (h : α ≃ᵢ β) : ⇑h.toEquiv = h :=
 #align isometry_equiv.coe_to_equiv IsometryEquiv.coe_toEquiv
 -/
 
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 protected theorem isometry (h : α ≃ᵢ β) : Isometry h :=
   h.isometry_toFun
 #align isometry_equiv.isometry IsometryEquiv.isometry
 
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 protected theorem bijective (h : α ≃ᵢ β) : Bijective h :=
   h.toEquiv.Bijective
 #align isometry_equiv.bijective IsometryEquiv.bijective
 
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 protected theorem injective (h : α ≃ᵢ β) : Injective h :=
   h.toEquiv.Injective
 #align isometry_equiv.injective IsometryEquiv.injective
 
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 protected theorem surjective (h : α ≃ᵢ β) : Surjective h :=
   h.toEquiv.Surjective
 #align isometry_equiv.surjective IsometryEquiv.surjective
 
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 protected theorem edist_eq (h : α ≃ᵢ β) (x y : α) : edist (h x) (h y) = edist x y :=
   h.Isometry.edist_eq x y
 #align isometry_equiv.edist_eq IsometryEquiv.edist_eq
 
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 protected theorem dist_eq {α β : Type _} [PseudoMetricSpace α] [PseudoMetricSpace β] (h : α ≃ᵢ β)
     (x y : α) : dist (h x) (h y) = dist x y :=
   h.Isometry.dist_eq x y
 #align isometry_equiv.dist_eq IsometryEquiv.dist_eq
 
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 protected theorem nndist_eq {α β : Type _} [PseudoMetricSpace α] [PseudoMetricSpace β] (h : α ≃ᵢ β)
     (x y : α) : nndist (h x) (h y) = nndist x y :=
   h.Isometry.nndist_eq x y
 #align isometry_equiv.nndist_eq IsometryEquiv.nndist_eq
 
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 protected theorem continuous (h : α ≃ᵢ β) : Continuous h :=
   h.Isometry.Continuous
 #align isometry_equiv.continuous IsometryEquiv.continuous
@@ -537,12 +441,6 @@ theorem toEquiv_injective : ∀ ⦃h₁ h₂ : α ≃ᵢ β⦄, h₁.toEquiv = h
 #align isometry_equiv.to_equiv_inj IsometryEquiv.toEquiv_injective
 -/
 
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 @[ext]
 theorem ext ⦃h₁ h₂ : α ≃ᵢ β⦄ (H : ∀ x, h₁ x = h₂ x) : h₁ = h₂ :=
   toEquiv_injective <| Equiv.ext H
@@ -576,12 +474,6 @@ protected def trans (h₁ : α ≃ᵢ β) (h₂ : β ≃ᵢ γ) : α ≃ᵢ γ :
 #align isometry_equiv.trans IsometryEquiv.trans
 -/
 
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 @[simp]
 theorem trans_apply (h₁ : α ≃ᵢ β) (h₂ : β ≃ᵢ γ) (x : α) : h₁.trans h₂ x = h₂ (h₁ x) :=
   rfl
@@ -620,105 +512,45 @@ theorem symm_symm (h : α ≃ᵢ β) : h.symm.symm = h :=
 #align isometry_equiv.symm_symm IsometryEquiv.symm_symm
 -/
 
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 @[simp]
 theorem apply_symm_apply (h : α ≃ᵢ β) (y : β) : h (h.symm y) = y :=
   h.toEquiv.apply_symm_apply y
 #align isometry_equiv.apply_symm_apply IsometryEquiv.apply_symm_apply
 
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 @[simp]
 theorem symm_apply_apply (h : α ≃ᵢ β) (x : α) : h.symm (h x) = x :=
   h.toEquiv.symm_apply_apply x
 #align isometry_equiv.symm_apply_apply IsometryEquiv.symm_apply_apply
 
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 theorem symm_apply_eq (h : α ≃ᵢ β) {x : α} {y : β} : h.symm y = x ↔ y = h x :=
   h.toEquiv.symm_apply_eq
 #align isometry_equiv.symm_apply_eq IsometryEquiv.symm_apply_eq
 
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 theorem eq_symm_apply (h : α ≃ᵢ β) {x : α} {y : β} : x = h.symm y ↔ h x = y :=
   h.toEquiv.eq_symm_apply
 #align isometry_equiv.eq_symm_apply IsometryEquiv.eq_symm_apply
 
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 theorem symm_comp_self (h : α ≃ᵢ β) : ⇑h.symm ∘ ⇑h = id :=
   funext fun a => h.toEquiv.left_inv a
 #align isometry_equiv.symm_comp_self IsometryEquiv.symm_comp_self
 
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 theorem self_comp_symm (h : α ≃ᵢ β) : ⇑h ∘ ⇑h.symm = id :=
   funext fun a => h.toEquiv.right_inv a
 #align isometry_equiv.self_comp_symm IsometryEquiv.self_comp_symm
 
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 @[simp]
 theorem range_eq_univ (h : α ≃ᵢ β) : range h = univ :=
   h.toEquiv.range_eq_univ
 #align isometry_equiv.range_eq_univ IsometryEquiv.range_eq_univ
 
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 theorem image_symm (h : α ≃ᵢ β) : image h.symm = preimage h :=
   image_eq_preimage_of_inverse h.symm.toEquiv.left_inv h.symm.toEquiv.right_inv
 #align isometry_equiv.image_symm IsometryEquiv.image_symm
 
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 theorem preimage_symm (h : α ≃ᵢ β) : preimage h.symm = image h :=
   (image_eq_preimage_of_inverse h.toEquiv.left_inv h.toEquiv.right_inv).symm
 #align isometry_equiv.preimage_symm IsometryEquiv.preimage_symm
 
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 @[simp]
 theorem symm_trans_apply (h₁ : α ≃ᵢ β) (h₂ : β ≃ᵢ γ) (x : γ) :
     (h₁.trans h₂).symm x = h₁.symm (h₂.symm x) :=
@@ -738,48 +570,24 @@ theorem ediam_preimage (h : α ≃ᵢ β) (s : Set β) : EMetric.diam (h ⁻¹'
 #align isometry_equiv.ediam_preimage IsometryEquiv.ediam_preimage
 -/
 
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 @[simp]
 theorem preimage_emetric_ball (h : α ≃ᵢ β) (x : β) (r : ℝ≥0∞) :
     h ⁻¹' EMetric.ball x r = EMetric.ball (h.symm x) r := by
   rw [← h.isometry.preimage_emetric_ball (h.symm x) r, h.apply_symm_apply]
 #align isometry_equiv.preimage_emetric_ball IsometryEquiv.preimage_emetric_ball
 
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 @[simp]
 theorem preimage_emetric_closedBall (h : α ≃ᵢ β) (x : β) (r : ℝ≥0∞) :
     h ⁻¹' EMetric.closedBall x r = EMetric.closedBall (h.symm x) r := by
   rw [← h.isometry.preimage_emetric_closed_ball (h.symm x) r, h.apply_symm_apply]
 #align isometry_equiv.preimage_emetric_closed_ball IsometryEquiv.preimage_emetric_closedBall
 
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 @[simp]
 theorem image_emetric_ball (h : α ≃ᵢ β) (x : α) (r : ℝ≥0∞) :
     h '' EMetric.ball x r = EMetric.ball (h x) r := by
   rw [← h.preimage_symm, h.symm.preimage_emetric_ball, symm_symm]
 #align isometry_equiv.image_emetric_ball IsometryEquiv.image_emetric_ball
 
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 @[simp]
 theorem image_emetric_closedBall (h : α ≃ᵢ β) (x : α) (r : ℝ≥0∞) :
     h '' EMetric.closedBall x r = EMetric.closedBall (h x) r := by
@@ -797,58 +605,28 @@ protected def toHomeomorph (h : α ≃ᵢ β) : α ≃ₜ β
 #align isometry_equiv.to_homeomorph IsometryEquiv.toHomeomorph
 -/
 
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 @[simp]
 theorem coe_toHomeomorph (h : α ≃ᵢ β) : ⇑h.toHomeomorph = h :=
   rfl
 #align isometry_equiv.coe_to_homeomorph IsometryEquiv.coe_toHomeomorph
 
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 @[simp]
 theorem coe_toHomeomorph_symm (h : α ≃ᵢ β) : ⇑h.toHomeomorph.symm = h.symm :=
   rfl
 #align isometry_equiv.coe_to_homeomorph_symm IsometryEquiv.coe_toHomeomorph_symm
 
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 @[simp]
 theorem comp_continuousOn_iff {γ} [TopologicalSpace γ] (h : α ≃ᵢ β) {f : γ → α} {s : Set γ} :
     ContinuousOn (h ∘ f) s ↔ ContinuousOn f s :=
   h.toHomeomorph.comp_continuousOn_iff _ _
 #align isometry_equiv.comp_continuous_on_iff IsometryEquiv.comp_continuousOn_iff
 
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 @[simp]
 theorem comp_continuous_iff {γ} [TopologicalSpace γ] (h : α ≃ᵢ β) {f : γ → α} :
     Continuous (h ∘ f) ↔ Continuous f :=
   h.toHomeomorph.comp_continuous_iff
 #align isometry_equiv.comp_continuous_iff IsometryEquiv.comp_continuous_iff
 
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 @[simp]
 theorem comp_continuous_iff' {γ} [TopologicalSpace γ] (h : α ≃ᵢ β) {f : β → γ} :
     Continuous (f ∘ h) ↔ Continuous f :=
@@ -865,55 +643,25 @@ instance : Group (α ≃ᵢ α) where
   mul_one e := ext fun _ => rfl
   mul_left_inv e := ext e.symm_apply_apply
 
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 @[simp]
 theorem coe_one : ⇑(1 : α ≃ᵢ α) = id :=
   rfl
 #align isometry_equiv.coe_one IsometryEquiv.coe_one
 
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 @[simp]
 theorem coe_mul (e₁ e₂ : α ≃ᵢ α) : ⇑(e₁ * e₂) = e₁ ∘ e₂ :=
   rfl
 #align isometry_equiv.coe_mul IsometryEquiv.coe_mul
 
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 theorem mul_apply (e₁ e₂ : α ≃ᵢ α) (x : α) : (e₁ * e₂) x = e₁ (e₂ x) :=
   rfl
 #align isometry_equiv.mul_apply IsometryEquiv.mul_apply
 
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 @[simp]
 theorem inv_apply_self (e : α ≃ᵢ α) (x : α) : e⁻¹ (e x) = x :=
   e.symm_apply_apply x
 #align isometry_equiv.inv_apply_self IsometryEquiv.inv_apply_self
 
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 @[simp]
 theorem apply_inv_self (e : α ≃ᵢ α) (x : α) : e (e⁻¹ x) = x :=
   e.apply_symm_apply x
@@ -945,12 +693,6 @@ def funUnique [Unique ι] [Fintype ι] : (ι → α) ≃ᵢ α
 #align isometry_equiv.fun_unique IsometryEquiv.funUnique
 -/
 
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 /-- `pi_fin_two_equiv` as an `isometry_equiv`. -/
 @[simps]
 def piFinTwo (α : Fin 2 → Type _) [∀ i, PseudoEMetricSpace (α i)] : (∀ i, α i) ≃ᵢ α 0 × α 1
@@ -985,71 +727,35 @@ theorem diam_univ : Metric.diam (univ : Set α) = Metric.diam (univ : Set β) :=
 #align isometry_equiv.diam_univ IsometryEquiv.diam_univ
 -/
 
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 @[simp]
 theorem preimage_ball (h : α ≃ᵢ β) (x : β) (r : ℝ) :
     h ⁻¹' Metric.ball x r = Metric.ball (h.symm x) r := by
   rw [← h.isometry.preimage_ball (h.symm x) r, h.apply_symm_apply]
 #align isometry_equiv.preimage_ball IsometryEquiv.preimage_ball
 
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 @[simp]
 theorem preimage_sphere (h : α ≃ᵢ β) (x : β) (r : ℝ) :
     h ⁻¹' Metric.sphere x r = Metric.sphere (h.symm x) r := by
   rw [← h.isometry.preimage_sphere (h.symm x) r, h.apply_symm_apply]
 #align isometry_equiv.preimage_sphere IsometryEquiv.preimage_sphere
 
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 @[simp]
 theorem preimage_closedBall (h : α ≃ᵢ β) (x : β) (r : ℝ) :
     h ⁻¹' Metric.closedBall x r = Metric.closedBall (h.symm x) r := by
   rw [← h.isometry.preimage_closed_ball (h.symm x) r, h.apply_symm_apply]
 #align isometry_equiv.preimage_closed_ball IsometryEquiv.preimage_closedBall
 
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 @[simp]
 theorem image_ball (h : α ≃ᵢ β) (x : α) (r : ℝ) : h '' Metric.ball x r = Metric.ball (h x) r := by
   rw [← h.preimage_symm, h.symm.preimage_ball, symm_symm]
 #align isometry_equiv.image_ball IsometryEquiv.image_ball
 
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 @[simp]
 theorem image_sphere (h : α ≃ᵢ β) (x : α) (r : ℝ) :
     h '' Metric.sphere x r = Metric.sphere (h x) r := by
   rw [← h.preimage_symm, h.symm.preimage_sphere, symm_symm]
 #align isometry_equiv.image_sphere IsometryEquiv.image_sphere
 
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 @[simp]
 theorem image_closedBall (h : α ≃ᵢ β) (x : α) (r : ℝ) :
     h '' Metric.closedBall x r = Metric.closedBall (h x) r := by

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

@@ -188,19 +188,13 @@ theorem right_inv {f : α → β} {g : β → α} (h : Isometry f) (hg : RightIn
 
 #print Isometry.preimage_emetric_closedBall /-
 theorem preimage_emetric_closedBall (h : Isometry f) (x : α) (r : ℝ≥0∞) :
-    f ⁻¹' EMetric.closedBall (f x) r = EMetric.closedBall x r :=
-  by
-  ext y
-  simp [h.edist_eq]
+    f ⁻¹' EMetric.closedBall (f x) r = EMetric.closedBall x r := by ext y; simp [h.edist_eq]
 #align isometry.preimage_emetric_closed_ball Isometry.preimage_emetric_closedBall
 -/
 
 #print Isometry.preimage_emetric_ball /-
 theorem preimage_emetric_ball (h : Isometry f) (x : α) (r : ℝ≥0∞) :
-    f ⁻¹' EMetric.ball (f x) r = EMetric.ball x r :=
-  by
-  ext y
-  simp [h.edist_eq]
+    f ⁻¹' EMetric.ball (f x) r = EMetric.ball x r := by ext y; simp [h.edist_eq]
 #align isometry.preimage_emetric_ball Isometry.preimage_emetric_ball
 -/
 
@@ -212,10 +206,8 @@ theorem ediam_image (hf : Isometry f) (s : Set α) : EMetric.diam (f '' s) = EMe
 -/
 
 #print Isometry.ediam_range /-
-theorem ediam_range (hf : Isometry f) : EMetric.diam (range f) = EMetric.diam (univ : Set α) :=
-  by
-  rw [← image_univ]
-  exact hf.ediam_image univ
+theorem ediam_range (hf : Isometry f) : EMetric.diam (range f) = EMetric.diam (univ : Set α) := by
+  rw [← image_univ]; exact hf.ediam_image univ
 #align isometry.ediam_range Isometry.ediam_range
 -/
 
@@ -312,19 +304,14 @@ theorem diam_image (hf : Isometry f) (s : Set α) : Metric.diam (f '' s) = Metri
 -/
 
 #print Isometry.diam_range /-
-theorem diam_range (hf : Isometry f) : Metric.diam (range f) = Metric.diam (univ : Set α) :=
-  by
-  rw [← image_univ]
-  exact hf.diam_image univ
+theorem diam_range (hf : Isometry f) : Metric.diam (range f) = Metric.diam (univ : Set α) := by
+  rw [← image_univ]; exact hf.diam_image univ
 #align isometry.diam_range Isometry.diam_range
 -/
 
 #print Isometry.preimage_setOf_dist /-
 theorem preimage_setOf_dist (hf : Isometry f) (x : α) (p : ℝ → Prop) :
-    f ⁻¹' { y | p (dist y (f x)) } = { y | p (dist y x) } :=
-  by
-  ext y
-  simp [hf.dist_eq]
+    f ⁻¹' { y | p (dist y (f x)) } = { y | p (dist y x) } := by ext y; simp [hf.dist_eq]
 #align isometry.preimage_set_of_dist Isometry.preimage_setOf_dist
 -/
 
@@ -546,9 +533,7 @@ theorem ediam_image (h : α ≃ᵢ β) (s : Set α) : EMetric.diam (h '' s) = EM
 
 #print IsometryEquiv.toEquiv_injective /-
 theorem toEquiv_injective : ∀ ⦃h₁ h₂ : α ≃ᵢ β⦄, h₁.toEquiv = h₂.toEquiv → h₁ = h₂
-  | ⟨e₁, h₁⟩, ⟨e₂, h₂⟩, H => by
-    dsimp at H
-    subst e₁
+  | ⟨e₁, h₁⟩, ⟨e₂, h₂⟩, H => by dsimp at H; subst e₁
 #align isometry_equiv.to_equiv_inj IsometryEquiv.toEquiv_injective
 -/
 
@@ -943,10 +928,8 @@ protected theorem completeSpace [CompleteSpace β] (e : α ≃ᵢ β) : Complete
 -/
 
 #print IsometryEquiv.completeSpace_iff /-
-theorem completeSpace_iff (e : α ≃ᵢ β) : CompleteSpace α ↔ CompleteSpace β :=
-  by
-  constructor <;> intro H
-  exacts[e.symm.complete_space, e.complete_space]
+theorem completeSpace_iff (e : α ≃ᵢ β) : CompleteSpace α ↔ CompleteSpace β := by
+  constructor <;> intro H; exacts[e.symm.complete_space, e.complete_space]
 #align isometry_equiv.complete_space_iff IsometryEquiv.completeSpace_iff
 -/
 

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

@@ -442,7 +442,7 @@ instance : CoeFun (α ≃ᵢ β) fun _ => α → β :=
 lean 3 declaration is
   forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : PseudoEMetricSpace.{u1} α] [_inst_2 : PseudoEMetricSpace.{u2} β] (h : IsometryEquiv.{u1, u2} α β _inst_1 _inst_2) (a : α), Eq.{succ u2} β (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (IsometryEquiv.{u1, u2} α β _inst_1 _inst_2) (fun (_x : IsometryEquiv.{u1, u2} α β _inst_1 _inst_2) => α -> β) (IsometryEquiv.hasCoeToFun.{u1, u2} α β _inst_1 _inst_2) h a) (coeFn.{max 1 (max (succ u1) (succ u2)) (succ u2) (succ u1), max (succ u1) (succ u2)} (Equiv.{succ u1, succ u2} α β) (fun (_x : Equiv.{succ u1, succ u2} α β) => α -> β) (Equiv.hasCoeToFun.{succ u1, succ u2} α β) (IsometryEquiv.toEquiv.{u1, u2} α β _inst_1 _inst_2 h) a)
 but is expected to have type
-  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : PseudoEMetricSpace.{u1} α] [_inst_2 : PseudoEMetricSpace.{u2} β] (h : IsometryEquiv.{u1, u2} α β _inst_1 _inst_2) (a : α), Eq.{succ u2} ((fun (x._@.Mathlib.Data.FunLike.Embedding._hyg.19 : α) => β) a) (FunLike.coe.{max (succ u1) (succ u2), succ u1, succ u2} (IsometryEquiv.{u1, u2} α β _inst_1 _inst_2) α (fun (_x : α) => (fun (x._@.Mathlib.Data.FunLike.Embedding._hyg.19 : α) => β) _x) (EmbeddingLike.toFunLike.{max (succ u1) (succ u2), succ u1, succ u2} (IsometryEquiv.{u1, u2} α β _inst_1 _inst_2) α β (EquivLike.toEmbeddingLike.{max (succ u1) (succ u2), succ u1, succ u2} (IsometryEquiv.{u1, u2} α β _inst_1 _inst_2) α β (IsometryEquiv.instEquivLikeIsometryEquiv.{u1, u2} α β _inst_1 _inst_2))) h a) (FunLike.coe.{max (succ u1) (succ u2), succ u1, succ u2} (Equiv.{succ u1, succ u2} α β) α (fun (_x : α) => (fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : α) => β) _x) (Equiv.instFunLikeEquiv.{succ u1, succ u2} α β) (IsometryEquiv.toEquiv.{u1, u2} α β _inst_1 _inst_2 h) a)
+  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : PseudoEMetricSpace.{u1} α] [_inst_2 : PseudoEMetricSpace.{u2} β] (h : IsometryEquiv.{u1, u2} α β _inst_1 _inst_2) (a : α), Eq.{succ u2} ((fun (x._@.Mathlib.Data.FunLike.Embedding._hyg.19 : α) => β) a) (FunLike.coe.{max (succ u1) (succ u2), succ u1, succ u2} (IsometryEquiv.{u1, u2} α β _inst_1 _inst_2) α (fun (_x : α) => (fun (x._@.Mathlib.Data.FunLike.Embedding._hyg.19 : α) => β) _x) (EmbeddingLike.toFunLike.{max (succ u1) (succ u2), succ u1, succ u2} (IsometryEquiv.{u1, u2} α β _inst_1 _inst_2) α β (EquivLike.toEmbeddingLike.{max (succ u1) (succ u2), succ u1, succ u2} (IsometryEquiv.{u1, u2} α β _inst_1 _inst_2) α β (IsometryEquiv.instEquivLikeIsometryEquiv.{u1, u2} α β _inst_1 _inst_2))) h a) (FunLike.coe.{max (succ u1) (succ u2), succ u1, succ u2} (Equiv.{succ u1, succ u2} α β) α (fun (_x : α) => (fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.812 : α) => β) _x) (Equiv.instFunLikeEquiv.{succ u1, succ u2} α β) (IsometryEquiv.toEquiv.{u1, u2} α β _inst_1 _inst_2 h) a)
 Case conversion may be inaccurate. Consider using '#align isometry_equiv.coe_eq_to_equiv IsometryEquiv.coe_eq_toEquivₓ'. -/
 theorem coe_eq_toEquiv (h : α ≃ᵢ β) (a : α) : h a = h.toEquiv a :=
   rfl

mathlib commit https://github.com/leanprover-community/mathlib/commit/c89fe2d59ae06402c3f55f978016d1ada444f57e

@@ -952,6 +952,7 @@ theorem completeSpace_iff (e : α ≃ᵢ β) : CompleteSpace α ↔ CompleteSpac
 
 variable (ι α)
 
+#print IsometryEquiv.funUnique /-
 /-- `equiv.fun_unique` as an `isometry_equiv`. -/
 @[simps]
 def funUnique [Unique ι] [Fintype ι] : (ι → α) ≃ᵢ α
@@ -959,7 +960,14 @@ def funUnique [Unique ι] [Fintype ι] : (ι → α) ≃ᵢ α
   toEquiv := Equiv.funUnique ι α
   isometry_toFun x hx := by simp [edist_pi_def, Finset.univ_unique, Finset.sup_singleton]
 #align isometry_equiv.fun_unique IsometryEquiv.funUnique
+-/
 
+/- warning: isometry_equiv.pi_fin_two -> IsometryEquiv.piFinTwo is a dubious translation:
+lean 3 declaration is
+  forall (α : (Fin (OfNat.ofNat.{0} Nat 2 (OfNat.mk.{0} Nat 2 (bit0.{0} Nat Nat.hasAdd (One.one.{0} Nat Nat.hasOne))))) -> Type.{u1}) [_inst_4 : forall (i : Fin (OfNat.ofNat.{0} Nat 2 (OfNat.mk.{0} Nat 2 (bit0.{0} Nat Nat.hasAdd (One.one.{0} Nat Nat.hasOne))))), PseudoEMetricSpace.{u1} (α i)], IsometryEquiv.{u1, u1} (forall (i : Fin (OfNat.ofNat.{0} Nat 2 (OfNat.mk.{0} Nat 2 (bit0.{0} Nat Nat.hasAdd (One.one.{0} Nat Nat.hasOne))))), α i) (Prod.{u1, u1} (α (OfNat.ofNat.{0} (Fin (bit0.{0} Nat Nat.hasAdd (One.one.{0} Nat Nat.hasOne))) 0 (OfNat.mk.{0} (Fin (bit0.{0} Nat Nat.hasAdd (One.one.{0} Nat Nat.hasOne))) 0 (Zero.zero.{0} (Fin (bit0.{0} Nat Nat.hasAdd (One.one.{0} Nat Nat.hasOne))) (Fin.hasZeroOfNeZero (bit0.{0} Nat Nat.hasAdd (One.one.{0} Nat Nat.hasOne)) IsometryEquiv.piFinTwo._proof_1))))) (α (OfNat.ofNat.{0} (Fin (bit0.{0} Nat Nat.hasAdd (One.one.{0} Nat Nat.hasOne))) 1 (OfNat.mk.{0} (Fin (bit0.{0} Nat Nat.hasAdd (One.one.{0} Nat Nat.hasOne))) 1 (One.one.{0} (Fin (bit0.{0} Nat Nat.hasAdd (One.one.{0} Nat Nat.hasOne))) (Fin.hasOneOfNeZero (bit0.{0} Nat Nat.hasAdd (One.one.{0} Nat Nat.hasOne)) IsometryEquiv.piFinTwo._proof_2)))))) (pseudoEMetricSpacePi.{0, u1} (Fin (OfNat.ofNat.{0} Nat 2 (OfNat.mk.{0} Nat 2 (bit0.{0} Nat Nat.hasAdd (One.one.{0} Nat Nat.hasOne))))) (fun (i : Fin (OfNat.ofNat.{0} Nat 2 (OfNat.mk.{0} Nat 2 (bit0.{0} Nat Nat.hasAdd (One.one.{0} Nat Nat.hasOne))))) => α i) (Fin.fintype (OfNat.ofNat.{0} Nat 2 (OfNat.mk.{0} Nat 2 (bit0.{0} Nat Nat.hasAdd (One.one.{0} Nat Nat.hasOne))))) (fun (b : Fin (OfNat.ofNat.{0} Nat 2 (OfNat.mk.{0} Nat 2 (bit0.{0} Nat Nat.hasAdd (One.one.{0} Nat Nat.hasOne))))) => _inst_4 b)) (Prod.pseudoEMetricSpaceMax.{u1, u1} (α (OfNat.ofNat.{0} (Fin (bit0.{0} Nat Nat.hasAdd (One.one.{0} Nat Nat.hasOne))) 0 (OfNat.mk.{0} (Fin (bit0.{0} Nat Nat.hasAdd (One.one.{0} Nat Nat.hasOne))) 0 (Zero.zero.{0} (Fin (bit0.{0} Nat Nat.hasAdd (One.one.{0} Nat Nat.hasOne))) (Fin.hasZeroOfNeZero (bit0.{0} Nat Nat.hasAdd (One.one.{0} Nat Nat.hasOne)) IsometryEquiv.piFinTwo._proof_1))))) (α (OfNat.ofNat.{0} (Fin (bit0.{0} Nat Nat.hasAdd (One.one.{0} Nat Nat.hasOne))) 1 (OfNat.mk.{0} (Fin (bit0.{0} Nat Nat.hasAdd (One.one.{0} Nat Nat.hasOne))) 1 (One.one.{0} (Fin (bit0.{0} Nat Nat.hasAdd (One.one.{0} Nat Nat.hasOne))) (Fin.hasOneOfNeZero (bit0.{0} Nat Nat.hasAdd (One.one.{0} Nat Nat.hasOne)) IsometryEquiv.piFinTwo._proof_2))))) (_inst_4 (OfNat.ofNat.{0} (Fin (bit0.{0} Nat Nat.hasAdd (One.one.{0} Nat Nat.hasOne))) 0 (OfNat.mk.{0} (Fin (bit0.{0} Nat Nat.hasAdd (One.one.{0} Nat Nat.hasOne))) 0 (Zero.zero.{0} (Fin (bit0.{0} Nat Nat.hasAdd (One.one.{0} Nat Nat.hasOne))) (Fin.hasZeroOfNeZero (bit0.{0} Nat Nat.hasAdd (One.one.{0} Nat Nat.hasOne)) IsometryEquiv.piFinTwo._proof_1))))) (_inst_4 (OfNat.ofNat.{0} (Fin (bit0.{0} Nat Nat.hasAdd (One.one.{0} Nat Nat.hasOne))) 1 (OfNat.mk.{0} (Fin (bit0.{0} Nat Nat.hasAdd (One.one.{0} Nat Nat.hasOne))) 1 (One.one.{0} (Fin (bit0.{0} Nat Nat.hasAdd (One.one.{0} Nat Nat.hasOne))) (Fin.hasOneOfNeZero (bit0.{0} Nat Nat.hasAdd (One.one.{0} Nat Nat.hasOne)) IsometryEquiv.piFinTwo._proof_2))))))
+but is expected to have type
+  forall (α : (Fin (OfNat.ofNat.{0} Nat 2 (instOfNatNat 2))) -> Type.{u1}) [_inst_4 : forall (i : Fin (OfNat.ofNat.{0} Nat 2 (instOfNatNat 2))), PseudoEMetricSpace.{u1} (α i)], IsometryEquiv.{u1, u1} (forall (i : Fin (OfNat.ofNat.{0} Nat 2 (instOfNatNat 2))), α i) (Prod.{u1, u1} (α (OfNat.ofNat.{0} (Fin (OfNat.ofNat.{0} Nat 2 (instOfNatNat 2))) 0 (Fin.instOfNatFin (OfNat.ofNat.{0} Nat 2 (instOfNatNat 2)) 0 (NeZero.succ (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))))) (α (OfNat.ofNat.{0} (Fin (OfNat.ofNat.{0} Nat 2 (instOfNatNat 2))) 1 (Fin.instOfNatFin (OfNat.ofNat.{0} Nat 2 (instOfNatNat 2)) 1 (NeZero.succ (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))))))) (pseudoEMetricSpacePi.{0, u1} (Fin (OfNat.ofNat.{0} Nat 2 (instOfNatNat 2))) (fun (i : Fin (OfNat.ofNat.{0} Nat 2 (instOfNatNat 2))) => α i) (Fin.fintype (OfNat.ofNat.{0} Nat 2 (instOfNatNat 2))) (fun (b : Fin (OfNat.ofNat.{0} Nat 2 (instOfNatNat 2))) => _inst_4 b)) (Prod.pseudoEMetricSpaceMax.{u1, u1} (α (OfNat.ofNat.{0} (Fin (OfNat.ofNat.{0} Nat 2 (instOfNatNat 2))) 0 (Fin.instOfNatFin (OfNat.ofNat.{0} Nat 2 (instOfNatNat 2)) 0 (NeZero.succ (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))))) (α (OfNat.ofNat.{0} (Fin (OfNat.ofNat.{0} Nat 2 (instOfNatNat 2))) 1 (Fin.instOfNatFin (OfNat.ofNat.{0} Nat 2 (instOfNatNat 2)) 1 (NeZero.succ (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))))) (_inst_4 (OfNat.ofNat.{0} (Fin (OfNat.ofNat.{0} Nat 2 (instOfNatNat 2))) 0 (Fin.instOfNatFin (OfNat.ofNat.{0} Nat 2 (instOfNatNat 2)) 0 (NeZero.succ (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))))) (_inst_4 (OfNat.ofNat.{0} (Fin (OfNat.ofNat.{0} Nat 2 (instOfNatNat 2))) 1 (Fin.instOfNatFin (OfNat.ofNat.{0} Nat 2 (instOfNatNat 2)) 1 (NeZero.succ (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))))))
+Case conversion may be inaccurate. Consider using '#align isometry_equiv.pi_fin_two IsometryEquiv.piFinTwoₓ'. -/
 /-- `pi_fin_two_equiv` as an `isometry_equiv`. -/
 @[simps]
 def piFinTwo (α : Fin 2 → Type _) [∀ i, PseudoEMetricSpace (α i)] : (∀ i, α i) ≃ᵢ α 0 × α 1

mathlib commit https://github.com/leanprover-community/mathlib/commit/c89fe2d59ae06402c3f55f978016d1ada444f57e

@@ -5,7 +5,7 @@ Isometries of emetric and metric spaces
 Authors: Sébastien Gouëzel
 
 ! This file was ported from Lean 3 source module topology.metric_space.isometry
-! leanprover-community/mathlib commit 69c6a5a12d8a2b159f20933e60115a4f2de62b58
+! leanprover-community/mathlib commit b1859b6d4636fdbb78c5d5cefd24530653cfd3eb
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
@@ -30,7 +30,7 @@ noncomputable section
 
 universe u v w
 
-variable {α : Type u} {β : Type v} {γ : Type w}
+variable {ι : Type _} {α : Type u} {β : Type v} {γ : Type w}
 
 open Function Set
 
@@ -950,6 +950,24 @@ theorem completeSpace_iff (e : α ≃ᵢ β) : CompleteSpace α ↔ CompleteSpac
 #align isometry_equiv.complete_space_iff IsometryEquiv.completeSpace_iff
 -/
 
+variable (ι α)
+
+/-- `equiv.fun_unique` as an `isometry_equiv`. -/
+@[simps]
+def funUnique [Unique ι] [Fintype ι] : (ι → α) ≃ᵢ α
+    where
+  toEquiv := Equiv.funUnique ι α
+  isometry_toFun x hx := by simp [edist_pi_def, Finset.univ_unique, Finset.sup_singleton]
+#align isometry_equiv.fun_unique IsometryEquiv.funUnique
+
+/-- `pi_fin_two_equiv` as an `isometry_equiv`. -/
+@[simps]
+def piFinTwo (α : Fin 2 → Type _) [∀ i, PseudoEMetricSpace (α i)] : (∀ i, α i) ≃ᵢ α 0 × α 1
+    where
+  toEquiv := piFinTwoEquiv α
+  isometry_toFun x hx := by simp [edist_pi_def, Fin.univ_succ, Prod.edist_eq]
+#align isometry_equiv.pi_fin_two IsometryEquiv.piFinTwo
+
 end PseudoEMetricSpace
 
 section PseudoMetricSpace

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

@@ -816,7 +816,7 @@ protected def toHomeomorph (h : α ≃ᵢ β) : α ≃ₜ β
 lean 3 declaration is
   forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : PseudoEMetricSpace.{u1} α] [_inst_2 : PseudoEMetricSpace.{u2} β] (h : IsometryEquiv.{u1, u2} α β _inst_1 _inst_2), Eq.{max (succ u1) (succ u2)} (α -> β) (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (Homeomorph.{u1, u2} α β (UniformSpace.toTopologicalSpace.{u1} α (PseudoEMetricSpace.toUniformSpace.{u1} α _inst_1)) (UniformSpace.toTopologicalSpace.{u2} β (PseudoEMetricSpace.toUniformSpace.{u2} β _inst_2))) (fun (_x : Homeomorph.{u1, u2} α β (UniformSpace.toTopologicalSpace.{u1} α (PseudoEMetricSpace.toUniformSpace.{u1} α _inst_1)) (UniformSpace.toTopologicalSpace.{u2} β (PseudoEMetricSpace.toUniformSpace.{u2} β _inst_2))) => α -> β) (Homeomorph.hasCoeToFun.{u1, u2} α β (UniformSpace.toTopologicalSpace.{u1} α (PseudoEMetricSpace.toUniformSpace.{u1} α _inst_1)) (UniformSpace.toTopologicalSpace.{u2} β (PseudoEMetricSpace.toUniformSpace.{u2} β _inst_2))) (IsometryEquiv.toHomeomorph.{u1, u2} α β _inst_1 _inst_2 h)) (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (IsometryEquiv.{u1, u2} α β _inst_1 _inst_2) (fun (_x : IsometryEquiv.{u1, u2} α β _inst_1 _inst_2) => α -> β) (IsometryEquiv.hasCoeToFun.{u1, u2} α β _inst_1 _inst_2) h)
 but is expected to have type
-  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : PseudoEMetricSpace.{u1} α] [_inst_2 : PseudoEMetricSpace.{u2} β] (h : IsometryEquiv.{u1, u2} α β _inst_1 _inst_2), Eq.{max (succ u1) (succ u2)} (forall (ᾰ : α), (fun (x._@.Mathlib.Data.FunLike.Embedding._hyg.19 : α) => β) ᾰ) (FunLike.coe.{max (succ u1) (succ u2), succ u1, succ u2} (Homeomorph.{u1, u2} α β (UniformSpace.toTopologicalSpace.{u1} α (PseudoEMetricSpace.toUniformSpace.{u1} α _inst_1)) (UniformSpace.toTopologicalSpace.{u2} β (PseudoEMetricSpace.toUniformSpace.{u2} β _inst_2))) α (fun (_x : α) => (fun (x._@.Mathlib.Data.FunLike.Embedding._hyg.19 : α) => β) _x) (EmbeddingLike.toFunLike.{max (succ u1) (succ u2), succ u1, succ u2} (Homeomorph.{u1, u2} α β (UniformSpace.toTopologicalSpace.{u1} α (PseudoEMetricSpace.toUniformSpace.{u1} α _inst_1)) (UniformSpace.toTopologicalSpace.{u2} β (PseudoEMetricSpace.toUniformSpace.{u2} β _inst_2))) α β (EquivLike.toEmbeddingLike.{max (succ u1) (succ u2), succ u1, succ u2} (Homeomorph.{u1, u2} α β (UniformSpace.toTopologicalSpace.{u1} α (PseudoEMetricSpace.toUniformSpace.{u1} α _inst_1)) (UniformSpace.toTopologicalSpace.{u2} β (PseudoEMetricSpace.toUniformSpace.{u2} β _inst_2))) α β (Homeomorph.instEquivLikeHomeomorph.{u1, u2} α β (UniformSpace.toTopologicalSpace.{u1} α (PseudoEMetricSpace.toUniformSpace.{u1} α _inst_1)) (UniformSpace.toTopologicalSpace.{u2} β (PseudoEMetricSpace.toUniformSpace.{u2} β _inst_2))))) (IsometryEquiv.toHomeomorph.{u1, u2} α β _inst_1 _inst_2 h)) (FunLike.coe.{max (succ u1) (succ u2), succ u1, succ u2} (IsometryEquiv.{u1, u2} α β _inst_1 _inst_2) α (fun (_x : α) => (fun (x._@.Mathlib.Data.FunLike.Embedding._hyg.19 : α) => β) _x) (EmbeddingLike.toFunLike.{max (succ u1) (succ u2), succ u1, succ u2} (IsometryEquiv.{u1, u2} α β _inst_1 _inst_2) α β (EquivLike.toEmbeddingLike.{max (succ u1) (succ u2), succ u1, succ u2} (IsometryEquiv.{u1, u2} α β _inst_1 _inst_2) α β (IsometryEquiv.instEquivLikeIsometryEquiv.{u1, u2} α β _inst_1 _inst_2))) h)
+  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : PseudoEMetricSpace.{u1} α] [_inst_2 : PseudoEMetricSpace.{u2} β] (h : IsometryEquiv.{u1, u2} α β _inst_1 _inst_2), Eq.{max (succ u1) (succ u2)} (α -> β) (FunLike.coe.{max (succ u1) (succ u2), succ u1, succ u2} (Homeomorph.{u1, u2} α β (UniformSpace.toTopologicalSpace.{u1} α (PseudoEMetricSpace.toUniformSpace.{u1} α _inst_1)) (UniformSpace.toTopologicalSpace.{u2} β (PseudoEMetricSpace.toUniformSpace.{u2} β _inst_2))) α (fun (_x : α) => β) (EmbeddingLike.toFunLike.{max (succ u1) (succ u2), succ u1, succ u2} (Homeomorph.{u1, u2} α β (UniformSpace.toTopologicalSpace.{u1} α (PseudoEMetricSpace.toUniformSpace.{u1} α _inst_1)) (UniformSpace.toTopologicalSpace.{u2} β (PseudoEMetricSpace.toUniformSpace.{u2} β _inst_2))) α β (EquivLike.toEmbeddingLike.{max (succ u1) (succ u2), succ u1, succ u2} (Homeomorph.{u1, u2} α β (UniformSpace.toTopologicalSpace.{u1} α (PseudoEMetricSpace.toUniformSpace.{u1} α _inst_1)) (UniformSpace.toTopologicalSpace.{u2} β (PseudoEMetricSpace.toUniformSpace.{u2} β _inst_2))) α β (Homeomorph.instEquivLikeHomeomorph.{u1, u2} α β (UniformSpace.toTopologicalSpace.{u1} α (PseudoEMetricSpace.toUniformSpace.{u1} α _inst_1)) (UniformSpace.toTopologicalSpace.{u2} β (PseudoEMetricSpace.toUniformSpace.{u2} β _inst_2))))) (IsometryEquiv.toHomeomorph.{u1, u2} α β _inst_1 _inst_2 h)) (FunLike.coe.{max (succ u1) (succ u2), succ u1, succ u2} (IsometryEquiv.{u1, u2} α β _inst_1 _inst_2) α (fun (_x : α) => (fun (x._@.Mathlib.Data.FunLike.Embedding._hyg.19 : α) => β) _x) (EmbeddingLike.toFunLike.{max (succ u1) (succ u2), succ u1, succ u2} (IsometryEquiv.{u1, u2} α β _inst_1 _inst_2) α β (EquivLike.toEmbeddingLike.{max (succ u1) (succ u2), succ u1, succ u2} (IsometryEquiv.{u1, u2} α β _inst_1 _inst_2) α β (IsometryEquiv.instEquivLikeIsometryEquiv.{u1, u2} α β _inst_1 _inst_2))) h)
 Case conversion may be inaccurate. Consider using '#align isometry_equiv.coe_to_homeomorph IsometryEquiv.coe_toHomeomorphₓ'. -/
 @[simp]
 theorem coe_toHomeomorph (h : α ≃ᵢ β) : ⇑h.toHomeomorph = h :=
@@ -827,7 +827,7 @@ theorem coe_toHomeomorph (h : α ≃ᵢ β) : ⇑h.toHomeomorph = h :=
 lean 3 declaration is
   forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : PseudoEMetricSpace.{u1} α] [_inst_2 : PseudoEMetricSpace.{u2} β] (h : IsometryEquiv.{u1, u2} α β _inst_1 _inst_2), Eq.{max (succ u2) (succ u1)} (β -> α) (coeFn.{max (succ u2) (succ u1), max (succ u2) (succ u1)} (Homeomorph.{u2, u1} β α (UniformSpace.toTopologicalSpace.{u2} β (PseudoEMetricSpace.toUniformSpace.{u2} β _inst_2)) (UniformSpace.toTopologicalSpace.{u1} α (PseudoEMetricSpace.toUniformSpace.{u1} α _inst_1))) (fun (_x : Homeomorph.{u2, u1} β α (UniformSpace.toTopologicalSpace.{u2} β (PseudoEMetricSpace.toUniformSpace.{u2} β _inst_2)) (UniformSpace.toTopologicalSpace.{u1} α (PseudoEMetricSpace.toUniformSpace.{u1} α _inst_1))) => β -> α) (Homeomorph.hasCoeToFun.{u2, u1} β α (UniformSpace.toTopologicalSpace.{u2} β (PseudoEMetricSpace.toUniformSpace.{u2} β _inst_2)) (UniformSpace.toTopologicalSpace.{u1} α (PseudoEMetricSpace.toUniformSpace.{u1} α _inst_1))) (Homeomorph.symm.{u1, u2} α β (UniformSpace.toTopologicalSpace.{u1} α (PseudoEMetricSpace.toUniformSpace.{u1} α _inst_1)) (UniformSpace.toTopologicalSpace.{u2} β (PseudoEMetricSpace.toUniformSpace.{u2} β _inst_2)) (IsometryEquiv.toHomeomorph.{u1, u2} α β _inst_1 _inst_2 h))) (coeFn.{max (succ u2) (succ u1), max (succ u2) (succ u1)} (IsometryEquiv.{u2, u1} β α _inst_2 _inst_1) (fun (_x : IsometryEquiv.{u2, u1} β α _inst_2 _inst_1) => β -> α) (IsometryEquiv.hasCoeToFun.{u2, u1} β α _inst_2 _inst_1) (IsometryEquiv.symm.{u1, u2} α β _inst_1 _inst_2 h))
 but is expected to have type
-  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : PseudoEMetricSpace.{u1} α] [_inst_2 : PseudoEMetricSpace.{u2} β] (h : IsometryEquiv.{u1, u2} α β _inst_1 _inst_2), Eq.{max (succ u1) (succ u2)} (forall (ᾰ : β), (fun (x._@.Mathlib.Data.FunLike.Embedding._hyg.19 : β) => α) ᾰ) (FunLike.coe.{max (succ u1) (succ u2), succ u2, succ u1} (Homeomorph.{u2, u1} β α (UniformSpace.toTopologicalSpace.{u2} β (PseudoEMetricSpace.toUniformSpace.{u2} β _inst_2)) (UniformSpace.toTopologicalSpace.{u1} α (PseudoEMetricSpace.toUniformSpace.{u1} α _inst_1))) β (fun (_x : β) => (fun (x._@.Mathlib.Data.FunLike.Embedding._hyg.19 : β) => α) _x) (EmbeddingLike.toFunLike.{max (succ u1) (succ u2), succ u2, succ u1} (Homeomorph.{u2, u1} β α (UniformSpace.toTopologicalSpace.{u2} β (PseudoEMetricSpace.toUniformSpace.{u2} β _inst_2)) (UniformSpace.toTopologicalSpace.{u1} α (PseudoEMetricSpace.toUniformSpace.{u1} α _inst_1))) β α (EquivLike.toEmbeddingLike.{max (succ u1) (succ u2), succ u2, succ u1} (Homeomorph.{u2, u1} β α (UniformSpace.toTopologicalSpace.{u2} β (PseudoEMetricSpace.toUniformSpace.{u2} β _inst_2)) (UniformSpace.toTopologicalSpace.{u1} α (PseudoEMetricSpace.toUniformSpace.{u1} α _inst_1))) β α (Homeomorph.instEquivLikeHomeomorph.{u2, u1} β α (UniformSpace.toTopologicalSpace.{u2} β (PseudoEMetricSpace.toUniformSpace.{u2} β _inst_2)) (UniformSpace.toTopologicalSpace.{u1} α (PseudoEMetricSpace.toUniformSpace.{u1} α _inst_1))))) (Homeomorph.symm.{u1, u2} α β (UniformSpace.toTopologicalSpace.{u1} α (PseudoEMetricSpace.toUniformSpace.{u1} α _inst_1)) (UniformSpace.toTopologicalSpace.{u2} β (PseudoEMetricSpace.toUniformSpace.{u2} β _inst_2)) (IsometryEquiv.toHomeomorph.{u1, u2} α β _inst_1 _inst_2 h))) (FunLike.coe.{max (succ u1) (succ u2), succ u2, succ u1} (IsometryEquiv.{u2, u1} β α _inst_2 _inst_1) β (fun (_x : β) => (fun (x._@.Mathlib.Data.FunLike.Embedding._hyg.19 : β) => α) _x) (EmbeddingLike.toFunLike.{max (succ u1) (succ u2), succ u2, succ u1} (IsometryEquiv.{u2, u1} β α _inst_2 _inst_1) β α (EquivLike.toEmbeddingLike.{max (succ u1) (succ u2), succ u2, succ u1} (IsometryEquiv.{u2, u1} β α _inst_2 _inst_1) β α (IsometryEquiv.instEquivLikeIsometryEquiv.{u2, u1} β α _inst_2 _inst_1))) (IsometryEquiv.symm.{u1, u2} α β _inst_1 _inst_2 h))
+  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : PseudoEMetricSpace.{u1} α] [_inst_2 : PseudoEMetricSpace.{u2} β] (h : IsometryEquiv.{u1, u2} α β _inst_1 _inst_2), Eq.{max (succ u1) (succ u2)} (β -> α) (FunLike.coe.{max (succ u2) (succ u1), succ u2, succ u1} (Homeomorph.{u2, u1} β α (UniformSpace.toTopologicalSpace.{u2} β (PseudoEMetricSpace.toUniformSpace.{u2} β _inst_2)) (UniformSpace.toTopologicalSpace.{u1} α (PseudoEMetricSpace.toUniformSpace.{u1} α _inst_1))) β (fun (_x : β) => α) (EmbeddingLike.toFunLike.{max (succ u2) (succ u1), succ u2, succ u1} (Homeomorph.{u2, u1} β α (UniformSpace.toTopologicalSpace.{u2} β (PseudoEMetricSpace.toUniformSpace.{u2} β _inst_2)) (UniformSpace.toTopologicalSpace.{u1} α (PseudoEMetricSpace.toUniformSpace.{u1} α _inst_1))) β α (EquivLike.toEmbeddingLike.{max (succ u2) (succ u1), succ u2, succ u1} (Homeomorph.{u2, u1} β α (UniformSpace.toTopologicalSpace.{u2} β (PseudoEMetricSpace.toUniformSpace.{u2} β _inst_2)) (UniformSpace.toTopologicalSpace.{u1} α (PseudoEMetricSpace.toUniformSpace.{u1} α _inst_1))) β α (Homeomorph.instEquivLikeHomeomorph.{u2, u1} β α (UniformSpace.toTopologicalSpace.{u2} β (PseudoEMetricSpace.toUniformSpace.{u2} β _inst_2)) (UniformSpace.toTopologicalSpace.{u1} α (PseudoEMetricSpace.toUniformSpace.{u1} α _inst_1))))) (Homeomorph.symm.{u1, u2} α β (UniformSpace.toTopologicalSpace.{u1} α (PseudoEMetricSpace.toUniformSpace.{u1} α _inst_1)) (UniformSpace.toTopologicalSpace.{u2} β (PseudoEMetricSpace.toUniformSpace.{u2} β _inst_2)) (IsometryEquiv.toHomeomorph.{u1, u2} α β _inst_1 _inst_2 h))) (FunLike.coe.{max (succ u1) (succ u2), succ u2, succ u1} (IsometryEquiv.{u2, u1} β α _inst_2 _inst_1) β (fun (_x : β) => (fun (x._@.Mathlib.Data.FunLike.Embedding._hyg.19 : β) => α) _x) (EmbeddingLike.toFunLike.{max (succ u1) (succ u2), succ u2, succ u1} (IsometryEquiv.{u2, u1} β α _inst_2 _inst_1) β α (EquivLike.toEmbeddingLike.{max (succ u1) (succ u2), succ u2, succ u1} (IsometryEquiv.{u2, u1} β α _inst_2 _inst_1) β α (IsometryEquiv.instEquivLikeIsometryEquiv.{u2, u1} β α _inst_2 _inst_1))) (IsometryEquiv.symm.{u1, u2} α β _inst_1 _inst_2 h))
 Case conversion may be inaccurate. Consider using '#align isometry_equiv.coe_to_homeomorph_symm IsometryEquiv.coe_toHomeomorph_symmₓ'. -/
 @[simp]
 theorem coe_toHomeomorph_symm (h : α ≃ᵢ β) : ⇑h.toHomeomorph.symm = h.symm :=

mathlib commit https://github.com/leanprover-community/mathlib/commit/2196ab363eb097c008d4497125e0dde23fb36db2

@@ -5,7 +5,7 @@ Isometries of emetric and metric spaces
 Authors: Sébastien Gouëzel
 
 ! This file was ported from Lean 3 source module topology.metric_space.isometry
-! leanprover-community/mathlib commit 832a8ba8f10f11fea99367c469ff802e69a5b8ec
+! leanprover-community/mathlib commit 69c6a5a12d8a2b159f20933e60115a4f2de62b58
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
@@ -14,6 +14,9 @@ import Mathbin.Topology.MetricSpace.Antilipschitz
 /-!
 # Isometries
 
+> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.
+> Any changes to this file require a corresponding PR to mathlib4.
+
 We define isometries, i.e., maps between emetric spaces that preserve
 the edistance (on metric spaces, these are exactly the maps that preserve distances),
 and prove their basic properties. We also introduce isometric bijections.

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

@@ -439,7 +439,7 @@ instance : CoeFun (α ≃ᵢ β) fun _ => α → β :=
 lean 3 declaration is
   forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : PseudoEMetricSpace.{u1} α] [_inst_2 : PseudoEMetricSpace.{u2} β] (h : IsometryEquiv.{u1, u2} α β _inst_1 _inst_2) (a : α), Eq.{succ u2} β (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (IsometryEquiv.{u1, u2} α β _inst_1 _inst_2) (fun (_x : IsometryEquiv.{u1, u2} α β _inst_1 _inst_2) => α -> β) (IsometryEquiv.hasCoeToFun.{u1, u2} α β _inst_1 _inst_2) h a) (coeFn.{max 1 (max (succ u1) (succ u2)) (succ u2) (succ u1), max (succ u1) (succ u2)} (Equiv.{succ u1, succ u2} α β) (fun (_x : Equiv.{succ u1, succ u2} α β) => α -> β) (Equiv.hasCoeToFun.{succ u1, succ u2} α β) (IsometryEquiv.toEquiv.{u1, u2} α β _inst_1 _inst_2 h) a)
 but is expected to have type
-  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : PseudoEMetricSpace.{u1} α] [_inst_2 : PseudoEMetricSpace.{u2} β] (h : IsometryEquiv.{u1, u2} α β _inst_1 _inst_2) (a : α), Eq.{succ u2} ((fun (x._@.Mathlib.Data.FunLike.Embedding._hyg.19 : α) => β) a) (FunLike.coe.{max (succ u1) (succ u2), succ u1, succ u2} (IsometryEquiv.{u1, u2} α β _inst_1 _inst_2) α (fun (_x : α) => (fun (x._@.Mathlib.Data.FunLike.Embedding._hyg.19 : α) => β) _x) (EmbeddingLike.toFunLike.{max (succ u1) (succ u2), succ u1, succ u2} (IsometryEquiv.{u1, u2} α β _inst_1 _inst_2) α β (EquivLike.toEmbeddingLike.{max (succ u1) (succ u2), succ u1, succ u2} (IsometryEquiv.{u1, u2} α β _inst_1 _inst_2) α β (IsometryEquiv.instEquivLikeIsometryEquiv.{u1, u2} α β _inst_1 _inst_2))) h a) (FunLike.coe.{max (succ u1) (succ u2), succ u1, succ u2} (Equiv.{succ u1, succ u2} α β) α (fun (_x : α) => (fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.805 : α) => β) _x) (Equiv.instFunLikeEquiv.{succ u1, succ u2} α β) (IsometryEquiv.toEquiv.{u1, u2} α β _inst_1 _inst_2 h) a)
+  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : PseudoEMetricSpace.{u1} α] [_inst_2 : PseudoEMetricSpace.{u2} β] (h : IsometryEquiv.{u1, u2} α β _inst_1 _inst_2) (a : α), Eq.{succ u2} ((fun (x._@.Mathlib.Data.FunLike.Embedding._hyg.19 : α) => β) a) (FunLike.coe.{max (succ u1) (succ u2), succ u1, succ u2} (IsometryEquiv.{u1, u2} α β _inst_1 _inst_2) α (fun (_x : α) => (fun (x._@.Mathlib.Data.FunLike.Embedding._hyg.19 : α) => β) _x) (EmbeddingLike.toFunLike.{max (succ u1) (succ u2), succ u1, succ u2} (IsometryEquiv.{u1, u2} α β _inst_1 _inst_2) α β (EquivLike.toEmbeddingLike.{max (succ u1) (succ u2), succ u1, succ u2} (IsometryEquiv.{u1, u2} α β _inst_1 _inst_2) α β (IsometryEquiv.instEquivLikeIsometryEquiv.{u1, u2} α β _inst_1 _inst_2))) h a) (FunLike.coe.{max (succ u1) (succ u2), succ u1, succ u2} (Equiv.{succ u1, succ u2} α β) α (fun (_x : α) => (fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : α) => β) _x) (Equiv.instFunLikeEquiv.{succ u1, succ u2} α β) (IsometryEquiv.toEquiv.{u1, u2} α β _inst_1 _inst_2 h) a)
 Case conversion may be inaccurate. Consider using '#align isometry_equiv.coe_eq_to_equiv IsometryEquiv.coe_eq_toEquivₓ'. -/
 theorem coe_eq_toEquiv (h : α ≃ᵢ β) (a : α) : h a = h.toEquiv a :=
   rfl

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

@@ -33,24 +33,30 @@ open Function Set
 
 open Topology ENNReal
 
+#print Isometry /-
 /-- An isometry (also known as isometric embedding) is a map preserving the edistance
 between pseudoemetric spaces, or equivalently the distance between pseudometric space.  -/
 def Isometry [PseudoEMetricSpace α] [PseudoEMetricSpace β] (f : α → β) : Prop :=
   ∀ x1 x2 : α, edist (f x1) (f x2) = edist x1 x2
 #align isometry Isometry
+-/
 
+#print isometry_iff_nndist_eq /-
 /-- On pseudometric spaces, a map is an isometry if and only if it preserves nonnegative
 distances. -/
 theorem isometry_iff_nndist_eq [PseudoMetricSpace α] [PseudoMetricSpace β] {f : α → β} :
     Isometry f ↔ ∀ x y, nndist (f x) (f y) = nndist x y := by
   simp only [Isometry, edist_nndist, ENNReal.coe_eq_coe]
 #align isometry_iff_nndist_eq isometry_iff_nndist_eq
+-/
 
+#print isometry_iff_dist_eq /-
 /-- On pseudometric spaces, a map is an isometry if and only if it preserves distances. -/
 theorem isometry_iff_dist_eq [PseudoMetricSpace α] [PseudoMetricSpace β] {f : α → β} :
     Isometry f ↔ ∀ x y, dist (f x) (f y) = dist x y := by
   simp only [isometry_iff_nndist_eq, ← coe_nndist, NNReal.coe_eq]
 #align isometry_iff_dist_eq isometry_iff_dist_eq
+-/
 
 /-- An isometry preserves distances. -/
 alias isometry_iff_dist_eq ↔ Isometry.dist_eq _
@@ -76,113 +82,177 @@ variable [PseudoEMetricSpace α] [PseudoEMetricSpace β] [PseudoEMetricSpace γ]
 
 variable {f : α → β} {x y z : α} {s : Set α}
 
+#print Isometry.edist_eq /-
 /-- An isometry preserves edistances. -/
 theorem edist_eq (hf : Isometry f) (x y : α) : edist (f x) (f y) = edist x y :=
   hf x y
 #align isometry.edist_eq Isometry.edist_eq
+-/
 
+#print Isometry.lipschitz /-
 theorem lipschitz (h : Isometry f) : LipschitzWith 1 f :=
   LipschitzWith.of_edist_le fun x y => (h x y).le
 #align isometry.lipschitz Isometry.lipschitz
+-/
 
+#print Isometry.antilipschitz /-
 theorem antilipschitz (h : Isometry f) : AntilipschitzWith 1 f := fun x y => by
   simp only [h x y, ENNReal.coe_one, one_mul, le_refl]
 #align isometry.antilipschitz Isometry.antilipschitz
+-/
 
+#print isometry_subsingleton /-
 /-- Any map on a subsingleton is an isometry -/
 @[nontriviality]
 theorem isometry_subsingleton [Subsingleton α] : Isometry f := fun x y => by
   rw [Subsingleton.elim x y] <;> simp
 #align isometry_subsingleton isometry_subsingleton
+-/
 
+#print isometry_id /-
 /-- The identity is an isometry -/
 theorem isometry_id : Isometry (id : α → α) := fun x y => rfl
 #align isometry_id isometry_id
+-/
 
+/- warning: isometry.prod_map -> Isometry.prod_map is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} {β : Type.{u2}} {γ : Type.{u3}} [_inst_1 : PseudoEMetricSpace.{u1} α] [_inst_2 : PseudoEMetricSpace.{u2} β] [_inst_3 : PseudoEMetricSpace.{u3} γ] {δ : Type.{u4}} [_inst_4 : PseudoEMetricSpace.{u4} δ] {f : α -> β} {g : γ -> δ}, (Isometry.{u1, u2} α β _inst_1 _inst_2 f) -> (Isometry.{u3, u4} γ δ _inst_3 _inst_4 g) -> (Isometry.{max u1 u3, max u2 u4} (Prod.{u1, u3} α γ) (Prod.{u2, u4} β δ) (Prod.pseudoEMetricSpaceMax.{u1, u3} α γ _inst_1 _inst_3) (Prod.pseudoEMetricSpaceMax.{u2, u4} β δ _inst_2 _inst_4) (Prod.map.{u1, u2, u3, u4} α β γ δ f g))
+but is expected to have type
+  forall {α : Type.{u2}} {β : Type.{u3}} {γ : Type.{u4}} [_inst_1 : PseudoEMetricSpace.{u2} α] [_inst_2 : PseudoEMetricSpace.{u3} β] [_inst_3 : PseudoEMetricSpace.{u4} γ] {δ : Type.{u1}} [_inst_4 : PseudoEMetricSpace.{u1} δ] {f : α -> β} {g : γ -> δ}, (Isometry.{u2, u3} α β _inst_1 _inst_2 f) -> (Isometry.{u4, u1} γ δ _inst_3 _inst_4 g) -> (Isometry.{max u4 u2, max u1 u3} (Prod.{u2, u4} α γ) (Prod.{u3, u1} β δ) (Prod.pseudoEMetricSpaceMax.{u2, u4} α γ _inst_1 _inst_3) (Prod.pseudoEMetricSpaceMax.{u3, u1} β δ _inst_2 _inst_4) (Prod.map.{u2, u3, u4, u1} α β γ δ f g))
+Case conversion may be inaccurate. Consider using '#align isometry.prod_map Isometry.prod_mapₓ'. -/
 theorem prod_map {δ} [PseudoEMetricSpace δ] {f : α → β} {g : γ → δ} (hf : Isometry f)
     (hg : Isometry g) : Isometry (Prod.map f g) := fun x y => by
   simp only [Prod.edist_eq, hf.edist_eq, hg.edist_eq, Prod_map]
 #align isometry.prod_map Isometry.prod_map
 
+/- warning: isometry_dcomp -> isometry_dcomp is a dubious translation:
+lean 3 declaration is
+  forall {ι : Type.{u1}} [_inst_4 : Fintype.{u1} ι] {α : ι -> Type.{u2}} {β : ι -> Type.{u3}} [_inst_5 : forall (i : ι), PseudoEMetricSpace.{u2} (α i)] [_inst_6 : forall (i : ι), PseudoEMetricSpace.{u3} (β i)] (f : forall (i : ι), (α i) -> (β i)), (forall (i : ι), Isometry.{u2, u3} (α i) (β i) (_inst_5 i) (_inst_6 i) (f i)) -> (Isometry.{max u1 u2, max u1 u3} (forall (x : ι), α x) (forall (x : ι), β x) (pseudoEMetricSpacePi.{u1, u2} ι (fun (x : ι) => α x) _inst_4 (fun (b : ι) => _inst_5 b)) (pseudoEMetricSpacePi.{u1, u3} ι (fun (x : ι) => β x) _inst_4 (fun (b : ι) => _inst_6 b)) (Function.dcomp.{succ u1, succ u2, succ u3} ι (fun (i : ι) => α i) (fun (i : ι) (ᾰ : α i) => β i) f))
+but is expected to have type
+  forall {ι : Type.{u3}} [_inst_4 : Fintype.{u3} ι] {α : ι -> Type.{u2}} {β : ι -> Type.{u1}} [_inst_5 : forall (i : ι), PseudoEMetricSpace.{u2} (α i)] [_inst_6 : forall (i : ι), PseudoEMetricSpace.{u1} (β i)] (f : forall (i : ι), (α i) -> (β i)), (forall (i : ι), Isometry.{u2, u1} (α i) (β i) (_inst_5 i) (_inst_6 i) (f i)) -> (Isometry.{max u3 u2, max u3 u1} (forall (x : ι), α x) (forall (x : ι), β x) (pseudoEMetricSpacePi.{u3, u2} ι (fun (x : ι) => α x) _inst_4 (fun (b : ι) => _inst_5 b)) (pseudoEMetricSpacePi.{u3, u1} ι (fun (x : ι) => β x) _inst_4 (fun (b : ι) => _inst_6 b)) (fun (g : forall (i : ι), α i) (i : ι) => f i (g i)))
+Case conversion may be inaccurate. Consider using '#align isometry_dcomp isometry_dcompₓ'. -/
 theorem isometry_dcomp {ι} [Fintype ι] {α β : ι → Type _} [∀ i, PseudoEMetricSpace (α i)]
     [∀ i, PseudoEMetricSpace (β i)] (f : ∀ i, α i → β i) (hf : ∀ i, Isometry (f i)) :
     Isometry (dcomp f) := fun x y => by simp only [edist_pi_def, (hf _).edist_eq]
 #align isometry_dcomp isometry_dcomp
 
+#print Isometry.comp /-
 /-- The composition of isometries is an isometry. -/
 theorem comp {g : β → γ} {f : α → β} (hg : Isometry g) (hf : Isometry f) : Isometry (g ∘ f) :=
   fun x y => (hg _ _).trans (hf _ _)
 #align isometry.comp Isometry.comp
+-/
 
+#print Isometry.uniformContinuous /-
 /-- An isometry from a metric space is a uniform continuous map -/
 protected theorem uniformContinuous (hf : Isometry f) : UniformContinuous f :=
   hf.lipschitz.UniformContinuous
 #align isometry.uniform_continuous Isometry.uniformContinuous
+-/
 
+#print Isometry.uniformInducing /-
 /-- An isometry from a metric space is a uniform inducing map -/
 protected theorem uniformInducing (hf : Isometry f) : UniformInducing f :=
   hf.antilipschitz.UniformInducing hf.UniformContinuous
 #align isometry.uniform_inducing Isometry.uniformInducing
+-/
 
+/- warning: isometry.tendsto_nhds_iff -> Isometry.tendsto_nhds_iff is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : PseudoEMetricSpace.{u1} α] [_inst_2 : PseudoEMetricSpace.{u2} β] {ι : Type.{u3}} {f : α -> β} {g : ι -> α} {a : Filter.{u3} ι} {b : α}, (Isometry.{u1, u2} α β _inst_1 _inst_2 f) -> (Iff (Filter.Tendsto.{u3, u1} ι α g a (nhds.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoEMetricSpace.toUniformSpace.{u1} α _inst_1)) b)) (Filter.Tendsto.{u3, u2} ι β (Function.comp.{succ u3, succ u1, succ u2} ι α β f g) a (nhds.{u2} β (UniformSpace.toTopologicalSpace.{u2} β (PseudoEMetricSpace.toUniformSpace.{u2} β _inst_2)) (f b))))
+but is expected to have type
+  forall {α : Type.{u2}} {β : Type.{u3}} [_inst_1 : PseudoEMetricSpace.{u2} α] [_inst_2 : PseudoEMetricSpace.{u3} β] {ι : Type.{u1}} {f : α -> β} {g : ι -> α} {a : Filter.{u1} ι} {b : α}, (Isometry.{u2, u3} α β _inst_1 _inst_2 f) -> (Iff (Filter.Tendsto.{u1, u2} ι α g a (nhds.{u2} α (UniformSpace.toTopologicalSpace.{u2} α (PseudoEMetricSpace.toUniformSpace.{u2} α _inst_1)) b)) (Filter.Tendsto.{u1, u3} ι β (Function.comp.{succ u1, succ u2, succ u3} ι α β f g) a (nhds.{u3} β (UniformSpace.toTopologicalSpace.{u3} β (PseudoEMetricSpace.toUniformSpace.{u3} β _inst_2)) (f b))))
+Case conversion may be inaccurate. Consider using '#align isometry.tendsto_nhds_iff Isometry.tendsto_nhds_iffₓ'. -/
 theorem tendsto_nhds_iff {ι : Type _} {f : α → β} {g : ι → α} {a : Filter ι} {b : α}
     (hf : Isometry f) : Filter.Tendsto g a (𝓝 b) ↔ Filter.Tendsto (f ∘ g) a (𝓝 (f b)) :=
   hf.UniformInducing.Inducing.tendsto_nhds_iff
 #align isometry.tendsto_nhds_iff Isometry.tendsto_nhds_iff
 
+#print Isometry.continuous /-
 /-- An isometry is continuous. -/
 protected theorem continuous (hf : Isometry f) : Continuous f :=
   hf.lipschitz.Continuous
 #align isometry.continuous Isometry.continuous
+-/
 
+#print Isometry.right_inv /-
 /-- The right inverse of an isometry is an isometry. -/
 theorem right_inv {f : α → β} {g : β → α} (h : Isometry f) (hg : RightInverse g f) : Isometry g :=
   fun x y => by rw [← h, hg _, hg _]
 #align isometry.right_inv Isometry.right_inv
+-/
 
+#print Isometry.preimage_emetric_closedBall /-
 theorem preimage_emetric_closedBall (h : Isometry f) (x : α) (r : ℝ≥0∞) :
     f ⁻¹' EMetric.closedBall (f x) r = EMetric.closedBall x r :=
   by
   ext y
   simp [h.edist_eq]
 #align isometry.preimage_emetric_closed_ball Isometry.preimage_emetric_closedBall
+-/
 
+#print Isometry.preimage_emetric_ball /-
 theorem preimage_emetric_ball (h : Isometry f) (x : α) (r : ℝ≥0∞) :
     f ⁻¹' EMetric.ball (f x) r = EMetric.ball x r :=
   by
   ext y
   simp [h.edist_eq]
 #align isometry.preimage_emetric_ball Isometry.preimage_emetric_ball
+-/
 
+#print Isometry.ediam_image /-
 /-- Isometries preserve the diameter in pseudoemetric spaces. -/
 theorem ediam_image (hf : Isometry f) (s : Set α) : EMetric.diam (f '' s) = EMetric.diam s :=
   eq_of_forall_ge_iff fun d => by simp only [EMetric.diam_le_iff, ball_image_iff, hf.edist_eq]
 #align isometry.ediam_image Isometry.ediam_image
+-/
 
+#print Isometry.ediam_range /-
 theorem ediam_range (hf : Isometry f) : EMetric.diam (range f) = EMetric.diam (univ : Set α) :=
   by
   rw [← image_univ]
   exact hf.ediam_image univ
 #align isometry.ediam_range Isometry.ediam_range
+-/
 
+#print Isometry.mapsTo_emetric_ball /-
 theorem mapsTo_emetric_ball (hf : Isometry f) (x : α) (r : ℝ≥0∞) :
     MapsTo f (EMetric.ball x r) (EMetric.ball (f x) r) :=
   (hf.preimage_emetric_ball x r).ge
 #align isometry.maps_to_emetric_ball Isometry.mapsTo_emetric_ball
+-/
 
+#print Isometry.mapsTo_emetric_closedBall /-
 theorem mapsTo_emetric_closedBall (hf : Isometry f) (x : α) (r : ℝ≥0∞) :
     MapsTo f (EMetric.closedBall x r) (EMetric.closedBall (f x) r) :=
   (hf.preimage_emetric_closedBall x r).ge
 #align isometry.maps_to_emetric_closed_ball Isometry.mapsTo_emetric_closedBall
+-/
 
+#print isometry_subtype_coe /-
 /-- The injection from a subtype is an isometry -/
 theorem isometry_subtype_coe {s : Set α} : Isometry (coe : s → α) := fun x y => rfl
 #align isometry_subtype_coe isometry_subtype_coe
+-/
 
+/- warning: isometry.comp_continuous_on_iff -> Isometry.comp_continuousOn_iff is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : PseudoEMetricSpace.{u1} α] [_inst_2 : PseudoEMetricSpace.{u2} β] {f : α -> β} {γ : Type.{u3}} [_inst_4 : TopologicalSpace.{u3} γ], (Isometry.{u1, u2} α β _inst_1 _inst_2 f) -> (forall {g : γ -> α} {s : Set.{u3} γ}, Iff (ContinuousOn.{u3, u2} γ β _inst_4 (UniformSpace.toTopologicalSpace.{u2} β (PseudoEMetricSpace.toUniformSpace.{u2} β _inst_2)) (Function.comp.{succ u3, succ u1, succ u2} γ α β f g) s) (ContinuousOn.{u3, u1} γ α _inst_4 (UniformSpace.toTopologicalSpace.{u1} α (PseudoEMetricSpace.toUniformSpace.{u1} α _inst_1)) g s))
+but is expected to have type
+  forall {α : Type.{u2}} {β : Type.{u3}} [_inst_1 : PseudoEMetricSpace.{u2} α] [_inst_2 : PseudoEMetricSpace.{u3} β] {f : α -> β} {γ : Type.{u1}} [_inst_4 : TopologicalSpace.{u1} γ], (Isometry.{u2, u3} α β _inst_1 _inst_2 f) -> (forall {g : γ -> α} {s : Set.{u1} γ}, Iff (ContinuousOn.{u1, u3} γ β _inst_4 (UniformSpace.toTopologicalSpace.{u3} β (PseudoEMetricSpace.toUniformSpace.{u3} β _inst_2)) (Function.comp.{succ u1, succ u2, succ u3} γ α β f g) s) (ContinuousOn.{u1, u2} γ α _inst_4 (UniformSpace.toTopologicalSpace.{u2} α (PseudoEMetricSpace.toUniformSpace.{u2} α _inst_1)) g s))
+Case conversion may be inaccurate. Consider using '#align isometry.comp_continuous_on_iff Isometry.comp_continuousOn_iffₓ'. -/
 theorem comp_continuousOn_iff {γ} [TopologicalSpace γ] (hf : Isometry f) {g : γ → α} {s : Set γ} :
     ContinuousOn (f ∘ g) s ↔ ContinuousOn g s :=
   hf.UniformInducing.Inducing.continuousOn_iff.symm
 #align isometry.comp_continuous_on_iff Isometry.comp_continuousOn_iff
 
+/- warning: isometry.comp_continuous_iff -> Isometry.comp_continuous_iff is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : PseudoEMetricSpace.{u1} α] [_inst_2 : PseudoEMetricSpace.{u2} β] {f : α -> β} {γ : Type.{u3}} [_inst_4 : TopologicalSpace.{u3} γ], (Isometry.{u1, u2} α β _inst_1 _inst_2 f) -> (forall {g : γ -> α}, Iff (Continuous.{u3, u2} γ β _inst_4 (UniformSpace.toTopologicalSpace.{u2} β (PseudoEMetricSpace.toUniformSpace.{u2} β _inst_2)) (Function.comp.{succ u3, succ u1, succ u2} γ α β f g)) (Continuous.{u3, u1} γ α _inst_4 (UniformSpace.toTopologicalSpace.{u1} α (PseudoEMetricSpace.toUniformSpace.{u1} α _inst_1)) g))
+but is expected to have type
+  forall {α : Type.{u2}} {β : Type.{u3}} [_inst_1 : PseudoEMetricSpace.{u2} α] [_inst_2 : PseudoEMetricSpace.{u3} β] {f : α -> β} {γ : Type.{u1}} [_inst_4 : TopologicalSpace.{u1} γ], (Isometry.{u2, u3} α β _inst_1 _inst_2 f) -> (forall {g : γ -> α}, Iff (Continuous.{u1, u3} γ β _inst_4 (UniformSpace.toTopologicalSpace.{u3} β (PseudoEMetricSpace.toUniformSpace.{u3} β _inst_2)) (Function.comp.{succ u1, succ u2, succ u3} γ α β f g)) (Continuous.{u1, u2} γ α _inst_4 (UniformSpace.toTopologicalSpace.{u2} α (PseudoEMetricSpace.toUniformSpace.{u2} α _inst_1)) g))
+Case conversion may be inaccurate. Consider using '#align isometry.comp_continuous_iff Isometry.comp_continuous_iffₓ'. -/
 theorem comp_continuous_iff {γ} [TopologicalSpace γ] (hf : Isometry f) {g : γ → α} :
     Continuous (f ∘ g) ↔ Continuous g :=
   hf.UniformInducing.Inducing.continuous_iff.symm
@@ -195,26 +265,34 @@ section EmetricIsometry
 
 variable [EMetricSpace α] [PseudoEMetricSpace β] {f : α → β}
 
+#print Isometry.injective /-
 /-- An isometry from an emetric space is injective -/
 protected theorem injective (h : Isometry f) : Injective f :=
   h.antilipschitz.Injective
 #align isometry.injective Isometry.injective
+-/
 
+#print Isometry.uniformEmbedding /-
 /-- An isometry from an emetric space is a uniform embedding -/
 protected theorem uniformEmbedding (hf : Isometry f) : UniformEmbedding f :=
   hf.antilipschitz.UniformEmbedding hf.lipschitz.UniformContinuous
 #align isometry.uniform_embedding Isometry.uniformEmbedding
+-/
 
+#print Isometry.embedding /-
 /-- An isometry from an emetric space is an embedding -/
 protected theorem embedding (hf : Isometry f) : Embedding f :=
   hf.UniformEmbedding.Embedding
 #align isometry.embedding Isometry.embedding
+-/
 
+#print Isometry.closedEmbedding /-
 /-- An isometry from a complete emetric space is a closed embedding -/
 theorem closedEmbedding [CompleteSpace α] [EMetricSpace γ] {f : α → γ} (hf : Isometry f) :
     ClosedEmbedding f :=
   hf.antilipschitz.ClosedEmbedding hf.lipschitz.UniformContinuous
 #align isometry.closed_embedding Isometry.closedEmbedding
+-/
 
 end EmetricIsometry
 
@@ -223,59 +301,83 @@ section PseudoMetricIsometry
 
 variable [PseudoMetricSpace α] [PseudoMetricSpace β] {f : α → β}
 
+#print Isometry.diam_image /-
 /-- An isometry preserves the diameter in pseudometric spaces. -/
 theorem diam_image (hf : Isometry f) (s : Set α) : Metric.diam (f '' s) = Metric.diam s := by
   rw [Metric.diam, Metric.diam, hf.ediam_image]
 #align isometry.diam_image Isometry.diam_image
+-/
 
+#print Isometry.diam_range /-
 theorem diam_range (hf : Isometry f) : Metric.diam (range f) = Metric.diam (univ : Set α) :=
   by
   rw [← image_univ]
   exact hf.diam_image univ
 #align isometry.diam_range Isometry.diam_range
+-/
 
+#print Isometry.preimage_setOf_dist /-
 theorem preimage_setOf_dist (hf : Isometry f) (x : α) (p : ℝ → Prop) :
     f ⁻¹' { y | p (dist y (f x)) } = { y | p (dist y x) } :=
   by
   ext y
   simp [hf.dist_eq]
 #align isometry.preimage_set_of_dist Isometry.preimage_setOf_dist
+-/
 
+#print Isometry.preimage_closedBall /-
 theorem preimage_closedBall (hf : Isometry f) (x : α) (r : ℝ) :
     f ⁻¹' Metric.closedBall (f x) r = Metric.closedBall x r :=
   hf.preimage_setOf_dist x (· ≤ r)
 #align isometry.preimage_closed_ball Isometry.preimage_closedBall
+-/
 
+#print Isometry.preimage_ball /-
 theorem preimage_ball (hf : Isometry f) (x : α) (r : ℝ) :
     f ⁻¹' Metric.ball (f x) r = Metric.ball x r :=
   hf.preimage_setOf_dist x (· < r)
 #align isometry.preimage_ball Isometry.preimage_ball
+-/
 
+#print Isometry.preimage_sphere /-
 theorem preimage_sphere (hf : Isometry f) (x : α) (r : ℝ) :
     f ⁻¹' Metric.sphere (f x) r = Metric.sphere x r :=
   hf.preimage_setOf_dist x (· = r)
 #align isometry.preimage_sphere Isometry.preimage_sphere
+-/
 
+#print Isometry.mapsTo_ball /-
 theorem mapsTo_ball (hf : Isometry f) (x : α) (r : ℝ) :
     MapsTo f (Metric.ball x r) (Metric.ball (f x) r) :=
   (hf.preimage_ball x r).ge
 #align isometry.maps_to_ball Isometry.mapsTo_ball
+-/
 
+#print Isometry.mapsTo_sphere /-
 theorem mapsTo_sphere (hf : Isometry f) (x : α) (r : ℝ) :
     MapsTo f (Metric.sphere x r) (Metric.sphere (f x) r) :=
   (hf.preimage_sphere x r).ge
 #align isometry.maps_to_sphere Isometry.mapsTo_sphere
+-/
 
+#print Isometry.mapsTo_closedBall /-
 theorem mapsTo_closedBall (hf : Isometry f) (x : α) (r : ℝ) :
     MapsTo f (Metric.closedBall x r) (Metric.closedBall (f x) r) :=
   (hf.preimage_closedBall x r).ge
 #align isometry.maps_to_closed_ball Isometry.mapsTo_closedBall
+-/
 
 end PseudoMetricIsometry
 
 -- section
 end Isometry
 
+/- warning: uniform_embedding.to_isometry -> UniformEmbedding.to_isometry is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : UniformSpace.{u1} α] [_inst_2 : MetricSpace.{u2} β] {f : α -> β} (h : UniformEmbedding.{u1, u2} α β _inst_1 (PseudoMetricSpace.toUniformSpace.{u2} β (MetricSpace.toPseudoMetricSpace.{u2} β _inst_2)) f), Isometry.{u1, u2} α β (PseudoMetricSpace.toPseudoEMetricSpace.{u1} α (MetricSpace.toPseudoMetricSpace.{u1} α (UniformEmbedding.comapMetricSpace.{u1, u2} α β _inst_1 _inst_2 f h))) (PseudoMetricSpace.toPseudoEMetricSpace.{u2} β (MetricSpace.toPseudoMetricSpace.{u2} β _inst_2)) f
+but is expected to have type
+  forall {α : Type.{u2}} {β : Type.{u1}} [_inst_1 : UniformSpace.{u2} α] [_inst_2 : MetricSpace.{u1} β] {f : α -> β} (h : UniformEmbedding.{u2, u1} α β _inst_1 (PseudoMetricSpace.toUniformSpace.{u1} β (MetricSpace.toPseudoMetricSpace.{u1} β _inst_2)) f), Isometry.{u2, u1} α β (EMetricSpace.toPseudoEMetricSpace.{u2} α (MetricSpace.toEMetricSpace.{u2} α (UniformEmbedding.comapMetricSpace.{u2, u1} α β _inst_1 _inst_2 f h))) (EMetricSpace.toPseudoEMetricSpace.{u1} β (MetricSpace.toEMetricSpace.{u1} β _inst_2)) f
+Case conversion may be inaccurate. Consider using '#align uniform_embedding.to_isometry UniformEmbedding.to_isometryₓ'. -/
 -- namespace
 /-- A uniform embedding from a uniform space to a metric space is an isometry with respect to the
 induced metric space structure on the source space. -/
@@ -291,6 +393,12 @@ theorem UniformEmbedding.to_isometry {α β} [UniformSpace α] [MetricSpace β]
   rfl
 #align uniform_embedding.to_isometry UniformEmbedding.to_isometry
 
+/- warning: embedding.to_isometry -> Embedding.to_isometry is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : TopologicalSpace.{u1} α] [_inst_2 : MetricSpace.{u2} β] {f : α -> β} (h : Embedding.{u1, u2} α β _inst_1 (UniformSpace.toTopologicalSpace.{u2} β (PseudoMetricSpace.toUniformSpace.{u2} β (MetricSpace.toPseudoMetricSpace.{u2} β _inst_2))) f), Isometry.{u1, u2} α β (PseudoMetricSpace.toPseudoEMetricSpace.{u1} α (MetricSpace.toPseudoMetricSpace.{u1} α (Embedding.comapMetricSpace.{u1, u2} α β _inst_1 _inst_2 f h))) (PseudoMetricSpace.toPseudoEMetricSpace.{u2} β (MetricSpace.toPseudoMetricSpace.{u2} β _inst_2)) f
+but is expected to have type
+  forall {α : Type.{u2}} {β : Type.{u1}} [_inst_1 : TopologicalSpace.{u2} α] [_inst_2 : MetricSpace.{u1} β] {f : α -> β} (h : Embedding.{u2, u1} α β _inst_1 (UniformSpace.toTopologicalSpace.{u1} β (PseudoMetricSpace.toUniformSpace.{u1} β (MetricSpace.toPseudoMetricSpace.{u1} β _inst_2))) f), Isometry.{u2, u1} α β (EMetricSpace.toPseudoEMetricSpace.{u2} α (MetricSpace.toEMetricSpace.{u2} α (Embedding.comapMetricSpace.{u2, u1} α β _inst_1 _inst_2 f h))) (EMetricSpace.toPseudoEMetricSpace.{u1} β (MetricSpace.toEMetricSpace.{u1} β _inst_2)) f
+Case conversion may be inaccurate. Consider using '#align embedding.to_isometry Embedding.to_isometryₓ'. -/
 /-- An embedding from a topological space to a metric space is an isometry with respect to the
 induced metric space structure on the source space. -/
 theorem Embedding.to_isometry {α β} [TopologicalSpace α] [MetricSpace β] {f : α → β}
@@ -305,6 +413,7 @@ theorem Embedding.to_isometry {α β} [TopologicalSpace α] [MetricSpace β] {f
   rfl
 #align embedding.to_isometry Embedding.to_isometry
 
+#print IsometryEquiv /-
 -- such a bijection need not exist
 /-- `α` and `β` are isometric if there is an isometric bijection between them. -/
 @[nolint has_nonempty_instance]
@@ -312,6 +421,7 @@ structure IsometryEquiv (α β : Type _) [PseudoEMetricSpace α] [PseudoEMetricS
   α ≃ β where
   isometry_toFun : Isometry to_fun
 #align isometry_equiv IsometryEquiv
+-/
 
 -- mathport name: «expr ≃ᵢ »
 infixl:25 " ≃ᵢ " => IsometryEquiv
@@ -325,65 +435,132 @@ variable [PseudoEMetricSpace α] [PseudoEMetricSpace β] [PseudoEMetricSpace γ]
 instance : CoeFun (α ≃ᵢ β) fun _ => α → β :=
   ⟨fun e => e.toEquiv⟩
 
+/- warning: isometry_equiv.coe_eq_to_equiv -> IsometryEquiv.coe_eq_toEquiv is a dubious translation:
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+Case conversion may be inaccurate. Consider using '#align isometry_equiv.coe_eq_to_equiv IsometryEquiv.coe_eq_toEquivₓ'. -/
 theorem coe_eq_toEquiv (h : α ≃ᵢ β) (a : α) : h a = h.toEquiv a :=
   rfl
 #align isometry_equiv.coe_eq_to_equiv IsometryEquiv.coe_eq_toEquiv
 
+#print IsometryEquiv.coe_toEquiv /-
 @[simp]
 theorem coe_toEquiv (h : α ≃ᵢ β) : ⇑h.toEquiv = h :=
   rfl
 #align isometry_equiv.coe_to_equiv IsometryEquiv.coe_toEquiv
+-/
 
+/- warning: isometry_equiv.isometry -> IsometryEquiv.isometry is a dubious translation:
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+Case conversion may be inaccurate. Consider using '#align isometry_equiv.isometry IsometryEquiv.isometryₓ'. -/
 protected theorem isometry (h : α ≃ᵢ β) : Isometry h :=
   h.isometry_toFun
 #align isometry_equiv.isometry IsometryEquiv.isometry
 
+/- warning: isometry_equiv.bijective -> IsometryEquiv.bijective is a dubious translation:
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 protected theorem bijective (h : α ≃ᵢ β) : Bijective h :=
   h.toEquiv.Bijective
 #align isometry_equiv.bijective IsometryEquiv.bijective
 
+/- warning: isometry_equiv.injective -> IsometryEquiv.injective is a dubious translation:
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 protected theorem injective (h : α ≃ᵢ β) : Injective h :=
   h.toEquiv.Injective
 #align isometry_equiv.injective IsometryEquiv.injective
 
+/- warning: isometry_equiv.surjective -> IsometryEquiv.surjective is a dubious translation:
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 protected theorem surjective (h : α ≃ᵢ β) : Surjective h :=
   h.toEquiv.Surjective
 #align isometry_equiv.surjective IsometryEquiv.surjective
 
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 protected theorem edist_eq (h : α ≃ᵢ β) (x y : α) : edist (h x) (h y) = edist x y :=
   h.Isometry.edist_eq x y
 #align isometry_equiv.edist_eq IsometryEquiv.edist_eq
 
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 protected theorem dist_eq {α β : Type _} [PseudoMetricSpace α] [PseudoMetricSpace β] (h : α ≃ᵢ β)
     (x y : α) : dist (h x) (h y) = dist x y :=
   h.Isometry.dist_eq x y
 #align isometry_equiv.dist_eq IsometryEquiv.dist_eq
 
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 protected theorem nndist_eq {α β : Type _} [PseudoMetricSpace α] [PseudoMetricSpace β] (h : α ≃ᵢ β)
     (x y : α) : nndist (h x) (h y) = nndist x y :=
   h.Isometry.nndist_eq x y
 #align isometry_equiv.nndist_eq IsometryEquiv.nndist_eq
 
+/- warning: isometry_equiv.continuous -> IsometryEquiv.continuous is a dubious translation:
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 protected theorem continuous (h : α ≃ᵢ β) : Continuous h :=
   h.Isometry.Continuous
 #align isometry_equiv.continuous IsometryEquiv.continuous
 
+#print IsometryEquiv.ediam_image /-
 @[simp]
 theorem ediam_image (h : α ≃ᵢ β) (s : Set α) : EMetric.diam (h '' s) = EMetric.diam s :=
   h.Isometry.ediam_image s
 #align isometry_equiv.ediam_image IsometryEquiv.ediam_image
+-/
 
-theorem toEquiv_inj : ∀ ⦃h₁ h₂ : α ≃ᵢ β⦄, h₁.toEquiv = h₂.toEquiv → h₁ = h₂
+#print IsometryEquiv.toEquiv_injective /-
+theorem toEquiv_injective : ∀ ⦃h₁ h₂ : α ≃ᵢ β⦄, h₁.toEquiv = h₂.toEquiv → h₁ = h₂
   | ⟨e₁, h₁⟩, ⟨e₂, h₂⟩, H => by
     dsimp at H
     subst e₁
-#align isometry_equiv.to_equiv_inj IsometryEquiv.toEquiv_inj
+#align isometry_equiv.to_equiv_inj IsometryEquiv.toEquiv_injective
+-/
 
+/- warning: isometry_equiv.ext -> IsometryEquiv.ext is a dubious translation:
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 @[ext]
 theorem ext ⦃h₁ h₂ : α ≃ᵢ β⦄ (H : ∀ x, h₁ x = h₂ x) : h₁ = h₂ :=
-  toEquiv_inj <| Equiv.ext H
+  toEquiv_injective <| Equiv.ext H
 #align isometry_equiv.ext IsometryEquiv.ext
 
+#print IsometryEquiv.mk' /-
 /-- Alternative constructor for isometric bijections,
 taking as input an isometry, and a right inverse. -/
 def mk' {α : Type u} [EMetricSpace α] (f : α → β) (g : β → α) (hfg : ∀ x, f (g x) = x)
@@ -394,126 +571,234 @@ def mk' {α : Type u} [EMetricSpace α] (f : α → β) (g : β → α) (hfg : 
   right_inv := hfg
   isometry_toFun := hf
 #align isometry_equiv.mk' IsometryEquiv.mk'
+-/
 
+#print IsometryEquiv.refl /-
 /-- The identity isometry of a space. -/
 protected def refl (α : Type _) [PseudoEMetricSpace α] : α ≃ᵢ α :=
   { Equiv.refl α with isometry_toFun := isometry_id }
 #align isometry_equiv.refl IsometryEquiv.refl
+-/
 
+#print IsometryEquiv.trans /-
 /-- The composition of two isometric isomorphisms, as an isometric isomorphism. -/
 protected def trans (h₁ : α ≃ᵢ β) (h₂ : β ≃ᵢ γ) : α ≃ᵢ γ :=
   { Equiv.trans h₁.toEquiv h₂.toEquiv with
     isometry_toFun := h₂.isometry_toFun.comp h₁.isometry_toFun }
 #align isometry_equiv.trans IsometryEquiv.trans
+-/
 
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 @[simp]
 theorem trans_apply (h₁ : α ≃ᵢ β) (h₂ : β ≃ᵢ γ) (x : α) : h₁.trans h₂ x = h₂ (h₁ x) :=
   rfl
 #align isometry_equiv.trans_apply IsometryEquiv.trans_apply
 
+#print IsometryEquiv.symm /-
 /-- The inverse of an isometric isomorphism, as an isometric isomorphism. -/
 protected def symm (h : α ≃ᵢ β) : β ≃ᵢ α
     where
   isometry_toFun := h.Isometry.right_inv h.right_inv
   toEquiv := h.toEquiv.symm
 #align isometry_equiv.symm IsometryEquiv.symm
+-/
 
+#print IsometryEquiv.Simps.apply /-
 /-- See Note [custom simps projection]. We need to specify this projection explicitly in this case,
   because it is a composition of multiple projections. -/
 def Simps.apply (h : α ≃ᵢ β) : α → β :=
   h
 #align isometry_equiv.simps.apply IsometryEquiv.Simps.apply
+-/
 
+#print IsometryEquiv.Simps.symm_apply /-
 /-- See Note [custom simps projection] -/
-def Simps.symmApply (h : α ≃ᵢ β) : β → α :=
+def Simps.symm_apply (h : α ≃ᵢ β) : β → α :=
   h.symm
-#align isometry_equiv.simps.symm_apply IsometryEquiv.Simps.symmApply
+#align isometry_equiv.simps.symm_apply IsometryEquiv.Simps.symm_apply
+-/
 
 initialize_simps_projections IsometryEquiv (to_equiv_to_fun → apply, to_equiv_inv_fun → symm_apply)
 
+#print IsometryEquiv.symm_symm /-
 @[simp]
 theorem symm_symm (h : α ≃ᵢ β) : h.symm.symm = h :=
-  toEquiv_inj h.toEquiv.symm_symm
+  toEquiv_injective h.toEquiv.symm_symm
 #align isometry_equiv.symm_symm IsometryEquiv.symm_symm
+-/
 
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 @[simp]
 theorem apply_symm_apply (h : α ≃ᵢ β) (y : β) : h (h.symm y) = y :=
   h.toEquiv.apply_symm_apply y
 #align isometry_equiv.apply_symm_apply IsometryEquiv.apply_symm_apply
 
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 @[simp]
 theorem symm_apply_apply (h : α ≃ᵢ β) (x : α) : h.symm (h x) = x :=
   h.toEquiv.symm_apply_apply x
 #align isometry_equiv.symm_apply_apply IsometryEquiv.symm_apply_apply
 
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 theorem symm_apply_eq (h : α ≃ᵢ β) {x : α} {y : β} : h.symm y = x ↔ y = h x :=
   h.toEquiv.symm_apply_eq
 #align isometry_equiv.symm_apply_eq IsometryEquiv.symm_apply_eq
 
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 theorem eq_symm_apply (h : α ≃ᵢ β) {x : α} {y : β} : x = h.symm y ↔ h x = y :=
   h.toEquiv.eq_symm_apply
 #align isometry_equiv.eq_symm_apply IsometryEquiv.eq_symm_apply
 
+/- warning: isometry_equiv.symm_comp_self -> IsometryEquiv.symm_comp_self is a dubious translation:
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 theorem symm_comp_self (h : α ≃ᵢ β) : ⇑h.symm ∘ ⇑h = id :=
   funext fun a => h.toEquiv.left_inv a
 #align isometry_equiv.symm_comp_self IsometryEquiv.symm_comp_self
 
+/- warning: isometry_equiv.self_comp_symm -> IsometryEquiv.self_comp_symm is a dubious translation:
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 theorem self_comp_symm (h : α ≃ᵢ β) : ⇑h ∘ ⇑h.symm = id :=
   funext fun a => h.toEquiv.right_inv a
 #align isometry_equiv.self_comp_symm IsometryEquiv.self_comp_symm
 
+/- warning: isometry_equiv.range_eq_univ -> IsometryEquiv.range_eq_univ is a dubious translation:
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 @[simp]
 theorem range_eq_univ (h : α ≃ᵢ β) : range h = univ :=
   h.toEquiv.range_eq_univ
 #align isometry_equiv.range_eq_univ IsometryEquiv.range_eq_univ
 
+/- warning: isometry_equiv.image_symm -> IsometryEquiv.image_symm is a dubious translation:
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 theorem image_symm (h : α ≃ᵢ β) : image h.symm = preimage h :=
   image_eq_preimage_of_inverse h.symm.toEquiv.left_inv h.symm.toEquiv.right_inv
 #align isometry_equiv.image_symm IsometryEquiv.image_symm
 
+/- warning: isometry_equiv.preimage_symm -> IsometryEquiv.preimage_symm is a dubious translation:
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 theorem preimage_symm (h : α ≃ᵢ β) : preimage h.symm = image h :=
   (image_eq_preimage_of_inverse h.toEquiv.left_inv h.toEquiv.right_inv).symm
 #align isometry_equiv.preimage_symm IsometryEquiv.preimage_symm
 
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 @[simp]
 theorem symm_trans_apply (h₁ : α ≃ᵢ β) (h₂ : β ≃ᵢ γ) (x : γ) :
     (h₁.trans h₂).symm x = h₁.symm (h₂.symm x) :=
   rfl
 #align isometry_equiv.symm_trans_apply IsometryEquiv.symm_trans_apply
 
+#print IsometryEquiv.ediam_univ /-
 theorem ediam_univ (h : α ≃ᵢ β) : EMetric.diam (univ : Set α) = EMetric.diam (univ : Set β) := by
   rw [← h.range_eq_univ, h.isometry.ediam_range]
 #align isometry_equiv.ediam_univ IsometryEquiv.ediam_univ
+-/
 
+#print IsometryEquiv.ediam_preimage /-
 @[simp]
 theorem ediam_preimage (h : α ≃ᵢ β) (s : Set β) : EMetric.diam (h ⁻¹' s) = EMetric.diam s := by
   rw [← image_symm, ediam_image]
 #align isometry_equiv.ediam_preimage IsometryEquiv.ediam_preimage
+-/
 
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 @[simp]
 theorem preimage_emetric_ball (h : α ≃ᵢ β) (x : β) (r : ℝ≥0∞) :
     h ⁻¹' EMetric.ball x r = EMetric.ball (h.symm x) r := by
   rw [← h.isometry.preimage_emetric_ball (h.symm x) r, h.apply_symm_apply]
 #align isometry_equiv.preimage_emetric_ball IsometryEquiv.preimage_emetric_ball
 
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 @[simp]
 theorem preimage_emetric_closedBall (h : α ≃ᵢ β) (x : β) (r : ℝ≥0∞) :
     h ⁻¹' EMetric.closedBall x r = EMetric.closedBall (h.symm x) r := by
   rw [← h.isometry.preimage_emetric_closed_ball (h.symm x) r, h.apply_symm_apply]
 #align isometry_equiv.preimage_emetric_closed_ball IsometryEquiv.preimage_emetric_closedBall
 
+/- warning: isometry_equiv.image_emetric_ball -> IsometryEquiv.image_emetric_ball is a dubious translation:
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 @[simp]
 theorem image_emetric_ball (h : α ≃ᵢ β) (x : α) (r : ℝ≥0∞) :
     h '' EMetric.ball x r = EMetric.ball (h x) r := by
   rw [← h.preimage_symm, h.symm.preimage_emetric_ball, symm_symm]
 #align isometry_equiv.image_emetric_ball IsometryEquiv.image_emetric_ball
 
+/- warning: isometry_equiv.image_emetric_closed_ball -> IsometryEquiv.image_emetric_closedBall is a dubious translation:
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 @[simp]
 theorem image_emetric_closedBall (h : α ≃ᵢ β) (x : α) (r : ℝ≥0∞) :
     h '' EMetric.closedBall x r = EMetric.closedBall (h x) r := by
   rw [← h.preimage_symm, h.symm.preimage_emetric_closed_ball, symm_symm]
 #align isometry_equiv.image_emetric_closed_ball IsometryEquiv.image_emetric_closedBall
 
+#print IsometryEquiv.toHomeomorph /-
 /-- The (bundled) homeomorphism associated to an isometric isomorphism. -/
 @[simps toEquiv]
 protected def toHomeomorph (h : α ≃ᵢ β) : α ≃ₜ β
@@ -522,29 +807,60 @@ protected def toHomeomorph (h : α ≃ᵢ β) : α ≃ₜ β
   continuous_invFun := h.symm.Continuous
   toEquiv := h.toEquiv
 #align isometry_equiv.to_homeomorph IsometryEquiv.toHomeomorph
+-/
 
+/- warning: isometry_equiv.coe_to_homeomorph -> IsometryEquiv.coe_toHomeomorph is a dubious translation:
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 @[simp]
 theorem coe_toHomeomorph (h : α ≃ᵢ β) : ⇑h.toHomeomorph = h :=
   rfl
 #align isometry_equiv.coe_to_homeomorph IsometryEquiv.coe_toHomeomorph
 
+/- warning: isometry_equiv.coe_to_homeomorph_symm -> IsometryEquiv.coe_toHomeomorph_symm is a dubious translation:
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 @[simp]
 theorem coe_toHomeomorph_symm (h : α ≃ᵢ β) : ⇑h.toHomeomorph.symm = h.symm :=
   rfl
 #align isometry_equiv.coe_to_homeomorph_symm IsometryEquiv.coe_toHomeomorph_symm
 
+/- warning: isometry_equiv.comp_continuous_on_iff -> IsometryEquiv.comp_continuousOn_iff is a dubious translation:
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+Case conversion may be inaccurate. Consider using '#align isometry_equiv.comp_continuous_on_iff IsometryEquiv.comp_continuousOn_iffₓ'. -/
 @[simp]
 theorem comp_continuousOn_iff {γ} [TopologicalSpace γ] (h : α ≃ᵢ β) {f : γ → α} {s : Set γ} :
     ContinuousOn (h ∘ f) s ↔ ContinuousOn f s :=
   h.toHomeomorph.comp_continuousOn_iff _ _
 #align isometry_equiv.comp_continuous_on_iff IsometryEquiv.comp_continuousOn_iff
 
+/- warning: isometry_equiv.comp_continuous_iff -> IsometryEquiv.comp_continuous_iff is a dubious translation:
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 @[simp]
 theorem comp_continuous_iff {γ} [TopologicalSpace γ] (h : α ≃ᵢ β) {f : γ → α} :
     Continuous (h ∘ f) ↔ Continuous f :=
   h.toHomeomorph.comp_continuous_iff
 #align isometry_equiv.comp_continuous_iff IsometryEquiv.comp_continuous_iff
 
+/- warning: isometry_equiv.comp_continuous_iff' -> IsometryEquiv.comp_continuous_iff' is a dubious translation:
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 @[simp]
 theorem comp_continuous_iff' {γ} [TopologicalSpace γ] (h : α ≃ᵢ β) {f : β → γ} :
     Continuous (f ∘ h) ↔ Continuous f :=
@@ -561,41 +877,75 @@ instance : Group (α ≃ᵢ α) where
   mul_one e := ext fun _ => rfl
   mul_left_inv e := ext e.symm_apply_apply
 
+/- warning: isometry_equiv.coe_one -> IsometryEquiv.coe_one is a dubious translation:
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 @[simp]
 theorem coe_one : ⇑(1 : α ≃ᵢ α) = id :=
   rfl
 #align isometry_equiv.coe_one IsometryEquiv.coe_one
 
+/- warning: isometry_equiv.coe_mul -> IsometryEquiv.coe_mul is a dubious translation:
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+Case conversion may be inaccurate. Consider using '#align isometry_equiv.coe_mul IsometryEquiv.coe_mulₓ'. -/
 @[simp]
 theorem coe_mul (e₁ e₂ : α ≃ᵢ α) : ⇑(e₁ * e₂) = e₁ ∘ e₂ :=
   rfl
 #align isometry_equiv.coe_mul IsometryEquiv.coe_mul
 
+/- warning: isometry_equiv.mul_apply -> IsometryEquiv.mul_apply 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 isometry_equiv.mul_apply IsometryEquiv.mul_applyₓ'. -/
 theorem mul_apply (e₁ e₂ : α ≃ᵢ α) (x : α) : (e₁ * e₂) x = e₁ (e₂ x) :=
   rfl
 #align isometry_equiv.mul_apply IsometryEquiv.mul_apply
 
+/- warning: isometry_equiv.inv_apply_self -> IsometryEquiv.inv_apply_self is a dubious translation:
+lean 3 declaration is
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+but is expected to have type
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+Case conversion may be inaccurate. Consider using '#align isometry_equiv.inv_apply_self IsometryEquiv.inv_apply_selfₓ'. -/
 @[simp]
 theorem inv_apply_self (e : α ≃ᵢ α) (x : α) : e⁻¹ (e x) = x :=
   e.symm_apply_apply x
 #align isometry_equiv.inv_apply_self IsometryEquiv.inv_apply_self
 
+/- warning: isometry_equiv.apply_inv_self -> IsometryEquiv.apply_inv_self is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} [_inst_1 : PseudoEMetricSpace.{u1} α] (e : IsometryEquiv.{u1, u1} α α _inst_1 _inst_1) (x : α), Eq.{succ u1} α (coeFn.{succ u1, succ u1} (IsometryEquiv.{u1, u1} α α _inst_1 _inst_1) (fun (_x : IsometryEquiv.{u1, u1} α α _inst_1 _inst_1) => α -> α) (IsometryEquiv.hasCoeToFun.{u1, u1} α α _inst_1 _inst_1) e (coeFn.{succ u1, succ u1} (IsometryEquiv.{u1, u1} α α _inst_1 _inst_1) (fun (_x : IsometryEquiv.{u1, u1} α α _inst_1 _inst_1) => α -> α) (IsometryEquiv.hasCoeToFun.{u1, u1} α α _inst_1 _inst_1) (Inv.inv.{u1} (IsometryEquiv.{u1, u1} α α _inst_1 _inst_1) (DivInvMonoid.toHasInv.{u1} (IsometryEquiv.{u1, u1} α α _inst_1 _inst_1) (Group.toDivInvMonoid.{u1} (IsometryEquiv.{u1, u1} α α _inst_1 _inst_1) (IsometryEquiv.group.{u1} α _inst_1))) e) x)) x
+but is expected to have type
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+Case conversion may be inaccurate. Consider using '#align isometry_equiv.apply_inv_self IsometryEquiv.apply_inv_selfₓ'. -/
 @[simp]
 theorem apply_inv_self (e : α ≃ᵢ α) (x : α) : e (e⁻¹ x) = x :=
   e.apply_symm_apply x
 #align isometry_equiv.apply_inv_self IsometryEquiv.apply_inv_self
 
+#print IsometryEquiv.completeSpace /-
 protected theorem completeSpace [CompleteSpace β] (e : α ≃ᵢ β) : CompleteSpace α :=
   completeSpace_of_isComplete_univ <|
     isComplete_of_complete_image e.Isometry.UniformInducing <| by
       rwa [Set.image_univ, IsometryEquiv.range_eq_univ, ← completeSpace_iff_isComplete_univ]
 #align isometry_equiv.complete_space IsometryEquiv.completeSpace
+-/
 
+#print IsometryEquiv.completeSpace_iff /-
 theorem completeSpace_iff (e : α ≃ᵢ β) : CompleteSpace α ↔ CompleteSpace β :=
   by
   constructor <;> intro H
   exacts[e.symm.complete_space, e.complete_space]
 #align isometry_equiv.complete_space_iff IsometryEquiv.completeSpace_iff
+-/
 
 end PseudoEMetricSpace
 
@@ -603,49 +953,91 @@ section PseudoMetricSpace
 
 variable [PseudoMetricSpace α] [PseudoMetricSpace β] (h : α ≃ᵢ β)
 
+#print IsometryEquiv.diam_image /-
 @[simp]
 theorem diam_image (s : Set α) : Metric.diam (h '' s) = Metric.diam s :=
   h.Isometry.diam_image s
 #align isometry_equiv.diam_image IsometryEquiv.diam_image
+-/
 
+#print IsometryEquiv.diam_preimage /-
 @[simp]
 theorem diam_preimage (s : Set β) : Metric.diam (h ⁻¹' s) = Metric.diam s := by
   rw [← image_symm, diam_image]
 #align isometry_equiv.diam_preimage IsometryEquiv.diam_preimage
+-/
 
+#print IsometryEquiv.diam_univ /-
 theorem diam_univ : Metric.diam (univ : Set α) = Metric.diam (univ : Set β) :=
   congr_arg ENNReal.toReal h.ediam_univ
 #align isometry_equiv.diam_univ IsometryEquiv.diam_univ
+-/
 
+/- warning: isometry_equiv.preimage_ball -> IsometryEquiv.preimage_ball is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : PseudoMetricSpace.{u1} α] [_inst_2 : PseudoMetricSpace.{u2} β] (h : IsometryEquiv.{u1, u2} α β (PseudoMetricSpace.toPseudoEMetricSpace.{u1} α _inst_1) (PseudoMetricSpace.toPseudoEMetricSpace.{u2} β _inst_2)) (x : β) (r : Real), Eq.{succ u1} (Set.{u1} α) (Set.preimage.{u1, u2} α β (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (IsometryEquiv.{u1, u2} α β (PseudoMetricSpace.toPseudoEMetricSpace.{u1} α _inst_1) (PseudoMetricSpace.toPseudoEMetricSpace.{u2} β _inst_2)) (fun (_x : IsometryEquiv.{u1, u2} α β (PseudoMetricSpace.toPseudoEMetricSpace.{u1} α _inst_1) (PseudoMetricSpace.toPseudoEMetricSpace.{u2} β _inst_2)) => α -> β) (IsometryEquiv.hasCoeToFun.{u1, u2} α β (PseudoMetricSpace.toPseudoEMetricSpace.{u1} α _inst_1) (PseudoMetricSpace.toPseudoEMetricSpace.{u2} β _inst_2)) h) (Metric.ball.{u2} β _inst_2 x r)) (Metric.ball.{u1} α _inst_1 (coeFn.{max (succ u2) (succ u1), max (succ u2) (succ u1)} (IsometryEquiv.{u2, u1} β α (PseudoMetricSpace.toPseudoEMetricSpace.{u2} β _inst_2) (PseudoMetricSpace.toPseudoEMetricSpace.{u1} α _inst_1)) (fun (_x : IsometryEquiv.{u2, u1} β α (PseudoMetricSpace.toPseudoEMetricSpace.{u2} β _inst_2) (PseudoMetricSpace.toPseudoEMetricSpace.{u1} α _inst_1)) => β -> α) (IsometryEquiv.hasCoeToFun.{u2, u1} β α (PseudoMetricSpace.toPseudoEMetricSpace.{u2} β _inst_2) (PseudoMetricSpace.toPseudoEMetricSpace.{u1} α _inst_1)) (IsometryEquiv.symm.{u1, u2} α β (PseudoMetricSpace.toPseudoEMetricSpace.{u1} α _inst_1) (PseudoMetricSpace.toPseudoEMetricSpace.{u2} β _inst_2) h) x) r)
+but is expected to have type
+  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : PseudoMetricSpace.{u1} α] [_inst_2 : PseudoMetricSpace.{u2} β] (h : IsometryEquiv.{u1, u2} α β (PseudoMetricSpace.toPseudoEMetricSpace.{u1} α _inst_1) (PseudoMetricSpace.toPseudoEMetricSpace.{u2} β _inst_2)) (x : β) (r : Real), Eq.{succ u1} (Set.{u1} α) (Set.preimage.{u1, u2} α β (FunLike.coe.{max (succ u1) (succ u2), succ u1, succ u2} (IsometryEquiv.{u1, u2} α β (PseudoMetricSpace.toPseudoEMetricSpace.{u1} α _inst_1) (PseudoMetricSpace.toPseudoEMetricSpace.{u2} β _inst_2)) α (fun (_x : α) => (fun (x._@.Mathlib.Data.FunLike.Embedding._hyg.19 : α) => β) _x) (EmbeddingLike.toFunLike.{max (succ u1) (succ u2), succ u1, succ u2} (IsometryEquiv.{u1, u2} α β (PseudoMetricSpace.toPseudoEMetricSpace.{u1} α _inst_1) (PseudoMetricSpace.toPseudoEMetricSpace.{u2} β _inst_2)) α β (EquivLike.toEmbeddingLike.{max (succ u1) (succ u2), succ u1, succ u2} (IsometryEquiv.{u1, u2} α β (PseudoMetricSpace.toPseudoEMetricSpace.{u1} α _inst_1) (PseudoMetricSpace.toPseudoEMetricSpace.{u2} β _inst_2)) α β (IsometryEquiv.instEquivLikeIsometryEquiv.{u1, u2} α β (PseudoMetricSpace.toPseudoEMetricSpace.{u1} α _inst_1) (PseudoMetricSpace.toPseudoEMetricSpace.{u2} β _inst_2)))) h) (Metric.ball.{u2} β _inst_2 x r)) (Metric.ball.{u1} ((fun (x._@.Mathlib.Data.FunLike.Embedding._hyg.19 : β) => α) x) _inst_1 (FunLike.coe.{max (succ u1) (succ u2), succ u2, succ u1} (IsometryEquiv.{u2, u1} β α (PseudoMetricSpace.toPseudoEMetricSpace.{u2} β _inst_2) (PseudoMetricSpace.toPseudoEMetricSpace.{u1} α _inst_1)) β (fun (_x : β) => (fun (x._@.Mathlib.Data.FunLike.Embedding._hyg.19 : β) => α) _x) (EmbeddingLike.toFunLike.{max (succ u1) (succ u2), succ u2, succ u1} (IsometryEquiv.{u2, u1} β α (PseudoMetricSpace.toPseudoEMetricSpace.{u2} β _inst_2) (PseudoMetricSpace.toPseudoEMetricSpace.{u1} α _inst_1)) β α (EquivLike.toEmbeddingLike.{max (succ u1) (succ u2), succ u2, succ u1} (IsometryEquiv.{u2, u1} β α (PseudoMetricSpace.toPseudoEMetricSpace.{u2} β _inst_2) (PseudoMetricSpace.toPseudoEMetricSpace.{u1} α _inst_1)) β α (IsometryEquiv.instEquivLikeIsometryEquiv.{u2, u1} β α (PseudoMetricSpace.toPseudoEMetricSpace.{u2} β _inst_2) (PseudoMetricSpace.toPseudoEMetricSpace.{u1} α _inst_1)))) (IsometryEquiv.symm.{u1, u2} α β (PseudoMetricSpace.toPseudoEMetricSpace.{u1} α _inst_1) (PseudoMetricSpace.toPseudoEMetricSpace.{u2} β _inst_2) h) x) r)
+Case conversion may be inaccurate. Consider using '#align isometry_equiv.preimage_ball IsometryEquiv.preimage_ballₓ'. -/
 @[simp]
 theorem preimage_ball (h : α ≃ᵢ β) (x : β) (r : ℝ) :
     h ⁻¹' Metric.ball x r = Metric.ball (h.symm x) r := by
   rw [← h.isometry.preimage_ball (h.symm x) r, h.apply_symm_apply]
 #align isometry_equiv.preimage_ball IsometryEquiv.preimage_ball
 
+/- warning: isometry_equiv.preimage_sphere -> IsometryEquiv.preimage_sphere is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : PseudoMetricSpace.{u1} α] [_inst_2 : PseudoMetricSpace.{u2} β] (h : IsometryEquiv.{u1, u2} α β (PseudoMetricSpace.toPseudoEMetricSpace.{u1} α _inst_1) (PseudoMetricSpace.toPseudoEMetricSpace.{u2} β _inst_2)) (x : β) (r : Real), Eq.{succ u1} (Set.{u1} α) (Set.preimage.{u1, u2} α β (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (IsometryEquiv.{u1, u2} α β (PseudoMetricSpace.toPseudoEMetricSpace.{u1} α _inst_1) (PseudoMetricSpace.toPseudoEMetricSpace.{u2} β _inst_2)) (fun (_x : IsometryEquiv.{u1, u2} α β (PseudoMetricSpace.toPseudoEMetricSpace.{u1} α _inst_1) (PseudoMetricSpace.toPseudoEMetricSpace.{u2} β _inst_2)) => α -> β) (IsometryEquiv.hasCoeToFun.{u1, u2} α β (PseudoMetricSpace.toPseudoEMetricSpace.{u1} α _inst_1) (PseudoMetricSpace.toPseudoEMetricSpace.{u2} β _inst_2)) h) (Metric.sphere.{u2} β _inst_2 x r)) (Metric.sphere.{u1} α _inst_1 (coeFn.{max (succ u2) (succ u1), max (succ u2) (succ u1)} (IsometryEquiv.{u2, u1} β α (PseudoMetricSpace.toPseudoEMetricSpace.{u2} β _inst_2) (PseudoMetricSpace.toPseudoEMetricSpace.{u1} α _inst_1)) (fun (_x : IsometryEquiv.{u2, u1} β α (PseudoMetricSpace.toPseudoEMetricSpace.{u2} β _inst_2) (PseudoMetricSpace.toPseudoEMetricSpace.{u1} α _inst_1)) => β -> α) (IsometryEquiv.hasCoeToFun.{u2, u1} β α (PseudoMetricSpace.toPseudoEMetricSpace.{u2} β _inst_2) (PseudoMetricSpace.toPseudoEMetricSpace.{u1} α _inst_1)) (IsometryEquiv.symm.{u1, u2} α β (PseudoMetricSpace.toPseudoEMetricSpace.{u1} α _inst_1) (PseudoMetricSpace.toPseudoEMetricSpace.{u2} β _inst_2) h) x) r)
+but is expected to have type
+  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : PseudoMetricSpace.{u1} α] [_inst_2 : PseudoMetricSpace.{u2} β] (h : IsometryEquiv.{u1, u2} α β (PseudoMetricSpace.toPseudoEMetricSpace.{u1} α _inst_1) (PseudoMetricSpace.toPseudoEMetricSpace.{u2} β _inst_2)) (x : β) (r : Real), Eq.{succ u1} (Set.{u1} α) (Set.preimage.{u1, u2} α β (FunLike.coe.{max (succ u1) (succ u2), succ u1, succ u2} (IsometryEquiv.{u1, u2} α β (PseudoMetricSpace.toPseudoEMetricSpace.{u1} α _inst_1) (PseudoMetricSpace.toPseudoEMetricSpace.{u2} β _inst_2)) α (fun (_x : α) => (fun (x._@.Mathlib.Data.FunLike.Embedding._hyg.19 : α) => β) _x) (EmbeddingLike.toFunLike.{max (succ u1) (succ u2), succ u1, succ u2} (IsometryEquiv.{u1, u2} α β (PseudoMetricSpace.toPseudoEMetricSpace.{u1} α _inst_1) (PseudoMetricSpace.toPseudoEMetricSpace.{u2} β _inst_2)) α β (EquivLike.toEmbeddingLike.{max (succ u1) (succ u2), succ u1, succ u2} (IsometryEquiv.{u1, u2} α β (PseudoMetricSpace.toPseudoEMetricSpace.{u1} α _inst_1) (PseudoMetricSpace.toPseudoEMetricSpace.{u2} β _inst_2)) α β (IsometryEquiv.instEquivLikeIsometryEquiv.{u1, u2} α β (PseudoMetricSpace.toPseudoEMetricSpace.{u1} α _inst_1) (PseudoMetricSpace.toPseudoEMetricSpace.{u2} β _inst_2)))) h) (Metric.sphere.{u2} β _inst_2 x r)) (Metric.sphere.{u1} ((fun (x._@.Mathlib.Data.FunLike.Embedding._hyg.19 : β) => α) x) _inst_1 (FunLike.coe.{max (succ u1) (succ u2), succ u2, succ u1} (IsometryEquiv.{u2, u1} β α (PseudoMetricSpace.toPseudoEMetricSpace.{u2} β _inst_2) (PseudoMetricSpace.toPseudoEMetricSpace.{u1} α _inst_1)) β (fun (_x : β) => (fun (x._@.Mathlib.Data.FunLike.Embedding._hyg.19 : β) => α) _x) (EmbeddingLike.toFunLike.{max (succ u1) (succ u2), succ u2, succ u1} (IsometryEquiv.{u2, u1} β α (PseudoMetricSpace.toPseudoEMetricSpace.{u2} β _inst_2) (PseudoMetricSpace.toPseudoEMetricSpace.{u1} α _inst_1)) β α (EquivLike.toEmbeddingLike.{max (succ u1) (succ u2), succ u2, succ u1} (IsometryEquiv.{u2, u1} β α (PseudoMetricSpace.toPseudoEMetricSpace.{u2} β _inst_2) (PseudoMetricSpace.toPseudoEMetricSpace.{u1} α _inst_1)) β α (IsometryEquiv.instEquivLikeIsometryEquiv.{u2, u1} β α (PseudoMetricSpace.toPseudoEMetricSpace.{u2} β _inst_2) (PseudoMetricSpace.toPseudoEMetricSpace.{u1} α _inst_1)))) (IsometryEquiv.symm.{u1, u2} α β (PseudoMetricSpace.toPseudoEMetricSpace.{u1} α _inst_1) (PseudoMetricSpace.toPseudoEMetricSpace.{u2} β _inst_2) h) x) r)
+Case conversion may be inaccurate. Consider using '#align isometry_equiv.preimage_sphere IsometryEquiv.preimage_sphereₓ'. -/
 @[simp]
 theorem preimage_sphere (h : α ≃ᵢ β) (x : β) (r : ℝ) :
     h ⁻¹' Metric.sphere x r = Metric.sphere (h.symm x) r := by
   rw [← h.isometry.preimage_sphere (h.symm x) r, h.apply_symm_apply]
 #align isometry_equiv.preimage_sphere IsometryEquiv.preimage_sphere
 
+/- warning: isometry_equiv.preimage_closed_ball -> IsometryEquiv.preimage_closedBall is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : PseudoMetricSpace.{u1} α] [_inst_2 : PseudoMetricSpace.{u2} β] (h : IsometryEquiv.{u1, u2} α β (PseudoMetricSpace.toPseudoEMetricSpace.{u1} α _inst_1) (PseudoMetricSpace.toPseudoEMetricSpace.{u2} β _inst_2)) (x : β) (r : Real), Eq.{succ u1} (Set.{u1} α) (Set.preimage.{u1, u2} α β (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (IsometryEquiv.{u1, u2} α β (PseudoMetricSpace.toPseudoEMetricSpace.{u1} α _inst_1) (PseudoMetricSpace.toPseudoEMetricSpace.{u2} β _inst_2)) (fun (_x : IsometryEquiv.{u1, u2} α β (PseudoMetricSpace.toPseudoEMetricSpace.{u1} α _inst_1) (PseudoMetricSpace.toPseudoEMetricSpace.{u2} β _inst_2)) => α -> β) (IsometryEquiv.hasCoeToFun.{u1, u2} α β (PseudoMetricSpace.toPseudoEMetricSpace.{u1} α _inst_1) (PseudoMetricSpace.toPseudoEMetricSpace.{u2} β _inst_2)) h) (Metric.closedBall.{u2} β _inst_2 x r)) (Metric.closedBall.{u1} α _inst_1 (coeFn.{max (succ u2) (succ u1), max (succ u2) (succ u1)} (IsometryEquiv.{u2, u1} β α (PseudoMetricSpace.toPseudoEMetricSpace.{u2} β _inst_2) (PseudoMetricSpace.toPseudoEMetricSpace.{u1} α _inst_1)) (fun (_x : IsometryEquiv.{u2, u1} β α (PseudoMetricSpace.toPseudoEMetricSpace.{u2} β _inst_2) (PseudoMetricSpace.toPseudoEMetricSpace.{u1} α _inst_1)) => β -> α) (IsometryEquiv.hasCoeToFun.{u2, u1} β α (PseudoMetricSpace.toPseudoEMetricSpace.{u2} β _inst_2) (PseudoMetricSpace.toPseudoEMetricSpace.{u1} α _inst_1)) (IsometryEquiv.symm.{u1, u2} α β (PseudoMetricSpace.toPseudoEMetricSpace.{u1} α _inst_1) (PseudoMetricSpace.toPseudoEMetricSpace.{u2} β _inst_2) h) x) r)
+but is expected to have type
+  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : PseudoMetricSpace.{u1} α] [_inst_2 : PseudoMetricSpace.{u2} β] (h : IsometryEquiv.{u1, u2} α β (PseudoMetricSpace.toPseudoEMetricSpace.{u1} α _inst_1) (PseudoMetricSpace.toPseudoEMetricSpace.{u2} β _inst_2)) (x : β) (r : Real), Eq.{succ u1} (Set.{u1} α) (Set.preimage.{u1, u2} α β (FunLike.coe.{max (succ u1) (succ u2), succ u1, succ u2} (IsometryEquiv.{u1, u2} α β (PseudoMetricSpace.toPseudoEMetricSpace.{u1} α _inst_1) (PseudoMetricSpace.toPseudoEMetricSpace.{u2} β _inst_2)) α (fun (_x : α) => (fun (x._@.Mathlib.Data.FunLike.Embedding._hyg.19 : α) => β) _x) (EmbeddingLike.toFunLike.{max (succ u1) (succ u2), succ u1, succ u2} (IsometryEquiv.{u1, u2} α β (PseudoMetricSpace.toPseudoEMetricSpace.{u1} α _inst_1) (PseudoMetricSpace.toPseudoEMetricSpace.{u2} β _inst_2)) α β (EquivLike.toEmbeddingLike.{max (succ u1) (succ u2), succ u1, succ u2} (IsometryEquiv.{u1, u2} α β (PseudoMetricSpace.toPseudoEMetricSpace.{u1} α _inst_1) (PseudoMetricSpace.toPseudoEMetricSpace.{u2} β _inst_2)) α β (IsometryEquiv.instEquivLikeIsometryEquiv.{u1, u2} α β (PseudoMetricSpace.toPseudoEMetricSpace.{u1} α _inst_1) (PseudoMetricSpace.toPseudoEMetricSpace.{u2} β _inst_2)))) h) (Metric.closedBall.{u2} β _inst_2 x r)) (Metric.closedBall.{u1} ((fun (x._@.Mathlib.Data.FunLike.Embedding._hyg.19 : β) => α) x) _inst_1 (FunLike.coe.{max (succ u1) (succ u2), succ u2, succ u1} (IsometryEquiv.{u2, u1} β α (PseudoMetricSpace.toPseudoEMetricSpace.{u2} β _inst_2) (PseudoMetricSpace.toPseudoEMetricSpace.{u1} α _inst_1)) β (fun (_x : β) => (fun (x._@.Mathlib.Data.FunLike.Embedding._hyg.19 : β) => α) _x) (EmbeddingLike.toFunLike.{max (succ u1) (succ u2), succ u2, succ u1} (IsometryEquiv.{u2, u1} β α (PseudoMetricSpace.toPseudoEMetricSpace.{u2} β _inst_2) (PseudoMetricSpace.toPseudoEMetricSpace.{u1} α _inst_1)) β α (EquivLike.toEmbeddingLike.{max (succ u1) (succ u2), succ u2, succ u1} (IsometryEquiv.{u2, u1} β α (PseudoMetricSpace.toPseudoEMetricSpace.{u2} β _inst_2) (PseudoMetricSpace.toPseudoEMetricSpace.{u1} α _inst_1)) β α (IsometryEquiv.instEquivLikeIsometryEquiv.{u2, u1} β α (PseudoMetricSpace.toPseudoEMetricSpace.{u2} β _inst_2) (PseudoMetricSpace.toPseudoEMetricSpace.{u1} α _inst_1)))) (IsometryEquiv.symm.{u1, u2} α β (PseudoMetricSpace.toPseudoEMetricSpace.{u1} α _inst_1) (PseudoMetricSpace.toPseudoEMetricSpace.{u2} β _inst_2) h) x) r)
+Case conversion may be inaccurate. Consider using '#align isometry_equiv.preimage_closed_ball IsometryEquiv.preimage_closedBallₓ'. -/
 @[simp]
 theorem preimage_closedBall (h : α ≃ᵢ β) (x : β) (r : ℝ) :
     h ⁻¹' Metric.closedBall x r = Metric.closedBall (h.symm x) r := by
   rw [← h.isometry.preimage_closed_ball (h.symm x) r, h.apply_symm_apply]
 #align isometry_equiv.preimage_closed_ball IsometryEquiv.preimage_closedBall
 
+/- warning: isometry_equiv.image_ball -> IsometryEquiv.image_ball is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : PseudoMetricSpace.{u1} α] [_inst_2 : PseudoMetricSpace.{u2} β] (h : IsometryEquiv.{u1, u2} α β (PseudoMetricSpace.toPseudoEMetricSpace.{u1} α _inst_1) (PseudoMetricSpace.toPseudoEMetricSpace.{u2} β _inst_2)) (x : α) (r : Real), Eq.{succ u2} (Set.{u2} β) (Set.image.{u1, u2} α β (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (IsometryEquiv.{u1, u2} α β (PseudoMetricSpace.toPseudoEMetricSpace.{u1} α _inst_1) (PseudoMetricSpace.toPseudoEMetricSpace.{u2} β _inst_2)) (fun (_x : IsometryEquiv.{u1, u2} α β (PseudoMetricSpace.toPseudoEMetricSpace.{u1} α _inst_1) (PseudoMetricSpace.toPseudoEMetricSpace.{u2} β _inst_2)) => α -> β) (IsometryEquiv.hasCoeToFun.{u1, u2} α β (PseudoMetricSpace.toPseudoEMetricSpace.{u1} α _inst_1) (PseudoMetricSpace.toPseudoEMetricSpace.{u2} β _inst_2)) h) (Metric.ball.{u1} α _inst_1 x r)) (Metric.ball.{u2} β _inst_2 (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (IsometryEquiv.{u1, u2} α β (PseudoMetricSpace.toPseudoEMetricSpace.{u1} α _inst_1) (PseudoMetricSpace.toPseudoEMetricSpace.{u2} β _inst_2)) (fun (_x : IsometryEquiv.{u1, u2} α β (PseudoMetricSpace.toPseudoEMetricSpace.{u1} α _inst_1) (PseudoMetricSpace.toPseudoEMetricSpace.{u2} β _inst_2)) => α -> β) (IsometryEquiv.hasCoeToFun.{u1, u2} α β (PseudoMetricSpace.toPseudoEMetricSpace.{u1} α _inst_1) (PseudoMetricSpace.toPseudoEMetricSpace.{u2} β _inst_2)) h x) r)
+but is expected to have type
+  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : PseudoMetricSpace.{u1} α] [_inst_2 : PseudoMetricSpace.{u2} β] (h : IsometryEquiv.{u1, u2} α β (PseudoMetricSpace.toPseudoEMetricSpace.{u1} α _inst_1) (PseudoMetricSpace.toPseudoEMetricSpace.{u2} β _inst_2)) (x : α) (r : Real), Eq.{succ u2} (Set.{u2} β) (Set.image.{u1, u2} α β (FunLike.coe.{max (succ u1) (succ u2), succ u1, succ u2} (IsometryEquiv.{u1, u2} α β (PseudoMetricSpace.toPseudoEMetricSpace.{u1} α _inst_1) (PseudoMetricSpace.toPseudoEMetricSpace.{u2} β _inst_2)) α (fun (_x : α) => (fun (x._@.Mathlib.Data.FunLike.Embedding._hyg.19 : α) => β) _x) (EmbeddingLike.toFunLike.{max (succ u1) (succ u2), succ u1, succ u2} (IsometryEquiv.{u1, u2} α β (PseudoMetricSpace.toPseudoEMetricSpace.{u1} α _inst_1) (PseudoMetricSpace.toPseudoEMetricSpace.{u2} β _inst_2)) α β (EquivLike.toEmbeddingLike.{max (succ u1) (succ u2), succ u1, succ u2} (IsometryEquiv.{u1, u2} α β (PseudoMetricSpace.toPseudoEMetricSpace.{u1} α _inst_1) (PseudoMetricSpace.toPseudoEMetricSpace.{u2} β _inst_2)) α β (IsometryEquiv.instEquivLikeIsometryEquiv.{u1, u2} α β (PseudoMetricSpace.toPseudoEMetricSpace.{u1} α _inst_1) (PseudoMetricSpace.toPseudoEMetricSpace.{u2} β _inst_2)))) h) (Metric.ball.{u1} α _inst_1 x r)) (Metric.ball.{u2} ((fun (x._@.Mathlib.Data.FunLike.Embedding._hyg.19 : α) => β) x) _inst_2 (FunLike.coe.{max (succ u1) (succ u2), succ u1, succ u2} (IsometryEquiv.{u1, u2} α β (PseudoMetricSpace.toPseudoEMetricSpace.{u1} α _inst_1) (PseudoMetricSpace.toPseudoEMetricSpace.{u2} β _inst_2)) α (fun (_x : α) => (fun (x._@.Mathlib.Data.FunLike.Embedding._hyg.19 : α) => β) _x) (EmbeddingLike.toFunLike.{max (succ u1) (succ u2), succ u1, succ u2} (IsometryEquiv.{u1, u2} α β (PseudoMetricSpace.toPseudoEMetricSpace.{u1} α _inst_1) (PseudoMetricSpace.toPseudoEMetricSpace.{u2} β _inst_2)) α β (EquivLike.toEmbeddingLike.{max (succ u1) (succ u2), succ u1, succ u2} (IsometryEquiv.{u1, u2} α β (PseudoMetricSpace.toPseudoEMetricSpace.{u1} α _inst_1) (PseudoMetricSpace.toPseudoEMetricSpace.{u2} β _inst_2)) α β (IsometryEquiv.instEquivLikeIsometryEquiv.{u1, u2} α β (PseudoMetricSpace.toPseudoEMetricSpace.{u1} α _inst_1) (PseudoMetricSpace.toPseudoEMetricSpace.{u2} β _inst_2)))) h x) r)
+Case conversion may be inaccurate. Consider using '#align isometry_equiv.image_ball IsometryEquiv.image_ballₓ'. -/
 @[simp]
 theorem image_ball (h : α ≃ᵢ β) (x : α) (r : ℝ) : h '' Metric.ball x r = Metric.ball (h x) r := by
   rw [← h.preimage_symm, h.symm.preimage_ball, symm_symm]
 #align isometry_equiv.image_ball IsometryEquiv.image_ball
 
+/- warning: isometry_equiv.image_sphere -> IsometryEquiv.image_sphere is a dubious translation:
+lean 3 declaration is
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+but is expected to have type
+  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : PseudoMetricSpace.{u1} α] [_inst_2 : PseudoMetricSpace.{u2} β] (h : IsometryEquiv.{u1, u2} α β (PseudoMetricSpace.toPseudoEMetricSpace.{u1} α _inst_1) (PseudoMetricSpace.toPseudoEMetricSpace.{u2} β _inst_2)) (x : α) (r : Real), Eq.{succ u2} (Set.{u2} β) (Set.image.{u1, u2} α β (FunLike.coe.{max (succ u1) (succ u2), succ u1, succ u2} (IsometryEquiv.{u1, u2} α β (PseudoMetricSpace.toPseudoEMetricSpace.{u1} α _inst_1) (PseudoMetricSpace.toPseudoEMetricSpace.{u2} β _inst_2)) α (fun (_x : α) => (fun (x._@.Mathlib.Data.FunLike.Embedding._hyg.19 : α) => β) _x) (EmbeddingLike.toFunLike.{max (succ u1) (succ u2), succ u1, succ u2} (IsometryEquiv.{u1, u2} α β (PseudoMetricSpace.toPseudoEMetricSpace.{u1} α _inst_1) (PseudoMetricSpace.toPseudoEMetricSpace.{u2} β _inst_2)) α β (EquivLike.toEmbeddingLike.{max (succ u1) (succ u2), succ u1, succ u2} (IsometryEquiv.{u1, u2} α β (PseudoMetricSpace.toPseudoEMetricSpace.{u1} α _inst_1) (PseudoMetricSpace.toPseudoEMetricSpace.{u2} β _inst_2)) α β (IsometryEquiv.instEquivLikeIsometryEquiv.{u1, u2} α β (PseudoMetricSpace.toPseudoEMetricSpace.{u1} α _inst_1) (PseudoMetricSpace.toPseudoEMetricSpace.{u2} β _inst_2)))) h) (Metric.sphere.{u1} α _inst_1 x r)) (Metric.sphere.{u2} ((fun (x._@.Mathlib.Data.FunLike.Embedding._hyg.19 : α) => β) x) _inst_2 (FunLike.coe.{max (succ u1) (succ u2), succ u1, succ u2} (IsometryEquiv.{u1, u2} α β (PseudoMetricSpace.toPseudoEMetricSpace.{u1} α _inst_1) (PseudoMetricSpace.toPseudoEMetricSpace.{u2} β _inst_2)) α (fun (_x : α) => (fun (x._@.Mathlib.Data.FunLike.Embedding._hyg.19 : α) => β) _x) (EmbeddingLike.toFunLike.{max (succ u1) (succ u2), succ u1, succ u2} (IsometryEquiv.{u1, u2} α β (PseudoMetricSpace.toPseudoEMetricSpace.{u1} α _inst_1) (PseudoMetricSpace.toPseudoEMetricSpace.{u2} β _inst_2)) α β (EquivLike.toEmbeddingLike.{max (succ u1) (succ u2), succ u1, succ u2} (IsometryEquiv.{u1, u2} α β (PseudoMetricSpace.toPseudoEMetricSpace.{u1} α _inst_1) (PseudoMetricSpace.toPseudoEMetricSpace.{u2} β _inst_2)) α β (IsometryEquiv.instEquivLikeIsometryEquiv.{u1, u2} α β (PseudoMetricSpace.toPseudoEMetricSpace.{u1} α _inst_1) (PseudoMetricSpace.toPseudoEMetricSpace.{u2} β _inst_2)))) h x) r)
+Case conversion may be inaccurate. Consider using '#align isometry_equiv.image_sphere IsometryEquiv.image_sphereₓ'. -/
 @[simp]
 theorem image_sphere (h : α ≃ᵢ β) (x : α) (r : ℝ) :
     h '' Metric.sphere x r = Metric.sphere (h x) r := by
   rw [← h.preimage_symm, h.symm.preimage_sphere, symm_symm]
 #align isometry_equiv.image_sphere IsometryEquiv.image_sphere
 
+/- warning: isometry_equiv.image_closed_ball -> IsometryEquiv.image_closedBall is a dubious translation:
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+  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : PseudoMetricSpace.{u1} α] [_inst_2 : PseudoMetricSpace.{u2} β] (h : IsometryEquiv.{u1, u2} α β (PseudoMetricSpace.toPseudoEMetricSpace.{u1} α _inst_1) (PseudoMetricSpace.toPseudoEMetricSpace.{u2} β _inst_2)) (x : α) (r : Real), Eq.{succ u2} (Set.{u2} β) (Set.image.{u1, u2} α β (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (IsometryEquiv.{u1, u2} α β (PseudoMetricSpace.toPseudoEMetricSpace.{u1} α _inst_1) (PseudoMetricSpace.toPseudoEMetricSpace.{u2} β _inst_2)) (fun (_x : IsometryEquiv.{u1, u2} α β (PseudoMetricSpace.toPseudoEMetricSpace.{u1} α _inst_1) (PseudoMetricSpace.toPseudoEMetricSpace.{u2} β _inst_2)) => α -> β) (IsometryEquiv.hasCoeToFun.{u1, u2} α β (PseudoMetricSpace.toPseudoEMetricSpace.{u1} α _inst_1) (PseudoMetricSpace.toPseudoEMetricSpace.{u2} β _inst_2)) h) (Metric.closedBall.{u1} α _inst_1 x r)) (Metric.closedBall.{u2} β _inst_2 (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (IsometryEquiv.{u1, u2} α β (PseudoMetricSpace.toPseudoEMetricSpace.{u1} α _inst_1) (PseudoMetricSpace.toPseudoEMetricSpace.{u2} β _inst_2)) (fun (_x : IsometryEquiv.{u1, u2} α β (PseudoMetricSpace.toPseudoEMetricSpace.{u1} α _inst_1) (PseudoMetricSpace.toPseudoEMetricSpace.{u2} β _inst_2)) => α -> β) (IsometryEquiv.hasCoeToFun.{u1, u2} α β (PseudoMetricSpace.toPseudoEMetricSpace.{u1} α _inst_1) (PseudoMetricSpace.toPseudoEMetricSpace.{u2} β _inst_2)) h x) r)
+but is expected to have type
+  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : PseudoMetricSpace.{u1} α] [_inst_2 : PseudoMetricSpace.{u2} β] (h : IsometryEquiv.{u1, u2} α β (PseudoMetricSpace.toPseudoEMetricSpace.{u1} α _inst_1) (PseudoMetricSpace.toPseudoEMetricSpace.{u2} β _inst_2)) (x : α) (r : Real), Eq.{succ u2} (Set.{u2} β) (Set.image.{u1, u2} α β (FunLike.coe.{max (succ u1) (succ u2), succ u1, succ u2} (IsometryEquiv.{u1, u2} α β (PseudoMetricSpace.toPseudoEMetricSpace.{u1} α _inst_1) (PseudoMetricSpace.toPseudoEMetricSpace.{u2} β _inst_2)) α (fun (_x : α) => (fun (x._@.Mathlib.Data.FunLike.Embedding._hyg.19 : α) => β) _x) (EmbeddingLike.toFunLike.{max (succ u1) (succ u2), succ u1, succ u2} (IsometryEquiv.{u1, u2} α β (PseudoMetricSpace.toPseudoEMetricSpace.{u1} α _inst_1) (PseudoMetricSpace.toPseudoEMetricSpace.{u2} β _inst_2)) α β (EquivLike.toEmbeddingLike.{max (succ u1) (succ u2), succ u1, succ u2} (IsometryEquiv.{u1, u2} α β (PseudoMetricSpace.toPseudoEMetricSpace.{u1} α _inst_1) (PseudoMetricSpace.toPseudoEMetricSpace.{u2} β _inst_2)) α β (IsometryEquiv.instEquivLikeIsometryEquiv.{u1, u2} α β (PseudoMetricSpace.toPseudoEMetricSpace.{u1} α _inst_1) (PseudoMetricSpace.toPseudoEMetricSpace.{u2} β _inst_2)))) h) (Metric.closedBall.{u1} α _inst_1 x r)) (Metric.closedBall.{u2} ((fun (x._@.Mathlib.Data.FunLike.Embedding._hyg.19 : α) => β) x) _inst_2 (FunLike.coe.{max (succ u1) (succ u2), succ u1, succ u2} (IsometryEquiv.{u1, u2} α β (PseudoMetricSpace.toPseudoEMetricSpace.{u1} α _inst_1) (PseudoMetricSpace.toPseudoEMetricSpace.{u2} β _inst_2)) α (fun (_x : α) => (fun (x._@.Mathlib.Data.FunLike.Embedding._hyg.19 : α) => β) _x) (EmbeddingLike.toFunLike.{max (succ u1) (succ u2), succ u1, succ u2} (IsometryEquiv.{u1, u2} α β (PseudoMetricSpace.toPseudoEMetricSpace.{u1} α _inst_1) (PseudoMetricSpace.toPseudoEMetricSpace.{u2} β _inst_2)) α β (EquivLike.toEmbeddingLike.{max (succ u1) (succ u2), succ u1, succ u2} (IsometryEquiv.{u1, u2} α β (PseudoMetricSpace.toPseudoEMetricSpace.{u1} α _inst_1) (PseudoMetricSpace.toPseudoEMetricSpace.{u2} β _inst_2)) α β (IsometryEquiv.instEquivLikeIsometryEquiv.{u1, u2} α β (PseudoMetricSpace.toPseudoEMetricSpace.{u1} α _inst_1) (PseudoMetricSpace.toPseudoEMetricSpace.{u2} β _inst_2)))) h x) r)
+Case conversion may be inaccurate. Consider using '#align isometry_equiv.image_closed_ball IsometryEquiv.image_closedBallₓ'. -/
 @[simp]
 theorem image_closedBall (h : α ≃ᵢ β) (x : α) (r : ℝ) :
     h '' Metric.closedBall x r = Metric.closedBall (h x) r := by
@@ -656,6 +1048,7 @@ end PseudoMetricSpace
 
 end IsometryEquiv
 
+#print Isometry.isometryEquivOnRange /-
 /-- An isometry induces an isometric isomorphism between the source space and the
 range of the isometry. -/
 @[simps (config := { simpRhs := true }) toEquiv apply]
@@ -665,4 +1058,5 @@ def Isometry.isometryEquivOnRange [EMetricSpace α] [PseudoEMetricSpace β] {f :
   isometry_toFun x y := by simpa [Subtype.edist_eq] using h x y
   toEquiv := Equiv.ofInjective f h.Injective
 #align isometry.isometry_equiv_on_range Isometry.isometryEquivOnRange
+-/
 

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

@@ -282,7 +282,7 @@ induced metric space structure on the source space. -/
 theorem UniformEmbedding.to_isometry {α β} [UniformSpace α] [MetricSpace β] {f : α → β}
     (h : UniformEmbedding f) :
     @Isometry α β
-      (@PseudoMetricSpace.toPseudoEmetricSpace α
+      (@PseudoMetricSpace.toPseudoEMetricSpace α
         (@MetricSpace.toPseudoMetricSpace α (h.comapMetricSpace f)))
       (by infer_instance) f :=
   by
@@ -296,7 +296,7 @@ induced metric space structure on the source space. -/
 theorem Embedding.to_isometry {α β} [TopologicalSpace α] [MetricSpace β] {f : α → β}
     (h : Embedding f) :
     @Isometry α β
-      (@PseudoMetricSpace.toPseudoEmetricSpace α
+      (@PseudoMetricSpace.toPseudoEMetricSpace α
         (@MetricSpace.toPseudoMetricSpace α (h.comapMetricSpace f)))
       (by infer_instance) f :=
   by

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

@@ -35,7 +35,7 @@ open Topology ENNReal
 
 /-- An isometry (also known as isometric embedding) is a map preserving the edistance
 between pseudoemetric spaces, or equivalently the distance between pseudometric space.  -/
-def Isometry [PseudoEmetricSpace α] [PseudoEmetricSpace β] (f : α → β) : Prop :=
+def Isometry [PseudoEMetricSpace α] [PseudoEMetricSpace β] (f : α → β) : Prop :=
   ∀ x1 x2 : α, edist (f x1) (f x2) = edist x1 x2
 #align isometry Isometry
 
@@ -72,7 +72,7 @@ namespace Isometry
 
 section PseudoEmetricIsometry
 
-variable [PseudoEmetricSpace α] [PseudoEmetricSpace β] [PseudoEmetricSpace γ]
+variable [PseudoEMetricSpace α] [PseudoEMetricSpace β] [PseudoEMetricSpace γ]
 
 variable {f : α → β} {x y z : α} {s : Set α}
 
@@ -99,13 +99,13 @@ theorem isometry_subsingleton [Subsingleton α] : Isometry f := fun x y => by
 theorem isometry_id : Isometry (id : α → α) := fun x y => rfl
 #align isometry_id isometry_id
 
-theorem prod_map {δ} [PseudoEmetricSpace δ] {f : α → β} {g : γ → δ} (hf : Isometry f)
+theorem prod_map {δ} [PseudoEMetricSpace δ] {f : α → β} {g : γ → δ} (hf : Isometry f)
     (hg : Isometry g) : Isometry (Prod.map f g) := fun x y => by
   simp only [Prod.edist_eq, hf.edist_eq, hg.edist_eq, Prod_map]
 #align isometry.prod_map Isometry.prod_map
 
-theorem isometry_dcomp {ι} [Fintype ι] {α β : ι → Type _} [∀ i, PseudoEmetricSpace (α i)]
-    [∀ i, PseudoEmetricSpace (β i)] (f : ∀ i, α i → β i) (hf : ∀ i, Isometry (f i)) :
+theorem isometry_dcomp {ι} [Fintype ι] {α β : ι → Type _} [∀ i, PseudoEMetricSpace (α i)]
+    [∀ i, PseudoEMetricSpace (β i)] (f : ∀ i, α i → β i) (hf : ∀ i, Isometry (f i)) :
     Isometry (dcomp f) := fun x y => by simp only [edist_pi_def, (hf _).edist_eq]
 #align isometry_dcomp isometry_dcomp
 
@@ -140,37 +140,37 @@ theorem right_inv {f : α → β} {g : β → α} (h : Isometry f) (hg : RightIn
 #align isometry.right_inv Isometry.right_inv
 
 theorem preimage_emetric_closedBall (h : Isometry f) (x : α) (r : ℝ≥0∞) :
-    f ⁻¹' Emetric.closedBall (f x) r = Emetric.closedBall x r :=
+    f ⁻¹' EMetric.closedBall (f x) r = EMetric.closedBall x r :=
   by
   ext y
   simp [h.edist_eq]
 #align isometry.preimage_emetric_closed_ball Isometry.preimage_emetric_closedBall
 
 theorem preimage_emetric_ball (h : Isometry f) (x : α) (r : ℝ≥0∞) :
-    f ⁻¹' Emetric.ball (f x) r = Emetric.ball x r :=
+    f ⁻¹' EMetric.ball (f x) r = EMetric.ball x r :=
   by
   ext y
   simp [h.edist_eq]
 #align isometry.preimage_emetric_ball Isometry.preimage_emetric_ball
 
 /-- Isometries preserve the diameter in pseudoemetric spaces. -/
-theorem ediam_image (hf : Isometry f) (s : Set α) : Emetric.diam (f '' s) = Emetric.diam s :=
-  eq_of_forall_ge_iff fun d => by simp only [Emetric.diam_le_iff, ball_image_iff, hf.edist_eq]
+theorem ediam_image (hf : Isometry f) (s : Set α) : EMetric.diam (f '' s) = EMetric.diam s :=
+  eq_of_forall_ge_iff fun d => by simp only [EMetric.diam_le_iff, ball_image_iff, hf.edist_eq]
 #align isometry.ediam_image Isometry.ediam_image
 
-theorem ediam_range (hf : Isometry f) : Emetric.diam (range f) = Emetric.diam (univ : Set α) :=
+theorem ediam_range (hf : Isometry f) : EMetric.diam (range f) = EMetric.diam (univ : Set α) :=
   by
   rw [← image_univ]
   exact hf.ediam_image univ
 #align isometry.ediam_range Isometry.ediam_range
 
 theorem mapsTo_emetric_ball (hf : Isometry f) (x : α) (r : ℝ≥0∞) :
-    MapsTo f (Emetric.ball x r) (Emetric.ball (f x) r) :=
+    MapsTo f (EMetric.ball x r) (EMetric.ball (f x) r) :=
   (hf.preimage_emetric_ball x r).ge
 #align isometry.maps_to_emetric_ball Isometry.mapsTo_emetric_ball
 
 theorem mapsTo_emetric_closedBall (hf : Isometry f) (x : α) (r : ℝ≥0∞) :
-    MapsTo f (Emetric.closedBall x r) (Emetric.closedBall (f x) r) :=
+    MapsTo f (EMetric.closedBall x r) (EMetric.closedBall (f x) r) :=
   (hf.preimage_emetric_closedBall x r).ge
 #align isometry.maps_to_emetric_closed_ball Isometry.mapsTo_emetric_closedBall
 
@@ -193,7 +193,7 @@ end PseudoEmetricIsometry
 --section
 section EmetricIsometry
 
-variable [EmetricSpace α] [PseudoEmetricSpace β] {f : α → β}
+variable [EMetricSpace α] [PseudoEMetricSpace β] {f : α → β}
 
 /-- An isometry from an emetric space is injective -/
 protected theorem injective (h : Isometry f) : Injective f :=
@@ -211,7 +211,7 @@ protected theorem embedding (hf : Isometry f) : Embedding f :=
 #align isometry.embedding Isometry.embedding
 
 /-- An isometry from a complete emetric space is a closed embedding -/
-theorem closedEmbedding [CompleteSpace α] [EmetricSpace γ] {f : α → γ} (hf : Isometry f) :
+theorem closedEmbedding [CompleteSpace α] [EMetricSpace γ] {f : α → γ} (hf : Isometry f) :
     ClosedEmbedding f :=
   hf.antilipschitz.ClosedEmbedding hf.lipschitz.UniformContinuous
 #align isometry.closed_embedding Isometry.closedEmbedding
@@ -308,7 +308,7 @@ theorem Embedding.to_isometry {α β} [TopologicalSpace α] [MetricSpace β] {f
 -- such a bijection need not exist
 /-- `α` and `β` are isometric if there is an isometric bijection between them. -/
 @[nolint has_nonempty_instance]
-structure IsometryEquiv (α β : Type _) [PseudoEmetricSpace α] [PseudoEmetricSpace β] extends
+structure IsometryEquiv (α β : Type _) [PseudoEMetricSpace α] [PseudoEMetricSpace β] extends
   α ≃ β where
   isometry_toFun : Isometry to_fun
 #align isometry_equiv IsometryEquiv
@@ -318,9 +318,9 @@ infixl:25 " ≃ᵢ " => IsometryEquiv
 
 namespace IsometryEquiv
 
-section PseudoEmetricSpace
+section PseudoEMetricSpace
 
-variable [PseudoEmetricSpace α] [PseudoEmetricSpace β] [PseudoEmetricSpace γ]
+variable [PseudoEMetricSpace α] [PseudoEMetricSpace β] [PseudoEMetricSpace γ]
 
 instance : CoeFun (α ≃ᵢ β) fun _ => α → β :=
   ⟨fun e => e.toEquiv⟩
@@ -369,7 +369,7 @@ protected theorem continuous (h : α ≃ᵢ β) : Continuous h :=
 #align isometry_equiv.continuous IsometryEquiv.continuous
 
 @[simp]
-theorem ediam_image (h : α ≃ᵢ β) (s : Set α) : Emetric.diam (h '' s) = Emetric.diam s :=
+theorem ediam_image (h : α ≃ᵢ β) (s : Set α) : EMetric.diam (h '' s) = EMetric.diam s :=
   h.Isometry.ediam_image s
 #align isometry_equiv.ediam_image IsometryEquiv.ediam_image
 
@@ -386,7 +386,7 @@ theorem ext ⦃h₁ h₂ : α ≃ᵢ β⦄ (H : ∀ x, h₁ x = h₂ x) : h₁ =
 
 /-- Alternative constructor for isometric bijections,
 taking as input an isometry, and a right inverse. -/
-def mk' {α : Type u} [EmetricSpace α] (f : α → β) (g : β → α) (hfg : ∀ x, f (g x) = x)
+def mk' {α : Type u} [EMetricSpace α] (f : α → β) (g : β → α) (hfg : ∀ x, f (g x) = x)
     (hf : Isometry f) : α ≃ᵢ β where
   toFun := f
   invFun := g
@@ -396,7 +396,7 @@ def mk' {α : Type u} [EmetricSpace α] (f : α → β) (g : β → α) (hfg : 
 #align isometry_equiv.mk' IsometryEquiv.mk'
 
 /-- The identity isometry of a space. -/
-protected def refl (α : Type _) [PseudoEmetricSpace α] : α ≃ᵢ α :=
+protected def refl (α : Type _) [PseudoEMetricSpace α] : α ≃ᵢ α :=
   { Equiv.refl α with isometry_toFun := isometry_id }
 #align isometry_equiv.refl IsometryEquiv.refl
 
@@ -481,36 +481,36 @@ theorem symm_trans_apply (h₁ : α ≃ᵢ β) (h₂ : β ≃ᵢ γ) (x : γ) :
   rfl
 #align isometry_equiv.symm_trans_apply IsometryEquiv.symm_trans_apply
 
-theorem ediam_univ (h : α ≃ᵢ β) : Emetric.diam (univ : Set α) = Emetric.diam (univ : Set β) := by
+theorem ediam_univ (h : α ≃ᵢ β) : EMetric.diam (univ : Set α) = EMetric.diam (univ : Set β) := by
   rw [← h.range_eq_univ, h.isometry.ediam_range]
 #align isometry_equiv.ediam_univ IsometryEquiv.ediam_univ
 
 @[simp]
-theorem ediam_preimage (h : α ≃ᵢ β) (s : Set β) : Emetric.diam (h ⁻¹' s) = Emetric.diam s := by
+theorem ediam_preimage (h : α ≃ᵢ β) (s : Set β) : EMetric.diam (h ⁻¹' s) = EMetric.diam s := by
   rw [← image_symm, ediam_image]
 #align isometry_equiv.ediam_preimage IsometryEquiv.ediam_preimage
 
 @[simp]
 theorem preimage_emetric_ball (h : α ≃ᵢ β) (x : β) (r : ℝ≥0∞) :
-    h ⁻¹' Emetric.ball x r = Emetric.ball (h.symm x) r := by
+    h ⁻¹' EMetric.ball x r = EMetric.ball (h.symm x) r := by
   rw [← h.isometry.preimage_emetric_ball (h.symm x) r, h.apply_symm_apply]
 #align isometry_equiv.preimage_emetric_ball IsometryEquiv.preimage_emetric_ball
 
 @[simp]
 theorem preimage_emetric_closedBall (h : α ≃ᵢ β) (x : β) (r : ℝ≥0∞) :
-    h ⁻¹' Emetric.closedBall x r = Emetric.closedBall (h.symm x) r := by
+    h ⁻¹' EMetric.closedBall x r = EMetric.closedBall (h.symm x) r := by
   rw [← h.isometry.preimage_emetric_closed_ball (h.symm x) r, h.apply_symm_apply]
 #align isometry_equiv.preimage_emetric_closed_ball IsometryEquiv.preimage_emetric_closedBall
 
 @[simp]
 theorem image_emetric_ball (h : α ≃ᵢ β) (x : α) (r : ℝ≥0∞) :
-    h '' Emetric.ball x r = Emetric.ball (h x) r := by
+    h '' EMetric.ball x r = EMetric.ball (h x) r := by
   rw [← h.preimage_symm, h.symm.preimage_emetric_ball, symm_symm]
 #align isometry_equiv.image_emetric_ball IsometryEquiv.image_emetric_ball
 
 @[simp]
 theorem image_emetric_closedBall (h : α ≃ᵢ β) (x : α) (r : ℝ≥0∞) :
-    h '' Emetric.closedBall x r = Emetric.closedBall (h x) r := by
+    h '' EMetric.closedBall x r = EMetric.closedBall (h x) r := by
   rw [← h.preimage_symm, h.symm.preimage_emetric_closed_ball, symm_symm]
 #align isometry_equiv.image_emetric_closed_ball IsometryEquiv.image_emetric_closedBall
 
@@ -597,7 +597,7 @@ theorem completeSpace_iff (e : α ≃ᵢ β) : CompleteSpace α ↔ CompleteSpac
   exacts[e.symm.complete_space, e.complete_space]
 #align isometry_equiv.complete_space_iff IsometryEquiv.completeSpace_iff
 
-end PseudoEmetricSpace
+end PseudoEMetricSpace
 
 section PseudoMetricSpace
 
@@ -659,7 +659,7 @@ end IsometryEquiv
 /-- An isometry induces an isometric isomorphism between the source space and the
 range of the isometry. -/
 @[simps (config := { simpRhs := true }) toEquiv apply]
-def Isometry.isometryEquivOnRange [EmetricSpace α] [PseudoEmetricSpace β] {f : α → β}
+def Isometry.isometryEquivOnRange [EMetricSpace α] [PseudoEMetricSpace β] {f : α → β}
     (h : Isometry f) : α ≃ᵢ range f
     where
   isometry_toFun x y := by simpa [Subtype.edist_eq] using h x y

mathlib commit https://github.com/leanprover-community/mathlib/commit/eb0cb4511aaef0da2462207b67358a0e1fe1e2ee

@@ -31,7 +31,7 @@ variable {α : Type u} {β : Type v} {γ : Type w}
 
 open Function Set
 
-open Topology Ennreal
+open Topology ENNReal
 
 /-- An isometry (also known as isometric embedding) is a map preserving the edistance
 between pseudoemetric spaces, or equivalently the distance between pseudometric space.  -/
@@ -43,7 +43,7 @@ def Isometry [PseudoEmetricSpace α] [PseudoEmetricSpace β] (f : α → β) : P
 distances. -/
 theorem isometry_iff_nndist_eq [PseudoMetricSpace α] [PseudoMetricSpace β] {f : α → β} :
     Isometry f ↔ ∀ x y, nndist (f x) (f y) = nndist x y := by
-  simp only [Isometry, edist_nndist, Ennreal.coe_eq_coe]
+  simp only [Isometry, edist_nndist, ENNReal.coe_eq_coe]
 #align isometry_iff_nndist_eq isometry_iff_nndist_eq
 
 /-- On pseudometric spaces, a map is an isometry if and only if it preserves distances. -/
@@ -86,7 +86,7 @@ theorem lipschitz (h : Isometry f) : LipschitzWith 1 f :=
 #align isometry.lipschitz Isometry.lipschitz
 
 theorem antilipschitz (h : Isometry f) : AntilipschitzWith 1 f := fun x y => by
-  simp only [h x y, Ennreal.coe_one, one_mul, le_refl]
+  simp only [h x y, ENNReal.coe_one, one_mul, le_refl]
 #align isometry.antilipschitz Isometry.antilipschitz
 
 /-- Any map on a subsingleton is an isometry -/
@@ -614,7 +614,7 @@ theorem diam_preimage (s : Set β) : Metric.diam (h ⁻¹' s) = Metric.diam s :=
 #align isometry_equiv.diam_preimage IsometryEquiv.diam_preimage
 
 theorem diam_univ : Metric.diam (univ : Set α) = Metric.diam (univ : Set β) :=
-  congr_arg Ennreal.toReal h.ediam_univ
+  congr_arg ENNReal.toReal h.ediam_univ
 #align isometry_equiv.diam_univ IsometryEquiv.diam_univ
 
 @[simp]

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

Reference the newly created issues #12094 and #12096, as well as the pre-existing #5171. Change all references to #10927 to #5171. Some of these changes were not labelled as "porting note"; change this for good measure.

@@ -287,7 +287,7 @@ theorem Embedding.to_isometry {α β} [TopologicalSpace α] [MetricSpace β] {f
 
 -- such a bijection need not exist
 /-- `α` and `β` are isometric if there is an isometric bijection between them. -/
--- Porting note: was @[nolint has_nonempty_instance]
+-- Porting note(#5171): was @[nolint has_nonempty_instance]
 structure IsometryEquiv (α : Type u) (β : Type v) [PseudoEMetricSpace α] [PseudoEMetricSpace β]
     extends α ≃ β where
   isometry_toFun : Isometry toFun

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)

@@ -70,7 +70,6 @@ namespace Isometry
 section PseudoEmetricIsometry
 
 variable [PseudoEMetricSpace α] [PseudoEMetricSpace β] [PseudoEMetricSpace γ]
-
 variable {f : α → β} {x y z : α} {s : Set α}
 
 /-- An isometry preserves edistances. -/

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

@@ -303,7 +303,7 @@ section PseudoEMetricSpace
 
 variable [PseudoEMetricSpace α] [PseudoEMetricSpace β] [PseudoEMetricSpace γ]
 
--- Porting note: TODO: add `IsometryEquivClass`
+-- Porting note (#11215): TODO: add `IsometryEquivClass`
 
 theorem toEquiv_injective : Injective (toEquiv : (α ≃ᵢ β) → (α ≃ β))
   | ⟨_, _⟩, ⟨_, _⟩, rfl => rfl

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.

@@ -151,7 +151,7 @@ theorem preimage_emetric_ball (h : Isometry f) (x : α) (r : ℝ≥0∞) :
 
 /-- Isometries preserve the diameter in pseudoemetric spaces. -/
 theorem ediam_image (hf : Isometry f) (s : Set α) : EMetric.diam (f '' s) = EMetric.diam s :=
-  eq_of_forall_ge_iff fun d => by simp only [EMetric.diam_le_iff, ball_image_iff, hf.edist_eq]
+  eq_of_forall_ge_iff fun d => by simp only [EMetric.diam_le_iff, forall_mem_image, hf.edist_eq]
 #align isometry.ediam_image Isometry.ediam_image
 
 theorem ediam_range (hf : Isometry f) : EMetric.diam (range f) = EMetric.diam (univ : Set α) := by

Homogenises porting notes via capitalisation and addition of whitespace.

It makes the following changes:

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

@@ -288,7 +288,7 @@ theorem Embedding.to_isometry {α β} [TopologicalSpace α] [MetricSpace β] {f
 
 -- such a bijection need not exist
 /-- `α` and `β` are isometric if there is an isometric bijection between them. -/
--- porting note: was @[nolint has_nonempty_instance]
+-- Porting note: was @[nolint has_nonempty_instance]
 structure IsometryEquiv (α : Type u) (β : Type v) [PseudoEMetricSpace α] [PseudoEMetricSpace β]
     extends α ≃ β where
   isometry_toFun : Isometry toFun

Add a few missing lemmas about the coercion NNReal → Real. Remove a bunch of protected on the existing coercion lemmas (so that it matches the convention for other coercions). Rename NNReal.coe_eq to NNReal.coe_inj

From LeanAPAP

@@ -46,7 +46,7 @@ theorem isometry_iff_nndist_eq [PseudoMetricSpace α] [PseudoMetricSpace β] {f
 /-- On pseudometric spaces, a map is an isometry if and only if it preserves distances. -/
 theorem isometry_iff_dist_eq [PseudoMetricSpace α] [PseudoMetricSpace β] {f : α → β} :
     Isometry f ↔ ∀ x y, dist (f x) (f y) = dist x y := by
-  simp only [isometry_iff_nndist_eq, ← coe_nndist, NNReal.coe_eq]
+  simp only [isometry_iff_nndist_eq, ← coe_nndist, NNReal.coe_inj]
 #align isometry_iff_dist_eq isometry_iff_dist_eq
 
 /-- An isometry preserves distances. -/

by tweaking the definition of the AddMonoid and MonoidWithZero instances for WithTop. Also unprotect ENNReal.coe_injective and rename ENNReal.coe_eq_coe → ENNReal.coe_inj.

From LeanAPAP

@@ -40,7 +40,7 @@ def Isometry [PseudoEMetricSpace α] [PseudoEMetricSpace β] (f : α → β) : P
 distances. -/
 theorem isometry_iff_nndist_eq [PseudoMetricSpace α] [PseudoMetricSpace β] {f : α → β} :
     Isometry f ↔ ∀ x y, nndist (f x) (f y) = nndist x y := by
-  simp only [Isometry, edist_nndist, ENNReal.coe_eq_coe]
+  simp only [Isometry, edist_nndist, ENNReal.coe_inj]
 #align isometry_iff_nndist_eq isometry_iff_nndist_eq
 
 /-- On pseudometric spaces, a map is an isometry if and only if it preserves distances. -/

This prepares for the introduction of a non-dependent synonym of FunLike, which helps a lot with keeping #8386 readable.

This is entirely search-and-replace in 680197f combined with manual fixes in 4145626, e900597 and b8428f8. The commands that generated this change:

sed -i 's/\bFunLike\b/DFunLike/g' {Archive,Counterexamples,Mathlib,test}/**/*.lean
sed -i 's/\btoFunLike\b/toDFunLike/g' {Archive,Counterexamples,Mathlib,test}/**/*.lean
sed -i 's/import Mathlib.Data.DFunLike/import Mathlib.Data.FunLike/g' {Archive,Counterexamples,Mathlib,test}/**/*.lean
sed -i 's/\bHom_FunLike\b/Hom_DFunLike/g' {Archive,Counterexamples,Mathlib,test}/**/*.lean     
sed -i 's/\binstFunLike\b/instDFunLike/g' {Archive,Counterexamples,Mathlib,test}/**/*.lean
sed -i 's/\bfunLike\b/instDFunLike/g' {Archive,Counterexamples,Mathlib,test}/**/*.lean
sed -i 's/\btoo many metavariables to apply `fun_like.has_coe_to_fun`/too many metavariables to apply `DFunLike.hasCoeToFun`/g' {Archive,Counterexamples,Mathlib,test}/**/*.lean

Co-authored-by: Anne Baanen <Vierkantor@users.noreply.github.com>

@@ -317,7 +317,7 @@ instance : EquivLike (α ≃ᵢ β) α β where
   inv e := e.toEquiv.symm
   left_inv e := e.left_inv
   right_inv e := e.right_inv
-  coe_injective' _ _ h _ := toEquiv_injective <| FunLike.ext' h
+  coe_injective' _ _ h _ := toEquiv_injective <| DFunLike.ext' h
 
 theorem coe_eq_toEquiv (h : α ≃ᵢ β) (a : α) : h a = h.toEquiv a := rfl
 #align isometry_equiv.coe_eq_to_equiv IsometryEquiv.coe_eq_toEquiv
@@ -368,7 +368,7 @@ theorem ediam_image (h : α ≃ᵢ β) (s : Set α) : EMetric.diam (h '' s) = EM
 
 @[ext]
 theorem ext ⦃h₁ h₂ : α ≃ᵢ β⦄ (H : ∀ x, h₁ x = h₂ x) : h₁ = h₂ :=
-  FunLike.ext _ _ H
+  DFunLike.ext _ _ H
 #align isometry_equiv.ext IsometryEquiv.ext
 
 /-- Alternative constructor for isometric bijections,

Co-authored-by: Eric Wieser <wieser.eric@gmail.com> Co-authored-by: lines <34025592+linesthatinterlace@users.noreply.github.com>

@@ -420,6 +420,9 @@ initialize_simps_projections IsometryEquiv (toEquiv_toFun → apply, toEquiv_inv
 theorem symm_symm (h : α ≃ᵢ β) : h.symm.symm = h := rfl
 #align isometry_equiv.symm_symm IsometryEquiv.symm_symm
 
+theorem symm_bijective : Bijective (IsometryEquiv.symm : (α ≃ᵢ β) → β ≃ᵢ α) :=
+  Function.bijective_iff_has_inverse.mpr ⟨_, symm_symm, symm_symm⟩
+
 @[simp]
 theorem apply_symm_apply (h : α ≃ᵢ β) (y : β) : h (h.symm y) = y :=
   h.toEquiv.apply_symm_apply y

We have turned to Type* instead of Type _, but many of them remained in mathlib because the straight replacement did not work. In general, having Type _ before the colon is a code smell, though, as it hides which types should be in the same universe and which shouldn't, and is not very robust.

This PR replaces most of the remaining Type _ before the colon (except those in category theory) by Type* or Type u. This has uncovered a few bugs (where declarations were not as polymorphic as they should be).

I had to increase heartbeats at two places when replacing Type _ by Type*, but I think it's worth it as it's really more robust.

@@ -289,8 +289,8 @@ theorem Embedding.to_isometry {α β} [TopologicalSpace α] [MetricSpace β] {f
 -- such a bijection need not exist
 /-- `α` and `β` are isometric if there is an isometric bijection between them. -/
 -- porting note: was @[nolint has_nonempty_instance]
-structure IsometryEquiv (α β : Type _) [PseudoEMetricSpace α] [PseudoEMetricSpace β] extends
-  α ≃ β where
+structure IsometryEquiv (α : Type u) (β : Type v) [PseudoEMetricSpace α] [PseudoEMetricSpace β]
+    extends α ≃ β where
   isometry_toFun : Isometry toFun
 #align isometry_equiv IsometryEquiv
 

@@ -50,19 +50,19 @@ theorem isometry_iff_dist_eq [PseudoMetricSpace α] [PseudoMetricSpace β] {f :
 #align isometry_iff_dist_eq isometry_iff_dist_eq
 
 /-- An isometry preserves distances. -/
-alias isometry_iff_dist_eq ↔ Isometry.dist_eq _
+alias ⟨Isometry.dist_eq, _⟩ := isometry_iff_dist_eq
 #align isometry.dist_eq Isometry.dist_eq
 
 /-- A map that preserves distances is an isometry -/
-alias isometry_iff_dist_eq ↔ _ Isometry.of_dist_eq
+alias ⟨_, Isometry.of_dist_eq⟩ := isometry_iff_dist_eq
 #align isometry.of_dist_eq Isometry.of_dist_eq
 
 /-- An isometry preserves non-negative distances. -/
-alias isometry_iff_nndist_eq ↔ Isometry.nndist_eq _
+alias ⟨Isometry.nndist_eq, _⟩ := isometry_iff_nndist_eq
 #align isometry.nndist_eq Isometry.nndist_eq
 
 /-- A map that preserves non-negative distances is an isometry. -/
-alias isometry_iff_nndist_eq ↔ _ Isometry.of_nndist_eq
+alias ⟨_, Isometry.of_nndist_eq⟩ := isometry_iff_nndist_eq
 #align isometry.of_nndist_eq Isometry.of_nndist_eq
 
 namespace Isometry

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

This has nice performance benefits.

@@ -24,7 +24,7 @@ noncomputable section
 
 universe u v w
 
-variable {ι : Type _} {α : Type u} {β : Type v} {γ : Type w}
+variable {ι : Type*} {α : Type u} {β : Type v} {γ : Type w}
 
 open Function Set
 
@@ -101,7 +101,7 @@ theorem prod_map {δ} [PseudoEMetricSpace δ] {f : α → β} {g : γ → δ} (h
   simp only [Prod.edist_eq, hf.edist_eq, hg.edist_eq, Prod_map]
 #align isometry.prod_map Isometry.prod_map
 
-theorem _root_.isometry_dcomp {ι} [Fintype ι] {α β : ι → Type _} [∀ i, PseudoEMetricSpace (α i)]
+theorem _root_.isometry_dcomp {ι} [Fintype ι] {α β : ι → Type*} [∀ i, PseudoEMetricSpace (α i)]
     [∀ i, PseudoEMetricSpace (β i)] (f : ∀ i, α i → β i) (hf : ∀ i, Isometry (f i)) :
     Isometry (fun g : (i : ι) → α i => fun i => f i (g i)) := fun x y => by
   simp only [edist_pi_def, (hf _).edist_eq]
@@ -122,7 +122,7 @@ protected theorem uniformInducing (hf : Isometry f) : UniformInducing f :=
   hf.antilipschitz.uniformInducing hf.uniformContinuous
 #align isometry.uniform_inducing Isometry.uniformInducing
 
-theorem tendsto_nhds_iff {ι : Type _} {f : α → β} {g : ι → α} {a : Filter ι} {b : α}
+theorem tendsto_nhds_iff {ι : Type*} {f : α → β} {g : ι → α} {a : Filter ι} {b : α}
     (hf : Isometry f) : Filter.Tendsto g a (𝓝 b) ↔ Filter.Tendsto (f ∘ g) a (𝓝 (f b)) :=
   hf.uniformInducing.inducing.tendsto_nhds_iff
 #align isometry.tendsto_nhds_iff Isometry.tendsto_nhds_iff
@@ -347,12 +347,12 @@ protected theorem edist_eq (h : α ≃ᵢ β) (x y : α) : edist (h x) (h y) = e
   h.isometry.edist_eq x y
 #align isometry_equiv.edist_eq IsometryEquiv.edist_eq
 
-protected theorem dist_eq {α β : Type _} [PseudoMetricSpace α] [PseudoMetricSpace β] (h : α ≃ᵢ β)
+protected theorem dist_eq {α β : Type*} [PseudoMetricSpace α] [PseudoMetricSpace β] (h : α ≃ᵢ β)
     (x y : α) : dist (h x) (h y) = dist x y :=
   h.isometry.dist_eq x y
 #align isometry_equiv.dist_eq IsometryEquiv.dist_eq
 
-protected theorem nndist_eq {α β : Type _} [PseudoMetricSpace α] [PseudoMetricSpace β] (h : α ≃ᵢ β)
+protected theorem nndist_eq {α β : Type*} [PseudoMetricSpace α] [PseudoMetricSpace β] (h : α ≃ᵢ β)
     (x y : α) : nndist (h x) (h y) = nndist x y :=
   h.isometry.nndist_eq x y
 #align isometry_equiv.nndist_eq IsometryEquiv.nndist_eq
@@ -383,7 +383,7 @@ def mk' {α : Type u} [EMetricSpace α] (f : α → β) (g : β → α) (hfg : 
 #align isometry_equiv.mk' IsometryEquiv.mk'
 
 /-- The identity isometry of a space. -/
-protected def refl (α : Type _) [PseudoEMetricSpace α] : α ≃ᵢ α :=
+protected def refl (α : Type*) [PseudoEMetricSpace α] : α ≃ᵢ α :=
   { Equiv.refl α with isometry_toFun := isometry_id }
 #align isometry_equiv.refl IsometryEquiv.refl
 
@@ -578,7 +578,7 @@ def funUnique [Unique ι] [Fintype ι] : (ι → α) ≃ᵢ α where
 
 /-- `piFinTwoEquiv` as an `IsometryEquiv`. -/
 @[simps!]
-def piFinTwo (α : Fin 2 → Type _) [∀ i, PseudoEMetricSpace (α i)] : (∀ i, α i) ≃ᵢ α 0 × α 1 where
+def piFinTwo (α : Fin 2 → Type*) [∀ i, PseudoEMetricSpace (α i)] : (∀ i, α i) ≃ᵢ α 0 × α 1 where
   toEquiv := piFinTwoEquiv α
   isometry_toFun x hx := by simp [edist_pi_def, Fin.univ_succ, Prod.edist_eq]
 #align isometry_equiv.pi_fin_two IsometryEquiv.piFinTwo

• depends on: #5966

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

@@ -3,14 +3,11 @@ Copyright (c) 2018 Sébastien Gouëzel. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Isometries of emetric and metric spaces
 Authors: Sébastien Gouëzel
-
-! This file was ported from Lean 3 source module topology.metric_space.isometry
-! leanprover-community/mathlib commit b1859b6d4636fdbb78c5d5cefd24530653cfd3eb
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
 -/
 import Mathlib.Topology.MetricSpace.Antilipschitz
 
+#align_import topology.metric_space.isometry from "leanprover-community/mathlib"@"b1859b6d4636fdbb78c5d5cefd24530653cfd3eb"
+
 /-!
 # Isometries
 

• leanprover-community/mathlib#19025 (already changed while porting)
• leanprover-community/mathlib#18982

@@ -5,7 +5,7 @@ Isometries of emetric and metric spaces
 Authors: Sébastien Gouëzel
 
 ! This file was ported from Lean 3 source module topology.metric_space.isometry
-! leanprover-community/mathlib commit 832a8ba8f10f11fea99367c469ff802e69a5b8ec
+! leanprover-community/mathlib commit b1859b6d4636fdbb78c5d5cefd24530653cfd3eb
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
@@ -19,7 +19,7 @@ the edistance (on metric spaces, these are exactly the maps that preserve distan
 and prove their basic properties. We also introduce isometric bijections.
 
 Since a lot of elementary properties don't require `eq_of_dist_eq_zero` we start setting up the
-theory for `pseudo_metric_space` and we specialize to `metric_space` when needed.
+theory for `PseudoMetricSpace` and we specialize to `MetricSpace` when needed.
 -/
 
 
@@ -27,7 +27,7 @@ noncomputable section
 
 universe u v w
 
-variable {α : Type u} {β : Type v} {γ : Type w}
+variable {ι : Type _} {α : Type u} {β : Type v} {γ : Type w}
 
 open Function Set
 
@@ -570,6 +570,22 @@ protected theorem completeSpace [CompleteSpace β] (e : α ≃ᵢ β) : Complete
   e.completeSpace_iff.2 ‹_›
 #align isometry_equiv.complete_space IsometryEquiv.completeSpace
 
+variable (ι α)
+
+/-- `Equiv.funUnique` as an `IsometryEquiv`. -/
+@[simps!]
+def funUnique [Unique ι] [Fintype ι] : (ι → α) ≃ᵢ α where
+  toEquiv := Equiv.funUnique ι α
+  isometry_toFun x hx := by simp [edist_pi_def, Finset.univ_unique, Finset.sup_singleton]
+#align isometry_equiv.fun_unique IsometryEquiv.funUnique
+
+/-- `piFinTwoEquiv` as an `IsometryEquiv`. -/
+@[simps!]
+def piFinTwo (α : Fin 2 → Type _) [∀ i, PseudoEMetricSpace (α i)] : (∀ i, α i) ≃ᵢ α 0 × α 1 where
+  toEquiv := piFinTwoEquiv α
+  isometry_toFun x hx := by simp [edist_pi_def, Fin.univ_succ, Prod.edist_eq]
+#align isometry_equiv.pi_fin_two IsometryEquiv.piFinTwo
+
 end PseudoEMetricSpace
 
 section PseudoMetricSpace