ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Isometry.mapsTo_sphere
theorem mapsTo_sphere (hf : Isometry f) (x : α) (r : ℝ) :
    MapsTo f (Metric.sphere x r) (Metric.sphere (f x) r) 
Arguments: {α : Type u} {β : Type v} [PseudoMetricSpace α] [PseudoMetricSpace β] {f : α → β} (hf : Isometry f) (x : α) (r : ℝ)
Goal: Set.MapsTo f (Metric.sphere x r) (Metric.sphere (f x) r)
-- Isometry.mapsTo_closedBall
theorem mapsTo_closedBall (hf : Isometry f) (x : α) (r : ℝ) :
    MapsTo f (Metric.closedBall x r) (Metric.closedBall (f x) r) 
Arguments: {α : Type u} {β : Type v} [PseudoMetricSpace α] [PseudoMetricSpace β] {f : α → β} (hf : Isometry f) (x : α) (r : ℝ)
Goal: Set.MapsTo f (Metric.closedBall x r) (Metric.closedBall (f x) r)
-- UniformEmbedding.to_isometry
theorem UniformEmbedding.to_isometry {α β} [UniformSpace α] [MetricSpace β] {f : α → β}
    (h : UniformEmbedding f) : (letI := h.comapMetricSpace f; Isometry f) 
Arguments: {α : Type u_2} {β : Type u_3} [UniformSpace α] [MetricSpace β] {f : α → β} (h : UniformEmbedding f)
Goal: Isometry f
-- Embedding.to_isometry
theorem Embedding.to_isometry {α β} [TopologicalSpace α] [MetricSpace β] {f : α → β}
    (h : Embedding f) : (letI := h.comapMetricSpace f; Isometry f) 
Arguments: {α : Type u_2} {β : Type u_3} [TopologicalSpace α] [MetricSpace β] {f : α → β} (h : Embedding f)
Goal: Isometry f
-- IsometryEquiv.isometry_toFun
-- 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
Arguments: {α : Type u} {β : Type v} [PseudoEMetricSpace α] [PseudoEMetricSpace β] (self : α ≃ᵢ β)
Goal: Isometry self.toFun
-- IsometryEquiv.toEquiv_injective
theorem toEquiv_injective : Injective (toEquiv : (α ≃ᵢ β) → (α ≃ β))
  
Arguments: {α : Type u} {β : Type v} [PseudoEMetricSpace α] [PseudoEMetricSpace β]
Goal: Function.Injective IsometryEquiv.toEquiv
-- IsometryEquiv.toEquiv_inj
@[simp] theorem toEquiv_inj {e₁ e₂ : α ≃ᵢ β} : e₁.toEquiv = e₂.toEquiv ↔ e₁ = e₂ 
Arguments: {α : Type u} {β : Type v} [PseudoEMetricSpace α] [PseudoEMetricSpace β] {e₁ e₂ : α ≃ᵢ β}
Goal: e₁.toEquiv = e₂.toEquiv ↔ e₁ = e₂
-- IsometryEquiv.coe_eq_toEquiv
theorem coe_eq_toEquiv (h : α ≃ᵢ β) (a : α) : h a = h.toEquiv a 
Arguments: {α : Type u} {β : Type v} [PseudoEMetricSpace α] [PseudoEMetricSpace β] (h : α ≃ᵢ β) (a : α)
Goal: h a = h.toEquiv a
-- IsometryEquiv.coe_toEquiv
@[simp] theorem coe_toEquiv (h : α ≃ᵢ β) : ⇑h.toEquiv = h 
Arguments: {α : Type u} {β : Type v} [PseudoEMetricSpace α] [PseudoEMetricSpace β] (h : α ≃ᵢ β)
Goal: ⇑h.toEquiv = ⇑h
-- IsometryEquiv.coe_mk
@[simp] theorem coe_mk (e : α ≃ β) (h) : ⇑(mk e h) = e 
Arguments: {α : Type u} {β : Type v} [PseudoEMetricSpace α] [PseudoEMetricSpace β] (e : α ≃ β) (h : Isometry e.toFun)
Goal: ⇑{ toEquiv := e, isometry_toFun := h } = ⇑e
-- IsometryEquiv.isometry
protected theorem isometry (h : α ≃ᵢ β) : Isometry h 
Arguments: {α : Type u} {β : Type v} [PseudoEMetricSpace α] [PseudoEMetricSpace β] (h : α ≃ᵢ β)
Goal: Isometry ⇑h
-- IsometryEquiv.bijective
protected theorem bijective (h : α ≃ᵢ β) : Bijective h 
Arguments: {α : Type u} {β : Type v} [PseudoEMetricSpace α] [PseudoEMetricSpace β] (h : α ≃ᵢ β)
Goal: Function.Bijective ⇑h
-- IsometryEquiv.injective
protected theorem injective (h : α ≃ᵢ β) : Injective h 
Arguments: {α : Type u} {β : Type v} [PseudoEMetricSpace α] [PseudoEMetricSpace β] (h : α ≃ᵢ β)
Goal: Function.Injective ⇑h
-- IsometryEquiv.surjective
protected theorem surjective (h : α ≃ᵢ β) : Surjective h 
Arguments: {α : Type u} {β : Type v} [PseudoEMetricSpace α] [PseudoEMetricSpace β] (h : α ≃ᵢ β)
Goal: Function.Surjective ⇑h
-- IsometryEquiv.edist_eq
protected theorem edist_eq (h : α ≃ᵢ β) (x y : α) : edist (h x) (h y) = edist x y 
Arguments: {α : Type u} {β : Type v} [PseudoEMetricSpace α] [PseudoEMetricSpace β] (h : α ≃ᵢ β) (x y : α)
Goal: edist (h x) (h y) = edist x y
-- 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 
Arguments: {α : Type u_2} {β : Type u_3} [PseudoMetricSpace α] [PseudoMetricSpace β] (h : α ≃ᵢ β) (x y : α)
Goal: dist (h x) (h y) = dist x y
-- 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 
Arguments: {α : Type u_2} {β : Type u_3} [PseudoMetricSpace α] [PseudoMetricSpace β] (h : α ≃ᵢ β) (x y : α)
Goal: nndist (h x) (h y) = nndist x y
-- IsometryEquiv.continuous
protected theorem continuous (h : α ≃ᵢ β) : Continuous h 
Arguments: {α : Type u} {β : Type v} [PseudoEMetricSpace α] [PseudoEMetricSpace β] (h : α ≃ᵢ β)
Goal: Continuous ⇑h
-- IsometryEquiv.ediam_image
@[simp]
theorem ediam_image (h : α ≃ᵢ β) (s : Set α) : EMetric.diam (h '' s) = EMetric.diam s 
Arguments: {α : Type u} {β : Type v} [PseudoEMetricSpace α] [PseudoEMetricSpace β] (h : α ≃ᵢ β) (s : Set α)
Goal: EMetric.diam (⇑h '' s) = EMetric.diam s
-- IsometryEquiv.ext
@[ext]
theorem ext ⦃h₁ h₂ : α ≃ᵢ β⦄ (H : ∀ x, h₁ x = h₂ x) : h₁ = h₂ 
Arguments: {α : Type u} {β : Type v} [PseudoEMetricSpace α] [PseudoEMetricSpace β] ⦃h₁ h₂ : α ≃ᵢ β⦄ (H : ∀ (x : α), h₁ x = h₂ x)
Goal: h₁ = h₂
-- IsometryEquiv.trans_apply
@[simp]
theorem trans_apply (h₁ : α ≃ᵢ β) (h₂ : β ≃ᵢ γ) (x : α) : h₁.trans h₂ x = h₂ (h₁ x) 
Arguments: {α : Type u} {β : Type v} {γ : Type w} [PseudoEMetricSpace α] [PseudoEMetricSpace β] [PseudoEMetricSpace γ] (h₁ : α ≃ᵢ β) (h₂ : β ≃ᵢ γ) (x : α)
Goal: (h₁.trans h₂) x = h₂ (h₁ x)
-- IsometryEquiv.symm_symm
@[simp]
theorem symm_symm (h : α ≃ᵢ β) : h.symm.symm = h 
Arguments: {α : Type u} {β : Type v} [PseudoEMetricSpace α] [PseudoEMetricSpace β] (h : α ≃ᵢ β)
Goal: h.symm.symm = h
-- IsometryEquiv.symm_bijective
theorem symm_bijective : Bijective (IsometryEquiv.symm : (α ≃ᵢ β) → β ≃ᵢ α) 
Arguments: {α : Type u} {β : Type v} [PseudoEMetricSpace α] [PseudoEMetricSpace β]
Goal: Function.Bijective IsometryEquiv.symm
-- IsometryEquiv.apply_symm_apply
@[simp]
theorem apply_symm_apply (h : α ≃ᵢ β) (y : β) : h (h.symm y) = y 
Arguments: {α : Type u} {β : Type v} [PseudoEMetricSpace α] [PseudoEMetricSpace β] (h : α ≃ᵢ β) (y : β)
Goal: h (h.symm y) = y
-- IsometryEquiv.symm_apply_apply
@[simp]
theorem symm_apply_apply (h : α ≃ᵢ β) (x : α) : h.symm (h x) = x 
Arguments: {α : Type u} {β : Type v} [PseudoEMetricSpace α] [PseudoEMetricSpace β] (h : α ≃ᵢ β) (x : α)
Goal: h.symm (h x) = x
-- IsometryEquiv.symm_apply_eq
theorem symm_apply_eq (h : α ≃ᵢ β) {x : α} {y : β} : h.symm y = x ↔ y = h x 
Arguments: {α : Type u} {β : Type v} [PseudoEMetricSpace α] [PseudoEMetricSpace β] (h : α ≃ᵢ β) {x : α} {y : β}
Goal: h.symm y = x ↔ y = h x
-- IsometryEquiv.eq_symm_apply
theorem eq_symm_apply (h : α ≃ᵢ β) {x : α} {y : β} : x = h.symm y ↔ h x = y 
Arguments: {α : Type u} {β : Type v} [PseudoEMetricSpace α] [PseudoEMetricSpace β] (h : α ≃ᵢ β) {x : α} {y : β}
Goal: x = h.symm y ↔ h x = y
-- IsometryEquiv.symm_comp_self
theorem symm_comp_self (h : α ≃ᵢ β) : (h.symm : β → α) ∘ h = id 
Arguments: {α : Type u} {β : Type v} [PseudoEMetricSpace α] [PseudoEMetricSpace β] (h : α ≃ᵢ β)
Goal: ⇑h.symm ∘ ⇑h = id
-- IsometryEquiv.self_comp_symm
theorem self_comp_symm (h : α ≃ᵢ β) : (h : α → β) ∘ h.symm = id 
Arguments: {α : Type u} {β : Type v} [PseudoEMetricSpace α] [PseudoEMetricSpace β] (h : α ≃ᵢ β)
Goal: ⇑h ∘ ⇑h.symm = id
-- IsometryEquiv.range_eq_univ
@[simp]
theorem range_eq_univ (h : α ≃ᵢ β) : range h = univ 
Arguments: {α : Type u} {β : Type v} [PseudoEMetricSpace α] [PseudoEMetricSpace β] (h : α ≃ᵢ β)
Goal: Set.range ⇑h = Set.univ
-- IsometryEquiv.image_symm
theorem image_symm (h : α ≃ᵢ β) : image h.symm = preimage h 
Arguments: {α : Type u} {β : Type v} [PseudoEMetricSpace α] [PseudoEMetricSpace β] (h : α ≃ᵢ β)
Goal: Set.image ⇑h.symm = Set.preimage ⇑h
-- IsometryEquiv.preimage_symm
theorem preimage_symm (h : α ≃ᵢ β) : preimage h.symm = image h 
Arguments: {α : Type u} {β : Type v} [PseudoEMetricSpace α] [PseudoEMetricSpace β] (h : α ≃ᵢ β)
Goal: Set.preimage ⇑h.symm = Set.image ⇑h
-- 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) 
Arguments: {α : Type u} {β : Type v} {γ : Type w} [PseudoEMetricSpace α] [PseudoEMetricSpace β] [PseudoEMetricSpace γ] (h₁ : α ≃ᵢ β) (h₂ : β ≃ᵢ γ) (x : γ)
Goal: (h₁.trans h₂).symm x = h₁.symm (h₂.symm x)
-- IsometryEquiv.ediam_univ
theorem ediam_univ (h : α ≃ᵢ β) : EMetric.diam (univ : Set α) = EMetric.diam (univ : Set β) 
Arguments: {α : Type u} {β : Type v} [PseudoEMetricSpace α] [PseudoEMetricSpace β] (h : α ≃ᵢ β)
Goal: EMetric.diam Set.univ = EMetric.diam Set.univ
-- IsometryEquiv.ediam_preimage
@[simp]
theorem ediam_preimage (h : α ≃ᵢ β) (s : Set β) : EMetric.diam (h ⁻¹' s) = EMetric.diam s 
Arguments: {α : Type u} {β : Type v} [PseudoEMetricSpace α] [PseudoEMetricSpace β] (h : α ≃ᵢ β) (s : Set β)
Goal: EMetric.diam (⇑h ⁻¹' s) = EMetric.diam s
-- 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 
Arguments: {α : Type u} {β : Type v} [PseudoEMetricSpace α] [PseudoEMetricSpace β] (h : α ≃ᵢ β) (x : β) (r : ℝ≥0∞)
Goal: ⇑h ⁻¹' EMetric.ball x r = EMetric.ball (h.symm x) r
-- 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 
Arguments: {α : Type u} {β : Type v} [PseudoEMetricSpace α] [PseudoEMetricSpace β] (h : α ≃ᵢ β) (x : β) (r : ℝ≥0∞)
Goal: ⇑h ⁻¹' EMetric.closedBall x r = EMetric.closedBall (h.symm x) r
-- 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 
Arguments: {α : Type u} {β : Type v} [PseudoEMetricSpace α] [PseudoEMetricSpace β] (h : α ≃ᵢ β) (x : α) (r : ℝ≥0∞)
Goal: ⇑h '' EMetric.ball x r = EMetric.ball (h x) r
-- 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 
Arguments: {α : Type u} {β : Type v} [PseudoEMetricSpace α] [PseudoEMetricSpace β] (h : α ≃ᵢ β) (x : α) (r : ℝ≥0∞)
Goal: ⇑h '' EMetric.closedBall x r = EMetric.closedBall (h x) r
-- IsometryEquiv.toHomeomorph_toEquiv
@[simps toEquiv]
protected def toHomeomorph (h : α ≃ᵢ β) : α ≃ₜ β 
Arguments: {α : Type u} {β : Type v} [PseudoEMetricSpace α] [PseudoEMetricSpace β] (h : α ≃ᵢ β)
Goal: h.toHomeomorph.toEquiv = h.toEquiv
-- IsometryEquiv.coe_toHomeomorph
@[simp]
theorem coe_toHomeomorph (h : α ≃ᵢ β) : ⇑h.toHomeomorph = h 
Arguments: {α : Type u} {β : Type v} [PseudoEMetricSpace α] [PseudoEMetricSpace β] (h : α ≃ᵢ β)
Goal: ⇑h.toHomeomorph = ⇑h
-- IsometryEquiv.coe_toHomeomorph_symm
@[simp]
theorem coe_toHomeomorph_symm (h : α ≃ᵢ β) : ⇑h.toHomeomorph.symm = h.symm 
Arguments: {α : Type u} {β : Type v} [PseudoEMetricSpace α] [PseudoEMetricSpace β] (h : α ≃ᵢ β)
Goal: ⇑h.toHomeomorph.symm = ⇑h.symm
-- 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 
Arguments: {α : Type u} {β : Type v} [PseudoEMetricSpace α] [PseudoEMetricSpace β] {γ : Type u_2} [TopologicalSpace γ] (h : α ≃ᵢ β) {f : γ → α} {s : Set γ}
Goal: ContinuousOn (⇑h ∘ f) s ↔ ContinuousOn f s
-- IsometryEquiv.comp_continuous_iff
@[simp]
theorem comp_continuous_iff {γ} [TopologicalSpace γ] (h : α ≃ᵢ β) {f : γ → α} :
    Continuous (h ∘ f) ↔ Continuous f 
Arguments: {α : Type u} {β : Type v} [PseudoEMetricSpace α] [PseudoEMetricSpace β] {γ : Type u_2} [TopologicalSpace γ] (h : α ≃ᵢ β) {f : γ → α}
Goal: Continuous (⇑h ∘ f) ↔ Continuous f
-- IsometryEquiv.comp_continuous_iff'
@[simp]
theorem comp_continuous_iff' {γ} [TopologicalSpace γ] (h : α ≃ᵢ β) {f : β → γ} :
    Continuous (f ∘ h) ↔ Continuous f 
Arguments: {α : Type u} {β : Type v} [PseudoEMetricSpace α] [PseudoEMetricSpace β] {γ : Type u_2} [TopologicalSpace γ] (h : α ≃ᵢ β) {f : β → γ}
Goal: Continuous (f ∘ ⇑h) ↔ Continuous f
-- IsometryEquiv.coe_one
@[simp] theorem coe_one : ⇑(1 : α ≃ᵢ α) = id 
Arguments: {α : Type u} [PseudoEMetricSpace α]
Goal: ⇑1 = id
-- IsometryEquiv.coe_mul
@[simp] theorem coe_mul (e₁ e₂ : α ≃ᵢ α) : ⇑(e₁ * e₂) = e₁ ∘ e₂ 
Arguments: {α : Type u} [PseudoEMetricSpace α] (e₁ e₂ : α ≃ᵢ α)
Goal: ⇑(e₁ * e₂) = ⇑e₁ ∘ ⇑e₂
-- IsometryEquiv.mul_apply
theorem mul_apply (e₁ e₂ : α ≃ᵢ α) (x : α) : (e₁ * e₂) x = e₁ (e₂ x) 
Arguments: {α : Type u} [PseudoEMetricSpace α] (e₁ e₂ : α ≃ᵢ α) (x : α)
Goal: (e₁ * e₂) x = e₁ (e₂ x)
-- IsometryEquiv.inv_apply_self
@[simp] theorem inv_apply_self (e : α ≃ᵢ α) (x : α) : e⁻¹ (e x) = x 
Arguments: {α : Type u} [PseudoEMetricSpace α] (e : α ≃ᵢ α) (x : α)
Goal: e⁻¹ (e x) = x
-- IsometryEquiv.apply_inv_self
@[simp] theorem apply_inv_self (e : α ≃ᵢ α) (x : α) : e (e⁻¹ x) = x 
Arguments: {α : Type u} [PseudoEMetricSpace α] (e : α ≃ᵢ α) (x : α)
Goal: e (e⁻¹ x) = x
-- IsometryEquiv.completeSpace_iff
theorem completeSpace_iff (e : α ≃ᵢ β) : CompleteSpace α ↔ CompleteSpace β 
Arguments: {α : Type u} {β : Type v} [PseudoEMetricSpace α] [PseudoEMetricSpace β] (e : α ≃ᵢ β)
Goal: CompleteSpace α ↔ CompleteSpace β
-- IsometryEquiv.completeSpace
protected theorem completeSpace [CompleteSpace β] (e : α ≃ᵢ β) : CompleteSpace α 
Arguments: {α : Type u} {β : Type v} [PseudoEMetricSpace α] [PseudoEMetricSpace β] [CompleteSpace β] (e : α ≃ᵢ β)
Goal: CompleteSpace α
-- IsometryEquiv.funUnique_symm_apply
@[simps!]
def funUnique [Unique ι] [Fintype ι] : (ι → α) ≃ᵢ α 
Arguments: (ι : Type u_1) (α : Type u) [PseudoEMetricSpace α] [Unique ι] [Fintype ι] (x : α) (i : ι)
Goal: (IsometryEquiv.funUnique ι α).symm x i = x
-- IsometryEquiv.funUnique_apply
@[simps!]
def funUnique [Unique ι] [Fintype ι] : (ι → α) ≃ᵢ α 
Arguments: (ι : Type u_1) (α : Type u) [PseudoEMetricSpace α] [Unique ι] [Fintype ι] (f : (i : ι) → (fun a => α) i)
Goal: (IsometryEquiv.funUnique ι α) f = f default
-- IsometryEquiv.funUnique_toFun
@[simps!]
def funUnique [Unique ι] [Fintype ι] : (ι → α) ≃ᵢ α 
Arguments: (ι : Type u_1) (α : Type u) [PseudoEMetricSpace α] [Unique ι] [Fintype ι] (f : (i : ι) → (fun a => α) i)
Goal: (IsometryEquiv.funUnique ι α) f = f default
-- IsometryEquiv.funUnique_invFun
@[simps!]
def funUnique [Unique ι] [Fintype ι] : (ι → α) ≃ᵢ α 
Arguments: (ι : Type u_1) (α : Type u) [PseudoEMetricSpace α] [Unique ι] [Fintype ι] (x : α) (i : ι)
Goal: (IsometryEquiv.funUnique ι α).invFun x i = x
-- IsometryEquiv.piFinTwo_toFun
@[simps!]
def piFinTwo (α : Fin 2 → Type*) [∀ i, PseudoEMetricSpace (α i)] : (∀ i, α i) ≃ᵢ α 0 × α 1 
Arguments: (α : Fin 2 → Type u_2) [(i : Fin 2) → PseudoEMetricSpace (α i)] (f : (i : Fin 2) → α i)
Goal: (IsometryEquiv.piFinTwo α) f = (f 0, f 1)
-- IsometryEquiv.piFinTwo_invFun
@[simps!]
def piFinTwo (α : Fin 2 → Type*) [∀ i, PseudoEMetricSpace (α i)] : (∀ i, α i) ≃ᵢ α 0 × α 1 
Arguments: (α : Fin 2 → Type u_2) [(i : Fin 2) → PseudoEMetricSpace (α i)] (p : α 0 × α 1) (i : Fin (1 + 1))
Goal: (IsometryEquiv.piFinTwo α).invFun p i = Fin.cons p.1 (Fin.cons p.2 finZeroElim) i
-- IsometryEquiv.piFinTwo_symm_apply
@[simps!]
def piFinTwo (α : Fin 2 → Type*) [∀ i, PseudoEMetricSpace (α i)] : (∀ i, α i) ≃ᵢ α 0 × α 1 
Arguments: (α : Fin 2 → Type u_2) [(i : Fin 2) → PseudoEMetricSpace (α i)] (p : α 0 × α 1) (i : Fin (1 + 1))
Goal: (IsometryEquiv.piFinTwo α).symm p i = Fin.cons p.1 (Fin.cons p.2 finZeroElim) i
-- IsometryEquiv.piFinTwo_apply
@[simps!]
def piFinTwo (α : Fin 2 → Type*) [∀ i, PseudoEMetricSpace (α i)] : (∀ i, α i) ≃ᵢ α 0 × α 1 
Arguments: (α : Fin 2 → Type u_2) [(i : Fin 2) → PseudoEMetricSpace (α i)] (f : (i : Fin 2) → α i)
Goal: (IsometryEquiv.piFinTwo α) f = (f 0, f 1)
-- IsometryEquiv.diam_image
@[simp]
theorem diam_image (s : Set α) : Metric.diam (h '' s) = Metric.diam s 
Arguments: {α : Type u} {β : Type v} [PseudoMetricSpace α] [PseudoMetricSpace β] (h : α ≃ᵢ β) (s : Set α)
Goal: Metric.diam (⇑h '' s) = Metric.diam s
-- IsometryEquiv.diam_preimage
@[simp]
theorem diam_preimage (s : Set β) : Metric.diam (h ⁻¹' s) = Metric.diam s 
Arguments: {α : Type u} {β : Type v} [PseudoMetricSpace α] [PseudoMetricSpace β] (h : α ≃ᵢ β) (s : Set β)
Goal: Metric.diam (⇑h ⁻¹' s) = Metric.diam s
-- IsometryEquiv.diam_univ
theorem diam_univ : Metric.diam (univ : Set α) = Metric.diam (univ : Set β) 
Arguments: {α : Type u} {β : Type v} [PseudoMetricSpace α] [PseudoMetricSpace β] (h : α ≃ᵢ β)
Goal: Metric.diam Set.univ = Metric.diam Set.univ
-- 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 
Arguments: {α : Type u} {β : Type v} [PseudoMetricSpace α] [PseudoMetricSpace β] (h : α ≃ᵢ β) (x : β) (r : ℝ)
Goal: ⇑h ⁻¹' Metric.ball x r = Metric.ball (h.symm x) r
-- 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 
Arguments: {α : Type u} {β : Type v} [PseudoMetricSpace α] [PseudoMetricSpace β] (h : α ≃ᵢ β) (x : β) (r : ℝ)
Goal: ⇑h ⁻¹' Metric.sphere x r = Metric.sphere (h.symm x) r
-- 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 
Arguments: {α : Type u} {β : Type v} [PseudoMetricSpace α] [PseudoMetricSpace β] (h : α ≃ᵢ β) (x : β) (r : ℝ)
Goal: ⇑h ⁻¹' Metric.closedBall x r = Metric.closedBall (h.symm x) r
-- 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 
Arguments: {α : Type u} {β : Type v} [PseudoMetricSpace α] [PseudoMetricSpace β] (h : α ≃ᵢ β) (x : α) (r : ℝ)
Goal: ⇑h '' Metric.ball x r = Metric.ball (h x) r
-- 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 
Arguments: {α : Type u} {β : Type v} [PseudoMetricSpace α] [PseudoMetricSpace β] (h : α ≃ᵢ β) (x : α) (r : ℝ)
Goal: ⇑h '' Metric.sphere x r = Metric.sphere (h x) r
-- 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 
Arguments: {α : Type u} {β : Type v} [PseudoMetricSpace α] [PseudoMetricSpace β] (h : α ≃ᵢ β) (x : α) (r : ℝ)
Goal: ⇑h '' Metric.closedBall x r = Metric.closedBall (h x) r
-- Isometry.isometryEquivOnRange_toEquiv
@[simps! (config := { simpRhs := true }) toEquiv apply]
def Isometry.isometryEquivOnRange [EMetricSpace α] [PseudoEMetricSpace β] {f : α → β}
    (h : Isometry f) : α ≃ᵢ range f 
Arguments: {α : Type u} {β : Type v} [EMetricSpace α] [PseudoEMetricSpace β] {f : α → β} (h : Isometry f)
Goal: h.isometryEquivOnRange.toEquiv = Equiv.ofInjective f ⋯
-- Isometry.isometryEquivOnRange_apply
@[simps! (config := { simpRhs := true }) toEquiv apply]
def Isometry.isometryEquivOnRange [EMetricSpace α] [PseudoEMetricSpace β] {f : α → β}
    (h : Isometry f) : α ≃ᵢ range f 
Arguments: {α : Type u} {β : Type v} [EMetricSpace α] [PseudoEMetricSpace β] {f : α → β} (h : Isometry f) (a : α)
Goal: h.isometryEquivOnRange a = ⟨f a, ⋯⟩
-- Isometry.lipschitzWith_iff
lemma Isometry.lipschitzWith_iff {α β γ : Type*} [PseudoEMetricSpace α] [PseudoEMetricSpace β]
    [PseudoEMetricSpace γ] {f : α → β} {g : β → γ} (K : ℝ≥0) (h : Isometry g) :
    LipschitzWith K (g ∘ f) ↔ LipschitzWith K f 
Arguments: {α : Type u_2} {β : Type u_3} {γ : Type u_4} [PseudoEMetricSpace α] [PseudoEMetricSpace β] [PseudoEMetricSpace γ] {f : α → β} {g : β → γ} (K : NNReal) (h : Isometry g)
Goal: LipschitzWith K (g ∘ f) ↔ LipschitzWith K f
-- CauSeq.tendsto_limit
theorem CauSeq.tendsto_limit [NormedRing β] [hn : IsAbsoluteValue (norm : β → ℝ)]
    (f : CauSeq β norm) [CauSeq.IsComplete β norm] : Tendsto f atTop (𝓝 f.lim) 
Arguments: {β : Type v} [NormedRing β] [hn : IsAbsoluteValue norm] (f : CauSeq β norm) [CauSeq.IsComplete β norm]
Goal: Filter.Tendsto (↑f) Filter.atTop (𝓝 f.lim)
-- CauchySeq.isCauSeq
theorem CauchySeq.isCauSeq {f : ℕ → β} (hf : CauchySeq f) : IsCauSeq norm f 
Arguments: {β : Type v} [NormedField β] {f : ℕ → β} (hf : CauchySeq f)
Goal: IsCauSeq norm f
-- CauSeq.cauchySeq
theorem CauSeq.cauchySeq (f : CauSeq β norm) : CauchySeq f 
Arguments: {β : Type v} [NormedField β] (f : CauSeq β norm)
Goal: CauchySeq ↑f
-- isCauSeq_iff_cauchySeq
theorem isCauSeq_iff_cauchySeq {α : Type u} [NormedField α] {u : ℕ → α} :
    IsCauSeq norm u ↔ CauchySeq u 
Arguments: {α : Type u} [NormedField α] {u : ℕ → α}
Goal: IsCauSeq norm u ↔ CauchySeq u
-- ProperSpace.isCompact_closedBall
class ProperSpace (α : Type u) [PseudoMetricSpace α] : Prop where
  isCompact_closedBall : ∀ x : α, ∀ r, IsCompact (closedBall x r)
Arguments: {α : Type u} [PseudoMetricSpace α] [self : ProperSpace α] (x : α) (r : ℝ)
Goal: IsCompact (Metric.closedBall x r)
-- isCompact_sphere
theorem isCompact_sphere {α : Type*} [PseudoMetricSpace α] [ProperSpace α] (x : α) (r : ℝ) :
    IsCompact (sphere x r) 
Arguments: {α : Type u_3} [PseudoMetricSpace α] [ProperSpace α] (x : α) (r : ℝ)
Goal: IsCompact (Metric.sphere x r)
-- ProperSpace.of_isCompact_closedBall_of_le
theorem ProperSpace.of_isCompact_closedBall_of_le (R : ℝ)
    (h : ∀ x : α, ∀ r, R ≤ r → IsCompact (closedBall x r)) : ProperSpace α 
Arguments: {α : Type u} [PseudoMetricSpace α] (R : ℝ) (h : ∀ (x : α) (r : ℝ), R ≤ r → IsCompact (Metric.closedBall x r))
Goal: ProperSpace α
-- ProperSpace.of_seq_closedBall
theorem ProperSpace.of_seq_closedBall {β : Type*} {l : Filter β} [NeBot l] {x : α} {r : β → ℝ}
    (hr : Tendsto r l atTop) (hc : ∀ᶠ i in l, IsCompact (closedBall x (r i))) :
    ProperSpace α 
Arguments: {α : Type u} [PseudoMetricSpace α] {β : Type u_3} {l : Filter β} [l.NeBot] {x : α} {r : β → ℝ} (hr : Filter.Tendsto r l Filter.atTop) (hc : ∀ᶠ (i : β) in l, IsCompact (Metric.closedBall x (r i)))
Goal: ProperSpace α
-- Metric.complete_of_convergent_controlled_sequences
theorem Metric.complete_of_convergent_controlled_sequences (B : ℕ → Real) (hB : ∀ n, 0 < B n)
    (H : ∀ u : ℕ → α, (∀ N n m : ℕ, N ≤ n → N ≤ m → dist (u n) (u m) < B N) →
      ∃ x, Tendsto u atTop (𝓝 x)) :
    CompleteSpace α 
Arguments: {α : Type u} [PseudoMetricSpace α] (B : ℕ → ℝ) (hB : ∀ (n : ℕ), 0 < B n) (H : ∀ (u : ℕ → α), (∀ (N n m : ℕ), N ≤ n → N ≤ m → dist (u n) (u m) < B N) → ∃ x, Filter.Tendsto u Filter.atTop (𝓝 x))
Goal: CompleteSpace α
-- Metric.complete_of_cauchySeq_tendsto
theorem Metric.complete_of_cauchySeq_tendsto :
    (∀ u : ℕ → α, CauchySeq u → ∃ a, Tendsto u atTop (𝓝 a)) → CompleteSpace α 
Arguments: {α : Type u} [PseudoMetricSpace α]
Goal: (∀ (u : ℕ → α), CauchySeq u → ∃ a, Filter.Tendsto u Filter.atTop (𝓝 a)) → CompleteSpace α
-- Metric.cauchySeq_iff
-- Porting note: @[nolint ge_or_gt] doesn't exist
theorem Metric.cauchySeq_iff {u : β → α} :
    CauchySeq u ↔ ∀ ε > 0, ∃ N, ∀ m ≥ N, ∀ n ≥ N, dist (u m) (u n) < ε 
Arguments: {α : Type u} {β : Type v} [PseudoMetricSpace α] [Nonempty β] [SemilatticeSup β] {u : β → α}
Goal: CauchySeq u ↔ ∀ ε > 0, ∃ N, ∀ m ≥ N, ∀ n ≥ N, dist (u m) (u n) < ε
-- Metric.cauchySeq_iff'
theorem Metric.cauchySeq_iff' {u : β → α} :
    CauchySeq u ↔ ∀ ε > 0, ∃ N, ∀ n ≥ N, dist (u n) (u N) < ε 
Arguments: {α : Type u} {β : Type v} [PseudoMetricSpace α] [Nonempty β] [SemilatticeSup β] {u : β → α}
Goal: CauchySeq u ↔ ∀ ε > 0, ∃ N, ∀ n ≥ N, dist (u n) (u N) < ε
-- Metric.uniformCauchySeqOn_iff
-- Porting note: no attr @[nolint ge_or_gt]
theorem Metric.uniformCauchySeqOn_iff {γ : Type*} {F : β → γ → α} {s : Set γ} :
    UniformCauchySeqOn F atTop s ↔ ∀ ε > (0 : ℝ),
      ∃ N : β, ∀ m ≥ N, ∀ n ≥ N, ∀ x ∈ s, dist (F m x) (F n x) < ε 
Arguments: {α : Type u} {β : Type v} [PseudoMetricSpace α] [Nonempty β] [SemilatticeSup β] {γ : Type u_3} {F : β → γ → α} {s : Set γ}
Goal: UniformCauchySeqOn F Filter.atTop s ↔ ∀ ε > 0, ∃ N, ∀ m ≥ N, ∀ n ≥ N, ∀ x ∈ s, dist (F m x) (F n x) < ε
-- cauchySeq_of_le_tendsto_0'
theorem cauchySeq_of_le_tendsto_0' {s : β → α} (b : β → ℝ)
    (h : ∀ n m : β, n ≤ m → dist (s n) (s m) ≤ b n) (h₀ : Tendsto b atTop (𝓝 0)) : CauchySeq s 
Arguments: {α : Type u} {β : Type v} [PseudoMetricSpace α] [Nonempty β] [SemilatticeSup β] {s : β → α} (b : β → ℝ) (h : ∀ (n m : β), n ≤ m → dist (s n) (s m) ≤ b n) (h₀ : Filter.Tendsto b Filter.atTop (𝓝 0))
Goal: CauchySeq s