ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Perfect.exists_nat_bool_injection
theorem Perfect.exists_nat_bool_injection [CompleteSpace α] :
    ∃ f : (ℕ → Bool) → α, range f ⊆ C ∧ Continuous f ∧ Injective f 
Arguments: {α : Type u_1} [MetricSpace α] {C : Set α} (hC : Perfect C) (hnonempty : C.Nonempty) [CompleteSpace α]
Goal: ∃ f, Set.range f ⊆ C ∧ Continuous f ∧ Function.Injective f
-- IsClosed.exists_nat_bool_injection_of_not_countable
theorem IsClosed.exists_nat_bool_injection_of_not_countable {α : Type*} [TopologicalSpace α]
    [PolishSpace α] {C : Set α} (hC : IsClosed C) (hunc : ¬C.Countable) :
    ∃ f : (ℕ → Bool) → α, range f ⊆ C ∧ Continuous f ∧ Function.Injective f 
Arguments: {α : Type u_1} [TopologicalSpace α] [PolishSpace α] {C : Set α} (hC : IsClosed C) (hunc : ¬C.Countable)
Goal: ∃ f, Set.range f ⊆ C ∧ Continuous f ∧ Function.Injective f
-- DilationEquivClass.edist_eq'
class DilationEquivClass [EquivLike F X Y] : Prop where
  edist_eq' : ∀ f : F, ∃ r : ℝ≥0, r ≠ 0 ∧ ∀ x y : X, edist (f x) (f y) = r * edist x y
Arguments: {F : Type u_1} {X : outParam (Type u_2)} {Y : outParam (Type u_3)} [PseudoEMetricSpace X] [PseudoEMetricSpace Y] [EquivLike F X Y] [self : DilationEquivClass F X Y] (f : F)
Goal: ∃ r, r ≠ 0 ∧ ∀ (x y : X), edist (f x) (f y) = ↑r * edist x y
-- DilationEquiv.coe_toEquiv
@[simp] theorem coe_toEquiv (e : X ≃ᵈ Y) : ⇑e.toEquiv = e 
Arguments: {X : Type u_1} {Y : Type u_2} [PseudoEMetricSpace X] [PseudoEMetricSpace Y] (e : X ≃ᵈ Y)
Goal: ⇑e.toEquiv = ⇑e
-- DilationEquiv.ext
@[ext]
protected theorem ext {e e' : X ≃ᵈ Y} (h : ∀ x, e x = e' x) : e = e' 
Arguments: {X : Type u_1} {Y : Type u_2} [PseudoEMetricSpace X] [PseudoEMetricSpace Y] {e e' : X ≃ᵈ Y} (h : ∀ (x : X), e x = e' x)
Goal: e = e'
-- DilationEquiv.symm_symm
@[simp] theorem symm_symm (e : X ≃ᵈ Y) : e.symm.symm = e 
Arguments: {X : Type u_1} {Y : Type u_2} [PseudoEMetricSpace X] [PseudoEMetricSpace Y] (e : X ≃ᵈ Y)
Goal: e.symm.symm = e
-- DilationEquiv.symm_bijective
theorem symm_bijective : Function.Bijective (DilationEquiv.symm : (X ≃ᵈ Y) → Y ≃ᵈ X) 
Arguments: {X : Type u_1} {Y : Type u_2} [PseudoEMetricSpace X] [PseudoEMetricSpace Y]
Goal: Function.Bijective DilationEquiv.symm
-- DilationEquiv.apply_symm_apply
@[simp] theorem apply_symm_apply (e : X ≃ᵈ Y) (x : Y) : e (e.symm x) = x 
Arguments: {X : Type u_1} {Y : Type u_2} [PseudoEMetricSpace X] [PseudoEMetricSpace Y] (e : X ≃ᵈ Y) (x : Y)
Goal: e (e.symm x) = x
-- DilationEquiv.symm_apply_apply
@[simp] theorem symm_apply_apply (e : X ≃ᵈ Y) (x : X) : e.symm (e x) = x 
Arguments: {X : Type u_1} {Y : Type u_2} [PseudoEMetricSpace X] [PseudoEMetricSpace Y] (e : X ≃ᵈ Y) (x : X)
Goal: e.symm (e x) = x
-- DilationEquiv.ratio_toDilation
lemma ratio_toDilation (e : X ≃ᵈ Y) : ratio e.toDilation = ratio e 
Arguments: {X : Type u_1} {Y : Type u_2} [PseudoEMetricSpace X] [PseudoEMetricSpace Y] (e : X ≃ᵈ Y)
Goal: Dilation.ratio e.toDilation = Dilation.ratio e
-- DilationEquiv.refl_apply
@[simps! (config := .asFn) apply]
def refl (X : Type*) [PseudoEMetricSpace X] : X ≃ᵈ X 
Arguments: (X : Type u_4) [PseudoEMetricSpace X]
Goal: ⇑(DilationEquiv.refl X) = id
-- DilationEquiv.refl_symm
@[simp] theorem refl_symm : (refl X).symm = refl X 
Arguments: {X : Type u_1} [PseudoEMetricSpace X]
Goal: (DilationEquiv.refl X).symm = DilationEquiv.refl X
-- DilationEquiv.ratio_refl
@[simp] theorem ratio_refl : ratio (refl X) = 1 
Arguments: {X : Type u_1} [PseudoEMetricSpace X]
Goal: Dilation.ratio (DilationEquiv.refl X) = 1
-- DilationEquiv.trans_apply
@[simps! (config := .asFn) apply]
def trans (e₁ : X ≃ᵈ Y) (e₂ : Y ≃ᵈ Z) : X ≃ᵈ Z 
Arguments: {X : Type u_1} {Y : Type u_2} {Z : Type u_3} [PseudoEMetricSpace X] [PseudoEMetricSpace Y] [PseudoEMetricSpace Z] (e₁ : X ≃ᵈ Y) (e₂ : Y ≃ᵈ Z)
Goal: ⇑(e₁.trans e₂) = ⇑e₂ ∘ ⇑e₁
-- DilationEquiv.refl_trans
@[simp] theorem refl_trans (e : X ≃ᵈ Y) : (refl X).trans e = e 
Arguments: {X : Type u_1} {Y : Type u_2} [PseudoEMetricSpace X] [PseudoEMetricSpace Y] (e : X ≃ᵈ Y)
Goal: (DilationEquiv.refl X).trans e = e
-- DilationEquiv.trans_refl
@[simp] theorem trans_refl (e : X ≃ᵈ Y) : e.trans (refl Y) = e 
Arguments: {X : Type u_1} {Y : Type u_2} [PseudoEMetricSpace X] [PseudoEMetricSpace Y] (e : X ≃ᵈ Y)
Goal: e.trans (DilationEquiv.refl Y) = e
-- DilationEquiv.symm_trans_self
@[simp] theorem symm_trans_self (e : X ≃ᵈ Y) : e.symm.trans e = refl Y 
Arguments: {X : Type u_1} {Y : Type u_2} [PseudoEMetricSpace X] [PseudoEMetricSpace Y] (e : X ≃ᵈ Y)
Goal: e.symm.trans e = DilationEquiv.refl Y
-- DilationEquiv.self_trans_symm
@[simp] theorem self_trans_symm (e : X ≃ᵈ Y) : e.trans e.symm = refl X 
Arguments: {X : Type u_1} {Y : Type u_2} [PseudoEMetricSpace X] [PseudoEMetricSpace Y] (e : X ≃ᵈ Y)
Goal: e.trans e.symm = DilationEquiv.refl X
-- DilationEquiv.surjective
protected theorem surjective (e : X ≃ᵈ Y) : Surjective e 
Arguments: {X : Type u_1} {Y : Type u_2} [PseudoEMetricSpace X] [PseudoEMetricSpace Y] (e : X ≃ᵈ Y)
Goal: Function.Surjective ⇑e
-- DilationEquiv.bijective
protected theorem bijective (e : X ≃ᵈ Y) : Bijective e 
Arguments: {X : Type u_1} {Y : Type u_2} [PseudoEMetricSpace X] [PseudoEMetricSpace Y] (e : X ≃ᵈ Y)
Goal: Function.Bijective ⇑e
-- DilationEquiv.injective
protected theorem injective (e : X ≃ᵈ Y) : Injective e 
Arguments: {X : Type u_1} {Y : Type u_2} [PseudoEMetricSpace X] [PseudoEMetricSpace Y] (e : X ≃ᵈ Y)
Goal: Function.Injective ⇑e
-- DilationEquiv.ratio_trans
@[simp]
theorem ratio_trans (e : X ≃ᵈ Y) (e' : Y ≃ᵈ Z) : ratio (e.trans e') = ratio e * ratio e' 
Arguments: {X : Type u_1} {Y : Type u_2} {Z : Type u_3} [PseudoEMetricSpace X] [PseudoEMetricSpace Y] [PseudoEMetricSpace Z] (e : X ≃ᵈ Y) (e' : Y ≃ᵈ Z)
Goal: Dilation.ratio (e.trans e') = Dilation.ratio e * Dilation.ratio e'
-- DilationEquiv.ratio_symm
@[simp]
theorem ratio_symm (e : X ≃ᵈ Y) : ratio e.symm = (ratio e)⁻¹ 
Arguments: {X : Type u_1} {Y : Type u_2} [PseudoEMetricSpace X] [PseudoEMetricSpace Y] (e : X ≃ᵈ Y)
Goal: Dilation.ratio e.symm = (Dilation.ratio e)⁻¹
-- DilationEquiv.mul_def
theorem mul_def (e e' : X ≃ᵈ X) : e * e' = e'.trans e 
Arguments: {X : Type u_1} [PseudoEMetricSpace X] (e e' : X ≃ᵈ X)
Goal: e * e' = e'.trans e
-- DilationEquiv.one_def
theorem one_def : (1 : X ≃ᵈ X) = refl X 
Arguments: {X : Type u_1} [PseudoEMetricSpace X]
Goal: 1 = DilationEquiv.refl X
-- DilationEquiv.inv_def
theorem inv_def (e : X ≃ᵈ X) : e⁻¹ = e.symm 
Arguments: {X : Type u_1} [PseudoEMetricSpace X] (e : X ≃ᵈ X)
Goal: e⁻¹ = e.symm
-- DilationEquiv.coe_mul
@[simp] theorem coe_mul (e e' : X ≃ᵈ X) : ⇑(e * e') = e ∘ e' 
Arguments: {X : Type u_1} [PseudoEMetricSpace X] (e e' : X ≃ᵈ X)
Goal: ⇑(e * e') = ⇑e ∘ ⇑e'
-- DilationEquiv.coe_one
@[simp] theorem coe_one : ⇑(1 : X ≃ᵈ X) = id 
Arguments: {X : Type u_1} [PseudoEMetricSpace X]
Goal: ⇑1 = id
-- DilationEquiv.coe_inv
theorem coe_inv (e : X ≃ᵈ X) : ⇑(e⁻¹) = e.symm 
Arguments: {X : Type u_1} [PseudoEMetricSpace X] (e : X ≃ᵈ X)
Goal: ⇑e⁻¹ = ⇑e.symm
-- DilationEquiv.ratio_inv
@[simp]
theorem ratio_inv (e : X ≃ᵈ X) : ratio (e⁻¹) = (ratio e)⁻¹ 
Arguments: {X : Type u_1} [PseudoEMetricSpace X] (e : X ≃ᵈ X)
Goal: Dilation.ratio e⁻¹ = (Dilation.ratio e)⁻¹
-- DilationEquiv.ratio_pow
@[simp]
theorem ratio_pow (e : X ≃ᵈ X) (n : ℕ) : ratio (e ^ n) = ratio e ^ n 
Arguments: {X : Type u_1} [PseudoEMetricSpace X] (e : X ≃ᵈ X) (n : ℕ)
Goal: Dilation.ratio (e ^ n) = Dilation.ratio e ^ n
-- DilationEquiv.ratio_zpow
@[simp]
theorem ratio_zpow (e : X ≃ᵈ X) (n : ℤ) : ratio (e ^ n) = ratio e ^ n 
Arguments: {X : Type u_1} [PseudoEMetricSpace X] (e : X ≃ᵈ X) (n : ℤ)
Goal: Dilation.ratio (e ^ n) = Dilation.ratio e ^ n
-- DilationEquiv.toPerm_apply
@[simps]
def toPerm : (X ≃ᵈ X) →* Equiv.Perm X 
Arguments: {X : Type u_1} [PseudoEMetricSpace X] (e : X ≃ᵈ X)
Goal: DilationEquiv.toPerm e = e.toEquiv
-- DilationEquiv.coe_pow
@[norm_cast]
theorem coe_pow (e : X ≃ᵈ X) (n : ℕ) : ⇑(e ^ n) = e^[n] 
Arguments: {X : Type u_1} [PseudoEMetricSpace X] (e : X ≃ᵈ X) (n : ℕ)
Goal: ⇑(e ^ n) = (⇑e)^[n]
-- IsometryEquiv.toDilationEquiv_apply
@[simp]
lemma _root_.IsometryEquiv.toDilationEquiv_apply (e : X ≃ᵢ Y) (x : X) :
    e.toDilationEquiv x = e x 
Arguments: {X : Type u_1} {Y : Type u_2} [PseudoEMetricSpace X] [PseudoEMetricSpace Y] (e : X ≃ᵢ Y) (x : X)
Goal: e.toDilationEquiv x = e x
-- IsometryEquiv.toDilationEquiv_symm
@[simp]
lemma _root_.IsometryEquiv.toDilationEquiv_symm (e : X ≃ᵢ Y) :
    e.toDilationEquiv.symm = e.symm.toDilationEquiv 
Arguments: {X : Type u_1} {Y : Type u_2} [PseudoEMetricSpace X] [PseudoEMetricSpace Y] (e : X ≃ᵢ Y)
Goal: e.toDilationEquiv.symm = e.symm.toDilationEquiv
-- IsometryEquiv.toDilationEquiv_toDilation
@[simp]
lemma _root_.IsometryEquiv.toDilationEquiv_toDilation (e : X ≃ᵢ Y) :
    (e.toDilationEquiv.toDilation : X →ᵈ Y) = e.isometry.toDilation 
Arguments: {X : Type u_1} {Y : Type u_2} [PseudoEMetricSpace X] [PseudoEMetricSpace Y] (e : X ≃ᵢ Y)
Goal: e.toDilationEquiv.toDilation = Isometry.toDilation ⇑e ⋯
-- IsometryEquiv.toDilationEquiv_ratio
@[simp]
lemma _root_.IsometryEquiv.toDilationEquiv_ratio (e : X ≃ᵢ Y) : ratio e.toDilationEquiv = 1 
Arguments: {X : Type u_1} {Y : Type u_2} [PseudoEMetricSpace X] [PseudoEMetricSpace Y] (e : X ≃ᵢ Y)
Goal: Dilation.ratio e.toDilationEquiv = 1
-- DilationEquiv.coe_toHomeomorph
@[simp]
lemma coe_toHomeomorph (e : X ≃ᵈ Y) : ⇑e.toHomeomorph = e 
Arguments: {X : Type u_1} {Y : Type u_2} [PseudoEMetricSpace X] [PseudoEMetricSpace Y] (e : X ≃ᵈ Y)
Goal: ⇑e.toHomeomorph = ⇑e
-- DilationEquiv.toHomeomorph_symm
@[simp]
lemma toHomeomorph_symm (e : X ≃ᵈ Y) : e.toHomeomorph.symm = e.symm.toHomeomorph 
Arguments: {X : Type u_1} {Y : Type u_2} [PseudoEMetricSpace X] [PseudoEMetricSpace Y] (e : X ≃ᵈ Y)
Goal: e.toHomeomorph.symm = e.symm.toHomeomorph
-- DilationEquiv.map_cobounded
@[simp]
lemma map_cobounded (e : F) : map e (cobounded X) = cobounded Y 
Arguments: {X : Type u_1} {Y : Type u_2} {F : Type u_3} [PseudoMetricSpace X] [PseudoMetricSpace Y] [EquivLike F X Y] [DilationEquivClass F X Y] (e : F)
Goal: Filter.map (⇑e) (Bornology.cobounded X) = Bornology.cobounded Y
-- AntilipschitzWith.edist_lt_top
theorem AntilipschitzWith.edist_lt_top [PseudoEMetricSpace α] [PseudoMetricSpace β] {K : ℝ≥0}
    {f : α → β} (h : AntilipschitzWith K f) (x y : α) : edist x y < ⊤ 
Arguments: {α : Type u_1} {β : Type u_2} [PseudoEMetricSpace α] [PseudoMetricSpace β] {K : ℝ≥0} {f : α → β} (h : AntilipschitzWith K f) (x y : α)
Goal: edist x y < ⊤
-- AntilipschitzWith.edist_ne_top
theorem AntilipschitzWith.edist_ne_top [PseudoEMetricSpace α] [PseudoMetricSpace β] {K : ℝ≥0}
    {f : α → β} (h : AntilipschitzWith K f) (x y : α) : edist x y ≠ ⊤ 
Arguments: {α : Type u_1} {β : Type u_2} [PseudoEMetricSpace α] [PseudoMetricSpace β] {K : ℝ≥0} {f : α → β} (h : AntilipschitzWith K f) (x y : α)
Goal: edist x y ≠ ⊤
-- antilipschitzWith_iff_le_mul_nndist
theorem antilipschitzWith_iff_le_mul_nndist :
    AntilipschitzWith K f ↔ ∀ x y, nndist x y ≤ K * nndist (f x) (f y) 
Arguments: {α : Type u_1} {β : Type u_2} [PseudoMetricSpace α] [PseudoMetricSpace β] {K : ℝ≥0} {f : α → β}
Goal: AntilipschitzWith K f ↔ ∀ (x y : α), nndist x y ≤ K * nndist (f x) (f y)
-- antilipschitzWith_iff_le_mul_dist
theorem antilipschitzWith_iff_le_mul_dist :
    AntilipschitzWith K f ↔ ∀ x y, dist x y ≤ K * dist (f x) (f y) 
Arguments: {α : Type u_1} {β : Type u_2} [PseudoMetricSpace α] [PseudoMetricSpace β] {K : ℝ≥0} {f : α → β}
Goal: AntilipschitzWith K f ↔ ∀ (x y : α), dist x y ≤ ↑K * dist (f x) (f y)
-- AntilipschitzWith.mul_le_nndist
theorem mul_le_nndist (hf : AntilipschitzWith K f) (x y : α) :
    K⁻¹ * nndist x y ≤ nndist (f x) (f y) 
Arguments: {α : Type u_1} {β : Type u_2} [PseudoMetricSpace α] [PseudoMetricSpace β] {K : ℝ≥0} {f : α → β} (hf : AntilipschitzWith K f) (x y : α)
Goal: K⁻¹ * nndist x y ≤ nndist (f x) (f y)
-- AntilipschitzWith.mul_le_dist
theorem mul_le_dist (hf : AntilipschitzWith K f) (x y : α) :
    (K⁻¹ * dist x y : ℝ) ≤ dist (f x) (f y) 
Arguments: {α : Type u_1} {β : Type u_2} [PseudoMetricSpace α] [PseudoMetricSpace β] {K : ℝ≥0} {f : α → β} (hf : AntilipschitzWith K f) (x y : α)
Goal: ↑K⁻¹ * dist x y ≤ dist (f x) (f y)
-- AntilipschitzWith.injective
protected theorem injective {α : Type*} {β : Type*} [EMetricSpace α] [PseudoEMetricSpace β]
    {K : ℝ≥0} {f : α → β} (hf : AntilipschitzWith K f) : Function.Injective f 
Arguments: {α : Type u_4} {β : Type u_5} [EMetricSpace α] [PseudoEMetricSpace β] {K : ℝ≥0} {f : α → β} (hf : AntilipschitzWith K f)
Goal: Function.Injective f
-- AntilipschitzWith.mul_le_edist
theorem mul_le_edist (hf : AntilipschitzWith K f) (x y : α) :
    (K : ℝ≥0∞)⁻¹ * edist x y ≤ edist (f x) (f y) 
Arguments: {α : Type u_1} {β : Type u_2} [PseudoEMetricSpace α] [PseudoEMetricSpace β] {K : ℝ≥0} {f : α → β} (hf : AntilipschitzWith K f) (x y : α)
Goal: (↑K)⁻¹ * edist x y ≤ edist (f x) (f y)
-- AntilipschitzWith.ediam_preimage_le
theorem ediam_preimage_le (hf : AntilipschitzWith K f) (s : Set β) : diam (f ⁻¹' s) ≤ K * diam s 
Arguments: {α : Type u_1} {β : Type u_2} [PseudoEMetricSpace α] [PseudoEMetricSpace β] {K : ℝ≥0} {f : α → β} (hf : AntilipschitzWith K f) (s : Set β)
Goal: EMetric.diam (f ⁻¹' s) ≤ ↑K * EMetric.diam s
-- AntilipschitzWith.le_mul_ediam_image
theorem le_mul_ediam_image (hf : AntilipschitzWith K f) (s : Set α) : diam s ≤ K * diam (f '' s) 
Arguments: {α : Type u_1} {β : Type u_2} [PseudoEMetricSpace α] [PseudoEMetricSpace β] {K : ℝ≥0} {f : α → β} (hf : AntilipschitzWith K f) (s : Set α)
Goal: EMetric.diam s ≤ ↑K * EMetric.diam (f '' s)
-- AntilipschitzWith.id
protected theorem id : AntilipschitzWith 1 (id : α → α) 
Arguments: {α : Type u_1} [PseudoEMetricSpace α]
Goal: AntilipschitzWith 1 id
-- AntilipschitzWith.comp
theorem comp {Kg : ℝ≥0} {g : β → γ} (hg : AntilipschitzWith Kg g) {Kf : ℝ≥0} {f : α → β}
    (hf : AntilipschitzWith Kf f) : AntilipschitzWith (Kf * Kg) (g ∘ f) 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} [PseudoEMetricSpace α] [PseudoEMetricSpace β] [PseudoEMetricSpace γ] {Kg : ℝ≥0} {g : β → γ} (hg : AntilipschitzWith Kg g) {Kf : ℝ≥0} {f : α → β} (hf : AntilipschitzWith Kf f)
Goal: AntilipschitzWith (Kf * Kg) (g ∘ f)
-- AntilipschitzWith.restrict
theorem restrict (hf : AntilipschitzWith K f) (s : Set α) : AntilipschitzWith K (s.restrict f) 
Arguments: {α : Type u_1} {β : Type u_2} [PseudoEMetricSpace α] [PseudoEMetricSpace β] {K : ℝ≥0} {f : α → β} (hf : AntilipschitzWith K f) (s : Set α)
Goal: AntilipschitzWith K (s.restrict f)
-- AntilipschitzWith.codRestrict
theorem codRestrict (hf : AntilipschitzWith K f) {s : Set β} (hs : ∀ x, f x ∈ s) :
    AntilipschitzWith K (s.codRestrict f hs) 
Arguments: {α : Type u_1} {β : Type u_2} [PseudoEMetricSpace α] [PseudoEMetricSpace β] {K : ℝ≥0} {f : α → β} (hf : AntilipschitzWith K f) {s : Set β} (hs : ∀ (x : α), f x ∈ s)
Goal: AntilipschitzWith K (Set.codRestrict f s hs)
-- AntilipschitzWith.to_rightInvOn'
theorem to_rightInvOn' {s : Set α} (hf : AntilipschitzWith K (s.restrict f)) {g : β → α}
    {t : Set β} (g_maps : MapsTo g t s) (g_inv : RightInvOn g f t) :
    LipschitzWith K (t.restrict g) 
Arguments: {α : Type u_1} {β : Type u_2} [PseudoEMetricSpace α] [PseudoEMetricSpace β] {K : ℝ≥0} {f : α → β} {s : Set α} (hf : AntilipschitzWith K (s.restrict f)) {g : β → α} {t : Set β} (g_maps : Set.MapsTo g t s) (g_inv : Set.RightInvOn g f t)
Goal: LipschitzWith K (t.restrict g)
-- AntilipschitzWith.to_rightInvOn
theorem to_rightInvOn (hf : AntilipschitzWith K f) {g : β → α} {t : Set β} (h : RightInvOn g f t) :
    LipschitzWith K (t.restrict g) 
Arguments: {α : Type u_1} {β : Type u_2} [PseudoEMetricSpace α] [PseudoEMetricSpace β] {K : ℝ≥0} {f : α → β} (hf : AntilipschitzWith K f) {g : β → α} {t : Set β} (h : Set.RightInvOn g f t)
Goal: LipschitzWith K (t.restrict g)
-- AntilipschitzWith.to_rightInverse
theorem to_rightInverse (hf : AntilipschitzWith K f) {g : β → α} (hg : Function.RightInverse g f) :
    LipschitzWith K g 
Arguments: {α : Type u_1} {β : Type u_2} [PseudoEMetricSpace α] [PseudoEMetricSpace β] {K : ℝ≥0} {f : α → β} (hf : AntilipschitzWith K f) {g : β → α} (hg : Function.RightInverse g f)
Goal: LipschitzWith K g
-- AntilipschitzWith.comap_uniformity_le
theorem comap_uniformity_le (hf : AntilipschitzWith K f) : (𝓤 β).comap (Prod.map f f) ≤ 𝓤 α 
Arguments: {α : Type u_1} {β : Type u_2} [PseudoEMetricSpace α] [PseudoEMetricSpace β] {K : ℝ≥0} {f : α → β} (hf : AntilipschitzWith K f)
Goal: Filter.comap (Prod.map f f) (𝓤 β) ≤ 𝓤 α
-- AntilipschitzWith.uniformInducing
protected theorem uniformInducing (hf : AntilipschitzWith K f) (hfc : UniformContinuous f) :
    UniformInducing f 
Arguments: {α : Type u_1} {β : Type u_2} [PseudoEMetricSpace α] [PseudoEMetricSpace β] {K : ℝ≥0} {f : α → β} (hf : AntilipschitzWith K f) (hfc : UniformContinuous f)
Goal: UniformInducing f
-- AntilipschitzWith.uniformEmbedding
protected theorem uniformEmbedding {α : Type*} {β : Type*} [EMetricSpace α] [PseudoEMetricSpace β]
    {K : ℝ≥0} {f : α → β} (hf : AntilipschitzWith K f) (hfc : UniformContinuous f) :
    UniformEmbedding f 
Arguments: {α : Type u_4} {β : Type u_5} [EMetricSpace α] [PseudoEMetricSpace β] {K : ℝ≥0} {f : α → β} (hf : AntilipschitzWith K f) (hfc : UniformContinuous f)
Goal: UniformEmbedding f
-- AntilipschitzWith.isComplete_range
theorem isComplete_range [CompleteSpace α] (hf : AntilipschitzWith K f)
    (hfc : UniformContinuous f) : IsComplete (range f) 
Arguments: {α : Type u_1} {β : Type u_2} [PseudoEMetricSpace α] [PseudoEMetricSpace β] {K : ℝ≥0} {f : α → β} [CompleteSpace α] (hf : AntilipschitzWith K f) (hfc : UniformContinuous f)
Goal: IsComplete (Set.range f)
-- AntilipschitzWith.isClosed_range
theorem isClosed_range {α β : Type*} [PseudoEMetricSpace α] [EMetricSpace β] [CompleteSpace α]
    {f : α → β} {K : ℝ≥0} (hf : AntilipschitzWith K f) (hfc : UniformContinuous f) :
    IsClosed (range f) 
Arguments: {α : Type u_4} {β : Type u_5} [PseudoEMetricSpace α] [EMetricSpace β] [CompleteSpace α] {f : α → β} {K : ℝ≥0} (hf : AntilipschitzWith K f) (hfc : UniformContinuous f)
Goal: IsClosed (Set.range f)
-- AntilipschitzWith.closedEmbedding
theorem closedEmbedding {α : Type*} {β : Type*} [EMetricSpace α] [EMetricSpace β] {K : ℝ≥0}
    {f : α → β} [CompleteSpace α] (hf : AntilipschitzWith K f) (hfc : UniformContinuous f) :
    ClosedEmbedding f 
Arguments: {α : Type u_4} {β : Type u_5} [EMetricSpace α] [EMetricSpace β] {K : ℝ≥0} {f : α → β} [CompleteSpace α] (hf : AntilipschitzWith K f) (hfc : UniformContinuous f)
Goal: ClosedEmbedding f
-- AntilipschitzWith.subtype_coe
theorem subtype_coe (s : Set α) : AntilipschitzWith 1 ((↑) : s → α) 
Arguments: {α : Type u_1} [PseudoEMetricSpace α] (s : Set α)
Goal: AntilipschitzWith 1 Subtype.val
-- AntilipschitzWith.of_subsingleton
@[nontriviality] -- Porting note: added `nontriviality`
theorem of_subsingleton [Subsingleton α] {K : ℝ≥0} : AntilipschitzWith K f 
Arguments: {α : Type u_1} {β : Type u_2} [PseudoEMetricSpace α] [PseudoEMetricSpace β] {f : α → β} [Subsingleton α] {K : ℝ≥0}
Goal: AntilipschitzWith K f
-- AntilipschitzWith.subsingleton
protected theorem subsingleton {α β} [EMetricSpace α] [PseudoEMetricSpace β] {f : α → β}
    (h : AntilipschitzWith 0 f) : Subsingleton α 
Arguments: {α : Type u_4} {β : Type u_5} [EMetricSpace α] [PseudoEMetricSpace β] {f : α → β} (h : AntilipschitzWith 0 f)
Goal: Subsingleton α
-- AntilipschitzWith.isBounded_preimage
theorem isBounded_preimage (hf : AntilipschitzWith K f) {s : Set β} (hs : IsBounded s) :
    IsBounded (f ⁻¹' s) 
Arguments: {α : Type u_1} {β : Type u_2} [PseudoMetricSpace α] [PseudoMetricSpace β] {K : ℝ≥0} {f : α → β} (hf : AntilipschitzWith K f) {s : Set β} (hs : Bornology.IsBounded s)
Goal: Bornology.IsBounded (f ⁻¹' s)
-- AntilipschitzWith.tendsto_cobounded
theorem tendsto_cobounded (hf : AntilipschitzWith K f) : Tendsto f (cobounded α) (cobounded β) 
Arguments: {α : Type u_1} {β : Type u_2} [PseudoMetricSpace α] [PseudoMetricSpace β] {K : ℝ≥0} {f : α → β} (hf : AntilipschitzWith K f)
Goal: Filter.Tendsto f (Bornology.cobounded α) (Bornology.cobounded β)
-- AntilipschitzWith.properSpace
protected theorem properSpace {α : Type*} [MetricSpace α] {K : ℝ≥0} {f : α → β} [ProperSpace α]
    (hK : AntilipschitzWith K f) (f_cont : Continuous f) (hf : Function.Surjective f) :
    ProperSpace β 
Arguments: {β : Type u_2} [PseudoMetricSpace β] {α : Type u_4} [MetricSpace α] {K : ℝ≥0} {f : α → β} [ProperSpace α] (hK : AntilipschitzWith K f) (f_cont : Continuous f) (hf : Function.Surjective f)
Goal: ProperSpace β
-- AntilipschitzWith.isBounded_of_image2_left
theorem isBounded_of_image2_left (f : α → β → γ) {K₁ : ℝ≥0}
    (hf : ∀ b, AntilipschitzWith K₁ fun a => f a b) {s : Set α} {t : Set β}
    (hst : IsBounded (Set.image2 f s t)) : IsBounded s ∨ IsBounded t 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} [PseudoMetricSpace α] [PseudoMetricSpace β] [PseudoMetricSpace γ] (f : α → β → γ) {K₁ : ℝ≥0} (hf : ∀ (b : β), AntilipschitzWith K₁ fun a => f a b) {s : Set α} {t : Set β} (hst : Bornology.IsBounded (Set.image2 f s t))
Goal: Bornology.IsBounded s ∨ Bornology.IsBounded t
-- AntilipschitzWith.isBounded_of_image2_right
theorem isBounded_of_image2_right {f : α → β → γ} {K₂ : ℝ≥0} (hf : ∀ a, AntilipschitzWith K₂ (f a))
    {s : Set α} {t : Set β} (hst : IsBounded (Set.image2 f s t)) : IsBounded s ∨ IsBounded t 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} [PseudoMetricSpace α] [PseudoMetricSpace β] [PseudoMetricSpace γ] {f : α → β → γ} {K₂ : ℝ≥0} (hf : ∀ (a : α), AntilipschitzWith K₂ (f a)) {s : Set α} {t : Set β} (hst : Bornology.IsBounded (Set.image2 f s t))
Goal: Bornology.IsBounded s ∨ Bornology.IsBounded t
-- LipschitzWith.to_rightInverse
theorem LipschitzWith.to_rightInverse [PseudoEMetricSpace α] [PseudoEMetricSpace β] {K : ℝ≥0}
    {f : α → β} (hf : LipschitzWith K f) {g : β → α} (hg : Function.RightInverse g f) :
    AntilipschitzWith K g 
Arguments: {α : Type u_1} {β : Type u_2} [PseudoEMetricSpace α] [PseudoEMetricSpace β] {K : ℝ≥0} {f : α → β} (hf : LipschitzWith K f) {g : β → α} (hg : Function.RightInverse g f)
Goal: AntilipschitzWith K g
-- LipschitzWith.properSpace
protected theorem LipschitzWith.properSpace [PseudoMetricSpace α] [MetricSpace β] [ProperSpace β]
    {K : ℝ≥0} {f : α ≃ₜ β} (hK : LipschitzWith K f) : ProperSpace α 
Arguments: {α : Type u_1} {β : Type u_2} [PseudoMetricSpace α] [MetricSpace β] [ProperSpace β] {K : ℝ≥0} {f : α ≃ₜ β} (hK : LipschitzWith K ⇑f)
Goal: ProperSpace α
-- Pretrivialization.IsLinear.linear
protected class Pretrivialization.IsLinear [AddCommMonoid F] [Module R F] [∀ x, AddCommMonoid (E x)]
  [∀ x, Module R (E x)] (e : Pretrivialization F (π F E)) : Prop where
  linear : ∀ b ∈ e.baseSet, IsLinearMap R fun x : E b => (e ⟨b, x⟩).2
Arguments: {R : Type u_1} {B : Type u_2} {F : Type u_3} {E : B → Type u_4} [Semiring R] [TopologicalSpace F] [TopologicalSpace B] [AddCommMonoid F] [Module R F] [(x : B) → AddCommMonoid (E x)] [(x : B) → Module R (E x)] {e : Pretrivialization F Bundle.TotalSpace.proj} [self : Pretrivialization.IsLinear R e] (b : B)
Goal: b ∈ e.baseSet → IsLinearMap R fun x => (↑e { proj := b, snd := x }).2
-- Pretrivialization.linear
theorem linear [AddCommMonoid F] [Module R F] [∀ x, AddCommMonoid (E x)] [∀ x, Module R (E x)]
    [e.IsLinear R] {b : B} (hb : b ∈ e.baseSet) :
    IsLinearMap R fun x : E b => (e ⟨b, x⟩).2 
Arguments: (R : Type u_1) {B : Type u_2} {F : Type u_3} {E : B → Type u_4} [Semiring R] [TopologicalSpace F] [TopologicalSpace B] (e : Pretrivialization F Bundle.TotalSpace.proj) [AddCommMonoid F] [Module R F] [(x : B) → AddCommMonoid (E x)] [(x : B) → Module R (E x)] [Pretrivialization.IsLinear R e] {b : B} (hb : b ∈ e.baseSet)
Goal: IsLinearMap R fun x => (↑e { proj := b, snd := x }).2
-- Pretrivialization.symmₗ_apply
@[simps!]
protected def symmₗ (e : Pretrivialization F (π F E)) [e.IsLinear R] (b : B) : F →ₗ[R] E b 
Arguments: (R : Type u_1) {B : Type u_2} {F : Type u_3} {E : B → Type u_4} [Semiring R] [TopologicalSpace F] [TopologicalSpace B] [AddCommMonoid F] [Module R F] [(x : B) → AddCommMonoid (E x)] [(x : B) → Module R (E x)] (e : Pretrivialization F Bundle.TotalSpace.proj) [Pretrivialization.IsLinear R e] (b : B) (y : F)
Goal: (Pretrivialization.symmₗ R e b) y = e.symm b y