ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Dilation.nndist_eq
@[simp]
theorem nndist_eq {α β F : Type*} [PseudoMetricSpace α] [PseudoMetricSpace β] [FunLike F α β]
    [DilationClass F α β] (f : F) (x y : α) :
    nndist (f x) (f y) = ratio f * nndist x y 
Arguments: {α : Type u_6} {β : Type u_7} {F : Type u_8} [PseudoMetricSpace α] [PseudoMetricSpace β] [FunLike F α β] [DilationClass F α β] (f : F) (x y : α)
Goal: nndist (f x) (f y) = Dilation.ratio f * nndist x y
-- Dilation.dist_eq
@[simp]
theorem dist_eq {α β F : Type*} [PseudoMetricSpace α] [PseudoMetricSpace β] [FunLike F α β]
    [DilationClass F α β] (f : F) (x y : α) :
    dist (f x) (f y) = ratio f * dist x y 
Arguments: {α : Type u_6} {β : Type u_7} {F : Type u_8} [PseudoMetricSpace α] [PseudoMetricSpace β] [FunLike F α β] [DilationClass F α β] (f : F) (x y : α)
Goal: dist (f x) (f y) = ↑(Dilation.ratio f) * dist x y
-- Dilation.ratio_unique
theorem ratio_unique [DilationClass F α β] {f : F} {x y : α} {r : ℝ≥0} (h₀ : edist x y ≠ 0)
    (htop : edist x y ≠ ⊤) (hr : edist (f x) (f y) = r * edist x y) : r = ratio f 
Arguments: {α : Type u_1} {β : Type u_2} {F : Type u_4} [PseudoEMetricSpace α] [PseudoEMetricSpace β] [FunLike F α β] [DilationClass F α β] {f : F} {x y : α} {r : ℝ≥0} (h₀ : edist x y ≠ 0) (htop : edist x y ≠ ⊤) (hr : edist (f x) (f y) = ↑r * edist x y)
Goal: r = Dilation.ratio f
-- Dilation.ratio_unique_of_nndist_ne_zero
theorem ratio_unique_of_nndist_ne_zero {α β F : Type*} [PseudoMetricSpace α] [PseudoMetricSpace β]
    [FunLike F α β] [DilationClass F α β] {f : F} {x y : α} {r : ℝ≥0} (hxy : nndist x y ≠ 0)
    (hr : nndist (f x) (f y) = r * nndist x y) : r = ratio f 
Arguments: {α : Type u_6} {β : Type u_7} {F : Type u_8} [PseudoMetricSpace α] [PseudoMetricSpace β] [FunLike F α β] [DilationClass F α β] {f : F} {x y : α} {r : ℝ≥0} (hxy : nndist x y ≠ 0) (hr : nndist (f x) (f y) = r * nndist x y)
Goal: r = Dilation.ratio f
-- Dilation.ratio_unique_of_dist_ne_zero
theorem ratio_unique_of_dist_ne_zero {α β} {F : Type*} [PseudoMetricSpace α] [PseudoMetricSpace β]
    [FunLike F α β] [DilationClass F α β] {f : F} {x y : α} {r : ℝ≥0} (hxy : dist x y ≠ 0)
    (hr : dist (f x) (f y) = r * dist x y) : r = ratio f 
Arguments: {α : Type u_7} {β : Type u_8} {F : Type u_6} [PseudoMetricSpace α] [PseudoMetricSpace β] [FunLike F α β] [DilationClass F α β] {f : F} {x y : α} {r : ℝ≥0} (hxy : dist x y ≠ 0) (hr : dist (f x) (f y) = ↑r * dist x y)
Goal: r = Dilation.ratio f
-- Dilation.coe_mkOfNNDistEq
@[simp]
theorem coe_mkOfNNDistEq {α β} [PseudoMetricSpace α] [PseudoMetricSpace β] (f : α → β) (h) :
    ⇑(mkOfNNDistEq f h : α →ᵈ β) = f 
Arguments: {α : Type u_6} {β : Type u_7} [PseudoMetricSpace α] [PseudoMetricSpace β] (f : α → β) (h : ∃ r, r ≠ 0 ∧ ∀ (x y : α), nndist (f x) (f y) = r * nndist x y)
Goal: ⇑(Dilation.mkOfNNDistEq f h) = f
-- Dilation.mk_coe_of_nndist_eq
@[simp]
theorem mk_coe_of_nndist_eq {α β} [PseudoMetricSpace α] [PseudoMetricSpace β] (f : α →ᵈ β)
    (h) : Dilation.mkOfNNDistEq f h = f 
Arguments: {α : Type u_6} {β : Type u_7} [PseudoMetricSpace α] [PseudoMetricSpace β] (f : α →ᵈ β) (h : ∃ r, r ≠ 0 ∧ ∀ (x y : α), nndist (f x) (f y) = r * nndist x y)
Goal: Dilation.mkOfNNDistEq (⇑f) h = f
-- Dilation.coe_mkOfDistEq
@[simp]
theorem coe_mkOfDistEq {α β} [PseudoMetricSpace α] [PseudoMetricSpace β] (f : α → β) (h) :
    ⇑(mkOfDistEq f h : α →ᵈ β) = f 
Arguments: {α : Type u_6} {β : Type u_7} [PseudoMetricSpace α] [PseudoMetricSpace β] (f : α → β) (h : ∃ r, r ≠ 0 ∧ ∀ (x y : α), dist (f x) (f y) = ↑r * dist x y)
Goal: ⇑(Dilation.mkOfDistEq f h) = f
-- Dilation.mk_coe_of_dist_eq
@[simp]
theorem mk_coe_of_dist_eq {α β} [PseudoMetricSpace α] [PseudoMetricSpace β] (f : α →ᵈ β) (h) :
    Dilation.mkOfDistEq f h = f 
Arguments: {α : Type u_6} {β : Type u_7} [PseudoMetricSpace α] [PseudoMetricSpace β] (f : α →ᵈ β) (h : ∃ r, r ≠ 0 ∧ ∀ (x y : α), dist (f x) (f y) = ↑r * dist x y)
Goal: Dilation.mkOfDistEq (⇑f) h = f
-- Isometry.toDilation_toFun
@[simps]
def _root_.Isometry.toDilation (f : α → β) (hf : Isometry f) : α →ᵈ β 
Arguments: {α : Type u_1} {β : Type u_2} [PseudoEMetricSpace α] [PseudoEMetricSpace β] (f : α → β) (hf : Isometry f)
Goal: ∀ (a : α), (Isometry.toDilation f hf) a = f a
-- Isometry.toDilation_ratio
@[simp]
lemma _root_.Isometry.toDilation_ratio {f : α → β} {hf : Isometry f} : ratio hf.toDilation = 1 
Arguments: {α : Type u_1} {β : Type u_2} [PseudoEMetricSpace α] [PseudoEMetricSpace β] {f : α → β} {hf : Isometry f}
Goal: Dilation.ratio (Isometry.toDilation f hf) = 1
-- Dilation.lipschitz
theorem lipschitz : LipschitzWith (ratio f) (f : α → β) 
Arguments: {α : Type u_1} {β : Type u_2} {F : Type u_4} [PseudoEMetricSpace α] [PseudoEMetricSpace β] [FunLike F α β] [DilationClass F α β] (f : F)
Goal: LipschitzWith (Dilation.ratio f) ⇑f
-- Dilation.antilipschitz
theorem antilipschitz : AntilipschitzWith (ratio f)⁻¹ (f : α → β) 
Arguments: {α : Type u_1} {β : Type u_2} {F : Type u_4} [PseudoEMetricSpace α] [PseudoEMetricSpace β] [FunLike F α β] [DilationClass F α β] (f : F)
Goal: AntilipschitzWith (Dilation.ratio f)⁻¹ ⇑f
-- Dilation.injective
protected theorem injective {α : Type*} [EMetricSpace α] [FunLike F α β]  [DilationClass F α β]
    (f : F) :
    Injective f 
Arguments: {β : Type u_2} {F : Type u_4} [PseudoEMetricSpace β] {α : Type u_6} [EMetricSpace α] [FunLike F α β] [DilationClass F α β] (f : F)
Goal: Function.Injective ⇑f
-- Dilation.coe_id
@[simp] -- Porting note: Removed `@[protected]`
theorem coe_id : ⇑(Dilation.id α) = id 
Arguments: {α : Type u_1} [PseudoEMetricSpace α]
Goal: ⇑(Dilation.id α) = id
-- Dilation.ratio_id
theorem ratio_id : ratio (Dilation.id α) = 1 
Arguments: {α : Type u_1} [PseudoEMetricSpace α]
Goal: Dilation.ratio (Dilation.id α) = 1
-- Dilation.comp_assoc
theorem comp_assoc {δ : Type*} [PseudoEMetricSpace δ] (f : α →ᵈ β) (g : β →ᵈ γ)
    (h : γ →ᵈ δ) : (h.comp g).comp f = h.comp (g.comp f) 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} [PseudoEMetricSpace α] [PseudoEMetricSpace β] [PseudoEMetricSpace γ] {δ : Type u_6} [PseudoEMetricSpace δ] (f : α →ᵈ β) (g : β →ᵈ γ) (h : γ →ᵈ δ)
Goal: (h.comp g).comp f = h.comp (g.comp f)
-- Dilation.coe_comp
@[simp]
theorem coe_comp (g : β →ᵈ γ) (f : α →ᵈ β) : (g.comp f : α → γ) = g ∘ f 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} [PseudoEMetricSpace α] [PseudoEMetricSpace β] [PseudoEMetricSpace γ] (g : β →ᵈ γ) (f : α →ᵈ β)
Goal: ⇑(g.comp f) = ⇑g ∘ ⇑f
-- Dilation.comp_apply
theorem comp_apply (g : β →ᵈ γ) (f : α →ᵈ β) (x : α) : (g.comp f : α → γ) x = g (f x) 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} [PseudoEMetricSpace α] [PseudoEMetricSpace β] [PseudoEMetricSpace γ] (g : β →ᵈ γ) (f : α →ᵈ β) (x : α)
Goal: (g.comp f) x = g (f x)
-- Dilation.ratio_comp'
theorem ratio_comp' {g : β →ᵈ γ} {f : α →ᵈ β}
    (hne : ∃ x y : α, edist x y ≠ 0 ∧ edist x y ≠ ⊤) : ratio (g.comp f) = ratio g * ratio f 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} [PseudoEMetricSpace α] [PseudoEMetricSpace β] [PseudoEMetricSpace γ] {g : β →ᵈ γ} {f : α →ᵈ β} (hne : ∃ x y, edist x y ≠ 0 ∧ edist x y ≠ ⊤)
Goal: Dilation.ratio (g.comp f) = Dilation.ratio g * Dilation.ratio f
-- Dilation.comp_id
@[simp]
theorem comp_id (f : α →ᵈ β) : f.comp (Dilation.id α) = f 
Arguments: {α : Type u_1} {β : Type u_2} [PseudoEMetricSpace α] [PseudoEMetricSpace β] (f : α →ᵈ β)
Goal: f.comp (Dilation.id α) = f
-- Dilation.id_comp
@[simp]
theorem id_comp (f : α →ᵈ β) : (Dilation.id β).comp f = f 
Arguments: {α : Type u_1} {β : Type u_2} [PseudoEMetricSpace α] [PseudoEMetricSpace β] (f : α →ᵈ β)
Goal: (Dilation.id β).comp f = f
-- Dilation.one_def
theorem one_def : (1 : α →ᵈ α) = Dilation.id α 
Arguments: {α : Type u_1} [PseudoEMetricSpace α]
Goal: 1 = Dilation.id α
-- Dilation.mul_def
theorem mul_def (f g : α →ᵈ α) : f * g = f.comp g 
Arguments: {α : Type u_1} [PseudoEMetricSpace α] (f g : α →ᵈ α)
Goal: f * g = f.comp g
-- Dilation.coe_one
@[simp]
theorem coe_one : ⇑(1 : α →ᵈ α) = id 
Arguments: {α : Type u_1} [PseudoEMetricSpace α]
Goal: ⇑1 = id
-- Dilation.coe_mul
@[simp]
theorem coe_mul (f g : α →ᵈ α) : ⇑(f * g) = f ∘ g 
Arguments: {α : Type u_1} [PseudoEMetricSpace α] (f g : α →ᵈ α)
Goal: ⇑(f * g) = ⇑f ∘ ⇑g
-- Dilation.ratio_one
@[simp] theorem ratio_one : ratio (1 : α →ᵈ α) = 1 
Arguments: {α : Type u_1} [PseudoEMetricSpace α]
Goal: Dilation.ratio 1 = 1
-- Dilation.ratio_mul
@[simp]
theorem ratio_mul (f g : α →ᵈ α) : ratio (f * g) = ratio f * ratio g 
Arguments: {α : Type u_1} [PseudoEMetricSpace α] (f g : α →ᵈ α)
Goal: Dilation.ratio (f * g) = Dilation.ratio f * Dilation.ratio g
-- Dilation.ratioHom_apply
@[simps]
def ratioHom : (α →ᵈ α) →* ℝ≥0 
Arguments: {α : Type u_1} [PseudoEMetricSpace α] (f : α →ᵈ α)
Goal: Dilation.ratioHom f = Dilation.ratio f
-- Dilation.ratio_pow
@[simp]
theorem ratio_pow (f : α →ᵈ α) (n : ℕ) : ratio (f ^ n) = ratio f ^ n 
Arguments: {α : Type u_1} [PseudoEMetricSpace α] (f : α →ᵈ α) (n : ℕ)
Goal: Dilation.ratio (f ^ n) = Dilation.ratio f ^ n
-- Dilation.cancel_right
@[simp]
theorem cancel_right {g₁ g₂ : β →ᵈ γ} {f : α →ᵈ β} (hf : Surjective f) :
    g₁.comp f = g₂.comp f ↔ g₁ = g₂ 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} [PseudoEMetricSpace α] [PseudoEMetricSpace β] [PseudoEMetricSpace γ] {g₁ g₂ : β →ᵈ γ} {f : α →ᵈ β} (hf : Function.Surjective ⇑f)
Goal: g₁.comp f = g₂.comp f ↔ g₁ = g₂
-- Dilation.cancel_left
@[simp]
theorem cancel_left {g : β →ᵈ γ} {f₁ f₂ : α →ᵈ β} (hg : Injective g) :
    g.comp f₁ = g.comp f₂ ↔ f₁ = f₂ 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} [PseudoEMetricSpace α] [PseudoEMetricSpace β] [PseudoEMetricSpace γ] {g : β →ᵈ γ} {f₁ f₂ : α →ᵈ β} (hg : Function.Injective ⇑g)
Goal: g.comp f₁ = g.comp f₂ ↔ f₁ = f₂
-- Dilation.uniformInducing
protected theorem uniformInducing : UniformInducing (f : α → β) 
Arguments: {α : Type u_1} {β : Type u_2} {F : Type u_4} [PseudoEMetricSpace α] [PseudoEMetricSpace β] [FunLike F α β] [DilationClass F α β] (f : F)
Goal: UniformInducing ⇑f
-- Dilation.tendsto_nhds_iff
theorem tendsto_nhds_iff {ι : Type*} {g : ι → α} {a : Filter ι} {b : α} :
    Filter.Tendsto g a (𝓝 b) ↔ Filter.Tendsto ((f : α → β) ∘ g) a (𝓝 (f b)) 
Arguments: {α : Type u_1} {β : Type u_2} {F : Type u_4} [PseudoEMetricSpace α] [PseudoEMetricSpace β] [FunLike F α β] [DilationClass F α β] (f : F) {ι : Type u_6} {g : ι → α} {a : Filter ι} {b : α}
Goal: Filter.Tendsto g a (𝓝 b) ↔ Filter.Tendsto (⇑f ∘ g) a (𝓝 (f b))
-- Dilation.toContinuous
theorem toContinuous : Continuous (f : α → β) 
Arguments: {α : Type u_1} {β : Type u_2} {F : Type u_4} [PseudoEMetricSpace α] [PseudoEMetricSpace β] [FunLike F α β] [DilationClass F α β] (f : F)
Goal: Continuous ⇑f
-- Dilation.ediam_image
theorem ediam_image (s : Set α) : EMetric.diam ((f : α → β) '' s) = ratio f * EMetric.diam s 
Arguments: {α : Type u_1} {β : Type u_2} {F : Type u_4} [PseudoEMetricSpace α] [PseudoEMetricSpace β] [FunLike F α β] [DilationClass F α β] (f : F) (s : Set α)
Goal: EMetric.diam (⇑f '' s) = ↑(Dilation.ratio f) * EMetric.diam s
-- Dilation.ediam_range
theorem ediam_range : EMetric.diam (range (f : α → β)) = ratio f * EMetric.diam (univ : Set α) 
Arguments: {α : Type u_1} {β : Type u_2} {F : Type u_4} [PseudoEMetricSpace α] [PseudoEMetricSpace β] [FunLike F α β] [DilationClass F α β] (f : F)
Goal: EMetric.diam (Set.range ⇑f) = ↑(Dilation.ratio f) * EMetric.diam Set.univ
-- Dilation.mapsTo_emetric_ball
theorem mapsTo_emetric_ball (x : α) (r : ℝ≥0∞) :
    MapsTo (f : α → β) (EMetric.ball x r) (EMetric.ball (f x) (ratio f * r)) 
Arguments: {α : Type u_1} {β : Type u_2} {F : Type u_4} [PseudoEMetricSpace α] [PseudoEMetricSpace β] [FunLike F α β] [DilationClass F α β] (f : F) (x : α) (r : ℝ≥0∞)
Goal: Set.MapsTo (⇑f) (EMetric.ball x r) (EMetric.ball (f x) (↑(Dilation.ratio f) * r))
-- Dilation.mapsTo_emetric_closedBall
theorem mapsTo_emetric_closedBall (x : α) (r' : ℝ≥0∞) :
    MapsTo (f : α → β) (EMetric.closedBall x r') (EMetric.closedBall (f x) (ratio f * r')) 
Arguments: {α : Type u_1} {β : Type u_2} {F : Type u_4} [PseudoEMetricSpace α] [PseudoEMetricSpace β] [FunLike F α β] [DilationClass F α β] (f : F) (x : α) (r' : ℝ≥0∞)
Goal: Set.MapsTo (⇑f) (EMetric.closedBall x r') (EMetric.closedBall (f x) (↑(Dilation.ratio f) * r'))
-- Dilation.comp_continuousOn_iff
theorem comp_continuousOn_iff {γ} [TopologicalSpace γ] {g : γ → α} {s : Set γ} :
    ContinuousOn ((f : α → β) ∘ g) s ↔ ContinuousOn g s 
Arguments: {α : Type u_1} {β : Type u_2} {F : Type u_4} [PseudoEMetricSpace α] [PseudoEMetricSpace β] [FunLike F α β] [DilationClass F α β] (f : F) {γ : Type u_6} [TopologicalSpace γ] {g : γ → α} {s : Set γ}
Goal: ContinuousOn (⇑f ∘ g) s ↔ ContinuousOn g s
-- Dilation.comp_continuous_iff
theorem comp_continuous_iff {γ} [TopologicalSpace γ] {g : γ → α} :
    Continuous ((f : α → β) ∘ g) ↔ Continuous g 
Arguments: {α : Type u_1} {β : Type u_2} {F : Type u_4} [PseudoEMetricSpace α] [PseudoEMetricSpace β] [FunLike F α β] [DilationClass F α β] (f : F) {γ : Type u_6} [TopologicalSpace γ] {g : γ → α}
Goal: Continuous (⇑f ∘ g) ↔ Continuous g
-- Dilation.uniformEmbedding
protected theorem uniformEmbedding [PseudoEMetricSpace β] [DilationClass F α β] (f : F) :
    UniformEmbedding f 
Arguments: {α : Type u_1} {β : Type u_2} {F : Type u_4} [EMetricSpace α] [FunLike F α β] [PseudoEMetricSpace β] [DilationClass F α β] (f : F)
Goal: UniformEmbedding ⇑f
-- Dilation.embedding
protected theorem embedding [PseudoEMetricSpace β] [DilationClass F α β] (f : F) :
    Embedding (f : α → β) 
Arguments: {α : Type u_1} {β : Type u_2} {F : Type u_4} [EMetricSpace α] [FunLike F α β] [PseudoEMetricSpace β] [DilationClass F α β] (f : F)
Goal: Embedding ⇑f
-- Dilation.closedEmbedding
protected theorem closedEmbedding [CompleteSpace α] [EMetricSpace β] [DilationClass F α β] (f : F) :
    ClosedEmbedding f 
Arguments: {α : Type u_1} {β : Type u_2} {F : Type u_4} [EMetricSpace α] [FunLike F α β] [CompleteSpace α] [EMetricSpace β] [DilationClass F α β] (f : F)
Goal: ClosedEmbedding ⇑f
-- Dilation.ratio_comp
@[simp]
theorem ratio_comp [MetricSpace α] [Nontrivial α] [PseudoEMetricSpace β]
    [PseudoEMetricSpace γ] {g : β →ᵈ γ} {f : α →ᵈ β} : ratio (g.comp f) = ratio g * ratio f 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} [MetricSpace α] [Nontrivial α] [PseudoEMetricSpace β] [PseudoEMetricSpace γ] {g : β →ᵈ γ} {f : α →ᵈ β}
Goal: Dilation.ratio (g.comp f) = Dilation.ratio g * Dilation.ratio f
-- Dilation.diam_image
theorem diam_image (s : Set α) : Metric.diam ((f : α → β) '' s) = ratio f * Metric.diam s 
Arguments: {α : Type u_1} {β : Type u_2} {F : Type u_4} [PseudoMetricSpace α] [PseudoMetricSpace β] [FunLike F α β] [DilationClass F α β] (f : F) (s : Set α)
Goal: Metric.diam (⇑f '' s) = ↑(Dilation.ratio f) * Metric.diam s
-- Dilation.diam_range
theorem diam_range : Metric.diam (range (f : α → β)) = ratio f * Metric.diam (univ : Set α) 
Arguments: {α : Type u_1} {β : Type u_2} {F : Type u_4} [PseudoMetricSpace α] [PseudoMetricSpace β] [FunLike F α β] [DilationClass F α β] (f : F)
Goal: Metric.diam (Set.range ⇑f) = ↑(Dilation.ratio f) * Metric.diam Set.univ
-- Dilation.mapsTo_ball
theorem mapsTo_ball (x : α) (r' : ℝ) :
    MapsTo (f : α → β) (Metric.ball x r') (Metric.ball (f x) (ratio f * r')) 
Arguments: {α : Type u_1} {β : Type u_2} {F : Type u_4} [PseudoMetricSpace α] [PseudoMetricSpace β] [FunLike F α β] [DilationClass F α β] (f : F) (x : α) (r' : ℝ)
Goal: Set.MapsTo (⇑f) (Metric.ball x r') (Metric.ball (f x) (↑(Dilation.ratio f) * r'))
-- Dilation.mapsTo_sphere
theorem mapsTo_sphere (x : α) (r' : ℝ) :
    MapsTo (f : α → β) (Metric.sphere x r') (Metric.sphere (f x) (ratio f * r')) 
Arguments: {α : Type u_1} {β : Type u_2} {F : Type u_4} [PseudoMetricSpace α] [PseudoMetricSpace β] [FunLike F α β] [DilationClass F α β] (f : F) (x : α) (r' : ℝ)
Goal: Set.MapsTo (⇑f) (Metric.sphere x r') (Metric.sphere (f x) (↑(Dilation.ratio f) * r'))
-- Dilation.mapsTo_closedBall
theorem mapsTo_closedBall (x : α) (r' : ℝ) :
    MapsTo (f : α → β) (Metric.closedBall x r') (Metric.closedBall (f x) (ratio f * r')) 
Arguments: {α : Type u_1} {β : Type u_2} {F : Type u_4} [PseudoMetricSpace α] [PseudoMetricSpace β] [FunLike F α β] [DilationClass F α β] (f : F) (x : α) (r' : ℝ)
Goal: Set.MapsTo (⇑f) (Metric.closedBall x r') (Metric.closedBall (f x) (↑(Dilation.ratio f) * r'))
-- Dilation.tendsto_cobounded
lemma tendsto_cobounded : Filter.Tendsto f (cobounded α) (cobounded β) 
Arguments: {α : Type u_1} {β : Type u_2} {F : Type u_4} [PseudoMetricSpace α] [PseudoMetricSpace β] [FunLike F α β] [DilationClass F α β] (f : F)
Goal: Filter.Tendsto (⇑f) (Bornology.cobounded α) (Bornology.cobounded β)
-- Dilation.comap_cobounded
@[simp]
lemma comap_cobounded : Filter.comap f (cobounded β) = cobounded α 
Arguments: {α : Type u_1} {β : Type u_2} {F : Type u_4} [PseudoMetricSpace α] [PseudoMetricSpace β] [FunLike F α β] [DilationClass F α β] (f : F)
Goal: Filter.comap (⇑f) (Bornology.cobounded β) = Bornology.cobounded α
-- TopCat.presheafToTypes_obj
@[simp]
theorem presheafToTypes_obj {T : X → Type v} {U : (Opens X)ᵒᵖ} :
    (presheafToTypes X T).obj U = ∀ x : U.unop, T x 
Arguments: (X : TopCat) {T : ↑X → Type v} {U : (TopologicalSpace.Opens ↑X)ᵒᵖ}
Goal: (X.presheafToTypes T).obj U = ((x : ↥(Opposite.unop U)) → T ↑x)
-- TopCat.presheafToTypes_map
@[simp]
theorem presheafToTypes_map {T : X → Type v} {U V : (Opens X)ᵒᵖ} {i : U ⟶ V} {f} :
    (presheafToTypes X T).map i f = fun x => f (i.unop x) 
Arguments: (X : TopCat) {T : ↑X → Type v} {U V : (TopologicalSpace.Opens ↑X)ᵒᵖ} {i : U ⟶ V} {f : (X.presheafToTypes T).obj U}
Goal: (X.presheafToTypes T).map i f = fun x => f ((fun x => ⟨↑x, ⋯⟩) x)
-- TopCat.presheafToType_obj
@[simp]
theorem presheafToType_obj {T : Type v} {U : (Opens X)ᵒᵖ} :
    (presheafToType X T).obj U = (U.unop → T) 
Arguments: (X : TopCat) {T : Type v} {U : (TopologicalSpace.Opens ↑X)ᵒᵖ}
Goal: (X.presheafToType T).obj U = (↥(Opposite.unop U) → T)
-- TopCat.presheafToType_map
@[simp]
theorem presheafToType_map {T : Type v} {U V : (Opens X)ᵒᵖ} {i : U ⟶ V} {f} :
    (presheafToType X T).map i f = f ∘ i.unop 
Arguments: (X : TopCat) {T : Type v} {U V : (TopologicalSpace.Opens ↑X)ᵒᵖ} {i : U ⟶ V} {f : (X.presheafToType T).obj U}
Goal: (X.presheafToType T).map i f = f ∘ fun x => ⟨↑x, ⋯⟩
-- TopCat.presheafToTop_obj
@[simp]
theorem presheafToTop_obj (T : TopCat.{v}) (U : (Opens X)ᵒᵖ) :
    (presheafToTop X T).obj U = ((Opens.toTopCat X).obj (unop U) ⟶ T) 
Arguments: (X T : TopCat) (U : (TopologicalSpace.Opens ↑X)ᵒᵖ)
Goal: (X.presheafToTop T).obj U = ((TopologicalSpace.Opens.toTopCat X).obj (Opposite.unop U) ⟶ T)
-- specializes_TFAE
theorem specializes_TFAE (x y : X) :
    TFAE [x ⤳ y,
      pure x ≤ 𝓝 y,
      ∀ s : Set X , IsOpen s → y ∈ s → x ∈ s,
      ∀ s : Set X , IsClosed s → x ∈ s → y ∈ s,
      y ∈ closure ({ x } : Set X),
      closure ({ y } : Set X) ⊆ closure { x },
      ClusterPt y (pure x)] 
Arguments: {X : Type u_1} [TopologicalSpace X] (x y : X)
Goal: [x ⤳ y, pure x ≤ 𝓝 y, ∀ (s : Set X), IsOpen s → y ∈ s → x ∈ s, ∀ (s : Set X), IsClosed s → x ∈ s → y ∈ s,
    y ∈ closure {x}, closure {y} ⊆ closure {x}, ClusterPt y (pure x)].TFAE
-- specializes_iff_nhds
theorem specializes_iff_nhds : x ⤳ y ↔ 𝓝 x ≤ 𝓝 y 
Arguments: {X : Type u_1} [TopologicalSpace X] {x y : X}
Goal: x ⤳ y ↔ 𝓝 x ≤ 𝓝 y
-- Specializes.not_disjoint
theorem Specializes.not_disjoint (h : x ⤳ y) : ¬Disjoint (𝓝 x) (𝓝 y) 
Arguments: {X : Type u_1} [TopologicalSpace X] {x y : X} (h : x ⤳ y)
Goal: ¬Disjoint (𝓝 x) (𝓝 y)
-- specializes_iff_pure
theorem specializes_iff_pure : x ⤳ y ↔ pure x ≤ 𝓝 y 
Arguments: {X : Type u_1} [TopologicalSpace X] {x y : X}
Goal: x ⤳ y ↔ pure x ≤ 𝓝 y
-- ker_nhds_eq_specializes
theorem ker_nhds_eq_specializes : (𝓝 x).ker = {y | y ⤳ x} 
Arguments: {X : Type u_1} [TopologicalSpace X] {x : X}
Goal: (𝓝 x).ker = {y | y ⤳ x}
-- specializes_iff_forall_open
theorem specializes_iff_forall_open : x ⤳ y ↔ ∀ s : Set X, IsOpen s → y ∈ s → x ∈ s 
Arguments: {X : Type u_1} [TopologicalSpace X] {x y : X}
Goal: x ⤳ y ↔ ∀ (s : Set X), IsOpen s → y ∈ s → x ∈ s
-- Specializes.mem_open
theorem Specializes.mem_open (h : x ⤳ y) (hs : IsOpen s) (hy : y ∈ s) : x ∈ s 
Arguments: {X : Type u_1} [TopologicalSpace X] {x y : X} {s : Set X} (h : x ⤳ y) (hs : IsOpen s) (hy : y ∈ s)
Goal: x ∈ s
-- IsOpen.not_specializes
theorem IsOpen.not_specializes (hs : IsOpen s) (hx : x ∉ s) (hy : y ∈ s) : ¬x ⤳ y 
Arguments: {X : Type u_1} [TopologicalSpace X] {x y : X} {s : Set X} (hs : IsOpen s) (hx : x ∉ s) (hy : y ∈ s)
Goal: ¬x ⤳ y
-- specializes_iff_forall_closed
theorem specializes_iff_forall_closed : x ⤳ y ↔ ∀ s : Set X, IsClosed s → x ∈ s → y ∈ s 
Arguments: {X : Type u_1} [TopologicalSpace X] {x y : X}
Goal: x ⤳ y ↔ ∀ (s : Set X), IsClosed s → x ∈ s → y ∈ s
-- Specializes.mem_closed
theorem Specializes.mem_closed (h : x ⤳ y) (hs : IsClosed s) (hx : x ∈ s) : y ∈ s 
Arguments: {X : Type u_1} [TopologicalSpace X] {x y : X} {s : Set X} (h : x ⤳ y) (hs : IsClosed s) (hx : x ∈ s)
Goal: y ∈ s
-- IsClosed.not_specializes
theorem IsClosed.not_specializes (hs : IsClosed s) (hx : x ∈ s) (hy : y ∉ s) : ¬x ⤳ y 
Arguments: {X : Type u_1} [TopologicalSpace X] {x y : X} {s : Set X} (hs : IsClosed s) (hx : x ∈ s) (hy : y ∉ s)
Goal: ¬x ⤳ y
-- specializes_iff_mem_closure
theorem specializes_iff_mem_closure : x ⤳ y ↔ y ∈ closure ({x} : Set X) 
Arguments: {X : Type u_1} [TopologicalSpace X] {x y : X}
Goal: x ⤳ y ↔ y ∈ closure {x}
-- specializes_iff_closure_subset
theorem specializes_iff_closure_subset : x ⤳ y ↔ closure ({y} : Set X) ⊆ closure {x} 
Arguments: {X : Type u_1} [TopologicalSpace X] {x y : X}
Goal: x ⤳ y ↔ closure {y} ⊆ closure {x}
-- specializes_iff_clusterPt
theorem specializes_iff_clusterPt : x ⤳ y ↔ ClusterPt y (pure x) 
Arguments: {X : Type u_1} [TopologicalSpace X] {x y : X}
Goal: x ⤳ y ↔ ClusterPt y (pure x)
-- Filter.HasBasis.specializes_iff
theorem Filter.HasBasis.specializes_iff {ι} {p : ι → Prop} {s : ι → Set X}
    (h : (𝓝 y).HasBasis p s) : x ⤳ y ↔ ∀ i, p i → x ∈ s i 
Arguments: {X : Type u_1} [TopologicalSpace X] {x y : X} {ι : Sort u_7} {p : ι → Prop} {s : ι → Set X} (h : (𝓝 y).HasBasis p s)
Goal: x ⤳ y ↔ ∀ (i : ι), p i → x ∈ s i
-- specializes_rfl
theorem specializes_rfl : x ⤳ x 
Arguments: {X : Type u_1} [TopologicalSpace X] {x : X}
Goal: x ⤳ x
-- specializes_refl
@[refl]
theorem specializes_refl (x : X) : x ⤳ x 
Arguments: {X : Type u_1} [TopologicalSpace X] (x : X)
Goal: x ⤳ x
-- Specializes.trans
@[trans]
theorem Specializes.trans : x ⤳ y → y ⤳ z → x ⤳ z 
Arguments: {X : Type u_1} [TopologicalSpace X] {x y z : X}
Goal: x ⤳ y → y ⤳ z → x ⤳ z
-- specializes_of_eq
theorem specializes_of_eq (e : x = y) : x ⤳ y 
Arguments: {X : Type u_1} [TopologicalSpace X] {x y : X} (e : x = y)
Goal: x ⤳ y
-- specializes_of_nhdsWithin
theorem specializes_of_nhdsWithin (h₁ : 𝓝[s] x ≤ 𝓝[s] y) (h₂ : x ∈ s) : x ⤳ y 
Arguments: {X : Type u_1} [TopologicalSpace X] {x y : X} {s : Set X} (h₁ : 𝓝[s] x ≤ 𝓝[s] y) (h₂ : x ∈ s)
Goal: x ⤳ y
-- Specializes.map_of_continuousAt
theorem Specializes.map_of_continuousAt (h : x ⤳ y) (hy : ContinuousAt f y) : f x ⤳ f y 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {x y : X} {f : X → Y} (h : x ⤳ y) (hy : ContinuousAt f y)
Goal: f x ⤳ f y
-- Specializes.map
theorem Specializes.map (h : x ⤳ y) (hf : Continuous f) : f x ⤳ f y 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {x y : X} {f : X → Y} (h : x ⤳ y) (hf : Continuous f)
Goal: f x ⤳ f y
-- Inducing.specializes_iff
theorem Inducing.specializes_iff (hf : Inducing f) : f x ⤳ f y ↔ x ⤳ y 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {x y : X} {f : X → Y} (hf : Inducing f)
Goal: f x ⤳ f y ↔ x ⤳ y
-- subtype_specializes_iff
theorem subtype_specializes_iff {p : X → Prop} (x y : Subtype p) : x ⤳ y ↔ (x : X) ⤳ y 
Arguments: {X : Type u_1} [TopologicalSpace X] {p : X → Prop} (x y : Subtype p)
Goal: x ⤳ y ↔ ↑x ⤳ ↑y
-- specializes_prod
@[simp]
theorem specializes_prod {x₁ x₂ : X} {y₁ y₂ : Y} : (x₁, y₁) ⤳ (x₂, y₂) ↔ x₁ ⤳ x₂ ∧ y₁ ⤳ y₂ 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {x₁ x₂ : X} {y₁ y₂ : Y}
Goal: (x₁, y₁) ⤳ (x₂, y₂) ↔ x₁ ⤳ x₂ ∧ y₁ ⤳ y₂
-- Specializes.prod
theorem Specializes.prod {x₁ x₂ : X} {y₁ y₂ : Y} (hx : x₁ ⤳ x₂) (hy : y₁ ⤳ y₂) :
    (x₁, y₁) ⤳ (x₂, y₂) 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {x₁ x₂ : X} {y₁ y₂ : Y} (hx : x₁ ⤳ x₂) (hy : y₁ ⤳ y₂)
Goal: (x₁, y₁) ⤳ (x₂, y₂)