ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Trivialization.coordChange_apply_snd
theorem coordChange_apply_snd (e₁ e₂ : Trivialization F proj) {p : Z} (h : proj p ∈ e₁.baseSet) :
    e₁.coordChange e₂ (proj p) (e₁ p).snd = (e₂ p).snd 
Arguments: {B : Type u_2} {F : Type u_3} {Z : Type u_5} [TopologicalSpace B] [TopologicalSpace F] {proj : Z → B} [TopologicalSpace Z] (e₁ e₂ : Trivialization F proj) {p : Z} (h : proj p ∈ e₁.baseSet)
Goal: e₁.coordChange e₂ (proj p) (↑e₁ p).2 = (↑e₂ p).2
-- Trivialization.coordChange_same_apply
theorem coordChange_same_apply (e : Trivialization F proj) {b : B} (h : b ∈ e.baseSet) (x : F) :
    e.coordChange e b x = x 
Arguments: {B : Type u_2} {F : Type u_3} {Z : Type u_5} [TopologicalSpace B] [TopologicalSpace F] {proj : Z → B} [TopologicalSpace Z] (e : Trivialization F proj) {b : B} (h : b ∈ e.baseSet) (x : F)
Goal: e.coordChange e b x = x
-- Trivialization.coordChange_same
theorem coordChange_same (e : Trivialization F proj) {b : B} (h : b ∈ e.baseSet) :
    e.coordChange e b = id 
Arguments: {B : Type u_2} {F : Type u_3} {Z : Type u_5} [TopologicalSpace B] [TopologicalSpace F] {proj : Z → B} [TopologicalSpace Z] (e : Trivialization F proj) {b : B} (h : b ∈ e.baseSet)
Goal: e.coordChange e b = id
-- Trivialization.coordChange_coordChange
theorem coordChange_coordChange (e₁ e₂ e₃ : Trivialization F proj) {b : B} (h₁ : b ∈ e₁.baseSet)
    (h₂ : b ∈ e₂.baseSet) (x : F) :
    e₂.coordChange e₃ b (e₁.coordChange e₂ b x) = e₁.coordChange e₃ b x 
Arguments: {B : Type u_2} {F : Type u_3} {Z : Type u_5} [TopologicalSpace B] [TopologicalSpace F] {proj : Z → B} [TopologicalSpace Z] (e₁ e₂ e₃ : Trivialization F proj) {b : B} (h₁ : b ∈ e₁.baseSet) (h₂ : b ∈ e₂.baseSet) (x : F)
Goal: e₂.coordChange e₃ b (e₁.coordChange e₂ b x) = e₁.coordChange e₃ b x
-- Trivialization.continuous_coordChange
theorem continuous_coordChange (e₁ e₂ : Trivialization F proj) {b : B} (h₁ : b ∈ e₁.baseSet)
    (h₂ : b ∈ e₂.baseSet) : Continuous (e₁.coordChange e₂ b) 
Arguments: {B : Type u_2} {F : Type u_3} {Z : Type u_5} [TopologicalSpace B] [TopologicalSpace F] {proj : Z → B} [TopologicalSpace Z] (e₁ e₂ : Trivialization F proj) {b : B} (h₁ : b ∈ e₁.baseSet) (h₂ : b ∈ e₂.baseSet)
Goal: Continuous (e₁.coordChange e₂ b)
-- Trivialization.coordChangeHomeomorph_coe
@[simp]
theorem coordChangeHomeomorph_coe (e₁ e₂ : Trivialization F proj) {b : B} (h₁ : b ∈ e₁.baseSet)
    (h₂ : b ∈ e₂.baseSet) : ⇑(e₁.coordChangeHomeomorph e₂ h₁ h₂) = e₁.coordChange e₂ b 
Arguments: {B : Type u_2} {F : Type u_3} {Z : Type u_5} [TopologicalSpace B] [TopologicalSpace F] {proj : Z → B} [TopologicalSpace Z] (e₁ e₂ : Trivialization F proj) {b : B} (h₁ : b ∈ e₁.baseSet) (h₂ : b ∈ e₂.baseSet)
Goal: ⇑(e₁.coordChangeHomeomorph e₂ h₁ h₂) = e₁.coordChange e₂ b
-- Trivialization.isImage_preimage_prod
theorem isImage_preimage_prod (e : Trivialization F proj) (s : Set B) :
    e.toPartialHomeomorph.IsImage (proj ⁻¹' s) (s ×ˢ univ) 
Arguments: {B : Type u_2} {F : Type u_3} {Z : Type u_5} [TopologicalSpace B] [TopologicalSpace F] {proj : Z → B} [TopologicalSpace Z] (e : Trivialization F proj) (s : Set B)
Goal: e.IsImage (proj ⁻¹' s) (s ×ˢ Set.univ)
-- Trivialization.frontier_preimage
theorem frontier_preimage (e : Trivialization F proj) (s : Set B) :
    e.source ∩ frontier (proj ⁻¹' s) = proj ⁻¹' (e.baseSet ∩ frontier s) 
Arguments: {B : Type u_2} {F : Type u_3} {Z : Type u_5} [TopologicalSpace B] [TopologicalSpace F] {proj : Z → B} [TopologicalSpace Z] (e : Trivialization F proj) (s : Set B)
Goal: e.source ∩ frontier (proj ⁻¹' s) = proj ⁻¹' (e.baseSet ∩ frontier s)
-- Profinite.exists_isClopen_of_cofiltered
theorem exists_isClopen_of_cofiltered {U : Set C.pt} (hC : IsLimit C) (hU : IsClopen U) :
    ∃ (j : J) (V : Set (F.obj j)), IsClopen V ∧ U = C.π.app j ⁻¹' V 
Arguments: {J : Type v} [CategoryTheory.SmallCategory J] [CategoryTheory.IsCofiltered J] {F : CategoryTheory.Functor J Profinite} (C : CategoryTheory.Limits.Cone F) {U : Set ↑C.pt.toTop} (hC : CategoryTheory.Limits.IsLimit C) (hU : IsClopen U)
Goal: ∃ j V, IsClopen V ∧ U = ⇑(C.π.app j) ⁻¹' V
-- Profinite.exists_locallyConstant_fin_two
theorem exists_locallyConstant_fin_two (hC : IsLimit C) (f : LocallyConstant C.pt (Fin 2)) :
    ∃ (j : J) (g : LocallyConstant (F.obj j) (Fin 2)), f = g.comap (C.π.app _) 
Arguments: {J : Type v} [CategoryTheory.SmallCategory J] [CategoryTheory.IsCofiltered J] {F : CategoryTheory.Functor J Profinite} (C : CategoryTheory.Limits.Cone F) (hC : CategoryTheory.Limits.IsLimit C) (f : LocallyConstant (↑C.pt.toTop) (Fin 2))
Goal: ∃ j g, f = LocallyConstant.comap (C.π.app j) g
-- Profinite.exists_locallyConstant_finite_aux
theorem exists_locallyConstant_finite_aux {α : Type*} [Finite α] (hC : IsLimit C)
    (f : LocallyConstant C.pt α) : ∃ (j : J) (g : LocallyConstant (F.obj j) (α → Fin 2)),
      (f.map fun a b => if a = b then (0 : Fin 2) else 1) = g.comap (C.π.app _) 
Arguments: {J : Type v} [CategoryTheory.SmallCategory J] [CategoryTheory.IsCofiltered J] {F : CategoryTheory.Functor J Profinite} (C : CategoryTheory.Limits.Cone F) {α : Type u_1} [Finite α] (hC : CategoryTheory.Limits.IsLimit C) (f : LocallyConstant (↑C.pt.toTop) α)
Goal: ∃ j g, LocallyConstant.map (fun a b => if a = b then 0 else 1) f = LocallyConstant.comap (C.π.app j) g
-- Profinite.exists_locallyConstant_finite_nonempty
theorem exists_locallyConstant_finite_nonempty {α : Type*} [Finite α] [Nonempty α]
    (hC : IsLimit C) (f : LocallyConstant C.pt α) :
    ∃ (j : J) (g : LocallyConstant (F.obj j) α), f = g.comap (C.π.app _) 
Arguments: {J : Type v} [CategoryTheory.SmallCategory J] [CategoryTheory.IsCofiltered J] {F : CategoryTheory.Functor J Profinite} (C : CategoryTheory.Limits.Cone F) {α : Type u_1} [Finite α] [Nonempty α] (hC : CategoryTheory.Limits.IsLimit C) (f : LocallyConstant (↑C.pt.toTop) α)
Goal: ∃ j g, f = LocallyConstant.comap (C.π.app j) g
-- Profinite.exists_locallyConstant
theorem exists_locallyConstant {α : Type*} (hC : IsLimit C) (f : LocallyConstant C.pt α) :
    ∃ (j : J) (g : LocallyConstant (F.obj j) α), f = g.comap (C.π.app _) 
Arguments: {J : Type v} [CategoryTheory.SmallCategory J] [CategoryTheory.IsCofiltered J] {F : CategoryTheory.Functor J Profinite} (C : CategoryTheory.Limits.Cone F) {α : Type u_1} (hC : CategoryTheory.Limits.IsLimit C) (f : LocallyConstant (↑C.pt.toTop) α)
Goal: ∃ j g, f = LocallyConstant.comap (C.π.app j) g
-- Real.dist_left_le_of_mem_uIcc
lemma Real.dist_left_le_of_mem_uIcc {x y z : ℝ} (h : y ∈ uIcc x z) : dist x y ≤ dist x z 
Arguments: {x y z : ℝ} (h : y ∈ Set.uIcc x z)
Goal: dist x y ≤ dist x z
-- Real.dist_right_le_of_mem_uIcc
lemma Real.dist_right_le_of_mem_uIcc {x y z : ℝ} (h : y ∈ uIcc x z) : dist y z ≤ dist x z 
Arguments: {x y z : ℝ} (h : y ∈ Set.uIcc x z)
Goal: dist y z ≤ dist x z
-- Real.dist_le_of_mem_uIcc
lemma Real.dist_le_of_mem_uIcc {x y x' y' : ℝ} (hx : x ∈ uIcc x' y') (hy : y ∈ uIcc x' y') :
    dist x y ≤ dist x' y' 
Arguments: {x y x' y' : ℝ} (hx : x ∈ Set.uIcc x' y') (hy : y ∈ Set.uIcc x' y')
Goal: dist x y ≤ dist x' y'
-- Real.dist_le_of_mem_Icc
lemma Real.dist_le_of_mem_Icc {x y x' y' : ℝ} (hx : x ∈ Icc x' y') (hy : y ∈ Icc x' y') :
    dist x y ≤ y' - x' 
Arguments: {x y x' y' : ℝ} (hx : x ∈ Set.Icc x' y') (hy : y ∈ Set.Icc x' y')
Goal: dist x y ≤ y' - x'
-- Real.dist_le_of_mem_Icc_01
lemma Real.dist_le_of_mem_Icc_01 {x y : ℝ} (hx : x ∈ Icc (0 : ℝ) 1) (hy : y ∈ Icc (0 : ℝ) 1) :
    dist x y ≤ 1 
Arguments: {x y : ℝ} (hx : x ∈ Set.Icc 0 1) (hy : y ∈ Set.Icc 0 1)
Goal: dist x y ≤ 1
-- Real.singleton_eq_inter_Icc
lemma Real.singleton_eq_inter_Icc (b : ℝ) : {b} = ⋂ (r > 0), Icc (b - r) (b + r) 
Arguments: (b : ℝ)
Goal: {b} = ⋂ r, ⋂ (_ : r > 0), Set.Icc (b - r) (b + r)
-- squeeze_zero'
lemma squeeze_zero' {α} {f g : α → ℝ} {t₀ : Filter α} (hf : ∀ᶠ t in t₀, 0 ≤ f t)
    (hft : ∀ᶠ t in t₀, f t ≤ g t) (g0 : Tendsto g t₀ (𝓝 0)) : Tendsto f t₀ (𝓝 0) 
Arguments: {α : Type u_3} {f g : α → ℝ} {t₀ : Filter α} (hf : ∀ᶠ (t : α) in t₀, 0 ≤ f t) (hft : ∀ᶠ (t : α) in t₀, f t ≤ g t) (g0 : Filter.Tendsto g t₀ (𝓝 0))
Goal: Filter.Tendsto f t₀ (𝓝 0)
-- squeeze_zero
lemma squeeze_zero {α} {f g : α → ℝ} {t₀ : Filter α} (hf : ∀ t, 0 ≤ f t) (hft : ∀ t, f t ≤ g t)
    (g0 : Tendsto g t₀ (𝓝 0)) : Tendsto f t₀ (𝓝 0) 
Arguments: {α : Type u_3} {f g : α → ℝ} {t₀ : Filter α} (hf : ∀ (t : α), 0 ≤ f t) (hft : ∀ (t : α), f t ≤ g t) (g0 : Filter.Tendsto g t₀ (𝓝 0))
Goal: Filter.Tendsto f t₀ (𝓝 0)
-- eventually_closedBall_subset
lemma eventually_closedBall_subset {x : α} {u : Set α} (hu : u ∈ 𝓝 x) :
    ∀ᶠ r in 𝓝 (0 : ℝ), closedBall x r ⊆ u 
Arguments: {α : Type u_2} [PseudoMetricSpace α] {x : α} {u : Set α} (hu : u ∈ 𝓝 x)
Goal: ∀ᶠ (r : ℝ) in 𝓝 0, Metric.closedBall x r ⊆ u
-- tendsto_closedBall_smallSets
lemma tendsto_closedBall_smallSets (x : α) : Tendsto (closedBall x) (𝓝 0) (𝓝 x).smallSets 
Arguments: {α : Type u_2} [PseudoMetricSpace α] (x : α)
Goal: Filter.Tendsto (Metric.closedBall x) (𝓝 0) (𝓝 x).smallSets
-- Metric.isClosed_ball
lemma isClosed_ball : IsClosed (closedBall x ε) 
Arguments: {α : Type u_2} [PseudoMetricSpace α] {x : α} {ε : ℝ}
Goal: IsClosed (Metric.closedBall x ε)
-- Metric.isClosed_sphere
lemma isClosed_sphere : IsClosed (sphere x ε) 
Arguments: {α : Type u_2} [PseudoMetricSpace α] {x : α} {ε : ℝ}
Goal: IsClosed (Metric.sphere x ε)
-- Metric.closure_closedBall
@[simp]
lemma closure_closedBall : closure (closedBall x ε) = closedBall x ε 
Arguments: {α : Type u_2} [PseudoMetricSpace α] {x : α} {ε : ℝ}
Goal: closure (Metric.closedBall x ε) = Metric.closedBall x ε
-- Metric.closure_sphere
@[simp]
lemma closure_sphere : closure (sphere x ε) = sphere x ε 
Arguments: {α : Type u_2} [PseudoMetricSpace α] {x : α} {ε : ℝ}
Goal: closure (Metric.sphere x ε) = Metric.sphere x ε
-- Metric.closure_ball_subset_closedBall
lemma closure_ball_subset_closedBall : closure (ball x ε) ⊆ closedBall x ε 
Arguments: {α : Type u_2} [PseudoMetricSpace α] {x : α} {ε : ℝ}
Goal: closure (Metric.ball x ε) ⊆ Metric.closedBall x ε
-- Metric.frontier_ball_subset_sphere
lemma frontier_ball_subset_sphere : frontier (ball x ε) ⊆ sphere x ε 
Arguments: {α : Type u_2} [PseudoMetricSpace α] {x : α} {ε : ℝ}
Goal: frontier (Metric.ball x ε) ⊆ Metric.sphere x ε
-- Metric.frontier_closedBall_subset_sphere
lemma frontier_closedBall_subset_sphere : frontier (closedBall x ε) ⊆ sphere x ε 
Arguments: {α : Type u_2} [PseudoMetricSpace α] {x : α} {ε : ℝ}
Goal: frontier (Metric.closedBall x ε) ⊆ Metric.sphere x ε
-- Metric.closedBall_zero'
lemma closedBall_zero' (x : α) : closedBall x 0 = closure {x} 
Arguments: {α : Type u_2} [PseudoMetricSpace α] (x : α)
Goal: Metric.closedBall x 0 = closure {x}
-- Metric.eventually_isCompact_closedBall
lemma eventually_isCompact_closedBall [WeaklyLocallyCompactSpace α] (x : α) :
    ∀ᶠ r in 𝓝 (0 : ℝ), IsCompact (closedBall x r) 
Arguments: {α : Type u_2} [PseudoMetricSpace α] [WeaklyLocallyCompactSpace α] (x : α)
Goal: ∀ᶠ (r : ℝ) in 𝓝 0, IsCompact (Metric.closedBall x r)
-- Metric.exists_isCompact_closedBall
lemma exists_isCompact_closedBall [WeaklyLocallyCompactSpace α] (x : α) :
    ∃ r, 0 < r ∧ IsCompact (closedBall x r) 
Arguments: {α : Type u_2} [PseudoMetricSpace α] [WeaklyLocallyCompactSpace α] (x : α)
Goal: ∃ r, 0 < r ∧ IsCompact (Metric.closedBall x r)
-- Real.dist_le_of_mem_pi_Icc
lemma dist_le_of_mem_pi_Icc (hx : x ∈ Icc x' y') (hy : y ∈ Icc x' y') : dist x y ≤ dist x' y' 
Arguments: {ι : Type u_1} [Fintype ι] {x y x' y' : ι → ℝ} (hx : x ∈ Set.Icc x' y') (hy : y ∈ Set.Icc x' y')
Goal: dist x y ≤ dist x' y'
-- MetricSpace.eq_of_dist_eq_zero
class MetricSpace (α : Type u) extends PseudoMetricSpace α : Type u where
  eq_of_dist_eq_zero : ∀ {x y : α}, dist x y = 0 → x = y
Arguments: {α : Type u} [self : MetricSpace α] {x y : α}
Goal: dist x y = 0 → x = y
-- MetricSpace.ext
@[ext]
theorem MetricSpace.ext {α : Type*} {m m' : MetricSpace α} (h : m.toDist = m'.toDist) :
    m = m' 
Arguments: {α : Type u_3} {m m' : MetricSpace α} (h : PseudoMetricSpace.toDist = PseudoMetricSpace.toDist)
Goal: m = m'
-- eq_of_dist_eq_zero
theorem eq_of_dist_eq_zero {x y : γ} : dist x y = 0 → x = y 
Arguments: {γ : Type w} [MetricSpace γ] {x y : γ}
Goal: dist x y = 0 → x = y
-- dist_eq_zero
@[simp]
theorem dist_eq_zero {x y : γ} : dist x y = 0 ↔ x = y 
Arguments: {γ : Type w} [MetricSpace γ] {x y : γ}
Goal: dist x y = 0 ↔ x = y
-- zero_eq_dist
@[simp]
theorem zero_eq_dist {x y : γ} : 0 = dist x y ↔ x = y 
Arguments: {γ : Type w} [MetricSpace γ] {x y : γ}
Goal: 0 = dist x y ↔ x = y
-- dist_ne_zero
theorem dist_ne_zero {x y : γ} : dist x y ≠ 0 ↔ x ≠ y 
Arguments: {γ : Type w} [MetricSpace γ] {x y : γ}
Goal: dist x y ≠ 0 ↔ x ≠ y
-- dist_le_zero
@[simp]
theorem dist_le_zero {x y : γ} : dist x y ≤ 0 ↔ x = y 
Arguments: {γ : Type w} [MetricSpace γ] {x y : γ}
Goal: dist x y ≤ 0 ↔ x = y
-- dist_pos
@[simp]
theorem dist_pos {x y : γ} : 0 < dist x y ↔ x ≠ y 
Arguments: {γ : Type w} [MetricSpace γ] {x y : γ}
Goal: 0 < dist x y ↔ x ≠ y
-- eq_of_forall_dist_le
theorem eq_of_forall_dist_le {x y : γ} (h : ∀ ε > 0, dist x y ≤ ε) : x = y 
Arguments: {γ : Type w} [MetricSpace γ] {x y : γ} (h : ∀ ε > 0, dist x y ≤ ε)
Goal: x = y
-- eq_of_nndist_eq_zero
theorem eq_of_nndist_eq_zero {x y : γ} : nndist x y = 0 → x = y 
Arguments: {γ : Type w} [MetricSpace γ] {x y : γ}
Goal: nndist x y = 0 → x = y
-- nndist_eq_zero
@[simp]
theorem nndist_eq_zero {x y : γ} : nndist x y = 0 ↔ x = y 
Arguments: {γ : Type w} [MetricSpace γ] {x y : γ}
Goal: nndist x y = 0 ↔ x = y
-- zero_eq_nndist
@[simp]
theorem zero_eq_nndist {x y : γ} : 0 = nndist x y ↔ x = y 
Arguments: {γ : Type w} [MetricSpace γ] {x y : γ}
Goal: 0 = nndist x y ↔ x = y
-- Metric.closedBall_zero
@[simp] theorem closedBall_zero : closedBall x 0 = {x} 
Arguments: {γ : Type w} [MetricSpace γ] {x : γ}
Goal: Metric.closedBall x 0 = {x}
-- Metric.sphere_zero
@[simp] theorem sphere_zero : sphere x 0 = {x} 
Arguments: {γ : Type w} [MetricSpace γ] {x : γ}
Goal: Metric.sphere x 0 = {x}
-- Metric.subsingleton_closedBall
theorem subsingleton_closedBall (x : γ) {r : ℝ} (hr : r ≤ 0) : (closedBall x r).Subsingleton 
Arguments: {γ : Type w} [MetricSpace γ] (x : γ) {r : ℝ} (hr : r ≤ 0)
Goal: (Metric.closedBall x r).Subsingleton
-- Metric.subsingleton_sphere
theorem subsingleton_sphere (x : γ) {r : ℝ} (hr : r ≤ 0) : (sphere x r).Subsingleton 
Arguments: {γ : Type w} [MetricSpace γ] (x : γ) {r : ℝ} (hr : r ≤ 0)
Goal: (Metric.sphere x r).Subsingleton
-- Metric.uniformEmbedding_iff'
theorem uniformEmbedding_iff' [MetricSpace β] {f : γ → β} :
    UniformEmbedding f ↔
      (∀ ε > 0, ∃ δ > 0, ∀ {a b : γ}, dist a b < δ → dist (f a) (f b) < ε) ∧
        ∀ δ > 0, ∃ ε > 0, ∀ {a b : γ}, dist (f a) (f b) < ε → dist a b < δ 
Arguments: {β : Type v} {γ : Type w} [MetricSpace γ] [MetricSpace β] {f : γ → β}
Goal: UniformEmbedding f ↔
  (∀ ε > 0, ∃ δ > 0, ∀ {a b : γ}, dist a b < δ → dist (f a) (f b) < ε) ∧
    ∀ δ > 0, ∃ ε > 0, ∀ {a b : γ}, dist (f a) (f b) < ε → dist a b < δ
-- Metric.isClosed_of_pairwise_le_dist
theorem isClosed_of_pairwise_le_dist {s : Set γ} {ε : ℝ} (hε : 0 < ε)
    (hs : s.Pairwise fun x y => ε ≤ dist x y) : IsClosed s 
Arguments: {γ : Type w} [MetricSpace γ] {s : Set γ} {ε : ℝ} (hε : 0 < ε) (hs : s.Pairwise fun x y => ε ≤ dist x y)
Goal: IsClosed s
-- Metric.closedEmbedding_of_pairwise_le_dist
theorem closedEmbedding_of_pairwise_le_dist {α : Type*} [TopologicalSpace α] [DiscreteTopology α]
    {ε : ℝ} (hε : 0 < ε) {f : α → γ} (hf : Pairwise fun x y => ε ≤ dist (f x) (f y)) :
    ClosedEmbedding f 
Arguments: {γ : Type w} [MetricSpace γ] {α : Type u_3} [TopologicalSpace α] [DiscreteTopology α] {ε : ℝ} (hε : 0 < ε) {f : α → γ} (hf : Pairwise fun x y => ε ≤ dist (f x) (f y))
Goal: ClosedEmbedding f
-- Metric.uniformEmbedding_bot_of_pairwise_le_dist
theorem uniformEmbedding_bot_of_pairwise_le_dist {β : Type*} {ε : ℝ} (hε : 0 < ε) {f : β → α}
    (hf : Pairwise fun x y => ε ≤ dist (f x) (f y)) :
    @UniformEmbedding _ _ ⊥ (by infer_instance) f 
Arguments: {α : Type u} [PseudoMetricSpace α] {β : Type u_3} {ε : ℝ} (hε : 0 < ε) {f : β → α} (hf : Pairwise fun x y => ε ≤ dist (f x) (f y))
Goal: UniformEmbedding f
-- MetricSpace.replaceUniformity_eq
theorem MetricSpace.replaceUniformity_eq {γ} [U : UniformSpace γ] (m : MetricSpace γ)
    (H : 𝓤[U] = 𝓤[PseudoEMetricSpace.toUniformSpace]) : m.replaceUniformity H = m 
Arguments: {γ : Type u_3} [U : UniformSpace γ] (m : MetricSpace γ) (H : 𝓤 γ = 𝓤 γ)
Goal: m.replaceUniformity H = m
-- MetricSpace.replaceTopology_eq
theorem MetricSpace.replaceTopology_eq {γ} [U : TopologicalSpace γ] (m : MetricSpace γ)
    (H : U = m.toPseudoMetricSpace.toUniformSpace.toTopologicalSpace) :
    m.replaceTopology H = m 
Arguments: {γ : Type u_3} [U : TopologicalSpace γ] (m : MetricSpace γ) (H : U = UniformSpace.toTopologicalSpace)
Goal: m.replaceTopology H = m
-- MetricSpace.replaceBornology_eq
theorem MetricSpace.replaceBornology_eq {α} [m : MetricSpace α] [B : Bornology α]
    (H : ∀ s, @IsBounded _ B s ↔ @IsBounded _ PseudoMetricSpace.toBornology s) :
    MetricSpace.replaceBornology _ H = m 
Arguments: {α : Type u_3} [m : MetricSpace α] [B : Bornology α] (H : ∀ (s : Set α), Bornology.IsBounded s ↔ Bornology.IsBounded s)
Goal: m.replaceBornology H = m
-- Metric.secondCountable_of_countable_discretization
theorem secondCountable_of_countable_discretization {α : Type u} [MetricSpace α]
    (H : ∀ ε > (0 : ℝ), ∃ (β : Type*) (_ : Encodable β) (F : α → β),
      ∀ x y, F x = F y → dist x y ≤ ε) :
    SecondCountableTopology α 
Arguments: {α : Type u} [MetricSpace α] (H : ∀ ε > 0, ∃ β x F, ∀ (x y : α), F x = F y → dist x y ≤ ε)
Goal: SecondCountableTopology α
-- SeparationQuotient.dist_mk
theorem SeparationQuotient.dist_mk {α : Type u} [PseudoMetricSpace α] (p q : α) :
    dist (mk p) (mk q) = dist p q 
Arguments: {α : Type u} [PseudoMetricSpace α] (p q : α)
Goal: dist (SeparationQuotient.mk p) (SeparationQuotient.mk q) = dist p q
-- dist_ofMul
@[simp] theorem dist_ofMul (a b : X) : dist (ofMul a) (ofMul b) = dist a b 
Arguments: {X : Type u_1} [Dist X] (a b : X)
Goal: dist (Additive.ofMul a) (Additive.ofMul b) = dist a b
-- dist_ofAdd
@[simp] theorem dist_ofAdd (a b : X) : dist (ofAdd a) (ofAdd b) = dist a b 
Arguments: {X : Type u_1} [Dist X] (a b : X)
Goal: dist (Multiplicative.ofAdd a) (Multiplicative.ofAdd b) = dist a b
-- dist_toMul
@[simp] theorem dist_toMul (a b : Additive X) : dist (toMul a) (toMul b) = dist a b 
Arguments: {X : Type u_1} [Dist X] (a b : Additive X)
Goal: dist (Additive.toMul a) (Additive.toMul b) = dist a b
-- dist_toAdd
@[simp] theorem dist_toAdd (a b : Multiplicative X) : dist (toAdd a) (toAdd b) = dist a b 
Arguments: {X : Type u_1} [Dist X] (a b : Multiplicative X)
Goal: dist (Multiplicative.toAdd a) (Multiplicative.toAdd b) = dist a b
-- nndist_ofMul
@[simp] theorem nndist_ofMul (a b : X) : nndist (ofMul a) (ofMul b) = nndist a b 
Arguments: {X : Type u_1} [PseudoMetricSpace X] (a b : X)
Goal: nndist (Additive.ofMul a) (Additive.ofMul b) = nndist a b
-- nndist_ofAdd
@[simp] theorem nndist_ofAdd (a b : X) : nndist (ofAdd a) (ofAdd b) = nndist a b 
Arguments: {X : Type u_1} [PseudoMetricSpace X] (a b : X)
Goal: nndist (Multiplicative.ofAdd a) (Multiplicative.ofAdd b) = nndist a b
-- nndist_toMul
@[simp] theorem nndist_toMul (a b : Additive X) : nndist (toMul a) (toMul b) = nndist a b 
Arguments: {X : Type u_1} [PseudoMetricSpace X] (a b : Additive X)
Goal: nndist (Additive.toMul a) (Additive.toMul b) = nndist a b
-- nndist_toAdd
@[simp]
theorem nndist_toAdd (a b : Multiplicative X) : nndist (toAdd a) (toAdd b) = nndist a b 
Arguments: {X : Type u_1} [PseudoMetricSpace X] (a b : Multiplicative X)
Goal: nndist (Multiplicative.toAdd a) (Multiplicative.toAdd b) = nndist a b
-- dist_toDual
@[simp] theorem dist_toDual (a b : X) : dist (toDual a) (toDual b) = dist a b 
Arguments: {X : Type u_1} [Dist X] (a b : X)
Goal: dist (OrderDual.toDual a) (OrderDual.toDual b) = dist a b
-- dist_ofDual
@[simp] theorem dist_ofDual (a b : Xᵒᵈ) : dist (ofDual a) (ofDual b) = dist a b 
Arguments: {X : Type u_1} [Dist X] (a b : Xᵒᵈ)
Goal: dist (OrderDual.ofDual a) (OrderDual.ofDual b) = dist a b
-- nndist_toDual
@[simp] theorem nndist_toDual (a b : X) : nndist (toDual a) (toDual b) = nndist a b 
Arguments: {X : Type u_1} [PseudoMetricSpace X] (a b : X)
Goal: nndist (OrderDual.toDual a) (OrderDual.toDual b) = nndist a b
-- nndist_ofDual
@[simp] theorem nndist_ofDual (a b : Xᵒᵈ) : nndist (ofDual a) (ofDual b) = nndist a b 
Arguments: {X : Type u_1} [PseudoMetricSpace X] (a b : Xᵒᵈ)
Goal: nndist (OrderDual.ofDual a) (OrderDual.ofDual b) = nndist a b
-- Urysohns.CU.open_U
structure CU {X : Type*} [TopologicalSpace X] (P : Set X → Prop) where
    protected C : Set X
    protected U : Set X
  protected P_C : P C
  protected closed_C : IsClosed C
  protected open_U : IsOpen U
  protected subset : C ⊆ U
  protected hP : ∀ {c u : Set X}, IsClosed c → P c → IsOpen u → c ⊆ u →
    ∃ v, IsOpen v ∧ c ⊆ v ∧ closure v ⊆ u ∧ P (closure v)
Arguments: {X : Type u_2} [TopologicalSpace X] {P : Set X → Prop} (self : Urysohns.CU P)
Goal: IsOpen self.U
-- Urysohns.CU.subset
structure CU {X : Type*} [TopologicalSpace X] (P : Set X → Prop) where
    protected C : Set X
    protected U : Set X
  protected P_C : P C
  protected closed_C : IsClosed C
  protected open_U : IsOpen U
  protected subset : C ⊆ U
  protected hP : ∀ {c u : Set X}, IsClosed c → P c → IsOpen u → c ⊆ u →
    ∃ v, IsOpen v ∧ c ⊆ v ∧ closure v ⊆ u ∧ P (closure v)
Arguments: {X : Type u_2} [TopologicalSpace X] {P : Set X → Prop} (self : Urysohns.CU P)
Goal: self.C ⊆ self.U
-- Urysohns.CU.P_C
structure CU {X : Type*} [TopologicalSpace X] (P : Set X → Prop) where
    protected C : Set X
    protected U : Set X
  protected P_C : P C
  protected closed_C : IsClosed C
  protected open_U : IsOpen U
  protected subset : C ⊆ U
  protected hP : ∀ {c u : Set X}, IsClosed c → P c → IsOpen u → c ⊆ u →
    ∃ v, IsOpen v ∧ c ⊆ v ∧ closure v ⊆ u ∧ P (closure v)
Arguments: {X : Type u_2} [TopologicalSpace X] {P : Set X → Prop} (self : Urysohns.CU P)
Goal: P self.C
-- Urysohns.CU.hP
structure CU {X : Type*} [TopologicalSpace X] (P : Set X → Prop) where
    protected C : Set X
    protected U : Set X
  protected P_C : P C
  protected closed_C : IsClosed C
  protected open_U : IsOpen U
  protected subset : C ⊆ U
  protected hP : ∀ {c u : Set X}, IsClosed c → P c → IsOpen u → c ⊆ u →
    ∃ v, IsOpen v ∧ c ⊆ v ∧ closure v ⊆ u ∧ P (closure v)
Arguments: {X : Type u_2} [TopologicalSpace X] {P : Set X → Prop} (self : Urysohns.CU P) {c u : Set X}
Goal: IsClosed c → P c → IsOpen u → c ⊆ u → ∃ v, IsOpen v ∧ c ⊆ v ∧ closure v ⊆ u ∧ P (closure v)
-- Urysohns.CU.closed_C
structure CU {X : Type*} [TopologicalSpace X] (P : Set X → Prop) where
    protected C : Set X
    protected U : Set X
  protected P_C : P C
  protected closed_C : IsClosed C
  protected open_U : IsOpen U
  protected subset : C ⊆ U
  protected hP : ∀ {c u : Set X}, IsClosed c → P c → IsOpen u → c ⊆ u →
    ∃ v, IsOpen v ∧ c ⊆ v ∧ closure v ⊆ u ∧ P (closure v)
Arguments: {X : Type u_2} [TopologicalSpace X] {P : Set X → Prop} (self : Urysohns.CU P)
Goal: IsClosed self.C
-- Urysohns.CU.left_C
@[simps C]
def left (c : CU P) : CU P 
Arguments: {X : Type u_1} [TopologicalSpace X] {P : Set X → Prop} (c : Urysohns.CU P)
Goal: c.left.C = c.C
-- Urysohns.CU.right_U
@[simps U]
def right (c : CU P) : CU P 
Arguments: {X : Type u_1} [TopologicalSpace X] {P : Set X → Prop} (c : Urysohns.CU P)
Goal: c.right.U = c.U
-- Urysohns.CU.left_U_subset_right_C
theorem left_U_subset_right_C (c : CU P) : c.left.U ⊆ c.right.C 
Arguments: {X : Type u_1} [TopologicalSpace X] {P : Set X → Prop} (c : Urysohns.CU P)
Goal: c.left.U ⊆ c.right.C
-- Urysohns.CU.left_U_subset
theorem left_U_subset (c : CU P) : c.left.U ⊆ c.U 
Arguments: {X : Type u_1} [TopologicalSpace X] {P : Set X → Prop} (c : Urysohns.CU P)
Goal: c.left.U ⊆ c.U
-- Urysohns.CU.subset_right_C
theorem subset_right_C (c : CU P) : c.C ⊆ c.right.C 
Arguments: {X : Type u_1} [TopologicalSpace X] {P : Set X → Prop} (c : Urysohns.CU P)
Goal: c.C ⊆ c.right.C
-- Urysohns.CU.approx_of_mem_C
theorem approx_of_mem_C (c : CU P) (n : ℕ) {x : X} (hx : x ∈ c.C) : c.approx n x = 0 
Arguments: {X : Type u_1} [TopologicalSpace X] {P : Set X → Prop} (c : Urysohns.CU P) (n : ℕ) {x : X} (hx : x ∈ c.C)
Goal: Urysohns.CU.approx n c x = 0