ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- dist_le_coe
@[simp, norm_cast]
theorem dist_le_coe {x y : α} {c : ℝ≥0} : dist x y ≤ c ↔ nndist x y ≤ c 
Arguments: {α : Type u} [PseudoMetricSpace α] {x y : α} {c : ℝ≥0}
Goal: dist x y ≤ ↑c ↔ nndist x y ≤ c
-- edist_lt_ofReal
@[simp]
theorem edist_lt_ofReal {x y : α} {r : ℝ} : edist x y < ENNReal.ofReal r ↔ dist x y < r 
Arguments: {α : Type u} [PseudoMetricSpace α] {x y : α} {r : ℝ}
Goal: edist x y < ENNReal.ofReal r ↔ dist x y < r
-- edist_le_ofReal
@[simp]
theorem edist_le_ofReal {x y : α} {r : ℝ} (hr : 0 ≤ r) :
    edist x y ≤ ENNReal.ofReal r ↔ dist x y ≤ r 
Arguments: {α : Type u} [PseudoMetricSpace α] {x y : α} {r : ℝ} (hr : 0 ≤ r)
Goal: edist x y ≤ ENNReal.ofReal r ↔ dist x y ≤ r
-- nndist_dist
theorem nndist_dist (x y : α) : nndist x y = Real.toNNReal (dist x y) 
Arguments: {α : Type u} [PseudoMetricSpace α] (x y : α)
Goal: nndist x y = (dist x y).toNNReal
-- nndist_comm
theorem nndist_comm (x y : α) : nndist x y = nndist y x 
Arguments: {α : Type u} [PseudoMetricSpace α] (x y : α)
Goal: nndist x y = nndist y x
-- nndist_triangle
theorem nndist_triangle (x y z : α) : nndist x z ≤ nndist x y + nndist y z 
Arguments: {α : Type u} [PseudoMetricSpace α] (x y z : α)
Goal: nndist x z ≤ nndist x y + nndist y z
-- nndist_triangle_left
theorem nndist_triangle_left (x y z : α) : nndist x y ≤ nndist z x + nndist z y 
Arguments: {α : Type u} [PseudoMetricSpace α] (x y z : α)
Goal: nndist x y ≤ nndist z x + nndist z y
-- nndist_triangle_right
theorem nndist_triangle_right (x y z : α) : nndist x y ≤ nndist x z + nndist y z 
Arguments: {α : Type u} [PseudoMetricSpace α] (x y z : α)
Goal: nndist x y ≤ nndist x z + nndist y z
-- dist_edist
theorem dist_edist (x y : α) : dist x y = (edist x y).toReal 
Arguments: {α : Type u} [PseudoMetricSpace α] (x y : α)
Goal: dist x y = (edist x y).toReal
-- Metric.mem_ball
@[simp]
theorem mem_ball : y ∈ ball x ε ↔ dist y x < ε 
Arguments: {α : Type u} [PseudoMetricSpace α] {x y : α} {ε : ℝ}
Goal: y ∈ Metric.ball x ε ↔ dist y x < ε
-- Metric.mem_ball'
theorem mem_ball' : y ∈ ball x ε ↔ dist x y < ε 
Arguments: {α : Type u} [PseudoMetricSpace α] {x y : α} {ε : ℝ}
Goal: y ∈ Metric.ball x ε ↔ dist x y < ε
-- Metric.pos_of_mem_ball
theorem pos_of_mem_ball (hy : y ∈ ball x ε) : 0 < ε 
Arguments: {α : Type u} [PseudoMetricSpace α] {x y : α} {ε : ℝ} (hy : y ∈ Metric.ball x ε)
Goal: 0 < ε
-- Metric.mem_ball_self
theorem mem_ball_self (h : 0 < ε) : x ∈ ball x ε 
Arguments: {α : Type u} [PseudoMetricSpace α] {x : α} {ε : ℝ} (h : 0 < ε)
Goal: x ∈ Metric.ball x ε
-- Metric.nonempty_ball
@[simp]
theorem nonempty_ball : (ball x ε).Nonempty ↔ 0 < ε 
Arguments: {α : Type u} [PseudoMetricSpace α] {x : α} {ε : ℝ}
Goal: (Metric.ball x ε).Nonempty ↔ 0 < ε
-- Metric.ball_eq_empty
@[simp]
theorem ball_eq_empty : ball x ε = ∅ ↔ ε ≤ 0 
Arguments: {α : Type u} [PseudoMetricSpace α] {x : α} {ε : ℝ}
Goal: Metric.ball x ε = ∅ ↔ ε ≤ 0
-- Metric.ball_zero
@[simp]
theorem ball_zero : ball x 0 = ∅ 
Arguments: {α : Type u} [PseudoMetricSpace α] {x : α}
Goal: Metric.ball x 0 = ∅
-- Metric.exists_lt_mem_ball_of_mem_ball
theorem exists_lt_mem_ball_of_mem_ball (h : x ∈ ball y ε) : ∃ ε' < ε, x ∈ ball y ε' 
Arguments: {α : Type u} [PseudoMetricSpace α] {x y : α} {ε : ℝ} (h : x ∈ Metric.ball y ε)
Goal: ∃ ε' < ε, x ∈ Metric.ball y ε'
-- Metric.ball_eq_ball
theorem ball_eq_ball (ε : ℝ) (x : α) :
    UniformSpace.ball x { p | dist p.2 p.1 < ε } = Metric.ball x ε 
Arguments: {α : Type u} [PseudoMetricSpace α] (ε : ℝ) (x : α)
Goal: UniformSpace.ball x {p | dist p.2 p.1 < ε} = Metric.ball x ε
-- Metric.ball_eq_ball'
theorem ball_eq_ball' (ε : ℝ) (x : α) :
    UniformSpace.ball x { p | dist p.1 p.2 < ε } = Metric.ball x ε 
Arguments: {α : Type u} [PseudoMetricSpace α] (ε : ℝ) (x : α)
Goal: UniformSpace.ball x {p | dist p.1 p.2 < ε} = Metric.ball x ε
-- Metric.iUnion_ball_nat
@[simp]
theorem iUnion_ball_nat (x : α) : ⋃ n : ℕ, ball x n = univ 
Arguments: {α : Type u} [PseudoMetricSpace α] (x : α)
Goal: ⋃ n, Metric.ball x ↑n = Set.univ
-- Metric.iUnion_ball_nat_succ
@[simp]
theorem iUnion_ball_nat_succ (x : α) : ⋃ n : ℕ, ball x (n + 1) = univ 
Arguments: {α : Type u} [PseudoMetricSpace α] (x : α)
Goal: ⋃ n, Metric.ball x (↑n + 1) = Set.univ
-- Metric.mem_closedBall
@[simp] theorem mem_closedBall : y ∈ closedBall x ε ↔ dist y x ≤ ε 
Arguments: {α : Type u} [PseudoMetricSpace α] {x y : α} {ε : ℝ}
Goal: y ∈ Metric.closedBall x ε ↔ dist y x ≤ ε
-- Metric.mem_closedBall'
theorem mem_closedBall' : y ∈ closedBall x ε ↔ dist x y ≤ ε 
Arguments: {α : Type u} [PseudoMetricSpace α] {x y : α} {ε : ℝ}
Goal: y ∈ Metric.closedBall x ε ↔ dist x y ≤ ε
-- Metric.mem_sphere
@[simp] theorem mem_sphere : y ∈ sphere x ε ↔ dist y x = ε 
Arguments: {α : Type u} [PseudoMetricSpace α] {x y : α} {ε : ℝ}
Goal: y ∈ Metric.sphere x ε ↔ dist y x = ε
-- Metric.mem_sphere'
theorem mem_sphere' : y ∈ sphere x ε ↔ dist x y = ε 
Arguments: {α : Type u} [PseudoMetricSpace α] {x y : α} {ε : ℝ}
Goal: y ∈ Metric.sphere x ε ↔ dist x y = ε
-- Metric.ne_of_mem_sphere
theorem ne_of_mem_sphere (h : y ∈ sphere x ε) (hε : ε ≠ 0) : y ≠ x 
Arguments: {α : Type u} [PseudoMetricSpace α] {x y : α} {ε : ℝ} (h : y ∈ Metric.sphere x ε) (hε : ε ≠ 0)
Goal: y ≠ x
-- Metric.nonneg_of_mem_sphere
theorem nonneg_of_mem_sphere (hy : y ∈ sphere x ε) : 0 ≤ ε 
Arguments: {α : Type u} [PseudoMetricSpace α] {x y : α} {ε : ℝ} (hy : y ∈ Metric.sphere x ε)
Goal: 0 ≤ ε
-- Metric.sphere_eq_empty_of_neg
@[simp]
theorem sphere_eq_empty_of_neg (hε : ε < 0) : sphere x ε = ∅ 
Arguments: {α : Type u} [PseudoMetricSpace α] {x : α} {ε : ℝ} (hε : ε < 0)
Goal: Metric.sphere x ε = ∅
-- Metric.sphere_eq_empty_of_subsingleton
theorem sphere_eq_empty_of_subsingleton [Subsingleton α] (hε : ε ≠ 0) : sphere x ε = ∅ 
Arguments: {α : Type u} [PseudoMetricSpace α] {x : α} {ε : ℝ} [Subsingleton α] (hε : ε ≠ 0)
Goal: Metric.sphere x ε = ∅
-- Metric.mem_closedBall_self
theorem mem_closedBall_self (h : 0 ≤ ε) : x ∈ closedBall x ε 
Arguments: {α : Type u} [PseudoMetricSpace α] {x : α} {ε : ℝ} (h : 0 ≤ ε)
Goal: x ∈ Metric.closedBall x ε
-- Metric.nonempty_closedBall
@[simp]
theorem nonempty_closedBall : (closedBall x ε).Nonempty ↔ 0 ≤ ε 
Arguments: {α : Type u} [PseudoMetricSpace α] {x : α} {ε : ℝ}
Goal: (Metric.closedBall x ε).Nonempty ↔ 0 ≤ ε
-- Metric.closedBall_eq_empty
@[simp]
theorem closedBall_eq_empty : closedBall x ε = ∅ ↔ ε < 0 
Arguments: {α : Type u} [PseudoMetricSpace α] {x : α} {ε : ℝ}
Goal: Metric.closedBall x ε = ∅ ↔ ε < 0
-- Metric.closedBall_eq_sphere_of_nonpos
theorem closedBall_eq_sphere_of_nonpos (hε : ε ≤ 0) : closedBall x ε = sphere x ε 
Arguments: {α : Type u} [PseudoMetricSpace α] {x : α} {ε : ℝ} (hε : ε ≤ 0)
Goal: Metric.closedBall x ε = Metric.sphere x ε
-- Metric.ball_subset_closedBall
theorem ball_subset_closedBall : ball x ε ⊆ closedBall x ε 
Arguments: {α : Type u} [PseudoMetricSpace α] {x : α} {ε : ℝ}
Goal: Metric.ball x ε ⊆ Metric.closedBall x ε
-- Metric.sphere_subset_closedBall
theorem sphere_subset_closedBall : sphere x ε ⊆ closedBall x ε 
Arguments: {α : Type u} [PseudoMetricSpace α] {x : α} {ε : ℝ}
Goal: Metric.sphere x ε ⊆ Metric.closedBall x ε
-- Metric.sphere_subset_ball
lemma sphere_subset_ball {r R : ℝ} (h : r < R) : sphere x r ⊆ ball x R 
Arguments: {α : Type u} [PseudoMetricSpace α] {x : α} {r R : ℝ} (h : r < R)
Goal: Metric.sphere x r ⊆ Metric.ball x R
-- Metric.closedBall_disjoint_ball
theorem closedBall_disjoint_ball (h : δ + ε ≤ dist x y) : Disjoint (closedBall x δ) (ball y ε) 
Arguments: {α : Type u} [PseudoMetricSpace α] {x y : α} {δ ε : ℝ} (h : δ + ε ≤ dist x y)
Goal: Disjoint (Metric.closedBall x δ) (Metric.ball y ε)
-- Metric.ball_disjoint_closedBall
theorem ball_disjoint_closedBall (h : δ + ε ≤ dist x y) : Disjoint (ball x δ) (closedBall y ε) 
Arguments: {α : Type u} [PseudoMetricSpace α] {x y : α} {δ ε : ℝ} (h : δ + ε ≤ dist x y)
Goal: Disjoint (Metric.ball x δ) (Metric.closedBall y ε)
-- Metric.ball_disjoint_ball
theorem ball_disjoint_ball (h : δ + ε ≤ dist x y) : Disjoint (ball x δ) (ball y ε) 
Arguments: {α : Type u} [PseudoMetricSpace α] {x y : α} {δ ε : ℝ} (h : δ + ε ≤ dist x y)
Goal: Disjoint (Metric.ball x δ) (Metric.ball y ε)
-- Metric.closedBall_disjoint_closedBall
theorem closedBall_disjoint_closedBall (h : δ + ε < dist x y) :
    Disjoint (closedBall x δ) (closedBall y ε) 
Arguments: {α : Type u} [PseudoMetricSpace α] {x y : α} {δ ε : ℝ} (h : δ + ε < dist x y)
Goal: Disjoint (Metric.closedBall x δ) (Metric.closedBall y ε)
-- Metric.sphere_disjoint_ball
theorem sphere_disjoint_ball : Disjoint (sphere x ε) (ball x ε) 
Arguments: {α : Type u} [PseudoMetricSpace α] {x : α} {ε : ℝ}
Goal: Disjoint (Metric.sphere x ε) (Metric.ball x ε)
-- Metric.ball_union_sphere
@[simp]
theorem ball_union_sphere : ball x ε ∪ sphere x ε = closedBall x ε 
Arguments: {α : Type u} [PseudoMetricSpace α] {x : α} {ε : ℝ}
Goal: Metric.ball x ε ∪ Metric.sphere x ε = Metric.closedBall x ε
-- Metric.sphere_union_ball
@[simp]
theorem sphere_union_ball : sphere x ε ∪ ball x ε = closedBall x ε 
Arguments: {α : Type u} [PseudoMetricSpace α] {x : α} {ε : ℝ}
Goal: Metric.sphere x ε ∪ Metric.ball x ε = Metric.closedBall x ε
-- Metric.closedBall_diff_sphere
@[simp]
theorem closedBall_diff_sphere : closedBall x ε \ sphere x ε = ball x ε 
Arguments: {α : Type u} [PseudoMetricSpace α] {x : α} {ε : ℝ}
Goal: Metric.closedBall x ε \ Metric.sphere x ε = Metric.ball x ε
-- Metric.closedBall_diff_ball
@[simp]
theorem closedBall_diff_ball : closedBall x ε \ ball x ε = sphere x ε 
Arguments: {α : Type u} [PseudoMetricSpace α] {x : α} {ε : ℝ}
Goal: Metric.closedBall x ε \ Metric.ball x ε = Metric.sphere x ε
-- Metric.mem_ball_comm
theorem mem_ball_comm : x ∈ ball y ε ↔ y ∈ ball x ε 
Arguments: {α : Type u} [PseudoMetricSpace α] {x y : α} {ε : ℝ}
Goal: x ∈ Metric.ball y ε ↔ y ∈ Metric.ball x ε
-- Metric.mem_closedBall_comm
theorem mem_closedBall_comm : x ∈ closedBall y ε ↔ y ∈ closedBall x ε 
Arguments: {α : Type u} [PseudoMetricSpace α] {x y : α} {ε : ℝ}
Goal: x ∈ Metric.closedBall y ε ↔ y ∈ Metric.closedBall x ε
-- Metric.mem_sphere_comm
theorem mem_sphere_comm : x ∈ sphere y ε ↔ y ∈ sphere x ε 
Arguments: {α : Type u} [PseudoMetricSpace α] {x y : α} {ε : ℝ}
Goal: x ∈ Metric.sphere y ε ↔ y ∈ Metric.sphere x ε
-- Metric.ball_subset_ball
@[gcongr]
theorem ball_subset_ball (h : ε₁ ≤ ε₂) : ball x ε₁ ⊆ ball x ε₂ 
Arguments: {α : Type u} [PseudoMetricSpace α] {x : α} {ε₁ ε₂ : ℝ} (h : ε₁ ≤ ε₂)
Goal: Metric.ball x ε₁ ⊆ Metric.ball x ε₂
-- Metric.closedBall_eq_bInter_ball
theorem closedBall_eq_bInter_ball : closedBall x ε = ⋂ δ > ε, ball x δ 
Arguments: {α : Type u} [PseudoMetricSpace α] {x : α} {ε : ℝ}
Goal: Metric.closedBall x ε = ⋂ δ, ⋂ (_ : δ > ε), Metric.ball x δ
-- Metric.ball_subset_ball'
theorem ball_subset_ball' (h : ε₁ + dist x y ≤ ε₂) : ball x ε₁ ⊆ ball y ε₂ 
Arguments: {α : Type u} [PseudoMetricSpace α] {x y : α} {ε₁ ε₂ : ℝ} (h : ε₁ + dist x y ≤ ε₂)
Goal: Metric.ball x ε₁ ⊆ Metric.ball y ε₂
-- Metric.closedBall_subset_closedBall
@[gcongr]
theorem closedBall_subset_closedBall (h : ε₁ ≤ ε₂) : closedBall x ε₁ ⊆ closedBall x ε₂ 
Arguments: {α : Type u} [PseudoMetricSpace α] {x : α} {ε₁ ε₂ : ℝ} (h : ε₁ ≤ ε₂)
Goal: Metric.closedBall x ε₁ ⊆ Metric.closedBall x ε₂
-- Metric.closedBall_subset_closedBall'
theorem closedBall_subset_closedBall' (h : ε₁ + dist x y ≤ ε₂) :
    closedBall x ε₁ ⊆ closedBall y ε₂ 
Arguments: {α : Type u} [PseudoMetricSpace α] {x y : α} {ε₁ ε₂ : ℝ} (h : ε₁ + dist x y ≤ ε₂)
Goal: Metric.closedBall x ε₁ ⊆ Metric.closedBall y ε₂
-- Metric.closedBall_subset_ball
theorem closedBall_subset_ball (h : ε₁ < ε₂) : closedBall x ε₁ ⊆ ball x ε₂ 
Arguments: {α : Type u} [PseudoMetricSpace α] {x : α} {ε₁ ε₂ : ℝ} (h : ε₁ < ε₂)
Goal: Metric.closedBall x ε₁ ⊆ Metric.ball x ε₂
-- Metric.closedBall_subset_ball'
theorem closedBall_subset_ball' (h : ε₁ + dist x y < ε₂) :
    closedBall x ε₁ ⊆ ball y ε₂ 
Arguments: {α : Type u} [PseudoMetricSpace α] {x y : α} {ε₁ ε₂ : ℝ} (h : ε₁ + dist x y < ε₂)
Goal: Metric.closedBall x ε₁ ⊆ Metric.ball y ε₂
-- Metric.dist_le_add_of_nonempty_closedBall_inter_closedBall
theorem dist_le_add_of_nonempty_closedBall_inter_closedBall
    (h : (closedBall x ε₁ ∩ closedBall y ε₂).Nonempty) : dist x y ≤ ε₁ + ε₂ 
Arguments: {α : Type u} [PseudoMetricSpace α] {x y : α} {ε₁ ε₂ : ℝ} (h : (Metric.closedBall x ε₁ ∩ Metric.closedBall y ε₂).Nonempty)
Goal: dist x y ≤ ε₁ + ε₂
-- Metric.dist_lt_add_of_nonempty_closedBall_inter_ball
theorem dist_lt_add_of_nonempty_closedBall_inter_ball (h : (closedBall x ε₁ ∩ ball y ε₂).Nonempty) :
    dist x y < ε₁ + ε₂ 
Arguments: {α : Type u} [PseudoMetricSpace α] {x y : α} {ε₁ ε₂ : ℝ} (h : (Metric.closedBall x ε₁ ∩ Metric.ball y ε₂).Nonempty)
Goal: dist x y < ε₁ + ε₂
-- Metric.dist_lt_add_of_nonempty_ball_inter_closedBall
theorem dist_lt_add_of_nonempty_ball_inter_closedBall (h : (ball x ε₁ ∩ closedBall y ε₂).Nonempty) :
    dist x y < ε₁ + ε₂ 
Arguments: {α : Type u} [PseudoMetricSpace α] {x y : α} {ε₁ ε₂ : ℝ} (h : (Metric.ball x ε₁ ∩ Metric.closedBall y ε₂).Nonempty)
Goal: dist x y < ε₁ + ε₂
-- Metric.dist_lt_add_of_nonempty_ball_inter_ball
theorem dist_lt_add_of_nonempty_ball_inter_ball (h : (ball x ε₁ ∩ ball y ε₂).Nonempty) :
    dist x y < ε₁ + ε₂ 
Arguments: {α : Type u} [PseudoMetricSpace α] {x y : α} {ε₁ ε₂ : ℝ} (h : (Metric.ball x ε₁ ∩ Metric.ball y ε₂).Nonempty)
Goal: dist x y < ε₁ + ε₂
-- Metric.iUnion_closedBall_nat
@[simp]
theorem iUnion_closedBall_nat (x : α) : ⋃ n : ℕ, closedBall x n = univ 
Arguments: {α : Type u} [PseudoMetricSpace α] (x : α)
Goal: ⋃ n, Metric.closedBall x ↑n = Set.univ
-- Metric.iUnion_inter_closedBall_nat
theorem iUnion_inter_closedBall_nat (s : Set α) (x : α) : ⋃ n : ℕ, s ∩ closedBall x n = s 
Arguments: {α : Type u} [PseudoMetricSpace α] (s : Set α) (x : α)
Goal: ⋃ n, s ∩ Metric.closedBall x ↑n = s
-- Metric.ball_subset
theorem ball_subset (h : dist x y ≤ ε₂ - ε₁) : ball x ε₁ ⊆ ball y ε₂ 
Arguments: {α : Type u} [PseudoMetricSpace α] {x y : α} {ε₁ ε₂ : ℝ} (h : dist x y ≤ ε₂ - ε₁)
Goal: Metric.ball x ε₁ ⊆ Metric.ball y ε₂
-- Metric.ball_half_subset
theorem ball_half_subset (y) (h : y ∈ ball x (ε / 2)) : ball y (ε / 2) ⊆ ball x ε 
Arguments: {α : Type u} [PseudoMetricSpace α] {x : α} {ε : ℝ} (y : α) (h : y ∈ Metric.ball x (ε / 2))
Goal: Metric.ball y (ε / 2) ⊆ Metric.ball x ε
-- Metric.exists_ball_subset_ball
theorem exists_ball_subset_ball (h : y ∈ ball x ε) : ∃ ε' > 0, ball y ε' ⊆ ball x ε 
Arguments: {α : Type u} [PseudoMetricSpace α] {x y : α} {ε : ℝ} (h : y ∈ Metric.ball x ε)
Goal: ∃ ε' > 0, Metric.ball y ε' ⊆ Metric.ball x ε
-- Metric.forall_of_forall_mem_closedBall
theorem forall_of_forall_mem_closedBall (p : α → Prop) (x : α)
    (H : ∃ᶠ R : ℝ in atTop, ∀ y ∈ closedBall x R, p y) (y : α) : p y 
Arguments: {α : Type u} [PseudoMetricSpace α] (p : α → Prop) (x : α) (H : ∃ᶠ (R : ℝ) in Filter.atTop, ∀ y ∈ Metric.closedBall x R, p y) (y : α)
Goal: p y
-- Metric.forall_of_forall_mem_ball
theorem forall_of_forall_mem_ball (p : α → Prop) (x : α)
    (H : ∃ᶠ R : ℝ in atTop, ∀ y ∈ ball x R, p y) (y : α) : p y 
Arguments: {α : Type u} [PseudoMetricSpace α] (p : α → Prop) (x : α) (H : ∃ᶠ (R : ℝ) in Filter.atTop, ∀ y ∈ Metric.ball x R, p y) (y : α)
Goal: p y
-- Metric.isBounded_iff
theorem isBounded_iff {s : Set α} :
    IsBounded s ↔ ∃ C : ℝ, ∀ ⦃x⦄, x ∈ s → ∀ ⦃y⦄, y ∈ s → dist x y ≤ C 
Arguments: {α : Type u} [PseudoMetricSpace α] {s : Set α}
Goal: Bornology.IsBounded s ↔ ∃ C, ∀ ⦃x : α⦄, x ∈ s → ∀ ⦃y : α⦄, y ∈ s → dist x y ≤ C
-- Metric.isBounded_iff_eventually
theorem isBounded_iff_eventually {s : Set α} :
    IsBounded s ↔ ∀ᶠ C in atTop, ∀ ⦃x⦄, x ∈ s → ∀ ⦃y⦄, y ∈ s → dist x y ≤ C 
Arguments: {α : Type u} [PseudoMetricSpace α] {s : Set α}
Goal: Bornology.IsBounded s ↔ ∀ᶠ (C : ℝ) in Filter.atTop, ∀ ⦃x : α⦄, x ∈ s → ∀ ⦃y : α⦄, y ∈ s → dist x y ≤ C
-- Metric.isBounded_iff_exists_ge
theorem isBounded_iff_exists_ge {s : Set α} (c : ℝ) :
    IsBounded s ↔ ∃ C, c ≤ C ∧ ∀ ⦃x⦄, x ∈ s → ∀ ⦃y⦄, y ∈ s → dist x y ≤ C 
Arguments: {α : Type u} [PseudoMetricSpace α] {s : Set α} (c : ℝ)
Goal: Bornology.IsBounded s ↔ ∃ C, c ≤ C ∧ ∀ ⦃x : α⦄, x ∈ s → ∀ ⦃y : α⦄, y ∈ s → dist x y ≤ C
-- Metric.isBounded_iff_nndist
theorem isBounded_iff_nndist {s : Set α} :
    IsBounded s ↔ ∃ C : ℝ≥0, ∀ ⦃x⦄, x ∈ s → ∀ ⦃y⦄, y ∈ s → nndist x y ≤ C 
Arguments: {α : Type u} [PseudoMetricSpace α] {s : Set α}
Goal: Bornology.IsBounded s ↔ ∃ C, ∀ ⦃x : α⦄, x ∈ s → ∀ ⦃y : α⦄, y ∈ s → nndist x y ≤ C
-- Metric.toUniformSpace_eq
theorem toUniformSpace_eq :
    ‹PseudoMetricSpace α›.toUniformSpace = .ofDist dist dist_self dist_comm dist_triangle 
Arguments: {α : Type u} [PseudoMetricSpace α]
Goal: PseudoMetricSpace.toUniformSpace = UniformSpace.ofDist dist ⋯ ⋯ ⋯
-- Metric.uniformity_basis_dist
theorem uniformity_basis_dist :
    (𝓤 α).HasBasis (fun ε : ℝ => 0 < ε) fun ε => { p : α × α | dist p.1 p.2 < ε } 
Arguments: {α : Type u} [PseudoMetricSpace α]
Goal: (𝓤 α).HasBasis (fun ε => 0 < ε) fun ε => {p | dist p.1 p.2 < ε}
-- Metric.mk_uniformity_basis
protected theorem mk_uniformity_basis {β : Type*} {p : β → Prop} {f : β → ℝ}
    (hf₀ : ∀ i, p i → 0 < f i) (hf : ∀ ⦃ε⦄, 0 < ε → ∃ i, p i ∧ f i ≤ ε) :
    (𝓤 α).HasBasis p fun i => { p : α × α | dist p.1 p.2 < f i } 
Arguments: {α : Type u} [PseudoMetricSpace α] {β : Type u_3} {p : β → Prop} {f : β → ℝ} (hf₀ : ∀ (i : β), p i → 0 < f i) (hf : ∀ ⦃ε : ℝ⦄, 0 < ε → ∃ i, p i ∧ f i ≤ ε)
Goal: (𝓤 α).HasBasis p fun i => {p | dist p.1 p.2 < f i}
-- Metric.uniformity_basis_dist_rat
theorem uniformity_basis_dist_rat :
    (𝓤 α).HasBasis (fun r : ℚ => 0 < r) fun r => { p : α × α | dist p.1 p.2 < r } 
Arguments: {α : Type u} [PseudoMetricSpace α]
Goal: (𝓤 α).HasBasis (fun r => 0 < r) fun r => {p | dist p.1 p.2 < ↑r}
-- Metric.uniformity_basis_dist_inv_nat_succ
theorem uniformity_basis_dist_inv_nat_succ :
    (𝓤 α).HasBasis (fun _ => True) fun n : ℕ => { p : α × α | dist p.1 p.2 < 1 / (↑n + 1) } 
Arguments: {α : Type u} [PseudoMetricSpace α]
Goal: (𝓤 α).HasBasis (fun x => True) fun n => {p | dist p.1 p.2 < 1 / (↑n + 1)}
-- Metric.uniformity_basis_dist_inv_nat_pos
theorem uniformity_basis_dist_inv_nat_pos :
    (𝓤 α).HasBasis (fun n : ℕ => 0 < n) fun n : ℕ => { p : α × α | dist p.1 p.2 < 1 / ↑n } 
Arguments: {α : Type u} [PseudoMetricSpace α]
Goal: (𝓤 α).HasBasis (fun n => 0 < n) fun n => {p | dist p.1 p.2 < 1 / ↑n}
-- Metric.uniformity_basis_dist_pow
theorem uniformity_basis_dist_pow {r : ℝ} (h0 : 0 < r) (h1 : r < 1) :
    (𝓤 α).HasBasis (fun _ : ℕ => True) fun n : ℕ => { p : α × α | dist p.1 p.2 < r ^ n } 
Arguments: {α : Type u} [PseudoMetricSpace α] {r : ℝ} (h0 : 0 < r) (h1 : r < 1)
Goal: (𝓤 α).HasBasis (fun x => True) fun n => {p | dist p.1 p.2 < r ^ n}
-- Metric.uniformity_basis_dist_lt
theorem uniformity_basis_dist_lt {R : ℝ} (hR : 0 < R) :
    (𝓤 α).HasBasis (fun r : ℝ => 0 < r ∧ r < R) fun r => { p : α × α | dist p.1 p.2 < r } 
Arguments: {α : Type u} [PseudoMetricSpace α] {R : ℝ} (hR : 0 < R)
Goal: (𝓤 α).HasBasis (fun r => 0 < r ∧ r < R) fun r => {p | dist p.1 p.2 < r}
-- Metric.mk_uniformity_basis_le
protected theorem mk_uniformity_basis_le {β : Type*} {p : β → Prop} {f : β → ℝ}
    (hf₀ : ∀ x, p x → 0 < f x) (hf : ∀ ε, 0 < ε → ∃ x, p x ∧ f x ≤ ε) :
    (𝓤 α).HasBasis p fun x => { p : α × α | dist p.1 p.2 ≤ f x } 
Arguments: {α : Type u} [PseudoMetricSpace α] {β : Type u_3} {p : β → Prop} {f : β → ℝ} (hf₀ : ∀ (x : β), p x → 0 < f x) (hf : ∀ (ε : ℝ), 0 < ε → ∃ x, p x ∧ f x ≤ ε)
Goal: (𝓤 α).HasBasis p fun x => {p | dist p.1 p.2 ≤ f x}
-- Metric.uniformity_basis_dist_le
theorem uniformity_basis_dist_le :
    (𝓤 α).HasBasis ((0 : ℝ) < ·) fun ε => { p : α × α | dist p.1 p.2 ≤ ε } 
Arguments: {α : Type u} [PseudoMetricSpace α]
Goal: (𝓤 α).HasBasis (fun x => 0 < x) fun ε => {p | dist p.1 p.2 ≤ ε}
-- Metric.uniformity_basis_dist_le_pow
theorem uniformity_basis_dist_le_pow {r : ℝ} (h0 : 0 < r) (h1 : r < 1) :
    (𝓤 α).HasBasis (fun _ : ℕ => True) fun n : ℕ => { p : α × α | dist p.1 p.2 ≤ r ^ n } 
Arguments: {α : Type u} [PseudoMetricSpace α] {r : ℝ} (h0 : 0 < r) (h1 : r < 1)
Goal: (𝓤 α).HasBasis (fun x => True) fun n => {p | dist p.1 p.2 ≤ r ^ n}
-- Metric.mem_uniformity_dist
theorem mem_uniformity_dist {s : Set (α × α)} :
    s ∈ 𝓤 α ↔ ∃ ε > 0, ∀ {a b : α}, dist a b < ε → (a, b) ∈ s 
Arguments: {α : Type u} [PseudoMetricSpace α] {s : Set (α × α)}
Goal: s ∈ 𝓤 α ↔ ∃ ε > 0, ∀ {a b : α}, dist a b < ε → (a, b) ∈ s
-- Metric.dist_mem_uniformity
theorem dist_mem_uniformity {ε : ℝ} (ε0 : 0 < ε) : { p : α × α | dist p.1 p.2 < ε } ∈ 𝓤 α 
Arguments: {α : Type u} [PseudoMetricSpace α] {ε : ℝ} (ε0 : 0 < ε)
Goal: {p | dist p.1 p.2 < ε} ∈ 𝓤 α
-- Metric.uniformContinuous_iff
theorem uniformContinuous_iff [PseudoMetricSpace β] {f : α → β} :
    UniformContinuous f ↔ ∀ ε > 0, ∃ δ > 0, ∀ {a b : α}, dist a b < δ → dist (f a) (f b) < ε 
Arguments: {α : Type u} {β : Type v} [PseudoMetricSpace α] [PseudoMetricSpace β] {f : α → β}
Goal: UniformContinuous f ↔ ∀ ε > 0, ∃ δ > 0, ∀ {a b : α}, dist a b < δ → dist (f a) (f b) < ε
-- Metric.uniformContinuousOn_iff
theorem uniformContinuousOn_iff [PseudoMetricSpace β] {f : α → β} {s : Set α} :
    UniformContinuousOn f s ↔
      ∀ ε > 0, ∃ δ > 0, ∀ x ∈ s, ∀ y ∈ s, dist x y < δ → dist (f x) (f y) < ε 
Arguments: {α : Type u} {β : Type v} [PseudoMetricSpace α] [PseudoMetricSpace β] {f : α → β} {s : Set α}
Goal: UniformContinuousOn f s ↔ ∀ ε > 0, ∃ δ > 0, ∀ x ∈ s, ∀ y ∈ s, dist x y < δ → dist (f x) (f y) < ε
-- Metric.uniformContinuousOn_iff_le
theorem uniformContinuousOn_iff_le [PseudoMetricSpace β] {f : α → β} {s : Set α} :
    UniformContinuousOn f s ↔
      ∀ ε > 0, ∃ δ > 0, ∀ x ∈ s, ∀ y ∈ s, dist x y ≤ δ → dist (f x) (f y) ≤ ε 
Arguments: {α : Type u} {β : Type v} [PseudoMetricSpace α] [PseudoMetricSpace β] {f : α → β} {s : Set α}
Goal: UniformContinuousOn f s ↔ ∀ ε > 0, ∃ δ > 0, ∀ x ∈ s, ∀ y ∈ s, dist x y ≤ δ → dist (f x) (f y) ≤ ε
-- Metric.uniformInducing_iff
nonrec theorem uniformInducing_iff [PseudoMetricSpace β] {f : α → β} :
    UniformInducing f ↔ UniformContinuous f ∧
      ∀ δ > 0, ∃ ε > 0, ∀ {a b : α}, dist (f a) (f b) < ε → dist a b < δ 
Arguments: {α : Type u} {β : Type v} [PseudoMetricSpace α] [PseudoMetricSpace β] {f : α → β}
Goal: UniformInducing f ↔ UniformContinuous f ∧ ∀ δ > 0, ∃ ε > 0, ∀ {a b : α}, dist (f a) (f b) < ε → dist a b < δ
-- Metric.uniformEmbedding_iff
nonrec theorem uniformEmbedding_iff [PseudoMetricSpace β] {f : α → β} :
    UniformEmbedding f ↔ Function.Injective f ∧ UniformContinuous f ∧
      ∀ δ > 0, ∃ ε > 0, ∀ {a b : α}, dist (f a) (f b) < ε → dist a b < δ 
Arguments: {α : Type u} {β : Type v} [PseudoMetricSpace α] [PseudoMetricSpace β] {f : α → β}
Goal: UniformEmbedding f ↔
  Function.Injective f ∧ UniformContinuous f ∧ ∀ δ > 0, ∃ ε > 0, ∀ {a b : α}, dist (f a) (f b) < ε → dist a b < δ
-- Metric.controlled_of_uniformEmbedding
theorem controlled_of_uniformEmbedding [PseudoMetricSpace β] {f : α → β} (h : UniformEmbedding f) :
    (∀ ε > 0, ∃ δ > 0, ∀ {a b : α}, dist a b < δ → dist (f a) (f b) < ε) ∧
      ∀ δ > 0, ∃ ε > 0, ∀ {a b : α}, dist (f a) (f b) < ε → dist a b < δ 
Arguments: {α : Type u} {β : Type v} [PseudoMetricSpace α] [PseudoMetricSpace β] {f : α → β} (h : UniformEmbedding f)
Goal: (∀ ε > 0, ∃ δ > 0, ∀ {a b : α}, dist a b < δ → dist (f a) (f b) < ε) ∧
  ∀ δ > 0, ∃ ε > 0, ∀ {a b : α}, dist (f a) (f b) < ε → dist a b < δ
-- Metric.totallyBounded_iff
theorem totallyBounded_iff {s : Set α} :
    TotallyBounded s ↔ ∀ ε > 0, ∃ t : Set α, t.Finite ∧ s ⊆ ⋃ y ∈ t, ball y ε 
Arguments: {α : Type u} [PseudoMetricSpace α] {s : Set α}
Goal: TotallyBounded s ↔ ∀ ε > 0, ∃ t, t.Finite ∧ s ⊆ ⋃ y ∈ t, Metric.ball y ε
-- Metric.totallyBounded_of_finite_discretization
theorem totallyBounded_of_finite_discretization {s : Set α}
    (H : ∀ ε > (0 : ℝ),
        ∃ (β : Type u) (_ : Fintype β) (F : s → β), ∀ x y, F x = F y → dist (x : α) y < ε) :
    TotallyBounded s 
Arguments: {α : Type u} [PseudoMetricSpace α] {s : Set α} (H : ∀ ε > 0, ∃ β x F, ∀ (x y : ↑s), F x = F y → dist ↑x ↑y < ε)
Goal: TotallyBounded s
-- Metric.finite_approx_of_totallyBounded
theorem finite_approx_of_totallyBounded {s : Set α} (hs : TotallyBounded s) :
    ∀ ε > 0, ∃ t, t ⊆ s ∧ Set.Finite t ∧ s ⊆ ⋃ y ∈ t, ball y ε 
Arguments: {α : Type u} [PseudoMetricSpace α] {s : Set α} (hs : TotallyBounded s) (ε : ℝ)
Goal: ε > 0 → ∃ t ⊆ s, t.Finite ∧ s ⊆ ⋃ y ∈ t, Metric.ball y ε