ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- NNReal.ball_zero_eq_Ico
lemma NNReal.ball_zero_eq_Ico (c : ℝ) :
    Metric.ball (0 : ℝ≥0) c = Set.Ico 0 c.toNNReal 
Arguments: (c : ℝ)
Goal: Metric.ball 0 c = Set.Ico 0 c.toNNReal
-- NNReal.closedBall_zero_eq_Icc'
lemma NNReal.closedBall_zero_eq_Icc' (c : ℝ≥0) :
    Metric.closedBall (0 : ℝ≥0) c.toReal = Set.Icc 0 c 
Arguments: (c : ℝ≥0)
Goal: Metric.closedBall 0 ↑c = Set.Icc 0 c
-- NNReal.closedBall_zero_eq_Icc
lemma NNReal.closedBall_zero_eq_Icc {c : ℝ} (c_nn : 0 ≤ c) :
    Metric.closedBall (0 : ℝ≥0) c = Set.Icc 0 c.toNNReal 
Arguments: {c : ℝ} (c_nn : 0 ≤ c)
Goal: Metric.closedBall 0 c = Set.Icc 0 c.toNNReal
-- ULift.dist_eq
lemma dist_eq (x y : ULift β) : dist x y = dist x.down y.down 
Arguments: {β : Type u_2} [PseudoMetricSpace β] (x y : ULift.{u_3, u_2} β)
Goal: dist x y = dist x.down y.down
-- ULift.nndist_eq
lemma nndist_eq (x y : ULift β) : nndist x y = nndist x.down y.down 
Arguments: {β : Type u_2} [PseudoMetricSpace β] (x y : ULift.{u_3, u_2} β)
Goal: nndist x y = nndist x.down y.down
-- ULift.dist_up_up
@[simp] lemma dist_up_up (x y : β) : dist (ULift.up x) (ULift.up y) = dist x y 
Arguments: {β : Type u_2} [PseudoMetricSpace β] (x y : β)
Goal: dist { down := x } { down := y } = dist x y
-- ULift.nndist_up_up
@[simp] lemma nndist_up_up (x y : β) : nndist (ULift.up x) (ULift.up y) = nndist x y 
Arguments: {β : Type u_2} [PseudoMetricSpace β] (x y : β)
Goal: nndist { down := x } { down := y } = nndist x y
-- Prod.dist_eq
lemma Prod.dist_eq {x y : α × β} : dist x y = max (dist x.1 y.1) (dist x.2 y.2) 
Arguments: {α : Type u_1} {β : Type u_2} [PseudoMetricSpace α] [PseudoMetricSpace β] {x y : α × β}
Goal: dist x y = max (dist x.1 y.1) (dist x.2 y.2)
-- dist_prod_same_left
@[simp]
lemma dist_prod_same_left {x : α} {y₁ y₂ : β} : dist (x, y₁) (x, y₂) = dist y₁ y₂ 
Arguments: {α : Type u_1} {β : Type u_2} [PseudoMetricSpace α] [PseudoMetricSpace β] {x : α} {y₁ y₂ : β}
Goal: dist (x, y₁) (x, y₂) = dist y₁ y₂
-- dist_prod_same_right
@[simp]
lemma dist_prod_same_right {x₁ x₂ : α} {y : β} : dist (x₁, y) (x₂, y) = dist x₁ x₂ 
Arguments: {α : Type u_1} {β : Type u_2} [PseudoMetricSpace α] [PseudoMetricSpace β] {x₁ x₂ : α} {y : β}
Goal: dist (x₁, y) (x₂, y) = dist x₁ x₂
-- ball_prod_same
lemma ball_prod_same (x : α) (y : β) (r : ℝ) : ball x r ×ˢ ball y r = ball (x, y) r 
Arguments: {α : Type u_1} {β : Type u_2} [PseudoMetricSpace α] [PseudoMetricSpace β] (x : α) (y : β) (r : ℝ)
Goal: Metric.ball x r ×ˢ Metric.ball y r = Metric.ball (x, y) r
-- closedBall_prod_same
lemma closedBall_prod_same (x : α) (y : β) (r : ℝ) :
    closedBall x r ×ˢ closedBall y r = closedBall (x, y) r 
Arguments: {α : Type u_1} {β : Type u_2} [PseudoMetricSpace α] [PseudoMetricSpace β] (x : α) (y : β) (r : ℝ)
Goal: Metric.closedBall x r ×ˢ Metric.closedBall y r = Metric.closedBall (x, y) r
-- sphere_prod
lemma sphere_prod (x : α × β) (r : ℝ) :
    sphere x r = sphere x.1 r ×ˢ closedBall x.2 r ∪ closedBall x.1 r ×ˢ sphere x.2 r 
Arguments: {α : Type u_1} {β : Type u_2} [PseudoMetricSpace α] [PseudoMetricSpace β] (x : α × β) (r : ℝ)
Goal: Metric.sphere x r = Metric.sphere x.1 r ×ˢ Metric.closedBall x.2 r ∪ Metric.closedBall x.1 r ×ˢ Metric.sphere x.2 r
-- uniformContinuous_dist
lemma uniformContinuous_dist : UniformContinuous fun p : α × α => dist p.1 p.2 
Arguments: {α : Type u_1} [PseudoMetricSpace α]
Goal: UniformContinuous fun p => dist p.1 p.2
-- UniformContinuous.dist
protected lemma UniformContinuous.dist [UniformSpace β] {f g : β → α} (hf : UniformContinuous f)
    (hg : UniformContinuous g) : UniformContinuous fun b => dist (f b) (g b) 
Arguments: {α : Type u_1} {β : Type u_2} [PseudoMetricSpace α] [UniformSpace β] {f g : β → α} (hf : UniformContinuous f) (hg : UniformContinuous g)
Goal: UniformContinuous fun b => dist (f b) (g b)
-- continuous_dist
@[continuity]
lemma continuous_dist : Continuous fun p : α × α ↦ dist p.1 p.2 
Arguments: {α : Type u_1} [PseudoMetricSpace α]
Goal: Continuous fun p => dist p.1 p.2
-- Continuous.dist
@[continuity, fun_prop]
protected lemma Continuous.dist [TopologicalSpace β] {f g : β → α} (hf : Continuous f)
    (hg : Continuous g) : Continuous fun b => dist (f b) (g b) 
Arguments: {α : Type u_1} {β : Type u_2} [PseudoMetricSpace α] [TopologicalSpace β] {f g : β → α} (hf : Continuous f) (hg : Continuous g)
Goal: Continuous fun b => dist (f b) (g b)
-- Filter.Tendsto.dist
protected lemma Filter.Tendsto.dist {f g : β → α} {x : Filter β} {a b : α}
    (hf : Tendsto f x (𝓝 a)) (hg : Tendsto g x (𝓝 b)) :
    Tendsto (fun x => dist (f x) (g x)) x (𝓝 (dist a b)) 
Arguments: {α : Type u_1} {β : Type u_2} [PseudoMetricSpace α] {f g : β → α} {x : Filter β} {a b : α} (hf : Filter.Tendsto f x (𝓝 a)) (hg : Filter.Tendsto g x (𝓝 b))
Goal: Filter.Tendsto (fun x => dist (f x) (g x)) x (𝓝 (dist a b))
-- continuous_iff_continuous_dist
lemma continuous_iff_continuous_dist [TopologicalSpace β] {f : β → α} :
    Continuous f ↔ Continuous fun x : β × β => dist (f x.1) (f x.2) 
Arguments: {α : Type u_1} {β : Type u_2} [PseudoMetricSpace α] [TopologicalSpace β] {f : β → α}
Goal: Continuous f ↔ Continuous fun x => dist (f x.1) (f x.2)
-- uniformContinuous_nndist
lemma uniformContinuous_nndist : UniformContinuous fun p : α × α => nndist p.1 p.2 
Arguments: {α : Type u_1} [PseudoMetricSpace α]
Goal: UniformContinuous fun p => nndist p.1 p.2
-- UniformContinuous.nndist
protected lemma UniformContinuous.nndist [UniformSpace β] {f g : β → α} (hf : UniformContinuous f)
    (hg : UniformContinuous g) : UniformContinuous fun b => nndist (f b) (g b) 
Arguments: {α : Type u_1} {β : Type u_2} [PseudoMetricSpace α] [UniformSpace β] {f g : β → α} (hf : UniformContinuous f) (hg : UniformContinuous g)
Goal: UniformContinuous fun b => nndist (f b) (g b)
-- continuous_nndist
lemma continuous_nndist : Continuous fun p : α × α => nndist p.1 p.2 
Arguments: {α : Type u_1} [PseudoMetricSpace α]
Goal: Continuous fun p => nndist p.1 p.2
-- Continuous.nndist
@[fun_prop]
protected lemma Continuous.nndist [TopologicalSpace β] {f g : β → α} (hf : Continuous f)
    (hg : Continuous g) : Continuous fun b => nndist (f b) (g b) 
Arguments: {α : Type u_1} {β : Type u_2} [PseudoMetricSpace α] [TopologicalSpace β] {f g : β → α} (hf : Continuous f) (hg : Continuous g)
Goal: Continuous fun b => nndist (f b) (g b)
-- Filter.Tendsto.nndist
protected lemma Filter.Tendsto.nndist {f g : β → α} {x : Filter β} {a b : α}
    (hf : Tendsto f x (𝓝 a)) (hg : Tendsto g x (𝓝 b)) :
    Tendsto (fun x => nndist (f x) (g x)) x (𝓝 (nndist a b)) 
Arguments: {α : Type u_1} {β : Type u_2} [PseudoMetricSpace α] {f g : β → α} {x : Filter β} {a b : α} (hf : Filter.Tendsto f x (𝓝 a)) (hg : Filter.Tendsto g x (𝓝 b))
Goal: Filter.Tendsto (fun x => nndist (f x) (g x)) x (𝓝 (nndist a b))
-- nndist_pi_def
lemma nndist_pi_def (f g : ∀ b, π b) : nndist f g = sup univ fun b => nndist (f b) (g b) 
Arguments: {β : Type u_2} {π : β → Type u_3} [Fintype β] [(b : β) → PseudoMetricSpace (π b)] (f g : (b : β) → π b)
Goal: nndist f g = Finset.univ.sup fun b => nndist (f b) (g b)
-- dist_pi_def
lemma dist_pi_def (f g : ∀ b, π b) : dist f g = (sup univ fun b => nndist (f b) (g b) : ℝ≥0) 
Arguments: {β : Type u_2} {π : β → Type u_3} [Fintype β] [(b : β) → PseudoMetricSpace (π b)] (f g : (b : β) → π b)
Goal: dist f g = ↑(Finset.univ.sup fun b => nndist (f b) (g b))
-- nndist_pi_le_iff
lemma nndist_pi_le_iff {f g : ∀ b, π b} {r : ℝ≥0} :
    nndist f g ≤ r ↔ ∀ b, nndist (f b) (g b) ≤ r 
Arguments: {β : Type u_2} {π : β → Type u_3} [Fintype β] [(b : β) → PseudoMetricSpace (π b)] {f g : (b : β) → π b} {r : ℝ≥0}
Goal: nndist f g ≤ r ↔ ∀ (b : β), nndist (f b) (g b) ≤ r
-- nndist_pi_lt_iff
lemma nndist_pi_lt_iff {f g : ∀ b, π b} {r : ℝ≥0} (hr : 0 < r) :
    nndist f g < r ↔ ∀ b, nndist (f b) (g b) < r 
Arguments: {β : Type u_2} {π : β → Type u_3} [Fintype β] [(b : β) → PseudoMetricSpace (π b)] {f g : (b : β) → π b} {r : ℝ≥0} (hr : 0 < r)
Goal: nndist f g < r ↔ ∀ (b : β), nndist (f b) (g b) < r
-- nndist_pi_eq_iff
lemma nndist_pi_eq_iff {f g : ∀ b, π b} {r : ℝ≥0} (hr : 0 < r) :
    nndist f g = r ↔ (∃ i, nndist (f i) (g i) = r) ∧ ∀ b, nndist (f b) (g b) ≤ r 
Arguments: {β : Type u_2} {π : β → Type u_3} [Fintype β] [(b : β) → PseudoMetricSpace (π b)] {f g : (b : β) → π b} {r : ℝ≥0} (hr : 0 < r)
Goal: nndist f g = r ↔ (∃ i, nndist (f i) (g i) = r) ∧ ∀ (b : β), nndist (f b) (g b) ≤ r
-- dist_pi_lt_iff
lemma dist_pi_lt_iff {f g : ∀ b, π b} {r : ℝ} (hr : 0 < r) :
    dist f g < r ↔ ∀ b, dist (f b) (g b) < r 
Arguments: {β : Type u_2} {π : β → Type u_3} [Fintype β] [(b : β) → PseudoMetricSpace (π b)] {f g : (b : β) → π b} {r : ℝ} (hr : 0 < r)
Goal: dist f g < r ↔ ∀ (b : β), dist (f b) (g b) < r
-- dist_pi_le_iff
lemma dist_pi_le_iff {f g : ∀ b, π b} {r : ℝ} (hr : 0 ≤ r) :
    dist f g ≤ r ↔ ∀ b, dist (f b) (g b) ≤ r 
Arguments: {β : Type u_2} {π : β → Type u_3} [Fintype β] [(b : β) → PseudoMetricSpace (π b)] {f g : (b : β) → π b} {r : ℝ} (hr : 0 ≤ r)
Goal: dist f g ≤ r ↔ ∀ (b : β), dist (f b) (g b) ≤ r
-- dist_pi_eq_iff
lemma dist_pi_eq_iff {f g : ∀ b, π b} {r : ℝ} (hr : 0 < r) :
    dist f g = r ↔ (∃ i, dist (f i) (g i) = r) ∧ ∀ b, dist (f b) (g b) ≤ r 
Arguments: {β : Type u_2} {π : β → Type u_3} [Fintype β] [(b : β) → PseudoMetricSpace (π b)] {f g : (b : β) → π b} {r : ℝ} (hr : 0 < r)
Goal: dist f g = r ↔ (∃ i, dist (f i) (g i) = r) ∧ ∀ (b : β), dist (f b) (g b) ≤ r
-- dist_pi_le_iff'
lemma dist_pi_le_iff' [Nonempty β] {f g : ∀ b, π b} {r : ℝ} :
    dist f g ≤ r ↔ ∀ b, dist (f b) (g b) ≤ r 
Arguments: {β : Type u_2} {π : β → Type u_3} [Fintype β] [(b : β) → PseudoMetricSpace (π b)] [Nonempty β] {f g : (b : β) → π b} {r : ℝ}
Goal: dist f g ≤ r ↔ ∀ (b : β), dist (f b) (g b) ≤ r
-- dist_pi_const_le
lemma dist_pi_const_le (a b : α) : (dist (fun _ : β => a) fun _ => b) ≤ dist a b 
Arguments: {α : Type u_1} {β : Type u_2} [PseudoMetricSpace α] [Fintype β] (a b : α)
Goal: (dist (fun x => a) fun x => b) ≤ dist a b
-- nndist_pi_const_le
lemma nndist_pi_const_le (a b : α) : (nndist (fun _ : β => a) fun _ => b) ≤ nndist a b 
Arguments: {α : Type u_1} {β : Type u_2} [PseudoMetricSpace α] [Fintype β] (a b : α)
Goal: (nndist (fun x => a) fun x => b) ≤ nndist a b
-- dist_pi_const
@[simp]
lemma dist_pi_const [Nonempty β] (a b : α) : (dist (fun _ : β => a) fun _ => b) = dist a b 
Arguments: {α : Type u_1} {β : Type u_2} [PseudoMetricSpace α] [Fintype β] [Nonempty β] (a b : α)
Goal: (dist (fun x => a) fun x => b) = dist a b
-- nndist_pi_const
@[simp]
lemma nndist_pi_const [Nonempty β] (a b : α) : (nndist (fun _ : β => a) fun _ => b) = nndist a b 
Arguments: {α : Type u_1} {β : Type u_2} [PseudoMetricSpace α] [Fintype β] [Nonempty β] (a b : α)
Goal: (nndist (fun x => a) fun x => b) = nndist a b
-- nndist_le_pi_nndist
lemma nndist_le_pi_nndist (f g : ∀ b, π b) (b : β) : nndist (f b) (g b) ≤ nndist f g 
Arguments: {β : Type u_2} {π : β → Type u_3} [Fintype β] [(b : β) → PseudoMetricSpace (π b)] (f g : (b : β) → π b) (b : β)
Goal: nndist (f b) (g b) ≤ nndist f g
-- dist_le_pi_dist
lemma dist_le_pi_dist (f g : ∀ b, π b) (b : β) : dist (f b) (g b) ≤ dist f g 
Arguments: {β : Type u_2} {π : β → Type u_3} [Fintype β] [(b : β) → PseudoMetricSpace (π b)] (f g : (b : β) → π b) (b : β)
Goal: dist (f b) (g b) ≤ dist f g
-- ball_pi
lemma ball_pi (x : ∀ b, π b) {r : ℝ} (hr : 0 < r) :
    ball x r = Set.pi univ fun b => ball (x b) r 
Arguments: {β : Type u_2} {π : β → Type u_3} [Fintype β] [(b : β) → PseudoMetricSpace (π b)] (x : (b : β) → π b) {r : ℝ} (hr : 0 < r)
Goal: Metric.ball x r = Set.univ.pi fun b => Metric.ball (x b) r
-- ball_pi'
lemma ball_pi' [Nonempty β] (x : ∀ b, π b) (r : ℝ) :
    ball x r = Set.pi univ fun b => ball (x b) r 
Arguments: {β : Type u_2} {π : β → Type u_3} [Fintype β] [(b : β) → PseudoMetricSpace (π b)] [Nonempty β] (x : (b : β) → π b) (r : ℝ)
Goal: Metric.ball x r = Set.univ.pi fun b => Metric.ball (x b) r
-- closedBall_pi
lemma closedBall_pi (x : ∀ b, π b) {r : ℝ} (hr : 0 ≤ r) :
    closedBall x r = Set.pi univ fun b => closedBall (x b) r 
Arguments: {β : Type u_2} {π : β → Type u_3} [Fintype β] [(b : β) → PseudoMetricSpace (π b)] (x : (b : β) → π b) {r : ℝ} (hr : 0 ≤ r)
Goal: Metric.closedBall x r = Set.univ.pi fun b => Metric.closedBall (x b) r
-- closedBall_pi'
lemma closedBall_pi' [Nonempty β] (x : ∀ b, π b) (r : ℝ) :
    closedBall x r = Set.pi univ fun b => closedBall (x b) r 
Arguments: {β : Type u_2} {π : β → Type u_3} [Fintype β] [(b : β) → PseudoMetricSpace (π b)] [Nonempty β] (x : (b : β) → π b) (r : ℝ)
Goal: Metric.closedBall x r = Set.univ.pi fun b => Metric.closedBall (x b) r
-- sphere_pi
lemma sphere_pi (x : ∀ b, π b) {r : ℝ} (h : 0 < r ∨ Nonempty β) :
    sphere x r = (⋃ i : β, Function.eval i ⁻¹' sphere (x i) r) ∩ closedBall x r 
Arguments: {β : Type u_2} {π : β → Type u_3} [Fintype β] [(b : β) → PseudoMetricSpace (π b)] (x : (b : β) → π b) {r : ℝ} (h : 0 < r ∨ Nonempty β)
Goal: Metric.sphere x r = (⋃ i, Function.eval i ⁻¹' Metric.sphere (x i) r) ∩ Metric.closedBall x r
-- Fin.nndist_insertNth_insertNth
@[simp]
lemma Fin.nndist_insertNth_insertNth {n : ℕ} {α : Fin (n + 1) → Type*}
    [∀ i, PseudoMetricSpace (α i)] (i : Fin (n + 1)) (x y : α i) (f g : ∀ j, α (i.succAbove j)) :
    nndist (i.insertNth x f) (i.insertNth y g) = max (nndist x y) (nndist f g) 
Arguments: {n : ℕ} {α : Fin (n + 1) → Type u_4} [(i : Fin (n + 1)) → PseudoMetricSpace (α i)] (i : Fin (n + 1)) (x y : α i) (f g : (j : Fin n) → α (i.succAbove j))
Goal: nndist (i.insertNth x f) (i.insertNth y g) = max (nndist x y) (nndist f g)
-- Fin.dist_insertNth_insertNth
@[simp]
lemma Fin.dist_insertNth_insertNth {n : ℕ} {α : Fin (n + 1) → Type*}
    [∀ i, PseudoMetricSpace (α i)] (i : Fin (n + 1)) (x y : α i) (f g : ∀ j, α (i.succAbove j)) :
    dist (i.insertNth x f) (i.insertNth y g) = max (dist x y) (dist f g) 
Arguments: {n : ℕ} {α : Fin (n + 1) → Type u_4} [(i : Fin (n + 1)) → PseudoMetricSpace (α i)] (i : Fin (n + 1)) (x y : α i) (f g : (j : Fin n) → α (i.succAbove j))
Goal: dist (i.insertNth x f) (i.insertNth y g) = max (dist x y) (dist f g)
-- isometry_iff_nndist_eq
theorem isometry_iff_nndist_eq [PseudoMetricSpace α] [PseudoMetricSpace β] {f : α → β} :
    Isometry f ↔ ∀ x y, nndist (f x) (f y) = nndist x y 
Arguments: {α : Type u} {β : Type v} [PseudoMetricSpace α] [PseudoMetricSpace β] {f : α → β}
Goal: Isometry f ↔ ∀ (x y : α), nndist (f x) (f y) = nndist x y
-- isometry_iff_dist_eq
theorem isometry_iff_dist_eq [PseudoMetricSpace α] [PseudoMetricSpace β] {f : α → β} :
    Isometry f ↔ ∀ x y, dist (f x) (f y) = dist x y 
Arguments: {α : Type u} {β : Type v} [PseudoMetricSpace α] [PseudoMetricSpace β] {f : α → β}
Goal: Isometry f ↔ ∀ (x y : α), dist (f x) (f y) = dist x y
-- Isometry.edist_eq
theorem edist_eq (hf : Isometry f) (x y : α) : edist (f x) (f y) = edist x y 
Arguments: {α : Type u} {β : Type v} [PseudoEMetricSpace α] [PseudoEMetricSpace β] {f : α → β} (hf : Isometry f) (x y : α)
Goal: edist (f x) (f y) = edist x y
-- Isometry.lipschitz
theorem lipschitz (h : Isometry f) : LipschitzWith 1 f 
Arguments: {α : Type u} {β : Type v} [PseudoEMetricSpace α] [PseudoEMetricSpace β] {f : α → β} (h : Isometry f)
Goal: LipschitzWith 1 f
-- Isometry.antilipschitz
theorem antilipschitz (h : Isometry f) : AntilipschitzWith 1 f 
Arguments: {α : Type u} {β : Type v} [PseudoEMetricSpace α] [PseudoEMetricSpace β] {f : α → β} (h : Isometry f)
Goal: AntilipschitzWith 1 f
-- isometry_subsingleton
@[nontriviality]
theorem _root_.isometry_subsingleton [Subsingleton α] : Isometry f 
Arguments: {α : Type u} {β : Type v} [PseudoEMetricSpace α] [PseudoEMetricSpace β] {f : α → β} [Subsingleton α]
Goal: Isometry f
-- isometry_id
theorem _root_.isometry_id : Isometry (id : α → α) 
Arguments: {α : Type u} [PseudoEMetricSpace α]
Goal: Isometry id
-- Isometry.prod_map
theorem prod_map {δ} [PseudoEMetricSpace δ] {f : α → β} {g : γ → δ} (hf : Isometry f)
    (hg : Isometry g) : Isometry (Prod.map f g) 
Arguments: {α : Type u} {β : Type v} {γ : Type w} [PseudoEMetricSpace α] [PseudoEMetricSpace β] [PseudoEMetricSpace γ] {δ : Type u_2} [PseudoEMetricSpace δ] {f : α → β} {g : γ → δ} (hf : Isometry f) (hg : Isometry g)
Goal: Isometry (Prod.map f g)
-- isometry_dcomp
theorem _root_.isometry_dcomp {ι} [Fintype ι] {α β : ι → Type*} [∀ i, PseudoEMetricSpace (α i)]
    [∀ i, PseudoEMetricSpace (β i)] (f : ∀ i, α i → β i) (hf : ∀ i, Isometry (f i)) :
    Isometry (fun g : (i : ι) → α i => fun i => f i (g i)) 
Arguments: {ι : Type u_4} [Fintype ι] {α : ι → Type u_2} {β : ι → Type u_3} [(i : ι) → PseudoEMetricSpace (α i)] [(i : ι) → PseudoEMetricSpace (β i)] (f : (i : ι) → α i → β i) (hf : ∀ (i : ι), Isometry (f i))
Goal: Isometry fun g i => f i (g i)
-- Isometry.comp
theorem comp {g : β → γ} {f : α → β} (hg : Isometry g) (hf : Isometry f) : Isometry (g ∘ f) 
Arguments: {α : Type u} {β : Type v} {γ : Type w} [PseudoEMetricSpace α] [PseudoEMetricSpace β] [PseudoEMetricSpace γ] {g : β → γ} {f : α → β} (hg : Isometry g) (hf : Isometry f)
Goal: Isometry (g ∘ f)
-- Isometry.uniformContinuous
protected theorem uniformContinuous (hf : Isometry f) : UniformContinuous f 
Arguments: {α : Type u} {β : Type v} [PseudoEMetricSpace α] [PseudoEMetricSpace β] {f : α → β} (hf : Isometry f)
Goal: UniformContinuous f
-- Isometry.uniformInducing
protected theorem uniformInducing (hf : Isometry f) : UniformInducing f 
Arguments: {α : Type u} {β : Type v} [PseudoEMetricSpace α] [PseudoEMetricSpace β] {f : α → β} (hf : Isometry f)
Goal: UniformInducing f
-- Isometry.tendsto_nhds_iff
theorem tendsto_nhds_iff {ι : Type*} {f : α → β} {g : ι → α} {a : Filter ι} {b : α}
    (hf : Isometry f) : Filter.Tendsto g a (𝓝 b) ↔ Filter.Tendsto (f ∘ g) a (𝓝 (f b)) 
Arguments: {α : Type u} {β : Type v} [PseudoEMetricSpace α] [PseudoEMetricSpace β] {ι : Type u_2} {f : α → β} {g : ι → α} {a : Filter ι} {b : α} (hf : Isometry f)
Goal: Filter.Tendsto g a (𝓝 b) ↔ Filter.Tendsto (f ∘ g) a (𝓝 (f b))
-- Isometry.continuous
protected theorem continuous (hf : Isometry f) : Continuous f 
Arguments: {α : Type u} {β : Type v} [PseudoEMetricSpace α] [PseudoEMetricSpace β] {f : α → β} (hf : Isometry f)
Goal: Continuous f
-- Isometry.right_inv
theorem right_inv {f : α → β} {g : β → α} (h : Isometry f) (hg : RightInverse g f) : Isometry g 
Arguments: {α : Type u} {β : Type v} [PseudoEMetricSpace α] [PseudoEMetricSpace β] {f : α → β} {g : β → α} (h : Isometry f) (hg : Function.RightInverse g f)
Goal: Isometry g
-- Isometry.preimage_emetric_closedBall
theorem preimage_emetric_closedBall (h : Isometry f) (x : α) (r : ℝ≥0∞) :
    f ⁻¹' EMetric.closedBall (f x) r = EMetric.closedBall x r 
Arguments: {α : Type u} {β : Type v} [PseudoEMetricSpace α] [PseudoEMetricSpace β] {f : α → β} (h : Isometry f) (x : α) (r : ℝ≥0∞)
Goal: f ⁻¹' EMetric.closedBall (f x) r = EMetric.closedBall x r
-- Isometry.preimage_emetric_ball
theorem preimage_emetric_ball (h : Isometry f) (x : α) (r : ℝ≥0∞) :
    f ⁻¹' EMetric.ball (f x) r = EMetric.ball x r 
Arguments: {α : Type u} {β : Type v} [PseudoEMetricSpace α] [PseudoEMetricSpace β] {f : α → β} (h : Isometry f) (x : α) (r : ℝ≥0∞)
Goal: f ⁻¹' EMetric.ball (f x) r = EMetric.ball x r
-- Isometry.ediam_image
theorem ediam_image (hf : Isometry f) (s : Set α) : EMetric.diam (f '' s) = EMetric.diam s 
Arguments: {α : Type u} {β : Type v} [PseudoEMetricSpace α] [PseudoEMetricSpace β] {f : α → β} (hf : Isometry f) (s : Set α)
Goal: EMetric.diam (f '' s) = EMetric.diam s
-- Isometry.ediam_range
theorem ediam_range (hf : Isometry f) : EMetric.diam (range f) = EMetric.diam (univ : Set α) 
Arguments: {α : Type u} {β : Type v} [PseudoEMetricSpace α] [PseudoEMetricSpace β] {f : α → β} (hf : Isometry f)
Goal: EMetric.diam (Set.range f) = EMetric.diam Set.univ
-- Isometry.mapsTo_emetric_ball
theorem mapsTo_emetric_ball (hf : Isometry f) (x : α) (r : ℝ≥0∞) :
    MapsTo f (EMetric.ball x r) (EMetric.ball (f x) r) 
Arguments: {α : Type u} {β : Type v} [PseudoEMetricSpace α] [PseudoEMetricSpace β] {f : α → β} (hf : Isometry f) (x : α) (r : ℝ≥0∞)
Goal: Set.MapsTo f (EMetric.ball x r) (EMetric.ball (f x) r)
-- Isometry.mapsTo_emetric_closedBall
theorem mapsTo_emetric_closedBall (hf : Isometry f) (x : α) (r : ℝ≥0∞) :
    MapsTo f (EMetric.closedBall x r) (EMetric.closedBall (f x) r) 
Arguments: {α : Type u} {β : Type v} [PseudoEMetricSpace α] [PseudoEMetricSpace β] {f : α → β} (hf : Isometry f) (x : α) (r : ℝ≥0∞)
Goal: Set.MapsTo f (EMetric.closedBall x r) (EMetric.closedBall (f x) r)
-- isometry_subtype_coe
theorem _root_.isometry_subtype_coe {s : Set α} : Isometry ((↑) : s → α) 
Arguments: {α : Type u} [PseudoEMetricSpace α] {s : Set α}
Goal: Isometry Subtype.val
-- Isometry.comp_continuousOn_iff
theorem comp_continuousOn_iff {γ} [TopologicalSpace γ] (hf : Isometry f) {g : γ → α} {s : Set γ} :
    ContinuousOn (f ∘ g) s ↔ ContinuousOn g s 
Arguments: {α : Type u} {β : Type v} [PseudoEMetricSpace α] [PseudoEMetricSpace β] {f : α → β} {γ : Type u_2} [TopologicalSpace γ] (hf : Isometry f) {g : γ → α} {s : Set γ}
Goal: ContinuousOn (f ∘ g) s ↔ ContinuousOn g s
-- Isometry.comp_continuous_iff
theorem comp_continuous_iff {γ} [TopologicalSpace γ] (hf : Isometry f) {g : γ → α} :
    Continuous (f ∘ g) ↔ Continuous g 
Arguments: {α : Type u} {β : Type v} [PseudoEMetricSpace α] [PseudoEMetricSpace β] {f : α → β} {γ : Type u_2} [TopologicalSpace γ] (hf : Isometry f) {g : γ → α}
Goal: Continuous (f ∘ g) ↔ Continuous g
-- Isometry.injective
protected theorem injective (h : Isometry f) : Injective f 
Arguments: {α : Type u} {β : Type v} [EMetricSpace α] [PseudoEMetricSpace β] {f : α → β} (h : Isometry f)
Goal: Function.Injective f
-- Isometry.uniformEmbedding
protected theorem uniformEmbedding (hf : Isometry f) : UniformEmbedding f 
Arguments: {α : Type u} {β : Type v} [EMetricSpace α] [PseudoEMetricSpace β] {f : α → β} (hf : Isometry f)
Goal: UniformEmbedding f
-- Isometry.embedding
protected theorem embedding (hf : Isometry f) : Embedding f 
Arguments: {α : Type u} {β : Type v} [EMetricSpace α] [PseudoEMetricSpace β] {f : α → β} (hf : Isometry f)
Goal: Embedding f
-- Isometry.closedEmbedding
theorem closedEmbedding [CompleteSpace α] [EMetricSpace γ] {f : α → γ} (hf : Isometry f) :
    ClosedEmbedding f 
Arguments: {α : Type u} {γ : Type w} [EMetricSpace α] [CompleteSpace α] [EMetricSpace γ] {f : α → γ} (hf : Isometry f)
Goal: ClosedEmbedding f
-- Isometry.diam_image
theorem diam_image (hf : Isometry f) (s : Set α) : Metric.diam (f '' s) = Metric.diam s 
Arguments: {α : Type u} {β : Type v} [PseudoMetricSpace α] [PseudoMetricSpace β] {f : α → β} (hf : Isometry f) (s : Set α)
Goal: Metric.diam (f '' s) = Metric.diam s
-- Isometry.diam_range
theorem diam_range (hf : Isometry f) : Metric.diam (range f) = Metric.diam (univ : Set α) 
Arguments: {α : Type u} {β : Type v} [PseudoMetricSpace α] [PseudoMetricSpace β] {f : α → β} (hf : Isometry f)
Goal: Metric.diam (Set.range f) = Metric.diam Set.univ
-- Isometry.preimage_setOf_dist
theorem preimage_setOf_dist (hf : Isometry f) (x : α) (p : ℝ → Prop) :
    f ⁻¹' { y | p (dist y (f x)) } = { y | p (dist y x) } 
Arguments: {α : Type u} {β : Type v} [PseudoMetricSpace α] [PseudoMetricSpace β] {f : α → β} (hf : Isometry f) (x : α) (p : ℝ → Prop)
Goal: f ⁻¹' {y | p (dist y (f x))} = {y | p (dist y x)}
-- Isometry.preimage_closedBall
theorem preimage_closedBall (hf : Isometry f) (x : α) (r : ℝ) :
    f ⁻¹' Metric.closedBall (f x) r = Metric.closedBall x r 
Arguments: {α : Type u} {β : Type v} [PseudoMetricSpace α] [PseudoMetricSpace β] {f : α → β} (hf : Isometry f) (x : α) (r : ℝ)
Goal: f ⁻¹' Metric.closedBall (f x) r = Metric.closedBall x r
-- Isometry.preimage_ball
theorem preimage_ball (hf : Isometry f) (x : α) (r : ℝ) :
    f ⁻¹' Metric.ball (f x) r = Metric.ball x r 
Arguments: {α : Type u} {β : Type v} [PseudoMetricSpace α] [PseudoMetricSpace β] {f : α → β} (hf : Isometry f) (x : α) (r : ℝ)
Goal: f ⁻¹' Metric.ball (f x) r = Metric.ball x r
-- Isometry.preimage_sphere
theorem preimage_sphere (hf : Isometry f) (x : α) (r : ℝ) :
    f ⁻¹' Metric.sphere (f x) r = Metric.sphere x r 
Arguments: {α : Type u} {β : Type v} [PseudoMetricSpace α] [PseudoMetricSpace β] {f : α → β} (hf : Isometry f) (x : α) (r : ℝ)
Goal: f ⁻¹' Metric.sphere (f x) r = Metric.sphere x r
-- Isometry.mapsTo_ball
theorem mapsTo_ball (hf : Isometry f) (x : α) (r : ℝ) :
    MapsTo f (Metric.ball x r) (Metric.ball (f x) r) 
Arguments: {α : Type u} {β : Type v} [PseudoMetricSpace α] [PseudoMetricSpace β] {f : α → β} (hf : Isometry f) (x : α) (r : ℝ)
Goal: Set.MapsTo f (Metric.ball x r) (Metric.ball (f x) r)