ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- ContinuousWithinAt.snd
theorem ContinuousWithinAt.snd {f : α → β × γ} {s : Set α} {a : α} (h : ContinuousWithinAt f s a) :
    ContinuousWithinAt (fun x => (f x).snd) s a 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} [TopologicalSpace α] [TopologicalSpace β] [TopologicalSpace γ] {f : α → β × γ} {s : Set α} {a : α} (h : ContinuousWithinAt f s a)
Goal: ContinuousWithinAt (fun x => (f x).2) s a
-- continuousWithinAt_prod_iff
theorem continuousWithinAt_prod_iff {f : α → β × γ} {s : Set α} {x : α} :
    ContinuousWithinAt f s x ↔
      ContinuousWithinAt (Prod.fst ∘ f) s x ∧ ContinuousWithinAt (Prod.snd ∘ f) s x 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} [TopologicalSpace α] [TopologicalSpace β] [TopologicalSpace γ] {f : α → β × γ} {s : Set α} {x : α}
Goal: ContinuousWithinAt f s x ↔ ContinuousWithinAt (Prod.fst ∘ f) s x ∧ ContinuousWithinAt (Prod.snd ∘ f) s x
-- TopCat.epi_iff_surjective
theorem epi_iff_surjective {X Y : TopCat.{u}} (f : X ⟶ Y) : Epi f ↔ Function.Surjective f 
Arguments: {X Y : TopCat} (f : X ⟶ Y)
Goal: CategoryTheory.Epi f ↔ Function.Surjective ⇑f
-- TopCat.mono_iff_injective
theorem mono_iff_injective {X Y : TopCat.{u}} (f : X ⟶ Y) : Mono f ↔ Function.Injective f 
Arguments: {X Y : TopCat} (f : X ⟶ Y)
Goal: CategoryTheory.Mono f ↔ Function.Injective ⇑f
-- lipschitzOnWith_empty
@[simp]
theorem lipschitzOnWith_empty (K : ℝ≥0) (f : α → β) : LipschitzOnWith K f ∅ 
Arguments: {α : Type u} {β : Type v} [PseudoEMetricSpace α] [PseudoEMetricSpace β] (K : ℝ≥0) (f : α → β)
Goal: LipschitzOnWith K f ∅
-- locallyLipschitzOn_empty
@[simp] lemma locallyLipschitzOn_empty (f : α → β) : LocallyLipschitzOn ∅ f 
Arguments: {α : Type u} {β : Type v} [PseudoEMetricSpace α] [PseudoEMetricSpace β] (f : α → β)
Goal: LocallyLipschitzOn ∅ f
-- LipschitzOnWith.mono
theorem LipschitzOnWith.mono (hf : LipschitzOnWith K f t) (h : s ⊆ t) : LipschitzOnWith K f s 
Arguments: {α : Type u} {β : Type v} [PseudoEMetricSpace α] [PseudoEMetricSpace β] {K : ℝ≥0} {s t : Set α} {f : α → β} (hf : LipschitzOnWith K f t) (h : s ⊆ t)
Goal: LipschitzOnWith K f s
-- LocallyLipschitzOn.mono
lemma LocallyLipschitzOn.mono (hf : LocallyLipschitzOn t f) (h : s ⊆ t) : LocallyLipschitzOn s f 
Arguments: {α : Type u} {β : Type v} [PseudoEMetricSpace α] [PseudoEMetricSpace β] {s t : Set α} {f : α → β} (hf : LocallyLipschitzOn t f) (h : s ⊆ t)
Goal: LocallyLipschitzOn s f
-- lipschitzOnWith_univ
@[simp] lemma lipschitzOnWith_univ : LipschitzOnWith K f univ ↔ LipschitzWith K f 
Arguments: {α : Type u} {β : Type v} [PseudoEMetricSpace α] [PseudoEMetricSpace β] {K : ℝ≥0} {f : α → β}
Goal: LipschitzOnWith K f Set.univ ↔ LipschitzWith K f
-- locallyLipschitzOn_univ
@[simp] lemma locallyLipschitzOn_univ : LocallyLipschitzOn univ f ↔ LocallyLipschitz f 
Arguments: {α : Type u} {β : Type v} [PseudoEMetricSpace α] [PseudoEMetricSpace β] {f : α → β}
Goal: LocallyLipschitzOn Set.univ f ↔ LocallyLipschitz f
-- lipschitzOnWith_iff_restrict
theorem lipschitzOnWith_iff_restrict : LipschitzOnWith K f s ↔ LipschitzWith K (s.restrict f) 
Arguments: {α : Type u} {β : Type v} [PseudoEMetricSpace α] [PseudoEMetricSpace β] {K : ℝ≥0} {s : Set α} {f : α → β}
Goal: LipschitzOnWith K f s ↔ LipschitzWith K (s.restrict f)
-- lipschitzOnWith_restrict
lemma lipschitzOnWith_restrict {t : Set s} :
    LipschitzOnWith K (s.restrict f) t ↔ LipschitzOnWith K f (s ∩ Subtype.val '' t) 
Arguments: {α : Type u} {β : Type v} [PseudoEMetricSpace α] [PseudoEMetricSpace β] {K : ℝ≥0} {s : Set α} {f : α → β} {t : Set ↑s}
Goal: LipschitzOnWith K (s.restrict f) t ↔ LipschitzOnWith K f (s ∩ Subtype.val '' t)
-- locallyLipschitzOn_iff_restrict
lemma locallyLipschitzOn_iff_restrict :
    LocallyLipschitzOn s f ↔ LocallyLipschitz (s.restrict f) 
Arguments: {α : Type u} {β : Type v} [PseudoEMetricSpace α] [PseudoEMetricSpace β] {s : Set α} {f : α → β}
Goal: LocallyLipschitzOn s f ↔ LocallyLipschitz (s.restrict f)
-- Set.MapsTo.lipschitzOnWith_iff_restrict
lemma Set.MapsTo.lipschitzOnWith_iff_restrict {t : Set β} (h : MapsTo f s t) :
    LipschitzOnWith K f s ↔ LipschitzWith K (h.restrict f s t) 
Arguments: {α : Type u} {β : Type v} [PseudoEMetricSpace α] [PseudoEMetricSpace β] {K : ℝ≥0} {s : Set α} {f : α → β} {t : Set β} (h : Set.MapsTo f s t)
Goal: LipschitzOnWith K f s ↔ LipschitzWith K (Set.MapsTo.restrict f s t h)
-- LipschitzWith.lipschitzOnWith
protected theorem lipschitzOnWith (h : LipschitzWith K f) (s : Set α) : LipschitzOnWith K f s 
Arguments: {α : Type u} {β : Type v} [PseudoEMetricSpace α] [PseudoEMetricSpace β] {K : ℝ≥0} {f : α → β} (h : LipschitzWith K f) (s : Set α)
Goal: LipschitzOnWith K f s
-- LipschitzWith.edist_le_mul
theorem edist_le_mul (h : LipschitzWith K f) (x y : α) : edist (f x) (f y) ≤ K * edist x y 
Arguments: {α : Type u} {β : Type v} [PseudoEMetricSpace α] [PseudoEMetricSpace β] {K : ℝ≥0} {f : α → β} (h : LipschitzWith K f) (x y : α)
Goal: edist (f x) (f y) ≤ ↑K * edist x y
-- LipschitzWith.edist_le_mul_of_le
theorem edist_le_mul_of_le (h : LipschitzWith K f) (hr : edist x y ≤ r) :
    edist (f x) (f y) ≤ K * r 
Arguments: {α : Type u} {β : Type v} [PseudoEMetricSpace α] [PseudoEMetricSpace β] {K : ℝ≥0} {f : α → β} {x y : α} {r : ℝ≥0∞} (h : LipschitzWith K f) (hr : edist x y ≤ r)
Goal: edist (f x) (f y) ≤ ↑K * r
-- LipschitzWith.edist_lt_mul_of_lt
theorem edist_lt_mul_of_lt (h : LipschitzWith K f) (hK : K ≠ 0) (hr : edist x y < r) :
    edist (f x) (f y) < K * r 
Arguments: {α : Type u} {β : Type v} [PseudoEMetricSpace α] [PseudoEMetricSpace β] {K : ℝ≥0} {f : α → β} {x y : α} {r : ℝ≥0∞} (h : LipschitzWith K f) (hK : K ≠ 0) (hr : edist x y < r)
Goal: edist (f x) (f y) < ↑K * r
-- LipschitzWith.mapsTo_emetric_closedBall
theorem mapsTo_emetric_closedBall (h : LipschitzWith K f) (x : α) (r : ℝ≥0∞) :
    MapsTo f (closedBall x r) (closedBall (f x) (K * r)) 
Arguments: {α : Type u} {β : Type v} [PseudoEMetricSpace α] [PseudoEMetricSpace β] {K : ℝ≥0} {f : α → β} (h : LipschitzWith K f) (x : α) (r : ℝ≥0∞)
Goal: Set.MapsTo f (EMetric.closedBall x r) (EMetric.closedBall (f x) (↑K * r))
-- LipschitzWith.mapsTo_emetric_ball
theorem mapsTo_emetric_ball (h : LipschitzWith K f) (hK : K ≠ 0) (x : α) (r : ℝ≥0∞) :
    MapsTo f (ball x r) (ball (f x) (K * r)) 
Arguments: {α : Type u} {β : Type v} [PseudoEMetricSpace α] [PseudoEMetricSpace β] {K : ℝ≥0} {f : α → β} (h : LipschitzWith K f) (hK : K ≠ 0) (x : α) (r : ℝ≥0∞)
Goal: Set.MapsTo f (EMetric.ball x r) (EMetric.ball (f x) (↑K * r))
-- LipschitzWith.edist_lt_top
theorem edist_lt_top (hf : LipschitzWith K f) {x y : α} (h : edist x y ≠ ⊤) :
    edist (f x) (f y) < ⊤ 
Arguments: {α : Type u} {β : Type v} [PseudoEMetricSpace α] [PseudoEMetricSpace β] {K : ℝ≥0} {f : α → β} (hf : LipschitzWith K f) {x y : α} (h : edist x y ≠ ⊤)
Goal: edist (f x) (f y) < ⊤
-- LipschitzWith.mul_edist_le
theorem mul_edist_le (h : LipschitzWith K f) (x y : α) :
    (K⁻¹ : ℝ≥0∞) * edist (f x) (f y) ≤ edist x y 
Arguments: {α : Type u} {β : Type v} [PseudoEMetricSpace α] [PseudoEMetricSpace β] {K : ℝ≥0} {f : α → β} (h : LipschitzWith K f) (x y : α)
Goal: (↑K)⁻¹ * edist (f x) (f y) ≤ edist x y
-- LipschitzWith.of_edist_le
protected theorem of_edist_le (h : ∀ x y, edist (f x) (f y) ≤ edist x y) : LipschitzWith 1 f 
Arguments: {α : Type u} {β : Type v} [PseudoEMetricSpace α] [PseudoEMetricSpace β] {f : α → β} (h : ∀ (x y : α), edist (f x) (f y) ≤ edist x y)
Goal: LipschitzWith 1 f
-- LipschitzWith.weaken
protected theorem weaken (hf : LipschitzWith K f) {K' : ℝ≥0} (h : K ≤ K') : LipschitzWith K' f 
Arguments: {α : Type u} {β : Type v} [PseudoEMetricSpace α] [PseudoEMetricSpace β] {K : ℝ≥0} {f : α → β} (hf : LipschitzWith K f) {K' : ℝ≥0} (h : K ≤ K')
Goal: LipschitzWith K' f
-- LipschitzWith.ediam_image_le
theorem ediam_image_le (hf : LipschitzWith K f) (s : Set α) :
    EMetric.diam (f '' s) ≤ K * EMetric.diam s 
Arguments: {α : Type u} {β : Type v} [PseudoEMetricSpace α] [PseudoEMetricSpace β] {K : ℝ≥0} {f : α → β} (hf : LipschitzWith K f) (s : Set α)
Goal: EMetric.diam (f '' s) ≤ ↑K * EMetric.diam s
-- LipschitzWith.edist_lt_of_edist_lt_div
theorem edist_lt_of_edist_lt_div (hf : LipschitzWith K f) {x y : α} {d : ℝ≥0∞}
    (h : edist x y < d / K) : edist (f x) (f y) < d 
Arguments: {α : Type u} {β : Type v} [PseudoEMetricSpace α] [PseudoEMetricSpace β] {K : ℝ≥0} {f : α → β} (hf : LipschitzWith K f) {x y : α} {d : ℝ≥0∞} (h : edist x y < d / ↑K)
Goal: edist (f x) (f y) < d
-- LipschitzWith.uniformContinuous
protected theorem uniformContinuous (hf : LipschitzWith K f) : UniformContinuous f 
Arguments: {α : Type u} {β : Type v} [PseudoEMetricSpace α] [PseudoEMetricSpace β] {K : ℝ≥0} {f : α → β} (hf : LipschitzWith K f)
Goal: UniformContinuous f
-- LipschitzWith.continuous
protected theorem continuous (hf : LipschitzWith K f) : Continuous f 
Arguments: {α : Type u} {β : Type v} [PseudoEMetricSpace α] [PseudoEMetricSpace β] {K : ℝ≥0} {f : α → β} (hf : LipschitzWith K f)
Goal: Continuous f
-- LipschitzWith.const
protected theorem const (b : β) : LipschitzWith 0 fun _ : α => b 
Arguments: {α : Type u} {β : Type v} [PseudoEMetricSpace α] [PseudoEMetricSpace β] (b : β)
Goal: LipschitzWith 0 fun x => b
-- LipschitzWith.const'
protected theorem const' (b : β) {K : ℝ≥0} : LipschitzWith K fun _ : α => b 
Arguments: {α : Type u} {β : Type v} [PseudoEMetricSpace α] [PseudoEMetricSpace β] (b : β) {K : ℝ≥0}
Goal: LipschitzWith K fun x => b
-- LipschitzWith.id
protected theorem id : LipschitzWith 1 (@id α) 
Arguments: {α : Type u} [PseudoEMetricSpace α]
Goal: LipschitzWith 1 id
-- LipschitzWith.subtype_val
protected theorem subtype_val (s : Set α) : LipschitzWith 1 (Subtype.val : s → α) 
Arguments: {α : Type u} [PseudoEMetricSpace α] (s : Set α)
Goal: LipschitzWith 1 Subtype.val
-- LipschitzWith.subtype_mk
theorem subtype_mk (hf : LipschitzWith K f) {p : β → Prop} (hp : ∀ x, p (f x)) :
    LipschitzWith K (fun x => ⟨f x, hp x⟩ : α → { y // p y }) 
Arguments: {α : Type u} {β : Type v} [PseudoEMetricSpace α] [PseudoEMetricSpace β] {K : ℝ≥0} {f : α → β} (hf : LipschitzWith K f) {p : β → Prop} (hp : ∀ (x : α), p (f x))
Goal: LipschitzWith K fun x => ⟨f x, ⋯⟩
-- LipschitzWith.eval
protected theorem eval {α : ι → Type u} [∀ i, PseudoEMetricSpace (α i)] [Fintype ι] (i : ι) :
    LipschitzWith 1 (Function.eval i : (∀ i, α i) → α i) 
Arguments: {ι : Type x} {α : ι → Type u} [(i : ι) → PseudoEMetricSpace (α i)] [Fintype ι] (i : ι)
Goal: LipschitzWith 1 (Function.eval i)
-- LipschitzWith.restrict
protected theorem restrict (hf : LipschitzWith K f) (s : Set α) : LipschitzWith K (s.restrict f) 
Arguments: {α : Type u} {β : Type v} [PseudoEMetricSpace α] [PseudoEMetricSpace β] {K : ℝ≥0} {f : α → β} (hf : LipschitzWith K f) (s : Set α)
Goal: LipschitzWith K (s.restrict f)
-- LipschitzWith.comp
protected theorem comp {Kf Kg : ℝ≥0} {f : β → γ} {g : α → β} (hf : LipschitzWith Kf f)
    (hg : LipschitzWith Kg g) : LipschitzWith (Kf * Kg) (f ∘ g) 
Arguments: {α : Type u} {β : Type v} {γ : Type w} [PseudoEMetricSpace α] [PseudoEMetricSpace β] [PseudoEMetricSpace γ] {Kf Kg : ℝ≥0} {f : β → γ} {g : α → β} (hf : LipschitzWith Kf f) (hg : LipschitzWith Kg g)
Goal: LipschitzWith (Kf * Kg) (f ∘ g)
-- LipschitzWith.comp_lipschitzOnWith
theorem comp_lipschitzOnWith {Kf Kg : ℝ≥0} {f : β → γ} {g : α → β} {s : Set α}
    (hf : LipschitzWith Kf f) (hg : LipschitzOnWith Kg g s) : LipschitzOnWith (Kf * Kg) (f ∘ g) s 
Arguments: {α : Type u} {β : Type v} {γ : Type w} [PseudoEMetricSpace α] [PseudoEMetricSpace β] [PseudoEMetricSpace γ] {Kf Kg : ℝ≥0} {f : β → γ} {g : α → β} {s : Set α} (hf : LipschitzWith Kf f) (hg : LipschitzOnWith Kg g s)
Goal: LipschitzOnWith (Kf * Kg) (f ∘ g) s
-- LipschitzWith.prod_fst
protected theorem prod_fst : LipschitzWith 1 (@Prod.fst α β) 
Arguments: {α : Type u} {β : Type v} [PseudoEMetricSpace α] [PseudoEMetricSpace β]
Goal: LipschitzWith 1 Prod.fst
-- LipschitzWith.prod_snd
protected theorem prod_snd : LipschitzWith 1 (@Prod.snd α β) 
Arguments: {α : Type u} {β : Type v} [PseudoEMetricSpace α] [PseudoEMetricSpace β]
Goal: LipschitzWith 1 Prod.snd
-- LipschitzWith.prod
protected theorem prod {f : α → β} {Kf : ℝ≥0} (hf : LipschitzWith Kf f) {g : α → γ} {Kg : ℝ≥0}
    (hg : LipschitzWith Kg g) : LipschitzWith (max Kf Kg) fun x => (f x, g x) 
Arguments: {α : Type u} {β : Type v} {γ : Type w} [PseudoEMetricSpace α] [PseudoEMetricSpace β] [PseudoEMetricSpace γ] {f : α → β} {Kf : ℝ≥0} (hf : LipschitzWith Kf f) {g : α → γ} {Kg : ℝ≥0} (hg : LipschitzWith Kg g)
Goal: LipschitzWith (max Kf Kg) fun x => (f x, g x)
-- LipschitzWith.prod_mk_left
protected theorem prod_mk_left (a : α) : LipschitzWith 1 (Prod.mk a : β → α × β) 
Arguments: {α : Type u} {β : Type v} [PseudoEMetricSpace α] [PseudoEMetricSpace β] (a : α)
Goal: LipschitzWith 1 (Prod.mk a)
-- LipschitzWith.prod_mk_right
protected theorem prod_mk_right (b : β) : LipschitzWith 1 fun a : α => (a, b) 
Arguments: {α : Type u} {β : Type v} [PseudoEMetricSpace α] [PseudoEMetricSpace β] (b : β)
Goal: LipschitzWith 1 fun a => (a, b)
-- LipschitzWith.uncurry
protected theorem uncurry {f : α → β → γ} {Kα Kβ : ℝ≥0} (hα : ∀ b, LipschitzWith Kα fun a => f a b)
    (hβ : ∀ a, LipschitzWith Kβ (f a)) : LipschitzWith (Kα + Kβ) (Function.uncurry f) 
Arguments: {α : Type u} {β : Type v} {γ : Type w} [PseudoEMetricSpace α] [PseudoEMetricSpace β] [PseudoEMetricSpace γ] {f : α → β → γ} {Kα Kβ : ℝ≥0} (hα : ∀ (b : β), LipschitzWith Kα fun a => f a b) (hβ : ∀ (a : α), LipschitzWith Kβ (f a))
Goal: LipschitzWith (Kα + Kβ) (Function.uncurry f)
-- LipschitzWith.iterate
protected theorem iterate {f : α → α} (hf : LipschitzWith K f) : ∀ n, LipschitzWith (K ^ n) f^[n]
  
Arguments: {α : Type u} [PseudoEMetricSpace α] {K : ℝ≥0} {f : α → α} (hf : LipschitzWith K f) (n : ℕ)
Goal: LipschitzWith (K ^ n) f^[n]
-- LipschitzWith.edist_iterate_succ_le_geometric
theorem edist_iterate_succ_le_geometric {f : α → α} (hf : LipschitzWith K f) (x n) :
    edist (f^[n] x) (f^[n + 1] x) ≤ edist x (f x) * (K : ℝ≥0∞) ^ n 
Arguments: {α : Type u} [PseudoEMetricSpace α] {K : ℝ≥0} {f : α → α} (hf : LipschitzWith K f) (x : α) (n : ℕ)
Goal: edist (f^[n] x) (f^[n + 1] x) ≤ edist x (f x) * ↑K ^ n
-- LipschitzWith.mul_end
protected theorem mul_end {f g : Function.End α} {Kf Kg} (hf : LipschitzWith Kf f)
    (hg : LipschitzWith Kg g) : LipschitzWith (Kf * Kg) (f * g : Function.End α) 
Arguments: {α : Type u} [PseudoEMetricSpace α] {f g : Function.End α} {Kf Kg : ℝ≥0} (hf : LipschitzWith Kf f) (hg : LipschitzWith Kg g)
Goal: LipschitzWith (Kf * Kg) (f * g)
-- LipschitzWith.list_prod
protected theorem list_prod (f : ι → Function.End α) (K : ι → ℝ≥0)
    (h : ∀ i, LipschitzWith (K i) (f i)) : ∀ l : List ι, LipschitzWith (l.map K).prod (l.map f).prod
  
Arguments: {α : Type u} {ι : Type x} [PseudoEMetricSpace α] (f : ι → Function.End α) (K : ι → ℝ≥0) (h : ∀ (i : ι), LipschitzWith (K i) (f i)) (l : List ι)
Goal: LipschitzWith (List.map K l).prod (List.map f l).prod
-- LipschitzWith.pow_end
protected theorem pow_end {f : Function.End α} {K} (h : LipschitzWith K f) :
    ∀ n : ℕ, LipschitzWith (K ^ n) (f ^ n : Function.End α)
  
Arguments: {α : Type u} [PseudoEMetricSpace α] {f : Function.End α} {K : ℝ≥0} (h : LipschitzWith K f) (n : ℕ)
Goal: LipschitzWith (K ^ n) (f ^ n)
-- LipschitzOnWith.uniformContinuousOn
protected theorem uniformContinuousOn (hf : LipschitzOnWith K f s) : UniformContinuousOn f s 
Arguments: {α : Type u} {β : Type v} [PseudoEMetricSpace α] [PseudoEMetricSpace β] {K : ℝ≥0} {s : Set α} {f : α → β} (hf : LipschitzOnWith K f s)
Goal: UniformContinuousOn f s
-- LipschitzOnWith.continuousOn
protected theorem continuousOn (hf : LipschitzOnWith K f s) : ContinuousOn f s 
Arguments: {α : Type u} {β : Type v} [PseudoEMetricSpace α] [PseudoEMetricSpace β] {K : ℝ≥0} {s : Set α} {f : α → β} (hf : LipschitzOnWith K f s)
Goal: ContinuousOn f s
-- LipschitzOnWith.edist_le_mul_of_le
theorem edist_le_mul_of_le (h : LipschitzOnWith K f s) {x y : α} (hx : x ∈ s) (hy : y ∈ s)
    {r : ℝ≥0∞} (hr : edist x y ≤ r) :
    edist (f x) (f y) ≤ K * r 
Arguments: {α : Type u} {β : Type v} [PseudoEMetricSpace α] [PseudoEMetricSpace β] {K : ℝ≥0} {s : Set α} {f : α → β} (h : LipschitzOnWith K f s) {x y : α} (hx : x ∈ s) (hy : y ∈ s) {r : ℝ≥0∞} (hr : edist x y ≤ r)
Goal: edist (f x) (f y) ≤ ↑K * r
-- LipschitzOnWith.edist_lt_of_edist_lt_div
theorem edist_lt_of_edist_lt_div (hf : LipschitzOnWith K f s) {x y : α} (hx : x ∈ s) (hy : y ∈ s)
    {d : ℝ≥0∞} (hd : edist x y < d / K) : edist (f x) (f y) < d 
Arguments: {α : Type u} {β : Type v} [PseudoEMetricSpace α] [PseudoEMetricSpace β] {K : ℝ≥0} {s : Set α} {f : α → β} (hf : LipschitzOnWith K f s) {x y : α} (hx : x ∈ s) (hy : y ∈ s) {d : ℝ≥0∞} (hd : edist x y < d / ↑K)
Goal: edist (f x) (f y) < d
-- LipschitzOnWith.comp
protected theorem comp {g : β → γ} {t : Set β} {Kg : ℝ≥0} (hg : LipschitzOnWith Kg g t)
    (hf : LipschitzOnWith K f s) (hmaps : MapsTo f s t) : LipschitzOnWith (Kg * K) (g ∘ f) s 
Arguments: {α : Type u} {β : Type v} {γ : Type w} [PseudoEMetricSpace α] [PseudoEMetricSpace β] [PseudoEMetricSpace γ] {K : ℝ≥0} {s : Set α} {f : α → β} {g : β → γ} {t : Set β} {Kg : ℝ≥0} (hg : LipschitzOnWith Kg g t) (hf : LipschitzOnWith K f s) (hmaps : Set.MapsTo f s t)
Goal: LipschitzOnWith (Kg * K) (g ∘ f) s
-- LipschitzOnWith.prod
protected theorem prod {g : α → γ} {Kf Kg : ℝ≥0} (hf : LipschitzOnWith Kf f s)
    (hg : LipschitzOnWith Kg g s) : LipschitzOnWith (max Kf Kg) (fun x => (f x, g x)) s 
Arguments: {α : Type u} {β : Type v} {γ : Type w} [PseudoEMetricSpace α] [PseudoEMetricSpace β] [PseudoEMetricSpace γ] {s : Set α} {f : α → β} {g : α → γ} {Kf Kg : ℝ≥0} (hf : LipschitzOnWith Kf f s) (hg : LipschitzOnWith Kg g s)
Goal: LipschitzOnWith (max Kf Kg) (fun x => (f x, g x)) s
-- LipschitzOnWith.ediam_image2_le
theorem ediam_image2_le (f : α → β → γ) {K₁ K₂ : ℝ≥0} (s : Set α) (t : Set β)
    (hf₁ : ∀ b ∈ t, LipschitzOnWith K₁ (f · b) s) (hf₂ : ∀ a ∈ s, LipschitzOnWith K₂ (f a) t) :
    EMetric.diam (Set.image2 f s t) ≤ ↑K₁ * EMetric.diam s + ↑K₂ * EMetric.diam t 
Arguments: {α : Type u} {β : Type v} {γ : Type w} [PseudoEMetricSpace α] [PseudoEMetricSpace β] [PseudoEMetricSpace γ] (f : α → β → γ) {K₁ K₂ : ℝ≥0} (s : Set α) (t : Set β) (hf₁ : ∀ b ∈ t, LipschitzOnWith K₁ (fun x => f x b) s) (hf₂ : ∀ a ∈ s, LipschitzOnWith K₂ (f a) t)
Goal: EMetric.diam (Set.image2 f s t) ≤ ↑K₁ * EMetric.diam s + ↑K₂ * EMetric.diam t
-- LipschitzWith.locallyLipschitz
protected lemma _root_.LipschitzWith.locallyLipschitz {K : ℝ≥0} (hf : LipschitzWith K f) :
    LocallyLipschitz f 
Arguments: {α : Type u} {β : Type v} [PseudoEMetricSpace α] [PseudoEMetricSpace β] {f : α → β} {K : ℝ≥0} (hf : LipschitzWith K f)
Goal: LocallyLipschitz f
-- LocallyLipschitz.id
protected lemma id : LocallyLipschitz (@id α) 
Arguments: {α : Type u} [PseudoEMetricSpace α]
Goal: LocallyLipschitz id
-- LocallyLipschitz.const
protected lemma const (b : β) : LocallyLipschitz (fun _ : α ↦ b) 
Arguments: {α : Type u} {β : Type v} [PseudoEMetricSpace α] [PseudoEMetricSpace β] (b : β)
Goal: LocallyLipschitz fun x => b
-- LocallyLipschitz.continuous
protected theorem continuous {f : α → β} (hf : LocallyLipschitz f) : Continuous f 
Arguments: {α : Type u} {β : Type v} [PseudoEMetricSpace α] [PseudoEMetricSpace β] {f : α → β} (hf : LocallyLipschitz f)
Goal: Continuous f
-- LocallyLipschitz.comp
protected lemma comp  {f : β → γ} {g : α → β}
    (hf : LocallyLipschitz f) (hg : LocallyLipschitz g) : LocallyLipschitz (f ∘ g) 
Arguments: {α : Type u} {β : Type v} {γ : Type w} [PseudoEMetricSpace α] [PseudoEMetricSpace β] [PseudoEMetricSpace γ] {f : β → γ} {g : α → β} (hf : LocallyLipschitz f) (hg : LocallyLipschitz g)
Goal: LocallyLipschitz (f ∘ g)
-- LocallyLipschitz.prod
protected lemma prod {f : α → β} (hf : LocallyLipschitz f) {g : α → γ} (hg : LocallyLipschitz g) :
    LocallyLipschitz fun x => (f x, g x) 
Arguments: {α : Type u} {β : Type v} {γ : Type w} [PseudoEMetricSpace α] [PseudoEMetricSpace β] [PseudoEMetricSpace γ] {f : α → β} (hf : LocallyLipschitz f) {g : α → γ} (hg : LocallyLipschitz g)
Goal: LocallyLipschitz fun x => (f x, g x)
-- LocallyLipschitz.prod_mk_left
protected theorem prod_mk_left (a : α) : LocallyLipschitz (Prod.mk a : β → α × β) 
Arguments: {α : Type u} {β : Type v} [PseudoEMetricSpace α] [PseudoEMetricSpace β] (a : α)
Goal: LocallyLipschitz (Prod.mk a)
-- LocallyLipschitz.prod_mk_right
protected theorem prod_mk_right (b : β) : LocallyLipschitz (fun a : α => (a, b)) 
Arguments: {α : Type u} {β : Type v} [PseudoEMetricSpace α] [PseudoEMetricSpace β] (b : β)
Goal: LocallyLipschitz fun a => (a, b)
-- LocallyLipschitz.iterate
protected theorem iterate {f : α → α} (hf : LocallyLipschitz f) : ∀ n, LocallyLipschitz f^[n]
  
Arguments: {α : Type u} [PseudoEMetricSpace α] {f : α → α} (hf : LocallyLipschitz f) (n : ℕ)
Goal: LocallyLipschitz f^[n]
-- LocallyLipschitz.mul_end
protected theorem mul_end {f g : Function.End α} (hf : LocallyLipschitz f)
    (hg : LocallyLipschitz g) : LocallyLipschitz (f * g : Function.End α) 
Arguments: {α : Type u} [PseudoEMetricSpace α] {f g : Function.End α} (hf : LocallyLipschitz f) (hg : LocallyLipschitz g)
Goal: LocallyLipschitz (f * g)
-- LocallyLipschitz.pow_end
protected theorem pow_end {f : Function.End α} (h : LocallyLipschitz f) :
    ∀ n : ℕ, LocallyLipschitz (f ^ n : Function.End α)
  
Arguments: {α : Type u} [PseudoEMetricSpace α] {f : Function.End α} (h : LocallyLipschitz f) (n : ℕ)
Goal: LocallyLipschitz (f ^ n)
-- LocallyLipschitzOn.continuousOn
protected lemma continuousOn (hf : LocallyLipschitzOn s f) : ContinuousOn f s 
Arguments: {α : Type u} {β : Type v} [PseudoEMetricSpace α] [PseudoEMetricSpace β] {f : α → β} {s : Set α} (hf : LocallyLipschitzOn s f)
Goal: ContinuousOn f s
-- continuousOn_prod_of_subset_closure_continuousOn_lipschitzOnWith
theorem continuousOn_prod_of_subset_closure_continuousOn_lipschitzOnWith [PseudoEMetricSpace α]
    [TopologicalSpace β] [PseudoEMetricSpace γ] (f : α × β → γ) {s s' : Set α} {t : Set β}
    (hs' : s' ⊆ s) (hss' : s ⊆ closure s') (K : ℝ≥0)
    (ha : ∀ a ∈ s', ContinuousOn (fun y => f (a, y)) t)
    (hb : ∀ b ∈ t, LipschitzOnWith K (fun x => f (x, b)) s) : ContinuousOn f (s ×ˢ t) 
Arguments: {α : Type u} {β : Type v} {γ : Type w} [PseudoEMetricSpace α] [TopologicalSpace β] [PseudoEMetricSpace γ] (f : α × β → γ) {s s' : Set α} {t : Set β} (hs' : s' ⊆ s) (hss' : s ⊆ closure s') (K : ℝ≥0) (ha : ∀ a ∈ s', ContinuousOn (fun y => f (a, y)) t) (hb : ∀ b ∈ t, LipschitzOnWith K (fun x => f (x, b)) s)
Goal: ContinuousOn f (s ×ˢ t)
-- continuousOn_prod_of_continuousOn_lipschitzOnWith
theorem continuousOn_prod_of_continuousOn_lipschitzOnWith [PseudoEMetricSpace α]
    [TopologicalSpace β] [PseudoEMetricSpace γ] (f : α × β → γ) {s : Set α} {t : Set β} (K : ℝ≥0)
    (ha : ∀ a ∈ s, ContinuousOn (fun y => f (a, y)) t)
    (hb : ∀ b ∈ t, LipschitzOnWith K (fun x => f (x, b)) s) : ContinuousOn f (s ×ˢ t) 
Arguments: {α : Type u} {β : Type v} {γ : Type w} [PseudoEMetricSpace α] [TopologicalSpace β] [PseudoEMetricSpace γ] (f : α × β → γ) {s : Set α} {t : Set β} (K : ℝ≥0) (ha : ∀ a ∈ s, ContinuousOn (fun y => f (a, y)) t) (hb : ∀ b ∈ t, LipschitzOnWith K (fun x => f (x, b)) s)
Goal: ContinuousOn f (s ×ˢ t)
-- continuous_prod_of_dense_continuous_lipschitzWith
theorem continuous_prod_of_dense_continuous_lipschitzWith [PseudoEMetricSpace α]
    [TopologicalSpace β] [PseudoEMetricSpace γ] (f : α × β → γ) (K : ℝ≥0) {s : Set α}
    (hs : Dense s) (ha : ∀ a ∈ s, Continuous fun y => f (a, y))
    (hb : ∀ b, LipschitzWith K fun x => f (x, b)) : Continuous f 
Arguments: {α : Type u} {β : Type v} {γ : Type w} [PseudoEMetricSpace α] [TopologicalSpace β] [PseudoEMetricSpace γ] (f : α × β → γ) (K : ℝ≥0) {s : Set α} (hs : Dense s) (ha : ∀ a ∈ s, Continuous fun y => f (a, y)) (hb : ∀ (b : β), LipschitzWith K fun x => f (x, b))
Goal: Continuous f
-- continuous_prod_of_continuous_lipschitzWith
theorem continuous_prod_of_continuous_lipschitzWith [PseudoEMetricSpace α] [TopologicalSpace β]
    [PseudoEMetricSpace γ] (f : α × β → γ) (K : ℝ≥0) (ha : ∀ a, Continuous fun y => f (a, y))
    (hb : ∀ b, LipschitzWith K fun x => f (x, b)) : Continuous f 
Arguments: {α : Type u} {β : Type v} {γ : Type w} [PseudoEMetricSpace α] [TopologicalSpace β] [PseudoEMetricSpace γ] (f : α × β → γ) (K : ℝ≥0) (ha : ∀ (a : α), Continuous fun y => f (a, y)) (hb : ∀ (b : β), LipschitzWith K fun x => f (x, b))
Goal: Continuous f