Loogle!

Result

Found 129 declarations mentioning AntilipschitzWith.

• AntilipschitzWith 📋 Mathlib.Topology.MetricSpace.Antilipschitz
  {α : Type u_1} {β : Type u_2} [PseudoEMetricSpace α] [PseudoEMetricSpace β] (K : NNReal) (f : α → β) : Prop
• AntilipschitzWith.id 📋 Mathlib.Topology.MetricSpace.Antilipschitz
  {α : Type u_1} [PseudoEMetricSpace α] : AntilipschitzWith 1 id
• AntilipschitzWith.k 📋 Mathlib.Topology.MetricSpace.Antilipschitz
  {α : Type u_1} {β : Type u_2} [PseudoEMetricSpace α] [PseudoEMetricSpace β] {K : NNReal} {f : α → β} (_hf : AntilipschitzWith K f) : NNReal
• AntilipschitzWith.of_subsingleton 📋 Mathlib.Topology.MetricSpace.Antilipschitz
  {α : Type u_1} {β : Type u_2} [PseudoEMetricSpace α] [PseudoEMetricSpace β] {f : α → β} [Subsingleton α] {K : NNReal} : AntilipschitzWith K f
• AntilipschitzWith.injective 📋 Mathlib.Topology.MetricSpace.Antilipschitz
  {α : Type u_4} {β : Type u_5} [EMetricSpace α] [PseudoEMetricSpace β] {K : NNReal} {f : α → β} (hf : AntilipschitzWith K f) : Function.Injective f
• AntilipschitzWith.subsingleton 📋 Mathlib.Topology.MetricSpace.Antilipschitz
  {α : Type u_4} {β : Type u_5} [EMetricSpace α] [PseudoEMetricSpace β] {f : α → β} (h : AntilipschitzWith 0 f) : Subsingleton α
• AntilipschitzWith.to_rightInverse 📋 Mathlib.Topology.MetricSpace.Antilipschitz
  {α : Type u_1} {β : Type u_2} [PseudoEMetricSpace α] [PseudoEMetricSpace β] {K : NNReal} {f : α → β} (hf : AntilipschitzWith K f) {g : β → α} (hg : Function.RightInverse g f) : LipschitzWith K g
• LipschitzWith.to_rightInverse 📋 Mathlib.Topology.MetricSpace.Antilipschitz
  {α : Type u_1} {β : Type u_2} [PseudoEMetricSpace α] [PseudoEMetricSpace β] {K : NNReal} {f : α → β} (hf : LipschitzWith K f) {g : β → α} (hg : Function.RightInverse g f) : AntilipschitzWith K g
• AntilipschitzWith.pos 📋 Mathlib.Topology.MetricSpace.Antilipschitz
  {β : Type u_2} [PseudoEMetricSpace β] {K : NNReal} {α : Type u_4} [EMetricSpace α] [Nontrivial α] {f : α → β} (hf : AntilipschitzWith K f) : 0 < K
• AntilipschitzWith.tendsto_cobounded 📋 Mathlib.Topology.MetricSpace.Antilipschitz
  {α : Type u_1} {β : Type u_2} [PseudoMetricSpace α] [PseudoMetricSpace β] {K : NNReal} {f : α → β} (hf : AntilipschitzWith K f) : Filter.Tendsto f (Bornology.cobounded α) (Bornology.cobounded β)
• AntilipschitzWith.isUniformInducing 📋 Mathlib.Topology.MetricSpace.Antilipschitz
  {α : Type u_1} {β : Type u_2} [PseudoEMetricSpace α] [PseudoEMetricSpace β] {K : NNReal} {f : α → β} (hf : AntilipschitzWith K f) (hfc : UniformContinuous f) : IsUniformInducing f
• AntilipschitzWith.isBounded_preimage 📋 Mathlib.Topology.MetricSpace.Antilipschitz
  {α : Type u_1} {β : Type u_2} [PseudoMetricSpace α] [PseudoMetricSpace β] {K : NNReal} {f : α → β} (hf : AntilipschitzWith K f) {s : Set β} (hs : Bornology.IsBounded s) : Bornology.IsBounded (f ⁻¹' s)
• AntilipschitzWith.edist_ne_top 📋 Mathlib.Topology.MetricSpace.Antilipschitz
  {α : Type u_1} {β : Type u_2} [PseudoEMetricSpace α] [PseudoMetricSpace β] {K : NNReal} {f : α → β} (h : AntilipschitzWith K f) (x y : α) : edist x y ≠ ⊤
• AntilipschitzWith.isComplete_range 📋 Mathlib.Topology.MetricSpace.Antilipschitz
  {α : Type u_1} {β : Type u_2} [PseudoEMetricSpace α] [PseudoEMetricSpace β] {K : NNReal} {f : α → β} [CompleteSpace α] (hf : AntilipschitzWith K f) (hfc : UniformContinuous f) : IsComplete (Set.range f)
• AntilipschitzWith.isUniformEmbedding 📋 Mathlib.Topology.MetricSpace.Antilipschitz
  {α : Type u_4} {β : Type u_5} [EMetricSpace α] [PseudoEMetricSpace β] {K : NNReal} {f : α → β} (hf : AntilipschitzWith K f) (hfc : UniformContinuous f) : IsUniformEmbedding f
• AntilipschitzWith.domRestrict 📋 Mathlib.Topology.MetricSpace.Antilipschitz
  {α : Type u_1} {β : Type u_2} [PseudoEMetricSpace α] [PseudoEMetricSpace β] {K : NNReal} {f : α → β} (hf : AntilipschitzWith K f) (s : Set α) : AntilipschitzWith K (s.domRestrict f)
• AntilipschitzWith.restrict 📋 Mathlib.Topology.MetricSpace.Antilipschitz
  {α : Type u_1} {β : Type u_2} [PseudoEMetricSpace α] [PseudoEMetricSpace β] {K : NNReal} {f : α → β} (hf : AntilipschitzWith K f) (s : Set α) : AntilipschitzWith K (s.domRestrict f)
• AntilipschitzWith.edist_lt_top 📋 Mathlib.Topology.MetricSpace.Antilipschitz
  {α : Type u_1} {β : Type u_2} [PseudoEMetricSpace α] [PseudoMetricSpace β] {K : NNReal} {f : α → β} (h : AntilipschitzWith K f) (x y : α) : edist x y < ⊤
• AntilipschitzWith.isInducing 📋 Mathlib.Topology.MetricSpace.Antilipschitz
  {α : Type u_1} {β : Type u_2} [PseudoEMetricSpace α] [PseudoEMetricSpace β] {K : NNReal} {f : α → β} (hf : AntilipschitzWith K f) (hfc : Continuous f) : Topology.IsInducing f
• AntilipschitzWith.isClosed_range 📋 Mathlib.Topology.MetricSpace.Antilipschitz
  {α : Type u_4} {β : Type u_5} [PseudoEMetricSpace α] [EMetricSpace β] [CompleteSpace α] {f : α → β} {K : NNReal} (hf : AntilipschitzWith K f) (hfc : UniformContinuous f) : IsClosed (Set.range f)
• AntilipschitzWith.subtype_coe 📋 Mathlib.Topology.MetricSpace.Antilipschitz
  {α : Type u_1} [PseudoEMetricSpace α] (s : Set α) : AntilipschitzWith 1 Subtype.val
• AntilipschitzWith.comap_nhds_le 📋 Mathlib.Topology.MetricSpace.Antilipschitz
  {α : Type u_1} {β : Type u_2} [PseudoEMetricSpace α] [PseudoEMetricSpace β] {K : NNReal} {f : α → β} (hf : AntilipschitzWith K f) (x : α) : Filter.comap f (nhds (f x)) ≤ nhds x
• AntilipschitzWith.isEmbedding 📋 Mathlib.Topology.MetricSpace.Antilipschitz
  {α : Type u_4} {β : Type u_5} [EMetricSpace α] [PseudoEMetricSpace β] {K : NNReal} {f : α → β} (hf : AntilipschitzWith K f) (hfc : Continuous f) : Topology.IsEmbedding f
• AntilipschitzWith.le_mul_dist 📋 Mathlib.Topology.MetricSpace.Antilipschitz
  {α : Type u_1} {β : Type u_2} [PseudoMetricSpace α] [PseudoMetricSpace β] {K : NNReal} {f : α → β} : AntilipschitzWith K f → ∀ (x y : α), dist x y ≤ ↑K * dist (f x) (f y)
• AntilipschitzWith.of_le_mul_dist 📋 Mathlib.Topology.MetricSpace.Antilipschitz
  {α : Type u_1} {β : Type u_2} [PseudoMetricSpace α] [PseudoMetricSpace β] {K : NNReal} {f : α → β} : (∀ (x y : α), dist x y ≤ ↑K * dist (f x) (f y)) → AntilipschitzWith K f
• AntilipschitzWith.properSpace 📋 Mathlib.Topology.MetricSpace.Antilipschitz
  {β : Type u_2} [PseudoMetricSpace β] {α : Type u_4} [MetricSpace α] {K : NNReal} {f : α → β} [ProperSpace α] (hK : AntilipschitzWith K f) (f_cont : Continuous f) (hf : Function.Surjective f) : ProperSpace β
• antilipschitzWith_iff_le_mul_dist 📋 Mathlib.Topology.MetricSpace.Antilipschitz
  {α : Type u_1} {β : Type u_2} [PseudoMetricSpace α] [PseudoMetricSpace β] {K : NNReal} {f : α → β} : AntilipschitzWith K f ↔ ∀ (x y : α), dist x y ≤ ↑K * dist (f x) (f y)
• AntilipschitzWith.to_rightInvOn 📋 Mathlib.Topology.MetricSpace.Antilipschitz
  {α : Type u_1} {β : Type u_2} [PseudoEMetricSpace α] [PseudoEMetricSpace β] {K : NNReal} {f : α → β} (hf : AntilipschitzWith K f) {g : β → α} {t : Set β} (h : Set.RightInvOn g f t) : LipschitzWith K (t.domRestrict g)
• AntilipschitzWith.codRestrict 📋 Mathlib.Topology.MetricSpace.Antilipschitz
  {α : Type u_1} {β : Type u_2} [PseudoEMetricSpace α] [PseudoEMetricSpace β] {K : NNReal} {f : α → β} (hf : AntilipschitzWith K f) {s : Set β} (hs : ∀ (x : α), f x ∈ s) : AntilipschitzWith K (Set.codRestrict f s hs)
• AntilipschitzWith.comp 📋 Mathlib.Topology.MetricSpace.Antilipschitz
  {α : Type u_1} {β : Type u_2} {γ : Type u_3} [PseudoEMetricSpace α] [PseudoEMetricSpace β] [PseudoEMetricSpace γ] {Kg : NNReal} {g : β → γ} (hg : AntilipschitzWith Kg g) {Kf : NNReal} {f : α → β} (hf : AntilipschitzWith Kf f) : AntilipschitzWith (Kf * Kg) (g ∘ f)
• AntilipschitzWith.mul_le_dist 📋 Mathlib.Topology.MetricSpace.Antilipschitz
  {α : Type u_1} {β : Type u_2} [PseudoMetricSpace α] [PseudoMetricSpace β] {K : NNReal} {f : α → β} (hf : AntilipschitzWith K f) (x y : α) : ↑K⁻¹ * dist x y ≤ dist (f x) (f y)
• AntilipschitzWith.isBounded_of_image2_right 📋 Mathlib.Topology.MetricSpace.Antilipschitz
  {α : Type u_1} {β : Type u_2} {γ : Type u_3} [PseudoMetricSpace α] [PseudoMetricSpace β] [PseudoMetricSpace γ] {f : α → β → γ} {K₂ : NNReal} (hf : ∀ (a : α), AntilipschitzWith K₂ (f a)) {s : Set α} {t : Set β} (hst : Bornology.IsBounded (Set.image2 f s t)) : Bornology.IsBounded s ∨ Bornology.IsBounded t
• AntilipschitzWith.isBounded_of_image2_left 📋 Mathlib.Topology.MetricSpace.Antilipschitz
  {α : Type u_1} {β : Type u_2} {γ : Type u_3} [PseudoMetricSpace α] [PseudoMetricSpace β] [PseudoMetricSpace γ] (f : α → β → γ) {K₁ : NNReal} (hf : ∀ (b : β), AntilipschitzWith K₁ fun a => f a b) {s : Set α} {t : Set β} (hst : Bornology.IsBounded (Set.image2 f s t)) : Bornology.IsBounded s ∨ Bornology.IsBounded t
• AntilipschitzWith.le_mul_nndist 📋 Mathlib.Topology.MetricSpace.Antilipschitz
  {α : Type u_1} {β : Type u_2} [PseudoMetricSpace α] [PseudoMetricSpace β] {K : NNReal} {f : α → β} : AntilipschitzWith K f → ∀ (x y : α), nndist x y ≤ K * nndist (f x) (f y)
• AntilipschitzWith.of_le_mul_nndist 📋 Mathlib.Topology.MetricSpace.Antilipschitz
  {α : Type u_1} {β : Type u_2} [PseudoMetricSpace α] [PseudoMetricSpace β] {K : NNReal} {f : α → β} : (∀ (x y : α), nndist x y ≤ K * nndist (f x) (f y)) → AntilipschitzWith K f
• antilipschitzWith_iff_le_mul_nndist 📋 Mathlib.Topology.MetricSpace.Antilipschitz
  {α : Type u_1} {β : Type u_2} [PseudoMetricSpace α] [PseudoMetricSpace β] {K : NNReal} {f : α → β} : AntilipschitzWith K f ↔ ∀ (x y : α), nndist x y ≤ K * nndist (f x) (f y)
• AntilipschitzWith.comap_uniformity_le 📋 Mathlib.Topology.MetricSpace.Antilipschitz
  {α : Type u_1} {β : Type u_2} [PseudoEMetricSpace α] [PseudoEMetricSpace β] {K : NNReal} {f : α → β} (hf : AntilipschitzWith K f) : Filter.comap (Prod.map f f) (uniformity β) ≤ uniformity α
• AntilipschitzWith.isClosedEmbedding 📋 Mathlib.Topology.MetricSpace.Antilipschitz
  {α : Type u_4} {β : Type u_5} [EMetricSpace α] [EMetricSpace β] {K : NNReal} {f : α → β} [CompleteSpace α] (hf : AntilipschitzWith K f) (hfc : UniformContinuous f) : Topology.IsClosedEmbedding f
• AntilipschitzWith.mul_le_nndist 📋 Mathlib.Topology.MetricSpace.Antilipschitz
  {α : Type u_1} {β : Type u_2} [PseudoMetricSpace α] [PseudoMetricSpace β] {K : NNReal} {f : α → β} (hf : AntilipschitzWith K f) (x y : α) : K⁻¹ * nndist x y ≤ nndist (f x) (f y)
• AntilipschitzWith.ediam_preimage_le 📋 Mathlib.Topology.MetricSpace.Antilipschitz
  {α : Type u_1} {β : Type u_2} [PseudoEMetricSpace α] [PseudoEMetricSpace β] {K : NNReal} {f : α → β} (hf : AntilipschitzWith K f) (s : Set β) : Metric.ediam (f ⁻¹' s) ≤ ↑K * Metric.ediam s
• AntilipschitzWith.le_mul_ediam_image 📋 Mathlib.Topology.MetricSpace.Antilipschitz
  {α : Type u_1} {β : Type u_2} [PseudoEMetricSpace α] [PseudoEMetricSpace β] {K : NNReal} {f : α → β} (hf : AntilipschitzWith K f) (s : Set α) : Metric.ediam s ≤ ↑K * Metric.ediam (f '' s)
• AntilipschitzWith.mul_le_edist 📋 Mathlib.Topology.MetricSpace.Antilipschitz
  {α : Type u_1} {β : Type u_2} [PseudoEMetricSpace α] [PseudoEMetricSpace β] {K : NNReal} {f : α → β} (hf : AntilipschitzWith K f) (x y : α) : (↑K)⁻¹ * edist x y ≤ edist (f x) (f y)
• AntilipschitzWith.to_rightInvOn' 📋 Mathlib.Topology.MetricSpace.Antilipschitz
  {α : Type u_1} {β : Type u_2} [PseudoEMetricSpace α] [PseudoEMetricSpace β] {K : NNReal} {f : α → β} {s : Set α} (hf : AntilipschitzWith K (s.domRestrict f)) {g : β → α} {t : Set β} (g_maps : Set.MapsTo g t s) (g_inv : Set.RightInvOn g f t) : LipschitzWith K (t.domRestrict g)
• Isometry.antilipschitz 📋 Mathlib.Topology.MetricSpace.Isometry
  {α : Type u} {β : Type v} [PseudoEMetricSpace α] [PseudoEMetricSpace β] {f : α → β} (h : Isometry f) : AntilipschitzWith 1 f
• Isometry.antilipschitzWith 📋 Mathlib.Topology.MetricSpace.Isometry
  {α : Type u} {β : Type v} [PseudoEMetricSpace α] [PseudoEMetricSpace β] {f : α → β} (h : Isometry f) : AntilipschitzWith 1 f
• IsometryClass.antilipschitz 📋 Mathlib.Topology.MetricSpace.Isometry
  {F : Type u_1} {α : Type u} {β : Type v} [PseudoEMetricSpace α] [PseudoEMetricSpace β] [FunLike F α β] [IsometryClass F α β] (f : F) : AntilipschitzWith 1 ⇑f
• AntilipschitzWith.inv 📋 Mathlib.Analysis.Normed.Group.Uniform
  {α : Type u_4} {E : Type u_5} [SeminormedCommGroup E] [PseudoEMetricSpace α] {K : NNReal} {f : α → E} : AntilipschitzWith K f → AntilipschitzWith K f⁻¹
• AntilipschitzWith.neg 📋 Mathlib.Analysis.Normed.Group.Uniform
  {α : Type u_4} {E : Type u_5} [SeminormedAddCommGroup E] [PseudoEMetricSpace α] {K : NNReal} {f : α → E} : AntilipschitzWith K f → AntilipschitzWith K (-f)
• AntilipschitzWith.of_inv 📋 Mathlib.Analysis.Normed.Group.Uniform
  {α : Type u_4} {E : Type u_5} [SeminormedCommGroup E] [PseudoEMetricSpace α] {K : NNReal} {f : α → E} : AntilipschitzWith K f⁻¹ → AntilipschitzWith K f
• AntilipschitzWith.of_neg 📋 Mathlib.Analysis.Normed.Group.Uniform
  {α : Type u_4} {E : Type u_5} [SeminormedAddCommGroup E] [PseudoEMetricSpace α] {K : NNReal} {f : α → E} : AntilipschitzWith K (-f) → AntilipschitzWith K f
• antilipschitzWith_inv_iff 📋 Mathlib.Analysis.Normed.Group.Uniform
  {α : Type u_4} {E : Type u_5} [SeminormedCommGroup E] [PseudoEMetricSpace α] {K : NNReal} {f : α → E} : AntilipschitzWith K f⁻¹ ↔ AntilipschitzWith K f
• antilipschitzWith_neg_iff 📋 Mathlib.Analysis.Normed.Group.Uniform
  {α : Type u_4} {E : Type u_5} [SeminormedAddCommGroup E] [PseudoEMetricSpace α] {K : NNReal} {f : α → E} : AntilipschitzWith K (-f) ↔ AntilipschitzWith K f
• AntilipschitzWith.le_mul_norm 📋 Mathlib.Analysis.Normed.Group.Uniform
  {E : Type u_2} {F : Type u_3} [SeminormedAddGroup E] [SeminormedAddGroup F] {f : E → F} {K : NNReal} (h : AntilipschitzWith K f) (hf : f 0 = 0) (x : E) : ‖x‖ ≤ ↑K * ‖f x‖
• AntilipschitzWith.le_mul_norm' 📋 Mathlib.Analysis.Normed.Group.Uniform
  {E : Type u_2} {F : Type u_3} [SeminormedGroup E] [SeminormedGroup F] {f : E → F} {K : NNReal} (h : AntilipschitzWith K f) (hf : f 1 = 1) (x : E) : ‖x‖ ≤ ↑K * ‖f x‖
• AntilipschitzWith.le_mul_nnnorm 📋 Mathlib.Analysis.Normed.Group.Uniform
  {E : Type u_2} {F : Type u_3} [SeminormedAddGroup E] [SeminormedAddGroup F] {f : E → F} {K : NNReal} (h : AntilipschitzWith K f) (hf : f 0 = 0) (x : E) : ‖x‖₊ ≤ K * ‖f x‖₊
• AntilipschitzWith.le_mul_nnnorm' 📋 Mathlib.Analysis.Normed.Group.Uniform
  {E : Type u_2} {F : Type u_3} [SeminormedGroup E] [SeminormedGroup F] {f : E → F} {K : NNReal} (h : AntilipschitzWith K f) (hf : f 1 = 1) (x : E) : ‖x‖₊ ≤ K * ‖f x‖₊
• AddMonoidHomClass.antilipschitz_of_bound 📋 Mathlib.Analysis.Normed.Group.Uniform
  {𝓕 : Type u_1} {E : Type u_2} {F : Type u_3} [SeminormedAddGroup E] [SeminormedAddGroup F] [FunLike 𝓕 E F] [AddMonoidHomClass 𝓕 E F] (f : 𝓕) {K : NNReal} (h : ∀ (x : E), ‖x‖ ≤ ↑K * ‖f x‖) : AntilipschitzWith K ⇑f
• MonoidHomClass.antilipschitz_of_bound 📋 Mathlib.Analysis.Normed.Group.Uniform
  {𝓕 : Type u_1} {E : Type u_2} {F : Type u_3} [SeminormedGroup E] [SeminormedGroup F] [FunLike 𝓕 E F] [MonoidHomClass 𝓕 E F] (f : 𝓕) {K : NNReal} (h : ∀ (x : E), ‖x‖ ≤ ↑K * ‖f x‖) : AntilipschitzWith K ⇑f
• OneHomClass.bound_of_antilipschitz 📋 Mathlib.Analysis.Normed.Group.Uniform
  {𝓕 : Type u_1} {E : Type u_2} {F : Type u_3} [SeminormedGroup E] [SeminormedGroup F] [FunLike 𝓕 E F] [OneHomClass 𝓕 E F] (f : 𝓕) {K : NNReal} (h : AntilipschitzWith K ⇑f) (x : E) : ‖x‖ ≤ ↑K * ‖f x‖
• ZeroHomClass.bound_of_antilipschitz 📋 Mathlib.Analysis.Normed.Group.Uniform
  {𝓕 : Type u_1} {E : Type u_2} {F : Type u_3} [SeminormedAddGroup E] [SeminormedAddGroup F] [FunLike 𝓕 E F] [ZeroHomClass 𝓕 E F] (f : 𝓕) {K : NNReal} (h : AntilipschitzWith K ⇑f) (x : E) : ‖x‖ ≤ ↑K * ‖f x‖
• AntilipschitzWith.add_lipschitzWith 📋 Mathlib.Analysis.Normed.Group.Uniform
  {α : Type u_4} {E : Type u_5} [SeminormedAddCommGroup E] [PseudoEMetricSpace α] {Kf Kg : NNReal} {f g : α → E} (hf : AntilipschitzWith Kf f) (hg : LipschitzWith Kg g) (hK : Kg < Kf⁻¹) : AntilipschitzWith (Kf⁻¹ - Kg)⁻¹ fun x => f x + g x
• AntilipschitzWith.add_sub_lipschitzWith 📋 Mathlib.Analysis.Normed.Group.Uniform
  {α : Type u_4} {E : Type u_5} [SeminormedAddCommGroup E] [PseudoEMetricSpace α] {Kf Kg : NNReal} {f g : α → E} (hf : AntilipschitzWith Kf f) (hg : LipschitzWith Kg (g - f)) (hK : Kg < Kf⁻¹) : AntilipschitzWith (Kf⁻¹ - Kg)⁻¹ g
• AntilipschitzWith.mul_div_lipschitzWith 📋 Mathlib.Analysis.Normed.Group.Uniform
  {α : Type u_4} {E : Type u_5} [SeminormedCommGroup E] [PseudoEMetricSpace α] {Kf Kg : NNReal} {f g : α → E} (hf : AntilipschitzWith Kf f) (hg : LipschitzWith Kg (g / f)) (hK : Kg < Kf⁻¹) : AntilipschitzWith (Kf⁻¹ - Kg)⁻¹ g
• AntilipschitzWith.mul_lipschitzWith 📋 Mathlib.Analysis.Normed.Group.Uniform
  {α : Type u_4} {E : Type u_5} [SeminormedCommGroup E] [PseudoEMetricSpace α] {Kf Kg : NNReal} {f g : α → E} (hf : AntilipschitzWith Kf f) (hg : LipschitzWith Kg g) (hK : Kg < Kf⁻¹) : AntilipschitzWith (Kf⁻¹ - Kg)⁻¹ fun x => f x * g x
• antilipschitzWith_iff_exists_mul_le_norm 📋 Mathlib.Analysis.Normed.Group.Uniform
  {𝓕 : Type u_1} {E : Type u_2} {F : Type u_3} [SeminormedAddGroup E] [SeminormedAddGroup F] [FunLike 𝓕 E F] [AddMonoidHomClass 𝓕 E F] {f : 𝓕} : (∃ K, AntilipschitzWith K ⇑f) ↔ ∃ c > 0, ∀ (x : E), c * ‖x‖ ≤ ‖f x‖
• antilipschitzWith_iff_exists_mul_le_norm' 📋 Mathlib.Analysis.Normed.Group.Uniform
  {𝓕 : Type u_1} {E : Type u_2} {F : Type u_3} [SeminormedGroup E] [SeminormedGroup F] [FunLike 𝓕 E F] [MonoidHomClass 𝓕 E F] {f : 𝓕} : (∃ K, AntilipschitzWith K ⇑f) ↔ ∃ c > 0, ∀ (x : E), c * ‖x‖ ≤ ‖f x‖
• AntilipschitzWith.le_mul_norm_div 📋 Mathlib.Analysis.Normed.Group.Uniform
  {F : Type u_3} [SeminormedCommGroup F] {E : Type u_5} [SeminormedCommGroup E] {K : NNReal} {f : E → F} (hf : AntilipschitzWith K f) (x y : E) : ‖x⁻¹ * y‖ ≤ ↑K * ‖(f x)⁻¹ * f y‖
• AntilipschitzWith.le_mul_norm_sub 📋 Mathlib.Analysis.Normed.Group.Uniform
  {F : Type u_3} [SeminormedAddCommGroup F] {E : Type u_5} [SeminormedAddCommGroup E] {K : NNReal} {f : E → F} (hf : AntilipschitzWith K f) (x y : E) : ‖-x + y‖ ≤ ↑K * ‖-f x + f y‖
• Dilation.antilipschitz 📋 Mathlib.Topology.MetricSpace.Dilation
  {α : Type u_1} {β : Type u_2} {F : Type u_4} [PseudoEMetricSpace α] [PseudoEMetricSpace β] [FunLike F α β] [DilationClass F α β] (f : F) : AntilipschitzWith (Dilation.ratio f)⁻¹ ⇑f
• antilipschitzWith_mul_left 📋 Mathlib.Analysis.Normed.Ring.Lemmas
  {α : Type u_1} [NonUnitalNormedRing α] [NormMulClass α] {a : α} (ha : a ≠ 0) : AntilipschitzWith ‖a‖₊⁻¹ fun x => a * x
• antilipschitzWith_mul_right 📋 Mathlib.Analysis.Normed.Ring.Lemmas
  {α : Type u_1} [NonUnitalNormedRing α] [NormMulClass α] {a : α} (ha : a ≠ 0) : AntilipschitzWith ‖a‖₊⁻¹ fun x => x * a
• NormedAddGroupHom.antilipschitz_of_norm_ge 📋 Mathlib.Analysis.Normed.Group.Hom
  {V₁ : Type u_2} {V₂ : Type u_3} [SeminormedAddCommGroup V₁] [SeminormedAddCommGroup V₂] (f : NormedAddGroupHom V₁ V₂) {K : NNReal} (h : ∀ (x : V₁), ‖x‖ ≤ ↑K * ‖f x‖) : AntilipschitzWith K ⇑f
• SemilinearIsometryClass.antilipschitz 📋 Mathlib.Analysis.Normed.Operator.LinearIsometry
  {R : Type u_1} {R₂ : Type u_2} {E : Type u_4} {E₂ : Type u_5} {𝓕 : Type u_8} [Semiring R] [Semiring R₂] {σ₁₂ : R →+* R₂} [SeminormedAddCommGroup E] [SeminormedAddCommGroup E₂] [Module R E] [Module R₂ E₂] [FunLike 𝓕 E E₂] [SemilinearIsometryClass 𝓕 σ₁₂ E E₂] (f : 𝓕) : AntilipschitzWith 1 ⇑f
• LinearIsometry.antilipschitz 📋 Mathlib.Analysis.Normed.Operator.LinearIsometry
  {R : Type u_1} {R₂ : Type u_2} {E : Type u_4} {E₂ : Type u_5} [Semiring R] [Semiring R₂] {σ₁₂ : R →+* R₂} [SeminormedAddCommGroup E] [SeminormedAddCommGroup E₂] [Module R E] [Module R₂ E₂] (f : E →ₛₗᵢ[σ₁₂] E₂) : AntilipschitzWith 1 ⇑f
• LinearIsometryEquiv.antilipschitz 📋 Mathlib.Analysis.Normed.Operator.LinearIsometry
  {R : Type u_1} {R₂ : Type u_2} {E : Type u_4} {E₂ : Type u_5} [Semiring R] [Semiring R₂] {σ₁₂ : R →+* R₂} {σ₂₁ : R₂ →+* R} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] [SeminormedAddCommGroup E] [SeminormedAddCommGroup E₂] [Module R E] [Module R₂ E₂] (e : E ≃ₛₗᵢ[σ₁₂] E₂) : AntilipschitzWith 1 ⇑e
• ContinuousLinearMap.antilipschitz_of_bound 📋 Mathlib.Analysis.Normed.Operator.ContinuousLinearMap
  {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_3} {F : Type u_4} [Ring 𝕜] [Ring 𝕜₂] [SeminormedAddCommGroup E] [SeminormedAddCommGroup F] [Module 𝕜 E] [Module 𝕜₂ F] {σ : 𝕜 →+* 𝕜₂} (f : E →SL[σ] F) {K : NNReal} (h : ∀ (x : E), ‖x‖ ≤ ↑K * ‖f x‖) : AntilipschitzWith K ⇑f
• ContinuousLinearMap.bound_of_antilipschitz 📋 Mathlib.Analysis.Normed.Operator.ContinuousLinearMap
  {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_3} {F : Type u_4} [Ring 𝕜] [Ring 𝕜₂] [SeminormedAddCommGroup E] [SeminormedAddCommGroup F] [Module 𝕜 E] [Module 𝕜₂ F] {σ : 𝕜 →+* 𝕜₂} (f : E →SL[σ] F) {K : NNReal} (h : AntilipschitzWith K ⇑f) (x : E) : ‖x‖ ≤ ↑K * ‖f x‖
• Complex.antilipschitz_equivRealProd 📋 Mathlib.Analysis.Complex.Basic
  : AntilipschitzWith (NNReal.sqrt 2) ⇑Complex.equivRealProd
• MeasureTheory.MemLp.of_comp_antilipschitzWith 📋 Mathlib.MeasureTheory.Function.LpSpace.Basic
  {p : ENNReal} {α : Type u_6} {E : Type u_7} {F : Type u_8} {K' : NNReal} [MeasurableSpace α] {μ : MeasureTheory.Measure α} [NormedAddCommGroup E] [NormedAddCommGroup F] {f : α → E} {g : E → F} (hL : MeasureTheory.MemLp (g ∘ f) p μ) (hg : UniformContinuous g) (hg' : AntilipschitzWith K' g) (g0 : g 0 = 0) : MeasureTheory.MemLp f p μ
• LipschitzWith.memLp_comp_iff_of_antilipschitz 📋 Mathlib.MeasureTheory.Function.LpSpace.Basic
  {p : ENNReal} {α : Type u_6} {E : Type u_7} {F : Type u_8} {K K' : NNReal} [MeasurableSpace α] {μ : MeasureTheory.Measure α} [NormedAddCommGroup E] [NormedAddCommGroup F] {f : α → E} {g : E → F} (hg : LipschitzWith K g) (hg' : AntilipschitzWith K' g) (g0 : g 0 = 0) : MeasureTheory.MemLp (g ∘ f) p μ ↔ MeasureTheory.MemLp f p μ
• LinearMap.antilipschitz_of_comap_nhds_le 📋 Mathlib.Analysis.Normed.Operator.NormedSpace
  {𝕜 : Type u_1} {𝕜₂ : Type u_3} {E : Type u_5} {F : Type u_6} [NormedAddCommGroup E] [NormedAddCommGroup F] [NontriviallyNormedField 𝕜] [NontriviallyNormedField 𝕜₂] [NormedSpace 𝕜 E] [NormedSpace 𝕜₂ F] {σ₁₂ : 𝕜 →+* 𝕜₂} [h : RingHomIsometric σ₁₂] (f : E →ₛₗ[σ₁₂] F) (hf : Filter.comap (⇑f) (nhds 0) ≤ nhds 0) : ∃ K, AntilipschitzWith K ⇑f
• ContinuousLinearMap.antilipschitz_of_isEmbedding 📋 Mathlib.Analysis.Normed.Operator.NormedSpace
  {𝕜 : Type u_1} {E : Type u_5} {Fₗ : Type u_7} [NormedAddCommGroup E] [NormedAddCommGroup Fₗ] [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 E] [NormedSpace 𝕜 Fₗ] (f : E →L[𝕜] Fₗ) (hf : Topology.IsEmbedding ⇑f) : ∃ K, AntilipschitzWith K ⇑f
• ContinuousLinearEquiv.antilipschitz 📋 Mathlib.Analysis.Normed.Operator.NormedSpace
  {𝕜 : Type u_1} {𝕜₂ : Type u_3} {E : Type u_5} {F : Type u_6} [NormedAddCommGroup E] [NormedAddCommGroup F] [NontriviallyNormedField 𝕜] [NontriviallyNormedField 𝕜₂] [NormedSpace 𝕜 E] [NormedSpace 𝕜₂ F] {σ₁₂ : 𝕜 →+* 𝕜₂} {σ₂₁ : 𝕜₂ →+* 𝕜} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] [RingHomIsometric σ₂₁] (e : E ≃SL[σ₁₂] F) : AntilipschitzWith ‖↑e.symm‖₊ ⇑e
• MeasureTheory.LipschitzWith.integrable_comp_iff_of_antilipschitz 📋 Mathlib.MeasureTheory.Function.L1Space.Integrable
  {α : Type u_1} {β : Type u_2} {γ : Type u_3} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [NormedAddCommGroup β] [NormedAddCommGroup γ] {K K' : NNReal} {f : α → β} {g : β → γ} (hg : LipschitzWith K g) (hg' : AntilipschitzWith K' g) (g0 : g 0 = 0) : MeasureTheory.Integrable (g ∘ f) μ ↔ MeasureTheory.Integrable f μ
• AffineIsometry.antilipschitz 📋 Mathlib.Analysis.Normed.Affine.Isometry
  {𝕜 : Type u_1} {V : Type u_2} {V₂ : Type u_5} {P : Type u_10} {P₂ : Type u_11} [NormedField 𝕜] [SeminormedAddCommGroup V] [NormedSpace 𝕜 V] [PseudoMetricSpace P] [NormedAddTorsor V P] [SeminormedAddCommGroup V₂] [NormedSpace 𝕜 V₂] [PseudoMetricSpace P₂] [NormedAddTorsor V₂ P₂] (f : P →ᵃⁱ[𝕜] P₂) : AntilipschitzWith 1 ⇑f
• AffineIsometryEquiv.antilipschitz 📋 Mathlib.Analysis.Normed.Affine.Isometry
  {𝕜 : Type u_1} {V : Type u_2} {V₂ : Type u_5} {P : Type u_10} {P₂ : Type u_11} [NormedField 𝕜] [SeminormedAddCommGroup V] [NormedSpace 𝕜 V] [PseudoMetricSpace P] [NormedAddTorsor V P] [SeminormedAddCommGroup V₂] [NormedSpace 𝕜 V₂] [PseudoMetricSpace P₂] [NormedAddTorsor V₂ P₂] (e : P ≃ᵃⁱ[𝕜] P₂) : AntilipschitzWith 1 ⇑e
• antilipschitz_of_bound_of_norm_one 📋 Mathlib.Analysis.Normed.Module.RCLike.Basic
  {𝕜 : Type u_1} [RCLike 𝕜] {𝓕 : Type u_3} {E : Type u_4} {F : Type u_5} [NormedAddCommGroup E] [NormedAddCommGroup F] [NormedSpace 𝕜 E] [NormedSpace 𝕜 F] [FunLike 𝓕 E F] [AddMonoidHomClass 𝓕 E F] [MulActionHomClass 𝓕 𝕜 E F] (f : 𝓕) {K : NNReal} (h : ∀ (x : E), ‖x‖ = 1 → 1 ≤ ↑K * ‖f x‖) : AntilipschitzWith K ⇑f
• AffineMap.antilipschitzWith_of_finiteDimensional 📋 Mathlib.Analysis.Normed.Module.FiniteDimension
  {𝕜 : Type u} [NontriviallyNormedField 𝕜] {E : Type v} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type w} [NormedAddCommGroup F] [NormedSpace 𝕜 F] [CompleteSpace 𝕜] {PE : Type u_1} {PF : Type u_2} [MetricSpace PE] [NormedAddTorsor E PE] [MetricSpace PF] [NormedAddTorsor F PF] [FiniteDimensional 𝕜 E] {f : PE →ᵃ[𝕜] PF} (hf : Function.Injective ⇑f) : ∃ K, AntilipschitzWith K ⇑f
• LinearMap.injective_iff_antilipschitz 📋 Mathlib.Analysis.Normed.Module.FiniteDimension
  {𝕜 : Type u} [NontriviallyNormedField 𝕜] {E : Type v} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type w} [NormedAddCommGroup F] [NormedSpace 𝕜 F] [CompleteSpace 𝕜] [FiniteDimensional 𝕜 E] (f : E →ₗ[𝕜] F) : Function.Injective ⇑f ↔ ∃ K > 0, AntilipschitzWith K ⇑f
• LinearMap.exists_antilipschitzWith 📋 Mathlib.Analysis.Normed.Module.FiniteDimension
  {𝕜 : Type u} [NontriviallyNormedField 𝕜] {E : Type v} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type w} [NormedAddCommGroup F] [NormedSpace 𝕜 F] [CompleteSpace 𝕜] [FiniteDimensional 𝕜 E] (f : E →ₗ[𝕜] F) (hf : f.ker = ⊥) : ∃ K > 0, AntilipschitzWith K ⇑f
• ContinuousLinearMap.antilipschitz_of_injective_of_isClosed_range 📋 Mathlib.Analysis.Normed.Operator.Banach
  {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_5} {F : Type u_6} [NormedAddCommGroup E] [NormedSpace 𝕜 E] [NormedAddCommGroup F] [NormedSpace 𝕜 F] [CompleteSpace E] [CompleteSpace F] (f : E →L[𝕜] F) (hf : Function.Injective ⇑f) (hf' : IsClosed (Set.range ⇑f)) : ∃ K, AntilipschitzWith K ⇑f
• ContinuousLinearMap.isClosed_range_iff_antilipschitz_of_injective 📋 Mathlib.Analysis.Normed.Operator.Banach
  {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_5} {F : Type u_6} [NormedAddCommGroup E] [NormedSpace 𝕜 E] [NormedAddCommGroup F] [NormedSpace 𝕜 F] [CompleteSpace E] [CompleteSpace F] (f : E →L[𝕜] F) (hf : Function.Injective ⇑f) : IsClosed (Set.range ⇑f) ↔ ∃ K, AntilipschitzWith K ⇑f
• ContinuousLinearMap.antilipschitz_antiLipschitzConstant_of_injective_of_isClosed_range 📋 Mathlib.Analysis.Normed.Operator.Banach
  {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_5} {F : Type u_6} [NormedAddCommGroup E] [NormedSpace 𝕜 E] [NormedAddCommGroup F] [NormedSpace 𝕜 F] [CompleteSpace E] [CompleteSpace F] (f : E →L[𝕜] F) (hf : Function.Injective ⇑f) (hf' : IsClosed (Set.range ⇑f)) : AntilipschitzWith (f.antilipschitzConstant_of_injective_of_isClosed_range hf hf') ⇑f
• AntilipschitzWith.completeSpace_range_clm 📋 Mathlib.Analysis.Normed.Operator.Banach
  {𝕜 : Type u_1} {𝕜' : Type u_2} [NontriviallyNormedField 𝕜] [NontriviallyNormedField 𝕜'] {E : Type u_3} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {σ : 𝕜 →+* 𝕜'} {σ' : 𝕜' →+* 𝕜} [RingHomInvPair σ σ'] {F : Type u_4} [NormedAddCommGroup F] [NormedSpace 𝕜' F] [CompleteSpace E] [CompleteSpace F] {f : E →SL[σ] F} {c : NNReal} (hf : AntilipschitzWith c ⇑f) : CompleteSpace ↥(↑f).range
• ContinuousLinearMap.closed_range_of_antilipschitz 📋 Mathlib.Analysis.Normed.Operator.Banach
  {𝕜 : Type u_1} {𝕜' : Type u_2} [NontriviallyNormedField 𝕜] [NontriviallyNormedField 𝕜'] {E : Type u_3} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {σ : 𝕜 →+* 𝕜'} {σ' : 𝕜' →+* 𝕜} [RingHomInvPair σ σ'] {F : Type u_4} [NormedAddCommGroup F] [NormedSpace 𝕜' F] [CompleteSpace E] {f : E →SL[σ] F} {c : NNReal} (hf : AntilipschitzWith c ⇑f) : (↑f).range.topologicalClosure = (↑f).range
• ContinuousLinearMap.bijective_iff_dense_range_and_antilipschitz 📋 Mathlib.Analysis.Normed.Operator.Banach
  {𝕜 : Type u_1} {𝕜' : Type u_2} [NontriviallyNormedField 𝕜] [NontriviallyNormedField 𝕜'] {E : Type u_3} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {σ : 𝕜 →+* 𝕜'} {σ' : 𝕜' →+* 𝕜} [RingHomInvPair σ σ'] {F : Type u_4} [NormedAddCommGroup F] [NormedSpace 𝕜' F] [CompleteSpace E] [CompleteSpace F] [RingHomInvPair σ' σ] [RingHomIsometric σ] [RingHomIsometric σ'] (f : E →SL[σ] F) : Function.Bijective ⇑f ↔ (↑f).range.topologicalClosure = ⊤ ∧ ∃ c, AntilipschitzWith c ⇑f
• WithLp.prod_antilipschitzWith_toLp 📋 Mathlib.Analysis.Normed.Lp.ProdLp
  (p : ENNReal) (α : Type u_2) (β : Type u_3) [hp : Fact (1 ≤ p)] [PseudoEMetricSpace α] [PseudoEMetricSpace β] : AntilipschitzWith 1 (WithLp.toLp p)
• WithLp.prod_antilipschitzWith_ofLp 📋 Mathlib.Analysis.Normed.Lp.ProdLp
  (p : ENNReal) (α : Type u_2) (β : Type u_3) [hp : Fact (1 ≤ p)] [PseudoEMetricSpace α] [PseudoEMetricSpace β] : AntilipschitzWith (2 ^ (1 / p).toReal) WithLp.ofLp
• PiLp.antilipschitzWith_toLp 📋 Mathlib.Analysis.Normed.Lp.PiLp
  (p : ENNReal) {ι : Type u_2} (β : ι → Type u_4) [hp : Fact (1 ≤ p)] [Fintype ι] [(i : ι) → PseudoEMetricSpace (β i)] : AntilipschitzWith 1 (WithLp.toLp p)
• PiLp.antilipschitzWith_ofLp 📋 Mathlib.Analysis.Normed.Lp.PiLp
  (p : ENNReal) {ι : Type u_2} (β : ι → Type u_4) [hp : Fact (1 ≤ p)] [Fintype ι] [(i : ι) → PseudoEMetricSpace (β i)] : AntilipschitzWith (↑(Fintype.card ι) ^ (1 / p).toReal) WithLp.ofLp
• HasFDerivAt.eventually_ne 📋 Mathlib.Analysis.Calculus.FDeriv.Equiv
  {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : E → F} {f' : E →L[𝕜] F} {x : E} {c : F} (h : HasFDerivAt f f' x) (hf' : ∃ C, AntilipschitzWith C ⇑f') : ∀ᶠ (z : E) in nhdsWithin x {x}ᶜ, f z ≠ c
• HasFDerivWithinAt.eventually_ne 📋 Mathlib.Analysis.Calculus.FDeriv.Equiv
  {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : E → F} {f' : E →L[𝕜] F} {x : E} {s : Set E} {c : F} (h : HasFDerivWithinAt f f' s x) (hf' : ∃ C, AntilipschitzWith C ⇑f') : ∀ᶠ (z : E) in nhdsWithin x (s \ {x}), f z ≠ c
• HasFDerivAt.tendsto_nhdsNE 📋 Mathlib.Analysis.Calculus.FDeriv.Equiv
  {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : E → F} {f' : E →L[𝕜] F} {x : E} (h : HasFDerivAt f f' x) (hf' : ∃ C, AntilipschitzWith C ⇑f') : Filter.Tendsto f (nhdsWithin x {x}ᶜ) (nhdsWithin (f x) {f x}ᶜ)
• HasFDerivAt.eventually_notMem 📋 Mathlib.Analysis.Calculus.FDeriv.Equiv
  {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : E → F} {f' : E →L[𝕜] F} {x : E} (h : HasFDerivAt f f' x) (hf' : ∃ C, AntilipschitzWith C ⇑f') (t : Set F) (ht : ¬AccPt (f x) (Filter.principal t)) : ∀ᶠ (z : E) in nhdsWithin x {x}ᶜ, f z ∉ t
• HasFDerivWithinAt.tendsto_nhdsWithin_nhdsNE 📋 Mathlib.Analysis.Calculus.FDeriv.Equiv
  {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : E → F} {f' : E →L[𝕜] F} {x : E} {s : Set E} (h : HasFDerivWithinAt f f' s x) (hf' : ∃ C, AntilipschitzWith C ⇑f') : Filter.Tendsto f (nhdsWithin x (s \ {x})) (nhdsWithin (f x) {f x}ᶜ)
• HasFDerivWithinAt.eventually_notMem 📋 Mathlib.Analysis.Calculus.FDeriv.Equiv
  {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : E → F} {f' : E →L[𝕜] F} {x : E} {s : Set E} (h : HasFDerivWithinAt f f' s x) (hf' : ∃ C, AntilipschitzWith C ⇑f') (t : Set F) (ht : ¬AccPt (f x) (Filter.principal t)) : ∀ᶠ (z : E) in nhdsWithin x (s \ {x}), f z ∉ t
• ContinuousLinearMap.integral_comp_comm' 📋 Mathlib.MeasureTheory.Integral.Bochner.ContinuousLinearMap
  {X : Type u_1} {E : Type u_3} {Fₗ : Type u_5} [MeasurableSpace X] {μ : MeasureTheory.Measure X} {𝕜 : Type u_6} [RCLike 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [NormedAddCommGroup Fₗ] [NormedSpace 𝕜 Fₗ] [NormedSpace ℝ Fₗ] [CompleteSpace Fₗ] [NormedSpace ℝ E] [CompleteSpace E] (L : E →L[𝕜] Fₗ) {K : NNReal} (hL : AntilipschitzWith K ⇑L) (φ : X → E) : ∫ (x : X), L (φ x) ∂μ = L (∫ (x : X), φ x ∂μ)
• ApproximatesLinearOn.antilipschitz 📋 Mathlib.Analysis.Calculus.InverseFunctionTheorem.ApproximatesLinearOn
  {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : E → F} {f' : E ≃L[𝕜] F} {s : Set E} {c : NNReal} (hf : ApproximatesLinearOn f (↑f') s c) (hc : Subsingleton E ∨ c < ‖↑f'.symm‖₊⁻¹) : AntilipschitzWith (‖↑f'.symm‖₊⁻¹ - c)⁻¹ (s.domRestrict f)
• Metric.IsSeparated.image_antilipschitz 📋 Mathlib.Topology.MetricSpace.MetricSeparated
  {X : Type u_1} {Y : Type u_2} [PseudoEMetricSpace X] [PseudoEMetricSpace Y] {s : Set X} {ε K₁ : NNReal} {f : X → Y} (hs : Metric.IsSeparated (↑ε) s) (hf : AntilipschitzWith K₁ f) (hK₁ : 0 < K₁) : Metric.IsSeparated (↑(ε / K₁)) (f '' s)
• Delone.DeloneSet.mapBilipschitz 📋 Mathlib.Analysis.AperiodicOrder.Delone.Basic
  {X : Type u_1} {Y : Type u_2} [MetricSpace X] [MetricSpace Y] (f : X ≃ Y) (K₁ K₂ : NNReal) (hK₁ : 0 < K₁) (hK₂ : 0 < K₂) (hf₁ : AntilipschitzWith K₁ ⇑f) (hf₂ : LipschitzWith K₂ ⇑f) (D : Delone.DeloneSet X) : Delone.DeloneSet Y
• Delone.DeloneSet.mapBilipschitz_packingRadius 📋 Mathlib.Analysis.AperiodicOrder.Delone.Basic
  {X : Type u_1} {Y : Type u_2} [MetricSpace X] [MetricSpace Y] (f : X ≃ Y) (K₁ K₂ : NNReal) (hK₁ : 0 < K₁) (hK₂ : 0 < K₂) (hf₁ : AntilipschitzWith K₁ ⇑f) (hf₂ : LipschitzWith K₂ ⇑f) (D : Delone.DeloneSet X) : (Delone.DeloneSet.mapBilipschitz f K₁ K₂ hK₁ hK₂ hf₁ hf₂ D).packingRadius = D.packingRadius / K₁
• Delone.DeloneSet.mapBilipschitz_coveringRadius 📋 Mathlib.Analysis.AperiodicOrder.Delone.Basic
  {X : Type u_1} {Y : Type u_2} [MetricSpace X] [MetricSpace Y] (f : X ≃ Y) (K₁ K₂ : NNReal) (hK₁ : 0 < K₁) (hK₂ : 0 < K₂) (hf₁ : AntilipschitzWith K₁ ⇑f) (hf₂ : LipschitzWith K₂ ⇑f) (D : Delone.DeloneSet X) : (Delone.DeloneSet.mapBilipschitz f K₁ K₂ hK₁ hK₂ hf₁ hf₂ D).coveringRadius = K₂ * D.coveringRadius
• Delone.DeloneSet.mapBilipschitz_carrier 📋 Mathlib.Analysis.AperiodicOrder.Delone.Basic
  {X : Type u_1} {Y : Type u_2} [MetricSpace X] [MetricSpace Y] (f : X ≃ Y) (K₁ K₂ : NNReal) (hK₁ : 0 < K₁) (hK₂ : 0 < K₂) (hf₁ : AntilipschitzWith K₁ ⇑f) (hf₂ : LipschitzWith K₂ ⇑f) (D : Delone.DeloneSet X) : (Delone.DeloneSet.mapBilipschitz f K₁ K₂ hK₁ hK₂ hf₁ hf₂ D).carrier = ⇑f '' D.carrier
• Delone.DeloneSet.mapBilipschitz_refl 📋 Mathlib.Analysis.AperiodicOrder.Delone.Basic
  {X : Type u_1} [MetricSpace X] (D : Delone.DeloneSet X) (hK1 hK2 : 0 < 1) (hA : AntilipschitzWith 1 ⇑(Equiv.refl X)) (hL : LipschitzWith 1 ⇑(Equiv.refl X)) : Delone.DeloneSet.mapBilipschitz (Equiv.refl X) 1 1 hK1 hK2 hA hL D = D
• Delone.DeloneSet.mapBilipschitz_trans 📋 Mathlib.Analysis.AperiodicOrder.Delone.Basic
  {X : Type u_1} {Y : Type u_2} [MetricSpace X] [MetricSpace Y] {Z : Type u_3} [MetricSpace Z] (D : Delone.DeloneSet X) (f : X ≃ Y) (g : Y ≃ Z) (K₁f K₂f K₁g K₂g : NNReal) (hf₁_pos : 0 < K₁f) (hf₂_pos : 0 < K₂f) (hg₁_pos : 0 < K₁g) (hg₂_pos : 0 < K₂g) (hf_anti : AntilipschitzWith K₁f ⇑f) (hf_lip : LipschitzWith K₂f ⇑f) (hg_anti : AntilipschitzWith K₁g ⇑g) (hg_lip : LipschitzWith K₂g ⇑g) : Delone.DeloneSet.mapBilipschitz g K₁g K₂g hg₁_pos hg₂_pos hg_anti hg_lip (Delone.DeloneSet.mapBilipschitz f K₁f K₂f hf₁_pos hf₂_pos hf_anti hf_lip D) = Delone.DeloneSet.mapBilipschitz (f.trans g) (K₁f * K₁g) (K₂g * K₂f) ⋯ ⋯ ⋯ ⋯ D
• Unitization.antilipschitzWith_addEquiv 📋 Mathlib.Analysis.Normed.Algebra.Unitization
  {𝕜 : Type u_1} {A : Type u_2} [NontriviallyNormedField 𝕜] [NonUnitalNormedRing A] [NormedSpace 𝕜 A] [IsScalarTower 𝕜 A A] [SMulCommClass 𝕜 A A] [RegularNormedAlgebra 𝕜 A] : AntilipschitzWith 2 ⇑(Unitization.addEquiv 𝕜 A)
• antilipschitzWith_lineMap 📋 Mathlib.Analysis.Normed.Affine.AddTorsor
  {W : Type u_3} {Q : Type u_4} [NormedAddCommGroup W] [MetricSpace Q] [NormedAddTorsor W Q] {𝕜 : Type u_5} [NormedField 𝕜] [NormedSpace 𝕜 W] {p₁ p₂ : Q} (h : p₁ ≠ p₂) : AntilipschitzWith (nndist p₁ p₂)⁻¹ ⇑(AffineMap.lineMap p₁ p₂)
• uniformity_eq_of_bilipschitz 📋 Mathlib.Topology.MetricSpace.Bilipschitz
  {α : Type u_1} {β : Type u_2} [PseudoEMetricSpace α] [PseudoEMetricSpace β] {K₁ K₂ : NNReal} {f : α → β} (hf₁ : AntilipschitzWith K₁ f) (hf₂ : LipschitzWith K₂ f) : uniformity α = uniformity α
• bornology_eq_of_bilipschitz 📋 Mathlib.Topology.MetricSpace.Bilipschitz
  {α : Type u_1} {β : Type u_2} [PseudoMetricSpace α] [PseudoMetricSpace β] {K₁ K₂ : NNReal} {f : α → β} (hf₁ : AntilipschitzWith K₁ f) (hf₂ : LipschitzWith K₂ f) : Bornology.cobounded α = Bornology.cobounded α
• isBounded_iff_of_bilipschitz 📋 Mathlib.Topology.MetricSpace.Bilipschitz
  {α : Type u_1} {β : Type u_2} [PseudoMetricSpace α] [PseudoMetricSpace β] {K₁ K₂ : NNReal} {f : α → β} (hf₁ : AntilipschitzWith K₁ f) (hf₂ : LipschitzWith K₂ f) (s : Set α) : Bornology.IsBounded s ↔ Bornology.IsBounded s
• IsCompactOperator.antilipschitz_of_not_hasEigenvalue 📋 Mathlib.Analysis.Normed.Operator.Compact.FredholmAlternative
  {𝕜 : Type u_1} {X : Type u_2} [NontriviallyNormedField 𝕜] [NormedAddCommGroup X] [NormedSpace 𝕜 X] {T : X →L[𝕜] X} {μ : 𝕜} (hT : IsCompactOperator ⇑T) (hμ : μ ≠ 0) (h : ¬Module.End.HasEigenvalue (↑T) μ) : ∃ K, AntilipschitzWith K ⇑(T - μ • 1)
• SchwartzMap.compCLMOfAntilipschitz 📋 Mathlib.Analysis.Distribution.SchwartzSpace.Basic
  (𝕜 : Type u_2) {D : Type u_4} {E : Type u_5} {F : Type u_6} [NormedAddCommGroup E] [NormedSpace ℝ E] [NormedAddCommGroup F] [NormedSpace ℝ F] [RCLike 𝕜] [NormedAddCommGroup D] [NormedSpace ℝ D] [NormedSpace 𝕜 F] {K : NNReal} {g : D → E} (hg : Function.HasTemperateGrowth g) (h'g : AntilipschitzWith K g) : SchwartzMap E F →L[𝕜] SchwartzMap D F
• SchwartzMap.compCLMOfAntilipschitz_apply 📋 Mathlib.Analysis.Distribution.SchwartzSpace.Basic
  (𝕜 : Type u_2) {D : Type u_4} {E : Type u_5} {F : Type u_6} [NormedAddCommGroup E] [NormedSpace ℝ E] [NormedAddCommGroup F] [NormedSpace ℝ F] [RCLike 𝕜] [NormedAddCommGroup D] [NormedSpace ℝ D] [NormedSpace 𝕜 F] {K : NNReal} {g : D → E} (hg : Function.HasTemperateGrowth g) (h'g : AntilipschitzWith K g) (f : SchwartzMap E F) : ⇑((SchwartzMap.compCLMOfAntilipschitz 𝕜 hg h'g) f) = ⇑f ∘ g
• ContinuousLinearMap.antilipschitz_of_forall_le_inner_map 📋 Mathlib.Analysis.InnerProductSpace.Positive
  {𝕜 : Type u_1} [RCLike 𝕜] {H : Type u_4} [NormedAddCommGroup H] [InnerProductSpace 𝕜 H] (f : H →L[𝕜] H) {c : NNReal} (hc : 0 < c) (h : ∀ (x : H), ‖x‖ ^ 2 * ↑c ≤ ‖inner 𝕜 (f x) x‖) : AntilipschitzWith c⁻¹ ⇑f
• IsCoercive.antilipschitz 📋 Mathlib.Analysis.InnerProductSpace.LaxMilgram
  {V : Type u} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [CompleteSpace V] {B : V →L[ℝ] V →L[ℝ] ℝ} (coercive : IsCoercive B) : ∃ C, 0 < C ∧ AntilipschitzWith C ⇑(InnerProductSpace.continuousLinearMapOfBilin B)
• AntilipschitzWith.hausdorffMeasure_preimage_le 📋 Mathlib.MeasureTheory.Measure.Hausdorff
  {X : Type u_2} {Y : Type u_3} [EMetricSpace X] [EMetricSpace Y] [MeasurableSpace X] [BorelSpace X] [MeasurableSpace Y] [BorelSpace Y] {f : X → Y} {K : NNReal} {d : ℝ} (hf : AntilipschitzWith K f) (hd : 0 ≤ d) (s : Set Y) : (MeasureTheory.Measure.hausdorffMeasure d) (f ⁻¹' s) ≤ ↑K ^ d * (MeasureTheory.Measure.hausdorffMeasure d) s
• AntilipschitzWith.le_hausdorffMeasure_image 📋 Mathlib.MeasureTheory.Measure.Hausdorff
  {X : Type u_2} {Y : Type u_3} [EMetricSpace X] [EMetricSpace Y] [MeasurableSpace X] [BorelSpace X] [MeasurableSpace Y] [BorelSpace Y] {f : X → Y} {K : NNReal} {d : ℝ} (hf : AntilipschitzWith K f) (hd : 0 ≤ d) (s : Set X) : (MeasureTheory.Measure.hausdorffMeasure d) s ≤ ↑K ^ d * (MeasureTheory.Measure.hausdorffMeasure d) (f '' s)
• AntilipschitzWith.dimH_preimage_le 📋 Mathlib.Topology.MetricSpace.HausdorffDimension
  {X : Type u_2} {Y : Type u_3} [EMetricSpace X] [EMetricSpace Y] {K : NNReal} {f : X → Y} (hf : AntilipschitzWith K f) (s : Set Y) : dimH (f ⁻¹' s) ≤ dimH s
• AntilipschitzWith.le_dimH_image 📋 Mathlib.Topology.MetricSpace.HausdorffDimension
  {X : Type u_2} {Y : Type u_3} [EMetricSpace X] [EMetricSpace Y] {K : NNReal} {f : X → Y} (hf : AntilipschitzWith K f) (s : Set X) : dimH s ≤ dimH (f '' s)

About

Loogle searches Lean and Mathlib definitions and theorems.

You can use Loogle from within the Lean4 VSCode language extension using the Loogle command from the command palette. You can also try the #loogle command from LeanSearchClient, the CLI version, the Loogle VS Code extension, the lean.nvim integration or the Zulip bot.

Usage

Loogle finds definitions and lemmas in various ways:

1. By constant:
   🔍 Real.sin
   finds all lemmas whose statement somehow mentions the sine function.

2. By lemma name substring:
   🔍 "differ"
   finds all lemmas that have "differ" somewhere in their lemma name.

3. By subexpression:
   🔍 _ * (_ ^ _)
   finds all lemmas whose statements somewhere include a product where the second argument is raised to some power.

   The pattern can also be non-linear, as in
   🔍 Real.sqrt ?a * Real.sqrt ?a

   If the pattern has parameters, they are matched in any order. Both of these will find List.map:
   🔍 (?a -> ?b) -> List ?a -> List ?b
   🔍 List ?a -> (?a -> ?b) -> List ?b

4. By main conclusion:
   🔍 |- tsum _ = _ * tsum _
   finds all lemmas where the conclusion (the subexpression to the right of all → and ∀) has the given shape.

   As before, if the pattern has parameters, they are matched against the hypotheses of the lemma in any order; for example,
   🔍 |- _ < _ → tsum _ < tsum _
   will find tsum_lt_tsum even though the hypothesis f i < g i is not the last.

5. You can filter for definitions vs theorems: Using ⊢ (_ : Type _) finds all definitions which provide data while ⊢ (_ : Prop) finds all theorems (and definitions of proofs).

If you pass more than one such search filter, separated by commas Loogle will return lemmas which match all of them. The search
🔍 Real.sin, "two", tsum, _ * _, _ ^ _, |- _ < _ → _
would find all lemmas which mention the constants Real.sin and tsum, have "two" as a substring of the lemma name, include a product and a power somewhere in the type, and have a hypothesis of the form _ < _ (if there were any such lemmas). Metavariables (?a) are assigned independently in each filter.

The #lucky button will directly send you to the documentation of the first hit.

Source code

You can find the source code for this service at https://github.com/nomeata/loogle. The https://loogle.lean-lang.org/ service is provided by the Lean FRO. Please review the Lean FRO Terms of Use and Privacy Policy.

This is Loogle revision 9f11169 serving mathlib revision 69fae59