Documentation

Mathlib.Topology.MetricSpace.Antilipschitz

Antilipschitz functions #

We say that a map f : α → β between two (extended) metric spaces is AntilipschitzWith K, K ≥ 0, if for all x, y we have edist x y ≤ K * edist (f x) (f y). For a metric space, the latter inequality is equivalent to dist x y ≤ K * dist (f x) (f y).

Implementation notes #

The parameter K has type ℝ≥0. This way we avoid conjunction in the definition and have coercions both to and ℝ≥0∞. We do not require 0 < K in the definition, mostly because we do not have a posreal type.

def AntilipschitzWith {α : Type u_1} {β : Type u_2} [PseudoEMetricSpace α] [PseudoEMetricSpace β] (K : NNReal) (f : αβ) :

We say that f : α → β is AntilipschitzWith K if for any two points x, y we have edist x y ≤ K * edist (f x) (f y).

Equations
Instances For
    theorem AntilipschitzWith.edist_lt_top {α : Type u_1} {β : Type u_2} [PseudoEMetricSpace α] [PseudoMetricSpace β] {K : NNReal} {f : αβ} (h : AntilipschitzWith K f) (x : α) (y : α) :
    edist x y <
    theorem AntilipschitzWith.edist_ne_top {α : Type u_1} {β : Type u_2} [PseudoEMetricSpace α] [PseudoMetricSpace β] {K : NNReal} {f : αβ} (h : AntilipschitzWith K f) (x : α) (y : α) :
    theorem antilipschitzWith_iff_le_mul_nndist {α : Type u_1} {β : Type u_2} [PseudoMetricSpace α] [PseudoMetricSpace β] {K : NNReal} {f : αβ} :
    AntilipschitzWith K f ∀ (x y : α), nndist x y K * nndist (f x) (f y)
    theorem AntilipschitzWith.le_mul_nndist {α : Type u_1} {β : Type u_2} [PseudoMetricSpace α] [PseudoMetricSpace β] {K : NNReal} {f : αβ} :
    AntilipschitzWith K f∀ (x y : α), nndist x y K * nndist (f x) (f y)

    Alias of the forward direction of antilipschitzWith_iff_le_mul_nndist.

    theorem AntilipschitzWith.of_le_mul_nndist {α : Type u_1} {β : Type u_2} [PseudoMetricSpace α] [PseudoMetricSpace β] {K : NNReal} {f : αβ} :
    (∀ (x y : α), nndist x y K * nndist (f x) (f y))AntilipschitzWith K f

    Alias of the reverse direction of antilipschitzWith_iff_le_mul_nndist.

    theorem antilipschitzWith_iff_le_mul_dist {α : Type u_1} {β : Type u_2} [PseudoMetricSpace α] [PseudoMetricSpace β] {K : NNReal} {f : αβ} :
    AntilipschitzWith K f ∀ (x y : α), dist x y K * dist (f x) (f y)
    theorem AntilipschitzWith.le_mul_dist {α : Type u_1} {β : Type u_2} [PseudoMetricSpace α] [PseudoMetricSpace β] {K : NNReal} {f : αβ} :
    AntilipschitzWith K f∀ (x y : α), dist x y K * dist (f x) (f y)

    Alias of the forward direction of antilipschitzWith_iff_le_mul_dist.

    theorem AntilipschitzWith.of_le_mul_dist {α : Type u_1} {β : Type u_2} [PseudoMetricSpace α] [PseudoMetricSpace β] {K : NNReal} {f : αβ} :
    (∀ (x y : α), dist x y K * dist (f x) (f y))AntilipschitzWith K f

    Alias of the reverse direction of antilipschitzWith_iff_le_mul_dist.

    theorem AntilipschitzWith.mul_le_nndist {α : Type u_1} {β : Type u_2} [PseudoMetricSpace α] [PseudoMetricSpace β] {K : NNReal} {f : αβ} (hf : AntilipschitzWith K f) (x : α) (y : α) :
    K⁻¹ * nndist x y nndist (f x) (f y)
    theorem AntilipschitzWith.mul_le_dist {α : Type u_1} {β : Type u_2} [PseudoMetricSpace α] [PseudoMetricSpace β] {K : NNReal} {f : αβ} (hf : AntilipschitzWith K f) (x : α) (y : α) :
    K⁻¹ * dist x y dist (f x) (f y)
    def AntilipschitzWith.k {α : Type u_1} {β : Type u_2} [PseudoEMetricSpace α] [PseudoEMetricSpace β] {K : NNReal} {f : αβ} (_hf : AntilipschitzWith K f) :

    Extract the constant from hf : AntilipschitzWith K f. This is useful, e.g., if K is given by a long formula, and we want to reuse this value.

    Equations
    Instances For
      theorem AntilipschitzWith.injective {α : Type u_4} {β : Type u_5} [EMetricSpace α] [PseudoEMetricSpace β] {K : NNReal} {f : αβ} (hf : AntilipschitzWith K f) :
      theorem AntilipschitzWith.mul_le_edist {α : Type u_1} {β : Type u_2} [PseudoEMetricSpace α] [PseudoEMetricSpace β] {K : NNReal} {f : αβ} (hf : AntilipschitzWith K f) (x : α) (y : α) :
      (K)⁻¹ * edist x y edist (f x) (f y)
      theorem AntilipschitzWith.ediam_preimage_le {α : Type u_1} {β : Type u_2} [PseudoEMetricSpace α] [PseudoEMetricSpace β] {K : NNReal} {f : αβ} (hf : AntilipschitzWith K f) (s : Set β) :
      theorem AntilipschitzWith.le_mul_ediam_image {α : Type u_1} {β : Type u_2} [PseudoEMetricSpace α] [PseudoEMetricSpace β] {K : NNReal} {f : αβ} (hf : AntilipschitzWith K f) (s : Set α) :
      theorem AntilipschitzWith.comp {α : Type u_1} {β : Type u_2} {γ : Type u_3} [PseudoEMetricSpace α] [PseudoEMetricSpace β] [PseudoEMetricSpace γ] {Kg : NNReal} {g : βγ} (hg : AntilipschitzWith Kg g) {Kf : NNReal} {f : αβ} (hf : AntilipschitzWith Kf f) :
      AntilipschitzWith (Kf * Kg) (g f)
      theorem AntilipschitzWith.restrict {α : Type u_1} {β : Type u_2} [PseudoEMetricSpace α] [PseudoEMetricSpace β] {K : NNReal} {f : αβ} (hf : AntilipschitzWith K f) (s : Set α) :
      theorem AntilipschitzWith.codRestrict {α : Type u_1} {β : Type u_2} [PseudoEMetricSpace α] [PseudoEMetricSpace β] {K : NNReal} {f : αβ} (hf : AntilipschitzWith K f) {s : Set β} (hs : ∀ (x : α), f x s) :
      theorem AntilipschitzWith.to_rightInvOn' {α : Type u_1} {β : Type u_2} [PseudoEMetricSpace α] [PseudoEMetricSpace β] {K : NNReal} {f : αβ} {s : Set α} (hf : AntilipschitzWith K (Set.restrict s f)) {g : βα} {t : Set β} (g_maps : Set.MapsTo g t s) (g_inv : Set.RightInvOn g f t) :
      theorem AntilipschitzWith.to_rightInvOn {α : Type u_1} {β : Type u_2} [PseudoEMetricSpace α] [PseudoEMetricSpace β] {K : NNReal} {f : αβ} (hf : AntilipschitzWith K f) {g : βα} {t : Set β} (h : Set.RightInvOn g f t) :
      theorem AntilipschitzWith.to_rightInverse {α : Type u_1} {β : Type u_2} [PseudoEMetricSpace α] [PseudoEMetricSpace β] {K : NNReal} {f : αβ} (hf : AntilipschitzWith K f) {g : βα} (hg : Function.RightInverse g f) :
      theorem AntilipschitzWith.uniformInducing {α : Type u_1} {β : Type u_2} [PseudoEMetricSpace α] [PseudoEMetricSpace β] {K : NNReal} {f : αβ} (hf : AntilipschitzWith K f) (hfc : UniformContinuous f) :
      theorem AntilipschitzWith.uniformEmbedding {α : Type u_4} {β : Type u_5} [EMetricSpace α] [PseudoEMetricSpace β] {K : NNReal} {f : αβ} (hf : AntilipschitzWith K f) (hfc : UniformContinuous f) :
      theorem AntilipschitzWith.isClosed_range {α : Type u_4} {β : Type u_5} [PseudoEMetricSpace α] [EMetricSpace β] [CompleteSpace α] {f : αβ} {K : NNReal} (hf : AntilipschitzWith K f) (hfc : UniformContinuous f) :
      theorem AntilipschitzWith.closedEmbedding {α : Type u_4} {β : Type u_5} [EMetricSpace α] [EMetricSpace β] {K : NNReal} {f : αβ} [CompleteSpace α] (hf : AntilipschitzWith K f) (hfc : UniformContinuous f) :
      theorem AntilipschitzWith.subtype_coe {α : Type u_1} [PseudoEMetricSpace α] (s : Set α) :
      AntilipschitzWith 1 Subtype.val
      theorem AntilipschitzWith.of_subsingleton {α : Type u_1} {β : Type u_2} [PseudoEMetricSpace α] [PseudoEMetricSpace β] {f : αβ} [Subsingleton α] {K : NNReal} :
      theorem AntilipschitzWith.subsingleton {α : Type u_4} {β : Type u_5} [EMetricSpace α] [PseudoEMetricSpace β] {f : αβ} (h : AntilipschitzWith 0 f) :

      If f : α → β is 0-antilipschitz, then α is a subsingleton.

      theorem AntilipschitzWith.isBounded_preimage {α : Type u_1} {β : Type u_2} [PseudoMetricSpace α] [PseudoMetricSpace β] {K : NNReal} {f : αβ} (hf : AntilipschitzWith K f) {s : Set β} (hs : Bornology.IsBounded s) :
      theorem AntilipschitzWith.properSpace {β : Type u_2} [PseudoMetricSpace β] {α : Type u_4} [MetricSpace α] {K : NNReal} {f : αβ} [ProperSpace α] (hK : AntilipschitzWith K f) (f_cont : Continuous f) (hf : Function.Surjective f) :

      The image of a proper space under an expanding onto map is proper.

      theorem AntilipschitzWith.isBounded_of_image2_left {α : Type u_1} {β : Type u_2} {γ : Type u_3} [PseudoMetricSpace α] [PseudoMetricSpace β] [PseudoMetricSpace γ] (f : αβγ) {K₁ : NNReal} (hf : ∀ (b : β), AntilipschitzWith K₁ fun (a : α) => f a b) {s : Set α} {t : Set β} (hst : Bornology.IsBounded (Set.image2 f s t)) :
      theorem AntilipschitzWith.isBounded_of_image2_right {α : Type u_1} {β : Type u_2} {γ : Type u_3} [PseudoMetricSpace α] [PseudoMetricSpace β] [PseudoMetricSpace γ] {f : αβγ} {K₂ : NNReal} (hf : ∀ (a : α), AntilipschitzWith K₂ (f a)) {s : Set α} {t : Set β} (hst : Bornology.IsBounded (Set.image2 f s t)) :
      theorem LipschitzWith.to_rightInverse {α : Type u_1} {β : Type u_2} [PseudoEMetricSpace α] [PseudoEMetricSpace β] {K : NNReal} {f : αβ} (hf : LipschitzWith K f) {g : βα} (hg : Function.RightInverse g f) :
      theorem LipschitzWith.properSpace {α : Type u_1} {β : Type u_2} [PseudoMetricSpace α] [MetricSpace β] [ProperSpace β] {K : NNReal} {f : α ≃ₜ β} (hK : LipschitzWith K f) :

      The preimage of a proper space under a Lipschitz homeomorphism is proper.