mathlib documentation

topology.metric_space.antilipschitz

Antilipschitz functions #

We say that a map f : α → β between two (extended) metric spaces is antilipschitz_with 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 conjuction 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 antilipschitz_with {α : Type u_1} {β : Type u_2} [pseudo_emetric_space α] [pseudo_emetric_space β] (K : ℝ≥0) (f : α → β) :
Prop

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

Equations
theorem antilipschitz_with_iff_le_mul_dist {α : Type u_1} {β : Type u_2} [pseudo_metric_space α] [pseudo_metric_space β] {K : ℝ≥0} {f : α → β} :
antilipschitz_with K f ∀ (x y : α), dist x y (K) * dist (f x) (f y)
theorem antilipschitz_with.le_mul_dist {α : Type u_1} {β : Type u_2} [pseudo_metric_space α] [pseudo_metric_space β] {K : ℝ≥0} {f : α → β} :
antilipschitz_with K f∀ (x y : α), dist x y (K) * dist (f x) (f y)

Alias of antilipschitz_with_iff_le_mul_dist.

theorem antilipschitz_with.of_le_mul_dist {α : Type u_1} {β : Type u_2} [pseudo_metric_space α] [pseudo_metric_space β] {K : ℝ≥0} {f : α → β} :
(∀ (x y : α), dist x y (K) * dist (f x) (f y))antilipschitz_with K f

Alias of antilipschitz_with_iff_le_mul_dist.

theorem antilipschitz_with.mul_le_dist {α : Type u_1} {β : Type u_2} [pseudo_metric_space α] [pseudo_metric_space β] {K : ℝ≥0} {f : α → β} (hf : antilipschitz_with K f) (x y : α) :
(K⁻¹) * dist x y dist (f x) (f y)
@[nolint]
def antilipschitz_with.K {α : Type u_1} {β : Type u_2} [pseudo_emetric_space α] [pseudo_emetric_space β] {K : ℝ≥0} {f : α → β} (hf : antilipschitz_with K f) :

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

Equations
  • hf.K = K
theorem antilipschitz_with.injective {α : Type u_1} {β : Type u_2} [emetric_space α] [pseudo_emetric_space β] {K : ℝ≥0} {f : α → β} (hf : antilipschitz_with K f) :
theorem antilipschitz_with.mul_le_edist {α : Type u_1} {β : Type u_2} [pseudo_emetric_space α] [pseudo_emetric_space β] {K : ℝ≥0} {f : α → β} (hf : antilipschitz_with K f) (x y : α) :
(K⁻¹) * edist x y edist (f x) (f y)
theorem antilipschitz_with.comp {α : Type u_1} {β : Type u_2} {γ : Type u_3} [pseudo_emetric_space α] [pseudo_emetric_space β] [pseudo_emetric_space γ] {Kg : ℝ≥0} {g : β → γ} (hg : antilipschitz_with Kg g) {Kf : ℝ≥0} {f : α → β} (hf : antilipschitz_with Kf f) :
antilipschitz_with (Kf * Kg) (g f)
theorem antilipschitz_with.restrict {α : Type u_1} {β : Type u_2} [pseudo_emetric_space α] [pseudo_emetric_space β] {K : ℝ≥0} {f : α → β} (hf : antilipschitz_with K f) (s : set α) :
theorem antilipschitz_with.cod_restrict {α : Type u_1} {β : Type u_2} [pseudo_emetric_space α] [pseudo_emetric_space β] {K : ℝ≥0} {f : α → β} (hf : antilipschitz_with K f) {s : set β} (hs : ∀ (x : α), f x s) :
theorem antilipschitz_with.to_right_inv_on' {α : Type u_1} {β : Type u_2} [pseudo_emetric_space α] [pseudo_emetric_space β] {K : ℝ≥0} {f : α → β} {s : set α} (hf : antilipschitz_with K (set.restrict f s)) {g : β → α} {t : set β} (g_maps : set.maps_to g t s) (g_inv : set.right_inv_on g f t) :
theorem antilipschitz_with.to_right_inv_on {α : Type u_1} {β : Type u_2} [pseudo_emetric_space α] [pseudo_emetric_space β] {K : ℝ≥0} {f : α → β} (hf : antilipschitz_with K f) {g : β → α} {t : set β} (h : set.right_inv_on g f t) :
theorem antilipschitz_with.to_right_inverse {α : Type u_1} {β : Type u_2} [pseudo_emetric_space α] [pseudo_emetric_space β] {K : ℝ≥0} {f : α → β} (hf : antilipschitz_with K f) {g : β → α} (hg : function.right_inverse g f) :
theorem antilipschitz_with.comap_uniformity_le {α : Type u_1} {β : Type u_2} [pseudo_emetric_space α] [pseudo_emetric_space β] {K : ℝ≥0} {f : α → β} (hf : antilipschitz_with K f) :
theorem antilipschitz_with.uniform_inducing {α : Type u_1} {β : Type u_2} [pseudo_emetric_space α] [pseudo_emetric_space β] {K : ℝ≥0} {f : α → β} (hf : antilipschitz_with K f) (hfc : uniform_continuous f) :
theorem antilipschitz_with.uniform_embedding {α : Type u_1} {β : Type u_2} [emetric_space α] [pseudo_emetric_space β] {K : ℝ≥0} {f : α → β} (hf : antilipschitz_with K f) (hfc : uniform_continuous f) :
theorem antilipschitz_with.is_complete_range {α : Type u_1} {β : Type u_2} [pseudo_emetric_space α] [pseudo_emetric_space β] {K : ℝ≥0} {f : α → β} [complete_space α] (hf : antilipschitz_with K f) (hfc : uniform_continuous f) :
theorem antilipschitz_with.is_closed_range {α : Type u_1} {β : Type u_2} [pseudo_emetric_space α] [emetric_space β] [complete_space α] {f : α → β} {K : ℝ≥0} (hf : antilipschitz_with K f) (hfc : uniform_continuous f) :
theorem antilipschitz_with.closed_embedding {α : Type u_1} {β : Type u_2} [emetric_space α] [emetric_space β] {K : ℝ≥0} {f : α → β} [complete_space α] (hf : antilipschitz_with K f) (hfc : uniform_continuous f) :
theorem antilipschitz_with.of_subsingleton {α : Type u_1} {β : Type u_2} [pseudo_emetric_space α] [pseudo_emetric_space β] {f : α → β} [subsingleton α] {K : ℝ≥0} :
theorem antilipschitz_with.bounded_preimage {α : Type u_1} {β : Type u_2} [pseudo_metric_space α] [pseudo_metric_space β] {K : ℝ≥0} {f : α → β} (hf : antilipschitz_with K f) {s : set β} (hs : metric.bounded s) :
theorem antilipschitz_with.proper_space {β : Type u_2} [pseudo_metric_space β] {α : Type u_1} [metric_space α] {K : ℝ≥0} {f : α → β} [proper_space α] (hK : antilipschitz_with K f) (f_cont : continuous f) (hf : function.surjective f) :

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

theorem lipschitz_with.to_right_inverse {α : Type u_1} {β : Type u_2} [pseudo_emetric_space α] [pseudo_emetric_space β] {K : ℝ≥0} {f : α → β} (hf : lipschitz_with K f) {g : β → α} (hg : function.right_inverse g f) :