statement stringlengths 1 2.88k | proof stringlengths 0 13.9k | type stringclasses 10
values | symbolic_name stringlengths 1 131 | library stringclasses 417
values | filename stringlengths 17 80 | imports listlengths 0 16 | deps listlengths 0 64 | docstring stringlengths 0 10.2k | source_url stringclasses 1
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lipschitz_with_iff_dist_le_mul [pseudo_metric_space α] [pseudo_metric_space β] {K : ℝ≥0}
{f : α → β} : lipschitz_with K f ↔ ∀ x y, dist (f x) (f y) ≤ K * dist x y | by { simp only [lipschitz_with, edist_nndist, dist_nndist], norm_cast } | lemma | lipschitz_with_iff_dist_le_mul | topology.metric_space | src/topology/metric_space/lipschitz.lean | [
"logic.function.iterate",
"data.set.intervals.proj_Icc",
"topology.algebra.order.field",
"topology.metric_space.basic",
"topology.bornology.hom"
] | [
"dist_nndist",
"edist_nndist",
"lipschitz_with",
"pseudo_metric_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lipschitz_on_with [pseudo_emetric_space α] [pseudo_emetric_space β] (K : ℝ≥0) (f : α → β)
(s : set α) | ∀ ⦃x⦄ (hx : x ∈ s) ⦃y⦄ (hy : y ∈ s), edist (f x) (f y) ≤ K * edist x y | def | lipschitz_on_with | topology.metric_space | src/topology/metric_space/lipschitz.lean | [
"logic.function.iterate",
"data.set.intervals.proj_Icc",
"topology.algebra.order.field",
"topology.metric_space.basic",
"topology.bornology.hom"
] | [
"pseudo_emetric_space"
] | A function `f` is Lipschitz continuous with constant `K ≥ 0` on `s` if for all `x, y` in `s`
we have `dist (f x) (f y) ≤ K * dist x y` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lipschitz_on_with_empty [pseudo_emetric_space α] [pseudo_emetric_space β] (K : ℝ≥0)
(f : α → β) : lipschitz_on_with K f ∅ | λ x x_in y y_in, false.elim x_in | lemma | lipschitz_on_with_empty | topology.metric_space | src/topology/metric_space/lipschitz.lean | [
"logic.function.iterate",
"data.set.intervals.proj_Icc",
"topology.algebra.order.field",
"topology.metric_space.basic",
"topology.bornology.hom"
] | [
"lipschitz_on_with",
"pseudo_emetric_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lipschitz_on_with.mono [pseudo_emetric_space α] [pseudo_emetric_space β] {K : ℝ≥0}
{s t : set α} {f : α → β} (hf : lipschitz_on_with K f t) (h : s ⊆ t) : lipschitz_on_with K f s | λ x x_in y y_in, hf (h x_in) (h y_in) | lemma | lipschitz_on_with.mono | topology.metric_space | src/topology/metric_space/lipschitz.lean | [
"logic.function.iterate",
"data.set.intervals.proj_Icc",
"topology.algebra.order.field",
"topology.metric_space.basic",
"topology.bornology.hom"
] | [
"lipschitz_on_with",
"pseudo_emetric_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lipschitz_on_with_iff_dist_le_mul [pseudo_metric_space α] [pseudo_metric_space β] {K : ℝ≥0}
{s : set α} {f : α → β} :
lipschitz_on_with K f s ↔ ∀ (x ∈ s) (y ∈ s), dist (f x) (f y) ≤ K * dist x y | by { simp only [lipschitz_on_with, edist_nndist, dist_nndist], norm_cast } | lemma | lipschitz_on_with_iff_dist_le_mul | topology.metric_space | src/topology/metric_space/lipschitz.lean | [
"logic.function.iterate",
"data.set.intervals.proj_Icc",
"topology.algebra.order.field",
"topology.metric_space.basic",
"topology.bornology.hom"
] | [
"dist_nndist",
"edist_nndist",
"lipschitz_on_with",
"pseudo_metric_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lipschitz_on_univ [pseudo_emetric_space α] [pseudo_emetric_space β] {K : ℝ≥0}
{f : α → β} : lipschitz_on_with K f univ ↔ lipschitz_with K f | by simp [lipschitz_on_with, lipschitz_with] | lemma | lipschitz_on_univ | topology.metric_space | src/topology/metric_space/lipschitz.lean | [
"logic.function.iterate",
"data.set.intervals.proj_Icc",
"topology.algebra.order.field",
"topology.metric_space.basic",
"topology.bornology.hom"
] | [
"lipschitz_on_with",
"lipschitz_with",
"pseudo_emetric_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lipschitz_on_with_iff_restrict [pseudo_emetric_space α] [pseudo_emetric_space β] {K : ℝ≥0}
{f : α → β} {s : set α} : lipschitz_on_with K f s ↔ lipschitz_with K (s.restrict f) | by simp only [lipschitz_on_with, lipschitz_with, set_coe.forall', restrict, subtype.edist_eq] | lemma | lipschitz_on_with_iff_restrict | topology.metric_space | src/topology/metric_space/lipschitz.lean | [
"logic.function.iterate",
"data.set.intervals.proj_Icc",
"topology.algebra.order.field",
"topology.metric_space.basic",
"topology.bornology.hom"
] | [
"lipschitz_on_with",
"lipschitz_with",
"pseudo_emetric_space",
"set_coe.forall'",
"subtype.edist_eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
maps_to.lipschitz_on_with_iff_restrict [pseudo_emetric_space α] [pseudo_emetric_space β]
{K : ℝ≥0} {f : α → β} {s : set α} {t : set β} (h : maps_to f s t) :
lipschitz_on_with K f s ↔ lipschitz_with K (h.restrict f s t) | lipschitz_on_with_iff_restrict | lemma | maps_to.lipschitz_on_with_iff_restrict | topology.metric_space | src/topology/metric_space/lipschitz.lean | [
"logic.function.iterate",
"data.set.intervals.proj_Icc",
"topology.algebra.order.field",
"topology.metric_space.basic",
"topology.bornology.hom"
] | [
"lipschitz_on_with",
"lipschitz_on_with_iff_restrict",
"lipschitz_with",
"pseudo_emetric_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lipschitz_on_with (h : lipschitz_with K f) (s : set α) : lipschitz_on_with K f s | λ x _ y _, h x y | lemma | lipschitz_with.lipschitz_on_with | topology.metric_space | src/topology/metric_space/lipschitz.lean | [
"logic.function.iterate",
"data.set.intervals.proj_Icc",
"topology.algebra.order.field",
"topology.metric_space.basic",
"topology.bornology.hom"
] | [
"lipschitz_on_with",
"lipschitz_with"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
edist_le_mul (h : lipschitz_with K f) (x y : α) : edist (f x) (f y) ≤ K * edist x y | h x y | lemma | lipschitz_with.edist_le_mul | topology.metric_space | src/topology/metric_space/lipschitz.lean | [
"logic.function.iterate",
"data.set.intervals.proj_Icc",
"topology.algebra.order.field",
"topology.metric_space.basic",
"topology.bornology.hom"
] | [
"lipschitz_with"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
edist_le_mul_of_le (h : lipschitz_with K f) (hr : edist x y ≤ r) :
edist (f x) (f y) ≤ K * r | (h x y).trans $ ennreal.mul_left_mono hr | lemma | lipschitz_with.edist_le_mul_of_le | topology.metric_space | src/topology/metric_space/lipschitz.lean | [
"logic.function.iterate",
"data.set.intervals.proj_Icc",
"topology.algebra.order.field",
"topology.metric_space.basic",
"topology.bornology.hom"
] | [
"ennreal.mul_left_mono",
"lipschitz_with"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
edist_lt_mul_of_lt (h : lipschitz_with K f) (hK : K ≠ 0) (hr : edist x y < r) :
edist (f x) (f y) < K * r | (h x y).trans_lt $ (ennreal.mul_lt_mul_left (ennreal.coe_ne_zero.2 hK) ennreal.coe_ne_top).2 hr | lemma | lipschitz_with.edist_lt_mul_of_lt | topology.metric_space | src/topology/metric_space/lipschitz.lean | [
"logic.function.iterate",
"data.set.intervals.proj_Icc",
"topology.algebra.order.field",
"topology.metric_space.basic",
"topology.bornology.hom"
] | [
"ennreal.coe_ne_top",
"ennreal.mul_lt_mul_left",
"lipschitz_with"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
maps_to_emetric_closed_ball (h : lipschitz_with K f) (x : α) (r : ℝ≥0∞) :
maps_to f (closed_ball x r) (closed_ball (f x) (K * r)) | λ y hy, h.edist_le_mul_of_le hy | lemma | lipschitz_with.maps_to_emetric_closed_ball | topology.metric_space | src/topology/metric_space/lipschitz.lean | [
"logic.function.iterate",
"data.set.intervals.proj_Icc",
"topology.algebra.order.field",
"topology.metric_space.basic",
"topology.bornology.hom"
] | [
"lipschitz_with"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
maps_to_emetric_ball (h : lipschitz_with K f) (hK : K ≠ 0) (x : α) (r : ℝ≥0∞) :
maps_to f (ball x r) (ball (f x) (K * r)) | λ y hy, h.edist_lt_mul_of_lt hK hy | lemma | lipschitz_with.maps_to_emetric_ball | topology.metric_space | src/topology/metric_space/lipschitz.lean | [
"logic.function.iterate",
"data.set.intervals.proj_Icc",
"topology.algebra.order.field",
"topology.metric_space.basic",
"topology.bornology.hom"
] | [
"lipschitz_with"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
edist_lt_top (hf : lipschitz_with K f) {x y : α} (h : edist x y ≠ ⊤) :
edist (f x) (f y) < ⊤ | (hf x y).trans_lt $ ennreal.mul_lt_top ennreal.coe_ne_top h | lemma | lipschitz_with.edist_lt_top | topology.metric_space | src/topology/metric_space/lipschitz.lean | [
"logic.function.iterate",
"data.set.intervals.proj_Icc",
"topology.algebra.order.field",
"topology.metric_space.basic",
"topology.bornology.hom"
] | [
"edist_lt_top",
"ennreal.coe_ne_top",
"ennreal.mul_lt_top",
"lipschitz_with"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_edist_le (h : lipschitz_with K f) (x y : α) :
(K⁻¹ : ℝ≥0∞) * edist (f x) (f y) ≤ edist x y | begin
rw [mul_comm, ← div_eq_mul_inv],
exact ennreal.div_le_of_le_mul' (h x y)
end | lemma | lipschitz_with.mul_edist_le | topology.metric_space | src/topology/metric_space/lipschitz.lean | [
"logic.function.iterate",
"data.set.intervals.proj_Icc",
"topology.algebra.order.field",
"topology.metric_space.basic",
"topology.bornology.hom"
] | [
"div_eq_mul_inv",
"ennreal.div_le_of_le_mul'",
"lipschitz_with",
"mul_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_edist_le (h : ∀ x y, edist (f x) (f y) ≤ edist x y) :
lipschitz_with 1 f | λ x y, by simp only [ennreal.coe_one, one_mul, h] | lemma | lipschitz_with.of_edist_le | topology.metric_space | src/topology/metric_space/lipschitz.lean | [
"logic.function.iterate",
"data.set.intervals.proj_Icc",
"topology.algebra.order.field",
"topology.metric_space.basic",
"topology.bornology.hom"
] | [
"ennreal.coe_one",
"lipschitz_with",
"one_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
weaken (hf : lipschitz_with K f) {K' : ℝ≥0} (h : K ≤ K') :
lipschitz_with K' f | assume x y, le_trans (hf x y) $ ennreal.mul_right_mono (ennreal.coe_le_coe.2 h) | lemma | lipschitz_with.weaken | topology.metric_space | src/topology/metric_space/lipschitz.lean | [
"logic.function.iterate",
"data.set.intervals.proj_Icc",
"topology.algebra.order.field",
"topology.metric_space.basic",
"topology.bornology.hom"
] | [
"ennreal.mul_right_mono",
"lipschitz_with"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ediam_image_le (hf : lipschitz_with K f) (s : set α) :
emetric.diam (f '' s) ≤ K * emetric.diam s | begin
apply emetric.diam_le,
rintros _ ⟨x, hx, rfl⟩ _ ⟨y, hy, rfl⟩,
exact hf.edist_le_mul_of_le (emetric.edist_le_diam_of_mem hx hy)
end | lemma | lipschitz_with.ediam_image_le | topology.metric_space | src/topology/metric_space/lipschitz.lean | [
"logic.function.iterate",
"data.set.intervals.proj_Icc",
"topology.algebra.order.field",
"topology.metric_space.basic",
"topology.bornology.hom"
] | [
"emetric.diam",
"emetric.diam_le",
"emetric.edist_le_diam_of_mem",
"lipschitz_with"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
edist_lt_of_edist_lt_div (hf : lipschitz_with K f) {x y : α} {d : ℝ≥0∞}
(h : edist x y < d / K) : edist (f x) (f y) < d | calc edist (f x) (f y) ≤ K * edist x y : hf x y
... < d : ennreal.mul_lt_of_lt_div' h | lemma | lipschitz_with.edist_lt_of_edist_lt_div | topology.metric_space | src/topology/metric_space/lipschitz.lean | [
"logic.function.iterate",
"data.set.intervals.proj_Icc",
"topology.algebra.order.field",
"topology.metric_space.basic",
"topology.bornology.hom"
] | [
"ennreal.mul_lt_of_lt_div'",
"lipschitz_with"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
uniform_continuous (hf : lipschitz_with K f) :
uniform_continuous f | begin
refine emetric.uniform_continuous_iff.2 (λε εpos, _),
use [ε / K, ennreal.div_pos_iff.2 ⟨ne_of_gt εpos, ennreal.coe_ne_top⟩],
exact λ x y, hf.edist_lt_of_edist_lt_div
end | lemma | lipschitz_with.uniform_continuous | topology.metric_space | src/topology/metric_space/lipschitz.lean | [
"logic.function.iterate",
"data.set.intervals.proj_Icc",
"topology.algebra.order.field",
"topology.metric_space.basic",
"topology.bornology.hom"
] | [
"lipschitz_with",
"uniform_continuous"
] | A Lipschitz function is uniformly continuous | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuous (hf : lipschitz_with K f) :
continuous f | hf.uniform_continuous.continuous | lemma | lipschitz_with.continuous | topology.metric_space | src/topology/metric_space/lipschitz.lean | [
"logic.function.iterate",
"data.set.intervals.proj_Icc",
"topology.algebra.order.field",
"topology.metric_space.basic",
"topology.bornology.hom"
] | [
"continuous",
"lipschitz_with"
] | A Lipschitz function is continuous | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
const (b : β) : lipschitz_with 0 (λa:α, b) | assume x y, by simp only [edist_self, zero_le] | lemma | lipschitz_with.const | topology.metric_space | src/topology/metric_space/lipschitz.lean | [
"logic.function.iterate",
"data.set.intervals.proj_Icc",
"topology.algebra.order.field",
"topology.metric_space.basic",
"topology.bornology.hom"
] | [
"lipschitz_with"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
id : lipschitz_with 1 (@id α) | lipschitz_with.of_edist_le $ assume x y, le_rfl | lemma | lipschitz_with.id | topology.metric_space | src/topology/metric_space/lipschitz.lean | [
"logic.function.iterate",
"data.set.intervals.proj_Icc",
"topology.algebra.order.field",
"topology.metric_space.basic",
"topology.bornology.hom"
] | [
"le_rfl",
"lipschitz_with",
"lipschitz_with.of_edist_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
subtype_val (s : set α) : lipschitz_with 1 (subtype.val : s → α) | lipschitz_with.of_edist_le $ assume x y, le_rfl | lemma | lipschitz_with.subtype_val | topology.metric_space | src/topology/metric_space/lipschitz.lean | [
"logic.function.iterate",
"data.set.intervals.proj_Icc",
"topology.algebra.order.field",
"topology.metric_space.basic",
"topology.bornology.hom"
] | [
"le_rfl",
"lipschitz_with",
"lipschitz_with.of_edist_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
subtype_coe (s : set α) : lipschitz_with 1 (coe : s → α) | lipschitz_with.subtype_val s | lemma | lipschitz_with.subtype_coe | topology.metric_space | src/topology/metric_space/lipschitz.lean | [
"logic.function.iterate",
"data.set.intervals.proj_Icc",
"topology.algebra.order.field",
"topology.metric_space.basic",
"topology.bornology.hom"
] | [
"lipschitz_with",
"lipschitz_with.subtype_val"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
subtype_mk (hf : lipschitz_with K f) {p : β → Prop} (hp : ∀ x, p (f x)) :
lipschitz_with K (λ x, ⟨f x, hp x⟩ : α → {y // p y}) | hf | lemma | lipschitz_with.subtype_mk | topology.metric_space | src/topology/metric_space/lipschitz.lean | [
"logic.function.iterate",
"data.set.intervals.proj_Icc",
"topology.algebra.order.field",
"topology.metric_space.basic",
"topology.bornology.hom"
] | [
"lipschitz_with"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eval {α : ι → Type u} [Π i, pseudo_emetric_space (α i)] [fintype ι] (i : ι) :
lipschitz_with 1 (function.eval i : (Π i, α i) → α i) | lipschitz_with.of_edist_le $ λ f g, by convert edist_le_pi_edist f g i | lemma | lipschitz_with.eval | topology.metric_space | src/topology/metric_space/lipschitz.lean | [
"logic.function.iterate",
"data.set.intervals.proj_Icc",
"topology.algebra.order.field",
"topology.metric_space.basic",
"topology.bornology.hom"
] | [
"edist_le_pi_edist",
"fintype",
"function.eval",
"lipschitz_with",
"lipschitz_with.of_edist_le",
"pseudo_emetric_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
restrict (hf : lipschitz_with K f) (s : set α) :
lipschitz_with K (s.restrict f) | λ x y, hf x y | lemma | lipschitz_with.restrict | topology.metric_space | src/topology/metric_space/lipschitz.lean | [
"logic.function.iterate",
"data.set.intervals.proj_Icc",
"topology.algebra.order.field",
"topology.metric_space.basic",
"topology.bornology.hom"
] | [
"lipschitz_with"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp {Kf Kg : ℝ≥0} {f : β → γ} {g : α → β}
(hf : lipschitz_with Kf f) (hg : lipschitz_with Kg g) : lipschitz_with (Kf * Kg) (f ∘ g) | assume x y,
calc edist (f (g x)) (f (g y)) ≤ Kf * edist (g x) (g y) : hf _ _
... ≤ Kf * (Kg * edist x y) : ennreal.mul_left_mono (hg _ _)
... = (Kf * Kg : ℝ≥0) * edist x y : by rw [← mul_assoc, ennreal.coe_mul] | lemma | lipschitz_with.comp | topology.metric_space | src/topology/metric_space/lipschitz.lean | [
"logic.function.iterate",
"data.set.intervals.proj_Icc",
"topology.algebra.order.field",
"topology.metric_space.basic",
"topology.bornology.hom"
] | [
"ennreal.coe_mul",
"ennreal.mul_left_mono",
"lipschitz_with",
"mul_assoc"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_lipschitz_on_with {Kf Kg : ℝ≥0} {f : β → γ} {g : α → β} {s : set α}
(hf : lipschitz_with Kf f) (hg : lipschitz_on_with Kg g s) :
lipschitz_on_with (Kf * Kg) (f ∘ g) s | lipschitz_on_with_iff_restrict.mpr $ hf.comp hg.to_restrict | lemma | lipschitz_with.comp_lipschitz_on_with | topology.metric_space | src/topology/metric_space/lipschitz.lean | [
"logic.function.iterate",
"data.set.intervals.proj_Icc",
"topology.algebra.order.field",
"topology.metric_space.basic",
"topology.bornology.hom"
] | [
"lipschitz_on_with",
"lipschitz_with"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_fst : lipschitz_with 1 (@prod.fst α β) | lipschitz_with.of_edist_le $ assume x y, le_max_left _ _ | lemma | lipschitz_with.prod_fst | topology.metric_space | src/topology/metric_space/lipschitz.lean | [
"logic.function.iterate",
"data.set.intervals.proj_Icc",
"topology.algebra.order.field",
"topology.metric_space.basic",
"topology.bornology.hom"
] | [
"lipschitz_with",
"lipschitz_with.of_edist_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_snd : lipschitz_with 1 (@prod.snd α β) | lipschitz_with.of_edist_le $ assume x y, le_max_right _ _ | lemma | lipschitz_with.prod_snd | topology.metric_space | src/topology/metric_space/lipschitz.lean | [
"logic.function.iterate",
"data.set.intervals.proj_Icc",
"topology.algebra.order.field",
"topology.metric_space.basic",
"topology.bornology.hom"
] | [
"lipschitz_with",
"lipschitz_with.of_edist_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod {f : α → β} {Kf : ℝ≥0} (hf : lipschitz_with Kf f)
{g : α → γ} {Kg : ℝ≥0} (hg : lipschitz_with Kg g) :
lipschitz_with (max Kf Kg) (λ x, (f x, g x)) | begin
assume x y,
rw [ennreal.coe_mono.map_max, prod.edist_eq, ennreal.max_mul],
exact max_le_max (hf x y) (hg x y)
end | lemma | lipschitz_with.prod | topology.metric_space | src/topology/metric_space/lipschitz.lean | [
"logic.function.iterate",
"data.set.intervals.proj_Icc",
"topology.algebra.order.field",
"topology.metric_space.basic",
"topology.bornology.hom"
] | [
"ennreal.max_mul",
"lipschitz_with",
"max_le_max",
"prod.edist_eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_mk_left (a : α) : lipschitz_with 1 (prod.mk a : β → α × β) | by simpa only [max_eq_right zero_le_one] using (lipschitz_with.const a).prod lipschitz_with.id | lemma | lipschitz_with.prod_mk_left | topology.metric_space | src/topology/metric_space/lipschitz.lean | [
"logic.function.iterate",
"data.set.intervals.proj_Icc",
"topology.algebra.order.field",
"topology.metric_space.basic",
"topology.bornology.hom"
] | [
"lipschitz_with",
"lipschitz_with.const",
"lipschitz_with.id",
"zero_le_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_mk_right (b : β) : lipschitz_with 1 (λ a : α, (a, b)) | by simpa only [max_eq_left zero_le_one] using lipschitz_with.id.prod (lipschitz_with.const b) | lemma | lipschitz_with.prod_mk_right | topology.metric_space | src/topology/metric_space/lipschitz.lean | [
"logic.function.iterate",
"data.set.intervals.proj_Icc",
"topology.algebra.order.field",
"topology.metric_space.basic",
"topology.bornology.hom"
] | [
"lipschitz_with",
"lipschitz_with.const",
"zero_le_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
uncurry {f : α → β → γ} {Kα Kβ : ℝ≥0} (hα : ∀ b, lipschitz_with Kα (λ a, f a b))
(hβ : ∀ a, lipschitz_with Kβ (f a)) :
lipschitz_with (Kα + Kβ) (function.uncurry f) | begin
rintros ⟨a₁, b₁⟩ ⟨a₂, b₂⟩,
simp only [function.uncurry, ennreal.coe_add, add_mul],
apply le_trans (edist_triangle _ (f a₂ b₁) _),
exact add_le_add (le_trans (hα _ _ _) $ ennreal.mul_left_mono $ le_max_left _ _)
(le_trans (hβ _ _ _) $ ennreal.mul_left_mono $ le_max_right _ _)
end | lemma | lipschitz_with.uncurry | topology.metric_space | src/topology/metric_space/lipschitz.lean | [
"logic.function.iterate",
"data.set.intervals.proj_Icc",
"topology.algebra.order.field",
"topology.metric_space.basic",
"topology.bornology.hom"
] | [
"ennreal.coe_add",
"ennreal.mul_left_mono",
"lipschitz_with"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
iterate {f : α → α} (hf : lipschitz_with K f) :
∀n, lipschitz_with (K ^ n) (f^[n]) | | 0 := by simpa only [pow_zero] using lipschitz_with.id
| (n + 1) := by rw [pow_succ']; exact (iterate n).comp hf | lemma | lipschitz_with.iterate | topology.metric_space | src/topology/metric_space/lipschitz.lean | [
"logic.function.iterate",
"data.set.intervals.proj_Icc",
"topology.algebra.order.field",
"topology.metric_space.basic",
"topology.bornology.hom"
] | [
"lipschitz_with",
"lipschitz_with.id",
"pow_succ'",
"pow_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
edist_iterate_succ_le_geometric {f : α → α} (hf : lipschitz_with K f) (x n) :
edist (f^[n] x) (f^[n + 1] x) ≤ edist x (f x) * K ^ n | begin
rw [iterate_succ, mul_comm],
simpa only [ennreal.coe_pow] using (hf.iterate n) x (f x)
end | lemma | lipschitz_with.edist_iterate_succ_le_geometric | topology.metric_space | src/topology/metric_space/lipschitz.lean | [
"logic.function.iterate",
"data.set.intervals.proj_Icc",
"topology.algebra.order.field",
"topology.metric_space.basic",
"topology.bornology.hom"
] | [
"ennreal.coe_pow",
"lipschitz_with",
"mul_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul {f g : function.End α} {Kf Kg} (hf : lipschitz_with Kf f)
(hg : lipschitz_with Kg g) :
lipschitz_with (Kf * Kg) (f * g : function.End α) | hf.comp hg | lemma | lipschitz_with.mul | topology.metric_space | src/topology/metric_space/lipschitz.lean | [
"logic.function.iterate",
"data.set.intervals.proj_Icc",
"topology.algebra.order.field",
"topology.metric_space.basic",
"topology.bornology.hom"
] | [
"function.End",
"lipschitz_with"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
list_prod (f : ι → function.End α) (K : ι → ℝ≥0)
(h : ∀ i, lipschitz_with (K i) (f i)) :
∀ l : list ι, lipschitz_with (l.map K).prod (l.map f).prod | | [] := by simpa using lipschitz_with.id
| (i :: l) := by { simp only [list.map_cons, list.prod_cons], exact (h i).mul (list_prod l) } | lemma | lipschitz_with.list_prod | topology.metric_space | src/topology/metric_space/lipschitz.lean | [
"logic.function.iterate",
"data.set.intervals.proj_Icc",
"topology.algebra.order.field",
"topology.metric_space.basic",
"topology.bornology.hom"
] | [
"function.End",
"lipschitz_with",
"lipschitz_with.id",
"list.prod_cons"
] | The product of a list of Lipschitz continuous endomorphisms is a Lipschitz continuous
endomorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pow {f : function.End α} {K} (h : lipschitz_with K f) :
∀ n : ℕ, lipschitz_with (K^n) (f^n : function.End α) | | 0 := by simpa only [pow_zero] using lipschitz_with.id
| (n + 1) := by { rw [pow_succ, pow_succ], exact h.mul (pow n) } | lemma | lipschitz_with.pow | topology.metric_space | src/topology/metric_space/lipschitz.lean | [
"logic.function.iterate",
"data.set.intervals.proj_Icc",
"topology.algebra.order.field",
"topology.metric_space.basic",
"topology.bornology.hom"
] | [
"function.End",
"lipschitz_with",
"lipschitz_with.id",
"pow_succ",
"pow_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_dist_le' {K : ℝ} (h : ∀ x y, dist (f x) (f y) ≤ K * dist x y) :
lipschitz_with (real.to_nnreal K) f | of_dist_le_mul $ λ x y, le_trans (h x y) $
mul_le_mul_of_nonneg_right (real.le_coe_to_nnreal K) dist_nonneg | lemma | lipschitz_with.of_dist_le' | topology.metric_space | src/topology/metric_space/lipschitz.lean | [
"logic.function.iterate",
"data.set.intervals.proj_Icc",
"topology.algebra.order.field",
"topology.metric_space.basic",
"topology.bornology.hom"
] | [
"dist_nonneg",
"lipschitz_with",
"mul_le_mul_of_nonneg_right",
"real.le_coe_to_nnreal",
"real.to_nnreal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_one (h : ∀ x y, dist (f x) (f y) ≤ dist x y) :
lipschitz_with 1 f | of_dist_le_mul $ by simpa only [nnreal.coe_one, one_mul] using h | lemma | lipschitz_with.mk_one | topology.metric_space | src/topology/metric_space/lipschitz.lean | [
"logic.function.iterate",
"data.set.intervals.proj_Icc",
"topology.algebra.order.field",
"topology.metric_space.basic",
"topology.bornology.hom"
] | [
"lipschitz_with",
"nnreal.coe_one",
"one_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_le_add_mul' {f : α → ℝ} (K : ℝ) (h : ∀x y, f x ≤ f y + K * dist x y) :
lipschitz_with (real.to_nnreal K) f | have I : ∀ x y, f x - f y ≤ K * dist x y,
from assume x y, sub_le_iff_le_add'.2 (h x y),
lipschitz_with.of_dist_le' $
assume x y,
abs_sub_le_iff.2 ⟨I x y, dist_comm y x ▸ I y x⟩ | lemma | lipschitz_with.of_le_add_mul' | topology.metric_space | src/topology/metric_space/lipschitz.lean | [
"logic.function.iterate",
"data.set.intervals.proj_Icc",
"topology.algebra.order.field",
"topology.metric_space.basic",
"topology.bornology.hom"
] | [
"dist_comm",
"lipschitz_with",
"lipschitz_with.of_dist_le'",
"real.to_nnreal"
] | For functions to `ℝ`, it suffices to prove `f x ≤ f y + K * dist x y`; this version
doesn't assume `0≤K`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of_le_add_mul {f : α → ℝ} (K : ℝ≥0) (h : ∀x y, f x ≤ f y + K * dist x y) :
lipschitz_with K f | by simpa only [real.to_nnreal_coe] using lipschitz_with.of_le_add_mul' K h | lemma | lipschitz_with.of_le_add_mul | topology.metric_space | src/topology/metric_space/lipschitz.lean | [
"logic.function.iterate",
"data.set.intervals.proj_Icc",
"topology.algebra.order.field",
"topology.metric_space.basic",
"topology.bornology.hom"
] | [
"lipschitz_with",
"lipschitz_with.of_le_add_mul'",
"real.to_nnreal_coe"
] | For functions to `ℝ`, it suffices to prove `f x ≤ f y + K * dist x y`; this version
assumes `0≤K`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of_le_add {f : α → ℝ} (h : ∀ x y, f x ≤ f y + dist x y) :
lipschitz_with 1 f | lipschitz_with.of_le_add_mul 1 $ by simpa only [nnreal.coe_one, one_mul] | lemma | lipschitz_with.of_le_add | topology.metric_space | src/topology/metric_space/lipschitz.lean | [
"logic.function.iterate",
"data.set.intervals.proj_Icc",
"topology.algebra.order.field",
"topology.metric_space.basic",
"topology.bornology.hom"
] | [
"lipschitz_with",
"lipschitz_with.of_le_add_mul",
"nnreal.coe_one",
"one_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_add_mul {f : α → ℝ} {K : ℝ≥0} (h : lipschitz_with K f) (x y) :
f x ≤ f y + K * dist x y | sub_le_iff_le_add'.1 $ le_trans (le_abs_self _) $ h.dist_le_mul x y | lemma | lipschitz_with.le_add_mul | topology.metric_space | src/topology/metric_space/lipschitz.lean | [
"logic.function.iterate",
"data.set.intervals.proj_Icc",
"topology.algebra.order.field",
"topology.metric_space.basic",
"topology.bornology.hom"
] | [
"le_abs_self",
"lipschitz_with"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
iff_le_add_mul {f : α → ℝ} {K : ℝ≥0} :
lipschitz_with K f ↔ ∀ x y, f x ≤ f y + K * dist x y | ⟨lipschitz_with.le_add_mul, lipschitz_with.of_le_add_mul K⟩ | lemma | lipschitz_with.iff_le_add_mul | topology.metric_space | src/topology/metric_space/lipschitz.lean | [
"logic.function.iterate",
"data.set.intervals.proj_Icc",
"topology.algebra.order.field",
"topology.metric_space.basic",
"topology.bornology.hom"
] | [
"lipschitz_with",
"lipschitz_with.of_le_add_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nndist_le (hf : lipschitz_with K f) (x y : α) :
nndist (f x) (f y) ≤ K * nndist x y | hf.dist_le_mul x y | lemma | lipschitz_with.nndist_le | topology.metric_space | src/topology/metric_space/lipschitz.lean | [
"logic.function.iterate",
"data.set.intervals.proj_Icc",
"topology.algebra.order.field",
"topology.metric_space.basic",
"topology.bornology.hom"
] | [
"lipschitz_with"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dist_le_mul_of_le (hf : lipschitz_with K f) (hr : dist x y ≤ r) :
dist (f x) (f y) ≤ K * r | (hf.dist_le_mul x y).trans $ mul_le_mul_of_nonneg_left hr K.coe_nonneg | lemma | lipschitz_with.dist_le_mul_of_le | topology.metric_space | src/topology/metric_space/lipschitz.lean | [
"logic.function.iterate",
"data.set.intervals.proj_Icc",
"topology.algebra.order.field",
"topology.metric_space.basic",
"topology.bornology.hom"
] | [
"lipschitz_with",
"mul_le_mul_of_nonneg_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
maps_to_closed_ball (hf : lipschitz_with K f) (x : α) (r : ℝ) :
maps_to f (metric.closed_ball x r) (metric.closed_ball (f x) (K * r)) | λ y hy, hf.dist_le_mul_of_le hy | lemma | lipschitz_with.maps_to_closed_ball | topology.metric_space | src/topology/metric_space/lipschitz.lean | [
"logic.function.iterate",
"data.set.intervals.proj_Icc",
"topology.algebra.order.field",
"topology.metric_space.basic",
"topology.bornology.hom"
] | [
"lipschitz_with",
"metric.closed_ball"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dist_lt_mul_of_lt (hf : lipschitz_with K f) (hK : K ≠ 0) (hr : dist x y < r) :
dist (f x) (f y) < K * r | (hf.dist_le_mul x y).trans_lt $ (mul_lt_mul_left $ nnreal.coe_pos.2 hK.bot_lt).2 hr | lemma | lipschitz_with.dist_lt_mul_of_lt | topology.metric_space | src/topology/metric_space/lipschitz.lean | [
"logic.function.iterate",
"data.set.intervals.proj_Icc",
"topology.algebra.order.field",
"topology.metric_space.basic",
"topology.bornology.hom"
] | [
"lipschitz_with",
"mul_lt_mul_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
maps_to_ball (hf : lipschitz_with K f) (hK : K ≠ 0) (x : α) (r : ℝ) :
maps_to f (metric.ball x r) (metric.ball (f x) (K * r)) | λ y hy, hf.dist_lt_mul_of_lt hK hy | lemma | lipschitz_with.maps_to_ball | topology.metric_space | src/topology/metric_space/lipschitz.lean | [
"logic.function.iterate",
"data.set.intervals.proj_Icc",
"topology.algebra.order.field",
"topology.metric_space.basic",
"topology.bornology.hom"
] | [
"lipschitz_with",
"metric.ball"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_locally_bounded_map (f : α → β) (hf : lipschitz_with K f) :
locally_bounded_map α β | locally_bounded_map.of_map_bounded f $ λ s hs, let ⟨C, hC⟩ := metric.is_bounded_iff.1 hs
in metric.is_bounded_iff.2 ⟨K * C, ball_image_iff.2 $ λ x hx, ball_image_iff.2 $ λ y hy,
hf.dist_le_mul_of_le (hC hx hy)⟩ | def | lipschitz_with.to_locally_bounded_map | topology.metric_space | src/topology/metric_space/lipschitz.lean | [
"logic.function.iterate",
"data.set.intervals.proj_Icc",
"topology.algebra.order.field",
"topology.metric_space.basic",
"topology.bornology.hom"
] | [
"lipschitz_with",
"locally_bounded_map",
"locally_bounded_map.of_map_bounded"
] | A Lipschitz continuous map is a locally bounded map. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_to_locally_bounded_map (hf : lipschitz_with K f) :
⇑(hf.to_locally_bounded_map f) = f | rfl | lemma | lipschitz_with.coe_to_locally_bounded_map | topology.metric_space | src/topology/metric_space/lipschitz.lean | [
"logic.function.iterate",
"data.set.intervals.proj_Icc",
"topology.algebra.order.field",
"topology.metric_space.basic",
"topology.bornology.hom"
] | [
"lipschitz_with"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comap_cobounded_le (hf : lipschitz_with K f) :
comap f (bornology.cobounded β) ≤ bornology.cobounded α | (hf.to_locally_bounded_map f).2 | lemma | lipschitz_with.comap_cobounded_le | topology.metric_space | src/topology/metric_space/lipschitz.lean | [
"logic.function.iterate",
"data.set.intervals.proj_Icc",
"topology.algebra.order.field",
"topology.metric_space.basic",
"topology.bornology.hom"
] | [
"lipschitz_with"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bounded_image (hf : lipschitz_with K f) {s : set α} (hs : metric.bounded s) :
metric.bounded (f '' s) | metric.bounded_iff_ediam_ne_top.2 $ ne_top_of_le_ne_top
(ennreal.mul_ne_top ennreal.coe_ne_top hs.ediam_ne_top) (hf.ediam_image_le s) | lemma | lipschitz_with.bounded_image | topology.metric_space | src/topology/metric_space/lipschitz.lean | [
"logic.function.iterate",
"data.set.intervals.proj_Icc",
"topology.algebra.order.field",
"topology.metric_space.basic",
"topology.bornology.hom"
] | [
"ennreal.coe_ne_top",
"ennreal.mul_ne_top",
"lipschitz_with",
"metric.bounded",
"ne_top_of_le_ne_top"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
diam_image_le (hf : lipschitz_with K f) (s : set α) (hs : metric.bounded s) :
metric.diam (f '' s) ≤ K * metric.diam s | metric.diam_le_of_forall_dist_le (mul_nonneg K.coe_nonneg metric.diam_nonneg) $
ball_image_iff.2 $ λ x hx, ball_image_iff.2 $ λ y hy, hf.dist_le_mul_of_le $
metric.dist_le_diam_of_mem hs hx hy | lemma | lipschitz_with.diam_image_le | topology.metric_space | src/topology/metric_space/lipschitz.lean | [
"logic.function.iterate",
"data.set.intervals.proj_Icc",
"topology.algebra.order.field",
"topology.metric_space.basic",
"topology.bornology.hom"
] | [
"lipschitz_with",
"metric.bounded",
"metric.diam",
"metric.diam_le_of_forall_dist_le",
"metric.diam_nonneg",
"metric.dist_le_diam_of_mem"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dist_left (y : α) : lipschitz_with 1 (λ x, dist x y) | lipschitz_with.of_le_add $ assume x z, by { rw [add_comm], apply dist_triangle } | lemma | lipschitz_with.dist_left | topology.metric_space | src/topology/metric_space/lipschitz.lean | [
"logic.function.iterate",
"data.set.intervals.proj_Icc",
"topology.algebra.order.field",
"topology.metric_space.basic",
"topology.bornology.hom"
] | [
"dist_triangle",
"lipschitz_with",
"lipschitz_with.of_le_add"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dist_right (x : α) : lipschitz_with 1 (dist x) | lipschitz_with.of_le_add $ assume y z, dist_triangle_right _ _ _ | lemma | lipschitz_with.dist_right | topology.metric_space | src/topology/metric_space/lipschitz.lean | [
"logic.function.iterate",
"data.set.intervals.proj_Icc",
"topology.algebra.order.field",
"topology.metric_space.basic",
"topology.bornology.hom"
] | [
"dist_triangle_right",
"lipschitz_with",
"lipschitz_with.of_le_add"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dist : lipschitz_with 2 (function.uncurry $ @dist α _) | lipschitz_with.uncurry lipschitz_with.dist_left lipschitz_with.dist_right | lemma | lipschitz_with.dist | topology.metric_space | src/topology/metric_space/lipschitz.lean | [
"logic.function.iterate",
"data.set.intervals.proj_Icc",
"topology.algebra.order.field",
"topology.metric_space.basic",
"topology.bornology.hom"
] | [
"lipschitz_with",
"lipschitz_with.dist_left",
"lipschitz_with.dist_right",
"lipschitz_with.uncurry"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dist_iterate_succ_le_geometric {f : α → α} (hf : lipschitz_with K f) (x n) :
dist (f^[n] x) (f^[n + 1] x) ≤ dist x (f x) * K ^ n | begin
rw [iterate_succ, mul_comm],
simpa only [nnreal.coe_pow] using (hf.iterate n).dist_le_mul x (f x)
end | lemma | lipschitz_with.dist_iterate_succ_le_geometric | topology.metric_space | src/topology/metric_space/lipschitz.lean | [
"logic.function.iterate",
"data.set.intervals.proj_Icc",
"topology.algebra.order.field",
"topology.metric_space.basic",
"topology.bornology.hom"
] | [
"lipschitz_with",
"mul_comm",
"nnreal.coe_pow"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.lipschitz_with_max : lipschitz_with 1 (λ p : ℝ × ℝ, max p.1 p.2) | lipschitz_with.of_le_add $ λ p₁ p₂, sub_le_iff_le_add'.1 $
(le_abs_self _).trans (abs_max_sub_max_le_max _ _ _ _) | lemma | lipschitz_with_max | topology.metric_space | src/topology/metric_space/lipschitz.lean | [
"logic.function.iterate",
"data.set.intervals.proj_Icc",
"topology.algebra.order.field",
"topology.metric_space.basic",
"topology.bornology.hom"
] | [
"abs_max_sub_max_le_max",
"le_abs_self",
"lipschitz_with",
"lipschitz_with.of_le_add"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.lipschitz_with_min : lipschitz_with 1 (λ p : ℝ × ℝ, min p.1 p.2) | lipschitz_with.of_le_add $ λ p₁ p₂, sub_le_iff_le_add'.1 $
(le_abs_self _).trans (abs_min_sub_min_le_max _ _ _ _) | lemma | lipschitz_with_min | topology.metric_space | src/topology/metric_space/lipschitz.lean | [
"logic.function.iterate",
"data.set.intervals.proj_Icc",
"topology.algebra.order.field",
"topology.metric_space.basic",
"topology.bornology.hom"
] | [
"abs_min_sub_min_le_max",
"le_abs_self",
"lipschitz_with",
"lipschitz_with.of_le_add"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
max (hf : lipschitz_with Kf f) (hg : lipschitz_with Kg g) :
lipschitz_with (max Kf Kg) (λ x, max (f x) (g x)) | by simpa only [(∘), one_mul] using lipschitz_with_max.comp (hf.prod hg) | lemma | lipschitz_with.max | topology.metric_space | src/topology/metric_space/lipschitz.lean | [
"logic.function.iterate",
"data.set.intervals.proj_Icc",
"topology.algebra.order.field",
"topology.metric_space.basic",
"topology.bornology.hom"
] | [
"lipschitz_with",
"one_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
min (hf : lipschitz_with Kf f) (hg : lipschitz_with Kg g) :
lipschitz_with (max Kf Kg) (λ x, min (f x) (g x)) | by simpa only [(∘), one_mul] using lipschitz_with_min.comp (hf.prod hg) | lemma | lipschitz_with.min | topology.metric_space | src/topology/metric_space/lipschitz.lean | [
"logic.function.iterate",
"data.set.intervals.proj_Icc",
"topology.algebra.order.field",
"topology.metric_space.basic",
"topology.bornology.hom"
] | [
"lipschitz_with",
"one_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
max_const (hf : lipschitz_with Kf f) (a : ℝ) : lipschitz_with Kf (λ x, max (f x) a) | by simpa only [max_eq_left (zero_le Kf)] using hf.max (lipschitz_with.const a) | lemma | lipschitz_with.max_const | topology.metric_space | src/topology/metric_space/lipschitz.lean | [
"logic.function.iterate",
"data.set.intervals.proj_Icc",
"topology.algebra.order.field",
"topology.metric_space.basic",
"topology.bornology.hom"
] | [
"lipschitz_with",
"lipschitz_with.const"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
const_max (hf : lipschitz_with Kf f) (a : ℝ) : lipschitz_with Kf (λ x, max a (f x)) | by simpa only [max_comm] using hf.max_const a | lemma | lipschitz_with.const_max | topology.metric_space | src/topology/metric_space/lipschitz.lean | [
"logic.function.iterate",
"data.set.intervals.proj_Icc",
"topology.algebra.order.field",
"topology.metric_space.basic",
"topology.bornology.hom"
] | [
"lipschitz_with"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
min_const (hf : lipschitz_with Kf f) (a : ℝ) : lipschitz_with Kf (λ x, min (f x) a) | by simpa only [max_eq_left (zero_le Kf)] using hf.min (lipschitz_with.const a) | lemma | lipschitz_with.min_const | topology.metric_space | src/topology/metric_space/lipschitz.lean | [
"logic.function.iterate",
"data.set.intervals.proj_Icc",
"topology.algebra.order.field",
"topology.metric_space.basic",
"topology.bornology.hom"
] | [
"lipschitz_with",
"lipschitz_with.const"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
const_min (hf : lipschitz_with Kf f) (a : ℝ) : lipschitz_with Kf (λ x, min a (f x)) | by simpa only [min_comm] using hf.min_const a | lemma | lipschitz_with.const_min | topology.metric_space | src/topology/metric_space/lipschitz.lean | [
"logic.function.iterate",
"data.set.intervals.proj_Icc",
"topology.algebra.order.field",
"topology.metric_space.basic",
"topology.bornology.hom"
] | [
"lipschitz_with"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
proj_Icc {a b : ℝ} (h : a ≤ b) :
lipschitz_with 1 (proj_Icc a b h) | ((lipschitz_with.id.const_min _).const_max _).subtype_mk _ | lemma | lipschitz_with.proj_Icc | topology.metric_space | src/topology/metric_space/lipschitz.lean | [
"logic.function.iterate",
"data.set.intervals.proj_Icc",
"topology.algebra.order.field",
"topology.metric_space.basic",
"topology.bornology.hom"
] | [
"lipschitz_with"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bounded.left_of_prod (h : bounded (s ×ˢ t)) (ht : t.nonempty) : bounded s | by simpa only [fst_image_prod s ht] using (@lipschitz_with.prod_fst α β _ _).bounded_image h | lemma | metric.bounded.left_of_prod | topology.metric_space | src/topology/metric_space/lipschitz.lean | [
"logic.function.iterate",
"data.set.intervals.proj_Icc",
"topology.algebra.order.field",
"topology.metric_space.basic",
"topology.bornology.hom"
] | [
"lipschitz_with.prod_fst"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bounded.right_of_prod (h : bounded (s ×ˢ t)) (hs : s.nonempty) : bounded t | by simpa only [snd_image_prod hs t] using (@lipschitz_with.prod_snd α β _ _).bounded_image h | lemma | metric.bounded.right_of_prod | topology.metric_space | src/topology/metric_space/lipschitz.lean | [
"logic.function.iterate",
"data.set.intervals.proj_Icc",
"topology.algebra.order.field",
"topology.metric_space.basic",
"topology.bornology.hom"
] | [
"lipschitz_with.prod_snd"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bounded_prod_of_nonempty (hs : s.nonempty) (ht : t.nonempty) :
bounded (s ×ˢ t) ↔ bounded s ∧ bounded t | ⟨λ h, ⟨h.left_of_prod ht, h.right_of_prod hs⟩, λ h, h.1.prod h.2⟩ | lemma | metric.bounded_prod_of_nonempty | topology.metric_space | src/topology/metric_space/lipschitz.lean | [
"logic.function.iterate",
"data.set.intervals.proj_Icc",
"topology.algebra.order.field",
"topology.metric_space.basic",
"topology.bornology.hom"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bounded_prod : bounded (s ×ˢ t) ↔ s = ∅ ∨ t = ∅ ∨ bounded s ∧ bounded t | begin
rcases s.eq_empty_or_nonempty with rfl|hs, { simp },
rcases t.eq_empty_or_nonempty with rfl|ht, { simp },
simp only [bounded_prod_of_nonempty hs ht, hs.ne_empty, ht.ne_empty, false_or]
end | lemma | metric.bounded_prod | topology.metric_space | src/topology/metric_space/lipschitz.lean | [
"logic.function.iterate",
"data.set.intervals.proj_Icc",
"topology.algebra.order.field",
"topology.metric_space.basic",
"topology.bornology.hom"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
uniform_continuous_on (hf : lipschitz_on_with K f s) : uniform_continuous_on f s | uniform_continuous_on_iff_restrict.mpr (lipschitz_on_with_iff_restrict.mp hf).uniform_continuous | lemma | lipschitz_on_with.uniform_continuous_on | topology.metric_space | src/topology/metric_space/lipschitz.lean | [
"logic.function.iterate",
"data.set.intervals.proj_Icc",
"topology.algebra.order.field",
"topology.metric_space.basic",
"topology.bornology.hom"
] | [
"lipschitz_on_with",
"uniform_continuous",
"uniform_continuous_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_on (hf : lipschitz_on_with K f s) : continuous_on f s | hf.uniform_continuous_on.continuous_on | lemma | lipschitz_on_with.continuous_on | topology.metric_space | src/topology/metric_space/lipschitz.lean | [
"logic.function.iterate",
"data.set.intervals.proj_Icc",
"topology.algebra.order.field",
"topology.metric_space.basic",
"topology.bornology.hom"
] | [
"continuous_on",
"lipschitz_on_with"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
edist_lt_of_edist_lt_div (hf : lipschitz_on_with 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 | (lipschitz_on_with_iff_restrict.mp hf).edist_lt_of_edist_lt_div $
show edist (⟨x, hx⟩ : s) ⟨y, hy⟩ < d / K, from hd | lemma | lipschitz_on_with.edist_lt_of_edist_lt_div | topology.metric_space | src/topology/metric_space/lipschitz.lean | [
"logic.function.iterate",
"data.set.intervals.proj_Icc",
"topology.algebra.order.field",
"topology.metric_space.basic",
"topology.bornology.hom"
] | [
"lipschitz_on_with"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp {g : β → γ} {t : set β} {Kg : ℝ≥0} (hg : lipschitz_on_with Kg g t)
(hf : lipschitz_on_with K f s) (hmaps : maps_to f s t) :
lipschitz_on_with (Kg * K) (g ∘ f) s | lipschitz_on_with_iff_restrict.mpr $ hg.to_restrict.comp (hf.to_restrict_maps_to hmaps) | lemma | lipschitz_on_with.comp | topology.metric_space | src/topology/metric_space/lipschitz.lean | [
"logic.function.iterate",
"data.set.intervals.proj_Icc",
"topology.algebra.order.field",
"topology.metric_space.basic",
"topology.bornology.hom"
] | [
"lipschitz_on_with"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ediam_image2_le (f : α → β → γ) {K₁ K₂ : ℝ≥0}
(s : set α) (t : set β)
(hf₁ : ∀ b ∈ t, lipschitz_on_with K₁ (λ a, f a b) s)
(hf₂ : ∀ a ∈ s, lipschitz_on_with K₂ (f a) t) :
emetric.diam (set.image2 f s t) ≤ ↑K₁ * emetric.diam s + ↑K₂ * emetric.diam t | begin
apply emetric.diam_le,
rintros _ ⟨a₁, b₁, ha₁, hb₁, rfl⟩ _ ⟨a₂, b₂, ha₂, hb₂, rfl⟩,
refine (edist_triangle _ (f a₂ b₁) _).trans _,
exact add_le_add
((hf₁ b₁ hb₁ ha₁ ha₂).trans $ ennreal.mul_left_mono $ emetric.edist_le_diam_of_mem ha₁ ha₂)
((hf₂ a₂ ha₂ hb₁ hb₂).trans $ ennreal.mul_left_mono $ eme... | lemma | lipschitz_on_with.ediam_image2_le | topology.metric_space | src/topology/metric_space/lipschitz.lean | [
"logic.function.iterate",
"data.set.intervals.proj_Icc",
"topology.algebra.order.field",
"topology.metric_space.basic",
"topology.bornology.hom"
] | [
"emetric.diam",
"emetric.diam_le",
"emetric.edist_le_diam_of_mem",
"ennreal.mul_left_mono",
"lipschitz_on_with",
"set.image2"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_dist_le' {K : ℝ} (h : ∀ (x ∈ s) (y ∈ s), dist (f x) (f y) ≤ K * dist x y) :
lipschitz_on_with (real.to_nnreal K) f s | of_dist_le_mul $ λ x hx y hy, le_trans (h x hx y hy) $
mul_le_mul_of_nonneg_right (real.le_coe_to_nnreal K) dist_nonneg | lemma | lipschitz_on_with.of_dist_le' | topology.metric_space | src/topology/metric_space/lipschitz.lean | [
"logic.function.iterate",
"data.set.intervals.proj_Icc",
"topology.algebra.order.field",
"topology.metric_space.basic",
"topology.bornology.hom"
] | [
"dist_nonneg",
"lipschitz_on_with",
"mul_le_mul_of_nonneg_right",
"real.le_coe_to_nnreal",
"real.to_nnreal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_one (h : ∀ (x ∈ s) (y ∈ s), dist (f x) (f y) ≤ dist x y) :
lipschitz_on_with 1 f s | of_dist_le_mul $ by simpa only [nnreal.coe_one, one_mul] using h | lemma | lipschitz_on_with.mk_one | topology.metric_space | src/topology/metric_space/lipschitz.lean | [
"logic.function.iterate",
"data.set.intervals.proj_Icc",
"topology.algebra.order.field",
"topology.metric_space.basic",
"topology.bornology.hom"
] | [
"lipschitz_on_with",
"nnreal.coe_one",
"one_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_le_add_mul' {f : α → ℝ} (K : ℝ)
(h : ∀ (x ∈ s) (y ∈ s), f x ≤ f y + K * dist x y) :
lipschitz_on_with (real.to_nnreal K) f s | have I : ∀ (x ∈ s) (y ∈ s), f x - f y ≤ K * dist x y,
from assume x hx y hy, sub_le_iff_le_add'.2 (h x hx y hy),
lipschitz_on_with.of_dist_le' $
assume x hx y hy,
abs_sub_le_iff.2 ⟨I x hx y hy, dist_comm y x ▸ I y hy x hx⟩ | lemma | lipschitz_on_with.of_le_add_mul' | topology.metric_space | src/topology/metric_space/lipschitz.lean | [
"logic.function.iterate",
"data.set.intervals.proj_Icc",
"topology.algebra.order.field",
"topology.metric_space.basic",
"topology.bornology.hom"
] | [
"dist_comm",
"lipschitz_on_with",
"lipschitz_on_with.of_dist_le'",
"real.to_nnreal"
] | For functions to `ℝ`, it suffices to prove `f x ≤ f y + K * dist x y`; this version
doesn't assume `0≤K`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of_le_add_mul {f : α → ℝ} (K : ℝ≥0)
(h : ∀ (x ∈ s) (y ∈ s), f x ≤ f y + K * dist x y) :
lipschitz_on_with K f s | by simpa only [real.to_nnreal_coe] using lipschitz_on_with.of_le_add_mul' K h | lemma | lipschitz_on_with.of_le_add_mul | topology.metric_space | src/topology/metric_space/lipschitz.lean | [
"logic.function.iterate",
"data.set.intervals.proj_Icc",
"topology.algebra.order.field",
"topology.metric_space.basic",
"topology.bornology.hom"
] | [
"lipschitz_on_with",
"lipschitz_on_with.of_le_add_mul'",
"real.to_nnreal_coe"
] | For functions to `ℝ`, it suffices to prove `f x ≤ f y + K * dist x y`; this version
assumes `0≤K`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of_le_add {f : α → ℝ} (h : ∀ (x ∈ s) (y ∈ s), f x ≤ f y + dist x y) :
lipschitz_on_with 1 f s | lipschitz_on_with.of_le_add_mul 1 $ by simpa only [nnreal.coe_one, one_mul] | lemma | lipschitz_on_with.of_le_add | topology.metric_space | src/topology/metric_space/lipschitz.lean | [
"logic.function.iterate",
"data.set.intervals.proj_Icc",
"topology.algebra.order.field",
"topology.metric_space.basic",
"topology.bornology.hom"
] | [
"lipschitz_on_with",
"lipschitz_on_with.of_le_add_mul",
"nnreal.coe_one",
"one_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_add_mul {f : α → ℝ} {K : ℝ≥0} (h : lipschitz_on_with K f s)
{x : α} (hx : x ∈ s) {y : α} (hy : y ∈ s) :
f x ≤ f y + K * dist x y | sub_le_iff_le_add'.1 $ le_trans (le_abs_self _) $ h.dist_le_mul x hx y hy | lemma | lipschitz_on_with.le_add_mul | topology.metric_space | src/topology/metric_space/lipschitz.lean | [
"logic.function.iterate",
"data.set.intervals.proj_Icc",
"topology.algebra.order.field",
"topology.metric_space.basic",
"topology.bornology.hom"
] | [
"le_abs_self",
"lipschitz_on_with"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
iff_le_add_mul {f : α → ℝ} {K : ℝ≥0} :
lipschitz_on_with K f s ↔ ∀ (x ∈ s) (y ∈ s), f x ≤ f y + K * dist x y | ⟨lipschitz_on_with.le_add_mul, lipschitz_on_with.of_le_add_mul K⟩ | lemma | lipschitz_on_with.iff_le_add_mul | topology.metric_space | src/topology/metric_space/lipschitz.lean | [
"logic.function.iterate",
"data.set.intervals.proj_Icc",
"topology.algebra.order.field",
"topology.metric_space.basic",
"topology.bornology.hom"
] | [
"lipschitz_on_with",
"lipschitz_on_with.of_le_add_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bounded_image2 (f : α → β → γ) {K₁ K₂ : ℝ≥0}
{s : set α} {t : set β} (hs : metric.bounded s) (ht : metric.bounded t)
(hf₁ : ∀ b ∈ t, lipschitz_on_with K₁ (λ a, f a b) s)
(hf₂ : ∀ a ∈ s, lipschitz_on_with K₂ (f a) t) :
metric.bounded (set.image2 f s t) | metric.bounded_iff_ediam_ne_top.2 $ ne_top_of_le_ne_top
(ennreal.add_ne_top.mpr ⟨
ennreal.mul_ne_top ennreal.coe_ne_top hs.ediam_ne_top,
ennreal.mul_ne_top ennreal.coe_ne_top ht.ediam_ne_top⟩)
(ediam_image2_le _ _ _ hf₁ hf₂) | lemma | lipschitz_on_with.bounded_image2 | topology.metric_space | src/topology/metric_space/lipschitz.lean | [
"logic.function.iterate",
"data.set.intervals.proj_Icc",
"topology.algebra.order.field",
"topology.metric_space.basic",
"topology.bornology.hom"
] | [
"ennreal.coe_ne_top",
"ennreal.mul_ne_top",
"lipschitz_on_with",
"metric.bounded",
"ne_top_of_le_ne_top",
"set.image2"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_on_prod_of_continuous_on_lipschitz_on [pseudo_emetric_space α]
[topological_space β] [pseudo_emetric_space γ] (f : α × β → γ) {s : set α} {t : set β}
(K : ℝ≥0) (ha : ∀ a ∈ s, continuous_on (λ y, f (a, y)) t)
(hb : ∀ b ∈ t, lipschitz_on_with K (λ x, f (x, b)) s) :
continuous_on f (s ×ˢ t) | begin
rintro ⟨x, y⟩ ⟨hx : x ∈ s, hy : y ∈ t⟩,
refine emetric.tendsto_nhds.2 (λ ε (ε0 : 0 < ε), _),
replace ε0 : 0 < ε / 2 := ennreal.half_pos (ne_of_gt ε0),
have εK : 0 < ε / 2 / K := ennreal.div_pos_iff.2 ⟨ε0.ne', ennreal.coe_ne_top⟩,
have A : s ∩ emetric.ball x (ε / 2 / K) ∈ 𝓝[s] x :=
inter_mem_nhds_wi... | lemma | continuous_on_prod_of_continuous_on_lipschitz_on | topology.metric_space | src/topology/metric_space/lipschitz.lean | [
"logic.function.iterate",
"data.set.intervals.proj_Icc",
"topology.algebra.order.field",
"topology.metric_space.basic",
"topology.bornology.hom"
] | [
"continuous_on",
"emetric.ball",
"emetric.ball_mem_nhds",
"ennreal.add_halves",
"ennreal.add_lt_add",
"ennreal.half_pos",
"inter_mem_nhds_within",
"lipschitz_on_with",
"nhds_within_prod",
"pseudo_emetric_space",
"self_mem_nhds_within",
"topological_space"
] | Consider a function `f : α × β → γ`. Suppose that it is continuous on each “vertical fiber”
`{a} × t`, `a ∈ s`, and is Lipschitz continuous on each “horizontal fiber” `s × {b}`, `b ∈ t`
with the same Lipschitz constant `K`. Then it is continuous on `s × t`.
The actual statement uses (Lipschitz) continuity of `λ y, f (... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuous_prod_of_continuous_lipschitz [pseudo_emetric_space α]
[topological_space β] [pseudo_emetric_space γ] (f : α × β → γ) (K : ℝ≥0)
(ha : ∀ a, continuous (λ y, f (a, y))) (hb : ∀ b, lipschitz_with K (λ x, f (x, b))) :
continuous f | begin
simp only [continuous_iff_continuous_on_univ, ← univ_prod_univ, ← lipschitz_on_univ] at *,
exact continuous_on_prod_of_continuous_on_lipschitz_on f K (λ a _, ha a) (λ b _, hb b)
end | lemma | continuous_prod_of_continuous_lipschitz | topology.metric_space | src/topology/metric_space/lipschitz.lean | [
"logic.function.iterate",
"data.set.intervals.proj_Icc",
"topology.algebra.order.field",
"topology.metric_space.basic",
"topology.bornology.hom"
] | [
"continuous",
"continuous_iff_continuous_on_univ",
"continuous_on_prod_of_continuous_on_lipschitz_on",
"lipschitz_on_univ",
"lipschitz_with",
"pseudo_emetric_space",
"topological_space"
] | Consider a function `f : α × β → γ`. Suppose that it is continuous on each “vertical section”
`{a} × univ`, `a : α`, and is Lipschitz continuous on each “horizontal section”
`univ × {b}`, `b : β` with the same Lipschitz constant `K`. Then it is continuous.
The actual statement uses (Lipschitz) continuity of `λ y, f (a... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuous_at_of_locally_lipschitz [pseudo_metric_space α] [pseudo_metric_space β]
{f : α → β} {x : α}
{r : ℝ} (hr : 0 < r) (K : ℝ) (h : ∀ y, dist y x < r → dist (f y) (f x) ≤ K * dist y x) :
continuous_at f x | begin
-- We use `h` to squeeze `dist (f y) (f x)` between `0` and `K * dist y x`
refine tendsto_iff_dist_tendsto_zero.2
(squeeze_zero' (eventually_of_forall $ λ _, dist_nonneg)
(mem_of_superset (ball_mem_nhds _ hr) h) _),
-- Then show that `K * dist y x` tends to zero as `y → x`
refine (continuous_const... | lemma | continuous_at_of_locally_lipschitz | topology.metric_space | src/topology/metric_space/lipschitz.lean | [
"logic.function.iterate",
"data.set.intervals.proj_Icc",
"topology.algebra.order.field",
"topology.metric_space.basic",
"topology.bornology.hom"
] | [
"continuous_at",
"continuous_const",
"dist_nonneg",
"pseudo_metric_space",
"squeeze_zero'"
] | If a function is locally Lipschitz around a point, then it is continuous at this point. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lipschitz_on_with.extend_real [pseudo_metric_space α] {f : α → ℝ} {s : set α} {K : ℝ≥0}
(hf : lipschitz_on_with K f s) :
∃ g : α → ℝ, lipschitz_with K g ∧ eq_on f g s | begin
/- An extension is given by `g y = Inf {f x + K * dist y x | x ∈ s}`. Taking `x = y`, one has
`g y ≤ f y` for `y ∈ s`, and the other inequality holds because `f` is `K`-Lipschitz, so that it
can not counterbalance the growth of `K * dist y x`. One readily checks from the formula that the
extended function... | lemma | lipschitz_on_with.extend_real | topology.metric_space | src/topology/metric_space/lipschitz.lean | [
"logic.function.iterate",
"data.set.intervals.proj_Icc",
"topology.algebra.order.field",
"topology.metric_space.basic",
"topology.bornology.hom"
] | [
"bdd_below",
"cinfi_le",
"dist_self",
"dist_triangle",
"dist_triangle_left",
"infi",
"le_cinfi",
"lipschitz_on_with",
"lipschitz_with",
"lipschitz_with.const",
"lipschitz_with.of_le_add_mul",
"mul_le_mul_of_nonneg_left",
"mul_zero",
"pseudo_metric_space",
"subtype.coe_mk"
] | A function `f : α → ℝ` which is `K`-Lipschitz on a subset `s` admits a `K`-Lipschitz extension
to the whole space. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lipschitz_on_with.extend_pi [pseudo_metric_space α] [fintype ι]
{f : α → (ι → ℝ)} {s : set α} {K : ℝ≥0}
(hf : lipschitz_on_with K f s) :
∃ g : α → (ι → ℝ), lipschitz_with K g ∧ eq_on f g s | begin
have : ∀ i, ∃ g : α → ℝ, lipschitz_with K g ∧ eq_on (λ x, f x i) g s,
{ assume i,
have : lipschitz_on_with K (λ (x : α), f x i) s,
{ apply lipschitz_on_with.of_dist_le_mul (λ x hx y hy, _),
exact (dist_le_pi_dist _ _ i).trans (hf.dist_le_mul x hx y hy) },
exact this.extend_real },
choose g... | lemma | lipschitz_on_with.extend_pi | topology.metric_space | src/topology/metric_space/lipschitz.lean | [
"logic.function.iterate",
"data.set.intervals.proj_Icc",
"topology.algebra.order.field",
"topology.metric_space.basic",
"topology.bornology.hom"
] | [
"dist_le_pi_dist",
"dist_nonneg",
"dist_pi_le_iff",
"fintype",
"lipschitz_on_with",
"lipschitz_with",
"pseudo_metric_space"
] | A function `f : α → (ι → ℝ)` which is `K`-Lipschitz on a subset `s` admits a `K`-Lipschitz
extension to the whole space.
TODO: state the same result (with the same proof) for the space `ℓ^∞ (ι, ℝ)` over a possibly
infinite type `ι`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_metric_separated {X : Type*} [emetric_space X] (s t : set X) | ∃ r ≠ 0, ∀ (x ∈ s) (y ∈ t), r ≤ edist x y | def | is_metric_separated | topology.metric_space | src/topology/metric_space/metric_separated.lean | [
"topology.metric_space.emetric_space"
] | [
"emetric_space"
] | Two sets in an (extended) metric space are called *metric separated* if the (extended) distance
between `x ∈ s` and `y ∈ t` is bounded from below by a positive constant. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
symm (h : is_metric_separated s t) : is_metric_separated t s | let ⟨r, r0, hr⟩ := h in ⟨r, r0, λ y hy x hx, edist_comm x y ▸ hr x hx y hy⟩ | lemma | is_metric_separated.symm | topology.metric_space | src/topology/metric_space/metric_separated.lean | [
"topology.metric_space.emetric_space"
] | [
"is_metric_separated"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comm : is_metric_separated s t ↔ is_metric_separated t s | ⟨symm, symm⟩ | lemma | is_metric_separated.comm | topology.metric_space | src/topology/metric_space/metric_separated.lean | [
"topology.metric_space.emetric_space"
] | [
"comm",
"is_metric_separated"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
empty_left (s : set X) : is_metric_separated ∅ s | ⟨1, one_ne_zero, λ x, false.elim⟩ | lemma | is_metric_separated.empty_left | topology.metric_space | src/topology/metric_space/metric_separated.lean | [
"topology.metric_space.emetric_space"
] | [
"is_metric_separated",
"one_ne_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
empty_right (s : set X) : is_metric_separated s ∅ | (empty_left s).symm | lemma | is_metric_separated.empty_right | topology.metric_space | src/topology/metric_space/metric_separated.lean | [
"topology.metric_space.emetric_space"
] | [
"is_metric_separated"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
disjoint (h : is_metric_separated s t) : disjoint s t | let ⟨r, r0, hr⟩ := h in
set.disjoint_left.mpr $ λ x hx1 hx2, r0 $ by simpa using hr x hx1 x hx2 | lemma | is_metric_separated.disjoint | topology.metric_space | src/topology/metric_space/metric_separated.lean | [
"topology.metric_space.emetric_space"
] | [
"disjoint",
"is_metric_separated"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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