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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|---|---|---|---|---|---|---|---|---|---|---|
eq_of_nndist_eq_zero {x y : γ} : nndist x y = 0 → x = y | by simp only [← nnreal.eq_iff, ← dist_nndist, imp_self, nnreal.coe_zero, dist_eq_zero] | theorem | eq_of_nndist_eq_zero | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"dist_eq_zero",
"dist_nndist",
"imp_self",
"nnreal.coe_zero",
"nnreal.eq_iff"
] | Deduce the equality of points with the vanishing of the nonnegative distance | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
nndist_eq_zero {x y : γ} : nndist x y = 0 ↔ x = y | by simp only [← nnreal.eq_iff, ← dist_nndist, imp_self, nnreal.coe_zero, dist_eq_zero] | theorem | nndist_eq_zero | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"dist_eq_zero",
"dist_nndist",
"imp_self",
"nnreal.coe_zero",
"nnreal.eq_iff"
] | Characterize the equality of points with the vanishing of the nonnegative distance | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
zero_eq_nndist {x y : γ} : 0 = nndist x y ↔ x = y | by simp only [← nnreal.eq_iff, ← dist_nndist, imp_self, nnreal.coe_zero, zero_eq_dist] | theorem | zero_eq_nndist | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"dist_nndist",
"imp_self",
"nnreal.coe_zero",
"nnreal.eq_iff",
"zero_eq_dist"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
closed_ball_zero : closed_ball x 0 = {x} | set.ext $ λ y, dist_le_zero | lemma | metric.closed_ball_zero | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"dist_le_zero",
"set.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sphere_zero : sphere x 0 = {x} | set.ext $ λ y, dist_eq_zero | lemma | metric.sphere_zero | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"dist_eq_zero",
"set.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
subsingleton_closed_ball (x : γ) {r : ℝ} (hr : r ≤ 0) : (closed_ball x r).subsingleton | begin
rcases hr.lt_or_eq with hr|rfl,
{ rw closed_ball_eq_empty.2 hr, exact subsingleton_empty },
{ rw closed_ball_zero, exact subsingleton_singleton }
end | lemma | metric.subsingleton_closed_ball | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
subsingleton_sphere (x : γ) {r : ℝ} (hr : r ≤ 0) : (sphere x r).subsingleton | (subsingleton_closed_ball x hr).anti sphere_subset_closed_ball | lemma | metric.subsingleton_sphere | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.metric_space.to_separated : separated_space γ | separated_def.2 $ λ x y h, eq_of_forall_dist_le $
λ ε ε0, le_of_lt (h _ (dist_mem_uniformity ε0)) | instance | metric_space.to_separated | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"eq_of_forall_dist_le",
"separated_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
uniform_embedding_iff' [metric_space β] {f : γ → β} :
uniform_embedding f ↔
(∀ ε > 0, ∃ δ > 0, ∀ {a b : γ}, dist a b < δ → dist (f a) (f b) < ε) ∧
(∀ δ > 0, ∃ ε > 0, ∀ {a b : γ}, dist (f a) (f b) < ε → dist a b < δ) | begin
simp only [uniform_embedding_iff_uniform_inducing,
uniformity_basis_dist.uniform_inducing_iff uniformity_basis_dist, exists_prop],
refl
end | theorem | metric.uniform_embedding_iff' | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"exists_prop",
"metric_space",
"uniform_embedding",
"uniform_embedding_iff'",
"uniform_embedding_iff_uniform_inducing"
] | A map between metric spaces is a uniform embedding if and only if the distance between `f x`
and `f y` is controlled in terms of the distance between `x` and `y` and conversely. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
_root_.metric_space.of_t0_pseudo_metric_space (α : Type*) [pseudo_metric_space α] [t0_space α] :
metric_space α | { eq_of_dist_eq_zero := λ x y hdist, inseparable.eq $ metric.inseparable_iff.2 hdist,
..‹pseudo_metric_space α› } | def | metric_space.of_t0_pseudo_metric_space | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"eq_of_dist_eq_zero",
"inseparable.eq",
"metric_space",
"pseudo_metric_space",
"t0_space"
] | If a `pseudo_metric_space` is a T₀ space, then it is a `metric_space`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
_root_.metric_space.to_emetric_space : emetric_space γ | emetric_space.of_t0_pseudo_emetric_space γ | instance | metric_space.to_emetric_space | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"emetric_space",
"emetric_space.of_t0_pseudo_emetric_space"
] | A metric space induces an emetric space | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_closed_of_pairwise_le_dist {s : set γ} {ε : ℝ} (hε : 0 < ε)
(hs : s.pairwise (λ x y, ε ≤ dist x y)) : is_closed s | is_closed_of_spaced_out (dist_mem_uniformity hε) $ by simpa using hs | lemma | metric.is_closed_of_pairwise_le_dist | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"is_closed",
"is_closed_of_spaced_out"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
closed_embedding_of_pairwise_le_dist {α : Type*} [topological_space α] [discrete_topology α]
{ε : ℝ} (hε : 0 < ε) {f : α → γ} (hf : pairwise (λ x y, ε ≤ dist (f x) (f y))) :
closed_embedding f | closed_embedding_of_spaced_out (dist_mem_uniformity hε) $ by simpa using hf | lemma | metric.closed_embedding_of_pairwise_le_dist | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"closed_embedding",
"closed_embedding_of_spaced_out",
"discrete_topology",
"pairwise",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
uniform_embedding_bot_of_pairwise_le_dist {β : Type*} {ε : ℝ} (hε : 0 < ε) {f : β → α}
(hf : pairwise (λ x y, ε ≤ dist (f x) (f y))) : @uniform_embedding _ _ ⊥ (by apply_instance) f | uniform_embedding_of_spaced_out (dist_mem_uniformity hε) $ by simpa using hf | lemma | metric.uniform_embedding_bot_of_pairwise_le_dist | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"pairwise",
"uniform_embedding",
"uniform_embedding_of_spaced_out"
] | If `f : β → α` sends any two distinct points to points at distance at least `ε > 0`, then
`f` is a uniform embedding with respect to the discrete uniformity on `β`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
metric_space.replace_uniformity {γ} [U : uniform_space γ] (m : metric_space γ)
(H : 𝓤[U] = 𝓤[pseudo_emetric_space.to_uniform_space]) :
metric_space γ | { eq_of_dist_eq_zero := @eq_of_dist_eq_zero _ _,
..pseudo_metric_space.replace_uniformity m.to_pseudo_metric_space H, } | def | metric_space.replace_uniformity | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"eq_of_dist_eq_zero",
"metric_space",
"pseudo_metric_space.replace_uniformity",
"uniform_space"
] | Build a new metric space from an old one where the bundled uniform structure is provably
(but typically non-definitionaly) equal to some given uniform structure.
See Note [forgetful inheritance]. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
metric_space.replace_uniformity_eq {γ} [U : uniform_space γ] (m : metric_space γ)
(H : 𝓤[U] = 𝓤[pseudo_emetric_space.to_uniform_space]) :
m.replace_uniformity H = m | by { ext, refl } | lemma | metric_space.replace_uniformity_eq | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"metric_space",
"uniform_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
metric_space.replace_topology {γ} [U : topological_space γ] (m : metric_space γ)
(H : U = m.to_pseudo_metric_space.to_uniform_space.to_topological_space) :
metric_space γ | @metric_space.replace_uniformity γ (m.to_uniform_space.replace_topology H) m rfl | def | metric_space.replace_topology | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"metric_space",
"metric_space.replace_uniformity",
"topological_space"
] | Build a new metric space from an old one where the bundled topological structure is provably
(but typically non-definitionaly) equal to some given topological structure.
See Note [forgetful inheritance]. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
metric_space.replace_topology_eq {γ} [U : topological_space γ] (m : metric_space γ)
(H : U = m.to_pseudo_metric_space.to_uniform_space.to_topological_space) :
m.replace_topology H = m | by { ext, refl } | lemma | metric_space.replace_topology_eq | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"metric_space",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
emetric_space.to_metric_space_of_dist {α : Type u} [e : emetric_space α]
(dist : α → α → ℝ)
(edist_ne_top : ∀x y: α, edist x y ≠ ⊤)
(h : ∀x y, dist x y = ennreal.to_real (edist x y)) :
metric_space α | @metric_space.of_t0_pseudo_metric_space α
(pseudo_emetric_space.to_pseudo_metric_space_of_dist dist edist_ne_top h) _ | def | emetric_space.to_metric_space_of_dist | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"edist_ne_top",
"emetric_space",
"ennreal.to_real",
"metric_space",
"metric_space.of_t0_pseudo_metric_space",
"pseudo_emetric_space.to_pseudo_metric_space_of_dist"
] | One gets a metric space from an emetric space if the edistance
is everywhere finite, by pushing the edistance to reals. We set it up so that the edist and the
uniformity are defeq in the metric space and the emetric space. In this definition, the distance
is given separately, to be able to prescribe some expression whi... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
emetric_space.to_metric_space {α : Type u} [emetric_space α] (h : ∀ x y : α, edist x y ≠ ⊤) :
metric_space α | emetric_space.to_metric_space_of_dist (λx y, ennreal.to_real (edist x y)) h (λ x y, rfl) | def | emetric_space.to_metric_space | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"emetric_space",
"emetric_space.to_metric_space_of_dist",
"ennreal.to_real",
"metric_space"
] | One gets a metric space from an emetric space if the edistance
is everywhere finite, by pushing the edistance to reals. We set it up so that the edist and the
uniformity are defeq in the metric space and the emetric space. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
metric_space.replace_bornology {α} [B : bornology α] (m : metric_space α)
(H : ∀ s, @is_bounded _ B s ↔ @is_bounded _ pseudo_metric_space.to_bornology s) :
metric_space α | { to_bornology := B,
.. pseudo_metric_space.replace_bornology _ H,
.. m } | def | metric_space.replace_bornology | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"bornology",
"metric_space",
"pseudo_metric_space.replace_bornology"
] | Build a new metric space from an old one where the bundled bornology structure is provably
(but typically non-definitionaly) equal to some given bornology structure.
See Note [forgetful inheritance]. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
metric_space.replace_bornology_eq {α} [m : metric_space α] [B : bornology α]
(H : ∀ s, @is_bounded _ B s ↔ @is_bounded _ pseudo_metric_space.to_bornology s) :
metric_space.replace_bornology _ H = m | by { ext, refl } | lemma | metric_space.replace_bornology_eq | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"bornology",
"metric_space",
"metric_space.replace_bornology"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
metric_space.induced {γ β} (f : γ → β) (hf : function.injective f)
(m : metric_space β) : metric_space γ | { eq_of_dist_eq_zero := λ x y h, hf (dist_eq_zero.1 h),
..pseudo_metric_space.induced f m.to_pseudo_metric_space } | def | metric_space.induced | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"eq_of_dist_eq_zero",
"metric_space",
"pseudo_metric_space.induced"
] | Metric space structure pulled back by an injective function. Injectivity is necessary to
ensure that `dist x y = 0` only if `x = y`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
uniform_embedding.comap_metric_space
{α β} [uniform_space α] [metric_space β] (f : α → β) (h : uniform_embedding f) :
metric_space α | (metric_space.induced f h.inj ‹_›).replace_uniformity h.comap_uniformity.symm | def | uniform_embedding.comap_metric_space | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"metric_space",
"metric_space.induced",
"uniform_embedding",
"uniform_space"
] | Pull back a metric space structure by a uniform embedding. This is a version of
`metric_space.induced` useful in case if the domain already has a `uniform_space` structure. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
embedding.comap_metric_space
{α β} [topological_space α] [metric_space β] (f : α → β) (h : embedding f) :
metric_space α | begin
letI : uniform_space α := embedding.comap_uniform_space f h,
exact uniform_embedding.comap_metric_space f (h.to_uniform_embedding f),
end | def | embedding.comap_metric_space | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"embedding",
"embedding.comap_uniform_space",
"metric_space",
"topological_space",
"uniform_embedding.comap_metric_space",
"uniform_space"
] | Pull back a metric space structure by an embedding. This is a version of
`metric_space.induced` useful in case if the domain already has a `topological_space` structure. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
subtype.metric_space {α : Type*} {p : α → Prop} [metric_space α] :
metric_space (subtype p) | metric_space.induced coe subtype.coe_injective ‹_› | instance | subtype.metric_space | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"metric_space",
"metric_space.induced",
"subtype.coe_injective"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
real.metric_space : metric_space ℝ | { eq_of_dist_eq_zero := λ x y h, by simpa [dist, sub_eq_zero] using h,
..real.pseudo_metric_space } | instance | real.metric_space | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"eq_of_dist_eq_zero",
"metric_space",
"real.pseudo_metric_space"
] | Instantiate the reals as a metric space. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
prod.metric_space_max [metric_space β] : metric_space (γ × β) | { eq_of_dist_eq_zero := λ x y h, begin
cases max_le_iff.1 (le_of_eq h) with h₁ h₂,
exact prod.ext_iff.2 ⟨dist_le_zero.1 h₁, dist_le_zero.1 h₂⟩
end,
..prod.pseudo_metric_space_max, } | instance | prod.metric_space_max | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"eq_of_dist_eq_zero",
"metric_space",
"prod.pseudo_metric_space_max"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
metric_space_pi : metric_space (Πb, π b) | /- we construct the instance from the emetric space instance to avoid checking again that the
uniformity is the same as the product uniformity, but we register nevertheless a nice formula
for the distance -/
{ eq_of_dist_eq_zero := assume f g eq0,
begin
have eq1 : edist f g = 0 := by simp only [edist_dist, eq... | instance | metric_space_pi | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"edist_dist",
"ennreal.of_real_zero",
"eq_of_dist_eq_zero",
"finset.sup_le_iff",
"metric_space",
"pseudo_metric_space_pi"
] | A finite product of metric spaces is a metric space, with the sup distance. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
second_countable_of_countable_discretization {α : Type u} [metric_space α]
(H : ∀ε > (0 : ℝ), ∃ (β : Type*) (_ : encodable β) (F : α → β), ∀x y, F x = F y → dist x y ≤ ε) :
second_countable_topology α | begin
cases (univ : set α).eq_empty_or_nonempty with hs hs,
{ haveI : compact_space α := ⟨by rw hs; exact is_compact_empty⟩, by apply_instance },
rcases hs with ⟨x0, hx0⟩,
letI : inhabited α := ⟨x0⟩,
refine second_countable_of_almost_dense_set (λε ε0, _),
rcases H ε ε0 with ⟨β, fβ, F, hF⟩,
resetI,
let F... | lemma | metric.second_countable_of_countable_discretization | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"compact_space",
"encodable",
"metric_space"
] | A metric space is second countable if one can reconstruct up to any `ε>0` any element of the
space from countably many data. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
uniform_space.separation_quotient.dist_mk {α : Type u} [pseudo_metric_space α] (p q : α) :
@dist (uniform_space.separation_quotient α) _ (quot.mk _ p) (quot.mk _ q) = dist p q | rfl | lemma | uniform_space.separation_quotient.dist_mk | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"pseudo_metric_space",
"uniform_space.separation_quotient"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dist_of_mul (a b : X) : dist (of_mul a) (of_mul b) = dist a b | rfl | lemma | dist_of_mul | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dist_of_add (a b : X) : dist (of_add a) (of_add b) = dist a b | rfl | lemma | dist_of_add | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dist_to_mul (a b : additive X) : dist (to_mul a) (to_mul b) = dist a b | rfl | lemma | dist_to_mul | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"additive"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dist_to_add (a b : multiplicative X) : dist (to_add a) (to_add b) = dist a b | rfl | lemma | dist_to_add | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"multiplicative"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nndist_of_mul (a b : X) : nndist (of_mul a) (of_mul b) = nndist a b | rfl | lemma | nndist_of_mul | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nndist_of_add (a b : X) : nndist (of_add a) (of_add b) = nndist a b | rfl | lemma | nndist_of_add | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nndist_to_mul (a b : additive X) : nndist (to_mul a) (to_mul b) = nndist a b | rfl | lemma | nndist_to_mul | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"additive"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nndist_to_add (a b : multiplicative X) : nndist (to_add a) (to_add b) = nndist a b | rfl | lemma | nndist_to_add | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"multiplicative"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dist_to_dual (a b : X) : dist (to_dual a) (to_dual b) = dist a b | rfl | lemma | dist_to_dual | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dist_of_dual (a b : Xᵒᵈ) : dist (of_dual a) (of_dual b) = dist a b | rfl | lemma | dist_of_dual | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nndist_to_dual (a b : X) : nndist (to_dual a) (to_dual b) = nndist a b | rfl | lemma | nndist_to_dual | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nndist_of_dual (a b : Xᵒᵈ) : nndist (of_dual a) (of_dual b) = nndist a b | rfl | lemma | nndist_of_dual | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
induced_map : Σ s : set (ℕ → β), s → α | ⟨λ x, set.nonempty ⋂ n : ℕ, A (res x n), λ x, x.property.some⟩ | def | cantor_scheme.induced_map | topology.metric_space | src/topology/metric_space/cantor_scheme.lean | [
"topology.metric_space.pi_nat"
] | [
"induced_map",
"set.nonempty"
] | From a `β`-scheme on `α` `A`, we define a partial function from `(ℕ → β)` to `α`
which sends each infinite sequence `x` to an element of the intersection along the
branch corresponding to `x`, if it exists.
We call this the map induced by the scheme. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
antitone : Prop | ∀ l : list β, ∀ a : β, A (a :: l) ⊆ A l | def | cantor_scheme.antitone | topology.metric_space | src/topology/metric_space/cantor_scheme.lean | [
"topology.metric_space.pi_nat"
] | [
"antitone"
] | A scheme is antitone if each set contains its children. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
closure_antitone [topological_space α] : Prop | ∀ l : list β, ∀ a : β, closure (A (a :: l)) ⊆ A l | def | cantor_scheme.closure_antitone | topology.metric_space | src/topology/metric_space/cantor_scheme.lean | [
"topology.metric_space.pi_nat"
] | [
"closure",
"topological_space"
] | A useful strengthening of being antitone is to require that each set contains
the closure of each of its children. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
disjoint : Prop | ∀ l : list β, _root_.pairwise $ λ a b, disjoint (A (a :: l)) (A (b :: l)) | def | cantor_scheme.disjoint | topology.metric_space | src/topology/metric_space/cantor_scheme.lean | [
"topology.metric_space.pi_nat"
] | [
"disjoint"
] | A scheme is disjoint if the children of each set of pairwise disjoint. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
map_mem (x : (induced_map A).1) (n : ℕ) :
(induced_map A).2 x ∈ A (res x n) | begin
have := x.property.some_mem,
rw mem_Inter at this,
exact this n,
end | lemma | cantor_scheme.map_mem | topology.metric_space | src/topology/metric_space/cantor_scheme.lean | [
"topology.metric_space.pi_nat"
] | [
"induced_map"
] | If `x` is in the domain of the induced map of a scheme `A`,
its image under this map is in each set along the corresponding branch. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
closure_antitone.antitone [topological_space α] (hA : closure_antitone A) :
cantor_scheme.antitone A | λ l a, subset_closure.trans (hA l a) | lemma | cantor_scheme.closure_antitone.antitone | topology.metric_space | src/topology/metric_space/cantor_scheme.lean | [
"topology.metric_space.pi_nat"
] | [
"cantor_scheme.antitone",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
antitone.closure_antitone [topological_space α] (hanti : cantor_scheme.antitone A)
(hclosed : ∀ l, is_closed (A l)) : closure_antitone A | λ l a, (hclosed _).closure_eq.subset.trans (hanti _ _) | lemma | cantor_scheme.antitone.closure_antitone | topology.metric_space | src/topology/metric_space/cantor_scheme.lean | [
"topology.metric_space.pi_nat"
] | [
"cantor_scheme.antitone",
"is_closed",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
disjoint.map_injective (hA : cantor_scheme.disjoint A) : injective (induced_map A).2 | begin
rintros ⟨x, hx⟩ ⟨y, hy⟩ hxy,
refine subtype.coe_injective (res_injective _),
dsimp,
ext n : 1,
induction n with n ih, { simp },
simp only [res_succ],
refine ⟨_, ih⟩,
contrapose hA,
simp only [cantor_scheme.disjoint, _root_.pairwise, ne.def, not_forall, exists_prop],
refine ⟨res x n, _, _, hA, ... | theorem | cantor_scheme.disjoint.map_injective | topology.metric_space | src/topology/metric_space/cantor_scheme.lean | [
"topology.metric_space.pi_nat"
] | [
"cantor_scheme.disjoint",
"exists_prop",
"ih",
"induced_map",
"not_forall",
"subtype.coe_injective"
] | A scheme where the children of each set are pairwise disjoint induces an injective map. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
vanishing_diam : Prop | ∀ x : ℕ → β, tendsto (λ n : ℕ, emetric.diam (A (res x n))) at_top (𝓝 0) | def | cantor_scheme.vanishing_diam | topology.metric_space | src/topology/metric_space/cantor_scheme.lean | [
"topology.metric_space.pi_nat"
] | [
"emetric.diam"
] | A scheme on a metric space has vanishing diameter if diameter approaches 0 along each branch. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
vanishing_diam.dist_lt (hA : vanishing_diam A) (ε : ℝ) (ε_pos : 0 < ε) (x : ℕ → β) :
∃ n : ℕ, ∀ y z ∈ A (res x n), dist y z < ε | begin
specialize hA x,
rw ennreal.tendsto_at_top_zero at hA,
cases hA (ennreal.of_real (ε / 2))
(by { simp only [gt_iff_lt, ennreal.of_real_pos], linarith }) with n hn,
use n,
intros y hy z hz,
rw [← ennreal.of_real_lt_of_real_iff ε_pos, ← edist_dist],
apply lt_of_le_of_lt (emetric.edist_le_diam_of_me... | lemma | cantor_scheme.vanishing_diam.dist_lt | topology.metric_space | src/topology/metric_space/cantor_scheme.lean | [
"topology.metric_space.pi_nat"
] | [
"edist_dist",
"emetric.edist_le_diam_of_mem",
"ennreal.of_real",
"ennreal.of_real_lt_of_real_iff",
"ennreal.of_real_pos",
"ennreal.tendsto_at_top_zero",
"gt_iff_lt"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
vanishing_diam.map_continuous [topological_space β] [discrete_topology β]
(hA : vanishing_diam A) : continuous (induced_map A).2 | begin
rw metric.continuous_iff',
rintros ⟨x, hx⟩ ε ε_pos,
cases hA.dist_lt _ ε_pos x with n hn,
rw _root_.eventually_nhds_iff,
refine ⟨coe ⁻¹' cylinder x n, _, _, by simp⟩,
{ rintros ⟨y, hy⟩ hyx,
rw [mem_preimage, subtype.coe_mk, cylinder_eq_res, mem_set_of] at hyx,
apply hn,
{ rw ← hyx,
a... | theorem | cantor_scheme.vanishing_diam.map_continuous | topology.metric_space | src/topology/metric_space/cantor_scheme.lean | [
"topology.metric_space.pi_nat"
] | [
"continuous",
"discrete_topology",
"induced_map",
"metric.continuous_iff'",
"subtype.coe_mk",
"topological_space"
] | A scheme with vanishing diameter along each branch induces a continuous map. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
closure_antitone.map_of_vanishing_diam [complete_space α]
(hdiam : vanishing_diam A) (hanti : closure_antitone A) (hnonempty : ∀ l, (A l).nonempty) :
(induced_map A).1 = univ | begin
rw eq_univ_iff_forall,
intro x,
choose u hu using λ n, hnonempty (res x n),
have umem : ∀ n m : ℕ, n ≤ m → u m ∈ A (res x n),
{ have : antitone (λ n : ℕ, A (res x n)),
{ refine antitone_nat_of_succ_le _,
intro n,
apply hanti.antitone, },
intros n m hnm,
exact this hnm (hu _), },
... | theorem | cantor_scheme.closure_antitone.map_of_vanishing_diam | topology.metric_space | src/topology/metric_space/cantor_scheme.lean | [
"topology.metric_space.pi_nat"
] | [
"antitone",
"antitone_nat_of_succ_le",
"cauchy_seq",
"cauchy_seq_tendsto_of_complete",
"complete_space",
"induced_map",
"mem_closure_of_tendsto",
"metric.cauchy_seq_iff"
] | A scheme on a complete space with vanishing diameter
such that each set contains the closure of its children
induces a total map. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cau_seq.tendsto_limit [normed_ring β] [hn : is_absolute_value (norm : β → ℝ)]
(f : cau_seq β norm) [cau_seq.is_complete β norm] :
tendsto f at_top (𝓝 f.lim) | _root_.tendsto_nhds.mpr
begin
intros s os lfs,
suffices : ∃ (a : ℕ), ∀ (b : ℕ), b ≥ a → f b ∈ s, by simpa using this,
rcases metric.is_open_iff.1 os _ lfs with ⟨ε, ⟨hε, hεs⟩⟩,
cases setoid.symm (cau_seq.equiv_lim f) _ hε with N hN,
existsi N,
intros b hb,
apply hεs,
dsimp [metric.ball], rw [dist_comm, d... | lemma | cau_seq.tendsto_limit | topology.metric_space | src/topology/metric_space/cau_seq_filter.lean | [
"analysis.normed.field.basic"
] | [
"cau_seq",
"cau_seq.equiv_lim",
"cau_seq.is_complete",
"dist_comm",
"is_absolute_value",
"metric.ball",
"normed_ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cauchy_seq.is_cau_seq {f : ℕ → β} (hf : cauchy_seq f) :
is_cau_seq norm f | begin
cases cauchy_iff.1 hf with hf1 hf2,
intros ε hε,
rcases hf2 {x | dist x.1 x.2 < ε} (dist_mem_uniformity hε) with ⟨t, ⟨ht, htsub⟩⟩,
simp at ht, cases ht with N hN,
existsi N,
intros j hj,
rw ←dist_eq_norm,
apply @htsub (f j, f N),
apply set.mk_mem_prod; solve_by_elim [le_refl]
end | lemma | cauchy_seq.is_cau_seq | topology.metric_space | src/topology/metric_space/cau_seq_filter.lean | [
"analysis.normed.field.basic"
] | [
"cauchy_seq",
"is_cau_seq",
"set.mk_mem_prod"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cau_seq.cauchy_seq (f : cau_seq β norm) : cauchy_seq f | begin
refine cauchy_iff.2 ⟨by apply_instance, λ s hs, _⟩,
rcases mem_uniformity_dist.1 hs with ⟨ε, ⟨hε, hεs⟩⟩,
cases cau_seq.cauchy₂ f hε with N hN,
existsi {n | n ≥ N}.image f,
simp only [exists_prop, mem_at_top_sets, mem_map, mem_image, ge_iff_le, mem_set_of_eq],
split,
{ existsi N, intros b hb, existsi... | lemma | cau_seq.cauchy_seq | topology.metric_space | src/topology/metric_space/cau_seq_filter.lean | [
"analysis.normed.field.basic"
] | [
"cau_seq",
"cau_seq.cauchy₂",
"cauchy_seq",
"exists_prop",
"ge_iff_le",
"mem_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cau_seq_iff_cauchy_seq {α : Type u} [normed_field α] {u : ℕ → α} :
is_cau_seq norm u ↔ cauchy_seq u | ⟨λh, cau_seq.cauchy_seq ⟨u, h⟩,
λh, h.is_cau_seq⟩ | lemma | cau_seq_iff_cauchy_seq | topology.metric_space | src/topology/metric_space/cau_seq_filter.lean | [
"analysis.normed.field.basic"
] | [
"cau_seq.cauchy_seq",
"cauchy_seq",
"is_cau_seq",
"normed_field"
] | In a normed field, `cau_seq` coincides with the usual notion of Cauchy sequences. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
complete_space_of_cau_seq_complete [cau_seq.is_complete β norm] : complete_space β | begin
apply complete_of_cauchy_seq_tendsto,
assume u hu,
have C : is_cau_seq norm u := cau_seq_iff_cauchy_seq.2 hu,
existsi cau_seq.lim ⟨u, C⟩,
rw metric.tendsto_at_top,
assume ε εpos,
cases (cau_seq.equiv_lim ⟨u, C⟩) _ εpos with N hN,
existsi N,
simpa [dist_eq_norm] using hN
end | instance | complete_space_of_cau_seq_complete | topology.metric_space | src/topology/metric_space/cau_seq_filter.lean | [
"analysis.normed.field.basic"
] | [
"cau_seq.equiv_lim",
"cau_seq.is_complete",
"cau_seq.lim",
"complete_space",
"is_cau_seq",
"metric.tendsto_at_top"
] | A complete normed field is complete as a metric space, as Cauchy sequences converge by
assumption and this suffices to characterize completeness. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
closeds.emetric_space : emetric_space (closeds α) | { edist := λs t, Hausdorff_edist (s : set α) t,
edist_self := λs, Hausdorff_edist_self,
edist_comm := λs t, Hausdorff_edist_comm,
edist_triangle := λs t u, Hausdorff_edist_triangle,
eq_of_edist_eq_zero :=
λ s t h, closeds.ext $ (Hausdorff_edist_zero_iff_eq_of_closed s.cl... | instance | emetric.closeds.emetric_space | topology.metric_space | src/topology/metric_space/closeds.lean | [
"analysis.specific_limits.basic",
"topology.metric_space.hausdorff_distance",
"topology.sets.compacts"
] | [
"emetric_space"
] | In emetric spaces, the Hausdorff edistance defines an emetric space structure
on the type of closed subsets | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuous_inf_edist_Hausdorff_edist :
continuous (λ p : α × (closeds α), inf_edist p.1 p.2) | begin
refine continuous_of_le_add_edist 2 (by simp) _,
rintros ⟨x, s⟩ ⟨y, t⟩,
calc inf_edist x s ≤ inf_edist x t + Hausdorff_edist (t : set α) s :
inf_edist_le_inf_edist_add_Hausdorff_edist
... ≤ inf_edist y t + edist x y + Hausdorff_edist (t : set α) s :
add_le_add_right inf_edist_le_inf_edist_add_edis... | lemma | emetric.continuous_inf_edist_Hausdorff_edist | topology.metric_space | src/topology/metric_space/closeds.lean | [
"analysis.specific_limits.basic",
"topology.metric_space.hausdorff_distance",
"topology.sets.compacts"
] | [
"continuous",
"continuous_of_le_add_edist",
"mul_comm",
"mul_two"
] | The edistance to a closed set depends continuously on the point and the set | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_closed_subsets_of_is_closed (hs : is_closed s) :
is_closed {t : closeds α | (t : set α) ⊆ s} | begin
refine is_closed_of_closure_subset (λt ht x hx, _),
-- t : closeds α, ht : t ∈ closure {t : closeds α | t ⊆ s},
-- x : α, hx : x ∈ t
-- goal : x ∈ s
have : x ∈ closure s,
{ refine mem_closure_iff.2 (λε εpos, _),
rcases mem_closure_iff.1 ht ε εpos with ⟨u, hu, Dtu⟩,
-- u : closeds α, hu : u ... | lemma | emetric.is_closed_subsets_of_is_closed | topology.metric_space | src/topology/metric_space/closeds.lean | [
"analysis.specific_limits.basic",
"topology.metric_space.hausdorff_distance",
"topology.sets.compacts"
] | [
"closure",
"is_closed",
"is_closed_of_closure_subset"
] | Subsets of a given closed subset form a closed set | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
closeds.edist_eq {s t : closeds α} : edist s t = Hausdorff_edist (s : set α) t | rfl | lemma | emetric.closeds.edist_eq | topology.metric_space | src/topology/metric_space/closeds.lean | [
"analysis.specific_limits.basic",
"topology.metric_space.hausdorff_distance",
"topology.sets.compacts"
] | [] | By definition, the edistance on `closeds α` is given by the Hausdorff edistance | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
closeds.complete_space [complete_space α] : complete_space (closeds α) | begin
/- We will show that, if a sequence of sets `s n` satisfies
`edist (s n) (s (n+1)) < 2^{-n}`, then it converges. This is enough to guarantee
completeness, by a standard completeness criterion.
We use the shorthand `B n = 2^{-n}` in ennreal. -/
let B : ℕ → ℝ≥0∞ := λ n, (2⁻¹)^n,
have B_pos : ∀ n, (0:ℝ≥0... | instance | emetric.closeds.complete_space | topology.metric_space | src/topology/metric_space/closeds.lean | [
"analysis.specific_limits.basic",
"topology.metric_space.hausdorff_distance",
"topology.sets.compacts"
] | [
"cauchy_seq",
"cauchy_seq_of_edist_le_geometric_two",
"cauchy_seq_tendsto_of_complete",
"closure",
"complete_space",
"div_eq_mul_inv",
"edist_le_of_edist_le_geometric_two_of_tendsto₀",
"ennreal.inv_pow",
"ennreal.one_lt_two",
"ennreal.pow_ne_top",
"ennreal.pow_pos",
"ennreal.tendsto.const_mul"... | In a complete space, the type of closed subsets is complete for the
Hausdorff edistance. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
closeds.compact_space [compact_space α] : compact_space (closeds α) | ⟨begin
/- by completeness, it suffices to show that it is totally bounded,
i.e., for all ε>0, there is a finite set which is ε-dense.
start from a set `s` which is ε-dense in α. Then the subsets of `s`
are finitely many, and ε-dense for the Hausdorff distance. -/
refine is_compact_of_totally_bounded_is_... | instance | emetric.closeds.compact_space | topology.metric_space | src/topology/metric_space/closeds.lean | [
"analysis.specific_limits.basic",
"topology.metric_space.hausdorff_distance",
"topology.sets.compacts"
] | [
"and_imp",
"compact_space",
"exists_between",
"exists_imp_distrib",
"is_closed",
"is_closed_univ",
"is_compact.is_closed",
"is_compact_of_totally_bounded_is_closed",
"is_compact_univ",
"set.mem_image"
] | In a compact space, the type of closed subsets is compact. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
nonempty_compacts.emetric_space : emetric_space (nonempty_compacts α) | { edist := λ s t, Hausdorff_edist (s : set α) t,
edist_self := λs, Hausdorff_edist_self,
edist_comm := λs t, Hausdorff_edist_comm,
edist_triangle := λs t u, Hausdorff_edist_triangle,
eq_of_edist_eq_zero := λ s t h, nonempty_compacts.ext $ begin
have : closure (s : set α)... | instance | emetric.nonempty_compacts.emetric_space | topology.metric_space | src/topology/metric_space/closeds.lean | [
"analysis.specific_limits.basic",
"topology.metric_space.hausdorff_distance",
"topology.sets.compacts"
] | [
"closure",
"emetric_space"
] | In an emetric space, the type of non-empty compact subsets is an emetric space,
where the edistance is the Hausdorff edistance | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
nonempty_compacts.to_closeds.uniform_embedding :
uniform_embedding (@nonempty_compacts.to_closeds α _ _) | isometry.uniform_embedding $ λx y, rfl | lemma | emetric.nonempty_compacts.to_closeds.uniform_embedding | topology.metric_space | src/topology/metric_space/closeds.lean | [
"analysis.specific_limits.basic",
"topology.metric_space.hausdorff_distance",
"topology.sets.compacts"
] | [
"isometry.uniform_embedding",
"uniform_embedding"
] | `nonempty_compacts.to_closeds` is a uniform embedding (as it is an isometry) | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
nonempty_compacts.is_closed_in_closeds [complete_space α] :
is_closed (range $ @nonempty_compacts.to_closeds α _ _) | begin
have : range nonempty_compacts.to_closeds =
{s : closeds α | (s : set α).nonempty ∧ is_compact (s : set α) },
{ ext s,
refine ⟨_, λ h, ⟨⟨⟨s, h.2⟩, h.1⟩, closeds.ext rfl⟩⟩,
rintro ⟨s, hs, rfl⟩,
exact ⟨s.nonempty, s.is_compact⟩ },
rw this,
refine is_closed_of_closure_subset (λs hs, ⟨_, _⟩),
... | lemma | emetric.nonempty_compacts.is_closed_in_closeds | topology.metric_space | src/topology/metric_space/closeds.lean | [
"analysis.specific_limits.basic",
"topology.metric_space.hausdorff_distance",
"topology.sets.compacts"
] | [
"complete_space",
"ennreal.add_halves",
"ennreal.add_lt_add",
"ennreal.coe_lt_top",
"ennreal.half_pos",
"is_closed",
"is_closed_of_closure_subset",
"is_compact"
] | The range of `nonempty_compacts.to_closeds` is closed in a complete space | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
nonempty_compacts.complete_space [complete_space α] :
complete_space (nonempty_compacts α) | (complete_space_iff_is_complete_range
nonempty_compacts.to_closeds.uniform_embedding.to_uniform_inducing).2 $
nonempty_compacts.is_closed_in_closeds.is_complete | instance | emetric.nonempty_compacts.complete_space | topology.metric_space | src/topology/metric_space/closeds.lean | [
"analysis.specific_limits.basic",
"topology.metric_space.hausdorff_distance",
"topology.sets.compacts"
] | [
"complete_space",
"complete_space_iff_is_complete_range"
] | In a complete space, the type of nonempty compact subsets is complete. This follows
from the same statement for closed subsets | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
nonempty_compacts.compact_space [compact_space α] : compact_space (nonempty_compacts α) | ⟨begin
rw nonempty_compacts.to_closeds.uniform_embedding.embedding.is_compact_iff_is_compact_image,
rw [image_univ],
exact nonempty_compacts.is_closed_in_closeds.is_compact
end⟩ | instance | emetric.nonempty_compacts.compact_space | topology.metric_space | src/topology/metric_space/closeds.lean | [
"analysis.specific_limits.basic",
"topology.metric_space.hausdorff_distance",
"topology.sets.compacts"
] | [
"compact_space"
] | In a compact space, the type of nonempty compact subsets is compact. This follows from
the same statement for closed subsets | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
nonempty_compacts.second_countable_topology [second_countable_topology α] :
second_countable_topology (nonempty_compacts α) | begin
haveI : separable_space (nonempty_compacts α) :=
begin
/- To obtain a countable dense subset of `nonempty_compacts α`, start from
a countable dense subset `s` of α, and then consider all its finite nonempty subsets.
This set is countable and made of nonempty compact sets. It turns out to be dense:... | instance | emetric.nonempty_compacts.second_countable_topology | topology.metric_space | src/topology/metric_space/closeds.lean | [
"analysis.specific_limits.basic",
"topology.metric_space.hausdorff_distance",
"topology.sets.compacts"
] | [
"ennreal.add_halves",
"ennreal.add_lt_add",
"ennreal.half_pos",
"exists_between",
"ne_top_of_lt",
"set_like.coe_injective",
"totally_bounded",
"uniform_space.second_countable_of_separable"
] | In a second countable space, the type of nonempty compact subsets is second countable | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
nonempty_compacts.metric_space : metric_space (nonempty_compacts α) | emetric_space.to_metric_space $ λ x y, Hausdorff_edist_ne_top_of_nonempty_of_bounded
x.nonempty y.nonempty x.is_compact.bounded y.is_compact.bounded | instance | metric.nonempty_compacts.metric_space | topology.metric_space | src/topology/metric_space/closeds.lean | [
"analysis.specific_limits.basic",
"topology.metric_space.hausdorff_distance",
"topology.sets.compacts"
] | [
"emetric_space.to_metric_space",
"metric_space"
] | `nonempty_compacts α` inherits a metric space structure, as the Hausdorff
edistance between two such sets is finite. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
nonempty_compacts.dist_eq {x y : nonempty_compacts α} :
dist x y = Hausdorff_dist (x : set α) y | rfl | lemma | metric.nonempty_compacts.dist_eq | topology.metric_space | src/topology/metric_space/closeds.lean | [
"analysis.specific_limits.basic",
"topology.metric_space.hausdorff_distance",
"topology.sets.compacts"
] | [] | The distance on `nonempty_compacts α` is the Hausdorff distance, by construction | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lipschitz_inf_dist_set (x : α) : lipschitz_with 1 (λ s : nonempty_compacts α, inf_dist x s) | lipschitz_with.of_le_add $ assume s t,
by { rw dist_comm,
exact inf_dist_le_inf_dist_add_Hausdorff_dist (edist_ne_top t s) } | lemma | metric.lipschitz_inf_dist_set | topology.metric_space | src/topology/metric_space/closeds.lean | [
"analysis.specific_limits.basic",
"topology.metric_space.hausdorff_distance",
"topology.sets.compacts"
] | [
"dist_comm",
"edist_ne_top",
"lipschitz_with",
"lipschitz_with.of_le_add"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lipschitz_inf_dist : lipschitz_with 2 (λ p : α × (nonempty_compacts α), inf_dist p.1 p.2) | @lipschitz_with.uncurry _ _ _ _ _ _ (λ (x : α) (s : nonempty_compacts α), inf_dist x s) 1 1
(λ s, lipschitz_inf_dist_pt s) lipschitz_inf_dist_set | lemma | metric.lipschitz_inf_dist | topology.metric_space | src/topology/metric_space/closeds.lean | [
"analysis.specific_limits.basic",
"topology.metric_space.hausdorff_distance",
"topology.sets.compacts"
] | [
"lipschitz_with",
"lipschitz_with.uncurry"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
uniform_continuous_inf_dist_Hausdorff_dist :
uniform_continuous (λ p : α × (nonempty_compacts α), inf_dist p.1 p.2) | lipschitz_inf_dist.uniform_continuous | lemma | metric.uniform_continuous_inf_dist_Hausdorff_dist | topology.metric_space | src/topology/metric_space/closeds.lean | [
"analysis.specific_limits.basic",
"topology.metric_space.hausdorff_distance",
"topology.sets.compacts"
] | [
"uniform_continuous"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
uniform_continuous_dist :
uniform_continuous (λp:completion α × completion α, dist p.1 p.2) | uniform_continuous_extension₂ dist | lemma | uniform_space.completion.uniform_continuous_dist | topology.metric_space | src/topology/metric_space/completion.lean | [
"topology.uniform_space.completion",
"topology.metric_space.isometry",
"topology.instances.real"
] | [
"uniform_continuous",
"uniform_continuous_dist"
] | The new distance is uniformly continuous. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuous_dist [topological_space β] {f g : β → completion α} (hf : continuous f)
(hg : continuous g) :
continuous (λ x, dist (f x) (g x)) | completion.uniform_continuous_dist.continuous.comp (hf.prod_mk hg : _) | lemma | uniform_space.completion.continuous_dist | topology.metric_space | src/topology/metric_space/completion.lean | [
"topology.uniform_space.completion",
"topology.metric_space.isometry",
"topology.instances.real"
] | [
"continuous",
"continuous_dist",
"topological_space"
] | The new distance is continuous. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
dist_eq (x y : α) : dist (x : completion α) y = dist x y | completion.extension₂_coe_coe uniform_continuous_dist _ _ | lemma | uniform_space.completion.dist_eq | topology.metric_space | src/topology/metric_space/completion.lean | [
"topology.uniform_space.completion",
"topology.metric_space.isometry",
"topology.instances.real"
] | [
"uniform_continuous_dist"
] | The new distance is an extension of the original distance. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
dist_self (x : completion α) : dist x x = 0 | begin
apply induction_on x,
{ refine is_closed_eq _ continuous_const,
exact completion.continuous_dist continuous_id continuous_id },
{ assume a,
rw [completion.dist_eq, dist_self] }
end | lemma | uniform_space.completion.dist_self | topology.metric_space | src/topology/metric_space/completion.lean | [
"topology.uniform_space.completion",
"topology.metric_space.isometry",
"topology.instances.real"
] | [
"continuous_const",
"continuous_id",
"dist_self",
"is_closed_eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dist_comm (x y : completion α) : dist x y = dist y x | begin
apply induction_on₂ x y,
{ exact is_closed_eq (completion.continuous_dist continuous_fst continuous_snd)
(completion.continuous_dist continuous_snd continuous_fst) },
{ assume a b,
rw [completion.dist_eq, completion.dist_eq, dist_comm] }
end | lemma | uniform_space.completion.dist_comm | topology.metric_space | src/topology/metric_space/completion.lean | [
"topology.uniform_space.completion",
"topology.metric_space.isometry",
"topology.instances.real"
] | [
"continuous_fst",
"continuous_snd",
"dist_comm",
"is_closed_eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dist_triangle (x y z : completion α) : dist x z ≤ dist x y + dist y z | begin
apply induction_on₃ x y z,
{ refine is_closed_le _ (continuous.add _ _);
apply_rules [completion.continuous_dist, continuous.fst, continuous.snd, continuous_id] },
{ assume a b c,
rw [completion.dist_eq, completion.dist_eq, completion.dist_eq],
exact dist_triangle a b c }
end | lemma | uniform_space.completion.dist_triangle | topology.metric_space | src/topology/metric_space/completion.lean | [
"topology.uniform_space.completion",
"topology.metric_space.isometry",
"topology.instances.real"
] | [
"continuous.fst",
"continuous.snd",
"continuous_id",
"dist_triangle",
"is_closed_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_uniformity_dist (s : set (completion α × completion α)) :
s ∈ 𝓤 (completion α) ↔ (∃ε>0, ∀{a b}, dist a b < ε → (a, b) ∈ s) | begin
split,
{ /- Start from an entourage `s`. It contains a closed entourage `t`. Its pullback in `α` is an
entourage, so it contains an `ε`-neighborhood of the diagonal by definition of the entourages
in metric spaces. Then `t` contains an `ε`-neighborhood of the diagonal in `completion α`, as
closed ... | lemma | uniform_space.completion.mem_uniformity_dist | topology.metric_space | src/topology/metric_space/completion.lean | [
"topology.uniform_space.completion",
"topology.metric_space.isometry",
"topology.instances.real"
] | [
"continuous_const",
"exists_prop",
"filter.mem_map",
"is_closed.union",
"is_closed_le",
"le_abs_self",
"mem_uniformity_is_closed",
"metric.dist_mem_uniformity",
"real.dist_eq",
"refl_mem_uniformity",
"uniformity",
"uniformity_prod_eq_prod"
] | Elements of the uniformity (defined generally for completions) can be characterized in terms
of the distance. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
eq_of_dist_eq_zero (x y : completion α) (h : dist x y = 0) : x = y | begin
/- This follows from the separation of `completion α` and from the description of
entourages in terms of the distance. -/
have : separated_space (completion α) := by apply_instance,
refine separated_def.1 this x y (λs hs, _),
rcases (completion.mem_uniformity_dist s).1 hs with ⟨ε, εpos, hε⟩,
rw ← h at... | lemma | uniform_space.completion.eq_of_dist_eq_zero | topology.metric_space | src/topology/metric_space/completion.lean | [
"topology.uniform_space.completion",
"topology.metric_space.isometry",
"topology.instances.real"
] | [
"eq_of_dist_eq_zero",
"separated_space"
] | If two points are at distance 0, then they coincide. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
uniformity_dist' :
𝓤 (completion α) = (⨅ε:{ε : ℝ // 0 < ε}, 𝓟 {p | dist p.1 p.2 < ε.val}) | begin
ext s, rw mem_infi_of_directed,
{ simp [completion.mem_uniformity_dist, subset_def] },
{ rintro ⟨r, hr⟩ ⟨p, hp⟩, use ⟨min r p, lt_min hr hp⟩,
simp [lt_min_iff, (≥)] {contextual := tt} }
end | lemma | uniform_space.completion.uniformity_dist' | topology.metric_space | src/topology/metric_space/completion.lean | [
"topology.uniform_space.completion",
"topology.metric_space.isometry",
"topology.instances.real"
] | [
"lt_min_iff"
] | Reformulate `completion.mem_uniformity_dist` in terms that are suitable for the definition
of the metric space structure. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
uniformity_dist :
𝓤 (completion α) = (⨅ ε>0, 𝓟 {p | dist p.1 p.2 < ε}) | by simpa [infi_subtype] using @completion.uniformity_dist' α _ | lemma | uniform_space.completion.uniformity_dist | topology.metric_space | src/topology/metric_space/completion.lean | [
"topology.uniform_space.completion",
"topology.metric_space.isometry",
"topology.instances.real"
] | [
"infi_subtype"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_isometry : isometry (coe : α → completion α) | isometry.of_dist_eq completion.dist_eq | lemma | uniform_space.completion.coe_isometry | topology.metric_space | src/topology/metric_space/completion.lean | [
"topology.uniform_space.completion",
"topology.metric_space.isometry",
"topology.instances.real"
] | [
"isometry"
] | The embedding of a metric space in its completion is an isometry. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
edist_eq (x y : α) : edist (x : completion α) y = edist x y | coe_isometry x y | lemma | uniform_space.completion.edist_eq | topology.metric_space | src/topology/metric_space/completion.lean | [
"topology.uniform_space.completion",
"topology.metric_space.isometry",
"topology.instances.real"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
contracting_with [emetric_space α] (K : ℝ≥0) (f : α → α) | (K < 1) ∧ lipschitz_with K f | def | contracting_with | topology.metric_space | src/topology/metric_space/contracting.lean | [
"analysis.specific_limits.basic",
"data.setoid.basic",
"dynamics.fixed_points.topology"
] | [
"emetric_space",
"lipschitz_with"
] | A map is said to be `contracting_with K`, if `K < 1` and `f` is `lipschitz_with K`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_lipschitz_with (hf : contracting_with K f) : lipschitz_with K f | hf.2 | lemma | contracting_with.to_lipschitz_with | topology.metric_space | src/topology/metric_space/contracting.lean | [
"analysis.specific_limits.basic",
"data.setoid.basic",
"dynamics.fixed_points.topology"
] | [
"contracting_with",
"lipschitz_with"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_sub_K_pos' (hf : contracting_with K f) : (0:ℝ≥0∞) < 1 - K | by simp [hf.1] | lemma | contracting_with.one_sub_K_pos' | topology.metric_space | src/topology/metric_space/contracting.lean | [
"analysis.specific_limits.basic",
"data.setoid.basic",
"dynamics.fixed_points.topology"
] | [
"contracting_with"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_sub_K_ne_zero (hf : contracting_with K f) : (1:ℝ≥0∞) - K ≠ 0 | ne_of_gt hf.one_sub_K_pos' | lemma | contracting_with.one_sub_K_ne_zero | topology.metric_space | src/topology/metric_space/contracting.lean | [
"analysis.specific_limits.basic",
"data.setoid.basic",
"dynamics.fixed_points.topology"
] | [
"contracting_with"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_sub_K_ne_top : (1:ℝ≥0∞) - K ≠ ∞ | by { norm_cast, exact ennreal.coe_ne_top } | lemma | contracting_with.one_sub_K_ne_top | topology.metric_space | src/topology/metric_space/contracting.lean | [
"analysis.specific_limits.basic",
"data.setoid.basic",
"dynamics.fixed_points.topology"
] | [
"ennreal.coe_ne_top"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
edist_inequality (hf : contracting_with K f) {x y} (h : edist x y ≠ ∞) :
edist x y ≤ (edist x (f x) + edist y (f y)) / (1 - K) | suffices edist x y ≤ edist x (f x) + edist y (f y) + K * edist x y,
by rwa [ennreal.le_div_iff_mul_le (or.inl hf.one_sub_K_ne_zero) (or.inl one_sub_K_ne_top),
mul_comm, ennreal.sub_mul (λ _ _, h), one_mul, tsub_le_iff_right],
calc edist x y ≤ edist x (f x) + edist (f x) (f y) + edist (f y) y : edist_triangle4 _ _... | lemma | contracting_with.edist_inequality | topology.metric_space | src/topology/metric_space/contracting.lean | [
"analysis.specific_limits.basic",
"data.setoid.basic",
"dynamics.fixed_points.topology"
] | [
"contracting_with",
"edist_triangle4",
"ennreal.le_div_iff_mul_le",
"ennreal.sub_mul",
"le_rfl",
"mul_comm",
"one_mul",
"tsub_le_iff_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
edist_le_of_fixed_point (hf : contracting_with K f) {x y}
(h : edist x y ≠ ∞) (hy : is_fixed_pt f y) :
edist x y ≤ (edist x (f x)) / (1 - K) | by simpa only [hy.eq, edist_self, add_zero] using hf.edist_inequality h | lemma | contracting_with.edist_le_of_fixed_point | topology.metric_space | src/topology/metric_space/contracting.lean | [
"analysis.specific_limits.basic",
"data.setoid.basic",
"dynamics.fixed_points.topology"
] | [
"contracting_with"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_or_edist_eq_top_of_fixed_points (hf : contracting_with K f) {x y}
(hx : is_fixed_pt f x) (hy : is_fixed_pt f y) :
x = y ∨ edist x y = ∞ | begin
refine or_iff_not_imp_right.2 (λ h, edist_le_zero.1 _),
simpa only [hx.eq, edist_self, add_zero, ennreal.zero_div]
using hf.edist_le_of_fixed_point h hy
end | lemma | contracting_with.eq_or_edist_eq_top_of_fixed_points | topology.metric_space | src/topology/metric_space/contracting.lean | [
"analysis.specific_limits.basic",
"data.setoid.basic",
"dynamics.fixed_points.topology"
] | [
"contracting_with",
"ennreal.zero_div"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
restrict (hf : contracting_with K f) {s : set α} (hs : maps_to f s s) :
contracting_with K (hs.restrict f s s) | ⟨hf.1, λ x y, hf.2 x y⟩ | lemma | contracting_with.restrict | topology.metric_space | src/topology/metric_space/contracting.lean | [
"analysis.specific_limits.basic",
"data.setoid.basic",
"dynamics.fixed_points.topology"
] | [
"contracting_with"
] | If a map `f` is `contracting_with K`, and `s` is a forward-invariant set, then
restriction of `f` to `s` is `contracting_with K` as well. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
exists_fixed_point (hf : contracting_with K f) (x : α) (hx : edist x (f x) ≠ ∞) :
∃ y, is_fixed_pt f y ∧ tendsto (λ n, f^[n] x) at_top (𝓝 y) ∧
∀ n:ℕ, edist (f^[n] x) y ≤ (edist x (f x)) * K^n / (1 - K) | have cauchy_seq (λ n, f^[n] x),
from cauchy_seq_of_edist_le_geometric K (edist x (f x)) (ennreal.coe_lt_one_iff.2 hf.1)
hx (hf.to_lipschitz_with.edist_iterate_succ_le_geometric x),
let ⟨y, hy⟩ := cauchy_seq_tendsto_of_complete this in
⟨y, is_fixed_pt_of_tendsto_iterate hy hf.2.continuous.continuous_at, hy,
edist_le... | theorem | contracting_with.exists_fixed_point | topology.metric_space | src/topology/metric_space/contracting.lean | [
"analysis.specific_limits.basic",
"data.setoid.basic",
"dynamics.fixed_points.topology"
] | [
"cauchy_seq",
"cauchy_seq_of_edist_le_geometric",
"cauchy_seq_tendsto_of_complete",
"contracting_with",
"edist_le_of_edist_le_geometric_of_tendsto",
"is_fixed_pt_of_tendsto_iterate"
] | Banach fixed-point theorem, contraction mapping theorem, `emetric_space` version.
A contracting map on a complete metric space has a fixed point.
We include more conclusions in this theorem to avoid proving them again later.
The main API for this theorem are the functions `efixed_point` and `fixed_point`,
and lemmas a... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
efixed_point (hf : contracting_with K f) (x : α) (hx : edist x (f x) ≠ ∞) :
α | classical.some $ hf.exists_fixed_point x hx | def | contracting_with.efixed_point | topology.metric_space | src/topology/metric_space/contracting.lean | [
"analysis.specific_limits.basic",
"data.setoid.basic",
"dynamics.fixed_points.topology"
] | [
"contracting_with"
] | Let `x` be a point of a complete emetric space. Suppose that `f` is a contracting map,
and `edist x (f x) ≠ ∞`. Then `efixed_point` is the unique fixed point of `f`
in `emetric.ball x ∞`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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