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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exists_lipschitz_retraction_of_is_closed
{s : set (Π n, E n)} (hs : is_closed s) (hne : s.nonempty) :
∃ f : (Π n, E n) → (Π n, E n), (∀ x ∈ s, f x = x) ∧ (range f = s) ∧ lipschitz_with 1 f | begin
/- The map `f` is defined as follows. For `x ∈ s`, let `f x = x`. Otherwise, consider the longest
prefix `w` that `x` shares with an element of `s`, and let `f x = z_w` where `z_w` is an element
of `s` starting with `w`. All the desired properties are clear, except the fact that `f`
is `1`-Lipschitz: if t... | theorem | pi_nat.exists_lipschitz_retraction_of_is_closed | topology.metric_space | src/topology/metric_space/pi_nat.lean | [
"tactic.ring_exp",
"topology.metric_space.hausdorff_distance"
] | [
"dist_nonneg",
"eq_or_ne",
"is_closed",
"lipschitz_with",
"lipschitz_with.mk_one"
] | Given a closed nonempty subset `s` of `Π (n : ℕ), E n`, there exists a Lipschitz retraction
onto this set, i.e., a Lipschitz map with range equal to `s`, equal to the identity on `s`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
exists_retraction_of_is_closed
{s : set (Π n, E n)} (hs : is_closed s) (hne : s.nonempty) :
∃ f : (Π n, E n) → (Π n, E n), (∀ x ∈ s, f x = x) ∧ (range f = s) ∧ continuous f | begin
rcases exists_lipschitz_retraction_of_is_closed hs hne with ⟨f, fs, frange, hf⟩,
exact ⟨f, fs, frange, hf.continuous⟩
end | theorem | pi_nat.exists_retraction_of_is_closed | topology.metric_space | src/topology/metric_space/pi_nat.lean | [
"tactic.ring_exp",
"topology.metric_space.hausdorff_distance"
] | [
"continuous",
"is_closed"
] | Given a closed nonempty subset `s` of `Π (n : ℕ), E n`, there exists a retraction onto this
set, i.e., a continuous map with range equal to `s`, equal to the identity on `s`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
exists_retraction_subtype_of_is_closed
{s : set (Π n, E n)} (hs : is_closed s) (hne : s.nonempty) :
∃ f : (Π n, E n) → s, (∀ x : s, f x = x) ∧ surjective f ∧ continuous f | begin
obtain ⟨f, fs, f_range, f_cont⟩ : ∃ f : (Π n, E n) → (Π n, E n),
(∀ x ∈ s, f x = x) ∧ (range f = s) ∧ continuous f :=
exists_retraction_of_is_closed hs hne,
have A : ∀ x, f x ∈ s, by simp [← f_range],
have B : ∀ (x : s), cod_restrict f s A x = x,
{ assume x,
apply subtype.coe_injective.eq_if... | theorem | pi_nat.exists_retraction_subtype_of_is_closed | topology.metric_space | src/topology/metric_space/pi_nat.lean | [
"tactic.ring_exp",
"topology.metric_space.hausdorff_distance"
] | [
"continuous",
"is_closed"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exists_nat_nat_continuous_surjective_of_complete_space
(α : Type*) [metric_space α] [complete_space α] [second_countable_topology α] [nonempty α] :
∃ (f : (ℕ → ℕ) → α), continuous f ∧ surjective f | begin
/- First, we define a surjective map from a closed subset `s` of `ℕ → ℕ`. Then, we compose
this map with a retraction of `ℕ → ℕ` onto `s` to obtain the desired map.
Let us consider a dense sequence `u` in `α`. Then `s` is the set of sequences `xₙ` such that the
balls `closed_ball (u xₙ) (1/2^n)` have a no... | lemma | exists_nat_nat_continuous_surjective_of_complete_space | topology.metric_space | src/topology/metric_space/pi_nat.lean | [
"tactic.ring_exp",
"topology.metric_space.hausdorff_distance"
] | [
"complete_space",
"continuous",
"continuous_at_of_locally_lipschitz",
"dist_triangle_right",
"eq_or_ne",
"ge_of_tendsto'",
"is_closed",
"le_rfl",
"metric_space",
"mul_zero",
"pi_nat.metric_space_nat_nat",
"pow_nonneg",
"pow_pos",
"pow_zero",
"squeeze_zero",
"tendsto_pow_at_top_nhds_0_o... | Any nonempty complete second countable metric space is the continuous image of the
fundamental space `ℕ → ℕ`. For a version of this theorem in the context of Polish spaces, see
`exists_nat_nat_continuous_surjective_of_polish_space`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_dist : has_dist (Π i, F i) | ⟨λ x y, ∑' (i : ι), min ((1/2)^(encode i)) (dist (x i) (y i))⟩ | def | pi_countable.has_dist | topology.metric_space | src/topology/metric_space/pi_nat.lean | [
"tactic.ring_exp",
"topology.metric_space.hausdorff_distance"
] | [
"has_dist"
] | Given a countable family of metric spaces, one may put a distance on their product `Π i, E i`.
It is highly non-canonical, though, and therefore not registered as a global instance.
The distance we use here is `dist x y = ∑' i, min (1/2)^(encode i) (dist (x i) (y i))`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
dist_eq_tsum (x y : Π i, F i) :
dist x y = ∑' (i : ι), min ((1/2)^(encode i)) (dist (x i) (y i)) | rfl | lemma | pi_countable.dist_eq_tsum | topology.metric_space | src/topology/metric_space/pi_nat.lean | [
"tactic.ring_exp",
"topology.metric_space.hausdorff_distance"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dist_summable (x y : Π i, F i) :
summable (λ (i : ι), min ((1/2)^(encode i)) (dist (x i) (y i))) | begin
refine summable_of_nonneg_of_le (λ i, _) (λ i, min_le_left _ _) summable_geometric_two_encode,
exact le_min (pow_nonneg (by norm_num) _) (dist_nonneg)
end | lemma | pi_countable.dist_summable | topology.metric_space | src/topology/metric_space/pi_nat.lean | [
"tactic.ring_exp",
"topology.metric_space.hausdorff_distance"
] | [
"dist_nonneg",
"pow_nonneg",
"summable",
"summable_geometric_two_encode",
"summable_of_nonneg_of_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
min_dist_le_dist_pi (x y : Π i, F i) (i : ι) :
min ((1/2)^(encode i)) (dist (x i) (y i)) ≤ dist x y | le_tsum (dist_summable x y) i (λ j hj, le_min (by simp) (dist_nonneg)) | lemma | pi_countable.min_dist_le_dist_pi | topology.metric_space | src/topology/metric_space/pi_nat.lean | [
"tactic.ring_exp",
"topology.metric_space.hausdorff_distance"
] | [
"dist_nonneg",
"le_tsum"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dist_le_dist_pi_of_dist_lt {x y : Π i, F i} {i : ι} (h : dist x y < (1/2)^(encode i)) :
dist (x i) (y i) ≤ dist x y | by simpa only [not_le.2 h, false_or] using min_le_iff.1 (min_dist_le_dist_pi x y i) | lemma | pi_countable.dist_le_dist_pi_of_dist_lt | topology.metric_space | src/topology/metric_space/pi_nat.lean | [
"tactic.ring_exp",
"topology.metric_space.hausdorff_distance"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
metric_space : metric_space (Π i, F i) | { dist_self := λ x, by simp [dist_eq_tsum],
dist_comm := λ x y, by simp [dist_eq_tsum, dist_comm],
dist_triangle := λ x y z,
begin
have I : ∀ i, min ((1/2)^(encode i)) (dist (x i) (z i)) ≤
min ((1/2)^(encode i)) (dist (x i) (y i)) + min ((1/2)^(encode i)) (dist (y i) (z i)) :=
λ i, calc
min ((... | def | pi_countable.metric_space | topology.metric_space | src/topology/metric_space/pi_nat.lean | [
"tactic.ring_exp",
"topology.metric_space.hausdorff_distance"
] | [
"Pi.uniform_space",
"Pi.uniformity",
"add_halves",
"and_imp",
"dist_comm",
"dist_le_zero",
"dist_self",
"dist_triangle",
"div_pos",
"eq_of_dist_eq_zero",
"finset",
"finset.mem_coe",
"finset.mem_range",
"gt_iff_lt",
"half_pos",
"imp_self",
"le_infi_iff",
"le_rfl",
"lt_min_iff",
... | Given a countable family of metric spaces, one may put a distance on their product `Π i, E i`,
defining the right topology and uniform structure. It is highly non-canonical, though, and therefore
not registered as a global instance.
The distance we use here is `dist x y = ∑' n, min (1/2)^(encode i) (dist (x n) (y n))`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
polish_space (α : Type*) [h : topological_space α] : Prop | (second_countable [] : second_countable_topology α)
(complete : ∃ m : metric_space α, m.to_uniform_space.to_topological_space = h ∧
@complete_space α m.to_uniform_space) | class | polish_space | topology.metric_space | src/topology/metric_space/polish.lean | [
"topology.metric_space.pi_nat",
"topology.metric_space.isometry",
"topology.metric_space.gluing",
"analysis.normed.field.basic"
] | [
"complete_space",
"metric_space",
"topological_space"
] | A Polish space is a topological space with second countable topology, that can be endowed
with a metric for which it is complete.
We register an instance from complete second countable metric space to polish space, and not the
other way around as this is the most common use case.
To endow a Polish space with a complet... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
upgraded_polish_space (α : Type*) extends metric_space α, second_countable_topology α,
complete_space α | class | upgraded_polish_space | topology.metric_space | src/topology/metric_space/polish.lean | [
"topology.metric_space.pi_nat",
"topology.metric_space.isometry",
"topology.metric_space.gluing",
"analysis.normed.field.basic"
] | [
"complete_space",
"metric_space"
] | A convenience class, for a Polish space endowed with a complete metric. No instance of this
class should be registered: It should be used as `letI := upgrade_polish_space α` to endow a Polish
space with a complete metric. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
polish_space_of_complete_second_countable
[m : metric_space α] [h : second_countable_topology α] [h' : complete_space α] :
polish_space α | { second_countable := h,
complete := ⟨m, rfl, h'⟩ } | instance | polish_space_of_complete_second_countable | topology.metric_space | src/topology/metric_space/polish.lean | [
"topology.metric_space.pi_nat",
"topology.metric_space.isometry",
"topology.metric_space.gluing",
"analysis.normed.field.basic"
] | [
"complete_space",
"metric_space",
"polish_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
polish_space_metric (α : Type*) [ht : topological_space α] [h : polish_space α] :
metric_space α | h.complete.some.replace_topology h.complete.some_spec.1.symm | def | polish_space_metric | topology.metric_space | src/topology/metric_space/polish.lean | [
"topology.metric_space.pi_nat",
"topology.metric_space.isometry",
"topology.metric_space.gluing",
"analysis.normed.field.basic"
] | [
"metric_space",
"polish_space",
"topological_space"
] | Construct on a Polish space a metric (compatible with the topology) which is complete. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
complete_polish_space_metric (α : Type*) [ht : topological_space α] [h : polish_space α] :
@complete_space α (polish_space_metric α).to_uniform_space | begin
convert h.complete.some_spec.2,
exact metric_space.replace_topology_eq _ _
end | lemma | complete_polish_space_metric | topology.metric_space | src/topology/metric_space/polish.lean | [
"topology.metric_space.pi_nat",
"topology.metric_space.isometry",
"topology.metric_space.gluing",
"analysis.normed.field.basic"
] | [
"complete_space",
"metric_space.replace_topology_eq",
"polish_space",
"polish_space_metric",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
upgrade_polish_space (α : Type*) [ht : topological_space α] [h : polish_space α] :
upgraded_polish_space α | begin
letI := polish_space_metric α,
exact { .. complete_polish_space_metric α, .. polish_space.second_countable α }
end | def | upgrade_polish_space | topology.metric_space | src/topology/metric_space/polish.lean | [
"topology.metric_space.pi_nat",
"topology.metric_space.isometry",
"topology.metric_space.gluing",
"analysis.normed.field.basic"
] | [
"complete_polish_space_metric",
"polish_space",
"polish_space_metric",
"topological_space",
"upgraded_polish_space"
] | This definition endows a Polish space with a complete metric. Use it as:
`letI := upgrade_polish_space α`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
t2_space (α : Type*) [topological_space α] [polish_space α] : t2_space α | by { letI := upgrade_polish_space α, apply_instance } | instance | polish_space.t2_space | topology.metric_space | src/topology/metric_space/polish.lean | [
"topology.metric_space.pi_nat",
"topology.metric_space.isometry",
"topology.metric_space.gluing",
"analysis.normed.field.basic"
] | [
"polish_space",
"t2_space",
"topological_space",
"upgrade_polish_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pi_countable {ι : Type*} [countable ι] {E : ι → Type*}
[∀ i, topological_space (E i)] [∀ i, polish_space (E i)] :
polish_space (Π i, E i) | begin
casesI nonempty_encodable ι,
letI := λ i, upgrade_polish_space (E i),
letI : metric_space (Π i, E i) := pi_countable.metric_space,
apply_instance,
end | instance | polish_space.pi_countable | topology.metric_space | src/topology/metric_space/polish.lean | [
"topology.metric_space.pi_nat",
"topology.metric_space.isometry",
"topology.metric_space.gluing",
"analysis.normed.field.basic"
] | [
"countable",
"metric_space",
"nonempty_encodable",
"pi_countable.metric_space",
"polish_space",
"topological_space",
"upgrade_polish_space"
] | A countable product of Polish spaces is Polish. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
nat_fun [topological_space α] [polish_space α] :
polish_space (ℕ → α) | by apply_instance | instance | polish_space.nat_fun | topology.metric_space | src/topology/metric_space/polish.lean | [
"topology.metric_space.pi_nat",
"topology.metric_space.isometry",
"topology.metric_space.gluing",
"analysis.normed.field.basic"
] | [
"polish_space",
"topological_space"
] | Without this instance, `polish_space (ℕ → ℕ)` is not found by typeclass inference. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
sigma {ι : Type*} [countable ι]
{E : ι → Type*} [∀ n, topological_space (E n)] [∀ n, polish_space (E n)] :
polish_space (Σ n, E n) | begin
letI := λ n, upgrade_polish_space (E n),
letI : metric_space (Σ n, E n) := sigma.metric_space,
haveI : complete_space (Σ n, E n) := sigma.complete_space,
apply_instance
end | instance | polish_space.sigma | topology.metric_space | src/topology/metric_space/polish.lean | [
"topology.metric_space.pi_nat",
"topology.metric_space.isometry",
"topology.metric_space.gluing",
"analysis.normed.field.basic"
] | [
"complete_space",
"countable",
"metric_space",
"polish_space",
"topological_space",
"upgrade_polish_space"
] | A countable disjoint union of Polish spaces is Polish. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
sum [topological_space α] [polish_space α] [topological_space β] [polish_space β] :
polish_space (α ⊕ β) | begin
letI := upgrade_polish_space α,
letI := upgrade_polish_space β,
letI : metric_space (α ⊕ β) := metric_space_sum,
apply_instance
end | instance | polish_space.sum | topology.metric_space | src/topology/metric_space/polish.lean | [
"topology.metric_space.pi_nat",
"topology.metric_space.isometry",
"topology.metric_space.gluing",
"analysis.normed.field.basic"
] | [
"metric_space",
"polish_space",
"topological_space",
"upgrade_polish_space"
] | The disjoint union of two Polish spaces is Polish. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
exists_nat_nat_continuous_surjective
(α : Type*) [topological_space α] [polish_space α] [nonempty α] :
∃ (f : (ℕ → ℕ) → α), continuous f ∧ surjective f | begin
letI := upgrade_polish_space α,
exact exists_nat_nat_continuous_surjective_of_complete_space α
end | lemma | polish_space.exists_nat_nat_continuous_surjective | topology.metric_space | src/topology/metric_space/polish.lean | [
"topology.metric_space.pi_nat",
"topology.metric_space.isometry",
"topology.metric_space.gluing",
"analysis.normed.field.basic"
] | [
"continuous",
"exists_nat_nat_continuous_surjective_of_complete_space",
"polish_space",
"topological_space",
"upgrade_polish_space"
] | Any nonempty Polish space is the continuous image of the fundamental space `ℕ → ℕ`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
_root_.closed_embedding.polish_space [topological_space α] [topological_space β]
[polish_space β] {f : α → β} (hf : closed_embedding f) :
polish_space α | begin
letI := upgrade_polish_space β,
letI : metric_space α := hf.to_embedding.comap_metric_space f,
haveI : second_countable_topology α := hf.to_embedding.second_countable_topology,
haveI : complete_space α,
{ rw complete_space_iff_is_complete_range hf.to_embedding.to_isometry.uniform_inducing,
apply is_... | lemma | closed_embedding.polish_space | topology.metric_space | src/topology/metric_space/polish.lean | [
"topology.metric_space.pi_nat",
"topology.metric_space.isometry",
"topology.metric_space.gluing",
"analysis.normed.field.basic"
] | [
"closed_embedding",
"complete_space",
"complete_space_iff_is_complete_range",
"is_closed.is_complete",
"metric_space",
"polish_space",
"topological_space",
"upgrade_polish_space"
] | Given a closed embedding into a Polish space, the source space is also Polish. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
_root_.equiv.polish_space_induced [t : topological_space β] [polish_space β]
(f : α ≃ β) :
@polish_space α (t.induced f) | begin
letI : topological_space α := t.induced f,
exact (f.to_homeomorph_of_inducing ⟨rfl⟩).closed_embedding.polish_space,
end | lemma | equiv.polish_space_induced | topology.metric_space | src/topology/metric_space/polish.lean | [
"topology.metric_space.pi_nat",
"topology.metric_space.isometry",
"topology.metric_space.gluing",
"analysis.normed.field.basic"
] | [
"closed_embedding.polish_space",
"polish_space",
"topological_space"
] | Pulling back a Polish topology under an equiv gives again a Polish topology. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
_root_.is_closed.polish_space {α : Type*} [topological_space α] [polish_space α] {s : set α}
(hs : is_closed s) :
polish_space s | (is_closed.closed_embedding_subtype_coe hs).polish_space | lemma | is_closed.polish_space | topology.metric_space | src/topology/metric_space/polish.lean | [
"topology.metric_space.pi_nat",
"topology.metric_space.isometry",
"topology.metric_space.gluing",
"analysis.normed.field.basic"
] | [
"is_closed",
"is_closed.closed_embedding_subtype_coe",
"polish_space",
"topological_space"
] | A closed subset of a Polish space is also Polish. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
aux_copy (α : Type*) {ι : Type*} (i : ι) : Type* | α | def | polish_space.aux_copy | topology.metric_space | src/topology/metric_space/polish.lean | [
"topology.metric_space.pi_nat",
"topology.metric_space.isometry",
"topology.metric_space.gluing",
"analysis.normed.field.basic"
] | [] | A sequence of type synonyms of a given type `α`, useful in the proof of
`exists_polish_space_forall_le` to endow each copy with a different topology. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
exists_polish_space_forall_le {ι : Type*} [countable ι]
[t : topological_space α] [p : polish_space α]
(m : ι → topological_space α) (hm : ∀ n, m n ≤ t) (h'm : ∀ n, @polish_space α (m n)) :
∃ (t' : topological_space α), (∀ n, t' ≤ m n) ∧ (t' ≤ t) ∧ @polish_space α t' | begin
rcases is_empty_or_nonempty ι with hι|hι,
{ exact ⟨t, λ i, (is_empty.elim hι i : _), le_rfl, p⟩ },
unfreezingI { inhabit ι },
/- Consider the product of infinitely many copies of `α`, each endowed with the topology `m n`.
This is a Polish space, as a product of Polish spaces. Pulling back this topology ... | lemma | polish_space.exists_polish_space_forall_le | topology.metric_space | src/topology/metric_space/polish.lean | [
"topology.metric_space.pi_nat",
"topology.metric_space.isometry",
"topology.metric_space.gluing",
"analysis.normed.field.basic"
] | [
"closed_embedding",
"closed_embedding.polish_space",
"continuous",
"continuous.comp",
"continuous_apply",
"countable",
"function.injective.embedding_induced",
"is_closed",
"is_closed_Inter",
"is_closed_eq",
"is_empty.elim",
"is_empty_or_nonempty",
"is_open_set_pi",
"le_rfl",
"polish_spac... | Given a Polish space, and countably many finer Polish topologies, there exists another Polish
topology which is finer than all of them. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
complete_copy {α : Type*} (s : set α) : Type* | s | def | polish_space.complete_copy | topology.metric_space | src/topology/metric_space/polish.lean | [
"topology.metric_space.pi_nat",
"topology.metric_space.isometry",
"topology.metric_space.gluing",
"analysis.normed.field.basic"
] | [] | A type synonym for a subset `s` of a metric space, on which we will construct another metric
for which it will be complete. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_dist_complete_copy (s : set α) : has_dist (complete_copy s) | ⟨λ x y, dist x.1 y.1 + abs (1 / inf_dist x.1 sᶜ - 1 / inf_dist y.1 sᶜ)⟩ | def | polish_space.has_dist_complete_copy | topology.metric_space | src/topology/metric_space/polish.lean | [
"topology.metric_space.pi_nat",
"topology.metric_space.isometry",
"topology.metric_space.gluing",
"analysis.normed.field.basic"
] | [
"has_dist"
] | A distance on a subset `s` of a metric space, designed to make it complete if `s` is open.
It is given by `dist' x y = dist x y + |1 / dist x sᶜ - 1 / dist y sᶜ|`, where the second term
blows up close to the boundary to ensure that Cauchy sequences for `dist'` remain well
inside `s`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
dist_complete_copy_eq (x y : complete_copy s) :
dist x y = dist x.1 y.1 + abs (1/inf_dist x.1 sᶜ - 1 / inf_dist y.1 sᶜ) | rfl | lemma | polish_space.dist_complete_copy_eq | topology.metric_space | src/topology/metric_space/polish.lean | [
"topology.metric_space.pi_nat",
"topology.metric_space.isometry",
"topology.metric_space.gluing",
"analysis.normed.field.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dist_le_dist_complete_copy (x y : complete_copy s) :
dist x.1 y.1 ≤ dist x y | (le_add_iff_nonneg_right _).2 (abs_nonneg _) | lemma | polish_space.dist_le_dist_complete_copy | topology.metric_space | src/topology/metric_space/polish.lean | [
"topology.metric_space.pi_nat",
"topology.metric_space.isometry",
"topology.metric_space.gluing",
"analysis.normed.field.basic"
] | [
"abs_nonneg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
complete_copy_metric_space (s : set α) : metric_space (complete_copy s) | { dist_self := λ x, by simp [dist_complete_copy_eq],
dist_comm := λ x y, by simp [dist_complete_copy_eq, dist_comm, abs_sub_comm],
dist_triangle := λ x y z, calc
dist x z = dist x.1 z.1 + abs (1 / inf_dist x.1 sᶜ - 1 / inf_dist z.1 sᶜ) : rfl
... ≤ (dist x.1 y.1 + dist y.1 z.1) +
(abs (1 / inf_dist x.1... | def | polish_space.complete_copy_metric_space | topology.metric_space | src/topology/metric_space/polish.lean | [
"topology.metric_space.pi_nat",
"topology.metric_space.isometry",
"topology.metric_space.gluing",
"analysis.normed.field.basic"
] | [
"abs_sub_comm",
"dist_comm",
"dist_self",
"dist_triangle",
"eq_of_dist_eq_zero",
"metric_space",
"real.dist_eq",
"subtype.coe_injective"
] | A metric space structure on a subset `s` of a metric space, designed to make it complete
if `s` is open. It is given by `dist' x y = dist x y + |1 / dist x sᶜ - 1 / dist y sᶜ|`, where the
second term blows up close to the boundary to ensure that Cauchy sequences for `dist'` remain well
inside `s`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
complete_copy_id_homeo (hs : is_open s) (h's : sᶜ.nonempty) : complete_copy s ≃ₜ s | { to_fun := id,
inv_fun := id,
left_inv := λ x, rfl,
right_inv := λ x, rfl,
continuous_to_fun :=
begin
have : lipschitz_with 1 (λ (x : complete_copy s), (id x : s)),
{ apply lipschitz_with.mk_one,
exact dist_le_dist_complete_copy },
exact this.continuous,
end,
continuous_inv_fun :=
beg... | def | polish_space.complete_copy_id_homeo | topology.metric_space | src/topology/metric_space/polish.lean | [
"topology.metric_space.pi_nat",
"topology.metric_space.isometry",
"topology.metric_space.gluing",
"analysis.normed.field.basic"
] | [
"abs_zero",
"continuous.tendsto",
"continuous_at",
"continuous_const",
"continuous_subtype_coe",
"dist_self",
"inv_fun",
"is_open",
"lipschitz_with",
"lipschitz_with.mk_one",
"tendsto_const_nhds",
"tendsto_iff_dist_tendsto_zero"
] | The identity between the type synonym `complete_copy s` (with its modified metric) and the
original subtype `s` is a homeomorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
complete_space_complete_copy [complete_space α] (hs : is_open s) (h's : sᶜ.nonempty) :
complete_space (complete_copy s) | begin
refine metric.complete_of_convergent_controlled_sequences (λ n, (1/2)^n) (by simp) _,
assume u hu,
have A : cauchy_seq (λ n, (u n).1),
{ apply cauchy_seq_of_le_tendsto_0 (λ (n : ℕ), (1/2)^n) (λ n m N hNn hNm, _) _,
{ exact (dist_le_dist_complete_copy (u n) (u m)).trans (hu N n m hNn hNm).le },
{ e... | lemma | polish_space.complete_space_complete_copy | topology.metric_space | src/topology/metric_space/polish.lean | [
"topology.metric_space.pi_nat",
"topology.metric_space.isometry",
"topology.metric_space.gluing",
"analysis.normed.field.basic"
] | [
"cauchy_seq",
"cauchy_seq_of_le_tendsto_0",
"complete_space",
"dist_nonneg",
"dist_triangle",
"div_le_iff",
"div_le_iff'",
"ge_of_tendsto'",
"is_open",
"le_abs_self",
"le_rfl",
"metric.complete_of_convergent_controlled_sequences",
"real.dist_0_eq_abs",
"tendsto_pow_at_top_nhds_0_of_lt_1"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.is_open.polish_space {α : Type*} [topological_space α] [polish_space α] {s : set α}
(hs : is_open s) :
polish_space s | begin
rcases eq_empty_or_nonempty sᶜ with h's|h's,
{ simp at h's,
apply is_closed.polish_space,
rw h's,
exact is_closed_univ },
{ letI := upgrade_polish_space α,
haveI : complete_space (complete_copy s) := complete_space_complete_copy hs h's,
haveI : second_countable_topology (complete_copy s)... | lemma | is_open.polish_space | topology.metric_space | src/topology/metric_space/polish.lean | [
"topology.metric_space.pi_nat",
"topology.metric_space.isometry",
"topology.metric_space.gluing",
"analysis.normed.field.basic"
] | [
"complete_space",
"embedding.second_countable_topology",
"is_closed.polish_space",
"is_closed_univ",
"is_open",
"polish_space",
"topological_space",
"upgrade_polish_space"
] | An open subset of a Polish space is also Polish. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_clopenable [t : topological_space α] (s : set α) : Prop | ∃ (t' : topological_space α), t' ≤ t ∧ @polish_space α t' ∧ is_closed[t'] s ∧ is_open[t'] s | def | polish_space.is_clopenable | topology.metric_space | src/topology/metric_space/polish.lean | [
"topology.metric_space.pi_nat",
"topology.metric_space.isometry",
"topology.metric_space.gluing",
"analysis.normed.field.basic"
] | [
"is_closed",
"is_open",
"polish_space",
"topological_space"
] | A set in a topological space is clopenable if there exists a finer Polish topology for which
this set is open and closed. It turns out that this notion is equivalent to being Borel-measurable,
but this is nontrivial (see `is_clopenable_iff_measurable_set`). | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
_root_.is_closed.is_clopenable [topological_space α] [polish_space α] {s : set α}
(hs : is_closed s) : is_clopenable s | begin
/- Both sets `s` and `sᶜ` admit a Polish topology. So does their disjoint union `s ⊕ sᶜ`.
Pulling back this topology by the canonical bijection with `α` gives the desired Polish
topology in which `s` is both open and closed. -/
haveI : polish_space s := hs.polish_space,
let t : set α := sᶜ,
haveI : po... | lemma | is_closed.is_clopenable | topology.metric_space | src/topology/metric_space/polish.lean | [
"topology.metric_space.pi_nat",
"topology.metric_space.isometry",
"topology.metric_space.gluing",
"analysis.normed.field.basic"
] | [
"continuous_subtype_coe",
"equiv.set.sum_compl",
"equiv.set.sum_compl_apply_inl",
"equiv.set.sum_compl_apply_inr",
"equiv.set.sum_compl_symm_apply_of_mem",
"equiv.set.sum_compl_symm_apply_of_not_mem",
"equiv.symm_preimage_preimage",
"equiv.symm_symm",
"equiv.to_fun_as_coe",
"equiv.to_homeomorph_of... | Given a closed set `s` in a Polish space, one can construct a finer Polish topology for
which `s` is both open and closed. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_clopenable.compl [topological_space α] {s : set α} (hs : is_clopenable s) :
is_clopenable sᶜ | begin
rcases hs with ⟨t, t_le, t_polish, h, h'⟩,
exact ⟨t, t_le, t_polish, @is_open.is_closed_compl α t s h', @is_closed.is_open_compl α t s h⟩,
end | lemma | polish_space.is_clopenable.compl | topology.metric_space | src/topology/metric_space/polish.lean | [
"topology.metric_space.pi_nat",
"topology.metric_space.isometry",
"topology.metric_space.gluing",
"analysis.normed.field.basic"
] | [
"is_open.is_closed_compl",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.is_open.is_clopenable [topological_space α] [polish_space α] {s : set α}
(hs : is_open s) : is_clopenable s | by simpa using hs.is_closed_compl.is_clopenable.compl | lemma | is_open.is_clopenable | topology.metric_space | src/topology/metric_space/polish.lean | [
"topology.metric_space.pi_nat",
"topology.metric_space.isometry",
"topology.metric_space.gluing",
"analysis.normed.field.basic"
] | [
"is_open",
"polish_space",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_clopenable.Union [t : topological_space α] [polish_space α] {s : ℕ → set α}
(hs : ∀ n, is_clopenable (s n)) :
is_clopenable (⋃ n, s n) | begin
choose m mt m_polish m_closed m_open using hs,
obtain ⟨t', t'm, -, t'_polish⟩ :
∃ (t' : topological_space α), (∀ (n : ℕ), t' ≤ m n) ∧ (t' ≤ t) ∧ @polish_space α t' :=
exists_polish_space_forall_le m mt m_polish,
have A : is_open[t'] (⋃ n, s n),
{ apply is_open_Union,
assume n,
apply t'm ... | lemma | polish_space.is_clopenable.Union | topology.metric_space | src/topology/metric_space/polish.lean | [
"topology.metric_space.pi_nat",
"topology.metric_space.isometry",
"topology.metric_space.gluing",
"analysis.normed.field.basic"
] | [
"is_closed",
"is_open",
"is_open.is_clopenable",
"is_open_Union",
"polish_space",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exists_subset_Union_ball_radius_lt {r : ι → ℝ} (hs : is_closed s)
(uf : ∀ x ∈ s, {i | x ∈ ball (c i) (r i)}.finite) (us : s ⊆ ⋃ i, ball (c i) (r i)) :
∃ r' : ι → ℝ, s ⊆ (⋃ i, ball (c i) (r' i)) ∧ ∀ i, r' i < r i | begin
rcases exists_subset_Union_closed_subset hs (λ i, @is_open_ball _ _ (c i) (r i)) uf us
with ⟨v, hsv, hvc, hcv⟩,
have := λ i, exists_lt_subset_ball (hvc i) (hcv i),
choose r' hlt hsub,
exact ⟨r', hsv.trans $ Union_mono $ hsub, hlt⟩
end | lemma | exists_subset_Union_ball_radius_lt | topology.metric_space | src/topology/metric_space/shrinking_lemma.lean | [
"topology.metric_space.basic",
"topology.metric_space.emetric_paracompact",
"topology.shrinking_lemma"
] | [
"exists_lt_subset_ball",
"exists_subset_Union_closed_subset",
"finite",
"is_closed"
] | Shrinking lemma for coverings by open balls in a proper metric space. A point-finite open cover
of a closed subset of a proper metric space by open balls can be shrunk to a new cover by open balls
so that each of the new balls has strictly smaller radius than the old one. This version assumes
that `λ x, ball (c i) (r i... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
exists_Union_ball_eq_radius_lt {r : ι → ℝ} (uf : ∀ x, {i | x ∈ ball (c i) (r i)}.finite)
(uU : (⋃ i, ball (c i) (r i)) = univ) :
∃ r' : ι → ℝ, (⋃ i, ball (c i) (r' i)) = univ ∧ ∀ i, r' i < r i | let ⟨r', hU, hv⟩ := exists_subset_Union_ball_radius_lt is_closed_univ (λ x _, uf x) uU.ge
in ⟨r', univ_subset_iff.1 hU, hv⟩ | lemma | exists_Union_ball_eq_radius_lt | topology.metric_space | src/topology/metric_space/shrinking_lemma.lean | [
"topology.metric_space.basic",
"topology.metric_space.emetric_paracompact",
"topology.shrinking_lemma"
] | [
"exists_subset_Union_ball_radius_lt",
"finite",
"is_closed_univ"
] | Shrinking lemma for coverings by open balls in a proper metric space. A point-finite open cover
of a proper metric space by open balls can be shrunk to a new cover by open balls so that each of
the new balls has strictly smaller radius than the old one. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
exists_subset_Union_ball_radius_pos_lt {r : ι → ℝ} (hr : ∀ i, 0 < r i) (hs : is_closed s)
(uf : ∀ x ∈ s, {i | x ∈ ball (c i) (r i)}.finite) (us : s ⊆ ⋃ i, ball (c i) (r i)) :
∃ r' : ι → ℝ, s ⊆ (⋃ i, ball (c i) (r' i)) ∧ ∀ i, r' i ∈ Ioo 0 (r i) | begin
rcases exists_subset_Union_closed_subset hs (λ i, @is_open_ball _ _ (c i) (r i)) uf us
with ⟨v, hsv, hvc, hcv⟩,
have := λ i, exists_pos_lt_subset_ball (hr i) (hvc i) (hcv i),
choose r' hlt hsub,
exact ⟨r', hsv.trans $ Union_mono hsub, hlt⟩
end | lemma | exists_subset_Union_ball_radius_pos_lt | topology.metric_space | src/topology/metric_space/shrinking_lemma.lean | [
"topology.metric_space.basic",
"topology.metric_space.emetric_paracompact",
"topology.shrinking_lemma"
] | [
"exists_pos_lt_subset_ball",
"exists_subset_Union_closed_subset",
"finite",
"is_closed"
] | Shrinking lemma for coverings by open balls in a proper metric space. A point-finite open cover
of a closed subset of a proper metric space by nonempty open balls can be shrunk to a new cover by
nonempty open balls so that each of the new balls has strictly smaller radius than the old one. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
exists_Union_ball_eq_radius_pos_lt {r : ι → ℝ} (hr : ∀ i, 0 < r i)
(uf : ∀ x, {i | x ∈ ball (c i) (r i)}.finite) (uU : (⋃ i, ball (c i) (r i)) = univ) :
∃ r' : ι → ℝ, (⋃ i, ball (c i) (r' i)) = univ ∧ ∀ i, r' i ∈ Ioo 0 (r i) | let ⟨r', hU, hv⟩ := exists_subset_Union_ball_radius_pos_lt hr is_closed_univ (λ x _, uf x) uU.ge
in ⟨r', univ_subset_iff.1 hU, hv⟩ | lemma | exists_Union_ball_eq_radius_pos_lt | topology.metric_space | src/topology/metric_space/shrinking_lemma.lean | [
"topology.metric_space.basic",
"topology.metric_space.emetric_paracompact",
"topology.shrinking_lemma"
] | [
"exists_subset_Union_ball_radius_pos_lt",
"finite",
"is_closed_univ"
] | Shrinking lemma for coverings by open balls in a proper metric space. A point-finite open cover
of a proper metric space by nonempty open balls can be shrunk to a new cover by nonempty open balls
so that each of the new balls has strictly smaller radius than the old one. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
exists_locally_finite_subset_Union_ball_radius_lt (hs : is_closed s)
{R : α → ℝ} (hR : ∀ x ∈ s, 0 < R x) :
∃ (ι : Type u) (c : ι → α) (r r' : ι → ℝ),
(∀ i, c i ∈ s ∧ 0 < r i ∧ r i < r' i ∧ r' i < R (c i)) ∧
locally_finite (λ i, ball (c i) (r' i)) ∧ s ⊆ ⋃ i, ball (c i) (r i) | begin
have : ∀ x ∈ s, (𝓝 x).has_basis (λ r : ℝ, 0 < r ∧ r < R x) (λ r, ball x r),
from λ x hx, nhds_basis_uniformity (uniformity_basis_dist_lt (hR x hx)),
rcases refinement_of_locally_compact_sigma_compact_of_nhds_basis_set hs this
with ⟨ι, c, r', hr', hsub', hfin⟩,
rcases exists_subset_Union_ball_radius... | lemma | exists_locally_finite_subset_Union_ball_radius_lt | topology.metric_space | src/topology/metric_space/shrinking_lemma.lean | [
"topology.metric_space.basic",
"topology.metric_space.emetric_paracompact",
"topology.shrinking_lemma"
] | [
"exists_subset_Union_ball_radius_pos_lt",
"is_closed",
"locally_finite",
"nhds_basis_uniformity",
"refinement_of_locally_compact_sigma_compact_of_nhds_basis_set"
] | Let `R : α → ℝ` be a (possibly discontinuous) function on a proper metric space.
Let `s` be a closed set in `α` such that `R` is positive on `s`. Then there exists a collection of
pairs of balls `metric.ball (c i) (r i)`, `metric.ball (c i) (r' i)` such that
* all centers belong to `s`;
* for all `i` we have `0 < r i ... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
exists_locally_finite_Union_eq_ball_radius_lt {R : α → ℝ} (hR : ∀ x, 0 < R x) :
∃ (ι : Type u) (c : ι → α) (r r' : ι → ℝ), (∀ i, 0 < r i ∧ r i < r' i ∧ r' i < R (c i)) ∧
locally_finite (λ i, ball (c i) (r' i)) ∧ (⋃ i, ball (c i) (r i)) = univ | let ⟨ι, c, r, r', hlt, hfin, hsub⟩ := exists_locally_finite_subset_Union_ball_radius_lt
is_closed_univ (λ x _, hR x)
in ⟨ι, c, r, r', λ i, (hlt i).2, hfin, univ_subset_iff.1 hsub⟩ | lemma | exists_locally_finite_Union_eq_ball_radius_lt | topology.metric_space | src/topology/metric_space/shrinking_lemma.lean | [
"topology.metric_space.basic",
"topology.metric_space.emetric_paracompact",
"topology.shrinking_lemma"
] | [
"exists_locally_finite_subset_Union_ball_radius_lt",
"is_closed_univ",
"locally_finite"
] | Let `R : α → ℝ` be a (possibly discontinuous) positive function on a proper metric space. Then
there exists a collection of pairs of balls `metric.ball (c i) (r i)`, `metric.ball (c i) (r' i)`
such that
* for all `i` we have `0 < r i < r' i < R (c i)`;
* the family of balls `metric.ball (c i) (r' i)` is locally finite... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
thickened_indicator_aux (δ : ℝ) (E : set α) : α → ℝ≥0∞ | λ (x : α), (1 : ℝ≥0∞) - (inf_edist x E) / (ennreal.of_real δ) | def | thickened_indicator_aux | topology.metric_space | src/topology/metric_space/thickened_indicator.lean | [
"data.real.ennreal",
"topology.continuous_function.bounded",
"topology.metric_space.hausdorff_distance"
] | [
"ennreal.of_real"
] | The `δ`-thickened indicator of a set `E` is the function that equals `1` on `E`
and `0` outside a `δ`-thickening of `E` and interpolates (continuously) between
these values using `inf_edist _ E`.
`thickened_indicator_aux` is the unbundled `ℝ≥0∞`-valued function. See `thickened_indicator`
for the (bundled) bounded cont... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuous_thickened_indicator_aux {δ : ℝ} (δ_pos : 0 < δ) (E : set α) :
continuous (thickened_indicator_aux δ E) | begin
unfold thickened_indicator_aux,
let f := λ (x : α), (⟨1, (inf_edist x E) / (ennreal.of_real δ)⟩ : ℝ≥0 × ℝ≥0∞),
let sub := λ (p : ℝ≥0 × ℝ≥0∞), ((p.1 : ℝ≥0∞) - p.2),
rw (show (λ (x : α), ((1 : ℝ≥0∞)) - (inf_edist x E) / (ennreal.of_real δ)) = sub ∘ f, by refl),
apply (@ennreal.continuous_nnreal_sub 1).com... | lemma | continuous_thickened_indicator_aux | topology.metric_space | src/topology/metric_space/thickened_indicator.lean | [
"data.real.ennreal",
"topology.continuous_function.bounded",
"topology.metric_space.hausdorff_distance"
] | [
"continuous",
"ennreal.continuous_div_const",
"ennreal.continuous_nnreal_sub",
"ennreal.of_real",
"thickened_indicator_aux"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
thickened_indicator_aux_le_one (δ : ℝ) (E : set α) (x : α) :
thickened_indicator_aux δ E x ≤ 1 | by apply @tsub_le_self _ _ _ _ (1 : ℝ≥0∞) | lemma | thickened_indicator_aux_le_one | topology.metric_space | src/topology/metric_space/thickened_indicator.lean | [
"data.real.ennreal",
"topology.continuous_function.bounded",
"topology.metric_space.hausdorff_distance"
] | [
"thickened_indicator_aux",
"tsub_le_self"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
thickened_indicator_aux_lt_top {δ : ℝ} {E : set α} {x : α} :
thickened_indicator_aux δ E x < ∞ | lt_of_le_of_lt (thickened_indicator_aux_le_one _ _ _) one_lt_top | lemma | thickened_indicator_aux_lt_top | topology.metric_space | src/topology/metric_space/thickened_indicator.lean | [
"data.real.ennreal",
"topology.continuous_function.bounded",
"topology.metric_space.hausdorff_distance"
] | [
"thickened_indicator_aux",
"thickened_indicator_aux_le_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
thickened_indicator_aux_closure_eq (δ : ℝ) (E : set α) :
thickened_indicator_aux δ (closure E) = thickened_indicator_aux δ E | by simp_rw [thickened_indicator_aux, inf_edist_closure] | lemma | thickened_indicator_aux_closure_eq | topology.metric_space | src/topology/metric_space/thickened_indicator.lean | [
"data.real.ennreal",
"topology.continuous_function.bounded",
"topology.metric_space.hausdorff_distance"
] | [
"closure",
"thickened_indicator_aux"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
thickened_indicator_aux_one (δ : ℝ) (E : set α) {x : α} (x_in_E : x ∈ E) :
thickened_indicator_aux δ E x = 1 | by simp [thickened_indicator_aux, inf_edist_zero_of_mem x_in_E, tsub_zero] | lemma | thickened_indicator_aux_one | topology.metric_space | src/topology/metric_space/thickened_indicator.lean | [
"data.real.ennreal",
"topology.continuous_function.bounded",
"topology.metric_space.hausdorff_distance"
] | [
"thickened_indicator_aux",
"tsub_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
thickened_indicator_aux_one_of_mem_closure
(δ : ℝ) (E : set α) {x : α} (x_mem : x ∈ closure E) :
thickened_indicator_aux δ E x = 1 | by rw [←thickened_indicator_aux_closure_eq, thickened_indicator_aux_one δ (closure E) x_mem] | lemma | thickened_indicator_aux_one_of_mem_closure | topology.metric_space | src/topology/metric_space/thickened_indicator.lean | [
"data.real.ennreal",
"topology.continuous_function.bounded",
"topology.metric_space.hausdorff_distance"
] | [
"closure",
"thickened_indicator_aux",
"thickened_indicator_aux_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
thickened_indicator_aux_zero
{δ : ℝ} (δ_pos : 0 < δ) (E : set α) {x : α} (x_out : x ∉ thickening δ E) :
thickened_indicator_aux δ E x = 0 | begin
rw [thickening, mem_set_of_eq, not_lt] at x_out,
unfold thickened_indicator_aux,
apply le_antisymm _ bot_le,
have key := tsub_le_tsub (@rfl _ (1 : ℝ≥0∞)).le (ennreal.div_le_div x_out rfl.le),
rw [ennreal.div_self (ne_of_gt (ennreal.of_real_pos.mpr δ_pos)) of_real_ne_top] at key,
simpa using key,
end | lemma | thickened_indicator_aux_zero | topology.metric_space | src/topology/metric_space/thickened_indicator.lean | [
"data.real.ennreal",
"topology.continuous_function.bounded",
"topology.metric_space.hausdorff_distance"
] | [
"bot_le",
"ennreal.div_le_div",
"ennreal.div_self",
"thickened_indicator_aux",
"tsub_le_tsub"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
thickened_indicator_aux_mono {δ₁ δ₂ : ℝ} (hle : δ₁ ≤ δ₂) (E : set α) :
thickened_indicator_aux δ₁ E ≤ thickened_indicator_aux δ₂ E | λ _, tsub_le_tsub (@rfl ℝ≥0∞ 1).le (ennreal.div_le_div rfl.le (of_real_le_of_real hle)) | lemma | thickened_indicator_aux_mono | topology.metric_space | src/topology/metric_space/thickened_indicator.lean | [
"data.real.ennreal",
"topology.continuous_function.bounded",
"topology.metric_space.hausdorff_distance"
] | [
"ennreal.div_le_div",
"thickened_indicator_aux",
"tsub_le_tsub"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
indicator_le_thickened_indicator_aux (δ : ℝ) (E : set α) :
E.indicator (λ _, (1 : ℝ≥0∞)) ≤ thickened_indicator_aux δ E | begin
intro a,
by_cases a ∈ E,
{ simp only [h, indicator_of_mem, thickened_indicator_aux_one δ E h, le_refl], },
{ simp only [h, indicator_of_not_mem, not_false_iff, zero_le], },
end | lemma | indicator_le_thickened_indicator_aux | topology.metric_space | src/topology/metric_space/thickened_indicator.lean | [
"data.real.ennreal",
"topology.continuous_function.bounded",
"topology.metric_space.hausdorff_distance"
] | [
"thickened_indicator_aux",
"thickened_indicator_aux_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
thickened_indicator_aux_subset (δ : ℝ) {E₁ E₂ : set α} (subset : E₁ ⊆ E₂) :
thickened_indicator_aux δ E₁ ≤ thickened_indicator_aux δ E₂ | λ _, tsub_le_tsub (@rfl ℝ≥0∞ 1).le (ennreal.div_le_div (inf_edist_anti subset) rfl.le) | lemma | thickened_indicator_aux_subset | topology.metric_space | src/topology/metric_space/thickened_indicator.lean | [
"data.real.ennreal",
"topology.continuous_function.bounded",
"topology.metric_space.hausdorff_distance"
] | [
"ennreal.div_le_div",
"thickened_indicator_aux",
"tsub_le_tsub"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
thickened_indicator_aux_tendsto_indicator_closure
{δseq : ℕ → ℝ} (δseq_lim : tendsto δseq at_top (𝓝 0)) (E : set α) :
tendsto (λ n, (thickened_indicator_aux (δseq n) E)) at_top
(𝓝 (indicator (closure E) (λ x, (1 : ℝ≥0∞)))) | begin
rw tendsto_pi_nhds,
intro x,
by_cases x_mem_closure : x ∈ closure E,
{ simp_rw [thickened_indicator_aux_one_of_mem_closure _ E x_mem_closure],
rw (show (indicator (closure E) (λ _, (1 : ℝ≥0∞))) x = 1,
by simp only [x_mem_closure, indicator_of_mem]),
exact tendsto_const_nhds, },
{ rw (sho... | lemma | thickened_indicator_aux_tendsto_indicator_closure | topology.metric_space | src/topology/metric_space/thickened_indicator.lean | [
"data.real.ennreal",
"topology.continuous_function.bounded",
"topology.metric_space.hausdorff_distance"
] | [
"bot_le",
"closure",
"ge_iff_le",
"lt_of_abs_lt",
"metric.tendsto_nhds",
"real.norm_eq_abs",
"tendsto_at_top_of_eventually_const",
"tendsto_const_nhds",
"tendsto_pi_nhds",
"thickened_indicator_aux",
"thickened_indicator_aux_mono",
"thickened_indicator_aux_one_of_mem_closure",
"thickened_indi... | As the thickening radius δ tends to 0, the δ-thickened indicator of a set E (in α) tends
pointwise (i.e., w.r.t. the product topology on `α → ℝ≥0∞`) to the indicator function of the
closure of E.
This statement is for the unbundled `ℝ≥0∞`-valued functions `thickened_indicator_aux δ E`, see
`thickened_indicator_tendsto... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
thickened_indicator {δ : ℝ} (δ_pos : 0 < δ) (E : set α) : α →ᵇ ℝ≥0 | { to_fun := λ (x : α), (thickened_indicator_aux δ E x).to_nnreal,
continuous_to_fun := begin
apply continuous_on.comp_continuous
continuous_on_to_nnreal (continuous_thickened_indicator_aux δ_pos E),
intro x,
exact (lt_of_le_of_lt (@thickened_indicator_aux_le_one _ _ δ E x) one_lt_top).ne,
en... | def | thickened_indicator | topology.metric_space | src/topology/metric_space/thickened_indicator.lean | [
"data.real.ennreal",
"topology.continuous_function.bounded",
"topology.metric_space.hausdorff_distance"
] | [
"abs_sub",
"continuous_on.comp_continuous",
"continuous_thickened_indicator_aux",
"nnreal.abs_eq",
"nnreal.dist_eq",
"thickened_indicator_aux",
"thickened_indicator_aux_le_one"
] | The `δ`-thickened indicator of a set `E` is the function that equals `1` on `E`
and `0` outside a `δ`-thickening of `E` and interpolates (continuously) between
these values using `inf_edist _ E`.
`thickened_indicator` is the (bundled) bounded continuous function with `ℝ≥0`-values.
See `thickened_indicator_aux` for the... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
thickened_indicator.coe_fn_eq_comp {δ : ℝ} (δ_pos : 0 < δ) (E : set α) :
⇑(thickened_indicator δ_pos E) = ennreal.to_nnreal ∘ thickened_indicator_aux δ E | rfl | lemma | thickened_indicator.coe_fn_eq_comp | topology.metric_space | src/topology/metric_space/thickened_indicator.lean | [
"data.real.ennreal",
"topology.continuous_function.bounded",
"topology.metric_space.hausdorff_distance"
] | [
"ennreal.to_nnreal",
"thickened_indicator",
"thickened_indicator_aux"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
thickened_indicator_le_one {δ : ℝ} (δ_pos : 0 < δ) (E : set α) (x : α) :
thickened_indicator δ_pos E x ≤ 1 | begin
rw [thickened_indicator.coe_fn_eq_comp],
simpa using (to_nnreal_le_to_nnreal thickened_indicator_aux_lt_top.ne one_ne_top).mpr
(thickened_indicator_aux_le_one δ E x),
end | lemma | thickened_indicator_le_one | topology.metric_space | src/topology/metric_space/thickened_indicator.lean | [
"data.real.ennreal",
"topology.continuous_function.bounded",
"topology.metric_space.hausdorff_distance"
] | [
"thickened_indicator",
"thickened_indicator.coe_fn_eq_comp",
"thickened_indicator_aux_le_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
thickened_indicator_one_of_mem_closure
{δ : ℝ} (δ_pos : 0 < δ) (E : set α) {x : α} (x_mem : x ∈ closure E) :
thickened_indicator δ_pos E x = 1 | by rw [thickened_indicator_apply,
thickened_indicator_aux_one_of_mem_closure δ E x_mem, one_to_nnreal] | lemma | thickened_indicator_one_of_mem_closure | topology.metric_space | src/topology/metric_space/thickened_indicator.lean | [
"data.real.ennreal",
"topology.continuous_function.bounded",
"topology.metric_space.hausdorff_distance"
] | [
"closure",
"thickened_indicator",
"thickened_indicator_aux_one_of_mem_closure"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
thickened_indicator_one {δ : ℝ} (δ_pos : 0 < δ) (E : set α) {x : α} (x_in_E : x ∈ E) :
thickened_indicator δ_pos E x = 1 | thickened_indicator_one_of_mem_closure _ _ (subset_closure x_in_E) | lemma | thickened_indicator_one | topology.metric_space | src/topology/metric_space/thickened_indicator.lean | [
"data.real.ennreal",
"topology.continuous_function.bounded",
"topology.metric_space.hausdorff_distance"
] | [
"subset_closure",
"thickened_indicator",
"thickened_indicator_one_of_mem_closure"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
thickened_indicator_zero
{δ : ℝ} (δ_pos : 0 < δ) (E : set α) {x : α} (x_out : x ∉ thickening δ E) :
thickened_indicator δ_pos E x = 0 | by rw [thickened_indicator_apply, thickened_indicator_aux_zero δ_pos E x_out, zero_to_nnreal] | lemma | thickened_indicator_zero | topology.metric_space | src/topology/metric_space/thickened_indicator.lean | [
"data.real.ennreal",
"topology.continuous_function.bounded",
"topology.metric_space.hausdorff_distance"
] | [
"thickened_indicator",
"thickened_indicator_aux_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
indicator_le_thickened_indicator {δ : ℝ} (δ_pos : 0 < δ) (E : set α) :
E.indicator (λ _, (1 : ℝ≥0)) ≤ thickened_indicator δ_pos E | begin
intro a,
by_cases a ∈ E,
{ simp only [h, indicator_of_mem, thickened_indicator_one δ_pos E h, le_refl], },
{ simp only [h, indicator_of_not_mem, not_false_iff, zero_le], },
end | lemma | indicator_le_thickened_indicator | topology.metric_space | src/topology/metric_space/thickened_indicator.lean | [
"data.real.ennreal",
"topology.continuous_function.bounded",
"topology.metric_space.hausdorff_distance"
] | [
"thickened_indicator",
"thickened_indicator_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
thickened_indicator_mono {δ₁ δ₂ : ℝ}
(δ₁_pos : 0 < δ₁) (δ₂_pos : 0 < δ₂) (hle : δ₁ ≤ δ₂) (E : set α) :
⇑(thickened_indicator δ₁_pos E) ≤ thickened_indicator δ₂_pos E | begin
intro x,
apply (to_nnreal_le_to_nnreal thickened_indicator_aux_lt_top.ne
thickened_indicator_aux_lt_top.ne).mpr,
apply thickened_indicator_aux_mono hle,
end | lemma | thickened_indicator_mono | topology.metric_space | src/topology/metric_space/thickened_indicator.lean | [
"data.real.ennreal",
"topology.continuous_function.bounded",
"topology.metric_space.hausdorff_distance"
] | [
"thickened_indicator",
"thickened_indicator_aux_mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
thickened_indicator_subset {δ : ℝ} (δ_pos : 0 < δ) {E₁ E₂ : set α} (subset : E₁ ⊆ E₂) :
⇑(thickened_indicator δ_pos E₁) ≤ thickened_indicator δ_pos E₂ | λ x, (to_nnreal_le_to_nnreal thickened_indicator_aux_lt_top.ne
thickened_indicator_aux_lt_top.ne).mpr (thickened_indicator_aux_subset δ subset x) | lemma | thickened_indicator_subset | topology.metric_space | src/topology/metric_space/thickened_indicator.lean | [
"data.real.ennreal",
"topology.continuous_function.bounded",
"topology.metric_space.hausdorff_distance"
] | [
"thickened_indicator",
"thickened_indicator_aux_subset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
thickened_indicator_tendsto_indicator_closure
{δseq : ℕ → ℝ} (δseq_pos : ∀ n, 0 < δseq n) (δseq_lim : tendsto δseq at_top (𝓝 0)) (E : set α) :
tendsto (λ (n : ℕ), (coe_fn : (α →ᵇ ℝ≥0) → (α → ℝ≥0)) (thickened_indicator (δseq_pos n) E))
at_top (𝓝 (indicator (closure E) (λ x, (1 : ℝ≥0)))) | begin
have key := thickened_indicator_aux_tendsto_indicator_closure δseq_lim E,
rw tendsto_pi_nhds at *,
intro x,
rw (show indicator (closure E) (λ x, (1 : ℝ≥0)) x
= (indicator (closure E) (λ x, (1 : ℝ≥0∞)) x).to_nnreal,
by refine (congr_fun (comp_indicator_const 1 ennreal.to_nnreal zero_to_nnrea... | lemma | thickened_indicator_tendsto_indicator_closure | topology.metric_space | src/topology/metric_space/thickened_indicator.lean | [
"data.real.ennreal",
"topology.continuous_function.bounded",
"topology.metric_space.hausdorff_distance"
] | [
"closure",
"ennreal.to_nnreal",
"tendsto_pi_nhds",
"thickened_indicator",
"thickened_indicator_aux_tendsto_indicator_closure"
] | As the thickening radius δ tends to 0, the δ-thickened indicator of a set E (in α) tends
pointwise to the indicator function of the closure of E.
Note: This version is for the bundled bounded continuous functions, but the topology is not
the topology on `α →ᵇ ℝ≥0`. Coercions to functions `α → ℝ≥0` are done first, so t... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
order_closed_topology (α : Type*) [topological_space α] [preorder α] : Prop | (is_closed_le' : is_closed {p : α × α | p.1 ≤ p.2}) | class | order_closed_topology | topology.order | src/topology/order/basic.lean | [
"data.set.intervals.pi",
"data.set.pointwise.interval",
"order.filter.interval",
"topology.support",
"topology.algebra.order.left_right"
] | [
"is_closed",
"is_closed_le'",
"topological_space"
] | A topology on a set which is both a topological space and a preorder is _order-closed_ if the
set of points `(x, y)` with `x ≤ y` is closed in the product space. We introduce this as a mixin.
This property is satisfied for the order topology on a linear order, but it can be satisfied more
generally, and suffices to der... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
dense.order_dual [topological_space α] {s : set α} (hs : dense s) :
dense (order_dual.of_dual ⁻¹' s) | hs | lemma | dense.order_dual | topology.order | src/topology/order/basic.lean | [
"data.set.intervals.pi",
"data.set.pointwise.interval",
"order.filter.interval",
"topology.support",
"topology.algebra.order.left_right"
] | [
"dense",
"order_dual.of_dual",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_closed_le_prod : is_closed {p : α × α | p.1 ≤ p.2} | t.is_closed_le' | lemma | is_closed_le_prod | topology.order | src/topology/order/basic.lean | [
"data.set.intervals.pi",
"data.set.pointwise.interval",
"order.filter.interval",
"topology.support",
"topology.algebra.order.left_right"
] | [
"is_closed"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_closed_le [topological_space β] {f g : β → α} (hf : continuous f) (hg : continuous g) :
is_closed {b | f b ≤ g b} | continuous_iff_is_closed.mp (hf.prod_mk hg) _ is_closed_le_prod | lemma | is_closed_le | topology.order | src/topology/order/basic.lean | [
"data.set.intervals.pi",
"data.set.pointwise.interval",
"order.filter.interval",
"topology.support",
"topology.algebra.order.left_right"
] | [
"continuous",
"is_closed",
"is_closed_le_prod",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_closed_le' (a : α) : is_closed {b | b ≤ a} | is_closed_le continuous_id continuous_const | lemma | is_closed_le' | topology.order | src/topology/order/basic.lean | [
"data.set.intervals.pi",
"data.set.pointwise.interval",
"order.filter.interval",
"topology.support",
"topology.algebra.order.left_right"
] | [
"continuous_const",
"continuous_id",
"is_closed",
"is_closed_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_closed_Iic {a : α} : is_closed (Iic a) | is_closed_le' a | lemma | is_closed_Iic | topology.order | src/topology/order/basic.lean | [
"data.set.intervals.pi",
"data.set.pointwise.interval",
"order.filter.interval",
"topology.support",
"topology.algebra.order.left_right"
] | [
"is_closed",
"is_closed_le'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_closed_ge' (a : α) : is_closed {b | a ≤ b} | is_closed_le continuous_const continuous_id | lemma | is_closed_ge' | topology.order | src/topology/order/basic.lean | [
"data.set.intervals.pi",
"data.set.pointwise.interval",
"order.filter.interval",
"topology.support",
"topology.algebra.order.left_right"
] | [
"continuous_const",
"continuous_id",
"is_closed",
"is_closed_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_closed_Ici {a : α} : is_closed (Ici a) | is_closed_ge' a | lemma | is_closed_Ici | topology.order | src/topology/order/basic.lean | [
"data.set.intervals.pi",
"data.set.pointwise.interval",
"order.filter.interval",
"topology.support",
"topology.algebra.order.left_right"
] | [
"is_closed",
"is_closed_ge'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_closed_Icc {a b : α} : is_closed (Icc a b) | is_closed.inter is_closed_Ici is_closed_Iic | lemma | is_closed_Icc | topology.order | src/topology/order/basic.lean | [
"data.set.intervals.pi",
"data.set.pointwise.interval",
"order.filter.interval",
"topology.support",
"topology.algebra.order.left_right"
] | [
"is_closed",
"is_closed.inter",
"is_closed_Ici",
"is_closed_Iic"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
closure_Icc (a b : α) : closure (Icc a b) = Icc a b | is_closed_Icc.closure_eq | lemma | closure_Icc | topology.order | src/topology/order/basic.lean | [
"data.set.intervals.pi",
"data.set.pointwise.interval",
"order.filter.interval",
"topology.support",
"topology.algebra.order.left_right"
] | [
"closure"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
closure_Iic (a : α) : closure (Iic a) = Iic a | is_closed_Iic.closure_eq | lemma | closure_Iic | topology.order | src/topology/order/basic.lean | [
"data.set.intervals.pi",
"data.set.pointwise.interval",
"order.filter.interval",
"topology.support",
"topology.algebra.order.left_right"
] | [
"closure"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
closure_Ici (a : α) : closure (Ici a) = Ici a | is_closed_Ici.closure_eq | lemma | closure_Ici | topology.order | src/topology/order/basic.lean | [
"data.set.intervals.pi",
"data.set.pointwise.interval",
"order.filter.interval",
"topology.support",
"topology.algebra.order.left_right"
] | [
"closure"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_of_tendsto_of_tendsto {f g : β → α} {b : filter β} {a₁ a₂ : α} [ne_bot b]
(hf : tendsto f b (𝓝 a₁)) (hg : tendsto g b (𝓝 a₂)) (h : f ≤ᶠ[b] g) :
a₁ ≤ a₂ | have tendsto (λb, (f b, g b)) b (𝓝 (a₁, a₂)),
by rw [nhds_prod_eq]; exact hf.prod_mk hg,
show (a₁, a₂) ∈ {p:α×α | p.1 ≤ p.2},
from t.is_closed_le'.mem_of_tendsto this h | lemma | le_of_tendsto_of_tendsto | topology.order | src/topology/order/basic.lean | [
"data.set.intervals.pi",
"data.set.pointwise.interval",
"order.filter.interval",
"topology.support",
"topology.algebra.order.left_right"
] | [
"filter",
"nhds_prod_eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_of_tendsto_of_tendsto' {f g : β → α} {b : filter β} {a₁ a₂ : α} [ne_bot b]
(hf : tendsto f b (𝓝 a₁)) (hg : tendsto g b (𝓝 a₂)) (h : ∀ x, f x ≤ g x) :
a₁ ≤ a₂ | le_of_tendsto_of_tendsto hf hg (eventually_of_forall h) | lemma | le_of_tendsto_of_tendsto' | topology.order | src/topology/order/basic.lean | [
"data.set.intervals.pi",
"data.set.pointwise.interval",
"order.filter.interval",
"topology.support",
"topology.algebra.order.left_right"
] | [
"filter",
"le_of_tendsto_of_tendsto"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_of_tendsto {f : β → α} {a b : α} {x : filter β}
[ne_bot x] (lim : tendsto f x (𝓝 a)) (h : ∀ᶠ c in x, f c ≤ b) : a ≤ b | le_of_tendsto_of_tendsto lim tendsto_const_nhds h | lemma | le_of_tendsto | topology.order | src/topology/order/basic.lean | [
"data.set.intervals.pi",
"data.set.pointwise.interval",
"order.filter.interval",
"topology.support",
"topology.algebra.order.left_right"
] | [
"filter",
"le_of_tendsto_of_tendsto",
"lim",
"tendsto_const_nhds"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_of_tendsto' {f : β → α} {a b : α} {x : filter β}
[ne_bot x] (lim : tendsto f x (𝓝 a)) (h : ∀ c, f c ≤ b) : a ≤ b | le_of_tendsto lim (eventually_of_forall h) | lemma | le_of_tendsto' | topology.order | src/topology/order/basic.lean | [
"data.set.intervals.pi",
"data.set.pointwise.interval",
"order.filter.interval",
"topology.support",
"topology.algebra.order.left_right"
] | [
"filter",
"le_of_tendsto",
"lim"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ge_of_tendsto {f : β → α} {a b : α} {x : filter β} [ne_bot x]
(lim : tendsto f x (𝓝 a)) (h : ∀ᶠ c in x, b ≤ f c) : b ≤ a | le_of_tendsto_of_tendsto tendsto_const_nhds lim h | lemma | ge_of_tendsto | topology.order | src/topology/order/basic.lean | [
"data.set.intervals.pi",
"data.set.pointwise.interval",
"order.filter.interval",
"topology.support",
"topology.algebra.order.left_right"
] | [
"filter",
"le_of_tendsto_of_tendsto",
"lim",
"tendsto_const_nhds"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ge_of_tendsto' {f : β → α} {a b : α} {x : filter β} [ne_bot x]
(lim : tendsto f x (𝓝 a)) (h : ∀ c, b ≤ f c) : b ≤ a | ge_of_tendsto lim (eventually_of_forall h) | lemma | ge_of_tendsto' | topology.order | src/topology/order/basic.lean | [
"data.set.intervals.pi",
"data.set.pointwise.interval",
"order.filter.interval",
"topology.support",
"topology.algebra.order.left_right"
] | [
"filter",
"ge_of_tendsto",
"lim"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
closure_le_eq [topological_space β] {f g : β → α} (hf : continuous f) (hg : continuous g) :
closure {b | f b ≤ g b} = {b | f b ≤ g b} | (is_closed_le hf hg).closure_eq | lemma | closure_le_eq | topology.order | src/topology/order/basic.lean | [
"data.set.intervals.pi",
"data.set.pointwise.interval",
"order.filter.interval",
"topology.support",
"topology.algebra.order.left_right"
] | [
"closure",
"continuous",
"is_closed_le",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
closure_lt_subset_le [topological_space β] {f g : β → α} (hf : continuous f)
(hg : continuous g) :
closure {b | f b < g b} ⊆ {b | f b ≤ g b} | closure_minimal (λ x, le_of_lt) $ is_closed_le hf hg | lemma | closure_lt_subset_le | topology.order | src/topology/order/basic.lean | [
"data.set.intervals.pi",
"data.set.pointwise.interval",
"order.filter.interval",
"topology.support",
"topology.algebra.order.left_right"
] | [
"closure",
"closure_minimal",
"continuous",
"is_closed_le",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_within_at.closure_le [topological_space β]
{f g : β → α} {s : set β} {x : β} (hx : x ∈ closure s)
(hf : continuous_within_at f s x)
(hg : continuous_within_at g s x)
(h : ∀ y ∈ s, f y ≤ g y) : f x ≤ g x | show (f x, g x) ∈ {p : α × α | p.1 ≤ p.2},
from order_closed_topology.is_closed_le'.closure_subset ((hf.prod hg).mem_closure hx h) | lemma | continuous_within_at.closure_le | topology.order | src/topology/order/basic.lean | [
"data.set.intervals.pi",
"data.set.pointwise.interval",
"order.filter.interval",
"topology.support",
"topology.algebra.order.left_right"
] | [
"closure",
"continuous_within_at",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_closed.is_closed_le [topological_space β] {f g : β → α} {s : set β} (hs : is_closed s)
(hf : continuous_on f s) (hg : continuous_on g s) :
is_closed {x ∈ s | f x ≤ g x} | (hf.prod hg).preimage_closed_of_closed hs order_closed_topology.is_closed_le' | lemma | is_closed.is_closed_le | topology.order | src/topology/order/basic.lean | [
"data.set.intervals.pi",
"data.set.pointwise.interval",
"order.filter.interval",
"topology.support",
"topology.algebra.order.left_right"
] | [
"continuous_on",
"is_closed",
"topological_space"
] | If `s` is a closed set and two functions `f` and `g` are continuous on `s`,
then the set `{x ∈ s | f x ≤ g x}` is a closed set. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
le_on_closure [topological_space β] {f g : β → α} {s : set β} (h : ∀ x ∈ s, f x ≤ g x)
(hf : continuous_on f (closure s)) (hg : continuous_on g (closure s)) ⦃x⦄ (hx : x ∈ closure s) :
f x ≤ g x | have s ⊆ {y ∈ closure s | f y ≤ g y}, from λ y hy, ⟨subset_closure hy, h y hy⟩,
(closure_minimal this (is_closed_closure.is_closed_le hf hg) hx).2 | lemma | le_on_closure | topology.order | src/topology/order/basic.lean | [
"data.set.intervals.pi",
"data.set.pointwise.interval",
"order.filter.interval",
"topology.support",
"topology.algebra.order.left_right"
] | [
"closure",
"closure_minimal",
"continuous_on",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_closed.epigraph [topological_space β] {f : β → α} {s : set β}
(hs : is_closed s) (hf : continuous_on f s) :
is_closed {p : β × α | p.1 ∈ s ∧ f p.1 ≤ p.2} | (hs.preimage continuous_fst).is_closed_le (hf.comp continuous_on_fst subset.rfl) continuous_on_snd | lemma | is_closed.epigraph | topology.order | src/topology/order/basic.lean | [
"data.set.intervals.pi",
"data.set.pointwise.interval",
"order.filter.interval",
"topology.support",
"topology.algebra.order.left_right"
] | [
"continuous_fst",
"continuous_on",
"continuous_on_fst",
"continuous_on_snd",
"is_closed",
"is_closed_le",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_closed.hypograph [topological_space β] {f : β → α} {s : set β}
(hs : is_closed s) (hf : continuous_on f s) :
is_closed {p : β × α | p.1 ∈ s ∧ p.2 ≤ f p.1} | (hs.preimage continuous_fst).is_closed_le continuous_on_snd (hf.comp continuous_on_fst subset.rfl) | lemma | is_closed.hypograph | topology.order | src/topology/order/basic.lean | [
"data.set.intervals.pi",
"data.set.pointwise.interval",
"order.filter.interval",
"topology.support",
"topology.algebra.order.left_right"
] | [
"continuous_fst",
"continuous_on",
"continuous_on_fst",
"continuous_on_snd",
"is_closed",
"is_closed_le",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nhds_within_Ici_ne_bot {a b : α} (H₂ : a ≤ b) :
ne_bot (𝓝[Ici a] b) | nhds_within_ne_bot_of_mem H₂ | lemma | nhds_within_Ici_ne_bot | topology.order | src/topology/order/basic.lean | [
"data.set.intervals.pi",
"data.set.pointwise.interval",
"order.filter.interval",
"topology.support",
"topology.algebra.order.left_right"
] | [
"nhds_within_ne_bot_of_mem"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nhds_within_Ici_self_ne_bot (a : α) :
ne_bot (𝓝[≥] a) | nhds_within_Ici_ne_bot (le_refl a) | lemma | nhds_within_Ici_self_ne_bot | topology.order | src/topology/order/basic.lean | [
"data.set.intervals.pi",
"data.set.pointwise.interval",
"order.filter.interval",
"topology.support",
"topology.algebra.order.left_right"
] | [
"nhds_within_Ici_ne_bot"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nhds_within_Iic_ne_bot {a b : α} (H : a ≤ b) :
ne_bot (𝓝[Iic b] a) | nhds_within_ne_bot_of_mem H | lemma | nhds_within_Iic_ne_bot | topology.order | src/topology/order/basic.lean | [
"data.set.intervals.pi",
"data.set.pointwise.interval",
"order.filter.interval",
"topology.support",
"topology.algebra.order.left_right"
] | [
"nhds_within_ne_bot_of_mem"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nhds_within_Iic_self_ne_bot (a : α) :
ne_bot (𝓝[≤] a) | nhds_within_Iic_ne_bot (le_refl a) | lemma | nhds_within_Iic_self_ne_bot | topology.order | src/topology/order/basic.lean | [
"data.set.intervals.pi",
"data.set.pointwise.interval",
"order.filter.interval",
"topology.support",
"topology.algebra.order.left_right"
] | [
"nhds_within_Iic_ne_bot"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
order_closed_topology.to_t2_space : t2_space α | t2_iff_is_closed_diagonal.2 $ by simpa only [diagonal, le_antisymm_iff] using
t.is_closed_le'.inter (is_closed_le continuous_snd continuous_fst) | instance | order_closed_topology.to_t2_space | topology.order | src/topology/order/basic.lean | [
"data.set.intervals.pi",
"data.set.pointwise.interval",
"order.filter.interval",
"topology.support",
"topology.algebra.order.left_right"
] | [
"continuous_fst",
"continuous_snd",
"is_closed_le",
"t2_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_open_lt_prod : is_open {p : α × α | p.1 < p.2} | by { simp_rw [← is_closed_compl_iff, compl_set_of, not_lt],
exact is_closed_le continuous_snd continuous_fst } | lemma | is_open_lt_prod | topology.order | src/topology/order/basic.lean | [
"data.set.intervals.pi",
"data.set.pointwise.interval",
"order.filter.interval",
"topology.support",
"topology.algebra.order.left_right"
] | [
"continuous_fst",
"continuous_snd",
"is_closed_compl_iff",
"is_closed_le",
"is_open"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_open_lt [topological_space β] {f g : β → α} (hf : continuous f) (hg : continuous g) :
is_open {b | f b < g b} | by simp [lt_iff_not_ge, -not_le]; exact (is_closed_le hg hf).is_open_compl | lemma | is_open_lt | topology.order | src/topology/order/basic.lean | [
"data.set.intervals.pi",
"data.set.pointwise.interval",
"order.filter.interval",
"topology.support",
"topology.algebra.order.left_right"
] | [
"continuous",
"is_closed_le",
"is_open",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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