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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continuous_at_lift₂ {f : X → Y → Z} {hf : ∀ a b c d, a ~ c → b ~ d → f a b = f c d}
{x : X} {y : Y} :
continuous_at (uncurry $ lift₂ f hf) (mk x, mk y) ↔ continuous_at (uncurry f) (x, y) | tendsto_lift₂_nhds | lemma | separation_quotient.continuous_at_lift₂ | topology | src/topology/inseparable.lean | [
"topology.continuous_on",
"data.setoid.basic",
"tactic.tfae"
] | [
"continuous_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_within_at_lift₂ {f : X → Y → Z}
{hf : ∀ a b c d, a ~ c → b ~ d → f a b = f c d}
{s : set (separation_quotient X × separation_quotient Y)} {x : X} {y : Y} :
continuous_within_at (uncurry $ lift₂ f hf) s (mk x, mk y) ↔
continuous_within_at (uncurry f) (prod.map mk mk ⁻¹' s) (x, y) | tendsto_lift₂_nhds_within | lemma | separation_quotient.continuous_within_at_lift₂ | topology | src/topology/inseparable.lean | [
"topology.continuous_on",
"data.setoid.basic",
"tactic.tfae"
] | [
"continuous_within_at",
"separation_quotient"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_on_lift₂ {f : X → Y → Z}
{hf : ∀ a b c d, a ~ c → b ~ d → f a b = f c d}
{s : set (separation_quotient X × separation_quotient Y)} :
continuous_on (uncurry $ lift₂ f hf) s ↔ continuous_on (uncurry f) (prod.map mk mk ⁻¹' s) | begin
simp_rw [continuous_on, (surjective_mk.prod_map surjective_mk).forall, prod.forall, prod.map,
continuous_within_at_lift₂],
refl
end | lemma | separation_quotient.continuous_on_lift₂ | topology | src/topology/inseparable.lean | [
"topology.continuous_on",
"data.setoid.basic",
"tactic.tfae"
] | [
"continuous_on",
"separation_quotient"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_lift₂ {f : X → Y → Z}
{hf : ∀ a b c d, a ~ c → b ~ d → f a b = f c d} :
continuous (uncurry $ lift₂ f hf) ↔ continuous (uncurry f) | by simp only [continuous_iff_continuous_on_univ, continuous_on_lift₂, preimage_univ] | lemma | separation_quotient.continuous_lift₂ | topology | src/topology/inseparable.lean | [
"topology.continuous_on",
"data.setoid.basic",
"tactic.tfae"
] | [
"continuous",
"continuous_iff_continuous_on_univ"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_locally_homeomorph_on | ∀ x ∈ s, ∃ e : local_homeomorph X Y, x ∈ e.source ∧ f = e | def | is_locally_homeomorph_on | topology | src/topology/is_locally_homeomorph.lean | [
"topology.local_homeomorph"
] | [
"local_homeomorph"
] | A function `f : X → Y` satisfies `is_locally_homeomorph_on f s` if each `x ∈ s` is contained in
the source of some `e : local_homeomorph X Y` with `f = e`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mk (h : ∀ x ∈ s, ∃ e : local_homeomorph X Y, x ∈ e.source ∧ ∀ y ∈ e.source, f y = e y) :
is_locally_homeomorph_on f s | begin
intros x hx,
obtain ⟨e, hx, he⟩ := h x hx,
exact ⟨{ to_fun := f,
map_source' := λ x hx, by rw he x hx; exact e.map_source' hx,
left_inv' := λ x hx, by rw he x hx; exact e.left_inv' hx,
right_inv' := λ y hy, by rw he _ (e.map_target' hy); exact e.right_inv' hy,
continuous_to_fun := (continuou... | lemma | is_locally_homeomorph_on.mk | topology | src/topology/is_locally_homeomorph.lean | [
"topology.local_homeomorph"
] | [
"continuous_on_congr",
"is_locally_homeomorph_on",
"local_homeomorph"
] | Proves that `f` satisfies `is_locally_homeomorph_on f s`. The condition `h` is weaker than the
definition of `is_locally_homeomorph_on f s`, since it only requires `e : local_homeomorph X Y` to
agree with `f` on its source `e.source`, as opposed to on the whole space `X`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
map_nhds_eq (hf : is_locally_homeomorph_on f s) {x : X} (hx : x ∈ s) :
(𝓝 x).map f = 𝓝 (f x) | let ⟨e, hx, he⟩ := hf x hx in he.symm ▸ e.map_nhds_eq hx | lemma | is_locally_homeomorph_on.map_nhds_eq | topology | src/topology/is_locally_homeomorph.lean | [
"topology.local_homeomorph"
] | [
"is_locally_homeomorph_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_at (hf : is_locally_homeomorph_on f s) {x : X} (hx : x ∈ s) :
continuous_at f x | (hf.map_nhds_eq hx).le | lemma | is_locally_homeomorph_on.continuous_at | topology | src/topology/is_locally_homeomorph.lean | [
"topology.local_homeomorph"
] | [
"continuous_at",
"is_locally_homeomorph_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_on (hf : is_locally_homeomorph_on f s) : continuous_on f s | continuous_at.continuous_on (λ x, hf.continuous_at) | lemma | is_locally_homeomorph_on.continuous_on | topology | src/topology/is_locally_homeomorph.lean | [
"topology.local_homeomorph"
] | [
"continuous_at.continuous_on",
"continuous_on",
"is_locally_homeomorph_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp (hg : is_locally_homeomorph_on g t) (hf : is_locally_homeomorph_on f s)
(h : set.maps_to f s t) : is_locally_homeomorph_on (g ∘ f) s | begin
intros x hx,
obtain ⟨eg, hxg, rfl⟩ := hg (f x) (h hx),
obtain ⟨ef, hxf, rfl⟩ := hf x hx,
exact ⟨ef.trans eg, ⟨hxf, hxg⟩, rfl⟩,
end | lemma | is_locally_homeomorph_on.comp | topology | src/topology/is_locally_homeomorph.lean | [
"topology.local_homeomorph"
] | [
"is_locally_homeomorph_on",
"set.maps_to"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_locally_homeomorph | ∀ x : X, ∃ e : local_homeomorph X Y, x ∈ e.source ∧ f = e | def | is_locally_homeomorph | topology | src/topology/is_locally_homeomorph.lean | [
"topology.local_homeomorph"
] | [
"local_homeomorph"
] | A function `f : X → Y` satisfies `is_locally_homeomorph f` if each `x : x` is contained in
the source of some `e : local_homeomorph X Y` with `f = e`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_locally_homeomorph_iff_is_locally_homeomorph_on_univ :
is_locally_homeomorph f ↔ is_locally_homeomorph_on f set.univ | by simp only [is_locally_homeomorph, is_locally_homeomorph_on, set.mem_univ, forall_true_left] | lemma | is_locally_homeomorph_iff_is_locally_homeomorph_on_univ | topology | src/topology/is_locally_homeomorph.lean | [
"topology.local_homeomorph"
] | [
"forall_true_left",
"is_locally_homeomorph",
"is_locally_homeomorph_on",
"set.mem_univ"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_locally_homeomorph.is_locally_homeomorph_on (hf : is_locally_homeomorph f) :
is_locally_homeomorph_on f set.univ | is_locally_homeomorph_iff_is_locally_homeomorph_on_univ.mp hf | lemma | is_locally_homeomorph.is_locally_homeomorph_on | topology | src/topology/is_locally_homeomorph.lean | [
"topology.local_homeomorph"
] | [
"is_locally_homeomorph",
"is_locally_homeomorph_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk (h : ∀ x : X, ∃ e : local_homeomorph X Y, x ∈ e.source ∧ ∀ y ∈ e.source, f y = e y) :
is_locally_homeomorph f | is_locally_homeomorph_iff_is_locally_homeomorph_on_univ.mpr
(is_locally_homeomorph_on.mk f set.univ (λ x hx, h x)) | lemma | is_locally_homeomorph.mk | topology | src/topology/is_locally_homeomorph.lean | [
"topology.local_homeomorph"
] | [
"is_locally_homeomorph",
"is_locally_homeomorph_on.mk",
"local_homeomorph"
] | Proves that `f` satisfies `is_locally_homeomorph f`. The condition `h` is weaker than the
definition of `is_locally_homeomorph f`, since it only requires `e : local_homeomorph X Y` to
agree with `f` on its source `e.source`, as opposed to on the whole space `X`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
map_nhds_eq (hf : is_locally_homeomorph f) (x : X) : (𝓝 x).map f = 𝓝 (f x) | hf.is_locally_homeomorph_on.map_nhds_eq (set.mem_univ x) | lemma | is_locally_homeomorph.map_nhds_eq | topology | src/topology/is_locally_homeomorph.lean | [
"topology.local_homeomorph"
] | [
"is_locally_homeomorph",
"set.mem_univ"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous (hf : is_locally_homeomorph f) : continuous f | continuous_iff_continuous_on_univ.mpr hf.is_locally_homeomorph_on.continuous_on | lemma | is_locally_homeomorph.continuous | topology | src/topology/is_locally_homeomorph.lean | [
"topology.local_homeomorph"
] | [
"continuous",
"is_locally_homeomorph"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_open_map (hf : is_locally_homeomorph f) : is_open_map f | is_open_map.of_nhds_le (λ x, ge_of_eq (hf.map_nhds_eq x)) | lemma | is_locally_homeomorph.is_open_map | topology | src/topology/is_locally_homeomorph.lean | [
"topology.local_homeomorph"
] | [
"ge_of_eq",
"is_locally_homeomorph",
"is_open_map",
"is_open_map.of_nhds_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp (hg : is_locally_homeomorph g) (hf : is_locally_homeomorph f) :
is_locally_homeomorph (g ∘ f) | is_locally_homeomorph_iff_is_locally_homeomorph_on_univ.mpr
(hg.is_locally_homeomorph_on.comp hf.is_locally_homeomorph_on (set.univ.maps_to_univ f)) | lemma | is_locally_homeomorph.comp | topology | src/topology/is_locally_homeomorph.lean | [
"topology.local_homeomorph"
] | [
"is_locally_homeomorph"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nhds_list (as : list α) : 𝓝 as = traverse 𝓝 as | begin
refine nhds_mk_of_nhds _ _ _ _,
{ assume l, induction l,
case list.nil { exact le_rfl },
case list.cons : a l ih
{ suffices : list.cons <$> pure a <*> pure l ≤ list.cons <$> 𝓝 a <*> traverse 𝓝 l,
{ simpa only [] with functor_norm using this },
exact filter.seq_mono (filter.map_mono $... | lemma | nhds_list | topology | src/topology/list.lean | [
"topology.constructions",
"topology.algebra.monoid"
] | [
"exists_eq_left",
"filter.map_mono",
"filter.seq_mono",
"ih",
"is_open",
"is_open.mem_nhds",
"le_rfl",
"list.forall₂",
"list.forall₂.flip",
"list.forall₂_and_left",
"list.forall₂_nil_left_iff",
"list.mem_traverse",
"pure_le_nhds",
"sequence",
"set.image_subset",
"set.seq_mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nhds_nil : 𝓝 ([] : list α) = pure [] | by rw [nhds_list, list.traverse_nil _]; apply_instance | lemma | nhds_nil | topology | src/topology/list.lean | [
"topology.constructions",
"topology.algebra.monoid"
] | [
"list.traverse_nil",
"nhds_list"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nhds_cons (a : α) (l : list α) :
𝓝 (a :: l) = list.cons <$> 𝓝 a <*> 𝓝 l | by rw [nhds_list, list.traverse_cons _, ← nhds_list]; apply_instance | lemma | nhds_cons | topology | src/topology/list.lean | [
"topology.constructions",
"topology.algebra.monoid"
] | [
"list.traverse_cons",
"nhds_list"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
list.tendsto_cons {a : α} {l : list α} :
tendsto (λp:α×list α, list.cons p.1 p.2) (𝓝 a ×ᶠ 𝓝 l) (𝓝 (a :: l)) | by rw [nhds_cons, tendsto, filter.map_prod]; exact le_rfl | lemma | list.tendsto_cons | topology | src/topology/list.lean | [
"topology.constructions",
"topology.algebra.monoid"
] | [
"filter.map_prod",
"le_rfl",
"nhds_cons"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
filter.tendsto.cons {α : Type*} {f : α → β} {g : α → list β}
{a : _root_.filter α} {b : β} {l : list β} (hf : tendsto f a (𝓝 b)) (hg : tendsto g a (𝓝 l)) :
tendsto (λa, list.cons (f a) (g a)) a (𝓝 (b :: l)) | list.tendsto_cons.comp (tendsto.prod_mk hf hg) | lemma | filter.tendsto.cons | topology | src/topology/list.lean | [
"topology.constructions",
"topology.algebra.monoid"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_cons_iff {β : Type*} {f : list α → β} {b : _root_.filter β} {a : α} {l : list α} :
tendsto f (𝓝 (a :: l)) b ↔ tendsto (λp:α×list α, f (p.1 :: p.2)) (𝓝 a ×ᶠ 𝓝 l) b | have 𝓝 (a :: l) = (𝓝 a ×ᶠ 𝓝 l).map (λp:α×list α, (p.1 :: p.2)),
begin
simp only
[nhds_cons, filter.prod_eq, (filter.map_def _ _).symm, (filter.seq_eq_filter_seq _ _).symm],
simp [-filter.seq_eq_filter_seq, -filter.map_def, (∘)] with functor_norm,
end,
by rw [this, filter.tendsto_map'_iff] | lemma | list.tendsto_cons_iff | topology | src/topology/list.lean | [
"topology.constructions",
"topology.algebra.monoid"
] | [
"filter.map_def",
"filter.prod_eq",
"filter.seq_eq_filter_seq",
"filter.tendsto_map'_iff",
"nhds_cons"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_cons : continuous (λ x : α × list α, (x.1 :: x.2 : list α)) | continuous_iff_continuous_at.mpr $ λ ⟨x, y⟩, continuous_at_fst.cons continuous_at_snd | lemma | list.continuous_cons | topology | src/topology/list.lean | [
"topology.constructions",
"topology.algebra.monoid"
] | [
"continuous",
"continuous_at_snd"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_nhds {β : Type*} {f : list α → β} {r : list α → _root_.filter β}
(h_nil : tendsto f (pure []) (r []))
(h_cons : ∀l a, tendsto f (𝓝 l) (r l) →
tendsto (λp:α×list α, f (p.1 :: p.2)) (𝓝 a ×ᶠ 𝓝 l) (r (a::l))) :
∀l, tendsto f (𝓝 l) (r l) | | [] := by rwa [nhds_nil]
| (a::l) := by rw [tendsto_cons_iff]; exact h_cons l a (tendsto_nhds l) | lemma | list.tendsto_nhds | topology | src/topology/list.lean | [
"topology.constructions",
"topology.algebra.monoid"
] | [
"nhds_nil",
"tendsto_nhds"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_at_length :
∀(l : list α), continuous_at list.length l | begin
simp only [continuous_at, nhds_discrete],
refine tendsto_nhds _ _,
{ exact tendsto_pure_pure _ _ },
{ assume l a ih,
dsimp only [list.length],
refine tendsto.comp (tendsto_pure_pure (λx, x + 1) _) _,
refine tendsto.comp ih tendsto_snd }
end | lemma | list.continuous_at_length | topology | src/topology/list.lean | [
"topology.constructions",
"topology.algebra.monoid"
] | [
"continuous_at",
"ih",
"nhds_discrete",
"tendsto_nhds"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_insert_nth' {a : α} : ∀{n : ℕ} {l : list α},
tendsto (λp:α×list α, insert_nth n p.1 p.2) (𝓝 a ×ᶠ 𝓝 l) (𝓝 (insert_nth n a l)) | | 0 l := tendsto_cons
| (n+1) [] := by simp
| (n+1) (a'::l) :=
have 𝓝 a ×ᶠ 𝓝 (a' :: l) =
(𝓝 a ×ᶠ (𝓝 a' ×ᶠ 𝓝 l)).map (λp:α×α×list α, (p.1, p.2.1 :: p.2.2)),
begin
simp only [nhds_cons, filter.prod_eq, ← filter.map_def, ← filter.seq_eq_filter_seq],
simp [-filter.seq_eq_filter_seq, -filter.map_de... | lemma | list.tendsto_insert_nth' | topology | src/topology/list.lean | [
"topology.constructions",
"topology.algebra.monoid"
] | [
"filter.map_def",
"filter.prod_eq",
"filter.seq_eq_filter_seq",
"nhds_cons"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_insert_nth {β} {n : ℕ} {a : α} {l : list α} {f : β → α} {g : β → list α}
{b : _root_.filter β} (hf : tendsto f b (𝓝 a)) (hg : tendsto g b (𝓝 l)) :
tendsto (λb:β, insert_nth n (f b) (g b)) b (𝓝 (insert_nth n a l)) | tendsto_insert_nth'.comp (tendsto.prod_mk hf hg) | lemma | list.tendsto_insert_nth | topology | src/topology/list.lean | [
"topology.constructions",
"topology.algebra.monoid"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_insert_nth {n : ℕ} : continuous (λp:α×list α, insert_nth n p.1 p.2) | continuous_iff_continuous_at.mpr $
assume ⟨a, l⟩, by rw [continuous_at, nhds_prod_eq]; exact tendsto_insert_nth' | lemma | list.continuous_insert_nth | topology | src/topology/list.lean | [
"topology.constructions",
"topology.algebra.monoid"
] | [
"continuous",
"continuous_at",
"nhds_prod_eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_remove_nth : ∀{n : ℕ} {l : list α},
tendsto (λl, remove_nth l n) (𝓝 l) (𝓝 (remove_nth l n)) | | _ [] := by rw [nhds_nil]; exact tendsto_pure_nhds _ _
| 0 (a::l) := by rw [tendsto_cons_iff]; exact tendsto_snd
| (n+1) (a::l) :=
begin
rw [tendsto_cons_iff],
dsimp [remove_nth],
exact tendsto_fst.cons ((@tendsto_remove_nth n l).comp tendsto_snd)
end | lemma | list.tendsto_remove_nth | topology | src/topology/list.lean | [
"topology.constructions",
"topology.algebra.monoid"
] | [
"nhds_nil",
"tendsto_pure_nhds"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_remove_nth {n : ℕ} : continuous (λl : list α, remove_nth l n) | continuous_iff_continuous_at.mpr $ assume a, tendsto_remove_nth | lemma | list.continuous_remove_nth | topology | src/topology/list.lean | [
"topology.constructions",
"topology.algebra.monoid"
] | [
"continuous"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_prod [monoid α] [has_continuous_mul α] {l : list α} :
tendsto list.prod (𝓝 l) (𝓝 l.prod) | begin
induction l with x l ih,
{ simp [nhds_nil, mem_of_mem_nhds, tendsto_pure_left] {contextual := tt} },
simp_rw [tendsto_cons_iff, prod_cons],
have := continuous_iff_continuous_at.mp continuous_mul (x, l.prod),
rw [continuous_at, nhds_prod_eq] at this,
exact this.comp (tendsto_id.prod_map ih)
end | lemma | list.tendsto_prod | topology | src/topology/list.lean | [
"topology.constructions",
"topology.algebra.monoid"
] | [
"continuous_at",
"continuous_mul",
"has_continuous_mul",
"ih",
"list.prod",
"mem_of_mem_nhds",
"monoid",
"nhds_nil",
"nhds_prod_eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_prod [monoid α] [has_continuous_mul α] : continuous (prod : list α → α) | continuous_iff_continuous_at.mpr $ λ l, tendsto_prod | lemma | list.continuous_prod | topology | src/topology/list.lean | [
"topology.constructions",
"topology.algebra.monoid"
] | [
"continuous",
"has_continuous_mul",
"monoid"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_cons {n : ℕ} {a : α} {l : vector α n}:
tendsto (λp:α×vector α n, p.1 ::ᵥ p.2) (𝓝 a ×ᶠ 𝓝 l) (𝓝 (a ::ᵥ l)) | by { simp [tendsto_subtype_rng, ←subtype.val_eq_coe, cons_val],
exact tendsto_fst.cons (tendsto.comp continuous_at_subtype_coe tendsto_snd) } | lemma | vector.tendsto_cons | topology | src/topology/list.lean | [
"topology.constructions",
"topology.algebra.monoid"
] | [
"continuous_at_subtype_coe",
"tendsto_subtype_rng"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_insert_nth
{n : ℕ} {i : fin (n+1)} {a:α} :
∀{l:vector α n}, tendsto (λp:α×vector α n, insert_nth p.1 i p.2)
(𝓝 a ×ᶠ 𝓝 l) (𝓝 (insert_nth a i l)) | | ⟨l, hl⟩ :=
begin
rw [insert_nth, tendsto_subtype_rng],
simp [insert_nth_val],
exact list.tendsto_insert_nth tendsto_fst (tendsto.comp continuous_at_subtype_coe tendsto_snd : _)
end | lemma | vector.tendsto_insert_nth | topology | src/topology/list.lean | [
"topology.constructions",
"topology.algebra.monoid"
] | [
"continuous_at_subtype_coe",
"list.tendsto_insert_nth",
"tendsto_subtype_rng"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_insert_nth' {n : ℕ} {i : fin (n+1)} :
continuous (λp:α×vector α n, insert_nth p.1 i p.2) | continuous_iff_continuous_at.mpr $ assume ⟨a, l⟩,
by rw [continuous_at, nhds_prod_eq]; exact tendsto_insert_nth | lemma | vector.continuous_insert_nth' | topology | src/topology/list.lean | [
"topology.constructions",
"topology.algebra.monoid"
] | [
"continuous",
"continuous_at",
"nhds_prod_eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_insert_nth {n : ℕ} {i : fin (n+1)}
{f : β → α} {g : β → vector α n} (hf : continuous f) (hg : continuous g) :
continuous (λb, insert_nth (f b) i (g b)) | continuous_insert_nth'.comp (hf.prod_mk hg : _) | lemma | vector.continuous_insert_nth | topology | src/topology/list.lean | [
"topology.constructions",
"topology.algebra.monoid"
] | [
"continuous"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_at_remove_nth {n : ℕ} {i : fin (n+1)} :
∀{l:vector α (n+1)}, continuous_at (remove_nth i) l | | ⟨l, hl⟩ :=
-- ∀{l:vector α (n+1)}, tendsto (remove_nth i) (𝓝 l) (𝓝 (remove_nth i l))
--| ⟨l, hl⟩ :=
begin
rw [continuous_at, remove_nth, tendsto_subtype_rng],
simp only [← subtype.val_eq_coe, vector.remove_nth_val],
exact tendsto.comp list.tendsto_remove_nth continuous_at_subtype_coe,
end | lemma | vector.continuous_at_remove_nth | topology | src/topology/list.lean | [
"topology.constructions",
"topology.algebra.monoid"
] | [
"continuous_at",
"continuous_at_subtype_coe",
"list.tendsto_remove_nth",
"subtype.val_eq_coe",
"tendsto_subtype_rng",
"vector.remove_nth_val"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_remove_nth {n : ℕ} {i : fin (n+1)} :
continuous (remove_nth i : vector α (n+1) → vector α n) | continuous_iff_continuous_at.mpr $ assume ⟨a, l⟩, continuous_at_remove_nth | lemma | vector.continuous_remove_nth | topology | src/topology/list.lean | [
"topology.constructions",
"topology.algebra.monoid"
] | [
"continuous"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
locally_finite (f : ι → set X) | ∀ x : X, ∃t ∈ 𝓝 x, {i | (f i ∩ t).nonempty}.finite | def | locally_finite | topology | src/topology/locally_finite.lean | [
"topology.continuous_on",
"order.filter.small_sets"
] | [
"finite"
] | A family of sets in `set X` is locally finite if at every point `x : X`,
there is a neighborhood of `x` which meets only finitely many sets in the family. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
locally_finite_of_finite [finite ι] (f : ι → set X) : locally_finite f | assume x, ⟨univ, univ_mem, to_finite _⟩ | lemma | locally_finite_of_finite | topology | src/topology/locally_finite.lean | [
"topology.continuous_on",
"order.filter.small_sets"
] | [
"finite",
"locally_finite"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
point_finite (hf : locally_finite f) (x : X) : {b | x ∈ f b}.finite | let ⟨t, hxt, ht⟩ := hf x in ht.subset $ λ b hb, ⟨x, hb, mem_of_mem_nhds hxt⟩ | lemma | locally_finite.point_finite | topology | src/topology/locally_finite.lean | [
"topology.continuous_on",
"order.filter.small_sets"
] | [
"finite",
"locally_finite",
"mem_of_mem_nhds"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
subset (hf : locally_finite f) (hg : ∀ i, g i ⊆ f i) : locally_finite g | assume a,
let ⟨t, ht₁, ht₂⟩ := hf a in
⟨t, ht₁, ht₂.subset $ assume i hi, hi.mono $ inter_subset_inter (hg i) subset.rfl⟩ | lemma | locally_finite.subset | topology | src/topology/locally_finite.lean | [
"topology.continuous_on",
"order.filter.small_sets"
] | [
"locally_finite"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_inj_on {g : ι' → ι} (hf : locally_finite f)
(hg : inj_on g {i | (f (g i)).nonempty}) : locally_finite (f ∘ g) | λ x, let ⟨t, htx, htf⟩ := hf x in ⟨t, htx, htf.preimage $ hg.mono $ λ i hi,
hi.out.mono $ inter_subset_left _ _⟩ | lemma | locally_finite.comp_inj_on | topology | src/topology/locally_finite.lean | [
"topology.continuous_on",
"order.filter.small_sets"
] | [
"locally_finite"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_injective {g : ι' → ι} (hf : locally_finite f) (hg : injective g) :
locally_finite (f ∘ g) | hf.comp_inj_on (hg.inj_on _) | lemma | locally_finite.comp_injective | topology | src/topology/locally_finite.lean | [
"topology.continuous_on",
"order.filter.small_sets"
] | [
"locally_finite"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.locally_finite_iff_small_sets :
locally_finite f ↔ ∀ x, ∀ᶠ s in (𝓝 x).small_sets, {i | (f i ∩ s).nonempty}.finite | forall_congr $ λ x, iff.symm $ eventually_small_sets' $ λ s t hst ht, ht.subset $
λ i hi, hi.mono $ inter_subset_inter_right _ hst | lemma | locally_finite_iff_small_sets | topology | src/topology/locally_finite.lean | [
"topology.continuous_on",
"order.filter.small_sets"
] | [
"finite",
"locally_finite"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eventually_small_sets (hf : locally_finite f) (x : X) :
∀ᶠ s in (𝓝 x).small_sets, {i | (f i ∩ s).nonempty}.finite | locally_finite_iff_small_sets.mp hf x | lemma | locally_finite.eventually_small_sets | topology | src/topology/locally_finite.lean | [
"topology.continuous_on",
"order.filter.small_sets"
] | [
"finite",
"locally_finite"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exists_mem_basis {ι' : Sort*} (hf : locally_finite f) {p : ι' → Prop}
{s : ι' → set X} {x : X} (hb : (𝓝 x).has_basis p s) :
∃ i (hi : p i), {j | (f j ∩ s i).nonempty}.finite | let ⟨i, hpi, hi⟩ := hb.small_sets.eventually_iff.mp (hf.eventually_small_sets x)
in ⟨i, hpi, hi subset.rfl⟩ | lemma | locally_finite.exists_mem_basis | topology | src/topology/locally_finite.lean | [
"topology.continuous_on",
"order.filter.small_sets"
] | [
"finite",
"locally_finite"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nhds_within_Union (hf : locally_finite f) (a : X) :
𝓝[⋃ i, f i] a = ⨆ i, 𝓝[f i] a | begin
rcases hf a with ⟨U, haU, hfin⟩,
refine le_antisymm _ (supr_le $ λ i, nhds_within_mono _ (subset_Union _ _)),
calc 𝓝[⋃ i, f i] a = 𝓝[⋃ i, f i ∩ U] a :
by rw [← Union_inter, ← nhds_within_inter_of_mem' (nhds_within_le_nhds haU)]
... = 𝓝[⋃ i ∈ {j | (f j ∩ U).nonempty}, (f i ∩ U)] a :
by simp only... | theorem | locally_finite.nhds_within_Union | topology | src/topology/locally_finite.lean | [
"topology.continuous_on",
"order.filter.small_sets"
] | [
"locally_finite",
"nhds_within_Union",
"nhds_within_bUnion",
"nhds_within_inter_of_mem'",
"nhds_within_le_nhds",
"nhds_within_mono",
"supr_le",
"supr_mono",
"supr₂_le_supr"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_on_Union' {g : X → Y} (hf : locally_finite f)
(hc : ∀ i x, x ∈ closure (f i) → continuous_within_at g (f i) x) :
continuous_on g (⋃ i, f i) | begin
rintro x -,
rw [continuous_within_at, hf.nhds_within_Union, tendsto_supr],
intro i,
by_cases hx : x ∈ closure (f i),
{ exact hc i _ hx },
{ rw [mem_closure_iff_nhds_within_ne_bot, not_ne_bot] at hx,
rw [hx],
exact tendsto_bot }
end | lemma | locally_finite.continuous_on_Union' | topology | src/topology/locally_finite.lean | [
"topology.continuous_on",
"order.filter.small_sets"
] | [
"closure",
"continuous_on",
"continuous_within_at",
"locally_finite",
"mem_closure_iff_nhds_within_ne_bot"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_on_Union {g : X → Y} (hf : locally_finite f) (h_cl : ∀ i, is_closed (f i))
(h_cont : ∀ i, continuous_on g (f i)) :
continuous_on g (⋃ i, f i) | hf.continuous_on_Union' $ λ i x hx, h_cont i x $ (h_cl i).closure_subset hx | lemma | locally_finite.continuous_on_Union | topology | src/topology/locally_finite.lean | [
"topology.continuous_on",
"order.filter.small_sets"
] | [
"continuous_on",
"is_closed",
"locally_finite"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous' {g : X → Y} (hf : locally_finite f) (h_cov : (⋃ i, f i) = univ)
(hc : ∀ i x, x ∈ closure (f i) → continuous_within_at g (f i) x) :
continuous g | continuous_iff_continuous_on_univ.2 $ h_cov ▸ hf.continuous_on_Union' hc | lemma | locally_finite.continuous' | topology | src/topology/locally_finite.lean | [
"topology.continuous_on",
"order.filter.small_sets"
] | [
"closure",
"continuous",
"continuous_within_at",
"locally_finite"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous {g : X → Y} (hf : locally_finite f) (h_cov : (⋃ i, f i) = univ)
(h_cl : ∀ i, is_closed (f i)) (h_cont : ∀ i, continuous_on g (f i)) :
continuous g | continuous_iff_continuous_on_univ.2 $ h_cov ▸ hf.continuous_on_Union h_cl h_cont | lemma | locally_finite.continuous | topology | src/topology/locally_finite.lean | [
"topology.continuous_on",
"order.filter.small_sets"
] | [
"continuous",
"continuous_on",
"is_closed",
"locally_finite"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
closure (hf : locally_finite f) : locally_finite (λ i, closure (f i)) | begin
intro x,
rcases hf x with ⟨s, hsx, hsf⟩,
refine ⟨interior s, interior_mem_nhds.2 hsx, hsf.subset $ λ i hi, _⟩,
exact (hi.mono is_open_interior.closure_inter).of_closure.mono
(inter_subset_inter_right _ interior_subset)
end | lemma | locally_finite.closure | topology | src/topology/locally_finite.lean | [
"topology.continuous_on",
"order.filter.small_sets"
] | [
"closure",
"interior_subset",
"locally_finite"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
closure_Union (h : locally_finite f) : closure (⋃ i, f i) = ⋃ i, closure (f i) | begin
ext x,
simp only [mem_closure_iff_nhds_within_ne_bot, h.nhds_within_Union, supr_ne_bot, mem_Union]
end | lemma | locally_finite.closure_Union | topology | src/topology/locally_finite.lean | [
"topology.continuous_on",
"order.filter.small_sets"
] | [
"closure",
"closure_Union",
"locally_finite",
"mem_closure_iff_nhds_within_ne_bot"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_closed_Union (hf : locally_finite f) (hc : ∀ i, is_closed (f i)) :
is_closed (⋃ i, f i) | by simp only [← closure_eq_iff_is_closed, hf.closure_Union, (hc _).closure_eq] | lemma | locally_finite.is_closed_Union | topology | src/topology/locally_finite.lean | [
"topology.continuous_on",
"order.filter.small_sets"
] | [
"closure_eq_iff_is_closed",
"is_closed",
"is_closed_Union",
"locally_finite"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Inter_compl_mem_nhds (hf : locally_finite f) (hc : ∀ i, is_closed (f i)) (x : X) :
(⋂ i (hi : x ∉ f i), (f i)ᶜ) ∈ 𝓝 x | begin
refine is_open.mem_nhds _ (mem_Inter₂.2 $ λ i, id),
suffices : is_closed (⋃ i : {i // x ∉ f i}, f i),
by rwa [← is_open_compl_iff, compl_Union, Inter_subtype] at this,
exact (hf.comp_injective subtype.coe_injective).is_closed_Union (λ i, hc _)
end | lemma | locally_finite.Inter_compl_mem_nhds | topology | src/topology/locally_finite.lean | [
"topology.continuous_on",
"order.filter.small_sets"
] | [
"is_closed",
"is_closed_Union",
"is_open.mem_nhds",
"is_open_compl_iff",
"locally_finite",
"subtype.coe_injective"
] | If `f : β → set α` is a locally finite family of closed sets, then for any `x : α`, the
intersection of the complements to `f i`, `x ∉ f i`, is a neighbourhood of `x`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
exists_forall_eventually_eq_prod {π : X → Sort*} {f : ℕ → Π x : X, π x}
(hf : locally_finite (λ n, {x | f (n + 1) x ≠ f n x})) :
∃ F : Π x : X, π x, ∀ x, ∀ᶠ p : ℕ × X in at_top ×ᶠ 𝓝 x, f p.1 p.2 = F p.2 | begin
choose U hUx hU using hf,
choose N hN using λ x, (hU x).bdd_above,
replace hN : ∀ x (n > N x) (y ∈ U x), f (n + 1) y = f n y,
from λ x n hn y hy, by_contra (λ hne, hn.lt.not_le $ hN x ⟨y, hne, hy⟩),
replace hN : ∀ x (n ≥ N x + 1) (y ∈ U x), f n y = f (N x + 1) y,
from λ x n hn y hy, nat.le_inducti... | lemma | locally_finite.exists_forall_eventually_eq_prod | topology | src/topology/locally_finite.lean | [
"topology.continuous_on",
"order.filter.small_sets"
] | [
"bdd_above",
"by_contra",
"filter.prod_mem_prod",
"locally_finite",
"mem_of_mem_nhds",
"nat.le_induction"
] | Let `f : ℕ → Π a, β a` be a sequence of (dependent) functions on a topological space. Suppose
that the family of sets `s n = {x | f (n + 1) x ≠ f n x}` is locally finite. Then there exists a
function `F : Π a, β a` such that for any `x`, we have `f n x = F x` on the product of an infinite
interval `[N, +∞)` and a neigh... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
exists_forall_eventually_at_top_eventually_eq' {π : X → Sort*}
{f : ℕ → Π x : X, π x} (hf : locally_finite (λ n, {x | f (n + 1) x ≠ f n x})) :
∃ F : Π x : X, π x, ∀ x, ∀ᶠ n : ℕ in at_top, ∀ᶠ y : X in 𝓝 x, f n y = F y | hf.exists_forall_eventually_eq_prod.imp $ λ F hF x, (hF x).curry | lemma | locally_finite.exists_forall_eventually_at_top_eventually_eq' | topology | src/topology/locally_finite.lean | [
"topology.continuous_on",
"order.filter.small_sets"
] | [
"locally_finite"
] | Let `f : ℕ → Π a, β a` be a sequence of (dependent) functions on a topological space. Suppose
that the family of sets `s n = {x | f (n + 1) x ≠ f n x}` is locally finite. Then there exists a
function `F : Π a, β a` such that for any `x`, for sufficiently large values of `n`, we have
`f n y = F y` in a neighbourhood of ... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
exists_forall_eventually_at_top_eventually_eq {f : ℕ → X → α}
(hf : locally_finite (λ n, {x | f (n + 1) x ≠ f n x})) :
∃ F : X → α, ∀ x, ∀ᶠ n : ℕ in at_top, f n =ᶠ[𝓝 x] F | hf.exists_forall_eventually_at_top_eventually_eq' | lemma | locally_finite.exists_forall_eventually_at_top_eventually_eq | topology | src/topology/locally_finite.lean | [
"topology.continuous_on",
"order.filter.small_sets"
] | [
"locally_finite"
] | Let `f : ℕ → α → β` be a sequence of functions on a topological space. Suppose
that the family of sets `s n = {x | f (n + 1) x ≠ f n x}` is locally finite. Then there exists a
function `F : α → β` such that for any `x`, for sufficiently large values of `n`, we have
`f n =ᶠ[𝓝 x] F`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
preimage_continuous {g : Y → X} (hf : locally_finite f) (hg : continuous g) :
locally_finite (λ i, g ⁻¹' (f i)) | λ x, let ⟨s, hsx, hs⟩ := hf (g x)
in ⟨g ⁻¹' s, hg.continuous_at hsx, hs.subset $ λ i ⟨y, hy⟩, ⟨g y, hy⟩⟩ | lemma | locally_finite.preimage_continuous | topology | src/topology/locally_finite.lean | [
"topology.continuous_on",
"order.filter.small_sets"
] | [
"continuous",
"locally_finite"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
equiv.locally_finite_comp_iff (e : ι' ≃ ι) :
locally_finite (f ∘ e) ↔ locally_finite f | ⟨λ h, by simpa only [(∘), e.apply_symm_apply] using h.comp_injective e.symm.injective,
λ h, h.comp_injective e.injective⟩ | lemma | equiv.locally_finite_comp_iff | topology | src/topology/locally_finite.lean | [
"topology.continuous_on",
"order.filter.small_sets"
] | [
"locally_finite"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
locally_finite_sum {f : ι ⊕ ι' → set X} :
locally_finite f ↔ locally_finite (f ∘ sum.inl) ∧ locally_finite (f ∘ sum.inr) | by simp only [locally_finite_iff_small_sets, ← forall_and_distrib, ← finite_preimage_inl_and_inr,
preimage_set_of_eq, (∘), eventually_and] | lemma | locally_finite_sum | topology | src/topology/locally_finite.lean | [
"topology.continuous_on",
"order.filter.small_sets"
] | [
"forall_and_distrib",
"locally_finite",
"locally_finite_iff_small_sets"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
locally_finite.sum_elim {g : ι' → set X} (hf : locally_finite f) (hg : locally_finite g) :
locally_finite (sum.elim f g) | locally_finite_sum.mpr ⟨hf, hg⟩ | lemma | locally_finite.sum_elim | topology | src/topology/locally_finite.lean | [
"topology.continuous_on",
"order.filter.small_sets"
] | [
"locally_finite",
"sum.elim"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
locally_finite_option {f : option ι → set X} :
locally_finite f ↔ locally_finite (f ∘ some) | begin
simp only [← (equiv.option_equiv_sum_punit.{u} ι).symm.locally_finite_comp_iff,
locally_finite_sum, locally_finite_of_finite, and_true],
refl
end | lemma | locally_finite_option | topology | src/topology/locally_finite.lean | [
"topology.continuous_on",
"order.filter.small_sets"
] | [
"locally_finite",
"locally_finite_of_finite",
"locally_finite_sum"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
locally_finite.option_elim (hf : locally_finite f) (s : set X) :
locally_finite (option.elim s f) | locally_finite_option.2 hf | lemma | locally_finite.option_elim | topology | src/topology/locally_finite.lean | [
"topology.continuous_on",
"order.filter.small_sets"
] | [
"locally_finite",
"option.elim"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
set.restrict_preimage_inducing (s : set β) (h : inducing f) :
inducing (s.restrict_preimage f) | begin
simp_rw [inducing_coe.inducing_iff, inducing_iff_nhds, restrict_preimage, maps_to.coe_restrict,
restrict_eq, ← @filter.comap_comap _ _ _ _ coe f] at h ⊢,
intros a,
rw [← h, ← inducing_coe.nhds_eq_comap],
end | lemma | set.restrict_preimage_inducing | topology | src/topology/local_at_target.lean | [
"topology.sets.opens"
] | [
"filter.comap_comap",
"inducing",
"inducing_iff_nhds"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
set.restrict_preimage_embedding (s : set β) (h : embedding f) :
embedding (s.restrict_preimage f) | ⟨h.1.restrict_preimage s, h.2.restrict_preimage s⟩ | lemma | set.restrict_preimage_embedding | topology | src/topology/local_at_target.lean | [
"topology.sets.opens"
] | [
"embedding"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
set.restrict_preimage_open_embedding (s : set β) (h : open_embedding f) :
open_embedding (s.restrict_preimage f) | ⟨h.1.restrict_preimage s,
(s.range_restrict_preimage f).symm ▸ continuous_subtype_coe.is_open_preimage _ h.2⟩ | lemma | set.restrict_preimage_open_embedding | topology | src/topology/local_at_target.lean | [
"topology.sets.opens"
] | [
"open_embedding"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
set.restrict_preimage_closed_embedding (s : set β) (h : closed_embedding f) :
closed_embedding (s.restrict_preimage f) | ⟨h.1.restrict_preimage s,
(s.range_restrict_preimage f).symm ▸ inducing_coe.is_closed_preimage _ h.2⟩ | lemma | set.restrict_preimage_closed_embedding | topology | src/topology/local_at_target.lean | [
"topology.sets.opens"
] | [
"closed_embedding"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
set.restrict_preimage_is_closed_map (s : set β) (H : is_closed_map f) :
is_closed_map (s.restrict_preimage f) | begin
rintros t ⟨u, hu, e⟩,
refine ⟨⟨_, (H _ (is_open.is_closed_compl hu)).1, _⟩⟩,
rw ← (congr_arg has_compl.compl e).trans (compl_compl t),
simp only [set.preimage_compl, compl_inj_iff],
ext ⟨x, hx⟩,
suffices : (∃ y, y ∉ u ∧ f y = x) ↔ ∃ y, f y ∈ s ∧ y ∉ u ∧ f y = x,
{ simpa [set.restrict_preimage, ← sub... | lemma | set.restrict_preimage_is_closed_map | topology | src/topology/local_at_target.lean | [
"topology.sets.opens"
] | [
"compl_compl",
"compl_inj_iff",
"is_closed_map",
"is_open.is_closed_compl",
"set.preimage_compl",
"set.restrict_preimage",
"subtype.coe_inj"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_open_iff_inter_of_supr_eq_top (s : set β) :
is_open s ↔ ∀ i, is_open (s ∩ U i) | begin
split,
{ exact λ H i, H.inter (U i).2 },
{ intro H,
have : (⋃ i, (U i : set β)) = set.univ := by { convert (congr_arg coe hU), simp },
rw [← s.inter_univ, ← this, set.inter_Union],
exact is_open_Union H }
end | lemma | is_open_iff_inter_of_supr_eq_top | topology | src/topology/local_at_target.lean | [
"topology.sets.opens"
] | [
"is_open",
"is_open_Union",
"set.inter_Union"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_open_iff_coe_preimage_of_supr_eq_top (s : set β) :
is_open s ↔ ∀ i, is_open (coe ⁻¹' s : set (U i)) | begin
simp_rw [(U _).2.open_embedding_subtype_coe.open_iff_image_open,
set.image_preimage_eq_inter_range, subtype.range_coe],
apply is_open_iff_inter_of_supr_eq_top,
assumption
end | lemma | is_open_iff_coe_preimage_of_supr_eq_top | topology | src/topology/local_at_target.lean | [
"topology.sets.opens"
] | [
"is_open",
"is_open_iff_inter_of_supr_eq_top",
"set.image_preimage_eq_inter_range",
"subtype.range_coe"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_closed_iff_coe_preimage_of_supr_eq_top (s : set β) :
is_closed s ↔ ∀ i, is_closed (coe ⁻¹' s : set (U i)) | by simpa using is_open_iff_coe_preimage_of_supr_eq_top hU sᶜ | lemma | is_closed_iff_coe_preimage_of_supr_eq_top | topology | src/topology/local_at_target.lean | [
"topology.sets.opens"
] | [
"is_closed",
"is_open_iff_coe_preimage_of_supr_eq_top"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_closed_map_iff_is_closed_map_of_supr_eq_top :
is_closed_map f ↔ ∀ i, is_closed_map ((U i).1.restrict_preimage f) | begin
refine ⟨λ h i, set.restrict_preimage_is_closed_map _ h, _⟩,
rintros H s hs,
rw is_closed_iff_coe_preimage_of_supr_eq_top hU,
intro i,
convert H i _ ⟨⟨_, hs.1, eq_compl_comm.mpr rfl⟩⟩,
ext ⟨x, hx⟩,
suffices : (∃ y, y ∈ s ∧ f y = x) ↔ ∃ y, f y ∈ U i ∧ y ∈ s ∧ f y = x,
{ simpa [set.restrict_preimage,... | lemma | is_closed_map_iff_is_closed_map_of_supr_eq_top | topology | src/topology/local_at_target.lean | [
"topology.sets.opens"
] | [
"is_closed_iff_coe_preimage_of_supr_eq_top",
"is_closed_map",
"set.restrict_preimage",
"set.restrict_preimage_is_closed_map",
"subtype.coe_inj"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inducing_iff_inducing_of_supr_eq_top (h : continuous f) :
inducing f ↔ ∀ i, inducing ((U i).1.restrict_preimage f) | begin
simp_rw [inducing_coe.inducing_iff, inducing_iff_nhds, restrict_preimage, maps_to.coe_restrict,
restrict_eq, ← @filter.comap_comap _ _ _ _ coe f],
split,
{ intros H i x, rw [← H, ← inducing_coe.nhds_eq_comap] },
{ intros H x,
obtain ⟨i, hi⟩ := opens.mem_supr.mp (show f x ∈ supr U, by { rw hU, triv... | lemma | inducing_iff_inducing_of_supr_eq_top | topology | src/topology/local_at_target.lean | [
"topology.sets.opens"
] | [
"continuous",
"filter.comap_comap",
"filter.le_principal_iff",
"filter.preimage_mem_comap",
"filter.subtype_coe_map_comap",
"function.comp_apply",
"inducing",
"inducing_iff_nhds",
"inf_eq_left",
"open_embedding.map_nhds_eq",
"subtype.coe_mk",
"supr"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
embedding_iff_embedding_of_supr_eq_top (h : continuous f) :
embedding f ↔ ∀ i, embedding ((U i).1.restrict_preimage f) | begin
simp_rw embedding_iff,
rw forall_and_distrib,
apply and_congr,
{ apply inducing_iff_inducing_of_supr_eq_top; assumption },
{ apply set.injective_iff_injective_of_Union_eq_univ, convert (congr_arg coe hU), simp }
end | lemma | embedding_iff_embedding_of_supr_eq_top | topology | src/topology/local_at_target.lean | [
"topology.sets.opens"
] | [
"continuous",
"embedding",
"forall_and_distrib",
"inducing_iff_inducing_of_supr_eq_top",
"set.injective_iff_injective_of_Union_eq_univ"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
open_embedding_iff_open_embedding_of_supr_eq_top (h : continuous f) :
open_embedding f ↔ ∀ i, open_embedding ((U i).1.restrict_preimage f) | begin
simp_rw open_embedding_iff,
rw forall_and_distrib,
apply and_congr,
{ apply embedding_iff_embedding_of_supr_eq_top; assumption },
{ simp_rw set.range_restrict_preimage, apply is_open_iff_coe_preimage_of_supr_eq_top hU }
end | lemma | open_embedding_iff_open_embedding_of_supr_eq_top | topology | src/topology/local_at_target.lean | [
"topology.sets.opens"
] | [
"continuous",
"embedding_iff_embedding_of_supr_eq_top",
"forall_and_distrib",
"is_open_iff_coe_preimage_of_supr_eq_top",
"open_embedding",
"set.range_restrict_preimage"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
closed_embedding_iff_closed_embedding_of_supr_eq_top (h : continuous f) :
closed_embedding f ↔ ∀ i, closed_embedding ((U i).1.restrict_preimage f) | begin
simp_rw closed_embedding_iff,
rw forall_and_distrib,
apply and_congr,
{ apply embedding_iff_embedding_of_supr_eq_top; assumption },
{ simp_rw set.range_restrict_preimage, apply is_closed_iff_coe_preimage_of_supr_eq_top hU }
end | lemma | closed_embedding_iff_closed_embedding_of_supr_eq_top | topology | src/topology/local_at_target.lean | [
"topology.sets.opens"
] | [
"closed_embedding",
"continuous",
"embedding_iff_embedding_of_supr_eq_top",
"forall_and_distrib",
"is_closed_iff_coe_preimage_of_supr_eq_top",
"set.range_restrict_preimage"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_local_min_on | is_min_filter f (𝓝[s] a) a | def | is_local_min_on | topology | src/topology/local_extr.lean | [
"order.filter.extr",
"topology.continuous_on"
] | [
"is_min_filter"
] | `is_local_min_on f s a` means that `f a ≤ f x` for all `x ∈ s` in some neighborhood of `a`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_local_max_on | is_max_filter f (𝓝[s] a) a | def | is_local_max_on | topology | src/topology/local_extr.lean | [
"order.filter.extr",
"topology.continuous_on"
] | [
"is_max_filter"
] | `is_local_max_on f s a` means that `f x ≤ f a` for all `x ∈ s` in some neighborhood of `a`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_local_extr_on | is_extr_filter f (𝓝[s] a) a | def | is_local_extr_on | topology | src/topology/local_extr.lean | [
"order.filter.extr",
"topology.continuous_on"
] | [
"is_extr_filter"
] | `is_local_extr_on f s a` means `is_local_min_on f s a ∨ is_local_max_on f s a`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_local_min | is_min_filter f (𝓝 a) a | def | is_local_min | topology | src/topology/local_extr.lean | [
"order.filter.extr",
"topology.continuous_on"
] | [
"is_min_filter"
] | `is_local_min f a` means that `f a ≤ f x` for all `x` in some neighborhood of `a`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_local_max | is_max_filter f (𝓝 a) a | def | is_local_max | topology | src/topology/local_extr.lean | [
"order.filter.extr",
"topology.continuous_on"
] | [
"is_max_filter"
] | `is_local_max f a` means that `f x ≤ f a` for all `x ∈ s` in some neighborhood of `a`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_local_extr | is_extr_filter f (𝓝 a) a | def | is_local_extr | topology | src/topology/local_extr.lean | [
"order.filter.extr",
"topology.continuous_on"
] | [
"is_extr_filter"
] | `is_local_extr_on f s a` means `is_local_min_on f s a ∨ is_local_max_on f s a`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_local_extr_on.elim {p : Prop} :
is_local_extr_on f s a → (is_local_min_on f s a → p) → (is_local_max_on f s a → p) → p | or.elim | lemma | is_local_extr_on.elim | topology | src/topology/local_extr.lean | [
"order.filter.extr",
"topology.continuous_on"
] | [
"is_local_extr_on",
"is_local_max_on",
"is_local_min_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_local_extr.elim {p : Prop} :
is_local_extr f a → (is_local_min f a → p) → (is_local_max f a → p) → p | or.elim | lemma | is_local_extr.elim | topology | src/topology/local_extr.lean | [
"order.filter.extr",
"topology.continuous_on"
] | [
"is_local_extr",
"is_local_max",
"is_local_min"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_local_min.on (h : is_local_min f a) (s) : is_local_min_on f s a | h.filter_inf _ | lemma | is_local_min.on | topology | src/topology/local_extr.lean | [
"order.filter.extr",
"topology.continuous_on"
] | [
"is_local_min",
"is_local_min_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_local_max.on (h : is_local_max f a) (s) : is_local_max_on f s a | h.filter_inf _ | lemma | is_local_max.on | topology | src/topology/local_extr.lean | [
"order.filter.extr",
"topology.continuous_on"
] | [
"is_local_max",
"is_local_max_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_local_extr.on (h : is_local_extr f a) (s) : is_local_extr_on f s a | h.filter_inf _ | lemma | is_local_extr.on | topology | src/topology/local_extr.lean | [
"order.filter.extr",
"topology.continuous_on"
] | [
"is_local_extr",
"is_local_extr_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_local_min_on.on_subset {t : set α} (hf : is_local_min_on f t a) (h : s ⊆ t) :
is_local_min_on f s a | hf.filter_mono $ nhds_within_mono a h | lemma | is_local_min_on.on_subset | topology | src/topology/local_extr.lean | [
"order.filter.extr",
"topology.continuous_on"
] | [
"is_local_min_on",
"nhds_within_mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_local_max_on.on_subset {t : set α} (hf : is_local_max_on f t a) (h : s ⊆ t) :
is_local_max_on f s a | hf.filter_mono $ nhds_within_mono a h | lemma | is_local_max_on.on_subset | topology | src/topology/local_extr.lean | [
"order.filter.extr",
"topology.continuous_on"
] | [
"is_local_max_on",
"nhds_within_mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_local_extr_on.on_subset {t : set α} (hf : is_local_extr_on f t a) (h : s ⊆ t) :
is_local_extr_on f s a | hf.filter_mono $ nhds_within_mono a h | lemma | is_local_extr_on.on_subset | topology | src/topology/local_extr.lean | [
"order.filter.extr",
"topology.continuous_on"
] | [
"is_local_extr_on",
"nhds_within_mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_local_min_on.inter (hf : is_local_min_on f s a) (t) : is_local_min_on f (s ∩ t) a | hf.on_subset (inter_subset_left s t) | lemma | is_local_min_on.inter | topology | src/topology/local_extr.lean | [
"order.filter.extr",
"topology.continuous_on"
] | [
"is_local_min_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_local_max_on.inter (hf : is_local_max_on f s a) (t) : is_local_max_on f (s ∩ t) a | hf.on_subset (inter_subset_left s t) | lemma | is_local_max_on.inter | topology | src/topology/local_extr.lean | [
"order.filter.extr",
"topology.continuous_on"
] | [
"is_local_max_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_local_extr_on.inter (hf : is_local_extr_on f s a) (t) : is_local_extr_on f (s ∩ t) a | hf.on_subset (inter_subset_left s t) | lemma | is_local_extr_on.inter | topology | src/topology/local_extr.lean | [
"order.filter.extr",
"topology.continuous_on"
] | [
"is_local_extr_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_min_on.localize (hf : is_min_on f s a) : is_local_min_on f s a | hf.filter_mono $ inf_le_right | lemma | is_min_on.localize | topology | src/topology/local_extr.lean | [
"order.filter.extr",
"topology.continuous_on"
] | [
"inf_le_right",
"is_local_min_on",
"is_min_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_max_on.localize (hf : is_max_on f s a) : is_local_max_on f s a | hf.filter_mono $ inf_le_right | lemma | is_max_on.localize | topology | src/topology/local_extr.lean | [
"order.filter.extr",
"topology.continuous_on"
] | [
"inf_le_right",
"is_local_max_on",
"is_max_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_extr_on.localize (hf : is_extr_on f s a) : is_local_extr_on f s a | hf.filter_mono $ inf_le_right | lemma | is_extr_on.localize | topology | src/topology/local_extr.lean | [
"order.filter.extr",
"topology.continuous_on"
] | [
"inf_le_right",
"is_extr_on",
"is_local_extr_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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