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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tendsto_order_unbounded {f : β → α} {a : α} {x : filter β}
(hu : ∃u, a < u) (hl : ∃l, l < a) (h : ∀l u, l < a → a < u → ∀ᶠ b in x, l < f b ∧ f b < u) :
tendsto f x (𝓝 a) | by rw [nhds_order_unbounded hu hl];
from (tendsto_infi.2 $ assume l, tendsto_infi.2 $ assume hl,
tendsto_infi.2 $ assume u, tendsto_infi.2 $ assume hu, tendsto_principal.2 $ h l u hl hu) | lemma | tendsto_order_unbounded | 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_order_unbounded"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_Ixx_nhds_within {α : Type*} [preorder α] [topological_space α]
(a : α) {s t : set α} {Ixx}
[tendsto_Ixx_class Ixx (𝓝 a) (𝓝 a)] [tendsto_Ixx_class Ixx (𝓟 s) (𝓟 t)]:
tendsto_Ixx_class Ixx (𝓝[s] a) (𝓝[t] a) | filter.tendsto_Ixx_class_inf | instance | tendsto_Ixx_nhds_within | 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.tendsto_Ixx_class_inf",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_Icc_class_nhds_pi {ι : Type*} {α : ι → Type*}
[Π i, preorder (α i)] [Π i, topological_space (α i)] [∀ i, order_topology (α i)]
(f : Π i, α i) :
tendsto_Ixx_class Icc (𝓝 f) (𝓝 f) | begin
constructor,
conv in ((𝓝 f).small_sets) { rw [nhds_pi, filter.pi] },
simp only [small_sets_infi, small_sets_comap, tendsto_infi, tendsto_lift', (∘), mem_powerset_iff],
intros i s hs,
have : tendsto (λ g : Π i, α i, g i) (𝓝 f) (𝓝 (f i)) := ((continuous_apply i).tendsto f),
refine (tendsto_lift'.1 ((... | instance | tendsto_Icc_class_nhds_pi | 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_apply",
"filter.pi",
"nhds_pi",
"order_topology",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
induced_order_topology' {α : Type u} {β : Type v}
[preorder α] [ta : topological_space β] [preorder β] [order_topology β]
(f : α → β) (hf : ∀ {x y}, f x < f y ↔ x < y)
(H₁ : ∀ {a x}, x < f a → ∃ b < a, x ≤ f b)
(H₂ : ∀ {a x}, f a < x → ∃ b > a, f b ≤ x) :
@order_topology _ (induced f ta) _ | begin
letI := induced f ta,
refine ⟨eq_of_nhds_eq_nhds (λ a, _)⟩,
rw [nhds_induced, nhds_generate_from, nhds_eq_order (f a)],
apply le_antisymm,
{ refine le_infi (λ s, le_infi $ λ hs, le_principal_iff.2 _),
rcases hs with ⟨ab, b, rfl|rfl⟩,
{ exact mem_comap.2 ⟨{x | f b < x},
mem_inf_of_left $ ... | theorem | induced_order_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"
] | [
"le_inf",
"le_infi",
"nhds_eq_order",
"nhds_induced",
"order_topology",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
induced_order_topology {α : Type u} {β : Type v}
[preorder α] [ta : topological_space β] [preorder β] [order_topology β]
(f : α → β) (hf : ∀ {x y}, f x < f y ↔ x < y)
(H : ∀ {x y}, x < y → ∃ a, x < f a ∧ f a < y) :
@order_topology _ (induced f ta) _ | induced_order_topology' f @hf
(λ a x xa, let ⟨b, xb, ba⟩ := H xa in ⟨b, hf.1 ba, le_of_lt xb⟩)
(λ a x ax, let ⟨b, ab, bx⟩ := H ax in ⟨b, hf.1 ab, le_of_lt bx⟩) | theorem | induced_order_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"
] | [
"induced_order_topology'",
"order_topology",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
order_topology_of_ord_connected {α : Type u}
[ta : topological_space α] [linear_order α] [order_topology α]
{t : set α} [ht : ord_connected t] :
order_topology t | begin
letI := induced (coe : t → α) ta,
refine ⟨eq_of_nhds_eq_nhds (λ a, _)⟩,
rw [nhds_induced, nhds_generate_from, nhds_eq_order (a : α)],
apply le_antisymm,
{ refine le_infi (λ s, le_infi $ λ hs, le_principal_iff.2 _),
rcases hs with ⟨ab, b, rfl|rfl⟩,
{ refine ⟨Ioi b, _, λ _, id⟩,
refine mem_i... | instance | order_topology_of_ord_connected | 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"
] | [
"le_inf",
"le_infi",
"nhds_eq_order",
"nhds_induced",
"order_topology",
"set_coe.exists",
"topological_space"
] | On an `ord_connected` subset of a linear order, the order topology for the restriction of the
order is the same as the restriction to the subset of the order topology. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
nhds_within_Ici_eq'' [topological_space α] [preorder α] [order_topology α] (a : α) :
𝓝[≥] a = (⨅ u (hu : a < u), 𝓟 (Iio u)) ⊓ 𝓟 (Ici a) | begin
rw [nhds_within, nhds_eq_order],
refine le_antisymm (inf_le_inf_right _ inf_le_right) (le_inf (le_inf _ inf_le_left) inf_le_right),
exact inf_le_right.trans (le_infi₂ $ λ l hl, principal_mono.2 $ Ici_subset_Ioi.2 hl)
end | lemma | nhds_within_Ici_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"
] | [
"inf_le_inf_right",
"inf_le_left",
"inf_le_right",
"le_inf",
"le_infi₂",
"nhds_eq_order",
"nhds_within",
"order_topology",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nhds_within_Iic_eq'' [topological_space α] [preorder α] [order_topology α] (a : α) :
𝓝[≤] a = (⨅ l < a, 𝓟 (Ioi l)) ⊓ 𝓟 (Iic a) | nhds_within_Ici_eq'' (to_dual a) | lemma | nhds_within_Iic_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"
] | [
"nhds_within_Ici_eq''",
"order_topology",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nhds_within_Ici_eq' [topological_space α] [preorder α] [order_topology α] {a : α}
(ha : ∃ u, a < u) :
𝓝[≥] a = ⨅ u (hu : a < u), 𝓟 (Ico a u) | by simp only [nhds_within_Ici_eq'', binfi_inf ha, inf_principal, Iio_inter_Ici] | lemma | nhds_within_Ici_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"
] | [
"binfi_inf",
"nhds_within_Ici_eq''",
"order_topology",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nhds_within_Iic_eq' [topological_space α] [preorder α] [order_topology α] {a : α}
(ha : ∃ l, l < a) :
𝓝[≤] a = ⨅ l < a, 𝓟 (Ioc l a) | by simp only [nhds_within_Iic_eq'', binfi_inf ha, inf_principal, Ioi_inter_Iic] | lemma | nhds_within_Iic_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"
] | [
"binfi_inf",
"nhds_within_Iic_eq''",
"order_topology",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nhds_within_Ici_basis' [topological_space α] [linear_order α] [order_topology α] {a : α}
(ha : ∃ u, a < u) : (𝓝[≥] a).has_basis (λ u, a < u) (λ u, Ico a u) | (nhds_within_Ici_eq' ha).symm ▸ has_basis_binfi_principal (λ b hb c hc,
⟨min b c, lt_min hb hc, Ico_subset_Ico_right (min_le_left _ _),
Ico_subset_Ico_right (min_le_right _ _)⟩) ha | lemma | nhds_within_Ici_basis' | 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_eq'",
"order_topology",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nhds_within_Iic_basis' [topological_space α] [linear_order α] [order_topology α] {a : α}
(ha : ∃ l, l < a) : (𝓝[≤] a).has_basis (λ l, l < a) (λ l, Ioc l a) | by { convert @nhds_within_Ici_basis' αᵒᵈ _ _ _ (to_dual a) ha,
exact funext (λ x, (@dual_Ico _ _ _ _).symm) } | lemma | nhds_within_Iic_basis' | 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_basis'",
"order_topology",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nhds_within_Ici_basis [topological_space α] [linear_order α] [order_topology α]
[no_max_order α] (a : α) : (𝓝[≥] a).has_basis (λ u, a < u) (λ u, Ico a u) | nhds_within_Ici_basis' (exists_gt a) | lemma | nhds_within_Ici_basis | 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_basis'",
"no_max_order",
"order_topology",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nhds_within_Iic_basis [topological_space α] [linear_order α] [order_topology α]
[no_min_order α] (a : α) : (𝓝[≤] a).has_basis (λ l, l < a) (λ l, Ioc l a) | nhds_within_Iic_basis' (exists_lt a) | lemma | nhds_within_Iic_basis | 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_basis'",
"no_min_order",
"order_topology",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nhds_top_order [topological_space α] [preorder α] [order_top α] [order_topology α] :
𝓝 (⊤:α) = (⨅l (h₂ : l < ⊤), 𝓟 (Ioi l)) | by simp [nhds_eq_order (⊤:α)] | lemma | nhds_top_order | 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_eq_order",
"order_top",
"order_topology",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nhds_bot_order [topological_space α] [preorder α] [order_bot α] [order_topology α] :
𝓝 (⊥:α) = (⨅l (h₂ : ⊥ < l), 𝓟 (Iio l)) | by simp [nhds_eq_order (⊥:α)] | lemma | nhds_bot_order | 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_eq_order",
"order_bot",
"order_topology",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nhds_top_basis [topological_space α] [linear_order α] [order_top α] [order_topology α]
[nontrivial α] :
(𝓝 ⊤).has_basis (λ a : α, a < ⊤) (λ a : α, Ioi a) | have ∃ x : α, x < ⊤, from (exists_ne ⊤).imp $ λ x hx, hx.lt_top,
by simpa only [Iic_top, nhds_within_univ, Ioc_top] using nhds_within_Iic_basis' this | lemma | nhds_top_basis | 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"
] | [
"exists_ne",
"nhds_within_Iic_basis'",
"nhds_within_univ",
"nontrivial",
"order_top",
"order_topology",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nhds_bot_basis [topological_space α] [linear_order α] [order_bot α] [order_topology α]
[nontrivial α] :
(𝓝 ⊥).has_basis (λ a : α, ⊥ < a) (λ a : α, Iio a) | @nhds_top_basis αᵒᵈ _ _ _ _ _ | lemma | nhds_bot_basis | 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_top_basis",
"nontrivial",
"order_bot",
"order_topology",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nhds_top_basis_Ici [topological_space α] [linear_order α] [order_top α] [order_topology α]
[nontrivial α] [densely_ordered α] :
(𝓝 ⊤).has_basis (λ a : α, a < ⊤) Ici | nhds_top_basis.to_has_basis
(λ a ha, let ⟨b, hab, hb⟩ := exists_between ha in ⟨b, hb, Ici_subset_Ioi.mpr hab⟩)
(λ a ha, ⟨a, ha, Ioi_subset_Ici_self⟩) | lemma | nhds_top_basis_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"
] | [
"densely_ordered",
"exists_between",
"nontrivial",
"order_top",
"order_topology",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nhds_bot_basis_Iic [topological_space α] [linear_order α] [order_bot α] [order_topology α]
[nontrivial α] [densely_ordered α] :
(𝓝 ⊥).has_basis (λ a : α, ⊥ < a) Iic | @nhds_top_basis_Ici αᵒᵈ _ _ _ _ _ _ | lemma | nhds_bot_basis_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"
] | [
"densely_ordered",
"nhds_top_basis_Ici",
"nontrivial",
"order_bot",
"order_topology",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_nhds_top_mono [topological_space β] [preorder β] [order_top β] [order_topology β]
{l : filter α} {f g : α → β} (hf : tendsto f l (𝓝 ⊤)) (hg : f ≤ᶠ[l] g) :
tendsto g l (𝓝 ⊤) | begin
simp only [nhds_top_order, tendsto_infi, tendsto_principal] at hf ⊢,
intros x hx,
filter_upwards [hf x hx, hg] with _ using lt_of_lt_of_le,
end | lemma | tendsto_nhds_top_mono | 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_top_order",
"order_top",
"order_topology",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_nhds_bot_mono [topological_space β] [preorder β] [order_bot β] [order_topology β]
{l : filter α} {f g : α → β} (hf : tendsto f l (𝓝 ⊥)) (hg : g ≤ᶠ[l] f) :
tendsto g l (𝓝 ⊥) | @tendsto_nhds_top_mono α βᵒᵈ _ _ _ _ _ _ _ hf hg | lemma | tendsto_nhds_bot_mono | 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",
"order_bot",
"order_topology",
"tendsto_nhds_top_mono",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_nhds_top_mono' [topological_space β] [preorder β] [order_top β]
[order_topology β] {l : filter α} {f g : α → β} (hf : tendsto f l (𝓝 ⊤)) (hg : f ≤ g) :
tendsto g l (𝓝 ⊤) | tendsto_nhds_top_mono hf (eventually_of_forall hg) | lemma | tendsto_nhds_top_mono' | 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",
"order_top",
"order_topology",
"tendsto_nhds_top_mono",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_nhds_bot_mono' [topological_space β] [preorder β] [order_bot β]
[order_topology β] {l : filter α} {f g : α → β} (hf : tendsto f l (𝓝 ⊥)) (hg : g ≤ f) :
tendsto g l (𝓝 ⊥) | tendsto_nhds_bot_mono hf (eventually_of_forall hg) | lemma | tendsto_nhds_bot_mono' | 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",
"order_bot",
"order_topology",
"tendsto_nhds_bot_mono",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eventually_le_nhds (hab : a < b) : ∀ᶠ x in 𝓝 a, x ≤ b | eventually_iff.mpr (mem_nhds_iff.mpr ⟨Iio b, Iio_subset_Iic_self, is_open_Iio, hab⟩) | lemma | eventually_le_nhds | 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_open_Iio"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eventually_lt_nhds (hab : a < b) : ∀ᶠ x in 𝓝 a, x < b | eventually_iff.mpr (mem_nhds_iff.mpr ⟨Iio b, rfl.subset, is_open_Iio, hab⟩) | lemma | eventually_lt_nhds | 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_open_Iio"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eventually_ge_nhds (hab : b < a) : ∀ᶠ x in 𝓝 a, b ≤ x | eventually_iff.mpr (mem_nhds_iff.mpr ⟨Ioi b, Ioi_subset_Ici_self, is_open_Ioi, hab⟩) | lemma | eventually_ge_nhds | 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_open_Ioi"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eventually_gt_nhds (hab : b < a) : ∀ᶠ x in 𝓝 a, b < x | eventually_iff.mpr (mem_nhds_iff.mpr ⟨Ioi b, rfl.subset, is_open_Ioi, hab⟩) | lemma | eventually_gt_nhds | 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_open_Ioi"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
order_separated {a₁ a₂ : α} (h : a₁ < a₂) :
∃u v : set α, is_open u ∧ is_open v ∧ a₁ ∈ u ∧ a₂ ∈ v ∧ (∀b₁∈u, ∀b₂∈v, b₁ < b₂) | match dense_or_discrete a₁ a₂ with
| or.inl ⟨a, ha₁, ha₂⟩ := ⟨{a' | a' < a}, {a' | a < a'}, is_open_gt' a, is_open_lt' a, ha₁, ha₂,
assume b₁ h₁ b₂ h₂, lt_trans h₁ h₂⟩
| or.inr ⟨h₁, h₂⟩ := ⟨{a | a < a₂}, {a | a₁ < a}, is_open_gt' a₂, is_open_lt' a₁, h, h,
assume b₁ hb₁ b₂ hb₂,
calc b₁ ≤ a₁ : h₂ _ hb₁
... | lemma | order_separated | 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_or_discrete",
"is_open",
"is_open_gt'",
"is_open_lt'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
order_topology.to_order_closed_topology : order_closed_topology α | { is_closed_le' :=
is_open_compl_iff.1 $ is_open_prod_iff.mpr $ assume a₁ a₂ (h : ¬ a₁ ≤ a₂),
have h : a₂ < a₁, from lt_of_not_ge h,
let ⟨u, v, hu, hv, ha₁, ha₂, h⟩ := order_separated h in
⟨v, u, hv, hu, ha₂, ha₁, assume ⟨b₁, b₂⟩ ⟨h₁, h₂⟩, not_le_of_gt $ h b₂ h₂ b₁ h₁⟩ } | instance | order_topology.to_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_le'",
"order_closed_topology",
"order_separated"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exists_Ioc_subset_of_mem_nhds {a : α} {s : set α} (hs : s ∈ 𝓝 a) (h : ∃ l, l < a) :
∃ l < a, Ioc l a ⊆ s | (nhds_within_Iic_basis' h).mem_iff.mp (nhds_within_le_nhds hs) | lemma | exists_Ioc_subset_of_mem_nhds | 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_basis'",
"nhds_within_le_nhds"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exists_Ioc_subset_of_mem_nhds' {a : α} {s : set α} (hs : s ∈ 𝓝 a) {l : α} (hl : l < a) :
∃ l' ∈ Ico l a, Ioc l' a ⊆ s | let ⟨l', hl'a, hl's⟩ := exists_Ioc_subset_of_mem_nhds hs ⟨l, hl⟩
in ⟨max l l', ⟨le_max_left _ _, max_lt hl hl'a⟩,
(Ioc_subset_Ioc_left $ le_max_right _ _).trans hl's⟩ | lemma | exists_Ioc_subset_of_mem_nhds' | 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"
] | [
"exists_Ioc_subset_of_mem_nhds"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exists_Ico_subset_of_mem_nhds' {a : α} {s : set α} (hs : s ∈ 𝓝 a) {u : α} (hu : a < u) :
∃ u' ∈ Ioc a u, Ico a u' ⊆ s | by simpa only [order_dual.exists, exists_prop, dual_Ico, dual_Ioc]
using exists_Ioc_subset_of_mem_nhds' (show of_dual ⁻¹' s ∈ 𝓝 (to_dual a), from hs) hu.dual | lemma | exists_Ico_subset_of_mem_nhds' | 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"
] | [
"exists_Ioc_subset_of_mem_nhds'",
"exists_prop"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exists_Ico_subset_of_mem_nhds {a : α} {s : set α} (hs : s ∈ 𝓝 a) (h : ∃ u, a < u) :
∃ u (_ : a < u), Ico a u ⊆ s | let ⟨l', hl'⟩ := h, ⟨l, hl⟩ := exists_Ico_subset_of_mem_nhds' hs hl' in ⟨l, hl.fst.1, hl.snd⟩ | lemma | exists_Ico_subset_of_mem_nhds | 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"
] | [
"exists_Ico_subset_of_mem_nhds'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exists_Icc_mem_subset_of_mem_nhds_within_Ici {a : α} {s : set α} (hs : s ∈ 𝓝[≥] a) :
∃ b (_ : a ≤ b), Icc a b ∈ 𝓝[≥] a ∧ Icc a b ⊆ s | begin
rcases (em (is_max a)).imp_right not_is_max_iff.mp with ha|ha,
{ use a, simpa [ha.Ici_eq] using hs },
{ rcases (nhds_within_Ici_basis' ha).mem_iff.mp hs with ⟨b, hab, hbs⟩,
rcases eq_empty_or_nonempty (Ioo a b) with H|⟨c, hac, hcb⟩,
{ have : Ico a b = Icc a a,
{ rw [← Icc_union_Ioo_eq_Ico le_r... | lemma | exists_Icc_mem_subset_of_mem_nhds_within_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"
] | [
"Icc_mem_nhds_within_Ici",
"em",
"is_max",
"le_rfl",
"nhds_within_Ici_basis'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exists_Icc_mem_subset_of_mem_nhds_within_Iic {a : α} {s : set α} (hs : s ∈ 𝓝[≤] a) :
∃ b ≤ a, Icc b a ∈ 𝓝[≤] a ∧ Icc b a ⊆ s | by simpa only [dual_Icc, to_dual.surjective.exists]
using @exists_Icc_mem_subset_of_mem_nhds_within_Ici αᵒᵈ _ _ _ (to_dual a) _ hs | lemma | exists_Icc_mem_subset_of_mem_nhds_within_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"
] | [
"exists_Icc_mem_subset_of_mem_nhds_within_Ici"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exists_Icc_mem_subset_of_mem_nhds {a : α} {s : set α} (hs : s ∈ 𝓝 a) :
∃ b c, a ∈ Icc b c ∧ Icc b c ∈ 𝓝 a ∧ Icc b c ⊆ s | begin
rcases exists_Icc_mem_subset_of_mem_nhds_within_Iic (nhds_within_le_nhds hs)
with ⟨b, hba, hb_nhds, hbs⟩,
rcases exists_Icc_mem_subset_of_mem_nhds_within_Ici (nhds_within_le_nhds hs)
with ⟨c, hac, hc_nhds, hcs⟩,
refine ⟨b, c, ⟨hba, hac⟩, _⟩,
rw [← Icc_union_Icc_eq_Icc hba hac, ← nhds_left_sup_nhds... | lemma | exists_Icc_mem_subset_of_mem_nhds | 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"
] | [
"exists_Icc_mem_subset_of_mem_nhds_within_Ici",
"exists_Icc_mem_subset_of_mem_nhds_within_Iic",
"nhds_left_sup_nhds_right",
"nhds_within_le_nhds"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_open.exists_Ioo_subset [nontrivial α] {s : set α} (hs : is_open s) (h : s.nonempty) :
∃ a b, a < b ∧ Ioo a b ⊆ s | begin
obtain ⟨x, hx⟩ : ∃ x, x ∈ s := h,
obtain ⟨y, hy⟩ : ∃ y, y ≠ x := exists_ne x,
rcases lt_trichotomy x y with H|rfl|H,
{ obtain ⟨u, xu, hu⟩ : ∃ (u : α) (hu : x < u), Ico x u ⊆ s :=
exists_Ico_subset_of_mem_nhds (hs.mem_nhds hx) ⟨y, H⟩,
exact ⟨x, u, xu, Ioo_subset_Ico_self.trans hu⟩ },
{ exact (h... | lemma | is_open.exists_Ioo_subset | 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"
] | [
"exists_Ico_subset_of_mem_nhds",
"exists_Ioc_subset_of_mem_nhds",
"exists_ne",
"is_open",
"nontrivial"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dense_of_exists_between [nontrivial α] {s : set α}
(h : ∀ ⦃a b⦄, a < b → ∃ c ∈ s, a < c ∧ c < b) : dense s | begin
apply dense_iff_inter_open.2 (λ U U_open U_nonempty, _),
obtain ⟨a, b, hab, H⟩ : ∃ (a b : α), a < b ∧ Ioo a b ⊆ U := U_open.exists_Ioo_subset U_nonempty,
obtain ⟨x, xs, hx⟩ : ∃ (x : α) (H : x ∈ s), a < x ∧ x < b := h hab,
exact ⟨x, ⟨H hx, xs⟩⟩
end | lemma | dense_of_exists_between | 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",
"nontrivial"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dense_iff_exists_between [densely_ordered α] [nontrivial α] {s : set α} :
dense s ↔ ∀ a b, a < b → ∃ c ∈ s, a < c ∧ c < b | ⟨λ h a b hab, h.exists_between hab, dense_of_exists_between⟩ | lemma | dense_iff_exists_between | 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",
"densely_ordered",
"nontrivial"
] | A set in a nontrivial densely linear ordered type is dense in the sense of topology if and only
if for any `a < b` there exists `c ∈ s`, `a < c < b`. Each implication requires less typeclass
assumptions. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mem_nhds_iff_exists_Ioo_subset' {a : α} {s : set α} (hl : ∃ l, l < a) (hu : ∃ u, a < u) :
s ∈ 𝓝 a ↔ ∃l u, a ∈ Ioo l u ∧ Ioo l u ⊆ s | begin
split,
{ assume h,
rcases exists_Ico_subset_of_mem_nhds h hu with ⟨u, au, hu⟩,
rcases exists_Ioc_subset_of_mem_nhds h hl with ⟨l, la, hl⟩,
exact ⟨l, u, ⟨la, au⟩, Ioc_union_Ico_eq_Ioo la au ▸ union_subset hl hu⟩ },
{ rintros ⟨l, u, ha, h⟩,
apply mem_of_superset (Ioo_mem_nhds ha.1 ha.2) h }
en... | lemma | mem_nhds_iff_exists_Ioo_subset' | 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"
] | [
"Ioo_mem_nhds",
"exists_Ico_subset_of_mem_nhds",
"exists_Ioc_subset_of_mem_nhds"
] | A set is a neighborhood of `a` if and only if it contains an interval `(l, u)` containing `a`,
provided `a` is neither a bottom element nor a top element. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mem_nhds_iff_exists_Ioo_subset [no_max_order α] [no_min_order α] {a : α} {s : set α} :
s ∈ 𝓝 a ↔ ∃l u, a ∈ Ioo l u ∧ Ioo l u ⊆ s | mem_nhds_iff_exists_Ioo_subset' (exists_lt a) (exists_gt a) | lemma | mem_nhds_iff_exists_Ioo_subset | 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"
] | [
"mem_nhds_iff_exists_Ioo_subset'",
"no_max_order",
"no_min_order"
] | A set is a neighborhood of `a` if and only if it contains an interval `(l, u)` containing `a`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
nhds_basis_Ioo' {a : α} (hl : ∃ l, l < a) (hu : ∃ u, a < u) :
(𝓝 a).has_basis (λ b : α × α, b.1 < a ∧ a < b.2) (λ b, Ioo b.1 b.2) | ⟨λ s, (mem_nhds_iff_exists_Ioo_subset' hl hu).trans $ by simp⟩ | lemma | nhds_basis_Ioo' | 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"
] | [
"mem_nhds_iff_exists_Ioo_subset'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nhds_basis_Ioo [no_max_order α] [no_min_order α] (a : α) :
(𝓝 a).has_basis (λ b : α × α, b.1 < a ∧ a < b.2) (λ b, Ioo b.1 b.2) | nhds_basis_Ioo' (exists_lt a) (exists_gt a) | lemma | nhds_basis_Ioo | 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_basis_Ioo'",
"no_max_order",
"no_min_order"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
filter.eventually.exists_Ioo_subset [no_max_order α] [no_min_order α] {a : α} {p : α → Prop}
(hp : ∀ᶠ x in 𝓝 a, p x) :
∃ l u, a ∈ Ioo l u ∧ Ioo l u ⊆ {x | p x} | mem_nhds_iff_exists_Ioo_subset.1 hp | lemma | filter.eventually.exists_Ioo_subset | 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"
] | [
"no_max_order",
"no_min_order"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
countable_of_isolated_right [second_countable_topology α] :
set.countable {x : α | ∃ y, x < y ∧ Ioo x y = ∅} | begin
nontriviality α,
let s := {x : α | ∃ y, x < y ∧ Ioo x y = ∅},
have : ∀ x ∈ s, ∃ y, x < y ∧ Ioo x y = ∅ := λ x, id,
choose! y hy h'y using this,
have Hy : ∀ x z, x ∈ s → z < y x → z ≤ x,
{ assume x z xs hz,
have A : Ioo x (y x) = ∅ := h'y _ xs,
contrapose! A,
exact nonempty.ne_empty ⟨z, A, ... | lemma | countable_of_isolated_right | 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"
] | [
"countable",
"exists_Ioc_subset_of_mem_nhds",
"is_bot",
"is_open",
"is_open_Ioo",
"not_forall",
"set.countable"
] | The set of points which are isolated on the right is countable when the space is
second-countable. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
countable_of_isolated_left [second_countable_topology α] :
set.countable {x : α | ∃ y, y < x ∧ Ioo y x = ∅} | begin
convert @countable_of_isolated_right αᵒᵈ _ _ _ _,
have : ∀ (x y : α), Ioo x y = {z | z < y ∧ x < z},
{ simp_rw [and_comm, Ioo], simp only [eq_self_iff_true, forall_2_true_iff] },
simp_rw [this],
refl
end | lemma | countable_of_isolated_left | 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"
] | [
"countable_of_isolated_right",
"forall_2_true_iff",
"set.countable"
] | The set of points which are isolated on the left is countable when the space is
second-countable. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
set.pairwise_disjoint.countable_of_Ioo [second_countable_topology α]
{y : α → α} {s : set α} (h : pairwise_disjoint s (λ x, Ioo x (y x))) (h' : ∀ x ∈ s, x < y x) :
s.countable | begin
let t := {x | x ∈ s ∧ (Ioo x (y x)).nonempty},
have t_count : t.countable,
{ have : t ⊆ s := λ x hx, hx.1,
exact (h.subset this).countable_of_is_open (λ x hx, is_open_Ioo) (λ x hx, hx.2) },
have : s ⊆ t ∪ {x : α | ∃ x', x < x' ∧ Ioo x x' = ∅},
{ assume x hx,
by_cases h'x : (Ioo x (y x)).nonempty... | lemma | set.pairwise_disjoint.countable_of_Ioo | 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"
] | [
"countable_of_isolated_right",
"is_open_Ioo"
] | Consider a disjoint family of intervals `(x, y)` with `x < y` in a second-countable space.
Then the family is countable.
This is not a straightforward consequence of second-countability as some of these intervals might be
empty (but in fact this can happen only for countably many of them). | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pi_Iic_mem_nhds (ha : ∀ i, x i < a i) : Iic a ∈ 𝓝 x | pi_univ_Iic a ▸ set_pi_mem_nhds (set.to_finite _) (λ i _, Iic_mem_nhds (ha _)) | lemma | pi_Iic_mem_nhds | 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"
] | [
"Iic_mem_nhds",
"set.to_finite",
"set_pi_mem_nhds"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pi_Iic_mem_nhds' (ha : ∀ i, x' i < a' i) : Iic a' ∈ 𝓝 x' | pi_Iic_mem_nhds ha | lemma | pi_Iic_mem_nhds' | 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"
] | [
"pi_Iic_mem_nhds"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pi_Ici_mem_nhds (ha : ∀ i, a i < x i) : Ici a ∈ 𝓝 x | pi_univ_Ici a ▸ set_pi_mem_nhds (set.to_finite _) (λ i _, Ici_mem_nhds (ha _)) | lemma | pi_Ici_mem_nhds | 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"
] | [
"Ici_mem_nhds",
"set.to_finite",
"set_pi_mem_nhds"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pi_Ici_mem_nhds' (ha : ∀ i, a' i < x' i) : Ici a' ∈ 𝓝 x' | pi_Ici_mem_nhds ha | lemma | pi_Ici_mem_nhds' | 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"
] | [
"pi_Ici_mem_nhds"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pi_Icc_mem_nhds (ha : ∀ i, a i < x i) (hb : ∀ i, x i < b i) : Icc a b ∈ 𝓝 x | pi_univ_Icc a b ▸ set_pi_mem_nhds finite_univ (λ i _, Icc_mem_nhds (ha _) (hb _)) | lemma | pi_Icc_mem_nhds | 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"
] | [
"Icc_mem_nhds",
"set_pi_mem_nhds"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pi_Icc_mem_nhds' (ha : ∀ i, a' i < x' i) (hb : ∀ i, x' i < b' i) : Icc a' b' ∈ 𝓝 x' | pi_Icc_mem_nhds ha hb | lemma | pi_Icc_mem_nhds' | 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"
] | [
"pi_Icc_mem_nhds"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pi_Iio_mem_nhds (ha : ∀ i, x i < a i) : Iio a ∈ 𝓝 x | begin
refine mem_of_superset (set_pi_mem_nhds (set.to_finite _) (λ i _, _))
(pi_univ_Iio_subset a),
exact Iio_mem_nhds (ha i)
end | lemma | pi_Iio_mem_nhds | 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"
] | [
"Iio_mem_nhds",
"set.to_finite",
"set_pi_mem_nhds"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pi_Iio_mem_nhds' (ha : ∀ i, x' i < a' i) : Iio a' ∈ 𝓝 x' | pi_Iio_mem_nhds ha | lemma | pi_Iio_mem_nhds' | 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"
] | [
"pi_Iio_mem_nhds"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pi_Ioi_mem_nhds (ha : ∀ i, a i < x i) : Ioi a ∈ 𝓝 x | @pi_Iio_mem_nhds ι (λ i, (π i)ᵒᵈ) _ _ _ _ _ _ _ ha | lemma | pi_Ioi_mem_nhds | 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"
] | [
"pi_Iio_mem_nhds"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pi_Ioi_mem_nhds' (ha : ∀ i, a' i < x' i) : Ioi a' ∈ 𝓝 x' | pi_Ioi_mem_nhds ha | lemma | pi_Ioi_mem_nhds' | 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"
] | [
"pi_Ioi_mem_nhds"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pi_Ioc_mem_nhds (ha : ∀ i, a i < x i) (hb : ∀ i, x i < b i) : Ioc a b ∈ 𝓝 x | begin
refine mem_of_superset (set_pi_mem_nhds (set.to_finite _) (λ i _, _))
(pi_univ_Ioc_subset a b),
exact Ioc_mem_nhds (ha i) (hb i)
end | lemma | pi_Ioc_mem_nhds | 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"
] | [
"Ioc_mem_nhds",
"set.to_finite",
"set_pi_mem_nhds"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pi_Ioc_mem_nhds' (ha : ∀ i, a' i < x' i) (hb : ∀ i, x' i < b' i) : Ioc a' b' ∈ 𝓝 x' | pi_Ioc_mem_nhds ha hb | lemma | pi_Ioc_mem_nhds' | 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"
] | [
"pi_Ioc_mem_nhds"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pi_Ico_mem_nhds (ha : ∀ i, a i < x i) (hb : ∀ i, x i < b i) : Ico a b ∈ 𝓝 x | begin
refine mem_of_superset (set_pi_mem_nhds (set.to_finite _) (λ i _, _))
(pi_univ_Ico_subset a b),
exact Ico_mem_nhds (ha i) (hb i)
end | lemma | pi_Ico_mem_nhds | 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"
] | [
"Ico_mem_nhds",
"set.to_finite",
"set_pi_mem_nhds"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pi_Ico_mem_nhds' (ha : ∀ i, a' i < x' i) (hb : ∀ i, x' i < b' i) : Ico a' b' ∈ 𝓝 x' | pi_Ico_mem_nhds ha hb | lemma | pi_Ico_mem_nhds' | 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"
] | [
"pi_Ico_mem_nhds"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pi_Ioo_mem_nhds (ha : ∀ i, a i < x i) (hb : ∀ i, x i < b i) : Ioo a b ∈ 𝓝 x | begin
refine mem_of_superset (set_pi_mem_nhds (set.to_finite _) (λ i _, _))
(pi_univ_Ioo_subset a b),
exact Ioo_mem_nhds (ha i) (hb i)
end | lemma | pi_Ioo_mem_nhds | 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"
] | [
"Ioo_mem_nhds",
"set.to_finite",
"set_pi_mem_nhds"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pi_Ioo_mem_nhds' (ha : ∀ i, a' i < x' i) (hb : ∀ i, x' i < b' i) : Ioo a' b' ∈ 𝓝 x' | pi_Ioo_mem_nhds ha hb | lemma | pi_Ioo_mem_nhds' | 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"
] | [
"pi_Ioo_mem_nhds"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
disjoint_nhds_at_top [no_max_order α] (x : α) :
disjoint (𝓝 x) at_top | begin
rcases exists_gt x with ⟨y, hy : x < y⟩,
refine disjoint_of_disjoint_of_mem _ (Iio_mem_nhds hy) (mem_at_top y),
exact disjoint_left.mpr (λ z, not_le.2)
end | lemma | disjoint_nhds_at_top | 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"
] | [
"Iio_mem_nhds",
"disjoint",
"no_max_order"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inf_nhds_at_top [no_max_order α] (x : α) :
𝓝 x ⊓ at_top = ⊥ | disjoint_iff.1 (disjoint_nhds_at_top x) | lemma | inf_nhds_at_top | 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"
] | [
"disjoint_nhds_at_top",
"no_max_order"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
disjoint_nhds_at_bot [no_min_order α] (x : α) : disjoint (𝓝 x) at_bot | @disjoint_nhds_at_top αᵒᵈ _ _ _ _ x | lemma | disjoint_nhds_at_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"
] | [
"disjoint",
"disjoint_nhds_at_top",
"no_min_order"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inf_nhds_at_bot [no_min_order α] (x : α) : 𝓝 x ⊓ at_bot = ⊥ | @inf_nhds_at_top αᵒᵈ _ _ _ _ x | lemma | inf_nhds_at_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"
] | [
"inf_nhds_at_top",
"no_min_order"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
not_tendsto_nhds_of_tendsto_at_top [no_max_order α]
{F : filter β} [ne_bot F] {f : β → α} (hf : tendsto f F at_top) (x : α) :
¬ tendsto f F (𝓝 x) | hf.not_tendsto (disjoint_nhds_at_top x).symm | lemma | not_tendsto_nhds_of_tendsto_at_top | 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"
] | [
"disjoint_nhds_at_top",
"filter",
"no_max_order"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
not_tendsto_at_top_of_tendsto_nhds [no_max_order α]
{F : filter β} [ne_bot F] {f : β → α} {x : α} (hf : tendsto f F (𝓝 x)) :
¬ tendsto f F at_top | hf.not_tendsto (disjoint_nhds_at_top x) | lemma | not_tendsto_at_top_of_tendsto_nhds | 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"
] | [
"disjoint_nhds_at_top",
"filter",
"no_max_order"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
not_tendsto_nhds_of_tendsto_at_bot [no_min_order α]
{F : filter β} [ne_bot F] {f : β → α} (hf : tendsto f F at_bot) (x : α) :
¬ tendsto f F (𝓝 x) | hf.not_tendsto (disjoint_nhds_at_bot x).symm | lemma | not_tendsto_nhds_of_tendsto_at_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"
] | [
"disjoint_nhds_at_bot",
"filter",
"no_min_order"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
not_tendsto_at_bot_of_tendsto_nhds [no_min_order α]
{F : filter β} [ne_bot F] {f : β → α} {x : α} (hf : tendsto f F (𝓝 x)) :
¬ tendsto f F at_bot | hf.not_tendsto (disjoint_nhds_at_bot x) | lemma | not_tendsto_at_bot_of_tendsto_nhds | 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"
] | [
"disjoint_nhds_at_bot",
"filter",
"no_min_order"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tfae_mem_nhds_within_Ioi {a b : α} (hab : a < b) (s : set α) :
tfae [s ∈ 𝓝[>] a, -- 0 : `s` is a neighborhood of `a` within `(a, +∞)`
s ∈ 𝓝[Ioc a b] a, -- 1 : `s` is a neighborhood of `a` within `(a, b]`
s ∈ 𝓝[Ioo a b] a, -- 2 : `s` is a neighborhood of `a` within `(a, b)`
∃ u ∈ Ioc a b, Ioo a u ⊆ ... | -- 4 : `s` includes `(a, u)` for some `u > a`
begin
tfae_have : 1 ↔ 2, by rw [nhds_within_Ioc_eq_nhds_within_Ioi hab],
tfae_have : 1 ↔ 3, by rw [nhds_within_Ioo_eq_nhds_within_Ioi hab],
tfae_have : 4 → 5, from λ ⟨u, umem, hu⟩, ⟨u, umem.1, hu⟩,
tfae_have : 5 → 1,
{ rintros ⟨u, hau, hu⟩,
exact mem_of_supers... | lemma | tfae_mem_nhds_within_Ioi | 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"
] | [
"Ioo_mem_nhds_within_Ioi",
"exists_Ico_subset_of_mem_nhds'",
"nhds_within_Ioc_eq_nhds_within_Ioi",
"nhds_within_Ioo_eq_nhds_within_Ioi"
] | The following statements are equivalent:
0. `s` is a neighborhood of `a` within `(a, +∞)`
1. `s` is a neighborhood of `a` within `(a, b]`
2. `s` is a neighborhood of `a` within `(a, b)`
3. `s` includes `(a, u)` for some `u ∈ (a, b]`
4. `s` includes `(a, u)` for some `u > a` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mem_nhds_within_Ioi_iff_exists_mem_Ioc_Ioo_subset {a u' : α} {s : set α} (hu' : a < u') :
s ∈ 𝓝[>] a ↔ ∃u ∈ Ioc a u', Ioo a u ⊆ s | (tfae_mem_nhds_within_Ioi hu' s).out 0 3 | lemma | mem_nhds_within_Ioi_iff_exists_mem_Ioc_Ioo_subset | 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"
] | [
"tfae_mem_nhds_within_Ioi"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_nhds_within_Ioi_iff_exists_Ioo_subset' {a u' : α} {s : set α} (hu' : a < u') :
s ∈ 𝓝[>] a ↔ ∃u ∈ Ioi a, Ioo a u ⊆ s | (tfae_mem_nhds_within_Ioi hu' s).out 0 4 | lemma | mem_nhds_within_Ioi_iff_exists_Ioo_subset' | 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"
] | [
"tfae_mem_nhds_within_Ioi"
] | A set is a neighborhood of `a` within `(a, +∞)` if and only if it contains an interval `(a, u)`
with `a < u < u'`, provided `a` is not a top element. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mem_nhds_within_Ioi_iff_exists_Ioo_subset [no_max_order α] {a : α} {s : set α} :
s ∈ 𝓝[>] a ↔ ∃u ∈ Ioi a, Ioo a u ⊆ s | let ⟨u', hu'⟩ := exists_gt a in mem_nhds_within_Ioi_iff_exists_Ioo_subset' hu' | lemma | mem_nhds_within_Ioi_iff_exists_Ioo_subset | 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"
] | [
"mem_nhds_within_Ioi_iff_exists_Ioo_subset'",
"no_max_order"
] | A set is a neighborhood of `a` within `(a, +∞)` if and only if it contains an interval `(a, u)`
with `a < u`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mem_nhds_within_Ioi_iff_exists_Ioc_subset [no_max_order α] [densely_ordered α]
{a : α} {s : set α} : s ∈ 𝓝[>] a ↔ ∃u ∈ Ioi a, Ioc a u ⊆ s | begin
rw mem_nhds_within_Ioi_iff_exists_Ioo_subset,
split,
{ rintros ⟨u, au, as⟩,
rcases exists_between au with ⟨v, hv⟩,
exact ⟨v, hv.1, λx hx, as ⟨hx.1, lt_of_le_of_lt hx.2 hv.2⟩⟩ },
{ rintros ⟨u, au, as⟩,
exact ⟨u, au, subset.trans Ioo_subset_Ioc_self as⟩ }
end | lemma | mem_nhds_within_Ioi_iff_exists_Ioc_subset | 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"
] | [
"densely_ordered",
"exists_between",
"mem_nhds_within_Ioi_iff_exists_Ioo_subset",
"no_max_order"
] | A set is a neighborhood of `a` within `(a, +∞)` if and only if it contains an interval `(a, u]`
with `a < u`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
tfae_mem_nhds_within_Iio {a b : α} (h : a < b) (s : set α) :
tfae [s ∈ 𝓝[<] b, -- 0 : `s` is a neighborhood of `b` within `(-∞, b)`
s ∈ 𝓝[Ico a b] b, -- 1 : `s` is a neighborhood of `b` within `[a, b)`
s ∈ 𝓝[Ioo a b] b, -- 2 : `s` is a neighborhood of `b` within `(a, b)`
∃ l ∈ Ico a b, Ioo l b ⊆ s,... | -- 4 : `s` includes `(l, b)` for some `l < b`
by simpa only [exists_prop, order_dual.exists, dual_Ioi, dual_Ioc, dual_Ioo]
using tfae_mem_nhds_within_Ioi h.dual (of_dual ⁻¹' s) | lemma | tfae_mem_nhds_within_Iio | 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"
] | [
"exists_prop",
"tfae_mem_nhds_within_Ioi"
] | The following statements are equivalent:
0. `s` is a neighborhood of `b` within `(-∞, b)`
1. `s` is a neighborhood of `b` within `[a, b)`
2. `s` is a neighborhood of `b` within `(a, b)`
3. `s` includes `(l, b)` for some `l ∈ [a, b)`
4. `s` includes `(l, b)` for some `l < b` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mem_nhds_within_Iio_iff_exists_mem_Ico_Ioo_subset {a l' : α} {s : set α} (hl' : l' < a) :
s ∈ 𝓝[<] a ↔ ∃l ∈ Ico l' a, Ioo l a ⊆ s | (tfae_mem_nhds_within_Iio hl' s).out 0 3 | lemma | mem_nhds_within_Iio_iff_exists_mem_Ico_Ioo_subset | 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"
] | [
"tfae_mem_nhds_within_Iio"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_nhds_within_Iio_iff_exists_Ioo_subset' {a l' : α} {s : set α} (hl' : l' < a) :
s ∈ 𝓝[<] a ↔ ∃l ∈ Iio a, Ioo l a ⊆ s | (tfae_mem_nhds_within_Iio hl' s).out 0 4 | lemma | mem_nhds_within_Iio_iff_exists_Ioo_subset' | 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"
] | [
"tfae_mem_nhds_within_Iio"
] | A set is a neighborhood of `a` within `(-∞, a)` if and only if it contains an interval `(l, a)`
with `l < a`, provided `a` is not a bottom element. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mem_nhds_within_Iio_iff_exists_Ioo_subset [no_min_order α] {a : α} {s : set α} :
s ∈ 𝓝[<] a ↔ ∃l ∈ Iio a, Ioo l a ⊆ s | let ⟨l', hl'⟩ := exists_lt a in mem_nhds_within_Iio_iff_exists_Ioo_subset' hl' | lemma | mem_nhds_within_Iio_iff_exists_Ioo_subset | 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"
] | [
"mem_nhds_within_Iio_iff_exists_Ioo_subset'",
"no_min_order"
] | A set is a neighborhood of `a` within `(-∞, a)` if and only if it contains an interval `(l, a)`
with `l < a`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mem_nhds_within_Iio_iff_exists_Ico_subset [no_min_order α] [densely_ordered α]
{a : α} {s : set α} : s ∈ 𝓝[<] a ↔ ∃l ∈ Iio a, Ico l a ⊆ s | begin
have : of_dual ⁻¹' s ∈ 𝓝[>] (to_dual a) ↔ _ :=
mem_nhds_within_Ioi_iff_exists_Ioc_subset,
simpa only [order_dual.exists, exists_prop, dual_Ioc] using this,
end | lemma | mem_nhds_within_Iio_iff_exists_Ico_subset | 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"
] | [
"densely_ordered",
"exists_prop",
"mem_nhds_within_Ioi_iff_exists_Ioc_subset",
"no_min_order"
] | A set is a neighborhood of `a` within `(-∞, a)` if and only if it contains an interval `[l, a)`
with `l < a`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
tfae_mem_nhds_within_Ici {a b : α} (hab : a < b) (s : set α) :
tfae [s ∈ 𝓝[≥] a, -- 0 : `s` is a neighborhood of `a` within `[a, +∞)`
s ∈ 𝓝[Icc a b] a, -- 1 : `s` is a neighborhood of `a` within `[a, b]`
s ∈ 𝓝[Ico a b] a, -- 2 : `s` is a neighborhood of `a` within `[a, b)`
∃ u ∈ Ioc a b, Ico a u ⊆ ... | -- 4 : `s` includes `[a, u)` for some `u > a`
begin
tfae_have : 1 ↔ 2, by rw [nhds_within_Icc_eq_nhds_within_Ici hab],
tfae_have : 1 ↔ 3, by rw [nhds_within_Ico_eq_nhds_within_Ici hab],
tfae_have : 1 ↔ 5, from (nhds_within_Ici_basis' ⟨b, hab⟩).mem_iff,
tfae_have : 4 → 5, from λ ⟨u, umem, hu⟩, ⟨u, umem.1, hu⟩,
... | lemma | tfae_mem_nhds_within_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"
] | [
"nhds_within_Icc_eq_nhds_within_Ici",
"nhds_within_Ici_basis'",
"nhds_within_Ico_eq_nhds_within_Ici"
] | The following statements are equivalent:
0. `s` is a neighborhood of `a` within `[a, +∞)`
1. `s` is a neighborhood of `a` within `[a, b]`
2. `s` is a neighborhood of `a` within `[a, b)`
3. `s` includes `[a, u)` for some `u ∈ (a, b]`
4. `s` includes `[a, u)` for some `u > a` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mem_nhds_within_Ici_iff_exists_mem_Ioc_Ico_subset {a u' : α} {s : set α} (hu' : a < u') :
s ∈ 𝓝[≥] a ↔ ∃u ∈ Ioc a u', Ico a u ⊆ s | (tfae_mem_nhds_within_Ici hu' s).out 0 3 (by norm_num) (by norm_num) | lemma | mem_nhds_within_Ici_iff_exists_mem_Ioc_Ico_subset | 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"
] | [
"tfae_mem_nhds_within_Ici"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_nhds_within_Ici_iff_exists_Ico_subset' {a u' : α} {s : set α} (hu' : a < u') :
s ∈ 𝓝[≥] a ↔ ∃u ∈ Ioi a, Ico a u ⊆ s | (tfae_mem_nhds_within_Ici hu' s).out 0 4 (by norm_num) (by norm_num) | lemma | mem_nhds_within_Ici_iff_exists_Ico_subset' | 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"
] | [
"tfae_mem_nhds_within_Ici"
] | A set is a neighborhood of `a` within `[a, +∞)` if and only if it contains an interval `[a, u)`
with `a < u < u'`, provided `a` is not a top element. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mem_nhds_within_Ici_iff_exists_Ico_subset [no_max_order α] {a : α} {s : set α} :
s ∈ 𝓝[≥] a ↔ ∃u ∈ Ioi a, Ico a u ⊆ s | let ⟨u', hu'⟩ := exists_gt a in mem_nhds_within_Ici_iff_exists_Ico_subset' hu' | lemma | mem_nhds_within_Ici_iff_exists_Ico_subset | 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"
] | [
"mem_nhds_within_Ici_iff_exists_Ico_subset'",
"no_max_order"
] | A set is a neighborhood of `a` within `[a, +∞)` if and only if it contains an interval `[a, u)`
with `a < u`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
nhds_within_Ici_basis_Ico [no_max_order α] (a : α) :
(𝓝[≥] a).has_basis (λ u, a < u) (Ico a) | ⟨λ s, mem_nhds_within_Ici_iff_exists_Ico_subset⟩ | lemma | nhds_within_Ici_basis_Ico | 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"
] | [
"no_max_order"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_nhds_within_Ici_iff_exists_Icc_subset [no_max_order α] [densely_ordered α]
{a : α} {s : set α} : s ∈ 𝓝[≥] a ↔ ∃ u, a < u ∧ Icc a u ⊆ s | begin
rw mem_nhds_within_Ici_iff_exists_Ico_subset,
split,
{ rintros ⟨u, au, as⟩,
rcases exists_between au with ⟨v, hv⟩,
exact ⟨v, hv.1, λx hx, as ⟨hx.1, lt_of_le_of_lt hx.2 hv.2⟩⟩ },
{ rintros ⟨u, au, as⟩,
exact ⟨u, au, subset.trans Ico_subset_Icc_self as⟩ }
end | lemma | mem_nhds_within_Ici_iff_exists_Icc_subset | 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"
] | [
"densely_ordered",
"exists_between",
"mem_nhds_within_Ici_iff_exists_Ico_subset",
"no_max_order"
] | A set is a neighborhood of `a` within `[a, +∞)` if and only if it contains an interval `[a, u]`
with `a < u`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
tfae_mem_nhds_within_Iic {a b : α} (h : a < b) (s : set α) :
tfae [s ∈ 𝓝[≤] b, -- 0 : `s` is a neighborhood of `b` within `(-∞, b]`
s ∈ 𝓝[Icc a b] b, -- 1 : `s` is a neighborhood of `b` within `[a, b]`
s ∈ 𝓝[Ioc a b] b, -- 2 : `s` is a neighborhood of `b` within `(a, b]`
∃ l ∈ Ico a b, Ioc l b ⊆ s,... | -- 4 : `s` includes `(l, b]` for some `l < b`
by simpa only [exists_prop, order_dual.exists, dual_Ici, dual_Ioc, dual_Icc, dual_Ico]
using tfae_mem_nhds_within_Ici h.dual (of_dual ⁻¹' s) | lemma | tfae_mem_nhds_within_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"
] | [
"exists_prop",
"tfae_mem_nhds_within_Ici"
] | The following statements are equivalent:
0. `s` is a neighborhood of `b` within `(-∞, b]`
1. `s` is a neighborhood of `b` within `[a, b]`
2. `s` is a neighborhood of `b` within `(a, b]`
3. `s` includes `(l, b]` for some `l ∈ [a, b)`
4. `s` includes `(l, b]` for some `l < b` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mem_nhds_within_Iic_iff_exists_mem_Ico_Ioc_subset {a l' : α} {s : set α} (hl' : l' < a) :
s ∈ 𝓝[≤] a ↔ ∃l ∈ Ico l' a, Ioc l a ⊆ s | (tfae_mem_nhds_within_Iic hl' s).out 0 3 (by norm_num) (by norm_num) | lemma | mem_nhds_within_Iic_iff_exists_mem_Ico_Ioc_subset | 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"
] | [
"tfae_mem_nhds_within_Iic"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_nhds_within_Iic_iff_exists_Ioc_subset' {a l' : α} {s : set α} (hl' : l' < a) :
s ∈ 𝓝[≤] a ↔ ∃l ∈ Iio a, Ioc l a ⊆ s | (tfae_mem_nhds_within_Iic hl' s).out 0 4 (by norm_num) (by norm_num) | lemma | mem_nhds_within_Iic_iff_exists_Ioc_subset' | 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"
] | [
"tfae_mem_nhds_within_Iic"
] | A set is a neighborhood of `a` within `(-∞, a]` if and only if it contains an interval `(l, a]`
with `l < a`, provided `a` is not a bottom element. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mem_nhds_within_Iic_iff_exists_Ioc_subset [no_min_order α] {a : α} {s : set α} :
s ∈ 𝓝[≤] a ↔ ∃l ∈ Iio a, Ioc l a ⊆ s | let ⟨l', hl'⟩ := exists_lt a in mem_nhds_within_Iic_iff_exists_Ioc_subset' hl' | lemma | mem_nhds_within_Iic_iff_exists_Ioc_subset | 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"
] | [
"mem_nhds_within_Iic_iff_exists_Ioc_subset'",
"no_min_order"
] | A set is a neighborhood of `a` within `(-∞, a]` if and only if it contains an interval `(l, a]`
with `l < a`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mem_nhds_within_Iic_iff_exists_Icc_subset [no_min_order α] [densely_ordered α]
{a : α} {s : set α} : s ∈ 𝓝[≤] a ↔ ∃ l, l < a ∧ Icc l a ⊆ s | begin
convert @mem_nhds_within_Ici_iff_exists_Icc_subset αᵒᵈ _ _ _ _ _ _ _,
simp_rw (show ∀ u : αᵒᵈ, @Icc αᵒᵈ _ a u = @Icc α _ u a, from λ u, dual_Icc),
refl,
end | lemma | mem_nhds_within_Iic_iff_exists_Icc_subset | 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"
] | [
"densely_ordered",
"mem_nhds_within_Ici_iff_exists_Icc_subset",
"no_min_order"
] | A set is a neighborhood of `a` within `(-∞, a]` if and only if it contains an interval `[l, a]`
with `l < a`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
nhds_eq_infi_abs_sub (a : α) : 𝓝 a = (⨅r>0, 𝓟 {b | |a - b| < r}) | begin
simp only [le_antisymm_iff, nhds_eq_order, le_inf_iff, le_infi_iff, le_principal_iff, mem_Ioi,
mem_Iio, abs_sub_lt_iff, @sub_lt_iff_lt_add _ _ _ _ _ _ a, @sub_lt_comm _ _ _ _ a, set_of_and],
refine ⟨_, _, _⟩,
{ intros ε ε0,
exact inter_mem_inf
(mem_infi_of_mem (a - ε) $ mem_infi_of_mem (sub_lt... | lemma | nhds_eq_infi_abs_sub | 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"
] | [
"abs_sub_lt_iff",
"le_inf_iff",
"le_infi_iff",
"nhds_eq_order"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
order_topology_of_nhds_abs {α : Type*} [topological_space α] [linear_ordered_add_comm_group α]
(h_nhds : ∀a:α, 𝓝 a = (⨅r>0, 𝓟 {b | |a - b| < r})) : order_topology α | begin
refine ⟨eq_of_nhds_eq_nhds $ λ a, _⟩,
rw [h_nhds],
letI := preorder.topology α, letI : order_topology α := ⟨rfl⟩,
exact (nhds_eq_infi_abs_sub a).symm
end | lemma | order_topology_of_nhds_abs | 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"
] | [
"linear_ordered_add_comm_group",
"nhds_eq_infi_abs_sub",
"order_topology",
"preorder.topology",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
linear_ordered_add_comm_group.tendsto_nhds {x : filter β} {a : α} :
tendsto f x (𝓝 a) ↔ ∀ ε > (0 : α), ∀ᶠ b in x, |f b - a| < ε | by simp [nhds_eq_infi_abs_sub, abs_sub_comm a] | lemma | linear_ordered_add_comm_group.tendsto_nhds | 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"
] | [
"abs_sub_comm",
"filter",
"nhds_eq_infi_abs_sub"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eventually_abs_sub_lt (a : α) {ε : α} (hε : 0 < ε) : ∀ᶠ x in 𝓝 a, |x - a| < ε | (nhds_eq_infi_abs_sub a).symm ▸ mem_infi_of_mem ε
(mem_infi_of_mem hε $ by simp only [abs_sub_comm, mem_principal_self]) | lemma | eventually_abs_sub_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"
] | [
"abs_sub_comm",
"nhds_eq_infi_abs_sub"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
filter.tendsto.add_at_top {C : α} (hf : tendsto f l (𝓝 C)) (hg : tendsto g l at_top) :
tendsto (λ x, f x + g x) l at_top | begin
nontriviality α,
obtain ⟨C', hC'⟩ : ∃ C', C' < C := exists_lt C,
refine tendsto_at_top_add_left_of_le' _ C' _ hg,
exact (hf.eventually (lt_mem_nhds hC')).mono (λ x, le_of_lt)
end | lemma | filter.tendsto.add_at_top | 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"
] | [
"lt_mem_nhds"
] | In a linearly ordered additive commutative group with the order topology, if `f` tends to `C`
and `g` tends to `at_top` then `f + g` tends to `at_top`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
filter.tendsto.add_at_bot {C : α} (hf : tendsto f l (𝓝 C)) (hg : tendsto g l at_bot) :
tendsto (λ x, f x + g x) l at_bot | @filter.tendsto.add_at_top αᵒᵈ _ _ _ _ _ _ _ _ hf hg | lemma | filter.tendsto.add_at_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"
] | [
"filter.tendsto.add_at_top"
] | In a linearly ordered additive commutative group with the order topology, if `f` tends to `C`
and `g` tends to `at_bot` then `f + g` tends to `at_bot`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
filter.tendsto.at_top_add {C : α} (hf : tendsto f l at_top) (hg : tendsto g l (𝓝 C)) :
tendsto (λ x, f x + g x) l at_top | by { conv in (_ + _) { rw add_comm }, exact hg.add_at_top hf } | lemma | filter.tendsto.at_top_add | 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"
] | [] | In a linearly ordered additive commutative group with the order topology, if `f` tends to
`at_top` and `g` tends to `C` then `f + g` tends to `at_top`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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