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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filter.tendsto.at_bot_add {C : α} (hf : tendsto f l at_bot) (hg : tendsto g l (𝓝 C)) :
tendsto (λ x, f x + g x) l at_bot | by { conv in (_ + _) { rw add_comm }, exact hg.add_at_bot hf } | lemma | filter.tendsto.at_bot_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_bot` and `g` tends to `C` then `f + g` tends to `at_bot`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
nhds_basis_Ioo_pos [no_min_order α] [no_max_order α] (a : α) :
(𝓝 a).has_basis (λ ε : α, (0 : α) < ε) (λ ε, Ioo (a-ε) (a+ε)) | ⟨begin
refine λ t, (nhds_basis_Ioo a).mem_iff.trans ⟨_, _⟩,
{ rintros ⟨⟨l, u⟩, ⟨hl : l < a, hu : a < u⟩, h' : Ioo l u ⊆ t⟩,
refine ⟨min (a-l) (u-a), by apply lt_min; rwa sub_pos, _⟩,
rintros x ⟨hx, hx'⟩,
apply h',
rw [sub_lt_comm, lt_min_iff, sub_lt_sub_iff_left] at hx,
rw [← sub_lt_iff_lt_add',... | lemma | nhds_basis_Ioo_pos | 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_min_iff",
"nhds_basis_Ioo",
"no_max_order",
"no_min_order"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nhds_basis_abs_sub_lt [no_min_order α] [no_max_order α] (a : α) :
(𝓝 a).has_basis (λ ε : α, (0 : α) < ε) (λ ε, {b | |b - a| < ε}) | begin
convert nhds_basis_Ioo_pos a,
{ ext ε,
change |x - a| < ε ↔ a - ε < x ∧ x < a + ε,
simp [abs_lt, sub_lt_iff_lt_add, add_comm ε a, add_comm x ε] }
end | lemma | nhds_basis_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_lt",
"nhds_basis_Ioo_pos",
"no_max_order",
"no_min_order"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nhds_basis_zero_abs_sub_lt [no_min_order α] [no_max_order α] :
(𝓝 (0 : α)).has_basis (λ ε : α, (0 : α) < ε) (λ ε, {b | |b| < ε}) | by simpa using nhds_basis_abs_sub_lt (0 : α) | lemma | nhds_basis_zero_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"
] | [
"nhds_basis_abs_sub_lt",
"no_max_order",
"no_min_order"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nhds_basis_Ioo_pos_of_pos [no_min_order α] [no_max_order α]
{a : α} (ha : 0 < a) :
(𝓝 a).has_basis (λ ε : α, (0 : α) < ε ∧ ε ≤ a) (λ ε, Ioo (a-ε) (a+ε)) | ⟨ λ t, (nhds_basis_Ioo_pos a).mem_iff.trans
⟨λ h, let ⟨i, hi, hit⟩ := h in
⟨min i a, ⟨lt_min hi ha, min_le_right i a⟩, trans (Ioo_subset_Ioo
(sub_le_sub_left (min_le_left i a) a) (add_le_add_left (min_le_left i a) a)) hit⟩,
λ h, let ⟨i, hi, hit⟩ := h in ⟨i, hi.1, hit⟩ ⟩ ⟩ | lemma | nhds_basis_Ioo_pos_of_pos | 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_pos",
"no_max_order",
"no_min_order"
] | If `a` is positive we can form a basis from only nonnegative `Ioo` intervals | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
preimage_neg [add_group α] : preimage (has_neg.neg : α → α) = image (has_neg.neg : α → α) | (image_eq_preimage_of_inverse neg_neg neg_neg).symm | lemma | preimage_neg | 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"
] | [
"add_group"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
filter.map_neg_eq_comap_neg [add_group α] :
map (has_neg.neg : α → α) = comap (has_neg.neg : α → α) | funext $ assume f, map_eq_comap_of_inverse (funext neg_neg) (funext neg_neg) | lemma | filter.map_neg_eq_comap_neg | 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"
] | [
"add_group"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_lub.frequently_mem {a : α} {s : set α} (ha : is_lub s a) (hs : s.nonempty) :
∃ᶠ x in 𝓝[≤] a, x ∈ s | begin
rcases hs with ⟨a', ha'⟩,
intro h,
rcases (ha.1 ha').eq_or_lt with (rfl|ha'a),
{ exact h.self_of_nhds_within le_rfl ha' },
{ rcases (mem_nhds_within_Iic_iff_exists_Ioc_subset' ha'a).1 h
with ⟨b, hba, hb⟩,
rcases ha.exists_between hba with ⟨b', hb's, hb'⟩,
exact hb hb' hb's },
end | lemma | is_lub.frequently_mem | 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_lub",
"le_rfl",
"mem_nhds_within_Iic_iff_exists_Ioc_subset'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_lub.frequently_nhds_mem {a : α} {s : set α} (ha : is_lub s a) (hs : s.nonempty) :
∃ᶠ x in 𝓝 a, x ∈ s | (ha.frequently_mem hs).filter_mono inf_le_left | lemma | is_lub.frequently_nhds_mem | 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_left",
"is_lub"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_glb.frequently_mem {a : α} {s : set α} (ha : is_glb s a) (hs : s.nonempty) :
∃ᶠ x in 𝓝[≥] a, x ∈ s | @is_lub.frequently_mem αᵒᵈ _ _ _ _ _ ha hs | lemma | is_glb.frequently_mem | 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_glb",
"is_lub.frequently_mem"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_glb.frequently_nhds_mem {a : α} {s : set α} (ha : is_glb s a) (hs : s.nonempty) :
∃ᶠ x in 𝓝 a, x ∈ s | (ha.frequently_mem hs).filter_mono inf_le_left | lemma | is_glb.frequently_nhds_mem | 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_left",
"is_glb"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_lub.mem_closure {a : α} {s : set α} (ha : is_lub s a) (hs : s.nonempty) :
a ∈ closure s | (ha.frequently_nhds_mem hs).mem_closure | lemma | is_lub.mem_closure | topology.order | src/topology/order/basic.lean | [
"data.set.intervals.pi",
"data.set.pointwise.interval",
"order.filter.interval",
"topology.support",
"topology.algebra.order.left_right"
] | [
"closure",
"is_lub"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_glb.mem_closure {a : α} {s : set α} (ha : is_glb s a) (hs : s.nonempty) :
a ∈ closure s | (ha.frequently_nhds_mem hs).mem_closure | lemma | is_glb.mem_closure | topology.order | src/topology/order/basic.lean | [
"data.set.intervals.pi",
"data.set.pointwise.interval",
"order.filter.interval",
"topology.support",
"topology.algebra.order.left_right"
] | [
"closure",
"is_glb"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_lub.nhds_within_ne_bot {a : α} {s : set α} (ha : is_lub s a) (hs : s.nonempty) :
ne_bot (𝓝[s] a) | mem_closure_iff_nhds_within_ne_bot.1 (ha.mem_closure hs) | lemma | is_lub.nhds_within_ne_bot | topology.order | src/topology/order/basic.lean | [
"data.set.intervals.pi",
"data.set.pointwise.interval",
"order.filter.interval",
"topology.support",
"topology.algebra.order.left_right"
] | [
"is_lub"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_glb.nhds_within_ne_bot : ∀ {a : α} {s : set α}, is_glb s a → s.nonempty →
ne_bot (𝓝[s] a) | @is_lub.nhds_within_ne_bot αᵒᵈ _ _ _ | lemma | is_glb.nhds_within_ne_bot | topology.order | src/topology/order/basic.lean | [
"data.set.intervals.pi",
"data.set.pointwise.interval",
"order.filter.interval",
"topology.support",
"topology.algebra.order.left_right"
] | [
"is_glb",
"is_lub.nhds_within_ne_bot"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_lub_of_mem_nhds {s : set α} {a : α} {f : filter α}
(hsa : a ∈ upper_bounds s) (hsf : s ∈ f) [ne_bot (f ⊓ 𝓝 a)] : is_lub s a | ⟨hsa, assume b hb,
not_lt.1 $ assume hba,
have s ∩ {a | b < a} ∈ f ⊓ 𝓝 a,
from inter_mem_inf hsf (is_open.mem_nhds (is_open_lt' _) hba),
let ⟨x, ⟨hxs, hxb⟩⟩ := filter.nonempty_of_mem this in
have b < b, from lt_of_lt_of_le hxb $ hb hxs,
lt_irrefl b this⟩ | lemma | is_lub_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"
] | [
"filter",
"filter.nonempty_of_mem",
"is_lub",
"is_open.mem_nhds",
"is_open_lt'",
"upper_bounds"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_lub_of_mem_closure {s : set α} {a : α} (hsa : a ∈ upper_bounds s) (hsf : a ∈ closure s) :
is_lub s a | begin
rw [mem_closure_iff_cluster_pt, cluster_pt, inf_comm] at hsf,
haveI : (𝓟 s ⊓ 𝓝 a).ne_bot := hsf,
exact is_lub_of_mem_nhds hsa (mem_principal_self s),
end | lemma | is_lub_of_mem_closure | topology.order | src/topology/order/basic.lean | [
"data.set.intervals.pi",
"data.set.pointwise.interval",
"order.filter.interval",
"topology.support",
"topology.algebra.order.left_right"
] | [
"closure",
"cluster_pt",
"inf_comm",
"is_lub",
"is_lub_of_mem_nhds",
"mem_closure_iff_cluster_pt",
"upper_bounds"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_glb_of_mem_nhds : ∀ {s : set α} {a : α} {f : filter α},
a ∈ lower_bounds s → s ∈ f → ne_bot (f ⊓ 𝓝 a) → is_glb s a | @is_lub_of_mem_nhds αᵒᵈ _ _ _ | lemma | is_glb_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"
] | [
"filter",
"is_glb",
"is_lub_of_mem_nhds",
"lower_bounds"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_glb_of_mem_closure {s : set α} {a : α} (hsa : a ∈ lower_bounds s) (hsf : a ∈ closure s) :
is_glb s a | @is_lub_of_mem_closure αᵒᵈ _ _ _ s a hsa hsf | lemma | is_glb_of_mem_closure | topology.order | src/topology/order/basic.lean | [
"data.set.intervals.pi",
"data.set.pointwise.interval",
"order.filter.interval",
"topology.support",
"topology.algebra.order.left_right"
] | [
"closure",
"is_glb",
"is_lub_of_mem_closure",
"lower_bounds"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_lub.mem_upper_bounds_of_tendsto [preorder γ] [topological_space γ]
[order_closed_topology γ] {f : α → γ} {s : set α} {a : α} {b : γ}
(hf : monotone_on f s) (ha : is_lub s a)
(hb : tendsto f (𝓝[s] a) (𝓝 b)) : b ∈ upper_bounds (f '' s) | begin
rintro _ ⟨x, hx, rfl⟩,
replace ha := ha.inter_Ici_of_mem hx,
haveI := ha.nhds_within_ne_bot ⟨x, hx, le_rfl⟩,
refine ge_of_tendsto (hb.mono_left (nhds_within_mono _ (inter_subset_left s (Ici x)))) _,
exact mem_of_superset self_mem_nhds_within (λ y hy, hf hx hy.1 hy.2)
end | lemma | is_lub.mem_upper_bounds_of_tendsto | topology.order | src/topology/order/basic.lean | [
"data.set.intervals.pi",
"data.set.pointwise.interval",
"order.filter.interval",
"topology.support",
"topology.algebra.order.left_right"
] | [
"ge_of_tendsto",
"is_lub",
"monotone_on",
"nhds_within_mono",
"order_closed_topology",
"self_mem_nhds_within",
"topological_space",
"upper_bounds"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_lub.is_lub_of_tendsto [preorder γ] [topological_space γ]
[order_closed_topology γ] {f : α → γ} {s : set α} {a : α} {b : γ}
(hf : monotone_on f s) (ha : is_lub s a) (hs : s.nonempty)
(hb : tendsto f (𝓝[s] a) (𝓝 b)) : is_lub (f '' s) b | begin
haveI := ha.nhds_within_ne_bot hs,
exact ⟨ha.mem_upper_bounds_of_tendsto hf hb, λ b' hb', le_of_tendsto hb
(mem_of_superset self_mem_nhds_within $ λ x hx, hb' $ mem_image_of_mem _ hx)⟩
end | lemma | is_lub.is_lub_of_tendsto | topology.order | src/topology/order/basic.lean | [
"data.set.intervals.pi",
"data.set.pointwise.interval",
"order.filter.interval",
"topology.support",
"topology.algebra.order.left_right"
] | [
"is_lub",
"le_of_tendsto",
"monotone_on",
"order_closed_topology",
"self_mem_nhds_within",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_glb.mem_lower_bounds_of_tendsto [preorder γ] [topological_space γ]
[order_closed_topology γ] {f : α → γ} {s : set α} {a : α} {b : γ}
(hf : monotone_on f s) (ha : is_glb s a)
(hb : tendsto f (𝓝[s] a) (𝓝 b)) : b ∈ lower_bounds (f '' s) | @is_lub.mem_upper_bounds_of_tendsto αᵒᵈ γᵒᵈ _ _ _ _ _ _ _ _ _ _ hf.dual ha hb | lemma | is_glb.mem_lower_bounds_of_tendsto | topology.order | src/topology/order/basic.lean | [
"data.set.intervals.pi",
"data.set.pointwise.interval",
"order.filter.interval",
"topology.support",
"topology.algebra.order.left_right"
] | [
"is_glb",
"is_lub.mem_upper_bounds_of_tendsto",
"lower_bounds",
"monotone_on",
"order_closed_topology",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_glb.is_glb_of_tendsto [preorder γ] [topological_space γ]
[order_closed_topology γ] {f : α → γ} {s : set α} {a : α} {b : γ}
(hf : monotone_on f s) : is_glb s a → s.nonempty →
tendsto f (𝓝[s] a) (𝓝 b) → is_glb (f '' s) b | @is_lub.is_lub_of_tendsto αᵒᵈ γᵒᵈ _ _ _ _ _ _ f s a b hf.dual | lemma | is_glb.is_glb_of_tendsto | topology.order | src/topology/order/basic.lean | [
"data.set.intervals.pi",
"data.set.pointwise.interval",
"order.filter.interval",
"topology.support",
"topology.algebra.order.left_right"
] | [
"is_glb",
"is_lub.is_lub_of_tendsto",
"monotone_on",
"order_closed_topology",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_lub.mem_lower_bounds_of_tendsto [preorder γ] [topological_space γ]
[order_closed_topology γ] {f : α → γ} {s : set α} {a : α} {b : γ}
(hf : antitone_on f s) (ha : is_lub s a)
(hb : tendsto f (𝓝[s] a) (𝓝 b)) : b ∈ lower_bounds (f '' s) | @is_lub.mem_upper_bounds_of_tendsto α γᵒᵈ _ _ _ _ _ _ _ _ _ _ hf ha hb | lemma | is_lub.mem_lower_bounds_of_tendsto | topology.order | src/topology/order/basic.lean | [
"data.set.intervals.pi",
"data.set.pointwise.interval",
"order.filter.interval",
"topology.support",
"topology.algebra.order.left_right"
] | [
"antitone_on",
"is_lub",
"is_lub.mem_upper_bounds_of_tendsto",
"lower_bounds",
"order_closed_topology",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_lub.is_glb_of_tendsto [preorder γ] [topological_space γ]
[order_closed_topology γ] : ∀ {f : α → γ} {s : set α} {a : α} {b : γ},
(antitone_on f s) → is_lub s a → s.nonempty →
tendsto f (𝓝[s] a) (𝓝 b) → is_glb (f '' s) b | @is_lub.is_lub_of_tendsto α γᵒᵈ _ _ _ _ _ _ | lemma | is_lub.is_glb_of_tendsto | topology.order | src/topology/order/basic.lean | [
"data.set.intervals.pi",
"data.set.pointwise.interval",
"order.filter.interval",
"topology.support",
"topology.algebra.order.left_right"
] | [
"antitone_on",
"is_glb",
"is_lub",
"is_lub.is_lub_of_tendsto",
"order_closed_topology",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_glb.mem_upper_bounds_of_tendsto [preorder γ] [topological_space γ]
[order_closed_topology γ] {f : α → γ} {s : set α} {a : α} {b : γ}
(hf : antitone_on f s) (ha : is_glb s a)
(hb : tendsto f (𝓝[s] a) (𝓝 b)) : b ∈ upper_bounds (f '' s) | @is_glb.mem_lower_bounds_of_tendsto α γᵒᵈ _ _ _ _ _ _ _ _ _ _ hf ha hb | lemma | is_glb.mem_upper_bounds_of_tendsto | topology.order | src/topology/order/basic.lean | [
"data.set.intervals.pi",
"data.set.pointwise.interval",
"order.filter.interval",
"topology.support",
"topology.algebra.order.left_right"
] | [
"antitone_on",
"is_glb",
"is_glb.mem_lower_bounds_of_tendsto",
"order_closed_topology",
"topological_space",
"upper_bounds"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_glb.is_lub_of_tendsto [preorder γ] [topological_space γ]
[order_closed_topology γ] : ∀ {f : α → γ} {s : set α} {a : α} {b : γ},
(antitone_on f s) → is_glb s a → s.nonempty →
tendsto f (𝓝[s] a) (𝓝 b) → is_lub (f '' s) b | @is_glb.is_glb_of_tendsto α γᵒᵈ _ _ _ _ _ _ | lemma | is_glb.is_lub_of_tendsto | topology.order | src/topology/order/basic.lean | [
"data.set.intervals.pi",
"data.set.pointwise.interval",
"order.filter.interval",
"topology.support",
"topology.algebra.order.left_right"
] | [
"antitone_on",
"is_glb",
"is_glb.is_glb_of_tendsto",
"is_lub",
"order_closed_topology",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_lub.mem_of_is_closed {a : α} {s : set α} (ha : is_lub s a) (hs : s.nonempty)
(sc : is_closed s) : a ∈ s | sc.closure_subset $ ha.mem_closure hs | lemma | is_lub.mem_of_is_closed | topology.order | src/topology/order/basic.lean | [
"data.set.intervals.pi",
"data.set.pointwise.interval",
"order.filter.interval",
"topology.support",
"topology.algebra.order.left_right"
] | [
"is_closed",
"is_lub"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_glb.mem_of_is_closed {a : α} {s : set α} (ha : is_glb s a) (hs : s.nonempty)
(sc : is_closed s) : a ∈ s | sc.closure_subset $ ha.mem_closure hs | lemma | is_glb.mem_of_is_closed | topology.order | src/topology/order/basic.lean | [
"data.set.intervals.pi",
"data.set.pointwise.interval",
"order.filter.interval",
"topology.support",
"topology.algebra.order.left_right"
] | [
"is_closed",
"is_glb"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_lub.exists_seq_strict_mono_tendsto_of_not_mem {t : set α} {x : α}
[is_countably_generated (𝓝 x)] (htx : is_lub t x) (not_mem : x ∉ t) (ht : t.nonempty) :
∃ u : ℕ → α, strict_mono u ∧ (∀ n, u n < x) ∧ tendsto u at_top (𝓝 x) ∧ (∀ n, u n ∈ t) | begin
rcases ht with ⟨l, hl⟩,
have hl : l < x,
from (htx.1 hl).eq_or_lt.resolve_left (λ h, (not_mem $ h ▸ hl).elim),
obtain ⟨s, hs⟩ : ∃ s : ℕ → set α, (𝓝 x).has_basis (λ (_x : ℕ), true) s :=
let ⟨s, hs⟩ := (𝓝 x).exists_antitone_basis in ⟨s, hs.to_has_basis⟩,
have : ∀ n k, k < x → ∃ y, Icc y x ⊆ s n ∧ ... | lemma | is_lub.exists_seq_strict_mono_tendsto_of_not_mem | 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'",
"ge_iff_le",
"is_lub",
"strict_mono",
"strict_mono_nat_of_lt_succ"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_lub.exists_seq_monotone_tendsto {t : set α} {x : α} [is_countably_generated (𝓝 x)]
(htx : is_lub t x) (ht : t.nonempty) :
∃ u : ℕ → α, monotone u ∧ (∀ n, u n ≤ x) ∧ tendsto u at_top (𝓝 x) ∧ (∀ n, u n ∈ t) | begin
by_cases h : x ∈ t,
{ exact ⟨λ n, x, monotone_const, λ n, le_rfl, tendsto_const_nhds, λ n, h⟩ },
{ rcases htx.exists_seq_strict_mono_tendsto_of_not_mem h ht with ⟨u, hu⟩,
exact ⟨u, hu.1.monotone, λ n, (hu.2.1 n).le, hu.2.2⟩ }
end | lemma | is_lub.exists_seq_monotone_tendsto | topology.order | src/topology/order/basic.lean | [
"data.set.intervals.pi",
"data.set.pointwise.interval",
"order.filter.interval",
"topology.support",
"topology.algebra.order.left_right"
] | [
"is_lub",
"le_rfl",
"monotone",
"monotone_const",
"tendsto_const_nhds"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exists_seq_strict_mono_tendsto' {α : Type*} [linear_order α] [topological_space α]
[densely_ordered α] [order_topology α]
[first_countable_topology α] {x y : α} (hy : y < x) :
∃ u : ℕ → α, strict_mono u ∧ (∀ n, u n ∈ Ioo y x) ∧ tendsto u at_top (𝓝 x) | begin
have hx : x ∉ Ioo y x := λ h, (lt_irrefl x h.2).elim,
have ht : set.nonempty (Ioo y x) := nonempty_Ioo.2 hy,
rcases (is_lub_Ioo hy).exists_seq_strict_mono_tendsto_of_not_mem hx ht with ⟨u, hu⟩,
exact ⟨u, hu.1, hu.2.2.symm⟩
end | lemma | exists_seq_strict_mono_tendsto' | topology.order | src/topology/order/basic.lean | [
"data.set.intervals.pi",
"data.set.pointwise.interval",
"order.filter.interval",
"topology.support",
"topology.algebra.order.left_right"
] | [
"densely_ordered",
"is_lub_Ioo",
"order_topology",
"set.nonempty",
"strict_mono",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exists_seq_strict_mono_tendsto [densely_ordered α] [no_min_order α]
[first_countable_topology α] (x : α) :
∃ u : ℕ → α, strict_mono u ∧ (∀ n, u n < x) ∧ tendsto u at_top (𝓝 x) | begin
obtain ⟨y, hy⟩ : ∃ y, y < x := exists_lt x,
rcases exists_seq_strict_mono_tendsto' hy with ⟨u, hu_mono, hu_mem, hux⟩,
exact ⟨u, hu_mono, λ n, (hu_mem n).2, hux⟩
end | lemma | exists_seq_strict_mono_tendsto | topology.order | src/topology/order/basic.lean | [
"data.set.intervals.pi",
"data.set.pointwise.interval",
"order.filter.interval",
"topology.support",
"topology.algebra.order.left_right"
] | [
"densely_ordered",
"exists_seq_strict_mono_tendsto'",
"no_min_order",
"strict_mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exists_seq_strict_mono_tendsto_nhds_within [densely_ordered α] [no_min_order α]
[first_countable_topology α] (x : α) :
∃ u : ℕ → α, strict_mono u ∧ (∀ n, u n < x) ∧ tendsto u at_top (𝓝[<] x) | let ⟨u, hu, hx, h⟩ := exists_seq_strict_mono_tendsto x in ⟨u, hu, hx,
tendsto_nhds_within_mono_right (range_subset_iff.2 hx) $ tendsto_nhds_within_range.2 h⟩ | lemma | exists_seq_strict_mono_tendsto_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"
] | [
"densely_ordered",
"exists_seq_strict_mono_tendsto",
"no_min_order",
"strict_mono",
"tendsto_nhds_within_mono_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exists_seq_tendsto_Sup {α : Type*} [conditionally_complete_linear_order α]
[topological_space α] [order_topology α] [first_countable_topology α]
{S : set α} (hS : S.nonempty) (hS' : bdd_above S) :
∃ (u : ℕ → α), monotone u ∧ tendsto u at_top (𝓝 (Sup S)) ∧ (∀ n, u n ∈ S) | begin
rcases (is_lub_cSup hS hS').exists_seq_monotone_tendsto hS with ⟨u, hu⟩,
exact ⟨u, hu.1, hu.2.2⟩,
end | lemma | exists_seq_tendsto_Sup | 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"
] | [
"bdd_above",
"conditionally_complete_linear_order",
"is_lub_cSup",
"monotone",
"order_topology",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_glb.exists_seq_strict_anti_tendsto_of_not_mem {t : set α} {x : α}
[is_countably_generated (𝓝 x)] (htx : is_glb t x) (not_mem : x ∉ t) (ht : t.nonempty) :
∃ u : ℕ → α, strict_anti u ∧ (∀ n, x < u n) ∧
tendsto u at_top (𝓝 x) ∧ (∀ n, u n ∈ t) | @is_lub.exists_seq_strict_mono_tendsto_of_not_mem αᵒᵈ _ _ _ t x _ htx not_mem ht | lemma | is_glb.exists_seq_strict_anti_tendsto_of_not_mem | 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_glb",
"is_lub.exists_seq_strict_mono_tendsto_of_not_mem",
"strict_anti"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_glb.exists_seq_antitone_tendsto {t : set α} {x : α} [is_countably_generated (𝓝 x)]
(htx : is_glb t x) (ht : t.nonempty) :
∃ u : ℕ → α, antitone u ∧ (∀ n, x ≤ u n) ∧
tendsto u at_top (𝓝 x) ∧ (∀ n, u n ∈ t) | @is_lub.exists_seq_monotone_tendsto αᵒᵈ _ _ _ t x _ htx ht | lemma | is_glb.exists_seq_antitone_tendsto | topology.order | src/topology/order/basic.lean | [
"data.set.intervals.pi",
"data.set.pointwise.interval",
"order.filter.interval",
"topology.support",
"topology.algebra.order.left_right"
] | [
"antitone",
"is_glb",
"is_lub.exists_seq_monotone_tendsto"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exists_seq_strict_anti_tendsto' [densely_ordered α]
[first_countable_topology α] {x y : α} (hy : x < y) :
∃ u : ℕ → α, strict_anti u ∧ (∀ n, u n ∈ Ioo x y) ∧ tendsto u at_top (𝓝 x) | by simpa only [dual_Ioo] using exists_seq_strict_mono_tendsto' (order_dual.to_dual_lt_to_dual.2 hy) | lemma | exists_seq_strict_anti_tendsto' | topology.order | src/topology/order/basic.lean | [
"data.set.intervals.pi",
"data.set.pointwise.interval",
"order.filter.interval",
"topology.support",
"topology.algebra.order.left_right"
] | [
"densely_ordered",
"exists_seq_strict_mono_tendsto'",
"strict_anti"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exists_seq_strict_anti_tendsto [densely_ordered α] [no_max_order α]
[first_countable_topology α] (x : α) :
∃ u : ℕ → α, strict_anti u ∧ (∀ n, x < u n) ∧ tendsto u at_top (𝓝 x) | @exists_seq_strict_mono_tendsto αᵒᵈ _ _ _ _ _ _ x | lemma | exists_seq_strict_anti_tendsto | topology.order | src/topology/order/basic.lean | [
"data.set.intervals.pi",
"data.set.pointwise.interval",
"order.filter.interval",
"topology.support",
"topology.algebra.order.left_right"
] | [
"densely_ordered",
"exists_seq_strict_mono_tendsto",
"no_max_order",
"strict_anti"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exists_seq_strict_anti_tendsto_nhds_within [densely_ordered α] [no_max_order α]
[first_countable_topology α] (x : α) :
∃ u : ℕ → α, strict_anti u ∧ (∀ n, x < u n) ∧ tendsto u at_top (𝓝[>] x) | @exists_seq_strict_mono_tendsto_nhds_within αᵒᵈ _ _ _ _ _ _ _ | lemma | exists_seq_strict_anti_tendsto_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"
] | [
"densely_ordered",
"exists_seq_strict_mono_tendsto_nhds_within",
"no_max_order",
"strict_anti"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exists_seq_strict_anti_strict_mono_tendsto [densely_ordered α] [first_countable_topology α]
{x y : α} (h : x < y) :
∃ (u v : ℕ → α), strict_anti u ∧ strict_mono v ∧ (∀ k, u k ∈ Ioo x y) ∧ (∀ l, v l ∈ Ioo x y) ∧
(∀ k l, u k < v l) ∧ tendsto u at_top (𝓝 x) ∧ tendsto v at_top (𝓝 y) | begin
rcases exists_seq_strict_anti_tendsto' h with ⟨u, hu_anti, hu_mem, hux⟩,
rcases exists_seq_strict_mono_tendsto' (hu_mem 0).2 with ⟨v, hv_mono, hv_mem, hvy⟩,
exact ⟨u, v, hu_anti, hv_mono, hu_mem, λ l, ⟨(hu_mem 0).1.trans (hv_mem l).1, (hv_mem l).2⟩,
λ k l, (hu_anti.antitone (zero_le k)).trans_lt (hv_mem... | lemma | exists_seq_strict_anti_strict_mono_tendsto | topology.order | src/topology/order/basic.lean | [
"data.set.intervals.pi",
"data.set.pointwise.interval",
"order.filter.interval",
"topology.support",
"topology.algebra.order.left_right"
] | [
"densely_ordered",
"exists_seq_strict_anti_tendsto'",
"exists_seq_strict_mono_tendsto'",
"strict_anti",
"strict_mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exists_seq_tendsto_Inf {α : Type*} [conditionally_complete_linear_order α]
[topological_space α] [order_topology α] [first_countable_topology α]
{S : set α} (hS : S.nonempty) (hS' : bdd_below S) :
∃ (u : ℕ → α), antitone u ∧ tendsto u at_top (𝓝 (Inf S)) ∧ (∀ n, u n ∈ S) | @exists_seq_tendsto_Sup αᵒᵈ _ _ _ _ S hS hS' | lemma | exists_seq_tendsto_Inf | 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"
] | [
"antitone",
"bdd_below",
"conditionally_complete_linear_order",
"exists_seq_tendsto_Sup",
"order_topology",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
closure_Ioi' {a : α} (h : (Ioi a).nonempty) :
closure (Ioi a) = Ici a | begin
apply subset.antisymm,
{ exact closure_minimal Ioi_subset_Ici_self is_closed_Ici },
{ rw [← diff_subset_closure_iff, Ici_diff_Ioi_same, singleton_subset_iff],
exact is_glb_Ioi.mem_closure h }
end | lemma | closure_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"
] | [
"closure",
"closure_minimal",
"diff_subset_closure_iff",
"is_closed_Ici"
] | The closure of the interval `(a, +∞)` is the closed interval `[a, +∞)`, unless `a` is a top
element. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
closure_Ioi (a : α) [no_max_order α] :
closure (Ioi a) = Ici a | closure_Ioi' nonempty_Ioi | lemma | closure_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"
] | [
"closure",
"closure_Ioi'",
"no_max_order"
] | The closure of the interval `(a, +∞)` is the closed interval `[a, +∞)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
closure_Iio' (h : (Iio a).nonempty) : closure (Iio a) = Iic a | @closure_Ioi' αᵒᵈ _ _ _ _ _ h | lemma | closure_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"
] | [
"closure",
"closure_Ioi'"
] | The closure of the interval `(-∞, a)` is the closed interval `(-∞, a]`, unless `a` is a bottom
element. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
closure_Iio (a : α) [no_min_order α] :
closure (Iio a) = Iic a | closure_Iio' nonempty_Iio | lemma | closure_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"
] | [
"closure",
"closure_Iio'",
"no_min_order"
] | The closure of the interval `(-∞, a)` is the interval `(-∞, a]`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
closure_Ioo {a b : α} (hab : a ≠ b) :
closure (Ioo a b) = Icc a b | begin
apply subset.antisymm,
{ exact closure_minimal Ioo_subset_Icc_self is_closed_Icc },
{ cases hab.lt_or_lt with hab hab,
{ rw [← diff_subset_closure_iff, Icc_diff_Ioo_same hab.le],
have hab' : (Ioo a b).nonempty, from nonempty_Ioo.2 hab,
simp only [insert_subset, singleton_subset_iff],
e... | lemma | closure_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"
] | [
"closure",
"closure_minimal",
"diff_subset_closure_iff",
"is_closed_Icc",
"is_glb_Ioo",
"is_lub_Ioo"
] | The closure of the open interval `(a, b)` is the closed interval `[a, b]`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
closure_Ioc {a b : α} (hab : a ≠ b) :
closure (Ioc a b) = Icc a b | begin
apply subset.antisymm,
{ exact closure_minimal Ioc_subset_Icc_self is_closed_Icc },
{ apply subset.trans _ (closure_mono Ioo_subset_Ioc_self),
rw closure_Ioo hab }
end | lemma | closure_Ioc | topology.order | src/topology/order/basic.lean | [
"data.set.intervals.pi",
"data.set.pointwise.interval",
"order.filter.interval",
"topology.support",
"topology.algebra.order.left_right"
] | [
"closure",
"closure_Ioo",
"closure_minimal",
"closure_mono",
"is_closed_Icc"
] | The closure of the interval `(a, b]` is the closed interval `[a, b]`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
closure_Ico {a b : α} (hab : a ≠ b) :
closure (Ico a b) = Icc a b | begin
apply subset.antisymm,
{ exact closure_minimal Ico_subset_Icc_self is_closed_Icc },
{ apply subset.trans _ (closure_mono Ioo_subset_Ico_self),
rw closure_Ioo hab }
end | lemma | closure_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"
] | [
"closure",
"closure_Ioo",
"closure_minimal",
"closure_mono",
"is_closed_Icc"
] | The closure of the interval `[a, b)` is the closed interval `[a, b]`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
interior_Ici' {a : α} (ha : (Iio a).nonempty) : interior (Ici a) = Ioi a | by rw [← compl_Iio, interior_compl, closure_Iio' ha, compl_Iic] | lemma | interior_Ici' | topology.order | src/topology/order/basic.lean | [
"data.set.intervals.pi",
"data.set.pointwise.interval",
"order.filter.interval",
"topology.support",
"topology.algebra.order.left_right"
] | [
"closure_Iio'",
"interior",
"interior_compl"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
interior_Ici [no_min_order α] {a : α} : interior (Ici a) = Ioi a | interior_Ici' nonempty_Iio | lemma | interior_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"
] | [
"interior",
"interior_Ici'",
"no_min_order"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
interior_Iic' {a : α} (ha : (Ioi a).nonempty) : interior (Iic a) = Iio a | @interior_Ici' αᵒᵈ _ _ _ _ _ ha | lemma | interior_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"
] | [
"interior",
"interior_Ici'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
interior_Iic [no_max_order α] {a : α} : interior (Iic a) = Iio a | interior_Iic' nonempty_Ioi | lemma | interior_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"
] | [
"interior",
"interior_Iic'",
"no_max_order"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
interior_Icc [no_min_order α] [no_max_order α] {a b : α}:
interior (Icc a b) = Ioo a b | by rw [← Ici_inter_Iic, interior_inter, interior_Ici, interior_Iic, Ioi_inter_Iio] | lemma | interior_Icc | topology.order | src/topology/order/basic.lean | [
"data.set.intervals.pi",
"data.set.pointwise.interval",
"order.filter.interval",
"topology.support",
"topology.algebra.order.left_right"
] | [
"interior",
"interior_Ici",
"interior_Iic",
"interior_inter",
"no_max_order",
"no_min_order"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
interior_Ico [no_min_order α] {a b : α} : interior (Ico a b) = Ioo a b | by rw [← Ici_inter_Iio, interior_inter, interior_Ici, interior_Iio, Ioi_inter_Iio] | lemma | interior_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"
] | [
"interior",
"interior_Ici",
"interior_Iio",
"interior_inter",
"no_min_order"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
interior_Ioc [no_max_order α] {a b : α} : interior (Ioc a b) = Ioo a b | by rw [← Ioi_inter_Iic, interior_inter, interior_Ioi, interior_Iic, Ioi_inter_Iio] | lemma | interior_Ioc | 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"
] | [
"interior",
"interior_Iic",
"interior_Ioi",
"interior_inter",
"no_max_order"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
closure_interior_Icc {a b : α} (h : a ≠ b) : closure (interior (Icc a b)) = Icc a b | (closure_minimal interior_subset is_closed_Icc).antisymm $
calc Icc a b = closure (Ioo a b) : (closure_Ioo h).symm
... ⊆ closure (interior (Icc a b)) : closure_mono (interior_maximal Ioo_subset_Icc_self is_open_Ioo) | lemma | closure_interior_Icc | topology.order | src/topology/order/basic.lean | [
"data.set.intervals.pi",
"data.set.pointwise.interval",
"order.filter.interval",
"topology.support",
"topology.algebra.order.left_right"
] | [
"closure",
"closure_Ioo",
"closure_minimal",
"closure_mono",
"interior",
"interior_maximal",
"interior_subset",
"is_closed_Icc",
"is_open_Ioo"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Ioc_subset_closure_interior (a b : α) : Ioc a b ⊆ closure (interior (Ioc a b)) | begin
rcases eq_or_ne a b with rfl|h,
{ simp },
{ calc Ioc a b ⊆ Icc a b : Ioc_subset_Icc_self
... = closure (Ioo a b) : (closure_Ioo h).symm
... ⊆ closure (interior (Ioc a b)) :
closure_mono (interior_maximal Ioo_subset_Ioc_self is_open_Ioo) }
end | lemma | Ioc_subset_closure_interior | topology.order | src/topology/order/basic.lean | [
"data.set.intervals.pi",
"data.set.pointwise.interval",
"order.filter.interval",
"topology.support",
"topology.algebra.order.left_right"
] | [
"closure",
"closure_Ioo",
"closure_mono",
"eq_or_ne",
"interior",
"interior_maximal",
"is_open_Ioo"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Ico_subset_closure_interior (a b : α) : Ico a b ⊆ closure (interior (Ico a b)) | by simpa only [dual_Ioc]
using Ioc_subset_closure_interior (order_dual.to_dual b) (order_dual.to_dual a) | lemma | Ico_subset_closure_interior | 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_subset_closure_interior",
"closure",
"interior",
"order_dual.to_dual"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
frontier_Ici' {a : α} (ha : (Iio a).nonempty) : frontier (Ici a) = {a} | by simp [frontier, ha] | lemma | frontier_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"
] | [
"frontier"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
frontier_Ici [no_min_order α] {a : α} : frontier (Ici a) = {a} | frontier_Ici' nonempty_Iio | lemma | frontier_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"
] | [
"frontier",
"frontier_Ici'",
"no_min_order"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
frontier_Iic' {a : α} (ha : (Ioi a).nonempty) : frontier (Iic a) = {a} | by simp [frontier, ha] | lemma | frontier_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"
] | [
"frontier"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
frontier_Iic [no_max_order α] {a : α} : frontier (Iic a) = {a} | frontier_Iic' nonempty_Ioi | lemma | frontier_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"
] | [
"frontier",
"frontier_Iic'",
"no_max_order"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
frontier_Ioi' {a : α} (ha : (Ioi a).nonempty) : frontier (Ioi a) = {a} | by simp [frontier, closure_Ioi' ha, Iic_diff_Iio, Icc_self] | lemma | frontier_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"
] | [
"closure_Ioi'",
"frontier"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
frontier_Ioi [no_max_order α] {a : α} : frontier (Ioi a) = {a} | frontier_Ioi' nonempty_Ioi | lemma | frontier_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"
] | [
"frontier",
"frontier_Ioi'",
"no_max_order"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
frontier_Iio' {a : α} (ha : (Iio a).nonempty) : frontier (Iio a) = {a} | by simp [frontier, closure_Iio' ha, Iic_diff_Iio, Icc_self] | lemma | frontier_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"
] | [
"closure_Iio'",
"frontier"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
frontier_Iio [no_min_order α] {a : α} : frontier (Iio a) = {a} | frontier_Iio' nonempty_Iio | lemma | frontier_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"
] | [
"frontier",
"frontier_Iio'",
"no_min_order"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
frontier_Icc [no_min_order α] [no_max_order α] {a b : α} (h : a ≤ b) :
frontier (Icc a b) = {a, b} | by simp [frontier, h, Icc_diff_Ioo_same] | lemma | frontier_Icc | topology.order | src/topology/order/basic.lean | [
"data.set.intervals.pi",
"data.set.pointwise.interval",
"order.filter.interval",
"topology.support",
"topology.algebra.order.left_right"
] | [
"frontier",
"no_max_order",
"no_min_order"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
frontier_Ioo {a b : α} (h : a < b) : frontier (Ioo a b) = {a, b} | by rw [frontier, closure_Ioo h.ne, interior_Ioo, Icc_diff_Ioo_same h.le] | lemma | frontier_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"
] | [
"closure_Ioo",
"frontier",
"interior_Ioo"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
frontier_Ico [no_min_order α] {a b : α} (h : a < b) : frontier (Ico a b) = {a, b} | by rw [frontier, closure_Ico h.ne, interior_Ico, Icc_diff_Ioo_same h.le] | lemma | frontier_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"
] | [
"closure_Ico",
"frontier",
"interior_Ico",
"no_min_order"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
frontier_Ioc [no_max_order α] {a b : α} (h : a < b) : frontier (Ioc a b) = {a, b} | by rw [frontier, closure_Ioc h.ne, interior_Ioc, Icc_diff_Ioo_same h.le] | lemma | frontier_Ioc | topology.order | src/topology/order/basic.lean | [
"data.set.intervals.pi",
"data.set.pointwise.interval",
"order.filter.interval",
"topology.support",
"topology.algebra.order.left_right"
] | [
"closure_Ioc",
"frontier",
"interior_Ioc",
"no_max_order"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nhds_within_Ioi_ne_bot' {a b : α} (H₁ : (Ioi a).nonempty) (H₂ : a ≤ b) :
ne_bot (𝓝[Ioi a] b) | mem_closure_iff_nhds_within_ne_bot.1 $ by rwa [closure_Ioi' H₁] | lemma | nhds_within_Ioi_ne_bot' | topology.order | src/topology/order/basic.lean | [
"data.set.intervals.pi",
"data.set.pointwise.interval",
"order.filter.interval",
"topology.support",
"topology.algebra.order.left_right"
] | [
"closure_Ioi'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nhds_within_Ioi_ne_bot [no_max_order α] {a b : α} (H : a ≤ b) :
ne_bot (𝓝[Ioi a] b) | nhds_within_Ioi_ne_bot' nonempty_Ioi H | lemma | nhds_within_Ioi_ne_bot | topology.order | src/topology/order/basic.lean | [
"data.set.intervals.pi",
"data.set.pointwise.interval",
"order.filter.interval",
"topology.support",
"topology.algebra.order.left_right"
] | [
"nhds_within_Ioi_ne_bot'",
"no_max_order"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nhds_within_Ioi_self_ne_bot' {a : α} (H : (Ioi a).nonempty) :
ne_bot (𝓝[>] a) | nhds_within_Ioi_ne_bot' H (le_refl a) | lemma | nhds_within_Ioi_self_ne_bot' | topology.order | src/topology/order/basic.lean | [
"data.set.intervals.pi",
"data.set.pointwise.interval",
"order.filter.interval",
"topology.support",
"topology.algebra.order.left_right"
] | [
"nhds_within_Ioi_ne_bot'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nhds_within_Ioi_self_ne_bot [no_max_order α] (a : α) :
ne_bot (𝓝[>] a) | nhds_within_Ioi_ne_bot (le_refl a) | lemma | nhds_within_Ioi_self_ne_bot | topology.order | src/topology/order/basic.lean | [
"data.set.intervals.pi",
"data.set.pointwise.interval",
"order.filter.interval",
"topology.support",
"topology.algebra.order.left_right"
] | [
"nhds_within_Ioi_ne_bot",
"no_max_order"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
filter.eventually.exists_gt [no_max_order α] {a : α} {p : α → Prop} (h : ∀ᶠ x in 𝓝 a, p x) :
∃ b > a, p b | by simpa only [exists_prop, gt_iff_lt, and_comm]
using ((h.filter_mono (@nhds_within_le_nhds _ _ a (Ioi a))).and self_mem_nhds_within).exists | lemma | filter.eventually.exists_gt | 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",
"gt_iff_lt",
"nhds_within_le_nhds",
"no_max_order",
"self_mem_nhds_within"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nhds_within_Iio_ne_bot' {b c : α} (H₁ : (Iio c).nonempty) (H₂ : b ≤ c) :
ne_bot (𝓝[Iio c] b) | mem_closure_iff_nhds_within_ne_bot.1 $ by rwa closure_Iio' H₁ | lemma | nhds_within_Iio_ne_bot' | topology.order | src/topology/order/basic.lean | [
"data.set.intervals.pi",
"data.set.pointwise.interval",
"order.filter.interval",
"topology.support",
"topology.algebra.order.left_right"
] | [
"closure_Iio'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nhds_within_Iio_ne_bot [no_min_order α] {a b : α} (H : a ≤ b) :
ne_bot (𝓝[Iio b] a) | nhds_within_Iio_ne_bot' nonempty_Iio H | lemma | nhds_within_Iio_ne_bot | topology.order | src/topology/order/basic.lean | [
"data.set.intervals.pi",
"data.set.pointwise.interval",
"order.filter.interval",
"topology.support",
"topology.algebra.order.left_right"
] | [
"nhds_within_Iio_ne_bot'",
"no_min_order"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nhds_within_Iio_self_ne_bot' {b : α} (H : (Iio b).nonempty) :
ne_bot (𝓝[<] b) | nhds_within_Iio_ne_bot' H (le_refl b) | lemma | nhds_within_Iio_self_ne_bot' | topology.order | src/topology/order/basic.lean | [
"data.set.intervals.pi",
"data.set.pointwise.interval",
"order.filter.interval",
"topology.support",
"topology.algebra.order.left_right"
] | [
"nhds_within_Iio_ne_bot'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nhds_within_Iio_self_ne_bot [no_min_order α] (a : α) :
ne_bot (𝓝[<] a) | nhds_within_Iio_ne_bot (le_refl a) | lemma | nhds_within_Iio_self_ne_bot | topology.order | src/topology/order/basic.lean | [
"data.set.intervals.pi",
"data.set.pointwise.interval",
"order.filter.interval",
"topology.support",
"topology.algebra.order.left_right"
] | [
"nhds_within_Iio_ne_bot",
"no_min_order"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
filter.eventually.exists_lt [no_min_order α] {a : α} {p : α → Prop} (h : ∀ᶠ x in 𝓝 a, p x) :
∃ b < a, p b | @filter.eventually.exists_gt αᵒᵈ _ _ _ _ _ _ _ h | lemma | filter.eventually.exists_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"
] | [
"filter.eventually.exists_gt",
"no_min_order"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
right_nhds_within_Ico_ne_bot {a b : α} (H : a < b) : ne_bot (𝓝[Ico a b] b) | (is_lub_Ico H).nhds_within_ne_bot (nonempty_Ico.2 H) | lemma | right_nhds_within_Ico_ne_bot | topology.order | src/topology/order/basic.lean | [
"data.set.intervals.pi",
"data.set.pointwise.interval",
"order.filter.interval",
"topology.support",
"topology.algebra.order.left_right"
] | [
"is_lub_Ico"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
left_nhds_within_Ioc_ne_bot {a b : α} (H : a < b) : ne_bot (𝓝[Ioc a b] a) | (is_glb_Ioc H).nhds_within_ne_bot (nonempty_Ioc.2 H) | lemma | left_nhds_within_Ioc_ne_bot | topology.order | src/topology/order/basic.lean | [
"data.set.intervals.pi",
"data.set.pointwise.interval",
"order.filter.interval",
"topology.support",
"topology.algebra.order.left_right"
] | [
"is_glb_Ioc"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
left_nhds_within_Ioo_ne_bot {a b : α} (H : a < b) : ne_bot (𝓝[Ioo a b] a) | (is_glb_Ioo H).nhds_within_ne_bot (nonempty_Ioo.2 H) | lemma | left_nhds_within_Ioo_ne_bot | topology.order | src/topology/order/basic.lean | [
"data.set.intervals.pi",
"data.set.pointwise.interval",
"order.filter.interval",
"topology.support",
"topology.algebra.order.left_right"
] | [
"is_glb_Ioo"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
right_nhds_within_Ioo_ne_bot {a b : α} (H : a < b) : ne_bot (𝓝[Ioo a b] b) | (is_lub_Ioo H).nhds_within_ne_bot (nonempty_Ioo.2 H) | lemma | right_nhds_within_Ioo_ne_bot | topology.order | src/topology/order/basic.lean | [
"data.set.intervals.pi",
"data.set.pointwise.interval",
"order.filter.interval",
"topology.support",
"topology.algebra.order.left_right"
] | [
"is_lub_Ioo"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comap_coe_nhds_within_Iio_of_Ioo_subset (hb : s ⊆ Iio b)
(hs : s.nonempty → ∃ a < b, Ioo a b ⊆ s) :
comap (coe : s → α) (𝓝[<] b) = at_top | begin
nontriviality,
haveI : nonempty s := nontrivial_iff_nonempty.1 ‹_›,
rcases hs (nonempty_subtype.1 ‹_›) with ⟨a, h, hs⟩,
ext u, split,
{ rintros ⟨t, ht, hts⟩,
obtain ⟨x, ⟨hxa : a ≤ x, hxb : x < b⟩, hxt : Ioo x b ⊆ t⟩ :=
(mem_nhds_within_Iio_iff_exists_mem_Ico_Ioo_subset h).mp ht,
obtain ⟨y,... | lemma | comap_coe_nhds_within_Iio_of_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_within_Iio",
"exists_between",
"mem_nhds_within_Iio_iff_exists_mem_Ico_Ioo_subset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comap_coe_nhds_within_Ioi_of_Ioo_subset (ha : s ⊆ Ioi a)
(hs : s.nonempty → ∃ b > a, Ioo a b ⊆ s) :
comap (coe : s → α) (𝓝[>] a) = at_bot | comap_coe_nhds_within_Iio_of_Ioo_subset
(show of_dual ⁻¹' s ⊆ Iio (to_dual a), from ha)
(λ h, by simpa only [order_dual.exists, dual_Ioo] using hs h) | lemma | comap_coe_nhds_within_Ioi_of_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"
] | [
"comap_coe_nhds_within_Iio_of_Ioo_subset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_coe_at_top_of_Ioo_subset (hb : s ⊆ Iio b)
(hs : ∀ a' < b, ∃ a < b, Ioo a b ⊆ s) :
map (coe : s → α) at_top = 𝓝[<] b | begin
rcases eq_empty_or_nonempty (Iio b) with (hb'|⟨a, ha⟩),
{ rw [filter_eq_bot_of_is_empty at_top, filter.map_bot, hb', nhds_within_empty],
exact ⟨λ x, hb'.subset (hb x.2)⟩ },
{ rw [← comap_coe_nhds_within_Iio_of_Ioo_subset hb (λ _, hs a ha), map_comap_of_mem],
rw subtype.range_coe,
exact (mem_nhds... | lemma | map_coe_at_top_of_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"
] | [
"comap_coe_nhds_within_Iio_of_Ioo_subset",
"filter.map_bot",
"mem_nhds_within_Iio_iff_exists_Ioo_subset'",
"nhds_within_empty",
"subtype.range_coe"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_coe_at_bot_of_Ioo_subset (ha : s ⊆ Ioi a)
(hs : ∀ b' > a, ∃ b > a, Ioo a b ⊆ s) :
map (coe : s → α) at_bot = (𝓝[>] a) | begin
-- the elaborator gets stuck without `(... : _)`
refine (map_coe_at_top_of_Ioo_subset
(show of_dual ⁻¹' s ⊆ Iio (to_dual a), from ha) (λ b' hb', _) : _),
simpa only [order_dual.exists, dual_Ioo] using hs b' hb',
end | lemma | map_coe_at_bot_of_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"
] | [
"map_coe_at_top_of_Ioo_subset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comap_coe_Ioo_nhds_within_Iio (a b : α) :
comap (coe : Ioo a b → α) (𝓝[<] b) = at_top | comap_coe_nhds_within_Iio_of_Ioo_subset Ioo_subset_Iio_self $
λ h, ⟨a, nonempty_Ioo.1 h, subset.refl _⟩ | lemma | comap_coe_Ioo_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"
] | [
"comap_coe_nhds_within_Iio_of_Ioo_subset"
] | The `at_top` filter for an open interval `Ioo a b` comes from the left-neighbourhoods filter at
the right endpoint in the ambient order. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
comap_coe_Ioo_nhds_within_Ioi (a b : α) :
comap (coe : Ioo a b → α) (𝓝[>] a) = at_bot | comap_coe_nhds_within_Ioi_of_Ioo_subset Ioo_subset_Ioi_self $
λ h, ⟨b, nonempty_Ioo.1 h, subset.refl _⟩ | lemma | comap_coe_Ioo_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"
] | [
"comap_coe_nhds_within_Ioi_of_Ioo_subset"
] | The `at_bot` filter for an open interval `Ioo a b` comes from the right-neighbourhoods filter at
the left endpoint in the ambient order. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
comap_coe_Ioi_nhds_within_Ioi (a : α) : comap (coe : Ioi a → α) (𝓝[>] a) = at_bot | comap_coe_nhds_within_Ioi_of_Ioo_subset (subset.refl _) $
λ ⟨x, hx⟩, ⟨x, hx, Ioo_subset_Ioi_self⟩ | lemma | comap_coe_Ioi_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"
] | [
"comap_coe_nhds_within_Ioi_of_Ioo_subset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comap_coe_Iio_nhds_within_Iio (a : α) :
comap (coe : Iio a → α) (𝓝[<] a) = at_top | @comap_coe_Ioi_nhds_within_Ioi αᵒᵈ _ _ _ _ a | lemma | comap_coe_Iio_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"
] | [
"comap_coe_Ioi_nhds_within_Ioi"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_coe_Ioo_at_top {a b : α} (h : a < b) :
map (coe : Ioo a b → α) at_top = 𝓝[<] b | map_coe_at_top_of_Ioo_subset Ioo_subset_Iio_self $ λ _ _, ⟨_, h, subset.refl _⟩ | lemma | map_coe_Ioo_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"
] | [
"map_coe_at_top_of_Ioo_subset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_coe_Ioo_at_bot {a b : α} (h : a < b) :
map (coe : Ioo a b → α) at_bot = 𝓝[>] a | map_coe_at_bot_of_Ioo_subset Ioo_subset_Ioi_self $ λ _ _, ⟨_, h, subset.refl _⟩ | lemma | map_coe_Ioo_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"
] | [
"map_coe_at_bot_of_Ioo_subset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_coe_Ioi_at_bot (a : α) :
map (coe : Ioi a → α) at_bot = 𝓝[>] a | map_coe_at_bot_of_Ioo_subset (subset.refl _) $ λ b hb, ⟨b, hb, Ioo_subset_Ioi_self⟩ | lemma | map_coe_Ioi_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"
] | [
"map_coe_at_bot_of_Ioo_subset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_coe_Iio_at_top (a : α) :
map (coe : Iio a → α) at_top = 𝓝[<] a | @map_coe_Ioi_at_bot αᵒᵈ _ _ _ _ _ | lemma | map_coe_Iio_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"
] | [
"map_coe_Ioi_at_bot"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_comp_coe_Ioo_at_top (h : a < b) :
tendsto (λ x : Ioo a b, f x) at_top l ↔ tendsto f (𝓝[<] b) l | by rw [← map_coe_Ioo_at_top h, tendsto_map'_iff] | lemma | tendsto_comp_coe_Ioo_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"
] | [
"map_coe_Ioo_at_top"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_comp_coe_Ioo_at_bot (h : a < b) :
tendsto (λ x : Ioo a b, f x) at_bot l ↔ tendsto f (𝓝[>] a) l | by rw [← map_coe_Ioo_at_bot h, tendsto_map'_iff] | lemma | tendsto_comp_coe_Ioo_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"
] | [
"map_coe_Ioo_at_bot"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_comp_coe_Ioi_at_bot :
tendsto (λ x : Ioi a, f x) at_bot l ↔ tendsto f (𝓝[>] a) l | by rw [← map_coe_Ioi_at_bot, tendsto_map'_iff] | lemma | tendsto_comp_coe_Ioi_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"
] | [
"map_coe_Ioi_at_bot"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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