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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is_open_Iio : is_open (Iio a) | is_open_lt continuous_id continuous_const | lemma | is_open_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"
] | [
"continuous_const",
"continuous_id",
"is_open",
"is_open_lt"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_open_Ioi : is_open (Ioi a) | is_open_lt continuous_const continuous_id | lemma | is_open_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"
] | [
"continuous_const",
"continuous_id",
"is_open",
"is_open_lt"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_open_Ioo : is_open (Ioo a b) | is_open.inter is_open_Ioi is_open_Iio | lemma | is_open_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"
] | [
"is_open",
"is_open.inter",
"is_open_Iio",
"is_open_Ioi"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
interior_Ioi : interior (Ioi a) = Ioi a | is_open_Ioi.interior_eq | lemma | interior_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"
] | [
"interior"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
interior_Iio : interior (Iio a) = Iio a | is_open_Iio.interior_eq | lemma | interior_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"
] | [
"interior"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
interior_Ioo : interior (Ioo a b) = Ioo a b | is_open_Ioo.interior_eq | lemma | interior_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"
] | [
"interior"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Ioo_subset_closure_interior : Ioo a b ⊆ closure (interior (Ioo a b)) | by simp only [interior_Ioo, subset_closure] | lemma | Ioo_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",
"interior",
"interior_Ioo",
"subset_closure"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Iio_mem_nhds {a b : α} (h : a < b) : Iio b ∈ 𝓝 a | is_open.mem_nhds is_open_Iio h | lemma | 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"
] | [
"is_open.mem_nhds",
"is_open_Iio"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Ioi_mem_nhds {a b : α} (h : a < b) : Ioi a ∈ 𝓝 b | is_open.mem_nhds is_open_Ioi h | lemma | 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"
] | [
"is_open.mem_nhds",
"is_open_Ioi"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Iic_mem_nhds {a b : α} (h : a < b) : Iic b ∈ 𝓝 a | mem_of_superset (Iio_mem_nhds h) Iio_subset_Iic_self | lemma | 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"
] | [
"Iio_mem_nhds"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Ici_mem_nhds {a b : α} (h : a < b) : Ici a ∈ 𝓝 b | mem_of_superset (Ioi_mem_nhds h) Ioi_subset_Ici_self | lemma | 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"
] | [
"Ioi_mem_nhds"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Ioo_mem_nhds {a b x : α} (ha : a < x) (hb : x < b) : Ioo a b ∈ 𝓝 x | is_open.mem_nhds is_open_Ioo ⟨ha, hb⟩ | lemma | 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"
] | [
"is_open.mem_nhds",
"is_open_Ioo"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Ioc_mem_nhds {a b x : α} (ha : a < x) (hb : x < b) : Ioc a b ∈ 𝓝 x | mem_of_superset (Ioo_mem_nhds ha hb) Ioo_subset_Ioc_self | lemma | 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"
] | [
"Ioo_mem_nhds"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Ico_mem_nhds {a b x : α} (ha : a < x) (hb : x < b) : Ico a b ∈ 𝓝 x | mem_of_superset (Ioo_mem_nhds ha hb) Ioo_subset_Ico_self | lemma | 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"
] | [
"Ioo_mem_nhds"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Icc_mem_nhds {a b x : α} (ha : a < x) (hb : x < b) : Icc a b ∈ 𝓝 x | mem_of_superset (Ioo_mem_nhds ha hb) Ioo_subset_Icc_self | lemma | 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"
] | [
"Ioo_mem_nhds"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eventually_lt_of_tendsto_lt {l : filter γ} {f : γ → α} {u v : α} (hv : v < u)
(h : filter.tendsto f l (𝓝 v)) : ∀ᶠ a in l, f a < u | tendsto_nhds.1 h (< u) is_open_Iio hv | lemma | eventually_lt_of_tendsto_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",
"filter.tendsto",
"is_open_Iio"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eventually_gt_of_tendsto_gt {l : filter γ} {f : γ → α} {u v : α} (hv : u < v)
(h : filter.tendsto f l (𝓝 v)) : ∀ᶠ a in l, u < f a | tendsto_nhds.1 h (> u) is_open_Ioi hv | lemma | eventually_gt_of_tendsto_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"
] | [
"filter",
"filter.tendsto",
"is_open_Ioi"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eventually_le_of_tendsto_lt {l : filter γ} {f : γ → α} {u v : α} (hv : v < u)
(h : tendsto f l (𝓝 v)) : ∀ᶠ a in l, f a ≤ u | (eventually_lt_of_tendsto_lt hv h).mono (λ v, le_of_lt) | lemma | eventually_le_of_tendsto_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"
] | [
"eventually_lt_of_tendsto_lt",
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eventually_ge_of_tendsto_gt {l : filter γ} {f : γ → α} {u v : α} (hv : u < v)
(h : tendsto f l (𝓝 v)) : ∀ᶠ a in l, u ≤ f a | (eventually_gt_of_tendsto_gt hv h).mono (λ v, le_of_lt) | lemma | eventually_ge_of_tendsto_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"
] | [
"eventually_gt_of_tendsto_gt",
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Ioo_mem_nhds_within_Ioi {a b c : α} (H : b ∈ Ico a c) :
Ioo a c ∈ 𝓝[>] b | mem_nhds_within.2 ⟨Iio c, is_open_Iio, H.2,
by rw [inter_comm, Ioi_inter_Iio]; exact Ioo_subset_Ioo_left H.1⟩ | lemma | Ioo_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"
] | [
"is_open_Iio"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Ioc_mem_nhds_within_Ioi {a b c : α} (H : b ∈ Ico a c) :
Ioc a c ∈ 𝓝[>] b | mem_of_superset (Ioo_mem_nhds_within_Ioi H) Ioo_subset_Ioc_self | lemma | Ioc_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"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Ico_mem_nhds_within_Ioi {a b c : α} (H : b ∈ Ico a c) :
Ico a c ∈ 𝓝[>] b | mem_of_superset (Ioo_mem_nhds_within_Ioi H) Ioo_subset_Ico_self | lemma | Ico_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"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Icc_mem_nhds_within_Ioi {a b c : α} (H : b ∈ Ico a c) :
Icc a c ∈ 𝓝[>] b | mem_of_superset (Ioo_mem_nhds_within_Ioi H) Ioo_subset_Icc_self | lemma | Icc_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"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nhds_within_Ioc_eq_nhds_within_Ioi {a b : α} (h : a < b) :
𝓝[Ioc a b] a = 𝓝[>] a | le_antisymm (nhds_within_mono _ Ioc_subset_Ioi_self) $
nhds_within_le_of_mem $ Ioc_mem_nhds_within_Ioi $ left_mem_Ico.2 h | lemma | nhds_within_Ioc_eq_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"
] | [
"Ioc_mem_nhds_within_Ioi",
"nhds_within_le_of_mem",
"nhds_within_mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nhds_within_Ioo_eq_nhds_within_Ioi {a b : α} (h : a < b) :
𝓝[Ioo a b] a = 𝓝[>] a | le_antisymm (nhds_within_mono _ Ioo_subset_Ioi_self) $
nhds_within_le_of_mem $ Ioo_mem_nhds_within_Ioi $ left_mem_Ico.2 h | lemma | nhds_within_Ioo_eq_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",
"nhds_within_le_of_mem",
"nhds_within_mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_within_at_Ioc_iff_Ioi [topological_space β] {a b : α} {f : α → β} (h : a < b) :
continuous_within_at f (Ioc a b) a ↔ continuous_within_at f (Ioi a) a | by simp only [continuous_within_at, nhds_within_Ioc_eq_nhds_within_Ioi h] | lemma | continuous_within_at_Ioc_iff_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"
] | [
"continuous_within_at",
"nhds_within_Ioc_eq_nhds_within_Ioi",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_within_at_Ioo_iff_Ioi [topological_space β] {a b : α} {f : α → β} (h : a < b) :
continuous_within_at f (Ioo a b) a ↔ continuous_within_at f (Ioi a) a | by simp only [continuous_within_at, nhds_within_Ioo_eq_nhds_within_Ioi h] | lemma | continuous_within_at_Ioo_iff_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"
] | [
"continuous_within_at",
"nhds_within_Ioo_eq_nhds_within_Ioi",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Ioo_mem_nhds_within_Iio {a b c : α} (H : b ∈ Ioc a c) :
Ioo a c ∈ 𝓝[<] b | by simpa only [dual_Ioo] using Ioo_mem_nhds_within_Ioi
(show to_dual b ∈ Ico (to_dual c) (to_dual a), from H.symm) | lemma | Ioo_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"
] | [
"Ioo_mem_nhds_within_Ioi"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Ico_mem_nhds_within_Iio {a b c : α} (H : b ∈ Ioc a c) :
Ico a c ∈ 𝓝[<] b | mem_of_superset (Ioo_mem_nhds_within_Iio H) Ioo_subset_Ico_self | lemma | Ico_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"
] | [
"Ioo_mem_nhds_within_Iio"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Ioc_mem_nhds_within_Iio {a b c : α} (H : b ∈ Ioc a c) :
Ioc a c ∈ 𝓝[<] b | mem_of_superset (Ioo_mem_nhds_within_Iio H) Ioo_subset_Ioc_self | lemma | Ioc_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"
] | [
"Ioo_mem_nhds_within_Iio"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Icc_mem_nhds_within_Iio {a b c : α} (H : b ∈ Ioc a c) :
Icc a c ∈ 𝓝[<] b | mem_of_superset (Ioo_mem_nhds_within_Iio H) Ioo_subset_Icc_self | lemma | Icc_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"
] | [
"Ioo_mem_nhds_within_Iio"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nhds_within_Ico_eq_nhds_within_Iio {a b : α} (h : a < b) :
𝓝[Ico a b] b = 𝓝[<] b | by simpa only [dual_Ioc] using nhds_within_Ioc_eq_nhds_within_Ioi h.dual | lemma | nhds_within_Ico_eq_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"
] | [
"nhds_within_Ioc_eq_nhds_within_Ioi"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nhds_within_Ioo_eq_nhds_within_Iio {a b : α} (h : a < b) :
𝓝[Ioo a b] b = 𝓝[<] b | by simpa only [dual_Ioo] using nhds_within_Ioo_eq_nhds_within_Ioi h.dual | lemma | nhds_within_Ioo_eq_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"
] | [
"nhds_within_Ioo_eq_nhds_within_Ioi"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_within_at_Ico_iff_Iio {a b : α} {f : α → γ} (h : a < b) :
continuous_within_at f (Ico a b) b ↔ continuous_within_at f (Iio b) b | by simp only [continuous_within_at, nhds_within_Ico_eq_nhds_within_Iio h] | lemma | continuous_within_at_Ico_iff_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"
] | [
"continuous_within_at",
"nhds_within_Ico_eq_nhds_within_Iio"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_within_at_Ioo_iff_Iio {a b : α} {f : α → γ} (h : a < b) :
continuous_within_at f (Ioo a b) b ↔ continuous_within_at f (Iio b) b | by simp only [continuous_within_at, nhds_within_Ioo_eq_nhds_within_Iio h] | lemma | continuous_within_at_Ioo_iff_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"
] | [
"continuous_within_at",
"nhds_within_Ioo_eq_nhds_within_Iio"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Ioo_mem_nhds_within_Ici {a b c : α} (H : b ∈ Ioo a c) :
Ioo a c ∈ 𝓝[≥] b | mem_nhds_within_of_mem_nhds $ is_open.mem_nhds is_open_Ioo H | lemma | Ioo_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"
] | [
"is_open.mem_nhds",
"is_open_Ioo",
"mem_nhds_within_of_mem_nhds"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Ioc_mem_nhds_within_Ici {a b c : α} (H : b ∈ Ioo a c) :
Ioc a c ∈ 𝓝[≥] b | mem_of_superset (Ioo_mem_nhds_within_Ici H) Ioo_subset_Ioc_self | lemma | Ioc_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"
] | [
"Ioo_mem_nhds_within_Ici"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Ico_mem_nhds_within_Ici {a b c : α} (H : b ∈ Ico a c) :
Ico a c ∈ 𝓝[≥] b | mem_nhds_within.2 ⟨Iio c, is_open_Iio, H.2,
by simp only [inter_comm, Ici_inter_Iio, Ico_subset_Ico_left H.1]⟩ | lemma | Ico_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"
] | [
"is_open_Iio"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Icc_mem_nhds_within_Ici {a b c : α} (H : b ∈ Ico a c) :
Icc a c ∈ 𝓝[≥] b | mem_of_superset (Ico_mem_nhds_within_Ici H) Ico_subset_Icc_self | lemma | Icc_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"
] | [
"Ico_mem_nhds_within_Ici"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nhds_within_Icc_eq_nhds_within_Ici {a b : α} (h : a < b) :
𝓝[Icc a b] a = 𝓝[≥] a | le_antisymm (nhds_within_mono _ Icc_subset_Ici_self) $
nhds_within_le_of_mem $ Icc_mem_nhds_within_Ici $ left_mem_Ico.2 h | lemma | nhds_within_Icc_eq_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",
"nhds_within_le_of_mem",
"nhds_within_mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nhds_within_Ico_eq_nhds_within_Ici {a b : α} (h : a < b) :
𝓝[Ico a b] a = 𝓝[≥] a | le_antisymm (nhds_within_mono _ (λ x, and.left)) $
nhds_within_le_of_mem $ Ico_mem_nhds_within_Ici $ left_mem_Ico.2 h | lemma | nhds_within_Ico_eq_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"
] | [
"Ico_mem_nhds_within_Ici",
"nhds_within_le_of_mem",
"nhds_within_mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_within_at_Icc_iff_Ici [topological_space β] {a b : α} {f : α → β} (h : a < b) :
continuous_within_at f (Icc a b) a ↔ continuous_within_at f (Ici a) a | by simp only [continuous_within_at, nhds_within_Icc_eq_nhds_within_Ici h] | lemma | continuous_within_at_Icc_iff_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"
] | [
"continuous_within_at",
"nhds_within_Icc_eq_nhds_within_Ici",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_within_at_Ico_iff_Ici [topological_space β] {a b : α} {f : α → β} (h : a < b) :
continuous_within_at f (Ico a b) a ↔ continuous_within_at f (Ici a) a | by simp only [continuous_within_at, nhds_within_Ico_eq_nhds_within_Ici h] | lemma | continuous_within_at_Ico_iff_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"
] | [
"continuous_within_at",
"nhds_within_Ico_eq_nhds_within_Ici",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Ioo_mem_nhds_within_Iic {a b c : α} (H : b ∈ Ioo a c) :
Ioo a c ∈ 𝓝[≤] b | mem_nhds_within_of_mem_nhds $ is_open.mem_nhds is_open_Ioo H | lemma | Ioo_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"
] | [
"is_open.mem_nhds",
"is_open_Ioo",
"mem_nhds_within_of_mem_nhds"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Ico_mem_nhds_within_Iic {a b c : α} (H : b ∈ Ioo a c) :
Ico a c ∈ 𝓝[≤] b | mem_of_superset (Ioo_mem_nhds_within_Iic H) Ioo_subset_Ico_self | lemma | Ico_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"
] | [
"Ioo_mem_nhds_within_Iic"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Ioc_mem_nhds_within_Iic {a b c : α} (H : b ∈ Ioc a c) :
Ioc a c ∈ 𝓝[≤] b | by simpa only [dual_Ico] using Ico_mem_nhds_within_Ici
(show to_dual b ∈ Ico (to_dual c) (to_dual a), from H.symm) | lemma | Ioc_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"
] | [
"Ico_mem_nhds_within_Ici"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Icc_mem_nhds_within_Iic {a b c : α} (H : b ∈ Ioc a c) :
Icc a c ∈ 𝓝[≤] b | mem_of_superset (Ioc_mem_nhds_within_Iic H) Ioc_subset_Icc_self | lemma | Icc_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"
] | [
"Ioc_mem_nhds_within_Iic"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nhds_within_Icc_eq_nhds_within_Iic {a b : α} (h : a < b) :
𝓝[Icc a b] b = 𝓝[≤] b | by simpa only [dual_Icc] using nhds_within_Icc_eq_nhds_within_Ici h.dual | lemma | nhds_within_Icc_eq_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"
] | [
"nhds_within_Icc_eq_nhds_within_Ici"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nhds_within_Ioc_eq_nhds_within_Iic {a b : α} (h : a < b) :
𝓝[Ioc a b] b = 𝓝[≤] b | by simpa only [dual_Ico] using nhds_within_Ico_eq_nhds_within_Ici h.dual | lemma | nhds_within_Ioc_eq_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"
] | [
"nhds_within_Ico_eq_nhds_within_Ici"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_within_at_Icc_iff_Iic [topological_space β] {a b : α} {f : α → β} (h : a < b) :
continuous_within_at f (Icc a b) b ↔ continuous_within_at f (Iic b) b | by simp only [continuous_within_at, nhds_within_Icc_eq_nhds_within_Iic h] | lemma | continuous_within_at_Icc_iff_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"
] | [
"continuous_within_at",
"nhds_within_Icc_eq_nhds_within_Iic",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_within_at_Ioc_iff_Iic [topological_space β] {a b : α} {f : α → β} (h : a < b) :
continuous_within_at f (Ioc a b) b ↔ continuous_within_at f (Iic b) b | by simp only [continuous_within_at, nhds_within_Ioc_eq_nhds_within_Iic h] | lemma | continuous_within_at_Ioc_iff_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"
] | [
"continuous_within_at",
"nhds_within_Ioc_eq_nhds_within_Iic",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lt_subset_interior_le (hf : continuous f) (hg : continuous g) :
{b | f b < g b} ⊆ interior {b | f b ≤ g b} | interior_maximal (λ p, le_of_lt) $ is_open_lt hf hg | lemma | lt_subset_interior_le | topology.order | src/topology/order/basic.lean | [
"data.set.intervals.pi",
"data.set.pointwise.interval",
"order.filter.interval",
"topology.support",
"topology.algebra.order.left_right"
] | [
"continuous",
"interior",
"interior_maximal",
"is_open_lt"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
frontier_le_subset_eq (hf : continuous f) (hg : continuous g) :
frontier {b | f b ≤ g b} ⊆ {b | f b = g b} | begin
rw [frontier_eq_closure_inter_closure, closure_le_eq hf hg],
rintros b ⟨hb₁, hb₂⟩,
refine le_antisymm hb₁ (closure_lt_subset_le hg hf _),
convert hb₂ using 2, simp only [not_le.symm], refl
end | lemma | frontier_le_subset_eq | topology.order | src/topology/order/basic.lean | [
"data.set.intervals.pi",
"data.set.pointwise.interval",
"order.filter.interval",
"topology.support",
"topology.algebra.order.left_right"
] | [
"closure_le_eq",
"closure_lt_subset_le",
"continuous",
"frontier",
"frontier_eq_closure_inter_closure"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
frontier_Iic_subset (a : α) : frontier (Iic a) ⊆ {a} | frontier_le_subset_eq (@continuous_id α _) continuous_const | lemma | frontier_Iic_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"
] | [
"continuous_const",
"continuous_id",
"frontier",
"frontier_le_subset_eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
frontier_Ici_subset (a : α) : frontier (Ici a) ⊆ {a} | @frontier_Iic_subset αᵒᵈ _ _ _ _ | lemma | frontier_Ici_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"
] | [
"frontier",
"frontier_Iic_subset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
frontier_lt_subset_eq (hf : continuous f) (hg : continuous g) :
frontier {b | f b < g b} ⊆ {b | f b = g b} | by rw ← frontier_compl;
convert frontier_le_subset_eq hg hf; simp [ext_iff, eq_comm] | lemma | frontier_lt_subset_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"
] | [
"continuous",
"frontier",
"frontier_compl",
"frontier_le_subset_eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_if_le [topological_space γ] [Π x, decidable (f x ≤ g x)]
{f' g' : β → γ} (hf : continuous f) (hg : continuous g)
(hf' : continuous_on f' {x | f x ≤ g x}) (hg' : continuous_on g' {x | g x ≤ f x})
(hfg : ∀ x, f x = g x → f' x = g' x) :
continuous (λ x, if f x ≤ g x then f' x else g' x) | begin
refine continuous_if (λ a ha, hfg _ (frontier_le_subset_eq hf hg ha)) _ (hg'.mono _),
{ rwa [(is_closed_le hf hg).closure_eq] },
{ simp only [not_le], exact closure_lt_subset_le hg hf }
end | lemma | continuous_if_le | topology.order | src/topology/order/basic.lean | [
"data.set.intervals.pi",
"data.set.pointwise.interval",
"order.filter.interval",
"topology.support",
"topology.algebra.order.left_right"
] | [
"closure_lt_subset_le",
"continuous",
"continuous_if",
"continuous_on",
"frontier_le_subset_eq",
"is_closed_le",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous.if_le [topological_space γ] [Π x, decidable (f x ≤ g x)] {f' g' : β → γ}
(hf' : continuous f') (hg' : continuous g') (hf : continuous f) (hg : continuous g)
(hfg : ∀ x, f x = g x → f' x = g' x) :
continuous (λ x, if f x ≤ g x then f' x else g' x) | continuous_if_le hf hg hf'.continuous_on hg'.continuous_on hfg | lemma | continuous.if_le | topology.order | src/topology/order/basic.lean | [
"data.set.intervals.pi",
"data.set.pointwise.interval",
"order.filter.interval",
"topology.support",
"topology.algebra.order.left_right"
] | [
"continuous",
"continuous_if_le",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto.eventually_lt {l : filter γ} {f g : γ → α} {y z : α}
(hf : tendsto f l (𝓝 y)) (hg : tendsto g l (𝓝 z)) (hyz : y < z) : ∀ᶠ x in l, f x < g x | begin
by_cases h : y ⋖ z,
{ filter_upwards [hf (Iio_mem_nhds hyz), hg (Ioi_mem_nhds hyz)],
rw [h.Iio_eq],
exact λ x hfx hgx, lt_of_le_of_lt hfx hgx },
{ obtain ⟨w, hyw, hwz⟩ := (not_covby_iff hyz).mp h,
filter_upwards [hf (Iio_mem_nhds hyw), hg (Ioi_mem_nhds hwz)],
exact λ x, lt_trans },
end | lemma | tendsto.eventually_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"
] | [
"Iio_mem_nhds",
"Ioi_mem_nhds",
"filter",
"not_covby_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_at.eventually_lt {x₀ : β} (hf : continuous_at f x₀)
(hg : continuous_at g x₀) (hfg : f x₀ < g x₀) : ∀ᶠ x in 𝓝 x₀, f x < g x | tendsto.eventually_lt hf hg hfg | lemma | continuous_at.eventually_lt | topology.order | src/topology/order/basic.lean | [
"data.set.intervals.pi",
"data.set.pointwise.interval",
"order.filter.interval",
"topology.support",
"topology.algebra.order.left_right"
] | [
"continuous_at",
"tendsto.eventually_lt"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous.min (hf : continuous f) (hg : continuous g) :
continuous (λb, min (f b) (g b)) | by { simp only [min_def], exact hf.if_le hg hf hg (λ x, id) } | lemma | continuous.min | 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"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous.max (hf : continuous f) (hg : continuous g) :
continuous (λb, max (f b) (g b)) | @continuous.min αᵒᵈ _ _ _ _ _ _ _ hf hg | lemma | continuous.max | 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",
"continuous.min"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_min : continuous (λ p : α × α, min p.1 p.2) | continuous_fst.min continuous_snd | lemma | continuous_min | 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",
"continuous_snd"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_max : continuous (λ p : α × α, max p.1 p.2) | continuous_fst.max continuous_snd | lemma | continuous_max | 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",
"continuous_snd"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
filter.tendsto.max {b : filter β} {a₁ a₂ : α} (hf : tendsto f b (𝓝 a₁))
(hg : tendsto g b (𝓝 a₂)) :
tendsto (λb, max (f b) (g b)) b (𝓝 (max a₁ a₂)) | (continuous_max.tendsto (a₁, a₂)).comp (hf.prod_mk_nhds hg) | lemma | filter.tendsto.max | 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"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
filter.tendsto.min {b : filter β} {a₁ a₂ : α} (hf : tendsto f b (𝓝 a₁))
(hg : tendsto g b (𝓝 a₂)) :
tendsto (λb, min (f b) (g b)) b (𝓝 (min a₁ a₂)) | (continuous_min.tendsto (a₁, a₂)).comp (hf.prod_mk_nhds hg) | lemma | filter.tendsto.min | 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"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
filter.tendsto.max_right {l : filter β} {a : α} (h : tendsto f l (𝓝 a)) :
tendsto (λ i, max a (f i)) l (𝓝 a) | by { convert ((continuous_max.comp (@continuous.prod.mk α α _ _ a)).tendsto a).comp h, simp, } | lemma | filter.tendsto.max_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"
] | [
"continuous.prod.mk",
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
filter.tendsto.max_left {l : filter β} {a : α} (h : tendsto f l (𝓝 a)) :
tendsto (λ i, max (f i) a) l (𝓝 a) | by { simp_rw max_comm _ a, exact h.max_right, } | lemma | filter.tendsto.max_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"
] | [
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
filter.tendsto_nhds_max_right {l : filter β} {a : α} (h : tendsto f l (𝓝[>] a)) :
tendsto (λ i, max a (f i)) l (𝓝[>] a) | begin
obtain ⟨h₁ : tendsto f l (𝓝 a), h₂ : ∀ᶠ i in l, f i ∈ Ioi a⟩ := tendsto_nhds_within_iff.mp h,
exact tendsto_nhds_within_iff.mpr ⟨h₁.max_right, h₂.mono $ λ i hi, lt_max_of_lt_right hi⟩,
end | lemma | filter.tendsto_nhds_max_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"
] | [
"filter",
"lt_max_of_lt_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
filter.tendsto_nhds_max_left {l : filter β} {a : α} (h : tendsto f l (𝓝[>] a)) :
tendsto (λ i, max (f i) a) l (𝓝[>] a) | by { simp_rw max_comm _ a, exact filter.tendsto_nhds_max_right h, } | lemma | filter.tendsto_nhds_max_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"
] | [
"filter",
"filter.tendsto_nhds_max_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
filter.tendsto.min_right {l : filter β} {a : α} (h : tendsto f l (𝓝 a)) :
tendsto (λ i, min a (f i)) l (𝓝 a) | @filter.tendsto.max_right αᵒᵈ β _ _ _ f l a h | lemma | filter.tendsto.min_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"
] | [
"filter",
"filter.tendsto.max_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
filter.tendsto.min_left {l : filter β} {a : α} (h : tendsto f l (𝓝 a)) :
tendsto (λ i, min (f i) a) l (𝓝 a) | @filter.tendsto.max_left αᵒᵈ β _ _ _ f l a h | lemma | filter.tendsto.min_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"
] | [
"filter",
"filter.tendsto.max_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
filter.tendsto_nhds_min_right {l : filter β} {a : α} (h : tendsto f l (𝓝[<] a)) :
tendsto (λ i, min a (f i)) l (𝓝[<] a) | @filter.tendsto_nhds_max_right αᵒᵈ β _ _ _ f l a h | lemma | filter.tendsto_nhds_min_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"
] | [
"filter",
"filter.tendsto_nhds_max_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
filter.tendsto_nhds_min_left {l : filter β} {a : α} (h : tendsto f l (𝓝[<] a)) :
tendsto (λ i, min (f i) a) l (𝓝[<] a) | @filter.tendsto_nhds_max_left αᵒᵈ β _ _ _ f l a h | lemma | filter.tendsto_nhds_min_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"
] | [
"filter",
"filter.tendsto_nhds_max_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dense.exists_lt [no_min_order α] {s : set α} (hs : dense s) (x : α) : ∃ y ∈ s, y < x | hs.exists_mem_open is_open_Iio (exists_lt x) | lemma | dense.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"
] | [
"dense",
"is_open_Iio",
"no_min_order"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dense.exists_gt [no_max_order α] {s : set α} (hs : dense s) (x : α) : ∃ y ∈ s, x < y | hs.order_dual.exists_lt x | lemma | dense.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"
] | [
"dense",
"no_max_order"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dense.exists_le [no_min_order α] {s : set α} (hs : dense s) (x : α) : ∃ y ∈ s, y ≤ x | (hs.exists_lt x).imp $ λ y hy, ⟨hy.fst, hy.snd.le⟩ | lemma | dense.exists_le | topology.order | src/topology/order/basic.lean | [
"data.set.intervals.pi",
"data.set.pointwise.interval",
"order.filter.interval",
"topology.support",
"topology.algebra.order.left_right"
] | [
"dense",
"no_min_order"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dense.exists_ge [no_max_order α] {s : set α} (hs : dense s) (x : α) : ∃ y ∈ s, x ≤ y | hs.order_dual.exists_le x | lemma | dense.exists_ge | topology.order | src/topology/order/basic.lean | [
"data.set.intervals.pi",
"data.set.pointwise.interval",
"order.filter.interval",
"topology.support",
"topology.algebra.order.left_right"
] | [
"dense",
"no_max_order"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dense.exists_le' {s : set α} (hs : dense s) (hbot : ∀ x, is_bot x → x ∈ s) (x : α) :
∃ y ∈ s, y ≤ x | begin
by_cases hx : is_bot x,
{ exact ⟨x, hbot x hx, le_rfl⟩ },
{ simp only [is_bot, not_forall, not_le] at hx,
rcases hs.exists_mem_open is_open_Iio hx with ⟨y, hys, hy : y < x⟩,
exact ⟨y, hys, hy.le⟩ }
end | lemma | dense.exists_le' | topology.order | src/topology/order/basic.lean | [
"data.set.intervals.pi",
"data.set.pointwise.interval",
"order.filter.interval",
"topology.support",
"topology.algebra.order.left_right"
] | [
"dense",
"is_bot",
"is_open_Iio",
"not_forall"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dense.exists_ge' {s : set α} (hs : dense s) (htop : ∀ x, is_top x → x ∈ s) (x : α) :
∃ y ∈ s, x ≤ y | hs.order_dual.exists_le' htop x | lemma | dense.exists_ge' | topology.order | src/topology/order/basic.lean | [
"data.set.intervals.pi",
"data.set.pointwise.interval",
"order.filter.interval",
"topology.support",
"topology.algebra.order.left_right"
] | [
"dense",
"is_top"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dense.exists_between [densely_ordered α] {s : set α} (hs : dense s) {x y : α} (h : x < y) :
∃ z ∈ s, z ∈ Ioo x y | hs.exists_mem_open is_open_Ioo (nonempty_Ioo.2 h) | lemma | dense.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",
"is_open_Ioo"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pi.order_closed_topology' [preorder β] [topological_space β]
[order_closed_topology β] : order_closed_topology (α → β) | pi.order_closed_topology | instance | pi.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"
] | [
"order_closed_topology",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
order_topology (α : Type*) [t : topological_space α] [preorder α] : Prop | (topology_eq_generate_intervals : t = generate_from {s | ∃ a, s = Ioi a ∨ s = Iio a}) | class | 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"
] | [
"topological_space"
] | The order topology on an ordered type is the topology generated by open intervals. We register
it on a preorder, but it is mostly interesting in linear orders, where it is also order-closed.
We define it as a mixin. If you want to introduce the order topology on a preorder, use
`preorder.topology`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
preorder.topology (α : Type*) [preorder α] : topological_space α | generate_from {s : set α | ∃ (a : α), s = {b : α | a < b} ∨ s = {b : α | b < a}} | def | preorder.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"
] | [
"topological_space"
] | (Order) topology on a partial order `α` generated by the subbase of open intervals
`(a, ∞) = { x ∣ a < x }, (-∞ , b) = {x ∣ x < b}` for all `a, b` in `α`. We do not register it as an
instance as many ordered sets are already endowed with the same topology, most often in a non-defeq
way though. Register as a local insta... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_open_iff_generate_intervals {s : set α} :
is_open s ↔ generate_open {s | ∃ a, s = Ioi a ∨ s = Iio a} s | by rw [t.topology_eq_generate_intervals]; refl | lemma | is_open_iff_generate_intervals | 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"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_open_lt' (a : α) : is_open {b : α | a < b} | by rw [@is_open_iff_generate_intervals α _ _ t]; exact generate_open.basic _ ⟨a, or.inl rfl⟩ | lemma | is_open_lt' | topology.order | src/topology/order/basic.lean | [
"data.set.intervals.pi",
"data.set.pointwise.interval",
"order.filter.interval",
"topology.support",
"topology.algebra.order.left_right"
] | [
"is_open",
"is_open_iff_generate_intervals"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_open_gt' (a : α) : is_open {b : α | b < a} | by rw [@is_open_iff_generate_intervals α _ _ t]; exact generate_open.basic _ ⟨a, or.inr rfl⟩ | lemma | is_open_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"
] | [
"is_open",
"is_open_iff_generate_intervals"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lt_mem_nhds {a b : α} (h : a < b) : ∀ᶠ x in 𝓝 b, a < x | is_open.mem_nhds (is_open_lt' _) h | lemma | lt_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"
] | [
"is_open.mem_nhds",
"is_open_lt'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_mem_nhds {a b : α} (h : a < b) : ∀ᶠ x in 𝓝 b, a ≤ x | (𝓝 b).sets_of_superset (lt_mem_nhds h) $ assume b hb, le_of_lt hb | lemma | le_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"
] | [
"lt_mem_nhds"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
gt_mem_nhds {a b : α} (h : a < b) : ∀ᶠ x in 𝓝 a, x < b | is_open.mem_nhds (is_open_gt' _) h | lemma | gt_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"
] | [
"is_open.mem_nhds",
"is_open_gt'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ge_mem_nhds {a b : α} (h : a < b) : ∀ᶠ x in 𝓝 a, x ≤ b | (𝓝 a).sets_of_superset (gt_mem_nhds h) $ assume b hb, le_of_lt hb | lemma | ge_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"
] | [
"gt_mem_nhds"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nhds_eq_order (a : α) :
𝓝 a = (⨅ b ∈ Iio a, 𝓟 (Ioi b)) ⊓ (⨅ b ∈ Ioi a, 𝓟 (Iio b)) | by rw [t.topology_eq_generate_intervals, nhds_generate_from];
from le_antisymm
(le_inf
(le_infi₂ $ assume b hb, infi_le_of_le {c : α | b < c} $ infi_le _ ⟨hb, b, or.inl rfl⟩)
(le_infi₂ $ assume b hb, infi_le_of_le {c : α | c < b} $ infi_le _ ⟨hb, b, or.inr rfl⟩))
(le_infi $ assume s, le_infi $ assume ⟨ha, b... | lemma | nhds_eq_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"
] | [
"inf_le_of_left_le",
"inf_le_of_right_le",
"infi_le",
"infi_le_of_le",
"le_inf",
"le_infi",
"le_infi₂"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_order {f : β → α} {a : α} {x : filter β} :
tendsto f x (𝓝 a) ↔ (∀ a' < a, ∀ᶠ b in x, a' < f b) ∧ (∀ a' > a, ∀ᶠ b in x, f b < a') | by simp [nhds_eq_order a, tendsto_inf, tendsto_infi, tendsto_principal] | lemma | tendsto_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"
] | [
"filter",
"nhds_eq_order"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_Icc_class_nhds (a : α) : tendsto_Ixx_class Icc (𝓝 a) (𝓝 a) | begin
simp only [nhds_eq_order, infi_subtype'],
refine ((has_basis_infi_principal_finite _).inf
(has_basis_infi_principal_finite _)).tendsto_Ixx_class (λ s hs, _),
refine ((ord_connected_bInter _).inter (ord_connected_bInter _)).out; intros _ _,
exacts [ord_connected_Ioi, ord_connected_Iio]
end | instance | tendsto_Icc_class_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"
] | [
"infi_subtype'",
"nhds_eq_order"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_Ico_class_nhds (a : α) : tendsto_Ixx_class Ico (𝓝 a) (𝓝 a) | tendsto_Ixx_class_of_subset (λ _ _, Ico_subset_Icc_self) | instance | tendsto_Ico_class_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"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_Ioc_class_nhds (a : α) : tendsto_Ixx_class Ioc (𝓝 a) (𝓝 a) | tendsto_Ixx_class_of_subset (λ _ _, Ioc_subset_Icc_self) | instance | tendsto_Ioc_class_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"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_Ioo_class_nhds (a : α) : tendsto_Ixx_class Ioo (𝓝 a) (𝓝 a) | tendsto_Ixx_class_of_subset (λ _ _, Ioo_subset_Icc_self) | instance | tendsto_Ioo_class_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"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_of_tendsto_of_tendsto_of_le_of_le' {f g h : β → α} {b : filter β} {a : α}
(hg : tendsto g b (𝓝 a)) (hh : tendsto h b (𝓝 a))
(hgf : ∀ᶠ b in b, g b ≤ f b) (hfh : ∀ᶠ b in b, f b ≤ h b) :
tendsto f b (𝓝 a) | (hg.Icc hh).of_small_sets $ hgf.and hfh | lemma | tendsto_of_tendsto_of_tendsto_of_le_of_le' | topology.order | src/topology/order/basic.lean | [
"data.set.intervals.pi",
"data.set.pointwise.interval",
"order.filter.interval",
"topology.support",
"topology.algebra.order.left_right"
] | [
"filter"
] | **Squeeze theorem** (also known as **sandwich theorem**). This version assumes that inequalities
hold eventually for the filter. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
tendsto_of_tendsto_of_tendsto_of_le_of_le {f g h : β → α} {b : filter β} {a : α}
(hg : tendsto g b (𝓝 a)) (hh : tendsto h b (𝓝 a)) (hgf : g ≤ f) (hfh : f ≤ h) :
tendsto f b (𝓝 a) | tendsto_of_tendsto_of_tendsto_of_le_of_le' hg hh
(eventually_of_forall hgf) (eventually_of_forall hfh) | lemma | tendsto_of_tendsto_of_tendsto_of_le_of_le | topology.order | src/topology/order/basic.lean | [
"data.set.intervals.pi",
"data.set.pointwise.interval",
"order.filter.interval",
"topology.support",
"topology.algebra.order.left_right"
] | [
"filter",
"tendsto_of_tendsto_of_tendsto_of_le_of_le'"
] | **Squeeze theorem** (also known as **sandwich theorem**). This version assumes that inequalities
hold everywhere. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
nhds_order_unbounded {a : α} (hu : ∃u, a < u) (hl : ∃l, l < a) :
𝓝 a = (⨅l (h₂ : l < a) u (h₂ : a < u), 𝓟 (Ioo l u)) | have ∃ u, u ∈ Ioi a, from hu, have ∃ l, l ∈ Iio a, from hl,
by { simp only [nhds_eq_order, inf_binfi, binfi_inf, *, inf_principal, Ioi_inter_Iio], refl } | lemma | nhds_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"
] | [
"binfi_inf",
"inf_binfi",
"nhds_eq_order"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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