statement stringlengths 1 2.93k | proof stringlengths 0 19.2k | type stringclasses 13
values | symbolic_name stringlengths 1 131 | library stringlengths 4 62 | filename stringlengths 20 95 | imports listlengths 0 10 | deps listlengths 0 64 | docstring stringlengths 0 4.95k | source_url stringclasses 1
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ContinuousAt.finset_sup' (hne : s.Nonempty) (hs : ∀ i ∈ s, ContinuousAt (f i) x) :
ContinuousAt (s.sup' hne f) x | by
simpa only [← Finset.sup'_apply] using finset_sup'_apply hne hs | lemma | ContinuousAt.finset_sup' | Topology.Order | Mathlib/Topology/Order/Lattice.lean | [] | [
"ContinuousAt",
"Finset.sup'_apply"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
ContinuousWithinAt.finset_sup'_apply (hne : s.Nonempty)
(hs : ∀ i ∈ s, ContinuousWithinAt (f i) t x) :
ContinuousWithinAt (fun a ↦ s.sup' hne (f · a)) t x | Tendsto.finset_sup'_nhds_apply hne hs | lemma | ContinuousWithinAt.finset_sup'_apply | Topology.Order | Mathlib/Topology/Order/Lattice.lean | [] | [
"ContinuousWithinAt"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
ContinuousWithinAt.finset_sup' (hne : s.Nonempty)
(hs : ∀ i ∈ s, ContinuousWithinAt (f i) t x) : ContinuousWithinAt (s.sup' hne f) t x | by
simpa only [← Finset.sup'_apply] using finset_sup'_apply hne hs | lemma | ContinuousWithinAt.finset_sup' | Topology.Order | Mathlib/Topology/Order/Lattice.lean | [] | [
"ContinuousWithinAt",
"Finset.sup'_apply"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
ContinuousOn.finset_sup'_apply (hne : s.Nonempty) (hs : ∀ i ∈ s, ContinuousOn (f i) t) :
ContinuousOn (fun a ↦ s.sup' hne (f · a)) t | fun x hx ↦
ContinuousWithinAt.finset_sup'_apply hne fun i hi ↦ hs i hi x hx | lemma | ContinuousOn.finset_sup'_apply | Topology.Order | Mathlib/Topology/Order/Lattice.lean | [] | [
"ContinuousOn",
"ContinuousWithinAt.finset_sup'_apply"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
ContinuousOn.finset_sup' (hne : s.Nonempty) (hs : ∀ i ∈ s, ContinuousOn (f i) t) :
ContinuousOn (s.sup' hne f) t | fun x hx ↦
ContinuousWithinAt.finset_sup' hne fun i hi ↦ hs i hi x hx | lemma | ContinuousOn.finset_sup' | Topology.Order | Mathlib/Topology/Order/Lattice.lean | [] | [
"ContinuousOn",
"ContinuousWithinAt.finset_sup'"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Continuous.finset_sup'_apply (hne : s.Nonempty) (hs : ∀ i ∈ s, Continuous (f i)) :
Continuous (fun a ↦ s.sup' hne (f · a)) | continuous_iff_continuousAt.2 fun _ ↦ ContinuousAt.finset_sup'_apply _ fun i hi ↦
(hs i hi).continuousAt | lemma | Continuous.finset_sup'_apply | Topology.Order | Mathlib/Topology/Order/Lattice.lean | [] | [
"Continuous",
"ContinuousAt.finset_sup'_apply"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Continuous.finset_sup' (hne : s.Nonempty) (hs : ∀ i ∈ s, Continuous (f i)) :
Continuous (s.sup' hne f) | continuous_iff_continuousAt.2 fun _ ↦ ContinuousAt.finset_sup' _ fun i hi ↦ (hs i hi).continuousAt | lemma | Continuous.finset_sup' | Topology.Order | Mathlib/Topology/Order/Lattice.lean | [] | [
"Continuous",
"ContinuousAt.finset_sup'"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
ContinuousAt.finset_sup_apply (hs : ∀ i ∈ s, ContinuousAt (f i) x) :
ContinuousAt (fun a ↦ s.sup (f · a)) x | Tendsto.finset_sup_nhds_apply hs | lemma | ContinuousAt.finset_sup_apply | Topology.Order | Mathlib/Topology/Order/Lattice.lean | [] | [
"ContinuousAt"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
ContinuousAt.finset_sup (hs : ∀ i ∈ s, ContinuousAt (f i) x) :
ContinuousAt (s.sup f) x | by
simpa only [← Finset.sup_apply] using finset_sup_apply hs | lemma | ContinuousAt.finset_sup | Topology.Order | Mathlib/Topology/Order/Lattice.lean | [] | [
"ContinuousAt",
"Finset.sup_apply"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
ContinuousWithinAt.finset_sup_apply
(hs : ∀ i ∈ s, ContinuousWithinAt (f i) t x) :
ContinuousWithinAt (fun a ↦ s.sup (f · a)) t x | Tendsto.finset_sup_nhds_apply hs | lemma | ContinuousWithinAt.finset_sup_apply | Topology.Order | Mathlib/Topology/Order/Lattice.lean | [] | [
"ContinuousWithinAt"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
ContinuousWithinAt.finset_sup
(hs : ∀ i ∈ s, ContinuousWithinAt (f i) t x) : ContinuousWithinAt (s.sup f) t x | by
simpa only [← Finset.sup_apply] using finset_sup_apply hs | lemma | ContinuousWithinAt.finset_sup | Topology.Order | Mathlib/Topology/Order/Lattice.lean | [] | [
"ContinuousWithinAt",
"Finset.sup_apply"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
ContinuousOn.finset_sup_apply (hs : ∀ i ∈ s, ContinuousOn (f i) t) :
ContinuousOn (fun a ↦ s.sup (f · a)) t | fun x hx ↦
ContinuousWithinAt.finset_sup_apply fun i hi ↦ hs i hi x hx | lemma | ContinuousOn.finset_sup_apply | Topology.Order | Mathlib/Topology/Order/Lattice.lean | [] | [
"ContinuousOn",
"ContinuousWithinAt.finset_sup_apply"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
ContinuousOn.finset_sup (hs : ∀ i ∈ s, ContinuousOn (f i) t) :
ContinuousOn (s.sup f) t | fun x hx ↦
ContinuousWithinAt.finset_sup fun i hi ↦ hs i hi x hx | lemma | ContinuousOn.finset_sup | Topology.Order | Mathlib/Topology/Order/Lattice.lean | [] | [
"ContinuousOn",
"ContinuousWithinAt.finset_sup"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Continuous.finset_sup_apply (hs : ∀ i ∈ s, Continuous (f i)) :
Continuous (fun a ↦ s.sup (f · a)) | continuous_iff_continuousAt.2 fun _ ↦ ContinuousAt.finset_sup_apply fun i hi ↦
(hs i hi).continuousAt | lemma | Continuous.finset_sup_apply | Topology.Order | Mathlib/Topology/Order/Lattice.lean | [] | [
"Continuous",
"ContinuousAt.finset_sup_apply"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Continuous.finset_sup (hs : ∀ i ∈ s, Continuous (f i)) : Continuous (s.sup f) | continuous_iff_continuousAt.2 fun _ ↦ ContinuousAt.finset_sup fun i hi ↦ (hs i hi).continuousAt | lemma | Continuous.finset_sup | Topology.Order | Mathlib/Topology/Order/Lattice.lean | [] | [
"Continuous",
"ContinuousAt.finset_sup"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
ContinuousAt.finset_inf'_apply (hne : s.Nonempty) (hs : ∀ i ∈ s, ContinuousAt (f i) x) :
ContinuousAt (fun a ↦ s.inf' hne (f · a)) x | Tendsto.finset_inf'_nhds_apply hne hs | lemma | ContinuousAt.finset_inf'_apply | Topology.Order | Mathlib/Topology/Order/Lattice.lean | [] | [
"ContinuousAt"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
ContinuousAt.finset_inf' (hne : s.Nonempty) (hs : ∀ i ∈ s, ContinuousAt (f i) x) :
ContinuousAt (s.inf' hne f) x | by
simpa only [← Finset.inf'_apply] using finset_inf'_apply hne hs | lemma | ContinuousAt.finset_inf' | Topology.Order | Mathlib/Topology/Order/Lattice.lean | [] | [
"ContinuousAt"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
ContinuousWithinAt.finset_inf'_apply (hne : s.Nonempty)
(hs : ∀ i ∈ s, ContinuousWithinAt (f i) t x) :
ContinuousWithinAt (fun a ↦ s.inf' hne (f · a)) t x | Tendsto.finset_inf'_nhds_apply hne hs | lemma | ContinuousWithinAt.finset_inf'_apply | Topology.Order | Mathlib/Topology/Order/Lattice.lean | [] | [
"ContinuousWithinAt"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
ContinuousWithinAt.finset_inf' (hne : s.Nonempty)
(hs : ∀ i ∈ s, ContinuousWithinAt (f i) t x) : ContinuousWithinAt (s.inf' hne f) t x | by
simpa only [← Finset.inf'_apply] using finset_inf'_apply hne hs | lemma | ContinuousWithinAt.finset_inf' | Topology.Order | Mathlib/Topology/Order/Lattice.lean | [] | [
"ContinuousWithinAt"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
ContinuousOn.finset_inf'_apply (hne : s.Nonempty) (hs : ∀ i ∈ s, ContinuousOn (f i) t) :
ContinuousOn (fun a ↦ s.inf' hne (f · a)) t | fun x hx ↦
ContinuousWithinAt.finset_inf'_apply hne fun i hi ↦ hs i hi x hx | lemma | ContinuousOn.finset_inf'_apply | Topology.Order | Mathlib/Topology/Order/Lattice.lean | [] | [
"ContinuousOn",
"ContinuousWithinAt.finset_inf'_apply"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
ContinuousOn.finset_inf' (hne : s.Nonempty) (hs : ∀ i ∈ s, ContinuousOn (f i) t) :
ContinuousOn (s.inf' hne f) t | fun x hx ↦
ContinuousWithinAt.finset_inf' hne fun i hi ↦ hs i hi x hx | lemma | ContinuousOn.finset_inf' | Topology.Order | Mathlib/Topology/Order/Lattice.lean | [] | [
"ContinuousOn",
"ContinuousWithinAt.finset_inf'"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Continuous.finset_inf'_apply (hne : s.Nonempty) (hs : ∀ i ∈ s, Continuous (f i)) :
Continuous (fun a ↦ s.inf' hne (f · a)) | continuous_iff_continuousAt.2 fun _ ↦ ContinuousAt.finset_inf'_apply _ fun i hi ↦
(hs i hi).continuousAt | lemma | Continuous.finset_inf'_apply | Topology.Order | Mathlib/Topology/Order/Lattice.lean | [] | [
"Continuous",
"ContinuousAt.finset_inf'_apply"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Continuous.finset_inf' (hne : s.Nonempty) (hs : ∀ i ∈ s, Continuous (f i)) :
Continuous (s.inf' hne f) | continuous_iff_continuousAt.2 fun _ ↦ ContinuousAt.finset_inf' _ fun i hi ↦ (hs i hi).continuousAt | lemma | Continuous.finset_inf' | Topology.Order | Mathlib/Topology/Order/Lattice.lean | [] | [
"Continuous",
"ContinuousAt.finset_inf'"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
ContinuousAt.finset_inf_apply (hs : ∀ i ∈ s, ContinuousAt (f i) x) :
ContinuousAt (fun a ↦ s.inf (f · a)) x | Tendsto.finset_inf_nhds_apply hs | lemma | ContinuousAt.finset_inf_apply | Topology.Order | Mathlib/Topology/Order/Lattice.lean | [] | [
"ContinuousAt"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
ContinuousAt.finset_inf (hs : ∀ i ∈ s, ContinuousAt (f i) x) :
ContinuousAt (s.inf f) x | by
simpa only [← Finset.inf_apply] using finset_inf_apply hs | lemma | ContinuousAt.finset_inf | Topology.Order | Mathlib/Topology/Order/Lattice.lean | [] | [
"ContinuousAt"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
ContinuousWithinAt.finset_inf_apply
(hs : ∀ i ∈ s, ContinuousWithinAt (f i) t x) :
ContinuousWithinAt (fun a ↦ s.inf (f · a)) t x | Tendsto.finset_inf_nhds_apply hs | lemma | ContinuousWithinAt.finset_inf_apply | Topology.Order | Mathlib/Topology/Order/Lattice.lean | [] | [
"ContinuousWithinAt"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
ContinuousWithinAt.finset_inf
(hs : ∀ i ∈ s, ContinuousWithinAt (f i) t x) : ContinuousWithinAt (s.inf f) t x | by
simpa only [← Finset.inf_apply] using finset_inf_apply hs | lemma | ContinuousWithinAt.finset_inf | Topology.Order | Mathlib/Topology/Order/Lattice.lean | [] | [
"ContinuousWithinAt"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
ContinuousOn.finset_inf_apply (hs : ∀ i ∈ s, ContinuousOn (f i) t) :
ContinuousOn (fun a ↦ s.inf (f · a)) t | fun x hx ↦
ContinuousWithinAt.finset_inf_apply fun i hi ↦ hs i hi x hx | lemma | ContinuousOn.finset_inf_apply | Topology.Order | Mathlib/Topology/Order/Lattice.lean | [] | [
"ContinuousOn",
"ContinuousWithinAt.finset_inf_apply"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
ContinuousOn.finset_inf (hs : ∀ i ∈ s, ContinuousOn (f i) t) :
ContinuousOn (s.inf f) t | fun x hx ↦
ContinuousWithinAt.finset_inf fun i hi ↦ hs i hi x hx | lemma | ContinuousOn.finset_inf | Topology.Order | Mathlib/Topology/Order/Lattice.lean | [] | [
"ContinuousOn",
"ContinuousWithinAt.finset_inf"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Continuous.finset_inf_apply (hs : ∀ i ∈ s, Continuous (f i)) :
Continuous (fun a ↦ s.inf (f · a)) | continuous_iff_continuousAt.2 fun _ ↦ ContinuousAt.finset_inf_apply fun i hi ↦
(hs i hi).continuousAt | lemma | Continuous.finset_inf_apply | Topology.Order | Mathlib/Topology/Order/Lattice.lean | [] | [
"Continuous",
"ContinuousAt.finset_inf_apply"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Continuous.finset_inf (hs : ∀ i ∈ s, Continuous (f i)) : Continuous (s.inf f) | continuous_iff_continuousAt.2 fun _ ↦ ContinuousAt.finset_inf fun i hi ↦ (hs i hi).continuousAt | lemma | Continuous.finset_inf | Topology.Order | Mathlib/Topology/Order/Lattice.lean | [] | [
"Continuous",
"ContinuousAt.finset_inf"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
lawson (α : Type*) [Preorder α] : TopologicalSpace α | lower α ⊓ scott α univ | def | Topology.lawson | Topology.Order | Mathlib/Topology/Order/LawsonTopology.lean | [] | [
"Preorder",
"TopologicalSpace"
] | The Lawson topology is defined as the meet of `Topology.lower` and the `Topology.scott`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
IsLawson : Prop where
topology_eq_lawson : ‹TopologicalSpace α› = lawson α | class | Topology.IsLawson | Topology.Order | Mathlib/Topology/Order/LawsonTopology.lean | [] | [] | Predicate for an ordered topological space to be equipped with its Lawson topology.
The Lawson topology is defined as the meet of `Topology.lower` and the `Topology.scott`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
lawsonBasis | { s : Set α | ∃ t : Set α, t.Finite ∧ ∃ u : Set α, IsOpen[scott α univ] u ∧
u \ upperClosure t = s } | def | Topology.IsLawson.lawsonBasis | Topology.Order | Mathlib/Topology/Order/LawsonTopology.lean | [] | [
"IsOpen",
"Set",
"upperClosure"
] | The complements of the upper closures of finite sets intersected with Scott open sets form
a basis for the lawson topology. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
isTopologicalBasis : TopologicalSpace.IsTopologicalBasis (lawsonBasis α) | by
have lawsonBasis_image2 : lawsonBasis α =
(image2 (fun x x_1 ↦ ⇑WithLower.toLower ⁻¹' x ∩ ⇑WithScott.toScott ⁻¹' x_1)
(IsLower.lowerBasis (WithLower α)) {U | IsOpen[scott α univ] U}) := by
rw [lawsonBasis, image2, IsLower.lowerBasis]
simp_rw [diff_eq_compl_inter]
aesop
rw [lawsonBasis_i... | theorem | Topology.IsLawson.isTopologicalBasis | Topology.Order | Mathlib/Topology/Order/LawsonTopology.lean | [] | [
"IsOpen",
"TopologicalSpace.IsTopologicalBasis"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
WithLawson (α : Type*) | α | def | Topology.WithLawson | Topology.Order | Mathlib/Topology/Order/LawsonTopology.lean | [] | [] | Type synonym for a preorder equipped with the Lawson topology. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
toLawson : α ≃ WithLawson α | Equiv.refl _ | def | Topology.WithLawson.toLawson | Topology.Order | Mathlib/Topology/Order/LawsonTopology.lean | [] | [
"Equiv.refl"
] | `toLawson` is the identity function to the `WithLawson` of a type. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
ofLawson : WithLawson α ≃ α | Equiv.refl _ | def | Topology.WithLawson.ofLawson | Topology.Order | Mathlib/Topology/Order/LawsonTopology.lean | [] | [
"Equiv.refl"
] | `ofLawson` is the identity function from the `WithLawson` of a type. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
to_Lawson_symm_eq : (@toLawson α).symm = ofLawson | rfl | lemma | Topology.WithLawson.to_Lawson_symm_eq | Topology.Order | Mathlib/Topology/Order/LawsonTopology.lean | [] | [
"symm"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
of_Lawson_symm_eq : (@ofLawson α).symm = toLawson | rfl | lemma | Topology.WithLawson.of_Lawson_symm_eq | Topology.Order | Mathlib/Topology/Order/LawsonTopology.lean | [] | [
"symm"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
toLawson_ofLawson (a : WithLawson α) : toLawson (ofLawson a) = a | rfl | lemma | Topology.WithLawson.toLawson_ofLawson | Topology.Order | Mathlib/Topology/Order/LawsonTopology.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
ofLawson_toLawson (a : α) : ofLawson (toLawson a) = a | rfl | lemma | Topology.WithLawson.ofLawson_toLawson | Topology.Order | Mathlib/Topology/Order/LawsonTopology.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
toLawson_inj {a b : α} : toLawson a = toLawson b ↔ a = b | Iff.rfl | lemma | Topology.WithLawson.toLawson_inj | Topology.Order | Mathlib/Topology/Order/LawsonTopology.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
ofLawson_inj {a b : WithLawson α} : ofLawson a = ofLawson b ↔ a = b | Iff.rfl | lemma | Topology.WithLawson.ofLawson_inj | Topology.Order | Mathlib/Topology/Order/LawsonTopology.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
rec {β : WithLawson α → Sort*}
(h : ∀ a, β (toLawson a)) : ∀ a, β a | fun a => h (ofLawson a) | def | Topology.WithLawson.rec | Topology.Order | Mathlib/Topology/Order/LawsonTopology.lean | [] | [] | A recursor for `WithLawson`. Use as `induction x`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
instPreorder : Preorder (WithLawson α) | ‹Preorder α› | instance | Topology.WithLawson.instPreorder | Topology.Order | Mathlib/Topology/Order/LawsonTopology.lean | [] | [
"Preorder"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
instTopologicalSpace : TopologicalSpace (WithLawson α) | -- fast_instance% lawson α fails
letI : TopologicalSpace α := lawson α
inferInstanceAs <| TopologicalSpace α | instance | Topology.WithLawson.instTopologicalSpace | Topology.Order | Mathlib/Topology/Order/LawsonTopology.lean | [] | [
"TopologicalSpace"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
instIsLawson : IsLawson (WithLawson α) | ⟨rfl⟩ | instance | Topology.WithLawson.instIsLawson | Topology.Order | Mathlib/Topology/Order/LawsonTopology.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
homeomorph [TopologicalSpace α] [IsLawson α] : WithLawson α ≃ₜ α | ofLawson.toHomeomorphOfIsInducing ⟨IsLawson.topology_eq_lawson (α := α) ▸ induced_id.symm⟩ | def | Topology.WithLawson.homeomorph | Topology.Order | Mathlib/Topology/Order/LawsonTopology.lean | [] | [
"TopologicalSpace"
] | If `α` is equipped with the Lawson topology, then it is homeomorphic to `WithLawson α`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
isOpen_preimage_ofLawson {S : Set α} :
IsOpen (ofLawson ⁻¹' S) ↔ (lawson α).IsOpen S | Iff.rfl | theorem | Topology.WithLawson.isOpen_preimage_ofLawson | Topology.Order | Mathlib/Topology/Order/LawsonTopology.lean | [] | [
"IsOpen",
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
isClosed_preimage_ofLawson {S : Set α} :
IsClosed (ofLawson ⁻¹' S) ↔ IsClosed[lawson α] S | Iff.rfl | theorem | Topology.WithLawson.isClosed_preimage_ofLawson | Topology.Order | Mathlib/Topology/Order/LawsonTopology.lean | [] | [
"IsClosed",
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
isOpen_def {T : Set (WithLawson α)} :
IsOpen T ↔ (lawson α).IsOpen (toLawson ⁻¹' T) | Iff.rfl | theorem | Topology.WithLawson.isOpen_def | Topology.Order | Mathlib/Topology/Order/LawsonTopology.lean | [] | [
"IsOpen",
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
lawson_le_scott : lawson α ≤ scott α univ | inf_le_right | lemma | Topology.lawson_le_scott | Topology.Order | Mathlib/Topology/Order/LawsonTopology.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
lawson_le_lower : lawson α ≤ lower α | inf_le_left | lemma | Topology.lawson_le_lower | Topology.Order | Mathlib/Topology/Order/LawsonTopology.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
scottHausdorff_le_lawson : scottHausdorff α univ ≤ lawson α | le_inf scottHausdorff_le_lower scottHausdorff_le_scott | lemma | Topology.scottHausdorff_le_lawson | Topology.Order | Mathlib/Topology/Order/LawsonTopology.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
lawsonClosed_of_scottClosed (s : Set α) (h : IsClosed (WithScott.ofScott ⁻¹' s)) :
IsClosed (WithLawson.ofLawson ⁻¹' s) | h.mono lawson_le_scott | lemma | Topology.lawsonClosed_of_scottClosed | Topology.Order | Mathlib/Topology/Order/LawsonTopology.lean | [] | [
"IsClosed",
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
lawsonClosed_of_lowerClosed (s : Set α) (h : IsClosed (WithLower.ofLower ⁻¹' s)) :
IsClosed (WithLawson.ofLawson ⁻¹' s) | h.mono lawson_le_lower | lemma | Topology.lawsonClosed_of_lowerClosed | Topology.Order | Mathlib/Topology/Order/LawsonTopology.lean | [] | [
"IsClosed",
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
lawsonOpen_iff_scottOpen_of_isUpperSet {s : Set α} (h : IsUpperSet s) :
IsOpen (WithLawson.ofLawson ⁻¹' s) ↔ IsOpen (WithScott.ofScott ⁻¹' s) | ⟨fun hs => IsScott.isOpen_iff_isUpperSet_and_scottHausdorff_open (D := univ).mpr
⟨h, (scottHausdorff_le_lawson s) hs⟩, lawson_le_scott _⟩ | lemma | Topology.lawsonOpen_iff_scottOpen_of_isUpperSet | Topology.Order | Mathlib/Topology/Order/LawsonTopology.lean | [] | [
"IsOpen",
"IsUpperSet",
"Set"
] | An upper set is Lawson open if and only if it is Scott open | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
isLawson_le_isScott : L ≤ S | by
rw [@IsScott.topology_eq α univ _ S _, @IsLawson.topology_eq_lawson α _ L _]
exact inf_le_right | lemma | Topology.isLawson_le_isScott | Topology.Order | Mathlib/Topology/Order/LawsonTopology.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
scottHausdorff_le_isLawson : scottHausdorff α univ ≤ L | by
rw [@IsLawson.topology_eq_lawson α _ L _]
exact scottHausdorff_le_lawson | lemma | Topology.scottHausdorff_le_isLawson | Topology.Order | Mathlib/Topology/Order/LawsonTopology.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
lawsonOpen_iff_scottOpen_of_isUpperSet' (s : Set α) (h : IsUpperSet s) :
IsOpen[L] s ↔ IsOpen[S] s | by
rw [@IsLawson.topology_eq_lawson α _ L _, @IsScott.topology_eq α univ _ S _]
exact lawsonOpen_iff_scottOpen_of_isUpperSet h | lemma | Topology.lawsonOpen_iff_scottOpen_of_isUpperSet' | Topology.Order | Mathlib/Topology/Order/LawsonTopology.lean | [] | [
"IsOpen",
"IsUpperSet",
"Set"
] | An upper set is Lawson open if and only if it is Scott open | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
lawsonClosed_iff_scottClosed_of_isLowerSet (s : Set α) (h : IsLowerSet s) :
IsClosed[L] s ↔ IsClosed[S] s | by
rw [← @isOpen_compl_iff, ← isOpen_compl_iff,
(lawsonOpen_iff_scottOpen_of_isUpperSet' L S _ (isUpperSet_compl.mpr h))]
include S in | lemma | Topology.lawsonClosed_iff_scottClosed_of_isLowerSet | Topology.Order | Mathlib/Topology/Order/LawsonTopology.lean | [] | [
"IsClosed",
"Set",
"isOpen_compl_iff"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
lawsonClosed_iff_dirSupClosed_of_isLowerSet (s : Set α) (h : IsLowerSet s) :
IsClosed[L] s ↔ DirSupClosed s | by
rw [lawsonClosed_iff_scottClosed_of_isLowerSet L S _ h,
@IsScott.isClosed_iff_isLowerSet_and_dirSupClosed]
simp_all | lemma | Topology.lawsonClosed_iff_dirSupClosed_of_isLowerSet | Topology.Order | Mathlib/Topology/Order/LawsonTopology.lean | [] | [
"DirSupClosed",
"IsClosed",
"Set"
] | A lower set is Lawson closed if and only if it is closed under sups of directed sets | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
frequently_lt_nhds (a : α) [NeBot (𝓝[<] a)] : ∃ᶠ x in 𝓝 a, x < a | frequently_iff_neBot.2 ‹_› | lemma | frequently_lt_nhds | Topology.Order | Mathlib/Topology/Order/LeftRight.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Filter.Eventually.exists_lt {a : α} [NeBot (𝓝[<] a)] {p : α → Prop}
(h : ∀ᶠ x in 𝓝 a, p x) : ∃ b < a, p b | ((frequently_lt_nhds a).and_eventually h).exists | theorem | Filter.Eventually.exists_lt | Topology.Order | Mathlib/Topology/Order/LeftRight.lean | [] | [
"frequently_lt_nhds"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
nhdsWithin_Ici_neBot {a b : α} (H₂ : a ≤ b) : NeBot (𝓝[Ici a] b) | nhdsWithin_neBot_of_mem H₂ | theorem | nhdsWithin_Ici_neBot | Topology.Order | Mathlib/Topology/Order/LeftRight.lean | [] | [
"nhdsWithin_neBot_of_mem"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
nhdsLE_neBot (a : α) : NeBot (𝓝[≤] a) | nhdsWithin_Iic_neBot (le_refl a) | instance | nhdsLE_neBot | Topology.Order | Mathlib/Topology/Order/LeftRight.lean | [] | [
"le_refl"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
nhdsLT_le_nhdsNE (a : α) : 𝓝[<] a ≤ 𝓝[≠] a | nhdsWithin_mono a fun _ => ne_of_lt | theorem | nhdsLT_le_nhdsNE | Topology.Order | Mathlib/Topology/Order/LeftRight.lean | [] | [
"ne_of_lt",
"nhdsWithin_mono"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
IsAntichain.interior_eq_empty [∀ x : α, (𝓝[<] x).NeBot] {s : Set α}
(hs : IsAntichain (· ≤ ·) s) : interior s = ∅ | by
refine eq_empty_of_forall_notMem fun x hx ↦ ?_
have : ∀ᶠ y in 𝓝 x, y ∈ s := mem_interior_iff_mem_nhds.1 hx
rcases this.exists_lt with ⟨y, hyx, hys⟩
exact hs hys (interior_subset hx) hyx.ne hyx.le | lemma | IsAntichain.interior_eq_empty | Topology.Order | Mathlib/Topology/Order/LeftRight.lean | [] | [
"IsAntichain",
"Set",
"interior",
"interior_subset"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
IsAntichain.interior_eq_empty' [∀ x : α, (𝓝[>] x).NeBot] {s : Set α}
(hs : IsAntichain (· ≤ ·) s) : interior s = ∅ | have : ∀ x : αᵒᵈ, NeBot (𝓝[<] x) := ‹_›
hs.to_dual.interior_eq_empty | lemma | IsAntichain.interior_eq_empty' | Topology.Order | Mathlib/Topology/Order/LeftRight.lean | [] | [
"IsAntichain",
"Set",
"interior"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
continuousWithinAt_Ioi_iff_Ici {a : α} {f : α → β} :
ContinuousWithinAt f (Ioi a) a ↔ ContinuousWithinAt f (Ici a) a | by
simp only [← Ici_diff_left, continuousWithinAt_diff_self] | theorem | continuousWithinAt_Ioi_iff_Ici | Topology.Order | Mathlib/Topology/Order/LeftRight.lean | [] | [
"ContinuousWithinAt",
"continuousWithinAt_diff_self"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
continuousWithinAt_inter_Ioi_iff_Ici {a : α} {f : α → β} {s : Set α} :
ContinuousWithinAt f (s ∩ Ioi a) a ↔ ContinuousWithinAt f (s ∩ Ici a) a | by
simp [← Ici_diff_left, ← inter_diff_assoc, continuousWithinAt_diff_self] | theorem | continuousWithinAt_inter_Ioi_iff_Ici | Topology.Order | Mathlib/Topology/Order/LeftRight.lean | [] | [
"ContinuousWithinAt",
"Set",
"continuousWithinAt_diff_self"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
nhdsLE_sup_nhdsGE (a : α) : 𝓝[≤] a ⊔ 𝓝[≥] a = 𝓝 a | by
rw [← nhdsWithin_union, Iic_union_Ici, nhdsWithin_univ] | theorem | nhdsLE_sup_nhdsGE | Topology.Order | Mathlib/Topology/Order/LeftRight.lean | [] | [
"nhdsWithin_union",
"nhdsWithin_univ"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
nhdsWithinLE_sup_nhdsWithinGE (a : α) : 𝓝[s ∩ Iic a] a ⊔ 𝓝[s ∩ Ici a] a = 𝓝[s] a | by
rw [← nhdsWithin_union, ← inter_union_distrib_left, Iic_union_Ici, inter_univ] | theorem | nhdsWithinLE_sup_nhdsWithinGE | Topology.Order | Mathlib/Topology/Order/LeftRight.lean | [] | [
"nhdsWithin_union"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
nhdsLT_sup_nhdsGE (a : α) : 𝓝[<] a ⊔ 𝓝[≥] a = 𝓝 a | by
rw [← nhdsWithin_union, Iio_union_Ici, nhdsWithin_univ] | theorem | nhdsLT_sup_nhdsGE | Topology.Order | Mathlib/Topology/Order/LeftRight.lean | [] | [
"nhdsWithin_union",
"nhdsWithin_univ"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
nhdsWithinLT_sup_nhdsWithinGE (a : α) : 𝓝[s ∩ Iio a] a ⊔ 𝓝[s ∩ Ici a] a = 𝓝[s] a | by
rw [← nhdsWithin_union, ← inter_union_distrib_left, Iio_union_Ici, inter_univ] | theorem | nhdsWithinLT_sup_nhdsWithinGE | Topology.Order | Mathlib/Topology/Order/LeftRight.lean | [] | [
"nhdsWithin_union"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
nhdsLE_sup_nhdsGT (a : α) : 𝓝[≤] a ⊔ 𝓝[>] a = 𝓝 a | by
rw [← nhdsWithin_union, Iic_union_Ioi, nhdsWithin_univ] | theorem | nhdsLE_sup_nhdsGT | Topology.Order | Mathlib/Topology/Order/LeftRight.lean | [] | [
"nhdsWithin_union",
"nhdsWithin_univ"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
nhdsWithinLE_sup_nhdsWithinGT (a : α) : 𝓝[s ∩ Iic a] a ⊔ 𝓝[s ∩ Ioi a] a = 𝓝[s] a | by
rw [← nhdsWithin_union, ← inter_union_distrib_left, Iic_union_Ioi, inter_univ] | theorem | nhdsWithinLE_sup_nhdsWithinGT | Topology.Order | Mathlib/Topology/Order/LeftRight.lean | [] | [
"nhdsWithin_union"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
nhdsLT_sup_nhdsGT (a : α) : 𝓝[<] a ⊔ 𝓝[>] a = 𝓝[≠] a | by
rw [← nhdsWithin_union, Iio_union_Ioi] | theorem | nhdsLT_sup_nhdsGT | Topology.Order | Mathlib/Topology/Order/LeftRight.lean | [] | [
"nhdsWithin_union"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
nhdsWithinLT_sup_nhdsWithinGT (a : α) :
𝓝[s ∩ Iio a] a ⊔ 𝓝[s ∩ Ioi a] a = 𝓝[s \ {a}] a | by
rw [← nhdsWithin_union, ← inter_union_distrib_left, Iio_union_Ioi, compl_eq_univ_diff,
inter_sdiff_left_comm, univ_inter] | theorem | nhdsWithinLT_sup_nhdsWithinGT | Topology.Order | Mathlib/Topology/Order/LeftRight.lean | [] | [
"nhdsWithin_union"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
nhdsGT_sup_nhdsWithin_singleton (a : α) :
𝓝[>] a ⊔ 𝓝[{a}] a = 𝓝[≥] a | by
simp only [union_singleton, Ioi_insert, ← nhdsWithin_union] | lemma | nhdsGT_sup_nhdsWithin_singleton | Topology.Order | Mathlib/Topology/Order/LeftRight.lean | [] | [
"nhdsWithin_union"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
nhdsWithin_uIoo_left_le_nhdsNE {a b : α} : 𝓝[uIoo a b] a ≤ 𝓝[≠] a | nhdsWithin_mono _ (by simp) | lemma | nhdsWithin_uIoo_left_le_nhdsNE | Topology.Order | Mathlib/Topology/Order/LeftRight.lean | [] | [
"nhdsWithin_mono"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
nhdsWithin_uIoo_right_le_nhdsNE {a b : α} : 𝓝[uIoo a b] b ≤ 𝓝[≠] b | nhdsWithin_mono _ (by simp) | lemma | nhdsWithin_uIoo_right_le_nhdsNE | Topology.Order | Mathlib/Topology/Order/LeftRight.lean | [] | [
"nhdsWithin_mono"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
continuousAt_iff_continuous_left_right {a : α} {f : α → β} :
ContinuousAt f a ↔ ContinuousWithinAt f (Iic a) a ∧ ContinuousWithinAt f (Ici a) a | by
simp only [ContinuousWithinAt, ContinuousAt, ← tendsto_sup, nhdsLE_sup_nhdsGE] | theorem | continuousAt_iff_continuous_left_right | Topology.Order | Mathlib/Topology/Order/LeftRight.lean | [] | [
"ContinuousAt",
"ContinuousWithinAt",
"nhdsLE_sup_nhdsGE"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
continuousAt_iff_continuous_left'_right' {a : α} {f : α → β} :
ContinuousAt f a ↔ ContinuousWithinAt f (Iio a) a ∧ ContinuousWithinAt f (Ioi a) a | by
rw [continuousWithinAt_Ioi_iff_Ici, continuousWithinAt_Iio_iff_Iic,
continuousAt_iff_continuous_left_right] | theorem | continuousAt_iff_continuous_left'_right' | Topology.Order | Mathlib/Topology/Order/LeftRight.lean | [] | [
"ContinuousAt",
"ContinuousWithinAt",
"continuousAt_iff_continuous_left_right",
"continuousWithinAt_Ioi_iff_Ici"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
continuousWithinAt_iff_continuous_left_right {a : α} {f : α → β} :
ContinuousWithinAt f s a ↔
ContinuousWithinAt f (s ∩ Iic a) a ∧ ContinuousWithinAt f (s ∩ Ici a) a | by
simp only [ContinuousWithinAt, ← tendsto_sup, nhdsWithinLE_sup_nhdsWithinGE] | theorem | continuousWithinAt_iff_continuous_left_right | Topology.Order | Mathlib/Topology/Order/LeftRight.lean | [] | [
"ContinuousWithinAt",
"nhdsWithinLE_sup_nhdsWithinGE"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
continuousWithinAt_iff_continuous_left'_right' {a : α} {f : α → β} :
ContinuousWithinAt f s a ↔
ContinuousWithinAt f (s ∩ Iio a) a ∧ ContinuousWithinAt f (s ∩ Ioi a) a | by
rw [continuousWithinAt_inter_Ioi_iff_Ici, continuousWithinAt_inter_Iio_iff_Iic,
continuousWithinAt_iff_continuous_left_right] | theorem | continuousWithinAt_iff_continuous_left'_right' | Topology.Order | Mathlib/Topology/Order/LeftRight.lean | [] | [
"ContinuousWithinAt",
"continuousWithinAt_iff_continuous_left_right",
"continuousWithinAt_inter_Ioi_iff_Ici"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Function.leftLim (f : α → β) (a : α) : β | by
classical
haveI : Nonempty β := ⟨f a⟩
letI : TopologicalSpace α := Preorder.topology α
exact if 𝓝[<] a = ⊥ ∨ ¬∃ y, Tendsto f (𝓝[<] a) (𝓝 y) then f a else limUnder (𝓝[<] a) f | def | Function.leftLim | Topology.Order | Mathlib/Topology/Order/LeftRightLim.lean | [] | [
"Preorder.topology",
"TopologicalSpace"
] | Let `f : α → β` be a function from a linear order `α` to a topological space `β`, and
let `a : α`. The limit strictly to the left of `f` at `a`, denoted with `leftLim f a`, is defined
by using the order topology on `α`. If `a` is isolated to its left or the function has no left
limit, we use `f a` instead to guarantee ... | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
Function.rightLim (f : α → β) (a : α) : β | @Function.leftLim αᵒᵈ β _ _ f a | def | Function.rightLim | Topology.Order | Mathlib/Topology/Order/LeftRightLim.lean | [] | [
"Function.leftLim"
] | Let `f : α → β` be a function from a linear order `α` to a topological space `β`, and
let `a : α`. The limit strictly to the right of `f` at `a`, denoted with `rightLim f a`, is defined
by using the order topology on `α`. If `a` is isolated to its right or the function has no right
limit, we use `f a` instead to guaran... | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
leftLim_eq_of_tendsto [hα : TopologicalSpace α] [h'α : OrderTopology α] [T2Space β]
{f : α → β} {a : α} {y : β} (h : 𝓝[<] a ≠ ⊥) (h' : Tendsto f (𝓝[<] a) (𝓝 y)) :
leftLim f a = y | by
have h'' : ∃ y, Tendsto f (𝓝[<] a) (𝓝 y) := ⟨y, h'⟩
rw [h'α.topology_eq_generate_intervals] at h h' h''
simp only [leftLim, h, h'', not_true, or_self_iff, if_false]
haveI := neBot_iff.2 h
exact lim_eq h' | theorem | leftLim_eq_of_tendsto | Topology.Order | Mathlib/Topology/Order/LeftRightLim.lean | [] | [
"OrderTopology",
"T2Space",
"TopologicalSpace",
"lim_eq"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
rightLim_eq_of_tendsto [TopologicalSpace α] [OrderTopology α] [T2Space β]
{f : α → β} {a : α} {y : β} (h : 𝓝[>] a ≠ ⊥) (h' : Tendsto f (𝓝[>] a) (𝓝 y)) :
Function.rightLim f a = y | leftLim_eq_of_tendsto (α := αᵒᵈ) h h' | theorem | rightLim_eq_of_tendsto | Topology.Order | Mathlib/Topology/Order/LeftRightLim.lean | [] | [
"Function.rightLim",
"OrderTopology",
"T2Space",
"TopologicalSpace",
"leftLim_eq_of_tendsto"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
leftLim_eq_of_eq_bot [hα : TopologicalSpace α] [h'α : OrderTopology α] (f : α → β) {a : α}
(h : 𝓝[<] a = ⊥) : leftLim f a = f a | by
rw [h'α.topology_eq_generate_intervals] at h
simp [leftLim, h] | theorem | leftLim_eq_of_eq_bot | Topology.Order | Mathlib/Topology/Order/LeftRightLim.lean | [] | [
"OrderTopology",
"TopologicalSpace"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
rightLim_eq_of_eq_bot [TopologicalSpace α] [OrderTopology α] (f : α → β) {a : α}
(h : 𝓝[>] a = ⊥) : rightLim f a = f a | leftLim_eq_of_eq_bot (α := αᵒᵈ) f h | theorem | rightLim_eq_of_eq_bot | Topology.Order | Mathlib/Topology/Order/LeftRightLim.lean | [] | [
"OrderTopology",
"TopologicalSpace",
"leftLim_eq_of_eq_bot"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
leftLim_eq_of_not_tendsto
[hα : TopologicalSpace α] [h'α : OrderTopology α] (f : α → β) {a : α}
(h : ¬ ∃ y, Tendsto f (𝓝[<] a) (𝓝 y)) : leftLim f a = f a | by
rw [h'α.topology_eq_generate_intervals] at h
simp [leftLim, h] | theorem | leftLim_eq_of_not_tendsto | Topology.Order | Mathlib/Topology/Order/LeftRightLim.lean | [] | [
"OrderTopology",
"TopologicalSpace"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
rightLim_eq_of_not_tendsto
[hα : TopologicalSpace α] [h'α : OrderTopology α] (f : α → β) {a : α}
(h : ¬ ∃ y, Tendsto f (𝓝[>] a) (𝓝 y)) : rightLim f a = f a | leftLim_eq_of_not_tendsto (α := αᵒᵈ) f h | theorem | rightLim_eq_of_not_tendsto | Topology.Order | Mathlib/Topology/Order/LeftRightLim.lean | [] | [
"OrderTopology",
"TopologicalSpace",
"leftLim_eq_of_not_tendsto"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
leftLim_eq_of_isBot {f : α → β} {a : α} (ha : IsBot a) :
leftLim f a = f a | by
let A : TopologicalSpace α := Preorder.topology α
have : OrderTopology α := ⟨rfl⟩
apply leftLim_eq_of_eq_bot
have : Iio a = ∅ := by simp; grind [IsBot, IsMin]
simp [this] | theorem | leftLim_eq_of_isBot | Topology.Order | Mathlib/Topology/Order/LeftRightLim.lean | [] | [
"IsBot",
"IsMin",
"OrderTopology",
"Preorder.topology",
"TopologicalSpace",
"leftLim_eq_of_eq_bot"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
rightLim_eq_of_isTop {f : α → β} {a : α} (ha : IsTop a) :
rightLim f a = f a | leftLim_eq_of_isBot (α := αᵒᵈ) ha | theorem | rightLim_eq_of_isTop | Topology.Order | Mathlib/Topology/Order/LeftRightLim.lean | [] | [
"leftLim_eq_of_isBot"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
ContinuousWithinAt.leftLim_eq [TopologicalSpace α] [OrderTopology α] [T2Space β]
{f : α → β} {a : α} (hf : ContinuousWithinAt f (Iic a) a) : leftLim f a = f a | by
rcases eq_or_ne (𝓝[<] a) ⊥ with h' | h'
· simp [leftLim_eq_of_eq_bot f h']
apply leftLim_eq_of_tendsto h'
exact hf.tendsto.mono_left (nhdsWithin_mono _ Iio_subset_Iic_self) | theorem | ContinuousWithinAt.leftLim_eq | Topology.Order | Mathlib/Topology/Order/LeftRightLim.lean | [] | [
"ContinuousWithinAt",
"OrderTopology",
"T2Space",
"TopologicalSpace",
"eq_or_ne",
"leftLim_eq_of_eq_bot",
"leftLim_eq_of_tendsto",
"nhdsWithin_mono"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
ContinuousWithinAt.rightLim_eq [TopologicalSpace α] [OrderTopology α] [T2Space β]
{f : α → β} {a : α} (hf : ContinuousWithinAt f (Ici a) a) : rightLim f a = f a | ContinuousWithinAt.leftLim_eq (α := αᵒᵈ) hf | theorem | ContinuousWithinAt.rightLim_eq | Topology.Order | Mathlib/Topology/Order/LeftRightLim.lean | [] | [
"ContinuousWithinAt",
"ContinuousWithinAt.leftLim_eq",
"OrderTopology",
"T2Space",
"TopologicalSpace"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
tendsto_leftLim_of_tendsto [TopologicalSpace α] [h'α : OrderTopology α]
{f : α → β} {a : α} (h : ∃ y, Tendsto f (𝓝[<] a) (𝓝 y)) :
Tendsto f (𝓝[<] a) (𝓝 (f.leftLim a)) | by
rcases eq_or_neBot (𝓝[<] a) with h' | h'
· simp [h']
rw [h'α.topology_eq_generate_intervals] at h h' ⊢
simp only [leftLim, neBot_iff.1 h', h, not_true_eq_false, or_self, ↓reduceIte]
exact tendsto_nhds_limUnder h | theorem | tendsto_leftLim_of_tendsto | Topology.Order | Mathlib/Topology/Order/LeftRightLim.lean | [] | [
"OrderTopology",
"TopologicalSpace",
"tendsto_nhds_limUnder"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
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