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
value | commit stringclasses 1
value |
|---|---|---|---|---|---|---|---|---|---|---|
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 | theorem | Ioc_mem_nhds | Topology.Order | Mathlib/Topology/Order/OrderClosed.lean | [] | [
"Ioo_mem_nhds"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
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 | theorem | Icc_mem_nhds | Topology.Order | Mathlib/Topology/Order/OrderClosed.lean | [] | [
"Ioo_mem_nhds"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
DiscreteTopology.of_predOrder_succOrder [PredOrder α] [SuccOrder α] :
DiscreteTopology α | by
refine discreteTopology_iff_nhds.mpr fun a ↦ ?_
rw [← nhdsWithin_univ, ← Iic_union_Ioi, nhdsWithin_union, PredOrder.nhdsLE, SuccOrder.nhdsGT,
sup_bot_eq] | theorem | DiscreteTopology.of_predOrder_succOrder | Topology.Order | Mathlib/Topology/Order/OrderClosed.lean | [] | [
"DiscreteTopology",
"PredOrder",
"PredOrder.nhdsLE",
"SuccOrder",
"nhdsWithin_union",
"nhdsWithin_univ",
"sup_bot_eq"
] | The only order closed topology on a linear order which is a `PredOrder` and a `SuccOrder`
is the discrete topology.
This theorem is not an instance,
because it causes searches for `PredOrder` and `SuccOrder` with their `Preorder` arguments
and very rarely matches. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
lt_subset_interior_le (hf : Continuous f) (hg : Continuous g) :
{ b | f b < g b } ⊆ interior { b | f b ≤ g b } | (interior_maximal fun _ => le_of_lt) <| isOpen_lt hf hg | theorem | lt_subset_interior_le | Topology.Order | Mathlib/Topology/Order/OrderClosed.lean | [] | [
"Continuous",
"interior",
"interior_maximal",
"isOpen_lt",
"le_of_lt"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
frontier_le_subset_eq (hf : Continuous f) (hg : Continuous g) :
frontier { b | f b ≤ g b } ⊆ { b | f b = g b } | by
rw [frontier_eq_closure_inter_closure, closure_le_eq hf hg]
rintro b ⟨hb₁, hb₂⟩
refine le_antisymm hb₁ (closure_lt_subset_le hg hf ?_)
convert! hb₂ using 2; simp only [not_le.symm]; rfl | theorem | frontier_le_subset_eq | Topology.Order | Mathlib/Topology/Order/OrderClosed.lean | [] | [
"Continuous",
"closure_le_eq",
"closure_lt_subset_le",
"frontier",
"frontier_eq_closure_inter_closure",
"le_antisymm"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
frontier_Iic_subset (a : α) : frontier (Iic a) ⊆ {a} | frontier_le_subset_eq (@continuous_id α _) continuous_const | theorem | frontier_Iic_subset | Topology.Order | Mathlib/Topology/Order/OrderClosed.lean | [] | [
"continuous_const",
"continuous_id",
"frontier",
"frontier_le_subset_eq"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
frontier_lt_subset_eq (hf : Continuous f) (hg : Continuous g) :
frontier { b | f b < g b } ⊆ { b | f b = g b } | by
simpa only [← not_lt, ← compl_setOf, frontier_compl, eq_comm] using frontier_le_subset_eq hg hf | theorem | frontier_lt_subset_eq | Topology.Order | Mathlib/Topology/Order/OrderClosed.lean | [] | [
"Continuous",
"frontier",
"frontier_compl",
"frontier_le_subset_eq",
"not_lt"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
continuous_if_le [TopologicalSpace γ] [∀ x, Decidable (f x ≤ g x)] {f' g' : β → γ}
(hf : Continuous f) (hg : Continuous g) (hf' : ContinuousOn f' { x | f x ≤ g x })
(hg' : ContinuousOn g' { x | g x ≤ f x }) (hfg : ∀ x, f x = g x → f' x = g' x) :
Continuous fun x => if f x ≤ g x then f' x else g' x | by
refine continuous_if (fun a ha => hfg _ (frontier_le_subset_eq hf hg ha)) ?_ (hg'.mono ?_)
· rwa [(isClosed_le hf hg).closure_eq]
· simp only [not_le]
exact closure_lt_subset_le hg hf | theorem | continuous_if_le | Topology.Order | Mathlib/Topology/Order/OrderClosed.lean | [] | [
"Continuous",
"ContinuousOn",
"TopologicalSpace",
"closure_lt_subset_le",
"continuous_if",
"frontier_le_subset_eq",
"isClosed_le",
"not_le"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Continuous.if_le [TopologicalSpace γ] [∀ 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 fun x => if f x ≤ g x then f' x else g' x | continuous_if_le hf hg hf'.continuousOn hg'.continuousOn hfg | theorem | Continuous.if_le | Topology.Order | Mathlib/Topology/Order/OrderClosed.lean | [] | [
"Continuous",
"TopologicalSpace",
"continuous_if_le"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Filter.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 | let ⟨_a, ha, _b, hb, h⟩ := hyz.exists_disjoint_Iio_Ioi
(hg.eventually (Ioi_mem_nhds hb)).mp <| (hf.eventually (Iio_mem_nhds ha)).mono fun _ h₁ h₂ =>
h _ h₁ _ h₂ | theorem | Filter.Tendsto.eventually_lt | Topology.Order | Mathlib/Topology/Order/OrderClosed.lean | [] | [
"Filter",
"Ioi_mem_nhds"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
ContinuousAt.eventually_lt {x₀ : β} (hf : ContinuousAt f x₀) (hg : ContinuousAt g x₀)
(hfg : f x₀ < g x₀) : ∀ᶠ x in 𝓝 x₀, f x < g x | hf.eventually_lt hg hfg | theorem | ContinuousAt.eventually_lt | Topology.Order | Mathlib/Topology/Order/OrderClosed.lean | [] | [
"ContinuousAt"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Continuous.max (hf : Continuous f) (hg : Continuous g) :
Continuous fun b => max (f b) (g b) | by
simp only [max_def]
exact hg.if_ge hf hg hf fun x => id | theorem | Continuous.max | Topology.Order | Mathlib/Topology/Order/OrderClosed.lean | [] | [
"Continuous",
"max_def"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
continuous_max : Continuous fun p : α × α => max p.1 p.2 | continuous_fst.max continuous_snd | theorem | continuous_max | Topology.Order | Mathlib/Topology/Order/OrderClosed.lean | [] | [
"Continuous",
"continuous_snd"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Filter.Tendsto.max {b : Filter β} {a₁ a₂ : α} (hf : Tendsto f b (𝓝 a₁))
(hg : Tendsto g b (𝓝 a₂)) : Tendsto (fun b => max (f b) (g b)) b (𝓝 (max a₁ a₂)) | (continuous_max.tendsto (a₁, a₂)).comp (hf.prodMk_nhds hg) | theorem | Filter.Tendsto.max | Topology.Order | Mathlib/Topology/Order/OrderClosed.lean | [] | [
"Filter"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Filter.Tendsto.max_right {l : Filter β} {a : α} (h : Tendsto f l (𝓝 a)) :
Tendsto (fun i => max a (f i)) l (𝓝 a) | by
simpa only [sup_idem] using (tendsto_const_nhds (x := a)).max h | theorem | Filter.Tendsto.max_right | Topology.Order | Mathlib/Topology/Order/OrderClosed.lean | [] | [
"Filter",
"sup_idem",
"tendsto_const_nhds"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Filter.Tendsto.max_left {l : Filter β} {a : α} (h : Tendsto f l (𝓝 a)) :
Tendsto (fun i => max (f i) a) l (𝓝 a) | by
simp_rw [max_comm _ a]
exact h.max_right | theorem | Filter.Tendsto.max_left | Topology.Order | Mathlib/Topology/Order/OrderClosed.lean | [] | [
"Filter"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Filter.tendsto_nhds_max_right {l : Filter β} {a : α} (h : Tendsto f l (𝓝[>] a)) :
Tendsto (fun i => max a (f i)) l (𝓝[>] a) | by
obtain ⟨h₁, h₂⟩ := tendsto_nhdsWithin_iff.mp h
exact tendsto_nhdsWithin_iff.mpr ⟨h₁.max_right, h₂.mono fun i hi => lt_max_of_lt_right hi⟩ | theorem | Filter.tendsto_nhds_max_right | Topology.Order | Mathlib/Topology/Order/OrderClosed.lean | [] | [
"Filter",
"lt_max_of_lt_right"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Filter.tendsto_nhds_max_left {l : Filter β} {a : α} (h : Tendsto f l (𝓝[>] a)) :
Tendsto (fun i => max (f i) a) l (𝓝[>] a) | by
simp_rw [max_comm _ a]
exact Filter.tendsto_nhds_max_right h | theorem | Filter.tendsto_nhds_max_left | Topology.Order | Mathlib/Topology/Order/OrderClosed.lean | [] | [
"Filter",
"Filter.tendsto_nhds_max_right"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Dense.exists_between [DenselyOrdered α] {s : Set α} (hs : Dense s) {x y : α} (h : x < y) :
∃ z ∈ s, z ∈ Ioo x y | hs.exists_mem_open isOpen_Ioo (nonempty_Ioo.2 h) | theorem | Dense.exists_between | Topology.Order | Mathlib/Topology/Order/OrderClosed.lean | [] | [
"Dense",
"DenselyOrdered",
"Set",
"isOpen_Ioo"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Dense.Ioi_eq_biUnion [DenselyOrdered α] {s : Set α} (hs : Dense s) (x : α) :
Ioi x = ⋃ y ∈ s ∩ Ioi x, Ioi y | by
refine Subset.antisymm (fun z hz ↦ ?_) (iUnion₂_subset fun y hy ↦ Ioi_subset_Ioi (le_of_lt hy.2))
rcases hs.exists_between hz with ⟨y, hys, hy⟩
exact mem_iUnion₂.2 ⟨y, ⟨hys, hy.1⟩, hy.2⟩ | theorem | Dense.Ioi_eq_biUnion | Topology.Order | Mathlib/Topology/Order/OrderClosed.lean | [] | [
"Dense",
"DenselyOrdered",
"Set",
"le_of_lt"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Pi.orderClosedTopology' [Preorder β] [TopologicalSpace β] [OrderClosedTopology β] :
OrderClosedTopology (α → β) | inferInstance | instance | Pi.orderClosedTopology' | Topology.Order | Mathlib/Topology/Order/OrderClosed.lean | [] | [
"OrderClosedTopology",
"Preorder",
"TopologicalSpace"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
isLocalMax_of_mono_anti
{α : Type*} [TopologicalSpace α] [LinearOrder α] [OrderClosedTopology α]
{β : Type*} [Preorder β]
{a b c : α} (g₀ : a < b) (g₁ : b < c) {f : α → β}
(h₀ : MonotoneOn f (Ioc a b))
(h₁ : AntitoneOn f (Ico b c)) : IsLocalMax f b | (isMaxOn_Ioo_of_mono_anti h₀ h₁).isLocalMax (Ioo_mem_nhds g₀ g₁) | lemma | isLocalMax_of_mono_anti | Topology.Order | Mathlib/Topology/Order/OrderClosedExtr.lean | [] | [
"AntitoneOn",
"Ioo_mem_nhds",
"IsLocalMax",
"LinearOrder",
"MonotoneOn",
"OrderClosedTopology",
"Preorder",
"TopologicalSpace",
"isMaxOn_Ioo_of_mono_anti"
] | If `f` is monotone on `Ioc a b` and antitone on `Ico b c` then `f` has
a local maximum at `b`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
isLocalMin_of_anti_mono
{α : Type*} [TopologicalSpace α] [LinearOrder α] [OrderClosedTopology α]
{β : Type*} [Preorder β] {a b c : α} (g₀ : a < b) (g₁ : b < c) {f : α → β}
(h₀ : AntitoneOn f (Ioc a b)) (h₁ : MonotoneOn f (Ico b c)) : IsLocalMin f b | isLocalMax_of_mono_anti (β := βᵒᵈ) g₀ g₁ h₀ h₁ | lemma | isLocalMin_of_anti_mono | Topology.Order | Mathlib/Topology/Order/OrderClosedExtr.lean | [] | [
"AntitoneOn",
"IsLocalMin",
"LinearOrder",
"MonotoneOn",
"OrderClosedTopology",
"Preorder",
"TopologicalSpace",
"isLocalMax_of_mono_anti"
] | If `f` is antitone on `Ioc a b` and monotone on `Ico b c` then `f` has
a local minimum at `b`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
partialSups (hf : ∀ k ≤ n, Tendsto (f k) l (𝓝 (g k))) :
Tendsto (partialSups f n) l (𝓝 (partialSups g n)) | by
simp only [partialSups_eq_sup'_range]
refine finset_sup'_nhds _ ?_
simpa [Nat.lt_succ_iff] | lemma | Filter.Tendsto.partialSups | Topology.Order | Mathlib/Topology/Order/PartialSups.lean | [] | [
"partialSups",
"partialSups_eq_sup'_range"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
partialSups_apply (hf : ∀ k ≤ n, Tendsto (f k) l (𝓝 (g k))) :
Tendsto (fun a ↦ partialSups (f · a) n) l (𝓝 (partialSups g n)) | by
simpa only [← Pi.partialSups_apply] using Tendsto.partialSups hf | lemma | Filter.Tendsto.partialSups_apply | Topology.Order | Mathlib/Topology/Order/PartialSups.lean | [] | [
"Pi.partialSups_apply",
"partialSups",
"partialSups_apply"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
ContinuousAt.partialSups_apply (hf : ∀ k ≤ n, ContinuousAt (f k) x) :
ContinuousAt (fun a ↦ partialSups (f · a) n) x | Tendsto.partialSups_apply hf | lemma | ContinuousAt.partialSups_apply | Topology.Order | Mathlib/Topology/Order/PartialSups.lean | [] | [
"ContinuousAt",
"partialSups"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
ContinuousAt.partialSups (hf : ∀ k ≤ n, ContinuousAt (f k) x) :
ContinuousAt (partialSups f n) x | by
simpa only [← Pi.partialSups_apply] using ContinuousAt.partialSups_apply hf | lemma | ContinuousAt.partialSups | Topology.Order | Mathlib/Topology/Order/PartialSups.lean | [] | [
"ContinuousAt",
"ContinuousAt.partialSups_apply",
"Pi.partialSups_apply",
"partialSups"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
ContinuousWithinAt.partialSups_apply (hf : ∀ k ≤ n, ContinuousWithinAt (f k) s x) :
ContinuousWithinAt (fun a ↦ partialSups (f · a) n) s x | Tendsto.partialSups_apply hf | lemma | ContinuousWithinAt.partialSups_apply | Topology.Order | Mathlib/Topology/Order/PartialSups.lean | [] | [
"ContinuousWithinAt",
"partialSups"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
ContinuousWithinAt.partialSups (hf : ∀ k ≤ n, ContinuousWithinAt (f k) s x) :
ContinuousWithinAt (partialSups f n) s x | by
simpa only [← Pi.partialSups_apply] using ContinuousWithinAt.partialSups_apply hf | lemma | ContinuousWithinAt.partialSups | Topology.Order | Mathlib/Topology/Order/PartialSups.lean | [] | [
"ContinuousWithinAt",
"ContinuousWithinAt.partialSups_apply",
"Pi.partialSups_apply",
"partialSups"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
ContinuousOn.partialSups_apply (hf : ∀ k ≤ n, ContinuousOn (f k) s) :
ContinuousOn (fun a ↦ partialSups (f · a) n) s | fun x hx ↦
ContinuousWithinAt.partialSups_apply fun k hk ↦ hf k hk x hx | lemma | ContinuousOn.partialSups_apply | Topology.Order | Mathlib/Topology/Order/PartialSups.lean | [] | [
"ContinuousOn",
"ContinuousWithinAt.partialSups_apply",
"partialSups"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
ContinuousOn.partialSups (hf : ∀ k ≤ n, ContinuousOn (f k) s) :
ContinuousOn (partialSups f n) s | fun x hx ↦
ContinuousWithinAt.partialSups fun k hk ↦ hf k hk x hx | lemma | ContinuousOn.partialSups | Topology.Order | Mathlib/Topology/Order/PartialSups.lean | [] | [
"ContinuousOn",
"ContinuousWithinAt.partialSups",
"partialSups"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Continuous.partialSups_apply (hf : ∀ k ≤ n, Continuous (f k)) :
Continuous (fun a ↦ partialSups (f · a) n) | continuous_iff_continuousAt.2 fun _ ↦ ContinuousAt.partialSups_apply fun k hk ↦
(hf k hk).continuousAt | lemma | Continuous.partialSups_apply | Topology.Order | Mathlib/Topology/Order/PartialSups.lean | [] | [
"Continuous",
"ContinuousAt.partialSups_apply",
"partialSups"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Continuous.partialSups (hf : ∀ k ≤ n, Continuous (f k)) :
Continuous (partialSups f n) | continuous_iff_continuousAt.2 fun _ ↦ ContinuousAt.partialSups fun k hk ↦ (hf k hk).continuousAt | lemma | Continuous.partialSups | Topology.Order | Mathlib/Topology/Order/PartialSups.lean | [] | [
"Continuous",
"ContinuousAt.partialSups",
"partialSups"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
PriestleySpace (α : Type*) [Preorder α] [TopologicalSpace α] : Prop where
priestley {x y : α} : ¬x ≤ y → ∃ U : Set α, IsClopen U ∧ IsUpperSet U ∧ x ∈ U ∧ y ∉ U | class | PriestleySpace | Topology.Order | Mathlib/Topology/Order/Priestley.lean | [] | [
"IsClopen",
"IsUpperSet",
"Preorder",
"Set",
"TopologicalSpace"
] | A Priestley space is an ordered topological space such that any two distinct points can be
separated by a clopen upper set. Compactness is often assumed, but we do not include it here. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
exists_isClopen_upper_of_not_le :
¬x ≤ y → ∃ U : Set α, IsClopen U ∧ IsUpperSet U ∧ x ∈ U ∧ y ∉ U | PriestleySpace.priestley | theorem | exists_isClopen_upper_of_not_le | Topology.Order | Mathlib/Topology/Order/Priestley.lean | [] | [
"IsClopen",
"IsUpperSet",
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
exists_isClopen_lower_of_not_le (h : ¬x ≤ y) :
∃ U : Set α, IsClopen U ∧ IsLowerSet U ∧ x ∉ U ∧ y ∈ U | let ⟨U, hU, hU', hx, hy⟩ := exists_isClopen_upper_of_not_le h
⟨Uᶜ, hU.compl, hU'.compl, Classical.not_not.2 hx, hy⟩ | theorem | exists_isClopen_lower_of_not_le | Topology.Order | Mathlib/Topology/Order/Priestley.lean | [] | [
"IsClopen",
"Set",
"exists_isClopen_upper_of_not_le"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
exists_isClopen_upper_or_lower_of_ne (h : x ≠ y) :
∃ U : Set α, IsClopen U ∧ (IsUpperSet U ∨ IsLowerSet U) ∧ x ∈ U ∧ y ∉ U | by
obtain h | h := h.not_le_or_not_ge
· exact (exists_isClopen_upper_of_not_le h).imp fun _ ↦ And.imp_right <| And.imp_left Or.inl
· obtain ⟨U, hU, hU', hy, hx⟩ := exists_isClopen_lower_of_not_le h
exact ⟨U, hU, Or.inr hU', hx, hy⟩ | theorem | exists_isClopen_upper_or_lower_of_ne | Topology.Order | Mathlib/Topology/Order/Priestley.lean | [] | [
"IsClopen",
"IsUpperSet",
"Set",
"exists_isClopen_lower_of_not_le",
"exists_isClopen_upper_of_not_le"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Filter.Tendsto.IccExtend (f : γ → Icc a b → β) {la : Filter α} {lb : Filter β}
{lc : Filter γ} (hf : Tendsto ↿f (lc ×ˢ la.map (projIcc a b h)) lb) :
Tendsto (↿(IccExtend h ∘ f)) (lc ×ˢ la) lb | hf.comp <| tendsto_id.prodMap tendsto_map | theorem | Filter.Tendsto.IccExtend | Topology.Order | Mathlib/Topology/Order/ProjIcc.lean | [] | [
"Filter"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
continuous_projIcc : Continuous (projIcc a b h) | Continuous.subtype_mk (by fun_prop) _ | theorem | continuous_projIcc | Topology.Order | Mathlib/Topology/Order/ProjIcc.lean | [] | [
"Continuous",
"Continuous.subtype_mk"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
isQuotientMap_projIcc : IsQuotientMap (projIcc a b h) | where
surjective := projIcc_surjective h
isCoinducing := .of_isOpen_preimage_iff_isOpen fun s ↦
⟨fun hs => ⟨_, hs, by ext; simp⟩, fun hs => hs.preimage continuous_projIcc⟩ | theorem | isQuotientMap_projIcc | Topology.Order | Mathlib/Topology/Order/ProjIcc.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
continuous_IccExtend_iff {f : Icc a b → β} : Continuous (IccExtend h f) ↔ Continuous f | isQuotientMap_projIcc.continuous_iff.symm | theorem | continuous_IccExtend_iff | Topology.Order | Mathlib/Topology/Order/ProjIcc.lean | [] | [
"Continuous"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Continuous.IccExtend {f : γ → Icc a b → β} {g : γ → α} (hf : Continuous ↿f)
(hg : Continuous g) : Continuous fun a => IccExtend h (f a) (g a) | show Continuous (↿f ∘ fun x => (x, projIcc a b h (g x)))
from hf.comp <| continuous_id.prodMk <| continuous_projIcc.comp hg | theorem | Continuous.IccExtend | Topology.Order | Mathlib/Topology/Order/ProjIcc.lean | [] | [
"Continuous"
] | See Note [continuity lemma statement]. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
Continuous.Icc_extend' {f : Icc a b → β} (hf : Continuous f) :
Continuous (IccExtend h f) | hf.comp continuous_projIcc | theorem | Continuous.Icc_extend' | Topology.Order | Mathlib/Topology/Order/ProjIcc.lean | [] | [
"Continuous",
"continuous_projIcc"
] | A useful special case of `Continuous.IccExtend`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
ContinuousAt.IccExtend {x : γ} (f : γ → Icc a b → β) {g : γ → α}
(hf : ContinuousAt ↿f (x, projIcc a b h (g x))) (hg : ContinuousAt g x) :
ContinuousAt (fun a => IccExtend h (f a) (g a)) x | show ContinuousAt (↿f ∘ fun x => (x, projIcc a b h (g x))) x from
ContinuousAt.comp hf <| continuousAt_id.prodMk <| continuous_projIcc.continuousAt.comp hg | theorem | ContinuousAt.IccExtend | Topology.Order | Mathlib/Topology/Order/ProjIcc.lean | [] | [
"ContinuousAt",
"ContinuousAt.comp"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
denseRange_ratCast : DenseRange (fun r : ℚ ↦ ((r : ℝ) : EReal)) | dense_of_exists_between fun _ _ h => exists_range_iff.2 <| exists_rat_btwn_of_lt h | lemma | EReal.denseRange_ratCast | Topology.Order | Mathlib/Topology/Order/Real.lean | [] | [
"DenseRange",
"EReal",
"dense_of_exists_between"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
exists_Ioo_extr_on_Icc (hab : a < b) (hfc : ContinuousOn f (Icc a b)) (hfI : f a = f b) :
∃ c ∈ Ioo a b, IsExtrOn f (Icc a b) c | by
have ne : (Icc a b).Nonempty := nonempty_Icc.2 (le_of_lt hab)
-- Consider absolute min and max points
obtain ⟨c, cmem, cle⟩ : ∃ c ∈ Icc a b, ∀ x ∈ Icc a b, f c ≤ f x :=
isCompact_Icc.exists_isMinOn ne hfc
obtain ⟨C, Cmem, Cge⟩ : ∃ C ∈ Icc a b, ∀ x ∈ Icc a b, f x ≤ f C :=
isCompact_Icc.exists_isMaxOn ... | theorem | exists_Ioo_extr_on_Icc | Topology.Order | Mathlib/Topology/Order/Rolle.lean | [] | [
"ContinuousOn",
"IsExtrOn",
"by_cases",
"le_antisymm",
"le_of_lt",
"le_rfl",
"lt_of_le_of_ne"
] | A continuous function on a closed interval with `f a = f b`
takes either its maximum or its minimum value at a point in the interior of the interval. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
exists_isLocalExtr_Ioo (hab : a < b) (hfc : ContinuousOn f (Icc a b)) (hfI : f a = f b) :
∃ c ∈ Ioo a b, IsLocalExtr f c | let ⟨c, cmem, hc⟩ := exists_Ioo_extr_on_Icc hab hfc hfI
⟨c, cmem, hc.isLocalExtr <| Icc_mem_nhds cmem.1 cmem.2⟩ | theorem | exists_isLocalExtr_Ioo | Topology.Order | Mathlib/Topology/Order/Rolle.lean | [] | [
"ContinuousOn",
"Icc_mem_nhds",
"IsLocalExtr",
"exists_Ioo_extr_on_Icc"
] | A continuous function on a closed interval with `f a = f b`
has a local extremum at some point of the corresponding open interval. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
exists_isExtrOn_Ioo_of_tendsto (hab : a < b) (hfc : ContinuousOn f (Ioo a b))
(ha : Tendsto f (𝓝[>] a) (𝓝 l)) (hb : Tendsto f (𝓝[<] b) (𝓝 l)) :
∃ c ∈ Ioo a b, IsExtrOn f (Ioo a b) c | by
have h : EqOn (extendFrom (Ioo a b) f) f (Ioo a b) := extendFrom_extends hfc
obtain ⟨c, hc, hfc⟩ : ∃ c ∈ Ioo a b, IsExtrOn (extendFrom (Ioo a b) f) (Icc a b) c :=
exists_Ioo_extr_on_Icc hab (continuousOn_Icc_extendFrom_Ioo hfc ha hb)
((eq_lim_at_left_extendFrom_Ioo hab ha).trans (eq_lim_at_right_extend... | lemma | exists_isExtrOn_Ioo_of_tendsto | Topology.Order | Mathlib/Topology/Order/Rolle.lean | [] | [
"ContinuousOn",
"IsExtrOn",
"continuousOn_Icc_extendFrom_Ioo",
"eq_lim_at_left_extendFrom_Ioo",
"eq_lim_at_right_extendFrom_Ioo",
"exists_Ioo_extr_on_Icc",
"extendFrom",
"extendFrom_extends",
"symm",
"trans"
] | If a function `f` is continuous on an open interval
and tends to the same value at its endpoints, then it has an extremum on this open interval. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
exists_isLocalExtr_Ioo_of_tendsto (hab : a < b) (hfc : ContinuousOn f (Ioo a b))
(ha : Tendsto f (𝓝[>] a) (𝓝 l)) (hb : Tendsto f (𝓝[<] b) (𝓝 l)) :
∃ c ∈ Ioo a b, IsLocalExtr f c | let ⟨c, cmem, hc⟩ := exists_isExtrOn_Ioo_of_tendsto hab hfc ha hb
⟨c, cmem, hc.isLocalExtr <| Ioo_mem_nhds cmem.1 cmem.2⟩ | lemma | exists_isLocalExtr_Ioo_of_tendsto | Topology.Order | Mathlib/Topology/Order/Rolle.lean | [] | [
"ContinuousOn",
"Ioo_mem_nhds",
"IsLocalExtr",
"exists_isExtrOn_Ioo_of_tendsto"
] | If a function `f` is continuous on an open interval
and tends to the same value at its endpoints,
then it has a local extremum on this open interval. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
exists_uIoo_isExtrOn_uIcc (hab : a ≠ b) (hfc : ContinuousOn f (uIcc a b))
(hfI : f a = f b) :
∃ c ∈ uIoo a b, IsExtrOn f (uIcc a b) c | exists_Ioo_extr_on_Icc (by simp [hab.symm]) hfc (by grind) | theorem | exists_uIoo_isExtrOn_uIcc | Topology.Order | Mathlib/Topology/Order/Rolle.lean | [] | [
"ContinuousOn",
"IsExtrOn",
"exists_Ioo_extr_on_Icc"
] | A continuous function on an unordered closed interval with `f a = f b`
takes either its maximum or its minimum value at a point in the interior of the interval. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
exists_isLocalExtr_uIoo (hab : a ≠ b) (hfc : ContinuousOn f (uIcc a b)) (hfI : f a = f b) :
∃ c ∈ uIoo a b, IsLocalExtr f c | exists_isLocalExtr_Ioo (by simp [hab.symm]) hfc (by grind) | theorem | exists_isLocalExtr_uIoo | Topology.Order | Mathlib/Topology/Order/Rolle.lean | [] | [
"ContinuousOn",
"IsLocalExtr",
"exists_isLocalExtr_Ioo"
] | A continuous function on a unordered closed interval with `f a = f b`
has a local extremum at some point of the corresponding unordered open interval. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
exists_isExtrOn_uIoo_of_tendsto (hab : a ≠ b) (hfc : ContinuousOn f (uIoo a b))
(ha : Tendsto f (𝓝[uIoo a b] a) (𝓝 l)) (hb : Tendsto f (𝓝[uIoo a b] b) (𝓝 l)) :
∃ c ∈ uIoo a b, IsExtrOn f (uIoo a b) c | by
have h : EqOn (extendFrom (uIoo a b) f) f (uIoo a b) := extendFrom_extends hfc
obtain ⟨c, hc, hfc⟩ : ∃ c ∈ uIoo a b, IsExtrOn (extendFrom (uIoo a b) f) (uIcc a b) c :=
exists_uIoo_isExtrOn_uIcc hab (continuousOn_uIcc_extendFrom_uIoo hfc ha hb)
((eq_lim_at_left_extendFrom_uIoo hab ha).trans (eq_lim_at_r... | lemma | exists_isExtrOn_uIoo_of_tendsto | Topology.Order | Mathlib/Topology/Order/Rolle.lean | [] | [
"ContinuousOn",
"IsExtrOn",
"continuousOn_uIcc_extendFrom_uIoo",
"eq_lim_at_left_extendFrom_uIoo",
"eq_lim_at_right_extendFrom_uIoo",
"exists_uIoo_isExtrOn_uIcc",
"extendFrom",
"extendFrom_extends",
"symm",
"trans"
] | If a function `f` is continuous on an unordered open interval
and tends to the same value at its endpoints,
then it has an extremum on this unordered open interval. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
exists_isLocalExtr_uIoo_of_tendsto (hab : a ≠ b) (hfc : ContinuousOn f (uIoo a b))
(ha : Tendsto f (𝓝[uIoo a b] a) (𝓝 l)) (hb : Tendsto f (𝓝[uIoo a b] b) (𝓝 l)) :
∃ c ∈ uIoo a b, IsLocalExtr f c | let ⟨c, cmem, hc⟩ := exists_isExtrOn_uIoo_of_tendsto hab hfc ha hb
⟨c, cmem, hc.isLocalExtr <| Ioo_mem_nhds cmem.1 cmem.2⟩ | lemma | exists_isLocalExtr_uIoo_of_tendsto | Topology.Order | Mathlib/Topology/Order/Rolle.lean | [] | [
"ContinuousOn",
"Ioo_mem_nhds",
"IsLocalExtr",
"exists_isExtrOn_uIoo_of_tendsto"
] | If a function `f` is continuous on an unordered open interval
and tends to the same value at its endpoints,
then it has a local extremum on this unordered open interval. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
scottHausdorff (α : Type*) (D : Set (Set α)) [Preorder α] : TopologicalSpace α | where
IsOpen u := ∀ ⦃d : Set α⦄, d ∈ D → d.Nonempty → DirectedOn (· ≤ ·) d → ∀ ⦃a : α⦄, IsLUB d a →
a ∈ u → ∃ b ∈ d, Ici b ∩ d ⊆ u
isOpen_univ := fun d _ ⟨b, hb⟩ _ _ _ _ ↦ ⟨b, hb, (Ici b ∩ d).subset_univ⟩
isOpen_inter s t hs ht d hd₀ hd₁ hd₂ a hd₃ ha := by
obtain ⟨b₁, hb₁d, hb₁ds⟩ := hs hd₀ hd₁ hd₂ hd₃ ha... | def | Topology.scottHausdorff | Topology.Order | Mathlib/Topology/Order/ScottTopology.lean | [] | [
"DirectedOn",
"IsLUB",
"IsOpen",
"Preorder",
"Set",
"Set.subset_sUnion_of_subset",
"TopologicalSpace",
"isOpen_sUnion",
"isOpen_univ"
] | The Scott-Hausdorff topology.
A set `u` is open in the Scott-Hausdorff topology iff when the least upper bound of a directed set
`d` lies in `u` then there is a tail of `d` which is a subset of `u`.
For mild conditions on `D`, this is equivalent to saying that open sets are `DirSupInaccOn D`,
and closed sets are `Dir... | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
IsScottHausdorff (α) (D : Set (Set α)) [Preorder α] [TopologicalSpace α] : Prop where
topology_eq_scottHausdorff : ‹TopologicalSpace α› = scottHausdorff α D | class | Topology.IsScottHausdorff | Topology.Order | Mathlib/Topology/Order/ScottTopology.lean | [] | [
"Preorder",
"Set",
"TopologicalSpace"
] | Predicate for an ordered topological space to be equipped with its Scott-Hausdorff topology.
A set `u` is open in the Scott-Hausdorff topology iff when the least upper bound of a directed set
`d` lies in `u` then there is a tail of `d` which is a subset of `u`.
For mild conditions on `D`, this is equivalent to saying... | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
topology_eq : ‹_› = scottHausdorff α D | topology_eq_scottHausdorff | lemma | Topology.IsScottHausdorff.topology_eq | Topology.Order | Mathlib/Topology/Order/ScottTopology.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
isOpen_iff :
IsOpen s ↔ ∀ ⦃d : Set α⦄, d ∈ D → d.Nonempty → DirectedOn (· ≤ ·) d → ∀ ⦃a : α⦄, IsLUB d a →
a ∈ s → ∃ b ∈ d, Ici b ∩ d ⊆ s | by
simp +instances [topology_eq_scottHausdorff (α := α) (D := D), IsOpen, scottHausdorff] | lemma | Topology.IsScottHausdorff.isOpen_iff | Topology.Order | Mathlib/Topology/Order/ScottTopology.lean | [] | [
"DirectedOn",
"IsLUB",
"IsOpen",
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
dirSupInaccOn_of_isOpen (h : IsOpen s) : DirSupInaccOn D s | .of_inter_subset (isOpen_iff.1 h) | lemma | Topology.IsScottHausdorff.dirSupInaccOn_of_isOpen | Topology.Order | Mathlib/Topology/Order/ScottTopology.lean | [] | [
"DirSupInaccOn",
"IsOpen"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
dirSupClosedOn_of_isClosed (h : IsClosed s) : DirSupClosedOn D s | .of_compl (dirSupInaccOn_of_isOpen h.isOpen_compl) | lemma | Topology.IsScottHausdorff.dirSupClosedOn_of_isClosed | Topology.Order | Mathlib/Topology/Order/ScottTopology.lean | [] | [
"DirSupClosedOn",
"IsClosed"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
isOpen_iff_dirSupInaccOn (hDL : IsLowerSet D) : IsOpen s ↔ DirSupInaccOn D s | by
rw [isOpen_iff (D := D), dirSupInaccOn_iff_inter_subset hDL] | theorem | Topology.IsScottHausdorff.isOpen_iff_dirSupInaccOn | Topology.Order | Mathlib/Topology/Order/ScottTopology.lean | [] | [
"DirSupInaccOn",
"IsOpen",
"dirSupInaccOn_iff_inter_subset"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
isClosed_iff_dirSupClosedOn (hDL : IsLowerSet D) : IsClosed s ↔ DirSupClosedOn D s | by
rw [← isOpen_compl_iff, isOpen_iff_dirSupInaccOn hDL, dirSupInaccOn_compl] | theorem | Topology.IsScottHausdorff.isClosed_iff_dirSupClosedOn | Topology.Order | Mathlib/Topology/Order/ScottTopology.lean | [] | [
"DirSupClosedOn",
"IsClosed",
"dirSupInaccOn_compl",
"isOpen_compl_iff"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
isOpen_of_isLowerSet (hDL : IsLowerSet D) (h : IsLowerSet s) : IsOpen s | (isOpen_iff_dirSupInaccOn hDL).2 h.dirSupInaccOn | theorem | Topology.IsScottHausdorff.isOpen_of_isLowerSet | Topology.Order | Mathlib/Topology/Order/ScottTopology.lean | [] | [
"IsOpen"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
isClosed_of_isUpperSet (hDL : IsLowerSet D) (h : IsUpperSet s) : IsClosed s | (isClosed_iff_dirSupClosedOn hDL).2 h.dirSupClosedOn | theorem | Topology.IsScottHausdorff.isClosed_of_isUpperSet | Topology.Order | Mathlib/Topology/Order/ScottTopology.lean | [] | [
"IsClosed",
"IsUpperSet"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
isOpen_iff_dirSupInacc : IsOpen s ↔ DirSupInacc s | by
rw [isOpen_iff_dirSupInaccOn isLowerSet_univ, dirSupInaccOn_univ] | theorem | Topology.IsScottHausdorff.isOpen_iff_dirSupInacc | Topology.Order | Mathlib/Topology/Order/ScottTopology.lean | [] | [
"DirSupInacc",
"IsOpen",
"dirSupInaccOn_univ"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
isClosed_iff_dirSupClosed : IsClosed s ↔ DirSupClosed s | by
rw [isClosed_iff_dirSupClosedOn isLowerSet_univ, dirSupClosedOn_univ] | theorem | Topology.IsScottHausdorff.isClosed_iff_dirSupClosed | Topology.Order | Mathlib/Topology/Order/ScottTopology.lean | [] | [
"DirSupClosed",
"IsClosed",
"dirSupClosedOn_univ"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
scott (α : Type*) (D : Set (Set α)) [Preorder α] : TopologicalSpace α | upperSet α ⊔ scottHausdorff α D | def | Topology.scott | Topology.Order | Mathlib/Topology/Order/ScottTopology.lean | [] | [
"Preorder",
"Set",
"TopologicalSpace"
] | The Scott topology.
It is defined as the join of the topology of upper sets and the Scott-Hausdorff topology. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
upperSet_le_scott [Preorder α] : upperSet α ≤ scott α univ | le_sup_left | lemma | Topology.upperSet_le_scott | Topology.Order | Mathlib/Topology/Order/ScottTopology.lean | [] | [
"Preorder",
"le_sup_left"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
scottHausdorff_le_scott [Preorder α] : scottHausdorff α univ ≤ scott α univ | le_sup_right | lemma | Topology.scottHausdorff_le_scott | Topology.Order | Mathlib/Topology/Order/ScottTopology.lean | [] | [
"Preorder",
"le_sup_right"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
IsScott : Prop where
topology_eq_scott : ‹TopologicalSpace α› = scott α D | class | Topology.IsScott | Topology.Order | Mathlib/Topology/Order/ScottTopology.lean | [] | [] | Predicate for an ordered topological space to be equipped with its Scott topology.
The Scott topology is defined as the join of the topology of upper sets and the Scott Hausdorff
topology. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
topology_eq [IsScott α D] : ‹_› = scott α D | topology_eq_scott | lemma | Topology.IsScott.topology_eq | Topology.Order | Mathlib/Topology/Order/ScottTopology.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
isOpen_iff_isUpperSet_and_scottHausdorff_open [IsScott α D] :
IsOpen s ↔ IsUpperSet s ∧ IsOpen[scottHausdorff α D] s | by rw [topology_eq α D]; rfl | lemma | Topology.IsScott.isOpen_iff_isUpperSet_and_scottHausdorff_open | Topology.Order | Mathlib/Topology/Order/ScottTopology.lean | [] | [
"IsOpen",
"IsUpperSet"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
isOpen_iff_isUpperSet_and_dirSupInaccOn [IsScott α D] :
IsOpen s ↔ IsUpperSet s ∧ DirSupInaccOn D s | by
rw [isOpen_iff_isUpperSet_and_scottHausdorff_open (D := D)]
refine and_congr_right fun h ↦
⟨IsScottHausdorff.dirSupInaccOn_of_isOpen (t := scottHausdorff α D),
fun h' d d₀ d₁ d₂ _ d₃ ha ↦ ?_⟩
obtain ⟨b, hbd, hbu⟩ := h' d₀ d₁ d₂ d₃ ha
exact ⟨b, hbd, Subset.trans inter_subset_left (h.Ici_subset hbu)⟩ | lemma | Topology.IsScott.isOpen_iff_isUpperSet_and_dirSupInaccOn | Topology.Order | Mathlib/Topology/Order/ScottTopology.lean | [] | [
"DirSupInaccOn",
"IsOpen",
"IsUpperSet"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
isClosed_iff_isLowerSet_and_dirSupClosed [IsScott α univ] :
IsClosed s ↔ IsLowerSet s ∧ DirSupClosed s | by
rw [← isOpen_compl_iff, isOpen_iff_isUpperSet_and_dirSupInaccOn (D := univ), isUpperSet_compl,
dirSupInaccOn_univ, dirSupInacc_compl] | lemma | Topology.IsScott.isClosed_iff_isLowerSet_and_dirSupClosed | Topology.Order | Mathlib/Topology/Order/ScottTopology.lean | [] | [
"DirSupClosed",
"IsClosed",
"dirSupInaccOn_univ",
"dirSupInacc_compl",
"isOpen_compl_iff",
"isUpperSet_compl"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
isUpperSet_of_isOpen [IsScott α D] : IsOpen s → IsUpperSet s | fun h ↦
(isOpen_iff_isUpperSet_and_scottHausdorff_open (D := D).mp h).left | lemma | Topology.IsScott.isUpperSet_of_isOpen | Topology.Order | Mathlib/Topology/Order/ScottTopology.lean | [] | [
"IsOpen",
"IsUpperSet"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
isLowerSet_of_isClosed [IsScott α univ] : IsClosed s → IsLowerSet s | fun h ↦
(isClosed_iff_isLowerSet_and_dirSupClosed.mp h).left | lemma | Topology.IsScott.isLowerSet_of_isClosed | Topology.Order | Mathlib/Topology/Order/ScottTopology.lean | [] | [
"IsClosed"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
dirSupClosed_of_isClosed [IsScott α univ] : IsClosed s → DirSupClosed s | fun h ↦
(isClosed_iff_isLowerSet_and_dirSupClosed.mp h).right | lemma | Topology.IsScott.dirSupClosed_of_isClosed | Topology.Order | Mathlib/Topology/Order/ScottTopology.lean | [] | [
"DirSupClosed",
"IsClosed"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
lowerClosure_subset_closure [IsScott α univ] : ↑(lowerClosure s) ⊆ closure s | by
convert! closure.mono (@upperSet_le_scott α _)
· rw [@IsUpperSet.closure_eq_lowerClosure α _ (upperSet α) ?_ s]
infer_instance
· exact topology_eq α univ | lemma | Topology.IsScott.lowerClosure_subset_closure | Topology.Order | Mathlib/Topology/Order/ScottTopology.lean | [] | [
"closure",
"closure.mono"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
closure_singleton [IsScott α univ] : closure {a} = Iic a | le_antisymm
(closure_minimal (by rw [singleton_subset_iff, mem_Iic]) isClosed_Iic) <| by
rw [← LowerSet.coe_Iic, ← lowerClosure_singleton]
apply lowerClosure_subset_closure | lemma | Topology.IsScott.closure_singleton | Topology.Order | Mathlib/Topology/Order/ScottTopology.lean | [] | [
"closure",
"closure_minimal",
"closure_singleton",
"isClosed_Iic",
"le_antisymm"
] | The closure of a singleton `{a}` in the Scott topology is the right-closed left-infinite interval
`(-∞,a]`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
monotone_of_continuous [IsScott α D] (hf : Continuous f) : Monotone f | fun _ b hab ↦ by
by_contra h
simpa only [mem_compl_iff, mem_preimage, mem_Iic, le_refl, not_true]
using isUpperSet_of_isOpen (D := D) ((isOpen_compl_iff.2 isClosed_Iic).preimage hf) hab h | lemma | Topology.IsScott.monotone_of_continuous | Topology.Order | Mathlib/Topology/Order/ScottTopology.lean | [] | [
"Continuous",
"Monotone",
"isClosed_Iic",
"le_refl"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
scottContinuousOn_iff_continuous {D : Set (Set α)} [Topology.IsScott α D]
(hD : ∀ a b : α, a ≤ b → {a, b} ∈ D) : ScottContinuousOn D f ↔ Continuous f | by
refine ⟨fun h ↦ continuous_def.2 fun u hu ↦ ?_, ?_⟩
· rw [isOpen_iff_isUpperSet_and_dirSupInaccOn (D := D)]
exact ⟨(isUpperSet_of_isOpen (D := univ) hu).preimage (h.monotone D hD),
fun t h₀ hd₁ hd₂ a hd₃ ha ↦ image_inter_nonempty_iff.mp <|
(isOpen_iff_isUpperSet_and_dirSupInaccOn (D := univ).mp... | lemma | Topology.IsScott.scottContinuousOn_iff_continuous | Topology.Order | Mathlib/Topology/Order/ScottTopology.lean | [] | [
"Continuous",
"IsOpen",
"ScottContinuousOn",
"Set",
"Topology.IsScott",
"upperBounds"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
isOpen_iff_Iic_compl_or_univ [TopologicalSpace α] [Topology.IsScott α univ] (U : Set α) :
IsOpen U ↔ U = univ ∨ ∃ a, (Iic a)ᶜ = U | by
constructor
· intro hU
rcases eq_empty_or_nonempty Uᶜ with eUc | neUc
· exact Or.inl (compl_empty_iff.mp eUc)
· apply Or.inr
use sSup Uᶜ
rw [compl_eq_comm, le_antisymm_iff]
refine ⟨fun _ ha ↦ le_sSup ha, (isLowerSet_of_isClosed hU.isClosed_compl).Iic_subset ?_⟩
exact dirSupClo... | lemma | Topology.IsScott.isOpen_iff_Iic_compl_or_univ | Topology.Order | Mathlib/Topology/Order/ScottTopology.lean | [] | [
"IsOpen",
"Set",
"TopologicalSpace",
"Topology.IsScott",
"compl_eq_comm",
"isChain_of_trichotomous",
"isOpen_univ",
"le_antisymm_iff",
"le_rfl",
"le_sSup"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
scott_eq_upper_of_completeLinearOrder : scott α univ = upper α | by
letI := upper α
ext U
rw [@Topology.IsUpper.isTopologicalSpace_basis _ _ (upper α)
({ topology_eq_upperTopology := rfl }) U]
letI := scott α univ
rw [@isOpen_iff_Iic_compl_or_univ _ _ (scott α univ) ({ topology_eq_scott := rfl }) U] | lemma | Topology.IsScott.scott_eq_upper_of_completeLinearOrder | Topology.Order | Mathlib/Topology/Order/ScottTopology.lean | [] | [
"Topology.IsUpper.isTopologicalSpace_basis"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
isOpen_iff_scottContinuous_mem [Preorder α] {s : Set α} [TopologicalSpace α]
[IsScott α univ] : IsOpen s ↔ ScottContinuous fun x ↦ x ∈ s | by
rw [← scottContinuousOn_univ, scottContinuousOn_iff_continuous (fun _ _ _ ↦ by trivial)]
exact isOpen_iff_continuous_mem | lemma | Topology.IsScott.isOpen_iff_scottContinuous_mem | Topology.Order | Mathlib/Topology/Order/ScottTopology.lean | [] | [
"IsOpen",
"Preorder",
"ScottContinuous",
"Set",
"TopologicalSpace",
"isOpen_iff_continuous_mem",
"scottContinuousOn_univ"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
WithScott (α : Type*) | α | def | Topology.WithScott | Topology.Order | Mathlib/Topology/Order/ScottTopology.lean | [] | [] | Type synonym for a preorder equipped with the Scott topology | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
toScott : α ≃ WithScott α | Equiv.refl _ | def | Topology.WithScott.toScott | Topology.Order | Mathlib/Topology/Order/ScottTopology.lean | [] | [
"Equiv.refl"
] | `toScott` is the identity function to the `WithScott` of a type. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
ofScott : WithScott α ≃ α | Equiv.refl _ | def | Topology.WithScott.ofScott | Topology.Order | Mathlib/Topology/Order/ScottTopology.lean | [] | [
"Equiv.refl"
] | `ofScott` is the identity function from the `WithScott` of a type. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
toScott_symm_eq : (@toScott α).symm = ofScott | rfl | lemma | Topology.WithScott.toScott_symm_eq | Topology.Order | Mathlib/Topology/Order/ScottTopology.lean | [] | [
"symm"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
ofScott_symm_eq : (@ofScott α).symm = toScott | rfl | lemma | Topology.WithScott.ofScott_symm_eq | Topology.Order | Mathlib/Topology/Order/ScottTopology.lean | [] | [
"symm"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
toScott_ofScott (a : WithScott α) : toScott (ofScott a) = a | rfl | lemma | Topology.WithScott.toScott_ofScott | Topology.Order | Mathlib/Topology/Order/ScottTopology.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
ofScott_toScott (a : α) : ofScott (toScott a) = a | rfl | lemma | Topology.WithScott.ofScott_toScott | Topology.Order | Mathlib/Topology/Order/ScottTopology.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
toScott_inj {a b : α} : toScott a = toScott b ↔ a = b | Iff.rfl | lemma | Topology.WithScott.toScott_inj | Topology.Order | Mathlib/Topology/Order/ScottTopology.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
ofScott_inj {a b : WithScott α} : ofScott a = ofScott b ↔ a = b | Iff.rfl | lemma | Topology.WithScott.ofScott_inj | Topology.Order | Mathlib/Topology/Order/ScottTopology.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
rec {β : WithScott α → Sort _}
(h : ∀ a, β (toScott a)) : ∀ a, β a | fun a ↦ h (ofScott a) | def | Topology.WithScott.rec | Topology.Order | Mathlib/Topology/Order/ScottTopology.lean | [] | [] | A recursor for `WithScott`. Use as `induction x`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
isOpen_iff_isUpperSet_and_scottHausdorff_open' {u : Set α} :
IsOpen (WithScott.ofScott ⁻¹' u) ↔ IsUpperSet u ∧ (scottHausdorff α univ).IsOpen u | Iff.rfl | lemma | Topology.WithScott.isOpen_iff_isUpperSet_and_scottHausdorff_open' | Topology.Order | Mathlib/Topology/Order/ScottTopology.lean | [] | [
"IsOpen",
"IsUpperSet",
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
scottHausdorff_le_lower : scottHausdorff α univ ≤ lower α | let : TopologicalSpace α := scottHausdorff α univ
fun s h ↦ IsScottHausdorff.isOpen_of_isLowerSet isLowerSet_univ <|
@IsLower.isLowerSet_of_isOpen (Topology.WithLower α) _ _ _ s h | lemma | Topology.scottHausdorff_le_lower | Topology.Order | Mathlib/Topology/Order/ScottTopology.lean | [] | [
"TopologicalSpace",
"Topology.WithLower"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
IsScott.withScottHomeomorph [IsScott α univ] : WithScott α ≃ₜ α | WithScott.ofScott.toHomeomorphOfIsInducing ⟨IsScott.topology_eq α univ ▸ induced_id.symm⟩ | def | Topology.IsScott.withScottHomeomorph | Topology.Order | Mathlib/Topology/Order/ScottTopology.lean | [] | [] | If `α` is equipped with the Scott topology, then it is homeomorphic to `WithScott α`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
IsScott.scottHausdorff_le [IsScott α univ] :
scottHausdorff α univ ≤ ‹TopologicalSpace α› | by
rw [IsScott.topology_eq α univ, scott]; exact le_sup_right | lemma | Topology.IsScott.scottHausdorff_le | Topology.Order | Mathlib/Topology/Order/ScottTopology.lean | [] | [
"le_sup_right"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
IsLower.scottHausdorff_le [IsLower α] : scottHausdorff α univ ≤ ‹TopologicalSpace α› | by
rw [IsLower.topology_eq α]
exact scottHausdorff_le_lower | lemma | Topology.IsLower.scottHausdorff_le | Topology.Order | Mathlib/Topology/Order/ScottTopology.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
isOpen_singleton_of_not_isSuccPrelimit (ha : ¬ IsSuccPrelimit a) : IsOpen {a} | by
obtain ⟨b, hb⟩ := not_isSuccPrelimit_iff.1 ha
by_cases ha' : IsMax a
· convert! isOpen_Ioi (a := b) using 1
rw [hb.Ioi_eq]
grind [IsMax]
· convert! isOpen_Ioo (a := b) (b := Order.succ a) using 1
simp [(covBy_succ_of_not_isMax ha').Ioo_eq_Ioc, hb.Ioc_eq] | theorem | SuccOrder.isOpen_singleton_of_not_isSuccPrelimit | Topology.Order | Mathlib/Topology/Order/SuccPred.lean | [] | [
"IsOpen",
"Order.succ",
"by_cases",
"isOpen_Ioi",
"isOpen_Ioo"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
isOpen_singleton_iff : IsOpen {a} ↔ ¬ IsSuccLimit a | by
nontriviality α
refine ⟨fun h ha ↦ ?_, fun ha ↦ ?_⟩
· obtain ⟨l, u, h₁, h₂⟩ := mem_nhds_iff_exists_Ioo_subset' (by simpa using ha.not_isMin)
(by simpa only [not_isMax_iff] using not_isMax a) |>.mp (h.mem_nhds (mem_singleton a))
refine ha.isSuccPrelimit l ?_
rw [← succ_eq_iff_covBy]
simp only ... | theorem | SuccOrder.isOpen_singleton_iff | Topology.Order | Mathlib/Topology/Order/SuccPred.lean | [] | [
"IsOpen",
"Order.succ",
"mem_nhds_iff_exists_Ioo_subset'"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
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