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 |
|---|---|---|---|---|---|---|---|---|---|---|
toUpper_inj {a b : α} : toUpper a = toUpper b ↔ a = b | Iff.rfl | lemma | Topology.WithUpper.toUpper_inj | Topology.Order | Mathlib/Topology/Order/LowerUpperTopology.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
ofUpper_inj {a b : WithUpper α} : ofUpper a = ofUpper b ↔ a = b | Iff.rfl | lemma | Topology.WithUpper.ofUpper_inj | Topology.Order | Mathlib/Topology/Order/LowerUpperTopology.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
rec {motive : WithUpper α → Sort*} (toUpper : ∀ a, motive (toUpper a)) :
∀ a, motive a | fun a =>
toUpper (ofUpper a) | def | Topology.WithUpper.rec | Topology.Order | Mathlib/Topology/Order/LowerUpperTopology.lean | [] | [] | A recursor for `WithUpper`. Use as `induction x`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
toUpper_le_toUpper {x y : α} : toUpper x ≤ toUpper y ↔ x ≤ y | .rfl | lemma | Topology.WithUpper.toUpper_le_toUpper | Topology.Order | Mathlib/Topology/Order/LowerUpperTopology.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
toUpper_lt_toUpper {x y : α} : toUpper x < toUpper y ↔ x < y | .rfl | lemma | Topology.WithUpper.toUpper_lt_toUpper | Topology.Order | Mathlib/Topology/Order/LowerUpperTopology.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
ofUpper_le_ofUpper {x y : WithUpper α} : ofUpper x ≤ ofUpper y ↔ x ≤ y | .rfl | lemma | Topology.WithUpper.ofUpper_le_ofUpper | Topology.Order | Mathlib/Topology/Order/LowerUpperTopology.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
ofUpper_lt_ofUpper {x y : WithUpper α} : ofUpper x < ofUpper y ↔ x < y | .rfl | lemma | Topology.WithUpper.ofUpper_lt_ofUpper | Topology.Order | Mathlib/Topology/Order/LowerUpperTopology.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
isOpen_preimage_ofUpper : IsOpen (ofUpper ⁻¹' s) ↔ (upper α).IsOpen s | Iff.rfl | lemma | Topology.WithUpper.isOpen_preimage_ofUpper | Topology.Order | Mathlib/Topology/Order/LowerUpperTopology.lean | [] | [
"IsOpen"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
isOpen_def {s : Set (WithUpper α)} : IsOpen s ↔ (upper α).IsOpen (toUpper ⁻¹' s) | Iff.rfl | lemma | Topology.WithUpper.isOpen_def | Topology.Order | Mathlib/Topology/Order/LowerUpperTopology.lean | [] | [
"IsOpen",
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
continuous_toUpper [TopologicalSpace α] [ClosedIicTopology α] :
Continuous (toUpper : α → WithUpper α) | continuous_generateFrom_iff.mpr <| by rintro _ ⟨a, rfl⟩; exact isClosed_Iic.isOpen_compl | theorem | Topology.WithUpper.continuous_toUpper | Topology.Order | Mathlib/Topology/Order/LowerUpperTopology.lean | [] | [
"ClosedIicTopology",
"Continuous",
"TopologicalSpace"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
IsLower (α : Type*) [t : TopologicalSpace α] [Preorder α] : Prop where
topology_eq_lowerTopology : t = lower α | class | Topology.IsLower | Topology.Order | Mathlib/Topology/Order/LowerUpperTopology.lean | [] | [
"Preorder",
"TopologicalSpace"
] | The lower topology is the topology generated by the complements of the left-closed right-infinite
intervals. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
IsUpper (α : Type*) [t : TopologicalSpace α] [Preorder α] : Prop where
topology_eq_upperTopology : t = upper α | class | Topology.IsUpper | Topology.Order | Mathlib/Topology/Order/LowerUpperTopology.lean | [] | [
"Preorder",
"TopologicalSpace"
] | The upper topology is the topology generated by the complements of the right-closed left-infinite
intervals. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
WithLower.toDualHomeomorph [Preorder α] : WithLower α ≃ₜ WithUpper αᵒᵈ | where
toFun := OrderDual.toDual
invFun := OrderDual.ofDual
left_inv := OrderDual.toDual_ofDual
right_inv := OrderDual.ofDual_toDual
continuous_toFun := continuous_coinduced_rng
continuous_invFun := continuous_coinduced_rng | def | Topology.WithLower.toDualHomeomorph | Topology.Order | Mathlib/Topology/Order/LowerUpperTopology.lean | [] | [
"OrderDual.ofDual",
"OrderDual.ofDual_toDual",
"OrderDual.toDual",
"OrderDual.toDual_ofDual",
"Preorder",
"continuous_coinduced_rng"
] | The lower topology is homeomorphic to the upper topology on the dual order | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
lowerBasis (α : Type*) [Preorder α] | { s : Set α | ∃ t : Set α, t.Finite ∧ (upperClosure t : Set α)ᶜ = s } | def | Topology.IsLower.lowerBasis | Topology.Order | Mathlib/Topology/Order/LowerUpperTopology.lean | [] | [
"Preorder",
"Set",
"upperClosure"
] | The complements of the upper closures of finite sets are a collection of lower sets
which form a basis for the lower topology. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
topology_eq : ‹_› = lower α | topology_eq_lowerTopology | lemma | Topology.IsLower.topology_eq | Topology.Order | Mathlib/Topology/Order/LowerUpperTopology.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
withLowerHomeomorph : WithLower α ≃ₜ α | WithLower.ofLower.toHomeomorphOfIsInducing ⟨topology_eq α ▸ induced_id.symm⟩ | def | Topology.IsLower.withLowerHomeomorph | Topology.Order | Mathlib/Topology/Order/LowerUpperTopology.lean | [] | [] | If `α` is equipped with the lower topology, then it is homeomorphic to `WithLower α`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
isOpen_iff_generate_Ici_compl : IsOpen s ↔ GenerateOpen { t | ∃ a, (Ici a)ᶜ = t } s | by
rw [topology_eq α]; rfl | theorem | Topology.IsLower.isOpen_iff_generate_Ici_compl | Topology.Order | Mathlib/Topology/Order/LowerUpperTopology.lean | [] | [
"IsOpen"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
_root_.OrderDual.instIsUpper : IsUpper αᵒᵈ | where
topology_eq_upperTopology := topology_eq_lowerTopology (α := α) | instance | OrderDual.instIsUpper | Topology.Order | Mathlib/Topology/Order/LowerUpperTopology.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
isClosed_upperClosure (h : s.Finite) : IsClosed (upperClosure s : Set α) | by
simp only [← UpperSet.iInf_Ici, UpperSet.coe_iInf]
exact h.isClosed_biUnion fun _ _ => isClosed_Ici | theorem | Topology.IsLower.isClosed_upperClosure | Topology.Order | Mathlib/Topology/Order/LowerUpperTopology.lean | [] | [
"IsClosed",
"Set",
"UpperSet.coe_iInf",
"UpperSet.iInf_Ici",
"upperClosure"
] | The upper closure of a finite set is closed in the lower topology. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
isLowerSet_of_isOpen (h : IsOpen s) : IsLowerSet s | by
replace h := isOpen_iff_generate_Ici_compl.1 h
induction h with
| basic u h' => obtain ⟨a, rfl⟩ := h'; exact (isUpperSet_Ici a).compl
| univ => exact isLowerSet_univ
| inter u v _ _ hu2 hv2 => exact hu2.inter hv2
| sUnion _ _ ih => exact isLowerSet_sUnion ih | theorem | Topology.IsLower.isLowerSet_of_isOpen | Topology.Order | Mathlib/Topology/Order/LowerUpperTopology.lean | [] | [
"IsOpen",
"isUpperSet_Ici"
] | Every set open in the lower topology is a lower set. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
isUpperSet_of_isClosed (h : IsClosed s) : IsUpperSet s | isLowerSet_compl.1 <| isLowerSet_of_isOpen h.isOpen_compl | theorem | Topology.IsLower.isUpperSet_of_isClosed | Topology.Order | Mathlib/Topology/Order/LowerUpperTopology.lean | [] | [
"IsClosed",
"IsUpperSet"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
tendsto_nhds_iff_not_le {β : Type*} {f : β → α} {l : Filter β} {x : α} :
Filter.Tendsto f l (𝓝 x) ↔ ∀ y, ¬y ≤ x → ∀ᶠ z in l, ¬y ≤ f z | by
simp +instances [topology_eq_lowerTopology, tendsto_nhds_generateFrom_iff, Filter.Eventually, Ici,
compl_setOf] | theorem | Topology.IsLower.tendsto_nhds_iff_not_le | Topology.Order | Mathlib/Topology/Order/LowerUpperTopology.lean | [] | [
"Filter",
"Filter.Eventually",
"Filter.Tendsto"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
closure_singleton (a : α) : closure {a} = Ici a | Subset.antisymm ((closure_minimal fun _ h => h.ge) <| isClosed_Ici) <|
(isUpperSet_of_isClosed isClosed_closure).Ici_subset <| subset_closure rfl | theorem | Topology.IsLower.closure_singleton | Topology.Order | Mathlib/Topology/Order/LowerUpperTopology.lean | [] | [
"closure",
"closure_minimal",
"closure_singleton",
"isClosed_closure",
"subset_closure"
] | The closure of a singleton `{a}` in the lower topology is the left-closed right-infinite interval
$[a, ∞)$. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
isTopologicalBasis : IsTopologicalBasis (lowerBasis α) | by
convert! isTopologicalBasis_of_subbasis (topology_eq α)
simp_rw [lowerBasis, coe_upperClosure, compl_iUnion]
ext s
constructor
· rintro ⟨F, hF, rfl⟩
refine ⟨(fun a => (Ici a)ᶜ) '' F, ⟨hF.image _, image_subset_iff.2 fun _ _ => ⟨_, rfl⟩⟩, ?_⟩
simp only [sInter_image]
· rintro ⟨F, ⟨hF, hs⟩, rfl⟩
... | theorem | Topology.IsLower.isTopologicalBasis | Topology.Order | Mathlib/Topology/Order/LowerUpperTopology.lean | [] | [
"Subtype.forall'",
"coe_upperClosure"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
continuous_iff_Ici [TopologicalSpace β] {f : β → α} :
Continuous f ↔ ∀ a, IsClosed (f ⁻¹' (Ici a)) | by
obtain rfl := IsLower.topology_eq α
simp [continuous_generateFrom_iff] | lemma | Topology.IsLower.continuous_iff_Ici | Topology.Order | Mathlib/Topology/Order/LowerUpperTopology.lean | [] | [
"Continuous",
"IsClosed",
"TopologicalSpace",
"continuous_generateFrom_iff"
] | A function `f : β → α` with lower topology in the codomain is continuous
if and only if the preimage of every interval `Set.Ici a` is a closed set. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
isTopologicalBasis_insert_univ_subbasis :
IsTopologicalBasis (insert univ {s : Set α | ∃ a, (Ici a)ᶜ = s}) | isTopologicalBasis_of_subbasis_of_inter (by rw [topology_eq α, lower]) (by
rintro _ ⟨b, rfl⟩ _ ⟨c, rfl⟩
use b ⊓ c
rw [compl_Ici, compl_Ici, compl_Ici, Iio_inter_Iio]) | lemma | Topology.IsLower.isTopologicalBasis_insert_univ_subbasis | Topology.Order | Mathlib/Topology/Order/LowerUpperTopology.lean | [] | [
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
tendsto_nhds_iff_lt {β : Type*} {f : β → α} {l : Filter β} {x : α} :
Filter.Tendsto f l (𝓝 x) ↔ ∀ y, x < y → ∀ᶠ z in l, f z < y | by
simp only [tendsto_nhds_iff_not_le, not_le] | theorem | Topology.IsLower.tendsto_nhds_iff_lt | Topology.Order | Mathlib/Topology/Order/LowerUpperTopology.lean | [] | [
"Filter",
"Filter.Tendsto",
"not_le"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
isTopologicalSpace_basis (U : Set α) : IsOpen U ↔ U = univ ∨ ∃ a, (Ici a)ᶜ = U | by
by_cases hU : U = univ
· simp only [hU, isOpen_univ, compl_Ici, true_or]
refine ⟨?_, isTopologicalBasis_insert_univ_subbasis.isOpen⟩
intro hO
apply Or.inr
convert! IsTopologicalBasis.open_eq_sUnion isTopologicalBasis_insert_univ_subbasis hO
constructor
· intro ⟨a, ha⟩
use {U}
constructor
... | lemma | Topology.IsLower.isTopologicalSpace_basis | Topology.Order | Mathlib/Topology/Order/LowerUpperTopology.lean | [] | [
"IsOpen",
"Set",
"by_cases",
"compl_inj_iff",
"isOpen_univ",
"le_antisymm",
"not_lt",
"sSup_le_iff",
"subset_trans"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
upperBasis (α : Type*) [Preorder α] | { s : Set α | ∃ t : Set α, t.Finite ∧ (lowerClosure t : Set α)ᶜ = s } | def | Topology.IsUpper.upperBasis | Topology.Order | Mathlib/Topology/Order/LowerUpperTopology.lean | [] | [
"Preorder",
"Set"
] | The complements of the lower closures of finite sets are a collection of upper sets
which form a basis for the upper topology. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
topology_eq : ‹_› = upper α | topology_eq_upperTopology | lemma | Topology.IsUpper.topology_eq | Topology.Order | Mathlib/Topology/Order/LowerUpperTopology.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
withUpperHomeomorph : WithUpper α ≃ₜ α | WithUpper.ofUpper.toHomeomorphOfIsInducing ⟨topology_eq α ▸ induced_id.symm⟩ | def | Topology.IsUpper.withUpperHomeomorph | Topology.Order | Mathlib/Topology/Order/LowerUpperTopology.lean | [] | [] | If `α` is equipped with the upper topology, then it is homeomorphic to `WithUpper α`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
isOpen_iff_generate_Iic_compl : IsOpen s ↔ GenerateOpen { t | ∃ a, (Iic a)ᶜ = t } s | by
rw [topology_eq α]; rfl | theorem | Topology.IsUpper.isOpen_iff_generate_Iic_compl | Topology.Order | Mathlib/Topology/Order/LowerUpperTopology.lean | [] | [
"IsOpen"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
_root_.OrderDual.instIsLower : IsLower αᵒᵈ | where
topology_eq_lowerTopology := topology_eq_upperTopology (α := α) | instance | OrderDual.instIsLower | Topology.Order | Mathlib/Topology/Order/LowerUpperTopology.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
isClosed_lowerClosure (h : s.Finite) : IsClosed (lowerClosure s : Set α) | IsLower.isClosed_upperClosure (α := αᵒᵈ) h | theorem | Topology.IsUpper.isClosed_lowerClosure | Topology.Order | Mathlib/Topology/Order/LowerUpperTopology.lean | [] | [
"IsClosed",
"Set"
] | The lower closure of a finite set is closed in the upper topology. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
isUpperSet_of_isOpen (h : IsOpen s) : IsUpperSet s | IsLower.isLowerSet_of_isOpen (α := αᵒᵈ) h | theorem | Topology.IsUpper.isUpperSet_of_isOpen | Topology.Order | Mathlib/Topology/Order/LowerUpperTopology.lean | [] | [
"IsOpen",
"IsUpperSet"
] | Every set open in the upper topology is an upper set. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
isLowerSet_of_isClosed (h : IsClosed s) : IsLowerSet s | isUpperSet_compl.1 <| isUpperSet_of_isOpen h.isOpen_compl | theorem | Topology.IsUpper.isLowerSet_of_isClosed | Topology.Order | Mathlib/Topology/Order/LowerUpperTopology.lean | [] | [
"IsClosed"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
tendsto_nhds_iff_not_le {β : Type*} {f : β → α} {l : Filter β} {x : α} :
Filter.Tendsto f l (𝓝 x) ↔ ∀ y, ¬x ≤ y → ∀ᶠ z in l, ¬f z ≤ y | IsLower.tendsto_nhds_iff_not_le (α := αᵒᵈ) | theorem | Topology.IsUpper.tendsto_nhds_iff_not_le | Topology.Order | Mathlib/Topology/Order/LowerUpperTopology.lean | [] | [
"Filter",
"Filter.Tendsto"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
closure_singleton (a : α) : closure {a} = Iic a | IsLower.closure_singleton (α := αᵒᵈ) _ | theorem | Topology.IsUpper.closure_singleton | Topology.Order | Mathlib/Topology/Order/LowerUpperTopology.lean | [] | [
"closure",
"closure_singleton"
] | The closure of a singleton `{a}` in the upper topology is the left-infinite right-closed interval
$(-∞,a]$. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
isTopologicalBasis : IsTopologicalBasis (upperBasis α) | IsLower.isTopologicalBasis (α := αᵒᵈ) | theorem | Topology.IsUpper.isTopologicalBasis | Topology.Order | Mathlib/Topology/Order/LowerUpperTopology.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
continuous_iff_Iic [TopologicalSpace β] {f : β → α} :
Continuous f ↔ ∀ a, IsClosed (f ⁻¹' (Iic a)) | IsLower.continuous_iff_Ici (α := αᵒᵈ) | lemma | Topology.IsUpper.continuous_iff_Iic | Topology.Order | Mathlib/Topology/Order/LowerUpperTopology.lean | [] | [
"Continuous",
"IsClosed",
"TopologicalSpace"
] | A function `f : β → α` with upper topology in the codomain is continuous
if and only if the preimage of every interval `Set.Iic a` is a closed set. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
isTopologicalBasis_insert_univ_subbasis :
IsTopologicalBasis (insert univ {s : Set α | ∃ a, (Iic a)ᶜ = s}) | IsLower.isTopologicalBasis_insert_univ_subbasis (α := αᵒᵈ) | lemma | Topology.IsUpper.isTopologicalBasis_insert_univ_subbasis | Topology.Order | Mathlib/Topology/Order/LowerUpperTopology.lean | [] | [
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
tendsto_nhds_iff_lt {β : Type*} {f : β → α} {l : Filter β} {x : α} :
Filter.Tendsto f l (𝓝 x) ↔ ∀ y < x, ∀ᶠ z in l, y < f z | IsLower.tendsto_nhds_iff_lt (α := αᵒᵈ) | theorem | Topology.IsUpper.tendsto_nhds_iff_lt | Topology.Order | Mathlib/Topology/Order/LowerUpperTopology.lean | [] | [
"Filter",
"Filter.Tendsto"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
isTopologicalSpace_basis (U : Set α) : IsOpen U ↔ U = univ ∨ ∃ a, (Iic a)ᶜ = U | IsLower.isTopologicalSpace_basis (α := αᵒᵈ) U | lemma | Topology.IsUpper.isTopologicalSpace_basis | Topology.Order | Mathlib/Topology/Order/LowerUpperTopology.lean | [] | [
"IsOpen",
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
instIsLowerProd [Preorder α] [TopologicalSpace α] [IsLower α]
[OrderBot α] [Preorder β] [TopologicalSpace β] [IsLower β] [OrderBot β] :
IsLower (α × β) | where
topology_eq_lowerTopology := by
refine le_antisymm (le_generateFrom ?_) ?_
· rintro _ ⟨x, rfl⟩
exact (isClosed_Ici.prod isClosed_Ici).isOpen_compl
rw [(IsLower.isTopologicalBasis.prod
IsLower.isTopologicalBasis).eq_generateFrom, le_generateFrom_iff_subset_isOpen,
image2_subset_iff]... | instance | Topology.instIsLowerProd | Topology.Order | Mathlib/Topology/Order/LowerUpperTopology.lean | [] | [
"OrderBot",
"Preorder",
"TopologicalSpace",
"coe_upperClosure",
"le_antisymm",
"le_generateFrom"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
instIsUpperProd [Preorder α] [TopologicalSpace α] [IsUpper α]
[OrderTop α] [Preorder β] [TopologicalSpace β] [IsUpper β] [OrderTop β] :
IsUpper (α × β) | where
topology_eq_upperTopology := by
suffices IsLower (α × β)ᵒᵈ from IsLower.topology_eq_lowerTopology (α := (α × β)ᵒᵈ)
exact instIsLowerProd (α := αᵒᵈ) (β := βᵒᵈ) | instance | Topology.instIsUpperProd | Topology.Order | Mathlib/Topology/Order/LowerUpperTopology.lean | [] | [
"OrderTop",
"Preorder",
"TopologicalSpace"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
_root_.sInfHom.continuous (f : sInfHom α β) : Continuous f | by
refine IsLower.continuous_iff_Ici.2 fun b => ?_
convert! isClosed_Ici (a := sInf <| f ⁻¹' Ici b)
refine Subset.antisymm (fun a => sInf_le) fun a ha => le_trans ?_ <|
OrderHomClass.mono (f : α →o β) ha
refine LE.le.trans ?_ (map_sInf f _).ge
simp | lemma | sInfHom.continuous | Topology.Order | Mathlib/Topology/Order/LowerUpperTopology.lean | [] | [
"Continuous",
"OrderHomClass.mono",
"le_trans",
"sInfHom"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
_root_.sSupHom.continuous (f : sSupHom α β) : Continuous f | sInfHom.continuous (α := αᵒᵈ) (β := βᵒᵈ) (sSupHom.dual.toFun f) | lemma | sSupHom.continuous | Topology.Order | Mathlib/Topology/Order/LowerUpperTopology.lean | [] | [
"Continuous",
"sInfHom.continuous",
"sSupHom"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
isUpper_orderDual [Preorder α] [TopologicalSpace α] : IsUpper αᵒᵈ ↔ IsLower α | by
constructor
· apply OrderDual.instIsLower
· apply OrderDual.instIsUpper | lemma | Topology.isUpper_orderDual | Topology.Order | Mathlib/Topology/Order/LowerUpperTopology.lean | [] | [
"OrderDual.instIsLower",
"OrderDual.instIsUpper",
"Preorder",
"TopologicalSpace"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
isLower_orderDual [Preorder α] [TopologicalSpace α] : IsLower αᵒᵈ ↔ IsUpper α | isUpper_orderDual.symm | lemma | Topology.isLower_orderDual | Topology.Order | Mathlib/Topology/Order/LowerUpperTopology.lean | [] | [
"Preorder",
"TopologicalSpace"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
MonotoneOn.insert_of_continuousWithinAt [TopologicalSpace β] [OrderClosedTopology β]
(hf : MonotoneOn f s) (hx : ClusterPt x (𝓟 s)) (h'x : ContinuousWithinAt f s x) :
MonotoneOn f (insert x s) | by
have : (𝓝[s] x).NeBot := hx
apply monotoneOn_insert_iff.2 ⟨fun b hb hbx ↦ ?_, fun b hb hxb ↦ ?_, hf⟩
· rcases hbx.eq_or_lt with rfl | hbx
· exact le_rfl
simp only [ContinuousWithinAt] at h'x
apply ge_of_tendsto h'x
have : s ∩ Ioi b ∈ 𝓝[s] x := inter_mem_nhdsWithin _ (Ioi_mem_nhds hbx)
fil... | lemma | MonotoneOn.insert_of_continuousWithinAt | Topology.Order | Mathlib/Topology/Order/Monotone.lean | [] | [
"ClusterPt",
"ContinuousWithinAt",
"Ioi_mem_nhds",
"MonotoneOn",
"OrderClosedTopology",
"TopologicalSpace",
"inter_mem_nhdsWithin",
"le_of_lt",
"le_of_tendsto",
"le_rfl"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
MonotoneOn.countable_setOf_two_preimages [SecondCountableTopology α]
(hf : MonotoneOn f s) :
Set.Countable {c | ∃ x y, x ∈ s ∧ y ∈ s ∧ x < y ∧ f x = c ∧ f y = c} | by
nontriviality α
let t := {c | ∃ x, ∃ y, x ∈ s ∧ y ∈ s ∧ x < y ∧ f x = c ∧ f y = c}
have : ∀ c ∈ t, ∃ x, ∃ y, x ∈ s ∧ y ∈ s ∧ x < y ∧ f x = c ∧ f y = c := fun c hc ↦ hc
choose! x y hxs hys hxy hfx hfy using this
let u := x '' t
suffices H : Set.Countable (x '' t) by
have : Set.InjOn x t := by
in... | lemma | MonotoneOn.countable_setOf_two_preimages | Topology.Order | Mathlib/Topology/Order/Monotone.lean | [] | [
"MonotoneOn",
"SecondCountableTopology",
"Set.Countable",
"Set.InjOn",
"Set.Pairwise",
"Set.PairwiseDisjoint.countable_of_Ioo",
"symm"
] | If a function is monotone on a set in a second countable topological space, then there
are only countably many points that have several preimages. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
Monotone.countable_setOf_two_preimages [SecondCountableTopology α]
(hf : Monotone f) :
Set.Countable {c | ∃ x y, x < y ∧ f x = c ∧ f y = c} | by
rw [← monotoneOn_univ] at hf
simpa using hf.countable_setOf_two_preimages | lemma | Monotone.countable_setOf_two_preimages | Topology.Order | Mathlib/Topology/Order/Monotone.lean | [] | [
"Monotone",
"SecondCountableTopology",
"Set.Countable",
"monotoneOn_univ"
] | If a function is monotone in a second countable topological space, then there
are only countably many points that have several preimages. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
AntitoneOn.countable_setOf_two_preimages [SecondCountableTopology α]
(hf : AntitoneOn f s) :
Set.Countable {c | ∃ x y, x ∈ s ∧ y ∈ s ∧ x < y ∧ f x = c ∧ f y = c} | (MonotoneOn.countable_setOf_two_preimages hf.dual_right :) | lemma | AntitoneOn.countable_setOf_two_preimages | Topology.Order | Mathlib/Topology/Order/Monotone.lean | [] | [
"AntitoneOn",
"MonotoneOn.countable_setOf_two_preimages",
"SecondCountableTopology",
"Set.Countable"
] | If a function is antitone on a set in a second countable topological space, then there
are only countably many points that have several preimages. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
Antitone.countable_setOf_two_preimages [SecondCountableTopology α]
(hf : Antitone f) :
Set.Countable {c | ∃ x y, x < y ∧ f x = c ∧ f y = c} | (Monotone.countable_setOf_two_preimages hf.dual_right :) | lemma | Antitone.countable_setOf_two_preimages | Topology.Order | Mathlib/Topology/Order/Monotone.lean | [] | [
"Antitone",
"Monotone.countable_setOf_two_preimages",
"SecondCountableTopology",
"Set.Countable"
] | If a function is antitone in a second countable topological space, then there
are only countably many points that have several preimages. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
MonotoneOn.countable_not_continuousWithinAt_Ioi (hf : MonotoneOn f s) :
Set.Countable {x ∈ s | ¬ContinuousWithinAt f (s ∩ Ioi x) x} | by
apply (countable_image_lt_image_Ioi_within s f).mono
rintro x ⟨xs, hx : ¬ContinuousWithinAt f (s ∩ Ioi x) x⟩
dsimp only [mem_setOf_eq]
contrapose! hx
refine tendsto_order.2 ⟨fun m hm => ?_, fun u hu => ?_⟩
· filter_upwards [@self_mem_nhdsWithin _ _ x (s ∩ Ioi x)] with y hy
exact hm.trans_le (hf xs hy... | theorem | MonotoneOn.countable_not_continuousWithinAt_Ioi | Topology.Order | Mathlib/Topology/Order/Monotone.lean | [] | [
"ContinuousWithinAt",
"MonotoneOn",
"Set.Countable",
"countable_image_lt_image_Ioi_within",
"le_of_lt",
"nhdsWithin_inter",
"self_mem_nhdsWithin"
] | In a second countable space, the set of points where a monotone function is not right-continuous
within a set is at most countable. Superseded by `MonotoneOn.countable_not_continuousWithinAt`
which gives the two-sided version. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
MonotoneOn.countable_not_continuousWithinAt_Iio (hf : MonotoneOn f s) :
Set.Countable {x ∈ s | ¬ContinuousWithinAt f (s ∩ Iio x) x} | hf.dual.countable_not_continuousWithinAt_Ioi | theorem | MonotoneOn.countable_not_continuousWithinAt_Iio | Topology.Order | Mathlib/Topology/Order/Monotone.lean | [] | [
"ContinuousWithinAt",
"MonotoneOn",
"Set.Countable"
] | In a second countable space, the set of points where a monotone function is not left-continuous
within a set is at most countable. Superseded by `MonotoneOn.countable_not_continuousWithinAt`
which gives the two-sided version. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
MonotoneOn.countable_not_continuousWithinAt (hf : MonotoneOn f s) :
Set.Countable {x ∈ s | ¬ContinuousWithinAt f s x} | by
apply (hf.countable_not_continuousWithinAt_Ioi.union hf.countable_not_continuousWithinAt_Iio).mono
refine compl_subset_compl.1 ?_
simp only [compl_union]
rintro x ⟨hx, h'x⟩
simp only [mem_compl_iff, mem_setOf_eq, not_and, not_not] at hx h'x ⊢
intro xs
exact continuousWithinAt_iff_continuous_left'_right... | theorem | MonotoneOn.countable_not_continuousWithinAt | Topology.Order | Mathlib/Topology/Order/Monotone.lean | [] | [
"ContinuousWithinAt",
"MonotoneOn",
"Set.Countable"
] | In a second countable space, the set of points where a monotone function is not continuous
within a set is at most countable. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
Monotone.countable_not_continuousAt (hf : Monotone f) :
Set.Countable {x | ¬ContinuousAt f x} | by
simpa [continuousWithinAt_univ] using (hf.monotoneOn univ).countable_not_continuousWithinAt | theorem | Monotone.countable_not_continuousAt | Topology.Order | Mathlib/Topology/Order/Monotone.lean | [] | [
"ContinuousAt",
"Monotone",
"Set.Countable",
"continuousWithinAt_univ"
] | In a second countable space, the set of points where a monotone function is not continuous
is at most countable. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
_root_.AntitoneOn.countable_not_continuousWithinAt
{s : Set α} (hf : AntitoneOn f s) :
Set.Countable {x ∈ s | ¬ContinuousWithinAt f s x} | hf.dual_right.countable_not_continuousWithinAt | theorem | AntitoneOn.countable_not_continuousWithinAt | Topology.Order | Mathlib/Topology/Order/Monotone.lean | [] | [
"AntitoneOn",
"ContinuousWithinAt",
"Set",
"Set.Countable"
] | In a second countable space, the set of points where an antitone function is not continuous
within a set is at most countable. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
Antitone.countable_not_continuousAt (hf : Antitone f) :
Set.Countable {x | ¬ContinuousAt f x} | hf.dual_right.countable_not_continuousAt | theorem | Antitone.countable_not_continuousAt | Topology.Order | Mathlib/Topology/Order/Monotone.lean | [] | [
"Antitone",
"ContinuousAt",
"Set.Countable"
] | In a second countable space, the set of points where an antitone function is not continuous
is at most countable. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
MonotoneOn.map_csSup_of_continuousWithinAt {f : α → β} {A : Set α}
(Cf : ContinuousWithinAt f A (sSup A))
(Mf : MonotoneOn f A) (A_nonemp : A.Nonempty) (A_bdd : BddAbove A := by bddDefault) :
f (sSup A) = sSup (f '' A) | --This is a particular case of the more general `IsLUB.isLUB_of_tendsto`
.symm <| ((isLUB_csSup A_nonemp A_bdd).isLUB_of_tendsto Mf A_nonemp <|
Cf.mono_left fun ⦃_⦄ a ↦ a).csSup_eq (A_nonemp.image f) | theorem | MonotoneOn.map_csSup_of_continuousWithinAt | Topology.Order | Mathlib/Topology/Order/Monotone.lean | [] | [
"BddAbove",
"ContinuousWithinAt",
"MonotoneOn",
"Set",
"isLUB_csSup",
"symm"
] | A monotone function continuous at the supremum of a nonempty set sends this supremum to
the supremum of the image of this set. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
Monotone.map_csSup_of_continuousAt {f : α → β} {A : Set α}
(Cf : ContinuousAt f (sSup A)) (Mf : Monotone f) (A_nonemp : A.Nonempty)
(A_bdd : BddAbove A := by bddDefault) : f (sSup A) = sSup (f '' A) | MonotoneOn.map_csSup_of_continuousWithinAt Cf.continuousWithinAt
(Mf.monotoneOn _) A_nonemp A_bdd | theorem | Monotone.map_csSup_of_continuousAt | Topology.Order | Mathlib/Topology/Order/Monotone.lean | [] | [
"BddAbove",
"ContinuousAt",
"Monotone",
"MonotoneOn.map_csSup_of_continuousWithinAt",
"Set"
] | A monotone function continuous at the supremum of a nonempty set sends this supremum to
the supremum of the image of this set. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
Monotone.map_ciSup_of_continuousAt {ι : Sort*} [Nonempty ι] {f : α → β} {g : ι → α}
(Cf : ContinuousAt f (iSup g)) (Mf : Monotone f)
(bdd : BddAbove (range g) := by bddDefault) : f (⨆ i, g i) = ⨆ i, f (g i) | by
rw [iSup, Monotone.map_csSup_of_continuousAt Cf Mf (range_nonempty g) bdd, ← range_comp, iSup,
comp_def] | theorem | Monotone.map_ciSup_of_continuousAt | Topology.Order | Mathlib/Topology/Order/Monotone.lean | [] | [
"BddAbove",
"ContinuousAt",
"Monotone",
"Monotone.map_csSup_of_continuousAt",
"iSup"
] | A monotone function continuous at the indexed supremum over a nonempty `Sort` sends this indexed
supremum to the indexed supremum of the composition. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
MonotoneOn.map_csInf_of_continuousWithinAt {f : α → β} {A : Set α}
(Cf : ContinuousWithinAt f A (sInf A))
(Mf : MonotoneOn f A) (A_nonemp : A.Nonempty) (A_bdd : BddBelow A := by bddDefault) :
f (sInf A) = sInf (f '' A) | MonotoneOn.map_csSup_of_continuousWithinAt (α := αᵒᵈ) (β := βᵒᵈ) Cf Mf.dual A_nonemp A_bdd | theorem | MonotoneOn.map_csInf_of_continuousWithinAt | Topology.Order | Mathlib/Topology/Order/Monotone.lean | [] | [
"ContinuousWithinAt",
"MonotoneOn",
"MonotoneOn.map_csSup_of_continuousWithinAt",
"Set"
] | A monotone function continuous at the infimum of a nonempty set sends this infimum to
the infimum of the image of this set. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
Monotone.map_csInf_of_continuousAt {f : α → β} {A : Set α} (Cf : ContinuousAt f (sInf A))
(Mf : Monotone f) (A_nonemp : A.Nonempty) (A_bdd : BddBelow A := by bddDefault) :
f (sInf A) = sInf (f '' A) | Monotone.map_csSup_of_continuousAt (α := αᵒᵈ) (β := βᵒᵈ) Cf Mf.dual A_nonemp A_bdd | theorem | Monotone.map_csInf_of_continuousAt | Topology.Order | Mathlib/Topology/Order/Monotone.lean | [] | [
"ContinuousAt",
"Monotone",
"Monotone.map_csSup_of_continuousAt",
"Set"
] | A monotone function continuous at the infimum of a nonempty set sends this infimum to
the infimum of the image of this set. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
Monotone.map_ciInf_of_continuousAt {ι : Sort*} [Nonempty ι] {f : α → β} {g : ι → α}
(Cf : ContinuousAt f (iInf g)) (Mf : Monotone f)
(bdd : BddBelow (range g) := by bddDefault) : f (⨅ i, g i) = ⨅ i, f (g i) | by
rw [iInf, Monotone.map_csInf_of_continuousAt Cf Mf (range_nonempty g) bdd, ← range_comp, iInf,
comp_def] | theorem | Monotone.map_ciInf_of_continuousAt | Topology.Order | Mathlib/Topology/Order/Monotone.lean | [] | [
"ContinuousAt",
"Monotone",
"Monotone.map_csInf_of_continuousAt"
] | A monotone function continuous at the indexed infimum over a nonempty `Sort` sends this indexed
infimum to the indexed infimum of the composition. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
AntitoneOn.map_csInf_of_continuousWithinAt {f : α → β} {A : Set α}
(Cf : ContinuousWithinAt f A (sInf A))
(Af : AntitoneOn f A) (A_nonemp : A.Nonempty) (A_bdd : BddBelow A := by bddDefault) :
f (sInf A) = sSup (f '' A) | MonotoneOn.map_csInf_of_continuousWithinAt (β := βᵒᵈ) Cf Af.dual_right A_nonemp A_bdd | theorem | AntitoneOn.map_csInf_of_continuousWithinAt | Topology.Order | Mathlib/Topology/Order/Monotone.lean | [] | [
"AntitoneOn",
"ContinuousWithinAt",
"MonotoneOn.map_csInf_of_continuousWithinAt",
"Set"
] | An antitone function continuous at the infimum of a nonempty set sends this infimum to
the supremum of the image of this set. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
Antitone.map_csInf_of_continuousAt {f : α → β} {A : Set α} (Cf : ContinuousAt f (sInf A))
(Af : Antitone f) (A_nonemp : A.Nonempty) (A_bdd : BddBelow A := by bddDefault) :
f (sInf A) = sSup (f '' A) | Monotone.map_csInf_of_continuousAt (β := βᵒᵈ) Cf Af.dual_right A_nonemp A_bdd | theorem | Antitone.map_csInf_of_continuousAt | Topology.Order | Mathlib/Topology/Order/Monotone.lean | [] | [
"Antitone",
"ContinuousAt",
"Monotone.map_csInf_of_continuousAt",
"Set"
] | An antitone function continuous at the infimum of a nonempty set sends this infimum to
the supremum of the image of this set. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
Antitone.map_ciInf_of_continuousAt {ι : Sort*} [Nonempty ι] {f : α → β} {g : ι → α}
(Cf : ContinuousAt f (iInf g)) (Af : Antitone f)
(bdd : BddBelow (range g) := by bddDefault) : f (⨅ i, g i) = ⨆ i, f (g i) | by
rw [iInf, Antitone.map_csInf_of_continuousAt Cf Af (range_nonempty g) bdd, ← range_comp, iSup,
comp_def] | theorem | Antitone.map_ciInf_of_continuousAt | Topology.Order | Mathlib/Topology/Order/Monotone.lean | [] | [
"Antitone",
"Antitone.map_csInf_of_continuousAt",
"ContinuousAt",
"iSup"
] | An antitone function continuous at the indexed infimum over a nonempty `Sort` sends this indexed
infimum to the indexed supremum of the composition. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
AntitoneOn.map_csSup_of_continuousWithinAt {f : α → β} {A : Set α}
(Cf : ContinuousWithinAt f A (sSup A))
(Af : AntitoneOn f A) (A_nonemp : A.Nonempty) (A_bdd : BddAbove A := by bddDefault) :
f (sSup A) = sInf (f '' A) | MonotoneOn.map_csSup_of_continuousWithinAt (β := βᵒᵈ) Cf Af.dual_right A_nonemp A_bdd | theorem | AntitoneOn.map_csSup_of_continuousWithinAt | Topology.Order | Mathlib/Topology/Order/Monotone.lean | [] | [
"AntitoneOn",
"BddAbove",
"ContinuousWithinAt",
"MonotoneOn.map_csSup_of_continuousWithinAt",
"Set"
] | An antitone function continuous at the supremum of a nonempty set sends this supremum to
the infimum of the image of this set. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
Antitone.map_csSup_of_continuousAt {f : α → β} {A : Set α} (Cf : ContinuousAt f (sSup A))
(Af : Antitone f) (A_nonemp : A.Nonempty) (A_bdd : BddAbove A := by bddDefault) :
f (sSup A) = sInf (f '' A) | Monotone.map_csSup_of_continuousAt (β := βᵒᵈ) Cf Af.dual_right A_nonemp A_bdd | theorem | Antitone.map_csSup_of_continuousAt | Topology.Order | Mathlib/Topology/Order/Monotone.lean | [] | [
"Antitone",
"BddAbove",
"ContinuousAt",
"Monotone.map_csSup_of_continuousAt",
"Set"
] | An antitone function continuous at the supremum of a nonempty set sends this supremum to
the infimum of the image of this set. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
Antitone.map_ciSup_of_continuousAt {ι : Sort*} [Nonempty ι] {f : α → β} {g : ι → α}
(Cf : ContinuousAt f (iSup g)) (Af : Antitone f)
(bdd : BddAbove (range g) := by bddDefault) : f (⨆ i, g i) = ⨅ i, f (g i) | by
rw [iSup, Antitone.map_csSup_of_continuousAt Cf Af (range_nonempty g) bdd, ← range_comp, iInf,
comp_def] | theorem | Antitone.map_ciSup_of_continuousAt | Topology.Order | Mathlib/Topology/Order/Monotone.lean | [] | [
"Antitone",
"Antitone.map_csSup_of_continuousAt",
"BddAbove",
"ContinuousAt",
"iSup"
] | An antitone function continuous at the indexed supremum over a nonempty `Sort` sends this
indexed supremum to the indexed infimum of the composition. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
sSup_mem_closure {s : Set α} (hs : s.Nonempty) : sSup s ∈ closure s | (isLUB_sSup s).mem_closure hs | theorem | sSup_mem_closure | Topology.Order | Mathlib/Topology/Order/Monotone.lean | [] | [
"Set",
"closure",
"isLUB_sSup"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
sInf_mem_closure {s : Set α} (hs : s.Nonempty) : sInf s ∈ closure s | (isGLB_sInf s).mem_closure hs | theorem | sInf_mem_closure | Topology.Order | Mathlib/Topology/Order/Monotone.lean | [] | [
"Set",
"closure"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
IsClosed.sSup_mem {s : Set α} (hs : s.Nonempty) (hc : IsClosed s) : sSup s ∈ s | (isLUB_sSup s).mem_of_isClosed hs hc | theorem | IsClosed.sSup_mem | Topology.Order | Mathlib/Topology/Order/Monotone.lean | [] | [
"IsClosed",
"Set",
"isLUB_sSup"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
IsClosed.sInf_mem {s : Set α} (hs : s.Nonempty) (hc : IsClosed s) : sInf s ∈ s | (isGLB_sInf s).mem_of_isClosed hs hc | theorem | IsClosed.sInf_mem | Topology.Order | Mathlib/Topology/Order/Monotone.lean | [] | [
"IsClosed",
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
MonotoneOn.map_sSup_of_continuousWithinAt {f : α → β} {s : Set α}
(Cf : ContinuousWithinAt f s (sSup s))
(Mf : MonotoneOn f s) (fbot : f ⊥ = ⊥) : f (sSup s) = sSup (f '' s) | by
rcases s.eq_empty_or_nonempty with h | h
· simp [h, fbot]
· exact Mf.map_csSup_of_continuousWithinAt Cf h | theorem | MonotoneOn.map_sSup_of_continuousWithinAt | Topology.Order | Mathlib/Topology/Order/Monotone.lean | [] | [
"ContinuousWithinAt",
"MonotoneOn",
"Set"
] | A monotone function `f` sending `bot` to `bot` and continuous at the supremum of a set sends
this supremum to the supremum of the image of this set. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
Monotone.map_sSup_of_continuousAt {f : α → β} {s : Set α} (Cf : ContinuousAt f (sSup s))
(Mf : Monotone f) (fbot : f ⊥ = ⊥) : f (sSup s) = sSup (f '' s) | MonotoneOn.map_sSup_of_continuousWithinAt Cf.continuousWithinAt (Mf.monotoneOn _) fbot | theorem | Monotone.map_sSup_of_continuousAt | Topology.Order | Mathlib/Topology/Order/Monotone.lean | [] | [
"ContinuousAt",
"Monotone",
"MonotoneOn.map_sSup_of_continuousWithinAt",
"Set"
] | A monotone function `f` sending `bot` to `bot` and continuous at the supremum of a set sends
this supremum to the supremum of the image of this set. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
Monotone.map_iSup_of_continuousAt {ι : Sort*} {f : α → β} {g : ι → α}
(Cf : ContinuousAt f (iSup g)) (Mf : Monotone f) (fbot : f ⊥ = ⊥) :
f (⨆ i, g i) = ⨆ i, f (g i) | by
rw [iSup, Mf.map_sSup_of_continuousAt Cf fbot, ← range_comp, iSup, comp_def] | theorem | Monotone.map_iSup_of_continuousAt | Topology.Order | Mathlib/Topology/Order/Monotone.lean | [] | [
"ContinuousAt",
"Monotone",
"iSup"
] | If a monotone function sending `bot` to `bot` is continuous at the indexed supremum over
a `Sort`, then it sends this indexed supremum to the indexed supremum of the composition. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
MonotoneOn.map_sInf_of_continuousWithinAt {f : α → β} {s : Set α}
(Cf : ContinuousWithinAt f s (sInf s)) (Mf : MonotoneOn f s) (ftop : f ⊤ = ⊤) :
f (sInf s) = sInf (f '' s) | MonotoneOn.map_sSup_of_continuousWithinAt (α := αᵒᵈ) (β := βᵒᵈ) Cf Mf.dual ftop | theorem | MonotoneOn.map_sInf_of_continuousWithinAt | Topology.Order | Mathlib/Topology/Order/Monotone.lean | [] | [
"ContinuousWithinAt",
"MonotoneOn",
"MonotoneOn.map_sSup_of_continuousWithinAt",
"Set"
] | A monotone function `f` sending `top` to `top` and continuous at the infimum of a set sends
this infimum to the infimum of the image of this set. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
Monotone.map_sInf_of_continuousAt {f : α → β} {s : Set α} (Cf : ContinuousAt f (sInf s))
(Mf : Monotone f) (ftop : f ⊤ = ⊤) : f (sInf s) = sInf (f '' s) | Monotone.map_sSup_of_continuousAt (α := αᵒᵈ) (β := βᵒᵈ) Cf Mf.dual ftop | theorem | Monotone.map_sInf_of_continuousAt | Topology.Order | Mathlib/Topology/Order/Monotone.lean | [] | [
"ContinuousAt",
"Monotone",
"Monotone.map_sSup_of_continuousAt",
"Set"
] | A monotone function `f` sending `top` to `top` and continuous at the infimum of a set sends
this infimum to the infimum of the image of this set. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
Monotone.map_iInf_of_continuousAt {ι : Sort*} {f : α → β} {g : ι → α}
(Cf : ContinuousAt f (iInf g)) (Mf : Monotone f) (ftop : f ⊤ = ⊤) : f (iInf g) = iInf (f ∘ g) | Monotone.map_iSup_of_continuousAt (α := αᵒᵈ) (β := βᵒᵈ) Cf Mf.dual ftop | theorem | Monotone.map_iInf_of_continuousAt | Topology.Order | Mathlib/Topology/Order/Monotone.lean | [] | [
"ContinuousAt",
"Monotone",
"Monotone.map_iSup_of_continuousAt"
] | If a monotone function sending `top` to `top` is continuous at the indexed infimum over
a `Sort`, then it sends this indexed infimum to the indexed infimum of the composition. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
AntitoneOn.map_sSup_of_continuousWithinAt {f : α → β} {s : Set α}
(Cf : ContinuousWithinAt f s (sSup s)) (Af : AntitoneOn f s) (fbot : f ⊥ = ⊤) :
f (sSup s) = sInf (f '' s) | MonotoneOn.map_sSup_of_continuousWithinAt
(show ContinuousWithinAt (OrderDual.toDual ∘ f) s (sSup s) from Cf) Af fbot | theorem | AntitoneOn.map_sSup_of_continuousWithinAt | Topology.Order | Mathlib/Topology/Order/Monotone.lean | [] | [
"AntitoneOn",
"ContinuousWithinAt",
"MonotoneOn.map_sSup_of_continuousWithinAt",
"OrderDual.toDual",
"Set"
] | An antitone function `f` sending `bot` to `top` and continuous at the supremum of a set sends
this supremum to the infimum of the image of this set. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
Antitone.map_sSup_of_continuousAt {f : α → β} {s : Set α} (Cf : ContinuousAt f (sSup s))
(Af : Antitone f) (fbot : f ⊥ = ⊤) : f (sSup s) = sInf (f '' s) | Monotone.map_sSup_of_continuousAt (show ContinuousAt (OrderDual.toDual ∘ f) (sSup s) from Cf) Af
fbot | theorem | Antitone.map_sSup_of_continuousAt | Topology.Order | Mathlib/Topology/Order/Monotone.lean | [] | [
"Antitone",
"ContinuousAt",
"Monotone.map_sSup_of_continuousAt",
"OrderDual.toDual",
"Set"
] | An antitone function `f` sending `bot` to `top` and continuous at the supremum of a set sends
this supremum to the infimum of the image of this set. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
Antitone.map_iSup_of_continuousAt {ι : Sort*} {f : α → β} {g : ι → α}
(Cf : ContinuousAt f (iSup g)) (Af : Antitone f) (fbot : f ⊥ = ⊤) :
f (⨆ i, g i) = ⨅ i, f (g i) | Monotone.map_iSup_of_continuousAt (show ContinuousAt (OrderDual.toDual ∘ f) (iSup g) from Cf) Af
fbot | theorem | Antitone.map_iSup_of_continuousAt | Topology.Order | Mathlib/Topology/Order/Monotone.lean | [] | [
"Antitone",
"ContinuousAt",
"Monotone.map_iSup_of_continuousAt",
"OrderDual.toDual",
"iSup"
] | An antitone function sending `bot` to `top` is continuous at the indexed supremum over
a `Sort`, then it sends this indexed supremum to the indexed supremum of the composition. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
AntitoneOn.map_sInf_of_continuousWithinAt {f : α → β} {s : Set α}
(Cf : ContinuousWithinAt f s (sInf s)) (Af : AntitoneOn f s) (ftop : f ⊤ = ⊥) :
f (sInf s) = sSup (f '' s) | MonotoneOn.map_sInf_of_continuousWithinAt
(show ContinuousWithinAt (OrderDual.toDual ∘ f) s (sInf s) from Cf) Af ftop | theorem | AntitoneOn.map_sInf_of_continuousWithinAt | Topology.Order | Mathlib/Topology/Order/Monotone.lean | [] | [
"AntitoneOn",
"ContinuousWithinAt",
"MonotoneOn.map_sInf_of_continuousWithinAt",
"OrderDual.toDual",
"Set"
] | An antitone function `f` sending `top` to `bot` and continuous at the infimum of a set sends
this infimum to the supremum of the image of this set. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
Antitone.map_sInf_of_continuousAt {f : α → β} {s : Set α} (Cf : ContinuousAt f (sInf s))
(Af : Antitone f) (ftop : f ⊤ = ⊥) : f (sInf s) = sSup (f '' s) | Monotone.map_sInf_of_continuousAt (show ContinuousAt (OrderDual.toDual ∘ f) (sInf s) from Cf) Af
ftop | theorem | Antitone.map_sInf_of_continuousAt | Topology.Order | Mathlib/Topology/Order/Monotone.lean | [] | [
"Antitone",
"ContinuousAt",
"Monotone.map_sInf_of_continuousAt",
"OrderDual.toDual",
"Set"
] | An antitone function `f` sending `top` to `bot` and continuous at the infimum of a set sends
this infimum to the supremum of the image of this set. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
Antitone.map_iInf_of_continuousAt {ι : Sort*} {f : α → β} {g : ι → α}
(Cf : ContinuousAt f (iInf g)) (Af : Antitone f) (ftop : f ⊤ = ⊥) : f (iInf g) = iSup (f ∘ g) | Monotone.map_iInf_of_continuousAt (show ContinuousAt (OrderDual.toDual ∘ f) (iInf g) from Cf) Af
ftop | theorem | Antitone.map_iInf_of_continuousAt | Topology.Order | Mathlib/Topology/Order/Monotone.lean | [] | [
"Antitone",
"ContinuousAt",
"Monotone.map_iInf_of_continuousAt",
"OrderDual.toDual",
"iSup"
] | If an antitone function sending `top` to `bot` is continuous at the indexed infimum over
a `Sort`, then it sends this indexed infimum to the indexed supremum of the composition. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
csSup_mem_closure {s : Set α} (hs : s.Nonempty) (B : BddAbove s) : sSup s ∈ closure s | (isLUB_csSup hs B).mem_closure hs | theorem | csSup_mem_closure | Topology.Order | Mathlib/Topology/Order/Monotone.lean | [] | [
"BddAbove",
"Set",
"closure",
"isLUB_csSup"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
csInf_mem_closure {s : Set α} (hs : s.Nonempty) (B : BddBelow s) : sInf s ∈ closure s | (isGLB_csInf hs B).mem_closure hs | theorem | csInf_mem_closure | Topology.Order | Mathlib/Topology/Order/Monotone.lean | [] | [
"Set",
"closure"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
IsClosed.csSup_mem {s : Set α} (hc : IsClosed s) (hs : s.Nonempty) (B : BddAbove s) :
sSup s ∈ s | (isLUB_csSup hs B).mem_of_isClosed hs hc | theorem | IsClosed.csSup_mem | Topology.Order | Mathlib/Topology/Order/Monotone.lean | [] | [
"BddAbove",
"IsClosed",
"Set",
"isLUB_csSup"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
IsClosed.csInf_mem {s : Set α} (hc : IsClosed s) (hs : s.Nonempty) (B : BddBelow s) :
sInf s ∈ s | (isGLB_csInf hs B).mem_of_isClosed hs hc | theorem | IsClosed.csInf_mem | Topology.Order | Mathlib/Topology/Order/Monotone.lean | [] | [
"IsClosed",
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
IsClosed.isLeast_csInf {s : Set α} (hc : IsClosed s) (hs : s.Nonempty) (B : BddBelow s) :
IsLeast s (sInf s) | ⟨hc.csInf_mem hs B, (isGLB_csInf hs B).1⟩ | theorem | IsClosed.isLeast_csInf | Topology.Order | Mathlib/Topology/Order/Monotone.lean | [] | [
"IsClosed",
"IsLeast",
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
IsClosed.isGreatest_csSup {s : Set α} (hc : IsClosed s) (hs : s.Nonempty) (B : BddAbove s) :
IsGreatest s (sSup s) | IsClosed.isLeast_csInf (α := αᵒᵈ) hc hs B | theorem | IsClosed.isGreatest_csSup | Topology.Order | Mathlib/Topology/Order/Monotone.lean | [] | [
"BddAbove",
"IsClosed",
"IsClosed.isLeast_csInf",
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
MonotoneOn.tendsto_nhdsWithin_Ioo_left {α β : Type*} [LinearOrder α] [TopologicalSpace α]
[OrderTopology α] [ConditionallyCompleteLinearOrder β] [TopologicalSpace β] [OrderTopology β]
{f : α → β} {x y : α} (h_nonempty : (Ioo y x).Nonempty) (Mf : MonotoneOn f (Ioo y x))
(h_bdd : BddAbove (f '' Ioo y x)) :
... | by
refine tendsto_order.2 ⟨fun l hl => ?_, fun m hm => ?_⟩
· obtain ⟨z, ⟨yz, zx⟩, lz⟩ : ∃ a : α, a ∈ Ioo y x ∧ l < f a := by
simpa only [mem_image, exists_prop, exists_exists_and_eq_and] using
exists_lt_of_lt_csSup (h_nonempty.image _) hl
filter_upwards [Ioo_mem_nhdsLT zx] with w hw
exact lz.t... | lemma | MonotoneOn.tendsto_nhdsWithin_Ioo_left | Topology.Order | Mathlib/Topology/Order/Monotone.lean | [] | [
"BddAbove",
"ConditionallyCompleteLinearOrder",
"Ioo_mem_nhdsLT",
"LinearOrder",
"MonotoneOn",
"OrderTopology",
"TopologicalSpace",
"exists_exists_and_eq_and",
"exists_lt_of_lt_csSup",
"le_csSup"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
MonotoneOn.tendsto_nhdsWithin_Ioo_right {α β : Type*} [LinearOrder α] [TopologicalSpace α]
[OrderTopology α] [ConditionallyCompleteLinearOrder β] [TopologicalSpace β] [OrderTopology β]
{f : α → β} {x y : α} (h_nonempty : (Ioo x y).Nonempty) (Mf : MonotoneOn f (Ioo x y))
(h_bdd : BddBelow (f '' Ioo x y)) :
... | by
refine tendsto_order.2 ⟨fun l hl => ?_, fun m hm => ?_⟩
· rcases h_nonempty with ⟨p, hy, hx⟩
filter_upwards [Ioo_mem_nhdsGT (hy.trans hx)] with w hw
exact hl.trans_le <| csInf_le h_bdd (mem_image_of_mem _ hw)
· obtain ⟨z, ⟨xz, zy⟩, zm⟩ : ∃ a : α, a ∈ Ioo x y ∧ f a < m := by
simpa [mem_image, exis... | lemma | MonotoneOn.tendsto_nhdsWithin_Ioo_right | Topology.Order | Mathlib/Topology/Order/Monotone.lean | [] | [
"ConditionallyCompleteLinearOrder",
"LinearOrder",
"MonotoneOn",
"OrderTopology",
"TopologicalSpace",
"exists_exists_and_eq_and",
"exists_lt_of_csInf_lt"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
MonotoneOn.tendsto_nhdsLT {α β : Type*} [LinearOrder α] [TopologicalSpace α] [OrderTopology α]
[ConditionallyCompleteLinearOrder β] [TopologicalSpace β] [OrderTopology β] {f : α → β} {x : α}
(Mf : MonotoneOn f (Iio x)) (h_bdd : BddAbove (f '' Iio x)) :
Tendsto f (𝓝[<] x) (𝓝 (sSup (f '' Iio x))) | by
rcases eq_empty_or_nonempty (Iio x) with (h | h); · simp [h]
refine tendsto_order.2 ⟨fun l hl => ?_, fun m hm => ?_⟩
· obtain ⟨z, zx, lz⟩ : ∃ a : α, a < x ∧ l < f a := by
simpa only [mem_image, exists_prop, exists_exists_and_eq_and] using!
exists_lt_of_lt_csSup (h.image _) hl
filter_upwards [... | lemma | MonotoneOn.tendsto_nhdsLT | Topology.Order | Mathlib/Topology/Order/Monotone.lean | [] | [
"BddAbove",
"ConditionallyCompleteLinearOrder",
"Ioo_mem_nhdsLT",
"LinearOrder",
"MonotoneOn",
"OrderTopology",
"TopologicalSpace",
"exists_exists_and_eq_and",
"exists_lt_of_lt_csSup",
"le_csSup",
"lt_of_le_of_lt",
"self_mem_nhdsWithin"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
MonotoneOn.tendsto_nhdsGT {α β : Type*} [LinearOrder α] [TopologicalSpace α] [OrderTopology α]
[ConditionallyCompleteLinearOrder β] [TopologicalSpace β] [OrderTopology β] {f : α → β} {x : α}
(Mf : MonotoneOn f (Ioi x)) (h_bdd : BddBelow (f '' Ioi x)) :
Tendsto f (𝓝[>] x) (𝓝 (sInf (f '' Ioi x))) | MonotoneOn.tendsto_nhdsLT (α := αᵒᵈ) (β := βᵒᵈ) Mf.dual h_bdd | lemma | MonotoneOn.tendsto_nhdsGT | Topology.Order | Mathlib/Topology/Order/Monotone.lean | [] | [
"ConditionallyCompleteLinearOrder",
"LinearOrder",
"MonotoneOn",
"MonotoneOn.tendsto_nhdsLT",
"OrderTopology",
"TopologicalSpace"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Monotone.tendsto_nhdsLT {α β : Type*} [LinearOrder α] [TopologicalSpace α] [OrderTopology α]
[ConditionallyCompleteLinearOrder β] [TopologicalSpace β] [OrderTopology β] {f : α → β}
(Mf : Monotone f) (x : α) : Tendsto f (𝓝[<] x) (𝓝 (sSup (f '' Iio x))) | MonotoneOn.tendsto_nhdsLT (Mf.monotoneOn _) (Mf.map_bddAbove bddAbove_Iio) | theorem | Monotone.tendsto_nhdsLT | Topology.Order | Mathlib/Topology/Order/Monotone.lean | [] | [
"ConditionallyCompleteLinearOrder",
"LinearOrder",
"Monotone",
"MonotoneOn.tendsto_nhdsLT",
"OrderTopology",
"TopologicalSpace",
"bddAbove_Iio"
] | A monotone map has a limit to the left of any point `x`, equal to `sSup (f '' (Iio x))`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
Monotone.tendsto_nhdsGT {α β : Type*} [LinearOrder α] [TopologicalSpace α] [OrderTopology α]
[ConditionallyCompleteLinearOrder β] [TopologicalSpace β] [OrderTopology β] {f : α → β}
(Mf : Monotone f) (x : α) : Tendsto f (𝓝[>] x) (𝓝 (sInf (f '' Ioi x))) | Monotone.tendsto_nhdsLT (α := αᵒᵈ) (β := βᵒᵈ) Mf.dual x | theorem | Monotone.tendsto_nhdsGT | Topology.Order | Mathlib/Topology/Order/Monotone.lean | [] | [
"ConditionallyCompleteLinearOrder",
"LinearOrder",
"Monotone",
"Monotone.tendsto_nhdsLT",
"OrderTopology",
"TopologicalSpace"
] | A monotone map has a limit to the right of any point `x`, equal to `sInf (f '' (Ioi x))`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.