statement stringlengths 1 2.93k | proof stringlengths 0 19.2k | type stringclasses 13
values | symbolic_name stringlengths 1 131 | library stringlengths 4 62 | filename stringlengths 20 95 | imports listlengths 0 10 | deps listlengths 0 64 | docstring stringlengths 0 4.95k | source_url stringclasses 1
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SecondCountableTopology.of_separableSpace_orderTopology [DenselyOrdered α]
[SeparableSpace α] : SecondCountableTopology α | by
rcases exists_countable_dense α with ⟨s, hc, hd⟩
refine ⟨⟨_, ?_, hd.topology_eq_generateFrom⟩⟩
exact (hc.image _).union (hc.image _) | theorem | SecondCountableTopology.of_separableSpace_orderTopology | Topology.Order | Mathlib/Topology/Order/Basic.lean | [] | [
"DenselyOrdered",
"SecondCountableTopology"
] | Let `α` be a densely ordered linear order with order topology. If `α` is a separable space, then
it has second countable topology. Note that the "densely ordered" assumption cannot be dropped, see
[double arrow space](https://topology.pi-base.org/spaces/S000093) for a counterexample. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
countable_setOf_covBy_right [SecondCountableTopology α] :
Set.Countable { x : α | ∃ y, x ⋖ y } | by
nontriviality α
let s := { x : α | ∃ y, x ⋖ y }
have : ∀ x ∈ s, ∃ y, x ⋖ y := fun x => id
choose! y hy using this
have Hy : ∀ x z, x ∈ s → z < y x → z ≤ x := fun x z hx => (hy x hx).le_of_lt
suffices H : ∀ a : Set α, IsOpen a → Set.Countable { x | x ∈ s ∧ x ∈ a ∧ y x ∉ a } by
have : s ⊆ ⋃ a ∈ countab... | theorem | countable_setOf_covBy_right | Topology.Order | Mathlib/Topology/Order/Basic.lean | [] | [
"IsBot",
"IsOpen",
"SecondCountableTopology",
"Set",
"Set.Countable",
"Set.Countable.mono",
"antisymm",
"exists_Ioc_subset_of_mem_nhds",
"isOpen_Ioo",
"le_of_lt",
"lt_irrefl",
"not_le"
] | The set of points which are isolated on the right is countable when the space is
second-countable. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
countable_of_isolated_left' [SecondCountableTopology α] :
Set.Countable { x : α | ∃ y, y < x ∧ Ioo y x = ∅ } | by
simpa only [← covBy_iff_Ioo_eq] using countable_setOf_covBy_left | theorem | countable_of_isolated_left' | Topology.Order | Mathlib/Topology/Order/Basic.lean | [] | [
"SecondCountableTopology",
"Set.Countable",
"covBy_iff_Ioo_eq"
] | The set of points which are isolated on the left is countable when the space is
second-countable. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
Set.PairwiseDisjoint.countable_of_Ioo [SecondCountableTopology α]
{y : α → α} {s : Set α} (h : PairwiseDisjoint s fun x => Ioo x (y x))
(h' : ∀ x ∈ s, x < y x) : s.Countable | have : (s \ { x | ∃ y, x ⋖ y }).Countable :=
(h.subset diff_subset).countable_of_isOpen (fun _ _ ↦ isOpen_Ioo) fun x hx ↦
(not_covBy_iff_exists_mem_Ioo (h' _ hx.1)).1 <| mt (Exists.intro (y x)) hx.2
this.of_diff countable_setOf_covBy_right | theorem | Set.PairwiseDisjoint.countable_of_Ioo | Topology.Order | Mathlib/Topology/Order/Basic.lean | [] | [
"Countable",
"SecondCountableTopology",
"Set",
"countable_setOf_covBy_right",
"isOpen_Ioo",
"not_covBy_iff_exists_mem_Ioo"
] | Consider a disjoint family of intervals `(x, y)` with `x < y` in a second-countable space.
Then the family is countable.
This is not a straightforward consequence of second-countability as some of these intervals might be
empty (but in fact this can happen only for countably many of them). | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
countable_image_lt_image_Ioi_within
[LinearOrder β] [SecondCountableTopology α] (t : Set β) (f : β → α) :
Set.Countable {x ∈ t | ∃ z, f x < z ∧ ∀ y ∈ t, x < y → z ≤ f y} | by
/- If the values of `f` are separated above on the right of `x`, there is an interval `(f x, z x)`
which is not reached by `f`. This gives a family of disjoint open intervals in `α`. Such a
family can only be countable as `α` is second-countable. -/
nontriviality β
have : Nonempty α := Nonempty.map f (... | theorem | countable_image_lt_image_Ioi_within | Topology.Order | Mathlib/Topology/Order/Basic.lean | [] | [
"Countable",
"LinearOrder",
"Nonempty.map",
"SecondCountableTopology",
"Set",
"Set.Countable",
"Set.PairwiseDisjoint.countable_of_Ioo",
"StrictMonoOn.injOn",
"le_of_not_ge",
"lt_irrefl",
"symm",
"trans"
] | For a function taking values in a second countable space, the set of points `x` for
which the image under `f` of `(x, ∞)` is separated above from `f x` is countable. We give
here a version relative to a set `t`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
countable_image_lt_image_Ioi [LinearOrder β] (f : β → α)
[SecondCountableTopology α] : Set.Countable {x | ∃ z, f x < z ∧ ∀ y, x < y → z ≤ f y} | by
simpa using countable_image_lt_image_Ioi_within univ f | theorem | countable_image_lt_image_Ioi | Topology.Order | Mathlib/Topology/Order/Basic.lean | [] | [
"LinearOrder",
"SecondCountableTopology",
"Set.Countable",
"countable_image_lt_image_Ioi_within"
] | For a function taking values in a second countable space, the set of points `x` for
which the image under `f` of `(x, ∞)` is separated above from `f x` is countable. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
countable_image_gt_image_Ioi_within
[LinearOrder β] [SecondCountableTopology α] (t : Set β) (f : β → α) :
Set.Countable {x ∈ t | ∃ z, z < f x ∧ ∀ y ∈ t, x < y → f y ≤ z} | countable_image_lt_image_Ioi_within (α := αᵒᵈ) t f | theorem | countable_image_gt_image_Ioi_within | Topology.Order | Mathlib/Topology/Order/Basic.lean | [] | [
"LinearOrder",
"SecondCountableTopology",
"Set",
"Set.Countable",
"countable_image_lt_image_Ioi_within"
] | For a function taking values in a second countable space, the set of points `x` for
which the image under `f` of `(x, ∞)` is separated below from `f x` is countable. We give
here a version relative to a set `t`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
countable_image_gt_image_Ioi [LinearOrder β] (f : β → α)
[SecondCountableTopology α] : Set.Countable {x | ∃ z, z < f x ∧ ∀ y, x < y → f y ≤ z} | countable_image_lt_image_Ioi (α := αᵒᵈ) f | theorem | countable_image_gt_image_Ioi | Topology.Order | Mathlib/Topology/Order/Basic.lean | [] | [
"LinearOrder",
"SecondCountableTopology",
"Set.Countable",
"countable_image_lt_image_Ioi"
] | For a function taking values in a second countable space, the set of points `x` for
which the image under `f` of `(x, ∞)` is separated below from `f x` is countable. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
instIsCountablyGenerated_atTop [SeparableSpace α] :
IsCountablyGenerated (atTop : Filter α) | by
obtain (h | ⟨x, hx⟩) := Set.eq_empty_or_nonempty {x : α | IsTop x}
· obtain ⟨s, s_count, hs⟩ := exists_countable_dense α
have : atTop = generate (Ici '' s) := by
refine atTop_eq_generate_of_not_bddAbove fun ⟨x, hx⟩ ↦ ?_
simp only [eq_empty_iff_forall_notMem, IsTop, mem_setOf_eq, not_forall, not_l... | instance | instIsCountablyGenerated_atTop | Topology.Order | Mathlib/Topology/Order/Basic.lean | [] | [
"Filter",
"Set.eq_empty_or_nonempty",
"isOpen_Ioi",
"not_le"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
pi_Iic_mem_nhds (ha : ∀ i, x i < a i) : Iic a ∈ 𝓝 x | pi_univ_Iic a ▸ set_pi_mem_nhds (Set.toFinite _) fun _ _ => Iic_mem_nhds (ha _) | theorem | pi_Iic_mem_nhds | Topology.Order | Mathlib/Topology/Order/Basic.lean | [] | [
"Set.toFinite",
"set_pi_mem_nhds"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
pi_Iic_mem_nhds' (ha : ∀ i, x' i < a' i) : Iic a' ∈ 𝓝 x' | pi_Iic_mem_nhds ha | theorem | pi_Iic_mem_nhds' | Topology.Order | Mathlib/Topology/Order/Basic.lean | [] | [
"pi_Iic_mem_nhds"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
pi_Icc_mem_nhds (ha : ∀ i, a i < x i) (hb : ∀ i, x i < b i) : Icc a b ∈ 𝓝 x | pi_univ_Icc a b ▸ set_pi_mem_nhds finite_univ fun _ _ => Icc_mem_nhds (ha _) (hb _) | theorem | pi_Icc_mem_nhds | Topology.Order | Mathlib/Topology/Order/Basic.lean | [] | [
"Icc_mem_nhds",
"set_pi_mem_nhds"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
pi_Icc_mem_nhds' (ha : ∀ i, a' i < x' i) (hb : ∀ i, x' i < b' i) : Icc a' b' ∈ 𝓝 x' | pi_Icc_mem_nhds ha hb | theorem | pi_Icc_mem_nhds' | Topology.Order | Mathlib/Topology/Order/Basic.lean | [] | [
"pi_Icc_mem_nhds"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
pi_Iio_mem_nhds (ha : ∀ i, x i < a i) : Iio a ∈ 𝓝 x | mem_of_superset
(set_pi_mem_nhds finite_univ fun i _ ↦ Iio_mem_nhds (ha i)) (pi_univ_Iio_subset a) | theorem | pi_Iio_mem_nhds | Topology.Order | Mathlib/Topology/Order/Basic.lean | [] | [
"set_pi_mem_nhds"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
pi_Iio_mem_nhds' (ha : ∀ i, x' i < a' i) : Iio a' ∈ 𝓝 x' | pi_Iio_mem_nhds ha | theorem | pi_Iio_mem_nhds' | Topology.Order | Mathlib/Topology/Order/Basic.lean | [] | [
"pi_Iio_mem_nhds"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
pi_Ioc_mem_nhds (ha : ∀ i, a i < x i) (hb : ∀ i, x i < b i) : Ioc a b ∈ 𝓝 x | by
refine mem_of_superset (set_pi_mem_nhds Set.finite_univ fun i _ => ?_) (pi_univ_Ioc_subset a b)
exact Ioc_mem_nhds (ha i) (hb i) | theorem | pi_Ioc_mem_nhds | Topology.Order | Mathlib/Topology/Order/Basic.lean | [] | [
"Ioc_mem_nhds",
"Set.finite_univ",
"set_pi_mem_nhds"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
pi_Ioc_mem_nhds' (ha : ∀ i, a' i < x' i) (hb : ∀ i, x' i < b' i) : Ioc a' b' ∈ 𝓝 x' | pi_Ioc_mem_nhds ha hb | theorem | pi_Ioc_mem_nhds' | Topology.Order | Mathlib/Topology/Order/Basic.lean | [] | [
"pi_Ioc_mem_nhds"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
pi_Ioo_mem_nhds (ha : ∀ i, a i < x i) (hb : ∀ i, x i < b i) : Ioo a b ∈ 𝓝 x | by
refine mem_of_superset (set_pi_mem_nhds Set.finite_univ fun i _ => ?_) (pi_univ_Ioo_subset a b)
exact Ioo_mem_nhds (ha i) (hb i) | theorem | pi_Ioo_mem_nhds | Topology.Order | Mathlib/Topology/Order/Basic.lean | [] | [
"Ioo_mem_nhds",
"Set.finite_univ",
"set_pi_mem_nhds"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
pi_Ioo_mem_nhds' (ha : ∀ i, a' i < x' i) (hb : ∀ i, x' i < b' i) : Ioo a' b' ∈ 𝓝 x' | pi_Ioo_mem_nhds ha hb | theorem | pi_Ioo_mem_nhds' | Topology.Order | Mathlib/Topology/Order/Basic.lean | [] | [
"pi_Ioo_mem_nhds"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
LeftOrdContinuous.continuousWithinAt_Iic (hf : LeftOrdContinuous f) :
ContinuousWithinAt f (Iic x) x | by
rw [ContinuousWithinAt, OrderTopology.topology_eq_generate_intervals (α := Y)]
simp_rw [TopologicalSpace.tendsto_nhds_generateFrom_iff, mem_nhdsWithin]
rintro V ⟨z, rfl | rfl⟩ hxz
-- The case `V = Ioi z`.
· obtain ⟨_, ⟨a, hax, rfl⟩, hza⟩ := (lt_isLUB_iff <| hf isLUB_Iio).mp hxz
exact ⟨Ioi a, isOpen_Ioi... | lemma | LeftOrdContinuous.continuousWithinAt_Iic | Topology.Order | Mathlib/Topology/Order/Basic.lean | [] | [
"ContinuousWithinAt",
"LeftOrdContinuous",
"TopologicalSpace.tendsto_nhds_generateFrom_iff",
"isLUB_Iio",
"isOpen_Ioi",
"isOpen_univ",
"lt_isLUB_iff",
"mem_nhdsWithin"
] | An order-theoretically left-continuous function is topologically left-continuous, assuming
the function is between conditionally complete linear orders with order topologies, and the domain
is densely ordered. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
RightOrdContinuous.continuousWithinAt_Ici (hf : RightOrdContinuous f) :
ContinuousWithinAt f (Ici x) x | hf.dual.continuousWithinAt_Iic | lemma | RightOrdContinuous.continuousWithinAt_Ici | Topology.Order | Mathlib/Topology/Order/Basic.lean | [] | [
"ContinuousWithinAt"
] | An order-theoretically right-continuous function is topologically right-continuous, assuming
the function is between conditionally complete linear orders with order topologies, and the domain
is densely ordered. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
orderBornology : Bornology α | .ofBounded
{s | BddBelow s ∧ BddAbove s}
(by simp)
(fun _ hs _ hst ↦ ⟨hs.1.mono hst, hs.2.mono hst⟩)
(fun _ hs _ ht ↦ ⟨hs.1.union ht.1, hs.2.union ht.2⟩)
(by simp) | def | orderBornology | Topology.Order | Mathlib/Topology/Order/Bornology.lean | [] | [
"BddAbove",
"Bornology"
] | Order-bornology on a nonempty lattice. The bounded sets are the sets that are bounded both above
and below. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
orderBornology_isBounded : orderBornology.IsBounded s ↔ BddBelow s ∧ BddAbove s | by
simp [IsBounded, IsCobounded, -isCobounded_compl_iff] | lemma | orderBornology_isBounded | Topology.Order | Mathlib/Topology/Order/Bornology.lean | [] | [
"BddAbove"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
IsOrderBornology : Prop where
protected isBounded_iff_bddBelow_bddAbove (s : Set α) : IsBounded s ↔ BddBelow s ∧ BddAbove s | class | IsOrderBornology | Topology.Order | Mathlib/Topology/Order/Bornology.lean | [] | [
"BddAbove",
"Set",
"isBounded_iff_bddBelow_bddAbove"
] | Predicate for a preorder to be equipped with its order-bornology, namely for its bounded sets
to be the ones that are bounded both above and below. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
isOrderBornology_iff_eq_orderBornology [Lattice α] [Nonempty α] :
IsOrderBornology α ↔ ‹Bornology α› = orderBornology | by
refine ⟨fun h ↦ ?_, fun h ↦ ⟨fun s ↦ by rw [h, orderBornology_isBounded]⟩⟩
ext s
exact isBounded_compl_iff.symm.trans (h.1 _) | lemma | isOrderBornology_iff_eq_orderBornology | Topology.Order | Mathlib/Topology/Order/Bornology.lean | [] | [
"IsOrderBornology",
"Lattice",
"orderBornology",
"orderBornology_isBounded"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
isBounded_iff_bddBelow_bddAbove : IsBounded s ↔ BddBelow s ∧ BddAbove s | IsOrderBornology.isBounded_iff_bddBelow_bddAbove _ | lemma | isBounded_iff_bddBelow_bddAbove | Topology.Order | Mathlib/Topology/Order/Bornology.lean | [] | [
"BddAbove"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Bornology.IsBounded.bddBelow (hs : IsBounded s) : BddBelow s | (isBounded_iff_bddBelow_bddAbove.1 hs).1 | lemma | Bornology.IsBounded.bddBelow | Topology.Order | Mathlib/Topology/Order/Bornology.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Bornology.IsBounded.bddAbove (hs : IsBounded s) : BddAbove s | (isBounded_iff_bddBelow_bddAbove.1 hs).2 | lemma | Bornology.IsBounded.bddAbove | Topology.Order | Mathlib/Topology/Order/Bornology.lean | [] | [
"BddAbove"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
BddBelow.isBounded (hs₀ : BddBelow s) (hs₁ : BddAbove s) : IsBounded s | isBounded_iff_bddBelow_bddAbove.2 ⟨hs₀, hs₁⟩ | lemma | BddBelow.isBounded | Topology.Order | Mathlib/Topology/Order/Bornology.lean | [] | [
"BddAbove"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
BddAbove.isBounded (hs₀ : BddAbove s) (hs₁ : BddBelow s) : IsBounded s | isBounded_iff_bddBelow_bddAbove.2 ⟨hs₁, hs₀⟩ | lemma | BddAbove.isBounded | Topology.Order | Mathlib/Topology/Order/Bornology.lean | [] | [
"BddAbove"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
BddBelow.isBounded_inter (hs : BddBelow s) (ht : BddAbove t) : IsBounded (s ∩ t) | (hs.mono inter_subset_left).isBounded <| ht.mono inter_subset_right | lemma | BddBelow.isBounded_inter | Topology.Order | Mathlib/Topology/Order/Bornology.lean | [] | [
"BddAbove"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
BddAbove.isBounded_inter (hs : BddAbove s) (ht : BddBelow t) : IsBounded (s ∩ t) | (hs.mono inter_subset_left).isBounded <| ht.mono inter_subset_right | lemma | BddAbove.isBounded_inter | Topology.Order | Mathlib/Topology/Order/Bornology.lean | [] | [
"BddAbove"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
OrderDual.instIsOrderBornology : IsOrderBornology αᵒᵈ | where
isBounded_iff_bddBelow_bddAbove s := by
rw [← isBounded_preimage_toDual, ← bddBelow_preimage_toDual, ← bddAbove_preimage_toDual,
isBounded_iff_bddBelow_bddAbove, and_comm] | instance | OrderDual.instIsOrderBornology | Topology.Order | Mathlib/Topology/Order/Bornology.lean | [] | [
"IsOrderBornology",
"bddAbove_preimage_toDual",
"isBounded_iff_bddBelow_bddAbove"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Prod.instIsOrderBornology {β : Type*} [Preorder β] [Bornology β] [IsOrderBornology β] :
IsOrderBornology (α × β) | where
isBounded_iff_bddBelow_bddAbove s := by
rw [← isBounded_image_fst_and_snd, bddBelow_prod, bddAbove_prod, and_and_and_comm,
isBounded_iff_bddBelow_bddAbove, isBounded_iff_bddBelow_bddAbove] | instance | Prod.instIsOrderBornology | Topology.Order | Mathlib/Topology/Order/Bornology.lean | [] | [
"Bornology",
"IsOrderBornology",
"Preorder",
"bddAbove_prod",
"isBounded_iff_bddBelow_bddAbove"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Pi.instIsOrderBornology {ι : Type*} {α : ι → Type*} [∀ i, Preorder (α i)]
[∀ i, Bornology (α i)] [∀ i, IsOrderBornology (α i)] : IsOrderBornology (∀ i, α i) | where
isBounded_iff_bddBelow_bddAbove s := by
simp_rw [← forall_isBounded_image_eval_iff, bddBelow_pi, bddAbove_pi, ← forall_and,
isBounded_iff_bddBelow_bddAbove] | instance | Pi.instIsOrderBornology | Topology.Order | Mathlib/Topology/Order/Bornology.lean | [] | [
"Bornology",
"IsOrderBornology",
"Preorder",
"bddAbove_pi",
"isBounded_iff_bddBelow_bddAbove"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
IsOrderBornology.atTop_le_cobounded [NoMaxOrder α] : .atTop ≤ Bornology.cobounded α | by
intro s
rw [← compl_compl s, ← isBounded_def, isBounded_iff_bddBelow_bddAbove, compl_compl s,
Filter.atTop_basis_Ioi.mem_iff]
intro ⟨_, b, hb⟩
rw [mem_upperBounds_iff_subset_Iic, ← compl_compl (Iic b), compl_subset_compl, compl_Iic] at hb
use b | lemma | IsOrderBornology.atTop_le_cobounded | Topology.Order | Mathlib/Topology/Order/Bornology.lean | [] | [
"NoMaxOrder",
"compl_compl",
"isBounded_iff_bddBelow_bddAbove",
"mem_upperBounds_iff_subset_Iic"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
IsOrderBornology.atBot_le_cobounded [NoMinOrder α] : .atBot ≤ Bornology.cobounded α | atTop_le_cobounded (α := αᵒᵈ) | lemma | IsOrderBornology.atBot_le_cobounded | Topology.Order | Mathlib/Topology/Order/Bornology.lean | [] | [
"NoMinOrder"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
IsOrderBornology.cobounded_le_atBot_sup_atTop :
cobounded α ≤ .atBot ⊔ .atTop | by
intro s
rw [Filter.mem_sup, Filter.atTop_basis.mem_iff, Filter.atBot_basis.mem_iff,
← compl_compl s, ← isBounded_def, isBounded_iff_bddBelow_bddAbove, compl_compl s]
intro ⟨⟨b, _, hb⟩, ⟨a, _, ha⟩⟩
refine ⟨⟨b, fun x hx ↦ ?_⟩, ⟨a, fun x hx ↦ ?_⟩⟩ <;> by_contra! hx'
· exact hx (hb hx'.le)
· exact hx (ha... | lemma | IsOrderBornology.cobounded_le_atBot_sup_atTop | Topology.Order | Mathlib/Topology/Order/Bornology.lean | [] | [
"Filter.mem_sup",
"compl_compl",
"isBounded_iff_bddBelow_bddAbove"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
IsOrderBornology.cobounded_eq [NoMaxOrder α] [NoMinOrder α] :
Bornology.cobounded α = .atBot ⊔ .atTop | cobounded_le_atBot_sup_atTop.antisymm <|
sup_le IsOrderBornology.atBot_le_cobounded IsOrderBornology.atTop_le_cobounded | lemma | IsOrderBornology.cobounded_eq | Topology.Order | Mathlib/Topology/Order/Bornology.lean | [] | [
"IsOrderBornology.atBot_le_cobounded",
"IsOrderBornology.atTop_le_cobounded",
"NoMaxOrder",
"NoMinOrder",
"sup_le"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
IsOrderBornology.cobounded_eq_atTop [NoMaxOrder α] [OrderBot α] :
Bornology.cobounded α = .atTop | by
refine atTop_le_cobounded.antisymm' fun s ↦ ?_
rw [Filter.atTop_basis.mem_iff,
← compl_compl s, ← isBounded_def, isBounded_iff_bddBelow_bddAbove, compl_compl s]
refine fun ⟨b, _, hb⟩ ↦ ⟨⟨⊥, fun x hx ↦ by simp⟩, ⟨b, fun x hx ↦ ?_⟩⟩
by_contra! hx'
exact hx (hb hx'.le) | lemma | IsOrderBornology.cobounded_eq_atTop | Topology.Order | Mathlib/Topology/Order/Bornology.lean | [] | [
"NoMaxOrder",
"OrderBot",
"compl_compl",
"isBounded_iff_bddBelow_bddAbove"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
IsOrderBornology.cobounded_eq_atBot [NoMinOrder α] [OrderTop α] :
Bornology.cobounded α = .atBot | cobounded_eq_atTop (α := αᵒᵈ) | lemma | IsOrderBornology.cobounded_eq_atBot | Topology.Order | Mathlib/Topology/Order/Bornology.lean | [] | [
"NoMinOrder",
"OrderTop"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Bornology.IsBounded.subset_Icc_sInf_sSup (hs : IsBounded s) :
s ⊆ Icc (sInf s) (sSup s) | subset_Icc_csInf_csSup hs.bddBelow hs.bddAbove | lemma | Bornology.IsBounded.subset_Icc_sInf_sSup | Topology.Order | Mathlib/Topology/Order/Bornology.lean | [] | [
"subset_Icc_csInf_csSup"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
CompactIccSpace (α : Type*) [TopologicalSpace α] [Preorder α] : Prop where
/-- A closed interval `Set.Icc a b` is a compact set for all `a` and `b`. -/
isCompact_Icc : ∀ {a b : α}, IsCompact (Icc a b) | class | CompactIccSpace | Topology.Order | Mathlib/Topology/Order/Compact.lean | [] | [
"IsCompact",
"Preorder",
"TopologicalSpace"
] | This typeclass says that all closed intervals in `α` are compact. This is true for all
conditionally complete linear orders with order topology and products (finite or infinite)
of such spaces. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
CompactIccSpace.mk' [TopologicalSpace α] [Preorder α]
(h : ∀ {a b : α}, a ≤ b → IsCompact (Icc a b)) : CompactIccSpace α | where
isCompact_Icc {a b} := by_cases h fun hab => by rw [Icc_eq_empty hab]; exact isCompact_empty | lemma | CompactIccSpace.mk' | Topology.Order | Mathlib/Topology/Order/Compact.lean | [] | [
"CompactIccSpace",
"IsCompact",
"Preorder",
"TopologicalSpace",
"by_cases",
"isCompact_empty"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
CompactIccSpace.mk'' [TopologicalSpace α] [PartialOrder α]
(h : ∀ {a b : α}, a < b → IsCompact (Icc a b)) : CompactIccSpace α | .mk' fun hab => hab.eq_or_lt.elim (by rintro rfl; simp) h | lemma | CompactIccSpace.mk'' | Topology.Order | Mathlib/Topology/Order/Compact.lean | [] | [
"CompactIccSpace",
"IsCompact",
"PartialOrder",
"TopologicalSpace"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Pi.compact_Icc_space' {α β : Type*} [Preorder β] [TopologicalSpace β]
[CompactIccSpace β] : CompactIccSpace (α → β) | inferInstance | instance | Pi.compact_Icc_space' | Topology.Order | Mathlib/Topology/Order/Compact.lean | [] | [
"CompactIccSpace",
"Preorder",
"TopologicalSpace"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
isCompact_uIcc {α : Type*} [LinearOrder α] [TopologicalSpace α] [CompactIccSpace α]
{a b : α} : IsCompact (uIcc a b) | isCompact_Icc | theorem | isCompact_uIcc | Topology.Order | Mathlib/Topology/Order/Compact.lean | [] | [
"CompactIccSpace",
"IsCompact",
"LinearOrder",
"TopologicalSpace"
] | An unordered closed interval is compact. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
compactSpace_Icc (a b : α) : CompactSpace (Icc a b) | isCompact_iff_compactSpace.mp isCompact_Icc | instance | compactSpace_Icc | Topology.Order | Mathlib/Topology/Order/Compact.lean | [] | [
"CompactSpace"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
isCompact_Ico_iff {a b : α} : IsCompact (Set.Ico a b) ↔ b ≤ a | ⟨fun h => isClosed_Ico_iff.mp h.isClosed, by simp_all⟩ | theorem | isCompact_Ico_iff | Topology.Order | Mathlib/Topology/Order/Compact.lean | [] | [
"IsCompact",
"Set.Ico"
] | `Set.Ico a b` is only compact if it is empty. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
isCompact_Ioc_iff {a b : α} : IsCompact (Set.Ioc a b) ↔ b ≤ a | ⟨fun h => isClosed_Ioc_iff.mp h.isClosed, by simp_all⟩ | theorem | isCompact_Ioc_iff | Topology.Order | Mathlib/Topology/Order/Compact.lean | [] | [
"IsCompact",
"Set.Ioc"
] | `Set.Ioc a b` is only compact if it is empty. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
isCompact_Ioo_iff {a b : α} : IsCompact (Set.Ioo a b) ↔ b ≤ a | ⟨fun h => isClosed_Ioo_iff.mp h.isClosed, by simp_all⟩ | theorem | isCompact_Ioo_iff | Topology.Order | Mathlib/Topology/Order/Compact.lean | [] | [
"IsCompact",
"Set.Ioo"
] | `Set.Ioo a b` is only compact if it is empty. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
IsCompact.exists_isLeast [ClosedIicTopology α] {s : Set α} (hs : IsCompact s)
(ne_s : s.Nonempty) : ∃ x, IsLeast s x | by
haveI : Nonempty s := ne_s.to_subtype
suffices (s ∩ ⋂ x ∈ s, Iic x).Nonempty from
⟨this.choose, this.choose_spec.1, mem_iInter₂.mp this.choose_spec.2⟩
rw [biInter_eq_iInter]
by_contra H
rw [not_nonempty_iff_eq_empty] at H
rcases hs.elim_directed_family_closed (fun x : s => Iic ↑x) (fun x => isClosed_... | theorem | IsCompact.exists_isLeast | Topology.Order | Mathlib/Topology/Order/Compact.lean | [] | [
"ClosedIicTopology",
"IsCompact",
"IsLeast",
"Set",
"isClosed_Iic"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
IsCompact.exists_isGreatest [ClosedIciTopology α] {s : Set α} (hs : IsCompact s)
(ne_s : s.Nonempty) : ∃ x, IsGreatest s x | IsCompact.exists_isLeast (α := αᵒᵈ) hs ne_s | theorem | IsCompact.exists_isGreatest | Topology.Order | Mathlib/Topology/Order/Compact.lean | [] | [
"ClosedIciTopology",
"IsCompact",
"IsCompact.exists_isLeast",
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
IsCompact.exists_isGLB [ClosedIicTopology α] {s : Set α} (hs : IsCompact s)
(ne_s : s.Nonempty) : ∃ x ∈ s, IsGLB s x | (hs.exists_isLeast ne_s).imp (fun x (hx : IsLeast s x) => ⟨hx.1, hx.isGLB⟩) | theorem | IsCompact.exists_isGLB | Topology.Order | Mathlib/Topology/Order/Compact.lean | [] | [
"ClosedIicTopology",
"IsCompact",
"IsLeast",
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
IsCompact.exists_isLUB [ClosedIciTopology α] {s : Set α} (hs : IsCompact s)
(ne_s : s.Nonempty) : ∃ x ∈ s, IsLUB s x | IsCompact.exists_isGLB (α := αᵒᵈ) hs ne_s | theorem | IsCompact.exists_isLUB | Topology.Order | Mathlib/Topology/Order/Compact.lean | [] | [
"ClosedIciTopology",
"IsCompact",
"IsCompact.exists_isGLB",
"IsLUB",
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
cocompact_le_atBot_atTop [CompactIccSpace α] :
cocompact α ≤ atBot ⊔ atTop | by
refine fun s hs ↦ mem_cocompact.mpr <| (isEmpty_or_nonempty α).casesOn ?_ ?_ <;> intro
· exact ⟨∅, isCompact_empty, fun x _ ↦ (IsEmpty.false x).elim⟩
· obtain ⟨t, ht⟩ := mem_atBot_sets.mp hs.1
obtain ⟨u, hu⟩ := mem_atTop_sets.mp hs.2
refine ⟨Icc t u, isCompact_Icc, fun x hx ↦ ?_⟩
exact (not_and_or.... | theorem | cocompact_le_atBot_atTop | Topology.Order | Mathlib/Topology/Order/Compact.lean | [] | [
"CompactIccSpace",
"isCompact_empty",
"isEmpty_or_nonempty",
"le_of_not_ge"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
cocompact_le_atBot [OrderTop α] [CompactIccSpace α] :
cocompact α ≤ atBot | by
refine fun _ hs ↦ mem_cocompact.mpr <| (isEmpty_or_nonempty α).casesOn ?_ ?_ <;> intro
· exact ⟨∅, isCompact_empty, fun x _ ↦ (IsEmpty.false x).elim⟩
· obtain ⟨t, ht⟩ := mem_atBot_sets.mp hs
refine ⟨Icc t ⊤, isCompact_Icc, fun _ hx ↦ ?_⟩
exact (not_and_or.mp hx).casesOn (fun h ↦ ht _ (le_of_not_ge h)) ... | theorem | cocompact_le_atBot | Topology.Order | Mathlib/Topology/Order/Compact.lean | [] | [
"CompactIccSpace",
"OrderTop",
"isCompact_empty",
"isEmpty_or_nonempty",
"le_of_not_ge",
"le_top"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
cocompact_le_atTop [OrderBot α] [CompactIccSpace α] :
cocompact α ≤ atTop | cocompact_le_atBot (α := αᵒᵈ) | theorem | cocompact_le_atTop | Topology.Order | Mathlib/Topology/Order/Compact.lean | [] | [
"CompactIccSpace",
"OrderBot",
"cocompact_le_atBot"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
atBot_le_cocompact [NoMinOrder α] [ClosedIicTopology α] :
atBot ≤ cocompact α | by
refine fun s hs ↦ ?_
obtain ⟨t, ht, hts⟩ := mem_cocompact.mp hs
refine (Set.eq_empty_or_nonempty t).casesOn (fun h_empty ↦ ?_) (fun h_nonempty ↦ ?_)
· rewrite [compl_univ_iff.mpr h_empty, univ_subset_iff] at hts
convert! univ_mem
· haveI := h_nonempty.nonempty
obtain ⟨a, ha⟩ := ht.exists_isLeast h_... | theorem | atBot_le_cocompact | Topology.Order | Mathlib/Topology/Order/Compact.lean | [] | [
"ClosedIicTopology",
"LT.lt.false",
"NoMinOrder",
"Set.eq_empty_or_nonempty"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
atTop_le_cocompact [NoMaxOrder α] [ClosedIciTopology α] :
atTop ≤ cocompact α | atBot_le_cocompact (α := αᵒᵈ) | theorem | atTop_le_cocompact | Topology.Order | Mathlib/Topology/Order/Compact.lean | [] | [
"ClosedIciTopology",
"NoMaxOrder",
"atBot_le_cocompact"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
atBot_atTop_le_cocompact [NoMinOrder α] [NoMaxOrder α]
[OrderClosedTopology α] : atBot ⊔ atTop ≤ cocompact α | sup_le atBot_le_cocompact atTop_le_cocompact | theorem | atBot_atTop_le_cocompact | Topology.Order | Mathlib/Topology/Order/Compact.lean | [] | [
"NoMaxOrder",
"NoMinOrder",
"OrderClosedTopology",
"atBot_le_cocompact",
"atTop_le_cocompact",
"sup_le"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
cocompact_eq_atBot_atTop [NoMaxOrder α] [NoMinOrder α]
[OrderClosedTopology α] [CompactIccSpace α] : cocompact α = atBot ⊔ atTop | cocompact_le_atBot_atTop.antisymm atBot_atTop_le_cocompact | theorem | cocompact_eq_atBot_atTop | Topology.Order | Mathlib/Topology/Order/Compact.lean | [] | [
"CompactIccSpace",
"NoMaxOrder",
"NoMinOrder",
"OrderClosedTopology",
"atBot_atTop_le_cocompact"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
cocompact_eq_atBot [NoMinOrder α] [OrderTop α]
[ClosedIicTopology α] [CompactIccSpace α] : cocompact α = atBot | cocompact_le_atBot.antisymm atBot_le_cocompact | theorem | cocompact_eq_atBot | Topology.Order | Mathlib/Topology/Order/Compact.lean | [] | [
"ClosedIicTopology",
"CompactIccSpace",
"NoMinOrder",
"OrderTop",
"atBot_le_cocompact"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
cocompact_eq_atTop [NoMaxOrder α] [OrderBot α]
[ClosedIciTopology α] [CompactIccSpace α] : cocompact α = atTop | cocompact_le_atTop.antisymm atTop_le_cocompact | theorem | cocompact_eq_atTop | Topology.Order | Mathlib/Topology/Order/Compact.lean | [] | [
"ClosedIciTopology",
"CompactIccSpace",
"NoMaxOrder",
"OrderBot",
"atTop_le_cocompact"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
IsCompact.exists_isMinOn [ClosedIicTopology α] {s : Set β} (hs : IsCompact s)
(ne_s : s.Nonempty) {f : β → α} (hf : ContinuousOn f s) : ∃ x ∈ s, IsMinOn f s x | by
rcases (hs.image_of_continuousOn hf).exists_isLeast (ne_s.image f) with ⟨_, ⟨x, hxs, rfl⟩, hx⟩
refine ⟨x, hxs, forall_mem_image.1 (fun _ hb => hx <| mem_image_of_mem f ?_)⟩
rwa [(image_id' s).symm] | theorem | IsCompact.exists_isMinOn | Topology.Order | Mathlib/Topology/Order/Compact.lean | [] | [
"ClosedIicTopology",
"ContinuousOn",
"IsCompact",
"IsMinOn",
"Set",
"symm"
] | The **extreme value theorem**: a continuous function realizes its minimum on a compact set. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
IsCompact.exists_forall_le' [ClosedIicTopology α] [NoMaxOrder α] {f : β → α}
{s : Set β} (hs : IsCompact s) (hf : ContinuousOn f s) {a : α} (hf' : ∀ b ∈ s, a < f b) :
∃ a', a < a' ∧ ∀ b ∈ s, a' ≤ f b | by
rcases s.eq_empty_or_nonempty with (rfl | hs')
· obtain ⟨a', ha'⟩ := exists_gt a
exact ⟨a', ha', fun _ a ↦ a.elim⟩
· obtain ⟨x, hx, hx'⟩ := hs.exists_isMinOn hs' hf
exact ⟨f x, hf' x hx, hx'⟩ | theorem | IsCompact.exists_forall_le' | Topology.Order | Mathlib/Topology/Order/Compact.lean | [] | [
"ClosedIicTopology",
"ContinuousOn",
"IsCompact",
"NoMaxOrder",
"Set"
] | If a continuous function lies strictly above `a` on a compact set,
it has a lower bound strictly above `a`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
IsCompact.exists_isMaxOn [ClosedIciTopology α] {s : Set β} (hs : IsCompact s)
(ne_s : s.Nonempty) {f : β → α} (hf : ContinuousOn f s) : ∃ x ∈ s, IsMaxOn f s x | IsCompact.exists_isMinOn (α := αᵒᵈ) hs ne_s hf | theorem | IsCompact.exists_isMaxOn | Topology.Order | Mathlib/Topology/Order/Compact.lean | [] | [
"ClosedIciTopology",
"ContinuousOn",
"IsCompact",
"IsCompact.exists_isMinOn",
"IsMaxOn",
"Set"
] | The **extreme value theorem**: a continuous function realizes its maximum on a compact set. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
ContinuousOn.exists_isMinOn' [ClosedIicTopology α] {s : Set β} {f : β → α}
(hf : ContinuousOn f s) (hsc : IsClosed s) {x₀ : β} (h₀ : x₀ ∈ s)
(hc : ∀ᶠ x in cocompact β ⊓ 𝓟 s, f x₀ ≤ f x) : ∃ x ∈ s, IsMinOn f s x | by
rcases (hasBasis_cocompact.inf_principal _).eventually_iff.1 hc with ⟨K, hK, hKf⟩
have hsub : insert x₀ (K ∩ s) ⊆ s := insert_subset_iff.2 ⟨h₀, inter_subset_right⟩
obtain ⟨x, hx, hxf⟩ : ∃ x ∈ insert x₀ (K ∩ s), ∀ y ∈ insert x₀ (K ∩ s), f x ≤ f y :=
((hK.inter_right hsc).insert x₀).exists_isMinOn (insert_no... | theorem | ContinuousOn.exists_isMinOn' | Topology.Order | Mathlib/Topology/Order/Compact.lean | [] | [
"ClosedIicTopology",
"ContinuousOn",
"IsClosed",
"IsMinOn",
"Set",
"by_cases",
"trans"
] | The **extreme value theorem**: if a function `f` is continuous on a closed set `s` and it is
larger than a value in its image away from compact sets, then it has a minimum on this set. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
ContinuousOn.exists_isMaxOn' [ClosedIciTopology α] {s : Set β} {f : β → α}
(hf : ContinuousOn f s) (hsc : IsClosed s) {x₀ : β} (h₀ : x₀ ∈ s)
(hc : ∀ᶠ x in cocompact β ⊓ 𝓟 s, f x ≤ f x₀) : ∃ x ∈ s, IsMaxOn f s x | ContinuousOn.exists_isMinOn' (α := αᵒᵈ) hf hsc h₀ hc | theorem | ContinuousOn.exists_isMaxOn' | Topology.Order | Mathlib/Topology/Order/Compact.lean | [] | [
"ClosedIciTopology",
"ContinuousOn",
"ContinuousOn.exists_isMinOn'",
"IsClosed",
"IsMaxOn",
"Set"
] | The **extreme value theorem**: if a function `f` is continuous on a closed set `s` and it is
smaller than a value in its image away from compact sets, then it has a maximum on this set. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
Continuous.exists_forall_le' [ClosedIicTopology α] {f : β → α} (hf : Continuous f)
(x₀ : β) (h : ∀ᶠ x in cocompact β, f x₀ ≤ f x) : ∃ x : β, ∀ y : β, f x ≤ f y | let ⟨x, _, hx⟩ := hf.continuousOn.exists_isMinOn' isClosed_univ (mem_univ x₀)
(by rwa [principal_univ, inf_top_eq])
⟨x, fun y => hx (mem_univ y)⟩ | theorem | Continuous.exists_forall_le' | Topology.Order | Mathlib/Topology/Order/Compact.lean | [] | [
"ClosedIicTopology",
"Continuous",
"isClosed_univ"
] | The **extreme value theorem**: if a continuous function `f` is larger than a value in its range
away from compact sets, then it has a global minimum. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
Continuous.exists_forall_ge' [ClosedIciTopology α] {f : β → α} (hf : Continuous f)
(x₀ : β) (h : ∀ᶠ x in cocompact β, f x ≤ f x₀) : ∃ x : β, ∀ y : β, f y ≤ f x | Continuous.exists_forall_le' (α := αᵒᵈ) hf x₀ h | theorem | Continuous.exists_forall_ge' | Topology.Order | Mathlib/Topology/Order/Compact.lean | [] | [
"ClosedIciTopology",
"Continuous",
"Continuous.exists_forall_le'"
] | The **extreme value theorem**: if a continuous function `f` is smaller than a value in its range
away from compact sets, then it has a global maximum. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
Continuous.exists_forall_le [ClosedIicTopology α] [Nonempty β] {f : β → α}
(hf : Continuous f) (hlim : Tendsto f (cocompact β) atTop) : ∃ x, ∀ y, f x ≤ f y | by
inhabit β
exact hf.exists_forall_le' default (hlim.eventually <| eventually_ge_atTop _) | theorem | Continuous.exists_forall_le | Topology.Order | Mathlib/Topology/Order/Compact.lean | [] | [
"ClosedIicTopology",
"Continuous"
] | The **extreme value theorem**: if a continuous function `f` tends to infinity away from compact
sets, then it has a global minimum. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
Continuous.exists_forall_ge [ClosedIciTopology α] [Nonempty β] {f : β → α}
(hf : Continuous f) (hlim : Tendsto f (cocompact β) atBot) : ∃ x, ∀ y, f y ≤ f x | Continuous.exists_forall_le (α := αᵒᵈ) hf hlim | theorem | Continuous.exists_forall_ge | Topology.Order | Mathlib/Topology/Order/Compact.lean | [] | [
"ClosedIciTopology",
"Continuous",
"Continuous.exists_forall_le"
] | The **extreme value theorem**: if a continuous function `f` tends to negative infinity away from
compact sets, then it has a global maximum. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
Continuous.exists_forall_le_of_hasCompactMulSupport [ClosedIicTopology α] [Nonempty β]
[One α] {f : β → α} (hf : Continuous f) (h : HasCompactMulSupport f) :
∃ x : β, ∀ y : β, f x ≤ f y | by
obtain ⟨_, ⟨x, rfl⟩, hx⟩ := (h.isCompact_range hf).exists_isLeast (range_nonempty _)
rw [mem_lowerBounds, forall_mem_range] at hx
exact ⟨x, hx⟩ | theorem | Continuous.exists_forall_le_of_hasCompactMulSupport | Topology.Order | Mathlib/Topology/Order/Compact.lean | [] | [
"ClosedIicTopology",
"Continuous",
"HasCompactMulSupport"
] | A continuous function with compact support has a global minimum. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
Continuous.exists_forall_ge_of_hasCompactMulSupport [ClosedIciTopology α] [Nonempty β]
[One α] {f : β → α} (hf : Continuous f) (h : HasCompactMulSupport f) :
∃ x : β, ∀ y : β, f y ≤ f x | Continuous.exists_forall_le_of_hasCompactMulSupport (α := αᵒᵈ) hf h | theorem | Continuous.exists_forall_ge_of_hasCompactMulSupport | Topology.Order | Mathlib/Topology/Order/Compact.lean | [] | [
"ClosedIciTopology",
"Continuous",
"Continuous.exists_forall_le_of_hasCompactMulSupport",
"HasCompactMulSupport"
] | A continuous function with compact support has a global maximum. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
IsCompact.bddBelow [ClosedIicTopology α] [Nonempty α] {s : Set α} (hs : IsCompact s) :
BddBelow s | by
rcases s.eq_empty_or_nonempty with rfl | hne
· exact bddBelow_empty
· obtain ⟨a, -, has⟩ := hs.exists_isLeast hne
exact ⟨a, has⟩ | theorem | IsCompact.bddBelow | Topology.Order | Mathlib/Topology/Order/Compact.lean | [] | [
"ClosedIicTopology",
"IsCompact",
"Set"
] | A compact set is bounded below | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
IsCompact.bddAbove [ClosedIciTopology α] [Nonempty α] {s : Set α} (hs : IsCompact s) :
BddAbove s | IsCompact.bddBelow (α := αᵒᵈ) hs | theorem | IsCompact.bddAbove | Topology.Order | Mathlib/Topology/Order/Compact.lean | [] | [
"BddAbove",
"ClosedIciTopology",
"IsCompact",
"IsCompact.bddBelow",
"Set"
] | A compact set is bounded above | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
IsCompact.bddBelow_image [ClosedIicTopology α] [Nonempty α] {f : β → α} {K : Set β}
(hK : IsCompact K) (hf : ContinuousOn f K) : BddBelow (f '' K) | (hK.image_of_continuousOn hf).bddBelow | theorem | IsCompact.bddBelow_image | Topology.Order | Mathlib/Topology/Order/Compact.lean | [] | [
"ClosedIicTopology",
"ContinuousOn",
"IsCompact",
"Set"
] | A continuous function is bounded below on a compact set. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
IsCompact.bddAbove_image [ClosedIciTopology α] [Nonempty α] {f : β → α} {K : Set β}
(hK : IsCompact K) (hf : ContinuousOn f K) : BddAbove (f '' K) | IsCompact.bddBelow_image (α := αᵒᵈ) hK hf | theorem | IsCompact.bddAbove_image | Topology.Order | Mathlib/Topology/Order/Compact.lean | [] | [
"BddAbove",
"ClosedIciTopology",
"ContinuousOn",
"IsCompact",
"IsCompact.bddBelow_image",
"Set"
] | A continuous function is bounded above on a compact set. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
Continuous.bddBelow_range_of_hasCompactMulSupport [ClosedIicTopology α] [One α]
{f : β → α} (hf : Continuous f) (h : HasCompactMulSupport f) : BddBelow (range f) | (h.isCompact_range hf).bddBelow | theorem | Continuous.bddBelow_range_of_hasCompactMulSupport | Topology.Order | Mathlib/Topology/Order/Compact.lean | [] | [
"ClosedIicTopology",
"Continuous",
"HasCompactMulSupport"
] | A continuous function with compact support is bounded below. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
Continuous.bddAbove_range_of_hasCompactMulSupport [ClosedIciTopology α] [One α]
{f : β → α} (hf : Continuous f) (h : HasCompactMulSupport f) : BddAbove (range f) | Continuous.bddBelow_range_of_hasCompactMulSupport (α := αᵒᵈ) hf h | theorem | Continuous.bddAbove_range_of_hasCompactMulSupport | Topology.Order | Mathlib/Topology/Order/Compact.lean | [] | [
"BddAbove",
"ClosedIciTopology",
"Continuous",
"Continuous.bddBelow_range_of_hasCompactMulSupport",
"HasCompactMulSupport"
] | A continuous function with compact support is bounded above. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
IsCompact.sSup_lt_iff_of_continuous [ClosedIciTopology α] {f : β → α} {K : Set β}
(hK : IsCompact K) (h0K : K.Nonempty) (hf : ContinuousOn f K) (y : α) :
sSup (f '' K) < y ↔ ∀ x ∈ K, f x < y | by
refine ⟨fun h x hx => (le_csSup (hK.bddAbove_image hf) <| mem_image_of_mem f hx).trans_lt h,
fun h => ?_⟩
obtain ⟨x, hx, h2x⟩ := hK.exists_isMaxOn h0K hf
refine (csSup_le (h0K.image f) ?_).trans_lt (h x hx)
rintro _ ⟨x', hx', rfl⟩; exact h2x hx' | theorem | IsCompact.sSup_lt_iff_of_continuous | Topology.Order | Mathlib/Topology/Order/Compact.lean | [] | [
"ClosedIciTopology",
"ContinuousOn",
"IsCompact",
"Set",
"csSup_le",
"le_csSup"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
IsCompact.lt_sInf_iff_of_continuous [ClosedIicTopology α] {f : β → α} {K : Set β}
(hK : IsCompact K) (h0K : K.Nonempty) (hf : ContinuousOn f K) (y : α) :
y < sInf (f '' K) ↔ ∀ x ∈ K, y < f x | IsCompact.sSup_lt_iff_of_continuous (α := αᵒᵈ) hK h0K hf y | theorem | IsCompact.lt_sInf_iff_of_continuous | Topology.Order | Mathlib/Topology/Order/Compact.lean | [] | [
"ClosedIicTopology",
"ContinuousOn",
"IsCompact",
"IsCompact.sSup_lt_iff_of_continuous",
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
IsCompact.sInf_mem [ClosedIicTopology α] {s : Set α} (hs : IsCompact s)
(ne_s : s.Nonempty) : sInf s ∈ s | let ⟨_a, ha⟩ := hs.exists_isLeast ne_s
ha.csInf_mem | theorem | IsCompact.sInf_mem | Topology.Order | Mathlib/Topology/Order/Compact.lean | [] | [
"ClosedIicTopology",
"IsCompact",
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
IsCompact.sSup_mem [ClosedIciTopology α] {s : Set α} (hs : IsCompact s)
(ne_s : s.Nonempty) : sSup s ∈ s | IsCompact.sInf_mem (α := αᵒᵈ) hs ne_s | theorem | IsCompact.sSup_mem | Topology.Order | Mathlib/Topology/Order/Compact.lean | [] | [
"ClosedIciTopology",
"IsCompact",
"IsCompact.sInf_mem",
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
IsCompact.isGLB_sInf [ClosedIicTopology α] {s : Set α} (hs : IsCompact s)
(ne_s : s.Nonempty) : IsGLB s (sInf s) | isGLB_csInf ne_s hs.bddBelow | theorem | IsCompact.isGLB_sInf | Topology.Order | Mathlib/Topology/Order/Compact.lean | [] | [
"ClosedIicTopology",
"IsCompact",
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
IsCompact.isLUB_sSup [ClosedIciTopology α] {s : Set α} (hs : IsCompact s)
(ne_s : s.Nonempty) : IsLUB s (sSup s) | IsCompact.isGLB_sInf (α := αᵒᵈ) hs ne_s | theorem | IsCompact.isLUB_sSup | Topology.Order | Mathlib/Topology/Order/Compact.lean | [] | [
"ClosedIciTopology",
"IsCompact",
"IsCompact.isGLB_sInf",
"IsLUB",
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
IsCompact.isLeast_sInf [ClosedIicTopology α] {s : Set α} (hs : IsCompact s)
(ne_s : s.Nonempty) : IsLeast s (sInf s) | ⟨hs.sInf_mem ne_s, (hs.isGLB_sInf ne_s).1⟩ | theorem | IsCompact.isLeast_sInf | Topology.Order | Mathlib/Topology/Order/Compact.lean | [] | [
"ClosedIicTopology",
"IsCompact",
"IsLeast",
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
IsCompact.isGreatest_sSup [ClosedIciTopology α] {s : Set α} (hs : IsCompact s)
(ne_s : s.Nonempty) : IsGreatest s (sSup s) | IsCompact.isLeast_sInf (α := αᵒᵈ) hs ne_s | theorem | IsCompact.isGreatest_sSup | Topology.Order | Mathlib/Topology/Order/Compact.lean | [] | [
"ClosedIciTopology",
"IsCompact",
"IsCompact.isLeast_sInf",
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
IsCompact.exists_sInf_image_eq_and_le [ClosedIicTopology α] {s : Set β}
(hs : IsCompact s) (ne_s : s.Nonempty) {f : β → α} (hf : ContinuousOn f s) :
∃ x ∈ s, sInf (f '' s) = f x ∧ ∀ y ∈ s, f x ≤ f y | let ⟨x, hxs, hx⟩ := (hs.image_of_continuousOn hf).sInf_mem (ne_s.image f)
⟨x, hxs, hx.symm, fun _y hy =>
hx.trans_le <| csInf_le (hs.image_of_continuousOn hf).bddBelow <| mem_image_of_mem f hy⟩ | theorem | IsCompact.exists_sInf_image_eq_and_le | Topology.Order | Mathlib/Topology/Order/Compact.lean | [] | [
"ClosedIicTopology",
"ContinuousOn",
"IsCompact",
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
IsCompact.exists_sSup_image_eq_and_ge [ClosedIciTopology α] {s : Set β}
(hs : IsCompact s) (ne_s : s.Nonempty) {f : β → α} (hf : ContinuousOn f s) :
∃ x ∈ s, sSup (f '' s) = f x ∧ ∀ y ∈ s, f y ≤ f x | IsCompact.exists_sInf_image_eq_and_le (α := αᵒᵈ) hs ne_s hf | theorem | IsCompact.exists_sSup_image_eq_and_ge | Topology.Order | Mathlib/Topology/Order/Compact.lean | [] | [
"ClosedIciTopology",
"ContinuousOn",
"IsCompact",
"IsCompact.exists_sInf_image_eq_and_le",
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
IsCompact.exists_sInf_image_eq [ClosedIicTopology α] {s : Set β} (hs : IsCompact s)
(ne_s : s.Nonempty) {f : β → α} (hf : ContinuousOn f s) : ∃ x ∈ s, sInf (f '' s) = f x | let ⟨x, hxs, hx, _⟩ := hs.exists_sInf_image_eq_and_le ne_s hf
⟨x, hxs, hx⟩ | theorem | IsCompact.exists_sInf_image_eq | Topology.Order | Mathlib/Topology/Order/Compact.lean | [] | [
"ClosedIicTopology",
"ContinuousOn",
"IsCompact",
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
IsCompact.exists_sSup_image_eq [ClosedIciTopology α] {s : Set β} (hs : IsCompact s)
(ne_s : s.Nonempty) : ∀ {f : β → α}, ContinuousOn f s → ∃ x ∈ s, sSup (f '' s) = f x | IsCompact.exists_sInf_image_eq (α := αᵒᵈ) hs ne_s | theorem | IsCompact.exists_sSup_image_eq | Topology.Order | Mathlib/Topology/Order/Compact.lean | [] | [
"ClosedIciTopology",
"ContinuousOn",
"IsCompact",
"IsCompact.exists_sInf_image_eq",
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
IsCompact.exists_isMinOn_mem_subset [ClosedIicTopology α] {f : β → α} {s t : Set β}
{z : β} (ht : IsCompact t) (hf : ContinuousOn f t) (hz : z ∈ t)
(hfz : ∀ z' ∈ t \ s, f z < f z') : ∃ x ∈ s, IsMinOn f t x | let ⟨x, hxt, hfx⟩ := ht.exists_isMinOn ⟨z, hz⟩ hf
⟨x, by_contra fun hxs => (hfz x ⟨hxt, hxs⟩).not_ge (hfx hz), hfx⟩ | theorem | IsCompact.exists_isMinOn_mem_subset | Topology.Order | Mathlib/Topology/Order/Compact.lean | [] | [
"ClosedIicTopology",
"ContinuousOn",
"IsCompact",
"IsMinOn",
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
IsCompact.exists_isMaxOn_mem_subset [ClosedIciTopology α] {f : β → α} {s t : Set β}
{z : β} (ht : IsCompact t) (hf : ContinuousOn f t) (hz : z ∈ t)
(hfz : ∀ z' ∈ t \ s, f z' < f z) : ∃ x ∈ s, IsMaxOn f t x | let ⟨x, hxt, hfx⟩ := ht.exists_isMaxOn ⟨z, hz⟩ hf
⟨x, by_contra fun hxs => (hfz x ⟨hxt, hxs⟩).not_ge (hfx hz), hfx⟩ | theorem | IsCompact.exists_isMaxOn_mem_subset | Topology.Order | Mathlib/Topology/Order/Compact.lean | [] | [
"ClosedIciTopology",
"ContinuousOn",
"IsCompact",
"IsMaxOn",
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
IsCompact.exists_isLocalMin_mem_open [ClosedIicTopology α] {f : β → α} {s t : Set β}
{z : β} (ht : IsCompact t) (hst : s ⊆ t) (hf : ContinuousOn f t) (hz : z ∈ t)
(hfz : ∀ z' ∈ t \ s, f z < f z') (hs : IsOpen s) : ∃ x ∈ s, IsLocalMin f x | let ⟨x, hxs, h⟩ := ht.exists_isMinOn_mem_subset hf hz hfz
⟨x, hxs, h.isLocalMin <| mem_nhds_iff.2 ⟨s, hst, hs, hxs⟩⟩ | theorem | IsCompact.exists_isLocalMin_mem_open | Topology.Order | Mathlib/Topology/Order/Compact.lean | [] | [
"ClosedIicTopology",
"ContinuousOn",
"IsCompact",
"IsLocalMin",
"IsOpen",
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
IsCompact.exists_isLocalMax_mem_open [ClosedIciTopology α] {f : β → α} {s t : Set β}
{z : β} (ht : IsCompact t) (hst : s ⊆ t) (hf : ContinuousOn f t) (hz : z ∈ t)
(hfz : ∀ z' ∈ t \ s, f z' < f z) (hs : IsOpen s) : ∃ x ∈ s, IsLocalMax f x | let ⟨x, hxs, h⟩ := ht.exists_isMaxOn_mem_subset hf hz hfz
⟨x, hxs, h.isLocalMax <| mem_nhds_iff.2 ⟨s, hst, hs, hxs⟩⟩ | theorem | IsCompact.exists_isLocalMax_mem_open | Topology.Order | Mathlib/Topology/Order/Compact.lean | [] | [
"ClosedIciTopology",
"ContinuousOn",
"IsCompact",
"IsLocalMax",
"IsOpen",
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
eq_Icc_of_connected_compact {s : Set α} (h₁ : IsConnected s) (h₂ : IsCompact s) :
s = Icc (sInf s) (sSup s) | eq_Icc_csInf_csSup_of_connected_bdd_closed h₁ h₂.bddBelow h₂.bddAbove h₂.isClosed | theorem | eq_Icc_of_connected_compact | Topology.Order | Mathlib/Topology/Order/Compact.lean | [] | [
"IsCompact",
"IsConnected",
"Set",
"eq_Icc_csInf_csSup_of_connected_bdd_closed"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
IsCompact.continuous_sSup {f : γ → β → α} {K : Set β} (hK : IsCompact K)
(hf : Continuous ↿f) : Continuous fun x => sSup (f x '' K) | by
rcases eq_empty_or_nonempty K with (rfl | h0K)
· simp_rw [image_empty]
exact continuous_const
rw [continuous_iff_continuousAt]
intro x
obtain ⟨y, hyK, h2y, hy⟩ :=
hK.exists_sSup_image_eq_and_ge h0K
(show Continuous (f x) from hf.comp <| .prodMk_right x).continuousOn
rw [ContinuousAt, h2y, t... | theorem | IsCompact.continuous_sSup | Topology.Order | Mathlib/Topology/Order/Compact.lean | [] | [
"Continuous",
"ContinuousAt",
"IsCompact",
"Set",
"continuous_const",
"continuous_iff_continuousAt",
"generalized_tube_lemma",
"isCompact_singleton",
"le_csSup",
"tendsto_order"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
IsCompact.continuous_sInf {f : γ → β → α} {K : Set β} (hK : IsCompact K)
(hf : Continuous ↿f) : Continuous fun x => sInf (f x '' K) | IsCompact.continuous_sSup (α := αᵒᵈ) hK hf | theorem | IsCompact.continuous_sInf | Topology.Order | Mathlib/Topology/Order/Compact.lean | [] | [
"Continuous",
"IsCompact",
"IsCompact.continuous_sSup",
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
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