fact stringlengths 6 3.84k | type stringclasses 11 values | library stringclasses 32 values | imports listlengths 1 14 | filename stringlengths 20 95 | symbolic_name stringlengths 1 90 | docstring stringlengths 7 20k ⌀ |
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csSup_mem_closure {s : Set α} (hs : s.Nonempty) (B : BddAbove s) : sSup s ∈ closure s :=
(isLUB_csSup hs B).mem_closure hs | theorem | Topology | [
"Mathlib.Tactic.Order",
"Mathlib.Topology.Order.IsLUB"
] | Mathlib/Topology/Order/Monotone.lean | csSup_mem_closure | null |
csInf_mem_closure {s : Set α} (hs : s.Nonempty) (B : BddBelow s) : sInf s ∈ closure s :=
(isGLB_csInf hs B).mem_closure hs | theorem | Topology | [
"Mathlib.Tactic.Order",
"Mathlib.Topology.Order.IsLUB"
] | Mathlib/Topology/Order/Monotone.lean | csInf_mem_closure | null |
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 | Topology | [
"Mathlib.Tactic.Order",
"Mathlib.Topology.Order.IsLUB"
] | Mathlib/Topology/Order/Monotone.lean | IsClosed.csSup_mem | null |
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 | Topology | [
"Mathlib.Tactic.Order",
"Mathlib.Topology.Order.IsLUB"
] | Mathlib/Topology/Order/Monotone.lean | IsClosed.csInf_mem | null |
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 | Topology | [
"Mathlib.Tactic.Order",
"Mathlib.Topology.Order.IsLUB"
] | Mathlib/Topology/Order/Monotone.lean | IsClosed.isLeast_csInf | null |
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 | Topology | [
"Mathlib.Tactic.Order",
"Mathlib.Topology.Order.IsLUB"
] | Mathlib/Topology/Order/Monotone.lean | IsClosed.isGreatest_csSup | null |
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)) :
Tendsto f (𝓝[<] x) (𝓝 (sSup (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.trans_le <| Mf ⟨yz, zx⟩ ⟨yz.trans_le hw.1.le, hw.2⟩ hw.1.le
· rcases h_nonempty with ⟨_, hy, hx⟩
filter_upwards [Ioo_mem_nhdsLT (hy.trans hx)] with w hw
exact (le_csSup h_bdd (mem_image_of_mem _ hw)).trans_lt hm | lemma | Topology | [
"Mathlib.Tactic.Order",
"Mathlib.Topology.Order.IsLUB"
] | Mathlib/Topology/Order/Monotone.lean | MonotoneOn.tendsto_nhdsWithin_Ioo_left | null |
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)) :
Tendsto f (𝓝[>] x) (𝓝 (sInf (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, exists_prop, exists_exists_and_eq_and] using
exists_lt_of_csInf_lt (h_nonempty.image _) hm
filter_upwards [Ioo_mem_nhdsGT xz] with w hw
exact (Mf ⟨hw.1, hw.2.trans zy⟩ ⟨xz, zy⟩ hw.2.le).trans_lt zm | lemma | Topology | [
"Mathlib.Tactic.Order",
"Mathlib.Topology.Order.IsLUB"
] | Mathlib/Topology/Order/Monotone.lean | MonotoneOn.tendsto_nhdsWithin_Ioo_right | null |
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 [Ioo_mem_nhdsLT zx] with y hy using lz.trans_le (Mf zx hy.2 hy.1.le)
· refine mem_of_superset self_mem_nhdsWithin fun y hy => lt_of_le_of_lt ?_ hm
exact le_csSup h_bdd (mem_image_of_mem _ hy) | lemma | Topology | [
"Mathlib.Tactic.Order",
"Mathlib.Topology.Order.IsLUB"
] | Mathlib/Topology/Order/Monotone.lean | MonotoneOn.tendsto_nhdsLT | null |
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 | Topology | [
"Mathlib.Tactic.Order",
"Mathlib.Topology.Order.IsLUB"
] | Mathlib/Topology/Order/Monotone.lean | MonotoneOn.tendsto_nhdsGT | null |
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 | Topology | [
"Mathlib.Tactic.Order",
"Mathlib.Topology.Order.IsLUB"
] | Mathlib/Topology/Order/Monotone.lean | Monotone.tendsto_nhdsLT | A monotone map has a limit to the left of any point `x`, equal to `sSup (f '' (Iio x))`. |
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 | Topology | [
"Mathlib.Tactic.Order",
"Mathlib.Topology.Order.IsLUB"
] | Mathlib/Topology/Order/Monotone.lean | Monotone.tendsto_nhdsGT | A monotone map has a limit to the right of any point `x`, equal to `sInf (f '' (Ioi x))`. |
AntitoneOn.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) (Af : AntitoneOn f (Ioo y x))
(h_bdd : BddBelow (f '' Ioo y x)) :
Tendsto f (𝓝[<] x) (𝓝 (sInf (f '' Ioo y x))) :=
MonotoneOn.tendsto_nhdsWithin_Ioo_left h_nonempty Af.dual_right h_bdd | lemma | Topology | [
"Mathlib.Tactic.Order",
"Mathlib.Topology.Order.IsLUB"
] | Mathlib/Topology/Order/Monotone.lean | AntitoneOn.tendsto_nhdsWithin_Ioo_left | null |
AntitoneOn.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) (Af : AntitoneOn f (Ioo x y))
(h_bdd : BddAbove (f '' Ioo x y)) :
Tendsto f (𝓝[>] x) (𝓝 (sSup (f '' Ioo x y))) :=
MonotoneOn.tendsto_nhdsWithin_Ioo_right h_nonempty Af.dual_right h_bdd | lemma | Topology | [
"Mathlib.Tactic.Order",
"Mathlib.Topology.Order.IsLUB"
] | Mathlib/Topology/Order/Monotone.lean | AntitoneOn.tendsto_nhdsWithin_Ioo_right | null |
AntitoneOn.tendsto_nhdsLT {α β : Type*} [LinearOrder α] [TopologicalSpace α] [OrderTopology α]
[ConditionallyCompleteLinearOrder β] [TopologicalSpace β] [OrderTopology β] {f : α → β} {x : α}
(Af : AntitoneOn f (Iio x)) (h_bdd : BddBelow (f '' Iio x)) :
Tendsto f (𝓝[<] x) (𝓝 (sInf (f '' Iio x))) :=
MonotoneOn.tendsto_nhdsLT Af.dual_right h_bdd | lemma | Topology | [
"Mathlib.Tactic.Order",
"Mathlib.Topology.Order.IsLUB"
] | Mathlib/Topology/Order/Monotone.lean | AntitoneOn.tendsto_nhdsLT | null |
AntitoneOn.tendsto_nhdsGT {α β : Type*} [LinearOrder α] [TopologicalSpace α] [OrderTopology α]
[ConditionallyCompleteLinearOrder β] [TopologicalSpace β] [OrderTopology β] {f : α → β} {x : α}
(Af : AntitoneOn f (Ioi x)) (h_bdd : BddAbove (f '' Ioi x)) :
Tendsto f (𝓝[>] x) (𝓝 (sSup (f '' Ioi x))) :=
MonotoneOn.tendsto_nhdsGT Af.dual_right h_bdd | lemma | Topology | [
"Mathlib.Tactic.Order",
"Mathlib.Topology.Order.IsLUB"
] | Mathlib/Topology/Order/Monotone.lean | AntitoneOn.tendsto_nhdsGT | null |
Antitone.tendsto_nhdsLT {α β : Type*} [LinearOrder α] [TopologicalSpace α] [OrderTopology α]
[ConditionallyCompleteLinearOrder β] [TopologicalSpace β] [OrderTopology β] {f : α → β}
(Af : Antitone f) (x : α) : Tendsto f (𝓝[<] x) (𝓝 (sInf (f '' Iio x))) :=
Monotone.tendsto_nhdsLT Af.dual_right x | theorem | Topology | [
"Mathlib.Tactic.Order",
"Mathlib.Topology.Order.IsLUB"
] | Mathlib/Topology/Order/Monotone.lean | Antitone.tendsto_nhdsLT | An antitone map has a limit to the left of any point `x`, equal to `sInf (f '' (Iio x))`. |
Antitone.tendsto_nhdsGT {α β : Type*} [LinearOrder α] [TopologicalSpace α] [OrderTopology α]
[ConditionallyCompleteLinearOrder β] [TopologicalSpace β] [OrderTopology β] {f : α → β}
(Af : Antitone f) (x : α) : Tendsto f (𝓝[>] x) (𝓝 (sSup (f '' Ioi x))) :=
Monotone.tendsto_nhdsGT Af.dual_right x | theorem | Topology | [
"Mathlib.Tactic.Order",
"Mathlib.Topology.Order.IsLUB"
] | Mathlib/Topology/Order/Monotone.lean | Antitone.tendsto_nhdsGT | An antitone map has a limit to the right of any point `x`, equal to `sSup (f '' (Ioi x))`. |
StrictMonoOn.continuousWithinAt_right_of_exists_between {f : α → β} {s : Set α} {a : α}
(h_mono : StrictMonoOn f s) (hs : s ∈ 𝓝[≥] a) (hfs : ∀ b > f a, ∃ c ∈ s, f c ∈ Ioc (f a) b) :
ContinuousWithinAt f (Ici a) a := by
have ha : a ∈ Ici a := left_mem_Ici
have has : a ∈ s := mem_of_mem_nhdsWithin ha hs
refine tendsto_order.2 ⟨fun b hb => ?_, fun b hb => ?_⟩
· filter_upwards [hs, @self_mem_nhdsWithin _ _ a (Ici a)] with _ hxs hxa using hb.trans_le
((h_mono.le_iff_le has hxs).2 hxa)
· rcases hfs b hb with ⟨c, hcs, hac, hcb⟩
rw [h_mono.lt_iff_lt has hcs] at hac
filter_upwards [hs, Ico_mem_nhdsGE hac]
rintro x hx ⟨_, hxc⟩
exact ((h_mono.lt_iff_lt hx hcs).2 hxc).trans_le hcb | theorem | Topology | [
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Order.LeftRightNhds"
] | Mathlib/Topology/Order/MonotoneContinuity.lean | StrictMonoOn.continuousWithinAt_right_of_exists_between | If `f` is a function strictly monotone on a right neighborhood of `a` and the
image of this neighborhood under `f` meets every interval `(f a, b]`, `b > f a`, then `f` is
continuous at `a` from the right.
The assumption `hfs : ∀ b > f a, ∃ c ∈ s, f c ∈ Ioc (f a) b` is required because otherwise the
function `f : ℝ → ℝ` given by `f x = if x ≤ 0 then x else x + 1` would be a counter-example at
`a = 0`. |
continuousWithinAt_right_of_monotoneOn_of_exists_between {f : α → β} {s : Set α} {a : α}
(h_mono : MonotoneOn f s) (hs : s ∈ 𝓝[≥] a) (hfs : ∀ b > f a, ∃ c ∈ s, f c ∈ Ioo (f a) b) :
ContinuousWithinAt f (Ici a) a := by
have ha : a ∈ Ici a := left_mem_Ici
have has : a ∈ s := mem_of_mem_nhdsWithin ha hs
refine tendsto_order.2 ⟨fun b hb => ?_, fun b hb => ?_⟩
· filter_upwards [hs, @self_mem_nhdsWithin _ _ a (Ici a)] with _ hxs hxa using hb.trans_le
(h_mono has hxs hxa)
· rcases hfs b hb with ⟨c, hcs, hac, hcb⟩
have : a < c := not_le.1 fun h => hac.not_ge <| h_mono hcs has h
filter_upwards [hs, Ico_mem_nhdsGE this]
rintro x hx ⟨_, hxc⟩
exact (h_mono hx hcs hxc.le).trans_lt hcb | theorem | Topology | [
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Order.LeftRightNhds"
] | Mathlib/Topology/Order/MonotoneContinuity.lean | continuousWithinAt_right_of_monotoneOn_of_exists_between | If `f` is a monotone function on a right neighborhood of `a` and the image of this neighborhood
under `f` meets every interval `(f a, b)`, `b > f a`, then `f` is continuous at `a` from the right.
The assumption `hfs : ∀ b > f a, ∃ c ∈ s, f c ∈ Ioo (f a) b` cannot be replaced by the weaker
assumption `hfs : ∀ b > f a, ∃ c ∈ s, f c ∈ Ioc (f a) b` we use for strictly monotone functions
because otherwise the function `ceil : ℝ → ℤ` would be a counter-example at `a = 0`. |
continuousWithinAt_right_of_monotoneOn_of_closure_image_mem_nhdsWithin [DenselyOrdered β]
{f : α → β} {s : Set α} {a : α} (h_mono : MonotoneOn f s) (hs : s ∈ 𝓝[≥] a)
(hfs : closure (f '' s) ∈ 𝓝[≥] f a) : ContinuousWithinAt f (Ici a) a := by
refine continuousWithinAt_right_of_monotoneOn_of_exists_between h_mono hs fun b hb => ?_
rcases (mem_nhdsGE_iff_exists_mem_Ioc_Ico_subset hb).1 hfs with ⟨b', ⟨hab', hbb'⟩, hb'⟩
rcases exists_between hab' with ⟨c', hc'⟩
rcases mem_closure_iff.1 (hb' ⟨hc'.1.le, hc'.2⟩) (Ioo (f a) b') isOpen_Ioo hc' with
⟨_, hc, ⟨c, hcs, rfl⟩⟩
exact ⟨c, hcs, hc.1, hc.2.trans_le hbb'⟩ | theorem | Topology | [
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Order.LeftRightNhds"
] | Mathlib/Topology/Order/MonotoneContinuity.lean | continuousWithinAt_right_of_monotoneOn_of_closure_image_mem_nhdsWithin | If a function `f` with a densely ordered codomain is monotone on a right neighborhood of `a` and
the closure of the image of this neighborhood under `f` is a right neighborhood of `f a`, then `f`
is continuous at `a` from the right. |
continuousWithinAt_right_of_monotoneOn_of_image_mem_nhdsWithin [DenselyOrdered β]
{f : α → β} {s : Set α} {a : α} (h_mono : MonotoneOn f s) (hs : s ∈ 𝓝[≥] a)
(hfs : f '' s ∈ 𝓝[≥] f a) : ContinuousWithinAt f (Ici a) a :=
continuousWithinAt_right_of_monotoneOn_of_closure_image_mem_nhdsWithin h_mono hs <|
mem_of_superset hfs subset_closure | theorem | Topology | [
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Order.LeftRightNhds"
] | Mathlib/Topology/Order/MonotoneContinuity.lean | continuousWithinAt_right_of_monotoneOn_of_image_mem_nhdsWithin | If a function `f` with a densely ordered codomain is monotone on a right neighborhood of `a` and
the image of this neighborhood under `f` is a right neighborhood of `f a`, then `f` is continuous at
`a` from the right. |
StrictMonoOn.continuousWithinAt_right_of_closure_image_mem_nhdsWithin [DenselyOrdered β]
{f : α → β} {s : Set α} {a : α} (h_mono : StrictMonoOn f s) (hs : s ∈ 𝓝[≥] a)
(hfs : closure (f '' s) ∈ 𝓝[≥] f a) : ContinuousWithinAt f (Ici a) a :=
continuousWithinAt_right_of_monotoneOn_of_closure_image_mem_nhdsWithin
(fun _ hx _ hy => (h_mono.le_iff_le hx hy).2) hs hfs | theorem | Topology | [
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Order.LeftRightNhds"
] | Mathlib/Topology/Order/MonotoneContinuity.lean | StrictMonoOn.continuousWithinAt_right_of_closure_image_mem_nhdsWithin | If a function `f` with a densely ordered codomain is strictly monotone on a right neighborhood
of `a` and the closure of the image of this neighborhood under `f` is a right neighborhood of `f a`,
then `f` is continuous at `a` from the right. |
StrictMonoOn.continuousWithinAt_right_of_image_mem_nhdsWithin [DenselyOrdered β] {f : α → β}
{s : Set α} {a : α} (h_mono : StrictMonoOn f s) (hs : s ∈ 𝓝[≥] a) (hfs : f '' s ∈ 𝓝[≥] f a) :
ContinuousWithinAt f (Ici a) a :=
h_mono.continuousWithinAt_right_of_closure_image_mem_nhdsWithin hs
(mem_of_superset hfs subset_closure) | theorem | Topology | [
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Order.LeftRightNhds"
] | Mathlib/Topology/Order/MonotoneContinuity.lean | StrictMonoOn.continuousWithinAt_right_of_image_mem_nhdsWithin | If a function `f` with a densely ordered codomain is strictly monotone on a right neighborhood
of `a` and the image of this neighborhood under `f` is a right neighborhood of `f a`, then `f` is
continuous at `a` from the right. |
StrictMonoOn.continuousWithinAt_right_of_surjOn {f : α → β} {s : Set α} {a : α}
(h_mono : StrictMonoOn f s) (hs : s ∈ 𝓝[≥] a) (hfs : SurjOn f s (Ioi (f a))) :
ContinuousWithinAt f (Ici a) a :=
h_mono.continuousWithinAt_right_of_exists_between hs fun _ hb =>
let ⟨c, hcs, hcb⟩ := hfs hb
⟨c, hcs, hcb.symm ▸ hb, hcb.le⟩ | theorem | Topology | [
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Order.LeftRightNhds"
] | Mathlib/Topology/Order/MonotoneContinuity.lean | StrictMonoOn.continuousWithinAt_right_of_surjOn | If a function `f` is strictly monotone on a right neighborhood of `a` and the image of this
neighborhood under `f` includes `Ioi (f a)`, then `f` is continuous at `a` from the right. |
StrictMonoOn.continuousWithinAt_left_of_exists_between {f : α → β} {s : Set α} {a : α}
(h_mono : StrictMonoOn f s) (hs : s ∈ 𝓝[≤] a) (hfs : ∀ b < f a, ∃ c ∈ s, f c ∈ Ico b (f a)) :
ContinuousWithinAt f (Iic a) a :=
h_mono.dual.continuousWithinAt_right_of_exists_between hs fun b hb =>
let ⟨c, hcs, hcb, hca⟩ := hfs b hb
⟨c, hcs, hca, hcb⟩ | theorem | Topology | [
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Order.LeftRightNhds"
] | Mathlib/Topology/Order/MonotoneContinuity.lean | StrictMonoOn.continuousWithinAt_left_of_exists_between | If `f` is a strictly monotone function on a left neighborhood of `a` and the image of this
neighborhood under `f` meets every interval `[b, f a)`, `b < f a`, then `f` is continuous at `a`
from the left.
The assumption `hfs : ∀ b < f a, ∃ c ∈ s, f c ∈ Ico b (f a)` is required because otherwise the
function `f : ℝ → ℝ` given by `f x = if x < 0 then x else x + 1` would be a counter-example at
`a = 0`. |
continuousWithinAt_left_of_monotoneOn_of_exists_between {f : α → β} {s : Set α} {a : α}
(hf : MonotoneOn f s) (hs : s ∈ 𝓝[≤] a) (hfs : ∀ b < f a, ∃ c ∈ s, f c ∈ Ioo b (f a)) :
ContinuousWithinAt f (Iic a) a :=
@continuousWithinAt_right_of_monotoneOn_of_exists_between αᵒᵈ βᵒᵈ _ _ _ _ _ _ f s a hf.dual hs
fun b hb =>
let ⟨c, hcs, hcb, hca⟩ := hfs b hb
⟨c, hcs, hca, hcb⟩ | theorem | Topology | [
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Order.LeftRightNhds"
] | Mathlib/Topology/Order/MonotoneContinuity.lean | continuousWithinAt_left_of_monotoneOn_of_exists_between | If `f` is a monotone function on a left neighborhood of `a` and the image of this neighborhood
under `f` meets every interval `(b, f a)`, `b < f a`, then `f` is continuous at `a` from the left.
The assumption `hfs : ∀ b < f a, ∃ c ∈ s, f c ∈ Ioo b (f a)` cannot be replaced by the weaker
assumption `hfs : ∀ b < f a, ∃ c ∈ s, f c ∈ Ico b (f a)` we use for strictly monotone functions
because otherwise the function `floor : ℝ → ℤ` would be a counter-example at `a = 0`. |
continuousWithinAt_left_of_monotoneOn_of_closure_image_mem_nhdsWithin [DenselyOrdered β]
{f : α → β} {s : Set α} {a : α} (hf : MonotoneOn f s) (hs : s ∈ 𝓝[≤] a)
(hfs : closure (f '' s) ∈ 𝓝[≤] f a) : ContinuousWithinAt f (Iic a) a :=
@continuousWithinAt_right_of_monotoneOn_of_closure_image_mem_nhdsWithin αᵒᵈ βᵒᵈ _ _ _ _ _ _ _ f s
a hf.dual hs hfs | theorem | Topology | [
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Order.LeftRightNhds"
] | Mathlib/Topology/Order/MonotoneContinuity.lean | continuousWithinAt_left_of_monotoneOn_of_closure_image_mem_nhdsWithin | If a function `f` with a densely ordered codomain is monotone on a left neighborhood of `a` and
the closure of the image of this neighborhood under `f` is a left neighborhood of `f a`, then `f` is
continuous at `a` from the left |
continuousWithinAt_left_of_monotoneOn_of_image_mem_nhdsWithin [DenselyOrdered β] {f : α → β}
{s : Set α} {a : α} (h_mono : MonotoneOn f s) (hs : s ∈ 𝓝[≤] a) (hfs : f '' s ∈ 𝓝[≤] f a) :
ContinuousWithinAt f (Iic a) a :=
continuousWithinAt_left_of_monotoneOn_of_closure_image_mem_nhdsWithin h_mono hs
(mem_of_superset hfs subset_closure) | theorem | Topology | [
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Order.LeftRightNhds"
] | Mathlib/Topology/Order/MonotoneContinuity.lean | continuousWithinAt_left_of_monotoneOn_of_image_mem_nhdsWithin | If a function `f` with a densely ordered codomain is monotone on a left neighborhood of `a` and
the image of this neighborhood under `f` is a left neighborhood of `f a`, then `f` is continuous at
`a` from the left. |
StrictMonoOn.continuousWithinAt_left_of_closure_image_mem_nhdsWithin [DenselyOrdered β]
{f : α → β} {s : Set α} {a : α} (h_mono : StrictMonoOn f s) (hs : s ∈ 𝓝[≤] a)
(hfs : closure (f '' s) ∈ 𝓝[≤] f a) : ContinuousWithinAt f (Iic a) a :=
h_mono.dual.continuousWithinAt_right_of_closure_image_mem_nhdsWithin hs hfs | theorem | Topology | [
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Order.LeftRightNhds"
] | Mathlib/Topology/Order/MonotoneContinuity.lean | StrictMonoOn.continuousWithinAt_left_of_closure_image_mem_nhdsWithin | If a function `f` with a densely ordered codomain is strictly monotone on a left neighborhood of
`a` and the closure of the image of this neighborhood under `f` is a left neighborhood of `f a`,
then `f` is continuous at `a` from the left. |
StrictMonoOn.continuousWithinAt_left_of_image_mem_nhdsWithin [DenselyOrdered β] {f : α → β}
{s : Set α} {a : α} (h_mono : StrictMonoOn f s) (hs : s ∈ 𝓝[≤] a) (hfs : f '' s ∈ 𝓝[≤] f a) :
ContinuousWithinAt f (Iic a) a :=
h_mono.dual.continuousWithinAt_right_of_image_mem_nhdsWithin hs hfs | theorem | Topology | [
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Order.LeftRightNhds"
] | Mathlib/Topology/Order/MonotoneContinuity.lean | StrictMonoOn.continuousWithinAt_left_of_image_mem_nhdsWithin | If a function `f` with a densely ordered codomain is strictly monotone on a left neighborhood of
`a` and the image of this neighborhood under `f` is a left neighborhood of `f a`, then `f` is
continuous at `a` from the left. |
StrictMonoOn.continuousWithinAt_left_of_surjOn {f : α → β} {s : Set α} {a : α}
(h_mono : StrictMonoOn f s) (hs : s ∈ 𝓝[≤] a) (hfs : SurjOn f s (Iio (f a))) :
ContinuousWithinAt f (Iic a) a :=
h_mono.dual.continuousWithinAt_right_of_surjOn hs hfs | theorem | Topology | [
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Order.LeftRightNhds"
] | Mathlib/Topology/Order/MonotoneContinuity.lean | StrictMonoOn.continuousWithinAt_left_of_surjOn | If a function `f` is strictly monotone on a left neighborhood of `a` and the image of this
neighborhood under `f` includes `Iio (f a)`, then `f` is continuous at `a` from the left. |
StrictMonoOn.continuousAt_of_exists_between {f : α → β} {s : Set α} {a : α}
(h_mono : StrictMonoOn f s) (hs : s ∈ 𝓝 a) (hfs_l : ∀ b < f a, ∃ c ∈ s, f c ∈ Ico b (f a))
(hfs_r : ∀ b > f a, ∃ c ∈ s, f c ∈ Ioc (f a) b) : ContinuousAt f a :=
continuousAt_iff_continuous_left_right.2
⟨h_mono.continuousWithinAt_left_of_exists_between (mem_nhdsWithin_of_mem_nhds hs) hfs_l,
h_mono.continuousWithinAt_right_of_exists_between (mem_nhdsWithin_of_mem_nhds hs) hfs_r⟩ | theorem | Topology | [
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Order.LeftRightNhds"
] | Mathlib/Topology/Order/MonotoneContinuity.lean | StrictMonoOn.continuousAt_of_exists_between | If a function `f` is strictly monotone on a neighborhood of `a` and the image of this
neighborhood under `f` meets every interval `[b, f a)`, `b < f a`, and every interval
`(f a, b]`, `b > f a`, then `f` is continuous at `a`. |
StrictMonoOn.continuousAt_of_closure_image_mem_nhds [DenselyOrdered β] {f : α → β}
{s : Set α} {a : α} (h_mono : StrictMonoOn f s) (hs : s ∈ 𝓝 a)
(hfs : closure (f '' s) ∈ 𝓝 (f a)) : ContinuousAt f a :=
continuousAt_iff_continuous_left_right.2
⟨h_mono.continuousWithinAt_left_of_closure_image_mem_nhdsWithin (mem_nhdsWithin_of_mem_nhds hs)
(mem_nhdsWithin_of_mem_nhds hfs),
h_mono.continuousWithinAt_right_of_closure_image_mem_nhdsWithin
(mem_nhdsWithin_of_mem_nhds hs) (mem_nhdsWithin_of_mem_nhds hfs)⟩ | theorem | Topology | [
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Order.LeftRightNhds"
] | Mathlib/Topology/Order/MonotoneContinuity.lean | StrictMonoOn.continuousAt_of_closure_image_mem_nhds | If a function `f` with a densely ordered codomain is strictly monotone on a neighborhood of `a`
and the closure of the image of this neighborhood under `f` is a neighborhood of `f a`, then `f` is
continuous at `a`. |
StrictMonoOn.continuousAt_of_image_mem_nhds [DenselyOrdered β] {f : α → β} {s : Set α}
{a : α} (h_mono : StrictMonoOn f s) (hs : s ∈ 𝓝 a) (hfs : f '' s ∈ 𝓝 (f a)) :
ContinuousAt f a :=
h_mono.continuousAt_of_closure_image_mem_nhds hs (mem_of_superset hfs subset_closure) | theorem | Topology | [
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Order.LeftRightNhds"
] | Mathlib/Topology/Order/MonotoneContinuity.lean | StrictMonoOn.continuousAt_of_image_mem_nhds | If a function `f` with a densely ordered codomain is strictly monotone on a neighborhood of `a`
and the image of this set under `f` is a neighborhood of `f a`, then `f` is continuous at `a`. |
continuousAt_of_monotoneOn_of_exists_between {f : α → β} {s : Set α} {a : α}
(h_mono : MonotoneOn f s) (hs : s ∈ 𝓝 a) (hfs_l : ∀ b < f a, ∃ c ∈ s, f c ∈ Ioo b (f a))
(hfs_r : ∀ b > f a, ∃ c ∈ s, f c ∈ Ioo (f a) b) : ContinuousAt f a :=
continuousAt_iff_continuous_left_right.2
⟨continuousWithinAt_left_of_monotoneOn_of_exists_between h_mono (mem_nhdsWithin_of_mem_nhds hs)
hfs_l,
continuousWithinAt_right_of_monotoneOn_of_exists_between h_mono
(mem_nhdsWithin_of_mem_nhds hs) hfs_r⟩ | theorem | Topology | [
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Order.LeftRightNhds"
] | Mathlib/Topology/Order/MonotoneContinuity.lean | continuousAt_of_monotoneOn_of_exists_between | If `f` is a monotone function on a neighborhood of `a` and the image of this neighborhood under
`f` meets every interval `(b, f a)`, `b < f a`, and every interval `(f a, b)`, `b > f a`, then `f`
is continuous at `a`. |
continuousAt_of_monotoneOn_of_closure_image_mem_nhds [DenselyOrdered β] {f : α → β}
{s : Set α} {a : α} (h_mono : MonotoneOn f s) (hs : s ∈ 𝓝 a)
(hfs : closure (f '' s) ∈ 𝓝 (f a)) : ContinuousAt f a :=
continuousAt_iff_continuous_left_right.2
⟨continuousWithinAt_left_of_monotoneOn_of_closure_image_mem_nhdsWithin h_mono
(mem_nhdsWithin_of_mem_nhds hs) (mem_nhdsWithin_of_mem_nhds hfs),
continuousWithinAt_right_of_monotoneOn_of_closure_image_mem_nhdsWithin h_mono
(mem_nhdsWithin_of_mem_nhds hs) (mem_nhdsWithin_of_mem_nhds hfs)⟩ | theorem | Topology | [
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Order.LeftRightNhds"
] | Mathlib/Topology/Order/MonotoneContinuity.lean | continuousAt_of_monotoneOn_of_closure_image_mem_nhds | If a function `f` with a densely ordered codomain is monotone on a neighborhood of `a` and the
closure of the image of this neighborhood under `f` is a neighborhood of `f a`, then `f` is
continuous at `a`. |
continuousAt_of_monotoneOn_of_image_mem_nhds [DenselyOrdered β] {f : α → β} {s : Set α}
{a : α} (h_mono : MonotoneOn f s) (hs : s ∈ 𝓝 a) (hfs : f '' s ∈ 𝓝 (f a)) : ContinuousAt f a :=
continuousAt_of_monotoneOn_of_closure_image_mem_nhds h_mono hs
(mem_of_superset hfs subset_closure) | theorem | Topology | [
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Order.LeftRightNhds"
] | Mathlib/Topology/Order/MonotoneContinuity.lean | continuousAt_of_monotoneOn_of_image_mem_nhds | If a function `f` with a densely ordered codomain is monotone on a neighborhood of `a` and the
image of this neighborhood under `f` is a neighborhood of `f a`, then `f` is continuous at `a`. |
Monotone.continuous_of_denseRange [DenselyOrdered β] {f : α → β} (h_mono : Monotone f)
(h_dense : DenseRange f) : Continuous f :=
continuous_iff_continuousAt.mpr fun a =>
continuousAt_of_monotoneOn_of_closure_image_mem_nhds (fun _ _ _ _ hxy => h_mono hxy)
univ_mem <|
by simp only [image_univ, h_dense.closure_eq, univ_mem] | theorem | Topology | [
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Order.LeftRightNhds"
] | Mathlib/Topology/Order/MonotoneContinuity.lean | Monotone.continuous_of_denseRange | A monotone function with densely ordered codomain and a dense range is continuous. |
Monotone.continuous_of_surjective [DenselyOrdered β] {f : α → β} (h_mono : Monotone f)
(h_surj : Function.Surjective f) : Continuous f :=
h_mono.continuous_of_denseRange h_surj.denseRange | theorem | Topology | [
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Order.LeftRightNhds"
] | Mathlib/Topology/Order/MonotoneContinuity.lean | Monotone.continuous_of_surjective | A monotone surjective function with a densely ordered codomain is continuous. |
protected continuous (e : α ≃o β) : Continuous e := by
rw [‹OrderTopology β›.topology_eq_generate_intervals, continuous_generateFrom_iff]
rintro s ⟨a, rfl | rfl⟩
· rw [e.preimage_Ioi]
apply isOpen_lt'
· rw [e.preimage_Iio]
apply isOpen_gt' | theorem | Topology | [
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Order.LeftRightNhds"
] | Mathlib/Topology/Order/MonotoneContinuity.lean | continuous | null |
toHomeomorph (e : α ≃o β) : α ≃ₜ β :=
HomeomorphClass.toHomeomorph e | abbrev | Topology | [
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Order.LeftRightNhds"
] | Mathlib/Topology/Order/MonotoneContinuity.lean | toHomeomorph | An order isomorphism between two linear order `OrderTopology` spaces is a homeomorphism. |
coe_toHomeomorph (e : α ≃o β) : ⇑e.toHomeomorph = e :=
rfl --Simp can prove this too
@[simp] | theorem | Topology | [
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Order.LeftRightNhds"
] | Mathlib/Topology/Order/MonotoneContinuity.lean | coe_toHomeomorph | null |
coe_toHomeomorph_symm (e : α ≃o β) : ⇑e.toHomeomorph.symm = e.symm :=
rfl | theorem | Topology | [
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Order.LeftRightNhds"
] | Mathlib/Topology/Order/MonotoneContinuity.lean | coe_toHomeomorph_symm | null |
SupConvergenceClass (α : Type*) [Preorder α] [TopologicalSpace α] : Prop where
/-- proof that a monotone function tends to `𝓝 a` as `x → ∞` -/
tendsto_coe_atTop_isLUB :
∀ (a : α) (s : Set α), IsLUB s a → Tendsto ((↑) : s → α) atTop (𝓝 a) | class | Topology | [
"Mathlib.Topology.Order.Basic"
] | Mathlib/Topology/Order/MonotoneConvergence.lean | SupConvergenceClass | We say that `α` is a `SupConvergenceClass` if the following holds. Let `f : ι → α` be a
monotone function, let `a : α` be a least upper bound of `Set.range f`. Then `f x` tends to `𝓝 a`
as `x → ∞` (formally, at the filter `Filter.atTop`). We require this for `ι = (s : Set α)`,
`f = (↑)` in the definition, then prove it for any `f` in `tendsto_atTop_isLUB`.
This property holds for linear orders with order topology as well as their products. |
InfConvergenceClass (α : Type*) [Preorder α] [TopologicalSpace α] : Prop where
/-- proof that a monotone function tends to `𝓝 a` as `x → -∞` -/
tendsto_coe_atBot_isGLB :
∀ (a : α) (s : Set α), IsGLB s a → Tendsto ((↑) : s → α) atBot (𝓝 a) | class | Topology | [
"Mathlib.Topology.Order.Basic"
] | Mathlib/Topology/Order/MonotoneConvergence.lean | InfConvergenceClass | We say that `α` is an `InfConvergenceClass` if the following holds. Let `f : ι → α` be a
monotone function, let `a : α` be a greatest lower bound of `Set.range f`. Then `f x` tends to `𝓝 a`
as `x → -∞` (formally, at the filter `Filter.atBot`). We require this for `ι = (s : Set α)`,
`f = (↑)` in the definition, then prove it for any `f` in `tendsto_atBot_isGLB`.
This property holds for linear orders with order topology as well as their products. |
OrderDual.supConvergenceClass [Preorder α] [TopologicalSpace α] [InfConvergenceClass α] :
SupConvergenceClass αᵒᵈ :=
⟨‹InfConvergenceClass α›.1⟩ | instance | Topology | [
"Mathlib.Topology.Order.Basic"
] | Mathlib/Topology/Order/MonotoneConvergence.lean | OrderDual.supConvergenceClass | null |
OrderDual.infConvergenceClass [Preorder α] [TopologicalSpace α] [SupConvergenceClass α] :
InfConvergenceClass αᵒᵈ :=
⟨‹SupConvergenceClass α›.1⟩ | instance | Topology | [
"Mathlib.Topology.Order.Basic"
] | Mathlib/Topology/Order/MonotoneConvergence.lean | OrderDual.infConvergenceClass | null |
tendsto_atTop_isLUB (h_mono : Monotone f) (ha : IsLUB (Set.range f) a) :
Tendsto f atTop (𝓝 a) := by
suffices Tendsto (rangeFactorization f) atTop atTop from
(SupConvergenceClass.tendsto_coe_atTop_isLUB _ _ ha).comp this
exact h_mono.rangeFactorization.tendsto_atTop_atTop fun b => b.2.imp fun a ha => ha.ge | theorem | Topology | [
"Mathlib.Topology.Order.Basic"
] | Mathlib/Topology/Order/MonotoneConvergence.lean | tendsto_atTop_isLUB | null |
tendsto_atBot_isLUB (h_anti : Antitone f) (ha : IsLUB (Set.range f) a) :
Tendsto f atBot (𝓝 a) := by convert tendsto_atTop_isLUB h_anti.dual_left ha using 1 | theorem | Topology | [
"Mathlib.Topology.Order.Basic"
] | Mathlib/Topology/Order/MonotoneConvergence.lean | tendsto_atBot_isLUB | null |
tendsto_atBot_isGLB (h_mono : Monotone f) (ha : IsGLB (Set.range f) a) :
Tendsto f atBot (𝓝 a) := by convert tendsto_atTop_isLUB h_mono.dual ha.dual using 1 | theorem | Topology | [
"Mathlib.Topology.Order.Basic"
] | Mathlib/Topology/Order/MonotoneConvergence.lean | tendsto_atBot_isGLB | null |
tendsto_atTop_isGLB (h_anti : Antitone f) (ha : IsGLB (Set.range f) a) :
Tendsto f atTop (𝓝 a) := by convert tendsto_atBot_isLUB h_anti.dual ha.dual using 1 | theorem | Topology | [
"Mathlib.Topology.Order.Basic"
] | Mathlib/Topology/Order/MonotoneConvergence.lean | tendsto_atTop_isGLB | null |
tendsto_atTop_ciSup (h_mono : Monotone f) (hbdd : BddAbove <| range f) :
Tendsto f atTop (𝓝 (⨆ i, f i)) := by
cases isEmpty_or_nonempty ι
exacts [tendsto_of_isEmpty, tendsto_atTop_isLUB h_mono (isLUB_ciSup hbdd)] | theorem | Topology | [
"Mathlib.Topology.Order.Basic"
] | Mathlib/Topology/Order/MonotoneConvergence.lean | tendsto_atTop_ciSup | null |
tendsto_atBot_ciSup (h_anti : Antitone f) (hbdd : BddAbove <| range f) :
Tendsto f atBot (𝓝 (⨆ i, f i)) := by convert tendsto_atTop_ciSup h_anti.dual hbdd.dual using 1 | theorem | Topology | [
"Mathlib.Topology.Order.Basic"
] | Mathlib/Topology/Order/MonotoneConvergence.lean | tendsto_atBot_ciSup | null |
tendsto_atBot_ciInf (h_mono : Monotone f) (hbdd : BddBelow <| range f) :
Tendsto f atBot (𝓝 (⨅ i, f i)) := by convert tendsto_atTop_ciSup h_mono.dual hbdd.dual using 1 | theorem | Topology | [
"Mathlib.Topology.Order.Basic"
] | Mathlib/Topology/Order/MonotoneConvergence.lean | tendsto_atBot_ciInf | null |
tendsto_atTop_ciInf (h_anti : Antitone f) (hbdd : BddBelow <| range f) :
Tendsto f atTop (𝓝 (⨅ i, f i)) := by convert tendsto_atBot_ciSup h_anti.dual hbdd.dual using 1 | theorem | Topology | [
"Mathlib.Topology.Order.Basic"
] | Mathlib/Topology/Order/MonotoneConvergence.lean | tendsto_atTop_ciInf | null |
tendsto_atTop_iSup (h_mono : Monotone f) : Tendsto f atTop (𝓝 (⨆ i, f i)) :=
tendsto_atTop_ciSup h_mono (OrderTop.bddAbove _) | theorem | Topology | [
"Mathlib.Topology.Order.Basic"
] | Mathlib/Topology/Order/MonotoneConvergence.lean | tendsto_atTop_iSup | null |
tendsto_atBot_iSup (h_anti : Antitone f) : Tendsto f atBot (𝓝 (⨆ i, f i)) :=
tendsto_atBot_ciSup h_anti (OrderTop.bddAbove _) | theorem | Topology | [
"Mathlib.Topology.Order.Basic"
] | Mathlib/Topology/Order/MonotoneConvergence.lean | tendsto_atBot_iSup | null |
tendsto_atBot_iInf (h_mono : Monotone f) : Tendsto f atBot (𝓝 (⨅ i, f i)) :=
tendsto_atBot_ciInf h_mono (OrderBot.bddBelow _) | theorem | Topology | [
"Mathlib.Topology.Order.Basic"
] | Mathlib/Topology/Order/MonotoneConvergence.lean | tendsto_atBot_iInf | null |
tendsto_atTop_iInf (h_anti : Antitone f) : Tendsto f atTop (𝓝 (⨅ i, f i)) :=
tendsto_atTop_ciInf h_anti (OrderBot.bddBelow _) | theorem | Topology | [
"Mathlib.Topology.Order.Basic"
] | Mathlib/Topology/Order/MonotoneConvergence.lean | tendsto_atTop_iInf | null |
Prod.supConvergenceClass
[Preorder α] [Preorder β] [TopologicalSpace α] [TopologicalSpace β]
[SupConvergenceClass α] [SupConvergenceClass β] : SupConvergenceClass (α × β) := by
constructor
rintro ⟨a, b⟩ s h
rw [isLUB_prod, ← range_restrict, ← range_restrict] at h
have A : Tendsto (fun x : s => (x : α × β).1) atTop (𝓝 a) :=
tendsto_atTop_isLUB (monotone_fst.restrict s) h.1
have B : Tendsto (fun x : s => (x : α × β).2) atTop (𝓝 b) :=
tendsto_atTop_isLUB (monotone_snd.restrict s) h.2
exact A.prodMk_nhds B | instance | Topology | [
"Mathlib.Topology.Order.Basic"
] | Mathlib/Topology/Order/MonotoneConvergence.lean | Prod.supConvergenceClass | null |
Pi.supConvergenceClass
{ι : Type*} {α : ι → Type*} [∀ i, Preorder (α i)] [∀ i, TopologicalSpace (α i)]
[∀ i, SupConvergenceClass (α i)] : SupConvergenceClass (∀ i, α i) := by
refine ⟨fun f s h => ?_⟩
simp only [isLUB_pi, ← range_restrict] at h
exact tendsto_pi_nhds.2 fun i => tendsto_atTop_isLUB ((monotone_eval _).restrict _) (h i) | instance | Topology | [
"Mathlib.Topology.Order.Basic"
] | Mathlib/Topology/Order/MonotoneConvergence.lean | Pi.supConvergenceClass | null |
Pi.infConvergenceClass
{ι : Type*} {α : ι → Type*} [∀ i, Preorder (α i)] [∀ i, TopologicalSpace (α i)]
[∀ i, InfConvergenceClass (α i)] : InfConvergenceClass (∀ i, α i) :=
show InfConvergenceClass (∀ i, (α i)ᵒᵈ)ᵒᵈ from OrderDual.infConvergenceClass | instance | Topology | [
"Mathlib.Topology.Order.Basic"
] | Mathlib/Topology/Order/MonotoneConvergence.lean | Pi.infConvergenceClass | null |
Pi.supConvergenceClass' {ι : Type*} [Preorder α] [TopologicalSpace α]
[SupConvergenceClass α] : SupConvergenceClass (ι → α) :=
supConvergenceClass | instance | Topology | [
"Mathlib.Topology.Order.Basic"
] | Mathlib/Topology/Order/MonotoneConvergence.lean | Pi.supConvergenceClass' | null |
Pi.infConvergenceClass' {ι : Type*} [Preorder α] [TopologicalSpace α]
[InfConvergenceClass α] : InfConvergenceClass (ι → α) :=
Pi.infConvergenceClass | instance | Topology | [
"Mathlib.Topology.Order.Basic"
] | Mathlib/Topology/Order/MonotoneConvergence.lean | Pi.infConvergenceClass' | null |
tendsto_of_monotone {ι α : Type*} [Preorder ι] [TopologicalSpace α]
[ConditionallyCompleteLinearOrder α] [OrderTopology α] {f : ι → α} (h_mono : Monotone f) :
Tendsto f atTop atTop ∨ ∃ l, Tendsto f atTop (𝓝 l) := by
classical
exact if H : BddAbove (range f) then Or.inr ⟨_, tendsto_atTop_ciSup h_mono H⟩
else Or.inl <| tendsto_atTop_atTop_of_monotone' h_mono H | theorem | Topology | [
"Mathlib.Topology.Order.Basic"
] | Mathlib/Topology/Order/MonotoneConvergence.lean | tendsto_of_monotone | null |
tendsto_of_antitone {ι α : Type*} [Preorder ι] [TopologicalSpace α]
[ConditionallyCompleteLinearOrder α] [OrderTopology α] {f : ι → α} (h_mono : Antitone f) :
Tendsto f atTop atBot ∨ ∃ l, Tendsto f atTop (𝓝 l) :=
@tendsto_of_monotone ι αᵒᵈ _ _ _ _ _ h_mono | theorem | Topology | [
"Mathlib.Topology.Order.Basic"
] | Mathlib/Topology/Order/MonotoneConvergence.lean | tendsto_of_antitone | null |
tendsto_iff_tendsto_subseq_of_monotone {ι₁ ι₂ α : Type*} [SemilatticeSup ι₁] [Preorder ι₂]
[Nonempty ι₁] [TopologicalSpace α] [ConditionallyCompleteLinearOrder α] [OrderTopology α]
[NoMaxOrder α] {f : ι₂ → α} {φ : ι₁ → ι₂} {l : α} (hf : Monotone f)
(hg : Tendsto φ atTop atTop) : Tendsto f atTop (𝓝 l) ↔ Tendsto (f ∘ φ) atTop (𝓝 l) := by
constructor <;> intro h
· exact h.comp hg
· rcases tendsto_of_monotone hf with (h' | ⟨l', hl'⟩)
· exact (not_tendsto_atTop_of_tendsto_nhds h (h'.comp hg)).elim
· rwa [tendsto_nhds_unique h (hl'.comp hg)] | theorem | Topology | [
"Mathlib.Topology.Order.Basic"
] | Mathlib/Topology/Order/MonotoneConvergence.lean | tendsto_iff_tendsto_subseq_of_monotone | null |
tendsto_iff_tendsto_subseq_of_antitone {ι₁ ι₂ α : Type*} [SemilatticeSup ι₁] [Preorder ι₂]
[Nonempty ι₁] [TopologicalSpace α] [ConditionallyCompleteLinearOrder α] [OrderTopology α]
[NoMinOrder α] {f : ι₂ → α} {φ : ι₁ → ι₂} {l : α} (hf : Antitone f)
(hg : Tendsto φ atTop atTop) : Tendsto f atTop (𝓝 l) ↔ Tendsto (f ∘ φ) atTop (𝓝 l) :=
tendsto_iff_tendsto_subseq_of_monotone (α := αᵒᵈ) hf hg
/-! The next family of results, such as `isLUB_of_tendsto_atTop` and `iSup_eq_of_tendsto`, are
converses to the standard fact that bounded monotone functions converge. They state, that if a
monotone function `f` tends to `a` along `Filter.atTop`, then that value `a` is a least upper bound
for the range of `f`.
Related theorems above (`IsLUB.isLUB_of_tendsto`, `IsGLB.isGLB_of_tendsto` etc) cover the case
when `f x` tends to `a` as `x` tends to some point `b` in the domain. -/ | theorem | Topology | [
"Mathlib.Topology.Order.Basic"
] | Mathlib/Topology/Order/MonotoneConvergence.lean | tendsto_iff_tendsto_subseq_of_antitone | null |
Monotone.ge_of_tendsto [TopologicalSpace α] [Preorder α] [OrderClosedTopology α]
[SemilatticeSup β] {f : β → α} {a : α} (hf : Monotone f) (ha : Tendsto f atTop (𝓝 a)) (b : β) :
f b ≤ a :=
haveI : Nonempty β := Nonempty.intro b
_root_.ge_of_tendsto ha ((eventually_ge_atTop b).mono fun _ hxy => hf hxy) | theorem | Topology | [
"Mathlib.Topology.Order.Basic"
] | Mathlib/Topology/Order/MonotoneConvergence.lean | Monotone.ge_of_tendsto | null |
Monotone.le_of_tendsto [TopologicalSpace α] [Preorder α] [OrderClosedTopology α]
[SemilatticeInf β] {f : β → α} {a : α} (hf : Monotone f) (ha : Tendsto f atBot (𝓝 a)) (b : β) :
a ≤ f b :=
hf.dual.ge_of_tendsto ha b | theorem | Topology | [
"Mathlib.Topology.Order.Basic"
] | Mathlib/Topology/Order/MonotoneConvergence.lean | Monotone.le_of_tendsto | null |
Antitone.le_of_tendsto [TopologicalSpace α] [Preorder α] [OrderClosedTopology α]
[SemilatticeSup β] {f : β → α} {a : α} (hf : Antitone f) (ha : Tendsto f atTop (𝓝 a)) (b : β) :
a ≤ f b :=
hf.dual_right.ge_of_tendsto ha b | theorem | Topology | [
"Mathlib.Topology.Order.Basic"
] | Mathlib/Topology/Order/MonotoneConvergence.lean | Antitone.le_of_tendsto | null |
Antitone.ge_of_tendsto [TopologicalSpace α] [Preorder α] [OrderClosedTopology α]
[SemilatticeInf β] {f : β → α} {a : α} (hf : Antitone f) (ha : Tendsto f atBot (𝓝 a)) (b : β) :
f b ≤ a :=
hf.dual_right.le_of_tendsto ha b | theorem | Topology | [
"Mathlib.Topology.Order.Basic"
] | Mathlib/Topology/Order/MonotoneConvergence.lean | Antitone.ge_of_tendsto | null |
isLUB_of_tendsto_atTop [TopologicalSpace α] [Preorder α] [OrderClosedTopology α]
[Nonempty β] [SemilatticeSup β] {f : β → α} {a : α} (hf : Monotone f)
(ha : Tendsto f atTop (𝓝 a)) : IsLUB (Set.range f) a := by
constructor
· rintro _ ⟨b, rfl⟩
exact hf.ge_of_tendsto ha b
· exact fun _ hb => le_of_tendsto' ha fun x => hb (Set.mem_range_self x) | theorem | Topology | [
"Mathlib.Topology.Order.Basic"
] | Mathlib/Topology/Order/MonotoneConvergence.lean | isLUB_of_tendsto_atTop | null |
isGLB_of_tendsto_atBot [TopologicalSpace α] [Preorder α] [OrderClosedTopology α]
[Nonempty β] [SemilatticeInf β] {f : β → α} {a : α} (hf : Monotone f)
(ha : Tendsto f atBot (𝓝 a)) : IsGLB (Set.range f) a :=
@isLUB_of_tendsto_atTop αᵒᵈ βᵒᵈ _ _ _ _ _ _ _ hf.dual ha | theorem | Topology | [
"Mathlib.Topology.Order.Basic"
] | Mathlib/Topology/Order/MonotoneConvergence.lean | isGLB_of_tendsto_atBot | null |
isLUB_of_tendsto_atBot [TopologicalSpace α] [Preorder α] [OrderClosedTopology α]
[Nonempty β] [SemilatticeInf β] {f : β → α} {a : α} (hf : Antitone f)
(ha : Tendsto f atBot (𝓝 a)) : IsLUB (Set.range f) a :=
@isLUB_of_tendsto_atTop α βᵒᵈ _ _ _ _ _ _ _ hf.dual_left ha | theorem | Topology | [
"Mathlib.Topology.Order.Basic"
] | Mathlib/Topology/Order/MonotoneConvergence.lean | isLUB_of_tendsto_atBot | null |
isGLB_of_tendsto_atTop [TopologicalSpace α] [Preorder α] [OrderClosedTopology α]
[Nonempty β] [SemilatticeSup β] {f : β → α} {a : α} (hf : Antitone f)
(ha : Tendsto f atTop (𝓝 a)) : IsGLB (Set.range f) a :=
@isGLB_of_tendsto_atBot α βᵒᵈ _ _ _ _ _ _ _ hf.dual_left ha | theorem | Topology | [
"Mathlib.Topology.Order.Basic"
] | Mathlib/Topology/Order/MonotoneConvergence.lean | isGLB_of_tendsto_atTop | null |
iSup_eq_of_tendsto {α β} [TopologicalSpace α] [CompleteLinearOrder α] [OrderTopology α]
[Nonempty β] [SemilatticeSup β] {f : β → α} {a : α} (hf : Monotone f) :
Tendsto f atTop (𝓝 a) → iSup f = a :=
tendsto_nhds_unique (tendsto_atTop_iSup hf) | theorem | Topology | [
"Mathlib.Topology.Order.Basic"
] | Mathlib/Topology/Order/MonotoneConvergence.lean | iSup_eq_of_tendsto | null |
iInf_eq_of_tendsto {α} [TopologicalSpace α] [CompleteLinearOrder α] [OrderTopology α]
[Nonempty β] [SemilatticeSup β] {f : β → α} {a : α} (hf : Antitone f) :
Tendsto f atTop (𝓝 a) → iInf f = a :=
tendsto_nhds_unique (tendsto_atTop_iInf hf) | theorem | Topology | [
"Mathlib.Topology.Order.Basic"
] | Mathlib/Topology/Order/MonotoneConvergence.lean | iInf_eq_of_tendsto | null |
iSup_eq_iSup_subseq_of_monotone {ι₁ ι₂ α : Type*} [Preorder ι₂] [CompleteLattice α]
{l : Filter ι₁} [l.NeBot] {f : ι₂ → α} {φ : ι₁ → ι₂} (hf : Monotone f)
(hφ : Tendsto φ l atTop) : ⨆ i, f i = ⨆ i, f (φ i) :=
le_antisymm
(iSup_mono' fun i =>
Exists.imp (fun j (hj : i ≤ φ j) => hf hj) (hφ.eventually <| eventually_ge_atTop i).exists)
(iSup_mono' fun i => ⟨φ i, le_rfl⟩) | theorem | Topology | [
"Mathlib.Topology.Order.Basic"
] | Mathlib/Topology/Order/MonotoneConvergence.lean | iSup_eq_iSup_subseq_of_monotone | null |
iSup_eq_iSup_subseq_of_antitone {ι₁ ι₂ α : Type*} [Preorder ι₂] [CompleteLattice α]
{l : Filter ι₁} [l.NeBot] {f : ι₂ → α} {φ : ι₁ → ι₂} (hf : Antitone f)
(hφ : Tendsto φ l atBot) : ⨆ i, f i = ⨆ i, f (φ i) :=
le_antisymm
(iSup_mono' fun i =>
Exists.imp (fun j (hj : φ j ≤ i) => hf hj) (hφ.eventually <| eventually_le_atBot i).exists)
(iSup_mono' fun i => ⟨φ i, le_rfl⟩) | theorem | Topology | [
"Mathlib.Topology.Order.Basic"
] | Mathlib/Topology/Order/MonotoneConvergence.lean | iSup_eq_iSup_subseq_of_antitone | null |
iInf_eq_iInf_subseq_of_monotone {ι₁ ι₂ α : Type*} [Preorder ι₂] [CompleteLattice α]
{l : Filter ι₁} [l.NeBot] {f : ι₂ → α} {φ : ι₁ → ι₂} (hf : Monotone f)
(hφ : Tendsto φ l atBot) : ⨅ i, f i = ⨅ i, f (φ i) :=
iSup_eq_iSup_subseq_of_monotone hf.dual hφ | theorem | Topology | [
"Mathlib.Topology.Order.Basic"
] | Mathlib/Topology/Order/MonotoneConvergence.lean | iInf_eq_iInf_subseq_of_monotone | null |
iInf_eq_iInf_subseq_of_antitone {ι₁ ι₂ α : Type*} [Preorder ι₂] [CompleteLattice α]
{l : Filter ι₁} [l.NeBot] {f : ι₂ → α} {φ : ι₁ → ι₂} (hf : Antitone f)
(hφ : Tendsto φ l atTop) : ⨅ i, f i = ⨅ i, f (φ i) :=
iSup_eq_iSup_subseq_of_antitone hf.dual hφ | theorem | Topology | [
"Mathlib.Topology.Order.Basic"
] | Mathlib/Topology/Order/MonotoneConvergence.lean | iInf_eq_iInf_subseq_of_antitone | null |
@[simp] nhdsSet_Ioi : 𝓝ˢ (Ioi a) = 𝓟 (Ioi a) := isOpen_Ioi.nhdsSet_eq
@[simp] theorem nhdsSet_Iio : 𝓝ˢ (Iio a) = 𝓟 (Iio a) := isOpen_Iio.nhdsSet_eq
@[simp] theorem nhdsSet_Ioo : 𝓝ˢ (Ioo a b) = 𝓟 (Ioo a b) := isOpen_Ioo.nhdsSet_eq | theorem | Topology | [
"Mathlib.Topology.Order.Basic"
] | Mathlib/Topology/Order/NhdsSet.lean | nhdsSet_Ioi | null |
nhdsSet_Ici : 𝓝ˢ (Ici a) = 𝓝 a ⊔ 𝓟 (Ioi a) := by
rw [← Ioi_insert, nhdsSet_insert, nhdsSet_Ioi] | theorem | Topology | [
"Mathlib.Topology.Order.Basic"
] | Mathlib/Topology/Order/NhdsSet.lean | nhdsSet_Ici | null |
nhdsSet_Iic : 𝓝ˢ (Iic a) = 𝓝 a ⊔ 𝓟 (Iio a) := nhdsSet_Ici (α := αᵒᵈ) | theorem | Topology | [
"Mathlib.Topology.Order.Basic"
] | Mathlib/Topology/Order/NhdsSet.lean | nhdsSet_Iic | null |
nhdsSet_Ico (h : a < b) : 𝓝ˢ (Ico a b) = 𝓝 a ⊔ 𝓟 (Ioo a b) := by
rw [← Ioo_insert_left h, nhdsSet_insert, nhdsSet_Ioo] | theorem | Topology | [
"Mathlib.Topology.Order.Basic"
] | Mathlib/Topology/Order/NhdsSet.lean | nhdsSet_Ico | null |
nhdsSet_Ioc (h : a < b) : 𝓝ˢ (Ioc a b) = 𝓝 b ⊔ 𝓟 (Ioo a b) := by
rw [← Ioo_insert_right h, nhdsSet_insert, nhdsSet_Ioo] | theorem | Topology | [
"Mathlib.Topology.Order.Basic"
] | Mathlib/Topology/Order/NhdsSet.lean | nhdsSet_Ioc | null |
nhdsSet_Icc (h : a ≤ b) : 𝓝ˢ (Icc a b) = 𝓝 a ⊔ 𝓝 b ⊔ 𝓟 (Ioo a b) := by
rcases h.eq_or_lt with rfl | hlt
· simp
· rw [← Ioc_insert_left h, nhdsSet_insert, nhdsSet_Ioc hlt, sup_assoc]
/-! | theorem | Topology | [
"Mathlib.Topology.Order.Basic"
] | Mathlib/Topology/Order/NhdsSet.lean | nhdsSet_Icc | null |
@[simp]
Ioi_mem_nhdsSet_Ici_iff : Ioi a ∈ 𝓝ˢ (Ici b) ↔ a < b := by
rw [isOpen_Ioi.mem_nhdsSet, Ici_subset_Ioi]
alias ⟨_, Ioi_mem_nhdsSet_Ici⟩ := Ioi_mem_nhdsSet_Ici_iff | theorem | Topology | [
"Mathlib.Topology.Order.Basic"
] | Mathlib/Topology/Order/NhdsSet.lean | Ioi_mem_nhdsSet_Ici_iff | null |
Ici_mem_nhdsSet_Ici (h : a < b) : Ici a ∈ 𝓝ˢ (Ici b) :=
mem_of_superset (Ioi_mem_nhdsSet_Ici h) Ioi_subset_Ici_self
/-! | theorem | Topology | [
"Mathlib.Topology.Order.Basic"
] | Mathlib/Topology/Order/NhdsSet.lean | Ici_mem_nhdsSet_Ici | null |
Iio_mem_nhdsSet_Iic_iff : Iio b ∈ 𝓝ˢ (Iic a) ↔ a < b :=
Ioi_mem_nhdsSet_Ici_iff (α := αᵒᵈ)
alias ⟨_, Iio_mem_nhdsSet_Iic⟩ := Iio_mem_nhdsSet_Iic_iff | theorem | Topology | [
"Mathlib.Topology.Order.Basic"
] | Mathlib/Topology/Order/NhdsSet.lean | Iio_mem_nhdsSet_Iic_iff | null |
Iic_mem_nhdsSet_Iic (h : a < b) : Iic b ∈ 𝓝ˢ (Iic a) :=
Ici_mem_nhdsSet_Ici (α := αᵒᵈ) h
/-! | theorem | Topology | [
"Mathlib.Topology.Order.Basic"
] | Mathlib/Topology/Order/NhdsSet.lean | Iic_mem_nhdsSet_Iic | null |
Ioi_mem_nhdsSet_Icc (h : a < b) : Ioi a ∈ 𝓝ˢ (Icc b c) :=
nhdsSet_mono Icc_subset_Ici_self <| Ioi_mem_nhdsSet_Ici h | theorem | Topology | [
"Mathlib.Topology.Order.Basic"
] | Mathlib/Topology/Order/NhdsSet.lean | Ioi_mem_nhdsSet_Icc | null |
Ici_mem_nhdsSet_Icc (h : a < b) : Ici a ∈ 𝓝ˢ (Icc b c) :=
mem_of_superset (Ioi_mem_nhdsSet_Icc h) Ioi_subset_Ici_self | theorem | Topology | [
"Mathlib.Topology.Order.Basic"
] | Mathlib/Topology/Order/NhdsSet.lean | Ici_mem_nhdsSet_Icc | null |
Iio_mem_nhdsSet_Icc (h : b < c) : Iio c ∈ 𝓝ˢ (Icc a b) :=
nhdsSet_mono Icc_subset_Iic_self <| Iio_mem_nhdsSet_Iic h | theorem | Topology | [
"Mathlib.Topology.Order.Basic"
] | Mathlib/Topology/Order/NhdsSet.lean | Iio_mem_nhdsSet_Icc | null |
Iic_mem_nhdsSet_Icc (h : b < c) : Iic c ∈ 𝓝ˢ (Icc a b) :=
mem_of_superset (Iio_mem_nhdsSet_Icc h) Iio_subset_Iic_self | theorem | Topology | [
"Mathlib.Topology.Order.Basic"
] | Mathlib/Topology/Order/NhdsSet.lean | Iic_mem_nhdsSet_Icc | null |
Ioo_mem_nhdsSet_Icc (h : a < b) (h' : c < d) : Ioo a d ∈ 𝓝ˢ (Icc b c) :=
inter_mem (Ioi_mem_nhdsSet_Icc h) (Iio_mem_nhdsSet_Icc h') | theorem | Topology | [
"Mathlib.Topology.Order.Basic"
] | Mathlib/Topology/Order/NhdsSet.lean | Ioo_mem_nhdsSet_Icc | null |
Ico_mem_nhdsSet_Icc (h : a < b) (h' : c < d) : Ico a d ∈ 𝓝ˢ (Icc b c) :=
inter_mem (Ici_mem_nhdsSet_Icc h) (Iio_mem_nhdsSet_Icc h') | theorem | Topology | [
"Mathlib.Topology.Order.Basic"
] | Mathlib/Topology/Order/NhdsSet.lean | Ico_mem_nhdsSet_Icc | null |
Ioc_mem_nhdsSet_Icc (h : a < b) (h' : c < d) : Ioc a d ∈ 𝓝ˢ (Icc b c) :=
inter_mem (Ioi_mem_nhdsSet_Icc h) (Iic_mem_nhdsSet_Icc h') | theorem | Topology | [
"Mathlib.Topology.Order.Basic"
] | Mathlib/Topology/Order/NhdsSet.lean | Ioc_mem_nhdsSet_Icc | null |
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