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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@[to_additive /-- If `a` is positive, then the intervals `(a - ε, a + ε)`, `0 < ε ≤ a`,
form a basis of neighborhoods of `a`.
This upper bound for `ε` guarantees that all elements of these intervals are positive. -/]
nhds_basis_Ioo_one_lt_of_one_lt [NoMaxOrder α] {a : α} (ha : 1 < a) :
(𝓝 a).HasBasis (fun ε : α => (1 : α) < ε ∧ ε ≤ a) fun ε => Ioo (a / ε) (a * ε) :=
(nhds_basis_Ioo_one_lt a).restrict fun ε hε ↦
⟨min a ε, lt_min ha hε, min_le_left _ _, by gcongr <;> apply min_le_right⟩ | theorem | Topology | [
"Mathlib.Algebra.Ring.Pointwise.Set",
"Mathlib.Order.Filter.AtTopBot.CompleteLattice",
"Mathlib.Order.Filter.AtTopBot.Group",
"Mathlib.Topology.Order.Basic"
] | Mathlib/Topology/Order/LeftRightNhds.lean | nhds_basis_Ioo_one_lt_of_one_lt | If `a > 1`, then open intervals `(a / ε, aε)`, `1 < ε ≤ a`,
form a basis of neighborhoods of `a`.
This upper bound for `ε` guarantees that all elements of these intervals are greater than one. |
mem_nhdsGE (hS : OrdConnected S) (hx : x ∈ S) (hy : y ∈ S) (hxy : x < y) : S ∈ 𝓝[≥] x :=
mem_of_superset (Icc_mem_nhdsGE hxy) <| hS.out hx hy | lemma | Topology | [
"Mathlib.Algebra.Ring.Pointwise.Set",
"Mathlib.Order.Filter.AtTopBot.CompleteLattice",
"Mathlib.Order.Filter.AtTopBot.Group",
"Mathlib.Topology.Order.Basic"
] | Mathlib/Topology/Order/LeftRightNhds.lean | mem_nhdsGE | If `S` is order-connected and contains two points `x < y`,
then `S` is a right neighbourhood of `x`. |
mem_nhdsGT (hS : OrdConnected S) (hx : x ∈ S) (hy : y ∈ S) (hxy : x < y) : S ∈ 𝓝[>] x :=
nhdsWithin_mono _ Ioi_subset_Ici_self <| hS.mem_nhdsGE hx hy hxy | lemma | Topology | [
"Mathlib.Algebra.Ring.Pointwise.Set",
"Mathlib.Order.Filter.AtTopBot.CompleteLattice",
"Mathlib.Order.Filter.AtTopBot.Group",
"Mathlib.Topology.Order.Basic"
] | Mathlib/Topology/Order/LeftRightNhds.lean | mem_nhdsGT | If `S` is order-connected and contains two points `x < y`,
then `S` is a punctured right neighbourhood of `x`. |
mem_nhdsLE (hS : OrdConnected S) (hx : x ∈ S) (hy : y ∈ S) (hxy : x < y) : S ∈ 𝓝[≤] y :=
hS.dual.mem_nhdsGE hy hx hxy | lemma | Topology | [
"Mathlib.Algebra.Ring.Pointwise.Set",
"Mathlib.Order.Filter.AtTopBot.CompleteLattice",
"Mathlib.Order.Filter.AtTopBot.Group",
"Mathlib.Topology.Order.Basic"
] | Mathlib/Topology/Order/LeftRightNhds.lean | mem_nhdsLE | If `S` is order-connected and contains two points `x < y`, then `S` is a left neighbourhood
of `y`. |
mem_nhdsLT (hS : OrdConnected S) (hx : x ∈ S) (hy : y ∈ S) (hxy : x < y) : S ∈ 𝓝[<] y :=
hS.dual.mem_nhdsGT hy hx hxy | lemma | Topology | [
"Mathlib.Algebra.Ring.Pointwise.Set",
"Mathlib.Order.Filter.AtTopBot.CompleteLattice",
"Mathlib.Order.Filter.AtTopBot.Group",
"Mathlib.Topology.Order.Basic"
] | Mathlib/Topology/Order/LeftRightNhds.lean | mem_nhdsLT | If `S` is order-connected and contains two points `x < y`, then `S` is a punctured left
neighbourhood of `y`. |
BoundedLENhdsClass (α : Type*) [Preorder α] [TopologicalSpace α] : Prop where
isBounded_le_nhds (a : α) : (𝓝 a).IsBounded (· ≤ ·) | class | Topology | [
"Mathlib.Order.Filter.CountableInter",
"Mathlib.Order.LiminfLimsup",
"Mathlib.Topology.Order.Monotone"
] | Mathlib/Topology/Order/LiminfLimsup.lean | BoundedLENhdsClass | Ad hoc typeclass stating that neighborhoods are eventually bounded above. |
BoundedGENhdsClass (α : Type*) [Preorder α] [TopologicalSpace α] : Prop where
isBounded_ge_nhds (a : α) : (𝓝 a).IsBounded (· ≥ ·) | class | Topology | [
"Mathlib.Order.Filter.CountableInter",
"Mathlib.Order.LiminfLimsup",
"Mathlib.Topology.Order.Monotone"
] | Mathlib/Topology/Order/LiminfLimsup.lean | BoundedGENhdsClass | Ad hoc typeclass stating that neighborhoods are eventually bounded below. |
isBounded_le_nhds (a : α) : (𝓝 a).IsBounded (· ≤ ·) :=
BoundedLENhdsClass.isBounded_le_nhds _ | theorem | Topology | [
"Mathlib.Order.Filter.CountableInter",
"Mathlib.Order.LiminfLimsup",
"Mathlib.Topology.Order.Monotone"
] | Mathlib/Topology/Order/LiminfLimsup.lean | isBounded_le_nhds | null |
Filter.Tendsto.isBoundedUnder_le (h : Tendsto u f (𝓝 a)) : f.IsBoundedUnder (· ≤ ·) u :=
(isBounded_le_nhds a).mono h | theorem | Topology | [
"Mathlib.Order.Filter.CountableInter",
"Mathlib.Order.LiminfLimsup",
"Mathlib.Topology.Order.Monotone"
] | Mathlib/Topology/Order/LiminfLimsup.lean | Filter.Tendsto.isBoundedUnder_le | null |
Filter.Tendsto.bddAbove_range_of_cofinite [IsDirected α (· ≤ ·)]
(h : Tendsto u cofinite (𝓝 a)) : BddAbove (Set.range u) :=
h.isBoundedUnder_le.bddAbove_range_of_cofinite | theorem | Topology | [
"Mathlib.Order.Filter.CountableInter",
"Mathlib.Order.LiminfLimsup",
"Mathlib.Topology.Order.Monotone"
] | Mathlib/Topology/Order/LiminfLimsup.lean | Filter.Tendsto.bddAbove_range_of_cofinite | null |
Filter.Tendsto.bddAbove_range [IsDirected α (· ≤ ·)] {u : ℕ → α}
(h : Tendsto u atTop (𝓝 a)) : BddAbove (Set.range u) :=
h.isBoundedUnder_le.bddAbove_range | theorem | Topology | [
"Mathlib.Order.Filter.CountableInter",
"Mathlib.Order.LiminfLimsup",
"Mathlib.Topology.Order.Monotone"
] | Mathlib/Topology/Order/LiminfLimsup.lean | Filter.Tendsto.bddAbove_range | null |
isCobounded_ge_nhds (a : α) : (𝓝 a).IsCobounded (· ≥ ·) :=
(isBounded_le_nhds a).isCobounded_flip | theorem | Topology | [
"Mathlib.Order.Filter.CountableInter",
"Mathlib.Order.LiminfLimsup",
"Mathlib.Topology.Order.Monotone"
] | Mathlib/Topology/Order/LiminfLimsup.lean | isCobounded_ge_nhds | null |
Filter.Tendsto.isCoboundedUnder_ge [NeBot f] (h : Tendsto u f (𝓝 a)) :
f.IsCoboundedUnder (· ≥ ·) u :=
h.isBoundedUnder_le.isCobounded_flip | theorem | Topology | [
"Mathlib.Order.Filter.CountableInter",
"Mathlib.Order.LiminfLimsup",
"Mathlib.Topology.Order.Monotone"
] | Mathlib/Topology/Order/LiminfLimsup.lean | Filter.Tendsto.isCoboundedUnder_ge | null |
Prod.instBoundedLENhdsClass : BoundedLENhdsClass (α × β) := by
refine ⟨fun x ↦ ?_⟩
obtain ⟨a, ha⟩ := isBounded_le_nhds x.1
obtain ⟨b, hb⟩ := isBounded_le_nhds x.2
rw [← @Prod.mk.eta _ _ x, nhds_prod_eq]
exact ⟨(a, b), ha.prod_mk hb⟩ | instance | Topology | [
"Mathlib.Order.Filter.CountableInter",
"Mathlib.Order.LiminfLimsup",
"Mathlib.Topology.Order.Monotone"
] | Mathlib/Topology/Order/LiminfLimsup.lean | Prod.instBoundedLENhdsClass | null |
Pi.instBoundedLENhdsClass [Finite ι] [∀ i, Preorder (π i)] [∀ i, TopologicalSpace (π i)]
[∀ i, BoundedLENhdsClass (π i)] : BoundedLENhdsClass (∀ i, π i) := by
refine ⟨fun x ↦ ?_⟩
rw [nhds_pi]
choose f hf using fun i ↦ isBounded_le_nhds (x i)
exact ⟨f, eventually_pi hf⟩ | instance | Topology | [
"Mathlib.Order.Filter.CountableInter",
"Mathlib.Order.LiminfLimsup",
"Mathlib.Topology.Order.Monotone"
] | Mathlib/Topology/Order/LiminfLimsup.lean | Pi.instBoundedLENhdsClass | null |
isBounded_ge_nhds (a : α) : (𝓝 a).IsBounded (· ≥ ·) :=
BoundedGENhdsClass.isBounded_ge_nhds _ | theorem | Topology | [
"Mathlib.Order.Filter.CountableInter",
"Mathlib.Order.LiminfLimsup",
"Mathlib.Topology.Order.Monotone"
] | Mathlib/Topology/Order/LiminfLimsup.lean | isBounded_ge_nhds | null |
Filter.Tendsto.isBoundedUnder_ge (h : Tendsto u f (𝓝 a)) : f.IsBoundedUnder (· ≥ ·) u :=
(isBounded_ge_nhds a).mono h | theorem | Topology | [
"Mathlib.Order.Filter.CountableInter",
"Mathlib.Order.LiminfLimsup",
"Mathlib.Topology.Order.Monotone"
] | Mathlib/Topology/Order/LiminfLimsup.lean | Filter.Tendsto.isBoundedUnder_ge | null |
Filter.Tendsto.bddBelow_range_of_cofinite [IsDirected α (· ≥ ·)]
(h : Tendsto u cofinite (𝓝 a)) : BddBelow (Set.range u) :=
h.isBoundedUnder_ge.bddBelow_range_of_cofinite | theorem | Topology | [
"Mathlib.Order.Filter.CountableInter",
"Mathlib.Order.LiminfLimsup",
"Mathlib.Topology.Order.Monotone"
] | Mathlib/Topology/Order/LiminfLimsup.lean | Filter.Tendsto.bddBelow_range_of_cofinite | null |
Filter.Tendsto.bddBelow_range [IsDirected α (· ≥ ·)] {u : ℕ → α}
(h : Tendsto u atTop (𝓝 a)) : BddBelow (Set.range u) :=
h.isBoundedUnder_ge.bddBelow_range | theorem | Topology | [
"Mathlib.Order.Filter.CountableInter",
"Mathlib.Order.LiminfLimsup",
"Mathlib.Topology.Order.Monotone"
] | Mathlib/Topology/Order/LiminfLimsup.lean | Filter.Tendsto.bddBelow_range | null |
isCobounded_le_nhds (a : α) : (𝓝 a).IsCobounded (· ≤ ·) :=
(isBounded_ge_nhds a).isCobounded_flip | theorem | Topology | [
"Mathlib.Order.Filter.CountableInter",
"Mathlib.Order.LiminfLimsup",
"Mathlib.Topology.Order.Monotone"
] | Mathlib/Topology/Order/LiminfLimsup.lean | isCobounded_le_nhds | null |
Filter.Tendsto.isCoboundedUnder_le [NeBot f] (h : Tendsto u f (𝓝 a)) :
f.IsCoboundedUnder (· ≤ ·) u :=
h.isBoundedUnder_ge.isCobounded_flip | theorem | Topology | [
"Mathlib.Order.Filter.CountableInter",
"Mathlib.Order.LiminfLimsup",
"Mathlib.Topology.Order.Monotone"
] | Mathlib/Topology/Order/LiminfLimsup.lean | Filter.Tendsto.isCoboundedUnder_le | null |
Prod.instBoundedGENhdsClass : BoundedGENhdsClass (α × β) :=
⟨(Prod.instBoundedLENhdsClass (α := αᵒᵈ) (β := βᵒᵈ)).isBounded_le_nhds⟩ | instance | Topology | [
"Mathlib.Order.Filter.CountableInter",
"Mathlib.Order.LiminfLimsup",
"Mathlib.Topology.Order.Monotone"
] | Mathlib/Topology/Order/LiminfLimsup.lean | Prod.instBoundedGENhdsClass | null |
Pi.instBoundedGENhdsClass [Finite ι] [∀ i, Preorder (π i)] [∀ i, TopologicalSpace (π i)]
[∀ i, BoundedGENhdsClass (π i)] : BoundedGENhdsClass (∀ i, π i) :=
⟨(Pi.instBoundedLENhdsClass (π := fun i ↦ (π i)ᵒᵈ)).isBounded_le_nhds⟩ | instance | Topology | [
"Mathlib.Order.Filter.CountableInter",
"Mathlib.Order.LiminfLimsup",
"Mathlib.Topology.Order.Monotone"
] | Mathlib/Topology/Order/LiminfLimsup.lean | Pi.instBoundedGENhdsClass | null |
le_nhds_of_limsSup_eq_limsInf {f : Filter α} {a : α} (hl : f.IsBounded (· ≤ ·))
(hg : f.IsBounded (· ≥ ·)) (hs : f.limsSup = a) (hi : f.limsInf = a) : f ≤ 𝓝 a :=
tendsto_order.2 ⟨fun _ hb ↦ gt_mem_sets_of_limsInf_gt hg <| hi.symm ▸ hb,
fun _ hb ↦ lt_mem_sets_of_limsSup_lt hl <| hs.symm ▸ hb⟩ | theorem | Topology | [
"Mathlib.Order.Filter.CountableInter",
"Mathlib.Order.LiminfLimsup",
"Mathlib.Topology.Order.Monotone"
] | Mathlib/Topology/Order/LiminfLimsup.lean | le_nhds_of_limsSup_eq_limsInf | If the liminf and the limsup of a filter coincide, then this filter converges to
their common value, at least if the filter is eventually bounded above and below. |
limsSup_nhds (a : α) : limsSup (𝓝 a) = a :=
csInf_eq_of_forall_ge_of_forall_gt_exists_lt (isBounded_le_nhds a)
(fun a' (h : { n : α | n ≤ a' } ∈ 𝓝 a) ↦ show a ≤ a' from @mem_of_mem_nhds _ _ a _ h)
fun b (hba : a < b) ↦
show ∃ c, { n : α | n ≤ c } ∈ 𝓝 a ∧ c < b from
match dense_or_discrete a b with
| Or.inl ⟨c, hac, hcb⟩ => ⟨c, ge_mem_nhds hac, hcb⟩
| Or.inr ⟨_, h⟩ => ⟨a, (𝓝 a).sets_of_superset (gt_mem_nhds hba) h, hba⟩ | theorem | Topology | [
"Mathlib.Order.Filter.CountableInter",
"Mathlib.Order.LiminfLimsup",
"Mathlib.Topology.Order.Monotone"
] | Mathlib/Topology/Order/LiminfLimsup.lean | limsSup_nhds | null |
limsInf_nhds (a : α) : limsInf (𝓝 a) = a :=
limsSup_nhds (α := αᵒᵈ) a | theorem | Topology | [
"Mathlib.Order.Filter.CountableInter",
"Mathlib.Order.LiminfLimsup",
"Mathlib.Topology.Order.Monotone"
] | Mathlib/Topology/Order/LiminfLimsup.lean | limsInf_nhds | null |
limsInf_eq_of_le_nhds {f : Filter α} {a : α} [NeBot f] (h : f ≤ 𝓝 a) : f.limsInf = a :=
have hb_ge : IsBounded (· ≥ ·) f := (isBounded_ge_nhds a).mono h
have hb_le : IsBounded (· ≤ ·) f := (isBounded_le_nhds a).mono h
le_antisymm
(calc
f.limsInf ≤ f.limsSup := limsInf_le_limsSup hb_le hb_ge
_ ≤ (𝓝 a).limsSup := limsSup_le_limsSup_of_le h hb_ge.isCobounded_flip (isBounded_le_nhds a)
_ = a := limsSup_nhds a)
(calc
a = (𝓝 a).limsInf := (limsInf_nhds a).symm
_ ≤ f.limsInf := limsInf_le_limsInf_of_le h (isBounded_ge_nhds a) hb_le.isCobounded_flip) | theorem | Topology | [
"Mathlib.Order.Filter.CountableInter",
"Mathlib.Order.LiminfLimsup",
"Mathlib.Topology.Order.Monotone"
] | Mathlib/Topology/Order/LiminfLimsup.lean | limsInf_eq_of_le_nhds | If a filter is converging, its limsup coincides with its limit. |
limsSup_eq_of_le_nhds {f : Filter α} {a : α} [NeBot f] (h : f ≤ 𝓝 a) : f.limsSup = a :=
limsInf_eq_of_le_nhds (α := αᵒᵈ) h | theorem | Topology | [
"Mathlib.Order.Filter.CountableInter",
"Mathlib.Order.LiminfLimsup",
"Mathlib.Topology.Order.Monotone"
] | Mathlib/Topology/Order/LiminfLimsup.lean | limsSup_eq_of_le_nhds | If a filter is converging, its liminf coincides with its limit. |
Filter.Tendsto.limsup_eq {f : Filter β} {u : β → α} {a : α} [NeBot f]
(h : Tendsto u f (𝓝 a)) : limsup u f = a :=
limsSup_eq_of_le_nhds h | theorem | Topology | [
"Mathlib.Order.Filter.CountableInter",
"Mathlib.Order.LiminfLimsup",
"Mathlib.Topology.Order.Monotone"
] | Mathlib/Topology/Order/LiminfLimsup.lean | Filter.Tendsto.limsup_eq | If a function has a limit, then its limsup coincides with its limit. |
Filter.Tendsto.liminf_eq {f : Filter β} {u : β → α} {a : α} [NeBot f]
(h : Tendsto u f (𝓝 a)) : liminf u f = a :=
limsInf_eq_of_le_nhds h | theorem | Topology | [
"Mathlib.Order.Filter.CountableInter",
"Mathlib.Order.LiminfLimsup",
"Mathlib.Topology.Order.Monotone"
] | Mathlib/Topology/Order/LiminfLimsup.lean | Filter.Tendsto.liminf_eq | If a function has a limit, then its liminf coincides with its limit. |
tendsto_of_liminf_eq_limsup {f : Filter β} {u : β → α} {a : α} (hinf : liminf u f = a)
(hsup : limsup u f = a) (h : f.IsBoundedUnder (· ≤ ·) u := by isBoundedDefault)
(h' : f.IsBoundedUnder (· ≥ ·) u := by isBoundedDefault) : Tendsto u f (𝓝 a) :=
le_nhds_of_limsSup_eq_limsInf h h' hsup hinf | theorem | Topology | [
"Mathlib.Order.Filter.CountableInter",
"Mathlib.Order.LiminfLimsup",
"Mathlib.Topology.Order.Monotone"
] | Mathlib/Topology/Order/LiminfLimsup.lean | tendsto_of_liminf_eq_limsup | If the liminf and the limsup of a function coincide, then the limit of the function
exists and has the same value. |
tendsto_of_le_liminf_of_limsup_le {f : Filter β} {u : β → α} {a : α} (hinf : a ≤ liminf u f)
(hsup : limsup u f ≤ a) (h : f.IsBoundedUnder (· ≤ ·) u := by isBoundedDefault)
(h' : f.IsBoundedUnder (· ≥ ·) u := by isBoundedDefault) : Tendsto u f (𝓝 a) := by
rcases f.eq_or_neBot with rfl | _
· exact tendsto_bot
· exact tendsto_of_liminf_eq_limsup (le_antisymm (le_trans (liminf_le_limsup h h') hsup) hinf)
(le_antisymm hsup (le_trans hinf (liminf_le_limsup h h'))) h h' | theorem | Topology | [
"Mathlib.Order.Filter.CountableInter",
"Mathlib.Order.LiminfLimsup",
"Mathlib.Topology.Order.Monotone"
] | Mathlib/Topology/Order/LiminfLimsup.lean | tendsto_of_le_liminf_of_limsup_le | If a number `a` is less than or equal to the `liminf` of a function `f` at some filter
and is greater than or equal to the `limsup` of `f`, then `f` tends to `a` along this filter. |
tendsto_of_no_upcrossings [DenselyOrdered α] {f : Filter β} {u : β → α} {s : Set α}
(hs : Dense s) (H : ∀ a ∈ s, ∀ b ∈ s, a < b → ¬((∃ᶠ n in f, u n < a) ∧ ∃ᶠ n in f, b < u n))
(h : f.IsBoundedUnder (· ≤ ·) u := by isBoundedDefault)
(h' : f.IsBoundedUnder (· ≥ ·) u := by isBoundedDefault) :
∃ c : α, Tendsto u f (𝓝 c) := by
rcases f.eq_or_neBot with rfl | hbot
· exact ⟨sInf ∅, tendsto_bot⟩
refine ⟨limsup u f, ?_⟩
apply tendsto_of_le_liminf_of_limsup_le _ le_rfl h h'
by_contra! hlt
obtain ⟨a, ⟨⟨la, au⟩, as⟩⟩ : ∃ a, (f.liminf u < a ∧ a < f.limsup u) ∧ a ∈ s :=
dense_iff_inter_open.1 hs (Set.Ioo (f.liminf u) (f.limsup u)) isOpen_Ioo
(Set.nonempty_Ioo.2 hlt)
obtain ⟨b, ⟨⟨ab, bu⟩, bs⟩⟩ : ∃ b, (a < b ∧ b < f.limsup u) ∧ b ∈ s :=
dense_iff_inter_open.1 hs (Set.Ioo a (f.limsup u)) isOpen_Ioo (Set.nonempty_Ioo.2 au)
have A : ∃ᶠ n in f, u n < a := frequently_lt_of_liminf_lt (IsBounded.isCobounded_ge h) la
have B : ∃ᶠ n in f, b < u n := frequently_lt_of_lt_limsup (IsBounded.isCobounded_le h') bu
exact H a as b bs ab ⟨A, B⟩
variable [FirstCountableTopology α] {f : Filter β} [CountableInterFilter f] {u : β → α} | theorem | Topology | [
"Mathlib.Order.Filter.CountableInter",
"Mathlib.Order.LiminfLimsup",
"Mathlib.Topology.Order.Monotone"
] | Mathlib/Topology/Order/LiminfLimsup.lean | tendsto_of_no_upcrossings | Assume that, for any `a < b`, a sequence cannot be infinitely many times below `a` and
above `b`. If it is also ultimately bounded above and below, then it has to converge. This even
works if `a` and `b` are restricted to a dense subset. |
eventually_le_limsup (hf : IsBoundedUnder (· ≤ ·) f u := by isBoundedDefault) :
∀ᶠ b in f, u b ≤ f.limsup u := by
obtain ha | ha := isTop_or_exists_gt (f.limsup u)
· exact Eventually.of_forall fun _ => ha _
by_cases H : IsGLB (Set.Ioi (f.limsup u)) (f.limsup u)
· obtain ⟨u, -, -, hua, hu⟩ := H.exists_seq_antitone_tendsto ha
have := fun n => eventually_lt_of_limsup_lt (hu n) hf
exact
(eventually_countable_forall.2 this).mono fun b hb =>
ge_of_tendsto hua <| Eventually.of_forall fun n => (hb _).le
· obtain ⟨x, hx, xa⟩ : ∃ x, (∀ ⦃b⦄, f.limsup u < b → x ≤ b) ∧ f.limsup u < x := by
simp only [IsGLB, IsGreatest, lowerBounds, upperBounds, Set.mem_Ioi, Set.mem_setOf_eq,
not_and, not_forall, not_le, exists_prop] at H
exact H fun x => le_of_lt
filter_upwards [eventually_lt_of_limsup_lt xa hf] with y hy
contrapose! hy
exact hx hy | theorem | Topology | [
"Mathlib.Order.Filter.CountableInter",
"Mathlib.Order.LiminfLimsup",
"Mathlib.Topology.Order.Monotone"
] | Mathlib/Topology/Order/LiminfLimsup.lean | eventually_le_limsup | null |
eventually_liminf_le (hf : IsBoundedUnder (· ≥ ·) f u := by isBoundedDefault) :
∀ᶠ b in f, f.liminf u ≤ u b :=
eventually_le_limsup (α := αᵒᵈ) hf | theorem | Topology | [
"Mathlib.Order.Filter.CountableInter",
"Mathlib.Order.LiminfLimsup",
"Mathlib.Topology.Order.Monotone"
] | Mathlib/Topology/Order/LiminfLimsup.lean | eventually_liminf_le | null |
@[simp]
limsup_eq_bot : f.limsup u = ⊥ ↔ u =ᶠ[f] ⊥ :=
⟨fun h =>
(EventuallyLE.trans eventually_le_limsup <| Eventually.of_forall fun _ => h.le).mono fun _ hx =>
le_antisymm hx bot_le,
fun h => by
rw [limsup_congr h]
exact limsup_const_bot⟩
@[simp] | theorem | Topology | [
"Mathlib.Order.Filter.CountableInter",
"Mathlib.Order.LiminfLimsup",
"Mathlib.Topology.Order.Monotone"
] | Mathlib/Topology/Order/LiminfLimsup.lean | limsup_eq_bot | null |
liminf_eq_top : f.liminf u = ⊤ ↔ u =ᶠ[f] ⊤ :=
limsup_eq_bot (α := αᵒᵈ) | theorem | Topology | [
"Mathlib.Order.Filter.CountableInter",
"Mathlib.Order.LiminfLimsup",
"Mathlib.Topology.Order.Monotone"
] | Mathlib/Topology/Order/LiminfLimsup.lean | liminf_eq_top | null |
Antitone.map_limsSup_of_continuousAt {F : Filter R} [NeBot F] {f : R → S}
(f_decr : Antitone f) (f_cont : ContinuousAt f F.limsSup)
(bdd_above : F.IsBounded (· ≤ ·) := by isBoundedDefault)
(cobdd : F.IsCobounded (· ≤ ·) := by isBoundedDefault) :
f F.limsSup = F.liminf f := by
apply le_antisymm
· rw [limsSup, f_decr.map_csInf_of_continuousAt f_cont bdd_above cobdd]
apply le_of_forall_lt
intro c hc
simp only [liminf, limsInf, eventually_map] at hc ⊢
obtain ⟨d, hd, h'd⟩ :=
exists_lt_of_lt_csSup (bdd_above.recOn fun x hx ↦ ⟨f x, Set.mem_image_of_mem f hx⟩) hc
apply lt_csSup_of_lt ?_ ?_ h'd
· simpa only [BddAbove, upperBounds]
using Antitone.isCoboundedUnder_ge_of_isCobounded f_decr cobdd
· rcases hd with ⟨e, ⟨he, fe_eq_d⟩⟩
filter_upwards [he] with x hx using (fe_eq_d.symm ▸ f_decr hx)
· by_cases h' : ∃ c, c < F.limsSup ∧ Set.Ioo c F.limsSup = ∅
· rcases h' with ⟨c, c_lt, hc⟩
have B : ∃ᶠ n in F, F.limsSup ≤ n := by
apply (frequently_lt_of_lt_limsSup cobdd c_lt).mono
intro x hx
by_contra!
have : (Set.Ioo c F.limsSup).Nonempty := ⟨x, ⟨hx, this⟩⟩
simp only [hc, Set.not_nonempty_empty] at this
apply liminf_le_of_frequently_le _ (bdd_above.isBoundedUnder f_decr)
exact B.mono fun x hx ↦ f_decr hx
push_neg at h'
by_contra! H
have not_bot : ¬ IsBot F.limsSup := fun maybe_bot ↦
lt_irrefl (F.liminf f) <| lt_of_le_of_lt
(liminf_le_of_frequently_le (Frequently.of_forall (fun r ↦ f_decr (maybe_bot r)))
(bdd_above.isBoundedUnder f_decr)) H
obtain ⟨l, l_lt, h'l⟩ :
∃ l < F.limsSup, Set.Ioc l F.limsSup ⊆ { x : R | f x < F.liminf f } := by
apply exists_Ioc_subset_of_mem_nhds ((tendsto_order.1 f_cont.tendsto).2 _ H)
simpa [IsBot] using not_bot
obtain ⟨m, l_m, m_lt⟩ : (Set.Ioo l F.limsSup).Nonempty := by
contrapose! h'
exact ⟨l, l_lt, h'⟩
have B : F.liminf f ≤ f m := by
apply liminf_le_of_frequently_le _ _
· apply (frequently_lt_of_lt_limsSup cobdd m_lt).mono
exact fun x hx ↦ f_decr hx.le
· exact IsBounded.isBoundedUnder f_decr bdd_above
have I : f m < F.liminf f := h'l ⟨l_m, m_lt.le⟩
exact lt_irrefl _ (B.trans_lt I) | theorem | Topology | [
"Mathlib.Order.Filter.CountableInter",
"Mathlib.Order.LiminfLimsup",
"Mathlib.Topology.Order.Monotone"
] | Mathlib/Topology/Order/LiminfLimsup.lean | Antitone.map_limsSup_of_continuousAt | An antitone function between (conditionally) complete linear ordered spaces sends a
`Filter.limsSup` to the `Filter.liminf` of the image if the function is continuous at the `limsSup`
(and the filter is bounded from above and frequently bounded from below). |
Antitone.map_limsup_of_continuousAt {f : R → S} (f_decr : Antitone f) (a : ι → R)
(f_cont : ContinuousAt f (F.limsup a))
(bdd_above : F.IsBoundedUnder (· ≤ ·) a := by isBoundedDefault)
(cobdd : F.IsCoboundedUnder (· ≤ ·) a := by isBoundedDefault) :
f (F.limsup a) = F.liminf (f ∘ a) :=
f_decr.map_limsSup_of_continuousAt f_cont bdd_above cobdd | theorem | Topology | [
"Mathlib.Order.Filter.CountableInter",
"Mathlib.Order.LiminfLimsup",
"Mathlib.Topology.Order.Monotone"
] | Mathlib/Topology/Order/LiminfLimsup.lean | Antitone.map_limsup_of_continuousAt | A continuous antitone function between (conditionally) complete linear ordered spaces sends a
`Filter.limsup` to the `Filter.liminf` of the images (if the filter is bounded from above and
frequently bounded from below). |
Antitone.map_limsInf_of_continuousAt {F : Filter R} [NeBot F] {f : R → S}
(f_decr : Antitone f) (f_cont : ContinuousAt f F.limsInf)
(cobdd : F.IsCobounded (· ≥ ·) := by isBoundedDefault)
(bdd_below : F.IsBounded (· ≥ ·) := by isBoundedDefault) : f F.limsInf = F.limsup f :=
Antitone.map_limsSup_of_continuousAt (R := Rᵒᵈ) (S := Sᵒᵈ) f_decr.dual f_cont bdd_below cobdd | theorem | Topology | [
"Mathlib.Order.Filter.CountableInter",
"Mathlib.Order.LiminfLimsup",
"Mathlib.Topology.Order.Monotone"
] | Mathlib/Topology/Order/LiminfLimsup.lean | Antitone.map_limsInf_of_continuousAt | An antitone function between (conditionally) complete linear ordered spaces sends a
`Filter.limsInf` to the `Filter.limsup` of the image if the function is continuous at the `limsInf`
(and the filter is bounded from below and frequently bounded from above). |
Antitone.map_liminf_of_continuousAt {f : R → S} (f_decr : Antitone f) (a : ι → R)
(f_cont : ContinuousAt f (F.liminf a))
(cobdd : F.IsCoboundedUnder (· ≥ ·) a := by isBoundedDefault)
(bdd_below : F.IsBoundedUnder (· ≥ ·) a := by isBoundedDefault) :
f (F.liminf a) = F.limsup (f ∘ a) :=
f_decr.map_limsInf_of_continuousAt f_cont cobdd bdd_below | theorem | Topology | [
"Mathlib.Order.Filter.CountableInter",
"Mathlib.Order.LiminfLimsup",
"Mathlib.Topology.Order.Monotone"
] | Mathlib/Topology/Order/LiminfLimsup.lean | Antitone.map_liminf_of_continuousAt | A continuous antitone function between (conditionally) complete linear ordered spaces sends a
`Filter.liminf` to the `Filter.limsup` of the images (if the filter is bounded from below and
frequently bounded from above). |
Monotone.map_limsSup_of_continuousAt {F : Filter R} [NeBot F] {f : R → S}
(f_incr : Monotone f) (f_cont : ContinuousAt f F.limsSup)
(bdd_above : F.IsBounded (· ≤ ·) := by isBoundedDefault)
(cobdd : F.IsCobounded (· ≤ ·) := by isBoundedDefault) : f F.limsSup = F.limsup f :=
Antitone.map_limsSup_of_continuousAt (S := Sᵒᵈ) f_incr f_cont bdd_above cobdd | theorem | Topology | [
"Mathlib.Order.Filter.CountableInter",
"Mathlib.Order.LiminfLimsup",
"Mathlib.Topology.Order.Monotone"
] | Mathlib/Topology/Order/LiminfLimsup.lean | Monotone.map_limsSup_of_continuousAt | A monotone function between (conditionally) complete linear ordered spaces sends a
`Filter.limsSup` to the `Filter.limsup` of the image if the function is continuous at the `limsSup`
(and the filter is bounded from above and frequently bounded from below). |
Monotone.map_limsup_of_continuousAt {f : R → S} (f_incr : Monotone f) (a : ι → R)
(f_cont : ContinuousAt f (F.limsup a))
(bdd_above : F.IsBoundedUnder (· ≤ ·) a := by isBoundedDefault)
(cobdd : F.IsCoboundedUnder (· ≤ ·) a := by isBoundedDefault) :
f (F.limsup a) = F.limsup (f ∘ a) :=
f_incr.map_limsSup_of_continuousAt f_cont bdd_above cobdd | theorem | Topology | [
"Mathlib.Order.Filter.CountableInter",
"Mathlib.Order.LiminfLimsup",
"Mathlib.Topology.Order.Monotone"
] | Mathlib/Topology/Order/LiminfLimsup.lean | Monotone.map_limsup_of_continuousAt | A continuous monotone function between (conditionally) complete linear ordered spaces sends a
`Filter.limsup` to the `Filter.limsup` of the images (if the filter is bounded from above and
frequently bounded from below). |
Monotone.map_limsInf_of_continuousAt {F : Filter R} [NeBot F] {f : R → S}
(f_incr : Monotone f) (f_cont : ContinuousAt f F.limsInf)
(cobdd : F.IsCobounded (· ≥ ·) := by isBoundedDefault)
(bdd_below : F.IsBounded (· ≥ ·) := by isBoundedDefault) : f F.limsInf = F.liminf f :=
Antitone.map_limsSup_of_continuousAt (R := Rᵒᵈ) f_incr.dual f_cont bdd_below cobdd | theorem | Topology | [
"Mathlib.Order.Filter.CountableInter",
"Mathlib.Order.LiminfLimsup",
"Mathlib.Topology.Order.Monotone"
] | Mathlib/Topology/Order/LiminfLimsup.lean | Monotone.map_limsInf_of_continuousAt | A monotone function between (conditionally) complete linear ordered spaces sends a
`Filter.limsInf` to the `Filter.liminf` of the image if the function is continuous at the `limsInf`
(and the filter is bounded from below and frequently bounded from above). |
Monotone.map_liminf_of_continuousAt {f : R → S} (f_incr : Monotone f) (a : ι → R)
(f_cont : ContinuousAt f (F.liminf a))
(cobdd : F.IsCoboundedUnder (· ≥ ·) a := by isBoundedDefault)
(bdd_below : F.IsBoundedUnder (· ≥ ·) a := by isBoundedDefault) :
f (F.liminf a) = F.liminf (f ∘ a) :=
f_incr.map_limsInf_of_continuousAt f_cont cobdd bdd_below | theorem | Topology | [
"Mathlib.Order.Filter.CountableInter",
"Mathlib.Order.LiminfLimsup",
"Mathlib.Topology.Order.Monotone"
] | Mathlib/Topology/Order/LiminfLimsup.lean | Monotone.map_liminf_of_continuousAt | A continuous monotone function between (conditionally) complete linear ordered spaces sends a
`Filter.liminf` to the `Filter.liminf` of the images (if the filter is bounded from below and
frequently bounded from above). |
IsLocalMinOn :=
IsMinFilter f (𝓝[s] a) a | def | Topology | [
"Mathlib.Algebra.Group.Defs",
"Mathlib.Order.Filter.Extr",
"Mathlib.Topology.ContinuousOn"
] | Mathlib/Topology/Order/LocalExtr.lean | IsLocalMinOn | `IsLocalMinOn f s a` means that `f a ≤ f x` for all `x ∈ s` in some neighborhood of `a`. |
IsLocalMaxOn :=
IsMaxFilter f (𝓝[s] a) a | def | Topology | [
"Mathlib.Algebra.Group.Defs",
"Mathlib.Order.Filter.Extr",
"Mathlib.Topology.ContinuousOn"
] | Mathlib/Topology/Order/LocalExtr.lean | IsLocalMaxOn | `IsLocalMaxOn f s a` means that `f x ≤ f a` for all `x ∈ s` in some neighborhood of `a`. |
IsLocalExtrOn :=
IsExtrFilter f (𝓝[s] a) a | def | Topology | [
"Mathlib.Algebra.Group.Defs",
"Mathlib.Order.Filter.Extr",
"Mathlib.Topology.ContinuousOn"
] | Mathlib/Topology/Order/LocalExtr.lean | IsLocalExtrOn | `IsLocalExtrOn f s a` means `IsLocalMinOn f s a ∨ IsLocalMaxOn f s a`. |
IsLocalMin :=
IsMinFilter f (𝓝 a) a | def | Topology | [
"Mathlib.Algebra.Group.Defs",
"Mathlib.Order.Filter.Extr",
"Mathlib.Topology.ContinuousOn"
] | Mathlib/Topology/Order/LocalExtr.lean | IsLocalMin | `IsLocalMin f a` means that `f a ≤ f x` for all `x` in some neighborhood of `a`. |
IsLocalMax :=
IsMaxFilter f (𝓝 a) a | def | Topology | [
"Mathlib.Algebra.Group.Defs",
"Mathlib.Order.Filter.Extr",
"Mathlib.Topology.ContinuousOn"
] | Mathlib/Topology/Order/LocalExtr.lean | IsLocalMax | `IsLocalMax f a` means that `f x ≤ f a` for all `x ∈ s` in some neighborhood of `a`. |
IsLocalExtr :=
IsExtrFilter f (𝓝 a) a
variable {f s a} | def | Topology | [
"Mathlib.Algebra.Group.Defs",
"Mathlib.Order.Filter.Extr",
"Mathlib.Topology.ContinuousOn"
] | Mathlib/Topology/Order/LocalExtr.lean | IsLocalExtr | `IsLocalExtr f s a` means `IsLocalMin f s a ∨ IsLocalMax f s a`. |
IsLocalExtrOn.elim {p : Prop} :
IsLocalExtrOn f s a → (IsLocalMinOn f s a → p) → (IsLocalMaxOn f s a → p) → p :=
Or.elim | theorem | Topology | [
"Mathlib.Algebra.Group.Defs",
"Mathlib.Order.Filter.Extr",
"Mathlib.Topology.ContinuousOn"
] | Mathlib/Topology/Order/LocalExtr.lean | IsLocalExtrOn.elim | null |
IsLocalExtr.elim {p : Prop} :
IsLocalExtr f a → (IsLocalMin f a → p) → (IsLocalMax f a → p) → p :=
Or.elim
/-! ### Restriction to (sub)sets -/ | theorem | Topology | [
"Mathlib.Algebra.Group.Defs",
"Mathlib.Order.Filter.Extr",
"Mathlib.Topology.ContinuousOn"
] | Mathlib/Topology/Order/LocalExtr.lean | IsLocalExtr.elim | null |
IsLocalMin.on (h : IsLocalMin f a) (s) : IsLocalMinOn f s a :=
h.filter_inf _ | theorem | Topology | [
"Mathlib.Algebra.Group.Defs",
"Mathlib.Order.Filter.Extr",
"Mathlib.Topology.ContinuousOn"
] | Mathlib/Topology/Order/LocalExtr.lean | IsLocalMin.on | null |
IsLocalMax.on (h : IsLocalMax f a) (s) : IsLocalMaxOn f s a :=
h.filter_inf _ | theorem | Topology | [
"Mathlib.Algebra.Group.Defs",
"Mathlib.Order.Filter.Extr",
"Mathlib.Topology.ContinuousOn"
] | Mathlib/Topology/Order/LocalExtr.lean | IsLocalMax.on | null |
IsLocalExtr.on (h : IsLocalExtr f a) (s) : IsLocalExtrOn f s a :=
h.filter_inf _ | theorem | Topology | [
"Mathlib.Algebra.Group.Defs",
"Mathlib.Order.Filter.Extr",
"Mathlib.Topology.ContinuousOn"
] | Mathlib/Topology/Order/LocalExtr.lean | IsLocalExtr.on | null |
IsLocalMinOn.on_subset {t : Set α} (hf : IsLocalMinOn f t a) (h : s ⊆ t) :
IsLocalMinOn f s a :=
hf.filter_mono <| nhdsWithin_mono a h | theorem | Topology | [
"Mathlib.Algebra.Group.Defs",
"Mathlib.Order.Filter.Extr",
"Mathlib.Topology.ContinuousOn"
] | Mathlib/Topology/Order/LocalExtr.lean | IsLocalMinOn.on_subset | null |
IsLocalMaxOn.on_subset {t : Set α} (hf : IsLocalMaxOn f t a) (h : s ⊆ t) :
IsLocalMaxOn f s a :=
hf.filter_mono <| nhdsWithin_mono a h | theorem | Topology | [
"Mathlib.Algebra.Group.Defs",
"Mathlib.Order.Filter.Extr",
"Mathlib.Topology.ContinuousOn"
] | Mathlib/Topology/Order/LocalExtr.lean | IsLocalMaxOn.on_subset | null |
IsLocalExtrOn.on_subset {t : Set α} (hf : IsLocalExtrOn f t a) (h : s ⊆ t) :
IsLocalExtrOn f s a :=
hf.filter_mono <| nhdsWithin_mono a h | theorem | Topology | [
"Mathlib.Algebra.Group.Defs",
"Mathlib.Order.Filter.Extr",
"Mathlib.Topology.ContinuousOn"
] | Mathlib/Topology/Order/LocalExtr.lean | IsLocalExtrOn.on_subset | null |
IsLocalMinOn.inter (hf : IsLocalMinOn f s a) (t) : IsLocalMinOn f (s ∩ t) a :=
hf.on_subset inter_subset_left | theorem | Topology | [
"Mathlib.Algebra.Group.Defs",
"Mathlib.Order.Filter.Extr",
"Mathlib.Topology.ContinuousOn"
] | Mathlib/Topology/Order/LocalExtr.lean | IsLocalMinOn.inter | null |
IsLocalMaxOn.inter (hf : IsLocalMaxOn f s a) (t) : IsLocalMaxOn f (s ∩ t) a :=
hf.on_subset inter_subset_left | theorem | Topology | [
"Mathlib.Algebra.Group.Defs",
"Mathlib.Order.Filter.Extr",
"Mathlib.Topology.ContinuousOn"
] | Mathlib/Topology/Order/LocalExtr.lean | IsLocalMaxOn.inter | null |
IsLocalExtrOn.inter (hf : IsLocalExtrOn f s a) (t) : IsLocalExtrOn f (s ∩ t) a :=
hf.on_subset inter_subset_left | theorem | Topology | [
"Mathlib.Algebra.Group.Defs",
"Mathlib.Order.Filter.Extr",
"Mathlib.Topology.ContinuousOn"
] | Mathlib/Topology/Order/LocalExtr.lean | IsLocalExtrOn.inter | null |
IsMinOn.localize (hf : IsMinOn f s a) : IsLocalMinOn f s a :=
hf.filter_mono <| inf_le_right | theorem | Topology | [
"Mathlib.Algebra.Group.Defs",
"Mathlib.Order.Filter.Extr",
"Mathlib.Topology.ContinuousOn"
] | Mathlib/Topology/Order/LocalExtr.lean | IsMinOn.localize | null |
IsMaxOn.localize (hf : IsMaxOn f s a) : IsLocalMaxOn f s a :=
hf.filter_mono <| inf_le_right | theorem | Topology | [
"Mathlib.Algebra.Group.Defs",
"Mathlib.Order.Filter.Extr",
"Mathlib.Topology.ContinuousOn"
] | Mathlib/Topology/Order/LocalExtr.lean | IsMaxOn.localize | null |
IsExtrOn.localize (hf : IsExtrOn f s a) : IsLocalExtrOn f s a :=
hf.filter_mono <| inf_le_right | theorem | Topology | [
"Mathlib.Algebra.Group.Defs",
"Mathlib.Order.Filter.Extr",
"Mathlib.Topology.ContinuousOn"
] | Mathlib/Topology/Order/LocalExtr.lean | IsExtrOn.localize | null |
IsLocalMinOn.isLocalMin (hf : IsLocalMinOn f s a) (hs : s ∈ 𝓝 a) : IsLocalMin f a :=
have : 𝓝 a ≤ 𝓟 s := le_principal_iff.2 hs
hf.filter_mono <| le_inf le_rfl this | theorem | Topology | [
"Mathlib.Algebra.Group.Defs",
"Mathlib.Order.Filter.Extr",
"Mathlib.Topology.ContinuousOn"
] | Mathlib/Topology/Order/LocalExtr.lean | IsLocalMinOn.isLocalMin | null |
IsLocalMaxOn.isLocalMax (hf : IsLocalMaxOn f s a) (hs : s ∈ 𝓝 a) : IsLocalMax f a :=
have : 𝓝 a ≤ 𝓟 s := le_principal_iff.2 hs
hf.filter_mono <| le_inf le_rfl this | theorem | Topology | [
"Mathlib.Algebra.Group.Defs",
"Mathlib.Order.Filter.Extr",
"Mathlib.Topology.ContinuousOn"
] | Mathlib/Topology/Order/LocalExtr.lean | IsLocalMaxOn.isLocalMax | null |
IsLocalExtrOn.isLocalExtr (hf : IsLocalExtrOn f s a) (hs : s ∈ 𝓝 a) : IsLocalExtr f a :=
hf.elim (fun hf => (hf.isLocalMin hs).isExtr) fun hf => (hf.isLocalMax hs).isExtr | theorem | Topology | [
"Mathlib.Algebra.Group.Defs",
"Mathlib.Order.Filter.Extr",
"Mathlib.Topology.ContinuousOn"
] | Mathlib/Topology/Order/LocalExtr.lean | IsLocalExtrOn.isLocalExtr | null |
isLocalMinOn_univ_iff : IsLocalMinOn f univ a ↔ IsLocalMin f a := by
simp only [IsLocalMinOn, IsLocalMin, nhdsWithin_univ] | lemma | Topology | [
"Mathlib.Algebra.Group.Defs",
"Mathlib.Order.Filter.Extr",
"Mathlib.Topology.ContinuousOn"
] | Mathlib/Topology/Order/LocalExtr.lean | isLocalMinOn_univ_iff | null |
isLocalMaxOn_univ_iff : IsLocalMaxOn f univ a ↔ IsLocalMax f a := by
simp only [IsLocalMaxOn, IsLocalMax, nhdsWithin_univ] | lemma | Topology | [
"Mathlib.Algebra.Group.Defs",
"Mathlib.Order.Filter.Extr",
"Mathlib.Topology.ContinuousOn"
] | Mathlib/Topology/Order/LocalExtr.lean | isLocalMaxOn_univ_iff | null |
isLocalExtrOn_univ_iff : IsLocalExtrOn f univ a ↔ IsLocalExtr f a :=
isLocalMinOn_univ_iff.or isLocalMaxOn_univ_iff | lemma | Topology | [
"Mathlib.Algebra.Group.Defs",
"Mathlib.Order.Filter.Extr",
"Mathlib.Topology.ContinuousOn"
] | Mathlib/Topology/Order/LocalExtr.lean | isLocalExtrOn_univ_iff | null |
IsMinOn.isLocalMin (hf : IsMinOn f s a) (hs : s ∈ 𝓝 a) : IsLocalMin f a :=
hf.localize.isLocalMin hs | theorem | Topology | [
"Mathlib.Algebra.Group.Defs",
"Mathlib.Order.Filter.Extr",
"Mathlib.Topology.ContinuousOn"
] | Mathlib/Topology/Order/LocalExtr.lean | IsMinOn.isLocalMin | null |
IsMaxOn.isLocalMax (hf : IsMaxOn f s a) (hs : s ∈ 𝓝 a) : IsLocalMax f a :=
hf.localize.isLocalMax hs | theorem | Topology | [
"Mathlib.Algebra.Group.Defs",
"Mathlib.Order.Filter.Extr",
"Mathlib.Topology.ContinuousOn"
] | Mathlib/Topology/Order/LocalExtr.lean | IsMaxOn.isLocalMax | null |
IsExtrOn.isLocalExtr (hf : IsExtrOn f s a) (hs : s ∈ 𝓝 a) : IsLocalExtr f a :=
hf.localize.isLocalExtr hs | theorem | Topology | [
"Mathlib.Algebra.Group.Defs",
"Mathlib.Order.Filter.Extr",
"Mathlib.Topology.ContinuousOn"
] | Mathlib/Topology/Order/LocalExtr.lean | IsExtrOn.isLocalExtr | null |
IsLocalMinOn.not_nhds_le_map [TopologicalSpace β] (hf : IsLocalMinOn f s a)
[NeBot (𝓝[<] f a)] : ¬𝓝 (f a) ≤ map f (𝓝[s] a) := fun hle =>
have : ∀ᶠ y in 𝓝[<] f a, f a ≤ y := (eventually_map.2 hf).filter_mono (inf_le_left.trans hle)
let ⟨_y, hy⟩ := (this.and self_mem_nhdsWithin).exists
hy.1.not_gt hy.2 | theorem | Topology | [
"Mathlib.Algebra.Group.Defs",
"Mathlib.Order.Filter.Extr",
"Mathlib.Topology.ContinuousOn"
] | Mathlib/Topology/Order/LocalExtr.lean | IsLocalMinOn.not_nhds_le_map | null |
IsLocalMaxOn.not_nhds_le_map [TopologicalSpace β] (hf : IsLocalMaxOn f s a)
[NeBot (𝓝[>] f a)] : ¬𝓝 (f a) ≤ map f (𝓝[s] a) :=
@IsLocalMinOn.not_nhds_le_map α βᵒᵈ _ _ _ _ _ ‹_› hf ‹_› | theorem | Topology | [
"Mathlib.Algebra.Group.Defs",
"Mathlib.Order.Filter.Extr",
"Mathlib.Topology.ContinuousOn"
] | Mathlib/Topology/Order/LocalExtr.lean | IsLocalMaxOn.not_nhds_le_map | null |
IsLocalExtrOn.not_nhds_le_map [TopologicalSpace β] (hf : IsLocalExtrOn f s a)
[NeBot (𝓝[<] f a)] [NeBot (𝓝[>] f a)] : ¬𝓝 (f a) ≤ map f (𝓝[s] a) :=
hf.elim (fun h => h.not_nhds_le_map) fun h => h.not_nhds_le_map
/-! ### Constant -/ | theorem | Topology | [
"Mathlib.Algebra.Group.Defs",
"Mathlib.Order.Filter.Extr",
"Mathlib.Topology.ContinuousOn"
] | Mathlib/Topology/Order/LocalExtr.lean | IsLocalExtrOn.not_nhds_le_map | null |
isLocalMinOn_const {b : β} : IsLocalMinOn (fun _ => b) s a :=
isMinFilter_const | theorem | Topology | [
"Mathlib.Algebra.Group.Defs",
"Mathlib.Order.Filter.Extr",
"Mathlib.Topology.ContinuousOn"
] | Mathlib/Topology/Order/LocalExtr.lean | isLocalMinOn_const | null |
isLocalMaxOn_const {b : β} : IsLocalMaxOn (fun _ => b) s a :=
isMaxFilter_const | theorem | Topology | [
"Mathlib.Algebra.Group.Defs",
"Mathlib.Order.Filter.Extr",
"Mathlib.Topology.ContinuousOn"
] | Mathlib/Topology/Order/LocalExtr.lean | isLocalMaxOn_const | null |
isLocalExtrOn_const {b : β} : IsLocalExtrOn (fun _ => b) s a :=
isExtrFilter_const | theorem | Topology | [
"Mathlib.Algebra.Group.Defs",
"Mathlib.Order.Filter.Extr",
"Mathlib.Topology.ContinuousOn"
] | Mathlib/Topology/Order/LocalExtr.lean | isLocalExtrOn_const | null |
isLocalMin_const {b : β} : IsLocalMin (fun _ => b) a :=
isMinFilter_const | theorem | Topology | [
"Mathlib.Algebra.Group.Defs",
"Mathlib.Order.Filter.Extr",
"Mathlib.Topology.ContinuousOn"
] | Mathlib/Topology/Order/LocalExtr.lean | isLocalMin_const | null |
isLocalMax_const {b : β} : IsLocalMax (fun _ => b) a :=
isMaxFilter_const | theorem | Topology | [
"Mathlib.Algebra.Group.Defs",
"Mathlib.Order.Filter.Extr",
"Mathlib.Topology.ContinuousOn"
] | Mathlib/Topology/Order/LocalExtr.lean | isLocalMax_const | null |
isLocalExtr_const {b : β} : IsLocalExtr (fun _ => b) a :=
isExtrFilter_const
/-! ### Composition with (anti)monotone functions -/
nonrec theorem IsLocalMin.comp_mono (hf : IsLocalMin f a) {g : β → γ} (hg : Monotone g) :
IsLocalMin (g ∘ f) a :=
hf.comp_mono hg
nonrec theorem IsLocalMax.comp_mono (hf : IsLocalMax f a) {g : β → γ} (hg : Monotone g) :
IsLocalMax (g ∘ f) a :=
hf.comp_mono hg
nonrec theorem IsLocalExtr.comp_mono (hf : IsLocalExtr f a) {g : β → γ} (hg : Monotone g) :
IsLocalExtr (g ∘ f) a :=
hf.comp_mono hg
nonrec theorem IsLocalMin.comp_antitone (hf : IsLocalMin f a) {g : β → γ} (hg : Antitone g) :
IsLocalMax (g ∘ f) a :=
hf.comp_antitone hg
nonrec theorem IsLocalMax.comp_antitone (hf : IsLocalMax f a) {g : β → γ} (hg : Antitone g) :
IsLocalMin (g ∘ f) a :=
hf.comp_antitone hg
nonrec theorem IsLocalExtr.comp_antitone (hf : IsLocalExtr f a) {g : β → γ} (hg : Antitone g) :
IsLocalExtr (g ∘ f) a :=
hf.comp_antitone hg
nonrec theorem IsLocalMinOn.comp_mono (hf : IsLocalMinOn f s a) {g : β → γ} (hg : Monotone g) :
IsLocalMinOn (g ∘ f) s a :=
hf.comp_mono hg
nonrec theorem IsLocalMaxOn.comp_mono (hf : IsLocalMaxOn f s a) {g : β → γ} (hg : Monotone g) :
IsLocalMaxOn (g ∘ f) s a :=
hf.comp_mono hg
nonrec theorem IsLocalExtrOn.comp_mono (hf : IsLocalExtrOn f s a) {g : β → γ} (hg : Monotone g) :
IsLocalExtrOn (g ∘ f) s a :=
hf.comp_mono hg
nonrec theorem IsLocalMinOn.comp_antitone (hf : IsLocalMinOn f s a) {g : β → γ} (hg : Antitone g) :
IsLocalMaxOn (g ∘ f) s a :=
hf.comp_antitone hg
nonrec theorem IsLocalMaxOn.comp_antitone (hf : IsLocalMaxOn f s a) {g : β → γ} (hg : Antitone g) :
IsLocalMinOn (g ∘ f) s a :=
hf.comp_antitone hg
nonrec theorem IsLocalExtrOn.comp_antitone (hf : IsLocalExtrOn f s a) {g : β → γ}
(hg : Antitone g) : IsLocalExtrOn (g ∘ f) s a :=
hf.comp_antitone hg
open scoped Relator
nonrec theorem IsLocalMin.bicomp_mono [Preorder δ] {op : β → γ → δ}
(hop : ((· ≤ ·) ⇒ (· ≤ ·) ⇒ (· ≤ ·)) op op) (hf : IsLocalMin f a) {g : α → γ}
(hg : IsLocalMin g a) : IsLocalMin (fun x => op (f x) (g x)) a :=
hf.bicomp_mono hop hg
nonrec theorem IsLocalMax.bicomp_mono [Preorder δ] {op : β → γ → δ}
(hop : ((· ≤ ·) ⇒ (· ≤ ·) ⇒ (· ≤ ·)) op op) (hf : IsLocalMax f a) {g : α → γ}
(hg : IsLocalMax g a) : IsLocalMax (fun x => op (f x) (g x)) a :=
hf.bicomp_mono hop hg
nonrec theorem IsLocalMinOn.bicomp_mono [Preorder δ] {op : β → γ → δ}
(hop : ((· ≤ ·) ⇒ (· ≤ ·) ⇒ (· ≤ ·)) op op) (hf : IsLocalMinOn f s a) {g : α → γ}
(hg : IsLocalMinOn g s a) : IsLocalMinOn (fun x => op (f x) (g x)) s a :=
... | theorem | Topology | [
"Mathlib.Algebra.Group.Defs",
"Mathlib.Order.Filter.Extr",
"Mathlib.Topology.ContinuousOn"
] | Mathlib/Topology/Order/LocalExtr.lean | isLocalExtr_const | null |
IsLocalMin.comp_continuous [TopologicalSpace δ] {g : δ → α} {b : δ}
(hf : IsLocalMin f (g b)) (hg : ContinuousAt g b) : IsLocalMin (f ∘ g) b :=
hg hf | theorem | Topology | [
"Mathlib.Algebra.Group.Defs",
"Mathlib.Order.Filter.Extr",
"Mathlib.Topology.ContinuousOn"
] | Mathlib/Topology/Order/LocalExtr.lean | IsLocalMin.comp_continuous | null |
IsLocalMax.comp_continuous [TopologicalSpace δ] {g : δ → α} {b : δ}
(hf : IsLocalMax f (g b)) (hg : ContinuousAt g b) : IsLocalMax (f ∘ g) b :=
hg hf | theorem | Topology | [
"Mathlib.Algebra.Group.Defs",
"Mathlib.Order.Filter.Extr",
"Mathlib.Topology.ContinuousOn"
] | Mathlib/Topology/Order/LocalExtr.lean | IsLocalMax.comp_continuous | null |
IsLocalExtr.comp_continuous [TopologicalSpace δ] {g : δ → α} {b : δ}
(hf : IsLocalExtr f (g b)) (hg : ContinuousAt g b) : IsLocalExtr (f ∘ g) b :=
hf.comp_tendsto hg | theorem | Topology | [
"Mathlib.Algebra.Group.Defs",
"Mathlib.Order.Filter.Extr",
"Mathlib.Topology.ContinuousOn"
] | Mathlib/Topology/Order/LocalExtr.lean | IsLocalExtr.comp_continuous | null |
IsLocalMin.comp_continuousOn [TopologicalSpace δ] {s : Set δ} {g : δ → α} {b : δ}
(hf : IsLocalMin f (g b)) (hg : ContinuousOn g s) (hb : b ∈ s) : IsLocalMinOn (f ∘ g) s b :=
hf.comp_tendsto (hg b hb) | theorem | Topology | [
"Mathlib.Algebra.Group.Defs",
"Mathlib.Order.Filter.Extr",
"Mathlib.Topology.ContinuousOn"
] | Mathlib/Topology/Order/LocalExtr.lean | IsLocalMin.comp_continuousOn | null |
IsLocalMax.comp_continuousOn [TopologicalSpace δ] {s : Set δ} {g : δ → α} {b : δ}
(hf : IsLocalMax f (g b)) (hg : ContinuousOn g s) (hb : b ∈ s) : IsLocalMaxOn (f ∘ g) s b :=
hf.comp_tendsto (hg b hb) | theorem | Topology | [
"Mathlib.Algebra.Group.Defs",
"Mathlib.Order.Filter.Extr",
"Mathlib.Topology.ContinuousOn"
] | Mathlib/Topology/Order/LocalExtr.lean | IsLocalMax.comp_continuousOn | null |
IsLocalExtr.comp_continuousOn [TopologicalSpace δ] {s : Set δ} (g : δ → α) {b : δ}
(hf : IsLocalExtr f (g b)) (hg : ContinuousOn g s) (hb : b ∈ s) : IsLocalExtrOn (f ∘ g) s b :=
hf.elim (fun hf => (hf.comp_continuousOn hg hb).isExtr) fun hf =>
(IsLocalMax.comp_continuousOn hf hg hb).isExtr | theorem | Topology | [
"Mathlib.Algebra.Group.Defs",
"Mathlib.Order.Filter.Extr",
"Mathlib.Topology.ContinuousOn"
] | Mathlib/Topology/Order/LocalExtr.lean | IsLocalExtr.comp_continuousOn | null |
IsLocalMinOn.comp_continuousOn [TopologicalSpace δ] {t : Set α} {s : Set δ} {g : δ → α}
{b : δ} (hf : IsLocalMinOn f t (g b)) (hst : s ⊆ g ⁻¹' t) (hg : ContinuousOn g s) (hb : b ∈ s) :
IsLocalMinOn (f ∘ g) s b :=
hf.comp_tendsto
(tendsto_nhdsWithin_mono_right (image_subset_iff.mpr hst)
(ContinuousWithinAt.tendsto_nhdsWithin_image (hg b hb))) | theorem | Topology | [
"Mathlib.Algebra.Group.Defs",
"Mathlib.Order.Filter.Extr",
"Mathlib.Topology.ContinuousOn"
] | Mathlib/Topology/Order/LocalExtr.lean | IsLocalMinOn.comp_continuousOn | null |
IsLocalMaxOn.comp_continuousOn [TopologicalSpace δ] {t : Set α} {s : Set δ} {g : δ → α}
{b : δ} (hf : IsLocalMaxOn f t (g b)) (hst : s ⊆ g ⁻¹' t) (hg : ContinuousOn g s) (hb : b ∈ s) :
IsLocalMaxOn (f ∘ g) s b :=
hf.comp_tendsto
(tendsto_nhdsWithin_mono_right (image_subset_iff.mpr hst)
(ContinuousWithinAt.tendsto_nhdsWithin_image (hg b hb))) | theorem | Topology | [
"Mathlib.Algebra.Group.Defs",
"Mathlib.Order.Filter.Extr",
"Mathlib.Topology.ContinuousOn"
] | Mathlib/Topology/Order/LocalExtr.lean | IsLocalMaxOn.comp_continuousOn | null |
IsLocalExtrOn.comp_continuousOn [TopologicalSpace δ] {t : Set α} {s : Set δ} (g : δ → α)
{b : δ} (hf : IsLocalExtrOn f t (g b)) (hst : s ⊆ g ⁻¹' t) (hg : ContinuousOn g s)
(hb : b ∈ s) : IsLocalExtrOn (f ∘ g) s b :=
hf.elim (fun hf => (hf.comp_continuousOn hst hg hb).isExtr) fun hf =>
(IsLocalMaxOn.comp_continuousOn hf hst hg hb).isExtr | theorem | Topology | [
"Mathlib.Algebra.Group.Defs",
"Mathlib.Order.Filter.Extr",
"Mathlib.Topology.ContinuousOn"
] | Mathlib/Topology/Order/LocalExtr.lean | IsLocalExtrOn.comp_continuousOn | null |
Filter.EventuallyLE.isLocalMaxOn {f g : α → β} {a : α} (hle : g ≤ᶠ[𝓝[s] a] f)
(hfga : f a = g a) (h : IsLocalMaxOn f s a) : IsLocalMaxOn g s a :=
hle.isMaxFilter hfga h
nonrec theorem IsLocalMaxOn.congr {f g : α → β} {a : α} (h : IsLocalMaxOn f s a)
(heq : f =ᶠ[𝓝[s] a] g) (hmem : a ∈ s) : IsLocalMaxOn g s a :=
h.congr heq <| heq.eq_of_nhdsWithin hmem | theorem | Topology | [
"Mathlib.Algebra.Group.Defs",
"Mathlib.Order.Filter.Extr",
"Mathlib.Topology.ContinuousOn"
] | Mathlib/Topology/Order/LocalExtr.lean | Filter.EventuallyLE.isLocalMaxOn | null |
Filter.EventuallyEq.isLocalMaxOn_iff {f g : α → β} {a : α} (heq : f =ᶠ[𝓝[s] a] g)
(hmem : a ∈ s) : IsLocalMaxOn f s a ↔ IsLocalMaxOn g s a :=
heq.isMaxFilter_iff <| heq.eq_of_nhdsWithin hmem | theorem | Topology | [
"Mathlib.Algebra.Group.Defs",
"Mathlib.Order.Filter.Extr",
"Mathlib.Topology.ContinuousOn"
] | Mathlib/Topology/Order/LocalExtr.lean | Filter.EventuallyEq.isLocalMaxOn_iff | null |
Filter.EventuallyLE.isLocalMinOn {f g : α → β} {a : α} (hle : f ≤ᶠ[𝓝[s] a] g)
(hfga : f a = g a) (h : IsLocalMinOn f s a) : IsLocalMinOn g s a :=
hle.isMinFilter hfga h
nonrec theorem IsLocalMinOn.congr {f g : α → β} {a : α} (h : IsLocalMinOn f s a)
(heq : f =ᶠ[𝓝[s] a] g) (hmem : a ∈ s) : IsLocalMinOn g s a :=
h.congr heq <| heq.eq_of_nhdsWithin hmem
nonrec theorem Filter.EventuallyEq.isLocalMinOn_iff {f g : α → β} {a : α} (heq : f =ᶠ[𝓝[s] a] g)
(hmem : a ∈ s) : IsLocalMinOn f s a ↔ IsLocalMinOn g s a :=
heq.isMinFilter_iff <| heq.eq_of_nhdsWithin hmem
nonrec theorem IsLocalExtrOn.congr {f g : α → β} {a : α} (h : IsLocalExtrOn f s a)
(heq : f =ᶠ[𝓝[s] a] g) (hmem : a ∈ s) : IsLocalExtrOn g s a :=
h.congr heq <| heq.eq_of_nhdsWithin hmem | theorem | Topology | [
"Mathlib.Algebra.Group.Defs",
"Mathlib.Order.Filter.Extr",
"Mathlib.Topology.ContinuousOn"
] | Mathlib/Topology/Order/LocalExtr.lean | Filter.EventuallyLE.isLocalMinOn | null |
Filter.EventuallyEq.isLocalExtrOn_iff {f g : α → β} {a : α} (heq : f =ᶠ[𝓝[s] a] g)
(hmem : a ∈ s) : IsLocalExtrOn f s a ↔ IsLocalExtrOn g s a :=
heq.isExtrFilter_iff <| heq.eq_of_nhdsWithin hmem | theorem | Topology | [
"Mathlib.Algebra.Group.Defs",
"Mathlib.Order.Filter.Extr",
"Mathlib.Topology.ContinuousOn"
] | Mathlib/Topology/Order/LocalExtr.lean | Filter.EventuallyEq.isLocalExtrOn_iff | null |
Filter.EventuallyLE.isLocalMax {f g : α → β} {a : α} (hle : g ≤ᶠ[𝓝 a] f) (hfga : f a = g a)
(h : IsLocalMax f a) : IsLocalMax g a :=
hle.isMaxFilter hfga h
nonrec theorem IsLocalMax.congr {f g : α → β} {a : α} (h : IsLocalMax f a) (heq : f =ᶠ[𝓝 a] g) :
IsLocalMax g a :=
h.congr heq heq.eq_of_nhds | theorem | Topology | [
"Mathlib.Algebra.Group.Defs",
"Mathlib.Order.Filter.Extr",
"Mathlib.Topology.ContinuousOn"
] | Mathlib/Topology/Order/LocalExtr.lean | Filter.EventuallyLE.isLocalMax | null |
Filter.EventuallyEq.isLocalMax_iff {f g : α → β} {a : α} (heq : f =ᶠ[𝓝 a] g) :
IsLocalMax f a ↔ IsLocalMax g a :=
heq.isMaxFilter_iff heq.eq_of_nhds | theorem | Topology | [
"Mathlib.Algebra.Group.Defs",
"Mathlib.Order.Filter.Extr",
"Mathlib.Topology.ContinuousOn"
] | Mathlib/Topology/Order/LocalExtr.lean | Filter.EventuallyEq.isLocalMax_iff | null |
Filter.EventuallyLE.isLocalMin {f g : α → β} {a : α} (hle : f ≤ᶠ[𝓝 a] g) (hfga : f a = g a)
(h : IsLocalMin f a) : IsLocalMin g a :=
hle.isMinFilter hfga h
nonrec theorem IsLocalMin.congr {f g : α → β} {a : α} (h : IsLocalMin f a) (heq : f =ᶠ[𝓝 a] g) :
IsLocalMin g a :=
h.congr heq heq.eq_of_nhds | theorem | Topology | [
"Mathlib.Algebra.Group.Defs",
"Mathlib.Order.Filter.Extr",
"Mathlib.Topology.ContinuousOn"
] | Mathlib/Topology/Order/LocalExtr.lean | Filter.EventuallyLE.isLocalMin | null |
Filter.EventuallyEq.isLocalMin_iff {f g : α → β} {a : α} (heq : f =ᶠ[𝓝 a] g) :
IsLocalMin f a ↔ IsLocalMin g a :=
heq.isMinFilter_iff heq.eq_of_nhds
nonrec theorem IsLocalExtr.congr {f g : α → β} {a : α} (h : IsLocalExtr f a) (heq : f =ᶠ[𝓝 a] g) :
IsLocalExtr g a :=
h.congr heq heq.eq_of_nhds | theorem | Topology | [
"Mathlib.Algebra.Group.Defs",
"Mathlib.Order.Filter.Extr",
"Mathlib.Topology.ContinuousOn"
] | Mathlib/Topology/Order/LocalExtr.lean | Filter.EventuallyEq.isLocalMin_iff | null |
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