statement stringlengths 1 2.93k | proof stringlengths 0 19.2k | type stringclasses 13
values | symbolic_name stringlengths 1 131 | library stringlengths 4 62 | filename stringlengths 20 95 | imports listlengths 0 10 | deps listlengths 0 64 | docstring stringlengths 0 4.95k | source_url stringclasses 1
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|---|---|---|---|---|---|---|---|---|---|---|
Filter.Tendsto.isBoundedUnder_le (h : Tendsto u f (𝓝 a)) : f.IsBoundedUnder (· ≤ ·) u | (isBounded_le_nhds a).mono h | theorem | Filter.Tendsto.isBoundedUnder_le | Topology.Order | Mathlib/Topology/Order/LiminfLimsup.lean | [
"Mathlib.Data.Fintype.Order",
"Mathlib.Topology.Order.MonotoneConvergence"
] | [
"isBounded_le_nhds"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Filter.Tendsto.bddAbove_range_of_cofinite [IsDirectedOrder α]
(h : Tendsto u cofinite (𝓝 a)) : BddAbove (Set.range u) | h.isBoundedUnder_le.bddAbove_range_of_cofinite | theorem | Filter.Tendsto.bddAbove_range_of_cofinite | Topology.Order | Mathlib/Topology/Order/LiminfLimsup.lean | [
"Mathlib.Data.Fintype.Order",
"Mathlib.Topology.Order.MonotoneConvergence"
] | [
"BddAbove",
"IsDirectedOrder",
"Set.range"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Filter.Tendsto.bddAbove_range [IsDirectedOrder α] {u : ℕ → α}
(h : Tendsto u atTop (𝓝 a)) : BddAbove (Set.range u) | h.isBoundedUnder_le.bddAbove_range | theorem | Filter.Tendsto.bddAbove_range | Topology.Order | Mathlib/Topology/Order/LiminfLimsup.lean | [
"Mathlib.Data.Fintype.Order",
"Mathlib.Topology.Order.MonotoneConvergence"
] | [
"BddAbove",
"IsDirectedOrder",
"Set.range"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
isCobounded_ge_nhds (a : α) : (𝓝 a).IsCobounded (· ≥ ·) | (isBounded_le_nhds a).isCobounded_flip | theorem | isCobounded_ge_nhds | Topology.Order | Mathlib/Topology/Order/LiminfLimsup.lean | [
"Mathlib.Data.Fintype.Order",
"Mathlib.Topology.Order.MonotoneConvergence"
] | [
"isBounded_le_nhds"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Filter.Tendsto.isCoboundedUnder_ge [NeBot f] (h : Tendsto u f (𝓝 a)) :
f.IsCoboundedUnder (· ≥ ·) u | h.isBoundedUnder_le.isCobounded_flip | theorem | Filter.Tendsto.isCoboundedUnder_ge | Topology.Order | Mathlib/Topology/Order/LiminfLimsup.lean | [
"Mathlib.Data.Fintype.Order",
"Mathlib.Topology.Order.MonotoneConvergence"
] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
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 | Prod.instBoundedLENhdsClass | Topology.Order | Mathlib/Topology/Order/LiminfLimsup.lean | [
"Mathlib.Data.Fintype.Order",
"Mathlib.Topology.Order.MonotoneConvergence"
] | [
"BoundedLENhdsClass",
"Prod.mk.eta",
"isBounded_le_nhds",
"nhds_prod_eq"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
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 | Pi.instBoundedLENhdsClass | Topology.Order | Mathlib/Topology/Order/LiminfLimsup.lean | [
"Mathlib.Data.Fintype.Order",
"Mathlib.Topology.Order.MonotoneConvergence"
] | [
"BoundedLENhdsClass",
"Finite",
"Preorder",
"TopologicalSpace",
"isBounded_le_nhds",
"nhds_pi"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
isBounded_ge_nhds (a : α) : (𝓝 a).IsBounded (· ≥ ·) | BoundedGENhdsClass.isBounded_ge_nhds _ | theorem | isBounded_ge_nhds | Topology.Order | Mathlib/Topology/Order/LiminfLimsup.lean | [
"Mathlib.Data.Fintype.Order",
"Mathlib.Topology.Order.MonotoneConvergence"
] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Filter.Tendsto.isBoundedUnder_ge (h : Tendsto u f (𝓝 a)) : f.IsBoundedUnder (· ≥ ·) u | (isBounded_ge_nhds a).mono h | theorem | Filter.Tendsto.isBoundedUnder_ge | Topology.Order | Mathlib/Topology/Order/LiminfLimsup.lean | [
"Mathlib.Data.Fintype.Order",
"Mathlib.Topology.Order.MonotoneConvergence"
] | [
"isBounded_ge_nhds"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Filter.Tendsto.bddBelow_range_of_cofinite [IsCodirectedOrder α]
(h : Tendsto u cofinite (𝓝 a)) : BddBelow (Set.range u) | h.isBoundedUnder_ge.bddBelow_range_of_cofinite | theorem | Filter.Tendsto.bddBelow_range_of_cofinite | Topology.Order | Mathlib/Topology/Order/LiminfLimsup.lean | [
"Mathlib.Data.Fintype.Order",
"Mathlib.Topology.Order.MonotoneConvergence"
] | [
"Set.range"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Filter.Tendsto.bddBelow_range [IsCodirectedOrder α] {u : ℕ → α}
(h : Tendsto u atTop (𝓝 a)) : BddBelow (Set.range u) | h.isBoundedUnder_ge.bddBelow_range | theorem | Filter.Tendsto.bddBelow_range | Topology.Order | Mathlib/Topology/Order/LiminfLimsup.lean | [
"Mathlib.Data.Fintype.Order",
"Mathlib.Topology.Order.MonotoneConvergence"
] | [
"Set.range"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
isCobounded_le_nhds (a : α) : (𝓝 a).IsCobounded (· ≤ ·) | (isBounded_ge_nhds a).isCobounded_flip | theorem | isCobounded_le_nhds | Topology.Order | Mathlib/Topology/Order/LiminfLimsup.lean | [
"Mathlib.Data.Fintype.Order",
"Mathlib.Topology.Order.MonotoneConvergence"
] | [
"isBounded_ge_nhds"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Filter.Tendsto.isCoboundedUnder_le [NeBot f] (h : Tendsto u f (𝓝 a)) :
f.IsCoboundedUnder (· ≤ ·) u | h.isBoundedUnder_ge.isCobounded_flip | theorem | Filter.Tendsto.isCoboundedUnder_le | Topology.Order | Mathlib/Topology/Order/LiminfLimsup.lean | [
"Mathlib.Data.Fintype.Order",
"Mathlib.Topology.Order.MonotoneConvergence"
] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Prod.instBoundedGENhdsClass : BoundedGENhdsClass (α × β) | ⟨(Prod.instBoundedLENhdsClass (α := αᵒᵈ) (β := βᵒᵈ)).isBounded_le_nhds⟩ | instance | Prod.instBoundedGENhdsClass | Topology.Order | Mathlib/Topology/Order/LiminfLimsup.lean | [
"Mathlib.Data.Fintype.Order",
"Mathlib.Topology.Order.MonotoneConvergence"
] | [
"BoundedGENhdsClass",
"Prod.instBoundedLENhdsClass"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
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 | Pi.instBoundedGENhdsClass | Topology.Order | Mathlib/Topology/Order/LiminfLimsup.lean | [
"Mathlib.Data.Fintype.Order",
"Mathlib.Topology.Order.MonotoneConvergence"
] | [
"BoundedGENhdsClass",
"Finite",
"Pi.instBoundedLENhdsClass",
"Preorder",
"TopologicalSpace"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
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 | le_nhds_of_limsSup_eq_limsInf | Topology.Order | Mathlib/Topology/Order/LiminfLimsup.lean | [
"Mathlib.Data.Fintype.Order",
"Mathlib.Topology.Order.MonotoneConvergence"
] | [
"Filter"
] | 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. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
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_nh... | theorem | limsSup_nhds | Topology.Order | Mathlib/Topology/Order/LiminfLimsup.lean | [
"Mathlib.Data.Fintype.Order",
"Mathlib.Topology.Order.MonotoneConvergence"
] | [
"csInf_eq_of_forall_ge_of_forall_gt_exists_lt",
"dense_or_discrete",
"isBounded_le_nhds",
"mem_of_mem_nhds"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
limsInf_nhds (a : α) : limsInf (𝓝 a) = a | limsSup_nhds (α := αᵒᵈ) a | theorem | limsInf_nhds | Topology.Order | Mathlib/Topology/Order/LiminfLimsup.lean | [
"Mathlib.Data.Fintype.Order",
"Mathlib.Topology.Order.MonotoneConvergence"
] | [
"limsSup_nhds"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
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)
... | theorem | limsInf_eq_of_le_nhds | Topology.Order | Mathlib/Topology/Order/LiminfLimsup.lean | [
"Mathlib.Data.Fintype.Order",
"Mathlib.Topology.Order.MonotoneConvergence"
] | [
"Filter",
"isBounded_ge_nhds",
"isBounded_le_nhds",
"le_antisymm",
"limsInf_nhds",
"limsSup_nhds",
"symm"
] | If a filter is converging, its limsup coincides with its limit. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
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 | limsSup_eq_of_le_nhds | Topology.Order | Mathlib/Topology/Order/LiminfLimsup.lean | [
"Mathlib.Data.Fintype.Order",
"Mathlib.Topology.Order.MonotoneConvergence"
] | [
"Filter",
"limsInf_eq_of_le_nhds"
] | If a filter is converging, its liminf coincides with its limit. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
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 | Filter.Tendsto.limsup_eq | Topology.Order | Mathlib/Topology/Order/LiminfLimsup.lean | [
"Mathlib.Data.Fintype.Order",
"Mathlib.Topology.Order.MonotoneConvergence"
] | [
"Filter",
"limsSup_eq_of_le_nhds"
] | If a function has a limit, then its limsup coincides with its limit. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
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 | Filter.Tendsto.liminf_eq | Topology.Order | Mathlib/Topology/Order/LiminfLimsup.lean | [
"Mathlib.Data.Fintype.Order",
"Mathlib.Topology.Order.MonotoneConvergence"
] | [
"Filter",
"limsInf_eq_of_le_nhds"
] | If a function has a limit, then its liminf coincides with its limit. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
ClusterPt.limsSup {f : Filter α} [NeBot f]
(hc : f.IsCobounded (· ≤ ·) := by isBoundedDefault)
(hb : f.IsBounded (· ≤ ·) := by isBoundedDefault) : ClusterPt f.limsSup f | by
by_cases! hn : Nontrivial α
· by_cases! htop : ∀ x, x ≤ f.limsSup
· let : OrderTop α := { top := f.limsSup, le_top := htop }
exact nhds_top_basis.clusterPt_iff_frequently |>.mpr fun a => frequently_lt_of_lt_limsSup hc
· by_cases! hbot : ∀ x, f.limsSup ≤ x
· let : OrderBot α := { bot := f.lims... | theorem | ClusterPt.limsSup | Topology.Order | Mathlib/Topology/Order/LiminfLimsup.lean | [
"Mathlib.Data.Fintype.Order",
"Mathlib.Topology.Order.MonotoneConvergence"
] | [
"ClusterPt",
"Filter",
"Filter.eq_top_of_neBot",
"Nontrivial",
"OrderBot",
"OrderTop",
"clusterPt_iff_frequently",
"le_top",
"nhds_basis_Ioo'"
] | The `limsSup` of a filter `f` is a cluster point of `f`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
ClusterPt.limsInf {f : Filter α} [NeBot f]
(hc : f.IsCobounded (· ≥ ·) := by isBoundedDefault)
(hb : f.IsBounded (· ≥ ·) := by isBoundedDefault) : ClusterPt f.limsInf f | ClusterPt.limsSup (α := αᵒᵈ) hc hb | theorem | ClusterPt.limsInf | Topology.Order | Mathlib/Topology/Order/LiminfLimsup.lean | [
"Mathlib.Data.Fintype.Order",
"Mathlib.Topology.Order.MonotoneConvergence"
] | [
"ClusterPt",
"ClusterPt.limsSup",
"Filter"
] | The `limsInf` of a filter `f` is a cluster point of `f`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
ClusterPt.le_limsSup {f : Filter α} {x : α} (hx : ClusterPt x f)
(hb : f.IsBounded (· ≤ ·) := by isBoundedDefault) :
x ≤ f.limsSup | by
simp only [ClusterPt] at hx
have : (𝓝 x ⊓ f).limsSup = x := limsSup_eq_of_le_nhds inf_le_left
refine this ▸ limsSup_le_limsSup_of_le inf_le_right ?_ hb
exact (IsBounded.mono inf_le_left (isBounded_ge_nhds x)).isCobounded_le | theorem | ClusterPt.le_limsSup | Topology.Order | Mathlib/Topology/Order/LiminfLimsup.lean | [
"Mathlib.Data.Fintype.Order",
"Mathlib.Topology.Order.MonotoneConvergence"
] | [
"ClusterPt",
"Filter",
"isBounded_ge_nhds",
"limsSup_eq_of_le_nhds"
] | Every cluster point `x` of a filter `f` is less than or equal to `f.limsSup`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
ClusterPt.limsInf_le {f : Filter α} {x : α} (hx : ClusterPt x f)
(hb : f.IsBounded (· ≥ ·) := by isBoundedDefault) :
f.limsInf ≤ x | hx.le_limsSup (α := αᵒᵈ) | theorem | ClusterPt.limsInf_le | Topology.Order | Mathlib/Topology/Order/LiminfLimsup.lean | [
"Mathlib.Data.Fintype.Order",
"Mathlib.Topology.Order.MonotoneConvergence"
] | [
"ClusterPt",
"Filter"
] | Every cluster point `x` of a filter `f` is greater than or equal to `f.limsInf`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
isGreatest_clusterPt_limsSup {f : Filter α} [NeBot f]
(hc : f.IsCobounded (· ≤ ·) := by isBoundedDefault)
(hb : f.IsBounded (· ≤ ·) := by isBoundedDefault) :
IsGreatest {x | ClusterPt x f} f.limsSup | ⟨ClusterPt.limsSup, fun a ha => ha.le_limsSup⟩ | theorem | isGreatest_clusterPt_limsSup | Topology.Order | Mathlib/Topology/Order/LiminfLimsup.lean | [
"Mathlib.Data.Fintype.Order",
"Mathlib.Topology.Order.MonotoneConvergence"
] | [
"ClusterPt",
"Filter"
] | The `limsSup` of a filter `f` is the greatest cluster point of `f`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
isLeast_clusterPt_limsInf {f : Filter α} [NeBot f]
(hc : f.IsCobounded (· ≥ ·) := by isBoundedDefault)
(hb : f.IsBounded (· ≥ ·) := by isBoundedDefault) :
IsLeast {x | ClusterPt x f} f.limsInf | isGreatest_clusterPt_limsSup (α := αᵒᵈ) | theorem | isLeast_clusterPt_limsInf | Topology.Order | Mathlib/Topology/Order/LiminfLimsup.lean | [
"Mathlib.Data.Fintype.Order",
"Mathlib.Topology.Order.MonotoneConvergence"
] | [
"ClusterPt",
"Filter",
"IsLeast",
"isGreatest_clusterPt_limsSup"
] | The `limsInf` of a filter `f` is the least cluster point of `f`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
MapClusterPt.limsup {u : β → α} {f : Filter β} [NeBot f]
(hc : IsCoboundedUnder (· ≤ ·) f u := by isBoundedDefault)
(hb : IsBoundedUnder (· ≤ ·) f u := by isBoundedDefault) :
MapClusterPt (f.limsup u) f u | ClusterPt.limsSup | theorem | MapClusterPt.limsup | Topology.Order | Mathlib/Topology/Order/LiminfLimsup.lean | [
"Mathlib.Data.Fintype.Order",
"Mathlib.Topology.Order.MonotoneConvergence"
] | [
"ClusterPt.limsSup",
"Filter",
"MapClusterPt"
] | The `limsup` of a function `u` along a filter `f` is a cluster point of `u` along `f`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
MapClusterPt.liminf {u : β → α} {f : Filter β} [NeBot f]
(hc : IsCoboundedUnder (· ≥ ·) f u := by isBoundedDefault)
(hb : IsBoundedUnder (· ≥ ·) f u := by isBoundedDefault) :
MapClusterPt (liminf u f) f u | ClusterPt.limsInf | theorem | MapClusterPt.liminf | Topology.Order | Mathlib/Topology/Order/LiminfLimsup.lean | [
"Mathlib.Data.Fintype.Order",
"Mathlib.Topology.Order.MonotoneConvergence"
] | [
"ClusterPt.limsInf",
"Filter",
"MapClusterPt"
] | The `liminf` of a function `u` along a filter `f` is a cluster point of `u` along `f`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
MapClusterPt.le_limsup {u : β → α} {f : Filter β}
{x : α} (hx : MapClusterPt x f u) (hb : IsBoundedUnder (· ≤ ·) f u := by isBoundedDefault) :
x ≤ f.limsup u | hx.le_limsSup | theorem | MapClusterPt.le_limsup | Topology.Order | Mathlib/Topology/Order/LiminfLimsup.lean | [
"Mathlib.Data.Fintype.Order",
"Mathlib.Topology.Order.MonotoneConvergence"
] | [
"Filter",
"MapClusterPt"
] | Every cluster point `x` of a function `u` along a filter `f` is less than or equal to
`limsup u f`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
MapClusterPt.liminf_le {u : β → α} {f : Filter β}
{x : α} (hx : MapClusterPt x f u) (hb : IsBoundedUnder (· ≥ ·) f u := by isBoundedDefault) :
f.liminf u ≤ x | hx.limsInf_le | theorem | MapClusterPt.liminf_le | Topology.Order | Mathlib/Topology/Order/LiminfLimsup.lean | [
"Mathlib.Data.Fintype.Order",
"Mathlib.Topology.Order.MonotoneConvergence"
] | [
"Filter",
"MapClusterPt"
] | Every cluster point `x` of a function `u` along a filter `f` is greater than or equal to
`liminf u f`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
isGreatest_mapClusterPt_limsup {u : β → α} {f : Filter β} [NeBot f]
(hc : IsCoboundedUnder (· ≤ ·) f u := by isBoundedDefault)
(hb : IsBoundedUnder (· ≤ ·) f u := by isBoundedDefault) :
IsGreatest {x | MapClusterPt x f u} (limsup u f) | isGreatest_clusterPt_limsSup | theorem | isGreatest_mapClusterPt_limsup | Topology.Order | Mathlib/Topology/Order/LiminfLimsup.lean | [
"Mathlib.Data.Fintype.Order",
"Mathlib.Topology.Order.MonotoneConvergence"
] | [
"Filter",
"MapClusterPt",
"isGreatest_clusterPt_limsSup"
] | The `limsup` of a function `u` along a filter `f` is the greatest cluster point of `u` along
`f`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
isLeast_mapClusterPt_liminf {u : β → α} {f : Filter β} [NeBot f]
(hc : IsCoboundedUnder (· ≥ ·) f u := by isBoundedDefault)
(hb : IsBoundedUnder (· ≥ ·) f u := by isBoundedDefault) :
IsLeast {x | MapClusterPt x f u} (liminf u f) | isLeast_clusterPt_limsInf | theorem | isLeast_mapClusterPt_liminf | Topology.Order | Mathlib/Topology/Order/LiminfLimsup.lean | [
"Mathlib.Data.Fintype.Order",
"Mathlib.Topology.Order.MonotoneConvergence"
] | [
"Filter",
"IsLeast",
"MapClusterPt",
"isLeast_clusterPt_limsInf"
] | The `liminf` of a function `u` along a filter `f` is the least cluster point of `u` along
`f`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
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 | tendsto_of_liminf_eq_limsup | Topology.Order | Mathlib/Topology/Order/LiminfLimsup.lean | [
"Mathlib.Data.Fintype.Order",
"Mathlib.Topology.Order.MonotoneConvergence"
] | [
"Filter",
"le_nhds_of_limsSup_eq_limsInf"
] | If the liminf and the limsup of a function coincide, then the limit of the function
exists and has the same value. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
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 | tendsto_of_le_liminf_of_limsup_le | Topology.Order | Mathlib/Topology/Order/LiminfLimsup.lean | [
"Mathlib.Data.Fintype.Order",
"Mathlib.Topology.Order.MonotoneConvergence"
] | [
"Filter",
"le_antisymm",
"le_trans",
"tendsto_of_liminf_eq_limsup"
] | 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. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
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, Tend... | 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)) ... | theorem | tendsto_of_no_upcrossings | Topology.Order | Mathlib/Topology/Order/LiminfLimsup.lean | [
"Mathlib.Data.Fintype.Order",
"Mathlib.Topology.Order.MonotoneConvergence"
] | [
"Dense",
"DenselyOrdered",
"Filter",
"Set",
"Set.Ioo",
"isOpen_Ioo",
"le_rfl",
"tendsto_of_le_liminf_of_limsup_le"
] | 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. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
exists_seq_tendsto_limsSup [NeBot f] [IsCountablyGenerated f]
(hc : f.IsCobounded (· ≤ ·) := by isBoundedDefault)
(hb : f.IsBounded (· ≤ ·) := by isBoundedDefault) :
∃ x : ℕ → α, Tendsto x atTop (𝓝 f.limsSup) ∧ Tendsto x atTop f | (ClusterPt.limsSup).exists_seq_tendsto | theorem | exists_seq_tendsto_limsSup | Topology.Order | Mathlib/Topology/Order/LiminfLimsup.lean | [
"Mathlib.Data.Fintype.Order",
"Mathlib.Topology.Order.MonotoneConvergence"
] | [
"ClusterPt.limsSup"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
exists_seq_tendsto_limsInf [NeBot f] [IsCountablyGenerated f]
(hc : f.IsCobounded (· ≥ ·) := by isBoundedDefault)
(hb : f.IsBounded (· ≥ ·) := by isBoundedDefault) :
∃ x : ℕ → α, Tendsto x atTop (𝓝 f.limsInf) ∧ Tendsto x atTop f | (ClusterPt.limsInf).exists_seq_tendsto | theorem | exists_seq_tendsto_limsInf | Topology.Order | Mathlib/Topology/Order/LiminfLimsup.lean | [
"Mathlib.Data.Fintype.Order",
"Mathlib.Topology.Order.MonotoneConvergence"
] | [
"ClusterPt.limsInf"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
exists_seq_tendsto_limsup [NeBot f] [IsCountablyGenerated f] {u : β → α}
(hc : IsCoboundedUnder (· ≤ ·) f u := by isBoundedDefault)
(hb : IsBoundedUnder (· ≤ ·) f u := by isBoundedDefault) :
∃ x : ℕ → β, Tendsto (u ∘ x) atTop (𝓝 (limsup u f)) ∧ Tendsto x atTop f | (MapClusterPt.limsup).exists_seq_tendsto | theorem | exists_seq_tendsto_limsup | Topology.Order | Mathlib/Topology/Order/LiminfLimsup.lean | [
"Mathlib.Data.Fintype.Order",
"Mathlib.Topology.Order.MonotoneConvergence"
] | [
"MapClusterPt.limsup"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
exists_seq_tendsto_liminf [NeBot f] {u : β → α} [IsCountablyGenerated f]
(hc : IsCoboundedUnder (· ≥ ·) f u := by isBoundedDefault)
(hb : IsBoundedUnder (· ≥ ·) f u := by isBoundedDefault) :
∃ x : ℕ → β, Tendsto (u ∘ x) atTop (𝓝 (liminf u f)) ∧ Tendsto x atTop f | (MapClusterPt.liminf).exists_seq_tendsto | theorem | exists_seq_tendsto_liminf | Topology.Order | Mathlib/Topology/Order/LiminfLimsup.lean | [
"Mathlib.Data.Fintype.Order",
"Mathlib.Topology.Order.MonotoneConvergence"
] | [
"MapClusterPt.liminf"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
eventually_le_limsup (hf : IsBoundedUnder (· ≤ ·) f u := by isBoundedDefault) :
∀ᶠ b in f, u b ≤ f.limsup u | by
rw [eventually_le_const_iff_forall_gt_eventually_lt_const]
exact fun _ hc ↦ eventually_lt_of_limsup_lt hc | theorem | eventually_le_limsup | Topology.Order | Mathlib/Topology/Order/LiminfLimsup.lean | [
"Mathlib.Data.Fintype.Order",
"Mathlib.Topology.Order.MonotoneConvergence"
] | [
"eventually_le_const_iff_forall_gt_eventually_lt_const"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
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 | eventually_liminf_le | Topology.Order | Mathlib/Topology/Order/LiminfLimsup.lean | [
"Mathlib.Data.Fintype.Order",
"Mathlib.Topology.Order.MonotoneConvergence"
] | [
"eventually_le_limsup"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
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⟩ | theorem | limsup_eq_bot | Topology.Order | Mathlib/Topology/Order/LiminfLimsup.lean | [
"Mathlib.Data.Fintype.Order",
"Mathlib.Topology.Order.MonotoneConvergence"
] | [
"eventually_le_limsup",
"le_antisymm"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
liminf_eq_top : f.liminf u = ⊤ ↔ u =ᶠ[f] ⊤ | limsup_eq_bot (α := αᵒᵈ) | theorem | liminf_eq_top | Topology.Order | Mathlib/Topology/Order/LiminfLimsup.lean | [
"Mathlib.Data.Fintype.Order",
"Mathlib.Topology.Order.MonotoneConvergence"
] | [
"limsup_eq_bot"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
tendsto_iSup_of_tendsto_limsup {α β : Type*} [ConditionallyCompleteLattice α]
[CompleteLinearOrder β] [TopologicalSpace β] [OrderTopology β]
{u : ι → α → β} {c : β}
(h_all : ∀ i, Tendsto (u i) atTop (𝓝 c))
(h_limsup : Tendsto (fun r : α ↦ limsup (fun i ↦ u i r) cofinite) atTop (𝓝 c))
(h_anti : ∀ i... | by
classical
rcases isEmpty_or_nonempty ι with hι | ⟨⟨n0⟩⟩
· simpa using! h_limsup
refine tendsto_order.mpr ⟨fun b hb ↦ ?_, fun b hb ↦ ?_⟩
· filter_upwards with r
have : c ≤ u n0 r := (h_anti n0).le_of_tendsto (h_all n0) r
exact hb.trans_le (this.trans (le_iSup_iff.mpr fun b a ↦ a n0))
-- `⊢ ∀ᶠ (b_1... | lemma | tendsto_iSup_of_tendsto_limsup | Topology.Order | Mathlib/Topology/Order/LiminfLimsup.lean | [
"Mathlib.Data.Fintype.Order",
"Mathlib.Topology.Order.MonotoneConvergence"
] | [
"Antitone",
"CompleteLinearOrder",
"ConditionallyCompleteLattice",
"Finite",
"Finite.bddAbove_range",
"OrderTopology",
"Set.Ico",
"Set.Ioo",
"Set.eq_empty_or_nonempty",
"TopologicalSpace",
"by_cases",
"iSup_le",
"isEmpty_or_nonempty",
"le_ciSup",
"le_of_tendsto",
"le_rfl",
"le_sup_le... | Let `u : ι → α → β` be a sequence of antitone functions `α → β` indexed by `ι`. Suppose that for
all `i : ι`, `u i` tends to `c` at infinity, and that furthermore the limsup of `i ↦ u i r` along
the cofinite filter tends to the same `c` as `r` tends to infinity.
Then the supremum function `r ↦ ⨆ i, u i r` also tends to... | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
Nat.tendsto_iSup_of_tendsto_limsup {α β : Type*} [ConditionallyCompleteLattice α]
[CompleteLinearOrder β] [TopologicalSpace β] [OrderTopology β]
{u : ℕ → α → β} {c : β}
(h_all : ∀ n, Tendsto (u n) atTop (𝓝 c))
(h_limsup : Tendsto (fun r : α ↦ limsup (fun n ↦ u n r) atTop) atTop (𝓝 c))
(h_anti : ∀ ... | by
rw [← cofinite_eq_atTop] at h_limsup
exact _root_.tendsto_iSup_of_tendsto_limsup h_all h_limsup h_anti | lemma | Nat.tendsto_iSup_of_tendsto_limsup | Topology.Order | Mathlib/Topology/Order/LiminfLimsup.lean | [
"Mathlib.Data.Fintype.Order",
"Mathlib.Topology.Order.MonotoneConvergence"
] | [
"Antitone",
"CompleteLinearOrder",
"ConditionallyCompleteLattice",
"OrderTopology",
"TopologicalSpace"
] | Let `u : ℕ → α → β` be a sequence of antitone functions `α → β` indexed by `ℕ`. Suppose that for
all `n : ℕ`, `u n` tends to `c` at infinity, and that furthermore the limsup of `n ↦ u n r`
tends to the same `c` as `r` tends to infinity.
Then the supremum function `r ↦ ⨆ n, u n r` also tends to `c` at infinity. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
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
ap... | theorem | Antitone.map_limsSup_of_continuousAt | Topology.Order | Mathlib/Topology/Order/LiminfLimsup.lean | [
"Mathlib.Data.Fintype.Order",
"Mathlib.Topology.Order.MonotoneConvergence"
] | [
"Antitone",
"Antitone.isCoboundedUnder_ge_of_isCobounded",
"BddAbove",
"ContinuousAt",
"Filter",
"IsBot",
"Set.Ioc",
"Set.Ioo",
"Set.mem_image_of_mem",
"Set.not_nonempty_empty",
"exists_Ioc_subset_of_mem_nhds",
"exists_lt_of_lt_csSup",
"le_antisymm",
"le_of_forall_lt",
"lt_csSup_of_lt",
... | 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). | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
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 | Antitone.map_limsup_of_continuousAt | Topology.Order | Mathlib/Topology/Order/LiminfLimsup.lean | [
"Mathlib.Data.Fintype.Order",
"Mathlib.Topology.Order.MonotoneConvergence"
] | [
"Antitone",
"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). | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
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 | Antitone.map_limsInf_of_continuousAt | Topology.Order | Mathlib/Topology/Order/LiminfLimsup.lean | [
"Mathlib.Data.Fintype.Order",
"Mathlib.Topology.Order.MonotoneConvergence"
] | [
"Antitone",
"Antitone.map_limsSup_of_continuousAt",
"ContinuousAt",
"Filter"
] | 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). | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
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 | Antitone.map_liminf_of_continuousAt | Topology.Order | Mathlib/Topology/Order/LiminfLimsup.lean | [
"Mathlib.Data.Fintype.Order",
"Mathlib.Topology.Order.MonotoneConvergence"
] | [
"Antitone",
"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). | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
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 | Monotone.map_limsSup_of_continuousAt | Topology.Order | Mathlib/Topology/Order/LiminfLimsup.lean | [
"Mathlib.Data.Fintype.Order",
"Mathlib.Topology.Order.MonotoneConvergence"
] | [
"Antitone.map_limsSup_of_continuousAt",
"ContinuousAt",
"Filter",
"Monotone"
] | 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). | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
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 | Monotone.map_limsup_of_continuousAt | Topology.Order | Mathlib/Topology/Order/LiminfLimsup.lean | [
"Mathlib.Data.Fintype.Order",
"Mathlib.Topology.Order.MonotoneConvergence"
] | [
"ContinuousAt",
"Monotone"
] | 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). | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
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 | Monotone.map_limsInf_of_continuousAt | Topology.Order | Mathlib/Topology/Order/LiminfLimsup.lean | [
"Mathlib.Data.Fintype.Order",
"Mathlib.Topology.Order.MonotoneConvergence"
] | [
"Antitone.map_limsSup_of_continuousAt",
"ContinuousAt",
"Filter",
"Monotone"
] | 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). | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
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 | Monotone.map_liminf_of_continuousAt | Topology.Order | Mathlib/Topology/Order/LiminfLimsup.lean | [
"Mathlib.Data.Fintype.Order",
"Mathlib.Topology.Order.MonotoneConvergence"
] | [
"ContinuousAt",
"Monotone"
] | 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). | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
Antitone.liminf_nhdsGT_eq_iSup₂_of_exists_gt (hf : Antitone f) (a : α) (hb : ∃ b, a < b) :
(𝓝[>] a).liminf f = ⨆ r > a, f r | by
rw [(nhdsGT_basis_of_exists_gt hb).liminf_eq_iSup_iInf]
refine le_antisymm (iSup₂_mono' fun r hr ↦ ?_)
(iSup₂_mono' fun r hr ↦ ⟨r, hr, le_iInf₂ fun i hi ↦ hf (Set.mem_Ioo.1 hi).2.le⟩)
obtain ⟨b, hb⟩ := exists_between hr
exact ⟨b, hb.1, iInf₂_le b hb⟩ | lemma | Antitone.liminf_nhdsGT_eq_iSup₂_of_exists_gt | Topology.Order | Mathlib/Topology/Order/LiminfLimsup.lean | [
"Mathlib.Data.Fintype.Order",
"Mathlib.Topology.Order.MonotoneConvergence"
] | [
"Antitone",
"exists_between",
"iSup₂_mono'",
"le_antisymm",
"nhdsGT_basis_of_exists_gt"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Antitone.liminf_nhdsGT_eq_iSup₂ [NoMaxOrder α] (hf : Antitone f) (a : α) :
(𝓝[>] a).liminf f = ⨆ r > a, f r | hf.liminf_nhdsGT_eq_iSup₂_of_exists_gt a (exists_gt a) | lemma | Antitone.liminf_nhdsGT_eq_iSup₂ | Topology.Order | Mathlib/Topology/Order/LiminfLimsup.lean | [
"Mathlib.Data.Fintype.Order",
"Mathlib.Topology.Order.MonotoneConvergence"
] | [
"Antitone",
"NoMaxOrder"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Monotone.liminf_nhdsLT_eq_iSup₂_of_exists_lt (hf : Monotone f) (a : α) (hb : ∃ b, b < a) :
(𝓝[<] a).liminf f = ⨆ r < a, f r | by
rw [(nhdsLT_basis_of_exists_lt hb).liminf_eq_iSup_iInf]
refine le_antisymm (iSup₂_mono' fun r hr ↦ ?_)
(iSup₂_mono' fun r hr ↦ ⟨r, hr, le_iInf₂ fun i hi ↦ hf (Set.mem_Ioo.1 hi).1.le⟩)
obtain ⟨b, hb⟩ := exists_between hr
exact ⟨b, hb.2, iInf₂_le b hb⟩ | lemma | Monotone.liminf_nhdsLT_eq_iSup₂_of_exists_lt | Topology.Order | Mathlib/Topology/Order/LiminfLimsup.lean | [
"Mathlib.Data.Fintype.Order",
"Mathlib.Topology.Order.MonotoneConvergence"
] | [
"Monotone",
"exists_between",
"iSup₂_mono'",
"le_antisymm",
"nhdsLT_basis_of_exists_lt"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Monotone.liminf_nhdsLT_eq_iSup₂ [NoMinOrder α] (hf : Monotone f) (a : α) :
(𝓝[<] a).liminf f = ⨆ r < a, f r | hf.liminf_nhdsLT_eq_iSup₂_of_exists_lt a (exists_lt a) | lemma | Monotone.liminf_nhdsLT_eq_iSup₂ | Topology.Order | Mathlib/Topology/Order/LiminfLimsup.lean | [
"Mathlib.Data.Fintype.Order",
"Mathlib.Topology.Order.MonotoneConvergence"
] | [
"Monotone",
"NoMinOrder"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Monotone.limsup_nhdsGT_eq_iInf₂_of_exists_gt (hf : Monotone f) (a : α) (hb : ∃ b, a < b) :
(𝓝[>] a).limsup f = ⨅ r > a, f r | by
rw [(nhdsGT_basis_of_exists_gt hb).limsup_eq_iInf_iSup]
refine le_antisymm
(iInf₂_mono' fun r hr ↦ ⟨r, hr, iSup₂_le fun i hi ↦ hf (Set.mem_Ioo.1 hi).2.le⟩)
(iInf₂_mono' fun r hr ↦ ?_)
obtain ⟨b, hb⟩ := exists_between hr
exact ⟨b, hb.1, le_iSup₂_of_le b hb le_rfl⟩ | lemma | Monotone.limsup_nhdsGT_eq_iInf₂_of_exists_gt | Topology.Order | Mathlib/Topology/Order/LiminfLimsup.lean | [
"Mathlib.Data.Fintype.Order",
"Mathlib.Topology.Order.MonotoneConvergence"
] | [
"Monotone",
"exists_between",
"iSup₂_le",
"le_antisymm",
"le_iSup₂_of_le",
"nhdsGT_basis_of_exists_gt"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Monotone.limsup_nhdsGT_eq_iInf₂ [NoMaxOrder α] (hf : Monotone f) (a : α) :
(𝓝[>] a).limsup f = ⨅ r > a, f r | hf.limsup_nhdsGT_eq_iInf₂_of_exists_gt a (exists_gt a) | lemma | Monotone.limsup_nhdsGT_eq_iInf₂ | Topology.Order | Mathlib/Topology/Order/LiminfLimsup.lean | [
"Mathlib.Data.Fintype.Order",
"Mathlib.Topology.Order.MonotoneConvergence"
] | [
"Monotone",
"NoMaxOrder"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Antitone.limsup_nhdsLT_eq_iInf₂_of_exists_lt (hf : Antitone f) (a : α) (hb : ∃ b, b < a) :
(𝓝[<] a).limsup f = ⨅ r < a, f r | by
rw [(nhdsLT_basis_of_exists_lt hb).limsup_eq_iInf_iSup]
refine le_antisymm
(iInf₂_mono' fun r hr ↦ ⟨r, hr, iSup₂_le fun i hi ↦ hf (Set.mem_Ioo.1 hi).1.le⟩)
(iInf₂_mono' fun r hr ↦ ?_)
obtain ⟨b, hb⟩ := exists_between hr
exact ⟨b, hb.2, le_iSup₂_of_le b hb le_rfl⟩ | lemma | Antitone.limsup_nhdsLT_eq_iInf₂_of_exists_lt | Topology.Order | Mathlib/Topology/Order/LiminfLimsup.lean | [
"Mathlib.Data.Fintype.Order",
"Mathlib.Topology.Order.MonotoneConvergence"
] | [
"Antitone",
"exists_between",
"iSup₂_le",
"le_antisymm",
"le_iSup₂_of_le",
"nhdsLT_basis_of_exists_lt"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Antitone.limsup_nhdsLT_eq_iInf₂ [NoMinOrder α] (hf : Antitone f) (a : α) :
(𝓝[<] a).limsup f = ⨅ r < a, f r | hf.limsup_nhdsLT_eq_iInf₂_of_exists_lt a (exists_lt a) | lemma | Antitone.limsup_nhdsLT_eq_iInf₂ | Topology.Order | Mathlib/Topology/Order/LiminfLimsup.lean | [
"Mathlib.Data.Fintype.Order",
"Mathlib.Topology.Order.MonotoneConvergence"
] | [
"Antitone",
"NoMinOrder"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
IsLocalMinOn | IsMinFilter f (𝓝[s] a) a | def | IsLocalMinOn | Topology.Order | Mathlib/Topology/Order/LocalExtr.lean | [] | [
"IsMinFilter"
] | `IsLocalMinOn f s a` means that `f a ≤ f x` for all `x ∈ s` in some neighborhood of `a`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
IsLocalMaxOn | IsMaxFilter f (𝓝[s] a) a | def | IsLocalMaxOn | Topology.Order | Mathlib/Topology/Order/LocalExtr.lean | [] | [
"IsMaxFilter"
] | `IsLocalMaxOn f s a` means that `f x ≤ f a` for all `x ∈ s` in some neighborhood of `a`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
IsLocalExtrOn | IsExtrFilter f (𝓝[s] a) a | def | IsLocalExtrOn | Topology.Order | Mathlib/Topology/Order/LocalExtr.lean | [] | [
"IsExtrFilter"
] | `IsLocalExtrOn f s a` means `IsLocalMinOn f s a ∨ IsLocalMaxOn f s a`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
IsLocalMin | IsMinFilter f (𝓝 a) a | def | IsLocalMin | Topology.Order | Mathlib/Topology/Order/LocalExtr.lean | [] | [
"IsMinFilter"
] | `IsLocalMin f a` means that `f a ≤ f x` for all `x` in some neighborhood of `a`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
IsLocalMax | IsMaxFilter f (𝓝 a) a | def | IsLocalMax | Topology.Order | Mathlib/Topology/Order/LocalExtr.lean | [] | [
"IsMaxFilter"
] | `IsLocalMax f a` means that `f x ≤ f a` for all `x ∈ s` in some neighborhood of `a`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
IsLocalExtr | IsExtrFilter f (𝓝 a) a | def | IsLocalExtr | Topology.Order | Mathlib/Topology/Order/LocalExtr.lean | [] | [
"IsExtrFilter"
] | `IsLocalExtr f s a` means `IsLocalMin f s a ∨ IsLocalMax f s a`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
IsLocalExtrOn.elim {p : Prop} :
IsLocalExtrOn f s a → (IsLocalMinOn f s a → p) → (IsLocalMaxOn f s a → p) → p | Or.elim | theorem | IsLocalExtrOn.elim | Topology.Order | Mathlib/Topology/Order/LocalExtr.lean | [] | [
"IsLocalExtrOn",
"IsLocalMaxOn",
"IsLocalMinOn"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
IsLocalExtr.elim {p : Prop} :
IsLocalExtr f a → (IsLocalMin f a → p) → (IsLocalMax f a → p) → p | Or.elim | theorem | IsLocalExtr.elim | Topology.Order | Mathlib/Topology/Order/LocalExtr.lean | [] | [
"IsLocalExtr",
"IsLocalMax",
"IsLocalMin"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
IsLocalMin.on (h : IsLocalMin f a) (s) : IsLocalMinOn f s a | h.filter_inf _ | theorem | IsLocalMin.on | Topology.Order | Mathlib/Topology/Order/LocalExtr.lean | [] | [
"IsLocalMin",
"IsLocalMinOn"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
IsLocalMax.on (h : IsLocalMax f a) (s) : IsLocalMaxOn f s a | h.filter_inf _ | theorem | IsLocalMax.on | Topology.Order | Mathlib/Topology/Order/LocalExtr.lean | [] | [
"IsLocalMax",
"IsLocalMaxOn"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
IsLocalExtr.on (h : IsLocalExtr f a) (s) : IsLocalExtrOn f s a | h.filter_inf _ | theorem | IsLocalExtr.on | Topology.Order | Mathlib/Topology/Order/LocalExtr.lean | [] | [
"IsLocalExtr",
"IsLocalExtrOn"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
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 | IsLocalMinOn.on_subset | Topology.Order | Mathlib/Topology/Order/LocalExtr.lean | [] | [
"IsLocalMinOn",
"Set",
"nhdsWithin_mono"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
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 | IsLocalMaxOn.on_subset | Topology.Order | Mathlib/Topology/Order/LocalExtr.lean | [] | [
"IsLocalMaxOn",
"Set",
"nhdsWithin_mono"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
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 | IsLocalExtrOn.on_subset | Topology.Order | Mathlib/Topology/Order/LocalExtr.lean | [] | [
"IsLocalExtrOn",
"Set",
"nhdsWithin_mono"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
IsLocalMinOn.inter (hf : IsLocalMinOn f s a) (t) : IsLocalMinOn f (s ∩ t) a | hf.on_subset inter_subset_left | theorem | IsLocalMinOn.inter | Topology.Order | Mathlib/Topology/Order/LocalExtr.lean | [] | [
"IsLocalMinOn"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
IsLocalMaxOn.inter (hf : IsLocalMaxOn f s a) (t) : IsLocalMaxOn f (s ∩ t) a | hf.on_subset inter_subset_left | theorem | IsLocalMaxOn.inter | Topology.Order | Mathlib/Topology/Order/LocalExtr.lean | [] | [
"IsLocalMaxOn"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
IsLocalExtrOn.inter (hf : IsLocalExtrOn f s a) (t) : IsLocalExtrOn f (s ∩ t) a | hf.on_subset inter_subset_left | theorem | IsLocalExtrOn.inter | Topology.Order | Mathlib/Topology/Order/LocalExtr.lean | [] | [
"IsLocalExtrOn"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
IsMinOn.localize (hf : IsMinOn f s a) : IsLocalMinOn f s a | hf.filter_mono <| inf_le_right | theorem | IsMinOn.localize | Topology.Order | Mathlib/Topology/Order/LocalExtr.lean | [] | [
"IsLocalMinOn",
"IsMinOn"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
IsMaxOn.localize (hf : IsMaxOn f s a) : IsLocalMaxOn f s a | hf.filter_mono <| inf_le_right | theorem | IsMaxOn.localize | Topology.Order | Mathlib/Topology/Order/LocalExtr.lean | [] | [
"IsLocalMaxOn",
"IsMaxOn"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
IsExtrOn.localize (hf : IsExtrOn f s a) : IsLocalExtrOn f s a | hf.filter_mono <| inf_le_right | theorem | IsExtrOn.localize | Topology.Order | Mathlib/Topology/Order/LocalExtr.lean | [] | [
"IsExtrOn",
"IsLocalExtrOn"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
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 | IsLocalMinOn.isLocalMin | Topology.Order | Mathlib/Topology/Order/LocalExtr.lean | [] | [
"IsLocalMin",
"IsLocalMinOn",
"le_rfl"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
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 | IsLocalMaxOn.isLocalMax | Topology.Order | Mathlib/Topology/Order/LocalExtr.lean | [] | [
"IsLocalMax",
"IsLocalMaxOn",
"le_rfl"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
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 | IsLocalExtrOn.isLocalExtr | Topology.Order | Mathlib/Topology/Order/LocalExtr.lean | [] | [
"IsLocalExtr",
"IsLocalExtrOn"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
isLocalMinOn_univ_iff : IsLocalMinOn f univ a ↔ IsLocalMin f a | by
simp only [IsLocalMinOn, IsLocalMin, nhdsWithin_univ] | lemma | isLocalMinOn_univ_iff | Topology.Order | Mathlib/Topology/Order/LocalExtr.lean | [] | [
"IsLocalMin",
"IsLocalMinOn",
"nhdsWithin_univ"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
isLocalMaxOn_univ_iff : IsLocalMaxOn f univ a ↔ IsLocalMax f a | by
simp only [IsLocalMaxOn, IsLocalMax, nhdsWithin_univ] | lemma | isLocalMaxOn_univ_iff | Topology.Order | Mathlib/Topology/Order/LocalExtr.lean | [] | [
"IsLocalMax",
"IsLocalMaxOn",
"nhdsWithin_univ"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
isLocalExtrOn_univ_iff : IsLocalExtrOn f univ a ↔ IsLocalExtr f a | isLocalMinOn_univ_iff.or isLocalMaxOn_univ_iff | lemma | isLocalExtrOn_univ_iff | Topology.Order | Mathlib/Topology/Order/LocalExtr.lean | [] | [
"IsLocalExtr",
"IsLocalExtrOn",
"isLocalMaxOn_univ_iff"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
IsMinOn.isLocalMin (hf : IsMinOn f s a) (hs : s ∈ 𝓝 a) : IsLocalMin f a | hf.localize.isLocalMin hs | theorem | IsMinOn.isLocalMin | Topology.Order | Mathlib/Topology/Order/LocalExtr.lean | [] | [
"IsLocalMin",
"IsMinOn"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
IsMaxOn.isLocalMax (hf : IsMaxOn f s a) (hs : s ∈ 𝓝 a) : IsLocalMax f a | hf.localize.isLocalMax hs | theorem | IsMaxOn.isLocalMax | Topology.Order | Mathlib/Topology/Order/LocalExtr.lean | [] | [
"IsLocalMax",
"IsMaxOn"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
IsExtrOn.isLocalExtr (hf : IsExtrOn f s a) (hs : s ∈ 𝓝 a) : IsLocalExtr f a | hf.localize.isLocalExtr hs | theorem | IsExtrOn.isLocalExtr | Topology.Order | Mathlib/Topology/Order/LocalExtr.lean | [] | [
"IsExtrOn",
"IsLocalExtr"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
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 | IsLocalMinOn.not_nhds_le_map | Topology.Order | Mathlib/Topology/Order/LocalExtr.lean | [] | [
"IsLocalMinOn",
"TopologicalSpace",
"self_mem_nhdsWithin"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
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 | IsLocalMaxOn.not_nhds_le_map | Topology.Order | Mathlib/Topology/Order/LocalExtr.lean | [] | [
"IsLocalMaxOn",
"IsLocalMinOn.not_nhds_le_map",
"TopologicalSpace"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
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 | theorem | IsLocalExtrOn.not_nhds_le_map | Topology.Order | Mathlib/Topology/Order/LocalExtr.lean | [] | [
"IsLocalExtrOn",
"TopologicalSpace"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
isLocalMinOn_const {b : β} : IsLocalMinOn (fun _ => b) s a | isMinFilter_const | theorem | isLocalMinOn_const | Topology.Order | Mathlib/Topology/Order/LocalExtr.lean | [] | [
"IsLocalMinOn",
"isMinFilter_const"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
isLocalMaxOn_const {b : β} : IsLocalMaxOn (fun _ => b) s a | isMaxFilter_const | theorem | isLocalMaxOn_const | Topology.Order | Mathlib/Topology/Order/LocalExtr.lean | [] | [
"IsLocalMaxOn",
"isMaxFilter_const"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
isLocalExtrOn_const {b : β} : IsLocalExtrOn (fun _ => b) s a | isExtrFilter_const | theorem | isLocalExtrOn_const | Topology.Order | Mathlib/Topology/Order/LocalExtr.lean | [] | [
"IsLocalExtrOn",
"isExtrFilter_const"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
isLocalMin_const {b : β} : IsLocalMin (fun _ => b) a | isMinFilter_const | theorem | isLocalMin_const | Topology.Order | Mathlib/Topology/Order/LocalExtr.lean | [] | [
"IsLocalMin",
"isMinFilter_const"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
isLocalMax_const {b : β} : IsLocalMax (fun _ => b) a | isMaxFilter_const | theorem | isLocalMax_const | Topology.Order | Mathlib/Topology/Order/LocalExtr.lean | [] | [
"IsLocalMax",
"isMaxFilter_const"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
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