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 ⌀ |
|---|---|---|---|---|---|---|
liminf_le_liminf_of_le {α β} [ConditionallyCompleteLattice β] {f g : Filter α} (h : g ≤ f)
{u : α → β}
(hf : f.IsBoundedUnder (· ≥ ·) u := by isBoundedDefault)
(hg : g.IsCoboundedUnder (· ≥ ·) u := by isBoundedDefault) :
liminf u f ≤ liminf u g :=
limsInf_le_limsInf_of_le (map_mono h) hf hg | theorem | Order | [
"Mathlib.Order.ConditionallyCompleteLattice.Indexed",
"Mathlib.Order.Filter.IsBounded",
"Mathlib.Order.Hom.CompleteLattice"
] | Mathlib/Order/LiminfLimsup.lean | liminf_le_liminf_of_le | null |
limsSup_principal_eq_csSup (h : BddAbove s) (hs : s.Nonempty) : limsSup (𝓟 s) = sSup s := by
simp only [limsSup, eventually_principal]; exact csInf_upperBounds_eq_csSup h hs | lemma | Order | [
"Mathlib.Order.ConditionallyCompleteLattice.Indexed",
"Mathlib.Order.Filter.IsBounded",
"Mathlib.Order.Hom.CompleteLattice"
] | Mathlib/Order/LiminfLimsup.lean | limsSup_principal_eq_csSup | null |
limsInf_principal_eq_csSup (h : BddBelow s) (hs : s.Nonempty) : limsInf (𝓟 s) = sInf s :=
limsSup_principal_eq_csSup (α := αᵒᵈ) h hs | lemma | Order | [
"Mathlib.Order.ConditionallyCompleteLattice.Indexed",
"Mathlib.Order.Filter.IsBounded",
"Mathlib.Order.Hom.CompleteLattice"
] | Mathlib/Order/LiminfLimsup.lean | limsInf_principal_eq_csSup | null |
limsup_top_eq_ciSup [Nonempty β] (hu : BddAbove (range u)) : limsup u ⊤ = ⨆ i, u i := by
rw [limsup, map_top, limsSup_principal_eq_csSup hu (range_nonempty _), sSup_range] | lemma | Order | [
"Mathlib.Order.ConditionallyCompleteLattice.Indexed",
"Mathlib.Order.Filter.IsBounded",
"Mathlib.Order.Hom.CompleteLattice"
] | Mathlib/Order/LiminfLimsup.lean | limsup_top_eq_ciSup | null |
liminf_top_eq_ciInf [Nonempty β] (hu : BddBelow (range u)) : liminf u ⊤ = ⨅ i, u i := by
rw [liminf, map_top, limsInf_principal_eq_csSup hu (range_nonempty _), sInf_range] | lemma | Order | [
"Mathlib.Order.ConditionallyCompleteLattice.Indexed",
"Mathlib.Order.Filter.IsBounded",
"Mathlib.Order.Hom.CompleteLattice"
] | Mathlib/Order/LiminfLimsup.lean | liminf_top_eq_ciInf | null |
limsup_congr {α : Type*} [ConditionallyCompleteLattice β] {f : Filter α} {u v : α → β}
(h : ∀ᶠ a in f, u a = v a) : limsup u f = limsup v f := by
rw [limsup_eq]
congr with b
exact eventually_congr (h.mono fun x hx => by simp [hx]) | theorem | Order | [
"Mathlib.Order.ConditionallyCompleteLattice.Indexed",
"Mathlib.Order.Filter.IsBounded",
"Mathlib.Order.Hom.CompleteLattice"
] | Mathlib/Order/LiminfLimsup.lean | limsup_congr | null |
blimsup_congr {f : Filter β} {u v : β → α} {p : β → Prop} (h : ∀ᶠ a in f, p a → u a = v a) :
blimsup u f p = blimsup v f p := by
simpa only [blimsup_eq_limsup] using limsup_congr <| eventually_inf_principal.2 h | theorem | Order | [
"Mathlib.Order.ConditionallyCompleteLattice.Indexed",
"Mathlib.Order.Filter.IsBounded",
"Mathlib.Order.Hom.CompleteLattice"
] | Mathlib/Order/LiminfLimsup.lean | blimsup_congr | null |
bliminf_congr {f : Filter β} {u v : β → α} {p : β → Prop} (h : ∀ᶠ a in f, p a → u a = v a) :
bliminf u f p = bliminf v f p :=
blimsup_congr (α := αᵒᵈ) h | theorem | Order | [
"Mathlib.Order.ConditionallyCompleteLattice.Indexed",
"Mathlib.Order.Filter.IsBounded",
"Mathlib.Order.Hom.CompleteLattice"
] | Mathlib/Order/LiminfLimsup.lean | bliminf_congr | null |
liminf_congr {α : Type*} [ConditionallyCompleteLattice β] {f : Filter α} {u v : α → β}
(h : ∀ᶠ a in f, u a = v a) : liminf u f = liminf v f :=
limsup_congr (β := βᵒᵈ) h
@[simp] | theorem | Order | [
"Mathlib.Order.ConditionallyCompleteLattice.Indexed",
"Mathlib.Order.Filter.IsBounded",
"Mathlib.Order.Hom.CompleteLattice"
] | Mathlib/Order/LiminfLimsup.lean | liminf_congr | null |
limsup_const {α : Type*} [ConditionallyCompleteLattice β] {f : Filter α} [NeBot f]
(b : β) : limsup (fun _ => b) f = b := by
simpa only [limsup_eq, eventually_const] using csInf_Ici
@[simp] | theorem | Order | [
"Mathlib.Order.ConditionallyCompleteLattice.Indexed",
"Mathlib.Order.Filter.IsBounded",
"Mathlib.Order.Hom.CompleteLattice"
] | Mathlib/Order/LiminfLimsup.lean | limsup_const | null |
liminf_const {α : Type*} [ConditionallyCompleteLattice β] {f : Filter α} [NeBot f]
(b : β) : liminf (fun _ => b) f = b :=
limsup_const (β := βᵒᵈ) b | theorem | Order | [
"Mathlib.Order.ConditionallyCompleteLattice.Indexed",
"Mathlib.Order.Filter.IsBounded",
"Mathlib.Order.Hom.CompleteLattice"
] | Mathlib/Order/LiminfLimsup.lean | liminf_const | null |
HasBasis.liminf_eq_sSup_iUnion_iInter {ι ι' : Type*} {f : ι → α} {v : Filter ι}
{p : ι' → Prop} {s : ι' → Set ι} (hv : v.HasBasis p s) :
liminf f v = sSup (⋃ (j : Subtype p), ⋂ (i : s j), Iic (f i)) := by
simp_rw [liminf_eq, hv.eventually_iff]
congr 1
ext x
simp only [mem_setOf_eq, iInter_coe_set, mem_i... | theorem | Order | [
"Mathlib.Order.ConditionallyCompleteLattice.Indexed",
"Mathlib.Order.Filter.IsBounded",
"Mathlib.Order.Hom.CompleteLattice"
] | Mathlib/Order/LiminfLimsup.lean | HasBasis.liminf_eq_sSup_iUnion_iInter | null |
HasBasis.liminf_eq_sSup_univ_of_empty {f : ι → α} {v : Filter ι}
{p : ι' → Prop} {s : ι' → Set ι} (hv : v.HasBasis p s) (i : ι') (hi : p i) (h'i : s i = ∅) :
liminf f v = sSup univ := by
simp [hv.eq_bot_iff.2 ⟨i, hi, h'i⟩, liminf_eq] | theorem | Order | [
"Mathlib.Order.ConditionallyCompleteLattice.Indexed",
"Mathlib.Order.Filter.IsBounded",
"Mathlib.Order.Hom.CompleteLattice"
] | Mathlib/Order/LiminfLimsup.lean | HasBasis.liminf_eq_sSup_univ_of_empty | null |
HasBasis.limsup_eq_sInf_iUnion_iInter {ι ι' : Type*} {f : ι → α} {v : Filter ι}
{p : ι' → Prop} {s : ι' → Set ι} (hv : v.HasBasis p s) :
limsup f v = sInf (⋃ (j : Subtype p), ⋂ (i : s j), Ici (f i)) :=
HasBasis.liminf_eq_sSup_iUnion_iInter (α := αᵒᵈ) hv | theorem | Order | [
"Mathlib.Order.ConditionallyCompleteLattice.Indexed",
"Mathlib.Order.Filter.IsBounded",
"Mathlib.Order.Hom.CompleteLattice"
] | Mathlib/Order/LiminfLimsup.lean | HasBasis.limsup_eq_sInf_iUnion_iInter | null |
HasBasis.limsup_eq_sInf_univ_of_empty {f : ι → α} {v : Filter ι}
{p : ι' → Prop} {s : ι' → Set ι} (hv : v.HasBasis p s) (i : ι') (hi : p i) (h'i : s i = ∅) :
limsup f v = sInf univ :=
HasBasis.liminf_eq_sSup_univ_of_empty (α := αᵒᵈ) hv i hi h'i
@[simp] | theorem | Order | [
"Mathlib.Order.ConditionallyCompleteLattice.Indexed",
"Mathlib.Order.Filter.IsBounded",
"Mathlib.Order.Hom.CompleteLattice"
] | Mathlib/Order/LiminfLimsup.lean | HasBasis.limsup_eq_sInf_univ_of_empty | null |
liminf_nat_add (f : ℕ → α) (k : ℕ) :
liminf (fun i => f (i + k)) atTop = liminf f atTop := by
rw [← Function.comp_def, liminf, liminf, ← map_map, map_add_atTop_eq_nat]
@[simp] | theorem | Order | [
"Mathlib.Order.ConditionallyCompleteLattice.Indexed",
"Mathlib.Order.Filter.IsBounded",
"Mathlib.Order.Hom.CompleteLattice"
] | Mathlib/Order/LiminfLimsup.lean | liminf_nat_add | null |
limsup_nat_add (f : ℕ → α) (k : ℕ) : limsup (fun i => f (i + k)) atTop = limsup f atTop :=
@liminf_nat_add αᵒᵈ _ f k | theorem | Order | [
"Mathlib.Order.ConditionallyCompleteLattice.Indexed",
"Mathlib.Order.Filter.IsBounded",
"Mathlib.Order.Hom.CompleteLattice"
] | Mathlib/Order/LiminfLimsup.lean | limsup_nat_add | null |
@[simp]
limsSup_bot : limsSup (⊥ : Filter α) = ⊥ :=
bot_unique <| sInf_le <| by simp
@[simp] theorem limsup_bot (f : β → α) : limsup f ⊥ = ⊥ := by simp [limsup]
@[simp] | theorem | Order | [
"Mathlib.Order.ConditionallyCompleteLattice.Indexed",
"Mathlib.Order.Filter.IsBounded",
"Mathlib.Order.Hom.CompleteLattice"
] | Mathlib/Order/LiminfLimsup.lean | limsSup_bot | null |
limsInf_bot : limsInf (⊥ : Filter α) = ⊤ :=
top_unique <| le_sSup <| by simp
@[simp] theorem liminf_bot (f : β → α) : liminf f ⊥ = ⊤ := by simp [liminf]
@[simp] | theorem | Order | [
"Mathlib.Order.ConditionallyCompleteLattice.Indexed",
"Mathlib.Order.Filter.IsBounded",
"Mathlib.Order.Hom.CompleteLattice"
] | Mathlib/Order/LiminfLimsup.lean | limsInf_bot | null |
limsSup_top : limsSup (⊤ : Filter α) = ⊤ :=
top_unique <| le_sInf <| by simpa [eq_univ_iff_forall] using fun b hb => top_unique <| hb _
@[simp] | theorem | Order | [
"Mathlib.Order.ConditionallyCompleteLattice.Indexed",
"Mathlib.Order.Filter.IsBounded",
"Mathlib.Order.Hom.CompleteLattice"
] | Mathlib/Order/LiminfLimsup.lean | limsSup_top | null |
limsInf_top : limsInf (⊤ : Filter α) = ⊥ :=
bot_unique <| sSup_le <| by simpa [eq_univ_iff_forall] using fun b hb => bot_unique <| hb _
@[simp] | theorem | Order | [
"Mathlib.Order.ConditionallyCompleteLattice.Indexed",
"Mathlib.Order.Filter.IsBounded",
"Mathlib.Order.Hom.CompleteLattice"
] | Mathlib/Order/LiminfLimsup.lean | limsInf_top | null |
blimsup_false {f : Filter β} {u : β → α} : (blimsup u f fun _ => False) = ⊥ := by
simp [blimsup_eq]
@[simp] | theorem | Order | [
"Mathlib.Order.ConditionallyCompleteLattice.Indexed",
"Mathlib.Order.Filter.IsBounded",
"Mathlib.Order.Hom.CompleteLattice"
] | Mathlib/Order/LiminfLimsup.lean | blimsup_false | null |
bliminf_false {f : Filter β} {u : β → α} : (bliminf u f fun _ => False) = ⊤ := by
simp [bliminf_eq] | theorem | Order | [
"Mathlib.Order.ConditionallyCompleteLattice.Indexed",
"Mathlib.Order.Filter.IsBounded",
"Mathlib.Order.Hom.CompleteLattice"
] | Mathlib/Order/LiminfLimsup.lean | bliminf_false | null |
@[simp]
limsup_const_bot {f : Filter β} : limsup (fun _ : β => (⊥ : α)) f = (⊥ : α) := by
rw [limsup_eq, eq_bot_iff]
exact sInf_le (Eventually.of_forall fun _ => le_rfl) | theorem | Order | [
"Mathlib.Order.ConditionallyCompleteLattice.Indexed",
"Mathlib.Order.Filter.IsBounded",
"Mathlib.Order.Hom.CompleteLattice"
] | Mathlib/Order/LiminfLimsup.lean | limsup_const_bot | Same as limsup_const applied to `⊥` but without the `NeBot f` assumption |
@[simp]
liminf_const_top {f : Filter β} : liminf (fun _ : β => (⊤ : α)) f = (⊤ : α) :=
limsup_const_bot (α := αᵒᵈ) | theorem | Order | [
"Mathlib.Order.ConditionallyCompleteLattice.Indexed",
"Mathlib.Order.Filter.IsBounded",
"Mathlib.Order.Hom.CompleteLattice"
] | Mathlib/Order/LiminfLimsup.lean | liminf_const_top | Same as limsup_const applied to `⊤` but without the `NeBot f` assumption |
HasBasis.limsSup_eq_iInf_sSup {ι} {p : ι → Prop} {s} {f : Filter α} (h : f.HasBasis p s) :
limsSup f = ⨅ (i) (_ : p i), sSup (s i) :=
le_antisymm (le_iInf₂ fun i hi => sInf_le <| h.eventually_iff.2 ⟨i, hi, fun _ => le_sSup⟩)
(le_sInf fun _ ha =>
let ⟨_, hi, ha⟩ := h.eventually_iff.1 ha
iInf₂_le_of... | theorem | Order | [
"Mathlib.Order.ConditionallyCompleteLattice.Indexed",
"Mathlib.Order.Filter.IsBounded",
"Mathlib.Order.Hom.CompleteLattice"
] | Mathlib/Order/LiminfLimsup.lean | HasBasis.limsSup_eq_iInf_sSup | null |
HasBasis.limsInf_eq_iSup_sInf {p : ι → Prop} {s : ι → Set α} {f : Filter α}
(h : f.HasBasis p s) : limsInf f = ⨆ (i) (_ : p i), sInf (s i) :=
HasBasis.limsSup_eq_iInf_sSup (α := αᵒᵈ) h | theorem | Order | [
"Mathlib.Order.ConditionallyCompleteLattice.Indexed",
"Mathlib.Order.Filter.IsBounded",
"Mathlib.Order.Hom.CompleteLattice"
] | Mathlib/Order/LiminfLimsup.lean | HasBasis.limsInf_eq_iSup_sInf | null |
limsSup_eq_iInf_sSup {f : Filter α} : limsSup f = ⨅ s ∈ f, sSup s :=
f.basis_sets.limsSup_eq_iInf_sSup | theorem | Order | [
"Mathlib.Order.ConditionallyCompleteLattice.Indexed",
"Mathlib.Order.Filter.IsBounded",
"Mathlib.Order.Hom.CompleteLattice"
] | Mathlib/Order/LiminfLimsup.lean | limsSup_eq_iInf_sSup | null |
limsInf_eq_iSup_sInf {f : Filter α} : limsInf f = ⨆ s ∈ f, sInf s :=
limsSup_eq_iInf_sSup (α := αᵒᵈ) | theorem | Order | [
"Mathlib.Order.ConditionallyCompleteLattice.Indexed",
"Mathlib.Order.Filter.IsBounded",
"Mathlib.Order.Hom.CompleteLattice"
] | Mathlib/Order/LiminfLimsup.lean | limsInf_eq_iSup_sInf | null |
limsup_le_iSup {f : Filter β} {u : β → α} : limsup u f ≤ ⨆ n, u n :=
limsup_le_of_le (by isBoundedDefault) (Eventually.of_forall (le_iSup u)) | theorem | Order | [
"Mathlib.Order.ConditionallyCompleteLattice.Indexed",
"Mathlib.Order.Filter.IsBounded",
"Mathlib.Order.Hom.CompleteLattice"
] | Mathlib/Order/LiminfLimsup.lean | limsup_le_iSup | null |
iInf_le_liminf {f : Filter β} {u : β → α} : ⨅ n, u n ≤ liminf u f :=
le_liminf_of_le (by isBoundedDefault) (Eventually.of_forall (iInf_le u)) | theorem | Order | [
"Mathlib.Order.ConditionallyCompleteLattice.Indexed",
"Mathlib.Order.Filter.IsBounded",
"Mathlib.Order.Hom.CompleteLattice"
] | Mathlib/Order/LiminfLimsup.lean | iInf_le_liminf | null |
limsup_eq_iInf_iSup {f : Filter β} {u : β → α} : limsup u f = ⨅ s ∈ f, ⨆ a ∈ s, u a :=
(f.basis_sets.map u).limsSup_eq_iInf_sSup.trans <| by simp only [sSup_image, id] | theorem | Order | [
"Mathlib.Order.ConditionallyCompleteLattice.Indexed",
"Mathlib.Order.Filter.IsBounded",
"Mathlib.Order.Hom.CompleteLattice"
] | Mathlib/Order/LiminfLimsup.lean | limsup_eq_iInf_iSup | In a complete lattice, the limsup of a function is the infimum over sets `s` in the filter
of the supremum of the function over `s` |
limsup_eq_iInf_iSup_of_nat {u : ℕ → α} : limsup u atTop = ⨅ n : ℕ, ⨆ i ≥ n, u i :=
(atTop_basis.map u).limsSup_eq_iInf_sSup.trans <| by simp only [sSup_image, iInf_const]; rfl | theorem | Order | [
"Mathlib.Order.ConditionallyCompleteLattice.Indexed",
"Mathlib.Order.Filter.IsBounded",
"Mathlib.Order.Hom.CompleteLattice"
] | Mathlib/Order/LiminfLimsup.lean | limsup_eq_iInf_iSup_of_nat | null |
limsup_eq_iInf_iSup_of_nat' {u : ℕ → α} : limsup u atTop = ⨅ n : ℕ, ⨆ i : ℕ, u (i + n) := by
simp only [limsup_eq_iInf_iSup_of_nat, iSup_ge_eq_iSup_nat_add] | theorem | Order | [
"Mathlib.Order.ConditionallyCompleteLattice.Indexed",
"Mathlib.Order.Filter.IsBounded",
"Mathlib.Order.Hom.CompleteLattice"
] | Mathlib/Order/LiminfLimsup.lean | limsup_eq_iInf_iSup_of_nat' | null |
HasBasis.limsup_eq_iInf_iSup {p : ι → Prop} {s : ι → Set β} {f : Filter β} {u : β → α}
(h : f.HasBasis p s) : limsup u f = ⨅ (i) (_ : p i), ⨆ a ∈ s i, u a :=
(h.map u).limsSup_eq_iInf_sSup.trans <| by simp only [sSup_image] | theorem | Order | [
"Mathlib.Order.ConditionallyCompleteLattice.Indexed",
"Mathlib.Order.Filter.IsBounded",
"Mathlib.Order.Hom.CompleteLattice"
] | Mathlib/Order/LiminfLimsup.lean | HasBasis.limsup_eq_iInf_iSup | null |
limsSup_principal_eq_sSup (s : Set α) : limsSup (𝓟 s) = sSup s := by
simpa only [limsSup, eventually_principal] using sInf_upperBounds_eq_csSup s | lemma | Order | [
"Mathlib.Order.ConditionallyCompleteLattice.Indexed",
"Mathlib.Order.Filter.IsBounded",
"Mathlib.Order.Hom.CompleteLattice"
] | Mathlib/Order/LiminfLimsup.lean | limsSup_principal_eq_sSup | null |
limsInf_principal_eq_sInf (s : Set α) : limsInf (𝓟 s) = sInf s := by
simpa only [limsInf, eventually_principal] using sSup_lowerBounds_eq_sInf s
@[simp] lemma limsup_top_eq_iSup (u : β → α) : limsup u ⊤ = ⨆ i, u i := by
rw [limsup, map_top, limsSup_principal_eq_sSup, sSup_range]
@[simp] lemma liminf_top_eq_iInf (u... | lemma | Order | [
"Mathlib.Order.ConditionallyCompleteLattice.Indexed",
"Mathlib.Order.Filter.IsBounded",
"Mathlib.Order.Hom.CompleteLattice"
] | Mathlib/Order/LiminfLimsup.lean | limsInf_principal_eq_sInf | null |
blimsup_congr' {f : Filter β} {p q : β → Prop} {u : β → α}
(h : ∀ᶠ x in f, u x ≠ ⊥ → (p x ↔ q x)) : blimsup u f p = blimsup u f q := by
simp only [blimsup_eq]
congr with a
refine eventually_congr (h.mono fun b hb => ?_)
rcases eq_or_ne (u b) ⊥ with hu | hu; · simp [hu]
rw [hb hu] | theorem | Order | [
"Mathlib.Order.ConditionallyCompleteLattice.Indexed",
"Mathlib.Order.Filter.IsBounded",
"Mathlib.Order.Hom.CompleteLattice"
] | Mathlib/Order/LiminfLimsup.lean | blimsup_congr' | null |
bliminf_congr' {f : Filter β} {p q : β → Prop} {u : β → α}
(h : ∀ᶠ x in f, u x ≠ ⊤ → (p x ↔ q x)) : bliminf u f p = bliminf u f q :=
blimsup_congr' (α := αᵒᵈ) h | theorem | Order | [
"Mathlib.Order.ConditionallyCompleteLattice.Indexed",
"Mathlib.Order.Filter.IsBounded",
"Mathlib.Order.Hom.CompleteLattice"
] | Mathlib/Order/LiminfLimsup.lean | bliminf_congr' | null |
HasBasis.blimsup_eq_iInf_iSup {p : ι → Prop} {s : ι → Set β} {f : Filter β} {u : β → α}
(hf : f.HasBasis p s) {q : β → Prop} :
blimsup u f q = ⨅ (i) (_ : p i), ⨆ a ∈ s i, ⨆ (_ : q a), u a := by
simp only [blimsup_eq_limsup, (hf.inf_principal _).limsup_eq_iInf_iSup, mem_inter_iff, iSup_and,
mem_setOf_eq] | lemma | Order | [
"Mathlib.Order.ConditionallyCompleteLattice.Indexed",
"Mathlib.Order.Filter.IsBounded",
"Mathlib.Order.Hom.CompleteLattice"
] | Mathlib/Order/LiminfLimsup.lean | HasBasis.blimsup_eq_iInf_iSup | null |
blimsup_eq_iInf_biSup {f : Filter β} {p : β → Prop} {u : β → α} :
blimsup u f p = ⨅ s ∈ f, ⨆ (b) (_ : p b ∧ b ∈ s), u b := by
simp only [f.basis_sets.blimsup_eq_iInf_iSup, iSup_and', id, and_comm] | theorem | Order | [
"Mathlib.Order.ConditionallyCompleteLattice.Indexed",
"Mathlib.Order.Filter.IsBounded",
"Mathlib.Order.Hom.CompleteLattice"
] | Mathlib/Order/LiminfLimsup.lean | blimsup_eq_iInf_biSup | null |
blimsup_eq_iInf_biSup_of_nat {p : ℕ → Prop} {u : ℕ → α} :
blimsup u atTop p = ⨅ i, ⨆ (j) (_ : p j ∧ i ≤ j), u j := by
simp only [atTop_basis.blimsup_eq_iInf_iSup, @and_comm (p _), iSup_and, mem_Ici, iInf_true] | theorem | Order | [
"Mathlib.Order.ConditionallyCompleteLattice.Indexed",
"Mathlib.Order.Filter.IsBounded",
"Mathlib.Order.Hom.CompleteLattice"
] | Mathlib/Order/LiminfLimsup.lean | blimsup_eq_iInf_biSup_of_nat | null |
liminf_eq_iSup_iInf {f : Filter β} {u : β → α} : liminf u f = ⨆ s ∈ f, ⨅ a ∈ s, u a :=
limsup_eq_iInf_iSup (α := αᵒᵈ) | theorem | Order | [
"Mathlib.Order.ConditionallyCompleteLattice.Indexed",
"Mathlib.Order.Filter.IsBounded",
"Mathlib.Order.Hom.CompleteLattice"
] | Mathlib/Order/LiminfLimsup.lean | liminf_eq_iSup_iInf | In a complete lattice, the liminf of a function is the infimum over sets `s` in the filter
of the supremum of the function over `s` |
liminf_eq_iSup_iInf_of_nat {u : ℕ → α} : liminf u atTop = ⨆ n : ℕ, ⨅ i ≥ n, u i :=
@limsup_eq_iInf_iSup_of_nat αᵒᵈ _ u | theorem | Order | [
"Mathlib.Order.ConditionallyCompleteLattice.Indexed",
"Mathlib.Order.Filter.IsBounded",
"Mathlib.Order.Hom.CompleteLattice"
] | Mathlib/Order/LiminfLimsup.lean | liminf_eq_iSup_iInf_of_nat | null |
liminf_eq_iSup_iInf_of_nat' {u : ℕ → α} : liminf u atTop = ⨆ n : ℕ, ⨅ i : ℕ, u (i + n) :=
@limsup_eq_iInf_iSup_of_nat' αᵒᵈ _ _ | theorem | Order | [
"Mathlib.Order.ConditionallyCompleteLattice.Indexed",
"Mathlib.Order.Filter.IsBounded",
"Mathlib.Order.Hom.CompleteLattice"
] | Mathlib/Order/LiminfLimsup.lean | liminf_eq_iSup_iInf_of_nat' | null |
HasBasis.liminf_eq_iSup_iInf {p : ι → Prop} {s : ι → Set β} {f : Filter β} {u : β → α}
(h : f.HasBasis p s) : liminf u f = ⨆ (i) (_ : p i), ⨅ a ∈ s i, u a :=
HasBasis.limsup_eq_iInf_iSup (α := αᵒᵈ) h | theorem | Order | [
"Mathlib.Order.ConditionallyCompleteLattice.Indexed",
"Mathlib.Order.Filter.IsBounded",
"Mathlib.Order.Hom.CompleteLattice"
] | Mathlib/Order/LiminfLimsup.lean | HasBasis.liminf_eq_iSup_iInf | null |
bliminf_eq_iSup_biInf {f : Filter β} {p : β → Prop} {u : β → α} :
bliminf u f p = ⨆ s ∈ f, ⨅ (b) (_ : p b ∧ b ∈ s), u b :=
@blimsup_eq_iInf_biSup αᵒᵈ β _ f p u | theorem | Order | [
"Mathlib.Order.ConditionallyCompleteLattice.Indexed",
"Mathlib.Order.Filter.IsBounded",
"Mathlib.Order.Hom.CompleteLattice"
] | Mathlib/Order/LiminfLimsup.lean | bliminf_eq_iSup_biInf | null |
bliminf_eq_iSup_biInf_of_nat {p : ℕ → Prop} {u : ℕ → α} :
bliminf u atTop p = ⨆ i, ⨅ (j) (_ : p j ∧ i ≤ j), u j :=
@blimsup_eq_iInf_biSup_of_nat αᵒᵈ _ p u | theorem | Order | [
"Mathlib.Order.ConditionallyCompleteLattice.Indexed",
"Mathlib.Order.Filter.IsBounded",
"Mathlib.Order.Hom.CompleteLattice"
] | Mathlib/Order/LiminfLimsup.lean | bliminf_eq_iSup_biInf_of_nat | null |
limsup_eq_sInf_sSup {ι R : Type*} (F : Filter ι) [CompleteLattice R] (a : ι → R) :
limsup a F = sInf ((fun I => sSup (a '' I)) '' F.sets) := by
apply le_antisymm
· rw [limsup_eq]
refine sInf_le_sInf fun x hx => ?_
rcases (mem_image _ F.sets x).mp hx with ⟨I, ⟨I_mem_F, hI⟩⟩
filter_upwards [I_mem_F] w... | theorem | Order | [
"Mathlib.Order.ConditionallyCompleteLattice.Indexed",
"Mathlib.Order.Filter.IsBounded",
"Mathlib.Order.Hom.CompleteLattice"
] | Mathlib/Order/LiminfLimsup.lean | limsup_eq_sInf_sSup | null |
liminf_eq_sSup_sInf {ι R : Type*} (F : Filter ι) [CompleteLattice R] (a : ι → R) :
liminf a F = sSup ((fun I => sInf (a '' I)) '' F.sets) :=
@Filter.limsup_eq_sInf_sSup ι (OrderDual R) _ _ a | theorem | Order | [
"Mathlib.Order.ConditionallyCompleteLattice.Indexed",
"Mathlib.Order.Filter.IsBounded",
"Mathlib.Order.Hom.CompleteLattice"
] | Mathlib/Order/LiminfLimsup.lean | liminf_eq_sSup_sInf | null |
liminf_le_of_frequently_le' {α β} [CompleteLattice β] {f : Filter α} {u : α → β} {x : β}
(h : ∃ᶠ a in f, u a ≤ x) : liminf u f ≤ x := by
rw [liminf_eq]
refine sSup_le fun b hb => ?_
have hbx : ∃ᶠ _ in f, b ≤ x := by
revert h
rw [← not_imp_not, not_frequently, not_frequently]
exact fun h => hb.mp (... | theorem | Order | [
"Mathlib.Order.ConditionallyCompleteLattice.Indexed",
"Mathlib.Order.Filter.IsBounded",
"Mathlib.Order.Hom.CompleteLattice"
] | Mathlib/Order/LiminfLimsup.lean | liminf_le_of_frequently_le' | null |
le_limsup_of_frequently_le' {α β} [CompleteLattice β] {f : Filter α} {u : α → β} {x : β}
(h : ∃ᶠ a in f, x ≤ u a) : x ≤ limsup u f :=
liminf_le_of_frequently_le' (β := βᵒᵈ) h | theorem | Order | [
"Mathlib.Order.ConditionallyCompleteLattice.Indexed",
"Mathlib.Order.Filter.IsBounded",
"Mathlib.Order.Hom.CompleteLattice"
] | Mathlib/Order/LiminfLimsup.lean | le_limsup_of_frequently_le' | null |
@[simp]
_root_.CompleteLatticeHom.apply_limsup_iterate (f : CompleteLatticeHom α α) (a : α) :
f (limsup (fun n => f^[n] a) atTop) = limsup (fun n => f^[n] a) atTop := by
rw [limsup_eq_iInf_iSup_of_nat', map_iInf]
simp_rw [_root_.map_iSup, ← Function.comp_apply (f := f), ← Function.iterate_succ' f,
← Nat.add... | theorem | Order | [
"Mathlib.Order.ConditionallyCompleteLattice.Indexed",
"Mathlib.Order.Filter.IsBounded",
"Mathlib.Order.Hom.CompleteLattice"
] | Mathlib/Order/LiminfLimsup.lean | _root_.CompleteLatticeHom.apply_limsup_iterate | If `f : α → α` is a morphism of complete lattices, then the limsup of its iterates of any
`a : α` is a fixed point. |
_root_.CompleteLatticeHom.apply_liminf_iterate (f : CompleteLatticeHom α α) (a : α) :
f (liminf (fun n => f^[n] a) atTop) = liminf (fun n => f^[n] a) atTop :=
(CompleteLatticeHom.dual f).apply_limsup_iterate _
variable {f g : Filter β} {p q : β → Prop} {u v : β → α} | theorem | Order | [
"Mathlib.Order.ConditionallyCompleteLattice.Indexed",
"Mathlib.Order.Filter.IsBounded",
"Mathlib.Order.Hom.CompleteLattice"
] | Mathlib/Order/LiminfLimsup.lean | _root_.CompleteLatticeHom.apply_liminf_iterate | If `f : α → α` is a morphism of complete lattices, then the liminf of its iterates of any
`a : α` is a fixed point. |
blimsup_mono (h : ∀ x, p x → q x) : blimsup u f p ≤ blimsup u f q :=
sInf_le_sInf fun a ha => ha.mono <| by tauto | theorem | Order | [
"Mathlib.Order.ConditionallyCompleteLattice.Indexed",
"Mathlib.Order.Filter.IsBounded",
"Mathlib.Order.Hom.CompleteLattice"
] | Mathlib/Order/LiminfLimsup.lean | blimsup_mono | null |
bliminf_antitone (h : ∀ x, p x → q x) : bliminf u f q ≤ bliminf u f p :=
sSup_le_sSup fun a ha => ha.mono <| by tauto | theorem | Order | [
"Mathlib.Order.ConditionallyCompleteLattice.Indexed",
"Mathlib.Order.Filter.IsBounded",
"Mathlib.Order.Hom.CompleteLattice"
] | Mathlib/Order/LiminfLimsup.lean | bliminf_antitone | null |
mono_blimsup' (h : ∀ᶠ x in f, p x → u x ≤ v x) : blimsup u f p ≤ blimsup v f p :=
sInf_le_sInf fun _ ha => (ha.and h).mono fun _ hx hx' => (hx.2 hx').trans (hx.1 hx') | theorem | Order | [
"Mathlib.Order.ConditionallyCompleteLattice.Indexed",
"Mathlib.Order.Filter.IsBounded",
"Mathlib.Order.Hom.CompleteLattice"
] | Mathlib/Order/LiminfLimsup.lean | mono_blimsup' | null |
mono_blimsup (h : ∀ x, p x → u x ≤ v x) : blimsup u f p ≤ blimsup v f p :=
mono_blimsup' <| Eventually.of_forall h | theorem | Order | [
"Mathlib.Order.ConditionallyCompleteLattice.Indexed",
"Mathlib.Order.Filter.IsBounded",
"Mathlib.Order.Hom.CompleteLattice"
] | Mathlib/Order/LiminfLimsup.lean | mono_blimsup | null |
mono_bliminf' (h : ∀ᶠ x in f, p x → u x ≤ v x) : bliminf u f p ≤ bliminf v f p :=
sSup_le_sSup fun _ ha => (ha.and h).mono fun _ hx hx' => (hx.1 hx').trans (hx.2 hx') | theorem | Order | [
"Mathlib.Order.ConditionallyCompleteLattice.Indexed",
"Mathlib.Order.Filter.IsBounded",
"Mathlib.Order.Hom.CompleteLattice"
] | Mathlib/Order/LiminfLimsup.lean | mono_bliminf' | null |
mono_bliminf (h : ∀ x, p x → u x ≤ v x) : bliminf u f p ≤ bliminf v f p :=
mono_bliminf' <| Eventually.of_forall h | theorem | Order | [
"Mathlib.Order.ConditionallyCompleteLattice.Indexed",
"Mathlib.Order.Filter.IsBounded",
"Mathlib.Order.Hom.CompleteLattice"
] | Mathlib/Order/LiminfLimsup.lean | mono_bliminf | null |
bliminf_antitone_filter (h : f ≤ g) : bliminf u g p ≤ bliminf u f p :=
sSup_le_sSup fun _ ha => ha.filter_mono h | theorem | Order | [
"Mathlib.Order.ConditionallyCompleteLattice.Indexed",
"Mathlib.Order.Filter.IsBounded",
"Mathlib.Order.Hom.CompleteLattice"
] | Mathlib/Order/LiminfLimsup.lean | bliminf_antitone_filter | null |
blimsup_monotone_filter (h : f ≤ g) : blimsup u f p ≤ blimsup u g p :=
sInf_le_sInf fun _ ha => ha.filter_mono h | theorem | Order | [
"Mathlib.Order.ConditionallyCompleteLattice.Indexed",
"Mathlib.Order.Filter.IsBounded",
"Mathlib.Order.Hom.CompleteLattice"
] | Mathlib/Order/LiminfLimsup.lean | blimsup_monotone_filter | null |
blimsup_and_le_inf : (blimsup u f fun x => p x ∧ q x) ≤ blimsup u f p ⊓ blimsup u f q :=
le_inf (blimsup_mono <| by tauto) (blimsup_mono <| by tauto)
@[simp] | theorem | Order | [
"Mathlib.Order.ConditionallyCompleteLattice.Indexed",
"Mathlib.Order.Filter.IsBounded",
"Mathlib.Order.Hom.CompleteLattice"
] | Mathlib/Order/LiminfLimsup.lean | blimsup_and_le_inf | null |
bliminf_sup_le_inf_aux_left :
(blimsup u f fun x => p x ∧ q x) ≤ blimsup u f p :=
blimsup_and_le_inf.trans inf_le_left
@[simp] | theorem | Order | [
"Mathlib.Order.ConditionallyCompleteLattice.Indexed",
"Mathlib.Order.Filter.IsBounded",
"Mathlib.Order.Hom.CompleteLattice"
] | Mathlib/Order/LiminfLimsup.lean | bliminf_sup_le_inf_aux_left | null |
bliminf_sup_le_inf_aux_right :
(blimsup u f fun x => p x ∧ q x) ≤ blimsup u f q :=
blimsup_and_le_inf.trans inf_le_right | theorem | Order | [
"Mathlib.Order.ConditionallyCompleteLattice.Indexed",
"Mathlib.Order.Filter.IsBounded",
"Mathlib.Order.Hom.CompleteLattice"
] | Mathlib/Order/LiminfLimsup.lean | bliminf_sup_le_inf_aux_right | null |
bliminf_sup_le_and : bliminf u f p ⊔ bliminf u f q ≤ bliminf u f fun x => p x ∧ q x :=
blimsup_and_le_inf (α := αᵒᵈ)
@[simp] | theorem | Order | [
"Mathlib.Order.ConditionallyCompleteLattice.Indexed",
"Mathlib.Order.Filter.IsBounded",
"Mathlib.Order.Hom.CompleteLattice"
] | Mathlib/Order/LiminfLimsup.lean | bliminf_sup_le_and | null |
bliminf_sup_le_and_aux_left : bliminf u f p ≤ bliminf u f fun x => p x ∧ q x :=
le_sup_left.trans bliminf_sup_le_and
@[simp] | theorem | Order | [
"Mathlib.Order.ConditionallyCompleteLattice.Indexed",
"Mathlib.Order.Filter.IsBounded",
"Mathlib.Order.Hom.CompleteLattice"
] | Mathlib/Order/LiminfLimsup.lean | bliminf_sup_le_and_aux_left | null |
bliminf_sup_le_and_aux_right : bliminf u f q ≤ bliminf u f fun x => p x ∧ q x :=
le_sup_right.trans bliminf_sup_le_and | theorem | Order | [
"Mathlib.Order.ConditionallyCompleteLattice.Indexed",
"Mathlib.Order.Filter.IsBounded",
"Mathlib.Order.Hom.CompleteLattice"
] | Mathlib/Order/LiminfLimsup.lean | bliminf_sup_le_and_aux_right | null |
blimsup_sup_le_or : blimsup u f p ⊔ blimsup u f q ≤ blimsup u f fun x => p x ∨ q x :=
sup_le (blimsup_mono <| by tauto) (blimsup_mono <| by tauto)
@[simp] | theorem | Order | [
"Mathlib.Order.ConditionallyCompleteLattice.Indexed",
"Mathlib.Order.Filter.IsBounded",
"Mathlib.Order.Hom.CompleteLattice"
] | Mathlib/Order/LiminfLimsup.lean | blimsup_sup_le_or | See also `Filter.blimsup_or_eq_sup`. |
bliminf_sup_le_or_aux_left : blimsup u f p ≤ blimsup u f fun x => p x ∨ q x :=
le_sup_left.trans blimsup_sup_le_or
@[simp] | theorem | Order | [
"Mathlib.Order.ConditionallyCompleteLattice.Indexed",
"Mathlib.Order.Filter.IsBounded",
"Mathlib.Order.Hom.CompleteLattice"
] | Mathlib/Order/LiminfLimsup.lean | bliminf_sup_le_or_aux_left | null |
bliminf_sup_le_or_aux_right : blimsup u f q ≤ blimsup u f fun x => p x ∨ q x :=
le_sup_right.trans blimsup_sup_le_or | theorem | Order | [
"Mathlib.Order.ConditionallyCompleteLattice.Indexed",
"Mathlib.Order.Filter.IsBounded",
"Mathlib.Order.Hom.CompleteLattice"
] | Mathlib/Order/LiminfLimsup.lean | bliminf_sup_le_or_aux_right | null |
bliminf_or_le_inf : (bliminf u f fun x => p x ∨ q x) ≤ bliminf u f p ⊓ bliminf u f q :=
blimsup_sup_le_or (α := αᵒᵈ)
@[simp] | theorem | Order | [
"Mathlib.Order.ConditionallyCompleteLattice.Indexed",
"Mathlib.Order.Filter.IsBounded",
"Mathlib.Order.Hom.CompleteLattice"
] | Mathlib/Order/LiminfLimsup.lean | bliminf_or_le_inf | See also `Filter.bliminf_or_eq_inf`. |
bliminf_or_le_inf_aux_left : (bliminf u f fun x => p x ∨ q x) ≤ bliminf u f p :=
bliminf_or_le_inf.trans inf_le_left
@[simp] | theorem | Order | [
"Mathlib.Order.ConditionallyCompleteLattice.Indexed",
"Mathlib.Order.Filter.IsBounded",
"Mathlib.Order.Hom.CompleteLattice"
] | Mathlib/Order/LiminfLimsup.lean | bliminf_or_le_inf_aux_left | null |
bliminf_or_le_inf_aux_right : (bliminf u f fun x => p x ∨ q x) ≤ bliminf u f q :=
bliminf_or_le_inf.trans inf_le_right | theorem | Order | [
"Mathlib.Order.ConditionallyCompleteLattice.Indexed",
"Mathlib.Order.Filter.IsBounded",
"Mathlib.Order.Hom.CompleteLattice"
] | Mathlib/Order/LiminfLimsup.lean | bliminf_or_le_inf_aux_right | null |
_root_.OrderIso.apply_blimsup [CompleteLattice γ] (e : α ≃o γ) :
e (blimsup u f p) = blimsup (e ∘ u) f p := by
simp only [blimsup_eq, map_sInf, Function.comp_apply, e.image_eq_preimage,
Set.preimage_setOf_eq, e.le_symm_apply] | theorem | Order | [
"Mathlib.Order.ConditionallyCompleteLattice.Indexed",
"Mathlib.Order.Filter.IsBounded",
"Mathlib.Order.Hom.CompleteLattice"
] | Mathlib/Order/LiminfLimsup.lean | _root_.OrderIso.apply_blimsup | null |
_root_.OrderIso.apply_bliminf [CompleteLattice γ] (e : α ≃o γ) :
e (bliminf u f p) = bliminf (e ∘ u) f p :=
e.dual.apply_blimsup | theorem | Order | [
"Mathlib.Order.ConditionallyCompleteLattice.Indexed",
"Mathlib.Order.Filter.IsBounded",
"Mathlib.Order.Hom.CompleteLattice"
] | Mathlib/Order/LiminfLimsup.lean | _root_.OrderIso.apply_bliminf | null |
_root_.sSupHom.apply_blimsup_le [CompleteLattice γ] (g : sSupHom α γ) :
g (blimsup u f p) ≤ blimsup (g ∘ u) f p := by
simp only [blimsup_eq_iInf_biSup, Function.comp]
refine ((OrderHomClass.mono g).map_iInf₂_le _).trans ?_
simp only [_root_.map_iSup, le_refl] | theorem | Order | [
"Mathlib.Order.ConditionallyCompleteLattice.Indexed",
"Mathlib.Order.Filter.IsBounded",
"Mathlib.Order.Hom.CompleteLattice"
] | Mathlib/Order/LiminfLimsup.lean | _root_.sSupHom.apply_blimsup_le | null |
_root_.sInfHom.le_apply_bliminf [CompleteLattice γ] (g : sInfHom α γ) :
bliminf (g ∘ u) f p ≤ g (bliminf u f p) :=
(sInfHom.dual g).apply_blimsup_le | theorem | Order | [
"Mathlib.Order.ConditionallyCompleteLattice.Indexed",
"Mathlib.Order.Filter.IsBounded",
"Mathlib.Order.Hom.CompleteLattice"
] | Mathlib/Order/LiminfLimsup.lean | _root_.sInfHom.le_apply_bliminf | null |
limsup_sup_filter {g} : limsup u (f ⊔ g) = limsup u f ⊔ limsup u g := by
refine le_antisymm ?_
(sup_le (limsup_le_limsup_of_le le_sup_left) (limsup_le_limsup_of_le le_sup_right))
simp_rw [limsup_eq, sInf_sup_eq, sup_sInf_eq, mem_setOf_eq, le_iInf₂_iff]
intro a ha b hb
exact sInf_le ⟨ha.mono fun _ h ↦ h.tran... | lemma | Order | [
"Mathlib.Order.ConditionallyCompleteLattice.Indexed",
"Mathlib.Order.Filter.IsBounded",
"Mathlib.Order.Hom.CompleteLattice"
] | Mathlib/Order/LiminfLimsup.lean | limsup_sup_filter | null |
liminf_sup_filter {g} : liminf u (f ⊔ g) = liminf u f ⊓ liminf u g :=
limsup_sup_filter (α := αᵒᵈ)
@[simp] | lemma | Order | [
"Mathlib.Order.ConditionallyCompleteLattice.Indexed",
"Mathlib.Order.Filter.IsBounded",
"Mathlib.Order.Hom.CompleteLattice"
] | Mathlib/Order/LiminfLimsup.lean | liminf_sup_filter | null |
blimsup_or_eq_sup : (blimsup u f fun x => p x ∨ q x) = blimsup u f p ⊔ blimsup u f q := by
simp only [blimsup_eq_limsup, ← limsup_sup_filter, ← inf_sup_left, sup_principal, setOf_or]
@[simp] | theorem | Order | [
"Mathlib.Order.ConditionallyCompleteLattice.Indexed",
"Mathlib.Order.Filter.IsBounded",
"Mathlib.Order.Hom.CompleteLattice"
] | Mathlib/Order/LiminfLimsup.lean | blimsup_or_eq_sup | null |
bliminf_or_eq_inf : (bliminf u f fun x => p x ∨ q x) = bliminf u f p ⊓ bliminf u f q :=
blimsup_or_eq_sup (α := αᵒᵈ)
@[simp] | theorem | Order | [
"Mathlib.Order.ConditionallyCompleteLattice.Indexed",
"Mathlib.Order.Filter.IsBounded",
"Mathlib.Order.Hom.CompleteLattice"
] | Mathlib/Order/LiminfLimsup.lean | bliminf_or_eq_inf | null |
blimsup_sup_not : blimsup u f p ⊔ blimsup u f (¬p ·) = limsup u f := by
simp_rw [← blimsup_or_eq_sup, or_not, blimsup_true]
@[simp] | lemma | Order | [
"Mathlib.Order.ConditionallyCompleteLattice.Indexed",
"Mathlib.Order.Filter.IsBounded",
"Mathlib.Order.Hom.CompleteLattice"
] | Mathlib/Order/LiminfLimsup.lean | blimsup_sup_not | null |
bliminf_inf_not : bliminf u f p ⊓ bliminf u f (¬p ·) = liminf u f :=
blimsup_sup_not (α := αᵒᵈ)
@[simp] | lemma | Order | [
"Mathlib.Order.ConditionallyCompleteLattice.Indexed",
"Mathlib.Order.Filter.IsBounded",
"Mathlib.Order.Hom.CompleteLattice"
] | Mathlib/Order/LiminfLimsup.lean | bliminf_inf_not | null |
blimsup_not_sup : blimsup u f (¬p ·) ⊔ blimsup u f p = limsup u f := by
simpa only [not_not] using blimsup_sup_not (p := (¬p ·))
@[simp] | lemma | Order | [
"Mathlib.Order.ConditionallyCompleteLattice.Indexed",
"Mathlib.Order.Filter.IsBounded",
"Mathlib.Order.Hom.CompleteLattice"
] | Mathlib/Order/LiminfLimsup.lean | blimsup_not_sup | null |
bliminf_not_inf : bliminf u f (¬p ·) ⊓ bliminf u f p = liminf u f :=
blimsup_not_sup (α := αᵒᵈ) | lemma | Order | [
"Mathlib.Order.ConditionallyCompleteLattice.Indexed",
"Mathlib.Order.Filter.IsBounded",
"Mathlib.Order.Hom.CompleteLattice"
] | Mathlib/Order/LiminfLimsup.lean | bliminf_not_inf | null |
limsup_piecewise {s : Set β} [DecidablePred (· ∈ s)] {v} :
limsup (s.piecewise u v) f = blimsup u f (· ∈ s) ⊔ blimsup v f (· ∉ s) := by
rw [← blimsup_sup_not (p := (· ∈ s))]
refine congr_arg₂ _ (blimsup_congr ?_) (blimsup_congr ?_) <;>
filter_upwards with _ h using by simp [h] | lemma | Order | [
"Mathlib.Order.ConditionallyCompleteLattice.Indexed",
"Mathlib.Order.Filter.IsBounded",
"Mathlib.Order.Hom.CompleteLattice"
] | Mathlib/Order/LiminfLimsup.lean | limsup_piecewise | null |
liminf_piecewise {s : Set β} [DecidablePred (· ∈ s)] {v} :
liminf (s.piecewise u v) f = bliminf u f (· ∈ s) ⊓ bliminf v f (· ∉ s) :=
limsup_piecewise (α := αᵒᵈ) | lemma | Order | [
"Mathlib.Order.ConditionallyCompleteLattice.Indexed",
"Mathlib.Order.Filter.IsBounded",
"Mathlib.Order.Hom.CompleteLattice"
] | Mathlib/Order/LiminfLimsup.lean | liminf_piecewise | null |
sup_limsup [NeBot f] (a : α) : a ⊔ limsup u f = limsup (fun x => a ⊔ u x) f := by
simp only [limsup_eq_iInf_iSup, iSup_sup_eq, sup_iInf₂_eq]
congr; ext s; congr; ext hs; congr
exact (biSup_const (nonempty_of_mem hs)).symm | theorem | Order | [
"Mathlib.Order.ConditionallyCompleteLattice.Indexed",
"Mathlib.Order.Filter.IsBounded",
"Mathlib.Order.Hom.CompleteLattice"
] | Mathlib/Order/LiminfLimsup.lean | sup_limsup | null |
inf_liminf [NeBot f] (a : α) : a ⊓ liminf u f = liminf (fun x => a ⊓ u x) f :=
sup_limsup (α := αᵒᵈ) a | theorem | Order | [
"Mathlib.Order.ConditionallyCompleteLattice.Indexed",
"Mathlib.Order.Filter.IsBounded",
"Mathlib.Order.Hom.CompleteLattice"
] | Mathlib/Order/LiminfLimsup.lean | inf_liminf | null |
sup_liminf (a : α) : a ⊔ liminf u f = liminf (fun x => a ⊔ u x) f := by
simp only [liminf_eq_iSup_iInf]
rw [sup_comm, biSup_sup (⟨univ, univ_mem⟩ : ∃ i : Set β, i ∈ f)]
simp_rw [iInf₂_sup_eq, sup_comm (a := a)] | theorem | Order | [
"Mathlib.Order.ConditionallyCompleteLattice.Indexed",
"Mathlib.Order.Filter.IsBounded",
"Mathlib.Order.Hom.CompleteLattice"
] | Mathlib/Order/LiminfLimsup.lean | sup_liminf | null |
inf_limsup (a : α) : a ⊓ limsup u f = limsup (fun x => a ⊓ u x) f :=
sup_liminf (α := αᵒᵈ) a | theorem | Order | [
"Mathlib.Order.ConditionallyCompleteLattice.Indexed",
"Mathlib.Order.Filter.IsBounded",
"Mathlib.Order.Hom.CompleteLattice"
] | Mathlib/Order/LiminfLimsup.lean | inf_limsup | null |
limsup_compl : (limsup u f)ᶜ = liminf (compl ∘ u) f := by
simp only [limsup_eq_iInf_iSup, compl_iInf, compl_iSup, liminf_eq_iSup_iInf, Function.comp_apply] | theorem | Order | [
"Mathlib.Order.ConditionallyCompleteLattice.Indexed",
"Mathlib.Order.Filter.IsBounded",
"Mathlib.Order.Hom.CompleteLattice"
] | Mathlib/Order/LiminfLimsup.lean | limsup_compl | null |
liminf_compl : (liminf u f)ᶜ = limsup (compl ∘ u) f := by
simp only [limsup_eq_iInf_iSup, compl_iInf, compl_iSup, liminf_eq_iSup_iInf, Function.comp_apply] | theorem | Order | [
"Mathlib.Order.ConditionallyCompleteLattice.Indexed",
"Mathlib.Order.Filter.IsBounded",
"Mathlib.Order.Hom.CompleteLattice"
] | Mathlib/Order/LiminfLimsup.lean | liminf_compl | null |
limsup_sdiff (a : α) : limsup u f \ a = limsup (fun b => u b \ a) f := by
simp only [limsup_eq_iInf_iSup, sdiff_eq]
rw [biInf_inf (⟨univ, univ_mem⟩ : ∃ i : Set β, i ∈ f)]
simp_rw [inf_comm, inf_iSup₂_eq, inf_comm] | theorem | Order | [
"Mathlib.Order.ConditionallyCompleteLattice.Indexed",
"Mathlib.Order.Filter.IsBounded",
"Mathlib.Order.Hom.CompleteLattice"
] | Mathlib/Order/LiminfLimsup.lean | limsup_sdiff | null |
liminf_sdiff [NeBot f] (a : α) : liminf u f \ a = liminf (fun b => u b \ a) f := by
simp only [sdiff_eq, inf_comm _ aᶜ, inf_liminf] | theorem | Order | [
"Mathlib.Order.ConditionallyCompleteLattice.Indexed",
"Mathlib.Order.Filter.IsBounded",
"Mathlib.Order.Hom.CompleteLattice"
] | Mathlib/Order/LiminfLimsup.lean | liminf_sdiff | null |
sdiff_limsup [NeBot f] (a : α) : a \ limsup u f = liminf (fun b => a \ u b) f := by
rw [← compl_inj_iff]
simp only [sdiff_eq, liminf_compl, comp_def, compl_inf, compl_compl, sup_limsup] | theorem | Order | [
"Mathlib.Order.ConditionallyCompleteLattice.Indexed",
"Mathlib.Order.Filter.IsBounded",
"Mathlib.Order.Hom.CompleteLattice"
] | Mathlib/Order/LiminfLimsup.lean | sdiff_limsup | null |
sdiff_liminf (a : α) : a \ liminf u f = limsup (fun b => a \ u b) f := by
rw [← compl_inj_iff]
simp only [sdiff_eq, limsup_compl, comp_def, compl_inf, compl_compl, sup_liminf] | theorem | Order | [
"Mathlib.Order.ConditionallyCompleteLattice.Indexed",
"Mathlib.Order.Filter.IsBounded",
"Mathlib.Order.Hom.CompleteLattice"
] | Mathlib/Order/LiminfLimsup.lean | sdiff_liminf | null |
mem_liminf_iff_eventually_mem : (a ∈ liminf s 𝓕) ↔ (∀ᶠ i in 𝓕, a ∈ s i) := by
simpa only [liminf_eq_iSup_iInf, iSup_eq_iUnion, iInf_eq_iInter, mem_iUnion, mem_iInter]
using ⟨fun ⟨S, hS, hS'⟩ ↦ mem_of_superset hS (by tauto), fun h ↦ ⟨{i | a ∈ s i}, h, by tauto⟩⟩ | lemma | Order | [
"Mathlib.Order.ConditionallyCompleteLattice.Indexed",
"Mathlib.Order.Filter.IsBounded",
"Mathlib.Order.Hom.CompleteLattice"
] | Mathlib/Order/LiminfLimsup.lean | mem_liminf_iff_eventually_mem | null |
mem_limsup_iff_frequently_mem : (a ∈ limsup s 𝓕) ↔ (∃ᶠ i in 𝓕, a ∈ s i) := by
simp only [Filter.Frequently, iff_not_comm, ← mem_compl_iff, limsup_compl, comp_apply,
mem_liminf_iff_eventually_mem] | lemma | Order | [
"Mathlib.Order.ConditionallyCompleteLattice.Indexed",
"Mathlib.Order.Filter.IsBounded",
"Mathlib.Order.Hom.CompleteLattice"
] | Mathlib/Order/LiminfLimsup.lean | mem_limsup_iff_frequently_mem | null |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.