statement stringlengths 1 2.88k | proof stringlengths 0 13.9k | type stringclasses 10
values | symbolic_name stringlengths 1 131 | library stringclasses 417
values | filename stringlengths 17 80 | imports listlengths 0 16 | deps listlengths 0 64 | docstring stringlengths 0 10.2k | source_url stringclasses 1
value | commit stringclasses 1
value |
|---|---|---|---|---|---|---|---|---|---|---|
small_sets_top : (⊤ : filter α).small_sets = ⊤ | by rw [small_sets, lift'_top, powerset_univ, principal_univ] | lemma | filter.small_sets_top | order.filter | src/order/filter/small_sets.lean | [
"order.filter.lift",
"order.filter.at_top_bot"
] | [
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
small_sets_principal (s : set α) : (𝓟 s).small_sets = 𝓟(𝒫 s) | lift'_principal monotone_powerset | lemma | filter.small_sets_principal | order.filter | src/order/filter/small_sets.lean | [
"order.filter.lift",
"order.filter.at_top_bot"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
small_sets_comap (l : filter β) (f : α → β) :
(comap f l).small_sets = l.lift' (powerset ∘ preimage f) | comap_lift'_eq2 monotone_powerset | lemma | filter.small_sets_comap | order.filter | src/order/filter/small_sets.lean | [
"order.filter.lift",
"order.filter.at_top_bot"
] | [
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comap_small_sets (l : filter β) (f : α → set β) :
comap f l.small_sets = l.lift' (preimage f ∘ powerset) | comap_lift'_eq | lemma | filter.comap_small_sets | order.filter | src/order/filter/small_sets.lean | [
"order.filter.lift",
"order.filter.at_top_bot"
] | [
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
small_sets_infi {f : ι → filter α} :
(infi f).small_sets = (⨅ i, (f i).small_sets) | lift'_infi_of_map_univ powerset_inter powerset_univ | lemma | filter.small_sets_infi | order.filter | src/order/filter/small_sets.lean | [
"order.filter.lift",
"order.filter.at_top_bot"
] | [
"filter",
"infi"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
small_sets_inf (l₁ l₂ : filter α) :
(l₁ ⊓ l₂).small_sets = l₁.small_sets ⊓ l₂.small_sets | lift'_inf _ _ powerset_inter | lemma | filter.small_sets_inf | order.filter | src/order/filter/small_sets.lean | [
"order.filter.lift",
"order.filter.at_top_bot"
] | [
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
small_sets_ne_bot (l : filter α) : ne_bot l.small_sets | (lift'_ne_bot_iff monotone_powerset).2 $ λ _ _, powerset_nonempty | instance | filter.small_sets_ne_bot | order.filter | src/order/filter/small_sets.lean | [
"order.filter.lift",
"order.filter.at_top_bot"
] | [
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto.small_sets_mono {s t : α → set β}
(ht : tendsto t la lb.small_sets) (hst : ∀ᶠ x in la, s x ⊆ t x) :
tendsto s la lb.small_sets | begin
rw [tendsto_small_sets_iff] at ht ⊢,
exact λ u hu, (ht u hu).mp (hst.mono $ λ a hst ht, subset.trans hst ht)
end | lemma | filter.tendsto.small_sets_mono | order.filter | src/order/filter/small_sets.lean | [
"order.filter.lift",
"order.filter.at_top_bot"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto.of_small_sets {s : α → set β} {f : α → β} (hs : tendsto s la lb.small_sets)
(hf : ∀ᶠ x in la, f x ∈ s x) : tendsto f la lb | λ t ht, hf.mp $ (tendsto_small_sets_iff.mp hs t ht).mono $ λ x h₁ h₂, h₁ h₂ | lemma | filter.tendsto.of_small_sets | order.filter | src/order/filter/small_sets.lean | [
"order.filter.lift",
"order.filter.at_top_bot"
] | [] | Generalized **squeeze theorem** (also known as **sandwich theorem**). If `s : α → set β` is a
family of sets that tends to `filter.small_sets lb` along `la` and `f : α → β` is a function such
that `f x ∈ s x` eventually along `la`, then `f` tends to `lb` along `la`.
If `s x` is the closed interval `[g x, h x]` for som... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
eventually_small_sets_eventually {p : α → Prop} :
(∀ᶠ s in l.small_sets, ∀ᶠ x in l', x ∈ s → p x) ↔ ∀ᶠ x in l ⊓ l', p x | calc _ ↔ ∃ s ∈ l, ∀ᶠ x in l', x ∈ s → p x :
eventually_small_sets' $ λ s t hst ht, ht.mono $ λ x hx hs, hx (hst hs)
... ↔ ∃ (s ∈ l) (t ∈ l'), ∀ x, x ∈ t → x ∈ s → p x :
by simp only [eventually_iff_exists_mem]
... ↔ ∀ᶠ x in l ⊓ l', p x : by simp only [eventually_inf, and_comm, mem_inter_iff, ← and_imp] | lemma | filter.eventually_small_sets_eventually | order.filter | src/order/filter/small_sets.lean | [
"order.filter.lift",
"order.filter.at_top_bot"
] | [
"and_imp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eventually_small_sets_forall {p : α → Prop} :
(∀ᶠ s in l.small_sets, ∀ x ∈ s, p x) ↔ ∀ᶠ x in l, p x | by simpa only [inf_top_eq, eventually_top] using @eventually_small_sets_eventually α l ⊤ p | lemma | filter.eventually_small_sets_forall | order.filter | src/order/filter/small_sets.lean | [
"order.filter.lift",
"order.filter.at_top_bot"
] | [
"inf_top_eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eventually_small_sets_subset {s : set α} :
(∀ᶠ t in l.small_sets, t ⊆ s) ↔ s ∈ l | eventually_small_sets_forall | lemma | filter.eventually_small_sets_subset | order.filter | src/order/filter/small_sets.lean | [
"order.filter.lift",
"order.filter.at_top_bot"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ultrafilter (α : Type*) extends filter α | (ne_bot' : ne_bot to_filter)
(le_of_le : ∀ g, filter.ne_bot g → g ≤ to_filter → to_filter ≤ g) | structure | ultrafilter | order.filter | src/order/filter/ultrafilter.lean | [
"order.filter.cofinite",
"order.zorn_atoms"
] | [
"filter",
"filter.ne_bot"
] | An ultrafilter is a minimal (maximal in the set order) proper filter. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
unique (f : ultrafilter α) {g : filter α} (h : g ≤ f)
(hne : ne_bot g . tactic.apply_instance) : g = f | le_antisymm h $ f.le_of_le g hne h | lemma | ultrafilter.unique | order.filter | src/order/filter/ultrafilter.lean | [
"order.filter.cofinite",
"order.zorn_atoms"
] | [
"filter",
"ultrafilter",
"unique"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ne_bot (f : ultrafilter α) : ne_bot (f : filter α) | f.ne_bot' | instance | ultrafilter.ne_bot | order.filter | src/order/filter/ultrafilter.lean | [
"order.filter.cofinite",
"order.zorn_atoms"
] | [
"filter",
"ultrafilter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_atom (f : ultrafilter α) : is_atom (f : filter α) | ⟨f.ne_bot.ne, λ g hgf, by_contra $ λ hg, hgf.ne $ f.unique hgf.le ⟨hg⟩⟩ | lemma | ultrafilter.is_atom | order.filter | src/order/filter/ultrafilter.lean | [
"order.filter.cofinite",
"order.zorn_atoms"
] | [
"by_contra",
"filter",
"is_atom",
"ultrafilter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_coe : s ∈ (f : filter α) ↔ s ∈ f | iff.rfl | lemma | ultrafilter.mem_coe | order.filter | src/order/filter/ultrafilter.lean | [
"order.filter.cofinite",
"order.zorn_atoms"
] | [
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_injective : injective (coe : ultrafilter α → filter α) | | ⟨f, h₁, h₂⟩ ⟨g, h₃, h₄⟩ rfl := by congr | lemma | ultrafilter.coe_injective | order.filter | src/order/filter/ultrafilter.lean | [
"order.filter.cofinite",
"order.zorn_atoms"
] | [
"filter",
"ultrafilter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_of_le {f g : ultrafilter α} (h : (f : filter α) ≤ g) : f = g | coe_injective (g.unique h) | lemma | ultrafilter.eq_of_le | order.filter | src/order/filter/ultrafilter.lean | [
"order.filter.cofinite",
"order.zorn_atoms"
] | [
"filter",
"ultrafilter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_le_coe {f g : ultrafilter α} : (f : filter α) ≤ g ↔ f = g | ⟨λ h, eq_of_le h, λ h, h ▸ le_rfl⟩ | lemma | ultrafilter.coe_le_coe | order.filter | src/order/filter/ultrafilter.lean | [
"order.filter.cofinite",
"order.zorn_atoms"
] | [
"filter",
"ultrafilter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_inj : (f : filter α) = g ↔ f = g | coe_injective.eq_iff | lemma | ultrafilter.coe_inj | order.filter | src/order/filter/ultrafilter.lean | [
"order.filter.cofinite",
"order.zorn_atoms"
] | [
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ext ⦃f g : ultrafilter α⦄ (h : ∀ s, s ∈ f ↔ s ∈ g) : f = g | coe_injective $ filter.ext h | lemma | ultrafilter.ext | order.filter | src/order/filter/ultrafilter.lean | [
"order.filter.cofinite",
"order.zorn_atoms"
] | [
"filter.ext",
"ultrafilter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_of_inf_ne_bot (f : ultrafilter α) {g : filter α} (hg : ne_bot (↑f ⊓ g)) : ↑f ≤ g | le_of_inf_eq (f.unique inf_le_left hg) | lemma | ultrafilter.le_of_inf_ne_bot | order.filter | src/order/filter/ultrafilter.lean | [
"order.filter.cofinite",
"order.zorn_atoms"
] | [
"filter",
"inf_le_left",
"ultrafilter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_of_inf_ne_bot' (f : ultrafilter α) {g : filter α} (hg : ne_bot (g ⊓ f)) : ↑f ≤ g | f.le_of_inf_ne_bot $ by rwa inf_comm | lemma | ultrafilter.le_of_inf_ne_bot' | order.filter | src/order/filter/ultrafilter.lean | [
"order.filter.cofinite",
"order.zorn_atoms"
] | [
"filter",
"inf_comm",
"ultrafilter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inf_ne_bot_iff {f : ultrafilter α} {g : filter α} : ne_bot (↑f ⊓ g) ↔ ↑f ≤ g | ⟨le_of_inf_ne_bot f, λ h, (inf_of_le_left h).symm ▸ f.ne_bot⟩ | lemma | ultrafilter.inf_ne_bot_iff | order.filter | src/order/filter/ultrafilter.lean | [
"order.filter.cofinite",
"order.zorn_atoms"
] | [
"filter",
"ultrafilter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
disjoint_iff_not_le {f : ultrafilter α} {g : filter α} : disjoint ↑f g ↔ ¬↑f ≤ g | by rw [← inf_ne_bot_iff, ne_bot_iff, ne.def, not_not, disjoint_iff] | lemma | ultrafilter.disjoint_iff_not_le | order.filter | src/order/filter/ultrafilter.lean | [
"order.filter.cofinite",
"order.zorn_atoms"
] | [
"disjoint",
"disjoint_iff",
"filter",
"not_not",
"ultrafilter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
compl_not_mem_iff : sᶜ ∉ f ↔ s ∈ f | ⟨λ hsc, le_principal_iff.1 $ f.le_of_inf_ne_bot
⟨λ h, hsc $ mem_of_eq_bot$ by rwa compl_compl⟩, compl_not_mem⟩ | lemma | ultrafilter.compl_not_mem_iff | order.filter | src/order/filter/ultrafilter.lean | [
"order.filter.cofinite",
"order.zorn_atoms"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
frequently_iff_eventually : (∃ᶠ x in f, p x) ↔ ∀ᶠ x in f, p x | compl_not_mem_iff | lemma | ultrafilter.frequently_iff_eventually | order.filter | src/order/filter/ultrafilter.lean | [
"order.filter.cofinite",
"order.zorn_atoms"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
compl_mem_iff_not_mem : sᶜ ∈ f ↔ s ∉ f | by rw [← compl_not_mem_iff, compl_compl] | lemma | ultrafilter.compl_mem_iff_not_mem | order.filter | src/order/filter/ultrafilter.lean | [
"order.filter.cofinite",
"order.zorn_atoms"
] | [
"compl_compl"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
diff_mem_iff (f : ultrafilter α) : s \ t ∈ f ↔ s ∈ f ∧ t ∉ f | inter_mem_iff.trans $ and_congr iff.rfl compl_mem_iff_not_mem | lemma | ultrafilter.diff_mem_iff | order.filter | src/order/filter/ultrafilter.lean | [
"order.filter.cofinite",
"order.zorn_atoms"
] | [
"ultrafilter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_compl_not_mem_iff (f : filter α) (h : ∀ s, sᶜ ∉ f ↔ s ∈ f) : ultrafilter α | { to_filter := f,
ne_bot' := ⟨λ hf, by simpa [hf] using h⟩,
le_of_le := λ g hg hgf s hs, (h s).1 $ λ hsc, by exactI compl_not_mem hs (hgf hsc) } | def | ultrafilter.of_compl_not_mem_iff | order.filter | src/order/filter/ultrafilter.lean | [
"order.filter.cofinite",
"order.zorn_atoms"
] | [
"filter",
"ultrafilter"
] | If `sᶜ ∉ f ↔ s ∈ f`, then `f` is an ultrafilter. The other implication is given by
`ultrafilter.compl_not_mem_iff`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of_atom (f : filter α) (hf : is_atom f) : ultrafilter α | { to_filter := f,
ne_bot' := ⟨hf.1⟩,
le_of_le := λ g hg, (_root_.is_atom_iff.1 hf).2 g hg.ne } | def | ultrafilter.of_atom | order.filter | src/order/filter/ultrafilter.lean | [
"order.filter.cofinite",
"order.zorn_atoms"
] | [
"filter",
"is_atom",
"ultrafilter"
] | If `f : filter α` is an atom, then it is an ultrafilter. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
nonempty_of_mem (hs : s ∈ f) : s.nonempty | nonempty_of_mem hs | lemma | ultrafilter.nonempty_of_mem | order.filter | src/order/filter/ultrafilter.lean | [
"order.filter.cofinite",
"order.zorn_atoms"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ne_empty_of_mem (hs : s ∈ f) : s ≠ ∅ | (nonempty_of_mem hs).ne_empty | lemma | ultrafilter.ne_empty_of_mem | order.filter | src/order/filter/ultrafilter.lean | [
"order.filter.cofinite",
"order.zorn_atoms"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
empty_not_mem : ∅ ∉ f | empty_not_mem f | lemma | ultrafilter.empty_not_mem | order.filter | src/order/filter/ultrafilter.lean | [
"order.filter.cofinite",
"order.zorn_atoms"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_sup_iff {u : ultrafilter α} {f g : filter α} : ↑u ≤ f ⊔ g ↔ ↑u ≤ f ∨ ↑u ≤ g | not_iff_not.1 $ by simp only [← disjoint_iff_not_le, not_or_distrib, disjoint_sup_right] | lemma | ultrafilter.le_sup_iff | order.filter | src/order/filter/ultrafilter.lean | [
"order.filter.cofinite",
"order.zorn_atoms"
] | [
"disjoint_sup_right",
"filter",
"le_sup_iff",
"not_or_distrib",
"ultrafilter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
union_mem_iff : s ∪ t ∈ f ↔ s ∈ f ∨ t ∈ f | by simp only [← mem_coe, ← le_principal_iff, ← sup_principal, le_sup_iff] | lemma | ultrafilter.union_mem_iff | order.filter | src/order/filter/ultrafilter.lean | [
"order.filter.cofinite",
"order.zorn_atoms"
] | [
"le_sup_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_or_compl_mem (f : ultrafilter α) (s : set α) : s ∈ f ∨ sᶜ ∈ f | or_iff_not_imp_left.2 compl_mem_iff_not_mem.2 | lemma | ultrafilter.mem_or_compl_mem | order.filter | src/order/filter/ultrafilter.lean | [
"order.filter.cofinite",
"order.zorn_atoms"
] | [
"ultrafilter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
em (f : ultrafilter α) (p : α → Prop) :
(∀ᶠ x in f, p x) ∨ ∀ᶠ x in f, ¬p x | f.mem_or_compl_mem {x | p x} | lemma | ultrafilter.em | order.filter | src/order/filter/ultrafilter.lean | [
"order.filter.cofinite",
"order.zorn_atoms"
] | [
"em",
"ultrafilter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eventually_or : (∀ᶠ x in f, p x ∨ q x) ↔ (∀ᶠ x in f, p x) ∨ ∀ᶠ x in f, q x | union_mem_iff | lemma | ultrafilter.eventually_or | order.filter | src/order/filter/ultrafilter.lean | [
"order.filter.cofinite",
"order.zorn_atoms"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eventually_not : (∀ᶠ x in f, ¬p x) ↔ ¬∀ᶠ x in f, p x | compl_mem_iff_not_mem | lemma | ultrafilter.eventually_not | order.filter | src/order/filter/ultrafilter.lean | [
"order.filter.cofinite",
"order.zorn_atoms"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eventually_imp : (∀ᶠ x in f, p x → q x) ↔ (∀ᶠ x in f, p x) → ∀ᶠ x in f, q x | by simp only [imp_iff_not_or, eventually_or, eventually_not] | lemma | ultrafilter.eventually_imp | order.filter | src/order/filter/ultrafilter.lean | [
"order.filter.cofinite",
"order.zorn_atoms"
] | [
"imp_iff_not_or"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
finite_sUnion_mem_iff {s : set (set α)} (hs : s.finite) : ⋃₀ s ∈ f ↔ ∃t∈s, t ∈ f | finite.induction_on hs (by simp) $ λ a s ha hs his,
by simp [union_mem_iff, his, or_and_distrib_right, exists_or_distrib] | lemma | ultrafilter.finite_sUnion_mem_iff | order.filter | src/order/filter/ultrafilter.lean | [
"order.filter.cofinite",
"order.zorn_atoms"
] | [
"exists_or_distrib",
"or_and_distrib_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
finite_bUnion_mem_iff {is : set β} {s : β → set α} (his : is.finite) :
(⋃i∈is, s i) ∈ f ↔ ∃i∈is, s i ∈ f | by simp only [← sUnion_image, finite_sUnion_mem_iff (his.image s), bex_image_iff] | lemma | ultrafilter.finite_bUnion_mem_iff | order.filter | src/order/filter/ultrafilter.lean | [
"order.filter.cofinite",
"order.zorn_atoms"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map (m : α → β) (f : ultrafilter α) : ultrafilter β | of_compl_not_mem_iff (map m f) $ λ s, @compl_not_mem_iff _ f (m ⁻¹' s) | def | ultrafilter.map | order.filter | src/order/filter/ultrafilter.lean | [
"order.filter.cofinite",
"order.zorn_atoms"
] | [
"ultrafilter"
] | Pushforward for ultrafilters. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_map (m : α → β) (f : ultrafilter α) :
(map m f : filter β) = filter.map m ↑f | rfl | lemma | ultrafilter.coe_map | order.filter | src/order/filter/ultrafilter.lean | [
"order.filter.cofinite",
"order.zorn_atoms"
] | [
"filter",
"filter.map",
"ultrafilter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_map {m : α → β} {f : ultrafilter α} {s : set β} :
s ∈ map m f ↔ m ⁻¹' s ∈ f | iff.rfl | lemma | ultrafilter.mem_map | order.filter | src/order/filter/ultrafilter.lean | [
"order.filter.cofinite",
"order.zorn_atoms"
] | [
"mem_map",
"ultrafilter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_id (f : ultrafilter α) : f.map id = f | coe_injective map_id | lemma | ultrafilter.map_id | order.filter | src/order/filter/ultrafilter.lean | [
"order.filter.cofinite",
"order.zorn_atoms"
] | [
"map_id",
"ultrafilter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_id' (f : ultrafilter α) : f.map (λ x, x) = f | map_id _ | lemma | ultrafilter.map_id' | order.filter | src/order/filter/ultrafilter.lean | [
"order.filter.cofinite",
"order.zorn_atoms"
] | [
"map_id",
"ultrafilter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_map (f : ultrafilter α) (m : α → β) (n : β → γ) :
(f.map m).map n = f.map (n ∘ m) | coe_injective map_map | lemma | ultrafilter.map_map | order.filter | src/order/filter/ultrafilter.lean | [
"order.filter.cofinite",
"order.zorn_atoms"
] | [
"ultrafilter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comap {m : α → β} (u : ultrafilter β) (inj : injective m)
(large : set.range m ∈ u) : ultrafilter α | { to_filter := comap m u,
ne_bot' := u.ne_bot'.comap_of_range_mem large,
le_of_le := λ g hg hgu, by { resetI,
simp only [← u.unique (map_le_iff_le_comap.2 hgu), comap_map inj, le_rfl] } } | def | ultrafilter.comap | order.filter | src/order/filter/ultrafilter.lean | [
"order.filter.cofinite",
"order.zorn_atoms"
] | [
"le_rfl",
"set.range",
"ultrafilter"
] | The pullback of an ultrafilter along an injection whose range is large with respect to the given
ultrafilter. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mem_comap {m : α → β} (u : ultrafilter β) (inj : injective m)
(large : set.range m ∈ u) {s : set α} :
s ∈ u.comap inj large ↔ m '' s ∈ u | mem_comap_iff inj large | lemma | ultrafilter.mem_comap | order.filter | src/order/filter/ultrafilter.lean | [
"order.filter.cofinite",
"order.zorn_atoms"
] | [
"set.range",
"ultrafilter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_comap {m : α → β} (u : ultrafilter β) (inj : injective m)
(large : set.range m ∈ u) : (u.comap inj large : filter α) = filter.comap m u | rfl | lemma | ultrafilter.coe_comap | order.filter | src/order/filter/ultrafilter.lean | [
"order.filter.cofinite",
"order.zorn_atoms"
] | [
"filter",
"filter.comap",
"set.range",
"ultrafilter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comap_id (f : ultrafilter α) (h₀ : injective (id : α → α) := injective_id)
(h₁ : range id ∈ f := by { rw range_id, exact univ_mem}) :
f.comap h₀ h₁ = f | coe_injective comap_id | lemma | ultrafilter.comap_id | order.filter | src/order/filter/ultrafilter.lean | [
"order.filter.cofinite",
"order.zorn_atoms"
] | [
"ultrafilter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comap_comap (f : ultrafilter γ) {m : α → β} {n : β → γ} (inj₀ : injective n)
(large₀ : range n ∈ f) (inj₁ : injective m) (large₁ : range m ∈ f.comap inj₀ large₀)
(inj₂ : injective (n ∘ m) := inj₀.comp inj₁)
(large₂ : range (n ∘ m) ∈ f := by { rw range_comp, exact image_mem_of_mem_comap large₀ large₁ }) :
(f.com... | coe_injective comap_comap | lemma | ultrafilter.comap_comap | order.filter | src/order/filter/ultrafilter.lean | [
"order.filter.cofinite",
"order.zorn_atoms"
] | [
"ultrafilter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_pure {a : α} {s : set α} : s ∈ (pure a : ultrafilter α) ↔ a ∈ s | iff.rfl | lemma | ultrafilter.mem_pure | order.filter | src/order/filter/ultrafilter.lean | [
"order.filter.cofinite",
"order.zorn_atoms"
] | [
"ultrafilter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_pure (a : α) : ↑(pure a : ultrafilter α) = (pure a : filter α) | rfl | lemma | ultrafilter.coe_pure | order.filter | src/order/filter/ultrafilter.lean | [
"order.filter.cofinite",
"order.zorn_atoms"
] | [
"filter",
"ultrafilter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_pure (m : α → β) (a : α) : map m (pure a) = pure (m a) | rfl | lemma | ultrafilter.map_pure | order.filter | src/order/filter/ultrafilter.lean | [
"order.filter.cofinite",
"order.zorn_atoms"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comap_pure {m : α → β} (a : α) (inj : injective m) (large) :
comap (pure $ m a) inj large = pure a | coe_injective $ comap_pure.trans $
by rw [coe_pure, ←principal_singleton, ←image_singleton, preimage_image_eq _ inj] | lemma | ultrafilter.comap_pure | order.filter | src/order/filter/ultrafilter.lean | [
"order.filter.cofinite",
"order.zorn_atoms"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pure_injective : injective (pure : α → ultrafilter α) | λ a b h, filter.pure_injective (congr_arg ultrafilter.to_filter h : _) | lemma | ultrafilter.pure_injective | order.filter | src/order/filter/ultrafilter.lean | [
"order.filter.cofinite",
"order.zorn_atoms"
] | [
"filter.pure_injective",
"ultrafilter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_pure_of_finite_mem (h : s.finite) (h' : s ∈ f) : ∃ x ∈ s, f = pure x | begin
rw ← bUnion_of_singleton s at h',
rcases (ultrafilter.finite_bUnion_mem_iff h).mp h' with ⟨a, has, haf⟩,
exact ⟨a, has, eq_of_le (filter.le_pure_iff.2 haf)⟩
end | lemma | ultrafilter.eq_pure_of_finite_mem | order.filter | src/order/filter/ultrafilter.lean | [
"order.filter.cofinite",
"order.zorn_atoms"
] | [
"ultrafilter.finite_bUnion_mem_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_pure_of_finite [finite α] (f : ultrafilter α) : ∃ a, f = pure a | (eq_pure_of_finite_mem finite_univ univ_mem).imp $ λ a ⟨_, ha⟩, ha | lemma | ultrafilter.eq_pure_of_finite | order.filter | src/order/filter/ultrafilter.lean | [
"order.filter.cofinite",
"order.zorn_atoms"
] | [
"finite",
"ultrafilter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_cofinite_or_eq_pure (f : ultrafilter α) : (f : filter α) ≤ cofinite ∨ ∃ a, f = pure a | or_iff_not_imp_left.2 $ λ h,
let ⟨s, hs, hfin⟩ := filter.disjoint_cofinite_right.1 (disjoint_iff_not_le.2 h),
⟨a, has, hf⟩ := eq_pure_of_finite_mem hfin hs
in ⟨a, hf⟩ | lemma | ultrafilter.le_cofinite_or_eq_pure | order.filter | src/order/filter/ultrafilter.lean | [
"order.filter.cofinite",
"order.zorn_atoms"
] | [
"filter",
"ultrafilter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bind (f : ultrafilter α) (m : α → ultrafilter β) : ultrafilter β | of_compl_not_mem_iff (bind ↑f (λ x, ↑(m x))) $ λ s,
by simp only [mem_bind', mem_coe, ← compl_mem_iff_not_mem, compl_set_of, compl_compl] | def | ultrafilter.bind | order.filter | src/order/filter/ultrafilter.lean | [
"order.filter.cofinite",
"order.zorn_atoms"
] | [
"compl_compl",
"ultrafilter"
] | Monadic bind for ultrafilters, coming from the one on filters
defined in terms of map and join. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_bind : has_bind ultrafilter | ⟨@ultrafilter.bind⟩ | instance | ultrafilter.has_bind | order.filter | src/order/filter/ultrafilter.lean | [
"order.filter.cofinite",
"order.zorn_atoms"
] | [
"ultrafilter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
functor : functor ultrafilter | { map := @ultrafilter.map } | instance | ultrafilter.functor | order.filter | src/order/filter/ultrafilter.lean | [
"order.filter.cofinite",
"order.zorn_atoms"
] | [
"ultrafilter",
"ultrafilter.map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
monad : monad ultrafilter | { map := @ultrafilter.map } | instance | ultrafilter.monad | order.filter | src/order/filter/ultrafilter.lean | [
"order.filter.cofinite",
"order.zorn_atoms"
] | [
"ultrafilter",
"ultrafilter.map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_lawful_monad : is_lawful_monad ultrafilter | { id_map := assume α f, coe_injective (id_map f.1),
pure_bind := assume α β a f, coe_injective (pure_bind a (coe ∘ f)),
bind_assoc := assume α β γ f m₁ m₂, coe_injective (filter_eq rfl),
bind_pure_comp_eq_map := assume α β f x, coe_injective (bind_pure_comp_eq_map f x.1) } | instance | ultrafilter.is_lawful_monad | order.filter | src/order/filter/ultrafilter.lean | [
"order.filter.cofinite",
"order.zorn_atoms"
] | [
"bind_assoc",
"ultrafilter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exists_le (f : filter α) [h : ne_bot f] : ∃ u : ultrafilter α, ↑u ≤ f | let ⟨u, hu, huf⟩ := (eq_bot_or_exists_atom_le f).resolve_left h.ne in ⟨of_atom u hu, huf⟩ | lemma | ultrafilter.exists_le | order.filter | src/order/filter/ultrafilter.lean | [
"order.filter.cofinite",
"order.zorn_atoms"
] | [
"filter",
"ultrafilter"
] | The ultrafilter lemma: Any proper filter is contained in an ultrafilter. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of (f : filter α) [ne_bot f] : ultrafilter α | classical.some (exists_le f) | def | ultrafilter.of | order.filter | src/order/filter/ultrafilter.lean | [
"order.filter.cofinite",
"order.zorn_atoms"
] | [
"filter",
"ultrafilter"
] | Construct an ultrafilter extending a given filter.
The ultrafilter lemma is the assertion that such a filter exists;
we use the axiom of choice to pick one. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of_le (f : filter α) [ne_bot f] : ↑(of f) ≤ f | classical.some_spec (exists_le f) | lemma | ultrafilter.of_le | order.filter | src/order/filter/ultrafilter.lean | [
"order.filter.cofinite",
"order.zorn_atoms"
] | [
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_coe (f : ultrafilter α) : of ↑f = f | coe_inj.1 $ f.unique (of_le f) | lemma | ultrafilter.of_coe | order.filter | src/order/filter/ultrafilter.lean | [
"order.filter.cofinite",
"order.zorn_atoms"
] | [
"ultrafilter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exists_ultrafilter_of_finite_inter_nonempty (S : set (set α))
(cond : ∀ T : finset (set α), (↑T : set (set α)) ⊆ S → (⋂₀ (↑T : set (set α))).nonempty) :
∃ F : ultrafilter α, S ⊆ F.sets | begin
haveI : ne_bot (generate S) := generate_ne_bot_iff.2
(λ t hts ht, ht.coe_to_finset ▸ cond ht.to_finset (ht.coe_to_finset.symm ▸ hts)),
exact ⟨of (generate S), λ t ht, (of_le $ generate S) $ generate_sets.basic ht⟩
end | lemma | ultrafilter.exists_ultrafilter_of_finite_inter_nonempty | order.filter | src/order/filter/ultrafilter.lean | [
"order.filter.cofinite",
"order.zorn_atoms"
] | [
"finset",
"ultrafilter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_atom_pure : is_atom (pure a : filter α) | (pure a : ultrafilter α).is_atom | lemma | filter.is_atom_pure | order.filter | src/order/filter/ultrafilter.lean | [
"order.filter.cofinite",
"order.zorn_atoms"
] | [
"filter",
"is_atom",
"ultrafilter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ne_bot.le_pure_iff (hf : f.ne_bot) : f ≤ pure a ↔ f = pure a | ⟨ultrafilter.unique (pure a), le_of_eq⟩ | lemma | filter.ne_bot.le_pure_iff | order.filter | src/order/filter/ultrafilter.lean | [
"order.filter.cofinite",
"order.zorn_atoms"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lt_pure_iff : f < pure a ↔ f = ⊥ | is_atom_pure.lt_iff | lemma | filter.lt_pure_iff | order.filter | src/order/filter/ultrafilter.lean | [
"order.filter.cofinite",
"order.zorn_atoms"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_pure_iff' : f ≤ pure a ↔ f = ⊥ ∨ f = pure a | is_atom_pure.le_iff | lemma | filter.le_pure_iff' | order.filter | src/order/filter/ultrafilter.lean | [
"order.filter.cofinite",
"order.zorn_atoms"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Iic_pure (a : α) : Iic (pure a : filter α) = {⊥, pure a} | is_atom_pure.Iic_eq | lemma | filter.Iic_pure | order.filter | src/order/filter/ultrafilter.lean | [
"order.filter.cofinite",
"order.zorn_atoms"
] | [
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_iff_ultrafilter : s ∈ f ↔ ∀ g : ultrafilter α, ↑g ≤ f → s ∈ g | begin
refine ⟨λ hf g hg, hg hf, λ H, by_contra $ λ hf, _⟩,
set g : filter ↥sᶜ := comap coe f,
haveI : ne_bot g := comap_ne_bot_iff_compl_range.2 (by simpa [compl_set_of]),
simpa using H ((of g).map coe) (map_le_iff_le_comap.mpr (of_le g))
end | lemma | filter.mem_iff_ultrafilter | order.filter | src/order/filter/ultrafilter.lean | [
"order.filter.cofinite",
"order.zorn_atoms"
] | [
"by_contra",
"filter",
"ultrafilter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_iff_ultrafilter {f₁ f₂ : filter α} : f₁ ≤ f₂ ↔ ∀ g : ultrafilter α, ↑g ≤ f₁ → ↑g ≤ f₂ | ⟨λ h g h₁, h₁.trans h, λ h s hs, mem_iff_ultrafilter.2 $ λ g hg, h g hg hs⟩ | lemma | filter.le_iff_ultrafilter | order.filter | src/order/filter/ultrafilter.lean | [
"order.filter.cofinite",
"order.zorn_atoms"
] | [
"filter",
"ultrafilter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
supr_ultrafilter_le_eq (f : filter α) :
(⨆ (g : ultrafilter α) (hg : ↑g ≤ f), (g : filter α)) = f | eq_of_forall_ge_iff $ λ f', by simp only [supr_le_iff, ← le_iff_ultrafilter] | lemma | filter.supr_ultrafilter_le_eq | order.filter | src/order/filter/ultrafilter.lean | [
"order.filter.cofinite",
"order.zorn_atoms"
] | [
"eq_of_forall_ge_iff",
"filter",
"supr_le_iff",
"ultrafilter"
] | A filter equals the intersection of all the ultrafilters which contain it. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
tendsto_iff_ultrafilter (f : α → β) (l₁ : filter α) (l₂ : filter β) :
tendsto f l₁ l₂ ↔ ∀ g : ultrafilter α, ↑g ≤ l₁ → tendsto f g l₂ | by simpa only [tendsto_iff_comap] using le_iff_ultrafilter | lemma | filter.tendsto_iff_ultrafilter | order.filter | src/order/filter/ultrafilter.lean | [
"order.filter.cofinite",
"order.zorn_atoms"
] | [
"filter",
"ultrafilter"
] | The `tendsto` relation can be checked on ultrafilters. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
exists_ultrafilter_iff {f : filter α} : (∃ (u : ultrafilter α), ↑u ≤ f) ↔ ne_bot f | ⟨λ ⟨u, uf⟩, ne_bot_of_le uf, λ h, @exists_ultrafilter_le _ _ h⟩ | lemma | filter.exists_ultrafilter_iff | order.filter | src/order/filter/ultrafilter.lean | [
"order.filter.cofinite",
"order.zorn_atoms"
] | [
"filter",
"ultrafilter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
forall_ne_bot_le_iff {g : filter α} {p : filter α → Prop} (hp : monotone p) :
(∀ f : filter α, ne_bot f → f ≤ g → p f) ↔ ∀ f : ultrafilter α, ↑f ≤ g → p f | begin
refine ⟨λ H f hf, H f f.ne_bot hf, _⟩,
introsI H f hf hfg,
exact hp (of_le f) (H _ ((of_le f).trans hfg))
end | lemma | filter.forall_ne_bot_le_iff | order.filter | src/order/filter/ultrafilter.lean | [
"order.filter.cofinite",
"order.zorn_atoms"
] | [
"filter",
"monotone",
"ultrafilter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
hyperfilter : ultrafilter α | ultrafilter.of cofinite | def | filter.hyperfilter | order.filter | src/order/filter/ultrafilter.lean | [
"order.filter.cofinite",
"order.zorn_atoms"
] | [
"ultrafilter",
"ultrafilter.of"
] | The ultrafilter extending the cofinite filter. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
hyperfilter_le_cofinite : ↑(hyperfilter α) ≤ @cofinite α | ultrafilter.of_le cofinite | lemma | filter.hyperfilter_le_cofinite | order.filter | src/order/filter/ultrafilter.lean | [
"order.filter.cofinite",
"order.zorn_atoms"
] | [
"ultrafilter.of_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bot_ne_hyperfilter : (⊥ : filter α) ≠ hyperfilter α | (by apply_instance : ne_bot ↑(hyperfilter α)).1.symm | lemma | filter.bot_ne_hyperfilter | order.filter | src/order/filter/ultrafilter.lean | [
"order.filter.cofinite",
"order.zorn_atoms"
] | [
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nmem_hyperfilter_of_finite {s : set α} (hf : s.finite) : s ∉ hyperfilter α | λ hy, compl_not_mem hy $ hyperfilter_le_cofinite hf.compl_mem_cofinite | theorem | filter.nmem_hyperfilter_of_finite | order.filter | src/order/filter/ultrafilter.lean | [
"order.filter.cofinite",
"order.zorn_atoms"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
compl_mem_hyperfilter_of_finite {s : set α} (hf : set.finite s) :
sᶜ ∈ hyperfilter α | compl_mem_iff_not_mem.2 hf.nmem_hyperfilter | theorem | filter.compl_mem_hyperfilter_of_finite | order.filter | src/order/filter/ultrafilter.lean | [
"order.filter.cofinite",
"order.zorn_atoms"
] | [
"set.finite"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_hyperfilter_of_finite_compl {s : set α} (hf : set.finite sᶜ) :
s ∈ hyperfilter α | compl_compl s ▸ hf.compl_mem_hyperfilter | theorem | filter.mem_hyperfilter_of_finite_compl | order.filter | src/order/filter/ultrafilter.lean | [
"order.filter.cofinite",
"order.zorn_atoms"
] | [
"compl_compl",
"set.finite"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comap_inf_principal_ne_bot_of_image_mem (h : m '' s ∈ g) :
(filter.comap m g ⊓ 𝓟 s).ne_bot | filter.comap_inf_principal_ne_bot_of_image_mem g.ne_bot h | lemma | ultrafilter.comap_inf_principal_ne_bot_of_image_mem | order.filter | src/order/filter/ultrafilter.lean | [
"order.filter.cofinite",
"order.zorn_atoms"
] | [
"filter.comap",
"filter.comap_inf_principal_ne_bot_of_image_mem"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_comap_inf_principal (h : m '' s ∈ g) : ultrafilter α | @of _ (filter.comap m g ⊓ 𝓟 s) (comap_inf_principal_ne_bot_of_image_mem h) | def | ultrafilter.of_comap_inf_principal | order.filter | src/order/filter/ultrafilter.lean | [
"order.filter.cofinite",
"order.zorn_atoms"
] | [
"filter.comap",
"ultrafilter"
] | Ultrafilter extending the inf of a comapped ultrafilter and a principal ultrafilter. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of_comap_inf_principal_mem (h : m '' s ∈ g) : s ∈ of_comap_inf_principal h | begin
let f := filter.comap m g ⊓ 𝓟 s,
haveI : f.ne_bot := comap_inf_principal_ne_bot_of_image_mem h,
have : s ∈ f := mem_inf_of_right (mem_principal_self s),
exact le_def.mp (of_le _) s this
end | lemma | ultrafilter.of_comap_inf_principal_mem | order.filter | src/order/filter/ultrafilter.lean | [
"order.filter.cofinite",
"order.zorn_atoms"
] | [
"filter.comap"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_comap_inf_principal_eq_of_map (h : m '' s ∈ g) :
(of_comap_inf_principal h).map m = g | begin
let f := filter.comap m g ⊓ 𝓟 s,
haveI : f.ne_bot := comap_inf_principal_ne_bot_of_image_mem h,
apply eq_of_le,
calc filter.map m (of f) ≤ filter.map m f : map_mono (of_le _)
... ≤ (filter.map m $ filter.comap m g) ⊓ filter.map m (𝓟 s) : map_inf_le
... = (filter.map m $ filter.comap m g) ⊓ (𝓟 $ m '... | lemma | ultrafilter.of_comap_inf_principal_eq_of_map | order.filter | src/order/filter/ultrafilter.lean | [
"order.filter.cofinite",
"order.zorn_atoms"
] | [
"filter.comap",
"filter.map",
"inf_le_inf_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zero_at_filter [has_zero β] [topological_space β] (l : filter α) (f : α → β) : Prop | filter.tendsto f l (𝓝 0) | def | filter.zero_at_filter | order.filter | src/order/filter/zero_and_bounded_at_filter.lean | [
"algebra.module.submodule.basic",
"topology.algebra.monoid",
"analysis.asymptotics.asymptotics"
] | [
"filter",
"filter.tendsto",
"topological_space"
] | If `l` is a filter on `α`, then a function `f : α → β` is `zero_at_filter l`
if it tends to zero along `l`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
zero_zero_at_filter [has_zero β] [topological_space β] (l : filter α) :
zero_at_filter l (0 : α → β) | tendsto_const_nhds | lemma | filter.zero_zero_at_filter | order.filter | src/order/filter/zero_and_bounded_at_filter.lean | [
"algebra.module.submodule.basic",
"topology.algebra.monoid",
"analysis.asymptotics.asymptotics"
] | [
"filter",
"tendsto_const_nhds",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zero_at_filter.add [topological_space β] [add_zero_class β] [has_continuous_add β]
{l : filter α} {f g : α → β} (hf : zero_at_filter l f) (hg : zero_at_filter l g) :
zero_at_filter l (f + g) | by simpa using hf.add hg | lemma | filter.zero_at_filter.add | order.filter | src/order/filter/zero_and_bounded_at_filter.lean | [
"algebra.module.submodule.basic",
"topology.algebra.monoid",
"analysis.asymptotics.asymptotics"
] | [
"add_zero_class",
"filter",
"has_continuous_add",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zero_at_filter.neg [topological_space β] [add_group β] [has_continuous_neg β] {l : filter α}
{f : α → β} (hf : zero_at_filter l f) :
zero_at_filter l (-f) | by simpa using hf.neg | lemma | filter.zero_at_filter.neg | order.filter | src/order/filter/zero_and_bounded_at_filter.lean | [
"algebra.module.submodule.basic",
"topology.algebra.monoid",
"analysis.asymptotics.asymptotics"
] | [
"add_group",
"filter",
"has_continuous_neg",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zero_at_filter.smul {𝕜 : Type*} [topological_space 𝕜] [topological_space β] [has_zero 𝕜]
[has_zero β] [smul_with_zero 𝕜 β] [has_continuous_smul 𝕜 β]
{l : filter α} {f : α → β} (c : 𝕜) (hf : zero_at_filter l f) :
zero_at_filter l (c • f) | by simpa using hf.const_smul c | lemma | filter.zero_at_filter.smul | order.filter | src/order/filter/zero_and_bounded_at_filter.lean | [
"algebra.module.submodule.basic",
"topology.algebra.monoid",
"analysis.asymptotics.asymptotics"
] | [
"filter",
"has_continuous_smul",
"smul_with_zero",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zero_at_filter_submodule [topological_space β] [semiring β]
[has_continuous_add β] [has_continuous_mul β] (l : filter α) : submodule β (α → β) | { carrier := zero_at_filter l,
zero_mem' := zero_zero_at_filter l,
add_mem' := λ a b ha hb, ha.add hb,
smul_mem' := λ c f hf, hf.smul c } | def | filter.zero_at_filter_submodule | order.filter | src/order/filter/zero_and_bounded_at_filter.lean | [
"algebra.module.submodule.basic",
"topology.algebra.monoid",
"analysis.asymptotics.asymptotics"
] | [
"filter",
"has_continuous_add",
"has_continuous_mul",
"semiring",
"submodule",
"topological_space"
] | `zero_at_filter_submodule l` is the submodule of `f : α → β` which
tend to zero along `l`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.