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
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|---|---|---|---|---|---|---|---|---|---|---|
tendsto_inf_left {f : α → β} {x₁ x₂ : filter α} {y : filter β}
(h : tendsto f x₁ y) : tendsto f (x₁ ⊓ x₂) y | le_trans (map_mono inf_le_left) h | lemma | filter.tendsto_inf_left | order.filter | src/order/filter/basic.lean | [
"control.traversable.instances",
"data.set.finite",
"order.copy",
"tactic.monotonicity"
] | [
"filter",
"inf_le_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_inf_right {f : α → β} {x₁ x₂ : filter α} {y : filter β}
(h : tendsto f x₂ y) : tendsto f (x₁ ⊓ x₂) y | le_trans (map_mono inf_le_right) h | lemma | filter.tendsto_inf_right | order.filter | src/order/filter/basic.lean | [
"control.traversable.instances",
"data.set.finite",
"order.copy",
"tactic.monotonicity"
] | [
"filter",
"inf_le_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto.inf {f : α → β} {x₁ x₂ : filter α} {y₁ y₂ : filter β}
(h₁ : tendsto f x₁ y₁) (h₂ : tendsto f x₂ y₂) : tendsto f (x₁ ⊓ x₂) (y₁ ⊓ y₂) | tendsto_inf.2 ⟨tendsto_inf_left h₁, tendsto_inf_right h₂⟩ | lemma | filter.tendsto.inf | order.filter | src/order/filter/basic.lean | [
"control.traversable.instances",
"data.set.finite",
"order.copy",
"tactic.monotonicity"
] | [
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_infi {f : α → β} {x : filter α} {y : ι → filter β} :
tendsto f x (⨅ i, y i) ↔ ∀ i, tendsto f x (y i) | by simp only [tendsto, iff_self, le_infi_iff] | lemma | filter.tendsto_infi | order.filter | src/order/filter/basic.lean | [
"control.traversable.instances",
"data.set.finite",
"order.copy",
"tactic.monotonicity"
] | [
"filter",
"le_infi_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_infi' {f : α → β} {x : ι → filter α} {y : filter β} (i : ι) (hi : tendsto f (x i) y) :
tendsto f (⨅ i, x i) y | hi.mono_left $ infi_le _ _ | lemma | filter.tendsto_infi' | order.filter | src/order/filter/basic.lean | [
"control.traversable.instances",
"data.set.finite",
"order.copy",
"tactic.monotonicity"
] | [
"filter",
"infi_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_infi_infi {f : α → β} {x : ι → filter α} {y : ι → filter β}
(h : ∀ i, tendsto f (x i) (y i)) : tendsto f (infi x) (infi y) | tendsto_infi.2 $ λ i, tendsto_infi' i (h i) | theorem | filter.tendsto_infi_infi | order.filter | src/order/filter/basic.lean | [
"control.traversable.instances",
"data.set.finite",
"order.copy",
"tactic.monotonicity"
] | [
"filter",
"infi"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_sup {f : α → β} {x₁ x₂ : filter α} {y : filter β} :
tendsto f (x₁ ⊔ x₂) y ↔ tendsto f x₁ y ∧ tendsto f x₂ y | by simp only [tendsto, map_sup, sup_le_iff] | lemma | filter.tendsto_sup | order.filter | src/order/filter/basic.lean | [
"control.traversable.instances",
"data.set.finite",
"order.copy",
"tactic.monotonicity"
] | [
"filter",
"sup_le_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto.sup {f : α → β} {x₁ x₂ : filter α} {y : filter β} :
tendsto f x₁ y → tendsto f x₂ y → tendsto f (x₁ ⊔ x₂) y | λ h₁ h₂, tendsto_sup.mpr ⟨ h₁, h₂ ⟩ | lemma | filter.tendsto.sup | order.filter | src/order/filter/basic.lean | [
"control.traversable.instances",
"data.set.finite",
"order.copy",
"tactic.monotonicity"
] | [
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_supr {f : α → β} {x : ι → filter α} {y : filter β} :
tendsto f (⨆ i, x i) y ↔ ∀ i, tendsto f (x i) y | by simp only [tendsto, map_supr, supr_le_iff] | lemma | filter.tendsto_supr | order.filter | src/order/filter/basic.lean | [
"control.traversable.instances",
"data.set.finite",
"order.copy",
"tactic.monotonicity"
] | [
"filter",
"map_supr",
"supr_le_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_supr_supr {f : α → β} {x : ι → filter α} {y : ι → filter β}
(h : ∀ i, tendsto f (x i) (y i)) : tendsto f (supr x) (supr y) | tendsto_supr.2 $ λ i, (h i).mono_right $ le_supr _ _ | theorem | filter.tendsto_supr_supr | order.filter | src/order/filter/basic.lean | [
"control.traversable.instances",
"data.set.finite",
"order.copy",
"tactic.monotonicity"
] | [
"filter",
"le_supr",
"supr"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_principal {f : α → β} {l : filter α} {s : set β} :
tendsto f l (𝓟 s) ↔ ∀ᶠ a in l, f a ∈ s | by simp only [tendsto, le_principal_iff, mem_map', filter.eventually] | lemma | filter.tendsto_principal | order.filter | src/order/filter/basic.lean | [
"control.traversable.instances",
"data.set.finite",
"order.copy",
"tactic.monotonicity"
] | [
"filter",
"filter.eventually"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_principal_principal {f : α → β} {s : set α} {t : set β} :
tendsto f (𝓟 s) (𝓟 t) ↔ ∀ a ∈ s, f a ∈ t | by simp only [tendsto_principal, eventually_principal] | lemma | filter.tendsto_principal_principal | order.filter | src/order/filter/basic.lean | [
"control.traversable.instances",
"data.set.finite",
"order.copy",
"tactic.monotonicity"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_pure {f : α → β} {a : filter α} {b : β} :
tendsto f a (pure b) ↔ ∀ᶠ x in a, f x = b | by simp only [tendsto, le_pure_iff, mem_map', mem_singleton_iff, filter.eventually] | lemma | filter.tendsto_pure | order.filter | src/order/filter/basic.lean | [
"control.traversable.instances",
"data.set.finite",
"order.copy",
"tactic.monotonicity"
] | [
"filter",
"filter.eventually"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_pure_pure (f : α → β) (a : α) :
tendsto f (pure a) (pure (f a)) | tendsto_pure.2 rfl | lemma | filter.tendsto_pure_pure | order.filter | src/order/filter/basic.lean | [
"control.traversable.instances",
"data.set.finite",
"order.copy",
"tactic.monotonicity"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_const_pure {a : filter α} {b : β} : tendsto (λ x, b) a (pure b) | tendsto_pure.2 $ univ_mem' $ λ _, rfl | lemma | filter.tendsto_const_pure | order.filter | src/order/filter/basic.lean | [
"control.traversable.instances",
"data.set.finite",
"order.copy",
"tactic.monotonicity"
] | [
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pure_le_iff {a : α} {l : filter α} : pure a ≤ l ↔ ∀ s ∈ l, a ∈ s | iff.rfl | lemma | filter.pure_le_iff | order.filter | src/order/filter/basic.lean | [
"control.traversable.instances",
"data.set.finite",
"order.copy",
"tactic.monotonicity"
] | [
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_pure_left {f : α → β} {a : α} {l : filter β} :
tendsto f (pure a) l ↔ ∀ s ∈ l, f a ∈ s | iff.rfl | lemma | filter.tendsto_pure_left | order.filter | src/order/filter/basic.lean | [
"control.traversable.instances",
"data.set.finite",
"order.copy",
"tactic.monotonicity"
] | [
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_inf_principal_preimage {f : α → β} {s : set β} {l : filter α} :
map f (l ⊓ 𝓟 (f ⁻¹' s)) = map f l ⊓ 𝓟 s | filter.ext $ λ t, by simp only [mem_map', mem_inf_principal, mem_set_of_eq, mem_preimage] | lemma | filter.map_inf_principal_preimage | order.filter | src/order/filter/basic.lean | [
"control.traversable.instances",
"data.set.finite",
"order.copy",
"tactic.monotonicity"
] | [
"filter",
"filter.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto.not_tendsto {f : α → β} {a : filter α} {b₁ b₂ : filter β} (hf : tendsto f a b₁)
[ne_bot a] (hb : disjoint b₁ b₂) :
¬ tendsto f a b₂ | λ hf', (tendsto_inf.2 ⟨hf, hf'⟩).ne_bot.ne hb.eq_bot | lemma | filter.tendsto.not_tendsto | order.filter | src/order/filter/basic.lean | [
"control.traversable.instances",
"data.set.finite",
"order.copy",
"tactic.monotonicity"
] | [
"disjoint",
"filter"
] | If two filters are disjoint, then a function cannot tend to both of them along a non-trivial
filter. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
tendsto.if {l₁ : filter α} {l₂ : filter β} {f g : α → β} {p : α → Prop}
[∀ x, decidable (p x)] (h₀ : tendsto f (l₁ ⊓ 𝓟 {x | p x}) l₂)
(h₁ : tendsto g (l₁ ⊓ 𝓟 { x | ¬ p x }) l₂) :
tendsto (λ x, if p x then f x else g x) l₁ l₂ | begin
simp only [tendsto_def, mem_inf_principal] at *,
intros s hs,
filter_upwards [h₀ s hs, h₁ s hs],
simp only [mem_preimage],
intros x hp₀ hp₁,
split_ifs,
exacts [hp₀ h, hp₁ h],
end | lemma | filter.tendsto.if | order.filter | src/order/filter/basic.lean | [
"control.traversable.instances",
"data.set.finite",
"order.copy",
"tactic.monotonicity"
] | [
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto.if' {α β : Type*} {l₁ : filter α} {l₂ : filter β} {f g : α → β}
{p : α → Prop} [decidable_pred p] (hf : tendsto f l₁ l₂) (hg : tendsto g l₁ l₂) :
tendsto (λ a, if p a then f a else g a) l₁ l₂ | begin
replace hf : tendsto f (l₁ ⊓ 𝓟 {x | p x}) l₂ := tendsto_inf_left hf,
replace hg : tendsto g (l₁ ⊓ 𝓟 {x | ¬ p x}) l₂ := tendsto_inf_left hg,
exact hf.if hg,
end | lemma | filter.tendsto.if' | order.filter | src/order/filter/basic.lean | [
"control.traversable.instances",
"data.set.finite",
"order.copy",
"tactic.monotonicity"
] | [
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto.piecewise {l₁ : filter α} {l₂ : filter β} {f g : α → β}
{s : set α} [∀ x, decidable (x ∈ s)]
(h₀ : tendsto f (l₁ ⊓ 𝓟 s) l₂) (h₁ : tendsto g (l₁ ⊓ 𝓟 sᶜ) l₂) :
tendsto (piecewise s f g) l₁ l₂ | h₀.if h₁ | lemma | filter.tendsto.piecewise | order.filter | src/order/filter/basic.lean | [
"control.traversable.instances",
"data.set.finite",
"order.copy",
"tactic.monotonicity"
] | [
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
set.eq_on.eventually_eq {α β} {s : set α} {f g : α → β} (h : eq_on f g s) :
f =ᶠ[𝓟 s] g | h | lemma | set.eq_on.eventually_eq | order.filter | src/order/filter/basic.lean | [
"control.traversable.instances",
"data.set.finite",
"order.copy",
"tactic.monotonicity"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
set.eq_on.eventually_eq_of_mem {α β} {s : set α} {l : filter α} {f g : α → β}
(h : eq_on f g s) (hl : s ∈ l) :
f =ᶠ[l] g | h.eventually_eq.filter_mono $ filter.le_principal_iff.2 hl | lemma | set.eq_on.eventually_eq_of_mem | order.filter | src/order/filter/basic.lean | [
"control.traversable.instances",
"data.set.finite",
"order.copy",
"tactic.monotonicity"
] | [
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_subset.subset.eventually_le {α} {l : filter α} {s t : set α} (h : s ⊆ t) : s ≤ᶠ[l] t | filter.eventually_of_forall h | lemma | has_subset.subset.eventually_le | order.filter | src/order/filter/basic.lean | [
"control.traversable.instances",
"data.set.finite",
"order.copy",
"tactic.monotonicity"
] | [
"filter",
"filter.eventually_of_forall"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
set.maps_to.tendsto {α β} {s : set α} {t : set β} {f : α → β} (h : maps_to f s t) :
filter.tendsto f (𝓟 s) (𝓟 t) | filter.tendsto_principal_principal.2 h | lemma | set.maps_to.tendsto | order.filter | src/order/filter/basic.lean | [
"control.traversable.instances",
"data.set.finite",
"order.copy",
"tactic.monotonicity"
] | [
"filter.tendsto"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cofinite : filter α | { sets := {s | sᶜ.finite},
univ_sets := by simp only [compl_univ, finite_empty, mem_set_of_eq],
sets_of_superset := assume s t (hs : sᶜ.finite) (st: s ⊆ t),
hs.subset $ compl_subset_compl.2 st,
inter_sets := assume s t (hs : sᶜ.finite) (ht : tᶜ.finite),
by simp only [compl_inter, ... | def | filter.cofinite | order.filter | src/order/filter/cofinite.lean | [
"order.filter.at_top_bot",
"order.filter.pi"
] | [
"filter"
] | The cofinite filter is the filter of subsets whose complements are finite. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mem_cofinite {s : set α} : s ∈ (@cofinite α) ↔ sᶜ.finite | iff.rfl | lemma | filter.mem_cofinite | order.filter | src/order/filter/cofinite.lean | [
"order.filter.at_top_bot",
"order.filter.pi"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eventually_cofinite {p : α → Prop} :
(∀ᶠ x in cofinite, p x) ↔ {x | ¬p x}.finite | iff.rfl | lemma | filter.eventually_cofinite | order.filter | src/order/filter/cofinite.lean | [
"order.filter.at_top_bot",
"order.filter.pi"
] | [
"finite"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_basis_cofinite : has_basis cofinite (λ s : set α, s.finite) compl | ⟨λ s, ⟨λ h, ⟨sᶜ, h, (compl_compl s).subset⟩, λ ⟨t, htf, hts⟩, htf.subset $ compl_subset_comm.2 hts⟩⟩ | lemma | filter.has_basis_cofinite | order.filter | src/order/filter/cofinite.lean | [
"order.filter.at_top_bot",
"order.filter.pi"
] | [
"compl_compl"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cofinite_ne_bot [infinite α] : ne_bot (@cofinite α) | has_basis_cofinite.ne_bot_iff.2 $ λ s hs, hs.infinite_compl.nonempty | instance | filter.cofinite_ne_bot | order.filter | src/order/filter/cofinite.lean | [
"order.filter.at_top_bot",
"order.filter.pi"
] | [
"infinite"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
frequently_cofinite_iff_infinite {p : α → Prop} :
(∃ᶠ x in cofinite, p x) ↔ set.infinite {x | p x} | by simp only [filter.frequently, filter.eventually, mem_cofinite, compl_set_of, not_not,
set.infinite] | lemma | filter.frequently_cofinite_iff_infinite | order.filter | src/order/filter/cofinite.lean | [
"order.filter.at_top_bot",
"order.filter.pi"
] | [
"filter.eventually",
"filter.frequently",
"not_not",
"set.infinite"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.set.finite.compl_mem_cofinite {s : set α} (hs : s.finite) : sᶜ ∈ (@cofinite α) | mem_cofinite.2 $ (compl_compl s).symm ▸ hs | lemma | set.finite.compl_mem_cofinite | order.filter | src/order/filter/cofinite.lean | [
"order.filter.at_top_bot",
"order.filter.pi"
] | [
"compl_compl"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.set.finite.eventually_cofinite_nmem {s : set α} (hs : s.finite) :
∀ᶠ x in cofinite, x ∉ s | hs.compl_mem_cofinite | lemma | set.finite.eventually_cofinite_nmem | order.filter | src/order/filter/cofinite.lean | [
"order.filter.at_top_bot",
"order.filter.pi"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.finset.eventually_cofinite_nmem (s : finset α) : ∀ᶠ x in cofinite, x ∉ s | s.finite_to_set.eventually_cofinite_nmem | lemma | finset.eventually_cofinite_nmem | order.filter | src/order/filter/cofinite.lean | [
"order.filter.at_top_bot",
"order.filter.pi"
] | [
"finset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.set.infinite_iff_frequently_cofinite {s : set α} :
set.infinite s ↔ (∃ᶠ x in cofinite, x ∈ s) | frequently_cofinite_iff_infinite.symm | lemma | set.infinite_iff_frequently_cofinite | order.filter | src/order/filter/cofinite.lean | [
"order.filter.at_top_bot",
"order.filter.pi"
] | [
"set.infinite"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eventually_cofinite_ne (x : α) : ∀ᶠ a in cofinite, a ≠ x | (set.finite_singleton x).eventually_cofinite_nmem | lemma | filter.eventually_cofinite_ne | order.filter | src/order/filter/cofinite.lean | [
"order.filter.at_top_bot",
"order.filter.pi"
] | [
"set.finite_singleton"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_cofinite_iff_compl_singleton_mem : l ≤ cofinite ↔ ∀ x, {x}ᶜ ∈ l | begin
refine ⟨λ h x, h (finite_singleton x).compl_mem_cofinite, λ h s (hs : sᶜ.finite), _⟩,
rw [← compl_compl s, ← bUnion_of_singleton sᶜ, compl_Union₂,filter.bInter_mem hs],
exact λ x _, h x
end | lemma | filter.le_cofinite_iff_compl_singleton_mem | order.filter | src/order/filter/cofinite.lean | [
"order.filter.at_top_bot",
"order.filter.pi"
] | [
"compl_compl",
"filter.bInter_mem"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_cofinite_iff_eventually_ne : l ≤ cofinite ↔ ∀ x, ∀ᶠ y in l, y ≠ x | le_cofinite_iff_compl_singleton_mem | lemma | filter.le_cofinite_iff_eventually_ne | order.filter | src/order/filter/cofinite.lean | [
"order.filter.at_top_bot",
"order.filter.pi"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
at_top_le_cofinite [preorder α] [no_max_order α] : (at_top : filter α) ≤ cofinite | le_cofinite_iff_eventually_ne.mpr eventually_ne_at_top | lemma | filter.at_top_le_cofinite | order.filter | src/order/filter/cofinite.lean | [
"order.filter.at_top_bot",
"order.filter.pi"
] | [
"filter",
"no_max_order"
] | If `α` is a preorder with no maximal element, then `at_top ≤ cofinite`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
comap_cofinite_le (f : α → β) : comap f cofinite ≤ cofinite | le_cofinite_iff_eventually_ne.mpr $ λ x,
mem_comap.2 ⟨{f x}ᶜ, (finite_singleton _).compl_mem_cofinite, λ y, ne_of_apply_ne f⟩ | lemma | filter.comap_cofinite_le | order.filter | src/order/filter/cofinite.lean | [
"order.filter.at_top_bot",
"order.filter.pi"
] | [
"ne_of_apply_ne"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coprod_cofinite : (cofinite : filter α).coprod (cofinite : filter β) = cofinite | filter.coext $ λ s, by simp only [compl_mem_coprod, mem_cofinite, compl_compl,
finite_image_fst_and_snd_iff] | lemma | filter.coprod_cofinite | order.filter | src/order/filter/cofinite.lean | [
"order.filter.at_top_bot",
"order.filter.pi"
] | [
"compl_compl",
"filter",
"filter.coext"
] | The coproduct of the cofinite filters on two types is the cofinite filter on their product. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Coprod_cofinite {α : ι → Type*} [finite ι] :
filter.Coprod (λ i, (cofinite : filter (α i))) = cofinite | filter.coext $ λ s, by simp only [compl_mem_Coprod, mem_cofinite, compl_compl,
forall_finite_image_eval_iff] | lemma | filter.Coprod_cofinite | order.filter | src/order/filter/cofinite.lean | [
"order.filter.at_top_bot",
"order.filter.pi"
] | [
"compl_compl",
"filter",
"filter.Coprod",
"filter.coext",
"finite"
] | Finite product of finite sets is finite | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
disjoint_cofinite_left : disjoint cofinite l ↔ ∃ s ∈ l, set.finite s | begin
simp only [has_basis_cofinite.disjoint_iff l.basis_sets, id, disjoint_compl_left_iff_subset],
exact ⟨λ ⟨s, hs, t, ht, hts⟩, ⟨t, ht, hs.subset hts⟩, λ ⟨s, hs, hsf⟩, ⟨s, hsf, s, hs, subset.rfl⟩⟩
end | lemma | filter.disjoint_cofinite_left | order.filter | src/order/filter/cofinite.lean | [
"order.filter.at_top_bot",
"order.filter.pi"
] | [
"disjoint",
"set.finite"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
disjoint_cofinite_right : disjoint l cofinite ↔ ∃ s ∈ l, set.finite s | disjoint.comm.trans disjoint_cofinite_left | lemma | filter.disjoint_cofinite_right | order.filter | src/order/filter/cofinite.lean | [
"order.filter.at_top_bot",
"order.filter.pi"
] | [
"disjoint",
"set.finite"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nat.cofinite_eq_at_top : @cofinite ℕ = at_top | begin
refine le_antisymm _ at_top_le_cofinite,
refine at_top_basis.ge_iff.2 (λ N hN, _),
simpa only [mem_cofinite, compl_Ici] using finite_lt_nat N
end | lemma | nat.cofinite_eq_at_top | order.filter | src/order/filter/cofinite.lean | [
"order.filter.at_top_bot",
"order.filter.pi"
] | [] | For natural numbers the filters `cofinite` and `at_top` coincide. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
nat.frequently_at_top_iff_infinite {p : ℕ → Prop} :
(∃ᶠ n in at_top, p n) ↔ set.infinite {n | p n} | by rw [← nat.cofinite_eq_at_top, frequently_cofinite_iff_infinite] | lemma | nat.frequently_at_top_iff_infinite | order.filter | src/order/filter/cofinite.lean | [
"order.filter.at_top_bot",
"order.filter.pi"
] | [
"nat.cofinite_eq_at_top",
"set.infinite"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
filter.tendsto.exists_within_forall_le {α β : Type*} [linear_order β] {s : set α}
(hs : s.nonempty)
{f : α → β} (hf : filter.tendsto f filter.cofinite filter.at_top) :
∃ a₀ ∈ s, ∀ a ∈ s, f a₀ ≤ f a | begin
rcases em (∃ y ∈ s, ∃ x, f y < x) with ⟨y, hys, x, hx⟩|not_all_top,
{ -- the set of points `{y | f y < x}` is nonempty and finite, so we take `min` over this set
have : {y | ¬x ≤ f y}.finite := (filter.eventually_cofinite.mp (tendsto_at_top.1 hf x)),
simp only [not_le] at this,
obtain ⟨a₀, ⟨ha₀ : ... | lemma | filter.tendsto.exists_within_forall_le | order.filter | src/order/filter/cofinite.lean | [
"order.filter.at_top_bot",
"order.filter.pi"
] | [
"em",
"filter.at_top",
"filter.cofinite",
"filter.tendsto",
"finite"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
filter.tendsto.exists_forall_le [nonempty α] [linear_order β] {f : α → β}
(hf : tendsto f cofinite at_top) :
∃ a₀, ∀ a, f a₀ ≤ f a | let ⟨a₀, _, ha₀⟩ := hf.exists_within_forall_le univ_nonempty in ⟨a₀, λ a, ha₀ a (mem_univ _)⟩ | lemma | filter.tendsto.exists_forall_le | order.filter | src/order/filter/cofinite.lean | [
"order.filter.at_top_bot",
"order.filter.pi"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
filter.tendsto.exists_within_forall_ge [linear_order β] {s : set α} (hs : s.nonempty)
{f : α → β} (hf : filter.tendsto f filter.cofinite filter.at_bot) :
∃ a₀ ∈ s, ∀ a ∈ s, f a ≤ f a₀ | @filter.tendsto.exists_within_forall_le _ βᵒᵈ _ _ hs _ hf | lemma | filter.tendsto.exists_within_forall_ge | order.filter | src/order/filter/cofinite.lean | [
"order.filter.at_top_bot",
"order.filter.pi"
] | [
"filter.at_bot",
"filter.cofinite",
"filter.tendsto",
"filter.tendsto.exists_within_forall_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
filter.tendsto.exists_forall_ge [nonempty α] [linear_order β] {f : α → β}
(hf : tendsto f cofinite at_bot) :
∃ a₀, ∀ a, f a ≤ f a₀ | @filter.tendsto.exists_forall_le _ βᵒᵈ _ _ _ hf | lemma | filter.tendsto.exists_forall_ge | order.filter | src/order/filter/cofinite.lean | [
"order.filter.at_top_bot",
"order.filter.pi"
] | [
"filter.tendsto.exists_forall_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
function.injective.tendsto_cofinite {f : α → β} (hf : injective f) :
tendsto f cofinite cofinite | λ s h, h.preimage (hf.inj_on _) | lemma | function.injective.tendsto_cofinite | order.filter | src/order/filter/cofinite.lean | [
"order.filter.at_top_bot",
"order.filter.pi"
] | [] | For an injective function `f`, inverse images of finite sets are finite. See also
`filter.comap_cofinite_le` and `function.injective.comap_cofinite_eq`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
function.injective.comap_cofinite_eq {f : α → β} (hf : injective f) :
comap f cofinite = cofinite | (comap_cofinite_le f).antisymm hf.tendsto_cofinite.le_comap | lemma | function.injective.comap_cofinite_eq | order.filter | src/order/filter/cofinite.lean | [
"order.filter.at_top_bot",
"order.filter.pi"
] | [] | The pullback of the `filter.cofinite` under an injective function is equal to `filter.cofinite`.
See also `filter.comap_cofinite_le` and `function.injective.tendsto_cofinite`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
function.injective.nat_tendsto_at_top {f : ℕ → ℕ} (hf : injective f) :
tendsto f at_top at_top | nat.cofinite_eq_at_top ▸ hf.tendsto_cofinite | lemma | function.injective.nat_tendsto_at_top | order.filter | src/order/filter/cofinite.lean | [
"order.filter.at_top_bot",
"order.filter.pi"
] | [
"nat.cofinite_eq_at_top"
] | An injective sequence `f : ℕ → ℕ` tends to infinity at infinity. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
countable_Inter_filter (l : filter α) : Prop | (countable_sInter_mem' :
∀ {S : set (set α)} (hSc : S.countable) (hS : ∀ s ∈ S, s ∈ l), ⋂₀ S ∈ l) | class | countable_Inter_filter | order.filter | src/order/filter/countable_Inter.lean | [
"order.filter.basic",
"data.set.countable"
] | [
"filter"
] | A filter `l` has the countable intersection property if for any countable collection
of sets `s ∈ l` their intersection belongs to `l` as well. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
countable_sInter_mem {S : set (set α)} (hSc : S.countable) :
⋂₀ S ∈ l ↔ ∀ s ∈ S, s ∈ l | ⟨λ hS s hs, mem_of_superset hS (sInter_subset_of_mem hs),
countable_Inter_filter.countable_sInter_mem' hSc⟩ | lemma | countable_sInter_mem | order.filter | src/order/filter/countable_Inter.lean | [
"order.filter.basic",
"data.set.countable"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
countable_Inter_mem [countable ι] {s : ι → set α} : (⋂ i, s i) ∈ l ↔ ∀ i, s i ∈ l | sInter_range s ▸ (countable_sInter_mem (countable_range _)).trans forall_range_iff | lemma | countable_Inter_mem | order.filter | src/order/filter/countable_Inter.lean | [
"order.filter.basic",
"data.set.countable"
] | [
"countable",
"countable_sInter_mem"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
countable_bInter_mem {ι : Type*} {S : set ι} (hS : S.countable) {s : Π i ∈ S, set α} :
(⋂ i ∈ S, s i ‹_›) ∈ l ↔ ∀ i ∈ S, s i ‹_› ∈ l | begin
rw [bInter_eq_Inter],
haveI := hS.to_encodable,
exact countable_Inter_mem.trans subtype.forall
end | lemma | countable_bInter_mem | order.filter | src/order/filter/countable_Inter.lean | [
"order.filter.basic",
"data.set.countable"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eventually_countable_forall [countable ι] {p : α → ι → Prop} :
(∀ᶠ x in l, ∀ i, p x i) ↔ ∀ i, ∀ᶠ x in l, p x i | by simpa only [filter.eventually, set_of_forall]
using @countable_Inter_mem _ _ l _ _ (λ i, {x | p x i}) | lemma | eventually_countable_forall | order.filter | src/order/filter/countable_Inter.lean | [
"order.filter.basic",
"data.set.countable"
] | [
"countable",
"countable_Inter_mem",
"filter.eventually"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eventually_countable_ball {ι : Type*} {S : set ι} (hS : S.countable)
{p : Π (x : α) (i ∈ S), Prop} :
(∀ᶠ x in l, ∀ i ∈ S, p x i ‹_›) ↔ ∀ i ∈ S, ∀ᶠ x in l, p x i ‹_› | by simpa only [filter.eventually, set_of_forall]
using @countable_bInter_mem _ l _ _ _ hS (λ i hi, {x | p x i hi}) | lemma | eventually_countable_ball | order.filter | src/order/filter/countable_Inter.lean | [
"order.filter.basic",
"data.set.countable"
] | [
"countable_bInter_mem",
"filter.eventually"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eventually_le.countable_Union [countable ι] {s t : ι → set α} (h : ∀ i, s i ≤ᶠ[l] t i) :
(⋃ i, s i) ≤ᶠ[l] ⋃ i, t i | (eventually_countable_forall.2 h).mono $ λ x hst hs, mem_Union.2 $
(mem_Union.1 hs).imp hst | lemma | eventually_le.countable_Union | order.filter | src/order/filter/countable_Inter.lean | [
"order.filter.basic",
"data.set.countable"
] | [
"countable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eventually_eq.countable_Union [countable ι] {s t : ι → set α} (h : ∀ i, s i =ᶠ[l] t i) :
(⋃ i, s i) =ᶠ[l] ⋃ i, t i | (eventually_le.countable_Union (λ i, (h i).le)).antisymm
(eventually_le.countable_Union (λ i, (h i).symm.le)) | lemma | eventually_eq.countable_Union | order.filter | src/order/filter/countable_Inter.lean | [
"order.filter.basic",
"data.set.countable"
] | [
"countable",
"eventually_le.countable_Union"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eventually_le.countable_bUnion {ι : Type*} {S : set ι} (hS : S.countable)
{s t : Π i ∈ S, set α} (h : ∀ i ∈ S, s i ‹_› ≤ᶠ[l] t i ‹_›) :
(⋃ i ∈ S, s i ‹_›) ≤ᶠ[l] ⋃ i ∈ S, t i ‹_› | begin
simp only [bUnion_eq_Union],
haveI := hS.to_encodable,
exact eventually_le.countable_Union (λ i, h i i.2)
end | lemma | eventually_le.countable_bUnion | order.filter | src/order/filter/countable_Inter.lean | [
"order.filter.basic",
"data.set.countable"
] | [
"eventually_le.countable_Union"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eventually_eq.countable_bUnion {ι : Type*} {S : set ι} (hS : S.countable)
{s t : Π i ∈ S, set α} (h : ∀ i ∈ S, s i ‹_› =ᶠ[l] t i ‹_›) :
(⋃ i ∈ S, s i ‹_›) =ᶠ[l] ⋃ i ∈ S, t i ‹_› | (eventually_le.countable_bUnion hS (λ i hi, (h i hi).le)).antisymm
(eventually_le.countable_bUnion hS (λ i hi, (h i hi).symm.le)) | lemma | eventually_eq.countable_bUnion | order.filter | src/order/filter/countable_Inter.lean | [
"order.filter.basic",
"data.set.countable"
] | [
"eventually_le.countable_bUnion"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eventually_le.countable_Inter [countable ι] {s t : ι → set α} (h : ∀ i, s i ≤ᶠ[l] t i) :
(⋂ i, s i) ≤ᶠ[l] ⋂ i, t i | (eventually_countable_forall.2 h).mono $ λ x hst hs, mem_Inter.2 $ λ i, hst _ (mem_Inter.1 hs i) | lemma | eventually_le.countable_Inter | order.filter | src/order/filter/countable_Inter.lean | [
"order.filter.basic",
"data.set.countable"
] | [
"countable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eventually_eq.countable_Inter [countable ι] {s t : ι → set α} (h : ∀ i, s i =ᶠ[l] t i) :
(⋂ i, s i) =ᶠ[l] ⋂ i, t i | (eventually_le.countable_Inter (λ i, (h i).le)).antisymm
(eventually_le.countable_Inter (λ i, (h i).symm.le)) | lemma | eventually_eq.countable_Inter | order.filter | src/order/filter/countable_Inter.lean | [
"order.filter.basic",
"data.set.countable"
] | [
"countable",
"eventually_le.countable_Inter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eventually_le.countable_bInter {ι : Type*} {S : set ι} (hS : S.countable)
{s t : Π i ∈ S, set α} (h : ∀ i ∈ S, s i ‹_› ≤ᶠ[l] t i ‹_›) :
(⋂ i ∈ S, s i ‹_›) ≤ᶠ[l] ⋂ i ∈ S, t i ‹_› | begin
simp only [bInter_eq_Inter],
haveI := hS.to_encodable,
exact eventually_le.countable_Inter (λ i, h i i.2)
end | lemma | eventually_le.countable_bInter | order.filter | src/order/filter/countable_Inter.lean | [
"order.filter.basic",
"data.set.countable"
] | [
"eventually_le.countable_Inter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eventually_eq.countable_bInter {ι : Type*} {S : set ι} (hS : S.countable)
{s t : Π i ∈ S, set α} (h : ∀ i ∈ S, s i ‹_› =ᶠ[l] t i ‹_›) :
(⋂ i ∈ S, s i ‹_›) =ᶠ[l] ⋂ i ∈ S, t i ‹_› | (eventually_le.countable_bInter hS (λ i hi, (h i hi).le)).antisymm
(eventually_le.countable_bInter hS (λ i hi, (h i hi).symm.le)) | lemma | eventually_eq.countable_bInter | order.filter | src/order/filter/countable_Inter.lean | [
"order.filter.basic",
"data.set.countable"
] | [
"eventually_le.countable_bInter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
filter.of_countable_Inter (l : set (set α))
(hp : ∀ S : set (set α), S.countable → S ⊆ l → (⋂₀ S) ∈ l)
(h_mono : ∀ s t, s ∈ l → s ⊆ t → t ∈ l) :
filter α | { sets := l,
univ_sets := @sInter_empty α ▸ hp _ countable_empty (empty_subset _),
sets_of_superset := h_mono,
inter_sets := λ s t hs ht, sInter_pair s t ▸
hp _ ((countable_singleton _).insert _) (insert_subset.2 ⟨hs, singleton_subset_iff.2 ht⟩) } | def | filter.of_countable_Inter | order.filter | src/order/filter/countable_Inter.lean | [
"order.filter.basic",
"data.set.countable"
] | [
"filter"
] | Construct a filter with countable intersection property. This constructor deduces
`filter.univ_sets` and `filter.inter_sets` from the countable intersection property. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
filter.countable_Inter_of_countable_Inter (l : set (set α))
(hp : ∀ S : set (set α), S.countable → S ⊆ l → (⋂₀ S) ∈ l)
(h_mono : ∀ s t, s ∈ l → s ⊆ t → t ∈ l) :
countable_Inter_filter (filter.of_countable_Inter l hp h_mono) | ⟨hp⟩ | instance | filter.countable_Inter_of_countable_Inter | order.filter | src/order/filter/countable_Inter.lean | [
"order.filter.basic",
"data.set.countable"
] | [
"countable_Inter_filter",
"filter.of_countable_Inter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
filter.mem_of_countable_Inter {l : set (set α)}
(hp : ∀ S : set (set α), S.countable → S ⊆ l → (⋂₀ S) ∈ l)
(h_mono : ∀ s t, s ∈ l → s ⊆ t → t ∈ l) {s : set α} :
s ∈ filter.of_countable_Inter l hp h_mono ↔ s ∈ l | iff.rfl | lemma | filter.mem_of_countable_Inter | order.filter | src/order/filter/countable_Inter.lean | [
"order.filter.basic",
"data.set.countable"
] | [
"filter.of_countable_Inter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
countable_Inter_filter_principal (s : set α) : countable_Inter_filter (𝓟 s) | ⟨λ S hSc hS, subset_sInter hS⟩ | instance | countable_Inter_filter_principal | order.filter | src/order/filter/countable_Inter.lean | [
"order.filter.basic",
"data.set.countable"
] | [
"countable_Inter_filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
countable_Inter_filter_bot : countable_Inter_filter (⊥ : filter α) | by { rw ← principal_empty, apply countable_Inter_filter_principal } | instance | countable_Inter_filter_bot | order.filter | src/order/filter/countable_Inter.lean | [
"order.filter.basic",
"data.set.countable"
] | [
"countable_Inter_filter",
"countable_Inter_filter_principal",
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
countable_Inter_filter_top : countable_Inter_filter (⊤ : filter α) | by { rw ← principal_univ, apply countable_Inter_filter_principal } | instance | countable_Inter_filter_top | order.filter | src/order/filter/countable_Inter.lean | [
"order.filter.basic",
"data.set.countable"
] | [
"countable_Inter_filter",
"countable_Inter_filter_principal",
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
countable_Inter_filter_inf (l₁ l₂ : filter α) [countable_Inter_filter l₁]
[countable_Inter_filter l₂] :
countable_Inter_filter (l₁ ⊓ l₂) | begin
refine ⟨λ S hSc hS, _⟩,
choose s hs t ht hst using hS,
replace hs : (⋂ i ∈ S, s i ‹_›) ∈ l₁ := (countable_bInter_mem hSc).2 hs,
replace ht : (⋂ i ∈ S, t i ‹_›) ∈ l₂ := (countable_bInter_mem hSc).2 ht,
refine mem_of_superset (inter_mem_inf hs ht) (subset_sInter $ λ i hi, _),
rw hst i hi,
apply inter_... | instance | countable_Inter_filter_inf | order.filter | src/order/filter/countable_Inter.lean | [
"order.filter.basic",
"data.set.countable"
] | [
"countable_Inter_filter",
"countable_bInter_mem",
"filter"
] | Infimum of two `countable_Inter_filter`s is a `countable_Inter_filter`. This is useful, e.g.,
to automatically get an instance for `residual α ⊓ 𝓟 s`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
countable_Inter_filter_sup (l₁ l₂ : filter α) [countable_Inter_filter l₁]
[countable_Inter_filter l₂] :
countable_Inter_filter (l₁ ⊔ l₂) | begin
refine ⟨λ S hSc hS, ⟨_, _⟩⟩; refine (countable_sInter_mem hSc).2 (λ s hs, _),
exacts [(hS s hs).1, (hS s hs).2]
end | instance | countable_Inter_filter_sup | order.filter | src/order/filter/countable_Inter.lean | [
"order.filter.basic",
"data.set.countable"
] | [
"countable_Inter_filter",
"countable_sInter_mem",
"filter"
] | Supremum of two `countable_Inter_filter`s is a `countable_Inter_filter`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
countable_generate_sets : set α → Prop
| basic {s : set α} : s ∈ g → countable_generate_sets s
| univ : countable_generate_sets univ
| superset {s t : set α} : countable_generate_sets s → s ⊆ t → countable_generate_sets t
| Inter {S : set (set α)} : S.countable →
(∀ s ∈ S, countable_generate... | inductive | filter.countable_generate_sets | order.filter | src/order/filter/countable_Inter.lean | [
"order.filter.basic",
"data.set.countable"
] | [] | `filter.countable_generate_sets g` is the (sets of the)
greatest `countable_Inter_filter` containing `g`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
countable_generate : filter α | of_countable_Inter (countable_generate_sets g) (λ S, countable_generate_sets.Inter)
(λ s t, countable_generate_sets.superset) | def | filter.countable_generate | order.filter | src/order/filter/countable_Inter.lean | [
"order.filter.basic",
"data.set.countable"
] | [
"filter"
] | `filter.countable_generate g` is the greatest `countable_Inter_filter` containing `g`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mem_countable_generate_iff {s : set α} : s ∈ countable_generate g ↔
∃ (S : set (set α)), S ⊆ g ∧ S.countable ∧ ⋂₀ S ⊆ s | begin
split; intro h,
{ induction h with s hs s t hs st ih S Sct hS ih,
{ exact ⟨{s}, by simp[hs]⟩ },
{ exact ⟨∅, by simp⟩ },
{ refine exists_imp_exists (λ S, _) ih,
tauto },
choose T Tg Tct hT using ih,
refine ⟨⋃ s (H : s ∈ S), T s H, by simpa, Sct.bUnion Tct, _⟩,
apply subset_sInter,... | lemma | filter.mem_countable_generate_iff | order.filter | src/order/filter/countable_Inter.lean | [
"order.filter.basic",
"data.set.countable"
] | [
"countable_sInter_mem",
"ih",
"subset_trans"
] | A set is in the `countable_Inter_filter` generated by `g` if and only if
it contains a countable intersection of elements of `g`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
le_countable_generate_iff_of_countable_Inter_filter {f : filter α}
[countable_Inter_filter f] : f ≤ countable_generate g ↔ g ⊆ f.sets | begin
split; intro h,
{ exact subset_trans (λ s, countable_generate_sets.basic) h },
intros s hs,
induction hs with s hs s t hs st ih S Sct hS ih,
{ exact h hs },
{ exact univ_mem },
{ exact mem_of_superset ih st, },
exact (countable_sInter_mem Sct).mpr ih,
end | lemma | filter.le_countable_generate_iff_of_countable_Inter_filter | order.filter | src/order/filter/countable_Inter.lean | [
"order.filter.basic",
"data.set.countable"
] | [
"countable_Inter_filter",
"countable_sInter_mem",
"filter",
"ih",
"subset_trans"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
countable_generate_is_greatest : is_greatest
{f : filter α | countable_Inter_filter f ∧ g ⊆ f.sets} (countable_generate g) | begin
refine ⟨⟨infer_instance, λ s, countable_generate_sets.basic⟩, _⟩,
rintros f ⟨fct, hf⟩,
rwa @le_countable_generate_iff_of_countable_Inter_filter _ _ _ fct,
end | lemma | filter.countable_generate_is_greatest | order.filter | src/order/filter/countable_Inter.lean | [
"order.filter.basic",
"data.set.countable"
] | [
"countable_Inter_filter",
"filter",
"is_greatest"
] | `countable_generate g` is the greatest `countable_Inter_filter` containing `g`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
curry (f : filter α) (g : filter β) : filter (α × β) | { sets := { s | ∀ᶠ (a : α) in f, ∀ᶠ (b : β) in g, (a, b) ∈ s },
univ_sets := (by simp only [set.mem_set_of_eq, set.mem_univ, eventually_true]),
sets_of_superset := begin
intros x y hx hxy,
simp only [set.mem_set_of_eq] at hx ⊢,
exact hx.mono (λ a ha, ha.mono(λ b hb, set.mem_of_subset_of_mem hxy hb)),
... | def | filter.curry | order.filter | src/order/filter/curry.lean | [
"order.filter.prod"
] | [
"filter",
"set.mem_inter_iff",
"set.mem_of_subset_of_mem",
"set.mem_univ"
] | This filter is characterized by `filter.eventually_curry_iff`:
`(∀ᶠ (x : α × β) in f.curry g, p x) ↔ ∀ᶠ (x : α) in f, ∀ᶠ (y : β) in g, p (x, y)`. Useful
in adding quantifiers to the middle of `tendsto`s. See
`has_fderiv_at_of_tendsto_uniformly_on_filter`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
eventually_curry_iff {f : filter α} {g : filter β} {p : α × β → Prop} :
(∀ᶠ (x : α × β) in f.curry g, p x) ↔ ∀ᶠ (x : α) in f, ∀ᶠ (y : β) in g, p (x, y) | iff.rfl | lemma | filter.eventually_curry_iff | order.filter | src/order/filter/curry.lean | [
"order.filter.prod"
] | [
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
curry_le_prod {f : filter α} {g : filter β} :
f.curry g ≤ f.prod g | begin
intros u hu,
rw ←eventually_mem_set at hu ⊢,
rw eventually_curry_iff,
exact hu.curry,
end | lemma | filter.curry_le_prod | order.filter | src/order/filter/curry.lean | [
"order.filter.prod"
] | [
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto.curry {f : α → β → γ} {la : filter α} {lb : filter β} {lc : filter γ} :
(∀ᶠ a in la, tendsto (λ b : β, f a b) lb lc) → tendsto ↿f (la.curry lb) lc | begin
intros h,
rw tendsto_def,
simp only [curry, filter.mem_mk, set.mem_set_of_eq, set.mem_preimage],
simp_rw tendsto_def at h,
refine (λ s hs, h.mono (λ a ha, eventually_iff.mpr _)),
simpa [function.has_uncurry.uncurry, set.preimage] using ha s hs,
end | lemma | filter.tendsto.curry | order.filter | src/order/filter/curry.lean | [
"order.filter.prod"
] | [
"filter",
"filter.mem_mk",
"set.mem_preimage",
"set.preimage"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eventually_le_limsup [countable_Inter_filter f] (u : α → ℝ≥0∞) :
∀ᶠ y in f, u y ≤ f.limsup u | eventually_le_limsup | lemma | ennreal.eventually_le_limsup | order.filter | src/order/filter/ennreal.lean | [
"topology.instances.ennreal"
] | [
"countable_Inter_filter",
"eventually_le_limsup"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
limsup_eq_zero_iff [countable_Inter_filter f] {u : α → ℝ≥0∞} :
f.limsup u = 0 ↔ u =ᶠ[f] 0 | limsup_eq_bot | lemma | ennreal.limsup_eq_zero_iff | order.filter | src/order/filter/ennreal.lean | [
"topology.instances.ennreal"
] | [
"countable_Inter_filter",
"limsup_eq_bot"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
limsup_const_mul_of_ne_top {u : α → ℝ≥0∞} {a : ℝ≥0∞} (ha_top : a ≠ ⊤) :
f.limsup (λ (x : α), a * (u x)) = a * f.limsup u | begin
by_cases ha_zero : a = 0,
{ simp_rw [ha_zero, zero_mul, ←ennreal.bot_eq_zero],
exact limsup_const_bot, },
let g := λ x : ℝ≥0∞, a * x,
have hg_bij : function.bijective g,
from function.bijective_iff_has_inverse.mpr ⟨(λ x, a⁻¹ * x),
⟨λ x, by simp [←mul_assoc, ennreal.inv_mul_cancel ha_zero ha_top]... | lemma | ennreal.limsup_const_mul_of_ne_top | order.filter | src/order/filter/ennreal.lean | [
"topology.instances.ennreal"
] | [
"ennreal.inv_mul_cancel",
"ennreal.mul_inv_cancel",
"monotone.strict_mono_of_injective",
"mul_le_mul_left",
"order_iso.limsup_apply",
"strict_mono",
"strict_mono.order_iso_of_surjective",
"zero_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
limsup_const_mul [countable_Inter_filter f] {u : α → ℝ≥0∞} {a : ℝ≥0∞} :
f.limsup (λ (x : α), a * (u x)) = a * f.limsup u | begin
by_cases ha_top : a ≠ ⊤,
{ exact limsup_const_mul_of_ne_top ha_top, },
push_neg at ha_top,
by_cases hu : u =ᶠ[f] 0,
{ have hau : (λ x, a * (u x)) =ᶠ[f] 0,
{ refine hu.mono (λ x hx, _),
rw pi.zero_apply at hx,
simp [hx], },
simp only [limsup_congr hu, limsup_congr hau, pi.zero_apply, ... | lemma | ennreal.limsup_const_mul | order.filter | src/order/filter/ennreal.lean | [
"topology.instances.ennreal"
] | [
"countable_Inter_filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
limsup_mul_le [countable_Inter_filter f] (u v : α → ℝ≥0∞) :
f.limsup (u * v) ≤ f.limsup u * f.limsup v | calc f.limsup (u * v) ≤ f.limsup (λ x, (f.limsup u) * v x) :
begin
refine limsup_le_limsup _ _,
{ filter_upwards [@eventually_le_limsup _ f _ u] with x hx using mul_le_mul_right' hx _ },
{ is_bounded_default, },
end
... = f.limsup u * f.limsup v : limsup_const_mul | lemma | ennreal.limsup_mul_le | order.filter | src/order/filter/ennreal.lean | [
"topology.instances.ennreal"
] | [
"countable_Inter_filter",
"eventually_le_limsup",
"mul_le_mul_right'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
limsup_add_le [countable_Inter_filter f] (u v : α → ℝ≥0∞) :
f.limsup (u + v) ≤ f.limsup u + f.limsup v | Inf_le ((eventually_le_limsup u).mp ((eventually_le_limsup v).mono
(λ _ hxg hxf, add_le_add hxf hxg))) | lemma | ennreal.limsup_add_le | order.filter | src/order/filter/ennreal.lean | [
"topology.instances.ennreal"
] | [
"Inf_le",
"countable_Inter_filter",
"eventually_le_limsup"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
limsup_liminf_le_liminf_limsup {β} [countable β] {f : filter α} [countable_Inter_filter f]
{g : filter β} (u : α → β → ℝ≥0∞) :
f.limsup (λ (a : α), g.liminf (λ (b : β), u a b)) ≤ g.liminf (λ b, f.limsup (λ a, u a b)) | begin
have h1 : ∀ᶠ a in f, ∀ b, u a b ≤ f.limsup (λ a', u a' b),
by { rw eventually_countable_forall, exact λ b, ennreal.eventually_le_limsup (λ a, u a b), },
refine Inf_le (h1.mono (λ x hx, filter.liminf_le_liminf (filter.eventually_of_forall hx) _)),
filter.is_bounded_default,
end | lemma | ennreal.limsup_liminf_le_liminf_limsup | order.filter | src/order/filter/ennreal.lean | [
"topology.instances.ennreal"
] | [
"Inf_le",
"countable",
"countable_Inter_filter",
"ennreal.eventually_le_limsup",
"eventually_countable_forall",
"filter",
"filter.eventually_of_forall",
"filter.is_bounded_default",
"filter.liminf_le_liminf"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_min_filter : Prop | ∀ᶠ x in l, f a ≤ f x | def | is_min_filter | order.filter | src/order/filter/extr.lean | [
"order.filter.basic"
] | [] | `is_min_filter f l a` means that `f a ≤ f x` in some `l`-neighborhood of `a` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_max_filter : Prop | ∀ᶠ x in l, f x ≤ f a | def | is_max_filter | order.filter | src/order/filter/extr.lean | [
"order.filter.basic"
] | [] | `is_max_filter f l a` means that `f x ≤ f a` in some `l`-neighborhood of `a` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_extr_filter : Prop | is_min_filter f l a ∨ is_max_filter f l a | def | is_extr_filter | order.filter | src/order/filter/extr.lean | [
"order.filter.basic"
] | [
"is_max_filter",
"is_min_filter"
] | `is_extr_filter f l a` means `is_min_filter f l a` or `is_max_filter f l a` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_min_on | is_min_filter f (𝓟 s) a | def | is_min_on | order.filter | src/order/filter/extr.lean | [
"order.filter.basic"
] | [
"is_min_filter"
] | `is_min_on f s a` means that `f a ≤ f x` for all `x ∈ a`. Note that we do not assume `a ∈ s`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_max_on | is_max_filter f (𝓟 s) a | def | is_max_on | order.filter | src/order/filter/extr.lean | [
"order.filter.basic"
] | [
"is_max_filter"
] | `is_max_on f s a` means that `f x ≤ f a` for all `x ∈ a`. Note that we do not assume `a ∈ s`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_extr_on : Prop | is_extr_filter f (𝓟 s) a | def | is_extr_on | order.filter | src/order/filter/extr.lean | [
"order.filter.basic"
] | [
"is_extr_filter"
] | `is_extr_on f s a` means `is_min_on f s a` or `is_max_on f s a` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_extr_on.elim {p : Prop} :
is_extr_on f s a → (is_min_on f s a → p) → (is_max_on f s a → p) → p | or.elim | lemma | is_extr_on.elim | order.filter | src/order/filter/extr.lean | [
"order.filter.basic"
] | [
"is_extr_on",
"is_max_on",
"is_min_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_min_on_iff : is_min_on f s a ↔ ∀ x ∈ s, f a ≤ f x | iff.rfl | lemma | is_min_on_iff | order.filter | src/order/filter/extr.lean | [
"order.filter.basic"
] | [
"is_min_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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