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
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|---|---|---|---|---|---|---|---|---|---|---|
const_lt [preorder β] {x y : β} : x < y → (↑x : β*) < ↑y | coe_lt.mpr ∘ lift_rel_const | lemma | filter.germ.const_lt | order.filter | src/order/filter/filter_product.lean | [
"order.filter.ultrafilter",
"order.filter.germ"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
const_lt_iff [preorder β] {x y : β} : (↑x : β*) < ↑y ↔ x < y | coe_lt.trans lift_rel_const_iff | lemma | filter.germ.const_lt_iff | order.filter | src/order/filter/filter_product.lean | [
"order.filter.ultrafilter",
"order.filter.germ"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lt_def [preorder β] : ((<) : β* → β* → Prop) = lift_rel (<) | by { ext ⟨f⟩ ⟨g⟩, exact coe_lt } | lemma | filter.germ.lt_def | order.filter | src/order/filter/filter_product.lean | [
"order.filter.ultrafilter",
"order.filter.germ"
] | [
"lift_rel"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
const_sup [has_sup β] (a b : β) : ↑(a ⊔ b) = (↑a ⊔ ↑b : β*) | rfl | lemma | filter.germ.const_sup | order.filter | src/order/filter/filter_product.lean | [
"order.filter.ultrafilter",
"order.filter.germ"
] | [
"has_sup"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
const_inf [has_inf β] (a b : β) : ↑(a ⊓ b) = (↑a ⊓ ↑b : β*) | rfl | lemma | filter.germ.const_inf | order.filter | src/order/filter/filter_product.lean | [
"order.filter.ultrafilter",
"order.filter.germ"
] | [
"has_inf"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
max_def [linear_order β] (x y : β*) : max x y = map₂ max x y | induction_on₂ x y $ λ a b,
begin
cases le_total (a : β*) b,
{ rw [max_eq_right h, map₂_coe, coe_eq], exact h.mono (λ i hi, (max_eq_right hi).symm) },
{ rw [max_eq_left h, map₂_coe, coe_eq], exact h.mono (λ i hi, (max_eq_left hi).symm) }
end | lemma | filter.germ.max_def | order.filter | src/order/filter/filter_product.lean | [
"order.filter.ultrafilter",
"order.filter.germ"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
min_def [K : linear_order β] (x y : β*) : min x y = map₂ min x y | induction_on₂ x y $ λ a b,
begin
cases le_total (a : β*) b,
{ rw [min_eq_left h, map₂_coe, coe_eq], exact h.mono (λ i hi, (min_eq_left hi).symm) },
{ rw [min_eq_right h, map₂_coe, coe_eq], exact h.mono (λ i hi, (min_eq_right hi).symm) }
end | lemma | filter.germ.min_def | order.filter | src/order/filter/filter_product.lean | [
"order.filter.ultrafilter",
"order.filter.germ"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
abs_def [linear_ordered_add_comm_group β] (x : β*) : |x| = map abs x | induction_on x $ λ a, by exact rfl | lemma | filter.germ.abs_def | order.filter | src/order/filter/filter_product.lean | [
"order.filter.ultrafilter",
"order.filter.germ"
] | [
"linear_ordered_add_comm_group"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
const_max [linear_order β] (x y : β) : (↑(max x y : β) : β*) = max ↑x ↑y | by rw [max_def, map₂_const] | lemma | filter.germ.const_max | order.filter | src/order/filter/filter_product.lean | [
"order.filter.ultrafilter",
"order.filter.germ"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
const_min [linear_order β] (x y : β) : (↑(min x y : β) : β*) = min ↑x ↑y | by rw [min_def, map₂_const] | lemma | filter.germ.const_min | order.filter | src/order/filter/filter_product.lean | [
"order.filter.ultrafilter",
"order.filter.germ"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
const_abs [linear_ordered_add_comm_group β] (x : β) :
(↑(|x|) : β*) = |↑x| | by rw [abs_def, map_const] | lemma | filter.germ.const_abs | order.filter | src/order/filter/filter_product.lean | [
"order.filter.ultrafilter",
"order.filter.germ"
] | [
"linear_ordered_add_comm_group"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
const_eventually_eq' [ne_bot l] {a b : β} : (∀ᶠ x in l, a = b) ↔ a = b | eventually_const | lemma | filter.const_eventually_eq' | order.filter | src/order/filter/germ.lean | [
"order.filter.basic",
"algebra.module.pi"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
const_eventually_eq [ne_bot l] {a b : β} : ((λ _, a) =ᶠ[l] (λ _, b)) ↔ a = b | @const_eventually_eq' _ _ _ _ a b | lemma | filter.const_eventually_eq | order.filter | src/order/filter/germ.lean | [
"order.filter.basic",
"algebra.module.pi"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eventually_eq.comp_tendsto {f' : α → β} (H : f =ᶠ[l] f') {g : γ → α} {lc : filter γ}
(hg : tendsto g lc l) :
f ∘ g =ᶠ[lc] f' ∘ g | hg.eventually H | lemma | filter.eventually_eq.comp_tendsto | order.filter | src/order/filter/germ.lean | [
"order.filter.basic",
"algebra.module.pi"
] | [
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
germ_setoid (l : filter α) (β : Type*) : setoid (α → β) | { r := eventually_eq l,
iseqv := ⟨eventually_eq.refl _, λ _ _, eventually_eq.symm, λ _ _ _, eventually_eq.trans⟩ } | def | filter.germ_setoid | order.filter | src/order/filter/germ.lean | [
"order.filter.basic",
"algebra.module.pi"
] | [
"filter"
] | Setoid used to define the space of germs. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
germ (l : filter α) (β : Type*) : Type* | quotient (germ_setoid l β) | def | filter.germ | order.filter | src/order/filter/germ.lean | [
"order.filter.basic",
"algebra.module.pi"
] | [
"filter"
] | The space of germs of functions `α → β` at a filter `l`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
product_setoid (l : filter α) (ε : α → Type*) : setoid (Π a, ε a) | { r := λ f g, ∀ᶠ a in l, f a = g a,
iseqv := ⟨λ _, eventually_of_forall (λ _, rfl),
λ _ _ h, h.mono (λ _, eq.symm),
λ x y z h1 h2, h1.congr (h2.mono (λ x hx, hx ▸ iff.rfl))⟩ } | def | filter.product_setoid | order.filter | src/order/filter/germ.lean | [
"order.filter.basic",
"algebra.module.pi"
] | [
"filter"
] | Setoid used to define the filter product. This is a dependent version of
`filter.germ_setoid`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
product (l : filter α) (ε : α → Type*) : Type* | quotient (product_setoid l ε) | def | filter.product | order.filter | src/order/filter/germ.lean | [
"order.filter.basic",
"algebra.module.pi"
] | [
"filter"
] | The filter product `Π (a : α), ε a` at a filter `l`. This is a dependent version of
`filter.germ`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
quot_mk_eq_coe (l : filter α) (f : α → β) : quot.mk _ f = (f : germ l β) | rfl | lemma | filter.germ.quot_mk_eq_coe | order.filter | src/order/filter/germ.lean | [
"order.filter.basic",
"algebra.module.pi"
] | [
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk'_eq_coe (l : filter α) (f : α → β) : quotient.mk' f = (f : germ l β) | rfl | lemma | filter.germ.mk'_eq_coe | order.filter | src/order/filter/germ.lean | [
"order.filter.basic",
"algebra.module.pi"
] | [
"filter",
"quotient.mk'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
induction_on (f : germ l β) {p : germ l β → Prop} (h : ∀ f : α → β, p f) : p f | quotient.induction_on' f h | lemma | filter.germ.induction_on | order.filter | src/order/filter/germ.lean | [
"order.filter.basic",
"algebra.module.pi"
] | [
"quotient.induction_on'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
induction_on₂ (f : germ l β) (g : germ l γ) {p : germ l β → germ l γ → Prop}
(h : ∀ (f : α → β) (g : α → γ), p f g) : p f g | quotient.induction_on₂' f g h | lemma | filter.germ.induction_on₂ | order.filter | src/order/filter/germ.lean | [
"order.filter.basic",
"algebra.module.pi"
] | [
"quotient.induction_on₂'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
induction_on₃ (f : germ l β) (g : germ l γ) (h : germ l δ)
{p : germ l β → germ l γ → germ l δ → Prop}
(H : ∀ (f : α → β) (g : α → γ) (h : α → δ), p f g h) :
p f g h | quotient.induction_on₃' f g h H | lemma | filter.germ.induction_on₃ | order.filter | src/order/filter/germ.lean | [
"order.filter.basic",
"algebra.module.pi"
] | [
"quotient.induction_on₃'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map' {lc : filter γ} (F : (α → β) → (γ → δ)) (hF : (l.eventually_eq ⇒ lc.eventually_eq) F F) :
germ l β → germ lc δ | quotient.map' F hF | def | filter.germ.map' | order.filter | src/order/filter/germ.lean | [
"order.filter.basic",
"algebra.module.pi"
] | [
"filter",
"quotient.map'"
] | Given a map `F : (α → β) → (γ → δ)` that sends functions eventually equal at `l` to functions
eventually equal at `lc`, returns a map from `germ l β` to `germ lc δ`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lift_on {γ : Sort*} (f : germ l β) (F : (α → β) → γ) (hF : (l.eventually_eq ⇒ (=)) F F) : γ | quotient.lift_on' f F hF | def | filter.germ.lift_on | order.filter | src/order/filter/germ.lean | [
"order.filter.basic",
"algebra.module.pi"
] | [
"quotient.lift_on'"
] | Given a germ `f : germ l β` and a function `F : (α → β) → γ` sending eventually equal functions
to the same value, returns the value `F` takes on functions having germ `f` at `l`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
map'_coe {lc : filter γ} (F : (α → β) → (γ → δ))
(hF : (l.eventually_eq ⇒ lc.eventually_eq) F F) (f : α → β) :
map' F hF f = F f | rfl | lemma | filter.germ.map'_coe | order.filter | src/order/filter/germ.lean | [
"order.filter.basic",
"algebra.module.pi"
] | [
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_eq : (f : germ l β) = g ↔ (f =ᶠ[l] g) | quotient.eq' | lemma | filter.germ.coe_eq | order.filter | src/order/filter/germ.lean | [
"order.filter.basic",
"algebra.module.pi"
] | [
"quotient.eq'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map (op : β → γ) : germ l β → germ l γ | map' ((∘) op) $ λ f g H, H.mono $ λ x H, congr_arg op H | def | filter.germ.map | order.filter | src/order/filter/germ.lean | [
"order.filter.basic",
"algebra.module.pi"
] | [] | Lift a function `β → γ` to a function `germ l β → germ l γ`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
map_coe (op : β → γ) (f : α → β) : map op (f : germ l β) = op ∘ f | rfl | lemma | filter.germ.map_coe | order.filter | src/order/filter/germ.lean | [
"order.filter.basic",
"algebra.module.pi"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_id : map id = (id : germ l β → germ l β) | by { ext ⟨f⟩, refl } | lemma | filter.germ.map_id | order.filter | src/order/filter/germ.lean | [
"order.filter.basic",
"algebra.module.pi"
] | [
"map_id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_map (op₁ : γ → δ) (op₂ : β → γ) (f : germ l β) :
map op₁ (map op₂ f) = map (op₁ ∘ op₂) f | induction_on f $ λ f, rfl | lemma | filter.germ.map_map | order.filter | src/order/filter/germ.lean | [
"order.filter.basic",
"algebra.module.pi"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map₂ (op : β → γ → δ) : germ l β → germ l γ → germ l δ | quotient.map₂' (λ f g x, op (f x) (g x)) $ λ f f' Hf g g' Hg,
Hg.mp $ Hf.mono $ λ x Hf Hg, by simp only [Hf, Hg] | def | filter.germ.map₂ | order.filter | src/order/filter/germ.lean | [
"order.filter.basic",
"algebra.module.pi"
] | [
"quotient.map₂'"
] | Lift a binary function `β → γ → δ` to a function `germ l β → germ l γ → germ l δ`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
map₂_coe (op : β → γ → δ) (f : α → β) (g : α → γ) :
map₂ op (f : germ l β) g = λ x, op (f x) (g x) | rfl | lemma | filter.germ.map₂_coe | order.filter | src/order/filter/germ.lean | [
"order.filter.basic",
"algebra.module.pi"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto (f : germ l β) (lb : filter β) : Prop | lift_on f (λ f, tendsto f l lb) $ λ f g H, propext (tendsto_congr' H) | def | filter.germ.tendsto | order.filter | src/order/filter/germ.lean | [
"order.filter.basic",
"algebra.module.pi"
] | [
"filter"
] | A germ at `l` of maps from `α` to `β` tends to `lb : filter β` if it is represented by a map
which tends to `lb` along `l`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_tendsto {f : α → β} {lb : filter β} :
(f : germ l β).tendsto lb ↔ tendsto f l lb | iff.rfl | lemma | filter.germ.coe_tendsto | order.filter | src/order/filter/germ.lean | [
"order.filter.basic",
"algebra.module.pi"
] | [
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_tendsto' (f : germ l β) {lc : filter γ} (g : germ lc α) (hg : g.tendsto l) :
germ lc β | lift_on f (λ f, g.map f) $ λ f₁ f₂ hF, (induction_on g $ λ g hg, coe_eq.2 $ hg.eventually hF) hg | def | filter.germ.comp_tendsto' | order.filter | src/order/filter/germ.lean | [
"order.filter.basic",
"algebra.module.pi"
] | [
"filter"
] | Given two germs `f : germ l β`, and `g : germ lc α`, where `l : filter α`, if `g` tends to `l`,
then the composition `f ∘ g` is well-defined as a germ at `lc`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_comp_tendsto' (f : α → β) {lc : filter γ} {g : germ lc α} (hg : g.tendsto l) :
(f : germ l β).comp_tendsto' g hg = g.map f | rfl | lemma | filter.germ.coe_comp_tendsto' | order.filter | src/order/filter/germ.lean | [
"order.filter.basic",
"algebra.module.pi"
] | [
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_tendsto (f : germ l β) {lc : filter γ} (g : γ → α) (hg : tendsto g lc l) :
germ lc β | f.comp_tendsto' _ hg.germ_tendsto | def | filter.germ.comp_tendsto | order.filter | src/order/filter/germ.lean | [
"order.filter.basic",
"algebra.module.pi"
] | [
"filter"
] | Given a germ `f : germ l β` and a function `g : γ → α`, where `l : filter α`, if `g` tends
to `l` along `lc : filter γ`, then the composition `f ∘ g` is well-defined as a germ at `lc`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_comp_tendsto (f : α → β) {lc : filter γ} {g : γ → α} (hg : tendsto g lc l) :
(f : germ l β).comp_tendsto g hg = f ∘ g | rfl | lemma | filter.germ.coe_comp_tendsto | order.filter | src/order/filter/germ.lean | [
"order.filter.basic",
"algebra.module.pi"
] | [
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_tendsto'_coe (f : germ l β) {lc : filter γ} {g : γ → α} (hg : tendsto g lc l) :
f.comp_tendsto' _ hg.germ_tendsto = f.comp_tendsto g hg | rfl | lemma | filter.germ.comp_tendsto'_coe | order.filter | src/order/filter/germ.lean | [
"order.filter.basic",
"algebra.module.pi"
] | [
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
const_inj [ne_bot l] {a b : β} : (↑a : germ l β) = ↑b ↔ a = b | coe_eq.trans $ const_eventually_eq | lemma | filter.germ.const_inj | order.filter | src/order/filter/germ.lean | [
"order.filter.basic",
"algebra.module.pi"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_const (l : filter α) (a : β) (f : β → γ) :
(↑a : germ l β).map f = ↑(f a) | rfl | lemma | filter.germ.map_const | order.filter | src/order/filter/germ.lean | [
"order.filter.basic",
"algebra.module.pi"
] | [
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map₂_const (l : filter α) (b : β) (c : γ) (f : β → γ → δ) :
map₂ f (↑b : germ l β) ↑c = ↑(f b c) | rfl | lemma | filter.germ.map₂_const | order.filter | src/order/filter/germ.lean | [
"order.filter.basic",
"algebra.module.pi"
] | [
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
const_comp_tendsto {l : filter α} (b : β) {lc : filter γ} {g : γ → α}
(hg : tendsto g lc l) :
(↑b : germ l β).comp_tendsto g hg = ↑b | rfl | lemma | filter.germ.const_comp_tendsto | order.filter | src/order/filter/germ.lean | [
"order.filter.basic",
"algebra.module.pi"
] | [
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
const_comp_tendsto' {l : filter α} (b : β) {lc : filter γ} {g : germ lc α}
(hg : g.tendsto l) :
(↑b : germ l β).comp_tendsto' g hg = ↑b | induction_on g (λ _ _, rfl) hg | lemma | filter.germ.const_comp_tendsto' | order.filter | src/order/filter/germ.lean | [
"order.filter.basic",
"algebra.module.pi"
] | [
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift_pred (p : β → Prop) (f : germ l β) : Prop | lift_on f (λ f, ∀ᶠ x in l, p (f x)) $
λ f g H, propext $ eventually_congr $ H.mono $ λ x hx, hx ▸ iff.rfl | def | filter.germ.lift_pred | order.filter | src/order/filter/germ.lean | [
"order.filter.basic",
"algebra.module.pi"
] | [] | Lift a predicate on `β` to `germ l β`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lift_pred_coe {p : β → Prop} {f : α → β} :
lift_pred p (f : germ l β) ↔ ∀ᶠ x in l, p (f x) | iff.rfl | lemma | filter.germ.lift_pred_coe | order.filter | src/order/filter/germ.lean | [
"order.filter.basic",
"algebra.module.pi"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift_pred_const {p : β → Prop} {x : β} (hx : p x) :
lift_pred p (↑x : germ l β) | eventually_of_forall $ λ y, hx | lemma | filter.germ.lift_pred_const | order.filter | src/order/filter/germ.lean | [
"order.filter.basic",
"algebra.module.pi"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift_pred_const_iff [ne_bot l] {p : β → Prop} {x : β} :
lift_pred p (↑x : germ l β) ↔ p x | @eventually_const _ _ _ (p x) | lemma | filter.germ.lift_pred_const_iff | order.filter | src/order/filter/germ.lean | [
"order.filter.basic",
"algebra.module.pi"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift_rel (r : β → γ → Prop) (f : germ l β) (g : germ l γ) : Prop | quotient.lift_on₂' f g (λ f g, ∀ᶠ x in l, r (f x) (g x)) $
λ f g f' g' Hf Hg, propext $ eventually_congr $ Hg.mp $ Hf.mono $ λ x hf hg, hf ▸ hg ▸ iff.rfl | def | filter.germ.lift_rel | order.filter | src/order/filter/germ.lean | [
"order.filter.basic",
"algebra.module.pi"
] | [
"lift_rel",
"quotient.lift_on₂'"
] | Lift a relation `r : β → γ → Prop` to `germ l β → germ l γ → Prop`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lift_rel_coe {r : β → γ → Prop} {f : α → β} {g : α → γ} :
lift_rel r (f : germ l β) g ↔ ∀ᶠ x in l, r (f x) (g x) | iff.rfl | lemma | filter.germ.lift_rel_coe | order.filter | src/order/filter/germ.lean | [
"order.filter.basic",
"algebra.module.pi"
] | [
"lift_rel"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift_rel_const {r : β → γ → Prop} {x : β} {y : γ} (h : r x y) :
lift_rel r (↑x : germ l β) ↑y | eventually_of_forall $ λ _, h | lemma | filter.germ.lift_rel_const | order.filter | src/order/filter/germ.lean | [
"order.filter.basic",
"algebra.module.pi"
] | [
"lift_rel"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift_rel_const_iff [ne_bot l] {r : β → γ → Prop} {x : β} {y : γ} :
lift_rel r (↑x : germ l β) ↑y ↔ r x y | @eventually_const _ _ _ (r x y) | lemma | filter.germ.lift_rel_const_iff | order.filter | src/order/filter/germ.lean | [
"order.filter.basic",
"algebra.module.pi"
] | [
"lift_rel"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_mul [has_mul M] (f g : α → M) : ↑(f * g) = (f * g : germ l M) | rfl | lemma | filter.germ.coe_mul | order.filter | src/order/filter/germ.lean | [
"order.filter.basic",
"algebra.module.pi"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_one [has_one M] : ↑(1 : α → M) = (1 : germ l M) | rfl | lemma | filter.germ.coe_one | order.filter | src/order/filter/germ.lean | [
"order.filter.basic",
"algebra.module.pi"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_smul [has_smul M G] (n : M) (f : α → G) : ↑(n • f) = (n • f : germ l G) | rfl | lemma | filter.germ.coe_smul | order.filter | src/order/filter/germ.lean | [
"order.filter.basic",
"algebra.module.pi"
] | [
"has_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
const_smul [has_smul M G] (n : M) (a : G) : (↑(n • a) : germ l G) = n • ↑a | rfl | lemma | filter.germ.const_smul | order.filter | src/order/filter/germ.lean | [
"order.filter.basic",
"algebra.module.pi"
] | [
"has_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_pow [has_pow G M] (f : α → G) (n : M) : ↑(f ^ n) = (f ^ n : germ l G) | rfl | lemma | filter.germ.coe_pow | order.filter | src/order/filter/germ.lean | [
"order.filter.basic",
"algebra.module.pi"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
const_pow [has_pow G M] (a : G) (n : M) : (↑(a ^ n) : germ l G) = ↑a ^ n | rfl | lemma | filter.germ.const_pow | order.filter | src/order/filter/germ.lean | [
"order.filter.basic",
"algebra.module.pi"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_mul_hom [monoid M] (l : filter α) : (α → M) →* germ l M | ⟨coe, rfl, λ f g, rfl⟩ | def | filter.germ.coe_mul_hom | order.filter | src/order/filter/germ.lean | [
"order.filter.basic",
"algebra.module.pi"
] | [
"filter",
"monoid"
] | Coercion from functions to germs as a monoid homomorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_coe_mul_hom [monoid M] : (coe_mul_hom l : (α → M) → germ l M) = coe | rfl | lemma | filter.germ.coe_coe_mul_hom | order.filter | src/order/filter/germ.lean | [
"order.filter.basic",
"algebra.module.pi"
] | [
"monoid"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_inv [has_inv G] (f : α → G) : ↑f⁻¹ = (f⁻¹ : germ l G) | rfl | lemma | filter.germ.coe_inv | order.filter | src/order/filter/germ.lean | [
"order.filter.basic",
"algebra.module.pi"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
const_inv [has_inv G] (a : G) : (↑a⁻¹ : germ l G) = (↑a)⁻¹ | rfl | lemma | filter.germ.const_inv | order.filter | src/order/filter/germ.lean | [
"order.filter.basic",
"algebra.module.pi"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_div [has_div M] (f g : α → M) : ↑(f / g) = (f / g : germ l M) | rfl | lemma | filter.germ.coe_div | order.filter | src/order/filter/germ.lean | [
"order.filter.basic",
"algebra.module.pi"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
const_div [has_div M] (a b : M) : (↑(a / b) : germ l M) = ↑a / ↑b | rfl | lemma | filter.germ.const_div | order.filter | src/order/filter/germ.lean | [
"order.filter.basic",
"algebra.module.pi"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nontrivial [nontrivial R] [ne_bot l] : nontrivial (germ l R) | let ⟨x, y, h⟩ := exists_pair_ne R in ⟨⟨↑x, ↑y, mt const_inj.1 h⟩⟩ | instance | filter.germ.nontrivial | order.filter | src/order/filter/germ.lean | [
"order.filter.basic",
"algebra.module.pi"
] | [
"exists_pair_ne",
"nontrivial"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_ring_hom [semiring R] (l : filter α) : (α → R) →+* germ l R | { to_fun := coe, .. (coe_mul_hom l : _ →* germ l R), .. (coe_add_hom l : _ →+ germ l R) } | def | filter.germ.coe_ring_hom | order.filter | src/order/filter/germ.lean | [
"order.filter.basic",
"algebra.module.pi"
] | [
"filter",
"semiring"
] | Coercion `(α → R) → germ l R` as a `ring_hom`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_coe_ring_hom [semiring R] : (coe_ring_hom l : (α → R) → germ l R) = coe | rfl | lemma | filter.germ.coe_coe_ring_hom | order.filter | src/order/filter/germ.lean | [
"order.filter.basic",
"algebra.module.pi"
] | [
"semiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_smul' [has_smul M β] : has_smul (germ l M) (germ l β) | ⟨map₂ (•)⟩ | instance | filter.germ.has_smul' | order.filter | src/order/filter/germ.lean | [
"order.filter.basic",
"algebra.module.pi"
] | [
"has_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_smul' [has_smul M β] (c : α → M) (f : α → β) :
↑(c • f) = (c : germ l M) • (f : germ l β) | rfl | lemma | filter.germ.coe_smul' | order.filter | src/order/filter/germ.lean | [
"order.filter.basic",
"algebra.module.pi"
] | [
"has_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_action' [monoid M] [mul_action M β] : mul_action (germ l M) (germ l β) | { one_smul := λ f, induction_on f $ λ f, by simp only [← coe_one, ← coe_smul', one_smul],
mul_smul := λ c₁ c₂ f, induction_on₃ c₁ c₂ f $ λ c₁ c₂ f, by { norm_cast, simp only [mul_smul] } } | instance | filter.germ.mul_action' | order.filter | src/order/filter/germ.lean | [
"order.filter.basic",
"algebra.module.pi"
] | [
"monoid",
"mul_action",
"one_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
distrib_mul_action' [monoid M] [add_monoid N] [distrib_mul_action M N] :
distrib_mul_action (germ l M) (germ l N) | { smul_add := λ c f g, induction_on₃ c f g $ λ c f g, by { norm_cast, simp only [smul_add] },
smul_zero := λ c, induction_on c $ λ c, by simp only [← coe_zero, ← coe_smul', smul_zero] } | instance | filter.germ.distrib_mul_action' | order.filter | src/order/filter/germ.lean | [
"order.filter.basic",
"algebra.module.pi"
] | [
"add_monoid",
"distrib_mul_action",
"monoid",
"smul_add",
"smul_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
module' [semiring R] [add_comm_monoid M] [module R M] :
module (germ l R) (germ l M) | { add_smul := λ c₁ c₂ f, induction_on₃ c₁ c₂ f $ λ c₁ c₂ f, by { norm_cast, simp only [add_smul] },
zero_smul := λ f, induction_on f $ λ f, by simp only [← coe_zero, ← coe_smul', zero_smul] } | instance | filter.germ.module' | order.filter | src/order/filter/germ.lean | [
"order.filter.basic",
"algebra.module.pi"
] | [
"add_comm_monoid",
"add_smul",
"module",
"semiring",
"zero_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_def [has_le β] : ((≤) : germ l β → germ l β → Prop) = lift_rel (≤) | rfl | lemma | filter.germ.le_def | order.filter | src/order/filter/germ.lean | [
"order.filter.basic",
"algebra.module.pi"
] | [
"lift_rel"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_le [has_le β] : (f : germ l β) ≤ g ↔ f ≤ᶠ[l] g | iff.rfl | lemma | filter.germ.coe_le | order.filter | src/order/filter/germ.lean | [
"order.filter.basic",
"algebra.module.pi"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_nonneg [has_le β] [has_zero β] {f : α → β} : 0 ≤ (f : germ l β) ↔ ∀ᶠ x in l, 0 ≤ f x | iff.rfl | lemma | filter.germ.coe_nonneg | order.filter | src/order/filter/germ.lean | [
"order.filter.basic",
"algebra.module.pi"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
const_le [has_le β] {x y : β} : x ≤ y → (↑x : germ l β) ≤ ↑y | lift_rel_const | lemma | filter.germ.const_le | order.filter | src/order/filter/germ.lean | [
"order.filter.basic",
"algebra.module.pi"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
const_le_iff [has_le β] [ne_bot l] {x y : β} : (↑x : germ l β) ≤ ↑y ↔ x ≤ y | lift_rel_const_iff | lemma | filter.germ.const_le_iff | order.filter | src/order/filter/germ.lean | [
"order.filter.basic",
"algebra.module.pi"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
const_bot [has_bot β] : (↑(⊥ : β) : germ l β) = ⊥ | rfl | lemma | filter.germ.const_bot | order.filter | src/order/filter/germ.lean | [
"order.filter.basic",
"algebra.module.pi"
] | [
"has_bot"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
const_top [has_top β] : (↑(⊤ : β) : germ l β) = ⊤ | rfl | lemma | filter.germ.const_top | order.filter | src/order/filter/germ.lean | [
"order.filter.basic",
"algebra.module.pi"
] | [
"has_top"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
indicator_eventually_eq (hf : f =ᶠ[l ⊓ 𝓟 s] g) (hs : s =ᶠ[l] t) :
indicator s f =ᶠ[l] indicator t g | (eventually_inf_principal.1 hf).mp $ hs.mem_iff.mono $ λ x hst hfg,
by_cases (λ hxs : x ∈ s, by simp only [*, hst.1 hxs, indicator_of_mem])
(λ hxs, by simp only [indicator_of_not_mem hxs, indicator_of_not_mem (mt hst.2 hxs)]) | lemma | indicator_eventually_eq | order.filter | src/order/filter/indicator_function.lean | [
"algebra.indicator_function",
"order.filter.at_top_bot"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
indicator_union_eventually_eq (h : ∀ᶠ a in l, a ∉ s ∩ t) :
indicator (s ∪ t) f =ᶠ[l] indicator s f + indicator t f | h.mono $ λ a ha, indicator_union_of_not_mem_inter ha _ | lemma | indicator_union_eventually_eq | order.filter | src/order/filter/indicator_function.lean | [
"algebra.indicator_function",
"order.filter.at_top_bot"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
indicator_eventually_le_indicator (h : f ≤ᶠ[l ⊓ 𝓟 s] g) :
indicator s f ≤ᶠ[l] indicator s g | (eventually_inf_principal.1 h).mono $ assume a h,
indicator_rel_indicator le_rfl h | lemma | indicator_eventually_le_indicator | order.filter | src/order/filter/indicator_function.lean | [
"algebra.indicator_function",
"order.filter.at_top_bot"
] | [
"le_rfl"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
monotone.tendsto_indicator {ι} [preorder ι] [has_zero β]
(s : ι → set α) (hs : monotone s) (f : α → β) (a : α) :
tendsto (λi, indicator (s i) f a) at_top (pure $ indicator (⋃ i, s i) f a) | begin
by_cases h : ∃i, a ∈ s i,
{ rcases h with ⟨i, hi⟩,
refine tendsto_pure.2 ((eventually_ge_at_top i).mono $ assume n hn, _),
rw [indicator_of_mem (hs hn hi) _, indicator_of_mem ((subset_Union _ _) hi) _] },
{ rw [not_exists] at h,
simp only [indicator_of_not_mem (h _)],
convert tendsto_const_p... | lemma | monotone.tendsto_indicator | order.filter | src/order/filter/indicator_function.lean | [
"algebra.indicator_function",
"order.filter.at_top_bot"
] | [
"monotone",
"not_exists"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
antitone.tendsto_indicator {ι} [preorder ι] [has_zero β]
(s : ι → set α) (hs : antitone s) (f : α → β) (a : α) :
tendsto (λi, indicator (s i) f a) at_top (pure $ indicator (⋂ i, s i) f a) | begin
by_cases h : ∃i, a ∉ s i,
{ rcases h with ⟨i, hi⟩,
refine tendsto_pure.2 ((eventually_ge_at_top i).mono $ assume n hn, _),
rw [indicator_of_not_mem _ _, indicator_of_not_mem _ _],
{ simp only [mem_Inter, not_forall], exact ⟨i, hi⟩ },
{ assume h, have := hs hn h, contradiction } },
{ push_neg... | lemma | antitone.tendsto_indicator | order.filter | src/order/filter/indicator_function.lean | [
"algebra.indicator_function",
"order.filter.at_top_bot"
] | [
"antitone",
"not_forall"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_indicator_bUnion_finset {ι} [has_zero β] (s : ι → set α) (f : α → β) (a : α) :
tendsto (λ (n : finset ι), indicator (⋃i∈n, s i) f a) at_top (pure $ indicator (Union s) f a) | begin
rw Union_eq_Union_finset s,
refine monotone.tendsto_indicator (λ n : finset ι, ⋃ i ∈ n, s i) _ f a,
exact λ t₁ t₂, bUnion_subset_bUnion_left
end | lemma | tendsto_indicator_bUnion_finset | order.filter | src/order/filter/indicator_function.lean | [
"algebra.indicator_function",
"order.filter.at_top_bot"
] | [
"finset",
"monotone.tendsto_indicator"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
filter.eventually_eq.support [has_zero β] {f g : α → β} {l : filter α}
(h : f =ᶠ[l] g) :
function.support f =ᶠ[l] function.support g | begin
filter_upwards [h] with x hx,
rw eq_iff_iff,
change f x ≠ 0 ↔ g x ≠ 0,
rw hx,
end | lemma | filter.eventually_eq.support | order.filter | src/order/filter/indicator_function.lean | [
"algebra.indicator_function",
"order.filter.at_top_bot"
] | [
"eq_iff_iff",
"filter",
"function.support"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
filter.eventually_eq.indicator [has_zero β] {l : filter α} {f g : α → β} {s : set α}
(hfg : f =ᶠ[l] g) :
s.indicator f =ᶠ[l] s.indicator g | begin
filter_upwards [hfg] with x hx,
by_cases x ∈ s,
{ rwa [indicator_of_mem h, indicator_of_mem h] },
{ rw [indicator_of_not_mem h, indicator_of_not_mem h] }
end | lemma | filter.eventually_eq.indicator | order.filter | src/order/filter/indicator_function.lean | [
"algebra.indicator_function",
"order.filter.at_top_bot"
] | [
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
filter.eventually_eq.indicator_zero [has_zero β] {l : filter α}
{f : α → β} {s : set α} (hf : f =ᶠ[l] 0) :
s.indicator f =ᶠ[l] 0 | begin
refine hf.indicator.trans _,
rw indicator_zero'
end | lemma | filter.eventually_eq.indicator_zero | order.filter | src/order/filter/indicator_function.lean | [
"algebra.indicator_function",
"order.filter.at_top_bot"
] | [
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_Ixx_class (Ixx : α → α → set α) (l₁ : filter α) (l₂ : out_param $ filter α) : Prop | (tendsto_Ixx : tendsto (λ p : α × α, Ixx p.1 p.2) (l₁ ×ᶠ l₁) l₂.small_sets) | class | filter.tendsto_Ixx_class | order.filter | src/order/filter/interval.lean | [
"data.set.intervals.ord_connected",
"order.filter.small_sets",
"order.filter.at_top_bot"
] | [
"filter"
] | A pair of filters `l₁`, `l₂` has `tendsto_Ixx_class Ixx` property if `Ixx a b` tends to
`l₂.small_sets` as `a` and `b` tend to `l₁`. In all instances `Ixx` is one of `Icc`, `Ico`, `Ioc`,
or `Ioo`. The instances provide the best `l₂` for a given `l₁`. In many cases `l₁ = l₂` but
sometimes we can drop an endpoint from an... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
tendsto.Icc {l₁ l₂ : filter α} [tendsto_Ixx_class Icc l₁ l₂]
{lb : filter β} {u₁ u₂ : β → α} (h₁ : tendsto u₁ lb l₁) (h₂ : tendsto u₂ lb l₁) :
tendsto (λ x, Icc (u₁ x) (u₂ x)) lb l₂.small_sets | tendsto_Ixx_class.tendsto_Ixx.comp $ h₁.prod_mk h₂ | lemma | filter.tendsto.Icc | order.filter | src/order/filter/interval.lean | [
"data.set.intervals.ord_connected",
"order.filter.small_sets",
"order.filter.at_top_bot"
] | [
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto.Ioc {l₁ l₂ : filter α} [tendsto_Ixx_class Ioc l₁ l₂]
{lb : filter β} {u₁ u₂ : β → α} (h₁ : tendsto u₁ lb l₁) (h₂ : tendsto u₂ lb l₁) :
tendsto (λ x, Ioc (u₁ x) (u₂ x)) lb l₂.small_sets | tendsto_Ixx_class.tendsto_Ixx.comp $ h₁.prod_mk h₂ | lemma | filter.tendsto.Ioc | order.filter | src/order/filter/interval.lean | [
"data.set.intervals.ord_connected",
"order.filter.small_sets",
"order.filter.at_top_bot"
] | [
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto.Ico {l₁ l₂ : filter α} [tendsto_Ixx_class Ico l₁ l₂]
{lb : filter β} {u₁ u₂ : β → α} (h₁ : tendsto u₁ lb l₁) (h₂ : tendsto u₂ lb l₁) :
tendsto (λ x, Ico (u₁ x) (u₂ x)) lb l₂.small_sets | tendsto_Ixx_class.tendsto_Ixx.comp $ h₁.prod_mk h₂ | lemma | filter.tendsto.Ico | order.filter | src/order/filter/interval.lean | [
"data.set.intervals.ord_connected",
"order.filter.small_sets",
"order.filter.at_top_bot"
] | [
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto.Ioo {l₁ l₂ : filter α} [tendsto_Ixx_class Ioo l₁ l₂]
{lb : filter β} {u₁ u₂ : β → α} (h₁ : tendsto u₁ lb l₁) (h₂ : tendsto u₂ lb l₁) :
tendsto (λ x, Ioo (u₁ x) (u₂ x)) lb l₂.small_sets | tendsto_Ixx_class.tendsto_Ixx.comp $ h₁.prod_mk h₂ | lemma | filter.tendsto.Ioo | order.filter | src/order/filter/interval.lean | [
"data.set.intervals.ord_connected",
"order.filter.small_sets",
"order.filter.at_top_bot"
] | [
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_Ixx_class_principal {s t : set α} {Ixx : α → α → set α} :
tendsto_Ixx_class Ixx (𝓟 s) (𝓟 t) ↔ ∀ x y ∈ s, Ixx x y ⊆ t | iff.trans ⟨λ h, h.1, λ h, ⟨h⟩⟩ $ by simp only [small_sets_principal, prod_principal_principal,
tendsto_principal_principal, forall_prod_set, mem_powerset_iff, mem_principal] | lemma | filter.tendsto_Ixx_class_principal | order.filter | src/order/filter/interval.lean | [
"data.set.intervals.ord_connected",
"order.filter.small_sets",
"order.filter.at_top_bot"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_Ixx_class_inf {l₁ l₁' l₂ l₂' : filter α} {Ixx}
[h : tendsto_Ixx_class Ixx l₁ l₂] [h' : tendsto_Ixx_class Ixx l₁' l₂'] :
tendsto_Ixx_class Ixx (l₁ ⊓ l₁') (l₂ ⊓ l₂') | ⟨by simpa only [prod_inf_prod, small_sets_inf] using h.1.inf h'.1⟩ | lemma | filter.tendsto_Ixx_class_inf | order.filter | src/order/filter/interval.lean | [
"data.set.intervals.ord_connected",
"order.filter.small_sets",
"order.filter.at_top_bot"
] | [
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_Ixx_class_of_subset {l₁ l₂ : filter α} {Ixx Ixx' : α → α → set α}
(h : ∀ a b, Ixx a b ⊆ Ixx' a b) [h' : tendsto_Ixx_class Ixx' l₁ l₂] :
tendsto_Ixx_class Ixx l₁ l₂ | ⟨h'.1.small_sets_mono $ eventually_of_forall $ prod.forall.2 h⟩ | lemma | filter.tendsto_Ixx_class_of_subset | order.filter | src/order/filter/interval.lean | [
"data.set.intervals.ord_connected",
"order.filter.small_sets",
"order.filter.at_top_bot"
] | [
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_basis.tendsto_Ixx_class {ι : Type*} {p : ι → Prop} {s} {l : filter α}
(hl : l.has_basis p s) {Ixx : α → α → set α}
(H : ∀ i, p i → ∀ (x ∈ s i) (y ∈ s i), Ixx x y ⊆ s i) :
tendsto_Ixx_class Ixx l l | ⟨(hl.prod_self.tendsto_iff hl.small_sets).2 $ λ i hi,
⟨i, hi, λ x hx, H i hi _ hx.1 _ hx.2⟩⟩ | lemma | filter.has_basis.tendsto_Ixx_class | order.filter | src/order/filter/interval.lean | [
"data.set.intervals.ord_connected",
"order.filter.small_sets",
"order.filter.at_top_bot"
] | [
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_Icc_at_top_at_top : tendsto_Ixx_class Icc (at_top : filter α) at_top | (has_basis_infi_principal_finite _).tendsto_Ixx_class $ λ s hs,
set.ord_connected.out $ ord_connected_bInter $ λ i hi, ord_connected_Ici | instance | filter.tendsto_Icc_at_top_at_top | order.filter | src/order/filter/interval.lean | [
"data.set.intervals.ord_connected",
"order.filter.small_sets",
"order.filter.at_top_bot"
] | [
"filter",
"set.ord_connected.out"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_Ico_at_top_at_top : tendsto_Ixx_class Ico (at_top : filter α) at_top | tendsto_Ixx_class_of_subset (λ _ _, Ico_subset_Icc_self) | instance | filter.tendsto_Ico_at_top_at_top | order.filter | src/order/filter/interval.lean | [
"data.set.intervals.ord_connected",
"order.filter.small_sets",
"order.filter.at_top_bot"
] | [
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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