mathlib documentation

order.filter.pointwise

Pointwise operations on filters #

This file defines pointwise operations on filters. This is useful because usual algebraic operations distribute over pointwise operations. For example,

Main declarations #

Tags #

filter multiplication, filter addition, pointwise addition, pointwise multiplication,

0/1 as filters #

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@[protected, instance]
def filter.has_zero {α : Type u_2} [has_zero α] :
has_zero (filter α)

0 : filter α is the set of sets containing 0 : α.

Equations
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@[protected, instance]
def filter.has_one {α : Type u_2} [has_one α] :
has_one (filter α)

1 : filter α is the set of sets containing 1 : α.

Equations
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@[simp]
theorem filter.mem_zero {α : Type u_2} [has_zero α] {s : set α} :
s ∈ 0 ↔ 0 ∈ s
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@[simp]
theorem filter.mem_one {α : Type u_2} [has_one α] {s : set α} :
s ∈ 1 ↔ 1 ∈ s
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theorem filter.zero_mem_zero {α : Type u_2} [has_zero α] :
0 ∈ 0
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theorem filter.one_mem_one {α : Type u_2} [has_one α] :
1 ∈ 1
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@[simp]
theorem filter.principal_one {α : Type u_2} [has_one α] :
𝓟 1 = 1
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@[simp]
theorem filter.principal_zero {α : Type u_2} [has_zero α] :
𝓟 0 = 0
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@[simp]
theorem filter.pure_zero {α : Type u_2} [has_zero α] :
pure 0 = 0
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@[simp]
theorem filter.pure_one {α : Type u_2} [has_one α] :
pure 1 = 1
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@[simp]
theorem filter.nonpos_iff {α : Type u_2} [has_zero α] {f : filter α} :
f ≤ 0 ↔ 0 ∈ f
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@[simp]
theorem filter.le_one_iff {α : Type u_2} [has_one α] {f : filter α} :
f ≤ 1 ↔ 1 ∈ f
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@[simp]
theorem filter.eventually_one {α : Type u_2} [has_one α] {p : α → Prop} :
(∀ᶠ (x : α) in 1, p x) ↔ p 1
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@[simp]
theorem filter.eventually_zero {α : Type u_2} [has_zero α] {p : α → Prop} :
(∀ᶠ (x : α) in 0, p x) ↔ p 0
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@[simp]
theorem filter.tendsto_one {α : Type u_2} {β : Type u_3} [has_one α] {a : filter β} {f : β → α} :
filter.tendsto f a 1 ↔ ∀ᶠ (x : β) in a, f x = 1
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@[simp]
theorem filter.tendsto_zero {α : Type u_2} {β : Type u_3} [has_zero α] {a : filter β} {f : β → α} :
filter.tendsto f a 0 ↔ ∀ᶠ (x : β) in a, f x = 0
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@[protected, simp]
theorem filter.map_one {F : Type u_1} {α : Type u_2} {β : Type u_3} [has_one α] [has_one β] [one_hom_class F α β] (φ : F) :
filter.map ⇑φ 1 = 1
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@[protected, simp]
theorem filter.map_zero {F : Type u_1} {α : Type u_2} {β : Type u_3} [has_zero α] [has_zero β] [zero_hom_class F α β] (φ : F) :
filter.map ⇑φ 0 = 0

Filter addition/multiplication #

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@[protected, instance]
def filter.has_add {α : Type u_2} [has_add α] :
has_add (filter α)
Equations
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@[protected, instance]
def filter.has_mul {α : Type u_2} [has_mul α] :
has_mul (filter α)
Equations
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@[simp]
theorem filter.map₂_mul {α : Type u_2} [has_mul α] {f g : filter α} :
filter.map₂ has_mul.mul f g = f * g
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@[simp]
theorem filter.map₂_add {α : Type u_2} [has_add α] {f g : filter α} :
filter.map₂ has_add.add f g = f + g
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theorem filter.mem_mul_iff {α : Type u_2} [has_mul α] {f g : filter α} {s : set α} :
s ∈ f * g ↔ ∃ (t₁ t₂ : set α), t₁ ∈ f ∧ t₂ ∈ g ∧ t₁ * t₂ ⊆ s
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theorem filter.mem_add_iff {α : Type u_2} [has_add α] {f g : filter α} {s : set α} :
s ∈ f + g ↔ ∃ (t₁ t₂ : set α), t₁ ∈ f ∧ t₂ ∈ g ∧ t₁ + t₂ ⊆ s
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theorem filter.mul_mem_mul {α : Type u_2} [has_mul α] {f g : filter α} {s t : set α} :
s ∈ f → t ∈ g → s * t ∈ f * g
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theorem filter.add_mem_add {α : Type u_2} [has_add α] {f g : filter α} {s t : set α} :
s ∈ f → t ∈ g → s + t ∈ f + g
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@[simp]
theorem filter.bot_add {α : Type u_2} [has_add α] {g : filter α} :
⊥ + g = ⊥
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@[simp]
theorem filter.bot_mul {α : Type u_2} [has_mul α] {g : filter α} :
⊥ * g = ⊥
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@[simp]
theorem filter.add_bot {α : Type u_2} [has_add α] {f : filter α} :
f + ⊥ = ⊥
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@[simp]
theorem filter.mul_bot {α : Type u_2} [has_mul α] {f : filter α} :
f * ⊥ = ⊥
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@[simp]
theorem filter.add_eq_bot_iff {α : Type u_2} [has_add α] {f g : filter α} :
f + g = ⊥ ↔ f = ⊥ ∨ g = ⊥
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@[simp]
theorem filter.mul_eq_bot_iff {α : Type u_2} [has_mul α] {f g : filter α} :
f * g = ⊥ ↔ f = ⊥ ∨ g = ⊥
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@[simp]
theorem filter.mul_ne_bot_iff {α : Type u_2} [has_mul α] {f g : filter α} :
(f * g).ne_bot ↔ f.ne_bot ∧ g.ne_bot
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@[simp]
theorem filter.add_ne_bot_iff {α : Type u_2} [has_add α] {f g : filter α} :
(f + g).ne_bot ↔ f.ne_bot ∧ g.ne_bot
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theorem filter.ne_bot.mul {α : Type u_2} [has_mul α] {f g : filter α} :
f.ne_bot → g.ne_bot → (f * g).ne_bot
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theorem filter.ne_bot.add {α : Type u_2} [has_add α] {f g : filter α} :
f.ne_bot → g.ne_bot → (f + g).ne_bot
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@[simp]
theorem filter.le_mul_iff {α : Type u_2} [has_mul α] {f g h : filter α} :
h ≤ f * g ↔ ∀ ⦃s : set α⦄, s ∈ f → ∀ ⦃t : set α⦄, t ∈ g → s * t ∈ h
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@[simp]
theorem filter.le_add_iff {α : Type u_2} [has_add α] {f g h : filter α} :
h ≤ f + g ↔ ∀ ⦃s : set α⦄, s ∈ f → ∀ ⦃t : set α⦄, t ∈ g → s + t ∈ h
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@[protected, instance]
def filter.covariant_add {α : Type u_2} [has_add α] :
covariant_class (filter α) (filter α) has_add.add has_le.le
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@[protected, instance]
def filter.covariant_mul {α : Type u_2} [has_mul α] :
covariant_class (filter α) (filter α) has_mul.mul has_le.le
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@[protected, instance]
def filter.covariant_swap_add {α : Type u_2} [has_add α] :
covariant_class (filter α) (filter α) (function.swap has_add.add) has_le.le
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@[protected, instance]
def filter.covariant_swap_mul {α : Type u_2} [has_mul α] :
covariant_class (filter α) (filter α) (function.swap has_mul.mul) has_le.le
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@[protected]
theorem filter.map_mul {F : Type u_1} {α : Type u_2} {β : Type u_3} [has_mul α] [has_mul β] {f₁ f₂ : filter α} [mul_hom_class F α β] (m : F) :
filter.map ⇑m (f₁ * f₂) = (filter.map ⇑m f₁) * filter.map ⇑m f₂
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@[protected]
theorem filter.map_add {F : Type u_1} {α : Type u_2} {β : Type u_3} [has_add α] [has_add β] {f₁ f₂ : filter α} [add_hom_class F α β] (m : F) :
filter.map ⇑m (f₁ + f₂) = filter.map ⇑m f₁ + filter.map ⇑m f₂
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@[protected, instance]
def filter.semigroup {α : Type u_2} [semigroup α] :
semigroup (filter α)
Equations
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@[protected, instance]
def filter.add_semigroup {α : Type u_2} [add_semigroup α] :
add_semigroup (filter α)
Equations
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@[protected, instance]
def filter.comm_semigroup {α : Type u_2} [comm_semigroup α] :
comm_semigroup (filter α)
Equations
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@[protected, instance]
def filter.add_comm_semigroup {α : Type u_2} [add_comm_semigroup α] :
add_comm_semigroup (filter α)
Equations
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@[protected, instance]
def filter.add_zero_class {α : Type u_2} [add_zero_class α] :
add_zero_class (filter α)
Equations
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@[protected, instance]
def filter.mul_one_class {α : Type u_2} [mul_one_class α] :
mul_one_class (filter α)
Equations
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@[protected, instance]
def filter.add_monoid {α : Type u_2} [add_monoid α] :
add_monoid (filter α)
Equations
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@[protected, instance]
def filter.monoid {α : Type u_2} [monoid α] :
monoid (filter α)
Equations
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@[protected, instance]
def filter.comm_monoid {α : Type u_2} [comm_monoid α] :
comm_monoid (filter α)
Equations
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@[protected, instance]
def filter.add_comm_monoid {α : Type u_2} [add_comm_monoid α] :
add_comm_monoid (filter α)
Equations
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def filter.map_add_monoid_hom {F : Type u_1} {α : Type u_2} {β : Type u_3} [add_zero_class α] [add_zero_class β] [add_monoid_hom_class F α β] (φ : F) :
filter α →+ filter β

If φ : α →+ β then map_add_monoid_hom φ is the monoid homomorphism filter α →+ filter β induced by map φ.

Equations
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def filter.map_monoid_hom {F : Type u_1} {α : Type u_2} {β : Type u_3} [mul_one_class α] [mul_one_class β] [monoid_hom_class F α β] (φ : F) :
filter α →* filter β

If φ : α →* β then map_monoid_hom φ is the monoid homomorphism filter α →* filter β induced by map φ.

Equations
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theorem filter.comap_add_comap_le {F : Type u_1} {α : Type u_2} {β : Type u_3} [add_zero_class α] [add_zero_class β] [add_hom_class F α β] (m : F) {f₁ f₂ : filter β} :
filter.comap ⇑m f₁ + filter.comap ⇑m f₂ ≤ filter.comap ⇑m (f₁ + f₂)
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theorem filter.comap_mul_comap_le {F : Type u_1} {α : Type u_2} {β : Type u_3} [mul_one_class α] [mul_one_class β] [mul_hom_class F α β] (m : F) {f₁ f₂ : filter β} :
(filter.comap ⇑m f₁) * filter.comap ⇑m f₂ ≤ filter.comap ⇑m (f₁ * f₂)
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theorem filter.tendsto.add_add {F : Type u_1} {α : Type u_2} {β : Type u_3} [add_zero_class α] [add_zero_class β] [add_hom_class F α β] (m : F) {f₁ g₁ : filter α} {f₂ g₂ : filter β} :
filter.tendsto ⇑m f₁ f₂ → filter.tendsto ⇑m g₁ g₂ → filter.tendsto ⇑m (f₁ + g₁) (f₂ + g₂)
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theorem filter.tendsto.mul_mul {F : Type u_1} {α : Type u_2} {β : Type u_3} [mul_one_class α] [mul_one_class β] [mul_hom_class F α β] (m : F) {f₁ g₁ : filter α} {f₂ g₂ : filter β} :
filter.tendsto ⇑m f₁ f₂ → filter.tendsto ⇑m g₁ g₂ → filter.tendsto ⇑m (f₁ * g₁) (f₂ * g₂)

Filter negation/inversion #

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@[protected, instance]
def filter.has_inv {α : Type u_2} [has_inv α] :
has_inv (filter α)

The inverse of a filter is the pointwise preimage under ⁻¹ of its sets.

Equations
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@[protected, instance]
def filter.has_neg {α : Type u_2} [has_neg α] :
has_neg (filter α)

The negation of a filter is the pointwise preimage under - of its sets.

Equations
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@[protected, simp]
theorem filter.map_neg {α : Type u_2} [has_neg α] {f : filter α} :
filter.map has_neg.neg f = -f
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@[protected, simp]
theorem filter.map_inv {α : Type u_2} [has_inv α] {f : filter α} :
filter.map has_inv.inv f = f⁻¹
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theorem filter.mem_inv {α : Type u_2} [has_inv α] {f : filter α} {s : set α} :
s ∈ f⁻¹ ↔ has_inv.inv ⁻¹' s ∈ f
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theorem filter.mem_neg {α : Type u_2} [has_neg α] {f : filter α} {s : set α} :
s ∈ -f ↔ has_neg.neg ⁻¹' s ∈ f
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@[protected]
theorem filter.neg_le_neg {α : Type u_2} [has_neg α] {f g : filter α} (hf : f ≤ g) :
-f ≤ -g
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@[protected]
theorem filter.inv_le_inv {α : Type u_2} [has_inv α] {f g : filter α} (hf : f ≤ g) :
f⁻¹ ≤ g⁻¹
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@[simp]
theorem filter.ne_bot_inv_iff {α : Type u_2} [has_inv α] {f : filter α} :
f⁻¹.ne_bot ↔ f.ne_bot
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@[simp]
theorem filter.ne_bot_neg_iff {α : Type u_2} [has_neg α] {f : filter α} :
(-f).ne_bot ↔ f.ne_bot
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theorem filter.ne_bot.neg {α : Type u_2} [has_neg α] {f : filter α} :
f.ne_bot → (-f).ne_bot
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theorem filter.ne_bot.inv {α : Type u_2} [has_inv α] {f : filter α} :
f.ne_bot → f⁻¹.ne_bot
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theorem filter.neg_mem_neg {α : Type u_2} [has_involutive_neg α] {f : filter α} {s : set α} (hs : s ∈ f) :
-s ∈ -f
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theorem filter.inv_mem_inv {α : Type u_2} [has_involutive_inv α] {f : filter α} {s : set α} (hs : s ∈ f) :
s⁻¹ ∈ f⁻¹
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@[protected, instance]
def filter.has_involutive_inv {α : Type u_2} [has_involutive_inv α] :
has_involutive_inv (filter α)
Equations
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theorem filter.map_neg' {F : Type u_1} {α : Type u_2} {β : Type u_3} [add_group α] [add_group β] [add_monoid_hom_class F α β] (m : F) {f : filter α} :
filter.map ⇑m (-f) = -filter.map ⇑m f
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theorem filter.map_inv' {F : Type u_1} {α : Type u_2} {β : Type u_3} [group α] [group β] [monoid_hom_class F α β] (m : F) {f : filter α} :
filter.map ⇑m f⁻¹ = (filter.map ⇑m f)⁻¹
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theorem filter.tendsto.neg_neg {F : Type u_1} {α : Type u_2} {β : Type u_3} [add_group α] [add_group β] [add_monoid_hom_class F α β] (m : F) {f₁ : filter α} {f₂ : filter β} :
filter.tendsto ⇑m f₁ f₂ → filter.tendsto ⇑m (-f₁) (-f₂)
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theorem filter.tendsto.inv_inv {F : Type u_1} {α : Type u_2} {β : Type u_3} [group α] [group β] [monoid_hom_class F α β] (m : F) {f₁ : filter α} {f₂ : filter β} :
filter.tendsto ⇑m f₁ f₂ → filter.tendsto ⇑m f₁⁻¹ f₂⁻¹

Filter subtraction/division #

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@[protected, instance]
def filter.has_sub {α : Type u_2} [has_sub α] :
has_sub (filter α)
Equations
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@[protected, instance]
def filter.has_div {α : Type u_2} [has_div α] :
has_div (filter α)
Equations
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@[simp]
theorem filter.map₂_sub {α : Type u_2} [has_sub α] {f g : filter α} :
filter.map₂ has_sub.sub f g = f - g
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@[simp]
theorem filter.map₂_div {α : Type u_2} [has_div α] {f g : filter α} :
filter.map₂ has_div.div f g = f / g
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theorem filter.mem_sub {α : Type u_2} [has_sub α] {f g : filter α} {s : set α} :
s ∈ f - g ↔ ∃ (t₁ t₂ : set α), t₁ ∈ f ∧ t₂ ∈ g ∧ t₁ - t₂ ⊆ s
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theorem filter.mem_div {α : Type u_2} [has_div α] {f g : filter α} {s : set α} :
s ∈ f / g ↔ ∃ (t₁ t₂ : set α), t₁ ∈ f ∧ t₂ ∈ g ∧ t₁ / t₂ ⊆ s
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theorem filter.div_mem_div {α : Type u_2} [has_div α] {f g : filter α} {s t : set α} :
s ∈ f → t ∈ g → s / t ∈ f / g
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theorem filter.sub_mem_sub {α : Type u_2} [has_sub α] {f g : filter α} {s t : set α} :
s ∈ f → t ∈ g → s - t ∈ f - g
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@[simp]
theorem filter.bot_sub {α : Type u_2} [has_sub α] {g : filter α} :
⊥ - g = ⊥
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@[simp]
theorem filter.bot_div {α : Type u_2} [has_div α] {g : filter α} :
⊥ / g = ⊥
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@[simp]
theorem filter.div_bot {α : Type u_2} [has_div α] {f : filter α} :
f / ⊥ = ⊥
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@[simp]
theorem filter.sub_bot {α : Type u_2} [has_sub α] {f : filter α} :
f - ⊥ = ⊥
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@[simp]
theorem filter.sub_eq_bot_iff {α : Type u_2} [has_sub α] {f g : filter α} :
f - g = ⊥ ↔ f = ⊥ ∨ g = ⊥
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@[simp]
theorem filter.div_eq_bot_iff {α : Type u_2} [has_div α] {f g : filter α} :
f / g = ⊥ ↔ f = ⊥ ∨ g = ⊥
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@[simp]
theorem filter.div_ne_bot_iff {α : Type u_2} [has_div α] {f g : filter α} :
(f / g).ne_bot ↔ f.ne_bot ∧ g.ne_bot
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@[simp]
theorem filter.sub_ne_bot_iff {α : Type u_2} [has_sub α] {f g : filter α} :
(f - g).ne_bot ↔ f.ne_bot ∧ g.ne_bot
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theorem filter.ne_bot.sub {α : Type u_2} [has_sub α] {f g : filter α} :
f.ne_bot → g.ne_bot → (f - g).ne_bot
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theorem filter.ne_bot.div {α : Type u_2} [has_div α] {f g : filter α} :
f.ne_bot → g.ne_bot → (f / g).ne_bot
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@[protected, simp]
theorem filter.le_sub_iff {α : Type u_2} [has_sub α] {f g h : filter α} :
h ≤ f - g ↔ ∀ ⦃s : set α⦄, s ∈ f → ∀ ⦃t : set α⦄, t ∈ g → s - t ∈ h
source
@[protected, simp]
theorem filter.le_div_iff {α : Type u_2} [has_div α] {f g h : filter α} :
h ≤ f / g ↔ ∀ ⦃s : set α⦄, s ∈ f → ∀ ⦃t : set α⦄, t ∈ g → s / t ∈ h
source
@[protected]
theorem filter.div_le_div {α : Type u_2} [has_div α] {f₁ f₂ g₁ g₂ : filter α} :
f₁ ≤ f₂ → g₁ ≤ g₂ → f₁ / g₁ ≤ f₂ / g₂
source
@[protected]
theorem filter.sub_le_sub {α : Type u_2} [has_sub α] {f₁ f₂ g₁ g₂ : filter α} :
f₁ ≤ f₂ → g₁ ≤ g₂ → f₁ - g₁ ≤ f₂ - g₂
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@[protected]
theorem filter.div_le_div_left {α : Type u_2} [has_div α] {f g₁ g₂ : filter α} :
g₁ ≤ g₂ → f / g₁ ≤ f / g₂
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@[protected]
theorem filter.sub_le_sub_left {α : Type u_2} [has_sub α] {f g₁ g₂ : filter α} :
g₁ ≤ g₂ → f - g₁ ≤ f - g₂
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@[protected]
theorem filter.div_le_div_right {α : Type u_2} [has_div α] {f₁ f₂ g : filter α} :
f₁ ≤ f₂ → f₁ / g ≤ f₂ / g
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@[protected]
theorem filter.sub_le_sub_right {α : Type u_2} [has_sub α] {f₁ f₂ g : filter α} :
f₁ ≤ f₂ → f₁ - g ≤ f₂ - g
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@[protected, instance]
def filter.covariant_sub {α : Type u_2} [has_sub α] :
covariant_class (filter α) (filter α) has_sub.sub has_le.le
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@[protected, instance]
def filter.covariant_div {α : Type u_2} [has_div α] :
covariant_class (filter α) (filter α) has_div.div has_le.le
source
@[protected, instance]
def filter.covariant_swap_sub {α : Type u_2} [has_sub α] :
covariant_class (filter α) (filter α) (function.swap has_sub.sub) has_le.le
source
@[protected, instance]
def filter.covariant_swap_div {α : Type u_2} [has_div α] :
covariant_class (filter α) (filter α) (function.swap has_div.div) has_le.le
source
@[protected]
theorem filter.map_sub {F : Type u_1} {α : Type u_2} {β : Type u_3} [add_group α] [add_group β] {f g : filter α} [add_monoid_hom_class F α β] (m : F) :
filter.map ⇑m (f - g) = filter.map ⇑m f - filter.map ⇑m g
source
@[protected]
theorem filter.map_div {F : Type u_1} {α : Type u_2} {β : Type u_3} [group α] [group β] {f g : filter α} [monoid_hom_class F α β] (m : F) :
filter.map ⇑m (f / g) = filter.map ⇑m f / filter.map ⇑m g
source
theorem filter.tendsto.div_div {F : Type u_1} {α : Type u_2} {β : Type u_3} [group α] [group β] [monoid_hom_class F α β] (m : F) {f₁ g₁ : filter α} {f₂ g₂ : filter β} :
filter.tendsto ⇑m f₁ f₂ → filter.tendsto ⇑m g₁ g₂ → filter.tendsto ⇑m (f₁ / g₁) (f₂ / g₂)
source
theorem filter.tendsto.sub_sub {F : Type u_1} {α : Type u_2} {β : Type u_3} [add_group α] [add_group β] [add_monoid_hom_class F α β] (m : F) {f₁ g₁ : filter α} {f₂ g₂ : filter β} :
filter.tendsto ⇑m f₁ f₂ → filter.tendsto ⇑m g₁ g₂ → filter.tendsto ⇑m (f₁ - g₁) (f₂ - g₂)
source
@[protected, instance]
def filter.sub_neg_add_monoid {α : Type u_2} [add_group α] :
sub_neg_monoid (filter α)

f - g = f + -g for all f g : filter α if a - b = a + -b for all a b : α.

Equations
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@[protected, instance]
def filter.div_inv_monoid {α : Type u_2} [group α] :
div_inv_monoid (filter α)

f / g = f * g⁻¹ for all f g : filter α if a / b = a * b⁻¹ for all a b : α.

Equations
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@[protected, instance]
def filter.div_inv_monoid' {α : Type u_2} [group_with_zero α] :
div_inv_monoid (filter α)

f / g = f * g⁻¹ for all f g : filter α if a / b = a * b⁻¹ for all a b : α.

Equations

Scalar addition/multiplication of filters #

source
@[protected, instance]
def filter.has_vadd {α : Type u_2} {β : Type u_3} [has_vadd α β] :
has_vadd (filter α) (filter β)
Equations
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@[protected, instance]
def filter.has_scalar {α : Type u_2} {β : Type u_3} [has_scalar α β] :
has_scalar (filter α) (filter β)
Equations
source
@[simp]
theorem filter.map₂_smul {α : Type u_2} {β : Type u_3} [has_scalar α β] {f : filter α} {g : filter β} :
filter.map₂ has_scalar.smul f g = f • g
source
@[simp]
theorem filter.map₂_vadd {α : Type u_2} {β : Type u_3} [has_vadd α β] {f : filter α} {g : filter β} :
filter.map₂ has_vadd.vadd f g = f +ᵥ g
source
theorem filter.mem_smul {α : Type u_2} {β : Type u_3} [has_scalar α β] {f : filter α} {g : filter β} {t : set β} :
t ∈ f • g ↔ ∃ (t₁ : set α) (t₂ : set β), t₁ ∈ f ∧ t₂ ∈ g ∧ t₁ • t₂ ⊆ t
source
theorem filter.mem_vadd {α : Type u_2} {β : Type u_3} [has_vadd α β] {f : filter α} {g : filter β} {t : set β} :
t ∈ f +ᵥ g ↔ ∃ (t₁ : set α) (t₂ : set β), t₁ ∈ f ∧ t₂ ∈ g ∧ t₁ +ᵥ t₂ ⊆ t
source
theorem filter.vadd_mem_vadd {α : Type u_2} {β : Type u_3} [has_vadd α β] {f : filter α} {g : filter β} {s : set α} {t : set β} :
s ∈ f → t ∈ g → s +ᵥ t ∈ f +ᵥ g
source
theorem filter.smul_mem_smul {α : Type u_2} {β : Type u_3} [has_scalar α β] {f : filter α} {g : filter β} {s : set α} {t : set β} :
s ∈ f → t ∈ g → s • t ∈ f • g
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@[simp]
theorem filter.bot_smul {α : Type u_2} {β : Type u_3} [has_scalar α β] {g : filter β} :
⊥ • g = ⊥
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@[simp]
theorem filter.bot_vadd {α : Type u_2} {β : Type u_3} [has_vadd α β] {g : filter β} :
⊥ +ᵥ g = ⊥
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@[simp]
theorem filter.vadd_bot {α : Type u_2} {β : Type u_3} [has_vadd α β] {f : filter α} :
f +ᵥ ⊥ = ⊥
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@[simp]
theorem filter.smul_bot {α : Type u_2} {β : Type u_3} [has_scalar α β] {f : filter α} :
f • ⊥ = ⊥
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@[simp]
theorem filter.smul_eq_bot_iff {α : Type u_2} {β : Type u_3} [has_scalar α β] {f : filter α} {g : filter β} :
f • g = ⊥ ↔ f = ⊥ ∨ g = ⊥
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@[simp]
theorem filter.vadd_eq_bot_iff {α : Type u_2} {β : Type u_3} [has_vadd α β] {f : filter α} {g : filter β} :
f +ᵥ g = ⊥ ↔ f = ⊥ ∨ g = ⊥
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@[simp]
theorem filter.smul_ne_bot_iff {α : Type u_2} {β : Type u_3} [has_scalar α β] {f : filter α} {g : filter β} :
(f • g).ne_bot ↔ f.ne_bot ∧ g.ne_bot
source
@[simp]
theorem filter.vadd_ne_bot_iff {α : Type u_2} {β : Type u_3} [has_vadd α β] {f : filter α} {g : filter β} :
(f +ᵥ g).ne_bot ↔ f.ne_bot ∧ g.ne_bot
source
theorem filter.ne_bot.smul {α : Type u_2} {β : Type u_3} [has_scalar α β] {f : filter α} {g : filter β} :
f.ne_bot → g.ne_bot → (f • g).ne_bot
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theorem filter.ne_bot.vadd {α : Type u_2} {β : Type u_3} [has_vadd α β] {f : filter α} {g : filter β} :
f.ne_bot → g.ne_bot → (f +ᵥ g).ne_bot
source
@[simp]
theorem filter.le_vadd_iff {α : Type u_2} {β : Type u_3} [has_vadd α β] {f : filter α} {g h : filter β} :
h ≤ f +ᵥ g ↔ ∀ ⦃s : set α⦄, s ∈ f → ∀ ⦃t : set β⦄, t ∈ g → s +ᵥ t ∈ h
source
@[simp]
theorem filter.le_smul_iff {α : Type u_2} {β : Type u_3} [has_scalar α β] {f : filter α} {g h : filter β} :
h ≤ f • g ↔ ∀ ⦃s : set α⦄, s ∈ f → ∀ ⦃t : set β⦄, t ∈ g → s • t ∈ h
source
theorem filter.smul_le_smul {α : Type u_2} {β : Type u_3} [has_scalar α β] {f₁ f₂ : filter α} {g₁ g₂ : filter β} :
f₁ ≤ f₂ → g₁ ≤ g₂ → f₁ • g₁ ≤ f₂ • g₂
source
theorem filter.vadd_le_vadd {α : Type u_2} {β : Type u_3} [has_vadd α β] {f₁ f₂ : filter α} {g₁ g₂ : filter β} :
f₁ ≤ f₂ → g₁ ≤ g₂ → f₁ +ᵥ g₁ ≤ f₂ +ᵥ g₂
source
theorem filter.vadd_le_vadd_left {α : Type u_2} {β : Type u_3} [has_vadd α β] {f : filter α} {g₁ g₂ : filter β} :
g₁ ≤ g₂ → f +ᵥ g₁ ≤ f +ᵥ g₂
source
theorem filter.smul_le_smul_left {α : Type u_2} {β : Type u_3} [has_scalar α β] {f : filter α} {g₁ g₂ : filter β} :
g₁ ≤ g₂ → f • g₁ ≤ f • g₂
source
theorem filter.smul_le_smul_right {α : Type u_2} {β : Type u_3} [has_scalar α β] {f₁ f₂ : filter α} {g : filter β} :
f₁ ≤ f₂ → f₁ • g ≤ f₂ • g
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theorem filter.vadd_le_vadd_right {α : Type u_2} {β : Type u_3} [has_vadd α β] {f₁ f₂ : filter α} {g : filter β} :
f₁ ≤ f₂ → f₁ +ᵥ g ≤ f₂ +ᵥ g
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@[protected, instance]
def filter.covariant_vadd {α : Type u_2} {β : Type u_3} [has_vadd α β] :
covariant_class (filter α) (filter β) has_vadd.vadd has_le.le
source
@[protected, instance]
def filter.covariant_smul {α : Type u_2} {β : Type u_3} [has_scalar α β] :
covariant_class (filter α) (filter β) has_scalar.smul has_le.le
source
@[protected, instance]
def filter.add_action {α : Type u_2} {β : Type u_3} [add_monoid α] [add_action α β] :
add_action (filter α) (filter β)
Equations
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@[protected, instance]
def filter.mul_action {α : Type u_2} {β : Type u_3} [monoid α] [mul_action α β] :
mul_action (filter α) (filter β)
Equations

Scalar subtraction of filters #

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@[protected, instance]
def filter.has_vsub {α : Type u_2} {β : Type u_3} [has_vsub α β] :
has_vsub (filter α) (filter β)
Equations
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@[simp]
theorem filter.map₂_vsub {α : Type u_2} {β : Type u_3} [has_vsub α β] {f g : filter β} :
filter.map₂ has_vsub.vsub f g = f -ᵥ g
source
theorem filter.mem_vsub {α : Type u_2} {β : Type u_3} [has_vsub α β] {f g : filter β} {s : set α} :
s ∈ f -ᵥ g ↔ ∃ (t₁ t₂ : set β), t₁ ∈ f ∧ t₂ ∈ g ∧ t₁ -ᵥ t₂ ⊆ s
source
theorem filter.vsub_mem_vsub {α : Type u_2} {β : Type u_3} [has_vsub α β] {f g : filter β} {s t : set β} :
s ∈ f → t ∈ g → s -ᵥ t ∈ f -ᵥ g
source
@[simp]
theorem filter.bot_vsub {α : Type u_2} {β : Type u_3} [has_vsub α β] {g : filter β} :
⊥ -ᵥ g = ⊥
source
@[simp]
theorem filter.vsub_bot {α : Type u_2} {β : Type u_3} [has_vsub α β] {f : filter β} :
f -ᵥ ⊥ = ⊥
source
@[simp]
theorem filter.vsub_eq_bot_iff {α : Type u_2} {β : Type u_3} [has_vsub α β] {f g : filter β} :
f -ᵥ g = ⊥ ↔ f = ⊥ ∨ g = ⊥
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@[simp]
theorem filter.vsub_ne_bot_iff {α : Type u_2} {β : Type u_3} [has_vsub α β] {f g : filter β} :
(f -ᵥ g).ne_bot ↔ f.ne_bot ∧ g.ne_bot
source
theorem filter.ne_bot.vsub {α : Type u_2} {β : Type u_3} [has_vsub α β] {f g : filter β} :
f.ne_bot → g.ne_bot → (f -ᵥ g).ne_bot
source
@[simp]
theorem filter.le_vsub_iff {α : Type u_2} {β : Type u_3} [has_vsub α β] {f g : filter β} {h : filter α} :
h ≤ f -ᵥ g ↔ ∀ ⦃s : set β⦄, s ∈ f → ∀ ⦃t : set β⦄, t ∈ g → s -ᵥ t ∈ h
source
theorem filter.vsub_le_vsub {α : Type u_2} {β : Type u_3} [has_vsub α β] {f₁ f₂ g₁ g₂ : filter β} :
f₁ ≤ f₂ → g₁ ≤ g₂ → f₁ -ᵥ g₁ ≤ f₂ -ᵥ g₂
source
theorem filter.vsub_le_vsub_left {α : Type u_2} {β : Type u_3} [has_vsub α β] {f g₁ g₂ : filter β} :
g₁ ≤ g₂ → f -ᵥ g₁ ≤ f -ᵥ g₂
source
theorem filter.vsub_le_vsub_right {α : Type u_2} {β : Type u_3} [has_vsub α β] {f₁ f₂ g : filter β} :
f₁ ≤ f₂ → f₁ -ᵥ g ≤ f₂ -ᵥ g

Translation/scaling of filters #

source
@[protected, instance]
def filter.has_vadd_filter {α : Type u_2} {β : Type u_3} [has_vadd α β] :
has_vadd α (filter β)
Equations
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@[protected, instance]
def filter.has_scalar_filter {α : Type u_2} {β : Type u_3} [has_scalar α β] :
has_scalar α (filter β)
Equations
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@[simp]
theorem filter.map_smul {α : Type u_2} {β : Type u_3} [has_scalar α β] {f : filter β} {a : α} :
filter.map (λ (b : β), a • b) f = a • f
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@[simp]
theorem filter.map_vadd {α : Type u_2} {β : Type u_3} [has_vadd α β] {f : filter β} {a : α} :
filter.map (λ (b : β), a +ᵥ b) f = a +ᵥ f
source
theorem filter.mem_smul_filter {α : Type u_2} {β : Type u_3} [has_scalar α β] {f : filter β} {s : set β} {a : α} :
s ∈ a • f ↔ has_scalar.smul a ⁻¹' s ∈ f
source
theorem filter.mem_vadd_filter {α : Type u_2} {β : Type u_3} [has_vadd α β] {f : filter β} {s : set β} {a : α} :
s ∈ a +ᵥ f ↔ has_vadd.vadd a ⁻¹' s ∈ f
source
theorem filter.vadd_set_mem_vadd_filter {α : Type u_2} {β : Type u_3} [has_vadd α β] {f : filter β} {s : set β} {a : α} :
s ∈ f → a +ᵥ s ∈ a +ᵥ f
source
theorem filter.smul_set_mem_smul_filter {α : Type u_2} {β : Type u_3} [has_scalar α β] {f : filter β} {s : set β} {a : α} :
s ∈ f → a • s ∈ a • f
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@[simp]
theorem filter.smul_filter_bot {α : Type u_2} {β : Type u_3} [has_scalar α β] {a : α} :
a • ⊥ = ⊥
source
@[simp]
theorem filter.vadd_filter_bot {α : Type u_2} {β : Type u_3} [has_vadd α β] {a : α} :
a +ᵥ ⊥ = ⊥
source
@[simp]
theorem filter.vadd_filter_eq_bot_iff {α : Type u_2} {β : Type u_3} [has_vadd α β] {f : filter β} {a : α} :
a +ᵥ f = ⊥ ↔ f = ⊥
source
@[simp]
theorem filter.smul_filter_eq_bot_iff {α : Type u_2} {β : Type u_3} [has_scalar α β] {f : filter β} {a : α} :
a • f = ⊥ ↔ f = ⊥
source
@[simp]
theorem filter.smul_filter_ne_bot_iff {α : Type u_2} {β : Type u_3} [has_scalar α β] {f : filter β} {a : α} :
(a • f).ne_bot ↔ f.ne_bot
source
@[simp]
theorem filter.vadd_filter_ne_bot_iff {α : Type u_2} {β : Type u_3} [has_vadd α β] {f : filter β} {a : α} :
(a +ᵥ f).ne_bot ↔ f.ne_bot
source
theorem filter.ne_bot.vadd_filter {α : Type u_2} {β : Type u_3} [has_vadd α β] {f : filter β} {a : α} :
f.ne_bot → (a +ᵥ f).ne_bot
source
theorem filter.ne_bot.smul_filter {α : Type u_2} {β : Type u_3} [has_scalar α β] {f : filter β} {a : α} :
f.ne_bot → (a • f).ne_bot
source
theorem filter.smul_filter_le_smul_filter {α : Type u_2} {β : Type u_3} [has_scalar α β] {f₁ f₂ : filter β} {a : α} (hf : f₁ ≤ f₂) :
a • f₁ ≤ a • f₂
source
theorem filter.vadd_filter_le_vadd_filter {α : Type u_2} {β : Type u_3} [has_vadd α β] {f₁ f₂ : filter β} {a : α} (hf : f₁ ≤ f₂) :
a +ᵥ f₁ ≤ a +ᵥ f₂
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@[protected, instance]
def filter.covariant_smul_filter {α : Type u_2} {β : Type u_3} [has_scalar α β] :
covariant_class α (filter β) has_scalar.smul has_le.le
source
@[protected, instance]
def filter.covariant_vadd_filter {α : Type u_2} {β : Type u_3} [has_vadd α β] :
covariant_class α (filter β) has_vadd.vadd has_le.le
source
@[protected, instance]
def filter.smul_comm_class_filter {α : Type u_2} {β : Type u_3} {γ : Type u_4} [has_scalar α γ] [has_scalar β γ] [smul_comm_class α β γ] :
smul_comm_class α (filter β) (filter γ)
source
@[protected, instance]
def filter.vadd_comm_class_filter {α : Type u_2} {β : Type u_3} {γ : Type u_4} [has_vadd α γ] [has_vadd β γ] [vadd_comm_class α β γ] :
vadd_comm_class α (filter β) (filter γ)
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@[protected, instance]
def filter.smul_comm_class_filter' {α : Type u_2} {β : Type u_3} {γ : Type u_4} [has_scalar α γ] [has_scalar β γ] [smul_comm_class α β γ] :
smul_comm_class (filter α) β (filter γ)
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@[protected, instance]
def filter.vadd_comm_class_filter' {α : Type u_2} {β : Type u_3} {γ : Type u_4} [has_vadd α γ] [has_vadd β γ] [vadd_comm_class α β γ] :
vadd_comm_class (filter α) β (filter γ)
source
@[protected, instance]
def filter.vadd_comm_class {α : Type u_2} {β : Type u_3} {γ : Type u_4} [has_vadd α γ] [has_vadd β γ] [vadd_comm_class α β γ] :
vadd_comm_class (filter α) (filter β) (filter γ)
source
@[protected, instance]
def filter.smul_comm_class {α : Type u_2} {β : Type u_3} {γ : Type u_4} [has_scalar α γ] [has_scalar β γ] [smul_comm_class α β γ] :
smul_comm_class (filter α) (filter β) (filter γ)
source
@[protected, instance]
def filter.is_scalar_tower {α : Type u_2} {β : Type u_3} {γ : Type u_4} [has_scalar α β] [has_scalar α γ] [has_scalar β γ] [is_scalar_tower α β γ] :
is_scalar_tower α β (filter γ)
source
@[protected, instance]
def filter.is_scalar_tower' {α : Type u_2} {β : Type u_3} {γ : Type u_4} [has_scalar α β] [has_scalar α γ] [has_scalar β γ] [is_scalar_tower α β γ] :
is_scalar_tower α (filter β) (filter γ)
source
@[protected, instance]
def filter.is_scalar_tower'' {α : Type u_2} {β : Type u_3} {γ : Type u_4} [has_scalar α β] [has_scalar α γ] [has_scalar β γ] [is_scalar_tower α β γ] :
is_scalar_tower (filter α) (filter β) (filter γ)
source
@[protected, instance]
def filter.is_central_scalar {α : Type u_2} {β : Type u_3} [has_scalar α β] [has_scalar αᵐᵒᵖ β] [is_central_scalar α β] :
is_central_scalar α (filter β)