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 #

For α a semigroup/monoid, filter α is a semigroup/monoid. As an unfortunate side effect, this means that n • f, where n : ℕ, is ambiguous between pointwise scaling and repeated pointwise addition. See note [pointwise nat action].

Implementation notes #

We put all instances in the locale pointwise, so that these instances are not available by default. Note that we do not mark them as reducible (as argued by note [reducible non-instances]) since we expect the locale to be open whenever the instances are actually used (and making the instances reducible changes the behavior of simp.

Tags #

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

0/1 as filters #

@[protected]
def filter.has_zero {α : Type u_2} [has_zero α] :

0 : filter α is defined as the filter of sets containing 0 : α in locale pointwise.

Equations
@[protected]
def filter.has_one {α : Type u_2} [has_one α] :

1 : filter α is defined as the filter of sets containing 1 : α in locale pointwise.

Equations
@[simp]
theorem filter.mem_zero {α : Type u_2} [has_zero α] {s : set α} :
s 0 0 s
@[simp]
theorem filter.mem_one {α : Type u_2} [has_one α] {s : set α} :
s 1 1 s
theorem filter.zero_mem_zero {α : Type u_2} [has_zero α] :
0 0
theorem filter.one_mem_one {α : Type u_2} [has_one α] :
1 1
@[simp]
theorem filter.pure_zero {α : Type u_2} [has_zero α] :
@[simp]
theorem filter.pure_one {α : Type u_2} [has_one α] :
@[simp]
theorem filter.principal_one {α : Type u_2} [has_one α] :
@[simp]
theorem filter.principal_zero {α : Type u_2} [has_zero α] :
theorem filter.one_ne_bot {α : Type u_2} [has_one α] :
theorem filter.zero_ne_bot {α : Type u_2} [has_zero α] :
@[protected, simp]
theorem filter.map_one' {α : Type u_2} {β : Type u_3} [has_one α] (f : α → β) :
@[protected, simp]
theorem filter.map_zero' {α : Type u_2} {β : Type u_3} [has_zero α] (f : α → β) :
@[simp]
theorem filter.nonpos_iff {α : Type u_2} [has_zero α] {f : filter α} :
f 0 0 f
@[simp]
theorem filter.le_one_iff {α : Type u_2} [has_one α] {f : filter α} :
f 1 1 f
@[protected]
theorem filter.ne_bot.le_one_iff {α : Type u_2} [has_one α] {f : filter α} (h : f.ne_bot) :
f 1 f = 1
@[protected]
theorem filter.ne_bot.nonpos_iff {α : Type u_2} [has_zero α] {f : filter α} (h : f.ne_bot) :
f 0 f = 0
@[simp]
theorem filter.eventually_one {α : Type u_2} [has_one α] {p : α → Prop} :
(∀ᶠ (x : α) in 1, p x) p 1
@[simp]
theorem filter.eventually_zero {α : Type u_2} [has_zero α] {p : α → Prop} :
(∀ᶠ (x : α) in 0, p x) p 0
@[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
@[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
def filter.pure_zero_hom {α : Type u_2} [has_zero α] :

pure as a zero_hom.

Equations
def filter.pure_one_hom {α : Type u_2} [has_one α] :
one_hom α (filter α)

pure as a one_hom.

Equations
@[simp]
theorem filter.pure_zero_hom_apply {α : Type u_2} [has_zero α] (a : α) :
@[simp]
theorem filter.pure_one_hom_apply {α : Type u_2} [has_one α] (a : α) :
@[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) :
@[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 negation/inversion #

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

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

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

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

Equations
@[protected, simp]
theorem filter.map_neg {α : Type u_2} [has_neg α] {f : filter α} :
@[protected, simp]
theorem filter.map_inv {α : Type u_2} [has_inv α] {f : filter α} :
theorem filter.mem_inv {α : Type u_2} [has_inv α] {f : filter α} {s : set α} :
theorem filter.mem_neg {α : Type u_2} [has_neg α] {f : filter α} {s : set α} :
@[protected]
theorem filter.neg_le_neg {α : Type u_2} [has_neg α] {f g : filter α} (hf : f g) :
-f -g
@[protected]
theorem filter.inv_le_inv {α : Type u_2} [has_inv α] {f g : filter α} (hf : f g) :
@[simp]
theorem filter.inv_pure {α : Type u_2} [has_inv α] {a : α} :
@[simp]
theorem filter.neg_pure {α : Type u_2} [has_neg α] {a : α} :
@[simp]
theorem filter.neg_eq_bot_iff {α : Type u_2} [has_neg α] {f : filter α} :
-f = f =
@[simp]
theorem filter.inv_eq_bot_iff {α : Type u_2} [has_inv α] {f : filter α} :
@[simp]
theorem filter.ne_bot_inv_iff {α : Type u_2} [has_inv α] {f : filter α} :
@[simp]
theorem filter.ne_bot_neg_iff {α : Type u_2} [has_neg α] {f : filter α} :
theorem filter.ne_bot.neg {α : Type u_2} [has_neg α] {f : filter α} :
f.ne_bot → (-f).ne_bot
theorem filter.ne_bot.inv {α : Type u_2} [has_inv α] {f : filter α} :
theorem filter.neg_mem_neg {α : Type u_2} [has_involutive_neg α] {f : filter α} {s : set α} (hs : s f) :
-s -f
theorem filter.inv_mem_inv {α : Type u_2} [has_involutive_inv α] {f : filter α} {s : set α} (hs : s f) :
@[protected]

Negation is involutive on filter α if it is on α.

Equations
@[protected]

Inversion is involutive on filter α if it is on α.

Equations

Filter addition/multiplication #

@[protected]
def filter.has_add {α : Type u_2} [has_add α] :

The filter f + g is generated by {s + t | s ∈ f, t ∈ g} in locale pointwise.

Equations
@[protected]
def filter.has_mul {α : Type u_2} [has_mul α] :

The filter f * g is generated by {s * t | s ∈ f, t ∈ g} in locale pointwise.

Equations
@[simp]
theorem filter.map₂_mul {α : Type u_2} [has_mul α] {f g : filter α} :
@[simp]
theorem filter.map₂_add {α : Type u_2} [has_add α] {f g : filter α} :
theorem filter.mem_mul {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
theorem filter.mem_add {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
theorem filter.mul_mem_mul {α : Type u_2} [has_mul α] {f g : filter α} {s t : set α} :
s ft gs * t f * g
theorem filter.add_mem_add {α : Type u_2} [has_add α] {f g : filter α} {s t : set α} :
s ft gs + t f + g
@[simp]
theorem filter.bot_add {α : Type u_2} [has_add α] {g : filter α} :
@[simp]
theorem filter.bot_mul {α : Type u_2} [has_mul α] {g : filter α} :
@[simp]
theorem filter.add_bot {α : Type u_2} [has_add α] {f : filter α} :
@[simp]
theorem filter.mul_bot {α : Type u_2} [has_mul α] {f : filter α} :
@[simp]
theorem filter.add_eq_bot_iff {α : Type u_2} [has_add α] {f g : filter α} :
f + g = f = g =
@[simp]
theorem filter.mul_eq_bot_iff {α : Type u_2} [has_mul α] {f g : filter α} :
f * g = f = g =
@[simp]
theorem filter.mul_ne_bot_iff {α : Type u_2} [has_mul α] {f g : filter α} :
@[simp]
theorem filter.add_ne_bot_iff {α : Type u_2} [has_add α] {f g : filter α} :
theorem filter.ne_bot.mul {α : Type u_2} [has_mul α] {f g : filter α} :
f.ne_botg.ne_bot(f * g).ne_bot
theorem filter.ne_bot.add {α : Type u_2} [has_add α] {f g : filter α} :
f.ne_botg.ne_bot(f + g).ne_bot
theorem filter.ne_bot.of_add_left {α : Type u_2} [has_add α] {f g : filter α} :
(f + g).ne_bot → f.ne_bot
theorem filter.ne_bot.of_mul_left {α : Type u_2} [has_mul α] {f g : filter α} :
(f * g).ne_bot → f.ne_bot
theorem filter.ne_bot.of_add_right {α : Type u_2} [has_add α] {f g : filter α} :
(f + g).ne_bot → g.ne_bot
theorem filter.ne_bot.of_mul_right {α : Type u_2} [has_mul α] {f g : filter α} :
(f * g).ne_bot → g.ne_bot
@[simp]
theorem filter.pure_mul {α : Type u_2} [has_mul α] {g : filter α} {a : α} :
@[simp]
theorem filter.pure_add {α : Type u_2} [has_add α] {g : filter α} {a : α} :
@[simp]
theorem filter.add_pure {α : Type u_2} [has_add α] {f : filter α} {b : α} :
f + has_pure.pure b = filter.map (λ (_x : α), _x + b) f
@[simp]
theorem filter.mul_pure {α : Type u_2} [has_mul α] {f : filter α} {b : α} :
f * has_pure.pure b = filter.map (λ (_x : α), _x * b) f
@[simp]
theorem filter.pure_mul_pure {α : Type u_2} [has_mul α] {a b : α} :
@[simp]
theorem filter.pure_add_pure {α : Type u_2} [has_add α] {a b : α} :
@[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 gs * t h
@[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 gs + t h
@[protected, instance]
@[protected, instance]
@[protected, instance]
@[protected, instance]
@[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
@[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
def filter.pure_add_hom {α : Type u_2} [has_add α] :
add_hom α (filter α)

The singleton operation as an add_hom.

Equations
def filter.pure_mul_hom {α : Type u_2} [has_mul α] :

pure operation as a mul_hom.

Equations
@[simp]
theorem filter.pure_mul_hom_apply {α : Type u_2} [has_mul α] (a : α) :
@[simp]
theorem filter.pure_add_hom_apply {α : Type u_2} [has_add α] (a : α) :

Filter subtraction/division #

@[protected]
def filter.has_sub {α : Type u_2} [has_sub α] :

The filter f - g is generated by {s - t | s ∈ f, t ∈ g} in locale pointwise.

Equations
@[protected]
def filter.has_div {α : Type u_2} [has_div α] :

The filter f / g is generated by {s / t | s ∈ f, t ∈ g} in locale pointwise.

Equations
@[simp]
theorem filter.map₂_sub {α : Type u_2} [has_sub α] {f g : filter α} :
@[simp]
theorem filter.map₂_div {α : Type u_2} [has_div α] {f g : filter α} :
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
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
theorem filter.div_mem_div {α : Type u_2} [has_div α] {f g : filter α} {s t : set α} :
s ft gs / t f / g
theorem filter.sub_mem_sub {α : Type u_2} [has_sub α] {f g : filter α} {s t : set α} :
s ft gs - t f - g
@[simp]
theorem filter.bot_sub {α : Type u_2} [has_sub α] {g : filter α} :
@[simp]
theorem filter.bot_div {α : Type u_2} [has_div α] {g : filter α} :
@[simp]
theorem filter.div_bot {α : Type u_2} [has_div α] {f : filter α} :
@[simp]
theorem filter.sub_bot {α : Type u_2} [has_sub α] {f : filter α} :
@[simp]
theorem filter.sub_eq_bot_iff {α : Type u_2} [has_sub α] {f g : filter α} :
f - g = f = g =
@[simp]
theorem filter.div_eq_bot_iff {α : Type u_2} [has_div α] {f g : filter α} :
f / g = f = g =
@[simp]
theorem filter.div_ne_bot_iff {α : Type u_2} [has_div α] {f g : filter α} :
@[simp]
theorem filter.sub_ne_bot_iff {α : Type u_2} [has_sub α] {f g : filter α} :
theorem filter.ne_bot.sub {α : Type u_2} [has_sub α] {f g : filter α} :
f.ne_botg.ne_bot(f - g).ne_bot
theorem filter.ne_bot.div {α : Type u_2} [has_div α] {f g : filter α} :
f.ne_botg.ne_bot(f / g).ne_bot
theorem filter.ne_bot.of_div_left {α : Type u_2} [has_div α] {f g : filter α} :
(f / g).ne_bot → f.ne_bot
theorem filter.ne_bot.of_sub_left {α : Type u_2} [has_sub α] {f g : filter α} :
(f - g).ne_bot → f.ne_bot
theorem filter.ne_bot.of_div_right {α : Type u_2} [has_div α] {f g : filter α} :
(f / g).ne_bot → g.ne_bot
theorem filter.ne_bot.of_sub_right {α : Type u_2} [has_sub α] {f g : filter α} :
(f - g).ne_bot → g.ne_bot
@[simp]
theorem filter.pure_div {α : Type u_2} [has_div α] {g : filter α} {a : α} :
@[simp]
theorem filter.pure_sub {α : Type u_2} [has_sub α] {g : filter α} {a : α} :
@[simp]
theorem filter.div_pure {α : Type u_2} [has_div α] {f : filter α} {b : α} :
f / has_pure.pure b = filter.map (λ (_x : α), _x / b) f
@[simp]
theorem filter.sub_pure {α : Type u_2} [has_sub α] {f : filter α} {b : α} :
f - has_pure.pure b = filter.map (λ (_x : α), _x - b) f
@[simp]
theorem filter.pure_sub_pure {α : Type u_2} [has_sub α] {a b : α} :
@[simp]
theorem filter.pure_div_pure {α : Type u_2} [has_div α] {a b : α} :
@[protected]
theorem filter.div_le_div {α : Type u_2} [has_div α] {f₁ f₂ g₁ g₂ : filter α} :
f₁ f₂g₁ g₂f₁ / g₁ f₂ / g₂
@[protected]
theorem filter.sub_le_sub {α : Type u_2} [has_sub α] {f₁ f₂ g₁ g₂ : filter α} :
f₁ f₂g₁ g₂f₁ - g₁ f₂ - g₂
@[protected]
theorem filter.div_le_div_left {α : Type u_2} [has_div α] {f g₁ g₂ : filter α} :
g₁ g₂f / g₁ f / g₂
@[protected]
theorem filter.sub_le_sub_left {α : Type u_2} [has_sub α] {f g₁ g₂ : filter α} :
g₁ g₂f - g₁ f - g₂
@[protected]
theorem filter.div_le_div_right {α : Type u_2} [has_div α] {f₁ f₂ g : filter α} :
f₁ f₂f₁ / g f₂ / g
@[protected]
theorem filter.sub_le_sub_right {α : Type u_2} [has_sub α] {f₁ f₂ g : filter α} :
f₁ f₂f₁ - g f₂ - g
@[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 gs - t h
@[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 gs / t h
@[protected, instance]
@[protected, instance]
@[protected, instance]
@[protected, instance]
@[protected]
def filter.has_nsmul {α : Type u_2} [has_zero α] [has_add α] :

Repeated pointwise addition (not the same as pointwise repeated addition!) of a filter. See Note [pointwise nat action].

Equations
@[protected]
def filter.has_npow {α : Type u_2} [has_one α] [has_mul α] :

Repeated pointwise multiplication (not the same as pointwise repeated multiplication!) of a filter. See Note [pointwise nat action].

Equations
@[protected]
def filter.has_zsmul {α : Type u_2} [has_zero α] [has_add α] [has_neg α] :

Repeated pointwise addition/subtraction (not the same as pointwise repeated addition/subtraction!) of a filter. See Note [pointwise nat action].

Equations
@[protected]
def filter.has_zpow {α : Type u_2} [has_one α] [has_mul α] [has_inv α] :

Repeated pointwise multiplication/division (not the same as pointwise repeated multiplication/division!) of a filter. See Note [pointwise nat action].

Equations
@[protected]
def filter.semigroup {α : Type u_2} [semigroup α] :

filter α is a semigroup under pointwise operations if α is.

Equations
@[protected]
def filter.add_semigroup {α : Type u_2} [add_semigroup α] :

filter α is an add_semigroup under pointwise operations if α is.

Equations
@[protected]
def filter.comm_semigroup {α : Type u_2} [comm_semigroup α] :

filter α is a comm_semigroup under pointwise operations if α is.

Equations
@[protected]

filter α is an add_comm_semigroup under pointwise operations if α is.

Equations
@[protected]
def filter.add_zero_class {α : Type u_2} [add_zero_class α] :

filter α is an add_zero_class under pointwise operations if α is.

Equations
@[protected]
def filter.mul_one_class {α : Type u_2} [mul_one_class α] :

filter α is a mul_one_class under pointwise operations if α is.

Equations
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) :

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

Equations
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) :

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

Equations
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) {f g : filter β} :
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) {f g : filter β} :
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 f₂filter.tendsto m g₁ g₂filter.tendsto m (f₁ + g₁) (f₂ + g₂)
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 f₂filter.tendsto m g₁ g₂filter.tendsto m (f₁ * g₁) (f₂ * g₂)
@[simp]
@[protected]
def filter.add_monoid {α : Type u_2} [add_monoid α] :

filter α is an add_monoid under pointwise operations if α is.

Equations
@[protected]
def filter.monoid {α : Type u_2} [monoid α] :

filter α is a monoid under pointwise operations if α is.

Equations
theorem filter.pow_mem_pow {α : Type u_2} [monoid α] {f : filter α} {s : set α} (hs : s f) (n : ) :
s ^ n f ^ n
theorem filter.nsmul_mem_nsmul {α : Type u_2} [add_monoid α] {f : filter α} {s : set α} (hs : s f) (n : ) :
n s n f
@[simp]
theorem filter.nsmul_bot {α : Type u_2} [add_monoid α] {n : } (hn : n 0) :
@[simp]
theorem filter.bot_pow {α : Type u_2} [monoid α] {n : } (hn : n 0) :
theorem filter.mul_top_of_one_le {α : Type u_2} [monoid α] {f : filter α} (hf : 1 f) :
theorem filter.add_top_of_nonneg {α : Type u_2} [add_monoid α] {f : filter α} (hf : 0 f) :
theorem filter.top_mul_of_one_le {α : Type u_2} [monoid α] {f : filter α} (hf : 1 f) :
theorem filter.top_add_of_nonneg {α : Type u_2} [add_monoid α] {f : filter α} (hf : 0 f) :
@[simp]
theorem filter.top_mul_top {α : Type u_2} [monoid α] :
@[simp]
theorem filter.top_add_top {α : Type u_2} [add_monoid α] :
theorem filter.nsmul_top {α : Type u_1} [add_monoid α] {n : } :
n 0n =
theorem filter.top_pow {α : Type u_2} [monoid α] {n : } :
n 0 ^ n =
@[protected]
theorem is_unit.filter {α : Type u_2} [monoid α] {a : α} :
@[protected]
theorem is_add_unit.filter {α : Type u_2} [add_monoid α] {a : α} :
@[protected]
def filter.comm_monoid {α : Type u_2} [comm_monoid α] :

filter α is a comm_monoid under pointwise operations if α is.

Equations
@[protected]

filter α is an add_comm_monoid under pointwise operations if α is.

Equations
@[protected]
theorem filter.mul_eq_one_iff {α : Type u_2} [division_monoid α] {f g : filter α} :
f * g = 1 ∃ (a b : α), f = has_pure.pure a g = has_pure.pure b a * b = 1
@[protected]
theorem filter.add_eq_zero_iff {α : Type u_2} [subtraction_monoid α] {f g : filter α} :
f + g = 0 ∃ (a b : α), f = has_pure.pure a g = has_pure.pure b a + b = 0
@[protected]

filter α is a division monoid under pointwise operations if α is.

Equations
@[protected]

filter α is a subtraction monoid under pointwise operations if α is.

Equations
theorem filter.is_unit_iff {α : Type u_2} [division_monoid α] {f : filter α} :
is_unit f ∃ (a : α), f = has_pure.pure a is_unit a
theorem filter.is_add_unit_iff {α : Type u_2} [subtraction_monoid α] {f : filter α} :
@[protected]
def filter.has_distrib_neg {α : Type u_2} [has_mul α] [has_distrib_neg α] :

filter α has distributive negation if α has.

Equations

Note that filter α is not a distrib because f * g + f * h has cross terms that f * (g + h) lacks.

theorem filter.mul_add_subset {α : Type u_2} [distrib α] {f g h : filter α} :
f * (g + h) f * g + f * h
theorem filter.add_mul_subset {α : Type u_2} [distrib α] {f g h : filter α} :
(f + g) * h f * h + g * h

Note that filter is not a mul_zero_class because 0 * ⊥ ≠ 0.

theorem filter.ne_bot.mul_zero_nonneg {α : Type u_2} [mul_zero_class α] {f : filter α} (hf : f.ne_bot) :
0 f * 0
theorem filter.ne_bot.zero_mul_nonneg {α : Type u_2} [mul_zero_class α] {g : filter α} (hg : g.ne_bot) :
0 0 * g

Note that filter α is not a group because f / f ≠ 1 in general

@[protected, simp]
theorem filter.nonneg_sub_iff {α : Type u_2} [add_group α] {f g : filter α} :
0 f - g ¬disjoint f g
@[protected, simp]
theorem filter.one_le_div_iff {α : Type u_2} [group α] {f g : filter α} :
1 f / g ¬disjoint f g
theorem filter.not_nonneg_sub_iff {α : Type u_2} [add_group α] {f g : filter α} :
¬0 f - g disjoint f g
theorem filter.not_one_le_div_iff {α : Type u_2} [group α] {f g : filter α} :
¬1 f / g disjoint f g
theorem filter.ne_bot.one_le_div {α : Type u_2} [group α] {f : filter α} (h : f.ne_bot) :
1 f / f
theorem filter.ne_bot.nonneg_sub {α : Type u_2} [add_group α] {f : filter α} (h : f.ne_bot) :
0 f - f
theorem filter.is_unit_pure {α : Type u_2} [group α] (a : α) :
theorem filter.is_add_unit_pure {α : Type u_2} [add_group α] (a : α) :
@[simp]
theorem filter.is_unit_iff_singleton {α : Type u_2} [group α] {f : filter α} :
is_unit f ∃ (a : α), f = has_pure.pure a
theorem filter.map_neg' {F : Type u_1} {α : Type u_2} {β : Type u_3} [add_group α] [subtraction_monoid β] [add_monoid_hom_class F α β] (m : F) {f : filter α} :
theorem filter.map_inv' {F : Type u_1} {α : Type u_2} {β : Type u_3} [group α] [division_monoid β] [monoid_hom_class F α β] (m : F) {f : filter α} :
theorem filter.tendsto.neg_neg {F : Type u_1} {α : Type u_2} {β : Type u_3} [add_group α] [subtraction_monoid β] [add_monoid_hom_class F α β] (m : F) {f₁ : filter α} {f₂ : filter β} :
filter.tendsto m f₁ f₂filter.tendsto m (-f₁) (-f₂)
theorem filter.tendsto.inv_inv {F : Type u_1} {α : Type u_2} {β : Type u_3} [group α] [division_monoid β] [monoid_hom_class F α β] (m : F) {f₁ : filter α} {f₂ : filter β} :
@[protected]
theorem filter.map_sub {F : Type u_1} {α : Type u_2} {β : Type u_3} [add_group α] [subtraction_monoid β] [add_monoid_hom_class F α β] (m : F) {f g : filter α} :
@[protected]
theorem filter.map_div {F : Type u_1} {α : Type u_2} {β : Type u_3} [group α] [division_monoid β] [monoid_hom_class F α β] (m : F) {f g : filter α} :
theorem filter.tendsto.div_div {F : Type u_1} {α : Type u_2} {β : Type u_3} [group α] [division_monoid β] [monoid_hom_class F α β] (m : F) {f₁ g₁ : filter α} {f₂ g₂ : filter β} :
filter.tendsto m f₁ f₂filter.tendsto m g₁ g₂filter.tendsto m (f₁ / g₁) (f₂ / g₂)
theorem filter.tendsto.sub_sub {F : Type u_1} {α : Type u_2} {β : Type u_3} [add_group α] [subtraction_monoid β] [add_monoid_hom_class F α β] (m : F) {f₁ g₁ : filter α} {f₂ g₂ : filter β} :
filter.tendsto m f₁ f₂filter.tendsto m g₁ g₂filter.tendsto m (f₁ - g₁) (f₂ - g₂)
theorem filter.ne_bot.div_zero_nonneg {α : Type u_2} [group_with_zero α] {f : filter α} (hf : f.ne_bot) :
0 f / 0
theorem filter.ne_bot.zero_div_nonneg {α : Type u_2} [group_with_zero α] {g : filter α} (hg : g.ne_bot) :
0 0 / g

Scalar addition/multiplication of filters #

@[protected]
def filter.has_smul {α : Type u_2} {β : Type u_3} [has_smul α β] :

The filter f • g is generated by {s • t | s ∈ f, t ∈ g} in locale pointwise.

Equations
@[protected]
def filter.has_vadd {α : Type u_2} {β : Type u_3} [has_vadd α β] :

The filter f +ᵥ g is generated by {s +ᵥ t | s ∈ f, t ∈ g} in locale pointwise.

Equations
@[simp]
theorem filter.map₂_smul {α : Type u_2} {β : Type u_3} [has_smul α β] {f : filter α} {g : filter β} :
@[simp]
theorem filter.map₂_vadd {α : Type u_2} {β : Type u_3} [has_vadd α β] {f : filter α} {g : filter β} :
theorem filter.mem_smul {α : Type u_2} {β : Type u_3} [has_smul 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
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
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 ft gs +ᵥ t f +ᵥ g
theorem filter.smul_mem_smul {α : Type u_2} {β : Type u_3} [has_smul α β] {f : filter α} {g : filter β} {s : set α} {t : set β} :
s ft gs t f g
@[simp]
theorem filter.bot_smul {α : Type u_2} {β : Type u_3} [has_smul α β] {g : filter β} :
@[simp]
theorem filter.bot_vadd {α : Type u_2} {β : Type u_3} [has_vadd α β] {g : filter β} :
@[simp]
theorem filter.vadd_bot {α : Type u_2} {β : Type u_3} [has_vadd α β] {f : filter α} :
@[simp]
theorem filter.smul_bot {α : Type u_2} {β : Type u_3} [has_smul α β] {f : filter α} :
@[simp]
theorem filter.smul_eq_bot_iff {α : Type u_2} {β : Type u_3} [has_smul α β] {f : filter α} {g : filter β} :
f g = f = g =
@[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 =
@[simp]
theorem filter.smul_ne_bot_iff {α : Type u_2} {β : Type u_3} [has_smul α β] {f : filter α} {g : filter β} :
@[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} :
theorem filter.ne_bot.smul {α : Type u_2} {β : Type u_3} [has_smul α β] {f : filter α} {g : filter β} :
f.ne_botg.ne_bot(f g).ne_bot
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_botg.ne_bot(f +ᵥ g).ne_bot
theorem filter.ne_bot.of_smul_left {α : Type u_2} {β : Type u_3} [has_smul α β] {f : filter α} {g : filter β} :
(f g).ne_bot → f.ne_bot
theorem filter.ne_bot.of_vadd_left {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
theorem filter.ne_bot.of_vadd_right {α : Type u_2} {β : Type u_3} [has_vadd α β] {f : filter α} {g : filter β} :
(f +ᵥ g).ne_bot → g.ne_bot
theorem filter.ne_bot.of_smul_right {α : Type u_2} {β : Type u_3} [has_smul α β] {f : filter α} {g : filter β} :
(f g).ne_bot → g.ne_bot
@[simp]
theorem filter.pure_vadd {α : Type u_2} {β : Type u_3} [has_vadd α β] {g : filter β} {a : α} :
@[simp]
theorem filter.pure_smul {α : Type u_2} {β : Type u_3} [has_smul α β] {g : filter β} {a : α} :
@[simp]
theorem filter.smul_pure {α : Type u_2} {β : Type u_3} [has_smul α β] {f : filter α} {b : β} :
f has_pure.pure b = filter.map (λ (_x : α), _x b) f
@[simp]
theorem filter.vadd_pure {α : Type u_2} {β : Type u_3} [has_vadd α β] {f : filter α} {b : β} :
f +ᵥ has_pure.pure b = filter.map (λ (_x : α), _x +ᵥ b) f
@[simp]
theorem filter.pure_vadd_pure {α : Type u_2} {β : Type u_3} [has_vadd α β] {a : α} {b : β} :
@[simp]
theorem filter.pure_smul_pure {α : Type u_2} {β : Type u_3} [has_smul α β] {a : α} {b : β} :
theorem filter.smul_le_smul {α : Type u_2} {β : Type u_3} [has_smul α β] {f₁ f₂ : filter α} {g₁ g₂ : filter β} :
f₁ f₂g₁ g₂f₁ g₁ f₂ g₂
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 f₂g₁ g₂f₁ +ᵥ g₁ f₂ +ᵥ g₂
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 g₂f +ᵥ g₁ f +ᵥ g₂
theorem filter.smul_le_smul_left {α : Type u_2} {β : Type u_3} [has_smul α β] {f : filter α} {g₁ g₂ : filter β} :
g₁ g₂f g₁ f g₂
theorem filter.smul_le_smul_right {α : Type u_2} {β : Type u_3} [has_smul α β] {f₁ f₂ : filter α} {g : filter β} :
f₁ f₂f₁ g f₂ g
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 f₂f₁ +ᵥ g f₂ +ᵥ g
@[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 gs +ᵥ t h
@[simp]
theorem filter.le_smul_iff {α : Type u_2} {β : Type u_3} [has_smul α β] {f : filter α} {g h : filter β} :
h f g ∀ ⦃s : set α⦄, s f∀ ⦃t : set β⦄, t gs t h
@[protected, instance]
def filter.covariant_vadd {α : Type u_2} {β : Type u_3} [has_vadd α β] :
@[protected, instance]
def filter.covariant_smul {α : Type u_2} {β : Type u_3} [has_smul α β] :

Scalar subtraction of filters #

@[protected]
def filter.has_vsub {α : Type u_2} {β : Type u_3} [has_vsub α β] :

The filter f -ᵥ g is generated by {s -ᵥ t | s ∈ f, t ∈ g} in locale pointwise.

Equations
@[simp]
theorem filter.map₂_vsub {α : Type u_2} {β : Type u_3} [has_vsub α β] {f g : filter β} :
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
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 ft gs -ᵥ t f -ᵥ g
@[simp]
theorem filter.bot_vsub {α : Type u_2} {β : Type u_3} [has_vsub α β] {g : filter β} :
@[simp]
theorem filter.vsub_bot {α : Type u_2} {β : Type u_3} [has_vsub α β] {f : filter β} :
@[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 =
@[simp]
theorem filter.vsub_ne_bot_iff {α : Type u_2} {β : Type u_3} [has_vsub α β] {f g : filter β} :
theorem filter.ne_bot.vsub {α : Type u_2} {β : Type u_3} [has_vsub α β] {f g : filter β} :
f.ne_botg.ne_bot(f -ᵥ g).ne_bot
theorem filter.ne_bot.of_vsub_left {α : Type u_2} {β : Type u_3} [has_vsub α β] {f g : filter β} :
(f -ᵥ g).ne_bot → f.ne_bot
theorem filter.ne_bot.of_vsub_right {α : Type u_2} {β : Type u_3} [has_vsub α β] {f g : filter β} :
(f -ᵥ g).ne_bot → g.ne_bot
@[simp]
theorem filter.pure_vsub {α : Type u_2} {β : Type u_3} [has_vsub α β] {g : filter β} {a : β} :
@[simp]
theorem filter.vsub_pure {α : Type u_2} {β : Type u_3} [has_vsub α β] {f : filter β} {b : β} :
f -ᵥ has_pure.pure b = filter.map (λ (_x : β), _x -ᵥ b) f
@[simp]
theorem filter.pure_vsub_pure {α : Type u_2} {β : Type u_3} [has_vsub α β] {a b : β} :
theorem filter.vsub_le_vsub {α : Type u_2} {β : Type u_3} [has_vsub α β] {f₁ f₂ g₁ g₂ : filter β} :
f₁ f₂g₁ g₂f₁ -ᵥ g₁ f₂ -ᵥ g₂
theorem filter.vsub_le_vsub_left {α : Type u_2} {β : Type u_3} [has_vsub α β] {f g₁ g₂ : filter β} :
g₁ g₂f -ᵥ g₁ f -ᵥ g₂
theorem filter.vsub_le_vsub_right {α : Type u_2} {β : Type u_3} [has_vsub α β] {f₁ f₂ g : filter β} :
f₁ f₂f₁ -ᵥ g f₂ -ᵥ g
@[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 gs -ᵥ t h

Translation/scaling of filters #

@[protected]
def filter.has_smul_filter {α : Type u_2} {β : Type u_3} [has_smul α β] :

a • f is the map of f under a • in locale pointwise.

Equations
@[protected]
def filter.has_vadd_filter {α : Type u_2} {β : Type u_3} [has_vadd α β] :

a +ᵥ f is the map of f under a +ᵥ in locale pointwise.

Equations
@[simp]
theorem filter.map_smul {α : Type u_2} {β : Type u_3} [has_smul α β] {f : filter β} {a : α} :
filter.map (λ (b : β), a b) f = a f
@[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
theorem filter.mem_smul_filter {α : Type u_2} {β : Type u_3} [has_smul α β] {f : filter β} {s : set β} {a : α} :
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} :
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 fa +ᵥ s a +ᵥ f
theorem filter.smul_set_mem_smul_filter {α : Type u_2} {β : Type u_3} [has_smul α β] {f : filter β} {s : set β} {a : α} :
s fa s a f
@[simp]
theorem filter.smul_filter_bot {α : Type u_2} {β : Type u_3} [has_smul α β] {a : α} :
@[simp]
theorem filter.vadd_filter_bot {α : Type u_2} {β : Type u_3} [has_vadd α β] {a : α} :
@[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 =
@[simp]
theorem filter.smul_filter_eq_bot_iff {α : Type u_2} {β : Type u_3} [has_smul α β] {f : filter β} {a : α} :
a f = f =
@[simp]
theorem filter.smul_filter_ne_bot_iff {α : Type u_2} {β : Type u_3} [has_smul α β] {f : filter β} {a : α} :
@[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} :
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
theorem filter.ne_bot.smul_filter {α : Type u_2} {β : Type u_3} [has_smul α β] {f : filter β} {a : α} :
f.ne_bot(a f).ne_bot
theorem filter.ne_bot.of_smul_filter {α : Type u_2} {β : Type u_3} [has_smul α β] {f : filter β} {a : α} :
(a f).ne_bot → f.ne_bot
theorem filter.ne_bot.of_vadd_filter {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
theorem filter.smul_filter_le_smul_filter {α : Type u_2} {β : Type u_3} [has_smul α β] {f₁ f₂ : filter β} {a : α} (hf : f₁ f₂) :
a f₁ a f₂
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
@[protected, instance]
def filter.covariant_smul_filter {α : Type u_2} {β : Type u_3} [has_smul α β] :
@[protected, instance]
def filter.covariant_vadd_filter {α : Type u_2} {β : Type u_3} [has_vadd α β] :
@[protected, instance]
def filter.smul_comm_class_filter {α : Type u_2} {β : Type u_3} {γ : Type u_4} [has_smul α γ] [has_smul β γ] [smul_comm_class α β γ] :
@[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] :
@[protected, instance]
def filter.smul_comm_class_filter' {α : Type u_2} {β : Type u_3} {γ : Type u_4} [has_smul α γ] [has_smul β γ] [smul_comm_class α β γ] :
@[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] :
@[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] :
@[protected, instance]
def filter.smul_comm_class_filter'' {α : Type u_2} {β : Type u_3} {γ : Type u_4} [has_smul α γ] [has_smul β γ] [smul_comm_class α β γ] :
@[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] :
@[protected, instance]
def filter.smul_comm_class {α : Type u_2} {β : Type u_3} {γ : Type u_4} [has_smul α γ] [has_smul β γ] [smul_comm_class α β γ] :
@[protected, instance]
def filter.is_scalar_tower {α : Type u_2} {β : Type u_3} {γ : Type u_4} [has_smul α β] [has_smul α γ] [has_smul β γ] [is_scalar_tower α β γ] :
@[protected, instance]
def filter.is_scalar_tower' {α : Type u_2} {β : Type u_3} {γ : Type u_4} [has_smul α β] [has_smul α γ] [has_smul β γ] [is_scalar_tower α β γ] :
@[protected, instance]
def filter.is_scalar_tower'' {α : Type u_2} {β : Type u_3} {γ : Type u_4} [has_smul α β] [has_smul α γ] [has_smul β γ] [is_scalar_tower α β γ] :
@[protected, instance]
def filter.is_central_scalar {α : Type u_2} {β : Type u_3} [has_smul α β] [has_smul αᵐᵒᵖ β] [is_central_scalar α β] :
@[protected]
def filter.add_action {α : Type u_2} {β : Type u_3} [add_monoid α] [add_action α β] :

An additive action of an additive monoid α on a type β gives an additive action of filter α on filter β

Equations
@[protected]
def filter.mul_action {α : Type u_2} {β : Type u_3} [monoid α] [mul_action α β] :

A multiplicative action of a monoid α on a type β gives a multiplicative action of filter α on filter β.

Equations
@[protected]
def filter.mul_action_filter {α : Type u_2} {β : Type u_3} [monoid α] [mul_action α β] :

A multiplicative action of a monoid on a type β gives a multiplicative action on filter β.

Equations
@[protected]
def filter.add_action_filter {α : Type u_2} {β : Type u_3} [add_monoid α] [add_action α β] :

An additive action of an additive monoid on a type β gives an additive action on filter β.

Equations
@[protected]
def filter.distrib_mul_action_filter {α : Type u_2} {β : Type u_3} [monoid α] [add_monoid β] [distrib_mul_action α β] :

A distributive multiplicative action of a monoid on an additive monoid β gives a distributive multiplicative action on filter β.

Equations
@[protected]
def filter.mul_distrib_mul_action_filter {α : Type u_2} {β : Type u_3} [monoid α] [monoid β] [mul_distrib_mul_action α β] :

A multiplicative action of a monoid on a monoid β gives a multiplicative action on set β.

Equations

Note that we have neither smul_with_zero α (filter β) nor smul_with_zero (filter α) (filter β) because 0 * ⊥ ≠ 0.

theorem filter.ne_bot.smul_zero_nonneg {α : Type u_2} {β : Type u_3} [has_zero α] [has_zero β] [smul_with_zero α β] {f : filter α} (hf : f.ne_bot) :
0 f 0
theorem filter.ne_bot.zero_smul_nonneg {α : Type u_2} {β : Type u_3} [has_zero α] [has_zero β] [smul_with_zero α β] {g : filter β} (hg : g.ne_bot) :
0 0 g
theorem filter.zero_smul_filter_nonpos {α : Type u_2} {β : Type u_3} [has_zero α] [has_zero β] [smul_with_zero α β] {g : filter β} :
0 g 0
theorem filter.zero_smul_filter {α : Type u_2} {β : Type u_3} [has_zero α] [has_zero β] [smul_with_zero α β] {g : filter β} (hg : g.ne_bot) :
0 g = 0