mathlib documentation

analysis.asymptotics

Asymptotics

We introduce these relations:

Here l is any filter on the domain of f and g, which are assumed to be the same. The codomains of f and g do not need to be the same; all that is needed that there is a norm associated with these types, and it is the norm that is compared asymptotically.

The relation is_O_with c is introduced to factor out common algebraic arguments in the proofs of similar properties of is_O and is_o. Usually proofs outside of this file should use is_O instead.

Often the ranges of f and g will be the real numbers, in which case the norm is the absolute value. In general, we have

is_O f g l ↔ is_O (λ x, ∥f x∥) (λ x, ∥g x∥) l,

and similarly for is_o. But our setup allows us to use the notions e.g. with functions to the integers, rationals, complex numbers, or any normed vector space without mentioning the norm explicitly.

If f and g are functions to a normed field like the reals or complex numbers and g is always nonzero, we have

is_o f g l ↔ tendsto (λ x, f x / (g x)) l (𝓝 0).

In fact, the right-to-left direction holds without the hypothesis on g, and in the other direction it suffices to assume that f is zero wherever g is. (This generalization is useful in defining the Fréchet derivative.)

Definitions

def asymptotics.is_O_with {α : Type u_1} {E : Type u_3} {F : Type u_4} [has_norm E] [has_norm F] :
(α → E)(α → F)filter α → Prop

This version of the Landau notation is_O_with C f g l where f and g are two functions on a type α and l is a filter on α, means that eventually for l, ∥f∥ is bounded by C * ∥g∥. In other words, ∥f∥ / ∥g∥ is eventually bounded by C, modulo division by zero issues that are avoided by this definition. Probably you want to use is_O instead of this relation.

Equations
theorem asymptotics.is_O_with_iff {α : Type u_1} {E : Type u_3} {F : Type u_4} [has_norm E] [has_norm F] {c : } {f : α → E} {g : α → F} {l : filter α} :
asymptotics.is_O_with c f g l ∀ᶠ (x : α) in l, f x c * g x

Definition of is_O_with. We record it in a lemma as we will set is_O_with to be irreducible at the end of this file.

theorem asymptotics.is_O_with.of_bound {α : Type u_1} {E : Type u_3} {F : Type u_4} [has_norm E] [has_norm F] {c : } {f : α → E} {g : α → F} {l : filter α} :
(∀ᶠ (x : α) in l, f x c * g x)asymptotics.is_O_with c f g l

def asymptotics.is_O {α : Type u_1} {E : Type u_3} {F : Type u_4} [has_norm E] [has_norm F] :
(α → E)(α → F)filter α → Prop

The Landau notation is_O f g l where f and g are two functions on a type α and l is a filter on α, means that eventually for l, ∥f∥ is bounded by a constant multiple of ∥g∥. In other words, ∥f∥ / ∥g∥ is eventually bounded, modulo division by zero issues that are avoided by this definition.

Equations
theorem asymptotics.is_O_iff_is_O_with {α : Type u_1} {E : Type u_3} {F : Type u_4} [has_norm E] [has_norm F] {f : α → E} {g : α → F} {l : filter α} :

Definition of is_O in terms of is_O_with. We record it in a lemma as we will set is_O to be irreducible at the end of this file.

theorem asymptotics.is_O_iff {α : Type u_1} {E : Type u_3} {F : Type u_4} [has_norm E] [has_norm F] {f : α → E} {g : α → F} {l : filter α} :
asymptotics.is_O f g l ∃ (c : ), ∀ᶠ (x : α) in l, f x c * g x

Definition of is_O in terms of filters. We record it in a lemma as we will set is_O to be irreducible at the end of this file.

theorem asymptotics.is_O.of_bound {α : Type u_1} {E : Type u_3} {F : Type u_4} [has_norm E] [has_norm F] (c : ) {f : α → E} {g : α → F} {l : filter α} :
(∀ᶠ (x : α) in l, f x c * g x)asymptotics.is_O f g l

def asymptotics.is_o {α : Type u_1} {E : Type u_3} {F : Type u_4} [has_norm E] [has_norm F] :
(α → E)(α → F)filter α → Prop

The Landau notation is_o f g l where f and g are two functions on a type α and l is a filter on α, means that eventually for l, ∥f∥ is bounded by an arbitrarily small constant multiple of ∥g∥. In other words, ∥f∥ / ∥g∥ tends to 0 along l, modulo division by zero issues that are avoided by this definition.

Equations
theorem asymptotics.is_o_iff_forall_is_O_with {α : Type u_1} {E : Type u_3} {F : Type u_4} [has_norm E] [has_norm F] {f : α → E} {g : α → F} {l : filter α} :
asymptotics.is_o f g l ∀ ⦃c : ⦄, 0 < casymptotics.is_O_with c f g l

Definition of is_o in terms of is_O_with. We record it in a lemma as we will set is_o to be irreducible at the end of this file.

theorem asymptotics.is_o_iff {α : Type u_1} {E : Type u_3} {F : Type u_4} [has_norm E] [has_norm F] {f : α → E} {g : α → F} {l : filter α} :
asymptotics.is_o f g l ∀ ⦃c : ⦄, 0 < c(∀ᶠ (x : α) in l, f x c * g x)

Definition of is_o in terms of filters. We record it in a lemma as we will set is_o to be irreducible at the end of this file.

theorem asymptotics.is_o.def {α : Type u_1} {E : Type u_3} {F : Type u_4} [has_norm E] [has_norm F] {f : α → E} {g : α → F} {l : filter α} (h : asymptotics.is_o f g l) {c : } :
0 < c(∀ᶠ (x : α) in l, f x c * g x)

theorem asymptotics.is_o.def' {α : Type u_1} {E : Type u_3} {F : Type u_4} [has_norm E] [has_norm F] {f : α → E} {g : α → F} {l : filter α} (h : asymptotics.is_o f g l) {c : } :
0 < casymptotics.is_O_with c f g l

Conversions

theorem asymptotics.is_O_with.is_O {α : Type u_1} {E : Type u_3} {F : Type u_4} [has_norm E] [has_norm F] {c : } {f : α → E} {g : α → F} {l : filter α} :

theorem asymptotics.is_o.is_O_with {α : Type u_1} {E : Type u_3} {F : Type u_4} [has_norm E] [has_norm F] {f : α → E} {g : α → F} {l : filter α} :

theorem asymptotics.is_o.is_O {α : Type u_1} {E : Type u_3} {F : Type u_4} [has_norm E] [has_norm F] {f : α → E} {g : α → F} {l : filter α} :

theorem asymptotics.is_O_with.weaken {α : Type u_1} {E : Type u_3} {F' : Type u_7} [has_norm E] [normed_group F'] {c c' : } {f : α → E} {g' : α → F'} {l : filter α} :
asymptotics.is_O_with c f g' lc c'asymptotics.is_O_with c' f g' l

theorem asymptotics.is_O_with.exists_pos {α : Type u_1} {E : Type u_3} {F' : Type u_7} [has_norm E] [normed_group F'] {c : } {f : α → E} {g' : α → F'} {l : filter α} :
asymptotics.is_O_with c f g' l(∃ (c' : ) (H : 0 < c'), asymptotics.is_O_with c' f g' l)

theorem asymptotics.is_O.exists_pos {α : Type u_1} {E : Type u_3} {F' : Type u_7} [has_norm E] [normed_group F'] {f : α → E} {g' : α → F'} {l : filter α} :
asymptotics.is_O f g' l(∃ (c : ) (H : 0 < c), asymptotics.is_O_with c f g' l)

theorem asymptotics.is_O_with.exists_nonneg {α : Type u_1} {E : Type u_3} {F' : Type u_7} [has_norm E] [normed_group F'] {c : } {f : α → E} {g' : α → F'} {l : filter α} :
asymptotics.is_O_with c f g' l(∃ (c' : ) (H : 0 c'), asymptotics.is_O_with c' f g' l)

theorem asymptotics.is_O.exists_nonneg {α : Type u_1} {E : Type u_3} {F' : Type u_7} [has_norm E] [normed_group F'] {f : α → E} {g' : α → F'} {l : filter α} :
asymptotics.is_O f g' l(∃ (c : ) (H : 0 c), asymptotics.is_O_with c f g' l)

Subsingleton

theorem asymptotics.is_o_of_subsingleton {α : Type u_1} {E' : Type u_6} {F' : Type u_7} [normed_group E'] [normed_group F'] {f' : α → E'} {g' : α → F'} {l : filter α} [subsingleton E'] :

theorem asymptotics.is_O_of_subsingleton {α : Type u_1} {E' : Type u_6} {F' : Type u_7} [normed_group E'] [normed_group F'] {f' : α → E'} {g' : α → F'} {l : filter α} [subsingleton E'] :

Congruence

theorem asymptotics.is_O_with_congr {α : Type u_1} {E : Type u_3} {F : Type u_4} [has_norm E] [has_norm F] {c₁ c₂ : } {f₁ f₂ : α → E} {g₁ g₂ : α → F} {l : filter α} :
c₁ = c₂f₁ =ᶠ[l] f₂g₁ =ᶠ[l] g₂(asymptotics.is_O_with c₁ f₁ g₁ l asymptotics.is_O_with c₂ f₂ g₂ l)

theorem asymptotics.is_O_with.congr' {α : Type u_1} {E : Type u_3} {F : Type u_4} [has_norm E] [has_norm F] {c₁ c₂ : } {f₁ f₂ : α → E} {g₁ g₂ : α → F} {l : filter α} :
c₁ = c₂f₁ =ᶠ[l] f₂g₁ =ᶠ[l] g₂asymptotics.is_O_with c₁ f₁ g₁ lasymptotics.is_O_with c₂ f₂ g₂ l

theorem asymptotics.is_O_with.congr {α : Type u_1} {E : Type u_3} {F : Type u_4} [has_norm E] [has_norm F] {c₁ c₂ : } {f₁ f₂ : α → E} {g₁ g₂ : α → F} {l : filter α} :
c₁ = c₂(∀ (x : α), f₁ x = f₂ x)(∀ (x : α), g₁ x = g₂ x)asymptotics.is_O_with c₁ f₁ g₁ lasymptotics.is_O_with c₂ f₂ g₂ l

theorem asymptotics.is_O_with.congr_left {α : Type u_1} {E : Type u_3} {F : Type u_4} [has_norm E] [has_norm F] {c : } {g : α → F} {f₁ f₂ : α → E} {l : filter α} :
(∀ (x : α), f₁ x = f₂ x)asymptotics.is_O_with c f₁ g lasymptotics.is_O_with c f₂ g l

theorem asymptotics.is_O_with.congr_right {α : Type u_1} {E : Type u_3} {F : Type u_4} [has_norm E] [has_norm F] {c : } {f : α → E} {g₁ g₂ : α → F} {l : filter α} :
(∀ (x : α), g₁ x = g₂ x)asymptotics.is_O_with c f g₁ lasymptotics.is_O_with c f g₂ l

theorem asymptotics.is_O_with.congr_const {α : Type u_1} {E : Type u_3} {F : Type u_4} [has_norm E] [has_norm F] {f : α → E} {g : α → F} {c₁ c₂ : } {l : filter α} :
c₁ = c₂asymptotics.is_O_with c₁ f g lasymptotics.is_O_with c₂ f g l

theorem asymptotics.is_O_congr {α : Type u_1} {E : Type u_3} {F : Type u_4} [has_norm E] [has_norm F] {f₁ f₂ : α → E} {g₁ g₂ : α → F} {l : filter α} :
f₁ =ᶠ[l] f₂g₁ =ᶠ[l] g₂(asymptotics.is_O f₁ g₁ l asymptotics.is_O f₂ g₂ l)

theorem asymptotics.is_O.congr' {α : Type u_1} {E : Type u_3} {F : Type u_4} [has_norm E] [has_norm F] {f₁ f₂ : α → E} {g₁ g₂ : α → F} {l : filter α} :
f₁ =ᶠ[l] f₂g₁ =ᶠ[l] g₂asymptotics.is_O f₁ g₁ lasymptotics.is_O f₂ g₂ l

theorem asymptotics.is_O.congr {α : Type u_1} {E : Type u_3} {F : Type u_4} [has_norm E] [has_norm F] {f₁ f₂ : α → E} {g₁ g₂ : α → F} {l : filter α} :
(∀ (x : α), f₁ x = f₂ x)(∀ (x : α), g₁ x = g₂ x)asymptotics.is_O f₁ g₁ lasymptotics.is_O f₂ g₂ l

theorem asymptotics.is_O.congr_left {α : Type u_1} {E : Type u_3} {F : Type u_4} [has_norm E] [has_norm F] {g : α → F} {f₁ f₂ : α → E} {l : filter α} :
(∀ (x : α), f₁ x = f₂ x)asymptotics.is_O f₁ g lasymptotics.is_O f₂ g l

theorem asymptotics.is_O.congr_right {α : Type u_1} {E : Type u_3} [has_norm E] {f g₁ g₂ : α → E} {l : filter α} :
(∀ (x : α), g₁ x = g₂ x)asymptotics.is_O f g₁ lasymptotics.is_O f g₂ l

theorem asymptotics.is_o_congr {α : Type u_1} {E : Type u_3} {F : Type u_4} [has_norm E] [has_norm F] {f₁ f₂ : α → E} {g₁ g₂ : α → F} {l : filter α} :
f₁ =ᶠ[l] f₂g₁ =ᶠ[l] g₂(asymptotics.is_o f₁ g₁ l asymptotics.is_o f₂ g₂ l)

theorem asymptotics.is_o.congr' {α : Type u_1} {E : Type u_3} {F : Type u_4} [has_norm E] [has_norm F] {f₁ f₂ : α → E} {g₁ g₂ : α → F} {l : filter α} :
f₁ =ᶠ[l] f₂g₁ =ᶠ[l] g₂asymptotics.is_o f₁ g₁ lasymptotics.is_o f₂ g₂ l

theorem asymptotics.is_o.congr {α : Type u_1} {E : Type u_3} {F : Type u_4} [has_norm E] [has_norm F] {f₁ f₂ : α → E} {g₁ g₂ : α → F} {l : filter α} :
(∀ (x : α), f₁ x = f₂ x)(∀ (x : α), g₁ x = g₂ x)asymptotics.is_o f₁ g₁ lasymptotics.is_o f₂ g₂ l

theorem asymptotics.is_o.congr_left {α : Type u_1} {E : Type u_3} {F : Type u_4} [has_norm E] [has_norm F] {g : α → F} {f₁ f₂ : α → E} {l : filter α} :
(∀ (x : α), f₁ x = f₂ x)asymptotics.is_o f₁ g lasymptotics.is_o f₂ g l

theorem asymptotics.is_o.congr_right {α : Type u_1} {E : Type u_3} [has_norm E] {f g₁ g₂ : α → E} {l : filter α} :
(∀ (x : α), g₁ x = g₂ x)asymptotics.is_o f g₁ lasymptotics.is_o f g₂ l

Filter operations and transitivity

theorem asymptotics.is_O_with.comp_tendsto {α : Type u_1} {β : Type u_2} {E : Type u_3} {F : Type u_4} [has_norm E] [has_norm F] {c : } {f : α → E} {g : α → F} {l : filter α} (hcfg : asymptotics.is_O_with c f g l) {k : β → α} {l' : filter β} :
filter.tendsto k l' lasymptotics.is_O_with c (f k) (g k) l'

theorem asymptotics.is_O.comp_tendsto {α : Type u_1} {β : Type u_2} {E : Type u_3} {F : Type u_4} [has_norm E] [has_norm F] {f : α → E} {g : α → F} {l : filter α} (hfg : asymptotics.is_O f g l) {k : β → α} {l' : filter β} :
filter.tendsto k l' lasymptotics.is_O (f k) (g k) l'

theorem asymptotics.is_o.comp_tendsto {α : Type u_1} {β : Type u_2} {E : Type u_3} {F : Type u_4} [has_norm E] [has_norm F] {f : α → E} {g : α → F} {l : filter α} (hfg : asymptotics.is_o f g l) {k : β → α} {l' : filter β} :
filter.tendsto k l' lasymptotics.is_o (f k) (g k) l'

theorem asymptotics.is_O_with.mono {α : Type u_1} {E : Type u_3} {F : Type u_4} [has_norm E] [has_norm F] {c : } {f : α → E} {g : α → F} {l l' : filter α} :
asymptotics.is_O_with c f g l'l l'asymptotics.is_O_with c f g l

theorem asymptotics.is_O.mono {α : Type u_1} {E : Type u_3} {F : Type u_4} [has_norm E] [has_norm F] {f : α → E} {g : α → F} {l l' : filter α} :
asymptotics.is_O f g l'l l'asymptotics.is_O f g l

theorem asymptotics.is_o.mono {α : Type u_1} {E : Type u_3} {F : Type u_4} [has_norm E] [has_norm F] {f : α → E} {g : α → F} {l l' : filter α} :
asymptotics.is_o f g l'l l'asymptotics.is_o f g l

theorem asymptotics.is_O_with.trans {α : Type u_1} {E : Type u_3} {F : Type u_4} {G : Type u_5} [has_norm E] [has_norm F] [has_norm G] {c c' : } {f : α → E} {g : α → F} {k : α → G} {l : filter α} :
asymptotics.is_O_with c f g lasymptotics.is_O_with c' g k l0 casymptotics.is_O_with (c * c') f k l

theorem asymptotics.is_O.trans {α : Type u_1} {E : Type u_3} {G : Type u_5} {F' : Type u_7} [has_norm E] [has_norm G] [normed_group F'] {f : α → E} {k : α → G} {g' : α → F'} {l : filter α} :

theorem asymptotics.is_o.trans_is_O_with {α : Type u_1} {E : Type u_3} {F : Type u_4} {G : Type u_5} [has_norm E] [has_norm F] [has_norm G] {c : } {f : α → E} {g : α → F} {k : α → G} {l : filter α} :
asymptotics.is_o f g lasymptotics.is_O_with c g k l0 < casymptotics.is_o f k l

theorem asymptotics.is_o.trans_is_O {α : Type u_1} {E : Type u_3} {F : Type u_4} {G' : Type u_8} [has_norm E] [has_norm F] [normed_group G'] {f : α → E} {g : α → F} {k' : α → G'} {l : filter α} :

theorem asymptotics.is_O_with.trans_is_o {α : Type u_1} {E : Type u_3} {F : Type u_4} {G : Type u_5} [has_norm E] [has_norm F] [has_norm G] {c : } {f : α → E} {g : α → F} {k : α → G} {l : filter α} :
asymptotics.is_O_with c f g lasymptotics.is_o g k l0 < casymptotics.is_o f k l

theorem asymptotics.is_O.trans_is_o {α : Type u_1} {E : Type u_3} {G : Type u_5} {F' : Type u_7} [has_norm E] [has_norm G] [normed_group F'] {f : α → E} {k : α → G} {g' : α → F'} {l : filter α} :

theorem asymptotics.is_o.trans {α : Type u_1} {E : Type u_3} {F : Type u_4} {G' : Type u_8} [has_norm E] [has_norm F] [normed_group G'] {f : α → E} {g : α → F} {k' : α → G'} {l : filter α} :

theorem asymptotics.is_o.trans' {α : Type u_1} {E : Type u_3} {G : Type u_5} {F' : Type u_7} [has_norm E] [has_norm G] [normed_group F'] {f : α → E} {k : α → G} {g' : α → F'} {l : filter α} :

theorem asymptotics.is_O_with_of_le' {α : Type u_1} {E : Type u_3} {F : Type u_4} [has_norm E] [has_norm F] {c : } {f : α → E} {g : α → F} (l : filter α) :
(∀ (x : α), f x c * g x)asymptotics.is_O_with c f g l

theorem asymptotics.is_O_with_of_le {α : Type u_1} {E : Type u_3} {F : Type u_4} [has_norm E] [has_norm F] {f : α → E} {g : α → F} (l : filter α) :
(∀ (x : α), f x g x)asymptotics.is_O_with 1 f g l

theorem asymptotics.is_O_of_le' {α : Type u_1} {E : Type u_3} {F : Type u_4} [has_norm E] [has_norm F] {c : } {f : α → E} {g : α → F} (l : filter α) :
(∀ (x : α), f x c * g x)asymptotics.is_O f g l

theorem asymptotics.is_O_of_le {α : Type u_1} {E : Type u_3} {F : Type u_4} [has_norm E] [has_norm F] {f : α → E} {g : α → F} (l : filter α) :
(∀ (x : α), f x g x)asymptotics.is_O f g l

theorem asymptotics.is_O_with_refl {α : Type u_1} {E : Type u_3} [has_norm E] (f : α → E) (l : filter α) :

theorem asymptotics.is_O_refl {α : Type u_1} {E : Type u_3} [has_norm E] (f : α → E) (l : filter α) :

theorem asymptotics.is_O_with.trans_le {α : Type u_1} {E : Type u_3} {F : Type u_4} {G : Type u_5} [has_norm E] [has_norm F] [has_norm G] {c : } {f : α → E} {g : α → F} {k : α → G} {l : filter α} :
asymptotics.is_O_with c f g l(∀ (x : α), g x k x)0 casymptotics.is_O_with c f k l

theorem asymptotics.is_O.trans_le {α : Type u_1} {E : Type u_3} {G : Type u_5} {F' : Type u_7} [has_norm E] [has_norm G] [normed_group F'] {f : α → E} {k : α → G} {g' : α → F'} {l : filter α} :
asymptotics.is_O f g' l(∀ (x : α), g' x k x)asymptotics.is_O f k l

theorem asymptotics.is_o.trans_le {α : Type u_1} {E : Type u_3} {F : Type u_4} {G : Type u_5} [has_norm E] [has_norm F] [has_norm G] {f : α → E} {g : α → F} {k : α → G} {l : filter α} :
asymptotics.is_o f g l(∀ (x : α), g x k x)asymptotics.is_o f k l

theorem asymptotics.is_O_with_bot {α : Type u_1} {E : Type u_3} {F : Type u_4} [has_norm E] [has_norm F] (c : ) (f : α → E) (g : α → F) :

theorem asymptotics.is_O_bot {α : Type u_1} {E : Type u_3} {F : Type u_4} [has_norm E] [has_norm F] (c : ) (f : α → E) (g : α → F) :

theorem asymptotics.is_o_bot {α : Type u_1} {E : Type u_3} {F : Type u_4} [has_norm E] [has_norm F] (f : α → E) (g : α → F) :

theorem asymptotics.is_O_with.join {α : Type u_1} {E : Type u_3} {F : Type u_4} [has_norm E] [has_norm F] {c : } {f : α → E} {g : α → F} {l l' : filter α} :

theorem asymptotics.is_O_with.join' {α : Type u_1} {E : Type u_3} {F' : Type u_7} [has_norm E] [normed_group F'] {c c' : } {f : α → E} {g' : α → F'} {l l' : filter α} :
asymptotics.is_O_with c f g' lasymptotics.is_O_with c' f g' l'asymptotics.is_O_with (max c c') f g' (l l')

theorem asymptotics.is_O.join {α : Type u_1} {E : Type u_3} {F' : Type u_7} [has_norm E] [normed_group F'] {f : α → E} {g' : α → F'} {l l' : filter α} :
asymptotics.is_O f g' lasymptotics.is_O f g' l'asymptotics.is_O f g' (l l')

theorem asymptotics.is_o.join {α : Type u_1} {E : Type u_3} {F : Type u_4} [has_norm E] [has_norm F] {f : α → E} {g : α → F} {l l' : filter α} :
asymptotics.is_o f g lasymptotics.is_o f g l'asymptotics.is_o f g (l l')

Simplification : norm

@[simp]
theorem asymptotics.is_O_with_norm_right {α : Type u_1} {E : Type u_3} {F' : Type u_7} [has_norm E] [normed_group F'] {c : } {f : α → E} {g' : α → F'} {l : filter α} :
asymptotics.is_O_with c f (λ (x : α), g' x) l asymptotics.is_O_with c f g' l

theorem asymptotics.is_O_with.norm_right {α : Type u_1} {E : Type u_3} {F' : Type u_7} [has_norm E] [normed_group F'] {c : } {f : α → E} {g' : α → F'} {l : filter α} :
asymptotics.is_O_with c f g' lasymptotics.is_O_with c f (λ (x : α), g' x) l

Alias of is_O_with_norm_right.

theorem asymptotics.is_O_with.of_norm_right {α : Type u_1} {E : Type u_3} {F' : Type u_7} [has_norm E] [normed_group F'] {c : } {f : α → E} {g' : α → F'} {l : filter α} :
asymptotics.is_O_with c f (λ (x : α), g' x) lasymptotics.is_O_with c f g' l

Alias of is_O_with_norm_right.

@[simp]
theorem asymptotics.is_O_norm_right {α : Type u_1} {E : Type u_3} {F' : Type u_7} [has_norm E] [normed_group F'] {f : α → E} {g' : α → F'} {l : filter α} :
asymptotics.is_O f (λ (x : α), g' x) l asymptotics.is_O f g' l

theorem asymptotics.is_O.norm_right {α : Type u_1} {E : Type u_3} {F' : Type u_7} [has_norm E] [normed_group F'] {f : α → E} {g' : α → F'} {l : filter α} :
asymptotics.is_O f g' lasymptotics.is_O f (λ (x : α), g' x) l

Alias of is_O_norm_right.

theorem asymptotics.is_O.of_norm_right {α : Type u_1} {E : Type u_3} {F' : Type u_7} [has_norm E] [normed_group F'] {f : α → E} {g' : α → F'} {l : filter α} :
asymptotics.is_O f (λ (x : α), g' x) lasymptotics.is_O f g' l

Alias of is_O_norm_right.

@[simp]
theorem asymptotics.is_o_norm_right {α : Type u_1} {E : Type u_3} {F' : Type u_7} [has_norm E] [normed_group F'] {f : α → E} {g' : α → F'} {l : filter α} :
asymptotics.is_o f (λ (x : α), g' x) l asymptotics.is_o f g' l

theorem asymptotics.is_o.norm_right {α : Type u_1} {E : Type u_3} {F' : Type u_7} [has_norm E] [normed_group F'] {f : α → E} {g' : α → F'} {l : filter α} :
asymptotics.is_o f g' lasymptotics.is_o f (λ (x : α), g' x) l

Alias of is_o_norm_right.

theorem asymptotics.is_o.of_norm_right {α : Type u_1} {E : Type u_3} {F' : Type u_7} [has_norm E] [normed_group F'] {f : α → E} {g' : α → F'} {l : filter α} :
asymptotics.is_o f (λ (x : α), g' x) lasymptotics.is_o f g' l

Alias of is_o_norm_right.

@[simp]
theorem asymptotics.is_O_with_norm_left {α : Type u_1} {F : Type u_4} {E' : Type u_6} [has_norm F] [normed_group E'] {c : } {g : α → F} {f' : α → E'} {l : filter α} :
asymptotics.is_O_with c (λ (x : α), f' x) g l asymptotics.is_O_with c f' g l

theorem asymptotics.is_O_with.of_norm_left {α : Type u_1} {F : Type u_4} {E' : Type u_6} [has_norm F] [normed_group E'] {c : } {g : α → F} {f' : α → E'} {l : filter α} :
asymptotics.is_O_with c (λ (x : α), f' x) g lasymptotics.is_O_with c f' g l

Alias of is_O_with_norm_left.

theorem asymptotics.is_O_with.norm_left {α : Type u_1} {F : Type u_4} {E' : Type u_6} [has_norm F] [normed_group E'] {c : } {g : α → F} {f' : α → E'} {l : filter α} :
asymptotics.is_O_with c f' g lasymptotics.is_O_with c (λ (x : α), f' x) g l

Alias of is_O_with_norm_left.

@[simp]
theorem asymptotics.is_O_norm_left {α : Type u_1} {F : Type u_4} {E' : Type u_6} [has_norm F] [normed_group E'] {g : α → F} {f' : α → E'} {l : filter α} :
asymptotics.is_O (λ (x : α), f' x) g l asymptotics.is_O f' g l

theorem asymptotics.is_O.of_norm_left {α : Type u_1} {F : Type u_4} {E' : Type u_6} [has_norm F] [normed_group E'] {g : α → F} {f' : α → E'} {l : filter α} :
asymptotics.is_O (λ (x : α), f' x) g lasymptotics.is_O f' g l

Alias of is_O_norm_left.

theorem asymptotics.is_O.norm_left {α : Type u_1} {F : Type u_4} {E' : Type u_6} [has_norm F] [normed_group E'] {g : α → F} {f' : α → E'} {l : filter α} :
asymptotics.is_O f' g lasymptotics.is_O (λ (x : α), f' x) g l

Alias of is_O_norm_left.

@[simp]
theorem asymptotics.is_o_norm_left {α : Type u_1} {F : Type u_4} {E' : Type u_6} [has_norm F] [normed_group E'] {g : α → F} {f' : α → E'} {l : filter α} :
asymptotics.is_o (λ (x : α), f' x) g l asymptotics.is_o f' g l

theorem asymptotics.is_o.norm_left {α : Type u_1} {F : Type u_4} {E' : Type u_6} [has_norm F] [normed_group E'] {g : α → F} {f' : α → E'} {l : filter α} :
asymptotics.is_o f' g lasymptotics.is_o (λ (x : α), f' x) g l

Alias of is_o_norm_left.

theorem asymptotics.is_o.of_norm_left {α : Type u_1} {F : Type u_4} {E' : Type u_6} [has_norm F] [normed_group E'] {g : α → F} {f' : α → E'} {l : filter α} :
asymptotics.is_o (λ (x : α), f' x) g lasymptotics.is_o f' g l

Alias of is_o_norm_left.

theorem asymptotics.is_O_with_norm_norm {α : Type u_1} {E' : Type u_6} {F' : Type u_7} [normed_group E'] [normed_group F'] {c : } {f' : α → E'} {g' : α → F'} {l : filter α} :
asymptotics.is_O_with c (λ (x : α), f' x) (λ (x : α), g' x) l asymptotics.is_O_with c f' g' l

theorem asymptotics.is_O_with.of_norm_norm {α : Type u_1} {E' : Type u_6} {F' : Type u_7} [normed_group E'] [normed_group F'] {c : } {f' : α → E'} {g' : α → F'} {l : filter α} :
asymptotics.is_O_with c (λ (x : α), f' x) (λ (x : α), g' x) lasymptotics.is_O_with c f' g' l

Alias of is_O_with_norm_norm.

theorem asymptotics.is_O_with.norm_norm {α : Type u_1} {E' : Type u_6} {F' : Type u_7} [normed_group E'] [normed_group F'] {c : } {f' : α → E'} {g' : α → F'} {l : filter α} :
asymptotics.is_O_with c f' g' lasymptotics.is_O_with c (λ (x : α), f' x) (λ (x : α), g' x) l

Alias of is_O_with_norm_norm.

theorem asymptotics.is_O_norm_norm {α : Type u_1} {E' : Type u_6} {F' : Type u_7} [normed_group E'] [normed_group F'] {f' : α → E'} {g' : α → F'} {l : filter α} :
asymptotics.is_O (λ (x : α), f' x) (λ (x : α), g' x) l asymptotics.is_O f' g' l

theorem asymptotics.is_O.norm_norm {α : Type u_1} {E' : Type u_6} {F' : Type u_7} [normed_group E'] [normed_group F'] {f' : α → E'} {g' : α → F'} {l : filter α} :
asymptotics.is_O f' g' lasymptotics.is_O (λ (x : α), f' x) (λ (x : α), g' x) l

Alias of is_O_norm_norm.

theorem asymptotics.is_O.of_norm_norm {α : Type u_1} {E' : Type u_6} {F' : Type u_7} [normed_group E'] [normed_group F'] {f' : α → E'} {g' : α → F'} {l : filter α} :
asymptotics.is_O (λ (x : α), f' x) (λ (x : α), g' x) lasymptotics.is_O f' g' l

Alias of is_O_norm_norm.

theorem asymptotics.is_o_norm_norm {α : Type u_1} {E' : Type u_6} {F' : Type u_7} [normed_group E'] [normed_group F'] {f' : α → E'} {g' : α → F'} {l : filter α} :
asymptotics.is_o (λ (x : α), f' x) (λ (x : α), g' x) l asymptotics.is_o f' g' l

theorem asymptotics.is_o.of_norm_norm {α : Type u_1} {E' : Type u_6} {F' : Type u_7} [normed_group E'] [normed_group F'] {f' : α → E'} {g' : α → F'} {l : filter α} :
asymptotics.is_o (λ (x : α), f' x) (λ (x : α), g' x) lasymptotics.is_o f' g' l

Alias of is_o_norm_norm.

theorem asymptotics.is_o.norm_norm {α : Type u_1} {E' : Type u_6} {F' : Type u_7} [normed_group E'] [normed_group F'] {f' : α → E'} {g' : α → F'} {l : filter α} :
asymptotics.is_o f' g' lasymptotics.is_o (λ (x : α), f' x) (λ (x : α), g' x) l

Alias of is_o_norm_norm.

Simplification: negate

@[simp]
theorem asymptotics.is_O_with_neg_right {α : Type u_1} {E : Type u_3} {F' : Type u_7} [has_norm E] [normed_group F'] {c : } {f : α → E} {g' : α → F'} {l : filter α} :
asymptotics.is_O_with c f (λ (x : α), -g' x) l asymptotics.is_O_with c f g' l

theorem asymptotics.is_O_with.neg_right {α : Type u_1} {E : Type u_3} {F' : Type u_7} [has_norm E] [normed_group F'] {c : } {f : α → E} {g' : α → F'} {l : filter α} :
asymptotics.is_O_with c f g' lasymptotics.is_O_with c f (λ (x : α), -g' x) l

Alias of is_O_with_neg_right.

theorem asymptotics.is_O_with.of_neg_right {α : Type u_1} {E : Type u_3} {F' : Type u_7} [has_norm E] [normed_group F'] {c : } {f : α → E} {g' : α → F'} {l : filter α} :
asymptotics.is_O_with c f (λ (x : α), -g' x) lasymptotics.is_O_with c f g' l

Alias of is_O_with_neg_right.

@[simp]
theorem asymptotics.is_O_neg_right {α : Type u_1} {E : Type u_3} {F' : Type u_7} [has_norm E] [normed_group F'] {f : α → E} {g' : α → F'} {l : filter α} :
asymptotics.is_O f (λ (x : α), -g' x) l asymptotics.is_O f g' l

theorem asymptotics.is_O.neg_right {α : Type u_1} {E : Type u_3} {F' : Type u_7} [has_norm E] [normed_group F'] {f : α → E} {g' : α → F'} {l : filter α} :
asymptotics.is_O f g' lasymptotics.is_O f (λ (x : α), -g' x) l

Alias of is_O_neg_right.

theorem asymptotics.is_O.of_neg_right {α : Type u_1} {E : Type u_3} {F' : Type u_7} [has_norm E] [normed_group F'] {f : α → E} {g' : α → F'} {l : filter α} :
asymptotics.is_O f (λ (x : α), -g' x) lasymptotics.is_O f g' l

Alias of is_O_neg_right.

@[simp]
theorem asymptotics.is_o_neg_right {α : Type u_1} {E : Type u_3} {F' : Type u_7} [has_norm E] [normed_group F'] {f : α → E} {g' : α → F'} {l : filter α} :
asymptotics.is_o f (λ (x : α), -g' x) l asymptotics.is_o f g' l

theorem asymptotics.is_o.of_neg_right {α : Type u_1} {E : Type u_3} {F' : Type u_7} [has_norm E] [normed_group F'] {f : α → E} {g' : α → F'} {l : filter α} :
asymptotics.is_o f (λ (x : α), -g' x) lasymptotics.is_o f g' l

Alias of is_o_neg_right.

theorem asymptotics.is_o.neg_right {α : Type u_1} {E : Type u_3} {F' : Type u_7} [has_norm E] [normed_group F'] {f : α → E} {g' : α → F'} {l : filter α} :
asymptotics.is_o f g' lasymptotics.is_o f (λ (x : α), -g' x) l

Alias of is_o_neg_right.

@[simp]
theorem asymptotics.is_O_with_neg_left {α : Type u_1} {F : Type u_4} {E' : Type u_6} [has_norm F] [normed_group E'] {c : } {g : α → F} {f' : α → E'} {l : filter α} :
asymptotics.is_O_with c (λ (x : α), -f' x) g l asymptotics.is_O_with c f' g l

theorem asymptotics.is_O_with.of_neg_left {α : Type u_1} {F : Type u_4} {E' : Type u_6} [has_norm F] [normed_group E'] {c : } {g : α → F} {f' : α → E'} {l : filter α} :
asymptotics.is_O_with c (λ (x : α), -f' x) g lasymptotics.is_O_with c f' g l

Alias of is_O_with_neg_left.

theorem asymptotics.is_O_with.neg_left {α : Type u_1} {F : Type u_4} {E' : Type u_6} [has_norm F] [normed_group E'] {c : } {g : α → F} {f' : α → E'} {l : filter α} :
asymptotics.is_O_with c f' g lasymptotics.is_O_with c (λ (x : α), -f' x) g l

Alias of is_O_with_neg_left.

@[simp]
theorem asymptotics.is_O_neg_left {α : Type u_1} {F : Type u_4} {E' : Type u_6} [has_norm F] [normed_group E'] {g : α → F} {f' : α → E'} {l : filter α} :
asymptotics.is_O (λ (x : α), -f' x) g l asymptotics.is_O f' g l

theorem asymptotics.is_O.neg_left {α : Type u_1} {F : Type u_4} {E' : Type u_6} [has_norm F] [normed_group E'] {g : α → F} {f' : α → E'} {l : filter α} :
asymptotics.is_O f' g lasymptotics.is_O (λ (x : α), -f' x) g l

Alias of is_O_neg_left.

theorem asymptotics.is_O.of_neg_left {α : Type u_1} {F : Type u_4} {E' : Type u_6} [has_norm F] [normed_group E'] {g : α → F} {f' : α → E'} {l : filter α} :
asymptotics.is_O (λ (x : α), -f' x) g lasymptotics.is_O f' g l

Alias of is_O_neg_left.

@[simp]
theorem asymptotics.is_o_neg_left {α : Type u_1} {F : Type u_4} {E' : Type u_6} [has_norm F] [normed_group E'] {g : α → F} {f' : α → E'} {l : filter α} :
asymptotics.is_o (λ (x : α), -f' x) g l asymptotics.is_o f' g l

theorem asymptotics.is_o.neg_left {α : Type u_1} {F : Type u_4} {E' : Type u_6} [has_norm F] [normed_group E'] {g : α → F} {f' : α → E'} {l : filter α} :
asymptotics.is_o f' g lasymptotics.is_o (λ (x : α), -f' x) g l

Alias of is_o_neg_left.

Product of functions (right)

theorem asymptotics.is_O_with_fst_prod {α : Type u_1} {E' : Type u_6} {F' : Type u_7} [normed_group E'] [normed_group F'] {f' : α → E'} {g' : α → F'} {l : filter α} :
asymptotics.is_O_with 1 f' (λ (x : α), (f' x, g' x)) l

theorem asymptotics.is_O_with_snd_prod {α : Type u_1} {E' : Type u_6} {F' : Type u_7} [normed_group E'] [normed_group F'] {f' : α → E'} {g' : α → F'} {l : filter α} :
asymptotics.is_O_with 1 g' (λ (x : α), (f' x, g' x)) l

theorem asymptotics.is_O_fst_prod {α : Type u_1} {E' : Type u_6} {F' : Type u_7} [normed_group E'] [normed_group F'] {f' : α → E'} {g' : α → F'} {l : filter α} :
asymptotics.is_O f' (λ (x : α), (f' x, g' x)) l

theorem asymptotics.is_O_snd_prod {α : Type u_1} {E' : Type u_6} {F' : Type u_7} [normed_group E'] [normed_group F'] {f' : α → E'} {g' : α → F'} {l : filter α} :
asymptotics.is_O g' (λ (x : α), (f' x, g' x)) l

theorem asymptotics.is_O_fst_prod' {α : Type u_1} {E' : Type u_6} {F' : Type u_7} [normed_group E'] [normed_group F'] {l : filter α} {f' : α → E' × F'} :
asymptotics.is_O (λ (x : α), (f' x).fst) f' l

theorem asymptotics.is_O_snd_prod' {α : Type u_1} {E' : Type u_6} {F' : Type u_7} [normed_group E'] [normed_group F'] {l : filter α} {f' : α → E' × F'} :
asymptotics.is_O (λ (x : α), (f' x).snd) f' l

theorem asymptotics.is_O_with.prod_rightl {α : Type u_1} {E : Type u_3} {F' : Type u_7} {G' : Type u_8} [has_norm E] [normed_group F'] [normed_group G'] {c : } {f : α → E} {g' : α → F'} (k' : α → G') {l : filter α} :
asymptotics.is_O_with c f g' l0 casymptotics.is_O_with c f (λ (x : α), (g' x, k' x)) l

theorem asymptotics.is_O.prod_rightl {α : Type u_1} {E : Type u_3} {F' : Type u_7} {G' : Type u_8} [has_norm E] [normed_group F'] [normed_group G'] {f : α → E} {g' : α → F'} (k' : α → G') {l : filter α} :
asymptotics.is_O f g' lasymptotics.is_O f (λ (x : α), (g' x, k' x)) l

theorem asymptotics.is_o.prod_rightl {α : Type u_1} {E : Type u_3} {F' : Type u_7} {G' : Type u_8} [has_norm E] [normed_group F'] [normed_group G'] {f : α → E} {g' : α → F'} (k' : α → G') {l : filter α} :
asymptotics.is_o f g' lasymptotics.is_o f (λ (x : α), (g' x, k' x)) l

theorem asymptotics.is_O_with.prod_rightr {α : Type u_1} {E : Type u_3} {E' : Type u_6} {F' : Type u_7} [has_norm E] [normed_group E'] [normed_group F'] {c : } {f : α → E} (f' : α → E') {g' : α → F'} {l : filter α} :
asymptotics.is_O_with c f g' l0 casymptotics.is_O_with c f (λ (x : α), (f' x, g' x)) l

theorem asymptotics.is_O.prod_rightr {α : Type u_1} {E : Type u_3} {E' : Type u_6} {F' : Type u_7} [has_norm E] [normed_group E'] [normed_group F'] {f : α → E} (f' : α → E') {g' : α → F'} {l : filter α} :
asymptotics.is_O f g' lasymptotics.is_O f (λ (x : α), (f' x, g' x)) l

theorem asymptotics.is_o.prod_rightr {α : Type u_1} {E : Type u_3} {E' : Type u_6} {F' : Type u_7} [has_norm E] [normed_group E'] [normed_group F'] {f : α → E} (f' : α → E') {g' : α → F'} {l : filter α} :
asymptotics.is_o f g' lasymptotics.is_o f (λ (x : α), (f' x, g' x)) l

theorem asymptotics.is_O_with.prod_left_same {α : Type u_1} {E' : Type u_6} {F' : Type u_7} {G' : Type u_8} [normed_group E'] [normed_group F'] [normed_group G'] {c : } {f' : α → E'} {g' : α → F'} {k' : α → G'} {l : filter α} :
asymptotics.is_O_with c f' k' lasymptotics.is_O_with c g' k' lasymptotics.is_O_with c (λ (x : α), (f' x, g' x)) k' l

theorem asymptotics.is_O_with.prod_left {α : Type u_1} {E' : Type u_6} {F' : Type u_7} {G' : Type u_8} [normed_group E'] [normed_group F'] [normed_group G'] {c c' : } {f' : α → E'} {g' : α → F'} {k' : α → G'} {l : filter α} :
asymptotics.is_O_with c f' k' lasymptotics.is_O_with c' g' k' lasymptotics.is_O_with (max c c') (λ (x : α), (f' x, g' x)) k' l

theorem asymptotics.is_O_with.prod_left_fst {α : Type u_1} {E' : Type u_6} {F' : Type u_7} {G' : Type u_8} [normed_group E'] [normed_group F'] [normed_group G'] {c : } {f' : α → E'} {g' : α → F'} {k' : α → G'} {l : filter α} :
asymptotics.is_O_with c (λ (x : α), (f' x, g' x)) k' lasymptotics.is_O_with c f' k' l

theorem asymptotics.is_O_with.prod_left_snd {α : Type u_1} {E' : Type u_6} {F' : Type u_7} {G' : Type u_8} [normed_group E'] [normed_group F'] [normed_group G'] {c : } {f' : α → E'} {g' : α → F'} {k' : α → G'} {l : filter α} :
asymptotics.is_O_with c (λ (x : α), (f' x, g' x)) k' lasymptotics.is_O_with c g' k' l

theorem asymptotics.is_O_with_prod_left {α : Type u_1} {E' : Type u_6} {F' : Type u_7} {G' : Type u_8} [normed_group E'] [normed_group F'] [normed_group G'] {c : } {f' : α → E'} {g' : α → F'} {k' : α → G'} {l : filter α} :
asymptotics.is_O_with c (λ (x : α), (f' x, g' x)) k' l asymptotics.is_O_with c f' k' l asymptotics.is_O_with c g' k' l

theorem asymptotics.is_O.prod_left {α : Type u_1} {E' : Type u_6} {F' : Type u_7} {G' : Type u_8} [normed_group E'] [normed_group F'] [normed_group G'] {f' : α → E'} {g' : α → F'} {k' : α → G'} {l : filter α} :
asymptotics.is_O f' k' lasymptotics.is_O g' k' lasymptotics.is_O (λ (x : α), (f' x, g' x)) k' l

theorem asymptotics.is_O.prod_left_fst {α : Type u_1} {E' : Type u_6} {F' : Type u_7} {G' : Type u_8} [normed_group E'] [normed_group F'] [normed_group G'] {f' : α → E'} {g' : α → F'} {k' : α → G'} {l : filter α} :
asymptotics.is_O (λ (x : α), (f' x, g' x)) k' lasymptotics.is_O f' k' l

theorem asymptotics.is_O.prod_left_snd {α : Type u_1} {E' : Type u_6} {F' : Type u_7} {G' : Type u_8} [normed_group E'] [normed_group F'] [normed_group G'] {f' : α → E'} {g' : α → F'} {k' : α → G'} {l : filter α} :
asymptotics.is_O (λ (x : α), (f' x, g' x)) k' lasymptotics.is_O g' k' l

@[simp]
theorem asymptotics.is_O_prod_left {α : Type u_1} {E' : Type u_6} {F' : Type u_7} {G' : Type u_8} [normed_group E'] [normed_group F'] [normed_group G'] {f' : α → E'} {g' : α → F'} {k' : α → G'} {l : filter α} :
asymptotics.is_O (λ (x : α), (f' x, g' x)) k' l asymptotics.is_O f' k' l asymptotics.is_O g' k' l

theorem asymptotics.is_o.prod_left {α : Type u_1} {E' : Type u_6} {F' : Type u_7} {G' : Type u_8} [normed_group E'] [normed_group F'] [normed_group G'] {f' : α → E'} {g' : α → F'} {k' : α → G'} {l : filter α} :
asymptotics.is_o f' k' lasymptotics.is_o g' k' lasymptotics.is_o (λ (x : α), (f' x, g' x)) k' l

theorem asymptotics.is_o.prod_left_fst {α : Type u_1} {E' : Type u_6} {F' : Type u_7} {G' : Type u_8} [normed_group E'] [normed_group F'] [normed_group G'] {f' : α → E'} {g' : α → F'} {k' : α → G'} {l : filter α} :
asymptotics.is_o (λ (x : α), (f' x, g' x)) k' lasymptotics.is_o f' k' l

theorem asymptotics.is_o.prod_left_snd {α : Type u_1} {E' : Type u_6} {F' : Type u_7} {G' : Type u_8} [normed_group E'] [normed_group F'] [normed_group G'] {f' : α → E'} {g' : α → F'} {k' : α → G'} {l : filter α} :
asymptotics.is_o (λ (x : α), (f' x, g' x)) k' lasymptotics.is_o g' k' l

@[simp]
theorem asymptotics.is_o_prod_left {α : Type u_1} {E' : Type u_6} {F' : Type u_7} {G' : Type u_8} [normed_group E'] [normed_group F'] [normed_group G'] {f' : α → E'} {g' : α → F'} {k' : α → G'} {l : filter α} :
asymptotics.is_o (λ (x : α), (f' x, g' x)) k' l asymptotics.is_o f' k' l asymptotics.is_o g' k' l

Addition and subtraction

theorem asymptotics.is_O_with.add {α : Type u_1} {F : Type u_4} {E' : Type u_6} [has_norm F] [normed_group E'] {g : α → F} {l : filter α} {c₁ c₂ : } {f₁ f₂ : α → E'} :
asymptotics.is_O_with c₁ f₁ g lasymptotics.is_O_with c₂ f₂ g lasymptotics.is_O_with (c₁ + c₂) (λ (x : α), f₁ x + f₂ x) g l

theorem asymptotics.is_O.add {α : Type u_1} {F : Type u_4} {E' : Type u_6} [has_norm F] [normed_group E'] {g : α → F} {l : filter α} {f₁ f₂ : α → E'} :
asymptotics.is_O f₁ g lasymptotics.is_O f₂ g lasymptotics.is_O (λ (x : α), f₁ x + f₂ x) g l

theorem asymptotics.is_o.add {α : Type u_1} {F : Type u_4} {E' : Type u_6} [has_norm F] [normed_group E'] {g : α → F} {l : filter α} {f₁ f₂ : α → E'} :
asymptotics.is_o f₁ g lasymptotics.is_o f₂ g lasymptotics.is_o (λ (x : α), f₁ x + f₂ x) g l

theorem asymptotics.is_o.add_add {α : Type u_1} {E' : Type u_6} {F' : Type u_7} [normed_group E'] [normed_group F'] {l : filter α} {f₁ f₂ : α → E'} {g₁ g₂ : α → F'} :
asymptotics.is_o f₁ g₁ lasymptotics.is_o f₂ g₂ lasymptotics.is_o (λ (x : α), f₁ x + f₂ x) (λ (x : α), g₁ x + g₂ x) l

theorem asymptotics.is_O.add_is_o {α : Type u_1} {F : Type u_4} {E' : Type u_6} [has_norm F] [normed_group E'] {g : α → F} {l : filter α} {f₁ f₂ : α → E'} :
asymptotics.is_O f₁ g lasymptotics.is_o f₂ g lasymptotics.is_O (λ (x : α), f₁ x + f₂ x) g l

theorem asymptotics.is_o.add_is_O {α : Type u_1} {F : Type u_4} {E' : Type u_6} [has_norm F] [normed_group E'] {g : α → F} {l : filter α} {f₁ f₂ : α → E'} :
asymptotics.is_o f₁ g lasymptotics.is_O f₂ g lasymptotics.is_O (λ (x : α), f₁ x + f₂ x) g l

theorem asymptotics.is_O_with.add_is_o {α : Type u_1} {F : Type u_4} {E' : Type u_6} [has_norm F] [normed_group E'] {g : α → F} {l : filter α} {c₁ c₂ : } {f₁ f₂ : α → E'} :
asymptotics.is_O_with c₁ f₁ g lasymptotics.is_o f₂ g lc₁ < c₂asymptotics.is_O_with c₂ (λ (x : α), f₁ x + f₂ x) g l

theorem asymptotics.is_o.add_is_O_with {α : Type u_1} {F : Type u_4} {E' : Type u_6} [has_norm F] [normed_group E'] {g : α → F} {l : filter α} {c₁ c₂ : } {f₁ f₂ : α → E'} :
asymptotics.is_o f₁ g lasymptotics.is_O_with c₁ f₂ g lc₁ < c₂asymptotics.is_O_with c₂ (λ (x : α), f₁ x + f₂ x) g l

theorem asymptotics.is_O_with.sub {α : Type u_1} {F : Type u_4} {E' : Type u_6} [has_norm F] [normed_group E'] {g : α → F} {l : filter α} {c₁ c₂ : } {f₁ f₂ : α → E'} :
asymptotics.is_O_with c₁ f₁ g lasymptotics.is_O_with c₂ f₂ g lasymptotics.is_O_with (c₁ + c₂) (λ (x : α), f₁ x - f₂ x) g l

theorem asymptotics.is_O_with.sub_is_o {α : Type u_1} {F : Type u_4} {E' : Type u_6} [has_norm F] [normed_group E'] {g : α → F} {l : filter α} {c₁ c₂ : } {f₁ f₂ : α → E'} :
asymptotics.is_O_with c₁ f₁ g lasymptotics.is_o f₂ g lc₁ < c₂asymptotics.is_O_with c₂ (λ (x : α), f₁ x - f₂ x) g l

theorem asymptotics.is_O.sub {α : Type u_1} {F : Type u_4} {E' : Type u_6} [has_norm F] [normed_group E'] {g : α → F} {l : filter α} {f₁ f₂ : α → E'} :
asymptotics.is_O f₁ g lasymptotics.is_O f₂ g lasymptotics.is_O (λ (x : α), f₁ x - f₂ x) g l

theorem asymptotics.is_o.sub {α : Type u_1} {F : Type u_4} {E' : Type u_6} [has_norm F] [normed_group E'] {g : α → F} {l : filter α} {f₁ f₂ : α → E'} :
asymptotics.is_o f₁ g lasymptotics.is_o f₂ g lasymptotics.is_o (λ (x : α), f₁ x - f₂ x) g l

Lemmas about is_O (f₁ - f₂) g l / is_o (f₁ - f₂) g l treated as a binary relation

theorem asymptotics.is_O_with.symm {α : Type u_1} {F : Type u_4} {E' : Type u_6} [has_norm F] [normed_group E'] {c : } {g : α → F} {l : filter α} {f₁ f₂ : α → E'} :
asymptotics.is_O_with c (λ (x : α), f₁ x - f₂ x) g lasymptotics.is_O_with c (λ (x : α), f₂ x - f₁ x) g l

theorem asymptotics.is_O_with_comm {α : Type u_1} {F : Type u_4} {E' : Type u_6} [has_norm F] [normed_group E'] {c : } {g : α → F} {l : filter α} {f₁ f₂ : α → E'} :
asymptotics.is_O_with c (λ (x : α), f₁ x - f₂ x) g l asymptotics.is_O_with c (λ (x : α), f₂ x - f₁ x) g l

theorem asymptotics.is_O.symm {α : Type u_1} {F : Type u_4} {E' : Type u_6} [has_norm F] [normed_group E'] {g : α → F} {l : filter α} {f₁ f₂ : α → E'} :
asymptotics.is_O (λ (x : α), f₁ x - f₂ x) g lasymptotics.is_O (λ (x : α), f₂ x - f₁ x) g l

theorem asymptotics.is_O_comm {α : Type u_1} {F : Type u_4} {E' : Type u_6} [has_norm F] [normed_group E'] {g : α → F} {l : filter α} {f₁ f₂ : α → E'} :
asymptotics.is_O (λ (x : α), f₁ x - f₂ x) g l asymptotics.is_O (λ (x : α), f₂ x - f₁ x) g l

theorem asymptotics.is_o.symm {α : Type u_1} {F : Type u_4} {E' : Type u_6} [has_norm F] [normed_group E'] {g : α → F} {l : filter α} {f₁ f₂ : α → E'} :
asymptotics.is_o (λ (x : α), f₁ x - f₂ x) g lasymptotics.is_o (λ (x : α), f₂ x - f₁ x) g l

theorem asymptotics.is_o_comm {α : Type u_1} {F : Type u_4} {E' : Type u_6} [has_norm F] [normed_group E'] {g : α → F} {l : filter α} {f₁ f₂ : α → E'} :
asymptotics.is_o (λ (x : α), f₁ x - f₂ x) g l asymptotics.is_o (λ (x : α), f₂ x - f₁ x) g l

theorem asymptotics.is_O_with.triangle {α : Type u_1} {F : Type u_4} {E' : Type u_6} [has_norm F] [normed_group E'] {c c' : } {g : α → F} {l : filter α} {f₁ f₂ f₃ : α → E'} :
asymptotics.is_O_with c (λ (x : α), f₁ x - f₂ x) g lasymptotics.is_O_with c' (λ (x : α), f₂ x - f₃ x) g lasymptotics.is_O_with (c + c') (λ (x : α), f₁ x - f₃ x) g l

theorem asymptotics.is_O.triangle {α : Type u_1} {F : Type u_4} {E' : Type u_6} [has_norm F] [normed_group E'] {g : α → F} {l : filter α} {f₁ f₂ f₃ : α → E'} :
asymptotics.is_O (λ (x : α), f₁ x - f₂ x) g lasymptotics.is_O (λ (x : α), f₂ x - f₃ x) g lasymptotics.is_O (λ (x : α), f₁ x - f₃ x) g l

theorem asymptotics.is_o.triangle {α : Type u_1} {F : Type u_4} {E' : Type u_6} [has_norm F] [normed_group E'] {g : α → F} {l : filter α} {f₁ f₂ f₃ : α → E'} :
asymptotics.is_o (λ (x : α), f₁ x - f₂ x) g lasymptotics.is_o (λ (x : α), f₂ x - f₃ x) g lasymptotics.is_o (λ (x : α), f₁ x - f₃ x) g l

theorem asymptotics.is_O.congr_of_sub {α : Type u_1} {F : Type u_4} {E' : Type u_6} [has_norm F] [normed_group E'] {g : α → F} {l : filter α} {f₁ f₂ : α → E'} :
asymptotics.is_O (λ (x : α), f₁ x - f₂ x) g l(asymptotics.is_O f₁ g l asymptotics.is_O f₂ g l)

theorem asymptotics.is_o.congr_of_sub {α : Type u_1} {F : Type u_4} {E' : Type u_6} [has_norm F] [normed_group E'] {g : α → F} {l : filter α} {f₁ f₂ : α → E'} :
asymptotics.is_o (λ (x : α), f₁ x - f₂ x) g l(asymptotics.is_o f₁ g l asymptotics.is_o f₂ g l)

Zero, one, and other constants

theorem asymptotics.is_o_zero {α : Type u_1} {E' : Type u_6} {F' : Type u_7} [normed_group E'] [normed_group F'] (g' : α → F') (l : filter α) :
asymptotics.is_o (λ (x : α), 0) g' l

theorem asymptotics.is_O_with_zero {α : Type u_1} {E' : Type u_6} {F' : Type u_7} [normed_group E'] [normed_group F'] {c : } (g' : α → F') (l : filter α) :
0 casymptotics.is_O_with c (λ (x : α), 0) g' l

theorem asymptotics.is_O_with_zero' {α : Type u_1} {F : Type u_4} {E' : Type u_6} [has_norm F] [normed_group E'] (g : α → F) (l : filter α) :
asymptotics.is_O_with 0 (λ (x : α), 0) g l

theorem asymptotics.is_O_zero {α : Type u_1} {F : Type u_4} {E' : Type u_6} [has_norm F] [normed_group E'] (g : α → F) (l : filter α) :
asymptotics.is_O (λ (x : α), 0) g l

theorem asymptotics.is_O_refl_left {α : Type u_1} {E' : Type u_6} {F' : Type u_7} [normed_group E'] [normed_group F'] {f' : α → E'} (g' : α → F') (l : filter α) :
asymptotics.is_O (λ (x : α), f' x - f' x) g' l

theorem asymptotics.is_o_refl_left {α : Type u_1} {E' : Type u_6} {F' : Type u_7} [normed_group E'] [normed_group F'] {f' : α → E'} (g' : α → F') (l : filter α) :
asymptotics.is_o (λ (x : α), f' x - f' x) g' l

theorem asymptotics.is_O_with_zero_right_iff {α : Type u_1} {E' : Type u_6} {F' : Type u_7} [normed_group E'] [normed_group F'] {c : } {f' : α → E'} {l : filter α} :
asymptotics.is_O_with c f' (λ (x : α), 0) l ∀ᶠ (x : α) in l, f' x = 0

theorem asymptotics.is_O_zero_right_iff {α : Type u_1} {E' : Type u_6} {F' : Type u_7} [normed_group E'] [normed_group F'] {f' : α → E'} {l : filter α} :
asymptotics.is_O f' (λ (x : α), 0) l ∀ᶠ (x : α) in l, f' x = 0

theorem asymptotics.is_o_zero_right_iff {α : Type u_1} {E' : Type u_6} {F' : Type u_7} [normed_group E'] [normed_group F'] {f' : α → E'} {l : filter α} :
asymptotics.is_o f' (λ (x : α), 0) l ∀ᶠ (x : α) in l, f' x = 0

theorem asymptotics.is_O_with_const_const {α : Type u_1} {E : Type u_3} {F' : Type u_7} [has_norm E] [normed_group F'] (c : E) {c' : F'} (hc' : c' 0) (l : filter α) :
asymptotics.is_O_with (c / c') (λ (x : α), c) (λ (x : α), c') l

theorem asymptotics.is_O_const_const {α : Type u_1} {E : Type u_3} {F' : Type u_7} [has_norm E] [normed_group F'] (c : E) {c' : F'} (hc' : c' 0) (l : filter α) :
asymptotics.is_O (λ (x : α), c) (λ (x : α), c') l

theorem asymptotics.is_O_with_const_one {α : Type u_1} {E : Type u_3} {𝕜 : Type u_11} [has_norm E] [normed_field 𝕜] (c : E) (l : filter α) :
asymptotics.is_O_with c (λ (x : α), c) (λ (x : α), 1) l

theorem asymptotics.is_O_const_one {α : Type u_1} {E : Type u_3} {𝕜 : Type u_11} [has_norm E] [normed_field 𝕜] (c : E) (l : filter α) :
asymptotics.is_O (λ (x : α), c) (λ (x : α), 1) l

theorem asymptotics.is_o_const_iff_is_o_one {α : Type u_1} {E : Type u_3} {F' : Type u_7} (𝕜 : Type u_11) [has_norm E] [normed_group F'] [normed_field 𝕜] {f : α → E} {l : filter α} {c : F'} :
c 0(asymptotics.is_o f (λ (x : α), c) l asymptotics.is_o f (λ (x : α), 1) l)

theorem asymptotics.is_o_const_iff {α : Type u_1} {E' : Type u_6} {F' : Type u_7} [normed_group E'] [normed_group F'] {f' : α → E'} {l : filter α} {c : F'} :
c 0(asymptotics.is_o f' (λ (x : α), c) l filter.tendsto f' l (𝓝 0))

theorem asymptotics.is_o_id_const {E' : Type u_6} {F' : Type u_7} [normed_group E'] [normed_group F'] {c : F'} :
c 0asymptotics.is_o (λ (x : E'), x) (λ (x : E'), c) (𝓝 0)

theorem asymptotics.is_O_const_of_tendsto {α : Type u_1} {E' : Type u_6} {F' : Type u_7} [normed_group E'] [normed_group F'] {f' : α → E'} {l : filter α} {y : E'} (h : filter.tendsto f' l (𝓝 y)) {c : F'} :
c 0asymptotics.is_O f' (λ (x : α), c) l

theorem asymptotics.is_o_one_iff {α : Type u_1} {E' : Type u_6} (𝕜 : Type u_11) [normed_group E'] [normed_field 𝕜] {f' : α → E'} {l : filter α} :
asymptotics.is_o f' (λ (x : α), 1) l filter.tendsto f' l (𝓝 0)

theorem asymptotics.is_O_one_of_tendsto {α : Type u_1} {E' : Type u_6} (𝕜 : Type u_11) [normed_group E'] [normed_field 𝕜] {f' : α → E'} {l : filter α} {y : E'} :
filter.tendsto f' l (𝓝 y)asymptotics.is_O f' (λ (x : α), 1) l

theorem asymptotics.is_O.trans_tendsto_nhds {α : Type u_1} {E : Type u_3} {F' : Type u_7} (𝕜 : Type u_11) [has_norm E] [normed_group F'] [normed_field 𝕜] {f : α → E} {g' : α → F'} {l : filter α} (hfg : asymptotics.is_O f g' l) {y : F'} :
filter.tendsto g' l (𝓝 y)asymptotics.is_O f (λ (x : α), 1) l

theorem asymptotics.is_O.trans_tendsto {α : Type u_1} {E' : Type u_6} {F' : Type u_7} [normed_group E'] [normed_group F'] {f' : α → E'} {g' : α → F'} {l : filter α} :
asymptotics.is_O f' g' lfilter.tendsto g' l (𝓝 0)filter.tendsto f' l (𝓝 0)

theorem asymptotics.is_o.trans_tendsto {α : Type u_1} {E' : Type u_6} {F' : Type u_7} [normed_group E'] [normed_group F'] {f' : α → E'} {g' : α → F'} {l : filter α} :
asymptotics.is_o f' g' lfilter.tendsto g' l (𝓝 0)filter.tendsto f' l (𝓝 0)

Multiplication by a constant

theorem asymptotics.is_O_with_const_mul_self {α : Type u_1} {R : Type u_9} [normed_ring R] (c : R) (f : α → R) (l : filter α) :
asymptotics.is_O_with c (λ (x : α), c * f x) f l

theorem asymptotics.is_O_const_mul_self {α : Type u_1} {R : Type u_9} [normed_ring R] (c : R) (f : α → R) (l : filter α) :
asymptotics.is_O (λ (x : α), c * f x) f l

theorem asymptotics.is_O_with.const_mul_left {α : Type u_1} {F : Type u_4} {R : Type u_9} [has_norm F] [normed_ring R] {c : } {g : α → F} {l : filter α} {f : α → R} (h : asymptotics.is_O_with c f g l) (c' : R) :
asymptotics.is_O_with (c' * c) (λ (x : α), c' * f x) g l

theorem asymptotics.is_O.const_mul_left {α : Type u_1} {F : Type u_4} {R : Type u_9} [has_norm F] [normed_ring R] {g : α → F} {l : filter α} {f : α → R} (h : asymptotics.is_O f g l) (c' : R) :
asymptotics.is_O (λ (x : α), c' * f x) g l

theorem asymptotics.is_O_with_self_const_mul' {α : Type u_1} {R : Type u_9} [normed_ring R] (u : units R) (f : α → R) (l : filter α) :
asymptotics.is_O_with u⁻¹ f (λ (x : α), (u) * f x) l

theorem asymptotics.is_O_with_self_const_mul {α : Type u_1} {𝕜 : Type u_11} [normed_field 𝕜] (c : 𝕜) (hc : c 0) (f : α → 𝕜) (l : filter α) :
asymptotics.is_O_with c⁻¹ f (λ (x : α), c * f x) l

theorem asymptotics.is_O_self_const_mul' {α : Type u_1} {R : Type u_9} [normed_ring R] {c : R} (hc : is_unit c) (f : α → R) (l : filter α) :
asymptotics.is_O f (λ (x : α), c * f x) l

theorem asymptotics.is_O_self_const_mul {α : Type u_1} {𝕜 : Type u_11} [normed_field 𝕜] (c : 𝕜) (hc : c 0) (f : α → 𝕜) (l : filter α) :
asymptotics.is_O f (λ (x : α), c * f x) l

theorem asymptotics.is_O_const_mul_left_iff' {α : Type u_1} {F : Type u_4} {R : Type u_9} [has_norm F] [normed_ring R] {g : α → F} {l : filter α} {f : α → R} {c : R} :
is_unit c(asymptotics.is_O (λ (x : α), c * f x) g l asymptotics.is_O f g l)

theorem asymptotics.is_O_const_mul_left_iff {α : Type u_1} {F : Type u_4} {𝕜 : Type u_11} [has_norm F] [normed_field 𝕜] {g : α → F} {l : filter α} {f : α → 𝕜} {c : 𝕜} :
c 0(asymptotics.is_O (λ (x : α), c * f x) g l asymptotics.is_O f g l)

theorem asymptotics.is_o.const_mul_left {α : Type u_1} {F : Type u_4} {R : Type u_9} [has_norm F] [normed_ring R] {g : α → F} {l : filter α} {f : α → R} (h : asymptotics.is_o f g l) (c : R) :
asymptotics.is_o (λ (x : α), c * f x) g l

theorem asymptotics.is_o_const_mul_left_iff' {α : Type u_1} {F : Type u_4} {R : Type u_9} [has_norm F] [normed_ring R] {g : α → F} {l : filter α} {f : α → R} {c : R} :
is_unit c(asymptotics.is_o (λ (x : α), c * f x) g l asymptotics.is_o f g l)

theorem asymptotics.is_o_const_mul_left_iff {α : Type u_1} {F : Type u_4} {𝕜 : Type u_11} [has_norm F] [normed_field 𝕜] {g : α → F} {l : filter α} {f : α → 𝕜} {c : 𝕜} :
c 0(asymptotics.is_o (λ (x : α), c * f x) g l asymptotics.is_o f g l)

theorem asymptotics.is_O_with.of_const_mul_right {α : Type u_1} {E : Type u_3} {R : Type u_9} [has_norm E] [normed_ring R] {c' : } {f : α → E} {l : filter α} {g : α → R} {c : R} :
0 c'asymptotics.is_O_with c' f (λ (x : α), c * g x) lasymptotics.is_O_with (c' * c) f g l

theorem asymptotics.is_O.of_const_mul_right {α : Type u_1} {E : Type u_3} {R : Type u_9} [has_norm E] [normed_ring R] {f : α → E} {l : filter α} {g : α → R} {c : R} :
asymptotics.is_O f (λ (x : α), c * g x) lasymptotics.is_O f g l

theorem asymptotics.is_O_with.const_mul_right' {α : Type u_1} {E : Type u_3} {R : Type u_9} [has_norm E] [normed_ring R] {f : α → E} {l : filter α} {g : α → R} {u : units R} {c' : } :
0 c'asymptotics.is_O_with c' f g lasymptotics.is_O_with (c' * u⁻¹) f (λ (x : α), (u) * g x) l

theorem asymptotics.is_O_with.const_mul_right {α : Type u_1} {E : Type u_3} {𝕜 : Type u_11} [has_norm E] [normed_field 𝕜] {f : α → E} {l : filter α} {g : α → 𝕜} {c : 𝕜} (hc : c 0) {c' : } :
0 c'asymptotics.is_O_with c' f g lasymptotics.is_O_with (c' * c⁻¹) f (λ (x : α), c * g x) l

theorem asymptotics.is_O.const_mul_right' {α : Type u_1} {E : Type u_3} {R : Type u_9} [has_norm E] [normed_ring R] {f : α → E} {l : filter α} {g : α → R} {c : R} :
is_unit casymptotics.is_O f g lasymptotics.is_O f (λ (x : α), c * g x) l

theorem asymptotics.is_O.const_mul_right {α : Type u_1} {E : Type u_3} {𝕜 : Type u_11} [has_norm E] [normed_field 𝕜] {f : α → E} {l : filter α} {g : α → 𝕜} {c : 𝕜} :
c 0asymptotics.is_O f g lasymptotics.is_O f (λ (x : α), c * g x) l

theorem asymptotics.is_O_const_mul_right_iff' {α : Type u_1} {E : Type u_3} {R : Type u_9} [has_norm E] [normed_ring R] {f : α → E} {l : filter α} {g : α → R} {c : R} :
is_unit c(asymptotics.is_O f (λ (x : α), c * g x) l asymptotics.is_O f g l)

theorem asymptotics.is_O_const_mul_right_iff {α : Type u_1} {E : Type u_3} {𝕜 : Type u_11} [has_norm E] [normed_field 𝕜] {f : α → E} {l : filter α} {g : α → 𝕜} {c : 𝕜} :
c 0(asymptotics.is_O f (λ (x : α), c * g x) l asymptotics.is_O f g l)

theorem asymptotics.is_o.of_const_mul_right {α : Type u_1} {E : Type u_3} {R : Type u_9} [has_norm E] [normed_ring R] {f : α → E} {l : filter α} {g : α → R} {c : R} :
asymptotics.is_o f (λ (x : α), c * g x) lasymptotics.is_o f g l

theorem asymptotics.is_o.const_mul_right' {α : Type u_1} {E : Type u_3} {R : Type u_9} [has_norm E] [normed_ring R] {f : α → E} {l : filter α} {g : α → R} {c : R} :
is_unit casymptotics.is_o f g lasymptotics.is_o f (λ (x : α), c * g x) l

theorem asymptotics.is_o.const_mul_right {α : Type u_1} {E : Type u_3} {𝕜 : Type u_11} [has_norm E] [normed_field 𝕜] {f : α → E} {l : filter α} {g : α → 𝕜} {c : 𝕜} :
c 0asymptotics.is_o f g lasymptotics.is_o f (λ (x : α), c * g x) l

theorem asymptotics.is_o_const_mul_right_iff' {α : Type u_1} {E : Type u_3} {R : Type u_9} [has_norm E] [normed_ring R] {f : α → E} {l : filter α} {g : α → R} {c : R} :
is_unit c(asymptotics.is_o f (λ (x : α), c * g x) l asymptotics.is_o f g l)

theorem asymptotics.is_o_const_mul_right_iff {α : Type u_1} {E : Type u_3} {𝕜 : Type u_11} [has_norm E] [normed_field 𝕜] {f : α → E} {l : filter α} {g : α → 𝕜} {c : 𝕜} :
c 0(asymptotics.is_o f (λ (x : α), c * g x) l asymptotics.is_o f g l)

Multiplication

theorem asymptotics.is_O_with.mul {α : Type u_1} {R : Type u_9} {𝕜 : Type u_11} [normed_ring R] [normed_field 𝕜] {l : filter α} {f₁ f₂ : α → R} {g₁ g₂ : α → 𝕜} {c₁ c₂ : } :
asymptotics.is_O_with c₁ f₁ g₁ lasymptotics.is_O_with c₂ f₂ g₂ lasymptotics.is_O_with (c₁ * c₂) (λ (x : α), (f₁ x) * f₂ x) (λ (x : α), (g₁ x) * g₂ x) l

theorem asymptotics.is_O.mul {α : Type u_1} {R : Type u_9} {𝕜 : Type u_11} [normed_ring R] [normed_field 𝕜] {l : filter α} {f₁ f₂ : α → R} {g₁ g₂ : α → 𝕜} :
asymptotics.is_O f₁ g₁ lasymptotics.is_O f₂ g₂ lasymptotics.is_O (λ (x : α), (f₁ x) * f₂ x) (λ (x : α), (g₁ x) * g₂ x) l

theorem asymptotics.is_O.mul_is_o {α : Type u_1} {R : Type u_9} {𝕜 : Type u_11} [normed_ring R] [normed_field 𝕜] {l : filter α} {f₁ f₂ : α → R} {g₁ g₂ : α → 𝕜} :
asymptotics.is_O f₁ g₁ lasymptotics.is_o f₂ g₂ lasymptotics.is_o (λ (x : α), (f₁ x) * f₂ x) (λ (x : α), (g₁ x) * g₂ x) l

theorem asymptotics.is_o.mul_is_O {α : Type u_1} {R : Type u_9} {𝕜 : Type u_11} [normed_ring R] [normed_field 𝕜] {l : filter α} {f₁ f₂ : α → R} {g₁ g₂ : α → 𝕜} :
asymptotics.is_o f₁ g₁ lasymptotics.is_O f₂ g₂ lasymptotics.is_o (λ (x : α), (f₁ x) * f₂ x) (λ (x : α), (g₁ x) * g₂ x) l

theorem asymptotics.is_o.mul {α : Type u_1} {R : Type u_9} {𝕜 : Type u_11} [normed_ring R] [normed_field 𝕜] {l : filter α} {f₁ f₂ : α → R} {g₁ g₂ : α → 𝕜} :
asymptotics.is_o f₁ g₁ lasymptotics.is_o f₂ g₂ lasymptotics.is_o (λ (x : α), (f₁ x) * f₂ x) (λ (x : α), (g₁ x) * g₂ x) l

Scalar multiplication

theorem asymptotics.is_O_with.const_smul_left {α : Type u_1} {F : Type u_4} {E' : Type u_6} {𝕜 : Type u_11} [has_norm F] [normed_group E'] [normed_field 𝕜] {c : } {g : α → F} {f' : α → E'} {l : filter α} [normed_space 𝕜 E'] (h : asymptotics.is_O_with c f' g l) (c' : 𝕜) :
asymptotics.is_O_with (c' * c) (λ (x : α), c' f' x) g l

theorem asymptotics.is_O_const_smul_left_iff {α : Type u_1} {F : Type u_4} {E' : Type u_6} {𝕜 : Type u_11} [has_norm F] [normed_group E'] [normed_field 𝕜] {g : α → F} {f' : α → E'} {l : filter α} [normed_space 𝕜 E'] {c : 𝕜} :
c 0(asymptotics.is_O (λ (x : α), c f' x) g l asymptotics.is_O f' g l)

theorem asymptotics.is_o_const_smul_left {α : Type u_1} {F : Type u_4} {E' : Type u_6} {𝕜 : Type u_11} [has_norm F] [normed_group E'] [normed_field 𝕜] {g : α → F} {f' : α → E'} {l : filter α} [normed_space 𝕜 E'] (h : asymptotics.is_o f' g l) (c : 𝕜) :
asymptotics.is_o (λ (x : α), c f' x) g l

theorem asymptotics.is_o_const_smul_left_iff {α : Type u_1} {F : Type u_4} {E' : Type u_6} {𝕜 : Type u_11} [has_norm F] [normed_group E'] [normed_field 𝕜] {g : α → F} {f' : α → E'} {l : filter α} [normed_space 𝕜 E'] {c : 𝕜} :
c 0(asymptotics.is_o (λ (x : α), c f' x) g l asymptotics.is_o f' g l)

theorem asymptotics.is_O_const_smul_right {α : Type u_1} {E : Type u_3} {E' : Type u_6} {𝕜 : Type u_11} [has_norm E] [normed_group E'] [normed_field 𝕜] {f : α → E} {f' : α → E'} {l : filter α} [normed_space 𝕜 E'] {c : 𝕜} :
c 0(asymptotics.is_O f (λ (x : α), c f' x) l asymptotics.is_O f f' l)

theorem asymptotics.is_o_const_smul_right {α : Type u_1} {E : Type u_3} {E' : Type u_6} {𝕜 : Type u_11} [has_norm E] [normed_group E'] [normed_field 𝕜] {f : α → E} {f' : α → E'} {l : filter α} [normed_space 𝕜 E'] {c : 𝕜} :
c 0(asymptotics.is_o f (λ (x : α), c f' x) l asymptotics.is_o f f' l)

theorem asymptotics.is_O_with.smul {α : Type u_1} {E' : Type u_6} {F' : Type u_7} {𝕜 : Type u_11} [normed_group E'] [normed_group F'] [normed_field 𝕜] {c c' : } {f' : α → E'} {g' : α → F'} {l : filter α} [normed_space 𝕜 E'] [normed_space 𝕜 F'] {k₁ k₂ : α → 𝕜} :
asymptotics.is_O_with c k₁ k₂ lasymptotics.is_O_with c' f' g' lasymptotics.is_O_with (c * c') (λ (x : α), k₁ x f' x) (λ (x : α), k₂ x g' x) l

theorem asymptotics.is_O.smul {α : Type u_1} {E' : Type u_6} {F' : Type u_7} {𝕜 : Type u_11} [normed_group E'] [normed_group F'] [normed_field 𝕜] {f' : α → E'} {g' : α → F'} {l : filter α} [normed_space 𝕜 E'] [normed_space 𝕜 F'] {k₁ k₂ : α → 𝕜} :
asymptotics.is_O k₁ k₂ lasymptotics.is_O f' g' lasymptotics.is_O (λ (x : α), k₁ x f' x) (λ (x : α), k₂ x g' x) l

theorem asymptotics.is_O.smul_is_o {α : Type u_1} {E' : Type u_6} {F' : Type u_7} {𝕜 : Type u_11} [normed_group E'] [normed_group F'] [normed_field 𝕜] {f' : α → E'} {g' : α → F'} {l : filter α} [normed_space 𝕜 E'] [normed_space 𝕜 F'] {k₁ k₂ : α → 𝕜} :
asymptotics.is_O k₁ k₂ lasymptotics.is_o f' g' lasymptotics.is_o (λ (x : α), k₁ x f' x) (λ (x : α), k₂ x g' x) l

theorem asymptotics.is_o.smul_is_O {α : Type u_1} {E' : Type u_6} {F' : Type u_7} {𝕜 : Type u_11} [normed_group E'] [normed_group F'] [normed_field 𝕜] {f' : α → E'} {g' : α → F'} {l : filter α} [normed_space 𝕜 E'] [normed_space 𝕜 F'] {k₁ k₂ : α → 𝕜} :
asymptotics.is_o k₁ k₂ lasymptotics.is_O f' g' lasymptotics.is_o (λ (x : α), k₁ x f' x) (λ (x : α), k₂ x g' x) l

theorem asymptotics.is_o.smul {α : Type u_1} {E' : Type u_6} {F' : Type u_7} {𝕜 : Type u_11} [normed_group E'] [normed_group F'] [normed_field 𝕜] {f' : α → E'} {g' : α → F'} {l : filter α} [normed_space 𝕜 E'] [normed_space 𝕜 F'] {k₁ k₂ : α → 𝕜} :
asymptotics.is_o k₁ k₂ lasymptotics.is_o f' g' lasymptotics.is_o (λ (x : α), k₁ x f' x) (λ (x : α), k₂ x g' x) l

Sum

theorem asymptotics.is_O_with.sum {α : Type u_1} {F : Type u_4} {E' : Type u_6} [has_norm F] [normed_group E'] {g : α → F} {l : filter α} {ι : Type u_13} {A : ι → α → E'} {C : ι → } {s : finset ι} :
(∀ (i : ι), i sasymptotics.is_O_with (C i) (A i) g l)asymptotics.is_O_with (∑ (i : ι) in s, C i) (λ (x : α), ∑ (i : ι) in s, A i x) g l

theorem asymptotics.is_O.sum {α : Type u_1} {F : Type u_4} {E' : Type u_6} [has_norm F] [normed_group E'] {g : α → F} {l : filter α} {ι : Type u_13} {A : ι → α → E'} {s : finset ι} :
(∀ (i : ι), i sasymptotics.is_O (A i) g l)asymptotics.is_O (λ (x : α), ∑ (i : ι) in s, A i x) g l

theorem asymptotics.is_o.sum {α : Type u_1} {E' : Type u_6} {F' : Type u_7} [normed_group E'] [normed_group F'] {g' : α → F'} {l : filter α} {ι : Type u_13} {A : ι → α → E'} {s : finset ι} :
(∀ (i : ι), i sasymptotics.is_o (A i) g' l)asymptotics.is_o (λ (x : α), ∑ (i : ι) in s, A i x) g' l

Relation between f = o(g) and f / g → 0

theorem asymptotics.is_o.tendsto_0 {α : Type u_1} {𝕜 : Type u_11} [normed_field 𝕜] {f g : α → 𝕜} {l : filter α} :
asymptotics.is_o f g lfilter.tendsto (λ (x : α), f x / g x) l (𝓝 0)

theorem asymptotics.is_o_iff_tendsto {α : Type u_1} {𝕜 : Type u_11} [normed_field 𝕜] {f g : α → 𝕜} {l : filter α} :
(∀ (x : α), g x = 0f x = 0)(asymptotics.is_o f g l filter.tendsto (λ (x : α), f x / g x) l (𝓝 0))

Eventually (u / v) * v = u

If u and v are linked by an is_O_with relation, then we eventually have (u / v) * v = u, even if v vanishes.

theorem asymptotics.is_O_with.eventually_mul_div_cancel {α : Type u_1} {𝕜 : Type u_11} [normed_field 𝕜] {c : } {l : filter α} {u v : α → 𝕜} :
asymptotics.is_O_with c u v l(u / v) * v =ᶠ[l] u

theorem asymptotics.is_O.eventually_mul_div_cancel {α : Type u_1} {𝕜 : Type u_11} [normed_field 𝕜] {l : filter α} {u v : α → 𝕜} :
asymptotics.is_O u v l(u / v) * v =ᶠ[l] u

If u = O(v) along l, then (u / v) * v = u eventually at l.

theorem asymptotics.is_o.eventually_mul_div_cancel {α : Type u_1} {𝕜 : Type u_11} [normed_field 𝕜] {l : filter α} {u v : α → 𝕜} :
asymptotics.is_o u v l(u / v) * v =ᶠ[l] u

If u = o(v) along l, then (u / v) * v = u eventually at l.

Equivalent definitions of the form ∃ φ, u =ᶠ[l] φ * v in a normed_field.

theorem asymptotics.is_O_with_of_eq_mul {α : Type u_1} {𝕜 : Type u_11} [normed_field 𝕜] {c : } {l : filter α} {u v : α → 𝕜} (φ : α → 𝕜) :
(∀ᶠ (x : α) in l, φ x c)u =ᶠ[l] φ * vasymptotics.is_O_with c u v l

If ∥φ∥ is eventually bounded by c, and u =ᶠ[l] φ * v, then we have is_O_with c u v l. This does not require any assumptions on c, which is why we keep this version along with is_O_with_iff_exists_eq_mul.

theorem asymptotics.is_O_with_iff_exists_eq_mul {α : Type u_1} {𝕜 : Type u_11} [normed_field 𝕜] {c : } {l : filter α} {u v : α → 𝕜} :
0 c(asymptotics.is_O_with c u v l ∃ (φ : α → 𝕜) (hφ : ∀ᶠ (x : α) in l, φ x c), u =ᶠ[l] φ * v)

theorem asymptotics.is_O_with.exists_eq_mul {α : Type u_1} {𝕜 : Type u_11} [normed_field 𝕜] {c : } {l : filter α} {u v : α → 𝕜} :
asymptotics.is_O_with c u v l0 c(∃ (φ : α → 𝕜) (hφ : ∀ᶠ (x : α) in l, φ x c), u =ᶠ[l] φ * v)

theorem asymptotics.is_O_iff_exists_eq_mul {α : Type u_1} {𝕜 : Type u_11} [normed_field 𝕜] {l : filter α} {u v : α → 𝕜} :
asymptotics.is_O u v l ∃ (φ : α → 𝕜) (hφ : filter.is_bounded_under has_le.le l (norm φ)), u =ᶠ[l] φ * v

theorem asymptotics.is_O.exists_eq_mul {α : Type u_1} {𝕜 : Type u_11} [normed_field 𝕜] {l : filter α} {u v : α → 𝕜} :
asymptotics.is_O u v l(∃ (φ : α → 𝕜) (hφ : filter.is_bounded_under has_le.le l (norm φ)), u =ᶠ[l] φ * v)

Alias of is_O_iff_exists_eq_mul.

theorem asymptotics.is_o_iff_exists_eq_mul {α : Type u_1} {𝕜 : Type u_11} [normed_field 𝕜] {l : filter α} {u v : α → 𝕜} :
asymptotics.is_o u v l ∃ (φ : α → 𝕜) (hφ : filter.tendsto φ l (𝓝 0)), u =ᶠ[l] φ * v

theorem asymptotics.is_o.exists_eq_mul {α : Type u_1} {𝕜 : Type u_11} [normed_field 𝕜] {l : filter α} {u v : α → 𝕜} :
asymptotics.is_o u v l(∃ (φ : α → 𝕜) (hφ : filter.tendsto φ l (𝓝 0)), u =ᶠ[l] φ * v)

Alias of is_o_iff_exists_eq_mul.

Miscellanous lemmas

theorem asymptotics.is_o_pow_pow {𝕜 : Type u_11} [normed_field 𝕜] {m n : } :
m < nasymptotics.is_o (λ (x : 𝕜), x ^ n) (λ (x : 𝕜), x ^ m) (𝓝 0)

theorem asymptotics.is_o_pow_id {𝕜 : Type u_11} [normed_field 𝕜] {n : } :
1 < nasymptotics.is_o (λ (x : 𝕜), x ^ n) (λ (x : 𝕜), x) (𝓝 0)

theorem asymptotics.is_O_with.right_le_sub_of_lt_1 {α : Type u_1} {E' : Type u_6} [normed_group E'] {c : } {l : filter α} {f₁ f₂ : α → E'} :
asymptotics.is_O_with c f₁ f₂ lc < 1asymptotics.is_O_with (1 / (1 - c)) f₂ (λ (x : α), f₂ x - f₁ x) l

theorem asymptotics.is_O_with.right_le_add_of_lt_1 {α : Type u_1} {E' : Type u_6} [normed_group E'] {c : } {l : filter α} {f₁ f₂ : α → E'} :
asymptotics.is_O_with c f₁ f₂ lc < 1asymptotics.is_O_with (1 / (1 - c)) f₂ (λ (x : α), f₁ x + f₂ x) l

theorem asymptotics.is_o.right_is_O_sub {α : Type u_1} {E' : Type u_6} [normed_group E'] {l : filter α} {f₁ f₂ : α → E'} :
asymptotics.is_o f₁ f₂ lasymptotics.is_O f₂ (λ (x : α), f₂ x - f₁ x) l

theorem asymptotics.is_o.right_is_O_add {α : Type u_1} {E' : Type u_6} [normed_group E'] {l : filter α} {f₁ f₂ : α → E'} :
asymptotics.is_o f₁ f₂ lasymptotics.is_O f₂ (λ (x : α), f₁ x + f₂ x) l

theorem local_homeomorph.is_O_with_congr {α : Type u_1} {β : Type u_2} [topological_space α] [topological_space β] {E : Type u_3} [has_norm E] {F : Type u_4} [has_norm F] (e : local_homeomorph α β) {b : β} (hb : b e.to_local_equiv.target) {f : β → E} {g : β → F} {C : } :

Transfer is_O_with over a local_homeomorph.

theorem local_homeomorph.is_O_congr {α : Type u_1} {β : Type u_2} [topological_space α] [topological_space β] {E : Type u_3} [has_norm E] {F : Type u_4} [has_norm F] (e : local_homeomorph α β) {b : β} (hb : b e.to_local_equiv.target) {f : β → E} {g : β → F} :

Transfer is_O over a local_homeomorph.

theorem local_homeomorph.is_o_congr {α : Type u_1} {β : Type u_2} [topological_space α] [topological_space β] {E : Type u_3} [has_norm E] {F : Type u_4} [has_norm F] (e : local_homeomorph α β) {b : β} (hb : b e.to_local_equiv.target) {f : β → E} {g : β → F} :

Transfer is_o over a local_homeomorph.

theorem homeomorph.is_O_with_congr {α : Type u_1} {β : Type u_2} [topological_space α] [topological_space β] {E : Type u_3} [has_norm E] {F : Type u_4} [has_norm F] (e : α ≃ₜ β) {b : β} {f : β → E} {g : β → F} {C : } :

Transfer is_O_with over a homeomorph.

theorem homeomorph.is_O_congr {α : Type u_1} {β : Type u_2} [topological_space α] [topological_space β] {E : Type u_3} [has_norm E] {F : Type u_4} [has_norm F] (e : α ≃ₜ β) {b : β} {f : β → E} {g : β → F} :

Transfer is_O over a homeomorph.

theorem homeomorph.is_o_congr {α : Type u_1} {β : Type u_2} [topological_space α] [topological_space β] {E : Type u_3} [has_norm E] {F : Type u_4} [has_norm F] (e : α ≃ₜ β) {b : β} {f : β → E} {g : β → F} :

Transfer is_o over a homeomorph.