ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Asymptotics.IsLittleO.congr'
theorem IsLittleO.congr' (h : f₁ =o[l] g₁) (hf : f₁ =ᶠ[l] f₂) (hg : g₁ =ᶠ[l] g₂) : f₂ =o[l] g₂ 
Arguments: {α : Type u_1} {E : Type u_3} {F : Type u_4} [Norm E] [Norm F] {l : Filter α} {f₁ f₂ : α → E} {g₁ g₂ : α → F} (h : f₁ =o[l] g₁) (hf : f₁ =ᶠ[l] f₂) (hg : g₁ =ᶠ[l] g₂)
Goal: f₂ =o[l] g₂
-- Asymptotics.IsLittleO.congr
theorem IsLittleO.congr (h : f₁ =o[l] g₁) (hf : ∀ x, f₁ x = f₂ x) (hg : ∀ x, g₁ x = g₂ x) :
    f₂ =o[l] g₂ 
Arguments: {α : Type u_1} {E : Type u_3} {F : Type u_4} [Norm E] [Norm F] {l : Filter α} {f₁ f₂ : α → E} {g₁ g₂ : α → F} (h : f₁ =o[l] g₁) (hf : ∀ (x : α), f₁ x = f₂ x) (hg : ∀ (x : α), g₁ x = g₂ x)
Goal: f₂ =o[l] g₂
-- Asymptotics.IsLittleO.congr_left
theorem IsLittleO.congr_left (h : f₁ =o[l] g) (hf : ∀ x, f₁ x = f₂ x) : f₂ =o[l] g 
Arguments: {α : Type u_1} {E : Type u_3} {F : Type u_4} [Norm E] [Norm F] {g : α → F} {l : Filter α} {f₁ f₂ : α → E} (h : f₁ =o[l] g) (hf : ∀ (x : α), f₁ x = f₂ x)
Goal: f₂ =o[l] g
-- Asymptotics.IsLittleO.congr_right
theorem IsLittleO.congr_right (h : f =o[l] g₁) (hg : ∀ x, g₁ x = g₂ x) : f =o[l] g₂ 
Arguments: {α : Type u_1} {E : Type u_3} {F : Type u_4} [Norm E] [Norm F] {f : α → E} {l : Filter α} {g₁ g₂ : α → F} (h : f =o[l] g₁) (hg : ∀ (x : α), g₁ x = g₂ x)
Goal: f =o[l] g₂
-- Filter.EventuallyEq.trans_isBigO
@[trans]
theorem _root_.Filter.EventuallyEq.trans_isBigO {f₁ f₂ : α → E} {g : α → F} (hf : f₁ =ᶠ[l] f₂)
    (h : f₂ =O[l] g) : f₁ =O[l] g 
Arguments: {α : Type u_1} {E : Type u_3} {F : Type u_4} [Norm E] [Norm F] {l : Filter α} {f₁ f₂ : α → E} {g : α → F} (hf : f₁ =ᶠ[l] f₂) (h : f₂ =O[l] g)
Goal: f₁ =O[l] g
-- Filter.EventuallyEq.trans_isLittleO
@[trans]
theorem _root_.Filter.EventuallyEq.trans_isLittleO {f₁ f₂ : α → E} {g : α → F} (hf : f₁ =ᶠ[l] f₂)
    (h : f₂ =o[l] g) : f₁ =o[l] g 
Arguments: {α : Type u_1} {E : Type u_3} {F : Type u_4} [Norm E] [Norm F] {l : Filter α} {f₁ f₂ : α → E} {g : α → F} (hf : f₁ =ᶠ[l] f₂) (h : f₂ =o[l] g)
Goal: f₁ =o[l] g
-- Asymptotics.IsBigO.trans_eventuallyEq
@[trans]
theorem IsBigO.trans_eventuallyEq {f : α → E} {g₁ g₂ : α → F} (h : f =O[l] g₁) (hg : g₁ =ᶠ[l] g₂) :
    f =O[l] g₂ 
Arguments: {α : Type u_1} {E : Type u_3} {F : Type u_4} [Norm E] [Norm F] {l : Filter α} {f : α → E} {g₁ g₂ : α → F} (h : f =O[l] g₁) (hg : g₁ =ᶠ[l] g₂)
Goal: f =O[l] g₂
-- Asymptotics.IsLittleO.trans_eventuallyEq
@[trans]
theorem IsLittleO.trans_eventuallyEq {f : α → E} {g₁ g₂ : α → F} (h : f =o[l] g₁)
    (hg : g₁ =ᶠ[l] g₂) : f =o[l] g₂ 
Arguments: {α : Type u_1} {E : Type u_3} {F : Type u_4} [Norm E] [Norm F] {l : Filter α} {f : α → E} {g₁ g₂ : α → F} (h : f =o[l] g₁) (hg : g₁ =ᶠ[l] g₂)
Goal: f =o[l] g₂
-- Asymptotics.IsBigOWith.comp_tendsto
theorem IsBigOWith.comp_tendsto (hcfg : IsBigOWith c l f g) {k : β → α} {l' : Filter β}
    (hk : Tendsto k l' l) : IsBigOWith c l' (f ∘ k) (g ∘ k) 
Arguments: {α : Type u_1} {β : Type u_2} {E : Type u_3} {F : Type u_4} [Norm E] [Norm F] {c : ℝ} {f : α → E} {g : α → F} {l : Filter α} (hcfg : Asymptotics.IsBigOWith c l f g) {k : β → α} {l' : Filter β} (hk : Filter.Tendsto k l' l)
Goal: Asymptotics.IsBigOWith c l' (f ∘ k) (g ∘ k)
-- Asymptotics.IsBigO.comp_tendsto
theorem IsBigO.comp_tendsto (hfg : f =O[l] g) {k : β → α} {l' : Filter β} (hk : Tendsto k l' l) :
    (f ∘ k) =O[l'] (g ∘ k) 
Arguments: {α : Type u_1} {β : Type u_2} {E : Type u_3} {F : Type u_4} [Norm E] [Norm F] {f : α → E} {g : α → F} {l : Filter α} (hfg : f =O[l] g) {k : β → α} {l' : Filter β} (hk : Filter.Tendsto k l' l)
Goal: (f ∘ k) =O[l'] (g ∘ k)
-- Asymptotics.IsLittleO.comp_tendsto
theorem IsLittleO.comp_tendsto (hfg : f =o[l] g) {k : β → α} {l' : Filter β} (hk : Tendsto k l' l) :
    (f ∘ k) =o[l'] (g ∘ k) 
Arguments: {α : Type u_1} {β : Type u_2} {E : Type u_3} {F : Type u_4} [Norm E] [Norm F] {f : α → E} {g : α → F} {l : Filter α} (hfg : f =o[l] g) {k : β → α} {l' : Filter β} (hk : Filter.Tendsto k l' l)
Goal: (f ∘ k) =o[l'] (g ∘ k)
-- Asymptotics.isBigOWith_map
@[simp]
theorem isBigOWith_map {k : β → α} {l : Filter β} :
    IsBigOWith c (map k l) f g ↔ IsBigOWith c l (f ∘ k) (g ∘ k) 
Arguments: {α : Type u_1} {β : Type u_2} {E : Type u_3} {F : Type u_4} [Norm E] [Norm F] {c : ℝ} {f : α → E} {g : α → F} {k : β → α} {l : Filter β}
Goal: Asymptotics.IsBigOWith c (Filter.map k l) f g ↔ Asymptotics.IsBigOWith c l (f ∘ k) (g ∘ k)
-- Asymptotics.isBigO_map
@[simp]
theorem isBigO_map {k : β → α} {l : Filter β} : f =O[map k l] g ↔ (f ∘ k) =O[l] (g ∘ k) 
Arguments: {α : Type u_1} {β : Type u_2} {E : Type u_3} {F : Type u_4} [Norm E] [Norm F] {f : α → E} {g : α → F} {k : β → α} {l : Filter β}
Goal: f =O[Filter.map k l] g ↔ (f ∘ k) =O[l] (g ∘ k)
-- Asymptotics.isLittleO_map
@[simp]
theorem isLittleO_map {k : β → α} {l : Filter β} : f =o[map k l] g ↔ (f ∘ k) =o[l] (g ∘ k) 
Arguments: {α : Type u_1} {β : Type u_2} {E : Type u_3} {F : Type u_4} [Norm E] [Norm F] {f : α → E} {g : α → F} {k : β → α} {l : Filter β}
Goal: f =o[Filter.map k l] g ↔ (f ∘ k) =o[l] (g ∘ k)
-- Asymptotics.IsBigOWith.mono
theorem IsBigOWith.mono (h : IsBigOWith c l' f g) (hl : l ≤ l') : IsBigOWith c l f g 
Arguments: {α : Type u_1} {E : Type u_3} {F : Type u_4} [Norm E] [Norm F] {c : ℝ} {f : α → E} {g : α → F} {l l' : Filter α} (h : Asymptotics.IsBigOWith c l' f g) (hl : l ≤ l')
Goal: Asymptotics.IsBigOWith c l f g
-- Asymptotics.IsBigO.mono
theorem IsBigO.mono (h : f =O[l'] g) (hl : l ≤ l') : f =O[l] g 
Arguments: {α : Type u_1} {E : Type u_3} {F : Type u_4} [Norm E] [Norm F] {f : α → E} {g : α → F} {l l' : Filter α} (h : f =O[l'] g) (hl : l ≤ l')
Goal: f =O[l] g
-- Asymptotics.IsLittleO.mono
theorem IsLittleO.mono (h : f =o[l'] g) (hl : l ≤ l') : f =o[l] g 
Arguments: {α : Type u_1} {E : Type u_3} {F : Type u_4} [Norm E] [Norm F] {f : α → E} {g : α → F} {l l' : Filter α} (h : f =o[l'] g) (hl : l ≤ l')
Goal: f =o[l] g
-- Asymptotics.IsBigOWith.trans
theorem IsBigOWith.trans (hfg : IsBigOWith c l f g) (hgk : IsBigOWith c' l g k) (hc : 0 ≤ c) :
    IsBigOWith (c * c') l f k 
Arguments: {α : Type u_1} {E : Type u_3} {F : Type u_4} {G : Type u_5} [Norm E] [Norm F] [Norm G] {c c' : ℝ} {f : α → E} {g : α → F} {k : α → G} {l : Filter α} (hfg : Asymptotics.IsBigOWith c l f g) (hgk : Asymptotics.IsBigOWith c' l g k) (hc : 0 ≤ c)
Goal: Asymptotics.IsBigOWith (c * c') l f k
-- Asymptotics.IsBigO.trans
@[trans]
theorem IsBigO.trans {f : α → E} {g : α → F'} {k : α → G} (hfg : f =O[l] g) (hgk : g =O[l] k) :
    f =O[l] k 
Arguments: {α : Type u_1} {E : Type u_3} {G : Type u_5} {F' : Type u_7} [Norm E] [Norm G] [SeminormedAddCommGroup F'] {l : Filter α} {f : α → E} {g : α → F'} {k : α → G} (hfg : f =O[l] g) (hgk : g =O[l] k)
Goal: f =O[l] k
-- Asymptotics.IsLittleO.trans_isBigOWith
theorem IsLittleO.trans_isBigOWith (hfg : f =o[l] g) (hgk : IsBigOWith c l g k) (hc : 0 < c) :
    f =o[l] k 
Arguments: {α : Type u_1} {E : Type u_3} {F : Type u_4} {G : Type u_5} [Norm E] [Norm F] [Norm G] {c : ℝ} {f : α → E} {g : α → F} {k : α → G} {l : Filter α} (hfg : f =o[l] g) (hgk : Asymptotics.IsBigOWith c l g k) (hc : 0 < c)
Goal: f =o[l] k
-- Asymptotics.IsLittleO.trans_isBigO
@[trans]
theorem IsLittleO.trans_isBigO {f : α → E} {g : α → F} {k : α → G'} (hfg : f =o[l] g)
    (hgk : g =O[l] k) : f =o[l] k 
Arguments: {α : Type u_1} {E : Type u_3} {F : Type u_4} {G' : Type u_8} [Norm E] [Norm F] [SeminormedAddCommGroup G'] {l : Filter α} {f : α → E} {g : α → F} {k : α → G'} (hfg : f =o[l] g) (hgk : g =O[l] k)
Goal: f =o[l] k
-- Asymptotics.IsBigOWith.trans_isLittleO
theorem IsBigOWith.trans_isLittleO (hfg : IsBigOWith c l f g) (hgk : g =o[l] k) (hc : 0 < c) :
    f =o[l] k 
Arguments: {α : Type u_1} {E : Type u_3} {F : Type u_4} {G : Type u_5} [Norm E] [Norm F] [Norm G] {c : ℝ} {f : α → E} {g : α → F} {k : α → G} {l : Filter α} (hfg : Asymptotics.IsBigOWith c l f g) (hgk : g =o[l] k) (hc : 0 < c)
Goal: f =o[l] k
-- Asymptotics.IsBigO.trans_isLittleO
@[trans]
theorem IsBigO.trans_isLittleO {f : α → E} {g : α → F'} {k : α → G} (hfg : f =O[l] g)
    (hgk : g =o[l] k) : f =o[l] k 
Arguments: {α : Type u_1} {E : Type u_3} {G : Type u_5} {F' : Type u_7} [Norm E] [Norm G] [SeminormedAddCommGroup F'] {l : Filter α} {f : α → E} {g : α → F'} {k : α → G} (hfg : f =O[l] g) (hgk : g =o[l] k)
Goal: f =o[l] k
-- Asymptotics.IsLittleO.trans
@[trans]
theorem IsLittleO.trans {f : α → E} {g : α → F} {k : α → G} (hfg : f =o[l] g) (hgk : g =o[l] k) :
    f =o[l] k 
Arguments: {α : Type u_1} {E : Type u_3} {F : Type u_4} {G : Type u_5} [Norm E] [Norm F] [Norm G] {l : Filter α} {f : α → E} {g : α → F} {k : α → G} (hfg : f =o[l] g) (hgk : g =o[l] k)
Goal: f =o[l] k
-- Filter.Eventually.trans_isBigO
theorem _root_.Filter.Eventually.trans_isBigO {f : α → E} {g : α → F'} {k : α → G}
    (hfg : ∀ᶠ x in l, ‖f x‖ ≤ ‖g x‖) (hgk : g =O[l] k) : f =O[l] k 
Arguments: {α : Type u_1} {E : Type u_3} {G : Type u_5} {F' : Type u_7} [Norm E] [Norm G] [SeminormedAddCommGroup F'] {l : Filter α} {f : α → E} {g : α → F'} {k : α → G} (hfg : ∀ᶠ (x : α) in l, ‖f x‖ ≤ ‖g x‖) (hgk : g =O[l] k)
Goal: f =O[l] k
-- Filter.Eventually.isBigO
theorem _root_.Filter.Eventually.isBigO {f : α → E} {g : α → ℝ} {l : Filter α}
    (hfg : ∀ᶠ x in l, ‖f x‖ ≤ g x) : f =O[l] g 
Arguments: {α : Type u_1} {E : Type u_3} [Norm E] {f : α → E} {g : α → ℝ} {l : Filter α} (hfg : ∀ᶠ (x : α) in l, ‖f x‖ ≤ g x)
Goal: f =O[l] g
-- Asymptotics.isBigOWith_of_le'
theorem isBigOWith_of_le' (hfg : ∀ x, ‖f x‖ ≤ c * ‖g x‖) : IsBigOWith c l f g 
Arguments: {α : Type u_1} {E : Type u_3} {F : Type u_4} [Norm E] [Norm F] {c : ℝ} {f : α → E} {g : α → F} (l : Filter α) (hfg : ∀ (x : α), ‖f x‖ ≤ c * ‖g x‖)
Goal: Asymptotics.IsBigOWith c l f g
-- Asymptotics.isBigOWith_of_le
theorem isBigOWith_of_le (hfg : ∀ x, ‖f x‖ ≤ ‖g x‖) : IsBigOWith 1 l f g 
Arguments: {α : Type u_1} {E : Type u_3} {F : Type u_4} [Norm E] [Norm F] {f : α → E} {g : α → F} (l : Filter α) (hfg : ∀ (x : α), ‖f x‖ ≤ ‖g x‖)
Goal: Asymptotics.IsBigOWith 1 l f g
-- Asymptotics.isBigO_of_le'
theorem isBigO_of_le' (hfg : ∀ x, ‖f x‖ ≤ c * ‖g x‖) : f =O[l] g 
Arguments: {α : Type u_1} {E : Type u_3} {F : Type u_4} [Norm E] [Norm F] {c : ℝ} {f : α → E} {g : α → F} (l : Filter α) (hfg : ∀ (x : α), ‖f x‖ ≤ c * ‖g x‖)
Goal: f =O[l] g
-- Asymptotics.isBigO_of_le
theorem isBigO_of_le (hfg : ∀ x, ‖f x‖ ≤ ‖g x‖) : f =O[l] g 
Arguments: {α : Type u_1} {E : Type u_3} {F : Type u_4} [Norm E] [Norm F] {f : α → E} {g : α → F} (l : Filter α) (hfg : ∀ (x : α), ‖f x‖ ≤ ‖g x‖)
Goal: f =O[l] g
-- Asymptotics.isBigOWith_refl
theorem isBigOWith_refl (f : α → E) (l : Filter α) : IsBigOWith 1 l f f 
Arguments: {α : Type u_1} {E : Type u_3} [Norm E] (f : α → E) (l : Filter α)
Goal: Asymptotics.IsBigOWith 1 l f f
-- Asymptotics.isBigO_refl
theorem isBigO_refl (f : α → E) (l : Filter α) : f =O[l] f 
Arguments: {α : Type u_1} {E : Type u_3} [Norm E] (f : α → E) (l : Filter α)
Goal: f =O[l] f
-- Filter.EventuallyEq.isBigO
theorem _root_.Filter.EventuallyEq.isBigO {f₁ f₂ : α → E} (hf : f₁ =ᶠ[l] f₂) : f₁ =O[l] f₂ 
Arguments: {α : Type u_1} {E : Type u_3} [Norm E] {l : Filter α} {f₁ f₂ : α → E} (hf : f₁ =ᶠ[l] f₂)
Goal: f₁ =O[l] f₂
-- Asymptotics.IsBigOWith.trans_le
theorem IsBigOWith.trans_le (hfg : IsBigOWith c l f g) (hgk : ∀ x, ‖g x‖ ≤ ‖k x‖) (hc : 0 ≤ c) :
    IsBigOWith c l f k 
Arguments: {α : Type u_1} {E : Type u_3} {F : Type u_4} {G : Type u_5} [Norm E] [Norm F] [Norm G] {c : ℝ} {f : α → E} {g : α → F} {k : α → G} {l : Filter α} (hfg : Asymptotics.IsBigOWith c l f g) (hgk : ∀ (x : α), ‖g x‖ ≤ ‖k x‖) (hc : 0 ≤ c)
Goal: Asymptotics.IsBigOWith c l f k
-- Asymptotics.IsBigO.trans_le
theorem IsBigO.trans_le (hfg : f =O[l] g') (hgk : ∀ x, ‖g' x‖ ≤ ‖k x‖) : f =O[l] k 
Arguments: {α : Type u_1} {E : Type u_3} {G : Type u_5} {F' : Type u_7} [Norm E] [Norm G] [SeminormedAddCommGroup F'] {f : α → E} {k : α → G} {g' : α → F'} {l : Filter α} (hfg : f =O[l] g') (hgk : ∀ (x : α), ‖g' x‖ ≤ ‖k x‖)
Goal: f =O[l] k
-- Asymptotics.IsLittleO.trans_le
theorem IsLittleO.trans_le (hfg : f =o[l] g) (hgk : ∀ x, ‖g x‖ ≤ ‖k x‖) : f =o[l] k 
Arguments: {α : Type u_1} {E : Type u_3} {F : Type u_4} {G : Type u_5} [Norm E] [Norm F] [Norm G] {f : α → E} {g : α → F} {k : α → G} {l : Filter α} (hfg : f =o[l] g) (hgk : ∀ (x : α), ‖g x‖ ≤ ‖k x‖)
Goal: f =o[l] k
-- Asymptotics.isLittleO_irrefl'
theorem isLittleO_irrefl' (h : ∃ᶠ x in l, ‖f' x‖ ≠ 0) : ¬f' =o[l] f' 
Arguments: {α : Type u_1} {E' : Type u_6} [SeminormedAddCommGroup E'] {f' : α → E'} {l : Filter α} (h : ∃ᶠ (x : α) in l, ‖f' x‖ ≠ 0)
Goal: ¬f' =o[l] f'
-- Asymptotics.isLittleO_irrefl
theorem isLittleO_irrefl (h : ∃ᶠ x in l, f'' x ≠ 0) : ¬f'' =o[l] f'' 
Arguments: {α : Type u_1} {E'' : Type u_9} [NormedAddCommGroup E''] {f'' : α → E''} {l : Filter α} (h : ∃ᶠ (x : α) in l, f'' x ≠ 0)
Goal: ¬f'' =o[l] f''
-- Asymptotics.IsBigO.not_isLittleO
theorem IsBigO.not_isLittleO (h : f'' =O[l] g') (hf : ∃ᶠ x in l, f'' x ≠ 0) :
    ¬g' =o[l] f'' 
Arguments: {α : Type u_1} {F' : Type u_7} {E'' : Type u_9} [SeminormedAddCommGroup F'] [NormedAddCommGroup E''] {g' : α → F'} {f'' : α → E''} {l : Filter α} (h : f'' =O[l] g') (hf : ∃ᶠ (x : α) in l, f'' x ≠ 0)
Goal: ¬g' =o[l] f''
-- Asymptotics.IsLittleO.not_isBigO
theorem IsLittleO.not_isBigO (h : f'' =o[l] g') (hf : ∃ᶠ x in l, f'' x ≠ 0) :
    ¬g' =O[l] f'' 
Arguments: {α : Type u_1} {F' : Type u_7} {E'' : Type u_9} [SeminormedAddCommGroup F'] [NormedAddCommGroup E''] {g' : α → F'} {f'' : α → E''} {l : Filter α} (h : f'' =o[l] g') (hf : ∃ᶠ (x : α) in l, f'' x ≠ 0)
Goal: ¬g' =O[l] f''
-- Asymptotics.isBigOWith_bot
@[simp]
theorem isBigOWith_bot : IsBigOWith c ⊥ f g 
Arguments: {α : Type u_1} {E : Type u_3} {F : Type u_4} [Norm E] [Norm F] (c : ℝ) (f : α → E) (g : α → F)
Goal: Asymptotics.IsBigOWith c ⊥ f g
-- Asymptotics.isBigO_bot
@[simp]
theorem isBigO_bot : f =O[⊥] g 
Arguments: {α : Type u_1} {E : Type u_3} {F : Type u_4} [Norm E] [Norm F] (f : α → E) (g : α → F)
Goal: f =O[⊥] g
-- Asymptotics.isLittleO_bot
@[simp]
theorem isLittleO_bot : f =o[⊥] g 
Arguments: {α : Type u_1} {E : Type u_3} {F : Type u_4} [Norm E] [Norm F] (f : α → E) (g : α → F)
Goal: f =o[⊥] g
-- Asymptotics.isBigOWith_pure
@[simp]
theorem isBigOWith_pure {x} : IsBigOWith c (pure x) f g ↔ ‖f x‖ ≤ c * ‖g x‖ 
Arguments: {α : Type u_1} {E : Type u_3} {F : Type u_4} [Norm E] [Norm F] {c : ℝ} {f : α → E} {g : α → F} {x : α}
Goal: Asymptotics.IsBigOWith c (pure x) f g ↔ ‖f x‖ ≤ c * ‖g x‖
-- Asymptotics.IsBigOWith.sup
theorem IsBigOWith.sup (h : IsBigOWith c l f g) (h' : IsBigOWith c l' f g) :
    IsBigOWith c (l ⊔ l') f g 
Arguments: {α : Type u_1} {E : Type u_3} {F : Type u_4} [Norm E] [Norm F] {c : ℝ} {f : α → E} {g : α → F} {l l' : Filter α} (h : Asymptotics.IsBigOWith c l f g) (h' : Asymptotics.IsBigOWith c l' f g)
Goal: Asymptotics.IsBigOWith c (l ⊔ l') f g
-- Asymptotics.IsBigOWith.sup'
theorem IsBigOWith.sup' (h : IsBigOWith c l f g') (h' : IsBigOWith c' l' f g') :
    IsBigOWith (max c c') (l ⊔ l') f g' 
Arguments: {α : Type u_1} {E : Type u_3} {F' : Type u_7} [Norm E] [SeminormedAddCommGroup F'] {c c' : ℝ} {f : α → E} {g' : α → F'} {l l' : Filter α} (h : Asymptotics.IsBigOWith c l f g') (h' : Asymptotics.IsBigOWith c' l' f g')
Goal: Asymptotics.IsBigOWith (max c c') (l ⊔ l') f g'
-- Asymptotics.IsBigO.sup
theorem IsBigO.sup (h : f =O[l] g') (h' : f =O[l'] g') : f =O[l ⊔ l'] g' 
Arguments: {α : Type u_1} {E : Type u_3} {F' : Type u_7} [Norm E] [SeminormedAddCommGroup F'] {f : α → E} {g' : α → F'} {l l' : Filter α} (h : f =O[l] g') (h' : f =O[l'] g')
Goal: f =O[l ⊔ l'] g'
-- Asymptotics.IsLittleO.sup
theorem IsLittleO.sup (h : f =o[l] g) (h' : f =o[l'] g) : f =o[l ⊔ l'] g 
Arguments: {α : Type u_1} {E : Type u_3} {F : Type u_4} [Norm E] [Norm F] {f : α → E} {g : α → F} {l l' : Filter α} (h : f =o[l] g) (h' : f =o[l'] g)
Goal: f =o[l ⊔ l'] g
-- Asymptotics.isBigO_sup
@[simp]
theorem isBigO_sup : f =O[l ⊔ l'] g' ↔ f =O[l] g' ∧ f =O[l'] g' 
Arguments: {α : Type u_1} {E : Type u_3} {F' : Type u_7} [Norm E] [SeminormedAddCommGroup F'] {f : α → E} {g' : α → F'} {l l' : Filter α}
Goal: f =O[l ⊔ l'] g' ↔ f =O[l] g' ∧ f =O[l'] g'
-- Asymptotics.isLittleO_sup
@[simp]
theorem isLittleO_sup : f =o[l ⊔ l'] g ↔ f =o[l] g ∧ f =o[l'] g 
Arguments: {α : Type u_1} {E : Type u_3} {F : Type u_4} [Norm E] [Norm F] {f : α → E} {g : α → F} {l l' : Filter α}
Goal: f =o[l ⊔ l'] g ↔ f =o[l] g ∧ f =o[l'] g
-- Asymptotics.isBigOWith_insert
theorem isBigOWith_insert [TopologicalSpace α] {x : α} {s : Set α} {C : ℝ} {g : α → E} {g' : α → F}
    (h : ‖g x‖ ≤ C * ‖g' x‖) : IsBigOWith C (𝓝[insert x s] x) g g' ↔
    IsBigOWith C (𝓝[s] x) g g' 
Arguments: {α : Type u_1} {E : Type u_3} {F : Type u_4} [Norm E] [Norm F] [TopologicalSpace α] {x : α} {s : Set α} {C : ℝ} {g : α → E} {g' : α → F} (h : ‖g x‖ ≤ C * ‖g' x‖)
Goal: Asymptotics.IsBigOWith C (𝓝[insert x s] x) g g' ↔ Asymptotics.IsBigOWith C (𝓝[s] x) g g'
-- Asymptotics.IsBigOWith.insert
protected theorem IsBigOWith.insert [TopologicalSpace α] {x : α} {s : Set α} {C : ℝ} {g : α → E}
    {g' : α → F} (h1 : IsBigOWith C (𝓝[s] x) g g') (h2 : ‖g x‖ ≤ C * ‖g' x‖) :
    IsBigOWith C (𝓝[insert x s] x) g g' 
Arguments: {α : Type u_1} {E : Type u_3} {F : Type u_4} [Norm E] [Norm F] [TopologicalSpace α] {x : α} {s : Set α} {C : ℝ} {g : α → E} {g' : α → F} (h1 : Asymptotics.IsBigOWith C (𝓝[s] x) g g') (h2 : ‖g x‖ ≤ C * ‖g' x‖)
Goal: Asymptotics.IsBigOWith C (𝓝[insert x s] x) g g'
-- Asymptotics.isLittleO_insert
theorem isLittleO_insert [TopologicalSpace α] {x : α} {s : Set α} {g : α → E'} {g' : α → F'}
    (h : g x = 0) : g =o[𝓝[insert x s] x] g' ↔ g =o[𝓝[s] x] g' 
Arguments: {α : Type u_1} {E' : Type u_6} {F' : Type u_7} [SeminormedAddCommGroup E'] [SeminormedAddCommGroup F'] [TopologicalSpace α] {x : α} {s : Set α} {g : α → E'} {g' : α → F'} (h : g x = 0)
Goal: g =o[𝓝[insert x s] x] g' ↔ g =o[𝓝[s] x] g'
-- Asymptotics.IsLittleO.insert
protected theorem IsLittleO.insert [TopologicalSpace α] {x : α} {s : Set α} {g : α → E'}
    {g' : α → F'} (h1 : g =o[𝓝[s] x] g') (h2 : g x = 0) : g =o[𝓝[insert x s] x] g' 
Arguments: {α : Type u_1} {E' : Type u_6} {F' : Type u_7} [SeminormedAddCommGroup E'] [SeminormedAddCommGroup F'] [TopologicalSpace α] {x : α} {s : Set α} {g : α → E'} {g' : α → F'} (h1 : g =o[𝓝[s] x] g') (h2 : g x = 0)
Goal: g =o[𝓝[insert x s] x] g'
-- Asymptotics.isBigOWith_norm_right
@[simp]
theorem isBigOWith_norm_right : (IsBigOWith c l f fun x => ‖g' x‖) ↔ IsBigOWith c l f g' 
Arguments: {α : Type u_1} {E : Type u_3} {F' : Type u_7} [Norm E] [SeminormedAddCommGroup F'] {c : ℝ} {f : α → E} {g' : α → F'} {l : Filter α}
Goal: (Asymptotics.IsBigOWith c l f fun x => ‖g' x‖) ↔ Asymptotics.IsBigOWith c l f g'
-- Asymptotics.isBigOWith_abs_right
@[simp]
theorem isBigOWith_abs_right : (IsBigOWith c l f fun x => |u x|) ↔ IsBigOWith c l f u 
Arguments: {α : Type u_1} {E : Type u_3} [Norm E] {c : ℝ} {f : α → E} {l : Filter α} {u : α → ℝ}
Goal: (Asymptotics.IsBigOWith c l f fun x => |u x|) ↔ Asymptotics.IsBigOWith c l f u
-- Asymptotics.isBigO_norm_right
@[simp]
theorem isBigO_norm_right : (f =O[l] fun x => ‖g' x‖) ↔ f =O[l] g' 
Arguments: {α : Type u_1} {E : Type u_3} {F' : Type u_7} [Norm E] [SeminormedAddCommGroup F'] {f : α → E} {g' : α → F'} {l : Filter α}
Goal: (f =O[l] fun x => ‖g' x‖) ↔ f =O[l] g'
-- Asymptotics.isBigO_abs_right
@[simp]
theorem isBigO_abs_right : (f =O[l] fun x => |u x|) ↔ f =O[l] u 
Arguments: {α : Type u_1} {E : Type u_3} [Norm E] {f : α → E} {l : Filter α} {u : α → ℝ}
Goal: (f =O[l] fun x => |u x|) ↔ f =O[l] u
-- Asymptotics.isLittleO_norm_right
@[simp]
theorem isLittleO_norm_right : (f =o[l] fun x => ‖g' x‖) ↔ f =o[l] g' 
Arguments: {α : Type u_1} {E : Type u_3} {F' : Type u_7} [Norm E] [SeminormedAddCommGroup F'] {f : α → E} {g' : α → F'} {l : Filter α}
Goal: (f =o[l] fun x => ‖g' x‖) ↔ f =o[l] g'
-- Asymptotics.isLittleO_abs_right
@[simp]
theorem isLittleO_abs_right : (f =o[l] fun x => |u x|) ↔ f =o[l] u 
Arguments: {α : Type u_1} {E : Type u_3} [Norm E] {f : α → E} {l : Filter α} {u : α → ℝ}
Goal: (f =o[l] fun x => |u x|) ↔ f =o[l] u
-- Asymptotics.isBigOWith_norm_left
@[simp]
theorem isBigOWith_norm_left : IsBigOWith c l (fun x => ‖f' x‖) g ↔ IsBigOWith c l f' g 
Arguments: {α : Type u_1} {F : Type u_4} {E' : Type u_6} [Norm F] [SeminormedAddCommGroup E'] {c : ℝ} {g : α → F} {f' : α → E'} {l : Filter α}
Goal: Asymptotics.IsBigOWith c l (fun x => ‖f' x‖) g ↔ Asymptotics.IsBigOWith c l f' g
-- Asymptotics.isBigOWith_abs_left
@[simp]
theorem isBigOWith_abs_left : IsBigOWith c l (fun x => |u x|) g ↔ IsBigOWith c l u g 
Arguments: {α : Type u_1} {F : Type u_4} [Norm F] {c : ℝ} {g : α → F} {l : Filter α} {u : α → ℝ}
Goal: Asymptotics.IsBigOWith c l (fun x => |u x|) g ↔ Asymptotics.IsBigOWith c l u g
-- Asymptotics.isBigO_norm_left
@[simp]
theorem isBigO_norm_left : (fun x => ‖f' x‖) =O[l] g ↔ f' =O[l] g 
Arguments: {α : Type u_1} {F : Type u_4} {E' : Type u_6} [Norm F] [SeminormedAddCommGroup E'] {g : α → F} {f' : α → E'} {l : Filter α}
Goal: (fun x => ‖f' x‖) =O[l] g ↔ f' =O[l] g
-- Asymptotics.isBigO_abs_left
@[simp]
theorem isBigO_abs_left : (fun x => |u x|) =O[l] g ↔ u =O[l] g 
Arguments: {α : Type u_1} {F : Type u_4} [Norm F] {g : α → F} {l : Filter α} {u : α → ℝ}
Goal: (fun x => |u x|) =O[l] g ↔ u =O[l] g
-- Asymptotics.isLittleO_norm_left
@[simp]
theorem isLittleO_norm_left : (fun x => ‖f' x‖) =o[l] g ↔ f' =o[l] g 
Arguments: {α : Type u_1} {F : Type u_4} {E' : Type u_6} [Norm F] [SeminormedAddCommGroup E'] {g : α → F} {f' : α → E'} {l : Filter α}
Goal: (fun x => ‖f' x‖) =o[l] g ↔ f' =o[l] g
-- Asymptotics.isLittleO_abs_left
@[simp]
theorem isLittleO_abs_left : (fun x => |u x|) =o[l] g ↔ u =o[l] g 
Arguments: {α : Type u_1} {F : Type u_4} [Norm F] {g : α → F} {l : Filter α} {u : α → ℝ}
Goal: (fun x => |u x|) =o[l] g ↔ u =o[l] g
-- Asymptotics.isBigOWith_norm_norm
theorem isBigOWith_norm_norm :
    (IsBigOWith c l (fun x => ‖f' x‖) fun x => ‖g' x‖) ↔ IsBigOWith c l f' g' 
Arguments: {α : Type u_1} {E' : Type u_6} {F' : Type u_7} [SeminormedAddCommGroup E'] [SeminormedAddCommGroup F'] {c : ℝ} {f' : α → E'} {g' : α → F'} {l : Filter α}
Goal: (Asymptotics.IsBigOWith c l (fun x => ‖f' x‖) fun x => ‖g' x‖) ↔ Asymptotics.IsBigOWith c l f' g'
-- Asymptotics.isBigOWith_abs_abs
theorem isBigOWith_abs_abs :
    (IsBigOWith c l (fun x => |u x|) fun x => |v x|) ↔ IsBigOWith c l u v 
Arguments: {α : Type u_1} {c : ℝ} {l : Filter α} {u v : α → ℝ}
Goal: (Asymptotics.IsBigOWith c l (fun x => |u x|) fun x => |v x|) ↔ Asymptotics.IsBigOWith c l u v
-- Asymptotics.isBigO_norm_norm
theorem isBigO_norm_norm : ((fun x => ‖f' x‖) =O[l] fun x => ‖g' x‖) ↔ f' =O[l] g' 
Arguments: {α : Type u_1} {E' : Type u_6} {F' : Type u_7} [SeminormedAddCommGroup E'] [SeminormedAddCommGroup F'] {f' : α → E'} {g' : α → F'} {l : Filter α}
Goal: ((fun x => ‖f' x‖) =O[l] fun x => ‖g' x‖) ↔ f' =O[l] g'
-- Asymptotics.isBigO_abs_abs
theorem isBigO_abs_abs : ((fun x => |u x|) =O[l] fun x => |v x|) ↔ u =O[l] v 
Arguments: {α : Type u_1} {l : Filter α} {u v : α → ℝ}
Goal: ((fun x => |u x|) =O[l] fun x => |v x|) ↔ u =O[l] v
-- Asymptotics.isLittleO_norm_norm
theorem isLittleO_norm_norm : ((fun x => ‖f' x‖) =o[l] fun x => ‖g' x‖) ↔ f' =o[l] g' 
Arguments: {α : Type u_1} {E' : Type u_6} {F' : Type u_7} [SeminormedAddCommGroup E'] [SeminormedAddCommGroup F'] {f' : α → E'} {g' : α → F'} {l : Filter α}
Goal: ((fun x => ‖f' x‖) =o[l] fun x => ‖g' x‖) ↔ f' =o[l] g'
-- Asymptotics.isLittleO_abs_abs
theorem isLittleO_abs_abs : ((fun x => |u x|) =o[l] fun x => |v x|) ↔ u =o[l] v 
Arguments: {α : Type u_1} {l : Filter α} {u v : α → ℝ}
Goal: ((fun x => |u x|) =o[l] fun x => |v x|) ↔ u =o[l] v
-- Asymptotics.isBigOWith_neg_right
@[simp]
theorem isBigOWith_neg_right : (IsBigOWith c l f fun x => -g' x) ↔ IsBigOWith c l f g' 
Arguments: {α : Type u_1} {E : Type u_3} {F' : Type u_7} [Norm E] [SeminormedAddCommGroup F'] {c : ℝ} {f : α → E} {g' : α → F'} {l : Filter α}
Goal: (Asymptotics.IsBigOWith c l f fun x => -g' x) ↔ Asymptotics.IsBigOWith c l f g'
-- Asymptotics.isBigO_neg_right
@[simp]
theorem isBigO_neg_right : (f =O[l] fun x => -g' x) ↔ f =O[l] g' 
Arguments: {α : Type u_1} {E : Type u_3} {F' : Type u_7} [Norm E] [SeminormedAddCommGroup F'] {f : α → E} {g' : α → F'} {l : Filter α}
Goal: (f =O[l] fun x => -g' x) ↔ f =O[l] g'
-- Asymptotics.isLittleO_neg_right
@[simp]
theorem isLittleO_neg_right : (f =o[l] fun x => -g' x) ↔ f =o[l] g' 
Arguments: {α : Type u_1} {E : Type u_3} {F' : Type u_7} [Norm E] [SeminormedAddCommGroup F'] {f : α → E} {g' : α → F'} {l : Filter α}
Goal: (f =o[l] fun x => -g' x) ↔ f =o[l] g'