ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Asymptotics.isBigOWith_neg_left
@[simp]
theorem isBigOWith_neg_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.isBigO_neg_left
@[simp]
theorem isBigO_neg_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_neg_left
@[simp]
theorem isLittleO_neg_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.isBigOWith_fst_prod
theorem isBigOWith_fst_prod : IsBigOWith 1 l f' fun x => (f' x, g' x) 
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: Asymptotics.IsBigOWith 1 l f' fun x => (f' x, g' x)
-- Asymptotics.isBigOWith_snd_prod
theorem isBigOWith_snd_prod : IsBigOWith 1 l g' fun x => (f' x, g' x) 
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: Asymptotics.IsBigOWith 1 l g' fun x => (f' x, g' x)
-- Asymptotics.isBigO_fst_prod
theorem isBigO_fst_prod : f' =O[l] fun x => (f' x, g' x) 
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: f' =O[l] fun x => (f' x, g' x)
-- Asymptotics.isBigO_snd_prod
theorem isBigO_snd_prod : g' =O[l] fun x => (f' x, g' x) 
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: g' =O[l] fun x => (f' x, g' x)
-- Asymptotics.isBigO_fst_prod'
theorem isBigO_fst_prod' {f' : α → E' × F'} : (fun x => (f' x).1) =O[l] f' 
Arguments: {α : Type u_1} {E' : Type u_6} {F' : Type u_7} [SeminormedAddCommGroup E'] [SeminormedAddCommGroup F'] {l : Filter α} {f' : α → E' × F'}
Goal: (fun x => (f' x).1) =O[l] f'
-- Asymptotics.isBigO_snd_prod'
theorem isBigO_snd_prod' {f' : α → E' × F'} : (fun x => (f' x).2) =O[l] f' 
Arguments: {α : Type u_1} {E' : Type u_6} {F' : Type u_7} [SeminormedAddCommGroup E'] [SeminormedAddCommGroup F'] {l : Filter α} {f' : α → E' × F'}
Goal: (fun x => (f' x).2) =O[l] f'
-- Asymptotics.IsBigOWith.prod_rightl
theorem IsBigOWith.prod_rightl (h : IsBigOWith c l f g') (hc : 0 ≤ c) :
    IsBigOWith c l f fun x => (g' x, k' x) 
Arguments: {α : Type u_1} {E : Type u_3} {F' : Type u_7} {G' : Type u_8} [Norm E] [SeminormedAddCommGroup F'] [SeminormedAddCommGroup G'] {c : ℝ} {f : α → E} {g' : α → F'} (k' : α → G') {l : Filter α} (h : Asymptotics.IsBigOWith c l f g') (hc : 0 ≤ c)
Goal: Asymptotics.IsBigOWith c l f fun x => (g' x, k' x)
-- Asymptotics.IsBigO.prod_rightl
theorem IsBigO.prod_rightl (h : f =O[l] g') : f =O[l] fun x => (g' x, k' x) 
Arguments: {α : Type u_1} {E : Type u_3} {F' : Type u_7} {G' : Type u_8} [Norm E] [SeminormedAddCommGroup F'] [SeminormedAddCommGroup G'] {f : α → E} {g' : α → F'} (k' : α → G') {l : Filter α} (h : f =O[l] g')
Goal: f =O[l] fun x => (g' x, k' x)
-- Asymptotics.IsLittleO.prod_rightl
theorem IsLittleO.prod_rightl (h : f =o[l] g') : f =o[l] fun x => (g' x, k' x) 
Arguments: {α : Type u_1} {E : Type u_3} {F' : Type u_7} {G' : Type u_8} [Norm E] [SeminormedAddCommGroup F'] [SeminormedAddCommGroup G'] {f : α → E} {g' : α → F'} (k' : α → G') {l : Filter α} (h : f =o[l] g')
Goal: f =o[l] fun x => (g' x, k' x)
-- Asymptotics.IsBigOWith.prod_rightr
theorem IsBigOWith.prod_rightr (h : IsBigOWith c l f g') (hc : 0 ≤ c) :
    IsBigOWith c l f fun x => (f' x, g' x) 
Arguments: {α : Type u_1} {E : Type u_3} {E' : Type u_6} {F' : Type u_7} [Norm E] [SeminormedAddCommGroup E'] [SeminormedAddCommGroup F'] {c : ℝ} {f : α → E} (f' : α → E') {g' : α → F'} {l : Filter α} (h : Asymptotics.IsBigOWith c l f g') (hc : 0 ≤ c)
Goal: Asymptotics.IsBigOWith c l f fun x => (f' x, g' x)
-- Asymptotics.IsBigO.prod_rightr
theorem IsBigO.prod_rightr (h : f =O[l] g') : f =O[l] fun x => (f' x, g' x) 
Arguments: {α : Type u_1} {E : Type u_3} {E' : Type u_6} {F' : Type u_7} [Norm E] [SeminormedAddCommGroup E'] [SeminormedAddCommGroup F'] {f : α → E} (f' : α → E') {g' : α → F'} {l : Filter α} (h : f =O[l] g')
Goal: f =O[l] fun x => (f' x, g' x)
-- Asymptotics.IsLittleO.prod_rightr
theorem IsLittleO.prod_rightr (h : f =o[l] g') : f =o[l] fun x => (f' x, g' x) 
Arguments: {α : Type u_1} {E : Type u_3} {E' : Type u_6} {F' : Type u_7} [Norm E] [SeminormedAddCommGroup E'] [SeminormedAddCommGroup F'] {f : α → E} (f' : α → E') {g' : α → F'} {l : Filter α} (h : f =o[l] g')
Goal: f =o[l] fun x => (f' x, g' x)
-- Asymptotics.IsBigOWith.prod_left_same
theorem IsBigOWith.prod_left_same (hf : IsBigOWith c l f' k') (hg : IsBigOWith c l g' k') :
    IsBigOWith c l (fun x => (f' x, g' x)) k' 
Arguments: {α : Type u_1} {E' : Type u_6} {F' : Type u_7} {G' : Type u_8} [SeminormedAddCommGroup E'] [SeminormedAddCommGroup F'] [SeminormedAddCommGroup G'] {c : ℝ} {f' : α → E'} {g' : α → F'} {k' : α → G'} {l : Filter α} (hf : Asymptotics.IsBigOWith c l f' k') (hg : Asymptotics.IsBigOWith c l g' k')
Goal: Asymptotics.IsBigOWith c l (fun x => (f' x, g' x)) k'
-- Asymptotics.IsBigOWith.prod_left
theorem IsBigOWith.prod_left (hf : IsBigOWith c l f' k') (hg : IsBigOWith c' l g' k') :
    IsBigOWith (max c c') l (fun x => (f' x, g' x)) k' 
Arguments: {α : Type u_1} {E' : Type u_6} {F' : Type u_7} {G' : Type u_8} [SeminormedAddCommGroup E'] [SeminormedAddCommGroup F'] [SeminormedAddCommGroup G'] {c c' : ℝ} {f' : α → E'} {g' : α → F'} {k' : α → G'} {l : Filter α} (hf : Asymptotics.IsBigOWith c l f' k') (hg : Asymptotics.IsBigOWith c' l g' k')
Goal: Asymptotics.IsBigOWith (max c c') l (fun x => (f' x, g' x)) k'
-- Asymptotics.IsBigOWith.prod_left_fst
theorem IsBigOWith.prod_left_fst (h : IsBigOWith c l (fun x => (f' x, g' x)) k') :
    IsBigOWith c l f' k' 
Arguments: {α : Type u_1} {E' : Type u_6} {F' : Type u_7} {G' : Type u_8} [SeminormedAddCommGroup E'] [SeminormedAddCommGroup F'] [SeminormedAddCommGroup G'] {c : ℝ} {f' : α → E'} {g' : α → F'} {k' : α → G'} {l : Filter α} (h : Asymptotics.IsBigOWith c l (fun x => (f' x, g' x)) k')
Goal: Asymptotics.IsBigOWith c l f' k'
-- Asymptotics.IsBigOWith.prod_left_snd
theorem IsBigOWith.prod_left_snd (h : IsBigOWith c l (fun x => (f' x, g' x)) k') :
    IsBigOWith c l g' k' 
Arguments: {α : Type u_1} {E' : Type u_6} {F' : Type u_7} {G' : Type u_8} [SeminormedAddCommGroup E'] [SeminormedAddCommGroup F'] [SeminormedAddCommGroup G'] {c : ℝ} {f' : α → E'} {g' : α → F'} {k' : α → G'} {l : Filter α} (h : Asymptotics.IsBigOWith c l (fun x => (f' x, g' x)) k')
Goal: Asymptotics.IsBigOWith c l g' k'
-- Asymptotics.isBigOWith_prod_left
theorem isBigOWith_prod_left :
    IsBigOWith c l (fun x => (f' x, g' x)) k' ↔ IsBigOWith c l f' k' ∧ IsBigOWith c l g' k' 
Arguments: {α : Type u_1} {E' : Type u_6} {F' : Type u_7} {G' : Type u_8} [SeminormedAddCommGroup E'] [SeminormedAddCommGroup F'] [SeminormedAddCommGroup G'] {c : ℝ} {f' : α → E'} {g' : α → F'} {k' : α → G'} {l : Filter α}
Goal: Asymptotics.IsBigOWith c l (fun x => (f' x, g' x)) k' ↔
  Asymptotics.IsBigOWith c l f' k' ∧ Asymptotics.IsBigOWith c l g' k'
-- Asymptotics.IsBigO.prod_left
theorem IsBigO.prod_left (hf : f' =O[l] k') (hg : g' =O[l] k') : (fun x => (f' x, g' x)) =O[l] k' 
Arguments: {α : Type u_1} {E' : Type u_6} {F' : Type u_7} {G' : Type u_8} [SeminormedAddCommGroup E'] [SeminormedAddCommGroup F'] [SeminormedAddCommGroup G'] {f' : α → E'} {g' : α → F'} {k' : α → G'} {l : Filter α} (hf : f' =O[l] k') (hg : g' =O[l] k')
Goal: (fun x => (f' x, g' x)) =O[l] k'
-- Asymptotics.IsBigO.prod_left_fst
theorem IsBigO.prod_left_fst : (fun x => (f' x, g' x)) =O[l] k' → f' =O[l] k' 
Arguments: {α : Type u_1} {E' : Type u_6} {F' : Type u_7} {G' : Type u_8} [SeminormedAddCommGroup E'] [SeminormedAddCommGroup F'] [SeminormedAddCommGroup G'] {f' : α → E'} {g' : α → F'} {k' : α → G'} {l : Filter α}
Goal: (fun x => (f' x, g' x)) =O[l] k' → f' =O[l] k'
-- Asymptotics.IsBigO.prod_left_snd
theorem IsBigO.prod_left_snd : (fun x => (f' x, g' x)) =O[l] k' → g' =O[l] k' 
Arguments: {α : Type u_1} {E' : Type u_6} {F' : Type u_7} {G' : Type u_8} [SeminormedAddCommGroup E'] [SeminormedAddCommGroup F'] [SeminormedAddCommGroup G'] {f' : α → E'} {g' : α → F'} {k' : α → G'} {l : Filter α}
Goal: (fun x => (f' x, g' x)) =O[l] k' → g' =O[l] k'
-- Asymptotics.isBigO_prod_left
@[simp]
theorem isBigO_prod_left : (fun x => (f' x, g' x)) =O[l] k' ↔ f' =O[l] k' ∧ g' =O[l] k' 
Arguments: {α : Type u_1} {E' : Type u_6} {F' : Type u_7} {G' : Type u_8} [SeminormedAddCommGroup E'] [SeminormedAddCommGroup F'] [SeminormedAddCommGroup G'] {f' : α → E'} {g' : α → F'} {k' : α → G'} {l : Filter α}
Goal: (fun x => (f' x, g' x)) =O[l] k' ↔ f' =O[l] k' ∧ g' =O[l] k'
-- Asymptotics.IsLittleO.prod_left
theorem IsLittleO.prod_left (hf : f' =o[l] k') (hg : g' =o[l] k') :
    (fun x => (f' x, g' x)) =o[l] k' 
Arguments: {α : Type u_1} {E' : Type u_6} {F' : Type u_7} {G' : Type u_8} [SeminormedAddCommGroup E'] [SeminormedAddCommGroup F'] [SeminormedAddCommGroup G'] {f' : α → E'} {g' : α → F'} {k' : α → G'} {l : Filter α} (hf : f' =o[l] k') (hg : g' =o[l] k')
Goal: (fun x => (f' x, g' x)) =o[l] k'
-- Asymptotics.IsLittleO.prod_left_fst
theorem IsLittleO.prod_left_fst : (fun x => (f' x, g' x)) =o[l] k' → f' =o[l] k' 
Arguments: {α : Type u_1} {E' : Type u_6} {F' : Type u_7} {G' : Type u_8} [SeminormedAddCommGroup E'] [SeminormedAddCommGroup F'] [SeminormedAddCommGroup G'] {f' : α → E'} {g' : α → F'} {k' : α → G'} {l : Filter α}
Goal: (fun x => (f' x, g' x)) =o[l] k' → f' =o[l] k'
-- Asymptotics.IsLittleO.prod_left_snd
theorem IsLittleO.prod_left_snd : (fun x => (f' x, g' x)) =o[l] k' → g' =o[l] k' 
Arguments: {α : Type u_1} {E' : Type u_6} {F' : Type u_7} {G' : Type u_8} [SeminormedAddCommGroup E'] [SeminormedAddCommGroup F'] [SeminormedAddCommGroup G'] {f' : α → E'} {g' : α → F'} {k' : α → G'} {l : Filter α}
Goal: (fun x => (f' x, g' x)) =o[l] k' → g' =o[l] k'
-- Asymptotics.isLittleO_prod_left
@[simp]
theorem isLittleO_prod_left : (fun x => (f' x, g' x)) =o[l] k' ↔ f' =o[l] k' ∧ g' =o[l] k' 
Arguments: {α : Type u_1} {E' : Type u_6} {F' : Type u_7} {G' : Type u_8} [SeminormedAddCommGroup E'] [SeminormedAddCommGroup F'] [SeminormedAddCommGroup G'] {f' : α → E'} {g' : α → F'} {k' : α → G'} {l : Filter α}
Goal: (fun x => (f' x, g' x)) =o[l] k' ↔ f' =o[l] k' ∧ g' =o[l] k'
-- Asymptotics.IsBigOWith.eq_zero_imp
theorem IsBigOWith.eq_zero_imp (h : IsBigOWith c l f'' g'') : ∀ᶠ x in l, g'' x = 0 → f'' x = 0 
Arguments: {α : Type u_1} {E'' : Type u_9} {F'' : Type u_10} [NormedAddCommGroup E''] [NormedAddCommGroup F''] {c : ℝ} {f'' : α → E''} {g'' : α → F''} {l : Filter α} (h : Asymptotics.IsBigOWith c l f'' g'')
Goal: ∀ᶠ (x : α) in l, g'' x = 0 → f'' x = 0
-- Asymptotics.IsBigO.eq_zero_imp
theorem IsBigO.eq_zero_imp (h : f'' =O[l] g'') : ∀ᶠ x in l, g'' x = 0 → f'' x = 0 
Arguments: {α : Type u_1} {E'' : Type u_9} {F'' : Type u_10} [NormedAddCommGroup E''] [NormedAddCommGroup F''] {f'' : α → E''} {g'' : α → F''} {l : Filter α} (h : f'' =O[l] g'')
Goal: ∀ᶠ (x : α) in l, g'' x = 0 → f'' x = 0
-- Asymptotics.IsBigOWith.add
theorem IsBigOWith.add (h₁ : IsBigOWith c₁ l f₁ g) (h₂ : IsBigOWith c₂ l f₂ g) :
    IsBigOWith (c₁ + c₂) l (fun x => f₁ x + f₂ x) g 
Arguments: {α : Type u_1} {F : Type u_4} {E' : Type u_6} [Norm F] [SeminormedAddCommGroup E'] {c₁ c₂ : ℝ} {g : α → F} {l : Filter α} {f₁ f₂ : α → E'} (h₁ : Asymptotics.IsBigOWith c₁ l f₁ g) (h₂ : Asymptotics.IsBigOWith c₂ l f₂ g)
Goal: Asymptotics.IsBigOWith (c₁ + c₂) l (fun x => f₁ x + f₂ x) g
-- Asymptotics.IsBigO.add
theorem IsBigO.add (h₁ : f₁ =O[l] g) (h₂ : f₂ =O[l] g) : (fun x => f₁ x + f₂ x) =O[l] g 
Arguments: {α : Type u_1} {F : Type u_4} {E' : Type u_6} [Norm F] [SeminormedAddCommGroup E'] {g : α → F} {l : Filter α} {f₁ f₂ : α → E'} (h₁ : f₁ =O[l] g) (h₂ : f₂ =O[l] g)
Goal: (fun x => f₁ x + f₂ x) =O[l] g
-- Asymptotics.IsLittleO.add
theorem IsLittleO.add (h₁ : f₁ =o[l] g) (h₂ : f₂ =o[l] g) : (fun x => f₁ x + f₂ x) =o[l] g 
Arguments: {α : Type u_1} {F : Type u_4} {E' : Type u_6} [Norm F] [SeminormedAddCommGroup E'] {g : α → F} {l : Filter α} {f₁ f₂ : α → E'} (h₁ : f₁ =o[l] g) (h₂ : f₂ =o[l] g)
Goal: (fun x => f₁ x + f₂ x) =o[l] g
-- Asymptotics.IsLittleO.add_add
theorem IsLittleO.add_add (h₁ : f₁ =o[l] g₁) (h₂ : f₂ =o[l] g₂) :
    (fun x => f₁ x + f₂ x) =o[l] fun x => ‖g₁ x‖ + ‖g₂ x‖ 
Arguments: {α : Type u_1} {E' : Type u_6} {F' : Type u_7} [SeminormedAddCommGroup E'] [SeminormedAddCommGroup F'] {l : Filter α} {f₁ f₂ : α → E'} {g₁ g₂ : α → F'} (h₁ : f₁ =o[l] g₁) (h₂ : f₂ =o[l] g₂)
Goal: (fun x => f₁ x + f₂ x) =o[l] fun x => ‖g₁ x‖ + ‖g₂ x‖
-- Asymptotics.IsBigO.add_isLittleO
theorem IsBigO.add_isLittleO (h₁ : f₁ =O[l] g) (h₂ : f₂ =o[l] g) : (fun x => f₁ x + f₂ x) =O[l] g 
Arguments: {α : Type u_1} {F : Type u_4} {E' : Type u_6} [Norm F] [SeminormedAddCommGroup E'] {g : α → F} {l : Filter α} {f₁ f₂ : α → E'} (h₁ : f₁ =O[l] g) (h₂ : f₂ =o[l] g)
Goal: (fun x => f₁ x + f₂ x) =O[l] g
-- Asymptotics.IsLittleO.add_isBigO
theorem IsLittleO.add_isBigO (h₁ : f₁ =o[l] g) (h₂ : f₂ =O[l] g) : (fun x => f₁ x + f₂ x) =O[l] g 
Arguments: {α : Type u_1} {F : Type u_4} {E' : Type u_6} [Norm F] [SeminormedAddCommGroup E'] {g : α → F} {l : Filter α} {f₁ f₂ : α → E'} (h₁ : f₁ =o[l] g) (h₂ : f₂ =O[l] g)
Goal: (fun x => f₁ x + f₂ x) =O[l] g
-- Asymptotics.IsBigOWith.add_isLittleO
theorem IsBigOWith.add_isLittleO (h₁ : IsBigOWith c₁ l f₁ g) (h₂ : f₂ =o[l] g) (hc : c₁ < c₂) :
    IsBigOWith c₂ l (fun x => f₁ x + f₂ x) g 
Arguments: {α : Type u_1} {F : Type u_4} {E' : Type u_6} [Norm F] [SeminormedAddCommGroup E'] {c₁ c₂ : ℝ} {g : α → F} {l : Filter α} {f₁ f₂ : α → E'} (h₁ : Asymptotics.IsBigOWith c₁ l f₁ g) (h₂ : f₂ =o[l] g) (hc : c₁ < c₂)
Goal: Asymptotics.IsBigOWith c₂ l (fun x => f₁ x + f₂ x) g
-- Asymptotics.IsLittleO.add_isBigOWith
theorem IsLittleO.add_isBigOWith (h₁ : f₁ =o[l] g) (h₂ : IsBigOWith c₁ l f₂ g) (hc : c₁ < c₂) :
    IsBigOWith c₂ l (fun x => f₁ x + f₂ x) g 
Arguments: {α : Type u_1} {F : Type u_4} {E' : Type u_6} [Norm F] [SeminormedAddCommGroup E'] {c₁ c₂ : ℝ} {g : α → F} {l : Filter α} {f₁ f₂ : α → E'} (h₁ : f₁ =o[l] g) (h₂ : Asymptotics.IsBigOWith c₁ l f₂ g) (hc : c₁ < c₂)
Goal: Asymptotics.IsBigOWith c₂ l (fun x => f₁ x + f₂ x) g
-- Asymptotics.IsBigOWith.sub
theorem IsBigOWith.sub (h₁ : IsBigOWith c₁ l f₁ g) (h₂ : IsBigOWith c₂ l f₂ g) :
    IsBigOWith (c₁ + c₂) l (fun x => f₁ x - f₂ x) g 
Arguments: {α : Type u_1} {F : Type u_4} {E' : Type u_6} [Norm F] [SeminormedAddCommGroup E'] {c₁ c₂ : ℝ} {g : α → F} {l : Filter α} {f₁ f₂ : α → E'} (h₁ : Asymptotics.IsBigOWith c₁ l f₁ g) (h₂ : Asymptotics.IsBigOWith c₂ l f₂ g)
Goal: Asymptotics.IsBigOWith (c₁ + c₂) l (fun x => f₁ x - f₂ x) g
-- Asymptotics.IsBigOWith.sub_isLittleO
theorem IsBigOWith.sub_isLittleO (h₁ : IsBigOWith c₁ l f₁ g) (h₂ : f₂ =o[l] g) (hc : c₁ < c₂) :
    IsBigOWith c₂ l (fun x => f₁ x - f₂ x) g 
Arguments: {α : Type u_1} {F : Type u_4} {E' : Type u_6} [Norm F] [SeminormedAddCommGroup E'] {c₁ c₂ : ℝ} {g : α → F} {l : Filter α} {f₁ f₂ : α → E'} (h₁ : Asymptotics.IsBigOWith c₁ l f₁ g) (h₂ : f₂ =o[l] g) (hc : c₁ < c₂)
Goal: Asymptotics.IsBigOWith c₂ l (fun x => f₁ x - f₂ x) g
-- Asymptotics.IsBigO.sub
theorem IsBigO.sub (h₁ : f₁ =O[l] g) (h₂ : f₂ =O[l] g) : (fun x => f₁ x - f₂ x) =O[l] g 
Arguments: {α : Type u_1} {F : Type u_4} {E' : Type u_6} [Norm F] [SeminormedAddCommGroup E'] {g : α → F} {l : Filter α} {f₁ f₂ : α → E'} (h₁ : f₁ =O[l] g) (h₂ : f₂ =O[l] g)
Goal: (fun x => f₁ x - f₂ x) =O[l] g
-- Asymptotics.IsLittleO.sub
theorem IsLittleO.sub (h₁ : f₁ =o[l] g) (h₂ : f₂ =o[l] g) : (fun x => f₁ x - f₂ x) =o[l] g 
Arguments: {α : Type u_1} {F : Type u_4} {E' : Type u_6} [Norm F] [SeminormedAddCommGroup E'] {g : α → F} {l : Filter α} {f₁ f₂ : α → E'} (h₁ : f₁ =o[l] g) (h₂ : f₂ =o[l] g)
Goal: (fun x => f₁ x - f₂ x) =o[l] g
-- Asymptotics.IsBigOWith.symm
theorem IsBigOWith.symm (h : IsBigOWith c l (fun x => f₁ x - f₂ x) g) :
    IsBigOWith c l (fun x => f₂ x - f₁ x) g 
Arguments: {α : Type u_1} {F : Type u_4} {E' : Type u_6} [Norm F] [SeminormedAddCommGroup E'] {c : ℝ} {g : α → F} {l : Filter α} {f₁ f₂ : α → E'} (h : Asymptotics.IsBigOWith c l (fun x => f₁ x - f₂ x) g)
Goal: Asymptotics.IsBigOWith c l (fun x => f₂ x - f₁ x) g
-- Asymptotics.isBigOWith_comm
theorem isBigOWith_comm :
    IsBigOWith c l (fun x => f₁ x - f₂ x) g ↔ IsBigOWith c l (fun x => f₂ x - f₁ x) g 
Arguments: {α : Type u_1} {F : Type u_4} {E' : Type u_6} [Norm F] [SeminormedAddCommGroup E'] {c : ℝ} {g : α → F} {l : Filter α} {f₁ f₂ : α → E'}
Goal: Asymptotics.IsBigOWith c l (fun x => f₁ x - f₂ x) g ↔ Asymptotics.IsBigOWith c l (fun x => f₂ x - f₁ x) g
-- Asymptotics.IsBigO.symm
theorem IsBigO.symm (h : (fun x => f₁ x - f₂ x) =O[l] g) : (fun x => f₂ x - f₁ x) =O[l] g 
Arguments: {α : Type u_1} {F : Type u_4} {E' : Type u_6} [Norm F] [SeminormedAddCommGroup E'] {g : α → F} {l : Filter α} {f₁ f₂ : α → E'} (h : (fun x => f₁ x - f₂ x) =O[l] g)
Goal: (fun x => f₂ x - f₁ x) =O[l] g
-- Asymptotics.isBigO_comm
theorem isBigO_comm : (fun x => f₁ x - f₂ x) =O[l] g ↔ (fun x => f₂ x - f₁ x) =O[l] g 
Arguments: {α : Type u_1} {F : Type u_4} {E' : Type u_6} [Norm F] [SeminormedAddCommGroup E'] {g : α → F} {l : Filter α} {f₁ f₂ : α → E'}
Goal: (fun x => f₁ x - f₂ x) =O[l] g ↔ (fun x => f₂ x - f₁ x) =O[l] g
-- Asymptotics.IsLittleO.symm
theorem IsLittleO.symm (h : (fun x => f₁ x - f₂ x) =o[l] g) : (fun x => f₂ x - f₁ x) =o[l] g 
Arguments: {α : Type u_1} {F : Type u_4} {E' : Type u_6} [Norm F] [SeminormedAddCommGroup E'] {g : α → F} {l : Filter α} {f₁ f₂ : α → E'} (h : (fun x => f₁ x - f₂ x) =o[l] g)
Goal: (fun x => f₂ x - f₁ x) =o[l] g
-- Asymptotics.isLittleO_comm
theorem isLittleO_comm : (fun x => f₁ x - f₂ x) =o[l] g ↔ (fun x => f₂ x - f₁ x) =o[l] g 
Arguments: {α : Type u_1} {F : Type u_4} {E' : Type u_6} [Norm F] [SeminormedAddCommGroup E'] {g : α → F} {l : Filter α} {f₁ f₂ : α → E'}
Goal: (fun x => f₁ x - f₂ x) =o[l] g ↔ (fun x => f₂ x - f₁ x) =o[l] g
-- Asymptotics.IsBigOWith.triangle
theorem IsBigOWith.triangle (h₁ : IsBigOWith c l (fun x => f₁ x - f₂ x) g)
    (h₂ : IsBigOWith c' l (fun x => f₂ x - f₃ x) g) :
    IsBigOWith (c + c') l (fun x => f₁ x - f₃ x) g 
Arguments: {α : Type u_1} {F : Type u_4} {E' : Type u_6} [Norm F] [SeminormedAddCommGroup E'] {c c' : ℝ} {g : α → F} {l : Filter α} {f₁ f₂ f₃ : α → E'} (h₁ : Asymptotics.IsBigOWith c l (fun x => f₁ x - f₂ x) g) (h₂ : Asymptotics.IsBigOWith c' l (fun x => f₂ x - f₃ x) g)
Goal: Asymptotics.IsBigOWith (c + c') l (fun x => f₁ x - f₃ x) g
-- Asymptotics.IsBigO.triangle
theorem IsBigO.triangle (h₁ : (fun x => f₁ x - f₂ x) =O[l] g)
    (h₂ : (fun x => f₂ x - f₃ x) =O[l] g) : (fun x => f₁ x - f₃ x) =O[l] g 
Arguments: {α : Type u_1} {F : Type u_4} {E' : Type u_6} [Norm F] [SeminormedAddCommGroup E'] {g : α → F} {l : Filter α} {f₁ f₂ f₃ : α → E'} (h₁ : (fun x => f₁ x - f₂ x) =O[l] g) (h₂ : (fun x => f₂ x - f₃ x) =O[l] g)
Goal: (fun x => f₁ x - f₃ x) =O[l] g
-- Asymptotics.IsLittleO.triangle
theorem IsLittleO.triangle (h₁ : (fun x => f₁ x - f₂ x) =o[l] g)
    (h₂ : (fun x => f₂ x - f₃ x) =o[l] g) : (fun x => f₁ x - f₃ x) =o[l] g 
Arguments: {α : Type u_1} {F : Type u_4} {E' : Type u_6} [Norm F] [SeminormedAddCommGroup E'] {g : α → F} {l : Filter α} {f₁ f₂ f₃ : α → E'} (h₁ : (fun x => f₁ x - f₂ x) =o[l] g) (h₂ : (fun x => f₂ x - f₃ x) =o[l] g)
Goal: (fun x => f₁ x - f₃ x) =o[l] g
-- Asymptotics.IsBigO.congr_of_sub
theorem IsBigO.congr_of_sub (h : (fun x => f₁ x - f₂ x) =O[l] g) : f₁ =O[l] g ↔ f₂ =O[l] g 
Arguments: {α : Type u_1} {F : Type u_4} {E' : Type u_6} [Norm F] [SeminormedAddCommGroup E'] {g : α → F} {l : Filter α} {f₁ f₂ : α → E'} (h : (fun x => f₁ x - f₂ x) =O[l] g)
Goal: f₁ =O[l] g ↔ f₂ =O[l] g
-- Asymptotics.IsLittleO.congr_of_sub
theorem IsLittleO.congr_of_sub (h : (fun x => f₁ x - f₂ x) =o[l] g) : f₁ =o[l] g ↔ f₂ =o[l] g 
Arguments: {α : Type u_1} {F : Type u_4} {E' : Type u_6} [Norm F] [SeminormedAddCommGroup E'] {g : α → F} {l : Filter α} {f₁ f₂ : α → E'} (h : (fun x => f₁ x - f₂ x) =o[l] g)
Goal: f₁ =o[l] g ↔ f₂ =o[l] g
-- Asymptotics.isLittleO_zero
theorem isLittleO_zero : (fun _x => (0 : E')) =o[l] g' 
Arguments: {α : Type u_1} {E' : Type u_6} {F' : Type u_7} [SeminormedAddCommGroup E'] [SeminormedAddCommGroup F'] (g' : α → F') (l : Filter α)
Goal: (fun _x => 0) =o[l] g'
-- Asymptotics.isBigOWith_zero
theorem isBigOWith_zero (hc : 0 ≤ c) : IsBigOWith c l (fun _x => (0 : E')) g' 
Arguments: {α : Type u_1} {E' : Type u_6} {F' : Type u_7} [SeminormedAddCommGroup E'] [SeminormedAddCommGroup F'] {c : ℝ} (g' : α → F') (l : Filter α) (hc : 0 ≤ c)
Goal: Asymptotics.IsBigOWith c l (fun _x => 0) g'
-- Asymptotics.isBigOWith_zero'
theorem isBigOWith_zero' : IsBigOWith 0 l (fun _x => (0 : E')) g 
Arguments: {α : Type u_1} {F : Type u_4} {E' : Type u_6} [Norm F] [SeminormedAddCommGroup E'] (g : α → F) (l : Filter α)
Goal: Asymptotics.IsBigOWith 0 l (fun _x => 0) g
-- Asymptotics.isBigO_zero
theorem isBigO_zero : (fun _x => (0 : E')) =O[l] g 
Arguments: {α : Type u_1} {F : Type u_4} {E' : Type u_6} [Norm F] [SeminormedAddCommGroup E'] (g : α → F) (l : Filter α)
Goal: (fun _x => 0) =O[l] g
-- Asymptotics.isBigO_refl_left
theorem isBigO_refl_left : (fun x => f' x - f' x) =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 - f' x) =O[l] g'
-- Asymptotics.isLittleO_refl_left
theorem isLittleO_refl_left : (fun x => f' x - f' x) =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 - f' x) =o[l] g'
-- Asymptotics.isBigOWith_zero_right_iff
@[simp]
theorem isBigOWith_zero_right_iff : (IsBigOWith c l f'' fun _x => (0 : F')) ↔ f'' =ᶠ[l] 0 
Arguments: {α : Type u_1} {F' : Type u_7} {E'' : Type u_9} [SeminormedAddCommGroup F'] [NormedAddCommGroup E''] {c : ℝ} {f'' : α → E''} {l : Filter α}
Goal: (Asymptotics.IsBigOWith c l f'' fun _x => 0) ↔ f'' =ᶠ[l] 0
-- Asymptotics.isBigO_zero_right_iff
@[simp]
theorem isBigO_zero_right_iff : (f'' =O[l] fun _x => (0 : F')) ↔ f'' =ᶠ[l] 0 
Arguments: {α : Type u_1} {F' : Type u_7} {E'' : Type u_9} [SeminormedAddCommGroup F'] [NormedAddCommGroup E''] {f'' : α → E''} {l : Filter α}
Goal: (f'' =O[l] fun _x => 0) ↔ f'' =ᶠ[l] 0
-- Asymptotics.isLittleO_zero_right_iff
@[simp]
theorem isLittleO_zero_right_iff : (f'' =o[l] fun _x => (0 : F')) ↔ f'' =ᶠ[l] 0 
Arguments: {α : Type u_1} {F' : Type u_7} {E'' : Type u_9} [SeminormedAddCommGroup F'] [NormedAddCommGroup E''] {f'' : α → E''} {l : Filter α}
Goal: (f'' =o[l] fun _x => 0) ↔ f'' =ᶠ[l] 0
-- Asymptotics.isBigOWith_const_const
theorem isBigOWith_const_const (c : E) {c' : F''} (hc' : c' ≠ 0) (l : Filter α) :
    IsBigOWith (‖c‖ / ‖c'‖) l (fun _x : α => c) fun _x => c' 
Arguments: {α : Type u_1} {E : Type u_3} {F'' : Type u_10} [Norm E] [NormedAddCommGroup F''] (c : E) {c' : F''} (hc' : c' ≠ 0) (l : Filter α)
Goal: Asymptotics.IsBigOWith (‖c‖ / ‖c'‖) l (fun _x => c) fun _x => c'
-- Asymptotics.isBigO_const_const
theorem isBigO_const_const (c : E) {c' : F''} (hc' : c' ≠ 0) (l : Filter α) :
    (fun _x : α => c) =O[l] fun _x => c' 
Arguments: {α : Type u_1} {E : Type u_3} {F'' : Type u_10} [Norm E] [NormedAddCommGroup F''] (c : E) {c' : F''} (hc' : c' ≠ 0) (l : Filter α)
Goal: (fun _x => c) =O[l] fun _x => c'
-- Asymptotics.isBigO_const_const_iff
@[simp]
theorem isBigO_const_const_iff {c : E''} {c' : F''} (l : Filter α) [l.NeBot] :
    ((fun _x : α => c) =O[l] fun _x => c') ↔ c' = 0 → c = 0 
Arguments: {α : Type u_1} {E'' : Type u_9} {F'' : Type u_10} [NormedAddCommGroup E''] [NormedAddCommGroup F''] {c : E''} {c' : F''} (l : Filter α) [l.NeBot]
Goal: ((fun _x => c) =O[l] fun _x => c') ↔ c' = 0 → c = 0