ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Asymptotics.IsBigO.inv_rev
theorem IsBigO.inv_rev {f : α → 𝕜} {g : α → 𝕜'} (h : f =O[l] g)
    (h₀ : ∀ᶠ x in l, f x = 0 → g x = 0) : (fun x => (g x)⁻¹) =O[l] fun x => (f x)⁻¹ 
Arguments: {α : Type u_1} {𝕜 : Type u_15} {𝕜' : Type u_16} [NormedDivisionRing 𝕜] [NormedDivisionRing 𝕜'] {l : Filter α} {f : α → 𝕜} {g : α → 𝕜'} (h : f =O[l] g) (h₀ : ∀ᶠ (x : α) in l, f x = 0 → g x = 0)
Goal: (fun x => (g x)⁻¹) =O[l] fun x => (f x)⁻¹
-- Asymptotics.IsLittleO.inv_rev
theorem IsLittleO.inv_rev {f : α → 𝕜} {g : α → 𝕜'} (h : f =o[l] g)
    (h₀ : ∀ᶠ x in l, f x = 0 → g x = 0) : (fun x => (g x)⁻¹) =o[l] fun x => (f x)⁻¹ 
Arguments: {α : Type u_1} {𝕜 : Type u_15} {𝕜' : Type u_16} [NormedDivisionRing 𝕜] [NormedDivisionRing 𝕜'] {l : Filter α} {f : α → 𝕜} {g : α → 𝕜'} (h : f =o[l] g) (h₀ : ∀ᶠ (x : α) in l, f x = 0 → g x = 0)
Goal: (fun x => (g x)⁻¹) =o[l] fun x => (f x)⁻¹
-- Asymptotics.IsBigOWith.const_smul_self
theorem IsBigOWith.const_smul_self (c' : R) :
    IsBigOWith (‖c'‖) l (fun x => c' • f' x) f' 
Arguments: {α : Type u_1} {E' : Type u_6} {R : Type u_13} [SeminormedAddCommGroup E'] [SeminormedRing R] {f' : α → E'} {l : Filter α} [Module R E'] [BoundedSMul R E'] (c' : R)
Goal: Asymptotics.IsBigOWith ‖c'‖ l (fun x => c' • f' x) f'
-- Asymptotics.IsBigO.const_smul_self
theorem IsBigO.const_smul_self (c' : R) : (fun x => c' • f' x) =O[l] f' 
Arguments: {α : Type u_1} {E' : Type u_6} {R : Type u_13} [SeminormedAddCommGroup E'] [SeminormedRing R] {f' : α → E'} {l : Filter α} [Module R E'] [BoundedSMul R E'] (c' : R)
Goal: (fun x => c' • f' x) =O[l] f'
-- Asymptotics.IsBigOWith.const_smul_left
theorem IsBigOWith.const_smul_left (h : IsBigOWith c l f' g) (c' : R) :
    IsBigOWith (‖c'‖ * c) l (fun x => c' • f' x) g 
Arguments: {α : Type u_1} {F : Type u_4} {E' : Type u_6} {R : Type u_13} [Norm F] [SeminormedAddCommGroup E'] [SeminormedRing R] {c : ℝ} {g : α → F} {f' : α → E'} {l : Filter α} [Module R E'] [BoundedSMul R E'] (h : Asymptotics.IsBigOWith c l f' g) (c' : R)
Goal: Asymptotics.IsBigOWith (‖c'‖ * c) l (fun x => c' • f' x) g
-- Asymptotics.IsBigO.const_smul_left
theorem IsBigO.const_smul_left (h : f' =O[l] g) (c : R) : (c • f') =O[l] g 
Arguments: {α : Type u_1} {F : Type u_4} {E' : Type u_6} {R : Type u_13} [Norm F] [SeminormedAddCommGroup E'] [SeminormedRing R] {g : α → F} {f' : α → E'} {l : Filter α} [Module R E'] [BoundedSMul R E'] (h : f' =O[l] g) (c : R)
Goal: (c • f') =O[l] g
-- Asymptotics.IsLittleO.const_smul_left
theorem IsLittleO.const_smul_left (h : f' =o[l] g) (c : R) : (c • f') =o[l] g 
Arguments: {α : Type u_1} {F : Type u_4} {E' : Type u_6} {R : Type u_13} [Norm F] [SeminormedAddCommGroup E'] [SeminormedRing R] {g : α → F} {f' : α → E'} {l : Filter α} [Module R E'] [BoundedSMul R E'] (h : f' =o[l] g) (c : R)
Goal: (c • f') =o[l] g
-- Asymptotics.isBigO_const_smul_left
theorem isBigO_const_smul_left {c : 𝕜} (hc : c ≠ 0) : (fun x => c • f' x) =O[l] g ↔ f' =O[l] g 
Arguments: {α : Type u_1} {F : Type u_4} {E' : Type u_6} {𝕜 : Type u_15} [Norm F] [SeminormedAddCommGroup E'] [NormedDivisionRing 𝕜] {g : α → F} {f' : α → E'} {l : Filter α} [Module 𝕜 E'] [BoundedSMul 𝕜 E'] {c : 𝕜} (hc : c ≠ 0)
Goal: (fun x => c • f' x) =O[l] g ↔ f' =O[l] g
-- Asymptotics.isLittleO_const_smul_left
theorem isLittleO_const_smul_left {c : 𝕜} (hc : c ≠ 0) :
    (fun x => c • f' x) =o[l] g ↔ f' =o[l] g 
Arguments: {α : Type u_1} {F : Type u_4} {E' : Type u_6} {𝕜 : Type u_15} [Norm F] [SeminormedAddCommGroup E'] [NormedDivisionRing 𝕜] {g : α → F} {f' : α → E'} {l : Filter α} [Module 𝕜 E'] [BoundedSMul 𝕜 E'] {c : 𝕜} (hc : c ≠ 0)
Goal: (fun x => c • f' x) =o[l] g ↔ f' =o[l] g
-- Asymptotics.isBigO_const_smul_right
theorem isBigO_const_smul_right {c : 𝕜} (hc : c ≠ 0) :
    (f =O[l] fun x => c • f' x) ↔ f =O[l] f' 
Arguments: {α : Type u_1} {E : Type u_3} {E' : Type u_6} {𝕜 : Type u_15} [Norm E] [SeminormedAddCommGroup E'] [NormedDivisionRing 𝕜] {f : α → E} {f' : α → E'} {l : Filter α} [Module 𝕜 E'] [BoundedSMul 𝕜 E'] {c : 𝕜} (hc : c ≠ 0)
Goal: (f =O[l] fun x => c • f' x) ↔ f =O[l] f'
-- Asymptotics.isLittleO_const_smul_right
theorem isLittleO_const_smul_right {c : 𝕜} (hc : c ≠ 0) :
    (f =o[l] fun x => c • f' x) ↔ f =o[l] f' 
Arguments: {α : Type u_1} {E : Type u_3} {E' : Type u_6} {𝕜 : Type u_15} [Norm E] [SeminormedAddCommGroup E'] [NormedDivisionRing 𝕜] {f : α → E} {f' : α → E'} {l : Filter α} [Module 𝕜 E'] [BoundedSMul 𝕜 E'] {c : 𝕜} (hc : c ≠ 0)
Goal: (f =o[l] fun x => c • f' x) ↔ f =o[l] f'
-- Asymptotics.IsBigOWith.smul
theorem IsBigOWith.smul (h₁ : IsBigOWith c l k₁ k₂) (h₂ : IsBigOWith c' l f' g') :
    IsBigOWith (c * c') l (fun x => k₁ x • f' x) fun x => k₂ x • g' x 
Arguments: {α : Type u_1} {E' : Type u_6} {F' : Type u_7} {R : Type u_13} {𝕜' : Type u_16} [SeminormedAddCommGroup E'] [SeminormedAddCommGroup F'] [SeminormedRing R] [NormedDivisionRing 𝕜'] {c c' : ℝ} {f' : α → E'} {g' : α → F'} {l : Filter α} [Module R E'] [BoundedSMul R E'] [Module 𝕜' F'] [BoundedSMul 𝕜' F'] {k₁ : α → R} {k₂ : α → 𝕜'} (h₁ : Asymptotics.IsBigOWith c l k₁ k₂) (h₂ : Asymptotics.IsBigOWith c' l f' g')
Goal: Asymptotics.IsBigOWith (c * c') l (fun x => k₁ x • f' x) fun x => k₂ x • g' x
-- Asymptotics.IsBigO.smul
theorem IsBigO.smul (h₁ : k₁ =O[l] k₂) (h₂ : f' =O[l] g') :
    (fun x => k₁ x • f' x) =O[l] fun x => k₂ x • g' x 
Arguments: {α : Type u_1} {E' : Type u_6} {F' : Type u_7} {R : Type u_13} {𝕜' : Type u_16} [SeminormedAddCommGroup E'] [SeminormedAddCommGroup F'] [SeminormedRing R] [NormedDivisionRing 𝕜'] {f' : α → E'} {g' : α → F'} {l : Filter α} [Module R E'] [BoundedSMul R E'] [Module 𝕜' F'] [BoundedSMul 𝕜' F'] {k₁ : α → R} {k₂ : α → 𝕜'} (h₁ : k₁ =O[l] k₂) (h₂ : f' =O[l] g')
Goal: (fun x => k₁ x • f' x) =O[l] fun x => k₂ x • g' x
-- Asymptotics.IsBigO.smul_isLittleO
theorem IsBigO.smul_isLittleO (h₁ : k₁ =O[l] k₂) (h₂ : f' =o[l] g') :
    (fun x => k₁ x • f' x) =o[l] fun x => k₂ x • g' x 
Arguments: {α : Type u_1} {E' : Type u_6} {F' : Type u_7} {R : Type u_13} {𝕜' : Type u_16} [SeminormedAddCommGroup E'] [SeminormedAddCommGroup F'] [SeminormedRing R] [NormedDivisionRing 𝕜'] {f' : α → E'} {g' : α → F'} {l : Filter α} [Module R E'] [BoundedSMul R E'] [Module 𝕜' F'] [BoundedSMul 𝕜' F'] {k₁ : α → R} {k₂ : α → 𝕜'} (h₁ : k₁ =O[l] k₂) (h₂ : f' =o[l] g')
Goal: (fun x => k₁ x • f' x) =o[l] fun x => k₂ x • g' x
-- Asymptotics.IsLittleO.smul_isBigO
theorem IsLittleO.smul_isBigO (h₁ : k₁ =o[l] k₂) (h₂ : f' =O[l] g') :
    (fun x => k₁ x • f' x) =o[l] fun x => k₂ x • g' x 
Arguments: {α : Type u_1} {E' : Type u_6} {F' : Type u_7} {R : Type u_13} {𝕜' : Type u_16} [SeminormedAddCommGroup E'] [SeminormedAddCommGroup F'] [SeminormedRing R] [NormedDivisionRing 𝕜'] {f' : α → E'} {g' : α → F'} {l : Filter α} [Module R E'] [BoundedSMul R E'] [Module 𝕜' F'] [BoundedSMul 𝕜' F'] {k₁ : α → R} {k₂ : α → 𝕜'} (h₁ : k₁ =o[l] k₂) (h₂ : f' =O[l] g')
Goal: (fun x => k₁ x • f' x) =o[l] fun x => k₂ x • g' x
-- Asymptotics.IsLittleO.smul
theorem IsLittleO.smul (h₁ : k₁ =o[l] k₂) (h₂ : f' =o[l] g') :
    (fun x => k₁ x • f' x) =o[l] fun x => k₂ x • g' x 
Arguments: {α : Type u_1} {E' : Type u_6} {F' : Type u_7} {R : Type u_13} {𝕜' : Type u_16} [SeminormedAddCommGroup E'] [SeminormedAddCommGroup F'] [SeminormedRing R] [NormedDivisionRing 𝕜'] {f' : α → E'} {g' : α → F'} {l : Filter α} [Module R E'] [BoundedSMul R E'] [Module 𝕜' F'] [BoundedSMul 𝕜' F'] {k₁ : α → R} {k₂ : α → 𝕜'} (h₁ : k₁ =o[l] k₂) (h₂ : f' =o[l] g')
Goal: (fun x => k₁ x • f' x) =o[l] fun x => k₂ x • g' x
-- Asymptotics.IsBigOWith.sum
theorem IsBigOWith.sum (h : ∀ i ∈ s, IsBigOWith (C i) l (A i) g) :
    IsBigOWith (∑ i ∈ s, C i) l (fun x => ∑ i ∈ s, A i x) g 
Arguments: {α : Type u_1} {F : Type u_4} {E' : Type u_6} [Norm F] [SeminormedAddCommGroup E'] {g : α → F} {l : Filter α} {ι : Type u_17} {A : ι → α → E'} {C : ι → ℝ} {s : Finset ι} (h : ∀ i ∈ s, Asymptotics.IsBigOWith (C i) l (A i) g)
Goal: Asymptotics.IsBigOWith (∑ i ∈ s, C i) l (fun x => ∑ i ∈ s, A i x) g
-- Asymptotics.IsBigO.sum
theorem IsBigO.sum (h : ∀ i ∈ s, A i =O[l] g) : (fun x => ∑ i ∈ s, A i 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} {ι : Type u_17} {A : ι → α → E'} {s : Finset ι} (h : ∀ i ∈ s, A i =O[l] g)
Goal: (fun x => ∑ i ∈ s, A i x) =O[l] g
-- Asymptotics.IsLittleO.sum
theorem IsLittleO.sum (h : ∀ i ∈ s, A i =o[l] g') : (fun x => ∑ i ∈ s, A i x) =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} {ι : Type u_17} {A : ι → α → E'} {s : Finset ι} (h : ∀ i ∈ s, A i =o[l] g')
Goal: (fun x => ∑ i ∈ s, A i x) =o[l] g'
-- Asymptotics.IsLittleO.tendsto_div_nhds_zero
theorem IsLittleO.tendsto_div_nhds_zero {f g : α → 𝕜} (h : f =o[l] g) :
    Tendsto (fun x => f x / g x) l (𝓝 0) 
Arguments: {α : Type u_1} {𝕜 : Type u_15} [NormedDivisionRing 𝕜] {l : Filter α} {f g : α → 𝕜} (h : f =o[l] g)
Goal: Filter.Tendsto (fun x => f x / g x) l (𝓝 0)
-- Asymptotics.IsLittleO.tendsto_inv_smul_nhds_zero
theorem IsLittleO.tendsto_inv_smul_nhds_zero [Module 𝕜 E'] [BoundedSMul 𝕜 E']
    {f : α → E'} {g : α → 𝕜}
    {l : Filter α} (h : f =o[l] g) : Tendsto (fun x => (g x)⁻¹ • f x) l (𝓝 0) 
Arguments: {α : Type u_1} {E' : Type u_6} {𝕜 : Type u_15} [SeminormedAddCommGroup E'] [NormedDivisionRing 𝕜] [Module 𝕜 E'] [BoundedSMul 𝕜 E'] {f : α → E'} {g : α → 𝕜} {l : Filter α} (h : f =o[l] g)
Goal: Filter.Tendsto (fun x => (g x)⁻¹ • f x) l (𝓝 0)
-- Asymptotics.isLittleO_iff_tendsto'
theorem isLittleO_iff_tendsto' {f g : α → 𝕜} (hgf : ∀ᶠ x in l, g x = 0 → f x = 0) :
    f =o[l] g ↔ Tendsto (fun x => f x / g x) l (𝓝 0) 
Arguments: {α : Type u_1} {𝕜 : Type u_15} [NormedDivisionRing 𝕜] {l : Filter α} {f g : α → 𝕜} (hgf : ∀ᶠ (x : α) in l, g x = 0 → f x = 0)
Goal: f =o[l] g ↔ Filter.Tendsto (fun x => f x / g x) l (𝓝 0)
-- Asymptotics.isLittleO_iff_tendsto
theorem isLittleO_iff_tendsto {f g : α → 𝕜} (hgf : ∀ x, g x = 0 → f x = 0) :
    f =o[l] g ↔ Tendsto (fun x => f x / g x) l (𝓝 0) 
Arguments: {α : Type u_1} {𝕜 : Type u_15} [NormedDivisionRing 𝕜] {l : Filter α} {f g : α → 𝕜} (hgf : ∀ (x : α), g x = 0 → f x = 0)
Goal: f =o[l] g ↔ Filter.Tendsto (fun x => f x / g x) l (𝓝 0)
-- Asymptotics.isLittleO_const_left_of_ne
theorem isLittleO_const_left_of_ne {c : E''} (hc : c ≠ 0) :
    (fun _x => c) =o[l] g ↔ Tendsto (fun x => ‖g x‖) l atTop 
Arguments: {α : Type u_1} {F : Type u_4} {E'' : Type u_9} [Norm F] [NormedAddCommGroup E''] {g : α → F} {l : Filter α} {c : E''} (hc : c ≠ 0)
Goal: (fun _x => c) =o[l] g ↔ Filter.Tendsto (fun x => ‖g x‖) l Filter.atTop
-- Asymptotics.isLittleO_const_left
@[simp]
theorem isLittleO_const_left {c : E''} :
    (fun _x => c) =o[l] g'' ↔ c = 0 ∨ Tendsto (norm ∘ g'') l atTop 
Arguments: {α : Type u_1} {E'' : Type u_9} {F'' : Type u_10} [NormedAddCommGroup E''] [NormedAddCommGroup F''] {g'' : α → F''} {l : Filter α} {c : E''}
Goal: (fun _x => c) =o[l] g'' ↔ c = 0 ∨ Filter.Tendsto (norm ∘ g'') l Filter.atTop
-- Asymptotics.isLittleO_const_const_iff
@[simp 1001] -- Porting note: increase priority so that this triggers before `isLittleO_const_left`
theorem isLittleO_const_const_iff [NeBot l] {d : E''} {c : F''} :
    ((fun _x => d) =o[l] fun _x => c) ↔ d = 0 
Arguments: {α : Type u_1} {E'' : Type u_9} {F'' : Type u_10} [NormedAddCommGroup E''] [NormedAddCommGroup F''] {l : Filter α} [l.NeBot] {d : E''} {c : F''}
Goal: ((fun _x => d) =o[l] fun _x => c) ↔ d = 0
-- Asymptotics.isLittleO_pure
@[simp]
theorem isLittleO_pure {x} : f'' =o[pure x] g'' ↔ 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''} {x : α}
Goal: f'' =o[pure x] g'' ↔ f'' x = 0
-- Asymptotics.isLittleO_const_id_cobounded
theorem isLittleO_const_id_cobounded (c : F'') :
    (fun _ => c) =o[Bornology.cobounded E''] id 
Arguments: {E'' : Type u_9} {F'' : Type u_10} [NormedAddCommGroup E''] [NormedAddCommGroup F''] (c : F'')
Goal: (fun x => c) =o[Bornology.cobounded E''] id
-- Asymptotics.isLittleO_const_id_atTop
theorem isLittleO_const_id_atTop (c : E'') : (fun _x : ℝ => c) =o[atTop] id 
Arguments: {E'' : Type u_9} [NormedAddCommGroup E''] (c : E'')
Goal: (fun _x => c) =o[Filter.atTop] id
-- Asymptotics.isLittleO_const_id_atBot
theorem isLittleO_const_id_atBot (c : E'') : (fun _x : ℝ => c) =o[atBot] id 
Arguments: {E'' : Type u_9} [NormedAddCommGroup E''] (c : E'')
Goal: (fun _x => c) =o[Filter.atBot] id
-- Asymptotics.IsBigOWith.eventually_mul_div_cancel
theorem IsBigOWith.eventually_mul_div_cancel (h : IsBigOWith c l u v) : u / v * v =ᶠ[l] u 
Arguments: {α : Type u_1} {𝕜 : Type u_15} [NormedDivisionRing 𝕜] {c : ℝ} {l : Filter α} {u v : α → 𝕜} (h : Asymptotics.IsBigOWith c l u v)
Goal: u / v * v =ᶠ[l] u
-- Asymptotics.IsBigO.eventually_mul_div_cancel
theorem IsBigO.eventually_mul_div_cancel (h : u =O[l] v) : u / v * v =ᶠ[l] u 
Arguments: {α : Type u_1} {𝕜 : Type u_15} [NormedDivisionRing 𝕜] {l : Filter α} {u v : α → 𝕜} (h : u =O[l] v)
Goal: u / v * v =ᶠ[l] u
-- Asymptotics.IsLittleO.eventually_mul_div_cancel
theorem IsLittleO.eventually_mul_div_cancel (h : u =o[l] v) : u / v * v =ᶠ[l] u 
Arguments: {α : Type u_1} {𝕜 : Type u_15} [NormedDivisionRing 𝕜] {l : Filter α} {u v : α → 𝕜} (h : u =o[l] v)
Goal: u / v * v =ᶠ[l] u
-- Asymptotics.isBigOWith_of_eq_mul
theorem isBigOWith_of_eq_mul {u v : α → R} (φ : α → R) (hφ : ∀ᶠ x in l, ‖φ x‖ ≤ c)
    (h : u =ᶠ[l] φ * v) :
    IsBigOWith c l u v 
Arguments: {α : Type u_1} {R : Type u_13} [SeminormedRing R] {c : ℝ} {l : Filter α} {u v : α → R} (φ : α → R) (hφ : ∀ᶠ (x : α) in l, ‖φ x‖ ≤ c) (h : u =ᶠ[l] φ * v)
Goal: Asymptotics.IsBigOWith c l u v
-- Asymptotics.isBigOWith_iff_exists_eq_mul
theorem isBigOWith_iff_exists_eq_mul (hc : 0 ≤ c) :
    IsBigOWith c l u v ↔ ∃ φ : α → 𝕜, (∀ᶠ x in l, ‖φ x‖ ≤ c) ∧ u =ᶠ[l] φ * v 
Arguments: {α : Type u_1} {𝕜 : Type u_15} [NormedDivisionRing 𝕜] {c : ℝ} {l : Filter α} {u v : α → 𝕜} (hc : 0 ≤ c)
Goal: Asymptotics.IsBigOWith c l u v ↔ ∃ φ, (∀ᶠ (x : α) in l, ‖φ x‖ ≤ c) ∧ u =ᶠ[l] φ * v
-- Asymptotics.IsBigOWith.exists_eq_mul
theorem IsBigOWith.exists_eq_mul (h : IsBigOWith c l u v) (hc : 0 ≤ c) :
    ∃ φ : α → 𝕜, (∀ᶠ x in l, ‖φ x‖ ≤ c) ∧ u =ᶠ[l] φ * v 
Arguments: {α : Type u_1} {𝕜 : Type u_15} [NormedDivisionRing 𝕜] {c : ℝ} {l : Filter α} {u v : α → 𝕜} (h : Asymptotics.IsBigOWith c l u v) (hc : 0 ≤ c)
Goal: ∃ φ, (∀ᶠ (x : α) in l, ‖φ x‖ ≤ c) ∧ u =ᶠ[l] φ * v
-- Asymptotics.isBigO_iff_exists_eq_mul
theorem isBigO_iff_exists_eq_mul :
    u =O[l] v ↔ ∃ φ : α → 𝕜, l.IsBoundedUnder (· ≤ ·) (norm ∘ φ) ∧ u =ᶠ[l] φ * v 
Arguments: {α : Type u_1} {𝕜 : Type u_15} [NormedDivisionRing 𝕜] {l : Filter α} {u v : α → 𝕜}
Goal: u =O[l] v ↔ ∃ φ, Filter.IsBoundedUnder (fun x x_1 => x ≤ x_1) l (norm ∘ φ) ∧ u =ᶠ[l] φ * v
-- Asymptotics.isLittleO_iff_exists_eq_mul
theorem isLittleO_iff_exists_eq_mul :
    u =o[l] v ↔ ∃ φ : α → 𝕜, Tendsto φ l (𝓝 0) ∧ u =ᶠ[l] φ * v 
Arguments: {α : Type u_1} {𝕜 : Type u_15} [NormedDivisionRing 𝕜] {l : Filter α} {u v : α → 𝕜}
Goal: u =o[l] v ↔ ∃ φ, Filter.Tendsto φ l (𝓝 0) ∧ u =ᶠ[l] φ * v
-- Asymptotics.div_isBoundedUnder_of_isBigO
theorem div_isBoundedUnder_of_isBigO {α : Type*} {l : Filter α} {f g : α → 𝕜} (h : f =O[l] g) :
    IsBoundedUnder (· ≤ ·) l fun x => ‖f x / g x‖ 
Arguments: {𝕜 : Type u_15} [NormedDivisionRing 𝕜] {α : Type u_17} {l : Filter α} {f g : α → 𝕜} (h : f =O[l] g)
Goal: Filter.IsBoundedUnder (fun x x_1 => x ≤ x_1) l fun x => ‖f x / g x‖
-- Asymptotics.isBigO_iff_div_isBoundedUnder
theorem isBigO_iff_div_isBoundedUnder {α : Type*} {l : Filter α} {f g : α → 𝕜}
    (hgf : ∀ᶠ x in l, g x = 0 → f x = 0) :
    f =O[l] g ↔ IsBoundedUnder (· ≤ ·) l fun x => ‖f x / g x‖ 
Arguments: {𝕜 : Type u_15} [NormedDivisionRing 𝕜] {α : Type u_17} {l : Filter α} {f g : α → 𝕜} (hgf : ∀ᶠ (x : α) in l, g x = 0 → f x = 0)
Goal: f =O[l] g ↔ Filter.IsBoundedUnder (fun x x_1 => x ≤ x_1) l fun x => ‖f x / g x‖
-- Asymptotics.isBigO_of_div_tendsto_nhds
theorem isBigO_of_div_tendsto_nhds {α : Type*} {l : Filter α} {f g : α → 𝕜}
    (hgf : ∀ᶠ x in l, g x = 0 → f x = 0) (c : 𝕜) (H : Filter.Tendsto (f / g) l (𝓝 c)) :
    f =O[l] g 
Arguments: {𝕜 : Type u_15} [NormedDivisionRing 𝕜] {α : Type u_17} {l : Filter α} {f g : α → 𝕜} (hgf : ∀ᶠ (x : α) in l, g x = 0 → f x = 0) (c : 𝕜) (H : Filter.Tendsto (f / g) l (𝓝 c))
Goal: f =O[l] g
-- Asymptotics.IsLittleO.tendsto_zero_of_tendsto
theorem IsLittleO.tendsto_zero_of_tendsto {α E 𝕜 : Type*} [NormedAddCommGroup E] [NormedField 𝕜]
    {u : α → E} {v : α → 𝕜} {l : Filter α} {y : 𝕜} (huv : u =o[l] v) (hv : Tendsto v l (𝓝 y)) :
    Tendsto u l (𝓝 0) 
Arguments: {α : Type u_17} {E : Type u_18} {𝕜 : Type u_19} [NormedAddCommGroup E] [NormedField 𝕜] {u : α → E} {v : α → 𝕜} {l : Filter α} {y : 𝕜} (huv : u =o[l] v) (hv : Filter.Tendsto v l (𝓝 y))
Goal: Filter.Tendsto u l (𝓝 0)
-- Asymptotics.isLittleO_pow_pow
theorem isLittleO_pow_pow {m n : ℕ} (h : m < n) : (fun x : 𝕜 => x ^ n) =o[𝓝 0] fun x => x ^ m 
Arguments: {𝕜 : Type u_15} [NormedDivisionRing 𝕜] {m n : ℕ} (h : m < n)
Goal: (fun x => x ^ n) =o[𝓝 0] fun x => x ^ m
-- Asymptotics.isLittleO_norm_pow_norm_pow
theorem isLittleO_norm_pow_norm_pow {m n : ℕ} (h : m < n) :
    (fun x : E' => ‖x‖ ^ n) =o[𝓝 0] fun x => ‖x‖ ^ m 
Arguments: {E' : Type u_6} [SeminormedAddCommGroup E'] {m n : ℕ} (h : m < n)
Goal: (fun x => ‖x‖ ^ n) =o[𝓝 0] fun x => ‖x‖ ^ m
-- Asymptotics.isLittleO_pow_id
theorem isLittleO_pow_id {n : ℕ} (h : 1 < n) : (fun x : 𝕜 => x ^ n) =o[𝓝 0] fun x => x 
Arguments: {𝕜 : Type u_15} [NormedDivisionRing 𝕜] {n : ℕ} (h : 1 < n)
Goal: (fun x => x ^ n) =o[𝓝 0] fun x => x
-- Asymptotics.isLittleO_norm_pow_id
theorem isLittleO_norm_pow_id {n : ℕ} (h : 1 < n) :
    (fun x : E' => ‖x‖ ^ n) =o[𝓝 0] fun x => x 
Arguments: {E' : Type u_6} [SeminormedAddCommGroup E'] {n : ℕ} (h : 1 < n)
Goal: (fun x => ‖x‖ ^ n) =o[𝓝 0] fun x => x
-- Asymptotics.IsBigO.eq_zero_of_norm_pow_within
theorem IsBigO.eq_zero_of_norm_pow_within {f : E'' → F''} {s : Set E''} {x₀ : E''} {n : ℕ}
    (h : f =O[𝓝[s] x₀] fun x => ‖x - x₀‖ ^ n) (hx₀ : x₀ ∈ s) (hn : n ≠ 0) : f x₀ = 0 
Arguments: {E'' : Type u_9} {F'' : Type u_10} [NormedAddCommGroup E''] [NormedAddCommGroup F''] {f : E'' → F''} {s : Set E''} {x₀ : E''} {n : ℕ} (h : f =O[𝓝[s] x₀] fun x => ‖x - x₀‖ ^ n) (hx₀ : x₀ ∈ s) (hn : n ≠ 0)
Goal: f x₀ = 0
-- Asymptotics.IsBigO.eq_zero_of_norm_pow
theorem IsBigO.eq_zero_of_norm_pow {f : E'' → F''} {x₀ : E''} {n : ℕ}
    (h : f =O[𝓝 x₀] fun x => ‖x - x₀‖ ^ n) (hn : n ≠ 0) : f x₀ = 0 
Arguments: {E'' : Type u_9} {F'' : Type u_10} [NormedAddCommGroup E''] [NormedAddCommGroup F''] {f : E'' → F''} {x₀ : E''} {n : ℕ} (h : f =O[𝓝 x₀] fun x => ‖x - x₀‖ ^ n) (hn : n ≠ 0)
Goal: f x₀ = 0
-- Asymptotics.isLittleO_pow_sub_pow_sub
theorem isLittleO_pow_sub_pow_sub (x₀ : E') {n m : ℕ} (h : n < m) :
    (fun x => ‖x - x₀‖ ^ m) =o[𝓝 x₀] fun x => ‖x - x₀‖ ^ n 
Arguments: {E' : Type u_6} [SeminormedAddCommGroup E'] (x₀ : E') {n m : ℕ} (h : n < m)
Goal: (fun x => ‖x - x₀‖ ^ m) =o[𝓝 x₀] fun x => ‖x - x₀‖ ^ n
-- Asymptotics.isLittleO_pow_sub_sub
theorem isLittleO_pow_sub_sub (x₀ : E') {m : ℕ} (h : 1 < m) :
    (fun x => ‖x - x₀‖ ^ m) =o[𝓝 x₀] fun x => x - x₀ 
Arguments: {E' : Type u_6} [SeminormedAddCommGroup E'] (x₀ : E') {m : ℕ} (h : 1 < m)
Goal: (fun x => ‖x - x₀‖ ^ m) =o[𝓝 x₀] fun x => x - x₀
-- Asymptotics.IsBigOWith.right_le_sub_of_lt_one
theorem IsBigOWith.right_le_sub_of_lt_one {f₁ f₂ : α → E'} (h : IsBigOWith c l f₁ f₂) (hc : c < 1) :
    IsBigOWith (1 / (1 - c)) l f₂ fun x => f₂ x - f₁ x 
Arguments: {α : Type u_1} {E' : Type u_6} [SeminormedAddCommGroup E'] {c : ℝ} {l : Filter α} {f₁ f₂ : α → E'} (h : Asymptotics.IsBigOWith c l f₁ f₂) (hc : c < 1)
Goal: Asymptotics.IsBigOWith (1 / (1 - c)) l f₂ fun x => f₂ x - f₁ x
-- Asymptotics.IsBigOWith.right_le_add_of_lt_one
theorem IsBigOWith.right_le_add_of_lt_one {f₁ f₂ : α → E'} (h : IsBigOWith c l f₁ f₂) (hc : c < 1) :
    IsBigOWith (1 / (1 - c)) l f₂ fun x => f₁ x + f₂ x 
Arguments: {α : Type u_1} {E' : Type u_6} [SeminormedAddCommGroup E'] {c : ℝ} {l : Filter α} {f₁ f₂ : α → E'} (h : Asymptotics.IsBigOWith c l f₁ f₂) (hc : c < 1)
Goal: Asymptotics.IsBigOWith (1 / (1 - c)) l f₂ fun x => f₁ x + f₂ x
-- Asymptotics.IsLittleO.right_isBigO_sub
theorem IsLittleO.right_isBigO_sub {f₁ f₂ : α → E'} (h : f₁ =o[l] f₂) :
    f₂ =O[l] fun x => f₂ x - f₁ x 
Arguments: {α : Type u_1} {E' : Type u_6} [SeminormedAddCommGroup E'] {l : Filter α} {f₁ f₂ : α → E'} (h : f₁ =o[l] f₂)
Goal: f₂ =O[l] fun x => f₂ x - f₁ x
-- Asymptotics.IsLittleO.right_isBigO_add
theorem IsLittleO.right_isBigO_add {f₁ f₂ : α → E'} (h : f₁ =o[l] f₂) :
    f₂ =O[l] fun x => f₁ x + f₂ x 
Arguments: {α : Type u_1} {E' : Type u_6} [SeminormedAddCommGroup E'] {l : Filter α} {f₁ f₂ : α → E'} (h : f₁ =o[l] f₂)
Goal: f₂ =O[l] fun x => f₁ x + f₂ x
-- Asymptotics.IsLittleO.right_isBigO_add'
theorem IsLittleO.right_isBigO_add' {f₁ f₂ : α → E'} (h : f₁ =o[l] f₂) :
    f₂ =O[l] (f₂ + f₁) 
Arguments: {α : Type u_1} {E' : Type u_6} [SeminormedAddCommGroup E'] {l : Filter α} {f₁ f₂ : α → E'} (h : f₁ =o[l] f₂)
Goal: f₂ =O[l] (f₂ + f₁)
-- Asymptotics.bound_of_isBigO_cofinite
theorem bound_of_isBigO_cofinite (h : f =O[cofinite] g'') :
    ∃ C > 0, ∀ ⦃x⦄, g'' x ≠ 0 → ‖f x‖ ≤ C * ‖g'' x‖ 
Arguments: {α : Type u_1} {E : Type u_3} {F'' : Type u_10} [Norm E] [NormedAddCommGroup F''] {f : α → E} {g'' : α → F''} (h : f =O[Filter.cofinite] g'')
Goal: ∃ C > 0, ∀ ⦃x : α⦄, g'' x ≠ 0 → ‖f x‖ ≤ C * ‖g'' x‖
-- Asymptotics.isBigO_cofinite_iff
theorem isBigO_cofinite_iff (h : ∀ x, g'' x = 0 → f'' x = 0) :
    f'' =O[cofinite] g'' ↔ ∃ C, ∀ x, ‖f'' x‖ ≤ C * ‖g'' x‖ 
Arguments: {α : Type u_1} {E'' : Type u_9} {F'' : Type u_10} [NormedAddCommGroup E''] [NormedAddCommGroup F''] {f'' : α → E''} {g'' : α → F''} (h : ∀ (x : α), g'' x = 0 → f'' x = 0)
Goal: f'' =O[Filter.cofinite] g'' ↔ ∃ C, ∀ (x : α), ‖f'' x‖ ≤ C * ‖g'' x‖
-- Asymptotics.bound_of_isBigO_nat_atTop
theorem bound_of_isBigO_nat_atTop {f : ℕ → E} {g'' : ℕ → E''} (h : f =O[atTop] g'') :
    ∃ C > 0, ∀ ⦃x⦄, g'' x ≠ 0 → ‖f x‖ ≤ C * ‖g'' x‖ 
Arguments: {E : Type u_3} {E'' : Type u_9} [Norm E] [NormedAddCommGroup E''] {f : ℕ → E} {g'' : ℕ → E''} (h : f =O[Filter.atTop] g'')
Goal: ∃ C > 0, ∀ ⦃x : ℕ⦄, g'' x ≠ 0 → ‖f x‖ ≤ C * ‖g'' x‖
-- Asymptotics.isBigO_nat_atTop_iff
theorem isBigO_nat_atTop_iff {f : ℕ → E''} {g : ℕ → F''} (h : ∀ x, g x = 0 → f x = 0) :
    f =O[atTop] g ↔ ∃ C, ∀ x, ‖f x‖ ≤ C * ‖g x‖ 
Arguments: {E'' : Type u_9} {F'' : Type u_10} [NormedAddCommGroup E''] [NormedAddCommGroup F''] {f : ℕ → E''} {g : ℕ → F''} (h : ∀ (x : ℕ), g x = 0 → f x = 0)
Goal: f =O[Filter.atTop] g ↔ ∃ C, ∀ (x : ℕ), ‖f x‖ ≤ C * ‖g x‖
-- Asymptotics.isBigO_one_nat_atTop_iff
theorem isBigO_one_nat_atTop_iff {f : ℕ → E''} :
    f =O[atTop] (fun _n => 1 : ℕ → ℝ) ↔ ∃ C, ∀ n, ‖f n‖ ≤ C 
Arguments: {E'' : Type u_9} [NormedAddCommGroup E''] {f : ℕ → E''}
Goal: (f =O[Filter.atTop] fun _n => 1) ↔ ∃ C, ∀ (n : ℕ), ‖f n‖ ≤ C
-- Asymptotics.isBigOWith_pi
theorem isBigOWith_pi {ι : Type*} [Fintype ι] {E' : ι → Type*} [∀ i, NormedAddCommGroup (E' i)]
    {f : α → ∀ i, E' i} {C : ℝ} (hC : 0 ≤ C) :
    IsBigOWith C l f g' ↔ ∀ i, IsBigOWith C l (fun x => f x i) g' 
Arguments: {α : Type u_1} {F' : Type u_7} [SeminormedAddCommGroup F'] {g' : α → F'} {l : Filter α} {ι : Type u_17} [Fintype ι] {E' : ι → Type u_18} [(i : ι) → NormedAddCommGroup (E' i)] {f : α → (i : ι) → E' i} {C : ℝ} (hC : 0 ≤ C)
Goal: Asymptotics.IsBigOWith C l f g' ↔ ∀ (i : ι), Asymptotics.IsBigOWith C l (fun x => f x i) g'
-- Asymptotics.isBigO_pi
@[simp]
theorem isBigO_pi {ι : Type*} [Fintype ι] {E' : ι → Type*} [∀ i, NormedAddCommGroup (E' i)]
    {f : α → ∀ i, E' i} : f =O[l] g' ↔ ∀ i, (fun x => f x i) =O[l] g' 
Arguments: {α : Type u_1} {F' : Type u_7} [SeminormedAddCommGroup F'] {g' : α → F'} {l : Filter α} {ι : Type u_17} [Fintype ι] {E' : ι → Type u_18} [(i : ι) → NormedAddCommGroup (E' i)] {f : α → (i : ι) → E' i}
Goal: f =O[l] g' ↔ ∀ (i : ι), (fun x => f x i) =O[l] g'
-- Asymptotics.isLittleO_pi
@[simp]
theorem isLittleO_pi {ι : Type*} [Fintype ι] {E' : ι → Type*} [∀ i, NormedAddCommGroup (E' i)]
    {f : α → ∀ i, E' i} : f =o[l] g' ↔ ∀ i, (fun x => f x i) =o[l] g' 
Arguments: {α : Type u_1} {F' : Type u_7} [SeminormedAddCommGroup F'] {g' : α → F'} {l : Filter α} {ι : Type u_17} [Fintype ι] {E' : ι → Type u_18} [(i : ι) → NormedAddCommGroup (E' i)] {f : α → (i : ι) → E' i}
Goal: f =o[l] g' ↔ ∀ (i : ι), (fun x => f x i) =o[l] g'