ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Asymptotics.isTheta_norm_right
@[simp]
theorem isTheta_norm_right : (f =Θ[l] fun x ↦ ‖g' x‖) ↔ f =Θ[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 =Θ[l] fun x => ‖g' x‖) ↔ f =Θ[l] g'
-- Asymptotics.isTheta_of_norm_eventuallyEq
theorem isTheta_of_norm_eventuallyEq (h : (fun x ↦ ‖f x‖) =ᶠ[l] fun x ↦ ‖g x‖) : f =Θ[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} (h : (fun x => ‖f x‖) =ᶠ[l] fun x => ‖g x‖)
Goal: f =Θ[l] g
-- Asymptotics.isTheta_of_norm_eventuallyEq'
theorem isTheta_of_norm_eventuallyEq' {g : α → ℝ} (h : (fun x ↦ ‖f' x‖) =ᶠ[l] g) : f' =Θ[l] g 
Arguments: {α : Type u_1} {E' : Type u_6} [SeminormedAddCommGroup E'] {f' : α → E'} {l : Filter α} {g : α → ℝ} (h : (fun x => ‖f' x‖) =ᶠ[l] g)
Goal: f' =Θ[l] g
-- Asymptotics.IsTheta.isLittleO_congr_left
theorem IsTheta.isLittleO_congr_left (h : f' =Θ[l] g') : f' =o[l] k ↔ g' =o[l] k 
Arguments: {α : Type u_1} {G : Type u_5} {E' : Type u_6} {F' : Type u_7} [Norm G] [SeminormedAddCommGroup E'] [SeminormedAddCommGroup F'] {k : α → G} {f' : α → E'} {g' : α → F'} {l : Filter α} (h : f' =Θ[l] g')
Goal: f' =o[l] k ↔ g' =o[l] k
-- Asymptotics.IsTheta.isLittleO_congr_right
theorem IsTheta.isLittleO_congr_right (h : g' =Θ[l] k') : f =o[l] g' ↔ f =o[l] k' 
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 : g' =Θ[l] k')
Goal: f =o[l] g' ↔ f =o[l] k'
-- Asymptotics.IsTheta.isBigO_congr_left
theorem IsTheta.isBigO_congr_left (h : f' =Θ[l] g') : f' =O[l] k ↔ g' =O[l] k 
Arguments: {α : Type u_1} {G : Type u_5} {E' : Type u_6} {F' : Type u_7} [Norm G] [SeminormedAddCommGroup E'] [SeminormedAddCommGroup F'] {k : α → G} {f' : α → E'} {g' : α → F'} {l : Filter α} (h : f' =Θ[l] g')
Goal: f' =O[l] k ↔ g' =O[l] k
-- Asymptotics.IsTheta.isBigO_congr_right
theorem IsTheta.isBigO_congr_right (h : g' =Θ[l] k') : f =O[l] g' ↔ f =O[l] k' 
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 : g' =Θ[l] k')
Goal: f =O[l] g' ↔ f =O[l] k'
-- Asymptotics.IsTheta.isTheta_congr_left
lemma IsTheta.isTheta_congr_left (h : f' =Θ[l] g') : f' =Θ[l] k ↔ g' =Θ[l] k 
Arguments: {α : Type u_1} {G : Type u_5} {E' : Type u_6} {F' : Type u_7} [Norm G] [SeminormedAddCommGroup E'] [SeminormedAddCommGroup F'] {k : α → G} {f' : α → E'} {g' : α → F'} {l : Filter α} (h : f' =Θ[l] g')
Goal: f' =Θ[l] k ↔ g' =Θ[l] k
-- Asymptotics.IsTheta.isTheta_congr_right
lemma IsTheta.isTheta_congr_right (h : f' =Θ[l] g') : k =Θ[l] f' ↔ k =Θ[l] g' 
Arguments: {α : Type u_1} {G : Type u_5} {E' : Type u_6} {F' : Type u_7} [Norm G] [SeminormedAddCommGroup E'] [SeminormedAddCommGroup F'] {k : α → G} {f' : α → E'} {g' : α → F'} {l : Filter α} (h : f' =Θ[l] g')
Goal: k =Θ[l] f' ↔ k =Θ[l] g'
-- Asymptotics.IsTheta.mono
theorem IsTheta.mono (h : f =Θ[l] g) (hl : l' ≤ l) : f =Θ[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 =Θ[l] g) (hl : l' ≤ l)
Goal: f =Θ[l'] g
-- Asymptotics.IsTheta.sup
theorem IsTheta.sup (h : f' =Θ[l] g') (h' : f' =Θ[l'] g') : f' =Θ[l ⊔ 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 l' : Filter α} (h : f' =Θ[l] g') (h' : f' =Θ[l'] g')
Goal: f' =Θ[l ⊔ l'] g'
-- Asymptotics.isTheta_sup
@[simp]
theorem isTheta_sup : f' =Θ[l ⊔ l'] g' ↔ f' =Θ[l] g' ∧ f' =Θ[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 l' : Filter α}
Goal: f' =Θ[l ⊔ l'] g' ↔ f' =Θ[l] g' ∧ f' =Θ[l'] g'
-- Asymptotics.IsTheta.eq_zero_iff
theorem IsTheta.eq_zero_iff (h : f'' =Θ[l] g'') : ∀ᶠ x in l, f'' x = 0 ↔ g'' 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'' =Θ[l] g'')
Goal: ∀ᶠ (x : α) in l, f'' x = 0 ↔ g'' x = 0
-- Asymptotics.IsTheta.tendsto_zero_iff
theorem IsTheta.tendsto_zero_iff (h : f'' =Θ[l] g'') :
    Tendsto f'' l (𝓝 0) ↔ Tendsto g'' l (𝓝 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'' =Θ[l] g'')
Goal: Filter.Tendsto f'' l (𝓝 0) ↔ Filter.Tendsto g'' l (𝓝 0)
-- Asymptotics.IsTheta.tendsto_norm_atTop_iff
theorem IsTheta.tendsto_norm_atTop_iff (h : f' =Θ[l] g') :
    Tendsto (norm ∘ f') l atTop ↔ Tendsto (norm ∘ g') l atTop 
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} (h : f' =Θ[l] g')
Goal: Filter.Tendsto (norm ∘ f') l Filter.atTop ↔ Filter.Tendsto (norm ∘ g') l Filter.atTop
-- Asymptotics.IsTheta.isBoundedUnder_le_iff
theorem IsTheta.isBoundedUnder_le_iff (h : f' =Θ[l] g') :
    IsBoundedUnder (· ≤ ·) l (norm ∘ f') ↔ IsBoundedUnder (· ≤ ·) l (norm ∘ 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} (h : f' =Θ[l] g')
Goal: Filter.IsBoundedUnder (fun x x_1 => x ≤ x_1) l (norm ∘ f') ↔ Filter.IsBoundedUnder (fun x x_1 => x ≤ x_1) l (norm ∘ g')
-- Asymptotics.IsTheta.smul
theorem IsTheta.smul [NormedSpace 𝕜 E'] [NormedSpace 𝕜' F'] {f₁ : α → 𝕜} {f₂ : α → 𝕜'} {g₁ : α → E'}
    {g₂ : α → F'} (hf : f₁ =Θ[l] f₂) (hg : g₁ =Θ[l] g₂) :
    (fun x ↦ f₁ x • g₁ x) =Θ[l] fun x ↦ f₂ x • g₂ x 
Arguments: {α : Type u_1} {E' : Type u_6} {F' : Type u_7} {𝕜 : Type u_14} {𝕜' : Type u_15} [SeminormedAddCommGroup E'] [SeminormedAddCommGroup F'] [NormedField 𝕜] [NormedField 𝕜'] {l : Filter α} [NormedSpace 𝕜 E'] [NormedSpace 𝕜' F'] {f₁ : α → 𝕜} {f₂ : α → 𝕜'} {g₁ : α → E'} {g₂ : α → F'} (hf : f₁ =Θ[l] f₂) (hg : g₁ =Θ[l] g₂)
Goal: (fun x => f₁ x • g₁ x) =Θ[l] fun x => f₂ x • g₂ x
-- Asymptotics.IsTheta.mul
theorem IsTheta.mul {f₁ f₂ : α → 𝕜} {g₁ g₂ : α → 𝕜'} (h₁ : f₁ =Θ[l] g₁) (h₂ : f₂ =Θ[l] g₂) :
    (fun x ↦ f₁ x * f₂ x) =Θ[l] fun x ↦ g₁ x * g₂ x 
Arguments: {α : Type u_1} {𝕜 : Type u_14} {𝕜' : Type u_15} [NormedField 𝕜] [NormedField 𝕜'] {l : Filter α} {f₁ f₂ : α → 𝕜} {g₁ g₂ : α → 𝕜'} (h₁ : f₁ =Θ[l] g₁) (h₂ : f₂ =Θ[l] g₂)
Goal: (fun x => f₁ x * f₂ x) =Θ[l] fun x => g₁ x * g₂ x
-- Asymptotics.IsTheta.inv
theorem IsTheta.inv {f : α → 𝕜} {g : α → 𝕜'} (h : f =Θ[l] g) :
    (fun x ↦ (f x)⁻¹) =Θ[l] fun x ↦ (g x)⁻¹ 
Arguments: {α : Type u_1} {𝕜 : Type u_14} {𝕜' : Type u_15} [NormedField 𝕜] [NormedField 𝕜'] {l : Filter α} {f : α → 𝕜} {g : α → 𝕜'} (h : f =Θ[l] g)
Goal: (fun x => (f x)⁻¹) =Θ[l] fun x => (g x)⁻¹
-- Asymptotics.isTheta_inv
@[simp]
theorem isTheta_inv {f : α → 𝕜} {g : α → 𝕜'} :
    ((fun x ↦ (f x)⁻¹) =Θ[l] fun x ↦ (g x)⁻¹) ↔ f =Θ[l] g 
Arguments: {α : Type u_1} {𝕜 : Type u_14} {𝕜' : Type u_15} [NormedField 𝕜] [NormedField 𝕜'] {l : Filter α} {f : α → 𝕜} {g : α → 𝕜'}
Goal: ((fun x => (f x)⁻¹) =Θ[l] fun x => (g x)⁻¹) ↔ f =Θ[l] g
-- Asymptotics.IsTheta.div
theorem IsTheta.div {f₁ f₂ : α → 𝕜} {g₁ g₂ : α → 𝕜'} (h₁ : f₁ =Θ[l] g₁) (h₂ : f₂ =Θ[l] g₂) :
    (fun x ↦ f₁ x / f₂ x) =Θ[l] fun x ↦ g₁ x / g₂ x 
Arguments: {α : Type u_1} {𝕜 : Type u_14} {𝕜' : Type u_15} [NormedField 𝕜] [NormedField 𝕜'] {l : Filter α} {f₁ f₂ : α → 𝕜} {g₁ g₂ : α → 𝕜'} (h₁ : f₁ =Θ[l] g₁) (h₂ : f₂ =Θ[l] g₂)
Goal: (fun x => f₁ x / f₂ x) =Θ[l] fun x => g₁ x / g₂ x
-- Asymptotics.IsTheta.pow
theorem IsTheta.pow {f : α → 𝕜} {g : α → 𝕜'} (h : f =Θ[l] g) (n : ℕ) :
    (fun x ↦ f x ^ n) =Θ[l] fun x ↦ g x ^ n 
Arguments: {α : Type u_1} {𝕜 : Type u_14} {𝕜' : Type u_15} [NormedField 𝕜] [NormedField 𝕜'] {l : Filter α} {f : α → 𝕜} {g : α → 𝕜'} (h : f =Θ[l] g) (n : ℕ)
Goal: (fun x => f x ^ n) =Θ[l] fun x => g x ^ n
-- Asymptotics.IsTheta.zpow
theorem IsTheta.zpow {f : α → 𝕜} {g : α → 𝕜'} (h : f =Θ[l] g) (n : ℤ) :
    (fun x ↦ f x ^ n) =Θ[l] fun x ↦ g x ^ n 
Arguments: {α : Type u_1} {𝕜 : Type u_14} {𝕜' : Type u_15} [NormedField 𝕜] [NormedField 𝕜'] {l : Filter α} {f : α → 𝕜} {g : α → 𝕜'} (h : f =Θ[l] g) (n : ℤ)
Goal: (fun x => f x ^ n) =Θ[l] fun x => g x ^ n
-- Asymptotics.isTheta_const_const
theorem isTheta_const_const {c₁ : E''} {c₂ : F''} (h₁ : c₁ ≠ 0) (h₂ : c₂ ≠ 0) :
    (fun _ : α ↦ c₁) =Θ[l] fun _ ↦ c₂ 
Arguments: {α : Type u_1} {E'' : Type u_9} {F'' : Type u_10} [NormedAddCommGroup E''] [NormedAddCommGroup F''] {l : Filter α} {c₁ : E''} {c₂ : F''} (h₁ : c₁ ≠ 0) (h₂ : c₂ ≠ 0)
Goal: (fun x => c₁) =Θ[l] fun x => c₂
-- Asymptotics.isTheta_const_const_iff
@[simp]
theorem isTheta_const_const_iff [NeBot l] {c₁ : E''} {c₂ : F''} :
    ((fun _ : α ↦ c₁) =Θ[l] fun _ ↦ c₂) ↔ (c₁ = 0 ↔ c₂ = 0) 
Arguments: {α : Type u_1} {E'' : Type u_9} {F'' : Type u_10} [NormedAddCommGroup E''] [NormedAddCommGroup F''] {l : Filter α} [l.NeBot] {c₁ : E''} {c₂ : F''}
Goal: ((fun x => c₁) =Θ[l] fun x => c₂) ↔ (c₁ = 0 ↔ c₂ = 0)
-- Asymptotics.isTheta_zero_left
@[simp]
theorem isTheta_zero_left : (fun _ ↦ (0 : E')) =Θ[l] g'' ↔ g'' =ᶠ[l] 0 
Arguments: {α : Type u_1} {E' : Type u_6} {F'' : Type u_10} [SeminormedAddCommGroup E'] [NormedAddCommGroup F''] {g'' : α → F''} {l : Filter α}
Goal: (fun x => 0) =Θ[l] g'' ↔ g'' =ᶠ[l] 0
-- Asymptotics.isTheta_zero_right
@[simp]
theorem isTheta_zero_right : (f'' =Θ[l] fun _ ↦ (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'' =Θ[l] fun x => 0) ↔ f'' =ᶠ[l] 0
-- Asymptotics.isTheta_const_smul_left
theorem isTheta_const_smul_left [NormedSpace 𝕜 E'] {c : 𝕜} (hc : c ≠ 0) :
    (fun x ↦ c • f' x) =Θ[l] g ↔ f' =Θ[l] g 
Arguments: {α : Type u_1} {F : Type u_4} {E' : Type u_6} {𝕜 : Type u_14} [Norm F] [SeminormedAddCommGroup E'] [NormedField 𝕜] {g : α → F} {f' : α → E'} {l : Filter α} [NormedSpace 𝕜 E'] {c : 𝕜} (hc : c ≠ 0)
Goal: (fun x => c • f' x) =Θ[l] g ↔ f' =Θ[l] g
-- Asymptotics.isTheta_const_smul_right
theorem isTheta_const_smul_right [NormedSpace 𝕜 F'] {c : 𝕜} (hc : c ≠ 0) :
    (f =Θ[l] fun x ↦ c • g' x) ↔ f =Θ[l] g' 
Arguments: {α : Type u_1} {E : Type u_3} {F' : Type u_7} {𝕜 : Type u_14} [Norm E] [SeminormedAddCommGroup F'] [NormedField 𝕜] {f : α → E} {g' : α → F'} {l : Filter α} [NormedSpace 𝕜 F'] {c : 𝕜} (hc : c ≠ 0)
Goal: (f =Θ[l] fun x => c • g' x) ↔ f =Θ[l] g'
-- Asymptotics.isTheta_const_mul_left
theorem isTheta_const_mul_left {c : 𝕜} {f : α → 𝕜} (hc : c ≠ 0) :
    (fun x ↦ c * f x) =Θ[l] g ↔ f =Θ[l] g 
Arguments: {α : Type u_1} {F : Type u_4} {𝕜 : Type u_14} [Norm F] [NormedField 𝕜] {g : α → F} {l : Filter α} {c : 𝕜} {f : α → 𝕜} (hc : c ≠ 0)
Goal: (fun x => c * f x) =Θ[l] g ↔ f =Θ[l] g
-- Asymptotics.isTheta_const_mul_right
theorem isTheta_const_mul_right {c : 𝕜} {g : α → 𝕜} (hc : c ≠ 0) :
    (f =Θ[l] fun x ↦ c * g x) ↔ f =Θ[l] g 
Arguments: {α : Type u_1} {E : Type u_3} {𝕜 : Type u_14} [Norm E] [NormedField 𝕜] {f : α → E} {l : Filter α} {c : 𝕜} {g : α → 𝕜} (hc : c ≠ 0)
Goal: (f =Θ[l] fun x => c * g x) ↔ f =Θ[l] g
-- Asymptotics.IsLittleO.right_isTheta_add
theorem IsLittleO.right_isTheta_add {f₁ f₂ : α → E'} (h : f₁ =o[l] f₂) :
    f₂ =Θ[l] (f₁ + f₂) 
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 =Θ[l] (f₁ + f₂)
-- Asymptotics.IsLittleO.right_isTheta_add'
theorem IsLittleO.right_isTheta_add' {f₁ f₂ : α → E'} (h : f₁ =o[l] f₂) :
    f₂ =Θ[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 =Θ[l] (f₂ + f₁)
-- Asymptotics.IsTheta.add_isLittleO
lemma IsTheta.add_isLittleO {f₁ f₂ : α → E'} {g : α → F}
    (hΘ : f₁ =Θ[l] g) (ho : f₂ =o[l] g) : (f₁ + f₂) =Θ[l] g 
Arguments: {α : Type u_1} {F : Type u_4} {E' : Type u_6} [Norm F] [SeminormedAddCommGroup E'] {l : Filter α} {f₁ f₂ : α → E'} {g : α → F} (hΘ : f₁ =Θ[l] g) (ho : f₂ =o[l] g)
Goal: (f₁ + f₂) =Θ[l] g
-- Asymptotics.IsLittleO.add_isTheta
lemma IsLittleO.add_isTheta {f₁ f₂ : α → E'} {g : α → F}
    (ho : f₁ =o[l] g) (hΘ : f₂ =Θ[l] g) : (f₁ + f₂) =Θ[l] g 
Arguments: {α : Type u_1} {F : Type u_4} {E' : Type u_6} [Norm F] [SeminormedAddCommGroup E'] {l : Filter α} {f₁ f₂ : α → E'} {g : α → F} (ho : f₁ =o[l] g) (hΘ : f₂ =Θ[l] g)
Goal: (f₁ + f₂) =Θ[l] g
-- Filter.IsBoundedUnder.isLittleO_sub_self_inv
theorem Filter.IsBoundedUnder.isLittleO_sub_self_inv {𝕜 E : Type*} [NormedField 𝕜] [Norm E] {a : 𝕜}
    {f : 𝕜 → E} (h : IsBoundedUnder (· ≤ ·) (𝓝[≠] a) (norm ∘ f)) :
    f =o[𝓝[≠] a] fun x => (x - a)⁻¹ 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [NormedField 𝕜] [Norm E] {a : 𝕜} {f : 𝕜 → E} (h : Filter.IsBoundedUnder (fun x x_1 => x ≤ x_1) (𝓝[≠] a) (norm ∘ f))
Goal: f =o[𝓝[≠] a] fun x => (x - a)⁻¹
-- pow_div_pow_eventuallyEq_atTop
theorem pow_div_pow_eventuallyEq_atTop {p q : ℕ} :
    (fun x : 𝕜 => x ^ p / x ^ q) =ᶠ[atTop] fun x => x ^ ((p : ℤ) - q) 
Arguments: {𝕜 : Type u_1} [LinearOrderedField 𝕜] {p q : ℕ}
Goal: (fun x => x ^ p / x ^ q) =ᶠ[Filter.atTop] fun x => x ^ (↑p - ↑q)
-- pow_div_pow_eventuallyEq_atBot
theorem pow_div_pow_eventuallyEq_atBot {p q : ℕ} :
    (fun x : 𝕜 => x ^ p / x ^ q) =ᶠ[atBot] fun x => x ^ ((p : ℤ) - q) 
Arguments: {𝕜 : Type u_1} [LinearOrderedField 𝕜] {p q : ℕ}
Goal: (fun x => x ^ p / x ^ q) =ᶠ[Filter.atBot] fun x => x ^ (↑p - ↑q)
-- tendsto_pow_div_pow_atTop_atTop
theorem tendsto_pow_div_pow_atTop_atTop {p q : ℕ} (hpq : q < p) :
    Tendsto (fun x : 𝕜 => x ^ p / x ^ q) atTop atTop 
Arguments: {𝕜 : Type u_1} [LinearOrderedField 𝕜] {p q : ℕ} (hpq : q < p)
Goal: Filter.Tendsto (fun x => x ^ p / x ^ q) Filter.atTop Filter.atTop
-- tendsto_pow_div_pow_atTop_zero
theorem tendsto_pow_div_pow_atTop_zero [TopologicalSpace 𝕜] [OrderTopology 𝕜] {p q : ℕ}
    (hpq : p < q) : Tendsto (fun x : 𝕜 => x ^ p / x ^ q) atTop (𝓝 0) 
Arguments: {𝕜 : Type u_1} [LinearOrderedField 𝕜] [TopologicalSpace 𝕜] [OrderTopology 𝕜] {p q : ℕ} (hpq : p < q)
Goal: Filter.Tendsto (fun x => x ^ p / x ^ q) Filter.atTop (𝓝 0)
-- Asymptotics.isLittleO_pow_pow_atTop_of_lt
theorem Asymptotics.isLittleO_pow_pow_atTop_of_lt [OrderTopology 𝕜] {p q : ℕ} (hpq : p < q) :
    (fun x : 𝕜 => x ^ p) =o[atTop] fun x => x ^ q 
Arguments: {𝕜 : Type u_1} [NormedLinearOrderedField 𝕜] [OrderTopology 𝕜] {p q : ℕ} (hpq : p < q)
Goal: (fun x => x ^ p) =o[Filter.atTop] fun x => x ^ q
-- Asymptotics.IsBigO.trans_tendsto_norm_atTop
theorem Asymptotics.IsBigO.trans_tendsto_norm_atTop {α : Type*} {u v : α → 𝕜} {l : Filter α}
    (huv : u =O[l] v) (hu : Tendsto (fun x => ‖u x‖) l atTop) :
    Tendsto (fun x => ‖v x‖) l atTop 
Arguments: {𝕜 : Type u_1} [NormedLinearOrderedField 𝕜] {α : Type u_2} {u v : α → 𝕜} {l : Filter α} (huv : u =O[l] v) (hu : Filter.Tendsto (fun x => ‖u x‖) l Filter.atTop)
Goal: Filter.Tendsto (fun x => ‖v x‖) l Filter.atTop
-- Asymptotics.IsLittleO.sum_range
theorem Asymptotics.IsLittleO.sum_range {α : Type*} [NormedAddCommGroup α] {f : ℕ → α} {g : ℕ → ℝ}
    (h : f =o[atTop] g) (hg : 0 ≤ g) (h'g : Tendsto (fun n => ∑ i ∈ range n, g i) atTop atTop) :
    (fun n => ∑ i ∈ range n, f i) =o[atTop] fun n => ∑ i ∈ range n, g i 
Arguments: {α : Type u_1} [NormedAddCommGroup α] {f : ℕ → α} {g : ℕ → ℝ} (h : f =o[Filter.atTop] g) (hg : 0 ≤ g) (h'g : Filter.Tendsto (fun n => ∑ i ∈ Finset.range n, g i) Filter.atTop Filter.atTop)
Goal: (fun n => ∑ i ∈ Finset.range n, f i) =o[Filter.atTop] fun n => ∑ i ∈ Finset.range n, g i
-- Asymptotics.isLittleO_sum_range_of_tendsto_zero
theorem Asymptotics.isLittleO_sum_range_of_tendsto_zero {α : Type*} [NormedAddCommGroup α]
    {f : ℕ → α} (h : Tendsto f atTop (𝓝 0)) :
    (fun n => ∑ i ∈ range n, f i) =o[atTop] fun n => (n : ℝ) 
Arguments: {α : Type u_1} [NormedAddCommGroup α] {f : ℕ → α} (h : Filter.Tendsto f Filter.atTop (𝓝 0))
Goal: (fun n => ∑ i ∈ Finset.range n, f i) =o[Filter.atTop] fun n => ↑n
-- Filter.Tendsto.cesaro_smul
theorem Filter.Tendsto.cesaro_smul {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] {u : ℕ → E}
    {l : E} (h : Tendsto u atTop (𝓝 l)) :
    Tendsto (fun n : ℕ => (n⁻¹ : ℝ) • ∑ i ∈ range n, u i) atTop (𝓝 l) 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {u : ℕ → E} {l : E} (h : Filter.Tendsto u Filter.atTop (𝓝 l))
Goal: Filter.Tendsto (fun n => (↑n)⁻¹ • ∑ i ∈ Finset.range n, u i) Filter.atTop (𝓝 l)
-- Filter.Tendsto.cesaro
theorem Filter.Tendsto.cesaro {u : ℕ → ℝ} {l : ℝ} (h : Tendsto u atTop (𝓝 l)) :
    Tendsto (fun n : ℕ => (n⁻¹ : ℝ) * ∑ i ∈ range n, u i) atTop (𝓝 l) 
Arguments: {u : ℕ → ℝ} {l : ℝ} (h : Filter.Tendsto u Filter.atTop (𝓝 l))
Goal: Filter.Tendsto (fun n => (↑n)⁻¹ * ∑ i ∈ Finset.range n, u i) Filter.atTop (𝓝 l)
-- Real.tendsto_integral_cexp_sq_smul
lemma tendsto_integral_cexp_sq_smul (hf : Integrable f) :
    Tendsto (fun (c : ℝ) ↦ (∫ v : V, cexp (- c⁻¹ * ‖v‖^2) • f v))
      atTop (𝓝 (∫ v : V, f v)) 
Arguments: {V : Type u_1} {E : Type u_2} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [MeasurableSpace V] [BorelSpace V] [FiniteDimensional ℝ V] [NormedAddCommGroup E] [NormedSpace ℂ E] {f : V → E} (hf : MeasureTheory.Integrable f MeasureTheory.volume)
Goal: Filter.Tendsto (fun c => ∫ (v : V), cexp (-↑c⁻¹ * ↑‖v‖ ^ 2) • f v) Filter.atTop (𝓝 (∫ (v : V), f v))
-- Real.tendsto_integral_gaussian_smul
lemma tendsto_integral_gaussian_smul (hf : Integrable f) (h'f : Integrable (𝓕 f)) (v : V) :
    Tendsto (fun (c : ℝ) ↦
      ∫ w : V, ((π * c) ^ (finrank ℝ V / 2 : ℂ) * cexp (-π ^ 2 * c * ‖v - w‖ ^ 2)) • f w)
    atTop (𝓝 (𝓕⁻ (𝓕 f) v)) 
Arguments: {V : Type u_1} {E : Type u_2} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [MeasurableSpace V] [BorelSpace V] [FiniteDimensional ℝ V] [NormedAddCommGroup E] [NormedSpace ℂ E] {f : V → E} [CompleteSpace E] (hf : MeasureTheory.Integrable f MeasureTheory.volume) (h'f : MeasureTheory.Integrable (𝓕 f) MeasureTheory.volume) (v : V)
Goal: Filter.Tendsto
  (fun c => ∫ (w : V), ((↑π * ↑c) ^ (↑(FiniteDimensional.finrank ℝ V) / 2) * cexp (-↑π ^ 2 * ↑c * ↑‖v - w‖ ^ 2)) • f w)
  Filter.atTop (𝓝 (𝓕⁻ (𝓕 f) v))
-- Real.tendsto_integral_gaussian_smul'
lemma tendsto_integral_gaussian_smul' (hf : Integrable f) {v : V} (h'f : ContinuousAt f v) :
    Tendsto (fun (c : ℝ) ↦
      ∫ w : V, ((π * c : ℂ) ^ (finrank ℝ V / 2 : ℂ) * cexp (-π ^ 2 * c * ‖v - w‖ ^ 2)) • f w)
    atTop (𝓝 (f v)) 
Arguments: {V : Type u_1} {E : Type u_2} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [MeasurableSpace V] [BorelSpace V] [FiniteDimensional ℝ V] [NormedAddCommGroup E] [NormedSpace ℂ E] {f : V → E} [CompleteSpace E] (hf : MeasureTheory.Integrable f MeasureTheory.volume) {v : V} (h'f : ContinuousAt f v)
Goal: Filter.Tendsto
  (fun c => ∫ (w : V), ((↑π * ↑c) ^ (↑(FiniteDimensional.finrank ℝ V) / 2) * cexp (-↑π ^ 2 * ↑c * ↑‖v - w‖ ^ 2)) • f w)
  Filter.atTop (𝓝 (f v))
-- MeasureTheory.Integrable.fourier_inversion
theorem MeasureTheory.Integrable.fourier_inversion
    (hf : Integrable f) (h'f : Integrable (𝓕 f)) {v : V}
    (hv : ContinuousAt f v) : 𝓕⁻ (𝓕 f) v = f v 
Arguments: {V : Type u_1} {E : Type u_2} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [MeasurableSpace V] [BorelSpace V] [FiniteDimensional ℝ V] [NormedAddCommGroup E] [NormedSpace ℂ E] {f : V → E} [CompleteSpace E] (hf : MeasureTheory.Integrable f MeasureTheory.volume) (h'f : MeasureTheory.Integrable (𝓕 f) MeasureTheory.volume) {v : V} (hv : ContinuousAt f v)
Goal: 𝓕⁻ (𝓕 f) v = f v
-- Continuous.fourier_inversion
theorem Continuous.fourier_inversion (h : Continuous f)
    (hf : Integrable f) (h'f : Integrable (𝓕 f)) :
    𝓕⁻ (𝓕 f) = f 
Arguments: {V : Type u_1} {E : Type u_2} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [MeasurableSpace V] [BorelSpace V] [FiniteDimensional ℝ V] [NormedAddCommGroup E] [NormedSpace ℂ E] {f : V → E} [CompleteSpace E] (h : Continuous f) (hf : MeasureTheory.Integrable f MeasureTheory.volume) (h'f : MeasureTheory.Integrable (𝓕 f) MeasureTheory.volume)
Goal: 𝓕⁻ (𝓕 f) = f
-- MeasureTheory.Integrable.fourier_inversion_inv
theorem MeasureTheory.Integrable.fourier_inversion_inv
    (hf : Integrable f) (h'f : Integrable (𝓕 f)) {v : V}
    (hv : ContinuousAt f v) : 𝓕 (𝓕⁻ f) v = f v 
Arguments: {V : Type u_1} {E : Type u_2} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [MeasurableSpace V] [BorelSpace V] [FiniteDimensional ℝ V] [NormedAddCommGroup E] [NormedSpace ℂ E] {f : V → E} [CompleteSpace E] (hf : MeasureTheory.Integrable f MeasureTheory.volume) (h'f : MeasureTheory.Integrable (𝓕 f) MeasureTheory.volume) {v : V} (hv : ContinuousAt f v)
Goal: 𝓕 (𝓕⁻ f) v = f v
-- Continuous.fourier_inversion_inv
theorem Continuous.fourier_inversion_inv (h : Continuous f)
    (hf : Integrable f) (h'f : Integrable (𝓕 f)) :
    𝓕 (𝓕⁻ f) = f 
Arguments: {V : Type u_1} {E : Type u_2} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [MeasurableSpace V] [BorelSpace V] [FiniteDimensional ℝ V] [NormedAddCommGroup E] [NormedSpace ℂ E] {f : V → E} [CompleteSpace E] (h : Continuous f) (hf : MeasureTheory.Integrable f MeasureTheory.volume) (h'f : MeasureTheory.Integrable (𝓕 f) MeasureTheory.volume)
Goal: 𝓕 (𝓕⁻ f) = f
-- NormedOrderedAddGroup.dist_eq
class NormedOrderedAddGroup (α : Type*) extends OrderedAddCommGroup α, Norm α, MetricSpace α where
    dist_eq : ∀ x y, dist x y = ‖x - y‖ := by aesop
Arguments: {α : Type u_2} [self : NormedOrderedAddGroup α] (x y : α)
Goal: dist x y = ‖x - y‖
-- NormedOrderedGroup.dist_eq
@[to_additive]
class NormedOrderedGroup (α : Type*) extends OrderedCommGroup α, Norm α, MetricSpace α where
    dist_eq : ∀ x y, dist x y = ‖x / y‖ := by aesop
Arguments: {α : Type u_2} [self : NormedOrderedGroup α] (x y : α)
Goal: dist x y = ‖x / y‖
-- NormedLinearOrderedAddGroup.dist_eq
class NormedLinearOrderedAddGroup (α : Type*) extends LinearOrderedAddCommGroup α, Norm α,
  MetricSpace α where
    dist_eq : ∀ x y, dist x y = ‖x - y‖ := by aesop
Arguments: {α : Type u_2} [self : NormedLinearOrderedAddGroup α] (x y : α)
Goal: dist x y = ‖x - y‖
-- NormedLinearOrderedGroup.dist_eq
@[to_additive]
class NormedLinearOrderedGroup (α : Type*) extends LinearOrderedCommGroup α, Norm α,
  MetricSpace α where
    dist_eq : ∀ x y, dist x y = ‖x / y‖ := by aesop
Arguments: {α : Type u_2} [self : NormedLinearOrderedGroup α] (x y : α)
Goal: dist x y = ‖x / y‖
-- NormedLinearOrderedField.norm_mul'
class NormedLinearOrderedField (α : Type*) extends LinearOrderedField α, Norm α,
  MetricSpace α where
    dist_eq : ∀ x y, dist x y = ‖x - y‖ := by aesop
    norm_mul' : ∀ x y : α, ‖x * y‖ = ‖x‖ * ‖y‖
Arguments: {α : Type u_2} [self : NormedLinearOrderedField α] (x y : α)
Goal: ‖x * y‖ = ‖x‖ * ‖y‖
-- NormedLinearOrderedField.dist_eq
class NormedLinearOrderedField (α : Type*) extends LinearOrderedField α, Norm α,
  MetricSpace α where
    dist_eq : ∀ x y, dist x y = ‖x - y‖ := by aesop
    norm_mul' : ∀ x y : α, ‖x * y‖ = ‖x‖ * ‖y‖
Arguments: {α : Type u_2} [self : NormedLinearOrderedField α] (x y : α)
Goal: dist x y = ‖x - y‖
-- Asymptotics.SuperpolynomialDecay.congr'
theorem SuperpolynomialDecay.congr' (hf : SuperpolynomialDecay l k f) (hfg : f =ᶠ[l] g) :
    SuperpolynomialDecay l k g 
Arguments: {α : Type u_1} {β : Type u_2} {l : Filter α} {k f g : α → β} [TopologicalSpace β] [CommSemiring β] (hf : Asymptotics.SuperpolynomialDecay l k f) (hfg : f =ᶠ[l] g)
Goal: Asymptotics.SuperpolynomialDecay l k g
-- Asymptotics.SuperpolynomialDecay.congr
theorem SuperpolynomialDecay.congr (hf : SuperpolynomialDecay l k f) (hfg : ∀ x, f x = g x) :
    SuperpolynomialDecay l k g 
Arguments: {α : Type u_1} {β : Type u_2} {l : Filter α} {k f g : α → β} [TopologicalSpace β] [CommSemiring β] (hf : Asymptotics.SuperpolynomialDecay l k f) (hfg : ∀ (x : α), f x = g x)
Goal: Asymptotics.SuperpolynomialDecay l k g
-- Asymptotics.superpolynomialDecay_zero
@[simp]
theorem superpolynomialDecay_zero (l : Filter α) (k : α → β) : SuperpolynomialDecay l k 0 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace β] [CommSemiring β] (l : Filter α) (k : α → β)
Goal: Asymptotics.SuperpolynomialDecay l k 0