ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- DilationEquiv.mulRight_symm_apply
@[simps!]
def DilationEquiv.mulRight (a : α) (ha : a ≠ 0) : α ≃ᵈ α 
Arguments: {α : Type u_1} [NormedDivisionRing α] (a : α) (ha : a ≠ 0) (x : α)
Goal: (DilationEquiv.mulRight a ha).symm x = x * a⁻¹
-- Filter.comap_mul_left_cobounded
@[simp]
lemma comap_mul_left_cobounded {a : α} (ha : a ≠ 0) :
    comap (a * ·) (cobounded α) = cobounded α 
Arguments: {α : Type u_1} [NormedDivisionRing α] {a : α} (ha : a ≠ 0)
Goal: Filter.comap (fun x => a * x) (Bornology.cobounded α) = Bornology.cobounded α
-- Filter.map_mul_left_cobounded
@[simp]
lemma map_mul_left_cobounded {a : α} (ha : a ≠ 0) :
    map (a * ·) (cobounded α) = cobounded α 
Arguments: {α : Type u_1} [NormedDivisionRing α] {a : α} (ha : a ≠ 0)
Goal: Filter.map (fun x => a * x) (Bornology.cobounded α) = Bornology.cobounded α
-- Filter.comap_mul_right_cobounded
@[simp]
lemma comap_mul_right_cobounded {a : α} (ha : a ≠ 0) :
    comap (· * a) (cobounded α) = cobounded α 
Arguments: {α : Type u_1} [NormedDivisionRing α] {a : α} (ha : a ≠ 0)
Goal: Filter.comap (fun x => x * a) (Bornology.cobounded α) = Bornology.cobounded α
-- Filter.map_mul_right_cobounded
@[simp]
lemma map_mul_right_cobounded {a : α} (ha : a ≠ 0) :
    map (· * a) (cobounded α) = cobounded α 
Arguments: {α : Type u_1} [NormedDivisionRing α] {a : α} (ha : a ≠ 0)
Goal: Filter.map (fun x => x * a) (Bornology.cobounded α) = Bornology.cobounded α
-- Filter.tendsto_mul_left_cobounded
theorem tendsto_mul_left_cobounded {a : α} (ha : a ≠ 0) :
    Tendsto (a * ·) (cobounded α) (cobounded α) 
Arguments: {α : Type u_1} [NormedDivisionRing α] {a : α} (ha : a ≠ 0)
Goal: Filter.Tendsto (fun x => a * x) (Bornology.cobounded α) (Bornology.cobounded α)
-- Filter.tendsto_mul_right_cobounded
theorem tendsto_mul_right_cobounded {a : α} (ha : a ≠ 0) :
    Tendsto (· * a) (cobounded α) (cobounded α) 
Arguments: {α : Type u_1} [NormedDivisionRing α] {a : α} (ha : a ≠ 0)
Goal: Filter.Tendsto (fun x => x * a) (Bornology.cobounded α) (Bornology.cobounded α)
-- Filter.inv_cobounded₀
@[simp]
lemma inv_cobounded₀ : (cobounded α)⁻¹ = 𝓝[≠] 0 
Arguments: {α : Type u_1} [NormedDivisionRing α]
Goal: (Bornology.cobounded α)⁻¹ = 𝓝[≠] 0
-- Filter.inv_nhdsWithin_ne_zero
@[simp]
lemma inv_nhdsWithin_ne_zero : (𝓝[≠] (0 : α))⁻¹ = cobounded α 
Arguments: {α : Type u_1} [NormedDivisionRing α]
Goal: (𝓝[≠] 0)⁻¹ = Bornology.cobounded α
-- Filter.tendsto_inv₀_cobounded'
lemma tendsto_inv₀_cobounded' : Tendsto Inv.inv (cobounded α) (𝓝[≠] 0) 
Arguments: {α : Type u_1} [NormedDivisionRing α]
Goal: Filter.Tendsto Inv.inv (Bornology.cobounded α) (𝓝[≠] 0)
-- Filter.tendsto_inv₀_cobounded
theorem tendsto_inv₀_cobounded : Tendsto Inv.inv (cobounded α) (𝓝 0) 
Arguments: {α : Type u_1} [NormedDivisionRing α]
Goal: Filter.Tendsto Inv.inv (Bornology.cobounded α) (𝓝 0)
-- Filter.tendsto_inv₀_nhdsWithin_ne_zero
lemma tendsto_inv₀_nhdsWithin_ne_zero : Tendsto Inv.inv (𝓝[≠] 0) (cobounded α) 
Arguments: {α : Type u_1} [NormedDivisionRing α]
Goal: Filter.Tendsto Inv.inv (𝓝[≠] 0) (Bornology.cobounded α)
-- IsOfFinOrder.norm_eq_one
protected lemma IsOfFinOrder.norm_eq_one (ha : IsOfFinOrder a) : ‖a‖ = 1 
Arguments: {α : Type u_1} [NormedDivisionRing α] {a : α} (ha : IsOfFinOrder a)
Goal: ‖a‖ = 1
-- NormedField.norm_mul'
class NormedField (α : Type*) extends Norm α, Field α, MetricSpace α where
    dist_eq : ∀ x y, dist x y = norm (x - y)
    norm_mul' : ∀ a b, norm (a * b) = norm a * norm b
Arguments: {α : Type u_5} [self : NormedField α] (a b : α)
Goal: ‖a * b‖ = ‖a‖ * ‖b‖
-- NormedField.dist_eq
class NormedField (α : Type*) extends Norm α, Field α, MetricSpace α where
    dist_eq : ∀ x y, dist x y = norm (x - y)
    norm_mul' : ∀ a b, norm (a * b) = norm a * norm b
Arguments: {α : Type u_5} [self : NormedField α] (x y : α)
Goal: dist x y = ‖x - y‖
-- NontriviallyNormedField.non_trivial
class NontriviallyNormedField (α : Type*) extends NormedField α where
    non_trivial : ∃ x : α, 1 < ‖x‖
Arguments: {α : Type u_5} [self : NontriviallyNormedField α]
Goal: ∃ x, 1 < ‖x‖
-- DenselyNormedField.lt_norm_lt
class DenselyNormedField (α : Type*) extends NormedField α where
    lt_norm_lt : ∀ x y : ℝ, 0 ≤ x → x < y → ∃ a : α, x < ‖a‖ ∧ ‖a‖ < y
Arguments: {α : Type u_5} [self : DenselyNormedField α] (x y : ℝ)
Goal: 0 ≤ x → x < y → ∃ a, x < ‖a‖ ∧ ‖a‖ < y
-- norm_prod
@[simp]
theorem norm_prod (s : Finset β) (f : β → α) : ‖∏ b ∈ s, f b‖ = ∏ b ∈ s, ‖f b‖ 
Arguments: {α : Type u_1} {β : Type u_2} [NormedField α] (s : Finset β) (f : β → α)
Goal: ‖∏ b ∈ s, f b‖ = ∏ b ∈ s, ‖f b‖
-- nnnorm_prod
@[simp]
theorem nnnorm_prod (s : Finset β) (f : β → α) : ‖∏ b ∈ s, f b‖₊ = ∏ b ∈ s, ‖f b‖₊ 
Arguments: {α : Type u_1} {β : Type u_2} [NormedField α] (s : Finset β) (f : β → α)
Goal: ‖∏ b ∈ s, f b‖₊ = ∏ b ∈ s, ‖f b‖₊
-- NormedField.exists_one_lt_norm
theorem exists_one_lt_norm : ∃ x : α, 1 < ‖x‖ 
Arguments: (α : Type u_1) [NontriviallyNormedField α]
Goal: ∃ x, 1 < ‖x‖
-- NormedField.exists_lt_norm
theorem exists_lt_norm (r : ℝ) : ∃ x : α, r < ‖x‖ 
Arguments: (α : Type u_1) [NontriviallyNormedField α] (r : ℝ)
Goal: ∃ x, r < ‖x‖
-- NormedField.exists_norm_lt
theorem exists_norm_lt {r : ℝ} (hr : 0 < r) : ∃ x : α, 0 < ‖x‖ ∧ ‖x‖ < r 
Arguments: (α : Type u_1) [NontriviallyNormedField α] {r : ℝ} (hr : 0 < r)
Goal: ∃ x, 0 < ‖x‖ ∧ ‖x‖ < r
-- NormedField.exists_norm_lt_one
theorem exists_norm_lt_one : ∃ x : α, 0 < ‖x‖ ∧ ‖x‖ < 1 
Arguments: (α : Type u_1) [NontriviallyNormedField α]
Goal: ∃ x, 0 < ‖x‖ ∧ ‖x‖ < 1
-- NormedField.punctured_nhds_neBot
@[instance]
theorem punctured_nhds_neBot (x : α) : NeBot (𝓝[≠] x) 
Arguments: {α : Type u_1} [NontriviallyNormedField α] (x : α)
Goal: (𝓝[≠] x).NeBot
-- NormedField.nhdsWithin_isUnit_neBot
@[instance]
theorem nhdsWithin_isUnit_neBot : NeBot (𝓝[{ x : α | IsUnit x }] 0) 
Arguments: {α : Type u_1} [NontriviallyNormedField α]
Goal: (𝓝[{x | IsUnit x}] 0).NeBot
-- NormedField.exists_lt_norm_lt
theorem exists_lt_norm_lt {r₁ r₂ : ℝ} (h₀ : 0 ≤ r₁) (h : r₁ < r₂) : ∃ x : α, r₁ < ‖x‖ ∧ ‖x‖ < r₂ 
Arguments: (α : Type u_1) [DenselyNormedField α] {r₁ r₂ : ℝ} (h₀ : 0 ≤ r₁) (h : r₁ < r₂)
Goal: ∃ x, r₁ < ‖x‖ ∧ ‖x‖ < r₂
-- NormedField.exists_lt_nnnorm_lt
theorem exists_lt_nnnorm_lt {r₁ r₂ : ℝ≥0} (h : r₁ < r₂) : ∃ x : α, r₁ < ‖x‖₊ ∧ ‖x‖₊ < r₂ 
Arguments: (α : Type u_1) [DenselyNormedField α] {r₁ r₂ : ℝ≥0} (h : r₁ < r₂)
Goal: ∃ x, r₁ < ‖x‖₊ ∧ ‖x‖₊ < r₂
-- NormedField.denseRange_nnnorm
theorem denseRange_nnnorm : DenseRange (nnnorm : α → ℝ≥0) 
Arguments: (α : Type u_1) [DenselyNormedField α]
Goal: DenseRange nnnorm
-- Real.toNNReal_mul_nnnorm
theorem toNNReal_mul_nnnorm {x : ℝ} (y : ℝ) (hx : 0 ≤ x) : x.toNNReal * ‖y‖₊ = ‖x * y‖₊ 
Arguments: {x : ℝ} (y : ℝ) (hx : 0 ≤ x)
Goal: x.toNNReal * ‖y‖₊ = ‖x * y‖₊
-- Real.nnnorm_mul_toNNReal
theorem nnnorm_mul_toNNReal (x : ℝ) {y : ℝ} (hy : 0 ≤ y) : ‖x‖₊ * y.toNNReal = ‖x * y‖₊ 
Arguments: (x : ℝ) {y : ℝ} (hy : 0 ≤ y)
Goal: ‖x‖₊ * y.toNNReal = ‖x * y‖₊
-- NNReal.norm_eq
theorem norm_eq (x : ℝ≥0) : ‖(x : ℝ)‖ = x 
Arguments: (x : ℝ≥0)
Goal: ‖↑x‖ = ↑x
-- NNReal.nnnorm_eq
@[simp]
theorem nnnorm_eq (x : ℝ≥0) : ‖(x : ℝ)‖₊ = x 
Arguments: (x : ℝ≥0)
Goal: ‖↑x‖₊ = x
-- NNReal.lipschitzWith_sub
lemma lipschitzWith_sub : LipschitzWith 2 (fun (p : ℝ≥0 × ℝ≥0) ↦ p.1 - p.2) 
Goal: LipschitzWith 2 fun p => p.1 - p.2
-- norm_norm
@[simp 1001] -- Porting note: increase priority so that the LHS doesn't simplify
theorem norm_norm [SeminormedAddCommGroup α] (x : α) : ‖‖x‖‖ = ‖x‖ 
Arguments: {α : Type u_1} [SeminormedAddCommGroup α] (x : α)
Goal: ‖‖x‖‖ = ‖x‖
-- nnnorm_norm
@[simp]
theorem nnnorm_norm [SeminormedAddCommGroup α] (a : α) : ‖‖a‖‖₊ = ‖a‖₊ 
Arguments: {α : Type u_1} [SeminormedAddCommGroup α] (a : α)
Goal: ‖‖a‖‖₊ = ‖a‖₊
-- NormedAddCommGroup.tendsto_atTop
theorem NormedAddCommGroup.tendsto_atTop [Nonempty α] [SemilatticeSup α] {β : Type*}
    [SeminormedAddCommGroup β] {f : α → β} {b : β} :
    Tendsto f atTop (𝓝 b) ↔ ∀ ε, 0 < ε → ∃ N, ∀ n, N ≤ n → ‖f n - b‖ < ε 
Arguments: {α : Type u_1} [Nonempty α] [SemilatticeSup α] {β : Type u_5} [SeminormedAddCommGroup β] {f : α → β} {b : β}
Goal: Filter.Tendsto f Filter.atTop (𝓝 b) ↔ ∀ (ε : ℝ), 0 < ε → ∃ N, ∀ (n : α), N ≤ n → ‖f n - b‖ < ε
-- NormedAddCommGroup.tendsto_atTop'
theorem NormedAddCommGroup.tendsto_atTop' [Nonempty α] [SemilatticeSup α] [NoMaxOrder α]
    {β : Type*} [SeminormedAddCommGroup β] {f : α → β} {b : β} :
    Tendsto f atTop (𝓝 b) ↔ ∀ ε, 0 < ε → ∃ N, ∀ n, N < n → ‖f n - b‖ < ε 
Arguments: {α : Type u_1} [Nonempty α] [SemilatticeSup α] [NoMaxOrder α] {β : Type u_5} [SeminormedAddCommGroup β] {f : α → β} {b : β}
Goal: Filter.Tendsto f Filter.atTop (𝓝 b) ↔ ∀ (ε : ℝ), 0 < ε → ∃ N, ∀ (n : α), N < n → ‖f n - b‖ < ε
-- RingHomIsometric.is_iso
class RingHomIsometric [Semiring R₁] [Semiring R₂] [Norm R₁] [Norm R₂] (σ : R₁ →+* R₂) : Prop where
    is_iso : ∀ {x : R₁}, ‖σ x‖ = ‖x‖
Arguments: {R₁ : Type u_5} {R₂ : Type u_6} [Semiring R₁] [Semiring R₂] [Norm R₁] [Norm R₂] {σ : R₁ →+* R₂} [self : RingHomIsometric σ] {x : R₁}
Goal: ‖σ x‖ = ‖x‖
-- NormOneClass.induced
theorem NormOneClass.induced {F : Type*} (R S : Type*) [Ring R] [SeminormedRing S]
    [NormOneClass S] [FunLike F R S] [RingHomClass F R S] (f : F) :
    @NormOneClass R (SeminormedRing.induced R S f).toNorm _ 
Arguments: {F : Type u_8} (R : Type u_9) (S : Type u_10) [Ring R] [SeminormedRing S] [NormOneClass S] [FunLike F R S] [RingHomClass F R S] (f : F)
Goal: NormOneClass R
-- Asymptotics.IsBigOWith_def
irreducible_def IsBigOWith (c : ℝ) (l : Filter α) (f : α → E) (g : α → F) : Prop 
Arguments: {α : Type u_17} {E : Type u_18} {F : Type u_19} [Norm E] [Norm F] (c : ℝ) (l : Filter α) (f : α → E) (g : α → F)
Goal: Asymptotics.IsBigOWith c l f g = ∀ᶠ (x : α) in l, ‖f x‖ ≤ c * ‖g x‖
-- Asymptotics.isBigOWith_iff
theorem isBigOWith_iff : IsBigOWith c l f g ↔ ∀ᶠ x in l, ‖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} {l : Filter α}
Goal: Asymptotics.IsBigOWith c l f g ↔ ∀ᶠ (x : α) in l, ‖f x‖ ≤ c * ‖g x‖
-- Asymptotics.IsBigO_def
irreducible_def IsBigO (l : Filter α) (f : α → E) (g : α → F) : Prop 
Arguments: {α : Type u_17} {E : Type u_18} {F : Type u_19} [Norm E] [Norm F] (l : Filter α) (f : α → E) (g : α → F)
Goal: f =O[l] g = ∃ c, Asymptotics.IsBigOWith c l f g
-- Asymptotics.isBigO_iff_isBigOWith
theorem isBigO_iff_isBigOWith : f =O[l] g ↔ ∃ c : ℝ, IsBigOWith c 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}
Goal: f =O[l] g ↔ ∃ c, Asymptotics.IsBigOWith c l f g
-- Asymptotics.isBigO_iff
theorem isBigO_iff : f =O[l] g ↔ ∃ c : ℝ, ∀ᶠ x in l, ‖f x‖ ≤ c * ‖g x‖ 
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}
Goal: f =O[l] g ↔ ∃ c, ∀ᶠ (x : α) in l, ‖f x‖ ≤ c * ‖g x‖
-- Asymptotics.isBigO_iff'
theorem isBigO_iff' {g : α → E'''} :
    f =O[l] g ↔ ∃ c > 0, ∀ᶠ x in l, ‖f x‖ ≤ c * ‖g x‖ 
Arguments: {α : Type u_1} {E : Type u_3} {E''' : Type u_12} [Norm E] [SeminormedAddGroup E'''] {f : α → E} {l : Filter α} {g : α → E'''}
Goal: f =O[l] g ↔ ∃ c > 0, ∀ᶠ (x : α) in l, ‖f x‖ ≤ c * ‖g x‖
-- Asymptotics.isBigO_iff''
theorem isBigO_iff'' {g : α → E'''} :
    f =O[l] g ↔ ∃ c > 0, ∀ᶠ x in l, c * ‖f x‖ ≤ ‖g x‖ 
Arguments: {α : Type u_1} {E : Type u_3} {E''' : Type u_12} [Norm E] [SeminormedAddGroup E'''] {f : α → E} {l : Filter α} {g : α → E'''}
Goal: f =O[l] g ↔ ∃ c > 0, ∀ᶠ (x : α) in l, c * ‖f x‖ ≤ ‖g x‖
-- Asymptotics.IsBigO.of_bound
theorem IsBigO.of_bound (c : ℝ) (h : ∀ᶠ x in l, ‖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] {f : α → E} {g : α → F} {l : Filter α} (c : ℝ) (h : ∀ᶠ (x : α) in l, ‖f x‖ ≤ c * ‖g x‖)
Goal: f =O[l] g
-- Asymptotics.IsBigO.of_bound'
theorem IsBigO.of_bound' (h : ∀ᶠ x in l, ‖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} (h : ∀ᶠ (x : α) in l, ‖f x‖ ≤ ‖g x‖)
Goal: f =O[l] g
-- Asymptotics.IsBigO.bound
theorem IsBigO.bound : f =O[l] g → ∃ c : ℝ, ∀ᶠ x in l, ‖f x‖ ≤ c * ‖g x‖ 
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}
Goal: f =O[l] g → ∃ c, ∀ᶠ (x : α) in l, ‖f x‖ ≤ c * ‖g x‖
-- Asymptotics.IsLittleO_def
irreducible_def IsLittleO (l : Filter α) (f : α → E) (g : α → F) : Prop 
Arguments: {α : Type u_17} {E : Type u_18} {F : Type u_19} [Norm E] [Norm F] (l : Filter α) (f : α → E) (g : α → F)
Goal: f =o[l] g = ∀ ⦃c : ℝ⦄, 0 < c → Asymptotics.IsBigOWith c l f g
-- Asymptotics.isLittleO_iff_forall_isBigOWith
theorem isLittleO_iff_forall_isBigOWith : f =o[l] g ↔ ∀ ⦃c : ℝ⦄, 0 < c → IsBigOWith c 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}
Goal: f =o[l] g ↔ ∀ ⦃c : ℝ⦄, 0 < c → Asymptotics.IsBigOWith c l f g
-- Asymptotics.isLittleO_iff
theorem isLittleO_iff : f =o[l] g ↔ ∀ ⦃c : ℝ⦄, 0 < c → ∀ᶠ x in l, ‖f x‖ ≤ c * ‖g x‖ 
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}
Goal: f =o[l] g ↔ ∀ ⦃c : ℝ⦄, 0 < c → ∀ᶠ (x : α) in l, ‖f x‖ ≤ c * ‖g x‖
-- Asymptotics.IsLittleO.def
theorem IsLittleO.def (h : f =o[l] g) (hc : 0 < c) : ∀ᶠ x in l, ‖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} {l : Filter α} (h : f =o[l] g) (hc : 0 < c)
Goal: ∀ᶠ (x : α) in l, ‖f x‖ ≤ c * ‖g x‖
-- Asymptotics.IsLittleO.def'
theorem IsLittleO.def' (h : f =o[l] g) (hc : 0 < c) : 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} (h : f =o[l] g) (hc : 0 < c)
Goal: Asymptotics.IsBigOWith c l f g
-- Asymptotics.IsLittleO.eventuallyLE
theorem IsLittleO.eventuallyLE (h : f =o[l] g) : ∀ᶠ x in l, ‖f x‖ ≤ ‖g x‖ 
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 : f =o[l] g)
Goal: ∀ᶠ (x : α) in l, ‖f x‖ ≤ ‖g x‖
-- Asymptotics.IsBigOWith.isBigO
theorem IsBigOWith.isBigO (h : IsBigOWith c l f g) : 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} (h : Asymptotics.IsBigOWith c l f g)
Goal: f =O[l] g
-- Asymptotics.IsLittleO.isBigOWith
theorem IsLittleO.isBigOWith (hgf : f =o[l] g) : 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} (hgf : f =o[l] g)
Goal: Asymptotics.IsBigOWith 1 l f g
-- Asymptotics.IsLittleO.isBigO
theorem IsLittleO.isBigO (hgf : 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 : Filter α} (hgf : f =o[l] g)
Goal: f =O[l] g
-- Asymptotics.IsBigO.isBigOWith
theorem IsBigO.isBigOWith : f =O[l] g → ∃ c : ℝ, IsBigOWith c 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}
Goal: f =O[l] g → ∃ c, Asymptotics.IsBigOWith c l f g
-- Asymptotics.IsBigOWith.weaken
theorem IsBigOWith.weaken (h : IsBigOWith c l f g') (hc : c ≤ c') : IsBigOWith c' 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 : Filter α} (h : Asymptotics.IsBigOWith c l f g') (hc : c ≤ c')
Goal: Asymptotics.IsBigOWith c' l f g'
-- Asymptotics.IsBigOWith.exists_pos
theorem IsBigOWith.exists_pos (h : IsBigOWith c l f g') :
    ∃ c' > 0, 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} (h : Asymptotics.IsBigOWith c l f g')
Goal: ∃ c' > 0, Asymptotics.IsBigOWith c' l f g'
-- Asymptotics.IsBigO.exists_pos
theorem IsBigO.exists_pos (h : f =O[l] g') : ∃ c > 0, IsBigOWith c l f 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} (h : f =O[l] g')
Goal: ∃ c > 0, Asymptotics.IsBigOWith c l f g'
-- Asymptotics.IsBigOWith.exists_nonneg
theorem IsBigOWith.exists_nonneg (h : IsBigOWith c l f g') :
    ∃ c' ≥ 0, 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} (h : Asymptotics.IsBigOWith c l f g')
Goal: ∃ c' ≥ 0, Asymptotics.IsBigOWith c' l f g'
-- Asymptotics.IsBigO.exists_nonneg
theorem IsBigO.exists_nonneg (h : f =O[l] g') : ∃ c ≥ 0, IsBigOWith c l f 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} (h : f =O[l] g')
Goal: ∃ c ≥ 0, Asymptotics.IsBigOWith c l f g'
-- Asymptotics.isBigO_iff_eventually_isBigOWith
theorem isBigO_iff_eventually_isBigOWith : f =O[l] g' ↔ ∀ᶠ c in atTop, IsBigOWith c l f 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] g' ↔ ∀ᶠ (c : ℝ) in Filter.atTop, Asymptotics.IsBigOWith c l f g'
-- Asymptotics.isBigO_iff_eventually
theorem isBigO_iff_eventually : f =O[l] g' ↔ ∀ᶠ c in atTop, ∀ᶠ x in l, ‖f x‖ ≤ c * ‖g' x‖ 
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] g' ↔ ∀ᶠ (c : ℝ) in Filter.atTop, ∀ᶠ (x : α) in l, ‖f x‖ ≤ c * ‖g' x‖
-- Asymptotics.IsBigO.exists_mem_basis
theorem IsBigO.exists_mem_basis {ι} {p : ι → Prop} {s : ι → Set α} (h : f =O[l] g')
    (hb : l.HasBasis p s) :
    ∃ c > 0, ∃ i : ι, p i ∧ ∀ x ∈ s i, ‖f x‖ ≤ c * ‖g' x‖ 
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} {ι : Sort u_17} {p : ι → Prop} {s : ι → Set α} (h : f =O[l] g') (hb : l.HasBasis p s)
Goal: ∃ c > 0, ∃ i, p i ∧ ∀ x ∈ s i, ‖f x‖ ≤ c * ‖g' x‖
-- Asymptotics.isBigOWith_inv
theorem isBigOWith_inv (hc : 0 < c) : IsBigOWith c⁻¹ l f g ↔ ∀ᶠ x in l, c * ‖f x‖ ≤ ‖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} {l : Filter α} (hc : 0 < c)
Goal: Asymptotics.IsBigOWith c⁻¹ l f g ↔ ∀ᶠ (x : α) in l, c * ‖f x‖ ≤ ‖g x‖
-- Asymptotics.isLittleO_iff_nat_mul_le_aux
theorem isLittleO_iff_nat_mul_le_aux (h₀ : (∀ x, 0 ≤ ‖f x‖) ∨ ∀ x, 0 ≤ ‖g x‖) :
    f =o[l] g ↔ ∀ n : ℕ, ∀ᶠ x in l, ↑n * ‖f x‖ ≤ ‖g x‖ 
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} (h₀ : (∀ (x : α), 0 ≤ ‖f x‖) ∨ ∀ (x : α), 0 ≤ ‖g x‖)
Goal: f =o[l] g ↔ ∀ (n : ℕ), ∀ᶠ (x : α) in l, ↑n * ‖f x‖ ≤ ‖g x‖
-- Asymptotics.isLittleO_iff_nat_mul_le
theorem isLittleO_iff_nat_mul_le : f =o[l] g' ↔ ∀ n : ℕ, ∀ᶠ x in l, ↑n * ‖f x‖ ≤ ‖g' x‖ 
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] g' ↔ ∀ (n : ℕ), ∀ᶠ (x : α) in l, ↑n * ‖f x‖ ≤ ‖g' x‖
-- Asymptotics.isLittleO_iff_nat_mul_le'
theorem isLittleO_iff_nat_mul_le' : f' =o[l] g ↔ ∀ n : ℕ, ∀ᶠ x in l, ↑n * ‖f' x‖ ≤ ‖g x‖ 
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: f' =o[l] g ↔ ∀ (n : ℕ), ∀ᶠ (x : α) in l, ↑n * ‖f' x‖ ≤ ‖g x‖
-- Asymptotics.isLittleO_of_subsingleton
@[nontriviality]
theorem isLittleO_of_subsingleton [Subsingleton E'] : 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} [Subsingleton E']
Goal: f' =o[l] g'
-- Asymptotics.isBigO_of_subsingleton
@[nontriviality]
theorem isBigO_of_subsingleton [Subsingleton E'] : 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} [Subsingleton E']
Goal: f' =O[l] g'
-- Asymptotics.isBigOWith_congr
theorem isBigOWith_congr (hc : c₁ = c₂) (hf : f₁ =ᶠ[l] f₂) (hg : g₁ =ᶠ[l] g₂) :
    IsBigOWith c₁ l f₁ g₁ ↔ IsBigOWith c₂ l f₂ g₂ 
Arguments: {α : Type u_1} {E : Type u_3} {F : Type u_4} [Norm E] [Norm F] {c₁ c₂ : ℝ} {l : Filter α} {f₁ f₂ : α → E} {g₁ g₂ : α → F} (hc : c₁ = c₂) (hf : f₁ =ᶠ[l] f₂) (hg : g₁ =ᶠ[l] g₂)
Goal: Asymptotics.IsBigOWith c₁ l f₁ g₁ ↔ Asymptotics.IsBigOWith c₂ l f₂ g₂
-- Asymptotics.IsBigOWith.congr'
theorem IsBigOWith.congr' (h : IsBigOWith c₁ l f₁ g₁) (hc : c₁ = c₂) (hf : f₁ =ᶠ[l] f₂)
    (hg : g₁ =ᶠ[l] g₂) : IsBigOWith c₂ l f₂ g₂ 
Arguments: {α : Type u_1} {E : Type u_3} {F : Type u_4} [Norm E] [Norm F] {c₁ c₂ : ℝ} {l : Filter α} {f₁ f₂ : α → E} {g₁ g₂ : α → F} (h : Asymptotics.IsBigOWith c₁ l f₁ g₁) (hc : c₁ = c₂) (hf : f₁ =ᶠ[l] f₂) (hg : g₁ =ᶠ[l] g₂)
Goal: Asymptotics.IsBigOWith c₂ l f₂ g₂
-- Asymptotics.IsBigOWith.congr
theorem IsBigOWith.congr (h : IsBigOWith c₁ l f₁ g₁) (hc : c₁ = c₂) (hf : ∀ x, f₁ x = f₂ x)
    (hg : ∀ x, g₁ x = g₂ x) : IsBigOWith c₂ l f₂ g₂ 
Arguments: {α : Type u_1} {E : Type u_3} {F : Type u_4} [Norm E] [Norm F] {c₁ c₂ : ℝ} {l : Filter α} {f₁ f₂ : α → E} {g₁ g₂ : α → F} (h : Asymptotics.IsBigOWith c₁ l f₁ g₁) (hc : c₁ = c₂) (hf : ∀ (x : α), f₁ x = f₂ x) (hg : ∀ (x : α), g₁ x = g₂ x)
Goal: Asymptotics.IsBigOWith c₂ l f₂ g₂
-- Asymptotics.IsBigOWith.congr_left
theorem IsBigOWith.congr_left (h : IsBigOWith c l f₁ g) (hf : ∀ x, f₁ x = f₂ x) :
    IsBigOWith c l f₂ g 
Arguments: {α : Type u_1} {E : Type u_3} {F : Type u_4} [Norm E] [Norm F] {c : ℝ} {g : α → F} {l : Filter α} {f₁ f₂ : α → E} (h : Asymptotics.IsBigOWith c l f₁ g) (hf : ∀ (x : α), f₁ x = f₂ x)
Goal: Asymptotics.IsBigOWith c l f₂ g
-- Asymptotics.IsBigOWith.congr_right
theorem IsBigOWith.congr_right (h : IsBigOWith c l f g₁) (hg : ∀ x, g₁ x = g₂ x) :
    IsBigOWith c l f g₂ 
Arguments: {α : Type u_1} {E : Type u_3} {F : Type u_4} [Norm E] [Norm F] {c : ℝ} {f : α → E} {l : Filter α} {g₁ g₂ : α → F} (h : Asymptotics.IsBigOWith c l f g₁) (hg : ∀ (x : α), g₁ x = g₂ x)
Goal: Asymptotics.IsBigOWith c l f g₂
-- Asymptotics.IsBigOWith.congr_const
theorem IsBigOWith.congr_const (h : IsBigOWith c₁ l f g) (hc : c₁ = c₂) : IsBigOWith c₂ l f g 
Arguments: {α : Type u_1} {E : Type u_3} {F : Type u_4} [Norm E] [Norm F] {c₁ c₂ : ℝ} {f : α → E} {g : α → F} {l : Filter α} (h : Asymptotics.IsBigOWith c₁ l f g) (hc : c₁ = c₂)
Goal: Asymptotics.IsBigOWith c₂ l f g
-- Asymptotics.isBigO_congr
theorem isBigO_congr (hf : f₁ =ᶠ[l] f₂) (hg : g₁ =ᶠ[l] g₂) : f₁ =O[l] g₁ ↔ 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} (hf : f₁ =ᶠ[l] f₂) (hg : g₁ =ᶠ[l] g₂)
Goal: f₁ =O[l] g₁ ↔ f₂ =O[l] g₂
-- Asymptotics.IsBigO.congr'
theorem IsBigO.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.IsBigO.congr
theorem IsBigO.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.IsBigO.congr_left
theorem IsBigO.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.IsBigO.congr_right
theorem IsBigO.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
-- Asymptotics.isLittleO_congr
theorem isLittleO_congr (hf : f₁ =ᶠ[l] f₂) (hg : g₁ =ᶠ[l] g₂) : f₁ =o[l] g₁ ↔ 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} (hf : f₁ =ᶠ[l] f₂) (hg : g₁ =ᶠ[l] g₂)
Goal: f₁ =o[l] g₁ ↔ f₂ =o[l] g₂