ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Asymptotics.isBigO_pure
@[simp]
theorem isBigO_pure {x} : f'' =O[pure x] g'' ↔ 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''} {x : α}
Goal: f'' =O[pure x] g'' ↔ g'' x = 0 → f'' x = 0
-- Asymptotics.isBigOWith_principal
@[simp]
theorem isBigOWith_principal {s : Set α} : IsBigOWith c (𝓟 s) f g ↔ ∀ x ∈ s, ‖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} {s : Set α}
Goal: Asymptotics.IsBigOWith c (𝓟 s) f g ↔ ∀ x ∈ s, ‖f x‖ ≤ c * ‖g x‖
-- Asymptotics.isBigO_principal
theorem isBigO_principal {s : Set α} : f =O[𝓟 s] g ↔ ∃ c, ∀ x ∈ s, ‖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} {s : Set α}
Goal: f =O[𝓟 s] g ↔ ∃ c, ∀ x ∈ s, ‖f x‖ ≤ c * ‖g x‖
-- Asymptotics.isLittleO_principal
@[simp]
theorem isLittleO_principal {s : Set α} : f'' =o[𝓟 s] g' ↔ ∀ x ∈ s, f'' x = 0 
Arguments: {α : Type u_1} {F' : Type u_7} {E'' : Type u_9} [SeminormedAddCommGroup F'] [NormedAddCommGroup E''] {g' : α → F'} {f'' : α → E''} {s : Set α}
Goal: f'' =o[𝓟 s] g' ↔ ∀ x ∈ s, f'' x = 0
-- Asymptotics.isBigOWith_top
@[simp]
theorem isBigOWith_top : IsBigOWith c ⊤ f g ↔ ∀ x, ‖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}
Goal: Asymptotics.IsBigOWith c ⊤ f g ↔ ∀ (x : α), ‖f x‖ ≤ c * ‖g x‖
-- Asymptotics.isBigO_top
@[simp]
theorem isBigO_top : f =O[⊤] g ↔ ∃ C, ∀ x, ‖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}
Goal: f =O[⊤] g ↔ ∃ C, ∀ (x : α), ‖f x‖ ≤ C * ‖g x‖
-- Asymptotics.isLittleO_top
@[simp]
theorem isLittleO_top : f'' =o[⊤] g' ↔ ∀ x, f'' x = 0 
Arguments: {α : Type u_1} {F' : Type u_7} {E'' : Type u_9} [SeminormedAddCommGroup F'] [NormedAddCommGroup E''] {g' : α → F'} {f'' : α → E''}
Goal: f'' =o[⊤] g' ↔ ∀ (x : α), f'' x = 0
-- Asymptotics.isBigOWith_const_one
theorem isBigOWith_const_one (c : E) (l : Filter α) :
    IsBigOWith ‖c‖ l (fun _x : α => c) fun _x => (1 : F) 
Arguments: {α : Type u_1} {E : Type u_3} (F : Type u_4) [Norm E] [Norm F] [One F] [NormOneClass F] (c : E) (l : Filter α)
Goal: Asymptotics.IsBigOWith ‖c‖ l (fun _x => c) fun _x => 1
-- Asymptotics.isBigO_const_one
theorem isBigO_const_one (c : E) (l : Filter α) : (fun _x : α => c) =O[l] fun _x => (1 : F) 
Arguments: {α : Type u_1} {E : Type u_3} (F : Type u_4) [Norm E] [Norm F] [One F] [NormOneClass F] (c : E) (l : Filter α)
Goal: (fun _x => c) =O[l] fun _x => 1
-- Asymptotics.isLittleO_const_iff_isLittleO_one
theorem isLittleO_const_iff_isLittleO_one {c : F''} (hc : c ≠ 0) :
    (f =o[l] fun _x => c) ↔ f =o[l] fun _x => (1 : F) 
Arguments: {α : Type u_1} {E : Type u_3} (F : Type u_4) {F'' : Type u_10} [Norm E] [Norm F] [NormedAddCommGroup F''] {f : α → E} {l : Filter α} [One F] [NormOneClass F] {c : F''} (hc : c ≠ 0)
Goal: (f =o[l] fun _x => c) ↔ f =o[l] fun _x => 1
-- Asymptotics.isLittleO_one_iff
@[simp]
theorem isLittleO_one_iff : f' =o[l] (fun _x => 1 : α → F) ↔ Tendsto f' l (𝓝 0) 
Arguments: {α : Type u_1} (F : Type u_4) {E' : Type u_6} [Norm F] [SeminormedAddCommGroup E'] {f' : α → E'} {l : Filter α} [One F] [NormOneClass F]
Goal: (f' =o[l] fun _x => 1) ↔ Filter.Tendsto f' l (𝓝 0)
-- Asymptotics.isBigO_one_iff
@[simp]
theorem isBigO_one_iff : f =O[l] (fun _x => 1 : α → F) ↔
    IsBoundedUnder (· ≤ ·) l fun x => ‖f x‖ 
Arguments: {α : Type u_1} {E : Type u_3} (F : Type u_4) [Norm E] [Norm F] {f : α → E} {l : Filter α} [One F] [NormOneClass F]
Goal: (f =O[l] fun _x => 1) ↔ Filter.IsBoundedUnder (fun x x_1 => x ≤ x_1) l fun x => ‖f x‖
-- Asymptotics.isLittleO_one_left_iff
@[simp]
theorem isLittleO_one_left_iff : (fun _x => 1 : α → F) =o[l] f ↔ Tendsto (fun x => ‖f x‖) l atTop 
Arguments: {α : Type u_1} {E : Type u_3} (F : Type u_4) [Norm E] [Norm F] {f : α → E} {l : Filter α} [One F] [NormOneClass F]
Goal: (fun _x => 1) =o[l] f ↔ Filter.Tendsto (fun x => ‖f x‖) l Filter.atTop
-- Filter.Tendsto.isBigO_one
theorem _root_.Filter.Tendsto.isBigO_one {c : E'} (h : Tendsto f' l (𝓝 c)) :
    f' =O[l] (fun _x => 1 : α → F) 
Arguments: {α : Type u_1} (F : Type u_4) {E' : Type u_6} [Norm F] [SeminormedAddCommGroup E'] {f' : α → E'} {l : Filter α} [One F] [NormOneClass F] {c : E'} (h : Filter.Tendsto f' l (𝓝 c))
Goal: f' =O[l] fun _x => 1
-- Asymptotics.IsBigO.trans_tendsto_nhds
theorem IsBigO.trans_tendsto_nhds (hfg : f =O[l] g') {y : F'} (hg : Tendsto g' l (𝓝 y)) :
    f =O[l] (fun _x => 1 : α → F) 
Arguments: {α : Type u_1} {E : Type u_3} (F : Type u_4) {F' : Type u_7} [Norm E] [Norm F] [SeminormedAddCommGroup F'] {f : α → E} {g' : α → F'} {l : Filter α} [One F] [NormOneClass F] (hfg : f =O[l] g') {y : F'} (hg : Filter.Tendsto g' l (𝓝 y))
Goal: f =O[l] fun _x => 1
-- Asymptotics.isBigO_one_nhds_ne_iff
lemma isBigO_one_nhds_ne_iff [TopologicalSpace α] {a : α} :
    f =O[𝓝[≠] a] (fun _ ↦ 1 : α → F) ↔ f =O[𝓝 a] (fun _ ↦ 1 : α → F) 
Arguments: {α : Type u_1} {E : Type u_3} (F : Type u_4) [Norm E] [Norm F] {f : α → E} [One F] [NormOneClass F] [TopologicalSpace α] {a : α}
Goal: (f =O[𝓝[≠] a] fun x => 1) ↔ f =O[𝓝 a] fun x => 1
-- Asymptotics.isLittleO_const_iff
theorem isLittleO_const_iff {c : F''} (hc : c ≠ 0) :
    (f'' =o[l] fun _x => c) ↔ Tendsto f'' l (𝓝 0) 
Arguments: {α : Type u_1} {E'' : Type u_9} {F'' : Type u_10} [NormedAddCommGroup E''] [NormedAddCommGroup F''] {f'' : α → E''} {l : Filter α} {c : F''} (hc : c ≠ 0)
Goal: (f'' =o[l] fun _x => c) ↔ Filter.Tendsto f'' l (𝓝 0)
-- Asymptotics.isLittleO_id_const
theorem isLittleO_id_const {c : F''} (hc : c ≠ 0) : (fun x : E'' => x) =o[𝓝 0] fun _x => c 
Arguments: {E'' : Type u_9} {F'' : Type u_10} [NormedAddCommGroup E''] [NormedAddCommGroup F''] {c : F''} (hc : c ≠ 0)
Goal: (fun x => x) =o[𝓝 0] fun _x => c
-- Filter.IsBoundedUnder.isBigO_const
theorem _root_.Filter.IsBoundedUnder.isBigO_const (h : IsBoundedUnder (· ≤ ·) l (norm ∘ f))
    {c : F''} (hc : c ≠ 0) : f =O[l] fun _x => c 
Arguments: {α : Type u_1} {E : Type u_3} {F'' : Type u_10} [Norm E] [NormedAddCommGroup F''] {f : α → E} {l : Filter α} (h : Filter.IsBoundedUnder (fun x x_1 => x ≤ x_1) l (norm ∘ f)) {c : F''} (hc : c ≠ 0)
Goal: f =O[l] fun _x => c
-- Asymptotics.isBigO_const_of_tendsto
theorem isBigO_const_of_tendsto {y : E''} (h : Tendsto f'' l (𝓝 y)) {c : F''} (hc : c ≠ 0) :
    f'' =O[l] fun _x => c 
Arguments: {α : Type u_1} {E'' : Type u_9} {F'' : Type u_10} [NormedAddCommGroup E''] [NormedAddCommGroup F''] {f'' : α → E''} {l : Filter α} {y : E''} (h : Filter.Tendsto f'' l (𝓝 y)) {c : F''} (hc : c ≠ 0)
Goal: f'' =O[l] fun _x => c
-- Asymptotics.IsBigO.isBoundedUnder_le
theorem IsBigO.isBoundedUnder_le {c : F} (h : f =O[l] fun _x => c) :
    IsBoundedUnder (· ≤ ·) l (norm ∘ f) 
Arguments: {α : Type u_1} {E : Type u_3} {F : Type u_4} [Norm E] [Norm F] {f : α → E} {l : Filter α} {c : F} (h : f =O[l] fun _x => c)
Goal: Filter.IsBoundedUnder (fun x x_1 => x ≤ x_1) l (norm ∘ f)
-- Asymptotics.isBigO_const_of_ne
theorem isBigO_const_of_ne {c : F''} (hc : c ≠ 0) :
    (f =O[l] fun _x => c) ↔ IsBoundedUnder (· ≤ ·) l (norm ∘ f) 
Arguments: {α : Type u_1} {E : Type u_3} {F'' : Type u_10} [Norm E] [NormedAddCommGroup F''] {f : α → E} {l : Filter α} {c : F''} (hc : c ≠ 0)
Goal: (f =O[l] fun _x => c) ↔ Filter.IsBoundedUnder (fun x x_1 => x ≤ x_1) l (norm ∘ f)
-- Asymptotics.isBigO_const_iff
theorem isBigO_const_iff {c : F''} : (f'' =O[l] fun _x => c) ↔
    (c = 0 → f'' =ᶠ[l] 0) ∧ IsBoundedUnder (· ≤ ·) l fun x => ‖f'' x‖ 
Arguments: {α : Type u_1} {E'' : Type u_9} {F'' : Type u_10} [NormedAddCommGroup E''] [NormedAddCommGroup F''] {f'' : α → E''} {l : Filter α} {c : F''}
Goal: (f'' =O[l] fun _x => c) ↔ (c = 0 → f'' =ᶠ[l] 0) ∧ Filter.IsBoundedUnder (fun x x_1 => x ≤ x_1) l fun x => ‖f'' x‖
-- Asymptotics.isBigO_iff_isBoundedUnder_le_div
theorem isBigO_iff_isBoundedUnder_le_div (h : ∀ᶠ x in l, g'' x ≠ 0) :
    f =O[l] g'' ↔ IsBoundedUnder (· ≤ ·) l fun x => ‖f x‖ / ‖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''} {l : Filter α} (h : ∀ᶠ (x : α) in l, g'' 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_const_left_iff_pos_le_norm
theorem isBigO_const_left_iff_pos_le_norm {c : E''} (hc : c ≠ 0) :
    (fun _x => c) =O[l] f' ↔ ∃ b, 0 < b ∧ ∀ᶠ x in l, b ≤ ‖f' x‖ 
Arguments: {α : Type u_1} {E' : Type u_6} {E'' : Type u_9} [SeminormedAddCommGroup E'] [NormedAddCommGroup E''] {f' : α → E'} {l : Filter α} {c : E''} (hc : c ≠ 0)
Goal: (fun _x => c) =O[l] f' ↔ ∃ b, 0 < b ∧ ∀ᶠ (x : α) in l, b ≤ ‖f' x‖
-- Asymptotics.IsBigO.trans_tendsto
theorem IsBigO.trans_tendsto (hfg : f'' =O[l] g'') (hg : Tendsto g'' l (𝓝 0)) :
    Tendsto f'' 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} (hfg : f'' =O[l] g'') (hg : Filter.Tendsto g'' l (𝓝 0))
Goal: Filter.Tendsto f'' l (𝓝 0)
-- Asymptotics.IsLittleO.trans_tendsto
theorem IsLittleO.trans_tendsto (hfg : f'' =o[l] g'') (hg : Tendsto g'' l (𝓝 0)) :
    Tendsto f'' 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} (hfg : f'' =o[l] g'') (hg : Filter.Tendsto g'' l (𝓝 0))
Goal: Filter.Tendsto f'' l (𝓝 0)
-- Asymptotics.isBigOWith_const_mul_self
theorem isBigOWith_const_mul_self (c : R) (f : α → R) (l : Filter α) :
    IsBigOWith ‖c‖ l (fun x => c * f x) f 
Arguments: {α : Type u_1} {R : Type u_13} [SeminormedRing R] (c : R) (f : α → R) (l : Filter α)
Goal: Asymptotics.IsBigOWith ‖c‖ l (fun x => c * f x) f
-- Asymptotics.isBigO_const_mul_self
theorem isBigO_const_mul_self (c : R) (f : α → R) (l : Filter α) : (fun x => c * f x) =O[l] f 
Arguments: {α : Type u_1} {R : Type u_13} [SeminormedRing R] (c : R) (f : α → R) (l : Filter α)
Goal: (fun x => c * f x) =O[l] f
-- Asymptotics.IsBigOWith.const_mul_left
theorem IsBigOWith.const_mul_left {f : α → R} (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} {R : Type u_13} [Norm F] [SeminormedRing R] {c : ℝ} {g : α → F} {l : Filter α} {f : α → R} (h : Asymptotics.IsBigOWith c l f g) (c' : R)
Goal: Asymptotics.IsBigOWith (‖c'‖ * c) l (fun x => c' * f x) g
-- Asymptotics.IsBigO.const_mul_left
theorem IsBigO.const_mul_left {f : α → R} (h : f =O[l] g) (c' : R) : (fun x => c' * f x) =O[l] g 
Arguments: {α : Type u_1} {F : Type u_4} {R : Type u_13} [Norm F] [SeminormedRing R] {g : α → F} {l : Filter α} {f : α → R} (h : f =O[l] g) (c' : R)
Goal: (fun x => c' * f x) =O[l] g
-- Asymptotics.isBigOWith_self_const_mul'
theorem isBigOWith_self_const_mul' (u : Rˣ) (f : α → R) (l : Filter α) :
    IsBigOWith ‖(↑u⁻¹ : R)‖ l f fun x => ↑u * f x 
Arguments: {α : Type u_1} {R : Type u_13} [SeminormedRing R] (u : Rˣ) (f : α → R) (l : Filter α)
Goal: Asymptotics.IsBigOWith ‖↑u⁻¹‖ l f fun x => ↑u * f x
-- Asymptotics.isBigOWith_self_const_mul
theorem isBigOWith_self_const_mul (c : 𝕜) (hc : c ≠ 0) (f : α → 𝕜) (l : Filter α) :
    IsBigOWith ‖c‖⁻¹ l f fun x => c * f x 
Arguments: {α : Type u_1} {𝕜 : Type u_15} [NormedDivisionRing 𝕜] (c : 𝕜) (hc : c ≠ 0) (f : α → 𝕜) (l : Filter α)
Goal: Asymptotics.IsBigOWith ‖c‖⁻¹ l f fun x => c * f x
-- Asymptotics.isBigO_self_const_mul'
theorem isBigO_self_const_mul' {c : R} (hc : IsUnit c) (f : α → R) (l : Filter α) :
    f =O[l] fun x => c * f x 
Arguments: {α : Type u_1} {R : Type u_13} [SeminormedRing R] {c : R} (hc : IsUnit c) (f : α → R) (l : Filter α)
Goal: f =O[l] fun x => c * f x
-- Asymptotics.isBigO_self_const_mul
theorem isBigO_self_const_mul (c : 𝕜) (hc : c ≠ 0) (f : α → 𝕜) (l : Filter α) :
    f =O[l] fun x => c * f x 
Arguments: {α : Type u_1} {𝕜 : Type u_15} [NormedDivisionRing 𝕜] (c : 𝕜) (hc : c ≠ 0) (f : α → 𝕜) (l : Filter α)
Goal: f =O[l] fun x => c * f x
-- Asymptotics.isBigO_const_mul_left_iff'
theorem isBigO_const_mul_left_iff' {f : α → R} {c : R} (hc : IsUnit c) :
    (fun x => c * f x) =O[l] g ↔ f =O[l] g 
Arguments: {α : Type u_1} {F : Type u_4} {R : Type u_13} [Norm F] [SeminormedRing R] {g : α → F} {l : Filter α} {f : α → R} {c : R} (hc : IsUnit c)
Goal: (fun x => c * f x) =O[l] g ↔ f =O[l] g
-- Asymptotics.isBigO_const_mul_left_iff
theorem isBigO_const_mul_left_iff {f : α → 𝕜} {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} {𝕜 : Type u_15} [Norm F] [NormedDivisionRing 𝕜] {g : α → F} {l : Filter α} {f : α → 𝕜} {c : 𝕜} (hc : c ≠ 0)
Goal: (fun x => c * f x) =O[l] g ↔ f =O[l] g
-- Asymptotics.IsLittleO.const_mul_left
theorem IsLittleO.const_mul_left {f : α → R} (h : f =o[l] g) (c : R) : (fun x => c * f x) =o[l] g 
Arguments: {α : Type u_1} {F : Type u_4} {R : Type u_13} [Norm F] [SeminormedRing R] {g : α → F} {l : Filter α} {f : α → R} (h : f =o[l] g) (c : R)
Goal: (fun x => c * f x) =o[l] g
-- Asymptotics.isLittleO_const_mul_left_iff'
theorem isLittleO_const_mul_left_iff' {f : α → R} {c : R} (hc : IsUnit c) :
    (fun x => c * f x) =o[l] g ↔ f =o[l] g 
Arguments: {α : Type u_1} {F : Type u_4} {R : Type u_13} [Norm F] [SeminormedRing R] {g : α → F} {l : Filter α} {f : α → R} {c : R} (hc : IsUnit c)
Goal: (fun x => c * f x) =o[l] g ↔ f =o[l] g
-- Asymptotics.isLittleO_const_mul_left_iff
theorem isLittleO_const_mul_left_iff {f : α → 𝕜} {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} {𝕜 : Type u_15} [Norm F] [NormedDivisionRing 𝕜] {g : α → F} {l : Filter α} {f : α → 𝕜} {c : 𝕜} (hc : c ≠ 0)
Goal: (fun x => c * f x) =o[l] g ↔ f =o[l] g
-- Asymptotics.IsBigOWith.of_const_mul_right
theorem IsBigOWith.of_const_mul_right {g : α → R} {c : R} (hc' : 0 ≤ c')
    (h : IsBigOWith c' l f fun x => c * g x) : IsBigOWith (c' * ‖c‖) l f g 
Arguments: {α : Type u_1} {E : Type u_3} {R : Type u_13} [Norm E] [SeminormedRing R] {c' : ℝ} {f : α → E} {l : Filter α} {g : α → R} {c : R} (hc' : 0 ≤ c') (h : Asymptotics.IsBigOWith c' l f fun x => c * g x)
Goal: Asymptotics.IsBigOWith (c' * ‖c‖) l f g
-- Asymptotics.IsBigO.of_const_mul_right
theorem IsBigO.of_const_mul_right {g : α → R} {c : R} (h : f =O[l] fun x => c * g x) : f =O[l] g 
Arguments: {α : Type u_1} {E : Type u_3} {R : Type u_13} [Norm E] [SeminormedRing R] {f : α → E} {l : Filter α} {g : α → R} {c : R} (h : f =O[l] fun x => c * g x)
Goal: f =O[l] g
-- Asymptotics.IsBigOWith.const_mul_right'
theorem IsBigOWith.const_mul_right' {g : α → R} {u : Rˣ} {c' : ℝ} (hc' : 0 ≤ c')
    (h : IsBigOWith c' l f g) : IsBigOWith (c' * ‖(↑u⁻¹ : R)‖) l f fun x => ↑u * g x 
Arguments: {α : Type u_1} {E : Type u_3} {R : Type u_13} [Norm E] [SeminormedRing R] {f : α → E} {l : Filter α} {g : α → R} {u : Rˣ} {c' : ℝ} (hc' : 0 ≤ c') (h : Asymptotics.IsBigOWith c' l f g)
Goal: Asymptotics.IsBigOWith (c' * ‖↑u⁻¹‖) l f fun x => ↑u * g x
-- Asymptotics.IsBigOWith.const_mul_right
theorem IsBigOWith.const_mul_right {g : α → 𝕜} {c : 𝕜} (hc : c ≠ 0) {c' : ℝ} (hc' : 0 ≤ c')
    (h : IsBigOWith c' l f g) : IsBigOWith (c' * ‖c‖⁻¹) l f fun x => c * g x 
Arguments: {α : Type u_1} {E : Type u_3} {𝕜 : Type u_15} [Norm E] [NormedDivisionRing 𝕜] {f : α → E} {l : Filter α} {g : α → 𝕜} {c : 𝕜} (hc : c ≠ 0) {c' : ℝ} (hc' : 0 ≤ c') (h : Asymptotics.IsBigOWith c' l f g)
Goal: Asymptotics.IsBigOWith (c' * ‖c‖⁻¹) l f fun x => c * g x
-- Asymptotics.IsBigO.const_mul_right'
theorem IsBigO.const_mul_right' {g : α → R} {c : R} (hc : IsUnit c) (h : f =O[l] g) :
    f =O[l] fun x => c * g x 
Arguments: {α : Type u_1} {E : Type u_3} {R : Type u_13} [Norm E] [SeminormedRing R] {f : α → E} {l : Filter α} {g : α → R} {c : R} (hc : IsUnit c) (h : f =O[l] g)
Goal: f =O[l] fun x => c * g x
-- Asymptotics.IsBigO.const_mul_right
theorem IsBigO.const_mul_right {g : α → 𝕜} {c : 𝕜} (hc : c ≠ 0) (h : f =O[l] g) :
    f =O[l] fun x => c * g x 
Arguments: {α : Type u_1} {E : Type u_3} {𝕜 : Type u_15} [Norm E] [NormedDivisionRing 𝕜] {f : α → E} {l : Filter α} {g : α → 𝕜} {c : 𝕜} (hc : c ≠ 0) (h : f =O[l] g)
Goal: f =O[l] fun x => c * g x
-- Asymptotics.isBigO_const_mul_right_iff'
theorem isBigO_const_mul_right_iff' {g : α → R} {c : R} (hc : IsUnit c) :
    (f =O[l] fun x => c * g x) ↔ f =O[l] g 
Arguments: {α : Type u_1} {E : Type u_3} {R : Type u_13} [Norm E] [SeminormedRing R] {f : α → E} {l : Filter α} {g : α → R} {c : R} (hc : IsUnit c)
Goal: (f =O[l] fun x => c * g x) ↔ f =O[l] g
-- Asymptotics.isBigO_const_mul_right_iff
theorem isBigO_const_mul_right_iff {g : α → 𝕜} {c : 𝕜} (hc : c ≠ 0) :
    (f =O[l] fun x => c * g x) ↔ f =O[l] g 
Arguments: {α : Type u_1} {E : Type u_3} {𝕜 : Type u_15} [Norm E] [NormedDivisionRing 𝕜] {f : α → E} {l : Filter α} {g : α → 𝕜} {c : 𝕜} (hc : c ≠ 0)
Goal: (f =O[l] fun x => c * g x) ↔ f =O[l] g
-- Asymptotics.IsLittleO.of_const_mul_right
theorem IsLittleO.of_const_mul_right {g : α → R} {c : R} (h : f =o[l] fun x => c * g x) :
    f =o[l] g 
Arguments: {α : Type u_1} {E : Type u_3} {R : Type u_13} [Norm E] [SeminormedRing R] {f : α → E} {l : Filter α} {g : α → R} {c : R} (h : f =o[l] fun x => c * g x)
Goal: f =o[l] g
-- Asymptotics.IsLittleO.const_mul_right'
theorem IsLittleO.const_mul_right' {g : α → R} {c : R} (hc : IsUnit c) (h : f =o[l] g) :
    f =o[l] fun x => c * g x 
Arguments: {α : Type u_1} {E : Type u_3} {R : Type u_13} [Norm E] [SeminormedRing R] {f : α → E} {l : Filter α} {g : α → R} {c : R} (hc : IsUnit c) (h : f =o[l] g)
Goal: f =o[l] fun x => c * g x
-- Asymptotics.IsLittleO.const_mul_right
theorem IsLittleO.const_mul_right {g : α → 𝕜} {c : 𝕜} (hc : c ≠ 0) (h : f =o[l] g) :
    f =o[l] fun x => c * g x 
Arguments: {α : Type u_1} {E : Type u_3} {𝕜 : Type u_15} [Norm E] [NormedDivisionRing 𝕜] {f : α → E} {l : Filter α} {g : α → 𝕜} {c : 𝕜} (hc : c ≠ 0) (h : f =o[l] g)
Goal: f =o[l] fun x => c * g x
-- Asymptotics.isLittleO_const_mul_right_iff'
theorem isLittleO_const_mul_right_iff' {g : α → R} {c : R} (hc : IsUnit c) :
    (f =o[l] fun x => c * g x) ↔ f =o[l] g 
Arguments: {α : Type u_1} {E : Type u_3} {R : Type u_13} [Norm E] [SeminormedRing R] {f : α → E} {l : Filter α} {g : α → R} {c : R} (hc : IsUnit c)
Goal: (f =o[l] fun x => c * g x) ↔ f =o[l] g
-- Asymptotics.isLittleO_const_mul_right_iff
theorem isLittleO_const_mul_right_iff {g : α → 𝕜} {c : 𝕜} (hc : c ≠ 0) :
    (f =o[l] fun x => c * g x) ↔ f =o[l] g 
Arguments: {α : Type u_1} {E : Type u_3} {𝕜 : Type u_15} [Norm E] [NormedDivisionRing 𝕜] {f : α → E} {l : Filter α} {g : α → 𝕜} {c : 𝕜} (hc : c ≠ 0)
Goal: (f =o[l] fun x => c * g x) ↔ f =o[l] g
-- Asymptotics.IsBigOWith.mul
theorem IsBigOWith.mul {f₁ f₂ : α → R} {g₁ g₂ : α → 𝕜} {c₁ c₂ : ℝ} (h₁ : IsBigOWith c₁ l f₁ g₁)
    (h₂ : IsBigOWith c₂ l f₂ g₂) :
    IsBigOWith (c₁ * c₂) l (fun x => f₁ x * f₂ x) fun x => g₁ x * g₂ x 
Arguments: {α : Type u_1} {R : Type u_13} {𝕜 : Type u_15} [SeminormedRing R] [NormedDivisionRing 𝕜] {l : Filter α} {f₁ f₂ : α → R} {g₁ g₂ : α → 𝕜} {c₁ c₂ : ℝ} (h₁ : Asymptotics.IsBigOWith c₁ l f₁ g₁) (h₂ : Asymptotics.IsBigOWith c₂ l f₂ g₂)
Goal: Asymptotics.IsBigOWith (c₁ * c₂) l (fun x => f₁ x * f₂ x) fun x => g₁ x * g₂ x
-- Asymptotics.IsBigO.mul
theorem IsBigO.mul {f₁ f₂ : α → R} {g₁ g₂ : α → 𝕜} (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} {R : Type u_13} {𝕜 : Type u_15} [SeminormedRing R] [NormedDivisionRing 𝕜] {l : Filter α} {f₁ f₂ : α → R} {g₁ g₂ : α → 𝕜} (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.mul_isLittleO
theorem IsBigO.mul_isLittleO {f₁ f₂ : α → R} {g₁ g₂ : α → 𝕜} (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} {R : Type u_13} {𝕜 : Type u_15} [SeminormedRing R] [NormedDivisionRing 𝕜] {l : Filter α} {f₁ f₂ : α → R} {g₁ g₂ : α → 𝕜} (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.IsLittleO.mul_isBigO
theorem IsLittleO.mul_isBigO {f₁ f₂ : α → R} {g₁ g₂ : α → 𝕜} (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} {R : Type u_13} {𝕜 : Type u_15} [SeminormedRing R] [NormedDivisionRing 𝕜] {l : Filter α} {f₁ f₂ : α → R} {g₁ g₂ : α → 𝕜} (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.IsLittleO.mul
theorem IsLittleO.mul {f₁ f₂ : α → R} {g₁ g₂ : α → 𝕜} (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} {R : Type u_13} {𝕜 : Type u_15} [SeminormedRing R] [NormedDivisionRing 𝕜] {l : Filter α} {f₁ f₂ : α → R} {g₁ g₂ : α → 𝕜} (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.IsBigOWith.pow'
theorem IsBigOWith.pow' {f : α → R} {g : α → 𝕜} (h : IsBigOWith c l f g) :
    ∀ n : ℕ, IsBigOWith (Nat.casesOn n ‖(1 : R)‖ fun n => c ^ (n + 1))
      l (fun x => f x ^ n) fun x => g x ^ n
  
Arguments: {α : Type u_1} {R : Type u_13} {𝕜 : Type u_15} [SeminormedRing R] [NormedDivisionRing 𝕜] {c : ℝ} {l : Filter α} {f : α → R} {g : α → 𝕜} (h : Asymptotics.IsBigOWith c l f g) (n : ℕ)
Goal: Asymptotics.IsBigOWith (Nat.casesOn n ‖1‖ fun n => c ^ (n + 1)) l (fun x => f x ^ n) fun x => g x ^ n
-- Asymptotics.IsBigOWith.pow
theorem IsBigOWith.pow [NormOneClass R] {f : α → R} {g : α → 𝕜} (h : IsBigOWith c l f g) :
    ∀ n : ℕ, IsBigOWith (c ^ n) l (fun x => f x ^ n) fun x => g x ^ n
  
Arguments: {α : Type u_1} {R : Type u_13} {𝕜 : Type u_15} [SeminormedRing R] [NormedDivisionRing 𝕜] {c : ℝ} {l : Filter α} [NormOneClass R] {f : α → R} {g : α → 𝕜} (h : Asymptotics.IsBigOWith c l f g) (n : ℕ)
Goal: Asymptotics.IsBigOWith (c ^ n) l (fun x => f x ^ n) fun x => g x ^ n
-- Asymptotics.IsBigOWith.of_pow
theorem IsBigOWith.of_pow {n : ℕ} {f : α → 𝕜} {g : α → R} (h : IsBigOWith c l (f ^ n) (g ^ n))
    (hn : n ≠ 0) (hc : c ≤ c' ^ n) (hc' : 0 ≤ c') : IsBigOWith c' l f g 
Arguments: {α : Type u_1} {R : Type u_13} {𝕜 : Type u_15} [SeminormedRing R] [NormedDivisionRing 𝕜] {c c' : ℝ} {l : Filter α} {n : ℕ} {f : α → 𝕜} {g : α → R} (h : Asymptotics.IsBigOWith c l (f ^ n) (g ^ n)) (hn : n ≠ 0) (hc : c ≤ c' ^ n) (hc' : 0 ≤ c')
Goal: Asymptotics.IsBigOWith c' l f g
-- Asymptotics.IsBigO.pow
theorem IsBigO.pow {f : α → R} {g : α → 𝕜} (h : f =O[l] g) (n : ℕ) :
    (fun x => f x ^ n) =O[l] fun x => g x ^ n 
Arguments: {α : Type u_1} {R : Type u_13} {𝕜 : Type u_15} [SeminormedRing R] [NormedDivisionRing 𝕜] {l : Filter α} {f : α → R} {g : α → 𝕜} (h : f =O[l] g) (n : ℕ)
Goal: (fun x => f x ^ n) =O[l] fun x => g x ^ n
-- Asymptotics.IsBigO.of_pow
theorem IsBigO.of_pow {f : α → 𝕜} {g : α → R} {n : ℕ} (hn : n ≠ 0) (h : (f ^ n) =O[l] (g ^ n)) :
    f =O[l] g 
Arguments: {α : Type u_1} {R : Type u_13} {𝕜 : Type u_15} [SeminormedRing R] [NormedDivisionRing 𝕜] {l : Filter α} {f : α → 𝕜} {g : α → R} {n : ℕ} (hn : n ≠ 0) (h : (f ^ n) =O[l] (g ^ n))
Goal: f =O[l] g
-- Asymptotics.IsLittleO.pow
theorem IsLittleO.pow {f : α → R} {g : α → 𝕜} (h : f =o[l] g) {n : ℕ} (hn : 0 < n) :
    (fun x => f x ^ n) =o[l] fun x => g x ^ n 
Arguments: {α : Type u_1} {R : Type u_13} {𝕜 : Type u_15} [SeminormedRing R] [NormedDivisionRing 𝕜] {l : Filter α} {f : α → R} {g : α → 𝕜} (h : f =o[l] g) {n : ℕ} (hn : 0 < n)
Goal: (fun x => f x ^ n) =o[l] fun x => g x ^ n
-- Asymptotics.IsLittleO.of_pow
theorem IsLittleO.of_pow {f : α → 𝕜} {g : α → R} {n : ℕ} (h : (f ^ n) =o[l] (g ^ n)) (hn : n ≠ 0) :
    f =o[l] g 
Arguments: {α : Type u_1} {R : Type u_13} {𝕜 : Type u_15} [SeminormedRing R] [NormedDivisionRing 𝕜] {l : Filter α} {f : α → 𝕜} {g : α → R} {n : ℕ} (h : (f ^ n) =o[l] (g ^ n)) (hn : n ≠ 0)
Goal: f =o[l] g
-- Asymptotics.IsBigOWith.inv_rev
theorem IsBigOWith.inv_rev {f : α → 𝕜} {g : α → 𝕜'} (h : IsBigOWith c l f g)
    (h₀ : ∀ᶠ x in l, f x = 0 → g x = 0) : IsBigOWith c l (fun x => (g x)⁻¹) fun x => (f x)⁻¹ 
Arguments: {α : Type u_1} {𝕜 : Type u_15} {𝕜' : Type u_16} [NormedDivisionRing 𝕜] [NormedDivisionRing 𝕜'] {c : ℝ} {l : Filter α} {f : α → 𝕜} {g : α → 𝕜'} (h : Asymptotics.IsBigOWith c l f g) (h₀ : ∀ᶠ (x : α) in l, f x = 0 → g x = 0)
Goal: Asymptotics.IsBigOWith c l (fun x => (g x)⁻¹) fun x => (f x)⁻¹