ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Asymptotics.SuperpolynomialDecay.add
theorem SuperpolynomialDecay.add [ContinuousAdd β] (hf : SuperpolynomialDecay l k f)
    (hg : SuperpolynomialDecay l k g) : SuperpolynomialDecay l k (f + g) 
Arguments: {α : Type u_1} {β : Type u_2} {l : Filter α} {k f g : α → β} [TopologicalSpace β] [CommSemiring β] [ContinuousAdd β] (hf : Asymptotics.SuperpolynomialDecay l k f) (hg : Asymptotics.SuperpolynomialDecay l k g)
Goal: Asymptotics.SuperpolynomialDecay l k (f + g)
-- Asymptotics.SuperpolynomialDecay.mul
theorem SuperpolynomialDecay.mul [ContinuousMul β] (hf : SuperpolynomialDecay l k f)
    (hg : SuperpolynomialDecay l k g) : SuperpolynomialDecay l k (f * g) 
Arguments: {α : Type u_1} {β : Type u_2} {l : Filter α} {k f g : α → β} [TopologicalSpace β] [CommSemiring β] [ContinuousMul β] (hf : Asymptotics.SuperpolynomialDecay l k f) (hg : Asymptotics.SuperpolynomialDecay l k g)
Goal: Asymptotics.SuperpolynomialDecay l k (f * g)
-- Asymptotics.SuperpolynomialDecay.mul_const
theorem SuperpolynomialDecay.mul_const [ContinuousMul β] (hf : SuperpolynomialDecay l k f) (c : β) :
    SuperpolynomialDecay l k fun n => f n * c 
Arguments: {α : Type u_1} {β : Type u_2} {l : Filter α} {k f : α → β} [TopologicalSpace β] [CommSemiring β] [ContinuousMul β] (hf : Asymptotics.SuperpolynomialDecay l k f) (c : β)
Goal: Asymptotics.SuperpolynomialDecay l k fun n => f n * c
-- Asymptotics.SuperpolynomialDecay.const_mul
theorem SuperpolynomialDecay.const_mul [ContinuousMul β] (hf : SuperpolynomialDecay l k f) (c : β) :
    SuperpolynomialDecay l k fun n => c * f n 
Arguments: {α : Type u_1} {β : Type u_2} {l : Filter α} {k f : α → β} [TopologicalSpace β] [CommSemiring β] [ContinuousMul β] (hf : Asymptotics.SuperpolynomialDecay l k f) (c : β)
Goal: Asymptotics.SuperpolynomialDecay l k fun n => c * f n
-- Asymptotics.SuperpolynomialDecay.param_mul
theorem SuperpolynomialDecay.param_mul (hf : SuperpolynomialDecay l k f) :
    SuperpolynomialDecay l k (k * f) 
Arguments: {α : Type u_1} {β : Type u_2} {l : Filter α} {k f : α → β} [TopologicalSpace β] [CommSemiring β] (hf : Asymptotics.SuperpolynomialDecay l k f)
Goal: Asymptotics.SuperpolynomialDecay l k (k * f)
-- Asymptotics.SuperpolynomialDecay.mul_param
theorem SuperpolynomialDecay.mul_param (hf : SuperpolynomialDecay l k f) :
    SuperpolynomialDecay l k (f * k) 
Arguments: {α : Type u_1} {β : Type u_2} {l : Filter α} {k f : α → β} [TopologicalSpace β] [CommSemiring β] (hf : Asymptotics.SuperpolynomialDecay l k f)
Goal: Asymptotics.SuperpolynomialDecay l k (f * k)
-- Asymptotics.SuperpolynomialDecay.param_pow_mul
theorem SuperpolynomialDecay.param_pow_mul (hf : SuperpolynomialDecay l k f) (n : ℕ) :
    SuperpolynomialDecay l k (k ^ n * f) 
Arguments: {α : Type u_1} {β : Type u_2} {l : Filter α} {k f : α → β} [TopologicalSpace β] [CommSemiring β] (hf : Asymptotics.SuperpolynomialDecay l k f) (n : ℕ)
Goal: Asymptotics.SuperpolynomialDecay l k (k ^ n * f)
-- Asymptotics.SuperpolynomialDecay.mul_param_pow
theorem SuperpolynomialDecay.mul_param_pow (hf : SuperpolynomialDecay l k f) (n : ℕ) :
    SuperpolynomialDecay l k (f * k ^ n) 
Arguments: {α : Type u_1} {β : Type u_2} {l : Filter α} {k f : α → β} [TopologicalSpace β] [CommSemiring β] (hf : Asymptotics.SuperpolynomialDecay l k f) (n : ℕ)
Goal: Asymptotics.SuperpolynomialDecay l k (f * k ^ n)
-- Asymptotics.SuperpolynomialDecay.polynomial_mul
theorem SuperpolynomialDecay.polynomial_mul [ContinuousAdd β] [ContinuousMul β]
    (hf : SuperpolynomialDecay l k f) (p : β[X]) :
    SuperpolynomialDecay l k fun x => (p.eval <| k x) * f x 
Arguments: {α : Type u_1} {β : Type u_2} {l : Filter α} {k f : α → β} [TopologicalSpace β] [CommSemiring β] [ContinuousAdd β] [ContinuousMul β] (hf : Asymptotics.SuperpolynomialDecay l k f) (p : Polynomial β)
Goal: Asymptotics.SuperpolynomialDecay l k fun x => Polynomial.eval (k x) p * f x
-- Asymptotics.SuperpolynomialDecay.mul_polynomial
theorem SuperpolynomialDecay.mul_polynomial [ContinuousAdd β] [ContinuousMul β]
    (hf : SuperpolynomialDecay l k f) (p : β[X]) :
    SuperpolynomialDecay l k fun x => f x * (p.eval <| k x) 
Arguments: {α : Type u_1} {β : Type u_2} {l : Filter α} {k f : α → β} [TopologicalSpace β] [CommSemiring β] [ContinuousAdd β] [ContinuousMul β] (hf : Asymptotics.SuperpolynomialDecay l k f) (p : Polynomial β)
Goal: Asymptotics.SuperpolynomialDecay l k fun x => f x * Polynomial.eval (k x) p
-- Asymptotics.SuperpolynomialDecay.trans_eventuallyLE
theorem SuperpolynomialDecay.trans_eventuallyLE (hk : 0 ≤ᶠ[l] k) (hg : SuperpolynomialDecay l k g)
    (hg' : SuperpolynomialDecay l k g') (hfg : g ≤ᶠ[l] f) (hfg' : f ≤ᶠ[l] g') :
    SuperpolynomialDecay l k f 
Arguments: {α : Type u_1} {β : Type u_2} {l : Filter α} {k f g g' : α → β} [TopologicalSpace β] [OrderedCommSemiring β] [OrderTopology β] (hk : 0 ≤ᶠ[l] k) (hg : Asymptotics.SuperpolynomialDecay l k g) (hg' : Asymptotics.SuperpolynomialDecay l k g') (hfg : g ≤ᶠ[l] f) (hfg' : f ≤ᶠ[l] g')
Goal: Asymptotics.SuperpolynomialDecay l k f
-- Asymptotics.superpolynomialDecay_iff_abs_tendsto_zero
theorem superpolynomialDecay_iff_abs_tendsto_zero :
    SuperpolynomialDecay l k f ↔ ∀ n : ℕ, Tendsto (fun a : α => |k a ^ n * f a|) l (𝓝 0) 
Arguments: {α : Type u_1} {β : Type u_2} (l : Filter α) (k f : α → β) [TopologicalSpace β] [LinearOrderedCommRing β] [OrderTopology β]
Goal: Asymptotics.SuperpolynomialDecay l k f ↔ ∀ (n : ℕ), Filter.Tendsto (fun a => |k a ^ n * f a|) l (nhds 0)
-- Asymptotics.superpolynomialDecay_iff_superpolynomialDecay_abs
theorem superpolynomialDecay_iff_superpolynomialDecay_abs :
    SuperpolynomialDecay l k f ↔ SuperpolynomialDecay l (fun a => |k a|) fun a => |f a| 
Arguments: {α : Type u_1} {β : Type u_2} (l : Filter α) (k f : α → β) [TopologicalSpace β] [LinearOrderedCommRing β] [OrderTopology β]
Goal: Asymptotics.SuperpolynomialDecay l k f ↔ Asymptotics.SuperpolynomialDecay l (fun a => |k a|) fun a => |f a|
-- Asymptotics.SuperpolynomialDecay.trans_eventually_abs_le
theorem SuperpolynomialDecay.trans_eventually_abs_le (hf : SuperpolynomialDecay l k f)
    (hfg : abs ∘ g ≤ᶠ[l] abs ∘ f) : SuperpolynomialDecay l k g 
Arguments: {α : Type u_1} {β : Type u_2} {l : Filter α} {k f g : α → β} [TopologicalSpace β] [LinearOrderedCommRing β] [OrderTopology β] (hf : Asymptotics.SuperpolynomialDecay l k f) (hfg : abs ∘ g ≤ᶠ[l] abs ∘ f)
Goal: Asymptotics.SuperpolynomialDecay l k g
-- Asymptotics.SuperpolynomialDecay.trans_abs_le
theorem SuperpolynomialDecay.trans_abs_le (hf : SuperpolynomialDecay l k f)
    (hfg : ∀ x, |g x| ≤ |f x|) : SuperpolynomialDecay l k g 
Arguments: {α : Type u_1} {β : Type u_2} {l : Filter α} {k f g : α → β} [TopologicalSpace β] [LinearOrderedCommRing β] [OrderTopology β] (hf : Asymptotics.SuperpolynomialDecay l k f) (hfg : ∀ (x : α), |g x| ≤ |f x|)
Goal: Asymptotics.SuperpolynomialDecay l k g
-- Asymptotics.superpolynomialDecay_mul_const_iff
theorem superpolynomialDecay_mul_const_iff [ContinuousMul β] {c : β} (hc0 : c ≠ 0) :
    (SuperpolynomialDecay l k fun n => f n * c) ↔ SuperpolynomialDecay l k f 
Arguments: {α : Type u_1} {β : Type u_2} (l : Filter α) (k f : α → β) [TopologicalSpace β] [Field β] [ContinuousMul β] {c : β} (hc0 : c ≠ 0)
Goal: (Asymptotics.SuperpolynomialDecay l k fun n => f n * c) ↔ Asymptotics.SuperpolynomialDecay l k f
-- Asymptotics.superpolynomialDecay_const_mul_iff
theorem superpolynomialDecay_const_mul_iff [ContinuousMul β] {c : β} (hc0 : c ≠ 0) :
    (SuperpolynomialDecay l k fun n => c * f n) ↔ SuperpolynomialDecay l k f 
Arguments: {α : Type u_1} {β : Type u_2} (l : Filter α) (k f : α → β) [TopologicalSpace β] [Field β] [ContinuousMul β] {c : β} (hc0 : c ≠ 0)
Goal: (Asymptotics.SuperpolynomialDecay l k fun n => c * f n) ↔ Asymptotics.SuperpolynomialDecay l k f
-- Asymptotics.superpolynomialDecay_iff_abs_isBoundedUnder
theorem superpolynomialDecay_iff_abs_isBoundedUnder (hk : Tendsto k l atTop) :
    SuperpolynomialDecay l k f ↔
    ∀ z : ℕ, IsBoundedUnder (· ≤ ·) l fun a : α => |k a ^ z * f a| 
Arguments: {α : Type u_1} {β : Type u_2} {l : Filter α} {k : α → β} (f : α → β) [TopologicalSpace β] [LinearOrderedField β] [OrderTopology β] (hk : Filter.Tendsto k l Filter.atTop)
Goal: Asymptotics.SuperpolynomialDecay l k f ↔
  ∀ (z : ℕ), Filter.IsBoundedUnder (fun x x_1 => x ≤ x_1) l fun a => |k a ^ z * f a|
-- Asymptotics.superpolynomialDecay_iff_zpow_tendsto_zero
theorem superpolynomialDecay_iff_zpow_tendsto_zero (hk : Tendsto k l atTop) :
    SuperpolynomialDecay l k f ↔ ∀ z : ℤ, Tendsto (fun a : α => k a ^ z * f a) l (𝓝 0) 
Arguments: {α : Type u_1} {β : Type u_2} {l : Filter α} {k : α → β} (f : α → β) [TopologicalSpace β] [LinearOrderedField β] [OrderTopology β] (hk : Filter.Tendsto k l Filter.atTop)
Goal: Asymptotics.SuperpolynomialDecay l k f ↔ ∀ (z : ℤ), Filter.Tendsto (fun a => k a ^ z * f a) l (nhds 0)
-- Asymptotics.SuperpolynomialDecay.param_zpow_mul
theorem SuperpolynomialDecay.param_zpow_mul (hk : Tendsto k l atTop)
    (hf : SuperpolynomialDecay l k f) (z : ℤ) :
    SuperpolynomialDecay l k fun a => k a ^ z * f a 
Arguments: {α : Type u_1} {β : Type u_2} {l : Filter α} {k f : α → β} [TopologicalSpace β] [LinearOrderedField β] [OrderTopology β] (hk : Filter.Tendsto k l Filter.atTop) (hf : Asymptotics.SuperpolynomialDecay l k f) (z : ℤ)
Goal: Asymptotics.SuperpolynomialDecay l k fun a => k a ^ z * f a
-- Asymptotics.SuperpolynomialDecay.mul_param_zpow
theorem SuperpolynomialDecay.mul_param_zpow (hk : Tendsto k l atTop)
    (hf : SuperpolynomialDecay l k f) (z : ℤ) : SuperpolynomialDecay l k fun a => f a * k a ^ z 
Arguments: {α : Type u_1} {β : Type u_2} {l : Filter α} {k f : α → β} [TopologicalSpace β] [LinearOrderedField β] [OrderTopology β] (hk : Filter.Tendsto k l Filter.atTop) (hf : Asymptotics.SuperpolynomialDecay l k f) (z : ℤ)
Goal: Asymptotics.SuperpolynomialDecay l k fun a => f a * k a ^ z
-- Asymptotics.SuperpolynomialDecay.inv_param_mul
theorem SuperpolynomialDecay.inv_param_mul (hk : Tendsto k l atTop)
    (hf : SuperpolynomialDecay l k f) : SuperpolynomialDecay l k (k⁻¹ * f) 
Arguments: {α : Type u_1} {β : Type u_2} {l : Filter α} {k f : α → β} [TopologicalSpace β] [LinearOrderedField β] [OrderTopology β] (hk : Filter.Tendsto k l Filter.atTop) (hf : Asymptotics.SuperpolynomialDecay l k f)
Goal: Asymptotics.SuperpolynomialDecay l k (k⁻¹ * f)
-- Asymptotics.SuperpolynomialDecay.param_inv_mul
theorem SuperpolynomialDecay.param_inv_mul (hk : Tendsto k l atTop)
    (hf : SuperpolynomialDecay l k f) : SuperpolynomialDecay l k (f * k⁻¹) 
Arguments: {α : Type u_1} {β : Type u_2} {l : Filter α} {k f : α → β} [TopologicalSpace β] [LinearOrderedField β] [OrderTopology β] (hk : Filter.Tendsto k l Filter.atTop) (hf : Asymptotics.SuperpolynomialDecay l k f)
Goal: Asymptotics.SuperpolynomialDecay l k (f * k⁻¹)
-- Asymptotics.superpolynomialDecay_param_mul_iff
theorem superpolynomialDecay_param_mul_iff (hk : Tendsto k l atTop) :
    SuperpolynomialDecay l k (k * f) ↔ SuperpolynomialDecay l k f 
Arguments: {α : Type u_1} {β : Type u_2} {l : Filter α} {k : α → β} (f : α → β) [TopologicalSpace β] [LinearOrderedField β] [OrderTopology β] (hk : Filter.Tendsto k l Filter.atTop)
Goal: Asymptotics.SuperpolynomialDecay l k (k * f) ↔ Asymptotics.SuperpolynomialDecay l k f
-- Asymptotics.superpolynomialDecay_mul_param_iff
theorem superpolynomialDecay_mul_param_iff (hk : Tendsto k l atTop) :
    SuperpolynomialDecay l k (f * k) ↔ SuperpolynomialDecay l k f 
Arguments: {α : Type u_1} {β : Type u_2} {l : Filter α} {k : α → β} (f : α → β) [TopologicalSpace β] [LinearOrderedField β] [OrderTopology β] (hk : Filter.Tendsto k l Filter.atTop)
Goal: Asymptotics.SuperpolynomialDecay l k (f * k) ↔ Asymptotics.SuperpolynomialDecay l k f
-- Asymptotics.superpolynomialDecay_param_pow_mul_iff
theorem superpolynomialDecay_param_pow_mul_iff (hk : Tendsto k l atTop) (n : ℕ) :
    SuperpolynomialDecay l k (k ^ n * f) ↔ SuperpolynomialDecay l k f 
Arguments: {α : Type u_1} {β : Type u_2} {l : Filter α} {k : α → β} (f : α → β) [TopologicalSpace β] [LinearOrderedField β] [OrderTopology β] (hk : Filter.Tendsto k l Filter.atTop) (n : ℕ)
Goal: Asymptotics.SuperpolynomialDecay l k (k ^ n * f) ↔ Asymptotics.SuperpolynomialDecay l k f
-- Asymptotics.superpolynomialDecay_mul_param_pow_iff
theorem superpolynomialDecay_mul_param_pow_iff (hk : Tendsto k l atTop) (n : ℕ) :
    SuperpolynomialDecay l k (f * k ^ n) ↔ SuperpolynomialDecay l k f 
Arguments: {α : Type u_1} {β : Type u_2} {l : Filter α} {k : α → β} (f : α → β) [TopologicalSpace β] [LinearOrderedField β] [OrderTopology β] (hk : Filter.Tendsto k l Filter.atTop) (n : ℕ)
Goal: Asymptotics.SuperpolynomialDecay l k (f * k ^ n) ↔ Asymptotics.SuperpolynomialDecay l k f
-- Asymptotics.superpolynomialDecay_iff_norm_tendsto_zero
theorem superpolynomialDecay_iff_norm_tendsto_zero :
    SuperpolynomialDecay l k f ↔ ∀ n : ℕ, Tendsto (fun a : α => ‖k a ^ n * f a‖) l (𝓝 0) 
Arguments: {α : Type u_1} {β : Type u_2} (l : Filter α) (k f : α → β) [NormedLinearOrderedField β]
Goal: Asymptotics.SuperpolynomialDecay l k f ↔ ∀ (n : ℕ), Filter.Tendsto (fun a => ‖k a ^ n * f a‖) l (nhds 0)
-- Asymptotics.superpolynomialDecay_iff_superpolynomialDecay_norm
theorem superpolynomialDecay_iff_superpolynomialDecay_norm :
    SuperpolynomialDecay l k f ↔ SuperpolynomialDecay l (fun a => ‖k a‖) fun a => ‖f a‖ 
Arguments: {α : Type u_1} {β : Type u_2} (l : Filter α) (k f : α → β) [NormedLinearOrderedField β]
Goal: Asymptotics.SuperpolynomialDecay l k f ↔ Asymptotics.SuperpolynomialDecay l (fun a => ‖k a‖) fun a => ‖f a‖
-- Asymptotics.superpolynomialDecay_iff_isBigO
theorem superpolynomialDecay_iff_isBigO (hk : Tendsto k l atTop) :
    SuperpolynomialDecay l k f ↔ ∀ z : ℤ, f =O[l] fun a : α => k a ^ z 
Arguments: {α : Type u_1} {β : Type u_2} {l : Filter α} {k : α → β} (f : α → β) [NormedLinearOrderedField β] [OrderTopology β] (hk : Filter.Tendsto k l Filter.atTop)
Goal: Asymptotics.SuperpolynomialDecay l k f ↔ ∀ (z : ℤ), f =O[l] fun a => k a ^ z
-- Asymptotics.superpolynomialDecay_iff_isLittleO
theorem superpolynomialDecay_iff_isLittleO (hk : Tendsto k l atTop) :
    SuperpolynomialDecay l k f ↔ ∀ z : ℤ, f =o[l] fun a : α => k a ^ z 
Arguments: {α : Type u_1} {β : Type u_2} {l : Filter α} {k : α → β} (f : α → β) [NormedLinearOrderedField β] [OrderTopology β] (hk : Filter.Tendsto k l Filter.atTop)
Goal: Asymptotics.SuperpolynomialDecay l k f ↔ ∀ (z : ℤ), f =o[l] fun a => k a ^ z
-- norm_smul_le
theorem norm_smul_le (r : α) (x : β) : ‖r • x‖ ≤ ‖r‖ * ‖x‖ 
Arguments: {α : Type u_1} {β : Type u_2} [SeminormedAddGroup α] [SeminormedAddGroup β] [SMulZeroClass α β] [BoundedSMul α β] (r : α) (x : β)
Goal: ‖r • x‖ ≤ ‖r‖ * ‖x‖
-- nnnorm_smul_le
theorem nnnorm_smul_le (r : α) (x : β) : ‖r • x‖₊ ≤ ‖r‖₊ * ‖x‖₊ 
Arguments: {α : Type u_1} {β : Type u_2} [SeminormedAddGroup α] [SeminormedAddGroup β] [SMulZeroClass α β] [BoundedSMul α β] (r : α) (x : β)
Goal: ‖r • x‖₊ ≤ ‖r‖₊ * ‖x‖₊
-- dist_smul_le
theorem dist_smul_le (s : α) (x y : β) : dist (s • x) (s • y) ≤ ‖s‖ * dist x y 
Arguments: {α : Type u_1} {β : Type u_2} [SeminormedAddGroup α] [SeminormedAddGroup β] [SMulZeroClass α β] [BoundedSMul α β] (s : α) (x y : β)
Goal: dist (s • x) (s • y) ≤ ‖s‖ * dist x y
-- nndist_smul_le
theorem nndist_smul_le (s : α) (x y : β) : nndist (s • x) (s • y) ≤ ‖s‖₊ * nndist x y 
Arguments: {α : Type u_1} {β : Type u_2} [SeminormedAddGroup α] [SeminormedAddGroup β] [SMulZeroClass α β] [BoundedSMul α β] (s : α) (x y : β)
Goal: nndist (s • x) (s • y) ≤ ‖s‖₊ * nndist x y
-- lipschitzWith_smul
theorem lipschitzWith_smul (s : α) : LipschitzWith ‖s‖₊ (s • · : β → β) 
Arguments: {α : Type u_1} {β : Type u_2} [SeminormedAddGroup α] [SeminormedAddGroup β] [SMulZeroClass α β] [BoundedSMul α β] (s : α)
Goal: LipschitzWith ‖s‖₊ fun x => s • x
-- edist_smul_le
theorem edist_smul_le (s : α) (x y : β) : edist (s • x) (s • y) ≤ ‖s‖₊ • edist x y 
Arguments: {α : Type u_1} {β : Type u_2} [SeminormedAddGroup α] [SeminormedAddGroup β] [SMulZeroClass α β] [BoundedSMul α β] (s : α) (x y : β)
Goal: edist (s • x) (s • y) ≤ ‖s‖₊ • edist x y
-- BoundedSMul.of_norm_smul_le
theorem BoundedSMul.of_norm_smul_le (h : ∀ (r : α) (x : β), ‖r • x‖ ≤ ‖r‖ * ‖x‖) :
    BoundedSMul α β 
Arguments: {α : Type u_1} {β : Type u_2} [SeminormedRing α] [SeminormedAddCommGroup β] [Module α β] (h : ∀ (r : α) (x : β), ‖r • x‖ ≤ ‖r‖ * ‖x‖)
Goal: BoundedSMul α β
-- BoundedSMul.of_nnnorm_smul_le
theorem BoundedSMul.of_nnnorm_smul_le (h : ∀ (r : α) (x : β), ‖r • x‖₊ ≤ ‖r‖₊ * ‖x‖₊) :
    BoundedSMul α β 
Arguments: {α : Type u_1} {β : Type u_2} [SeminormedRing α] [SeminormedAddCommGroup β] [Module α β] (h : ∀ (r : α) (x : β), ‖r • x‖₊ ≤ ‖r‖₊ * ‖x‖₊)
Goal: BoundedSMul α β
-- norm_smul
theorem norm_smul (r : α) (x : β) : ‖r • x‖ = ‖r‖ * ‖x‖ 
Arguments: {α : Type u_1} {β : Type u_2} [NormedDivisionRing α] [SeminormedAddGroup β] [MulActionWithZero α β] [BoundedSMul α β] (r : α) (x : β)
Goal: ‖r • x‖ = ‖r‖ * ‖x‖
-- nnnorm_smul
theorem nnnorm_smul (r : α) (x : β) : ‖r • x‖₊ = ‖r‖₊ * ‖x‖₊ 
Arguments: {α : Type u_1} {β : Type u_2} [NormedDivisionRing α] [SeminormedAddGroup β] [MulActionWithZero α β] [BoundedSMul α β] (r : α) (x : β)
Goal: ‖r • x‖₊ = ‖r‖₊ * ‖x‖₊
-- dist_smul₀
theorem dist_smul₀ (s : α) (x y : β) : dist (s • x) (s • y) = ‖s‖ * dist x y 
Arguments: {α : Type u_1} {β : Type u_2} [NormedDivisionRing α] [SeminormedAddCommGroup β] [Module α β] [BoundedSMul α β] (s : α) (x y : β)
Goal: dist (s • x) (s • y) = ‖s‖ * dist x y
-- nndist_smul₀
theorem nndist_smul₀ (s : α) (x y : β) : nndist (s • x) (s • y) = ‖s‖₊ * nndist x y 
Arguments: {α : Type u_1} {β : Type u_2} [NormedDivisionRing α] [SeminormedAddCommGroup β] [Module α β] [BoundedSMul α β] (s : α) (x y : β)
Goal: nndist (s • x) (s • y) = ‖s‖₊ * nndist x y
-- edist_smul₀
theorem edist_smul₀ (s : α) (x y : β) : edist (s • x) (s • y) = ‖s‖₊ • edist x y 
Arguments: {α : Type u_1} {β : Type u_2} [NormedDivisionRing α] [SeminormedAddCommGroup β] [Module α β] [BoundedSMul α β] (s : α) (x y : β)
Goal: edist (s • x) (s • y) = ‖s‖₊ • edist x y
-- LinearMap.clmOfExistsBoundedImage_coe
theorem LinearMap.clmOfExistsBoundedImage_coe {f : E →ₗ[𝕜] F}
    {h : ∃ V ∈ 𝓝 (0 : E), Bornology.IsVonNBounded 𝕜 (f '' V)} :
    (f.clmOfExistsBoundedImage h : E →ₗ[𝕜] F) = f 
Arguments: {𝕜 : Type u_1} {E : Type u_3} {F : Type u_4} [AddCommGroup E] [UniformSpace E] [UniformAddGroup E] [AddCommGroup F] [UniformSpace F] [UniformAddGroup F] [NontriviallyNormedField 𝕜] [Module 𝕜 E] [Module 𝕜 F] [ContinuousSMul 𝕜 E] {f : E →ₗ[𝕜] F} {h : ∃ V ∈ 𝓝 0, Bornology.IsVonNBounded 𝕜 (⇑f '' V)}
Goal: ↑(f.clmOfExistsBoundedImage h) = f
-- LinearMap.clmOfExistsBoundedImage_apply
@[simp]
theorem LinearMap.clmOfExistsBoundedImage_apply {f : E →ₗ[𝕜] F}
    {h : ∃ V ∈ 𝓝 (0 : E), Bornology.IsVonNBounded 𝕜 (f '' V)} {x : E} :
    f.clmOfExistsBoundedImage h x = f x 
Arguments: {𝕜 : Type u_1} {E : Type u_3} {F : Type u_4} [AddCommGroup E] [UniformSpace E] [UniformAddGroup E] [AddCommGroup F] [UniformSpace F] [UniformAddGroup F] [NontriviallyNormedField 𝕜] [Module 𝕜 E] [Module 𝕜 F] [ContinuousSMul 𝕜 E] {f : E →ₗ[𝕜] F} {h : ∃ V ∈ 𝓝 0, Bornology.IsVonNBounded 𝕜 (⇑f '' V)} {x : E}
Goal: (f.clmOfExistsBoundedImage h) x = f x
-- LinearMap.continuousAt_zero_of_locally_bounded
theorem LinearMap.continuousAt_zero_of_locally_bounded (f : E →ₛₗ[σ] F)
    (hf : ∀ s, IsVonNBounded 𝕜 s → IsVonNBounded 𝕜' (f '' s)) : ContinuousAt f 0 
Arguments: {𝕜 : Type u_1} {𝕜' : Type u_2} {E : Type u_3} {F : Type u_4} [AddCommGroup E] [UniformSpace E] [UniformAddGroup E] [AddCommGroup F] [UniformSpace F] [FirstCountableTopology E] [RCLike 𝕜] [Module 𝕜 E] [ContinuousSMul 𝕜 E] [RCLike 𝕜'] [Module 𝕜' F] [ContinuousSMul 𝕜' F] {σ : 𝕜 →+* 𝕜'} (f : E →ₛₗ[σ] F) (hf : ∀ (s : Set E), Bornology.IsVonNBounded 𝕜 s → Bornology.IsVonNBounded 𝕜' (⇑f '' s))
Goal: ContinuousAt (⇑f) 0
-- LinearMap.continuous_of_locally_bounded
theorem LinearMap.continuous_of_locally_bounded [UniformAddGroup F] (f : E →ₛₗ[σ] F)
    (hf : ∀ s, IsVonNBounded 𝕜 s → IsVonNBounded 𝕜' (f '' s)) : Continuous f 
Arguments: {𝕜 : Type u_1} {𝕜' : Type u_2} {E : Type u_3} {F : Type u_4} [AddCommGroup E] [UniformSpace E] [UniformAddGroup E] [AddCommGroup F] [UniformSpace F] [FirstCountableTopology E] [RCLike 𝕜] [Module 𝕜 E] [ContinuousSMul 𝕜 E] [RCLike 𝕜'] [Module 𝕜' F] [ContinuousSMul 𝕜' F] {σ : 𝕜 →+* 𝕜'} [UniformAddGroup F] (f : E →ₛₗ[σ] F) (hf : ∀ (s : Set E), Bornology.IsVonNBounded 𝕜 s → Bornology.IsVonNBounded 𝕜' (⇑f '' s))
Goal: Continuous ⇑f
-- ContinuousLinearMap.opNorm_ext
theorem opNorm_ext [RingHomIsometric σ₁₃] (f : E →SL[σ₁₂] F) (g : E →SL[σ₁₃] G)
    (h : ∀ x, ‖f x‖ = ‖g x‖) : ‖f‖ = ‖g‖ 
Arguments: {𝕜 : Type u_1} {𝕜₂ : Type u_2} {𝕜₃ : Type u_3} {E : Type u_4} {F : Type u_6} {G : Type u_8} [SeminormedAddCommGroup E] [SeminormedAddCommGroup F] [SeminormedAddCommGroup G] [NontriviallyNormedField 𝕜] [NontriviallyNormedField 𝕜₂] [NontriviallyNormedField 𝕜₃] [NormedSpace 𝕜 E] [NormedSpace 𝕜₂ F] [NormedSpace 𝕜₃ G] {σ₁₂ : 𝕜 →+* 𝕜₂} {σ₁₃ : 𝕜 →+* 𝕜₃} [RingHomIsometric σ₁₃] (f : E →SL[σ₁₂] F) (g : E →SL[σ₁₃] G) (h : ∀ (x : E), ‖f x‖ = ‖g x‖)
Goal: ‖f‖ = ‖g‖
-- ContinuousLinearMap.opNorm_le_bound₂
theorem opNorm_le_bound₂ (f : E →SL[σ₁₃] F →SL[σ₂₃] G) {C : ℝ} (h0 : 0 ≤ C)
    (hC : ∀ x y, ‖f x y‖ ≤ C * ‖x‖ * ‖y‖) : ‖f‖ ≤ C 
Arguments: {𝕜 : Type u_1} {𝕜₂ : Type u_2} {𝕜₃ : Type u_3} {E : Type u_4} {F : Type u_6} {G : Type u_8} [SeminormedAddCommGroup E] [SeminormedAddCommGroup F] [SeminormedAddCommGroup G] [NontriviallyNormedField 𝕜] [NontriviallyNormedField 𝕜₂] [NontriviallyNormedField 𝕜₃] [NormedSpace 𝕜 E] [NormedSpace 𝕜₂ F] [NormedSpace 𝕜₃ G] {σ₂₃ : 𝕜₂ →+* 𝕜₃} {σ₁₃ : 𝕜 →+* 𝕜₃} [RingHomIsometric σ₂₃] (f : E →SL[σ₁₃] F →SL[σ₂₃] G) {C : ℝ} (h0 : 0 ≤ C) (hC : ∀ (x : E) (y : F), ‖(f x) y‖ ≤ C * ‖x‖ * ‖y‖)
Goal: ‖f‖ ≤ C
-- ContinuousLinearMap.le_opNorm₂
theorem le_opNorm₂ [RingHomIsometric σ₁₃] (f : E →SL[σ₁₃] F →SL[σ₂₃] G) (x : E) (y : F) :
    ‖f x y‖ ≤ ‖f‖ * ‖x‖ * ‖y‖ 
Arguments: {𝕜 : Type u_1} {𝕜₂ : Type u_2} {𝕜₃ : Type u_3} {E : Type u_4} {F : Type u_6} {G : Type u_8} [SeminormedAddCommGroup E] [SeminormedAddCommGroup F] [SeminormedAddCommGroup G] [NontriviallyNormedField 𝕜] [NontriviallyNormedField 𝕜₂] [NontriviallyNormedField 𝕜₃] [NormedSpace 𝕜 E] [NormedSpace 𝕜₂ F] [NormedSpace 𝕜₃ G] {σ₂₃ : 𝕜₂ →+* 𝕜₃} {σ₁₃ : 𝕜 →+* 𝕜₃} [RingHomIsometric σ₂₃] [RingHomIsometric σ₁₃] (f : E →SL[σ₁₃] F →SL[σ₂₃] G) (x : E) (y : F)
Goal: ‖(f x) y‖ ≤ ‖f‖ * ‖x‖ * ‖y‖
-- ContinuousLinearMap.le_of_opNorm₂_le_of_le
theorem le_of_opNorm₂_le_of_le [RingHomIsometric σ₁₃] (f : E →SL[σ₁₃] F →SL[σ₂₃] G) {x : E} {y : F}
    {a b c : ℝ} (hf : ‖f‖ ≤ a) (hx : ‖x‖ ≤ b) (hy : ‖y‖ ≤ c) :
    ‖f x y‖ ≤ a * b * c 
Arguments: {𝕜 : Type u_1} {𝕜₂ : Type u_2} {𝕜₃ : Type u_3} {E : Type u_4} {F : Type u_6} {G : Type u_8} [SeminormedAddCommGroup E] [SeminormedAddCommGroup F] [SeminormedAddCommGroup G] [NontriviallyNormedField 𝕜] [NontriviallyNormedField 𝕜₂] [NontriviallyNormedField 𝕜₃] [NormedSpace 𝕜 E] [NormedSpace 𝕜₂ F] [NormedSpace 𝕜₃ G] {σ₂₃ : 𝕜₂ →+* 𝕜₃} {σ₁₃ : 𝕜 →+* 𝕜₃} [RingHomIsometric σ₂₃] [RingHomIsometric σ₁₃] (f : E →SL[σ₁₃] F →SL[σ₂₃] G) {x : E} {y : F} {a b c : ℝ} (hf : ‖f‖ ≤ a) (hx : ‖x‖ ≤ b) (hy : ‖y‖ ≤ c)
Goal: ‖(f x) y‖ ≤ a * b * c
-- LinearMap.norm_mkContinuous₂_aux
lemma norm_mkContinuous₂_aux (f : E →ₛₗ[σ₁₃] F →ₛₗ[σ₂₃] G) (C : ℝ)
    (h : ∀ x y, ‖f x y‖ ≤ C * ‖x‖ * ‖y‖) (x : E) :
    ‖(f x).mkContinuous (C * ‖x‖) (h x)‖ ≤ max C 0 * ‖x‖ 
Arguments: {𝕜 : Type u_1} {𝕜₂ : Type u_2} {𝕜₃ : Type u_3} {E : Type u_4} {F : Type u_6} {G : Type u_8} [SeminormedAddCommGroup E] [SeminormedAddCommGroup F] [SeminormedAddCommGroup G] [NontriviallyNormedField 𝕜] [NontriviallyNormedField 𝕜₂] [NontriviallyNormedField 𝕜₃] [NormedSpace 𝕜 E] [NormedSpace 𝕜₂ F] [NormedSpace 𝕜₃ G] {σ₂₃ : 𝕜₂ →+* 𝕜₃} {σ₁₃ : 𝕜 →+* 𝕜₃} (f : E →ₛₗ[σ₁₃] F →ₛₗ[σ₂₃] G) (C : ℝ) (h : ∀ (x : E) (y : F), ‖(f x) y‖ ≤ C * ‖x‖ * ‖y‖) (x : E)
Goal: ‖(f x).mkContinuous (C * ‖x‖) ⋯‖ ≤ max C 0 * ‖x‖