ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- hasDerivAt_fourier_neg
theorem hasDerivAt_fourier_neg (n : ℤ) (x : ℝ) :
    HasDerivAt (fun y : ℝ => fourier (-n) (y : AddCircle T))
      (-2 * π * I * n / T * fourier (-n) (x : AddCircle T)) x 
Arguments: (T : ℝ) (n : ℤ) (x : ℝ)
Goal: HasDerivAt (fun y => (fourier (-n)) ↑y) (-2 * ↑π * Complex.I * ↑n / ↑T * (fourier (-n)) ↑x) x
-- has_antideriv_at_fourier_neg
theorem has_antideriv_at_fourier_neg (hT : Fact (0 < T)) {n : ℤ} (hn : n ≠ 0) (x : ℝ) :
    HasDerivAt (fun y : ℝ => (T : ℂ) / (-2 * π * I * n) * fourier (-n) (y : AddCircle T))
      (fourier (-n) (x : AddCircle T)) x 
Arguments: {T : ℝ} (hT : Fact (0 < T)) {n : ℤ} (hn : n ≠ 0) (x : ℝ)
Goal: HasDerivAt (fun y => ↑T / (-2 * ↑π * Complex.I * ↑n) * (fourier (-n)) ↑y) ((fourier (-n)) ↑x) x
-- fourierCoeffOn_of_hasDeriv_right
theorem fourierCoeffOn_of_hasDeriv_right {a b : ℝ} (hab : a < b) {f f' : ℝ → ℂ}
    {n : ℤ} (hn : n ≠ 0)
    (hf : ContinuousOn f [[a, b]])
    (hff' : ∀ x, x ∈ Ioo (min a b) (max a b) → HasDerivWithinAt f (f' x) (Ioi x) x)
    (hf' : IntervalIntegrable f' volume a b) :
    fourierCoeffOn hab f n = 1 / (-2 * π * I * n) *
      (fourier (-n) (a : AddCircle (b - a)) * (f b - f a) - (b - a) * fourierCoeffOn hab f' n) 
Arguments: {a b : ℝ} (hab : a < b) {f f' : ℝ → ℂ} {n : ℤ} (hn : n ≠ 0) (hf : ContinuousOn f (Set.uIcc a b)) (hff' : ∀ x ∈ Set.Ioo (min a b) (max a b), HasDerivWithinAt f (f' x) (Set.Ioi x) x) (hf' : IntervalIntegrable f' MeasureTheory.volume a b)
Goal: fourierCoeffOn hab f n =
  1 / (-2 * ↑π * Complex.I * ↑n) * ((fourier (-n)) ↑a * (f b - f a) - (↑b - ↑a) * fourierCoeffOn hab f' n)
-- fourierCoeffOn_of_hasDerivAt_Ioo
theorem fourierCoeffOn_of_hasDerivAt_Ioo {a b : ℝ} (hab : a < b) {f f' : ℝ → ℂ}
    {n : ℤ} (hn : n ≠ 0)
    (hf : ContinuousOn f [[a, b]])
    (hff' : ∀ x, x ∈ Ioo (min a b) (max a b) → HasDerivAt f (f' x) x)
    (hf' : IntervalIntegrable f' volume a b) :
    fourierCoeffOn hab f n = 1 / (-2 * π * I * n) *
      (fourier (-n) (a : AddCircle (b - a)) * (f b - f a) - (b - a) * fourierCoeffOn hab f' n) 
Arguments: {a b : ℝ} (hab : a < b) {f f' : ℝ → ℂ} {n : ℤ} (hn : n ≠ 0) (hf : ContinuousOn f (Set.uIcc a b)) (hff' : ∀ x ∈ Set.Ioo (min a b) (max a b), HasDerivAt f (f' x) x) (hf' : IntervalIntegrable f' MeasureTheory.volume a b)
Goal: fourierCoeffOn hab f n =
  1 / (-2 * ↑π * Complex.I * ↑n) * ((fourier (-n)) ↑a * (f b - f a) - (↑b - ↑a) * fourierCoeffOn hab f' n)
-- fourierCoeffOn_of_hasDerivAt
theorem fourierCoeffOn_of_hasDerivAt {a b : ℝ} (hab : a < b) {f f' : ℝ → ℂ} {n : ℤ} (hn : n ≠ 0)
    (hf : ∀ x, x ∈ [[a, b]] → HasDerivAt f (f' x) x) (hf' : IntervalIntegrable f' volume a b) :
    fourierCoeffOn hab f n = 1 / (-2 * π * I * n) *
      (fourier (-n) (a : AddCircle (b - a)) * (f b - f a) - (b - a) * fourierCoeffOn hab f' n) 
Arguments: {a b : ℝ} (hab : a < b) {f f' : ℝ → ℂ} {n : ℤ} (hn : n ≠ 0) (hf : ∀ x ∈ Set.uIcc a b, HasDerivAt f (f' x) x) (hf' : IntervalIntegrable f' MeasureTheory.volume a b)
Goal: fourierCoeffOn hab f n =
  1 / (-2 * ↑π * Complex.I * ↑n) * ((fourier (-n)) ↑a * (f b - f a) - (↑b - ↑a) * fourierCoeffOn hab f' n)
-- eventually_nnnorm_sub_lt
theorem eventually_nnnorm_sub_lt (x₀ : E) {ε : ℝ≥0} (ε_pos : 0 < ε) :
    ∀ᶠ x in 𝓝 x₀, ‖x - x₀‖₊ < ε 
Arguments: {E : Type u_1} [SeminormedAddCommGroup E] (x₀ : E) {ε : ℝ≥0} (ε_pos : 0 < ε)
Goal: ∀ᶠ (x : E) in 𝓝 x₀, ‖x - x₀‖₊ < ε
-- eventually_norm_sub_lt
theorem eventually_norm_sub_lt (x₀ : E) {ε : ℝ} (ε_pos : 0 < ε) :
    ∀ᶠ x in 𝓝 x₀, ‖x - x₀‖ < ε 
Arguments: {E : Type u_1} [SeminormedAddCommGroup E] (x₀ : E) {ε : ℝ} (ε_pos : 0 < ε)
Goal: ∀ᶠ (x : E) in 𝓝 x₀, ‖x - x₀‖ < ε
-- continuous_of_linear_of_boundₛₗ
theorem continuous_of_linear_of_boundₛₗ {f : E → F} (h_add : ∀ x y, f (x + y) = f x + f y)
    (h_smul : ∀ (c : 𝕜) (x), f (c • x) = σ c • f x) {C : ℝ} (h_bound : ∀ x, ‖f x‖ ≤ C * ‖x‖) :
    Continuous f 
Arguments: {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_3} {F : Type u_4} [Ring 𝕜] [Ring 𝕜₂] [SeminormedAddCommGroup E] [SeminormedAddCommGroup F] [Module 𝕜 E] [Module 𝕜₂ F] {σ : 𝕜 →+* 𝕜₂} {f : E → F} (h_add : ∀ (x y : E), f (x + y) = f x + f y) (h_smul : ∀ (c : 𝕜) (x : E), f (c • x) = σ c • f x) {C : ℝ} (h_bound : ∀ (x : E), ‖f x‖ ≤ C * ‖x‖)
Goal: Continuous f
-- continuous_of_linear_of_bound
theorem continuous_of_linear_of_bound {f : E → G} (h_add : ∀ x y, f (x + y) = f x + f y)
    (h_smul : ∀ (c : 𝕜) (x), f (c • x) = c • f x) {C : ℝ} (h_bound : ∀ x, ‖f x‖ ≤ C * ‖x‖) :
    Continuous f 
Arguments: {𝕜 : Type u_1} {E : Type u_3} {G : Type u_5} [Ring 𝕜] [SeminormedAddCommGroup E] [SeminormedAddCommGroup G] [Module 𝕜 E] [Module 𝕜 G] {f : E → G} (h_add : ∀ (x y : E), f (x + y) = f x + f y) (h_smul : ∀ (c : 𝕜) (x : E), f (c • x) = c • f x) {C : ℝ} (h_bound : ∀ (x : E), ‖f x‖ ≤ C * ‖x‖)
Goal: Continuous f
-- LinearMap.mkContinuous_coe
@[simp, norm_cast]
theorem LinearMap.mkContinuous_coe (C : ℝ) (h : ∀ x, ‖f x‖ ≤ C * ‖x‖) :
    (f.mkContinuous C h : E →ₛₗ[σ] F) = f 
Arguments: {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_3} {F : Type u_4} [Ring 𝕜] [Ring 𝕜₂] [SeminormedAddCommGroup E] [SeminormedAddCommGroup F] [Module 𝕜 E] [Module 𝕜₂ F] {σ : 𝕜 →+* 𝕜₂} (f : E →ₛₗ[σ] F) (C : ℝ) (h : ∀ (x : E), ‖f x‖ ≤ C * ‖x‖)
Goal: ↑(f.mkContinuous C h) = f
-- LinearMap.mkContinuous_apply
@[simp]
theorem LinearMap.mkContinuous_apply (C : ℝ) (h : ∀ x, ‖f x‖ ≤ C * ‖x‖) (x : E) :
    f.mkContinuous C h x = f x 
Arguments: {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_3} {F : Type u_4} [Ring 𝕜] [Ring 𝕜₂] [SeminormedAddCommGroup E] [SeminormedAddCommGroup F] [Module 𝕜 E] [Module 𝕜₂ F] {σ : 𝕜 →+* 𝕜₂} (f : E →ₛₗ[σ] F) (C : ℝ) (h : ∀ (x : E), ‖f x‖ ≤ C * ‖x‖) (x : E)
Goal: (f.mkContinuous C h) x = f x
-- LinearMap.mkContinuousOfExistsBound_coe
@[simp, norm_cast]
theorem LinearMap.mkContinuousOfExistsBound_coe (h : ∃ C, ∀ x, ‖f x‖ ≤ C * ‖x‖) :
    (f.mkContinuousOfExistsBound h : E →ₛₗ[σ] F) = f 
Arguments: {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_3} {F : Type u_4} [Ring 𝕜] [Ring 𝕜₂] [SeminormedAddCommGroup E] [SeminormedAddCommGroup F] [Module 𝕜 E] [Module 𝕜₂ F] {σ : 𝕜 →+* 𝕜₂} (f : E →ₛₗ[σ] F) (h : ∃ C, ∀ (x : E), ‖f x‖ ≤ C * ‖x‖)
Goal: ↑(f.mkContinuousOfExistsBound h) = f
-- LinearMap.mkContinuousOfExistsBound_apply
@[simp]
theorem LinearMap.mkContinuousOfExistsBound_apply (h : ∃ C, ∀ x, ‖f x‖ ≤ C * ‖x‖) (x : E) :
    f.mkContinuousOfExistsBound h x = f x 
Arguments: {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_3} {F : Type u_4} [Ring 𝕜] [Ring 𝕜₂] [SeminormedAddCommGroup E] [SeminormedAddCommGroup F] [Module 𝕜 E] [Module 𝕜₂ F] {σ : 𝕜 →+* 𝕜₂} (f : E →ₛₗ[σ] F) (h : ∃ C, ∀ (x : E), ‖f x‖ ≤ C * ‖x‖) (x : E)
Goal: (f.mkContinuousOfExistsBound h) x = f x
-- ContinuousLinearMap.antilipschitz_of_bound
theorem antilipschitz_of_bound (f : E →SL[σ] F) {K : ℝ≥0} (h : ∀ x, ‖x‖ ≤ K * ‖f x‖) :
    AntilipschitzWith K f 
Arguments: {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_3} {F : Type u_4} [Ring 𝕜] [Ring 𝕜₂] [SeminormedAddCommGroup E] [SeminormedAddCommGroup F] [Module 𝕜 E] [Module 𝕜₂ F] {σ : 𝕜 →+* 𝕜₂} (f : E →SL[σ] F) {K : ℝ≥0} (h : ∀ (x : E), ‖x‖ ≤ ↑K * ‖f x‖)
Goal: AntilipschitzWith K ⇑f
-- ContinuousLinearMap.bound_of_antilipschitz
theorem bound_of_antilipschitz (f : E →SL[σ] F) {K : ℝ≥0} (h : AntilipschitzWith K f) (x) :
    ‖x‖ ≤ K * ‖f x‖ 
Arguments: {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_3} {F : Type u_4} [Ring 𝕜] [Ring 𝕜₂] [SeminormedAddCommGroup E] [SeminormedAddCommGroup F] [Module 𝕜 E] [Module 𝕜₂ F] {σ : 𝕜 →+* 𝕜₂} (f : E →SL[σ] F) {K : ℝ≥0} (h : AntilipschitzWith K ⇑f) (x : E)
Goal: ‖x‖ ≤ ↑K * ‖f x‖
-- LinearMap.toContinuousLinearMap₁_coe
@[simp]
theorem LinearMap.toContinuousLinearMap₁_coe (f : 𝕜 →ₗ[𝕜] E) :
    (f.toContinuousLinearMap₁ : 𝕜 →ₗ[𝕜] E) = f 
Arguments: {𝕜 : Type u_1} {E : Type u_3} [SeminormedRing 𝕜] [SeminormedAddCommGroup E] [Module 𝕜 E] [BoundedSMul 𝕜 E] (f : 𝕜 →ₗ[𝕜] E)
Goal: ↑f.toContinuousLinearMap₁ = f
-- LinearMap.toContinuousLinearMap₁_apply
@[simp]
theorem LinearMap.toContinuousLinearMap₁_apply (f : 𝕜 →ₗ[𝕜] E) (x) :
    f.toContinuousLinearMap₁ x = f x 
Arguments: {𝕜 : Type u_1} {E : Type u_3} [SeminormedRing 𝕜] [SeminormedAddCommGroup E] [Module 𝕜 E] [BoundedSMul 𝕜 E] (f : 𝕜 →ₗ[𝕜] E) (x : 𝕜)
Goal: f.toContinuousLinearMap₁ x = f x
-- ContinuousLinearMap.uniformEmbedding_of_bound
theorem ContinuousLinearMap.uniformEmbedding_of_bound {K : ℝ≥0} (hf : ∀ x, ‖x‖ ≤ K * ‖f x‖) :
    UniformEmbedding f 
Arguments: {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_3} {F : Type u_4} [Ring 𝕜] [Ring 𝕜₂] [NormedAddCommGroup E] [NormedAddCommGroup F] [Module 𝕜 E] [Module 𝕜₂ F] {σ : 𝕜 →+* 𝕜₂} (f : E →SL[σ] F) {K : ℝ≥0} (hf : ∀ (x : E), ‖x‖ ≤ ↑K * ‖f x‖)
Goal: UniformEmbedding ⇑f
-- ContinuousLinearEquiv.homothety_inverse
theorem ContinuousLinearEquiv.homothety_inverse (a : ℝ) (ha : 0 < a) (f : E ≃ₛₗ[σ] F) :
    (∀ x : E, ‖f x‖ = a * ‖x‖) → ∀ y : F, ‖f.symm y‖ = a⁻¹ * ‖y‖ 
Arguments: {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_3} {F : Type u_4} [Ring 𝕜] [Ring 𝕜₂] [SeminormedAddCommGroup E] [SeminormedAddCommGroup F] [Module 𝕜 E] [Module 𝕜₂ F] {σ : 𝕜 →+* 𝕜₂} {σ₂₁ : 𝕜₂ →+* 𝕜} [RingHomInvPair σ σ₂₁] [RingHomInvPair σ₂₁ σ] (a : ℝ) (ha : 0 < a) (f : E ≃ₛₗ[σ] F)
Goal: (∀ (x : E), ‖f x‖ = a * ‖x‖) → ∀ (y : F), ‖f.symm y‖ = a⁻¹ * ‖y‖
-- Summable.mul_of_nonneg
theorem Summable.mul_of_nonneg {f : ι → ℝ} {g : ι' → ℝ} (hf : Summable f) (hg : Summable g)
    (hf' : 0 ≤ f) (hg' : 0 ≤ g) : Summable fun x : ι × ι' => f x.1 * g x.2 
Arguments: {ι : Type u_2} {ι' : Type u_3} {f : ι → ℝ} {g : ι' → ℝ} (hf : Summable f) (hg : Summable g) (hf' : 0 ≤ f) (hg' : 0 ≤ g)
Goal: Summable fun x => f x.1 * g x.2
-- Summable.mul_norm
theorem Summable.mul_norm {f : ι → R} {g : ι' → R} (hf : Summable fun x => ‖f x‖)
    (hg : Summable fun x => ‖g x‖) : Summable fun x : ι × ι' => ‖f x.1 * g x.2‖ 
Arguments: {R : Type u_1} {ι : Type u_2} {ι' : Type u_3} [NormedRing R] {f : ι → R} {g : ι' → R} (hf : Summable fun x => ‖f x‖) (hg : Summable fun x => ‖g x‖)
Goal: Summable fun x => ‖f x.1 * g x.2‖
-- summable_mul_of_summable_norm
theorem summable_mul_of_summable_norm [CompleteSpace R] {f : ι → R} {g : ι' → R}
    (hf : Summable fun x => ‖f x‖) (hg : Summable fun x => ‖g x‖) :
    Summable fun x : ι × ι' => f x.1 * g x.2 
Arguments: {R : Type u_1} {ι : Type u_2} {ι' : Type u_3} [NormedRing R] [CompleteSpace R] {f : ι → R} {g : ι' → R} (hf : Summable fun x => ‖f x‖) (hg : Summable fun x => ‖g x‖)
Goal: Summable fun x => f x.1 * g x.2
-- tsum_mul_tsum_of_summable_norm
theorem tsum_mul_tsum_of_summable_norm [CompleteSpace R] {f : ι → R} {g : ι' → R}
    (hf : Summable fun x => ‖f x‖) (hg : Summable fun x => ‖g x‖) :
    ((∑' x, f x) * ∑' y, g y) = ∑' z : ι × ι', f z.1 * g z.2 
Arguments: {R : Type u_1} {ι : Type u_2} {ι' : Type u_3} [NormedRing R] [CompleteSpace R] {f : ι → R} {g : ι' → R} (hf : Summable fun x => ‖f x‖) (hg : Summable fun x => ‖g x‖)
Goal: (∑' (x : ι), f x) * ∑' (y : ι'), g y = ∑' (z : ι × ι'), f z.1 * g z.2
-- summable_norm_sum_mul_antidiagonal_of_summable_norm
theorem summable_norm_sum_mul_antidiagonal_of_summable_norm {f g : ℕ → R}
    (hf : Summable fun x => ‖f x‖) (hg : Summable fun x => ‖g x‖) :
    Summable fun n => ‖∑ kl ∈ antidiagonal n, f kl.1 * g kl.2‖ 
Arguments: {R : Type u_1} [NormedRing R] {f g : ℕ → R} (hf : Summable fun x => ‖f x‖) (hg : Summable fun x => ‖g x‖)
Goal: Summable fun n => ‖∑ kl ∈ Finset.antidiagonal n, f kl.1 * g kl.2‖
-- tsum_mul_tsum_eq_tsum_sum_antidiagonal_of_summable_norm
theorem tsum_mul_tsum_eq_tsum_sum_antidiagonal_of_summable_norm [CompleteSpace R] {f g : ℕ → R}
    (hf : Summable fun x => ‖f x‖) (hg : Summable fun x => ‖g x‖) :
    ((∑' n, f n) * ∑' n, g n) = ∑' n, ∑ kl ∈ antidiagonal n, f kl.1 * g kl.2 
Arguments: {R : Type u_1} [NormedRing R] [CompleteSpace R] {f g : ℕ → R} (hf : Summable fun x => ‖f x‖) (hg : Summable fun x => ‖g x‖)
Goal: (∑' (n : ℕ), f n) * ∑' (n : ℕ), g n = ∑' (n : ℕ), ∑ kl ∈ Finset.antidiagonal n, f kl.1 * g kl.2
-- summable_norm_sum_mul_range_of_summable_norm
theorem summable_norm_sum_mul_range_of_summable_norm {f g : ℕ → R} (hf : Summable fun x => ‖f x‖)
    (hg : Summable fun x => ‖g x‖) : Summable fun n => ‖∑ k ∈ range (n + 1), f k * g (n - k)‖ 
Arguments: {R : Type u_1} [NormedRing R] {f g : ℕ → R} (hf : Summable fun x => ‖f x‖) (hg : Summable fun x => ‖g x‖)
Goal: Summable fun n => ‖∑ k ∈ Finset.range (n + 1), f k * g (n - k)‖
-- tsum_mul_tsum_eq_tsum_sum_range_of_summable_norm
theorem tsum_mul_tsum_eq_tsum_sum_range_of_summable_norm [CompleteSpace R] {f g : ℕ → R}
    (hf : Summable fun x => ‖f x‖) (hg : Summable fun x => ‖g x‖) :
    ((∑' n, f n) * ∑' n, g n) = ∑' n, ∑ k ∈ range (n + 1), f k * g (n - k) 
Arguments: {R : Type u_1} [NormedRing R] [CompleteSpace R] {f g : ℕ → R} (hf : Summable fun x => ‖f x‖) (hg : Summable fun x => ‖g x‖)
Goal: (∑' (n : ℕ), f n) * ∑' (n : ℕ), g n = ∑' (n : ℕ), ∑ k ∈ Finset.range (n + 1), f k * g (n - k)
-- coe_mul_unitBall
@[simp, norm_cast]
theorem coe_mul_unitBall [NonUnitalSeminormedRing 𝕜] (x y : ball (0 : 𝕜) 1) :
    ↑(x * y) = (x * y : 𝕜) 
Arguments: {𝕜 : Type u_1} [NonUnitalSeminormedRing 𝕜] (x y : ↑(Metric.ball 0 1))
Goal: ↑(x * y) = ↑x * ↑y
-- coe_mul_unitClosedBall
@[simp, norm_cast]
theorem coe_mul_unitClosedBall [NonUnitalSeminormedRing 𝕜] (x y : closedBall (0 : 𝕜) 1) :
    ↑(x * y) = (x * y : 𝕜) 
Arguments: {𝕜 : Type u_1} [NonUnitalSeminormedRing 𝕜] (x y : ↑(Metric.closedBall 0 1))
Goal: ↑(x * y) = ↑x * ↑y
-- coe_one_unitClosedBall
@[simp, norm_cast]
theorem coe_one_unitClosedBall [SeminormedRing 𝕜] [NormOneClass 𝕜] :
    ((1 : closedBall (0 : 𝕜) 1) : 𝕜) = 1 
Arguments: {𝕜 : Type u_1} [SeminormedRing 𝕜] [NormOneClass 𝕜]
Goal: ↑1 = 1
-- coe_pow_unitClosedBall
@[simp, norm_cast]
theorem coe_pow_unitClosedBall [SeminormedRing 𝕜] [NormOneClass 𝕜] (x : closedBall (0 : 𝕜) 1)
    (n : ℕ) : ↑(x ^ n) = (x : 𝕜) ^ n 
Arguments: {𝕜 : Type u_1} [SeminormedRing 𝕜] [NormOneClass 𝕜] (x : ↑(Metric.closedBall 0 1)) (n : ℕ)
Goal: ↑(x ^ n) = ↑x ^ n
-- coe_inv_unitSphere
@[simp, norm_cast]
theorem coe_inv_unitSphere [NormedDivisionRing 𝕜] (x : sphere (0 : 𝕜) 1) : ↑x⁻¹ = (x⁻¹ : 𝕜) 
Arguments: {𝕜 : Type u_1} [NormedDivisionRing 𝕜] (x : ↑(Metric.sphere 0 1))
Goal: ↑x⁻¹ = (↑x)⁻¹
-- coe_div_unitSphere
@[simp, norm_cast]
theorem coe_div_unitSphere [NormedDivisionRing 𝕜] (x y : sphere (0 : 𝕜) 1) :
    ↑(x / y) = (x / y : 𝕜) 
Arguments: {𝕜 : Type u_1} [NormedDivisionRing 𝕜] (x y : ↑(Metric.sphere 0 1))
Goal: ↑(x / y) = ↑x / ↑y
-- coe_zpow_unitSphere
@[simp, norm_cast]
theorem coe_zpow_unitSphere [NormedDivisionRing 𝕜] (x : sphere (0 : 𝕜) 1) (n : ℤ) :
    ↑(x ^ n) = (x : 𝕜) ^ n 
Arguments: {𝕜 : Type u_1} [NormedDivisionRing 𝕜] (x : ↑(Metric.sphere 0 1)) (n : ℤ)
Goal: ↑(x ^ n) = ↑x ^ n
-- coe_one_unitSphere
@[simp, norm_cast]
theorem coe_one_unitSphere [NormedDivisionRing 𝕜] : ((1 : sphere (0 : 𝕜) 1) : 𝕜) = 1 
Arguments: {𝕜 : Type u_1} [NormedDivisionRing 𝕜]
Goal: ↑1 = 1
-- coe_mul_unitSphere
@[simp, norm_cast]
theorem coe_mul_unitSphere [NormedDivisionRing 𝕜] (x y : sphere (0 : 𝕜) 1) :
    ↑(x * y) = (x * y : 𝕜) 
Arguments: {𝕜 : Type u_1} [NormedDivisionRing 𝕜] (x y : ↑(Metric.sphere 0 1))
Goal: ↑(x * y) = ↑x * ↑y
-- coe_pow_unitSphere
@[simp, norm_cast]
theorem coe_pow_unitSphere [NormedDivisionRing 𝕜] (x : sphere (0 : 𝕜) 1) (n : ℕ) :
    ↑(x ^ n) = (x : 𝕜) ^ n 
Arguments: {𝕜 : Type u_1} [NormedDivisionRing 𝕜] (x : ↑(Metric.sphere 0 1)) (n : ℕ)
Goal: ↑(x ^ n) = ↑x ^ n
-- unitSphereToUnits_apply_coe
@[simp]
theorem unitSphereToUnits_apply_coe [NormedDivisionRing 𝕜] (x : sphere (0 : 𝕜) 1) :
    (unitSphereToUnits 𝕜 x : 𝕜) = x 
Arguments: {𝕜 : Type u_1} [NormedDivisionRing 𝕜] (x : ↑(Metric.sphere 0 1))
Goal: ↑((unitSphereToUnits 𝕜) x) = ↑x
-- unitSphereToUnits_injective
theorem unitSphereToUnits_injective [NormedDivisionRing 𝕜] :
    Function.Injective (unitSphereToUnits 𝕜) 
Arguments: {𝕜 : Type u_1} [NormedDivisionRing 𝕜]
Goal: Function.Injective ⇑(unitSphereToUnits 𝕜)
-- Int.norm_cast_real
@[norm_cast]
theorem norm_cast_real (m : ℤ) : ‖(m : ℝ)‖ = ‖m‖ 
Arguments: (m : ℤ)
Goal: ‖↑m‖ = ‖m‖
-- Int.norm_eq_abs
theorem norm_eq_abs (n : ℤ) : ‖n‖ = |(n : ℝ)| 
Arguments: (n : ℤ)
Goal: ‖n‖ = |↑n|
-- Int.norm_natCast
@[simp]
theorem norm_natCast (n : ℕ) : ‖(n : ℤ)‖ = n 
Arguments: (n : ℕ)
Goal: ‖↑n‖ = ↑n
-- NNReal.natCast_natAbs
theorem _root_.NNReal.natCast_natAbs (n : ℤ) : (n.natAbs : ℝ≥0) = ‖n‖₊ 
Arguments: (n : ℤ)
Goal: ↑n.natAbs = ‖n‖₊
-- Int.abs_le_floor_nnreal_iff
theorem abs_le_floor_nnreal_iff (z : ℤ) (c : ℝ≥0) : |z| ≤ ⌊c⌋₊ ↔ ‖z‖₊ ≤ c 
Arguments: (z : ℤ) (c : ℝ≥0)
Goal: |z| ≤ ↑⌊c⌋₊ ↔ ‖z‖₊ ≤ c
-- norm_zsmul_le
@[to_additive norm_zsmul_le]
theorem norm_zpow_le_mul_norm (n : ℤ) (a : α) : ‖a ^ n‖ ≤ ‖n‖ * ‖a‖ 
Arguments: {α : Type u_1} [SeminormedAddCommGroup α] (n : ℤ) (a : α)
Goal: ‖n • a‖ ≤ ‖n‖ * ‖a‖
-- norm_zpow_le_mul_norm
@[to_additive norm_zsmul_le]
theorem norm_zpow_le_mul_norm (n : ℤ) (a : α) : ‖a ^ n‖ ≤ ‖n‖ * ‖a‖ 
Arguments: {α : Type u_1} [SeminormedCommGroup α] (n : ℤ) (a : α)
Goal: ‖a ^ n‖ ≤ ‖n‖ * ‖a‖
-- nnnorm_zpow_le_mul_norm
@[to_additive nnnorm_zsmul_le]
theorem nnnorm_zpow_le_mul_norm (n : ℤ) (a : α) : ‖a ^ n‖₊ ≤ ‖n‖₊ * ‖a‖₊ 
Arguments: {α : Type u_1} [SeminormedCommGroup α] (n : ℤ) (a : α)
Goal: ‖a ^ n‖₊ ≤ ‖n‖₊ * ‖a‖₊
-- nnnorm_zsmul_le
@[to_additive nnnorm_zsmul_le]
theorem nnnorm_zpow_le_mul_norm (n : ℤ) (a : α) : ‖a ^ n‖₊ ≤ ‖n‖₊ * ‖a‖₊ 
Arguments: {α : Type u_1} [SeminormedAddCommGroup α] (n : ℤ) (a : α)
Goal: ‖n • a‖₊ ≤ ‖n‖₊ * ‖a‖₊
-- comap_norm_atTop
@[to_additive (attr := simp) comap_norm_atTop]
lemma comap_norm_atTop' : comap norm atTop = cobounded E 
Arguments: {E : Type u_3} [SeminormedAddGroup E]
Goal: Filter.comap norm Filter.atTop = Bornology.cobounded E
-- comap_norm_atTop'
@[to_additive (attr := simp) comap_norm_atTop]
lemma comap_norm_atTop' : comap norm atTop = cobounded E 
Arguments: {E : Type u_3} [SeminormedGroup E]
Goal: Filter.comap norm Filter.atTop = Bornology.cobounded E
-- Filter.HasBasis.cobounded_of_norm'
@[to_additive Filter.HasBasis.cobounded_of_norm]
lemma Filter.HasBasis.cobounded_of_norm' {ι : Sort*} {p : ι → Prop} {s : ι → Set ℝ}
    (h : HasBasis atTop p s) : HasBasis (cobounded E) p fun i ↦ norm ⁻¹' s i 
Arguments: {E : Type u_3} [SeminormedGroup E] {ι : Sort u_6} {p : ι → Prop} {s : ι → Set ℝ} (h : Filter.atTop.HasBasis p s)
Goal: (Bornology.cobounded E).HasBasis p fun i => norm ⁻¹' s i
-- Filter.HasBasis.cobounded_of_norm
@[to_additive Filter.HasBasis.cobounded_of_norm]
lemma Filter.HasBasis.cobounded_of_norm' {ι : Sort*} {p : ι → Prop} {s : ι → Set ℝ}
    (h : HasBasis atTop p s) : HasBasis (cobounded E) p fun i ↦ norm ⁻¹' s i 
Arguments: {E : Type u_3} [SeminormedAddGroup E] {ι : Sort u_6} {p : ι → Prop} {s : ι → Set ℝ} (h : Filter.atTop.HasBasis p s)
Goal: (Bornology.cobounded E).HasBasis p fun i => norm ⁻¹' s i
-- Filter.hasBasis_cobounded_norm
@[to_additive Filter.hasBasis_cobounded_norm]
lemma Filter.hasBasis_cobounded_norm' : HasBasis (cobounded E) (fun _ ↦ True) ({x | · ≤ ‖x‖}) 
Arguments: {E : Type u_3} [SeminormedAddGroup E]
Goal: (Bornology.cobounded E).HasBasis (fun x => True) fun x => {x_1 | x ≤ ‖x_1‖}
-- Filter.hasBasis_cobounded_norm'
@[to_additive Filter.hasBasis_cobounded_norm]
lemma Filter.hasBasis_cobounded_norm' : HasBasis (cobounded E) (fun _ ↦ True) ({x | · ≤ ‖x‖}) 
Arguments: {E : Type u_3} [SeminormedGroup E]
Goal: (Bornology.cobounded E).HasBasis (fun x => True) fun x => {x_1 | x ≤ ‖x_1‖}
-- tendsto_norm_atTop_iff_cobounded
@[to_additive (attr := simp) tendsto_norm_atTop_iff_cobounded]
lemma tendsto_norm_atTop_iff_cobounded' {f : α → E} {l : Filter α} :
    Tendsto (‖f ·‖) l atTop ↔ Tendsto f l (cobounded E) 
Arguments: {α : Type u_1} {E : Type u_3} [SeminormedAddGroup E] {f : α → E} {l : Filter α}
Goal: Filter.Tendsto (fun x => ‖f x‖) l Filter.atTop ↔ Filter.Tendsto f l (Bornology.cobounded E)
-- tendsto_norm_atTop_iff_cobounded'
@[to_additive (attr := simp) tendsto_norm_atTop_iff_cobounded]
lemma tendsto_norm_atTop_iff_cobounded' {f : α → E} {l : Filter α} :
    Tendsto (‖f ·‖) l atTop ↔ Tendsto f l (cobounded E) 
Arguments: {α : Type u_1} {E : Type u_3} [SeminormedGroup E] {f : α → E} {l : Filter α}
Goal: Filter.Tendsto (fun x => ‖f x‖) l Filter.atTop ↔ Filter.Tendsto f l (Bornology.cobounded E)
-- tendsto_norm_cobounded_atTop'
@[to_additive tendsto_norm_cobounded_atTop]
lemma tendsto_norm_cobounded_atTop' : Tendsto norm (cobounded E) atTop 
Arguments: {E : Type u_3} [SeminormedGroup E]
Goal: Filter.Tendsto norm (Bornology.cobounded E) Filter.atTop
-- tendsto_norm_cobounded_atTop
@[to_additive tendsto_norm_cobounded_atTop]
lemma tendsto_norm_cobounded_atTop' : Tendsto norm (cobounded E) atTop 
Arguments: {E : Type u_3} [SeminormedAddGroup E]
Goal: Filter.Tendsto norm (Bornology.cobounded E) Filter.atTop
-- eventually_cobounded_le_norm
@[to_additive eventually_cobounded_le_norm]
lemma eventually_cobounded_le_norm' (a : ℝ) : ∀ᶠ x in cobounded E, a ≤ ‖x‖ 
Arguments: {E : Type u_3} [SeminormedAddGroup E] (a : ℝ)
Goal: ∀ᶠ (x : E) in Bornology.cobounded E, a ≤ ‖x‖
-- eventually_cobounded_le_norm'
@[to_additive eventually_cobounded_le_norm]
lemma eventually_cobounded_le_norm' (a : ℝ) : ∀ᶠ x in cobounded E, a ≤ ‖x‖ 
Arguments: {E : Type u_3} [SeminormedGroup E] (a : ℝ)
Goal: ∀ᶠ (x : E) in Bornology.cobounded E, a ≤ ‖x‖
-- tendsto_norm_cocompact_atTop'
@[to_additive tendsto_norm_cocompact_atTop]
lemma tendsto_norm_cocompact_atTop' [ProperSpace E] : Tendsto norm (cocompact E) atTop 
Arguments: {E : Type u_3} [SeminormedGroup E] [ProperSpace E]
Goal: Filter.Tendsto norm (Filter.cocompact E) Filter.atTop
-- tendsto_norm_cocompact_atTop
@[to_additive tendsto_norm_cocompact_atTop]
lemma tendsto_norm_cocompact_atTop' [ProperSpace E] : Tendsto norm (cocompact E) atTop 
Arguments: {E : Type u_3} [SeminormedAddGroup E] [ProperSpace E]
Goal: Filter.Tendsto norm (Filter.cocompact E) Filter.atTop
-- Filter.neg_cobounded
@[to_additive (attr := simp)]
lemma Filter.inv_cobounded : (cobounded E)⁻¹ = cobounded E 
Arguments: {E : Type u_3} [SeminormedAddGroup E]
Goal: -Bornology.cobounded E = Bornology.cobounded E
-- Filter.inv_cobounded
@[to_additive (attr := simp)]
lemma Filter.inv_cobounded : (cobounded E)⁻¹ = cobounded E 
Arguments: {E : Type u_3} [SeminormedGroup E]
Goal: (Bornology.cobounded E)⁻¹ = Bornology.cobounded E
-- Filter.tendsto_inv_cobounded
@[to_additive "In a (semi)normed group, negation `x ↦ -x` tends to infinity at infinity."]
theorem Filter.tendsto_inv_cobounded : Tendsto Inv.inv (cobounded E) (cobounded E) 
Arguments: {E : Type u_3} [SeminormedGroup E]
Goal: Filter.Tendsto Inv.inv (Bornology.cobounded E) (Bornology.cobounded E)
-- Filter.tendsto_neg_cobounded
@[to_additive "In a (semi)normed group, negation `x ↦ -x` tends to infinity at infinity."]
theorem Filter.tendsto_inv_cobounded : Tendsto Inv.inv (cobounded E) (cobounded E) 
Arguments: {E : Type u_3} [SeminormedAddGroup E]
Goal: Filter.Tendsto Neg.neg (Bornology.cobounded E) (Bornology.cobounded E)
-- isBounded_iff_forall_norm_le'
@[to_additive isBounded_iff_forall_norm_le]
lemma isBounded_iff_forall_norm_le' : Bornology.IsBounded s ↔ ∃ C, ∀ x ∈ s, ‖x‖ ≤ C 
Arguments: {E : Type u_3} [SeminormedGroup E] {s : Set E}
Goal: Bornology.IsBounded s ↔ ∃ C, ∀ x ∈ s, ‖x‖ ≤ C
-- isBounded_iff_forall_norm_le
@[to_additive isBounded_iff_forall_norm_le]
lemma isBounded_iff_forall_norm_le' : Bornology.IsBounded s ↔ ∃ C, ∀ x ∈ s, ‖x‖ ≤ C 
Arguments: {E : Type u_3} [SeminormedAddGroup E] {s : Set E}
Goal: Bornology.IsBounded s ↔ ∃ C, ∀ x ∈ s, ‖x‖ ≤ C
-- Bornology.IsBounded.exists_pos_norm_le
@[to_additive exists_pos_norm_le]
lemma Bornology.IsBounded.exists_pos_norm_le' (hs : IsBounded s) : ∃ R > 0, ∀ x ∈ s, ‖x‖ ≤ R 
Arguments: {E : Type u_3} [SeminormedAddGroup E] {s : Set E} (hs : Bornology.IsBounded s)
Goal: ∃ R > 0, ∀ x ∈ s, ‖x‖ ≤ R
-- Bornology.IsBounded.exists_pos_norm_le'
@[to_additive exists_pos_norm_le]
lemma Bornology.IsBounded.exists_pos_norm_le' (hs : IsBounded s) : ∃ R > 0, ∀ x ∈ s, ‖x‖ ≤ R 
Arguments: {E : Type u_3} [SeminormedGroup E] {s : Set E} (hs : Bornology.IsBounded s)
Goal: ∃ R > 0, ∀ x ∈ s, ‖x‖ ≤ R
-- Bornology.IsBounded.exists_pos_norm_lt
@[to_additive Bornology.IsBounded.exists_pos_norm_lt]
lemma Bornology.IsBounded.exists_pos_norm_lt' (hs : IsBounded s) : ∃ R > 0, ∀ x ∈ s, ‖x‖ < R 
Arguments: {E : Type u_3} [SeminormedAddGroup E] {s : Set E} (hs : Bornology.IsBounded s)
Goal: ∃ R > 0, ∀ x ∈ s, ‖x‖ < R