ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- ContinuousLinearMap.isCompact_image_coe_of_bounded_of_weak_closed
theorem isCompact_image_coe_of_bounded_of_weak_closed [ProperSpace F] {s : Set (E' →SL[σ₁₂] F)}
    (hb : IsBounded s) (hc : ∀ f : E' →SL[σ₁₂] F,
      (⇑f : E' → F) ∈ closure (((↑) : (E' →SL[σ₁₂] F) → E' → F) '' s) → f ∈ s) :
    IsCompact (((↑) : (E' →SL[σ₁₂] F) → E' → F) '' s) 
Arguments: {𝕜 : Type u_1} {𝕜₂ : Type u_2} {F : Type u_6} [NormedAddCommGroup F] [NontriviallyNormedField 𝕜] [NontriviallyNormedField 𝕜₂] [NormedSpace 𝕜₂ F] {σ₁₂ : 𝕜 →+* 𝕜₂} {E' : Type u_11} [SeminormedAddCommGroup E'] [NormedSpace 𝕜 E'] [RingHomIsometric σ₁₂] [ProperSpace F] {s : Set (E' →SL[σ₁₂] F)} (hb : Bornology.IsBounded s) (hc : ∀ (f : E' →SL[σ₁₂] F), ⇑f ∈ closure (DFunLike.coe '' s) → f ∈ s)
Goal: IsCompact (DFunLike.coe '' s)
-- ContinuousLinearMap.is_weak_closed_closedBall
theorem is_weak_closed_closedBall (f₀ : E' →SL[σ₁₂] F) (r : ℝ) ⦃f : E' →SL[σ₁₂] F⦄
    (hf : ⇑f ∈ closure (((↑) : (E' →SL[σ₁₂] F) → E' → F) '' closedBall f₀ r)) :
    f ∈ closedBall f₀ r 
Arguments: {𝕜 : Type u_1} {𝕜₂ : Type u_2} {F : Type u_6} [NormedAddCommGroup F] [NontriviallyNormedField 𝕜] [NontriviallyNormedField 𝕜₂] [NormedSpace 𝕜₂ F] {σ₁₂ : 𝕜 →+* 𝕜₂} {E' : Type u_11} [SeminormedAddCommGroup E'] [NormedSpace 𝕜 E'] [RingHomIsometric σ₁₂] (f₀ : E' →SL[σ₁₂] F) (r : ℝ) ⦃f : E' →SL[σ₁₂] F⦄ (hf : ⇑f ∈ closure (DFunLike.coe '' Metric.closedBall f₀ r))
Goal: f ∈ Metric.closedBall f₀ r
-- ContinuousLinearMap.isClosed_image_coe_closedBall
theorem isClosed_image_coe_closedBall (f₀ : E →SL[σ₁₂] F) (r : ℝ) :
    IsClosed (((↑) : (E →SL[σ₁₂] F) → E → F) '' closedBall f₀ r) 
Arguments: {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_4} {F : Type u_6} [NormedAddCommGroup E] [NormedAddCommGroup F] [NontriviallyNormedField 𝕜] [NontriviallyNormedField 𝕜₂] [NormedSpace 𝕜 E] [NormedSpace 𝕜₂ F] {σ₁₂ : 𝕜 →+* 𝕜₂} [RingHomIsometric σ₁₂] (f₀ : E →SL[σ₁₂] F) (r : ℝ)
Goal: IsClosed (DFunLike.coe '' Metric.closedBall f₀ r)
-- ContinuousLinearMap.isCompact_image_coe_closedBall
theorem isCompact_image_coe_closedBall [ProperSpace F] (f₀ : E →SL[σ₁₂] F) (r : ℝ) :
    IsCompact (((↑) : (E →SL[σ₁₂] F) → E → F) '' closedBall f₀ r) 
Arguments: {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_4} {F : Type u_6} [NormedAddCommGroup E] [NormedAddCommGroup F] [NontriviallyNormedField 𝕜] [NontriviallyNormedField 𝕜₂] [NormedSpace 𝕜 E] [NormedSpace 𝕜₂ F] {σ₁₂ : 𝕜 →+* 𝕜₂} [RingHomIsometric σ₁₂] [ProperSpace F] (f₀ : E →SL[σ₁₂] F) (r : ℝ)
Goal: IsCompact (DFunLike.coe '' Metric.closedBall f₀ r)
-- ContinuousLinearMap.extend_eq
@[simp]
theorem extend_eq (x : E) : extend f e h_dense h_e (e x) = f x 
Arguments: {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_4} {F : Type u_6} {Fₗ : Type u_7} [NormedAddCommGroup E] [NormedAddCommGroup F] [NormedAddCommGroup Fₗ] [NontriviallyNormedField 𝕜] [NontriviallyNormedField 𝕜₂] [NormedSpace 𝕜 E] [NormedSpace 𝕜₂ F] [NormedSpace 𝕜 Fₗ] {σ₁₂ : 𝕜 →+* 𝕜₂} (f : E →SL[σ₁₂] F) [CompleteSpace F] (e : E →L[𝕜] Fₗ) (h_dense : DenseRange ⇑e) (h_e : UniformInducing ⇑e) (x : E)
Goal: (f.extend e h_dense h_e) (e x) = f x
-- ContinuousLinearMap.extend_unique
theorem extend_unique (g : Fₗ →SL[σ₁₂] F) (H : g.comp e = f) : extend f e h_dense h_e = g 
Arguments: {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_4} {F : Type u_6} {Fₗ : Type u_7} [NormedAddCommGroup E] [NormedAddCommGroup F] [NormedAddCommGroup Fₗ] [NontriviallyNormedField 𝕜] [NontriviallyNormedField 𝕜₂] [NormedSpace 𝕜 E] [NormedSpace 𝕜₂ F] [NormedSpace 𝕜 Fₗ] {σ₁₂ : 𝕜 →+* 𝕜₂} (f : E →SL[σ₁₂] F) [CompleteSpace F] (e : E →L[𝕜] Fₗ) (h_dense : DenseRange ⇑e) (h_e : UniformInducing ⇑e) (g : Fₗ →SL[σ₁₂] F) (H : g.comp e = f)
Goal: f.extend e h_dense h_e = g
-- ContinuousLinearMap.extend_zero
@[simp]
theorem extend_zero : extend (0 : E →SL[σ₁₂] F) e h_dense h_e = 0 
Arguments: {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_4} {F : Type u_6} {Fₗ : Type u_7} [NormedAddCommGroup E] [NormedAddCommGroup F] [NormedAddCommGroup Fₗ] [NontriviallyNormedField 𝕜] [NontriviallyNormedField 𝕜₂] [NormedSpace 𝕜 E] [NormedSpace 𝕜₂ F] [NormedSpace 𝕜 Fₗ] {σ₁₂ : 𝕜 →+* 𝕜₂} [CompleteSpace F] (e : E →L[𝕜] Fₗ) (h_dense : DenseRange ⇑e) (h_e : UniformInducing ⇑e)
Goal: ContinuousLinearMap.extend 0 e h_dense h_e = 0
-- ContinuousLinearMap.opNorm_extend_le
theorem opNorm_extend_le :
    ‖f.extend e h_dense (uniformEmbedding_of_bound _ h_e).toUniformInducing‖ ≤ N * ‖f‖ 
Arguments: {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_4} {F : Type u_6} {Fₗ : Type u_7} [NormedAddCommGroup E] [NormedAddCommGroup F] [NormedAddCommGroup Fₗ] [NontriviallyNormedField 𝕜] [NontriviallyNormedField 𝕜₂] [NormedSpace 𝕜 E] [NormedSpace 𝕜₂ F] [NormedSpace 𝕜 Fₗ] {σ₁₂ : 𝕜 →+* 𝕜₂} (f : E →SL[σ₁₂] F) [CompleteSpace F] (e : E →L[𝕜] Fₗ) (h_dense : DenseRange ⇑e) {N : ℝ≥0} (h_e : ∀ (x : E), ‖x‖ ≤ ↑N * ‖e x‖) [RingHomIsometric σ₁₂]
Goal: ‖f.extend e h_dense ⋯‖ ≤ ↑N * ‖f‖
-- Real.rpow_eq_pow
@[simp]
theorem rpow_eq_pow (x y : ℝ) : rpow x y = x ^ y 
Arguments: (x y : ℝ)
Goal: x.rpow y = x ^ y
-- Real.rpow_def
theorem rpow_def (x y : ℝ) : x ^ y = ((x : ℂ) ^ (y : ℂ)).re 
Arguments: (x y : ℝ)
Goal: x ^ y = (↑x ^ ↑y).re
-- Real.rpow_def_of_nonneg
theorem rpow_def_of_nonneg {x : ℝ} (hx : 0 ≤ x) (y : ℝ) :
    x ^ y = if x = 0 then if y = 0 then 1 else 0 else exp (log x * y) 
Arguments: {x : ℝ} (hx : 0 ≤ x) (y : ℝ)
Goal: x ^ y = if x = 0 then if y = 0 then 1 else 0 else rexp (Real.log x * y)
-- Real.rpow_def_of_pos
theorem rpow_def_of_pos {x : ℝ} (hx : 0 < x) (y : ℝ) : x ^ y = exp (log x * y) 
Arguments: {x : ℝ} (hx : 0 < x) (y : ℝ)
Goal: x ^ y = rexp (Real.log x * y)
-- Real.exp_mul
theorem exp_mul (x y : ℝ) : exp (x * y) = exp x ^ y 
Arguments: (x y : ℝ)
Goal: rexp (x * y) = rexp x ^ y
-- Real.rpow_intCast
@[simp, norm_cast]
theorem rpow_intCast (x : ℝ) (n : ℤ) : x ^ (n : ℝ) = x ^ n 
Arguments: (x : ℝ) (n : ℤ)
Goal: x ^ ↑n = x ^ n
-- Real.rpow_natCast
@[simp, norm_cast]
theorem rpow_natCast (x : ℝ) (n : ℕ) : x ^ (n : ℝ) = x ^ n 
Arguments: (x : ℝ) (n : ℕ)
Goal: x ^ ↑n = x ^ n
-- Real.exp_one_rpow
@[simp]
theorem exp_one_rpow (x : ℝ) : exp 1 ^ x = exp x 
Arguments: (x : ℝ)
Goal: rexp 1 ^ x = rexp x
-- Real.exp_one_pow
@[simp] lemma exp_one_pow (n : ℕ) : exp 1 ^ n = exp n 
Arguments: (n : ℕ)
Goal: rexp 1 ^ n = rexp ↑n
-- Real.rpow_eq_zero_iff_of_nonneg
theorem rpow_eq_zero_iff_of_nonneg (hx : 0 ≤ x) : x ^ y = 0 ↔ x = 0 ∧ y ≠ 0 
Arguments: {x y : ℝ} (hx : 0 ≤ x)
Goal: x ^ y = 0 ↔ x = 0 ∧ y ≠ 0
-- Real.rpow_eq_zero
@[simp]
lemma rpow_eq_zero (hx : 0 ≤ x) (hy : y ≠ 0) : x ^ y = 0 ↔ x = 0 
Arguments: {x y : ℝ} (hx : 0 ≤ x) (hy : y ≠ 0)
Goal: x ^ y = 0 ↔ x = 0
-- Real.rpow_ne_zero
@[simp]
lemma rpow_ne_zero (hx : 0 ≤ x) (hy : y ≠ 0) : x ^ y ≠ 0 ↔ x ≠ 0 
Arguments: {x y : ℝ} (hx : 0 ≤ x) (hy : y ≠ 0)
Goal: x ^ y ≠ 0 ↔ x ≠ 0
-- Real.rpow_def_of_neg
theorem rpow_def_of_neg {x : ℝ} (hx : x < 0) (y : ℝ) : x ^ y = exp (log x * y) * cos (y * π) 
Arguments: {x : ℝ} (hx : x < 0) (y : ℝ)
Goal: x ^ y = rexp (Real.log x * y) * Real.cos (y * π)
-- Real.rpow_def_of_nonpos
theorem rpow_def_of_nonpos {x : ℝ} (hx : x ≤ 0) (y : ℝ) :
    x ^ y = if x = 0 then if y = 0 then 1 else 0 else exp (log x * y) * cos (y * π) 
Arguments: {x : ℝ} (hx : x ≤ 0) (y : ℝ)
Goal: x ^ y = if x = 0 then if y = 0 then 1 else 0 else rexp (Real.log x * y) * Real.cos (y * π)
-- Real.rpow_pos_of_pos
theorem rpow_pos_of_pos {x : ℝ} (hx : 0 < x) (y : ℝ) : 0 < x ^ y 
Arguments: {x : ℝ} (hx : 0 < x) (y : ℝ)
Goal: 0 < x ^ y
-- Real.rpow_zero
@[simp]
theorem rpow_zero (x : ℝ) : x ^ (0 : ℝ) = 1 
Arguments: (x : ℝ)
Goal: x ^ 0 = 1
-- Real.rpow_zero_pos
theorem rpow_zero_pos (x : ℝ) : 0 < x ^ (0 : ℝ) 
Arguments: (x : ℝ)
Goal: 0 < x ^ 0
-- Real.zero_rpow
@[simp]
theorem zero_rpow {x : ℝ} (h : x ≠ 0) : (0 : ℝ) ^ x = 0 
Arguments: {x : ℝ} (h : x ≠ 0)
Goal: 0 ^ x = 0
-- Real.zero_rpow_eq_iff
theorem zero_rpow_eq_iff {x : ℝ} {a : ℝ} : 0 ^ x = a ↔ x ≠ 0 ∧ a = 0 ∨ x = 0 ∧ a = 1 
Arguments: {x a : ℝ}
Goal: 0 ^ x = a ↔ x ≠ 0 ∧ a = 0 ∨ x = 0 ∧ a = 1
-- Real.eq_zero_rpow_iff
theorem eq_zero_rpow_iff {x : ℝ} {a : ℝ} : a = 0 ^ x ↔ x ≠ 0 ∧ a = 0 ∨ x = 0 ∧ a = 1 
Arguments: {x a : ℝ}
Goal: a = 0 ^ x ↔ x ≠ 0 ∧ a = 0 ∨ x = 0 ∧ a = 1
-- Real.rpow_one
@[simp]
theorem rpow_one (x : ℝ) : x ^ (1 : ℝ) = x 
Arguments: (x : ℝ)
Goal: x ^ 1 = x
-- Real.one_rpow
@[simp]
theorem one_rpow (x : ℝ) : (1 : ℝ) ^ x = 1 
Arguments: (x : ℝ)
Goal: 1 ^ x = 1
-- Real.zero_rpow_le_one
theorem zero_rpow_le_one (x : ℝ) : (0 : ℝ) ^ x ≤ 1 
Arguments: (x : ℝ)
Goal: 0 ^ x ≤ 1
-- Real.zero_rpow_nonneg
theorem zero_rpow_nonneg (x : ℝ) : 0 ≤ (0 : ℝ) ^ x 
Arguments: (x : ℝ)
Goal: 0 ≤ 0 ^ x
-- Real.rpow_nonneg
theorem rpow_nonneg {x : ℝ} (hx : 0 ≤ x) (y : ℝ) : 0 ≤ x ^ y 
Arguments: {x : ℝ} (hx : 0 ≤ x) (y : ℝ)
Goal: 0 ≤ x ^ y
-- Real.abs_rpow_of_nonneg
theorem abs_rpow_of_nonneg {x y : ℝ} (hx_nonneg : 0 ≤ x) : |x ^ y| = |x| ^ y 
Arguments: {x y : ℝ} (hx_nonneg : 0 ≤ x)
Goal: |x ^ y| = |x| ^ y
-- Real.abs_rpow_le_abs_rpow
theorem abs_rpow_le_abs_rpow (x y : ℝ) : |x ^ y| ≤ |x| ^ y 
Arguments: (x y : ℝ)
Goal: |x ^ y| ≤ |x| ^ y
-- Real.abs_rpow_le_exp_log_mul
theorem abs_rpow_le_exp_log_mul (x y : ℝ) : |x ^ y| ≤ exp (log x * y) 
Arguments: (x y : ℝ)
Goal: |x ^ y| ≤ rexp (Real.log x * y)
-- Real.norm_rpow_of_nonneg
theorem norm_rpow_of_nonneg {x y : ℝ} (hx_nonneg : 0 ≤ x) : ‖x ^ y‖ = ‖x‖ ^ y 
Arguments: {x y : ℝ} (hx_nonneg : 0 ≤ x)
Goal: ‖x ^ y‖ = ‖x‖ ^ y
-- Real.rpow_add
theorem rpow_add (hx : 0 < x) (y z : ℝ) : x ^ (y + z) = x ^ y * x ^ z 
Arguments: {x : ℝ} (hx : 0 < x) (y z : ℝ)
Goal: x ^ (y + z) = x ^ y * x ^ z
-- Real.rpow_add'
theorem rpow_add' (hx : 0 ≤ x) (h : y + z ≠ 0) : x ^ (y + z) = x ^ y * x ^ z 
Arguments: {x y z : ℝ} (hx : 0 ≤ x) (h : y + z ≠ 0)
Goal: x ^ (y + z) = x ^ y * x ^ z
-- Real.rpow_of_add_eq
lemma rpow_of_add_eq (hx : 0 ≤ x) (hw : w ≠ 0) (h : y + z = w) : x ^ w = x ^ y * x ^ z 
Arguments: {w x y z : ℝ} (hx : 0 ≤ x) (hw : w ≠ 0) (h : y + z = w)
Goal: x ^ w = x ^ y * x ^ z
-- Real.rpow_add_of_nonneg
theorem rpow_add_of_nonneg (hx : 0 ≤ x) (hy : 0 ≤ y) (hz : 0 ≤ z) :
    x ^ (y + z) = x ^ y * x ^ z 
Arguments: {x y z : ℝ} (hx : 0 ≤ x) (hy : 0 ≤ y) (hz : 0 ≤ z)
Goal: x ^ (y + z) = x ^ y * x ^ z
-- Real.le_rpow_add
theorem le_rpow_add {x : ℝ} (hx : 0 ≤ x) (y z : ℝ) : x ^ y * x ^ z ≤ x ^ (y + z) 
Arguments: {x : ℝ} (hx : 0 ≤ x) (y z : ℝ)
Goal: x ^ y * x ^ z ≤ x ^ (y + z)
-- Real.rpow_sum_of_pos
theorem rpow_sum_of_pos {ι : Type*} {a : ℝ} (ha : 0 < a) (f : ι → ℝ) (s : Finset ι) :
    (a ^ ∑ x ∈ s, f x) = ∏ x ∈ s, a ^ f x 
Arguments: {ι : Type u_1} {a : ℝ} (ha : 0 < a) (f : ι → ℝ) (s : Finset ι)
Goal: a ^ ∑ x ∈ s, f x = ∏ x ∈ s, a ^ f x
-- Real.rpow_sum_of_nonneg
theorem rpow_sum_of_nonneg {ι : Type*} {a : ℝ} (ha : 0 ≤ a) {s : Finset ι} {f : ι → ℝ}
    (h : ∀ x ∈ s, 0 ≤ f x) : (a ^ ∑ x ∈ s, f x) = ∏ x ∈ s, a ^ f x 
Arguments: {ι : Type u_1} {a : ℝ} (ha : 0 ≤ a) {s : Finset ι} {f : ι → ℝ} (h : ∀ x ∈ s, 0 ≤ f x)
Goal: a ^ ∑ x ∈ s, f x = ∏ x ∈ s, a ^ f x
-- Real.rpow_neg
theorem rpow_neg {x : ℝ} (hx : 0 ≤ x) (y : ℝ) : x ^ (-y) = (x ^ y)⁻¹ 
Arguments: {x : ℝ} (hx : 0 ≤ x) (y : ℝ)
Goal: x ^ (-y) = (x ^ y)⁻¹
-- Real.rpow_sub
theorem rpow_sub {x : ℝ} (hx : 0 < x) (y z : ℝ) : x ^ (y - z) = x ^ y / x ^ z 
Arguments: {x : ℝ} (hx : 0 < x) (y z : ℝ)
Goal: x ^ (y - z) = x ^ y / x ^ z
-- Real.rpow_sub'
theorem rpow_sub' {x : ℝ} (hx : 0 ≤ x) {y z : ℝ} (h : y - z ≠ 0) : x ^ (y - z) = x ^ y / x ^ z 
Arguments: {x : ℝ} (hx : 0 ≤ x) {y z : ℝ} (h : y - z ≠ 0)
Goal: x ^ (y - z) = x ^ y / x ^ z
-- HasCompactSupport.rpow_const
protected theorem _root_.HasCompactSupport.rpow_const {α : Type*} [TopologicalSpace α] {f : α → ℝ}
    (hf : HasCompactSupport f) {r : ℝ} (hr : r ≠ 0) : HasCompactSupport (fun x ↦ f x ^ r) 
Arguments: {α : Type u_1} [TopologicalSpace α] {f : α → ℝ} (hf : HasCompactSupport f) {r : ℝ} (hr : r ≠ 0)
Goal: HasCompactSupport fun x => f x ^ r
-- Complex.ofReal_cpow
theorem ofReal_cpow {x : ℝ} (hx : 0 ≤ x) (y : ℝ) : ((x ^ y : ℝ) : ℂ) = (x : ℂ) ^ (y : ℂ) 
Arguments: {x : ℝ} (hx : 0 ≤ x) (y : ℝ)
Goal: ↑(x ^ y) = ↑x ^ ↑y
-- Complex.ofReal_cpow_of_nonpos
theorem ofReal_cpow_of_nonpos {x : ℝ} (hx : x ≤ 0) (y : ℂ) :
    (x : ℂ) ^ y = (-x : ℂ) ^ y * exp (π * I * y) 
Arguments: {x : ℝ} (hx : x ≤ 0) (y : ℂ)
Goal: ↑x ^ y = (-↑x) ^ y * Complex.exp (↑π * Complex.I * y)
-- Complex.cpow_ofReal
lemma cpow_ofReal (x : ℂ) (y : ℝ) :
    x ^ (y : ℂ) = ↑(abs x ^ y) * (Real.cos (arg x * y) + Real.sin (arg x * y) * I) 
Arguments: (x : ℂ) (y : ℝ)
Goal: x ^ ↑y = ↑(Complex.abs x ^ y) * (↑(Real.cos (x.arg * y)) + ↑(Real.sin (x.arg * y)) * Complex.I)
-- Complex.cpow_ofReal_re
lemma cpow_ofReal_re (x : ℂ) (y : ℝ) : (x ^ (y : ℂ)).re = (abs x) ^ y * Real.cos (arg x * y) 
Arguments: (x : ℂ) (y : ℝ)
Goal: (x ^ ↑y).re = Complex.abs x ^ y * Real.cos (x.arg * y)
-- Complex.cpow_ofReal_im
lemma cpow_ofReal_im (x : ℂ) (y : ℝ) : (x ^ (y : ℂ)).im = (abs x) ^ y * Real.sin (arg x * y) 
Arguments: (x : ℂ) (y : ℝ)
Goal: (x ^ ↑y).im = Complex.abs x ^ y * Real.sin (x.arg * y)
-- Complex.abs_cpow_of_ne_zero
theorem abs_cpow_of_ne_zero {z : ℂ} (hz : z ≠ 0) (w : ℂ) :
    abs (z ^ w) = abs z ^ w.re / Real.exp (arg z * im w) 
Arguments: {z : ℂ} (hz : z ≠ 0) (w : ℂ)
Goal: Complex.abs (z ^ w) = Complex.abs z ^ w.re / rexp (z.arg * w.im)
-- Complex.abs_cpow_of_imp
theorem abs_cpow_of_imp {z w : ℂ} (h : z = 0 → w.re = 0 → w = 0) :
    abs (z ^ w) = abs z ^ w.re / Real.exp (arg z * im w) 
Arguments: {z w : ℂ} (h : z = 0 → w.re = 0 → w = 0)
Goal: Complex.abs (z ^ w) = Complex.abs z ^ w.re / rexp (z.arg * w.im)
-- Complex.abs_cpow_le
theorem abs_cpow_le (z w : ℂ) : abs (z ^ w) ≤ abs z ^ w.re / Real.exp (arg z * im w) 
Arguments: (z w : ℂ)
Goal: Complex.abs (z ^ w) ≤ Complex.abs z ^ w.re / rexp (z.arg * w.im)
-- Complex.abs_cpow_real
@[simp]
theorem abs_cpow_real (x : ℂ) (y : ℝ) : abs (x ^ (y : ℂ)) = Complex.abs x ^ y 
Arguments: (x : ℂ) (y : ℝ)
Goal: Complex.abs (x ^ ↑y) = Complex.abs x ^ y
-- Complex.abs_cpow_inv_nat
@[simp]
theorem abs_cpow_inv_nat (x : ℂ) (n : ℕ) : abs (x ^ (n⁻¹ : ℂ)) = Complex.abs x ^ (n⁻¹ : ℝ) 
Arguments: (x : ℂ) (n : ℕ)
Goal: Complex.abs (x ^ (↑n)⁻¹) = Complex.abs x ^ (↑n)⁻¹
-- Complex.abs_cpow_eq_rpow_re_of_pos
theorem abs_cpow_eq_rpow_re_of_pos {x : ℝ} (hx : 0 < x) (y : ℂ) : abs (x ^ y) = x ^ y.re 
Arguments: {x : ℝ} (hx : 0 < x) (y : ℂ)
Goal: Complex.abs (↑x ^ y) = x ^ y.re
-- Complex.abs_cpow_eq_rpow_re_of_nonneg
theorem abs_cpow_eq_rpow_re_of_nonneg {x : ℝ} (hx : 0 ≤ x) {y : ℂ} (hy : re y ≠ 0) :
    abs (x ^ y) = x ^ re y 
Arguments: {x : ℝ} (hx : 0 ≤ x) {y : ℂ} (hy : y.re ≠ 0)
Goal: Complex.abs (↑x ^ y) = x ^ y.re
-- Complex.norm_natCast_cpow_of_re_ne_zero
lemma norm_natCast_cpow_of_re_ne_zero (n : ℕ) {s : ℂ} (hs : s.re ≠ 0) :
    ‖(n : ℂ) ^ s‖ = (n : ℝ) ^ (s.re) 
Arguments: (n : ℕ) {s : ℂ} (hs : s.re ≠ 0)
Goal: ‖↑n ^ s‖ = ↑n ^ s.re
-- Complex.norm_natCast_cpow_of_pos
lemma norm_natCast_cpow_of_pos {n : ℕ} (hn : 0 < n) (s : ℂ) :
    ‖(n : ℂ) ^ s‖ = (n : ℝ) ^ (s.re) 
Arguments: {n : ℕ} (hn : 0 < n) (s : ℂ)
Goal: ‖↑n ^ s‖ = ↑n ^ s.re
-- Complex.norm_natCast_cpow_pos_of_pos
lemma norm_natCast_cpow_pos_of_pos {n : ℕ} (hn : 0 < n) (s : ℂ) : 0 < ‖(n : ℂ) ^ s‖ 
Arguments: {n : ℕ} (hn : 0 < n) (s : ℂ)
Goal: 0 < ‖↑n ^ s‖
-- Complex.cpow_mul_ofReal_nonneg
theorem cpow_mul_ofReal_nonneg {x : ℝ} (hx : 0 ≤ x) (y : ℝ) (z : ℂ) :
    (x : ℂ) ^ (↑y * z) = (↑(x ^ y) : ℂ) ^ z 
Arguments: {x : ℝ} (hx : 0 ≤ x) (y : ℝ) (z : ℂ)
Goal: ↑x ^ (↑y * z) = ↑(x ^ y) ^ z
-- Real.rpow_mul
theorem rpow_mul {x : ℝ} (hx : 0 ≤ x) (y z : ℝ) : x ^ (y * z) = (x ^ y) ^ z 
Arguments: {x : ℝ} (hx : 0 ≤ x) (y z : ℝ)
Goal: x ^ (y * z) = (x ^ y) ^ z
-- Real.rpow_add_int
theorem rpow_add_int {x : ℝ} (hx : x ≠ 0) (y : ℝ) (n : ℤ) : x ^ (y + n) = x ^ y * x ^ n 
Arguments: {x : ℝ} (hx : x ≠ 0) (y : ℝ) (n : ℤ)
Goal: x ^ (y + ↑n) = x ^ y * x ^ n
-- Real.rpow_add_nat
theorem rpow_add_nat {x : ℝ} (hx : x ≠ 0) (y : ℝ) (n : ℕ) : x ^ (y + n) = x ^ y * x ^ n 
Arguments: {x : ℝ} (hx : x ≠ 0) (y : ℝ) (n : ℕ)
Goal: x ^ (y + ↑n) = x ^ y * x ^ n
-- Real.rpow_sub_int
theorem rpow_sub_int {x : ℝ} (hx : x ≠ 0) (y : ℝ) (n : ℕ) : x ^ (y - n) = x ^ y / x ^ n 
Arguments: {x : ℝ} (hx : x ≠ 0) (y : ℝ) (n : ℕ)
Goal: x ^ (y - ↑n) = x ^ y / x ^ n
-- Real.rpow_sub_nat
theorem rpow_sub_nat {x : ℝ} (hx : x ≠ 0) (y : ℝ) (n : ℕ) : x ^ (y - n) = x ^ y / x ^ n 
Arguments: {x : ℝ} (hx : x ≠ 0) (y : ℝ) (n : ℕ)
Goal: x ^ (y - ↑n) = x ^ y / x ^ n
-- Real.rpow_add_int'
lemma rpow_add_int' (hx : 0 ≤ x) {n : ℤ} (h : y + n ≠ 0) : x ^ (y + n) = x ^ y * x ^ n 
Arguments: {x y : ℝ} (hx : 0 ≤ x) {n : ℤ} (h : y + ↑n ≠ 0)
Goal: x ^ (y + ↑n) = x ^ y * x ^ n
-- Real.rpow_add_nat'
lemma rpow_add_nat' (hx : 0 ≤ x) (h : y + n ≠ 0) : x ^ (y + n) = x ^ y * x ^ n 
Arguments: {x y : ℝ} {n : ℕ} (hx : 0 ≤ x) (h : y + ↑n ≠ 0)
Goal: x ^ (y + ↑n) = x ^ y * x ^ n
-- Real.rpow_sub_int'
lemma rpow_sub_int' (hx : 0 ≤ x) {n : ℤ} (h : y - n ≠ 0) : x ^ (y - n) = x ^ y / x ^ n 
Arguments: {x y : ℝ} (hx : 0 ≤ x) {n : ℤ} (h : y - ↑n ≠ 0)
Goal: x ^ (y - ↑n) = x ^ y / x ^ n
-- Real.rpow_sub_nat'
lemma rpow_sub_nat' (hx : 0 ≤ x) (h : y - n ≠ 0) : x ^ (y - n) = x ^ y / x ^ n 
Arguments: {x y : ℝ} {n : ℕ} (hx : 0 ≤ x) (h : y - ↑n ≠ 0)
Goal: x ^ (y - ↑n) = x ^ y / x ^ n
-- Real.rpow_add_one
theorem rpow_add_one {x : ℝ} (hx : x ≠ 0) (y : ℝ) : x ^ (y + 1) = x ^ y * x 
Arguments: {x : ℝ} (hx : x ≠ 0) (y : ℝ)
Goal: x ^ (y + 1) = x ^ y * x
-- Real.rpow_sub_one
theorem rpow_sub_one {x : ℝ} (hx : x ≠ 0) (y : ℝ) : x ^ (y - 1) = x ^ y / x 
Arguments: {x : ℝ} (hx : x ≠ 0) (y : ℝ)
Goal: x ^ (y - 1) = x ^ y / x
-- Real.rpow_add_one'
lemma rpow_add_one' (hx : 0 ≤ x) (h : y + 1 ≠ 0) : x ^ (y + 1) = x ^ y * x 
Arguments: {x y : ℝ} (hx : 0 ≤ x) (h : y + 1 ≠ 0)
Goal: x ^ (y + 1) = x ^ y * x
-- Real.rpow_one_add'
lemma rpow_one_add' (hx : 0 ≤ x) (h : 1 + y ≠ 0) : x ^ (1 + y) = x * x ^ y 
Arguments: {x y : ℝ} (hx : 0 ≤ x) (h : 1 + y ≠ 0)
Goal: x ^ (1 + y) = x * x ^ y
-- Real.rpow_sub_one'
lemma rpow_sub_one' (hx : 0 ≤ x) (h : y - 1 ≠ 0) : x ^ (y - 1) = x ^ y / x 
Arguments: {x y : ℝ} (hx : 0 ≤ x) (h : y - 1 ≠ 0)
Goal: x ^ (y - 1) = x ^ y / x
-- Real.rpow_one_sub'
lemma rpow_one_sub' (hx : 0 ≤ x) (h : 1 - y ≠ 0) : x ^ (1 - y) = x / x ^ y 
Arguments: {x y : ℝ} (hx : 0 ≤ x) (h : 1 - y ≠ 0)
Goal: x ^ (1 - y) = x / x ^ y
-- Real.rpow_two
@[simp]
theorem rpow_two (x : ℝ) : x ^ (2 : ℝ) = x ^ 2 
Arguments: (x : ℝ)
Goal: x ^ 2 = x ^ 2
-- Real.rpow_neg_one
theorem rpow_neg_one (x : ℝ) : x ^ (-1 : ℝ) = x⁻¹ 
Arguments: (x : ℝ)
Goal: x ^ (-1) = x⁻¹
-- Real.mul_rpow
theorem mul_rpow (hx : 0 ≤ x) (hy : 0 ≤ y) : (x * y) ^ z = x ^ z * y ^ z 
Arguments: {x y z : ℝ} (hx : 0 ≤ x) (hy : 0 ≤ y)
Goal: (x * y) ^ z = x ^ z * y ^ z
-- Real.inv_rpow
theorem inv_rpow (hx : 0 ≤ x) (y : ℝ) : x⁻¹ ^ y = (x ^ y)⁻¹ 
Arguments: {x : ℝ} (hx : 0 ≤ x) (y : ℝ)
Goal: x⁻¹ ^ y = (x ^ y)⁻¹
-- Real.div_rpow
theorem div_rpow (hx : 0 ≤ x) (hy : 0 ≤ y) (z : ℝ) : (x / y) ^ z = x ^ z / y ^ z 
Arguments: {x y : ℝ} (hx : 0 ≤ x) (hy : 0 ≤ y) (z : ℝ)
Goal: (x / y) ^ z = x ^ z / y ^ z
-- Real.log_rpow
theorem log_rpow {x : ℝ} (hx : 0 < x) (y : ℝ) : log (x ^ y) = y * log x 
Arguments: {x : ℝ} (hx : 0 < x) (y : ℝ)
Goal: Real.log (x ^ y) = y * Real.log x
-- Real.mul_log_eq_log_iff
theorem mul_log_eq_log_iff {x y z : ℝ} (hx : 0 < x) (hz : 0 < z) :
    y * log x = log z ↔ x ^ y = z 
Arguments: {x y z : ℝ} (hx : 0 < x) (hz : 0 < z)
Goal: y * Real.log x = Real.log z ↔ x ^ y = z
-- Real.rpow_rpow_inv
@[simp] lemma rpow_rpow_inv (hx : 0 ≤ x) (hy : y ≠ 0) : (x ^ y) ^ y⁻¹ = x 
Arguments: {x y : ℝ} (hx : 0 ≤ x) (hy : y ≠ 0)
Goal: (x ^ y) ^ y⁻¹ = x
-- Real.rpow_inv_rpow
@[simp] lemma rpow_inv_rpow (hx : 0 ≤ x) (hy : y ≠ 0) : (x ^ y⁻¹) ^ y = x 
Arguments: {x y : ℝ} (hx : 0 ≤ x) (hy : y ≠ 0)
Goal: (x ^ y⁻¹) ^ y = x
-- Real.pow_rpow_inv_natCast
theorem pow_rpow_inv_natCast (hx : 0 ≤ x) (hn : n ≠ 0) : (x ^ n) ^ (n⁻¹ : ℝ) = x 
Arguments: {x : ℝ} {n : ℕ} (hx : 0 ≤ x) (hn : n ≠ 0)
Goal: (x ^ n) ^ (↑n)⁻¹ = x
-- Real.rpow_inv_natCast_pow
theorem rpow_inv_natCast_pow (hx : 0 ≤ x) (hn : n ≠ 0) : (x ^ (n⁻¹ : ℝ)) ^ n = x 
Arguments: {x : ℝ} {n : ℕ} (hx : 0 ≤ x) (hn : n ≠ 0)
Goal: (x ^ (↑n)⁻¹) ^ n = x
-- Real.rpow_natCast_mul
lemma rpow_natCast_mul (hx : 0 ≤ x) (n : ℕ) (z : ℝ) : x ^ (n * z) = (x ^ n) ^ z 
Arguments: {x : ℝ} (hx : 0 ≤ x) (n : ℕ) (z : ℝ)
Goal: x ^ (↑n * z) = (x ^ n) ^ z
-- Real.rpow_mul_natCast
lemma rpow_mul_natCast (hx : 0 ≤ x) (y : ℝ) (n : ℕ) : x ^ (y * n) = (x ^ y) ^ n 
Arguments: {x : ℝ} (hx : 0 ≤ x) (y : ℝ) (n : ℕ)
Goal: x ^ (y * ↑n) = (x ^ y) ^ n
-- Real.rpow_intCast_mul
lemma rpow_intCast_mul (hx : 0 ≤ x) (n : ℤ) (z : ℝ) : x ^ (n * z) = (x ^ n) ^ z 
Arguments: {x : ℝ} (hx : 0 ≤ x) (n : ℤ) (z : ℝ)
Goal: x ^ (↑n * z) = (x ^ n) ^ z
-- Real.rpow_mul_intCast
lemma rpow_mul_intCast (hx : 0 ≤ x) (y : ℝ) (n : ℤ) : x ^ (y * n) = (x ^ y) ^ n 
Arguments: {x : ℝ} (hx : 0 ≤ x) (y : ℝ) (n : ℤ)
Goal: x ^ (y * ↑n) = (x ^ y) ^ n
-- Real.rpow_lt_rpow
@[gcongr]
theorem rpow_lt_rpow (hx : 0 ≤ x) (hxy : x < y) (hz : 0 < z) : x ^ z < y ^ z 
Arguments: {x y z : ℝ} (hx : 0 ≤ x) (hxy : x < y) (hz : 0 < z)
Goal: x ^ z < y ^ z
-- Real.strictMonoOn_rpow_Ici_of_exponent_pos
theorem strictMonoOn_rpow_Ici_of_exponent_pos {r : ℝ} (hr : 0 < r) :
    StrictMonoOn (fun (x : ℝ) => x ^ r) (Set.Ici 0) 
Arguments: {r : ℝ} (hr : 0 < r)
Goal: StrictMonoOn (fun x => x ^ r) (Set.Ici 0)
-- Real.rpow_le_rpow
@[gcongr]
theorem rpow_le_rpow {x y z : ℝ} (h : 0 ≤ x) (h₁ : x ≤ y) (h₂ : 0 ≤ z) : x ^ z ≤ y ^ z 
Arguments: {x y z : ℝ} (h : 0 ≤ x) (h₁ : x ≤ y) (h₂ : 0 ≤ z)
Goal: x ^ z ≤ y ^ z
-- Real.monotoneOn_rpow_Ici_of_exponent_nonneg
theorem monotoneOn_rpow_Ici_of_exponent_nonneg {r : ℝ} (hr : 0 ≤ r) :
    MonotoneOn (fun (x : ℝ) => x ^ r) (Set.Ici 0) 
Arguments: {r : ℝ} (hr : 0 ≤ r)
Goal: MonotoneOn (fun x => x ^ r) (Set.Ici 0)
-- Real.rpow_lt_rpow_of_neg
lemma rpow_lt_rpow_of_neg (hx : 0 < x) (hxy : x < y) (hz : z < 0) : y ^ z < x ^ z 
Arguments: {x y z : ℝ} (hx : 0 < x) (hxy : x < y) (hz : z < 0)
Goal: y ^ z < x ^ z
-- Real.rpow_le_rpow_of_nonpos
lemma rpow_le_rpow_of_nonpos (hx : 0 < x) (hxy : x ≤ y) (hz : z ≤ 0) : y ^ z ≤ x ^ z 
Arguments: {x y z : ℝ} (hx : 0 < x) (hxy : x ≤ y) (hz : z ≤ 0)
Goal: y ^ z ≤ x ^ z
-- Real.rpow_lt_rpow_iff
theorem rpow_lt_rpow_iff (hx : 0 ≤ x) (hy : 0 ≤ y) (hz : 0 < z) : x ^ z < y ^ z ↔ x < y 
Arguments: {x y z : ℝ} (hx : 0 ≤ x) (hy : 0 ≤ y) (hz : 0 < z)
Goal: x ^ z < y ^ z ↔ x < y
-- Real.rpow_le_rpow_iff
theorem rpow_le_rpow_iff (hx : 0 ≤ x) (hy : 0 ≤ y) (hz : 0 < z) : x ^ z ≤ y ^ z ↔ x ≤ y 
Arguments: {x y z : ℝ} (hx : 0 ≤ x) (hy : 0 ≤ y) (hz : 0 < z)
Goal: x ^ z ≤ y ^ z ↔ x ≤ y
-- Real.rpow_lt_rpow_iff_of_neg
lemma rpow_lt_rpow_iff_of_neg (hx : 0 < x) (hy : 0 < y) (hz : z < 0) : x ^ z < y ^ z ↔ y < x 
Arguments: {x y z : ℝ} (hx : 0 < x) (hy : 0 < y) (hz : z < 0)
Goal: x ^ z < y ^ z ↔ y < x
-- Real.rpow_le_rpow_iff_of_neg
lemma rpow_le_rpow_iff_of_neg (hx : 0 < x) (hy : 0 < y) (hz : z < 0) : x ^ z ≤ y ^ z ↔ y ≤ x 
Arguments: {x y z : ℝ} (hx : 0 < x) (hy : 0 < y) (hz : z < 0)
Goal: x ^ z ≤ y ^ z ↔ y ≤ x
-- Real.le_rpow_inv_iff_of_pos
lemma le_rpow_inv_iff_of_pos (hx : 0 ≤ x) (hy : 0 ≤ y) (hz : 0 < z) : x ≤ y ^ z⁻¹ ↔ x ^ z ≤ y 
Arguments: {x y z : ℝ} (hx : 0 ≤ x) (hy : 0 ≤ y) (hz : 0 < z)
Goal: x ≤ y ^ z⁻¹ ↔ x ^ z ≤ y
-- Real.rpow_inv_le_iff_of_pos
lemma rpow_inv_le_iff_of_pos (hx : 0 ≤ x) (hy : 0 ≤ y) (hz : 0 < z) : x ^ z⁻¹ ≤ y ↔ x ≤ y ^ z 
Arguments: {x y z : ℝ} (hx : 0 ≤ x) (hy : 0 ≤ y) (hz : 0 < z)
Goal: x ^ z⁻¹ ≤ y ↔ x ≤ y ^ z
-- Real.lt_rpow_inv_iff_of_pos
lemma lt_rpow_inv_iff_of_pos (hx : 0 ≤ x) (hy : 0 ≤ y) (hz : 0 < z) : x < y ^ z⁻¹ ↔ x ^ z < y 
Arguments: {x y z : ℝ} (hx : 0 ≤ x) (hy : 0 ≤ y) (hz : 0 < z)
Goal: x < y ^ z⁻¹ ↔ x ^ z < y
-- Real.rpow_inv_lt_iff_of_pos
lemma rpow_inv_lt_iff_of_pos (hx : 0 ≤ x) (hy : 0 ≤ y) (hz : 0 < z) : x ^ z⁻¹ < y ↔ x < y ^ z 
Arguments: {x y z : ℝ} (hx : 0 ≤ x) (hy : 0 ≤ y) (hz : 0 < z)
Goal: x ^ z⁻¹ < y ↔ x < y ^ z
-- Real.le_rpow_inv_iff_of_neg
theorem le_rpow_inv_iff_of_neg (hx : 0 < x) (hy : 0 < y) (hz : z < 0) :
    x ≤ y ^ z⁻¹ ↔ y ≤ x ^ z 
Arguments: {x y z : ℝ} (hx : 0 < x) (hy : 0 < y) (hz : z < 0)
Goal: x ≤ y ^ z⁻¹ ↔ y ≤ x ^ z
-- Real.lt_rpow_inv_iff_of_neg
theorem lt_rpow_inv_iff_of_neg (hx : 0 < x) (hy : 0 < y) (hz : z < 0) :
    x < y ^ z⁻¹ ↔ y < x ^ z 
Arguments: {x y z : ℝ} (hx : 0 < x) (hy : 0 < y) (hz : z < 0)
Goal: x < y ^ z⁻¹ ↔ y < x ^ z
-- Real.rpow_inv_lt_iff_of_neg
theorem rpow_inv_lt_iff_of_neg (hx : 0 < x) (hy : 0 < y) (hz : z < 0) :
    x ^ z⁻¹ < y ↔ y ^ z < x 
Arguments: {x y z : ℝ} (hx : 0 < x) (hy : 0 < y) (hz : z < 0)
Goal: x ^ z⁻¹ < y ↔ y ^ z < x