ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- DifferentiableWithinAt.sinh
theorem DifferentiableWithinAt.sinh (hf : DifferentiableWithinAt ℝ f s x) :
    DifferentiableWithinAt ℝ (fun x => Real.sinh (f x)) s x 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {f : E → ℝ} {x : E} {s : Set E} (hf : DifferentiableWithinAt ℝ f s x)
Goal: DifferentiableWithinAt ℝ (fun x => Real.sinh (f x)) s x
-- DifferentiableAt.sinh
@[simp]
theorem DifferentiableAt.sinh (hc : DifferentiableAt ℝ f x) :
    DifferentiableAt ℝ (fun x => Real.sinh (f x)) x 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {f : E → ℝ} {x : E} (hc : DifferentiableAt ℝ f x)
Goal: DifferentiableAt ℝ (fun x => Real.sinh (f x)) x
-- DifferentiableOn.sinh
theorem DifferentiableOn.sinh (hc : DifferentiableOn ℝ f s) :
    DifferentiableOn ℝ (fun x => Real.sinh (f x)) s 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {f : E → ℝ} {s : Set E} (hc : DifferentiableOn ℝ f s)
Goal: DifferentiableOn ℝ (fun x => Real.sinh (f x)) s
-- Differentiable.sinh
@[simp]
theorem Differentiable.sinh (hc : Differentiable ℝ f) : Differentiable ℝ fun x => Real.sinh (f x) 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {f : E → ℝ} (hc : Differentiable ℝ f)
Goal: Differentiable ℝ fun x => Real.sinh (f x)
-- fderivWithin_sinh
theorem fderivWithin_sinh (hf : DifferentiableWithinAt ℝ f s x) (hxs : UniqueDiffWithinAt ℝ s x) :
    fderivWithin ℝ (fun x => Real.sinh (f x)) s x = Real.cosh (f x) • fderivWithin ℝ f s x 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {f : E → ℝ} {x : E} {s : Set E} (hf : DifferentiableWithinAt ℝ f s x) (hxs : UniqueDiffWithinAt ℝ s x)
Goal: fderivWithin ℝ (fun x => Real.sinh (f x)) s x = Real.cosh (f x) • fderivWithin ℝ f s x
-- fderiv_sinh
@[simp]
theorem fderiv_sinh (hc : DifferentiableAt ℝ f x) :
    fderiv ℝ (fun x => Real.sinh (f x)) x = Real.cosh (f x) • fderiv ℝ f x 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {f : E → ℝ} {x : E} (hc : DifferentiableAt ℝ f x)
Goal: fderiv ℝ (fun x => Real.sinh (f x)) x = Real.cosh (f x) • fderiv ℝ f x
-- ContDiff.sinh
theorem ContDiff.sinh {n} (h : ContDiff ℝ n f) : ContDiff ℝ n fun x => Real.sinh (f x) 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {f : E → ℝ} {n : ℕ∞} (h : ContDiff ℝ n f)
Goal: ContDiff ℝ n fun x => Real.sinh (f x)
-- ContDiffAt.sinh
theorem ContDiffAt.sinh {n} (hf : ContDiffAt ℝ n f x) :
    ContDiffAt ℝ n (fun x => Real.sinh (f x)) x 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {f : E → ℝ} {x : E} {n : ℕ∞} (hf : ContDiffAt ℝ n f x)
Goal: ContDiffAt ℝ n (fun x => Real.sinh (f x)) x
-- ContDiffOn.sinh
theorem ContDiffOn.sinh {n} (hf : ContDiffOn ℝ n f s) :
    ContDiffOn ℝ n (fun x => Real.sinh (f x)) s 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {f : E → ℝ} {s : Set E} {n : ℕ∞} (hf : ContDiffOn ℝ n f s)
Goal: ContDiffOn ℝ n (fun x => Real.sinh (f x)) s
-- ContDiffWithinAt.sinh
theorem ContDiffWithinAt.sinh {n} (hf : ContDiffWithinAt ℝ n f s x) :
    ContDiffWithinAt ℝ n (fun x => Real.sinh (f x)) s x 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {f : E → ℝ} {x : E} {s : Set E} {n : ℕ∞} (hf : ContDiffWithinAt ℝ n f s x)
Goal: ContDiffWithinAt ℝ n (fun x => Real.sinh (f x)) s x
-- Complex.logDeriv_sin
@[simp]
theorem Complex.logDeriv_sin : logDeriv (Complex.sin) = Complex.cot 
Goal: logDeriv Complex.sin = Complex.cot
-- Real.logDeriv_sin
@[simp]
theorem Real.logDeriv_sin : logDeriv (Real.sin) = Real.cot 
Goal: logDeriv Real.sin = Real.cot
-- Complex.logDeriv_cos
@[simp]
theorem Complex.logDeriv_cos : logDeriv (Complex.cos) = -Complex.tan 
Goal: logDeriv Complex.cos = -Complex.tan
-- Real.logDeriv_cos
@[simp]
theorem Real.logDeriv_cos : logDeriv (Real.cos) = -Real.tan 
Goal: logDeriv Real.cos = -Real.tan
-- Complex.logDeriv_cosh
@[simp]
theorem Complex.logDeriv_cosh : logDeriv (Complex.cosh) = Complex.tanh 
Goal: logDeriv Complex.cosh = Complex.tanh
-- Real.logDeriv_cosh
@[simp]
theorem Real.logDeriv_cosh : logDeriv (Real.cosh) = Real.tanh 
Goal: logDeriv Real.cosh = Real.tanh
-- Complex.LogDeriv_exp
@[simp]
theorem Complex.LogDeriv_exp : logDeriv (Complex.exp) = 1 
Goal: logDeriv Complex.exp = 1
-- Real.LogDeriv_exp
@[simp]
theorem Real.LogDeriv_exp : logDeriv (Real.exp) = 1 
Goal: logDeriv Real.exp = 1
-- polarCoord_source
@[simps]
def polarCoord : PartialHomeomorph (ℝ × ℝ) (ℝ × ℝ) 
Goal: polarCoord.source = {q | 0 < q.1} ∪ {q | q.2 ≠ 0}
-- polarCoord_target
@[simps]
def polarCoord : PartialHomeomorph (ℝ × ℝ) (ℝ × ℝ) 
Goal: polarCoord.target = Set.Ioi 0 ×ˢ Set.Ioo (-Real.pi) Real.pi
-- polarCoord_apply
@[simps]
def polarCoord : PartialHomeomorph (ℝ × ℝ) (ℝ × ℝ) 
Arguments: (q : ℝ × ℝ)
Goal: ↑polarCoord q = (√(q.1 ^ 2 + q.2 ^ 2), (Complex.equivRealProd.symm q).arg)
-- polarCoord_symm_apply
@[simps]
def polarCoord : PartialHomeomorph (ℝ × ℝ) (ℝ × ℝ) 
Arguments: (p : ℝ × ℝ)
Goal: ↑polarCoord.symm p = (p.1 * Real.cos p.2, p.1 * Real.sin p.2)
-- hasFDerivAt_polarCoord_symm
theorem hasFDerivAt_polarCoord_symm (p : ℝ × ℝ) :
    HasFDerivAt polarCoord.symm
      (LinearMap.toContinuousLinearMap (Matrix.toLin (Basis.finTwoProd ℝ) (Basis.finTwoProd ℝ)
        !![cos p.2, -p.1 * sin p.2; sin p.2, p.1 * cos p.2])) p 
Arguments: (p : ℝ × ℝ)
Goal: HasFDerivAt (↑polarCoord.symm)
  (LinearMap.toContinuousLinearMap
    ((Matrix.toLin (Basis.finTwoProd ℝ) (Basis.finTwoProd ℝ))
      !![Real.cos p.2, -p.1 * Real.sin p.2; Real.sin p.2, p.1 * Real.cos p.2]))
  p
-- polarCoord_source_ae_eq_univ
theorem polarCoord_source_ae_eq_univ : polarCoord.source =ᵐ[volume] univ 
Goal: polarCoord.source =ᵐ[MeasureTheory.volume] Set.univ
-- integral_comp_polarCoord_symm
theorem integral_comp_polarCoord_symm {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
    (f : ℝ × ℝ → E) :
    (∫ p in polarCoord.target, p.1 • f (polarCoord.symm p)) = ∫ p, f p 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] (f : ℝ × ℝ → E)
Goal: ∫ (p : ℝ × ℝ) in polarCoord.target, p.1 • f (↑polarCoord.symm p) = ∫ (p : ℝ × ℝ), f p
-- Complex.polarCoord_apply
protected theorem polarCoord_apply (a : ℂ) :
    Complex.polarCoord a = (Complex.abs a, Complex.arg a) 
Arguments: (a : ℂ)
Goal: ↑Complex.polarCoord a = (Complex.abs a, a.arg)
-- Complex.polarCoord_source
protected theorem polarCoord_source : Complex.polarCoord.source = slitPlane 
Goal: Complex.polarCoord.source = Complex.slitPlane
-- Complex.polarCoord_target
protected theorem polarCoord_target :
    Complex.polarCoord.target = Set.Ioi (0 : ℝ) ×ˢ Set.Ioo (-π) π 
Goal: Complex.polarCoord.target = Set.Ioi 0 ×ˢ Set.Ioo (-Real.pi) Real.pi
-- Complex.polarCoord_symm_apply
@[simp]
protected theorem polarCoord_symm_apply (p : ℝ × ℝ) :
    Complex.polarCoord.symm p = p.1 * (Real.cos p.2 + Real.sin p.2 * Complex.I) 
Arguments: (p : ℝ × ℝ)
Goal: ↑Complex.polarCoord.symm p = ↑p.1 * (↑(Real.cos p.2) + ↑(Real.sin p.2) * Complex.I)
-- Complex.polarCoord_symm_abs
theorem polarCoord_symm_abs (p : ℝ × ℝ) :
    Complex.abs (Complex.polarCoord.symm p) = |p.1| 
Arguments: (p : ℝ × ℝ)
Goal: Complex.abs (↑Complex.polarCoord.symm p) = |p.1|
-- Complex.integral_comp_polarCoord_symm
protected theorem integral_comp_polarCoord_symm {E : Type*} [NormedAddCommGroup E]
    [NormedSpace ℝ E] (f : ℂ → E) :
    (∫ p in polarCoord.target, p.1 • f (Complex.polarCoord.symm p)) = ∫ p, f p 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] (f : ℂ → E)
Goal: ∫ (p : ℝ × ℝ) in polarCoord.target, p.1 • f (↑Complex.polarCoord.symm p) = ∫ (p : ℂ), f p
-- hasStrictFDerivAt_exp_zero_of_radius_pos
theorem hasStrictFDerivAt_exp_zero_of_radius_pos (h : 0 < (expSeries 𝕂 𝔸).radius) :
    HasStrictFDerivAt (exp 𝕂) (1 : 𝔸 →L[𝕂] 𝔸) 0 
Arguments: {𝕂 : Type u_1} {𝔸 : Type u_2} [NontriviallyNormedField 𝕂] [NormedRing 𝔸] [NormedAlgebra 𝕂 𝔸] [CompleteSpace 𝔸] (h : 0 < (NormedSpace.expSeries 𝕂 𝔸).radius)
Goal: HasStrictFDerivAt (NormedSpace.exp 𝕂) 1 0
-- hasFDerivAt_exp_zero_of_radius_pos
theorem hasFDerivAt_exp_zero_of_radius_pos (h : 0 < (expSeries 𝕂 𝔸).radius) :
    HasFDerivAt (exp 𝕂) (1 : 𝔸 →L[𝕂] 𝔸) 0 
Arguments: {𝕂 : Type u_1} {𝔸 : Type u_2} [NontriviallyNormedField 𝕂] [NormedRing 𝔸] [NormedAlgebra 𝕂 𝔸] [CompleteSpace 𝔸] (h : 0 < (NormedSpace.expSeries 𝕂 𝔸).radius)
Goal: HasFDerivAt (NormedSpace.exp 𝕂) 1 0
-- hasFDerivAt_exp_of_mem_ball
theorem hasFDerivAt_exp_of_mem_ball [CharZero 𝕂] {x : 𝔸}
    (hx : x ∈ EMetric.ball (0 : 𝔸) (expSeries 𝕂 𝔸).radius) :
    HasFDerivAt (exp 𝕂) (exp 𝕂 x • (1 : 𝔸 →L[𝕂] 𝔸)) x 
Arguments: {𝕂 : Type u_1} {𝔸 : Type u_2} [NontriviallyNormedField 𝕂] [NormedCommRing 𝔸] [NormedAlgebra 𝕂 𝔸] [CompleteSpace 𝔸] [CharZero 𝕂] {x : 𝔸} (hx : x ∈ EMetric.ball 0 (NormedSpace.expSeries 𝕂 𝔸).radius)
Goal: HasFDerivAt (NormedSpace.exp 𝕂) (NormedSpace.exp 𝕂 x • 1) x
-- hasStrictFDerivAt_exp_of_mem_ball
theorem hasStrictFDerivAt_exp_of_mem_ball [CharZero 𝕂] {x : 𝔸}
    (hx : x ∈ EMetric.ball (0 : 𝔸) (expSeries 𝕂 𝔸).radius) :
    HasStrictFDerivAt (exp 𝕂) (exp 𝕂 x • (1 : 𝔸 →L[𝕂] 𝔸)) x 
Arguments: {𝕂 : Type u_1} {𝔸 : Type u_2} [NontriviallyNormedField 𝕂] [NormedCommRing 𝔸] [NormedAlgebra 𝕂 𝔸] [CompleteSpace 𝔸] [CharZero 𝕂] {x : 𝔸} (hx : x ∈ EMetric.ball 0 (NormedSpace.expSeries 𝕂 𝔸).radius)
Goal: HasStrictFDerivAt (NormedSpace.exp 𝕂) (NormedSpace.exp 𝕂 x • 1) x
-- hasStrictDerivAt_exp_of_mem_ball
theorem hasStrictDerivAt_exp_of_mem_ball [CharZero 𝕂] {x : 𝕂}
    (hx : x ∈ EMetric.ball (0 : 𝕂) (expSeries 𝕂 𝕂).radius) :
    HasStrictDerivAt (exp 𝕂) (exp 𝕂 x) x 
Arguments: {𝕂 : Type u_1} [NontriviallyNormedField 𝕂] [CompleteSpace 𝕂] [CharZero 𝕂] {x : 𝕂} (hx : x ∈ EMetric.ball 0 (NormedSpace.expSeries 𝕂 𝕂).radius)
Goal: HasStrictDerivAt (NormedSpace.exp 𝕂) (NormedSpace.exp 𝕂 x) x
-- hasDerivAt_exp_of_mem_ball
theorem hasDerivAt_exp_of_mem_ball [CharZero 𝕂] {x : 𝕂}
    (hx : x ∈ EMetric.ball (0 : 𝕂) (expSeries 𝕂 𝕂).radius) : HasDerivAt (exp 𝕂) (exp 𝕂 x) x 
Arguments: {𝕂 : Type u_1} [NontriviallyNormedField 𝕂] [CompleteSpace 𝕂] [CharZero 𝕂] {x : 𝕂} (hx : x ∈ EMetric.ball 0 (NormedSpace.expSeries 𝕂 𝕂).radius)
Goal: HasDerivAt (NormedSpace.exp 𝕂) (NormedSpace.exp 𝕂 x) x
-- hasStrictDerivAt_exp_zero_of_radius_pos
theorem hasStrictDerivAt_exp_zero_of_radius_pos (h : 0 < (expSeries 𝕂 𝕂).radius) :
    HasStrictDerivAt (exp 𝕂) (1 : 𝕂) 0 
Arguments: {𝕂 : Type u_1} [NontriviallyNormedField 𝕂] [CompleteSpace 𝕂] (h : 0 < (NormedSpace.expSeries 𝕂 𝕂).radius)
Goal: HasStrictDerivAt (NormedSpace.exp 𝕂) 1 0
-- hasDerivAt_exp_zero_of_radius_pos
theorem hasDerivAt_exp_zero_of_radius_pos (h : 0 < (expSeries 𝕂 𝕂).radius) :
    HasDerivAt (exp 𝕂) (1 : 𝕂) 0 
Arguments: {𝕂 : Type u_1} [NontriviallyNormedField 𝕂] [CompleteSpace 𝕂] (h : 0 < (NormedSpace.expSeries 𝕂 𝕂).radius)
Goal: HasDerivAt (NormedSpace.exp 𝕂) 1 0
-- hasStrictFDerivAt_exp_zero
theorem hasStrictFDerivAt_exp_zero : HasStrictFDerivAt (exp 𝕂) (1 : 𝔸 →L[𝕂] 𝔸) 0 
Arguments: {𝕂 : Type u_1} {𝔸 : Type u_2} [RCLike 𝕂] [NormedRing 𝔸] [NormedAlgebra 𝕂 𝔸] [CompleteSpace 𝔸]
Goal: HasStrictFDerivAt (NormedSpace.exp 𝕂) 1 0
-- hasFDerivAt_exp_zero
theorem hasFDerivAt_exp_zero : HasFDerivAt (exp 𝕂) (1 : 𝔸 →L[𝕂] 𝔸) 0 
Arguments: {𝕂 : Type u_1} {𝔸 : Type u_2} [RCLike 𝕂] [NormedRing 𝔸] [NormedAlgebra 𝕂 𝔸] [CompleteSpace 𝔸]
Goal: HasFDerivAt (NormedSpace.exp 𝕂) 1 0
-- hasStrictFDerivAt_exp
theorem hasStrictFDerivAt_exp {x : 𝔸} : HasStrictFDerivAt (exp 𝕂) (exp 𝕂 x • (1 : 𝔸 →L[𝕂] 𝔸)) x 
Arguments: {𝕂 : Type u_1} {𝔸 : Type u_2} [RCLike 𝕂] [NormedCommRing 𝔸] [NormedAlgebra 𝕂 𝔸] [CompleteSpace 𝔸] {x : 𝔸}
Goal: HasStrictFDerivAt (NormedSpace.exp 𝕂) (NormedSpace.exp 𝕂 x • 1) x
-- hasFDerivAt_exp
theorem hasFDerivAt_exp {x : 𝔸} : HasFDerivAt (exp 𝕂) (exp 𝕂 x • (1 : 𝔸 →L[𝕂] 𝔸)) x 
Arguments: {𝕂 : Type u_1} {𝔸 : Type u_2} [RCLike 𝕂] [NormedCommRing 𝔸] [NormedAlgebra 𝕂 𝔸] [CompleteSpace 𝔸] {x : 𝔸}
Goal: HasFDerivAt (NormedSpace.exp 𝕂) (NormedSpace.exp 𝕂 x • 1) x
-- hasStrictDerivAt_exp
theorem hasStrictDerivAt_exp {x : 𝕂} : HasStrictDerivAt (exp 𝕂) (exp 𝕂 x) x 
Arguments: {𝕂 : Type u_1} [RCLike 𝕂] {x : 𝕂}
Goal: HasStrictDerivAt (NormedSpace.exp 𝕂) (NormedSpace.exp 𝕂 x) x
-- hasDerivAt_exp
theorem hasDerivAt_exp {x : 𝕂} : HasDerivAt (exp 𝕂) (exp 𝕂 x) x 
Arguments: {𝕂 : Type u_1} [RCLike 𝕂] {x : 𝕂}
Goal: HasDerivAt (NormedSpace.exp 𝕂) (NormedSpace.exp 𝕂 x) x
-- hasStrictDerivAt_exp_zero
theorem hasStrictDerivAt_exp_zero : HasStrictDerivAt (exp 𝕂) (1 : 𝕂) 0 
Arguments: {𝕂 : Type u_1} [RCLike 𝕂]
Goal: HasStrictDerivAt (NormedSpace.exp 𝕂) 1 0
-- hasDerivAt_exp_zero
theorem hasDerivAt_exp_zero : HasDerivAt (exp 𝕂) (1 : 𝕂) 0 
Arguments: {𝕂 : Type u_1} [RCLike 𝕂]
Goal: HasDerivAt (NormedSpace.exp 𝕂) 1 0
-- Complex.exp_eq_exp_ℂ
theorem Complex.exp_eq_exp_ℂ : Complex.exp = NormedSpace.exp ℂ 
Goal: Complex.exp = NormedSpace.exp ℂ
-- Real.exp_eq_exp_ℝ
theorem Real.exp_eq_exp_ℝ : Real.exp = NormedSpace.exp ℝ 
Goal: Real.exp = NormedSpace.exp ℝ
-- hasFDerivAt_exp_smul_const_of_mem_ball
theorem hasFDerivAt_exp_smul_const_of_mem_ball (x : 𝔸) (t : 𝕊)
    (htx : t • x ∈ EMetric.ball (0 : 𝔸) (expSeries 𝕂 𝔸).radius) :
    HasFDerivAt (fun u : 𝕊 => exp 𝕂 (u • x)) (exp 𝕂 (t • x) • (1 : 𝕊 →L[𝕂] 𝕊).smulRight x) t 
Arguments: (𝕂 : Type u_1) {𝕊 : Type u_2} {𝔸 : Type u_3} [NontriviallyNormedField 𝕂] [CharZero 𝕂] [NormedCommRing 𝕊] [NormedRing 𝔸] [NormedSpace 𝕂 𝕊] [NormedAlgebra 𝕂 𝔸] [Algebra 𝕊 𝔸] [ContinuousSMul 𝕊 𝔸] [IsScalarTower 𝕂 𝕊 𝔸] [CompleteSpace 𝔸] (x : 𝔸) (t : 𝕊) (htx : t • x ∈ EMetric.ball 0 (NormedSpace.expSeries 𝕂 𝔸).radius)
Goal: HasFDerivAt (fun u => NormedSpace.exp 𝕂 (u • x)) (NormedSpace.exp 𝕂 (t • x) • ContinuousLinearMap.smulRight 1 x) t
-- hasFDerivAt_exp_smul_const_of_mem_ball'
theorem hasFDerivAt_exp_smul_const_of_mem_ball' (x : 𝔸) (t : 𝕊)
    (htx : t • x ∈ EMetric.ball (0 : 𝔸) (expSeries 𝕂 𝔸).radius) :
    HasFDerivAt (fun u : 𝕊 => exp 𝕂 (u • x))
      (((1 : 𝕊 →L[𝕂] 𝕊).smulRight x).smulRight (exp 𝕂 (t • x))) t 
Arguments: (𝕂 : Type u_1) {𝕊 : Type u_2} {𝔸 : Type u_3} [NontriviallyNormedField 𝕂] [CharZero 𝕂] [NormedCommRing 𝕊] [NormedRing 𝔸] [NormedSpace 𝕂 𝕊] [NormedAlgebra 𝕂 𝔸] [Algebra 𝕊 𝔸] [ContinuousSMul 𝕊 𝔸] [IsScalarTower 𝕂 𝕊 𝔸] [CompleteSpace 𝔸] (x : 𝔸) (t : 𝕊) (htx : t • x ∈ EMetric.ball 0 (NormedSpace.expSeries 𝕂 𝔸).radius)
Goal: HasFDerivAt (fun u => NormedSpace.exp 𝕂 (u • x))
  ((ContinuousLinearMap.smulRight 1 x).smulRight (NormedSpace.exp 𝕂 (t • x))) t
-- hasStrictFDerivAt_exp_smul_const_of_mem_ball
theorem hasStrictFDerivAt_exp_smul_const_of_mem_ball (x : 𝔸) (t : 𝕊)
    (htx : t • x ∈ EMetric.ball (0 : 𝔸) (expSeries 𝕂 𝔸).radius) :
    HasStrictFDerivAt (fun u : 𝕊 => exp 𝕂 (u • x))
      (exp 𝕂 (t • x) • (1 : 𝕊 →L[𝕂] 𝕊).smulRight x) t 
Arguments: (𝕂 : Type u_1) {𝕊 : Type u_2} {𝔸 : Type u_3} [NontriviallyNormedField 𝕂] [CharZero 𝕂] [NormedCommRing 𝕊] [NormedRing 𝔸] [NormedSpace 𝕂 𝕊] [NormedAlgebra 𝕂 𝔸] [Algebra 𝕊 𝔸] [ContinuousSMul 𝕊 𝔸] [IsScalarTower 𝕂 𝕊 𝔸] [CompleteSpace 𝔸] (x : 𝔸) (t : 𝕊) (htx : t • x ∈ EMetric.ball 0 (NormedSpace.expSeries 𝕂 𝔸).radius)
Goal: HasStrictFDerivAt (fun u => NormedSpace.exp 𝕂 (u • x)) (NormedSpace.exp 𝕂 (t • x) • ContinuousLinearMap.smulRight 1 x) t
-- hasStrictFDerivAt_exp_smul_const_of_mem_ball'
theorem hasStrictFDerivAt_exp_smul_const_of_mem_ball' (x : 𝔸) (t : 𝕊)
    (htx : t • x ∈ EMetric.ball (0 : 𝔸) (expSeries 𝕂 𝔸).radius) :
    HasStrictFDerivAt (fun u : 𝕊 => exp 𝕂 (u • x))
      (((1 : 𝕊 →L[𝕂] 𝕊).smulRight x).smulRight (exp 𝕂 (t • x))) t 
Arguments: (𝕂 : Type u_1) {𝕊 : Type u_2} {𝔸 : Type u_3} [NontriviallyNormedField 𝕂] [CharZero 𝕂] [NormedCommRing 𝕊] [NormedRing 𝔸] [NormedSpace 𝕂 𝕊] [NormedAlgebra 𝕂 𝔸] [Algebra 𝕊 𝔸] [ContinuousSMul 𝕊 𝔸] [IsScalarTower 𝕂 𝕊 𝔸] [CompleteSpace 𝔸] (x : 𝔸) (t : 𝕊) (htx : t • x ∈ EMetric.ball 0 (NormedSpace.expSeries 𝕂 𝔸).radius)
Goal: HasStrictFDerivAt (fun u => NormedSpace.exp 𝕂 (u • x))
  ((ContinuousLinearMap.smulRight 1 x).smulRight (NormedSpace.exp 𝕂 (t • x))) t
-- hasStrictDerivAt_exp_smul_const_of_mem_ball
theorem hasStrictDerivAt_exp_smul_const_of_mem_ball (x : 𝔸) (t : 𝕂)
    (htx : t • x ∈ EMetric.ball (0 : 𝔸) (expSeries 𝕂 𝔸).radius) :
    HasStrictDerivAt (fun u : 𝕂 => exp 𝕂 (u • x)) (exp 𝕂 (t • x) * x) t 
Arguments: {𝕂 : Type u_1} {𝔸 : Type u_3} [NontriviallyNormedField 𝕂] [CharZero 𝕂] [NormedRing 𝔸] [NormedAlgebra 𝕂 𝔸] [CompleteSpace 𝔸] (x : 𝔸) (t : 𝕂) (htx : t • x ∈ EMetric.ball 0 (NormedSpace.expSeries 𝕂 𝔸).radius)
Goal: HasStrictDerivAt (fun u => NormedSpace.exp 𝕂 (u • x)) (NormedSpace.exp 𝕂 (t • x) * x) t
-- hasStrictDerivAt_exp_smul_const_of_mem_ball'
theorem hasStrictDerivAt_exp_smul_const_of_mem_ball' (x : 𝔸) (t : 𝕂)
    (htx : t • x ∈ EMetric.ball (0 : 𝔸) (expSeries 𝕂 𝔸).radius) :
    HasStrictDerivAt (fun u : 𝕂 => exp 𝕂 (u • x)) (x * exp 𝕂 (t • x)) t 
Arguments: {𝕂 : Type u_1} {𝔸 : Type u_3} [NontriviallyNormedField 𝕂] [CharZero 𝕂] [NormedRing 𝔸] [NormedAlgebra 𝕂 𝔸] [CompleteSpace 𝔸] (x : 𝔸) (t : 𝕂) (htx : t • x ∈ EMetric.ball 0 (NormedSpace.expSeries 𝕂 𝔸).radius)
Goal: HasStrictDerivAt (fun u => NormedSpace.exp 𝕂 (u • x)) (x * NormedSpace.exp 𝕂 (t • x)) t
-- hasDerivAt_exp_smul_const_of_mem_ball
theorem hasDerivAt_exp_smul_const_of_mem_ball (x : 𝔸) (t : 𝕂)
    (htx : t • x ∈ EMetric.ball (0 : 𝔸) (expSeries 𝕂 𝔸).radius) :
    HasDerivAt (fun u : 𝕂 => exp 𝕂 (u • x)) (exp 𝕂 (t • x) * x) t 
Arguments: {𝕂 : Type u_1} {𝔸 : Type u_3} [NontriviallyNormedField 𝕂] [CharZero 𝕂] [NormedRing 𝔸] [NormedAlgebra 𝕂 𝔸] [CompleteSpace 𝔸] (x : 𝔸) (t : 𝕂) (htx : t • x ∈ EMetric.ball 0 (NormedSpace.expSeries 𝕂 𝔸).radius)
Goal: HasDerivAt (fun u => NormedSpace.exp 𝕂 (u • x)) (NormedSpace.exp 𝕂 (t • x) * x) t
-- hasDerivAt_exp_smul_const_of_mem_ball'
theorem hasDerivAt_exp_smul_const_of_mem_ball' (x : 𝔸) (t : 𝕂)
    (htx : t • x ∈ EMetric.ball (0 : 𝔸) (expSeries 𝕂 𝔸).radius) :
    HasDerivAt (fun u : 𝕂 => exp 𝕂 (u • x)) (x * exp 𝕂 (t • x)) t 
Arguments: {𝕂 : Type u_1} {𝔸 : Type u_3} [NontriviallyNormedField 𝕂] [CharZero 𝕂] [NormedRing 𝔸] [NormedAlgebra 𝕂 𝔸] [CompleteSpace 𝔸] (x : 𝔸) (t : 𝕂) (htx : t • x ∈ EMetric.ball 0 (NormedSpace.expSeries 𝕂 𝔸).radius)
Goal: HasDerivAt (fun u => NormedSpace.exp 𝕂 (u • x)) (x * NormedSpace.exp 𝕂 (t • x)) t
-- hasFDerivAt_exp_smul_const
theorem hasFDerivAt_exp_smul_const (x : 𝔸) (t : 𝕊) :
    HasFDerivAt (fun u : 𝕊 => exp 𝕂 (u • x)) (exp 𝕂 (t • x) • (1 : 𝕊 →L[𝕂] 𝕊).smulRight x) t 
Arguments: (𝕂 : Type u_1) {𝕊 : Type u_2} {𝔸 : Type u_3} [RCLike 𝕂] [NormedCommRing 𝕊] [NormedRing 𝔸] [NormedAlgebra 𝕂 𝕊] [NormedAlgebra 𝕂 𝔸] [Algebra 𝕊 𝔸] [ContinuousSMul 𝕊 𝔸] [IsScalarTower 𝕂 𝕊 𝔸] [CompleteSpace 𝔸] (x : 𝔸) (t : 𝕊)
Goal: HasFDerivAt (fun u => NormedSpace.exp 𝕂 (u • x)) (NormedSpace.exp 𝕂 (t • x) • ContinuousLinearMap.smulRight 1 x) t
-- hasFDerivAt_exp_smul_const'
theorem hasFDerivAt_exp_smul_const' (x : 𝔸) (t : 𝕊) :
    HasFDerivAt (fun u : 𝕊 => exp 𝕂 (u • x))
      (((1 : 𝕊 →L[𝕂] 𝕊).smulRight x).smulRight (exp 𝕂 (t • x))) t 
Arguments: (𝕂 : Type u_1) {𝕊 : Type u_2} {𝔸 : Type u_3} [RCLike 𝕂] [NormedCommRing 𝕊] [NormedRing 𝔸] [NormedAlgebra 𝕂 𝕊] [NormedAlgebra 𝕂 𝔸] [Algebra 𝕊 𝔸] [ContinuousSMul 𝕊 𝔸] [IsScalarTower 𝕂 𝕊 𝔸] [CompleteSpace 𝔸] (x : 𝔸) (t : 𝕊)
Goal: HasFDerivAt (fun u => NormedSpace.exp 𝕂 (u • x))
  ((ContinuousLinearMap.smulRight 1 x).smulRight (NormedSpace.exp 𝕂 (t • x))) t
-- hasStrictFDerivAt_exp_smul_const
theorem hasStrictFDerivAt_exp_smul_const (x : 𝔸) (t : 𝕊) :
    HasStrictFDerivAt (fun u : 𝕊 => exp 𝕂 (u • x))
      (exp 𝕂 (t • x) • (1 : 𝕊 →L[𝕂] 𝕊).smulRight x) t 
Arguments: (𝕂 : Type u_1) {𝕊 : Type u_2} {𝔸 : Type u_3} [RCLike 𝕂] [NormedCommRing 𝕊] [NormedRing 𝔸] [NormedAlgebra 𝕂 𝕊] [NormedAlgebra 𝕂 𝔸] [Algebra 𝕊 𝔸] [ContinuousSMul 𝕊 𝔸] [IsScalarTower 𝕂 𝕊 𝔸] [CompleteSpace 𝔸] (x : 𝔸) (t : 𝕊)
Goal: HasStrictFDerivAt (fun u => NormedSpace.exp 𝕂 (u • x)) (NormedSpace.exp 𝕂 (t • x) • ContinuousLinearMap.smulRight 1 x) t
-- hasStrictFDerivAt_exp_smul_const'
theorem hasStrictFDerivAt_exp_smul_const' (x : 𝔸) (t : 𝕊) :
    HasStrictFDerivAt (fun u : 𝕊 => exp 𝕂 (u • x))
      (((1 : 𝕊 →L[𝕂] 𝕊).smulRight x).smulRight (exp 𝕂 (t • x))) t 
Arguments: (𝕂 : Type u_1) {𝕊 : Type u_2} {𝔸 : Type u_3} [RCLike 𝕂] [NormedCommRing 𝕊] [NormedRing 𝔸] [NormedAlgebra 𝕂 𝕊] [NormedAlgebra 𝕂 𝔸] [Algebra 𝕊 𝔸] [ContinuousSMul 𝕊 𝔸] [IsScalarTower 𝕂 𝕊 𝔸] [CompleteSpace 𝔸] (x : 𝔸) (t : 𝕊)
Goal: HasStrictFDerivAt (fun u => NormedSpace.exp 𝕂 (u • x))
  ((ContinuousLinearMap.smulRight 1 x).smulRight (NormedSpace.exp 𝕂 (t • x))) t
-- hasStrictDerivAt_exp_smul_const
theorem hasStrictDerivAt_exp_smul_const (x : 𝔸) (t : 𝕂) :
    HasStrictDerivAt (fun u : 𝕂 => exp 𝕂 (u • x)) (exp 𝕂 (t • x) * x) t 
Arguments: {𝕂 : Type u_1} {𝔸 : Type u_3} [RCLike 𝕂] [NormedRing 𝔸] [NormedAlgebra 𝕂 𝔸] [CompleteSpace 𝔸] (x : 𝔸) (t : 𝕂)
Goal: HasStrictDerivAt (fun u => NormedSpace.exp 𝕂 (u • x)) (NormedSpace.exp 𝕂 (t • x) * x) t
-- hasStrictDerivAt_exp_smul_const'
theorem hasStrictDerivAt_exp_smul_const' (x : 𝔸) (t : 𝕂) :
    HasStrictDerivAt (fun u : 𝕂 => exp 𝕂 (u • x)) (x * exp 𝕂 (t • x)) t 
Arguments: {𝕂 : Type u_1} {𝔸 : Type u_3} [RCLike 𝕂] [NormedRing 𝔸] [NormedAlgebra 𝕂 𝔸] [CompleteSpace 𝔸] (x : 𝔸) (t : 𝕂)
Goal: HasStrictDerivAt (fun u => NormedSpace.exp 𝕂 (u • x)) (x * NormedSpace.exp 𝕂 (t • x)) t
-- hasDerivAt_exp_smul_const
theorem hasDerivAt_exp_smul_const (x : 𝔸) (t : 𝕂) :
    HasDerivAt (fun u : 𝕂 => exp 𝕂 (u • x)) (exp 𝕂 (t • x) * x) t 
Arguments: {𝕂 : Type u_1} {𝔸 : Type u_3} [RCLike 𝕂] [NormedRing 𝔸] [NormedAlgebra 𝕂 𝔸] [CompleteSpace 𝔸] (x : 𝔸) (t : 𝕂)
Goal: HasDerivAt (fun u => NormedSpace.exp 𝕂 (u • x)) (NormedSpace.exp 𝕂 (t • x) * x) t
-- hasDerivAt_exp_smul_const'
theorem hasDerivAt_exp_smul_const' (x : 𝔸) (t : 𝕂) :
    HasDerivAt (fun u : 𝕂 => exp 𝕂 (u • x)) (x * exp 𝕂 (t • x)) t 
Arguments: {𝕂 : Type u_1} {𝔸 : Type u_3} [RCLike 𝕂] [NormedRing 𝔸] [NormedAlgebra 𝕂 𝔸] [CompleteSpace 𝔸] (x : 𝔸) (t : 𝕂)
Goal: HasDerivAt (fun u => NormedSpace.exp 𝕂 (u • x)) (x * NormedSpace.exp 𝕂 (t • x)) t
-- HasSum.exp
lemma HasSum.exp {ι : Type*} {f : ι → 𝔸} {a : 𝔸} (h : HasSum f a) :
    HasProd (exp 𝕂 ∘ f) (exp 𝕂 a) 
Arguments: {𝕂 : Type u_1} {𝔸 : Type u_2} [RCLike 𝕂] [NormedCommRing 𝔸] [NormedAlgebra 𝕂 𝔸] [CompleteSpace 𝔸] {ι : Type u_3} {f : ι → 𝔸} {a : 𝔸} (h : HasSum f a)
Goal: HasProd (NormedSpace.exp 𝕂 ∘ f) (NormedSpace.exp 𝕂 a)
-- analyticOn_cexp
theorem analyticOn_cexp : AnalyticOn ℂ exp univ 
Goal: AnalyticOn ℂ cexp Set.univ
-- analyticAt_cexp
theorem analyticAt_cexp : AnalyticAt ℂ exp z 
Arguments: {z : ℂ}
Goal: AnalyticAt ℂ cexp z
-- AnalyticAt.cexp
theorem AnalyticAt.cexp (fa : AnalyticAt ℂ f x) : AnalyticAt ℂ (fun z ↦ exp (f z)) x 
Arguments: {E : Type} [NormedAddCommGroup E] [NormedSpace ℂ E] {f : E → ℂ} {x : E} (fa : AnalyticAt ℂ f x)
Goal: AnalyticAt ℂ (fun z => cexp (f z)) x
-- AnalyticOn.cexp
theorem AnalyticOn.cexp (fs : AnalyticOn ℂ f s) : AnalyticOn ℂ (fun z ↦ exp (f z)) s 
Arguments: {E : Type} [NormedAddCommGroup E] [NormedSpace ℂ E] {f : E → ℂ} {s : Set E} (fs : AnalyticOn ℂ f s)
Goal: AnalyticOn ℂ (fun z => cexp (f z)) s
-- analyticAt_clog
theorem analyticAt_clog (m : z ∈ slitPlane) : AnalyticAt ℂ log z 
Arguments: {z : ℂ} (m : z ∈ Complex.slitPlane)
Goal: AnalyticAt ℂ Complex.log z
-- AnalyticAt.clog
theorem AnalyticAt.clog (fa : AnalyticAt ℂ f x) (m : f x ∈ slitPlane) :
    AnalyticAt ℂ (fun z ↦ log (f z)) x 
Arguments: {E : Type} [NormedAddCommGroup E] [NormedSpace ℂ E] {f : E → ℂ} {x : E} (fa : AnalyticAt ℂ f x) (m : f x ∈ Complex.slitPlane)
Goal: AnalyticAt ℂ (fun z => Complex.log (f z)) x
-- AnalyticOn.clog
theorem AnalyticOn.clog (fs : AnalyticOn ℂ f s) (m : ∀ z ∈ s, f z ∈ slitPlane) :
    AnalyticOn ℂ (fun z ↦ log (f z)) s 
Arguments: {E : Type} [NormedAddCommGroup E] [NormedSpace ℂ E] {f : E → ℂ} {s : Set E} (fs : AnalyticOn ℂ f s) (m : ∀ z ∈ s, f z ∈ Complex.slitPlane)
Goal: AnalyticOn ℂ (fun z => Complex.log (f z)) s
-- AnalyticAt.cpow
theorem AnalyticAt.cpow (fa : AnalyticAt ℂ f x) (ga : AnalyticAt ℂ g x)
    (m : f x ∈ slitPlane) : AnalyticAt ℂ (fun z ↦ f z ^ g z) x 
Arguments: {E : Type} [NormedAddCommGroup E] [NormedSpace ℂ E] {f g : E → ℂ} {x : E} (fa : AnalyticAt ℂ f x) (ga : AnalyticAt ℂ g x) (m : f x ∈ Complex.slitPlane)
Goal: AnalyticAt ℂ (fun z => f z ^ g z) x
-- AnalyticOn.cpow
theorem AnalyticOn.cpow (fs : AnalyticOn ℂ f s) (gs : AnalyticOn ℂ g s)
    (m : ∀ z ∈ s, f z ∈ slitPlane) : AnalyticOn ℂ (fun z ↦ f z ^ g z) s 
Arguments: {E : Type} [NormedAddCommGroup E] [NormedSpace ℂ E] {f g : E → ℂ} {s : Set E} (fs : AnalyticOn ℂ f s) (gs : AnalyticOn ℂ g s) (m : ∀ z ∈ s, f z ∈ Complex.slitPlane)
Goal: AnalyticOn ℂ (fun z => f z ^ g z) s