ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- ContinuousWithinAt.clog
nonrec
theorem _root_.ContinuousWithinAt.clog {f : α → ℂ} {s : Set α} {x : α}
    (h₁ : ContinuousWithinAt f s x) (h₂ : f x ∈ slitPlane) :
    ContinuousWithinAt (fun t => log (f t)) s x 
Arguments: {α : Type u_1} [TopologicalSpace α] {f : α → ℂ} {s : Set α} {x : α} (h₁ : ContinuousWithinAt f s x) (h₂ : f x ∈ Complex.slitPlane)
Goal: ContinuousWithinAt (fun t => Complex.log (f t)) s x
-- ContinuousOn.clog
nonrec
theorem _root_.ContinuousOn.clog {f : α → ℂ} {s : Set α} (h₁ : ContinuousOn f s)
    (h₂ : ∀ x ∈ s, f x ∈ slitPlane) : ContinuousOn (fun t => log (f t)) s 
Arguments: {α : Type u_1} [TopologicalSpace α] {f : α → ℂ} {s : Set α} (h₁ : ContinuousOn f s) (h₂ : ∀ x ∈ s, f x ∈ Complex.slitPlane)
Goal: ContinuousOn (fun t => Complex.log (f t)) s
-- Continuous.clog
nonrec
theorem _root_.Continuous.clog {f : α → ℂ} (h₁ : Continuous f)
    (h₂ : ∀ x, f x ∈ slitPlane) : Continuous fun t => log (f t) 
Arguments: {α : Type u_1} [TopologicalSpace α] {f : α → ℂ} (h₁ : Continuous f) (h₂ : ∀ (x : α), f x ∈ Complex.slitPlane)
Goal: Continuous fun t => Complex.log (f t)
-- exp_neg_mul_rpow_isLittleO_exp_neg
theorem exp_neg_mul_rpow_isLittleO_exp_neg {p b : ℝ} (hb : 0 < b) (hp : 1 < p) :
    (fun x : ℝ => exp (- b * x ^ p)) =o[atTop] fun x : ℝ => exp (-x) 
Arguments: {p b : ℝ} (hb : 0 < b) (hp : 1 < p)
Goal: (fun x => rexp (-b * x ^ p)) =o[Filter.atTop] fun x => rexp (-x)
-- exp_neg_mul_sq_isLittleO_exp_neg
theorem exp_neg_mul_sq_isLittleO_exp_neg {b : ℝ} (hb : 0 < b) :
    (fun x : ℝ => exp (-b * x ^ 2)) =o[atTop] fun x : ℝ => exp (-x) 
Arguments: {b : ℝ} (hb : 0 < b)
Goal: (fun x => rexp (-b * x ^ 2)) =o[Filter.atTop] fun x => rexp (-x)
-- rpow_mul_exp_neg_mul_rpow_isLittleO_exp_neg
theorem rpow_mul_exp_neg_mul_rpow_isLittleO_exp_neg (s : ℝ) {b p : ℝ} (hp : 1 < p) (hb : 0 < b) :
    (fun x : ℝ => x ^ s * exp (- b * x ^ p)) =o[atTop] fun x : ℝ => exp (-(1 / 2) * x) 
Arguments: (s : ℝ) {b p : ℝ} (hp : 1 < p) (hb : 0 < b)
Goal: (fun x => x ^ s * rexp (-b * x ^ p)) =o[Filter.atTop] fun x => rexp (-(1 / 2) * x)
-- rpow_mul_exp_neg_mul_sq_isLittleO_exp_neg
theorem rpow_mul_exp_neg_mul_sq_isLittleO_exp_neg {b : ℝ} (hb : 0 < b) (s : ℝ) :
    (fun x : ℝ => x ^ s * exp (-b * x ^ 2)) =o[atTop] fun x : ℝ => exp (-(1 / 2) * x) 
Arguments: {b : ℝ} (hb : 0 < b) (s : ℝ)
Goal: (fun x => x ^ s * rexp (-b * x ^ 2)) =o[Filter.atTop] fun x => rexp (-(1 / 2) * x)
-- integrableOn_rpow_mul_exp_neg_rpow
theorem integrableOn_rpow_mul_exp_neg_rpow {p s : ℝ} (hs : -1 < s) (hp : 1 ≤ p) :
    IntegrableOn (fun x : ℝ => x ^ s * exp (- x ^ p)) (Ioi 0) 
Arguments: {p s : ℝ} (hs : -1 < s) (hp : 1 ≤ p)
Goal: MeasureTheory.IntegrableOn (fun x => x ^ s * rexp (-x ^ p)) (Set.Ioi 0) MeasureTheory.volume
-- integrableOn_rpow_mul_exp_neg_mul_rpow
theorem integrableOn_rpow_mul_exp_neg_mul_rpow {p s b : ℝ} (hs : -1 < s) (hp : 1 ≤ p) (hb : 0 < b) :
    IntegrableOn (fun x : ℝ => x ^ s * exp (- b * x ^ p)) (Ioi 0) 
Arguments: {p s b : ℝ} (hs : -1 < s) (hp : 1 ≤ p) (hb : 0 < b)
Goal: MeasureTheory.IntegrableOn (fun x => x ^ s * rexp (-b * x ^ p)) (Set.Ioi 0) MeasureTheory.volume
-- integrableOn_rpow_mul_exp_neg_mul_sq
theorem integrableOn_rpow_mul_exp_neg_mul_sq {b : ℝ} (hb : 0 < b) {s : ℝ} (hs : -1 < s) :
    IntegrableOn (fun x : ℝ => x ^ s * exp (-b * x ^ 2)) (Ioi 0) 
Arguments: {b : ℝ} (hb : 0 < b) {s : ℝ} (hs : -1 < s)
Goal: MeasureTheory.IntegrableOn (fun x => x ^ s * rexp (-b * x ^ 2)) (Set.Ioi 0) MeasureTheory.volume
-- integrable_rpow_mul_exp_neg_mul_sq
theorem integrable_rpow_mul_exp_neg_mul_sq {b : ℝ} (hb : 0 < b) {s : ℝ} (hs : -1 < s) :
    Integrable fun x : ℝ => x ^ s * exp (-b * x ^ 2) 
Arguments: {b : ℝ} (hb : 0 < b) {s : ℝ} (hs : -1 < s)
Goal: MeasureTheory.Integrable (fun x => x ^ s * rexp (-b * x ^ 2)) MeasureTheory.volume
-- integrable_exp_neg_mul_sq
theorem integrable_exp_neg_mul_sq {b : ℝ} (hb : 0 < b) :
    Integrable fun x : ℝ => exp (-b * x ^ 2) 
Arguments: {b : ℝ} (hb : 0 < b)
Goal: MeasureTheory.Integrable (fun x => rexp (-b * x ^ 2)) MeasureTheory.volume
-- integrableOn_Ioi_exp_neg_mul_sq_iff
theorem integrableOn_Ioi_exp_neg_mul_sq_iff {b : ℝ} :
    IntegrableOn (fun x : ℝ => exp (-b * x ^ 2)) (Ioi 0) ↔ 0 < b 
Arguments: {b : ℝ}
Goal: MeasureTheory.IntegrableOn (fun x => rexp (-b * x ^ 2)) (Set.Ioi 0) MeasureTheory.volume ↔ 0 < b
-- integrable_exp_neg_mul_sq_iff
theorem integrable_exp_neg_mul_sq_iff {b : ℝ} :
    (Integrable fun x : ℝ => exp (-b * x ^ 2)) ↔ 0 < b 
Arguments: {b : ℝ}
Goal: MeasureTheory.Integrable (fun x => rexp (-b * x ^ 2)) MeasureTheory.volume ↔ 0 < b
-- integrable_mul_exp_neg_mul_sq
theorem integrable_mul_exp_neg_mul_sq {b : ℝ} (hb : 0 < b) :
    Integrable fun x : ℝ => x * exp (-b * x ^ 2) 
Arguments: {b : ℝ} (hb : 0 < b)
Goal: MeasureTheory.Integrable (fun x => x * rexp (-b * x ^ 2)) MeasureTheory.volume
-- norm_cexp_neg_mul_sq
theorem norm_cexp_neg_mul_sq (b : ℂ) (x : ℝ) :
    ‖Complex.exp (-b * (x : ℂ) ^ 2)‖ = exp (-b.re * x ^ 2) 
Arguments: (b : ℂ) (x : ℝ)
Goal: ‖cexp (-b * ↑x ^ 2)‖ = rexp (-b.re * x ^ 2)
-- integrable_cexp_neg_mul_sq
theorem integrable_cexp_neg_mul_sq {b : ℂ} (hb : 0 < b.re) :
    Integrable fun x : ℝ => cexp (-b * (x : ℂ) ^ 2) 
Arguments: {b : ℂ} (hb : 0 < b.re)
Goal: MeasureTheory.Integrable (fun x => cexp (-b * ↑x ^ 2)) MeasureTheory.volume
-- integrable_mul_cexp_neg_mul_sq
theorem integrable_mul_cexp_neg_mul_sq {b : ℂ} (hb : 0 < b.re) :
    Integrable fun x : ℝ => ↑x * cexp (-b * (x : ℂ) ^ 2) 
Arguments: {b : ℂ} (hb : 0 < b.re)
Goal: MeasureTheory.Integrable (fun x => ↑x * cexp (-b * ↑x ^ 2)) MeasureTheory.volume
-- integral_mul_cexp_neg_mul_sq
theorem integral_mul_cexp_neg_mul_sq {b : ℂ} (hb : 0 < b.re) :
    ∫ r : ℝ in Ioi 0, (r : ℂ) * cexp (-b * (r : ℂ) ^ 2) = (2 * b)⁻¹ 
Arguments: {b : ℂ} (hb : 0 < b.re)
Goal: ∫ (r : ℝ) in Set.Ioi 0, ↑r * cexp (-b * ↑r ^ 2) = (2 * b)⁻¹
-- integral_gaussian_sq_complex
theorem integral_gaussian_sq_complex {b : ℂ} (hb : 0 < b.re) :
    (∫ x : ℝ, cexp (-b * (x : ℂ) ^ 2)) ^ 2 = π / b 
Arguments: {b : ℂ} (hb : 0 < b.re)
Goal: (∫ (x : ℝ), cexp (-b * ↑x ^ 2)) ^ 2 = ↑π / b
-- integral_gaussian
theorem integral_gaussian (b : ℝ) : ∫ x : ℝ, exp (-b * x ^ 2) = √(π / b) 
Arguments: (b : ℝ)
Goal: ∫ (x : ℝ), rexp (-b * x ^ 2) = √(π / b)
-- continuousAt_gaussian_integral
theorem continuousAt_gaussian_integral (b : ℂ) (hb : 0 < re b) :
    ContinuousAt (fun c : ℂ => ∫ x : ℝ, cexp (-c * (x : ℂ) ^ 2)) b 
Arguments: (b : ℂ) (hb : 0 < b.re)
Goal: ContinuousAt (fun c => ∫ (x : ℝ), cexp (-c * ↑x ^ 2)) b
-- integral_gaussian_complex
theorem integral_gaussian_complex {b : ℂ} (hb : 0 < re b) :
    ∫ x : ℝ, cexp (-b * (x : ℂ) ^ 2) = (π / b) ^ (1 / 2 : ℂ) 
Arguments: {b : ℂ} (hb : 0 < b.re)
Goal: ∫ (x : ℝ), cexp (-b * ↑x ^ 2) = (↑π / b) ^ (1 / 2)
-- integral_gaussian_complex_Ioi
theorem integral_gaussian_complex_Ioi {b : ℂ} (hb : 0 < re b) :
    ∫ x : ℝ in Ioi 0, cexp (-b * (x : ℂ) ^ 2) = (π / b) ^ (1 / 2 : ℂ) / 2 
Arguments: {b : ℂ} (hb : 0 < b.re)
Goal: ∫ (x : ℝ) in Set.Ioi 0, cexp (-b * ↑x ^ 2) = (↑π / b) ^ (1 / 2) / 2
-- integral_gaussian_Ioi
theorem integral_gaussian_Ioi (b : ℝ) :
    ∫ x in Ioi (0 : ℝ), exp (-b * x ^ 2) = √(π / b) / 2 
Arguments: (b : ℝ)
Goal: ∫ (x : ℝ) in Set.Ioi 0, rexp (-b * x ^ 2) = √(π / b) / 2
-- Real.Gamma_one_half_eq
theorem Real.Gamma_one_half_eq : Real.Gamma (1 / 2) = √π 
Goal: Real.Gamma (1 / 2) = √π
-- Complex.Gamma_one_half_eq
theorem Complex.Gamma_one_half_eq : Complex.Gamma (1 / 2) = (π : ℂ) ^ (1 / 2 : ℂ) 
Goal: Complex.Gamma (1 / 2) = ↑π ^ (1 / 2)
-- Int.nnnorm_coe_units
theorem nnnorm_coe_units (e : ℤˣ) : ‖(e : ℤ)‖₊ = 1 
Arguments: (e : ℤˣ)
Goal: ‖↑e‖₊ = 1
-- Int.norm_coe_units
theorem norm_coe_units (e : ℤˣ) : ‖(e : ℤ)‖ = 1 
Arguments: (e : ℤˣ)
Goal: ‖↑e‖ = 1
-- Int.nnnorm_natCast
@[simp]
theorem nnnorm_natCast (n : ℕ) : ‖(n : ℤ)‖₊ = n 
Arguments: (n : ℕ)
Goal: ‖↑n‖₊ = ↑n
-- Int.toNat_add_toNat_neg_eq_nnnorm
@[simp]
theorem toNat_add_toNat_neg_eq_nnnorm (n : ℤ) : ↑n.toNat + ↑(-n).toNat = ‖n‖₊ 
Arguments: (n : ℤ)
Goal: ↑n.toNat + ↑(-n).toNat = ‖n‖₊
-- Int.toNat_add_toNat_neg_eq_norm
@[simp]
theorem toNat_add_toNat_neg_eq_norm (n : ℤ) : ↑n.toNat + ↑(-n).toNat = ‖n‖ 
Arguments: (n : ℤ)
Goal: ↑n.toNat + ↑(-n).toNat = ‖n‖
-- Complex.hasDerivAt_exp
theorem hasDerivAt_exp (x : ℂ) : HasDerivAt exp (exp x) x 
Arguments: (x : ℂ)
Goal: HasDerivAt Complex.exp (Complex.exp x) x
-- Complex.differentiable_exp
theorem differentiable_exp : Differentiable 𝕜 exp 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] [NormedAlgebra 𝕜 ℂ]
Goal: Differentiable 𝕜 Complex.exp
-- Complex.differentiableAt_exp
theorem differentiableAt_exp {x : ℂ} : DifferentiableAt 𝕜 exp x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] [NormedAlgebra 𝕜 ℂ] {x : ℂ}
Goal: DifferentiableAt 𝕜 Complex.exp x
-- Complex.deriv_exp
@[simp]
theorem deriv_exp : deriv exp = exp 
Goal: deriv Complex.exp = Complex.exp
-- Complex.iter_deriv_exp
@[simp]
theorem iter_deriv_exp : ∀ n : ℕ, deriv^[n] exp = exp
  
Arguments: (n : ℕ)
Goal: deriv^[n] Complex.exp = Complex.exp
-- Complex.contDiff_exp
theorem contDiff_exp : ∀ {n}, ContDiff 𝕜 n exp 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] [NormedAlgebra 𝕜 ℂ] {n : ℕ∞}
Goal: ContDiff 𝕜 n Complex.exp
-- Complex.hasStrictDerivAt_exp
theorem hasStrictDerivAt_exp (x : ℂ) : HasStrictDerivAt exp (exp x) x 
Arguments: (x : ℂ)
Goal: HasStrictDerivAt Complex.exp (Complex.exp x) x
-- Complex.hasStrictFDerivAt_exp_real
theorem hasStrictFDerivAt_exp_real (x : ℂ) : HasStrictFDerivAt exp (exp x • (1 : ℂ →L[ℝ] ℂ)) x 
Arguments: (x : ℂ)
Goal: HasStrictFDerivAt Complex.exp (Complex.exp x • 1) x
-- HasStrictDerivAt.cexp
theorem HasStrictDerivAt.cexp (hf : HasStrictDerivAt f f' x) :
    HasStrictDerivAt (fun x => Complex.exp (f x)) (Complex.exp (f x) * f') x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] [NormedAlgebra 𝕜 ℂ] {f : 𝕜 → ℂ} {f' : ℂ} {x : 𝕜} (hf : HasStrictDerivAt f f' x)
Goal: HasStrictDerivAt (fun x => Complex.exp (f x)) (Complex.exp (f x) * f') x
-- HasDerivAt.cexp
theorem HasDerivAt.cexp (hf : HasDerivAt f f' x) :
    HasDerivAt (fun x => Complex.exp (f x)) (Complex.exp (f x) * f') x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] [NormedAlgebra 𝕜 ℂ] {f : 𝕜 → ℂ} {f' : ℂ} {x : 𝕜} (hf : HasDerivAt f f' x)
Goal: HasDerivAt (fun x => Complex.exp (f x)) (Complex.exp (f x) * f') x
-- HasDerivWithinAt.cexp
theorem HasDerivWithinAt.cexp (hf : HasDerivWithinAt f f' s x) :
    HasDerivWithinAt (fun x => Complex.exp (f x)) (Complex.exp (f x) * f') s x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] [NormedAlgebra 𝕜 ℂ] {f : 𝕜 → ℂ} {f' : ℂ} {x : 𝕜} {s : Set 𝕜} (hf : HasDerivWithinAt f f' s x)
Goal: HasDerivWithinAt (fun x => Complex.exp (f x)) (Complex.exp (f x) * f') s x
-- derivWithin_cexp
theorem derivWithin_cexp (hf : DifferentiableWithinAt 𝕜 f s x) (hxs : UniqueDiffWithinAt 𝕜 s x) :
    derivWithin (fun x => Complex.exp (f x)) s x = Complex.exp (f x) * derivWithin f s x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] [NormedAlgebra 𝕜 ℂ] {f : 𝕜 → ℂ} {x : 𝕜} {s : Set 𝕜} (hf : DifferentiableWithinAt 𝕜 f s x) (hxs : UniqueDiffWithinAt 𝕜 s x)
Goal: derivWithin (fun x => Complex.exp (f x)) s x = Complex.exp (f x) * derivWithin f s x
-- deriv_cexp
@[simp]
theorem deriv_cexp (hc : DifferentiableAt 𝕜 f x) :
    deriv (fun x => Complex.exp (f x)) x = Complex.exp (f x) * deriv f x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] [NormedAlgebra 𝕜 ℂ] {f : 𝕜 → ℂ} {x : 𝕜} (hc : DifferentiableAt 𝕜 f x)
Goal: deriv (fun x => Complex.exp (f x)) x = Complex.exp (f x) * deriv f x
-- HasStrictFDerivAt.cexp
theorem HasStrictFDerivAt.cexp (hf : HasStrictFDerivAt f f' x) :
    HasStrictFDerivAt (fun x => Complex.exp (f x)) (Complex.exp (f x) • f') x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] [NormedAlgebra 𝕜 ℂ] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {f : E → ℂ} {f' : E →L[𝕜] ℂ} {x : E} (hf : HasStrictFDerivAt f f' x)
Goal: HasStrictFDerivAt (fun x => Complex.exp (f x)) (Complex.exp (f x) • f') x
-- HasFDerivWithinAt.cexp
theorem HasFDerivWithinAt.cexp (hf : HasFDerivWithinAt f f' s x) :
    HasFDerivWithinAt (fun x => Complex.exp (f x)) (Complex.exp (f x) • f') s x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] [NormedAlgebra 𝕜 ℂ] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {f : E → ℂ} {f' : E →L[𝕜] ℂ} {x : E} {s : Set E} (hf : HasFDerivWithinAt f f' s x)
Goal: HasFDerivWithinAt (fun x => Complex.exp (f x)) (Complex.exp (f x) • f') s x
-- HasFDerivAt.cexp
theorem HasFDerivAt.cexp (hf : HasFDerivAt f f' x) :
    HasFDerivAt (fun x => Complex.exp (f x)) (Complex.exp (f x) • f') x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] [NormedAlgebra 𝕜 ℂ] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {f : E → ℂ} {f' : E →L[𝕜] ℂ} {x : E} (hf : HasFDerivAt f f' x)
Goal: HasFDerivAt (fun x => Complex.exp (f x)) (Complex.exp (f x) • f') x
-- DifferentiableWithinAt.cexp
theorem DifferentiableWithinAt.cexp (hf : DifferentiableWithinAt 𝕜 f s x) :
    DifferentiableWithinAt 𝕜 (fun x => Complex.exp (f x)) s x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] [NormedAlgebra 𝕜 ℂ] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {f : E → ℂ} {x : E} {s : Set E} (hf : DifferentiableWithinAt 𝕜 f s x)
Goal: DifferentiableWithinAt 𝕜 (fun x => Complex.exp (f x)) s x
-- DifferentiableAt.cexp
@[simp]
theorem DifferentiableAt.cexp (hc : DifferentiableAt 𝕜 f x) :
    DifferentiableAt 𝕜 (fun x => Complex.exp (f x)) x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] [NormedAlgebra 𝕜 ℂ] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {f : E → ℂ} {x : E} (hc : DifferentiableAt 𝕜 f x)
Goal: DifferentiableAt 𝕜 (fun x => Complex.exp (f x)) x
-- DifferentiableOn.cexp
theorem DifferentiableOn.cexp (hc : DifferentiableOn 𝕜 f s) :
    DifferentiableOn 𝕜 (fun x => Complex.exp (f x)) s 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] [NormedAlgebra 𝕜 ℂ] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {f : E → ℂ} {s : Set E} (hc : DifferentiableOn 𝕜 f s)
Goal: DifferentiableOn 𝕜 (fun x => Complex.exp (f x)) s
-- Differentiable.cexp
@[simp]
theorem Differentiable.cexp (hc : Differentiable 𝕜 f) :
    Differentiable 𝕜 fun x => Complex.exp (f x) 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] [NormedAlgebra 𝕜 ℂ] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {f : E → ℂ} (hc : Differentiable 𝕜 f)
Goal: Differentiable 𝕜 fun x => Complex.exp (f x)
-- ContDiff.cexp
theorem ContDiff.cexp {n} (h : ContDiff 𝕜 n f) : ContDiff 𝕜 n fun x => Complex.exp (f x) 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] [NormedAlgebra 𝕜 ℂ] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {f : E → ℂ} {n : ℕ∞} (h : ContDiff 𝕜 n f)
Goal: ContDiff 𝕜 n fun x => Complex.exp (f x)
-- ContDiffAt.cexp
theorem ContDiffAt.cexp {n} (hf : ContDiffAt 𝕜 n f x) :
    ContDiffAt 𝕜 n (fun x => Complex.exp (f x)) x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] [NormedAlgebra 𝕜 ℂ] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {f : E → ℂ} {x : E} {n : ℕ∞} (hf : ContDiffAt 𝕜 n f x)
Goal: ContDiffAt 𝕜 n (fun x => Complex.exp (f x)) x
-- ContDiffOn.cexp
theorem ContDiffOn.cexp {n} (hf : ContDiffOn 𝕜 n f s) :
    ContDiffOn 𝕜 n (fun x => Complex.exp (f x)) s 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] [NormedAlgebra 𝕜 ℂ] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {f : E → ℂ} {s : Set E} {n : ℕ∞} (hf : ContDiffOn 𝕜 n f s)
Goal: ContDiffOn 𝕜 n (fun x => Complex.exp (f x)) s
-- ContDiffWithinAt.cexp
theorem ContDiffWithinAt.cexp {n} (hf : ContDiffWithinAt 𝕜 n f s x) :
    ContDiffWithinAt 𝕜 n (fun x => Complex.exp (f x)) s x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] [NormedAlgebra 𝕜 ℂ] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {f : E → ℂ} {x : E} {s : Set E} {n : ℕ∞} (hf : ContDiffWithinAt 𝕜 n f s x)
Goal: ContDiffWithinAt 𝕜 n (fun x => Complex.exp (f x)) s x
-- iteratedDeriv_cexp_const_mul
@[simp]
theorem iteratedDeriv_cexp_const_mul (n : ℕ) (c : ℂ) :
    (iteratedDeriv n fun s : ℂ => exp (c * s)) = fun s => c ^ n * exp (c * s) 
Arguments: (n : ℕ) (c : ℂ)
Goal: (iteratedDeriv n fun s => Complex.exp (c * s)) = fun s => c ^ n * Complex.exp (c * s)
-- Real.hasStrictDerivAt_exp
theorem hasStrictDerivAt_exp (x : ℝ) : HasStrictDerivAt exp (exp x) x 
Arguments: (x : ℝ)
Goal: HasStrictDerivAt Real.exp (Real.exp x) x
-- Real.hasDerivAt_exp
theorem hasDerivAt_exp (x : ℝ) : HasDerivAt exp (exp x) x 
Arguments: (x : ℝ)
Goal: HasDerivAt Real.exp (Real.exp x) x
-- Real.contDiff_exp
theorem contDiff_exp {n} : ContDiff ℝ n exp 
Arguments: {n : ℕ∞}
Goal: ContDiff ℝ n Real.exp
-- Real.differentiable_exp
theorem differentiable_exp : Differentiable ℝ exp 
Goal: Differentiable ℝ Real.exp
-- Real.differentiableAt_exp
theorem differentiableAt_exp : DifferentiableAt ℝ exp x 
Arguments: {x : ℝ}
Goal: DifferentiableAt ℝ Real.exp x
-- Real.deriv_exp
@[simp]
theorem deriv_exp : deriv exp = exp 
Goal: deriv Real.exp = Real.exp
-- Real.iter_deriv_exp
@[simp]
theorem iter_deriv_exp : ∀ n : ℕ, deriv^[n] exp = exp
  
Arguments: (n : ℕ)
Goal: deriv^[n] Real.exp = Real.exp
-- HasStrictDerivAt.exp
theorem HasStrictDerivAt.exp (hf : HasStrictDerivAt f f' x) :
    HasStrictDerivAt (fun x => Real.exp (f x)) (Real.exp (f x) * f') x 
Arguments: {f : ℝ → ℝ} {f' x : ℝ} (hf : HasStrictDerivAt f f' x)
Goal: HasStrictDerivAt (fun x => Real.exp (f x)) (Real.exp (f x) * f') x
-- HasDerivAt.exp
theorem HasDerivAt.exp (hf : HasDerivAt f f' x) :
    HasDerivAt (fun x => Real.exp (f x)) (Real.exp (f x) * f') x 
Arguments: {f : ℝ → ℝ} {f' x : ℝ} (hf : HasDerivAt f f' x)
Goal: HasDerivAt (fun x => Real.exp (f x)) (Real.exp (f x) * f') x
-- HasDerivWithinAt.exp
theorem HasDerivWithinAt.exp (hf : HasDerivWithinAt f f' s x) :
    HasDerivWithinAt (fun x => Real.exp (f x)) (Real.exp (f x) * f') s x 
Arguments: {f : ℝ → ℝ} {f' x : ℝ} {s : Set ℝ} (hf : HasDerivWithinAt f f' s x)
Goal: HasDerivWithinAt (fun x => Real.exp (f x)) (Real.exp (f x) * f') s x
-- derivWithin_exp
theorem derivWithin_exp (hf : DifferentiableWithinAt ℝ f s x) (hxs : UniqueDiffWithinAt ℝ s x) :
    derivWithin (fun x => Real.exp (f x)) s x = Real.exp (f x) * derivWithin f s x 
Arguments: {f : ℝ → ℝ} {x : ℝ} {s : Set ℝ} (hf : DifferentiableWithinAt ℝ f s x) (hxs : UniqueDiffWithinAt ℝ s x)
Goal: derivWithin (fun x => Real.exp (f x)) s x = Real.exp (f x) * derivWithin f s x
-- deriv_exp
@[simp]
theorem deriv_exp (hc : DifferentiableAt ℝ f x) :
    deriv (fun x => Real.exp (f x)) x = Real.exp (f x) * deriv f x 
Arguments: {f : ℝ → ℝ} {x : ℝ} (hc : DifferentiableAt ℝ f x)
Goal: deriv (fun x => Real.exp (f x)) x = Real.exp (f x) * deriv f x
-- ContDiff.exp
theorem ContDiff.exp {n} (hf : ContDiff ℝ n f) : ContDiff ℝ n fun x => Real.exp (f x) 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {f : E → ℝ} {n : ℕ∞} (hf : ContDiff ℝ n f)
Goal: ContDiff ℝ n fun x => Real.exp (f x)
-- ContDiffAt.exp
theorem ContDiffAt.exp {n} (hf : ContDiffAt ℝ n f x) : ContDiffAt ℝ n (fun x => Real.exp (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.exp (f x)) x
-- ContDiffOn.exp
theorem ContDiffOn.exp {n} (hf : ContDiffOn ℝ n f s) : ContDiffOn ℝ n (fun x => Real.exp (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.exp (f x)) s
-- ContDiffWithinAt.exp
theorem ContDiffWithinAt.exp {n} (hf : ContDiffWithinAt ℝ n f s x) :
    ContDiffWithinAt ℝ n (fun x => Real.exp (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.exp (f x)) s x
-- HasFDerivWithinAt.exp
theorem HasFDerivWithinAt.exp (hf : HasFDerivWithinAt f f' s x) :
    HasFDerivWithinAt (fun x => Real.exp (f x)) (Real.exp (f x) • f') s x 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {f : E → ℝ} {f' : E →L[ℝ] ℝ} {x : E} {s : Set E} (hf : HasFDerivWithinAt f f' s x)
Goal: HasFDerivWithinAt (fun x => Real.exp (f x)) (Real.exp (f x) • f') s x
-- HasFDerivAt.exp
theorem HasFDerivAt.exp (hf : HasFDerivAt f f' x) :
    HasFDerivAt (fun x => Real.exp (f x)) (Real.exp (f x) • f') x 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {f : E → ℝ} {f' : E →L[ℝ] ℝ} {x : E} (hf : HasFDerivAt f f' x)
Goal: HasFDerivAt (fun x => Real.exp (f x)) (Real.exp (f x) • f') x
-- HasStrictFDerivAt.exp
theorem HasStrictFDerivAt.exp (hf : HasStrictFDerivAt f f' x) :
    HasStrictFDerivAt (fun x => Real.exp (f x)) (Real.exp (f x) • f') x 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {f : E → ℝ} {f' : E →L[ℝ] ℝ} {x : E} (hf : HasStrictFDerivAt f f' x)
Goal: HasStrictFDerivAt (fun x => Real.exp (f x)) (Real.exp (f x) • f') x
-- DifferentiableWithinAt.exp
theorem DifferentiableWithinAt.exp (hf : DifferentiableWithinAt ℝ f s x) :
    DifferentiableWithinAt ℝ (fun x => Real.exp (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.exp (f x)) s x
-- DifferentiableAt.exp
@[simp]
theorem DifferentiableAt.exp (hc : DifferentiableAt ℝ f x) :
    DifferentiableAt ℝ (fun x => Real.exp (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.exp (f x)) x
-- DifferentiableOn.exp
theorem DifferentiableOn.exp (hc : DifferentiableOn ℝ f s) :
    DifferentiableOn ℝ (fun x => Real.exp (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.exp (f x)) s
-- Differentiable.exp
@[simp]
theorem Differentiable.exp (hc : Differentiable ℝ f) : Differentiable ℝ fun x => Real.exp (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.exp (f x)
-- fderivWithin_exp
theorem fderivWithin_exp (hf : DifferentiableWithinAt ℝ f s x) (hxs : UniqueDiffWithinAt ℝ s x) :
    fderivWithin ℝ (fun x => Real.exp (f x)) s x = Real.exp (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.exp (f x)) s x = Real.exp (f x) • fderivWithin ℝ f s x
-- fderiv_exp
@[simp]
theorem fderiv_exp (hc : DifferentiableAt ℝ f x) :
    fderiv ℝ (fun x => Real.exp (f x)) x = Real.exp (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.exp (f x)) x = Real.exp (f x) • fderiv ℝ f x
-- iteratedDeriv_exp_const_mul
@[simp]
theorem iteratedDeriv_exp_const_mul (n : ℕ) (c : ℝ) :
    (iteratedDeriv n fun s => exp (c * s)) = fun s => c ^ n * exp (c * s) 
Arguments: (n : ℕ) (c : ℝ)
Goal: (iteratedDeriv n fun s => Real.exp (c * s)) = fun s => c ^ n * Real.exp (c * s)
-- circle.injective_arg
theorem injective_arg : Injective fun z : circle => arg z 
Goal: Function.Injective fun z => (↑z).arg
-- circle.arg_eq_arg
@[simp]
theorem arg_eq_arg {z w : circle} : arg z = arg w ↔ z = w 
Arguments: {z w : ↥circle}
Goal: (↑z).arg = (↑w).arg ↔ z = w
-- arg_expMapCircle
theorem arg_expMapCircle {x : ℝ} (h₁ : -π < x) (h₂ : x ≤ π) : arg (expMapCircle x) = x 
Arguments: {x : ℝ} (h₁ : -π < x) (h₂ : x ≤ π)
Goal: (↑(expMapCircle x)).arg = x
-- expMapCircle_arg
@[simp]
theorem expMapCircle_arg (z : circle) : expMapCircle (arg z) = z 
Arguments: (z : ↥circle)
Goal: expMapCircle (↑z).arg = z
-- circle.argPartialEquiv_source
@[simps (config := .asFn)]
noncomputable def argPartialEquiv : PartialEquiv circle ℝ 
Goal: circle.argPartialEquiv.source = Set.univ
-- circle.argPartialEquiv_target
@[simps (config := .asFn)]
noncomputable def argPartialEquiv : PartialEquiv circle ℝ 
Goal: circle.argPartialEquiv.target = Set.Ioc (-π) π
-- circle.argPartialEquiv_symm_apply
@[simps (config := .asFn)]
noncomputable def argPartialEquiv : PartialEquiv circle ℝ 
Goal: ↑circle.argPartialEquiv.symm = ⇑expMapCircle