ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- exists_forall_hasDerivAt_Ioo_eq_of_contDiff
theorem exists_forall_hasDerivAt_Ioo_eq_of_contDiff (hv : ContDiff ℝ 1 v) :
    ∃ f : ℝ → E, f t₀ = x₀ ∧
      ∃ ε > (0 : ℝ), ∀ t ∈ Ioo (t₀ - ε) (t₀ + ε), HasDerivAt f (v (f t)) t 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {v : E → E} (t₀ : ℝ) {x₀ : E} [CompleteSpace E] (hv : ContDiff ℝ 1 v)
Goal: ∃ f, f t₀ = x₀ ∧ ∃ ε > 0, ∀ t ∈ Set.Ioo (t₀ - ε) (t₀ + ε), HasDerivAt f (v (f t)) t
-- Complex.hasSum_cos'
theorem Complex.hasSum_cos' (z : ℂ) :
    HasSum (fun n : ℕ => (z * Complex.I) ^ (2 * n) / ↑(2 * n)!) (Complex.cos z) 
Arguments: (z : ℂ)
Goal: HasSum (fun n => (z * Complex.I) ^ (2 * n) / ↑(2 * n)!) (Complex.cos z)
-- Complex.hasSum_sin'
theorem Complex.hasSum_sin' (z : ℂ) :
    HasSum (fun n : ℕ => (z * Complex.I) ^ (2 * n + 1) / ↑(2 * n + 1)! / Complex.I)
      (Complex.sin z) 
Arguments: (z : ℂ)
Goal: HasSum (fun n => (z * Complex.I) ^ (2 * n + 1) / ↑(2 * n + 1)! / Complex.I) (Complex.sin z)
-- Complex.hasSum_cos
theorem Complex.hasSum_cos (z : ℂ) :
    HasSum (fun n : ℕ => (-1) ^ n * z ^ (2 * n) / ↑(2 * n)!) (Complex.cos z) 
Arguments: (z : ℂ)
Goal: HasSum (fun n => (-1) ^ n * z ^ (2 * n) / ↑(2 * n)!) (Complex.cos z)
-- Complex.hasSum_sin
theorem Complex.hasSum_sin (z : ℂ) :
    HasSum (fun n : ℕ => (-1) ^ n * z ^ (2 * n + 1) / ↑(2 * n + 1)!) (Complex.sin z) 
Arguments: (z : ℂ)
Goal: HasSum (fun n => (-1) ^ n * z ^ (2 * n + 1) / ↑(2 * n + 1)!) (Complex.sin z)
-- Complex.cos_eq_tsum'
theorem Complex.cos_eq_tsum' (z : ℂ) :
    Complex.cos z = ∑' n : ℕ, (z * Complex.I) ^ (2 * n) / ↑(2 * n)! 
Arguments: (z : ℂ)
Goal: Complex.cos z = ∑' (n : ℕ), (z * Complex.I) ^ (2 * n) / ↑(2 * n)!
-- Complex.sin_eq_tsum'
theorem Complex.sin_eq_tsum' (z : ℂ) :
    Complex.sin z = ∑' n : ℕ, (z * Complex.I) ^ (2 * n + 1) / ↑(2 * n + 1)! / Complex.I 
Arguments: (z : ℂ)
Goal: Complex.sin z = ∑' (n : ℕ), (z * Complex.I) ^ (2 * n + 1) / ↑(2 * n + 1)! / Complex.I
-- Complex.cos_eq_tsum
theorem Complex.cos_eq_tsum (z : ℂ) :
    Complex.cos z = ∑' n : ℕ, (-1) ^ n * z ^ (2 * n) / ↑(2 * n)! 
Arguments: (z : ℂ)
Goal: Complex.cos z = ∑' (n : ℕ), (-1) ^ n * z ^ (2 * n) / ↑(2 * n)!
-- Complex.sin_eq_tsum
theorem Complex.sin_eq_tsum (z : ℂ) :
    Complex.sin z = ∑' n : ℕ, (-1) ^ n * z ^ (2 * n + 1) / ↑(2 * n + 1)! 
Arguments: (z : ℂ)
Goal: Complex.sin z = ∑' (n : ℕ), (-1) ^ n * z ^ (2 * n + 1) / ↑(2 * n + 1)!
-- Real.hasSum_cos
theorem Real.hasSum_cos (r : ℝ) :
    HasSum (fun n : ℕ => (-1) ^ n * r ^ (2 * n) / ↑(2 * n)!) (Real.cos r) 
Arguments: (r : ℝ)
Goal: HasSum (fun n => (-1) ^ n * r ^ (2 * n) / ↑(2 * n)!) (Real.cos r)
-- Real.hasSum_sin
theorem Real.hasSum_sin (r : ℝ) :
    HasSum (fun n : ℕ => (-1) ^ n * r ^ (2 * n + 1) / ↑(2 * n + 1)!) (Real.sin r) 
Arguments: (r : ℝ)
Goal: HasSum (fun n => (-1) ^ n * r ^ (2 * n + 1) / ↑(2 * n + 1)!) (Real.sin r)
-- Real.cos_eq_tsum
theorem Real.cos_eq_tsum (r : ℝ) : Real.cos r = ∑' n : ℕ, (-1) ^ n * r ^ (2 * n) / ↑(2 * n)! 
Arguments: (r : ℝ)
Goal: Real.cos r = ∑' (n : ℕ), (-1) ^ n * r ^ (2 * n) / ↑(2 * n)!
-- Real.sin_eq_tsum
theorem Real.sin_eq_tsum (r : ℝ) :
    Real.sin r = ∑' n : ℕ, (-1) ^ n * r ^ (2 * n + 1) / ↑(2 * n + 1)! 
Arguments: (r : ℝ)
Goal: Real.sin r = ∑' (n : ℕ), (-1) ^ n * r ^ (2 * n + 1) / ↑(2 * n + 1)!
-- Complex.hasSum_cosh
lemma hasSum_cosh (z : ℂ) : HasSum (fun n ↦ z ^ (2 * n) / ↑(2 * n)!) (cosh z) 
Arguments: (z : ℂ)
Goal: HasSum (fun n => z ^ (2 * n) / ↑(2 * n)!) (Complex.cosh z)
-- Complex.hasSum_sinh
lemma hasSum_sinh (z : ℂ) : HasSum (fun n ↦ z ^ (2 * n + 1) / ↑(2 * n + 1)!) (sinh z) 
Arguments: (z : ℂ)
Goal: HasSum (fun n => z ^ (2 * n + 1) / ↑(2 * n + 1)!) (Complex.sinh z)
-- Complex.cosh_eq_tsum
lemma cosh_eq_tsum (z : ℂ) : cosh z = ∑' n, z ^ (2 * n) / ↑(2 * n)! 
Arguments: (z : ℂ)
Goal: Complex.cosh z = ∑' (n : ℕ), z ^ (2 * n) / ↑(2 * n)!
-- Complex.sinh_eq_tsum
lemma sinh_eq_tsum (z : ℂ) : sinh z = ∑' n, z ^ (2 * n + 1) / ↑(2 * n + 1)! 
Arguments: (z : ℂ)
Goal: Complex.sinh z = ∑' (n : ℕ), z ^ (2 * n + 1) / ↑(2 * n + 1)!
-- Real.hasSum_cosh
lemma hasSum_cosh (r : ℝ) : HasSum (fun n  ↦ r ^ (2 * n) / ↑(2 * n)!) (cosh r) 
Arguments: (r : ℝ)
Goal: HasSum (fun n => r ^ (2 * n) / ↑(2 * n)!) (Real.cosh r)
-- Real.hasSum_sinh
lemma hasSum_sinh (r : ℝ) : HasSum (fun n ↦ r ^ (2 * n + 1) / ↑(2 * n + 1)!) (sinh r) 
Arguments: (r : ℝ)
Goal: HasSum (fun n => r ^ (2 * n + 1) / ↑(2 * n + 1)!) (Real.sinh r)
-- Real.cosh_eq_tsum
lemma cosh_eq_tsum (r : ℝ) : cosh r = ∑' n, r ^ (2 * n) / ↑(2 * n)! 
Arguments: (r : ℝ)
Goal: Real.cosh r = ∑' (n : ℕ), r ^ (2 * n) / ↑(2 * n)!
-- Real.sinh_eq_tsum
lemma sinh_eq_tsum (r : ℝ) : sinh r = ∑' n, r ^ (2 * n + 1) / ↑(2 * n + 1)! 
Arguments: (r : ℝ)
Goal: Real.sinh r = ∑' (n : ℕ), r ^ (2 * n + 1) / ↑(2 * n + 1)!
-- Real.cosh_le_exp_half_sq
lemma cosh_le_exp_half_sq (x : ℝ) : cosh x ≤ exp (x ^ 2 / 2) 
Arguments: (x : ℝ)
Goal: Real.cosh x ≤ Real.exp (x ^ 2 / 2)
-- NormedSpace.exp_continuousMap_eq
lemma NormedSpace.exp_continuousMap_eq (f : C(α, 𝕜)) :
    exp 𝕜 f = (⟨exp 𝕜 ∘ f, exp_continuous.comp f.continuous⟩ : C(α, 𝕜)) 
Arguments: {𝕜 : Type u_1} {α : Type u_2} [RCLike 𝕜] [TopologicalSpace α] [CompactSpace α] (f : C(α, 𝕜))
Goal: NormedSpace.exp 𝕜 f = { toFun := NormedSpace.exp 𝕜 ∘ ⇑f, continuous_toFun := ⋯ }
-- CFC.exp_eq_normedSpace_exp
lemma exp_eq_normedSpace_exp {a : A} (ha : p a := by cfc_tac) :
    cfc (exp 𝕜 : 𝕜 → 𝕜) a = exp 𝕜 a 
Arguments: {𝕜 : Type u_1} {A : Type u_2} [RCLike 𝕜] {p : A → Prop} [NormedRing A] [StarRing A] [TopologicalRing A] [NormedAlgebra 𝕜 A] [CompleteSpace A] [ContinuousFunctionalCalculus 𝕜 p] {a : A} (ha : p a := by cfc_tac)
Goal: cfc (NormedSpace.exp 𝕜) a = NormedSpace.exp 𝕜 a
-- CFC.real_exp_eq_normedSpace_exp
lemma real_exp_eq_normedSpace_exp {a : A} (ha : p a := by cfc_tac) :
    cfc Real.exp a = exp ℝ a 
Arguments: {A : Type u_1} {p : A → Prop} [NormedRing A] [StarRing A] [TopologicalRing A] [NormedAlgebra ℝ A] [CompleteSpace A] [ContinuousFunctionalCalculus ℝ p] {a : A} (ha : p a := by cfc_tac)
Goal: cfc Real.exp a = NormedSpace.exp ℝ a
-- CFC.complex_exp_eq_normedSpace_exp
lemma complex_exp_eq_normedSpace_exp {a : A} (ha : p a := by cfc_tac) :
    cfc Complex.exp a = exp ℂ a 
Arguments: {A : Type u_1} {p : A → Prop} [NormedRing A] [StarRing A] [TopologicalRing A] [NormedAlgebra ℂ A] [CompleteSpace A] [ContinuousFunctionalCalculus ℂ p] {a : A} (ha : p a := by cfc_tac)
Goal: cfc Complex.exp a = NormedSpace.exp ℂ a
-- IsSelfAdjoint.log
@[simp]
protected lemma _root_.IsSelfAdjoint.log {a : A} : IsSelfAdjoint (log a) 
Arguments: {A : Type u_1} [NormedRing A] [StarRing A] [NormedAlgebra ℝ A] [ContinuousFunctionalCalculus ℝ IsSelfAdjoint] {a : A}
Goal: IsSelfAdjoint (CFC.log a)
-- CFC.log_exp
lemma log_exp (a : A) (ha : IsSelfAdjoint a := by cfc_tac) : log (NormedSpace.exp ℝ a) = a 
Arguments: {A : Type u_1} [NormedRing A] [StarRing A] [TopologicalRing A] [NormedAlgebra ℝ A] [CompleteSpace A] [ContinuousFunctionalCalculus ℝ IsSelfAdjoint] [UniqueContinuousFunctionalCalculus ℝ A] (a : A) (ha : IsSelfAdjoint a := by cfc_tac)
Goal: CFC.log (NormedSpace.exp ℝ a) = a
-- CFC.exp_log
lemma exp_log (a : A) (ha₂ : ∀ x ∈ spectrum ℝ a, 0 < x) (ha₁ : IsSelfAdjoint a := by cfc_tac) :
    NormedSpace.exp ℝ (log a) = a 
Arguments: {A : Type u_1} [NormedRing A] [StarRing A] [TopologicalRing A] [NormedAlgebra ℝ A] [CompleteSpace A] [ContinuousFunctionalCalculus ℝ IsSelfAdjoint] [UniqueContinuousFunctionalCalculus ℝ A] (a : A) (ha₂ : ∀ x ∈ spectrum ℝ a, 0 < x) (ha₁ : IsSelfAdjoint a := by cfc_tac)
Goal: NormedSpace.exp ℝ (CFC.log a) = a
-- CFC.log_zero
@[simp] lemma log_zero : log (0 : A) = 0 
Arguments: {A : Type u_1} [NormedRing A] [StarRing A] [NormedAlgebra ℝ A] [ContinuousFunctionalCalculus ℝ IsSelfAdjoint]
Goal: CFC.log 0 = 0
-- CFC.log_one
@[simp] lemma log_one : log (1 : A) = 0 
Arguments: {A : Type u_1} [NormedRing A] [StarRing A] [NormedAlgebra ℝ A] [ContinuousFunctionalCalculus ℝ IsSelfAdjoint]
Goal: CFC.log 1 = 0
-- CFC.log_algebraMap
@[simp]
lemma log_algebraMap {r : ℝ} : log (algebraMap ℝ A r) = algebraMap ℝ A (Real.log r) 
Arguments: {A : Type u_1} [NormedRing A] [StarRing A] [NormedAlgebra ℝ A] [ContinuousFunctionalCalculus ℝ IsSelfAdjoint] {r : ℝ}
Goal: CFC.log ((algebraMap ℝ A) r) = (algebraMap ℝ A) (Real.log r)
-- CFC.log_smul
lemma log_smul {r : ℝ} (a : A) (ha₂ : ∀ x ∈ spectrum ℝ a, 0 < x) (hr : 0 < r)
    (ha₁ : IsSelfAdjoint a := by cfc_tac) :
    log (r • a) = algebraMap ℝ A (Real.log r) + log a 
Arguments: {A : Type u_1} [NormedRing A] [StarRing A] [NormedAlgebra ℝ A] [ContinuousFunctionalCalculus ℝ IsSelfAdjoint] [UniqueContinuousFunctionalCalculus ℝ A] {r : ℝ} (a : A) (ha₂ : ∀ x ∈ spectrum ℝ a, 0 < x) (hr : 0 < r) (ha₁ : IsSelfAdjoint a := by cfc_tac)
Goal: CFC.log (r • a) = (algebraMap ℝ A) (Real.log r) + CFC.log a
-- CFC.log_pow
lemma log_pow (n : ℕ) (a : A) (ha₂ : ∀ x ∈ spectrum ℝ a, 0 < x)
    (ha₁ : IsSelfAdjoint a := by cfc_tac) : log (a ^ n) = n • log a 
Arguments: {A : Type u_1} [NormedRing A] [StarRing A] [NormedAlgebra ℝ A] [ContinuousFunctionalCalculus ℝ IsSelfAdjoint] [UniqueContinuousFunctionalCalculus ℝ A] (n : ℕ) (a : A) (ha₂ : ∀ x ∈ spectrum ℝ a, 0 < x) (ha₁ : IsSelfAdjoint a := by cfc_tac)
Goal: CFC.log (a ^ n) = n • CFC.log a
-- Polynomial.Chebyshev.aeval_T
@[simp]
theorem aeval_T (x : A) (n : ℤ) : aeval x (T R n) = (T A n).eval x 
Arguments: {R : Type u_1} {A : Type u_2} [CommRing R] [CommRing A] [Algebra R A] (x : A) (n : ℤ)
Goal: (Polynomial.aeval x) (Polynomial.Chebyshev.T R n) = Polynomial.eval x (Polynomial.Chebyshev.T A n)
-- Polynomial.Chebyshev.aeval_U
@[simp]
theorem aeval_U (x : A) (n : ℤ) : aeval x (U R n) = (U A n).eval x 
Arguments: {R : Type u_1} {A : Type u_2} [CommRing R] [CommRing A] [Algebra R A] (x : A) (n : ℤ)
Goal: (Polynomial.aeval x) (Polynomial.Chebyshev.U R n) = Polynomial.eval x (Polynomial.Chebyshev.U A n)
-- Polynomial.Chebyshev.algebraMap_eval_T
@[simp]
theorem algebraMap_eval_T (x : R) (n : ℤ) :
    algebraMap R A ((T R n).eval x) = (T A n).eval (algebraMap R A x) 
Arguments: {R : Type u_1} {A : Type u_2} [CommRing R] [CommRing A] [Algebra R A] (x : R) (n : ℤ)
Goal: (algebraMap R A) (Polynomial.eval x (Polynomial.Chebyshev.T R n)) =
  Polynomial.eval ((algebraMap R A) x) (Polynomial.Chebyshev.T A n)
-- Polynomial.Chebyshev.algebraMap_eval_U
@[simp]
theorem algebraMap_eval_U (x : R) (n : ℤ) :
    algebraMap R A ((U R n).eval x) = (U A n).eval (algebraMap R A x) 
Arguments: {R : Type u_1} {A : Type u_2} [CommRing R] [CommRing A] [Algebra R A] (x : R) (n : ℤ)
Goal: (algebraMap R A) (Polynomial.eval x (Polynomial.Chebyshev.U R n)) =
  Polynomial.eval ((algebraMap R A) x) (Polynomial.Chebyshev.U A n)
-- Polynomial.Chebyshev.complex_ofReal_eval_T
@[simp, norm_cast]
theorem complex_ofReal_eval_T : ∀ (x : ℝ) n, (((T ℝ n).eval x : ℝ) : ℂ) = (T ℂ n).eval (x : ℂ) 
Arguments: (x : ℝ) (n : ℤ)
Goal: ↑(Polynomial.eval x (Polynomial.Chebyshev.T ℝ n)) = Polynomial.eval (↑x) (Polynomial.Chebyshev.T ℂ n)
-- Polynomial.Chebyshev.complex_ofReal_eval_U
@[simp, norm_cast]
theorem complex_ofReal_eval_U : ∀ (x : ℝ) n, (((U ℝ n).eval x : ℝ) : ℂ) = (U ℂ n).eval (x : ℂ) 
Arguments: (x : ℝ) (n : ℤ)
Goal: ↑(Polynomial.eval x (Polynomial.Chebyshev.U ℝ n)) = Polynomial.eval (↑x) (Polynomial.Chebyshev.U ℂ n)
-- Polynomial.Chebyshev.T_complex_cos
@[simp]
theorem T_complex_cos (n : ℤ) : (T ℂ n).eval (cos θ) = cos (n * θ) 
Arguments: (θ : ℂ) (n : ℤ)
Goal: Polynomial.eval (Complex.cos θ) (Polynomial.Chebyshev.T ℂ n) = Complex.cos (↑n * θ)
-- Polynomial.Chebyshev.U_complex_cos
@[simp]
theorem U_complex_cos (n : ℤ) : (U ℂ n).eval (cos θ) * sin θ = sin ((n + 1) * θ) 
Arguments: (θ : ℂ) (n : ℤ)
Goal: Polynomial.eval (Complex.cos θ) (Polynomial.Chebyshev.U ℂ n) * Complex.sin θ = Complex.sin ((↑n + 1) * θ)
-- Polynomial.Chebyshev.T_real_cos
@[simp]
theorem T_real_cos : (T ℝ n).eval (cos θ) = cos (n * θ) 
Arguments: (θ : ℝ) (n : ℤ)
Goal: Polynomial.eval (Real.cos θ) (Polynomial.Chebyshev.T ℝ n) = Real.cos (↑n * θ)
-- Polynomial.Chebyshev.U_real_cos
@[simp]
theorem U_real_cos : (U ℝ n).eval (cos θ) * sin θ = sin ((n + 1) * θ) 
Arguments: (θ : ℝ) (n : ℤ)
Goal: Polynomial.eval (Real.cos θ) (Polynomial.Chebyshev.U ℝ n) * Real.sin θ = Real.sin ((↑n + 1) * θ)
-- Real.hasStrictDerivAt_tan
theorem hasStrictDerivAt_tan {x : ℝ} (h : cos x ≠ 0) : HasStrictDerivAt tan (1 / cos x ^ 2) x 
Arguments: {x : ℝ} (h : Real.cos x ≠ 0)
Goal: HasStrictDerivAt Real.tan (1 / Real.cos x ^ 2) x
-- Real.hasDerivAt_tan
theorem hasDerivAt_tan {x : ℝ} (h : cos x ≠ 0) : HasDerivAt tan (1 / cos x ^ 2) x 
Arguments: {x : ℝ} (h : Real.cos x ≠ 0)
Goal: HasDerivAt Real.tan (1 / Real.cos x ^ 2) x
-- Real.tendsto_abs_tan_of_cos_eq_zero
theorem tendsto_abs_tan_of_cos_eq_zero {x : ℝ} (hx : cos x = 0) :
    Tendsto (fun x => abs (tan x)) (𝓝[≠] x) atTop 
Arguments: {x : ℝ} (hx : Real.cos x = 0)
Goal: Filter.Tendsto (fun x => |Real.tan x|) (nhdsWithin x {x}ᶜ) Filter.atTop
-- Real.tendsto_abs_tan_atTop
theorem tendsto_abs_tan_atTop (k : ℤ) :
    Tendsto (fun x => abs (tan x)) (𝓝[≠] ((2 * k + 1) * π / 2)) atTop 
Arguments: (k : ℤ)
Goal: Filter.Tendsto (fun x => |Real.tan x|) (nhdsWithin ((2 * ↑k + 1) * Real.pi / 2) {(2 * ↑k + 1) * Real.pi / 2}ᶜ)
  Filter.atTop
-- Real.continuousAt_tan
theorem continuousAt_tan {x : ℝ} : ContinuousAt tan x ↔ cos x ≠ 0 
Arguments: {x : ℝ}
Goal: ContinuousAt Real.tan x ↔ Real.cos x ≠ 0
-- Real.differentiableAt_tan
theorem differentiableAt_tan {x : ℝ} : DifferentiableAt ℝ tan x ↔ cos x ≠ 0 
Arguments: {x : ℝ}
Goal: DifferentiableAt ℝ Real.tan x ↔ Real.cos x ≠ 0
-- Real.deriv_tan
@[simp]
theorem deriv_tan (x : ℝ) : deriv tan x = 1 / cos x ^ 2 
Arguments: (x : ℝ)
Goal: deriv Real.tan x = 1 / Real.cos x ^ 2
-- Real.contDiffAt_tan
@[simp]
theorem contDiffAt_tan {n x} : ContDiffAt ℝ n tan x ↔ cos x ≠ 0 
Arguments: {n : ℕ∞} {x : ℝ}
Goal: ContDiffAt ℝ n Real.tan x ↔ Real.cos x ≠ 0
-- Real.hasDerivAt_tan_of_mem_Ioo
theorem hasDerivAt_tan_of_mem_Ioo {x : ℝ} (h : x ∈ Ioo (-(π / 2) : ℝ) (π / 2)) :
    HasDerivAt tan (1 / cos x ^ 2) x 
Arguments: {x : ℝ} (h : x ∈ Set.Ioo (-(Real.pi / 2)) (Real.pi / 2))
Goal: HasDerivAt Real.tan (1 / Real.cos x ^ 2) x
-- Real.differentiableAt_tan_of_mem_Ioo
theorem differentiableAt_tan_of_mem_Ioo {x : ℝ} (h : x ∈ Ioo (-(π / 2) : ℝ) (π / 2)) :
    DifferentiableAt ℝ tan x 
Arguments: {x : ℝ} (h : x ∈ Set.Ioo (-(Real.pi / 2)) (Real.pi / 2))
Goal: DifferentiableAt ℝ Real.tan x
-- Real.hasStrictDerivAt_arctan
theorem hasStrictDerivAt_arctan (x : ℝ) : HasStrictDerivAt arctan (1 / (1 + x ^ 2)) x 
Arguments: (x : ℝ)
Goal: HasStrictDerivAt Real.arctan (1 / (1 + x ^ 2)) x
-- Real.hasDerivAt_arctan
theorem hasDerivAt_arctan (x : ℝ) : HasDerivAt arctan (1 / (1 + x ^ 2)) x 
Arguments: (x : ℝ)
Goal: HasDerivAt Real.arctan (1 / (1 + x ^ 2)) x
-- Real.hasDerivAt_arctan'
theorem hasDerivAt_arctan' (x : ℝ) : HasDerivAt arctan (1 + x ^ 2)⁻¹ x 
Arguments: (x : ℝ)
Goal: HasDerivAt Real.arctan (1 + x ^ 2)⁻¹ x
-- Real.differentiableAt_arctan
theorem differentiableAt_arctan (x : ℝ) : DifferentiableAt ℝ arctan x 
Arguments: (x : ℝ)
Goal: DifferentiableAt ℝ Real.arctan x
-- Real.differentiable_arctan
theorem differentiable_arctan : Differentiable ℝ arctan 
Goal: Differentiable ℝ Real.arctan
-- Real.deriv_arctan
@[simp]
theorem deriv_arctan : deriv arctan = fun (x : ℝ) => 1 / (1 + x ^ 2) 
Goal: deriv Real.arctan = fun x => 1 / (1 + x ^ 2)
-- Real.contDiff_arctan
theorem contDiff_arctan {n : ℕ∞} : ContDiff ℝ n arctan 
Arguments: {n : ℕ∞}
Goal: ContDiff ℝ n Real.arctan
-- HasStrictDerivAt.arctan
theorem HasStrictDerivAt.arctan (hf : HasStrictDerivAt f f' x) :
    HasStrictDerivAt (fun x => arctan (f x)) (1 / (1 + f x ^ 2) * f') x 
Arguments: {f : ℝ → ℝ} {f' x : ℝ} (hf : HasStrictDerivAt f f' x)
Goal: HasStrictDerivAt (fun x => Real.arctan (f x)) (1 / (1 + f x ^ 2) * f') x
-- HasDerivAt.arctan
theorem HasDerivAt.arctan (hf : HasDerivAt f f' x) :
    HasDerivAt (fun x => arctan (f x)) (1 / (1 + f x ^ 2) * f') x 
Arguments: {f : ℝ → ℝ} {f' x : ℝ} (hf : HasDerivAt f f' x)
Goal: HasDerivAt (fun x => Real.arctan (f x)) (1 / (1 + f x ^ 2) * f') x
-- HasDerivWithinAt.arctan
theorem HasDerivWithinAt.arctan (hf : HasDerivWithinAt f f' s x) :
    HasDerivWithinAt (fun x => arctan (f x)) (1 / (1 + f x ^ 2) * f') s x 
Arguments: {f : ℝ → ℝ} {f' x : ℝ} {s : Set ℝ} (hf : HasDerivWithinAt f f' s x)
Goal: HasDerivWithinAt (fun x => Real.arctan (f x)) (1 / (1 + f x ^ 2) * f') s x
-- derivWithin_arctan
theorem derivWithin_arctan (hf : DifferentiableWithinAt ℝ f s x) (hxs : UniqueDiffWithinAt ℝ s x) :
    derivWithin (fun x => arctan (f x)) s x = 1 / (1 + f x ^ 2) * 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.arctan (f x)) s x = 1 / (1 + f x ^ 2) * derivWithin f s x
-- deriv_arctan
@[simp]
theorem deriv_arctan (hc : DifferentiableAt ℝ f x) :
    deriv (fun x => arctan (f x)) x = 1 / (1 + f x ^ 2) * deriv f x 
Arguments: {f : ℝ → ℝ} {x : ℝ} (hc : DifferentiableAt ℝ f x)
Goal: deriv (fun x => Real.arctan (f x)) x = 1 / (1 + f x ^ 2) * deriv f x
-- HasStrictFDerivAt.arctan
theorem HasStrictFDerivAt.arctan (hf : HasStrictFDerivAt f f' x) :
    HasStrictFDerivAt (fun x => arctan (f x)) ((1 / (1 + f x ^ 2)) • 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.arctan (f x)) ((1 / (1 + f x ^ 2)) • f') x
-- HasFDerivAt.arctan
theorem HasFDerivAt.arctan (hf : HasFDerivAt f f' x) :
    HasFDerivAt (fun x => arctan (f x)) ((1 / (1 + f x ^ 2)) • 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.arctan (f x)) ((1 / (1 + f x ^ 2)) • f') x
-- HasFDerivWithinAt.arctan
theorem HasFDerivWithinAt.arctan (hf : HasFDerivWithinAt f f' s x) :
    HasFDerivWithinAt (fun x => arctan (f x)) ((1 / (1 + f x ^ 2)) • 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.arctan (f x)) ((1 / (1 + f x ^ 2)) • f') s x
-- fderivWithin_arctan
theorem fderivWithin_arctan (hf : DifferentiableWithinAt ℝ f s x) (hxs : UniqueDiffWithinAt ℝ s x) :
    fderivWithin ℝ (fun x => arctan (f x)) s x = (1 / (1 + f x ^ 2)) • 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.arctan (f x)) s x = (1 / (1 + f x ^ 2)) • fderivWithin ℝ f s x
-- fderiv_arctan
@[simp]
theorem fderiv_arctan (hc : DifferentiableAt ℝ f x) :
    fderiv ℝ (fun x => arctan (f x)) x = (1 / (1 + f x ^ 2)) • 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.arctan (f x)) x = (1 / (1 + f x ^ 2)) • fderiv ℝ f x
-- DifferentiableWithinAt.arctan
theorem DifferentiableWithinAt.arctan (hf : DifferentiableWithinAt ℝ f s x) :
    DifferentiableWithinAt ℝ (fun x => Real.arctan (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.arctan (f x)) s x
-- DifferentiableAt.arctan
@[simp]
theorem DifferentiableAt.arctan (hc : DifferentiableAt ℝ f x) :
    DifferentiableAt ℝ (fun x => arctan (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.arctan (f x)) x
-- DifferentiableOn.arctan
theorem DifferentiableOn.arctan (hc : DifferentiableOn ℝ f s) :
    DifferentiableOn ℝ (fun x => arctan (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.arctan (f x)) s
-- Differentiable.arctan
@[simp]
theorem Differentiable.arctan (hc : Differentiable ℝ f) : Differentiable ℝ fun x => arctan (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.arctan (f x)
-- ContDiffAt.arctan
theorem ContDiffAt.arctan (h : ContDiffAt ℝ n f x) : ContDiffAt ℝ n (fun x => arctan (f x)) x 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {f : E → ℝ} {x : E} {n : ℕ∞} (h : ContDiffAt ℝ n f x)
Goal: ContDiffAt ℝ n (fun x => Real.arctan (f x)) x
-- ContDiff.arctan
theorem ContDiff.arctan (h : ContDiff ℝ n f) : ContDiff ℝ n fun x => arctan (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.arctan (f x)
-- ContDiffWithinAt.arctan
theorem ContDiffWithinAt.arctan (h : ContDiffWithinAt ℝ n f s x) :
    ContDiffWithinAt ℝ n (fun x => arctan (f x)) s x 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {f : E → ℝ} {x : E} {s : Set E} {n : ℕ∞} (h : ContDiffWithinAt ℝ n f s x)
Goal: ContDiffWithinAt ℝ n (fun x => Real.arctan (f x)) s x
-- ContDiffOn.arctan
theorem ContDiffOn.arctan (h : ContDiffOn ℝ n f s) : ContDiffOn ℝ n (fun x => arctan (f x)) s 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {f : E → ℝ} {s : Set E} {n : ℕ∞} (h : ContDiffOn ℝ n f s)
Goal: ContDiffOn ℝ n (fun x => Real.arctan (f x)) s
-- Real.arcsin_mem_Icc
theorem arcsin_mem_Icc (x : ℝ) : arcsin x ∈ Icc (-(π / 2)) (π / 2) 
Arguments: (x : ℝ)
Goal: Real.arcsin x ∈ Set.Icc (-(π / 2)) (π / 2)
-- Real.range_arcsin
@[simp]
theorem range_arcsin : range arcsin = Icc (-(π / 2)) (π / 2) 
Goal: Set.range Real.arcsin = Set.Icc (-(π / 2)) (π / 2)
-- Real.arcsin_le_pi_div_two
theorem arcsin_le_pi_div_two (x : ℝ) : arcsin x ≤ π / 2 
Arguments: (x : ℝ)
Goal: Real.arcsin x ≤ π / 2
-- Real.neg_pi_div_two_le_arcsin
theorem neg_pi_div_two_le_arcsin (x : ℝ) : -(π / 2) ≤ arcsin x 
Arguments: (x : ℝ)
Goal: -(π / 2) ≤ Real.arcsin x
-- Real.arcsin_projIcc
theorem arcsin_projIcc (x : ℝ) :
    arcsin (projIcc (-1) 1 (neg_le_self zero_le_one) x) = arcsin x 
Arguments: (x : ℝ)
Goal: Real.arcsin ↑(Set.projIcc (-1) 1 ⋯ x) = Real.arcsin x
-- Real.sin_arcsin'
theorem sin_arcsin' {x : ℝ} (hx : x ∈ Icc (-1 : ℝ) 1) : sin (arcsin x) = x 
Arguments: {x : ℝ} (hx : x ∈ Set.Icc (-1) 1)
Goal: Real.sin (Real.arcsin x) = x
-- Real.sin_arcsin
theorem sin_arcsin {x : ℝ} (hx₁ : -1 ≤ x) (hx₂ : x ≤ 1) : sin (arcsin x) = x 
Arguments: {x : ℝ} (hx₁ : -1 ≤ x) (hx₂ : x ≤ 1)
Goal: Real.sin (Real.arcsin x) = x
-- Real.arcsin_sin'
theorem arcsin_sin' {x : ℝ} (hx : x ∈ Icc (-(π / 2)) (π / 2)) : arcsin (sin x) = x 
Arguments: {x : ℝ} (hx : x ∈ Set.Icc (-(π / 2)) (π / 2))
Goal: Real.arcsin (Real.sin x) = x
-- Real.arcsin_sin
theorem arcsin_sin {x : ℝ} (hx₁ : -(π / 2) ≤ x) (hx₂ : x ≤ π / 2) : arcsin (sin x) = x 
Arguments: {x : ℝ} (hx₁ : -(π / 2) ≤ x) (hx₂ : x ≤ π / 2)
Goal: Real.arcsin (Real.sin x) = x
-- Real.strictMonoOn_arcsin
theorem strictMonoOn_arcsin : StrictMonoOn arcsin (Icc (-1) 1) 
Goal: StrictMonoOn Real.arcsin (Set.Icc (-1) 1)
-- Real.monotone_arcsin
theorem monotone_arcsin : Monotone arcsin 
Goal: Monotone Real.arcsin
-- Real.injOn_arcsin
theorem injOn_arcsin : InjOn arcsin (Icc (-1) 1) 
Goal: Set.InjOn Real.arcsin (Set.Icc (-1) 1)
-- Real.arcsin_inj
theorem arcsin_inj {x y : ℝ} (hx₁ : -1 ≤ x) (hx₂ : x ≤ 1) (hy₁ : -1 ≤ y) (hy₂ : y ≤ 1) :
    arcsin x = arcsin y ↔ x = y 
Arguments: {x y : ℝ} (hx₁ : -1 ≤ x) (hx₂ : x ≤ 1) (hy₁ : -1 ≤ y) (hy₂ : y ≤ 1)
Goal: Real.arcsin x = Real.arcsin y ↔ x = y
-- Real.continuous_arcsin
@[continuity]
theorem continuous_arcsin : Continuous arcsin 
Goal: Continuous Real.arcsin
-- Real.continuousAt_arcsin
theorem continuousAt_arcsin {x : ℝ} : ContinuousAt arcsin x 
Arguments: {x : ℝ}
Goal: ContinuousAt Real.arcsin x
-- Real.arcsin_eq_of_sin_eq
theorem arcsin_eq_of_sin_eq {x y : ℝ} (h₁ : sin x = y) (h₂ : x ∈ Icc (-(π / 2)) (π / 2)) :
    arcsin y = x 
Arguments: {x y : ℝ} (h₁ : Real.sin x = y) (h₂ : x ∈ Set.Icc (-(π / 2)) (π / 2))
Goal: Real.arcsin y = x
-- Real.arcsin_zero
@[simp]
theorem arcsin_zero : arcsin 0 = 0 
Goal: Real.arcsin 0 = 0
-- Real.arcsin_one
@[simp]
theorem arcsin_one : arcsin 1 = π / 2 
Goal: Real.arcsin 1 = π / 2