ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- intervalIntegral.inv_mul_integral_comp_div
@[simp]
theorem inv_mul_integral_comp_div : (c⁻¹ * ∫ x in a..b, f (x / c)) = ∫ x in a / c..b / c, f x 
Arguments: {a b : ℝ} {f : ℝ → ℝ} (c : ℝ)
Goal: c⁻¹ * ∫ (x : ℝ) in a..b, f (x / c) = ∫ (x : ℝ) in a / c..b / c, f x
-- intervalIntegral.mul_integral_comp_mul_add
@[simp]
theorem mul_integral_comp_mul_add :
    (c * ∫ x in a..b, f (c * x + d)) = ∫ x in c * a + d..c * b + d, f x 
Arguments: {a b : ℝ} {f : ℝ → ℝ} (c d : ℝ)
Goal: c * ∫ (x : ℝ) in a..b, f (c * x + d) = ∫ (x : ℝ) in c * a + d..c * b + d, f x
-- intervalIntegral.mul_integral_comp_add_mul
@[simp]
theorem mul_integral_comp_add_mul :
    (c * ∫ x in a..b, f (d + c * x)) = ∫ x in d + c * a..d + c * b, f x 
Arguments: {a b : ℝ} {f : ℝ → ℝ} (c d : ℝ)
Goal: c * ∫ (x : ℝ) in a..b, f (d + c * x) = ∫ (x : ℝ) in d + c * a..d + c * b, f x
-- intervalIntegral.inv_mul_integral_comp_div_add
@[simp]
theorem inv_mul_integral_comp_div_add :
    (c⁻¹ * ∫ x in a..b, f (x / c + d)) = ∫ x in a / c + d..b / c + d, f x 
Arguments: {a b : ℝ} {f : ℝ → ℝ} (c d : ℝ)
Goal: c⁻¹ * ∫ (x : ℝ) in a..b, f (x / c + d) = ∫ (x : ℝ) in a / c + d..b / c + d, f x
-- intervalIntegral.inv_mul_integral_comp_add_div
@[simp]
theorem inv_mul_integral_comp_add_div :
    (c⁻¹ * ∫ x in a..b, f (d + x / c)) = ∫ x in d + a / c..d + b / c, f x 
Arguments: {a b : ℝ} {f : ℝ → ℝ} (c d : ℝ)
Goal: c⁻¹ * ∫ (x : ℝ) in a..b, f (d + x / c) = ∫ (x : ℝ) in d + a / c..d + b / c, f x
-- intervalIntegral.mul_integral_comp_mul_sub
@[simp]
theorem mul_integral_comp_mul_sub :
    (c * ∫ x in a..b, f (c * x - d)) = ∫ x in c * a - d..c * b - d, f x 
Arguments: {a b : ℝ} {f : ℝ → ℝ} (c d : ℝ)
Goal: c * ∫ (x : ℝ) in a..b, f (c * x - d) = ∫ (x : ℝ) in c * a - d..c * b - d, f x
-- intervalIntegral.mul_integral_comp_sub_mul
@[simp]
theorem mul_integral_comp_sub_mul :
    (c * ∫ x in a..b, f (d - c * x)) = ∫ x in d - c * b..d - c * a, f x 
Arguments: {a b : ℝ} {f : ℝ → ℝ} (c d : ℝ)
Goal: c * ∫ (x : ℝ) in a..b, f (d - c * x) = ∫ (x : ℝ) in d - c * b..d - c * a, f x
-- intervalIntegral.inv_mul_integral_comp_div_sub
@[simp]
theorem inv_mul_integral_comp_div_sub :
    (c⁻¹ * ∫ x in a..b, f (x / c - d)) = ∫ x in a / c - d..b / c - d, f x 
Arguments: {a b : ℝ} {f : ℝ → ℝ} (c d : ℝ)
Goal: c⁻¹ * ∫ (x : ℝ) in a..b, f (x / c - d) = ∫ (x : ℝ) in a / c - d..b / c - d, f x
-- intervalIntegral.inv_mul_integral_comp_sub_div
@[simp]
theorem inv_mul_integral_comp_sub_div :
    (c⁻¹ * ∫ x in a..b, f (d - x / c)) = ∫ x in d - b / c..d - a / c, f x 
Arguments: {a b : ℝ} {f : ℝ → ℝ} (c d : ℝ)
Goal: c⁻¹ * ∫ (x : ℝ) in a..b, f (d - x / c) = ∫ (x : ℝ) in d - b / c..d - a / c, f x
-- integral_cpow
theorem integral_cpow {r : ℂ} (h : -1 < r.re ∨ r ≠ -1 ∧ (0 : ℝ) ∉ [[a, b]]) :
    (∫ x : ℝ in a..b, (x : ℂ) ^ r) = ((b : ℂ) ^ (r + 1) - (a : ℂ) ^ (r + 1)) / (r + 1) 
Arguments: {a b : ℝ} {r : ℂ} (h : -1 < r.re ∨ r ≠ -1 ∧ 0 ∉ [[a, b]])
Goal: ∫ (x : ℝ) in a..b, ↑x ^ r = (↑b ^ (r + 1) - ↑a ^ (r + 1)) / (r + 1)
-- integral_rpow
theorem integral_rpow {r : ℝ} (h : -1 < r ∨ r ≠ -1 ∧ (0 : ℝ) ∉ [[a, b]]) :
    ∫ x in a..b, x ^ r = (b ^ (r + 1) - a ^ (r + 1)) / (r + 1) 
Arguments: {a b r : ℝ} (h : -1 < r ∨ r ≠ -1 ∧ 0 ∉ [[a, b]])
Goal: ∫ (x : ℝ) in a..b, x ^ r = (b ^ (r + 1) - a ^ (r + 1)) / (r + 1)
-- integral_zpow
theorem integral_zpow {n : ℤ} (h : 0 ≤ n ∨ n ≠ -1 ∧ (0 : ℝ) ∉ [[a, b]]) :
    ∫ x in a..b, x ^ n = (b ^ (n + 1) - a ^ (n + 1)) / (n + 1) 
Arguments: {a b : ℝ} {n : ℤ} (h : 0 ≤ n ∨ n ≠ -1 ∧ 0 ∉ [[a, b]])
Goal: ∫ (x : ℝ) in a..b, x ^ n = (b ^ (n + 1) - a ^ (n + 1)) / (↑n + 1)
-- integral_pow
@[simp]
theorem integral_pow : ∫ x in a..b, x ^ n = (b ^ (n + 1) - a ^ (n + 1)) / (n + 1) 
Arguments: {a b : ℝ} (n : ℕ)
Goal: ∫ (x : ℝ) in a..b, x ^ n = (b ^ (n + 1) - a ^ (n + 1)) / (↑n + 1)
-- integral_pow_abs_sub_uIoc
theorem integral_pow_abs_sub_uIoc : ∫ x in Ι a b, |x - a| ^ n = |b - a| ^ (n + 1) / (n + 1) 
Arguments: {a b : ℝ} (n : ℕ)
Goal: ∫ (x : ℝ) in Ι a b, |x - a| ^ n = |b - a| ^ (n + 1) / (↑n + 1)
-- integral_id
@[simp]
theorem integral_id : ∫ x in a..b, x = (b ^ 2 - a ^ 2) / 2 
Arguments: {a b : ℝ}
Goal: ∫ (x : ℝ) in a..b, x = (b ^ 2 - a ^ 2) / 2
-- integral_one
theorem integral_one : (∫ _ in a..b, (1 : ℝ)) = b - a 
Arguments: {a b : ℝ}
Goal: ∫ (x : ℝ) in a..b, 1 = b - a
-- integral_const_on_unit_interval
theorem integral_const_on_unit_interval : ∫ _ in a..a + 1, b = b 
Arguments: {a b : ℝ}
Goal: ∫ (x : ℝ) in a..a + 1, b = b
-- integral_inv
@[simp]
theorem integral_inv (h : (0 : ℝ) ∉ [[a, b]]) : ∫ x in a..b, x⁻¹ = log (b / a) 
Arguments: {a b : ℝ} (h : 0 ∉ [[a, b]])
Goal: ∫ (x : ℝ) in a..b, x⁻¹ = Real.log (b / a)
-- integral_inv_of_pos
@[simp]
theorem integral_inv_of_pos (ha : 0 < a) (hb : 0 < b) : ∫ x in a..b, x⁻¹ = log (b / a) 
Arguments: {a b : ℝ} (ha : 0 < a) (hb : 0 < b)
Goal: ∫ (x : ℝ) in a..b, x⁻¹ = Real.log (b / a)
-- integral_inv_of_neg
@[simp]
theorem integral_inv_of_neg (ha : a < 0) (hb : b < 0) : ∫ x in a..b, x⁻¹ = log (b / a) 
Arguments: {a b : ℝ} (ha : a < 0) (hb : b < 0)
Goal: ∫ (x : ℝ) in a..b, x⁻¹ = Real.log (b / a)
-- integral_one_div
theorem integral_one_div (h : (0 : ℝ) ∉ [[a, b]]) : ∫ x : ℝ in a..b, 1 / x = log (b / a) 
Arguments: {a b : ℝ} (h : 0 ∉ [[a, b]])
Goal: ∫ (x : ℝ) in a..b, 1 / x = Real.log (b / a)
-- integral_one_div_of_pos
theorem integral_one_div_of_pos (ha : 0 < a) (hb : 0 < b) :
    ∫ x : ℝ in a..b, 1 / x = log (b / a) 
Arguments: {a b : ℝ} (ha : 0 < a) (hb : 0 < b)
Goal: ∫ (x : ℝ) in a..b, 1 / x = Real.log (b / a)
-- integral_one_div_of_neg
theorem integral_one_div_of_neg (ha : a < 0) (hb : b < 0) :
    ∫ x : ℝ in a..b, 1 / x = log (b / a) 
Arguments: {a b : ℝ} (ha : a < 0) (hb : b < 0)
Goal: ∫ (x : ℝ) in a..b, 1 / x = Real.log (b / a)
-- integral_exp
@[simp]
theorem integral_exp : ∫ x in a..b, exp x = exp b - exp a 
Arguments: {a b : ℝ}
Goal: ∫ (x : ℝ) in a..b, rexp x = rexp b - rexp a
-- integral_exp_mul_complex
theorem integral_exp_mul_complex {c : ℂ} (hc : c ≠ 0) :
    (∫ x in a..b, Complex.exp (c * x)) = (Complex.exp (c * b) - Complex.exp (c * a)) / c 
Arguments: {a b : ℝ} {c : ℂ} (hc : c ≠ 0)
Goal: ∫ (x : ℝ) in a..b, Complex.exp (c * ↑x) = (Complex.exp (c * ↑b) - Complex.exp (c * ↑a)) / c
-- integral_log
@[simp]
theorem integral_log (h : (0 : ℝ) ∉ [[a, b]]) :
    ∫ x in a..b, log x = b * log b - a * log a - b + a 
Arguments: {a b : ℝ} (h : 0 ∉ [[a, b]])
Goal: ∫ (x : ℝ) in a..b, Real.log x = b * Real.log b - a * Real.log a - b + a
-- integral_log_of_pos
@[simp]
theorem integral_log_of_pos (ha : 0 < a) (hb : 0 < b) :
    ∫ x in a..b, log x = b * log b - a * log a - b + a 
Arguments: {a b : ℝ} (ha : 0 < a) (hb : 0 < b)
Goal: ∫ (x : ℝ) in a..b, Real.log x = b * Real.log b - a * Real.log a - b + a
-- integral_log_of_neg
@[simp]
theorem integral_log_of_neg (ha : a < 0) (hb : b < 0) :
    ∫ x in a..b, log x = b * log b - a * log a - b + a 
Arguments: {a b : ℝ} (ha : a < 0) (hb : b < 0)
Goal: ∫ (x : ℝ) in a..b, Real.log x = b * Real.log b - a * Real.log a - b + a
-- integral_sin
@[simp]
theorem integral_sin : ∫ x in a..b, sin x = cos a - cos b 
Arguments: {a b : ℝ}
Goal: ∫ (x : ℝ) in a..b, Real.sin x = Real.cos a - Real.cos b
-- integral_cos
@[simp]
theorem integral_cos : ∫ x in a..b, cos x = sin b - sin a 
Arguments: {a b : ℝ}
Goal: ∫ (x : ℝ) in a..b, Real.cos x = Real.sin b - Real.sin a
-- integral_cos_mul_complex
theorem integral_cos_mul_complex {z : ℂ} (hz : z ≠ 0) (a b : ℝ) :
    (∫ x in a..b, Complex.cos (z * x)) = Complex.sin (z * b) / z - Complex.sin (z * a) / z 
Arguments: {z : ℂ} (hz : z ≠ 0) (a b : ℝ)
Goal: ∫ (x : ℝ) in a..b, Complex.cos (z * ↑x) = Complex.sin (z * ↑b) / z - Complex.sin (z * ↑a) / z
-- integral_cos_sq_sub_sin_sq
theorem integral_cos_sq_sub_sin_sq :
    ∫ x in a..b, cos x ^ 2 - sin x ^ 2 = sin b * cos b - sin a * cos a 
Arguments: {a b : ℝ}
Goal: ∫ (x : ℝ) in a..b, Real.cos x ^ 2 - Real.sin x ^ 2 = Real.sin b * Real.cos b - Real.sin a * Real.cos a
-- integral_one_div_one_add_sq
theorem integral_one_div_one_add_sq :
    (∫ x : ℝ in a..b, ↑1 / (↑1 + x ^ 2)) = arctan b - arctan a 
Arguments: {a b : ℝ}
Goal: ∫ (x : ℝ) in a..b, 1 / (1 + x ^ 2) = Real.arctan b - Real.arctan a
-- integral_inv_one_add_sq
@[simp]
theorem integral_inv_one_add_sq : (∫ x : ℝ in a..b, (↑1 + x ^ 2)⁻¹) = arctan b - arctan a 
Arguments: {a b : ℝ}
Goal: ∫ (x : ℝ) in a..b, (1 + x ^ 2)⁻¹ = Real.arctan b - Real.arctan a
-- integral_mul_cpow_one_add_sq
theorem integral_mul_cpow_one_add_sq {t : ℂ} (ht : t ≠ -1) :
    (∫ x : ℝ in a..b, (x : ℂ) * ((1 : ℂ) + ↑x ^ 2) ^ t) =
      ((1 : ℂ) + (b : ℂ) ^ 2) ^ (t + 1) / (2 * (t + ↑1)) -
      ((1 : ℂ) + (a : ℂ) ^ 2) ^ (t + 1) / (2 * (t + ↑1)) 
Arguments: {a b : ℝ} {t : ℂ} (ht : t ≠ -1)
Goal: ∫ (x : ℝ) in a..b, ↑x * (1 + ↑x ^ 2) ^ t =
  (1 + ↑b ^ 2) ^ (t + 1) / (2 * (t + 1)) - (1 + ↑a ^ 2) ^ (t + 1) / (2 * (t + 1))
-- integral_mul_rpow_one_add_sq
theorem integral_mul_rpow_one_add_sq {t : ℝ} (ht : t ≠ -1) :
    (∫ x : ℝ in a..b, x * (↑1 + x ^ 2) ^ t) =
      (↑1 + b ^ 2) ^ (t + 1) / (↑2 * (t + ↑1)) - (↑1 + a ^ 2) ^ (t + 1) / (↑2 * (t + ↑1)) 
Arguments: {a b t : ℝ} (ht : t ≠ -1)
Goal: ∫ (x : ℝ) in a..b, x * (1 + x ^ 2) ^ t = (1 + b ^ 2) ^ (t + 1) / (2 * (t + 1)) - (1 + a ^ 2) ^ (t + 1) / (2 * (t + 1))
-- integral_sin_pow_aux
theorem integral_sin_pow_aux :
    (∫ x in a..b, sin x ^ (n + 2)) =
      (sin a ^ (n + 1) * cos a - sin b ^ (n + 1) * cos b + (↑n + 1) * ∫ x in a..b, sin x ^ n) -
        (↑n + 1) * ∫ x in a..b, sin x ^ (n + 2) 
Arguments: {a b : ℝ} (n : ℕ)
Goal: ∫ (x : ℝ) in a..b, Real.sin x ^ (n + 2) =
  (Real.sin a ^ (n + 1) * Real.cos a - Real.sin b ^ (n + 1) * Real.cos b +
      (↑n + 1) * ∫ (x : ℝ) in a..b, Real.sin x ^ n) -
    (↑n + 1) * ∫ (x : ℝ) in a..b, Real.sin x ^ (n + 2)
-- integral_sin_pow
theorem integral_sin_pow :
    (∫ x in a..b, sin x ^ (n + 2)) =
      (sin a ^ (n + 1) * cos a - sin b ^ (n + 1) * cos b) / (n + 2) +
        (n + 1) / (n + 2) * ∫ x in a..b, sin x ^ n 
Arguments: {a b : ℝ} (n : ℕ)
Goal: ∫ (x : ℝ) in a..b, Real.sin x ^ (n + 2) =
  (Real.sin a ^ (n + 1) * Real.cos a - Real.sin b ^ (n + 1) * Real.cos b) / (↑n + 2) +
    (↑n + 1) / (↑n + 2) * ∫ (x : ℝ) in a..b, Real.sin x ^ n
-- integral_sin_sq
@[simp]
theorem integral_sin_sq : ∫ x in a..b, sin x ^ 2 = (sin a * cos a - sin b * cos b + b - a) / 2 
Arguments: {a b : ℝ}
Goal: ∫ (x : ℝ) in a..b, Real.sin x ^ 2 = (Real.sin a * Real.cos a - Real.sin b * Real.cos b + b - a) / 2
-- integral_sin_pow_odd
theorem integral_sin_pow_odd :
    (∫ x in (0)..π, sin x ^ (2 * n + 1)) = 2 * ∏ i ∈ range n, (2 * (i : ℝ) + 2) / (2 * i + 3) 
Arguments: (n : ℕ)
Goal: ∫ (x : ℝ) in 0 ..π, Real.sin x ^ (2 * n + 1) = 2 * ∏ i ∈ Finset.range n, (2 * ↑i + 2) / (2 * ↑i + 3)
-- integral_sin_pow_even
theorem integral_sin_pow_even :
    (∫ x in (0)..π, sin x ^ (2 * n)) = π * ∏ i ∈ range n, (2 * (i : ℝ) + 1) / (2 * i + 2) 
Arguments: (n : ℕ)
Goal: ∫ (x : ℝ) in 0 ..π, Real.sin x ^ (2 * n) = π * ∏ i ∈ Finset.range n, (2 * ↑i + 1) / (2 * ↑i + 2)
-- integral_sin_pow_pos
theorem integral_sin_pow_pos : 0 < ∫ x in (0)..π, sin x ^ n 
Arguments: (n : ℕ)
Goal: 0 < ∫ (x : ℝ) in 0 ..π, Real.sin x ^ n
-- integral_sin_pow_succ_le
theorem integral_sin_pow_succ_le : (∫ x in (0)..π, sin x ^ (n + 1)) ≤ ∫ x in (0)..π, sin x ^ n 
Arguments: (n : ℕ)
Goal: ∫ (x : ℝ) in 0 ..π, Real.sin x ^ (n + 1) ≤ ∫ (x : ℝ) in 0 ..π, Real.sin x ^ n
-- integral_sin_pow_antitone
theorem integral_sin_pow_antitone : Antitone fun n : ℕ => ∫ x in (0)..π, sin x ^ n 
Goal: Antitone fun n => ∫ (x : ℝ) in 0 ..π, Real.sin x ^ n
-- integral_cos_pow_aux
theorem integral_cos_pow_aux :
    (∫ x in a..b, cos x ^ (n + 2)) =
      (cos b ^ (n + 1) * sin b - cos a ^ (n + 1) * sin a + (n + 1) * ∫ x in a..b, cos x ^ n) -
        (n + 1) * ∫ x in a..b, cos x ^ (n + 2) 
Arguments: {a b : ℝ} (n : ℕ)
Goal: ∫ (x : ℝ) in a..b, Real.cos x ^ (n + 2) =
  (Real.cos b ^ (n + 1) * Real.sin b - Real.cos a ^ (n + 1) * Real.sin a +
      (↑n + 1) * ∫ (x : ℝ) in a..b, Real.cos x ^ n) -
    (↑n + 1) * ∫ (x : ℝ) in a..b, Real.cos x ^ (n + 2)
-- integral_cos_pow
theorem integral_cos_pow :
    (∫ x in a..b, cos x ^ (n + 2)) =
      (cos b ^ (n + 1) * sin b - cos a ^ (n + 1) * sin a) / (n + 2) +
        (n + 1) / (n + 2) * ∫ x in a..b, cos x ^ n 
Arguments: {a b : ℝ} (n : ℕ)
Goal: ∫ (x : ℝ) in a..b, Real.cos x ^ (n + 2) =
  (Real.cos b ^ (n + 1) * Real.sin b - Real.cos a ^ (n + 1) * Real.sin a) / (↑n + 2) +
    (↑n + 1) / (↑n + 2) * ∫ (x : ℝ) in a..b, Real.cos x ^ n
-- integral_cos_sq
@[simp]
theorem integral_cos_sq : ∫ x in a..b, cos x ^ 2 = (cos b * sin b - cos a * sin a + b - a) / 2 
Arguments: {a b : ℝ}
Goal: ∫ (x : ℝ) in a..b, Real.cos x ^ 2 = (Real.cos b * Real.sin b - Real.cos a * Real.sin a + b - a) / 2
-- integral_sin_pow_mul_cos_pow_odd
theorem integral_sin_pow_mul_cos_pow_odd (m n : ℕ) :
    (∫ x in a..b, sin x ^ m * cos x ^ (2 * n + 1)) = ∫ u in sin a..sin b, u^m * (↑1 - u ^ 2) ^ n 
Arguments: {a b : ℝ} (m n : ℕ)
Goal: ∫ (x : ℝ) in a..b, Real.sin x ^ m * Real.cos x ^ (2 * n + 1) =
  ∫ (u : ℝ) in Real.sin a..Real.sin b, u ^ m * (1 - u ^ 2) ^ n
-- integral_sin_mul_cos₁
@[simp]
theorem integral_sin_mul_cos₁ : ∫ x in a..b, sin x * cos x = (sin b ^ 2 - sin a ^ 2) / 2 
Arguments: {a b : ℝ}
Goal: ∫ (x : ℝ) in a..b, Real.sin x * Real.cos x = (Real.sin b ^ 2 - Real.sin a ^ 2) / 2
-- integral_sin_sq_mul_cos
@[simp]
theorem integral_sin_sq_mul_cos :
    ∫ x in a..b, sin x ^ 2 * cos x = (sin b ^ 3 - sin a ^ 3) / 3 
Arguments: {a b : ℝ}
Goal: ∫ (x : ℝ) in a..b, Real.sin x ^ 2 * Real.cos x = (Real.sin b ^ 3 - Real.sin a ^ 3) / 3
-- integral_cos_pow_three
@[simp]
theorem integral_cos_pow_three :
    ∫ x in a..b, cos x ^ 3 = sin b - sin a - (sin b ^ 3 - sin a ^ 3) / 3 
Arguments: {a b : ℝ}
Goal: ∫ (x : ℝ) in a..b, Real.cos x ^ 3 = Real.sin b - Real.sin a - (Real.sin b ^ 3 - Real.sin a ^ 3) / 3
-- integral_sin_pow_odd_mul_cos_pow
theorem integral_sin_pow_odd_mul_cos_pow (m n : ℕ) :
    (∫ x in a..b, sin x ^ (2 * m + 1) * cos x ^ n) = ∫ u in cos b..cos a, u^n * (↑1 - u ^ 2) ^ m 
Arguments: {a b : ℝ} (m n : ℕ)
Goal: ∫ (x : ℝ) in a..b, Real.sin x ^ (2 * m + 1) * Real.cos x ^ n =
  ∫ (u : ℝ) in Real.cos b..Real.cos a, u ^ n * (1 - u ^ 2) ^ m
-- integral_sin_mul_cos₂
theorem integral_sin_mul_cos₂ : ∫ x in a..b, sin x * cos x = (cos a ^ 2 - cos b ^ 2) / 2 
Arguments: {a b : ℝ}
Goal: ∫ (x : ℝ) in a..b, Real.sin x * Real.cos x = (Real.cos a ^ 2 - Real.cos b ^ 2) / 2
-- integral_sin_mul_cos_sq
@[simp]
theorem integral_sin_mul_cos_sq :
    ∫ x in a..b, sin x * cos x ^ 2 = (cos a ^ 3 - cos b ^ 3) / 3 
Arguments: {a b : ℝ}
Goal: ∫ (x : ℝ) in a..b, Real.sin x * Real.cos x ^ 2 = (Real.cos a ^ 3 - Real.cos b ^ 3) / 3
-- integral_sin_pow_three
@[simp]
theorem integral_sin_pow_three :
    ∫ x in a..b, sin x ^ 3 = cos a - cos b - (cos a ^ 3 - cos b ^ 3) / 3 
Arguments: {a b : ℝ}
Goal: ∫ (x : ℝ) in a..b, Real.sin x ^ 3 = Real.cos a - Real.cos b - (Real.cos a ^ 3 - Real.cos b ^ 3) / 3
-- integral_sin_pow_even_mul_cos_pow_even
theorem integral_sin_pow_even_mul_cos_pow_even (m n : ℕ) :
    (∫ x in a..b, sin x ^ (2 * m) * cos x ^ (2 * n)) =
      ∫ x in a..b, ((1 - cos (2 * x)) / 2) ^ m * ((1 + cos (2 * x)) / 2) ^ n 
Arguments: {a b : ℝ} (m n : ℕ)
Goal: ∫ (x : ℝ) in a..b, Real.sin x ^ (2 * m) * Real.cos x ^ (2 * n) =
  ∫ (x : ℝ) in a..b, ((1 - Real.cos (2 * x)) / 2) ^ m * ((1 + Real.cos (2 * x)) / 2) ^ n
-- integral_sin_sq_mul_cos_sq
@[simp]
theorem integral_sin_sq_mul_cos_sq :
    ∫ x in a..b, sin x ^ 2 * cos x ^ 2 = (b - a) / 8 - (sin (4 * b) - sin (4 * a)) / 32 
Arguments: {a b : ℝ}
Goal: ∫ (x : ℝ) in a..b, Real.sin x ^ 2 * Real.cos x ^ 2 = (b - a) / 8 - (Real.sin (4 * b) - Real.sin (4 * a)) / 32
-- integral_sqrt_one_sub_sq
theorem integral_sqrt_one_sub_sq : ∫ x in (-1 : ℝ)..1, √(1 - x ^ 2 : ℝ) = π / 2 
Goal: ∫ (x : ℝ) in -1 ..1, √(1 - x ^ 2) = π / 2
-- Real.exists_extension_norm_eq
theorem exists_extension_norm_eq (p : Subspace ℝ E) (f : p →L[ℝ] ℝ) :
    ∃ g : E →L[ℝ] ℝ, (∀ x : p, g x = f x) ∧ ‖g‖ = ‖f‖ 
Arguments: {E : Type u_1} [SeminormedAddCommGroup E] [NormedSpace ℝ E] (p : Subspace ℝ E) (f : ↥p →L[ℝ] ℝ)
Goal: ∃ g, (∀ (x : ↥p), g ↑x = f x) ∧ ‖g‖ = ‖f‖
-- exists_extension_norm_eq
theorem exists_extension_norm_eq (p : Subspace 𝕜 E) (f : p →L[𝕜] 𝕜) :
    ∃ g : E →L[𝕜] 𝕜, (∀ x : p, g x = f x) ∧ ‖g‖ = ‖f‖ 
Arguments: {𝕜 : Type u_1} [RCLike 𝕜] {E : Type u_2} [SeminormedAddCommGroup E] [NormedSpace 𝕜 E] (p : Subspace 𝕜 E) (f : ↥p →L[𝕜] 𝕜)
Goal: ∃ g, (∀ (x : ↥p), g ↑x = f x) ∧ ‖g‖ = ‖f‖
-- ContinuousLinearMap.exist_extension_of_finiteDimensional_range
lemma ContinuousLinearMap.exist_extension_of_finiteDimensional_range {p : Submodule 𝕜 E}
    (f : p →L[𝕜] F) [FiniteDimensional 𝕜 (LinearMap.range f)] :
    ∃ g : E →L[𝕜] F, f = g.comp p.subtypeL 
Arguments: {𝕜 : Type u_1} [RCLike 𝕜] {E : Type u_2} {F : Type u_3} [SeminormedAddCommGroup E] [NormedSpace 𝕜 E] [NormedAddCommGroup F] [NormedSpace 𝕜 F] {p : Submodule 𝕜 E} (f : ↥p →L[𝕜] F) [FiniteDimensional 𝕜 ↥(LinearMap.range f)]
Goal: ∃ g, f = g.comp p.subtypeL
-- Submodule.ClosedComplemented.of_finiteDimensional
lemma Submodule.ClosedComplemented.of_finiteDimensional (p : Submodule 𝕜 F)
    [FiniteDimensional 𝕜 p] : p.ClosedComplemented 
Arguments: {𝕜 : Type u_1} [RCLike 𝕜] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] (p : Submodule 𝕜 F) [FiniteDimensional 𝕜 ↥p]
Goal: p.ClosedComplemented
-- coord_norm'
theorem coord_norm' {x : E} (h : x ≠ 0) : ‖(‖x‖ : 𝕜) • coord 𝕜 x h‖ = 1 
Arguments: (𝕜 : Type v) [RCLike 𝕜] {E : Type u} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {x : E} (h : x ≠ 0)
Goal: ‖↑‖x‖ • ContinuousLinearEquiv.coord 𝕜 x h‖ = 1
-- exists_dual_vector
theorem exists_dual_vector (x : E) (h : x ≠ 0) : ∃ g : E →L[𝕜] 𝕜, ‖g‖ = 1 ∧ g x = ‖x‖ 
Arguments: (𝕜 : Type v) [RCLike 𝕜] {E : Type u} [NormedAddCommGroup E] [NormedSpace 𝕜 E] (x : E) (h : x ≠ 0)
Goal: ∃ g, ‖g‖ = 1 ∧ g x = ↑‖x‖
-- exists_dual_vector'
theorem exists_dual_vector' [Nontrivial E] (x : E) : ∃ g : E →L[𝕜] 𝕜, ‖g‖ = 1 ∧ g x = ‖x‖ 
Arguments: (𝕜 : Type v) [RCLike 𝕜] {E : Type u} [NormedAddCommGroup E] [NormedSpace 𝕜 E] [Nontrivial E] (x : E)
Goal: ∃ g, ‖g‖ = 1 ∧ g x = ↑‖x‖
-- exists_dual_vector''
theorem exists_dual_vector'' (x : E) : ∃ g : E →L[𝕜] 𝕜, ‖g‖ ≤ 1 ∧ g x = ‖x‖ 
Arguments: (𝕜 : Type v) [RCLike 𝕜] {E : Type u} [NormedAddCommGroup E] [NormedSpace 𝕜 E] (x : E)
Goal: ∃ g, ‖g‖ ≤ 1 ∧ g x = ↑‖x‖
-- ContinuousAffineMap.coe_contLinear
@[simp]
theorem coe_contLinear (f : P →ᴬ[R] Q) : (f.contLinear : V → W) = f.linear 
Arguments: {R : Type u_2} {V : Type u_3} {W : Type u_4} {P : Type u_6} {Q : Type u_7} [NormedAddCommGroup V] [MetricSpace P] [NormedAddTorsor V P] [NormedAddCommGroup W] [MetricSpace Q] [NormedAddTorsor W Q] [NormedField R] [NormedSpace R V] [NormedSpace R W] (f : P →ᴬ[R] Q)
Goal: ⇑f.contLinear = ⇑f.linear
-- ContinuousAffineMap.coe_contLinear_eq_linear
@[simp]
theorem coe_contLinear_eq_linear (f : P →ᴬ[R] Q) :
    (f.contLinear : V →ₗ[R] W) = (f : P →ᵃ[R] Q).linear 
Arguments: {R : Type u_2} {V : Type u_3} {W : Type u_4} {P : Type u_6} {Q : Type u_7} [NormedAddCommGroup V] [MetricSpace P] [NormedAddTorsor V P] [NormedAddCommGroup W] [MetricSpace Q] [NormedAddTorsor W Q] [NormedField R] [NormedSpace R V] [NormedSpace R W] (f : P →ᴬ[R] Q)
Goal: ↑f.contLinear = f.linear
-- ContinuousAffineMap.coe_mk_const_linear_eq_linear
@[simp]
theorem coe_mk_const_linear_eq_linear (f : P →ᵃ[R] Q) (h) :
    ((⟨f, h⟩ : P →ᴬ[R] Q).contLinear : V → W) = f.linear 
Arguments: {R : Type u_2} {V : Type u_3} {W : Type u_4} {P : Type u_6} {Q : Type u_7} [NormedAddCommGroup V] [MetricSpace P] [NormedAddTorsor V P] [NormedAddCommGroup W] [MetricSpace Q] [NormedAddTorsor W Q] [NormedField R] [NormedSpace R V] [NormedSpace R W] (f : P →ᵃ[R] Q) (h : Continuous f.toFun)
Goal: ⇑{ toAffineMap := f, cont := h }.contLinear = ⇑f.linear
-- ContinuousAffineMap.coe_linear_eq_coe_contLinear
theorem coe_linear_eq_coe_contLinear (f : P →ᴬ[R] Q) :
    ((f : P →ᵃ[R] Q).linear : V → W) = (⇑f.contLinear : V → W) 
Arguments: {R : Type u_2} {V : Type u_3} {W : Type u_4} {P : Type u_6} {Q : Type u_7} [NormedAddCommGroup V] [MetricSpace P] [NormedAddTorsor V P] [NormedAddCommGroup W] [MetricSpace Q] [NormedAddTorsor W Q] [NormedField R] [NormedSpace R V] [NormedSpace R W] (f : P →ᴬ[R] Q)
Goal: ⇑f.linear = ⇑f.contLinear
-- ContinuousAffineMap.comp_contLinear
@[simp]
theorem comp_contLinear (f : P →ᴬ[R] Q) (g : Q →ᴬ[R] Q₂) :
    (g.comp f).contLinear = g.contLinear.comp f.contLinear 
Arguments: {R : Type u_2} {V : Type u_3} {W : Type u_4} {W₂ : Type u_5} {P : Type u_6} {Q : Type u_7} {Q₂ : Type u_8} [NormedAddCommGroup V] [MetricSpace P] [NormedAddTorsor V P] [NormedAddCommGroup W] [MetricSpace Q] [NormedAddTorsor W Q] [NormedAddCommGroup W₂] [MetricSpace Q₂] [NormedAddTorsor W₂ Q₂] [NormedField R] [NormedSpace R V] [NormedSpace R W] [NormedSpace R W₂] (f : P →ᴬ[R] Q) (g : Q →ᴬ[R] Q₂)
Goal: (g.comp f).contLinear = g.contLinear.comp f.contLinear
-- ContinuousAffineMap.map_vadd
@[simp]
theorem map_vadd (f : P →ᴬ[R] Q) (p : P) (v : V) : f (v +ᵥ p) = f.contLinear v +ᵥ f p 
Arguments: {R : Type u_2} {V : Type u_3} {W : Type u_4} {P : Type u_6} {Q : Type u_7} [NormedAddCommGroup V] [MetricSpace P] [NormedAddTorsor V P] [NormedAddCommGroup W] [MetricSpace Q] [NormedAddTorsor W Q] [NormedField R] [NormedSpace R V] [NormedSpace R W] (f : P →ᴬ[R] Q) (p : P) (v : V)
Goal: f (v +ᵥ p) = f.contLinear v +ᵥ f p
-- ContinuousAffineMap.contLinear_map_vsub
@[simp]
theorem contLinear_map_vsub (f : P →ᴬ[R] Q) (p₁ p₂ : P) : f.contLinear (p₁ -ᵥ p₂) = f p₁ -ᵥ f p₂ 
Arguments: {R : Type u_2} {V : Type u_3} {W : Type u_4} {P : Type u_6} {Q : Type u_7} [NormedAddCommGroup V] [MetricSpace P] [NormedAddTorsor V P] [NormedAddCommGroup W] [MetricSpace Q] [NormedAddTorsor W Q] [NormedField R] [NormedSpace R V] [NormedSpace R W] (f : P →ᴬ[R] Q) (p₁ p₂ : P)
Goal: f.contLinear (p₁ -ᵥ p₂) = f p₁ -ᵥ f p₂
-- ContinuousAffineMap.const_contLinear
@[simp]
theorem const_contLinear (q : Q) : (const R P q).contLinear = 0 
Arguments: {R : Type u_2} {V : Type u_3} {W : Type u_4} {P : Type u_6} {Q : Type u_7} [NormedAddCommGroup V] [MetricSpace P] [NormedAddTorsor V P] [NormedAddCommGroup W] [MetricSpace Q] [NormedAddTorsor W Q] [NormedField R] [NormedSpace R V] [NormedSpace R W] (q : Q)
Goal: (ContinuousAffineMap.const R P q).contLinear = 0
-- ContinuousAffineMap.contLinear_eq_zero_iff_exists_const
theorem contLinear_eq_zero_iff_exists_const (f : P →ᴬ[R] Q) :
    f.contLinear = 0 ↔ ∃ q, f = const R P q 
Arguments: {R : Type u_2} {V : Type u_3} {W : Type u_4} {P : Type u_6} {Q : Type u_7} [NormedAddCommGroup V] [MetricSpace P] [NormedAddTorsor V P] [NormedAddCommGroup W] [MetricSpace Q] [NormedAddTorsor W Q] [NormedField R] [NormedSpace R V] [NormedSpace R W] (f : P →ᴬ[R] Q)
Goal: f.contLinear = 0 ↔ ∃ q, f = ContinuousAffineMap.const R P q
-- ContinuousAffineMap.to_affine_map_contLinear
@[simp]
theorem to_affine_map_contLinear (f : V →L[R] W) : f.toContinuousAffineMap.contLinear = f 
Arguments: {R : Type u_2} {V : Type u_3} {W : Type u_4} [NormedAddCommGroup V] [NormedAddCommGroup W] [NormedField R] [NormedSpace R V] [NormedSpace R W] (f : V →L[R] W)
Goal: f.toContinuousAffineMap.contLinear = f
-- ContinuousAffineMap.zero_contLinear
@[simp]
theorem zero_contLinear : (0 : P →ᴬ[R] W).contLinear = 0 
Arguments: {R : Type u_2} {V : Type u_3} {W : Type u_4} {P : Type u_6} [NormedAddCommGroup V] [MetricSpace P] [NormedAddTorsor V P] [NormedAddCommGroup W] [NormedField R] [NormedSpace R V] [NormedSpace R W]
Goal: ContinuousAffineMap.contLinear 0 = 0
-- ContinuousAffineMap.add_contLinear
@[simp]
theorem add_contLinear (f g : P →ᴬ[R] W) : (f + g).contLinear = f.contLinear + g.contLinear 
Arguments: {R : Type u_2} {V : Type u_3} {W : Type u_4} {P : Type u_6} [NormedAddCommGroup V] [MetricSpace P] [NormedAddTorsor V P] [NormedAddCommGroup W] [NormedField R] [NormedSpace R V] [NormedSpace R W] (f g : P →ᴬ[R] W)
Goal: (f + g).contLinear = f.contLinear + g.contLinear
-- ContinuousAffineMap.sub_contLinear
@[simp]
theorem sub_contLinear (f g : P →ᴬ[R] W) : (f - g).contLinear = f.contLinear - g.contLinear 
Arguments: {R : Type u_2} {V : Type u_3} {W : Type u_4} {P : Type u_6} [NormedAddCommGroup V] [MetricSpace P] [NormedAddTorsor V P] [NormedAddCommGroup W] [NormedField R] [NormedSpace R V] [NormedSpace R W] (f g : P →ᴬ[R] W)
Goal: (f - g).contLinear = f.contLinear - g.contLinear