ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Complex.tanh_mul_I
theorem tanh_mul_I : tanh (x * I) = tan x * I 
Arguments: (x : ℂ)
Goal: Complex.tanh (x * Complex.I) = Complex.tan x * Complex.I
-- Complex.cos_mul_I
theorem cos_mul_I : cos (x * I) = cosh x 
Arguments: (x : ℂ)
Goal: Complex.cos (x * Complex.I) = Complex.cosh x
-- Complex.sin_mul_I
theorem sin_mul_I : sin (x * I) = sinh x * I 
Arguments: (x : ℂ)
Goal: Complex.sin (x * Complex.I) = Complex.sinh x * Complex.I
-- Complex.tan_mul_I
theorem tan_mul_I : tan (x * I) = tanh x * I 
Arguments: (x : ℂ)
Goal: Complex.tan (x * Complex.I) = Complex.tanh x * Complex.I
-- Complex.sin_add
theorem sin_add : sin (x + y) = sin x * cos y + cos x * sin y 
Arguments: (x y : ℂ)
Goal: Complex.sin (x + y) = Complex.sin x * Complex.cos y + Complex.cos x * Complex.sin y
-- Complex.cos_zero
@[simp]
theorem cos_zero : cos 0 = 1 
Goal: Complex.cos 0 = 1
-- Complex.cos_neg
@[simp]
theorem cos_neg : cos (-x) = cos x 
Arguments: (x : ℂ)
Goal: Complex.cos (-x) = Complex.cos x
-- _private.Mathlib.Data.Complex.Exponential.0.Complex.cos_add_aux
private theorem cos_add_aux {a b c d : ℂ} :
    (a + b) * (c + d) - (b - a) * (d - c) * -1 = 2 * (a * c + b * d) 
Arguments: {a b c d : ℂ}
Goal: (a + b) * (c + d) - (b - a) * (d - c) * -1 = 2 * (a * c + b * d)
-- Complex.cos_add
theorem cos_add : cos (x + y) = cos x * cos y - sin x * sin y 
Arguments: (x y : ℂ)
Goal: Complex.cos (x + y) = Complex.cos x * Complex.cos y - Complex.sin x * Complex.sin y
-- Complex.sin_sub
theorem sin_sub : sin (x - y) = sin x * cos y - cos x * sin y 
Arguments: (x y : ℂ)
Goal: Complex.sin (x - y) = Complex.sin x * Complex.cos y - Complex.cos x * Complex.sin y
-- Complex.cos_sub
theorem cos_sub : cos (x - y) = cos x * cos y + sin x * sin y 
Arguments: (x y : ℂ)
Goal: Complex.cos (x - y) = Complex.cos x * Complex.cos y + Complex.sin x * Complex.sin y
-- Complex.sin_add_mul_I
theorem sin_add_mul_I (x y : ℂ) : sin (x + y * I) = sin x * cosh y + cos x * sinh y * I 
Arguments: (x y : ℂ)
Goal: Complex.sin (x + y * Complex.I) = Complex.sin x * Complex.cosh y + Complex.cos x * Complex.sinh y * Complex.I
-- Complex.sin_eq
theorem sin_eq (z : ℂ) : sin z = sin z.re * cosh z.im + cos z.re * sinh z.im * I 
Arguments: (z : ℂ)
Goal: Complex.sin z = Complex.sin ↑z.re * Complex.cosh ↑z.im + Complex.cos ↑z.re * Complex.sinh ↑z.im * Complex.I
-- Complex.cos_add_mul_I
theorem cos_add_mul_I (x y : ℂ) : cos (x + y * I) = cos x * cosh y - sin x * sinh y * I 
Arguments: (x y : ℂ)
Goal: Complex.cos (x + y * Complex.I) = Complex.cos x * Complex.cosh y - Complex.sin x * Complex.sinh y * Complex.I
-- Complex.cos_eq
theorem cos_eq (z : ℂ) : cos z = cos z.re * cosh z.im - sin z.re * sinh z.im * I 
Arguments: (z : ℂ)
Goal: Complex.cos z = Complex.cos ↑z.re * Complex.cosh ↑z.im - Complex.sin ↑z.re * Complex.sinh ↑z.im * Complex.I
-- Complex.sin_sub_sin
theorem sin_sub_sin : sin x - sin y = 2 * sin ((x - y) / 2) * cos ((x + y) / 2) 
Arguments: (x y : ℂ)
Goal: Complex.sin x - Complex.sin y = 2 * Complex.sin ((x - y) / 2) * Complex.cos ((x + y) / 2)
-- Complex.cos_sub_cos
theorem cos_sub_cos : cos x - cos y = -2 * sin ((x + y) / 2) * sin ((x - y) / 2) 
Arguments: (x y : ℂ)
Goal: Complex.cos x - Complex.cos y = -2 * Complex.sin ((x + y) / 2) * Complex.sin ((x - y) / 2)
-- Complex.sin_add_sin
theorem sin_add_sin : sin x + sin y = 2 * sin ((x + y) / 2) * cos ((x - y) / 2) 
Arguments: (x y : ℂ)
Goal: Complex.sin x + Complex.sin y = 2 * Complex.sin ((x + y) / 2) * Complex.cos ((x - y) / 2)
-- Complex.cos_add_cos
theorem cos_add_cos : cos x + cos y = 2 * cos ((x + y) / 2) * cos ((x - y) / 2) 
Arguments: (x y : ℂ)
Goal: Complex.cos x + Complex.cos y = 2 * Complex.cos ((x + y) / 2) * Complex.cos ((x - y) / 2)
-- Complex.sin_conj
theorem sin_conj : sin (conj x) = conj (sin x) 
Arguments: (x : ℂ)
Goal: Complex.sin ((starRingEnd ℂ) x) = (starRingEnd ℂ) (Complex.sin x)
-- Complex.ofReal_sin_ofReal_re
@[simp]
theorem ofReal_sin_ofReal_re (x : ℝ) : ((sin x).re : ℂ) = sin x 
Arguments: (x : ℝ)
Goal: ↑(Complex.sin ↑x).re = Complex.sin ↑x
-- Complex.ofReal_sin
@[simp, norm_cast]
theorem ofReal_sin (x : ℝ) : (Real.sin x : ℂ) = sin x 
Arguments: (x : ℝ)
Goal: ↑(Real.sin x) = Complex.sin ↑x
-- Complex.sin_ofReal_im
@[simp]
theorem sin_ofReal_im (x : ℝ) : (sin x).im = 0 
Arguments: (x : ℝ)
Goal: (Complex.sin ↑x).im = 0
-- Complex.sin_ofReal_re
theorem sin_ofReal_re (x : ℝ) : (sin x).re = Real.sin x 
Arguments: (x : ℝ)
Goal: (Complex.sin ↑x).re = Real.sin x
-- Complex.cos_conj
theorem cos_conj : cos (conj x) = conj (cos x) 
Arguments: (x : ℂ)
Goal: Complex.cos ((starRingEnd ℂ) x) = (starRingEnd ℂ) (Complex.cos x)
-- Complex.ofReal_cos_ofReal_re
@[simp]
theorem ofReal_cos_ofReal_re (x : ℝ) : ((cos x).re : ℂ) = cos x 
Arguments: (x : ℝ)
Goal: ↑(Complex.cos ↑x).re = Complex.cos ↑x
-- Complex.ofReal_cos
@[simp, norm_cast]
theorem ofReal_cos (x : ℝ) : (Real.cos x : ℂ) = cos x 
Arguments: (x : ℝ)
Goal: ↑(Real.cos x) = Complex.cos ↑x
-- Complex.cos_ofReal_im
@[simp]
theorem cos_ofReal_im (x : ℝ) : (cos x).im = 0 
Arguments: (x : ℝ)
Goal: (Complex.cos ↑x).im = 0
-- Complex.cos_ofReal_re
theorem cos_ofReal_re (x : ℝ) : (cos x).re = Real.cos x 
Arguments: (x : ℝ)
Goal: (Complex.cos ↑x).re = Real.cos x
-- Complex.tan_zero
@[simp]
theorem tan_zero : tan 0 = 0 
Goal: Complex.tan 0 = 0
-- Complex.tan_eq_sin_div_cos
theorem tan_eq_sin_div_cos : tan x = sin x / cos x 
Arguments: (x : ℂ)
Goal: Complex.tan x = Complex.sin x / Complex.cos x
-- Complex.cot_eq_cos_div_sin
theorem cot_eq_cos_div_sin : cot x = cos x / sin x 
Arguments: (x : ℂ)
Goal: x.cot = Complex.cos x / Complex.sin x
-- Complex.tan_mul_cos
theorem tan_mul_cos {x : ℂ} (hx : cos x ≠ 0) : tan x * cos x = sin x 
Arguments: {x : ℂ} (hx : Complex.cos x ≠ 0)
Goal: Complex.tan x * Complex.cos x = Complex.sin x
-- Complex.tan_neg
@[simp]
theorem tan_neg : tan (-x) = -tan x 
Arguments: (x : ℂ)
Goal: Complex.tan (-x) = -Complex.tan x
-- Complex.tan_conj
theorem tan_conj : tan (conj x) = conj (tan x) 
Arguments: (x : ℂ)
Goal: Complex.tan ((starRingEnd ℂ) x) = (starRingEnd ℂ) (Complex.tan x)
-- Complex.cot_conj
theorem cot_conj : cot (conj x) = conj (cot x) 
Arguments: (x : ℂ)
Goal: ((starRingEnd ℂ) x).cot = (starRingEnd ℂ) x.cot
-- Complex.ofReal_tan_ofReal_re
@[simp]
theorem ofReal_tan_ofReal_re (x : ℝ) : ((tan x).re : ℂ) = tan x 
Arguments: (x : ℝ)
Goal: ↑(Complex.tan ↑x).re = Complex.tan ↑x
-- Complex.ofReal_cot_ofReal_re
@[simp]
theorem ofReal_cot_ofReal_re (x : ℝ) : ((cot x).re : ℂ) = cot x 
Arguments: (x : ℝ)
Goal: ↑(↑x).cot.re = (↑x).cot
-- Complex.ofReal_tan
@[simp, norm_cast]
theorem ofReal_tan (x : ℝ) : (Real.tan x : ℂ) = tan x 
Arguments: (x : ℝ)
Goal: ↑(Real.tan x) = Complex.tan ↑x
-- Complex.ofReal_cot
@[simp, norm_cast]
theorem ofReal_cot (x : ℝ) : (Real.cot x : ℂ) = cot x 
Arguments: (x : ℝ)
Goal: ↑x.cot = (↑x).cot
-- Complex.tan_ofReal_im
@[simp]
theorem tan_ofReal_im (x : ℝ) : (tan x).im = 0 
Arguments: (x : ℝ)
Goal: (Complex.tan ↑x).im = 0
-- Complex.tan_ofReal_re
theorem tan_ofReal_re (x : ℝ) : (tan x).re = Real.tan x 
Arguments: (x : ℝ)
Goal: (Complex.tan ↑x).re = Real.tan x
-- Complex.cos_add_sin_I
theorem cos_add_sin_I : cos x + sin x * I = exp (x * I) 
Arguments: (x : ℂ)
Goal: Complex.cos x + Complex.sin x * Complex.I = Complex.exp (x * Complex.I)
-- Complex.cos_sub_sin_I
theorem cos_sub_sin_I : cos x - sin x * I = exp (-x * I) 
Arguments: (x : ℂ)
Goal: Complex.cos x - Complex.sin x * Complex.I = Complex.exp (-x * Complex.I)
-- Complex.sin_sq_add_cos_sq
@[simp]
theorem sin_sq_add_cos_sq : sin x ^ 2 + cos x ^ 2 = 1 
Arguments: (x : ℂ)
Goal: Complex.sin x ^ 2 + Complex.cos x ^ 2 = 1
-- Complex.cos_sq_add_sin_sq
@[simp]
theorem cos_sq_add_sin_sq : cos x ^ 2 + sin x ^ 2 = 1 
Arguments: (x : ℂ)
Goal: Complex.cos x ^ 2 + Complex.sin x ^ 2 = 1
-- Complex.cos_two_mul'
theorem cos_two_mul' : cos (2 * x) = cos x ^ 2 - sin x ^ 2 
Arguments: (x : ℂ)
Goal: Complex.cos (2 * x) = Complex.cos x ^ 2 - Complex.sin x ^ 2
-- Complex.cos_two_mul
theorem cos_two_mul : cos (2 * x) = 2 * cos x ^ 2 - 1 
Arguments: (x : ℂ)
Goal: Complex.cos (2 * x) = 2 * Complex.cos x ^ 2 - 1
-- Complex.sin_two_mul
theorem sin_two_mul : sin (2 * x) = 2 * sin x * cos x 
Arguments: (x : ℂ)
Goal: Complex.sin (2 * x) = 2 * Complex.sin x * Complex.cos x
-- Complex.cos_sq
theorem cos_sq : cos x ^ 2 = 1 / 2 + cos (2 * x) / 2 
Arguments: (x : ℂ)
Goal: Complex.cos x ^ 2 = 1 / 2 + Complex.cos (2 * x) / 2
-- Complex.cos_sq'
theorem cos_sq' : cos x ^ 2 = 1 - sin x ^ 2 
Arguments: (x : ℂ)
Goal: Complex.cos x ^ 2 = 1 - Complex.sin x ^ 2
-- Complex.sin_sq
theorem sin_sq : sin x ^ 2 = 1 - cos x ^ 2 
Arguments: (x : ℂ)
Goal: Complex.sin x ^ 2 = 1 - Complex.cos x ^ 2
-- Complex.inv_one_add_tan_sq
theorem inv_one_add_tan_sq {x : ℂ} (hx : cos x ≠ 0) : (1 + tan x ^ 2)⁻¹ = cos x ^ 2 
Arguments: {x : ℂ} (hx : Complex.cos x ≠ 0)
Goal: (1 + Complex.tan x ^ 2)⁻¹ = Complex.cos x ^ 2
-- Complex.tan_sq_div_one_add_tan_sq
theorem tan_sq_div_one_add_tan_sq {x : ℂ} (hx : cos x ≠ 0) :
    tan x ^ 2 / (1 + tan x ^ 2) = sin x ^ 2 
Arguments: {x : ℂ} (hx : Complex.cos x ≠ 0)
Goal: Complex.tan x ^ 2 / (1 + Complex.tan x ^ 2) = Complex.sin x ^ 2
-- Complex.cos_three_mul
theorem cos_three_mul : cos (3 * x) = 4 * cos x ^ 3 - 3 * cos x 
Arguments: (x : ℂ)
Goal: Complex.cos (3 * x) = 4 * Complex.cos x ^ 3 - 3 * Complex.cos x
-- Complex.sin_three_mul
theorem sin_three_mul : sin (3 * x) = 3 * sin x - 4 * sin x ^ 3 
Arguments: (x : ℂ)
Goal: Complex.sin (3 * x) = 3 * Complex.sin x - 4 * Complex.sin x ^ 3
-- Complex.exp_mul_I
theorem exp_mul_I : exp (x * I) = cos x + sin x * I 
Arguments: (x : ℂ)
Goal: Complex.exp (x * Complex.I) = Complex.cos x + Complex.sin x * Complex.I
-- Complex.exp_add_mul_I
theorem exp_add_mul_I : exp (x + y * I) = exp x * (cos y + sin y * I) 
Arguments: (x y : ℂ)
Goal: Complex.exp (x + y * Complex.I) = Complex.exp x * (Complex.cos y + Complex.sin y * Complex.I)
-- Complex.exp_eq_exp_re_mul_sin_add_cos
theorem exp_eq_exp_re_mul_sin_add_cos : exp x = exp x.re * (cos x.im + sin x.im * I) 
Arguments: (x : ℂ)
Goal: Complex.exp x = Complex.exp ↑x.re * (Complex.cos ↑x.im + Complex.sin ↑x.im * Complex.I)
-- Complex.exp_re
theorem exp_re : (exp x).re = Real.exp x.re * Real.cos x.im 
Arguments: (x : ℂ)
Goal: (Complex.exp x).re = Real.exp x.re * Real.cos x.im
-- Complex.exp_im
theorem exp_im : (exp x).im = Real.exp x.re * Real.sin x.im 
Arguments: (x : ℂ)
Goal: (Complex.exp x).im = Real.exp x.re * Real.sin x.im
-- Complex.exp_ofReal_mul_I_re
@[simp]
theorem exp_ofReal_mul_I_re (x : ℝ) : (exp (x * I)).re = Real.cos x 
Arguments: (x : ℝ)
Goal: (Complex.exp (↑x * Complex.I)).re = Real.cos x
-- Complex.exp_ofReal_mul_I_im
@[simp]
theorem exp_ofReal_mul_I_im (x : ℝ) : (exp (x * I)).im = Real.sin x 
Arguments: (x : ℝ)
Goal: (Complex.exp (↑x * Complex.I)).im = Real.sin x
-- Complex.cos_add_sin_mul_I_pow
theorem cos_add_sin_mul_I_pow (n : ℕ) (z : ℂ) :
    (cos z + sin z * I) ^ n = cos (↑n * z) + sin (↑n * z) * I 
Arguments: (n : ℕ) (z : ℂ)
Goal: (Complex.cos z + Complex.sin z * Complex.I) ^ n = Complex.cos (↑n * z) + Complex.sin (↑n * z) * Complex.I
-- Real.exp_zero
@[simp]
theorem exp_zero : exp 0 = 1 
Goal: Real.exp 0 = 1
-- Real.exp_add
nonrec theorem exp_add : exp (x + y) = exp x * exp y 
Arguments: (x y : ℝ)
Goal: Real.exp (x + y) = Real.exp x * Real.exp y
-- Real.exp_list_sum
theorem exp_list_sum (l : List ℝ) : exp l.sum = (l.map exp).prod 
Arguments: (l : List ℝ)
Goal: Real.exp l.sum = (List.map Real.exp l).prod
-- Real.exp_multiset_sum
theorem exp_multiset_sum (s : Multiset ℝ) : exp s.sum = (s.map exp).prod 
Arguments: (s : Multiset ℝ)
Goal: Real.exp s.sum = (Multiset.map Real.exp s).prod
-- Real.exp_sum
theorem exp_sum {α : Type*} (s : Finset α) (f : α → ℝ) :
    exp (∑ x ∈ s, f x) = ∏ x ∈ s, exp (f x) 
Arguments: {α : Type u_1} (s : Finset α) (f : α → ℝ)
Goal: Real.exp (∑ x ∈ s, f x) = ∏ x ∈ s, Real.exp (f x)
-- Real.exp_nsmul
lemma exp_nsmul (x : ℝ) (n : ℕ) : exp (n • x) = exp x ^ n 
Arguments: (x : ℝ) (n : ℕ)
Goal: Real.exp (n • x) = Real.exp x ^ n
-- Real.exp_nat_mul
nonrec theorem exp_nat_mul (x : ℝ) (n : ℕ) : exp (n * x) = exp x ^ n 
Arguments: (x : ℝ) (n : ℕ)
Goal: Real.exp (↑n * x) = Real.exp x ^ n
-- Real.exp_ne_zero
@[simp]
nonrec theorem exp_ne_zero : exp x ≠ 0 
Arguments: (x : ℝ)
Goal: Real.exp x ≠ 0
-- Real.exp_neg
nonrec theorem exp_neg : exp (-x) = (exp x)⁻¹ 
Arguments: (x : ℝ)
Goal: Real.exp (-x) = (Real.exp x)⁻¹
-- Real.exp_sub
theorem exp_sub : exp (x - y) = exp x / exp y 
Arguments: (x y : ℝ)
Goal: Real.exp (x - y) = Real.exp x / Real.exp y
-- Real.sin_zero
@[simp]
theorem sin_zero : sin 0 = 0 
Goal: Real.sin 0 = 0
-- Real.sin_neg
@[simp]
theorem sin_neg : sin (-x) = -sin x 
Arguments: (x : ℝ)
Goal: Real.sin (-x) = -Real.sin x
-- Real.sin_add
nonrec theorem sin_add : sin (x + y) = sin x * cos y + cos x * sin y 
Arguments: (x y : ℝ)
Goal: Real.sin (x + y) = Real.sin x * Real.cos y + Real.cos x * Real.sin y
-- Real.cos_zero
@[simp]
theorem cos_zero : cos 0 = 1 
Goal: Real.cos 0 = 1
-- Real.cos_neg
@[simp]
theorem cos_neg : cos (-x) = cos x 
Arguments: (x : ℝ)
Goal: Real.cos (-x) = Real.cos x
-- Real.cos_abs
@[simp]
theorem cos_abs : cos |x| = cos x 
Arguments: (x : ℝ)
Goal: Real.cos |x| = Real.cos x
-- Real.cos_add
nonrec theorem cos_add : cos (x + y) = cos x * cos y - sin x * sin y 
Arguments: (x y : ℝ)
Goal: Real.cos (x + y) = Real.cos x * Real.cos y - Real.sin x * Real.sin y
-- Real.sin_sub
theorem sin_sub : sin (x - y) = sin x * cos y - cos x * sin y 
Arguments: (x y : ℝ)
Goal: Real.sin (x - y) = Real.sin x * Real.cos y - Real.cos x * Real.sin y
-- Real.cos_sub
theorem cos_sub : cos (x - y) = cos x * cos y + sin x * sin y 
Arguments: (x y : ℝ)
Goal: Real.cos (x - y) = Real.cos x * Real.cos y + Real.sin x * Real.sin y
-- Real.sin_sub_sin
nonrec theorem sin_sub_sin : sin x - sin y = 2 * sin ((x - y) / 2) * cos ((x + y) / 2) 
Arguments: (x y : ℝ)
Goal: Real.sin x - Real.sin y = 2 * Real.sin ((x - y) / 2) * Real.cos ((x + y) / 2)
-- Real.cos_sub_cos
nonrec theorem cos_sub_cos : cos x - cos y = -2 * sin ((x + y) / 2) * sin ((x - y) / 2) 
Arguments: (x y : ℝ)
Goal: Real.cos x - Real.cos y = -2 * Real.sin ((x + y) / 2) * Real.sin ((x - y) / 2)
-- Real.cos_add_cos
nonrec theorem cos_add_cos : cos x + cos y = 2 * cos ((x + y) / 2) * cos ((x - y) / 2) 
Arguments: (x y : ℝ)
Goal: Real.cos x + Real.cos y = 2 * Real.cos ((x + y) / 2) * Real.cos ((x - y) / 2)
-- Real.tan_eq_sin_div_cos
nonrec theorem tan_eq_sin_div_cos : tan x = sin x / cos x 
Arguments: (x : ℝ)
Goal: Real.tan x = Real.sin x / Real.cos x
-- Real.cot_eq_cos_div_sin
nonrec theorem cot_eq_cos_div_sin : cot x = cos x / sin x 
Arguments: (x : ℝ)
Goal: x.cot = Real.cos x / Real.sin x
-- Real.tan_mul_cos
theorem tan_mul_cos {x : ℝ} (hx : cos x ≠ 0) : tan x * cos x = sin x 
Arguments: {x : ℝ} (hx : Real.cos x ≠ 0)
Goal: Real.tan x * Real.cos x = Real.sin x
-- Real.tan_zero
@[simp]
theorem tan_zero : tan 0 = 0 
Goal: Real.tan 0 = 0
-- Real.tan_neg
@[simp]
theorem tan_neg : tan (-x) = -tan x 
Arguments: (x : ℝ)
Goal: Real.tan (-x) = -Real.tan x
-- Real.sin_sq_add_cos_sq
@[simp]
nonrec theorem sin_sq_add_cos_sq : sin x ^ 2 + cos x ^ 2 = 1 
Arguments: (x : ℝ)
Goal: Real.sin x ^ 2 + Real.cos x ^ 2 = 1
-- Real.cos_sq_add_sin_sq
@[simp]
theorem cos_sq_add_sin_sq : cos x ^ 2 + sin x ^ 2 = 1 
Arguments: (x : ℝ)
Goal: Real.cos x ^ 2 + Real.sin x ^ 2 = 1
-- Real.sin_sq_le_one
theorem sin_sq_le_one : sin x ^ 2 ≤ 1 
Arguments: (x : ℝ)
Goal: Real.sin x ^ 2 ≤ 1
-- Real.cos_sq_le_one
theorem cos_sq_le_one : cos x ^ 2 ≤ 1 
Arguments: (x : ℝ)
Goal: Real.cos x ^ 2 ≤ 1
-- Real.abs_sin_le_one
theorem abs_sin_le_one : |sin x| ≤ 1 
Arguments: (x : ℝ)
Goal: |Real.sin x| ≤ 1
-- Real.abs_cos_le_one
theorem abs_cos_le_one : |cos x| ≤ 1 
Arguments: (x : ℝ)
Goal: |Real.cos x| ≤ 1
-- Real.sin_le_one
theorem sin_le_one : sin x ≤ 1 
Arguments: (x : ℝ)
Goal: Real.sin x ≤ 1
-- Real.cos_le_one
theorem cos_le_one : cos x ≤ 1 
Arguments: (x : ℝ)
Goal: Real.cos x ≤ 1
-- Real.neg_one_le_sin
theorem neg_one_le_sin : -1 ≤ sin x 
Arguments: (x : ℝ)
Goal: -1 ≤ Real.sin x
-- Real.neg_one_le_cos
theorem neg_one_le_cos : -1 ≤ cos x 
Arguments: (x : ℝ)
Goal: -1 ≤ Real.cos x
-- Real.cos_two_mul
nonrec theorem cos_two_mul : cos (2 * x) = 2 * cos x ^ 2 - 1 
Arguments: (x : ℝ)
Goal: Real.cos (2 * x) = 2 * Real.cos x ^ 2 - 1
-- Real.cos_two_mul'
nonrec theorem cos_two_mul' : cos (2 * x) = cos x ^ 2 - sin x ^ 2 
Arguments: (x : ℝ)
Goal: Real.cos (2 * x) = Real.cos x ^ 2 - Real.sin x ^ 2
-- Real.sin_two_mul
nonrec theorem sin_two_mul : sin (2 * x) = 2 * sin x * cos x 
Arguments: (x : ℝ)
Goal: Real.sin (2 * x) = 2 * Real.sin x * Real.cos x
-- Real.cos_sq
nonrec theorem cos_sq : cos x ^ 2 = 1 / 2 + cos (2 * x) / 2 
Arguments: (x : ℝ)
Goal: Real.cos x ^ 2 = 1 / 2 + Real.cos (2 * x) / 2
-- Real.cos_sq'
theorem cos_sq' : cos x ^ 2 = 1 - sin x ^ 2 
Arguments: (x : ℝ)
Goal: Real.cos x ^ 2 = 1 - Real.sin x ^ 2
-- Real.sin_sq
theorem sin_sq : sin x ^ 2 = 1 - cos x ^ 2 
Arguments: (x : ℝ)
Goal: Real.sin x ^ 2 = 1 - Real.cos x ^ 2
-- Real.sin_sq_eq_half_sub
lemma sin_sq_eq_half_sub : sin x ^ 2 = 1 / 2 - cos (2 * x) / 2 
Arguments: (x : ℝ)
Goal: Real.sin x ^ 2 = 1 / 2 - Real.cos (2 * x) / 2
-- Real.abs_sin_eq_sqrt_one_sub_cos_sq
theorem abs_sin_eq_sqrt_one_sub_cos_sq (x : ℝ) : |sin x| = √(1 - cos x ^ 2) 
Arguments: (x : ℝ)
Goal: |Real.sin x| = √(1 - Real.cos x ^ 2)
-- Real.abs_cos_eq_sqrt_one_sub_sin_sq
theorem abs_cos_eq_sqrt_one_sub_sin_sq (x : ℝ) : |cos x| = √(1 - sin x ^ 2) 
Arguments: (x : ℝ)
Goal: |Real.cos x| = √(1 - Real.sin x ^ 2)
-- Real.inv_one_add_tan_sq
theorem inv_one_add_tan_sq {x : ℝ} (hx : cos x ≠ 0) : (1 + tan x ^ 2)⁻¹ = cos x ^ 2 
Arguments: {x : ℝ} (hx : Real.cos x ≠ 0)
Goal: (1 + Real.tan x ^ 2)⁻¹ = Real.cos x ^ 2
-- Real.tan_sq_div_one_add_tan_sq
theorem tan_sq_div_one_add_tan_sq {x : ℝ} (hx : cos x ≠ 0) :
    tan x ^ 2 / (1 + tan x ^ 2) = sin x ^ 2 
Arguments: {x : ℝ} (hx : Real.cos x ≠ 0)
Goal: Real.tan x ^ 2 / (1 + Real.tan x ^ 2) = Real.sin x ^ 2
-- Real.inv_sqrt_one_add_tan_sq
theorem inv_sqrt_one_add_tan_sq {x : ℝ} (hx : 0 < cos x) : (√(1 + tan x ^ 2))⁻¹ = cos x 
Arguments: {x : ℝ} (hx : 0 < Real.cos x)
Goal: (√(1 + Real.tan x ^ 2))⁻¹ = Real.cos x
-- Real.tan_div_sqrt_one_add_tan_sq
theorem tan_div_sqrt_one_add_tan_sq {x : ℝ} (hx : 0 < cos x) :
    tan x / √(1 + tan x ^ 2) = sin x 
Arguments: {x : ℝ} (hx : 0 < Real.cos x)
Goal: Real.tan x / √(1 + Real.tan x ^ 2) = Real.sin x
-- Real.cos_three_mul
nonrec theorem cos_three_mul : cos (3 * x) = 4 * cos x ^ 3 - 3 * cos x 
Arguments: (x : ℝ)
Goal: Real.cos (3 * x) = 4 * Real.cos x ^ 3 - 3 * Real.cos x
-- Real.sin_three_mul
nonrec theorem sin_three_mul : sin (3 * x) = 3 * sin x - 4 * sin x ^ 3 
Arguments: (x : ℝ)
Goal: Real.sin (3 * x) = 3 * Real.sin x - 4 * Real.sin x ^ 3
-- Real.sinh_eq
nonrec theorem sinh_eq (x : ℝ) : sinh x = (exp x - exp (-x)) / 2 
Arguments: (x : ℝ)
Goal: Real.sinh x = (Real.exp x - Real.exp (-x)) / 2
-- Real.sinh_zero
@[simp]
theorem sinh_zero : sinh 0 = 0 
Goal: Real.sinh 0 = 0
-- Real.sinh_neg
@[simp]
theorem sinh_neg : sinh (-x) = -sinh x 
Arguments: (x : ℝ)
Goal: Real.sinh (-x) = -Real.sinh x
-- Real.sinh_add
nonrec theorem sinh_add : sinh (x + y) = sinh x * cosh y + cosh x * sinh y 
Arguments: (x y : ℝ)
Goal: Real.sinh (x + y) = Real.sinh x * Real.cosh y + Real.cosh x * Real.sinh y
-- Real.cosh_eq
theorem cosh_eq (x : ℝ) : cosh x = (exp x + exp (-x)) / 2 
Arguments: (x : ℝ)
Goal: Real.cosh x = (Real.exp x + Real.exp (-x)) / 2
-- Real.cosh_zero
@[simp]
theorem cosh_zero : cosh 0 = 1 
Goal: Real.cosh 0 = 1
-- Real.cosh_neg
@[simp]
theorem cosh_neg : cosh (-x) = cosh x 
Arguments: (x : ℝ)
Goal: Real.cosh (-x) = Real.cosh x
-- Real.cosh_abs
@[simp]
theorem cosh_abs : cosh |x| = cosh x 
Arguments: (x : ℝ)
Goal: Real.cosh |x| = Real.cosh x
-- Real.cosh_add
nonrec theorem cosh_add : cosh (x + y) = cosh x * cosh y + sinh x * sinh y 
Arguments: (x y : ℝ)
Goal: Real.cosh (x + y) = Real.cosh x * Real.cosh y + Real.sinh x * Real.sinh y
-- Real.sinh_sub
theorem sinh_sub : sinh (x - y) = sinh x * cosh y - cosh x * sinh y 
Arguments: (x y : ℝ)
Goal: Real.sinh (x - y) = Real.sinh x * Real.cosh y - Real.cosh x * Real.sinh y
-- Real.cosh_sub
theorem cosh_sub : cosh (x - y) = cosh x * cosh y - sinh x * sinh y 
Arguments: (x y : ℝ)
Goal: Real.cosh (x - y) = Real.cosh x * Real.cosh y - Real.sinh x * Real.sinh y
-- Real.tanh_eq_sinh_div_cosh
nonrec theorem tanh_eq_sinh_div_cosh : tanh x = sinh x / cosh x 
Arguments: (x : ℝ)
Goal: Real.tanh x = Real.sinh x / Real.cosh x
-- Real.tanh_zero
@[simp]
theorem tanh_zero : tanh 0 = 0 
Goal: Real.tanh 0 = 0
-- Real.tanh_neg
@[simp]
theorem tanh_neg : tanh (-x) = -tanh x 
Arguments: (x : ℝ)
Goal: Real.tanh (-x) = -Real.tanh x
-- Real.cosh_add_sinh
@[simp]
theorem cosh_add_sinh : cosh x + sinh x = exp x 
Arguments: (x : ℝ)
Goal: Real.cosh x + Real.sinh x = Real.exp x
-- Real.sinh_add_cosh
@[simp]
theorem sinh_add_cosh : sinh x + cosh x = exp x 
Arguments: (x : ℝ)
Goal: Real.sinh x + Real.cosh x = Real.exp x
-- Real.exp_sub_cosh
@[simp]
theorem exp_sub_cosh : exp x - cosh x = sinh x 
Arguments: (x : ℝ)
Goal: Real.exp x - Real.cosh x = Real.sinh x
-- Real.exp_sub_sinh
@[simp]
theorem exp_sub_sinh : exp x - sinh x = cosh x 
Arguments: (x : ℝ)
Goal: Real.exp x - Real.sinh x = Real.cosh x
-- Real.cosh_sub_sinh
@[simp]
theorem cosh_sub_sinh : cosh x - sinh x = exp (-x) 
Arguments: (x : ℝ)
Goal: Real.cosh x - Real.sinh x = Real.exp (-x)
-- Real.sinh_sub_cosh
@[simp]
theorem sinh_sub_cosh : sinh x - cosh x = -exp (-x) 
Arguments: (x : ℝ)
Goal: Real.sinh x - Real.cosh x = -Real.exp (-x)
-- Real.cosh_sq_sub_sinh_sq
@[simp]
theorem cosh_sq_sub_sinh_sq (x : ℝ) : cosh x ^ 2 - sinh x ^ 2 = 1 
Arguments: (x : ℝ)
Goal: Real.cosh x ^ 2 - Real.sinh x ^ 2 = 1
-- Real.cosh_sq
nonrec theorem cosh_sq : cosh x ^ 2 = sinh x ^ 2 + 1 
Arguments: (x : ℝ)
Goal: Real.cosh x ^ 2 = Real.sinh x ^ 2 + 1
-- Real.cosh_sq'
theorem cosh_sq' : cosh x ^ 2 = 1 + sinh x ^ 2 
Arguments: (x : ℝ)
Goal: Real.cosh x ^ 2 = 1 + Real.sinh x ^ 2
-- Real.sinh_sq
nonrec theorem sinh_sq : sinh x ^ 2 = cosh x ^ 2 - 1 
Arguments: (x : ℝ)
Goal: Real.sinh x ^ 2 = Real.cosh x ^ 2 - 1
-- Real.cosh_two_mul
nonrec theorem cosh_two_mul : cosh (2 * x) = cosh x ^ 2 + sinh x ^ 2 
Arguments: (x : ℝ)
Goal: Real.cosh (2 * x) = Real.cosh x ^ 2 + Real.sinh x ^ 2
-- Real.sinh_two_mul
nonrec theorem sinh_two_mul : sinh (2 * x) = 2 * sinh x * cosh x 
Arguments: (x : ℝ)
Goal: Real.sinh (2 * x) = 2 * Real.sinh x * Real.cosh x
-- Real.cosh_three_mul
nonrec theorem cosh_three_mul : cosh (3 * x) = 4 * cosh x ^ 3 - 3 * cosh x 
Arguments: (x : ℝ)
Goal: Real.cosh (3 * x) = 4 * Real.cosh x ^ 3 - 3 * Real.cosh x
-- Real.sinh_three_mul
nonrec theorem sinh_three_mul : sinh (3 * x) = 4 * sinh x ^ 3 + 3 * sinh x 
Arguments: (x : ℝ)
Goal: Real.sinh (3 * x) = 4 * Real.sinh x ^ 3 + 3 * Real.sinh x
-- Real.sum_le_exp_of_nonneg
theorem sum_le_exp_of_nonneg {x : ℝ} (hx : 0 ≤ x) (n : ℕ) : ∑ i ∈ range n, x ^ i / i ! ≤ exp x 
Arguments: {x : ℝ} (hx : 0 ≤ x) (n : ℕ)
Goal: ∑ i ∈ Finset.range n, x ^ i / ↑i.factorial ≤ Real.exp x
-- Real.pow_div_factorial_le_exp
lemma pow_div_factorial_le_exp (hx : 0 ≤ x) (n : ℕ) : x ^ n / n ! ≤ exp x 
Arguments: (x : ℝ) (hx : 0 ≤ x) (n : ℕ)
Goal: x ^ n / ↑n.factorial ≤ Real.exp x
-- Real.quadratic_le_exp_of_nonneg
theorem quadratic_le_exp_of_nonneg {x : ℝ} (hx : 0 ≤ x) : 1 + x + x ^ 2 / 2 ≤ exp x 
Arguments: {x : ℝ} (hx : 0 ≤ x)
Goal: 1 + x + x ^ 2 / 2 ≤ Real.exp x
-- _private.Mathlib.Data.Complex.Exponential.0.Real.add_one_lt_exp_of_pos
private theorem add_one_lt_exp_of_pos {x : ℝ} (hx : 0 < x) : x + 1 < exp x 
Arguments: {x : ℝ} (hx : 0 < x)
Goal: x + 1 < Real.exp x
-- _private.Mathlib.Data.Complex.Exponential.0.Real.add_one_le_exp_of_nonneg
private theorem add_one_le_exp_of_nonneg {x : ℝ} (hx : 0 ≤ x) : x + 1 ≤ exp x 
Arguments: {x : ℝ} (hx : 0 ≤ x)
Goal: x + 1 ≤ Real.exp x
-- Real.one_le_exp
theorem one_le_exp {x : ℝ} (hx : 0 ≤ x) : 1 ≤ exp x 
Arguments: {x : ℝ} (hx : 0 ≤ x)
Goal: 1 ≤ Real.exp x
-- Real.exp_pos
theorem exp_pos (x : ℝ) : 0 < exp x 
Arguments: (x : ℝ)
Goal: 0 < Real.exp x
-- Real.exp_nonneg
lemma exp_nonneg (x : ℝ) : 0 ≤ exp x 
Arguments: (x : ℝ)
Goal: 0 ≤ Real.exp x
-- Real.abs_exp
@[simp]
theorem abs_exp (x : ℝ) : |exp x| = exp x 
Arguments: (x : ℝ)
Goal: |Real.exp x| = Real.exp x
-- Real.exp_abs_le
lemma exp_abs_le (x : ℝ) : exp |x| ≤ exp x + exp (-x) 
Arguments: (x : ℝ)
Goal: Real.exp |x| ≤ Real.exp x + Real.exp (-x)
-- Real.exp_strictMono
@[mono]
theorem exp_strictMono : StrictMono exp 
Goal: StrictMono Real.exp