ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Real.continuous_cosh
@[continuity, fun_prop]
theorem continuous_cosh : Continuous cosh 
Goal: Continuous Real.cosh
-- Real.exists_cos_eq_zero
theorem exists_cos_eq_zero : 0 ∈ cos '' Icc (1 : ℝ) 2 
Goal: 0 ∈ Real.cos '' Set.Icc 1 2
-- Real.cos_pi_div_two
@[simp]
theorem cos_pi_div_two : cos (π / 2) = 0 
Goal: Real.cos (Real.pi / 2) = 0
-- Real.one_le_pi_div_two
theorem one_le_pi_div_two : (1 : ℝ) ≤ π / 2 
Goal: 1 ≤ Real.pi / 2
-- Real.pi_div_two_le_two
theorem pi_div_two_le_two : π / 2 ≤ 2 
Goal: Real.pi / 2 ≤ 2
-- Real.two_le_pi
theorem two_le_pi : (2 : ℝ) ≤ π 
Goal: 2 ≤ Real.pi
-- Real.pi_le_four
theorem pi_le_four : π ≤ 4 
Goal: Real.pi ≤ 4
-- Real.pi_pos
theorem pi_pos : 0 < π 
Goal: 0 < Real.pi
-- Real.pi_nonneg
theorem pi_nonneg : 0 ≤ π 
Goal: 0 ≤ Real.pi
-- Real.pi_ne_zero
theorem pi_ne_zero : π ≠ 0 
Goal: Real.pi ≠ 0
-- Real.pi_div_two_pos
theorem pi_div_two_pos : 0 < π / 2 
Goal: 0 < Real.pi / 2
-- Real.two_pi_pos
theorem two_pi_pos : 0 < 2 * π 
Goal: 0 < 2 * Real.pi
-- NNReal.coe_real_pi
@[simp]
theorem coe_real_pi : (pi : ℝ) = π 
Goal: ↑NNReal.pi = Real.pi
-- NNReal.pi_pos
theorem pi_pos : 0 < pi 
Goal: 0 < NNReal.pi
-- NNReal.pi_ne_zero
theorem pi_ne_zero : pi ≠ 0 
Goal: NNReal.pi ≠ 0
-- Real.sin_pi
@[simp]
theorem sin_pi : sin π = 0 
Goal: Real.sin Real.pi = 0
-- Real.cos_pi
@[simp]
theorem cos_pi : cos π = -1 
Goal: Real.cos Real.pi = -1
-- Real.sin_two_pi
@[simp]
theorem sin_two_pi : sin (2 * π) = 0 
Goal: Real.sin (2 * Real.pi) = 0
-- Real.cos_two_pi
@[simp]
theorem cos_two_pi : cos (2 * π) = 1 
Goal: Real.cos (2 * Real.pi) = 1
-- Real.sin_antiperiodic
theorem sin_antiperiodic : Function.Antiperiodic sin π 
Goal: Function.Antiperiodic Real.sin Real.pi
-- Real.sin_periodic
theorem sin_periodic : Function.Periodic sin (2 * π) 
Goal: Function.Periodic Real.sin (2 * Real.pi)
-- Real.sin_add_pi
@[simp]
theorem sin_add_pi (x : ℝ) : sin (x + π) = -sin x 
Arguments: (x : ℝ)
Goal: Real.sin (x + Real.pi) = -Real.sin x
-- Real.sin_add_two_pi
@[simp]
theorem sin_add_two_pi (x : ℝ) : sin (x + 2 * π) = sin x 
Arguments: (x : ℝ)
Goal: Real.sin (x + 2 * Real.pi) = Real.sin x
-- Real.sin_sub_pi
@[simp]
theorem sin_sub_pi (x : ℝ) : sin (x - π) = -sin x 
Arguments: (x : ℝ)
Goal: Real.sin (x - Real.pi) = -Real.sin x
-- Real.sin_sub_two_pi
@[simp]
theorem sin_sub_two_pi (x : ℝ) : sin (x - 2 * π) = sin x 
Arguments: (x : ℝ)
Goal: Real.sin (x - 2 * Real.pi) = Real.sin x
-- Real.sin_pi_sub
@[simp]
theorem sin_pi_sub (x : ℝ) : sin (π - x) = sin x 
Arguments: (x : ℝ)
Goal: Real.sin (Real.pi - x) = Real.sin x
-- Real.sin_two_pi_sub
@[simp]
theorem sin_two_pi_sub (x : ℝ) : sin (2 * π - x) = -sin x 
Arguments: (x : ℝ)
Goal: Real.sin (2 * Real.pi - x) = -Real.sin x
-- Real.sin_nat_mul_pi
@[simp]
theorem sin_nat_mul_pi (n : ℕ) : sin (n * π) = 0 
Arguments: (n : ℕ)
Goal: Real.sin (↑n * Real.pi) = 0
-- Real.sin_int_mul_pi
@[simp]
theorem sin_int_mul_pi (n : ℤ) : sin (n * π) = 0 
Arguments: (n : ℤ)
Goal: Real.sin (↑n * Real.pi) = 0
-- Real.sin_add_nat_mul_two_pi
@[simp]
theorem sin_add_nat_mul_two_pi (x : ℝ) (n : ℕ) : sin (x + n * (2 * π)) = sin x 
Arguments: (x : ℝ) (n : ℕ)
Goal: Real.sin (x + ↑n * (2 * Real.pi)) = Real.sin x
-- Real.sin_add_int_mul_two_pi
@[simp]
theorem sin_add_int_mul_two_pi (x : ℝ) (n : ℤ) : sin (x + n * (2 * π)) = sin x 
Arguments: (x : ℝ) (n : ℤ)
Goal: Real.sin (x + ↑n * (2 * Real.pi)) = Real.sin x
-- Real.sin_sub_nat_mul_two_pi
@[simp]
theorem sin_sub_nat_mul_two_pi (x : ℝ) (n : ℕ) : sin (x - n * (2 * π)) = sin x 
Arguments: (x : ℝ) (n : ℕ)
Goal: Real.sin (x - ↑n * (2 * Real.pi)) = Real.sin x
-- Real.sin_sub_int_mul_two_pi
@[simp]
theorem sin_sub_int_mul_two_pi (x : ℝ) (n : ℤ) : sin (x - n * (2 * π)) = sin x 
Arguments: (x : ℝ) (n : ℤ)
Goal: Real.sin (x - ↑n * (2 * Real.pi)) = Real.sin x
-- Real.sin_nat_mul_two_pi_sub
@[simp]
theorem sin_nat_mul_two_pi_sub (x : ℝ) (n : ℕ) : sin (n * (2 * π) - x) = -sin x 
Arguments: (x : ℝ) (n : ℕ)
Goal: Real.sin (↑n * (2 * Real.pi) - x) = -Real.sin x
-- Real.sin_int_mul_two_pi_sub
@[simp]
theorem sin_int_mul_two_pi_sub (x : ℝ) (n : ℤ) : sin (n * (2 * π) - x) = -sin x 
Arguments: (x : ℝ) (n : ℤ)
Goal: Real.sin (↑n * (2 * Real.pi) - x) = -Real.sin x
-- Real.sin_add_int_mul_pi
theorem sin_add_int_mul_pi (x : ℝ) (n : ℤ) : sin (x + n * π) = (-1) ^ n * sin x 
Arguments: (x : ℝ) (n : ℤ)
Goal: Real.sin (x + ↑n * Real.pi) = (-1) ^ n * Real.sin x
-- Real.sin_add_nat_mul_pi
theorem sin_add_nat_mul_pi (x : ℝ) (n : ℕ) : sin (x + n * π) = (-1) ^ n * sin x 
Arguments: (x : ℝ) (n : ℕ)
Goal: Real.sin (x + ↑n * Real.pi) = (-1) ^ n * Real.sin x
-- Real.sin_sub_int_mul_pi
theorem sin_sub_int_mul_pi (x : ℝ) (n : ℤ) : sin (x - n * π) = (-1) ^ n * sin x 
Arguments: (x : ℝ) (n : ℤ)
Goal: Real.sin (x - ↑n * Real.pi) = (-1) ^ n * Real.sin x
-- Real.sin_sub_nat_mul_pi
theorem sin_sub_nat_mul_pi (x : ℝ) (n : ℕ) : sin (x - n * π) = (-1) ^ n * sin x 
Arguments: (x : ℝ) (n : ℕ)
Goal: Real.sin (x - ↑n * Real.pi) = (-1) ^ n * Real.sin x
-- Real.sin_int_mul_pi_sub
theorem sin_int_mul_pi_sub (x : ℝ) (n : ℤ) : sin (n * π - x) = -((-1) ^ n * sin x) 
Arguments: (x : ℝ) (n : ℤ)
Goal: Real.sin (↑n * Real.pi - x) = -((-1) ^ n * Real.sin x)
-- Real.sin_nat_mul_pi_sub
theorem sin_nat_mul_pi_sub (x : ℝ) (n : ℕ) : sin (n * π - x) = -((-1) ^ n * sin x) 
Arguments: (x : ℝ) (n : ℕ)
Goal: Real.sin (↑n * Real.pi - x) = -((-1) ^ n * Real.sin x)
-- Real.cos_antiperiodic
theorem cos_antiperiodic : Function.Antiperiodic cos π 
Goal: Function.Antiperiodic Real.cos Real.pi
-- Real.cos_periodic
theorem cos_periodic : Function.Periodic cos (2 * π) 
Goal: Function.Periodic Real.cos (2 * Real.pi)
-- Real.cos_add_pi
@[simp]
theorem cos_add_pi (x : ℝ) : cos (x + π) = -cos x 
Arguments: (x : ℝ)
Goal: Real.cos (x + Real.pi) = -Real.cos x
-- Real.cos_add_two_pi
@[simp]
theorem cos_add_two_pi (x : ℝ) : cos (x + 2 * π) = cos x 
Arguments: (x : ℝ)
Goal: Real.cos (x + 2 * Real.pi) = Real.cos x
-- Real.cos_sub_pi
@[simp]
theorem cos_sub_pi (x : ℝ) : cos (x - π) = -cos x 
Arguments: (x : ℝ)
Goal: Real.cos (x - Real.pi) = -Real.cos x
-- Real.cos_sub_two_pi
@[simp]
theorem cos_sub_two_pi (x : ℝ) : cos (x - 2 * π) = cos x 
Arguments: (x : ℝ)
Goal: Real.cos (x - 2 * Real.pi) = Real.cos x
-- Real.cos_pi_sub
@[simp]
theorem cos_pi_sub (x : ℝ) : cos (π - x) = -cos x 
Arguments: (x : ℝ)
Goal: Real.cos (Real.pi - x) = -Real.cos x
-- Real.cos_two_pi_sub
@[simp]
theorem cos_two_pi_sub (x : ℝ) : cos (2 * π - x) = cos x 
Arguments: (x : ℝ)
Goal: Real.cos (2 * Real.pi - x) = Real.cos x
-- Real.cos_nat_mul_two_pi
@[simp]
theorem cos_nat_mul_two_pi (n : ℕ) : cos (n * (2 * π)) = 1 
Arguments: (n : ℕ)
Goal: Real.cos (↑n * (2 * Real.pi)) = 1
-- Real.cos_int_mul_two_pi
@[simp]
theorem cos_int_mul_two_pi (n : ℤ) : cos (n * (2 * π)) = 1 
Arguments: (n : ℤ)
Goal: Real.cos (↑n * (2 * Real.pi)) = 1
-- Real.cos_add_nat_mul_two_pi
@[simp]
theorem cos_add_nat_mul_two_pi (x : ℝ) (n : ℕ) : cos (x + n * (2 * π)) = cos x 
Arguments: (x : ℝ) (n : ℕ)
Goal: Real.cos (x + ↑n * (2 * Real.pi)) = Real.cos x
-- Real.cos_add_int_mul_two_pi
@[simp]
theorem cos_add_int_mul_two_pi (x : ℝ) (n : ℤ) : cos (x + n * (2 * π)) = cos x 
Arguments: (x : ℝ) (n : ℤ)
Goal: Real.cos (x + ↑n * (2 * Real.pi)) = Real.cos x
-- Real.cos_sub_nat_mul_two_pi
@[simp]
theorem cos_sub_nat_mul_two_pi (x : ℝ) (n : ℕ) : cos (x - n * (2 * π)) = cos x 
Arguments: (x : ℝ) (n : ℕ)
Goal: Real.cos (x - ↑n * (2 * Real.pi)) = Real.cos x
-- Real.cos_sub_int_mul_two_pi
@[simp]
theorem cos_sub_int_mul_two_pi (x : ℝ) (n : ℤ) : cos (x - n * (2 * π)) = cos x 
Arguments: (x : ℝ) (n : ℤ)
Goal: Real.cos (x - ↑n * (2 * Real.pi)) = Real.cos x
-- Real.cos_nat_mul_two_pi_sub
@[simp]
theorem cos_nat_mul_two_pi_sub (x : ℝ) (n : ℕ) : cos (n * (2 * π) - x) = cos x 
Arguments: (x : ℝ) (n : ℕ)
Goal: Real.cos (↑n * (2 * Real.pi) - x) = Real.cos x
-- Real.cos_int_mul_two_pi_sub
@[simp]
theorem cos_int_mul_two_pi_sub (x : ℝ) (n : ℤ) : cos (n * (2 * π) - x) = cos x 
Arguments: (x : ℝ) (n : ℤ)
Goal: Real.cos (↑n * (2 * Real.pi) - x) = Real.cos x
-- Real.cos_add_int_mul_pi
theorem cos_add_int_mul_pi (x : ℝ) (n : ℤ) : cos (x + n * π) = (-1) ^ n * cos x 
Arguments: (x : ℝ) (n : ℤ)
Goal: Real.cos (x + ↑n * Real.pi) = (-1) ^ n * Real.cos x
-- Real.cos_add_nat_mul_pi
theorem cos_add_nat_mul_pi (x : ℝ) (n : ℕ) : cos (x + n * π) = (-1) ^ n * cos x 
Arguments: (x : ℝ) (n : ℕ)
Goal: Real.cos (x + ↑n * Real.pi) = (-1) ^ n * Real.cos x
-- Real.cos_sub_int_mul_pi
theorem cos_sub_int_mul_pi (x : ℝ) (n : ℤ) : cos (x - n * π) = (-1) ^ n * cos x 
Arguments: (x : ℝ) (n : ℤ)
Goal: Real.cos (x - ↑n * Real.pi) = (-1) ^ n * Real.cos x
-- Real.cos_sub_nat_mul_pi
theorem cos_sub_nat_mul_pi (x : ℝ) (n : ℕ) : cos (x - n * π) = (-1) ^ n * cos x 
Arguments: (x : ℝ) (n : ℕ)
Goal: Real.cos (x - ↑n * Real.pi) = (-1) ^ n * Real.cos x
-- Real.cos_int_mul_pi_sub
theorem cos_int_mul_pi_sub (x : ℝ) (n : ℤ) : cos (n * π - x) = (-1) ^ n * cos x 
Arguments: (x : ℝ) (n : ℤ)
Goal: Real.cos (↑n * Real.pi - x) = (-1) ^ n * Real.cos x
-- Real.cos_nat_mul_pi_sub
theorem cos_nat_mul_pi_sub (x : ℝ) (n : ℕ) : cos (n * π - x) = (-1) ^ n * cos x 
Arguments: (x : ℝ) (n : ℕ)
Goal: Real.cos (↑n * Real.pi - x) = (-1) ^ n * Real.cos x
-- Real.cos_nat_mul_two_pi_add_pi
theorem cos_nat_mul_two_pi_add_pi (n : ℕ) : cos (n * (2 * π) + π) = -1 
Arguments: (n : ℕ)
Goal: Real.cos (↑n * (2 * Real.pi) + Real.pi) = -1
-- Real.cos_int_mul_two_pi_add_pi
theorem cos_int_mul_two_pi_add_pi (n : ℤ) : cos (n * (2 * π) + π) = -1 
Arguments: (n : ℤ)
Goal: Real.cos (↑n * (2 * Real.pi) + Real.pi) = -1
-- Real.cos_nat_mul_two_pi_sub_pi
theorem cos_nat_mul_two_pi_sub_pi (n : ℕ) : cos (n * (2 * π) - π) = -1 
Arguments: (n : ℕ)
Goal: Real.cos (↑n * (2 * Real.pi) - Real.pi) = -1
-- Real.cos_int_mul_two_pi_sub_pi
theorem cos_int_mul_two_pi_sub_pi (n : ℤ) : cos (n * (2 * π) - π) = -1 
Arguments: (n : ℤ)
Goal: Real.cos (↑n * (2 * Real.pi) - Real.pi) = -1
-- Real.sin_pos_of_pos_of_lt_pi
theorem sin_pos_of_pos_of_lt_pi {x : ℝ} (h0x : 0 < x) (hxp : x < π) : 0 < sin x 
Arguments: {x : ℝ} (h0x : 0 < x) (hxp : x < Real.pi)
Goal: 0 < Real.sin x
-- Real.sin_pos_of_mem_Ioo
theorem sin_pos_of_mem_Ioo {x : ℝ} (hx : x ∈ Ioo 0 π) : 0 < sin x 
Arguments: {x : ℝ} (hx : x ∈ Set.Ioo 0 Real.pi)
Goal: 0 < Real.sin x
-- Real.sin_nonneg_of_mem_Icc
theorem sin_nonneg_of_mem_Icc {x : ℝ} (hx : x ∈ Icc 0 π) : 0 ≤ sin x 
Arguments: {x : ℝ} (hx : x ∈ Set.Icc 0 Real.pi)
Goal: 0 ≤ Real.sin x
-- Real.sin_nonneg_of_nonneg_of_le_pi
theorem sin_nonneg_of_nonneg_of_le_pi {x : ℝ} (h0x : 0 ≤ x) (hxp : x ≤ π) : 0 ≤ sin x 
Arguments: {x : ℝ} (h0x : 0 ≤ x) (hxp : x ≤ Real.pi)
Goal: 0 ≤ Real.sin x
-- Real.sin_neg_of_neg_of_neg_pi_lt
theorem sin_neg_of_neg_of_neg_pi_lt {x : ℝ} (hx0 : x < 0) (hpx : -π < x) : sin x < 0 
Arguments: {x : ℝ} (hx0 : x < 0) (hpx : -Real.pi < x)
Goal: Real.sin x < 0
-- Real.sin_nonpos_of_nonnpos_of_neg_pi_le
theorem sin_nonpos_of_nonnpos_of_neg_pi_le {x : ℝ} (hx0 : x ≤ 0) (hpx : -π ≤ x) : sin x ≤ 0 
Arguments: {x : ℝ} (hx0 : x ≤ 0) (hpx : -Real.pi ≤ x)
Goal: Real.sin x ≤ 0
-- Real.sin_pi_div_two
@[simp]
theorem sin_pi_div_two : sin (π / 2) = 1 
Goal: Real.sin (Real.pi / 2) = 1
-- Real.sin_add_pi_div_two
theorem sin_add_pi_div_two (x : ℝ) : sin (x + π / 2) = cos x 
Arguments: (x : ℝ)
Goal: Real.sin (x + Real.pi / 2) = Real.cos x
-- Real.sin_sub_pi_div_two
theorem sin_sub_pi_div_two (x : ℝ) : sin (x - π / 2) = -cos x 
Arguments: (x : ℝ)
Goal: Real.sin (x - Real.pi / 2) = -Real.cos x
-- Real.sin_pi_div_two_sub
theorem sin_pi_div_two_sub (x : ℝ) : sin (π / 2 - x) = cos x 
Arguments: (x : ℝ)
Goal: Real.sin (Real.pi / 2 - x) = Real.cos x
-- Real.cos_add_pi_div_two
theorem cos_add_pi_div_two (x : ℝ) : cos (x + π / 2) = -sin x 
Arguments: (x : ℝ)
Goal: Real.cos (x + Real.pi / 2) = -Real.sin x
-- Real.cos_sub_pi_div_two
theorem cos_sub_pi_div_two (x : ℝ) : cos (x - π / 2) = sin x 
Arguments: (x : ℝ)
Goal: Real.cos (x - Real.pi / 2) = Real.sin x
-- Real.cos_pi_div_two_sub
theorem cos_pi_div_two_sub (x : ℝ) : cos (π / 2 - x) = sin x 
Arguments: (x : ℝ)
Goal: Real.cos (Real.pi / 2 - x) = Real.sin x
-- Real.cos_pos_of_mem_Ioo
theorem cos_pos_of_mem_Ioo {x : ℝ} (hx : x ∈ Ioo (-(π / 2)) (π / 2)) : 0 < cos x 
Arguments: {x : ℝ} (hx : x ∈ Set.Ioo (-(Real.pi / 2)) (Real.pi / 2))
Goal: 0 < Real.cos x
-- Real.cos_nonneg_of_mem_Icc
theorem cos_nonneg_of_mem_Icc {x : ℝ} (hx : x ∈ Icc (-(π / 2)) (π / 2)) : 0 ≤ cos x 
Arguments: {x : ℝ} (hx : x ∈ Set.Icc (-(Real.pi / 2)) (Real.pi / 2))
Goal: 0 ≤ Real.cos x
-- Real.cos_nonneg_of_neg_pi_div_two_le_of_le
theorem cos_nonneg_of_neg_pi_div_two_le_of_le {x : ℝ} (hl : -(π / 2) ≤ x) (hu : x ≤ π / 2) :
    0 ≤ cos x 
Arguments: {x : ℝ} (hl : -(Real.pi / 2) ≤ x) (hu : x ≤ Real.pi / 2)
Goal: 0 ≤ Real.cos x
-- Real.cos_neg_of_pi_div_two_lt_of_lt
theorem cos_neg_of_pi_div_two_lt_of_lt {x : ℝ} (hx₁ : π / 2 < x) (hx₂ : x < π + π / 2) :
    cos x < 0 
Arguments: {x : ℝ} (hx₁ : Real.pi / 2 < x) (hx₂ : x < Real.pi + Real.pi / 2)
Goal: Real.cos x < 0
-- Real.cos_nonpos_of_pi_div_two_le_of_le
theorem cos_nonpos_of_pi_div_two_le_of_le {x : ℝ} (hx₁ : π / 2 ≤ x) (hx₂ : x ≤ π + π / 2) :
    cos x ≤ 0 
Arguments: {x : ℝ} (hx₁ : Real.pi / 2 ≤ x) (hx₂ : x ≤ Real.pi + Real.pi / 2)
Goal: Real.cos x ≤ 0
-- Real.sin_eq_sqrt_one_sub_cos_sq
theorem sin_eq_sqrt_one_sub_cos_sq {x : ℝ} (hl : 0 ≤ x) (hu : x ≤ π) :
    sin x = √(1 - cos x ^ 2) 
Arguments: {x : ℝ} (hl : 0 ≤ x) (hu : x ≤ Real.pi)
Goal: Real.sin x = √(1 - Real.cos x ^ 2)
-- Real.cos_eq_sqrt_one_sub_sin_sq
theorem cos_eq_sqrt_one_sub_sin_sq {x : ℝ} (hl : -(π / 2) ≤ x) (hu : x ≤ π / 2) :
    cos x = √(1 - sin x ^ 2) 
Arguments: {x : ℝ} (hl : -(Real.pi / 2) ≤ x) (hu : x ≤ Real.pi / 2)
Goal: Real.cos x = √(1 - Real.sin x ^ 2)
-- Real.cos_half
lemma cos_half {x : ℝ} (hl : -π ≤ x) (hr : x ≤ π) : cos (x / 2) = sqrt ((1 + cos x) / 2) 
Arguments: {x : ℝ} (hl : -Real.pi ≤ x) (hr : x ≤ Real.pi)
Goal: Real.cos (x / 2) = √((1 + Real.cos x) / 2)
-- Real.abs_sin_half
lemma abs_sin_half (x : ℝ) : |sin (x / 2)| = sqrt ((1 - cos x) / 2) 
Arguments: (x : ℝ)
Goal: |Real.sin (x / 2)| = √((1 - Real.cos x) / 2)
-- Real.sin_half_eq_sqrt
lemma sin_half_eq_sqrt {x : ℝ} (hl : 0 ≤ x) (hr : x ≤ 2 * π) :
    sin (x / 2) = sqrt ((1 - cos x) / 2) 
Arguments: {x : ℝ} (hl : 0 ≤ x) (hr : x ≤ 2 * Real.pi)
Goal: Real.sin (x / 2) = √((1 - Real.cos x) / 2)
-- Real.sin_half_eq_neg_sqrt
lemma sin_half_eq_neg_sqrt {x : ℝ} (hl : -(2 * π) ≤ x) (hr : x ≤ 0) :
    sin (x / 2) = -sqrt ((1 - cos x) / 2) 
Arguments: {x : ℝ} (hl : -(2 * Real.pi) ≤ x) (hr : x ≤ 0)
Goal: Real.sin (x / 2) = -√((1 - Real.cos x) / 2)
-- Real.sin_eq_zero_iff_of_lt_of_lt
theorem sin_eq_zero_iff_of_lt_of_lt {x : ℝ} (hx₁ : -π < x) (hx₂ : x < π) : sin x = 0 ↔ x = 0 
Arguments: {x : ℝ} (hx₁ : -Real.pi < x) (hx₂ : x < Real.pi)
Goal: Real.sin x = 0 ↔ x = 0
-- Real.sin_eq_zero_iff
theorem sin_eq_zero_iff {x : ℝ} : sin x = 0 ↔ ∃ n : ℤ, (n : ℝ) * π = x 
Arguments: {x : ℝ}
Goal: Real.sin x = 0 ↔ ∃ n, ↑n * Real.pi = x
-- Real.sin_ne_zero_iff
theorem sin_ne_zero_iff {x : ℝ} : sin x ≠ 0 ↔ ∀ n : ℤ, (n : ℝ) * π ≠ x 
Arguments: {x : ℝ}
Goal: Real.sin x ≠ 0 ↔ ∀ (n : ℤ), ↑n * Real.pi ≠ x
-- Real.sin_eq_zero_iff_cos_eq
theorem sin_eq_zero_iff_cos_eq {x : ℝ} : sin x = 0 ↔ cos x = 1 ∨ cos x = -1 
Arguments: {x : ℝ}
Goal: Real.sin x = 0 ↔ Real.cos x = 1 ∨ Real.cos x = -1
-- Real.cos_eq_one_iff
theorem cos_eq_one_iff (x : ℝ) : cos x = 1 ↔ ∃ n : ℤ, (n : ℝ) * (2 * π) = x 
Arguments: (x : ℝ)
Goal: Real.cos x = 1 ↔ ∃ n, ↑n * (2 * Real.pi) = x
-- Real.cos_eq_one_iff_of_lt_of_lt
theorem cos_eq_one_iff_of_lt_of_lt {x : ℝ} (hx₁ : -(2 * π) < x) (hx₂ : x < 2 * π) :
    cos x = 1 ↔ x = 0 
Arguments: {x : ℝ} (hx₁ : -(2 * Real.pi) < x) (hx₂ : x < 2 * Real.pi)
Goal: Real.cos x = 1 ↔ x = 0
-- Real.sin_lt_sin_of_lt_of_le_pi_div_two
theorem sin_lt_sin_of_lt_of_le_pi_div_two {x y : ℝ} (hx₁ : -(π / 2) ≤ x) (hy₂ : y ≤ π / 2)
    (hxy : x < y) : sin x < sin y 
Arguments: {x y : ℝ} (hx₁ : -(Real.pi / 2) ≤ x) (hy₂ : y ≤ Real.pi / 2) (hxy : x < y)
Goal: Real.sin x < Real.sin y
-- Real.strictMonoOn_sin
theorem strictMonoOn_sin : StrictMonoOn sin (Icc (-(π / 2)) (π / 2)) 
Goal: StrictMonoOn Real.sin (Set.Icc (-(Real.pi / 2)) (Real.pi / 2))
-- Real.cos_lt_cos_of_nonneg_of_le_pi
theorem cos_lt_cos_of_nonneg_of_le_pi {x y : ℝ} (hx₁ : 0 ≤ x) (hy₂ : y ≤ π) (hxy : x < y) :
    cos y < cos x 
Arguments: {x y : ℝ} (hx₁ : 0 ≤ x) (hy₂ : y ≤ Real.pi) (hxy : x < y)
Goal: Real.cos y < Real.cos x
-- Real.cos_lt_cos_of_nonneg_of_le_pi_div_two
theorem cos_lt_cos_of_nonneg_of_le_pi_div_two {x y : ℝ} (hx₁ : 0 ≤ x) (hy₂ : y ≤ π / 2)
    (hxy : x < y) : cos y < cos x 
Arguments: {x y : ℝ} (hx₁ : 0 ≤ x) (hy₂ : y ≤ Real.pi / 2) (hxy : x < y)
Goal: Real.cos y < Real.cos x
-- Real.strictAntiOn_cos
theorem strictAntiOn_cos : StrictAntiOn cos (Icc 0 π) 
Goal: StrictAntiOn Real.cos (Set.Icc 0 Real.pi)
-- Real.cos_le_cos_of_nonneg_of_le_pi
theorem cos_le_cos_of_nonneg_of_le_pi {x y : ℝ} (hx₁ : 0 ≤ x) (hy₂ : y ≤ π) (hxy : x ≤ y) :
    cos y ≤ cos x 
Arguments: {x y : ℝ} (hx₁ : 0 ≤ x) (hy₂ : y ≤ Real.pi) (hxy : x ≤ y)
Goal: Real.cos y ≤ Real.cos x
-- Real.sin_le_sin_of_le_of_le_pi_div_two
theorem sin_le_sin_of_le_of_le_pi_div_two {x y : ℝ} (hx₁ : -(π / 2) ≤ x) (hy₂ : y ≤ π / 2)
    (hxy : x ≤ y) : sin x ≤ sin y 
Arguments: {x y : ℝ} (hx₁ : -(Real.pi / 2) ≤ x) (hy₂ : y ≤ Real.pi / 2) (hxy : x ≤ y)
Goal: Real.sin x ≤ Real.sin y
-- Real.injOn_sin
theorem injOn_sin : InjOn sin (Icc (-(π / 2)) (π / 2)) 
Goal: Set.InjOn Real.sin (Set.Icc (-(Real.pi / 2)) (Real.pi / 2))
-- Real.injOn_cos
theorem injOn_cos : InjOn cos (Icc 0 π) 
Goal: Set.InjOn Real.cos (Set.Icc 0 Real.pi)
-- Real.surjOn_sin
theorem surjOn_sin : SurjOn sin (Icc (-(π / 2)) (π / 2)) (Icc (-1) 1) 
Goal: Set.SurjOn Real.sin (Set.Icc (-(Real.pi / 2)) (Real.pi / 2)) (Set.Icc (-1) 1)
-- Real.surjOn_cos
theorem surjOn_cos : SurjOn cos (Icc 0 π) (Icc (-1) 1) 
Goal: Set.SurjOn Real.cos (Set.Icc 0 Real.pi) (Set.Icc (-1) 1)
-- Real.sin_mem_Icc
theorem sin_mem_Icc (x : ℝ) : sin x ∈ Icc (-1 : ℝ) 1 
Arguments: (x : ℝ)
Goal: Real.sin x ∈ Set.Icc (-1) 1
-- Real.cos_mem_Icc
theorem cos_mem_Icc (x : ℝ) : cos x ∈ Icc (-1 : ℝ) 1 
Arguments: (x : ℝ)
Goal: Real.cos x ∈ Set.Icc (-1) 1
-- Real.mapsTo_sin
theorem mapsTo_sin (s : Set ℝ) : MapsTo sin s (Icc (-1 : ℝ) 1) 
Arguments: (s : Set ℝ)
Goal: Set.MapsTo Real.sin s (Set.Icc (-1) 1)
-- Real.mapsTo_cos
theorem mapsTo_cos (s : Set ℝ) : MapsTo cos s (Icc (-1 : ℝ) 1) 
Arguments: (s : Set ℝ)
Goal: Set.MapsTo Real.cos s (Set.Icc (-1) 1)
-- Real.bijOn_sin
theorem bijOn_sin : BijOn sin (Icc (-(π / 2)) (π / 2)) (Icc (-1) 1) 
Goal: Set.BijOn Real.sin (Set.Icc (-(Real.pi / 2)) (Real.pi / 2)) (Set.Icc (-1) 1)
-- Real.bijOn_cos
theorem bijOn_cos : BijOn cos (Icc 0 π) (Icc (-1) 1) 
Goal: Set.BijOn Real.cos (Set.Icc 0 Real.pi) (Set.Icc (-1) 1)
-- Real.range_cos
@[simp]
theorem range_cos : range cos = (Icc (-1) 1 : Set ℝ) 
Goal: Set.range Real.cos = Set.Icc (-1) 1
-- Real.range_sin
@[simp]
theorem range_sin : range sin = (Icc (-1) 1 : Set ℝ) 
Goal: Set.range Real.sin = Set.Icc (-1) 1
-- Real.range_cos_infinite
theorem range_cos_infinite : (range Real.cos).Infinite 
Goal: (Set.range Real.cos).Infinite
-- Real.range_sin_infinite
theorem range_sin_infinite : (range Real.sin).Infinite 
Goal: (Set.range Real.sin).Infinite
-- Real.sqrtTwoAddSeries_zero
theorem sqrtTwoAddSeries_zero : sqrtTwoAddSeries x 0 = x 
Arguments: (x : ℝ)
Goal: x.sqrtTwoAddSeries 0 = x
-- Real.sqrtTwoAddSeries_one
theorem sqrtTwoAddSeries_one : sqrtTwoAddSeries 0 1 = √2 
Goal: Real.sqrtTwoAddSeries 0 1 = √2
-- Real.sqrtTwoAddSeries_two
theorem sqrtTwoAddSeries_two : sqrtTwoAddSeries 0 2 = √(2 + √2) 
Goal: Real.sqrtTwoAddSeries 0 2 = √(2 + √2)
-- Real.sqrtTwoAddSeries_zero_nonneg
theorem sqrtTwoAddSeries_zero_nonneg : ∀ n : ℕ, 0 ≤ sqrtTwoAddSeries 0 n
  
Arguments: (n : ℕ)
Goal: 0 ≤ Real.sqrtTwoAddSeries 0 n
-- Real.sqrtTwoAddSeries_nonneg
theorem sqrtTwoAddSeries_nonneg {x : ℝ} (h : 0 ≤ x) : ∀ n : ℕ, 0 ≤ sqrtTwoAddSeries x n
  
Arguments: {x : ℝ} (h : 0 ≤ x) (n : ℕ)
Goal: 0 ≤ x.sqrtTwoAddSeries n
-- Real.sqrtTwoAddSeries_lt_two
theorem sqrtTwoAddSeries_lt_two : ∀ n : ℕ, sqrtTwoAddSeries 0 n < 2
  
Arguments: (n : ℕ)
Goal: Real.sqrtTwoAddSeries 0 n < 2
-- Real.sqrtTwoAddSeries_succ
theorem sqrtTwoAddSeries_succ (x : ℝ) :
    ∀ n : ℕ, sqrtTwoAddSeries x (n + 1) = sqrtTwoAddSeries (√(2 + x)) n
  
Arguments: (x : ℝ) (n : ℕ)
Goal: x.sqrtTwoAddSeries (n + 1) = (√(2 + x)).sqrtTwoAddSeries n
-- Real.sqrtTwoAddSeries_monotone_left
theorem sqrtTwoAddSeries_monotone_left {x y : ℝ} (h : x ≤ y) :
    ∀ n : ℕ, sqrtTwoAddSeries x n ≤ sqrtTwoAddSeries y n
  
Arguments: {x y : ℝ} (h : x ≤ y) (n : ℕ)
Goal: x.sqrtTwoAddSeries n ≤ y.sqrtTwoAddSeries n
-- Real.cos_pi_over_two_pow
@[simp]
theorem cos_pi_over_two_pow : ∀ n : ℕ, cos (π / 2 ^ (n + 1)) = sqrtTwoAddSeries 0 n / 2
  
Arguments: (n : ℕ)
Goal: Real.cos (Real.pi / 2 ^ (n + 1)) = Real.sqrtTwoAddSeries 0 n / 2
-- Real.sin_sq_pi_over_two_pow
theorem sin_sq_pi_over_two_pow (n : ℕ) :
    sin (π / 2 ^ (n + 1)) ^ 2 = 1 - (sqrtTwoAddSeries 0 n / 2) ^ 2 
Arguments: (n : ℕ)
Goal: Real.sin (Real.pi / 2 ^ (n + 1)) ^ 2 = 1 - (Real.sqrtTwoAddSeries 0 n / 2) ^ 2
-- Real.sin_sq_pi_over_two_pow_succ
theorem sin_sq_pi_over_two_pow_succ (n : ℕ) :
    sin (π / 2 ^ (n + 2)) ^ 2 = 1 / 2 - sqrtTwoAddSeries 0 n / 4 
Arguments: (n : ℕ)
Goal: Real.sin (Real.pi / 2 ^ (n + 2)) ^ 2 = 1 / 2 - Real.sqrtTwoAddSeries 0 n / 4
-- Real.sin_pi_over_two_pow_succ
@[simp]
theorem sin_pi_over_two_pow_succ (n : ℕ) :
    sin (π / 2 ^ (n + 2)) = √(2 - sqrtTwoAddSeries 0 n) / 2 
Arguments: (n : ℕ)
Goal: Real.sin (Real.pi / 2 ^ (n + 2)) = √(2 - Real.sqrtTwoAddSeries 0 n) / 2
-- Real.cos_pi_div_four
@[simp]
theorem cos_pi_div_four : cos (π / 4) = √2 / 2 
Goal: Real.cos (Real.pi / 4) = √2 / 2
-- Real.sin_pi_div_four
@[simp]
theorem sin_pi_div_four : sin (π / 4) = √2 / 2 
Goal: Real.sin (Real.pi / 4) = √2 / 2
-- Real.cos_pi_div_eight
@[simp]
theorem cos_pi_div_eight : cos (π / 8) = √(2 + √2) / 2 
Goal: Real.cos (Real.pi / 8) = √(2 + √2) / 2
-- Real.sin_pi_div_eight
@[simp]
theorem sin_pi_div_eight : sin (π / 8) = √(2 - √2) / 2 
Goal: Real.sin (Real.pi / 8) = √(2 - √2) / 2
-- Real.cos_pi_div_sixteen
@[simp]
theorem cos_pi_div_sixteen : cos (π / 16) = √(2 + √(2 + √2)) / 2 
Goal: Real.cos (Real.pi / 16) = √(2 + √(2 + √2)) / 2
-- Real.sin_pi_div_sixteen
@[simp]
theorem sin_pi_div_sixteen : sin (π / 16) = √(2 - √(2 + √2)) / 2 
Goal: Real.sin (Real.pi / 16) = √(2 - √(2 + √2)) / 2
-- Real.cos_pi_div_thirty_two
@[simp]
theorem cos_pi_div_thirty_two : cos (π / 32) = √(2 + √(2 + √(2 + √2))) / 2 
Goal: Real.cos (Real.pi / 32) = √(2 + √(2 + √(2 + √2))) / 2
-- Real.sin_pi_div_thirty_two
@[simp]
theorem sin_pi_div_thirty_two : sin (π / 32) = √(2 - √(2 + √(2 + √2))) / 2 
Goal: Real.sin (Real.pi / 32) = √(2 - √(2 + √(2 + √2))) / 2
-- Real.cos_pi_div_three
@[simp]
theorem cos_pi_div_three : cos (π / 3) = 1 / 2 
Goal: Real.cos (Real.pi / 3) = 1 / 2
-- Real.cos_pi_div_six
@[simp]
theorem cos_pi_div_six : cos (π / 6) = √3 / 2 
Goal: Real.cos (Real.pi / 6) = √3 / 2
-- Real.sq_cos_pi_div_six
theorem sq_cos_pi_div_six : cos (π / 6) ^ 2 = 3 / 4 
Goal: Real.cos (Real.pi / 6) ^ 2 = 3 / 4
-- Real.sin_pi_div_six
@[simp]
theorem sin_pi_div_six : sin (π / 6) = 1 / 2 
Goal: Real.sin (Real.pi / 6) = 1 / 2
-- Real.sq_sin_pi_div_three
theorem sq_sin_pi_div_three : sin (π / 3) ^ 2 = 3 / 4 
Goal: Real.sin (Real.pi / 3) ^ 2 = 3 / 4
-- Real.sin_pi_div_three
@[simp]
theorem sin_pi_div_three : sin (π / 3) = √3 / 2 
Goal: Real.sin (Real.pi / 3) = √3 / 2
-- Real.coe_sinOrderIso_apply
@[simp]
theorem coe_sinOrderIso_apply (x : Icc (-(π / 2)) (π / 2)) : (sinOrderIso x : ℝ) = sin x 
Arguments: (x : ↑(Set.Icc (-(Real.pi / 2)) (Real.pi / 2)))
Goal: ↑(Real.sinOrderIso x) = Real.sin ↑x
-- Real.sinOrderIso_apply
theorem sinOrderIso_apply (x : Icc (-(π / 2)) (π / 2)) : sinOrderIso x = ⟨sin x, sin_mem_Icc x⟩ 
Arguments: (x : ↑(Set.Icc (-(Real.pi / 2)) (Real.pi / 2)))
Goal: Real.sinOrderIso x = ⟨Real.sin ↑x, ⋯⟩
-- Real.tan_pi_div_four
@[simp]
theorem tan_pi_div_four : tan (π / 4) = 1 
Goal: Real.tan (Real.pi / 4) = 1
-- Real.tan_pi_div_two
@[simp]
theorem tan_pi_div_two : tan (π / 2) = 0 
Goal: Real.tan (Real.pi / 2) = 0
-- Real.tan_pi_div_six
@[simp]
theorem tan_pi_div_six : tan (π / 6) = 1 / sqrt 3 
Goal: Real.tan (Real.pi / 6) = 1 / √3
-- Real.tan_pi_div_three
@[simp]
theorem tan_pi_div_three : tan (π / 3) = sqrt 3 
Goal: Real.tan (Real.pi / 3) = √3