ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Real.tan_pos_of_pos_of_lt_pi_div_two
theorem tan_pos_of_pos_of_lt_pi_div_two {x : ℝ} (h0x : 0 < x) (hxp : x < π / 2) : 0 < tan x 
Arguments: {x : ℝ} (h0x : 0 < x) (hxp : x < Real.pi / 2)
Goal: 0 < Real.tan x
-- Real.tan_nonneg_of_nonneg_of_le_pi_div_two
theorem tan_nonneg_of_nonneg_of_le_pi_div_two {x : ℝ} (h0x : 0 ≤ x) (hxp : x ≤ π / 2) : 0 ≤ tan x 
Arguments: {x : ℝ} (h0x : 0 ≤ x) (hxp : x ≤ Real.pi / 2)
Goal: 0 ≤ Real.tan x
-- Real.tan_neg_of_neg_of_pi_div_two_lt
theorem tan_neg_of_neg_of_pi_div_two_lt {x : ℝ} (hx0 : x < 0) (hpx : -(π / 2) < x) : tan x < 0 
Arguments: {x : ℝ} (hx0 : x < 0) (hpx : -(Real.pi / 2) < x)
Goal: Real.tan x < 0
-- Real.tan_nonpos_of_nonpos_of_neg_pi_div_two_le
theorem tan_nonpos_of_nonpos_of_neg_pi_div_two_le {x : ℝ} (hx0 : x ≤ 0) (hpx : -(π / 2) ≤ x) :
    tan x ≤ 0 
Arguments: {x : ℝ} (hx0 : x ≤ 0) (hpx : -(Real.pi / 2) ≤ x)
Goal: Real.tan x ≤ 0
-- Real.strictMonoOn_tan
theorem strictMonoOn_tan : StrictMonoOn tan (Ioo (-(π / 2)) (π / 2)) 
Goal: StrictMonoOn Real.tan (Set.Ioo (-(Real.pi / 2)) (Real.pi / 2))
-- Real.tan_lt_tan_of_lt_of_lt_pi_div_two
theorem tan_lt_tan_of_lt_of_lt_pi_div_two {x y : ℝ} (hx₁ : -(π / 2) < x) (hy₂ : y < π / 2)
    (hxy : x < y) : tan x < tan y 
Arguments: {x y : ℝ} (hx₁ : -(Real.pi / 2) < x) (hy₂ : y < Real.pi / 2) (hxy : x < y)
Goal: Real.tan x < Real.tan y
-- Real.tan_lt_tan_of_nonneg_of_lt_pi_div_two
theorem tan_lt_tan_of_nonneg_of_lt_pi_div_two {x y : ℝ} (hx₁ : 0 ≤ x) (hy₂ : y < π / 2)
    (hxy : x < y) : tan x < tan y 
Arguments: {x y : ℝ} (hx₁ : 0 ≤ x) (hy₂ : y < Real.pi / 2) (hxy : x < y)
Goal: Real.tan x < Real.tan y
-- Real.injOn_tan
theorem injOn_tan : InjOn tan (Ioo (-(π / 2)) (π / 2)) 
Goal: Set.InjOn Real.tan (Set.Ioo (-(Real.pi / 2)) (Real.pi / 2))
-- Real.tan_inj_of_lt_of_lt_pi_div_two
theorem tan_inj_of_lt_of_lt_pi_div_two {x y : ℝ} (hx₁ : -(π / 2) < x) (hx₂ : x < π / 2)
    (hy₁ : -(π / 2) < y) (hy₂ : y < π / 2) (hxy : tan x = tan y) : x = y 
Arguments: {x y : ℝ} (hx₁ : -(Real.pi / 2) < x) (hx₂ : x < Real.pi / 2) (hy₁ : -(Real.pi / 2) < y) (hy₂ : y < Real.pi / 2) (hxy : Real.tan x = Real.tan y)
Goal: x = y
-- Real.tan_periodic
theorem tan_periodic : Function.Periodic tan π 
Goal: Function.Periodic Real.tan Real.pi
-- Real.tan_pi
@[simp]
theorem tan_pi : tan π = 0 
Goal: Real.tan Real.pi = 0
-- Real.tan_add_pi
theorem tan_add_pi (x : ℝ) : tan (x + π) = tan x 
Arguments: (x : ℝ)
Goal: Real.tan (x + Real.pi) = Real.tan x
-- Real.tan_sub_pi
theorem tan_sub_pi (x : ℝ) : tan (x - π) = tan x 
Arguments: (x : ℝ)
Goal: Real.tan (x - Real.pi) = Real.tan x
-- Real.tan_pi_sub
theorem tan_pi_sub (x : ℝ) : tan (π - x) = -tan x 
Arguments: (x : ℝ)
Goal: Real.tan (Real.pi - x) = -Real.tan x
-- Real.tan_pi_div_two_sub
theorem tan_pi_div_two_sub (x : ℝ) : tan (π / 2 - x) = (tan x)⁻¹ 
Arguments: (x : ℝ)
Goal: Real.tan (Real.pi / 2 - x) = (Real.tan x)⁻¹
-- Real.tan_nat_mul_pi
theorem tan_nat_mul_pi (n : ℕ) : tan (n * π) = 0 
Arguments: (n : ℕ)
Goal: Real.tan (↑n * Real.pi) = 0
-- Real.tan_int_mul_pi
theorem tan_int_mul_pi (n : ℤ) : tan (n * π) = 0 
Arguments: (n : ℤ)
Goal: Real.tan (↑n * Real.pi) = 0
-- Real.tan_add_nat_mul_pi
theorem tan_add_nat_mul_pi (x : ℝ) (n : ℕ) : tan (x + n * π) = tan x 
Arguments: (x : ℝ) (n : ℕ)
Goal: Real.tan (x + ↑n * Real.pi) = Real.tan x
-- Real.tan_add_int_mul_pi
theorem tan_add_int_mul_pi (x : ℝ) (n : ℤ) : tan (x + n * π) = tan x 
Arguments: (x : ℝ) (n : ℤ)
Goal: Real.tan (x + ↑n * Real.pi) = Real.tan x
-- Real.tan_sub_nat_mul_pi
theorem tan_sub_nat_mul_pi (x : ℝ) (n : ℕ) : tan (x - n * π) = tan x 
Arguments: (x : ℝ) (n : ℕ)
Goal: Real.tan (x - ↑n * Real.pi) = Real.tan x
-- Real.tan_sub_int_mul_pi
theorem tan_sub_int_mul_pi (x : ℝ) (n : ℤ) : tan (x - n * π) = tan x 
Arguments: (x : ℝ) (n : ℤ)
Goal: Real.tan (x - ↑n * Real.pi) = Real.tan x
-- Real.tan_nat_mul_pi_sub
theorem tan_nat_mul_pi_sub (x : ℝ) (n : ℕ) : tan (n * π - x) = -tan x 
Arguments: (x : ℝ) (n : ℕ)
Goal: Real.tan (↑n * Real.pi - x) = -Real.tan x
-- Real.tan_int_mul_pi_sub
theorem tan_int_mul_pi_sub (x : ℝ) (n : ℤ) : tan (n * π - x) = -tan x 
Arguments: (x : ℝ) (n : ℤ)
Goal: Real.tan (↑n * Real.pi - x) = -Real.tan x
-- Real.tendsto_sin_pi_div_two
theorem tendsto_sin_pi_div_two : Tendsto sin (𝓝[<] (π / 2)) (𝓝 1) 
Goal: Filter.Tendsto Real.sin (𝓝[<] (Real.pi / 2)) (𝓝 1)
-- Real.tendsto_cos_pi_div_two
theorem tendsto_cos_pi_div_two : Tendsto cos (𝓝[<] (π / 2)) (𝓝[>] 0) 
Goal: Filter.Tendsto Real.cos (𝓝[<] (Real.pi / 2)) (𝓝[>] 0)
-- Real.tendsto_tan_pi_div_two
theorem tendsto_tan_pi_div_two : Tendsto tan (𝓝[<] (π / 2)) atTop 
Goal: Filter.Tendsto Real.tan (𝓝[<] (Real.pi / 2)) Filter.atTop
-- Real.tendsto_sin_neg_pi_div_two
theorem tendsto_sin_neg_pi_div_two : Tendsto sin (𝓝[>] (-(π / 2))) (𝓝 (-1)) 
Goal: Filter.Tendsto Real.sin (𝓝[>] (-(Real.pi / 2))) (𝓝 (-1))
-- Real.tendsto_cos_neg_pi_div_two
theorem tendsto_cos_neg_pi_div_two : Tendsto cos (𝓝[>] (-(π / 2))) (𝓝[>] 0) 
Goal: Filter.Tendsto Real.cos (𝓝[>] (-(Real.pi / 2))) (𝓝[>] 0)
-- Real.tendsto_tan_neg_pi_div_two
theorem tendsto_tan_neg_pi_div_two : Tendsto tan (𝓝[>] (-(π / 2))) atBot 
Goal: Filter.Tendsto Real.tan (𝓝[>] (-(Real.pi / 2))) Filter.atBot
-- Complex.sin_eq_zero_iff_cos_eq
theorem sin_eq_zero_iff_cos_eq {z : ℂ} : sin z = 0 ↔ cos z = 1 ∨ cos z = -1 
Arguments: {z : ℂ}
Goal: Complex.sin z = 0 ↔ Complex.cos z = 1 ∨ Complex.cos z = -1
-- Complex.cos_pi_div_two
@[simp]
theorem cos_pi_div_two : cos (π / 2) = 0 
Goal: Complex.cos (↑Real.pi / 2) = 0
-- Complex.sin_pi_div_two
@[simp]
theorem sin_pi_div_two : sin (π / 2) = 1 
Goal: Complex.sin (↑Real.pi / 2) = 1
-- Complex.sin_pi
@[simp]
theorem sin_pi : sin π = 0 
Goal: Complex.sin ↑Real.pi = 0
-- Complex.cos_pi
@[simp]
theorem cos_pi : cos π = -1 
Goal: Complex.cos ↑Real.pi = -1
-- Complex.sin_two_pi
@[simp]
theorem sin_two_pi : sin (2 * π) = 0 
Goal: Complex.sin (2 * ↑Real.pi) = 0
-- Complex.cos_two_pi
@[simp]
theorem cos_two_pi : cos (2 * π) = 1 
Goal: Complex.cos (2 * ↑Real.pi) = 1
-- Complex.sin_antiperiodic
theorem sin_antiperiodic : Function.Antiperiodic sin π 
Goal: Function.Antiperiodic Complex.sin ↑Real.pi
-- Complex.sin_periodic
theorem sin_periodic : Function.Periodic sin (2 * π) 
Goal: Function.Periodic Complex.sin (2 * ↑Real.pi)
-- Complex.sin_add_pi
theorem sin_add_pi (x : ℂ) : sin (x + π) = -sin x 
Arguments: (x : ℂ)
Goal: Complex.sin (x + ↑Real.pi) = -Complex.sin x
-- Complex.sin_add_two_pi
theorem sin_add_two_pi (x : ℂ) : sin (x + 2 * π) = sin x 
Arguments: (x : ℂ)
Goal: Complex.sin (x + 2 * ↑Real.pi) = Complex.sin x
-- Complex.sin_sub_pi
theorem sin_sub_pi (x : ℂ) : sin (x - π) = -sin x 
Arguments: (x : ℂ)
Goal: Complex.sin (x - ↑Real.pi) = -Complex.sin x
-- Complex.sin_sub_two_pi
theorem sin_sub_two_pi (x : ℂ) : sin (x - 2 * π) = sin x 
Arguments: (x : ℂ)
Goal: Complex.sin (x - 2 * ↑Real.pi) = Complex.sin x
-- Complex.sin_pi_sub
theorem sin_pi_sub (x : ℂ) : sin (π - x) = sin x 
Arguments: (x : ℂ)
Goal: Complex.sin (↑Real.pi - x) = Complex.sin x
-- Complex.sin_two_pi_sub
theorem sin_two_pi_sub (x : ℂ) : sin (2 * π - x) = -sin x 
Arguments: (x : ℂ)
Goal: Complex.sin (2 * ↑Real.pi - x) = -Complex.sin x
-- Complex.sin_nat_mul_pi
theorem sin_nat_mul_pi (n : ℕ) : sin (n * π) = 0 
Arguments: (n : ℕ)
Goal: Complex.sin (↑n * ↑Real.pi) = 0
-- Complex.sin_int_mul_pi
theorem sin_int_mul_pi (n : ℤ) : sin (n * π) = 0 
Arguments: (n : ℤ)
Goal: Complex.sin (↑n * ↑Real.pi) = 0
-- Complex.sin_add_nat_mul_two_pi
theorem sin_add_nat_mul_two_pi (x : ℂ) (n : ℕ) : sin (x + n * (2 * π)) = sin x 
Arguments: (x : ℂ) (n : ℕ)
Goal: Complex.sin (x + ↑n * (2 * ↑Real.pi)) = Complex.sin x
-- Complex.sin_add_int_mul_two_pi
theorem sin_add_int_mul_two_pi (x : ℂ) (n : ℤ) : sin (x + n * (2 * π)) = sin x 
Arguments: (x : ℂ) (n : ℤ)
Goal: Complex.sin (x + ↑n * (2 * ↑Real.pi)) = Complex.sin x
-- Complex.sin_sub_nat_mul_two_pi
theorem sin_sub_nat_mul_two_pi (x : ℂ) (n : ℕ) : sin (x - n * (2 * π)) = sin x 
Arguments: (x : ℂ) (n : ℕ)
Goal: Complex.sin (x - ↑n * (2 * ↑Real.pi)) = Complex.sin x
-- Complex.sin_sub_int_mul_two_pi
theorem sin_sub_int_mul_two_pi (x : ℂ) (n : ℤ) : sin (x - n * (2 * π)) = sin x 
Arguments: (x : ℂ) (n : ℤ)
Goal: Complex.sin (x - ↑n * (2 * ↑Real.pi)) = Complex.sin x
-- Complex.sin_nat_mul_two_pi_sub
theorem sin_nat_mul_two_pi_sub (x : ℂ) (n : ℕ) : sin (n * (2 * π) - x) = -sin x 
Arguments: (x : ℂ) (n : ℕ)
Goal: Complex.sin (↑n * (2 * ↑Real.pi) - x) = -Complex.sin x
-- Complex.sin_int_mul_two_pi_sub
theorem sin_int_mul_two_pi_sub (x : ℂ) (n : ℤ) : sin (n * (2 * π) - x) = -sin x 
Arguments: (x : ℂ) (n : ℤ)
Goal: Complex.sin (↑n * (2 * ↑Real.pi) - x) = -Complex.sin x
-- Complex.cos_antiperiodic
theorem cos_antiperiodic : Function.Antiperiodic cos π 
Goal: Function.Antiperiodic Complex.cos ↑Real.pi
-- Complex.cos_periodic
theorem cos_periodic : Function.Periodic cos (2 * π) 
Goal: Function.Periodic Complex.cos (2 * ↑Real.pi)
-- Complex.cos_add_pi
theorem cos_add_pi (x : ℂ) : cos (x + π) = -cos x 
Arguments: (x : ℂ)
Goal: Complex.cos (x + ↑Real.pi) = -Complex.cos x
-- Complex.cos_add_two_pi
theorem cos_add_two_pi (x : ℂ) : cos (x + 2 * π) = cos x 
Arguments: (x : ℂ)
Goal: Complex.cos (x + 2 * ↑Real.pi) = Complex.cos x
-- Complex.cos_sub_pi
theorem cos_sub_pi (x : ℂ) : cos (x - π) = -cos x 
Arguments: (x : ℂ)
Goal: Complex.cos (x - ↑Real.pi) = -Complex.cos x
-- Complex.cos_sub_two_pi
theorem cos_sub_two_pi (x : ℂ) : cos (x - 2 * π) = cos x 
Arguments: (x : ℂ)
Goal: Complex.cos (x - 2 * ↑Real.pi) = Complex.cos x
-- Complex.cos_pi_sub
theorem cos_pi_sub (x : ℂ) : cos (π - x) = -cos x 
Arguments: (x : ℂ)
Goal: Complex.cos (↑Real.pi - x) = -Complex.cos x
-- Complex.cos_two_pi_sub
theorem cos_two_pi_sub (x : ℂ) : cos (2 * π - x) = cos x 
Arguments: (x : ℂ)
Goal: Complex.cos (2 * ↑Real.pi - x) = Complex.cos x
-- Complex.cos_nat_mul_two_pi
theorem cos_nat_mul_two_pi (n : ℕ) : cos (n * (2 * π)) = 1 
Arguments: (n : ℕ)
Goal: Complex.cos (↑n * (2 * ↑Real.pi)) = 1
-- Complex.cos_int_mul_two_pi
theorem cos_int_mul_two_pi (n : ℤ) : cos (n * (2 * π)) = 1 
Arguments: (n : ℤ)
Goal: Complex.cos (↑n * (2 * ↑Real.pi)) = 1
-- Complex.cos_add_nat_mul_two_pi
theorem cos_add_nat_mul_two_pi (x : ℂ) (n : ℕ) : cos (x + n * (2 * π)) = cos x 
Arguments: (x : ℂ) (n : ℕ)
Goal: Complex.cos (x + ↑n * (2 * ↑Real.pi)) = Complex.cos x
-- Complex.cos_add_int_mul_two_pi
theorem cos_add_int_mul_two_pi (x : ℂ) (n : ℤ) : cos (x + n * (2 * π)) = cos x 
Arguments: (x : ℂ) (n : ℤ)
Goal: Complex.cos (x + ↑n * (2 * ↑Real.pi)) = Complex.cos x
-- Complex.cos_sub_nat_mul_two_pi
theorem cos_sub_nat_mul_two_pi (x : ℂ) (n : ℕ) : cos (x - n * (2 * π)) = cos x 
Arguments: (x : ℂ) (n : ℕ)
Goal: Complex.cos (x - ↑n * (2 * ↑Real.pi)) = Complex.cos x
-- Complex.cos_sub_int_mul_two_pi
theorem cos_sub_int_mul_two_pi (x : ℂ) (n : ℤ) : cos (x - n * (2 * π)) = cos x 
Arguments: (x : ℂ) (n : ℤ)
Goal: Complex.cos (x - ↑n * (2 * ↑Real.pi)) = Complex.cos x
-- Complex.cos_nat_mul_two_pi_sub
theorem cos_nat_mul_two_pi_sub (x : ℂ) (n : ℕ) : cos (n * (2 * π) - x) = cos x 
Arguments: (x : ℂ) (n : ℕ)
Goal: Complex.cos (↑n * (2 * ↑Real.pi) - x) = Complex.cos x
-- Complex.cos_int_mul_two_pi_sub
theorem cos_int_mul_two_pi_sub (x : ℂ) (n : ℤ) : cos (n * (2 * π) - x) = cos x 
Arguments: (x : ℂ) (n : ℤ)
Goal: Complex.cos (↑n * (2 * ↑Real.pi) - x) = Complex.cos x
-- Complex.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: Complex.cos (↑n * (2 * ↑Real.pi) + ↑Real.pi) = -1
-- Complex.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: Complex.cos (↑n * (2 * ↑Real.pi) + ↑Real.pi) = -1
-- Complex.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: Complex.cos (↑n * (2 * ↑Real.pi) - ↑Real.pi) = -1
-- Complex.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: Complex.cos (↑n * (2 * ↑Real.pi) - ↑Real.pi) = -1
-- Complex.sin_add_pi_div_two
theorem sin_add_pi_div_two (x : ℂ) : sin (x + π / 2) = cos x 
Arguments: (x : ℂ)
Goal: Complex.sin (x + ↑Real.pi / 2) = Complex.cos x
-- Complex.sin_sub_pi_div_two
theorem sin_sub_pi_div_two (x : ℂ) : sin (x - π / 2) = -cos x 
Arguments: (x : ℂ)
Goal: Complex.sin (x - ↑Real.pi / 2) = -Complex.cos x
-- Complex.sin_pi_div_two_sub
theorem sin_pi_div_two_sub (x : ℂ) : sin (π / 2 - x) = cos x 
Arguments: (x : ℂ)
Goal: Complex.sin (↑Real.pi / 2 - x) = Complex.cos x
-- Complex.cos_add_pi_div_two
theorem cos_add_pi_div_two (x : ℂ) : cos (x + π / 2) = -sin x 
Arguments: (x : ℂ)
Goal: Complex.cos (x + ↑Real.pi / 2) = -Complex.sin x
-- Complex.cos_sub_pi_div_two
theorem cos_sub_pi_div_two (x : ℂ) : cos (x - π / 2) = sin x 
Arguments: (x : ℂ)
Goal: Complex.cos (x - ↑Real.pi / 2) = Complex.sin x
-- Complex.cos_pi_div_two_sub
theorem cos_pi_div_two_sub (x : ℂ) : cos (π / 2 - x) = sin x 
Arguments: (x : ℂ)
Goal: Complex.cos (↑Real.pi / 2 - x) = Complex.sin x
-- Complex.tan_periodic
theorem tan_periodic : Function.Periodic tan π 
Goal: Function.Periodic Complex.tan ↑Real.pi
-- Complex.tan_add_pi
theorem tan_add_pi (x : ℂ) : tan (x + π) = tan x 
Arguments: (x : ℂ)
Goal: Complex.tan (x + ↑Real.pi) = Complex.tan x
-- Complex.tan_sub_pi
theorem tan_sub_pi (x : ℂ) : tan (x - π) = tan x 
Arguments: (x : ℂ)
Goal: Complex.tan (x - ↑Real.pi) = Complex.tan x
-- Complex.tan_pi_sub
theorem tan_pi_sub (x : ℂ) : tan (π - x) = -tan x 
Arguments: (x : ℂ)
Goal: Complex.tan (↑Real.pi - x) = -Complex.tan x
-- Complex.tan_pi_div_two_sub
theorem tan_pi_div_two_sub (x : ℂ) : tan (π / 2 - x) = (tan x)⁻¹ 
Arguments: (x : ℂ)
Goal: Complex.tan (↑Real.pi / 2 - x) = (Complex.tan x)⁻¹
-- Complex.tan_nat_mul_pi
theorem tan_nat_mul_pi (n : ℕ) : tan (n * π) = 0 
Arguments: (n : ℕ)
Goal: Complex.tan (↑n * ↑Real.pi) = 0
-- Complex.tan_int_mul_pi
theorem tan_int_mul_pi (n : ℤ) : tan (n * π) = 0 
Arguments: (n : ℤ)
Goal: Complex.tan (↑n * ↑Real.pi) = 0
-- Complex.tan_add_nat_mul_pi
theorem tan_add_nat_mul_pi (x : ℂ) (n : ℕ) : tan (x + n * π) = tan x 
Arguments: (x : ℂ) (n : ℕ)
Goal: Complex.tan (x + ↑n * ↑Real.pi) = Complex.tan x
-- Complex.tan_add_int_mul_pi
theorem tan_add_int_mul_pi (x : ℂ) (n : ℤ) : tan (x + n * π) = tan x 
Arguments: (x : ℂ) (n : ℤ)
Goal: Complex.tan (x + ↑n * ↑Real.pi) = Complex.tan x
-- Complex.tan_sub_nat_mul_pi
theorem tan_sub_nat_mul_pi (x : ℂ) (n : ℕ) : tan (x - n * π) = tan x 
Arguments: (x : ℂ) (n : ℕ)
Goal: Complex.tan (x - ↑n * ↑Real.pi) = Complex.tan x
-- Complex.tan_sub_int_mul_pi
theorem tan_sub_int_mul_pi (x : ℂ) (n : ℤ) : tan (x - n * π) = tan x 
Arguments: (x : ℂ) (n : ℤ)
Goal: Complex.tan (x - ↑n * ↑Real.pi) = Complex.tan x
-- Complex.tan_nat_mul_pi_sub
theorem tan_nat_mul_pi_sub (x : ℂ) (n : ℕ) : tan (n * π - x) = -tan x 
Arguments: (x : ℂ) (n : ℕ)
Goal: Complex.tan (↑n * ↑Real.pi - x) = -Complex.tan x
-- Complex.tan_int_mul_pi_sub
theorem tan_int_mul_pi_sub (x : ℂ) (n : ℤ) : tan (n * π - x) = -tan x 
Arguments: (x : ℂ) (n : ℤ)
Goal: Complex.tan (↑n * ↑Real.pi - x) = -Complex.tan x
-- Complex.exp_antiperiodic
theorem exp_antiperiodic : Function.Antiperiodic exp (π * I) 
Goal: Function.Antiperiodic Complex.exp (↑Real.pi * Complex.I)
-- Complex.exp_periodic
theorem exp_periodic : Function.Periodic exp (2 * π * I) 
Goal: Function.Periodic Complex.exp (2 * ↑Real.pi * Complex.I)
-- Complex.exp_mul_I_antiperiodic
theorem exp_mul_I_antiperiodic : Function.Antiperiodic (fun x => exp (x * I)) π 
Goal: Function.Antiperiodic (fun x => Complex.exp (x * Complex.I)) ↑Real.pi
-- Complex.exp_mul_I_periodic
theorem exp_mul_I_periodic : Function.Periodic (fun x => exp (x * I)) (2 * π) 
Goal: Function.Periodic (fun x => Complex.exp (x * Complex.I)) (2 * ↑Real.pi)
-- Complex.exp_pi_mul_I
@[simp]
theorem exp_pi_mul_I : exp (π * I) = -1 
Goal: Complex.exp (↑Real.pi * Complex.I) = -1
-- Complex.exp_two_pi_mul_I
@[simp]
theorem exp_two_pi_mul_I : exp (2 * π * I) = 1 
Goal: Complex.exp (2 * ↑Real.pi * Complex.I) = 1
-- Complex.exp_nat_mul_two_pi_mul_I
@[simp]
theorem exp_nat_mul_two_pi_mul_I (n : ℕ) : exp (n * (2 * π * I)) = 1 
Arguments: (n : ℕ)
Goal: Complex.exp (↑n * (2 * ↑Real.pi * Complex.I)) = 1
-- Complex.exp_int_mul_two_pi_mul_I
@[simp]
theorem exp_int_mul_two_pi_mul_I (n : ℤ) : exp (n * (2 * π * I)) = 1 
Arguments: (n : ℤ)
Goal: Complex.exp (↑n * (2 * ↑Real.pi * Complex.I)) = 1
-- Complex.exp_add_pi_mul_I
@[simp]
theorem exp_add_pi_mul_I (z : ℂ) : exp (z + π * I) = -exp z 
Arguments: (z : ℂ)
Goal: Complex.exp (z + ↑Real.pi * Complex.I) = -Complex.exp z
-- Complex.exp_sub_pi_mul_I
@[simp]
theorem exp_sub_pi_mul_I (z : ℂ) : exp (z - π * I) = -exp z 
Arguments: (z : ℂ)
Goal: Complex.exp (z - ↑Real.pi * Complex.I) = -Complex.exp z
-- Complex.abs_exp_mul_exp_add_exp_neg_le_of_abs_im_le
theorem abs_exp_mul_exp_add_exp_neg_le_of_abs_im_le {a b : ℝ} (ha : a ≤ 0) {z : ℂ} (hz : |z.im| ≤ b)
    (hb : b ≤ π / 2) :
    abs (exp (a * (exp z + exp (-z)))) ≤ Real.exp (a * Real.cos b * Real.exp |z.re|) 
Arguments: {a b : ℝ} (ha : a ≤ 0) {z : ℂ} (hz : |z.im| ≤ b) (hb : b ≤ Real.pi / 2)
Goal: Complex.abs (Complex.exp (↑a * (Complex.exp z + Complex.exp (-z)))) ≤ Real.exp (a * Real.cos b * Real.exp |z.re|)
-- Complex.exp_bound_sq
theorem exp_bound_sq (x z : ℂ) (hz : ‖z‖ ≤ 1) :
    ‖exp (x + z) - exp x - z • exp x‖ ≤ ‖exp x‖ * ‖z‖ ^ 2 
Arguments: (x z : ℂ) (hz : ‖z‖ ≤ 1)
Goal: ‖Complex.exp (x + z) - Complex.exp x - z • Complex.exp x‖ ≤ ‖Complex.exp x‖ * ‖z‖ ^ 2
-- Complex.locally_lipschitz_exp
theorem locally_lipschitz_exp {r : ℝ} (hr_nonneg : 0 ≤ r) (hr_le : r ≤ 1) (x y : ℂ)
    (hyx : ‖y - x‖ < r) : ‖exp y - exp x‖ ≤ (1 + r) * ‖exp x‖ * ‖y - x‖ 
Arguments: {r : ℝ} (hr_nonneg : 0 ≤ r) (hr_le : r ≤ 1) (x y : ℂ) (hyx : ‖y - x‖ < r)
Goal: ‖Complex.exp y - Complex.exp x‖ ≤ (1 + r) * ‖Complex.exp x‖ * ‖y - x‖
-- Complex.continuous_exp
@[continuity]
theorem continuous_exp : Continuous exp 
Goal: Continuous Complex.exp
-- Complex.continuousOn_exp
theorem continuousOn_exp {s : Set ℂ} : ContinuousOn exp s 
Arguments: {s : Set ℂ}
Goal: ContinuousOn Complex.exp s
-- Complex.exp_sub_sum_range_isBigO_pow
lemma exp_sub_sum_range_isBigO_pow (n : ℕ) :
    (fun x ↦ exp x - ∑ i ∈ Finset.range n, x ^ i / i !) =O[𝓝 0] (· ^ n) 
Arguments: (n : ℕ)
Goal: (fun x => Complex.exp x - ∑ i ∈ Finset.range n, x ^ i / ↑i !) =O[𝓝 0] fun x => x ^ n
-- Complex.exp_sub_sum_range_succ_isLittleO_pow
lemma exp_sub_sum_range_succ_isLittleO_pow (n : ℕ) :
    (fun x ↦ exp x - ∑ i ∈ Finset.range (n + 1), x ^ i / i !) =o[𝓝 0] (· ^ n) 
Arguments: (n : ℕ)
Goal: (fun x => Complex.exp x - ∑ i ∈ Finset.range (n + 1), x ^ i / ↑i !) =o[𝓝 0] fun x => x ^ n
-- Filter.Tendsto.cexp
theorem Filter.Tendsto.cexp {l : Filter α} {f : α → ℂ} {z : ℂ} (hf : Tendsto f l (𝓝 z)) :
    Tendsto (fun x => exp (f x)) l (𝓝 (exp z)) 
Arguments: {α : Type u_1} {l : Filter α} {f : α → ℂ} {z : ℂ} (hf : Filter.Tendsto f l (𝓝 z))
Goal: Filter.Tendsto (fun x => Complex.exp (f x)) l (𝓝 (Complex.exp z))
-- ContinuousWithinAt.cexp
nonrec
theorem ContinuousWithinAt.cexp (h : ContinuousWithinAt f s x) :
    ContinuousWithinAt (fun y => exp (f y)) s x 
Arguments: {α : Type u_1} [TopologicalSpace α] {f : α → ℂ} {s : Set α} {x : α} (h : ContinuousWithinAt f s x)
Goal: ContinuousWithinAt (fun y => Complex.exp (f y)) s x
-- ContinuousAt.cexp
@[fun_prop]
nonrec
theorem ContinuousAt.cexp (h : ContinuousAt f x) : ContinuousAt (fun y => exp (f y)) x 
Arguments: {α : Type u_1} [TopologicalSpace α] {f : α → ℂ} {x : α} (h : ContinuousAt f x)
Goal: ContinuousAt (fun y => Complex.exp (f y)) x
-- ContinuousOn.cexp
@[fun_prop]
theorem ContinuousOn.cexp (h : ContinuousOn f s) : ContinuousOn (fun y => exp (f y)) s 
Arguments: {α : Type u_1} [TopologicalSpace α] {f : α → ℂ} {s : Set α} (h : ContinuousOn f s)
Goal: ContinuousOn (fun y => Complex.exp (f y)) s
-- Continuous.cexp
@[fun_prop]
theorem Continuous.cexp (h : Continuous f) : Continuous fun y => exp (f y) 
Arguments: {α : Type u_1} [TopologicalSpace α] {f : α → ℂ} (h : Continuous f)
Goal: Continuous fun y => Complex.exp (f y)
-- Real.continuous_exp
@[continuity]
theorem continuous_exp : Continuous exp 
Goal: Continuous Real.exp
-- Real.continuousOn_exp
theorem continuousOn_exp {s : Set ℝ} : ContinuousOn exp s 
Arguments: {s : Set ℝ}
Goal: ContinuousOn Real.exp s
-- Real.exp_sub_sum_range_isBigO_pow
lemma exp_sub_sum_range_isBigO_pow (n : ℕ) :
    (fun x ↦ exp x - ∑ i ∈ Finset.range n, x ^ i / i !) =O[𝓝 0] (· ^ n) 
Arguments: (n : ℕ)
Goal: (fun x => Real.exp x - ∑ i ∈ Finset.range n, x ^ i / ↑i !) =O[𝓝 0] fun x => x ^ n
-- Real.exp_sub_sum_range_succ_isLittleO_pow
lemma exp_sub_sum_range_succ_isLittleO_pow (n : ℕ) :
    (fun x ↦ exp x - ∑ i ∈ Finset.range (n + 1), x ^ i / i !) =o[𝓝 0] (· ^ n) 
Arguments: (n : ℕ)
Goal: (fun x => Real.exp x - ∑ i ∈ Finset.range (n + 1), x ^ i / ↑i !) =o[𝓝 0] fun x => x ^ n
-- Filter.Tendsto.rexp
theorem Filter.Tendsto.rexp {l : Filter α} {f : α → ℝ} {z : ℝ} (hf : Tendsto f l (𝓝 z)) :
    Tendsto (fun x => exp (f x)) l (𝓝 (exp z)) 
Arguments: {α : Type u_1} {l : Filter α} {f : α → ℝ} {z : ℝ} (hf : Filter.Tendsto f l (𝓝 z))
Goal: Filter.Tendsto (fun x => Real.exp (f x)) l (𝓝 (Real.exp z))
-- ContinuousWithinAt.rexp
nonrec
theorem ContinuousWithinAt.rexp (h : ContinuousWithinAt f s x) :
    ContinuousWithinAt (fun y ↦ exp (f y)) s x 
Arguments: {α : Type u_1} [TopologicalSpace α] {f : α → ℝ} {s : Set α} {x : α} (h : ContinuousWithinAt f s x)
Goal: ContinuousWithinAt (fun y => Real.exp (f y)) s x
-- ContinuousAt.rexp
@[fun_prop]
nonrec
theorem ContinuousAt.rexp (h : ContinuousAt f x) : ContinuousAt (fun y ↦ exp (f y)) x 
Arguments: {α : Type u_1} [TopologicalSpace α] {f : α → ℝ} {x : α} (h : ContinuousAt f x)
Goal: ContinuousAt (fun y => Real.exp (f y)) x
-- ContinuousOn.rexp
@[fun_prop]
theorem ContinuousOn.rexp (h : ContinuousOn f s) :
    ContinuousOn (fun y ↦ exp (f y)) s 
Arguments: {α : Type u_1} [TopologicalSpace α] {f : α → ℝ} {s : Set α} (h : ContinuousOn f s)
Goal: ContinuousOn (fun y => Real.exp (f y)) s
-- Continuous.rexp
@[fun_prop]
theorem Continuous.rexp (h : Continuous f) : Continuous fun y ↦ exp (f y) 
Arguments: {α : Type u_1} [TopologicalSpace α] {f : α → ℝ} (h : Continuous f)
Goal: Continuous fun y => Real.exp (f y)
-- Real.exp_half
theorem exp_half (x : ℝ) : exp (x / 2) = √(exp x) 
Arguments: (x : ℝ)
Goal: Real.exp (x / 2) = √(Real.exp x)
-- Real.tendsto_exp_atTop
theorem tendsto_exp_atTop : Tendsto exp atTop atTop 
Goal: Filter.Tendsto Real.exp Filter.atTop Filter.atTop
-- Real.tendsto_exp_neg_atTop_nhds_zero
theorem tendsto_exp_neg_atTop_nhds_zero : Tendsto (fun x => exp (-x)) atTop (𝓝 0) 
Goal: Filter.Tendsto (fun x => Real.exp (-x)) Filter.atTop (𝓝 0)
-- Real.tendsto_exp_nhds_zero_nhds_one
theorem tendsto_exp_nhds_zero_nhds_one : Tendsto exp (𝓝 0) (𝓝 1) 
Goal: Filter.Tendsto Real.exp (𝓝 0) (𝓝 1)
-- Real.tendsto_exp_atBot
theorem tendsto_exp_atBot : Tendsto exp atBot (𝓝 0) 
Goal: Filter.Tendsto Real.exp Filter.atBot (𝓝 0)
-- Real.tendsto_exp_atBot_nhdsWithin
theorem tendsto_exp_atBot_nhdsWithin : Tendsto exp atBot (𝓝[>] 0) 
Goal: Filter.Tendsto Real.exp Filter.atBot (𝓝[>] 0)
-- Real.isBoundedUnder_ge_exp_comp
@[simp]
theorem isBoundedUnder_ge_exp_comp (l : Filter α) (f : α → ℝ) :
    IsBoundedUnder (· ≥ ·) l fun x => exp (f x) 
Arguments: {α : Type u_1} (l : Filter α) (f : α → ℝ)
Goal: Filter.IsBoundedUnder (fun x x_1 => x ≥ x_1) l fun x => Real.exp (f x)
-- Real.isBoundedUnder_le_exp_comp
@[simp]
theorem isBoundedUnder_le_exp_comp {f : α → ℝ} :
    (IsBoundedUnder (· ≤ ·) l fun x => exp (f x)) ↔ IsBoundedUnder (· ≤ ·) l f 
Arguments: {α : Type u_1} {l : Filter α} {f : α → ℝ}
Goal: (Filter.IsBoundedUnder (fun x x_1 => x ≤ x_1) l fun x => Real.exp (f x)) ↔
  Filter.IsBoundedUnder (fun x x_1 => x ≤ x_1) l f
-- Real.tendsto_exp_div_pow_atTop
theorem tendsto_exp_div_pow_atTop (n : ℕ) : Tendsto (fun x => exp x / x ^ n) atTop atTop 
Arguments: (n : ℕ)
Goal: Filter.Tendsto (fun x => Real.exp x / x ^ n) Filter.atTop Filter.atTop
-- Real.tendsto_pow_mul_exp_neg_atTop_nhds_zero
theorem tendsto_pow_mul_exp_neg_atTop_nhds_zero (n : ℕ) :
    Tendsto (fun x => x ^ n * exp (-x)) atTop (𝓝 0) 
Arguments: (n : ℕ)
Goal: Filter.Tendsto (fun x => x ^ n * Real.exp (-x)) Filter.atTop (𝓝 0)