ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- dist_le_of_trajectories_ODE_of_mem
theorem dist_le_of_trajectories_ODE_of_mem
    (hf : ContinuousOn f (Icc a b))
    (hf' : ∀ t ∈ Ico a b, HasDerivWithinAt f (v t (f t)) (Ici t) t)
    (hfs : ∀ t ∈ Ico a b, f t ∈ s t)
    (hg : ContinuousOn g (Icc a b)) (hg' : ∀ t ∈ Ico a b, HasDerivWithinAt g (v t (g t)) (Ici t) t)
    (hgs : ∀ t ∈ Ico a b, g t ∈ s t) (ha : dist (f a) (g a) ≤ δ) :
    ∀ t ∈ Icc a b, dist (f t) (g t) ≤ δ * exp (K * (t - a)) 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {v : ℝ → E → E} {s : ℝ → Set E} {K : ℝ≥0} {f g : ℝ → E} {a b δ : ℝ} (hv : ∀ (t : ℝ), LipschitzOnWith K (v t) (s t)) (hf : ContinuousOn f (Set.Icc a b)) (hf' : ∀ t ∈ Set.Ico a b, HasDerivWithinAt f (v t (f t)) (Set.Ici t) t) (hfs : ∀ t ∈ Set.Ico a b, f t ∈ s t) (hg : ContinuousOn g (Set.Icc a b)) (hg' : ∀ t ∈ Set.Ico a b, HasDerivWithinAt g (v t (g t)) (Set.Ici t) t) (hgs : ∀ t ∈ Set.Ico a b, g t ∈ s t) (ha : dist (f a) (g a) ≤ δ) (t : ℝ)
Goal: t ∈ Set.Icc a b → dist (f t) (g t) ≤ δ * rexp (↑K * (t - a))
-- dist_le_of_trajectories_ODE
theorem dist_le_of_trajectories_ODE
    (hv : ∀ t, LipschitzWith K (v t))
    (hf : ContinuousOn f (Icc a b))
    (hf' : ∀ t ∈ Ico a b, HasDerivWithinAt f (v t (f t)) (Ici t) t)
    (hg : ContinuousOn g (Icc a b))
    (hg' : ∀ t ∈ Ico a b, HasDerivWithinAt g (v t (g t)) (Ici t) t)
    (ha : dist (f a) (g a) ≤ δ) :
    ∀ t ∈ Icc a b, dist (f t) (g t) ≤ δ * exp (K * (t - a)) 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {v : ℝ → E → E} {K : ℝ≥0} {f g : ℝ → E} {a b δ : ℝ} (hv : ∀ (t : ℝ), LipschitzWith K (v t)) (hf : ContinuousOn f (Set.Icc a b)) (hf' : ∀ t ∈ Set.Ico a b, HasDerivWithinAt f (v t (f t)) (Set.Ici t) t) (hg : ContinuousOn g (Set.Icc a b)) (hg' : ∀ t ∈ Set.Ico a b, HasDerivWithinAt g (v t (g t)) (Set.Ici t) t) (ha : dist (f a) (g a) ≤ δ) (t : ℝ)
Goal: t ∈ Set.Icc a b → dist (f t) (g t) ≤ δ * rexp (↑K * (t - a))
-- ODE_solution_unique_of_mem_Icc_right
theorem ODE_solution_unique_of_mem_Icc_right
    (hf : ContinuousOn f (Icc a b))
    (hf' : ∀ t ∈ Ico a b, HasDerivWithinAt f (v t (f t)) (Ici t) t)
    (hfs : ∀ t ∈ Ico a b, f t ∈ s t)
    (hg : ContinuousOn g (Icc a b))
    (hg' : ∀ t ∈ Ico a b, HasDerivWithinAt g (v t (g t)) (Ici t) t)
    (hgs : ∀ t ∈ Ico a b, g t ∈ s t)
    (ha : f a = g a) :
    EqOn f g (Icc a b) 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {v : ℝ → E → E} {s : ℝ → Set E} {K : ℝ≥0} {f g : ℝ → E} {a b : ℝ} (hv : ∀ (t : ℝ), LipschitzOnWith K (v t) (s t)) (hf : ContinuousOn f (Set.Icc a b)) (hf' : ∀ t ∈ Set.Ico a b, HasDerivWithinAt f (v t (f t)) (Set.Ici t) t) (hfs : ∀ t ∈ Set.Ico a b, f t ∈ s t) (hg : ContinuousOn g (Set.Icc a b)) (hg' : ∀ t ∈ Set.Ico a b, HasDerivWithinAt g (v t (g t)) (Set.Ici t) t) (hgs : ∀ t ∈ Set.Ico a b, g t ∈ s t) (ha : f a = g a)
Goal: Set.EqOn f g (Set.Icc a b)
-- ODE_solution_unique_of_mem_Icc_left
theorem ODE_solution_unique_of_mem_Icc_left
    (hf : ContinuousOn f (Icc a b))
    (hf' : ∀ t ∈ Ioc a b, HasDerivWithinAt f (v t (f t)) (Iic t) t)
    (hfs : ∀ t ∈ Ioc a b, f t ∈ s t)
    (hg : ContinuousOn g (Icc a b))
    (hg' : ∀ t ∈ Ioc a b, HasDerivWithinAt g (v t (g t)) (Iic t) t)
    (hgs : ∀ t ∈ Ioc a b, g t ∈ s t)
    (hb : f b = g b) :
    EqOn f g (Icc a b) 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {v : ℝ → E → E} {s : ℝ → Set E} {K : ℝ≥0} {f g : ℝ → E} {a b : ℝ} (hv : ∀ (t : ℝ), LipschitzOnWith K (v t) (s t)) (hf : ContinuousOn f (Set.Icc a b)) (hf' : ∀ t ∈ Set.Ioc a b, HasDerivWithinAt f (v t (f t)) (Set.Iic t) t) (hfs : ∀ t ∈ Set.Ioc a b, f t ∈ s t) (hg : ContinuousOn g (Set.Icc a b)) (hg' : ∀ t ∈ Set.Ioc a b, HasDerivWithinAt g (v t (g t)) (Set.Iic t) t) (hgs : ∀ t ∈ Set.Ioc a b, g t ∈ s t) (hb : f b = g b)
Goal: Set.EqOn f g (Set.Icc a b)
-- ODE_solution_unique_of_mem_Icc
theorem ODE_solution_unique_of_mem_Icc
    (ht : t₀ ∈ Ioo a b)
    (hf : ContinuousOn f (Icc a b))
    (hf' : ∀ t ∈ Ioo a b, HasDerivAt f (v t (f t)) t)
    (hfs : ∀ t ∈ Ioo a b, f t ∈ s t)
    (hg : ContinuousOn g (Icc a b))
    (hg' : ∀ t ∈ Ioo a b, HasDerivAt g (v t (g t)) t)
    (hgs : ∀ t ∈ Ioo a b, g t ∈ s t)
    (heq : f t₀ = g t₀) :
    EqOn f g (Icc a b) 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {v : ℝ → E → E} {s : ℝ → Set E} {K : ℝ≥0} {f g : ℝ → E} {a b t₀ : ℝ} (hv : ∀ (t : ℝ), LipschitzOnWith K (v t) (s t)) (ht : t₀ ∈ Set.Ioo a b) (hf : ContinuousOn f (Set.Icc a b)) (hf' : ∀ t ∈ Set.Ioo a b, HasDerivAt f (v t (f t)) t) (hfs : ∀ t ∈ Set.Ioo a b, f t ∈ s t) (hg : ContinuousOn g (Set.Icc a b)) (hg' : ∀ t ∈ Set.Ioo a b, HasDerivAt g (v t (g t)) t) (hgs : ∀ t ∈ Set.Ioo a b, g t ∈ s t) (heq : f t₀ = g t₀)
Goal: Set.EqOn f g (Set.Icc a b)
-- ODE_solution_unique_of_mem_Ioo
theorem ODE_solution_unique_of_mem_Ioo
    (ht : t₀ ∈ Ioo a b)
    (hf : ∀ t ∈ Ioo a b, HasDerivAt f (v t (f t)) t ∧ f t ∈ s t)
    (hg : ∀ t ∈ Ioo a b, HasDerivAt g (v t (g t)) t ∧ g t ∈ s t)
    (heq : f t₀ = g t₀) :
    EqOn f g (Ioo a b) 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {v : ℝ → E → E} {s : ℝ → Set E} {K : ℝ≥0} {f g : ℝ → E} {a b t₀ : ℝ} (hv : ∀ (t : ℝ), LipschitzOnWith K (v t) (s t)) (ht : t₀ ∈ Set.Ioo a b) (hf : ∀ t ∈ Set.Ioo a b, HasDerivAt f (v t (f t)) t ∧ f t ∈ s t) (hg : ∀ t ∈ Set.Ioo a b, HasDerivAt g (v t (g t)) t ∧ g t ∈ s t) (heq : f t₀ = g t₀)
Goal: Set.EqOn f g (Set.Ioo a b)
-- ODE_solution_unique_of_eventually
theorem ODE_solution_unique_of_eventually
    (hf : ∀ᶠ t in 𝓝 t₀, HasDerivAt f (v t (f t)) t ∧ f t ∈ s t)
    (hg : ∀ᶠ t in 𝓝 t₀, HasDerivAt g (v t (g t)) t ∧ g t ∈ s t)
    (heq : f t₀ = g t₀) : f =ᶠ[𝓝 t₀] g 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {v : ℝ → E → E} {s : ℝ → Set E} {K : ℝ≥0} {f g : ℝ → E} {t₀ : ℝ} (hv : ∀ (t : ℝ), LipschitzOnWith K (v t) (s t)) (hf : ∀ᶠ (t : ℝ) in 𝓝 t₀, HasDerivAt f (v t (f t)) t ∧ f t ∈ s t) (hg : ∀ᶠ (t : ℝ) in 𝓝 t₀, HasDerivAt g (v t (g t)) t ∧ g t ∈ s t) (heq : f t₀ = g t₀)
Goal: f =ᶠ[𝓝 t₀] g
-- ODE_solution_unique
theorem ODE_solution_unique
    (hv : ∀ t, LipschitzWith K (v t))
    (hf : ContinuousOn f (Icc a b))
    (hf' : ∀ t ∈ Ico a b, HasDerivWithinAt f (v t (f t)) (Ici t) t)
    (hg : ContinuousOn g (Icc a b))
    (hg' : ∀ t ∈ Ico a b, HasDerivWithinAt g (v t (g t)) (Ici t) t)
    (ha : f a = g a) :
    EqOn f g (Icc a b) 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {v : ℝ → E → E} {K : ℝ≥0} {f g : ℝ → E} {a b : ℝ} (hv : ∀ (t : ℝ), LipschitzWith K (v t)) (hf : ContinuousOn f (Set.Icc a b)) (hf' : ∀ t ∈ Set.Ico a b, HasDerivWithinAt f (v t (f t)) (Set.Ici t) t) (hg : ContinuousOn g (Set.Icc a b)) (hg' : ∀ t ∈ Set.Ico a b, HasDerivWithinAt g (v t (g t)) (Set.Ici t) t) (ha : f a = g a)
Goal: Set.EqOn f g (Set.Icc a b)
-- Complex.cos_eq_zero_iff
theorem cos_eq_zero_iff {θ : ℂ} : cos θ = 0 ↔ ∃ k : ℤ, θ = (2 * k + 1) * π / 2 
Arguments: {θ : ℂ}
Goal: Complex.cos θ = 0 ↔ ∃ k, θ = (2 * ↑k + 1) * ↑Real.pi / 2
-- Complex.cos_ne_zero_iff
theorem cos_ne_zero_iff {θ : ℂ} : cos θ ≠ 0 ↔ ∀ k : ℤ, θ ≠ (2 * k + 1) * π / 2 
Arguments: {θ : ℂ}
Goal: Complex.cos θ ≠ 0 ↔ ∀ (k : ℤ), θ ≠ (2 * ↑k + 1) * ↑Real.pi / 2
-- Complex.sin_eq_zero_iff
theorem sin_eq_zero_iff {θ : ℂ} : sin θ = 0 ↔ ∃ k : ℤ, θ = k * π 
Arguments: {θ : ℂ}
Goal: Complex.sin θ = 0 ↔ ∃ k, θ = ↑k * ↑Real.pi
-- Complex.sin_ne_zero_iff
theorem sin_ne_zero_iff {θ : ℂ} : sin θ ≠ 0 ↔ ∀ k : ℤ, θ ≠ k * π 
Arguments: {θ : ℂ}
Goal: Complex.sin θ ≠ 0 ↔ ∀ (k : ℤ), θ ≠ ↑k * ↑Real.pi
-- Complex.tan_eq_zero_iff
theorem tan_eq_zero_iff {θ : ℂ} : tan θ = 0 ↔ ∃ k : ℤ, k * π / 2 = θ 
Arguments: {θ : ℂ}
Goal: Complex.tan θ = 0 ↔ ∃ k, ↑k * ↑Real.pi / 2 = θ
-- Complex.tan_ne_zero_iff
theorem tan_ne_zero_iff {θ : ℂ} : tan θ ≠ 0 ↔ ∀ k : ℤ, (k * π / 2 : ℂ) ≠ θ 
Arguments: {θ : ℂ}
Goal: Complex.tan θ ≠ 0 ↔ ∀ (k : ℤ), ↑k * ↑Real.pi / 2 ≠ θ
-- Complex.tan_int_mul_pi_div_two
theorem tan_int_mul_pi_div_two (n : ℤ) : tan (n * π / 2) = 0 
Arguments: (n : ℤ)
Goal: Complex.tan (↑n * ↑Real.pi / 2) = 0
-- Complex.tan_eq_zero_iff'
theorem tan_eq_zero_iff' {θ : ℂ} (hθ : cos θ ≠ 0) : tan θ = 0 ↔ ∃ k : ℤ, k * π = θ 
Arguments: {θ : ℂ} (hθ : Complex.cos θ ≠ 0)
Goal: Complex.tan θ = 0 ↔ ∃ k, ↑k * ↑Real.pi = θ
-- Complex.cos_eq_cos_iff
theorem cos_eq_cos_iff {x y : ℂ} : cos x = cos y ↔ ∃ k : ℤ, y = 2 * k * π + x ∨ y = 2 * k * π - x 
Arguments: {x y : ℂ}
Goal: Complex.cos x = Complex.cos y ↔ ∃ k, y = 2 * ↑k * ↑Real.pi + x ∨ y = 2 * ↑k * ↑Real.pi - x
-- Complex.sin_eq_sin_iff
theorem sin_eq_sin_iff {x y : ℂ} :
    sin x = sin y ↔ ∃ k : ℤ, y = 2 * k * π + x ∨ y = (2 * k + 1) * π - x 
Arguments: {x y : ℂ}
Goal: Complex.sin x = Complex.sin y ↔ ∃ k, y = 2 * ↑k * ↑Real.pi + x ∨ y = (2 * ↑k + 1) * ↑Real.pi - x
-- Complex.cos_eq_one_iff
theorem cos_eq_one_iff {x : ℂ} : cos x = 1 ↔ ∃ k : ℤ, k * (2 * π) = x 
Arguments: {x : ℂ}
Goal: Complex.cos x = 1 ↔ ∃ k, ↑k * (2 * ↑Real.pi) = x
-- Complex.cos_eq_neg_one_iff
theorem cos_eq_neg_one_iff {x : ℂ} : cos x = -1 ↔ ∃ k : ℤ, π + k * (2 * π) = x 
Arguments: {x : ℂ}
Goal: Complex.cos x = -1 ↔ ∃ k, ↑Real.pi + ↑k * (2 * ↑Real.pi) = x
-- Complex.sin_eq_one_iff
theorem sin_eq_one_iff {x : ℂ} : sin x = 1 ↔ ∃ k : ℤ, π / 2 + k * (2 * π) = x 
Arguments: {x : ℂ}
Goal: Complex.sin x = 1 ↔ ∃ k, ↑Real.pi / 2 + ↑k * (2 * ↑Real.pi) = x
-- Complex.sin_eq_neg_one_iff
theorem sin_eq_neg_one_iff {x : ℂ} : sin x = -1 ↔ ∃ k : ℤ, -(π / 2) + k * (2 * π) = x 
Arguments: {x : ℂ}
Goal: Complex.sin x = -1 ↔ ∃ k, -(↑Real.pi / 2) + ↑k * (2 * ↑Real.pi) = x
-- Complex.tan_add
theorem tan_add {x y : ℂ}
    (h : ((∀ k : ℤ, x ≠ (2 * k + 1) * π / 2) ∧ ∀ l : ℤ, y ≠ (2 * l + 1) * π / 2) ∨
      (∃ k : ℤ, x = (2 * k + 1) * π / 2) ∧ ∃ l : ℤ, y = (2 * l + 1) * π / 2) :
    tan (x + y) = (tan x + tan y) / (1 - tan x * tan y) 
Arguments: {x y : ℂ} (h :
  ((∀ (k : ℤ), x ≠ (2 * ↑k + 1) * ↑Real.pi / 2) ∧ ∀ (l : ℤ), y ≠ (2 * ↑l + 1) * ↑Real.pi / 2) ∨
    (∃ k, x = (2 * ↑k + 1) * ↑Real.pi / 2) ∧ ∃ l, y = (2 * ↑l + 1) * ↑Real.pi / 2)
Goal: Complex.tan (x + y) = (Complex.tan x + Complex.tan y) / (1 - Complex.tan x * Complex.tan y)
-- Complex.tan_add'
theorem tan_add' {x y : ℂ}
    (h : (∀ k : ℤ, x ≠ (2 * k + 1) * π / 2) ∧ ∀ l : ℤ, y ≠ (2 * l + 1) * π / 2) :
    tan (x + y) = (tan x + tan y) / (1 - tan x * tan y) 
Arguments: {x y : ℂ} (h : (∀ (k : ℤ), x ≠ (2 * ↑k + 1) * ↑Real.pi / 2) ∧ ∀ (l : ℤ), y ≠ (2 * ↑l + 1) * ↑Real.pi / 2)
Goal: Complex.tan (x + y) = (Complex.tan x + Complex.tan y) / (1 - Complex.tan x * Complex.tan y)
-- Complex.tan_two_mul
theorem tan_two_mul {z : ℂ} : tan (2 * z) = (2 : ℂ) * tan z / ((1 : ℂ) - tan z ^ 2) 
Arguments: {z : ℂ}
Goal: Complex.tan (2 * z) = 2 * Complex.tan z / (1 - Complex.tan z ^ 2)
-- Complex.tan_add_mul_I
theorem tan_add_mul_I {x y : ℂ}
    (h :
      ((∀ k : ℤ, x ≠ (2 * k + 1) * π / 2) ∧ ∀ l : ℤ, y * I ≠ (2 * l + 1) * π / 2) ∨
        (∃ k : ℤ, x = (2 * k + 1) * π / 2) ∧ ∃ l : ℤ, y * I = (2 * l + 1) * π / 2) :
    tan (x + y * I) = (tan x + tanh y * I) / (1 - tan x * tanh y * I) 
Arguments: {x y : ℂ} (h :
  ((∀ (k : ℤ), x ≠ (2 * ↑k + 1) * ↑Real.pi / 2) ∧ ∀ (l : ℤ), y * Complex.I ≠ (2 * ↑l + 1) * ↑Real.pi / 2) ∨
    (∃ k, x = (2 * ↑k + 1) * ↑Real.pi / 2) ∧ ∃ l, y * Complex.I = (2 * ↑l + 1) * ↑Real.pi / 2)
Goal: Complex.tan (x + y * Complex.I) =
  (Complex.tan x + Complex.tanh y * Complex.I) / (1 - Complex.tan x * Complex.tanh y * Complex.I)
-- Complex.tan_eq
theorem tan_eq {z : ℂ}
    (h :
      ((∀ k : ℤ, (z.re : ℂ) ≠ (2 * k + 1) * π / 2) ∧
          ∀ l : ℤ, (z.im : ℂ) * I ≠ (2 * l + 1) * π / 2) ∨
        (∃ k : ℤ, (z.re : ℂ) = (2 * k + 1) * π / 2) ∧
          ∃ l : ℤ, (z.im : ℂ) * I = (2 * l + 1) * π / 2) :
    tan z = (tan z.re + tanh z.im * I) / (1 - tan z.re * tanh z.im * I) 
Arguments: {z : ℂ} (h :
  ((∀ (k : ℤ), ↑z.re ≠ (2 * ↑k + 1) * ↑Real.pi / 2) ∧ ∀ (l : ℤ), ↑z.im * Complex.I ≠ (2 * ↑l + 1) * ↑Real.pi / 2) ∨
    (∃ k, ↑z.re = (2 * ↑k + 1) * ↑Real.pi / 2) ∧ ∃ l, ↑z.im * Complex.I = (2 * ↑l + 1) * ↑Real.pi / 2)
Goal: Complex.tan z =
  (Complex.tan ↑z.re + Complex.tanh ↑z.im * Complex.I) / (1 - Complex.tan ↑z.re * Complex.tanh ↑z.im * Complex.I)
-- Complex.continuousOn_tan
theorem continuousOn_tan : ContinuousOn tan {x | cos x ≠ 0} 
Goal: ContinuousOn Complex.tan {x | Complex.cos x ≠ 0}
-- Complex.continuous_tan
@[continuity]
theorem continuous_tan : Continuous fun x : {x | cos x ≠ 0} => tan x 
Goal: Continuous fun x => Complex.tan ↑x
-- Complex.cos_eq_iff_quadratic
theorem cos_eq_iff_quadratic {z w : ℂ} :
    cos z = w ↔ exp (z * I) ^ 2 - 2 * w * exp (z * I) + 1 = 0 
Arguments: {z w : ℂ}
Goal: Complex.cos z = w ↔ Complex.exp (z * Complex.I) ^ 2 - 2 * w * Complex.exp (z * Complex.I) + 1 = 0
-- Complex.cos_surjective
theorem cos_surjective : Function.Surjective cos 
Goal: Function.Surjective Complex.cos
-- Complex.range_cos
@[simp]
theorem range_cos : Set.range cos = Set.univ 
Goal: Set.range Complex.cos = Set.univ
-- Complex.sin_surjective
theorem sin_surjective : Function.Surjective sin 
Goal: Function.Surjective Complex.sin
-- Complex.range_sin
@[simp]
theorem range_sin : Set.range sin = Set.univ 
Goal: Set.range Complex.sin = Set.univ
-- Real.cos_eq_zero_iff
theorem cos_eq_zero_iff {θ : ℝ} : cos θ = 0 ↔ ∃ k : ℤ, θ = (2 * k + 1) * π / 2 
Arguments: {θ : ℝ}
Goal: Real.cos θ = 0 ↔ ∃ k, θ = (2 * ↑k + 1) * Real.pi / 2
-- Real.cos_ne_zero_iff
theorem cos_ne_zero_iff {θ : ℝ} : cos θ ≠ 0 ↔ ∀ k : ℤ, θ ≠ (2 * k + 1) * π / 2 
Arguments: {θ : ℝ}
Goal: Real.cos θ ≠ 0 ↔ ∀ (k : ℤ), θ ≠ (2 * ↑k + 1) * Real.pi / 2
-- Real.cos_eq_cos_iff
theorem cos_eq_cos_iff {x y : ℝ} : cos x = cos y ↔ ∃ k : ℤ, y = 2 * k * π + x ∨ y = 2 * k * π - x 
Arguments: {x y : ℝ}
Goal: Real.cos x = Real.cos y ↔ ∃ k, y = 2 * ↑k * Real.pi + x ∨ y = 2 * ↑k * Real.pi - x
-- Real.sin_eq_sin_iff
theorem sin_eq_sin_iff {x y : ℝ} :
    sin x = sin y ↔ ∃ k : ℤ, y = 2 * k * π + x ∨ y = (2 * k + 1) * π - x 
Arguments: {x y : ℝ}
Goal: Real.sin x = Real.sin y ↔ ∃ k, y = 2 * ↑k * Real.pi + x ∨ y = (2 * ↑k + 1) * Real.pi - x
-- Real.cos_eq_neg_one_iff
theorem cos_eq_neg_one_iff {x : ℝ} : cos x = -1 ↔ ∃ k : ℤ, π + k * (2 * π) = x 
Arguments: {x : ℝ}
Goal: Real.cos x = -1 ↔ ∃ k, Real.pi + ↑k * (2 * Real.pi) = x
-- Real.sin_eq_one_iff
theorem sin_eq_one_iff {x : ℝ} : sin x = 1 ↔ ∃ k : ℤ, π / 2 + k * (2 * π) = x 
Arguments: {x : ℝ}
Goal: Real.sin x = 1 ↔ ∃ k, Real.pi / 2 + ↑k * (2 * Real.pi) = x
-- Real.sin_eq_neg_one_iff
theorem sin_eq_neg_one_iff {x : ℝ} : sin x = -1 ↔ ∃ k : ℤ, -(π / 2) + k * (2 * π) = x 
Arguments: {x : ℝ}
Goal: Real.sin x = -1 ↔ ∃ k, -(Real.pi / 2) + ↑k * (2 * Real.pi) = x
-- Real.tan_eq_zero_iff
theorem tan_eq_zero_iff {θ : ℝ} : tan θ = 0 ↔ ∃ k : ℤ, k * π / 2 = θ 
Arguments: {θ : ℝ}
Goal: Real.tan θ = 0 ↔ ∃ k, ↑k * Real.pi / 2 = θ
-- Real.tan_eq_zero_iff'
theorem tan_eq_zero_iff' {θ : ℝ} (hθ : cos θ ≠ 0) : tan θ = 0 ↔ ∃ k : ℤ, k * π = θ 
Arguments: {θ : ℝ} (hθ : Real.cos θ ≠ 0)
Goal: Real.tan θ = 0 ↔ ∃ k, ↑k * Real.pi = θ
-- Real.tan_ne_zero_iff
theorem tan_ne_zero_iff {θ : ℝ} : tan θ ≠ 0 ↔ ∀ k : ℤ, k * π / 2 ≠ θ 
Arguments: {θ : ℝ}
Goal: Real.tan θ ≠ 0 ↔ ∀ (k : ℤ), ↑k * Real.pi / 2 ≠ θ
-- Real.lt_sin_mul
theorem lt_sin_mul {x : ℝ} (hx : 0 < x) (hx' : x < 1) : x < sin (π / 2 * x) 
Arguments: {x : ℝ} (hx : 0 < x) (hx' : x < 1)
Goal: x < Real.sin (Real.pi / 2 * x)
-- Real.le_sin_mul
theorem le_sin_mul {x : ℝ} (hx : 0 ≤ x) (hx' : x ≤ 1) : x ≤ sin (π / 2 * x) 
Arguments: {x : ℝ} (hx : 0 ≤ x) (hx' : x ≤ 1)
Goal: x ≤ Real.sin (Real.pi / 2 * x)
-- Real.mul_lt_sin
theorem mul_lt_sin {x : ℝ} (hx : 0 < x) (hx' : x < π / 2) : 2 / π * x < sin x 
Arguments: {x : ℝ} (hx : 0 < x) (hx' : x < Real.pi / 2)
Goal: 2 / Real.pi * x < Real.sin x
-- Real.mul_le_sin
theorem mul_le_sin {x : ℝ} (hx : 0 ≤ x) (hx' : x ≤ π / 2) : 2 / π * x ≤ sin x 
Arguments: {x : ℝ} (hx : 0 ≤ x) (hx' : x ≤ Real.pi / 2)
Goal: 2 / Real.pi * x ≤ Real.sin x
-- Complex.cot_eq_exp_ratio
lemma Complex.cot_eq_exp_ratio (z : ℂ) :
    cot z = (Complex.exp (2 * I * z) + 1) / (I * (1 - Complex.exp (2 * I * z))) 
Arguments: (z : ℂ)
Goal: z.cot = (cexp (2 * Complex.I * z) + 1) / (Complex.I * (1 - cexp (2 * Complex.I * z)))
-- Complex.cot_pi_eq_exp_ratio
lemma Complex.cot_pi_eq_exp_ratio (z : ℂ) :
    cot (π * z) = (Complex.exp (2 * π * I * z) + 1) / (I * (1 - Complex.exp (2 * π * I * z))) 
Arguments: (z : ℂ)
Goal: (↑π * z).cot = (cexp (2 * ↑π * Complex.I * z) + 1) / (Complex.I * (1 - cexp (2 * ↑π * Complex.I * z)))
-- pi_mul_cot_pi_q_exp
theorem pi_mul_cot_pi_q_exp (z : ℍ) :
    π * cot (π * z) = π * I - 2 * π * I * ∑' n : ℕ, Complex.exp (2 * π * I * z) ^ n 
Arguments: (z : ℍ)
Goal: ↑π * (↑π * ↑z).cot = ↑π * Complex.I - 2 * ↑π * Complex.I * ∑' (n : ℕ), cexp (2 * ↑π * Complex.I * ↑z) ^ n
-- EReal.exp_bot
@[simp] lemma exp_bot : exp ⊥ = 0 
Goal: ⊥.exp = 0
-- EReal.exp_zero
@[simp] lemma exp_zero : exp 0 = 1 
Goal: EReal.exp 0 = 1
-- EReal.exp_top
@[simp] lemma exp_top : exp ⊤ = ∞ 
Goal: ⊤.exp = ⊤
-- EReal.exp_coe
@[simp] lemma exp_coe (x : ℝ) : exp x = ENNReal.ofReal (Real.exp x) 
Arguments: (x : ℝ)
Goal: (↑x).exp = ENNReal.ofReal (Real.exp x)
-- EReal.exp_eq_zero_iff
@[simp] lemma exp_eq_zero_iff {x : EReal} : exp x = 0 ↔ x = ⊥ 
Arguments: {x : EReal}
Goal: x.exp = 0 ↔ x = ⊥
-- EReal.exp_eq_top_iff
@[simp] lemma exp_eq_top_iff {x : EReal} : exp x = ∞ ↔ x = ⊤ 
Arguments: {x : EReal}
Goal: x.exp = ⊤ ↔ x = ⊤
-- EReal.exp_strictMono
lemma exp_strictMono : StrictMono exp 
Goal: StrictMono EReal.exp
-- EReal.exp_monotone
lemma exp_monotone : Monotone exp 
Goal: Monotone EReal.exp
-- EReal.exp_lt_exp_iff
@[simp] lemma exp_lt_exp_iff {a b : EReal} : exp a < exp b ↔ a < b 
Arguments: {a b : EReal}
Goal: a.exp < b.exp ↔ a < b
-- EReal.zero_lt_exp_iff
@[simp] lemma zero_lt_exp_iff {a : EReal} : 0 < exp a ↔ ⊥ < a 
Arguments: {a : EReal}
Goal: 0 < a.exp ↔ ⊥ < a
-- EReal.exp_lt_top_iff
@[simp] lemma exp_lt_top_iff {a : EReal} : exp a < ⊤ ↔ a < ⊤ 
Arguments: {a : EReal}
Goal: a.exp < ⊤ ↔ a < ⊤
-- EReal.exp_lt_one_iff
@[simp] lemma exp_lt_one_iff {a : EReal} : exp a < 1 ↔ a < 0 
Arguments: {a : EReal}
Goal: a.exp < 1 ↔ a < 0
-- EReal.one_lt_exp_iff
@[simp] lemma one_lt_exp_iff {a : EReal} : 1 < exp a ↔ 0 < a 
Arguments: {a : EReal}
Goal: 1 < a.exp ↔ 0 < a
-- EReal.exp_le_exp_iff
@[simp] lemma exp_le_exp_iff {a b : EReal} : exp a ≤ exp b ↔ a ≤ b 
Arguments: {a b : EReal}
Goal: a.exp ≤ b.exp ↔ a ≤ b
-- EReal.exp_le_one_iff
@[simp] lemma exp_le_one_iff {a : EReal} : exp a ≤ 1 ↔ a ≤ 0 
Arguments: {a : EReal}
Goal: a.exp ≤ 1 ↔ a ≤ 0
-- EReal.one_le_exp_iff
@[simp] lemma one_le_exp_iff {a : EReal} : 1 ≤ exp a ↔ 0 ≤ a 
Arguments: {a : EReal}
Goal: 1 ≤ a.exp ↔ 0 ≤ a
-- EReal.exp_neg
lemma exp_neg (x : EReal) : exp (-x) = (exp x)⁻¹ 
Arguments: (x : EReal)
Goal: (-x).exp = x.exp⁻¹
-- EReal.exp_add
lemma exp_add (x y : EReal) : exp (x + y) = exp x * exp y 
Arguments: (x y : EReal)
Goal: (x + y).exp = x.exp * y.exp
-- affineSegment_eq_segment
theorem affineSegment_eq_segment (x y : V) : affineSegment R x y = segment R x y 
Arguments: (R : Type u_1) {V : Type u_2} [OrderedRing R] [AddCommGroup V] [Module R V] (x y : V)
Goal: affineSegment R x y = segment R x y
-- affineSegment_comm
theorem affineSegment_comm (x y : P) : affineSegment R x y = affineSegment R y x 
Arguments: (R : Type u_1) {V : Type u_2} {P : Type u_4} [OrderedRing R] [AddCommGroup V] [Module R V] [AddTorsor V P] (x y : P)
Goal: affineSegment R x y = affineSegment R y x
-- left_mem_affineSegment
theorem left_mem_affineSegment (x y : P) : x ∈ affineSegment R x y 
Arguments: (R : Type u_1) {V : Type u_2} {P : Type u_4} [OrderedRing R] [AddCommGroup V] [Module R V] [AddTorsor V P] (x y : P)
Goal: x ∈ affineSegment R x y
-- right_mem_affineSegment
theorem right_mem_affineSegment (x y : P) : y ∈ affineSegment R x y 
Arguments: (R : Type u_1) {V : Type u_2} {P : Type u_4} [OrderedRing R] [AddCommGroup V] [Module R V] [AddTorsor V P] (x y : P)
Goal: y ∈ affineSegment R x y
-- affineSegment_same
@[simp]
theorem affineSegment_same (x : P) : affineSegment R x x = {x} 
Arguments: (R : Type u_1) {V : Type u_2} {P : Type u_4} [OrderedRing R] [AddCommGroup V] [Module R V] [AddTorsor V P] (x : P)
Goal: affineSegment R x x = {x}
-- affineSegment_image
@[simp]
theorem affineSegment_image (f : P →ᵃ[R] P') (x y : P) :
    f '' affineSegment R x y = affineSegment R (f x) (f y) 
Arguments: {R : Type u_1} {V : Type u_2} {V' : Type u_3} {P : Type u_4} {P' : Type u_5} [OrderedRing R] [AddCommGroup V] [Module R V] [AddTorsor V P] [AddCommGroup V'] [Module R V'] [AddTorsor V' P'] (f : P →ᵃ[R] P') (x y : P)
Goal: ⇑f '' affineSegment R x y = affineSegment R (f x) (f y)
-- affineSegment_const_vadd_image
@[simp]
theorem affineSegment_const_vadd_image (x y : P) (v : V) :
    (v +ᵥ ·) '' affineSegment R x y = affineSegment R (v +ᵥ x) (v +ᵥ y) 
Arguments: (R : Type u_1) {V : Type u_2} {P : Type u_4} [OrderedRing R] [AddCommGroup V] [Module R V] [AddTorsor V P] (x y : P) (v : V)
Goal: (fun x => v +ᵥ x) '' affineSegment R x y = affineSegment R (v +ᵥ x) (v +ᵥ y)
-- affineSegment_vadd_const_image
@[simp]
theorem affineSegment_vadd_const_image (x y : V) (p : P) :
    (· +ᵥ p) '' affineSegment R x y = affineSegment R (x +ᵥ p) (y +ᵥ p) 
Arguments: (R : Type u_1) {V : Type u_2} {P : Type u_4} [OrderedRing R] [AddCommGroup V] [Module R V] [AddTorsor V P] (x y : V) (p : P)
Goal: (fun x => x +ᵥ p) '' affineSegment R x y = affineSegment R (x +ᵥ p) (y +ᵥ p)
-- affineSegment_const_vsub_image
@[simp]
theorem affineSegment_const_vsub_image (x y p : P) :
    (p -ᵥ ·) '' affineSegment R x y = affineSegment R (p -ᵥ x) (p -ᵥ y) 
Arguments: (R : Type u_1) {V : Type u_2} {P : Type u_4} [OrderedRing R] [AddCommGroup V] [Module R V] [AddTorsor V P] (x y p : P)
Goal: (fun x => p -ᵥ x) '' affineSegment R x y = affineSegment R (p -ᵥ x) (p -ᵥ y)
-- affineSegment_vsub_const_image
@[simp]
theorem affineSegment_vsub_const_image (x y p : P) :
    (· -ᵥ p) '' affineSegment R x y = affineSegment R (x -ᵥ p) (y -ᵥ p) 
Arguments: (R : Type u_1) {V : Type u_2} {P : Type u_4} [OrderedRing R] [AddCommGroup V] [Module R V] [AddTorsor V P] (x y p : P)
Goal: (fun x => x -ᵥ p) '' affineSegment R x y = affineSegment R (x -ᵥ p) (y -ᵥ p)
-- mem_const_vadd_affineSegment
@[simp]
theorem mem_const_vadd_affineSegment {x y z : P} (v : V) :
    v +ᵥ z ∈ affineSegment R (v +ᵥ x) (v +ᵥ y) ↔ z ∈ affineSegment R x y 
Arguments: {R : Type u_1} {V : Type u_2} {P : Type u_4} [OrderedRing R] [AddCommGroup V] [Module R V] [AddTorsor V P] {x y z : P} (v : V)
Goal: v +ᵥ z ∈ affineSegment R (v +ᵥ x) (v +ᵥ y) ↔ z ∈ affineSegment R x y
-- mem_vadd_const_affineSegment
@[simp]
theorem mem_vadd_const_affineSegment {x y z : V} (p : P) :
    z +ᵥ p ∈ affineSegment R (x +ᵥ p) (y +ᵥ p) ↔ z ∈ affineSegment R x y 
Arguments: {R : Type u_1} {V : Type u_2} {P : Type u_4} [OrderedRing R] [AddCommGroup V] [Module R V] [AddTorsor V P] {x y z : V} (p : P)
Goal: z +ᵥ p ∈ affineSegment R (x +ᵥ p) (y +ᵥ p) ↔ z ∈ affineSegment R x y
-- mem_const_vsub_affineSegment
@[simp]
theorem mem_const_vsub_affineSegment {x y z : P} (p : P) :
    p -ᵥ z ∈ affineSegment R (p -ᵥ x) (p -ᵥ y) ↔ z ∈ affineSegment R x y 
Arguments: {R : Type u_1} {V : Type u_2} {P : Type u_4} [OrderedRing R] [AddCommGroup V] [Module R V] [AddTorsor V P] {x y z : P} (p : P)
Goal: p -ᵥ z ∈ affineSegment R (p -ᵥ x) (p -ᵥ y) ↔ z ∈ affineSegment R x y
-- mem_vsub_const_affineSegment
@[simp]
theorem mem_vsub_const_affineSegment {x y z : P} (p : P) :
    z -ᵥ p ∈ affineSegment R (x -ᵥ p) (y -ᵥ p) ↔ z ∈ affineSegment R x y 
Arguments: {R : Type u_1} {V : Type u_2} {P : Type u_4} [OrderedRing R] [AddCommGroup V] [Module R V] [AddTorsor V P] {x y z : P} (p : P)
Goal: z -ᵥ p ∈ affineSegment R (x -ᵥ p) (y -ᵥ p) ↔ z ∈ affineSegment R x y
-- mem_segment_iff_wbtw
lemma mem_segment_iff_wbtw {x y z : V} : y ∈ segment R x z ↔ Wbtw R x y z 
Arguments: {R : Type u_1} {V : Type u_2} [OrderedRing R] [AddCommGroup V] [Module R V] {x y z : V}
Goal: y ∈ segment R x z ↔ Wbtw R x y z
-- Wbtw.map
theorem Wbtw.map {x y z : P} (h : Wbtw R x y z) (f : P →ᵃ[R] P') : Wbtw R (f x) (f y) (f z) 
Arguments: {R : Type u_1} {V : Type u_2} {V' : Type u_3} {P : Type u_4} {P' : Type u_5} [OrderedRing R] [AddCommGroup V] [Module R V] [AddTorsor V P] [AddCommGroup V'] [Module R V'] [AddTorsor V' P'] {x y z : P} (h : Wbtw R x y z) (f : P →ᵃ[R] P')
Goal: Wbtw R (f x) (f y) (f z)