ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- strictConcaveOn_iff_div
theorem strictConcaveOn_iff_div {f : E → β} :
    StrictConcaveOn 𝕜 s f ↔
      Convex 𝕜 s ∧ ∀ ⦃x⦄, x ∈ s → ∀ ⦃y⦄, y ∈ s → x ≠ y → ∀ ⦃a b : 𝕜⦄, 0 < a → 0 < b →
        (a / (a + b)) • f x + (b / (a + b)) • f y < f ((a / (a + b)) • x + (b / (a + b)) • y) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {β : Type u_5} [LinearOrderedField 𝕜] [AddCommMonoid E] [OrderedAddCommMonoid β] [SMul 𝕜 E] [SMul 𝕜 β] {s : Set E} {f : E → β}
Goal: StrictConcaveOn 𝕜 s f ↔
  Convex 𝕜 s ∧
    ∀ ⦃x : E⦄,
      x ∈ s →
        ∀ ⦃y : E⦄,
          y ∈ s →
            x ≠ y →
              ∀ ⦃a b : 𝕜⦄,
                0 < a → 0 < b → (a / (a + b)) • f x + (b / (a + b)) • f y < f ((a / (a + b)) • x + (b / (a + b)) • y)
-- OrderIso.strictConvexOn_symm
theorem OrderIso.strictConvexOn_symm (f : α ≃o β) (hf : StrictConcaveOn 𝕜 univ f) :
    StrictConvexOn 𝕜 univ f.symm 
Arguments: {𝕜 : Type u_1} {α : Type u_4} {β : Type u_5} [OrderedSemiring 𝕜] [OrderedAddCommMonoid α] [SMul 𝕜 α] [OrderedAddCommMonoid β] [SMul 𝕜 β] (f : α ≃o β) (hf : StrictConcaveOn 𝕜 Set.univ ⇑f)
Goal: StrictConvexOn 𝕜 Set.univ ⇑f.symm
-- OrderIso.convexOn_symm
theorem OrderIso.convexOn_symm (f : α ≃o β) (hf : ConcaveOn 𝕜 univ f) :
    ConvexOn 𝕜 univ f.symm 
Arguments: {𝕜 : Type u_1} {α : Type u_4} {β : Type u_5} [OrderedSemiring 𝕜] [OrderedAddCommMonoid α] [SMul 𝕜 α] [OrderedAddCommMonoid β] [SMul 𝕜 β] (f : α ≃o β) (hf : ConcaveOn 𝕜 Set.univ ⇑f)
Goal: ConvexOn 𝕜 Set.univ ⇑f.symm
-- OrderIso.strictConcaveOn_symm
theorem OrderIso.strictConcaveOn_symm (f : α ≃o β) (hf : StrictConvexOn 𝕜 univ f) :
    StrictConcaveOn 𝕜 univ f.symm 
Arguments: {𝕜 : Type u_1} {α : Type u_4} {β : Type u_5} [OrderedSemiring 𝕜] [OrderedAddCommMonoid α] [SMul 𝕜 α] [OrderedAddCommMonoid β] [SMul 𝕜 β] (f : α ≃o β) (hf : StrictConvexOn 𝕜 Set.univ ⇑f)
Goal: StrictConcaveOn 𝕜 Set.univ ⇑f.symm
-- OrderIso.concaveOn_symm
theorem OrderIso.concaveOn_symm (f : α ≃o β) (hf : ConvexOn 𝕜 univ f) :
    ConcaveOn 𝕜 univ f.symm 
Arguments: {𝕜 : Type u_1} {α : Type u_4} {β : Type u_5} [OrderedSemiring 𝕜] [OrderedAddCommMonoid α] [SMul 𝕜 α] [OrderedAddCommMonoid β] [SMul 𝕜 β] (f : α ≃o β) (hf : ConvexOn 𝕜 Set.univ ⇑f)
Goal: ConcaveOn 𝕜 Set.univ ⇑f.symm
-- StrictConvexOn.eq_of_isMinOn
lemma StrictConvexOn.eq_of_isMinOn (hf : StrictConvexOn 𝕜 s f) (hfx : IsMinOn f s x)
    (hfy : IsMinOn f s y) (hx : x ∈ s) (hy : y ∈ s) : x = y 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {β : Type u_5} [LinearOrderedField 𝕜] [OrderedAddCommMonoid β] [AddCommMonoid E] [SMul 𝕜 E] [Module 𝕜 β] [OrderedSMul 𝕜 β] {f : E → β} {s : Set E} {x y : E} (hf : StrictConvexOn 𝕜 s f) (hfx : IsMinOn f s x) (hfy : IsMinOn f s y) (hx : x ∈ s) (hy : y ∈ s)
Goal: x = y
-- StrictConcaveOn.eq_of_isMaxOn
lemma StrictConcaveOn.eq_of_isMaxOn (hf : StrictConcaveOn 𝕜 s f) (hfx : IsMaxOn f s x)
    (hfy : IsMaxOn f s y) (hx : x ∈ s) (hy : y ∈ s) : x = y 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {β : Type u_5} [LinearOrderedField 𝕜] [OrderedAddCommMonoid β] [AddCommMonoid E] [SMul 𝕜 E] [Module 𝕜 β] [OrderedSMul 𝕜 β] {f : E → β} {s : Set E} {x y : E} (hf : StrictConcaveOn 𝕜 s f) (hfx : IsMaxOn f s x) (hfy : IsMaxOn f s y) (hx : x ∈ s) (hy : y ∈ s)
Goal: x = y
-- ConvexOn.le_right_of_left_le''
theorem ConvexOn.le_right_of_left_le'' (hf : ConvexOn 𝕜 s f) (hx : x ∈ s) (hz : z ∈ s) (hxy : x < y)
    (hyz : y ≤ z) (h : f x ≤ f y) : f y ≤ f z 
Arguments: {𝕜 : Type u_1} {β : Type u_5} [LinearOrderedField 𝕜] [LinearOrderedCancelAddCommMonoid β] [Module 𝕜 β] [OrderedSMul 𝕜 β] {x y z : 𝕜} {s : Set 𝕜} {f : 𝕜 → β} (hf : ConvexOn 𝕜 s f) (hx : x ∈ s) (hz : z ∈ s) (hxy : x < y) (hyz : y ≤ z) (h : f x ≤ f y)
Goal: f y ≤ f z
-- ConvexOn.le_left_of_right_le''
theorem ConvexOn.le_left_of_right_le'' (hf : ConvexOn 𝕜 s f) (hx : x ∈ s) (hz : z ∈ s) (hxy : x ≤ y)
    (hyz : y < z) (h : f z ≤ f y) : f y ≤ f x 
Arguments: {𝕜 : Type u_1} {β : Type u_5} [LinearOrderedField 𝕜] [LinearOrderedCancelAddCommMonoid β] [Module 𝕜 β] [OrderedSMul 𝕜 β] {x y z : 𝕜} {s : Set 𝕜} {f : 𝕜 → β} (hf : ConvexOn 𝕜 s f) (hx : x ∈ s) (hz : z ∈ s) (hxy : x ≤ y) (hyz : y < z) (h : f z ≤ f y)
Goal: f y ≤ f x
-- ConcaveOn.right_le_of_le_left''
theorem ConcaveOn.right_le_of_le_left'' (hf : ConcaveOn 𝕜 s f) (hx : x ∈ s) (hz : z ∈ s)
    (hxy : x < y) (hyz : y ≤ z) (h : f y ≤ f x) : f z ≤ f y 
Arguments: {𝕜 : Type u_1} {β : Type u_5} [LinearOrderedField 𝕜] [LinearOrderedCancelAddCommMonoid β] [Module 𝕜 β] [OrderedSMul 𝕜 β] {x y z : 𝕜} {s : Set 𝕜} {f : 𝕜 → β} (hf : ConcaveOn 𝕜 s f) (hx : x ∈ s) (hz : z ∈ s) (hxy : x < y) (hyz : y ≤ z) (h : f y ≤ f x)
Goal: f z ≤ f y
-- ConcaveOn.left_le_of_le_right''
theorem ConcaveOn.left_le_of_le_right'' (hf : ConcaveOn 𝕜 s f) (hx : x ∈ s) (hz : z ∈ s)
    (hxy : x ≤ y) (hyz : y < z) (h : f y ≤ f z) : f x ≤ f y 
Arguments: {𝕜 : Type u_1} {β : Type u_5} [LinearOrderedField 𝕜] [LinearOrderedCancelAddCommMonoid β] [Module 𝕜 β] [OrderedSMul 𝕜 β] {x y z : 𝕜} {s : Set 𝕜} {f : 𝕜 → β} (hf : ConcaveOn 𝕜 s f) (hx : x ∈ s) (hz : z ∈ s) (hxy : x ≤ y) (hyz : y < z) (h : f y ≤ f z)
Goal: f x ≤ f y
-- Real.Angle.continuous_coe
@[continuity]
theorem continuous_coe : Continuous ((↑) : ℝ → Angle) 
Goal: Continuous Real.Angle.coe
-- Real.Angle.coe_coeHom
@[simp]
theorem coe_coeHom : (coeHom : ℝ → Angle) = ((↑) : ℝ → Angle) 
Goal: ⇑Real.Angle.coeHom = Real.Angle.coe
-- Real.Angle.induction_on
@[elab_as_elim]
protected theorem induction_on {p : Angle → Prop} (θ : Angle) (h : ∀ x : ℝ, p x) : p θ 
Arguments: {p : Real.Angle → Prop} (θ : Real.Angle) (h : ∀ (x : ℝ), p ↑x)
Goal: p θ
-- Real.Angle.coe_zero
@[simp]
theorem coe_zero : ↑(0 : ℝ) = (0 : Angle) 
Goal: ↑0 = 0
-- Real.Angle.coe_add
@[simp]
theorem coe_add (x y : ℝ) : ↑(x + y : ℝ) = (↑x + ↑y : Angle) 
Arguments: (x y : ℝ)
Goal: ↑(x + y) = ↑x + ↑y
-- Real.Angle.coe_neg
@[simp]
theorem coe_neg (x : ℝ) : ↑(-x : ℝ) = -(↑x : Angle) 
Arguments: (x : ℝ)
Goal: ↑(-x) = -↑x
-- Real.Angle.coe_sub
@[simp]
theorem coe_sub (x y : ℝ) : ↑(x - y : ℝ) = (↑x - ↑y : Angle) 
Arguments: (x y : ℝ)
Goal: ↑(x - y) = ↑x - ↑y
-- Real.Angle.coe_nsmul
theorem coe_nsmul (n : ℕ) (x : ℝ) : ↑(n • x : ℝ) = n • (↑x : Angle) 
Arguments: (n : ℕ) (x : ℝ)
Goal: ↑(n • x) = n • ↑x
-- Real.Angle.coe_zsmul
theorem coe_zsmul (z : ℤ) (x : ℝ) : ↑(z • x : ℝ) = z • (↑x : Angle) 
Arguments: (z : ℤ) (x : ℝ)
Goal: ↑(z • x) = z • ↑x
-- Real.Angle.natCast_mul_eq_nsmul
@[simp, norm_cast]
theorem natCast_mul_eq_nsmul (x : ℝ) (n : ℕ) : ↑((n : ℝ) * x) = n • (↑x : Angle) 
Arguments: (x : ℝ) (n : ℕ)
Goal: ↑(↑n * x) = n • ↑x
-- Real.Angle.intCast_mul_eq_zsmul
@[simp, norm_cast]
theorem intCast_mul_eq_zsmul (x : ℝ) (n : ℤ) : ↑((n : ℝ) * x : ℝ) = n • (↑x : Angle) 
Arguments: (x : ℝ) (n : ℤ)
Goal: ↑(↑n * x) = n • ↑x
-- Real.Angle.angle_eq_iff_two_pi_dvd_sub
theorem angle_eq_iff_two_pi_dvd_sub {ψ θ : ℝ} : (θ : Angle) = ψ ↔ ∃ k : ℤ, θ - ψ = 2 * π * k 
Arguments: {ψ θ : ℝ}
Goal: ↑θ = ↑ψ ↔ ∃ k, θ - ψ = 2 * π * ↑k
-- Real.Angle.coe_two_pi
@[simp]
theorem coe_two_pi : ↑(2 * π : ℝ) = (0 : Angle) 
Goal: ↑(2 * π) = 0
-- Real.Angle.neg_coe_pi
@[simp]
theorem neg_coe_pi : -(π : Angle) = π 
Goal: -↑π = ↑π
-- Real.Angle.two_nsmul_coe_div_two
@[simp]
theorem two_nsmul_coe_div_two (θ : ℝ) : (2 : ℕ) • (↑(θ / 2) : Angle) = θ 
Arguments: (θ : ℝ)
Goal: 2 • ↑(θ / 2) = ↑θ
-- Real.Angle.two_zsmul_coe_div_two
@[simp]
theorem two_zsmul_coe_div_two (θ : ℝ) : (2 : ℤ) • (↑(θ / 2) : Angle) = θ 
Arguments: (θ : ℝ)
Goal: 2 • ↑(θ / 2) = ↑θ
-- Real.Angle.two_nsmul_neg_pi_div_two
theorem two_nsmul_neg_pi_div_two : (2 : ℕ) • (↑(-π / 2) : Angle) = π 
Goal: 2 • ↑(-π / 2) = ↑π
-- Real.Angle.two_zsmul_neg_pi_div_two
theorem two_zsmul_neg_pi_div_two : (2 : ℤ) • (↑(-π / 2) : Angle) = π 
Goal: 2 • ↑(-π / 2) = ↑π
-- Real.Angle.sub_coe_pi_eq_add_coe_pi
theorem sub_coe_pi_eq_add_coe_pi (θ : Angle) : θ - π = θ + π 
Arguments: (θ : Real.Angle)
Goal: θ - ↑π = θ + ↑π
-- Real.Angle.two_nsmul_coe_pi
@[simp]
theorem two_nsmul_coe_pi : (2 : ℕ) • (π : Angle) = 0 
Goal: 2 • ↑π = 0
-- Real.Angle.two_zsmul_coe_pi
@[simp]
theorem two_zsmul_coe_pi : (2 : ℤ) • (π : Angle) = 0 
Goal: 2 • ↑π = 0
-- Real.Angle.coe_pi_add_coe_pi
@[simp]
theorem coe_pi_add_coe_pi : (π : Real.Angle) + π = 0 
Goal: ↑π + ↑π = 0
-- Real.Angle.zsmul_eq_iff
theorem zsmul_eq_iff {ψ θ : Angle} {z : ℤ} (hz : z ≠ 0) :
    z • ψ = z • θ ↔ ∃ k : Fin z.natAbs, ψ = θ + (k : ℕ) • (2 * π / z : ℝ) 
Arguments: {ψ θ : Real.Angle} {z : ℤ} (hz : z ≠ 0)
Goal: z • ψ = z • θ ↔ ∃ k, ψ = θ + ↑k • ↑(2 * π / ↑z)
-- Real.Angle.nsmul_eq_iff
theorem nsmul_eq_iff {ψ θ : Angle} {n : ℕ} (hz : n ≠ 0) :
    n • ψ = n • θ ↔ ∃ k : Fin n, ψ = θ + (k : ℕ) • (2 * π / n : ℝ) 
Arguments: {ψ θ : Real.Angle} {n : ℕ} (hz : n ≠ 0)
Goal: n • ψ = n • θ ↔ ∃ k, ψ = θ + ↑k • ↑(2 * π / ↑n)
-- Real.Angle.two_zsmul_eq_iff
theorem two_zsmul_eq_iff {ψ θ : Angle} : (2 : ℤ) • ψ = (2 : ℤ) • θ ↔ ψ = θ ∨ ψ = θ + ↑π 
Arguments: {ψ θ : Real.Angle}
Goal: 2 • ψ = 2 • θ ↔ ψ = θ ∨ ψ = θ + ↑π
-- Real.Angle.two_nsmul_eq_iff
theorem two_nsmul_eq_iff {ψ θ : Angle} : (2 : ℕ) • ψ = (2 : ℕ) • θ ↔ ψ = θ ∨ ψ = θ + ↑π 
Arguments: {ψ θ : Real.Angle}
Goal: 2 • ψ = 2 • θ ↔ ψ = θ ∨ ψ = θ + ↑π
-- Real.Angle.two_nsmul_eq_zero_iff
theorem two_nsmul_eq_zero_iff {θ : Angle} : (2 : ℕ) • θ = 0 ↔ θ = 0 ∨ θ = π 
Arguments: {θ : Real.Angle}
Goal: 2 • θ = 0 ↔ θ = 0 ∨ θ = ↑π
-- Real.Angle.two_nsmul_ne_zero_iff
theorem two_nsmul_ne_zero_iff {θ : Angle} : (2 : ℕ) • θ ≠ 0 ↔ θ ≠ 0 ∧ θ ≠ π 
Arguments: {θ : Real.Angle}
Goal: 2 • θ ≠ 0 ↔ θ ≠ 0 ∧ θ ≠ ↑π
-- Real.Angle.two_zsmul_eq_zero_iff
theorem two_zsmul_eq_zero_iff {θ : Angle} : (2 : ℤ) • θ = 0 ↔ θ = 0 ∨ θ = π 
Arguments: {θ : Real.Angle}
Goal: 2 • θ = 0 ↔ θ = 0 ∨ θ = ↑π
-- Real.Angle.two_zsmul_ne_zero_iff
theorem two_zsmul_ne_zero_iff {θ : Angle} : (2 : ℤ) • θ ≠ 0 ↔ θ ≠ 0 ∧ θ ≠ π 
Arguments: {θ : Real.Angle}
Goal: 2 • θ ≠ 0 ↔ θ ≠ 0 ∧ θ ≠ ↑π
-- Real.Angle.eq_neg_self_iff
theorem eq_neg_self_iff {θ : Angle} : θ = -θ ↔ θ = 0 ∨ θ = π 
Arguments: {θ : Real.Angle}
Goal: θ = -θ ↔ θ = 0 ∨ θ = ↑π
-- Real.Angle.ne_neg_self_iff
theorem ne_neg_self_iff {θ : Angle} : θ ≠ -θ ↔ θ ≠ 0 ∧ θ ≠ π 
Arguments: {θ : Real.Angle}
Goal: θ ≠ -θ ↔ θ ≠ 0 ∧ θ ≠ ↑π
-- Real.Angle.neg_eq_self_iff
theorem neg_eq_self_iff {θ : Angle} : -θ = θ ↔ θ = 0 ∨ θ = π 
Arguments: {θ : Real.Angle}
Goal: -θ = θ ↔ θ = 0 ∨ θ = ↑π
-- Real.Angle.neg_ne_self_iff
theorem neg_ne_self_iff {θ : Angle} : -θ ≠ θ ↔ θ ≠ 0 ∧ θ ≠ π 
Arguments: {θ : Real.Angle}
Goal: -θ ≠ θ ↔ θ ≠ 0 ∧ θ ≠ ↑π
-- Real.Angle.two_nsmul_eq_pi_iff
theorem two_nsmul_eq_pi_iff {θ : Angle} : (2 : ℕ) • θ = π ↔ θ = (π / 2 : ℝ) ∨ θ = (-π / 2 : ℝ) 
Arguments: {θ : Real.Angle}
Goal: 2 • θ = ↑π ↔ θ = ↑(π / 2) ∨ θ = ↑(-π / 2)
-- Real.Angle.two_zsmul_eq_pi_iff
theorem two_zsmul_eq_pi_iff {θ : Angle} : (2 : ℤ) • θ = π ↔ θ = (π / 2 : ℝ) ∨ θ = (-π / 2 : ℝ) 
Arguments: {θ : Real.Angle}
Goal: 2 • θ = ↑π ↔ θ = ↑(π / 2) ∨ θ = ↑(-π / 2)
-- Real.Angle.cos_eq_iff_coe_eq_or_eq_neg
theorem cos_eq_iff_coe_eq_or_eq_neg {θ ψ : ℝ} :
    cos θ = cos ψ ↔ (θ : Angle) = ψ ∨ (θ : Angle) = -ψ 
Arguments: {θ ψ : ℝ}
Goal: Real.cos θ = Real.cos ψ ↔ ↑θ = ↑ψ ∨ ↑θ = -↑ψ
-- Real.Angle.sin_eq_iff_coe_eq_or_add_eq_pi
theorem sin_eq_iff_coe_eq_or_add_eq_pi {θ ψ : ℝ} :
    sin θ = sin ψ ↔ (θ : Angle) = ψ ∨ (θ : Angle) + ψ = π 
Arguments: {θ ψ : ℝ}
Goal: Real.sin θ = Real.sin ψ ↔ ↑θ = ↑ψ ∨ ↑θ + ↑ψ = ↑π
-- Real.Angle.cos_sin_inj
theorem cos_sin_inj {θ ψ : ℝ} (Hcos : cos θ = cos ψ) (Hsin : sin θ = sin ψ) : (θ : Angle) = ψ 
Arguments: {θ ψ : ℝ} (Hcos : Real.cos θ = Real.cos ψ) (Hsin : Real.sin θ = Real.sin ψ)
Goal: ↑θ = ↑ψ
-- Real.Angle.sin_coe
@[simp]
theorem sin_coe (x : ℝ) : sin (x : Angle) = Real.sin x 
Arguments: (x : ℝ)
Goal: (↑x).sin = Real.sin x
-- Real.Angle.continuous_sin
@[continuity]
theorem continuous_sin : Continuous sin 
Goal: Continuous Real.Angle.sin
-- Real.Angle.cos_coe
@[simp]
theorem cos_coe (x : ℝ) : cos (x : Angle) = Real.cos x 
Arguments: (x : ℝ)
Goal: (↑x).cos = Real.cos x
-- Real.Angle.continuous_cos
@[continuity]
theorem continuous_cos : Continuous cos 
Goal: Continuous Real.Angle.cos
-- Real.Angle.cos_eq_real_cos_iff_eq_or_eq_neg
theorem cos_eq_real_cos_iff_eq_or_eq_neg {θ : Angle} {ψ : ℝ} :
    cos θ = Real.cos ψ ↔ θ = ψ ∨ θ = -ψ 
Arguments: {θ : Real.Angle} {ψ : ℝ}
Goal: θ.cos = Real.cos ψ ↔ θ = ↑ψ ∨ θ = -↑ψ
-- Real.Angle.cos_eq_iff_eq_or_eq_neg
theorem cos_eq_iff_eq_or_eq_neg {θ ψ : Angle} : cos θ = cos ψ ↔ θ = ψ ∨ θ = -ψ 
Arguments: {θ ψ : Real.Angle}
Goal: θ.cos = ψ.cos ↔ θ = ψ ∨ θ = -ψ
-- Real.Angle.sin_eq_real_sin_iff_eq_or_add_eq_pi
theorem sin_eq_real_sin_iff_eq_or_add_eq_pi {θ : Angle} {ψ : ℝ} :
    sin θ = Real.sin ψ ↔ θ = ψ ∨ θ + ψ = π 
Arguments: {θ : Real.Angle} {ψ : ℝ}
Goal: θ.sin = Real.sin ψ ↔ θ = ↑ψ ∨ θ + ↑ψ = ↑π
-- Real.Angle.sin_eq_iff_eq_or_add_eq_pi
theorem sin_eq_iff_eq_or_add_eq_pi {θ ψ : Angle} : sin θ = sin ψ ↔ θ = ψ ∨ θ + ψ = π 
Arguments: {θ ψ : Real.Angle}
Goal: θ.sin = ψ.sin ↔ θ = ψ ∨ θ + ψ = ↑π
-- Real.Angle.sin_zero
@[simp]
theorem sin_zero : sin (0 : Angle) = 0 
Goal: Real.Angle.sin 0 = 0
-- Real.Angle.sin_coe_pi
theorem sin_coe_pi : sin (π : Angle) = 0 
Goal: (↑π).sin = 0
-- Real.Angle.sin_eq_zero_iff
theorem sin_eq_zero_iff {θ : Angle} : sin θ = 0 ↔ θ = 0 ∨ θ = π 
Arguments: {θ : Real.Angle}
Goal: θ.sin = 0 ↔ θ = 0 ∨ θ = ↑π
-- Real.Angle.sin_ne_zero_iff
theorem sin_ne_zero_iff {θ : Angle} : sin θ ≠ 0 ↔ θ ≠ 0 ∧ θ ≠ π 
Arguments: {θ : Real.Angle}
Goal: θ.sin ≠ 0 ↔ θ ≠ 0 ∧ θ ≠ ↑π
-- Real.Angle.sin_neg
@[simp]
theorem sin_neg (θ : Angle) : sin (-θ) = -sin θ 
Arguments: (θ : Real.Angle)
Goal: (-θ).sin = -θ.sin
-- Real.Angle.sin_antiperiodic
theorem sin_antiperiodic : Function.Antiperiodic sin (π : Angle) 
Goal: Function.Antiperiodic Real.Angle.sin ↑π
-- Real.Angle.sin_add_pi
@[simp]
theorem sin_add_pi (θ : Angle) : sin (θ + π) = -sin θ 
Arguments: (θ : Real.Angle)
Goal: (θ + ↑π).sin = -θ.sin
-- Real.Angle.sin_sub_pi
@[simp]
theorem sin_sub_pi (θ : Angle) : sin (θ - π) = -sin θ 
Arguments: (θ : Real.Angle)
Goal: (θ - ↑π).sin = -θ.sin
-- Real.Angle.cos_zero
@[simp]
theorem cos_zero : cos (0 : Angle) = 1 
Goal: Real.Angle.cos 0 = 1
-- Real.Angle.cos_coe_pi
theorem cos_coe_pi : cos (π : Angle) = -1 
Goal: (↑π).cos = -1
-- Real.Angle.cos_neg
@[simp]
theorem cos_neg (θ : Angle) : cos (-θ) = cos θ 
Arguments: (θ : Real.Angle)
Goal: (-θ).cos = θ.cos
-- Real.Angle.cos_antiperiodic
theorem cos_antiperiodic : Function.Antiperiodic cos (π : Angle) 
Goal: Function.Antiperiodic Real.Angle.cos ↑π
-- Real.Angle.cos_add_pi
@[simp]
theorem cos_add_pi (θ : Angle) : cos (θ + π) = -cos θ 
Arguments: (θ : Real.Angle)
Goal: (θ + ↑π).cos = -θ.cos
-- Real.Angle.cos_sub_pi
@[simp]
theorem cos_sub_pi (θ : Angle) : cos (θ - π) = -cos θ 
Arguments: (θ : Real.Angle)
Goal: (θ - ↑π).cos = -θ.cos
-- Real.Angle.cos_eq_zero_iff
theorem cos_eq_zero_iff {θ : Angle} : cos θ = 0 ↔ θ = (π / 2 : ℝ) ∨ θ = (-π / 2 : ℝ) 
Arguments: {θ : Real.Angle}
Goal: θ.cos = 0 ↔ θ = ↑(π / 2) ∨ θ = ↑(-π / 2)
-- Real.Angle.sin_add
theorem sin_add (θ₁ θ₂ : Real.Angle) : sin (θ₁ + θ₂) = sin θ₁ * cos θ₂ + cos θ₁ * sin θ₂ 
Arguments: (θ₁ θ₂ : Real.Angle)
Goal: (θ₁ + θ₂).sin = θ₁.sin * θ₂.cos + θ₁.cos * θ₂.sin
-- Real.Angle.cos_add
theorem cos_add (θ₁ θ₂ : Real.Angle) : cos (θ₁ + θ₂) = cos θ₁ * cos θ₂ - sin θ₁ * sin θ₂ 
Arguments: (θ₁ θ₂ : Real.Angle)
Goal: (θ₁ + θ₂).cos = θ₁.cos * θ₂.cos - θ₁.sin * θ₂.sin
-- Real.Angle.cos_sq_add_sin_sq
@[simp]
theorem cos_sq_add_sin_sq (θ : Real.Angle) : cos θ ^ 2 + sin θ ^ 2 = 1 
Arguments: (θ : Real.Angle)
Goal: θ.cos ^ 2 + θ.sin ^ 2 = 1
-- Real.Angle.sin_add_pi_div_two
theorem sin_add_pi_div_two (θ : Angle) : sin (θ + ↑(π / 2)) = cos θ 
Arguments: (θ : Real.Angle)
Goal: (θ + ↑(π / 2)).sin = θ.cos
-- Real.Angle.sin_sub_pi_div_two
theorem sin_sub_pi_div_two (θ : Angle) : sin (θ - ↑(π / 2)) = -cos θ 
Arguments: (θ : Real.Angle)
Goal: (θ - ↑(π / 2)).sin = -θ.cos
-- Real.Angle.sin_pi_div_two_sub
theorem sin_pi_div_two_sub (θ : Angle) : sin (↑(π / 2) - θ) = cos θ 
Arguments: (θ : Real.Angle)
Goal: (↑(π / 2) - θ).sin = θ.cos
-- Real.Angle.cos_add_pi_div_two
theorem cos_add_pi_div_two (θ : Angle) : cos (θ + ↑(π / 2)) = -sin θ 
Arguments: (θ : Real.Angle)
Goal: (θ + ↑(π / 2)).cos = -θ.sin
-- Real.Angle.cos_sub_pi_div_two
theorem cos_sub_pi_div_two (θ : Angle) : cos (θ - ↑(π / 2)) = sin θ 
Arguments: (θ : Real.Angle)
Goal: (θ - ↑(π / 2)).cos = θ.sin
-- Real.Angle.cos_pi_div_two_sub
theorem cos_pi_div_two_sub (θ : Angle) : cos (↑(π / 2) - θ) = sin θ 
Arguments: (θ : Real.Angle)
Goal: (↑(π / 2) - θ).cos = θ.sin
-- Real.Angle.abs_sin_eq_of_two_nsmul_eq
theorem abs_sin_eq_of_two_nsmul_eq {θ ψ : Angle} (h : (2 : ℕ) • θ = (2 : ℕ) • ψ) :
    |sin θ| = |sin ψ| 
Arguments: {θ ψ : Real.Angle} (h : 2 • θ = 2 • ψ)
Goal: |θ.sin| = |ψ.sin|
-- Real.Angle.abs_sin_eq_of_two_zsmul_eq
theorem abs_sin_eq_of_two_zsmul_eq {θ ψ : Angle} (h : (2 : ℤ) • θ = (2 : ℤ) • ψ) :
    |sin θ| = |sin ψ| 
Arguments: {θ ψ : Real.Angle} (h : 2 • θ = 2 • ψ)
Goal: |θ.sin| = |ψ.sin|
-- Real.Angle.abs_cos_eq_of_two_nsmul_eq
theorem abs_cos_eq_of_two_nsmul_eq {θ ψ : Angle} (h : (2 : ℕ) • θ = (2 : ℕ) • ψ) :
    |cos θ| = |cos ψ| 
Arguments: {θ ψ : Real.Angle} (h : 2 • θ = 2 • ψ)
Goal: |θ.cos| = |ψ.cos|
-- Real.Angle.abs_cos_eq_of_two_zsmul_eq
theorem abs_cos_eq_of_two_zsmul_eq {θ ψ : Angle} (h : (2 : ℤ) • θ = (2 : ℤ) • ψ) :
    |cos θ| = |cos ψ| 
Arguments: {θ ψ : Real.Angle} (h : 2 • θ = 2 • ψ)
Goal: |θ.cos| = |ψ.cos|
-- Real.Angle.coe_toIcoMod
@[simp]
theorem coe_toIcoMod (θ ψ : ℝ) : ↑(toIcoMod two_pi_pos ψ θ) = (θ : Angle) 
Arguments: (θ ψ : ℝ)
Goal: ↑(toIcoMod Real.two_pi_pos ψ θ) = ↑θ
-- Real.Angle.coe_toIocMod
@[simp]
theorem coe_toIocMod (θ ψ : ℝ) : ↑(toIocMod two_pi_pos ψ θ) = (θ : Angle) 
Arguments: (θ ψ : ℝ)
Goal: ↑(toIocMod Real.two_pi_pos ψ θ) = ↑θ
-- Real.Angle.toReal_coe
theorem toReal_coe (θ : ℝ) : (θ : Angle).toReal = toIocMod two_pi_pos (-π) θ 
Arguments: (θ : ℝ)
Goal: (↑θ).toReal = toIocMod Real.two_pi_pos (-π) θ
-- Real.Angle.toReal_coe_eq_self_iff
theorem toReal_coe_eq_self_iff {θ : ℝ} : (θ : Angle).toReal = θ ↔ -π < θ ∧ θ ≤ π 
Arguments: {θ : ℝ}
Goal: (↑θ).toReal = θ ↔ -π < θ ∧ θ ≤ π
-- Real.Angle.toReal_coe_eq_self_iff_mem_Ioc
theorem toReal_coe_eq_self_iff_mem_Ioc {θ : ℝ} : (θ : Angle).toReal = θ ↔ θ ∈ Set.Ioc (-π) π 
Arguments: {θ : ℝ}
Goal: (↑θ).toReal = θ ↔ θ ∈ Set.Ioc (-π) π
-- Real.Angle.toReal_injective
theorem toReal_injective : Function.Injective toReal 
Goal: Function.Injective Real.Angle.toReal
-- Real.Angle.toReal_inj
@[simp]
theorem toReal_inj {θ ψ : Angle} : θ.toReal = ψ.toReal ↔ θ = ψ 
Arguments: {θ ψ : Real.Angle}
Goal: θ.toReal = ψ.toReal ↔ θ = ψ
-- Real.Angle.coe_toReal
@[simp]
theorem coe_toReal (θ : Angle) : (θ.toReal : Angle) = θ 
Arguments: (θ : Real.Angle)
Goal: ↑θ.toReal = θ
-- Real.Angle.neg_pi_lt_toReal
theorem neg_pi_lt_toReal (θ : Angle) : -π < θ.toReal 
Arguments: (θ : Real.Angle)
Goal: -π < θ.toReal
-- Real.Angle.toReal_le_pi
theorem toReal_le_pi (θ : Angle) : θ.toReal ≤ π 
Arguments: (θ : Real.Angle)
Goal: θ.toReal ≤ π
-- Real.Angle.abs_toReal_le_pi
theorem abs_toReal_le_pi (θ : Angle) : |θ.toReal| ≤ π 
Arguments: (θ : Real.Angle)
Goal: |θ.toReal| ≤ π
-- Real.Angle.toReal_mem_Ioc
theorem toReal_mem_Ioc (θ : Angle) : θ.toReal ∈ Set.Ioc (-π) π 
Arguments: (θ : Real.Angle)
Goal: θ.toReal ∈ Set.Ioc (-π) π
-- Real.Angle.toIocMod_toReal
@[simp]
theorem toIocMod_toReal (θ : Angle) : toIocMod two_pi_pos (-π) θ.toReal = θ.toReal 
Arguments: (θ : Real.Angle)
Goal: toIocMod Real.two_pi_pos (-π) θ.toReal = θ.toReal
-- Real.Angle.toReal_zero
@[simp]
theorem toReal_zero : (0 : Angle).toReal = 0 
Goal: Real.Angle.toReal 0 = 0
-- Real.Angle.toReal_eq_zero_iff
@[simp]
theorem toReal_eq_zero_iff {θ : Angle} : θ.toReal = 0 ↔ θ = 0 
Arguments: {θ : Real.Angle}
Goal: θ.toReal = 0 ↔ θ = 0
-- Real.Angle.toReal_pi
@[simp]
theorem toReal_pi : (π : Angle).toReal = π 
Goal: (↑π).toReal = π
-- Real.Angle.toReal_eq_pi_iff
@[simp]
theorem toReal_eq_pi_iff {θ : Angle} : θ.toReal = π ↔ θ = π 
Arguments: {θ : Real.Angle}
Goal: θ.toReal = π ↔ θ = ↑π
-- Real.Angle.pi_ne_zero
theorem pi_ne_zero : (π : Angle) ≠ 0 
Goal: ↑π ≠ 0
-- Real.Angle.toReal_pi_div_two
@[simp]
theorem toReal_pi_div_two : ((π / 2 : ℝ) : Angle).toReal = π / 2 
Goal: (↑(π / 2)).toReal = π / 2
-- Real.Angle.toReal_eq_pi_div_two_iff
@[simp]
theorem toReal_eq_pi_div_two_iff {θ : Angle} : θ.toReal = π / 2 ↔ θ = (π / 2 : ℝ) 
Arguments: {θ : Real.Angle}
Goal: θ.toReal = π / 2 ↔ θ = ↑(π / 2)
-- Real.Angle.toReal_neg_pi_div_two
@[simp]
theorem toReal_neg_pi_div_two : ((-π / 2 : ℝ) : Angle).toReal = -π / 2 
Goal: (↑(-π / 2)).toReal = -π / 2
-- Real.Angle.toReal_eq_neg_pi_div_two_iff
@[simp]
theorem toReal_eq_neg_pi_div_two_iff {θ : Angle} : θ.toReal = -π / 2 ↔ θ = (-π / 2 : ℝ) 
Arguments: {θ : Real.Angle}
Goal: θ.toReal = -π / 2 ↔ θ = ↑(-π / 2)
-- Real.Angle.pi_div_two_ne_zero
theorem pi_div_two_ne_zero : ((π / 2 : ℝ) : Angle) ≠ 0 
Goal: ↑(π / 2) ≠ 0
-- Real.Angle.neg_pi_div_two_ne_zero
theorem neg_pi_div_two_ne_zero : ((-π / 2 : ℝ) : Angle) ≠ 0 
Goal: ↑(-π / 2) ≠ 0
-- Real.Angle.abs_toReal_coe_eq_self_iff
theorem abs_toReal_coe_eq_self_iff {θ : ℝ} : |(θ : Angle).toReal| = θ ↔ 0 ≤ θ ∧ θ ≤ π 
Arguments: {θ : ℝ}
Goal: |(↑θ).toReal| = θ ↔ 0 ≤ θ ∧ θ ≤ π
-- Real.Angle.abs_toReal_neg_coe_eq_self_iff
theorem abs_toReal_neg_coe_eq_self_iff {θ : ℝ} : |(-θ : Angle).toReal| = θ ↔ 0 ≤ θ ∧ θ ≤ π 
Arguments: {θ : ℝ}
Goal: |(-↑θ).toReal| = θ ↔ 0 ≤ θ ∧ θ ≤ π
-- Real.Angle.abs_toReal_eq_pi_div_two_iff
theorem abs_toReal_eq_pi_div_two_iff {θ : Angle} :
    |θ.toReal| = π / 2 ↔ θ = (π / 2 : ℝ) ∨ θ = (-π / 2 : ℝ) 
Arguments: {θ : Real.Angle}
Goal: |θ.toReal| = π / 2 ↔ θ = ↑(π / 2) ∨ θ = ↑(-π / 2)
-- Real.Angle.nsmul_toReal_eq_mul
theorem nsmul_toReal_eq_mul {n : ℕ} (h : n ≠ 0) {θ : Angle} :
    (n • θ).toReal = n * θ.toReal ↔ θ.toReal ∈ Set.Ioc (-π / n) (π / n) 
Arguments: {n : ℕ} (h : n ≠ 0) {θ : Real.Angle}
Goal: (n • θ).toReal = ↑n * θ.toReal ↔ θ.toReal ∈ Set.Ioc (-π / ↑n) (π / ↑n)
-- Real.Angle.two_nsmul_toReal_eq_two_mul
theorem two_nsmul_toReal_eq_two_mul {θ : Angle} :
    ((2 : ℕ) • θ).toReal = 2 * θ.toReal ↔ θ.toReal ∈ Set.Ioc (-π / 2) (π / 2) 
Arguments: {θ : Real.Angle}
Goal: (2 • θ).toReal = 2 * θ.toReal ↔ θ.toReal ∈ Set.Ioc (-π / 2) (π / 2)
-- Real.Angle.two_zsmul_toReal_eq_two_mul
theorem two_zsmul_toReal_eq_two_mul {θ : Angle} :
    ((2 : ℤ) • θ).toReal = 2 * θ.toReal ↔ θ.toReal ∈ Set.Ioc (-π / 2) (π / 2) 
Arguments: {θ : Real.Angle}
Goal: (2 • θ).toReal = 2 * θ.toReal ↔ θ.toReal ∈ Set.Ioc (-π / 2) (π / 2)
-- Real.Angle.toReal_coe_eq_self_sub_two_mul_int_mul_pi_iff
theorem toReal_coe_eq_self_sub_two_mul_int_mul_pi_iff {θ : ℝ} {k : ℤ} :
    (θ : Angle).toReal = θ - 2 * k * π ↔ θ ∈ Set.Ioc ((2 * k - 1 : ℝ) * π) ((2 * k + 1) * π) 
Arguments: {θ : ℝ} {k : ℤ}
Goal: (↑θ).toReal = θ - 2 * ↑k * π ↔ θ ∈ Set.Ioc ((2 * ↑k - 1) * π) ((2 * ↑k + 1) * π)
-- Real.Angle.toReal_coe_eq_self_sub_two_pi_iff
theorem toReal_coe_eq_self_sub_two_pi_iff {θ : ℝ} :
    (θ : Angle).toReal = θ - 2 * π ↔ θ ∈ Set.Ioc π (3 * π) 
Arguments: {θ : ℝ}
Goal: (↑θ).toReal = θ - 2 * π ↔ θ ∈ Set.Ioc π (3 * π)
-- Real.Angle.toReal_coe_eq_self_add_two_pi_iff
theorem toReal_coe_eq_self_add_two_pi_iff {θ : ℝ} :
    (θ : Angle).toReal = θ + 2 * π ↔ θ ∈ Set.Ioc (-3 * π) (-π) 
Arguments: {θ : ℝ}
Goal: (↑θ).toReal = θ + 2 * π ↔ θ ∈ Set.Ioc (-3 * π) (-π)
-- Real.Angle.two_nsmul_toReal_eq_two_mul_sub_two_pi
theorem two_nsmul_toReal_eq_two_mul_sub_two_pi {θ : Angle} :
    ((2 : ℕ) • θ).toReal = 2 * θ.toReal - 2 * π ↔ π / 2 < θ.toReal 
Arguments: {θ : Real.Angle}
Goal: (2 • θ).toReal = 2 * θ.toReal - 2 * π ↔ π / 2 < θ.toReal
-- Real.Angle.two_zsmul_toReal_eq_two_mul_sub_two_pi
theorem two_zsmul_toReal_eq_two_mul_sub_two_pi {θ : Angle} :
    ((2 : ℤ) • θ).toReal = 2 * θ.toReal - 2 * π ↔ π / 2 < θ.toReal 
Arguments: {θ : Real.Angle}
Goal: (2 • θ).toReal = 2 * θ.toReal - 2 * π ↔ π / 2 < θ.toReal
-- Real.Angle.two_nsmul_toReal_eq_two_mul_add_two_pi
theorem two_nsmul_toReal_eq_two_mul_add_two_pi {θ : Angle} :
    ((2 : ℕ) • θ).toReal = 2 * θ.toReal + 2 * π ↔ θ.toReal ≤ -π / 2 
Arguments: {θ : Real.Angle}
Goal: (2 • θ).toReal = 2 * θ.toReal + 2 * π ↔ θ.toReal ≤ -π / 2
-- Real.Angle.two_zsmul_toReal_eq_two_mul_add_two_pi
theorem two_zsmul_toReal_eq_two_mul_add_two_pi {θ : Angle} :
    ((2 : ℤ) • θ).toReal = 2 * θ.toReal + 2 * π ↔ θ.toReal ≤ -π / 2 
Arguments: {θ : Real.Angle}
Goal: (2 • θ).toReal = 2 * θ.toReal + 2 * π ↔ θ.toReal ≤ -π / 2
-- Real.Angle.sin_toReal
@[simp]
theorem sin_toReal (θ : Angle) : Real.sin θ.toReal = sin θ 
Arguments: (θ : Real.Angle)
Goal: Real.sin θ.toReal = θ.sin
-- Real.Angle.cos_toReal
@[simp]
theorem cos_toReal (θ : Angle) : Real.cos θ.toReal = cos θ 
Arguments: (θ : Real.Angle)
Goal: Real.cos θ.toReal = θ.cos