ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Orientation.rotationAux_apply
@[simp]
theorem rotationAux_apply (θ : Real.Angle) (x : V) :
    o.rotationAux θ x = Real.Angle.cos θ • x + Real.Angle.sin θ • J x 
Arguments: {V : Type u_1} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [Fact (FiniteDimensional.finrank ℝ V = 2)] (o : Orientation ℝ V (Fin 2)) (θ : Real.Angle) (x : V)
Goal: (o.rotationAux θ) x = θ.cos • x + θ.sin • o.rightAngleRotation x
-- Orientation.rotation_apply
theorem rotation_apply (θ : Real.Angle) (x : V) :
    o.rotation θ x = Real.Angle.cos θ • x + Real.Angle.sin θ • J x 
Arguments: {V : Type u_1} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [Fact (FiniteDimensional.finrank ℝ V = 2)] (o : Orientation ℝ V (Fin 2)) (θ : Real.Angle) (x : V)
Goal: (o.rotation θ) x = θ.cos • x + θ.sin • o.rightAngleRotation x
-- Orientation.rotation_symm_apply
theorem rotation_symm_apply (θ : Real.Angle) (x : V) :
    (o.rotation θ).symm x = Real.Angle.cos θ • x - Real.Angle.sin θ • J x 
Arguments: {V : Type u_1} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [Fact (FiniteDimensional.finrank ℝ V = 2)] (o : Orientation ℝ V (Fin 2)) (θ : Real.Angle) (x : V)
Goal: (o.rotation θ).symm x = θ.cos • x - θ.sin • o.rightAngleRotation x
-- Orientation.rotation_eq_matrix_toLin
theorem rotation_eq_matrix_toLin (θ : Real.Angle) {x : V} (hx : x ≠ 0) :
    (o.rotation θ).toLinearMap =
      Matrix.toLin (o.basisRightAngleRotation x hx) (o.basisRightAngleRotation x hx)
        !![θ.cos, -θ.sin; θ.sin, θ.cos] 
Arguments: {V : Type u_1} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [Fact (FiniteDimensional.finrank ℝ V = 2)] (o : Orientation ℝ V (Fin 2)) (θ : Real.Angle) {x : V} (hx : x ≠ 0)
Goal: ↑(o.rotation θ).toLinearEquiv =
  (Matrix.toLin (o.basisRightAngleRotation x hx) (o.basisRightAngleRotation x hx)) !![θ.cos, -θ.sin; θ.sin, θ.cos]
-- Orientation.det_rotation
@[simp]
theorem det_rotation (θ : Real.Angle) : LinearMap.det (o.rotation θ).toLinearMap = 1 
Arguments: {V : Type u_1} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [Fact (FiniteDimensional.finrank ℝ V = 2)] (o : Orientation ℝ V (Fin 2)) (θ : Real.Angle)
Goal: LinearMap.det ↑(o.rotation θ).toLinearEquiv = 1
-- Orientation.linearEquiv_det_rotation
@[simp]
theorem linearEquiv_det_rotation (θ : Real.Angle) :
    LinearEquiv.det (o.rotation θ).toLinearEquiv = 1 
Arguments: {V : Type u_1} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [Fact (FiniteDimensional.finrank ℝ V = 2)] (o : Orientation ℝ V (Fin 2)) (θ : Real.Angle)
Goal: LinearEquiv.det (o.rotation θ).toLinearEquiv = 1
-- Orientation.rotation_symm
@[simp]
theorem rotation_symm (θ : Real.Angle) : (o.rotation θ).symm = o.rotation (-θ) 
Arguments: {V : Type u_1} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [Fact (FiniteDimensional.finrank ℝ V = 2)] (o : Orientation ℝ V (Fin 2)) (θ : Real.Angle)
Goal: (o.rotation θ).symm = o.rotation (-θ)
-- Orientation.rotation_zero
@[simp]
theorem rotation_zero : o.rotation 0 = LinearIsometryEquiv.refl ℝ V 
Arguments: {V : Type u_1} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [Fact (FiniteDimensional.finrank ℝ V = 2)] (o : Orientation ℝ V (Fin 2))
Goal: o.rotation 0 = LinearIsometryEquiv.refl ℝ V
-- Orientation.rotation_pi
@[simp]
theorem rotation_pi : o.rotation π = LinearIsometryEquiv.neg ℝ 
Arguments: {V : Type u_1} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [Fact (FiniteDimensional.finrank ℝ V = 2)] (o : Orientation ℝ V (Fin 2))
Goal: o.rotation ↑π = LinearIsometryEquiv.neg ℝ
-- Orientation.rotation_pi_apply
theorem rotation_pi_apply (x : V) : o.rotation π x = -x 
Arguments: {V : Type u_1} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [Fact (FiniteDimensional.finrank ℝ V = 2)] (o : Orientation ℝ V (Fin 2)) (x : V)
Goal: (o.rotation ↑π) x = -x
-- Orientation.rotation_pi_div_two
theorem rotation_pi_div_two : o.rotation (π / 2 : ℝ) = J 
Arguments: {V : Type u_1} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [Fact (FiniteDimensional.finrank ℝ V = 2)] (o : Orientation ℝ V (Fin 2))
Goal: o.rotation ↑(π / 2) = o.rightAngleRotation
-- Orientation.rotation_rotation
@[simp]
theorem rotation_rotation (θ₁ θ₂ : Real.Angle) (x : V) :
    o.rotation θ₁ (o.rotation θ₂ x) = o.rotation (θ₁ + θ₂) x 
Arguments: {V : Type u_1} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [Fact (FiniteDimensional.finrank ℝ V = 2)] (o : Orientation ℝ V (Fin 2)) (θ₁ θ₂ : Real.Angle) (x : V)
Goal: (o.rotation θ₁) ((o.rotation θ₂) x) = (o.rotation (θ₁ + θ₂)) x
-- Orientation.rotation_trans
@[simp]
theorem rotation_trans (θ₁ θ₂ : Real.Angle) :
    (o.rotation θ₁).trans (o.rotation θ₂) = o.rotation (θ₂ + θ₁) 
Arguments: {V : Type u_1} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [Fact (FiniteDimensional.finrank ℝ V = 2)] (o : Orientation ℝ V (Fin 2)) (θ₁ θ₂ : Real.Angle)
Goal: (o.rotation θ₁).trans (o.rotation θ₂) = o.rotation (θ₂ + θ₁)
-- Orientation.kahler_rotation_left
@[simp]
theorem kahler_rotation_left (x y : V) (θ : Real.Angle) :
    o.kahler (o.rotation θ x) y = conj (θ.expMapCircle : ℂ) * o.kahler x y 
Arguments: {V : Type u_1} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [Fact (FiniteDimensional.finrank ℝ V = 2)] (o : Orientation ℝ V (Fin 2)) (x y : V) (θ : Real.Angle)
Goal: (o.kahler ((o.rotation θ) x)) y = (starRingEnd ℂ) ↑θ.expMapCircle * (o.kahler x) y
-- Orientation.neg_rotation
theorem neg_rotation (θ : Real.Angle) (x : V) : -o.rotation θ x = o.rotation (π + θ) x 
Arguments: {V : Type u_1} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [Fact (FiniteDimensional.finrank ℝ V = 2)] (o : Orientation ℝ V (Fin 2)) (θ : Real.Angle) (x : V)
Goal: -(o.rotation θ) x = (o.rotation (↑π + θ)) x
-- Orientation.neg_rotation_neg_pi_div_two
@[simp]
theorem neg_rotation_neg_pi_div_two (x : V) :
    -o.rotation (-π / 2 : ℝ) x = o.rotation (π / 2 : ℝ) x 
Arguments: {V : Type u_1} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [Fact (FiniteDimensional.finrank ℝ V = 2)] (o : Orientation ℝ V (Fin 2)) (x : V)
Goal: -(o.rotation ↑(-π / 2)) x = (o.rotation ↑(π / 2)) x
-- Orientation.neg_rotation_pi_div_two
theorem neg_rotation_pi_div_two (x : V) : -o.rotation (π / 2 : ℝ) x = o.rotation (-π / 2 : ℝ) x 
Arguments: {V : Type u_1} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [Fact (FiniteDimensional.finrank ℝ V = 2)] (o : Orientation ℝ V (Fin 2)) (x : V)
Goal: -(o.rotation ↑(π / 2)) x = (o.rotation ↑(-π / 2)) x
-- Orientation.kahler_rotation_left'
theorem kahler_rotation_left' (x y : V) (θ : Real.Angle) :
    o.kahler (o.rotation θ x) y = (-θ).expMapCircle * o.kahler x y 
Arguments: {V : Type u_1} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [Fact (FiniteDimensional.finrank ℝ V = 2)] (o : Orientation ℝ V (Fin 2)) (x y : V) (θ : Real.Angle)
Goal: (o.kahler ((o.rotation θ) x)) y = ↑(-θ).expMapCircle * (o.kahler x) y
-- Orientation.kahler_rotation_right
@[simp]
theorem kahler_rotation_right (x y : V) (θ : Real.Angle) :
    o.kahler x (o.rotation θ y) = θ.expMapCircle * o.kahler x y 
Arguments: {V : Type u_1} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [Fact (FiniteDimensional.finrank ℝ V = 2)] (o : Orientation ℝ V (Fin 2)) (x y : V) (θ : Real.Angle)
Goal: (o.kahler x) ((o.rotation θ) y) = ↑θ.expMapCircle * (o.kahler x) y
-- Orientation.oangle_rotation_left
@[simp]
theorem oangle_rotation_left {x y : V} (hx : x ≠ 0) (hy : y ≠ 0) (θ : Real.Angle) :
    o.oangle (o.rotation θ x) y = o.oangle x y - θ 
Arguments: {V : Type u_1} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [Fact (FiniteDimensional.finrank ℝ V = 2)] (o : Orientation ℝ V (Fin 2)) {x y : V} (hx : x ≠ 0) (hy : y ≠ 0) (θ : Real.Angle)
Goal: o.oangle ((o.rotation θ) x) y = o.oangle x y - θ
-- Orientation.oangle_rotation_right
@[simp]
theorem oangle_rotation_right {x y : V} (hx : x ≠ 0) (hy : y ≠ 0) (θ : Real.Angle) :
    o.oangle x (o.rotation θ y) = o.oangle x y + θ 
Arguments: {V : Type u_1} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [Fact (FiniteDimensional.finrank ℝ V = 2)] (o : Orientation ℝ V (Fin 2)) {x y : V} (hx : x ≠ 0) (hy : y ≠ 0) (θ : Real.Angle)
Goal: o.oangle x ((o.rotation θ) y) = o.oangle x y + θ
-- Orientation.oangle_rotation_self_left
@[simp]
theorem oangle_rotation_self_left {x : V} (hx : x ≠ 0) (θ : Real.Angle) :
    o.oangle (o.rotation θ x) x = -θ 
Arguments: {V : Type u_1} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [Fact (FiniteDimensional.finrank ℝ V = 2)] (o : Orientation ℝ V (Fin 2)) {x : V} (hx : x ≠ 0) (θ : Real.Angle)
Goal: o.oangle ((o.rotation θ) x) x = -θ
-- Orientation.oangle_rotation_self_right
@[simp]
theorem oangle_rotation_self_right {x : V} (hx : x ≠ 0) (θ : Real.Angle) :
    o.oangle x (o.rotation θ x) = θ 
Arguments: {V : Type u_1} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [Fact (FiniteDimensional.finrank ℝ V = 2)] (o : Orientation ℝ V (Fin 2)) {x : V} (hx : x ≠ 0) (θ : Real.Angle)
Goal: o.oangle x ((o.rotation θ) x) = θ
-- Orientation.oangle_rotation_oangle_left
@[simp]
theorem oangle_rotation_oangle_left (x y : V) : o.oangle (o.rotation (o.oangle x y) x) y = 0 
Arguments: {V : Type u_1} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [Fact (FiniteDimensional.finrank ℝ V = 2)] (o : Orientation ℝ V (Fin 2)) (x y : V)
Goal: o.oangle ((o.rotation (o.oangle x y)) x) y = 0
-- Orientation.oangle_rotation_oangle_right
@[simp]
theorem oangle_rotation_oangle_right (x y : V) : o.oangle y (o.rotation (o.oangle x y) x) = 0 
Arguments: {V : Type u_1} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [Fact (FiniteDimensional.finrank ℝ V = 2)] (o : Orientation ℝ V (Fin 2)) (x y : V)
Goal: o.oangle y ((o.rotation (o.oangle x y)) x) = 0
-- Orientation.oangle_rotation
@[simp]
theorem oangle_rotation (x y : V) (θ : Real.Angle) :
    o.oangle (o.rotation θ x) (o.rotation θ y) = o.oangle x y 
Arguments: {V : Type u_1} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [Fact (FiniteDimensional.finrank ℝ V = 2)] (o : Orientation ℝ V (Fin 2)) (x y : V) (θ : Real.Angle)
Goal: o.oangle ((o.rotation θ) x) ((o.rotation θ) y) = o.oangle x y
-- Orientation.rotation_eq_self_iff_angle_eq_zero
@[simp]
theorem rotation_eq_self_iff_angle_eq_zero {x : V} (hx : x ≠ 0) (θ : Real.Angle) :
    o.rotation θ x = x ↔ θ = 0 
Arguments: {V : Type u_1} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [Fact (FiniteDimensional.finrank ℝ V = 2)] (o : Orientation ℝ V (Fin 2)) {x : V} (hx : x ≠ 0) (θ : Real.Angle)
Goal: (o.rotation θ) x = x ↔ θ = 0
-- Orientation.eq_rotation_self_iff_angle_eq_zero
@[simp]
theorem eq_rotation_self_iff_angle_eq_zero {x : V} (hx : x ≠ 0) (θ : Real.Angle) :
    x = o.rotation θ x ↔ θ = 0 
Arguments: {V : Type u_1} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [Fact (FiniteDimensional.finrank ℝ V = 2)] (o : Orientation ℝ V (Fin 2)) {x : V} (hx : x ≠ 0) (θ : Real.Angle)
Goal: x = (o.rotation θ) x ↔ θ = 0
-- Orientation.rotation_eq_self_iff
theorem rotation_eq_self_iff (x : V) (θ : Real.Angle) : o.rotation θ x = x ↔ x = 0 ∨ θ = 0 
Arguments: {V : Type u_1} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [Fact (FiniteDimensional.finrank ℝ V = 2)] (o : Orientation ℝ V (Fin 2)) (x : V) (θ : Real.Angle)
Goal: (o.rotation θ) x = x ↔ x = 0 ∨ θ = 0
-- Orientation.eq_rotation_self_iff
theorem eq_rotation_self_iff (x : V) (θ : Real.Angle) : x = o.rotation θ x ↔ x = 0 ∨ θ = 0 
Arguments: {V : Type u_1} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [Fact (FiniteDimensional.finrank ℝ V = 2)] (o : Orientation ℝ V (Fin 2)) (x : V) (θ : Real.Angle)
Goal: x = (o.rotation θ) x ↔ x = 0 ∨ θ = 0
-- Orientation.rotation_oangle_eq_iff_norm_eq
@[simp]
theorem rotation_oangle_eq_iff_norm_eq (x y : V) : o.rotation (o.oangle x y) x = y ↔ ‖x‖ = ‖y‖ 
Arguments: {V : Type u_1} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [Fact (FiniteDimensional.finrank ℝ V = 2)] (o : Orientation ℝ V (Fin 2)) (x y : V)
Goal: (o.rotation (o.oangle x y)) x = y ↔ ‖x‖ = ‖y‖
-- Orientation.oangle_eq_iff_eq_norm_div_norm_smul_rotation_of_ne_zero
theorem oangle_eq_iff_eq_norm_div_norm_smul_rotation_of_ne_zero {x y : V} (hx : x ≠ 0) (hy : y ≠ 0)
    (θ : Real.Angle) : o.oangle x y = θ ↔ y = (‖y‖ / ‖x‖) • o.rotation θ x 
Arguments: {V : Type u_1} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [Fact (FiniteDimensional.finrank ℝ V = 2)] (o : Orientation ℝ V (Fin 2)) {x y : V} (hx : x ≠ 0) (hy : y ≠ 0) (θ : Real.Angle)
Goal: o.oangle x y = θ ↔ y = (‖y‖ / ‖x‖) • (o.rotation θ) x
-- Orientation.oangle_eq_iff_eq_pos_smul_rotation_of_ne_zero
theorem oangle_eq_iff_eq_pos_smul_rotation_of_ne_zero {x y : V} (hx : x ≠ 0) (hy : y ≠ 0)
    (θ : Real.Angle) : o.oangle x y = θ ↔ ∃ r : ℝ, 0 < r ∧ y = r • o.rotation θ x 
Arguments: {V : Type u_1} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [Fact (FiniteDimensional.finrank ℝ V = 2)] (o : Orientation ℝ V (Fin 2)) {x y : V} (hx : x ≠ 0) (hy : y ≠ 0) (θ : Real.Angle)
Goal: o.oangle x y = θ ↔ ∃ r, 0 < r ∧ y = r • (o.rotation θ) x
-- Orientation.oangle_eq_iff_eq_norm_div_norm_smul_rotation_or_eq_zero
theorem oangle_eq_iff_eq_norm_div_norm_smul_rotation_or_eq_zero {x y : V} (θ : Real.Angle) :
    o.oangle x y = θ ↔
      x ≠ 0 ∧ y ≠ 0 ∧ y = (‖y‖ / ‖x‖) • o.rotation θ x ∨ θ = 0 ∧ (x = 0 ∨ y = 0) 
Arguments: {V : Type u_1} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [Fact (FiniteDimensional.finrank ℝ V = 2)] (o : Orientation ℝ V (Fin 2)) {x y : V} (θ : Real.Angle)
Goal: o.oangle x y = θ ↔ x ≠ 0 ∧ y ≠ 0 ∧ y = (‖y‖ / ‖x‖) • (o.rotation θ) x ∨ θ = 0 ∧ (x = 0 ∨ y = 0)
-- Orientation.oangle_eq_iff_eq_pos_smul_rotation_or_eq_zero
theorem oangle_eq_iff_eq_pos_smul_rotation_or_eq_zero {x y : V} (θ : Real.Angle) :
    o.oangle x y = θ ↔
      (x ≠ 0 ∧ y ≠ 0 ∧ ∃ r : ℝ, 0 < r ∧ y = r • o.rotation θ x) ∨ θ = 0 ∧ (x = 0 ∨ y = 0) 
Arguments: {V : Type u_1} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [Fact (FiniteDimensional.finrank ℝ V = 2)] (o : Orientation ℝ V (Fin 2)) {x y : V} (θ : Real.Angle)
Goal: o.oangle x y = θ ↔ (x ≠ 0 ∧ y ≠ 0 ∧ ∃ r, 0 < r ∧ y = r • (o.rotation θ) x) ∨ θ = 0 ∧ (x = 0 ∨ y = 0)
-- Orientation.exists_linearIsometryEquiv_eq_of_det_pos
theorem exists_linearIsometryEquiv_eq_of_det_pos {f : V ≃ₗᵢ[ℝ] V}
    (hd : 0 < LinearMap.det (f.toLinearEquiv : V →ₗ[ℝ] V)) :
    ∃ θ : Real.Angle, f = o.rotation θ 
Arguments: {V : Type u_1} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [Fact (FiniteDimensional.finrank ℝ V = 2)] (o : Orientation ℝ V (Fin 2)) {f : V ≃ₗᵢ[ℝ] V} (hd : 0 < LinearMap.det ↑f.toLinearEquiv)
Goal: ∃ θ, f = o.rotation θ
-- Orientation.rotation_map
theorem rotation_map (θ : Real.Angle) (f : V ≃ₗᵢ[ℝ] V') (x : V') :
    (Orientation.map (Fin 2) f.toLinearEquiv o).rotation θ x = f (o.rotation θ (f.symm x)) 
Arguments: {V : Type u_1} {V' : Type u_2} [NormedAddCommGroup V] [NormedAddCommGroup V'] [InnerProductSpace ℝ V] [InnerProductSpace ℝ V'] [Fact (FiniteDimensional.finrank ℝ V = 2)] [Fact (FiniteDimensional.finrank ℝ V' = 2)] (o : Orientation ℝ V (Fin 2)) (θ : Real.Angle) (f : V ≃ₗᵢ[ℝ] V') (x : V')
Goal: (((Orientation.map (Fin 2) f.toLinearEquiv) o).rotation θ) x = f ((o.rotation θ) (f.symm x))
-- Complex.rotation
@[simp]
protected theorem _root_.Complex.rotation (θ : Real.Angle) (z : ℂ) :
    Complex.orientation.rotation θ z = θ.expMapCircle * z 
Arguments: (θ : Real.Angle) (z : ℂ)
Goal: (Complex.orientation.rotation θ) z = ↑θ.expMapCircle * z
-- Orientation.rotation_map_complex
theorem rotation_map_complex (θ : Real.Angle) (f : V ≃ₗᵢ[ℝ] ℂ)
    (hf : Orientation.map (Fin 2) f.toLinearEquiv o = Complex.orientation) (x : V) :
    f (o.rotation θ x) = θ.expMapCircle * f x 
Arguments: {V : Type u_1} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [Fact (FiniteDimensional.finrank ℝ V = 2)] (o : Orientation ℝ V (Fin 2)) (θ : Real.Angle) (f : V ≃ₗᵢ[ℝ] ℂ) (hf : (Orientation.map (Fin 2) f.toLinearEquiv) o = Complex.orientation) (x : V)
Goal: f ((o.rotation θ) x) = ↑θ.expMapCircle * f x
-- Orientation.rotation_neg_orientation_eq_neg
theorem rotation_neg_orientation_eq_neg (θ : Real.Angle) : (-o).rotation θ = o.rotation (-θ) 
Arguments: {V : Type u_1} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [Fact (FiniteDimensional.finrank ℝ V = 2)] (o : Orientation ℝ V (Fin 2)) (θ : Real.Angle)
Goal: (-o).rotation θ = o.rotation (-θ)
-- Orientation.inner_rotation_pi_div_two_left
@[simp]
theorem inner_rotation_pi_div_two_left (x : V) : ⟪o.rotation (π / 2 : ℝ) x, x⟫ = 0 
Arguments: {V : Type u_1} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [Fact (FiniteDimensional.finrank ℝ V = 2)] (o : Orientation ℝ V (Fin 2)) (x : V)
Goal: ⟪(o.rotation ↑(π / 2)) x, x⟫_ℝ = 0
-- Orientation.inner_rotation_pi_div_two_right
@[simp]
theorem inner_rotation_pi_div_two_right (x : V) : ⟪x, o.rotation (π / 2 : ℝ) x⟫ = 0 
Arguments: {V : Type u_1} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [Fact (FiniteDimensional.finrank ℝ V = 2)] (o : Orientation ℝ V (Fin 2)) (x : V)
Goal: ⟪x, (o.rotation ↑(π / 2)) x⟫_ℝ = 0
-- Orientation.inner_smul_rotation_pi_div_two_left
@[simp]
theorem inner_smul_rotation_pi_div_two_left (x : V) (r : ℝ) :
    ⟪r • o.rotation (π / 2 : ℝ) x, x⟫ = 0 
Arguments: {V : Type u_1} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [Fact (FiniteDimensional.finrank ℝ V = 2)] (o : Orientation ℝ V (Fin 2)) (x : V) (r : ℝ)
Goal: ⟪r • (o.rotation ↑(π / 2)) x, x⟫_ℝ = 0
-- Orientation.inner_smul_rotation_pi_div_two_right
@[simp]
theorem inner_smul_rotation_pi_div_two_right (x : V) (r : ℝ) :
    ⟪x, r • o.rotation (π / 2 : ℝ) x⟫ = 0 
Arguments: {V : Type u_1} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [Fact (FiniteDimensional.finrank ℝ V = 2)] (o : Orientation ℝ V (Fin 2)) (x : V) (r : ℝ)
Goal: ⟪x, r • (o.rotation ↑(π / 2)) x⟫_ℝ = 0
-- Orientation.inner_rotation_pi_div_two_left_smul
@[simp]
theorem inner_rotation_pi_div_two_left_smul (x : V) (r : ℝ) :
    ⟪o.rotation (π / 2 : ℝ) x, r • x⟫ = 0 
Arguments: {V : Type u_1} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [Fact (FiniteDimensional.finrank ℝ V = 2)] (o : Orientation ℝ V (Fin 2)) (x : V) (r : ℝ)
Goal: ⟪(o.rotation ↑(π / 2)) x, r • x⟫_ℝ = 0
-- Orientation.inner_rotation_pi_div_two_right_smul
@[simp]
theorem inner_rotation_pi_div_two_right_smul (x : V) (r : ℝ) :
    ⟪r • x, o.rotation (π / 2 : ℝ) x⟫ = 0 
Arguments: {V : Type u_1} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [Fact (FiniteDimensional.finrank ℝ V = 2)] (o : Orientation ℝ V (Fin 2)) (x : V) (r : ℝ)
Goal: ⟪r • x, (o.rotation ↑(π / 2)) x⟫_ℝ = 0
-- Orientation.inner_smul_rotation_pi_div_two_smul_left
@[simp]
theorem inner_smul_rotation_pi_div_two_smul_left (x : V) (r₁ r₂ : ℝ) :
    ⟪r₁ • o.rotation (π / 2 : ℝ) x, r₂ • x⟫ = 0 
Arguments: {V : Type u_1} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [Fact (FiniteDimensional.finrank ℝ V = 2)] (o : Orientation ℝ V (Fin 2)) (x : V) (r₁ r₂ : ℝ)
Goal: ⟪r₁ • (o.rotation ↑(π / 2)) x, r₂ • x⟫_ℝ = 0
-- Orientation.inner_smul_rotation_pi_div_two_smul_right
@[simp]
theorem inner_smul_rotation_pi_div_two_smul_right (x : V) (r₁ r₂ : ℝ) :
    ⟪r₂ • x, r₁ • o.rotation (π / 2 : ℝ) x⟫ = 0 
Arguments: {V : Type u_1} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [Fact (FiniteDimensional.finrank ℝ V = 2)] (o : Orientation ℝ V (Fin 2)) (x : V) (r₁ r₂ : ℝ)
Goal: ⟪r₂ • x, r₁ • (o.rotation ↑(π / 2)) x⟫_ℝ = 0
-- Orientation.inner_eq_zero_iff_eq_zero_or_eq_smul_rotation_pi_div_two
theorem inner_eq_zero_iff_eq_zero_or_eq_smul_rotation_pi_div_two {x y : V} :
    ⟪x, y⟫ = 0 ↔ x = 0 ∨ ∃ r : ℝ, r • o.rotation (π / 2 : ℝ) x = y 
Arguments: {V : Type u_1} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [Fact (FiniteDimensional.finrank ℝ V = 2)] (o : Orientation ℝ V (Fin 2)) {x y : V}
Goal: ⟪x, y⟫_ℝ = 0 ↔ x = 0 ∨ ∃ r, r • (o.rotation ↑(π / 2)) x = y
-- Orientation.oangle_eq_two_zsmul_oangle_sub_of_norm_eq
theorem oangle_eq_two_zsmul_oangle_sub_of_norm_eq {x y z : V} (hxyne : x ≠ y) (hxzne : x ≠ z)
    (hxy : ‖x‖ = ‖y‖) (hxz : ‖x‖ = ‖z‖) : o.oangle y z = (2 : ℤ) • o.oangle (y - x) (z - x) 
Arguments: {V : Type u_1} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [Fact (FiniteDimensional.finrank ℝ V = 2)] (o : Orientation ℝ V (Fin 2)) {x y z : V} (hxyne : x ≠ y) (hxzne : x ≠ z) (hxy : ‖x‖ = ‖y‖) (hxz : ‖x‖ = ‖z‖)
Goal: o.oangle y z = 2 • o.oangle (y - x) (z - x)
-- Orientation.oangle_eq_two_zsmul_oangle_sub_of_norm_eq_real
theorem oangle_eq_two_zsmul_oangle_sub_of_norm_eq_real {x y z : V} (hxyne : x ≠ y) (hxzne : x ≠ z)
    {r : ℝ} (hx : ‖x‖ = r) (hy : ‖y‖ = r) (hz : ‖z‖ = r) :
    o.oangle y z = (2 : ℤ) • o.oangle (y - x) (z - x) 
Arguments: {V : Type u_1} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [Fact (FiniteDimensional.finrank ℝ V = 2)] (o : Orientation ℝ V (Fin 2)) {x y z : V} (hxyne : x ≠ y) (hxzne : x ≠ z) {r : ℝ} (hx : ‖x‖ = r) (hy : ‖y‖ = r) (hz : ‖z‖ = r)
Goal: o.oangle y z = 2 • o.oangle (y - x) (z - x)
-- Orientation.two_zsmul_oangle_sub_eq_two_zsmul_oangle_sub_of_norm_eq
theorem two_zsmul_oangle_sub_eq_two_zsmul_oangle_sub_of_norm_eq {x₁ x₂ y z : V} (hx₁yne : x₁ ≠ y)
    (hx₁zne : x₁ ≠ z) (hx₂yne : x₂ ≠ y) (hx₂zne : x₂ ≠ z) {r : ℝ} (hx₁ : ‖x₁‖ = r) (hx₂ : ‖x₂‖ = r)
    (hy : ‖y‖ = r) (hz : ‖z‖ = r) :
    (2 : ℤ) • o.oangle (y - x₁) (z - x₁) = (2 : ℤ) • o.oangle (y - x₂) (z - x₂) 
Arguments: {V : Type u_1} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [Fact (FiniteDimensional.finrank ℝ V = 2)] (o : Orientation ℝ V (Fin 2)) {x₁ x₂ y z : V} (hx₁yne : x₁ ≠ y) (hx₁zne : x₁ ≠ z) (hx₂yne : x₂ ≠ y) (hx₂zne : x₂ ≠ z) {r : ℝ} (hx₁ : ‖x₁‖ = r) (hx₂ : ‖x₂‖ = r) (hy : ‖y‖ = r) (hz : ‖z‖ = r)
Goal: 2 • o.oangle (y - x₁) (z - x₁) = 2 • o.oangle (y - x₂) (z - x₂)
-- EuclideanGeometry.Sphere.oangle_center_eq_two_zsmul_oangle
theorem oangle_center_eq_two_zsmul_oangle {s : Sphere P} {p₁ p₂ p₃ : P} (hp₁ : p₁ ∈ s)
    (hp₂ : p₂ ∈ s) (hp₃ : p₃ ∈ s) (hp₂p₁ : p₂ ≠ p₁) (hp₂p₃ : p₂ ≠ p₃) :
    ∡ p₁ s.center p₃ = (2 : ℤ) • ∡ p₁ p₂ p₃ 
Arguments: {V : Type u_1} {P : Type u_2} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [MetricSpace P] [NormedAddTorsor V P] [hd2 : Fact (FiniteDimensional.finrank ℝ V = 2)] [Module.Oriented ℝ V (Fin 2)] {s : EuclideanGeometry.Sphere P} {p₁ p₂ p₃ : P} (hp₁ : p₁ ∈ s) (hp₂ : p₂ ∈ s) (hp₃ : p₃ ∈ s) (hp₂p₁ : p₂ ≠ p₁) (hp₂p₃ : p₂ ≠ p₃)
Goal: ∡ p₁ s.center p₃ = 2 • ∡ p₁ p₂ p₃
-- EuclideanGeometry.Sphere.two_zsmul_oangle_eq
theorem two_zsmul_oangle_eq {s : Sphere P} {p₁ p₂ p₃ p₄ : P} (hp₁ : p₁ ∈ s) (hp₂ : p₂ ∈ s)
    (hp₃ : p₃ ∈ s) (hp₄ : p₄ ∈ s) (hp₂p₁ : p₂ ≠ p₁) (hp₂p₄ : p₂ ≠ p₄) (hp₃p₁ : p₃ ≠ p₁)
    (hp₃p₄ : p₃ ≠ p₄) : (2 : ℤ) • ∡ p₁ p₂ p₄ = (2 : ℤ) • ∡ p₁ p₃ p₄ 
Arguments: {V : Type u_1} {P : Type u_2} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [MetricSpace P] [NormedAddTorsor V P] [hd2 : Fact (FiniteDimensional.finrank ℝ V = 2)] [Module.Oriented ℝ V (Fin 2)] {s : EuclideanGeometry.Sphere P} {p₁ p₂ p₃ p₄ : P} (hp₁ : p₁ ∈ s) (hp₂ : p₂ ∈ s) (hp₃ : p₃ ∈ s) (hp₄ : p₄ ∈ s) (hp₂p₁ : p₂ ≠ p₁) (hp₂p₄ : p₂ ≠ p₄) (hp₃p₁ : p₃ ≠ p₁) (hp₃p₄ : p₃ ≠ p₄)
Goal: 2 • ∡ p₁ p₂ p₄ = 2 • ∡ p₁ p₃ p₄
-- EuclideanGeometry.Cospherical.two_zsmul_oangle_eq
theorem Cospherical.two_zsmul_oangle_eq {p₁ p₂ p₃ p₄ : P}
    (h : Cospherical ({p₁, p₂, p₃, p₄} : Set P)) (hp₂p₁ : p₂ ≠ p₁) (hp₂p₄ : p₂ ≠ p₄)
    (hp₃p₁ : p₃ ≠ p₁) (hp₃p₄ : p₃ ≠ p₄) : (2 : ℤ) • ∡ p₁ p₂ p₄ = (2 : ℤ) • ∡ p₁ p₃ p₄ 
Arguments: {V : Type u_1} {P : Type u_2} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [MetricSpace P] [NormedAddTorsor V P] [hd2 : Fact (FiniteDimensional.finrank ℝ V = 2)] [Module.Oriented ℝ V (Fin 2)] {p₁ p₂ p₃ p₄ : P} (h : EuclideanGeometry.Cospherical {p₁, p₂, p₃, p₄}) (hp₂p₁ : p₂ ≠ p₁) (hp₂p₄ : p₂ ≠ p₄) (hp₃p₁ : p₃ ≠ p₁) (hp₃p₄ : p₃ ≠ p₄)
Goal: 2 • ∡ p₁ p₂ p₄ = 2 • ∡ p₁ p₃ p₄
-- EuclideanGeometry.Sphere.oangle_eq_pi_sub_two_zsmul_oangle_center_left
theorem oangle_eq_pi_sub_two_zsmul_oangle_center_left {s : Sphere P} {p₁ p₂ : P} (hp₁ : p₁ ∈ s)
    (hp₂ : p₂ ∈ s) (h : p₁ ≠ p₂) : ∡ p₁ s.center p₂ = π - (2 : ℤ) • ∡ s.center p₂ p₁ 
Arguments: {V : Type u_1} {P : Type u_2} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [MetricSpace P] [NormedAddTorsor V P] [hd2 : Fact (FiniteDimensional.finrank ℝ V = 2)] [Module.Oriented ℝ V (Fin 2)] {s : EuclideanGeometry.Sphere P} {p₁ p₂ : P} (hp₁ : p₁ ∈ s) (hp₂ : p₂ ∈ s) (h : p₁ ≠ p₂)
Goal: ∡ p₁ s.center p₂ = ↑π - 2 • ∡ s.center p₂ p₁
-- EuclideanGeometry.Sphere.oangle_eq_pi_sub_two_zsmul_oangle_center_right
theorem oangle_eq_pi_sub_two_zsmul_oangle_center_right {s : Sphere P} {p₁ p₂ : P} (hp₁ : p₁ ∈ s)
    (hp₂ : p₂ ∈ s) (h : p₁ ≠ p₂) : ∡ p₁ s.center p₂ = π - (2 : ℤ) • ∡ p₂ p₁ s.center 
Arguments: {V : Type u_1} {P : Type u_2} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [MetricSpace P] [NormedAddTorsor V P] [hd2 : Fact (FiniteDimensional.finrank ℝ V = 2)] [Module.Oriented ℝ V (Fin 2)] {s : EuclideanGeometry.Sphere P} {p₁ p₂ : P} (hp₁ : p₁ ∈ s) (hp₂ : p₂ ∈ s) (h : p₁ ≠ p₂)
Goal: ∡ p₁ s.center p₂ = ↑π - 2 • ∡ p₂ p₁ s.center