ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Orientation.two_zsmul_oangle_smul_smul_self
@[simp]
theorem two_zsmul_oangle_smul_smul_self (x : V) {r₁ r₂ : ℝ} :
    (2 : ℤ) • o.oangle (r₁ • 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: 2 • o.oangle (r₁ • x) (r₂ • x) = 0
-- Orientation.two_zsmul_oangle_left_of_span_eq
theorem two_zsmul_oangle_left_of_span_eq {x y : V} (z : V) (h : (ℝ ∙ x) = ℝ ∙ y) :
    (2 : ℤ) • o.oangle x z = (2 : ℤ) • o.oangle y z 
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} (z : V) (h : Submodule.span ℝ {x} = Submodule.span ℝ {y})
Goal: 2 • o.oangle x z = 2 • o.oangle y z
-- Orientation.two_zsmul_oangle_right_of_span_eq
theorem two_zsmul_oangle_right_of_span_eq (x : V) {y z : V} (h : (ℝ ∙ y) = ℝ ∙ z) :
    (2 : ℤ) • o.oangle x y = (2 : ℤ) • o.oangle x z 
Arguments: {V : Type u_1} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [Fact (FiniteDimensional.finrank ℝ V = 2)] (o : Orientation ℝ V (Fin 2)) (x : V) {y z : V} (h : Submodule.span ℝ {y} = Submodule.span ℝ {z})
Goal: 2 • o.oangle x y = 2 • o.oangle x z
-- Orientation.two_zsmul_oangle_of_span_eq_of_span_eq
theorem two_zsmul_oangle_of_span_eq_of_span_eq {w x y z : V} (hwx : (ℝ ∙ w) = ℝ ∙ x)
    (hyz : (ℝ ∙ y) = ℝ ∙ z) : (2 : ℤ) • o.oangle w y = (2 : ℤ) • o.oangle x z 
Arguments: {V : Type u_1} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [Fact (FiniteDimensional.finrank ℝ V = 2)] (o : Orientation ℝ V (Fin 2)) {w x y z : V} (hwx : Submodule.span ℝ {w} = Submodule.span ℝ {x}) (hyz : Submodule.span ℝ {y} = Submodule.span ℝ {z})
Goal: 2 • o.oangle w y = 2 • o.oangle x z
-- Orientation.oangle_eq_zero_iff_oangle_rev_eq_zero
theorem oangle_eq_zero_iff_oangle_rev_eq_zero {x y : V} : o.oangle x y = 0 ↔ o.oangle 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 x y = 0 ↔ o.oangle y x = 0
-- Orientation.oangle_eq_zero_iff_sameRay
theorem oangle_eq_zero_iff_sameRay {x y : V} : o.oangle x y = 0 ↔ SameRay 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: o.oangle x y = 0 ↔ SameRay ℝ x y
-- Orientation.oangle_eq_pi_iff_oangle_rev_eq_pi
theorem oangle_eq_pi_iff_oangle_rev_eq_pi {x y : V} : o.oangle x y = π ↔ o.oangle y x = π 
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 x y = ↑π ↔ o.oangle y x = ↑π
-- Orientation.oangle_eq_pi_iff_sameRay_neg
theorem oangle_eq_pi_iff_sameRay_neg {x y : V} :
    o.oangle x y = π ↔ x ≠ 0 ∧ y ≠ 0 ∧ SameRay 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: o.oangle x y = ↑π ↔ x ≠ 0 ∧ y ≠ 0 ∧ SameRay ℝ x (-y)
-- Orientation.oangle_eq_zero_or_eq_pi_iff_not_linearIndependent
theorem oangle_eq_zero_or_eq_pi_iff_not_linearIndependent {x y : V} :
    o.oangle x y = 0 ∨ o.oangle x y = π ↔ ¬LinearIndependent 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: o.oangle x y = 0 ∨ o.oangle x y = ↑π ↔ ¬LinearIndependent ℝ ![x, y]
-- Orientation.oangle_eq_zero_or_eq_pi_iff_right_eq_smul
theorem oangle_eq_zero_or_eq_pi_iff_right_eq_smul {x y : V} :
    o.oangle x y = 0 ∨ o.oangle x y = π ↔ x = 0 ∨ ∃ r : ℝ, y = 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 y : V}
Goal: o.oangle x y = 0 ∨ o.oangle x y = ↑π ↔ x = 0 ∨ ∃ r, y = r • x
-- Orientation.oangle_ne_zero_and_ne_pi_iff_linearIndependent
theorem oangle_ne_zero_and_ne_pi_iff_linearIndependent {x y : V} :
    o.oangle x y ≠ 0 ∧ o.oangle x y ≠ π ↔ LinearIndependent 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: o.oangle x y ≠ 0 ∧ o.oangle x y ≠ ↑π ↔ LinearIndependent ℝ ![x, y]
-- Orientation.eq_iff_norm_eq_and_oangle_eq_zero
theorem eq_iff_norm_eq_and_oangle_eq_zero (x y : V) : x = y ↔ ‖x‖ = ‖y‖ ∧ o.oangle 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: x = y ↔ ‖x‖ = ‖y‖ ∧ o.oangle x y = 0
-- Orientation.eq_iff_oangle_eq_zero_of_norm_eq
theorem eq_iff_oangle_eq_zero_of_norm_eq {x y : V} (h : ‖x‖ = ‖y‖) : x = y ↔ o.oangle 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} (h : ‖x‖ = ‖y‖)
Goal: x = y ↔ o.oangle x y = 0
-- Orientation.eq_iff_norm_eq_of_oangle_eq_zero
theorem eq_iff_norm_eq_of_oangle_eq_zero {x y : V} (h : o.oangle x y = 0) : 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} (h : o.oangle x y = 0)
Goal: x = y ↔ ‖x‖ = ‖y‖
-- Orientation.oangle_add
@[simp]
theorem oangle_add {x y z : V} (hx : x ≠ 0) (hy : y ≠ 0) (hz : z ≠ 0) :
    o.oangle x y + o.oangle y z = o.oangle x z 
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} (hx : x ≠ 0) (hy : y ≠ 0) (hz : z ≠ 0)
Goal: o.oangle x y + o.oangle y z = o.oangle x z
-- Orientation.oangle_add_swap
@[simp]
theorem oangle_add_swap {x y z : V} (hx : x ≠ 0) (hy : y ≠ 0) (hz : z ≠ 0) :
    o.oangle y z + o.oangle x y = o.oangle x z 
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} (hx : x ≠ 0) (hy : y ≠ 0) (hz : z ≠ 0)
Goal: o.oangle y z + o.oangle x y = o.oangle x z
-- Orientation.oangle_sub_left
@[simp]
theorem oangle_sub_left {x y z : V} (hx : x ≠ 0) (hy : y ≠ 0) (hz : z ≠ 0) :
    o.oangle x z - o.oangle x y = o.oangle y z 
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} (hx : x ≠ 0) (hy : y ≠ 0) (hz : z ≠ 0)
Goal: o.oangle x z - o.oangle x y = o.oangle y z
-- Orientation.oangle_sub_right
@[simp]
theorem oangle_sub_right {x y z : V} (hx : x ≠ 0) (hy : y ≠ 0) (hz : z ≠ 0) :
    o.oangle x z - o.oangle y z = 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 z : V} (hx : x ≠ 0) (hy : y ≠ 0) (hz : z ≠ 0)
Goal: o.oangle x z - o.oangle y z = o.oangle x y
-- Orientation.oangle_add_cyc3
@[simp]
theorem oangle_add_cyc3 {x y z : V} (hx : x ≠ 0) (hy : y ≠ 0) (hz : z ≠ 0) :
    o.oangle x y + o.oangle y z + o.oangle z 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 z : V} (hx : x ≠ 0) (hy : y ≠ 0) (hz : z ≠ 0)
Goal: o.oangle x y + o.oangle y z + o.oangle z x = 0
-- Orientation.oangle_add_cyc3_neg_left
@[simp]
theorem oangle_add_cyc3_neg_left {x y z : V} (hx : x ≠ 0) (hy : y ≠ 0) (hz : z ≠ 0) :
    o.oangle (-x) y + o.oangle (-y) z + o.oangle (-z) x = π 
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} (hx : x ≠ 0) (hy : y ≠ 0) (hz : z ≠ 0)
Goal: o.oangle (-x) y + o.oangle (-y) z + o.oangle (-z) x = ↑π
-- Orientation.oangle_add_cyc3_neg_right
@[simp]
theorem oangle_add_cyc3_neg_right {x y z : V} (hx : x ≠ 0) (hy : y ≠ 0) (hz : z ≠ 0) :
    o.oangle x (-y) + o.oangle y (-z) + o.oangle z (-x) = π 
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} (hx : x ≠ 0) (hy : y ≠ 0) (hz : z ≠ 0)
Goal: o.oangle x (-y) + o.oangle y (-z) + o.oangle z (-x) = ↑π
-- Orientation.oangle_sub_eq_oangle_sub_rev_of_norm_eq
theorem oangle_sub_eq_oangle_sub_rev_of_norm_eq {x y : V} (h : ‖x‖ = ‖y‖) :
    o.oangle x (x - y) = o.oangle (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} (h : ‖x‖ = ‖y‖)
Goal: o.oangle x (x - y) = o.oangle (y - x) y
-- Orientation.oangle_eq_pi_sub_two_zsmul_oangle_sub_of_norm_eq
theorem oangle_eq_pi_sub_two_zsmul_oangle_sub_of_norm_eq {x y : V} (hn : x ≠ y) (h : ‖x‖ = ‖y‖) :
    o.oangle y x = π - (2 : ℤ) • o.oangle (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} (hn : x ≠ y) (h : ‖x‖ = ‖y‖)
Goal: o.oangle y x = ↑π - 2 • o.oangle (y - x) y
-- Orientation.oangle_map
@[simp]
theorem oangle_map (x y : V') (f : V ≃ₗᵢ[ℝ] V') :
    (Orientation.map (Fin 2) f.toLinearEquiv o).oangle x y = o.oangle (f.symm x) (f.symm y) 
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)) (x y : V') (f : V ≃ₗᵢ[ℝ] V')
Goal: ((Orientation.map (Fin 2) f.toLinearEquiv) o).oangle x y = o.oangle (f.symm x) (f.symm y)
-- Complex.oangle
@[simp]
protected theorem _root_.Complex.oangle (w z : ℂ) :
    Complex.orientation.oangle w z = Complex.arg (conj w * z) 
Arguments: (w z : ℂ)
Goal: Complex.orientation.oangle w z = ↑((starRingEnd ℂ) w * z).arg
-- Orientation.oangle_map_complex
theorem oangle_map_complex (f : V ≃ₗᵢ[ℝ] ℂ)
    (hf : Orientation.map (Fin 2) f.toLinearEquiv o = Complex.orientation) (x y : V) :
    o.oangle x y = Complex.arg (conj (f x) * f y) 
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] ℂ) (hf : (Orientation.map (Fin 2) f.toLinearEquiv) o = Complex.orientation) (x y : V)
Goal: o.oangle x y = ↑((starRingEnd ℂ) (f x) * f y).arg
-- Orientation.oangle_neg_orientation_eq_neg
theorem oangle_neg_orientation_eq_neg (x y : V) : (-o).oangle x 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)
Goal: (-o).oangle x y = -o.oangle x y
-- Orientation.inner_eq_norm_mul_norm_mul_cos_oangle
theorem inner_eq_norm_mul_norm_mul_cos_oangle (x y : V) :
    ⟪x, y⟫ = ‖x‖ * ‖y‖ * Real.Angle.cos (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)
Goal: ⟪x, y⟫_ℝ = ‖x‖ * ‖y‖ * (o.oangle x y).cos
-- Orientation.cos_oangle_eq_inner_div_norm_mul_norm
theorem cos_oangle_eq_inner_div_norm_mul_norm {x y : V} (hx : x ≠ 0) (hy : y ≠ 0) :
    Real.Angle.cos (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} (hx : x ≠ 0) (hy : y ≠ 0)
Goal: (o.oangle x y).cos = ⟪x, y⟫_ℝ / (‖x‖ * ‖y‖)
-- Orientation.cos_oangle_eq_cos_angle
theorem cos_oangle_eq_cos_angle {x y : V} (hx : x ≠ 0) (hy : y ≠ 0) :
    Real.Angle.cos (o.oangle x y) = Real.cos (InnerProductGeometry.angle 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} (hx : x ≠ 0) (hy : y ≠ 0)
Goal: (o.oangle x y).cos = Real.cos (InnerProductGeometry.angle x y)
-- Orientation.oangle_eq_angle_or_eq_neg_angle
theorem oangle_eq_angle_or_eq_neg_angle {x y : V} (hx : x ≠ 0) (hy : y ≠ 0) :
    o.oangle x y = InnerProductGeometry.angle x y ∨
      o.oangle x y = -InnerProductGeometry.angle 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} (hx : x ≠ 0) (hy : y ≠ 0)
Goal: o.oangle x y = ↑(InnerProductGeometry.angle x y) ∨ o.oangle x y = -↑(InnerProductGeometry.angle x y)
-- Orientation.angle_eq_abs_oangle_toReal
theorem angle_eq_abs_oangle_toReal {x y : V} (hx : x ≠ 0) (hy : y ≠ 0) :
    InnerProductGeometry.angle x y = |(o.oangle x y).toReal| 
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)
Goal: InnerProductGeometry.angle x y = |(o.oangle x y).toReal|
-- Orientation.eq_zero_or_angle_eq_zero_or_pi_of_sign_oangle_eq_zero
theorem eq_zero_or_angle_eq_zero_or_pi_of_sign_oangle_eq_zero {x y : V}
    (h : (o.oangle x y).sign = 0) :
    x = 0 ∨ y = 0 ∨ InnerProductGeometry.angle x y = 0 ∨ InnerProductGeometry.angle x y = π 
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} (h : (o.oangle x y).sign = 0)
Goal: x = 0 ∨ y = 0 ∨ InnerProductGeometry.angle x y = 0 ∨ InnerProductGeometry.angle x y = π
-- Orientation.oangle_eq_of_angle_eq_of_sign_eq
theorem oangle_eq_of_angle_eq_of_sign_eq {w x y z : V}
    (h : InnerProductGeometry.angle w x = InnerProductGeometry.angle y z)
    (hs : (o.oangle w x).sign = (o.oangle y z).sign) : o.oangle w x = o.oangle y z 
Arguments: {V : Type u_1} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [Fact (FiniteDimensional.finrank ℝ V = 2)] (o : Orientation ℝ V (Fin 2)) {w x y z : V} (h : InnerProductGeometry.angle w x = InnerProductGeometry.angle y z) (hs : (o.oangle w x).sign = (o.oangle y z).sign)
Goal: o.oangle w x = o.oangle y z
-- Orientation.angle_eq_iff_oangle_eq_of_sign_eq
theorem angle_eq_iff_oangle_eq_of_sign_eq {w x y z : V} (hw : w ≠ 0) (hx : x ≠ 0) (hy : y ≠ 0)
    (hz : z ≠ 0) (hs : (o.oangle w x).sign = (o.oangle y z).sign) :
    InnerProductGeometry.angle w x = InnerProductGeometry.angle y z ↔
    o.oangle w x = o.oangle y z 
Arguments: {V : Type u_1} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [Fact (FiniteDimensional.finrank ℝ V = 2)] (o : Orientation ℝ V (Fin 2)) {w x y z : V} (hw : w ≠ 0) (hx : x ≠ 0) (hy : y ≠ 0) (hz : z ≠ 0) (hs : (o.oangle w x).sign = (o.oangle y z).sign)
Goal: InnerProductGeometry.angle w x = InnerProductGeometry.angle y z ↔ o.oangle w x = o.oangle y z
-- Orientation.oangle_eq_angle_of_sign_eq_one
theorem oangle_eq_angle_of_sign_eq_one {x y : V} (h : (o.oangle x y).sign = 1) :
    o.oangle x y = InnerProductGeometry.angle 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} (h : (o.oangle x y).sign = 1)
Goal: o.oangle x y = ↑(InnerProductGeometry.angle x y)
-- Orientation.oangle_eq_neg_angle_of_sign_eq_neg_one
theorem oangle_eq_neg_angle_of_sign_eq_neg_one {x y : V} (h : (o.oangle x y).sign = -1) :
    o.oangle x y = -InnerProductGeometry.angle 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} (h : (o.oangle x y).sign = -1)
Goal: o.oangle x y = -↑(InnerProductGeometry.angle x y)
-- Orientation.oangle_eq_zero_iff_angle_eq_zero
theorem oangle_eq_zero_iff_angle_eq_zero {x y : V} (hx : x ≠ 0) (hy : y ≠ 0) :
    o.oangle x y = 0 ↔ InnerProductGeometry.angle 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} (hx : x ≠ 0) (hy : y ≠ 0)
Goal: o.oangle x y = 0 ↔ InnerProductGeometry.angle x y = 0
-- Orientation.oangle_eq_pi_iff_angle_eq_pi
theorem oangle_eq_pi_iff_angle_eq_pi {x y : V} :
    o.oangle x y = π ↔ InnerProductGeometry.angle x y = π 
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 x y = ↑π ↔ InnerProductGeometry.angle x y = π
-- Orientation.eq_zero_or_oangle_eq_iff_inner_eq_zero
theorem eq_zero_or_oangle_eq_iff_inner_eq_zero {x y : V} :
    x = 0 ∨ y = 0 ∨ o.oangle x y = (π / 2 : ℝ) ∨ o.oangle x y = (-π / 2 : ℝ) ↔ ⟪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: x = 0 ∨ y = 0 ∨ o.oangle x y = ↑(π / 2) ∨ o.oangle x y = ↑(-π / 2) ↔ ⟪x, y⟫_ℝ = 0
-- Orientation.inner_eq_zero_of_oangle_eq_pi_div_two
theorem inner_eq_zero_of_oangle_eq_pi_div_two {x y : V} (h : o.oangle x y = (π / 2 : ℝ)) :
    ⟪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} (h : o.oangle x y = ↑(π / 2))
Goal: ⟪x, y⟫_ℝ = 0
-- Orientation.inner_rev_eq_zero_of_oangle_eq_pi_div_two
theorem inner_rev_eq_zero_of_oangle_eq_pi_div_two {x y : V} (h : o.oangle x y = (π / 2 : ℝ)) :
    ⟪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} (h : o.oangle x y = ↑(π / 2))
Goal: ⟪y, x⟫_ℝ = 0
-- Orientation.inner_eq_zero_of_oangle_eq_neg_pi_div_two
theorem inner_eq_zero_of_oangle_eq_neg_pi_div_two {x y : V} (h : o.oangle x y = (-π / 2 : ℝ)) :
    ⟪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} (h : o.oangle x y = ↑(-π / 2))
Goal: ⟪x, y⟫_ℝ = 0
-- Orientation.inner_rev_eq_zero_of_oangle_eq_neg_pi_div_two
theorem inner_rev_eq_zero_of_oangle_eq_neg_pi_div_two {x y : V} (h : o.oangle x y = (-π / 2 : ℝ)) :
    ⟪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} (h : o.oangle x y = ↑(-π / 2))
Goal: ⟪y, x⟫_ℝ = 0
-- Orientation.oangle_sign_neg_left
@[simp]
theorem oangle_sign_neg_left (x y : V) : (o.oangle (-x) y).sign = -(o.oangle x y).sign 
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 (-x) y).sign = -(o.oangle x y).sign
-- Orientation.oangle_sign_neg_right
@[simp]
theorem oangle_sign_neg_right (x y : V) : (o.oangle x (-y)).sign = -(o.oangle x y).sign 
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 x (-y)).sign = -(o.oangle x y).sign
-- Orientation.oangle_sign_smul_left
@[simp]
theorem oangle_sign_smul_left (x y : V) (r : ℝ) :
    (o.oangle (r • x) y).sign = SignType.sign r * (o.oangle x y).sign 
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) (r : ℝ)
Goal: (o.oangle (r • x) y).sign = SignType.sign r * (o.oangle x y).sign
-- Orientation.oangle_sign_smul_right
@[simp]
theorem oangle_sign_smul_right (x y : V) (r : ℝ) :
    (o.oangle x (r • y)).sign = SignType.sign r * (o.oangle x y).sign 
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) (r : ℝ)
Goal: (o.oangle x (r • y)).sign = SignType.sign r * (o.oangle x y).sign
-- Orientation.oangle_smul_add_right_eq_zero_or_eq_pi_iff
theorem oangle_smul_add_right_eq_zero_or_eq_pi_iff {x y : V} (r : ℝ) :
    o.oangle x (r • x + y) = 0 ∨ o.oangle x (r • x + y) = π ↔
    o.oangle x y = 0 ∨ o.oangle x y = π 
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} (r : ℝ)
Goal: o.oangle x (r • x + y) = 0 ∨ o.oangle x (r • x + y) = ↑π ↔ o.oangle x y = 0 ∨ o.oangle x y = ↑π
-- Orientation.oangle_sign_smul_add_right
@[simp]
theorem oangle_sign_smul_add_right (x y : V) (r : ℝ) :
    (o.oangle x (r • x + y)).sign = (o.oangle x y).sign 
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) (r : ℝ)
Goal: (o.oangle x (r • x + y)).sign = (o.oangle x y).sign
-- Orientation.oangle_sign_add_smul_left
@[simp]
theorem oangle_sign_add_smul_left (x y : V) (r : ℝ) :
    (o.oangle (x + r • y) y).sign = (o.oangle x y).sign 
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) (r : ℝ)
Goal: (o.oangle (x + r • y) y).sign = (o.oangle x y).sign
-- Orientation.oangle_sign_sub_smul_right
@[simp]
theorem oangle_sign_sub_smul_right (x y : V) (r : ℝ) :
    (o.oangle x (y - r • x)).sign = (o.oangle x y).sign 
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) (r : ℝ)
Goal: (o.oangle x (y - r • x)).sign = (o.oangle x y).sign
-- Orientation.oangle_sign_sub_smul_left
@[simp]
theorem oangle_sign_sub_smul_left (x y : V) (r : ℝ) :
    (o.oangle (x - r • y) y).sign = (o.oangle x y).sign 
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) (r : ℝ)
Goal: (o.oangle (x - r • y) y).sign = (o.oangle x y).sign
-- Orientation.oangle_sign_add_right
@[simp]
theorem oangle_sign_add_right (x y : V) : (o.oangle x (x + y)).sign = (o.oangle x y).sign 
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 x (x + y)).sign = (o.oangle x y).sign
-- Orientation.oangle_sign_add_left
@[simp]
theorem oangle_sign_add_left (x y : V) : (o.oangle (x + y) y).sign = (o.oangle x y).sign 
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 (x + y) y).sign = (o.oangle x y).sign
-- Orientation.oangle_sign_sub_right
@[simp]
theorem oangle_sign_sub_right (x y : V) : (o.oangle x (y - x)).sign = (o.oangle x y).sign 
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 x (y - x)).sign = (o.oangle x y).sign
-- Orientation.oangle_sign_sub_left
@[simp]
theorem oangle_sign_sub_left (x y : V) : (o.oangle (x - y) y).sign = (o.oangle x y).sign 
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 (x - y) y).sign = (o.oangle x y).sign
-- Orientation.oangle_sign_smul_sub_right
@[simp]
theorem oangle_sign_smul_sub_right (x y : V) (r : ℝ) :
    (o.oangle x (r • x - y)).sign = -(o.oangle x y).sign 
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) (r : ℝ)
Goal: (o.oangle x (r • x - y)).sign = -(o.oangle x y).sign
-- Orientation.oangle_sign_smul_sub_left
@[simp]
theorem oangle_sign_smul_sub_left (x y : V) (r : ℝ) :
    (o.oangle (r • y - x) y).sign = -(o.oangle x y).sign 
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) (r : ℝ)
Goal: (o.oangle (r • y - x) y).sign = -(o.oangle x y).sign