ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- EuclideanGeometry.right_ne_of_oangle_ne_zero
theorem right_ne_of_oangle_ne_zero {p₁ p₂ p₃ : P} (h : ∡ p₁ p₂ p₃ ≠ 0) : p₃ ≠ p₂ 
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 : P} (h : ∡ p₁ p₂ p₃ ≠ 0)
Goal: p₃ ≠ p₂
-- EuclideanGeometry.left_ne_right_of_oangle_ne_zero
theorem left_ne_right_of_oangle_ne_zero {p₁ p₂ p₃ : P} (h : ∡ p₁ p₂ p₃ ≠ 0) : p₁ ≠ 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)] {p₁ p₂ p₃ : P} (h : ∡ p₁ p₂ p₃ ≠ 0)
Goal: p₁ ≠ p₃
-- EuclideanGeometry.left_ne_of_oangle_eq_pi
theorem left_ne_of_oangle_eq_pi {p₁ p₂ p₃ : P} (h : ∡ p₁ p₂ p₃ = π) : p₁ ≠ p₂ 
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 : P} (h : ∡ p₁ p₂ p₃ = ↑π)
Goal: p₁ ≠ p₂
-- EuclideanGeometry.right_ne_of_oangle_eq_pi
theorem right_ne_of_oangle_eq_pi {p₁ p₂ p₃ : P} (h : ∡ p₁ p₂ p₃ = π) : p₃ ≠ p₂ 
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 : P} (h : ∡ p₁ p₂ p₃ = ↑π)
Goal: p₃ ≠ p₂
-- EuclideanGeometry.left_ne_right_of_oangle_eq_pi
theorem left_ne_right_of_oangle_eq_pi {p₁ p₂ p₃ : P} (h : ∡ p₁ p₂ p₃ = π) : p₁ ≠ 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)] {p₁ p₂ p₃ : P} (h : ∡ p₁ p₂ p₃ = ↑π)
Goal: p₁ ≠ p₃
-- EuclideanGeometry.left_ne_of_oangle_eq_pi_div_two
theorem left_ne_of_oangle_eq_pi_div_two {p₁ p₂ p₃ : P} (h : ∡ p₁ p₂ p₃ = (π / 2 : ℝ)) : p₁ ≠ p₂ 
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 : P} (h : ∡ p₁ p₂ p₃ = ↑(π / 2))
Goal: p₁ ≠ p₂
-- EuclideanGeometry.right_ne_of_oangle_eq_pi_div_two
theorem right_ne_of_oangle_eq_pi_div_two {p₁ p₂ p₃ : P} (h : ∡ p₁ p₂ p₃ = (π / 2 : ℝ)) : p₃ ≠ p₂ 
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 : P} (h : ∡ p₁ p₂ p₃ = ↑(π / 2))
Goal: p₃ ≠ p₂
-- EuclideanGeometry.left_ne_right_of_oangle_eq_pi_div_two
theorem left_ne_right_of_oangle_eq_pi_div_two {p₁ p₂ p₃ : P} (h : ∡ p₁ p₂ p₃ = (π / 2 : ℝ)) :
    p₁ ≠ 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)] {p₁ p₂ p₃ : P} (h : ∡ p₁ p₂ p₃ = ↑(π / 2))
Goal: p₁ ≠ p₃
-- EuclideanGeometry.left_ne_of_oangle_eq_neg_pi_div_two
theorem left_ne_of_oangle_eq_neg_pi_div_two {p₁ p₂ p₃ : P} (h : ∡ p₁ p₂ p₃ = (-π / 2 : ℝ)) :
    p₁ ≠ p₂ 
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 : P} (h : ∡ p₁ p₂ p₃ = ↑(-π / 2))
Goal: p₁ ≠ p₂
-- EuclideanGeometry.right_ne_of_oangle_eq_neg_pi_div_two
theorem right_ne_of_oangle_eq_neg_pi_div_two {p₁ p₂ p₃ : P} (h : ∡ p₁ p₂ p₃ = (-π / 2 : ℝ)) :
    p₃ ≠ p₂ 
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 : P} (h : ∡ p₁ p₂ p₃ = ↑(-π / 2))
Goal: p₃ ≠ p₂
-- EuclideanGeometry.left_ne_right_of_oangle_eq_neg_pi_div_two
theorem left_ne_right_of_oangle_eq_neg_pi_div_two {p₁ p₂ p₃ : P} (h : ∡ p₁ p₂ p₃ = (-π / 2 : ℝ)) :
    p₁ ≠ 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)] {p₁ p₂ p₃ : P} (h : ∡ p₁ p₂ p₃ = ↑(-π / 2))
Goal: p₁ ≠ p₃
-- EuclideanGeometry.left_ne_of_oangle_sign_ne_zero
theorem left_ne_of_oangle_sign_ne_zero {p₁ p₂ p₃ : P} (h : (∡ p₁ p₂ p₃).sign ≠ 0) : p₁ ≠ p₂ 
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 : P} (h : (∡ p₁ p₂ p₃).sign ≠ 0)
Goal: p₁ ≠ p₂
-- EuclideanGeometry.right_ne_of_oangle_sign_ne_zero
theorem right_ne_of_oangle_sign_ne_zero {p₁ p₂ p₃ : P} (h : (∡ p₁ p₂ p₃).sign ≠ 0) : p₃ ≠ p₂ 
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 : P} (h : (∡ p₁ p₂ p₃).sign ≠ 0)
Goal: p₃ ≠ p₂
-- EuclideanGeometry.left_ne_right_of_oangle_sign_ne_zero
theorem left_ne_right_of_oangle_sign_ne_zero {p₁ p₂ p₃ : P} (h : (∡ p₁ p₂ p₃).sign ≠ 0) : p₁ ≠ 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)] {p₁ p₂ p₃ : P} (h : (∡ p₁ p₂ p₃).sign ≠ 0)
Goal: p₁ ≠ p₃
-- EuclideanGeometry.left_ne_of_oangle_sign_eq_one
theorem left_ne_of_oangle_sign_eq_one {p₁ p₂ p₃ : P} (h : (∡ p₁ p₂ p₃).sign = 1) : p₁ ≠ p₂ 
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 : P} (h : (∡ p₁ p₂ p₃).sign = 1)
Goal: p₁ ≠ p₂
-- EuclideanGeometry.right_ne_of_oangle_sign_eq_one
theorem right_ne_of_oangle_sign_eq_one {p₁ p₂ p₃ : P} (h : (∡ p₁ p₂ p₃).sign = 1) : p₃ ≠ p₂ 
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 : P} (h : (∡ p₁ p₂ p₃).sign = 1)
Goal: p₃ ≠ p₂
-- EuclideanGeometry.left_ne_right_of_oangle_sign_eq_one
theorem left_ne_right_of_oangle_sign_eq_one {p₁ p₂ p₃ : P} (h : (∡ p₁ p₂ p₃).sign = 1) : p₁ ≠ 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)] {p₁ p₂ p₃ : P} (h : (∡ p₁ p₂ p₃).sign = 1)
Goal: p₁ ≠ p₃
-- EuclideanGeometry.left_ne_of_oangle_sign_eq_neg_one
theorem left_ne_of_oangle_sign_eq_neg_one {p₁ p₂ p₃ : P} (h : (∡ p₁ p₂ p₃).sign = -1) : p₁ ≠ p₂ 
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 : P} (h : (∡ p₁ p₂ p₃).sign = -1)
Goal: p₁ ≠ p₂
-- EuclideanGeometry.right_ne_of_oangle_sign_eq_neg_one
theorem right_ne_of_oangle_sign_eq_neg_one {p₁ p₂ p₃ : P} (h : (∡ p₁ p₂ p₃).sign = -1) : p₃ ≠ p₂ 
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 : P} (h : (∡ p₁ p₂ p₃).sign = -1)
Goal: p₃ ≠ p₂
-- EuclideanGeometry.left_ne_right_of_oangle_sign_eq_neg_one
theorem left_ne_right_of_oangle_sign_eq_neg_one {p₁ p₂ p₃ : P} (h : (∡ p₁ p₂ p₃).sign = -1) :
    p₁ ≠ 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)] {p₁ p₂ p₃ : P} (h : (∡ p₁ p₂ p₃).sign = -1)
Goal: p₁ ≠ p₃
-- EuclideanGeometry.oangle_rev
theorem oangle_rev (p₁ p₂ p₃ : P) : ∡ p₃ p₂ p₁ = -∡ 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)] (p₁ p₂ p₃ : P)
Goal: ∡ p₃ p₂ p₁ = -∡ p₁ p₂ p₃
-- EuclideanGeometry.oangle_add_oangle_rev
@[simp]
theorem oangle_add_oangle_rev (p₁ p₂ p₃ : P) : ∡ p₁ p₂ p₃ + ∡ p₃ p₂ p₁ = 0 
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 : P)
Goal: ∡ p₁ p₂ p₃ + ∡ p₃ p₂ p₁ = 0
-- EuclideanGeometry.oangle_eq_zero_iff_oangle_rev_eq_zero
theorem oangle_eq_zero_iff_oangle_rev_eq_zero {p₁ p₂ p₃ : P} : ∡ p₁ p₂ p₃ = 0 ↔ ∡ p₃ p₂ p₁ = 0 
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 : P}
Goal: ∡ p₁ p₂ p₃ = 0 ↔ ∡ p₃ p₂ p₁ = 0
-- EuclideanGeometry.oangle_eq_pi_iff_oangle_rev_eq_pi
theorem oangle_eq_pi_iff_oangle_rev_eq_pi {p₁ p₂ p₃ : P} : ∡ p₁ p₂ p₃ = π ↔ ∡ p₃ p₂ p₁ = π 
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 : P}
Goal: ∡ p₁ p₂ p₃ = ↑π ↔ ∡ p₃ p₂ p₁ = ↑π
-- EuclideanGeometry.oangle_ne_zero_and_ne_pi_iff_affineIndependent
theorem oangle_ne_zero_and_ne_pi_iff_affineIndependent {p₁ p₂ p₃ : P} :
    ∡ p₁ p₂ p₃ ≠ 0 ∧ ∡ p₁ p₂ p₃ ≠ π ↔ AffineIndependent ℝ ![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)] {p₁ p₂ p₃ : P}
Goal: ∡ p₁ p₂ p₃ ≠ 0 ∧ ∡ p₁ p₂ p₃ ≠ ↑π ↔ AffineIndependent ℝ ![p₁, p₂, p₃]
-- EuclideanGeometry.oangle_eq_zero_or_eq_pi_iff_collinear
theorem oangle_eq_zero_or_eq_pi_iff_collinear {p₁ p₂ p₃ : P} :
    ∡ p₁ p₂ p₃ = 0 ∨ ∡ p₁ p₂ p₃ = π ↔ Collinear ℝ ({p₁, p₂, p₃} : Set P) 
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 : P}
Goal: ∡ p₁ p₂ p₃ = 0 ∨ ∡ p₁ p₂ p₃ = ↑π ↔ Collinear ℝ {p₁, p₂, p₃}
-- EuclideanGeometry.oangle_sign_eq_zero_iff_collinear
theorem oangle_sign_eq_zero_iff_collinear {p₁ p₂ p₃ : P} :
    (∡ p₁ p₂ p₃).sign = 0 ↔ Collinear ℝ ({p₁, p₂, p₃} : Set P) 
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 : P}
Goal: (∡ p₁ p₂ p₃).sign = 0 ↔ Collinear ℝ {p₁, p₂, p₃}
-- EuclideanGeometry.affineIndependent_iff_of_two_zsmul_oangle_eq
theorem affineIndependent_iff_of_two_zsmul_oangle_eq {p₁ p₂ p₃ p₄ p₅ p₆ : P}
    (h : (2 : ℤ) • ∡ p₁ p₂ p₃ = (2 : ℤ) • ∡ p₄ p₅ p₆) :
    AffineIndependent ℝ ![p₁, p₂, p₃] ↔ AffineIndependent ℝ ![p₄, p₅, p₆] 
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 p₅ p₆ : P} (h : 2 • ∡ p₁ p₂ p₃ = 2 • ∡ p₄ p₅ p₆)
Goal: AffineIndependent ℝ ![p₁, p₂, p₃] ↔ AffineIndependent ℝ ![p₄, p₅, p₆]
-- EuclideanGeometry.collinear_iff_of_two_zsmul_oangle_eq
theorem collinear_iff_of_two_zsmul_oangle_eq {p₁ p₂ p₃ p₄ p₅ p₆ : P}
    (h : (2 : ℤ) • ∡ p₁ p₂ p₃ = (2 : ℤ) • ∡ p₄ p₅ p₆) :
    Collinear ℝ ({p₁, p₂, p₃} : Set P) ↔ Collinear ℝ ({p₄, p₅, p₆} : Set P) 
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 p₅ p₆ : P} (h : 2 • ∡ p₁ p₂ p₃ = 2 • ∡ p₄ p₅ p₆)
Goal: Collinear ℝ {p₁, p₂, p₃} ↔ Collinear ℝ {p₄, p₅, p₆}
-- EuclideanGeometry.two_zsmul_oangle_of_vectorSpan_eq
theorem two_zsmul_oangle_of_vectorSpan_eq {p₁ p₂ p₃ p₄ p₅ p₆ : P}
    (h₁₂₄₅ : vectorSpan ℝ ({p₁, p₂} : Set P) = vectorSpan ℝ ({p₄, p₅} : Set P))
    (h₃₂₆₅ : vectorSpan ℝ ({p₃, p₂} : Set P) = vectorSpan ℝ ({p₆, p₅} : Set P)) :
    (2 : ℤ) • ∡ p₁ p₂ p₃ = (2 : ℤ) • ∡ p₄ p₅ p₆ 
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 p₅ p₆ : P} (h₁₂₄₅ : vectorSpan ℝ {p₁, p₂} = vectorSpan ℝ {p₄, p₅}) (h₃₂₆₅ : vectorSpan ℝ {p₃, p₂} = vectorSpan ℝ {p₆, p₅})
Goal: 2 • ∡ p₁ p₂ p₃ = 2 • ∡ p₄ p₅ p₆
-- EuclideanGeometry.two_zsmul_oangle_of_parallel
theorem two_zsmul_oangle_of_parallel {p₁ p₂ p₃ p₄ p₅ p₆ : P}
    (h₁₂₄₅ : line[ℝ, p₁, p₂] ∥ line[ℝ, p₄, p₅]) (h₃₂₆₅ : line[ℝ, p₃, p₂] ∥ line[ℝ, p₆, p₅]) :
    (2 : ℤ) • ∡ p₁ p₂ p₃ = (2 : ℤ) • ∡ p₄ p₅ p₆ 
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 p₅ p₆ : P} (h₁₂₄₅ : affineSpan ℝ {p₁, p₂} ∥ affineSpan ℝ {p₄, p₅}) (h₃₂₆₅ : affineSpan ℝ {p₃, p₂} ∥ affineSpan ℝ {p₆, p₅})
Goal: 2 • ∡ p₁ p₂ p₃ = 2 • ∡ p₄ p₅ p₆
-- EuclideanGeometry.oangle_add
@[simp]
theorem oangle_add {p p₁ p₂ p₃ : P} (hp₁ : p₁ ≠ p) (hp₂ : p₂ ≠ p) (hp₃ : p₃ ≠ p) :
    ∡ p₁ p p₂ + ∡ p₂ p p₃ = ∡ p₁ p 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)] {p p₁ p₂ p₃ : P} (hp₁ : p₁ ≠ p) (hp₂ : p₂ ≠ p) (hp₃ : p₃ ≠ p)
Goal: ∡ p₁ p p₂ + ∡ p₂ p p₃ = ∡ p₁ p p₃
-- EuclideanGeometry.oangle_add_swap
@[simp]
theorem oangle_add_swap {p p₁ p₂ p₃ : P} (hp₁ : p₁ ≠ p) (hp₂ : p₂ ≠ p) (hp₃ : p₃ ≠ p) :
    ∡ p₂ p p₃ + ∡ p₁ p p₂ = ∡ p₁ p 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)] {p p₁ p₂ p₃ : P} (hp₁ : p₁ ≠ p) (hp₂ : p₂ ≠ p) (hp₃ : p₃ ≠ p)
Goal: ∡ p₂ p p₃ + ∡ p₁ p p₂ = ∡ p₁ p p₃
-- EuclideanGeometry.oangle_sub_left
@[simp]
theorem oangle_sub_left {p p₁ p₂ p₃ : P} (hp₁ : p₁ ≠ p) (hp₂ : p₂ ≠ p) (hp₃ : p₃ ≠ p) :
    ∡ p₁ p p₃ - ∡ p₁ p p₂ = ∡ p₂ p 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)] {p p₁ p₂ p₃ : P} (hp₁ : p₁ ≠ p) (hp₂ : p₂ ≠ p) (hp₃ : p₃ ≠ p)
Goal: ∡ p₁ p p₃ - ∡ p₁ p p₂ = ∡ p₂ p p₃
-- EuclideanGeometry.oangle_sub_right
@[simp]
theorem oangle_sub_right {p p₁ p₂ p₃ : P} (hp₁ : p₁ ≠ p) (hp₂ : p₂ ≠ p) (hp₃ : p₃ ≠ p) :
    ∡ p₁ p p₃ - ∡ p₂ p p₃ = ∡ p₁ p p₂ 
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)] {p p₁ p₂ p₃ : P} (hp₁ : p₁ ≠ p) (hp₂ : p₂ ≠ p) (hp₃ : p₃ ≠ p)
Goal: ∡ p₁ p p₃ - ∡ p₂ p p₃ = ∡ p₁ p p₂
-- EuclideanGeometry.oangle_add_cyc3
@[simp]
theorem oangle_add_cyc3 {p p₁ p₂ p₃ : P} (hp₁ : p₁ ≠ p) (hp₂ : p₂ ≠ p) (hp₃ : p₃ ≠ p) :
    ∡ p₁ p p₂ + ∡ p₂ p p₃ + ∡ p₃ p p₁ = 0 
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)] {p p₁ p₂ p₃ : P} (hp₁ : p₁ ≠ p) (hp₂ : p₂ ≠ p) (hp₃ : p₃ ≠ p)
Goal: ∡ p₁ p p₂ + ∡ p₂ p p₃ + ∡ p₃ p p₁ = 0
-- EuclideanGeometry.oangle_eq_oangle_of_dist_eq
theorem oangle_eq_oangle_of_dist_eq {p₁ p₂ p₃ : P} (h : dist p₁ p₂ = dist p₁ p₃) :
    ∡ p₁ p₂ p₃ = ∡ p₂ p₃ 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)] {p₁ p₂ p₃ : P} (h : dist p₁ p₂ = dist p₁ p₃)
Goal: ∡ p₁ p₂ p₃ = ∡ p₂ p₃ p₁
-- EuclideanGeometry.oangle_eq_pi_sub_two_zsmul_oangle_of_dist_eq
theorem oangle_eq_pi_sub_two_zsmul_oangle_of_dist_eq {p₁ p₂ p₃ : P} (hn : p₂ ≠ p₃)
    (h : dist p₁ p₂ = dist p₁ p₃) : ∡ p₃ p₁ p₂ = π - (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)] {p₁ p₂ p₃ : P} (hn : p₂ ≠ p₃) (h : dist p₁ p₂ = dist p₁ p₃)
Goal: ∡ p₃ p₁ p₂ = ↑π - 2 • ∡ p₁ p₂ p₃
-- EuclideanGeometry.abs_oangle_right_toReal_lt_pi_div_two_of_dist_eq
theorem abs_oangle_right_toReal_lt_pi_div_two_of_dist_eq {p₁ p₂ p₃ : P}
    (h : dist p₁ p₂ = dist p₁ p₃) : |(∡ p₁ p₂ p₃).toReal| < π / 2 
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 : P} (h : dist p₁ p₂ = dist p₁ p₃)
Goal: |(∡ p₁ p₂ p₃).toReal| < π / 2
-- EuclideanGeometry.abs_oangle_left_toReal_lt_pi_div_two_of_dist_eq
theorem abs_oangle_left_toReal_lt_pi_div_two_of_dist_eq {p₁ p₂ p₃ : P}
    (h : dist p₁ p₂ = dist p₁ p₃) : |(∡ p₂ p₃ p₁).toReal| < π / 2 
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 : P} (h : dist p₁ p₂ = dist p₁ p₃)
Goal: |(∡ p₂ p₃ p₁).toReal| < π / 2
-- EuclideanGeometry.cos_oangle_eq_cos_angle
theorem cos_oangle_eq_cos_angle {p p₁ p₂ : P} (hp₁ : p₁ ≠ p) (hp₂ : p₂ ≠ p) :
    Real.Angle.cos (∡ p₁ p p₂) = Real.cos (∠ p₁ p p₂) 
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)] {p p₁ p₂ : P} (hp₁ : p₁ ≠ p) (hp₂ : p₂ ≠ p)
Goal: (∡ p₁ p p₂).cos = Real.cos (∠ p₁ p p₂)
-- EuclideanGeometry.oangle_eq_angle_or_eq_neg_angle
theorem oangle_eq_angle_or_eq_neg_angle {p p₁ p₂ : P} (hp₁ : p₁ ≠ p) (hp₂ : p₂ ≠ p) :
    ∡ p₁ p p₂ = ∠ p₁ p p₂ ∨ ∡ p₁ p p₂ = -∠ p₁ p p₂ 
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)] {p p₁ p₂ : P} (hp₁ : p₁ ≠ p) (hp₂ : p₂ ≠ p)
Goal: ∡ p₁ p p₂ = ↑(∠ p₁ p p₂) ∨ ∡ p₁ p p₂ = -↑(∠ p₁ p p₂)
-- EuclideanGeometry.angle_eq_abs_oangle_toReal
theorem angle_eq_abs_oangle_toReal {p p₁ p₂ : P} (hp₁ : p₁ ≠ p) (hp₂ : p₂ ≠ p) :
    ∠ p₁ p p₂ = |(∡ p₁ p p₂).toReal| 
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)] {p p₁ p₂ : P} (hp₁ : p₁ ≠ p) (hp₂ : p₂ ≠ p)
Goal: ∠ p₁ p p₂ = |(∡ p₁ p p₂).toReal|
-- EuclideanGeometry.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 {p p₁ p₂ : P}
    (h : (∡ p₁ p p₂).sign = 0) : p₁ = p ∨ p₂ = p ∨ ∠ p₁ p p₂ = 0 ∨ ∠ p₁ p p₂ = π 
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)] {p p₁ p₂ : P} (h : (∡ p₁ p p₂).sign = 0)
Goal: p₁ = p ∨ p₂ = p ∨ ∠ p₁ p p₂ = 0 ∨ ∠ p₁ p p₂ = π
-- EuclideanGeometry.oangle_eq_of_angle_eq_of_sign_eq
theorem oangle_eq_of_angle_eq_of_sign_eq {p₁ p₂ p₃ p₄ p₅ p₆ : P} (h : ∠ p₁ p₂ p₃ = ∠ p₄ p₅ p₆)
    (hs : (∡ p₁ p₂ p₃).sign = (∡ p₄ p₅ p₆).sign) : ∡ p₁ p₂ p₃ = ∡ p₄ p₅ p₆ 
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 p₅ p₆ : P} (h : ∠ p₁ p₂ p₃ = ∠ p₄ p₅ p₆) (hs : (∡ p₁ p₂ p₃).sign = (∡ p₄ p₅ p₆).sign)
Goal: ∡ p₁ p₂ p₃ = ∡ p₄ p₅ p₆
-- EuclideanGeometry.angle_eq_iff_oangle_eq_of_sign_eq
theorem angle_eq_iff_oangle_eq_of_sign_eq {p₁ p₂ p₃ p₄ p₅ p₆ : P} (hp₁ : p₁ ≠ p₂) (hp₃ : p₃ ≠ p₂)
    (hp₄ : p₄ ≠ p₅) (hp₆ : p₆ ≠ p₅) (hs : (∡ p₁ p₂ p₃).sign = (∡ p₄ p₅ p₆).sign) :
    ∠ p₁ p₂ p₃ = ∠ p₄ p₅ p₆ ↔ ∡ p₁ p₂ p₃ = ∡ p₄ p₅ p₆ 
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 p₅ p₆ : P} (hp₁ : p₁ ≠ p₂) (hp₃ : p₃ ≠ p₂) (hp₄ : p₄ ≠ p₅) (hp₆ : p₆ ≠ p₅) (hs : (∡ p₁ p₂ p₃).sign = (∡ p₄ p₅ p₆).sign)
Goal: ∠ p₁ p₂ p₃ = ∠ p₄ p₅ p₆ ↔ ∡ p₁ p₂ p₃ = ∡ p₄ p₅ p₆
-- EuclideanGeometry.oangle_eq_angle_of_sign_eq_one
theorem oangle_eq_angle_of_sign_eq_one {p₁ p₂ p₃ : P} (h : (∡ p₁ p₂ p₃).sign = 1) :
    ∡ p₁ p₂ p₃ = ∠ 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)] {p₁ p₂ p₃ : P} (h : (∡ p₁ p₂ p₃).sign = 1)
Goal: ∡ p₁ p₂ p₃ = ↑(∠ p₁ p₂ p₃)
-- EuclideanGeometry.oangle_eq_neg_angle_of_sign_eq_neg_one
theorem oangle_eq_neg_angle_of_sign_eq_neg_one {p₁ p₂ p₃ : P} (h : (∡ p₁ p₂ p₃).sign = -1) :
    ∡ p₁ p₂ p₃ = -∠ 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)] {p₁ p₂ p₃ : P} (h : (∡ p₁ p₂ p₃).sign = -1)
Goal: ∡ p₁ p₂ p₃ = -↑(∠ p₁ p₂ p₃)
-- EuclideanGeometry.oangle_eq_zero_iff_angle_eq_zero
theorem oangle_eq_zero_iff_angle_eq_zero {p p₁ p₂ : P} (hp₁ : p₁ ≠ p) (hp₂ : p₂ ≠ p) :
    ∡ p₁ p p₂ = 0 ↔ ∠ p₁ p p₂ = 0 
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)] {p p₁ p₂ : P} (hp₁ : p₁ ≠ p) (hp₂ : p₂ ≠ p)
Goal: ∡ p₁ p p₂ = 0 ↔ ∠ p₁ p p₂ = 0
-- EuclideanGeometry.oangle_eq_pi_iff_angle_eq_pi
theorem oangle_eq_pi_iff_angle_eq_pi {p₁ p₂ p₃ : P} : ∡ p₁ p₂ p₃ = π ↔ ∠ p₁ p₂ p₃ = π 
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 : P}
Goal: ∡ p₁ p₂ p₃ = ↑π ↔ ∠ p₁ p₂ p₃ = π
-- EuclideanGeometry.angle_eq_pi_div_two_of_oangle_eq_pi_div_two
theorem angle_eq_pi_div_two_of_oangle_eq_pi_div_two {p₁ p₂ p₃ : P} (h : ∡ p₁ p₂ p₃ = ↑(π / 2)) :
    ∠ p₁ p₂ p₃ = π / 2 
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 : P} (h : ∡ p₁ p₂ p₃ = ↑(π / 2))
Goal: ∠ p₁ p₂ p₃ = π / 2
-- EuclideanGeometry.angle_rev_eq_pi_div_two_of_oangle_eq_pi_div_two
theorem angle_rev_eq_pi_div_two_of_oangle_eq_pi_div_two {p₁ p₂ p₃ : P} (h : ∡ p₁ p₂ p₃ = ↑(π / 2)) :
    ∠ p₃ p₂ p₁ = π / 2 
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 : P} (h : ∡ p₁ p₂ p₃ = ↑(π / 2))
Goal: ∠ p₃ p₂ p₁ = π / 2
-- EuclideanGeometry.angle_eq_pi_div_two_of_oangle_eq_neg_pi_div_two
theorem angle_eq_pi_div_two_of_oangle_eq_neg_pi_div_two {p₁ p₂ p₃ : P}
    (h : ∡ p₁ p₂ p₃ = ↑(-π / 2)) : ∠ p₁ p₂ p₃ = π / 2 
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 : P} (h : ∡ p₁ p₂ p₃ = ↑(-π / 2))
Goal: ∠ p₁ p₂ p₃ = π / 2