ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Orientation.sin_oangle_add_left_of_oangle_eq_pi_div_two
theorem sin_oangle_add_left_of_oangle_eq_pi_div_two {x y : V} (h : o.oangle x y = ↑(π / 2)) :
    Real.Angle.sin (o.oangle (x + y) y) = ‖x‖ / ‖x + y‖ 
Arguments: {V : Type u_1} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [hd2 : Fact (FiniteDimensional.finrank ℝ V = 2)] (o : Orientation ℝ V (Fin 2)) {x y : V} (h : o.oangle x y = ↑(π / 2))
Goal: (o.oangle (x + y) y).sin = ‖x‖ / ‖x + y‖
-- Orientation.tan_oangle_add_right_of_oangle_eq_pi_div_two
theorem tan_oangle_add_right_of_oangle_eq_pi_div_two {x y : V} (h : o.oangle x y = ↑(π / 2)) :
    Real.Angle.tan (o.oangle x (x + y)) = ‖y‖ / ‖x‖ 
Arguments: {V : Type u_1} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [hd2 : Fact (FiniteDimensional.finrank ℝ V = 2)] (o : Orientation ℝ V (Fin 2)) {x y : V} (h : o.oangle x y = ↑(π / 2))
Goal: (o.oangle x (x + y)).tan = ‖y‖ / ‖x‖
-- Orientation.tan_oangle_add_left_of_oangle_eq_pi_div_two
theorem tan_oangle_add_left_of_oangle_eq_pi_div_two {x y : V} (h : o.oangle x y = ↑(π / 2)) :
    Real.Angle.tan (o.oangle (x + y) y) = ‖x‖ / ‖y‖ 
Arguments: {V : Type u_1} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [hd2 : Fact (FiniteDimensional.finrank ℝ V = 2)] (o : Orientation ℝ V (Fin 2)) {x y : V} (h : o.oangle x y = ↑(π / 2))
Goal: (o.oangle (x + y) y).tan = ‖x‖ / ‖y‖
-- Orientation.cos_oangle_add_right_mul_norm_of_oangle_eq_pi_div_two
theorem cos_oangle_add_right_mul_norm_of_oangle_eq_pi_div_two {x y : V}
    (h : o.oangle x y = ↑(π / 2)) : Real.Angle.cos (o.oangle x (x + y)) * ‖x + y‖ = ‖x‖ 
Arguments: {V : Type u_1} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [hd2 : Fact (FiniteDimensional.finrank ℝ V = 2)] (o : Orientation ℝ V (Fin 2)) {x y : V} (h : o.oangle x y = ↑(π / 2))
Goal: (o.oangle x (x + y)).cos * ‖x + y‖ = ‖x‖
-- Orientation.cos_oangle_add_left_mul_norm_of_oangle_eq_pi_div_two
theorem cos_oangle_add_left_mul_norm_of_oangle_eq_pi_div_two {x y : V}
    (h : o.oangle x y = ↑(π / 2)) : Real.Angle.cos (o.oangle (x + y) y) * ‖x + y‖ = ‖y‖ 
Arguments: {V : Type u_1} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [hd2 : Fact (FiniteDimensional.finrank ℝ V = 2)] (o : Orientation ℝ V (Fin 2)) {x y : V} (h : o.oangle x y = ↑(π / 2))
Goal: (o.oangle (x + y) y).cos * ‖x + y‖ = ‖y‖
-- Orientation.sin_oangle_add_right_mul_norm_of_oangle_eq_pi_div_two
theorem sin_oangle_add_right_mul_norm_of_oangle_eq_pi_div_two {x y : V}
    (h : o.oangle x y = ↑(π / 2)) : Real.Angle.sin (o.oangle x (x + y)) * ‖x + y‖ = ‖y‖ 
Arguments: {V : Type u_1} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [hd2 : Fact (FiniteDimensional.finrank ℝ V = 2)] (o : Orientation ℝ V (Fin 2)) {x y : V} (h : o.oangle x y = ↑(π / 2))
Goal: (o.oangle x (x + y)).sin * ‖x + y‖ = ‖y‖
-- Orientation.sin_oangle_add_left_mul_norm_of_oangle_eq_pi_div_two
theorem sin_oangle_add_left_mul_norm_of_oangle_eq_pi_div_two {x y : V}
    (h : o.oangle x y = ↑(π / 2)) : Real.Angle.sin (o.oangle (x + y) y) * ‖x + y‖ = ‖x‖ 
Arguments: {V : Type u_1} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [hd2 : Fact (FiniteDimensional.finrank ℝ V = 2)] (o : Orientation ℝ V (Fin 2)) {x y : V} (h : o.oangle x y = ↑(π / 2))
Goal: (o.oangle (x + y) y).sin * ‖x + y‖ = ‖x‖
-- Orientation.tan_oangle_add_right_mul_norm_of_oangle_eq_pi_div_two
theorem tan_oangle_add_right_mul_norm_of_oangle_eq_pi_div_two {x y : V}
    (h : o.oangle x y = ↑(π / 2)) : Real.Angle.tan (o.oangle x (x + y)) * ‖x‖ = ‖y‖ 
Arguments: {V : Type u_1} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [hd2 : Fact (FiniteDimensional.finrank ℝ V = 2)] (o : Orientation ℝ V (Fin 2)) {x y : V} (h : o.oangle x y = ↑(π / 2))
Goal: (o.oangle x (x + y)).tan * ‖x‖ = ‖y‖
-- Orientation.tan_oangle_add_left_mul_norm_of_oangle_eq_pi_div_two
theorem tan_oangle_add_left_mul_norm_of_oangle_eq_pi_div_two {x y : V}
    (h : o.oangle x y = ↑(π / 2)) : Real.Angle.tan (o.oangle (x + y) y) * ‖y‖ = ‖x‖ 
Arguments: {V : Type u_1} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [hd2 : Fact (FiniteDimensional.finrank ℝ V = 2)] (o : Orientation ℝ V (Fin 2)) {x y : V} (h : o.oangle x y = ↑(π / 2))
Goal: (o.oangle (x + y) y).tan * ‖y‖ = ‖x‖
-- Orientation.norm_div_cos_oangle_add_right_of_oangle_eq_pi_div_two
theorem norm_div_cos_oangle_add_right_of_oangle_eq_pi_div_two {x y : V}
    (h : o.oangle x y = ↑(π / 2)) : ‖x‖ / Real.Angle.cos (o.oangle x (x + y)) = ‖x + y‖ 
Arguments: {V : Type u_1} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [hd2 : Fact (FiniteDimensional.finrank ℝ V = 2)] (o : Orientation ℝ V (Fin 2)) {x y : V} (h : o.oangle x y = ↑(π / 2))
Goal: ‖x‖ / (o.oangle x (x + y)).cos = ‖x + y‖
-- Orientation.norm_div_cos_oangle_add_left_of_oangle_eq_pi_div_two
theorem norm_div_cos_oangle_add_left_of_oangle_eq_pi_div_two {x y : V}
    (h : o.oangle x y = ↑(π / 2)) : ‖y‖ / Real.Angle.cos (o.oangle (x + y) y) = ‖x + y‖ 
Arguments: {V : Type u_1} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [hd2 : Fact (FiniteDimensional.finrank ℝ V = 2)] (o : Orientation ℝ V (Fin 2)) {x y : V} (h : o.oangle x y = ↑(π / 2))
Goal: ‖y‖ / (o.oangle (x + y) y).cos = ‖x + y‖
-- Orientation.norm_div_sin_oangle_add_right_of_oangle_eq_pi_div_two
theorem norm_div_sin_oangle_add_right_of_oangle_eq_pi_div_two {x y : V}
    (h : o.oangle x y = ↑(π / 2)) : ‖y‖ / Real.Angle.sin (o.oangle x (x + y)) = ‖x + y‖ 
Arguments: {V : Type u_1} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [hd2 : Fact (FiniteDimensional.finrank ℝ V = 2)] (o : Orientation ℝ V (Fin 2)) {x y : V} (h : o.oangle x y = ↑(π / 2))
Goal: ‖y‖ / (o.oangle x (x + y)).sin = ‖x + y‖
-- Orientation.norm_div_sin_oangle_add_left_of_oangle_eq_pi_div_two
theorem norm_div_sin_oangle_add_left_of_oangle_eq_pi_div_two {x y : V}
    (h : o.oangle x y = ↑(π / 2)) : ‖x‖ / Real.Angle.sin (o.oangle (x + y) y) = ‖x + y‖ 
Arguments: {V : Type u_1} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [hd2 : Fact (FiniteDimensional.finrank ℝ V = 2)] (o : Orientation ℝ V (Fin 2)) {x y : V} (h : o.oangle x y = ↑(π / 2))
Goal: ‖x‖ / (o.oangle (x + y) y).sin = ‖x + y‖
-- Orientation.norm_div_tan_oangle_add_right_of_oangle_eq_pi_div_two
theorem norm_div_tan_oangle_add_right_of_oangle_eq_pi_div_two {x y : V}
    (h : o.oangle x y = ↑(π / 2)) : ‖y‖ / Real.Angle.tan (o.oangle x (x + y)) = ‖x‖ 
Arguments: {V : Type u_1} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [hd2 : Fact (FiniteDimensional.finrank ℝ V = 2)] (o : Orientation ℝ V (Fin 2)) {x y : V} (h : o.oangle x y = ↑(π / 2))
Goal: ‖y‖ / (o.oangle x (x + y)).tan = ‖x‖
-- Orientation.norm_div_tan_oangle_add_left_of_oangle_eq_pi_div_two
theorem norm_div_tan_oangle_add_left_of_oangle_eq_pi_div_two {x y : V}
    (h : o.oangle x y = ↑(π / 2)) : ‖x‖ / Real.Angle.tan (o.oangle (x + y) y) = ‖y‖ 
Arguments: {V : Type u_1} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [hd2 : Fact (FiniteDimensional.finrank ℝ V = 2)] (o : Orientation ℝ V (Fin 2)) {x y : V} (h : o.oangle x y = ↑(π / 2))
Goal: ‖x‖ / (o.oangle (x + y) y).tan = ‖y‖
-- Orientation.oangle_sub_right_eq_arccos_of_oangle_eq_pi_div_two
theorem oangle_sub_right_eq_arccos_of_oangle_eq_pi_div_two {x y : V} (h : o.oangle x y = ↑(π / 2)) :
    o.oangle y (y - x) = Real.arccos (‖y‖ / ‖y - x‖) 
Arguments: {V : Type u_1} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [hd2 : Fact (FiniteDimensional.finrank ℝ V = 2)] (o : Orientation ℝ V (Fin 2)) {x y : V} (h : o.oangle x y = ↑(π / 2))
Goal: o.oangle y (y - x) = ↑(Real.arccos (‖y‖ / ‖y - x‖))
-- Orientation.oangle_sub_left_eq_arccos_of_oangle_eq_pi_div_two
theorem oangle_sub_left_eq_arccos_of_oangle_eq_pi_div_two {x y : V} (h : o.oangle x y = ↑(π / 2)) :
    o.oangle (x - y) x = Real.arccos (‖x‖ / ‖x - y‖) 
Arguments: {V : Type u_1} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [hd2 : Fact (FiniteDimensional.finrank ℝ V = 2)] (o : Orientation ℝ V (Fin 2)) {x y : V} (h : o.oangle x y = ↑(π / 2))
Goal: o.oangle (x - y) x = ↑(Real.arccos (‖x‖ / ‖x - y‖))
-- Orientation.oangle_sub_right_eq_arcsin_of_oangle_eq_pi_div_two
theorem oangle_sub_right_eq_arcsin_of_oangle_eq_pi_div_two {x y : V} (h : o.oangle x y = ↑(π / 2)) :
    o.oangle y (y - x) = Real.arcsin (‖x‖ / ‖y - x‖) 
Arguments: {V : Type u_1} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [hd2 : Fact (FiniteDimensional.finrank ℝ V = 2)] (o : Orientation ℝ V (Fin 2)) {x y : V} (h : o.oangle x y = ↑(π / 2))
Goal: o.oangle y (y - x) = ↑(Real.arcsin (‖x‖ / ‖y - x‖))
-- Orientation.oangle_sub_left_eq_arcsin_of_oangle_eq_pi_div_two
theorem oangle_sub_left_eq_arcsin_of_oangle_eq_pi_div_two {x y : V} (h : o.oangle x y = ↑(π / 2)) :
    o.oangle (x - y) x = Real.arcsin (‖y‖ / ‖x - y‖) 
Arguments: {V : Type u_1} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [hd2 : Fact (FiniteDimensional.finrank ℝ V = 2)] (o : Orientation ℝ V (Fin 2)) {x y : V} (h : o.oangle x y = ↑(π / 2))
Goal: o.oangle (x - y) x = ↑(Real.arcsin (‖y‖ / ‖x - y‖))
-- Orientation.oangle_sub_right_eq_arctan_of_oangle_eq_pi_div_two
theorem oangle_sub_right_eq_arctan_of_oangle_eq_pi_div_two {x y : V} (h : o.oangle x y = ↑(π / 2)) :
    o.oangle y (y - x) = Real.arctan (‖x‖ / ‖y‖) 
Arguments: {V : Type u_1} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [hd2 : Fact (FiniteDimensional.finrank ℝ V = 2)] (o : Orientation ℝ V (Fin 2)) {x y : V} (h : o.oangle x y = ↑(π / 2))
Goal: o.oangle y (y - x) = ↑(Real.arctan (‖x‖ / ‖y‖))
-- Orientation.oangle_sub_left_eq_arctan_of_oangle_eq_pi_div_two
theorem oangle_sub_left_eq_arctan_of_oangle_eq_pi_div_two {x y : V} (h : o.oangle x y = ↑(π / 2)) :
    o.oangle (x - y) x = Real.arctan (‖y‖ / ‖x‖) 
Arguments: {V : Type u_1} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [hd2 : Fact (FiniteDimensional.finrank ℝ V = 2)] (o : Orientation ℝ V (Fin 2)) {x y : V} (h : o.oangle x y = ↑(π / 2))
Goal: o.oangle (x - y) x = ↑(Real.arctan (‖y‖ / ‖x‖))
-- Orientation.cos_oangle_sub_right_of_oangle_eq_pi_div_two
theorem cos_oangle_sub_right_of_oangle_eq_pi_div_two {x y : V} (h : o.oangle x y = ↑(π / 2)) :
    Real.Angle.cos (o.oangle y (y - x)) = ‖y‖ / ‖y - x‖ 
Arguments: {V : Type u_1} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [hd2 : Fact (FiniteDimensional.finrank ℝ V = 2)] (o : Orientation ℝ V (Fin 2)) {x y : V} (h : o.oangle x y = ↑(π / 2))
Goal: (o.oangle y (y - x)).cos = ‖y‖ / ‖y - x‖
-- Orientation.cos_oangle_sub_left_of_oangle_eq_pi_div_two
theorem cos_oangle_sub_left_of_oangle_eq_pi_div_two {x y : V} (h : o.oangle x y = ↑(π / 2)) :
    Real.Angle.cos (o.oangle (x - y) x) = ‖x‖ / ‖x - y‖ 
Arguments: {V : Type u_1} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [hd2 : Fact (FiniteDimensional.finrank ℝ V = 2)] (o : Orientation ℝ V (Fin 2)) {x y : V} (h : o.oangle x y = ↑(π / 2))
Goal: (o.oangle (x - y) x).cos = ‖x‖ / ‖x - y‖
-- Orientation.sin_oangle_sub_right_of_oangle_eq_pi_div_two
theorem sin_oangle_sub_right_of_oangle_eq_pi_div_two {x y : V} (h : o.oangle x y = ↑(π / 2)) :
    Real.Angle.sin (o.oangle y (y - x)) = ‖x‖ / ‖y - x‖ 
Arguments: {V : Type u_1} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [hd2 : Fact (FiniteDimensional.finrank ℝ V = 2)] (o : Orientation ℝ V (Fin 2)) {x y : V} (h : o.oangle x y = ↑(π / 2))
Goal: (o.oangle y (y - x)).sin = ‖x‖ / ‖y - x‖
-- Orientation.sin_oangle_sub_left_of_oangle_eq_pi_div_two
theorem sin_oangle_sub_left_of_oangle_eq_pi_div_two {x y : V} (h : o.oangle x y = ↑(π / 2)) :
    Real.Angle.sin (o.oangle (x - y) x) = ‖y‖ / ‖x - y‖ 
Arguments: {V : Type u_1} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [hd2 : Fact (FiniteDimensional.finrank ℝ V = 2)] (o : Orientation ℝ V (Fin 2)) {x y : V} (h : o.oangle x y = ↑(π / 2))
Goal: (o.oangle (x - y) x).sin = ‖y‖ / ‖x - y‖
-- Orientation.tan_oangle_sub_right_of_oangle_eq_pi_div_two
theorem tan_oangle_sub_right_of_oangle_eq_pi_div_two {x y : V} (h : o.oangle x y = ↑(π / 2)) :
    Real.Angle.tan (o.oangle y (y - x)) = ‖x‖ / ‖y‖ 
Arguments: {V : Type u_1} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [hd2 : Fact (FiniteDimensional.finrank ℝ V = 2)] (o : Orientation ℝ V (Fin 2)) {x y : V} (h : o.oangle x y = ↑(π / 2))
Goal: (o.oangle y (y - x)).tan = ‖x‖ / ‖y‖
-- Orientation.tan_oangle_sub_left_of_oangle_eq_pi_div_two
theorem tan_oangle_sub_left_of_oangle_eq_pi_div_two {x y : V} (h : o.oangle x y = ↑(π / 2)) :
    Real.Angle.tan (o.oangle (x - y) x) = ‖y‖ / ‖x‖ 
Arguments: {V : Type u_1} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [hd2 : Fact (FiniteDimensional.finrank ℝ V = 2)] (o : Orientation ℝ V (Fin 2)) {x y : V} (h : o.oangle x y = ↑(π / 2))
Goal: (o.oangle (x - y) x).tan = ‖y‖ / ‖x‖
-- Orientation.cos_oangle_sub_right_mul_norm_of_oangle_eq_pi_div_two
theorem cos_oangle_sub_right_mul_norm_of_oangle_eq_pi_div_two {x y : V}
    (h : o.oangle x y = ↑(π / 2)) : Real.Angle.cos (o.oangle y (y - x)) * ‖y - x‖ = ‖y‖ 
Arguments: {V : Type u_1} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [hd2 : Fact (FiniteDimensional.finrank ℝ V = 2)] (o : Orientation ℝ V (Fin 2)) {x y : V} (h : o.oangle x y = ↑(π / 2))
Goal: (o.oangle y (y - x)).cos * ‖y - x‖ = ‖y‖
-- Orientation.cos_oangle_sub_left_mul_norm_of_oangle_eq_pi_div_two
theorem cos_oangle_sub_left_mul_norm_of_oangle_eq_pi_div_two {x y : V}
    (h : o.oangle x y = ↑(π / 2)) : Real.Angle.cos (o.oangle (x - y) x) * ‖x - y‖ = ‖x‖ 
Arguments: {V : Type u_1} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [hd2 : Fact (FiniteDimensional.finrank ℝ V = 2)] (o : Orientation ℝ V (Fin 2)) {x y : V} (h : o.oangle x y = ↑(π / 2))
Goal: (o.oangle (x - y) x).cos * ‖x - y‖ = ‖x‖
-- Orientation.sin_oangle_sub_right_mul_norm_of_oangle_eq_pi_div_two
theorem sin_oangle_sub_right_mul_norm_of_oangle_eq_pi_div_two {x y : V}
    (h : o.oangle x y = ↑(π / 2)) : Real.Angle.sin (o.oangle y (y - x)) * ‖y - x‖ = ‖x‖ 
Arguments: {V : Type u_1} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [hd2 : Fact (FiniteDimensional.finrank ℝ V = 2)] (o : Orientation ℝ V (Fin 2)) {x y : V} (h : o.oangle x y = ↑(π / 2))
Goal: (o.oangle y (y - x)).sin * ‖y - x‖ = ‖x‖
-- Orientation.sin_oangle_sub_left_mul_norm_of_oangle_eq_pi_div_two
theorem sin_oangle_sub_left_mul_norm_of_oangle_eq_pi_div_two {x y : V}
    (h : o.oangle x y = ↑(π / 2)) : Real.Angle.sin (o.oangle (x - y) x) * ‖x - y‖ = ‖y‖ 
Arguments: {V : Type u_1} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [hd2 : Fact (FiniteDimensional.finrank ℝ V = 2)] (o : Orientation ℝ V (Fin 2)) {x y : V} (h : o.oangle x y = ↑(π / 2))
Goal: (o.oangle (x - y) x).sin * ‖x - y‖ = ‖y‖
-- Orientation.tan_oangle_sub_right_mul_norm_of_oangle_eq_pi_div_two
theorem tan_oangle_sub_right_mul_norm_of_oangle_eq_pi_div_two {x y : V}
    (h : o.oangle x y = ↑(π / 2)) : Real.Angle.tan (o.oangle y (y - x)) * ‖y‖ = ‖x‖ 
Arguments: {V : Type u_1} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [hd2 : Fact (FiniteDimensional.finrank ℝ V = 2)] (o : Orientation ℝ V (Fin 2)) {x y : V} (h : o.oangle x y = ↑(π / 2))
Goal: (o.oangle y (y - x)).tan * ‖y‖ = ‖x‖
-- Orientation.tan_oangle_sub_left_mul_norm_of_oangle_eq_pi_div_two
theorem tan_oangle_sub_left_mul_norm_of_oangle_eq_pi_div_two {x y : V}
    (h : o.oangle x y = ↑(π / 2)) : Real.Angle.tan (o.oangle (x - y) x) * ‖x‖ = ‖y‖ 
Arguments: {V : Type u_1} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [hd2 : Fact (FiniteDimensional.finrank ℝ V = 2)] (o : Orientation ℝ V (Fin 2)) {x y : V} (h : o.oangle x y = ↑(π / 2))
Goal: (o.oangle (x - y) x).tan * ‖x‖ = ‖y‖
-- Orientation.norm_div_cos_oangle_sub_right_of_oangle_eq_pi_div_two
theorem norm_div_cos_oangle_sub_right_of_oangle_eq_pi_div_two {x y : V}
    (h : o.oangle x y = ↑(π / 2)) : ‖y‖ / Real.Angle.cos (o.oangle y (y - x)) = ‖y - x‖ 
Arguments: {V : Type u_1} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [hd2 : Fact (FiniteDimensional.finrank ℝ V = 2)] (o : Orientation ℝ V (Fin 2)) {x y : V} (h : o.oangle x y = ↑(π / 2))
Goal: ‖y‖ / (o.oangle y (y - x)).cos = ‖y - x‖
-- Orientation.norm_div_cos_oangle_sub_left_of_oangle_eq_pi_div_two
theorem norm_div_cos_oangle_sub_left_of_oangle_eq_pi_div_two {x y : V}
    (h : o.oangle x y = ↑(π / 2)) : ‖x‖ / Real.Angle.cos (o.oangle (x - y) x) = ‖x - y‖ 
Arguments: {V : Type u_1} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [hd2 : Fact (FiniteDimensional.finrank ℝ V = 2)] (o : Orientation ℝ V (Fin 2)) {x y : V} (h : o.oangle x y = ↑(π / 2))
Goal: ‖x‖ / (o.oangle (x - y) x).cos = ‖x - y‖
-- Orientation.norm_div_sin_oangle_sub_right_of_oangle_eq_pi_div_two
theorem norm_div_sin_oangle_sub_right_of_oangle_eq_pi_div_two {x y : V}
    (h : o.oangle x y = ↑(π / 2)) : ‖x‖ / Real.Angle.sin (o.oangle y (y - x)) = ‖y - x‖ 
Arguments: {V : Type u_1} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [hd2 : Fact (FiniteDimensional.finrank ℝ V = 2)] (o : Orientation ℝ V (Fin 2)) {x y : V} (h : o.oangle x y = ↑(π / 2))
Goal: ‖x‖ / (o.oangle y (y - x)).sin = ‖y - x‖
-- Orientation.norm_div_sin_oangle_sub_left_of_oangle_eq_pi_div_two
theorem norm_div_sin_oangle_sub_left_of_oangle_eq_pi_div_two {x y : V}
    (h : o.oangle x y = ↑(π / 2)) : ‖y‖ / Real.Angle.sin (o.oangle (x - y) x) = ‖x - y‖ 
Arguments: {V : Type u_1} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [hd2 : Fact (FiniteDimensional.finrank ℝ V = 2)] (o : Orientation ℝ V (Fin 2)) {x y : V} (h : o.oangle x y = ↑(π / 2))
Goal: ‖y‖ / (o.oangle (x - y) x).sin = ‖x - y‖
-- Orientation.norm_div_tan_oangle_sub_right_of_oangle_eq_pi_div_two
theorem norm_div_tan_oangle_sub_right_of_oangle_eq_pi_div_two {x y : V}
    (h : o.oangle x y = ↑(π / 2)) : ‖x‖ / Real.Angle.tan (o.oangle y (y - x)) = ‖y‖ 
Arguments: {V : Type u_1} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [hd2 : Fact (FiniteDimensional.finrank ℝ V = 2)] (o : Orientation ℝ V (Fin 2)) {x y : V} (h : o.oangle x y = ↑(π / 2))
Goal: ‖x‖ / (o.oangle y (y - x)).tan = ‖y‖
-- Orientation.norm_div_tan_oangle_sub_left_of_oangle_eq_pi_div_two
theorem norm_div_tan_oangle_sub_left_of_oangle_eq_pi_div_two {x y : V}
    (h : o.oangle x y = ↑(π / 2)) : ‖y‖ / Real.Angle.tan (o.oangle (x - y) x) = ‖x‖ 
Arguments: {V : Type u_1} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [hd2 : Fact (FiniteDimensional.finrank ℝ V = 2)] (o : Orientation ℝ V (Fin 2)) {x y : V} (h : o.oangle x y = ↑(π / 2))
Goal: ‖y‖ / (o.oangle (x - y) x).tan = ‖x‖
-- Orientation.oangle_add_right_smul_rotation_pi_div_two
theorem oangle_add_right_smul_rotation_pi_div_two {x : V} (h : x ≠ 0) (r : ℝ) :
    o.oangle x (x + r • o.rotation (π / 2 : ℝ) x) = Real.arctan r 
Arguments: {V : Type u_1} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [hd2 : Fact (FiniteDimensional.finrank ℝ V = 2)] (o : Orientation ℝ V (Fin 2)) {x : V} (h : x ≠ 0) (r : ℝ)
Goal: o.oangle x (x + r • (o.rotation ↑(π / 2)) x) = ↑(Real.arctan r)
-- Orientation.oangle_add_left_smul_rotation_pi_div_two
theorem oangle_add_left_smul_rotation_pi_div_two {x : V} (h : x ≠ 0) (r : ℝ) :
    o.oangle (x + r • o.rotation (π / 2 : ℝ) x) (r • o.rotation (π / 2 : ℝ) x)
      = Real.arctan r⁻¹ 
Arguments: {V : Type u_1} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [hd2 : Fact (FiniteDimensional.finrank ℝ V = 2)] (o : Orientation ℝ V (Fin 2)) {x : V} (h : x ≠ 0) (r : ℝ)
Goal: o.oangle (x + r • (o.rotation ↑(π / 2)) x) (r • (o.rotation ↑(π / 2)) x) = ↑(Real.arctan r⁻¹)
-- Orientation.tan_oangle_add_right_smul_rotation_pi_div_two
theorem tan_oangle_add_right_smul_rotation_pi_div_two {x : V} (h : x ≠ 0) (r : ℝ) :
    Real.Angle.tan (o.oangle x (x + r • o.rotation (π / 2 : ℝ) x)) = r 
Arguments: {V : Type u_1} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [hd2 : Fact (FiniteDimensional.finrank ℝ V = 2)] (o : Orientation ℝ V (Fin 2)) {x : V} (h : x ≠ 0) (r : ℝ)
Goal: (o.oangle x (x + r • (o.rotation ↑(π / 2)) x)).tan = r
-- Orientation.tan_oangle_add_left_smul_rotation_pi_div_two
theorem tan_oangle_add_left_smul_rotation_pi_div_two {x : V} (h : x ≠ 0) (r : ℝ) :
    Real.Angle.tan (o.oangle (x + r • o.rotation (π / 2 : ℝ) x) (r • o.rotation (π / 2 : ℝ) x)) =
      r⁻¹ 
Arguments: {V : Type u_1} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [hd2 : Fact (FiniteDimensional.finrank ℝ V = 2)] (o : Orientation ℝ V (Fin 2)) {x : V} (h : x ≠ 0) (r : ℝ)
Goal: (o.oangle (x + r • (o.rotation ↑(π / 2)) x) (r • (o.rotation ↑(π / 2)) x)).tan = r⁻¹
-- Orientation.oangle_sub_right_smul_rotation_pi_div_two
theorem oangle_sub_right_smul_rotation_pi_div_two {x : V} (h : x ≠ 0) (r : ℝ) :
    o.oangle (r • o.rotation (π / 2 : ℝ) x) (r • o.rotation (π / 2 : ℝ) x - x)
      = Real.arctan r⁻¹ 
Arguments: {V : Type u_1} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [hd2 : Fact (FiniteDimensional.finrank ℝ V = 2)] (o : Orientation ℝ V (Fin 2)) {x : V} (h : x ≠ 0) (r : ℝ)
Goal: o.oangle (r • (o.rotation ↑(π / 2)) x) (r • (o.rotation ↑(π / 2)) x - x) = ↑(Real.arctan r⁻¹)
-- Orientation.oangle_sub_left_smul_rotation_pi_div_two
theorem oangle_sub_left_smul_rotation_pi_div_two {x : V} (h : x ≠ 0) (r : ℝ) :
    o.oangle (x - r • o.rotation (π / 2 : ℝ) x) x = Real.arctan r 
Arguments: {V : Type u_1} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [hd2 : Fact (FiniteDimensional.finrank ℝ V = 2)] (o : Orientation ℝ V (Fin 2)) {x : V} (h : x ≠ 0) (r : ℝ)
Goal: o.oangle (x - r • (o.rotation ↑(π / 2)) x) x = ↑(Real.arctan r)
-- EuclideanGeometry.oangle_right_eq_arccos_of_oangle_eq_pi_div_two
theorem oangle_right_eq_arccos_of_oangle_eq_pi_div_two {p₁ p₂ p₃ : P} (h : ∡ p₁ p₂ p₃ = ↑(π / 2)) :
    ∡ p₂ p₃ p₁ = Real.arccos (dist p₃ p₂ / dist 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: ∡ p₂ p₃ p₁ = ↑(Real.arccos (dist p₃ p₂ / dist p₁ p₃))
-- EuclideanGeometry.oangle_left_eq_arccos_of_oangle_eq_pi_div_two
theorem oangle_left_eq_arccos_of_oangle_eq_pi_div_two {p₁ p₂ p₃ : P} (h : ∡ p₁ p₂ p₃ = ↑(π / 2)) :
    ∡ p₃ p₁ p₂ = Real.arccos (dist p₁ p₂ / dist 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: ∡ p₃ p₁ p₂ = ↑(Real.arccos (dist p₁ p₂ / dist p₁ p₃))
-- EuclideanGeometry.oangle_right_eq_arcsin_of_oangle_eq_pi_div_two
theorem oangle_right_eq_arcsin_of_oangle_eq_pi_div_two {p₁ p₂ p₃ : P} (h : ∡ p₁ p₂ p₃ = ↑(π / 2)) :
    ∡ p₂ p₃ p₁ = Real.arcsin (dist p₁ p₂ / dist 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: ∡ p₂ p₃ p₁ = ↑(Real.arcsin (dist p₁ p₂ / dist p₁ p₃))
-- EuclideanGeometry.oangle_left_eq_arcsin_of_oangle_eq_pi_div_two
theorem oangle_left_eq_arcsin_of_oangle_eq_pi_div_two {p₁ p₂ p₃ : P} (h : ∡ p₁ p₂ p₃ = ↑(π / 2)) :
    ∡ p₃ p₁ p₂ = Real.arcsin (dist p₃ p₂ / dist 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: ∡ p₃ p₁ p₂ = ↑(Real.arcsin (dist p₃ p₂ / dist p₁ p₃))
-- EuclideanGeometry.oangle_right_eq_arctan_of_oangle_eq_pi_div_two
theorem oangle_right_eq_arctan_of_oangle_eq_pi_div_two {p₁ p₂ p₃ : P} (h : ∡ p₁ p₂ p₃ = ↑(π / 2)) :
    ∡ p₂ p₃ p₁ = Real.arctan (dist p₁ p₂ / dist 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: ∡ p₂ p₃ p₁ = ↑(Real.arctan (dist p₁ p₂ / dist p₃ p₂))
-- EuclideanGeometry.oangle_left_eq_arctan_of_oangle_eq_pi_div_two
theorem oangle_left_eq_arctan_of_oangle_eq_pi_div_two {p₁ p₂ p₃ : P} (h : ∡ p₁ p₂ p₃ = ↑(π / 2)) :
    ∡ p₃ p₁ p₂ = Real.arctan (dist p₃ p₂ / dist 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: ∡ p₃ p₁ p₂ = ↑(Real.arctan (dist p₃ p₂ / dist p₁ p₂))