awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Module: Mathlib.Geometry.Euclidean.Angle.Oriented.RightAngle
Goal:
V : Type u_1
inst : NormedAddCommGroup V
inst_1 : InnerProductSpace ℝ V
hd2 : Fact (finrank ℝ V = 2)
o : Orientation ℝ V (Fin 2)
x y : V
h : o.oangle x y = (↑(π / 2) : Real.Angle)
hs : (o.oangle x (x + y)).sign = 1
rw : (↑(InnerProductGeometry.angle x (x + y)) : Real.Angle).cos * ‖x + y‖ = ‖x‖
rw₁ : Real.cos (InnerProductGeometry.angle x (x + y)) * ‖x + y‖ = ‖x‖
rw₂ rw₃ : ‖x‖ = ‖x‖
⊢ (o.oangle x (x + y)).cos * ‖x + y‖ = ‖x‖

Tactic:
rw [o.oangle_eq_angle_of_sign_eq_one hs, Real.Angle.cos_coe,
    InnerProductGeometry.cos_angle_add_mul_norm_of_inner_eq_zero
      (o.inner_eq_zero_of_oangle_eq_pi_div_two h)]

Premises used: Orientation.oangle_eq_angle_of_sign_eq_one, Orientation.inner_eq_zero_of_oangle_eq_pi_div_two, InnerProductGeometry.cos_angle_add_mul_norm_of_inner_eq_zero, Real.Angle.cos_coe
-- Module: Mathlib.Geometry.Euclidean.Angle.Oriented.RightAngle
Goal:
V : Type u_1
inst : NormedAddCommGroup V
inst_1 : InnerProductSpace ℝ V
hd2 : Fact (finrank ℝ V = 2)
o : Orientation ℝ V (Fin 2)
x y : V
h : o.oangle x y = (↑(π / 2) : Real.Angle)
rw : ∀ (h : (-o).oangle y x = (↑(π / 2) : Real.Angle)), ((-o).oangle y (x + y)).cos * ‖x + y‖ = ‖y‖
⊢ (o.oangle (x + y) y).cos * ‖x + y‖ = ‖y‖

Tactic:
rw [← neg_inj, oangle_rev, ← oangle_neg_orientation_eq_neg, neg_inj] at h ⊢

Premises used: neg_inj, Orientation.oangle_rev, Orientation.oangle_neg_orientation_eq_neg
-- Module: Mathlib.Geometry.Euclidean.Angle.Oriented.RightAngle
Goal:
V : Type u_1
inst : NormedAddCommGroup V
inst_1 : InnerProductSpace ℝ V
hd2 : Fact (finrank ℝ V = 2)
o : Orientation ℝ V (Fin 2)
x y : V
h : (-o).oangle y x = (↑(π / 2) : Real.Angle)
rw : ((-o).oangle y (y + x)).cos * ‖y + x‖ = ‖y‖
⊢ ((-o).oangle y (x + y)).cos * ‖x + y‖ = ‖y‖

Tactic:
rw [add_comm]

Premises used: add_comm
-- Module: Mathlib.Geometry.Euclidean.Angle.Oriented.RightAngle
Goal:
V : Type u_1
inst : NormedAddCommGroup V
inst_1 : InnerProductSpace ℝ V
hd2 : Fact (finrank ℝ V = 2)
o : Orientation ℝ V (Fin 2)
x y : V
h : (-o).oangle y x = (↑(π / 2) : Real.Angle)
⊢ ((-o).oangle y (y + x)).cos * ‖y + x‖ = ‖y‖

Tactic:
(-o).cos_oangle_add_right_mul_norm_of_oangle_eq_pi_div_two h

Premises used: Orientation.cos_oangle_add_right_mul_norm_of_oangle_eq_pi_div_two, Neg.neg
-- Module: Mathlib.Geometry.Euclidean.Angle.Oriented.RightAngle
Goal:
V : Type u_1
inst : NormedAddCommGroup V
inst_1 : InnerProductSpace ℝ V
hd2 : Fact (finrank ℝ V = 2)
o : Orientation ℝ V (Fin 2)
x y : V
h : (-o).oangle y x = (↑(π / 2) : Real.Angle)
⊢ ((-o).oangle y (y + x)).cos * ‖y + x‖ = ‖y‖

Tactic:
exact (-o).cos_oangle_add_right_mul_norm_of_oangle_eq_pi_div_two h

Premises used: Orientation.cos_oangle_add_right_mul_norm_of_oangle_eq_pi_div_two, Neg.neg
-- Module: Mathlib.Geometry.Euclidean.Angle.Oriented.RightAngle
Goal:
V : Type u_1
inst : NormedAddCommGroup V
inst_1 : InnerProductSpace ℝ V
hd2 : Fact (finrank ℝ V = 2)
o : Orientation ℝ V (Fin 2)
x y : V
h : o.oangle x y = (↑(π / 2) : Real.Angle)
hs : (o.oangle x (x + y)).sign = 1
rw : (↑(InnerProductGeometry.angle x (x + y)) : Real.Angle).sin * ‖x + y‖ = ‖y‖
rw₁ : Real.sin (InnerProductGeometry.angle x (x + y)) * ‖x + y‖ = ‖y‖
rw₂ rw₃ : ‖y‖ = ‖y‖
⊢ (o.oangle x (x + y)).sin * ‖x + y‖ = ‖y‖

Tactic:
rw [o.oangle_eq_angle_of_sign_eq_one hs, Real.Angle.sin_coe,
    InnerProductGeometry.sin_angle_add_mul_norm_of_inner_eq_zero
      (o.inner_eq_zero_of_oangle_eq_pi_div_two h)]

Premises used: Orientation.oangle_eq_angle_of_sign_eq_one, Orientation.inner_eq_zero_of_oangle_eq_pi_div_two, Real.Angle.sin_coe, InnerProductGeometry.sin_angle_add_mul_norm_of_inner_eq_zero
-- Module: Mathlib.Geometry.Euclidean.Angle.Oriented.RightAngle
Goal:
V : Type u_1
inst : NormedAddCommGroup V
inst_1 : InnerProductSpace ℝ V
hd2 : Fact (finrank ℝ V = 2)
o : Orientation ℝ V (Fin 2)
x y : V
h : o.oangle x y = (↑(π / 2) : Real.Angle)
rw : ∀ (h : (-o).oangle y x = (↑(π / 2) : Real.Angle)), ((-o).oangle y (x + y)).sin * ‖x + y‖ = ‖x‖
⊢ (o.oangle (x + y) y).sin * ‖x + y‖ = ‖x‖

Tactic:
rw [← neg_inj, oangle_rev, ← oangle_neg_orientation_eq_neg, neg_inj] at h ⊢

Premises used: neg_inj, Orientation.oangle_rev, Orientation.oangle_neg_orientation_eq_neg
-- Module: Mathlib.Geometry.Euclidean.Angle.Oriented.RightAngle
Goal:
V : Type u_1
inst : NormedAddCommGroup V
inst_1 : InnerProductSpace ℝ V
hd2 : Fact (finrank ℝ V = 2)
o : Orientation ℝ V (Fin 2)
x y : V
h : (-o).oangle y x = (↑(π / 2) : Real.Angle)
rw : ((-o).oangle y (y + x)).sin * ‖y + x‖ = ‖x‖
⊢ ((-o).oangle y (x + y)).sin * ‖x + y‖ = ‖x‖

Tactic:
rw [add_comm]

Premises used: add_comm
-- Module: Mathlib.Geometry.Euclidean.Angle.Oriented.RightAngle
Goal:
V : Type u_1
inst : NormedAddCommGroup V
inst_1 : InnerProductSpace ℝ V
hd2 : Fact (finrank ℝ V = 2)
o : Orientation ℝ V (Fin 2)
x y : V
h : (-o).oangle y x = (↑(π / 2) : Real.Angle)
⊢ ((-o).oangle y (y + x)).sin * ‖y + x‖ = ‖x‖

Tactic:
(-o).sin_oangle_add_right_mul_norm_of_oangle_eq_pi_div_two h

Premises used: Orientation.sin_oangle_add_right_mul_norm_of_oangle_eq_pi_div_two, Neg.neg
-- Module: Mathlib.Geometry.Euclidean.Angle.Oriented.RightAngle
Goal:
V : Type u_1
inst : NormedAddCommGroup V
inst_1 : InnerProductSpace ℝ V
hd2 : Fact (finrank ℝ V = 2)
o : Orientation ℝ V (Fin 2)
x y : V
h : (-o).oangle y x = (↑(π / 2) : Real.Angle)
⊢ ((-o).oangle y (y + x)).sin * ‖y + x‖ = ‖x‖

Tactic:
exact (-o).sin_oangle_add_right_mul_norm_of_oangle_eq_pi_div_two h

Premises used: Orientation.sin_oangle_add_right_mul_norm_of_oangle_eq_pi_div_two, Neg.neg
-- Module: Mathlib.Geometry.Euclidean.Angle.Oriented.RightAngle
Goal:
V : Type u_1
inst : NormedAddCommGroup V
inst_1 : InnerProductSpace ℝ V
hd2 : Fact (finrank ℝ V = 2)
o : Orientation ℝ V (Fin 2)
x y : V
h : o.oangle x y = (↑(π / 2) : Real.Angle)
hs : (o.oangle x (x + y)).sign = 1
rw : (↑(InnerProductGeometry.angle x (x + y)) : Real.Angle).tan * ‖x‖ = ‖y‖
rw₁ : Real.tan (InnerProductGeometry.angle x (x + y)) * ‖x‖ = ‖y‖
rw₂ rw₃ : ‖y‖ = ‖y‖
⊢ (o.oangle x (x + y)).tan * ‖x‖ = ‖y‖

Tactic:
rw [o.oangle_eq_angle_of_sign_eq_one hs, Real.Angle.tan_coe,
    InnerProductGeometry.tan_angle_add_mul_norm_of_inner_eq_zero
      (o.inner_eq_zero_of_oangle_eq_pi_div_two h)
      (Or.inl (o.left_ne_zero_of_oangle_eq_pi_div_two h))]

Premises used: Orientation.left_ne_zero_of_oangle_eq_pi_div_two, InnerProductGeometry.tan_angle_add_mul_norm_of_inner_eq_zero, Orientation.oangle_eq_angle_of_sign_eq_one, Orientation.inner_eq_zero_of_oangle_eq_pi_div_two, Or.inl, Real.Angle.tan_coe
-- Module: Mathlib.Geometry.Euclidean.Angle.Oriented.RightAngle
Goal:
V : Type u_1
inst : NormedAddCommGroup V
inst_1 : InnerProductSpace ℝ V
hd2 : Fact (finrank ℝ V = 2)
o : Orientation ℝ V (Fin 2)
x y : V
h : o.oangle x y = (↑(π / 2) : Real.Angle)
rw : ∀ (h : (-o).oangle y x = (↑(π / 2) : Real.Angle)), ((-o).oangle y (x + y)).tan * ‖y‖ = ‖x‖
⊢ (o.oangle (x + y) y).tan * ‖y‖ = ‖x‖

Tactic:
rw [← neg_inj, oangle_rev, ← oangle_neg_orientation_eq_neg, neg_inj] at h ⊢

Premises used: neg_inj, Orientation.oangle_rev, Orientation.oangle_neg_orientation_eq_neg
-- Module: Mathlib.Geometry.Euclidean.Angle.Oriented.RightAngle
Goal:
V : Type u_1
inst : NormedAddCommGroup V
inst_1 : InnerProductSpace ℝ V
hd2 : Fact (finrank ℝ V = 2)
o : Orientation ℝ V (Fin 2)
x y : V
h : (-o).oangle y x = (↑(π / 2) : Real.Angle)
rw : ((-o).oangle y (y + x)).tan * ‖y‖ = ‖x‖
⊢ ((-o).oangle y (x + y)).tan * ‖y‖ = ‖x‖

Tactic:
rw [add_comm]

Premises used: add_comm
-- Module: Mathlib.Geometry.Euclidean.Angle.Oriented.RightAngle
Goal:
V : Type u_1
inst : NormedAddCommGroup V
inst_1 : InnerProductSpace ℝ V
hd2 : Fact (finrank ℝ V = 2)
o : Orientation ℝ V (Fin 2)
x y : V
h : (-o).oangle y x = (↑(π / 2) : Real.Angle)
⊢ ((-o).oangle y (y + x)).tan * ‖y‖ = ‖x‖

Tactic:
(-o).tan_oangle_add_right_mul_norm_of_oangle_eq_pi_div_two h

Premises used: Orientation.tan_oangle_add_right_mul_norm_of_oangle_eq_pi_div_two, Neg.neg
-- Module: Mathlib.Geometry.Euclidean.Angle.Oriented.RightAngle
Goal:
V : Type u_1
inst : NormedAddCommGroup V
inst_1 : InnerProductSpace ℝ V
hd2 : Fact (finrank ℝ V = 2)
o : Orientation ℝ V (Fin 2)
x y : V
h : (-o).oangle y x = (↑(π / 2) : Real.Angle)
⊢ ((-o).oangle y (y + x)).tan * ‖y‖ = ‖x‖

Tactic:
exact (-o).tan_oangle_add_right_mul_norm_of_oangle_eq_pi_div_two h

Premises used: Orientation.tan_oangle_add_right_mul_norm_of_oangle_eq_pi_div_two, Neg.neg
-- Module: Mathlib.Geometry.Euclidean.Angle.Oriented.RightAngle
Goal:
V : Type u_1
inst : NormedAddCommGroup V
inst_1 : InnerProductSpace ℝ V
hd2 : Fact (finrank ℝ V = 2)
o : Orientation ℝ V (Fin 2)
x y : V
h : o.oangle x y = (↑(π / 2) : Real.Angle)
hs : (o.oangle x (x + y)).sign = 1
rw : ‖x‖ / (↑(InnerProductGeometry.angle x (x + y)) : Real.Angle).cos = ‖x + y‖
rw₁ : ‖x‖ / Real.cos (InnerProductGeometry.angle x (x + y)) = ‖x + y‖
rw₂ rw₃ : ‖x + y‖ = ‖x + y‖
⊢ ‖x‖ / (o.oangle x (x + y)).cos = ‖x + y‖

Tactic:
rw [o.oangle_eq_angle_of_sign_eq_one hs, Real.Angle.cos_coe,
    InnerProductGeometry.norm_div_cos_angle_add_of_inner_eq_zero
      (o.inner_eq_zero_of_oangle_eq_pi_div_two h)
      (Or.inl (o.left_ne_zero_of_oangle_eq_pi_div_two h))]

Premises used: Orientation.left_ne_zero_of_oangle_eq_pi_div_two, Orientation.oangle_eq_angle_of_sign_eq_one, Orientation.inner_eq_zero_of_oangle_eq_pi_div_two, Real.Angle.cos_coe, Or.inl, InnerProductGeometry.norm_div_cos_angle_add_of_inner_eq_zero
-- Module: Mathlib.Geometry.Euclidean.Angle.Oriented.RightAngle
Goal:
V : Type u_1
inst : NormedAddCommGroup V
inst_1 : InnerProductSpace ℝ V
hd2 : Fact (finrank ℝ V = 2)
o : Orientation ℝ V (Fin 2)
x y : V
h : o.oangle x y = (↑(π / 2) : Real.Angle)
rw : ∀ (h : (-o).oangle y x = (↑(π / 2) : Real.Angle)), ‖y‖ / ((-o).oangle y (x + y)).cos = ‖x + y‖
⊢ ‖y‖ / (o.oangle (x + y) y).cos = ‖x + y‖

Tactic:
rw [← neg_inj, oangle_rev, ← oangle_neg_orientation_eq_neg, neg_inj] at h ⊢

Premises used: neg_inj, Orientation.oangle_rev, Orientation.oangle_neg_orientation_eq_neg
-- Module: Mathlib.Geometry.Euclidean.Angle.Oriented.RightAngle
Goal:
V : Type u_1
inst : NormedAddCommGroup V
inst_1 : InnerProductSpace ℝ V
hd2 : Fact (finrank ℝ V = 2)
o : Orientation ℝ V (Fin 2)
x y : V
h : (-o).oangle y x = (↑(π / 2) : Real.Angle)
rw : ‖y‖ / ((-o).oangle y (y + x)).cos = ‖y + x‖
⊢ ‖y‖ / ((-o).oangle y (x + y)).cos = ‖x + y‖

Tactic:
rw [add_comm]

Premises used: add_comm
-- Module: Mathlib.Geometry.Euclidean.Angle.Oriented.RightAngle
Goal:
V : Type u_1
inst : NormedAddCommGroup V
inst_1 : InnerProductSpace ℝ V
hd2 : Fact (finrank ℝ V = 2)
o : Orientation ℝ V (Fin 2)
x y : V
h : (-o).oangle y x = (↑(π / 2) : Real.Angle)
⊢ ‖y‖ / ((-o).oangle y (y + x)).cos = ‖y + x‖

Tactic:
(-o).norm_div_cos_oangle_add_right_of_oangle_eq_pi_div_two h

Premises used: Orientation.norm_div_cos_oangle_add_right_of_oangle_eq_pi_div_two, Neg.neg
-- Module: Mathlib.Geometry.Euclidean.Angle.Oriented.RightAngle
Goal:
V : Type u_1
inst : NormedAddCommGroup V
inst_1 : InnerProductSpace ℝ V
hd2 : Fact (finrank ℝ V = 2)
o : Orientation ℝ V (Fin 2)
x y : V
h : (-o).oangle y x = (↑(π / 2) : Real.Angle)
⊢ ‖y‖ / ((-o).oangle y (y + x)).cos = ‖y + x‖

Tactic:
exact (-o).norm_div_cos_oangle_add_right_of_oangle_eq_pi_div_two h

Premises used: Orientation.norm_div_cos_oangle_add_right_of_oangle_eq_pi_div_two, Neg.neg
-- Module: Mathlib.Geometry.Euclidean.Angle.Oriented.RightAngle
Goal:
V : Type u_1
inst : NormedAddCommGroup V
inst_1 : InnerProductSpace ℝ V
hd2 : Fact (finrank ℝ V = 2)
o : Orientation ℝ V (Fin 2)
x y : V
h : o.oangle x y = (↑(π / 2) : Real.Angle)
hs : (o.oangle x (x + y)).sign = 1
rw : ‖y‖ / (↑(InnerProductGeometry.angle x (x + y)) : Real.Angle).sin = ‖x + y‖
rw₁ : ‖y‖ / Real.sin (InnerProductGeometry.angle x (x + y)) = ‖x + y‖
rw₂ rw₃ : ‖x + y‖ = ‖x + y‖
⊢ ‖y‖ / (o.oangle x (x + y)).sin = ‖x + y‖

Tactic:
rw [o.oangle_eq_angle_of_sign_eq_one hs, Real.Angle.sin_coe,
    InnerProductGeometry.norm_div_sin_angle_add_of_inner_eq_zero
      (o.inner_eq_zero_of_oangle_eq_pi_div_two h)
      (Or.inr (o.right_ne_zero_of_oangle_eq_pi_div_two h))]

Premises used: InnerProductGeometry.norm_div_sin_angle_add_of_inner_eq_zero, Orientation.oangle_eq_angle_of_sign_eq_one, Orientation.inner_eq_zero_of_oangle_eq_pi_div_two, Real.Angle.sin_coe, Orientation.right_ne_zero_of_oangle_eq_pi_div_two, Or.inr
-- Module: Mathlib.Geometry.Euclidean.Angle.Oriented.RightAngle
Goal:
V : Type u_1
inst : NormedAddCommGroup V
inst_1 : InnerProductSpace ℝ V
hd2 : Fact (finrank ℝ V = 2)
o : Orientation ℝ V (Fin 2)
x y : V
h : o.oangle x y = (↑(π / 2) : Real.Angle)
rw : ∀ (h : (-o).oangle y x = (↑(π / 2) : Real.Angle)), ‖x‖ / ((-o).oangle y (x + y)).sin = ‖x + y‖
⊢ ‖x‖ / (o.oangle (x + y) y).sin = ‖x + y‖

Tactic:
rw [← neg_inj, oangle_rev, ← oangle_neg_orientation_eq_neg, neg_inj] at h ⊢

Premises used: neg_inj, Orientation.oangle_rev, Orientation.oangle_neg_orientation_eq_neg
-- Module: Mathlib.Geometry.Euclidean.Angle.Oriented.RightAngle
Goal:
V : Type u_1
inst : NormedAddCommGroup V
inst_1 : InnerProductSpace ℝ V
hd2 : Fact (finrank ℝ V = 2)
o : Orientation ℝ V (Fin 2)
x y : V
h : (-o).oangle y x = (↑(π / 2) : Real.Angle)
rw : ‖x‖ / ((-o).oangle y (y + x)).sin = ‖y + x‖
⊢ ‖x‖ / ((-o).oangle y (x + y)).sin = ‖x + y‖

Tactic:
rw [add_comm]

Premises used: add_comm
-- Module: Mathlib.Geometry.Euclidean.Angle.Oriented.RightAngle
Goal:
V : Type u_1
inst : NormedAddCommGroup V
inst_1 : InnerProductSpace ℝ V
hd2 : Fact (finrank ℝ V = 2)
o : Orientation ℝ V (Fin 2)
x y : V
h : (-o).oangle y x = (↑(π / 2) : Real.Angle)
⊢ ‖x‖ / ((-o).oangle y (y + x)).sin = ‖y + x‖

Tactic:
(-o).norm_div_sin_oangle_add_right_of_oangle_eq_pi_div_two h

Premises used: Orientation.norm_div_sin_oangle_add_right_of_oangle_eq_pi_div_two, Neg.neg
-- Module: Mathlib.Geometry.Euclidean.Angle.Oriented.RightAngle
Goal:
V : Type u_1
inst : NormedAddCommGroup V
inst_1 : InnerProductSpace ℝ V
hd2 : Fact (finrank ℝ V = 2)
o : Orientation ℝ V (Fin 2)
x y : V
h : (-o).oangle y x = (↑(π / 2) : Real.Angle)
⊢ ‖x‖ / ((-o).oangle y (y + x)).sin = ‖y + x‖

Tactic:
exact (-o).norm_div_sin_oangle_add_right_of_oangle_eq_pi_div_two h

Premises used: Orientation.norm_div_sin_oangle_add_right_of_oangle_eq_pi_div_two, Neg.neg
-- Module: Mathlib.Geometry.Euclidean.Angle.Oriented.RightAngle
Goal:
V : Type u_1
inst : NormedAddCommGroup V
inst_1 : InnerProductSpace ℝ V
hd2 : Fact (finrank ℝ V = 2)
o : Orientation ℝ V (Fin 2)
x y : V
h : o.oangle x y = (↑(π / 2) : Real.Angle)
hs : (o.oangle x (x + y)).sign = 1
rw : ‖y‖ / (↑(InnerProductGeometry.angle x (x + y)) : Real.Angle).tan = ‖x‖
rw₁ : ‖y‖ / Real.tan (InnerProductGeometry.angle x (x + y)) = ‖x‖
rw₂ rw₃ : ‖x‖ = ‖x‖
⊢ ‖y‖ / (o.oangle x (x + y)).tan = ‖x‖

Tactic:
rw [o.oangle_eq_angle_of_sign_eq_one hs, Real.Angle.tan_coe,
    InnerProductGeometry.norm_div_tan_angle_add_of_inner_eq_zero
      (o.inner_eq_zero_of_oangle_eq_pi_div_two h)
      (Or.inr (o.right_ne_zero_of_oangle_eq_pi_div_two h))]

Premises used: InnerProductGeometry.norm_div_tan_angle_add_of_inner_eq_zero, Orientation.oangle_eq_angle_of_sign_eq_one, Orientation.inner_eq_zero_of_oangle_eq_pi_div_two, Orientation.right_ne_zero_of_oangle_eq_pi_div_two, Or.inr, Real.Angle.tan_coe
-- Module: Mathlib.Geometry.Euclidean.Angle.Oriented.RightAngle
Goal:
V : Type u_1
inst : NormedAddCommGroup V
inst_1 : InnerProductSpace ℝ V
hd2 : Fact (finrank ℝ V = 2)
o : Orientation ℝ V (Fin 2)
x y : V
h : o.oangle x y = (↑(π / 2) : Real.Angle)
rw : ∀ (h : (-o).oangle y x = (↑(π / 2) : Real.Angle)), ‖x‖ / ((-o).oangle y (x + y)).tan = ‖y‖
⊢ ‖x‖ / (o.oangle (x + y) y).tan = ‖y‖

Tactic:
rw [← neg_inj, oangle_rev, ← oangle_neg_orientation_eq_neg, neg_inj] at h ⊢

Premises used: neg_inj, Orientation.oangle_rev, Orientation.oangle_neg_orientation_eq_neg
-- Module: Mathlib.Geometry.Euclidean.Angle.Oriented.RightAngle
Goal:
V : Type u_1
inst : NormedAddCommGroup V
inst_1 : InnerProductSpace ℝ V
hd2 : Fact (finrank ℝ V = 2)
o : Orientation ℝ V (Fin 2)
x y : V
h : (-o).oangle y x = (↑(π / 2) : Real.Angle)
rw : ‖x‖ / ((-o).oangle y (y + x)).tan = ‖y‖
⊢ ‖x‖ / ((-o).oangle y (x + y)).tan = ‖y‖

Tactic:
rw [add_comm]

Premises used: add_comm
-- Module: Mathlib.Geometry.Euclidean.Angle.Oriented.RightAngle
Goal:
V : Type u_1
inst : NormedAddCommGroup V
inst_1 : InnerProductSpace ℝ V
hd2 : Fact (finrank ℝ V = 2)
o : Orientation ℝ V (Fin 2)
x y : V
h : (-o).oangle y x = (↑(π / 2) : Real.Angle)
⊢ ‖x‖ / ((-o).oangle y (y + x)).tan = ‖y‖

Tactic:
(-o).norm_div_tan_oangle_add_right_of_oangle_eq_pi_div_two h

Premises used: Orientation.norm_div_tan_oangle_add_right_of_oangle_eq_pi_div_two, Neg.neg
-- Module: Mathlib.Geometry.Euclidean.Angle.Oriented.RightAngle
Goal:
V : Type u_1
inst : NormedAddCommGroup V
inst_1 : InnerProductSpace ℝ V
hd2 : Fact (finrank ℝ V = 2)
o : Orientation ℝ V (Fin 2)
x y : V
h : (-o).oangle y x = (↑(π / 2) : Real.Angle)
⊢ ‖x‖ / ((-o).oangle y (y + x)).tan = ‖y‖

Tactic:
exact (-o).norm_div_tan_oangle_add_right_of_oangle_eq_pi_div_two h

Premises used: Orientation.norm_div_tan_oangle_add_right_of_oangle_eq_pi_div_two, Neg.neg
-- Module: Mathlib.Geometry.Euclidean.Angle.Oriented.RightAngle
Goal:
V : Type u_1
inst : NormedAddCommGroup V
inst_1 : InnerProductSpace ℝ V
hd2 : Fact (finrank ℝ V = 2)
o : Orientation ℝ V (Fin 2)
x y : V
h : o.oangle x y = (↑(π / 2) : Real.Angle)
rw : (o.oangle x y).sign = 1
rw₁ : (↑(π / 2) : Real.Angle).sign = 1
rw₂ rw₃ : Eq (α := SignType) 1 1
⊢ (o.oangle y (y - x)).sign = 1

Tactic:
rw [oangle_sign_sub_right_swap, h, Real.Angle.sign_coe_pi_div_two]

Premises used: Orientation.oangle_sign_sub_right_swap, Real.Angle.sign_coe_pi_div_two
-- Module: Mathlib.Geometry.Euclidean.Angle.Oriented.RightAngle
Goal:
V : Type u_1
inst : NormedAddCommGroup V
inst_1 : InnerProductSpace ℝ V
hd2 : Fact (finrank ℝ V = 2)
o : Orientation ℝ V (Fin 2)
x y : V
h : o.oangle x y = (↑(π / 2) : Real.Angle)
⊢ (o.oangle y (y - x)).sign = 1

Tactic:
have hs : (o.oangle y (y - x)).sign = 1 := by
    rw [oangle_sign_sub_right_swap, h, Real.Angle.sign_coe_pi_div_two]
-- Module: Mathlib.Geometry.Euclidean.Angle.Oriented.RightAngle
Goal:
V : Type u_1
inst : NormedAddCommGroup V
inst_1 : InnerProductSpace ℝ V
hd2 : Fact (finrank ℝ V = 2)
o : Orientation ℝ V (Fin 2)
x y : V
h : o.oangle x y = (↑(π / 2) : Real.Angle)
hs : (o.oangle y (y - x)).sign = 1
rw : (↑(InnerProductGeometry.angle y (y - x)) : Real.Angle) = (↑(Real.arccos (‖y‖ / ‖y - x‖)) : Real.Angle)
rw₁ rw₂ : (↑(Real.arccos (‖y‖ / ‖y - x‖)) : Real.Angle) = (↑(Real.arccos (‖y‖ / ‖y - x‖)) : Real.Angle)
⊢ o.oangle y (y - x) = (↑(Real.arccos (‖y‖ / ‖y - x‖)) : Real.Angle)

Tactic:
rw [o.oangle_eq_angle_of_sign_eq_one hs,
    InnerProductGeometry.angle_sub_eq_arccos_of_inner_eq_zero
      (o.inner_rev_eq_zero_of_oangle_eq_pi_div_two h)]

Premises used: InnerProductGeometry.angle_sub_eq_arccos_of_inner_eq_zero, Orientation.oangle_eq_angle_of_sign_eq_one, Orientation.inner_rev_eq_zero_of_oangle_eq_pi_div_two
-- Module: Mathlib.Geometry.Euclidean.Angle.Oriented.RightAngle
Goal:
V : Type u_1
inst : NormedAddCommGroup V
inst_1 : InnerProductSpace ℝ V
hd2 : Fact (finrank ℝ V = 2)
o : Orientation ℝ V (Fin 2)
x y : V
h : o.oangle x y = (↑(π / 2) : Real.Angle)
rw :
  ∀ (h : (-o).oangle y x = (↑(π / 2) : Real.Angle)),
    (-o).oangle x (x - y) = (↑(Real.arccos (‖x‖ / ‖x - y‖)) : Real.Angle)
⊢ o.oangle (x - y) x = (↑(Real.arccos (‖x‖ / ‖x - y‖)) : Real.Angle)

Tactic:
rw [← neg_inj, oangle_rev, ← oangle_neg_orientation_eq_neg, neg_inj] at h ⊢

Premises used: neg_inj, Orientation.oangle_rev, Orientation.oangle_neg_orientation_eq_neg
-- Module: Mathlib.Geometry.Euclidean.Angle.Oriented.RightAngle
Goal:
V : Type u_1
inst : NormedAddCommGroup V
inst_1 : InnerProductSpace ℝ V
hd2 : Fact (finrank ℝ V = 2)
o : Orientation ℝ V (Fin 2)
x y : V
h : (-o).oangle y x = (↑(π / 2) : Real.Angle)
⊢ (-o).oangle x (x - y) = (↑(Real.arccos (‖x‖ / ‖x - y‖)) : Real.Angle)

Tactic:
(-o).oangle_sub_right_eq_arccos_of_oangle_eq_pi_div_two h

Premises used: Orientation.oangle_sub_right_eq_arccos_of_oangle_eq_pi_div_two, Neg.neg
-- Module: Mathlib.Geometry.Euclidean.Angle.Oriented.RightAngle
Goal:
V : Type u_1
inst : NormedAddCommGroup V
inst_1 : InnerProductSpace ℝ V
hd2 : Fact (finrank ℝ V = 2)
o : Orientation ℝ V (Fin 2)
x y : V
h : (-o).oangle y x = (↑(π / 2) : Real.Angle)
⊢ (-o).oangle x (x - y) = (↑(Real.arccos (‖x‖ / ‖x - y‖)) : Real.Angle)

Tactic:
exact (-o).oangle_sub_right_eq_arccos_of_oangle_eq_pi_div_two h

Premises used: Orientation.oangle_sub_right_eq_arccos_of_oangle_eq_pi_div_two, Neg.neg
-- Module: Mathlib.Geometry.Euclidean.Angle.Oriented.RightAngle
Goal:
V : Type u_1
inst : NormedAddCommGroup V
inst_1 : InnerProductSpace ℝ V
hd2 : Fact (finrank ℝ V = 2)
o : Orientation ℝ V (Fin 2)
x y : V
h : o.oangle x y = (↑(π / 2) : Real.Angle)
hs : (o.oangle y (y - x)).sign = 1
rw : (↑(InnerProductGeometry.angle y (y - x)) : Real.Angle) = (↑(Real.arcsin (‖x‖ / ‖y - x‖)) : Real.Angle)
rw₁ rw₂ : (↑(Real.arcsin (‖x‖ / ‖y - x‖)) : Real.Angle) = (↑(Real.arcsin (‖x‖ / ‖y - x‖)) : Real.Angle)
⊢ o.oangle y (y - x) = (↑(Real.arcsin (‖x‖ / ‖y - x‖)) : Real.Angle)

Tactic:
rw [o.oangle_eq_angle_of_sign_eq_one hs,
    InnerProductGeometry.angle_sub_eq_arcsin_of_inner_eq_zero
      (o.inner_rev_eq_zero_of_oangle_eq_pi_div_two h)
      (Or.inl (o.right_ne_zero_of_oangle_eq_pi_div_two h))]

Premises used: Orientation.oangle_eq_angle_of_sign_eq_one, Orientation.right_ne_zero_of_oangle_eq_pi_div_two, InnerProductGeometry.angle_sub_eq_arcsin_of_inner_eq_zero, Or.inl, Orientation.inner_rev_eq_zero_of_oangle_eq_pi_div_two
-- Module: Mathlib.Geometry.Euclidean.Angle.Oriented.RightAngle
Goal:
V : Type u_1
inst : NormedAddCommGroup V
inst_1 : InnerProductSpace ℝ V
hd2 : Fact (finrank ℝ V = 2)
o : Orientation ℝ V (Fin 2)
x y : V
h : o.oangle x y = (↑(π / 2) : Real.Angle)
rw :
  ∀ (h : (-o).oangle y x = (↑(π / 2) : Real.Angle)),
    (-o).oangle x (x - y) = (↑(Real.arcsin (‖y‖ / ‖x - y‖)) : Real.Angle)
⊢ o.oangle (x - y) x = (↑(Real.arcsin (‖y‖ / ‖x - y‖)) : Real.Angle)

Tactic:
rw [← neg_inj, oangle_rev, ← oangle_neg_orientation_eq_neg, neg_inj] at h ⊢

Premises used: neg_inj, Orientation.oangle_rev, Orientation.oangle_neg_orientation_eq_neg
-- Module: Mathlib.Geometry.Euclidean.Angle.Oriented.RightAngle
Goal:
V : Type u_1
inst : NormedAddCommGroup V
inst_1 : InnerProductSpace ℝ V
hd2 : Fact (finrank ℝ V = 2)
o : Orientation ℝ V (Fin 2)
x y : V
h : (-o).oangle y x = (↑(π / 2) : Real.Angle)
⊢ (-o).oangle x (x - y) = (↑(Real.arcsin (‖y‖ / ‖x - y‖)) : Real.Angle)

Tactic:
(-o).oangle_sub_right_eq_arcsin_of_oangle_eq_pi_div_two h

Premises used: Orientation.oangle_sub_right_eq_arcsin_of_oangle_eq_pi_div_two, Neg.neg
-- Module: Mathlib.Geometry.Euclidean.Angle.Oriented.RightAngle
Goal:
V : Type u_1
inst : NormedAddCommGroup V
inst_1 : InnerProductSpace ℝ V
hd2 : Fact (finrank ℝ V = 2)
o : Orientation ℝ V (Fin 2)
x y : V
h : (-o).oangle y x = (↑(π / 2) : Real.Angle)
⊢ (-o).oangle x (x - y) = (↑(Real.arcsin (‖y‖ / ‖x - y‖)) : Real.Angle)

Tactic:
exact (-o).oangle_sub_right_eq_arcsin_of_oangle_eq_pi_div_two h

Premises used: Orientation.oangle_sub_right_eq_arcsin_of_oangle_eq_pi_div_two, Neg.neg
-- Module: Mathlib.Geometry.Euclidean.Angle.Oriented.RightAngle
Goal:
V : Type u_1
inst : NormedAddCommGroup V
inst_1 : InnerProductSpace ℝ V
hd2 : Fact (finrank ℝ V = 2)
o : Orientation ℝ V (Fin 2)
x y : V
h : o.oangle x y = (↑(π / 2) : Real.Angle)
hs : (o.oangle y (y - x)).sign = 1
rw : (↑(InnerProductGeometry.angle y (y - x)) : Real.Angle) = (↑(Real.arctan (‖x‖ / ‖y‖)) : Real.Angle)
rw₁ rw₂ : (↑(Real.arctan (‖x‖ / ‖y‖)) : Real.Angle) = (↑(Real.arctan (‖x‖ / ‖y‖)) : Real.Angle)
⊢ o.oangle y (y - x) = (↑(Real.arctan (‖x‖ / ‖y‖)) : Real.Angle)

Tactic:
rw [o.oangle_eq_angle_of_sign_eq_one hs,
    InnerProductGeometry.angle_sub_eq_arctan_of_inner_eq_zero
      (o.inner_rev_eq_zero_of_oangle_eq_pi_div_two h) (o.right_ne_zero_of_oangle_eq_pi_div_two h)]

Premises used: Orientation.oangle_eq_angle_of_sign_eq_one, Orientation.right_ne_zero_of_oangle_eq_pi_div_two, InnerProductGeometry.angle_sub_eq_arctan_of_inner_eq_zero, Orientation.inner_rev_eq_zero_of_oangle_eq_pi_div_two