mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Geometry.Euclidean.Angle.Oriented.Affine

Goal state before:
pp: V : Type u_1
P : Type u_2
inst✝⁴ : NormedAddCommGroup V
inst✝³ : InnerProductSpace ℝ V
inst✝² : MetricSpace P
inst✝¹ : NormedAddTorsor V P
hd2 : Fact (finrank ℝ V = 2)
inst✝ : Oriented ℝ V (Fin 2)
p₁ p₂ p₃ p₄ p₅ p₆ : P
h₂ : p₂ ∉ line[ℝ, p₁, p₃]
h₄ : p₄ ∈ line[ℝ, p₁, p₃]
h₆ : p₆ ∈ line[ℝ, p₁, p₃]
h₁₂₄₅ : line[ℝ, p₁, p₂] ∥ line[ℝ, p₄, p₅]
h₃₂₆₅ : line[ℝ, p₃, p₂] ∥ line[ℝ, p₆, p₅]
⊢ o.oangle (p₁ -ᵥ p₂) (p₃ -ᵥ p₂) = o.oangle (p₄ -ᵥ p₅) (p₆ -ᵥ p₅)

Tactic:
]

Goal state after:
V : Type u_1
P : Type u_2
inst✝⁴ : NormedAddCommGroup V
inst✝³ : InnerProductSpace ℝ V
inst✝² : MetricSpace P
inst✝¹ : NormedAddTorsor V P
hd2 : Fact (finrank ℝ V = 2)
inst✝ : Oriented ℝ V (Fin 2)
p₁ p₂ p₃ p₄ p₅ p₆ : P
h₂ : p₂ ∉ line[ℝ, p₁, p₃]
h₄ : p₄ ∈ line[ℝ, p₁, p₃]
h₆ : p₆ ∈ line[ℝ, p₁, p₃]
h₁₂₄₅ : line[ℝ, p₁, p₂] ∥ line[ℝ, p₄, p₅]
h₃₂₆₅ : line[ℝ, p₃, p₂] ∥ line[ℝ, p₆, p₅]
⊢ o.oangle (p₁ -ᵥ p₂) (p₃ -ᵥ p₂) = o.oangle (p₄ -ᵥ p₅) (p₆ -ᵥ p₅)
Module: Mathlib.Geometry.Euclidean.Angle.Unoriented.RightAngle

Goal state before:
pp: case neg
V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x y : V
h : ⟪x, y⟫ = 0
h0 : ¬x = 0 ∨ ¬y = 0
⊢ Real.sin (angle x (x + y)) * ‖x + y‖ = ‖y‖

Tactic:
]

Goal state after:
case neg
V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x y : V
h : ⟪x, y⟫ = 0
h0 : ¬x = 0 ∨ ¬y = 0
⊢ Real.sin (angle x (x + y)) * ‖x + y‖ = ‖y‖
Module: Mathlib.Geometry.Euclidean.Angle.Unoriented.RightAngle

Goal state before:
pp: case neg
V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x y : V
h : ⟪x, y⟫ = 0
h0 : ¬x = 0 ∨ ¬y = 0
⊢ ‖y‖ = ‖y‖

pp: case neg.h
V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x y : V
h : ⟪x, y⟫ = 0
h0 : ¬x = 0 ∨ ¬y = 0
⊢ ‖x + y‖ ≠ 0

Tactic:
]

Goal state after:
case neg.h
V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x y : V
h : ⟪x, y⟫ = 0
h0 : ¬x = 0 ∨ ¬y = 0
⊢ ‖x + y‖ ≠ 0
Module: Mathlib.Geometry.Euclidean.Angle.Unoriented.RightAngle

Goal state before:
pp: case neg.h
V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x y : V
h : ⟪x, y⟫ = 0
h0 : ¬x = 0 ∨ ¬y = 0
⊢ ‖x‖ * ‖x‖ + ‖y‖ * ‖y‖ ≠ 0

Tactic:
]

Goal state after:
case neg.h
V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x y : V
h : ⟪x, y⟫ = 0
h0 : ¬x = 0 ∨ ¬y = 0
⊢ ‖x‖ * ‖x‖ + ‖y‖ * ‖y‖ ≠ 0
Module: Mathlib.Geometry.Euclidean.Angle.Oriented.Affine

Goal state before:
pp: V : Type u_1
P : Type u_2
inst✝⁴ : NormedAddCommGroup V
inst✝³ : InnerProductSpace ℝ V
inst✝² : MetricSpace P
inst✝¹ : NormedAddTorsor V P
hd2 : Fact (finrank ℝ V = 2)
inst✝ : Oriented ℝ V (Fin 2)
p₁ p₂ p₃ p₄ p₅ p₆ : P
h₂ : p₂ ∉ line[ℝ, p₁, p₃]
h₄ : p₄ ∈ line[ℝ, p₁, p₃]
h₆ : p₆ ∈ line[ℝ, p₁, p₃]
h₁₂₄₅ : line[ℝ, p₁, p₂] ∥ line[ℝ, p₄, p₅]
h₃₂₆₅ : line[ℝ, p₃, p₂] ∥ line[ℝ, p₆, p₅]
⊢ line[ℝ, p₆, p₄].direction ≤ line[ℝ, p₁, p₃].direction

Tactic:
]

Goal state after:
V : Type u_1
P : Type u_2
inst✝⁴ : NormedAddCommGroup V
inst✝³ : InnerProductSpace ℝ V
inst✝² : MetricSpace P
inst✝¹ : NormedAddTorsor V P
hd2 : Fact (finrank ℝ V = 2)
inst✝ : Oriented ℝ V (Fin 2)
p₁ p₂ p₃ p₄ p₅ p₆ : P
h₂ : p₂ ∉ line[ℝ, p₁, p₃]
h₄ : p₄ ∈ line[ℝ, p₁, p₃]
h₆ : p₆ ∈ line[ℝ, p₁, p₃]
h₁₂₄₅ : line[ℝ, p₁, p₂] ∥ line[ℝ, p₄, p₅]
h₃₂₆₅ : line[ℝ, p₃, p₂] ∥ line[ℝ, p₆, p₅]
⊢ line[ℝ, p₆, p₄].direction ≤ line[ℝ, p₁, p₃].direction
Module: Mathlib.Geometry.Euclidean.Angle.Unoriented.RightAngle

Goal state before:
pp: V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x y : V
h : ⟪x, y⟫ = 0
h0 : x ≠ 0 ∨ y = 0
⊢ ‖y‖ / ‖x‖ * ‖x‖ = ‖y‖

Tactic:
]

Goal state after:
V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x y : V
h : ⟪x, y⟫ = 0
h0 : x ≠ 0 ∨ y = 0
⊢ ‖y‖ / ‖x‖ * ‖x‖ = ‖y‖
Module: Mathlib.Geometry.Euclidean.Angle.Unoriented.Affine

Goal state before:
pp: case neg
V : Type u_1
P : Type u_2
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
p₁ p₂ p₃ : P
h : Wbtw ℝ p₁ p₂ p₃ ∨ Wbtw ℝ p₂ p₃ p₁ ∨ Wbtw ℝ p₃ p₁ p₂
h₁₂ : ¬p₁ = p₂
h₃₂ : ¬p₃ = p₂
⊢ ∠ p₁ p₂ p₃ = 0 ∨ ∠ p₁ p₂ p₃ = π

Tactic:
]

Goal state after:
case neg
V : Type u_1
P : Type u_2
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
p₁ p₂ p₃ : P
h : Wbtw ℝ p₁ p₂ p₃ ∨ Wbtw ℝ p₂ p₃ p₁ ∨ Wbtw ℝ p₃ p₁ p₂
h₁₂ : ¬p₁ = p₂
h₃₂ : ¬p₃ = p₂
⊢ ∠ p₁ p₂ p₃ = 0 ∨ ∠ p₁ p₂ p₃ = π
Module: Mathlib.Geometry.Euclidean.Angle.Unoriented.RightAngle

Goal state before:
pp: V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x y : V
h : ⟪x, y⟫ = 0
h0 : x ≠ 0 ∨ y = 0
⊢ ‖x‖ / (‖x‖ / ‖x + y‖) = ‖x + y‖

Tactic:
]

Goal state after:
V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x y : V
h : ⟪x, y⟫ = 0
h0 : x ≠ 0 ∨ y = 0
⊢ ‖x‖ / (‖x‖ / ‖x + y‖) = ‖x + y‖
Module: Mathlib.Geometry.Euclidean.Angle.Unoriented.RightAngle

Goal state before:
pp: case inl
V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x y : V
h : ⟪x, y⟫ = 0
h0 : x ≠ 0
⊢ ‖x + y‖ = ‖x + y‖

Tactic:
]

Goal state after:
Module: Mathlib.Geometry.Euclidean.Angle.Unoriented.RightAngle

Goal state before:
pp: case inr
V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x y : V
h : ⟪x, y⟫ = 0
h0 : y ≠ 0
⊢ ‖x + y‖ = ‖x + y‖

Tactic:
]

Goal state after:
Module: Mathlib.Geometry.Euclidean.Angle.Unoriented.RightAngle

Goal state before:
pp: V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x y : V
h : ⟪x, y⟫ = 0
h0 : x = 0 ∨ y ≠ 0
⊢ ‖y‖ / (‖y‖ / ‖x‖) = ‖x‖

Tactic:
]

Goal state after:
V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x y : V
h : ⟪x, y⟫ = 0
h0 : x = 0 ∨ y ≠ 0
⊢ ‖y‖ / (‖y‖ / ‖x‖) = ‖x‖
Module: Mathlib.Geometry.Euclidean.Angle.Unoriented.RightAngle

Goal state before:
pp: case inr
V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x y : V
h : ⟪x, y⟫ = 0
h0 : y ≠ 0
⊢ ‖x‖ = ‖x‖

Tactic:
]

Goal state after:
Module: Mathlib.Geometry.Euclidean.Angle.Unoriented.RightAngle

Goal state before:
pp: V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x y : V
h : ⟪x, -y⟫ = 0
⊢ angle x (x - y) = Real.arccos (‖x‖ / ‖x - y‖)

Tactic:
]

Goal state after:
V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x y : V
h : ⟪x, -y⟫ = 0
⊢ angle x (x - y) = Real.arccos (‖x‖ / ‖x - y‖)
Module: Mathlib.Geometry.Euclidean.Angle.Unoriented.RightAngle

Goal state before:
pp: V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x y : V
h : ⟪x, -y⟫ = 0
⊢ Real.arccos (‖x‖ / ‖x + -y‖) = Real.arccos (‖x‖ / ‖x + -y‖)

Tactic:
]

Goal state after:
Module: Mathlib.Geometry.Euclidean.Angle.Unoriented.RightAngle

Goal state before:
pp: V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x y : V
h : ⟪x, -y⟫ = 0
h0 : x ≠ 0 ∨ y ≠ 0
⊢ angle x (x - y) = Real.arcsin (‖y‖ / ‖x - y‖)

Tactic:
]

Goal state after:
V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x y : V
h : ⟪x, -y⟫ = 0
h0 : x ≠ 0 ∨ y ≠ 0
⊢ angle x (x - y) = Real.arcsin (‖y‖ / ‖x - y‖)
Module: Mathlib.Geometry.Euclidean.Angle.Unoriented.RightAngle

Goal state before:
pp: V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x y : V
h : ⟪x, -y⟫ = 0
h0 : x ≠ 0 ∨ -y ≠ 0
⊢ angle x (x - y) = Real.arcsin (‖y‖ / ‖x - y‖)

Tactic:
]

Goal state after:
V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x y : V
h : ⟪x, -y⟫ = 0
h0 : x ≠ 0 ∨ -y ≠ 0
⊢ angle x (x - y) = Real.arcsin (‖y‖ / ‖x - y‖)
Module: Mathlib.Geometry.Euclidean.Angle.Oriented.Affine

Goal state before:
pp: V : Type u_1
P : Type u_2
inst✝⁴ : NormedAddCommGroup V
inst✝³ : InnerProductSpace ℝ V
inst✝² : MetricSpace P
inst✝¹ : NormedAddTorsor V P
hd2 : Fact (finrank ℝ V = 2)
inst✝ : Oriented ℝ V (Fin 2)
p₁ p₂ p₃ p₄ p₅ p₆ : P
h₂ : p₂ ∉ line[ℝ, p₁, p₃]
h₄ : p₄ ∈ line[ℝ, p₁, p₃]
h₆ : p₆ ∈ line[ℝ, p₁, p₃]
h₁₂₄₅ : line[ℝ, p₁, p₂] ∥ line[ℝ, p₄, p₅]
h₃₂₆₅ : line[ℝ, p₃, p₂] ∥ line[ℝ, p₆, p₅]
hd : line[ℝ, p₆, p₄].direction ≤ line[ℝ, p₃, p₁].direction
r : ℝ
hr : r ≠ 0
h₅₄ : p₄ -ᵥ p₅ = r • (p₁ -ᵥ p₂)
h₆₅ : p₆ -ᵥ p₅ = r • (p₃ -ᵥ p₂)
⊢ o.oangle (p₁ -ᵥ p₂) (p₃ -ᵥ p₂) = o.oangle (p₄ -ᵥ p₅) (p₆ -ᵥ p₅)

Tactic:
]

Goal state after:
V : Type u_1
P : Type u_2
inst✝⁴ : NormedAddCommGroup V
inst✝³ : InnerProductSpace ℝ V
inst✝² : MetricSpace P
inst✝¹ : NormedAddTorsor V P
hd2 : Fact (finrank ℝ V = 2)
inst✝ : Oriented ℝ V (Fin 2)
p₁ p₂ p₃ p₄ p₅ p₆ : P
h₂ : p₂ ∉ line[ℝ, p₁, p₃]
h₄ : p₄ ∈ line[ℝ, p₁, p₃]
h₆ : p₆ ∈ line[ℝ, p₁, p₃]
h₁₂₄₅ : line[ℝ, p₁, p₂] ∥ line[ℝ, p₄, p₅]
h₃₂₆₅ : line[ℝ, p₃, p₂] ∥ line[ℝ, p₆, p₅]
hd : line[ℝ, p₆, p₄].direction ≤ line[ℝ, p₃, p₁].direction
r : ℝ
hr : r ≠ 0
h₅₄ : p₄ -ᵥ p₅ = r • (p₁ -ᵥ p₂)
h₆₅ : p₆ -ᵥ p₅ = r • (p₃ -ᵥ p₂)
⊢ o.oangle (p₁ -ᵥ p₂) (p₃ -ᵥ p₂) = o.oangle (p₄ -ᵥ p₅) (p₆ -ᵥ p₅)
Module: Mathlib.Geometry.Euclidean.Angle.Unoriented.RightAngle

Goal state before:
pp: V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x y : V
h : ⟪x, -y⟫ = 0
h0 : x ≠ 0 ∨ -y ≠ 0
⊢ Real.arcsin (‖y‖ / ‖x + -y‖) = Real.arcsin (‖y‖ / ‖x + -y‖)

Tactic:
]

Goal state after:
Module: Mathlib.Geometry.Euclidean.Angle.Unoriented.RightAngle

Goal state before:
pp: V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x y : V
h : ⟪x, -y⟫ = 0
h0 : x ≠ 0
⊢ angle x (x - y) = Real.arctan (‖y‖ / ‖x‖)

Tactic:
]

Goal state after:
V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x y : V
h : ⟪x, -y⟫ = 0
h0 : x ≠ 0
⊢ angle x (x - y) = Real.arctan (‖y‖ / ‖x‖)
Module: Mathlib.Geometry.Euclidean.Angle.Unoriented.Affine

Goal state before:
pp: case refine_2.inr.inr.inl
V : Type u_1
P : Type u_2
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
p₁ p₂ p₃ : P
h : p₁ ≠ p₂ ∧ Wbtw ℝ p₂ p₁ p₃ ∨ p₃ ≠ p₂ ∧ Wbtw ℝ p₂ p₃ p₁
⊢ Collinear ℝ {p₁, p₂, p₃}

Tactic:
]

Goal state after:
case refine_2.inr.inr.inl
V : Type u_1
P : Type u_2
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
p₁ p₂ p₃ : P
h : p₁ ≠ p₂ ∧ Wbtw ℝ p₂ p₁ p₃ ∨ p₃ ≠ p₂ ∧ Wbtw ℝ p₂ p₃ p₁
⊢ Collinear ℝ {p₁, p₂, p₃}
Module: Mathlib.Geometry.Euclidean.Angle.Unoriented.RightAngle

Goal state before:
pp: V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x y : V
h : ⟪x, -y⟫ = 0
h0 : x ≠ 0
⊢ Real.arctan (‖y‖ / ‖x‖) = Real.arctan (‖y‖ / ‖x‖)

Tactic:
]

Goal state after:
Module: Mathlib.Geometry.Euclidean.Angle.Unoriented.RightAngle

Goal state before:
pp: V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x y : V
h : ⟪x, -y⟫ = 0
h0 : x = 0 ∨ y ≠ 0
⊢ 0 < angle x (x - y)

Tactic:
]

Goal state after:
V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x y : V
h : ⟪x, -y⟫ = 0
h0 : x = 0 ∨ y ≠ 0
⊢ 0 < angle x (x - y)
Module: Mathlib.Geometry.Euclidean.Angle.Unoriented.RightAngle

Goal state before:
pp: V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x y : V
h : ⟪x, -y⟫ = 0
h0 : x = 0 ∨ -y ≠ 0
⊢ 0 < angle x (x - y)

Tactic:
]

Goal state after:
V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x y : V
h : ⟪x, -y⟫ = 0
h0 : x = 0 ∨ -y ≠ 0
⊢ 0 < angle x (x - y)
Module: Mathlib.Geometry.Euclidean.Angle.Unoriented.RightAngle

Goal state before:
pp: V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x y : V
h : ⟪x, -y⟫ = 0
h0 : x = 0 ∨ -y ≠ 0
⊢ 0 < angle x (x + -y)

Tactic:
]

Goal state after:
V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x y : V
h : ⟪x, -y⟫ = 0
h0 : x = 0 ∨ -y ≠ 0
⊢ 0 < angle x (x + -y)
Module: Mathlib.Geometry.Euclidean.Angle.Unoriented.Affine

Goal state before:
pp: case refine_2.inr.inr.inl.inl
V : Type u_1
P : Type u_2
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
p₁ p₂ p₃ : P
h : Wbtw ℝ p₂ p₁ p₃
⊢ Collinear ℝ {p₂, p₁, p₃}

Tactic:
]

Goal state after:
case refine_2.inr.inr.inl.inl
V : Type u_1
P : Type u_2
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
p₁ p₂ p₃ : P
h : Wbtw ℝ p₂ p₁ p₃
⊢ Collinear ℝ {p₂, p₁, p₃}
Module: Mathlib.Geometry.Euclidean.Angle.Unoriented.RightAngle

Goal state before:
pp: V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x y : V
h : ⟪x, -y⟫ = 0
⊢ angle x (x - y) ≤ π / 2

Tactic:
]

Goal state after:
V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x y : V
h : ⟪x, -y⟫ = 0
⊢ angle x (x - y) ≤ π / 2
Module: Mathlib.Geometry.Euclidean.Angle.Unoriented.RightAngle

Goal state before:
pp: V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x y : V
h : ⟪x, -y⟫ = 0
⊢ angle x (x + -y) ≤ π / 2

Tactic:
]

Goal state after:
V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x y : V
h : ⟪x, -y⟫ = 0
⊢ angle x (x + -y) ≤ π / 2
Module: Mathlib.Geometry.Euclidean.Angle.Oriented.Basic

Goal state before:
pp: V : Type u_1
inst✝² : NormedAddCommGroup V
inst✝¹ : InnerProductSpace ℝ V
inst✝ : Fact (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
h0 : (w = 0 ∨ x = 0) ∨ y = 0 ∨ z = 0
hswx : (o.oangle w x).sign = 0
hsyz : (o.oangle y z).sign = 0
hwx : InnerProductGeometry.angle w x = π / 2
hyz : InnerProductGeometry.angle y z = π / 2
hpi : π = 0
⊢ False

pp: V : Type u_1
inst✝² : NormedAddCommGroup V
inst✝¹ : InnerProductSpace ℝ V
inst✝ : Fact (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
h0 : (w = 0 ∨ x = 0) ∨ y = 0 ∨ z = 0
hswx : (o.oangle w x).sign = 0
hsyz : (o.oangle y z).sign = 0
hwx : InnerProductGeometry.angle w x = π / 2
hyz : InnerProductGeometry.angle y z = π / 2
hpi : π / 2 = π
⊢ 2 ≠ 0

Tactic:
]

Goal state after:
V : Type u_1
inst✝² : NormedAddCommGroup V
inst✝¹ : InnerProductSpace ℝ V
inst✝ : Fact (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
h0 : (w = 0 ∨ x = 0) ∨ y = 0 ∨ z = 0
hswx : (o.oangle w x).sign = 0
hsyz : (o.oangle y z).sign = 0
hwx : InnerProductGeometry.angle w x = π / 2
hyz : InnerProductGeometry.angle y z = π / 2
hpi : π = 0
⊢ False

V : Type u_1
inst✝² : NormedAddCommGroup V
inst✝¹ : InnerProductSpace ℝ V
inst✝ : Fact (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
h0 : (w = 0 ∨ x = 0) ∨ y = 0 ∨ z = 0
hswx : (o.oangle w x).sign = 0
hsyz : (o.oangle y z).sign = 0
hwx : InnerProductGeometry.angle w x = π / 2
hyz : InnerProductGeometry.angle y z = π / 2
hpi : π / 2 = π
⊢ 2 ≠ 0
Module: Mathlib.Geometry.Euclidean.Angle.Unoriented.Affine

Goal state before:
pp: case refine_2.inr.inr.inl.inr
V : Type u_1
P : Type u_2
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
p₁ p₂ p₃ : P
h : Wbtw ℝ p₂ p₃ p₁
⊢ Collinear ℝ {p₂, p₃, p₁}

Tactic:
]

Goal state after:
case refine_2.inr.inr.inl.inr
V : Type u_1
P : Type u_2
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
p₁ p₂ p₃ : P
h : Wbtw ℝ p₂ p₃ p₁
⊢ Collinear ℝ {p₂, p₃, p₁}
Module: Mathlib.Geometry.Euclidean.Angle.Unoriented.RightAngle

Goal state before:
pp: V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x y : V
h : ⟪x, -y⟫ = 0
h0 : x ≠ 0
⊢ angle x (x - y) < π / 2

Tactic:
]

Goal state after:
V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x y : V
h : ⟪x, -y⟫ = 0
h0 : x ≠ 0
⊢ angle x (x - y) < π / 2
Module: Mathlib.Geometry.Euclidean.Angle.Unoriented.RightAngle

Goal state before:
pp: V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x y : V
h : ⟪x, -y⟫ = 0
h0 : x ≠ 0
⊢ angle x (x + -y) < π / 2

Tactic:
]

Goal state after:
V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x y : V
h : ⟪x, -y⟫ = 0
h0 : x ≠ 0
⊢ angle x (x + -y) < π / 2
Module: Mathlib.Geometry.Euclidean.Angle.Oriented.Affine

Goal state before:
pp: V : Type u_1
P : Type u_2
inst✝⁴ : NormedAddCommGroup V
inst✝³ : InnerProductSpace ℝ V
inst✝² : MetricSpace P
inst✝¹ : NormedAddTorsor V P
hd2 : Fact (finrank ℝ V = 2)
inst✝ : Oriented ℝ V (Fin 2)
p₁ p₂ p₃ p₄ p₅ p₆ : P
h₂ : p₂ ∉ line[ℝ, p₁, p₃]
h₄ : p₄ ∈ line[ℝ, p₁, p₃]
h₆ : p₆ ∈ line[ℝ, p₁, p₃]
h₁₂₄₅ : line[ℝ, p₁, p₂] ∥ line[ℝ, p₄, p₅]
h₃₂₆₅ : line[ℝ, p₃, p₂] ∥ line[ℝ, p₆, p₅]
hd : line[ℝ, p₆, p₄].direction ≤ line[ℝ, p₃, p₁].direction
r : ℝ
hr : r ≠ 0
h₅₄ : p₄ -ᵥ p₅ = r • (p₁ -ᵥ p₂)
h₆₅ : p₆ -ᵥ p₅ = r • (p₃ -ᵥ p₂)
⊢ o.oangle (p₁ -ᵥ p₂) (p₃ -ᵥ p₂) = o.oangle (r • (p₁ -ᵥ p₂)) (r • (p₃ -ᵥ p₂))

Tactic:
]

Goal state after:
V : Type u_1
P : Type u_2
inst✝⁴ : NormedAddCommGroup V
inst✝³ : InnerProductSpace ℝ V
inst✝² : MetricSpace P
inst✝¹ : NormedAddTorsor V P
hd2 : Fact (finrank ℝ V = 2)
inst✝ : Oriented ℝ V (Fin 2)
p₁ p₂ p₃ p₄ p₅ p₆ : P
h₂ : p₂ ∉ line[ℝ, p₁, p₃]
h₄ : p₄ ∈ line[ℝ, p₁, p₃]
h₆ : p₆ ∈ line[ℝ, p₁, p₃]
h₁₂₄₅ : line[ℝ, p₁, p₂] ∥ line[ℝ, p₄, p₅]
h₃₂₆₅ : line[ℝ, p₃, p₂] ∥ line[ℝ, p₆, p₅]
hd : line[ℝ, p₆, p₄].direction ≤ line[ℝ, p₃, p₁].direction
r : ℝ
hr : r ≠ 0
h₅₄ : p₄ -ᵥ p₅ = r • (p₁ -ᵥ p₂)
h₆₅ : p₆ -ᵥ p₅ = r • (p₃ -ᵥ p₂)
⊢ o.oangle (p₁ -ᵥ p₂) (p₃ -ᵥ p₂) = o.oangle (r • (p₁ -ᵥ p₂)) (r • (p₃ -ᵥ p₂))
Module: Mathlib.Geometry.Euclidean.Angle.Unoriented.RightAngle

Goal state before:
pp: V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x y : V
h : ⟪x, -y⟫ = 0
⊢ Real.cos (angle x (x - y)) = ‖x‖ / ‖x - y‖

Tactic:
]

Goal state after:
V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x y : V
h : ⟪x, -y⟫ = 0
⊢ Real.cos (angle x (x - y)) = ‖x‖ / ‖x - y‖
Module: Mathlib.Geometry.Euclidean.Angle.Unoriented.Affine

Goal state before:
pp: case refine_2.inr.inr.inr
V : Type u_1
P : Type u_2
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
p₁ p₂ p₃ : P
h : Sbtw ℝ p₁ p₂ p₃
⊢ Collinear ℝ {p₁, p₂, p₃}

Tactic:
]

Goal state after:
case refine_2.inr.inr.inr
V : Type u_1
P : Type u_2
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
p₁ p₂ p₃ : P
h : Sbtw ℝ p₁ p₂ p₃
⊢ Collinear ℝ {p₁, p₂, p₃}
Module: Mathlib.Geometry.Euclidean.Angle.Unoriented.RightAngle

Goal state before:
pp: V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x y : V
h : ⟪x, -y⟫ = 0
⊢ ‖x‖ / ‖x + -y‖ = ‖x‖ / ‖x + -y‖

Tactic:
]

Goal state after:
Module: Mathlib.Geometry.Euclidean.Angle.Unoriented.Affine

Goal state before:
pp: V : Type u_1
P : Type u_2
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
p₁ p₂ p₃ : P
⊢ p₁ = p₂ ∨ p₃ = p₂ ∨ ∠ p₁ p₂ p₃ = 0 ∨ ∠ p₁ p₂ p₃ = π ↔ p₁ = p₂ ∨ p₃ = p₂ ∨ ∠ p₁ p₂ p₃ = 0 ∨ ∠ p₁ p₂ p₃ = π

Tactic:
]

Goal state after:
Module: Mathlib.Geometry.Euclidean.Angle.Unoriented.RightAngle

Goal state before:
pp: V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x y : V
h : ⟪x, -y⟫ = 0
h0 : x ≠ 0 ∨ y ≠ 0
⊢ Real.sin (angle x (x - y)) = ‖y‖ / ‖x - y‖

Tactic:
]

Goal state after:
V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x y : V
h : ⟪x, -y⟫ = 0
h0 : x ≠ 0 ∨ y ≠ 0
⊢ Real.sin (angle x (x - y)) = ‖y‖ / ‖x - y‖
Module: Mathlib.Geometry.Euclidean.Angle.Unoriented.RightAngle

Goal state before:
pp: V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x y : V
h : ⟪x, -y⟫ = 0
h0 : x ≠ 0 ∨ -y ≠ 0
⊢ Real.sin (angle x (x - y)) = ‖y‖ / ‖x - y‖

Tactic:
]

Goal state after:
V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x y : V
h : ⟪x, -y⟫ = 0
h0 : x ≠ 0 ∨ -y ≠ 0
⊢ Real.sin (angle x (x - y)) = ‖y‖ / ‖x - y‖
Module: Mathlib.Geometry.Euclidean.Angle.Oriented.RightAngle

Goal state before:
pp: V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : 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)
⊢ 1 = 1

Tactic:
]

Goal state after:
Module: Mathlib.Geometry.Euclidean.Angle.Oriented.RightAngle

Goal state before:
pp: V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : 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)
hs : (o.oangle x (x + y)).sign = 1
⊢ ↑(Real.arccos (‖x‖ / ‖x + y‖)) = ↑(Real.arccos (‖x‖ / ‖x + y‖))

Tactic:
]

Goal state after:
Module: Mathlib.Geometry.Euclidean.Angle.Oriented.RightAngle

Goal state before:
pp: V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : 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)
⊢ (-o).oangle y (x + y) = ↑(Real.arccos (‖y‖ / ‖x + y‖))

Tactic:
]

Goal state after:
V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : 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)
⊢ (-o).oangle y (x + y) = ↑(Real.arccos (‖y‖ / ‖x + y‖))
Module: Mathlib.Geometry.Euclidean.Angle.Oriented.RightAngle

Goal state before:
pp: V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : 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)
⊢ (-o).oangle y (y + x) = ↑(Real.arccos (‖y‖ / ‖y + x‖))

Tactic:
]

Goal state after:
V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : 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)
⊢ (-o).oangle y (y + x) = ↑(Real.arccos (‖y‖ / ‖y + x‖))
Module: Mathlib.Geometry.Euclidean.Angle.Unoriented.RightAngle

Goal state before:
pp: V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x y : V
h : ⟪x, -y⟫ = 0
h0 : x ≠ 0 ∨ -y ≠ 0
⊢ ‖y‖ / ‖x + -y‖ = ‖y‖ / ‖x + -y‖

Tactic:
]

Goal state after:
Module: Mathlib.Geometry.Euclidean.Angle.Unoriented.RightAngle

Goal state before:
pp: V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x y : V
h : ⟪x, -y⟫ = 0
⊢ Real.tan (angle x (x - y)) = ‖y‖ / ‖x‖

Tactic:
]

Goal state after:
V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x y : V
h : ⟪x, -y⟫ = 0
⊢ Real.tan (angle x (x - y)) = ‖y‖ / ‖x‖
Module: Mathlib.Geometry.Euclidean.Angle.Unoriented.RightAngle

Goal state before:
pp: V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x y : V
h : ⟪x, -y⟫ = 0
⊢ ‖y‖ / ‖x‖ = ‖y‖ / ‖x‖

Tactic:
]

Goal state after:
Module: Mathlib.Geometry.Euclidean.Angle.Oriented.RightAngle

Goal state before:
pp: V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : 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)
hs : (o.oangle x (x + y)).sign = 1
⊢ ↑(Real.arcsin (‖y‖ / ‖x + y‖)) = ↑(Real.arcsin (‖y‖ / ‖x + y‖))

Tactic:
]

Goal state after:
Module: Mathlib.Geometry.Euclidean.Angle.Unoriented.RightAngle

Goal state before:
pp: V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x y : V
h : ⟪x, -y⟫ = 0
⊢ Real.cos (angle x (x - y)) * ‖x - y‖ = ‖x‖

Tactic:
]

Goal state after:
V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x y : V
h : ⟪x, -y⟫ = 0
⊢ Real.cos (angle x (x - y)) * ‖x - y‖ = ‖x‖
Module: Mathlib.Geometry.Euclidean.Angle.Unoriented.RightAngle

Goal state before:
pp: V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x y : V
h : ⟪x, -y⟫ = 0
⊢ ‖x‖ = ‖x‖

Tactic:
]

Goal state after:
Module: Mathlib.Geometry.Euclidean.Angle.Unoriented.RightAngle

Goal state before:
pp: V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x y : V
h : ⟪x, -y⟫ = 0
⊢ Real.sin (angle x (x - y)) * ‖x - y‖ = ‖y‖

Tactic:
]

Goal state after:
V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x y : V
h : ⟪x, -y⟫ = 0
⊢ Real.sin (angle x (x - y)) * ‖x - y‖ = ‖y‖