mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Geometry.Euclidean.Incenter

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
n : ℕ
inst✝ : NeZero n
s : Simplex ℝ P n
signs : Finset (Fin (n + 1))
h : s.ExcenterExists signs
i : Fin (n + 1)
⊢ (∑ i, (if i ∈ signs then -1 else 1) * (s.height i)⁻¹)⁻¹ • ((if i ∈ signs then -1 else 1) * (s.height i)⁻¹) *
      dist (s.points i) ↑((s.faceOpposite i).orthogonalProjectionSpan (s.points i)) =
    (if i ∈ signs then -1 else 1) * (∑ i, (if i ∈ signs then -1 else 1) * (s.height i)⁻¹)⁻¹

Tactic:
rw [← altitudeFoot, ← height]

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
n : ℕ
inst✝ : NeZero n
s : Simplex ℝ P n
signs : Finset (Fin (n + 1))
h : s.ExcenterExists signs
i : Fin (n + 1)
⊢ (∑ i, (if i ∈ signs then -1 else 1) * (s.height i)⁻¹)⁻¹ • ((if i ∈ signs then -1 else 1) * (s.height i)⁻¹) *
      s.height i =
    (if i ∈ signs then -1 else 1) * (∑ i, (if i ∈ signs then -1 else 1) * (s.height i)⁻¹)⁻¹
Module: Mathlib.Geometry.Euclidean.Triangle

Goal state before:
pp: case pos
V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x y : V
hpi : angle x y ≠ π
hxy : ¬x = y
h : ‖y‖ = 0
hx0 : x = 0
⊢ ‖x‖ = ‖y‖

Tactic:
rw [hx0, norm_zero, h]

Goal state after:
Module: Mathlib.Geometry.Euclidean.Circumcenter

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
ι : Type u_3
hne : Nonempty ι
inst✝ : Finite ι
p : ι → P
ha : AffineIndependent ℝ p
val✝ : Fintype ι
m : ℕ
hn : Fintype.card ι = m + 1 + 1
i : ι
ι2 : Type (max 0 u_3) := { x // x ≠ i }
hc : Fintype.card ι2 = m + 1
this : Nonempty ι2
ha2 : AffineIndependent ℝ fun i2 ↦ p ↑i2
hm :
  ∃! cs,
    cs.center ∈ affineSpan ℝ (Set.range fun i2 ↦ p ↑i2) ∧ (Set.range fun i2 ↦ p ↑i2) ⊆ Metric.sphere cs.center cs.radius
⊢ p '' Set.univ = insert (p i) (p '' {x | x ≠ i})

Tactic:
← Set.image_insert_eq

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
ι : Type u_3
hne : Nonempty ι
inst✝ : Finite ι
p : ι → P
ha : AffineIndependent ℝ p
val✝ : Fintype ι
m : ℕ
hn : Fintype.card ι = m + 1 + 1
i : ι
ι2 : Type (max 0 u_3) := { x // x ≠ i }
hc : Fintype.card ι2 = m + 1
this : Nonempty ι2
ha2 : AffineIndependent ℝ fun i2 ↦ p ↑i2
hm :
  ∃! cs,
    cs.center ∈ affineSpan ℝ (Set.range fun i2 ↦ p ↑i2) ∧ (Set.range fun i2 ↦ p ↑i2) ⊆ Metric.sphere cs.center cs.radius
⊢ p '' Set.univ = p '' insert i {x | x ≠ i}
Module: Mathlib.Geometry.Euclidean.Circumcenter

Goal state before:
pp: V : Type u_1
P : Type u_2
inst✝⁷ : NormedAddCommGroup V
inst✝⁶ : InnerProductSpace ℝ V
inst✝⁵ : MetricSpace P
inst✝⁴ : NormedAddTorsor V P
V₂ : Type u_3
P₂ : Type u_4
inst✝³ : NormedAddCommGroup V₂
inst✝² : InnerProductSpace ℝ V₂
inst✝¹ : MetricSpace P₂
inst✝ : NormedAddTorsor V₂ P₂
n : ℕ
s : Simplex ℝ P n
f : P →ᵃⁱ[ℝ] P₂
⊢ s.circumradius = (s.map f.toAffineMap ⋯).circumradius

Tactic:
refine (s.map f.toAffineMap f.injective).eq_circumradius_of_dist_eq (p := f s.circumcenter) ?_
    fun i ↦ by simp

Goal state after:
V : Type u_1
P : Type u_2
inst✝⁷ : NormedAddCommGroup V
inst✝⁶ : InnerProductSpace ℝ V
inst✝⁵ : MetricSpace P
inst✝⁴ : NormedAddTorsor V P
V₂ : Type u_3
P₂ : Type u_4
inst✝³ : NormedAddCommGroup V₂
inst✝² : InnerProductSpace ℝ V₂
inst✝¹ : MetricSpace P₂
inst✝ : NormedAddTorsor V₂ P₂
n : ℕ
s : Simplex ℝ P n
f : P →ᵃⁱ[ℝ] P₂
⊢ f s.circumcenter ∈ affineSpan ℝ (Set.range (s.map f.toAffineMap ⋯).points)
Module: Mathlib.Geometry.Euclidean.Triangle

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
a b c p x : P
hx : Sbtw ℝ a x c
hxb : SameRay ℝ (x -ᵥ p) (b -ᵥ p)
hb : b ≠ p
⊢ ∠ a p b + ∠ b p c = ∠ a p c

Tactic:
by
  rcases eq_or_ne p x with rfl | hpx
  · have hpi : ∠ a p c = π := hx.angle₁₂₃_eq_pi
    rw [hpi, angle_comm a p b]
    exact angle_add_angle_eq_pi_of_angle_eq_pi b hpi
  obtain ⟨r, hr, hrb⟩ := (exists_pos_left_iff_sameRay (by aesop) (by aesop)).2 hxb
  have hab : ∠ a p b = ∠ a p x := angle_smul_right_of_pos a hr hrb
  have hbc : ∠ b p c = ∠ x p c := angle_smul_left_of_pos c hr hrb
  rw [hab, hbc]
  exact angle_add_angle_eq_of_sbtw hx

Goal state after:
Module: Mathlib.Geometry.Euclidean.Triangle

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
a b c : P
h : ¬Collinear ℝ {a, b, c}
hsin : Real.sin (∠ c b a) * dist a b = Real.sin (∠ a c b) * dist a c
⊢ ∠ a c b < ∠ a b c → dist a b < dist a c

Tactic:
angle_comm c b a

Goal state after:
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
a b c : P
h : ¬Collinear ℝ {a, b, c}
hsin : Real.sin (∠ a b c) * dist a b = Real.sin (∠ a c b) * dist a c
⊢ ∠ a c b < ∠ a b c → dist a b < dist a c
Module: Mathlib.Geometry.Euclidean.Triangle

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
h₂₃₁ : ∠ p₂ p₃ p₁ ≤ ∠ p₁ p₂ p₃
h₃₁₂ : ∠ p₃ p₁ p₂ ≤ ∠ p₁ p₂ p₃
hne : ∠ p₁ p₂ p₃ ≠ ∠ p₂ p₃ p₁ ∨ ∠ p₁ p₂ p₃ ≠ ∠ p₃ p₁ p₂ ∨ ∠ p₂ p₃ p₁ ≠ ∠ p₃ p₁ p₂
h : ¬p₂ = p₁
⊢ π / 3 < ∠ p₁ p₂ p₃

Tactic:
rcases hne with hne | hne | hne

Goal state after:
case neg.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
h₂₃₁ : ∠ p₂ p₃ p₁ ≤ ∠ p₁ p₂ p₃
h₃₁₂ : ∠ p₃ p₁ p₂ ≤ ∠ p₁ p₂ p₃
h : ¬p₂ = p₁
hne : ∠ p₁ p₂ p₃ ≠ ∠ p₂ p₃ p₁
⊢ π / 3 < ∠ p₁ p₂ p₃

case neg.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
h₂₃₁ : ∠ p₂ p₃ p₁ ≤ ∠ p₁ p₂ p₃
h₃₁₂ : ∠ p₃ p₁ p₂ ≤ ∠ p₁ p₂ p₃
h : ¬p₂ = p₁
hne : ∠ p₁ p₂ p₃ ≠ ∠ p₃ p₁ p₂
⊢ π / 3 < ∠ p₁ p₂ p₃

case neg.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
h₂₃₁ : ∠ p₂ p₃ p₁ ≤ ∠ p₁ p₂ p₃
h₃₁₂ : ∠ p₃ p₁ p₂ ≤ ∠ p₁ p₂ p₃
h : ¬p₂ = p₁
hne : ∠ p₂ p₃ p₁ ≠ ∠ p₃ p₁ p₂
⊢ π / 3 < ∠ p₁ p₂ p₃
Module: Mathlib.Geometry.Euclidean.Triangle

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
h₂₃₁ : ∠ p₁ p₂ p₃ ≤ ∠ p₂ p₃ p₁
h₃₁₂ : ∠ p₁ p₂ p₃ ≤ ∠ p₃ p₁ p₂
hne : ∠ p₁ p₂ p₃ ≠ ∠ p₂ p₃ p₁ ∨ ∠ p₁ p₂ p₃ ≠ ∠ p₃ p₁ p₂ ∨ ∠ p₂ p₃ p₁ ≠ ∠ p₃ p₁ p₂
h : ¬p₂ = p₁
⊢ ∠ p₁ p₂ p₃ < π / 3

Tactic:
rcases hne with hne | hne | hne

Goal state after:
case neg.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
h₂₃₁ : ∠ p₁ p₂ p₃ ≤ ∠ p₂ p₃ p₁
h₃₁₂ : ∠ p₁ p₂ p₃ ≤ ∠ p₃ p₁ p₂
h : ¬p₂ = p₁
hne : ∠ p₁ p₂ p₃ ≠ ∠ p₂ p₃ p₁
⊢ ∠ p₁ p₂ p₃ < π / 3

case neg.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
h₂₃₁ : ∠ p₁ p₂ p₃ ≤ ∠ p₂ p₃ p₁
h₃₁₂ : ∠ p₁ p₂ p₃ ≤ ∠ p₃ p₁ p₂
h : ¬p₂ = p₁
hne : ∠ p₁ p₂ p₃ ≠ ∠ p₃ p₁ p₂
⊢ ∠ p₁ p₂ p₃ < π / 3

case neg.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
h₂₃₁ : ∠ p₁ p₂ p₃ ≤ ∠ p₂ p₃ p₁
h₃₁₂ : ∠ p₁ p₂ p₃ ≤ ∠ p₃ p₁ p₂
h : ¬p₂ = p₁
hne : ∠ p₂ p₃ p₁ ≠ ∠ p₃ p₁ p₂
⊢ ∠ p₁ p₂ p₃ < π / 3
Module: Mathlib.Geometry.Euclidean.Circumcenter

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
n : ℕ
s : Simplex ℝ P n
p p₁ p₂ : P
r : ℝ
h₁ : ∀ (i : Fin (n + 1)), dist (s.points i) p₁ = r
h₂ : ∀ (i : Fin (n + 1)), dist (s.points i) p₂ = r
span_s : AffineSubspace ℝ P := affineSpan ℝ (Set.range s.points)
h₁' : ↑(s.orthogonalProjectionSpan p₁) = s.circumcenter
h₂' : ↑(s.orthogonalProjectionSpan p₂) = s.circumcenter
r₁ r₂ : ℝ
hp₁o : ↑(s.orthogonalProjectionSpan p₁) ∈ affineSpan ℝ (Set.range s.points)
hp₁ : p₁ = r₁ • (p -ᵥ ↑((orthogonalProjection (affineSpan ℝ (Set.range s.points))) p)) +ᵥ s.circumcenter
hp₂o : ↑(s.orthogonalProjectionSpan p₂) ∈ affineSpan ℝ (Set.range s.points)
hp₂ : p₂ = r₂ • (p -ᵥ ↑((orthogonalProjection (affineSpan ℝ (Set.range s.points))) p)) +ᵥ s.circumcenter
h : s.points 0 ∈ span_s
hd₁ :
  ‖(r₁ • (p -ᵥ ↑((orthogonalProjection (affineSpan ℝ (Set.range s.points))) p)) +ᵥ s.circumcenter) -ᵥ s.circumcenter‖ *
      ‖(r₁ • (p -ᵥ ↑((orthogonalProjection (affineSpan ℝ (Set.range s.points))) p)) +ᵥ s.circumcenter) -ᵥ
          s.circumcenter‖ =
    ‖(r₂ • (p -ᵥ ↑((orthogonalProjection (affineSpan ℝ (Set.range s.points))) p)) +ᵥ s.circumcenter) -ᵥ
          s.circumcenter‖ *
      ‖(r₂ • (p -ᵥ ↑((orthogonalProjection (affineSpan ℝ (Set.range s.points))) p)) +ᵥ s.circumcenter) -ᵥ
          s.circumcenter‖
hd₂ : dist p₂ s.circumcenter * dist p₂ s.circumcenter = r * r - s.circumradius * s.circumradius
⊢ p₁ = p₂ ∨ p₁ = (reflection (affineSpan ℝ (Set.range s.points))) p₂

Tactic:
vadd_vsub,

Goal state after:
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
n : ℕ
s : Simplex ℝ P n
p p₁ p₂ : P
r : ℝ
h₁ : ∀ (i : Fin (n + 1)), dist (s.points i) p₁ = r
h₂ : ∀ (i : Fin (n + 1)), dist (s.points i) p₂ = r
span_s : AffineSubspace ℝ P := affineSpan ℝ (Set.range s.points)
h₁' : ↑(s.orthogonalProjectionSpan p₁) = s.circumcenter
h₂' : ↑(s.orthogonalProjectionSpan p₂) = s.circumcenter
r₁ r₂ : ℝ
hp₁o : ↑(s.orthogonalProjectionSpan p₁) ∈ affineSpan ℝ (Set.range s.points)
hp₁ : p₁ = r₁ • (p -ᵥ ↑((orthogonalProjection (affineSpan ℝ (Set.range s.points))) p)) +ᵥ s.circumcenter
hp₂o : ↑(s.orthogonalProjectionSpan p₂) ∈ affineSpan ℝ (Set.range s.points)
hp₂ : p₂ = r₂ • (p -ᵥ ↑((orthogonalProjection (affineSpan ℝ (Set.range s.points))) p)) +ᵥ s.circumcenter
h : s.points 0 ∈ span_s
hd₁ :
  ‖r₁ • (p -ᵥ ↑((orthogonalProjection (affineSpan ℝ (Set.range s.points))) p))‖ *
      ‖r₁ • (p -ᵥ ↑((orthogonalProjection (affineSpan ℝ (Set.range s.points))) p))‖ =
    ‖(r₂ • (p -ᵥ ↑((orthogonalProjection (affineSpan ℝ (Set.range s.points))) p)) +ᵥ s.circumcenter) -ᵥ
          s.circumcenter‖ *
      ‖(r₂ • (p -ᵥ ↑((orthogonalProjection (affineSpan ℝ (Set.range s.points))) p)) +ᵥ s.circumcenter) -ᵥ
          s.circumcenter‖
hd₂ : dist p₂ s.circumcenter * dist p₂ s.circumcenter = r * r - s.circumradius * s.circumradius
⊢ p₁ = p₂ ∨ p₁ = (reflection (affineSpan ℝ (Set.range s.points))) p₂
Module: Mathlib.Geometry.Euclidean.Circumcenter

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
n : ℕ
s : Simplex ℝ P n
p p₁ p₂ : P
r : ℝ
h₁ : ∀ (i : Fin (n + 1)), dist (s.points i) p₁ = r
h₂ : ∀ (i : Fin (n + 1)), dist (s.points i) p₂ = r
span_s : AffineSubspace ℝ P := affineSpan ℝ (Set.range s.points)
h₁' : ↑(s.orthogonalProjectionSpan p₁) = s.circumcenter
h₂' : ↑(s.orthogonalProjectionSpan p₂) = s.circumcenter
r₁ r₂ : ℝ
hp₁o : ↑(s.orthogonalProjectionSpan p₁) ∈ affineSpan ℝ (Set.range s.points)
hp₁ : p₁ = r₁ • (p -ᵥ ↑((orthogonalProjection (affineSpan ℝ (Set.range s.points))) p)) +ᵥ s.circumcenter
hp₂o : ↑(s.orthogonalProjectionSpan p₂) ∈ affineSpan ℝ (Set.range s.points)
hp₂ : p₂ = r₂ • (p -ᵥ ↑((orthogonalProjection (affineSpan ℝ (Set.range s.points))) p)) +ᵥ s.circumcenter
h : s.points 0 ∈ span_s
hd₁ :
  ‖r₁ • (p -ᵥ ↑((orthogonalProjection (affineSpan ℝ (Set.range s.points))) p))‖ *
      ‖r₁ • (p -ᵥ ↑((orthogonalProjection (affineSpan ℝ (Set.range s.points))) p))‖ =
    ‖(r₂ • (p -ᵥ ↑((orthogonalProjection (affineSpan ℝ (Set.range s.points))) p)) +ᵥ s.circumcenter) -ᵥ
          s.circumcenter‖ *
      ‖(r₂ • (p -ᵥ ↑((orthogonalProjection (affineSpan ℝ (Set.range s.points))) p)) +ᵥ s.circumcenter) -ᵥ
          s.circumcenter‖
hd₂ : dist p₂ s.circumcenter * dist p₂ s.circumcenter = r * r - s.circumradius * s.circumradius
⊢ p₁ = p₂ ∨ p₁ = (reflection (affineSpan ℝ (Set.range s.points))) p₂

Tactic:
vadd_vsub,

Goal state after:
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
n : ℕ
s : Simplex ℝ P n
p p₁ p₂ : P
r : ℝ
h₁ : ∀ (i : Fin (n + 1)), dist (s.points i) p₁ = r
h₂ : ∀ (i : Fin (n + 1)), dist (s.points i) p₂ = r
span_s : AffineSubspace ℝ P := affineSpan ℝ (Set.range s.points)
h₁' : ↑(s.orthogonalProjectionSpan p₁) = s.circumcenter
h₂' : ↑(s.orthogonalProjectionSpan p₂) = s.circumcenter
r₁ r₂ : ℝ
hp₁o : ↑(s.orthogonalProjectionSpan p₁) ∈ affineSpan ℝ (Set.range s.points)
hp₁ : p₁ = r₁ • (p -ᵥ ↑((orthogonalProjection (affineSpan ℝ (Set.range s.points))) p)) +ᵥ s.circumcenter
hp₂o : ↑(s.orthogonalProjectionSpan p₂) ∈ affineSpan ℝ (Set.range s.points)
hp₂ : p₂ = r₂ • (p -ᵥ ↑((orthogonalProjection (affineSpan ℝ (Set.range s.points))) p)) +ᵥ s.circumcenter
h : s.points 0 ∈ span_s
hd₁ :
  ‖r₁ • (p -ᵥ ↑((orthogonalProjection (affineSpan ℝ (Set.range s.points))) p))‖ *
      ‖r₁ • (p -ᵥ ↑((orthogonalProjection (affineSpan ℝ (Set.range s.points))) p))‖ =
    ‖r₂ • (p -ᵥ ↑((orthogonalProjection (affineSpan ℝ (Set.range s.points))) p))‖ *
      ‖r₂ • (p -ᵥ ↑((orthogonalProjection (affineSpan ℝ (Set.range s.points))) p))‖
hd₂ : dist p₂ s.circumcenter * dist p₂ s.circumcenter = r * r - s.circumradius * s.circumradius
⊢ p₁ = p₂ ∨ p₁ = (reflection (affineSpan ℝ (Set.range s.points))) p₂
Module: Mathlib.Geometry.Euclidean.Inversion.Calculus

Goal state before:
pp: F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : InnerProductSpace ℝ F
c x : F
R : ℝ
hx : x ≠ c
⊢ HasFDerivAt (inversion c R) ((R / dist x c) ^ 2 • ↑↑(ℝ ∙ (x - c))ᗮ.reflection) x

Tactic:
rcases add_left_surjective c x with ⟨x, rfl⟩

Goal state after:
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : InnerProductSpace ℝ F
c : F
R : ℝ
x : F
hx : (fun x ↦ c + x) x ≠ c
⊢ HasFDerivAt (inversion c R) ((R / dist ((fun x ↦ c + x) x) c) ^ 2 • ↑↑(ℝ ∙ ((fun x ↦ c + x) x - c))ᗮ.reflection)
    ((fun x ↦ c + x) x)
Module: Mathlib.Geometry.Euclidean.NinePointCircle

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
n : ℕ
hn : NeZero n
s : Simplex ℝ P n
i : Fin (n + 1)
⊢ ⅟2 • (s.faceOppositeCentroid i -ᵥ s.circumcenter) + ⅟2 • (s.eulerPoint i -ᵥ s.circumcenter) =
    (((↑n + 1) / ↑n) • (s.centroid -ᵥ s.circumcenter) +ᵥ s.circumcenter) -ᵥ s.circumcenter

Tactic:
vadd_vsub,

Goal state after:
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
n : ℕ
hn : NeZero n
s : Simplex ℝ P n
i : Fin (n + 1)
⊢ ⅟2 • (s.faceOppositeCentroid i -ᵥ s.circumcenter) + ⅟2 • (s.eulerPoint i -ᵥ s.circumcenter) =
    ((↑n + 1) / ↑n) • (s.centroid -ᵥ s.circumcenter)
Module: Mathlib.Geometry.Euclidean.NinePointCircle

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
n : ℕ
hn : NeZero n
s : Simplex ℝ P n
i : Fin (n + 1)
hn1 : ¬n = 1
hltn : 1 < n
hnsub1 : ↑(n - 1) = ↑n - 1
⊢ ⅟2 • (s.faceOppositeCentroid i -ᵥ s.circumcenter) +
      ⅟2 •
        ((((↑n)⁻¹ • (s.points i -ᵥ ((↑(n + 1) / ↑(n - 1)) • (s.centroid -ᵥ s.circumcenter) +ᵥ s.circumcenter)) +
              (↑(n + 1) / ↑(n - 1)) • (s.centroid -ᵥ s.circumcenter)) +ᵥ
            s.circumcenter) -ᵥ
          s.circumcenter) =
    ((↑n + 1) / ↑n) • (s.centroid -ᵥ s.circumcenter)

Tactic:
vadd_vsub,

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
n : ℕ
hn : NeZero n
s : Simplex ℝ P n
i : Fin (n + 1)
hn1 : ¬n = 1
hltn : 1 < n
hnsub1 : ↑(n - 1) = ↑n - 1
⊢ ⅟2 • (s.faceOppositeCentroid i -ᵥ s.circumcenter) +
      ⅟2 •
        ((↑n)⁻¹ • (s.points i -ᵥ ((↑(n + 1) / ↑(n - 1)) • (s.centroid -ᵥ s.circumcenter) +ᵥ s.circumcenter)) +
          (↑(n + 1) / ↑(n - 1)) • (s.centroid -ᵥ s.circumcenter)) =
    ((↑n + 1) / ↑n) • (s.centroid -ᵥ s.circumcenter)
Module: Mathlib.Geometry.Euclidean.MongePoint

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
n : ℕ
s : Simplex ℝ P (n + 2)
⊢ (n + 1) •
      (((↑(n + 2 + 1) / ↑(n + 2 - 1)) • (Finset.centroid ℝ univ s.points -ᵥ s.circumcenter) +ᵥ s.circumcenter) -ᵥ
        s.circumcenter) =
    ∑ i, (s.points i -ᵥ s.circumcenter)

Tactic:
vadd_vsub,

Goal state after:
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
n : ℕ
s : Simplex ℝ P (n + 2)
⊢ (n + 1) • (↑(n + 2 + 1) / ↑(n + 2 - 1)) • (Finset.centroid ℝ univ s.points -ᵥ s.circumcenter) =
    ∑ i, (s.points i -ᵥ s.circumcenter)
Module: Mathlib.Geometry.Euclidean.MongePoint

Goal state before:
pp: case circumcenterIndex
n : ℕ
i₁ i₂ : Fin (n + 3)
h : i₁ ≠ i₂
⊢ mongePointVSubFaceCentroidWeightsWithCircumcenter i₁ i₂ circumcenterIndex =
    (mongePointWeightsWithCircumcenter n - centroidWeightsWithCircumcenter {i₁, i₂}ᶜ) circumcenterIndex

Tactic:
· simp [mongePointWeightsWithCircumcenter, centroidWeightsWithCircumcenter,
      mongePointVSubFaceCentroidWeightsWithCircumcenter]

Goal state after:
Module: Mathlib.Geometry.Euclidean.Sphere.Power

Goal state before:
pp: V : Type u_1
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
P : Type u_2
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
a b p q : P
hp : p ∈ line[ℝ, a, b]
hq : dist a q = dist b q
m : P := midpoint ℝ a b
h1 : m -ᵥ p - (m -ᵥ a) = a -ᵥ p
h1' : m -ᵥ a - (m -ᵥ p) = p -ᵥ a
h2 : m -ᵥ q - (m -ᵥ p) = p -ᵥ q
h3 : m -ᵥ q - (m -ᵥ a) = a -ᵥ q
h : ∀ (r : P), b -ᵥ r = m -ᵥ r + (m -ᵥ a)
⊢ ‖m -ᵥ p - (m -ᵥ a)‖ * ‖b -ᵥ p‖ = |‖b -ᵥ q‖ ^ 2 - ‖p -ᵥ q‖ ^ 2|

Tactic:
← h2,

Goal state after:
V : Type u_1
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
P : Type u_2
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
a b p q : P
hp : p ∈ line[ℝ, a, b]
hq : dist a q = dist b q
m : P := midpoint ℝ a b
h1 : m -ᵥ p - (m -ᵥ a) = a -ᵥ p
h1' : m -ᵥ a - (m -ᵥ p) = p -ᵥ a
h2 : m -ᵥ q - (m -ᵥ p) = p -ᵥ q
h3 : m -ᵥ q - (m -ᵥ a) = a -ᵥ q
h : ∀ (r : P), b -ᵥ r = m -ᵥ r + (m -ᵥ a)
⊢ ‖m -ᵥ p - (m -ᵥ a)‖ * ‖b -ᵥ p‖ = |‖b -ᵥ q‖ ^ 2 - ‖m -ᵥ q - (m -ᵥ p)‖ ^ 2|
Module: Mathlib.Geometry.Euclidean.Similarity

Goal state before:
pp: V₁ : Type u_2
V₂ : Type u_3
P₁ : Type u_4
P₂ : Type u_5
inst✝⁷ : NormedAddCommGroup V₁
inst✝⁶ : NormedAddCommGroup V₂
inst✝⁵ : InnerProductSpace ℝ V₁
inst✝⁴ : InnerProductSpace ℝ V₂
inst✝³ : MetricSpace P₁
inst✝² : MetricSpace P₂
inst✝¹ : NormedAddTorsor V₁ P₁
inst✝ : NormedAddTorsor V₂ P₂
a b c : P₁
a' b' c' : P₂
h_not_col : ¬Collinear ℝ {a, b, c}
h_not_col' : ¬Collinear ℝ {a', b', c'}
h : ∠ a b c = ∠ a' b' c'
dist_a'b' : dist a' b' ≠ 0
dist_b'c' : dist b' c' ≠ 0
k : ℝ := dist a b / dist a' b'
hd : k = dist b c / dist b' c'
hk : k = dist a b / dist a' b'
k_pos : 0 < k
h_ab : dist a b = k * dist a' b'
h_bc : dist c b = k * dist c' b'
hcos :
  dist a c * dist a c =
    k * dist a' b' * (k * dist a' b') + k * dist c' b' * (k * dist c' b') -
      2 * (k * dist a' b') * (k * dist c' b') * Real.cos (∠ a b c)
⊢ ![a, b, c] ∼ ![a', b', c']

Tactic:
field_simp at hcos

Goal state after:
V₁ : Type u_2
V₂ : Type u_3
P₁ : Type u_4
P₂ : Type u_5
inst✝⁷ : NormedAddCommGroup V₁
inst✝⁶ : NormedAddCommGroup V₂
inst✝⁵ : InnerProductSpace ℝ V₁
inst✝⁴ : InnerProductSpace ℝ V₂
inst✝³ : MetricSpace P₁
inst✝² : MetricSpace P₂
inst✝¹ : NormedAddTorsor V₁ P₁
inst✝ : NormedAddTorsor V₂ P₂
a b c : P₁
a' b' c' : P₂
h_not_col : ¬Collinear ℝ {a, b, c}
h_not_col' : ¬Collinear ℝ {a', b', c'}
h : ∠ a b c = ∠ a' b' c'
dist_a'b' : dist a' b' ≠ 0
dist_b'c' : dist b' c' ≠ 0
k : ℝ := dist a b / dist a' b'
hd : k = dist b c / dist b' c'
hk : k = dist a b / dist a' b'
k_pos : 0 < k
h_ab : dist a b = k * dist a' b'
h_bc : dist c b = k * dist c' b'
hcos : dist a c ^ 2 = k ^ 2 * (dist a' b' ^ 2 + dist c' b' ^ 2 - dist a' b' * dist c' b' * 2 * Real.cos (∠ a b c))
⊢ ![a, b, c] ∼ ![a', b', c']
Module: Mathlib.Geometry.Euclidean.Sphere.Power

Goal state before:
pp: V : Type u_1
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
P : Type u_2
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
a b p q : P
m : P := midpoint ℝ a b
hq : ‖m -ᵥ q - (m -ᵥ a)‖ = ‖m -ᵥ q + (m -ᵥ a)‖
h1 : m -ᵥ p - (m -ᵥ a) = a -ᵥ p
h1' : m -ᵥ a - (m -ᵥ p) = p -ᵥ a
h2 : m -ᵥ q - (m -ᵥ p) = p -ᵥ q
h3 : m -ᵥ q - (m -ᵥ a) = a -ᵥ q
h : ∀ (r : P), b -ᵥ r = m -ᵥ r + (m -ᵥ a)
r : ℝ
hr : r • (b -ᵥ a) = p -ᵥ a
⊢ ∃ k, m -ᵥ p = k • (m -ᵥ a)

Tactic:
rw [h, ← h1', eq_sub_iff_add_eq, ← eq_sub_iff_add_eq'] at hr

Goal state after:
V : Type u_1
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
P : Type u_2
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
a b p q : P
m : P := midpoint ℝ a b
hq : ‖m -ᵥ q - (m -ᵥ a)‖ = ‖m -ᵥ q + (m -ᵥ a)‖
h1 : m -ᵥ p - (m -ᵥ a) = a -ᵥ p
h1' : m -ᵥ a - (m -ᵥ p) = p -ᵥ a
h2 : m -ᵥ q - (m -ᵥ p) = p -ᵥ q
h3 : m -ᵥ q - (m -ᵥ a) = a -ᵥ q
h : ∀ (r : P), b -ᵥ r = m -ᵥ r + (m -ᵥ a)
r : ℝ
hr : m -ᵥ p = m -ᵥ a - r • (m -ᵥ a + (m -ᵥ a))
⊢ ∃ k, m -ᵥ p = k • (m -ᵥ a)
Module: Mathlib.Geometry.Euclidean.Sphere.SecondInter

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
s : Sphere P
p : P
v : V
hv : ¬v = 0
hv' : ⟪v, v⟫ ≠ 0
⊢ (((-2 * (-2 * ⟪v, p -ᵥ s.center⟫ + ⟪v, p -ᵥ s.center⟫) / ⟪v, v⟫) • v + (-2 * ⟪v, p -ᵥ s.center⟫ / ⟪v, v⟫) • v) +ᵥ
        p) -ᵥ
      p =
    0

Tactic:
vadd_vsub,

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
s : Sphere P
p : P
v : V
hv : ¬v = 0
hv' : ⟪v, v⟫ ≠ 0
⊢ (-2 * (-2 * ⟪v, p -ᵥ s.center⟫ + ⟪v, p -ᵥ s.center⟫) / ⟪v, v⟫) • v + (-2 * ⟪v, p -ᵥ s.center⟫ / ⟪v, v⟫) • v = 0
Module: Mathlib.Geometry.Euclidean.Sphere.Power

Goal state before:
pp: V : Type u_1
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
P : Type u_2
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
s : Sphere P
p a b : P
hp : p ∈ line[ℝ, a, b]
ha : a ∈ s
hb : b ∈ s
hp_on : p ∈ s
hq : dist a s.center = dist b s.center
⊢ |0| = 0

Tactic:
abs_zero

Goal state after:
V : Type u_1
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
P : Type u_2
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
s : Sphere P
p a b : P
hp : p ∈ line[ℝ, a, b]
ha : a ∈ s
hb : b ∈ s
hp_on : p ∈ s
hq : dist a s.center = dist b s.center
⊢ 0 = 0
Module: Mathlib.Geometry.Manifold.IsManifold.InteriorBoundary

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁵ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
H : Type u_3
inst✝² : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝¹ : TopologicalSpace M
inst✝ : ChartedSpace H M
x : M
h : I.IsInteriorPoint x
⊢ range ↑I ∈ 𝓝 (↑(extChartAt I x) x)

Tactic:
by
  rw [mem_nhds_iff]
  exact ⟨interior (range I), interior_subset, isOpen_interior, h⟩

Goal state after: