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

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
hf2 : Fact (Module.finrank ℝ V = 2)
s : Sphere P
p : P
hpc : p ≠ s.center
⊢ (Metric.sphere s.center s.radius ∩ ↑(s.orthRadius p)).ncard ≤ 2

Tactic:
rcases lt_trichotomy (dist p s.center) s.radius with h | h | h

Goal state after:
case 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
hf2 : Fact (Module.finrank ℝ V = 2)
s : Sphere P
p : P
hpc : p ≠ s.center
h : dist p s.center < s.radius
⊢ (Metric.sphere s.center s.radius ∩ ↑(s.orthRadius p)).ncard ≤ 2

case 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
hf2 : Fact (Module.finrank ℝ V = 2)
s : Sphere P
p : P
hpc : p ≠ s.center
h : dist p s.center = s.radius
⊢ (Metric.sphere s.center s.radius ∩ ↑(s.orthRadius p)).ncard ≤ 2

case 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
hf2 : Fact (Module.finrank ℝ V = 2)
s : Sphere P
p : P
hpc : p ≠ s.center
h : s.radius < dist p s.center
⊢ (Metric.sphere s.center s.radius ∩ ↑(s.orthRadius p)).ncard ≤ 2
Module: Mathlib.Geometry.Euclidean.Angle.Sphere

Goal state before:
pp: V : Type u_3
P : Type u_4
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)
s : Sphere P
p₁ p₂ : P
hp₁ : p₁ ∈ s
hp₂ : p₂ ∈ s
h : p₁ ≠ p₂
⊢ ((∡ p₂ p₁ s.center).tan / 2) • («o».rotation ↑(π / 2)) (p₂ -ᵥ p₁) +ᵥ midpoint ℝ p₁ p₂ = s.center

Tactic:
obtain ⟨r, hr⟩ := (dist_eq_iff_eq_smul_rotation_pi_div_two_vadd_midpoint h).1
    (dist_center_eq_dist_center_of_mem_sphere hp₁ hp₂)

Goal state after:
V : Type u_3
P : Type u_4
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)
s : Sphere P
p₁ p₂ : P
hp₁ : p₁ ∈ s
hp₂ : p₂ ∈ s
h : p₁ ≠ p₂
r : ℝ
hr : r • («o».rotation ↑(π / 2)) (p₂ -ᵥ p₁) +ᵥ midpoint ℝ p₁ p₂ = s.center
⊢ ((∡ p₂ p₁ s.center).tan / 2) • («o».rotation ↑(π / 2)) (p₂ -ᵥ p₁) +ᵥ midpoint ℝ p₁ p₂ = s.center
Module: Mathlib.Geometry.Euclidean.Triangle

Goal state before:
pp: V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x y : V
hy : y ≠ 0
hx : x ≠ 0
h✝ : ↑(angle x y) = ↑(angle x (x + y) + angle y (x + y))
n : ℤ
this : -1 < n
h : π + π ≤ angle x (x + y) + angle y (x + y)
hn : 1 ≤ n
h' : π + π ≤ angle y (x + y) + angle x (x + y)
⊢ False

Tactic:
angle_le_pi

Goal state after:
V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x y : V
hy : y ≠ 0
hx : x ≠ 0
h✝ : ↑(angle x y) = ↑(angle x (x + y) + angle y (x + y))
n : ℤ
this : -1 < n
h : π + π ≤ π + angle y (x + y)
hn : 1 ≤ n
h' : π + π ≤ π + angle x (x + y)
⊢ False
Module: Mathlib.Geometry.Euclidean.Triangle

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)
x y : V
h : 2 • o.oangle x (x - y) = 2 • o.oangle (y - x) y
h0 : o.oangle x y ≠ 0
hpi : o.oangle x y ≠ ↑π
⊢ ‖x‖ = ‖y‖

Tactic:
have hs : (o.oangle x (x - y)).sign = (o.oangle (y - x) y).sign := by simp
  rw [Real.Angle.two_zsmul_eq_iff] at h
  rcases h with h | h
  · rw [← o.angle_eq_iff_oangle_eq_of_sign_eq (o.left_ne_zero_of_oangle_ne_zero h0)
      (sub_ne_zero_of_ne (o.ne_of_oangle_ne_zero h0))
      (sub_ne_zero_of_ne (o.ne_of_oangle_ne_zero h0).symm)
      (o.right_ne_zero_of_oangle_ne_zero h0) hs, angle_comm (y - x)] at h
    refine norm_eq_of_angle_sub_eq_angle_sub_rev_of_angle_ne_pi h ?_
    rw [ne_eq, ← o.oangle_eq_pi_iff_angle_eq_pi]
    exact hpi
  · rw [h, Real.Angle.sign_add_pi, SignType.neg_eq_self_iff, oangle_sign_sub_left_swap,
      o.oangle_rev, Real.Angle.sign_neg, SignType.neg_eq_zero_iff,
      Real.Angle.sign_eq_zero_iff] at hs
    simp [h0, hpi] at hs

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

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
t : Triangle ℝ P
i₁ i₂ i₃ : Fin 3
h₁₂ : i₁ ≠ i₂
h₁₃ : i₁ ≠ i₃
h₂₃ : i₂ ≠ i₃
⊢ dist (t.points i₁) (t.points i₃) / Real.sin (∠ (t.points i₁) (t.points i₂) (t.points i₃)) / 2 = Simplex.circumradius t

Tactic:
set S : AffineSubspace ℝ P := affineSpan ℝ (Set.range t.points) with hS

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
t : Triangle ℝ P
i₁ i₂ i₃ : Fin 3
h₁₂ : i₁ ≠ i₂
h₁₃ : i₁ ≠ i₃
h₂₃ : i₂ ≠ i₃
S : AffineSubspace ℝ P := affineSpan ℝ (Set.range t.points)
hS : S = affineSpan ℝ (Set.range t.points)
⊢ dist (t.points i₁) (t.points i₃) / Real.sin (∠ (t.points i₁) (t.points i₂) (t.points i₃)) / 2 = Simplex.circumradius t
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
s₁ s₂ : Simplex ℝ P n
h : Set.range s₁.points = Set.range s₂.points
hs : s₁.circumcenter ∈ affineSpan ℝ (Set.range s₂.points)
i : Fin (n + 1)
hi : s₂.points i ∈ Set.range s₁.points
⊢ dist (s₂.points i) s₁.circumcenter = s₁.circumradius

Tactic:
Set.mem_range

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
s₁ s₂ : Simplex ℝ P n
h : Set.range s₁.points = Set.range s₂.points
hs : s₁.circumcenter ∈ affineSpan ℝ (Set.range s₂.points)
i : Fin (n + 1)
hi : ∃ y, s₁.points y = s₂.points i
⊢ dist (s₂.points i) s₁.circumcenter = s₁.circumradius
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
fs : Finset (Fin (n + 1))
⊢ ∑ i, (↑fs).indicator (Function.const (Fin (n + 1)) (↑(#fs))⁻¹) i • (s.points i -ᵥ Classical.choice ⋯) +ᵥ
      Classical.choice ⋯ =
    ∑ i, (if i ∈ fs then (↑(#fs))⁻¹ else 0) • (s.points i -ᵥ Classical.choice ⋯) +ᵥ Classical.choice ⋯

Tactic:
Set.indicator_apply

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
fs : Finset (Fin (n + 1))
⊢ ∑ x, (if x ∈ ↑fs then Function.const (Fin (n + 1)) (↑(#fs))⁻¹ x else 0) • (s.points x -ᵥ Classical.choice ⋯) +ᵥ
      Classical.choice ⋯ =
    ∑ i, (if i ∈ fs then (↑(#fs))⁻¹ else 0) • (s.points i -ᵥ Classical.choice ⋯) +ᵥ Classical.choice ⋯
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
i₁ i₂ : Fin (n + 1)
h : i₁ ≠ i₂
hc : #{i₁, i₂} = 2
W : AffineSubspace ℝ P := affineSpan ℝ (s.points '' {i₁, i₂})
h_faces : ↑((orthogonalProjection W) s.circumcenter) = ↑((s.face hc).orthogonalProjectionSpan s.circumcenter)
⊢ (↑((orthogonalProjection (affineSpan ℝ (s.points '' {i₁, i₂}))) s.circumcenter) -ᵥ s.circumcenter) +ᵥ
      ↑((orthogonalProjection (affineSpan ℝ (s.points '' {i₁, i₂}))) s.circumcenter) =
    (affineCombination ℝ univ s.pointsWithCircumcenter) (reflectionCircumcenterWeightsWithCircumcenter i₁ i₂)

Tactic:
h_faces,

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
i₁ i₂ : Fin (n + 1)
h : i₁ ≠ i₂
hc : #{i₁, i₂} = 2
W : AffineSubspace ℝ P := affineSpan ℝ (s.points '' {i₁, i₂})
h_faces : ↑((orthogonalProjection W) s.circumcenter) = ↑((s.face hc).orthogonalProjectionSpan s.circumcenter)
⊢ (↑((s.face hc).orthogonalProjectionSpan s.circumcenter) -ᵥ s.circumcenter) +ᵥ
      ↑((s.face hc).orthogonalProjectionSpan s.circumcenter) =
    (affineCombination ℝ univ s.pointsWithCircumcenter) (reflectionCircumcenterWeightsWithCircumcenter i₁ i₂)
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
t : Triangle ℝ P
inst✝ : Fact (Module.finrank ℝ V = 2)
signs : Finset (Fin 3)
i₁ i₂ i₃ : Fin 3
h₁₂ : i₁ ≠ i₂
h₁₃ : i₁ ≠ i₃
h₂₃ : i₂ ≠ i₃
⊢ {t.points i₂, t.points i₃} = Set.range (Simplex.faceOpposite t i₁).points

Tactic:
have hc : {i₁}ᶜ = ({i₂, i₃} : Set (Fin 3)) := by grind

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
t : Triangle ℝ P
inst✝ : Fact (Module.finrank ℝ V = 2)
signs : Finset (Fin 3)
i₁ i₂ i₃ : Fin 3
h₁₂ : i₁ ≠ i₂
h₁₃ : i₁ ≠ i₃
h₂₃ : i₂ ≠ i₃
hc : {i₁}ᶜ = {i₂, i₃}
⊢ {t.points i₂, t.points i₃} = Set.range (Simplex.faceOpposite t i₁).points
Module: Mathlib.Geometry.Euclidean.Circumcenter

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
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 * r₁ *
      ⟪p -ᵥ ↑((orthogonalProjection (affineSpan ℝ (Set.range s.points))) p),
        p -ᵥ ↑((orthogonalProjection (affineSpan ℝ (Set.range s.points))) p)⟫ =
    r₂ * r₂ *
      ⟪p -ᵥ ↑((orthogonalProjection (affineSpan ℝ (Set.range s.points))) p),
        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
hp : ¬p = ↑(s.orthogonalProjectionSpan p)
⊢ p₁ = p₂ ∨ p₁ = (reflection (affineSpan ℝ (Set.range s.points))) p₂

Tactic:
have hz : ⟪p -ᵥ orthogonalProjection span_s p, p -ᵥ orthogonalProjection span_s p⟫ ≠ 0 := by
      simpa only [Ne, vsub_eq_zero_iff_eq, inner_self_eq_zero] using! hp
    rw [mul_left_inj' hz, mul_self_eq_mul_self_iff] at hd₁
    rw [hp₁, hp₂]
    rcases hd₁ with hd₁ | hd₁
    · left
      rw [hd₁]
    · right
      rw [hd₁, reflection_vadd_smul_vsub_orthogonalProjection p r₂ s.circumcenter_mem_affineSpan,
        neg_smul]

Goal state after:
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
m n : ℕ
s : Simplex ℝ P n
e : Fin (n + 1) ≃ Fin (m + 1)
⊢ (s.reindex e).eulerPoint = s.eulerPoint ∘ ⇑e.symm

Tactic:
have h : n = m := by simpa using Fin.equiv_iff_eq.mp ⟨e⟩
  ext i
  simp [eulerPoint, h]

Goal state after:
Module: Mathlib.Topology.MetricSpace.Similarity

Goal state before:
pp: ι : Type u_1
P₁ : Type u_3
P₂ : Type u_4
P₃ : Type u_5
v₁ : ι → P₁
v₂ : ι → P₂
inst✝⁴ : PseudoEMetricSpace P₁
inst✝³ : PseudoEMetricSpace P₂
inst✝² : PseudoEMetricSpace P₃
F : Type u_6
inst✝¹ : FunLike F P₂ P₃
inst✝ : DilationClass F P₂ P₃
f : F
⊢ v₂ ∼ v₁ ↔ v₁ ∼ v₂

Tactic:
similar_comm

Goal state after:
ι : Type u_1
P₁ : Type u_3
P₂ : Type u_4
P₃ : Type u_5
v₁ : ι → P₁
v₂ : ι → P₂
inst✝⁴ : PseudoEMetricSpace P₁
inst✝³ : PseudoEMetricSpace P₂
inst✝² : PseudoEMetricSpace P₃
F : Type u_6
inst✝¹ : FunLike F P₂ P₃
inst✝ : DilationClass F P₂ P₃
f : F
⊢ v₁ ∼ v₂ ↔ v₁ ∼ v₂
Module: Mathlib.Geometry.Euclidean.MongePoint

Goal state before:
pp: n : ℕ
⊢ ↑(n + 2 + 1) * (↑(n + 1))⁻¹ + -2 / ↑(n + 1) = 1

Tactic:
simp [field]

Goal state after:
n : ℕ
⊢ ↑n + 2 + 1 + -2 = ↑n + 1
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}
h₁ : ∠ a b c = ∠ a' b' c'
h₂ : ∠ b c a = ∠ b' c' a'
hne_pi_div_two : ∠ a b c ≠ Real.pi / 2 ∨ ∠ b c a ≠ Real.pi / 2
not_all_eq : a' ≠ b' ∨ b' ≠ c' ∨ a' ≠ c'
h_not_col' : ¬Collinear ℝ {a', b', c'}
h_pos1 : 0 < dist a b
h_pos1' : 0 < dist a' b'
h_pos2 : 0 < dist b c
h_pos2' : 0 < dist b' c'
h₃ : ∠ c a b = ∠ c' a' b'
h_sin_ne1 : Real.sin (∠ b c a) ≠ 0
h_sin_ne2 : Real.sin (∠ c a b) ≠ 0
h_sin1 : Real.sin (∠ c a b) = Real.sin (∠ b c a) * dist b c / dist a b
h_sin1' : Real.sin (∠ c a b) = Real.sin (∠ b' c' a') * dist b' c' / dist a' b'
⊢ ![a, b, c] ∼ ![a', b', c']

Tactic:
← h₂

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}
h₁ : ∠ a b c = ∠ a' b' c'
h₂ : ∠ b c a = ∠ b' c' a'
hne_pi_div_two : ∠ a b c ≠ Real.pi / 2 ∨ ∠ b c a ≠ Real.pi / 2
not_all_eq : a' ≠ b' ∨ b' ≠ c' ∨ a' ≠ c'
h_not_col' : ¬Collinear ℝ {a', b', c'}
h_pos1 : 0 < dist a b
h_pos1' : 0 < dist a' b'
h_pos2 : 0 < dist b c
h_pos2' : 0 < dist b' c'
h₃ : ∠ c a b = ∠ c' a' b'
h_sin_ne1 : Real.sin (∠ b c a) ≠ 0
h_sin_ne2 : Real.sin (∠ c a b) ≠ 0
h_sin1 : Real.sin (∠ c a b) = Real.sin (∠ b c a) * dist b c / dist a b
h_sin1' : Real.sin (∠ c a b) = Real.sin (∠ b c a) * dist b' c' / dist a' b'
⊢ ![a, b, c] ∼ ![a', b', c']
Module: Mathlib.Geometry.Euclidean.MongePoint

Goal state before:
pp: case e_6.circumcenterIndex
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 + 2 + 1) / ↑(n + 1) * (0 - 1) + 1 = -2 / ↑(n + 1)

Tactic:
simp [field]

Goal state after:
case e_6.circumcenterIndex
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)
⊢ -1 + (-2 + -↑n) + (↑n + 1) = -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}
h₁ : ∠ a b c = ∠ a' b' c'
h₂ : ∠ b c a = ∠ b' c' a'
hne_pi_div_two : ∠ a b c ≠ Real.pi / 2 ∨ ∠ b c a ≠ Real.pi / 2
not_all_eq : a' ≠ b' ∨ b' ≠ c' ∨ a' ≠ c'
h_not_col' : ¬Collinear ℝ {a', b', c'}
h_pos1 : 0 < dist a b
h_pos1' : 0 < dist a' b'
h_pos2 : 0 < dist b c
h_pos2' : 0 < dist b' c'
h₃ : ∠ c a b = ∠ c' a' b'
h_sin_ne1 : Real.sin (∠ b c a) ≠ 0
h_sin_ne2 : Real.sin (∠ c a b) ≠ 0
h_sin1 : dist a b * dist b' c' = dist b c * dist a' b'
h_sin1' : Real.sin (∠ c a b) = Real.sin (∠ b c a) * dist b' c' / dist a' b'
h_sin2 : Real.sin (∠ c a b) * (dist c' a' / dist b' c') = Real.sin (∠ c a b) * (dist c a / dist b c)
h_sin2' : Real.sin (∠ a b c) = Real.sin (∠ c a b) * dist c' a' / dist b' c'
⊢ ![a, b, c] ∼ ![a', b', c']

Tactic:
mul_right_inj' h_sin_ne2,

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}
h₁ : ∠ a b c = ∠ a' b' c'
h₂ : ∠ b c a = ∠ b' c' a'
hne_pi_div_two : ∠ a b c ≠ Real.pi / 2 ∨ ∠ b c a ≠ Real.pi / 2
not_all_eq : a' ≠ b' ∨ b' ≠ c' ∨ a' ≠ c'
h_not_col' : ¬Collinear ℝ {a', b', c'}
h_pos1 : 0 < dist a b
h_pos1' : 0 < dist a' b'
h_pos2 : 0 < dist b c
h_pos2' : 0 < dist b' c'
h₃ : ∠ c a b = ∠ c' a' b'
h_sin_ne1 : Real.sin (∠ b c a) ≠ 0
h_sin_ne2 : Real.sin (∠ c a b) ≠ 0
h_sin1 : dist a b * dist b' c' = dist b c * dist a' b'
h_sin1' : Real.sin (∠ c a b) = Real.sin (∠ b c a) * dist b' c' / dist a' b'
h_sin2 : dist c' a' / dist b' c' = dist c a / dist b c
h_sin2' : Real.sin (∠ a b c) = Real.sin (∠ c a b) * dist c' a' / dist 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 c d p : P
h : Cospherical {a, b, c, d}
hapb : ∠ a p b = π
hcpd : ∠ c p d = π
⊢ dist a p * dist b p = dist c p * dist d p

Tactic:
rw [EuclideanGeometry.angle_eq_pi_iff_sbtw] at hapb hcpd
  exact mul_dist_eq_mul_dist_of_cospherical h hapb.wbtw.mem_affineSpan hcpd.wbtw.mem_affineSpan

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
p₁ p₂ p₃ p₄ p : P
h : dist p₁ p * dist p₂ p = dist p₃ p * dist p₄ p
hp₁p₂ : ∠ p₁ p p₂ = π
hp₃p₄ : ∠ p₃ p p₄ = π
hn : ¬Collinear ℝ {p₁, p, p₃}
hp₁p₂_sbtw : Sbtw ℝ p₁ p p₂
hp₃p₄_sbtw : Sbtw ℝ p₃ p p₄
hindep : AffineIndependent ℝ ![p₁, p, p₃]
t : Affine.Triangle ℝ P := { points := ![p₁, p, p₃], independent := hindep }
ht : t = { points := ![p₁, p, p₃], independent := hindep }
⊢ Cospherical {p₁, p₂, p₃, p₄}

Tactic:
set S : AffineSubspace ℝ P := affineSpan ℝ (Set.range t.points) with hS

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
p₁ p₂ p₃ p₄ p : P
h : dist p₁ p * dist p₂ p = dist p₃ p * dist p₄ p
hp₁p₂ : ∠ p₁ p p₂ = π
hp₃p₄ : ∠ p₃ p p₄ = π
hn : ¬Collinear ℝ {p₁, p, p₃}
hp₁p₂_sbtw : Sbtw ℝ p₁ p p₂
hp₃p₄_sbtw : Sbtw ℝ p₃ p p₄
hindep : AffineIndependent ℝ ![p₁, p, p₃]
t : Affine.Triangle ℝ P := { points := ![p₁, p, p₃], independent := hindep }
ht : t = { points := ![p₁, p, p₃], independent := hindep }
S : AffineSubspace ℝ P := affineSpan ℝ (Set.range t.points)
hS : S = affineSpan ℝ (Set.range t.points)
⊢ Cospherical {p₁, p₂, p₃, p₄}
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
t p : P
s : Sphere P
ht : t ∈ s
h_dist_eq : dist p t ^ 2 = s.power p
⊢ ⟪p -ᵥ t, t -ᵥ s.center⟫ = 0

Tactic:
simp only [Sphere.power, ← mem_sphere.mp ht, dist_eq_norm_vsub V, sq,
                 ← vsub_add_vsub_cancel p t s.center] at h_dist_eq

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
t p : P
s : Sphere P
ht : t ∈ s
h_dist_eq :
  ‖p -ᵥ t‖ * ‖p -ᵥ t‖ = ‖p -ᵥ t + (t -ᵥ s.center)‖ * ‖p -ᵥ t + (t -ᵥ s.center)‖ - ‖t -ᵥ s.center‖ * ‖t -ᵥ s.center‖
⊢ ⟪p -ᵥ t, t -ᵥ s.center⟫ = 0
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
t : Triangle ℝ P
i₁ i₂ : Fin 3
h : i₁ ≠ i₂
⊢ dist (circumcenter t) ((reflection (affineSpan ℝ (t.points '' {i₁, i₂}))) t.orthocenter) = circumradius t

Tactic:
rw [EuclideanGeometry.dist_reflection, dist_comm, dist_orthocenter_reflection_circumcenter t h]

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

Goal state before:
pp: case h
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 : Set P
t : Triangle ℝ P
hto : t.orthocenter ∉ Set.range t.points
hts : s = insert t.orthocenter (Set.range t.points)
⊢ ∀ (t_1 : Triangle ℝ P), Set.range t_1.points ⊆ s → Simplex.circumradius t_1 = Simplex.circumradius t

Tactic:
intro t₂ ht₂

Goal state after:
case h
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 : Set P
t : Triangle ℝ P
hto : t.orthocenter ∉ Set.range t.points
hts : s = insert t.orthocenter (Set.range t.points)
t₂ : Triangle ℝ P
ht₂ : Set.range t₂.points ⊆ s
⊢ Simplex.circumradius t₂ = Simplex.circumradius t
Module: Mathlib.Geometry.Manifold.IsManifold.InteriorBoundary

Goal state before:
pp: 𝕜 : Type u_1
inst✝¹⁰ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace 𝕜 E
H : Type u_3
inst✝⁷ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝⁶ : TopologicalSpace M
inst✝⁵ : ChartedSpace H M
E' : Type u_5
inst✝⁴ : NormedAddCommGroup E'
inst✝³ : NormedSpace 𝕜 E'
H' : Type u_6
inst✝² : TopologicalSpace H'
N : Type u_7
inst✝¹ : TopologicalSpace N
inst✝ : ChartedSpace H' N
J : ModelWithCorners 𝕜 E' H'
p : M × N
⊢ interior (range (Prod.map ↑I ↑J)) = interior (range ↑(I.prod J))

Tactic:
modelWithCorners_prod_coe

Goal state after:
𝕜 : Type u_1
inst✝¹⁰ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace 𝕜 E
H : Type u_3
inst✝⁷ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝⁶ : TopologicalSpace M
inst✝⁵ : ChartedSpace H M
E' : Type u_5
inst✝⁴ : NormedAddCommGroup E'
inst✝³ : NormedSpace 𝕜 E'
H' : Type u_6
inst✝² : TopologicalSpace H'
N : Type u_7
inst✝¹ : TopologicalSpace N
inst✝ : ChartedSpace H' N
J : ModelWithCorners 𝕜 E' H'
p : M × N
⊢ interior (range (Prod.map ↑I ↑J)) = interior (range (Prod.map ↑I ↑J))
Module: Mathlib.Geometry.Manifold.VectorBundle.MDifferentiable

Goal state before:
pp: 𝕜 : Type u_1
B : Type u_2
F : Type u_4
E : B → Type u_6
inst✝¹³ : TopologicalSpace B
inst✝¹² : TopologicalSpace (TotalSpace F E)
inst✝¹¹ : (x : B) → TopologicalSpace (E x)
inst✝¹⁰ : NormedAddCommGroup F
inst✝⁹ : NontriviallyNormedField 𝕜
inst✝⁸ : NormedSpace 𝕜 F
inst✝⁷ : FiberBundle F E
inst✝⁶ : (x : B) → AddCommGroup (E x)
inst✝⁵ : (x : B) → Module 𝕜 (E x)
inst✝⁴ : VectorBundle 𝕜 F E
HB : Type u_7
inst✝³ : TopologicalSpace HB
inst✝² : ChartedSpace HB B
EB : Type u_8
inst✝¹ : NormedAddCommGroup EB
inst✝ : NormedSpace 𝕜 EB
I : ModelWithCorners 𝕜 EB HB
x₀ : B
ι : Type u_9
t : ι → (x : B) → E x
ht : LocallyFinite fun i ↦ {x | t i x ≠ 0}
ht' : ∀ (i : ι), MDiffAt (T% t) x₀
⊢ (MDiffAt fun x ↦ ⟨x, ∑' (i : ι), t i x⟩) x₀

Tactic:
simp_rw [← mdifferentiableWithinAt_univ] at ht' ⊢

Goal state after:
𝕜 : Type u_1
B : Type u_2
F : Type u_4
E : B → Type u_6
inst✝¹³ : TopologicalSpace B
inst✝¹² : TopologicalSpace (TotalSpace F E)
inst✝¹¹ : (x : B) → TopologicalSpace (E x)
inst✝¹⁰ : NormedAddCommGroup F
inst✝⁹ : NontriviallyNormedField 𝕜
inst✝⁸ : NormedSpace 𝕜 F
inst✝⁷ : FiberBundle F E
inst✝⁶ : (x : B) → AddCommGroup (E x)
inst✝⁵ : (x : B) → Module 𝕜 (E x)
inst✝⁴ : VectorBundle 𝕜 F E
HB : Type u_7
inst✝³ : TopologicalSpace HB
inst✝² : ChartedSpace HB B
EB : Type u_8
inst✝¹ : NormedAddCommGroup EB
inst✝ : NormedSpace 𝕜 EB
I : ModelWithCorners 𝕜 EB HB
x₀ : B
ι : Type u_9
t : ι → (x : B) → E x
ht : LocallyFinite fun i ↦ {x | t i x ≠ 0}
ht' : ∀ (i : ι), MDiffAt[univ] (T% t) x₀
⊢ (MDiffAt[univ] fun x ↦ ⟨x, ∑' (i : ι), t i x⟩) x₀