mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.FieldTheory.RatFunc.Luroth

Goal state before:
pp: K : Type u_1
inst✝ : Field K
E : IntermediateField K K⟮X⟯
h : E ≠ ⊥
this : (Polynomial.map Polynomial.C (Q₂ h)).natDegree + (Φ E).natDegree = (θ E).natDegree
⊢ (Φ E).natDegree = (θ E).natDegree

Tactic:
natDegree_map,

Goal state after:
K : Type u_1
inst✝ : Field K
E : IntermediateField K K⟮X⟯
h : E ≠ ⊥
this : (Q₂ h).natDegree + (Φ E).natDegree = (θ E).natDegree
⊢ (Φ E).natDegree = (θ E).natDegree
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)
p p₁ p₂ p₃ : P
r : ℝ
h : r ≠ 0
⊢ ∡ ((AffineMap.homothety p r) p₁) ((AffineMap.homothety p r) p₂) ((AffineMap.homothety p r) p₃) = ∡ p₁ p₂ p₃

Tactic:
by
  simp_rw [oangle, ← AffineMap.linearMap_vsub, AffineMap.homothety_linear, LinearMap.smul_apply,
    LinearMap.id_coe, id_eq]
  rcases h.lt_or_gt with hlt | hlt <;> simp [hlt, -neg_vsub_eq_vsub_rev]

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
h0 : ¬x = 0
h : ⟪x, y⟫ = 0
⊢ Real.arcsin (‖y‖ / √(‖x‖ * ‖x‖ + ‖y‖ * ‖y‖)) = Real.arcsin (‖y‖ / √(‖x‖ * ‖x‖ * (1 + ‖y‖ * ‖y‖ / (‖x‖ * ‖x‖))))

Tactic:
mul_add

Goal state after:
V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x y : V
h0 : ¬x = 0
h : ⟪x, y⟫ = 0
⊢ Real.arcsin (‖y‖ / √(‖x‖ * ‖x‖ + ‖y‖ * ‖y‖)) =
    Real.arcsin (‖y‖ / √(‖x‖ * ‖x‖ * 1 + ‖x‖ * ‖x‖ * (‖y‖ * ‖y‖ / (‖x‖ * ‖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
⊢ angle x (x + y) = Real.arctan (‖y‖ / ‖x‖)

Tactic:
by
  rw [angle_add_eq_arcsin_of_inner_eq_zero h (Or.inl h0), Real.arctan_eq_arcsin, ←
    div_mul_eq_div_div, norm_add_eq_sqrt_iff_real_inner_eq_zero.2 h]
  nth_rw 3 [← Real.sqrt_sq (norm_nonneg x)]
  rw_mod_cast [← Real.sqrt_mul (sq_nonneg _), div_pow, pow_two, pow_two, mul_add, mul_one, mul_div,
    mul_comm (‖x‖ * ‖x‖), ← mul_div, div_self (mul_self_pos.2 (norm_ne_zero_iff.2 h0)).ne', mul_one]

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

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

pp: case neg
V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x y : V
h : ⟪x, y⟫ = 0
hxy : ¬‖x + y‖ = 0
⊢ ‖x‖ / ‖x + y‖ * ‖x + y‖ = ‖x‖

Tactic:
· have h' := norm_add_sq_eq_norm_sq_add_norm_sq_real h
    rw [hxy, zero_mul, eq_comm,
      add_eq_zero_iff_of_nonneg (mul_self_nonneg ‖x‖) (mul_self_nonneg ‖y‖), mul_self_eq_zero] at h'
    simp [h'.1]

Goal state after:
case neg
V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x y : V
h : ⟪x, y⟫ = 0
hxy : ¬‖x + y‖ = 0
⊢ ‖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
⊢ Real.tan (angle x (x + -y)) = ‖y‖ / ‖x‖

Tactic:
tan_angle_add_of_inner_eq_zero h,

Goal state after:
V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x y : V
h : ⟪x, -y⟫ = 0
⊢ ‖-y‖ / ‖x‖ = ‖y‖ / ‖x‖
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 : P
⊢ (∡ p₂ p₃ p₁).sign = (∡ p₁ p₂ p₃).sign

Tactic:
by
  rw [← oangle_swap₁₂_sign, oangle_swap₁₃_sign]

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)
⊢ o.oangle y (y - x) = ↑(Real.arccos (‖y‖ / ‖y - x‖))

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]
  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)]

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

Goal state before:
pp: case refine_1
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
r : ℝ
h : ∃ g, g 0 • ![x, r • x + y] 0 + g 1 • ![x, r • x + y] 1 = 0 ∧ (g 0 ≠ 0 ∨ g 1 ≠ 0)
⊢ ∃ g, g 0 • ![x, y] 0 + g 1 • ![x, y] 1 = 0 ∧ (g 0 ≠ 0 ∨ g 1 ≠ 0)

pp: case refine_2
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
r : ℝ
h : ∃ g, g 0 • ![x, y] 0 + g 1 • ![x, y] 1 = 0 ∧ (g 0 ≠ 0 ∨ g 1 ≠ 0)
⊢ ∃ g, g 0 • ![x, r • x + y] 0 + g 1 • ![x, r • x + y] 1 = 0 ∧ (g 0 ≠ 0 ∨ g 1 ≠ 0)

Tactic:
· rcases h with ⟨m, h, hm⟩
    change m 0 • x + m 1 • (r • x + y) = 0 at h
    refine ⟨![m 0 + m 1 * r, m 1], ?_⟩
    change (m 0 + m 1 * r) • x + m 1 • y = 0 ∧ (m 0 + m 1 * r ≠ 0 ∨ m 1 ≠ 0)
    rw [smul_add, smul_smul, ← add_assoc, ← add_smul] at h
    refine ⟨h, not_and_or.1 fun h0 => ?_⟩
    obtain ⟨h0, h1⟩ := h0
    rw [h1] at h0 hm
    rw [zero_mul, add_zero] at h0
    simp [h0] at hm

Goal state after:
case refine_2
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
r : ℝ
h : ∃ g, g 0 • ![x, y] 0 + g 1 • ![x, y] 1 = 0 ∧ (g 0 ≠ 0 ∨ g 1 ≠ 0)
⊢ ∃ g, g 0 • ![x, r • x + y] 0 + g 1 • ![x, r • x + y] 1 = 0 ∧ (g 0 ≠ 0 ∨ g 1 ≠ 0)
Module: Mathlib.Geometry.Euclidean.Angle.Oriented.Basic

Goal state before:
pp: case refine_2
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
r : ℝ
h : ∃ g, g 0 • ![x, y] 0 + g 1 • ![x, y] 1 = 0 ∧ (g 0 ≠ 0 ∨ g 1 ≠ 0)
⊢ ∃ g, g 0 • ![x, r • x + y] 0 + g 1 • ![x, r • x + y] 1 = 0 ∧ (g 0 ≠ 0 ∨ g 1 ≠ 0)

Tactic:
· rcases h with ⟨m, h, hm⟩
    change m 0 • x + m 1 • y = 0 at h
    refine ⟨![m 0 - m 1 * r, m 1], ?_⟩
    change (m 0 - m 1 * r) • x + m 1 • (r • x + y) = 0 ∧ (m 0 - m 1 * r ≠ 0 ∨ m 1 ≠ 0)
    rw [sub_smul, smul_add, smul_smul, ← add_assoc, sub_add_cancel]
    refine ⟨h, not_and_or.1 fun h0 => ?_⟩
    obtain ⟨h0, h1⟩ := h0
    rw [h1] at h0 hm
    rw [zero_mul, sub_zero] at h0
    simp [h0] at hm

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

Goal state before:
pp: case neg
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 : ‖x‖ = ‖y‖
hn : ¬x = y
hs : (2 • o.oangle (y - x) y).sign = (o.oangle (y - x) y).sign
⊢ |(o.oangle (y - x) y).toReal| < π / 2

Tactic:
Real.Angle.sign_two_zsmul_eq_sign_iff

Goal state after:
case neg
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 : ‖x‖ = ‖y‖
hn : ¬x = y
hs : o.oangle (y - x) y = ↑π ∨ |(o.oangle (y - x) y).toReal| < π / 2
⊢ |(o.oangle (y - x) y).toReal| < π / 2
Module: Mathlib.Geometry.Euclidean.Angle.Bisector

Goal state before:
pp: V : Type u_1
P : Type u_2
inst✝⁵ : NormedAddCommGroup V
inst✝⁴ : InnerProductSpace ℝ V
inst✝³ : MetricSpace P
inst✝² : NormedAddTorsor V P
inst✝¹ : Fact (finrank ℝ V = 2)
inst✝ : Oriented ℝ V (Fin 2)
p p' : P
s₁ s₂ : AffineSubspace ℝ P
hp'₁ : p' ∈ s₁
hp'₂ : p' ∈ s₂
⊢ ↑((orthogonalProjection s₁) p) ≠ p' →
    ↑((orthogonalProjection s₂) p) ≠ p' →
      ∡ (↑((orthogonalProjection s₁) p)) p' p = ∡ p p' ↑((orthogonalProjection s₂) p) →
        dist p ↑((orthogonalProjection s₁) p) = dist p ↑((orthogonalProjection s₂) p)

Tactic:
intro hp₁ hp₂ h

Goal state after:
V : Type u_1
P : Type u_2
inst✝⁵ : NormedAddCommGroup V
inst✝⁴ : InnerProductSpace ℝ V
inst✝³ : MetricSpace P
inst✝² : NormedAddTorsor V P
inst✝¹ : Fact (finrank ℝ V = 2)
inst✝ : Oriented ℝ V (Fin 2)
p p' : P
s₁ s₂ : AffineSubspace ℝ P
hp'₁ : p' ∈ s₁
hp'₂ : p' ∈ s₂
hp₁ : ↑((orthogonalProjection s₁) p) ≠ p'
hp₂ : ↑((orthogonalProjection s₂) p) ≠ p'
h : ∡ (↑((orthogonalProjection s₁) p)) p' p = ∡ p p' ↑((orthogonalProjection s₂) p)
⊢ dist p ↑((orthogonalProjection s₁) p) = dist p ↑((orthogonalProjection s₂) p)
Module: Mathlib.Geometry.Euclidean.PerpBisector

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
c p₁ p₂ : P
⊢ ⟪c -ᵥ p₁, p₂ -ᵥ p₁⟫ = 2⁻¹ * ⟪p₂ -ᵥ p₁, p₂ -ᵥ p₁⟫ ↔ ⟪c -ᵥ p₁, p₂ -ᵥ p₁⟫ = dist p₁ p₂ ^ 2 / 2

Tactic:
real_inner_self_eq_norm_sq,

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
c p₁ p₂ : P
⊢ ⟪c -ᵥ p₁, p₂ -ᵥ p₁⟫ = 2⁻¹ * ‖p₂ -ᵥ p₁‖ ^ 2 ↔ ⟪c -ᵥ p₁, p₂ -ᵥ p₁⟫ = dist p₁ p₂ ^ 2 / 2
Module: Mathlib.Geometry.Euclidean.Angle.Bisector

Goal state before:
pp: V : Type u_1
P : Type u_2
inst✝⁵ : NormedAddCommGroup V
inst✝⁴ : InnerProductSpace ℝ V
inst✝³ : MetricSpace P
inst✝² : NormedAddTorsor V P
inst✝¹ : Fact (finrank ℝ V = 2)
inst✝ : Oriented ℝ V (Fin 2)
p p' : P
s₁ s₂ : AffineSubspace ℝ P
hp'₁ : p' ∈ s₁
hp'₂ : p' ∈ s₂
⊢ ↑((orthogonalProjection s₁) p) ≠ p' →
    ↑((orthogonalProjection s₂) p) ≠ p' →
      2 • ∡ (↑((orthogonalProjection s₁) p)) p' p = 2 • ∡ p p' ↑((orthogonalProjection s₂) p) →
        dist p ↑((orthogonalProjection s₁) p) = dist p ↑((orthogonalProjection s₂) p)

Tactic:
intro hp₁ hp₂ h

Goal state after:
V : Type u_1
P : Type u_2
inst✝⁵ : NormedAddCommGroup V
inst✝⁴ : InnerProductSpace ℝ V
inst✝³ : MetricSpace P
inst✝² : NormedAddTorsor V P
inst✝¹ : Fact (finrank ℝ V = 2)
inst✝ : Oriented ℝ V (Fin 2)
p p' : P
s₁ s₂ : AffineSubspace ℝ P
hp'₁ : p' ∈ s₁
hp'₂ : p' ∈ s₂
hp₁ : ↑((orthogonalProjection s₁) p) ≠ p'
hp₂ : ↑((orthogonalProjection s₂) p) ≠ p'
h : 2 • ∡ (↑((orthogonalProjection s₁) p)) p' p = 2 • ∡ p p' ↑((orthogonalProjection s₂) p)
⊢ dist p ↑((orthogonalProjection s₁) p) = dist p ↑((orthogonalProjection s₂) p)
Module: Mathlib.Geometry.Euclidean.Sphere.Basic

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 p : P
s : Sphere P
hp₁ : p₁ ∈ s
hp₂ : p₂ ∈ s
o : P := s.center
t : ℝ
hpt : (AffineMap.lineMap p₁ p₂) t = p
ht₀ : 0 ≤ t
ht₁ : t ≤ 1
hne₁ : p ≠ p₁
hne₂ : p ≠ p₂
ht₀' : 0 < t
ht₁' : t < 1
u : V := p₁ -ᵥ o
v : V := p₂ -ᵥ o
hu : ‖u‖ = s.radius
hv : ‖v‖ = s.radius
huv : u ≠ v
hpo : p -ᵥ o = (1 - t) • u + t • v
hmem : (1 - t) • u + t • v ∈ interior (Metric.closedBall 0 s.radius)
⊢ ‖(1 - t) • u + t • v‖ < s.radius

Tactic:
interior_closedBall _ (fun h : s.radius = 0 => huv <|
      (norm_eq_zero.mp (hu.trans h)).trans (norm_eq_zero.mp (hv.trans h)).symm),

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
p₁ p₂ p : P
s : Sphere P
hp₁ : p₁ ∈ s
hp₂ : p₂ ∈ s
o : P := s.center
t : ℝ
hpt : (AffineMap.lineMap p₁ p₂) t = p
ht₀ : 0 ≤ t
ht₁ : t ≤ 1
hne₁ : p ≠ p₁
hne₂ : p ≠ p₂
ht₀' : 0 < t
ht₁' : t < 1
u : V := p₁ -ᵥ o
v : V := p₂ -ᵥ o
hu : ‖u‖ = s.radius
hv : ‖v‖ = s.radius
huv : u ≠ v
hpo : p -ᵥ o = (1 - t) • u + t • v
hmem : (1 - t) • u + t • v ∈ Metric.ball 0 s.radius
⊢ ‖(1 - t) • u + t • v‖ < s.radius
Module: Mathlib.Geometry.Euclidean.Sphere.Tangent

Goal state before:
pp: case refine_2
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
as : AffineSubspace ℝ P
p : P
h : s.IsTangentAt p as
⊢ s.radius ≤ Metric.infDist s.center ↑as

Tactic:
· rw [Metric.infDist_eq_iInf]
    have : Nonempty as := ⟨⟨p, h.mem_space⟩⟩
    refine le_ciInf fun x ↦ ?_
    rw [dist_comm]
    exact h.isTangent.radius_le_dist_center x.property

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

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
s : Sphere P
as : AffineSubspace ℝ P
h : s.IsTangent as
⊢ Metric.infDist s.center ↑as = s.radius

Tactic:
by
  obtain ⟨p, h⟩ := h
  refine le_antisymm ?_ ?_
  · convert! Metric.infDist_le_dist_of_mem h.mem_space
    rw [mem_sphere'.1 h.mem_sphere]
  · rw [Metric.infDist_eq_iInf]
    have : Nonempty as := ⟨⟨p, h.mem_space⟩⟩
    refine le_ciInf fun x ↦ ?_
    rw [dist_comm]
    exact h.isTangent.radius_le_dist_center x.property

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

Goal state before:
pp: case mp
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 q : P
h : Metric.sphere s.center s.radius ∩ ↑(s.orthRadius p) = {q}
hq : q ∈ Metric.sphere s.center s.radius ∩ ↑(s.orthRadius p)
hr : 0 ≤ s.radius
h' : ∀ (q' : P), q' -ᵥ p ∈ (ℝ ∙ (p -ᵥ s.center))ᗮ ∧ ‖q' -ᵥ p‖ ^ 2 = s.radius ^ 2 - dist p s.center ^ 2 ↔ q' = q
hq' : q -ᵥ p ∈ (ℝ ∙ (p -ᵥ s.center))ᗮ ∧ ‖q -ᵥ p‖ ^ 2 = s.radius ^ 2 - dist p s.center ^ 2
hq'' : (-(q -ᵥ p) +ᵥ p) -ᵥ p ∈ (ℝ ∙ (p -ᵥ s.center))ᗮ ∧ ‖(-(q -ᵥ p) +ᵥ p) -ᵥ p‖ ^ 2 = s.radius ^ 2 - dist p s.center ^ 2
hqq : q = p
⊢ dist p s.center = s.radius

Tactic:
subst hqq

Goal state after:
case mp
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
q : P
hr : 0 ≤ s.radius
h : Metric.sphere s.center s.radius ∩ ↑(s.orthRadius q) = {q}
hq : q ∈ Metric.sphere s.center s.radius ∩ ↑(s.orthRadius q)
h' : ∀ (q' : P), q' -ᵥ q ∈ (ℝ ∙ (q -ᵥ s.center))ᗮ ∧ ‖q' -ᵥ q‖ ^ 2 = s.radius ^ 2 - dist q s.center ^ 2 ↔ q' = q
hq' : q -ᵥ q ∈ (ℝ ∙ (q -ᵥ s.center))ᗮ ∧ ‖q -ᵥ q‖ ^ 2 = s.radius ^ 2 - dist q s.center ^ 2
hq'' : (-(q -ᵥ q) +ᵥ q) -ᵥ q ∈ (ℝ ∙ (q -ᵥ s.center))ᗮ ∧ ‖(-(q -ᵥ q) +ᵥ q) -ᵥ q‖ ^ 2 = s.radius ^ 2 - dist q s.center ^ 2
⊢ dist q s.center = s.radius
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
p₁ p₂ p₃ : P
s : Sphere P
hd : s.IsDiameter p₁ p₃
o : P := s.center
h_center : o = midpoint ℝ p₁ p₃
h_opp : p₁ -ᵥ o = -(p₃ -ᵥ o)
⊢ -⟪p₃ -ᵥ o, p₃ -ᵥ o⟫_ℝ + -⟪p₃ -ᵥ o, o -ᵥ p₂⟫_ℝ + (⟪o -ᵥ p₂, p₃ -ᵥ o⟫_ℝ + ⟪o -ᵥ p₂, o -ᵥ p₂⟫_ℝ) = 0 ↔
    dist s.center p₂ = s.radius

Tactic:
real_inner_comm (p₃ -ᵥ o) (o -ᵥ p₂)

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
p₁ p₂ p₃ : P
s : Sphere P
hd : s.IsDiameter p₁ p₃
o : P := s.center
h_center : o = midpoint ℝ p₁ p₃
h_opp : p₁ -ᵥ o = -(p₃ -ᵥ o)
⊢ -⟪p₃ -ᵥ o, p₃ -ᵥ o⟫_ℝ + -⟪p₃ -ᵥ o, o -ᵥ p₂⟫_ℝ + (⟪p₃ -ᵥ o, o -ᵥ p₂⟫_ℝ + ⟪o -ᵥ p₂, o -ᵥ p₂⟫_ℝ) = 0 ↔
    dist s.center p₂ = s.radius
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
p₁ p₂ p₃ : P
s : Sphere P
hd : s.IsDiameter p₁ p₃
o : P := s.center
h_center : o = midpoint ℝ p₁ p₃
h_opp : p₁ -ᵥ o = -(p₃ -ᵥ o)
⊢ ⟪p₃ -ᵥ o, p₃ -ᵥ o⟫_ℝ = ⟪o -ᵥ p₂, o -ᵥ p₂⟫_ℝ ↔ dist s.center p₂ = s.radius

Tactic:
real_inner_self_eq_norm_sq,

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
p₁ p₂ p₃ : P
s : Sphere P
hd : s.IsDiameter p₁ p₃
o : P := s.center
h_center : o = midpoint ℝ p₁ p₃
h_opp : p₁ -ᵥ o = -(p₃ -ᵥ o)
⊢ ‖p₃ -ᵥ o‖ ^ 2 = ⟪o -ᵥ p₂, o -ᵥ p₂⟫_ℝ ↔ dist s.center p₂ = s.radius
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
p₁ p₂ p₃ : P
s : Sphere P
hd : s.IsDiameter p₁ p₃
o : P := s.center
h_center : o = midpoint ℝ p₁ p₃
h_opp : p₁ -ᵥ o = -(p₃ -ᵥ o)
⊢ dist p₃ o ^ 2 = ⟪o -ᵥ p₂, o -ᵥ p₂⟫_ℝ ↔ dist s.center p₂ = s.radius

Tactic:
real_inner_self_eq_norm_sq,

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
p₁ p₂ p₃ : P
s : Sphere P
hd : s.IsDiameter p₁ p₃
o : P := s.center
h_center : o = midpoint ℝ p₁ p₃
h_opp : p₁ -ᵥ o = -(p₃ -ᵥ o)
⊢ dist p₃ o ^ 2 = ‖o -ᵥ p₂‖ ^ 2 ↔ dist s.center p₂ = s.radius
Module: Mathlib.Geometry.Euclidean.Circumcenter

Goal state before:
pp: case h.right
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
s : AffineSubspace ℝ P
inst✝ : s.direction.HasOrthogonalProjection
ps : Set P
hnps : ps.Nonempty
p : P
hps : ps ⊆ ↑s
hp : p ∉ s
this : Nonempty ↥s
cc : P
cr : ℝ
hcccru : ∀ (y : Sphere P), y.center ∈ s ∧ ps ⊆ Metric.sphere y.center y.radius → y = { center := cc, radius := cr }
hcc : cc ∈ s
hcr : ps ⊆ Metric.sphere cc cr
x : ℝ := dist cc ↑((orthogonalProjection s) p)
y : ℝ := dist p ↑((orthogonalProjection s) p)
hy0 : y ≠ 0
ycc₂ : ℝ := (x * x + y * y - cr * cr) / (2 * y)
cc₂ : P := (ycc₂ / y) • (p -ᵥ ↑((orthogonalProjection s) p)) +ᵥ cc
cr₂ : ℝ := √(cr * cr + ycc₂ * ycc₂)
hpo : p = 1 • (p -ᵥ ↑((orthogonalProjection s) p)) +ᵥ ↑((orthogonalProjection s) p)
cc₃ : P
cr₃ : ℝ
hcc₃ : cc₃ ∈ affineSpan ℝ (insert p ↑s)
hcr₃ : insert p ps ⊆ Metric.sphere cc₃ cr₃
t₃ : ℝ
cc₃' : P
hcc₃' : cc₃' ∈ s
hcc₃'' : cc₃ = t₃ • (p -ᵥ ↑((orthogonalProjection s) p)) +ᵥ cc₃'
hcr₃' : ∃ r, ∀ p₁ ∈ ps, dist p₁ ↑⟨cc₃', hcc₃'⟩ = r
⊢ { center := cc₃, radius := cr₃ } = { center := cc₂, radius := cr₂ }

Tactic:
obtain ⟨cr₃', hcr₃'⟩ := hcr₃'

Goal state after:
case h.right
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
s : AffineSubspace ℝ P
inst✝ : s.direction.HasOrthogonalProjection
ps : Set P
hnps : ps.Nonempty
p : P
hps : ps ⊆ ↑s
hp : p ∉ s
this : Nonempty ↥s
cc : P
cr : ℝ
hcccru : ∀ (y : Sphere P), y.center ∈ s ∧ ps ⊆ Metric.sphere y.center y.radius → y = { center := cc, radius := cr }
hcc : cc ∈ s
hcr : ps ⊆ Metric.sphere cc cr
x : ℝ := dist cc ↑((orthogonalProjection s) p)
y : ℝ := dist p ↑((orthogonalProjection s) p)
hy0 : y ≠ 0
ycc₂ : ℝ := (x * x + y * y - cr * cr) / (2 * y)
cc₂ : P := (ycc₂ / y) • (p -ᵥ ↑((orthogonalProjection s) p)) +ᵥ cc
cr₂ : ℝ := √(cr * cr + ycc₂ * ycc₂)
hpo : p = 1 • (p -ᵥ ↑((orthogonalProjection s) p)) +ᵥ ↑((orthogonalProjection s) p)
cc₃ : P
cr₃ : ℝ
hcc₃ : cc₃ ∈ affineSpan ℝ (insert p ↑s)
hcr₃ : insert p ps ⊆ Metric.sphere cc₃ cr₃
t₃ : ℝ
cc₃' : P
hcc₃' : cc₃' ∈ s
hcc₃'' : cc₃ = t₃ • (p -ᵥ ↑((orthogonalProjection s) p)) +ᵥ cc₃'
cr₃' : ℝ
hcr₃' : ∀ p₁ ∈ ps, dist p₁ ↑⟨cc₃', hcc₃'⟩ = cr₃'
⊢ { center := cc₃, radius := cr₃ } = { center := cc₂, radius := cr₂ }
Module: Mathlib.Geometry.Euclidean.Incenter

Goal state before:
pp: V : Type u_1
P : Type u_2
inst✝⁵ : NormedAddCommGroup V
inst✝⁴ : InnerProductSpace ℝ V
inst✝³ : MetricSpace P
inst✝² : NormedAddTorsor V P
n : ℕ
inst✝¹ : NeZero n
s : Simplex ℝ P n
inst✝ : n.AtLeastTwo
i : Fin (n + 1)
⊢ SignType.sign ((∑ i_1, s.excenterWeightsUnnorm {i} i_1)⁻¹ * s.excenterWeightsUnnorm {i} i) = -1

Tactic:
sign_mul

Goal state after:
V : Type u_1
P : Type u_2
inst✝⁵ : NormedAddCommGroup V
inst✝⁴ : InnerProductSpace ℝ V
inst✝³ : MetricSpace P
inst✝² : NormedAddTorsor V P
n : ℕ
inst✝¹ : NeZero n
s : Simplex ℝ P n
inst✝ : n.AtLeastTwo
i : Fin (n + 1)
⊢ SignType.sign (∑ i_1, s.excenterWeightsUnnorm {i} i_1)⁻¹ * SignType.sign (s.excenterWeightsUnnorm {i} i) = -1
Module: Mathlib.Geometry.Euclidean.Incenter

Goal state before:
pp: V : Type u_1
P : Type u_2
inst✝⁵ : NormedAddCommGroup V
inst✝⁴ : InnerProductSpace ℝ V
inst✝³ : MetricSpace P
inst✝² : NormedAddTorsor V P
n : ℕ
inst✝¹ : NeZero n
s : Simplex ℝ P n
inst✝ : n.AtLeastTwo
i j : Fin (n + 1)
h : i ≠ j
⊢ SignType.sign ((∑ i_1, s.excenterWeightsUnnorm {i} i_1)⁻¹ * s.excenterWeightsUnnorm {i} j) = 1

Tactic:
sign_mul

Goal state after:
V : Type u_1
P : Type u_2
inst✝⁵ : NormedAddCommGroup V
inst✝⁴ : InnerProductSpace ℝ V
inst✝³ : MetricSpace P
inst✝² : NormedAddTorsor V P
n : ℕ
inst✝¹ : NeZero n
s : Simplex ℝ P n
inst✝ : n.AtLeastTwo
i j : Fin (n + 1)
h : i ≠ j
⊢ SignType.sign (∑ i_1, s.excenterWeightsUnnorm {i} i_1)⁻¹ * SignType.sign (s.excenterWeightsUnnorm {i} j) = 1
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)
⊢ SignType.sign
      (s.excenterWeights signs i * ‖s.points i -ᵥ ↑((s.faceOpposite i).orthogonalProjectionSpan (s.points i))‖) =
    SignType.sign (s.excenterWeights signs i)

Tactic:
sign_mul

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)
⊢ SignType.sign (s.excenterWeights signs i) *
      SignType.sign ‖s.points i -ᵥ ↑((s.faceOpposite i).orthogonalProjectionSpan (s.points i))‖ =
    SignType.sign (s.excenterWeights signs i)
Module: Mathlib.Geometry.Euclidean.Incenter

Goal state before:
pp: case refine_1
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
signs₁ signs₂ : Finset (Fin (n + 1))
h₁ : s.ExcenterExists signs₁
h₂ : s.ExcenterExists signs₂
h : s.excenterWeights signs₁ = s.excenterWeights signs₂
hi :
  ∀ (i : Fin (n + 1)),
    SignType.sign ((∑ i, s.excenterWeightsUnnorm signs₁ i)⁻¹ * s.excenterWeightsUnnorm signs₁ i) =
      SignType.sign ((∑ i, s.excenterWeightsUnnorm signs₂ i)⁻¹ * s.excenterWeightsUnnorm signs₂ i)
⊢ signs₁ = signs₂ ∨ signs₁ = signs₂ᶜ

Tactic:
sign_mul

Goal state after:
case refine_1
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
signs₁ signs₂ : Finset (Fin (n + 1))
h₁ : s.ExcenterExists signs₁
h₂ : s.ExcenterExists signs₂
h : s.excenterWeights signs₁ = s.excenterWeights signs₂
hi :
  ∀ (i : Fin (n + 1)),
    SignType.sign (∑ i, s.excenterWeightsUnnorm signs₁ i)⁻¹ * SignType.sign (s.excenterWeightsUnnorm signs₁ i) =
      SignType.sign (∑ i, s.excenterWeightsUnnorm signs₂ i)⁻¹ * SignType.sign (s.excenterWeightsUnnorm signs₂ i)
⊢ signs₁ = signs₂ ∨ signs₁ = signs₂ᶜ
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
s : Simplex ℝ P 0
i : Fin 1
h : s.circumcenter ∈ affineSpan ℝ (Set.range s.points)
this : Unique (Fin 1)
⊢ s.circumcenter = s.points i

Tactic:
simp only [Set.range_unique, AffineSubspace.mem_affineSpan_singleton] at h

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
s : Simplex ℝ P 0
i : Fin 1
this : Unique (Fin 1)
h : s.circumcenter = s.points default
⊢ s.circumcenter = s.points i