mathlib-initiative/mathlib-tactics huggingface dataset:
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
p : P
h₁₂ : i₁ ≠ i₂
h₂ : p ∈ altitude t i₂
i₃ : Fin 3
h₁ : p ∈ mongePlane t i₃ i₂
h₂₃ : i₂ ≠ i₃
h₁₃ : i₁ ≠ i₃
⊢ p = t.orthocenter

Tactic:
]

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 : Fin 3
p : P
h₁₂ : i₁ ≠ i₂
h₂ : p ∈ altitude t i₂
i₃ : Fin 3
h₁ : p ∈ mongePlane t i₃ i₂
h₂₃ : i₂ ≠ i₃
h₁₃ : i₁ ≠ i₃
⊢ p = t.orthocenter
Module: Mathlib.Geometry.Euclidean.Sphere.SecondInter

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
s : Sphere P
n : ℕ
inst✝ : n.AtLeastTwo
sx : Affine.Simplex ℝ P n
i : Fin (n + 1)
hi : sx.points i ∈ s
hsx : ∀ (j : Fin (n + 1)), dist (sx.points j) s.center ≤ s.radius
w : Fin (n + 1) → ℝ
hw : ∑ i, w i = 1
hw01 : ∀ (i : Fin (n + 1)), w i ∈ Set.Ioo 0 1
r : ℝ := (1 - w i)⁻¹
hrpos : 0 < r
p' : P := (AffineMap.lineMap (sx.points i) ((Finset.affineCombination ℝ Finset.univ sx.points) w)) r
hp' : p' -ᵥ sx.points i = r • ((Finset.affineCombination ℝ Finset.univ sx.points) w -ᵥ sx.points i)
this :
  (affineSpan ℝ (Set.range (sx.faceOpposite i).points)).SOppSide (sx.points i)
    (s.secondInter (sx.points i) ((Finset.affineCombination ℝ Finset.univ sx.points) w -ᵥ sx.points i))
⊢ (affineSpan ℝ (Set.range (sx.faceOpposite i).points)).SOppSide (sx.points i)
    (s.secondInter (sx.points i) ((Finset.affineCombination ℝ Finset.univ sx.points) w -ᵥ sx.points i))

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
s : Sphere P
n : ℕ
inst✝ : n.AtLeastTwo
sx : Affine.Simplex ℝ P n
i : Fin (n + 1)
hi : sx.points i ∈ s
hsx : ∀ (j : Fin (n + 1)), dist (sx.points j) s.center ≤ s.radius
w : Fin (n + 1) → ℝ
hw : ∑ i, w i = 1
hw01 : ∀ (i : Fin (n + 1)), w i ∈ Set.Ioo 0 1
r : ℝ := (1 - w i)⁻¹
hrpos : 0 < r
p' : P := (AffineMap.lineMap (sx.points i) ((Finset.affineCombination ℝ Finset.univ sx.points) w)) r
hp' : p' -ᵥ sx.points i = r • ((Finset.affineCombination ℝ Finset.univ sx.points) w -ᵥ sx.points i)
this :
  (affineSpan ℝ (Set.range (sx.faceOpposite i).points)).SOppSide (sx.points i)
    (s.secondInter (sx.points i) ((Finset.affineCombination ℝ Finset.univ sx.points) w -ᵥ sx.points i))
⊢ (affineSpan ℝ (Set.range (sx.faceOpposite i).points)).SOppSide (sx.points i)
    (s.secondInter (sx.points i) ((Finset.affineCombination ℝ Finset.univ sx.points) w -ᵥ sx.points i))
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
p : P
h₁₂ : i₁ ≠ i₂
i₃ : Fin 3
h₂ : p ∈ mongePlane t i₃ i₁
h₁ : p ∈ mongePlane t i₃ i₂
h₂₃ : i₂ ≠ i₃
h₁₃ : i₁ ≠ i₃
⊢ p = t.orthocenter

Tactic:
]

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 : Fin 3
p : P
h₁₂ : i₁ ≠ i₂
i₃ : Fin 3
h₂ : p ∈ mongePlane t i₃ i₁
h₁ : p ∈ mongePlane t i₃ i₂
h₂₃ : i₂ ≠ i₃
h₁₃ : i₁ ≠ i₃
⊢ p = t.orthocenter
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
p : P
h₁₂ : i₁ ≠ i₂
i₃ : Fin 3
h₂ : p ∈ mongePlane t i₃ i₁
h₁ : p ∈ mongePlane t i₃ i₂
h₂₃ : i₂ ≠ i₃
h₁₃ : i₁ ≠ i₃
⊢ p = mongePoint t

Tactic:
]

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 : Fin 3
p : P
h₁₂ : i₁ ≠ i₂
i₃ : Fin 3
h₂ : p ∈ mongePlane t i₃ i₁
h₁ : p ∈ mongePlane t i₃ i₂
h₂₃ : i₂ ≠ i₃
h₁₃ : i₁ ≠ i₃
⊢ p = mongePoint t
Module: Mathlib.Geometry.Euclidean.Sphere.Power

Goal state before:
pp: P : Type u_2
inst✝ : MetricSpace P
s : Sphere P
p : P
hr : 0 ≤ s.radius
⊢ dist p s.center = s.radius ↔ dist p s.center = s.radius

Tactic:
]

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

Goal state before:
pp: P : Type u_2
inst✝ : MetricSpace P
s : Sphere P
p : P
hr : 0 ≤ s.radius
⊢ s.radius < dist p s.center ↔ s.radius < dist p s.center

Tactic:
]

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

Goal state before:
pp: case pos.inr
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
r : ℝ
hr_pos : 0 < r
hdist :
  ∀ (i₁ i₂ : Fin (Nat.succ 0).succ.succ),
    dist (![a, b, c] i₁) (![a, b, c] i₂) = r * dist (![a', b', c'] i₁) (![a', b', c'] i₂)
h_ab : dist a b = r * dist a' b'
h_cb : dist c b = r * dist c' b'
h_ac : dist a c = r * dist a' c'
h_cos : dist a' b' * dist c' b' * 2 * Real.cos (∠ a' b' c') = dist a' b' * dist c' b' * 2 * Real.cos (∠ a b c)
h_cos' : dist a' c' ^ 2 = dist a' b' ^ 2 + dist c' b' ^ 2 - dist a' b' * dist c' b' * 2 * Real.cos (∠ a' b' c')
h2 : c' = b'
h_dist_cb : c = b
⊢ ∠ a b c = ∠ a' b' c'

Tactic:
]

Goal state after:
case pos.inr
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
r : ℝ
hr_pos : 0 < r
hdist :
  ∀ (i₁ i₂ : Fin (Nat.succ 0).succ.succ),
    dist (![a, b, c] i₁) (![a, b, c] i₂) = r * dist (![a', b', c'] i₁) (![a', b', c'] i₂)
h_ab : dist a b = r * dist a' b'
h_cb : dist c b = r * dist c' b'
h_ac : dist a c = r * dist a' c'
h_cos : dist a' b' * dist c' b' * 2 * Real.cos (∠ a' b' c') = dist a' b' * dist c' b' * 2 * Real.cos (∠ a b c)
h_cos' : dist a' c' ^ 2 = dist a' b' ^ 2 + dist c' b' ^ 2 - dist a' b' * dist c' b' * 2 * Real.cos (∠ a' b' c')
h2 : c' = b'
h_dist_cb : c = b
⊢ ∠ a b c = ∠ a' b' c'
Module: Mathlib.Geometry.Euclidean.Sphere.Power

Goal state before:
pp: P : Type u_2
inst✝ : MetricSpace P
s : Sphere P
p : P
hr : 0 ≤ s.radius
⊢ dist p s.center < s.radius ↔ dist p s.center < s.radius

Tactic:
]

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

Goal state before:
pp: P : Type u_2
inst✝ : MetricSpace P
s : Sphere P
p : P
hr : 0 ≤ s.radius
⊢ s.radius ≤ dist p s.center ↔ s.radius ≤ dist p s.center

Tactic:
]

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

Goal state before:
pp: P : Type u_2
inst✝ : MetricSpace P
s : Sphere P
p : P
hr : 0 ≤ s.radius
⊢ dist p s.center ≤ s.radius ↔ dist p s.center ≤ s.radius

Tactic:
]

Goal state after:
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
⊢ (-∑ i₁_1,
          ∑ i₂_1,
            (mongePointWeightsWithCircumcenter 0 - reflectionCircumcenterWeightsWithCircumcenter i₁ i₂) i₁_1 *
                (mongePointWeightsWithCircumcenter 0 - reflectionCircumcenterWeightsWithCircumcenter i₁ i₂) i₂_1 *
              (dist (pointsWithCircumcenter t i₁_1) (pointsWithCircumcenter t i₂_1) *
                dist (pointsWithCircumcenter t i₁_1) (pointsWithCircumcenter t i₂_1))) /
      2 =
    circumradius t * circumradius t

Tactic:
]

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 : Fin 3
h : i₁ ≠ i₂
⊢ (-∑ i₁_1,
          ∑ i₂_1,
            (mongePointWeightsWithCircumcenter 0 - reflectionCircumcenterWeightsWithCircumcenter i₁ i₂) i₁_1 *
                (mongePointWeightsWithCircumcenter 0 - reflectionCircumcenterWeightsWithCircumcenter i₁ i₂) i₂_1 *
              (dist (pointsWithCircumcenter t i₁_1) (pointsWithCircumcenter t i₂_1) *
                dist (pointsWithCircumcenter t i₁_1) (pointsWithCircumcenter t i₂_1))) /
      2 =
    circumradius t * circumradius t
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
⊢ s.radius = s.radius

Tactic:
]

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
s : Sphere P
p a b : P
hp : p ∈ line[ℝ, a, b]
ha : a ∈ s
hb : b ∈ s
hq : dist a s.center = dist b s.center
⊢ |dist p s.center ^ 2 - s.radius ^ 2| = |dist p s.center ^ 2 - s.radius ^ 2|

Tactic:
]

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
s : Sphere P
p a b : P
hp : p ∈ line[ℝ, a, b]
ha : a ∈ s
hb : b ∈ s
hp_on : p ∈ s
⊢ s.radius = s.radius

Tactic:
]

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
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:
]

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
s : Sphere P
p a b : P
hr : 0 ≤ s.radius
hp : p ∈ line[ℝ, a, b]
ha : a ∈ s
hb : b ∈ s
hle : s.radius ≤ dist p s.center
⊢ s.power p = s.power p

Tactic:
]

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
s : Sphere P
p a b : P
hr : 0 ≤ s.radius
hp : p ∈ line[ℝ, a, b]
ha : a ∈ s
hb : b ∈ s
hle : dist p s.center ≤ s.radius
⊢ -s.power p = -s.power p

Tactic:
]

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

Goal state before:
pp: case neg
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
r : ℝ
hr_pos : 0 < r
hdist :
  ∀ (i₁ i₂ : Fin (Nat.succ 0).succ.succ),
    dist (![a, b, c] i₁) (![a, b, c] i₂) = r * dist (![a', b', c'] i₁) (![a', b', c'] i₂)
h_ab : dist a b = r * dist a' b'
h_cb : dist c b = r * dist c' b'
h_ac : dist a c = r * dist a' c'
h_cos : Real.cos (∠ a' b' c') = Real.cos (∠ a b c)
h_cos' : dist a' c' ^ 2 = dist a' b' ^ 2 + dist c' b' ^ 2 - dist a' b' * dist c' b' * 2 * Real.cos (∠ a' b' c')
heq : ¬dist a' b' * dist c' b' * 2 = 0
⊢ ∠ a b c = ∠ a' b' c'

Tactic:
]

Goal state after:
case neg
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
r : ℝ
hr_pos : 0 < r
hdist :
  ∀ (i₁ i₂ : Fin (Nat.succ 0).succ.succ),
    dist (![a, b, c] i₁) (![a, b, c] i₂) = r * dist (![a', b', c'] i₁) (![a', b', c'] i₂)
h_ab : dist a b = r * dist a' b'
h_cb : dist c b = r * dist c' b'
h_ac : dist a c = r * dist a' c'
h_cos : Real.cos (∠ a' b' c') = Real.cos (∠ a b c)
h_cos' : dist a' c' ^ 2 = dist a' b' ^ 2 + dist c' b' ^ 2 - dist a' b' * dist c' b' * 2 * Real.cos (∠ a' b' c')
heq : ¬dist a' b' * dist c' b' * 2 = 0
⊢ ∠ 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 t p : P
s : Sphere P
ha : a ∈ s
hb : b ∈ s
hp : p ∈ line[ℝ, a, b]
h_tangent : s.IsTangentAt t line[ℝ, p, t]
hr : 0 ≤ s.radius
radius_le_dist : s.radius ≤ dist p s.center
⊢ dist p t ^ 2 = s.radius ^ 2 + dist p t ^ 2 - s.radius ^ 2

Tactic:
]

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 t p : P
s : Sphere P
ha : a ∈ s
hb : b ∈ s
hp : p ∈ line[ℝ, a, b]
h_tangent : s.IsTangentAt t line[ℝ, p, t]
hr : 0 ≤ s.radius
radius_le_dist : s.radius ≤ dist p s.center
⊢ dist p t ^ 2 = s.radius ^ 2 + dist p t ^ 2 - s.radius ^ 2
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
t p : P
h_tangent : s.IsTangentAt t line[ℝ, p, t]
⊢ s.radius ^ 2 + dist p t ^ 2 - s.radius ^ 2 = dist p t ^ 2

Tactic:
]

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
t p : P
h_tangent : s.IsTangentAt t line[ℝ, p, t]
⊢ s.radius ^ 2 + dist p t ^ 2 - s.radius ^ 2 = dist p t ^ 2
Module: Mathlib.Geometry.Euclidean.Sphere.SecondInter

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
s : Sphere P
n : ℕ
inst✝ : n.AtLeastTwo
sx : Affine.Simplex ℝ P n
i : Fin (n + 1)
hi : sx.points i ∈ s
hsx : ∀ (j : Fin (n + 1)), dist (sx.points j) s.center ≤ s.radius
w : Fin (n + 1) → ℝ
hw : ∑ i, w i = 1
hw01 : ∀ (i : Fin (n + 1)), w i ∈ Set.Ioo 0 1
r : ℝ := (1 - w i)⁻¹
hrpos : 0 < r
p' : P := (AffineMap.lineMap (sx.points i) ((Finset.affineCombination ℝ Finset.univ sx.points) w)) r
hp' : p' -ᵥ sx.points i = r • ((Finset.affineCombination ℝ Finset.univ sx.points) w -ᵥ sx.points i)
⊢ (∀ i_1 ∈ {i}ᶜ, 0 < (r • (w - Pi.single i 1) + Pi.single i 1) i_1) ∧
    ∀ i_1 ∉ {i}ᶜ, (r • (w - Pi.single i 1) + Pi.single i 1) i_1 = 0

pp: case hw
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 : Sphere P
n : ℕ
inst✝ : n.AtLeastTwo
sx : Affine.Simplex ℝ P n
i : Fin (n + 1)
hi : sx.points i ∈ s
hsx : ∀ (j : Fin (n + 1)), dist (sx.points j) s.center ≤ s.radius
w : Fin (n + 1) → ℝ
hw : ∑ i, w i = 1
hw01 : ∀ (i : Fin (n + 1)), w i ∈ Set.Ioo 0 1
r : ℝ := (1 - w i)⁻¹
hrpos : 0 < r
p' : P := (AffineMap.lineMap (sx.points i) ((Finset.affineCombination ℝ Finset.univ sx.points) w)) r
hp' : p' -ᵥ sx.points i = r • ((Finset.affineCombination ℝ Finset.univ sx.points) w -ᵥ sx.points i)
⊢ ∑ i_1, (r • (w - Pi.single i 1) + Pi.single i 1) i_1 = 1

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
s : Sphere P
n : ℕ
inst✝ : n.AtLeastTwo
sx : Affine.Simplex ℝ P n
i : Fin (n + 1)
hi : sx.points i ∈ s
hsx : ∀ (j : Fin (n + 1)), dist (sx.points j) s.center ≤ s.radius
w : Fin (n + 1) → ℝ
hw : ∑ i, w i = 1
hw01 : ∀ (i : Fin (n + 1)), w i ∈ Set.Ioo 0 1
r : ℝ := (1 - w i)⁻¹
hrpos : 0 < r
p' : P := (AffineMap.lineMap (sx.points i) ((Finset.affineCombination ℝ Finset.univ sx.points) w)) r
hp' : p' -ᵥ sx.points i = r • ((Finset.affineCombination ℝ Finset.univ sx.points) w -ᵥ sx.points i)
⊢ (∀ i_1 ∈ {i}ᶜ, 0 < (r • (w - Pi.single i 1) + Pi.single i 1) i_1) ∧
    ∀ i_1 ∉ {i}ᶜ, (r • (w - Pi.single i 1) + Pi.single i 1) i_1 = 0

case hw
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 : Sphere P
n : ℕ
inst✝ : n.AtLeastTwo
sx : Affine.Simplex ℝ P n
i : Fin (n + 1)
hi : sx.points i ∈ s
hsx : ∀ (j : Fin (n + 1)), dist (sx.points j) s.center ≤ s.radius
w : Fin (n + 1) → ℝ
hw : ∑ i, w i = 1
hw01 : ∀ (i : Fin (n + 1)), w i ∈ Set.Ioo 0 1
r : ℝ := (1 - w i)⁻¹
hrpos : 0 < r
p' : P := (AffineMap.lineMap (sx.points i) ((Finset.affineCombination ℝ Finset.univ sx.points) w)) r
hp' : p' -ᵥ sx.points i = r • ((Finset.affineCombination ℝ Finset.univ sx.points) w -ᵥ sx.points i)
⊢ ∑ i_1, (r • (w - Pi.single i 1) + Pi.single i 1) i_1 = 1
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
h_orth : ⟪p -ᵥ t, t -ᵥ s.center⟫ = 0
x : P
hx : x ∈ line[ℝ, p, t]
⊢ ⟪x -ᵥ t, t -ᵥ s.center⟫ = 0

Tactic:
]

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 : dist p t ^ 2 = s.power p
h_orth : ⟪p -ᵥ t, t -ᵥ s.center⟫ = 0
x : P
hx : x ∈ line[ℝ, p, t]
⊢ ⟪x -ᵥ t, t -ᵥ s.center⟫ = 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
t p : P
s : Sphere P
ht : t ∈ s
h_dist_eq : dist p t ^ 2 = s.power p
h_orth : ⟪p -ᵥ t, t -ᵥ s.center⟫ = 0
x : P
hx : x ∈ line[ℝ, p, t]
r : ℝ
hr : r • (p -ᵥ t) = x -ᵥ t
⊢ 0 = 0

Tactic:
]

Goal state after:
Module: Mathlib.Geometry.Group.Growth.LinearLowerBound

Goal state before:
pp: case inr
G : Type u_1
inst✝¹ : Group G
inst✝ : DecidableEq G
X : Finset G
n : ℕ
hX₁ : 1 ∈ X
hX : X.Nontrivial
hn : n ≠ 0
hXn : X ^ n = X ^ (n + 1)
⊢ ↑X ^ n = ↑(closure (↑X ^ n))

Tactic:
]

Goal state after:
case inr
G : Type u_1
inst✝¹ : Group G
inst✝ : DecidableEq G
X : Finset G
n : ℕ
hX₁ : 1 ∈ X
hX : X.Nontrivial
hn : n ≠ 0
hXn : X ^ n = X ^ (n + 1)
⊢ ↑X ^ n = ↑(closure (↑X ^ n))
Module: Mathlib.Geometry.Group.Growth.LinearLowerBound

Goal state before:
pp: case zero
G : Type u_1
inst✝¹ : Group G
inst✝ : DecidableEq G
X : Finset G
n : ℕ
hX₁ : 1 ∈ X
hX : X.Nontrivial
hn : n ≠ 0
hXn : X ^ n = X ^ (n + 1)
hn₁ : ¬n = 1
this :
  ∀ {G : Type u_1} [inst : Group G] [inst_1 : DecidableEq G] {X : Finset G} {n : ℕ},
    1 ∈ X → X.Nontrivial → n ≠ 0 → X ^ n = X ^ (n + 1) → n = 1 → ↑X = ↑(closure ↑X)
⊢ X ^ n = X ^ n

Tactic:
]

Goal state after:
Module: Mathlib.Geometry.Group.Growth.LinearLowerBound

Goal state before:
pp: case succ
G : Type u_1
inst✝¹ : Group G
inst✝ : DecidableEq G
X : Finset G
n : ℕ
hX₁ : 1 ∈ X
hX : X.Nontrivial
hn : n ≠ 0
hXn : X ^ n = X ^ n * X
hn₁ : ¬n = 1
this :
  ∀ {G : Type u_1} [inst : Group G] [inst_1 : DecidableEq G] {X : Finset G} {n : ℕ},
    1 ∈ X → X.Nontrivial → n ≠ 0 → X ^ n = X ^ (n + 1) → n = 1 → ↑X = ↑(closure ↑X)
d : ℕ
hd : X ^ (n + d) = X ^ n
⊢ X ^ (n + (d + 1)) = X ^ n

Tactic:
]

Goal state after:
case succ
G : Type u_1
inst✝¹ : Group G
inst✝ : DecidableEq G
X : Finset G
n : ℕ
hX₁ : 1 ∈ X
hX : X.Nontrivial
hn : n ≠ 0
hXn : X ^ n = X ^ n * X
hn₁ : ¬n = 1
this :
  ∀ {G : Type u_1} [inst : Group G] [inst_1 : DecidableEq G] {X : Finset G} {n : ℕ},
    1 ∈ X → X.Nontrivial → n ≠ 0 → X ^ n = X ^ (n + 1) → n = 1 → ↑X = ↑(closure ↑X)
d : ℕ
hd : X ^ (n + d) = X ^ n
⊢ X ^ (n + (d + 1)) = X ^ n
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
⊢ circumradius t = circumradius t

Tactic:
]

Goal state after:
Module: Mathlib.Geometry.Group.Growth.LinearLowerBound

Goal state before:
pp: case succ
G : Type u_1
inst✝¹ : Group G
inst✝ : DecidableEq G
X : Finset G
n : ℕ
hX₁ : 1 ∈ X
hX : X.Nontrivial
hn : n ≠ 0
hXn : X ^ n = X ^ n * X
hn₁ : ¬n = 1
this :
  ∀ {G : Type u_1} [inst : Group G] [inst_1 : DecidableEq G] {X : Finset G} {n : ℕ},
    1 ∈ X → X.Nontrivial → n ≠ 0 → X ^ n = X ^ (n + 1) → n = 1 → ↑X = ↑(closure ↑X)
d : ℕ
hd : X ^ (n + d) = X ^ n
⊢ X ^ n = X ^ n

Tactic:
]

Goal state after:
Module: Mathlib.Geometry.Group.Growth.QuotientInter

Goal state before:
pp: G : Type u_1
inst✝³ : Group G
inst✝² : DecidableEq G
H : Subgroup G
inst✝¹ : DecidablePred fun x ↦ x ∈ H
inst✝ : H.Normal
A : Finset G
m n : ℕ
π : G →* G ⧸ H := QuotientGroup.mk' H
φ : G ⧸ H → G := invFunOn (⇑π) (↑A ^ m)
hφ : Set.InjOn φ (⇑π '' ↑A ^ m)
hφA : ∀ {a : G ⧸ H}, a ∈ ⇑π '' ↑A ^ m → φ a ∈ A ^ m
hπφ : ∀ {a : G ⧸ H}, a ∈ ⇑π '' ↑A ^ m → π (φ a) = a
⊢ #(image (⇑π) (A ^ m)) * #({x ∈ A ^ n | x ∈ H}) = #(image (⇑π) (A ^ m)) * #({x ∈ A ^ n | x ∈ H})

Tactic:
]

Goal state after:
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
i : Fin 3
⊢ t.orthocenter ∈ ↑(altitude t i) ∧ t.points i ∈ ↑(altitude t i)

Tactic:
]

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
i : Fin 3
⊢ t.orthocenter ∈ ↑(altitude t i) ∧ t.points i ∈ ↑(altitude t i)
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
i : Fin 3
⊢ line[ℝ, t.orthocenter, t.points i] ≤ altitude t i

Tactic:
refine affineSpan_le_of_subset_coe ?_
  rw [Set.insert_subset_iff, Set.singleton_subset_iff]
  exact ⟨t.orthocenter_mem_altitude, t.mem_altitude i⟩

Goal state after:
Module: Mathlib.Geometry.Group.Growth.QuotientInter

Goal state before:
pp: G : Type u_1
inst✝³ : Group G
inst✝² : DecidableEq G
H : Subgroup G
inst✝¹ : DecidablePred fun x ↦ x ∈ H
inst✝ : H.Normal
A : Finset G
m n : ℕ
π : G →* G ⧸ H := QuotientGroup.mk' H
φ : G ⧸ H → G := invFunOn (⇑π) (↑A ^ m)
hφ : Set.InjOn φ (⇑π '' ↑A ^ m)
hφA : ∀ {a : G ⧸ H}, a ∈ ⇑π '' ↑A ^ m → φ a ∈ A ^ m
hπφ : ∀ {a : G ⧸ H}, a ∈ ⇑π '' ↑A ^ m → π (φ a) = a
⊢ #(image φ (image (⇑π) (A ^ m))) * #({x ∈ A ^ n | x ∈ H}) ≤ #(image φ (image (⇑π) (A ^ m))) * #({x ∈ A ^ n | x ∈ H})

pp: G : Type u_1
inst✝³ : Group G
inst✝² : DecidableEq G
H : Subgroup G
inst✝¹ : DecidablePred fun x ↦ x ∈ H
inst✝ : H.Normal
A : Finset G
m n : ℕ
π : G →* G ⧸ H := QuotientGroup.mk' H
φ : G ⧸ H → G := invFunOn (⇑π) (↑A ^ m)
hφ : Set.InjOn φ (⇑π '' ↑A ^ m)
hφA : ∀ {a : G ⧸ H}, a ∈ ⇑π '' ↑A ^ m → φ a ∈ A ^ m
hπφ : ∀ {a : G ⧸ H}, a ∈ ⇑π '' ↑A ^ m → π (φ a) = a
⊢ Set.InjOn (fun p ↦ p.1 * p.2) (↑(image φ (image (⇑π) (A ^ m))) ×ˢ ↑({x ∈ A ^ n | x ∈ H}))

Tactic:
]

Goal state after:
G : Type u_1
inst✝³ : Group G
inst✝² : DecidableEq G
H : Subgroup G
inst✝¹ : DecidablePred fun x ↦ x ∈ H
inst✝ : H.Normal
A : Finset G
m n : ℕ
π : G →* G ⧸ H := QuotientGroup.mk' H
φ : G ⧸ H → G := invFunOn (⇑π) (↑A ^ m)
hφ : Set.InjOn φ (⇑π '' ↑A ^ m)
hφA : ∀ {a : G ⧸ H}, a ∈ ⇑π '' ↑A ^ m → φ a ∈ A ^ m
hπφ : ∀ {a : G ⧸ H}, a ∈ ⇑π '' ↑A ^ m → π (φ a) = a
⊢ Set.InjOn (fun p ↦ p.1 * p.2) (↑(image φ (image (⇑π) (A ^ m))) ×ˢ ↑({x ∈ A ^ n | x ∈ H}))