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

Goal state before:
pp: case mpr
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
s : AffineSubspace ℝ P
ps : Set P
h : ps ⊆ ↑s
inst✝¹ : Nonempty ↥s
inst✝ : s.direction.HasOrthogonalProjection
⊢ (∃ center ∈ s, ∃ radius, ∀ p ∈ ps, dist p center = radius) → Cospherical ps

Tactic:
exact fun ⟨c, _, hd⟩ => ⟨c, hd⟩

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

Goal state before:
pp: case inr
V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x y z : V
hx : ‖x‖ = 1
hy : ‖y‖ = 1
hz : ‖z‖ = 1
H : angle x y + angle y z ≤ π
H1 :
  ⟪x, z⟫ =
    Real.cos (angle y z) * Real.cos (angle x y) +
      Real.sin (angle y z) *
        (Real.sin (angle x y) * ⟪NormedSpace.normalize (ortho y x), NormedSpace.normalize (ortho y z)⟫)
⊢ Real.cos (angle x y + angle y z) ≤ Real.cos (angle x z)

Tactic:
rw [mul_comm (Real.cos (angle y z))] at H1

Goal state after:
case inr
V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x y z : V
hx : ‖x‖ = 1
hy : ‖y‖ = 1
hz : ‖z‖ = 1
H : angle x y + angle y z ≤ π
H1 :
  ⟪x, z⟫ =
    Real.cos (angle x y) * Real.cos (angle y z) +
      Real.sin (angle y z) *
        (Real.sin (angle x y) * ⟪NormedSpace.normalize (ortho y x), NormedSpace.normalize (ortho y z)⟫)
⊢ Real.cos (angle x y + angle y z) ≤ Real.cos (angle x z)
Module: Mathlib.Geometry.Euclidean.Triangle

Goal state before:
pp: case pos
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

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:
· subst h
    by_cases h₂₃ : p₂ = p₃
    · subst h₂₃
      simp at hne
    · simp_all [angle_self_of_ne]
      linarith [Real.pi_pos]

Goal state after:
case neg
V : Type u_1
P : Type u_2
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
p₁ p₂ p₃ : P
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
Module: Mathlib.Topology.MetricSpace.Similarity

Goal state before:
pp: ι : Type u_1
P₁ : Type u_3
P₂ : Type u_4
inst✝¹ : PseudoEMetricSpace P₁
inst✝ : PseudoEMetricSpace P₂
v₁ : ι → P₁
v₂ : ι → P₂
h : v₁ ≅ v₂
i₁ i₂ : ι
⊢ edist (v₁ i₁) (v₁ i₂) = ↑1 * edist (v₂ i₁) (v₂ i₂)

Tactic:
simpa using h i₁ i₂

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
m n : ℕ
s : Simplex ℝ P n
e : Fin (n + 1) ≃ Fin (m + 1)
⊢ (↑(m + 1) / ↑(m - 1)) •
        ((affineCombination ℝ univ (s.points ∘ ⇑e.symm)) (centroidWeights ℝ univ) -ᵥ s.circumcenter) +ᵥ
      s.circumcenter =
    (↑(n + 1) / ↑(n - 1)) • ((affineCombination ℝ univ s.points) (centroidWeights ℝ univ) -ᵥ s.circumcenter) +ᵥ
      s.circumcenter

Tactic:
obtain rfl : n = m := by simpa using Fintype.card_eq.2 ⟨e⟩

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
e : Fin (n + 1) ≃ Fin (n + 1)
⊢ (↑(n + 1) / ↑(n - 1)) •
        ((affineCombination ℝ univ (s.points ∘ ⇑e.symm)) (centroidWeights ℝ univ) -ᵥ s.circumcenter) +ᵥ
      s.circumcenter =
    (↑(n + 1) / ↑(n - 1)) • ((affineCombination ℝ univ s.points) (centroidWeights ℝ univ) -ᵥ s.circumcenter) +ᵥ
      s.circumcenter
Module: Mathlib.Geometry.Euclidean.MongePoint

Goal state before:
pp: case e_6.pointIndex
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)
a✝ : Fin (n + 2 + 1)
⊢ ↑(n + 2 + 1) / ↑(n + (2 - 1)) * ((if a✝ ∈ univ then (↑(#univ))⁻¹ else 0) - 0) + 0 = (↑(n + 1))⁻¹

Tactic:
(by decide : 2 - 1 = 1)

Goal state after:
case e_6.pointIndex
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)
a✝ : Fin (n + 2 + 1)
⊢ ↑(n + 2 + 1) / ↑(n + 1) * ((if a✝ ∈ univ then (↑(#univ))⁻¹ else 0) - 0) + 0 = (↑(n + 1))⁻¹
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 + (2 - 1)) * (0 - 1) + 1 = -2 / ↑(n + 1)

Tactic:
(by decide : 2 - 1 = 1)

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)
⊢ ↑(n + 2 + 1) / ↑(n + 1) * (0 - 1) + 1 = -2 / ↑(n + 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
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)
⊢ dist a p * dist b p = |dist b q ^ 2 - dist p q ^ 2|

Tactic:
rw [dist_eq_norm_vsub V]

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)
⊢ ‖a -ᵥ p‖ * dist b p = |dist b q ^ 2 - dist p q ^ 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
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)
⊢ ‖a -ᵥ p‖ * dist b p = |dist b q ^ 2 - dist p q ^ 2|

Tactic:
rw [dist_eq_norm_vsub V]

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)
⊢ ‖a -ᵥ p‖ * ‖b -ᵥ p‖ = |dist b q ^ 2 - dist p q ^ 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
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)
⊢ ‖a -ᵥ p‖ * ‖b -ᵥ p‖ = |dist b q ^ 2 - dist p q ^ 2|

Tactic:
rw [dist_eq_norm_vsub V]

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)
⊢ ‖a -ᵥ p‖ * ‖b -ᵥ p‖ = |‖b -ᵥ q‖ ^ 2 - dist p q ^ 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
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)
⊢ ‖a -ᵥ p‖ * ‖b -ᵥ p‖ = |‖b -ᵥ q‖ ^ 2 - dist p q ^ 2|

Tactic:
rw [dist_eq_norm_vsub V]

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)
⊢ ‖a -ᵥ p‖ * ‖b -ᵥ p‖ = |‖b -ᵥ q‖ ^ 2 - ‖p -ᵥ q‖ ^ 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 b' c'
hcos : dist a c * dist a c = dist a b * dist a b + dist c b * dist c b - 2 * dist a b * dist c b * Real.cos (∠ a b c)
⊢ ![a, b, c] ∼ ![a', b', c']

Tactic:
dist_comm b' _

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 * dist a c = dist a b * dist a b + dist c b * dist c b - 2 * dist a b * dist c b * Real.cos (∠ a b c)
⊢ ![a, b, c] ∼ ![a', b', c']
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
m n : ℕ
s : Simplex ℝ P (n + 2)
e : Fin (n + 3) ≃ Fin (m + 3)
i₁ i₂ : Fin (m + 3)
⊢ (s.reindex e).mongePlane i₁ i₂ = s.mongePlane (e.symm i₁) (e.symm i₂)

Tactic:
obtain rfl : n = m := by simpa using Fintype.card_eq.2 ⟨e⟩

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)
e : Fin (n + 3) ≃ Fin (n + 3)
i₁ i₂ : Fin (n + 3)
⊢ (s.reindex e).mongePlane i₁ i₂ = s.mongePlane (e.symm i₁) (e.symm i₂)
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
⊢ |s.radius ^ 2 - dist p s.center ^ 2| = 0

Tactic:
mem_sphere.mp hp_on,

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
⊢ |s.radius ^ 2 - s.radius ^ 2| = 0
Module: Mathlib.Geometry.Group.Growth.QuotientInter

Goal state before:

Tactic:
#((A ^ m).image π) * #{x ∈ A ^ n | x ∈ H}
    _ = #(((A ^ m).image π).image φ) * #{x ∈ A ^ n | x ∈ H} := by
      rw [Finset.card_image_of_injOn (f := φ) (mod_cast hφ)]
    _ ≤ #(((A ^ m).image π).image φ * {x ∈ A ^ n | x ∈ H}) := by
      rw [Finset.card_mul_iff.2]
      simp only [Set.InjOn, coe_image, coe_pow, coe_filter, Set.mem_prod, Set.mem_image,
        exists_exists_and_eq_and, Set.mem_ofPred_eq, and_imp, forall_exists_index, Prod.forall,
        Prod.mk.injEq]
      rintro _ a₁ b₁ hb₁ rfl - ha₁ _ a₂ b₂ hb₂ rfl - ha₂ hab
      have hπa₁ : π a₁ = 1 := (QuotientGroup.eq_one_iff _).2 ha₁
      have hπa₂ : π a₂ = 1 := (QuotientGroup.eq_one_iff _).2 ha₂
      have hπb : π b₁ = π b₂ := by
        simpa [hπφ, Set.mem_image_of_mem π, hb₁, hb₂, hπa₁, hπa₂] using congr(π $hab)
      simp_all
    _ ≤ #(A ^ (m + n)) := by
      gcongr
      simp only [mul_subset_iff, mem_image, exists_exists_and_eq_and, Finset.mem_filter, and_imp,
        forall_exists_index, forall_apply_eq_imp_iff₂, pow_add]
      rintro a ha b hb -
      exact mul_mem_mul (hφA <| Set.mem_image_of_mem _ <| mod_cast ha) hb

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

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
t₁ t₂ : Triangle ℝ P
i₁ i₂ i₃ j₁ j₂ j₃ : Fin 3
hi₁₂ : i₁ ≠ i₂
hi₁₃ : i₁ ≠ i₃
hi₂₃ : i₂ ≠ i₃
hj₁₂ : j₁ ≠ j₂
hj₁₃ : j₁ ≠ j₃
hj₂₃ : j₂ ≠ j₃
h₁ : t₂.points j₁ = t₁.orthocenter
h₂ : t₂.points j₂ = t₁.points i₂
h₃ : t₂.points j₃ = t₁.points i₃
⊢ t₁.points i₁ ∈ altitude t₂ j₃

Tactic:
rw [altitude_replace_orthocenter_eq_affineSpan hi₁₃ hi₁₂ hi₂₃.symm hj₁₃ hj₁₂ hj₂₃.symm h₁ h₃ h₂]

Goal state after:
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
t₁ t₂ : Triangle ℝ P
i₁ i₂ i₃ j₁ j₂ j₃ : Fin 3
hi₁₂ : i₁ ≠ i₂
hi₁₃ : i₁ ≠ i₃
hi₂₃ : i₂ ≠ i₃
hj₁₂ : j₁ ≠ j₂
hj₁₃ : j₁ ≠ j₃
hj₂₃ : j₂ ≠ j₃
h₁ : t₂.points j₁ = t₁.orthocenter
h₂ : t₂.points j₂ = t₁.points i₂
h₃ : t₂.points j₃ = t₁.points i₃
⊢ t₁.points i₁ ∈ line[ℝ, t₁.points i₁, t₁.points i₃]
Module: Mathlib.Geometry.Manifold.Algebra.SmoothFunctions

Goal state before:
pp: 𝕜 : Type u_1
inst✝¹⁵ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹⁴ : NormedAddCommGroup E
inst✝¹³ : NormedSpace 𝕜 E
E' : Type u_3
inst✝¹² : NormedAddCommGroup E'
inst✝¹¹ : NormedSpace 𝕜 E'
H : Type u_4
inst✝¹⁰ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
H' : Type u_5
inst✝⁹ : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
N : Type u_6
inst✝⁸ : TopologicalSpace N
inst✝⁷ : ChartedSpace H N
E'' : Type u_7
inst✝⁶ : NormedAddCommGroup E''
inst✝⁵ : NormedSpace 𝕜 E''
H'' : Type u_8
inst✝⁴ : TopologicalSpace H''
I'' : ModelWithCorners 𝕜 E'' H''
N' : Type u_9
inst✝³ : TopologicalSpace N'
inst✝² : ChartedSpace H'' N'
n : ℕ∞ω
V : Type u_10
inst✝¹ : NormedAddCommGroup V
inst✝ : NormedSpace 𝕜 V
c₁ c₂ : C^n⟮I, N; 𝓘(𝕜, 𝕜), 𝕜⟯
f : C^n⟮I, N; 𝓘(𝕜, V), V⟯
x : N
⊢ ((c₁ * c₂) • f) x = (c₁ • c₂ • f) x

Tactic:
exact mul_smul (c₁ x) (c₂ x) (f x)

Goal state after:
Module: Mathlib.Geometry.Manifold.IsManifold.InteriorBoundary

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
H : Type u_3
inst✝³ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace H M
n : WithTop ℕ∞
inst✝ : IsManifold I n M
e e' : OpenPartialHomeomorph M H
x : M
hn : n ≠ 0
he : e ∈ atlas H M
he' : e' ∈ atlas H M
hex : x ∈ e.source
hex' : x ∈ e'.source
hx : ↑(e.extend I) x ∈ interior (e.extend I).target
φ : PartialEquiv E E := extendCoordChange e e'
hφ : ContDiffOn 𝕜 n (↑φ) φ.source
⊢ Surjective ⇑(fderivWithin 𝕜 (↑φ) φ.source (↑(e.extend I) x)) → ↑(e'.extend I) x ∈ interior (range ↑I)

Tactic:
intro hφx'

Goal state after:
𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
H : Type u_3
inst✝³ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace H M
n : WithTop ℕ∞
inst✝ : IsManifold I n M
e e' : OpenPartialHomeomorph M H
x : M
hn : n ≠ 0
he : e ∈ atlas H M
he' : e' ∈ atlas H M
hex : x ∈ e.source
hex' : x ∈ e'.source
hx : ↑(e.extend I) x ∈ interior (e.extend I).target
φ : PartialEquiv E E := extendCoordChange e e'
hφ : ContDiffOn 𝕜 n (↑φ) φ.source
hφx' : Surjective ⇑(fderivWithin 𝕜 (↑φ) φ.source (↑(e.extend I) x))
⊢ ↑(e'.extend I) x ∈ interior (range ↑I)
Module: Mathlib.Geometry.Manifold.Algebra.LeftInvariantDerivation

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁷ : NontriviallyNormedField 𝕜
n : ℕ∞ω
E : Type u_2
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace 𝕜 E
H : Type u_3
inst✝⁴ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
G : Type u_4
inst✝³ : TopologicalSpace G
inst✝² : ChartedSpace H G
inst✝¹ : Monoid G
inst✝ : ContMDiffMul I ∞ G
g h : G
r : 𝕜
X✝ Y✝ : LeftInvariantDerivation I G
f f' : C^∞⟮I, G; 𝓘(𝕜, 𝕜), 𝕜⟯
X Y Z : LeftInvariantDerivation I G
⊢ ⁅X + Y, Z⁆ = ⁅X, Z⁆ + ⁅Y, Z⁆

Tactic:
ext1
    simp only [commutator_apply, coe_add, Pi.add_apply, map_add]
    ring

Goal state after:
Module: Mathlib.Geometry.Manifold.Algebra.LeftInvariantDerivation

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁷ : NontriviallyNormedField 𝕜
n : ℕ∞ω
E : Type u_2
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace 𝕜 E
H : Type u_3
inst✝⁴ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
G : Type u_4
inst✝³ : TopologicalSpace G
inst✝² : ChartedSpace H G
inst✝¹ : Monoid G
inst✝ : ContMDiffMul I ∞ G
g h : G
r : 𝕜
X✝ Y✝ : LeftInvariantDerivation I G
f f' : C^∞⟮I, G; 𝓘(𝕜, 𝕜), 𝕜⟯
X Y Z : LeftInvariantDerivation I G
⊢ ⁅X, Y + Z⁆ = ⁅X, Y⁆ + ⁅X, Z⁆

Tactic:
ext1
    simp only [commutator_apply, coe_add, Pi.add_apply, map_add]
    ring

Goal state after:
Module: Mathlib.Geometry.Manifold.VectorBundle.Hom

Goal state before:
pp: 𝕜 : Type u_1
B : Type u_2
F₁ : Type u_3
F₂ : Type u_4
E₁ : B → Type u_6
E₂ : B → Type u_7
inst✝²¹ : NontriviallyNormedField 𝕜
inst✝²⁰ : (x : B) → AddCommGroup (E₁ x)
inst✝¹⁹ : (x : B) → Module 𝕜 (E₁ x)
inst✝¹⁸ : NormedAddCommGroup F₁
inst✝¹⁷ : NormedSpace 𝕜 F₁
inst✝¹⁶ : TopologicalSpace (TotalSpace F₁ E₁)
inst✝¹⁵ : (x : B) → TopologicalSpace (E₁ x)
inst✝¹⁴ : (x : B) → AddCommGroup (E₂ x)
inst✝¹³ : (x : B) → Module 𝕜 (E₂ x)
inst✝¹² : NormedAddCommGroup F₂
inst✝¹¹ : NormedSpace 𝕜 F₂
inst✝¹⁰ : TopologicalSpace (TotalSpace F₂ E₂)
inst✝⁹ : (x : B) → TopologicalSpace (E₂ x)
HB : Type u_9
inst✝⁸ : TopologicalSpace HB
inst✝⁷ : TopologicalSpace B
inst✝⁶ : ChartedSpace HB B
inst✝⁵ : FiberBundle F₁ E₁
inst✝⁴ : VectorBundle 𝕜 F₁ E₁
inst✝³ : FiberBundle F₂ E₂
inst✝² : VectorBundle 𝕜 F₂ E₂
inst✝¹ : ∀ (x : B), IsTopologicalAddGroup (E₂ x)
inst✝ : ∀ (x : B), ContinuousSMul 𝕜 (E₂ x)
y₀ y : TotalSpace (F₁ →L[𝕜] F₂) fun b ↦ E₁ b →L[𝕜] E₂ b
⊢ ↑(chartAt (ModelProd HB (F₁ →L[𝕜] F₂)) y₀) y =
    (↑(chartAt HB y₀.proj) y.proj, inCoordinates F₁ E₁ F₂ E₂ y₀.proj y.proj y₀.proj y.proj y.snd)

Tactic:
by
  rw [FiberBundle.chartedSpace_chartAt, trans_apply, OpenPartialHomeomorph.prod_apply,
    Trivialization.coe_coe, OpenPartialHomeomorph.refl_apply, Function.id_def,
    hom_trivializationAt_apply]

Goal state after:
Module: Mathlib.Geometry.Manifold.VectorBundle.MDifferentiable

Goal state before:
pp: case refine_1
𝕜 : Type u_1
inst✝¹¹ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : NormedSpace 𝕜 E
H : Type u_3
inst✝⁸ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝⁷ : TopologicalSpace M
inst✝⁶ : ChartedSpace H M
F : Type u_5
inst✝⁵ : NormedAddCommGroup F
V : M → Type u_6
inst✝⁴ : TopologicalSpace (TotalSpace F V)
inst✝³ : (x : M) → AddCommGroup (V x)
inst✝² : (x : M) → TopologicalSpace (V x)
inst✝¹ : FiberBundle F V
inst✝ : NormedSpace 𝕜 F
k : WithTop ℕ∞
x✝ : M
σ₀ : V x✝
t : Trivialization F TotalSpace.proj := trivializationAt F V x✝
w : F := (↑t ⟨x✝, σ₀⟩).2
this : ContMDiffAt I 𝓘(𝕜, F) k (fun _x ↦ w) x✝
x : M
hx : x ∈ t.baseSet
⊢ (fun x ↦ (↑t ⟨x, extend F σ₀ x⟩).2) x = (fun _x ↦ w) x

Tactic:
simp [extend, t, hx, w]

Goal state after:
Module: Mathlib.Geometry.Manifold.MFDeriv.NormedSpace

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁷ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace 𝕜 E
H : Type u_3
inst✝⁴ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace H M
F : Type u_8
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
x : M
⊢ MDiffAt 0 x

Tactic:
exact mdifferentiable_const ..

Goal state after:
Module: Mathlib.Geometry.Manifold.VectorField.LieBracket

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁵ : NontriviallyNormedField 𝕜
H : Type u_2
inst✝⁴ : TopologicalSpace H
E : Type u_3
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝¹ : TopologicalSpace M
inst✝ : ChartedSpace H M
V : (x : M) → TangentSpace I x
x : M
⊢ mlieBracketWithin I V V univ x = 0 x

Tactic:
mlieBracketWithin_self,

Goal state after:
𝕜 : Type u_1
inst✝⁵ : NontriviallyNormedField 𝕜
H : Type u_2
inst✝⁴ : TopologicalSpace H
E : Type u_3
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝¹ : TopologicalSpace M
inst✝ : ChartedSpace H M
V : (x : M) → TangentSpace I x
x : M
⊢ 0 univ x = 0 x
Module: Mathlib.Geometry.Manifold.GroupLieAlgebra

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁷ : NontriviallyNormedField 𝕜
H : Type u_2
inst✝⁶ : TopologicalSpace H
E : Type u_3
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
I : ModelWithCorners 𝕜 E H
G : Type u_4
inst✝³ : TopologicalSpace G
inst✝² : ChartedSpace H G
inst✝¹ : Group G
inst✝ : LieGroup I (minSmoothness 𝕜 3) G
g h : G
M : minSmoothness 𝕜 3 ≠ 0
A : (mfderiv% fun x ↦ g⁻¹ * x) (g * h) ∘SL (mfderiv% fun x ↦ g * x) h = ContinuousLinearMap.id 𝕜 (TangentSpace I h)
⊢ ((mfderiv% fun b ↦ g * b) h).inverse = (mfderiv% fun b ↦ g⁻¹ * b) (g * h)

Tactic:
have A' : mfderiv% ((fun x ↦ g * x) ∘ (fun x ↦ g⁻¹ * x)) (g * h) =
      ContinuousLinearMap.id _ _ := by
    have : (fun x ↦ g * x) ∘ (fun x ↦ g⁻¹ * x) = id := by ext x; simp
    rw [this, id_eq, mfderiv_id]

Goal state after:
𝕜 : Type u_1
inst✝⁷ : NontriviallyNormedField 𝕜
H : Type u_2
inst✝⁶ : TopologicalSpace H
E : Type u_3
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
I : ModelWithCorners 𝕜 E H
G : Type u_4
inst✝³ : TopologicalSpace G
inst✝² : ChartedSpace H G
inst✝¹ : Group G
inst✝ : LieGroup I (minSmoothness 𝕜 3) G
g h : G
M : minSmoothness 𝕜 3 ≠ 0
A : (mfderiv% fun x ↦ g⁻¹ * x) (g * h) ∘SL (mfderiv% fun x ↦ g * x) h = ContinuousLinearMap.id 𝕜 (TangentSpace I h)
A' : mfderiv% ((fun x ↦ g * x) ∘ fun x ↦ g⁻¹ * x) (g * h) = ContinuousLinearMap.id 𝕜 (TangentSpace I (g * h))
⊢ ((mfderiv% fun b ↦ g * b) h).inverse = (mfderiv% fun b ↦ g⁻¹ * b) (g * h)