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

Goal state before:
pp: case pos
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
p : P
hp : p ∈ affineSpan ℝ (Set.range s.points)
i₁ : Fin (n + 1)
h :
  ∀ (i₂ : Fin (n + 1)),
    i₂ ≠ i₁ →
      ∃ p' ∈ affineSpan ℝ (Set.range (s.faceOpposite i₁).points),
        p' ∈ affineSpan ℝ (Set.range (s.faceOpposite i₂).points) ∧
          ∠ p p' ↑((s.faceOpposite i₁).orthogonalProjectionSpan p) =
            ∠ p p' ↑((s.faceOpposite i₂).orthogonalProjectionSpan p)
i : Fin (n + 1)
hi : i = i₁
⊢ dist p ↑((s.faceOpposite i).orthogonalProjectionSpan p) = dist p ↑((s.faceOpposite i₁).orthogonalProjectionSpan p)

pp: case neg
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
p : P
hp : p ∈ affineSpan ℝ (Set.range s.points)
i₁ : Fin (n + 1)
h :
  ∀ (i₂ : Fin (n + 1)),
    i₂ ≠ i₁ →
      ∃ p' ∈ affineSpan ℝ (Set.range (s.faceOpposite i₁).points),
        p' ∈ affineSpan ℝ (Set.range (s.faceOpposite i₂).points) ∧
          ∠ p p' ↑((s.faceOpposite i₁).orthogonalProjectionSpan p) =
            ∠ p p' ↑((s.faceOpposite i₂).orthogonalProjectionSpan p)
i : Fin (n + 1)
hi : ¬i = i₁
⊢ dist p ↑((s.faceOpposite i).orthogonalProjectionSpan p) = dist p ↑((s.faceOpposite i₁).orthogonalProjectionSpan p)

Tactic:
· rw [hi]

Goal state after:
case neg
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
p : P
hp : p ∈ affineSpan ℝ (Set.range s.points)
i₁ : Fin (n + 1)
h :
  ∀ (i₂ : Fin (n + 1)),
    i₂ ≠ i₁ →
      ∃ p' ∈ affineSpan ℝ (Set.range (s.faceOpposite i₁).points),
        p' ∈ affineSpan ℝ (Set.range (s.faceOpposite i₂).points) ∧
          ∠ p p' ↑((s.faceOpposite i₁).orthogonalProjectionSpan p) =
            ∠ p p' ↑((s.faceOpposite i₂).orthogonalProjectionSpan p)
i : Fin (n + 1)
hi : ¬i = i₁
⊢ dist p ↑((s.faceOpposite i).orthogonalProjectionSpan p) = dist p ↑((s.faceOpposite i₁).orthogonalProjectionSpan p)
Module: Mathlib.Geometry.Euclidean.Sphere.Tangent

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
s₁ s₂ : Sphere P
as : AffineSubspace ℝ P
h : ∃ p ∈ as, Wbtw ℝ s₁.center p s₂.center
⊢ (∃ p ∈ as, Wbtw ℝ s₁.center p s₂.center) ∨ ∀ p ∈ as, ¬Sbtw ℝ s₁.center p s₂.center

Tactic:
exact .inl h

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

Goal state before:
pp: case e'_3
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
⊢ 1 = SignType.sign (s.excenterWeightsUnnorm {i} j)

Tactic:
· simp [excenterWeightsUnnorm, h.symm]

Goal state after:
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
⊢ dist p₁ (midpoint ℝ p₁ p₂) /
      (∡ p₂ p₁ (((∡ p₂ p₁ s.center).tan / 2) • («o».rotation ↑(π / 2)) (p₂ -ᵥ p₁) +ᵥ midpoint ℝ p₁ p₂)).cos =
    dist p₁ (((∡ p₂ p₁ s.center).tan / 2) • («o».rotation ↑(π / 2)) (p₂ -ᵥ p₁) +ᵥ midpoint ℝ p₁ p₂)

Tactic:
←
    oangle_midpoint_rev_left,

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
⊢ dist p₁ (midpoint ℝ p₁ p₂) /
      (∡ (midpoint ℝ p₁ p₂) p₁
          (((∡ p₂ p₁ s.center).tan / 2) • («o».rotation ↑(π / 2)) (p₂ -ᵥ p₁) +ᵥ midpoint ℝ p₁ p₂)).cos =
    dist p₁ (((∡ p₂ p₁ s.center).tan / 2) • («o».rotation ↑(π / 2)) (p₂ -ᵥ p₁) +ᵥ midpoint ℝ p₁ p₂)
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
fs : Finset (Fin (n + 1))
m : ℕ
hfs : #fs = m + 1
hne : m ≠ n
hm : s.excenter signs ∈ affineSpan ℝ (s.points '' ↑fs)
i : Fin (n + 1)
hi : i ∉ ↑fs
⊢ False

Tactic:
excenter_eq_affineCombination

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
fs : Finset (Fin (n + 1))
m : ℕ
hfs : #fs = m + 1
hne : m ≠ n
hm : (Finset.affineCombination ℝ Finset.univ s.points) (s.excenterWeights signs) ∈ affineSpan ℝ (s.points '' ↑fs)
i : Fin (n + 1)
hi : i ∉ ↑fs
⊢ False
Module: Mathlib.Geometry.Euclidean.Circumcenter

Goal state before:
pp: V : Type u_1
P : Type u_2
inst✝⁴ : NormedAddCommGroup V
inst✝³ : InnerProductSpace ℝ V
inst✝² : MetricSpace P
inst✝¹ : NormedAddTorsor V P
ι : Type u_3
hne : Nonempty ι
inst✝ : Finite ι
p : ι → P
ha : AffineIndependent ℝ p
val✝ : Fintype ι
m : ℕ
hm :
  ∀ {ι : Type u_3} [hne : Nonempty ι] [Finite ι] {p : ι → P},
    AffineIndependent ℝ p →
      ∀ (val : Fintype ι),
        Fintype.card ι = m + 1 →
          ∃! cs, cs.center ∈ affineSpan ℝ (Set.range p) ∧ Set.range p ⊆ Metric.sphere cs.center cs.radius
hn : Fintype.card ι = m + 1 + 1
i : ι
ι2 : Type (max 0 u_3) := { x // x ≠ i }
⊢ {x | x ≠ i}.card = m + 1

Tactic:
rw [Finset.filter_not, Finset.filter_eq' _ i, if_pos (Finset.mem_univ _),
            Finset.card_sdiff, Finset.card_univ, hn]
          simp

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

Goal state before:
pp: case intro.succ.succ
V : Type u_1
P : Type u_2
inst✝⁴ : NormedAddCommGroup V
inst✝³ : InnerProductSpace ℝ V
inst✝² : MetricSpace P
inst✝¹ : NormedAddTorsor V P
ι : Type u_3
hne : Nonempty ι
inst✝ : Finite ι
p : ι → P
ha : AffineIndependent ℝ p
val✝ : Fintype ι
m : ℕ
hn : Fintype.card ι = m + 1 + 1
i : ι
ι2 : Type (max 0 u_3) := { x // x ≠ i }
hc : Fintype.card ι2 = m + 1
this : Nonempty ι2
ha2 : AffineIndependent ℝ fun i2 ↦ p ↑i2
hm :
  ∃! cs,
    cs.center ∈ affineSpan ℝ (Set.range fun i2 ↦ p ↑i2) ∧ (Set.range fun i2 ↦ p ↑i2) ⊆ Metric.sphere cs.center cs.radius
hr : Set.range p = insert (p i) (Set.range fun i2 ↦ p ↑i2)
⊢ ∃! cs, cs.center ∈ affineSpan ℝ (Set.range p) ∧ Set.range p ⊆ Metric.sphere cs.center cs.radius

Tactic:
hr,

Goal state after:
case intro.succ.succ
V : Type u_1
P : Type u_2
inst✝⁴ : NormedAddCommGroup V
inst✝³ : InnerProductSpace ℝ V
inst✝² : MetricSpace P
inst✝¹ : NormedAddTorsor V P
ι : Type u_3
hne : Nonempty ι
inst✝ : Finite ι
p : ι → P
ha : AffineIndependent ℝ p
val✝ : Fintype ι
m : ℕ
hn : Fintype.card ι = m + 1 + 1
i : ι
ι2 : Type (max 0 u_3) := { x // x ≠ i }
hc : Fintype.card ι2 = m + 1
this : Nonempty ι2
ha2 : AffineIndependent ℝ fun i2 ↦ p ↑i2
hm :
  ∃! cs,
    cs.center ∈ affineSpan ℝ (Set.range fun i2 ↦ p ↑i2) ∧ (Set.range fun i2 ↦ p ↑i2) ⊆ Metric.sphere cs.center cs.radius
hr : Set.range p = insert (p i) (Set.range fun i2 ↦ p ↑i2)
⊢ ∃! cs,
    cs.center ∈ affineSpan ℝ (insert (p i) (Set.range fun i2 ↦ p ↑i2)) ∧
      insert (p i) (Set.range fun i2 ↦ p ↑i2) ⊆ Metric.sphere cs.center cs.radius
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 1
i : Fin 2
hi : i ∈ Set.univ
j : Fin 2
hj : j ∈ Set.univ
hij : i ≠ j
⊢ ‖s.points i -ᵥ (2⁻¹ • (s.points 1 -ᵥ s.points 0) +ᵥ s.points 0)‖ =
    ‖s.points j -ᵥ (2⁻¹ • (s.points 1 -ᵥ s.points 0) +ᵥ s.points 0)‖

Tactic:
vsub_vadd_eq_vsub_sub,

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 1
i : Fin 2
hi : i ∈ Set.univ
j : Fin 2
hj : j ∈ Set.univ
hij : i ≠ j
⊢ ‖s.points i -ᵥ s.points 0 - 2⁻¹ • (s.points 1 -ᵥ s.points 0)‖ =
    ‖s.points j -ᵥ (2⁻¹ • (s.points 1 -ᵥ s.points 0) +ᵥ s.points 0)‖
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 1
i : Fin 2
hi : i ∈ Set.univ
j : Fin 2
hj : j ∈ Set.univ
hij : i ≠ j
⊢ ‖s.points i -ᵥ s.points 0 - 2⁻¹ • (s.points 1 -ᵥ s.points 0)‖ =
    ‖s.points j -ᵥ (2⁻¹ • (s.points 1 -ᵥ s.points 0) +ᵥ s.points 0)‖

Tactic:
vsub_vadd_eq_vsub_sub,

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 1
i : Fin 2
hi : i ∈ Set.univ
j : Fin 2
hj : j ∈ Set.univ
hij : i ≠ j
⊢ ‖s.points i -ᵥ s.points 0 - 2⁻¹ • (s.points 1 -ᵥ s.points 0)‖ =
    ‖s.points j -ᵥ s.points 0 - 2⁻¹ • (s.points 1 -ᵥ s.points 0)‖
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)
⊢ ((affineCombination ℝ univ s.pointsWithCircumcenter) (centroidWeightsWithCircumcenter {i₁, i₂}) -ᵥ
        (affineCombination ℝ univ s.pointsWithCircumcenter) (circumcenterWeightsWithCircumcenter n)) +ᵥ
      (affineCombination ℝ univ s.pointsWithCircumcenter) (centroidWeightsWithCircumcenter {i₁, i₂}) =
    (affineCombination ℝ univ s.pointsWithCircumcenter) (reflectionCircumcenterWeightsWithCircumcenter i₁ i₂)

Tactic:
← @vsub_eq_zero_iff_eq V,

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)
⊢ (((affineCombination ℝ univ s.pointsWithCircumcenter) (centroidWeightsWithCircumcenter {i₁, i₂}) -ᵥ
          (affineCombination ℝ univ s.pointsWithCircumcenter) (circumcenterWeightsWithCircumcenter n)) +ᵥ
        (affineCombination ℝ univ s.pointsWithCircumcenter) (centroidWeightsWithCircumcenter {i₁, i₂})) -ᵥ
      (affineCombination ℝ univ s.pointsWithCircumcenter) (reflectionCircumcenterWeightsWithCircumcenter i₁ i₂) =
    0
Module: Mathlib.Geometry.Euclidean.Incenter

Goal state before:
pp: 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
t : Triangle ℝ P
signs : Finset (Fin 3)
i₁ i₂ i₃ : Fin 3
h₁₂ : i₁ ≠ i₂
h₁₃ : i₁ ≠ i₃
h₂₃ : i₂ ≠ i₃
h : Simplex.excenter t signs = Simplex.incenter t
⊢ Simplex.excenter t signs = Simplex.incenter t ∨
    Simplex.excenter t signs = Simplex.excenter t {i₁} ∨
      Simplex.excenter t signs = Simplex.excenter t {i₂} ∨ Simplex.excenter t signs = Simplex.excenter t {i₃}

Tactic:
exact .inl h

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

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
i : Fin (n + 1)
⊢ ‖↑n • (s.faceOppositeCentroid i -ᵥ (((↑n + 1) / ↑n) • (s.centroid -ᵥ s.circumcenter) +ᵥ s.circumcenter))‖ =
    ‖s.circumcenter -ᵥ s.points i‖

Tactic:
vsub_vadd_eq_vsub_sub,

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
i : Fin (n + 1)
⊢ ‖↑n • (s.faceOppositeCentroid i -ᵥ s.circumcenter - ((↑n + 1) / ↑n) • (s.centroid -ᵥ s.circumcenter))‖ =
    ‖s.circumcenter -ᵥ s.points i‖
Module: Mathlib.Geometry.Euclidean.NinePointCircle

Goal state before:
pp: case neg
V : Type u_1
P : Type u_2
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
n : ℕ
hn : NeZero n
s : Simplex ℝ P n
i : Fin (n + 1)
hn1 : ¬n = 1
hltn : 1 < n
hnsub1 : ↑(n - 1) = ↑n - 1
⊢ ⅟2 • (s.faceOppositeCentroid i -ᵥ s.circumcenter) +
      ⅟2 •
        ((↑n)⁻¹ • (s.points i -ᵥ ((↑(n + 1) / ↑(n - 1)) • (s.centroid -ᵥ s.circumcenter) +ᵥ s.circumcenter)) +
          (↑(n + 1) / ↑(n - 1)) • (s.centroid -ᵥ s.circumcenter)) =
    ((↑n + 1) / ↑n) • (s.centroid -ᵥ s.circumcenter)

Tactic:
vsub_vadd_eq_vsub_sub,

Goal state after:
case neg
V : Type u_1
P : Type u_2
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
n : ℕ
hn : NeZero n
s : Simplex ℝ P n
i : Fin (n + 1)
hn1 : ¬n = 1
hltn : 1 < n
hnsub1 : ↑(n - 1) = ↑n - 1
⊢ ⅟2 • (s.faceOppositeCentroid i -ᵥ s.circumcenter) +
      ⅟2 •
        ((↑n)⁻¹ • (s.points i -ᵥ s.circumcenter - (↑(n + 1) / ↑(n - 1)) • (s.centroid -ᵥ s.circumcenter)) +
          (↑(n + 1) / ↑(n - 1)) • (s.centroid -ᵥ s.circumcenter)) =
    ((↑n + 1) / ↑n) • (s.centroid -ᵥ s.circumcenter)
Module: Mathlib.Geometry.Euclidean.Sphere.Power

Goal state before:
pp: V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x y z : V
h₃ : ‖z - y‖ = ‖z + y‖
r : ℝ
hr : x = r • y
hzy : ⟪z, y⟫ = 0
⊢ ⟪z, x⟫ = 0

Tactic:
hr,

Goal state after:
V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x y z : V
h₃ : ‖z - y‖ = ‖z + y‖
r : ℝ
hr : x = r • y
hzy : ⟪z, y⟫ = 0
⊢ ⟪z, r • y⟫ = 0
Module: Mathlib.Geometry.Euclidean.Sphere.SecondInter

Goal state before:
pp: case neg
V : Type u_1
P : Type u_2
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
s : Sphere P
p : P
v : V
hv : ¬v = 0
hv' : ⟪v, v⟫ ≠ 0
⊢ ((-2 * (-2 * ⟪v, p -ᵥ s.center⟫ + ⟪v, p -ᵥ s.center⟫) / ⟪v, v⟫) • v + (-2 * ⟪v, p -ᵥ s.center⟫ / ⟪v, v⟫) • v) +ᵥ p = p

Tactic:
← @vsub_eq_zero_iff_eq V,

Goal state after:
case neg
V : Type u_1
P : Type u_2
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
s : Sphere P
p : P
v : V
hv : ¬v = 0
hv' : ⟪v, v⟫ ≠ 0
⊢ (((-2 * (-2 * ⟪v, p -ᵥ s.center⟫ + ⟪v, p -ᵥ s.center⟫) / ⟪v, v⟫) • v + (-2 * ⟪v, p -ᵥ s.center⟫ / ⟪v, v⟫) • v) +ᵥ
        p) -ᵥ
      p =
    0
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
⊢ #(image (⇑π) (A ^ m)) * #({x ∈ A ^ n | x ∈ H}) ≤ #(A ^ (m + n))

Tactic:
let φ := invFunOn π (A ^ m)

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)
⊢ #(image (⇑π) (A ^ m)) * #({x ∈ A ^ n | x ∈ H}) ≤ #(A ^ (m + n))
Module: Mathlib.Geometry.Group.Growth.LinearLowerBound

Goal state before:
pp: G : Type u_1
inst✝¹ : Group G
inst✝ : DecidableEq G
X : Finset G
hX₁ : 1 ∈ X
hXclosure : (↑(closure ↑X)).Infinite
hX : X.Nontrivial
n : ℕ
h : ↑X ^ (n - 1) = ↑(closure ↑X)
⊢ False

Tactic:
simp [← h, ← coe_pow] at hXclosure

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁵ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
H : Type u_3
inst✝² : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝¹ : TopologicalSpace M
inst✝ : ChartedSpace H M
⊢ (ModelWithCorners.boundary M)ᶜ = ModelWithCorners.interior M

Tactic:
rw [← compl_interior, compl_compl]

Goal state after:
Module: Mathlib.Geometry.Manifold.Bordism

Goal state before:
pp: X : Type u_1
Y : Type u_2
Z : Type u_3
inst✝¹⁸ : TopologicalSpace X
inst✝¹⁷ : TopologicalSpace Y
inst✝¹⁶ : TopologicalSpace Z
k : WithTop ℕ∞
E : Type u_4
H : Type u_5
M✝ : Type u_6
inst✝¹⁵ : NormedAddCommGroup E
inst✝¹⁴ : NormedSpace ℝ E
inst✝¹³ : FiniteDimensional ℝ E
inst✝¹² : TopologicalSpace H
I✝ : ModelWithCorners ℝ E H
inst✝¹¹ : TopologicalSpace M✝
inst✝¹⁰ : ChartedSpace H M✝
inst✝⁹ : IsManifold I✝ k M✝
inst✝⁸ : CompactSpace M✝
inst✝⁷ : BoundarylessManifold I✝ M✝
E' : Type u_7
H' : Type u_8
inst✝⁶ : NormedAddCommGroup E'
inst✝⁵ : NormedSpace ℝ E'
inst✝⁴ : TopologicalSpace H'
M : Type u
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace H M
I : ModelWithCorners ℝ E H
inst✝¹ : IsManifold I k M
inst✝ : IsEmpty M
⊢ Continuous fun x ↦ ⋯.elim

Tactic:
rw [continuous_iff_continuousAt]

Goal state after:
X : Type u_1
Y : Type u_2
Z : Type u_3
inst✝¹⁸ : TopologicalSpace X
inst✝¹⁷ : TopologicalSpace Y
inst✝¹⁶ : TopologicalSpace Z
k : WithTop ℕ∞
E : Type u_4
H : Type u_5
M✝ : Type u_6
inst✝¹⁵ : NormedAddCommGroup E
inst✝¹⁴ : NormedSpace ℝ E
inst✝¹³ : FiniteDimensional ℝ E
inst✝¹² : TopologicalSpace H
I✝ : ModelWithCorners ℝ E H
inst✝¹¹ : TopologicalSpace M✝
inst✝¹⁰ : ChartedSpace H M✝
inst✝⁹ : IsManifold I✝ k M✝
inst✝⁸ : CompactSpace M✝
inst✝⁷ : BoundarylessManifold I✝ M✝
E' : Type u_7
H' : Type u_8
inst✝⁶ : NormedAddCommGroup E'
inst✝⁵ : NormedSpace ℝ E'
inst✝⁴ : TopologicalSpace H'
M : Type u
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace H M
I : ModelWithCorners ℝ E H
inst✝¹ : IsManifold I k M
inst✝ : IsEmpty M
⊢ ∀ (x : M), ContinuousAt (fun x ↦ ⋯.elim) x
Module: Mathlib.Geometry.Euclidean.MongePoint

Goal state before:
pp: 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
s : Set P
t t₀ : Triangle ℝ P
ht : Set.range t.points ⊆ insert t₀.orthocenter (Set.range t₀.points)
ht₀o : t₀.orthocenter ∉ Set.range t₀.points
ht₀s : s = insert t₀.orthocenter (Set.range t₀.points)
i₁ i₂ i₃ j₂ j₃ : Fin 3
h₁₂ : i₁ ≠ i₂
h₁₃ : i₁ ≠ i₃
h₂₃ : i₂ ≠ i₃
h₁₂₃ : ∀ (i : Fin 3), i = i₁ ∨ i = i₂ ∨ i = i₃
h₁ : t.points i₁ = t₀.orthocenter
hj₂₃ : j₂ ≠ j₃
h₂ : t₀.points j₂ = t.points i₂
h₃ : t₀.points j₃ = t.points i₃
j₁ : Fin 3
hj₁₂ : j₁ ≠ j₂
hj₁₃ : j₁ ≠ j₃
hj₁₂₃ : ∀ (j : Fin 3), j = j₁ ∨ j = j₂ ∨ j = j₃
⊢ s = insert t.orthocenter (Set.range t.points)

Tactic:
suffices h : t₀.points j₁ = t.orthocenter by
      have hui : (Set.univ : Set (Fin 3)) = {i₁, i₂, i₃} := by ext x; simpa using h₁₂₃ x
      have huj : (Set.univ : Set (Fin 3)) = {j₁, j₂, j₃} := by ext x; simpa using hj₁₂₃ x
      rw [← h, ht₀s, ← Set.image_univ, huj, ← Set.image_univ, hui]
      simp_rw [Set.image_insert_eq, Set.image_singleton, h₁, ← h₂, ← h₃]
      rw [Set.insert_comm]

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
s : Set P
t t₀ : Triangle ℝ P
ht : Set.range t.points ⊆ insert t₀.orthocenter (Set.range t₀.points)
ht₀o : t₀.orthocenter ∉ Set.range t₀.points
ht₀s : s = insert t₀.orthocenter (Set.range t₀.points)
i₁ i₂ i₃ j₂ j₃ : Fin 3
h₁₂ : i₁ ≠ i₂
h₁₃ : i₁ ≠ i₃
h₂₃ : i₂ ≠ i₃
h₁₂₃ : ∀ (i : Fin 3), i = i₁ ∨ i = i₂ ∨ i = i₃
h₁ : t.points i₁ = t₀.orthocenter
hj₂₃ : j₂ ≠ j₃
h₂ : t₀.points j₂ = t.points i₂
h₃ : t₀.points j₃ = t.points i₃
j₁ : Fin 3
hj₁₂ : j₁ ≠ j₂
hj₁₃ : j₁ ≠ j₃
hj₁₂₃ : ∀ (j : Fin 3), j = j₁ ∨ j = j₂ ∨ j = j₃
⊢ t₀.points j₁ = t.orthocenter
Module: Mathlib.Geometry.Manifold.VectorBundle.MDifferentiable

Goal state before:
pp: case refine_1
𝕜 : Type u_1
B : Type u_2
F : Type u_4
M : Type u_5
E : B → Type u_6
inst✝¹⁵ : NontriviallyNormedField 𝕜
inst✝¹⁴ : NormedAddCommGroup F
inst✝¹³ : NormedSpace 𝕜 F
inst✝¹² : TopologicalSpace (TotalSpace F E)
inst✝¹¹ : (x : B) → TopologicalSpace (E x)
EB : Type u_7
inst✝¹⁰ : NormedAddCommGroup EB
inst✝⁹ : NormedSpace 𝕜 EB
HB : Type u_8
inst✝⁸ : TopologicalSpace HB
IB : ModelWithCorners 𝕜 EB HB
EM : Type u_10
inst✝⁷ : NormedAddCommGroup EM
inst✝⁶ : NormedSpace 𝕜 EM
HM : Type u_11
inst✝⁵ : TopologicalSpace HM
IM : ModelWithCorners 𝕜 EM HM
inst✝⁴ : TopologicalSpace M
inst✝³ : ChartedSpace HM M
inst✝² : TopologicalSpace B
inst✝¹ : ChartedSpace HB B
inst✝ : FiberBundle F E
f : M → TotalSpace F E
s : Set M
x₀ : M
hf : ContinuousWithinAt f s x₀
h1 : (fun x ↦ (f x).proj) ⁻¹' (trivializationAt F E (f x₀).proj).baseSet ∈ 𝓝[s] x₀
x : M
hx : x ∈ (fun x ↦ (f x).proj) ⁻¹' (trivializationAt F E (f x₀).proj).baseSet
⊢ (fun x ↦ ↑(extChartAt IB (f x₀).proj) (f x).proj) x =
    (Prod.fst ∘ fun x ↦
        (↑(extChartAt IB (f x₀).proj) (↑(trivializationAt F E (f x₀).proj).toPartialEquiv (f x)).1,
          (↑(trivializationAt F E (f x₀).proj).toPartialEquiv (f x)).2))
      x

Tactic:
simp_rw [Function.comp, OpenPartialHomeomorph.coe_toPartialEquiv, Trivialization.coe_coe]

Goal state after:
case refine_1
𝕜 : Type u_1
B : Type u_2
F : Type u_4
M : Type u_5
E : B → Type u_6
inst✝¹⁵ : NontriviallyNormedField 𝕜
inst✝¹⁴ : NormedAddCommGroup F
inst✝¹³ : NormedSpace 𝕜 F
inst✝¹² : TopologicalSpace (TotalSpace F E)
inst✝¹¹ : (x : B) → TopologicalSpace (E x)
EB : Type u_7
inst✝¹⁰ : NormedAddCommGroup EB
inst✝⁹ : NormedSpace 𝕜 EB
HB : Type u_8
inst✝⁸ : TopologicalSpace HB
IB : ModelWithCorners 𝕜 EB HB
EM : Type u_10
inst✝⁷ : NormedAddCommGroup EM
inst✝⁶ : NormedSpace 𝕜 EM
HM : Type u_11
inst✝⁵ : TopologicalSpace HM
IM : ModelWithCorners 𝕜 EM HM
inst✝⁴ : TopologicalSpace M
inst✝³ : ChartedSpace HM M
inst✝² : TopologicalSpace B
inst✝¹ : ChartedSpace HB B
inst✝ : FiberBundle F E
f : M → TotalSpace F E
s : Set M
x₀ : M
hf : ContinuousWithinAt f s x₀
h1 : (fun x ↦ (f x).proj) ⁻¹' (trivializationAt F E (f x₀).proj).baseSet ∈ 𝓝[s] x₀
x : M
hx : x ∈ (fun x ↦ (f x).proj) ⁻¹' (trivializationAt F E (f x₀).proj).baseSet
⊢ ↑(extChartAt IB (f x₀).proj) (f x).proj = ↑(extChartAt IB (f x₀).proj) (↑(trivializationAt F E (f x₀).proj) (f x)).1
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
s : (x : B) → E x
u : Set B
x₀ : B
hs : MDiffAt[u] (T% s) x₀
⊢ (MDiffAt[u] fun x ↦ ⟨x, (-s) x⟩) x₀

Tactic:
by
  rw [mdifferentiableWithinAt_section] at hs ⊢
  set e := trivializationAt F E x₀
  refine hs.neg.congr_of_eventuallyEq ?_ ?_
  · apply eventually_of_mem (U := e.baseSet)
    · exact mem_nhdsWithin_of_mem_nhds <|
        (e.open_baseSet.mem_nhds <| mem_baseSet_trivializationAt F E x₀)
    · exact fun x hx ↦ (e.linear 𝕜 hx).map_neg ..
  · exact (e.linear 𝕜 (FiberBundle.mem_baseSet_trivializationAt' x₀)).map_neg ..

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
s : Set M
x : M
V W V₁ W₁ : (x : M) → TangentSpace I x
hV : V₁ =ᶠ[𝓝[insert x s] x] V
hW : W₁ =ᶠ[𝓝[insert x s] x] W
⊢ mlieBracketWithin I V₁ W₁ s x = mlieBracketWithin I V W s x

Tactic:
apply mem_of_mem_nhdsWithin (mem_insert x s)
    (hV.mlieBracketWithin_vectorField' hW (subset_insert x s))

Goal state after: