mathlib-initiative/mathlib-tactics huggingface dataset:
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
h : ∡ p₁ p₂ p₃ ≠ 0
⊢ p₃ ≠ p₂

Tactic:
rw [← @vsub_ne_zero V]

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
hd2 : Fact (finrank ℝ V = 2)
inst✝ : Oriented ℝ V (Fin 2)
p₁ p₂ p₃ : P
h : ∡ p₁ p₂ p₃ ≠ 0
⊢ p₃ -ᵥ p₂ ≠ 0
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 ∨ y ≠ 0
⊢ 0 ≠ ‖x‖ * ‖x‖ + ‖y‖ * ‖y‖

Tactic:
refine ne_of_lt ?_

Goal state after:
V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x y : V
h : ⟪x, y⟫ = 0
h0 : x ≠ 0 ∨ y ≠ 0
⊢ 0 < ‖x‖ * ‖x‖ + ‖y‖ * ‖y‖
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)
w x y z : V
h : InnerProductGeometry.angle w x = InnerProductGeometry.angle y z
hs : (o.oangle w x).sign = (o.oangle y z).sign
h0 : (w ≠ 0 ∧ x ≠ 0) ∧ y ≠ 0 ∧ z ≠ 0
⊢ o.oangle w x = o.oangle y z

Tactic:
· rw [Real.Angle.eq_iff_abs_toReal_eq_of_sign_eq hs]
    rwa [o.angle_eq_abs_oangle_toReal h0.1.1 h0.1.2,
      o.angle_eq_abs_oangle_toReal h0.2.1 h0.2.2] at h

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

Goal state before:
pp: case neg.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
hx : ¬x = 0
hy : ¬y = 0
h : InnerProductGeometry.angle x y = π / 2
⊢ o.oangle x y = ↑(π / 2) ∨ o.oangle x y = ↑(-π / 2)

Tactic:
convert! o.oangle_eq_angle_or_eq_neg_angle hx hy using 2 <;> rw [h]
    simp only [neg_div, Real.Angle.coe_neg]

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

Goal state before:
pp: V : Type u_1
inst✝² : NormedAddCommGroup V
inst✝¹ : InnerProductSpace ℝ V
inst✝ : Fact (finrank ℝ V = 2)
o : Orientation ℝ V (Fin 2)
x y : V
⊢ (o.oangle x (y - x)).sign = (o.oangle x y).sign

Tactic:
by
  rw [← o.oangle_sign_sub_smul_right x y 1, one_smul]

Goal state after:
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
c₁ c₂ p₁ p₂ : P
hc₁ : dist p₁ c₁ = dist p₂ c₁
hc₂ : dist p₁ c₂ = dist p₂ c₂
⊢ ⟪c₂ -ᵥ c₁, p₂ -ᵥ p₁⟫ = 0

Tactic:
← Submodule.mem_orthogonal_singleton_iff_inner_left,

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
c₁ c₂ p₁ p₂ : P
hc₁ : dist p₁ c₁ = dist p₂ c₁
hc₂ : dist p₁ c₂ = dist p₂ c₂
⊢ c₂ -ᵥ c₁ ∈ (ℝ ∙ (p₂ -ᵥ p₁))ᗮ
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 p₄ : P
h₁₂ : p₁ ≠ p₂
ha : ∡ p₁ p₂ p₃ = ∡ p₃ p₂ p₄
hs : line[ℝ, p₁, p₂].SSameSide p₃ p₄
h₃₂ : p₃ ≠ p₂
h₄₂ : p₄ ≠ p₂
hadd : ∡ p₁ p₂ p₃ + ∡ p₃ p₂ p₄ = ∡ p₁ p₂ p₄
⊢ ((∡ p₁ p₂ p₃).toReal + (∡ p₃ p₂ p₄).toReal) / 2 = (∡ p₁ p₂ p₄).toReal / 2

Tactic:
rw [div_left_inj' (by norm_num), ← hadd]

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
hd2 : Fact (finrank ℝ V = 2)
inst✝ : Oriented ℝ V (Fin 2)
p₁ p₂ p₃ p₄ : P
h₁₂ : p₁ ≠ p₂
ha : ∡ p₁ p₂ p₃ = ∡ p₃ p₂ p₄
hs : line[ℝ, p₁, p₂].SSameSide p₃ p₄
h₃₂ : p₃ ≠ p₂
h₄₂ : p₄ ≠ p₂
hadd : ∡ p₁ p₂ p₃ + ∡ p₃ p₂ p₄ = ∡ p₁ p₂ p₄
⊢ (∡ p₁ p₂ p₃).toReal + (∡ p₃ p₂ p₄).toReal = (∡ p₁ p₂ p₃ + ∡ p₃ p₂ p₄).toReal
Module: Mathlib.Geometry.Euclidean.Sphere.OrthRadius

Goal state before:
pp: case mpr
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
h : dist q s.center = s.radius
⊢ Metric.sphere s.center s.radius ∩ ↑(s.orthRadius q) = {q}

Tactic:
exact inter_orthRadius_eq_singleton_of_dist_eq_radius h

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

Goal state before:
pp: V : Type u_1
P : Type u_2
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
s : Sphere P
p q : P
h : ∠ q p s.center = π / 2
hp : p ∈ s
hp_mem : p ∈ line[ℝ, p, q]
h_ortho : ⟪q -ᵥ p, p -ᵥ s.center⟫_ℝ = 0
hq : q ∈ s.orthRadius p
⊢ {p, q} ⊆ ↑(s.orthRadius p)

Tactic:
have hp : p ∈ s.orthRadius p := by
    simp [Sphere.self_mem_orthRadius]

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 : Sphere P
p q : P
h : ∠ q p s.center = π / 2
hp✝ : p ∈ s
hp_mem : p ∈ line[ℝ, p, q]
h_ortho : ⟪q -ᵥ p, p -ᵥ s.center⟫_ℝ = 0
hq : q ∈ s.orthRadius p
hp : p ∈ s.orthRadius p
⊢ {p, q} ⊆ ↑(s.orthRadius p)
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
i : Fin (n + 1)
⊢ SignType.sign ((s.signedInfDist i) s.incenter) = 1

Tactic:
convert! s.excenterExists_empty.sign_signedInfDist_excenter i

Goal state after:
case e'_3
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)
⊢ 1 = SignType.sign (s.excenterWeights ∅ i)
Module: Mathlib.Geometry.Euclidean.Triangle

Goal state before:
pp: V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x y : V
hy : y ≠ 0
hx : x ≠ 0
h✝ : ↑(angle x y) = ↑(angle x (x + y) + angle y (x + y))
n : ℤ
this : -1 < n
h : angle x y + 2 * π ≤ angle x (x + y) + angle y (x + y)
hn : 1 ≤ n
⊢ False

Tactic:
← angle_nonneg x y,

Goal state after:
V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x y : V
hy : y ≠ 0
hx : x ≠ 0
h✝ : ↑(angle x y) = ↑(angle x (x + y) + angle y (x + y))
n : ℤ
this : -1 < n
h : 0 + 2 * π ≤ angle x (x + y) + angle y (x + y)
hn : 1 ≤ n
⊢ False
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
signs₁ signs₂ : Finset (Fin (n + 1))
h₁ : s.ExcenterExists signs₁
h₂ : s.ExcenterExists signs₂
⊢ s.excenter signs₁ = s.excenter signs₂ ↔ signs₁ = signs₂ ∨ signs₁ = signs₂ᶜ

Tactic:
rw [excenter_eq_affineCombination,
    h₂.affineCombination_eq_excenter_iff (s.sum_excenterWeights_eq_one_iff.2 h₁)]

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
signs₁ signs₂ : Finset (Fin (n + 1))
h₁ : s.ExcenterExists signs₁
h₂ : s.ExcenterExists signs₂
⊢ s.excenterWeights signs₁ = s.excenterWeights signs₂ ↔ signs₁ = signs₂ ∨ signs₁ = signs₂ᶜ
Module: Mathlib.Geometry.Euclidean.Triangle

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
a b c p : P
hb : a ≠ b
hc : a ≠ c
pb : ¬p = b
pc : ¬p = c
ea : ∠ b a c + ∠ a c b + ∠ c b a = π
eb : ∠ b a p + ∠ a p b + ∠ c b a = π
ec : ∠ a c p + ∠ c p a + ∠ p a c = π
hp : ∠ b p c = π
hp' : ∠ c p b = π
⊢ ∠ b a p + ∠ p a c = ∠ b a c

Tactic:
angle_eq_angle_of_angle_eq_pi a hp',

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
a b c p : P
hb : a ≠ b
hc : a ≠ c
pb : ¬p = b
pc : ¬p = c
ea : ∠ b a c + ∠ a c b + ∠ c b a = π
eb : ∠ b a p + ∠ a p b + ∠ c b a = π
ec : ∠ a c b + ∠ c p a + ∠ p a c = π
hp : ∠ b p c = π
hp' : ∠ c p b = π
⊢ ∠ b a p + ∠ p a c = ∠ b a c
Module: Mathlib.Geometry.Euclidean.Angle.Sphere

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 p₄ : P
h : 2 • ∡ p₁ p₂ p₄ = 2 • ∡ p₁ p₃ p₄
hn : ¬Collinear ℝ {p₁, p₂, p₄}
hn' : ¬Collinear ℝ {p₁, p₃, p₄}
t₁ : Affine.Triangle ℝ P := { points := ![p₁, p₂, p₄], independent := ⋯ }
t₂ : Affine.Triangle ℝ P := { points := ![p₁, p₃, p₄], independent := ⋯ }
⊢ p₂ ∈ Metric.sphere (Affine.Simplex.circumsphere t₁).center (Affine.Simplex.circumsphere t₁).radius

Tactic:
exact t₁.mem_circumsphere 1

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

Goal state before:
pp: case neg
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 z ≠ π
H0 : angle x z = angle x y + angle y z
H1 : ¬angle x z = 0
Hxz : Real.sin (angle x z) ≠ 0
H2 : ¬angle x y = 0
H3 : ¬angle y z = 0
H4 : ¬angle x y = π
H5 : ¬angle y z = π
H6 :
  Real.cos (angle x y + angle y 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.sin (angle x z) • y = Real.sin (angle y z) • x + Real.sin (angle x y) • z

Tactic:
Real.cos_add

Goal state after:
case neg
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 z ≠ π
H0 : angle x z = angle x y + angle y z
H1 : ¬angle x z = 0
Hxz : Real.sin (angle x z) ≠ 0
H2 : ¬angle x y = 0
H3 : ¬angle y z = 0
H4 : ¬angle x y = π
H5 : ¬angle y z = π
H6 :
  Real.cos (angle x y) * Real.cos (angle y z) - Real.sin (angle x y) * Real.sin (angle y 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.sin (angle x z) • y = Real.sin (angle y z) • x + Real.sin (angle x y) • z
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
p p₁ p₂ : P
r : ℝ
h₁ : ∀ (i : Fin (n + 1)), dist (s.points i) p₁ = r
h₂ : ∀ (i : Fin (n + 1)), dist (s.points i) p₂ = r
span_s : AffineSubspace ℝ P := affineSpan ℝ (Set.range s.points)
h₁' : ↑(s.orthogonalProjectionSpan p₁) = s.circumcenter
h₂' : ↑(s.orthogonalProjectionSpan p₂) = s.circumcenter
r₁ r₂ : ℝ
hp₁o : ↑(s.orthogonalProjectionSpan p₁) ∈ affineSpan ℝ (Set.range s.points)
hp₁ : p₁ = r₁ • (p -ᵥ ↑((orthogonalProjection (affineSpan ℝ (Set.range s.points))) p)) +ᵥ s.circumcenter
hp₂o : ↑(s.orthogonalProjectionSpan p₂) ∈ affineSpan ℝ (Set.range s.points)
hp₂ : p₂ = r₂ • (p -ᵥ ↑((orthogonalProjection (affineSpan ℝ (Set.range s.points))) p)) +ᵥ s.circumcenter
h : s.points 0 ∈ span_s
hd₁ : dist p₁ s.circumcenter * dist p₁ s.circumcenter = r * r - s.circumradius * s.circumradius
hd₂ : dist p₂ s.circumcenter * dist p₂ s.circumcenter = r * r - s.circumradius * s.circumradius
⊢ p₁ = p₂ ∨ p₁ = (reflection (affineSpan ℝ (Set.range s.points))) p₂

Tactic:
rw [← hd₂, hp₁, hp₂, dist_eq_norm_vsub V _ s.circumcenter, dist_eq_norm_vsub V _ s.circumcenter,
    vadd_vsub, vadd_vsub, ← real_inner_self_eq_norm_mul_norm, ← real_inner_self_eq_norm_mul_norm,
    real_inner_smul_left, real_inner_smul_left, real_inner_smul_right, real_inner_smul_right, ←
    mul_assoc, ← mul_assoc] at hd₁

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
p p₁ p₂ : P
r : ℝ
h₁ : ∀ (i : Fin (n + 1)), dist (s.points i) p₁ = r
h₂ : ∀ (i : Fin (n + 1)), dist (s.points i) p₂ = r
span_s : AffineSubspace ℝ P := affineSpan ℝ (Set.range s.points)
h₁' : ↑(s.orthogonalProjectionSpan p₁) = s.circumcenter
h₂' : ↑(s.orthogonalProjectionSpan p₂) = s.circumcenter
r₁ r₂ : ℝ
hp₁o : ↑(s.orthogonalProjectionSpan p₁) ∈ affineSpan ℝ (Set.range s.points)
hp₁ : p₁ = r₁ • (p -ᵥ ↑((orthogonalProjection (affineSpan ℝ (Set.range s.points))) p)) +ᵥ s.circumcenter
hp₂o : ↑(s.orthogonalProjectionSpan p₂) ∈ affineSpan ℝ (Set.range s.points)
hp₂ : p₂ = r₂ • (p -ᵥ ↑((orthogonalProjection (affineSpan ℝ (Set.range s.points))) p)) +ᵥ s.circumcenter
h : s.points 0 ∈ span_s
hd₁ :
  r₁ * r₁ *
      ⟪p -ᵥ ↑((orthogonalProjection (affineSpan ℝ (Set.range s.points))) p),
        p -ᵥ ↑((orthogonalProjection (affineSpan ℝ (Set.range s.points))) p)⟫ =
    r₂ * r₂ *
      ⟪p -ᵥ ↑((orthogonalProjection (affineSpan ℝ (Set.range s.points))) p),
        p -ᵥ ↑((orthogonalProjection (affineSpan ℝ (Set.range s.points))) p)⟫
hd₂ : dist p₂ s.circumcenter * dist p₂ s.circumcenter = r * r - s.circumradius * s.circumradius
⊢ p₁ = p₂ ∨ p₁ = (reflection (affineSpan ℝ (Set.range s.points))) p₂
Module: Mathlib.Geometry.Euclidean.Incenter

Goal state before:
pp: V : Type u_1
P : Type u_2
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
t : Triangle ℝ P
i₁ i₂ i₃ : Fin 3
h₁₂ : i₁ ≠ i₂
h₁₃ : i₁ ≠ i₃
h₂₃ : i₂ ≠ i₃
⊢ Sbtw ℝ (Simplex.touchpoint t {i₁} i₂) (t.points i₃) (t.points i₁)

Tactic:
rw [← Affine.Simplex.affineCombination_touchpointWeights]
  have hw := t.sum_touchpointWeights {i₁} i₂
  rw [(by clear hw; decide +revert : (Finset.univ : Finset (Fin 3)) = {i₁, i₂, i₃})] at hw
  simp only [Nat.reduceAdd, Finset.mem_insert, h₁₂, Finset.mem_singleton, h₁₃, or_self,
    not_false_eq_true, Finset.sum_insert, h₂₃, Simplex.touchpointWeights_eq_zero,
    Finset.sum_singleton, zero_add] at hw
  have h : t.touchpointWeights {i₁} i₂ =
      Finset.affineCombinationLineMapWeights i₁ i₃ (t.touchpointWeights {i₁} i₂ i₃) := by
    ext i
    have h : i = i₁ ∨ i = i₂ ∨ i = i₃ := by clear hw; decide +revert
    rcases h with rfl | rfl | rfl
    · rw [Finset.affineCombinationLineMapWeights_apply_left h₁₃]
      simp [← hw]
    · simp [h₁₂.symm, h₂₃]
    · simp [h₁₃]
  rw [h, Finset.univ.affineCombination_affineCombinationLineMapWeights _ (Finset.mem_univ _)
    (Finset.mem_univ _), sbtw_iff_right_ne_and_left_mem_image_Ioi]
  simp [t.independent.injective.ne h₁₃, ← hw, t.touchpointWeights_singleton_neg h₁₂]

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

Goal state before:
pp: case 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
n : ℕ
inst✝ : NeZero n
s : Simplex ℝ P n
⊢ Set.range s.medial.points ⊆ Metric.sphere s.ninePointCircle.center s.ninePointCircle.radius

Tactic:
rw [Set.range_subset_iff]
    simpa [medial_points] using s.faceOppositeCentroid_mem_ninePointCircle

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

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)
⊢ dist (s.points i) s.centroid = ↑n * dist s.centroid (s.faceOppositeCentroid i)

Tactic:
by
  simp_rw [dist_eq_norm_vsub, s.point_vsub_centroid_eq_smul_vsub i, norm_smul, Real.norm_natCast]

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
⊢ t.orthocenter -ᵥ circumcenter t = ∑ i, (t.points i -ᵥ circumcenter t)

Tactic:
← t.smul_mongePoint_vsub_circumcenter_eq_sum_vsub,

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
⊢ t.orthocenter -ᵥ circumcenter t = (0 + 1) • (mongePoint t -ᵥ circumcenter t)
Module: Mathlib.Geometry.Manifold.LocalDiffeomorph

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
F : Type u_3
inst✝¹² : NormedAddCommGroup F
inst✝¹¹ : NormedSpace 𝕜 F
F' : Type u_4
inst✝¹⁰ : NormedAddCommGroup F'
inst✝⁹ : NormedSpace 𝕜 F'
H₁ : Type u_5
inst✝⁸ : TopologicalSpace H₁
H₂ : Type u_6
inst✝⁷ : TopologicalSpace H₂
H₃ : Type u_7
inst✝⁶ : TopologicalSpace H₃
I : ModelWithCorners 𝕜 E H₁
J : ModelWithCorners 𝕜 F H₂
K : ModelWithCorners 𝕜 F' H₃
M : Type u_8
inst✝⁵ : TopologicalSpace M
inst✝⁴ : ChartedSpace H₁ M
N : Type u_9
inst✝³ : TopologicalSpace N
inst✝² : ChartedSpace H₂ N
P : Type u_10
inst✝¹ : TopologicalSpace P
inst✝ : ChartedSpace H₃ P
n : WithTop ℕ∞
f : M → N
s : Set M
x : M
I' : ?m.25
hf : IsLocalDiffeomorphAt I J n f x
hn : n ≠ 0
⊢ mfderiv% f x ∘SL mfderiv% ↑hf.localInverse.toPartialEquiv (f x) = mfderiv% (f ∘ ↑hf.localInverse.toPartialEquiv) (f x)

Tactic:
have hf' : MDifferentiableAt I J f (hf.localInverse (f x)) := by
      rw [hf.localInverse_left_inv hf.localInverse_mem_target]
      exact hf.mdifferentiableAt hn

Goal state after:
𝕜 : Type u_1
inst✝¹⁵ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹⁴ : NormedAddCommGroup E
inst✝¹³ : NormedSpace 𝕜 E
F : Type u_3
inst✝¹² : NormedAddCommGroup F
inst✝¹¹ : NormedSpace 𝕜 F
F' : Type u_4
inst✝¹⁰ : NormedAddCommGroup F'
inst✝⁹ : NormedSpace 𝕜 F'
H₁ : Type u_5
inst✝⁸ : TopologicalSpace H₁
H₂ : Type u_6
inst✝⁷ : TopologicalSpace H₂
H₃ : Type u_7
inst✝⁶ : TopologicalSpace H₃
I : ModelWithCorners 𝕜 E H₁
J : ModelWithCorners 𝕜 F H₂
K : ModelWithCorners 𝕜 F' H₃
M : Type u_8
inst✝⁵ : TopologicalSpace M
inst✝⁴ : ChartedSpace H₁ M
N : Type u_9
inst✝³ : TopologicalSpace N
inst✝² : ChartedSpace H₂ N
P : Type u_10
inst✝¹ : TopologicalSpace P
inst✝ : ChartedSpace H₃ P
n : WithTop ℕ∞
f : M → N
s : Set M
x : M
I' : ?m.25
hf : IsLocalDiffeomorphAt I J n f x
hn : n ≠ 0
hf' : MDiffAt f (↑hf.localInverse.toPartialEquiv (f x))
⊢ mfderiv% f x ∘SL mfderiv% ↑hf.localInverse.toPartialEquiv (f x) = mfderiv% (f ∘ ↑hf.localInverse.toPartialEquiv) (f x)
Module: Mathlib.Geometry.Manifold.MFDeriv.Tangent

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
inst✝ : IsManifold I 1 M
p q : TangentBundle I M
h : q.proj ∈ (chartAt H p.proj).source
⊢ tangentMap I I (↑(chartAt H p.proj)) q = (TotalSpace.toProd H E).symm (↑(chartAt (ModelProd H E) p) q)

Tactic:
by
  dsimp [tangentMap]
  rw [MDifferentiableAt.mfderiv]
  · rfl
  · exact mdifferentiableAt_atlas (chart_mem_atlas _ _) h

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

Goal state before:
pp: 𝕜 : Type u_1
F₁ : Type u_2
F₂ : Type u_3
B₁ : Type u_4
B₂ : Type u_5
M : Type u_6
E₁ : B₁ → Type u_7
E₂ : B₂ → Type u_8
inst✝³¹ : NontriviallyNormedField 𝕜
inst✝³⁰ : (x : B₁) → AddCommGroup (E₁ x)
inst✝²⁹ : (x : B₁) → Module 𝕜 (E₁ x)
inst✝²⁸ : NormedAddCommGroup F₁
inst✝²⁷ : NormedSpace 𝕜 F₁
inst✝²⁶ : TopologicalSpace (TotalSpace F₁ E₁)
inst✝²⁵ : (x : B₁) → TopologicalSpace (E₁ x)
inst✝²⁴ : (x : B₂) → AddCommGroup (E₂ x)
inst✝²³ : (x : B₂) → Module 𝕜 (E₂ x)
inst✝²² : NormedAddCommGroup F₂
inst✝²¹ : NormedSpace 𝕜 F₂
inst✝²⁰ : TopologicalSpace (TotalSpace F₂ E₂)
inst✝¹⁹ : (x : B₂) → TopologicalSpace (E₂ x)
EB₁ : Type u_9
inst✝¹⁸ : NormedAddCommGroup EB₁
inst✝¹⁷ : NormedSpace 𝕜 EB₁
HB₁ : Type u_10
inst✝¹⁶ : TopologicalSpace HB₁
IB₁ : ModelWithCorners 𝕜 EB₁ HB₁
inst✝¹⁵ : TopologicalSpace B₁
inst✝¹⁴ : ChartedSpace HB₁ B₁
EB₂ : Type u_11
inst✝¹³ : NormedAddCommGroup EB₂
inst✝¹² : NormedSpace 𝕜 EB₂
HB₂ : Type u_12
inst✝¹¹ : TopologicalSpace HB₂
IB₂ : ModelWithCorners 𝕜 EB₂ HB₂
inst✝¹⁰ : TopologicalSpace B₂
inst✝⁹ : ChartedSpace HB₂ B₂
EM : Type u_13
inst✝⁸ : NormedAddCommGroup EM
inst✝⁷ : NormedSpace 𝕜 EM
HM : Type u_14
inst✝⁶ : TopologicalSpace HM
IM : ModelWithCorners 𝕜 EM HM
inst✝⁵ : TopologicalSpace M
inst✝⁴ : ChartedSpace HM M
n : WithTop ℕ∞
inst✝³ : FiberBundle F₁ E₁
inst✝² : VectorBundle 𝕜 F₁ E₁
inst✝¹ : FiberBundle F₂ E₂
inst✝ : VectorBundle 𝕜 F₂ E₂
b₁ : M → B₁
b₂ : M → B₂
m₀ : M
ϕ : (m : M) → E₁ (b₁ m) →L[𝕜] E₂ (b₂ m)
v : (m : M) → E₁ (b₁ m)
s : Set M
hϕ :
  ContMDiffWithinAt IM 𝓘(𝕜, F₁ →L[𝕜] F₂) n (fun m ↦ inCoordinates F₁ E₁ F₂ E₂ (b₁ m₀) (b₁ m) (b₂ m₀) (b₂ m) (ϕ m)) s m₀
hv : ContMDiffWithinAt IM (IB₁.prod 𝓘(𝕜, F₁)) n (fun m ↦ ⟨b₁ m, v m⟩) s m₀
hb₂ : ContMDiffWithinAt IM IB₂ n b₂ s m₀
⊢ ContMDiffWithinAt IM (IB₂.prod 𝓘(𝕜, F₂)) n (fun m ↦ ⟨b₂ m, (ϕ m) (v m)⟩) s m₀

Tactic:
by
  rw [← contMDiffWithinAt_insert_self] at hϕ hv hb₂ ⊢
  rw [contMDiffWithinAt_totalSpace] at hv ⊢
  refine ⟨hb₂, ?_⟩
  apply (ContMDiffWithinAt.clm_apply hϕ hv.2).congr_of_eventuallyEq_of_mem ?_ (mem_insert m₀ s)
  have A : ∀ᶠ m in 𝓝[insert m₀ s] m₀, b₁ m ∈ (trivializationAt F₁ E₁ (b₁ m₀)).baseSet := by
    apply hv.1.continuousWithinAt
    apply (trivializationAt F₁ E₁ (b₁ m₀)).open_baseSet.mem_nhds
    exact FiberBundle.mem_baseSet_trivializationAt' (b₁ m₀)
  have A' : ∀ᶠ m in 𝓝[insert m₀ s] m₀, b₂ m ∈ (trivializationAt F₂ E₂ (b₂ m₀)).baseSet := by
    apply hb₂.continuousWithinAt
    apply (trivializationAt F₂ E₂ (b₂ m₀)).open_baseSet.mem_nhds
    exact FiberBundle.mem_baseSet_trivializationAt' (b₂ m₀)
  filter_upwards [A, A'] with m hm h'm
  rw [inCoordinates_eq hm h'm]
  simp [*]

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝¹⁵ : NontriviallyNormedField 𝕜
m n : WithTop ℕ∞
E : Type u_2
inst✝¹⁴ : NormedAddCommGroup E
inst✝¹³ : NormedSpace 𝕜 E
H : Type u_3
inst✝¹² : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝¹¹ : TopologicalSpace M
inst✝¹⁰ : ChartedSpace H M
E' : Type u_5
inst✝⁹ : NormedAddCommGroup E'
inst✝⁸ : NormedSpace 𝕜 E'
H' : Type u_6
inst✝⁷ : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
M' : Type u_7
inst✝⁶ : TopologicalSpace M'
inst✝⁵ : ChartedSpace H' M'
F : Type u_8
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace 𝕜 F
G : Type u_9
inst✝² : TopologicalSpace G
J : ModelWithCorners 𝕜 F G
N : Type u_10
inst✝¹ : TopologicalSpace N
inst✝ : ChartedSpace G N
Js : IsManifold J 1 N
Is : IsManifold I 1 M
I's : IsManifold I' 1 M'
x₀ : N
f : N → M → M'
g : N → M
t : Set N
u : Set M
hf : ContMDiffWithinAt (J.prod I) I' n (uncurry f) (t ×ˢ u) (x₀, g x₀)
hg : ContMDiffWithinAt J I m g t x₀
hx₀ : x₀ ∈ t
hu : MapsTo g t u
hmn : m + 1 ≤ n
h'u : UniqueMDiff[u]
t' : Set N := t ∩ g ⁻¹' (extChartAt I (g x₀)).source
ht't : t' ⊆ t
⊢ (x₀, g x₀) ∈ t ×ˢ u

Tactic:
simp [hx₀, hu hx₀]

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
W : (x : M) → TangentSpace I x
x : M
⊢ mlieBracketWithin I 0 W s x = 0 x

Tactic:
simp [mlieBracketWithin]

Goal state after: