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

Goal state before:
pp: case refine_1.inr
V : Type u_1
P : Type u_2
inst✝⁴ : NormedAddCommGroup V
inst✝³ : InnerProductSpace ℝ V
inst✝² : MetricSpace P
inst✝¹ : NormedAddTorsor V P
n : ℕ
inst✝ : NeZero n
s : Simplex ℝ P n
signs₁ signs₂ : Finset (Fin (n + 1))
h₁ : s.ExcenterExists signs₁
h₂ : s.ExcenterExists signs₂
h : s.excenterWeights signs₁ = s.excenterWeights signs₂
hn₁ : ∑ i, s.excenterWeightsUnnorm signs₁ i ≠ 0
hn₂ : ∑ i, s.excenterWeightsUnnorm signs₂ i ≠ 0
hs : SignType.sign (∑ i, s.excenterWeightsUnnorm signs₁ i)⁻¹ = -SignType.sign (∑ i, s.excenterWeightsUnnorm signs₂ i)⁻¹
hi :
  ∀ (i : Fin (n + 1)),
    -(SignType.sign (∑ i, s.excenterWeightsUnnorm signs₂ i)⁻¹ * SignType.sign (s.excenterWeightsUnnorm signs₁ i)) =
      SignType.sign (∑ i, s.excenterWeightsUnnorm signs₂ i)⁻¹ * SignType.sign (s.excenterWeightsUnnorm signs₂ i)
⊢ signs₁ = signs₂ ∨ signs₁ = signs₂ᶜ

Tactic:
← mul_neg,

Goal state after:
case refine_1.inr
V : Type u_1
P : Type u_2
inst✝⁴ : NormedAddCommGroup V
inst✝³ : InnerProductSpace ℝ V
inst✝² : MetricSpace P
inst✝¹ : NormedAddTorsor V P
n : ℕ
inst✝ : NeZero n
s : Simplex ℝ P n
signs₁ signs₂ : Finset (Fin (n + 1))
h₁ : s.ExcenterExists signs₁
h₂ : s.ExcenterExists signs₂
h : s.excenterWeights signs₁ = s.excenterWeights signs₂
hn₁ : ∑ i, s.excenterWeightsUnnorm signs₁ i ≠ 0
hn₂ : ∑ i, s.excenterWeightsUnnorm signs₂ i ≠ 0
hs : SignType.sign (∑ i, s.excenterWeightsUnnorm signs₁ i)⁻¹ = -SignType.sign (∑ i, s.excenterWeightsUnnorm signs₂ i)⁻¹
hi :
  ∀ (i : Fin (n + 1)),
    SignType.sign (∑ i, s.excenterWeightsUnnorm signs₂ i)⁻¹ * -SignType.sign (s.excenterWeightsUnnorm signs₁ i) =
      SignType.sign (∑ i, s.excenterWeightsUnnorm signs₂ i)⁻¹ * SignType.sign (s.excenterWeightsUnnorm signs₂ i)
⊢ signs₁ = signs₂ ∨ signs₁ = signs₂ᶜ
Module: Mathlib.Geometry.Euclidean.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
h : n ≤ -1
⊢ π + -(2 * π) < 0 + 0

Tactic:
linear_combination Real.pi_pos

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

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
h : 2 • o.oangle x (x - y) = 2 • o.oangle (y - x) y
h0 : o.oangle x y ≠ 0
hpi : o.oangle x y ≠ ↑π
hs : (o.oangle x (x - y)).sign = (o.oangle (y - x) y).sign
⊢ ‖x‖ = ‖y‖

Tactic:
rw [Real.Angle.two_zsmul_eq_iff] at h

Goal state after:
V : Type u_1
inst✝² : NormedAddCommGroup V
inst✝¹ : InnerProductSpace ℝ V
inst✝ : Fact (finrank ℝ V = 2)
o : Orientation ℝ V (Fin 2)
x y : V
h : o.oangle x (x - y) = o.oangle (y - x) y ∨ o.oangle x (x - y) = o.oangle (y - x) y + ↑π
h0 : o.oangle x y ≠ 0
hpi : o.oangle x y ≠ ↑π
hs : (o.oangle x (x - y)).sign = (o.oangle (y - x) y).sign
⊢ ‖x‖ = ‖y‖
Module: Mathlib.Geometry.Euclidean.Triangle

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
a b c a' b' c' : P
r : ℝ
h : ∠ a' b' c' = ∠ a b c
hab : dist a' b' = r * dist a b
hcb : dist c' b' = r * dist c b
⊢ r ^ 2 * (dist a b ^ 2 + dist c b ^ 2 - 2 * dist a b * dist c b * Real.cos (∠ a b c)) = (r * dist a c) ^ 2

Tactic:
by simp [pow_two, ← law_cos a b c]; ring

Goal state after:
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
ps : Set P
n : ℕ
inst✝ : FiniteDimensional ℝ V
hd : finrank ℝ V = n
hc : Cospherical ps
⊢ ∃ c, ∀ (sx : Simplex ℝ P n), Set.range sx.points ⊆ ps → sx.circumsphere = c

Tactic:
rw [← finrank_top, ← direction_top ℝ V P] at hd
  refine exists_circumsphere_eq_of_cospherical_subset ?_ hd hc
  exact Set.subset_univ _

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) * Real.cos (angle y z) - Real.sin (angle x y) * Real.sin (angle y z) =
    Real.cos (angle x y) * Real.cos (angle y z) +
      Real.sin (angle x y) * Real.sin (angle y z) *
        ⟪NormedSpace.normalize (ortho y x), NormedSpace.normalize (ortho y z)⟫
Hw : Real.sin (angle x y) * Real.sin (angle y z) ≠ 0
H8 : ⟪NormedSpace.normalize (ortho y x), NormedSpace.normalize (ortho y z)⟫ = -1
H9 : ortho y x ≠ 0
H10 : ortho y z ≠ 0
H11 : ‖NormedSpace.normalize (ortho y x)‖ = 1
⊢ Real.sin (angle x z) • y = Real.sin (angle y z) • x + Real.sin (angle x y) • z

Tactic:
have H12 : ‖normalize (ortho y z)‖ = 1 := norm_normalize_eq_one_iff.mpr H10

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 x y) * Real.cos (angle y z) +
      Real.sin (angle x y) * Real.sin (angle y z) *
        ⟪NormedSpace.normalize (ortho y x), NormedSpace.normalize (ortho y z)⟫
Hw : Real.sin (angle x y) * Real.sin (angle y z) ≠ 0
H8 : ⟪NormedSpace.normalize (ortho y x), NormedSpace.normalize (ortho y z)⟫ = -1
H9 : ortho y x ≠ 0
H10 : ortho y z ≠ 0
H11 : ‖NormedSpace.normalize (ortho y x)‖ = 1
H12 : ‖NormedSpace.normalize (ortho y z)‖ = 1
⊢ Real.sin (angle x z) • y = Real.sin (angle y z) • x + Real.sin (angle x y) • z
Module: Mathlib.Topology.MetricSpace.Congruence

Goal state before:
pp: ι : Type u_1
P₁ : Type u_3
P₂ : Type u_4
v₁ : ι → P₁
v₂ : ι → P₂
inst✝¹ : PseudoMetricSpace P₁
inst✝ : PseudoMetricSpace P₂
x✝¹ x✝ : ι
⊢ ↑(nndist (v₁ x✝¹) (v₁ x✝)) = edist (v₂ x✝¹) (v₂ x✝) ↔ nndist (v₁ x✝¹) (v₁ x✝) = nndist (v₂ x✝¹) (v₂ x✝)

Tactic:
edist_nndist

Goal state after:
ι : Type u_1
P₁ : Type u_3
P₂ : Type u_4
v₁ : ι → P₁
v₂ : ι → P₂
inst✝¹ : PseudoMetricSpace P₁
inst✝ : PseudoMetricSpace P₂
x✝¹ x✝ : ι
⊢ ↑(nndist (v₁ x✝¹) (v₁ x✝)) = ↑(nndist (v₂ x✝¹) (v₂ x✝)) ↔ nndist (v₁ x✝¹) (v₁ x✝) = nndist (v₂ x✝¹) (v₂ x✝)
Module: Mathlib.Topology.MetricSpace.Congruence

Goal state before:
pp: ι : Type u_1
P₁ : Type u_3
P₂ : Type u_4
v₁ : ι → P₁
v₂ : ι → P₂
inst✝¹ : PseudoMetricSpace P₁
inst✝ : PseudoMetricSpace P₂
⊢ (Pairwise fun i₁ i₂ ↦ edist (v₁ i₁) (v₁ i₂) = edist (v₂ i₁) (v₂ i₂)) ↔
    Pairwise fun i₁ i₂ ↦ nndist (v₁ i₁) (v₁ i₂) = nndist (v₂ i₁) (v₂ i₂)

Tactic:
edist_nndist

Goal state after:
ι : Type u_1
P₁ : Type u_3
P₂ : Type u_4
v₁ : ι → P₁
v₂ : ι → P₂
inst✝¹ : PseudoMetricSpace P₁
inst✝ : PseudoMetricSpace P₂
⊢ (Pairwise fun i₁ i₂ ↦ ↑(nndist (v₁ i₁) (v₁ i₂)) = ↑(nndist (v₂ i₁) (v₂ i₂))) ↔
    Pairwise fun i₁ i₂ ↦ nndist (v₁ i₁) (v₁ i₂) = nndist (v₂ i₁) (v₂ i₂)
Module: Mathlib.Geometry.Euclidean.Incenter

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

Tactic:
s.sign_excenterWeights_singleton_pos hne

Goal state after:
V : Type u_1
P : Type u_2
inst✝⁵ : NormedAddCommGroup V
inst✝⁴ : InnerProductSpace ℝ V
inst✝³ : MetricSpace P
inst✝² : NormedAddTorsor V P
n : ℕ
inst✝¹ : NeZero n
s : Simplex ℝ P n
inst✝ : n.AtLeastTwo
i j : Fin (n + 1)
hne : i ≠ j
⊢ 1 = 1
Module: Mathlib.Geometry.Euclidean.Incenter

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

Tactic:
by
  rw [(s.excenterExists_singleton i).sign_touchpointWeights hne.symm,
    s.sign_excenterWeights_singleton_neg]

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:
show (n : ℝ) = ‖(n : ℝ)‖ by simp

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 (((↑n + 1) / ↑n) • (s.centroid -ᵥ s.circumcenter) +ᵥ s.circumcenter)‖ =
    ‖s.circumcenter -ᵥ s.points i‖
Module: Mathlib.Topology.MetricSpace.Similarity

Goal state before:
pp: ι : Type u_1
P₁ : Type u_3
P₂ : Type u_4
v₁ : ι → P₁
v₂ : ι → P₂
inst✝¹ : PseudoMetricSpace P₁
inst✝ : PseudoMetricSpace P₂
x✝² : ℝ≥0
x✝¹ x✝ : ι
⊢ ↑(nndist (v₁ x✝¹) (v₁ x✝)) = ↑x✝² * edist (v₂ x✝¹) (v₂ x✝) ↔ nndist (v₁ x✝¹) (v₁ x✝) = x✝² * nndist (v₂ x✝¹) (v₂ x✝)

Tactic:
edist_nndist

Goal state after:
ι : Type u_1
P₁ : Type u_3
P₂ : Type u_4
v₁ : ι → P₁
v₂ : ι → P₂
inst✝¹ : PseudoMetricSpace P₁
inst✝ : PseudoMetricSpace P₂
x✝² : ℝ≥0
x✝¹ x✝ : ι
⊢ ↑(nndist (v₁ x✝¹) (v₁ x✝)) = ↑x✝² * ↑(nndist (v₂ x✝¹) (v₂ x✝)) ↔
    nndist (v₁ x✝¹) (v₁ x✝) = x✝² * nndist (v₂ x✝¹) (v₂ x✝)
Module: Mathlib.Topology.MetricSpace.Similarity

Goal state before:
pp: ι : Type u_1
P₁ : Type u_3
P₂ : Type u_4
v₁ : ι → P₁
v₂ : ι → P₂
inst✝¹ : PseudoMetricSpace P₁
inst✝ : PseudoMetricSpace P₂
⊢ (∃ r, r ≠ 0 ∧ Pairwise fun i₁ i₂ ↦ edist (v₁ i₁) (v₁ i₂) = ↑r * edist (v₂ i₁) (v₂ i₂)) ↔
    ∃ r, r ≠ 0 ∧ Pairwise fun i₁ i₂ ↦ nndist (v₁ i₁) (v₁ i₂) = r * nndist (v₂ i₁) (v₂ i₂)

Tactic:
edist_nndist

Goal state after:
ι : Type u_1
P₁ : Type u_3
P₂ : Type u_4
v₁ : ι → P₁
v₂ : ι → P₂
inst✝¹ : PseudoMetricSpace P₁
inst✝ : PseudoMetricSpace P₂
⊢ (∃ r, r ≠ 0 ∧ Pairwise fun i₁ i₂ ↦ ↑(nndist (v₁ i₁) (v₁ i₂)) = ↑r * ↑(nndist (v₂ i₁) (v₂ i₂))) ↔
    ∃ r, r ≠ 0 ∧ Pairwise fun i₁ i₂ ↦ nndist (v₁ i₁) (v₁ i₂) = r * nndist (v₂ i₁) (v₂ i₂)
Module: Mathlib.Geometry.Euclidean.NinePointCircle

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 : Triangle ℝ P
i : Fin 3
⊢ (orthogonalProjection (affineSpan ℝ (Set.range (Simplex.faceOpposite s i).points))) (s.points i) =
    (orthogonalProjection (affineSpan ℝ (Set.range (Simplex.faceOpposite s i).points))) (Simplex.eulerPoint s i)

Tactic:
rw [orthogonalProjection_eq_orthogonalProjection_iff_vsub_mem,
    Simplex.points_vsub_eulerPoint, Submodule.smul_mem_iff _ (by norm_num),
    ← orthocenter_eq_mongePoint, direction_affineSpan, Simplex.range_faceOpposite_points]

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 : Triangle ℝ P
i : Fin 3
⊢ s.points i -ᵥ s.orthocenter ∈ (vectorSpan ℝ (s.points '' {i}ᶜ))ᗮ
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'
⊢ ![a, b, c] ∼ ![a', b', c']

Tactic:
have k_pos : 0 < k := by
    rw [hk]
    apply div_pos
    · simp [dist_pos, ne₁₂_of_not_collinear h_not_col]
    · simp [dist_pos, ne₁₂_of_not_collinear h_not_col']

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
⊢ ![a, b, c] ∼ ![a', b', c']
Module: Mathlib.Geometry.Euclidean.Sphere.SecondInter

Goal state before:
pp: case refine_2
V : Type u_1
P : Type u_2
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
s : Sphere P
p : P
v : V
hp : ⟪v, p -ᵥ s.center⟫ = 0
⊢ (-2 * 0 / ⟪v, v⟫) • v +ᵥ p = p

Tactic:
mul_zero,

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
s : Sphere P
p : P
v : V
hp : ⟪v, p -ᵥ s.center⟫ = 0
⊢ (0 / ⟪v, v⟫) • v +ᵥ p = p
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 ^ 2 ≤ s.radius ^ 2 ↔ dist p s.center ≤ s.radius

Tactic:
pow_le_pow_iff_left₀ dist_nonneg hr two_ne_zero

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

Goal state before:
pp: G✝ : Type u_1
inst✝³ : Group G✝
inst✝² : DecidableEq G✝
X✝ : Finset G✝
n : ℕ
G : Type u_1
inst✝¹ : Group G
inst✝ : DecidableEq G
X : Finset G
hX₁ : 1 ∈ X
hX : X.Nontrivial
hn : True
hXn : X = X ^ 2
x : G
hx : x ∈ ↑X
⊢ x⁻¹ ∈ ↑X

Tactic:
by
      norm_cast at *
      have : x • X ⊆ X := by
        simpa [← hXn, add_assoc, ← sq] using! smul_finset_subset_mul (t := X) hx
      have : x • X = X := eq_of_subset_of_card_le this (card_smul_finset ..).ge
      rw [← eq_inv_smul_iff] at this
      rw [this]
      simpa [mem_inv_smul_finset_iff]

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

Goal state before:
pp: case right
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
he : affineSpan ℝ (Set.range t₂.points) = affineSpan ℝ (Set.range t₁.points)
hu : {j₂}ᶜ = {j₁, j₃}
hle : (altitude t₁ i₃).directionᗮ ≤ line[ℝ, t₁.orthocenter, t₁.points i₃].directionᗮ
hui : {i₃}ᶜ = {i₁, i₂}
⊢ t₁.points i₁ -ᵥ t₁.points i₂ ∈ vectorSpan ℝ (t₁.points '' {i₃}ᶜ)

Tactic:
hui,

Goal state after:
case right
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
he : affineSpan ℝ (Set.range t₂.points) = affineSpan ℝ (Set.range t₁.points)
hu : {j₂}ᶜ = {j₁, j₃}
hle : (altitude t₁ i₃).directionᗮ ≤ line[ℝ, t₁.orthocenter, t₁.points i₃].directionᗮ
hui : {i₃}ᶜ = {i₁, i₂}
⊢ t₁.points i₁ -ᵥ t₁.points i₂ ∈ vectorSpan ℝ (t₁.points '' {i₁, i₂})
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:
dsimp [tangentMap]

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
inst✝ : IsManifold I 1 M
p q : TangentBundle I M
h : q.proj ∈ (chartAt H p.proj).source
⊢ ⟨↑(chartAt H p.proj) q.proj, (mfderiv% ↑(chartAt H p.proj) q.proj) q.snd⟩ =
    (TotalSpace.toProd H E).symm (↑(chartAt (ModelProd H E) p) q)
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 t : (x : B) → E x
u : Set B
x₀ : B
hs : MDiffAt[u] (T% s) x₀
ht : MDiffAt[u] (T% t) x₀
⊢ (MDiffAt[u] fun x ↦ ⟨x, (s + t) x⟩) x₀

Tactic:
rw [mdifferentiableWithinAt_section] at hs ht ⊢

Goal state after:
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 t : (x : B) → E x
u : Set B
x₀ : B
hs : (MDiffAt[u] fun b ↦ (↑(trivializationAt F E x₀) ⟨b, s b⟩).2) x₀
ht : (MDiffAt[u] fun b ↦ (↑(trivializationAt F E x₀) ⟨b, t b⟩).2) x₀
⊢ (MDiffAt[u] fun b ↦ (↑(trivializationAt F E x₀) ⟨b, (s + t) b⟩).2) x₀
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
x : M
V W V₁ W₁ : (x : M) → TangentSpace I x
hV : V₁ =ᶠ[𝓝 x] V
hW : W₁ =ᶠ[𝓝 x] W
⊢ mlieBracket I V₁ W₁ x = mlieBracket I V W x

Tactic:
by
  rw [← mlieBracketWithin_univ, ← mlieBracketWithin_univ,
    hV.mlieBracketWithin_vectorField_eq_nhds hW]

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

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
M : Type u_4
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace H M
s : Set M
x : M
V W : (x : M) → TangentSpace I x
inst✝¹ : IsManifold I 2 M
inst✝ : CompleteSpace E
f : M → 𝕜
hf : MDiffAt[s] f x
hW : (MDiffAt[s] fun x ↦ ⟨x, W x⟩) x
hs : UniqueMDiffAt[s] x
V' : (x : E) → TangentSpace 𝓘(𝕜, E) x := mpullbackWithin 𝓘(𝕜, E) I (↑(extChartAt I x).symm) V (range ↑I)
W' : (x : E) → TangentSpace 𝓘(𝕜, E) x := mpullbackWithin 𝓘(𝕜, E) I (↑(extChartAt I x).symm) W (range ↑I)
f' : E → 𝕜 := f ∘ ↑(extChartAt I x).symm
s' : Set E := ↑(extChartAt I x).symm ⁻¹' s ∩ range ↑I
A : (x₀ : E) → TangentSpace 𝓘(𝕜, E) x₀ := fun x₀ ↦ (fderivWithin 𝕜 f' s' x₀) (V' x₀) • W' x₀
⊢ mpullback I 𝓘(𝕜, E) (↑(extChartAt I x)) (lieBracketWithin 𝕜 V' (f' • W') s') x =
    (d[s] f x) (V x) • W x + f x • mpullback I 𝓘(𝕜, E) (↑(extChartAt I x)) (lieBracketWithin 𝕜 V' W' s') x

Tactic:
let B (x₀) : TangentSpace 𝓘(𝕜, E) x₀ := f' x₀ • lieBracketWithin 𝕜 V' W' s' x₀

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
M : Type u_4
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace H M
s : Set M
x : M
V W : (x : M) → TangentSpace I x
inst✝¹ : IsManifold I 2 M
inst✝ : CompleteSpace E
f : M → 𝕜
hf : MDiffAt[s] f x
hW : (MDiffAt[s] fun x ↦ ⟨x, W x⟩) x
hs : UniqueMDiffAt[s] x
V' : (x : E) → TangentSpace 𝓘(𝕜, E) x := mpullbackWithin 𝓘(𝕜, E) I (↑(extChartAt I x).symm) V (range ↑I)
W' : (x : E) → TangentSpace 𝓘(𝕜, E) x := mpullbackWithin 𝓘(𝕜, E) I (↑(extChartAt I x).symm) W (range ↑I)
f' : E → 𝕜 := f ∘ ↑(extChartAt I x).symm
s' : Set E := ↑(extChartAt I x).symm ⁻¹' s ∩ range ↑I
A : (x₀ : E) → TangentSpace 𝓘(𝕜, E) x₀ := fun x₀ ↦ (fderivWithin 𝕜 f' s' x₀) (V' x₀) • W' x₀
B : (x₀ : E) → TangentSpace 𝓘(𝕜, E) x₀ := fun x₀ ↦ f' x₀ • lieBracketWithin 𝕜 V' W' s' x₀
⊢ mpullback I 𝓘(𝕜, E) (↑(extChartAt I x)) (lieBracketWithin 𝕜 V' (f' • W') s') x =
    (d[s] f x) (V x) • W x + f x • mpullback I 𝓘(𝕜, E) (↑(extChartAt I x)) (lieBracketWithin 𝕜 V' W' s') x
Module: Mathlib.Geometry.Manifold.Instances.Icc

Goal state before:
pp: x y : ℝ
h : Fact (x < y)
n : WithTop ℕ∞
z : ↑(Icc x y)
φ₀ : (EuclideanSpace ℝ (Fin 1) × Unit) ≃L[ℝ] EuclideanSpace ℝ (Fin 1) :=
  ContinuousLinearEquiv.prodUnique ℝ (EuclideanSpace ℝ (Fin 1)) Unit
φ : (EuclideanSpace ℝ (Fin 1) × Unit) ≃L[ℝ] ℝ := φ₀.trans (PiLp.equivOfUnique 2 ℝ fun x ↦ ℝ)
hz : ↑z < y
z' : EuclideanSpace ℝ (Fin 1)
hz' : z' ∈ ((chartAt (EuclideanHalfSpace 1) z).extend (𝓡∂ 1)).target
⊢ (∃ y, ↑(𝓡∂ 1) y = z') ∧ ↑(𝓡∂ 1).symm z' ∈ (IccLeftChart x y).target

Tactic:
by
      simpa [hz] using! hz'

Goal state after:
Module: Mathlib.Geometry.Manifold.Instances.Icc

Goal state before:
pp: x y : ℝ
h : Fact (x < y)
n : WithTop ℕ∞
z : ↑(Icc x y)
φ₀ : (EuclideanSpace ℝ (Fin 1) × Unit) ≃L[ℝ] EuclideanSpace ℝ (Fin 1) :=
  ContinuousLinearEquiv.prodUnique ℝ (EuclideanSpace ℝ (Fin 1)) Unit
φ : (EuclideanSpace ℝ (Fin 1) × Unit) ≃L[ℝ] ℝ := φ₀.trans (PiLp.equivOfUnique 2 ℝ fun x ↦ ℝ)
hz : ¬↑z < y
z' : EuclideanSpace ℝ (Fin 1)
hz' : z' ∈ ((chartAt (EuclideanHalfSpace 1) z).extend (𝓡∂ 1)).target
⊢ (∃ y, ↑(𝓡∂ 1) y = z') ∧ ↑(𝓡∂ 1).symm z' ∈ (IccRightChart x y).target

Tactic:
by
      simpa [hz] using! hz'

Goal state after:
Module: Mathlib.Geometry.Manifold.Instances.Icc

Goal state before:
pp: case neg
x y : ℝ
h : Fact (x < y)
n : WithTop ℕ∞
z : ℝ
hz : z ∈ Icc x y
h'z : ¬↑(projIcc x y ⋯ z) < y
⊢ ContDiffWithinAt ℝ n ((Subtype.val ∘ ↑(IccRightChart x y)) ∘ projIcc x y ⋯ ∘ ↑(chartAt ℝ z).symm) (Icc x y) z

Tactic:
· have : ContDiff ℝ n (fun (w : ℝ) ↦
        (show EuclideanSpace ℝ (Fin 1) from toLp 2 fun (_ : Fin 1) ↦ y - w)) := by
      dsimp
      apply contDiff_euclidean.2 (fun i ↦ by fun_prop)
    apply this.contDiffWithinAt.congr_of_eventuallyEq_of_mem _ hz
    filter_upwards [self_mem_nhdsWithin] with w hw
    ext i
    suffices y - max x (min y w) = y - w by
      simpa [modelWithCornersEuclideanHalfSpace, IccRightChart]
    rw [max_eq_right, min_eq_right hw.2]
    simp [hw.1, h.out.le]

Goal state after: