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

Goal state before:
pp: case h
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 c d p : P
h : Cospherical {a, b, c, d}
hapc : ∠ a p c = π
hbpd : ∠ b p d = π
h' : Cospherical {a, c, b, d}
hmul : dist a p * dist c p = dist b p * dist d p
hbp : dist b p ≠ 0
h₁ : dist c d = dist c p / dist b p * dist a b
⊢ ∠ a p d = ∠ c p b

Tactic:
angle_eq_angle_of_angle_eq_pi_of_angle_eq_pi

Goal state after:
case h
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 c d p : P
h : Cospherical {a, b, c, d}
hapc : ∠ a p c = π
hbpd : ∠ b p d = π
h' : Cospherical {a, c, b, d}
hmul : dist a p * dist c p = dist b p * dist d p
hbp : dist b p ≠ 0
h₁ : dist c d = dist c p / dist b p * dist a b
⊢ ∠ ?m.219 p ?m.220 = ∠ c p b

case h.hapc
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 c d p : P
h : Cospherical {a, b, c, d}
hapc : ∠ a p c = π
hbpd : ∠ b p d = π
h' : Cospherical {a, c, b, d}
hmul : dist a p * dist c p = dist b p * dist d p
hbp : dist b p ≠ 0
h₁ : dist c d = dist c p / dist b p * dist a b
⊢ ∠ a p ?m.219 = π

case h.hbpd
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 c d p : P
h : Cospherical {a, b, c, d}
hapc : ∠ a p c = π
hbpd : ∠ b p d = π
h' : Cospherical {a, c, b, d}
hmul : dist a p * dist c p = dist b p * dist d p
hbp : dist b p ≠ 0
h₁ : dist c d = dist c p / dist b p * dist a b
⊢ ∠ d p ?m.220 = π

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 c d p : P
h : Cospherical {a, b, c, d}
hapc : ∠ a p c = π
hbpd : ∠ b p d = π
h' : Cospherical {a, c, b, d}
hmul : dist a p * dist c p = dist b p * dist d p
hbp : dist b p ≠ 0
h₁ : dist c d = dist c p / dist b p * dist a b
⊢ P

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 c d p : P
h : Cospherical {a, b, c, d}
hapc : ∠ a p c = π
hbpd : ∠ b p d = π
h' : Cospherical {a, c, b, d}
hmul : dist a p * dist c p = dist b p * dist d p
hbp : dist b p ≠ 0
h₁ : dist c d = dist c p / dist b p * dist a b
⊢ P
Module: Mathlib.Geometry.Euclidean.Similarity

Goal state before:
pp: case pos
V₁ : Type u_2
V₂ : Type u_3
P₁ : Type u_4
P₂ : Type u_5
inst✝⁷ : NormedAddCommGroup V₁
inst✝⁶ : NormedAddCommGroup V₂
inst✝⁵ : InnerProductSpace ℝ V₁
inst✝⁴ : InnerProductSpace ℝ V₂
inst✝³ : MetricSpace P₁
inst✝² : MetricSpace P₂
inst✝¹ : NormedAddTorsor V₁ P₁
inst✝ : NormedAddTorsor V₂ P₂
a b c : P₁
a' b' c' : P₂
r : ℝ
hr_pos : 0 < r
hdist :
  ∀ (i₁ i₂ : Fin (Nat.succ 0).succ.succ),
    dist (![a, b, c] i₁) (![a, b, c] i₂) = r * dist (![a', b', c'] i₁) (![a', b', c'] i₂)
h_ab : dist a b = r * dist a' b'
h_cb : dist c b = r * dist c' b'
h_ac : dist a c = r * dist a' c'
h_cos : dist a' b' * dist c' b' * 2 * Real.cos (∠ a' b' c') = dist a' b' * dist c' b' * 2 * Real.cos (∠ a b c)
h_cos' : dist a' c' ^ 2 = dist a' b' ^ 2 + dist c' b' ^ 2 - dist a' b' * dist c' b' * 2 * Real.cos (∠ a' b' c')
heq : dist a' b' = 0 ∨ dist c' b' = 0
⊢ ∠ a b c = ∠ a' b' c'

Tactic:
rcases heq with h1 | h2

Goal state after:
case pos.inl
V₁ : Type u_2
V₂ : Type u_3
P₁ : Type u_4
P₂ : Type u_5
inst✝⁷ : NormedAddCommGroup V₁
inst✝⁶ : NormedAddCommGroup V₂
inst✝⁵ : InnerProductSpace ℝ V₁
inst✝⁴ : InnerProductSpace ℝ V₂
inst✝³ : MetricSpace P₁
inst✝² : MetricSpace P₂
inst✝¹ : NormedAddTorsor V₁ P₁
inst✝ : NormedAddTorsor V₂ P₂
a b c : P₁
a' b' c' : P₂
r : ℝ
hr_pos : 0 < r
hdist :
  ∀ (i₁ i₂ : Fin (Nat.succ 0).succ.succ),
    dist (![a, b, c] i₁) (![a, b, c] i₂) = r * dist (![a', b', c'] i₁) (![a', b', c'] i₂)
h_ab : dist a b = r * dist a' b'
h_cb : dist c b = r * dist c' b'
h_ac : dist a c = r * dist a' c'
h_cos : dist a' b' * dist c' b' * 2 * Real.cos (∠ a' b' c') = dist a' b' * dist c' b' * 2 * Real.cos (∠ a b c)
h_cos' : dist a' c' ^ 2 = dist a' b' ^ 2 + dist c' b' ^ 2 - dist a' b' * dist c' b' * 2 * Real.cos (∠ a' b' c')
h1 : dist a' b' = 0
⊢ ∠ a b c = ∠ a' b' c'

case pos.inr
V₁ : Type u_2
V₂ : Type u_3
P₁ : Type u_4
P₂ : Type u_5
inst✝⁷ : NormedAddCommGroup V₁
inst✝⁶ : NormedAddCommGroup V₂
inst✝⁵ : InnerProductSpace ℝ V₁
inst✝⁴ : InnerProductSpace ℝ V₂
inst✝³ : MetricSpace P₁
inst✝² : MetricSpace P₂
inst✝¹ : NormedAddTorsor V₁ P₁
inst✝ : NormedAddTorsor V₂ P₂
a b c : P₁
a' b' c' : P₂
r : ℝ
hr_pos : 0 < r
hdist :
  ∀ (i₁ i₂ : Fin (Nat.succ 0).succ.succ),
    dist (![a, b, c] i₁) (![a, b, c] i₂) = r * dist (![a', b', c'] i₁) (![a', b', c'] i₂)
h_ab : dist a b = r * dist a' b'
h_cb : dist c b = r * dist c' b'
h_ac : dist a c = r * dist a' c'
h_cos : dist a' b' * dist c' b' * 2 * Real.cos (∠ a' b' c') = dist a' b' * dist c' b' * 2 * Real.cos (∠ a b c)
h_cos' : dist a' c' ^ 2 = dist a' b' ^ 2 + dist c' b' ^ 2 - dist a' b' * dist c' b' * 2 * Real.cos (∠ a' b' c')
h2 : dist c' b' = 0
⊢ ∠ 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
t : Triangle ℝ P
i₁ : Fin 3
⊢ t.orthocenter ∈ altitude t i₁

Tactic:
by
  obtain ⟨i₂, i₃, h₁₂, h₂₃, h₁₃⟩ : ∃ i₂ i₃, i₁ ≠ i₂ ∧ i₂ ≠ i₃ ∧ i₁ ≠ i₃ := by
    decide +revert
  rw [orthocenter_eq_mongePoint, t.altitude_eq_mongePlane h₁₂ h₁₃ h₂₃]
  exact t.mongePoint_mem_mongePlane

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

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
n : ℕ∞ω
x : M
k : 𝕜
f : C^∞⟮I, M; 𝕜⟯⟨x⟩
h : 𝕜
⊢ (k • f) x * h = k • (f x * h)

Tactic:
ContMDiffMap.coe_smul,

Goal state after:
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
n : ℕ∞ω
x : M
k : 𝕜
f : C^∞⟮I, M; 𝕜⟯⟨x⟩
h : 𝕜
⊢ (k • ⇑f) x * h = k • (f x * h)
Module: Mathlib.Geometry.Euclidean.MongePoint

Goal state before:
pp: case inr
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
ho : t.orthocenter ∉ Set.range t.points
p : Fin 3 → P
hps : Set.range p ⊆ insert t.orthocenter (Set.range t.points)
hpi : Function.Injective p
hs : Set.range p = Set.range t.points
⊢ ∃ c ∈ affineSpan ℝ (Set.range t.points), ∀ p₁ ∈ Set.range p, dist p₁ c = Simplex.circumradius t

Tactic:
· use t.circumcenter, t.circumcenter_mem_affineSpan
    intro p₁ hp₁
    rw [hs] at hp₁
    rcases hp₁ with ⟨i, rfl⟩
    exact t.dist_circumcenter_eq_circumradius _

Goal state after:
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
x₀ : B
ι : Type u_9
t : ι → (x : B) → E x
ht : LocallyFinite fun i ↦ {x | t i x ≠ 0}
ht' : ∀ (i : ι), MDiffAt (T% t) x₀
⊢ (MDiffAt fun x ↦ ⟨x, ∑' (i : ι), t i x⟩) x₀

Tactic:
simp_rw [← mdifferentiableWithinAt_univ] at ht' ⊢
  exact .sum_section_of_locallyFinite ht ht'

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
inst✝¹⁷ : NontriviallyNormedField 𝕜
n : WithTop ℕ∞
EB : Type u_4
inst✝¹⁶ : NormedAddCommGroup EB
inst✝¹⁵ : NormedSpace 𝕜 EB
HB : Type u_5
inst✝¹⁴ : TopologicalSpace HB
IB : ModelWithCorners 𝕜 EB HB
inst✝¹³ : TopologicalSpace B
inst✝¹² : ChartedSpace HB B
E₁ : B → Type u_6
inst✝¹¹ : (x : B) → AddCommGroup (E₁ x)
inst✝¹⁰ : (x : B) → Module 𝕜 (E₁ x)
inst✝⁹ : NormedAddCommGroup F₁
inst✝⁸ : NormedSpace 𝕜 F₁
inst✝⁷ : TopologicalSpace (TotalSpace F₁ E₁)
inst✝⁶ : (x : B) → TopologicalSpace (E₁ x)
inst✝⁵ : ∀ (x : B), IsTopologicalAddGroup (E₁ x)
inst✝⁴ : ∀ (x : B), ContinuousSMul 𝕜 (E₁ x)
inst✝³ : FiberBundle F₁ E₁
inst✝² : VectorBundle 𝕜 F₁ E₁
inst✝¹ : ContMDiffVectorBundle n F₁ E₁ IB
e : Trivialization F₁ TotalSpace.proj
inst✝ : MemTrivializationAtlas e
x : B
hx : x ∈ e.baseSet
⊢ ContMDiffAt IB (IB.prod 𝓘(𝕜, F₁ →L[𝕜] F₁)) n (fun m ↦ ⟨m, symmL 𝕜 e m⟩) x

Tactic:
have hx' : x ∈ (trivializationAt F₁ E₁ x).baseSet := mem_baseSet_trivializationAt F₁ E₁ x

Goal state after:
𝕜 : Type u_1
B : Type u_2
F₁ : Type u_3
inst✝¹⁷ : NontriviallyNormedField 𝕜
n : WithTop ℕ∞
EB : Type u_4
inst✝¹⁶ : NormedAddCommGroup EB
inst✝¹⁵ : NormedSpace 𝕜 EB
HB : Type u_5
inst✝¹⁴ : TopologicalSpace HB
IB : ModelWithCorners 𝕜 EB HB
inst✝¹³ : TopologicalSpace B
inst✝¹² : ChartedSpace HB B
E₁ : B → Type u_6
inst✝¹¹ : (x : B) → AddCommGroup (E₁ x)
inst✝¹⁰ : (x : B) → Module 𝕜 (E₁ x)
inst✝⁹ : NormedAddCommGroup F₁
inst✝⁸ : NormedSpace 𝕜 F₁
inst✝⁷ : TopologicalSpace (TotalSpace F₁ E₁)
inst✝⁶ : (x : B) → TopologicalSpace (E₁ x)
inst✝⁵ : ∀ (x : B), IsTopologicalAddGroup (E₁ x)
inst✝⁴ : ∀ (x : B), ContinuousSMul 𝕜 (E₁ x)
inst✝³ : FiberBundle F₁ E₁
inst✝² : VectorBundle 𝕜 F₁ E₁
inst✝¹ : ContMDiffVectorBundle n F₁ E₁ IB
e : Trivialization F₁ TotalSpace.proj
inst✝ : MemTrivializationAtlas e
x : B
hx : x ∈ e.baseSet
hx' : x ∈ (trivializationAt F₁ E₁ x).baseSet
⊢ ContMDiffAt IB (IB.prod 𝓘(𝕜, F₁ →L[𝕜] F₁)) n (fun m ↦ ⟨m, symmL 𝕜 e m⟩) x
Module: Mathlib.Geometry.Manifold.GroupLieAlgebra

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁸ : NontriviallyNormedField 𝕜
H : Type u_2
inst✝⁷ : TopologicalSpace H
E : Type u_3
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace 𝕜 E
I : ModelWithCorners 𝕜 E H
G : Type u_4
inst✝⁴ : TopologicalSpace G
inst✝³ : ChartedSpace H G
inst✝² : Group G
inst✝¹ : LieGroup I (minSmoothness 𝕜 3) G
inst✝ : CompleteSpace E
v w : GroupLieAlgebra I G
⊢ mulInvariantVectorField (mlieBracket I (mulInvariantVectorField v) (mulInvariantVectorField w) 1) =
    mlieBracket I (mulInvariantVectorField v) (mulInvariantVectorField w)

Tactic:
by
  ext g
  rw [mulInvariantVectorField_eq_mpullback, mpullback_mlieBracket (n := minSmoothness 𝕜 3),
    mpullback_mulInvariantVectorField, mpullback_mulInvariantVectorField]
  · exact mdifferentiableAt_mulInvariantVectorField _
  · exact mdifferentiableAt_mulInvariantVectorField _
  · exact contMDiffAt_mul_left
  · exact minSmoothness_monotone (by norm_cast)

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 : (x : M) → TangentSpace I x
inst✝¹ : IsManifold I 2 M
inst✝ : CompleteSpace E
hV : (MDiffAt[s] fun x ↦ ⟨x, V x⟩) x
⊢ DifferentiableWithinAt 𝕜 (mpullbackWithin 𝓘(𝕜, E) I (↑(extChartAt I x).symm) V (range ↑I))
    (↑(extChartAt I x).symm ⁻¹' s ∩ range ↑I) (↑(extChartAt I x) x)

Tactic:
by
  apply MDifferentiableWithinAt.differentiableWithinAt
  have := MDifferentiableWithinAt.mpullbackWithin_vectorField_inter_of_eq hV
    (contMDiffWithinAt_extChartAt_symm_range x (mem_extChartAt_target x))
    (isInvertible_mfderivWithin_extChartAt_symm (mem_extChartAt_target x)) (mem_range_self _)
    I.uniqueMDiffOn le_rfl (extChartAt_to_inv x).symm
  rw [inter_comm]
  exact (contMDiff_snd_tangentBundle_modelSpace E 𝓘(𝕜, E)).contMDiffAt.mdifferentiableAt one_ne_zero
    |>.comp_mdifferentiableWithinAt _ this

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁸ : NontriviallyNormedField 𝕜
H : Type u_2
inst✝⁷ : TopologicalSpace H
E : Type u_3
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace 𝕜 E
I : ModelWithCorners 𝕜 E H
G : Type u_4
inst✝⁴ : TopologicalSpace G
inst✝³ : ChartedSpace H G
inst✝² : Group G
inst✝¹ : LieGroup I (minSmoothness 𝕜 3) G
inst✝ : CompleteSpace E
c : 𝕜
v w : GroupLieAlgebra I G
⊢ ⁅v, c • w⁆ = c • ⁅v, w⁆

Tactic:
simp only [GroupLieAlgebra.bracket_def, mulInvariantVectorField_smul]
    rw [mlieBracket_const_smul_right]
    exact mdifferentiableAt_mulInvariantVectorField _

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
H : Type u_2
inst✝⁵ : TopologicalSpace H
E : Type u_3
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace H M
x : M
W : (x : M) → TangentSpace I x
inst✝ : IsManifold I 2 M
⊢ ((mfderiv% ↑(extChartAt I x) x).inverse ∘SL (mfderiv[range ↑I] ↑(extChartAt I x).symm (↑(extChartAt I x) x)).inverse)
      (W (↑(extChartAt I x).symm (↑(extChartAt I x) x))) =
    W x

Tactic:
by
  rw [mfderiv_extChartAt_inverse_comp_mfderivWithin_extChartAT_symm, extChartAt_to_inv]

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝¹⁴ : NontriviallyNormedField 𝕜
H : Type u_2
inst✝¹³ : TopologicalSpace H
E : Type u_3
inst✝¹² : NormedAddCommGroup E
inst✝¹¹ : NormedSpace 𝕜 E
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝¹⁰ : TopologicalSpace M
inst✝⁹ : ChartedSpace H M
H' : Type u_5
inst✝⁸ : TopologicalSpace H'
E' : Type u_6
inst✝⁷ : NormedAddCommGroup E'
inst✝⁶ : NormedSpace 𝕜 E'
I' : ModelWithCorners 𝕜 E' H'
M' : Type u_7
inst✝⁵ : TopologicalSpace M'
inst✝⁴ : ChartedSpace H' M'
inst✝³ : IsManifold I 2 M
inst✝² : IsManifold I' 2 M'
inst✝¹ : CompleteSpace E
inst✝ : CompleteSpace E'
f : M → M'
V W : (x : M') → TangentSpace I' x
x₀ : M
s : Set M
t : Set M'
hV : MDiffAt[t] (T% V) (f x₀)
hW : MDiffAt[t] (T% W) (f x₀)
hu : UniqueMDiff[s]
hf : ContMDiffOn I I' 2 f s
hx₀ : x₀ ∈ s
ht : t ⊆ (extChartAt I' (f x₀)).source
hst : MapsTo f s t
hsymm :
  IsSymmSndFDerivWithinAt 𝕜 (↑(extChartAt I' (f x₀)) ∘ f ∘ ↑(extChartAt I x₀).symm)
    (↑(extChartAt I x₀).symm ⁻¹' s ∩ range ↑I) (↑(extChartAt I x₀) x₀)
⊢ mpullbackWithin I I' f (mlieBracketWithin I' V W t) s x₀ =
    mlieBracketWithin I (mpullbackWithin I I' f V s) (mpullbackWithin I I' f W s) s x₀

Tactic:
have A : (extChartAt I x₀).symm (extChartAt I x₀ x₀) = x₀ := by simp

Goal state after:
𝕜 : Type u_1
inst✝¹⁴ : NontriviallyNormedField 𝕜
H : Type u_2
inst✝¹³ : TopologicalSpace H
E : Type u_3
inst✝¹² : NormedAddCommGroup E
inst✝¹¹ : NormedSpace 𝕜 E
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝¹⁰ : TopologicalSpace M
inst✝⁹ : ChartedSpace H M
H' : Type u_5
inst✝⁸ : TopologicalSpace H'
E' : Type u_6
inst✝⁷ : NormedAddCommGroup E'
inst✝⁶ : NormedSpace 𝕜 E'
I' : ModelWithCorners 𝕜 E' H'
M' : Type u_7
inst✝⁵ : TopologicalSpace M'
inst✝⁴ : ChartedSpace H' M'
inst✝³ : IsManifold I 2 M
inst✝² : IsManifold I' 2 M'
inst✝¹ : CompleteSpace E
inst✝ : CompleteSpace E'
f : M → M'
V W : (x : M') → TangentSpace I' x
x₀ : M
s : Set M
t : Set M'
hV : MDiffAt[t] (T% V) (f x₀)
hW : MDiffAt[t] (T% W) (f x₀)
hu : UniqueMDiff[s]
hf : ContMDiffOn I I' 2 f s
hx₀ : x₀ ∈ s
ht : t ⊆ (extChartAt I' (f x₀)).source
hst : MapsTo f s t
hsymm :
  IsSymmSndFDerivWithinAt 𝕜 (↑(extChartAt I' (f x₀)) ∘ f ∘ ↑(extChartAt I x₀).symm)
    (↑(extChartAt I x₀).symm ⁻¹' s ∩ range ↑I) (↑(extChartAt I x₀) x₀)
A : ↑(extChartAt I x₀).symm (↑(extChartAt I x₀) x₀) = x₀
⊢ mpullbackWithin I I' f (mlieBracketWithin I' V W t) s x₀ =
    mlieBracketWithin I (mpullbackWithin I I' f V s) (mpullbackWithin I I' f W s) s x₀
Module: Mathlib.Geometry.Manifold.Instances.Sphere

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace ℝ E
v : E
hv : ‖v‖ = 1
w : ↥(ℝ ∙ v)ᗮ
⊢ ↑(stereoInvFun hv w) ≠ ↑⟨v, ⋯⟩

Tactic:
rw [← inner_lt_one_iff_real_of_norm_eq_one _ hv]

Goal state after:
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace ℝ E
v : E
hv : ‖v‖ = 1
w : ↥(ℝ ∙ v)ᗮ
⊢ ⟪↑(stereoInvFun hv w), v⟫_ℝ < 1

E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace ℝ E
v : E
hv : ‖v‖ = 1
w : ↥(ℝ ∙ v)ᗮ
⊢ ‖↑(stereoInvFun hv w)‖ = 1
Module: Mathlib.Geometry.Manifold.Instances.UnitsOfNormedAlgebra

Goal state before:
pp: R : Type u_1
inst✝⁸ : NormedRing R
inst✝⁷ : CompleteSpace R
n : ℕ∞ω
𝕜 : Type u_2
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NormedAlgebra 𝕜 R
H : Type u_3
inst✝⁴ : TopologicalSpace H
E : Type u_4
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
I : ModelWithCorners 𝕜 E H
M : Type u_5
inst✝¹ : TopologicalSpace M
inst✝ : ChartedSpace H M
this : (val ∘ fun x ↦ x.1 * x.2) = (fun x ↦ x.1 * x.2) ∘ fun x ↦ (↑x.1, ↑x.2)
⊢ ContMDiff (𝓘(𝕜, R).prod 𝓘(𝕜, R)) 𝓘(𝕜, R) n ((fun x ↦ x.1 * x.2) ∘ fun x ↦ (↑x.1, ↑x.2))

Tactic:
have : ContMDiff (𝓘(𝕜, R).prod 𝓘(𝕜, R)) 𝓘(𝕜, R × R) n
      (fun x : Rˣ × Rˣ => ((x.1 : R), (x.2 : R))) :=
      (contMDiff_val.comp contMDiff_fst).prodMk_space (contMDiff_val.comp contMDiff_snd)

Goal state after:
R : Type u_1
inst✝⁸ : NormedRing R
inst✝⁷ : CompleteSpace R
n : ℕ∞ω
𝕜 : Type u_2
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NormedAlgebra 𝕜 R
H : Type u_3
inst✝⁴ : TopologicalSpace H
E : Type u_4
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
I : ModelWithCorners 𝕜 E H
M : Type u_5
inst✝¹ : TopologicalSpace M
inst✝ : ChartedSpace H M
this✝ : (val ∘ fun x ↦ x.1 * x.2) = (fun x ↦ x.1 * x.2) ∘ fun x ↦ (↑x.1, ↑x.2)
this : ContMDiff (𝓘(𝕜, R).prod 𝓘(𝕜, R)) 𝓘(𝕜, R × R) n fun x ↦ (↑x.1, ↑x.2)
⊢ ContMDiff (𝓘(𝕜, R).prod 𝓘(𝕜, R)) 𝓘(𝕜, R) n ((fun x ↦ x.1 * x.2) ∘ fun x ↦ (↑x.1, ↑x.2))
Module: Mathlib.Geometry.Manifold.IntegralCurve.Transform

Goal state before:
pp: E : Type u_1
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
H : Type u_2
inst✝² : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹ : TopologicalSpace M
inst✝ : ChartedSpace H M
γ : ℝ → M
v : (x : M) → TangentSpace I x
s : Set ℝ
a : ℝ
ha : a ≠ 0
⊢ IsMIntegralCurveOn (γ ∘ fun x ↦ x * a) (a • v) {t | t * a ∈ s} ↔ IsMIntegralCurveOn γ v s

Tactic:
refine ⟨fun hγ ↦ ?_, fun hγ ↦ hγ.comp_mul a⟩

Goal state after:
E : Type u_1
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
H : Type u_2
inst✝² : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹ : TopologicalSpace M
inst✝ : ChartedSpace H M
γ : ℝ → M
v : (x : M) → TangentSpace I x
s : Set ℝ
a : ℝ
ha : a ≠ 0
hγ : IsMIntegralCurveOn (γ ∘ fun x ↦ x * a) (a • v) {t | t * a ∈ s}
⊢ IsMIntegralCurveOn γ v s
Module: Mathlib.Geometry.Manifold.ContMDiffMFDeriv

Goal state before:
pp: 𝕜 : Type u_1
inst✝¹⁰ : NontriviallyNormedField 𝕜
m n : WithTop ℕ∞
E : Type u_2
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace 𝕜 E
H : Type u_3
inst✝⁷ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝⁶ : TopologicalSpace M
inst✝⁵ : ChartedSpace H M
E' : Type u_5
inst✝⁴ : NormedAddCommGroup E'
inst✝³ : NormedSpace 𝕜 E'
H' : Type u_6
inst✝² : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
M' : Type u_7
inst✝¹ : TopologicalSpace M'
inst✝ : ChartedSpace H' M'
f : M → M'
Is : IsManifold I 1 M
I's : IsManifold I' 1 M'
hf : ContMDiffOn I I' n f univ
hmn : m + 1 ≤ n
⊢ ContMDiffOn I.tangent I'.tangent m (tangentMap I I' f) univ

Tactic:
convert! hf.contMDiffOn_tangentMapWithin hmn uniqueMDiffOn_univ

Goal state after:
case e'_22
𝕜 : Type u_1
inst✝¹⁰ : NontriviallyNormedField 𝕜
m n : WithTop ℕ∞
E : Type u_2
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace 𝕜 E
H : Type u_3
inst✝⁷ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝⁶ : TopologicalSpace M
inst✝⁵ : ChartedSpace H M
E' : Type u_5
inst✝⁴ : NormedAddCommGroup E'
inst✝³ : NormedSpace 𝕜 E'
H' : Type u_6
inst✝² : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
M' : Type u_7
inst✝¹ : TopologicalSpace M'
inst✝ : ChartedSpace H' M'
f : M → M'
Is : IsManifold I 1 M
I's : IsManifold I' 1 M'
hf : ContMDiffOn I I' n f univ
hmn : m + 1 ≤ n
⊢ tangentMap I I' f = tangentMapWithin I I' f univ
Module: Mathlib.Geometry.Manifold.IntegralCurve.UniformTime

Goal state before:
pp: case h₁
E : Type u_1
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
H : Type u_2
inst✝⁵ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁴ : TopologicalSpace M
inst✝³ : ChartedSpace H M
inst✝² : IsManifold I 1 M
inst✝¹ : T2Space M
v : (x : M) → TangentSpace I x
inst✝ : BoundarylessManifold I M
hv : ContMDiff I I.tangent 1 fun x ↦ ⟨x, v x⟩
x : M
γ : ℝ → ℝ → M
hγx : ∀ (a : ℝ), γ a 0 = x
hγ : ∀ a > 0, IsMIntegralCurveOn (γ a) v (Ioo (-a) a)
t : ℝ
ht : t ∈ Ioo (-(|t| + 1)) (|t| + 1)
⊢ ∃ s ∈ 𝓝 t, EqOn (fun t ↦ γ (|t| + 1) t) (γ (|t| + 1)) s

Tactic:
refine ⟨Ioo (-(|t| + 1)) (|t| + 1), ?_,
      eqOn_abs_add_one_of_isMIntegralCurveOn_Ioo hv γ hγx hγ⟩

Goal state after:
case h₁
E : Type u_1
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
H : Type u_2
inst✝⁵ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁴ : TopologicalSpace M
inst✝³ : ChartedSpace H M
inst✝² : IsManifold I 1 M
inst✝¹ : T2Space M
v : (x : M) → TangentSpace I x
inst✝ : BoundarylessManifold I M
hv : ContMDiff I I.tangent 1 fun x ↦ ⟨x, v x⟩
x : M
γ : ℝ → ℝ → M
hγx : ∀ (a : ℝ), γ a 0 = x
hγ : ∀ a > 0, IsMIntegralCurveOn (γ a) v (Ioo (-a) a)
t : ℝ
ht : t ∈ Ioo (-(|t| + 1)) (|t| + 1)
⊢ Ioo (-(|t| + 1)) (|t| + 1) ∈ 𝓝 t
Module: Mathlib.Geometry.Manifold.VectorField.LieBracket

Goal state before:
pp: 𝕜 : Type u_1
inst✝¹³ : NontriviallyNormedField 𝕜
H : Type u_2
inst✝¹² : TopologicalSpace H
E : Type u_3
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace 𝕜 E
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝⁹ : TopologicalSpace M
inst✝⁸ : ChartedSpace H M
H' : Type u_5
inst✝⁷ : TopologicalSpace H'
E' : Type u_6
inst✝⁶ : NormedAddCommGroup E'
inst✝⁵ : NormedSpace 𝕜 E'
I' : ModelWithCorners 𝕜 E' H'
M' : Type u_7
inst✝⁴ : TopologicalSpace M'
inst✝³ : ChartedSpace H' M'
inst✝² : IsManifold I 2 M
inst✝¹ : IsManifold I' 2 M'
inst✝ : CompleteSpace E
f : M → M'
V W : (x : M') → TangentSpace I' x
x₀ : M
s : Set M
t : Set M'
hV : MDiffAt[t] (T% V) (f x₀)
hW : MDiffAt[t] (T% W) (f x₀)
hu : UniqueMDiff[s]
hf : ContMDiffWithinAt I I' 2 f s x₀
hx₀ : x₀ ∈ s
hst : f ⁻¹' t ∈ 𝓝[s] x₀
hsymm :
  IsSymmSndFDerivWithinAt 𝕜 (↑(extChartAt I' (f x₀)) ∘ f ∘ ↑(extChartAt I x₀).symm)
    (↑(extChartAt I x₀).symm ⁻¹' s ∩ range ↑I) (↑(extChartAt I x₀) x₀)
⊢ mpullbackWithin I I' f (mlieBracketWithin I' V W t) s x₀ =
    mlieBracketWithin I (mpullbackWithin I I' f V s) (mpullbackWithin I I' f W s) s x₀

Tactic:
have A : (extChartAt I x₀).symm (extChartAt I x₀ x₀) = x₀ := by simp

Goal state after:
𝕜 : Type u_1
inst✝¹³ : NontriviallyNormedField 𝕜
H : Type u_2
inst✝¹² : TopologicalSpace H
E : Type u_3
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace 𝕜 E
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝⁹ : TopologicalSpace M
inst✝⁸ : ChartedSpace H M
H' : Type u_5
inst✝⁷ : TopologicalSpace H'
E' : Type u_6
inst✝⁶ : NormedAddCommGroup E'
inst✝⁵ : NormedSpace 𝕜 E'
I' : ModelWithCorners 𝕜 E' H'
M' : Type u_7
inst✝⁴ : TopologicalSpace M'
inst✝³ : ChartedSpace H' M'
inst✝² : IsManifold I 2 M
inst✝¹ : IsManifold I' 2 M'
inst✝ : CompleteSpace E
f : M → M'
V W : (x : M') → TangentSpace I' x
x₀ : M
s : Set M
t : Set M'
hV : MDiffAt[t] (T% V) (f x₀)
hW : MDiffAt[t] (T% W) (f x₀)
hu : UniqueMDiff[s]
hf : ContMDiffWithinAt I I' 2 f s x₀
hx₀ : x₀ ∈ s
hst : f ⁻¹' t ∈ 𝓝[s] x₀
hsymm :
  IsSymmSndFDerivWithinAt 𝕜 (↑(extChartAt I' (f x₀)) ∘ f ∘ ↑(extChartAt I x₀).symm)
    (↑(extChartAt I x₀).symm ⁻¹' s ∩ range ↑I) (↑(extChartAt I x₀) x₀)
A : ↑(extChartAt I x₀).symm (↑(extChartAt I x₀) x₀) = x₀
⊢ mpullbackWithin I I' f (mlieBracketWithin I' V W t) s x₀ =
    mlieBracketWithin I (mpullbackWithin I I' f V s) (mpullbackWithin I I' f W s) s x₀