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

Goal state before:
pp: V : Type u_1
P : Type u_2
inst✝⁷ : NormedAddCommGroup V
inst✝⁶ : InnerProductSpace ℝ V
inst✝⁵ : MetricSpace P
inst✝⁴ : NormedAddTorsor V P
V₂ : Type u_3
P₂ : Type u_4
inst✝³ : NormedAddCommGroup V₂
inst✝² : InnerProductSpace ℝ V₂
inst✝¹ : MetricSpace P₂
inst✝ : NormedAddTorsor V₂ P₂
n : ℕ
s : Simplex ℝ P n
f : P →ᵃⁱ[ℝ] P₂
⊢ (s.map f.toAffineMap ⋯).mongePoint = f s.mongePoint

Tactic:
simp_rw [mongePoint_eq_smul_vsub_vadd_circumcenter]
  rw [← Simplex.centroid, ← Simplex.centroid]
  simp [centroid_map, circumcenter_map]

Goal state after:
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}
h₁ : ∠ a b c = ∠ a' b' c'
h₂ : ∠ b c a = ∠ b' c' a'
hne_pi_div_two : ∠ a b c ≠ Real.pi / 2 ∨ ∠ b c a ≠ Real.pi / 2
not_all_eq : a' ≠ b' ∨ b' ≠ c' ∨ a' ≠ c'
h_not_col' : ¬Collinear ℝ {a', b', c'}
h_pos1 : 0 < dist a b
h_pos1' : 0 < dist a' b'
h_pos2 : 0 < dist b c
h_pos2' : 0 < dist b' c'
h₃ : ∠ c a b = ∠ c' a' b'
h_sin_ne1 : Real.sin (∠ b c a) ≠ 0
h_sin_ne2 : Real.sin (∠ c a b) ≠ 0
h_sin1 : dist a b * dist b' c' = dist b c * dist a' b'
h_sin1' : Real.sin (∠ c a b) = Real.sin (∠ b c a) * dist b' c' / dist a' b'
h_sin2 : Real.sin (∠ a b c) * dist b c = Real.sin (∠ c a b) * dist c a
h_sin2' : Real.sin (∠ a' b' c') * dist b' c' = Real.sin (∠ c' a' b') * dist c' a'
⊢ ![a, b, c] ∼ ![a', b', c']

Tactic:
rw [← eq_div_iff_mul_eq (by positivity)] at h_sin2 h_sin2'

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}
h₁ : ∠ a b c = ∠ a' b' c'
h₂ : ∠ b c a = ∠ b' c' a'
hne_pi_div_two : ∠ a b c ≠ Real.pi / 2 ∨ ∠ b c a ≠ Real.pi / 2
not_all_eq : a' ≠ b' ∨ b' ≠ c' ∨ a' ≠ c'
h_not_col' : ¬Collinear ℝ {a', b', c'}
h_pos1 : 0 < dist a b
h_pos1' : 0 < dist a' b'
h_pos2 : 0 < dist b c
h_pos2' : 0 < dist b' c'
h₃ : ∠ c a b = ∠ c' a' b'
h_sin_ne1 : Real.sin (∠ b c a) ≠ 0
h_sin_ne2 : Real.sin (∠ c a b) ≠ 0
h_sin1 : dist a b * dist b' c' = dist b c * dist a' b'
h_sin1' : Real.sin (∠ c a b) = Real.sin (∠ b c a) * dist b' c' / dist a' b'
h_sin2 : Real.sin (∠ a b c) = Real.sin (∠ c a b) * dist c a / dist b c
h_sin2' : Real.sin (∠ a' b' c') = Real.sin (∠ c' a' b') * dist c' a' / dist b' c'
⊢ ![a, b, c] ∼ ![a', b', c']
Module: Mathlib.Geometry.Euclidean.MongePoint

Goal state before:
pp: V : Type u_1
P : Type u_2
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
n : ℕ
s : Simplex ℝ P (n + 2)
i₁ i₂ : Fin (n + 3)
⊢ s.mongePlane i₁ i₂ = s.mongePlane i₂ i₁

Tactic:
by
  simp_rw [mongePlane_def]
  congr 3
  · congr 1
    exact pair_comm _ _
  · ext
    simp_rw [Submodule.mem_span_singleton]
    constructor
    all_goals rintro ⟨r, rfl⟩; use -r; rw [neg_smul, ← smul_neg, neg_vsub_eq_vsub_rev]

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

Goal state before:
pp: V : Type u_1
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
P : Type u_2
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
s : Sphere P
p a b : P
hp : p ∈ line[ℝ, a, b]
ha : a ∈ s
hb : b ∈ s
hq : dist a s.center = dist b s.center
⊢ dist p a * dist p b = |s.power p|

Tactic:
dist_comm p a,

Goal state after:
V : Type u_1
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
P : Type u_2
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
s : Sphere P
p a b : P
hp : p ∈ line[ℝ, a, b]
ha : a ∈ s
hb : b ∈ s
hq : dist a s.center = dist b s.center
⊢ dist a p * dist p b = |s.power p|
Module: Mathlib.Geometry.Euclidean.Sphere.Power

Goal state before:
pp: V : Type u_1
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
P : Type u_2
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
s : Sphere P
p a b : P
hp : p ∈ line[ℝ, a, b]
ha : a ∈ s
hb : b ∈ s
hp_on : p ∈ s
hq : dist a s.center = dist b s.center
⊢ dist p a * dist p b = 0

Tactic:
dist_comm p a,

Goal state after:
V : Type u_1
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
P : Type u_2
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
s : Sphere P
p a b : P
hp : p ∈ line[ℝ, a, b]
ha : a ∈ s
hb : b ∈ s
hp_on : p ∈ s
hq : dist a s.center = dist b s.center
⊢ dist a p * dist p b = 0
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 i₂ : Fin 3
h : i₁ ≠ i₂
⊢ dist t.orthocenter ((reflection (affineSpan ℝ (t.points '' {i₁, i₂}))) (circumcenter t)) = circumradius t

Tactic:
rw [← mul_self_inj_of_nonneg dist_nonneg t.circumradius_nonneg,
    t.reflection_circumcenter_eq_affineCombination_of_pointsWithCircumcenter h,
    t.orthocenter_eq_mongePoint, mongePoint_eq_affineCombination_of_pointsWithCircumcenter,
    dist_affineCombination t.pointsWithCircumcenter (sum_mongePointWeightsWithCircumcenter _)
      (sum_reflectionCircumcenterWeightsWithCircumcenter h)]
  simp_rw [sum_pointsWithCircumcenter, Pi.sub_apply, mongePointWeightsWithCircumcenter,
    reflectionCircumcenterWeightsWithCircumcenter]
  have hu : ({i₁, i₂} : Finset (Fin 3)) ⊆ univ := subset_univ _
  obtain ⟨i₃, hi₃, hi₃₁, hi₃₂⟩ :
      ∃ i₃, univ \ ({i₁, i₂} : Finset (Fin 3)) = {i₃} ∧ i₃ ≠ i₁ ∧ i₃ ≠ i₂ := by
    decide +revert
  simp_rw [← sum_sdiff hu, hi₃]
  norm_num [hi₃₁, hi₃₂]

Goal state after:
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
hAsymm : A⁻¹ = A
π : G →* G ⧸ H := QuotientGroup.mk' H
a : G
ha : a ∈ A
⊢ #({x ∈ A⁻¹ * A | x ∈ H}) = #({x ∈ A ^ 2 | x ∈ H})

Tactic:
simp [hAsymm, sq]

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝¹⁰ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace 𝕜 E
F : Type u_3
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : NormedSpace 𝕜 F
H₁ : Type u_5
inst✝⁵ : TopologicalSpace H₁
H₂ : Type u_6
inst✝⁴ : TopologicalSpace H₂
I : ModelWithCorners 𝕜 E H₁
J : ModelWithCorners 𝕜 F H₂
M : Type u_8
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace H₁ M
N : Type u_9
inst✝¹ : TopologicalSpace N
inst✝ : ChartedSpace H₂ N
n : WithTop ℕ∞
f : M → N
x : M
Φ : PartialDiffeomorph I J M N n
hx : x ∈ Φ.source
heq : EqOn f (↑Φ.toPartialEquiv) Φ.source
⊢ ContMDiffAt I J n f x

Tactic:
exact ((Φ.contMDiffOn_toFun).congr heq).contMDiffAt (Φ.open_source.mem_nhds hx)

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

Goal state before:
pp: case h
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
u : Set B
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[u] (T% t) x₀
u' : Set B
hu' : u' ∈ 𝓝 x₀
hfin : {i | ((fun i ↦ {x | t i x ≠ 0}) i ∩ u').Nonempty}.Finite
s : Set ι := ⋯
this✝ : Fintype ↑{i | ((fun i ↦ {x | t i x ≠ 0}) i ∩ u').Nonempty}
this : (MDiffAt[u ∩ u'] fun x ↦ ⟨x, ∑ i ∈ s.toFinset, t i x⟩) x₀
y : B
hy : y ∈ u'
i : ι
hi : i ∉ ↑s.toFinset
h : t i y ≠ 0
⊢ y ∈ {x | ¬t i x = 0} ∩ u'

Tactic:
simp [h, hy]

Goal state after:
Module: Mathlib.Topology.VectorBundle.Hom

Goal state before:
pp: 𝕜₁ : Type u_1
inst✝¹⁸ : NontriviallyNormedField 𝕜₁
𝕜₂ : Type u_2
inst✝¹⁷ : NontriviallyNormedField 𝕜₂
σ : 𝕜₁ →+* 𝕜₂
B : Type u_3
F₁ : Type u_4
inst✝¹⁶ : NormedAddCommGroup F₁
inst✝¹⁵ : NormedSpace 𝕜₁ F₁
E₁ : B → Type u_5
inst✝¹⁴ : (x : B) → AddCommGroup (E₁ x)
inst✝¹³ : (x : B) → Module 𝕜₁ (E₁ x)
inst✝¹² : TopologicalSpace (TotalSpace F₁ E₁)
F₂ : Type u_6
inst✝¹¹ : NormedAddCommGroup F₂
inst✝¹⁰ : NormedSpace 𝕜₂ F₂
E₂ : B → Type u_7
inst✝⁹ : (x : B) → AddCommGroup (E₂ x)
inst✝⁸ : (x : B) → Module 𝕜₂ (E₂ x)
inst✝⁷ : TopologicalSpace (TotalSpace F₂ E₂)
inst✝⁶ : TopologicalSpace B
e₁ : Trivialization F₁ TotalSpace.proj
e₂ : Trivialization F₂ TotalSpace.proj
inst✝⁵ : (x : B) → TopologicalSpace (E₁ x)
inst✝⁴ : FiberBundle F₁ E₁
inst✝³ : (x : B) → TopologicalSpace (E₂ x)
inst✝² : FiberBundle F₂ E₂
inst✝¹ : Trivialization.IsLinear 𝕜₁ e₁
inst✝ : Trivialization.IsLinear 𝕜₂ e₂
b : B
hb : b ∈ e₁.baseSet ∩ e₂.baseSet
L : F₁ →SL[σ] F₂
⊢ (continuousLinearMap σ e₁ e₂).symm b L =
    Trivialization.symmL 𝕜₂ e₂ b ∘SL L ∘SL Trivialization.continuousLinearMapAt 𝕜₁ e₁ b

Tactic:
by
  rw [symm_apply]
  · rfl
  · exact hb

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

Goal state before:
pp: case refine_1
𝕜 : Type u_1
inst✝¹⁴ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹³ : NormedAddCommGroup E
inst✝¹² : NormedSpace 𝕜 E
H : Type u_3
inst✝¹¹ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝¹⁰ : TopologicalSpace M
inst✝⁹ : ChartedSpace H M
F : Type u_5
inst✝⁸ : NormedAddCommGroup F
V : M → Type u_6
inst✝⁷ : TopologicalSpace (TotalSpace F V)
inst✝⁶ : (x : M) → AddCommGroup (V x)
inst✝⁵ : (x : M) → TopologicalSpace (V x)
inst✝⁴ : FiberBundle F V
inst✝³ : NormedSpace 𝕜 F
k : WithTop ℕ∞
inst✝² : (x : M) → Module 𝕜 (V x)
inst✝¹ : VectorBundle 𝕜 F V
inst✝ : ContMDiffVectorBundle k F V I
x₀ : M
σ₀ : V x₀
t : Trivialization F TotalSpace.proj := trivializationAt F V x₀
⊢ t.baseSet ∈ 𝓝 x₀

Tactic:
refine t.open_baseSet.mem_nhds ?_

Goal state after:
case refine_1
𝕜 : Type u_1
inst✝¹⁴ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹³ : NormedAddCommGroup E
inst✝¹² : NormedSpace 𝕜 E
H : Type u_3
inst✝¹¹ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝¹⁰ : TopologicalSpace M
inst✝⁹ : ChartedSpace H M
F : Type u_5
inst✝⁸ : NormedAddCommGroup F
V : M → Type u_6
inst✝⁷ : TopologicalSpace (TotalSpace F V)
inst✝⁶ : (x : M) → AddCommGroup (V x)
inst✝⁵ : (x : M) → TopologicalSpace (V x)
inst✝⁴ : FiberBundle F V
inst✝³ : NormedSpace 𝕜 F
k : WithTop ℕ∞
inst✝² : (x : M) → Module 𝕜 (V x)
inst✝¹ : VectorBundle 𝕜 F V
inst✝ : ContMDiffVectorBundle k F V I
x₀ : M
σ₀ : V x₀
t : Trivialization F TotalSpace.proj := trivializationAt F V x₀
⊢ x₀ ∈ t.baseSet
Module: Mathlib.Geometry.Manifold.VectorField.Pullback

Goal state before:
pp: 𝕜 : Type u_1
inst✝¹⁵ : NontriviallyNormedField 𝕜
H : Type u_2
inst✝¹⁴ : TopologicalSpace H
E : Type u_3
inst✝¹³ : NormedAddCommGroup E
inst✝¹² : NormedSpace 𝕜 E
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝¹¹ : TopologicalSpace M
inst✝¹⁰ : ChartedSpace H M
H' : Type u_5
inst✝⁹ : TopologicalSpace H'
E' : Type u_6
inst✝⁸ : NormedAddCommGroup E'
inst✝⁷ : NormedSpace 𝕜 E'
I' : ModelWithCorners 𝕜 E' H'
M' : Type u_7
inst✝⁶ : TopologicalSpace M'
inst✝⁵ : ChartedSpace H' M'
H'' : Type u_8
inst✝⁴ : TopologicalSpace H''
E'' : Type u_9
inst✝³ : NormedAddCommGroup E''
inst✝² : NormedSpace 𝕜 E''
I'' : ModelWithCorners 𝕜 E'' H''
M'' : Type u_10
inst✝¹ : TopologicalSpace M''
inst✝ : ChartedSpace H'' M''
g : M' → M''
f : M → M'
V : (x : M'') → TangentSpace I'' x
s : Set M
t : Set M'
x₀ : M
hg : MDiffAt[t] g (f x₀)
h : MapsTo f s t
hu : UniqueMDiffAt[s] x₀
hf' : (mfderiv[s] f x₀).IsInvertible
hf : MDiffAt[s] f x₀
⊢ (mfderiv[s] f x₀).inverse ((mfderiv[t] g (f x₀)).inverse (V ((g ∘ f) x₀))) =
    (mfderiv[s] f x₀).inverse ((mfderiv[t] g (f x₀)).inverse (V (g (f x₀))))

Tactic:
Function.comp_apply

Goal state after:
𝕜 : Type u_1
inst✝¹⁵ : NontriviallyNormedField 𝕜
H : Type u_2
inst✝¹⁴ : TopologicalSpace H
E : Type u_3
inst✝¹³ : NormedAddCommGroup E
inst✝¹² : NormedSpace 𝕜 E
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝¹¹ : TopologicalSpace M
inst✝¹⁰ : ChartedSpace H M
H' : Type u_5
inst✝⁹ : TopologicalSpace H'
E' : Type u_6
inst✝⁸ : NormedAddCommGroup E'
inst✝⁷ : NormedSpace 𝕜 E'
I' : ModelWithCorners 𝕜 E' H'
M' : Type u_7
inst✝⁶ : TopologicalSpace M'
inst✝⁵ : ChartedSpace H' M'
H'' : Type u_8
inst✝⁴ : TopologicalSpace H''
E'' : Type u_9
inst✝³ : NormedAddCommGroup E''
inst✝² : NormedSpace 𝕜 E''
I'' : ModelWithCorners 𝕜 E'' H''
M'' : Type u_10
inst✝¹ : TopologicalSpace M''
inst✝ : ChartedSpace H'' M''
g : M' → M''
f : M → M'
V : (x : M'') → TangentSpace I'' x
s : Set M
t : Set M'
x₀ : M
hg : MDiffAt[t] g (f x₀)
h : MapsTo f s t
hu : UniqueMDiffAt[s] x₀
hf' : (mfderiv[s] f x₀).IsInvertible
hf : MDiffAt[s] f x₀
⊢ (mfderiv[s] f x₀).inverse ((mfderiv[t] g (f x₀)).inverse (V (g (f x₀)))) =
    (mfderiv[s] f x₀).inverse ((mfderiv[t] g (f x₀)).inverse (V (g (f x₀))))
Module: Mathlib.Geometry.Manifold.VectorField.LieBracket

Goal state before:
pp: case e_6
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[s] x] V
hxV : V₁ x = V x
hW : W₁ =ᶠ[𝓝[s] x] W
hxW : W₁ x = W x
I1 : NormedAddCommGroup (TangentSpace 𝓘(𝕜, E) (↑(extChartAt I x) x)) :=
  { toNorm := EventuallyEq.mlieBracketWithin_vectorField_eq._aux_1_1,
    toAddCommGroup := instAddCommGroupTangentSpace 𝓘(𝕜, E) (↑(extChartAt I x) x),
    toPseudoMetricSpace := EventuallyEq.mlieBracketWithin_vectorField_eq._aux_1_13, eq_of_dist_eq_zero := ⋯,
    dist_eq := ⋯ }
⊢ lieBracketWithin 𝕜 (mpullbackWithin 𝓘(𝕜, E) I (↑(extChartAt I x).symm) V₁ (range ↑I))
      (mpullbackWithin 𝓘(𝕜, E) I (↑(extChartAt I x).symm) W₁ (range ↑I)) (↑(extChartAt I x).symm ⁻¹' s ∩ range ↑I)
      (↑(extChartAt I x) x) =
    lieBracketWithin 𝕜 (mpullbackWithin 𝓘(𝕜, E) I (↑(extChartAt I x).symm) V (range ↑I))
      (mpullbackWithin 𝓘(𝕜, E) I (↑(extChartAt I x).symm) W (range ↑I)) (↑(extChartAt I x).symm ⁻¹' s ∩ range ↑I)
      (↑(extChartAt I x) x)

Tactic:
let _I2 : NormedSpace 𝕜 (TangentSpace 𝓘(𝕜, E) (extChartAt I x x)) :=
    ‹NormedSpace 𝕜 E›

Goal state after:
case e_6
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[s] x] V
hxV : V₁ x = V x
hW : W₁ =ᶠ[𝓝[s] x] W
hxW : W₁ x = W x
I1 : NormedAddCommGroup (TangentSpace 𝓘(𝕜, E) (↑(extChartAt I x) x)) :=
  { toNorm := EventuallyEq.mlieBracketWithin_vectorField_eq._aux_1_1,
    toAddCommGroup := instAddCommGroupTangentSpace 𝓘(𝕜, E) (↑(extChartAt I x) x),
    toPseudoMetricSpace := EventuallyEq.mlieBracketWithin_vectorField_eq._aux_1_13, eq_of_dist_eq_zero := ⋯,
    dist_eq := ⋯ }
_I2 : NormedSpace 𝕜 (TangentSpace 𝓘(𝕜, E) (↑(extChartAt I x) x)) := inst✝²
⊢ lieBracketWithin 𝕜 (mpullbackWithin 𝓘(𝕜, E) I (↑(extChartAt I x).symm) V₁ (range ↑I))
      (mpullbackWithin 𝓘(𝕜, E) I (↑(extChartAt I x).symm) W₁ (range ↑I)) (↑(extChartAt I x).symm ⁻¹' s ∩ range ↑I)
      (↑(extChartAt I x) x) =
    lieBracketWithin 𝕜 (mpullbackWithin 𝓘(𝕜, E) I (↑(extChartAt I x).symm) V (range ↑I))
      (mpullbackWithin 𝓘(𝕜, E) I (↑(extChartAt I x).symm) W (range ↑I)) (↑(extChartAt I x).symm ⁻¹' s ∩ range ↑I)
      (↑(extChartAt I x) x)
Module: Mathlib.Geometry.Manifold.GroupLieAlgebra

Goal state before:
pp: case hg
𝕜 : Type u_1
inst✝⁷ : NontriviallyNormedField 𝕜
H : Type u_2
inst✝⁶ : TopologicalSpace H
E : Type u_3
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
I : ModelWithCorners 𝕜 E H
G : Type u_4
inst✝³ : TopologicalSpace G
inst✝² : ChartedSpace H G
inst✝¹ : Group G
inst✝ : LieGroup I (minSmoothness 𝕜 3) G
g : G
v : GroupLieAlgebra I G
M : minSmoothness 𝕜 3 ≠ 0
h : G
D : (fun x ↦ h * x) = (fun b ↦ g⁻¹ * b) ∘ fun x ↦ g * h * x
⊢ (MDiffAt fun b ↦ g⁻¹ * b) (g * h * 1)

Tactic:
exact contMDiff_mul_left.contMDiffAt.mdifferentiableAt M

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

Goal state before:
pp: case hf
𝕜 : Type u_1
inst✝⁷ : NontriviallyNormedField 𝕜
H : Type u_2
inst✝⁶ : TopologicalSpace H
E : Type u_3
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
I : ModelWithCorners 𝕜 E H
G : Type u_4
inst✝³ : TopologicalSpace G
inst✝² : ChartedSpace H G
inst✝¹ : Group G
inst✝ : LieGroup I (minSmoothness 𝕜 3) G
g : G
v : GroupLieAlgebra I G
M : minSmoothness 𝕜 3 ≠ 0
h : G
D : (fun x ↦ h * x) = (fun b ↦ g⁻¹ * b) ∘ fun x ↦ g * h * x
⊢ (MDiffAt fun x ↦ g * h * x) 1

Tactic:
exact contMDiff_mul_left.contMDiffAt.mdifferentiableAt M

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

Goal state before:
pp: case e_6.hW
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[s] x] V
hxV : V₁ x = V x
hW : W₁ =ᶠ[𝓝[s] x] W
hxW : W₁ x = W x
I1 : NormedAddCommGroup (TangentSpace 𝓘(𝕜, E) (↑(extChartAt I x) x)) := ⋯
_I2 : NormedSpace 𝕜 (TangentSpace 𝓘(𝕜, E) (↑(extChartAt I x) x)) := inst✝²
⊢ mpullbackWithin 𝓘(𝕜, E) I (↑(extChartAt I x).symm) W₁
      (range ↑I) =ᶠ[𝓝[↑(extChartAt I x).symm ⁻¹' s ∩ range ↑I] ↑(extChartAt I x) x]
    mpullbackWithin 𝓘(𝕜, E) I (↑(extChartAt I x).symm) W (range ↑I)

pp: case hxW
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[s] x] V
hxV : V₁ x = V x
hW : W₁ =ᶠ[𝓝[s] x] W
hxW : W₁ x = W x
I1 : NormedAddCommGroup (TangentSpace 𝓘(𝕜, E) (↑(extChartAt I x) x)) := ⋯
_I2 : NormedSpace 𝕜 (TangentSpace 𝓘(𝕜, E) (↑(extChartAt I x) x)) := inst✝²
⊢ mpullbackWithin 𝓘(𝕜, E) I (↑(extChartAt I x).symm) W₁ (range ↑I) (↑(extChartAt I x) x) =
    mpullbackWithin 𝓘(𝕜, E) I (↑(extChartAt I x).symm) W (range ↑I) (↑(extChartAt I x) x)

Tactic:
· apply nhdsWithin_mono _ inter_subset_left
    filter_upwards [(continuousAt_extChartAt_symm x).continuousWithinAt.preimage_mem_nhdsWithin''
      hW (by simp)] with y hy
    simp only [mpullbackWithin_apply]
    congr 1

Goal state after:
case hxW
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[s] x] V
hxV : V₁ x = V x
hW : W₁ =ᶠ[𝓝[s] x] W
hxW : W₁ x = W x
I1 : NormedAddCommGroup (TangentSpace 𝓘(𝕜, E) (↑(extChartAt I x) x)) := ⋯
_I2 : NormedSpace 𝕜 (TangentSpace 𝓘(𝕜, E) (↑(extChartAt I x) x)) := inst✝²
⊢ mpullbackWithin 𝓘(𝕜, E) I (↑(extChartAt I x).symm) W₁ (range ↑I) (↑(extChartAt I x) x) =
    mpullbackWithin 𝓘(𝕜, E) I (↑(extChartAt I x).symm) W (range ↑I) (↑(extChartAt I x) x)
Module: Mathlib.Geometry.Manifold.Instances.Icc

Goal state before:
pp: case hφ
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) := ⋯
φ : (EuclideanSpace ℝ (Fin 1) × Unit) ≃L[ℝ] ℝ := ⋯
hz : ↑z < y
⊢ ContMDiffOn 𝓘(ℝ, ℝ) 𝓘(ℝ, ℝ) n (↑(Homeomorph.addLeft (-x)).toOpenPartialHomeomorph)
    (Homeomorph.addLeft (-x)).toOpenPartialHomeomorph.source

Tactic:
have : ContDiff ℝ n (fun y ↦ -x + y) := by fun_prop

Goal state after:
case hφ
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
this : ContDiff ℝ n fun y ↦ -x + y
⊢ ContMDiffOn 𝓘(ℝ, ℝ) 𝓘(ℝ, ℝ) n (↑(Homeomorph.addLeft (-x)).toOpenPartialHomeomorph)
    (Homeomorph.addLeft (-x)).toOpenPartialHomeomorph.source
Module: Mathlib.Geometry.Manifold.Instances.Icc

Goal state before:
pp: case pos
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
x y : ℝ
h : Fact (x < y)
f : ↑(Icc x y) → M
w : ↑(Icc x y)
hw : MDiffAt f w
this : w = projIcc x y ⋯ ↑w
⊢ (mfderiv% f (projIcc x y ⋯ ↑w)) ((mfderiv[Icc x y] (projIcc x y ⋯) ↑w) 1) = (mfderiv% f w) 1

Tactic:
rw [projIcc_of_mem _ w.2]

Goal state after:
case pos
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
x y : ℝ
h : Fact (x < y)
f : ↑(Icc x y) → M
w : ↑(Icc x y)
hw : MDiffAt f w
this : w = projIcc x y ⋯ ↑w
⊢ (mfderiv% f ⟨↑w, ⋯⟩) ((mfderiv[Icc x y] (projIcc x y ⋯) ↑w) 1) = (mfderiv% f w) 1