mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Geometry.Manifold.IntegralCurve.ExistUnique

Goal state before:
pp: E : Type u_1
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
H : Type u_2
inst✝⁴ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace H M
inst✝¹ : IsManifold I 1 M
γ γ' : ℝ → M
v : (x : M) → TangentSpace I x
t₀ : ℝ
inst✝ : T2Space M
a b : ℝ
ht₀ : t₀ ∈ Ioo a b
hγt : ∀ t ∈ Ioo a b, I.IsInteriorPoint (γ t)
hv : ContMDiff I I.tangent 1 fun x ↦ ⟨x, v x⟩
hγ : IsMIntegralCurveOn γ v (Ioo a b)
hγ' : IsMIntegralCurveOn γ' v (Ioo a b)
h : γ t₀ = γ' t₀
s : Set ℝ := {t | γ t = γ' t} ∩ Ioo a b
hs : s = {t | γ t = γ' t} ∩ Ioo a b
t₁ : ℝ
ht₁ : t₁ ∈ s
⊢ s ∈ 𝓝 t₁

Tactic:
have hmem := Ioo_mem_nhds ht₁.2.1 ht₁.2.2

Goal state after:
E : Type u_1
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
H : Type u_2
inst✝⁴ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace H M
inst✝¹ : IsManifold I 1 M
γ γ' : ℝ → M
v : (x : M) → TangentSpace I x
t₀ : ℝ
inst✝ : T2Space M
a b : ℝ
ht₀ : t₀ ∈ Ioo a b
hγt : ∀ t ∈ Ioo a b, I.IsInteriorPoint (γ t)
hv : ContMDiff I I.tangent 1 fun x ↦ ⟨x, v x⟩
hγ : IsMIntegralCurveOn γ v (Ioo a b)
hγ' : IsMIntegralCurveOn γ' v (Ioo a b)
h : γ t₀ = γ' t₀
s : Set ℝ := {t | γ t = γ' t} ∩ Ioo a b
hs : s = {t | γ t = γ' t} ∩ Ioo a b
t₁ : ℝ
ht₁ : t₁ ∈ s
hmem : Ioo a b ∈ 𝓝 t₁
⊢ s ∈ 𝓝 t₁
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)
A : ↑(extChartAt I x₀).symm (↑(extChartAt I x₀) x₀) = x₀
M : TangentSpace I x₀ ≃L[𝕜] TangentSpace I' (f x₀)
M' : E ≃L[𝕜] E' := M
⊢ CompleteSpace E'

Tactic:
exact (completeSpace_congr (e := M'.toEquiv) M'.isUniformEmbedding).1 (by assumption)

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

Goal state before:
pp: EB : Type u_1
inst✝¹⁶ : NormedAddCommGroup EB
inst✝¹⁵ : NormedSpace ℝ EB
HB : Type u_2
inst✝¹⁴ : TopologicalSpace HB
IB : ModelWithCorners ℝ EB HB
B : Type u_3
inst✝¹³ : TopologicalSpace B
inst✝¹² : ChartedSpace HB B
F : Type u_4
inst✝¹¹ : NormedAddCommGroup F
inst✝¹⁰ : NormedSpace ℝ F
E : B → Type u_5
inst✝⁹ : TopologicalSpace (TotalSpace F E)
inst✝⁸ : (x : B) → NormedAddCommGroup (E x)
inst✝⁷ : (x : B) → InnerProductSpace ℝ (E x)
inst✝⁶ : FiberBundle F E
inst✝⁵ : VectorBundle ℝ F E
EM : Type u_6
inst✝⁴ : NormedAddCommGroup EM
inst✝³ : NormedSpace ℝ EM
HM : Type u_7
inst✝² : TopologicalSpace HM
IM : ModelWithCorners ℝ EM HM
M : Type u_8
inst✝¹ : TopologicalSpace M
inst✝ : ChartedSpace HM M
h : IsContMDiffRiemannianBundle IB 1 F E
b : M → B
v w : (x : M) → E (b x)
s : Set M
x : M
hv : (MDiffAt[s] fun m ↦ ⟨b m, v m⟩) x
hw : (MDiffAt[s] fun m ↦ ⟨b m, w m⟩) x
g : (x : B) → E x →L[ℝ] E x →L[ℝ] ℝ
g_smooth : ContMDiff IB (IB.prod 𝓘(ℝ, F →L[ℝ] F →L[ℝ] ℝ)) 1 fun b ↦ ⟨b, g b⟩
hg : ∀ (x : B) (v w : E x), ⟪v, w⟫ = ((g x) v) w
⊢ (MDiffAt[s] fun m ↦ ⟪v m, w m⟫) x

Tactic:
have hb : MDiffAt[s] b x := by
    simp only [mdifferentiableWithinAt_totalSpace] at hv
    exact hv.1

Goal state after:
EB : Type u_1
inst✝¹⁶ : NormedAddCommGroup EB
inst✝¹⁵ : NormedSpace ℝ EB
HB : Type u_2
inst✝¹⁴ : TopologicalSpace HB
IB : ModelWithCorners ℝ EB HB
B : Type u_3
inst✝¹³ : TopologicalSpace B
inst✝¹² : ChartedSpace HB B
F : Type u_4
inst✝¹¹ : NormedAddCommGroup F
inst✝¹⁰ : NormedSpace ℝ F
E : B → Type u_5
inst✝⁹ : TopologicalSpace (TotalSpace F E)
inst✝⁸ : (x : B) → NormedAddCommGroup (E x)
inst✝⁷ : (x : B) → InnerProductSpace ℝ (E x)
inst✝⁶ : FiberBundle F E
inst✝⁵ : VectorBundle ℝ F E
EM : Type u_6
inst✝⁴ : NormedAddCommGroup EM
inst✝³ : NormedSpace ℝ EM
HM : Type u_7
inst✝² : TopologicalSpace HM
IM : ModelWithCorners ℝ EM HM
M : Type u_8
inst✝¹ : TopologicalSpace M
inst✝ : ChartedSpace HM M
h : IsContMDiffRiemannianBundle IB 1 F E
b : M → B
v w : (x : M) → E (b x)
s : Set M
x : M
hv : (MDiffAt[s] fun m ↦ ⟨b m, v m⟩) x
hw : (MDiffAt[s] fun m ↦ ⟨b m, w m⟩) x
g : (x : B) → E x →L[ℝ] E x →L[ℝ] ℝ
g_smooth : ContMDiff IB (IB.prod 𝓘(ℝ, F →L[ℝ] F →L[ℝ] ℝ)) 1 fun b ↦ ⟨b, g b⟩
hg : ∀ (x : B) (v w : E x), ⟪v, w⟫ = ((g x) v) w
hb : MDiffAt[s] b x
⊢ (MDiffAt[s] fun m ↦ ⟪v m, w m⟫) x
Module: Mathlib.Geometry.Manifold.Riemannian.Basic

Goal state before:
pp: 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 : RiemannianBundle fun x ↦ TangentSpace I x
inst✝¹ : IsManifold I 1 M
inst✝ : IsContinuousRiemannianBundle E fun x ↦ TangentSpace I x
x : M
C : ℝ≥0
C_pos : C > 0
hC : ∀ᶠ (y : E) in 𝓝[range ↑I] ↑(extChartAt I x) x, ‖mfderiv[range ↑I] ↑(extChartAt I x).symm y‖ₑ < ↑C
r : ℝ
r_pos : r > 0
hr :
  ball (↑(extChartAt I x) x) r ∩ range ↑I ⊆
    (extChartAt I x).target ∩ {y | ‖mfderiv[range ↑I] ↑(extChartAt I x).symm y‖ₑ < ↑C}
A : ↑(extChartAt I x) ⁻¹' (ball (↑(extChartAt I x) x) r ∩ range ↑I) ∈ 𝓝 x
y : M
hy : y ∈ ↑(extChartAt I x) ⁻¹' (ball (↑(extChartAt I x) x) r ∩ range ↑I)
h'y : y ∈ (chartAt H x).source
η : ℝ →ᴬ[ℝ] E := ContinuousAffineMap.lineMap (↑(extChartAt I x) x) (↑(extChartAt I x) y)
γ : ℝ → M := ↑(extChartAt I x).symm ∘ ⇑η
hη :
  Icc 0 1 ⊆ ⇑η ⁻¹' (extChartAt I x).target ∧ Icc 0 1 ⊆ ⇑η ⁻¹' {y | ‖mfderiv[range ↑I] ↑(extChartAt I x).symm y‖ₑ < ↑C}
η_smooth : ContMDiffOn 𝓘(ℝ, ℝ) 𝓘(ℝ, E) 1 (⇑η) (Icc 0 1)
this : riemannianEDist I x y ≤ pathELength I γ 0 1
⊢ ∫⁻ (t : ℝ) in Icc 0 1, ‖(mfderiv[Icc 0 1] γ t) 1‖ₑ ≤
    ∫⁻ (a : ℝ) in Icc 0 1,
      ↑C *
        ‖(fderivWithin ℝ (⇑(ContinuousAffineMap.lineMap (↑(extChartAt I x) x) (↑(extChartAt I x) y))) (Icc 0 1) a) 1‖ₑ

Tactic:
apply setLIntegral_mono' measurableSet_Icc (fun t ht ↦ ?_)

Goal state after:
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 : RiemannianBundle fun x ↦ TangentSpace I x
inst✝¹ : IsManifold I 1 M
inst✝ : IsContinuousRiemannianBundle E fun x ↦ TangentSpace I x
x : M
C : ℝ≥0
C_pos : C > 0
hC : ∀ᶠ (y : E) in 𝓝[range ↑I] ↑(extChartAt I x) x, ‖mfderiv[range ↑I] ↑(extChartAt I x).symm y‖ₑ < ↑C
r : ℝ
r_pos : r > 0
hr :
  ball (↑(extChartAt I x) x) r ∩ range ↑I ⊆
    (extChartAt I x).target ∩ {y | ‖mfderiv[range ↑I] ↑(extChartAt I x).symm y‖ₑ < ↑C}
A : ↑(extChartAt I x) ⁻¹' (ball (↑(extChartAt I x) x) r ∩ range ↑I) ∈ 𝓝 x
y : M
hy : y ∈ ↑(extChartAt I x) ⁻¹' (ball (↑(extChartAt I x) x) r ∩ range ↑I)
h'y : y ∈ (chartAt H x).source
η : ℝ →ᴬ[ℝ] E := ContinuousAffineMap.lineMap (↑(extChartAt I x) x) (↑(extChartAt I x) y)
γ : ℝ → M := ↑(extChartAt I x).symm ∘ ⇑η
hη :
  Icc 0 1 ⊆ ⇑η ⁻¹' (extChartAt I x).target ∧ Icc 0 1 ⊆ ⇑η ⁻¹' {y | ‖mfderiv[range ↑I] ↑(extChartAt I x).symm y‖ₑ < ↑C}
η_smooth : ContMDiffOn 𝓘(ℝ, ℝ) 𝓘(ℝ, E) 1 (⇑η) (Icc 0 1)
this : riemannianEDist I x y ≤ pathELength I γ 0 1
t : ℝ
ht : t ∈ Icc 0 1
⊢ ‖(mfderiv[Icc 0 1] γ t) 1‖ₑ ≤
    ↑C * ‖(fderivWithin ℝ (⇑(ContinuousAffineMap.lineMap (↑(extChartAt I x) x) (↑(extChartAt I x) y))) (Icc 0 1) t) 1‖ₑ
Module: Mathlib.Geometry.Manifold.VectorBundle.LocalFrame

Goal state before:
pp: 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
inst✝⁷ : NormedSpace 𝕜 F
n✝ : ℕ∞ω
V : M → Type u_6
inst✝⁶ : TopologicalSpace (TotalSpace F V)
inst✝⁵ : (x : M) → AddCommGroup (V x)
inst✝⁴ : (x : M) → Module 𝕜 (V x)
inst✝³ : (x : M) → TopologicalSpace (V x)
inst✝² : FiberBundle F V
ι : Type u_7
s s' : ι → (x : M) → V x
u u' : Set M
x : M
n : ℕ∞ω
inst✝¹ : VectorBundle 𝕜 F V
inst✝ : FiniteDimensional 𝕜 F
hs : IsLocalFrameOn I F n s u
hx : x ∈ u
⊢ Fintype ι

Tactic:
by
  have : FiniteDimensional 𝕜 (V x) := by
    let phi := (trivializationAt F V x).linearEquivAt 𝕜 x
      (FiberBundle.mem_baseSet_trivializationAt' x)
    exact Finite.equiv phi.symm
  exact FiniteDimensional.fintypeBasisIndex (hs.toBasisAt hx)

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝¹⁸ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹⁷ : NormedAddCommGroup E
inst✝¹⁶ : NormedSpace 𝕜 E
H : Type u_3
inst✝¹⁵ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝¹⁴ : TopologicalSpace M
inst✝¹³ : ChartedSpace H M
F : Type u_5
inst✝¹² : NormedAddCommGroup F
inst✝¹¹ : NormedSpace 𝕜 F
V : M → Type u_6
inst✝¹⁰ : TopologicalSpace (TotalSpace F V)
inst✝⁹ : (x : M) → AddCommGroup (V x)
inst✝⁸ : (x : M) → Module 𝕜 (V x)
inst✝⁷ : (x : M) → TopologicalSpace (V x)
inst✝⁶ : FiberBundle F V
inst✝⁵ : VectorBundle 𝕜 F V
inst✝⁴ : ContMDiffVectorBundle 1 F V I
e : Trivialization F TotalSpace.proj
inst✝³ : MemTrivializationAtlas e
ι : Type u_7
b : Basis ι 𝕜 F
s : (x : M) → V x
k : ℕ∞ω
x : M
inst✝² : FiniteDimensional 𝕜 F
inst✝¹ : CompleteSpace 𝕜
inst✝ : ContMDiffVectorBundle k F V I
hxe : x ∈ e.baseSet
hs : ContMDiffAt I (I.prod 𝓘(𝕜, F)) k (fun x ↦ ⟨x, s x⟩) x
i : ι
⊢ ContMDiffAt I 𝓘(𝕜, 𝕜) k ((LinearMap.piApply (Trivialization.localFrameCoeff I e b i)) s) x

Tactic:
let aux := fun x ↦ b.repr (e ((T% s) x)).2 i

Goal state after:
𝕜 : Type u_1
inst✝¹⁸ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹⁷ : NormedAddCommGroup E
inst✝¹⁶ : NormedSpace 𝕜 E
H : Type u_3
inst✝¹⁵ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝¹⁴ : TopologicalSpace M
inst✝¹³ : ChartedSpace H M
F : Type u_5
inst✝¹² : NormedAddCommGroup F
inst✝¹¹ : NormedSpace 𝕜 F
V : M → Type u_6
inst✝¹⁰ : TopologicalSpace (TotalSpace F V)
inst✝⁹ : (x : M) → AddCommGroup (V x)
inst✝⁸ : (x : M) → Module 𝕜 (V x)
inst✝⁷ : (x : M) → TopologicalSpace (V x)
inst✝⁶ : FiberBundle F V
inst✝⁵ : VectorBundle 𝕜 F V
inst✝⁴ : ContMDiffVectorBundle 1 F V I
e : Trivialization F TotalSpace.proj
inst✝³ : MemTrivializationAtlas e
ι : Type u_7
b : Basis ι 𝕜 F
s : (x : M) → V x
k : ℕ∞ω
x : M
inst✝² : FiniteDimensional 𝕜 F
inst✝¹ : CompleteSpace 𝕜
inst✝ : ContMDiffVectorBundle k F V I
hxe : x ∈ e.baseSet
hs : ContMDiffAt I (I.prod 𝓘(𝕜, F)) k (fun x ↦ ⟨x, s x⟩) x
i : ι
aux : M → 𝕜 := fun x ↦ (b.repr (↑e ((fun x ↦ ⟨x, s x⟩) x)).2) i
⊢ ContMDiffAt I 𝓘(𝕜, 𝕜) k ((LinearMap.piApply (Trivialization.localFrameCoeff I e b i)) s) x
Module: Mathlib.Geometry.Manifold.VectorBundle.LocalFrame

Goal state before:
pp: 𝕜 : Type u_1
inst✝¹⁷ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹⁶ : NormedAddCommGroup E
inst✝¹⁵ : NormedSpace 𝕜 E
H : Type u_3
inst✝¹⁴ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝¹³ : TopologicalSpace M
inst✝¹² : ChartedSpace H M
F : Type u_5
inst✝¹¹ : NormedAddCommGroup F
inst✝¹⁰ : NormedSpace 𝕜 F
V : M → Type u_6
inst✝⁹ : TopologicalSpace (TotalSpace F V)
inst✝⁸ : (x : M) → AddCommGroup (V x)
inst✝⁷ : (x : M) → Module 𝕜 (V x)
inst✝⁶ : (x : M) → TopologicalSpace (V x)
inst✝⁵ : FiberBundle F V
inst✝⁴ : VectorBundle 𝕜 F V
inst✝³ : ContMDiffVectorBundle 1 F V I
e : Trivialization F TotalSpace.proj
inst✝² : MemTrivializationAtlas e
ι : Type u_7
b : Basis ι 𝕜 F
s : (x : M) → V x
x : M
inst✝¹ : FiniteDimensional 𝕜 F
inst✝ : CompleteSpace 𝕜
hxe : x ∈ e.baseSet
hs : MDiffAt (T% s) x
i : ι
⊢ MDiffAt ((LinearMap.piApply (Trivialization.localFrameCoeff I e b i)) s) x

Tactic:
let aux := fun x ↦ b.repr (e ((T% s) x)).2 i

Goal state after:
𝕜 : Type u_1
inst✝¹⁷ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹⁶ : NormedAddCommGroup E
inst✝¹⁵ : NormedSpace 𝕜 E
H : Type u_3
inst✝¹⁴ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝¹³ : TopologicalSpace M
inst✝¹² : ChartedSpace H M
F : Type u_5
inst✝¹¹ : NormedAddCommGroup F
inst✝¹⁰ : NormedSpace 𝕜 F
V : M → Type u_6
inst✝⁹ : TopologicalSpace (TotalSpace F V)
inst✝⁸ : (x : M) → AddCommGroup (V x)
inst✝⁷ : (x : M) → Module 𝕜 (V x)
inst✝⁶ : (x : M) → TopologicalSpace (V x)
inst✝⁵ : FiberBundle F V
inst✝⁴ : VectorBundle 𝕜 F V
inst✝³ : ContMDiffVectorBundle 1 F V I
e : Trivialization F TotalSpace.proj
inst✝² : MemTrivializationAtlas e
ι : Type u_7
b : Basis ι 𝕜 F
s : (x : M) → V x
x : M
inst✝¹ : FiniteDimensional 𝕜 F
inst✝ : CompleteSpace 𝕜
hxe : x ∈ e.baseSet
hs : MDiffAt (T% s) x
i : ι
aux : M → 𝕜 := fun x ↦ (b.repr (↑e ((fun x ↦ ⟨x, s x⟩) x)).2) i
⊢ MDiffAt ((LinearMap.piApply (Trivialization.localFrameCoeff I e b i)) s) x
Module: Mathlib.Geometry.Manifold.VectorBundle.Tensoriality

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
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
F : Type u_5
inst✝⁹ : NormedAddCommGroup F
inst✝⁸ : NormedSpace 𝕜 F
V : M → Type u_6
inst✝⁷ : TopologicalSpace (TotalSpace F V)
inst✝⁶ : (x : M) → AddCommGroup (V x)
inst✝⁵ : (x : M) → Module 𝕜 (V x)
inst✝⁴ : (x : M) → TopologicalSpace (V x)
inst✝³ : FiberBundle F V
A : Type u_9
inst✝² : AddCommGroup A
inst✝¹ : Module 𝕜 A
Φ : ((x : M) → V x) → A
x : M
inst✝ : VectorBundle 𝕜 F V
hΦ : TensorialAt I F Φ x
⊢ Φ 0 = 0

Tactic:
by
  calc
    Φ 0 = Φ ((0 : M → 𝕜) • (0 : Π x, V x)) := by simp
    _   = 0 • Φ 0 := hΦ.smul mdifferentiableAt_const (mdifferentiable_zeroSection ..)
    _   = 0 := by simp

Goal state after:
Module: Mathlib.Geometry.Manifold.VectorBundle.CovariantDerivative.Basic

Goal state before:
pp: 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
inst✝⁷ : NormedSpace 𝕜 F
V : M → Type u_6
inst✝⁶ : TopologicalSpace (TotalSpace F V)
inst✝⁵ : (x : M) → AddCommGroup (V x)
inst✝⁴ : (x : M) → Module 𝕜 (V x)
inst✝³ : (x : M) → TopologicalSpace (V x)
inst✝² : ∀ (x : M), IsTopologicalAddGroup (V x)
inst✝¹ : ∀ (x : M), ContinuousSMul 𝕜 (V x)
inst✝ : FiberBundle F V
cov : ((x : M) → V x) → (x : M) → TangentSpace I x →L[𝕜] V x
s : Set M
hcov : IsCovariantDerivativeOn F cov s
σ σ' : (x : M) → V x
x : M
hσ : MDiffAt (T% σ) x
hσ' : MDiffAt (T% σ') x
hxs : s ∈ 𝓝 x
hσσ' : ∀ x ∈ s, σ x = σ' x
hxs' : x ∈ s
ψ : M → 𝕜 := fun x' ↦ if x' ∈ s then 1 else 0
hψx : ψ x = 1
⊢ cov σ x = cov σ' x

Tactic:
have H (x' : M) : ((ψ : M → 𝕜) • σ) x' = ((ψ : M → 𝕜) • σ') x' := by
    dsimp [ψ]
    split_ifs with hx's
    · simpa using hσσ' _ hx's
    · simp

Goal state after:
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
inst✝⁷ : NormedSpace 𝕜 F
V : M → Type u_6
inst✝⁶ : TopologicalSpace (TotalSpace F V)
inst✝⁵ : (x : M) → AddCommGroup (V x)
inst✝⁴ : (x : M) → Module 𝕜 (V x)
inst✝³ : (x : M) → TopologicalSpace (V x)
inst✝² : ∀ (x : M), IsTopologicalAddGroup (V x)
inst✝¹ : ∀ (x : M), ContinuousSMul 𝕜 (V x)
inst✝ : FiberBundle F V
cov : ((x : M) → V x) → (x : M) → TangentSpace I x →L[𝕜] V x
s : Set M
hcov : IsCovariantDerivativeOn F cov s
σ σ' : (x : M) → V x
x : M
hσ : MDiffAt (T% σ) x
hσ' : MDiffAt (T% σ') x
hxs : s ∈ 𝓝 x
hσσ' : ∀ x ∈ s, σ x = σ' x
hxs' : x ∈ s
ψ : M → 𝕜 := fun x' ↦ if x' ∈ s then 1 else 0
hψx : ψ x = 1
H : ∀ (x' : M), (ψ • σ) x' = (ψ • σ') x'
⊢ cov σ x = cov σ' x
Module: Mathlib.Geometry.Polygon.Basic

Goal state before:
pp: R : Type u_1
V : Type u_2
P : Type u_3
inst✝³ : Ring R
inst✝² : AddCommGroup V
inst✝¹ : Module R V
inst✝ : AddTorsor V P
poly : Polygon P 3
h : HasNondegenerateVertices R poly
⊢ Affine.Triangle.toPolygon (toTriangle R poly h) = poly

Tactic:
by
  rfl

Goal state after:
Module: Mathlib.Geometry.Polygon.Basic

Goal state before:
pp: R : Type u_1
V : Type u_2
P : Type u_3
inst✝³ : Ring R
inst✝² : AddCommGroup V
inst✝¹ : Module R V
inst✝ : AffineSpace V P
t : Triangle R P
⊢ Polygon.toTriangle R (toPolygon t) ⋯ = t

Tactic:
by
  rfl

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

Goal state before:
pp: case convert_2
ι : Type uι
E : Type uE
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
inst✝⁶ : FiniteDimensional ℝ E
H : Type uH
inst✝⁵ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type uM
inst✝⁴ : TopologicalSpace M
inst✝³ : ChartedSpace H M
inst✝² : IsManifold I ∞ M
inst✝¹ : T2Space M
inst✝ : Fintype ι
s : Set M
f : SmoothBumpCovering ι I M s
x : M
hx : x ∈ s
L : (ι → E × ℝ) →L[ℝ] E := ContinuousLinearMap.fst ℝ E ℝ ∘SL ContinuousLinearMap.proj (f.ind x hx)
this : HasMFDerivAt% (⇑L ∘ ⇑f.embeddingPiTangent) x (L ∘SL mfderiv% ⇑f.embeddingPiTangent x)
⊢ ⇑L ∘ ⇑f.embeddingPiTangent =ᶠ[𝓝 x] ↑(extChartAt I (f.c (f.ind x hx)))

Tactic:
refine (f.eventuallyEq_one x hx).mono fun y hy => ?_

Goal state after:
case convert_2
ι : Type uι
E : Type uE
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
inst✝⁶ : FiniteDimensional ℝ E
H : Type uH
inst✝⁵ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type uM
inst✝⁴ : TopologicalSpace M
inst✝³ : ChartedSpace H M
inst✝² : IsManifold I ∞ M
inst✝¹ : T2Space M
inst✝ : Fintype ι
s : Set M
f : SmoothBumpCovering ι I M s
x : M
hx : x ∈ s
L : (ι → E × ℝ) →L[ℝ] E := ContinuousLinearMap.fst ℝ E ℝ ∘SL ContinuousLinearMap.proj (f.ind x hx)
this : HasMFDerivAt% (⇑L ∘ ⇑f.embeddingPiTangent) x (L ∘SL mfderiv% ⇑f.embeddingPiTangent x)
y : M
hy : ↑(f.toFun (f.ind x hx)) y = 1 y
⊢ (⇑L ∘ ⇑f.embeddingPiTangent) y = ↑(extChartAt I (f.c (f.ind x hx))) y
Module: Mathlib.GroupTheory.Commutator.Finite

Goal state before:
pp: G : Type u_1
inst✝¹ : Group G
inst✝ : Finite ↑(commutatorSet G)
⊢ Group.FG ↥(closure (commutatorSet G))

Tactic:
apply Group.closure_finite_fg

Goal state after:
Module: Mathlib.GroupTheory.Commutator.Finite

Goal state before:
pp: G : Type u_1
inst✝² : Group G
inst✝¹ : Finite ↑(commutatorSet G)
inst✝ : Group.FG G
S : Finset G
hS1 : S.card = Group.rank G
hS2 : closure ↑S = ⊤
⊢ (center G).index ≤ Nat.card ↑(commutatorSet G) ^ Group.rank G

Tactic:
rw [← hS1, ← Fintype.card_coe, ← Nat.card_eq_fintype_card, ← Finset.coe_sort_coe, ← Nat.card_fun]

Goal state after:
G : Type u_1
inst✝² : Group G
inst✝¹ : Finite ↑(commutatorSet G)
inst✝ : Group.FG G
S : Finset G
hS1 : S.card = Group.rank G
hS2 : closure ↑S = ⊤
⊢ (center G).index ≤ Nat.card (↑↑S → ↑(commutatorSet G))
Module: Mathlib.GroupTheory.SpecificGroups.KleinFour

Goal state before:
pp: G : Type u_1
inst✝¹ : Group G
inst✝ : IsKleinFour G
x y z : G
hx : x ≠ 1
hy : y ≠ 1
hxy : x ≠ y
hz : z ≠ 1
hzx : z ≠ x
hzy : z ≠ y
⊢ z = x * y

Tactic:
let _ := Fintype.ofFinite G
  apply eq_of_mem_insert_of_notMem <| (eq_finset_univ hx hy hxy).symm ▸ mem_univ _
  simpa only [mem_singleton, mem_insert, not_or] using ⟨hzx, hzy, hz⟩

Goal state after:
Module: Mathlib.GroupTheory.SpecificGroups.Dihedral

Goal state before:
pp: case succ
n✝ : ℕ
⊢ Fintype.card (DihedralGroup (n✝ + 1)) = 2 * (n✝ + 1)

Tactic:
card

Goal state after:
case succ
n✝ : ℕ
⊢ 2 * (n✝ + 1) = 2 * (n✝ + 1)
Module: Mathlib.GroupTheory.Coprod.Basic

Goal state before:
pp: G : Type u_1
H : Type u_2
inst✝¹ : Group G
inst✝ : Group H
⊢ inl.range ⊔ inr.range = ⊤

Tactic:
rw [← closure_range_inl_union_inr, Subgroup.closure_union, ← MonoidHom.coe_range,
    ← MonoidHom.coe_range, Subgroup.closure_eq, Subgroup.closure_eq]

Goal state after:
Module: Mathlib.GroupTheory.Coprod.Basic

Goal state before:
pp: G : Type u_1
H : Type u_2
inst✝² : Group G
inst✝¹ : Group H
K : Type u_3
inst✝ : Group K
f : G ∗ H →* K
⊢ (f.comp inl).range ⊔ Subgroup.map f inr.range = (f.comp inl).range ⊔ (f.comp inr).range

Tactic:
MonoidHom.map_range

Goal state after:
G : Type u_1
H : Type u_2
inst✝² : Group G
inst✝¹ : Group H
K : Type u_3
inst✝ : Group K
f : G ∗ H →* K
⊢ (f.comp inl).range ⊔ (f.comp inr).range = (f.comp inl).range ⊔ (f.comp inr).range
Module: Mathlib.GroupTheory.CoprodI

Goal state before:
pp: ι : Type u_1
M : ι → Type u_2
inst✝¹ : (i : ι) → Monoid (M i)
N : Type u_3
inst✝ : Monoid N
i : ι
m : M i
w : Word M
hmw : w.fstIdx ≠ some i
h1 : m ≠ 1
⊢ ∀ l ∈ ⟨i, m⟩ :: w.toList, l.snd ≠ 1

Tactic:
simp only [List.mem_cons]

Goal state after:
ι : Type u_1
M : ι → Type u_2
inst✝¹ : (i : ι) → Monoid (M i)
N : Type u_3
inst✝ : Monoid N
i : ι
m : M i
w : Word M
hmw : w.fstIdx ≠ some i
h1 : m ≠ 1
⊢ ∀ (l : (i : ι) × M i), l = ⟨i, m⟩ ∨ l ∈ w.toList → l.snd ≠ 1
Module: Mathlib.GroupTheory.CoprodI

Goal state before:
pp: case pos
ι : Type u_1
M : ι → Type u_2
inst✝¹ : (i : ι) → Monoid (M i)
inst✝ : (i : ι) → DecidableEq (M i)
i : ι
m : M i
w : Word M
h : w.fstIdx ≠ some i
m' : M i
w' : Word M
h' : w'.fstIdx ≠ some i
he : rcons { head := m, tail := w, fstIdx_ne := h } = rcons { head := m', tail := w', fstIdx_ne := h' }
hm : m = 1
hm' : m' = 1
⊢ { head := m, tail := w, fstIdx_ne := h } = { head := m', tail := w', fstIdx_ne := h' }

Tactic:
simp only [rcons, dif_pos hm, dif_pos hm'] at he

Goal state after:
case pos
ι : Type u_1
M : ι → Type u_2
inst✝¹ : (i : ι) → Monoid (M i)
inst✝ : (i : ι) → DecidableEq (M i)
i : ι
m : M i
w : Word M
h : w.fstIdx ≠ some i
m' : M i
w' : Word M
h' : w'.fstIdx ≠ some i
hm : m = 1
hm' : m' = 1
he : w = w'
⊢ { head := m, tail := w, fstIdx_ne := h } = { head := m', tail := w', fstIdx_ne := h' }
Module: Mathlib.GroupTheory.Coxeter.Basic

Goal state before:
pp: case succ
B : Type u_1
i j : B
p k : ℕ
h' : k < 2 * p → take k (alternatingWord i j (2 * p)) = if Even k then alternatingWord i j k else alternatingWord j i k
h : k + 1 < 2 * p
⊢ take (k + 1) (alternatingWord i j (2 * p)) =
    if Even (k + 1) then alternatingWord i j (k + 1) else alternatingWord j i (k + 1)

Tactic:
have hk : k < 2 * p := by lia
      apply h' at hk
      by_cases h_even : Even k
      · simp only [h_even, ↓reduceIte] at hk
        simp only [Nat.not_even_iff_odd.mpr (Even.add_one h_even), ↓reduceIte]
        rw [← List.take_concat_get (by simp; lia), alternatingWord_succ, ← hk]
        apply congr_arg
        rw [getElem_alternatingWord i j (2 * p) k (by lia)]
        simp [(by apply Nat.even_add.mpr; simp [h_even] : Even (2 * p + k))]
      · simp only [h_even, ↓reduceIte] at hk
        simp only [Odd.add_one (by simpa using h_even), ↓reduceIte]
        rw [← List.take_concat_get (by simp; lia), alternatingWord_succ, hk]
        apply congr_arg
        rw [getElem_alternatingWord i j (2 * p) k (by lia)]
        simp [(by apply Nat.odd_add.mpr; simp [h_even] : Odd (2 * p + k))]

Goal state after: