mathlib-initiative/mathlib-tactics huggingface dataset:
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
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
⊢ (MDiffAt[u] fun x ↦ ⟨x, ∑' (i : ι), t i x⟩) x₀

Tactic:
let s := {i | ((fun i ↦ {x | t i x ≠ 0}) i ∩ u').Nonempty}

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
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 := {i | ((fun i ↦ {x | t i x ≠ 0}) i ∩ u').Nonempty}
⊢ (MDiffAt[u] fun x ↦ ⟨x, ∑' (i : ι), t i x⟩) x₀
Module: Mathlib.Topology.VectorBundle.Hom

Goal state before:
pp: 𝕜 : Type u_8
F₁ : Type u_9
F₂ : Type u_10
B₁ : Type u_11
B₂ : Type u_12
M : Type u_13
E₁ : B₁ → Type u_14
E₂ : B₂ → Type u_15
inst✝¹⁹ : NontriviallyNormedField 𝕜
inst✝¹⁸ : (x : B₁) → AddCommGroup (E₁ x)
inst✝¹⁷ : (x : B₁) → Module 𝕜 (E₁ x)
inst✝¹⁶ : NormedAddCommGroup F₁
inst✝¹⁵ : NormedSpace 𝕜 F₁
inst✝¹⁴ : TopologicalSpace (TotalSpace F₁ E₁)
inst✝¹³ : (x : B₁) → TopologicalSpace (E₁ x)
inst✝¹² : (x : B₂) → AddCommGroup (E₂ x)
inst✝¹¹ : (x : B₂) → Module 𝕜 (E₂ x)
inst✝¹⁰ : NormedAddCommGroup F₂
inst✝⁹ : NormedSpace 𝕜 F₂
inst✝⁸ : TopologicalSpace (TotalSpace F₂ E₂)
inst✝⁷ : (x : B₂) → TopologicalSpace (E₂ x)
inst✝⁶ : TopologicalSpace B₁
inst✝⁵ : TopologicalSpace B₂
inst✝⁴ : TopologicalSpace M
inst✝³ : FiberBundle F₁ E₁
inst✝² : VectorBundle 𝕜 F₁ E₁
inst✝¹ : FiberBundle F₂ E₂
inst✝ : VectorBundle 𝕜 F₂ E₂
b₁ : M → B₁
b₂ : M → B₂
m₀ : M
ϕ : (m : M) → E₁ (b₁ m) →L[𝕜] E₂ (b₂ m)
v : (m : M) → E₁ (b₁ m)
hϕ : ContinuousWithinAt (fun m ↦ inCoordinates F₁ E₁ F₂ E₂ (b₁ m₀) (b₁ m) (b₂ m₀) (b₂ m) (ϕ m)) univ m₀
hv : ContinuousWithinAt (fun m ↦ { proj := b₁ m, snd := v m }) univ m₀
hb₂ : ContinuousWithinAt b₂ univ m₀
⊢ ContinuousWithinAt (fun m ↦ { proj := b₂ m, snd := (ϕ m) (v m) }) univ m₀

Tactic:
exact hϕ.clm_apply_of_inCoordinates hv hb₂

Goal state after:
Module: Mathlib.Geometry.Manifold.MFDeriv.NormedSpace

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁹ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace 𝕜 E
H : Type u_3
inst✝⁶ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝⁵ : TopologicalSpace M
inst✝⁴ : ChartedSpace H M
F₁ : Type u_14
inst✝³ : NormedAddCommGroup F₁
inst✝² : NormedSpace 𝕜 F₁
F₂ : Type u_15
inst✝¹ : NormedAddCommGroup F₂
inst✝ : NormedSpace 𝕜 F₂
g : M → F₁ →L[𝕜] F₂
f : M → F₁
s : Set M
x : M
hg : MDiffAt[s] g x
hf : MDiffAt[s] f x
⊢ DifferentiableWithinAt 𝕜 (fun x ↦ x.1 x.2) univ (g x, f x)

Tactic:
by apply (Differentiable.differentiableAt _).differentiableWithinAt
        exact differentiable_fst.clm_apply differentiable_snd

Goal state after:
Module: Mathlib.Geometry.Manifold.MFDeriv.NormedSpace

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁹ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace 𝕜 E
H : Type u_3
inst✝⁶ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝⁵ : TopologicalSpace M
inst✝⁴ : ChartedSpace H M
F₁ : Type u_14
inst✝³ : NormedAddCommGroup F₁
inst✝² : NormedSpace 𝕜 F₁
F₂ : Type u_15
inst✝¹ : NormedAddCommGroup F₂
inst✝ : NormedSpace 𝕜 F₂
g : M → F₁ →L[𝕜] F₂
f : M → F₁
x : M
hg : MDiffAt g x
hf : MDiffAt f x
⊢ DifferentiableWithinAt 𝕜 (fun x ↦ x.1 x.2) univ (g x, f x)

Tactic:
by apply (Differentiable.differentiableAt _).differentiableWithinAt
        exact differentiable_fst.clm_apply differentiable_snd

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
⊢ mpullback I 𝓘(𝕜, E) (↑(extChartAt I x))
      (lieBracketWithin 𝕜 (mpullbackWithin 𝓘(𝕜, E) I (↑(extChartAt I x).symm) V (range ↑I))
        (f ∘ ↑(extChartAt I x).symm • mpullbackWithin 𝓘(𝕜, E) I (↑(extChartAt I x).symm) W (range ↑I))
        (↑(extChartAt I x).symm ⁻¹' s ∩ range ↑I))
      x =
    (d[s] f x) (V x) • W x +
      f x •
        mpullback I 𝓘(𝕜, E) (↑(extChartAt I 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))
          x

Tactic:
set V' := mpullbackWithin 𝓘(𝕜, E) I (extChartAt I x).symm V (range I)

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)
⊢ mpullback I 𝓘(𝕜, E) (↑(extChartAt I x))
      (lieBracketWithin 𝕜 V'
        (f ∘ ↑(extChartAt I x).symm • mpullbackWithin 𝓘(𝕜, E) I (↑(extChartAt I x).symm) W (range ↑I))
        (↑(extChartAt I x).symm ⁻¹' s ∩ range ↑I))
      x =
    (d[s] f x) (V x) • W x +
      f x •
        mpullback I 𝓘(𝕜, E) (↑(extChartAt I x))
          (lieBracketWithin 𝕜 V' (mpullbackWithin 𝓘(𝕜, E) I (↑(extChartAt I x).symm) W (range ↑I))
            (↑(extChartAt I x).symm ⁻¹' s ∩ range ↑I))
          x
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
t₀ : ℝ
hγ : IsMIntegralCurveAt γ v t₀
dt : ℝ
⊢ IsMIntegralCurveAt (γ ∘ fun x ↦ x + dt) v (t₀ - dt)

Tactic:
by
  rw [isMIntegralCurveAt_iff'] at *
  obtain ⟨ε, hε, h⟩ := hγ
  refine ⟨ε, hε, ?_⟩
  convert! h.comp_add dt
  rw [Metric.ball]
  simp_rw [Metric.mem_ball, Real.dist_eq, ← sub_add, add_sub_right_comm]

Goal state after:
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)
⊢ (fun t ↦ γ (|t| + 1) t) =ᶠ[𝓝 t] γ (|t| + 1)

Tactic:
Filter.eventuallyEq_iff_exists_mem

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)
⊢ ∃ s ∈ 𝓝 t, EqOn (fun t ↦ γ (|t| + 1) t) (γ (|t| + 1)) s
Module: Mathlib.Geometry.Manifold.IntegralCurve.UniformTime

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 : IsManifold I 1 M
inst✝¹ : T2Space M
γ γ' : ℝ → M
v : (x : M) → TangentSpace I x
inst✝ : BoundarylessManifold I M
hv : ContMDiff I I.tangent 1 fun x ↦ ⟨x, v x⟩
a b a' b' : ℝ
hγ : IsMIntegralCurveOn γ v (Ioo a b)
hγ' : IsMIntegralCurveOn γ' v (Ioo a' b')
t₀ : ℝ
ht₀ : t₀ ∈ Ioo a b ∩ Ioo a' b'
h : γ t₀ = γ' t₀
t : ℝ
ht : t ∈ Ioo a b ∪ Ioo a' b'
hmem : t ∈ Ioo a b
⊢ (Ioo a b).piecewise γ γ' =ᶠ[𝓝[Ioo a b ∪ Ioo a' b'] t] γ

Tactic:
Filter.eventuallyEq_iff_exists_mem

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 : IsManifold I 1 M
inst✝¹ : T2Space M
γ γ' : ℝ → M
v : (x : M) → TangentSpace I x
inst✝ : BoundarylessManifold I M
hv : ContMDiff I I.tangent 1 fun x ↦ ⟨x, v x⟩
a b a' b' : ℝ
hγ : IsMIntegralCurveOn γ v (Ioo a b)
hγ' : IsMIntegralCurveOn γ' v (Ioo a' b')
t₀ : ℝ
ht₀ : t₀ ∈ Ioo a b ∩ Ioo a' b'
h : γ t₀ = γ' t₀
t : ℝ
ht : t ∈ Ioo a b ∪ Ioo a' b'
hmem : t ∈ Ioo a b
⊢ ∃ s ∈ 𝓝[Ioo a b ∪ Ioo a' b'] t, EqOn ((Ioo a b).piecewise γ γ') γ s
Module: Mathlib.Geometry.Manifold.IntegralCurve.UniformTime

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 : IsManifold I 1 M
inst✝¹ : T2Space M
γ γ' : ℝ → M
v : (x : M) → TangentSpace I x
inst✝ : BoundarylessManifold I M
hv : ContMDiff I I.tangent 1 fun x ↦ ⟨x, v x⟩
a b a' b' : ℝ
hγ : IsMIntegralCurveOn γ v (Ioo a b)
hγ' : IsMIntegralCurveOn γ' v (Ioo a' b')
t₀ : ℝ
ht₀ : t₀ ∈ Ioo a b ∩ Ioo a' b'
h : γ t₀ = γ' t₀
t : ℝ
ht : t ∉ Ioo a b → t ∈ Ioo a' b'
hmem : t ∉ Ioo a b
ht' : t ∈ Ioo a b ∪ Ioo a' b'
⊢ (Ioo a b).piecewise γ γ' =ᶠ[𝓝[Ioo a b ∪ Ioo a' b'] t] γ'

Tactic:
Filter.eventuallyEq_iff_exists_mem

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 : IsManifold I 1 M
inst✝¹ : T2Space M
γ γ' : ℝ → M
v : (x : M) → TangentSpace I x
inst✝ : BoundarylessManifold I M
hv : ContMDiff I I.tangent 1 fun x ↦ ⟨x, v x⟩
a b a' b' : ℝ
hγ : IsMIntegralCurveOn γ v (Ioo a b)
hγ' : IsMIntegralCurveOn γ' v (Ioo a' b')
t₀ : ℝ
ht₀ : t₀ ∈ Ioo a b ∩ Ioo a' b'
h : γ t₀ = γ' t₀
t : ℝ
ht : t ∉ Ioo a b → t ∈ Ioo a' b'
hmem : t ∉ Ioo a b
ht' : t ∈ Ioo a b ∪ Ioo a' b'
⊢ ∃ s ∈ 𝓝[Ioo a b ∪ Ioo a' b'] t, EqOn ((Ioo a b).piecewise γ γ') γ' s
Module: Mathlib.Geometry.Manifold.Riemannian.PathELength

Goal state before:
pp: E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
H : Type u_2
inst✝³ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace H M
inst✝ : (x : M) → ENorm (TangentSpace I x)
a b : ℝ
γ γ' : ℝ → M
h : EqOn γ γ' (Ioo a b)
⊢ pathELength I γ a b = pathELength I γ' a b

Tactic:
simp only [pathELength_eq_lintegral_mfderiv_Ioo]
  apply setLIntegral_congr_fun measurableSet_Ioo (fun t ht ↦ ?_)
  have A : γ t = γ' t := h ht
  congr! 2
  apply Filter.EventuallyEq.mfderiv_eq
  filter_upwards [Ioo_mem_nhds ht.1 ht.2] with a ha using h ha

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

Goal state before:
pp: case hus
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
A' : x₀ = ↑(extChartAt I x₀).symm (↑(extChartAt I x₀) x₀)
h'f : MDiffAt[s] f x₀
this : (mfderiv[range ↑I] ↑(extChartAt I x₀).symm (↑(extChartAt I x₀) x₀)).inverse = mfderiv% ↑(extChartAt I x₀) x₀
V' : (x : E') → TangentSpace 𝓘(𝕜, E') x := mpullbackWithin 𝓘(𝕜, E') I' (↑(extChartAt I' (f x₀)).symm) V (range ↑I')
hV' : V' = mpullbackWithin 𝓘(𝕜, E') I' (↑(extChartAt I' (f x₀)).symm) V (range ↑I')
W' : (x : E') → TangentSpace 𝓘(𝕜, E') x := mpullbackWithin 𝓘(𝕜, E') I' (↑(extChartAt I' (f x₀)).symm) W (range ↑I')
hW' : W' = mpullbackWithin 𝓘(𝕜, E') I' (↑(extChartAt I' (f x₀)).symm) W (range ↑I')
F : E → E' := (↑(extChartAt I' (f x₀)) ∘ f) ∘ ↑(extChartAt I x₀).symm
hF : F = (↑(extChartAt I' (f x₀)) ∘ f) ∘ ↑(extChartAt I x₀).symm
hFx₀ : ↑(extChartAt I' (f x₀)) (f x₀) = F (↑(extChartAt I x₀) x₀)
P :
  ∀ (Y : (x : M') → TangentSpace I' x),
    mpullbackWithin 𝓘(𝕜, E) I (↑(extChartAt I x₀).symm) (mpullbackWithin I I' f Y s)
        (range ↑I) =ᶠ[𝓝[↑(extChartAt I x₀).symm ⁻¹' s ∩ range ↑I] ↑(extChartAt I x₀) x₀]
      mpullbackWithin 𝓘(𝕜, E) 𝓘(𝕜, E') F (mpullbackWithin 𝓘(𝕜, E') I' (↑(extChartAt I' (f x₀)).symm) Y (range ↑I'))
        (↑(extChartAt I x₀).symm ⁻¹' s ∩ range ↑I)
⊢ ↑(extChartAt I x₀).symm ⁻¹' s ∩ (extChartAt I x₀).target =ᶠ[𝓝 (↑(extChartAt I x₀) x₀)]
    ↑(extChartAt I x₀).symm ⁻¹' s ∩ range ↑I

Tactic:
rw [← nhdsWithin_eq_iff_eventuallyEq]
    apply le_antisymm
    · exact nhdsWithin_mono _ (inter_subset_inter_right _ (extChartAt_target_subset_range x₀))
    · rw [nhdsWithin_le_iff, nhdsWithin_inter]
      exact Filter.inter_mem_inf self_mem_nhdsWithin (extChartAt_target_mem_nhdsWithin x₀)

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

Goal state before:
pp: case inr
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 : (x : M) → ENorm (TangentSpace I x)
a b : ℝ
γ : ℝ → M
inst✝ : ∀ (x : M), ENormSMulClass ℝ (TangentSpace I x)
f : ℝ → ℝ
h✝ : a ≤ b
hf : AntitoneOn f (Icc a b)
h'f : DifferentiableOn ℝ f (Icc a b)
hγ : MDiff[Icc (f b) (f a)] γ
h : a < b
f_im : f '' Icc a b = Icc (f b) (f a)
B : ∀ t ∈ Icc a b, HasDerivWithinAt f (derivWithin f (Icc a b) t) (Icc a b) t
⊢ ∫⁻ (t : ℝ) in Icc a b, ‖(mfderiv[Icc a b] (γ ∘ f) t) 1‖ₑ =
    ∫⁻ (t : ℝ) in f '' Icc a b, ‖(mfderiv[f '' Icc a b] γ t) 1‖ₑ

Tactic:
rw [lintegral_image_eq_lintegral_deriv_mul_of_antitoneOn measurableSet_Icc B hf]

Goal state after:
case inr
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 : (x : M) → ENorm (TangentSpace I x)
a b : ℝ
γ : ℝ → M
inst✝ : ∀ (x : M), ENormSMulClass ℝ (TangentSpace I x)
f : ℝ → ℝ
h✝ : a ≤ b
hf : AntitoneOn f (Icc a b)
h'f : DifferentiableOn ℝ f (Icc a b)
hγ : MDiff[Icc (f b) (f a)] γ
h : a < b
f_im : f '' Icc a b = Icc (f b) (f a)
B : ∀ t ∈ Icc a b, HasDerivWithinAt f (derivWithin f (Icc a b) t) (Icc a b) t
⊢ ∫⁻ (t : ℝ) in Icc a b, ‖(mfderiv[Icc a b] (γ ∘ f) t) 1‖ₑ =
    ∫⁻ (x : ℝ) in Icc a b, ENNReal.ofReal (-derivWithin f (Icc a b) x) * ‖(mfderiv[f '' Icc a b] γ (f x)) 1‖ₑ
Module: Mathlib.Geometry.Manifold.VectorField.LieBracket

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
M : Type u_4
inst✝⁴ : TopologicalSpace M
inst✝³ : ChartedSpace H M
inst✝² : IsManifold I (minSmoothness 𝕜 2) M
inst✝¹ : CompleteSpace E
m n : ℕ∞
inst✝ : IsManifold I (↑n + 1) M
U V : (x : M) → TangentSpace I x
x : M
hU : ContMDiffAt I I.tangent (↑n) (fun x ↦ ⟨x, U x⟩) x
hV : ContMDiffAt I I.tangent (↑n) (fun x ↦ ⟨x, V x⟩) x
hmn : minSmoothness 𝕜 (↑m + 1) ≤ ↑n
⊢ ContMDiffAt I I.tangent (↑m) (fun x₀ ↦ ⟨x₀, mlieBracket I U V x₀⟩) x

Tactic:
simp only [← contMDiffWithinAt_univ, ← mlieBracketWithin_univ] at hU hV ⊢

Goal state after:
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
M : Type u_4
inst✝⁴ : TopologicalSpace M
inst✝³ : ChartedSpace H M
inst✝² : IsManifold I (minSmoothness 𝕜 2) M
inst✝¹ : CompleteSpace E
m n : ℕ∞
inst✝ : IsManifold I (↑n + 1) M
U V : (x : M) → TangentSpace I x
x : M
hmn : minSmoothness 𝕜 (↑m + 1) ≤ ↑n
hU : ContMDiffWithinAt I I.tangent (↑n) (fun x ↦ ⟨x, U x⟩) univ x
hV : ContMDiffWithinAt I I.tangent (↑n) (fun x ↦ ⟨x, V x⟩) univ x
⊢ ContMDiffWithinAt I I.tangent (↑m) (fun x₀ ↦ ⟨x₀, mlieBracketWithin I U V univ x₀⟩) univ x
Module: Mathlib.Geometry.Manifold.Submersion

Goal state before:
pp: case h
𝕜 : Type u_1
E'' : Type u_3
H : Type u_7
G : Type u_9
E : Type u
inst✝¹⁰ : NontriviallyNormedField 𝕜
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace 𝕜 E
inst✝⁷ : NormedAddCommGroup E''
inst✝⁶ : NormedSpace 𝕜 E''
inst✝⁵ : TopologicalSpace H
inst✝⁴ : TopologicalSpace G
I : ModelWithCorners 𝕜 E H
J : ModelWithCorners 𝕜 E'' G
M : Type u_11
N : Type u_13
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace H M
inst✝¹ : TopologicalSpace N
inst✝ : ChartedSpace G N
n : ℕ∞ω
f g : M → N
x : M
hf : IsSubmersionAt I J n f x
hfg : f =ᶠ[𝓝 x] g
⊢ IsSubmersionAtOfComplement hf.complement I J n g x

Tactic:
exact hf.isSubmersionAtOfComplement_complement.congr_of_eventuallyEq hfg

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

Goal state before:
pp: 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
y : M
a✝ : y ∈ s
h :
  ∀ (x_1 : ι),
    (↑(f.toFun x_1) x • ↑(extChartAt I (f.c x_1)) x, ↑(f.toFun x_1) x) =
      (↑(f.toFun x_1) y • ↑(extChartAt I (f.c x_1)) y, ↑(f.toFun x_1) y)
h₁ : ↑(f.toFun (f.ind x hx)) x • ↑(extChartAt I (f.c (f.ind x hx))) x = 1 • ↑(extChartAt I (f.c (f.ind x hx))) y
h₂ : 1 = ↑(f.toFun (f.ind x hx)) y
⊢ x = y

Tactic:
f.apply_ind x hx,

Goal state after:
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
y : M
a✝ : y ∈ s
h :
  ∀ (x_1 : ι),
    (↑(f.toFun x_1) x • ↑(extChartAt I (f.c x_1)) x, ↑(f.toFun x_1) x) =
      (↑(f.toFun x_1) y • ↑(extChartAt I (f.c x_1)) y, ↑(f.toFun x_1) y)
h₁ : 1 • ↑(extChartAt I (f.c (f.ind x hx))) x = 1 • ↑(extChartAt I (f.c (f.ind x hx))) y
h₂ : 1 = ↑(f.toFun (f.ind x hx)) y
⊢ x = y
Module: Mathlib.GroupTheory.Commutator.Finite

Goal state before:
pp: case h₁
η : Type u_2
inst✝¹ : Finite η
Gs : η → Type u_3
inst✝ : (i : η) → Group (Gs i)
H K : (i : η) → Subgroup (Gs i)
i : η
hi : (i : η) → Gs i
j : η
_hj : j ∈ Set.univ
⊢ map (MonoidHom.mulSingle Gs i) (H i) ≤ comap (Pi.evalMonoidHom Gs j) (H j)

Tactic:
rintro _ ⟨x, hx, rfl⟩

Goal state after:
case h₁
η : Type u_2
inst✝¹ : Finite η
Gs : η → Type u_3
inst✝ : (i : η) → Group (Gs i)
H K : (i : η) → Subgroup (Gs i)
i : η
hi : (i : η) → Gs i
j : η
_hj : j ∈ Set.univ
x : Gs i
hx : x ∈ ↑(H i)
⊢ (MonoidHom.mulSingle Gs i) x ∈ comap (Pi.evalMonoidHom Gs j) (H j)
Module: Mathlib.GroupTheory.Commutator.Finite

Goal state before:
pp: case h₂
η : Type u_2
inst✝¹ : Finite η
Gs : η → Type u_3
inst✝ : (i : η) → Group (Gs i)
H K : (i : η) → Subgroup (Gs i)
i : η
hi : (i : η) → Gs i
j : η
_hj : j ∈ Set.univ
⊢ map (MonoidHom.mulSingle Gs i) (K i) ≤ comap (Pi.evalMonoidHom Gs j) (K j)

Tactic:
rintro _ ⟨x, hx, rfl⟩

Goal state after:
case h₂
η : Type u_2
inst✝¹ : Finite η
Gs : η → Type u_3
inst✝ : (i : η) → Group (Gs i)
H K : (i : η) → Subgroup (Gs i)
i : η
hi : (i : η) → Gs i
j : η
_hj : j ∈ Set.univ
x : Gs i
hx : x ∈ ↑(K i)
⊢ (MonoidHom.mulSingle Gs i) x ∈ comap (Pi.evalMonoidHom Gs j) (K j)
Module: Mathlib.GroupTheory.SpecificGroups.Dihedral

Goal state before:
pp: case zero
n : ℕ
i : ZMod n
⊢ r i ^ 0 = r (i * ↑0)

Tactic:
simp only [pow_zero, Nat.cast_zero, mul_zero, r_zero]

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

Goal state before:
pp: B : Type u_1
W : Type u_3
inst✝ : Group W
M : CoxeterMatrix B
cs : CoxeterSystem M W
⊢ Set.range cs.simple = Set.range cs.simple ∪ (Set.range cs.simple)⁻¹

Tactic:
simp_rw [inv_range, inv_simple, union_self]

Goal state after:
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_4
inst✝ : Monoid N
f : (i : ι) → M i →* N
⊢ MonoidHom.mrange ((conGen (Rel M)).lift (FreeMonoid.lift fun p ↦ (f p.fst) p.snd) ⋯) = ⨆ i, MonoidHom.mrange (f i)

Tactic:
Con.lift_range,

Goal state after:
ι : Type u_1
M : ι → Type u_2
inst✝¹ : (i : ι) → Monoid (M i)
N : Type u_4
inst✝ : Monoid N
f : (i : ι) → M i →* N
⊢ MonoidHom.mrange (FreeMonoid.lift fun p ↦ (f p.fst) p.snd) = ⨆ i, MonoidHom.mrange (f i)
Module: Mathlib.GroupTheory.Coxeter.Basic

Goal state before:
pp: B : Type u_1
B' : Type u_2
e : B ≃ B'
W : Type u_3
H : Type u_4
inst✝¹ : Group W
inst✝ : Group H
M : CoxeterMatrix B
cs : CoxeterSystem M W
i j : B
p k : ℕ
h : k + 1 < p
⊢ k < (alternatingWord j i p).length

Tactic:
simp; lia

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

Goal state before:
pp: 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
h_even : Even k
hk : take k (alternatingWord i j (2 * p)) = alternatingWord i j k
⊢ k < (alternatingWord i j (2 * p)).length

Tactic:
simp; lia

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

Goal state before:
pp: 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
h_even : ¬Even k
hk : take k (alternatingWord i j (2 * p)) = alternatingWord j i k
⊢ Odd ?m.111

Tactic:
by simpa using h_even

Goal state after: