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

Goal state before:
pp: case hg
𝕜 : Type u_1
inst✝¹⁵ : NontriviallyNormedField 𝕜
m n : WithTop ℕ∞
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
E' : Type u_5
inst✝⁹ : NormedAddCommGroup E'
inst✝⁸ : NormedSpace 𝕜 E'
H' : Type u_6
inst✝⁷ : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
M' : Type u_7
inst✝⁶ : TopologicalSpace M'
inst✝⁵ : ChartedSpace H' M'
F : Type u_8
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace 𝕜 F
G : Type u_9
inst✝² : TopologicalSpace G
J : ModelWithCorners 𝕜 F G
N : Type u_10
inst✝¹ : TopologicalSpace N
inst✝ : ChartedSpace G N
Js : IsManifold J 1 N
Is : IsManifold I 1 M
I's : IsManifold I' 1 M'
x₀ : N
f : N → M → M'
g : N → M
t : Set N
u : Set M
hf : ContMDiffWithinAt (J.prod I) I' n (uncurry f) (t ×ˢ u) (x₀, g x₀)
hg : ContMDiffWithinAt J I m g t x₀
hx₀ : x₀ ∈ t
hu : MapsTo g t u
hmn : m + 1 ≤ n
h'u : UniqueMDiff[u]
t' : Set N := t ∩ g ⁻¹' (extChartAt I (g x₀)).source
ht't : t' ⊆ t
hx₀gx₀ : (x₀, g x₀) ∈ t ×ˢ u
h4f✝ : ContinuousWithinAt (fun x ↦ f x (g x)) t x₀
h4f : (fun x ↦ f x (g x)) ⁻¹' (extChartAt I' (f x₀ (g x₀))).source ∈ 𝓝[t] x₀
h3f : ∀ᶠ (x' : N × M) in 𝓝[t ×ˢ u] (x₀, g x₀), ContMDiffWithinAt (J.prod I) I' 1 (uncurry f) (t ×ˢ u) x'
h2f : ∀ᶠ (x₂ : N) in 𝓝[t] x₀, ContMDiffWithinAt I I' 1 (f x₂) u (g x₂)
h2g : g ⁻¹' (extChartAt I (g x₀)).source ∈ 𝓝[t] x₀
this✝ :
  ContDiffWithinAt 𝕜 m
    (fun x ↦
      fderivWithin 𝕜 (↑(extChartAt I' (f x₀ (g x₀))) ∘ f (↑(extChartAt J x₀).symm x) ∘ ↑(extChartAt I (g x₀)).symm)
        ((extChartAt I (g x₀)).target ∩ ↑(extChartAt I (g x₀)).symm ⁻¹' u)
        (↑(extChartAt I (g x₀)) (g (↑(extChartAt J x₀).symm x))))
    (↑(extChartAt J x₀).symm ⁻¹' t' ∩ range ↑J) (↑(extChartAt J x₀) x₀)
this :
  ContMDiffWithinAt J 𝓘(𝕜, E →L[𝕜] E') m
    (fun x ↦
      fderivWithin 𝕜 (↑(extChartAt I' (f x₀ (g x₀))) ∘ f x ∘ ↑(extChartAt I (g x₀)).symm)
        ((extChartAt I (g x₀)).target ∩ ↑(extChartAt I (g x₀)).symm ⁻¹' u) (↑(extChartAt I (g x₀)) (g x)))
    t' x₀
x : N
hx : ContMDiffWithinAt I I' 1 (f x) u (g x)
h'x : x ∈ (fun x ↦ f x (g x)) ⁻¹' (extChartAt I' (f x₀ (g x₀))).source
h2 : x ∈ g ⁻¹' (extChartAt I (g x₀)).source
hxt : x ∈ t
h1 : g x ∈ u
h3 : UniqueMDiffAt[(extChartAt I (g x₀)).target ∩ ↑(extChartAt I (g x₀)).symm ⁻¹' u] (↑(extChartAt I (g x₀)) (g x))
A :
  mfderiv[range ↑I] ↑(extChartAt I (g x₀)).symm (↑(extChartAt I (g x₀)) (g x)) =
    mfderiv[(extChartAt I (g x₀)).target ∩ ↑(extChartAt I (g x₀)).symm ⁻¹' u] ↑(extChartAt I (g x₀)).symm
      (↑(extChartAt I (g x₀)) (g x))
⊢ MDiffAt ↑(extChartAt I' (f x₀ (g x₀))) (f x (g x))

Tactic:
exact mdifferentiableAt_extChartAt (by simpa using h'x)

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E₁ : Type u_2
inst✝⁵ : NormedAddCommGroup E₁
inst✝⁴ : NormedSpace 𝕜 E₁
H : Type u_6
inst✝³ : TopologicalSpace H
I : ModelWithCorners 𝕜 E₁ H
M : Type u_10
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace H M
n : ℕ∞ω
inst✝ : IsManifold I n M
s : TopologicalSpace.Opens M
⊢ IsSmoothEmbedding I I n Subtype.val

Tactic:
rw [isSmoothEmbedding_iff]

Goal state after:
𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E₁ : Type u_2
inst✝⁵ : NormedAddCommGroup E₁
inst✝⁴ : NormedSpace 𝕜 E₁
H : Type u_6
inst✝³ : TopologicalSpace H
I : ModelWithCorners 𝕜 E₁ H
M : Type u_10
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace H M
n : ℕ∞ω
inst✝ : IsManifold I n M
s : TopologicalSpace.Opens M
⊢ IsImmersion I I n Subtype.val ∧ IsEmbedding Subtype.val
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:
← 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.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
hγ : IsMIntegralCurveOn γ v s
a t : ℝ
ht : t ∈ {t | t * a ∈ s}
this :
  ContinuousLinearMap.smulRight 1 (a • v (γ (t * a))) =
    ContinuousLinearMap.smulRight 1 (v (γ (t * a))) ∘SL ContinuousLinearMap.smulRight 1 a
⊢ HasMFDerivAt[{t | t * a ∈ s}] (γ ∘ fun x ↦ x * a) t
    (ContinuousLinearMap.smulRight 1 ((a • v) ((γ ∘ fun x ↦ x * a) t)))

Tactic:
rw [comp_apply, Pi.smul_apply, this]

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
hγ : IsMIntegralCurveOn γ v s
a t : ℝ
ht : t ∈ {t | t * a ∈ s}
this :
  ContinuousLinearMap.smulRight 1 (a • v (γ (t * a))) =
    ContinuousLinearMap.smulRight 1 (v (γ (t * a))) ∘SL ContinuousLinearMap.smulRight 1 a
⊢ HasMFDerivAt[{t | t * a ∈ s}] (γ ∘ fun x ↦ x * a) t
    (ContinuousLinearMap.smulRight 1 (v (γ (t * a))) ∘SL ContinuousLinearMap.smulRight 1 a)
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
inst✝ : BoundarylessManifold I M
v : (x : M) → TangentSpace I x
hv : ContMDiff I I.tangent 1 fun x ↦ ⟨x, v x⟩
ε : ℝ
hε : 0 < ε
h : ∀ (x : M), ∃ γ, γ 0 = x ∧ IsMIntegralCurveOn γ v (Ioo (-ε) ε)
x : M
s : Set ℝ := {a | ∃ γ, γ 0 = x ∧ IsMIntegralCurveOn γ v (Ioo (-a) a)}
hbdd : BddAbove s
asup : ℝ := sSup s
hasup : asup = sSup s
a : ℝ
hlt : asup - ε / 2 < a
γ : ℝ → M
h0 : γ 0 = x
hγ : IsMIntegralCurveOn γ v (Ioo (-a) a)
γ1_aux : ℝ → M
h1_aux : γ1_aux 0 = γ (-(asup - ε / 2))
γ1 : ℝ → M := γ1_aux ∘ fun x ↦ x + (asup - ε / 2)
γ1_def : γ1 = γ1_aux ∘ fun x ↦ x + (asup - ε / 2)
heq1 : γ1 (-(asup - ε / 2)) = γ (-(asup - ε / 2))
γ2_aux : ℝ → M
h2_aux : γ2_aux 0 = γ (asup - ε / 2)
γ2 : ℝ → M := γ2_aux ∘ fun x ↦ x - (asup - ε / 2)
γ2_def : γ2 = γ2_aux ∘ fun x ↦ x - (asup - ε / 2)
heq2 : γ2 (asup - ε / 2) = γ (asup - ε / 2)
hγ1 : IsMIntegralCurveOn γ1 v {t | t ∈ Ioo (-ε - (asup - ε / 2)) (ε - (asup - ε / 2))}
hγ2 : IsMIntegralCurveOn γ2 v {t | t ∈ Ioo (-ε + (asup - ε / 2)) (ε + (asup - ε / 2))}
hεle : ε ≤ asup
γ_ext : ℝ → M := (Ioo (-(asup + ε / 2)) a).piecewise ((Ioo (-a) a).piecewise γ γ1) γ2
γ_ext_def : γ_ext = (Ioo (-(asup + ε / 2)) a).piecewise ((Ioo (-a) a).piecewise γ γ1) γ2
⊢ γ 0 = x

Tactic:
h0

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
inst✝ : BoundarylessManifold I M
v : (x : M) → TangentSpace I x
hv : ContMDiff I I.tangent 1 fun x ↦ ⟨x, v x⟩
ε : ℝ
hε : 0 < ε
h : ∀ (x : M), ∃ γ, γ 0 = x ∧ IsMIntegralCurveOn γ v (Ioo (-ε) ε)
x : M
s : Set ℝ := {a | ∃ γ, γ 0 = x ∧ IsMIntegralCurveOn γ v (Ioo (-a) a)}
hbdd : BddAbove s
asup : ℝ := sSup s
hasup : asup = sSup s
a : ℝ
hlt : asup - ε / 2 < a
γ : ℝ → M
h0 : γ 0 = x
hγ : IsMIntegralCurveOn γ v (Ioo (-a) a)
γ1_aux : ℝ → M
h1_aux : γ1_aux 0 = γ (-(asup - ε / 2))
γ1 : ℝ → M := γ1_aux ∘ fun x ↦ x + (asup - ε / 2)
γ1_def : γ1 = γ1_aux ∘ fun x ↦ x + (asup - ε / 2)
heq1 : γ1 (-(asup - ε / 2)) = γ (-(asup - ε / 2))
γ2_aux : ℝ → M
h2_aux : γ2_aux 0 = γ (asup - ε / 2)
γ2 : ℝ → M := γ2_aux ∘ fun x ↦ x - (asup - ε / 2)
γ2_def : γ2 = γ2_aux ∘ fun x ↦ x - (asup - ε / 2)
heq2 : γ2 (asup - ε / 2) = γ (asup - ε / 2)
hγ1 : IsMIntegralCurveOn γ1 v {t | t ∈ Ioo (-ε - (asup - ε / 2)) (ε - (asup - ε / 2))}
hγ2 : IsMIntegralCurveOn γ2 v {t | t ∈ Ioo (-ε + (asup - ε / 2)) (ε + (asup - ε / 2))}
hεle : ε ≤ asup
γ_ext : ℝ → M := (Ioo (-(asup + ε / 2)) a).piecewise ((Ioo (-a) a).piecewise γ γ1) γ2
γ_ext_def : γ_ext = (Ioo (-(asup + ε / 2)) a).piecewise ((Ioo (-a) a).piecewise γ γ1) γ2
⊢ x = x
Module: Mathlib.MeasureTheory.Constructions.UnitInterval

Goal state before:
pp: x y : ↑I
⊢ volume (Ico x y) = ENNReal.ofReal (↑y - ↑x)

Tactic:
simp only [volume_apply, image_subtype_val_Ico, Real.volume_Ico]

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

Goal state before:
pp: case h
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 : MonotoneOn f (Icc a b)
h'f : DifferentiableOn ℝ f (Icc a b)
hγ : MDiff[f '' Icc a b] γ
h : a < b
f_im : f '' Icc a b = Icc (f a) (f b)
B : ∀ t ∈ Icc a b, HasDerivWithinAt f (derivWithin f (Icc a b) t) (Icc a b) t
t : ℝ
ht : t ∈ Icc a b
⊢ Icc a b ⊆ f ⁻¹' f '' Icc a b

pp: case hxs
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 : MonotoneOn f (Icc a b)
h'f : DifferentiableOn ℝ f (Icc a b)
hγ : MDiff[f '' Icc a b] γ
h : a < b
f_im : f '' Icc a b = Icc (f a) (f b)
B : ∀ t ∈ Icc a b, HasDerivWithinAt f (derivWithin f (Icc a b) t) (Icc a b) t
t : ℝ
ht : t ∈ Icc a b
⊢ UniqueMDiffAt[Icc a b] t

Tactic:
· exact subset_preimage_image _ _

Goal state after:
case hxs
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 : MonotoneOn f (Icc a b)
h'f : DifferentiableOn ℝ f (Icc a b)
hγ : MDiff[f '' Icc a b] γ
h : a < b
f_im : f '' Icc a b = Icc (f a) (f b)
B : ∀ t ∈ Icc a b, HasDerivWithinAt f (derivWithin f (Icc a b) t) (Icc a b) t
t : ℝ
ht : t ∈ Icc a b
⊢ UniqueMDiffAt[Icc a b] t
Module: Mathlib.Geometry.Manifold.Riemannian.PathELength

Goal state before:
pp: case h
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[f '' Icc a b] 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
ht : t ∈ Icc a b
⊢ Icc a b ⊆ f ⁻¹' f '' Icc a b

pp: case hxs
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[f '' Icc a b] 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
ht : t ∈ Icc a b
⊢ UniqueMDiffAt[Icc a b] t

Tactic:
· exact subset_preimage_image _ _

Goal state after:
case hxs
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[f '' Icc a b] 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
ht : t ∈ Icc a b
⊢ UniqueMDiffAt[Icc a b] t
Module: Mathlib.Geometry.Manifold.VectorField.LieBracket

Goal state before:
pp: case refine_1
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
hfi : (mfderiv[s] f x₀).IsInvertible
this✝ : CompleteSpace E'
u : Set M
u_open : IsOpen[inst✝⁹] u
x₀u : x₀ ∈ u
hu : ContMDiffOn I I' 2 f (insert x₀ s ∩ u)
this : f ⁻¹' (extChartAt I' (f x₀)).source ∈ 𝓝[s] x₀
w : Set M
w_open : IsOpen[inst✝⁹] w
x₀w : x₀ ∈ w
hw : w ∩ s ⊆ f ⁻¹' t ∩ f ⁻¹' (extChartAt I' (f x₀)).source
⊢ s ∩ (u ∩ w) ⊆ f ⁻¹' t

Tactic:
apply Subset.trans _ (hw.trans inter_subset_left)
      exact fun y hy ↦ ⟨hy.2.2, hy.1⟩

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

Goal state before:
pp: case hγ
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)
inst✝ : ∀ (x : M), ENormSMulClass ℝ (TangentSpace I x)
x✝ y✝ x y : M
r : ℝ≥0∞
hr : riemannianEDist I x y < r
γ : ℝ → M
γ0 : γ 0 = x
γ1 : γ 1 = y
γ_smooth : ContMDiff 𝓘(ℝ, ℝ) I 1 γ
hγ : pathELength I γ 0 1 < r
η : ℝ → ℝ := fun t ↦ -t
h_smooth : ContMDiff 𝓘(ℝ, ℝ) I 1 (γ ∘ η)
this : riemannianEDist I y x ≤ pathELength I (γ ∘ η) (η 1) (η 0)
⊢ MDiff[Icc (η 1) (η 0)] (γ ∘ η)

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 : (x : M) → ENorm (TangentSpace I x)
inst✝ : ∀ (x : M), ENormSMulClass ℝ (TangentSpace I x)
x✝ y✝ x y : M
r : ℝ≥0∞
hr : riemannianEDist I x y < r
γ : ℝ → M
γ0 : γ 0 = x
γ1 : γ 1 = y
γ_smooth : ContMDiff 𝓘(ℝ, ℝ) I 1 γ
hγ : pathELength I γ 0 1 < r
η : ℝ → ℝ := fun t ↦ -t
h_smooth : ContMDiff 𝓘(ℝ, ℝ) I 1 (γ ∘ η)
this : riemannianEDist I y x ≤ pathELength I ((γ ∘ η) ∘ η) 0 1
⊢ riemannianEDist I y x < r

Tactic:
· exact h_smooth.contMDiffOn.mdifferentiableOn one_ne_zero

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 : (x : M) → ENorm (TangentSpace I x)
inst✝ : ∀ (x : M), ENormSMulClass ℝ (TangentSpace I x)
x✝ y✝ x y : M
r : ℝ≥0∞
hr : riemannianEDist I x y < r
γ : ℝ → M
γ0 : γ 0 = x
γ1 : γ 1 = y
γ_smooth : ContMDiff 𝓘(ℝ, ℝ) I 1 γ
hγ : pathELength I γ 0 1 < r
η : ℝ → ℝ := fun t ↦ -t
h_smooth : ContMDiff 𝓘(ℝ, ℝ) I 1 (γ ∘ η)
this : riemannianEDist I y x ≤ pathELength I ((γ ∘ η) ∘ η) 0 1
⊢ riemannianEDist I y x < r
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 ≤
    ↑C *
      ∫⁻ (t : ℝ) in Icc 0 1,
        ‖(fderivWithin ℝ (⇑(ContinuousAffineMap.lineMap (↑(extChartAt I x) x) (↑(extChartAt I x) y))) (Icc 0 1) t) 1‖ₑ

Tactic:
← lintegral_const_mul' _ _ ENNReal.coe_ne_top

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) 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
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 := ⋯
γ : ℝ → M := ⋯
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
⊢ pathELength I γ 0 1 ≤ ↑C * edist (↑(extChartAt I x) x) (↑(extChartAt I x) y)

Tactic:
rw [← lintegral_fderiv_lineMap_eq_edist, pathELength_eq_lintegral_mfderivWithin_Icc,
    ← lintegral_const_mul' _ _ ENNReal.coe_ne_top]

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) 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
Module: Mathlib.Geometry.Manifold.VectorBundle.LocalFrame

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
inst✝² : VectorBundle 𝕜 F V
ι : Type u_7
e : Trivialization F TotalSpace.proj
inst✝¹ : MemTrivializationAtlas e
b : Basis ι 𝕜 F
inst✝ : ContMDiffVectorBundle 1 F V I
x : M
hx : x ∉ e.baseSet
i : ι
⊢ localFrameCoeff I e b i x = 0

Tactic:
simpa [localFrameCoeff] using
    (e.isLocalFrameOn_localFrame_baseSet I 1 b).coeff_apply_of_notMem hx i

Goal state after:
Module: Mathlib.GroupTheory.GroupAction.CardCommute

Goal state before:
pp: G : Type u_3
inst✝ : Group G
⊢ Nat.card { p // Commute p.1 p.2 } = Nat.card (ConjClasses G) * Nat.card G

Tactic:
rcases fintypeOrInfinite G

Goal state after:
case inl
G : Type u_3
inst✝ : Group G
val✝ : Fintype G
⊢ Nat.card { p // Commute p.1 p.2 } = Nat.card (ConjClasses G) * Nat.card G

case inr
G : Type u_3
inst✝ : Group G
val✝ : Infinite G
⊢ Nat.card { p // Commute p.1 p.2 } = Nat.card (ConjClasses G) * Nat.card G
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
fi : (i : ι) → M i →* N
⊢ (fun f x ↦ f.comp of) ((fun fi ↦ (conGen (Rel M)).lift (FreeMonoid.lift fun p ↦ (fi p.fst) p.snd) ⋯) fi) = fi

Tactic:
ext i x

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
fi : (i : ι) → M i →* N
i : ι
x : M i
⊢ ((fun f x ↦ f.comp of) ((fun fi ↦ (conGen (Rel M)).lift (FreeMonoid.lift fun p ↦ (fi p.fst) p.snd) ⋯) fi) i) x =
    (fi i) x