mathlib-initiative/mathlib-tactics huggingface dataset:
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
x : M
V W : (x : M) → TangentSpace I x
inst✝¹ : IsManifold I 2 M
inst✝ : CompleteSpace E
f : M → 𝕜
hf : MDiffAt f x
hV : (MDiffAt fun x ↦ ⟨x, V x⟩) x
⊢ mlieBracket I (f • V) W x = -(d% f x) (W x) • V x + f x • mlieBracket I V W x

Tactic:
by
  rw [← mdifferentiableWithinAt_univ] at hf hV
  rw [← mlieBracketWithin_univ, mvfderiv, ← mfderivWithin_univ]
  exact mlieBracketWithin_smul_left hf hV (uniqueMDiffWithinAt_univ I)

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

Goal state before:
pp: 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 := ⋯
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))
⊢ (extChartAt I (g x₀)).target ∩ ↑(extChartAt I (g x₀)).symm ⁻¹' u ⊆ range ↑I

Tactic:
exact inter_subset_left.trans (extChartAt_target_subset_range (g x₀))

Goal state after:
Module: Mathlib.Geometry.Manifold.Instances.Real

Goal state before:
pp: n : ℕ
p : ℝ≥0∞
a : ℝ
⊢ interior {y | ∀ (i : Fin n), a ≤ y.ofLp i} = {y | ∀ (i : Fin n), a < y.ofLp i}

Tactic:
by
  let f i : PiLp p (fun _ : Fin n ↦ ℝ) → ℝ := fun x ↦ x i
  have h : { y : PiLp p (fun _ : Fin n ↦ ℝ) | ∀ i : Fin n, a ≤ y i } = ⋂ i, (f i) ⁻¹' Ici a := by
    ext; simp; rfl
  have h' : { y : PiLp p (fun _ : Fin n ↦ ℝ) | ∀ i : Fin n, a < y i } = ⋂ i, (f i) ⁻¹' Ioi a := by
    ext; simp; rfl
  rw [h, h', interior_iInter_of_finite]
  apply iInter_congr fun i ↦ ?_
  rw [← (PiLp.isOpenMap_apply p _ i).preimage_interior_eq_interior_preimage, interior_Ici]
  fun_prop

Goal state after:
Module: Mathlib.Geometry.Manifold.Instances.Real

Goal state before:
pp: case inl.inr
x✝ y✝ : ℝ
hxy : Fact (x✝ < y✝)
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✝
x y : ℝ
inst✝ : Fact (x < y)
n : ℕ∞ω
M :
  ContDiff ℝ n
    (have this := fun z ↦ toLp 2 fun i ↦ -z.ofLp i + (y - x);
    this)
z : EuclideanSpace ℝ (Fin 1)
hz₀ : 0 ≤ z.ofLp 0
hz₁ : ↑(𝓡∂ 1).symm (↑(𝓡∂ 1) ⟨z, hz₀⟩) ∈ (IccLeftChart x y).symm.symm.target
hz₂ : ↑(𝓡∂ 1).symm (↑(𝓡∂ 1) ⟨z, hz₀⟩) ∈ ↑(IccLeftChart x y).symm.symm.symm ⁻¹' (IccRightChart x y).source
⊢ (↑(𝓡∂ 1) ∘ ↑((IccLeftChart x y).symm ≫ₕ IccRightChart x y) ∘ ↑(𝓡∂ 1).symm) (↑(𝓡∂ 1) ⟨z, hz₀⟩) =
    (have this := fun z ↦ toLp 2 fun i ↦ -z.ofLp i + (y - x);
      this)
      (↑(𝓡∂ 1) ⟨z, hz₀⟩)

Tactic:
simp only [modelWithCornersEuclideanHalfSpace, IccLeftChart, IccRightChart, update_self,
      max_eq_left, hz₀, lt_sub_iff_add_lt, mfld_simps] at hz₁ hz₂

Goal state after:
case inl.inr
x✝ y✝ : ℝ
hxy : Fact (x✝ < y✝)
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✝
x y : ℝ
inst✝ : Fact (x < y)
n : ℕ∞ω
M :
  ContDiff ℝ n
    (have this := fun z ↦ toLp 2 fun i ↦ -z.ofLp i + (y - x);
    this)
z : EuclideanSpace ℝ (Fin 1)
hz₀ : 0 ≤ z.ofLp 0
hz₁ : z.ofLp 0 + x < y
hz₂ : x < min (z.ofLp 0 + x) y
⊢ (↑(𝓡∂ 1) ∘ ↑((IccLeftChart x y).symm ≫ₕ IccRightChart x y) ∘ ↑(𝓡∂ 1).symm) (↑(𝓡∂ 1) ⟨z, hz₀⟩) =
    (have this := fun z ↦ toLp 2 fun i ↦ -z.ofLp i + (y - x);
      this)
      (↑(𝓡∂ 1) ⟨z, hz₀⟩)
Module: Mathlib.Geometry.Manifold.ContMDiffMFDeriv

Goal state before:
pp: 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 := ⋯
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))
⊢ (extChartAt I (g x₀)).target ∩ ↑(extChartAt I (g x₀)).symm ⁻¹' u ⊆ range ↑I

Tactic:
exact inter_subset_left.trans (extChartAt_target_subset_range (g x₀))

Goal state after:
Module: Mathlib.Geometry.Manifold.Instances.Icc

Goal state before:
pp: case refine_2
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)
hf : MDiffAt f w
⊢ MDiffAt[Icc x y] (f ∘ projIcc x y ⋯) ↑w

Tactic:
· have := (contMDiffOn_projIcc (x := x) (y := y) w w.2).mdifferentiableWithinAt one_ne_zero
    exact MDifferentiableAt.comp_mdifferentiableWithinAt_of_eq (w : ℝ) hf this (by simp)

Goal state after:
Module: Mathlib.Geometry.Manifold.IntegralCurve.Basic

Goal state before:
pp: case mpr
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
s : Set ℝ
hs : s ∈ 𝓝 t₀
h : IsMIntegralCurveOn γ v s
s' : Set ℝ
h1 : s' ⊆ s
h2 : IsOpen s'
h3 : t₀ ∈ s'
t : ℝ
ht : t ∈ s'
⊢ HasMFDerivAt% γ t (ContinuousLinearMap.smulRight 1 (v (γ t)))

Tactic:
apply (h t (h1 ht)).hasMFDerivAt

Goal state after:
case mpr
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
s : Set ℝ
hs : s ∈ 𝓝 t₀
h : IsMIntegralCurveOn γ v s
s' : Set ℝ
h1 : s' ⊆ s
h2 : IsOpen s'
h3 : t₀ ∈ s'
t : ℝ
ht : t ∈ s'
⊢ s ∈ 𝓝 t
Module: Mathlib.Geometry.Manifold.ContMDiffMFDeriv

Goal state before:
pp: case hf
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 := ⋯
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))
⊢ (extChartAt I (g x₀)).target ∩ ↑(extChartAt I (g x₀)).symm ⁻¹' u ⊆ range ↑I

Tactic:
exact inter_subset_left.trans (extChartAt_target_subset_range (g x₀))

Goal state after:
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 : ↑(sphere 0 1)
h : w ∈ {⟨v, ?m.125⟩}ᶜ
⊢ ↑w ∈ {x | ((innerSL ℝ) v) x ≠ 1}

Tactic:
dsimp
      exact
        h ∘ Subtype.ext ∘ Eq.symm ∘ (inner_eq_one_iff_of_norm_eq_one hv (by simp)).mp

Goal state after:
Module: Mathlib.Geometry.Manifold.Instances.Sphere

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace ℝ E
c : Circle → ℂ := Subtype.val
h₂ : ContMDiff (𝓘(ℝ, ℂ).prod 𝓘(ℝ, ℂ)) 𝓘(ℝ, ℂ) ω fun z ↦ z.1 * z.2
⊢ ContMDiff ((𝓡 1).prod (𝓡 1)) (𝓘(ℝ, ℂ).prod 𝓘(ℝ, ℂ)) ω (Prod.map c c)

Tactic:
apply ContMDiff.prodMap <;> exact contMDiff_coe_sphere

Goal state after:
Module: Mathlib.Geometry.Manifold.IntegralCurve.UniformTime

Goal state before:
pp: case pos
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)
a t : ℝ
ht : t ∈ Ioo (-a) a
hlt : |t| + 1 < a
⊢ (fun t ↦ γ (|t| + 1) t) t = γ a t

Tactic:
exact eqOn_of_isMIntegralCurveOn_Ioo hv γ hγx hγ
      (by positivity) hlt.le (abs_lt.mp <| lt_add_one _)

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

Goal state before:
pp: F : Type u_4
inst✝¹ : NormedAddCommGroup F
inst✝ : InnerProductSpace ℝ F
x : F
v : TangentSpace 𝓘(ℝ, F) x
⊢ ‖v‖₊ = ‖v‖₊

Tactic:
by
  simp [nnnorm, norm_tangentSpace_vectorSpace]

Goal state after:
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}
⊢ ↑(extChartAt I x) ⁻¹' (ball (↑(extChartAt I x) x) r ∩ range ↑I) ∈ 𝓝 x

Tactic:
apply extChartAt_preimage_mem_nhds_of_mem_nhdsWithin (by simp)

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}
⊢ ball (↑(extChartAt I x) x) r ∩ range ↑I ∈ 𝓝[range ↑I] ↑(extChartAt I x) x
Module: Mathlib.Geometry.Manifold.VectorBundle.LocalFrame

Goal state before:
pp: 𝕜 : Type u_1
inst✝¹² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace 𝕜 E
H : Type u_3
inst✝⁹ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝⁸ : TopologicalSpace M
inst✝⁷ : ChartedSpace H M
F : Type u_5
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace 𝕜 F
V : M → Type u_6
inst✝⁴ : TopologicalSpace (TotalSpace F V)
inst✝³ : (x : M) → AddCommGroup (V x)
inst✝² : (x : M) → Module 𝕜 (V x)
inst✝¹ : (x : M) → TopologicalSpace (V x)
inst✝ : FiberBundle F V
ι : Type u_7
s : ι → (x : M) → V x
u : Set M
n : ℕ∞ω
x : M
t t' : (x : M) → V x
hs : IsLocalFrameOn I F n s u
htt' : t x = t' x
i : ι
⊢ (hs.coeff i x) (t x) = (hs.coeff i x) (t' x)

Tactic:
by
  by_cases hxe : x ∈ u <;> simp [coeff, hxe, htt']

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝¹² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace 𝕜 E
H : Type u_3
inst✝⁹ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝⁸ : TopologicalSpace M
inst✝⁷ : ChartedSpace H M
F : Type u_5
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace 𝕜 F
V : M → Type u_6
inst✝⁴ : TopologicalSpace (TotalSpace F V)
inst✝³ : (x : M) → AddCommGroup (V x)
inst✝² : (x : M) → Module 𝕜 (V x)
inst✝¹ : (x : M) → TopologicalSpace (V x)
inst✝ : FiberBundle F V
ι : Type u_7
s s' : ι → (x : M) → V x
u : Set M
n : ℕ∞ω
x : M
hs : IsLocalFrameOn I F n s u
hs' : IsLocalFrameOn I F n s' u
hss' : ∀ (i : ι), s i x = s' i x
t : (x : M) → V x
i : ι
⊢ (hs.coeff i x) (t x) = (hs'.coeff i x) (t x)

Tactic:
by_cases hxe : x ∈ u
  · simp [coeff, hxe]
    simp_all only [toBasisAt]
  · simp [coeff, hxe]

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

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 : FiberBundle F V
A : Type u_9
inst✝¹ : AddCommGroup A
inst✝ : Module 𝕜 A
Φ : ((x : M) → V x) → A
x : M
hΦ : TensorialAt I F Φ x
σ σ' : (x : M) → V x
hσ : MDiffAt (T% σ) x
hσ' : MDiffAt (T% σ') x
hσσ' : ∀ᶠ (x' : M) in 𝓝 x, σ x' = σ' x'
⊢ Φ σ = Φ σ'

Tactic:
classical
  -- Introduce the indicator function of a neighbourhood `t` of `x` on which equality holds,
  -- and cut off the two sections `σ` and `σ'` using this indicator function.
  let ψ (x' : M) : 𝕜 := if σ x' = σ' x' then 1 else 0
  have hψx : ψ x = 1 := by simp [ψ, hσσ'.self_of_nhds]
  have (x' : M) : (ψ • σ) x' = (ψ • σ') x' := by
    dsimp [ψ]
    split_ifs with hx' <;> simp [hx']
  have hψ' : MDiffAt ψ x := by
    have : MDiffAt (fun (_x : M) ↦ (1 : 𝕜)) x := mdifferentiableAt_const
    exact this.congr_of_eventuallyEq (hσσ'.mono fun x' hx' ↦ by simp [ψ, hx'])
  calc Φ σ
    _ = Φ (ψ • σ) := by simp [hΦ.smul hψ' hσ, hψx]
    _ = Φ (ψ • σ') := by rw [funext this]
    _ = Φ σ' := by simp [hΦ.smul hψ' hσ', hψx]

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

Goal state before:
pp: B : Type u_1
inst✝⁷ : TopologicalSpace B
F : Type u_2
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace ℝ F
E : B → Type u_3
inst✝⁴ : TopologicalSpace (TotalSpace F E)
inst✝³ : (x : B) → NormedAddCommGroup (E x)
inst✝² : (x : B) → InnerProductSpace ℝ (E x)
inst✝¹ : FiberBundle F E
inst✝ : VectorBundle ℝ F E
h : IsContinuousRiemannianBundle F E
x : B
r : ℝ
hr : 1 < r
r' : ℝ
hr' : 1 < r'
r'r : r' < r
h'x : x ∈ (trivializationAt F E x).baseSet
G : E x ≃L[ℝ] F := Trivialization.continuousLinearEquivAt ℝ (trivializationAt F E x) x h'x
C : ℝ := ‖↑G‖ ^ 2
δ : ℝ
δpos : 0 < δ
hδ : (r' ^ 2)⁻¹ < 1 - δ * C
g : (x : B) → E x →L[ℝ] E x →L[ℝ] ℝ
g_cont :
  Continuous[inst✝⁷, topologicalSpaceTotalSpace (RingHom.id ℝ) F E (F →L[ℝ] ℝ) fun x ↦ E x →L[ℝ] ℝ] fun x ↦
    TotalSpace.mk' (F →L[ℝ] F →L[ℝ] ℝ) x (g x)
hg : ∀ (x : B) (v w : E x), ⟪v, w⟫ = ((g x) v) w
g' : B → F →L[ℝ] F →L[ℝ] ℝ := fun y ↦ inCoordinates F E (F →L[ℝ] ℝ) (fun x ↦ E x →L[ℝ] ℝ) x y x y (g y)
hg' : ContinuousAt g' x
this✝ : ∀ᶠ (y : B) in 𝓝 x, dist (g' y) (g' x) < δ
y : B
hy : dist (g' y) (g' x) < δ
h'y : y ∈ (trivializationAt F E x).baseSet
this : ‖g' x - g' y‖ ≤ δ
v : E y
w : F := (Trivialization.continuousLinearMapAt ℝ (trivializationAt F E x) y) v
hgy : ((g y) v) v = ((g' y) w) w
⊢ ((g x) ((trivializationAt F E x).symm x w)) ((trivializationAt F E x).symm x w) = ((g' x) w) w

Tactic:
rw [inCoordinates_apply_eq₂ h'x h'x (Set.mem_univ _)]

Goal state after:
B : Type u_1
inst✝⁷ : TopologicalSpace B
F : Type u_2
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace ℝ F
E : B → Type u_3
inst✝⁴ : TopologicalSpace (TotalSpace F E)
inst✝³ : (x : B) → NormedAddCommGroup (E x)
inst✝² : (x : B) → InnerProductSpace ℝ (E x)
inst✝¹ : FiberBundle F E
inst✝ : VectorBundle ℝ F E
h : IsContinuousRiemannianBundle F E
x : B
r : ℝ
hr : 1 < r
r' : ℝ
hr' : 1 < r'
r'r : r' < r
h'x : x ∈ (trivializationAt F E x).baseSet
G : E x ≃L[ℝ] F := Trivialization.continuousLinearEquivAt ℝ (trivializationAt F E x) x h'x
C : ℝ := ‖↑G‖ ^ 2
δ : ℝ
δpos : 0 < δ
hδ : (r' ^ 2)⁻¹ < 1 - δ * C
g : (x : B) → E x →L[ℝ] E x →L[ℝ] ℝ
g_cont :
  Continuous[inst✝⁷, topologicalSpaceTotalSpace (RingHom.id ℝ) F E (F →L[ℝ] ℝ) fun x ↦ E x →L[ℝ] ℝ] fun x ↦
    TotalSpace.mk' (F →L[ℝ] F →L[ℝ] ℝ) x (g x)
hg : ∀ (x : B) (v w : E x), ⟪v, w⟫ = ((g x) v) w
g' : B → F →L[ℝ] F →L[ℝ] ℝ := fun y ↦ inCoordinates F E (F →L[ℝ] ℝ) (fun x ↦ E x →L[ℝ] ℝ) x y x y (g y)
hg' : ContinuousAt g' x
this✝ : ∀ᶠ (y : B) in 𝓝 x, dist (g' y) (g' x) < δ
y : B
hy : dist (g' y) (g' x) < δ
h'y : y ∈ (trivializationAt F E x).baseSet
this : ‖g' x - g' y‖ ≤ δ
v : E y
w : F := (Trivialization.continuousLinearMapAt ℝ (trivializationAt F E x) y) v
hgy : ((g y) v) v = ((g' y) w) w
⊢ ((g x) ((trivializationAt F E x).symm x w)) ((trivializationAt F E x).symm x w) =
    (Trivialization.linearMapAt ℝ (trivializationAt ℝ (fun x ↦ ℝ) x) x)
      (((g x) ((trivializationAt F E x).symm x w)) ((trivializationAt F E x).symm x w))