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

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁷ : NontriviallyNormedField 𝕜
n : ℕ∞ω
E : Type u_2
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace 𝕜 E
H : Type u_3
inst✝⁴ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
G : Type u_4
inst✝³ : TopologicalSpace G
inst✝² : ChartedSpace H G
inst✝¹ : Monoid G
inst✝ : ContMDiffMul I ∞ G
g✝ h : G
r : 𝕜
X✝ Y✝ : LeftInvariantDerivation I G
f✝ f' : C^∞⟮I, G; 𝓘(𝕜, 𝕜), 𝕜⟯
X Y : LeftInvariantDerivation I G
g : G
f : C^∞⟮I, G; 𝕜⟯⟨g⟩
hX : (X (Y (ContMDiffMap.comp f (𝑳 I g)))) 1 = ((Derivation.evalAt g) ↑X) (Y f)
hY : (Y (X (ContMDiffMap.comp f (𝑳 I g)))) 1 = ((Derivation.evalAt g) ↑Y) (X f)
⊢ ((Derivation.evalAt g) ↑X) (Y f) - (Y (X (ContMDiffMap.comp f (𝑳 I g)))) 1 = ((Derivation.evalAt g) ⁅↑X, ↑Y⁆) f

Tactic:
hY

Goal state after:
𝕜 : Type u_1
inst✝⁷ : NontriviallyNormedField 𝕜
n : ℕ∞ω
E : Type u_2
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace 𝕜 E
H : Type u_3
inst✝⁴ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
G : Type u_4
inst✝³ : TopologicalSpace G
inst✝² : ChartedSpace H G
inst✝¹ : Monoid G
inst✝ : ContMDiffMul I ∞ G
g✝ h : G
r : 𝕜
X✝ Y✝ : LeftInvariantDerivation I G
f✝ f' : C^∞⟮I, G; 𝓘(𝕜, 𝕜), 𝕜⟯
X Y : LeftInvariantDerivation I G
g : G
f : C^∞⟮I, G; 𝕜⟯⟨g⟩
hX : (X (Y (ContMDiffMap.comp f (𝑳 I g)))) 1 = ((Derivation.evalAt g) ↑X) (Y f)
hY : (Y (X (ContMDiffMap.comp f (𝑳 I g)))) 1 = ((Derivation.evalAt g) ↑Y) (X f)
⊢ ((Derivation.evalAt g) ↑X) (Y f) - ((Derivation.evalAt g) ↑Y) (X f) = ((Derivation.evalAt g) ⁅↑X, ↑Y⁆) f
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✝¹¹ : NontriviallyNormedField 𝕜
inst✝¹⁰ : NormedAddCommGroup F
inst✝⁹ : NormedSpace 𝕜 F
inst✝⁸ : TopologicalSpace (TotalSpace F E)
inst✝⁷ : (x : B) → TopologicalSpace (E x)
EB : Type u_7
inst✝⁶ : NormedAddCommGroup EB
inst✝⁵ : NormedSpace 𝕜 EB
HB : Type u_8
inst✝⁴ : TopologicalSpace HB
IB : ModelWithCorners 𝕜 EB HB
inst✝³ : TopologicalSpace B
inst✝² : ChartedSpace HB B
inst✝¹ : FiberBundle F E
s : (x : B) → E x
x : B
inst✝ : Subsingleton F
⊢ MDiffAt (T% s) x

Tactic:
rw [← mdifferentiableWithinAt_univ]

Goal state after:
𝕜 : Type u_1
B : Type u_2
F : Type u_4
E : B → Type u_6
inst✝¹¹ : NontriviallyNormedField 𝕜
inst✝¹⁰ : NormedAddCommGroup F
inst✝⁹ : NormedSpace 𝕜 F
inst✝⁸ : TopologicalSpace (TotalSpace F E)
inst✝⁷ : (x : B) → TopologicalSpace (E x)
EB : Type u_7
inst✝⁶ : NormedAddCommGroup EB
inst✝⁵ : NormedSpace 𝕜 EB
HB : Type u_8
inst✝⁴ : TopologicalSpace HB
IB : ModelWithCorners 𝕜 EB HB
inst✝³ : TopologicalSpace B
inst✝² : ChartedSpace HB B
inst✝¹ : FiberBundle F E
s : (x : B) → E x
x : B
inst✝ : Subsingleton F
⊢ MDiffAt[univ] (T% s) x
Module: Mathlib.Geometry.Manifold.MFDeriv.NormedSpace

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁷ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace 𝕜 E
H : Type u_3
inst✝⁴ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace H M
F : Type u_8
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
x : M
⊢ d% 0 x = 0

Tactic:
by
  have : d% (0 : M → F) x + d% (0 : M → F) x = d% (0 : M → F) x := by
    rw [← mvfderiv_add (by exact mdifferentiable_const ..) (by exact mdifferentiable_const ..)]
    simp
  simpa using this

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁵ : NontriviallyNormedField 𝕜
H : Type u_2
inst✝⁴ : TopologicalSpace H
E : Type u_3
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝¹ : TopologicalSpace M
inst✝ : ChartedSpace H M
x : M
V W V₁ W₁ : (x : M) → TangentSpace I x
hV : V₁ =ᶠ[𝓝 x] V
hW : W₁ =ᶠ[𝓝 x] W
⊢ mlieBracketWithin I V₁ W₁ univ x = mlieBracketWithin I V W univ x

Tactic:
hV.mlieBracketWithin_vectorField_eq_nhds hW

Goal state after:
𝕜 : Type u_1
inst✝⁵ : NontriviallyNormedField 𝕜
H : Type u_2
inst✝⁴ : TopologicalSpace H
E : Type u_3
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝¹ : TopologicalSpace M
inst✝ : ChartedSpace H M
x : M
V W V₁ W₁ : (x : M) → TangentSpace I x
hV : V₁ =ᶠ[𝓝 x] V
hW : W₁ =ᶠ[𝓝 x] W
⊢ mlieBracketWithin I V W univ x = mlieBracketWithin I V W univ x
Module: Mathlib.Geometry.Manifold.LocalSourceTargetProperty

Goal state before:
pp: case codChart_mem_maximalAtlas
𝕜 : Type u_1
E : Type u_2
E' : Type u_3
F : Type u_4
F' : Type u_5
H : Type u_6
H' : Type u_7
G : Type u_8
G' : Type u_9
inst✝²⁴ : NontriviallyNormedField 𝕜
inst✝²³ : NormedAddCommGroup E
inst✝²² : NormedSpace 𝕜 E
inst✝²¹ : NormedAddCommGroup E'
inst✝²⁰ : NormedSpace 𝕜 E'
inst✝¹⁹ : NormedAddCommGroup F
inst✝¹⁸ : NormedSpace 𝕜 F
inst✝¹⁷ : NormedAddCommGroup F'
inst✝¹⁶ : NormedSpace 𝕜 F'
inst✝¹⁵ : TopologicalSpace H
inst✝¹⁴ : TopologicalSpace H'
inst✝¹³ : TopologicalSpace G
inst✝¹² : TopologicalSpace G'
I : ModelWithCorners 𝕜 E H
I' : ModelWithCorners 𝕜 E' H'
J : ModelWithCorners 𝕜 F G
J' : ModelWithCorners 𝕜 F' G'
M : Type u_10
M' : Type u_11
N : Type u_12
N' : Type u_13
inst✝¹¹ : TopologicalSpace M
inst✝¹⁰ : ChartedSpace H M
inst✝⁹ : TopologicalSpace M'
inst✝⁸ : ChartedSpace H' M'
inst✝⁷ : TopologicalSpace N
inst✝⁶ : ChartedSpace G N
inst✝⁵ : TopologicalSpace N'
inst✝⁴ : ChartedSpace G' N'
n : ℕ∞ω
f : M → N
x : M
P : (M → N) → OpenPartialHomeomorph M H → OpenPartialHomeomorph N G → Prop
inst✝³ : IsManifold I n M
inst✝² : IsManifold I' n M'
inst✝¹ : IsManifold J n N
inst✝ : IsManifold J' n N'
Q : (M' → N') → OpenPartialHomeomorph M' H' → OpenPartialHomeomorph N' G' → Prop
R : (M × M' → N × N') → OpenPartialHomeomorph (M × M') (H × H') → OpenPartialHomeomorph (N × N') (G × G') → Prop
hf : LiftSourceTargetPropertyAt I J n f x P
g : M' → N'
x' : M'
hg : LiftSourceTargetPropertyAt I' J' n g x' Q
h :
  ∀ {f : M → N} {φ₁ : OpenPartialHomeomorph M H} {ψ₁ : OpenPartialHomeomorph N G} {g : M' → N'}
    {φ₂ : OpenPartialHomeomorph M' H'} {ψ₂ : OpenPartialHomeomorph N' G'},
    P f φ₁ ψ₁ → Q g φ₂ ψ₂ → R (Prod.map f g) (φ₁.prod φ₂) (ψ₁.prod ψ₂)
⊢ hf.codChart.prod hg.codChart ∈ IsManifold.maximalAtlas (J.prod J') n (N × N')

Tactic:
apply IsManifold.mem_maximalAtlas_prod
      (codChart_mem_maximalAtlas hf) (codChart_mem_maximalAtlas hg)

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝¹² : NontriviallyNormedField 𝕜
E : Type u_2
E'' : Type u
F : Type u_5
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace 𝕜 E
inst✝⁹ : NormedAddCommGroup E''
inst✝⁸ : NormedSpace 𝕜 E''
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : NormedSpace 𝕜 F
H : Type u_7
inst✝⁵ : TopologicalSpace H
G : Type u_9
inst✝⁴ : TopologicalSpace G
I : ModelWithCorners 𝕜 E H
J : ModelWithCorners 𝕜 E'' G
M : Type u_11
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace H M
N : Type u_13
inst✝¹ : TopologicalSpace N
inst✝ : ChartedSpace G N
n : ℕ∞ω
f : M → N
x : M
equiv : (E × F) ≃L[𝕜] E''
domChart : OpenPartialHomeomorph M H
codChart : OpenPartialHomeomorph N G
hx : x ∈ domChart.source
hfx : f x ∈ codChart.source
hdomChart : domChart ∈ IsManifold.maximalAtlas I n M
hcodChart : codChart ∈ IsManifold.maximalAtlas J n N
hsource : domChart.source ⊆ f ⁻¹' codChart.source
hwrittenInExtend :
  EqOn (↑(codChart.extend J) ∘ f ∘ ↑(domChart.extend I).symm) (⇑equiv ∘ fun x ↦ (x, 0)) (domChart.extend I).target
⊢ IsImmersionAtOfComplement F I J n f x

Tactic:
use domChart, codChart
  use equiv

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

Goal state before:
pp: x y : ℝ
h : Fact (x < y)
n : WithTop ℕ∞
z : ℝ
hz : z ∈ Icc x y
⊢ ContinuousWithinAt (projIcc x y ⋯) (Icc x y) z

Tactic:
apply ContinuousAt.continuousWithinAt; fun_prop

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

Goal state before:
pp: x y : ℝ
h : Fact (x < y)
n : WithTop ℕ∞
⊢ ContMDiffOn 𝓘(ℝ, ℝ) (𝓡∂ 1) n (projIcc x y ⋯) (Icc x y)

Tactic:
by
  intro z hz
  rw [contMDiffWithinAt_iff]
  refine ⟨by apply ContinuousAt.continuousWithinAt; fun_prop, ?_⟩
  -- We come back to the definition: we should check that, in each chart, the map is smooth
  -- There are two charts, and we check things separately in each of them using the
  -- explicit formulas.
  suffices ContDiffWithinAt ℝ n _ (Icc x y) z by simpa
  split_ifs with h'z
  · have : ContDiff ℝ n (fun (w : ℝ) ↦
        (show EuclideanSpace ℝ (Fin 1) from toLp 2 fun (_ : Fin 1) ↦ w - x)) := by
      dsimp
      apply contDiff_euclidean.2 (fun i ↦ by fun_prop)
    apply this.contDiffWithinAt.congr_of_eventuallyEq_of_mem _ hz
    filter_upwards [self_mem_nhdsWithin] with w hw
    ext i
    suffices max x (min y w) - x = w - x by
      simpa [modelWithCornersEuclideanHalfSpace, IccLeftChart]
    rw [max_eq_right, min_eq_right hw.2]
    simp [hw.1, h.out.le]
  · have : ContDiff ℝ n (fun (w : ℝ) ↦
        (show EuclideanSpace ℝ (Fin 1) from toLp 2 fun (_ : Fin 1) ↦ y - w)) := by
      dsimp
      apply contDiff_euclidean.2 (fun i ↦ by fun_prop)
    apply this.contDiffWithinAt.congr_of_eventuallyEq_of_mem _ hz
    filter_upwards [self_mem_nhdsWithin] with w hw
    ext i
    suffices y - max x (min y w) = y - w by
      simpa [modelWithCornersEuclideanHalfSpace, IccRightChart]
    rw [max_eq_right, min_eq_right hw.2]
    simp [hw.1, h.out.le]

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)
⊢ Ioo (-(|t| + 1)) (|t| + 1) ∈ 𝓝 t

Tactic:
have : |t| < |t| + 1 := lt_add_of_pos_right |t| zero_lt_one

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)
this : |t| < |t| + 1
⊢ Ioo (-(|t| + 1)) (|t| + 1) ∈ 𝓝 t
Module: Mathlib.Geometry.Manifold.Riemannian.PathELength

Goal state before:
pp: case hf
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 : M
r : ℝ≥0∞
a b : ℝ
hr : riemannianEDist I x y < r
hab : a < b
γ : ℝ → M
hγx : γ 0 = x
hγy : γ 1 = y
γ_smooth : ContMDiffOn 𝓘(ℝ, ℝ) I 1 γ (Icc 0 1)
hγ : pathELength I γ 0 1 < r
a' : ℝ
haa' : a < a'
ha'b : a' < b
b' : ℝ
ha'b' : a' < b'
hb'b : b' < b
η : ℝ → ℝ := fun t ↦ ((b' - a')⁻¹ * (t - a')).smoothTransition
A : ∀ t < a', η t = 0
A' : ∀ t < a', (γ ∘ η) t = x
B : ∀ (t : ℝ), b' < t → η t = 1
B' : ∀ (t : ℝ), b' < t → (γ ∘ η) t = y
t : ℝ
⊢ ∀ ⦃b : ℝ⦄, t ≤ b → (fun t ↦ (b' - a')⁻¹ * (t - a')) t ≤ (fun t ↦ (b' - a')⁻¹ * (t - a')) b

Tactic:
u

Goal state after:
case hf
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 : M
r : ℝ≥0∞
a b : ℝ
hr : riemannianEDist I x y < r
hab : a < b
γ : ℝ → M
hγx : γ 0 = x
hγy : γ 1 = y
γ_smooth : ContMDiffOn 𝓘(ℝ, ℝ) I 1 γ (Icc 0 1)
hγ : pathELength I γ 0 1 < r
a' : ℝ
haa' : a < a'
ha'b : a' < b
b' : ℝ
ha'b' : a' < b'
hb'b : b' < b
η : ℝ → ℝ := fun t ↦ ((b' - a')⁻¹ * (t - a')).smoothTransition
A : ∀ t < a', η t = 0
A' : ∀ t < a', (γ ∘ η) t = x
B : ∀ (t : ℝ), b' < t → η t = 1
B' : ∀ (t : ℝ), b' < t → (γ ∘ η) t = y
t u : ℝ
⊢ t ≤ u → (fun t ↦ (b' - a')⁻¹ * (t - a')) t ≤ (fun t ↦ (b' - a')⁻¹ * (t - a')) u
Module: Mathlib.Geometry.Manifold.Riemannian.Basic

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
n : ℕ∞ω
M : Type u_3
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace H M
F : Type u_4
inst✝¹ : NormedAddCommGroup F
inst✝ : InnerProductSpace ℝ F
⊢ IsRiemannianManifold 𝓘(ℝ, F) F

Tactic:
refine ⟨fun x y ↦ le_antisymm ?_ ?_⟩
  · simp only [riemannianEDist, le_iInf_iff]
    intro γ hγ
    let e : ℝ → F := γ ∘ (projIcc 0 1 zero_le_one)
    have D : ContDiffOn ℝ 1 e (Icc 0 1) :=
      contMDiffOn_iff_contDiffOn.mp (hγ.comp_contMDiffOn contMDiffOn_projIcc)
    rw [lintegral_norm_mfderiv_Icc_eq_pathELength_projIcc,
      pathELength_eq_lintegral_mfderivWithin_Icc]
    simp only [mfderivWithin_eq_fderivWithin, enorm_tangentSpace_vectorSpace]
    conv_lhs =>
      rw [edist_comm, edist_eq_enorm_sub, show x = e 0 by simp [e], show y = e 1 by simp [e]]
    exact (enorm_sub_le_lintegral_derivWithin_Icc_of_contDiffOn_Icc D zero_le_one).trans_eq rfl
  · let γ := ContinuousAffineMap.lineMap (R := ℝ) x y
    have : riemannianEDist 𝓘(ℝ, F) x y ≤ pathELength 𝓘(ℝ, F) γ 0 1 := by
      apply riemannianEDist_le_pathELength ?_ (by simp [γ, ContinuousAffineMap.coe_lineMap_eq])
        (by simp [γ, ContinuousAffineMap.coe_lineMap_eq]) zero_le_one
      rw [contMDiffOn_iff_contDiffOn]
      exact γ.contDiff.contDiffOn
    apply this.trans_eq
    rw [pathELength_eq_lintegral_mfderivWithin_Icc]
    simp only [mfderivWithin_eq_fderivWithin, enorm_tangentSpace_vectorSpace]
    exact lintegral_fderiv_lineMap_eq_edist

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

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 : 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✝¹ : riemannianEDist I x y ≤ pathELength I γ 0 1
t : ℝ
ht : t ∈ Icc 0 1
this✝ : mfderiv[Icc 0 1] γ t = mfderiv[range ↑I] ↑(extChartAt I x).symm (η t) ∘SL mfderiv[Icc 0 1] ⇑η t
this : (mfderiv[Icc 0 1] γ t) 1 = (mfderiv[range ↑I] ↑(extChartAt I x).symm (η t)) ((mfderiv[Icc 0 1] ⇑η t) 1)
⊢ ‖mfderiv[range ↑I] ↑(extChartAt I x).symm (η t)‖ₑ ≤ ↑C

Tactic:
exact (hη.2 ht).le

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

Goal state before:
pp: 𝕜 : Type u_1
E'' : Type u_3
F : Type u_5
H : Type u_7
G : Type u_9
E : Type u
inst✝¹² : NontriviallyNormedField 𝕜
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace 𝕜 E
inst✝⁹ : NormedAddCommGroup E''
inst✝⁸ : NormedSpace 𝕜 E''
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : NormedSpace 𝕜 F
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 : M → N
x : M
equiv : E ≃L[𝕜] E'' × F
domChart : OpenPartialHomeomorph M H
codChart : OpenPartialHomeomorph N G
hx : x ∈ domChart.source
hfx : f x ∈ codChart.source
hdomChart : domChart ∈ IsManifold.maximalAtlas I n M
hcodChart : codChart ∈ IsManifold.maximalAtlas J n N
hsource : domChart.source ⊆ f ⁻¹' codChart.source
hwrittenInExtend :
  EqOn (↑(codChart.extend J) ∘ f ∘ ↑(domChart.extend I).symm) (Prod.fst ∘ ⇑equiv) (domChart.extend I).target
⊢ IsSubmersionAtOfComplement F I J n f x

Tactic:
use domChart, codChart
  use equiv

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

Goal state before:
pp: E : Type u_1
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
H : Type u_2
inst✝⁶ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁵ : TopologicalSpace M
inst✝⁴ : ChartedSpace H M
inst✝³ : RiemannianBundle fun x ↦ TangentSpace I x
inst✝² : IsManifold I 1 M
inst✝¹ : IsContinuousRiemannianBundle E fun x ↦ TangentSpace I x
inst✝ : RegularSpace M
x : M
s : Set M
hs : s ∈ 𝓝 x
C : ℝ≥0
C_pos : C > 0
hC✝ : ∀ᶠ (y : M) in 𝓝 x, ‖mfderiv% ↑(extChartAt I x) y‖ₑ < ↑C
u : Set M
u_mem : u ∈ 𝓝 x
u_closed : IsClosed[inst✝⁵] u
us : u ⊆ s
hu : u ⊆ {y | ‖mfderiv% ↑(extChartAt I x) y‖ₑ < ↑C}
uc : u ⊆ (extChartAt I x).source
uc' : u ⊆ (chartAt H x).source
v : Set M
hv : v ⊆ u
v_mem : v ∈ 𝓝 x
v_open : IsOpen[inst✝⁵] v
r : ℝ≥0
hr : ball (↑(extChartAt I x) x) ↑r ⊆ ↑(extChartAt I x).symm ⁻¹' v
r_pos : 0 < r
y : M
hy : y ∈ {y | riemannianEDist I x y < ↑(r / C)}
γ : ℝ → M
hγx : γ 0 = x
hγy : γ 1 = y
γ_smooth : ContMDiff 𝓘(ℝ, ℝ) I 1 γ
hγ : pathELength I γ 0 1 < ↑(r / C)
A : Set ℝ := γ ⁻¹' u
zero_mem : 0 ∈ A
A_closed : IsClosed (A ∩ Icc 0 1)
t₁ : ℝ
ht₁0 : 0 ≤ t₁
ht₁1 : t₁ < 1
t₁_mem : Icc 0 t₁ ⊆ A
γ' : ℝ → E := ↑(extChartAt I x) ∘ γ
hC : ContMDiffOn 𝓘(ℝ, ℝ) 𝓘(ℝ, E) 1 γ' (Icc 0 t₁)
h't' : 0 < t₁
⊢ ↑C ≠ 0

Tactic:
by simpa using C_pos.ne'

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

Goal state before:
pp: E : Type u_1
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
H : Type u_2
inst✝⁶ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁵ : TopologicalSpace M
inst✝⁴ : ChartedSpace H M
inst✝³ : RiemannianBundle fun x ↦ TangentSpace I x
inst✝² : IsManifold I 1 M
inst✝¹ : IsContinuousRiemannianBundle E fun x ↦ TangentSpace I x
inst✝ : RegularSpace M
x : M
s : Set M
hs : s ∈ 𝓝 x
C : ℝ≥0
C_pos : C > 0
hC✝ : ∀ᶠ (y : M) in 𝓝 x, ‖mfderiv% ↑(extChartAt I x) y‖ₑ < ↑C
u : Set M
u_mem : u ∈ 𝓝 x
u_closed : IsClosed[inst✝⁵] u
us : u ⊆ s
hu : u ⊆ {y | ‖mfderiv% ↑(extChartAt I x) y‖ₑ < ↑C}
uc : u ⊆ (extChartAt I x).source
uc' : u ⊆ (chartAt H x).source
v : Set M
hv : v ⊆ u
v_mem : v ∈ 𝓝 x
v_open : IsOpen[inst✝⁵] v
r : ℝ≥0
hr : ball (↑(extChartAt I x) x) ↑r ⊆ ↑(extChartAt I x).symm ⁻¹' v
r_pos : 0 < r
y : M
hy : y ∈ {y | riemannianEDist I x y < ↑(r / C)}
γ : ℝ → M
hγx : γ 0 = x
hγy : γ 1 = y
γ_smooth : ContMDiff 𝓘(ℝ, ℝ) I 1 γ
hγ : pathELength I γ 0 1 < ↑(r / C)
A : Set ℝ := ⋯
zero_mem : 0 ∈ A
A_closed : IsClosed (A ∩ Icc 0 1)
t₁ : ℝ
ht₁0 : 0 ≤ t₁
ht₁1 : t₁ < 1
t₁_mem : Icc 0 t₁ ⊆ A
γ' : ℝ → E := ⋯
hC : ContMDiffOn 𝓘(ℝ, ℝ) 𝓘(ℝ, E) 1 γ' (Icc 0 t₁)
this : ‖γ' t₁ - γ' 0‖ₑ < ↑r
⊢ γ' t₁ ∈ ball (↑(extChartAt I x) x) ↑r

Tactic:
rw [← Metric.eball_coe, Metric.mem_eball, edist_eq_enorm_sub]

Goal state after:
E : Type u_1
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
H : Type u_2
inst✝⁶ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁵ : TopologicalSpace M
inst✝⁴ : ChartedSpace H M
inst✝³ : RiemannianBundle fun x ↦ TangentSpace I x
inst✝² : IsManifold I 1 M
inst✝¹ : IsContinuousRiemannianBundle E fun x ↦ TangentSpace I x
inst✝ : RegularSpace M
x : M
s : Set M
hs : s ∈ 𝓝 x
C : ℝ≥0
C_pos : C > 0
hC✝ : ∀ᶠ (y : M) in 𝓝 x, ‖mfderiv% ↑(extChartAt I x) y‖ₑ < ↑C
u : Set M
u_mem : u ∈ 𝓝 x
u_closed : IsClosed[inst✝⁵] u
us : u ⊆ s
hu : u ⊆ {y | ‖mfderiv% ↑(extChartAt I x) y‖ₑ < ↑C}
uc : u ⊆ (extChartAt I x).source
uc' : u ⊆ (chartAt H x).source
v : Set M
hv : v ⊆ u
v_mem : v ∈ 𝓝 x
v_open : IsOpen[inst✝⁵] v
r : ℝ≥0
hr : ball (↑(extChartAt I x) x) ↑r ⊆ ↑(extChartAt I x).symm ⁻¹' v
r_pos : 0 < r
y : M
hy : y ∈ {y | riemannianEDist I x y < ↑(r / C)}
γ : ℝ → M
hγx : γ 0 = x
hγy : γ 1 = y
γ_smooth : ContMDiff 𝓘(ℝ, ℝ) I 1 γ
hγ : pathELength I γ 0 1 < ↑(r / C)
A : Set ℝ := γ ⁻¹' u
zero_mem : 0 ∈ A
A_closed : IsClosed (A ∩ Icc 0 1)
t₁ : ℝ
ht₁0 : 0 ≤ t₁
ht₁1 : t₁ < 1
t₁_mem : Icc 0 t₁ ⊆ A
γ' : ℝ → E := ↑(extChartAt I x) ∘ γ
hC : ContMDiffOn 𝓘(ℝ, ℝ) 𝓘(ℝ, E) 1 γ' (Icc 0 t₁)
this : ‖γ' t₁ - γ' 0‖ₑ < ↑r
⊢ ‖γ' t₁ - ↑(extChartAt I x) x‖ₑ < ↑r
Module: Mathlib.Geometry.Manifold.VectorBundle.Tensoriality

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
A : Type u_9
inst✝⁵ : AddCommGroup A
inst✝⁴ : Module 𝕜 A
Φ : ((x : M) → V x) → A
x : M
inst✝³ : VectorBundle 𝕜 F V
inst✝² : CompleteSpace 𝕜
inst✝¹ : FiniteDimensional 𝕜 F
inst✝ : ContMDiffVectorBundle 1 F V I
hΦ : TensorialAt I F Φ x
σ σ' : (x : M) → V x
hσ : MDiffAt (T% σ) x
hσ' : MDiffAt (T% σ') x
hσσ' : σ x = σ' x
⊢ Φ σ = Φ σ'

Tactic:
by
  -- Select a local frame `s` for the bundle `V` near `x`,
  -- and let `c` be the family of linear maps evaluating the coefficients of a section relative to
  -- this frame
  let t := trivializationAt F V x
  have x_mem : x ∈ t.baseSet := FiberBundle.mem_baseSet_trivializationAt F V x
  let b := Basis.ofVectorSpace 𝕜 F
  let s := t.localFrame b
  let c := t.localFrameCoeff I b
  have hs (i) : MDiffAt (T% (s i)) x :=
    (contMDiffAt_localFrame_of_mem 1 _ b i x_mem).mdifferentiableAt (by simp)
  have hc {σ : (x : M) → V x} (hσ : MDiffAt (T% σ) x) (i) :
      MDiffAt (LinearMap.piApply (c i) σ) x :=
    mdifferentiableAt_localFrameCoeff b x_mem hσ i
  -- By the locality of the operation `(Φ · x)`, its value on `σ` agrees with the value of `Φ` on
  -- the expansion of `σ` into coefficients relative to the frame.
  have hΦ_eq {σ : (x : M) → V x} (hσ : MDiffAt (T% σ) x) :
      Φ σ = Φ (fun x' ↦ ∑ i, c i x' (σ x') • s i x') :=
    hΦ.local hσ
      (.sum_section fun i _ ↦ (hc hσ i).smul_section (hs i))
      (t.eventually_eq_localFrame_sum_coeff_smul b x_mem)
  -- Now evaluate using the tensoriality properties.
  rw [hΦ_eq hσ, hΦ_eq hσ', hΦ.sum, hΦ.sum]
  · congr! 1 with i
    calc Φ ((LinearMap.piApply (c i) σ) • (s i))
        = c i x (σ x) • Φ (s i) := hΦ.smul (hc hσ i) (hs i)
      _ = c i x (σ' x) • Φ (s i) := by rw [hσσ']
      _ = Φ ((LinearMap.piApply (c i) σ') • (s i)) :=
          hΦ.smul (hc hσ' i) (hs i) |>.symm
  · exact fun i _ ↦ (hc hσ' i).smul_section (hs i)
  · exact fun i _ ↦ (hc hσ i).smul_section (hs i)

Goal state after: