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

Goal state before:
pp: 𝕜 : Type u_1
inst✝¹⁰ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace 𝕜 E
F : Type u_3
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : NormedSpace 𝕜 F
H₁ : Type u_5
inst✝⁵ : TopologicalSpace H₁
H₂ : Type u_6
inst✝⁴ : TopologicalSpace H₂
I : ModelWithCorners 𝕜 E H₁
J : ModelWithCorners 𝕜 F H₂
M : Type u_8
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace H₁ M
N : Type u_9
inst✝¹ : TopologicalSpace N
inst✝ : ChartedSpace H₂ N
n : WithTop ℕ∞
f : M → N
x : M
hf : IsLocalDiffeomorphAt I J n f x
y : N
hy : y ∈ hf.localInverse.source
⊢ f (↑hf.localInverse.toPartialEquiv y) = y

Tactic:
have : hf.localInverse y ∈ hf.choose.source := by
    rw [← hf.choose.symm_target]
    exact hf.choose.symm.map_source hy
  rw [hf.choose_spec.2 this]
  exact hf.choose.right_inv hy

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

Goal state before:
pp: case refine_2
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
f : B → 𝕜
s : (x : B) → E x
u : Set B
x₀ : B
hf : MDiffAt[u] f x₀
e : Trivialization F TotalSpace.proj := trivializationAt F E x₀
hs : (MDiffAt[u] fun b ↦ (↑e ⟨b, s b⟩).2) x₀
⊢ (↑e ⟨x₀, (f • s) x₀⟩).2 = f x₀ • (↑e ⟨x₀, s x₀⟩).2

Tactic:
apply (e.linear 𝕜 (FiberBundle.mem_baseSet_trivializationAt' x₀)).2

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

Goal state before:
pp: 𝕜₁ : Type u_1
inst✝²¹ : NontriviallyNormedField 𝕜₁
𝕜₂ : Type u_2
inst✝²⁰ : NontriviallyNormedField 𝕜₂
σ : 𝕜₁ →+* 𝕜₂
B : Type u_3
F₁ : Type u_4
inst✝¹⁹ : NormedAddCommGroup F₁
inst✝¹⁸ : NormedSpace 𝕜₁ F₁
E₁ : B → Type u_5
inst✝¹⁷ : (x : B) → AddCommGroup (E₁ x)
inst✝¹⁶ : (x : B) → Module 𝕜₁ (E₁ x)
inst✝¹⁵ : TopologicalSpace (TotalSpace F₁ E₁)
F₂ : Type u_6
inst✝¹⁴ : NormedAddCommGroup F₂
inst✝¹³ : NormedSpace 𝕜₂ F₂
E₂ : B → Type u_7
inst✝¹² : (x : B) → AddCommGroup (E₂ x)
inst✝¹¹ : (x : B) → Module 𝕜₂ (E₂ x)
inst✝¹⁰ : TopologicalSpace (TotalSpace F₂ E₂)
inst✝⁹ : TopologicalSpace B
e₁✝ e₁'✝ : Trivialization F₁ TotalSpace.proj
e₂✝ e₂'✝ : Trivialization F₂ TotalSpace.proj
inst✝⁸ : (x : B) → TopologicalSpace (E₁ x)
inst✝⁷ : FiberBundle F₁ E₁
inst✝⁶ : VectorBundle 𝕜₁ F₁ E₁
inst✝⁵ : (x : B) → TopologicalSpace (E₂ x)
inst✝⁴ : FiberBundle F₂ E₂
inst✝³ : VectorBundle 𝕜₂ F₂ E₂
inst✝² : ∀ (x : B), IsTopologicalAddGroup (E₂ x)
inst✝¹ : ∀ (x : B), ContinuousSMul 𝕜₂ (E₂ x)
inst✝ : RingHomIsometric σ
e₁ : Trivialization F₁ TotalSpace.proj
e₂ : Trivialization F₂ TotalSpace.proj
he₁ : MemTrivializationAtlas e₁
he₂ : MemTrivializationAtlas e₂
e₁' : Trivialization F₁ TotalSpace.proj
e₂' : Trivialization F₂ TotalSpace.proj
he₁' : MemTrivializationAtlas e₁'
he₂' : MemTrivializationAtlas e₂'
⊢ ∃ f,
    ContinuousOn f ((continuousLinearMap σ e₁ e₂).baseSet ∩ (continuousLinearMap σ e₁' e₂').baseSet) ∧
      ∀ b ∈ (continuousLinearMap σ e₁ e₂).baseSet ∩ (continuousLinearMap σ e₁' e₂').baseSet,
        ∀ (v : F₁ →SL[σ] F₂),
          (f b) v = (↑(continuousLinearMap σ e₁' e₂') { proj := b, snd := (continuousLinearMap σ e₁ e₂).symm b v }).2

Tactic:
exact ⟨continuousLinearMapCoordChange σ e₁ e₁' e₂ e₂',
      continuousOn_continuousLinearMapCoordChange,
      continuousLinearMapCoordChange_apply σ e₁ e₁' e₂ e₂'⟩

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

Goal state before:
pp: case refine_2
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
V : M → Type u_6
inst✝⁷ : TopologicalSpace (TotalSpace F V)
inst✝⁶ : (x : M) → AddCommGroup (V x)
inst✝⁵ : (x : M) → TopologicalSpace (V x)
inst✝⁴ : FiberBundle F V
inst✝³ : NormedSpace 𝕜 F
k : WithTop ℕ∞
inst✝² : (x : M) → Module 𝕜 (V x)
inst✝¹ : VectorBundle 𝕜 F V
inst✝ : ContMDiffVectorBundle k F V I
x₀ : M
σ₀ : V x₀
t : Trivialization F TotalSpace.proj := trivializationAt F V x₀
⊢ ContMDiffOn I (I.prod 𝓘(𝕜, F)) k (fun x' ↦ ⟨x', extend F σ₀ x'⟩) t.baseSet

Tactic:
suffices CMDiff[t.baseSet] k (fun x ↦ (t ⟨x, extend F σ₀ x⟩).2) by
    intro x hx
    rw [t.contMDiffWithinAt_section _ hx]
    exact this x hx

Goal state after:
case refine_2
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
V : M → Type u_6
inst✝⁷ : TopologicalSpace (TotalSpace F V)
inst✝⁶ : (x : M) → AddCommGroup (V x)
inst✝⁵ : (x : M) → TopologicalSpace (V x)
inst✝⁴ : FiberBundle F V
inst✝³ : NormedSpace 𝕜 F
k : WithTop ℕ∞
inst✝² : (x : M) → Module 𝕜 (V x)
inst✝¹ : VectorBundle 𝕜 F V
inst✝ : ContMDiffVectorBundle k F V I
x₀ : M
σ₀ : V x₀
t : Trivialization F TotalSpace.proj := trivializationAt F V x₀
⊢ ContMDiffOn I 𝓘(𝕜, F) k (fun x ↦ (↑t ⟨x, extend F σ₀ x⟩).2) t.baseSet
Module: Mathlib.Geometry.Manifold.LocalSourceTargetProperty

Goal state before:
pp: case refine_4
𝕜 : Type u_1
E : Type u_2
F : Type u_4
H : Type u_6
G : Type u_8
inst✝¹⁰ : NontriviallyNormedField 𝕜
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 F G
M : Type u_10
N : Type u_12
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
P : (M → N) → OpenPartialHomeomorph M H → OpenPartialHomeomorph N G → Prop
hP : IsLocalSourceTargetProperty P
hf : LiftSourceTargetPropertyAt I J n f x P
h' : f =ᶠ[𝓝 x] g
s' : Set M
hxs' : s' ∈ 𝓝 x
hfg : EqOn f g s'
s : Set M
hss' : s ⊆ s'
hs : IsOpen[inst✝³] s
hxs : x ∈ s
⊢ (hf.domChart.restr s).source ⊆ g ⁻¹' hf.codChart.source

Tactic:
trans s' ∩ f ⁻¹' hf.codChart.source
    · apply subset_inter
      · exact Subset.trans (by simp [interior_eq_iff_isOpen.mpr hs]) hss'
      · exact Subset.trans (by simp) hf.source_subset_preimage_source
    · rw [hfg.inter_preimage_eq]; exact inter_subset_right

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

Goal state before:
pp: n : ℕ
p : ℝ≥0∞
a : ℝ
i : Fin n
⊢ frontier {y | a ≤ y.ofLp i} = {y | a = y.ofLp i}

Tactic:
rw [frontier, closure_halfSpace, interior_halfSpace]

Goal state after:
n : ℕ
p : ℝ≥0∞
a : ℝ
i : Fin n
⊢ {y | a ≤ y.ofLp i} \ {y | a < y.ofLp i} = {y | a = y.ofLp i}
Module: Mathlib.Geometry.Manifold.Instances.Real

Goal state before:
pp: case pos
x✝ y✝ : ℝ
hxy : Fact (x✝ < y✝)
x y : ℝ
h : Fact (x < y)
z : ↑(Icc x y)
h' : ↑z < y
⊢ z ∈ (if ↑z < y then IccLeftChart x y else IccRightChart x y).source

Tactic:
simp only [h', if_true]
      exact h'

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

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
γ : ℝ → M
v : (x : M) → TangentSpace I x
s : Set ℝ
t₀ : ℝ
inst✝ : IsManifold I 1 M
hγ : IsMIntegralCurveOn γ v s
t : ℝ
ht : t ∈ s
hsrc : γ t ∈ (chartAt H (γ t₀)).source
⊢ mfderiv% ↑(chartAt H (γ t₀)) (γ t) ∘SL ContinuousLinearMap.smulRight 1 (v (γ t)) =
    ContinuousLinearMap.toSpanSingleton ℝ ((tangentCoordChange I (γ t) (γ t₀) (γ t)) (v (γ t)))

Tactic:
ContinuousLinearMap.ext_iff

Goal state after:
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
γ : ℝ → M
v : (x : M) → TangentSpace I x
s : Set ℝ
t₀ : ℝ
inst✝ : IsManifold I 1 M
hγ : IsMIntegralCurveOn γ v s
t : ℝ
ht : t ∈ s
hsrc : γ t ∈ (chartAt H (γ t₀)).source
⊢ ∀ (x : TangentSpace 𝓘(ℝ, ℝ) t),
    (mfderiv% ↑(chartAt H (γ t₀)) (γ t) ∘SL ContinuousLinearMap.smulRight 1 (v (γ t))) x =
      (ContinuousLinearMap.toSpanSingleton ℝ ((tangentCoordChange I (γ t) (γ t₀) (γ t)) (v (γ t)))) x
Module: Mathlib.Geometry.Manifold.IntegralCurve.Basic

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
γ : ℝ → M
v : (x : M) → TangentSpace I x
t₀ : ℝ
inst✝ : IsManifold I 1 M
hγ : IsMIntegralCurveAt γ v t₀
t : ℝ
ht1 : γ ⁻¹' (extChartAt I (γ t₀)).source ∈ 𝓝 t
ht2 : HasMFDerivAt% γ t (ContinuousLinearMap.smulRight 1 (v (γ t)))
hsrc : γ t ∈ (chartAt H (γ t₀)).source
⊢ mfderiv% ↑(chartAt H (γ t₀)) (γ t) ∘SL ContinuousLinearMap.smulRight 1 (v (γ t)) =
    ContinuousLinearMap.toSpanSingleton ℝ ((tangentCoordChange I (γ t) (γ t₀) (γ t)) (v (γ t)))

Tactic:
ContinuousLinearMap.ext_iff

Goal state after:
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
γ : ℝ → M
v : (x : M) → TangentSpace I x
t₀ : ℝ
inst✝ : IsManifold I 1 M
hγ : IsMIntegralCurveAt γ v t₀
t : ℝ
ht1 : γ ⁻¹' (extChartAt I (γ t₀)).source ∈ 𝓝 t
ht2 : HasMFDerivAt% γ t (ContinuousLinearMap.smulRight 1 (v (γ t)))
hsrc : γ t ∈ (chartAt H (γ t₀)).source
⊢ ∀ (x : TangentSpace 𝓘(ℝ, ℝ) t),
    (mfderiv% ↑(chartAt H (γ t₀)) (γ t) ∘SL ContinuousLinearMap.smulRight 1 (v (γ t))) x =
      (ContinuousLinearMap.toSpanSingleton ℝ ((tangentCoordChange I (γ t) (γ t₀) (γ t)) (v (γ t)))) x
Module: Mathlib.Geometry.Manifold.Instances.Sphere

Goal state before:
pp: E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
n : ℕ
inst✝ : Fact (finrank ℝ E = n + 1)
v : ↑(sphere 0 1)
⊢ (↑(d% Subtype.val v)).range = (ℝ ∙ ↑v)ᗮ

Tactic:
rw [((contMDiff_coe_sphere v).mdifferentiableAt one_ne_zero).mvfderiv]
  dsimp [chartAt]
  simp only [fderivWithin_univ, mfld_simps]
  let U := (OrthonormalBasis.fromOrthogonalSpanSingleton (𝕜 := ℝ) n
    (ne_zero_of_mem_unit_sphere (-v))).repr
  suffices
      (fderiv ℝ ((stereoInvFunAux (-v : E) ∘ (↑)) ∘ U.symm) 0).range = (ℝ ∙ (v : E))ᗮ by
    rw [← this]
    congr 3
    apply stereographic'_neg
  have :
    HasFDerivAt (stereoInvFunAux (-v : E) ∘ (Subtype.val : (ℝ ∙ (↑(-v) : E))ᗮ → E))
      (ℝ ∙ (↑(-v) : E))ᗮ.subtypeL (U.symm 0) := by
    convert! hasFDerivAt_stereoInvFunAux_comp_coe (-v : E)
    simp
  convert! congr($((this.comp 0 U.symm.toContinuousLinearEquiv.hasFDerivAt).fderiv).range)
  symm
  convert!
    (U.symm : EuclideanSpace ℝ (Fin n) ≃ₗᵢ[ℝ] (ℝ ∙ (↑(-v) : E))ᗮ).range_comp
      (ℝ ∙ (↑(-v) : E))ᗮ.subtype using 1
  simp only [Submodule.range_subtype, coe_neg_sphere]
  congr 1
  -- we must show `Submodule.span ℝ {v} = Submodule.span ℝ {-v}`
  apply Submodule.span_eq_span
  · simp only [Set.singleton_subset_iff, SetLike.mem_coe]
    rw [← Submodule.neg_mem_iff]
    exact Submodule.mem_span_singleton_self (-v : E)
  · simp only [Set.singleton_subset_iff, SetLike.mem_coe]
    rw [Submodule.neg_mem_iff]
    exact Submodule.mem_span_singleton_self (v : E)

Goal state after:
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
ha : a ∈ s
hlt : sSup s + -(ε / 2) < a
⊢ False

Tactic:
mem_ofPred

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
ha : ∃ γ, γ 0 = x ∧ IsMIntegralCurveOn γ v (Ioo (-a) a)
hlt : sSup s + -(ε / 2) < a
⊢ False
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_ext 0 = x

Tactic:
rw [γ_ext_def, piecewise, if_pos ⟨by linarith, by linarith⟩, piecewise,
      if_pos ⟨by linarith, by linarith⟩, h0]

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

Goal state before:
pp: case hW
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)
⊢ DifferentiableWithinAt 𝕜 W' (↑(extChartAt I' (f x₀)).symm ⁻¹' t ∩ range ↑I') (F (↑(extChartAt I x₀) x₀))

Tactic:
rw [← hFx₀]
    exact hW.differentiableWithinAt_mpullbackWithin_vectorField

Goal state after:
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 := ⋯
hbdd : BddAbove s
asup : ℝ := ⋯
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_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_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 := ⋯
γ_ext_def : γ_ext = (Ioo (-(asup + ε / 2)) a).piecewise ((Ioo (-a) a).piecewise γ γ1) γ2
heq_ext : γ_ext 0 = x
⊢ IsMIntegralCurveOn ((Ioo (-a) a).piecewise γ γ1) v (Ioo (-(asup + ε / 2)) a)

Tactic:
exact (isMIntegralCurveOn_piecewise (t₀ := -(asup - ε / 2)) hv hγ hγ1
      ⟨⟨neg_lt_neg hlt, by linarith⟩, ⟨by linarith, by linarith⟩⟩ heq1.symm).mono
    (union_comm _ _ ▸ Ioo_subset_Ioo_union_Ioo (by linarith) (by linarith) le_rfl)

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

Goal state before:
pp: E : Type u_1
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
H : Type u_2
inst✝⁴ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace H M
inst✝¹ : (x : M) → ENorm (TangentSpace I x)
inst✝ : ∀ (x : M), ENormSMulClass ℝ (TangentSpace I x)
x y z : M
⊢ riemannianEDist I x z ≤ riemannianEDist I x y + riemannianEDist I y z

Tactic:
apply le_of_forall_gt (fun r hr ↦ ?_)
  rcases ENNReal.exists_add_lt_of_add_lt hr with ⟨u, hu, v, hv, huv⟩
  rcases exists_lt_locally_constant_of_riemannianEDist_lt hu zero_lt_one with
    ⟨γ₁, hγ₁0, hγ₁1, hγ₁_smooth, hγ₁, -, hγ₁_const⟩
  rcases exists_lt_locally_constant_of_riemannianEDist_lt hv one_lt_two with
    ⟨γ₂, hγ₂1, hγ₂2, hγ₂_smooth, hγ₂, hγ₂_const, -⟩
  let γ := piecewise (Iic 1) γ₁ γ₂
  have : riemannianEDist I x z ≤ pathELength I γ 0 2 := by
    apply riemannianEDist_le_pathELength
    · apply ContMDiff.contMDiffOn
      exact ContMDiff.piecewise_Iic hγ₁_smooth hγ₂_smooth (hγ₁_const.trans hγ₂_const.symm)
    · simp [γ, hγ₁0]
    · simp [γ, hγ₂2]
    · exact zero_le_two
  apply this.trans_lt (lt_trans ?_ huv)
  rw [← pathELength_add zero_le_one one_le_two]
  gcongr
  · convert! hγ₁ using 1
    apply pathELength_congr
    intro t ht
    simp [γ, ht.2]
  · convert! hγ₂ using 1
    apply pathELength_congr_Ioo
    intro t ht
    simp [γ, ht.1]

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

Goal state before:
pp: H : Type u_1
inst✝⁵ : TopologicalSpace H
H' : Type u_2
inst✝⁴ : TopologicalSpace H'
G : StructureGroupoid H
G' : StructureGroupoid H'
P : (H → H') → Set H → H → Prop
M : Type u
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace H M
M' : Type u
inst✝¹ : TopologicalSpace M'
inst✝ : ChartedSpace H' M'
hG : G.LocalInvariantProp G' P
U V : Opens ↑(TopCat.of M)
i : U ⟶ V
⊢ ∀ (f : ↥V → M'), ChartedSpace.LiftProp P f → ChartedSpace.LiftProp P fun x ↦ f (i x)

Tactic:
f

Goal state after:
H : Type u_1
inst✝⁵ : TopologicalSpace H
H' : Type u_2
inst✝⁴ : TopologicalSpace H'
G : StructureGroupoid H
G' : StructureGroupoid H'
P : (H → H') → Set H → H → Prop
M : Type u
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace H M
M' : Type u
inst✝¹ : TopologicalSpace M'
inst✝ : ChartedSpace H' M'
hG : G.LocalInvariantProp G' P
U V : Opens ↑(TopCat.of M)
i : U ⟶ V
f : ↥V → M'
⊢ ChartedSpace.LiftProp P f → ChartedSpace.LiftProp P fun x ↦ f (i x)
Module: Mathlib.Geometry.Manifold.Sheaf.Basic

Goal state before:
pp: H : Type u_1
inst✝⁵ : TopologicalSpace H
H' : Type u_2
inst✝⁴ : TopologicalSpace H'
G : StructureGroupoid H
G' : StructureGroupoid H'
P : (H → H') → Set H → H → Prop
M : Type u
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace H M
M' : Type u
inst✝¹ : TopologicalSpace M'
inst✝ : ChartedSpace H' M'
hG : G.LocalInvariantProp G' P
V : Opens ↑(TopCat.of M)
⊢ ∀ (f : ↥V → M'),
    (∀ (x : ↥V), ∃ V_1, ∃ (_ : ↑x ∈ V_1), ∃ i, ChartedSpace.LiftProp P fun x ↦ f (i x)) → ChartedSpace.LiftProp P f

Tactic:
f

Goal state after:
H : Type u_1
inst✝⁵ : TopologicalSpace H
H' : Type u_2
inst✝⁴ : TopologicalSpace H'
G : StructureGroupoid H
G' : StructureGroupoid H'
P : (H → H') → Set H → H → Prop
M : Type u
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace H M
M' : Type u
inst✝¹ : TopologicalSpace M'
inst✝ : ChartedSpace H' M'
hG : G.LocalInvariantProp G' P
V : Opens ↑(TopCat.of M)
f : ↥V → M'
⊢ (∀ (x : ↥V), ∃ V_1, ∃ (_ : ↑x ∈ V_1), ∃ i, ChartedSpace.LiftProp P fun x ↦ f (i x)) → ChartedSpace.LiftProp P f
Module: Mathlib.Geometry.Manifold.Riemannian.Basic

Goal state before:
pp: case refine_1
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
x y : F
⊢ edist x y ≤ riemannianEDist 𝓘(ℝ, F) x y

Tactic:
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

Goal state after: