mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.MeasureTheory.Function.Jacobian

Goal state before:
pp: case inr
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : FiniteDimensional ℝ E
s : Set E
f : E → E
f' : E → E →L[ℝ] E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
hs : MeasurableSet s
hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x
hf : InjOn f s
ε : ℝ≥0
εpos : 0 < ε
A : E →L[ℝ] E
δ' : ℝ
δ'pos : 0 < δ'
hδ' : ∀ (B : E →L[ℝ] E), dist B A < δ' → dist B.det A.det < ↑ε
δ'' : ℝ≥0 := ⟨δ' / 2, ⋯⟩
I'' : ∀ (B : E →L[ℝ] E), ‖B - A‖ ≤ ↑δ'' → |B.det - A.det| ≤ ↑ε
hA : A.det ≠ 0
⊢ ∃ δ,
    0 < δ ∧
      (∀ (B : E →L[ℝ] E), ‖B - A‖ ≤ ↑δ → |B.det - A.det| ≤ ↑ε) ∧
        ∀ (t : Set E) (g : E → E), ApproximatesLinearOn g A t δ → ENNReal.ofReal |A.det| * μ t ≤ μ (g '' t) + ↑ε * μ t

Tactic:
let m : ℝ≥0 := Real.toNNReal |A.det| - ε

Goal state after:
case inr
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : FiniteDimensional ℝ E
s : Set E
f : E → E
f' : E → E →L[ℝ] E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
hs : MeasurableSet s
hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x
hf : InjOn f s
ε : ℝ≥0
εpos : 0 < ε
A : E →L[ℝ] E
δ' : ℝ
δ'pos : 0 < δ'
hδ' : ∀ (B : E →L[ℝ] E), dist B A < δ' → dist B.det A.det < ↑ε
δ'' : ℝ≥0 := ⟨δ' / 2, ⋯⟩
I'' : ∀ (B : E →L[ℝ] E), ‖B - A‖ ≤ ↑δ'' → |B.det - A.det| ≤ ↑ε
hA : A.det ≠ 0
m : ℝ≥0 := |A.det|.toNNReal - ε
⊢ ∃ δ,
    0 < δ ∧
      (∀ (B : E →L[ℝ] E), ‖B - A‖ ≤ ↑δ → |B.det - A.det| ≤ ↑ε) ∧
        ∀ (t : Set E) (g : E → E), ApproximatesLinearOn g A t δ → ENNReal.ofReal |A.det| * μ t ≤ μ (g '' t) + ↑ε * μ t
Module: Mathlib.Geometry.Manifold.LocalInvariantProperties

Goal state before:
pp: H : Type u_1
M : Type u_2
inst✝² : TopologicalSpace H
inst✝¹ : TopologicalSpace M
inst✝ : ChartedSpace H M
G : StructureGroupoid H
Q : (H → H) → Set H → H → Prop
hG : G.LocalInvariantProp G Q
hQ : ∀ (y : H), Q id univ y
U : Opens M
x : ↥U
⊢ LiftPropAt Q (id ∘ Subtype.val) x

Tactic:
rw [← hG.liftPropAt_iff_comp_subtype_val]

Goal state after:
H : Type u_1
M : Type u_2
inst✝² : TopologicalSpace H
inst✝¹ : TopologicalSpace M
inst✝ : ChartedSpace H M
G : StructureGroupoid H
Q : (H → H) → Set H → H → Prop
hG : G.LocalInvariantProp G Q
hQ : ∀ (y : H), Q id univ y
U : Opens M
x : ↥U
⊢ LiftPropAt Q id ↑x
Module: Mathlib.Geometry.Manifold.LocalInvariantProperties

Goal state before:
pp: H₁ : Type u_6
inst✝⁷ : TopologicalSpace H₁
H₂ : Type u_7
inst✝⁶ : TopologicalSpace H₂
H₃ : Type u_8
inst✝⁵ : TopologicalSpace H₃
inst✝⁴ : ChartedSpace H₁ H₂
inst✝³ : ChartedSpace H₂ H₃
G₁ : StructureGroupoid H₁
inst✝² : HasGroupoid H₂ G₁
inst✝¹ : ClosedUnderRestriction G₁
G₂ : StructureGroupoid H₂
inst✝ : HasGroupoid H₃ G₂
H : ∀ e ∈ G₂, LiftPropOn G₁.IsLocalStructomorphWithinAt (↑e) e.source
x✝ : ChartedSpace H₁ H₃ := ChartedSpace.comp H₁ H₂ H₃
e : OpenPartialHomeomorph H₃ H₂
he : e ∈ atlas H₂ H₃
f : OpenPartialHomeomorph H₂ H₁
hf : f ∈ atlas H₁ H₂
e' : OpenPartialHomeomorph H₃ H₂
he' : e' ∈ atlas H₂ H₃
f' : OpenPartialHomeomorph H₂ H₁
hf' : f' ∈ atlas H₁ H₂
x : H₁
hx : (x ∈ f.target ∧ ↑f.symm x ∈ e.target) ∧ ↑e.symm (↑f.symm x) ∈ e'.source ∧ ↑e' (↑e.symm (↑f.symm x)) ∈ f'.source
hxs : x ∈ ↑f.symm ⁻¹' (e.symm ≫ₕ e').source
hxs' : x ∈ f.target ∩ ↑f.symm ⁻¹' ((e.symm ≫ₕ e').source ∩ ↑(e.symm ≫ₕ e') ⁻¹' f'.source)
φ : OpenPartialHomeomorph H₁ H₁
hφG₁ : φ ∈ G₁
hφ : EqOn (↑f' ∘ ↑(e.symm ≫ₕ e') ∘ ↑f.symm) (↑φ.toPartialEquiv) (↑f.symm ⁻¹' (e.symm ≫ₕ e').source ∩ φ.source)
hφ_dom : x ∈ φ.source
⊢ ∃ s, IsOpen[inst✝⁷] s ∧ x ∈ s ∧ ((e ≫ₕ f).symm ≫ₕ e' ≫ₕ f').restr s ∈ G₁

Tactic:
simp_rw [← OpenPartialHomeomorph.coe_trans, OpenPartialHomeomorph.trans_assoc] at hφ

Goal state after:
H₁ : Type u_6
inst✝⁷ : TopologicalSpace H₁
H₂ : Type u_7
inst✝⁶ : TopologicalSpace H₂
H₃ : Type u_8
inst✝⁵ : TopologicalSpace H₃
inst✝⁴ : ChartedSpace H₁ H₂
inst✝³ : ChartedSpace H₂ H₃
G₁ : StructureGroupoid H₁
inst✝² : HasGroupoid H₂ G₁
inst✝¹ : ClosedUnderRestriction G₁
G₂ : StructureGroupoid H₂
inst✝ : HasGroupoid H₃ G₂
H : ∀ e ∈ G₂, LiftPropOn G₁.IsLocalStructomorphWithinAt (↑e) e.source
x✝ : ChartedSpace H₁ H₃ := ChartedSpace.comp H₁ H₂ H₃
e : OpenPartialHomeomorph H₃ H₂
he : e ∈ atlas H₂ H₃
f : OpenPartialHomeomorph H₂ H₁
hf : f ∈ atlas H₁ H₂
e' : OpenPartialHomeomorph H₃ H₂
he' : e' ∈ atlas H₂ H₃
f' : OpenPartialHomeomorph H₂ H₁
hf' : f' ∈ atlas H₁ H₂
x : H₁
hx : (x ∈ f.target ∧ ↑f.symm x ∈ e.target) ∧ ↑e.symm (↑f.symm x) ∈ e'.source ∧ ↑e' (↑e.symm (↑f.symm x)) ∈ f'.source
hxs : x ∈ ↑f.symm ⁻¹' (e.symm ≫ₕ e').source
hxs' : x ∈ f.target ∩ ↑f.symm ⁻¹' ((e.symm ≫ₕ e').source ∩ ↑(e.symm ≫ₕ e') ⁻¹' f'.source)
φ : OpenPartialHomeomorph H₁ H₁
hφG₁ : φ ∈ G₁
hφ_dom : x ∈ φ.source
hφ : EqOn (↑(f.symm ≫ₕ e.symm ≫ₕ e' ≫ₕ f')) (↑φ.toPartialEquiv) (↑f.symm ⁻¹' (e.symm ≫ₕ e').source ∩ φ.source)
⊢ ∃ s, IsOpen[inst✝⁷] s ∧ x ∈ s ∧ ((e ≫ₕ f).symm ≫ₕ e' ≫ₕ f').restr s ∈ G₁
Module: Mathlib.MeasureTheory.Function.Jacobian

Goal state before:
pp: case hn
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : FiniteDimensional ℝ E
s : Set E
f : E → E
f' : E → E →L[ℝ] E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
hs : MeasurableSet s
hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x
hf : InjOn f s
ε : ℝ≥0
εpos : 0 < ε
δ : (E →L[ℝ] E) → ℝ≥0
hδ :
  ∀ (A : E →L[ℝ] E),
    0 < δ A ∧
      (∀ (B : E →L[ℝ] E), ‖B - A‖ ≤ ↑(δ A) → |B.det - A.det| ≤ ↑ε) ∧
        ∀ (t : Set E) (g : E → E),
          ApproximatesLinearOn g A t (δ A) → ENNReal.ofReal |A.det| * μ t ≤ μ (g '' t) + ↑ε * μ t
t : ℕ → Set E
A : ℕ → E →L[ℝ] E
t_disj : Pairwise (Disjoint on t)
t_meas : ∀ (n : ℕ), MeasurableSet (t n)
t_cover : s ⊆ ⋃ n, t n
ht : ∀ (n : ℕ), ApproximatesLinearOn f (A n) (s ∩ t n) (δ (A n))
s_eq : s = ⋃ n, s ∩ t n
⊢ Pairwise (Disjoint on fun i ↦ f '' (s ∩ t i))

Tactic:
intro i j hij
        apply Disjoint.image _ hf inter_subset_left inter_subset_left
        exact Disjoint.mono inter_subset_right inter_subset_right (t_disj hij)

Goal state after:
Module: Mathlib.MeasureTheory.Function.Jacobian

Goal state before:
pp: case h
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : FiniteDimensional ℝ E
s : Set E
f : E → E
f' : E → E →L[ℝ] E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
hs : MeasurableSet s
hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x
hf : InjOn f s
ε : ℝ≥0
εpos : 0 < ε
δ : (E →L[ℝ] E) → ℝ≥0
hδ :
  ∀ (A : E →L[ℝ] E),
    0 < δ A ∧
      (∀ (B : E →L[ℝ] E), ‖B - A‖ ≤ ↑(δ A) → |B.det - A.det| ≤ ↑ε) ∧
        ∀ (t : Set E) (g : E → E),
          ApproximatesLinearOn g A t (δ A) → ENNReal.ofReal |A.det| * μ t ≤ μ (g '' t) + ↑ε * μ t
t : ℕ → Set E
A : ℕ → E →L[ℝ] E
t_disj : Pairwise (Disjoint on t)
t_meas : ∀ (n : ℕ), MeasurableSet (t n)
t_cover : s ⊆ ⋃ n, t n
ht : ∀ (n : ℕ), ApproximatesLinearOn f (A n) (s ∩ t n) (δ (A n))
s_eq : s = ⋃ n, s ∩ t n
⊢ ∀ (i : ℕ), MeasurableSet (f '' (s ∩ t i))

Tactic:
intro i
        exact
          measurable_image_of_fderivWithin (hs.inter (t_meas i))
            (fun x hx => (hf' x hx.1).mono inter_subset_left)
            (hf.mono inter_subset_left)

Goal state after:
Module: Mathlib.Geometry.Manifold.ContMDiff.Defs

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁵ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
E' : Type u_5
inst✝² : NormedAddCommGroup E'
inst✝¹ : NormedSpace 𝕜 E'
H' : Type u_6
inst✝ : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
n : ℕ∞ω
f : E → H'
s : Set E
x : E
⊢ ContDiffWithinAt 𝕜 n (↑I' ∘ f ∘ id) (id ⁻¹' s) (id x) ↔ ContDiffWithinAt 𝕜 n (↑I' ∘ f) s x

Tactic:
CompTriple.comp_eq,

Goal state after:
𝕜 : Type u_1
inst✝⁵ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
E' : Type u_5
inst✝² : NormedAddCommGroup E'
inst✝¹ : NormedSpace 𝕜 E'
H' : Type u_6
inst✝ : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
n : ℕ∞ω
f : E → H'
s : Set E
x : E
⊢ ContDiffWithinAt 𝕜 n (↑I' ∘ f) (id ⁻¹' s) (id x) ↔ ContDiffWithinAt 𝕜 n (↑I' ∘ f) s x
Module: Mathlib.Geometry.Manifold.ContMDiff.Constructions

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
H : Type u_3
inst✝⁷ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
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
n : WithTop ℕ∞
s : Set (M × N)
p : M × N
⊢ ContMDiffWithinAt (I.prod J) I n Prod.fst s p

Tactic:
by
  /- porting note: `simp` fails to apply lemmas to `ModelProd`. Was
  rw [contMDiffWithinAt_iff']
  refine' ⟨continuousWithinAt_fst, _⟩
  refine' contDiffWithinAt_fst.congr (fun y hy => _) _
  · simp only [mfld_simps] at hy
    simp only [hy, mfld_simps]
  · simp only [mfld_simps]
  -/
  rw [contMDiffWithinAt_iff']
  refine ⟨continuousWithinAt_fst, contDiffWithinAt_fst.congr (fun y hy => ?_) ?_⟩
  · exact (extChartAt I p.1).right_inv ⟨hy.1.1.1, hy.1.2.1⟩
  · exact (extChartAt I p.1).right_inv <| (extChartAt I p.1).map_source (mem_extChartAt_source _)

Goal state after:
Module: Mathlib.Geometry.Manifold.ContMDiff.Constructions

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
n : WithTop ℕ∞
E' : Type u_17
inst✝⁶ : NormedAddCommGroup E'
inst✝⁵ : NormedSpace 𝕜 E'
H' : Type u_18
inst✝⁴ : TopologicalSpace H'
J : ModelWithCorners 𝕜 E' H'
N : Type u_20
N' : Type u_21
inst✝³ : TopologicalSpace N
inst✝² : TopologicalSpace N'
inst✝¹ : ChartedSpace H' N
inst✝ : ChartedSpace H' N'
f : M → N
h : ContMDiff I J n (Sum.inl ∘ f)
a✝ : Nontrivial N
inhabited_h : Inhabited N
aux : N ⊕ N' → N := Sum.elim id fun x ↦ default
x✝ : M
⊢ (aux ∘ Sum.inl ∘ f) x✝ = f x✝

Tactic:
simp [aux]

Goal state after:
Module: Mathlib.Geometry.Manifold.ContMDiff.Constructions

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_16
inst✝⁸ : TopologicalSpace M'
inst✝⁷ : ChartedSpace H M'
n : WithTop ℕ∞
E' : Type u_17
inst✝⁶ : NormedAddCommGroup E'
inst✝⁵ : NormedSpace 𝕜 E'
H' : Type u_18
inst✝⁴ : TopologicalSpace H'
J : ModelWithCorners 𝕜 E' H'
N : Type u_20
N' : Type u_21
inst✝³ : TopologicalSpace N
inst✝² : TopologicalSpace N'
inst✝¹ : ChartedSpace H' N
inst✝ : ChartedSpace H' N'
g : M' → N'
h : ContMDiff I J n (Sum.inr ∘ g)
a✝ : Nontrivial N'
inhabited_h : Inhabited N'
aux : N ⊕ N' → N' := Sum.elim (fun x ↦ default) id
x✝ : M'
⊢ (aux ∘ Sum.inr ∘ g) x✝ = g x✝

Tactic:
simp [aux]

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁵ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
E' : Type u_5
inst✝² : NormedAddCommGroup E'
inst✝¹ : NormedSpace 𝕜 E'
H' : Type u_6
inst✝ : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
f : E → H'
s : Set E
x : E
⊢ DifferentiableWithinAt 𝕜 (↑I' ∘ f ∘ id) (id ⁻¹' s) (id x) ↔ DifferentiableWithinAt 𝕜 (↑I' ∘ f) s x

Tactic:
CompTriple.comp_eq,

Goal state after:
𝕜 : Type u_1
inst✝⁵ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
E' : Type u_5
inst✝² : NormedAddCommGroup E'
inst✝¹ : NormedSpace 𝕜 E'
H' : Type u_6
inst✝ : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
f : E → H'
s : Set E
x : E
⊢ DifferentiableWithinAt 𝕜 (↑I' ∘ f) (id ⁻¹' s) (id x) ↔ DifferentiableWithinAt 𝕜 (↑I' ∘ f) s x
Module: Mathlib.Geometry.Manifold.ContMDiff.Atlas

Goal state before:
pp: case refine_2
𝕜 : Type u_1
inst✝¹⁰ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace 𝕜 E
H : Type u_3
inst✝⁷ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝⁶ : TopologicalSpace M
inst✝⁵ : ChartedSpace H M
F : Type u_6
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace 𝕜 F
G : Type u_7
inst✝² : TopologicalSpace G
J : ModelWithCorners 𝕜 F G
N : Type u_8
inst✝¹ : TopologicalSpace N
inst✝ : ChartedSpace G N
n : ℕ∞ω
f : M → N
s : Set M
φ : OpenPartialHomeomorph M H
ψ : OpenPartialHomeomorph N G
y : M
hφ : φ ∈ maximalAtlas I n M
hψ : ψ ∈ maximalAtlas J n N
hy : y ∈ φ.source
hgy : f y ∈ ψ.source
hmaps : MapsTo f s ψ.source
h :
  ContMDiffWithinAt 𝓘(𝕜, E) 𝓘(𝕜, F) n (↑(ψ.extend J) ∘ f ∘ ↑(φ.extend I).symm) (↑(φ.extend I).symm ⁻¹' s ∩ range ↑I)
    (↑(φ.extend I) y)
⊢ ContDiffWithinAt 𝕜 n (↑(ψ.extend J) ∘ f ∘ ↑(φ.extend I).symm) (↑(φ.extend I).symm ⁻¹' s ∩ range ↑I) (↑(φ.extend I) y)

pp: case refine_3
𝕜 : Type u_1
inst✝¹⁰ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace 𝕜 E
H : Type u_3
inst✝⁷ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝⁶ : TopologicalSpace M
inst✝⁵ : ChartedSpace H M
F : Type u_6
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace 𝕜 F
G : Type u_7
inst✝² : TopologicalSpace G
J : ModelWithCorners 𝕜 F G
N : Type u_8
inst✝¹ : TopologicalSpace N
inst✝ : ChartedSpace G N
n : ℕ∞ω
f : M → N
s : Set M
φ : OpenPartialHomeomorph M H
ψ : OpenPartialHomeomorph N G
y : M
hφ : φ ∈ maximalAtlas I n M
hψ : ψ ∈ maximalAtlas J n N
hy : y ∈ φ.source
hgy : f y ∈ ψ.source
hmaps : MapsTo f s ψ.source
h :
  ContinuousWithinAt f s y ∧
    ContDiffWithinAt 𝕜 n (↑(ψ.extend J) ∘ f ∘ ↑(φ.extend I).symm) (↑(φ.extend I).symm ⁻¹' s ∩ range ↑I)
      (↑(φ.extend I) y)
⊢ ContMDiffWithinAt 𝓘(𝕜, E) 𝓘(𝕜, F) n (↑(ψ.extend J) ∘ f ∘ ↑(φ.extend I).symm) (↑(φ.extend I).symm ⁻¹' s ∩ range ↑I)
    (↑(φ.extend I) y)

Tactic:
· rwa [← contMDiffWithinAt_iff_contDiffWithinAt]

Goal state after:
case refine_3
𝕜 : Type u_1
inst✝¹⁰ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace 𝕜 E
H : Type u_3
inst✝⁷ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝⁶ : TopologicalSpace M
inst✝⁵ : ChartedSpace H M
F : Type u_6
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace 𝕜 F
G : Type u_7
inst✝² : TopologicalSpace G
J : ModelWithCorners 𝕜 F G
N : Type u_8
inst✝¹ : TopologicalSpace N
inst✝ : ChartedSpace G N
n : ℕ∞ω
f : M → N
s : Set M
φ : OpenPartialHomeomorph M H
ψ : OpenPartialHomeomorph N G
y : M
hφ : φ ∈ maximalAtlas I n M
hψ : ψ ∈ maximalAtlas J n N
hy : y ∈ φ.source
hgy : f y ∈ ψ.source
hmaps : MapsTo f s ψ.source
h :
  ContinuousWithinAt f s y ∧
    ContDiffWithinAt 𝕜 n (↑(ψ.extend J) ∘ f ∘ ↑(φ.extend I).symm) (↑(φ.extend I).symm ⁻¹' s ∩ range ↑I)
      (↑(φ.extend I) y)
⊢ ContMDiffWithinAt 𝓘(𝕜, E) 𝓘(𝕜, F) n (↑(ψ.extend J) ∘ f ∘ ↑(φ.extend I).symm) (↑(φ.extend I).symm ⁻¹' s ∩ range ↑I)
    (↑(φ.extend I) y)
Module: Mathlib.Topology.FiberBundle.Constructions

Goal state before:
pp: B : Type u
F : Type v
E : B → Type w₁
B' : Type w₂
f✝ : B' → B
inst✝⁶ : TopologicalSpace B'
inst✝⁵ : TopologicalSpace (TotalSpace F E)
inst✝⁴ : TopologicalSpace F
inst✝³ : TopologicalSpace B
inst✝² : ∀ (_b : B), Nonempty (E _b)
K : Type U
inst✝¹ : FunLike K B' B
inst✝ : ContinuousMapClass K B' B
e : Trivialization F TotalSpace.proj
f : K
x : B' × F
h : x ∈ (⇑f ⁻¹' e.baseSet) ×ˢ univ
⊢ ((TotalSpace.mk' F x.1 (e.symm (f x.1) x.2)).proj,
      (↑e (Pullback.lift (⇑f) (TotalSpace.mk' F x.1 (e.symm (f x.1) x.2)))).2) =
    x

Tactic:
simp_rw [mem_prod, mem_preimage, mem_univ, and_true] at h

Goal state after:
B : Type u
F : Type v
E : B → Type w₁
B' : Type w₂
f✝ : B' → B
inst✝⁶ : TopologicalSpace B'
inst✝⁵ : TopologicalSpace (TotalSpace F E)
inst✝⁴ : TopologicalSpace F
inst✝³ : TopologicalSpace B
inst✝² : ∀ (_b : B), Nonempty (E _b)
K : Type U
inst✝¹ : FunLike K B' B
inst✝ : ContinuousMapClass K B' B
e : Trivialization F TotalSpace.proj
f : K
x : B' × F
h : f x.1 ∈ e.baseSet
⊢ ((TotalSpace.mk' F x.1 (e.symm (f x.1) x.2)).proj,
      (↑e (Pullback.lift (⇑f) (TotalSpace.mk' F x.1 (e.symm (f x.1) x.2)))).2) =
    x
Module: Mathlib.Geometry.Manifold.MFDeriv.Basic

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
H : Type u_3
inst✝⁷ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝⁶ : TopologicalSpace M
inst✝⁵ : ChartedSpace H M
E' : Type u_5
inst✝⁴ : NormedAddCommGroup E'
inst✝³ : NormedSpace 𝕜 E'
H' : Type u_6
inst✝² : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
M' : Type u_7
inst✝¹ : TopologicalSpace M'
inst✝ : ChartedSpace H' M'
f : M → M'
x : M
s : Set M
⊢ MDiffAt[s] f x ↔ ∃ f', HasMFDerivAt[s] f x f'

Tactic:
refine ⟨fun h ↦ ⟨mfderiv[s] f x, h.hasMFDerivWithinAt⟩, ?_⟩

Goal state after:
𝕜 : Type u_1
inst✝¹⁰ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace 𝕜 E
H : Type u_3
inst✝⁷ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝⁶ : TopologicalSpace M
inst✝⁵ : ChartedSpace H M
E' : Type u_5
inst✝⁴ : NormedAddCommGroup E'
inst✝³ : NormedSpace 𝕜 E'
H' : Type u_6
inst✝² : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
M' : Type u_7
inst✝¹ : TopologicalSpace M'
inst✝ : ChartedSpace H' M'
f : M → M'
x : M
s : Set M
⊢ (∃ f', HasMFDerivAt[s] f x f') → MDiffAt[s] f x
Module: Mathlib.Geometry.Manifold.MFDeriv.Basic

Goal state before:
pp: case pos
𝕜 : Type u_1
inst✝¹⁰ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace 𝕜 E
H : Type u_3
inst✝⁷ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝⁶ : TopologicalSpace M
inst✝⁵ : ChartedSpace H M
E' : Type u_5
inst✝⁴ : NormedAddCommGroup E'
inst✝³ : NormedSpace 𝕜 E'
H' : Type u_6
inst✝² : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
M' : Type u_7
inst✝¹ : TopologicalSpace M'
inst✝ : ChartedSpace H' M'
f : M → M'
x : M
s t : Set M
y : M
h : s =ᶠ[𝓝[{y}ᶜ] x] t
hx : MDiffAt[s] f x
⊢ mfderiv[s] f x = mfderiv[t] f x

Tactic:
simp only [mfderivWithin, hx, (mdifferentiableWithinAt_congr_set' y h).1 hx, ↓reduceIte]

Goal state after:
case pos
𝕜 : Type u_1
inst✝¹⁰ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace 𝕜 E
H : Type u_3
inst✝⁷ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝⁶ : TopologicalSpace M
inst✝⁵ : ChartedSpace H M
E' : Type u_5
inst✝⁴ : NormedAddCommGroup E'
inst✝³ : NormedSpace 𝕜 E'
H' : Type u_6
inst✝² : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
M' : Type u_7
inst✝¹ : TopologicalSpace M'
inst✝ : ChartedSpace H' M'
f : M → M'
x : M
s t : Set M
y : M
h : s =ᶠ[𝓝[{y}ᶜ] x] t
hx : MDiffAt[s] f x
⊢ fderivWithin 𝕜 (writtenInExtChartAt I I' x f) (↑(extChartAt I x).symm ⁻¹' s ∩ range ↑I) (↑(extChartAt I x) x) =
    fderivWithin 𝕜 (writtenInExtChartAt I I' x f) (↑(extChartAt I x).symm ⁻¹' t ∩ range ↑I) (↑(extChartAt I x) x)
Module: Mathlib.Geometry.Manifold.MFDeriv.Atlas

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
H : Type u_3
inst✝⁷ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝⁶ : TopologicalSpace M
inst✝⁵ : ChartedSpace H M
E' : Type u_5
inst✝⁴ : NormedAddCommGroup E'
inst✝³ : NormedSpace 𝕜 E'
H' : Type u_6
inst✝² : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
M' : Type u_7
inst✝¹ : TopologicalSpace M'
inst✝ : ChartedSpace H' M'
e : OpenPartialHomeomorph M M'
he : MDifferentiable I I' e
x : M
hx : x ∈ e.source
⊢ mfderiv% ↑e.symm (↑e x) ∘SL mfderiv% ↑e x = ContinuousLinearMap.id 𝕜 (TangentSpace I x)

Tactic:
have : mfderiv% (e.symm ∘ e) x = (mfderiv% e.symm (e x)).comp (mfderiv% e x) :=
    mfderiv_comp x (he.mdifferentiableAt_symm (e.map_source hx)) (he.mdifferentiableAt hx)
  rw [← this]
  have : mfderiv% (_root_.id : M → M) x = ContinuousLinearMap.id _ _ := mfderiv_id
  rw [← this]
  apply Filter.EventuallyEq.mfderiv_eq
  have : e.source ∈ 𝓝 x := e.open_source.mem_nhds hx
  exact Filter.mem_of_superset this (by mfld_set_tac)

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
H : Type u_3
inst✝³ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace H M
inst✝ : IsManifold I 1 M
x : M
y : E
hy : y ∈ (extChartAt I x).target
h'y : ↑(extChartAt I x).symm y ∈ (extChartAt I x).source
h''y : ↑(extChartAt I x).symm y ∈ (chartAt H x).source
U' : UniqueMDiffAt[(extChartAt I x).source] (↑(extChartAt I x).symm y)
⊢ mfderiv% ↑(extChartAt I x) (↑(extChartAt I x).symm y) =
    mfderiv[(extChartAt I x).source] ↑(extChartAt I x) (↑(extChartAt I x).symm y)

Tactic:
mfderivWithin_eq_mfderiv U'

Goal state after:
𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
H : Type u_3
inst✝³ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace H M
inst✝ : IsManifold I 1 M
x : M
y : E
hy : y ∈ (extChartAt I x).target
h'y : ↑(extChartAt I x).symm y ∈ (extChartAt I x).source
h''y : ↑(extChartAt I x).symm y ∈ (chartAt H x).source
U' : UniqueMDiffAt[(extChartAt I x).source] (↑(extChartAt I x).symm y)
⊢ mfderiv% ↑(extChartAt I x) (↑(extChartAt I x).symm y) = mfderiv% ↑(extChartAt I x) (↑(extChartAt I x).symm y)

𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
H : Type u_3
inst✝³ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace H M
inst✝ : IsManifold I 1 M
x : M
y : E
hy : y ∈ (extChartAt I x).target
h'y : ↑(extChartAt I x).symm y ∈ (extChartAt I x).source
h''y : ↑(extChartAt I x).symm y ∈ (chartAt H x).source
U' : UniqueMDiffAt[(extChartAt I x).source] (↑(extChartAt I x).symm y)
⊢ MDiffAt ↑(extChartAt I x) (↑(extChartAt I x).symm y)
Module: Mathlib.Geometry.Manifold.MFDeriv.Atlas

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
H : Type u_3
inst✝³ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace H M
inst✝ : IsManifold I 1 M
x y : M
hy : y ∈ (extChartAt I x).source
this : y = ↑(extChartAt I x).symm (↑(extChartAt I x) y)
⊢ y ∈ (extChartAt I x).source

Tactic:
simpa using hy

Goal state after:
Module: Mathlib.Topology.PartitionOfUnity

Goal state before:
pp: ι : Type u
X : Type v
inst✝ : TopologicalSpace X
s : Set X
ρ : PartitionOfUnity ι X s
x₀ : X
⊢ ∃ I, (∀ᶠ (x : X) in 𝓝[s] x₀, ∑ i ∈ I, (ρ i) x = 1) ∧ ∀ᶠ (x : X) in 𝓝 x₀, (support fun x_1 ↦ (ρ x_1) x) ⊆ ↑I

Tactic:
by
  rcases ρ.locallyFinite.exists_finset_support x₀ with ⟨I, hI⟩
  refine ⟨I, eventually_nhdsWithin_iff.mpr (hI.mono fun x hx x_in ↦ ?_), hI⟩
  have : ∑ᶠ i : ι, ρ i x = ∑ i ∈ I, ρ i x := finsum_eq_sum_of_support_subset _ hx
  rwa [eq_comm, ρ.sum_eq_one x_in] at this

Goal state after:
Module: Mathlib.Topology.PartitionOfUnity

Goal state before:
pp: ι : Type u
X : Type v
inst✝ : TopologicalSpace X
s✝ : Set X
f : BumpCovering ι X s✝
i : ι
s : Set X
x : X
x✝ : x ∈ s
⊢ ⇑(Pi.single i 1 i) =ᶠ[𝓝 x] 1

Tactic:
Pi.single_eq_same,

Goal state after:
ι : Type u
X : Type v
inst✝ : TopologicalSpace X
s✝ : Set X
f : BumpCovering ι X s✝
i : ι
s : Set X
x : X
x✝ : x ∈ s
⊢ ⇑1 =ᶠ[𝓝 x] 1