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

Goal state before:
pp: F : Type u_1
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
s : Set ℝ
f f' : ℝ → ℝ
hs : MeasurableSet s
hf' : ∀ x ∈ s, HasDerivWithinAt f (f' x) s x
hf : AntitoneOn f s
g : ℝ → F
n : ℝ → ℝ := fun x ↦ -x
⊢ ∫ (x : ℝ) in f '' s, g x = ∫ (x : ℝ) in s, -f' x • g (f x)

Tactic:
let e := n ∘ f

Goal state after:
F : Type u_1
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
s : Set ℝ
f f' : ℝ → ℝ
hs : MeasurableSet s
hf' : ∀ x ∈ s, HasDerivWithinAt f (f' x) s x
hf : AntitoneOn f s
g : ℝ → F
n : ℝ → ℝ := fun x ↦ -x
e : ℝ → ℝ := n ∘ f
⊢ ∫ (x : ℝ) in f '' s, g x = ∫ (x : ℝ) in s, -f' x • g (f x)
Module: Mathlib.Analysis.Calculus.LocalExtr.LineDeriv

Goal state before:
pp: E : Type u_1
inst✝¹ : AddCommGroup E
inst✝ : Module ℝ E
f : E → ℝ
a b : E
l : Filter E
h : IsExtrFilter f l a
h' : Tendsto (fun t ↦ a + t • b) (𝓝 0) l
⊢ lineDeriv ℝ f a b = 0

Tactic:
exact if hd : LineDifferentiableAt ℝ f a b then
    h.hasLineDerivAt_eq_zero hd.hasLineDerivAt h'
  else
    lineDeriv_zero_of_not_lineDifferentiableAt hd

Goal state after:
Module: Mathlib.Analysis.Calculus.LineDeriv.IntegrationByParts

Goal state before:
pp: case pos
E : Type u_1
F : Type u_2
G : Type u_3
W : Type u_4
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace ℝ E
inst✝⁹ : NormedAddCommGroup F
inst✝⁸ : NormedSpace ℝ F
inst✝⁷ : NormedAddCommGroup G
inst✝⁶ : NormedSpace ℝ G
inst✝⁵ : NormedAddCommGroup W
inst✝⁴ : NormedSpace ℝ W
inst✝³ : MeasurableSpace E
μ : Measure E
inst✝² : BorelSpace E
inst✝¹ : FiniteDimensional ℝ E
inst✝ : μ.IsAddHaarMeasure
f f' : E → F
g g' : E → G
B : F →L[ℝ] G →L[ℝ] W
hf'g : Integrable (fun x ↦ (B (f' x)) (g x)) μ
hfg' : Integrable (fun x ↦ (B (f x)) (g' x)) μ
hfg : Integrable (fun x ↦ (B (f x)) (g x)) μ
hW : CompleteSpace W
hf : ∀ x ∈ tsupport g, HasLineDerivAt ℝ f (f' x) x 0
hg : ∀ x ∈ tsupport f, HasLineDerivAt ℝ g (g' x) x 0
Hf' : ∀ (x : E), (B (f' x)) (g x) = 0
x : E
hx : x ∈ tsupport f
⊢ (B (f x)) (g' x) = 0

Tactic:
simp [(hasLineDerivAt_zero (f := g) (x := x)).lineDeriv, (hg x hx).lineDeriv.symm]

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

Goal state before:
pp: 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
ε : ℝ
εpos : ε > 0
δ : ℝ≥0 := ⟨ε, ⋯⟩
δpos : 0 < δ
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
right✝ : s.Nonempty → ∀ (n : ℕ), ∃ y ∈ s, A n = f' y
⊢ ∃ f, AEMeasurable f (μ.restrict s) ∧ ∀ᵐ (x : E) ∂μ.restrict s, dist (f x) (f' x) ≤ ε

Tactic:
obtain ⟨g, g_meas, hg⟩ :
      ∃ g : E → E →L[ℝ] E, Measurable g ∧ ∀ (n : ℕ) (x : E), x ∈ t n → g x = A n :=
    exists_measurable_piecewise t t_meas (fun n _ => A n) (fun n => measurable_const) <|
      t_disj.mono fun i j h => by simp only [h.inter_eq, eqOn_empty]

Goal state after:
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
ε : ℝ
εpos : ε > 0
δ : ℝ≥0 := ⟨ε, ⋯⟩
δpos : 0 < δ
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
right✝ : s.Nonempty → ∀ (n : ℕ), ∃ y ∈ s, A n = f' y
g : E → E →L[ℝ] E
g_meas : Measurable g
hg : ∀ (n : ℕ), ∀ x ∈ t n, g x = A n
⊢ ∃ f, AEMeasurable f (μ.restrict s) ∧ ∀ᵐ (x : E) ∂μ.restrict s, dist (f x) (f' x) ≤ ε
Module: Mathlib.Analysis.Calculus.ParametricIntervalIntegral

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁵ : RCLike 𝕜
μ : Measure ℝ
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : NormedSpace 𝕜 E
H : Type u_3
inst✝¹ : NormedAddCommGroup H
inst✝ : NormedSpace 𝕜 H
s : Set H
a b : ℝ
bound : ℝ → ℝ
F : H → ℝ → E
F' : ℝ → H →L[𝕜] E
x₀ : H
hs : s ∈ 𝓝 x₀
hF_meas : ∀ᶠ (x : H) in 𝓝 x₀, AEStronglyMeasurable (F x) (μ.restrict (Ι a b))
hF_int : IntervalIntegrable (F x₀) μ a b
hF'_meas : AEStronglyMeasurable F' (μ.restrict (Ι a b))
h_lip : ∀ᵐ (t : ℝ) ∂μ, t ∈ Ι a b → LipschitzOnWith (Real.nnabs (bound t)) (fun x ↦ F x t) s
bound_integrable : IntervalIntegrable bound μ a b
h_diff : ∀ᵐ (t : ℝ) ∂μ, t ∈ Ι a b → HasFDerivAt (fun x ↦ F x t) (F' t) x₀
⊢ IntervalIntegrable F' μ a b ∧ HasFDerivAt (fun x ↦ ∫ (t : ℝ) in a..b, F x t ∂μ) (∫ (t : ℝ) in a..b, F' t ∂μ) x₀

Tactic:
rw [← ae_restrict_iff' measurableSet_uIoc] at h_lip h_diff
  simp only [intervalIntegrable_iff] at hF_int bound_integrable ⊢
  simp only [intervalIntegral_eq_integral_uIoc]
  have := hasFDerivAt_integral_of_dominated_loc_of_lip hs hF_meas hF_int hF'_meas h_lip
    bound_integrable h_diff
  exact ⟨this.1, this.2.const_smul _⟩

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

Goal state before:
pp: 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
⊢ AEMeasurable (fun x ↦ |(f' x).det|.toNNReal) (μ.restrict s)

Tactic:
by
  apply measurable_real_toNNReal.comp_aemeasurable
  refine continuous_abs.measurable.comp_aemeasurable ?_
  refine ContinuousLinearMap.continuous_det.measurable.comp_aemeasurable ?_
  exact aemeasurable_fderivWithin μ hs hf'

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

Goal state before:
pp: H : Type u
H' : Type u_1
M : Type u_2
M' : Type u_3
M'' : Type u_4
inst✝⁶ : TopologicalSpace H
inst✝⁵ : TopologicalSpace M
inst✝⁴ : ChartedSpace H M
inst✝³ : TopologicalSpace M'
inst✝² : TopologicalSpace M''
G : StructureGroupoid H
inst✝¹ : ChartedSpace H M'
inst✝ : ChartedSpace H M''
e : Structomorph G M M'
e' : Structomorph G M' M''
c : OpenPartialHomeomorph M H
c' : OpenPartialHomeomorph M'' H
hc : c ∈ atlas H M
hc' : c' ∈ atlas H M''
x : H
hx : x ∈ (c.symm ≫ₕ (e.trans e'.toHomeomorph).toOpenPartialHomeomorph ≫ₕ c').source
f₁ : OpenPartialHomeomorph M M' := e.toOpenPartialHomeomorph
f₂ : OpenPartialHomeomorph M' M'' := e'.toOpenPartialHomeomorph
f : OpenPartialHomeomorph M M'' := (e.trans e'.toHomeomorph).toOpenPartialHomeomorph
feq : f = f₁ ≫ₕ f₂
y : M' := ↑(c.symm ≫ₕ f₁) x
⊢ ∃ s, IsOpen[inst✝⁶] s ∧ x ∈ s ∧ (c.symm ≫ₕ (e.trans e'.toHomeomorph).toOpenPartialHomeomorph ≫ₕ c').restr s ∈ G

Tactic:
let g := chartAt (H := H) y

Goal state after:
H : Type u
H' : Type u_1
M : Type u_2
M' : Type u_3
M'' : Type u_4
inst✝⁶ : TopologicalSpace H
inst✝⁵ : TopologicalSpace M
inst✝⁴ : ChartedSpace H M
inst✝³ : TopologicalSpace M'
inst✝² : TopologicalSpace M''
G : StructureGroupoid H
inst✝¹ : ChartedSpace H M'
inst✝ : ChartedSpace H M''
e : Structomorph G M M'
e' : Structomorph G M' M''
c : OpenPartialHomeomorph M H
c' : OpenPartialHomeomorph M'' H
hc : c ∈ atlas H M
hc' : c' ∈ atlas H M''
x : H
hx : x ∈ (c.symm ≫ₕ (e.trans e'.toHomeomorph).toOpenPartialHomeomorph ≫ₕ c').source
f₁ : OpenPartialHomeomorph M M' := e.toOpenPartialHomeomorph
f₂ : OpenPartialHomeomorph M' M'' := e'.toOpenPartialHomeomorph
f : OpenPartialHomeomorph M M'' := (e.trans e'.toHomeomorph).toOpenPartialHomeomorph
feq : f = f₁ ≫ₕ f₂
y : M' := ↑(c.symm ≫ₕ f₁) x
g : OpenPartialHomeomorph M' H := chartAt H y
⊢ ∃ s, IsOpen[inst✝⁶] s ∧ x ∈ s ∧ (c.symm ≫ₕ (e.trans e'.toHomeomorph).toOpenPartialHomeomorph ≫ₕ c').restr s ∈ G
Module: Mathlib.MeasureTheory.Function.Jacobian

Goal state before:
pp: case inl
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:
refine ⟨δ'', half_pos δ'pos, I'', ?_⟩
      simp only [hA, forall_const, zero_mul, ENNReal.ofReal_zero, imp_true_iff,
        zero_le, abs_zero]

Goal state after:
Module: Mathlib.Geometry.Manifold.IsManifold.ExtChartAt

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
M : Type u_3
H : Type u_4
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : TopologicalSpace H
inst✝ : TopologicalSpace M
f : OpenPartialHomeomorph M H
I : ModelWithCorners 𝕜 E H
s : Set M
y : M
hy : y ∈ f.source
⊢ map (↑(f.extend I)) (𝓝[s] y) = 𝓝[↑(f.extend I).symm ⁻¹' s ∩ range ↑I] ↑(f.extend I) y

Tactic:
rw [map_extend_nhdsWithin_eq_image f hy, nhdsWithin_inter, ←
    nhdsWithin_extend_target_eq _ hy, ← nhdsWithin_inter, (f.extend I).image_source_inter_eq',
    inter_comm]

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

Goal state before:
pp: case pos
𝕜 : Type u
inst✝⁴ : NontriviallyNormedField 𝕜
ι : Type v
inst✝³ : Fintype ι
E : ι → Type w
inst✝² : (i : ι) → NormedAddCommGroup (E i)
inst✝¹ : (i : ι) → NormedSpace 𝕜 (E i)
H : ι → Type u'
inst✝ : (i : ι) → TopologicalSpace (H i)
I : (i : ι) → ModelWithCorners 𝕜 (E i) (H i)
h : IsRCLikeNormedField 𝕜
⊢ Convex ℝ (univ.pi fun i ↦ range ↑(I i).toPartialEquiv)

Tactic:
let := h.rclike
      let := fun i ↦ NormedSpace.restrictScalars ℝ 𝕜 (E i)
      exact convex_pi fun i _hi ↦ (I i).convex_range

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

Goal state before:
pp: case left
n : ℕ∞ω
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
E' : Type u_5
H' : Type u_6
inst✝² : NormedAddCommGroup E'
inst✝¹ : NormedSpace 𝕜 E'
inst✝ : TopologicalSpace H'
I : ModelWithCorners 𝕜 E H
I' : ModelWithCorners 𝕜 E' H'
e : OpenPartialHomeomorph H H
e' : OpenPartialHomeomorph H' H'
he : (contDiffPregroupoid n I).property (↑e) e.source
he_symm : (contDiffPregroupoid n I).property (↑e.symm) e.target
he' : (contDiffPregroupoid n I').property (↑e') e'.source
he'_symm : (contDiffPregroupoid n I').property (↑e'.symm) e'.target
h3 :
  ContDiffOn 𝕜 n (Prod.map (↑I ∘ ↑e ∘ ↑I.symm) (↑I' ∘ ↑e' ∘ ↑I'.symm))
    ((↑I '' e.source) ×ˢ (↑I'.symm ⁻¹' e'.source ∩ range ↑I'))
⊢ ContDiffOn 𝕜 n (↑(I.prod I') ∘ ↑(e.prod e') ∘ ↑(I.prod I').symm)
    (↑(I.prod I').symm ⁻¹' (e.prod e'.toPartialEquiv).source ∩ range ↑(I.prod I'))

Tactic:
← I'.image_eq,

Goal state after:
case left
n : ℕ∞ω
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
E' : Type u_5
H' : Type u_6
inst✝² : NormedAddCommGroup E'
inst✝¹ : NormedSpace 𝕜 E'
inst✝ : TopologicalSpace H'
I : ModelWithCorners 𝕜 E H
I' : ModelWithCorners 𝕜 E' H'
e : OpenPartialHomeomorph H H
e' : OpenPartialHomeomorph H' H'
he : (contDiffPregroupoid n I).property (↑e) e.source
he_symm : (contDiffPregroupoid n I).property (↑e.symm) e.target
he' : (contDiffPregroupoid n I').property (↑e') e'.source
he'_symm : (contDiffPregroupoid n I').property (↑e'.symm) e'.target
h3 : ContDiffOn 𝕜 n (Prod.map (↑I ∘ ↑e ∘ ↑I.symm) (↑I' ∘ ↑e' ∘ ↑I'.symm)) ((↑I '' e.source) ×ˢ (↑I' '' e'.source))
⊢ ContDiffOn 𝕜 n (↑(I.prod I') ∘ ↑(e.prod e') ∘ ↑(I.prod I').symm)
    (↑(I.prod I').symm ⁻¹' (e.prod e'.toPartialEquiv).source ∩ range ↑(I.prod I'))
Module: Mathlib.Geometry.Manifold.LocalInvariantProperties

Goal state before:
pp: H : Type u_1
inst✝¹ : TopologicalSpace H
G : StructureGroupoid H
inst✝ : ClosedUnderRestriction G
⊢ ∀ {s : Set H} {x : H} {f g : H → H},
    (∀ y ∈ s, f y = g y) → f x = g x → G.IsLocalStructomorphWithinAt f s x → G.IsLocalStructomorphWithinAt g s x

Tactic:
intro s x f g hfgs _ h hx

Goal state after:
H : Type u_1
inst✝¹ : TopologicalSpace H
G : StructureGroupoid H
inst✝ : ClosedUnderRestriction G
s : Set H
x : H
f g : H → H
hfgs : ∀ y ∈ s, f y = g y
a✝ : f x = g x
h : G.IsLocalStructomorphWithinAt f s x
hx : x ∈ s
⊢ ∃ e ∈ G, EqOn g (↑e.toPartialEquiv) (s ∩ e.source) ∧ x ∈ e.source
Module: Mathlib.Geometry.Manifold.IsManifold.Basic

Goal state before:
pp: case right
n : ℕ∞ω
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
E' : Type u_5
H' : Type u_6
inst✝² : NormedAddCommGroup E'
inst✝¹ : NormedSpace 𝕜 E'
inst✝ : TopologicalSpace H'
I : ModelWithCorners 𝕜 E H
I' : ModelWithCorners 𝕜 E' H'
e : OpenPartialHomeomorph H H
e' : OpenPartialHomeomorph H' H'
he : (contDiffPregroupoid n I).property (↑e) e.source
he_symm : (contDiffPregroupoid n I).property (↑e.symm) e.target
he' : (contDiffPregroupoid n I').property (↑e') e'.source
he'_symm : (contDiffPregroupoid n I').property (↑e'.symm) e'.target
⊢ ContDiffOn 𝕜 n (↑(I.prod I') ∘ ↑(e.prod e').symm ∘ ↑(I.prod I').symm)
    (↑(I.prod I').symm ⁻¹' (e.prod e'.toPartialEquiv).target ∩ range ↑(I.prod I'))

Tactic:
have h3 := ContDiffOn.prodMap he_symm he'_symm

Goal state after:
case right
n : ℕ∞ω
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
E' : Type u_5
H' : Type u_6
inst✝² : NormedAddCommGroup E'
inst✝¹ : NormedSpace 𝕜 E'
inst✝ : TopologicalSpace H'
I : ModelWithCorners 𝕜 E H
I' : ModelWithCorners 𝕜 E' H'
e : OpenPartialHomeomorph H H
e' : OpenPartialHomeomorph H' H'
he : (contDiffPregroupoid n I).property (↑e) e.source
he_symm : (contDiffPregroupoid n I).property (↑e.symm) e.target
he' : (contDiffPregroupoid n I').property (↑e') e'.source
he'_symm : (contDiffPregroupoid n I').property (↑e'.symm) e'.target
h3 :
  ContDiffOn 𝕜 n (Prod.map (↑I ∘ ↑e.symm ∘ ↑I.symm) (↑I' ∘ ↑e'.symm ∘ ↑I'.symm))
    ((↑I.symm ⁻¹' e.target ∩ range ↑I) ×ˢ (↑I'.symm ⁻¹' e'.target ∩ range ↑I'))
⊢ ContDiffOn 𝕜 n (↑(I.prod I') ∘ ↑(e.prod e').symm ∘ ↑(I.prod I').symm)
    (↑(I.prod I').symm ⁻¹' (e.prod e'.toPartialEquiv).target ∩ range ↑(I.prod I'))
Module: Mathlib.Geometry.Manifold.IsManifold.Basic

Goal state before:
pp: case right
n : ℕ∞ω
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
E' : Type u_5
H' : Type u_6
inst✝² : NormedAddCommGroup E'
inst✝¹ : NormedSpace 𝕜 E'
inst✝ : TopologicalSpace H'
I : ModelWithCorners 𝕜 E H
I' : ModelWithCorners 𝕜 E' H'
e : OpenPartialHomeomorph H H
e' : OpenPartialHomeomorph H' H'
he : (contDiffPregroupoid n I).property (↑e) e.source
he_symm : (contDiffPregroupoid n I).property (↑e.symm) e.target
he' : (contDiffPregroupoid n I').property (↑e') e'.source
he'_symm : (contDiffPregroupoid n I').property (↑e'.symm) e'.target
h3 :
  ContDiffOn 𝕜 n (Prod.map (↑I ∘ ↑e.symm ∘ ↑I.symm) (↑I' ∘ ↑e'.symm ∘ ↑I'.symm))
    ((↑I '' e.target) ×ˢ (↑I'.symm ⁻¹' e'.target ∩ range ↑I'))
⊢ ContDiffOn 𝕜 n (↑(I.prod I') ∘ ↑(e.prod e').symm ∘ ↑(I.prod I').symm)
    (↑(I.prod I').symm ⁻¹' (e.prod e'.toPartialEquiv).target ∩ range ↑(I.prod I'))

Tactic:
← I'.image_eq,

Goal state after:
case right
n : ℕ∞ω
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
E' : Type u_5
H' : Type u_6
inst✝² : NormedAddCommGroup E'
inst✝¹ : NormedSpace 𝕜 E'
inst✝ : TopologicalSpace H'
I : ModelWithCorners 𝕜 E H
I' : ModelWithCorners 𝕜 E' H'
e : OpenPartialHomeomorph H H
e' : OpenPartialHomeomorph H' H'
he : (contDiffPregroupoid n I).property (↑e) e.source
he_symm : (contDiffPregroupoid n I).property (↑e.symm) e.target
he' : (contDiffPregroupoid n I').property (↑e') e'.source
he'_symm : (contDiffPregroupoid n I').property (↑e'.symm) e'.target
h3 :
  ContDiffOn 𝕜 n (Prod.map (↑I ∘ ↑e.symm ∘ ↑I.symm) (↑I' ∘ ↑e'.symm ∘ ↑I'.symm))
    ((↑I '' e.target) ×ˢ (↑I' '' e'.target))
⊢ ContDiffOn 𝕜 n (↑(I.prod I') ∘ ↑(e.prod e').symm ∘ ↑(I.prod I').symm)
    (↑(I.prod I').symm ⁻¹' (e.prod e'.toPartialEquiv).target ∩ range ↑(I.prod I'))
Module: Mathlib.Geometry.Manifold.IsManifold.Basic

Goal state before:
pp: m n : ℕ∞ω
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
E' : Type u_5
H' : Type u_6
inst✝² : NormedAddCommGroup E'
inst✝¹ : NormedSpace 𝕜 E'
inst✝ : TopologicalSpace H'
⊢ idRestrGroupoid ≤ contDiffGroupoid n I

Tactic:
by
      rw [StructureGroupoid.le_iff]
      rintro e ⟨s, hs, hes⟩
      apply (contDiffGroupoid n I).mem_of_eqOnSource' _ _ _ hes
      exact ofSet_mem_contDiffGroupoid hs

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝¹⁵ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹⁴ : NormedAddCommGroup E
inst✝¹³ : NormedSpace 𝕜 E
H : Type u_3
inst✝¹² : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝¹¹ : TopologicalSpace M
E' : Type u_5
inst✝¹⁰ : NormedAddCommGroup E'
inst✝⁹ : NormedSpace 𝕜 E'
H' : Type u_6
inst✝⁸ : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
M' : Type u_7
inst✝⁷ : TopologicalSpace M'
E'' : Type u_8
inst✝⁶ : NormedAddCommGroup E''
inst✝⁵ : NormedSpace 𝕜 E''
H'' : Type u_9
inst✝⁴ : TopologicalSpace H''
I'' : ModelWithCorners 𝕜 E'' H''
M'' : Type u_10
inst✝³ : TopologicalSpace M''
inst✝² : ChartedSpace H M
inst✝¹ : ChartedSpace H' M'
inst✝ : ChartedSpace H'' M''
f : M → M'
s : Set M
n : ℕ∞ω
t : Set M'
g : M' → M''
x : M
st : MapsTo f s t
e : PartialEquiv M E := extChartAt I x
e' : PartialEquiv M' E' := extChartAt I' (f x)
hg :
  ContinuousWithinAt g t (f x) ∧
    ContDiffWithinAt 𝕜 n (↑(extChartAt I'' (g (f x))) ∘ g ∘ ↑e'.symm) (↑e'.symm ⁻¹' t ∩ range ↑I')
      (writtenInExtChartAt I I' x f (↑e x))
hf : ContinuousWithinAt f s x ∧ ContDiffWithinAt 𝕜 n (↑e' ∘ f ∘ ↑e.symm) (↑e.symm ⁻¹' s ∩ range ↑I) (↑e x)
this : ↑e' (f x) = writtenInExtChartAt I I' x f (↑e x)
A : ∀ᶠ (y : E) in 𝓝[↑e.symm ⁻¹' s ∩ range ↑I] ↑e x, f (↑e.symm y) ∈ t ∧ f (↑e.symm y) ∈ e'.source
x' : E
ht : f (↑e.symm x') ∈ t
hfx' : f (↑e.symm x') ∈ e'.source
⊢ ↑(extChartAt I' (f x)) (f (↑(extChartAt I x).symm x')) ∈ range ↑I'

Tactic:
exact mem_range_self _

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

Goal state before:
pp: 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
u : E → E
u_meas : Measurable u
uf : EqOn u f s
u' : ∀ x ∈ s, HasFDerivWithinAt u (f' x) s x
F : ↑s → E := u ∘ Subtype.val
hF : F = u ∘ Subtype.val
⊢ Measure.map u ((μ.withDensity fun x ↦ ENNReal.ofReal |(f' x).det|).restrict s) = μ.restrict (u '' s)

Tactic:
restrict_withDensity hs

Goal state after:
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
u : E → E
u_meas : Measurable u
uf : EqOn u f s
u' : ∀ x ∈ s, HasFDerivWithinAt u (f' x) s x
F : ↑s → E := u ∘ Subtype.val
hF : F = u ∘ Subtype.val
⊢ Measure.map u ((μ.restrict s).withDensity fun x ↦ ENNReal.ofReal |(f' x).det|) = μ.restrict (u '' s)
Module: Mathlib.Geometry.Manifold.Algebra.LieGroup

Goal state before:
pp: 𝕜 : Type u_1
inst✝ : NontriviallyNormedField 𝕜
n : ℕ∞ω
x : 𝕜
hx : x ≠ 0
⊢ ContMDiffAt 𝓘(𝕜, 𝕜) 𝓘(𝕜, 𝕜) n (fun y ↦ y⁻¹) x

Tactic:
by
    change ContMDiffAt 𝓘(𝕜) 𝓘(𝕜) n Inv.inv x
    rw [contMDiffAt_iff_contDiffAt]
    exact contDiffAt_inv 𝕜 hx

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

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
E' : Type u_5
inst✝⁶ : NormedAddCommGroup E'
inst✝⁵ : NormedSpace 𝕜 E'
H' : Type u_6
inst✝⁴ : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
M' : Type u_7
inst✝³ : TopologicalSpace M'
inst✝² : ChartedSpace H' M'
f : M → M'
s : Set M
x : M
m✝ n : ℕ∞ω
inst✝¹ : IsManifold I n M
inst✝ : IsManifold I' n M'
h : ContMDiffWithinAt I I' n f s x
m : ℕ
hm : ↑m ≤ n
h' : ↑m = ∞ → n = ω
this✝ : IsManifold I (↑m) M
this : IsManifold I' (↑m) M'
⊢ ∃ u, IsOpen[inst✝⁸] u ∧ x ∈ u ∧ ContMDiffOn I I' (↑m) f (insert x s ∩ u)

Tactic:
rcases (contMDiffWithinAt_iff_contMDiffOn_nhds (by simp)).1 (h.of_le hm) with ⟨t, ht, h't⟩
    rcases mem_nhdsWithin.1 ht with ⟨u, u_open, xu, hu⟩
    rw [inter_comm] at hu
    exact ⟨u, u_open, xu, h't.mono hu⟩

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

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
E' : Type u_5
inst✝⁶ : NormedAddCommGroup E'
inst✝⁵ : NormedSpace 𝕜 E'
H' : Type u_6
inst✝⁴ : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
M' : Type u_7
inst✝³ : TopologicalSpace M'
inst✝² : ChartedSpace H' M'
f : M → M'
s : Set M
x : M
m n : ℕ∞ω
inst✝¹ : IsManifold I n M
inst✝ : IsManifold I' n M'
h : ContMDiffWithinAt I I' n f s x
hm : ω ≤ n
h' : ω = ∞ → n = ω
this✝ : IsManifold I ω M
this : IsManifold I' ω M'
⊢ ∃ u, IsOpen[inst✝⁸] u ∧ x ∈ u ∧ ContMDiffOn I I' ω f (insert x s ∩ u)

Tactic:
rcases (contMDiffWithinAt_iff_contMDiffOn_nhds (by simp)).1 (h.of_le hm) with ⟨t, ht, h't⟩
    rcases mem_nhdsWithin.1 ht with ⟨u, u_open, xu, hu⟩
    rw [inter_comm] at hu
    exact ⟨u, u_open, xu, h't.mono hu⟩

Goal state after:
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'
s : Set M
e : OpenPartialHomeomorph M H
e' : OpenPartialHomeomorph M' H'
he : e ∈ maximalAtlas I 1 M
he' : e' ∈ maximalAtlas I' 1 M'
hs : s ⊆ e.source
h2s : MapsTo f s e'.source
⊢ MDiff[s] f ↔
    (∀ x ∈ s, ContinuousWithinAt f s x) ∧
      DifferentiableOn 𝕜 (↑(e'.extend I') ∘ f ∘ ↑(e.extend I).symm) (↑(e.extend I) '' s)

Tactic:
DifferentiableOn,

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'
s : Set M
e : OpenPartialHomeomorph M H
e' : OpenPartialHomeomorph M' H'
he : e ∈ maximalAtlas I 1 M
he' : e' ∈ maximalAtlas I' 1 M'
hs : s ⊆ e.source
h2s : MapsTo f s e'.source
⊢ MDiff[s] f ↔
    (∀ x ∈ s, ContinuousWithinAt f s x) ∧
      ∀ x ∈ ↑(e.extend I) '' s,
        DifferentiableWithinAt 𝕜 (↑(e'.extend I') ∘ f ∘ ↑(e.extend I).symm) (↑(e.extend I) '' s) x