mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Analysis.Calculus.FDeriv.Partial

Goal state before:

Tactic:
fun (v, w) => f v.1 v.2 - f w.1 w.2 - ((↿f₁ u).coprod (↿f₂ u)) (v - w)
    _ = fun (v, w) => (f v.1 w.2 - f w.1 w.2 - ↿f₁ u (v.1 - w.1))
          + (f v.1 v.2 - f v.1 w.2 - ↿f₂ u (v.2 - w.2)) := by
      ext
      dsimp only [ContinuousLinearMap.coprod_apply]
      abel
    _ =o[𝓝 (u, u)] fun (v, w) => v - w := by
      let : RCLike 𝕜 := IsRCLikeNormedField.rclike 𝕜
      rw [nhds_prod_eq, nhds_prod_eq]
      apply IsLittleO.add
      · calc
          fun (v, w) => f v.1 w.2 - f w.1 w.2 - ↿f₁ u (v.1 - w.1)
          _ =o[(𝓝 u.1 ×ˢ 𝓝 u.2) ×ˢ (𝓝 u.1 ×ˢ 𝓝 u.2)] (fun (v, w) => v.1 - w.1 : _ → E₁) := by
            have h := tendsto_snd.prodMk <| tendsto_snd.comp <| tendsto_snd.comp <|
              tendsto_fst (f := (𝓝 u.1 ×ˢ 𝓝 u.2) ×ˢ (𝓝 u.1 ×ˢ 𝓝 u.2)) (g := 𝓝 u.1)
            let : NormedSpace ℝ E₁ := RestrictScalars.normedSpace ℝ 𝕜 E₁
            apply isLittleO_sub_sub_fderiv (α := (E₁ × E₂) × (E₁ × E₂))
              (f := fun (v, w) x => f x w.2) (f' := fun (v, w) x => f₁ x w.2)
              (tendsto_fst.comp tendsto_fst) (tendsto_fst.comp tendsto_snd)
            · simpa using! h.eventually df₁
            · simpa using! cf₁.comp h
          _ =O[(𝓝 u.1 ×ˢ 𝓝 u.2) ×ˢ (𝓝 u.1 ×ˢ 𝓝 u.2)] (fun (v, w) => v - w : _ → E₁ × E₂) := by
            simp [isBigO_of_le]
      · calc
          fun (v, w) => f v.1 v.2 - f v.1 w.2 - ↿f₂ u (v.2 - w.2)
          _ =o[(𝓝 u.1 ×ˢ 𝓝 u.2) ×ˢ (𝓝 u.1 ×ˢ 𝓝 u.2)] (fun (v, w) => v.2 - w.2 : _ → E₂) := by
            have h := (tendsto_fst.comp <| tendsto_fst.comp tendsto_fst).prodMk <|
              tendsto_snd (f := (𝓝 u.1 ×ˢ 𝓝 u.2) ×ˢ (𝓝 u.1 ×ˢ 𝓝 u.2)) (g := 𝓝 u.2)
            let : NormedSpace ℝ E₂ := RestrictScalars.normedSpace ℝ 𝕜 E₂
            apply isLittleO_sub_sub_fderiv (f' := fun (v, w) y => f₂ v.1 y)
              (tendsto_snd.comp tendsto_fst) (tendsto_snd.comp tendsto_snd)
            · simpa using! h.eventually df₂
            · simpa using! cf₂.comp h
          _ =O[(𝓝 u.1 ×ˢ 𝓝 u.2) ×ˢ (𝓝 u.1 ×ˢ 𝓝 u.2)] (fun (v, w) => v - w : _ → E₁ × E₂) := by
            simp [isBigO_of_le]

Goal state after:
Module: Mathlib.Analysis.Normed.Module.Complemented

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
inst✝⁷ : NontriviallyNormedField 𝕜
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace 𝕜 E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace 𝕜 F
inst✝² : NormedAddCommGroup G
inst✝¹ : NormedSpace 𝕜 G
inst✝ : CompleteSpace E
p q : Subspace 𝕜 E
h : IsCompl p q
hp : IsClosed ↑p
hq : IsClosed ↑q
this : CompleteSpace ↑↑p
⊢ (↥p × ↥q) ≃L[𝕜] E

Tactic:
haveI := hq.completeSpace_coe

Goal state after:
𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
inst✝⁷ : NontriviallyNormedField 𝕜
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace 𝕜 E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace 𝕜 F
inst✝² : NormedAddCommGroup G
inst✝¹ : NormedSpace 𝕜 G
inst✝ : CompleteSpace E
p q : Subspace 𝕜 E
h : IsCompl p q
hp : IsClosed ↑p
hq : IsClosed ↑q
this✝ : CompleteSpace ↑↑p
this : CompleteSpace ↑↑q
⊢ (↥p × ↥q) ≃L[𝕜] E
Module: Mathlib.Analysis.Calculus.Implicit

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁹ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace 𝕜 E
inst✝⁶ : CompleteSpace E
F : Type u_3
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
inst✝³ : CompleteSpace F
G : Type u_4
inst✝² : NormedAddCommGroup G
inst✝¹ : NormedSpace 𝕜 G
inst✝ : CompleteSpace G
φ : ImplicitFunctionData 𝕜 E F G
⊢ map Prod.fst (𝓝 (φ.prodFun φ.pt)) = 𝓝 (φ.prodFun φ.pt).1

Tactic:
map_fst_nhds

Goal state after:
𝕜 : Type u_1
inst✝⁹ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace 𝕜 E
inst✝⁶ : CompleteSpace E
F : Type u_3
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
inst✝³ : CompleteSpace F
G : Type u_4
inst✝² : NormedAddCommGroup G
inst✝¹ : NormedSpace 𝕜 G
inst✝ : CompleteSpace G
φ : ImplicitFunctionData 𝕜 E F G
⊢ 𝓝 (φ.prodFun φ.pt).1 = 𝓝 (φ.prodFun φ.pt).1
Module: Mathlib.MeasureTheory.Function.JacobianOneDim

Goal state before:
pp: s : Set ℝ
f f' : ℝ → ℝ
hs : MeasurableSet s
hf' : ∀ x ∈ s, HasDerivWithinAt f (f' x) s x
hf : AntitoneOn f s
u : ℝ → ℝ≥0∞
n : ℝ → ℝ := fun x ↦ -x
e : ℝ → ℝ := n ∘ f
hg' : ∀ x ∈ s, HasDerivWithinAt e (-f' x) s x
⊢ ∫⁻ (x : ℝ) in e '' s, u (n x) = ∫⁻ (x : ℝ) in s, ENNReal.ofReal (-f' x) * (u ∘ n) (e x)

Tactic:
← lintegral_image_eq_lintegral_deriv_mul_of_monotoneOn hs hg' hf.neg (u ∘ n)

Goal state after:
s : Set ℝ
f f' : ℝ → ℝ
hs : MeasurableSet s
hf' : ∀ x ∈ s, HasDerivWithinAt f (f' x) s x
hf : AntitoneOn f s
u : ℝ → ℝ≥0∞
n : ℝ → ℝ := fun x ↦ -x
e : ℝ → ℝ := n ∘ f
hg' : ∀ x ∈ s, HasDerivWithinAt e (-f' x) s x
⊢ ∫⁻ (x : ℝ) in e '' s, u (n x) = ∫⁻ (x : ℝ) in e '' s, (u ∘ n) x
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
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
A : E →L[ℝ] E
m : ℝ≥0
hm : ↑m < ENNReal.ofReal |A.det|
mpos : 0 < m
hA : A.det ≠ 0
B : E ≃L[ℝ] E := A.toContinuousLinearEquivOfDetNeZero hA
I : ENNReal.ofReal |(↑B.symm).det| < ↑m⁻¹
δ₀ : ℝ≥0
δ₀pos : 0 < δ₀
hδ₀ : ∀ (t : Set E) (g : E → E), ApproximatesLinearOn g (↑B.symm) t δ₀ → μ (g '' t) ≤ ↑m⁻¹ * μ t
L1 : ∀ᶠ (δ : ℝ≥0) in 𝓝 0, Subsingleton E ∨ δ < ‖↑B.symm‖₊⁻¹
this : Tendsto (fun δ ↦ ‖↑B.symm‖₊ * (‖↑B.symm‖₊⁻¹ - δ)⁻¹ * δ) (𝓝 0) (𝓝 (‖↑B.symm‖₊ * (‖↑B.symm‖₊⁻¹ - 0)⁻¹ * 0))
⊢ ∀ᶠ (δ : ℝ≥0) in 𝓝 0, ‖↑B.symm‖₊ * (‖↑B.symm‖₊⁻¹ - δ)⁻¹ * δ < δ₀

Tactic:
simp only [mul_zero] at this

Goal state after:
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : FiniteDimensional ℝ E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
A : E →L[ℝ] E
m : ℝ≥0
hm : ↑m < ENNReal.ofReal |A.det|
mpos : 0 < m
hA : A.det ≠ 0
B : E ≃L[ℝ] E := A.toContinuousLinearEquivOfDetNeZero hA
I : ENNReal.ofReal |(↑B.symm).det| < ↑m⁻¹
δ₀ : ℝ≥0
δ₀pos : 0 < δ₀
hδ₀ : ∀ (t : Set E) (g : E → E), ApproximatesLinearOn g (↑B.symm) t δ₀ → μ (g '' t) ≤ ↑m⁻¹ * μ t
L1 : ∀ᶠ (δ : ℝ≥0) in 𝓝 0, Subsingleton E ∨ δ < ‖↑B.symm‖₊⁻¹
this : Tendsto (fun δ ↦ ‖↑B.symm‖₊ * (‖↑B.symm‖₊⁻¹ - δ)⁻¹ * δ) (𝓝 0) (𝓝 0)
⊢ ∀ᶠ (δ : ℝ≥0) in 𝓝 0, ‖↑B.symm‖₊ * (‖↑B.symm‖₊⁻¹ - δ)⁻¹ * δ < δ₀
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
hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x
R : ℝ
hs : s ⊆ closedBall 0 R
ε : ℝ≥0
εpos : 0 < ε
h'f' : ∀ x ∈ s, (f' x).det = 0
⊢ μ (f '' s) ≤ ↑ε * μ (closedBall 0 R)

Tactic:
rcases eq_empty_or_nonempty s with (rfl | h's)

Goal state after:
case inl
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : FiniteDimensional ℝ E
f : E → E
f' : E → E →L[ℝ] E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
R : ℝ
ε : ℝ≥0
εpos : 0 < ε
hf' : ∀ x ∈ ∅, HasFDerivWithinAt f (f' x) ∅ x
hs : ∅ ⊆ closedBall 0 R
h'f' : ∀ x ∈ ∅, (f' x).det = 0
⊢ μ (f '' ∅) ≤ ↑ε * μ (closedBall 0 R)

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
hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x
R : ℝ
hs : s ⊆ closedBall 0 R
ε : ℝ≥0
εpos : 0 < ε
h'f' : ∀ x ∈ s, (f' x).det = 0
h's : s.Nonempty
⊢ μ (f '' s) ≤ ↑ε * μ (closedBall 0 R)
Module: Mathlib.Analysis.Complex.LocallyUniformLimit

Goal state before:
pp: E : Type u_1
ι : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
U : Set ℂ
z : ℂ
F : ι → ℂ → E
inst✝ : CompleteSpace E
u : ι → ℝ
hu : Summable u
hf : ∀ (i : ι), DifferentiableOn ℂ (F i) U
hU : IsOpen U
hF_le : ∀ (i : ι), ∀ w ∈ U, ‖F i w‖ ≤ u i
hz : z ∈ U
hc : TendstoLocallyUniformlyOn (fun t x ↦ ∑ n ∈ t, F n x) (fun x ↦ ∑' (n : ι), F n x) atTop U
⊢ Tendsto (fun s ↦ ∑ b ∈ s, deriv (F b) z) (SummationFilter.unconditional ι).filter
    (𝓝 (deriv (fun w ↦ ∑' (i : ι), F i w) z))

Tactic:
convert!
    (hc.deriv (Eventually.of_forall fun s => DifferentiableOn.fun_sum fun i _ => hf i)
          hU).tendsto_at
      hz using 1

Goal state after:
case e'_3
E : Type u_1
ι : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
U : Set ℂ
z : ℂ
F : ι → ℂ → E
inst✝ : CompleteSpace E
u : ι → ℝ
hu : Summable u
hf : ∀ (i : ι), DifferentiableOn ℂ (F i) U
hU : IsOpen U
hF_le : ∀ (i : ι), ∀ w ∈ U, ‖F i w‖ ≤ u i
hz : z ∈ U
hc : TendstoLocallyUniformlyOn (fun t x ↦ ∑ n ∈ t, F n x) (fun x ↦ ∑' (n : ι), F n x) atTop U
⊢ (fun s ↦ ∑ b ∈ s, deriv (F b) z) = fun i ↦ (deriv ∘ fun t x ↦ ∑ n ∈ t, F n x) i z
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≥0
εpos : 0 < ε
A : E →L[ℝ] E
m : ℝ≥0 := |A.det|.toNNReal + ε
I : ENNReal.ofReal |A.det| < ↑m
δ : ℝ≥0
h : ∀ (s : Set E) (f : E → E), ApproximatesLinearOn f A s δ → μ (f '' s) ≤ ↑m * μ s
δpos : 0 < δ
⊢ ∃ 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 → μ (g '' t) ≤ (ENNReal.ofReal |A.det| + ↑ε) * μ t

Tactic:
obtain ⟨δ', δ'pos, hδ'⟩ : ∃ (δ' : ℝ), 0 < δ' ∧ ∀ B, dist B A < δ' → dist B.det A.det < ↑ε := by
      refine continuousAt_iff.1 ?_ ε εpos
      exact ContinuousLinearMap.continuous_det.continuousAt

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≥0
εpos : 0 < ε
A : E →L[ℝ] E
m : ℝ≥0 := |A.det|.toNNReal + ε
I : ENNReal.ofReal |A.det| < ↑m
δ : ℝ≥0
h : ∀ (s : Set E) (f : E → E), ApproximatesLinearOn f A s δ → μ (f '' s) ≤ ↑m * μ s
δpos : 0 < δ
δ' : ℝ
δ'pos : 0 < δ'
hδ' : ∀ (B : E →L[ℝ] E), dist B A < δ' → dist B.det A.det < ↑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 → μ (g '' t) ≤ (ENNReal.ofReal |A.det| + ↑ε) * μ t
Module: Mathlib.Geometry.Manifold.StructureGroupoid

Goal state before:
pp: H : Type u_1
inst✝ : TopologicalSpace H
S : Set (StructureGroupoid H)
e : OpenPartialHomeomorph H H
⊢ ∀ (e' : OpenPartialHomeomorph H H),
    (∀ i ∈ S, e ∈ i.members) → (∀ i ∈ S, e' ∈ i.members) → ∀ i ∈ S, e ≫ₕ e' ∈ i.members

Tactic:
e'

Goal state after:
H : Type u_1
inst✝ : TopologicalSpace H
S : Set (StructureGroupoid H)
e e' : OpenPartialHomeomorph H H
⊢ (∀ i ∈ S, e ∈ i.members) → (∀ i ∈ S, e' ∈ i.members) → ∀ i ∈ S, e ≫ₕ e' ∈ i.members
Module: Mathlib.Geometry.Manifold.StructureGroupoid

Goal state before:
pp: H : Type u_1
inst✝ : TopologicalSpace H
S : Set (StructureGroupoid H)
e : OpenPartialHomeomorph H H
⊢ ∀ (e' : OpenPartialHomeomorph H H), (∀ i ∈ S, e ∈ i.members) → e' ≈ e → ∀ i ∈ S, e' ∈ i.members

Tactic:
e'

Goal state after:
H : Type u_1
inst✝ : TopologicalSpace H
S : Set (StructureGroupoid H)
e e' : OpenPartialHomeomorph H H
⊢ (∀ i ∈ S, e ∈ i.members) → e' ≈ e → ∀ i ∈ S, e' ∈ i.members
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✝³ : TopologicalSpace H
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace H M
G : StructureGroupoid H
inst✝ : HasGroupoid M G
s : Opens M
x : ↥s
⊢ (chartAt H ↑x).subtypeRestr ⋯ ∈ ⋃ x, {(chartAt H ↑x).subtypeRestr ⋯}

Tactic:
simp only [mem_iUnion, mem_singleton_iff]

Goal state after:
H : Type u
H' : Type u_1
M : Type u_2
M' : Type u_3
M'' : Type u_4
inst✝³ : TopologicalSpace H
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace H M
G : StructureGroupoid H
inst✝ : HasGroupoid M G
s : Opens M
x : ↥s
⊢ ∃ i, (chartAt H ↑x).subtypeRestr ⋯ = (chartAt H ↑i).subtypeRestr ⋯
Module: Mathlib.Geometry.Manifold.ChartedSpace

Goal state before:
pp: H : Type u
H' : Type u_1
M : Type u_2
M' : Type u_3
M'' : Type u_4
inst✝ : TopologicalSpace H
c : ChartedSpaceCore H M
e : PartialEquiv M H
x : M
⊢ c.openPartialHomeomorph (c.chartAt x) ⋯ ∈ ⋃ e, ⋃ (he : e ∈ c.atlas), {c.openPartialHomeomorph e he}

Tactic:
simp only [mem_iUnion, mem_singleton_iff]

Goal state after:
H : Type u
H' : Type u_1
M : Type u_2
M' : Type u_3
M'' : Type u_4
inst✝ : TopologicalSpace H
c : ChartedSpaceCore H M
e : PartialEquiv M H
x : M
⊢ ∃ i, ∃ (h : i ∈ c.atlas), c.openPartialHomeomorph (c.chartAt x) ⋯ = c.openPartialHomeomorph i ⋯
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✝⁴ : TopologicalSpace H
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace H M
G : StructureGroupoid H
inst✝¹ : HasGroupoid M G
s : Opens M
inst✝ : ClosedUnderRestriction G
e e' : OpenPartialHomeomorph (↥s) H
w✝¹ : Set (OpenPartialHomeomorph (↥s) H)
x : ↥s
he : e = (chartAt H ↑x).subtypeRestr ⋯
hc : (fun x ↦ {(chartAt H ↑x).subtypeRestr ⋯}) x = w✝¹
w✝ : Set (OpenPartialHomeomorph (↥s) H)
he' : e' ∈ w✝
x' : ↥s
hc' : (fun x ↦ {(chartAt H ↑x).subtypeRestr ⋯}) x' = w✝
⊢ e.symm ≫ₕ e' ∈ G

Tactic:
rw [hc'.symm, mem_singleton_iff] at he'

Goal state after:
H : Type u
H' : Type u_1
M : Type u_2
M' : Type u_3
M'' : Type u_4
inst✝⁴ : TopologicalSpace H
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace H M
G : StructureGroupoid H
inst✝¹ : HasGroupoid M G
s : Opens M
inst✝ : ClosedUnderRestriction G
e e' : OpenPartialHomeomorph (↥s) H
w✝¹ : Set (OpenPartialHomeomorph (↥s) H)
x : ↥s
he : e = (chartAt H ↑x).subtypeRestr ⋯
hc : (fun x ↦ {(chartAt H ↑x).subtypeRestr ⋯}) x = w✝¹
w✝ : Set (OpenPartialHomeomorph (↥s) H)
x' : ↥s
he' : e' = (chartAt H ↑x').subtypeRestr ⋯
hc' : (fun x ↦ {(chartAt H ↑x).subtypeRestr ⋯}) x' = w✝
⊢ e.symm ≫ₕ e' ∈ G
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
ε : ℝ≥0
εpos : 0 < ε
A : E →L[ℝ] E
⊢ ∃ 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:
obtain ⟨δ', δ'pos, hδ'⟩ : ∃ (δ' : ℝ), 0 < δ' ∧ ∀ B, dist B A < δ' → dist B.det A.det < ↑ε := by
      refine continuousAt_iff.1 ?_ ε εpos
      exact ContinuousLinearMap.continuous_det.continuousAt

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
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,
    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.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
⊢ (f.extend I).source = f.source

Tactic:
extend,

Goal state after:
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
⊢ (f.toPartialEquiv ≫ I.toPartialEquiv).source = f.source
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
⊢ (f.extend I).target = ↑I.symm ⁻¹' f.target ∩ range ↑I

Tactic:
extend,

Goal state after:
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
⊢ (f.toPartialEquiv ≫ I.toPartialEquiv).target = ↑I.symm ⁻¹' f.target ∩ range ↑I
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
⊢ (f.extend I).target = ↑I '' f.target

Tactic:
extend,

Goal state after:
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
⊢ (f.toPartialEquiv ≫ I.toPartialEquiv).target = ↑I '' f.target
Module: Mathlib.Geometry.Manifold.LocalInvariantProperties

Goal state before:
pp: H : Type u_1
H' : Type u_3
M' : Type u_4
X : Type u_5
inst✝⁴ : TopologicalSpace H
inst✝³ : TopologicalSpace H'
inst✝² : TopologicalSpace M'
inst✝¹ : ChartedSpace H' M'
inst✝ : TopologicalSpace X
G : StructureGroupoid H
G' : StructureGroupoid H'
f : OpenPartialHomeomorph M' H'
P : (H → H') → Set H → H → Prop
hG : G.LocalInvariantProp G' P
g : X → M'
e : OpenPartialHomeomorph X H
x : X
s : Set X
xe : x ∈ e.source
hf : f ∈ maximalAtlas M' G'
xf : g (↑e.symm (↑e x)) ∈ f.source
hgs : ContinuousWithinAt g s (↑e.symm (↑e x))
⊢ ContinuousWithinAt (g ∘ ↑e.symm) (↑e.symm ⁻¹' s) (↑e x)

Tactic:
exact hgs.comp (e.symm.continuousAt <| e.mapsTo xe).continuousWithinAt Subset.rfl

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝³ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace 𝕜 E
H : Type u_3
inst✝ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
⊢ range ↑I ⊆ closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (interior (range ↑I))

Tactic:
by_cases h : IsRCLikeNormedField 𝕜
  · let := h.rclike 𝕜
    let := NormedSpace.restrictScalars ℝ 𝕜 E
    rw [Convex.closure_interior_eq_closure_of_nonempty_interior (𝕜 := ℝ)]
    · apply subset_closure
    · apply I.convex_range
    · apply I.nonempty_interior
  · simp [range_eq_univ_of_not_isRCLikeNormedField I h]

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

Goal state before:
pp: H : Type u_1
M : Type u_2
H' : Type u_3
M' : Type u_4
inst✝⁵ : TopologicalSpace H
inst✝⁴ : TopologicalSpace M
inst✝³ : ChartedSpace H M
inst✝² : TopologicalSpace H'
inst✝¹ : TopologicalSpace M'
inst✝ : ChartedSpace H' M'
G : StructureGroupoid H
G' : StructureGroupoid H'
e : OpenPartialHomeomorph M H
f : OpenPartialHomeomorph M' H'
P : (H → H') → Set H → H → Prop
g : M → M'
s : Set M
x : M
hG : G.LocalInvariantProp G' P
he : e ∈ maximalAtlas M G
xe : x ∈ e.source
hf : f ∈ maximalAtlas M' G'
xf : g x ∈ f.source
h : ContinuousWithinAt g s x ∧ P (↑f ∘ g ∘ ↑e.symm) (↑e.symm ⁻¹' s) (↑e x)
h1 : ContinuousWithinAt (g ∘ ↑e.symm) (↑e.symm ⁻¹' s) (↑e.symm.symm x)
⊢ ContinuousAt (↑f) (g x)

Tactic:
f.continuousAt xf

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

Goal state before:
pp: H : Type u_1
M : Type u_2
H' : Type u_3
M' : Type u_4
inst✝⁵ : TopologicalSpace H
inst✝⁴ : TopologicalSpace M
inst✝³ : ChartedSpace H M
inst✝² : TopologicalSpace H'
inst✝¹ : TopologicalSpace M'
inst✝ : ChartedSpace H' M'
G : StructureGroupoid H
G' : StructureGroupoid H'
P : (H → H') → Set H → H → Prop
g : M → M'
s t : Set M
x : M
hG : G.LocalInvariantProp G' P
ht : t ∈ 𝓝[s] x
⊢ LiftPropWithinAt P g (s ∩ t) x ↔ LiftPropWithinAt P g s x

Tactic:
rw [liftPropWithinAt_iff', liftPropWithinAt_iff', continuousWithinAt_inter' ht, hG.congr_set]
  simp_rw [eventuallyEq_set, mem_preimage,
    (chartAt _ x).eventually_nhds' (fun x ↦ x ∈ s ∩ t ↔ x ∈ s) (mem_chart_source _ x)]
  exact (mem_nhdsWithin_iff_eventuallyEq.mp ht).symm.mem_iff

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

Goal state before:
pp: H : Type u_1
M : Type u_2
H' : Type u_3
M' : Type u_4
inst✝⁵ : TopologicalSpace H
inst✝⁴ : TopologicalSpace M
inst✝³ : ChartedSpace H M
inst✝² : TopologicalSpace H'
inst✝¹ : TopologicalSpace M'
inst✝ : ChartedSpace H' M'
P : (H → H') → Set H → H → Prop
g : M → M'
s : Set M
mono : ∀ ⦃s : Set H⦄ ⦃x : H⦄ ⦃t : Set H⦄ ⦃f : H → H'⦄, t ⊆ s → P f s x → P f t x
h : LiftPropOn P g univ
⊢ LiftPropOn P g s

Tactic:
exact liftPropOn_mono mono h (subset_univ _)

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

Goal state before:
pp: m n : ℕ∞ω
𝕜 : Type u_1
inst✝³ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace 𝕜 E
H : Type u_3
inst✝ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
h : m ≤ n
⊢ ∀ (f : H → H) (s : Set H), (contDiffPregroupoid n I).property f s → (contDiffPregroupoid m I).property f s

Tactic:
intro f

Goal state after:
m n : ℕ∞ω
𝕜 : Type u_1
inst✝³ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace 𝕜 E
H : Type u_3
inst✝ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
h : m ≤ n
f : H → H
⊢ ∀ (s : Set H), (contDiffPregroupoid n I).property f s → (contDiffPregroupoid m I).property f s
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 : H → H
⊢ ∀ {e : OpenPartialHomeomorph H H},
    e ∈ G →
      x ∈ e.source →
        G.IsLocalStructomorphWithinAt f s x → G.IsLocalStructomorphWithinAt (f ∘ ↑e.symm) (↑e.symm ⁻¹' s) (↑e x)

Tactic:
e'

Goal state after:
H : Type u_1
inst✝¹ : TopologicalSpace H
G : StructureGroupoid H
inst✝ : ClosedUnderRestriction G
s : Set H
x : H
f : H → H
e' : OpenPartialHomeomorph H H
⊢ e' ∈ G →
    x ∈ e'.source →
      G.IsLocalStructomorphWithinAt f s x → G.IsLocalStructomorphWithinAt (f ∘ ↑e'.symm) (↑e'.symm ⁻¹' s) (↑e' x)
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:
hfgs

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
⊢ f x = g x → G.IsLocalStructomorphWithinAt f s x → G.IsLocalStructomorphWithinAt g s x
Module: Mathlib.MeasureTheory.Function.Jacobian

Goal state before:

Tactic:
ENNReal.ofReal |(f' x).det| ≤ ENNReal.ofReal (|(A n).det| + ε) :=
          ENNReal.ofReal_le_ofReal I
        _ = ENNReal.ofReal |(A n).det| + ε := by
          simp only [ENNReal.ofReal_add, abs_nonneg, NNReal.zero_le_coe, ENNReal.ofReal_coe_nnreal]

Goal state after: