awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Module: Mathlib.MeasureTheory.Function.Jacobian
Goal:
E : Type u_1
inst : NormedAddCommGroup E
inst_1 : NormedSpace ℝ E
inst_2 : FiniteDimensional ℝ E
s : Set E
f : E → E
inst_3 : MeasurableSpace E
inst_4 : BorelSpace E
μ : Measure E
inst_5 : μ.IsAddHaarMeasure
A : E →L[ℝ] E
δ : ℝ≥0
hf : ApproximatesLinearOn f A s δ
hs : MeasurableSet s
f' : E → E →L[ℝ] E
hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x
x : E
hx :
  Tendsto (β := ℝ≥0∞)
    (fun (r : ℝ) ↦
      HDiv.hDiv (α := ℝ≥0∞) (β := ℝ≥0∞) ((μ : (a : Set E) → ℝ≥0∞) (s ∩ closedBall x r) : ℝ≥0∞)
        ((μ : (a : Set E) → ℝ≥0∞) (closedBall x r) : ℝ≥0∞))
    (𝓝[>] 0) (𝓝 1)
xs : x ∈ s
z : E
suffices :
  ∀ (ε : ℝ), 0 < ε → norm (E := E) ((f' x - A : (a : E) → E) z : E) ≤ ((↑δ : ℝ) + ε) * (‖z‖ + ε) + ‖f' x - A‖ * ε
⊢ norm (E := E) ((f' x - A : (a : E) → E) z : E) ≤ (↑δ : ℝ) * ‖z‖

Tactic:
suffices H : ∀ ε, 0 < ε → ‖(f' x - A) z‖ ≤ (δ + ε) * (‖z‖ + ε) + ‖f' x - A‖ * ε by
    have :
      Tendsto (fun ε : ℝ => ((δ : ℝ) + ε) * (‖z‖ + ε) + ‖f' x - A‖ * ε) (𝓝[>] 0)
        (𝓝 ((δ + 0) * (‖z‖ + 0) + ‖f' x - A‖ * 0)) :=
      Tendsto.mono_left (Continuous.tendsto (by fun_prop) 0) nhdsWithin_le_nhds
    simp only [add_zero, mul_zero] at this
    apply le_of_tendsto_of_tendsto tendsto_const_nhds this
    filter_upwards [self_mem_nhdsWithin]
    exact H
-- Module: Mathlib.MeasureTheory.Function.Jacobian
Goal:
E : Type u_1
inst : NormedAddCommGroup E
inst_1 : NormedSpace ℝ E
inst_2 : FiniteDimensional ℝ E
s : Set E
f : E → E
inst_3 : MeasurableSpace E
inst_4 : BorelSpace E
μ : Measure E
inst_5 : μ.IsAddHaarMeasure
A : E →L[ℝ] E
δ : ℝ≥0
hf : ApproximatesLinearOn f A s δ
hs : MeasurableSet s
f' : E → E →L[ℝ] E
hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x
x : E
hx :
  Tendsto (β := ℝ≥0∞)
    (fun (r : ℝ) ↦
      HDiv.hDiv (α := ℝ≥0∞) (β := ℝ≥0∞) ((μ : (a : Set E) → ℝ≥0∞) (s ∩ closedBall x r) : ℝ≥0∞)
        ((μ : (a : Set E) → ℝ≥0∞) (closedBall x r) : ℝ≥0∞))
    (𝓝[>] 0) (𝓝 1)
xs : x ∈ s
z : E
ε : ℝ
εpos : 0 < ε
⊢ ∀ᶠ (r : ℝ) in 𝓝[>] 0, (s ∩ ({x} + r • closedBall z ε)).Nonempty

Tactic:
have B₁ : ∀ᶠ r in 𝓝[>] (0 : ℝ), (s ∩ ({x} + r • closedBall z ε)).Nonempty :=
    eventually_nonempty_inter_smul_of_density_one μ s x hx _ measurableSet_closedBall
      (measure_closedBall_pos μ z εpos).ne'
-- Module: Mathlib.MeasureTheory.Function.Jacobian
Goal:
E : Type u_1
inst : NormedAddCommGroup E
inst_1 : NormedSpace ℝ E
inst_2 : FiniteDimensional ℝ E
s : Set E
f : E → E
inst_3 : MeasurableSpace E
inst_4 : BorelSpace E
μ : Measure E
inst_5 : μ.IsAddHaarMeasure
A : E →L[ℝ] E
δ : ℝ≥0
hf : ApproximatesLinearOn f A s δ
hs : MeasurableSet s
f' : E → E →L[ℝ] E
hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x
x : E
hx :
  Tendsto (β := ℝ≥0∞)
    (fun (r : ℝ) ↦
      HDiv.hDiv (α := ℝ≥0∞) (β := ℝ≥0∞) ((μ : (a : Set E) → ℝ≥0∞) (s ∩ closedBall x r) : ℝ≥0∞)
        ((μ : (a : Set E) → ℝ≥0∞) (closedBall x r) : ℝ≥0∞))
    (𝓝[>] 0) (𝓝 1)
xs : x ∈ s
z : E
ε : ℝ
εpos : 0 < ε
B₁ : ∀ᶠ (r : ℝ) in 𝓝[>] 0, (s ∩ ({x} + r • closedBall z ε)).Nonempty
⊢ ∃ ρ > 0, ball x ρ ∩ s ⊆ {y : E | ‖HSub.hSub (β := E) (f y - f x) ((f' x : (a : E) → E) (y - x) : E)‖ ≤ ε * ‖y - x‖}

Tactic:
mem_nhdsWithin_iff.1 ((hf' x xs).isLittleO.def εpos)

Premises used: Metric.mem_nhdsWithin_iff, Iff.mp, Asymptotics.IsLittleO.def, HasFDerivWithinAt.isLittleO
-- Module: Mathlib.MeasureTheory.Function.Jacobian
Goal:
E : Type u_1
inst : NormedAddCommGroup E
inst_1 : NormedSpace ℝ E
inst_2 : FiniteDimensional ℝ E
s : Set E
f : E → E
inst_3 : MeasurableSpace E
inst_4 : BorelSpace E
μ : Measure E
inst_5 : μ.IsAddHaarMeasure
A : E →L[ℝ] E
δ : ℝ≥0
hf : ApproximatesLinearOn f A s δ
hs : MeasurableSet s
f' : E → E →L[ℝ] E
hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x
x : E
hx :
  Tendsto (β := ℝ≥0∞)
    (fun (r : ℝ) ↦
      HDiv.hDiv (α := ℝ≥0∞) (β := ℝ≥0∞) ((μ : (a : Set E) → ℝ≥0∞) (s ∩ closedBall x r) : ℝ≥0∞)
        ((μ : (a : Set E) → ℝ≥0∞) (closedBall x r) : ℝ≥0∞))
    (𝓝[>] 0) (𝓝 1)
xs : x ∈ s
z : E
ε : ℝ
εpos : 0 < ε
B₁ : ∀ᶠ (r : ℝ) in 𝓝[>] 0, (s ∩ ({x} + r • closedBall z ε)).Nonempty
obtain :
  ∀ (ρ : ℝ) (ρpos : ρ > 0)
    (hρ : ball x ρ ∩ s ⊆ {y : E | ‖HSub.hSub (β := E) (f y - f x) ((f' x : (a : E) → E) (y - x) : E)‖ ≤ ε * ‖y - x‖}),
    norm (E := E) ((f' x - A : (a : E) → E) z : E) ≤ ((↑δ : ℝ) + ε) * (‖z‖ + ε) + ‖f' x - A‖ * ε
⊢ norm (E := E) ((f' x - A : (a : E) → E) z : E) ≤ ((↑δ : ℝ) + ε) * (‖z‖ + ε) + ‖f' x - A‖ * ε

Tactic:
obtain ⟨ρ, ρpos, hρ⟩ :
    ∃ ρ > 0, ball x ρ ∩ s ⊆ {y : E | ‖f y - f x - (f' x) (y - x)‖ ≤ ε * ‖y - x‖} :=
    mem_nhdsWithin_iff.1 ((hf' x xs).isLittleO.def εpos)

Premises used: Metric.mem_nhdsWithin_iff, Iff.mp, Asymptotics.IsLittleO.def, HasFDerivWithinAt.isLittleO
-- Module: Mathlib.MeasureTheory.Function.Jacobian
Goal:
E : Type u_1
inst : NormedAddCommGroup E
inst_1 : NormedSpace ℝ E
inst_2 : FiniteDimensional ℝ E
s : Set E
f : E → E
inst_3 : MeasurableSpace E
inst_4 : BorelSpace E
μ : Measure E
inst_5 : μ.IsAddHaarMeasure
A : E →L[ℝ] E
δ : ℝ≥0
hf : ApproximatesLinearOn f A s δ
hs : MeasurableSet s
f' : E → E →L[ℝ] E
hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x
x : E
hx :
  Tendsto (β := ℝ≥0∞)
    (fun (r : ℝ) ↦
      HDiv.hDiv (α := ℝ≥0∞) (β := ℝ≥0∞) ((μ : (a : Set E) → ℝ≥0∞) (s ∩ closedBall x r) : ℝ≥0∞)
        ((μ : (a : Set E) → ℝ≥0∞) (closedBall x r) : ℝ≥0∞))
    (𝓝[>] 0) (𝓝 1)
xs : x ∈ s
z : E
ε : ℝ
εpos : 0 < ε
B₁ : ∀ᶠ (r : ℝ) in 𝓝[>] 0, (s ∩ ({x} + r • closedBall z ε)).Nonempty
ρ : ℝ
ρpos : ρ > 0
hρ : ball x ρ ∩ s ⊆ {y : E | ‖HSub.hSub (β := E) (f y - f x) ((f' x : (a : E) → E) (y - x) : E)‖ ≤ ε * ‖y - x‖}
a : Membership.mem (γ := Filter ℝ) (𝓝 0) {x_1 : ℝ | (fun (r : ℝ) ↦ {x} + r • closedBall z ε ⊆ ball x ρ) x_1}
⊢ ∀ᶠ (r : ℝ) in 𝓝[>] 0, {x} + r • closedBall z ε ⊆ ball x ρ

Tactic:
apply nhdsWithin_le_nhds

Premises used: nhdsWithin_le_nhds
-- Module: Mathlib.MeasureTheory.Function.Jacobian
Goal:
E : Type u_1
inst : NormedAddCommGroup E
inst_1 : NormedSpace ℝ E
inst_2 : FiniteDimensional ℝ E
s : Set E
f : E → E
inst_3 : MeasurableSpace E
inst_4 : BorelSpace E
μ : Measure E
inst_5 : μ.IsAddHaarMeasure
A : E →L[ℝ] E
δ : ℝ≥0
hf : ApproximatesLinearOn f A s δ
hs : MeasurableSet s
f' : E → E →L[ℝ] E
hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x
x : E
hx :
  Tendsto (β := ℝ≥0∞)
    (fun (r : ℝ) ↦
      HDiv.hDiv (α := ℝ≥0∞) (β := ℝ≥0∞) ((μ : (a : Set E) → ℝ≥0∞) (s ∩ closedBall x r) : ℝ≥0∞)
        ((μ : (a : Set E) → ℝ≥0∞) (closedBall x r) : ℝ≥0∞))
    (𝓝[>] 0) (𝓝 1)
xs : x ∈ s
z : E
ε : ℝ
εpos : 0 < ε
B₁ : ∀ᶠ (r : ℝ) in 𝓝[>] 0, (s ∩ ({x} + r • closedBall z ε)).Nonempty
ρ : ℝ
ρpos : ρ > 0
hρ : ball x ρ ∩ s ⊆ {y : E | ‖HSub.hSub (β := E) (f y - f x) ((f' x : (a : E) → E) (y - x) : E)‖ ≤ ε * ‖y - x‖}
⊢ Membership.mem (γ := Filter ℝ) (𝓝 0) {x_1 : ℝ | (fun (r : ℝ) ↦ {x} + r • closedBall z ε ⊆ ball x ρ) x_1}

Tactic:
eventually_singleton_add_smul_subset isBounded_closedBall (ball_mem_nhds x ρpos)

Premises used: Metric.isBounded_closedBall, Set.Mem, Filter.Eventually, setOf, eventually_singleton_add_smul_subset, Metric.ball_mem_nhds
-- Module: Mathlib.MeasureTheory.Function.Jacobian
Goal:
E : Type u_1
inst : NormedAddCommGroup E
inst_1 : NormedSpace ℝ E
inst_2 : FiniteDimensional ℝ E
s : Set E
f : E → E
inst_3 : MeasurableSpace E
inst_4 : BorelSpace E
μ : Measure E
inst_5 : μ.IsAddHaarMeasure
A : E →L[ℝ] E
δ : ℝ≥0
hf : ApproximatesLinearOn f A s δ
hs : MeasurableSet s
f' : E → E →L[ℝ] E
hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x
x : E
hx :
  Tendsto (β := ℝ≥0∞)
    (fun (r : ℝ) ↦
      HDiv.hDiv (α := ℝ≥0∞) (β := ℝ≥0∞) ((μ : (a : Set E) → ℝ≥0∞) (s ∩ closedBall x r) : ℝ≥0∞)
        ((μ : (a : Set E) → ℝ≥0∞) (closedBall x r) : ℝ≥0∞))
    (𝓝[>] 0) (𝓝 1)
xs : x ∈ s
z : E
ε : ℝ
εpos : 0 < ε
B₁ : ∀ᶠ (r : ℝ) in 𝓝[>] 0, (s ∩ ({x} + r • closedBall z ε)).Nonempty
ρ : ℝ
ρpos : ρ > 0
hρ : ball x ρ ∩ s ⊆ {y : E | ‖HSub.hSub (β := E) (f y - f x) ((f' x : (a : E) → E) (y - x) : E)‖ ≤ ε * ‖y - x‖}
⊢ Membership.mem (γ := Filter ℝ) (𝓝 0) {x_1 : ℝ | (fun (r : ℝ) ↦ {x} + r • closedBall z ε ⊆ ball x ρ) x_1}

Tactic:
exact eventually_singleton_add_smul_subset isBounded_closedBall (ball_mem_nhds x ρpos)

Premises used: Metric.isBounded_closedBall, eventually_singleton_add_smul_subset, Metric.ball_mem_nhds
-- Module: Mathlib.MeasureTheory.Function.Jacobian
Goal:
E : Type u_1
inst : NormedAddCommGroup E
inst_1 : NormedSpace ℝ E
inst_2 : FiniteDimensional ℝ E
s : Set E
f : E → E
inst_3 : MeasurableSpace E
inst_4 : BorelSpace E
μ : Measure E
inst_5 : μ.IsAddHaarMeasure
A : E →L[ℝ] E
δ : ℝ≥0
hf : ApproximatesLinearOn f A s δ
hs : MeasurableSet s
f' : E → E →L[ℝ] E
hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x
x : E
hx :
  Tendsto (β := ℝ≥0∞)
    (fun (r : ℝ) ↦
      HDiv.hDiv (α := ℝ≥0∞) (β := ℝ≥0∞) ((μ : (a : Set E) → ℝ≥0∞) (s ∩ closedBall x r) : ℝ≥0∞)
        ((μ : (a : Set E) → ℝ≥0∞) (closedBall x r) : ℝ≥0∞))
    (𝓝[>] 0) (𝓝 1)
xs : x ∈ s
z : E
ε : ℝ
εpos : 0 < ε
B₁ : ∀ᶠ (r : ℝ) in 𝓝[>] 0, (s ∩ ({x} + r • closedBall z ε)).Nonempty
ρ : ℝ
ρpos : ρ > 0
hρ : ball x ρ ∩ s ⊆ {y : E | ‖HSub.hSub (β := E) (f y - f x) ((f' x : (a : E) → E) (y - x) : E)‖ ≤ ε * ‖y - x‖}
⊢ ∀ᶠ (r : ℝ) in 𝓝[>] 0, {x} + r • closedBall z ε ⊆ ball x ρ

Tactic:
have B₂ : ∀ᶠ r in 𝓝[>] (0 : ℝ), {x} + r • closedBall z ε ⊆ ball x ρ := by
    apply nhdsWithin_le_nhds
    exact eventually_singleton_add_smul_subset isBounded_closedBall (ball_mem_nhds x ρpos)
-- Module: Mathlib.MeasureTheory.Function.Jacobian
Goal:
E : Type u_1
inst : NormedAddCommGroup E
inst_1 : NormedSpace ℝ E
inst_2 : FiniteDimensional ℝ E
s : Set E
f : E → E
inst_3 : MeasurableSpace E
inst_4 : BorelSpace E
μ : Measure E
inst_5 : μ.IsAddHaarMeasure
A : E →L[ℝ] E
δ : ℝ≥0
hf : ApproximatesLinearOn f A s δ
hs : MeasurableSet s
f' : E → E →L[ℝ] E
hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x
x : E
hx :
  Tendsto (β := ℝ≥0∞)
    (fun (r : ℝ) ↦
      HDiv.hDiv (α := ℝ≥0∞) (β := ℝ≥0∞) ((μ : (a : Set E) → ℝ≥0∞) (s ∩ closedBall x r) : ℝ≥0∞)
        ((μ : (a : Set E) → ℝ≥0∞) (closedBall x r) : ℝ≥0∞))
    (𝓝[>] 0) (𝓝 1)
xs : x ∈ s
z : E
ε : ℝ
εpos : 0 < ε
B₁ : ∀ᶠ (r : ℝ) in 𝓝[>] 0, (s ∩ ({x} + r • closedBall z ε)).Nonempty
ρ : ℝ
ρpos : ρ > 0
hρ : ball x ρ ∩ s ⊆ {y : E | ‖HSub.hSub (β := E) (f y - f x) ((f' x : (a : E) → E) (y - x) : E)‖ ≤ ε * ‖y - x‖}
B₂ : ∀ᶠ (r : ℝ) in 𝓝[>] 0, {x} + r • closedBall z ε ⊆ ball x ρ
⊢ ∃ (r : ℝ), (s ∩ ({x} + r • closedBall z ε)).Nonempty ∧ {x} + r • closedBall z ε ⊆ ball x ρ ∧ 0 < r

Tactic:
(B₁.and (B₂.and self_mem_nhdsWithin)).exists

Premises used: Filter.Eventually.and, Filter.Eventually.exists, self_mem_nhdsWithin
-- Module: Mathlib.MeasureTheory.Function.Jacobian
Goal:
E : Type u_1
inst : NormedAddCommGroup E
inst_1 : NormedSpace ℝ E
inst_2 : FiniteDimensional ℝ E
s : Set E
f : E → E
inst_3 : MeasurableSpace E
inst_4 : BorelSpace E
μ : Measure E
inst_5 : μ.IsAddHaarMeasure
A : E →L[ℝ] E
δ : ℝ≥0
hf : ApproximatesLinearOn f A s δ
hs : MeasurableSet s
f' : E → E →L[ℝ] E
hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x
x : E
hx :
  Tendsto (β := ℝ≥0∞)
    (fun (r : ℝ) ↦
      HDiv.hDiv (α := ℝ≥0∞) (β := ℝ≥0∞) ((μ : (a : Set E) → ℝ≥0∞) (s ∩ closedBall x r) : ℝ≥0∞)
        ((μ : (a : Set E) → ℝ≥0∞) (closedBall x r) : ℝ≥0∞))
    (𝓝[>] 0) (𝓝 1)
xs : x ∈ s
z : E
ε : ℝ
εpos : 0 < ε
B₁ : ∀ᶠ (r : ℝ) in 𝓝[>] 0, (s ∩ ({x} + r • closedBall z ε)).Nonempty
ρ : ℝ
ρpos : ρ > 0
hρ : ball x ρ ∩ s ⊆ {y : E | ‖HSub.hSub (β := E) (f y - f x) ((f' x : (a : E) → E) (y - x) : E)‖ ≤ ε * ‖y - x‖}
B₂ : ∀ᶠ (r : ℝ) in 𝓝[>] 0, {x} + r • closedBall z ε ⊆ ball x ρ
obtain :
  ∀ (r : ℝ) (y : E) (ys : y ∈ s) (hy : y ∈ {x} + r • closedBall z ε) (rρ : {x} + r • closedBall z ε ⊆ ball x ρ)
    (rpos : 0 < r), norm (E := E) ((f' x - A : (a : E) → E) z : E) ≤ ((↑δ : ℝ) + ε) * (‖z‖ + ε) + ‖f' x - A‖ * ε
⊢ norm (E := E) ((f' x - A : (a : E) → E) z : E) ≤ ((↑δ : ℝ) + ε) * (‖z‖ + ε) + ‖f' x - A‖ * ε

Tactic:
obtain ⟨r, ⟨y, ⟨ys, hy⟩⟩, rρ, rpos⟩ :
    ∃ r : ℝ,
      (s ∩ ({x} + r • closedBall z ε)).Nonempty ∧ {x} + r • closedBall z ε ⊆ ball x ρ ∧ 0 < r :=
    (B₁.and (B₂.and self_mem_nhdsWithin)).exists

Premises used: Filter.Eventually.and, Metric.ball, Filter.Eventually.exists, self_mem_nhdsWithin, Metric.closedBall
-- Module: Mathlib.MeasureTheory.Function.Jacobian
Goal:
E : Type u_1
inst : NormedAddCommGroup E
inst_1 : NormedSpace ℝ E
inst_2 : FiniteDimensional ℝ E
s : Set E
f : E → E
inst_3 : MeasurableSpace E
inst_4 : BorelSpace E
μ : Measure E
inst_5 : μ.IsAddHaarMeasure
A : E →L[ℝ] E
δ : ℝ≥0
hf : ApproximatesLinearOn f A s δ
hs : MeasurableSet s
f' : E → E →L[ℝ] E
hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x
x : E
hx :
  Tendsto (β := ℝ≥0∞)
    (fun (r : ℝ) ↦
      HDiv.hDiv (α := ℝ≥0∞) (β := ℝ≥0∞) ((μ : (a : Set E) → ℝ≥0∞) (s ∩ closedBall x r) : ℝ≥0∞)
        ((μ : (a : Set E) → ℝ≥0∞) (closedBall x r) : ℝ≥0∞))
    (𝓝[>] 0) (𝓝 1)
xs : x ∈ s
z : E
ε : ℝ
εpos : 0 < ε
B₁ : ∀ᶠ (r : ℝ) in 𝓝[>] 0, (s ∩ ({x} + r • closedBall z ε)).Nonempty
ρ : ℝ
ρpos : ρ > 0
hρ : ball x ρ ∩ s ⊆ {y : E | ‖HSub.hSub (β := E) (f y - f x) ((f' x : (a : E) → E) (y - x) : E)‖ ≤ ε * ‖y - x‖}
B₂ : ∀ᶠ (r : ℝ) in 𝓝[>] 0, {x} + r • closedBall z ε ⊆ ball x ρ
r : ℝ
y : E
ys : y ∈ s
hy : y ∈ {x} + r • closedBall z ε
rρ : {x} + r • closedBall z ε ⊆ ball x ρ
rpos : 0 < r
⊢ ∃ y_1 ∈ closedBall z ε, r • y_1 = -x + y

Tactic:
simp only [mem_smul_set, image_add_left, mem_preimage, singleton_add] at hy

Premises used: Set.mem_smul_set, Set.image_add_left, Set.singleton_add, Set.mem_preimage
-- Module: Mathlib.MeasureTheory.Function.Jacobian
Goal:
E : Type u_1
inst : NormedAddCommGroup E
inst_1 : NormedSpace ℝ E
inst_2 : FiniteDimensional ℝ E
s : Set E
f : E → E
inst_3 : MeasurableSpace E
inst_4 : BorelSpace E
μ : Measure E
inst_5 : μ.IsAddHaarMeasure
A : E →L[ℝ] E
δ : ℝ≥0
hf : ApproximatesLinearOn f A s δ
hs : MeasurableSet s
f' : E → E →L[ℝ] E
hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x
x : E
hx :
  Tendsto (β := ℝ≥0∞)
    (fun (r : ℝ) ↦
      HDiv.hDiv (α := ℝ≥0∞) (β := ℝ≥0∞) ((μ : (a : Set E) → ℝ≥0∞) (s ∩ closedBall x r) : ℝ≥0∞)
        ((μ : (a : Set E) → ℝ≥0∞) (closedBall x r) : ℝ≥0∞))
    (𝓝[>] 0) (𝓝 1)
xs : x ∈ s
z : E
ε : ℝ
εpos : 0 < ε
B₁ : ∀ᶠ (r : ℝ) in 𝓝[>] 0, (s ∩ ({x} + r • closedBall z ε)).Nonempty
ρ : ℝ
ρpos : ρ > 0
hρ : ball x ρ ∩ s ⊆ {y : E | ‖HSub.hSub (β := E) (f y - f x) ((f' x : (a : E) → E) (y - x) : E)‖ ≤ ε * ‖y - x‖}
B₂ : ∀ᶠ (r : ℝ) in 𝓝[>] 0, {x} + r • closedBall z ε ⊆ ball x ρ
r : ℝ
y : E
ys : y ∈ s
rρ : {x} + r • closedBall z ε ⊆ ball x ρ
rpos : 0 < r
hy : ∃ y_1 ∈ closedBall z ε, r • y_1 = -x + y
rcases : ∀ (a : E) (az : a ∈ closedBall z ε) (ha : r • a = -x + y), ∃ a ∈ closedBall z ε, y = x + r • a
⊢ ∃ a ∈ closedBall z ε, y = x + r • a

Tactic:
rcases hy with ⟨a, az, ha⟩
-- Module: Mathlib.MeasureTheory.Function.Jacobian
Goal:
E : Type u_1
inst : NormedAddCommGroup E
inst_1 : NormedSpace ℝ E
inst_2 : FiniteDimensional ℝ E
s : Set E
f : E → E
inst_3 : MeasurableSpace E
inst_4 : BorelSpace E
μ : Measure E
inst_5 : μ.IsAddHaarMeasure
A : E →L[ℝ] E
δ : ℝ≥0
hf : ApproximatesLinearOn f A s δ
hs : MeasurableSet s
f' : E → E →L[ℝ] E
hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x
x : E
hx :
  Tendsto (β := ℝ≥0∞)
    (fun (r : ℝ) ↦
      HDiv.hDiv (α := ℝ≥0∞) (β := ℝ≥0∞) ((μ : (a : Set E) → ℝ≥0∞) (s ∩ closedBall x r) : ℝ≥0∞)
        ((μ : (a : Set E) → ℝ≥0∞) (closedBall x r) : ℝ≥0∞))
    (𝓝[>] 0) (𝓝 1)
xs : x ∈ s
z : E
ε : ℝ
εpos : 0 < ε
B₁ : ∀ᶠ (r : ℝ) in 𝓝[>] 0, (s ∩ ({x} + r • closedBall z ε)).Nonempty
ρ : ℝ
ρpos : ρ > 0
hρ : ball x ρ ∩ s ⊆ {y : E | ‖HSub.hSub (β := E) (f y - f x) ((f' x : (a : E) → E) (y - x) : E)‖ ≤ ε * ‖y - x‖}
B₂ : ∀ᶠ (r : ℝ) in 𝓝[>] 0, {x} + r • closedBall z ε ⊆ ball x ρ
r : ℝ
y : E
ys : y ∈ s
rρ : {x} + r • closedBall z ε ⊆ ball x ρ
rpos : 0 < r
a : E
az : a ∈ closedBall z ε
ha : r • a = -x + y
⊢ y = x + r • a

Tactic:
simp only [ha, add_neg_cancel_left]

Premises used: add_neg_cancel_left
-- Module: Mathlib.MeasureTheory.Function.Jacobian
Goal:
E : Type u_1
inst : NormedAddCommGroup E
inst_1 : NormedSpace ℝ E
inst_2 : FiniteDimensional ℝ E
s : Set E
f : E → E
inst_3 : MeasurableSpace E
inst_4 : BorelSpace E
μ : Measure E
inst_5 : μ.IsAddHaarMeasure
A : E →L[ℝ] E
δ : ℝ≥0
hf : ApproximatesLinearOn f A s δ
hs : MeasurableSet s
f' : E → E →L[ℝ] E
hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x
x : E
hx :
  Tendsto (β := ℝ≥0∞)
    (fun (r : ℝ) ↦
      HDiv.hDiv (α := ℝ≥0∞) (β := ℝ≥0∞) ((μ : (a : Set E) → ℝ≥0∞) (s ∩ closedBall x r) : ℝ≥0∞)
        ((μ : (a : Set E) → ℝ≥0∞) (closedBall x r) : ℝ≥0∞))
    (𝓝[>] 0) (𝓝 1)
xs : x ∈ s
z : E
ε : ℝ
εpos : 0 < ε
B₁ : ∀ᶠ (r : ℝ) in 𝓝[>] 0, (s ∩ ({x} + r • closedBall z ε)).Nonempty
ρ : ℝ
ρpos : ρ > 0
hρ : ball x ρ ∩ s ⊆ {y : E | ‖HSub.hSub (β := E) (f y - f x) ((f' x : (a : E) → E) (y - x) : E)‖ ≤ ε * ‖y - x‖}
B₂ : ∀ᶠ (r : ℝ) in 𝓝[>] 0, {x} + r • closedBall z ε ⊆ ball x ρ
r : ℝ
y : E
ys : y ∈ s
rρ : {x} + r • closedBall z ε ⊆ ball x ρ
rpos : 0 < r
a : E
az : a ∈ closedBall z ε
ha : r • a = -x + y
⊢ ∃ a ∈ closedBall z ε, y = x + r • a

Tactic:
exact ⟨a, az, by simp only [ha, add_neg_cancel_left]⟩
-- Module: Mathlib.MeasureTheory.Function.Jacobian
Goal:
E : Type u_1
inst : NormedAddCommGroup E
inst_1 : NormedSpace ℝ E
inst_2 : FiniteDimensional ℝ E
s : Set E
f : E → E
inst_3 : MeasurableSpace E
inst_4 : BorelSpace E
μ : Measure E
inst_5 : μ.IsAddHaarMeasure
A : E →L[ℝ] E
δ : ℝ≥0
hf : ApproximatesLinearOn f A s δ
hs : MeasurableSet s
f' : E → E →L[ℝ] E
hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x
x : E
hx :
  Tendsto (β := ℝ≥0∞)
    (fun (r : ℝ) ↦
      HDiv.hDiv (α := ℝ≥0∞) (β := ℝ≥0∞) ((μ : (a : Set E) → ℝ≥0∞) (s ∩ closedBall x r) : ℝ≥0∞)
        ((μ : (a : Set E) → ℝ≥0∞) (closedBall x r) : ℝ≥0∞))
    (𝓝[>] 0) (𝓝 1)
xs : x ∈ s
z : E
ε : ℝ
εpos : 0 < ε
B₁ : ∀ᶠ (r : ℝ) in 𝓝[>] 0, (s ∩ ({x} + r • closedBall z ε)).Nonempty
ρ : ℝ
ρpos : ρ > 0
hρ : ball x ρ ∩ s ⊆ {y : E | ‖HSub.hSub (β := E) (f y - f x) ((f' x : (a : E) → E) (y - x) : E)‖ ≤ ε * ‖y - x‖}
B₂ : ∀ᶠ (r : ℝ) in 𝓝[>] 0, {x} + r • closedBall z ε ⊆ ball x ρ
r : ℝ
y : E
ys : y ∈ s
hy : y ∈ {x} + r • closedBall z ε
rρ : {x} + r • closedBall z ε ⊆ ball x ρ
rpos : 0 < r
obtain :
  ∀ (a : E) (az : a ∈ closedBall z ε) (ya : y = x + r • a),
    norm (E := E) ((f' x - A : (a : E) → E) z : E) ≤ ((↑δ : ℝ) + ε) * (‖z‖ + ε) + ‖f' x - A‖ * ε
⊢ norm (E := E) ((f' x - A : (a : E) → E) z : E) ≤ ((↑δ : ℝ) + ε) * (‖z‖ + ε) + ‖f' x - A‖ * ε

Tactic:
obtain ⟨a, az, ya⟩ : ∃ a, a ∈ closedBall z ε ∧ y = x + r • a := by
    simp only [mem_smul_set, image_add_left, mem_preimage, singleton_add] at hy
    rcases hy with ⟨a, az, ha⟩
    exact ⟨a, az, by simp only [ha, add_neg_cancel_left]⟩

Premises used: Metric.closedBall
-- Module: Mathlib.MeasureTheory.Function.Jacobian
Goal:
E : Type u_1
inst : NormedAddCommGroup E
inst_1 : NormedSpace ℝ E
inst_2 : FiniteDimensional ℝ E
s : Set E
f : E → E
inst_3 : MeasurableSpace E
inst_4 : BorelSpace E
μ : Measure E
inst_5 : μ.IsAddHaarMeasure
A : E →L[ℝ] E
δ : ℝ≥0
hf : ApproximatesLinearOn f A s δ
hs : MeasurableSet s
f' : E → E →L[ℝ] E
hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x
x : E
hx :
  Tendsto (β := ℝ≥0∞)
    (fun (r : ℝ) ↦
      HDiv.hDiv (α := ℝ≥0∞) (β := ℝ≥0∞) ((μ : (a : Set E) → ℝ≥0∞) (s ∩ closedBall x r) : ℝ≥0∞)
        ((μ : (a : Set E) → ℝ≥0∞) (closedBall x r) : ℝ≥0∞))
    (𝓝[>] 0) (𝓝 1)
xs : x ∈ s
z : E
ε : ℝ
εpos : 0 < ε
B₁ : ∀ᶠ (r : ℝ) in 𝓝[>] 0, (s ∩ ({x} + r • closedBall z ε)).Nonempty
ρ : ℝ
ρpos : ρ > 0
hρ : ball x ρ ∩ s ⊆ {y : E | ‖HSub.hSub (β := E) (f y - f x) ((f' x : (a : E) → E) (y - x) : E)‖ ≤ ε * ‖y - x‖}
B₂ : ∀ᶠ (r : ℝ) in 𝓝[>] 0, {x} + r • closedBall z ε ⊆ ball x ρ
r : ℝ
y : E
ys : y ∈ s
hy : y ∈ {x} + r • closedBall z ε
rρ : {x} + r • closedBall z ε ⊆ ball x ρ
rpos : 0 < r
a : E
az : a ∈ closedBall z ε
ya : y = x + r • a
⊢ ‖a‖ = ‖z + (a - z)‖

Tactic:
simp only [add_sub_cancel]

Premises used: add_sub_cancel
-- Module: Mathlib.MeasureTheory.Function.Jacobian
Goal:
E : Type u_1
inst : NormedAddCommGroup E
inst_1 : NormedSpace ℝ E
inst_2 : FiniteDimensional ℝ E
s : Set E
f : E → E
inst_3 : MeasurableSpace E
inst_4 : BorelSpace E
μ : Measure E
inst_5 : μ.IsAddHaarMeasure
A : E →L[ℝ] E
δ : ℝ≥0
hf : ApproximatesLinearOn f A s δ
hs : MeasurableSet s
f' : E → E →L[ℝ] E
hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x
x : E
hx :
  Tendsto (β := ℝ≥0∞)
    (fun (r : ℝ) ↦
      HDiv.hDiv (α := ℝ≥0∞) (β := ℝ≥0∞) ((μ : (a : Set E) → ℝ≥0∞) (s ∩ closedBall x r) : ℝ≥0∞)
        ((μ : (a : Set E) → ℝ≥0∞) (closedBall x r) : ℝ≥0∞))
    (𝓝[>] 0) (𝓝 1)
xs : x ∈ s
z : E
ε : ℝ
εpos : 0 < ε
B₁ : ∀ᶠ (r : ℝ) in 𝓝[>] 0, (s ∩ ({x} + r • closedBall z ε)).Nonempty
ρ : ℝ
ρpos : ρ > 0
hρ : ball x ρ ∩ s ⊆ {y : E | ‖HSub.hSub (β := E) (f y - f x) ((f' x : (a : E) → E) (y - x) : E)‖ ≤ ε * ‖y - x‖}
B₂ : ∀ᶠ (r : ℝ) in 𝓝[>] 0, {x} + r • closedBall z ε ⊆ ball x ρ
r : ℝ
y : E
ys : y ∈ s
hy : y ∈ {x} + r • closedBall z ε
rρ : {x} + r • closedBall z ε ⊆ ball x ρ
rpos : 0 < r
a : E
az : a ∈ closedBall z ε
ya : y = x + r • a
grw grw₁ : ‖z‖ + ε ≤ ‖z‖ + ε
⊢ ‖z‖ + ‖a - z‖ ≤ ‖z‖ + ε

Tactic:
grw [mem_closedBall_iff_norm.1 az]

Premises used: mem_closedBall_iff_norm, Iff.mp
-- Module: Mathlib.MeasureTheory.Function.Jacobian
Goal:
E : Type u_1
inst : NormedAddCommGroup E
inst_1 : NormedSpace ℝ E
inst_2 : FiniteDimensional ℝ E
s : Set E
f : E → E
inst_3 : MeasurableSpace E
inst_4 : BorelSpace E
μ : Measure E
inst_5 : μ.IsAddHaarMeasure
A : E →L[ℝ] E
δ : ℝ≥0
hf : ApproximatesLinearOn f A s δ
hs : MeasurableSet s
f' : E → E →L[ℝ] E
hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x
x : E
hx :
  Tendsto (β := ℝ≥0∞)
    (fun (r : ℝ) ↦
      HDiv.hDiv (α := ℝ≥0∞) (β := ℝ≥0∞) ((μ : (a : Set E) → ℝ≥0∞) (s ∩ closedBall x r) : ℝ≥0∞)
        ((μ : (a : Set E) → ℝ≥0∞) (closedBall x r) : ℝ≥0∞))
    (𝓝[>] 0) (𝓝 1)
xs : x ∈ s
z : E
ε : ℝ
εpos : 0 < ε
B₁ : ∀ᶠ (r : ℝ) in 𝓝[>] 0, (s ∩ ({x} + r • closedBall z ε)).Nonempty
ρ : ℝ
ρpos : ρ > 0
hρ : ball x ρ ∩ s ⊆ {y : E | ‖HSub.hSub (β := E) (f y - f x) ((f' x : (a : E) → E) (y - x) : E)‖ ≤ ε * ‖y - x‖}
B₂ : ∀ᶠ (r : ℝ) in 𝓝[>] 0, {x} + r • closedBall z ε ⊆ ball x ρ
r : ℝ
y : E
ys : y ∈ s
hy : y ∈ {x} + r • closedBall z ε
rρ : {x} + r • closedBall z ε ⊆ ball x ρ
rpos : 0 < r
a : E
az : a ∈ closedBall z ε
ya : y = x + r • a
⊢ ‖a‖ ≤ ‖z‖ + ε

Tactic:
have norm_a : ‖a‖ ≤ ‖z‖ + ε :=
    calc
      ‖a‖ = ‖z + (a - z)‖ := by simp only [add_sub_cancel]
      _ ≤ ‖z‖ + ‖a - z‖ := norm_add_le _ _
      _ ≤ ‖z‖ + ε := by grw [mem_closedBall_iff_norm.1 az]
-- Module: Mathlib.MeasureTheory.Function.Jacobian
Goal:
E : Type u_1
inst : NormedAddCommGroup E
inst_1 : NormedSpace ℝ E
inst_2 : FiniteDimensional ℝ E
s : Set E
f : E → E
inst_3 : MeasurableSpace E
inst_4 : BorelSpace E
μ : Measure E
inst_5 : μ.IsAddHaarMeasure
A : E →L[ℝ] E
δ : ℝ≥0
hf : ApproximatesLinearOn f A s δ
hs : MeasurableSet s
f' : E → E →L[ℝ] E
hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x
x : E
hx :
  Tendsto (β := ℝ≥0∞)
    (fun (r : ℝ) ↦
      HDiv.hDiv (α := ℝ≥0∞) (β := ℝ≥0∞) ((μ : (a : Set E) → ℝ≥0∞) (s ∩ closedBall x r) : ℝ≥0∞)
        ((μ : (a : Set E) → ℝ≥0∞) (closedBall x r) : ℝ≥0∞))
    (𝓝[>] 0) (𝓝 1)
xs : x ∈ s
z : E
ε : ℝ
εpos : 0 < ε
B₁ : ∀ᶠ (r : ℝ) in 𝓝[>] 0, (s ∩ ({x} + r • closedBall z ε)).Nonempty
ρ : ℝ
ρpos : ρ > 0
hρ : ball x ρ ∩ s ⊆ {y : E | ‖HSub.hSub (β := E) (f y - f x) ((f' x : (a : E) → E) (y - x) : E)‖ ≤ ε * ‖y - x‖}
B₂ : ∀ᶠ (r : ℝ) in 𝓝[>] 0, {x} + r • closedBall z ε ⊆ ball x ρ
r : ℝ
y : E
ys : y ∈ s
hy : y ∈ {x} + r • closedBall z ε
rρ : {x} + r • closedBall z ε ⊆ ball x ρ
rpos : 0 < r
a : E
az : a ∈ closedBall z ε
ya : y = x + r • a
norm_a : ‖a‖ ≤ ‖z‖ + ε
⊢ r * norm (E := E) ((f' x - A : (a : E) → E) a : E) = norm (E := E) ((f' x - A : (a : E) → E) (r • a) : E)

Tactic:
simp only [map_smul, norm_smul, Real.norm_eq_abs, abs_of_nonneg rpos.le]

Premises used: norm_smul, LT.lt.le, abs_of_nonneg, Real.norm_eq_abs, map_smul