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

Goal state before:
pp: E : Type u_1
F : Type u_2
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : FiniteDimensional ℝ E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace ℝ F
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
inst✝ : SecondCountableTopology F
f : E → F
s : Set E
f' : E → E →L[ℝ] F
hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x
r : (E →L[ℝ] F) → ℝ≥0
rpos : ∀ (A : E →L[ℝ] F), r A ≠ 0
⊢ ∃ t A,
    Pairwise (Disjoint on t) ∧
      (∀ (n : ℕ), MeasurableSet (t n)) ∧
        s ⊆ ⋃ n, t n ∧
          (∀ (n : ℕ), ApproximatesLinearOn f (A n) (s ∩ t n) (r (A n))) ∧ (s.Nonempty → ∀ (n : ℕ), ∃ y ∈ s, A n = f' y)

Tactic:
rcases exists_closed_cover_approximatesLinearOn_of_hasFDerivWithinAt f s f' hf' r rpos with
    ⟨t, A, t_closed, st, t_approx, ht⟩

Goal state after:
E : Type u_1
F : Type u_2
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : FiniteDimensional ℝ E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace ℝ F
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
inst✝ : SecondCountableTopology F
f : E → F
s : Set E
f' : E → E →L[ℝ] F
hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x
r : (E →L[ℝ] F) → ℝ≥0
rpos : ∀ (A : E →L[ℝ] F), r A ≠ 0
t : ℕ → Set E
A : ℕ → E →L[ℝ] F
t_closed : ∀ (n : ℕ), IsClosed[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (t n)
st : s ⊆ ⋃ n, t n
t_approx : ∀ (n : ℕ), ApproximatesLinearOn f (A n) (s ∩ t n) (r (A n))
ht : s.Nonempty → ∀ (n : ℕ), ∃ y ∈ s, A n = f' y
⊢ ∃ t A,
    Pairwise (Disjoint on t) ∧
      (∀ (n : ℕ), MeasurableSet (t n)) ∧
        s ⊆ ⋃ n, t n ∧
          (∀ (n : ℕ), ApproximatesLinearOn f (A n) (s ∩ t n) (r (A n))) ∧ (s.Nonempty → ∀ (n : ℕ), ∃ y ∈ s, A n = f' y)
Module: Mathlib.MeasureTheory.Integral.IntervalIntegral.IntegrationByParts

Goal state before:
pp: case inl.hff'
a b : ℝ
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f f' : ℝ → ℝ
g : ℝ → E
hf : ContinuousOn f [[a, b]]
hff' : ∀ x ∈ Ioo (min a b) (max a b), HasDerivAt f (f' x) x
hf' : ∀ x ∈ Ioo (min a b) (max a b), f' x ≤ 0
M : AntitoneOn f [[a, b]]
hab : a ≤ b
⊢ ∀ x ∈ Ioo a b, HasDerivAt f (f' x) x

pp: case inl.hf'
a b : ℝ
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f f' : ℝ → ℝ
g : ℝ → E
hf : ContinuousOn f [[a, b]]
hff' : ∀ x ∈ Ioo (min a b) (max a b), HasDerivAt f (f' x) x
hf' : ∀ x ∈ Ioo (min a b) (max a b), f' x ≤ 0
M : AntitoneOn f [[a, b]]
hab : a ≤ b
⊢ ∀ x ∈ Ioo a b, f' x ≤ 0

pp: case inl.hab
a b : ℝ
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f f' : ℝ → ℝ
g : ℝ → E
hf : ContinuousOn f [[a, b]]
hff' : ∀ x ∈ Ioo (min a b) (max a b), HasDerivAt f (f' x) x
hf' : ∀ x ∈ Ioo (min a b) (max a b), f' x ≤ 0
M : AntitoneOn f [[a, b]]
hab : a ≤ b
⊢ a ≤ b

Tactic:
· grind

Goal state after:
case inl.hf'
a b : ℝ
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f f' : ℝ → ℝ
g : ℝ → E
hf : ContinuousOn f [[a, b]]
hff' : ∀ x ∈ Ioo (min a b) (max a b), HasDerivAt f (f' x) x
hf' : ∀ x ∈ Ioo (min a b) (max a b), f' x ≤ 0
M : AntitoneOn f [[a, b]]
hab : a ≤ b
⊢ ∀ x ∈ Ioo a b, f' x ≤ 0

case inl.hab
a b : ℝ
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f f' : ℝ → ℝ
g : ℝ → E
hf : ContinuousOn f [[a, b]]
hff' : ∀ x ∈ Ioo (min a b) (max a b), HasDerivAt f (f' x) x
hf' : ∀ x ∈ Ioo (min a b) (max a b), f' x ≤ 0
M : AntitoneOn f [[a, b]]
hab : a ≤ b
⊢ a ≤ b
Module: Mathlib.MeasureTheory.Integral.IntervalIntegral.IntegrationByParts

Goal state before:
pp: case inl.hf'
a b : ℝ
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f f' : ℝ → ℝ
g : ℝ → E
hf : ContinuousOn f [[a, b]]
hff' : ∀ x ∈ Ioo (min a b) (max a b), HasDerivAt f (f' x) x
hf' : ∀ x ∈ Ioo (min a b) (max a b), f' x ≤ 0
M : AntitoneOn f [[a, b]]
hab : a ≤ b
⊢ ∀ x ∈ Ioo a b, f' x ≤ 0

pp: case inl.hab
a b : ℝ
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f f' : ℝ → ℝ
g : ℝ → E
hf : ContinuousOn f [[a, b]]
hff' : ∀ x ∈ Ioo (min a b) (max a b), HasDerivAt f (f' x) x
hf' : ∀ x ∈ Ioo (min a b) (max a b), f' x ≤ 0
M : AntitoneOn f [[a, b]]
hab : a ≤ b
⊢ a ≤ b

Tactic:
· grind

Goal state after:
case inl.hab
a b : ℝ
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f f' : ℝ → ℝ
g : ℝ → E
hf : ContinuousOn f [[a, b]]
hff' : ∀ x ∈ Ioo (min a b) (max a b), HasDerivAt f (f' x) x
hf' : ∀ x ∈ Ioo (min a b) (max a b), f' x ≤ 0
M : AntitoneOn f [[a, b]]
hab : a ≤ b
⊢ a ≤ b
Module: Mathlib.MeasureTheory.Integral.IntervalIntegral.IntegrationByParts

Goal state before:
pp: case inr.hff'
a b : ℝ
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f f' : ℝ → ℝ
g : ℝ → E
hf : ContinuousOn f [[a, b]]
hff' : ∀ x ∈ Ioo (min a b) (max a b), HasDerivAt f (f' x) x
hf' : ∀ x ∈ Ioo (min a b) (max a b), f' x ≤ 0
M : AntitoneOn f [[a, b]]
hab : b < a
⊢ ∀ x ∈ Ioo b a, HasDerivAt f (f' x) x

pp: case inr.hf'
a b : ℝ
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f f' : ℝ → ℝ
g : ℝ → E
hf : ContinuousOn f [[a, b]]
hff' : ∀ x ∈ Ioo (min a b) (max a b), HasDerivAt f (f' x) x
hf' : ∀ x ∈ Ioo (min a b) (max a b), f' x ≤ 0
M : AntitoneOn f [[a, b]]
hab : b < a
⊢ ∀ x ∈ Ioo b a, f' x ≤ 0

pp: case inr.hab
a b : ℝ
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f f' : ℝ → ℝ
g : ℝ → E
hf : ContinuousOn f [[a, b]]
hff' : ∀ x ∈ Ioo (min a b) (max a b), HasDerivAt f (f' x) x
hf' : ∀ x ∈ Ioo (min a b) (max a b), f' x ≤ 0
M : AntitoneOn f [[a, b]]
hab : b < a
⊢ b ≤ a

Tactic:
· grind

Goal state after:
case inr.hf'
a b : ℝ
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f f' : ℝ → ℝ
g : ℝ → E
hf : ContinuousOn f [[a, b]]
hff' : ∀ x ∈ Ioo (min a b) (max a b), HasDerivAt f (f' x) x
hf' : ∀ x ∈ Ioo (min a b) (max a b), f' x ≤ 0
M : AntitoneOn f [[a, b]]
hab : b < a
⊢ ∀ x ∈ Ioo b a, f' x ≤ 0

case inr.hab
a b : ℝ
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f f' : ℝ → ℝ
g : ℝ → E
hf : ContinuousOn f [[a, b]]
hff' : ∀ x ∈ Ioo (min a b) (max a b), HasDerivAt f (f' x) x
hf' : ∀ x ∈ Ioo (min a b) (max a b), f' x ≤ 0
M : AntitoneOn f [[a, b]]
hab : b < a
⊢ b ≤ a
Module: Mathlib.MeasureTheory.Integral.IntervalIntegral.IntegrationByParts

Goal state before:
pp: case inr.hf'
a b : ℝ
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f f' : ℝ → ℝ
g : ℝ → E
hf : ContinuousOn f [[a, b]]
hff' : ∀ x ∈ Ioo (min a b) (max a b), HasDerivAt f (f' x) x
hf' : ∀ x ∈ Ioo (min a b) (max a b), f' x ≤ 0
M : AntitoneOn f [[a, b]]
hab : b < a
⊢ ∀ x ∈ Ioo b a, f' x ≤ 0

pp: case inr.hab
a b : ℝ
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f f' : ℝ → ℝ
g : ℝ → E
hf : ContinuousOn f [[a, b]]
hff' : ∀ x ∈ Ioo (min a b) (max a b), HasDerivAt f (f' x) x
hf' : ∀ x ∈ Ioo (min a b) (max a b), f' x ≤ 0
M : AntitoneOn f [[a, b]]
hab : b < a
⊢ b ≤ a

Tactic:
· grind

Goal state after:
case inr.hab
a b : ℝ
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f f' : ℝ → ℝ
g : ℝ → E
hf : ContinuousOn f [[a, b]]
hff' : ∀ x ∈ Ioo (min a b) (max a b), HasDerivAt f (f' x) x
hf' : ∀ x ∈ Ioo (min a b) (max a b), f' x ≤ 0
M : AntitoneOn f [[a, b]]
hab : b < a
⊢ b ≤ a
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 : ENNReal.ofReal |A.det| < ↑m
d : ℝ≥0∞ := ENNReal.ofReal |A.det|
ε : ℝ
hε : μ (closedBall 0 ε + ⇑A '' closedBall 0 1) < ↑m * μ (closedBall 0 1)
εpos : 0 < ε
this✝ : Iio (NNReal.mk ε ⋯) ∈ 𝓝 0
δ : ℝ≥0
s : Set E
f : E → E
hf : ApproximatesLinearOn f A s δ
hδ : ↑δ < ε
I : ∀ (x : E) (r : ℝ), x ∈ s → 0 ≤ r → μ (f '' (s ∩ closedBall x r)) ≤ ↑m * μ (closedBall x r)
a : ℝ≥0∞
ha : 0 < a
t : Set E
r : E → ℝ
t_count : t.Countable
ts : t ⊆ s
rpos : ∀ x ∈ t, 0 < r x
st : s ⊆ ⋃ x ∈ t, closedBall x (r x)
μt : ∑' (x : ↑t), μ (closedBall (↑x) (r ↑x)) ≤ μ s + a
this : Encodable ↑t
⊢ ∑' (x : ↑t), ↑m * μ (closedBall (↑x) (r ↑x)) ≤ ↑m * (μ s + a)

Tactic:
ENNReal.tsum_mul_left

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 : ENNReal.ofReal |A.det| < ↑m
d : ℝ≥0∞ := ENNReal.ofReal |A.det|
ε : ℝ
hε : μ (closedBall 0 ε + ⇑A '' closedBall 0 1) < ↑m * μ (closedBall 0 1)
εpos : 0 < ε
this✝ : Iio (NNReal.mk ε ⋯) ∈ 𝓝 0
δ : ℝ≥0
s : Set E
f : E → E
hf : ApproximatesLinearOn f A s δ
hδ : ↑δ < ε
I : ∀ (x : E) (r : ℝ), x ∈ s → 0 ≤ r → μ (f '' (s ∩ closedBall x r)) ≤ ↑m * μ (closedBall x r)
a : ℝ≥0∞
ha : 0 < a
t : Set E
r : E → ℝ
t_count : t.Countable
ts : t ⊆ s
rpos : ∀ x ∈ t, 0 < r x
st : s ⊆ ⋃ x ∈ t, closedBall x (r x)
μt : ∑' (x : ↑t), μ (closedBall (↑x) (r ↑x)) ≤ μ s + a
this : Encodable ↑t
⊢ ↑m * ∑' (i : ↑t), μ (closedBall (↑i) (r ↑i)) ≤ ↑m * (μ s + a)
Module: Mathlib.MeasureTheory.Integral.IntegralEqImproper

Goal state before:
pp: case neg
ι : Type u_1
E : Type u_2
μ : Measure ℝ
l : Filter ι
inst✝² : l.IsCountablyGenerated
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
a : ι → ℝ
f : ℝ → E
ha : Tendsto a l atBot
b : ℝ
hb : IntegrableOn f (Iic b) μ
this : ∀ᶠ (i : ι) in l, ∫ (x : ℝ) in Iic b, f x ∂μ - ∫ (x : ℝ) in a i..b, f x ∂μ = ∫ (x : ℝ) in Iic (a i), f x ∂μ
⊢ Tendsto (fun i ↦ ∫ (x : ℝ) in Iic (a i), f x ∂μ) l (𝓝 0)

Tactic:
← sub_self (∫ x in Iic b, f x ∂μ)

Goal state after:
case neg
ι : Type u_1
E : Type u_2
μ : Measure ℝ
l : Filter ι
inst✝² : l.IsCountablyGenerated
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
a : ι → ℝ
f : ℝ → E
ha : Tendsto a l atBot
b : ℝ
hb : IntegrableOn f (Iic b) μ
this : ∀ᶠ (i : ι) in l, ∫ (x : ℝ) in Iic b, f x ∂μ - ∫ (x : ℝ) in a i..b, f x ∂μ = ∫ (x : ℝ) in Iic (a i), f x ∂μ
⊢ Tendsto (fun i ↦ ∫ (x : ℝ) in Iic (a i), f x ∂μ) l (𝓝 (∫ (x : ℝ) in Iic b, f x ∂μ - ∫ (x : ℝ) in Iic b, f x ∂μ))
Module: Mathlib.MeasureTheory.Integral.IntegralEqImproper

Goal state before:
pp: E : Type u_1
f f' : ℝ → E
a : ℝ
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
hderiv : ∀ x ∈ Iic a, HasDerivAt f (f' x) x
f'int : IntegrableOn f' (Iic a) volume
fint : IntegrableOn f (Iic a) volume
⊢ Tendsto f atBot (𝓝 0)

Tactic:
let F : E →L[ℝ] Completion E := Completion.toComplL
  have Fderiv : ∀ x ∈ Iic a, HasDerivAt (F ∘ f) (F (f' x)) x :=
    fun x hx ↦ F.hasFDerivAt.comp_hasDerivAt _ (hderiv x hx)
  have Fint : IntegrableOn (F ∘ f) (Iic a) := by apply F.integrable_comp fint
  have F'int : IntegrableOn (F ∘ f') (Iic a) := by apply F.integrable_comp f'int
  have A : Tendsto (F ∘ f) atBot (𝓝 (limUnder atBot (F ∘ f))) := by
    apply tendsto_limUnder_of_hasDerivAt_of_integrableOn_Iic Fderiv F'int
  have B : limUnder atBot (F ∘ f) = F 0 := by
    have : IntegrableAtFilter (F ∘ f) atBot := by exact ⟨Iic a, Iic_mem_atBot _, Fint⟩
    apply IntegrableAtFilter.eq_zero_of_tendsto this ?_ A
    intro s hs
    rcases mem_atBot_sets.1 hs with ⟨b, hb⟩
    apply le_antisymm (le_top)
    rw [← volume_Iic (a := b)]
    exact measure_mono hb
  rwa [B, ← IsEmbedding.tendsto_nhds_iff] at A
  exact (Completion.isUniformEmbedding_coe E).isEmbedding

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
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : Measure E
inst✝ : 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 (fun r ↦ μ (s ∩ closedBall x r) / μ (closedBall x r)) (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 | ‖f y - f x - (f' x) (y - x)‖ ≤ 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
⊢ ‖(f' x - A) (r • a)‖ = ‖f y - f x - A (y - x) - (f y - f x - (f' x) (y - x))‖

Tactic:
by
        simp only [ya, add_sub_cancel_left, sub_sub_sub_cancel_left, FunLike.coe_sub,
          Pi.sub_apply, map_smul, smul_sub]

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
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
δ : (E →L[ℝ] E) → ℝ≥0
hδ :
  ∀ (A : E →L[ℝ] E),
    0 < δ A ∧ ∀ (t : Set E), ApproximatesLinearOn f A t (δ A) → μ (f '' t) ≤ ↑(|A.det|.toNNReal + 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))
Af' : s.Nonempty → ∀ (n : ℕ), ∃ y ∈ s, A n = f' y
⊢ ∑' (n : ℕ), ↑ε * μ (s ∩ t n) ≤ ↑ε * ∑' (n : ℕ), μ (closedBall 0 R ∩ t n)

Tactic:
ENNReal.tsum_mul_left

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
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
δ : (E →L[ℝ] E) → ℝ≥0
hδ :
  ∀ (A : E →L[ℝ] E),
    0 < δ A ∧ ∀ (t : Set E), ApproximatesLinearOn f A t (δ A) → μ (f '' t) ≤ ↑(|A.det|.toNNReal + 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))
Af' : s.Nonempty → ∀ (n : ℕ), ∃ y ∈ s, A n = f' y
⊢ ↑ε * ∑' (i : ℕ), μ (s ∩ t i) ≤ ↑ε * ∑' (n : ℕ), μ (closedBall 0 R ∩ t n)
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
h'f' : ∀ x ∈ s, (f' x).det = 0
R : ℝ
A : ∀ (ε : ℝ≥0), 0 < ε → μ (f '' (s ∩ closedBall 0 R)) ≤ ↑ε * μ (closedBall 0 R)
B : Tendsto (fun ε ↦ ↑ε * μ (closedBall 0 R)) (𝓝[>] 0) (𝓝 0)
⊢ μ (f '' (s ∩ closedBall 0 R)) ≤ 0

Tactic:
apply ge_of_tendsto B

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
hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x
h'f' : ∀ x ∈ s, (f' x).det = 0
R : ℝ
A : ∀ (ε : ℝ≥0), 0 < ε → μ (f '' (s ∩ closedBall 0 R)) ≤ ↑ε * μ (closedBall 0 R)
B : Tendsto (fun ε ↦ ↑ε * μ (closedBall 0 R)) (𝓝[>] 0) (𝓝 0)
⊢ ∀ᶠ (c : ℝ≥0) in 𝓝[>] 0, μ (f '' (s ∩ closedBall 0 R)) ≤ ↑c * μ (closedBall 0 R)
Module: Mathlib.Geometry.Manifold.ChartedSpace

Goal state before:
pp: case refine_2
H : Type u
M : Type u_2
inst✝³ : TopologicalSpace H
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace H M
inst✝ : LocallyConnectedSpace H
e : M → OpenPartialHomeomorph M H := chartAt H
⊢ ∀ (x : M) (i : Set H),
    (IsOpen[inst✝³] i ∧ ↑(e x) x ∈ i ∧ IsConnected i) ∧ i ⊆ (e x).target → IsPreconnected (↑(e x).symm '' i)

Tactic:
rintro x s ⟨⟨-, -, hsconn⟩, hssubset⟩

Goal state after:
case refine_2
H : Type u
M : Type u_2
inst✝³ : TopologicalSpace H
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace H M
inst✝ : LocallyConnectedSpace H
e : M → OpenPartialHomeomorph M H := chartAt H
x : M
s : Set H
hssubset : s ⊆ (e x).target
hsconn : IsConnected s
⊢ IsPreconnected (↑(e x).symm '' s)
Module: Mathlib.Geometry.Manifold.ChartedSpace

Goal state before:
pp: case refine_1
H : Type u
M : Type u_2
inst✝³ : TopologicalSpace H
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace H M
inst✝ : LocallyPathConnectedSpace H
x : M
s : Set M
hs : s ∈ 𝓝 x
e : OpenPartialHomeomorph M H := chartAt H x
t : Set M := s ∩ e.source
ht : t ∈ 𝓝 x
⊢ ↑e.symm '' pathComponentIn (↑e '' t) (↑e x) ∈ 𝓝 x

Tactic:
← e.left_inv (mem_chart_source _ _)

Goal state after:
case refine_1
H : Type u
M : Type u_2
inst✝³ : TopologicalSpace H
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace H M
inst✝ : LocallyPathConnectedSpace H
x : M
s : Set M
hs : s ∈ 𝓝 x
e : OpenPartialHomeomorph M H := chartAt H x
t : Set M := s ∩ e.source
ht : t ∈ 𝓝 x
⊢ ↑e.symm '' pathComponentIn (↑e '' t) (↑e x) ∈ 𝓝 (↑e.symm (↑e x))
Module: Mathlib.Geometry.Manifold.StructureGroupoid

Goal state before:
pp: H✝ : Type u_1
inst✝¹ : TopologicalSpace H✝
H : Type u_2
inst✝ : TopologicalSpace H
e : OpenPartialHomeomorph H H
he : e ∈ {OpenPartialHomeomorph.refl H} ∪ {e | e.source = ∅}
⊢ e.symm ∈ {OpenPartialHomeomorph.refl H} ∪ {e | e.source = ∅}

Tactic:
rcases (mem_union _ _ _).1 he with E | E

Goal state after:
case inl
H✝ : Type u_1
inst✝¹ : TopologicalSpace H✝
H : Type u_2
inst✝ : TopologicalSpace H
e : OpenPartialHomeomorph H H
he : e ∈ {OpenPartialHomeomorph.refl H} ∪ {e | e.source = ∅}
E : e ∈ {OpenPartialHomeomorph.refl H}
⊢ e.symm ∈ {OpenPartialHomeomorph.refl H} ∪ {e | e.source = ∅}

case inr
H✝ : Type u_1
inst✝¹ : TopologicalSpace H✝
H : Type u_2
inst✝ : TopologicalSpace H
e : OpenPartialHomeomorph H H
he : e ∈ {OpenPartialHomeomorph.refl H} ∪ {e | e.source = ∅}
E : e ∈ {e | e.source = ∅}
⊢ e.symm ∈ {OpenPartialHomeomorph.refl H} ∪ {e | e.source = ∅}
Module: Mathlib.Geometry.Manifold.ChartedSpace

Goal state before:
pp: H : Type u
M : Type u_2
inst✝ : TopologicalSpace H
c : ChartedSpaceCore H M
e : PartialEquiv M H
he : e ∈ c.atlas
⊢ IsOpen[inst✝] e.target

Tactic:
have E : e.target ∩ e.symm ⁻¹' e.source = e.target :=
    Subset.antisymm inter_subset_left fun x hx ↦
      ⟨hx, PartialEquiv.target_subset_preimage_source _ hx⟩
  simpa [PartialEquiv.trans_source, E] using c.open_source e e he he

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✝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]
    use x

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

Goal state before:
pp: H✝ : Type u_1
inst✝¹ : TopologicalSpace H✝
H : Type u_2
inst✝ : TopologicalSpace H
e e' : OpenPartialHomeomorph H H
he : e ∈ {OpenPartialHomeomorph.refl H} ∪ {e | e.source = ∅}
he'e : e' ≈ e
⊢ e' ∈ {OpenPartialHomeomorph.refl H} ∪ {e | e.source = ∅}

Tactic:
rcases he with he | he
    · left
      have : e = e' := by
        refine eq_of_eqOnSource_univ (Setoid.symm he'e) ?_ ?_ <;>
          rw [Set.mem_singleton_iff.1 he] <;> rfl
      rwa [← this]
    · right
      have he : e.toPartialEquiv.source = ∅ := he
      rwa [Set.mem_ofPred_eq, EqOnSource.source_eq he'e]

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

Goal state before:
pp: case mpr
H : Type u_1
inst✝ : TopologicalSpace H
G : StructureGroupoid H
⊢ idRestrGroupoid ≤ G → ClosedUnderRestriction G

Tactic:
· intro h
    constructor
    intro e he s hs
    rw [← ofSet_trans (e : OpenPartialHomeomorph H H) hs]
    refine G.trans ?_ he
    apply StructureGroupoid.le_iff.mp h
    exact idRestrGroupoid_mem hs

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
⊢ ↑(chartAt H x).symm ⁻¹' (s ∩ t) =ᶠ[𝓝 (↑(chartAt H x) x)] ↑(chartAt H x).symm ⁻¹' s

Tactic:
simp_rw [eventuallyEq_set, mem_preimage,
    (chartAt _ x).eventually_nhds' (fun x ↦ x ∈ s ∩ t ↔ x ∈ s) (mem_chart_source _ x)]

Goal state after:
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
⊢ ∀ᶠ (x : M) in 𝓝 x, x ∈ s ∩ t ↔ x ∈ s
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✝⁵ : NontriviallyNormedField 𝕜
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
inst✝² : TopologicalSpace H
inst✝¹ : TopologicalSpace M
I : ModelWithCorners 𝕜 E H
inst✝ : ChartedSpace H M
x : M
⊢ UniqueDiffOn 𝕜 (extChartAt I x).target

Tactic:
rw [extChartAt_target]
  exact I.uniqueDiffOn_preimage (chartAt H x).open_target

Goal state after:
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 V : Opens M
hUV : U ≤ V
⊢ LiftProp Q (Opens.inclusion hUV)

Tactic:
intro x
  change LiftPropAt Q (id ∘ Opens.inclusion hUV) x
  rw [← hG.liftPropAt_iff_comp_inclusion hUV]
  apply hG.liftProp_id hQ

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

Goal state before:
pp: case refine_2
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
he'G : e' ∈ G
he'x : x ∈ e'.source
h : G.IsLocalStructomorphWithinAt f s x
hx : ↑e' x ∈ ↑e'.symm ⁻¹' s
hxs : x ∈ s
e : OpenPartialHomeomorph H H
heG : e ∈ G
hef : EqOn f (↑e.toPartialEquiv) (s ∩ e.source)
hex : x ∈ e.source
⊢ ↑e' x ∈ (e'.symm ≫ₕ e).source

Tactic:
· simp only [hex, he'x, mfld_simps]

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
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
⊢ ∑' (n : ℕ), (ENNReal.ofReal |(A n).det| * μ (s ∩ t n) + ↑ε * μ (s ∩ t n)) ≤
    ∑' (n : ℕ), (μ (f '' (s ∩ t n)) + ↑ε * μ (s ∩ t n) + ↑ε * μ (s ∩ t n))

Tactic:
by
      gcongr
      exact (hδ (A _)).2.2 _ _ (ht _)

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
h's : μ s ≠ ∞
hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x
hf : InjOn f s
this : Tendsto (fun ε ↦ μ (f '' s) + 2 * ↑ε * μ s) (𝓝[>] 0) (𝓝 (μ (f '' s)))
ε : ℝ≥0
εpos : 0 < ε
⊢ ∫⁻ (x : E) in s, ENNReal.ofReal |(f' x).det| ∂μ ≤ μ (f '' s) + 2 * ↑ε * μ s

Tactic:
exact lintegral_abs_det_fderiv_le_addHaar_image_aux1 μ hs hf' hf εpos

Goal state after: