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

Goal state before:
pp: case pos
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 : MonotoneOn f s
g : ℝ → F
H : IntegrableOn g (f '' s) volume
H' : IntegrableOn (fun x ↦ f' x • g (f x)) s volume
a b c : Set ℝ
h_union : a ∪ (b ∪ c) = s
ha : MeasurableSet a
hb : MeasurableSet b
hc : MeasurableSet c
h_disj : Disjoint a (b ∪ c)
h_disj' : Disjoint b c
a_count : a.Countable
fb_count : (f '' b).Countable
deriv_b : ∀ x ∈ b, f' x = 0
deriv_c : ∀ x ∈ c, 0 ≤ f' x
inj_c : InjOn f c
a_s : a ⊆ s
bc_s : b ∪ c ⊆ s
b_s : b ⊆ s
c_s : c ⊆ s
I : ∫ (x : ℝ) in s, f' x • g (f x) = ∫ (x : ℝ) in c, f' x • g (f x)
J : ∫ (x : ℝ) in f '' s, g x = ∫ (x : ℝ) in f '' c, g x
F' : ℝ → ℝ →L[ℝ] ℝ := fun x ↦ smulRight 1 (f' x)
hF' : ∀ x ∈ c, HasFDerivWithinAt f (F' x) c x
this : ∫ (x : ℝ) in c, f' x • g (f x) = ∫ (x : ℝ) in c, |(F' x).det| • g (f x)
⊢ ∫ (x : ℝ) in f '' c, g x = ∫ (x : ℝ) in c, |(F' x).det| • g (f x)

Tactic:
]

Goal state after:
case pos
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 : MonotoneOn f s
g : ℝ → F
H : IntegrableOn g (f '' s) volume
H' : IntegrableOn (fun x ↦ f' x • g (f x)) s volume
a b c : Set ℝ
h_union : a ∪ (b ∪ c) = s
ha : MeasurableSet a
hb : MeasurableSet b
hc : MeasurableSet c
h_disj : Disjoint a (b ∪ c)
h_disj' : Disjoint b c
a_count : a.Countable
fb_count : (f '' b).Countable
deriv_b : ∀ x ∈ b, f' x = 0
deriv_c : ∀ x ∈ c, 0 ≤ f' x
inj_c : InjOn f c
a_s : a ⊆ s
bc_s : b ∪ c ⊆ s
b_s : b ⊆ s
c_s : c ⊆ s
I : ∫ (x : ℝ) in s, f' x • g (f x) = ∫ (x : ℝ) in c, f' x • g (f x)
J : ∫ (x : ℝ) in f '' s, g x = ∫ (x : ℝ) in f '' c, g x
F' : ℝ → ℝ →L[ℝ] ℝ := fun x ↦ smulRight 1 (f' x)
hF' : ∀ x ∈ c, HasFDerivWithinAt f (F' x) c x
this : ∫ (x : ℝ) in c, f' x • g (f x) = ∫ (x : ℝ) in c, |(F' x).det| • g (f x)
⊢ ∫ (x : ℝ) in f '' c, g x = ∫ (x : ℝ) in c, |(F' x).det| • g (f 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 : 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
x : E
r : ℝ
xs : x ∈ s
r0 : 0 ≤ r
K : f '' (s ∩ closedBall x r) ⊆ {f x} + r • (⇑A '' closedBall 0 1 + closedBall 0 ε)
this : ⇑A '' closedBall 0 r + closedBall (f x) (ε * r) = {f x} + r • (⇑A '' closedBall 0 1 + closedBall 0 ε)
⊢ ENNReal.ofReal (r ^ finrank ℝ E) * μ (closedBall 0 ε + ⇑A '' closedBall 0 1) ≤
    ENNReal.ofReal (r ^ finrank ℝ E) * (↑m * μ (closedBall 0 1))

Tactic:
]

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
x : E
r : ℝ
xs : x ∈ s
r0 : 0 ≤ r
K : f '' (s ∩ closedBall x r) ⊆ {f x} + r • (⇑A '' closedBall 0 1 + closedBall 0 ε)
this : ⇑A '' closedBall 0 r + closedBall (f x) (ε * r) = {f x} + r • (⇑A '' closedBall 0 1 + closedBall 0 ε)
⊢ ENNReal.ofReal (r ^ finrank ℝ E) * μ (closedBall 0 ε + ⇑A '' closedBall 0 1) ≤
    ENNReal.ofReal (r ^ finrank ℝ E) * (↑m * μ (closedBall 0 1))
Module: Mathlib.MeasureTheory.Integral.IntegralEqImproper

Goal state before:
pp: ι : Type u_1
E : Type u_2
μ : Measure ℝ
l : Filter ι
inst✝² : l.NeBot
inst✝¹ : l.IsCountablyGenerated
inst✝ : NormedAddCommGroup E
a : ι → ℝ
f : ℝ → E
I b : ℝ
hfi✝ : ∀ (i : ι), IntegrableOn f (Ioc (a i) b) μ
ha : Tendsto a l atBot
h : ∀ᶠ (i : ι) in l, ∫ (x : ℝ) in a i..b, ‖f x‖ ∂μ ≤ I
hφ : AECover (μ.restrict (Iic b)) l fun i ↦ Ioi (a i)
hfi : ∀ (i : ι), IntegrableOn f (Ioi (a i)) (μ.restrict (Iic b))
i : ι
hai : a i ≤ b
⊢ ∫ (x : ℝ) in Ioc (a i) b, ‖f x‖ ∂μ ≤ I → ∫ (x : ℝ) in Ioi (a i) ∩ Iic b, ‖f x‖ ∂μ ≤ I

Tactic:
]

Goal state after:
ι : Type u_1
E : Type u_2
μ : Measure ℝ
l : Filter ι
inst✝² : l.NeBot
inst✝¹ : l.IsCountablyGenerated
inst✝ : NormedAddCommGroup E
a : ι → ℝ
f : ℝ → E
I b : ℝ
hfi✝ : ∀ (i : ι), IntegrableOn f (Ioc (a i) b) μ
ha : Tendsto a l atBot
h : ∀ᶠ (i : ι) in l, ∫ (x : ℝ) in a i..b, ‖f x‖ ∂μ ≤ I
hφ : AECover (μ.restrict (Iic b)) l fun i ↦ Ioi (a i)
hfi : ∀ (i : ι), IntegrableOn f (Ioi (a i)) (μ.restrict (Iic b))
i : ι
hai : a i ≤ b
⊢ ∫ (x : ℝ) in Ioc (a i) b, ‖f x‖ ∂μ ≤ I → ∫ (x : ℝ) in Ioi (a i) ∩ Iic b, ‖f x‖ ∂μ ≤ I
Module: Mathlib.MeasureTheory.Function.JacobianOneDim

Goal state before:
pp: case hf'
F : Type u_1
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f f' : ℝ → ℝ
a b : ℝ
g : ℝ → F
hf : ContinuousOn f (Icc a b)
hff' : ∀ x ∈ Ioo a b, HasDerivAt f (f' x) x
hf' : ∀ x ∈ Ioo a b, 0 ≤ f' x
hab : a ≤ b
⊢ DifferentiableOn ℝ f (Ioo a b)

Tactic:
]

Goal state after:
case hf'
F : Type u_1
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f f' : ℝ → ℝ
a b : ℝ
g : ℝ → F
hf : ContinuousOn f (Icc a b)
hff' : ∀ x ∈ Ioo a b, HasDerivAt f (f' x) x
hf' : ∀ x ∈ Ioo a b, 0 ≤ f' x
hab : a ≤ b
⊢ DifferentiableOn ℝ f (Ioo a b)
Module: Mathlib.MeasureTheory.Function.JacobianOneDim

Goal state before:
pp: case hf'_nonneg
F : Type u_1
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f f' : ℝ → ℝ
a b : ℝ
g : ℝ → F
hf : ContinuousOn f (Icc a b)
hff' : ∀ x ∈ Ioo a b, HasDerivAt f (f' x) x
hf' : ∀ x ∈ Ioo a b, 0 ≤ f' x
hab : a ≤ b
⊢ ∀ x ∈ Ioo a b, 0 ≤ deriv f x

Tactic:
]

Goal state after:
case hf'_nonneg
F : Type u_1
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f f' : ℝ → ℝ
a b : ℝ
g : ℝ → F
hf : ContinuousOn f (Icc a b)
hff' : ∀ x ∈ Ioo a b, HasDerivAt f (f' x) x
hf' : ∀ x ∈ Ioo a b, 0 ≤ f' x
hab : a ≤ b
⊢ ∀ x ∈ Ioo a b, 0 ≤ deriv f x
Module: Mathlib.MeasureTheory.Integral.IntegralEqImproper

Goal state before:
pp: ι : Type u_1
E : Type u_2
μ : Measure ℝ
l : Filter ι
inst✝² : l.NeBot
inst✝¹ : l.IsCountablyGenerated
inst✝ : NormedAddCommGroup E
b : ι → ℝ
f : ℝ → E
I a : ℝ
hfi : ∀ (i : ι), IntegrableOn f (Ioc a (b i)) μ
hb : Tendsto b l atTop
h : ∀ᶠ (i : ι) in l, ∫ (x : ℝ) in a..b i, ‖f x‖ ∂μ ≤ I
hφ : AECover (μ.restrict (Ioi a)) l fun i ↦ Iic (b i)
i : ι
⊢ Integrable f (μ.restrict (Ioi a ∩ Iic (b i)))

Tactic:
]

Goal state after:
ι : Type u_1
E : Type u_2
μ : Measure ℝ
l : Filter ι
inst✝² : l.NeBot
inst✝¹ : l.IsCountablyGenerated
inst✝ : NormedAddCommGroup E
b : ι → ℝ
f : ℝ → E
I a : ℝ
hfi : ∀ (i : ι), IntegrableOn f (Ioc a (b i)) μ
hb : Tendsto b l atTop
h : ∀ᶠ (i : ι) in l, ∫ (x : ℝ) in a..b i, ‖f x‖ ∂μ ≤ I
hφ : AECover (μ.restrict (Ioi a)) l fun i ↦ Iic (b i)
i : ι
⊢ Integrable f (μ.restrict (Ioi a ∩ Iic (b i)))
Module: Mathlib.MeasureTheory.Function.JacobianOneDim

Goal state before:
pp: F : Type u_1
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f f' : ℝ → ℝ
a b : ℝ
g : ℝ → F
hf : ContinuousOn f (Icc a b)
hff' : ∀ x ∈ Ioo a b, HasDerivAt f (f' x) x
hf' : ∀ x ∈ Ioo a b, 0 ≤ f' x
hab : a ≤ b
M : MonotoneOn f (Icc a b)
⊢ f '' Icc a b =ᵐ[volume] f '' Ioo a b

Tactic:
]

Goal state after:
F : Type u_1
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f f' : ℝ → ℝ
a b : ℝ
g : ℝ → F
hf : ContinuousOn f (Icc a b)
hff' : ∀ x ∈ Ioo a b, HasDerivAt f (f' x) x
hf' : ∀ x ∈ Ioo a b, 0 ≤ f' x
hab : a ≤ b
M : MonotoneOn f (Icc a b)
⊢ f '' Icc a b =ᵐ[volume] f '' Ioo a b
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 (Icc b a)
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
⊢ ContinuousOn f (Icc b a)

Tactic:
]

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 (Icc b a)
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
⊢ ContinuousOn f (Icc b a)
Module: Mathlib.MeasureTheory.Integral.IntegralEqImproper

Goal state before:
pp: ι : Type u_1
E : Type u_2
μ : Measure ℝ
l : Filter ι
inst✝² : l.NeBot
inst✝¹ : l.IsCountablyGenerated
inst✝ : NormedAddCommGroup E
b : ι → ℝ
f : ℝ → E
I a : ℝ
hfi✝ : ∀ (i : ι), IntegrableOn f (Ioc a (b i)) μ
hb : Tendsto b l atTop
h : ∀ᶠ (i : ι) in l, ∫ (x : ℝ) in a..b i, ‖f x‖ ∂μ ≤ I
hφ : AECover (μ.restrict (Ioi a)) l fun i ↦ Iic (b i)
hfi : ∀ (i : ι), IntegrableOn f (Iic (b i)) (μ.restrict (Ioi a))
i : ι
hbi : a ≤ b i
⊢ ∫ (x : ℝ) in Ioc a (b i), ‖f x‖ ∂μ ≤ I → ∫ (x : ℝ) in Ioi a ∩ Iic (b i), ‖f x‖ ∂μ ≤ I

Tactic:
]

Goal state after:
ι : Type u_1
E : Type u_2
μ : Measure ℝ
l : Filter ι
inst✝² : l.NeBot
inst✝¹ : l.IsCountablyGenerated
inst✝ : NormedAddCommGroup E
b : ι → ℝ
f : ℝ → E
I a : ℝ
hfi✝ : ∀ (i : ι), IntegrableOn f (Ioc a (b i)) μ
hb : Tendsto b l atTop
h : ∀ᶠ (i : ι) in l, ∫ (x : ℝ) in a..b i, ‖f x‖ ∂μ ≤ I
hφ : AECover (μ.restrict (Ioi a)) l fun i ↦ Iic (b i)
hfi : ∀ (i : ι), IntegrableOn f (Iic (b i)) (μ.restrict (Ioi a))
i : ι
hbi : a ≤ b i
⊢ ∫ (x : ℝ) in Ioc a (b i), ‖f x‖ ∂μ ≤ I → ∫ (x : ℝ) in Ioi a ∩ Iic (b i), ‖f x‖ ∂μ ≤ I
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
⊢ μ (f '' s) ≤ ↑m * (μ s + a)

Tactic:
]

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
⊢ μ (f '' s) ≤ ↑m * (μ s + a)
Module: Mathlib.MeasureTheory.Integral.IntegralEqImproper

Goal state before:
pp: ι : Type u_1
E : Type u_2
l : Filter ι
inst✝² : l.NeBot
inst✝¹ : l.IsCountablyGenerated
inst✝ : NormedAddCommGroup E
a b : ι → ℝ
f : ℝ → E
I a₀ b₀ : ℝ
hfi : ∀ (i : ι), IntegrableOn f (Ioc (a i) (b i)) volume
ha : Tendsto a l (𝓝 a₀)
hb : Tendsto b l (𝓝 b₀)
h : ∀ᶠ (i : ι) in l, ∫ (x : ℝ) in Ioc (a i) (b i), ‖f x‖ ≤ I
i : ι
hi : ∫ (x : ℝ) in Ioc (a i) (b i), ‖f x‖ ≤ I
⊢ ∫ (x : ℝ) in Ioc (a i) (b i) ∩ Ioc a₀ b₀, ‖f x‖ ∂volume ≤ I

Tactic:
]

Goal state after:
ι : Type u_1
E : Type u_2
l : Filter ι
inst✝² : l.NeBot
inst✝¹ : l.IsCountablyGenerated
inst✝ : NormedAddCommGroup E
a b : ι → ℝ
f : ℝ → E
I a₀ b₀ : ℝ
hfi : ∀ (i : ι), IntegrableOn f (Ioc (a i) (b i)) volume
ha : Tendsto a l (𝓝 a₀)
hb : Tendsto b l (𝓝 b₀)
h : ∀ᶠ (i : ι) in l, ∫ (x : ℝ) in Ioc (a i) (b i), ‖f x‖ ≤ I
i : ι
hi : ∫ (x : ℝ) in Ioc (a i) (b i), ‖f x‖ ≤ I
⊢ ∫ (x : ℝ) in Ioc (a i) (b i) ∩ Ioc a₀ b₀, ‖f x‖ ∂volume ≤ I
Module: Mathlib.MeasureTheory.Integral.IntegralEqImproper

Goal state before:
pp: ι : Type u_1
E : Type u_2
l : Filter ι
inst✝² : l.NeBot
inst✝¹ : l.IsCountablyGenerated
inst✝ : NormedAddCommGroup E
a b : ι → ℝ
f : ℝ → E
I a₀ b₀ : ℝ
hfi : ∀ (i : ι), IntegrableOn f (Ioc (a i) (b i)) volume
ha : Tendsto a l (𝓝 a₀)
hb : Tendsto b l (𝓝 b₀)
h : ∀ᶠ (i : ι) in l, ∫ (x : ℝ) in Ioc (a i) (b i), ‖f x‖ ≤ I
i : ι
hi : ∫ (x : ℝ) in Ioc (a i) (b i), ‖f x‖ ≤ I
⊢ ∫ (x : ℝ) in Ioc (a i) (b i) ∩ Ioc a₀ b₀, ‖f x‖ ∂volume ≤ ∫ (x : ℝ) in Ioc (a i) (b i), ‖f x‖

Tactic:
]

Goal state after:
ι : Type u_1
E : Type u_2
l : Filter ι
inst✝² : l.NeBot
inst✝¹ : l.IsCountablyGenerated
inst✝ : NormedAddCommGroup E
a b : ι → ℝ
f : ℝ → E
I a₀ b₀ : ℝ
hfi : ∀ (i : ι), IntegrableOn f (Ioc (a i) (b i)) volume
ha : Tendsto a l (𝓝 a₀)
hb : Tendsto b l (𝓝 b₀)
h : ∀ᶠ (i : ι) in l, ∫ (x : ℝ) in Ioc (a i) (b i), ‖f x‖ ≤ I
i : ι
hi : ∫ (x : ℝ) in Ioc (a i) (b i), ‖f x‖ ≤ I
⊢ ∫ (x : ℝ) in Ioc (a i) (b i) ∩ Ioc a₀ b₀, ‖f x‖ ∂volume ≤ ∫ (x : ℝ) in Ioc (a i) (b i), ‖f 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 : 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, closedBall (↑x) (r ↑x)
μt : ∑' (x : ↑t), μ (closedBall (↑x) (r ↑x)) ≤ μ s + a
this : Encodable ↑t
⊢ μ (f '' s) ≤ μ (⋃ x, f '' (s ∩ closedBall (↑x) (r ↑x)))

Tactic:
]

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, closedBall (↑x) (r ↑x)
μt : ∑' (x : ↑t), μ (closedBall (↑x) (r ↑x)) ≤ μ s + a
this : Encodable ↑t
⊢ μ (f '' s) ≤ μ (⋃ x, f '' (s ∩ closedBall (↑x) (r ↑x)))
Module: Mathlib.MeasureTheory.Integral.IntegralEqImproper

Goal state before:
pp: 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 b : ι → ℝ
f : ℝ → E
hfi : Integrable f μ
ha : Tendsto a l atBot
hb : Tendsto b l atTop
⊢ Tendsto (fun i ↦ ∫ (x : ℝ) in a i..b i, f x ∂μ) l (𝓝 (∫ (x : ℝ), f x ∂μ))

Tactic:
let φ i := Ioc (a i) (b i)

Goal state after:
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 b : ι → ℝ
f : ℝ → E
hfi : Integrable f μ
ha : Tendsto a l atBot
hb : Tendsto b l atTop
φ : ι → Set ℝ := fun i ↦ Ioc (a i) (b i)
⊢ Tendsto (fun i ↦ ∫ (x : ℝ) in a i..b i, f x ∂μ) l (𝓝 (∫ (x : ℝ), f x ∂μ))
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, closedBall (↑x) (r ↑x)
μt : ∑' (x : ↑t), μ (closedBall (↑x) (r ↑x)) ≤ μ s + a
this : Encodable ↑t
⊢ f '' s ⊆ f '' (s ∩ ⋃ i, closedBall (↑i) (r ↑i))

Tactic:
]

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, closedBall (↑x) (r ↑x)
μt : ∑' (x : ↑t), μ (closedBall (↑x) (r ↑x)) ≤ μ s + a
this : Encodable ↑t
⊢ f '' s ⊆ f '' (s ∩ ⋃ i, closedBall (↑i) (r ↑i))
Module: Mathlib.MeasureTheory.Function.JacobianOneDim

Goal state before:
pp: F : Type u_1
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f f' : ℝ → ℝ
a b : ℝ
g : ℝ → F
hf : ContinuousOn f (Icc a b)
hff' : ∀ x ∈ Ioo a b, HasDerivAt f (f' x) x
hf' : ∀ x ∈ Ioo a b, 0 ≤ f' x
hab : a ≤ b
M : MonotoneOn f (Icc a b)
A : ∫ (u : ℝ) in Icc (f a) (f b), g u = ∫ (u : ℝ) in f '' Ioo a b, g u
⊢ ∫ (t : ℝ) in Ioo a b, f' t • g (f t) = ∫ (x : ℝ) in Ioo a b, ?m.259 x • g (f x)

pp: case hf'
F : Type u_1
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f f' : ℝ → ℝ
a b : ℝ
g : ℝ → F
hf : ContinuousOn f (Icc a b)
hff' : ∀ x ∈ Ioo a b, HasDerivAt f (f' x) x
hf' : ∀ x ∈ Ioo a b, 0 ≤ f' x
hab : a ≤ b
M : MonotoneOn f (Icc a b)
A : ∫ (u : ℝ) in Icc (f a) (f b), g u = ∫ (u : ℝ) in f '' Ioo a b, g u
⊢ ∀ x ∈ Ioo a b, HasDerivWithinAt f (?m.259 x) (Ioo a b) x

pp: case hf
F : Type u_1
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f f' : ℝ → ℝ
a b : ℝ
g : ℝ → F
hf : ContinuousOn f (Icc a b)
hff' : ∀ x ∈ Ioo a b, HasDerivAt f (f' x) x
hf' : ∀ x ∈ Ioo a b, 0 ≤ f' x
hab : a ≤ b
M : MonotoneOn f (Icc a b)
A : ∫ (u : ℝ) in Icc (f a) (f b), g u = ∫ (u : ℝ) in f '' Ioo a b, g u
⊢ MonotoneOn f (Ioo a b)

pp: F : Type u_1
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f f' : ℝ → ℝ
a b : ℝ
g : ℝ → F
hf : ContinuousOn f (Icc a b)
hff' : ∀ x ∈ Ioo a b, HasDerivAt f (f' x) x
hf' : ∀ x ∈ Ioo a b, 0 ≤ f' x
hab : a ≤ b
M : MonotoneOn f (Icc a b)
A : ∫ (u : ℝ) in Icc (f a) (f b), g u = ∫ (u : ℝ) in f '' Ioo a b, g u
⊢ ℝ → ℝ

Tactic:
]

Goal state after:
case hf'
F : Type u_1
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f f' : ℝ → ℝ
a b : ℝ
g : ℝ → F
hf : ContinuousOn f (Icc a b)
hff' : ∀ x ∈ Ioo a b, HasDerivAt f (f' x) x
hf' : ∀ x ∈ Ioo a b, 0 ≤ f' x
hab : a ≤ b
M : MonotoneOn f (Icc a b)
A : ∫ (u : ℝ) in Icc (f a) (f b), g u = ∫ (u : ℝ) in f '' Ioo a b, g u
⊢ ∀ x ∈ Ioo a b, HasDerivWithinAt f (f' x) (Ioo a b) x

case hf
F : Type u_1
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f f' : ℝ → ℝ
a b : ℝ
g : ℝ → F
hf : ContinuousOn f (Icc a b)
hff' : ∀ x ∈ Ioo a b, HasDerivAt f (f' x) x
hf' : ∀ x ∈ Ioo a b, 0 ≤ f' x
hab : a ≤ b
M : MonotoneOn f (Icc a b)
A : ∫ (u : ℝ) in Icc (f a) (f b), g u = ∫ (u : ℝ) in f '' Ioo a b, g u
⊢ MonotoneOn f (Ioo a b)
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
⊢ ↑m * ∑' (i : ↑t), μ (closedBall (↑i) (r ↑i)) ≤ ↑m * (μ s + a)

Tactic:
]

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: 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
b : ℝ
hfi : IntegrableOn f (Iic b) μ
ha : Tendsto a l atBot
φ : ι → Set ℝ := fun i ↦ Ioi (a i)
hφ : AECover (μ.restrict (Iic b)) l φ
i : ι
hai : a i ≤ b
⊢ ∫ (x : ℝ) in φ i ∩ Iic b, f x ∂μ = ∫ (x : ℝ) in Ioc (a i) b, f x ∂μ

Tactic:
]

Goal state after:
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
b : ℝ
hfi : IntegrableOn f (Iic b) μ
ha : Tendsto a l atBot
φ : ι → Set ℝ := fun i ↦ Ioi (a i)
hφ : AECover (μ.restrict (Iic b)) l φ
i : ι
hai : a i ≤ b
⊢ ∫ (x : ℝ) in φ i ∩ Iic b, f x ∂μ = ∫ (x : ℝ) in Ioc (a i) b, f x ∂μ
Module: Mathlib.MeasureTheory.Integral.IntegralEqImproper

Goal state before:
pp: 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
i : ι
hi : a i ∈ Iic b
⊢ ∫ (x : ℝ) in a i..b, f x ∂μ = ∫ (x : ℝ) in a i..b, f x ∂μ

Tactic:
]

Goal state after:
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 (∫ (x : ℝ) in Iic b, f x ∂μ - ∫ (x : ℝ) in Iic b, f x ∂μ))

Tactic:
]

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))