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

Goal state before:
pp: F : Type u_1
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
s : Set ℝ
f f' : ℝ → ℝ
hs : MeasurableSet s
hf' : ∀ x ∈ s, HasDerivWithinAt f (f' x) s x
hf : 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 ⊆ a ∪ (b ∪ c)

Tactic:
]

Goal state after:
F : Type u_1
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
s : Set ℝ
f f' : ℝ → ℝ
hs : MeasurableSet s
hf' : ∀ x ∈ s, HasDerivWithinAt f (f' x) s x
hf : 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 ⊆ a ∪ (b ∪ c)
Module: Mathlib.MeasureTheory.Function.JacobianOneDim

Goal state before:
pp: F : Type u_1
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
s : Set ℝ
f f' : ℝ → ℝ
hs : MeasurableSet s
hf' : ∀ x ∈ s, HasDerivWithinAt f (f' x) s x
hf : 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 ⊆ a ∪ (b ∪ c)

Tactic:
]

Goal state after:
F : Type u_1
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
s : Set ℝ
f f' : ℝ → ℝ
hs : MeasurableSet s
hf' : ∀ x ∈ s, HasDerivWithinAt f (f' x) s x
hf : 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 ⊆ a ∪ (b ∪ c)
Module: Mathlib.MeasureTheory.Function.JacobianOneDim

Goal state before:
pp: F : Type u_1
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
s : Set ℝ
f f' : ℝ → ℝ
hs : MeasurableSet s
hf' : ∀ x ∈ s, HasDerivWithinAt f (f' x) s x
hf : 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
this : ∫ (x : ℝ) in a, f' x • g (f x) = 0
⊢ ∫ (x : ℝ) in b ∪ c, f' x • g (f x) ∂volume = ∫ (x : ℝ) in c, f' x • g (f x)

Tactic:
]

Goal state after:
F : Type u_1
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
s : Set ℝ
f f' : ℝ → ℝ
hs : MeasurableSet s
hf' : ∀ x ∈ s, HasDerivWithinAt f (f' x) s x
hf : 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
this : ∫ (x : ℝ) in a, f' x • g (f x) = 0
⊢ ∫ (x : ℝ) in b ∪ c, f' x • g (f x) ∂volume = ∫ (x : ℝ) in c, f' x • g (f x)
Module: Mathlib.MeasureTheory.Integral.IntegralEqImproper

Goal state before:
pp: α : Type u_1
ι : Type u_2
inst✝ : MeasurableSpace α
μ : Measure α
l : Filter ι
s : Set α
φ : ι → Set α
hs : MeasurableSet s
ae_eventually_mem : ∀ᵐ (x : α) ∂μ, x ∈ s → ∀ᶠ (n : ι) in l, x ∈ φ n
measurable : ∀ (n : ι), MeasurableSet (φ n)
⊢ ∀ᵐ (x : α) ∂μ, x ∈ s → ∀ᶠ (i : ι) in l, x ∈ φ i

Tactic:
]

Goal state after:
α : Type u_1
ι : Type u_2
inst✝ : MeasurableSpace α
μ : Measure α
l : Filter ι
s : Set α
φ : ι → Set α
hs : MeasurableSet s
ae_eventually_mem : ∀ᵐ (x : α) ∂μ, x ∈ s → ∀ᶠ (n : ι) in l, x ∈ φ n
measurable : ∀ (n : ι), MeasurableSet (φ n)
⊢ ∀ᵐ (x : α) ∂μ, x ∈ s → ∀ᶠ (i : ι) in l, x ∈ φ 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
x : E
r : ℝ
xs : x ∈ s
r0 : 0 ≤ r
z : E
zs : z ∈ s
zr : ‖z - x‖ ≤ r
⊢ f z ∈ ⇑A '' closedBall 0 r + closedBall (f x) (ε * r)

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
z : E
zs : z ∈ s
zr : ‖z - x‖ ≤ r
⊢ f z ∈ ⇑A '' closedBall 0 r + closedBall (f x) (ε * r)
Module: Mathlib.MeasureTheory.Integral.IntegralEqImproper

Goal state before:
pp: α : Type u_1
ι : Type u_2
inst✝³ : MeasurableSpace α
μ : Measure α
l : Filter ι
β : Type u_3
inst✝² : MeasurableSpace β
inst✝¹ : l.IsCountablyGenerated
inst✝ : l.NeBot
f : α → β
φ : ι → Set α
hφ : AECover μ l φ
hfm : ∀ (i : ι), AEMeasurable f (μ.restrict (φ i))
u : ℕ → ι
hu : Tendsto u atTop l
this : AEMeasurable f μ
⊢ AEMeasurable f μ

pp: case hs
α : Type u_1
ι : Type u_2
inst✝³ : MeasurableSpace α
μ : Measure α
l : Filter ι
β : Type u_3
inst✝² : MeasurableSpace β
inst✝¹ : l.IsCountablyGenerated
inst✝ : l.NeBot
f : α → β
φ : ι → Set α
hφ : AECover μ l φ
hfm : ∀ (i : ι), AEMeasurable f (μ.restrict (φ i))
u : ℕ → ι
hu : Tendsto u atTop l
this : AEMeasurable f (μ.restrict (⋃ i, φ (u i)))
⊢ ∀ᵐ (x : α) ∂μ, x ∈ ⋃ i, φ (u i)

Tactic:
]

Goal state after:
α : Type u_1
ι : Type u_2
inst✝³ : MeasurableSpace α
μ : Measure α
l : Filter ι
β : Type u_3
inst✝² : MeasurableSpace β
inst✝¹ : l.IsCountablyGenerated
inst✝ : l.NeBot
f : α → β
φ : ι → Set α
hφ : AECover μ l φ
hfm : ∀ (i : ι), AEMeasurable f (μ.restrict (φ i))
u : ℕ → ι
hu : Tendsto u atTop l
this : AEMeasurable f μ
⊢ AEMeasurable f μ

case hs
α : Type u_1
ι : Type u_2
inst✝³ : MeasurableSpace α
μ : Measure α
l : Filter ι
β : Type u_3
inst✝² : MeasurableSpace β
inst✝¹ : l.IsCountablyGenerated
inst✝ : l.NeBot
f : α → β
φ : ι → Set α
hφ : AECover μ l φ
hfm : ∀ (i : ι), AEMeasurable f (μ.restrict (φ i))
u : ℕ → ι
hu : Tendsto u atTop l
this : AEMeasurable f (μ.restrict (⋃ i, φ (u i)))
⊢ ∀ᵐ (x : α) ∂μ, x ∈ ⋃ i, φ (u i)
Module: Mathlib.MeasureTheory.Integral.IntegralEqImproper

Goal state before:
pp: α : Type u_1
ι : Type u_2
inst✝⁴ : MeasurableSpace α
μ : Measure α
l : Filter ι
β : Type u_3
inst✝³ : TopologicalSpace β
inst✝² : PseudoMetrizableSpace β
inst✝¹ : l.IsCountablyGenerated
inst✝ : l.NeBot
f : α → β
φ : ι → Set α
hφ : AECover μ l φ
hfm : ∀ (i : ι), AEStronglyMeasurable f (μ.restrict (φ i))
u : ℕ → ι
hu : Tendsto u atTop l
this : AEStronglyMeasurable f μ
⊢ AEStronglyMeasurable f μ

pp: case hs
α : Type u_1
ι : Type u_2
inst✝⁴ : MeasurableSpace α
μ : Measure α
l : Filter ι
β : Type u_3
inst✝³ : TopologicalSpace β
inst✝² : PseudoMetrizableSpace β
inst✝¹ : l.IsCountablyGenerated
inst✝ : l.NeBot
f : α → β
φ : ι → Set α
hφ : AECover μ l φ
hfm : ∀ (i : ι), AEStronglyMeasurable f (μ.restrict (φ i))
u : ℕ → ι
hu : Tendsto u atTop l
this : AEStronglyMeasurable f (μ.restrict (⋃ i, φ (u i)))
⊢ ∀ᵐ (x : α) ∂μ, x ∈ ⋃ i, φ (u i)

Tactic:
]

Goal state after:
α : Type u_1
ι : Type u_2
inst✝⁴ : MeasurableSpace α
μ : Measure α
l : Filter ι
β : Type u_3
inst✝³ : TopologicalSpace β
inst✝² : PseudoMetrizableSpace β
inst✝¹ : l.IsCountablyGenerated
inst✝ : l.NeBot
f : α → β
φ : ι → Set α
hφ : AECover μ l φ
hfm : ∀ (i : ι), AEStronglyMeasurable f (μ.restrict (φ i))
u : ℕ → ι
hu : Tendsto u atTop l
this : AEStronglyMeasurable f μ
⊢ AEStronglyMeasurable f μ

case hs
α : Type u_1
ι : Type u_2
inst✝⁴ : MeasurableSpace α
μ : Measure α
l : Filter ι
β : Type u_3
inst✝³ : TopologicalSpace β
inst✝² : PseudoMetrizableSpace β
inst✝¹ : l.IsCountablyGenerated
inst✝ : l.NeBot
f : α → β
φ : ι → Set α
hφ : AECover μ l φ
hfm : ∀ (i : ι), AEStronglyMeasurable f (μ.restrict (φ i))
u : ℕ → ι
hu : Tendsto u atTop l
this : AEStronglyMeasurable f (μ.restrict (⋃ i, φ (u i)))
⊢ ∀ᵐ (x : α) ∂μ, x ∈ ⋃ i, φ (u i)
Module: Mathlib.MeasureTheory.Function.JacobianOneDim

Goal state before:
pp: F : Type u_1
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
s : Set ℝ
f f' : ℝ → ℝ
hs : MeasurableSet s
hf' : ∀ x ∈ s, HasDerivWithinAt f (f' x) s x
hf : 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
this✝ : ∫ (x : ℝ) in a, f' x • g (f x) = 0
this : ∫ (x : ℝ) in b, f' x • g (f x) = 0
⊢ ∫ (x : ℝ) in c, f' x • g (f x) ∂volume = ∫ (x : ℝ) in c, f' x • g (f x)

Tactic:
]

Goal state after:
Module: Mathlib.MeasureTheory.Integral.IntegralEqImproper

Goal state before:
pp: α : Type u_1
inst✝ : MeasurableSpace α
μ : Measure α
φ : ℕ → Set α
hφ : AECover μ atTop φ
hmono : Monotone φ
f : α → ℝ≥0∞
hfm : AEMeasurable f μ
F : ℕ → α → ℝ≥0∞ := fun n ↦ (φ n).indicator f
key₁ : ∀ (n : ℕ), AEMeasurable (F n) μ
x : α
_i _j : ℕ
hij : _i ≤ _j
⊢ (φ _j).indicator f x ≤ (φ _j).indicator f x

Tactic:
]

Goal state after:
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 (Icc 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
⊢ ContinuousOn f (Icc a b)

Tactic:
]

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 (Icc 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
⊢ ContinuousOn f (Icc a b)
Module: Mathlib.MeasureTheory.Function.JacobianOneDim

Goal state before:
pp: F : Type u_1
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
s : Set ℝ
f f' : ℝ → ℝ
hs : MeasurableSet s
hf' : ∀ x ∈ s, HasDerivWithinAt f (f' x) s x
hf : 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)
⊢ f '' a ∪ (f '' b ∪ f '' c) =ᵐ[volume] f '' c

Tactic:
]

Goal state after:
F : Type u_1
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
s : Set ℝ
f f' : ℝ → ℝ
hs : MeasurableSet s
hf' : ∀ x ∈ s, HasDerivWithinAt f (f' x) s x
hf : 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)
⊢ f '' a ∪ (f '' b ∪ f '' c) =ᵐ[volume] f '' c
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
z : E
zs : z ∈ s
zr : ‖z - x‖ ≤ r
⊢ ‖f z - f x - A (z - x)‖ ≤ ε * r

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
z : E
zs : z ∈ s
zr : ‖z - x‖ ≤ r
⊢ ‖f z - f x - A (z - x)‖ ≤ ε * r
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
⊢ ∫ (x : ℝ) in f '' c, g x = ∫ (x : ℝ) in c, f' x • 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
⊢ ∫ (x : ℝ) in f '' c, g x = ∫ (x : ℝ) in c, f' x • g (f x)
Module: Mathlib.MeasureTheory.Integral.IntegralEqImproper

Goal state before:
pp: α : Type u_1
ι : Type u_2
E : Type u_3
inst✝³ : MeasurableSpace α
μ : Measure α
l : Filter ι
inst✝² : NormedAddCommGroup E
inst✝¹ : l.NeBot
inst✝ : l.IsCountablyGenerated
φ : ι → Set α
hφ : AECover μ l φ
f : α → E
I : ℝ
hfi : ∀ (i : ι), IntegrableOn f (φ i) μ
hbounded : ∀ᶠ (i : ι) in l, (∫⁻ (a : α) in φ i, ↑‖f a‖₊ ∂μ).toReal ≤ I
hfm : AEStronglyMeasurable f μ
i : ι
hi : (∫⁻ (a : α) in φ i, ↑‖f a‖₊ ∂μ).toReal ≤ I
⊢ ENNReal.ofReal (∫⁻ (a : α) in φ i, ‖f a‖ₑ ∂μ).toReal ≤ ENNReal.ofReal I

Tactic:
]

Goal state after:
α : Type u_1
ι : Type u_2
E : Type u_3
inst✝³ : MeasurableSpace α
μ : Measure α
l : Filter ι
inst✝² : NormedAddCommGroup E
inst✝¹ : l.NeBot
inst✝ : l.IsCountablyGenerated
φ : ι → Set α
hφ : AECover μ l φ
f : α → E
I : ℝ
hfi : ∀ (i : ι), IntegrableOn f (φ i) μ
hbounded : ∀ᶠ (i : ι) in l, (∫⁻ (a : α) in φ i, ↑‖f a‖₊ ∂μ).toReal ≤ I
hfm : AEStronglyMeasurable f μ
i : ι
hi : (∫⁻ (a : α) in φ i, ↑‖f a‖₊ ∂μ).toReal ≤ I
⊢ ENNReal.ofReal (∫⁻ (a : α) in φ i, ‖f a‖ₑ ∂μ).toReal ≤ ENNReal.ofReal I
Module: Mathlib.MeasureTheory.Integral.IntegralEqImproper

Goal state before:
pp: α : Type u_1
ι : Type u_2
E : Type u_3
inst✝³ : MeasurableSpace α
μ : Measure α
l : Filter ι
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : l.IsCountablyGenerated
φ : ι → Set α
hφ : AECover μ l φ
f : α → E
hfi : Integrable f μ
h : Tendsto (fun i ↦ ∫ (x : α), (φ i).indicator f x ∂μ) l (𝓝 (∫ (x : α), f x ∂μ))
x✝ : ι
⊢ ∫ (x : α) in φ x✝, f x ∂μ = ∫ (x : α) in φ x✝, f x ∂μ

Tactic:
]

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
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) ⊆ ⇑A '' closedBall 0 r + closedBall (f x) (ε * r)
⊢ ⇑A '' closedBall 0 r + closedBall (f x) (r * ε) = ⇑A '' closedBall 0 r + closedBall (f x) (r * ε)

Tactic:
]

Goal state after:
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 b : ι → ℝ
f : ℝ → E
I : ℝ
hfi : ∀ (i : ι), IntegrableOn f (Ioc (a i) (b i)) μ
ha : Tendsto a l atBot
hb : Tendsto b l atTop
h : ∀ᶠ (i : ι) in l, ∫ (x : ℝ) in a i..b i, ‖f x‖ ∂μ ≤ I
hφ : AECover μ l fun i ↦ Ioc (a i) (b i)
i : ι
hai : a i ≤ 0
hbi : 0 ≤ b i
ht : ∫ (x : ℝ) in a i..b i, ‖f x‖ ∂μ ≤ I
⊢ ∫ (x : ℝ) in a i..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
a b : ι → ℝ
f : ℝ → E
I : ℝ
hfi : ∀ (i : ι), IntegrableOn f (Ioc (a i) (b i)) μ
ha : Tendsto a l atBot
hb : Tendsto b l atTop
h : ∀ᶠ (i : ι) in l, ∫ (x : ℝ) in a i..b i, ‖f x‖ ∂μ ≤ I
hφ : AECover μ l fun i ↦ Ioc (a i) (b i)
i : ι
hai : a i ≤ 0
hbi : 0 ≤ b i
ht : ∫ (x : ℝ) in a i..b i, ‖f x‖ ∂μ ≤ I
⊢ ∫ (x : ℝ) in a i..b i, ‖f x‖ ∂μ ≤ I
Module: Mathlib.MeasureTheory.Function.JacobianOneDim

Goal state before:
pp: F : Type u_1
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
s : Set ℝ
f f' : ℝ → ℝ
hs : MeasurableSet s
hf' : ∀ x ∈ s, HasDerivWithinAt f (f' x) s x
hf : 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
x : ℝ
hx : x ∈ c
⊢ f' x • g (f x) = f' x • g (f x)

Tactic:
]

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
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)
⊢ μ (f '' (s ∩ closedBall x r)) ≤ ↑m * μ (closedBall x r)

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)
⊢ μ (f '' (s ∩ closedBall x r)) ≤ ↑m * μ (closedBall x r)
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)
i : ι
⊢ Integrable f (μ.restrict (Ioi (a i) ∩ Iic b))

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)
i : ι
⊢ Integrable f (μ.restrict (Ioi (a i) ∩ Iic b))