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

Goal state before:
pp: E : Type u_1
F : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : FiniteDimensional ℝ E
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℝ F
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
hs : s.Nonempty
T : Set ↑s
T_count : T.Countable
hT : ⋃ x ∈ T, ball (f' ↑x) ↑(r (f' ↑x)) = ⋃ x, ball (f' ↑x) ↑(r (f' ↑x))
u : ℕ → ℝ
left✝ : StrictAnti u
u_pos : ∀ (n : ℕ), 0 < u n
u_lim : Tendsto u atTop (𝓝 0)
M : ℕ → ↑T → Set E :=
  fun n z ↦ {x | x ∈ s ∧ ∀ y ∈ s ∩ ball x (u n), ‖f y - f x - (f' ↑↑z) (y - x)‖ ≤ ↑(r (f' ↑↑z)) * ‖y - x‖}
x : E
xs : x ∈ s
z : ↑s
zT : z ∈ T
hz : f' x ∈ ball (f' ↑z) ↑(r (f' ↑z))
ε : ℝ
εpos : 0 < ε
hε : ‖f' x - f' ↑z‖ + ε ≤ ↑(r (f' ↑z))
δ : ℝ
δpos : 0 < δ
hδ : ball x δ ∩ s ⊆ {y | ‖f y - f x - (f' x) (y - x)‖ ≤ ε * ‖y - x‖}
n : ℕ
hn : u n < δ
y : E
hy : y ∈ s ∩ ball x (u n)
⊢ y ∈ s ∩ ball x δ

Tactic:
]

Goal state after:
E : Type u_1
F : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : FiniteDimensional ℝ E
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℝ F
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
hs : s.Nonempty
T : Set ↑s
T_count : T.Countable
hT : ⋃ x ∈ T, ball (f' ↑x) ↑(r (f' ↑x)) = ⋃ x, ball (f' ↑x) ↑(r (f' ↑x))
u : ℕ → ℝ
left✝ : StrictAnti u
u_pos : ∀ (n : ℕ), 0 < u n
u_lim : Tendsto u atTop (𝓝 0)
M : ℕ → ↑T → Set E :=
  fun n z ↦ {x | x ∈ s ∧ ∀ y ∈ s ∩ ball x (u n), ‖f y - f x - (f' ↑↑z) (y - x)‖ ≤ ↑(r (f' ↑↑z)) * ‖y - x‖}
x : E
xs : x ∈ s
z : ↑s
zT : z ∈ T
hz : f' x ∈ ball (f' ↑z) ↑(r (f' ↑z))
ε : ℝ
εpos : 0 < ε
hε : ‖f' x - f' ↑z‖ + ε ≤ ↑(r (f' ↑z))
δ : ℝ
δpos : 0 < δ
hδ : ball x δ ∩ s ⊆ {y | ‖f y - f x - (f' x) (y - x)‖ ≤ ε * ‖y - x‖}
n : ℕ
hn : u n < δ
y : E
hy : y ∈ s ∩ ball x (u n)
⊢ y ∈ s ∩ ball x δ
Module: Mathlib.MeasureTheory.Function.JacobianOneDim

Goal state before:
pp: s : Set ℝ
f f' : ℝ → ℝ
hs : MeasurableSet s
hf : MonotoneOn f s
hf' : ∀ x ∈ s, HasDerivWithinAt f (f' x) s x
a : Set ℝ := {x | x ∈ s ∧ 𝓝[s ∩ Ioi x] x = ⊥} ∪ {x | x ∈ s ∧ 𝓝[s ∩ Iio x] x = ⊥}
a_count : a.Countable
s₁ : Set ℝ := s \ a
hs₁ : MeasurableSet s₁
u : Set ℝ := {c | ∃ x y, x ∈ s₁ ∧ y ∈ s₁ ∧ x < y ∧ f x = c ∧ f y = c}
hu : u.Countable
b : Set ℝ := s₁ ∩ f ⁻¹' u
hb : MeasurableSet b
c : Set ℝ := s₁ \ b
hc : MeasurableSet c
x : ℝ
hx : x ∈ b
p : ℝ
ps₁ : p ∈ s₁
px✝ : p ≠ x
fpx : f p = f x
px : x < p
y : ℝ
hy : y ∈ s ∩ Ioo x p
⊢ f y ≤ f p

Tactic:
]

Goal state after:
s : Set ℝ
f f' : ℝ → ℝ
hs : MeasurableSet s
hf : MonotoneOn f s
hf' : ∀ x ∈ s, HasDerivWithinAt f (f' x) s x
a : Set ℝ := {x | x ∈ s ∧ 𝓝[s ∩ Ioi x] x = ⊥} ∪ {x | x ∈ s ∧ 𝓝[s ∩ Iio x] x = ⊥}
a_count : a.Countable
s₁ : Set ℝ := s \ a
hs₁ : MeasurableSet s₁
u : Set ℝ := {c | ∃ x y, x ∈ s₁ ∧ y ∈ s₁ ∧ x < y ∧ f x = c ∧ f y = c}
hu : u.Countable
b : Set ℝ := s₁ ∩ f ⁻¹' u
hb : MeasurableSet b
c : Set ℝ := s₁ \ b
hc : MeasurableSet c
x : ℝ
hx : x ∈ b
p : ℝ
ps₁ : p ∈ s₁
px✝ : p ≠ x
fpx : f p = f x
px : x < p
y : ℝ
hy : y ∈ s ∩ Ioo x p
⊢ f y ≤ f p
Module: Mathlib.Analysis.Calculus.LHopital

Goal state before:
pp: a : ℝ
l : Filter ℝ
f f' g g' : ℝ → ℝ
hff' : ∃ v ∈ 𝓝[<] a, ∀ y ∈ v, HasDerivAt f (f' y) y
hgg' : ∃ v ∈ 𝓝[<] a, ∀ y ∈ v, HasDerivAt g (g' y) y
hg' : ∃ v ∈ 𝓝[<] a, ∀ y ∈ v, g' y ≠ 0
hfa : Tendsto f (𝓝[<] a) (𝓝 0)
hga : Tendsto g (𝓝[<] a) (𝓝 0)
hdiv : Tendsto (fun x ↦ f' x / g' x) (𝓝[<] a) l
⊢ Tendsto (fun x ↦ f x / g x) (𝓝[<] a) l

Tactic:
]

Goal state after:
a : ℝ
l : Filter ℝ
f f' g g' : ℝ → ℝ
hff' : ∃ v ∈ 𝓝[<] a, ∀ y ∈ v, HasDerivAt f (f' y) y
hgg' : ∃ v ∈ 𝓝[<] a, ∀ y ∈ v, HasDerivAt g (g' y) y
hg' : ∃ v ∈ 𝓝[<] a, ∀ y ∈ v, g' y ≠ 0
hfa : Tendsto f (𝓝[<] a) (𝓝 0)
hga : Tendsto g (𝓝[<] a) (𝓝 0)
hdiv : Tendsto (fun x ↦ f' x / g' x) (𝓝[<] a) l
⊢ Tendsto (fun x ↦ f x / g x) (𝓝[<] a) l
Module: Mathlib.MeasureTheory.Function.Jacobian

Goal state before:
pp: E : Type u_1
F : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : FiniteDimensional ℝ E
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℝ F
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
hs : s.Nonempty
T : Set ↑s
T_count : T.Countable
hT : ⋃ x ∈ T, ball (f' ↑x) ↑(r (f' ↑x)) = ⋃ x, ball (f' ↑x) ↑(r (f' ↑x))
u : ℕ → ℝ
left✝ : StrictAnti u
u_pos : ∀ (n : ℕ), 0 < u n
u_lim : Tendsto u atTop (𝓝 0)
M : ℕ → ↑T → Set E :=
  fun n z ↦ {x | x ∈ s ∧ ∀ y ∈ s ∩ ball x (u n), ‖f y - f x - (f' ↑↑z) (y - x)‖ ≤ ↑(r (f' ↑↑z)) * ‖y - x‖}
x : E
xs : x ∈ s
z : ↑s
zT : z ∈ T
hz : f' x ∈ ball (f' ↑z) ↑(r (f' ↑z))
ε : ℝ
εpos : 0 < ε
hε : ‖f' x - f' ↑z‖ + ε ≤ ↑(r (f' ↑z))
δ : ℝ
δpos : 0 < δ
hδ : ball x δ ∩ s ⊆ {y | ‖f y - f x - (f' x) (y - x)‖ ≤ ε * ‖y - x‖}
n : ℕ
hn : u n < δ
y : E
hy : y ∈ s ∩ ball x (u n)
⊢ (‖f' x - f' ↑z‖ + ε) * ‖y - x‖ ≤ ↑(r (f' ↑z)) * ‖y - x‖

Tactic:
]

Goal state after:
E : Type u_1
F : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : FiniteDimensional ℝ E
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℝ F
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
hs : s.Nonempty
T : Set ↑s
T_count : T.Countable
hT : ⋃ x ∈ T, ball (f' ↑x) ↑(r (f' ↑x)) = ⋃ x, ball (f' ↑x) ↑(r (f' ↑x))
u : ℕ → ℝ
left✝ : StrictAnti u
u_pos : ∀ (n : ℕ), 0 < u n
u_lim : Tendsto u atTop (𝓝 0)
M : ℕ → ↑T → Set E :=
  fun n z ↦ {x | x ∈ s ∧ ∀ y ∈ s ∩ ball x (u n), ‖f y - f x - (f' ↑↑z) (y - x)‖ ≤ ↑(r (f' ↑↑z)) * ‖y - x‖}
x : E
xs : x ∈ s
z : ↑s
zT : z ∈ T
hz : f' x ∈ ball (f' ↑z) ↑(r (f' ↑z))
ε : ℝ
εpos : 0 < ε
hε : ‖f' x - f' ↑z‖ + ε ≤ ↑(r (f' ↑z))
δ : ℝ
δpos : 0 < δ
hδ : ball x δ ∩ s ⊆ {y | ‖f y - f x - (f' x) (y - x)‖ ≤ ε * ‖y - x‖}
n : ℕ
hn : u n < δ
y : E
hy : y ∈ s ∩ ball x (u n)
⊢ (‖f' x - f' ↑z‖ + ε) * ‖y - x‖ ≤ ↑(r (f' ↑z)) * ‖y - x‖
Module: Mathlib.Analysis.Calculus.LHopital

Goal state before:
pp: a : ℝ
l : Filter ℝ
f f' g g' : ℝ → ℝ
hfa : Tendsto f (𝓝[<] a) (𝓝 0)
hga : Tendsto g (𝓝[<] a) (𝓝 0)
hdiv : Tendsto (fun x ↦ f' x / g' x) (𝓝[<] a) l
s₁ : Set ℝ
hs₁ : s₁ ∈ 𝓝[<] a
hff' : ∀ y ∈ s₁, HasDerivAt f (f' y) y
s₂ : Set ℝ
hs₂ : s₂ ∈ 𝓝[<] a
hgg' : ∀ y ∈ s₂, HasDerivAt g (g' y) y
s₃ : Set ℝ
hs₃ : s₃ ∈ 𝓝[<] a
hg' : ∀ y ∈ s₃, g' y ≠ 0
s : Set ℝ := s₁ ∩ s₂ ∩ s₃
hs : ∃ l ∈ Iio a, Ioo l a ⊆ s
⊢ Tendsto (fun x ↦ f x / g x) (𝓝[<] a) l

Tactic:
]

Goal state after:
a : ℝ
l : Filter ℝ
f f' g g' : ℝ → ℝ
hfa : Tendsto f (𝓝[<] a) (𝓝 0)
hga : Tendsto g (𝓝[<] a) (𝓝 0)
hdiv : Tendsto (fun x ↦ f' x / g' x) (𝓝[<] a) l
s₁ : Set ℝ
hs₁ : s₁ ∈ 𝓝[<] a
hff' : ∀ y ∈ s₁, HasDerivAt f (f' y) y
s₂ : Set ℝ
hs₂ : s₂ ∈ 𝓝[<] a
hgg' : ∀ y ∈ s₂, HasDerivAt g (g' y) y
s₃ : Set ℝ
hs₃ : s₃ ∈ 𝓝[<] a
hg' : ∀ y ∈ s₃, g' y ≠ 0
s : Set ℝ := s₁ ∩ s₂ ∩ s₃
hs : ∃ l ∈ Iio a, Ioo l a ⊆ s
⊢ Tendsto (fun x ↦ f x / g x) (𝓝[<] a) l
Module: Mathlib.MeasureTheory.Integral.IntervalIntegral.IntegrationByParts

Goal state before:
pp: 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), HasDerivWithinAt f (f' x) (Ioi x) x
hg1 : IntervalIntegrable g volume (sInf (f '' [[a, b]])) (sSup (f '' [[a, b]]))
hg2 : IntegrableOn (fun x ↦ f' x • (g ∘ f) x) [[a, b]] volume
hE : CompleteSpace E
h_cont : ContinuousOn (fun u ↦ ∫ (t : ℝ) in f a..f u, g t) [[a, b]]
x : ℝ
hx : x ∈ Ioo (min a b) (max a b)
c : ℝ
hc : c ∈ Ioo (min a b) x
d : ℝ
hd : d ∈ Ioo x (max a b)
cdsub : [[c, d]] ⊆ Ioo (min a b) (max a b)
J : Set ℝ := [[sInf (f '' [[c, d]]), sSup (f '' [[c, d]])]]
hg_cont : ContinuousOn g J
hJ : f '' [[c, d]] = J
h2x : f x ∈ J
h2g : IntervalIntegrable g volume (f a) (f x)
h3g : StronglyMeasurableAtFilter g (𝓝[J] f x) volume
this✝ : Fact (f x ∈ J)
this : HasDerivWithinAt (fun u ↦ ∫ (x : ℝ) in f a..u, g x) (g (f x)) J (f x)
⊢ MapsTo f (Ioo x d) (f '' [[c, d]])

Tactic:
]

Goal state after:
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), HasDerivWithinAt f (f' x) (Ioi x) x
hg1 : IntervalIntegrable g volume (sInf (f '' [[a, b]])) (sSup (f '' [[a, b]]))
hg2 : IntegrableOn (fun x ↦ f' x • (g ∘ f) x) [[a, b]] volume
hE : CompleteSpace E
h_cont : ContinuousOn (fun u ↦ ∫ (t : ℝ) in f a..f u, g t) [[a, b]]
x : ℝ
hx : x ∈ Ioo (min a b) (max a b)
c : ℝ
hc : c ∈ Ioo (min a b) x
d : ℝ
hd : d ∈ Ioo x (max a b)
cdsub : [[c, d]] ⊆ Ioo (min a b) (max a b)
J : Set ℝ := [[sInf (f '' [[c, d]]), sSup (f '' [[c, d]])]]
hg_cont : ContinuousOn g J
hJ : f '' [[c, d]] = J
h2x : f x ∈ J
h2g : IntervalIntegrable g volume (f a) (f x)
h3g : StronglyMeasurableAtFilter g (𝓝[J] f x) volume
this✝ : Fact (f x ∈ J)
this : HasDerivWithinAt (fun u ↦ ∫ (x : ℝ) in f a..u, g x) (g (f x)) J (f x)
⊢ MapsTo f (Ioo x d) (f '' [[c, d]])
Module: Mathlib.MeasureTheory.Integral.IntervalIntegral.IntegrationByParts

Goal state before:
pp: case pos
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), HasDerivWithinAt f (f' x) (Ioi x) x
hg_cont : ContinuousOn g (f '' Ioo (min a b) (max a b))
hg1 : IntervalIntegrable g volume (sInf (f '' [[a, b]])) (sSup (f '' [[a, b]]))
hg2 : IntervalIntegrable (fun x ↦ f' x • (g ∘ f) x) volume a b
hE : CompleteSpace E
h_cont : ContinuousOn (fun u ↦ ∫ (t : ℝ) in f a..f u, g t) [[a, b]]
h_der :
  ∀ x ∈ Ioo (min a b) (max a b), HasDerivWithinAt (fun u ↦ ∫ (t : ℝ) in f a..f u, g t) (f' x • (g ∘ f) x) (Ioi x) x
⊢ ∫ (x : ℝ) in a..b, f' x • (g ∘ f) x = ∫ (u : ℝ) in f a..f b, g u

Tactic:
]

Goal state after:
case pos
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), HasDerivWithinAt f (f' x) (Ioi x) x
hg_cont : ContinuousOn g (f '' Ioo (min a b) (max a b))
hg1 : IntervalIntegrable g volume (sInf (f '' [[a, b]])) (sSup (f '' [[a, b]]))
hg2 : IntervalIntegrable (fun x ↦ f' x • (g ∘ f) x) volume a b
hE : CompleteSpace E
h_cont : ContinuousOn (fun u ↦ ∫ (t : ℝ) in f a..f u, g t) [[a, b]]
h_der :
  ∀ x ∈ Ioo (min a b) (max a b), HasDerivWithinAt (fun u ↦ ∫ (t : ℝ) in f a..f u, g t) (f' x • (g ∘ f) x) (Ioi x) x
⊢ ∫ (x : ℝ) in a..b, f' x • (g ∘ f) x = ∫ (u : ℝ) in f a..f b, g u
Module: Mathlib.MeasureTheory.Function.JacobianOneDim

Goal state before:
pp: s : Set ℝ
f f' : ℝ → ℝ
hs : MeasurableSet s
hf : MonotoneOn f s
hf' : ∀ x ∈ s, HasDerivWithinAt f (f' x) s x
a : Set ℝ := {x | x ∈ s ∧ 𝓝[s ∩ Ioi x] x = ⊥} ∪ {x | x ∈ s ∧ 𝓝[s ∩ Iio x] x = ⊥}
a_count : a.Countable
s₁ : Set ℝ := s \ a
hs₁ : MeasurableSet s₁
u : Set ℝ := {c | ∃ x y, x ∈ s₁ ∧ y ∈ s₁ ∧ x < y ∧ f x = c ∧ f y = c}
hu : u.Countable
b : Set ℝ := s₁ ∩ f ⁻¹' u
hb : MeasurableSet b
c : Set ℝ := s₁ \ b
hc : MeasurableSet c
x : ℝ
hx : x ∈ b
p : ℝ
ps₁ : p ∈ s₁
px✝ : p ≠ x
fpx : f p = f x
px : x < p
K : HasDerivWithinAt f 0 (s ∩ Ioo x p) x
K' : HasDerivWithinAt f (f' x) (s ∩ Ioo x p) x
J1 : (s ∩ Ioo x p) \ {x} = s ∩ Ioo x p
J2 : 𝓝[s ∩ Ioo x p] x = 𝓝[s ∩ Ioi x] x
⊢ (𝓝[s ∩ Ioi x] x).NeBot

Tactic:
]

Goal state after:
s : Set ℝ
f f' : ℝ → ℝ
hs : MeasurableSet s
hf : MonotoneOn f s
hf' : ∀ x ∈ s, HasDerivWithinAt f (f' x) s x
a : Set ℝ := {x | x ∈ s ∧ 𝓝[s ∩ Ioi x] x = ⊥} ∪ {x | x ∈ s ∧ 𝓝[s ∩ Iio x] x = ⊥}
a_count : a.Countable
s₁ : Set ℝ := s \ a
hs₁ : MeasurableSet s₁
u : Set ℝ := {c | ∃ x y, x ∈ s₁ ∧ y ∈ s₁ ∧ x < y ∧ f x = c ∧ f y = c}
hu : u.Countable
b : Set ℝ := s₁ ∩ f ⁻¹' u
hb : MeasurableSet b
c : Set ℝ := s₁ \ b
hc : MeasurableSet c
x : ℝ
hx : x ∈ b
p : ℝ
ps₁ : p ∈ s₁
px✝ : p ≠ x
fpx : f p = f x
px : x < p
K : HasDerivWithinAt f 0 (s ∩ Ioo x p) x
K' : HasDerivWithinAt f (f' x) (s ∩ Ioo x p) x
J1 : (s ∩ Ioo x p) \ {x} = s ∩ Ioo x p
J2 : 𝓝[s ∩ Ioo x p] x = 𝓝[s ∩ Ioi x] x
⊢ (𝓝[s ∩ Ioi x] x).NeBot
Module: Mathlib.MeasureTheory.Integral.IntervalIntegral.IntegrationByParts

Goal state before:
pp: 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), HasDerivWithinAt f (f' x) (Ioi x) x
hf' : ContinuousOn f' [[a, b]]
hg : ContinuousOn g [[sInf (f '' [[a, b]]), sSup (f '' [[a, b]])]]
⊢ IntegrableOn g [[sInf (f '' [[a, b]]), sSup (f '' [[a, b]])]] volume

Tactic:
]

Goal state after:
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), HasDerivWithinAt f (f' x) (Ioi x) x
hf' : ContinuousOn f' [[a, b]]
hg : ContinuousOn g [[sInf (f '' [[a, b]]), sSup (f '' [[a, b]])]]
⊢ IntegrableOn g [[sInf (f '' [[a, b]]), sSup (f '' [[a, b]])]] volume
Module: Mathlib.MeasureTheory.Integral.IntervalIntegral.IntegrationByParts

Goal state before:
pp: case 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), 0 ≤ f' x
⊢ DifferentiableOn ℝ f (Ioo (min a b) (max a b))

Tactic:
]

Goal state after:
case 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), 0 ≤ f' x
⊢ DifferentiableOn ℝ f (Ioo (min a b) (max a b))
Module: Mathlib.MeasureTheory.Integral.IntervalIntegral.IntegrationByParts

Goal state before:
pp: case hf'_nonneg
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), 0 ≤ f' x
⊢ ∀ x ∈ Ioo (min a b) (max a b), 0 ≤ deriv f x

Tactic:
]

Goal state after:
case hf'_nonneg
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), 0 ≤ f' x
⊢ ∀ x ∈ Ioo (min a b) (max a b), 0 ≤ deriv f x
Module: Mathlib.MeasureTheory.Function.JacobianOneDim

Goal state before:
pp: s : Set ℝ
f f' : ℝ → ℝ
hs : MeasurableSet s
hf : MonotoneOn f s
hf' : ∀ x ∈ s, HasDerivWithinAt f (f' x) s x
a : Set ℝ := {x | x ∈ s ∧ 𝓝[s ∩ Ioi x] x = ⊥} ∪ {x | x ∈ s ∧ 𝓝[s ∩ Iio x] x = ⊥}
a_count : a.Countable
s₁ : Set ℝ := s \ a
hs₁ : MeasurableSet s₁
u : Set ℝ := {c | ∃ x y, x ∈ s₁ ∧ y ∈ s₁ ∧ x < y ∧ f x = c ∧ f y = c}
hu : u.Countable
b : Set ℝ := s₁ ∩ f ⁻¹' u
hb : MeasurableSet b
c : Set ℝ := s₁ \ b
hc : MeasurableSet c
x : ℝ
hx :
  (¬𝓝[s ∩ Ioi x] x = ⊥ ∧ ¬𝓝[s ∩ Iio x] x = ⊥) ∧
    x ∉ s \ ({x | x ∈ s ∧ 𝓝[s ∩ Ioi x] x = ⊥} ∪ {x | x ∈ s ∧ 𝓝[s ∩ Iio x] x = ⊥}) ∩ f ⁻¹' u
⊢ (𝓝[s \ {x}] x).NeBot

Tactic:
]

Goal state after:
s : Set ℝ
f f' : ℝ → ℝ
hs : MeasurableSet s
hf : MonotoneOn f s
hf' : ∀ x ∈ s, HasDerivWithinAt f (f' x) s x
a : Set ℝ := {x | x ∈ s ∧ 𝓝[s ∩ Ioi x] x = ⊥} ∪ {x | x ∈ s ∧ 𝓝[s ∩ Iio x] x = ⊥}
a_count : a.Countable
s₁ : Set ℝ := s \ a
hs₁ : MeasurableSet s₁
u : Set ℝ := {c | ∃ x y, x ∈ s₁ ∧ y ∈ s₁ ∧ x < y ∧ f x = c ∧ f y = c}
hu : u.Countable
b : Set ℝ := s₁ ∩ f ⁻¹' u
hb : MeasurableSet b
c : Set ℝ := s₁ \ b
hc : MeasurableSet c
x : ℝ
hx :
  (¬𝓝[s ∩ Ioi x] x = ⊥ ∧ ¬𝓝[s ∩ Iio x] x = ⊥) ∧
    x ∉ s \ ({x | x ∈ s ∧ 𝓝[s ∩ Ioi x] x = ⊥} ∪ {x | x ∈ s ∧ 𝓝[s ∩ Iio x] x = ⊥}) ∩ f ⁻¹' u
⊢ (𝓝[s \ {x}] x).NeBot
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), 0 ≤ f' x
M : MonotoneOn 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), 0 ≤ f' x
M : MonotoneOn f [[a, b]]
hab : a ≤ b
⊢ ContinuousOn f (Icc a b)
Module: Mathlib.MeasureTheory.Function.JacobianOneDim

Goal state before:
pp: s : Set ℝ
f f' : ℝ → ℝ
hs : MeasurableSet s
hf' : ∀ x ∈ s, HasDerivWithinAt f (f' x) s x
hf : MonotoneOn f s
u : ℝ → ℝ≥0∞
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
this : ∫⁻ (x : ℝ) in a, ENNReal.ofReal (f' x) * u (f x) = 0
⊢ ∫⁻ (a : ℝ) in b ∪ c, ENNReal.ofReal (f' a) * u (f a) ∂volume = ∫⁻ (x : ℝ) in c, ENNReal.ofReal (f' x) * u (f x)

Tactic:
]

Goal state after:
s : Set ℝ
f f' : ℝ → ℝ
hs : MeasurableSet s
hf' : ∀ x ∈ s, HasDerivWithinAt f (f' x) s x
hf : MonotoneOn f s
u : ℝ → ℝ≥0∞
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
this : ∫⁻ (x : ℝ) in a, ENNReal.ofReal (f' x) * u (f x) = 0
⊢ ∫⁻ (a : ℝ) in b ∪ c, ENNReal.ofReal (f' a) * u (f a) ∂volume = ∫⁻ (x : ℝ) in c, ENNReal.ofReal (f' x) * u (f x)
Module: Mathlib.MeasureTheory.Function.JacobianOneDim

Goal state before:
pp: s : Set ℝ
f f' : ℝ → ℝ
hs : MeasurableSet s
hf' : ∀ x ∈ s, HasDerivWithinAt f (f' x) s x
hf : MonotoneOn f s
u : ℝ → ℝ≥0∞
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
this✝ : ∫⁻ (x : ℝ) in a, ENNReal.ofReal (f' x) * u (f x) = 0
this : ∫⁻ (x : ℝ) in b, ENNReal.ofReal (f' x) * u (f x) = 0
⊢ ∫⁻ (a : ℝ) in c, ENNReal.ofReal (f' a) * u (f a) ∂volume = ∫⁻ (x : ℝ) in c, ENNReal.ofReal (f' x) * u (f x)

Tactic:
]

Goal state after:
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), 0 ≤ f' x
M : MonotoneOn 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), 0 ≤ f' x
M : MonotoneOn f [[a, b]]
hab : b < a
⊢ ContinuousOn f (Icc b a)
Module: Mathlib.MeasureTheory.Function.JacobianOneDim

Goal state before:
pp: s : Set ℝ
f f' : ℝ → ℝ
hs : MeasurableSet s
hf' : ∀ x ∈ s, HasDerivWithinAt f (f' x) s x
hf : MonotoneOn f s
u : ℝ → ℝ≥0∞
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
I : ∫⁻ (x : ℝ) in s, ENNReal.ofReal (f' x) * u (f x) = ∫⁻ (x : ℝ) in c, ENNReal.ofReal (f' x) * u (f x)
⊢ f '' a ∪ (f '' b ∪ f '' c) =ᵐ[volume] f '' c

Tactic:
]

Goal state after:
s : Set ℝ
f f' : ℝ → ℝ
hs : MeasurableSet s
hf' : ∀ x ∈ s, HasDerivWithinAt f (f' x) s x
hf : MonotoneOn f s
u : ℝ → ℝ≥0∞
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
I : ∫⁻ (x : ℝ) in s, ENNReal.ofReal (f' x) * u (f x) = ∫⁻ (x : ℝ) in c, ENNReal.ofReal (f' x) * u (f x)
⊢ f '' a ∪ (f '' b ∪ f '' c) =ᵐ[volume] f '' c
Module: Mathlib.Analysis.Calculus.LHopital

Goal state before:
pp: l : Filter ℝ
f f' g g' : ℝ → ℝ
hff' : ∃ v ∈ atTop, ∀ y ∈ v, HasDerivAt f (f' y) y
hgg' : ∃ v ∈ atTop, ∀ y ∈ v, HasDerivAt g (g' y) y
hg' : ∃ v ∈ atTop, ∀ y ∈ v, g' y ≠ 0
hftop : Tendsto f atTop (𝓝 0)
hgtop : Tendsto g atTop (𝓝 0)
hdiv : Tendsto (fun x ↦ f' x / g' x) atTop l
⊢ Tendsto (fun x ↦ f x / g x) atTop l

Tactic:
]

Goal state after:
l : Filter ℝ
f f' g g' : ℝ → ℝ
hff' : ∃ v ∈ atTop, ∀ y ∈ v, HasDerivAt f (f' y) y
hgg' : ∃ v ∈ atTop, ∀ y ∈ v, HasDerivAt g (g' y) y
hg' : ∃ v ∈ atTop, ∀ y ∈ v, g' y ≠ 0
hftop : Tendsto f atTop (𝓝 0)
hgtop : Tendsto g atTop (𝓝 0)
hdiv : Tendsto (fun x ↦ f' x / g' x) atTop l
⊢ Tendsto (fun x ↦ f x / g x) atTop l
Module: Mathlib.Analysis.Calculus.LHopital

Goal state before:
pp: l : Filter ℝ
f f' g g' : ℝ → ℝ
hftop : Tendsto f atTop (𝓝 0)
hgtop : Tendsto g atTop (𝓝 0)
hdiv : Tendsto (fun x ↦ f' x / g' x) atTop l
s₁ : Set ℝ
hs₁ : s₁ ∈ atTop
hff' : ∀ y ∈ s₁, HasDerivAt f (f' y) y
s₂ : Set ℝ
hs₂ : s₂ ∈ atTop
hgg' : ∀ y ∈ s₂, HasDerivAt g (g' y) y
s₃ : Set ℝ
hs₃ : s₃ ∈ atTop
hg' : ∀ y ∈ s₃, g' y ≠ 0
s : Set ℝ := s₁ ∩ s₂ ∩ s₃
hs : ∃ a, ∀ (b : ℝ), a ≤ b → b ∈ s
⊢ Tendsto (fun x ↦ f x / g x) atTop l

Tactic:
]

Goal state after:
l : Filter ℝ
f f' g g' : ℝ → ℝ
hftop : Tendsto f atTop (𝓝 0)
hgtop : Tendsto g atTop (𝓝 0)
hdiv : Tendsto (fun x ↦ f' x / g' x) atTop l
s₁ : Set ℝ
hs₁ : s₁ ∈ atTop
hff' : ∀ y ∈ s₁, HasDerivAt f (f' y) y
s₂ : Set ℝ
hs₂ : s₂ ∈ atTop
hgg' : ∀ y ∈ s₂, HasDerivAt g (g' y) y
s₃ : Set ℝ
hs₃ : s₃ ∈ atTop
hg' : ∀ y ∈ s₃, g' y ≠ 0
s : Set ℝ := s₁ ∩ s₂ ∩ s₃
hs : ∃ a, ∀ (b : ℝ), a ≤ b → b ∈ s
⊢ Tendsto (fun x ↦ f x / g x) atTop l
Module: Mathlib.MeasureTheory.Function.JacobianOneDim

Goal state before:
pp: s : Set ℝ
f f' : ℝ → ℝ
hs : MeasurableSet s
hf' : ∀ x ∈ s, HasDerivWithinAt f (f' x) s x
hf : MonotoneOn f s
u : ℝ → ℝ≥0∞
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
I : ∫⁻ (x : ℝ) in s, ENNReal.ofReal (f' x) * u (f x) = ∫⁻ (x : ℝ) in c, ENNReal.ofReal (f' x) * u (f x)
J : ∫⁻ (x : ℝ) in f '' s, u x = ∫⁻ (x : ℝ) in f '' c, u x
⊢ ∫⁻ (x : ℝ) in f '' c, u x = ∫⁻ (x : ℝ) in c, ENNReal.ofReal (f' x) * u (f x)

Tactic:
]

Goal state after:
s : Set ℝ
f f' : ℝ → ℝ
hs : MeasurableSet s
hf' : ∀ x ∈ s, HasDerivWithinAt f (f' x) s x
hf : MonotoneOn f s
u : ℝ → ℝ≥0∞
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
I : ∫⁻ (x : ℝ) in s, ENNReal.ofReal (f' x) * u (f x) = ∫⁻ (x : ℝ) in c, ENNReal.ofReal (f' x) * u (f x)
J : ∫⁻ (x : ℝ) in f '' s, u x = ∫⁻ (x : ℝ) in f '' c, u x
⊢ ∫⁻ (x : ℝ) in f '' c, u x = ∫⁻ (x : ℝ) in c, ENNReal.ofReal (f' x) * u (f x)