mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Analysis.Calculus.LHopital

Goal state before:
pp: case hdf
a : ℝ
l : Filter ℝ
f g : ℝ → ℝ
hdf : ∀ᶠ (x : ℝ) in 𝓝 a, DifferentiableAt ℝ f x
hg' : ∀ᶠ (x : ℝ) in 𝓝 a, deriv g x ≠ 0
hfa : Tendsto f (𝓝 a) (𝓝 0)
hga : Tendsto g (𝓝 a) (𝓝 0)
hdiv : Tendsto (fun x ↦ deriv f x / deriv g x) (𝓝 a) l
⊢ ∀ᶠ (x : ℝ) in 𝓝[≠] a, DifferentiableAt ℝ f x

Tactic:
first | apply eventually_nhdsWithin_of_eventually_nhds |
      apply tendsto_nhdsWithin_of_tendsto_nhds

Goal state after:
case hdf
a : ℝ
l : Filter ℝ
f g : ℝ → ℝ
hdf : ∀ᶠ (x : ℝ) in 𝓝 a, DifferentiableAt ℝ f x
hg' : ∀ᶠ (x : ℝ) in 𝓝 a, deriv g x ≠ 0
hfa : Tendsto f (𝓝 a) (𝓝 0)
hga : Tendsto g (𝓝 a) (𝓝 0)
hdiv : Tendsto (fun x ↦ deriv f x / deriv g x) (𝓝 a) l
⊢ ∀ᶠ (x : ℝ) in 𝓝 a, DifferentiableAt ℝ f x
Module: Mathlib.Analysis.Calculus.LHopital

Goal state before:
pp: case hg'
a : ℝ
l : Filter ℝ
f g : ℝ → ℝ
hdf : ∀ᶠ (x : ℝ) in 𝓝 a, DifferentiableAt ℝ f x
hg' : ∀ᶠ (x : ℝ) in 𝓝 a, deriv g x ≠ 0
hfa : Tendsto f (𝓝 a) (𝓝 0)
hga : Tendsto g (𝓝 a) (𝓝 0)
hdiv : Tendsto (fun x ↦ deriv f x / deriv g x) (𝓝 a) l
⊢ ∀ᶠ (x : ℝ) in 𝓝[≠] a, deriv g x ≠ 0

Tactic:
first | apply eventually_nhdsWithin_of_eventually_nhds |
      apply tendsto_nhdsWithin_of_tendsto_nhds

Goal state after:
case hg'
a : ℝ
l : Filter ℝ
f g : ℝ → ℝ
hdf : ∀ᶠ (x : ℝ) in 𝓝 a, DifferentiableAt ℝ f x
hg' : ∀ᶠ (x : ℝ) in 𝓝 a, deriv g x ≠ 0
hfa : Tendsto f (𝓝 a) (𝓝 0)
hga : Tendsto g (𝓝 a) (𝓝 0)
hdiv : Tendsto (fun x ↦ deriv f x / deriv g x) (𝓝 a) l
⊢ ∀ᶠ (x : ℝ) in 𝓝 a, deriv g x ≠ 0
Module: Mathlib.Analysis.Calculus.LHopital

Goal state before:
pp: case hfa
a : ℝ
l : Filter ℝ
f g : ℝ → ℝ
hdf : ∀ᶠ (x : ℝ) in 𝓝 a, DifferentiableAt ℝ f x
hg' : ∀ᶠ (x : ℝ) in 𝓝 a, deriv g x ≠ 0
hfa : Tendsto f (𝓝 a) (𝓝 0)
hga : Tendsto g (𝓝 a) (𝓝 0)
hdiv : Tendsto (fun x ↦ deriv f x / deriv g x) (𝓝 a) l
⊢ Tendsto f (𝓝[≠] a) (𝓝 0)

Tactic:
first | apply eventually_nhdsWithin_of_eventually_nhds |
      apply tendsto_nhdsWithin_of_tendsto_nhds

Goal state after:
case hfa
a : ℝ
l : Filter ℝ
f g : ℝ → ℝ
hdf : ∀ᶠ (x : ℝ) in 𝓝 a, DifferentiableAt ℝ f x
hg' : ∀ᶠ (x : ℝ) in 𝓝 a, deriv g x ≠ 0
hfa : Tendsto f (𝓝 a) (𝓝 0)
hga : Tendsto g (𝓝 a) (𝓝 0)
hdiv : Tendsto (fun x ↦ deriv f x / deriv g x) (𝓝 a) l
⊢ Tendsto f (𝓝 a) (𝓝 0)
Module: Mathlib.Analysis.Calculus.LHopital

Goal state before:
pp: case hga
a : ℝ
l : Filter ℝ
f g : ℝ → ℝ
hdf : ∀ᶠ (x : ℝ) in 𝓝 a, DifferentiableAt ℝ f x
hg' : ∀ᶠ (x : ℝ) in 𝓝 a, deriv g x ≠ 0
hfa : Tendsto f (𝓝 a) (𝓝 0)
hga : Tendsto g (𝓝 a) (𝓝 0)
hdiv : Tendsto (fun x ↦ deriv f x / deriv g x) (𝓝 a) l
⊢ Tendsto g (𝓝[≠] a) (𝓝 0)

Tactic:
first | apply eventually_nhdsWithin_of_eventually_nhds |
      apply tendsto_nhdsWithin_of_tendsto_nhds

Goal state after:
case hga
a : ℝ
l : Filter ℝ
f g : ℝ → ℝ
hdf : ∀ᶠ (x : ℝ) in 𝓝 a, DifferentiableAt ℝ f x
hg' : ∀ᶠ (x : ℝ) in 𝓝 a, deriv g x ≠ 0
hfa : Tendsto f (𝓝 a) (𝓝 0)
hga : Tendsto g (𝓝 a) (𝓝 0)
hdiv : Tendsto (fun x ↦ deriv f x / deriv g x) (𝓝 a) l
⊢ Tendsto g (𝓝 a) (𝓝 0)
Module: Mathlib.Analysis.Calculus.LHopital

Goal state before:
pp: case hdiv
a : ℝ
l : Filter ℝ
f g : ℝ → ℝ
hdf : ∀ᶠ (x : ℝ) in 𝓝 a, DifferentiableAt ℝ f x
hg' : ∀ᶠ (x : ℝ) in 𝓝 a, deriv g x ≠ 0
hfa : Tendsto f (𝓝 a) (𝓝 0)
hga : Tendsto g (𝓝 a) (𝓝 0)
hdiv : Tendsto (fun x ↦ deriv f x / deriv g x) (𝓝 a) l
⊢ Tendsto (fun x ↦ deriv f x / deriv g x) (𝓝[≠] a) l

Tactic:
first | apply eventually_nhdsWithin_of_eventually_nhds |
      apply tendsto_nhdsWithin_of_tendsto_nhds

Goal state after:
case hdiv
a : ℝ
l : Filter ℝ
f g : ℝ → ℝ
hdf : ∀ᶠ (x : ℝ) in 𝓝 a, DifferentiableAt ℝ f x
hg' : ∀ᶠ (x : ℝ) in 𝓝 a, deriv g x ≠ 0
hfa : Tendsto f (𝓝 a) (𝓝 0)
hga : Tendsto g (𝓝 a) (𝓝 0)
hdiv : Tendsto (fun x ↦ deriv f x / deriv g x) (𝓝 a) l
⊢ Tendsto (fun x ↦ deriv f x / deriv g x) (𝓝 a) l
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 : a ∈ Ioi 0
⊢ μ (f '' s) ≤ ↑m * (μ s + a)

Tactic:
mem_Ioi

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.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
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

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)

Tactic:
· exact fun z hz ↦ (hff' z hz).hasDerivWithinAt

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
⊢ MonotoneOn f (Ioo a b)
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
M : MonotoneOn f (Icc a b)
A : IntegrableOn g (Icc (f a) (f b)) volume ↔ IntegrableOn g (f '' Ioo a b) volume
⊢ ∀ x ∈ Ioo a b, HasDerivWithinAt f (f' 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 : IntegrableOn g (Icc (f a) (f b)) volume ↔ IntegrableOn g (f '' Ioo a b) volume
⊢ MonotoneOn f (Ioo a b)

Tactic:
· exact fun z hz ↦ (hff' z hz).hasDerivWithinAt

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 : IntegrableOn g (Icc (f a) (f b)) volume ↔ IntegrableOn g (f '' Ioo a b) volume
⊢ MonotoneOn f (Ioo 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 : AntitoneOn f s
g : ℝ → F
n : ℝ → ℝ := fun x ↦ -x
e : ℝ → ℝ := n ∘ f
hg' : ∀ x ∈ s, HasDerivWithinAt e (-f' x) s x
⊢ IntegrableOn (fun x ↦ g (n x)) (e '' s) volume ↔ IntegrableOn (fun x ↦ -f' x • (g ∘ n) (e x)) s volume

Tactic:
← integrableOn_image_iff_integrableOn_deriv_smul_of_monotoneOn hs hg' hf.neg (g ∘ n)

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 : AntitoneOn f s
g : ℝ → F
n : ℝ → ℝ := fun x ↦ -x
e : ℝ → ℝ := n ∘ f
hg' : ∀ x ∈ s, HasDerivWithinAt e (-f' x) s x
⊢ IntegrableOn (fun x ↦ g (n x)) (e '' s) volume ↔ IntegrableOn (g ∘ n) (e '' s) volume
Module: Mathlib.MeasureTheory.Integral.IntegralEqImproper

Goal state before:
pp: case hderiv
E : Type u_1
f f' : ℝ → E
a : ℝ
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
hderiv : ∀ x ∈ Ioi a, HasDerivAt f (f' x) x
f'int : IntegrableOn f' (Ioi a) volume
ε : ℝ
εpos : ε > 0
A : ∀ᶠ (n : ℕ) in atTop, ∫ (x : ℝ) in Ici ↑n, ‖f' x‖ < ε
B : ∀ᶠ (n : ℕ) in atTop, a < ↑n
N : ℕ
hN : ∫ (x : ℝ) in Ici ↑N, ‖f' x‖ < ε
h'N : a < ↑N
x : ℝ
hx : x ≥ ↑N
⊢ ∀ x_1 ∈ [[↑N, x]], HasDerivAt f (f' x_1) x_1

Tactic:
intro y hy

Goal state after:
case hderiv
E : Type u_1
f f' : ℝ → E
a : ℝ
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
hderiv : ∀ x ∈ Ioi a, HasDerivAt f (f' x) x
f'int : IntegrableOn f' (Ioi a) volume
ε : ℝ
εpos : ε > 0
A : ∀ᶠ (n : ℕ) in atTop, ∫ (x : ℝ) in Ici ↑n, ‖f' x‖ < ε
B : ∀ᶠ (n : ℕ) in atTop, a < ↑n
N : ℕ
hN : ∫ (x : ℝ) in Ici ↑N, ‖f' x‖ < ε
h'N : a < ↑N
x : ℝ
hx : x ≥ ↑N
y : ℝ
hy : y ∈ [[↑N, x]]
⊢ HasDerivAt f (f' y) y
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, f' x ≤ 0
hab : a ≤ b
M : AntitoneOn f (Icc a b)
A : ∫ (u : ℝ) in Icc (f b) (f a), g u = ∫ (u : ℝ) in f '' Ioo a b, g u
⊢ ∀ x ∈ Ioo a b, HasDerivWithinAt f (f' 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, f' x ≤ 0
hab : a ≤ b
M : AntitoneOn f (Icc a b)
A : ∫ (u : ℝ) in Icc (f b) (f a), g u = ∫ (u : ℝ) in f '' Ioo a b, g u
⊢ AntitoneOn f (Ioo a b)

Tactic:
· exact fun z hz ↦ (hff' z hz).hasDerivWithinAt

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, f' x ≤ 0
hab : a ≤ b
M : AntitoneOn f (Icc a b)
A : ∫ (u : ℝ) in Icc (f b) (f a), g u = ∫ (u : ℝ) in f '' Ioo a b, g u
⊢ AntitoneOn f (Ioo a b)
Module: Mathlib.MeasureTheory.Integral.IntegralEqImproper

Goal state before:
pp: g g' : ℝ → ℝ
a l : ℝ
hcont : ContinuousWithinAt g (Ici a) a
hderiv : ∀ x ∈ Ioi a, HasDerivAt g (g' x) x
g'pos : ∀ x ∈ Ioi a, 0 ≤ g' x
hg : Tendsto g atTop (𝓝 l)
⊢ IntegrableOn g' (Ioi a) volume

Tactic:
have hcont : ContinuousOn g (Ici a) := by
    intro x hx
    rcases hx.out.eq_or_lt with rfl | hx
    · exact hcont
    · exact (hderiv x hx).continuousAt.continuousWithinAt

Goal state after:
g g' : ℝ → ℝ
a l : ℝ
hcont✝ : ContinuousWithinAt g (Ici a) a
hderiv : ∀ x ∈ Ioi a, HasDerivAt g (g' x) x
g'pos : ∀ x ∈ Ioi a, 0 ≤ g' x
hg : Tendsto g atTop (𝓝 l)
hcont : ContinuousOn g (Ici a)
⊢ IntegrableOn g' (Ioi a) volume
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, f' x ≤ 0
hab : a ≤ b
M : AntitoneOn f (Icc a b)
A : IntegrableOn g (Icc (f b) (f a)) volume ↔ IntegrableOn g (f '' Ioo a b) volume
⊢ ∀ x ∈ Ioo a b, HasDerivWithinAt f (f' 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, f' x ≤ 0
hab : a ≤ b
M : AntitoneOn f (Icc a b)
A : IntegrableOn g (Icc (f b) (f a)) volume ↔ IntegrableOn g (f '' Ioo a b) volume
⊢ AntitoneOn f (Ioo a b)

Tactic:
· exact fun z hz ↦ (hff' z hz).hasDerivWithinAt

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, f' x ≤ 0
hab : a ≤ b
M : AntitoneOn f (Icc a b)
A : IntegrableOn g (Icc (f b) (f a)) volume ↔ IntegrableOn g (f '' Ioo a b) volume
⊢ AntitoneOn f (Ioo a b)
Module: Mathlib.MeasureTheory.Integral.IntegralEqImproper

Goal state before:
pp: E : Type u_1
f f' : ℝ → E
a : ℝ
m : E
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
hcont : ContinuousWithinAt f (Iic a) a
hderiv : ∀ x ∈ Iio a, HasDerivAt f (f' x) x
f'int : IntegrableOn f' (Iic a) volume
hf : Tendsto f atBot (𝓝 m)
⊢ ∫ (x : ℝ) in Iic a, f' x = f a - m

Tactic:
have hcont : ContinuousOn f (Iic a) := by
    intro x hx
    rcases hx.out.eq_or_lt with rfl | hx
    · exact hcont
    · exact (hderiv x hx).continuousAt.continuousWithinAt
  refine tendsto_nhds_unique (intervalIntegral_tendsto_integral_Iic a f'int tendsto_id) ?_
  apply Tendsto.congr' _ (hf.const_sub _)
  filter_upwards [Iic_mem_atBot a] with x hx
  symm
  apply intervalIntegral.integral_eq_sub_of_hasDerivAt_of_le hx
    (hcont.mono Icc_subset_Iic_self) fun y hy => hderiv y hy.2
  rw [intervalIntegrable_iff_integrableOn_Ioc_of_le hx]
  exact f'int.mono (fun y hy => hy.2) le_rfl

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

Goal state before:
pp: E : Type u_1
f f' : ℝ → E
a : ℝ
m : E
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
hderiv : ∀ x ∈ Iic a, HasDerivAt f (f' x) x
f'int : IntegrableOn f' (Iic a) volume
hf : Tendsto f atBot (𝓝 m)
⊢ ContinuousWithinAt f (Iic a) a

Tactic:
exact (hderiv a self_mem_Iic).continuousAt.continuousWithinAt

Goal state after:
Module: Mathlib.Analysis.Calculus.LineDeriv.IntegrationByParts

Goal state before:
pp: case hg
E : Type u_1
F : Type u_2
G : Type u_3
W : Type u_4
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace ℝ E
inst✝⁹ : NormedAddCommGroup F
inst✝⁸ : NormedSpace ℝ F
inst✝⁷ : NormedAddCommGroup G
inst✝⁶ : NormedSpace ℝ G
inst✝⁵ : NormedAddCommGroup W
inst✝⁴ : NormedSpace ℝ W
inst✝³ : MeasurableSpace E
μ : Measure E
inst✝² : BorelSpace E
inst✝¹ : FiniteDimensional ℝ E
inst✝ : μ.IsAddHaarMeasure
f f' : E → F
g g' : E → G
v : E
B : F →L[ℝ] G →L[ℝ] W
hf'g : Integrable (fun x ↦ (B (f' x)) (g x)) μ
hfg' : Integrable (fun x ↦ (B (f x)) (g' x)) μ
hfg : Integrable (fun x ↦ (B (f x)) (g x)) μ
hf : ∀ x ∈ tsupport g, HasLineDerivAt ℝ f (f' x) x v
hg : ∀ x ∈ tsupport f, HasLineDerivAt ℝ g (g' x) x v
hW : CompleteSpace W
hv : v ≠ 0
this✝ : Nontrivial E
n : ℕ := finrank ℝ E
E' : Type := Fin (n - 1) → ℝ
L : E ≃L[ℝ] E' × ℝ
hL : L v = (0, 1)
ν : Measure (E' × ℝ) := map (⇑L) μ
L_emb : MeasurableEmbedding ⇑L
x : E' × ℝ
hx : x ∈ tsupport fun x ↦ f (L.symm x)
this : g = (g ∘ ⇑L.symm) ∘ ⇑↑↑L
⊢ HasLineDerivAt ℝ (fun x ↦ g (L.symm x)) (g' (L.symm x)) x (0, 1)

Tactic:
have h2x : L.symm x ∈ tsupport f :=
      (Set.ext_iff.mp (tsupport_comp_eq_preimage f L.symm.toHomeomorph) x).mp hx

Goal state after:
case hg
E : Type u_1
F : Type u_2
G : Type u_3
W : Type u_4
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace ℝ E
inst✝⁹ : NormedAddCommGroup F
inst✝⁸ : NormedSpace ℝ F
inst✝⁷ : NormedAddCommGroup G
inst✝⁶ : NormedSpace ℝ G
inst✝⁵ : NormedAddCommGroup W
inst✝⁴ : NormedSpace ℝ W
inst✝³ : MeasurableSpace E
μ : Measure E
inst✝² : BorelSpace E
inst✝¹ : FiniteDimensional ℝ E
inst✝ : μ.IsAddHaarMeasure
f f' : E → F
g g' : E → G
v : E
B : F →L[ℝ] G →L[ℝ] W
hf'g : Integrable (fun x ↦ (B (f' x)) (g x)) μ
hfg' : Integrable (fun x ↦ (B (f x)) (g' x)) μ
hfg : Integrable (fun x ↦ (B (f x)) (g x)) μ
hf : ∀ x ∈ tsupport g, HasLineDerivAt ℝ f (f' x) x v
hg : ∀ x ∈ tsupport f, HasLineDerivAt ℝ g (g' x) x v
hW : CompleteSpace W
hv : v ≠ 0
this✝ : Nontrivial E
n : ℕ := finrank ℝ E
E' : Type := Fin (n - 1) → ℝ
L : E ≃L[ℝ] E' × ℝ
hL : L v = (0, 1)
ν : Measure (E' × ℝ) := map (⇑L) μ
L_emb : MeasurableEmbedding ⇑L
x : E' × ℝ
hx : x ∈ tsupport fun x ↦ f (L.symm x)
this : g = (g ∘ ⇑L.symm) ∘ ⇑↑↑L
h2x : L.symm x ∈ tsupport f
⊢ HasLineDerivAt ℝ (fun x ↦ g (L.symm x)) (g' (L.symm x)) x (0, 1)
Module: Mathlib.MeasureTheory.Integral.IntegralEqImproper

Goal state before:
pp: E : Type u_1
F : Type u_2
G : Type u_3
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace ℝ F
inst✝² : NormedAddCommGroup G
inst✝¹ : NormedSpace ℝ G
L : E →L[ℝ] F →L[ℝ] G
u : ℝ → E
v : ℝ → F
u' : ℝ → E
v' : ℝ → F
m n : G
inst✝ : CompleteSpace G
hu : ∀ x ∈ tsupport v, HasDerivAt u (u' x) x
hv : ∀ x ∈ tsupport u, HasDerivAt v (v' x) x
huv' : Integrable (fun x ↦ (L (u x)) (v' x)) volume
hu'v : Integrable (fun x ↦ (L (u' x)) (v x)) volume
h_bot : Tendsto (fun x ↦ (L (u x)) (v x)) atBot (𝓝 m)
h_top : Tendsto (fun x ↦ (L (u x)) (v x)) atTop (𝓝 n)
⊢ ∫ (x : ℝ), (L (u x)) (v' x) = n - m - ∫ (x : ℝ), (L (u' x)) (v x)

Tactic:
eq_sub_iff_add_eq,

Goal state after:
E : Type u_1
F : Type u_2
G : Type u_3
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace ℝ F
inst✝² : NormedAddCommGroup G
inst✝¹ : NormedSpace ℝ G
L : E →L[ℝ] F →L[ℝ] G
u : ℝ → E
v : ℝ → F
u' : ℝ → E
v' : ℝ → F
m n : G
inst✝ : CompleteSpace G
hu : ∀ x ∈ tsupport v, HasDerivAt u (u' x) x
hv : ∀ x ∈ tsupport u, HasDerivAt v (v' x) x
huv' : Integrable (fun x ↦ (L (u x)) (v' x)) volume
hu'v : Integrable (fun x ↦ (L (u' x)) (v x)) volume
h_bot : Tendsto (fun x ↦ (L (u x)) (v x)) atBot (𝓝 m)
h_top : Tendsto (fun x ↦ (L (u x)) (v x)) atTop (𝓝 n)
⊢ (∫ (x : ℝ), (L (u x)) (v' x)) + ∫ (x : ℝ), (L (u' x)) (v x) = n - m
Module: Mathlib.MeasureTheory.Integral.IntegralEqImproper

Goal state before:
pp: A : Type u_1
inst✝² : NormedRing A
inst✝¹ : NormedAlgebra ℝ A
a : ℝ
a' b' : A
u v u' v' : ℝ → A
inst✝ : CompleteSpace A
hu : ∀ x ∈ Ioi a, HasDerivAt u (u' x) x
hv : ∀ x ∈ Ioi a, HasDerivAt v (v' x) x
huv' : IntegrableOn (fun i ↦ u i * v' i) (Ioi a) volume
hu'v : IntegrableOn (fun i ↦ u' i * v i) (Ioi a) volume
h_zero : Tendsto (u * v) (𝓝[>] a) (𝓝 a')
h_infty : Tendsto (u * v) atTop (𝓝 b')
⊢ ∫ (x : ℝ) in Ioi a, u x * v' x = b' - a' - ∫ (x : ℝ) in Ioi a, u' x * v x

Tactic:
eq_sub_iff_add_eq,

Goal state after:
A : Type u_1
inst✝² : NormedRing A
inst✝¹ : NormedAlgebra ℝ A
a : ℝ
a' b' : A
u v u' v' : ℝ → A
inst✝ : CompleteSpace A
hu : ∀ x ∈ Ioi a, HasDerivAt u (u' x) x
hv : ∀ x ∈ Ioi a, HasDerivAt v (v' x) x
huv' : IntegrableOn (fun i ↦ u i * v' i) (Ioi a) volume
hu'v : IntegrableOn (fun i ↦ u' i * v i) (Ioi a) volume
h_zero : Tendsto (u * v) (𝓝[>] a) (𝓝 a')
h_infty : Tendsto (u * v) atTop (𝓝 b')
⊢ (∫ (x : ℝ) in Ioi a, u x * v' x) + ∫ (x : ℝ) in Ioi a, u' x * v x = b' - a'
Module: Mathlib.MeasureTheory.Integral.IntegralEqImproper

Goal state before:
pp: A : Type u_1
inst✝² : NormedRing A
inst✝¹ : NormedAlgebra ℝ A
a : ℝ
a' b' : A
u v u' v' : ℝ → A
inst✝ : CompleteSpace A
hu : ∀ x ∈ Iio a, HasDerivAt u (u' x) x
hv : ∀ x ∈ Iio a, HasDerivAt v (v' x) x
huv' : IntegrableOn (fun i ↦ u i * v' i) (Iic a) volume
hu'v : IntegrableOn (fun i ↦ u' i * v i) (Iic a) volume
h_zero : Tendsto (u * v) (𝓝[<] a) (𝓝 a')
h_infty : Tendsto (u * v) atBot (𝓝 b')
⊢ ∫ (x : ℝ) in Iic a, u x * v' x = a' - b' - ∫ (x : ℝ) in Iic a, u' x * v x

Tactic:
eq_sub_iff_add_eq,

Goal state after:
A : Type u_1
inst✝² : NormedRing A
inst✝¹ : NormedAlgebra ℝ A
a : ℝ
a' b' : A
u v u' v' : ℝ → A
inst✝ : CompleteSpace A
hu : ∀ x ∈ Iio a, HasDerivAt u (u' x) x
hv : ∀ x ∈ Iio a, HasDerivAt v (v' x) x
huv' : IntegrableOn (fun i ↦ u i * v' i) (Iic a) volume
hu'v : IntegrableOn (fun i ↦ u' i * v i) (Iic a) volume
h_zero : Tendsto (u * v) (𝓝[<] a) (𝓝 a')
h_infty : Tendsto (u * v) atBot (𝓝 b')
⊢ (∫ (x : ℝ) in Iic a, u x * v' x) + ∫ (x : ℝ) in Iic a, u' x * v x = 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
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
⊢ AEMeasurable (fun x ↦ (f' x).det) (μ.restrict s)

Tactic:
refine ContinuousLinearMap.continuous_det.measurable.comp_aemeasurable ?_

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
hs : MeasurableSet s
hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x
⊢ AEMeasurable f' (μ.restrict s)
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
⊢ AEMeasurable (fun x ↦ |(f' x).det|.toNNReal) (μ.restrict s)

Tactic:
apply measurable_real_toNNReal.comp_aemeasurable
  refine continuous_abs.measurable.comp_aemeasurable ?_
  refine ContinuousLinearMap.continuous_det.measurable.comp_aemeasurable ?_
  exact aemeasurable_fderivWithin μ hs hf'

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

Goal state before:
pp: case inl
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 ∈ {OpenPartialHomeomorph.refl H}
⊢ e ≫ₕ e' ∈ {OpenPartialHomeomorph.refl H} ∪ {e | e.source = ∅}

pp: case inr
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.source = ∅}
⊢ e ≫ₕ e' ∈ {OpenPartialHomeomorph.refl H} ∪ {e | e.source = ∅}

Tactic:
· simpa only [mem_singleton_iff.1 he, refl_trans]

Goal state after:
case inr
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.source = ∅}
⊢ e ≫ₕ e' ∈ {OpenPartialHomeomorph.refl H} ∪ {e | e.source = ∅}
Module: Mathlib.Geometry.Manifold.HasGroupoid

Goal state before:
pp: H : Type u
M : Type u_2
inst✝² : TopologicalSpace H
inst✝¹ : TopologicalSpace M
inst✝ : ChartedSpace H M
U : Opens M
x : ↥U
⊢ ↑(chartAt H ↑x).symm =ᶠ[𝓝 (↑(chartAt H ↑x) ↑x)] Subtype.val ∘ ↑(chartAt H x).symm

Tactic:
set e := chartAt H x.val

Goal state after:
H : Type u
M : Type u_2
inst✝² : TopologicalSpace H
inst✝¹ : TopologicalSpace M
inst✝ : ChartedSpace H M
U : Opens M
x : ↥U
e : OpenPartialHomeomorph M H := chartAt H ↑x
⊢ ↑e.symm =ᶠ[𝓝 (↑e ↑x)] Subtype.val ∘ ↑(chartAt H x).symm
Module: Mathlib.Geometry.Manifold.StructureGroupoid

Goal state before:
pp: H : Type u_1
inst✝ : TopologicalSpace H
PG₁ PG₂ : Pregroupoid H
h : ∀ (f : H → H) (s : Set H), PG₁.property f s → PG₂.property f s
⊢ PG₁.groupoid ≤ PG₂.groupoid

Tactic:
refine StructureGroupoid.le_iff.2 fun e he ↦ ?_
  rw [mem_groupoid_of_pregroupoid] at he ⊢
  exact ⟨h _ _ he.1, h _ _ he.2⟩

Goal state after: