mathlib-initiative/mathlib-tactics huggingface dataset:
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 ∈ interior (Icc (min a b) (max a b)), 0 ≤ deriv f x

Tactic:
interior_Icc

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.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), f' x ≤ 0
⊢ DifferentiableOn ℝ f (interior (Icc (min a b) (max a b)))

Tactic:
interior_Icc

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

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

Tactic:
interior_Icc

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

Tactic:
interior_Icc

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'
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 (interior (Icc a b))

Tactic:
rw [interior_Icc]

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 ∈ interior (Icc a b), 0 ≤ deriv f x

Tactic:
interior_Icc

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.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 ∈ interior (Icc a b), 0 ≤ deriv f x

Tactic:
rw [interior_Icc]

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.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 : ↑m < ENNReal.ofReal |A.det|
mpos : 0 < m
hA : A.det ≠ 0
B : E ≃L[ℝ] E := A.toContinuousLinearEquivOfDetNeZero hA
I : ENNReal.ofReal |(↑B.symm).det| < ↑m⁻¹
δ₀ : ℝ≥0
δ₀pos : 0 < δ₀
hδ₀ : ∀ (t : Set E) (g : E → E), ApproximatesLinearOn g (↑B.symm) t δ₀ → μ (g '' t) ≤ ↑m⁻¹ * μ t
L1 : ∀ᶠ (δ : ℝ≥0) in 𝓝 0, Subsingleton E ∨ δ < ‖↑B.symm‖₊⁻¹
L2 : ∀ᶠ (δ : ℝ≥0) in 𝓝 0, ‖↑B.symm‖₊ * (‖↑B.symm‖₊⁻¹ - δ)⁻¹ * δ < δ₀
δ : ℝ≥0
h1δ : Subsingleton E ∨ δ < ‖↑B.symm‖₊⁻¹
h2δ : ‖↑B.symm‖₊ * (‖↑B.symm‖₊⁻¹ - δ)⁻¹ * δ < δ₀
s : Set E
f : E → E
hf : ApproximatesLinearOn f A s δ
⊢ ↑m * μ s ≤ μ (f '' s)

Tactic:
have hf' : ApproximatesLinearOn f (B : E →L[ℝ] E) s δ := by convert! hf

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 : ↑m < ENNReal.ofReal |A.det|
mpos : 0 < m
hA : A.det ≠ 0
B : E ≃L[ℝ] E := A.toContinuousLinearEquivOfDetNeZero hA
I : ENNReal.ofReal |(↑B.symm).det| < ↑m⁻¹
δ₀ : ℝ≥0
δ₀pos : 0 < δ₀
hδ₀ : ∀ (t : Set E) (g : E → E), ApproximatesLinearOn g (↑B.symm) t δ₀ → μ (g '' t) ≤ ↑m⁻¹ * μ t
L1 : ∀ᶠ (δ : ℝ≥0) in 𝓝 0, Subsingleton E ∨ δ < ‖↑B.symm‖₊⁻¹
L2 : ∀ᶠ (δ : ℝ≥0) in 𝓝 0, ‖↑B.symm‖₊ * (‖↑B.symm‖₊⁻¹ - δ)⁻¹ * δ < δ₀
δ : ℝ≥0
h1δ : Subsingleton E ∨ δ < ‖↑B.symm‖₊⁻¹
h2δ : ‖↑B.symm‖₊ * (‖↑B.symm‖₊⁻¹ - δ)⁻¹ * δ < δ₀
s : Set E
f : E → E
hf : ApproximatesLinearOn f A s δ
hf' : ApproximatesLinearOn f (↑B) s δ
⊢ ↑m * μ s ≤ μ (f '' s)
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
⊢ DifferentiableOn ℝ f (interior (Icc a b))

Tactic:
interior_Icc

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
⊢ DifferentiableOn ℝ 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, f' x ≤ 0
hab : a ≤ b
⊢ DifferentiableOn ℝ f (interior (Icc a b))

Tactic:
rw [interior_Icc]

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
⊢ DifferentiableOn ℝ f (Ioo a b)
Module: Mathlib.MeasureTheory.Function.JacobianOneDim

Goal state before:
pp: case hf'_nonpos
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
⊢ ∀ x ∈ interior (Icc a b), deriv f x ≤ 0

Tactic:
interior_Icc

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

Goal state before:
pp: case hf'_nonpos
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
⊢ ∀ x ∈ interior (Icc a b), deriv f x ≤ 0

Tactic:
rw [interior_Icc]

Goal state after:
case hf'_nonpos
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
⊢ ∀ x ∈ Ioo a b, deriv f x ≤ 0
Module: Mathlib.MeasureTheory.Integral.IntervalIntegral.IntegrationByParts

Goal state before:
pp: a b : ℝ
f f' g g' : ℝ → ℝ
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 [[f a, f b]]
hgg' : ∀ x ∈ Ioo (min (f a) (f b)) (max (f a) (f b)), HasDerivWithinAt g (g' x) (Ioi x) x
hg' : ContinuousOn g' (f '' [[a, b]])
⊢ ∫ (x : ℝ) in a..b, (g' ∘ f) x * f' x = (g ∘ f) b - (g ∘ f) a

Tactic:
simpa [mul_comm] using integral_deriv_smul_deriv_comp' hf hff' hf' hg hgg' hg'

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

Goal state before:
pp: E : Type u_1
f f' : ℝ → E
a : ℝ
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
hderiv : ∀ x ∈ Ioi a, HasDerivAt f (f' x) x
f'int : IntegrableOn f' (Ioi a) volume
fint : IntegrableOn f (Ioi a) volume
F : E →L[ℝ] Completion E := Completion.toComplL
Fderiv : ∀ x ∈ Ioi a, HasDerivAt (⇑F ∘ f) (F (f' x)) x
Fint : IntegrableOn (⇑F ∘ f) (Ioi a) volume
F'int : IntegrableOn (⇑F ∘ f') (Ioi a) volume
A : Tendsto (⇑F ∘ f) atTop (𝓝 (atTop.limUnder (⇑F ∘ f)))
this : IntegrableAtFilter (⇑F ∘ f) atTop volume
⊢ ∀ s ∈ atTop, volume s = ∞

Tactic:
intro s hs

Goal state after:
E : Type u_1
f f' : ℝ → E
a : ℝ
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
hderiv : ∀ x ∈ Ioi a, HasDerivAt f (f' x) x
f'int : IntegrableOn f' (Ioi a) volume
fint : IntegrableOn f (Ioi a) volume
F : E →L[ℝ] Completion E := Completion.toComplL
Fderiv : ∀ x ∈ Ioi a, HasDerivAt (⇑F ∘ f) (F (f' x)) x
Fint : IntegrableOn (⇑F ∘ f) (Ioi a) volume
F'int : IntegrableOn (⇑F ∘ f') (Ioi a) volume
A : Tendsto (⇑F ∘ f) atTop (𝓝 (atTop.limUnder (⇑F ∘ f)))
this : IntegrableAtFilter (⇑F ∘ f) atTop volume
s : Set ℝ
hs : s ∈ atTop
⊢ volume s = ∞
Module: Mathlib.MeasureTheory.Integral.IntegralEqImproper

Goal state before:
pp: E : Type u_1
f f' : ℝ → E
a : ℝ
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
hderiv : ∀ x ∈ Iic a, HasDerivAt f (f' x) x
f'int : IntegrableOn f' (Iic a) volume
fint : IntegrableOn f (Iic a) volume
F : E →L[ℝ] Completion E := Completion.toComplL
Fderiv : ∀ x ∈ Iic a, HasDerivAt (⇑F ∘ f) (F (f' x)) x
Fint : IntegrableOn (⇑F ∘ f) (Iic a) volume
F'int : IntegrableOn (⇑F ∘ f') (Iic a) volume
A : Tendsto (⇑F ∘ f) atBot (𝓝 (atBot.limUnder (⇑F ∘ f)))
this : IntegrableAtFilter (⇑F ∘ f) atBot volume
⊢ ∀ s ∈ atBot, volume s = ∞

Tactic:
intro s hs

Goal state after:
E : Type u_1
f f' : ℝ → E
a : ℝ
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
hderiv : ∀ x ∈ Iic a, HasDerivAt f (f' x) x
f'int : IntegrableOn f' (Iic a) volume
fint : IntegrableOn f (Iic a) volume
F : E →L[ℝ] Completion E := Completion.toComplL
Fderiv : ∀ x ∈ Iic a, HasDerivAt (⇑F ∘ f) (F (f' x)) x
Fint : IntegrableOn (⇑F ∘ f) (Iic a) volume
F'int : IntegrableOn (⇑F ∘ f') (Iic a) volume
A : Tendsto (⇑F ∘ f) atBot (𝓝 (atBot.limUnder (⇑F ∘ f)))
this : IntegrableAtFilter (⇑F ∘ f) atBot volume
s : Set ℝ
hs : s ∈ atBot
⊢ volume s = ∞
Module: Mathlib.MeasureTheory.Integral.IntegralEqImproper

Goal state before:
pp: E : Type u_1
f f' : ℝ → E
m n : E
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
hderiv : ∀ (x : ℝ), HasDerivAt f (f' x) x
hf' : Integrable f' volume
hbot : Tendsto f atBot (𝓝 m)
htop : Tendsto f atTop (𝓝 n)
⊢ ∫ (x : ℝ) in Iic 0 ∪ Ioi 0, f' x = n - m

Tactic:
setIntegral_union (Iic_disjoint_Ioi le_rfl) measurableSet_Ioi hf'.integrableOn hf'.integrableOn,

Goal state after:
E : Type u_1
f f' : ℝ → E
m n : E
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
hderiv : ∀ (x : ℝ), HasDerivAt f (f' x) x
hf' : Integrable f' volume
hbot : Tendsto f atBot (𝓝 m)
htop : Tendsto f atTop (𝓝 n)
⊢ (∫ (x : ℝ) in Iic 0, f' x) + ∫ (x : ℝ) in Ioi 0, f' x = n - m
Module: Mathlib.Analysis.Calculus.ParametricIntervalIntegral

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁵ : RCLike 𝕜
μ : Measure ℝ
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : NormedSpace 𝕜 E
H : Type u_3
inst✝¹ : NormedAddCommGroup H
inst✝ : NormedSpace 𝕜 H
s : Set H
a b : ℝ
bound : ℝ → ℝ
F : H → ℝ → E
F' : ℝ → H →L[𝕜] E
x₀ : H
hs : s ∈ 𝓝 x₀
hF_meas : ∀ᶠ (x : H) in 𝓝 x₀, AEStronglyMeasurable (F x) (μ.restrict (Ι a b))
hF_int : IntervalIntegrable (F x₀) μ a b
hF'_meas : AEStronglyMeasurable F' (μ.restrict (Ι a b))
h_lip : ∀ᵐ (x : ℝ) ∂μ.restrict (Ι a b), LipschitzOnWith (Real.nnabs (bound x)) (fun x_1 ↦ F x_1 x) s
bound_integrable : IntervalIntegrable bound μ a b
h_diff : ∀ᵐ (x : ℝ) ∂μ.restrict (Ι a b), HasFDerivAt (fun x_1 ↦ F x_1 x) (F' x) x₀
⊢ IntervalIntegrable F' μ a b ∧ HasFDerivAt (fun x ↦ ∫ (t : ℝ) in a..b, F x t ∂μ) (∫ (t : ℝ) in a..b, F' t ∂μ) x₀

Tactic:
simp only [intervalIntegrable_iff] at hF_int bound_integrable ⊢

Goal state after:
𝕜 : Type u_1
inst✝⁵ : RCLike 𝕜
μ : Measure ℝ
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : NormedSpace 𝕜 E
H : Type u_3
inst✝¹ : NormedAddCommGroup H
inst✝ : NormedSpace 𝕜 H
s : Set H
a b : ℝ
bound : ℝ → ℝ
F : H → ℝ → E
F' : ℝ → H →L[𝕜] E
x₀ : H
hs : s ∈ 𝓝 x₀
hF_meas : ∀ᶠ (x : H) in 𝓝 x₀, AEStronglyMeasurable (F x) (μ.restrict (Ι a b))
hF'_meas : AEStronglyMeasurable F' (μ.restrict (Ι a b))
h_lip : ∀ᵐ (x : ℝ) ∂μ.restrict (Ι a b), LipschitzOnWith (Real.nnabs (bound x)) (fun x_1 ↦ F x_1 x) s
h_diff : ∀ᵐ (x : ℝ) ∂μ.restrict (Ι a b), HasFDerivAt (fun x_1 ↦ F x_1 x) (F' x) x₀
hF_int : IntegrableOn (F x₀) (Ι a b) μ
bound_integrable : IntegrableOn bound (Ι a b) μ
⊢ IntegrableOn F' (Ι a b) μ ∧ HasFDerivAt (fun x ↦ ∫ (t : ℝ) in a..b, F x t ∂μ) (∫ (t : ℝ) in a..b, F' t ∂μ) x₀
Module: Mathlib.Analysis.Complex.LocallyUniformLimit

Goal state before:
pp: E : Type u_1
ι : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
K : Set ℂ
δ : ℝ
φ : Filter ι
F : ι → ℂ → E
f : ℂ → E
hF : ∀ ε > 0, ∀ᶠ (n : ι) in φ, ∀ x ∈ cthickening δ K, dist (f x) (F n x) < ε
hδ : 0 < δ
hFn : ∀ᶠ (n : ι) in φ, ContinuousOn (F n) (cthickening δ K)
hne : φ.NeBot
e1 : ContinuousOn f (cthickening δ K)
ε : ℝ
hε : ε > 0
n : ι
h : ∀ x ∈ cthickening δ K, dist (f x) (F n x) < ε * δ
h' : ContinuousOn (F n) (cthickening δ K)
z : ℂ
hz : z ∈ K
⊢ dist (Complex.cderiv δ f z) ((Complex.cderiv δ ∘ F) n z) < ε

Tactic:
simp_rw [dist_eq_norm] at h ⊢

Goal state after:
E : Type u_1
ι : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
K : Set ℂ
δ : ℝ
φ : Filter ι
F : ι → ℂ → E
f : ℂ → E
hF : ∀ ε > 0, ∀ᶠ (n : ι) in φ, ∀ x ∈ cthickening δ K, dist (f x) (F n x) < ε
hδ : 0 < δ
hFn : ∀ᶠ (n : ι) in φ, ContinuousOn (F n) (cthickening δ K)
hne : φ.NeBot
e1 : ContinuousOn f (cthickening δ K)
ε : ℝ
hε : ε > 0
n : ι
h' : ContinuousOn (F n) (cthickening δ K)
z : ℂ
hz : z ∈ K
h : ∀ x ∈ cthickening δ K, ‖f x - F n x‖ < ε * δ
⊢ ‖Complex.cderiv δ f z - (Complex.cderiv δ ∘ F) n z‖ < ε
Module: Mathlib.Analysis.Calculus.ParametricIntervalIntegral

Goal state before:
pp: 𝕜 : Type u_1
inst✝³ : RCLike 𝕜
μ : Measure ℝ
E : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : NormedSpace 𝕜 E
a b : ℝ
bound : ℝ → ℝ
F : 𝕜 → ℝ → E
F' : ℝ → E
x₀ : 𝕜
s : Set 𝕜
hs : s ∈ 𝓝 x₀
hF_meas : ∀ᶠ (x : 𝕜) in 𝓝 x₀, AEStronglyMeasurable (F x) (μ.restrict (Ι a b))
hF_int : IntervalIntegrable (F x₀) μ a b
hF'_meas : AEStronglyMeasurable F' (μ.restrict (Ι a b))
h_lipsch : ∀ᵐ (x : ℝ) ∂μ.restrict (Ι a b), LipschitzOnWith (Real.nnabs (bound x)) (fun x_1 ↦ F x_1 x) s
bound_integrable : IntervalIntegrable bound μ a b
h_diff : ∀ᵐ (x : ℝ) ∂μ.restrict (Ι a b), HasDerivAt (fun x_1 ↦ F x_1 x) (F' x) x₀
⊢ IntervalIntegrable F' μ a b ∧ HasDerivAt (fun x ↦ ∫ (t : ℝ) in a..b, F x t ∂μ) (∫ (t : ℝ) in a..b, F' t ∂μ) x₀

Tactic:
simp only [intervalIntegrable_iff] at hF_int bound_integrable ⊢

Goal state after:
𝕜 : Type u_1
inst✝³ : RCLike 𝕜
μ : Measure ℝ
E : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : NormedSpace 𝕜 E
a b : ℝ
bound : ℝ → ℝ
F : 𝕜 → ℝ → E
F' : ℝ → E
x₀ : 𝕜
s : Set 𝕜
hs : s ∈ 𝓝 x₀
hF_meas : ∀ᶠ (x : 𝕜) in 𝓝 x₀, AEStronglyMeasurable (F x) (μ.restrict (Ι a b))
hF'_meas : AEStronglyMeasurable F' (μ.restrict (Ι a b))
h_lipsch : ∀ᵐ (x : ℝ) ∂μ.restrict (Ι a b), LipschitzOnWith (Real.nnabs (bound x)) (fun x_1 ↦ F x_1 x) s
h_diff : ∀ᵐ (x : ℝ) ∂μ.restrict (Ι a b), HasDerivAt (fun x_1 ↦ F x_1 x) (F' x) x₀
hF_int : IntegrableOn (F x₀) (Ι a b) μ
bound_integrable : IntegrableOn bound (Ι a b) μ
⊢ IntegrableOn F' (Ι a b) μ ∧ HasDerivAt (fun x ↦ ∫ (t : ℝ) in a..b, F x t ∂μ) (∫ (t : ℝ) in a..b, F' t ∂μ) x₀
Module: Mathlib.Analysis.Calculus.ParametricIntervalIntegral

Goal state before:
pp: 𝕜 : Type u_1
inst✝³ : RCLike 𝕜
μ : Measure ℝ
E : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : NormedSpace 𝕜 E
a b : ℝ
bound : ℝ → ℝ
F F' : 𝕜 → ℝ → E
x₀ : 𝕜
s : Set 𝕜
hs : s ∈ 𝓝 x₀
hF_meas : ∀ᶠ (x : 𝕜) in 𝓝 x₀, AEStronglyMeasurable (F x) (μ.restrict (Ι a b))
hF_int : IntervalIntegrable (F x₀) μ a b
hF'_meas : AEStronglyMeasurable (F' x₀) (μ.restrict (Ι a b))
h_bound : ∀ᵐ (x : ℝ) ∂μ.restrict (Ι a b), ∀ x_1 ∈ s, ‖F' x_1 x‖ ≤ bound x
bound_integrable : IntervalIntegrable bound μ a b
h_diff : ∀ᵐ (x : ℝ) ∂μ.restrict (Ι a b), ∀ x_1 ∈ s, HasDerivAt (fun x_2 ↦ F x_2 x) (F' x_1 x) x_1
⊢ IntervalIntegrable (F' x₀) μ a b ∧ HasDerivAt (fun x ↦ ∫ (t : ℝ) in a..b, F x t ∂μ) (∫ (t : ℝ) in a..b, F' x₀ t ∂μ) x₀

Tactic:
simp only [intervalIntegrable_iff] at hF_int bound_integrable ⊢

Goal state after:
𝕜 : Type u_1
inst✝³ : RCLike 𝕜
μ : Measure ℝ
E : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : NormedSpace 𝕜 E
a b : ℝ
bound : ℝ → ℝ
F F' : 𝕜 → ℝ → E
x₀ : 𝕜
s : Set 𝕜
hs : s ∈ 𝓝 x₀
hF_meas : ∀ᶠ (x : 𝕜) in 𝓝 x₀, AEStronglyMeasurable (F x) (μ.restrict (Ι a b))
hF'_meas : AEStronglyMeasurable (F' x₀) (μ.restrict (Ι a b))
h_bound : ∀ᵐ (x : ℝ) ∂μ.restrict (Ι a b), ∀ x_1 ∈ s, ‖F' x_1 x‖ ≤ bound x
h_diff : ∀ᵐ (x : ℝ) ∂μ.restrict (Ι a b), ∀ x_1 ∈ s, HasDerivAt (fun x_2 ↦ F x_2 x) (F' x_1 x) x_1
hF_int : IntegrableOn (F x₀) (Ι a b) μ
bound_integrable : IntegrableOn bound (Ι a b) μ
⊢ IntegrableOn (F' x₀) (Ι a b) μ ∧ HasDerivAt (fun x ↦ ∫ (t : ℝ) in a..b, F x t ∂μ) (∫ (t : ℝ) in a..b, F' x₀ t ∂μ) x₀
Module: Mathlib.Analysis.RCLike.TangentCone

Goal state before:
pp: 𝕜 : Type u_1
inst✝³ : NontriviallyNormedField 𝕜
h𝕜 : IsRCLikeNormedField 𝕜
E : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace 𝕜 E
inst✝ : NormedSpace ℝ E
s : Set E
x : E
⊢ tangentConeAt ℝ s x ⊆ tangentConeAt 𝕜 s x

Tactic:
by
  let := h𝕜.rclike
  exact tangentConeAt_mono_field

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

Goal state before:
pp: H : Type u
M : Type u_2
inst✝⁴ : TopologicalSpace H
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace H M
inst✝¹ : SecondCountableTopology H
inst✝ : SigmaCompactSpace M
s : Set M
hsc : s.Countable
hsU : ⋃ x ∈ s, (chartAt H x).source = univ
⊢ SecondCountableTopology M

Tactic:
exact ChartedSpace.secondCountable_of_countable_cover H hsU hsc

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

Goal state before:
pp: H : Type u
M : Type u_2
inst✝³ : TopologicalSpace H
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace H M
inst✝ : LocallyPathConnectedSpace H
x : M
s : Set M
hs : s ∈ 𝓝 x
e : OpenPartialHomeomorph M H := chartAt H x
t : Set M := s ∩ e.source
ht : t ∈ 𝓝 x
⊢ ∃ i, (i ∈ 𝓝 x ∧ IsPathConnected i) ∧ id i ⊆ s

Tactic:
refine ⟨e.symm '' pathComponentIn (e '' t) (e x), ⟨?_, ?_⟩, (?_ : _ ⊆ t).trans inter_subset_left⟩

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

case refine_2
H : Type u
M : Type u_2
inst✝³ : TopologicalSpace H
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace H M
inst✝ : LocallyPathConnectedSpace H
x : M
s : Set M
hs : s ∈ 𝓝 x
e : OpenPartialHomeomorph M H := chartAt H x
t : Set M := s ∩ e.source
ht : t ∈ 𝓝 x
⊢ IsPathConnected (↑e.symm '' pathComponentIn (↑e '' t) (↑e x))

case refine_3
H : Type u
M : Type u_2
inst✝³ : TopologicalSpace H
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace H M
inst✝ : LocallyPathConnectedSpace H
x : M
s : Set M
hs : s ∈ 𝓝 x
e : OpenPartialHomeomorph M H := chartAt H x
t : Set M := s ∩ e.source
ht : t ∈ 𝓝 x
⊢ id (↑e.symm '' pathComponentIn (↑e '' t) (↑e x)) ⊆ t
Module: Mathlib.Geometry.Manifold.StructureGroupoid

Goal state before:
pp: H✝ : Type u_1
inst✝¹ : TopologicalSpace H✝
H : Type u_2
inst✝ : TopologicalSpace H
e : OpenPartialHomeomorph H H
he : ∀ x ∈ e.source, ∃ s, IsOpen[inst✝] s ∧ x ∈ s ∧ e.restr s ∈ {OpenPartialHomeomorph.refl H} ∪ {e | e.source = ∅}
x : H
hx : x ∈ e.source
s : Set H
open_s : IsOpen[inst✝] s
xs : x ∈ s
x's : x ∈ (e.restr s).source
hs : e.restr s = OpenPartialHomeomorph.refl H
this✝ : (e.restr s).source = univ
this : e.source ∩ interior s = univ
⊢ e.source ∩ interior s ⊆ interior s

Tactic:
exact inter_subset_right

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

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

Tactic:
StructureGroupoid.le_iff

Goal state after:
case mp
H : Type u_1
inst✝ : TopologicalSpace H
G : StructureGroupoid H
_i : ClosedUnderRestriction G
⊢ ∀ e ∈ idRestrGroupoid, e ∈ G
Module: Mathlib.Geometry.Manifold.IsManifold.ExtChartAt

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
M : Type u_3
H : Type u_4
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : TopologicalSpace H
inst✝ : TopologicalSpace M
f : OpenPartialHomeomorph M H
I : ModelWithCorners 𝕜 E H
⊢ (f.extend I).target = ↑I '' f.target

Tactic:
rw [extend, PartialEquiv.trans_target'', I.source_eq, univ_inter, I.toPartialEquiv_coe]

Goal state after:
Module: Mathlib.Geometry.Manifold.IsManifold.ExtChartAt

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
M : Type u_3
H : Type u_4
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : TopologicalSpace H
inst✝ : TopologicalSpace M
f : OpenPartialHomeomorph M H
I : ModelWithCorners 𝕜 E H
⊢ ↑I.symm ⁻¹' f.target ∩ interior (range ↑I) ⊆ interior (range ↑I)

Tactic:
exact inter_subset_right

Goal state after: