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

Goal state before:
pp: f : ℝ ≃ᵐ ℝ
s : Set ℝ
hs : MeasurableSet s
f' : ℝ → ℝ
hf' : ∀ (x : ℝ), HasDerivAt (⇑f) (f' x) x
g : ℝ → ℝ
hg : 0 ≤ᵐ[volume] g
hg_int : Integrable g volume
⊢ (Measure.map (⇑f.symm) (volume.withDensity fun x ↦ ENNReal.ofReal (g x))) s =
    ENNReal.ofReal (∫ (x : ℝ) in s, |f' x| * g (f x))

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁸ : NontriviallyNormedField 𝕜
inst✝⁷ : LocallyCompactSpace 𝕜
E : Type u_2
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace 𝕜 E
inst✝⁴ : MeasurableSpace E
inst✝³ : OpensMeasurableSpace E
F : Type u_3
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
inst✝ : CompleteSpace F
f : E → F
v : E
hf : Continuous f
this✝¹ : MeasurableSpace 𝕜 := borel 𝕜
this✝ : BorelSpace 𝕜
g : E → 𝕜 → F := fun x t ↦ f (x + t • v)
⊢ Continuous (Function.uncurry g)

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁸ : NontriviallyNormedField 𝕜
inst✝⁷ : LocallyCompactSpace 𝕜
E : Type u_2
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace 𝕜 E
inst✝⁴ : MeasurableSpace E
inst✝³ : OpensMeasurableSpace E
F : Type u_3
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
inst✝ : CompleteSpace F
f : E → F
v : E
hf : Continuous f
⊢ MeasurableSet {x | LineDifferentiableAt 𝕜 f x v}

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝¹⁰ : NontriviallyNormedField 𝕜
inst✝⁹ : LocallyCompactSpace 𝕜
E : Type u_2
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace 𝕜 E
inst✝⁶ : MeasurableSpace E
inst✝⁵ : OpensMeasurableSpace E
F : Type u_3
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace 𝕜 F
inst✝² : CompleteSpace F
f : E → F
v : E
inst✝¹ : MeasurableSpace F
inst✝ : BorelSpace F
hf : Continuous f
this✝¹ : MeasurableSpace 𝕜 := borel 𝕜
this✝ : BorelSpace 𝕜
g : E → 𝕜 → F := fun x t ↦ f (x + t • v)
⊢ Continuous (Function.uncurry g)

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝¹⁰ : NontriviallyNormedField 𝕜
inst✝⁹ : LocallyCompactSpace 𝕜
E : Type u_2
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace 𝕜 E
inst✝⁶ : MeasurableSpace E
inst✝⁵ : OpensMeasurableSpace E
F : Type u_3
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace 𝕜 F
inst✝² : CompleteSpace F
f : E → F
v : E
inst✝¹ : MeasurableSpace F
inst✝ : BorelSpace F
hf : Continuous f
⊢ Measurable fun x ↦ lineDeriv 𝕜 f x v

Tactic:
by

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
s : Set E
f : E → E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
A : E →L[ℝ] E
δ : ℝ≥0
hf : ApproximatesLinearOn f A s δ
hs : MeasurableSet s
f' : E → E →L[ℝ] E
hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x
x : E
hx : Tendsto (fun r ↦ μ (s ∩ closedBall x r) / μ (closedBall x r)) (𝓝[>] 0) (𝓝 1)
xs : x ∈ s
z : E
ε : ℝ
εpos : 0 < ε
B₁ : ∀ᶠ (r : ℝ) in 𝓝[>] 0, (s ∩ ({x} + r • closedBall z ε)).Nonempty
ρ : ℝ
ρpos : ρ > 0
hρ : ball x ρ ∩ s ⊆ {y | ‖f y - f x - (f' x) (y - x)‖ ≤ ε * ‖y - x‖}
B₂ : ∀ᶠ (r : ℝ) in 𝓝[>] 0, {x} + r • closedBall z ε ⊆ ball x ρ
r : ℝ
y : E
ys : y ∈ s
hy : y ∈ {x} + r • closedBall z ε
rρ : {x} + r • closedBall z ε ⊆ ball x ρ
rpos : 0 < r
a : E
az : a ∈ closedBall z ε
ya : y = x + r • a
norm_a : ‖a‖ ≤ ‖z‖ + ε
⊢ r * ‖(f' x - A) a‖ = ‖(f' x - A) (r • a)‖

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁹ : NontriviallyNormedField 𝕜
inst✝⁸ : LocallyCompactSpace 𝕜
E : Type u_2
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace 𝕜 E
inst✝⁵ : MeasurableSpace E
inst✝⁴ : OpensMeasurableSpace E
F : Type u_3
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
inst✝¹ : CompleteSpace F
f : E → F
v : E
inst✝ : SecondCountableTopologyEither E F
hf : Continuous f
this✝¹ : MeasurableSpace 𝕜 := borel 𝕜
this✝ : BorelSpace 𝕜
g : E → 𝕜 → F := fun x t ↦ f (x + t • v)
⊢ Continuous (Function.uncurry g)

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁹ : NontriviallyNormedField 𝕜
inst✝⁸ : LocallyCompactSpace 𝕜
E : Type u_2
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace 𝕜 E
inst✝⁵ : MeasurableSpace E
inst✝⁴ : OpensMeasurableSpace E
F : Type u_3
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
inst✝¹ : CompleteSpace F
f : E → F
v : E
inst✝ : SecondCountableTopologyEither E F
hf : Continuous f
this✝¹ : MeasurableSpace 𝕜 := borel 𝕜
this✝ : BorelSpace 𝕜
g : E → 𝕜 → F := fun x t ↦ f (x + t • v)
hg : Continuous (Function.uncurry g)
⊢ StronglyMeasurable fun x ↦ lineDeriv 𝕜 f x v

Tactic:
exact (stronglyMeasurable_deriv_with_param hg).comp_measurable measurable_prodMk_right

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁹ : NontriviallyNormedField 𝕜
inst✝⁸ : LocallyCompactSpace 𝕜
E : Type u_2
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace 𝕜 E
inst✝⁵ : MeasurableSpace E
inst✝⁴ : OpensMeasurableSpace E
F : Type u_3
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
inst✝¹ : CompleteSpace F
f : E → F
v : E
inst✝ : SecondCountableTopologyEither E F
hf : Continuous f
⊢ StronglyMeasurable fun x ↦ lineDeriv 𝕜 f x v

Tactic:
by

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 ∈ Iic a, HasDerivAt f (f' x) x
f'int : IntegrableOn f' (Iic a) volume
fint : IntegrableOn f (Iic a) volume
⊢ Tendsto f atBot (𝓝 0)

Tactic:
by

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
s : Set E
f : E → E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
A : E →L[ℝ] E
δ : ℝ≥0
hf : ApproximatesLinearOn f A s δ
hs : MeasurableSet s
f' : E → E →L[ℝ] E
hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x
x : E
hx : Tendsto (fun r ↦ μ (s ∩ closedBall x r) / μ (closedBall x r)) (𝓝[>] 0) (𝓝 1)
xs : x ∈ s
z : E
ε : ℝ
εpos : 0 < ε
B₁ : ∀ᶠ (r : ℝ) in 𝓝[>] 0, (s ∩ ({x} + r • closedBall z ε)).Nonempty
ρ : ℝ
ρpos : ρ > 0
hρ : ball x ρ ∩ s ⊆ {y | ‖f y - f x - (f' x) (y - x)‖ ≤ ε * ‖y - x‖}
B₂ : ∀ᶠ (r : ℝ) in 𝓝[>] 0, {x} + r • closedBall z ε ⊆ ball x ρ
r : ℝ
y : E
ys : y ∈ s
hy : y ∈ {x} + r • closedBall z ε
rρ : {x} + r • closedBall z ε ⊆ ball x ρ
rpos : 0 < r
a : E
az : a ∈ closedBall z ε
ya : y = x + r • a
norm_a : ‖a‖ ≤ ‖z‖ + ε
⊢ ‖(f' x - A) (r • a)‖ = ‖f y - f x - A (y - x) - (f y - f x - (f' x) (y - x))‖

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁹ : NontriviallyNormedField 𝕜
inst✝⁸ : LocallyCompactSpace 𝕜
E : Type u_2
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace 𝕜 E
inst✝⁵ : MeasurableSpace E
inst✝⁴ : OpensMeasurableSpace E
F : Type u_3
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
inst✝¹ : CompleteSpace F
f : E → F
inst✝ : SecondCountableTopology E
hf : Continuous f
⊢ MeasurableSet {p | LineDifferentiableAt 𝕜 f p.1 p.2}

Tactic:
by

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

Goal state before:
pp: 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
inst✝² : BorelSpace E
inst✝¹ : FiniteDimensional ℝ E
μ : Measure (E × ℝ)
inst✝ : μ.IsAddHaarMeasure
f f' : E × ℝ → F
g g' : E × ℝ → G
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 (0, 1)
hg : ∀ x ∈ tsupport f, HasLineDerivAt ℝ g (g' x) x (0, 1)
⊢ ∫ (x : E × ℝ), (B (f x)) (g' x) ∂μ = -∫ (x : E × ℝ), (B (f' x)) (g x) ∂μ

Tactic:
by

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
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)
⊢ ContinuousOn f (Iic a)

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝¹¹ : NontriviallyNormedField 𝕜
inst✝¹⁰ : LocallyCompactSpace 𝕜
E : Type u_2
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace 𝕜 E
inst✝⁷ : MeasurableSpace E
inst✝⁶ : OpensMeasurableSpace E
F : Type u_3
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
inst✝³ : CompleteSpace F
f : E → F
inst✝² : SecondCountableTopology E
inst✝¹ : MeasurableSpace F
inst✝ : BorelSpace F
hf : Continuous f
⊢ Measurable fun p ↦ lineDeriv 𝕜 f p.1 p.2

Tactic:
by

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
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:
by

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁹ : NontriviallyNormedField 𝕜
inst✝⁸ : LocallyCompactSpace 𝕜
E : Type u_2
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace 𝕜 E
inst✝⁵ : MeasurableSpace E
inst✝⁴ : OpensMeasurableSpace E
F : Type u_3
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
inst✝¹ : CompleteSpace F
f : E → F
inst✝ : SecondCountableTopology E
hf : Continuous f
⊢ StronglyMeasurable fun p ↦ lineDeriv 𝕜 f p.1 p.2

Tactic:
by

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)
⊢ ∫ (x : ℝ) in Iic a, f' x = f a - m

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : AddCommGroup E
inst✝² : Module 𝕜 E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : QuadraticMap 𝕜 E F
a b : E
⊢ HasLineDerivAt 𝕜 (⇑f) (polar (⇑f) a b) a b

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : AddCommGroup E
inst✝² : Module 𝕜 E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : QuadraticMap 𝕜 E F
⊢ lineDeriv 𝕜 ⇑f = polar ⇑f

Tactic:
by

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

Goal state before:
pp: E : Type u_1
inst✝¹ : AddCommGroup E
inst✝ : Module ℝ E
f : E → ℝ
a b : E
f' : ℝ
l : Filter E
h : IsExtrFilter f l a
hd : HasLineDerivAt ℝ f f' a b
h' : Tendsto (fun t ↦ a + t • b) (𝓝 0) l
⊢ IsExtrFilter f l (a + 0 • b)

Tactic:
by

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

Goal state before:
pp: E : Type u_1
inst✝¹ : AddCommGroup E
inst✝ : Module ℝ E
f : E → ℝ
a b : E
l : Filter E
h : IsExtrFilter f l a
h' : Tendsto (fun t ↦ a + t • b) (𝓝 0) l
⊢ lineDeriv ℝ f a b = 0

Tactic:
by

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

Goal state before:
pp: E : Type u_1
inst✝¹ : AddCommGroup E
inst✝ : Module ℝ E
f : E → ℝ
s : Set E
a b : E
h : IsExtrOn f s a
h' : ∀ᶠ (t : ℝ) in 𝓝 0, a + t • b ∈ s
⊢ lineDerivWithin ℝ f s a b = 0

Tactic:
by

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

Goal state before:
pp: E : Type u_1
inst✝⁴ : AddCommGroup E
inst✝³ : Module ℝ E
inst✝² : TopologicalSpace E
inst✝¹ : ContinuousAdd E
inst✝ : ContinuousSMul ℝ E
f : E → ℝ
a b : E
f' : ℝ
h : IsLocalExtr f a
hd : HasLineDerivAt ℝ f f' a b
⊢ Continuous[_, inst✝²] fun t ↦ a + t • b

Tactic:
by

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

Goal state before:
pp: E : Type u_1
inst✝⁴ : AddCommGroup E
inst✝³ : Module ℝ E
inst✝² : TopologicalSpace E
inst✝¹ : ContinuousAdd E
inst✝ : ContinuousSMul ℝ E
f : E → ℝ
a b : E
f' : ℝ
h : IsLocalExtr f a
hd : HasLineDerivAt ℝ f f' a b
⊢ a + 0 • b = a

Tactic:
by

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

Goal state before:
pp: E : Type u_1
inst✝⁴ : AddCommGroup E
inst✝³ : Module ℝ E
inst✝² : TopologicalSpace E
inst✝¹ : ContinuousAdd E
inst✝ : ContinuousSMul ℝ E
f : E → ℝ
a : E
h : IsLocalExtr f a
b : E
⊢ Continuous[_, inst✝²] fun t ↦ a + t • b

Tactic:
by

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

Goal state before:
pp: E : Type u_1
inst✝⁴ : AddCommGroup E
inst✝³ : Module ℝ E
inst✝² : TopologicalSpace E
inst✝¹ : ContinuousAdd E
inst✝ : ContinuousSMul ℝ E
f : E → ℝ
a : E
h : IsLocalExtr f a
b : E
⊢ a + 0 • b = a

Tactic:
by

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
s : Set E
f : E → E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
A : E →L[ℝ] E
δ : ℝ≥0
hf : ApproximatesLinearOn f A s δ
hs : MeasurableSet s
f' : E → E →L[ℝ] E
hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x
x : E
hx : Tendsto (fun r ↦ μ (s ∩ closedBall x r) / μ (closedBall x r)) (𝓝[>] 0) (𝓝 1)
xs : x ∈ s
z : E
ε : ℝ
εpos : 0 < ε
B₁ : ∀ᶠ (r : ℝ) in 𝓝[>] 0, (s ∩ ({x} + r • closedBall z ε)).Nonempty
ρ : ℝ
ρpos : ρ > 0
hρ : ball x ρ ∩ s ⊆ {y | ‖f y - f x - (f' x) (y - x)‖ ≤ ε * ‖y - x‖}
B₂ : ∀ᶠ (r : ℝ) in 𝓝[>] 0, {x} + r • closedBall z ε ⊆ ball x ρ
r : ℝ
y : E
ys : y ∈ s
hy : y ∈ {x} + r • closedBall z ε
rρ : {x} + r • closedBall z ε ⊆ ball x ρ
rpos : 0 < r
a : E
az : a ∈ closedBall z ε
ya : y = x + r • a
norm_a : ‖a‖ ≤ ‖z‖ + ε
⊢ ↑δ * ‖y - x‖ + ε * ‖y - x‖ = r * (↑δ + ε) * ‖a‖

Tactic:
by

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
s : Set E
f : E → E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
A : E →L[ℝ] E
δ : ℝ≥0
hf : ApproximatesLinearOn f A s δ
hs : MeasurableSet s
f' : E → E →L[ℝ] E
hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x
x : E
hx : Tendsto (fun r ↦ μ (s ∩ closedBall x r) / μ (closedBall x r)) (𝓝[>] 0) (𝓝 1)
xs : x ∈ s
z : E
ε : ℝ
εpos : 0 < ε
B₁ : ∀ᶠ (r : ℝ) in 𝓝[>] 0, (s ∩ ({x} + r • closedBall z ε)).Nonempty
ρ : ℝ
ρpos : ρ > 0
hρ : ball x ρ ∩ s ⊆ {y | ‖f y - f x - (f' x) (y - x)‖ ≤ ε * ‖y - x‖}
B₂ : ∀ᶠ (r : ℝ) in 𝓝[>] 0, {x} + r • closedBall z ε ⊆ ball x ρ
r : ℝ
y : E
ys : y ∈ s
hy : y ∈ {x} + r • closedBall z ε
rρ : {x} + r • closedBall z ε ⊆ ball x ρ
rpos : 0 < r
a : E
az : a ∈ closedBall z ε
ya : y = x + r • a
norm_a : ‖a‖ ≤ ‖z‖ + ε
⊢ r * (↑δ + ε) * ‖a‖ ≤ r * (↑δ + ε) * (‖z‖ + ε)

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Calculus.LocalExtr.Polynomial

Goal state before:
pp: p : ℝ[X]
hp' : derivative p ≠ 0
⊢ derivative p ≠ derivative 0

Tactic:
by

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

Goal state before:
pp: E : Type u_1
f : ℝ → E
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
hf : ContDiff ℝ 1 f
h2f : HasCompactSupport f
b : ℝ
⊢ ∫ (x : ℝ) in Iic b, deriv f x = f b

Tactic:
by

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

Goal state before:
pp: 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
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
hW : CompleteSpace W
hf : ∀ x ∈ tsupport g, HasLineDerivAt ℝ f (f' x) x 0
hg : ∀ x ∈ tsupport f, HasLineDerivAt ℝ g (g' x) x 0
x : E
⊢ (B (f' x)) (g x) = 0

Tactic:
by

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

Goal state before:
pp: F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f : ℝ → F
hf : ContDiff ℝ 1 f
h'f : HasCompactSupport f
x : ℝ
I : F →L[ℝ] Completion F := Completion.toComplL
f' : ℝ → Completion F := ⇑I ∘ f
hf' : ContDiff ℝ 1 f'
h'f' : HasCompactSupport f'
⊢ ‖f' x‖ₑ ≤ ∫⁻ (y : ℝ) in Iic x, ‖deriv f' y‖ₑ

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Calculus.LocalExtr.Polynomial

Goal state before:
pp: p : ℝ[X]
⊢ p.roots.toFinset.card ≤ ((derivative p).roots.toFinset \ p.roots.toFinset).card + 1

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Calculus.LocalExtr.Polynomial

Goal state before:
pp: p : ℝ[X]
⊢ ((derivative p).roots.toFinset \ p.roots.toFinset).card + 1 ≤ (derivative p).roots.toFinset.card + 1

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Calculus.LocalExtr.Polynomial

Goal state before:
pp: p : ℝ[X]
⊢ ∑ x ∈ p.roots.toFinset, (Multiset.count x p.roots - 1 + 1) =
    ∑ x ∈ p.roots.toFinset, (rootMultiplicity x p - 1) + p.roots.toFinset.card

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Calculus.LocalExtr.Polynomial

Goal state before:
pp: p : ℝ[X]
⊢ ∑ x ∈ p.roots.toFinset, rootMultiplicity x (derivative p) +
      (((derivative p).roots.toFinset \ p.roots.toFinset).card + 1) ≤
    ∑ x ∈ p.roots.toFinset, Multiset.count x (derivative p).roots +
      (∑ x ∈ (derivative p).roots.toFinset \ p.roots.toFinset, Multiset.count x (derivative p).roots + 1)

Tactic:
by

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
s : Set E
f : E → E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
A : E →L[ℝ] E
δ : ℝ≥0
hf : ApproximatesLinearOn f A s δ
hs : MeasurableSet s
f' : E → E →L[ℝ] E
hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x
x : E
hx : Tendsto (fun r ↦ μ (s ∩ closedBall x r) / μ (closedBall x r)) (𝓝[>] 0) (𝓝 1)
xs : x ∈ s
z : E
ε : ℝ
εpos : 0 < ε
B₁ : ∀ᶠ (r : ℝ) in 𝓝[>] 0, (s ∩ ({x} + r • closedBall z ε)).Nonempty
ρ : ℝ
ρpos : ρ > 0
hρ : ball x ρ ∩ s ⊆ {y | ‖f y - f x - (f' x) (y - x)‖ ≤ ε * ‖y - x‖}
B₂ : ∀ᶠ (r : ℝ) in 𝓝[>] 0, {x} + r • closedBall z ε ⊆ ball x ρ
r : ℝ
y : E
ys : y ∈ s
hy : y ∈ {x} + r • closedBall z ε
rρ : {x} + r • closedBall z ε ⊆ ball x ρ
rpos : 0 < r
a : E
az : a ∈ closedBall z ε
ya : y = x + r • a
norm_a : ‖a‖ ≤ ‖z‖ + ε
I : r * ‖(f' x - A) a‖ ≤ r * (↑δ + ε) * (‖z‖ + ε)
⊢ ‖(f' x - A) z‖ = ‖(f' x - A) a + (f' x - A) (z - a)‖

Tactic:
by

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

Goal state before:
pp: F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f : ℝ → F
hf : ContDiff ℝ 1 f
h'f : HasCompactSupport f
x : ℝ
⊢ ‖f x‖ₑ ≤ ∫⁻ (y : ℝ) in Iic x, ‖deriv f y‖ₑ

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Calculus.LocalExtr.Polynomial

Goal state before:
pp: p : ℝ[X]
⊢ ∑ x ∈ p.roots.toFinset, Multiset.count x (derivative p).roots +
      (∑ x ∈ (derivative p).roots.toFinset \ p.roots.toFinset, Multiset.count x (derivative p).roots + 1) =
    (derivative p).roots.card + 1

Tactic:
by

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

Goal state before:
pp: 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
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
hW : CompleteSpace W
hf : ∀ x ∈ tsupport g, HasLineDerivAt ℝ f (f' x) x 0
hg : ∀ x ∈ tsupport f, HasLineDerivAt ℝ g (g' x) x 0
Hf' : ∀ (x : E), (B (f' x)) (g x) = 0
x : E
⊢ (B (f x)) (g' x) = 0

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Calculus.LocalExtr.Polynomial

Goal state before:
pp: F : Type u_1
inst✝¹ : CommRing F
inst✝ : Algebra F ℝ
p : F[X]
⊢ Fintype.card ↑(p.rootSet ℝ) ≤ Fintype.card ↑((derivative p).rootSet ℝ) + 1

Tactic:
by

Goal state after:
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), f' x = n - m

Tactic:
by

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

Goal state before:
pp: case neg
E : Type u_1
f f' : ℝ → E
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
hderiv : ∀ (x : ℝ), HasDerivAt f (f' x) x
hf' : Integrable f' volume
hf : Integrable f volume
hE : ¬CompleteSpace E
⊢ ∫ (x : ℝ), f' x = 0

Tactic:
simp [integral, hE]

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

Goal state before:
pp: E : Type u_1
f f' : ℝ → E
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
hderiv : ∀ (x : ℝ), HasDerivAt f (f' x) x
hf' : Integrable f' volume
hf : Integrable f volume
⊢ ∫ (x : ℝ), f' x = 0

Tactic:
by

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

Goal state before:
pp: 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
⊢ finrank ℝ (E' × ℝ) = n

Tactic:
by

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

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f f' : ℝ → ℝ
g : ℝ → E
a : ℝ
hf : ContinuousOn f (Ici a)
hft : Tendsto f atTop atTop
hff' : ∀ x ∈ Ioi a, HasDerivWithinAt f (f' x) (Ioi x) x
hg_cont : ContinuousOn g (f '' Ioi a)
hg1 : IntegrableOn g (f '' Ici a) volume
hg2 : IntegrableOn (fun x ↦ f' x • (g ∘ f) x) (Ici a) volume
b : ℝ
hb : a < b
⊢ Ioo (min a b) (max a b) ⊆ Ioi a

Tactic:
by

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

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f f' : ℝ → ℝ
g : ℝ → E
a : ℝ
hf : ContinuousOn f (Ici a)
hft : Tendsto f atTop atTop
hff' : ∀ x ∈ Ioi a, HasDerivWithinAt f (f' x) (Ioi x) x
hg_cont : ContinuousOn g (f '' Ioi a)
hg1 : IntegrableOn g (f '' Ici a) volume
hg2 : IntegrableOn (fun x ↦ f' x • (g ∘ f) x) (Ici a) volume
b : ℝ
hb : a < b
i1 : Ioo (min a b) (max a b) ⊆ Ioi a
⊢ [[a, b]] ⊆ Ici a

Tactic:
by

Goal state after: