mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Topology.Algebra.InfiniteSum.UniformOn

Goal state before:
pp: α : Type u_1
β : Type u_2
ι : Type u_3
inst✝² : CommMonoid α
s : Set β
inst✝¹ : UniformSpace α
inst✝ : TopologicalSpace β
f f' : ι → β → α
h : ∀ (i : ι), Set.EqOn (f i) (f' i) s
h2 : MultipliableLocallyUniformlyOn f s
⊢ MultipliableLocallyUniformlyOn f' s

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
f' : E → E →L[ℝ] E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x
h'f' : ∀ x ∈ s, (f' x).det = 0
H : ∀ (R : ℝ), μ (f '' (s ∩ closedBall 0 R)) = 0
⊢ μ (f '' ⋃ n, s ∩ closedBall 0 ↑n) ≤ ∑' (n : ℕ), μ (f '' (s ∩ closedBall 0 ↑n))

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Complex.LocallyUniformLimit

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
z : ℂ
r : ℝ
f g : ℂ → E
hr : 0 < r
hf : ContinuousOn f (sphere z r)
hg : ContinuousOn g (sphere z r)
⊢ cderiv r (f - g) z = cderiv r f z - cderiv r g z

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
f' : E → E →L[ℝ] E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x
h'f' : ∀ x ∈ s, (f' x).det = 0
H : ∀ (R : ℝ), μ (f '' (s ∩ closedBall 0 R)) = 0
⊢ ∑' (n : ℕ), μ (f '' (s ∩ closedBall 0 ↑n)) ≤ 0

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Complex.LocallyUniformLimit

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
z : ℂ
M r : ℝ
f : ℂ → E
hr : 0 < r
hfM : ∀ w ∈ sphere z r, ‖f w‖ < M
hf : ContinuousOn f (sphere z r)
⊢ ∃ L < M, ∀ w ∈ sphere z r, ‖f w‖ ≤ L

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Complex.LocallyUniformLimit

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
z : ℂ
M r : ℝ
f : ℂ → E
hr : 0 < r
hfM : ∀ w ∈ sphere z r, ‖f w‖ < M
hf : ContinuousOn f (sphere z r)
⊢ ‖cderiv r f z‖ < M / r

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
f' : E → E →L[ℝ] E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x
h'f' : ∀ x ∈ s, (f' x).det = 0
H : ∀ (R : ℝ), μ (f '' (s ∩ closedBall 0 R)) = 0
⊢ μ (f '' s) = 0

Tactic:
by

Goal state after:
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 (u' * v + u * v') (Ioi a) volume
h_zero : Tendsto (u * v) (𝓝[Ici a \ {a}] a) (𝓝 a')
h_infty : Tendsto (u * v) atTop (𝓝 b')
f : ℝ → A := Function.update (u * v) a a'
hderiv : ∀ x ∈ Ioi a, HasDerivAt f (u' x * v x + u x * v' x) x
⊢ Tendsto f atTop (𝓝 b')

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Algebra.InfiniteSum.UniformOn

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝² : CommMonoid α
g : β → α
s : Set β
inst✝¹ : UniformSpace α
inst✝ : TopologicalSpace β
f : ℕ → β → α
h : HasProdLocallyUniformlyOn f g s
⊢ TendstoLocallyUniformlyOn (fun N b ↦ ∏ i ∈ Finset.range N, f i b) g atTop s

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Algebra.InfiniteSum.UniformOn

Goal state before:
pp: α : Type u_1
β : Type u_2
ι : Type u_3
inst✝¹ : CommMonoid α
f : ι → β → α
g : β → α
inst✝ : UniformSpace α
⊢ HasProdUniformly f g ↔ TendstoUniformly (fun x1 x2 ↦ ∏ i ∈ x1, f i x2) g atTop

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Algebra.InfiniteSum.UniformOn

Goal state before:
pp: α : Type u_1
β : Type u_2
ι : Type u_3
inst✝¹ : CommMonoid α
f : ι → β → α
g : β → α
inst✝ : UniformSpace α
⊢ HasProdUniformlyOn f g Set.univ ↔ HasProdUniformly f g

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
f' : E → E →L[ℝ] E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x
h'f' : ∀ x ∈ s, (f' x).det = 0
R : ℝ
A : ∀ (ε : ℝ≥0), 0 < ε → μ (f '' (s ∩ closedBall 0 R)) ≤ ↑ε * μ (closedBall 0 R)
⊢ Tendsto (fun ε ↦ ↑ε * μ (closedBall 0 R)) (𝓝[>] 0) (𝓝 0)

Tactic:
by

Goal state after:
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 (u' * v + u * v') (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 + u x * v' x = b' - a'

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Algebra.InfiniteSum.UniformOn

Goal state before:
pp: α : Type u_1
β : Type u_2
ι : Type u_3
inst✝¹ : CommMonoid α
f : ι → β → α
g : β → α
inst✝ : UniformSpace α
f' : ι → β → α
h : TendstoUniformly (fun x1 x2 ↦ ∏ i ∈ x1, f i x2) g atTop
hff' : ∀ᶠ (n : Finset ι) in atTop, ∀ (b : β), ∏ i ∈ n, f i b = ∏ i ∈ n, f' i b
⊢ (fun x1 x2 ↦ ∏ i ∈ x1, f i x2) =ᶠ[atTop] fun x1 x2 ↦ ∏ i ∈ x1, f' i x2

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Algebra.InfiniteSum.UniformOn

Goal state before:
pp: α : Type u_1
β : Type u_2
ι : Type u_3
inst✝¹ : CommMonoid α
f : ι → β → α
g : β → α
inst✝ : UniformSpace α
f' : ι → β → α
h : HasProdUniformly f g
hff' : ∀ᶠ (n : Finset ι) in atTop, ∀ (b : β), ∏ i ∈ n, f i b = ∏ i ∈ n, f' i b
⊢ HasProdUniformly f' g

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Algebra.InfiniteSum.UniformOn

Goal state before:
pp: α : Type u_1
β : Type u_2
ι : Type u_3
inst✝² : CommMonoid α
f : ι → β → α
g : β → α
inst✝¹ : UniformSpace α
inst✝ : TopologicalSpace β
h : ∀ (x : β), ∃ t ∈ 𝓝 x, HasProdUniformlyOn f g t
⊢ ∀ (x : β), ∃ t ∈ 𝓝 x, TendstoUniformlyOn (fun x1 x2 ↦ ∏ i ∈ x1, f i x2) g atTop t

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
f' : E → E →L[ℝ] E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x
h'f' : ∀ x ∈ s, (f' x).det = 0
⊢ μ (f '' s) = 0

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Algebra.InfiniteSum.UniformOn

Goal state before:
pp: α : Type u_1
β : Type u_2
ι : Type u_3
inst✝² : CommMonoid α
f : ι → β → α
g : β → α
inst✝¹ : UniformSpace α
inst✝ : TopologicalSpace β
h : HasProdUniformly f g
⊢ HasProdLocallyUniformly f g

Tactic:
simp only [hasProdUniformly_iff_tendstoUniformly, HasProdLocallyUniformly] at *

Goal state after:
α : Type u_1
β : Type u_2
ι : Type u_3
inst✝² : CommMonoid α
f : ι → β → α
g : β → α
inst✝¹ : UniformSpace α
inst✝ : TopologicalSpace β
h : TendstoUniformly (fun x1 x2 ↦ ∏ i ∈ x1, f i x2) g atTop
⊢ TendstoLocallyUniformly (fun x1 x2 ↦ ∏ i ∈ x1, f i x2) g atTop
Module: Mathlib.Topology.Algebra.InfiniteSum.UniformOn

Goal state before:
pp: α : Type u_1
β : Type u_2
ι : Type u_3
inst✝² : CommMonoid α
f : ι → β → α
g : β → α
inst✝¹ : UniformSpace α
inst✝ : TopologicalSpace β
h : HasProdUniformly f g
⊢ HasProdLocallyUniformly f g

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Algebra.InfiniteSum.UniformOn

Goal state before:
pp: α : Type u_1
β : Type u_2
ι : Type u_3
inst✝³ : CommMonoid α
f : ι → β → α
g : β → α
inst✝² : UniformSpace α
inst✝¹ : TopologicalSpace β
inst✝ : LocallyCompactSpace β
h : ∀ (K : Set β), IsCompact K → HasProdUniformlyOn f g K
⊢ HasProdLocallyUniformly f g

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Algebra.InfiniteSum.UniformOn

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝² : CommMonoid α
g : β → α
inst✝¹ : UniformSpace α
inst✝ : TopologicalSpace β
f : ℕ → β → α
h : HasProdLocallyUniformly f g
⊢ TendstoLocallyUniformly (fun x1 x2 ↦ ∏ i ∈ Finset.range x1, f i x2) g atTop

Tactic:
by

Goal state after:
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 (u * v') (Ioi a) volume
hu'v : IntegrableOn (u' * v) (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:
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
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
ε : ℝ
εpos : ε > 0
δ : ℝ≥0 := ⟨ε, ⋯⟩
δpos : 0 < δ
t : ℕ → Set E
A : ℕ → E →L[ℝ] E
t_disj : Pairwise (Disjoint on t)
t_meas : ∀ (n : ℕ), MeasurableSet (t n)
t_cover : s ⊆ ⋃ n, t n
ht : ∀ (n : ℕ), ApproximatesLinearOn f (A n) (s ∩ t n) δ
right✝ : s.Nonempty → ∀ (n : ℕ), ∃ y ∈ s, A n = f' y
i j : ℕ
h : (Disjoint on t) i j
⊢ EqOn (fun x ↦ A i) (fun x ↦ A j) (t i ∩ t j)

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Calculus.LogDerivUniformlyOn

Goal state before:
pp: ι : Type u_1
s : Set ℂ
hs : IsOpen s
x : ℂ
hx : x ∈ s
f : ι → ℂ → ℂ
hf : ∀ (i : ι), f i x ≠ 0
hd : ∀ (i : ι), DifferentiableOn ℂ (f i) s
hm : Summable fun i ↦ logDeriv (f i) x
htend : MultipliableLocallyUniformlyOn f s
hnez : ∏' (i : ι), f i x ≠ 0
⊢ ∀ (x : Finset ι), DifferentiableOn ℂ (fun x2 ↦ ∏ i ∈ x, f i x2) s

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Calculus.LogDerivUniformlyOn

Goal state before:
pp: ι : Type u_1
s : Set ℂ
hs : IsOpen s
x : ℂ
hx : x ∈ s
f : ι → ℂ → ℂ
hf : ∀ (i : ι), f i x ≠ 0
hd : ∀ (i : ι), DifferentiableOn ℂ (f i) s
hm : Summable fun i ↦ logDeriv (f i) x
htend : MultipliableLocallyUniformlyOn f s
hnez : ∏' (i : ι), f i x ≠ 0
⊢ logDeriv (fun x ↦ ∏' (i : ι), f i x) x = ∑' (i : ι), logDeriv (f i) x

Tactic:
by

Goal state after:
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 (t : ℝ) ∂μ, t ∈ Ι a b → LipschitzOnWith (Real.nnabs (bound t)) (fun x ↦ F x t) s
bound_integrable : IntervalIntegrable bound μ a b
h_diff : ∀ᵐ (t : ℝ) ∂μ, t ∈ Ι a b → HasFDerivAt (fun x ↦ F x t) (F' t) 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:
by

Goal state after:
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 : TendstoUniformlyOn F f φ (cthickening δ K)
hδ : 0 < δ
hFn : ∀ᶠ (n : ι) in φ, ContinuousOn (F n) (cthickening δ K)
⊢ TendstoUniformlyOn (Complex.cderiv δ ∘ F) (Complex.cderiv δ f) φ K

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
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
ε : ℝ
εpos : ε > 0
δ : ℝ≥0 := ⟨ε, ⋯⟩
δpos : 0 < δ
t : ℕ → Set E
A : ℕ → E →L[ℝ] E
t_disj : Pairwise (Disjoint on t)
t_meas : ∀ (n : ℕ), MeasurableSet (t n)
t_cover : s ⊆ ⋃ n, t n
ht : ∀ (n : ℕ), ApproximatesLinearOn f (A n) (s ∩ t n) δ
right✝ : s.Nonempty → ∀ (n : ℕ), ∃ y ∈ s, A n = f' y
g : E → E →L[ℝ] E
g_meas : Measurable g
hg : ∀ (n : ℕ), ∀ x ∈ t n, g x = A n
H : ∀ᵐ (x : E) ∂sum fun n ↦ μ.restrict (s ∩ t n), dist (g x) (f' x) ≤ ε
⊢ s = ⋃ n, s ∩ t n

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Complex.LocallyUniformLimit

Goal state before:
pp: E : Type u_1
ι : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
U K : Set ℂ
φ : Filter ι
F : ι → ℂ → E
f : ℂ → E
inst✝ : CompleteSpace E
hf : TendstoLocallyUniformlyOn F f φ U
hF : ∀ᶠ (n : ι) in φ, DifferentiableOn ℂ (F n) U
δ : ℝ
hδ : 0 < δ
hK : IsCompact K
hU : IsOpen U
hKU : cthickening δ K ⊆ U
⊢ ∀ᶠ (n : ι) in φ, ContinuousOn (F n) (cthickening δ K)

Tactic:
by

Goal state after:
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' : H → 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' x₀) (μ.restrict (Ι a b))
h_bound : ∀ᵐ (t : ℝ) ∂μ, t ∈ Ι a b → ∀ x ∈ s, ‖F' x t‖ ≤ bound t
bound_integrable : IntervalIntegrable bound μ a b
h_diff : ∀ᵐ (t : ℝ) ∂μ, t ∈ Ι a b → ∀ x ∈ s, HasFDerivAt (fun x ↦ F x t) (F' x t) x
⊢ HasFDerivAt (fun x ↦ ∫ (t : ℝ) in a..b, F x t ∂μ) (∫ (t : ℝ) in a..b, F' x₀ t ∂μ) x₀

Tactic:
by

Goal state after:
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 (u' * v + u * v') (Iic a) volume
h_zero : Tendsto (u * v) (𝓝[Iic a \ {a}] a) (𝓝 a')
h_infty : Tendsto (u * v) atBot (𝓝 b')
f : ℝ → A := Function.update (u * v) a a'
⊢ ∀ x ∈ Iio a, HasDerivAt f (u' x * v x + u x * v' x) x

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Complex.LocallyUniformLimit

Goal state before:
pp: E : Type u_1
ι : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
U K : Set ℂ
φ : Filter ι
F : ι → ℂ → E
f : ℂ → E
inst✝ : CompleteSpace E
hf : TendstoLocallyUniformlyOn F f φ U
hF : ∀ᶠ (n : ι) in φ, DifferentiableOn ℂ (F n) U
δ : ℝ
hδ : 0 < δ
hK : IsCompact K
hU : IsOpen U
hKU : cthickening δ K ⊆ U
h1 : ∀ᶠ (n : ι) in φ, ContinuousOn (F n) (cthickening δ K)
h2 : IsCompact (cthickening δ K)
⊢ TendstoUniformlyOn (deriv ∘ F) (cderiv δ f) φ K

Tactic:
have h3 : TendstoUniformlyOn F f φ (cthickening δ K) :=
    (tendstoLocallyUniformlyOn_iff_forall_isCompact hU).mp hf (cthickening δ K) hKU h2

Goal state after:
E : Type u_1
ι : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
U K : Set ℂ
φ : Filter ι
F : ι → ℂ → E
f : ℂ → E
inst✝ : CompleteSpace E
hf : TendstoLocallyUniformlyOn F f φ U
hF : ∀ᶠ (n : ι) in φ, DifferentiableOn ℂ (F n) U
δ : ℝ
hδ : 0 < δ
hK : IsCompact K
hU : IsOpen U
hKU : cthickening δ K ⊆ U
h1 : ∀ᶠ (n : ι) in φ, ContinuousOn (F n) (cthickening δ K)
h2 : IsCompact (cthickening δ K)
h3 : TendstoUniformlyOn F f φ (cthickening δ K)
⊢ TendstoUniformlyOn (deriv ∘ F) (cderiv δ f) φ K
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 (u' * v + u * v') (Iic a) volume
h_zero : Tendsto (u * v) (𝓝[Iic a \ {a}] a) (𝓝 a')
h_infty : Tendsto (u * v) atBot (𝓝 b')
f : ℝ → A := Function.update (u * v) a a'
hderiv : ∀ x ∈ Iio a, HasDerivAt f (u' x * v x + u x * v' x) x
⊢ Tendsto f atBot (𝓝 b')

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Complex.LocallyUniformLimit

Goal state before:
pp: E : Type u_1
ι : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
U K : Set ℂ
φ : Filter ι
F : ι → ℂ → E
f : ℂ → E
inst✝ : CompleteSpace E
hf : TendstoLocallyUniformlyOn F f φ U
hF : ∀ᶠ (n : ι) in φ, DifferentiableOn ℂ (F n) U
δ : ℝ
hδ : 0 < δ
hK : IsCompact K
hU : IsOpen U
hKU : cthickening δ K ⊆ U
⊢ TendstoUniformlyOn (deriv ∘ F) (cderiv δ f) φ K

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Complex.LocallyUniformLimit

Goal state before:
pp: E : Type u_1
ι : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
U K : Set ℂ
φ : Filter ι
F : ι → ℂ → E
f : ℂ → E
inst✝ : CompleteSpace E
hf : TendstoLocallyUniformlyOn F f φ U
hF : ∀ᶠ (n : ι) in φ, DifferentiableOn ℂ (F n) U
hK : IsCompact K
hU : IsOpen U
hKU : K ⊆ U
⊢ ∃ δ > 0, cthickening δ K ⊆ U ∧ TendstoUniformlyOn (deriv ∘ F) (cderiv δ f) φ K

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Complex.LocallyUniformLimit

Goal state before:
pp: E : Type u_1
ι : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
U : Set ℂ
φ : Filter ι
F : ι → ℂ → E
f : ℂ → E
inst✝¹ : CompleteSpace E
inst✝ : φ.NeBot
hf : TendstoLocallyUniformlyOn F f φ U
hF : ∀ᶠ (n : ι) in φ, DifferentiableOn ℂ (F n) U
hU : IsOpen U
x : ℂ
hx : x ∈ U
K : Set ℂ
hKU : K ⊆ U
hKx : K ∈ 𝓝 x
hK : IsCompact K
δ : ℝ
left✝¹ : δ > 0
left✝ : cthickening δ K ⊆ U
h1 : TendstoUniformlyOn (deriv ∘ F) (cderiv δ f) φ K
h2 : interior K ⊆ U
⊢ ∀ᶠ (n : ι) in φ, DifferentiableOn ℂ (F n) (interior K)

Tactic:
by

Goal state after:
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 (u' * v + u * v') (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 + u x * v' x = a' - b'

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
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
ε : ℝ
εpos : ε > 0
δ : ℝ≥0 := ⟨ε, ⋯⟩
δpos : 0 < δ
t : ℕ → Set E
A : ℕ → E →L[ℝ] E
t_disj : Pairwise (Disjoint on t)
t_meas : ∀ (n : ℕ), MeasurableSet (t n)
t_cover : s ⊆ ⋃ n, t n
ht : ∀ (n : ℕ), ApproximatesLinearOn f (A n) (s ∩ t n) δ
right✝ : s.Nonempty → ∀ (n : ℕ), ∃ y ∈ s, A n = f' y
g : E → E →L[ℝ] E
g_meas : Measurable g
hg : ∀ (n : ℕ), ∀ x ∈ t n, g x = A n
H : ∀ᵐ (x : E) ∂sum fun n ↦ μ.restrict (s ∩ t n), dist (g x) (f' x) ≤ ε
⊢ μ.restrict s ≤ sum fun n ↦ μ.restrict (s ∩ t n)

Tactic:
by

Goal state after:
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 (u' * v + u * v') (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 + u x * v' x = a' - b'

Tactic:
by
  rw [← Iic_sdiff_right] at h_zero
  let f := Function.update (u * v) a a'
  have hderiv : ∀ x ∈ Iio a, HasDerivAt f (u' x * v x + u x * v' x) x := by
    intro x hx
    apply ((hu x hx).mul (hv x hx)).congr_of_eventuallyEq
    filter_upwards [Iio_mem_nhds hx] with x (hx : x < a)
    exact Function.update_of_ne (ne_of_lt hx) a' (u * v)
  have htendsto : Tendsto f atBot (𝓝 b') := by
    apply h_infty.congr'
    filter_upwards [Iio_mem_atBot a] with x (hx : x < a)
    exact (Function.update_of_ne (ne_of_lt hx) a' (u * v)).symm
  simpa using integral_Iic_of_hasDerivAt_of_tendsto
    (continuousWithinAt_update_same.mpr h_zero) hderiv huv htendsto

Goal state after:
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 (t : ℝ) ∂μ, t ∈ Ι a b → LipschitzOnWith (Real.nnabs (bound t)) (fun x ↦ F x t) s
bound_integrable : IntervalIntegrable bound μ a b
h_diff : ∀ᵐ (t : ℝ) ∂μ, t ∈ Ι a b → HasDerivAt (fun x ↦ F x t) (F' t) 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:
by

Goal state after:
Module: Mathlib.Analysis.Complex.LocallyUniformLimit

Goal state before:
pp: E : Type u_1
ι : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
U : Set ℂ
φ : Filter ι
F : ι → ℂ → E
f : ℂ → E
inst✝¹ : CompleteSpace E
inst✝ : φ.NeBot
hf : TendstoLocallyUniformlyOn F f φ U
hF : ∀ᶠ (n : ι) in φ, DifferentiableOn ℂ (F n) U
hU : IsOpen U
⊢ DifferentiableOn ℂ f U

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
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
ε : ℝ
εpos : ε > 0
δ : ℝ≥0 := ⟨ε, ⋯⟩
δpos : 0 < δ
t : ℕ → Set E
A : ℕ → E →L[ℝ] E
t_disj : Pairwise (Disjoint on t)
t_meas : ∀ (n : ℕ), MeasurableSet (t n)
t_cover : s ⊆ ⋃ n, t n
ht : ∀ (n : ℕ), ApproximatesLinearOn f (A n) (s ∩ t n) δ
right✝ : s.Nonempty → ∀ (n : ℕ), ∃ y ∈ s, A n = f' y
g : E → E →L[ℝ] E
g_meas : Measurable g
hg : ∀ (n : ℕ), ∀ x ∈ t n, g x = A n
H : ∀ᵐ (x : E) ∂sum fun n ↦ μ.restrict (s ∩ t n), dist (g x) (f' x) ≤ ε
⊢ ∀ᵐ (x : E) ∂μ.restrict s, dist (g x) (f' x) ≤ ε

Tactic:
by

Goal state after:
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 (t : ℝ) ∂μ, t ∈ Ι a b → ∀ x ∈ s, ‖F' x t‖ ≤ bound t
bound_integrable : IntervalIntegrable bound μ a b
h_diff : ∀ᵐ (t : ℝ) ∂μ, t ∈ Ι a b → ∀ x ∈ s, HasDerivAt (fun x ↦ F x t) (F' x t) x
⊢ 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:
by

Goal state after:
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 (u * v') (Iic a) volume
hu'v : IntegrableOn (u' * v) (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:
by

Goal state after: