mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.MeasureTheory.Integral.DominatedConvergence

Goal state before:
pp: E : Type u_1
X : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : TopologicalSpace X
μ : Measure ℝ
inst✝ : FirstCountableTopology X
F : X → ℝ → E
bound : ℝ → ℝ
a b a₀ b₀ : ℝ
x₀ : X
hF_meas : ∀ (x : X), AEStronglyMeasurable (F x) (μ.restrict (Ι a b))
h_bound : ∀ᶠ (x : X) in 𝓝 x₀, ∀ᵐ (t : ℝ) ∂μ.restrict (Ι a b), ‖F x t‖ ≤ bound t
bound_integrable : IntervalIntegrable bound μ a b
h_cont : ∀ᵐ (t : ℝ) ∂μ.restrict (Ι a b), ContinuousAt (fun x ↦ F x t) x₀
ha₀ : a₀ ∈ Ioo a b
hb₀ : b₀ ∈ Ioo a b
hμb₀ : μ {b₀} = 0
⊢ ContinuousAt (fun p ↦ ∫ (t : ℝ) in a₀..p.2, F p.1 t ∂μ) (x₀, b₀)

Tactic:
by

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

Goal state before:
pp: E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
a b : ℝ
μ : Measure ℝ
f : ℝ → E
inst✝ : NullSingletonClass μ
h_int : IntegrableOn f (Icc a b) μ
h : a ≤ b
⊢ ∀ x ∈ Icc a b, ∫ (t : ℝ) in Ioc a x, f t ∂μ = ∫ (t : ℝ) in a..x, f t ∂μ

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : LocallyCompactSpace E
F : Type u_3
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
α : Type u_4
inst✝ : TopologicalSpace α
f : α → E → F
r s t : ℝ
hf : Continuous[instTopologicalSpaceProd, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (Function.uncurry f)
K : Set (E →L[𝕜] F)
⊢ IsOpen[instTopologicalSpaceProd] {p | p.2 ∈ B (f p.1) K r s t}

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Integral.IntervalIntegral.FundThmCalculus

Goal state before:
pp: g' g φ : ℝ → ℝ
a b : ℝ
hab : a ≤ b
hcont : ContinuousOn g (Icc a b)
hderiv : ∀ x ∈ Ico a b, HasDerivWithinAt g (g' x) (Ioi x) x
φint : IntegrableOn φ (Icc a b) volume
hφg : ∀ x ∈ Ico a b, g' x ≤ φ x
ε : ℝ
εpos : 0 < ε
G' : ℝ → EReal
f_lt_G' : ∀ (x : ℝ), ↑(φ x) < G' x
G'cont : LowerSemicontinuous G'
G'int : Integrable (fun x ↦ (G' x).toReal) (volume.restrict (Icc a b))
G'lt_top : ∀ᵐ (x : ℝ) ∂volume.restrict (Icc a b), G' x < ⊤
hG' : ∫ (x : ℝ) in Icc a b, (G' x).toReal < (∫ (x : ℝ) in Icc a b, φ x) + ε
s : Set ℝ := {t | g t - g a ≤ ∫ (u : ℝ) in a..t, (G' u).toReal} ∩ Icc a b
s_closed : IsClosed s
t : ℝ
ht : t ∈ {t | g t - g a ≤ ∫ (u : ℝ) in a..t, (G' u).toReal} ∩ Ico a b
v : ℝ
t_lt_v : v ∈ Ioi t
y : ℝ
g'_lt_y' : ↑(g' t) < ↑y
y_lt_G' : ↑y < G' t
I1 : ∀ᶠ (u : ℝ) in 𝓝[>] t, (u - t) * y ≤ ∫ (w : ℝ) in t..u, (G' w).toReal
I2 : ∀ᶠ (u : ℝ) in 𝓝[>] t, g u - g t ≤ (u - t) * y
I3 : ∀ᶠ (u : ℝ) in 𝓝[>] t, g u - g t ≤ ∫ (w : ℝ) in t..u, (G' w).toReal
I4 : ∀ᶠ (u : ℝ) in 𝓝[>] t, u ∈ Ioc t (min v b)
x : ℝ
hx : g x - g t ≤ ∫ (w : ℝ) in t..x, (G' w).toReal
h'x : x ∈ Ioc t (min v b)
⊢ g x - g a = g t - g a + (g x - g t)

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝¹⁰ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace 𝕜 E
inst✝⁷ : LocallyCompactSpace E
F : Type u_3
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace 𝕜 F
α : Type u_4
inst✝⁴ : TopologicalSpace α
f : α → E → F
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : MeasurableSpace E
inst✝ : OpensMeasurableSpace E
hf : Continuous[instTopologicalSpaceProd, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (Function.uncurry f)
K : Set (E →L[𝕜] F)
hK : IsComplete K
⊢ {p | DifferentiableAt 𝕜 (f p.1) p.2 ∧ fderiv 𝕜 (f p.1) p.2 ∈ K} = {p | p.2 ∈ D (f p.1) K}

Tactic:
by

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

Goal state before:
pp: E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
a b : ℝ
μ : Measure ℝ
f : ℝ → E
inst✝ : NullSingletonClass μ
h_int : IntegrableOn f (Icc a b) μ
⊢ ContinuousOn (fun x ↦ ∫ (t : ℝ) in Ioc a x, f t ∂μ) (Icc a b)

Tactic:
by

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

Goal state before:
pp: E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
a b : ℝ
μ : Measure ℝ
f : ℝ → E
inst✝ : NullSingletonClass μ
h_int : IntegrableOn f (Icc a b) μ
⊢ (fun x ↦ ∫ (t : ℝ) in Icc a x, f t ∂μ) = fun x ↦ ∫ (t : ℝ) in Ioc a x, f t ∂μ

Tactic:
by

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

Goal state before:
pp: E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
a b : ℝ
μ : Measure ℝ
f : ℝ → E
inst✝ : NullSingletonClass μ
h_int : IntegrableOn f (Icc a b) μ
⊢ ContinuousOn (fun x ↦ ∫ (t : ℝ) in Icc a x, f t ∂μ) (Icc a b)

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic

Goal state before:
pp: E : Type u_5
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
a b c : ℝ
f : ℝ → E
μ : Measure ℝ
hab : IntervalIntegrable f μ a b
hbc : IntervalIntegrable f μ b c
hac : IntervalIntegrable f μ a c
this : Ioc a b ∪ Ioc b c ∪ Ioc c a = Ioc b a ∪ Ioc c b ∪ Ioc a c
⊢ ∫ (x : ℝ) in Ioc a b ∪ Ioc b c ∪ Ioc c a, f x ∂μ = ∫ (x : ℝ) in Ioc b a ∪ Ioc c b ∪ Ioc a c, f x ∂μ

Tactic:
by

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

Goal state before:
pp: E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
a b₁ b₂ : ℝ
μ : Measure ℝ
f : ℝ → E
inst✝ : NullSingletonClass μ
h_int : IntervalIntegrable f μ b₁ b₂
ha : a ∈ [[b₁, b₂]]
x✝¹ : ℝ
x✝ : x✝¹ ∈ [[b₁, b₂]]
⊢ ContinuousWithinAt (fun b ↦ ∫ (x : ℝ) in a..b, f x ∂μ) [[b₁, b₂]] x✝¹

Tactic:
by

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

Goal state before:
pp: E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
a b : ℝ
μ : Measure ℝ
f : ℝ → E
inst✝ : NullSingletonClass μ
h_int : IntegrableOn f [[a, b]] μ
⊢ ContinuousOn (fun x ↦ ∫ (t : ℝ) in x..b, f t ∂μ) [[a, b]]

Tactic:
by

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

Goal state before:
pp: E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
μ : Measure ℝ
f : ℝ → E
inst✝ : NullSingletonClass μ
h_int : ∀ (a b : ℝ), IntervalIntegrable f μ a b
a : ℝ
⊢ Continuous[_, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] fun b ↦ ∫ (x : ℝ) in a..b, f x ∂μ

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic

Goal state before:
pp: E : Type u_5
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
a b c : ℝ
f : ℝ → E
μ : Measure ℝ
hab : IntervalIntegrable f μ a b
hbc : IntervalIntegrable f μ b c
⊢ ∫ (x : ℝ) in a..b, f x ∂μ + ∫ (x : ℝ) in b..c, f x ∂μ + ∫ (x : ℝ) in c..a, f x ∂μ = 0

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝¹⁰ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace 𝕜 E
inst✝⁷ : LocallyCompactSpace E
F : Type u_3
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace 𝕜 F
α : Type u_4
inst✝⁴ : TopologicalSpace α
f : α → E → F
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : MeasurableSpace E
inst✝ : OpensMeasurableSpace E
hf : Continuous[instTopologicalSpaceProd, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (Function.uncurry f)
K : Set (E →L[𝕜] F)
hK : IsComplete K
⊢ MeasurableSet {p | DifferentiableAt 𝕜 (f p.1) p.2 ∧ fderiv 𝕜 (f p.1) p.2 ∈ K}

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic

Goal state before:
pp: E : Type u_5
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
a b c : ℝ
f : ℝ → E
μ : Measure ℝ
hab : IntervalIntegrable f μ a b
hbc : IntervalIntegrable f μ b c
⊢ ∫ (x : ℝ) in a..b, f x ∂μ + ∫ (x : ℝ) in b..c, f x ∂μ = ∫ (x : ℝ) in a..c, f x ∂μ

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝¹¹ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : NormedSpace 𝕜 E
inst✝⁸ : LocallyCompactSpace E
F : Type u_3
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : NormedSpace 𝕜 F
α : Type u_4
inst✝⁵ : TopologicalSpace α
f : α → E → F
inst✝⁴ : MeasurableSpace α
inst✝³ : OpensMeasurableSpace α
inst✝² : MeasurableSpace E
inst✝¹ : OpensMeasurableSpace E
inst✝ : CompleteSpace F
hf : Continuous[instTopologicalSpaceProd, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (Function.uncurry f)
⊢ MeasurableSet {p | DifferentiableAt 𝕜 (f p.1) p.2}

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝¹¹ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : NormedSpace 𝕜 E
inst✝⁸ : LocallyCompactSpace E
F : Type u_3
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : NormedSpace 𝕜 F
α : Type u_4
inst✝⁵ : TopologicalSpace α
f : α → E → F
inst✝⁴ : MeasurableSpace α
inst✝³ : OpensMeasurableSpace α
inst✝² : MeasurableSpace E
inst✝¹ : OpensMeasurableSpace E
inst✝ : CompleteSpace F
hf : Continuous[instTopologicalSpaceProd, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (Function.uncurry f)
⊢ Measurable fun p ↦ fderiv 𝕜 (f p.1) p.2

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Integral.IntervalIntegral.FundThmCalculus

Goal state before:
pp: g' g φ : ℝ → ℝ
a b : ℝ
hab : a ≤ b
hcont : ContinuousOn g (Icc a b)
hderiv : ∀ x ∈ Ico a b, HasDerivWithinAt g (g' x) (Ioi x) x
φint : IntegrableOn φ (Icc a b) volume
hφg : ∀ x ∈ Ico a b, g' x ≤ φ x
ε : ℝ
εpos : 0 < ε
G' : ℝ → EReal
f_lt_G' : ∀ (x : ℝ), ↑(φ x) < G' x
G'cont : LowerSemicontinuous G'
G'int : Integrable (fun x ↦ (G' x).toReal) (volume.restrict (Icc a b))
G'lt_top : ∀ᵐ (x : ℝ) ∂volume.restrict (Icc a b), G' x < ⊤
hG' : ∫ (x : ℝ) in Icc a b, (G' x).toReal < (∫ (x : ℝ) in Icc a b, φ x) + ε
s : Set ℝ := {t | g t - g a ≤ ∫ (u : ℝ) in a..t, (G' u).toReal} ∩ Icc a b
s_closed : IsClosed s
t : ℝ
ht : t ∈ {t | g t - g a ≤ ∫ (u : ℝ) in a..t, (G' u).toReal} ∩ Ico a b
v : ℝ
t_lt_v : v ∈ Ioi t
y : ℝ
g'_lt_y' : ↑(g' t) < ↑y
y_lt_G' : ↑y < G' t
I1 : ∀ᶠ (u : ℝ) in 𝓝[>] t, (u - t) * y ≤ ∫ (w : ℝ) in t..u, (G' w).toReal
I2 : ∀ᶠ (u : ℝ) in 𝓝[>] t, g u - g t ≤ (u - t) * y
I3 : ∀ᶠ (u : ℝ) in 𝓝[>] t, g u - g t ≤ ∫ (w : ℝ) in t..u, (G' w).toReal
I4 : ∀ᶠ (u : ℝ) in 𝓝[>] t, u ∈ Ioc t (min v b)
x : ℝ
hx : g x - g t ≤ ∫ (w : ℝ) in t..x, (G' w).toReal
h'x : x ∈ Ioc t (min v b)
⊢ (∫ (w : ℝ) in a..t, (G' w).toReal) + ∫ (w : ℝ) in t..x, (G' w).toReal = ∫ (w : ℝ) in a..x, (G' w).toReal

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝¹¹ : NontriviallyNormedField 𝕜
F : Type u_3
inst✝¹⁰ : NormedAddCommGroup F
inst✝⁹ : NormedSpace 𝕜 F
α : Type u_4
inst✝⁸ : TopologicalSpace α
inst✝⁷ : MeasurableSpace α
inst✝⁶ : OpensMeasurableSpace α
inst✝⁵ : CompleteSpace F
inst✝⁴ : LocallyCompactSpace 𝕜
inst✝³ : MeasurableSpace 𝕜
inst✝² : OpensMeasurableSpace 𝕜
inst✝¹ : MeasurableSpace F
inst✝ : BorelSpace F
f : α → 𝕜 → F
hf : Continuous[instTopologicalSpaceProd, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (Function.uncurry f)
⊢ Measurable fun p ↦ deriv (f p.1) p.2

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Integral.IntervalIntegral.FundThmCalculus

Goal state before:
pp: g' g φ : ℝ → ℝ
a b : ℝ
hab : a ≤ b
hcont : ContinuousOn g (Icc a b)
hderiv : ∀ x ∈ Ico a b, HasDerivWithinAt g (g' x) (Ioi x) x
φint : IntegrableOn φ (Icc a b) volume
hφg : ∀ x ∈ Ico a b, g' x ≤ φ x
ε : ℝ
εpos : 0 < ε
G' : ℝ → EReal
f_lt_G' : ∀ (x : ℝ), ↑(φ x) < G' x
G'cont : LowerSemicontinuous G'
G'int : Integrable (fun x ↦ (G' x).toReal) (volume.restrict (Icc a b))
G'lt_top : ∀ᵐ (x : ℝ) ∂volume.restrict (Icc a b), G' x < ⊤
hG' : ∫ (x : ℝ) in Icc a b, (G' x).toReal < (∫ (x : ℝ) in Icc a b, φ x) + ε
s : Set ℝ := {t | g t - g a ≤ ∫ (u : ℝ) in a..t, (G' u).toReal} ∩ Icc a b
s_closed : IsClosed s
⊢ Icc a b ⊆ {t | g t - g a ≤ ∫ (u : ℝ) in a..t, (G' u).toReal}

Tactic:
by

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

Goal state before:
pp: E : Type u_1
X : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : TopologicalSpace X
μ : Measure ℝ
inst✝¹ : NullSingletonClass μ
inst✝ : IsLocallyFiniteMeasure μ
f : X → ℝ → E
a₀ : ℝ
hf : Continuous[instTopologicalSpaceProd, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (Function.uncurry f)
q : X
b₀ ε : ℝ
εpos : ε > 0
a : ℝ
a_lt : a < a₀ ∧ a < b₀
b : ℝ
lt_b : a₀ < b ∧ b₀ < b
this : IsCompact ({q} ×ˢ Icc a b)
M : ℝ
hM : M ∈ upperBounds ((fun x ↦ ‖Function.uncurry f x‖) '' {q} ×ˢ Icc a b)
A : ∀ᶠ (δ : ℝ) in 𝓝[>] 0, δ ∈ Ioo 0 1
⊢ ∀ᶠ (δ : ℝ) in 𝓝 0, Icc (b₀ - δ) (b₀ + δ) ⊆ Icc a b

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁹ : NontriviallyNormedField 𝕜
F : Type u_3
inst✝⁸ : NormedAddCommGroup F
inst✝⁷ : NormedSpace 𝕜 F
α : Type u_4
inst✝⁶ : TopologicalSpace α
inst✝⁵ : MeasurableSpace α
inst✝⁴ : OpensMeasurableSpace α
inst✝³ : CompleteSpace F
inst✝² : LocallyCompactSpace 𝕜
inst✝¹ : MeasurableSpace 𝕜
inst✝ : OpensMeasurableSpace 𝕜
h : SecondCountableTopologyEither α F
f : α → 𝕜 → F
hf : Continuous[instTopologicalSpaceProd, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (Function.uncurry f)
this✝¹ : MeasurableSpace F := borel F
this✝ : BorelSpace F
hα : SecondCountableTopology α
this : ProperSpace 𝕜
p : α × 𝕜
⊢ deriv (f p.1) p.2 ∈ closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] ↑(Submodule.span 𝕜 (range (f p.1)))

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic

Goal state before:
pp: E : Type u_5
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℝ → E
μ : Measure ℝ
a : ℕ → ℝ
m n : ℕ
hmn : m ≤ n
hint : ∀ k ∈ Ico m n, IntervalIntegrable f μ (a k) (a (k + 1))
⊢ ∑ k ∈ Finset.Ico m n, ∫ (x : ℝ) in a k..a (k + 1), f x ∂μ = ∫ (x : ℝ) in a m..a n, f x ∂μ

Tactic:
by

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

Goal state before:
pp: E : Type u_1
X : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : TopologicalSpace X
μ : Measure ℝ
inst✝¹ : NullSingletonClass μ
inst✝ : IsLocallyFiniteMeasure μ
f : X → ℝ → E
a₀ : ℝ
hf : Continuous[instTopologicalSpaceProd, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (Function.uncurry f)
q : X
b₀ ε : ℝ
εpos : ε > 0
a : ℝ
a_lt : a < a₀ ∧ a < b₀
b : ℝ
lt_b : a₀ < b ∧ b₀ < b
this✝ : IsCompact ({q} ×ˢ Icc a b)
M : ℝ
hM : M ∈ upperBounds ((fun x ↦ ‖Function.uncurry f x‖) '' {q} ×ˢ Icc a b)
A : ∀ᶠ (δ : ℝ) in 𝓝[>] 0, δ ∈ Ioo 0 1
B : ∀ᶠ (δ : ℝ) in 𝓝 0, Icc (b₀ - δ) (b₀ + δ) ⊆ Icc a b
this :
  Tendsto (fun δ ↦ (M + 1) * μ.real (Icc (b₀ - δ) (b₀ + δ)) + δ * μ.real (Icc a b)) (𝓝 0)
    (𝓝 ((M + 1) * ENNReal.toReal 0 + 0 * μ.real (Icc a b)))
⊢ ∀ᶠ (δ : ℝ) in 𝓝 0, (M + 1) * μ.real (Icc (b₀ - δ) (b₀ + δ)) + δ * μ.real (Icc a b) < ε

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic

Goal state before:
pp: E : Type u_5
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℝ → E
μ : Measure ℝ
a : ℕ → ℝ
n : ℕ
hint : ∀ k < n, IntervalIntegrable f μ (a k) (a (k + 1))
⊢ ∑ k ∈ Finset.range n, ∫ (x : ℝ) in a k..a (k + 1), f x ∂μ = ∫ (x : ℝ) in a 0..a n, f x ∂μ

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁹ : NontriviallyNormedField 𝕜
F : Type u_3
inst✝⁸ : NormedAddCommGroup F
inst✝⁷ : NormedSpace 𝕜 F
α : Type u_4
inst✝⁶ : TopologicalSpace α
inst✝⁵ : MeasurableSpace α
inst✝⁴ : OpensMeasurableSpace α
inst✝³ : CompleteSpace F
inst✝² : LocallyCompactSpace 𝕜
inst✝¹ : MeasurableSpace 𝕜
inst✝ : OpensMeasurableSpace 𝕜
h : SecondCountableTopologyEither α F
f : α → 𝕜 → F
hf : Continuous[instTopologicalSpaceProd, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (Function.uncurry f)
this✝¹ : MeasurableSpace F := borel F
this✝ : BorelSpace F
hα : SecondCountableTopology α
this : ProperSpace 𝕜
p : α × 𝕜
A : deriv (f p.1) p.2 ∈ closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] ↑(Submodule.span 𝕜 (range (f p.1)))
⊢ range (f p.1) ⊆ range (Function.uncurry f)

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁹ : NontriviallyNormedField 𝕜
F : Type u_3
inst✝⁸ : NormedAddCommGroup F
inst✝⁷ : NormedSpace 𝕜 F
α : Type u_4
inst✝⁶ : TopologicalSpace α
inst✝⁵ : MeasurableSpace α
inst✝⁴ : OpensMeasurableSpace α
inst✝³ : CompleteSpace F
inst✝² : LocallyCompactSpace 𝕜
inst✝¹ : MeasurableSpace 𝕜
inst✝ : OpensMeasurableSpace 𝕜
h : SecondCountableTopologyEither α F
f : α → 𝕜 → F
hf : Continuous[instTopologicalSpaceProd, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (Function.uncurry f)
this✝¹ : MeasurableSpace F := borel F
this✝ : BorelSpace F
hα : SecondCountableTopology α
this : ProperSpace 𝕜
⊢ (range fun p ↦ deriv (f p.1) p.2) ⊆
    closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] ↑(Submodule.span 𝕜 (range (Function.uncurry f)))

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic

Goal state before:
pp: E : Type u_5
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
a b c d : ℝ
f : ℝ → E
μ : Measure ℝ
hab : IntervalIntegrable f μ a b
hcd : IntervalIntegrable f μ c d
hac : IntervalIntegrable f μ a c
⊢ ∫ (x : ℝ) in a..b, f x ∂μ + ∫ (x : ℝ) in c..d, f x ∂μ = ∫ (x : ℝ) in a..d, f x ∂μ + ∫ (x : ℝ) in c..b, f x ∂μ

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic

Goal state before:
pp: E : Type u_5
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
a b c d : ℝ
f : ℝ → E
μ : Measure ℝ
hab : IntervalIntegrable f μ a b
hcd : IntervalIntegrable f μ c d
hac : IntervalIntegrable f μ a c
⊢ ∫ (x : ℝ) in a..b, f x ∂μ - ∫ (x : ℝ) in c..d, f x ∂μ = ∫ (x : ℝ) in a..c, f x ∂μ - ∫ (x : ℝ) in b..d, f x ∂μ

Tactic:
by

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

Goal state before:
pp: E : Type u_1
X : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : TopologicalSpace X
μ : Measure ℝ
inst✝¹ : NullSingletonClass μ
inst✝ : IsLocallyFiniteMeasure μ
f : X → ℝ → E
a₀ : ℝ
hf : Continuous[instTopologicalSpaceProd, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (Function.uncurry f)
q : X
b₀ ε : ℝ
εpos : ε > 0
a : ℝ
a_lt : a < a₀ ∧ a < b₀
b : ℝ
lt_b : a₀ < b ∧ b₀ < b
this : IsCompact ({q} ×ˢ Icc a b)
M : ℝ
hM : M ∈ upperBounds ((fun x ↦ ‖Function.uncurry f x‖) '' {q} ×ˢ Icc a b)
A : ∀ᶠ (δ : ℝ) in 𝓝[>] 0, δ ∈ Ioo 0 1
B : ∀ᶠ (δ : ℝ) in 𝓝 0, Icc (b₀ - δ) (b₀ + δ) ⊆ Icc a b
⊢ ∀ᶠ (δ : ℝ) in 𝓝 0, (M + 1) * μ.real (Icc (b₀ - δ) (b₀ + δ)) + δ * μ.real (Icc a b) < ε

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic

Goal state before:
pp: E : Type u_5
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
a b c d : ℝ
f : ℝ → E
μ : Measure ℝ
hab : IntervalIntegrable f μ a b
hcd : IntervalIntegrable f μ c d
hac : IntervalIntegrable f μ a c
⊢ ∫ (x : ℝ) in a..b, f x ∂μ - ∫ (x : ℝ) in c..d, f x ∂μ = ∫ (x : ℝ) in d..b, f x ∂μ - ∫ (x : ℝ) in c..a, f x ∂μ

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Integral.IntervalIntegral.FundThmCalculus

Goal state before:
pp: g' g φ : ℝ → ℝ
a b : ℝ
hab : a ≤ b
hcont : ContinuousOn g (Icc a b)
hderiv : ∀ x ∈ Ico a b, HasDerivWithinAt g (g' x) (Ioi x) x
φint : IntegrableOn φ (Icc a b) volume
hφg : ∀ x ∈ Ico a b, g' x ≤ φ x
ε : ℝ
εpos : 0 < ε
G' : ℝ → EReal
f_lt_G' : ∀ (x : ℝ), ↑(φ x) < G' x
G'cont : LowerSemicontinuous G'
G'int : Integrable (fun x ↦ (G' x).toReal) (volume.restrict (Icc a b))
G'lt_top : ∀ᵐ (x : ℝ) ∂volume.restrict (Icc a b), G' x < ⊤
hG' : ∫ (x : ℝ) in Icc a b, (G' x).toReal < (∫ (x : ℝ) in Icc a b, φ x) + ε
s : Set ℝ := {t | g t - g a ≤ ∫ (u : ℝ) in a..t, (G' u).toReal} ∩ Icc a b
s_closed : IsClosed s
main : Icc a b ⊆ {t | g t - g a ≤ ∫ (u : ℝ) in a..t, (G' u).toReal}
⊢ ∫ (y : ℝ) in a..b, (G' y).toReal ≤ (∫ (y : ℝ) in a..b, φ y) + ε

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁹ : NontriviallyNormedField 𝕜
F : Type u_3
inst✝⁸ : NormedAddCommGroup F
inst✝⁷ : NormedSpace 𝕜 F
α : Type u_4
inst✝⁶ : TopologicalSpace α
inst✝⁵ : MeasurableSpace α
inst✝⁴ : OpensMeasurableSpace α
inst✝³ : CompleteSpace F
inst✝² : LocallyCompactSpace 𝕜
inst✝¹ : MeasurableSpace 𝕜
inst✝ : OpensMeasurableSpace 𝕜
h : SecondCountableTopologyEither α F
f : α → 𝕜 → F
hf : Continuous[instTopologicalSpaceProd, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (Function.uncurry f)
⊢ StronglyMeasurable fun p ↦ deriv (f p.1) p.2

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Integral.IntervalIntegral.FundThmCalculus

Goal state before:
pp: g' g φ : ℝ → ℝ
a b : ℝ
hab : a ≤ b
hcont : ContinuousOn g (Icc a b)
hderiv : ∀ x ∈ Ico a b, HasDerivWithinAt g (g' x) (Ioi x) x
φint : IntegrableOn φ (Icc a b) volume
hφg : ∀ x ∈ Ico a b, g' x ≤ φ x
⊢ g b - g a ≤ ∫ (y : ℝ) in a..b, φ y

Tactic:
by

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

Goal state before:
pp: E : Type u_1
X : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : TopologicalSpace X
μ : Measure ℝ
inst✝¹ : NullSingletonClass μ
inst✝ : IsLocallyFiniteMeasure μ
f : X → ℝ → E
a₀ : ℝ
hf : Continuous[instTopologicalSpaceProd, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (Function.uncurry f)
q : X
b₀ ε : ℝ
εpos : ε > 0
a : ℝ
a_lt : a < a₀ ∧ a < b₀
b : ℝ
lt_b : a₀ < b ∧ b₀ < b
this : IsCompact ({q} ×ˢ Icc a b)
M : ℝ
hM : M ∈ upperBounds ((fun x ↦ ‖Function.uncurry f x‖) '' {q} ×ˢ Icc a b)
⊢ ∃ δ,
    0 < δ ∧
      δ < 1 ∧ Icc (b₀ - δ) (b₀ + δ) ⊆ Icc a b ∧ (M + 1) * μ.real (Icc (b₀ - δ) (b₀ + δ)) + δ * μ.real (Icc a b) < ε

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.NonIntegrable

Goal state before:
pp: E : Type u_1
F : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : NormedAddCommGroup F
inst✝² : CompleteSpace E
f : ℝ → E
g : ℝ → F
k : Set ℝ
l : Filter ℝ
inst✝¹ : l.NeBot
inst✝ : TendstoIxxClass Icc l l
hl : k ∈ l
hd : ∀ᶠ (x : ℝ) in l, DifferentiableAt ℝ f x
hf : Tendsto (fun x ↦ ‖f x‖) l atTop
hfg : deriv f =O[l] g
hgi : IntegrableOn g k volume
⊢ ∃ C,
    ∃ (_ : 0 ≤ C),
      ∃ s ∈ l,
        (∀ x ∈ s, ∀ y ∈ s, [[x, y]] ⊆ k) ∧
          (∀ x ∈ s, ∀ y ∈ s, ∀ z ∈ [[x, y]], DifferentiableAt ℝ f z) ∧
            ∀ x ∈ s, ∀ y ∈ s, ∀ z ∈ [[x, y]], ‖deriv f z‖ ≤ C * ‖g z‖

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.NonIntegrable

Goal state before:
pp: E : Type u_1
F : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : NormedAddCommGroup F
inst✝² : CompleteSpace E
f : ℝ → E
g : ℝ → F
k : Set ℝ
l : Filter ℝ
inst✝¹ : l.NeBot
inst✝ : TendstoIxxClass Icc l l
hl : k ∈ l
hd : ∀ᶠ (x : ℝ) in l, DifferentiableAt ℝ f x
hf : Tendsto (fun x ↦ ‖f x‖) l atTop
hfg : deriv f =O[l] g
hgi : IntegrableOn g k volume
C : ℝ
hC₀ : 0 ≤ C
s : Set ℝ
hsl : s ∈ l
hsub : ∀ x ∈ s, ∀ y ∈ s, [[x, y]] ⊆ k
hfd : ∀ x ∈ s, ∀ y ∈ s, ∀ z ∈ [[x, y]], DifferentiableAt ℝ f z
hg : ∀ x ∈ s, ∀ y ∈ s, ∀ z ∈ [[x, y]], ‖deriv f z‖ ≤ C * ‖g z‖
⊢ IntegrableOn (fun x ↦ C * ‖g x‖) k volume

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Integral.IntervalIntegral.FundThmCalculus

Goal state before:
pp: g' g φ : ℝ → ℝ
a b : ℝ
hab : a ≤ b
hcont : ContinuousOn g (Icc a b)
hderiv : ∀ x ∈ Ioo a b, HasDerivWithinAt g (g' x) (Ioi x) x
φint : IntegrableOn φ (Icc a b) volume
hφg : ∀ x ∈ Ioo a b, g' x ≤ φ x
a_lt_b : a < b
s : Set ℝ := {t | g b - g t ≤ ∫ (u : ℝ) in t..b, φ u} ∩ Icc a b
⊢ ContinuousOn (fun t ↦ (g b - g t, ∫ (u : ℝ) in t..b, φ u)) (Icc a b)

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Integral.IntervalIntegral.FundThmCalculus

Goal state before:
pp: g' g φ : ℝ → ℝ
a b : ℝ
hab : a ≤ b
hcont : ContinuousOn g (Icc a b)
hderiv : ∀ x ∈ Ioo a b, HasDerivWithinAt g (g' x) (Ioi x) x
φint : IntegrableOn φ (Icc a b) volume
hφg : ∀ x ∈ Ioo a b, g' x ≤ φ x
a_lt_b : a < b
s : Set ℝ := {t | g b - g t ≤ ∫ (u : ℝ) in t..b, φ u} ∩ Icc a b
⊢ IsClosed s

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic

Goal state before:
pp: E : Type u_5
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
a b : ℝ
f : ℝ → E
μ : Measure ℝ
ha : IntegrableOn f (Iic a) μ
hb : IntegrableOn f (Iic b) μ
⊢ ∫ (x : ℝ) in Iic b, f x ∂μ - ∫ (x : ℝ) in Iic a, f x ∂μ = ∫ (x : ℝ) in a..b, f x ∂μ

Tactic:
by

Goal state after: