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

Goal state before:
pp: E : Type u_5
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f g : ℝ → E
μ : Measure ℝ
a b : ℝ
h : EqOn f g [[a, b]]
⊢ ∫ (x : ℝ) in a..b, f x ∂μ = ∫ (x : ℝ) in a..b, g x ∂μ

Tactic:
rcases le_total a b with hab | hab

Goal state after:
case inl
E : Type u_5
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f g : ℝ → E
μ : Measure ℝ
a b : ℝ
h : EqOn f g [[a, b]]
hab : a ≤ b
⊢ ∫ (x : ℝ) in a..b, f x ∂μ = ∫ (x : ℝ) in a..b, g x ∂μ

case inr
E : Type u_5
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f g : ℝ → E
μ : Measure ℝ
a b : ℝ
h : EqOn f g [[a, b]]
hab : b ≤ a
⊢ ∫ (x : ℝ) in a..b, f x ∂μ = ∫ (x : ℝ) in a..b, g x ∂μ
Module: Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic

Goal state before:
pp: case inl
E : Type u_5
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f g : ℝ → E
μ : Measure ℝ
a b : ℝ
h : EqOn f g [[a, b]]
hab : a ≤ b
⊢ ∫ (x : ℝ) in a..b, f x ∂μ = ∫ (x : ℝ) in a..b, g x ∂μ

Tactic:
simpa using

Goal state after:
case inl
E : Type u_5
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f g : ℝ → E
μ : Measure ℝ
a b : ℝ
h : EqOn f g [[a, b]]
hab : a ≤ b
⊢ ∫ (x : ℝ) in Ioc a b, f x ∂μ = ∫ (x : ℝ) in Ioc a b, g x ∂μ
Module: Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic

Goal state before:
pp: case inr
E : Type u_5
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f g : ℝ → E
μ : Measure ℝ
a b : ℝ
h : EqOn f g [[a, b]]
hab : b ≤ a
⊢ ∫ (x : ℝ) in a..b, f x ∂μ = ∫ (x : ℝ) in a..b, g x ∂μ

Tactic:
simpa using

Goal state after:
case inr
E : Type u_5
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f g : ℝ → E
μ : Measure ℝ
a b : ℝ
h : EqOn f g [[a, b]]
hab : b ≤ a
⊢ ∫ (x : ℝ) in Ioc b a, f x ∂μ = ∫ (x : ℝ) in Ioc b a, g x ∂μ
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
hsub : ∀ {a₀ b₀ : ℝ}, a₀ ∈ Ioo a b → b₀ ∈ Ioo a b → Ι a₀ b₀ ⊆ Ι a b
Ioo_nhds : Ioo a b ∈ 𝓝 b₀
Icc_nhds : Icc a b ∈ 𝓝 b₀
hx₀ : ∀ᵐ (t : ℝ) ∂μ.restrict (Ι a b), ‖F x₀ t‖ ≤ bound t
this✝ :
  ∀ᶠ (p : X × ℝ) in 𝓝 (x₀, b₀), ‖∫ (s : ℝ) in b₀..p.2, F p.1 s - F x₀ s ∂μ‖ ≤ |∫ (s : ℝ) in b₀..p.2, 2 * bound s ∂μ|
this : ContinuousAt (fun t ↦ ∫ (s : ℝ) in b₀..t, 2 * bound s ∂μ) b₀
⊢ Tendsto (fun t ↦ ∫ (s : ℝ) in b₀..t, 2 * bound s ∂μ) (𝓝 b₀) (𝓝 0)

Tactic:
simpa using

Goal state after:
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
hsub : ∀ {a₀ b₀ : ℝ}, a₀ ∈ Ioo a b → b₀ ∈ Ioo a b → Ι a₀ b₀ ⊆ Ι a b
Ioo_nhds : Ioo a b ∈ 𝓝 b₀
Icc_nhds : Icc a b ∈ 𝓝 b₀
hx₀ : ∀ᵐ (t : ℝ) ∂μ.restrict (Ι a b), ‖F x₀ t‖ ≤ bound t
this✝ :
  ∀ᶠ (p : X × ℝ) in 𝓝 (x₀, b₀), ‖∫ (s : ℝ) in b₀..p.2, F p.1 s - F x₀ s ∂μ‖ ≤ |∫ (s : ℝ) in b₀..p.2, 2 * bound s ∂μ|
this : ContinuousAt (fun t ↦ ∫ (s : ℝ) in b₀..t, 2 * bound s ∂μ) b₀
⊢ Tendsto (fun t ↦ 2 * ∫ (x : ℝ) in b₀..t, bound x ∂μ) (𝓝 b₀) (𝓝 0)
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]]
⊢ IntervalIntegrable f μ (min b₁ b₂) (max b₁ b₂)

Tactic:
simpa using

Goal state after:
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]]
⊢ IntegrableOn f (Ioc (min b₁ b₂) (max b₁ b₂)) μ
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:
simpa using

Goal state after:
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
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 : ℝ
δpos : 0 < δ
hδ : δ < 1
h'δ : Icc (b₀ - δ) (b₀ + δ) ⊆ Icc a b
h''δ : (M + 1) * μ.real (Icc (b₀ - δ) (b₀ + δ)) + δ * μ.real (Icc a b) < ε
v : Set X
v_mem : v ∈ 𝓝[univ] q
hv : ∀ p ∈ v, ∀ x ∈ Icc a b, dist (f p x) (f q x) < δ
this : v ×ˢ Ioo (b₀ - δ) (b₀ + δ) ∈ 𝓝 (q, b₀)
⊢ ∀ a ∈ v ×ˢ Ioo (b₀ - δ) (b₀ + δ), dist (∫ (t : ℝ) in a₀..a.2, f a.1 t ∂μ) (∫ (t : ℝ) in a₀..b₀, f q t ∂μ) < ε

Tactic:
rintro ⟨p, s⟩ ⟨hp : p ∈ v, hs : s ∈ Ioo (b₀ - δ) (b₀ + δ)⟩

Goal state after:
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 : ℝ
δpos : 0 < δ
hδ : δ < 1
h'δ : Icc (b₀ - δ) (b₀ + δ) ⊆ Icc a b
h''δ : (M + 1) * μ.real (Icc (b₀ - δ) (b₀ + δ)) + δ * μ.real (Icc a b) < ε
v : Set X
v_mem : v ∈ 𝓝[univ] q
hv : ∀ p ∈ v, ∀ x ∈ Icc a b, dist (f p x) (f q x) < δ
this : v ×ˢ Ioo (b₀ - δ) (b₀ + δ) ∈ 𝓝 (q, b₀)
p : X
s : ℝ
hp : p ∈ v
hs : s ∈ Ioo (b₀ - δ) (b₀ + δ)
⊢ dist (∫ (t : ℝ) in a₀..(p, s).2, f (p, s).1 t ∂μ) (∫ (t : ℝ) in a₀..b₀, f q t ∂μ) < ε
Module: Mathlib.MeasureTheory.Integral.IntervalIntegral.FundThmCalculus

Goal state before:
pp: E : Type u_3
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
a b : ℝ
inst✝ : CompleteSpace E
f f' : ℝ → E
hcont : ContinuousOn f [[a, b]]
hderiv : ∀ x ∈ Ioo (min a b) (max a b), HasDerivWithinAt f (f' x) (Ioi x) x
hint : IntervalIntegrable f' volume a b
⊢ ∫ (y : ℝ) in a..b, f' y = f b - f a

Tactic:
rcases le_total a b with hab | hab

Goal state after:
case inl
E : Type u_3
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
a b : ℝ
inst✝ : CompleteSpace E
f f' : ℝ → E
hcont : ContinuousOn f [[a, b]]
hderiv : ∀ x ∈ Ioo (min a b) (max a b), HasDerivWithinAt f (f' x) (Ioi x) x
hint : IntervalIntegrable f' volume a b
hab : a ≤ b
⊢ ∫ (y : ℝ) in a..b, f' y = f b - f a

case inr
E : Type u_3
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
a b : ℝ
inst✝ : CompleteSpace E
f f' : ℝ → E
hcont : ContinuousOn f [[a, b]]
hderiv : ∀ x ∈ Ioo (min a b) (max a b), HasDerivWithinAt f (f' x) (Ioi x) x
hint : IntervalIntegrable f' volume a b
hab : b ≤ a
⊢ ∫ (y : ℝ) in a..b, f' y = f b - f a
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
hf : IntegrableOn f (Ici a) μ
hab : a ≤ b
ha : IntegrableOn f (Ici b) μ
h : IntegrableOn f (Ico a b) μ
⊢ ∫ (x : ℝ) in Ici a, f x ∂μ = ∫ (x : ℝ) in Ico a b ∪ Ici b, f x ∂μ

Tactic:
Ico_union_Ici_eq_Ici hab

Goal state after:
E : Type u_5
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
a b : ℝ
f : ℝ → E
μ : Measure ℝ
hf : IntegrableOn f (Ici a) μ
hab : a ≤ b
ha : IntegrableOn f (Ici b) μ
h : IntegrableOn f (Ico a b) μ
⊢ ∫ (x : ℝ) in Ici a, f x ∂μ = ∫ (x : ℝ) in Ici a, f x ∂μ
Module: Mathlib.MeasureTheory.Integral.IntervalIntegral.FundThmCalculus

Goal state before:
pp: E : Type u_3
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
a b : ℝ
inst✝ : CompleteSpace E
f f' : ℝ → E
hab : a < b
fa fb : E
hderiv : ∀ x ∈ Ioo a b, HasDerivAt f (f' x) x
hint : IntervalIntegrable f' volume a b
ha : Tendsto f (𝓝[>] a) (𝓝 fa)
hb : Tendsto f (𝓝[<] b) (𝓝 fb)
F : ℝ → E := update (update f a fa) b fb
Fderiv : ∀ x ∈ Ioo a b, HasDerivAt F (f' x) x
hcont : ContinuousOn F (Icc a b)
⊢ ∫ (y : ℝ) in a..b, f' y = fb - fa

Tactic:
simpa using

Goal state after:
E : Type u_3
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
a b : ℝ
inst✝ : CompleteSpace E
f f' : ℝ → E
hab : a < b
fa fb : E
hderiv : ∀ x ∈ Ioo a b, HasDerivAt f (f' x) x
hint : IntervalIntegrable f' volume a b
ha : Tendsto f (𝓝[>] a) (𝓝 fa)
hb : Tendsto f (𝓝[<] b) (𝓝 fb)
F : ℝ → E := update (update f a fa) b fb
Fderiv : ∀ x ∈ Ioo a b, HasDerivAt F (f' x) x
hcont : ContinuousOn F (Icc a b)
⊢ ∫ (y : ℝ) in a..b, f' y = fb - fa
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 b : ℝ
h : support f ⊆ Ioc a b
⊢ ∫ (x : ℝ) in a..b, f x ∂μ = ∫ (x : ℝ), f x ∂μ

Tactic:
rcases le_total a b with hab | hab

Goal state after:
case inl
E : Type u_5
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℝ → E
μ : Measure ℝ
a b : ℝ
h : support f ⊆ Ioc a b
hab : a ≤ b
⊢ ∫ (x : ℝ) in a..b, f x ∂μ = ∫ (x : ℝ), f x ∂μ

case inr
E : Type u_5
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℝ → E
μ : Measure ℝ
a b : ℝ
h : support f ⊆ Ioc a b
hab : b ≤ a
⊢ ∫ (x : ℝ) in a..b, f x ∂μ = ∫ (x : ℝ), f x ∂μ
Module: Mathlib.MeasureTheory.Integral.IntervalIntegral.FundThmCalculus

Goal state before:
pp: E : Type u_3
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
a b : ℝ
inst✝ : CompleteSpace E
f : ℝ → E
hcont : ContinuousOn f [[a, b]]
hderiv : ∀ x ∈ uIoo a b, DifferentiableAt ℝ f x
hint : IntervalIntegrable (deriv f) volume a b
⊢ ∫ (y : ℝ) in a..b, deriv f y = f b - f a

Tactic:
rcases le_total a b with hab | hab

Goal state after:
case inl
E : Type u_3
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
a b : ℝ
inst✝ : CompleteSpace E
f : ℝ → E
hcont : ContinuousOn f [[a, b]]
hderiv : ∀ x ∈ uIoo a b, DifferentiableAt ℝ f x
hint : IntervalIntegrable (deriv f) volume a b
hab : a ≤ b
⊢ ∫ (y : ℝ) in a..b, deriv f y = f b - f a

case inr
E : Type u_3
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
a b : ℝ
inst✝ : CompleteSpace E
f : ℝ → E
hcont : ContinuousOn f [[a, b]]
hderiv : ∀ x ∈ uIoo a b, DifferentiableAt ℝ f x
hint : IntervalIntegrable (deriv f) volume a b
hab : b ≤ a
⊢ ∫ (y : ℝ) in a..b, deriv f y = f b - f a
Module: Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic

Goal state before:
pp: f : ℝ → ℝ
a b : ℝ
μ : Measure ℝ
hf : 0 ≤ᵐ[μ.restrict (Ioc a b ∪ Ioc b a)] f
hfi : IntervalIntegrable f μ a b
⊢ ∫ (x : ℝ) in a..b, f x ∂μ = 0 ↔ f =ᵐ[μ.restrict (Ioc a b ∪ Ioc b a)] 0

Tactic:
rcases le_total a b with hab | hab

Goal state after:
case inl
f : ℝ → ℝ
a b : ℝ
μ : Measure ℝ
hf : 0 ≤ᵐ[μ.restrict (Ioc a b ∪ Ioc b a)] f
hfi : IntervalIntegrable f μ a b
hab : a ≤ b
⊢ ∫ (x : ℝ) in a..b, f x ∂μ = 0 ↔ f =ᵐ[μ.restrict (Ioc a b ∪ Ioc b a)] 0

case inr
f : ℝ → ℝ
a b : ℝ
μ : Measure ℝ
hf : 0 ≤ᵐ[μ.restrict (Ioc a b ∪ Ioc b a)] f
hfi : IntervalIntegrable f μ a b
hab : b ≤ a
⊢ ∫ (x : ℝ) in a..b, f x ∂μ = 0 ↔ f =ᵐ[μ.restrict (Ioc a b ∪ Ioc b a)] 0
Module: Mathlib.MeasureTheory.Integral.IntervalIntegral.FundThmCalculus

Goal state before:
pp: g' g : ℝ → ℝ
a b : ℝ
hcont : ContinuousOn g (Icc a b)
hderiv : ∀ x ∈ Ioo a b, HasDerivWithinAt g (g' x) (Ioi x) x
g'pos : ∀ x ∈ Ioo a b, 0 ≤ g' x
hab : a < b
meas_g' : AEMeasurable g' (volume.restrict (Ioo a b))
H : ENNReal.ofReal (g b - g a) < ∫⁻ (x : ℝ) in Ioo a b, ↑‖g' x‖₊
f : SimpleFunc ℝ ℝ≥0
fle : ∀ (x : ℝ), f x ≤ ‖g' x‖₊
fint : ∫⁻ (x : ℝ) in Ioo a b, ↑(f x) < ∞
hf : ENNReal.ofReal (g b - g a) < ∫⁻ (x : ℝ) in Ioo a b, ↑(f x)
F : ℝ → ℝ := NNReal.toReal ∘ ⇑f
⊢ HasFiniteIntegral F (volume.restrict (Ioo a b))

Tactic:
simpa using

Goal state after:
g' g : ℝ → ℝ
a b : ℝ
hcont : ContinuousOn g (Icc a b)
hderiv : ∀ x ∈ Ioo a b, HasDerivWithinAt g (g' x) (Ioi x) x
g'pos : ∀ x ∈ Ioo a b, 0 ≤ g' x
hab : a < b
meas_g' : AEMeasurable g' (volume.restrict (Ioo a b))
H : ENNReal.ofReal (g b - g a) < ∫⁻ (x : ℝ) in Ioo a b, ↑‖g' x‖₊
f : SimpleFunc ℝ ℝ≥0
fle : ∀ (x : ℝ), f x ≤ ‖g' x‖₊
fint : ∫⁻ (x : ℝ) in Ioo a b, ↑(f x) < ∞
hf : ENNReal.ofReal (g b - g a) < ∫⁻ (x : ℝ) in Ioo a b, ↑(f x)
F : ℝ → ℝ := NNReal.toReal ∘ ⇑f
⊢ ∫⁻ (a : ℝ) in Ioo a b, ↑(f a) < ∞
Module: Mathlib.MeasureTheory.Integral.IntervalIntegral.FundThmCalculus

Goal state before:
pp: g' g : ℝ → ℝ
a b : ℝ
hcont : ContinuousOn g [[a, b]]
hderiv : ∀ x ∈ Ioo (min a b) (max a b), HasDerivAt g (g' x) x
hpos : ∀ x ∈ Ioo (min a b) (max a b), 0 ≤ g' x
⊢ IntervalIntegrable g' volume a b

Tactic:
rcases le_total a b with hab | hab

Goal state after:
case inl
g' g : ℝ → ℝ
a b : ℝ
hcont : ContinuousOn g [[a, b]]
hderiv : ∀ x ∈ Ioo (min a b) (max a b), HasDerivAt g (g' x) x
hpos : ∀ x ∈ Ioo (min a b) (max a b), 0 ≤ g' x
hab : a ≤ b
⊢ IntervalIntegrable g' volume a b

case inr
g' g : ℝ → ℝ
a b : ℝ
hcont : ContinuousOn g [[a, b]]
hderiv : ∀ x ∈ Ioo (min a b) (max a b), HasDerivAt g (g' x) x
hpos : ∀ x ∈ Ioo (min a b) (max a b), 0 ≤ g' x
hab : b ≤ a
⊢ IntervalIntegrable g' volume a b
Module: Mathlib.Analysis.SpecialFunctions.NonIntegrable

Goal state before:
pp: F : Type u_2
inst✝ : NormedAddCommGroup F
f : ℝ → F
a b c : ℝ
hf : (fun x ↦ (x - c)⁻¹) =O[𝓝[≠] c] f
hne : a ≠ b
hc : c ∈ [[a, b]]
x : ℝ
hx : x ∈ {c}ᶜ
⊢ HasDerivAt (fun x ↦ Real.log (x - c)) (x - c)⁻¹ x

Tactic:
simpa using

Goal state after:
F : Type u_2
inst✝ : NormedAddCommGroup F
f : ℝ → F
a b c : ℝ
hf : (fun x ↦ (x - c)⁻¹) =O[𝓝[≠] c] f
hne : a ≠ b
hc : c ∈ [[a, b]]
x : ℝ
hx : x ∈ {c}ᶜ
⊢ HasDerivAt (fun x ↦ Real.log (x - c)) (x - c)⁻¹ x
Module: Mathlib.MeasureTheory.Integral.IntervalIntegral.FundThmCalculus

Goal state before:
pp: case inr
g' g : ℝ → ℝ
a b : ℝ
hcont : ContinuousOn g [[a, b]]
hderiv : ∀ x ∈ Ioo (min a b) (max a b), HasDerivAt g (g' x) x
hpos : ∀ x ∈ Ioo (min a b) (max a b), 0 ≤ g' x
hab : b ≤ a
⊢ IntervalIntegrable g' volume a b

Tactic:
· simp only [uIcc_of_ge, min_eq_right, max_eq_left, hab, IntervalIntegrable, Ioc_eq_empty_of_le,
      integrableOn_empty, true_and] at hcont hderiv hpos ⊢
    exact integrableOn_deriv_of_nonneg hcont hderiv hpos

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

Goal state before:
pp: g' g : ℝ → ℝ
a b : ℝ
hcont : ContinuousOn g [[a, b]]
hderiv : ∀ x ∈ Ioo (min a b) (max a b), HasDerivAt g (g' x) x
hpos : ∀ x ∈ Ioo (min a b) (max a b), 0 ≤ g' x
⊢ IntervalIntegrable g' volume a b

Tactic:
by
  rcases le_total a b with hab | hab
  · simp only [uIcc_of_le, min_eq_left, max_eq_right, IntervalIntegrable, hab,
      Ioc_eq_empty_of_le, integrableOn_empty, and_true] at hcont hderiv hpos ⊢
    exact integrableOn_deriv_of_nonneg hcont hderiv hpos
  · simp only [uIcc_of_ge, min_eq_right, max_eq_left, hab, IntervalIntegrable, Ioc_eq_empty_of_le,
      integrableOn_empty, true_and] at hcont hderiv hpos ⊢
    exact integrableOn_deriv_of_nonneg hcont hderiv hpos

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

Goal state before:
pp: F : Type u_2
inst✝ : NormedAddCommGroup F
f : ℝ → F
a b c : ℝ
hf : (fun x ↦ (x - c)⁻¹) =O[𝓝[≠] c] f
hne : a ≠ b
hc : c ∈ [[a, b]]
A : ∀ᶠ (x : ℝ) in 𝓝[≠] c, HasDerivAt (fun x ↦ Real.log (x - c)) (x - c)⁻¹ x
⊢ Tendsto (fun x ↦ x - c) (𝓝[≠] c) (𝓝[≠] 0)

Tactic:
rw [← sub_self c]

Goal state after:
F : Type u_2
inst✝ : NormedAddCommGroup F
f : ℝ → F
a b c : ℝ
hf : (fun x ↦ (x - c)⁻¹) =O[𝓝[≠] c] f
hne : a ≠ b
hc : c ∈ [[a, b]]
A : ∀ᶠ (x : ℝ) in 𝓝[≠] c, HasDerivAt (fun x ↦ Real.log (x - c)) (x - c)⁻¹ x
⊢ Tendsto (fun x ↦ x - c) (𝓝[≠] c) (𝓝[≠] (c - c))
Module: Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic

Goal state before:
pp: f : ℝ → ℝ
a b : ℝ
μ : Measure ℝ
hab : a ≤ b
hf : 0 ≤ᵐ[μ.restrict (Icc a b)] f
H : ∀ᵐ (x : ℝ) ∂μ.restrict (Ioc a b), 0 x ≤ f x := ae_restrict_of_ae_restrict_of_subset Ioc_subset_Icc_self hf
⊢ 0 ≤ ∫ (u : ℝ) in a..b, f u ∂μ

Tactic:
simpa using

Goal state after:
f : ℝ → ℝ
a b : ℝ
μ : Measure ℝ
hab : a ≤ b
hf : 0 ≤ᵐ[μ.restrict (Icc a b)] f
H : ∀ᵐ (x : ℝ) ∂μ.restrict (Ioc a b), 0 x ≤ f x := ae_restrict_of_ae_restrict_of_subset Ioc_subset_Icc_self hf
⊢ 0 ≤ ∫ (u : ℝ) in Ioc a b, f u ∂μ
Module: Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic

Goal state before:
pp: f : ℝ → ℝ
a b : ℝ
μ : Measure ℝ
hab : a ≤ b
⊢ |∫ (x : ℝ) in a..b, f x ∂μ| ≤ ∫ (x : ℝ) in a..b, |f x| ∂μ

Tactic:
simpa using

Goal state after:
f : ℝ → ℝ
a b : ℝ
μ : Measure ℝ
hab : a ≤ b
⊢ ‖∫ (x : ℝ) in a..b, f x ∂μ‖ ≤ ∫ (x : ℝ) in a..b, ‖f x‖ ∂μ
Module: Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic

Goal state before:
pp: f g : ℝ → ℝ
a b : ℝ
μ : Measure ℝ
hab : a ≤ b
hf : IntervalIntegrable f μ a b
hg : IntervalIntegrable g μ a b
h : f ≤ᵐ[μ.restrict (Icc a b)] g
H : ∀ᵐ (x : ℝ) ∂μ.restrict (Ioc a b), f x ≤ g x :=
  Eventually.filter_mono (ae_mono (Measure.restrict_mono Ioc_subset_Icc_self (le_refl μ))) h
⊢ ∫ (u : ℝ) in a..b, f u ∂μ ≤ ∫ (u : ℝ) in a..b, g u ∂μ

Tactic:
simpa using

Goal state after:
f g : ℝ → ℝ
a b : ℝ
μ : Measure ℝ
hab : a ≤ b
hf : IntervalIntegrable f μ a b
hg : IntervalIntegrable g μ a b
h : f ≤ᵐ[μ.restrict (Icc a b)] g
H : ∀ᵐ (x : ℝ) ∂μ.restrict (Ioc a b), f x ≤ g x :=
  Eventually.filter_mono (ae_mono (Measure.restrict_mono Ioc_subset_Icc_self (le_refl μ))) h
⊢ ∫ (u : ℝ) in Ioc a b, f u ∂μ ≤ ∫ (u : ℝ) in Ioc a b, g u ∂μ
Module: Mathlib.Analysis.SpecialFunctions.NonIntegrable

Goal state before:
pp: a : ℝ
⊢ ¬IntegrableOn (fun x ↦ x⁻¹) (Ioi a) volume

Tactic:
simpa using

Goal state after:
a : ℝ
⊢ ¬Integrable (fun x ↦ x⁻¹) (volume.restrict (Ici a))
Module: Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic

Goal state before:
pp: f g : ℝ → ℝ
a b : ℝ
μ : Measure ℝ
hab : a ≤ b
hf : IntervalIntegrable f μ a b
hg : IntervalIntegrable g μ a b
h : f ≤ᵐ[μ] g
⊢ ∫ (u : ℝ) in a..b, f u ∂μ ≤ ∫ (u : ℝ) in a..b, g u ∂μ

Tactic:
simpa using

Goal state after:
f g : ℝ → ℝ
a b : ℝ
μ : Measure ℝ
hab : a ≤ b
hf : IntervalIntegrable f μ a b
hg : IntervalIntegrable g μ a b
h : f ≤ᵐ[μ] g
⊢ ∫ (u : ℝ) in Ioc a b, f u ∂μ ≤ ∫ (u : ℝ) in Ioc a b, g u ∂μ
Module: Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic

Goal state before:
pp: f g : ℝ → ℝ
a b : ℝ
μ : Measure ℝ
hab : a ≤ b
hf : IntervalIntegrable f μ a b
hg : IntervalIntegrable g μ a b
h : ∀ x ∈ Icc a b, f x ≤ g x
H : ∀ x ∈ Ioc a b, f x ≤ g x := fun x hx ↦ h x (Ioc_subset_Icc_self hx)
⊢ ∫ (u : ℝ) in a..b, f u ∂μ ≤ ∫ (u : ℝ) in a..b, g u ∂μ

Tactic:
simpa using

Goal state after:
f g : ℝ → ℝ
a b : ℝ
μ : Measure ℝ
hab : a ≤ b
hf : IntervalIntegrable f μ a b
hg : IntervalIntegrable g μ a b
h : ∀ x ∈ Icc a b, f x ≤ g x
H : ∀ x ∈ Ioc a b, f x ≤ g x := fun x hx ↦ h x (Ioc_subset_Icc_self hx)
⊢ ∫ (u : ℝ) in Ioc a b, f u ∂μ ≤ ∫ (u : ℝ) in Ioc a b, g u ∂μ
Module: Mathlib.Topology.Algebra.Order.Floor

Goal state before:
pp: K : Type u_1
inst✝⁵ : Field K
inst✝⁴ : LinearOrder K
inst✝³ : IsStrictOrderedRing K
inst✝² : FloorSemiring K
inst✝¹ : TopologicalSpace K
inst✝ : OrderTopology K
a c : K
d : ℕ
ε : K
hε : ε < 0
n : ℕ
h : a * c ^ n / ε < ↑(n - d)!
⊢ ε < a * c ^ n / ↑(n - d)!

pp: K : Type u_1
inst✝⁵ : Field K
inst✝⁴ : LinearOrder K
inst✝³ : IsStrictOrderedRing K
inst✝² : FloorSemiring K
inst✝¹ : TopologicalSpace K
inst✝ : OrderTopology K
a c : K
d : ℕ
ε : K
hε : ε > 0
n : ℕ
h : a * c ^ n / ε < ↑(n - d)!
⊢ a * c ^ n / ↑(n - d)! < ε

Tactic:
· rw [div_lt_iff_of_neg hε] at h
    rwa [lt_div_iff₀' (Nat.cast_pos.mpr (Nat.factorial_pos _))]

Goal state after:
K : Type u_1
inst✝⁵ : Field K
inst✝⁴ : LinearOrder K
inst✝³ : IsStrictOrderedRing K
inst✝² : FloorSemiring K
inst✝¹ : TopologicalSpace K
inst✝ : OrderTopology K
a c : K
d : ℕ
ε : K
hε : ε > 0
n : ℕ
h : a * c ^ n / ε < ↑(n - d)!
⊢ a * c ^ n / ↑(n - d)! < ε
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
hfi : Integrable f volume
⊢ HasSum (fun n ↦ ∫ (x : ℝ) in 0..1, f (x + ↑n)) (∫ (x : ℝ), f x)

Tactic:
simpa using

Goal state after:
E : Type u_5
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℝ → E
hfi : Integrable f volume
⊢ HasSum (fun n ↦ ∫ (x : ℝ) in ↑n..↑n + 1, f x) (∫ (x : ℝ), f x)
Module: Mathlib.Topology.Algebra.Order.Floor

Goal state before:
pp: α : Type u_1
inst✝³ : Ring α
inst✝² : LinearOrder α
inst✝¹ : FloorRing α
inst✝ : IsStrictOrderedRing α
b : ℤ
⊢ b - 1 ≤ b

Tactic:
exact (sub_one_lt _).le

Goal state after:
Module: Mathlib.Topology.Algebra.Order.Floor

Goal state before:
pp: α : Type u_1
inst✝⁵ : Ring α
inst✝⁴ : LinearOrder α
inst✝³ : FloorRing α
inst✝² : TopologicalSpace α
inst✝¹ : IsStrictOrderedRing α
inst✝ : OrderClosedTopology α
n : ℤ
⊢ Tendsto floor (𝓝[≥] ↑n) (pure n)

Tactic:
simpa using

Goal state after:
α : Type u_1
inst✝⁵ : Ring α
inst✝⁴ : LinearOrder α
inst✝³ : FloorRing α
inst✝² : TopologicalSpace α
inst✝¹ : IsStrictOrderedRing α
inst✝ : OrderClosedTopology α
n : ℤ
⊢ Tendsto floor (𝓝[≥] ↑n) (pure n)
Module: Mathlib.Topology.Algebra.Order.Floor

Goal state before:
pp: α : Type u_1
inst✝⁵ : Ring α
inst✝⁴ : LinearOrder α
inst✝³ : FloorRing α
inst✝² : TopologicalSpace α
inst✝¹ : IsStrictOrderedRing α
inst✝ : OrderClosedTopology α
n : ℤ
⊢ Tendsto ceil (𝓝[≤] ↑n) (pure n)

Tactic:
simpa using

Goal state after:
α : Type u_1
inst✝⁵ : Ring α
inst✝⁴ : LinearOrder α
inst✝³ : FloorRing α
inst✝² : TopologicalSpace α
inst✝¹ : IsStrictOrderedRing α
inst✝ : OrderClosedTopology α
n : ℤ
⊢ Tendsto ceil (𝓝[≤] ↑n) (pure n)
Module: Mathlib.Topology.Algebra.Order.Floor

Goal state before:
pp: α : Type u_1
inst✝⁵ : Ring α
inst✝⁴ : LinearOrder α
inst✝³ : FloorRing α
inst✝² : TopologicalSpace α
inst✝¹ : IsStrictOrderedRing α
inst✝ : OrderClosedTopology α
n : ℤ
⊢ Tendsto floor (𝓝[<] ↑n) (pure (n - 1))

Tactic:
simpa using

Goal state after:
α : Type u_1
inst✝⁵ : Ring α
inst✝⁴ : LinearOrder α
inst✝³ : FloorRing α
inst✝² : TopologicalSpace α
inst✝¹ : IsStrictOrderedRing α
inst✝ : OrderClosedTopology α
n : ℤ
⊢ Tendsto floor (𝓝[<] ↑n) (pure (n - 1))
Module: Mathlib.Topology.Algebra.Order.Floor

Goal state before:
pp: α : Type u_1
inst✝⁵ : Ring α
inst✝⁴ : LinearOrder α
inst✝³ : FloorRing α
inst✝² : TopologicalSpace α
inst✝¹ : IsStrictOrderedRing α
inst✝ : OrderClosedTopology α
n : ℤ
⊢ Tendsto ceil (𝓝[>] ↑n) (pure (n + 1))

Tactic:
simpa using

Goal state after:
α : Type u_1
inst✝⁵ : Ring α
inst✝⁴ : LinearOrder α
inst✝³ : FloorRing α
inst✝² : TopologicalSpace α
inst✝¹ : IsStrictOrderedRing α
inst✝ : OrderClosedTopology α
n : ℤ
⊢ Tendsto ceil (𝓝[>] ↑n) (pure (n + 1))