-- Mathlib/MeasureTheory/Integral/IntegralEqImproper.lean
theorem integrableOn_Ioi_deriv_of_nonneg (hcont : ContinuousWithinAt g (Ici a) a)
    (hderiv : ∀ x ∈ Ioi a, HasDerivAt g (g' x) x) (g'pos : ∀ x ∈ Ioi a, 0 ≤ g' x)
    (hg : Tendsto g atTop (𝓝 l)) : IntegrableOn g' (Ioi a)

Proof state:
g g' : ℝ → ℝ
a l : ℝ
hcont✝ : ContinuousWithinAt g (Ici a) a
hderiv : ∀ x ∈ Ioi a, HasDerivAt g (g' x) x
g'pos : ∀ x ∈ Ioi a, 0 ≤ g' x
hg : Tendsto g atTop (𝓝 l)
hcont : ContinuousOn g (Ici a)
x : ℝ
hx : x ∈ Ioi a
h'x : a ≤ id x
⊢ IntervalIntegrable g' volume a (id x)

Tactic:
rw [intervalIntegrable_iff_integrableOn_Ioc_of_le h'x]

Result:
g g' : ℝ → ℝ
a l : ℝ
hcont✝ : ContinuousWithinAt g (Ici a) a
hderiv : ∀ x ∈ Ioi a, HasDerivAt g (g' x) x
g'pos : ∀ x ∈ Ioi a, 0 ≤ g' x
hg : Tendsto g atTop (𝓝 l)
hcont : ContinuousOn g (Ici a)
x : ℝ
hx : x ∈ Ioi a
h'x : a ≤ id x
⊢ IntegrableOn g' (Ioc a (id x)) volume

Proof state:
g g' : ℝ → ℝ
a l : ℝ
hcont✝ : ContinuousWithinAt g (Ici a) a
hderiv : ∀ x ∈ Ioi a, HasDerivAt g (g' x) x
g'pos : ∀ x ∈ Ioi a, 0 ≤ g' x
hg : Tendsto g atTop (𝓝 l)
hcont : ContinuousOn g (Ici a)
⊢ (fun x => g x - g a) =ᶠ[atTop] fun i => ∫ (x : ℝ) in a..id i, ‖g' x‖

Tactic:
filter_upwards [Ioi_mem_atTop a] with x hx

Result:
case h
g g' : ℝ → ℝ
a l : ℝ
hcont✝ : ContinuousWithinAt g (Ici a) a
hderiv : ∀ x ∈ Ioi a, HasDerivAt g (g' x) x
g'pos : ∀ x ∈ Ioi a, 0 ≤ g' x
hg : Tendsto g atTop (𝓝 l)
hcont : ContinuousOn g (Ici a)
x : ℝ
hx : x ∈ Ioi a
⊢ g x - g a = ∫ (x : ℝ) in a..id x, ‖g' x‖

Proof state:
case inl
g g' : ℝ → ℝ
a l : ℝ
hcont : ContinuousWithinAt g (Ici a) a
hderiv : ∀ x ∈ Ioi a, HasDerivAt g (g' x) x
g'pos : ∀ x ∈ Ioi a, 0 ≤ g' x
hg : Tendsto g atTop (𝓝 l)
hx : a ∈ Ici a
⊢ ContinuousWithinAt g (Ici a) a

Tactic:
exact hcont

Result:
no goals

Proof state:
g g' : ℝ → ℝ
a l : ℝ
hcont✝ : ContinuousWithinAt g (Ici a) a
hderiv : ∀ x ∈ Ioi a, HasDerivAt g (g' x) x
g'pos : ∀ x ∈ Ioi a, 0 ≤ g' x
hg : Tendsto g atTop (𝓝 l)
hcont : ContinuousOn g (Ici a)
x : ℝ
hx : x ∈ Ioi a
h'x : a ≤ id x
⊢ ∫ (y : ℝ) in a..id x, g' y = g x - g a

Tactic:
apply intervalIntegral.integral_eq_sub_of_hasDerivAt_of_le h'x
  (hcont.mono Icc_subset_Ici_self) fun y hy => hderiv y hy.1

Result:
g g' : ℝ → ℝ
a l : ℝ
hcont✝ : ContinuousWithinAt g (Ici a) a
hderiv : ∀ x ∈ Ioi a, HasDerivAt g (g' x) x
g'pos : ∀ x ∈ Ioi a, 0 ≤ g' x
hg : Tendsto g atTop (𝓝 l)
hcont : ContinuousOn g (Ici a)
x : ℝ
hx : x ∈ Ioi a
h'x : a ≤ id x
⊢ IntervalIntegrable g' volume a (id x)

Proof state:
g g' : ℝ → ℝ
a l : ℝ
hcont : ContinuousWithinAt g (Ici a) a
hderiv : ∀ x ∈ Ioi a, HasDerivAt g (g' x) x
g'pos : ∀ x ∈ Ioi a, 0 ≤ g' x
hg : Tendsto g atTop (𝓝 l)
x : ℝ
hx : x ∈ Ici a
⊢ ContinuousWithinAt g (Ici a) x

Tactic:
rcases hx.out.eq_or_lt with rfl|hx

Result:
case inl
g g' : ℝ → ℝ
a l : ℝ
hcont : ContinuousWithinAt g (Ici a) a
hderiv : ∀ x ∈ Ioi a, HasDerivAt g (g' x) x
g'pos : ∀ x ∈ Ioi a, 0 ≤ g' x
hg : Tendsto g atTop (𝓝 l)
hx : a ∈ Ici a
⊢ ContinuousWithinAt g (Ici a) a

case inr
g g' : ℝ → ℝ
a l : ℝ
hcont : ContinuousWithinAt g (Ici a) a
hderiv : ∀ x ∈ Ioi a, HasDerivAt g (g' x) x
g'pos : ∀ x ∈ Ioi a, 0 ≤ g' x
hg : Tendsto g atTop (𝓝 l)
x : ℝ
hx✝ : x ∈ Ici a
hx : a < x
⊢ ContinuousWithinAt g (Ici a) x

Proof state:
case refine_2
g g' : ℝ → ℝ
a l : ℝ
hcont✝ : ContinuousWithinAt g (Ici a) a
hderiv : ∀ x ∈ Ioi a, HasDerivAt g (g' x) x
g'pos : ∀ x ∈ Ioi a, 0 ≤ g' x
hg : Tendsto g atTop (𝓝 l)
hcont : ContinuousOn g (Ici a)
⊢ Tendsto (fun i => ∫ (x : ℝ) in a..id i, ‖g' x‖) atTop (𝓝 (l - g a))

Tactic:
apply Tendsto.congr' _ (hg.sub_const _)

Result:
g g' : ℝ → ℝ
a l : ℝ
hcont✝ : ContinuousWithinAt g (Ici a) a
hderiv : ∀ x ∈ Ioi a, HasDerivAt g (g' x) x
g'pos : ∀ x ∈ Ioi a, 0 ≤ g' x
hg : Tendsto g atTop (𝓝 l)
hcont : ContinuousOn g (Ici a)
⊢ (fun x => g x - g a) =ᶠ[atTop] fun i => ∫ (x : ℝ) in a..id i, ‖g' x‖