/-- Hard part of FTC-2 for integrable derivatives, real-valued functions: one has
`g b - g a ≤ ∫ y in a..b, g' y` when `g'` is integrable.
Auxiliary lemma in the proof of `integral_eq_sub_of_hasDeriv_right_of_le`.
We give the slightly more general version that `g b - g a ≤ ∫ y in a..b, φ y` when `g' ≤ φ` and
`φ` is integrable (even if `g'` is not known to be integrable).
Version assuming that `g` is differentiable on `[a, b)`. -/
theorem sub_le_integral_of_hasDeriv_right_of_le_Ico (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)) (hφg : ∀ x ∈ Ico a b, g' x ≤ φ x) :
    g b - g a ≤ ∫ y in a..b, φ y := by
  refine' le_of_forall_pos_le_add fun ε εpos => _
  -- Bound from above `g'` by a lower-semicontinuous function `G'`.
  rcases exists_lt_lowerSemicontinuous_integral_lt φ φint εpos with
    ⟨G', f_lt_G', G'cont, G'int, G'lt_top, hG'⟩
  -- we will show by "induction" that `g t - g a ≤ ∫ u in a..t, G' u` for all `t ∈ [a, b]`.
  set s := {t | g t - g a ≤ ∫ u in a..t, (G' u).toReal} ∩ Icc a b
  -- the set `s` of points where this property holds is closed.
  have s_closed : IsClosed s := by
    have : ContinuousOn (fun t => (g t - g a, ∫ u in a..t, (G' u).toReal)) (Icc a b) := by
      rw [← uIcc_of_le hab] at G'int hcont ⊢
      exact (hcont.sub continuousOn_const).prod (continuousOn_primitive_interval G'int)
    simp only [inter_comm]
    exact this.preimage_isClosed_of_isClosed isClosed_Icc OrderClosedTopology.isClosed_le'
  have main : Icc a b ⊆ {t | g t - g a ≤ ∫ u in a..t, (G' u).toReal} := by
    -- to show that the set `s` is all `[a, b]`, it suffices to show that any point `t` in `s`
    -- with `t < b` admits another point in `s` slightly to its right
    -- (this is a sort of real induction).
    refine s_closed.Icc_subset_of_forall_exists_gt
      (by simp only [integral_same, mem_setOf_eq, sub_self, le_rfl]) fun t ht v t_lt_v => ?_
    obtain ⟨y, g'_lt_y', y_lt_G'⟩ : ∃ y : ℝ, (g' t : EReal) < y ∧ (y : EReal) < G' t :=
      EReal.lt_iff_exists_real_btwn.1 ((EReal.coe_le_coe_iff.2 (hφg t ht.2)).trans_lt (f_lt_G' t))
    -- bound from below the increase of `∫ x in a..u, G' x` on the right of `t`, using the lower
    -- semicontinuity of `G'`.
    have I1 : ∀ᶠ u in 𝓝[>] t, (u - t) * y ≤ ∫ w in t..u, (G' w).toReal := by
      have B : ∀ᶠ u in 𝓝 t, (y : EReal) < G' u := G'cont.lowerSemicontinuousAt _ _ y_lt_G'
      rcases mem_nhds_iff_exists_Ioo_subset.1 B with ⟨m, M, ⟨hm, hM⟩, H⟩
      have : Ioo t (min M b) ∈ 𝓝[>] t := Ioo_mem_nhdsWithin_Ioi' (lt_min hM ht.right.right)
      filter_upwards [this] with u hu
      have I : Icc t u ⊆ Icc a b := Icc_subset_Icc ht.2.1 (hu.2.le.trans (min_le_right _ _))
      calc
        (u - t) * y = ∫ _ in Icc t u, y := by
          simp only [hu.left.le, MeasureTheory.integral_const, Algebra.id.smul_eq_mul, sub_nonneg,
            MeasurableSet.univ, Real.volume_Icc, Measure.restrict_apply, univ_inter,
            ENNReal.toReal_ofReal]
        _ ≤ ∫ w in t..u, (G' w).toReal := by
          rw [intervalIntegral.integral_of_le hu.1.le, ← integral_Icc_eq_integral_Ioc]
          apply set_integral_mono_ae_restrict
          · simp only [integrableOn_const, Real.volume_Icc, ENNReal.ofReal_lt_top, or_true_iff]
          · exact IntegrableOn.mono_set G'int I
          · have C1 : ∀ᵐ x : ℝ ∂volume.restrict (Icc t u), G' x < ∞ :=
              ae_mono (Measure.restrict_mono I le_rfl) G'lt_top
            have C2 : ∀ᵐ x : ℝ ∂volume.restrict (Icc t u), x ∈ Icc t u :=
              ae_restrict_mem measurableSet_Icc
            filter_upwards [C1, C2] with x G'x hx
            apply EReal.coe_le_coe_iff.1
            have : x ∈ Ioo m M := by
              simp only [hm.trans_le hx.left,
                (hx.right.trans_lt hu.right).trans_le (min_le_left M b), mem_Ioo, and_self_iff]
            refine (H this).out.le.trans_eq ?_
            exact (EReal.coe_toReal G'x.ne (f_lt_G' x).ne_bot).symm
    -- bound from above the increase of `g u - g a` on the right of `t`, using the derivative at `t`
    have I2 : ∀ᶠ u in 𝓝[>] t, g u - g t ≤ (u - t) * y := by
      have g'_lt_y : g' t < y := EReal.coe_lt_coe_iff.1 g'_lt_y'
      filter_upwards [(hderiv t ⟨ht.2.1, ht.2.2⟩).limsup_slope_le' (not_mem_Ioi.2 le_rfl) g'_lt_y,
        self_mem_nhdsWithin] with u hu t_lt_u
      have := mul_le_mul_of_nonneg_left hu.le (sub_pos.2 t_lt_u.out).le
      rwa [← smul_eq_mul, sub_smul_slope] at this
    -- combine the previous two bounds to show that `g u - g a` increases less quickly than
    -- `∫ x in a..u, G' x`.
    have I3 : ∀ᶠ u in 𝓝[>] t, g u - g t ≤ ∫ w in t..u, (G' w).toReal := by
      filter_upwards [I1, I2] with u hu1 hu2 using hu2.trans hu1
    have I4 : ∀ᶠ u in 𝓝[>] t, u ∈ Ioc t (min v b) := by
      refine' mem_nhdsWithin_Ioi_iff_exists_Ioc_subset.2 ⟨min v b, _, Subset.rfl⟩
      simp only [lt_min_iff, mem_Ioi]
      exact ⟨t_lt_v, ht.2.2⟩
    -- choose a point `x` slightly to the right of `t` which satisfies the above bound
    rcases (I3.and I4).exists with ⟨x, hx, h'x⟩
    -- we check that it belongs to `s`, essentially by construction
    refine' ⟨x, _, Ioc_subset_Ioc le_rfl (min_le_left _ _) h'x⟩
    calc
      g x - g a = g t - g a + (g x - g t) := by abel
      _ ≤ (∫ w in a..t, (G' w).toReal) + ∫ w in t..x, (G' w).toReal := (add_le_add ht.1 hx)
      _ = ∫ w in a..x, (G' w).toReal := by
        apply integral_add_adjacent_intervals
        · rw [intervalIntegrable_iff_integrableOn_Ioc_of_le ht.2.1]
          exact IntegrableOn.mono_set G'int
            (Ioc_subset_Icc_self.trans (Icc_subset_Icc le_rfl ht.2.2.le))
        · rw [intervalIntegrable_iff_integrableOn_Ioc_of_le h'x.1.le]
          apply IntegrableOn.mono_set G'int
          

Proof state:
case hbc
ι : Type u_1
𝕜 : Type u_2
E : Type u_3
F : Type u_4
A : Type u_5
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℝ → E
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)
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 => EReal.toReal (G' x)
G'lt_top : ∀ᵐ (x : ℝ) ∂Measure.restrict volume (Icc a b), G' x < ⊤
hG' : ∫ (x : ℝ) in Icc a b, EReal.toReal (G' x) ∂volume < ∫ (x : ℝ) in Icc a b, φ x ∂volume + ε
s : Set ℝ := {t | g t - g a ≤ ∫ (u : ℝ) in a..t, EReal.toReal (G' u)} ∩ Icc a b
s_closed : IsClosed s
t : ℝ
ht : t ∈ {t | g t - g a ≤ ∫ (u : ℝ) in a..t, EReal.toReal (G' u)} ∩ 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, EReal.toReal (G' w)
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, EReal.toReal (G' w)
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, EReal.toReal (G' w)
h'x : x ∈ Ioc t (min v b)
⊢ Ioc t x ⊆ Icc a b

Next tactic:
exact Ioc_subset_Icc_self.trans (Icc_subset_Icc ht.2.1 (h'x.2.trans (min_le_right _ _)))