theorem extracted_formal_statement_9.{u_1} {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : InnerProductSpace ℝ E]
  {f : E → ℝ} (xm x₀ : E) {alg : subgradient_method f x₀} (hm : IsMinOn f Set.univ xm)
  (ha'' :
    ∀ (s : Set ℝ) (x : ℝ),
      (∀ (b : ℝ), x ≤ b → b ∈ s) → ∃ a, ∀ (b : ℕ), a ≤ b → (Finset.range (b + 1)).sum (subgradient_method.a f x₀) ∈ s)
  (s_1 : Set ℝ)
  (ha' :
    ∀ (s x : Set ℝ),
      x ⊆ s →
        (∀ x_1 ∈ x, ∃ ε, 0 < ε ∧ Metric.ball x_1 ε ⊆ x) →
          0 ∈ x → ∃ a, ∀ (b : ℕ), a ≤ b → subgradient_method.a f x₀ b ∈ s)
  (t : Set ℝ) (hs₁ : t ⊆ s_1) (hs₂ : ∀ x ∈ t, ∃ ε, 0 < ε ∧ Metric.ball x ε ⊆ t) (hs₀ : 0 ∈ t) (ε : ℝ) (εpos : 0 < ε)
  (hs₂₀ : Metric.ball 0 ε ⊆ t)
  (ha₁ : ∃ a₁, ∀ (b : ℕ), a₁ ≤ b → ‖x₀ - xm‖ ^ 2 / (2 * (Finset.range (b + 1)).sum (subgradient_method.a f x₀)) < ε / 2)
  (hG : ¬subgradient_method.G f x₀ = 0) (hpos' : ↑(subgradient_method.G f x₀) ^ 2 > 0) :
  let s := {x | |x| < ε / (2 * ↑(subgradient_method.G f x₀) ^ 2)};
  (∀ x_1 ∈ s, x_1 ∈ s) →
    (∀ x_1 ∈ s, ∃ ε, 0 < ε ∧ ∀ (x_2 : ℝ), dist x_2 x_1 < ε → x_2 ∈ s) →
      0 ∈ s →
        ∀ (a₂ : ℕ),
          (∀ (b : ℕ), a₂ ≤ b → subgradient_method.a f x₀ b ∈ s) →
            (∃ a₂,
                ∀ (b : ℕ),
                  a₂ ≤ b →
                    (↑(subgradient_method.G f x₀) ^ 2 * ∑ i ∈ Finset.range (b + 1), subgradient_method.a f x₀ i ^ 2) /
                        (2 * (Finset.range (b + 1)).sum (subgradient_method.a f x₀)) <
                      ε / 2) →
              ∀ (b : ℕ), a₂ ≤ b → |subgradient_method.a f x₀ b| < ε / (2 * ↑(subgradient_method.G f x₀) ^ 2) := sorry

State before:
case neg.intro
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace ℝ E
f : E → ℝ
xm x₀ : E
alg : subgradient_method f x₀
hm : IsMinOn f Set.univ xm
ha'' :
  ∀ (s : Set ℝ) (x : ℝ),
    (∀ (b : ℝ), x ≤ b → b ∈ s) → ∃ a, ∀ (b : ℕ), a ≤ b → (Finset.range (b + 1)).sum (subgradient_method.a f x₀) ∈ s
s✝ : Set ℝ
ha' :
  ∀ (s x : Set ℝ),
    x ⊆ s →
      (∀ x_1 ∈ x, ∃ ε, 0 < ε ∧ Metric.ball x_1 ε ⊆ x) → 0 ∈ x → ∃ a, ∀ (b : ℕ), a ≤ b → subgradient_method.a f x₀ b ∈ s
t : Set ℝ
hs₁ : t ⊆ s✝
hs₂ : ∀ x ∈ t, ∃ ε, 0 < ε ∧ Metric.ball x ε ⊆ t
hs₀ : 0 ∈ t
ε : ℝ
εpos : 0 < ε
hs₂₀ : Metric.ball 0 ε ⊆ t
ha₁ : ∃ a₁, ∀ (b : ℕ), a₁ ≤ b → ‖x₀ - xm‖ ^ 2 / (2 * (Finset.range (b + 1)).sum (subgradient_method.a f x₀)) < ε / 2
hG : ¬subgradient_method.G f x₀ = 0
hpos' : ↑(subgradient_method.G f x₀) ^ 2 > 0
s : Set ℝ := {x | |x| < ε / (2 * ↑(subgradient_method.G f x₀) ^ 2)}
c₁ : ∀ x_1 ∈ s, x_1 ∈ s
c₂ : ∀ x_1 ∈ s, ∃ ε, 0 < ε ∧ ∀ (x_2 : ℝ), dist x_2 x_1 < ε → x_2 ∈ s
c₃ : 0 ∈ s
a₂ : ℕ
ha₂ : ∀ (b : ℕ), a₂ ≤ b → subgradient_method.a f x₀ b ∈ s
⊢ ∃ a₂,
    ∀ (b : ℕ),
      a₂ ≤ b →
        (↑(subgradient_method.G f x₀) ^ 2 * ∑ i ∈ Finset.range (b + 1), subgradient_method.a f x₀ i ^ 2) /
            (2 * (Finset.range (b + 1)).sum (subgradient_method.a f x₀)) <
          ε / 2

Tactic: simp [s] at ha₂
    let A := (2 * alg.G ^ 2 * (Finset.range (a₂ + 1)).sum fun x => (alg.a x) ^ 2) / ε + 1
    let s₁ := {x | x > (2 * alg.G ^ 2 * (Finset.range (a₂ + 1)).sum fun x => (alg.a x) ^ 2) / ε}
    have : ∀ (b : ℝ), A ≤ b → b ∈ s₁ := by intro b hb; simp [s₁]; apply lt_of_lt_of_le _ hb; simp [A]
    rcases ha'' s₁ A this with ⟨a₁, hasA⟩

State after:
[case neg.intro.intro
 E : Type u_1
 inst✝¹ : NormedAddCommGroup E
 inst✝ : InnerProductSpace ℝ E
 f : E → ℝ
 xm x₀ : E
 alg : subgradient_method f x₀
 hm : IsMinOn f Set.univ xm
 ha'' :
   ∀ (s : Set ℝ) (x : ℝ),
     (∀ (b : ℝ), x ≤ b → b ∈ s) → ∃ a, ∀ (b : ℕ), a ≤ b → (Finset.range (b + 1)).sum (subgradient_method.a f x₀) ∈ s
 s✝ : Set ℝ
 ha' :
   ∀ (s x : Set ℝ),
     x ⊆ s →
       (∀ x_1 ∈ x, ∃ ε, 0 < ε ∧ Metric.ball x_1 ε ⊆ x) → 0 ∈ x → ∃ a, ∀ (b : ℕ), a ≤ b → subgradient_method.a f x₀ b ∈ s
 t : Set ℝ
 hs₁ : t ⊆ s✝
 hs₂ : ∀ x ∈ t, ∃ ε, 0 < ε ∧ Metric.ball x ε ⊆ t
 hs₀ : 0 ∈ t
 ε : ℝ
 εpos : 0 < ε
 hs₂₀ : Metric.ball 0 ε ⊆ t
 ha₁ : ∃ a₁, ∀ (b : ℕ), a₁ ≤ b → ‖x₀ - xm‖ ^ 2 / (2 * (Finset.range (b + 1)).sum (subgradient_method.a f x₀)) < ε / 2
 hG : ¬subgradient_method.G f x₀ = 0
 hpos' : ↑(subgradient_method.G f x₀) ^ 2 > 0
 s : Set ℝ := {x | |x| < ε / (2 * ↑(subgradient_method.G f x₀) ^ 2)}
 c₁ : ∀ x_1 ∈ s, x_1 ∈ s
 c₂ : ∀ x_1 ∈ s, ∃ ε, 0 < ε ∧ ∀ (x_2 : ℝ), dist x_2 x_1 < ε → x_2 ∈ s
 c₃ : 0 ∈ s
 a₂ : ℕ
 ha₂ : ∀ (b : ℕ), a₂ ≤ b → |subgradient_method.a f x₀ b| < ε / (2 * ↑(subgradient_method.G f x₀) ^ 2)
 A : ℝ := (2 * ↑(subgradient_method.G f x₀) ^ 2 * ∑ x ∈ Finset.range (a₂ + 1), subgradient_method.a f x₀ x ^ 2) / ε + 1
 s₁ : Set ℝ :=
   {x | x > (2 * ↑(subgradient_method.G f x₀) ^ 2 * ∑ x ∈ Finset.range (a₂ + 1), subgradient_method.a f x₀ x ^ 2) / ε}
 this : ∀ (b : ℝ), A ≤ b → b ∈ s₁
 a₁ : ℕ
 hasA : ∀ (b : ℕ), a₁ ≤ b → (Finset.range (b + 1)).sum (subgradient_method.a f x₀) ∈ s₁
 ⊢ ∃ a₂,
     ∀ (b : ℕ),
       a₂ ≤ b →
         (↑(subgradient_method.G f x₀) ^ 2 * ∑ i ∈ Finset.range (b + 1), subgradient_method.a f x₀ i ^ 2) /
             (2 * (Finset.range (b + 1)).sum (subgradient_method.a f x₀)) <
           ε / 2]

Relevant lemmas:
lt_of_lt_of_le : ∀ {α : Type u_1} [inst : Preorder α] {a b c : α}, a < b → b ≤ c → a < c