theorem extracted_formal_statement_43.{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 : 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) (a₂ : ℕ)
  (ha₂ : ∀ (b : ℕ), a₂ ≤ b → |subgradient_method.a f x₀ b| < ε / (2 * ↑(subgradient_method.G f x₀) ^ 2)) :
  let A :=
    (2 * ↑(subgradient_method.G f x₀) ^ 2 * ∑ x ∈ Finset.range (a₂ + 1), subgradient_method.a f x₀ x ^ 2) / ε + 1;
  let s₁ :=
    {x | x > (2 * ↑(subgradient_method.G f x₀) ^ 2 * ∑ x ∈ Finset.range (a₂ + 1), subgradient_method.a f x₀ x ^ 2) / ε};
  (∀ (b : ℝ), A ≤ b → b ∈ s₁) →
    ∀ (a₁ : ℕ),
      (∀ (b : ℕ), a₁ ≤ b → (Finset.range (b + 1)).sum (subgradient_method.a f x₀) ∈ s₁) →
        ∀ (b : ℕ),
          a₁ ⊔ (a₂ + 1) ≤ b →
            b ≥ a₁ →
              b ≥ a₂ + 1 →
                b ≥ a₂ →
                  2 * (Finset.range (b + 1)).sum (subgradient_method.a f x₀) > 0 →
                    (Finset.range (b + 1)).sum (subgradient_method.a f x₀) > 0 →
                      ↑(subgradient_method.G f x₀) ^ 2 * (ε / (2 * ↑(subgradient_method.G f x₀) ^ 2)) > 0 →
                        ∑ x ∈ Finset.range (a₂ + 1 + (b - a₂)), subgradient_method.a f x₀ x =
                            ∑ x ∈ Finset.range (a₂ + 1), subgradient_method.a f x₀ x +
                              ∑ x ∈ Finset.range (b - a₂), subgradient_method.a f x₀ (a₂ + 1 + x) →
                          b + 1 = a₂ + 1 + (b - a₂) →
                            ∑ x ∈ Finset.range (b - a₂), subgradient_method.a f x₀ (a₂ + 1 + x) <
                                ∑ x ∈ Finset.range (b + 1), subgradient_method.a f x₀ x →
                              (Finset.range (a₂ + 1)).Nonempty := sorry

State before:
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
b : ℕ
hb : a₁ ⊔ (a₂ + 1) ≤ b
hba₁ : b ≥ a₁
hba₂' : b ≥ a₂ + 1
hba₂ : b ≥ a₂
hpos : 2 * (Finset.range (b + 1)).sum (subgradient_method.a f x₀) > 0
hpos'' : (Finset.range (b + 1)).sum (subgradient_method.a f x₀) > 0
hposG : ↑(subgradient_method.G f x₀) ^ 2 * (ε / (2 * ↑(subgradient_method.G f x₀) ^ 2)) > 0
heq :
  ∑ x ∈ Finset.range (a₂ + 1 + (b - a₂)), subgradient_method.a f x₀ x =
    ∑ x ∈ Finset.range (a₂ + 1), subgradient_method.a f x₀ x +
      ∑ x ∈ Finset.range (b - a₂), subgradient_method.a f x₀ (a₂ + 1 + x)
h₃' : b + 1 = a₂ + 1 + (b - a₂)
⊢ ∑ x ∈ Finset.range (b - a₂), subgradient_method.a f x₀ (a₂ + 1 + x) <
    ∑ x ∈ Finset.range (b + 1), subgradient_method.a f x₀ x

Tactic: rw [h₃', heq]
    simp
    apply Finset.sum_pos

State after:
[case h
 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
 b : ℕ
 hb : a₁ ⊔ (a₂ + 1) ≤ b
 hba₁ : b ≥ a₁
 hba₂' : b ≥ a₂ + 1
 hba₂ : b ≥ a₂
 hpos : 2 * (Finset.range (b + 1)).sum (subgradient_method.a f x₀) > 0
 hpos'' : (Finset.range (b + 1)).sum (subgradient_method.a f x₀) > 0
 hposG : ↑(subgradient_method.G f x₀) ^ 2 * (ε / (2 * ↑(subgradient_method.G f x₀) ^ 2)) > 0
 heq :
   ∑ x ∈ Finset.range (a₂ + 1 + (b - a₂)), subgradient_method.a f x₀ x =
     ∑ x ∈ Finset.range (a₂ + 1), subgradient_method.a f x₀ x +
       ∑ x ∈ Finset.range (b - a₂), subgradient_method.a f x₀ (a₂ + 1 + x)
 h₃' : b + 1 = a₂ + 1 + (b - a₂)
 ⊢ ∀ i ∈ Finset.range (a₂ + 1), 0 < subgradient_method.a f x₀ i,
 case hs
 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
 b : ℕ
 hb : a₁ ⊔ (a₂ + 1) ≤ b
 hba₁ : b ≥ a₁
 hba₂' : b ≥ a₂ + 1
 hba₂ : b ≥ a₂
 hpos : 2 * (Finset.range (b + 1)).sum (subgradient_method.a f x₀) > 0
 hpos'' : (Finset.range (b + 1)).sum (subgradient_method.a f x₀) > 0
 hposG : ↑(subgradient_method.G f x₀) ^ 2 * (ε / (2 * ↑(subgradient_method.G f x₀) ^ 2)) > 0
 heq :
   ∑ x ∈ Finset.range (a₂ + 1 + (b - a₂)), subgradient_method.a f x₀ x =
     ∑ x ∈ Finset.range (a₂ + 1), subgradient_method.a f x₀ x +
       ∑ x ∈ Finset.range (b - a₂), subgradient_method.a f x₀ (a₂ + 1 + x)
 h₃' : b + 1 = a₂ + 1 + (b - a₂)
 ⊢ (Finset.range (a₂ + 1)).Nonempty]

Relevant lemmas:
Finset.sum_pos : ∀ {ι : Type u_1} {M : Type u_4} [inst : OrderedCancelAddCommMonoid M] {f : ι → M} {s : Finset ι},
  (∀ i ∈ s, 0 < f i) → s.Nonempty → 0 < ∑ i ∈ s, f i