theorem extracted_formal_statement_157.{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) :
  let A := ‖x₀ - xm‖ ^ 2 / ε + 1;
  let s := {x | x > ‖x₀ - xm‖ ^ 2 / ε};
  (∀ (b : ℝ), A ≤ b → b ∈ s) →
    ∀ (a_1 : ℕ),
      (∀ (b : ℕ), a_1 ≤ b → (Finset.range (b + 1)).sum (subgradient_method.a f x₀) ∈ s) →
        ∀ (b : ℕ),
          a_1 ≤ b →
            ‖x₀ - xm‖ ^ 2 / ε < (Finset.range (b + 1)).sum (subgradient_method.a f x₀) →
              ‖x₀ - xm‖ ^ 2 < (Finset.range (b + 1)).sum (subgradient_method.a f x₀) * ε →
                0 < (Finset.range (b + 1)).sum (subgradient_method.a f x₀) → (Finset.range (b + 1)).Nonempty := sorry

State before:
case hb
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
A : ℝ := ‖x₀ - xm‖ ^ 2 / ε + 1
s : Set ℝ := {x | x > ‖x₀ - xm‖ ^ 2 / ε}
this : ∀ (b : ℝ), A ≤ b → b ∈ s
a_1 : ℕ
hasA : ∀ (b : ℕ), a_1 ≤ b → (Finset.range (b + 1)).sum (subgradient_method.a f x₀) ∈ s
b : ℕ
hb : a_1 ≤ b
hb' : ‖x₀ - xm‖ ^ 2 / ε < (Finset.range (b + 1)).sum (subgradient_method.a f x₀)
h₂ : ‖x₀ - xm‖ ^ 2 < (Finset.range (b + 1)).sum (subgradient_method.a f x₀) * ε
⊢ 0 < (Finset.range (b + 1)).sum (subgradient_method.a f x₀)

Tactic: apply Finset.sum_pos

State after:
[case hb.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
 A : ℝ := ‖x₀ - xm‖ ^ 2 / ε + 1
 s : Set ℝ := {x | x > ‖x₀ - xm‖ ^ 2 / ε}
 this : ∀ (b : ℝ), A ≤ b → b ∈ s
 a_1 : ℕ
 hasA : ∀ (b : ℕ), a_1 ≤ b → (Finset.range (b + 1)).sum (subgradient_method.a f x₀) ∈ s
 b : ℕ
 hb : a_1 ≤ b
 hb' : ‖x₀ - xm‖ ^ 2 / ε < (Finset.range (b + 1)).sum (subgradient_method.a f x₀)
 h₂ : ‖x₀ - xm‖ ^ 2 < (Finset.range (b + 1)).sum (subgradient_method.a f x₀) * ε
 ⊢ ∀ i ∈ Finset.range (b + 1), 0 < subgradient_method.a f x₀ i,
 case hb.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
 A : ℝ := ‖x₀ - xm‖ ^ 2 / ε + 1
 s : Set ℝ := {x | x > ‖x₀ - xm‖ ^ 2 / ε}
 this : ∀ (b : ℝ), A ≤ b → b ∈ s
 a_1 : ℕ
 hasA : ∀ (b : ℕ), a_1 ≤ b → (Finset.range (b + 1)).sum (subgradient_method.a f x₀) ∈ s
 b : ℕ
 hb : a_1 ≤ b
 hb' : ‖x₀ - xm‖ ^ 2 / ε < (Finset.range (b + 1)).sum (subgradient_method.a f x₀)
 h₂ : ‖x₀ - xm‖ ^ 2 < (Finset.range (b + 1)).sum (subgradient_method.a f x₀) * ε
 ⊢ (Finset.range (b + 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