theorem extracted_formal_statement_256 : 0 < 2 := sorry

State before:
case ha
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
a₁ : ℕ
ha₁ : ∀ (b : ℕ), a₁ ≤ b → ‖x₀ - xm‖ ^ 2 / (2 * (Finset.range (b + 1)).sum (subgradient_method.a f x₀)) < ε / 2
a₂ : ℕ
ha₂ :
  ∀ (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
hb : a₁ ⊔ a₂ ≤ b
hba₁ : b ≥ a₁
hba₂ : b ≥ a₂
hne : sInf {x | ∃ i < b + 1, f (subgradient_method.x f x₀ i) = x} - f xm ≥ 0
h₁ :
  ∀ (k : ℕ),
    2 * (Finset.range (k + 1)).sum (subgradient_method.a f x₀) *
        (sInf {x | ∃ i ∈ Finset.range (k + 1), f (subgradient_method.x f x₀ i) = x} - f xm) ≤
      ‖x₀ - xm‖ ^ 2 + ↑(subgradient_method.G f x₀) ^ 2 * ∑ i ∈ Finset.range (k + 1), subgradient_method.a f x₀ i ^ 2
hb₁ :
  2 * (Finset.range (b + 1)).sum (subgradient_method.a f x₀) *
      (sInf {x | ∃ i ∈ Finset.range (b + 1), f (subgradient_method.x f x₀ i) = x} - f xm) ≤
    ‖x₀ - xm‖ ^ 2 + ↑(subgradient_method.G f x₀) ^ 2 * ∑ i ∈ Finset.range (b + 1), subgradient_method.a f x₀ i ^ 2
hba₁' : ‖x₀ - xm‖ ^ 2 / (2 * (Finset.range (b + 1)).sum (subgradient_method.a f x₀)) < ε / 2
hba₂' :
  (↑(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
⊢ 0 < 2

Tactic: simp

State after:

Relevant lemmas: