theorem extracted_formal_statement_24.{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} (k : ℕ)
  (h' : ∀ ⦃x g : E⦄, g ∈ SubderivAt f x → ‖g‖ ≤ ↑(subgradient_method.G f x₀)) (k₀ : k = 0)
  (h_original_goal :
    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) :
  2 * subgradient_method.a f x₀ 0 * (f (subgradient_method.x f x₀ 0) - f xm) ≤
    ‖x₀ - xm‖ ^ 2 + ↑(subgradient_method.G f x₀) ^ 2 * subgradient_method.a f x₀ 0 ^ 2 := sorry

State before:
case pos
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace ℝ E
f : E → ℝ
xm x₀ : E
alg : subgradient_method f x₀
k : ℕ
h' : ∀ ⦃x g : E⦄, g ∈ SubderivAt f x → ‖g‖ ≤ ↑(subgradient_method.G f x₀)
k₀ : k = 0
⊢ 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

Tactic: simp [k₀]

State after:
[case pos
 E : Type u_1
 inst✝¹ : NormedAddCommGroup E
 inst✝ : InnerProductSpace ℝ E
 f : E → ℝ
 xm x₀ : E
 alg : subgradient_method f x₀
 k : ℕ
 h' : ∀ ⦃x g : E⦄, g ∈ SubderivAt f x → ‖g‖ ≤ ↑(subgradient_method.G f x₀)
 k₀ : k = 0
 ⊢ 2 * subgradient_method.a f x₀ 0 * (f (subgradient_method.x f x₀ 0) - f xm) ≤
     ‖x₀ - xm‖ ^ 2 + ↑(subgradient_method.G f x₀) ^ 2 * subgradient_method.a f x₀ 0 ^ 2]

Relevant lemmas: