theorem extracted_formal_statement_80.{u_1} {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : InnerProductSpace ℝ E]
  [inst_2 : CompleteSpace E] {f : E → ℝ} (xm x₀ : E) {alg : subgradient_method f x₀} {t : ℝ}
  (ha' : ∀ (n : ℕ), subgradient_method.a f x₀ n = t) (ht : t > 0) (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 := sorry

State before:
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
f : E → ℝ
xm x₀ : E
alg : subgradient_method f x₀
t : ℝ
ha' : ∀ (n : ℕ), subgradient_method.a f x₀ n = t
k : ℕ
ht : t > 0
⊢ ∀ (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

Tactic: apply subgradient_method_converge

State after:

Relevant lemmas:
subgradient_method_converge : ∀ {E : Type u_1} [inst : NormedAddCommGroup E] [inst_3 : InnerProductSpace ℝ E]
  {f : E → ℝ} (xm x₀ : E) {alg : subgradient_method f x₀} (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