theorem extracted_formal_statement_113.{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} {s : ℝ}
  (hm : IsMinOn f Set.univ xm) (ha' : ∀ (n : ℕ), subgradient_method.a f x₀ n * ‖subgradient_method.g f x₀ n‖ = s)
  (hs : s > 0) (k : ℕ)
  (heq :
    (Set.range fun x => f (subgradient_method.x f x₀ ↑x)) =
      {x | ∃ i ∈ Finset.range (k + 1), f (subgradient_method.x f x₀ i) = x})
  (hnek : Nonempty { x // x ∈ Finset.range (k + 1) })
  (h' : ∀ ⦃x g : E⦄, g ∈ SubderivAt f x → ‖g‖ ≤ ↑(subgradient_method.G f x₀))
  (h₁ :
    ∀ ⦃i : ℕ⦄,
      i ≥ 0 ∧ i ≤ k →
        ‖subgradient_method.x f x₀ (i + 1) - xm‖ ^ 2 ≤
          ‖subgradient_method.x f x₀ i - xm‖ ^ 2 -
              2 * subgradient_method.a f x₀ i *
                (sInf {x | ∃ i ∈ Finset.range (k + 1), f (subgradient_method.x f x₀ i) = x} - f xm) +
            ‖subgradient_method.a f x₀ i‖ ^ 2 * ‖subgradient_method.g f x₀ i‖ ^ 2)
  (h₁' :
    ∀ ⦃i : ℕ⦄,
      i ≥ 0 ∧ i ≤ k →
        subgradient_method.a f x₀ i *
            (2 * (sInf {x | ∃ i ∈ Finset.range (k + 1), f (subgradient_method.x f x₀ i) = x} - f xm)) ≤
          ‖subgradient_method.x f x₀ i - xm‖ ^ 2 - ‖subgradient_method.x f x₀ (i + 1) - xm‖ ^ 2 + s ^ 2)
  (h_original_goal :
    ∑ i ∈ Finset.range (k + 1),
        subgradient_method.a f x₀ i *
          (2 * (sInf {x | ∃ i ∈ Finset.range (k + 1), f (subgradient_method.x f x₀ i) = x} - f xm)) ≤
      ∑ i ∈ Finset.range (k + 1),
        (‖subgradient_method.x f x₀ i - xm‖ ^ 2 - ‖subgradient_method.x f x₀ (i + 1) - xm‖ ^ 2 + s ^ 2))
  (i : ℕ) :
  i ∈ Finset.range (k + 1) →
    subgradient_method.a f x₀ i *
        (2 * (sInf {x | ∃ i ∈ Finset.range (k + 1), f (subgradient_method.x f x₀ i) = x} - f xm)) ≤
      ‖subgradient_method.x f x₀ i - xm‖ ^ 2 - ‖subgradient_method.x f x₀ (i + 1) - xm‖ ^ 2 + s ^ 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
s : ℝ
hm : IsMinOn f Set.univ xm
ha' : ∀ (n : ℕ), subgradient_method.a f x₀ n * ‖subgradient_method.g f x₀ n‖ = s
hs : s > 0
k : ℕ
heq :
  (Set.range fun x => f (subgradient_method.x f x₀ ↑x)) =
    {x | ∃ i ∈ Finset.range (k + 1), f (subgradient_method.x f x₀ i) = x}
hnek : Nonempty { x // x ∈ Finset.range (k + 1) }
h' : ∀ ⦃x g : E⦄, g ∈ SubderivAt f x → ‖g‖ ≤ ↑(subgradient_method.G f x₀)
h₁ :
  ∀ ⦃i : ℕ⦄,
    i ≥ 0 ∧ i ≤ k →
      ‖subgradient_method.x f x₀ (i + 1) - xm‖ ^ 2 ≤
        ‖subgradient_method.x f x₀ i - xm‖ ^ 2 -
            2 * subgradient_method.a f x₀ i *
              (sInf {x | ∃ i ∈ Finset.range (k + 1), f (subgradient_method.x f x₀ i) = x} - f xm) +
          ‖subgradient_method.a f x₀ i‖ ^ 2 * ‖subgradient_method.g f x₀ i‖ ^ 2
h₁' :
  ∀ ⦃i : ℕ⦄,
    i ≥ 0 ∧ i ≤ k →
      subgradient_method.a f x₀ i *
          (2 * (sInf {x | ∃ i ∈ Finset.range (k + 1), f (subgradient_method.x f x₀ i) = x} - f xm)) ≤
        ‖subgradient_method.x f x₀ i - xm‖ ^ 2 - ‖subgradient_method.x f x₀ (i + 1) - xm‖ ^ 2 + s ^ 2
⊢ ∑ i ∈ Finset.range (k + 1),
      subgradient_method.a f x₀ i *
        (2 * (sInf {x | ∃ i ∈ Finset.range (k + 1), f (subgradient_method.x f x₀ i) = x} - f xm)) ≤
    ∑ i ∈ Finset.range (k + 1),
      (‖subgradient_method.x f x₀ i - xm‖ ^ 2 - ‖subgradient_method.x f x₀ (i + 1) - xm‖ ^ 2 + s ^ 2)

Tactic: apply Finset.sum_le_sum

State after:
[case h
 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
 s : ℝ
 hm : IsMinOn f Set.univ xm
 ha' : ∀ (n : ℕ), subgradient_method.a f x₀ n * ‖subgradient_method.g f x₀ n‖ = s
 hs : s > 0
 k : ℕ
 heq :
   (Set.range fun x => f (subgradient_method.x f x₀ ↑x)) =
     {x | ∃ i ∈ Finset.range (k + 1), f (subgradient_method.x f x₀ i) = x}
 hnek : Nonempty { x // x ∈ Finset.range (k + 1) }
 h' : ∀ ⦃x g : E⦄, g ∈ SubderivAt f x → ‖g‖ ≤ ↑(subgradient_method.G f x₀)
 h₁ :
   ∀ ⦃i : ℕ⦄,
     i ≥ 0 ∧ i ≤ k →
       ‖subgradient_method.x f x₀ (i + 1) - xm‖ ^ 2 ≤
         ‖subgradient_method.x f x₀ i - xm‖ ^ 2 -
             2 * subgradient_method.a f x₀ i *
               (sInf {x | ∃ i ∈ Finset.range (k + 1), f (subgradient_method.x f x₀ i) = x} - f xm) +
           ‖subgradient_method.a f x₀ i‖ ^ 2 * ‖subgradient_method.g f x₀ i‖ ^ 2
 h₁' :
   ∀ ⦃i : ℕ⦄,
     i ≥ 0 ∧ i ≤ k →
       subgradient_method.a f x₀ i *
           (2 * (sInf {x | ∃ i ∈ Finset.range (k + 1), f (subgradient_method.x f x₀ i) = x} - f xm)) ≤
         ‖subgradient_method.x f x₀ i - xm‖ ^ 2 - ‖subgradient_method.x f x₀ (i + 1) - xm‖ ^ 2 + s ^ 2
 ⊢ ∀ i ∈ Finset.range (k + 1),
     subgradient_method.a f x₀ i *
         (2 * (sInf {x | ∃ i ∈ Finset.range (k + 1), f (subgradient_method.x f x₀ i) = x} - f xm)) ≤
       ‖subgradient_method.x f x₀ i - xm‖ ^ 2 - ‖subgradient_method.x f x₀ (i + 1) - xm‖ ^ 2 + s ^ 2]

Relevant lemmas:
Finset.sum_le_sum : ∀ {ι : Type u_1} {N : Type u_5} [inst : OrderedAddCommMonoid N] {f g : ι → N} {s : Finset ι},
  (∀ i ∈ s, f i ≤ g i) → ∑ i ∈ s, f i ≤ ∑ i ∈ s, g i