theorem extracted_formal_statement_90.{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)) ⦃i : ℕ⦄ (left : i ≥ 0) (hi₂ : i ≤ k)
  (h_original_goal :
    ‖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) :
  ‖subgradient_method.x f x₀ i - xm‖ ^ 2 +
        -(2 * ⟪subgradient_method.x f x₀ i - xm, subgradient_method.a f x₀ i • subgradient_method.g f x₀ i⟫_ℝ) +
      ‖subgradient_method.a f x₀ i‖ ^ 2 * ‖subgradient_method.g f x₀ i‖ ^ 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 := 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)
i : ℕ
left✝ : i ≥ 0
hi₂ : 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

Tactic: rw [alg.update i, sub_right_comm, norm_sub_sq_real, norm_smul, mul_pow, sub_eq_add_neg]

State after:
[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)
 i : ℕ
 left✝ : i ≥ 0
 hi₂ : i ≤ k
 ⊢ ‖subgradient_method.x f x₀ i - xm‖ ^ 2 +
         -(2 * ⟪subgradient_method.x f x₀ i - xm, subgradient_method.a f x₀ i • subgradient_method.g f x₀ i⟫_ℝ) +
       ‖subgradient_method.a f x₀ i‖ ^ 2 * ‖subgradient_method.g f x₀ i‖ ^ 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]

Relevant lemmas:
sub_eq_add_neg : ∀ {G : Type u_1} [inst : SubNegMonoid G] (a b : G), a - b = a + -b
norm_smul : ∀ {α : Type u_1} {β : Type u_2} [inst : NormedDivisionRing α] [inst_3 : SeminormedAddGroup β]
  [inst_4 : MulActionWithZero α β] [inst_5 : IsBoundedSMul α β] (r : α) (x : β), ‖r • x‖ = ‖r‖ * ‖x‖
sub_right_comm : ∀ {α : Type u_1} [inst : SubtractionCommMonoid α] (a b c : α), a - b - c = a - c - b
norm_sub_sq_real : ∀ {F : Type u_3} [inst : SeminormedAddCommGroup F] [inst_3 : InnerProductSpace ℝ F] (x y : F),
  ‖x - y‖ ^ 2 = ‖x‖ ^ 2 - 2 * ⟪x, y⟫_ℝ + ‖y‖ ^ 2
mul_pow : ∀ {M : Type u_4} [inst : CommMonoid M] (a b : M) (n : ℕ), (a * b) ^ n = a ^ n * b ^ n