theorem extracted_formal_statement_8.{u_1} {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : InnerProductSpace ℝ E]
  [inst_2 : CompleteSpace E] {f : E → ℝ} {f' : E → E} {xm x₀ : E} {alg : Gradient_Descent_fix_stepsize f f' x₀}
  (hfun : ConvexOn ℝ univ f)
  (step₂ : Gradient_Descent_fix_stepsize.a f f' x₀ ≤ 1 / ↑(Gradient_Descent_fix_stepsize.l f f' x₀)) (k : ℕ)
  (step1₂ : ↑(Gradient_Descent_fix_stepsize.l f f' x₀) ≤ 1 / Gradient_Descent_fix_stepsize.a f f' x₀)
  (this_1 : LipschitzWith (Gradient_Descent_fix_stepsize.l f f' x₀) f')
  (this : Gradient_Descent_fix_stepsize.l f f' x₀ > 0)
  (tα : 1 / (2 * (↑k + 1) * Gradient_Descent_fix_stepsize.a f f' x₀) > 0)
  (mono :
    ∀ (k : ℕ), f (Gradient_Descent_fix_stepsize.x f f' x₀ (k + 1)) ≤ f (Gradient_Descent_fix_stepsize.x f f' x₀ k))
  (j : ℕ)
  (IH :
    ∑ k ∈ Finset.range (j + 1), (f (Gradient_Descent_fix_stepsize.x f f' x₀ (k + 1)) - f xm) ≤
      1 / (2 * Gradient_Descent_fix_stepsize.a f f' x₀) *
        (‖x₀ - xm‖ ^ 2 - ‖Gradient_Descent_fix_stepsize.x f f' x₀ (j + 1) - xm‖ ^ 2))
  (xdescent :
    f (Gradient_Descent_fix_stepsize.x f f' x₀ (j + 1 + 1)) ≤
      f xm +
        1 / (2 * Gradient_Descent_fix_stepsize.a f f' x₀) *
          (‖Gradient_Descent_fix_stepsize.x f f' x₀ (j + 1) - xm‖ ^ 2 -
            ‖Gradient_Descent_fix_stepsize.x f f' x₀ (j + 1 + 1) - xm‖ ^ 2))
  (h_original_goal :
    ∑ k ∈ Finset.range (j + 1 + 1), (f (Gradient_Descent_fix_stepsize.x f f' x₀ (k + 1)) - f xm) =
      ∑ k ∈ Finset.range (j + 1), (f (Gradient_Descent_fix_stepsize.x f f' x₀ (k + 1)) - f xm) +
          f (Gradient_Descent_fix_stepsize.x f f' x₀ (j + 2)) -
        f xm) :
  ∑ x ∈ Finset.range (1 + j), (f (Gradient_Descent_fix_stepsize.x f f' x₀ (x + 1)) - f xm) +
      (f (Gradient_Descent_fix_stepsize.x f f' x₀ (2 + j)) - f xm) =
    ∑ x ∈ Finset.range (1 + j), (f (Gradient_Descent_fix_stepsize.x f f' x₀ (x + 1)) - f xm) +
        f (Gradient_Descent_fix_stepsize.x f f' x₀ (2 + j)) -
      f xm := sorry

State before:
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
f : E → ℝ
f' : E → E
xm x₀ : E
alg : Gradient_Descent_fix_stepsize f f' x₀
hfun : ConvexOn ℝ univ f
step₂ : Gradient_Descent_fix_stepsize.a f f' x₀ ≤ 1 / ↑(Gradient_Descent_fix_stepsize.l f f' x₀)
k : ℕ
step1₂ : ↑(Gradient_Descent_fix_stepsize.l f f' x₀) ≤ 1 / Gradient_Descent_fix_stepsize.a f f' x₀
this✝ : LipschitzWith (Gradient_Descent_fix_stepsize.l f f' x₀) f'
this : Gradient_Descent_fix_stepsize.l f f' x₀ > 0
tα : 1 / (2 * (↑k + 1) * Gradient_Descent_fix_stepsize.a f f' x₀) > 0
mono : ∀ (k : ℕ), f (Gradient_Descent_fix_stepsize.x f f' x₀ (k + 1)) ≤ f (Gradient_Descent_fix_stepsize.x f f' x₀ k)
j : ℕ
IH :
  ∑ k ∈ Finset.range (j + 1), (f (Gradient_Descent_fix_stepsize.x f f' x₀ (k + 1)) - f xm) ≤
    1 / (2 * Gradient_Descent_fix_stepsize.a f f' x₀) *
      (‖x₀ - xm‖ ^ 2 - ‖Gradient_Descent_fix_stepsize.x f f' x₀ (j + 1) - xm‖ ^ 2)
xdescent :
  f (Gradient_Descent_fix_stepsize.x f f' x₀ (j + 1 + 1)) ≤
    f xm +
      1 / (2 * Gradient_Descent_fix_stepsize.a f f' x₀) *
        (‖Gradient_Descent_fix_stepsize.x f f' x₀ (j + 1) - xm‖ ^ 2 -
          ‖Gradient_Descent_fix_stepsize.x f f' x₀ (j + 1 + 1) - xm‖ ^ 2)
⊢ ∑ k ∈ Finset.range (j + 1 + 1), (f (Gradient_Descent_fix_stepsize.x f f' x₀ (k + 1)) - f xm) =
    ∑ k ∈ Finset.range (j + 1), (f (Gradient_Descent_fix_stepsize.x f f' x₀ (k + 1)) - f xm) +
        f (Gradient_Descent_fix_stepsize.x f f' x₀ (j + 2)) -
      f xm

Tactic: rw [Finset.sum_range_succ (fun (k : ℕ) ↦ f (alg.x (k + 1)) - f (xm)) j.succ]
    rw [Nat.succ_eq_add_one j]
    ring_nf

State after:
[E : Type u_1
 inst✝² : NormedAddCommGroup E
 inst✝¹ : InnerProductSpace ℝ E
 inst✝ : CompleteSpace E
 f : E → ℝ
 f' : E → E
 xm x₀ : E
 alg : Gradient_Descent_fix_stepsize f f' x₀
 hfun : ConvexOn ℝ univ f
 step₂ : Gradient_Descent_fix_stepsize.a f f' x₀ ≤ 1 / ↑(Gradient_Descent_fix_stepsize.l f f' x₀)
 k : ℕ
 step1₂ : ↑(Gradient_Descent_fix_stepsize.l f f' x₀) ≤ 1 / Gradient_Descent_fix_stepsize.a f f' x₀
 this✝ : LipschitzWith (Gradient_Descent_fix_stepsize.l f f' x₀) f'
 this : Gradient_Descent_fix_stepsize.l f f' x₀ > 0
 tα : 1 / (2 * (↑k + 1) * Gradient_Descent_fix_stepsize.a f f' x₀) > 0
 mono : ∀ (k : ℕ), f (Gradient_Descent_fix_stepsize.x f f' x₀ (k + 1)) ≤ f (Gradient_Descent_fix_stepsize.x f f' x₀ k)
 j : ℕ
 IH :
   ∑ k ∈ Finset.range (j + 1), (f (Gradient_Descent_fix_stepsize.x f f' x₀ (k + 1)) - f xm) ≤
     1 / (2 * Gradient_Descent_fix_stepsize.a f f' x₀) *
       (‖x₀ - xm‖ ^ 2 - ‖Gradient_Descent_fix_stepsize.x f f' x₀ (j + 1) - xm‖ ^ 2)
 xdescent :
   f (Gradient_Descent_fix_stepsize.x f f' x₀ (j + 1 + 1)) ≤
     f xm +
       1 / (2 * Gradient_Descent_fix_stepsize.a f f' x₀) *
         (‖Gradient_Descent_fix_stepsize.x f f' x₀ (j + 1) - xm‖ ^ 2 -
           ‖Gradient_Descent_fix_stepsize.x f f' x₀ (j + 1 + 1) - xm‖ ^ 2)
 ⊢ ∑ x ∈ Finset.range (1 + j), (f (Gradient_Descent_fix_stepsize.x f f' x₀ (x + 1)) - f xm) +
       (f (Gradient_Descent_fix_stepsize.x f f' x₀ (2 + j)) - f xm) =
     ∑ x ∈ Finset.range (1 + j), (f (Gradient_Descent_fix_stepsize.x f f' x₀ (x + 1)) - f xm) +
         f (Gradient_Descent_fix_stepsize.x f f' x₀ (2 + j)) -
       f xm]

Relevant lemmas:
Nat.succ_eq_add_one : ∀ (n : ℕ), n.succ = n + 1
Finset.sum_range_succ : ∀ {β : Type u_4} [inst : AddCommMonoid β] (f : ℕ → β) (n : ℕ),
  ∑ x ∈ Finset.range (n + 1), f x = ∑ x ∈ Finset.range n, f x + f n