theorem extracted_formal_statement_41.{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_1 : Gradient_Descent_fix_stepsize.a f f' x₀ ≤ 1 / ↑(Gradient_Descent_fix_stepsize.l f f' x₀))
  (step₂ : ↑(Gradient_Descent_fix_stepsize.l f f' x₀) ≤ 1 / Gradient_Descent_fix_stepsize.a f f' x₀) (k : ℕ)
  (this_1 : LipschitzWith (Gradient_Descent_fix_stepsize.l f f' x₀) f')
  (this : Gradient_Descent_fix_stepsize.l f f' x₀ > 0)
  (descent :
    f
        (Gradient_Descent_fix_stepsize.x f f' x₀ k -
          Gradient_Descent_fix_stepsize.a f f' x₀ • f' (Gradient_Descent_fix_stepsize.x f f' x₀ k)) ≤
      f xm +
        1 / (2 * Gradient_Descent_fix_stepsize.a f f' x₀) *
          (‖Gradient_Descent_fix_stepsize.x f f' x₀ k - xm‖ ^ 2 -
            ‖Gradient_Descent_fix_stepsize.x f f' x₀ k -
                    Gradient_Descent_fix_stepsize.a f f' x₀ • f' (Gradient_Descent_fix_stepsize.x f f' x₀ k) -
                  xm‖ ^
              2))
  (h_original_goal :
    f (Gradient_Descent_fix_stepsize.x f f' x₀ (k + 1)) ≤
      f xm +
        1 / (2 * Gradient_Descent_fix_stepsize.a f f' x₀) *
          (‖Gradient_Descent_fix_stepsize.x f f' x₀ k - xm‖ ^ 2 -
            ‖Gradient_Descent_fix_stepsize.x f f' x₀ (k + 1) - xm‖ ^ 2)) :
  f
      (Gradient_Descent_fix_stepsize.x f f' x₀ k -
        Gradient_Descent_fix_stepsize.a f f' x₀ • f' (Gradient_Descent_fix_stepsize.x f f' x₀ k)) ≤
    f xm +
      1 / (2 * Gradient_Descent_fix_stepsize.a f f' x₀) *
        (‖Gradient_Descent_fix_stepsize.x f f' x₀ k - xm‖ ^ 2 -
          ‖Gradient_Descent_fix_stepsize.x f f' x₀ k -
                  Gradient_Descent_fix_stepsize.a f f' x₀ • f' (Gradient_Descent_fix_stepsize.x f f' x₀ k) -
                xm‖ ^
            2) := 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)
step₂ : ↑(Gradient_Descent_fix_stepsize.l f f' x₀) ≤ 1 / Gradient_Descent_fix_stepsize.a f f' x₀
k : ℕ
this✝ : LipschitzWith (Gradient_Descent_fix_stepsize.l f f' x₀) f'
this : Gradient_Descent_fix_stepsize.l f f' x₀ > 0
descent :
  f
      (Gradient_Descent_fix_stepsize.x f f' x₀ k -
        Gradient_Descent_fix_stepsize.a f f' x₀ • f' (Gradient_Descent_fix_stepsize.x f f' x₀ k)) ≤
    f xm +
      1 / (2 * Gradient_Descent_fix_stepsize.a f f' x₀) *
        (‖Gradient_Descent_fix_stepsize.x f f' x₀ k - xm‖ ^ 2 -
          ‖Gradient_Descent_fix_stepsize.x f f' x₀ k -
                  Gradient_Descent_fix_stepsize.a f f' x₀ • f' (Gradient_Descent_fix_stepsize.x f f' x₀ k) -
                xm‖ ^
            2)
⊢ f (Gradient_Descent_fix_stepsize.x f f' x₀ (k + 1)) ≤
    f xm +
      1 / (2 * Gradient_Descent_fix_stepsize.a f f' x₀) *
        (‖Gradient_Descent_fix_stepsize.x f f' x₀ k - xm‖ ^ 2 -
          ‖Gradient_Descent_fix_stepsize.x f f' x₀ (k + 1) - xm‖ ^ 2)

Tactic: rw [alg.update k]

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)
 step₂ : ↑(Gradient_Descent_fix_stepsize.l f f' x₀) ≤ 1 / Gradient_Descent_fix_stepsize.a f f' x₀
 k : ℕ
 this✝ : LipschitzWith (Gradient_Descent_fix_stepsize.l f f' x₀) f'
 this : Gradient_Descent_fix_stepsize.l f f' x₀ > 0
 descent :
   f
       (Gradient_Descent_fix_stepsize.x f f' x₀ k -
         Gradient_Descent_fix_stepsize.a f f' x₀ • f' (Gradient_Descent_fix_stepsize.x f f' x₀ k)) ≤
     f xm +
       1 / (2 * Gradient_Descent_fix_stepsize.a f f' x₀) *
         (‖Gradient_Descent_fix_stepsize.x f f' x₀ k - xm‖ ^ 2 -
           ‖Gradient_Descent_fix_stepsize.x f f' x₀ k -
                   Gradient_Descent_fix_stepsize.a f f' x₀ • f' (Gradient_Descent_fix_stepsize.x f f' x₀ k) -
                 xm‖ ^
             2)
 ⊢ f
       (Gradient_Descent_fix_stepsize.x f f' x₀ k -
         Gradient_Descent_fix_stepsize.a f f' x₀ • f' (Gradient_Descent_fix_stepsize.x f f' x₀ k)) ≤
     f xm +
       1 / (2 * Gradient_Descent_fix_stepsize.a f f' x₀) *
         (‖Gradient_Descent_fix_stepsize.x f f' x₀ k - xm‖ ^ 2 -
           ‖Gradient_Descent_fix_stepsize.x f f' x₀ k -
                   Gradient_Descent_fix_stepsize.a f f' x₀ • f' (Gradient_Descent_fix_stepsize.x f f' x₀ k) -
                 xm‖ ^
             2)]

Relevant lemmas: