theorem extracted_formal_statement_45.{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))
  (xdescent :
    f (Gradient_Descent_fix_stepsize.x f f' x₀ (0 + 1)) ≤
      f xm +
        1 / (2 * Gradient_Descent_fix_stepsize.a f f' x₀) *
          (‖Gradient_Descent_fix_stepsize.x f f' x₀ 0 - xm‖ ^ 2 -
            ‖Gradient_Descent_fix_stepsize.x f f' x₀ (0 + 1) - xm‖ ^ 2))
  (h_original_goal :
    f xm +
        1 / (2 * Gradient_Descent_fix_stepsize.a f f' x₀) *
          (‖x₀ - xm‖ ^ 2 - ‖Gradient_Descent_fix_stepsize.x f f' x₀ (0 + 1) - xm‖ ^ 2) =
      (Gradient_Descent_fix_stepsize.a f f' x₀)⁻¹ * 2⁻¹ *
          (‖x₀ - xm‖ ^ 2 - ‖Gradient_Descent_fix_stepsize.x f f' x₀ 1 - xm‖ ^ 2) +
        f xm) :
  f xm +
      (Gradient_Descent_fix_stepsize.a f f' x₀)⁻¹ * 2⁻¹ *
        (‖x₀ - xm‖ ^ 2 - ‖Gradient_Descent_fix_stepsize.x f f' x₀ 1 - xm‖ ^ 2) =
    (Gradient_Descent_fix_stepsize.a f f' x₀)⁻¹ * 2⁻¹ *
        (‖x₀ - xm‖ ^ 2 - ‖Gradient_Descent_fix_stepsize.x f f' x₀ 1 - xm‖ ^ 2) +
      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)
xdescent :
  f (Gradient_Descent_fix_stepsize.x f f' x₀ (0 + 1)) ≤
    f xm +
      1 / (2 * Gradient_Descent_fix_stepsize.a f f' x₀) *
        (‖Gradient_Descent_fix_stepsize.x f f' x₀ 0 - xm‖ ^ 2 -
          ‖Gradient_Descent_fix_stepsize.x f f' x₀ (0 + 1) - xm‖ ^ 2)
⊢ f xm +
      1 / (2 * Gradient_Descent_fix_stepsize.a f f' x₀) *
        (‖x₀ - xm‖ ^ 2 - ‖Gradient_Descent_fix_stepsize.x f f' x₀ (0 + 1) - xm‖ ^ 2) =
    (Gradient_Descent_fix_stepsize.a f f' x₀)⁻¹ * 2⁻¹ *
        (‖x₀ - xm‖ ^ 2 - ‖Gradient_Descent_fix_stepsize.x f f' x₀ 1 - xm‖ ^ 2) +
      f xm

Tactic: simp

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)
 xdescent :
   f (Gradient_Descent_fix_stepsize.x f f' x₀ (0 + 1)) ≤
     f xm +
       1 / (2 * Gradient_Descent_fix_stepsize.a f f' x₀) *
         (‖Gradient_Descent_fix_stepsize.x f f' x₀ 0 - xm‖ ^ 2 -
           ‖Gradient_Descent_fix_stepsize.x f f' x₀ (0 + 1) - xm‖ ^ 2)
 ⊢ f xm +
       (Gradient_Descent_fix_stepsize.a f f' x₀)⁻¹ * 2⁻¹ *
         (‖x₀ - xm‖ ^ 2 - ‖Gradient_Descent_fix_stepsize.x f f' x₀ 1 - xm‖ ^ 2) =
     (Gradient_Descent_fix_stepsize.a f f' x₀)⁻¹ * 2⁻¹ *
         (‖x₀ - xm‖ ^ 2 - ‖Gradient_Descent_fix_stepsize.x f f' x₀ 1 - xm‖ ^ 2) +
       f xm]

Relevant lemmas: