theorem extracted_formal_statement_74.{u_1} {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : InnerProductSpace ℝ E]
  [inst_2 : CompleteSpace E] {f : E → ℝ} {m : ℝ} {f' : E → E} {xm x₀ : E} {alg : Gradient_Descent_fix_stepsize f f' x₀}
  (hm : m > 0) (min : IsMinOn f univ xm)
  (step₂ : Gradient_Descent_fix_stepsize.a f f' x₀ ≤ 2 / (m + ↑(Gradient_Descent_fix_stepsize.l f f' x₀)))
  (hsc : StrongConvexOn univ m f) (this_1 : LipschitzWith (Gradient_Descent_fix_stepsize.l f f' x₀) f')
  (this_2 : ↑(Gradient_Descent_fix_stepsize.l f f' x₀) > 0) (k : ℕ)
  (this :
    inner (Gradient_Descent_fix_stepsize.x f f' x₀ k - xm) (f' (Gradient_Descent_fix_stepsize.x f f' x₀ k)) ≥
      m * ↑(Gradient_Descent_fix_stepsize.l f f' x₀) / (m + ↑(Gradient_Descent_fix_stepsize.l f f' x₀)) *
          ‖Gradient_Descent_fix_stepsize.x f f' x₀ k - xm‖ ^ 2 +
        1 / (m + ↑(Gradient_Descent_fix_stepsize.l f f' x₀)) * ‖f' (Gradient_Descent_fix_stepsize.x f f' x₀ k)‖ ^ 2)
  (h_original_goal :
    Gradient_Descent_fix_stepsize.a f f' x₀ * (2 * m * ↑(Gradient_Descent_fix_stepsize.l f f' x₀)) *
          ‖Gradient_Descent_fix_stepsize.x f f' x₀ k - xm‖ ^ 2 +
        Gradient_Descent_fix_stepsize.a f f' x₀ * 2 * ‖f' (Gradient_Descent_fix_stepsize.x f f' x₀ k)‖ ^ 2 =
      2 * Gradient_Descent_fix_stepsize.a f f' x₀ *
        (m * ↑(Gradient_Descent_fix_stepsize.l f f' x₀) * ‖Gradient_Descent_fix_stepsize.x f f' x₀ k - xm‖ ^ 2 +
          ‖f' (Gradient_Descent_fix_stepsize.x f f' x₀ k)‖ ^ 2)) :
  2 * Gradient_Descent_fix_stepsize.a f f' x₀ * m * ↑(Gradient_Descent_fix_stepsize.l f f' x₀) *
        ‖Gradient_Descent_fix_stepsize.x f f' x₀ k - xm‖ ^ 2 +
      2 * Gradient_Descent_fix_stepsize.a f f' x₀ * ‖f' (Gradient_Descent_fix_stepsize.x f f' x₀ k)‖ ^ 2 =
    2 * Gradient_Descent_fix_stepsize.a f f' x₀ *
        (m * ↑(Gradient_Descent_fix_stepsize.l f f' x₀) * ‖Gradient_Descent_fix_stepsize.x f f' x₀ k - xm‖ ^ 2) +
      2 * Gradient_Descent_fix_stepsize.a f f' x₀ * ‖f' (Gradient_Descent_fix_stepsize.x f f' x₀ k)‖ ^ 2 := sorry

State before:
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
f : E → ℝ
m : ℝ
f' : E → E
xm x₀ : E
alg : Gradient_Descent_fix_stepsize f f' x₀
hm : m > 0
min : IsMinOn f univ xm
step₂ : Gradient_Descent_fix_stepsize.a f f' x₀ ≤ 2 / (m + ↑(Gradient_Descent_fix_stepsize.l f f' x₀))
hsc : StrongConvexOn univ m f
this✝¹ : LipschitzWith (Gradient_Descent_fix_stepsize.l f f' x₀) f'
this✝ : ↑(Gradient_Descent_fix_stepsize.l f f' x₀) > 0
k : ℕ
this :
  inner (Gradient_Descent_fix_stepsize.x f f' x₀ k - xm) (f' (Gradient_Descent_fix_stepsize.x f f' x₀ k)) ≥
    m * ↑(Gradient_Descent_fix_stepsize.l f f' x₀) / (m + ↑(Gradient_Descent_fix_stepsize.l f f' x₀)) *
        ‖Gradient_Descent_fix_stepsize.x f f' x₀ k - xm‖ ^ 2 +
      1 / (m + ↑(Gradient_Descent_fix_stepsize.l f f' x₀)) * ‖f' (Gradient_Descent_fix_stepsize.x f f' x₀ k)‖ ^ 2
⊢ Gradient_Descent_fix_stepsize.a f f' x₀ * (2 * m * ↑(Gradient_Descent_fix_stepsize.l f f' x₀)) *
        ‖Gradient_Descent_fix_stepsize.x f f' x₀ k - xm‖ ^ 2 +
      Gradient_Descent_fix_stepsize.a f f' x₀ * 2 * ‖f' (Gradient_Descent_fix_stepsize.x f f' x₀ k)‖ ^ 2 =
    2 * Gradient_Descent_fix_stepsize.a f f' x₀ *
      (m * ↑(Gradient_Descent_fix_stepsize.l f f' x₀) * ‖Gradient_Descent_fix_stepsize.x f f' x₀ k - xm‖ ^ 2 +
        ‖f' (Gradient_Descent_fix_stepsize.x f f' x₀ k)‖ ^ 2)

Tactic: rw [mul_add, mul_comm alg.a 2, ← mul_assoc, ← mul_assoc, mul_comm alg.a 2]

State after:
[E : Type u_1
 inst✝² : NormedAddCommGroup E
 inst✝¹ : InnerProductSpace ℝ E
 inst✝ : CompleteSpace E
 f : E → ℝ
 m : ℝ
 f' : E → E
 xm x₀ : E
 alg : Gradient_Descent_fix_stepsize f f' x₀
 hm : m > 0
 min : IsMinOn f univ xm
 step₂ : Gradient_Descent_fix_stepsize.a f f' x₀ ≤ 2 / (m + ↑(Gradient_Descent_fix_stepsize.l f f' x₀))
 hsc : StrongConvexOn univ m f
 this✝¹ : LipschitzWith (Gradient_Descent_fix_stepsize.l f f' x₀) f'
 this✝ : ↑(Gradient_Descent_fix_stepsize.l f f' x₀) > 0
 k : ℕ
 this :
   inner (Gradient_Descent_fix_stepsize.x f f' x₀ k - xm) (f' (Gradient_Descent_fix_stepsize.x f f' x₀ k)) ≥
     m * ↑(Gradient_Descent_fix_stepsize.l f f' x₀) / (m + ↑(Gradient_Descent_fix_stepsize.l f f' x₀)) *
         ‖Gradient_Descent_fix_stepsize.x f f' x₀ k - xm‖ ^ 2 +
       1 / (m + ↑(Gradient_Descent_fix_stepsize.l f f' x₀)) * ‖f' (Gradient_Descent_fix_stepsize.x f f' x₀ k)‖ ^ 2
 ⊢ 2 * Gradient_Descent_fix_stepsize.a f f' x₀ * m * ↑(Gradient_Descent_fix_stepsize.l f f' x₀) *
         ‖Gradient_Descent_fix_stepsize.x f f' x₀ k - xm‖ ^ 2 +
       2 * Gradient_Descent_fix_stepsize.a f f' x₀ * ‖f' (Gradient_Descent_fix_stepsize.x f f' x₀ k)‖ ^ 2 =
     2 * Gradient_Descent_fix_stepsize.a f f' x₀ *
         (m * ↑(Gradient_Descent_fix_stepsize.l f f' x₀) * ‖Gradient_Descent_fix_stepsize.x f f' x₀ k - xm‖ ^ 2) +
       2 * Gradient_Descent_fix_stepsize.a f f' x₀ * ‖f' (Gradient_Descent_fix_stepsize.x f f' x₀ k)‖ ^ 2]

Relevant lemmas:
mul_add : ∀ {R : Type v} [inst : Mul R] [inst_3 : Add R] [inst_4 : LeftDistribClass R] (a b c : R),
  a * (b + c) = a * b + a * c
mul_comm : ∀ {G : Type u_1} [inst : CommMagma G] (a b : G), a * b = b * a
mul_assoc : ∀ {G : Type u_1} [inst : Semigroup G] (a b c : G), a * b * c = a * (b * c)