theorem extracted_formal_statement_103.{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 : ↑(Gradient_Descent_fix_stepsize.l f f' x₀) > 0)
  (reduction :
    ∀ (k : ℕ),
      ‖Gradient_Descent_fix_stepsize.x f f' x₀ (k + 1) - xm‖ ^ 2 ≤
        (1 -
            Gradient_Descent_fix_stepsize.a f f' x₀ *
              (2 * 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)
  (eq :
    0 ≤
      1 -
        Gradient_Descent_fix_stepsize.a f f' x₀ *
          (2 * m * ↑(Gradient_Descent_fix_stepsize.l f f' x₀) / (m + ↑(Gradient_Descent_fix_stepsize.l f f' x₀))))
  (q : ℕ)
  (IH1 :
    ‖Gradient_Descent_fix_stepsize.x f f' x₀ q - xm‖ ^ 2 ≤
      (1 -
            Gradient_Descent_fix_stepsize.a f f' x₀ *
              (2 * m * ↑(Gradient_Descent_fix_stepsize.l f f' x₀) / (m + ↑(Gradient_Descent_fix_stepsize.l f f' x₀)))) ^
          q *
        ‖x₀ - xm‖ ^ 2) :
  (1 -
        Gradient_Descent_fix_stepsize.a f f' x₀ *
          (2 * m * ↑(Gradient_Descent_fix_stepsize.l f f' x₀) / (m + ↑(Gradient_Descent_fix_stepsize.l f f' x₀)))) *
      (1 -
          Gradient_Descent_fix_stepsize.a f f' x₀ *
            (2 * m * ↑(Gradient_Descent_fix_stepsize.l f f' x₀) / (m + ↑(Gradient_Descent_fix_stepsize.l f f' x₀)))) ^
        q =
    (1 -
        Gradient_Descent_fix_stepsize.a f f' x₀ *
          (2 * m * ↑(Gradient_Descent_fix_stepsize.l f f' x₀) / (m + ↑(Gradient_Descent_fix_stepsize.l f f' x₀)))) ^
      (q + 1) := sorry

State before:
case h
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✝ : LipschitzWith (Gradient_Descent_fix_stepsize.l f f' x₀) f'
this : ↑(Gradient_Descent_fix_stepsize.l f f' x₀) > 0
reduction :
  ∀ (k : ℕ),
    ‖Gradient_Descent_fix_stepsize.x f f' x₀ (k + 1) - xm‖ ^ 2 ≤
      (1 -
          Gradient_Descent_fix_stepsize.a f f' x₀ *
            (2 * 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
eq :
  0 ≤
    1 -
      Gradient_Descent_fix_stepsize.a f f' x₀ *
        (2 * m * ↑(Gradient_Descent_fix_stepsize.l f f' x₀) / (m + ↑(Gradient_Descent_fix_stepsize.l f f' x₀)))
q : ℕ
IH1 :
  ‖Gradient_Descent_fix_stepsize.x f f' x₀ q - xm‖ ^ 2 ≤
    (1 -
          Gradient_Descent_fix_stepsize.a f f' x₀ *
            (2 * m * ↑(Gradient_Descent_fix_stepsize.l f f' x₀) / (m + ↑(Gradient_Descent_fix_stepsize.l f f' x₀)))) ^
        q *
      ‖x₀ - xm‖ ^ 2
⊢ (1 -
        Gradient_Descent_fix_stepsize.a f f' x₀ *
          (2 * m * ↑(Gradient_Descent_fix_stepsize.l f f' x₀) / (m + ↑(Gradient_Descent_fix_stepsize.l f f' x₀)))) *
      (1 -
          Gradient_Descent_fix_stepsize.a f f' x₀ *
            (2 * m * ↑(Gradient_Descent_fix_stepsize.l f f' x₀) / (m + ↑(Gradient_Descent_fix_stepsize.l f f' x₀)))) ^
        q =
    (1 -
        Gradient_Descent_fix_stepsize.a f f' x₀ *
          (2 * m * ↑(Gradient_Descent_fix_stepsize.l f f' x₀) / (m + ↑(Gradient_Descent_fix_stepsize.l f f' x₀)))) ^
      (q + 1)

Tactic: rw [pow_succ, pow_mul_comm']

State after:

Relevant lemmas:
pow_succ : ∀ {M : Type u_2} [inst : Monoid M] (a : M) (n : ℕ), a ^ (n + 1) = a ^ n * a
pow_mul_comm' : ∀ {M : Type u_2} [inst : Monoid M] (a : M) (n : ℕ), a ^ n * a = a * a ^ n