theorem extracted_formal_statement_100.{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)
  (h_original_goal :
    (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 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)))) *
        ((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)) :
  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))) := 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✝ : 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)))) *
      ‖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)))) *
      ((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)

Tactic: apply mul_le_mul_of_nonneg_left

State after:
[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
 ⊢ ‖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,
 case a0
 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
 ⊢ 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)))]

Relevant lemmas:
mul_le_mul_of_nonneg_left : ∀ {α : Type u_3} {a b c : α} [inst : Mul α] [inst_3 : Zero α] [inst_4 : Preorder α]
  [inst_5 : PosMulMono α], b ≤ c → 0 ≤ a → a * b ≤ a * c