theorem extracted_formal_statement_83.{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) (k : ℕ)
  (eq2 :
    Gradient_Descent_fix_stepsize.a f f' x₀ *
        (Gradient_Descent_fix_stepsize.a f f' x₀ - 2 / (m + ↑(Gradient_Descent_fix_stepsize.l f f' x₀))) ≤
      0)
  (eq3 : 0 ≤ ‖f' (Gradient_Descent_fix_stepsize.x f f' x₀ k)‖ ^ 2) :
  Gradient_Descent_fix_stepsize.a f f' x₀ *
        (Gradient_Descent_fix_stepsize.a f f' x₀ - 2 / (m + ↑(Gradient_Descent_fix_stepsize.l f f' x₀))) *
      ‖f' (Gradient_Descent_fix_stepsize.x f f' x₀ k)‖ ^ 2 ≤
    0 := 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
k : ℕ
eq2 :
  Gradient_Descent_fix_stepsize.a f f' x₀ *
      (Gradient_Descent_fix_stepsize.a f f' x₀ - 2 / (m + ↑(Gradient_Descent_fix_stepsize.l f f' x₀))) ≤
    0
eq3 : 0 ≤ ‖f' (Gradient_Descent_fix_stepsize.x f f' x₀ k)‖ ^ 2
⊢ Gradient_Descent_fix_stepsize.a f f' x₀ *
        (Gradient_Descent_fix_stepsize.a f f' x₀ - 2 / (m + ↑(Gradient_Descent_fix_stepsize.l f f' x₀))) *
      ‖f' (Gradient_Descent_fix_stepsize.x f f' x₀ k)‖ ^ 2 ≤
    0

Tactic: apply mul_nonpos_of_nonpos_of_nonneg eq2 eq3

State after:

Relevant lemmas:
mul_nonpos_of_nonpos_of_nonneg : ∀ {α : Type u_3} {a b : α} [inst : MulZeroClass α] [inst_3 : Preorder α]
  [inst_4 : MulPosMono α], a ≤ 0 → 0 ≤ b → a * b ≤ 0