theorem extracted_formal_statement_81.{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)
  (h_original_goal :
    0 ≤
      (2 / (m + ↑(Gradient_Descent_fix_stepsize.l f f' x₀)) - Gradient_Descent_fix_stepsize.a f f' x₀) *
        Gradient_Descent_fix_stepsize.a f f' x₀) :
  0 ≤ Gradient_Descent_fix_stepsize.a 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
k : ℕ
⊢ 0 ≤
    (2 / (m + ↑(Gradient_Descent_fix_stepsize.l f f' x₀)) - Gradient_Descent_fix_stepsize.a f f' x₀) *
      Gradient_Descent_fix_stepsize.a f f' x₀

Tactic: apply mul_nonneg

State after:
[case ha
 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
 ⊢ 0 ≤ 2 / (m + ↑(Gradient_Descent_fix_stepsize.l f f' x₀)) - Gradient_Descent_fix_stepsize.a f f' x₀,
 case hb
 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
 ⊢ 0 ≤ Gradient_Descent_fix_stepsize.a f f' x₀]

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