theorem extracted_formal_statement_88.{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)
  (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)
  (this : 0 < m + ↑(Gradient_Descent_fix_stepsize.l f f' x₀)) :
  0 < 2 * 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✝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
this : 0 < m + ↑(Gradient_Descent_fix_stepsize.l f f' x₀)
⊢ 0 < 2 * m * ↑(Gradient_Descent_fix_stepsize.l f f' x₀)

Tactic: (apply mul_pos) <;> linarith

State after:

Relevant lemmas:
mul_pos : ∀ {α : Type u_3} {a b : α} [inst : MulZeroClass α] [inst_3 : Preorder α] [inst_4 : PosMulStrictMono α],
  0 < a → 0 < b → 0 < a * b