-- File: optlib/Optlib/Algorithm/GD/GradientDescentStronglyConvex.lean
Theorem statement:
theorem extracted_formal_statement_78.{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 :
    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) :
  -0 ≤
    -(Gradient_Descent_fix_stepsize.a f f' x₀ - 2 / (m + ↑(Gradient_Descent_fix_stepsize.l f f' x₀))) *
      Gradient_Descent_fix_stepsize.a f f' x₀ := sorry

Tactic 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
⊢ 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

Tactic: rw [← neg_le_neg_iff, mul_comm, ← neg_mul]

Tactic state after:
[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 - 2 / (m + ↑(Gradient_Descent_fix_stepsize.l f f' x₀))) *
       Gradient_Descent_fix_stepsize.a f f' x₀]

Premises used:
neg_mul : ∀ {α : Type u} [inst : Mul α] [inst_3 : HasDistribNeg α] (a b : α), -a * b = -(a * b)
neg_le_neg_iff : ∀ {α : Type u} [inst : AddGroup α] [inst_3 : LE α] [inst_4 : AddLeftMono α] {a b : α}
  [inst_5 : AddRightMono α], -a ≤ -b ↔ b ≤ a
mul_comm : ∀ {G : Type u_1} [inst : CommMagma G] (a b : G), a * b = b * a