theorem extracted_formal_statement_69.{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) (k : ℕ)
  (this :
    inner (Gradient_Descent_fix_stepsize.x f f' x₀ k - xm) (f' (Gradient_Descent_fix_stepsize.x f f' x₀ k)) ≥
      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 +
        1 / (m + ↑(Gradient_Descent_fix_stepsize.l f f' x₀)) * ‖f' (Gradient_Descent_fix_stepsize.x f f' x₀ k)‖ ^ 2)
  (h_original_goal :
    ‖Gradient_Descent_fix_stepsize.x f f' x₀ k - xm‖ ^ 2 -
          2 * Gradient_Descent_fix_stepsize.a f f' x₀ *
            inner (Gradient_Descent_fix_stepsize.x f f' x₀ k - xm) (f' (Gradient_Descent_fix_stepsize.x f f' x₀ k)) +
        Gradient_Descent_fix_stepsize.a f f' x₀ ^ 2 * ‖f' (Gradient_Descent_fix_stepsize.x f f' x₀ k)‖ ^ 2 ≤
      ‖Gradient_Descent_fix_stepsize.x f f' x₀ k - xm‖ ^ 2 -
            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 -
          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 +
        Gradient_Descent_fix_stepsize.a f f' x₀ ^ 2 * ‖f' (Gradient_Descent_fix_stepsize.x f f' x₀ k)‖ ^ 2) :
  ‖Gradient_Descent_fix_stepsize.x f f' x₀ k - xm‖ ^ 2 -
      2 * Gradient_Descent_fix_stepsize.a f f' x₀ *
        inner (Gradient_Descent_fix_stepsize.x f f' x₀ k - xm) (f' (Gradient_Descent_fix_stepsize.x f f' x₀ k)) ≤
    ‖Gradient_Descent_fix_stepsize.x f f' x₀ k - xm‖ ^ 2 -
        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 -
      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 := 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
k : ℕ
this :
  inner (Gradient_Descent_fix_stepsize.x f f' x₀ k - xm) (f' (Gradient_Descent_fix_stepsize.x f f' x₀ k)) ≥
    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 +
      1 / (m + ↑(Gradient_Descent_fix_stepsize.l f f' x₀)) * ‖f' (Gradient_Descent_fix_stepsize.x f f' x₀ k)‖ ^ 2
⊢ ‖Gradient_Descent_fix_stepsize.x f f' x₀ k - xm‖ ^ 2 -
        2 * Gradient_Descent_fix_stepsize.a f f' x₀ *
          inner (Gradient_Descent_fix_stepsize.x f f' x₀ k - xm) (f' (Gradient_Descent_fix_stepsize.x f f' x₀ k)) +
      Gradient_Descent_fix_stepsize.a f f' x₀ ^ 2 * ‖f' (Gradient_Descent_fix_stepsize.x f f' x₀ k)‖ ^ 2 ≤
    ‖Gradient_Descent_fix_stepsize.x f f' x₀ k - xm‖ ^ 2 -
          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 -
        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 +
      Gradient_Descent_fix_stepsize.a f f' x₀ ^ 2 * ‖f' (Gradient_Descent_fix_stepsize.x f f' x₀ k)‖ ^ 2

Tactic: apply add_le_add_right

State after:
[case bc
 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
 k : ℕ
 this :
   inner (Gradient_Descent_fix_stepsize.x f f' x₀ k - xm) (f' (Gradient_Descent_fix_stepsize.x f f' x₀ k)) ≥
     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 +
       1 / (m + ↑(Gradient_Descent_fix_stepsize.l f f' x₀)) * ‖f' (Gradient_Descent_fix_stepsize.x f f' x₀ k)‖ ^ 2
 ⊢ ‖Gradient_Descent_fix_stepsize.x f f' x₀ k - xm‖ ^ 2 -
       2 * Gradient_Descent_fix_stepsize.a f f' x₀ *
         inner (Gradient_Descent_fix_stepsize.x f f' x₀ k - xm) (f' (Gradient_Descent_fix_stepsize.x f f' x₀ k)) ≤
     ‖Gradient_Descent_fix_stepsize.x f f' x₀ k - xm‖ ^ 2 -
         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 -
       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]

Relevant lemmas:
add_le_add_right : ∀ {α : Type u_1} [inst : Add α] [inst_3 : LE α] [i : AddRightMono α] {b c : α},
  b ≤ c → ∀ (a : α), b + a ≤ c + a