import Mathlib
import Optlib


open Set Finset

open InnerProductSpace Set

theorem extracted_formal_statement_34.{u_1} {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : InnerProductSpace ℝ E]
  [inst_2 : CompleteSpace E] {f : E → ℝ} {f' : E → E} {xm x₀ : E} {alg : Gradient_Descent_fix_stepsize f f' x₀}
  (hfun : ConvexOn ℝ univ f)
  (step₂_1 : Gradient_Descent_fix_stepsize.a f f' x₀ ≤ 1 / ↑(Gradient_Descent_fix_stepsize.l f f' x₀))
  (step₂ : ↑(Gradient_Descent_fix_stepsize.l f f' x₀) ≤ 1 / Gradient_Descent_fix_stepsize.a f f' x₀)
  (this_1 : LipschitzWith (Gradient_Descent_fix_stepsize.l f f' x₀) f')
  (this : Gradient_Descent_fix_stepsize.l f f' x₀ > 0) (x : E)
  (t1 : 1 / (2 * Gradient_Descent_fix_stepsize.a f f' x₀) * (2 * Gradient_Descent_fix_stepsize.a f f' x₀) = 1)
  (t2₁ :
    ‖x - Gradient_Descent_fix_stepsize.a f f' x₀ • f' x - xm‖ ^ 2 =
      ‖x - xm‖ ^ 2 - 2 * Gradient_Descent_fix_stepsize.a f f' x₀ * ⟪f' x, x - xm⟫_ℝ +
        ‖Gradient_Descent_fix_stepsize.a f f' x₀ • f' x‖ ^ 2)
  (h_original_goal :
    ⟪f' x, x - xm⟫_ℝ +
        1 / (2 * Gradient_Descent_fix_stepsize.a f f' x₀) * -‖Gradient_Descent_fix_stepsize.a f f' x₀ • f' x‖ ^ 2 =
      ⟪f' x, x - xm⟫_ℝ -
        1 / (2 * Gradient_Descent_fix_stepsize.a f f' x₀) *
            (Gradient_Descent_fix_stepsize.a f f' x₀ * Gradient_Descent_fix_stepsize.a f f' x₀) *
          ‖f' x‖ ^ 2) :
  ⟪f' x, x - xm⟫_ℝ +
      1 / (2 * Gradient_Descent_fix_stepsize.a f f' x₀) * -(‖Gradient_Descent_fix_stepsize.a f f' x₀‖ * ‖f' x‖) ^ 2 =
    ⟪f' x, x - xm⟫_ℝ -
      1 / (2 * Gradient_Descent_fix_stepsize.a f f' x₀) *
          (Gradient_Descent_fix_stepsize.a f f' x₀ * Gradient_Descent_fix_stepsize.a f f' x₀) *
        ‖f' x‖ ^ 2 := sorry

Tactic state before:
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
f : E → ℝ
f' : E → E
xm x₀ : E
alg : Gradient_Descent_fix_stepsize f f' x₀
hfun : ConvexOn ℝ univ f
step₂✝ : Gradient_Descent_fix_stepsize.a f f' x₀ ≤ 1 / ↑(Gradient_Descent_fix_stepsize.l f f' x₀)
step₂ : ↑(Gradient_Descent_fix_stepsize.l f f' x₀) ≤ 1 / Gradient_Descent_fix_stepsize.a f f' x₀
k : ℕ
this✝ : LipschitzWith (Gradient_Descent_fix_stepsize.l f f' x₀) f'
this : Gradient_Descent_fix_stepsize.l f f' x₀ > 0
x : E
t1 : 1 / (2 * Gradient_Descent_fix_stepsize.a f f' x₀) * (2 * Gradient_Descent_fix_stepsize.a f f' x₀) = 1
t2₁ :
  ‖x - Gradient_Descent_fix_stepsize.a f f' x₀ • f' x - xm‖ ^ 2 =
    ‖x - xm‖ ^ 2 - 2 * Gradient_Descent_fix_stepsize.a f f' x₀ * ⟪f' x, x - xm⟫_ℝ +
      ‖Gradient_Descent_fix_stepsize.a f f' x₀ • f' x‖ ^ 2
⊢ ⟪f' x, x - xm⟫_ℝ +
      1 / (2 * Gradient_Descent_fix_stepsize.a f f' x₀) * -‖Gradient_Descent_fix_stepsize.a f f' x₀ • f' x‖ ^ 2 =
    ⟪f' x, x - xm⟫_ℝ -
      1 / (2 * Gradient_Descent_fix_stepsize.a f f' x₀) *
          (Gradient_Descent_fix_stepsize.a f f' x₀ * Gradient_Descent_fix_stepsize.a f f' x₀) *
        ‖f' x‖ ^ 2

Tactic:
rw [norm_smul _ _]

Tactic state after:
[E : Type u_1
 inst✝² : NormedAddCommGroup E
 inst✝¹ : InnerProductSpace ℝ E
 inst✝ : CompleteSpace E
 f : E → ℝ
 f' : E → E
 xm x₀ : E
 alg : Gradient_Descent_fix_stepsize f f' x₀
 hfun : ConvexOn ℝ univ f
 step₂✝ : Gradient_Descent_fix_stepsize.a f f' x₀ ≤ 1 / ↑(Gradient_Descent_fix_stepsize.l f f' x₀)
 step₂ : ↑(Gradient_Descent_fix_stepsize.l f f' x₀) ≤ 1 / Gradient_Descent_fix_stepsize.a f f' x₀
 k : ℕ
 this✝ : LipschitzWith (Gradient_Descent_fix_stepsize.l f f' x₀) f'
 this : Gradient_Descent_fix_stepsize.l f f' x₀ > 0
 x : E
 t1 : 1 / (2 * Gradient_Descent_fix_stepsize.a f f' x₀) * (2 * Gradient_Descent_fix_stepsize.a f f' x₀) = 1
 t2₁ :
   ‖x - Gradient_Descent_fix_stepsize.a f f' x₀ • f' x - xm‖ ^ 2 =
     ‖x - xm‖ ^ 2 - 2 * Gradient_Descent_fix_stepsize.a f f' x₀ * ⟪f' x, x - xm⟫_ℝ +
       ‖Gradient_Descent_fix_stepsize.a f f' x₀ • f' x‖ ^ 2
 ⊢ ⟪f' x, x - xm⟫_ℝ +
       1 / (2 * Gradient_Descent_fix_stepsize.a f f' x₀) * -(‖Gradient_Descent_fix_stepsize.a f f' x₀‖ * ‖f' x‖) ^ 2 =
     ⟪f' x, x - xm⟫_ℝ -
       1 / (2 * Gradient_Descent_fix_stepsize.a f f' x₀) *
           (Gradient_Descent_fix_stepsize.a f f' x₀ * Gradient_Descent_fix_stepsize.a f f' x₀) *
         ‖f' x‖ ^ 2]