import Mathlib
import Optlib


open Set Finset

open InnerProductSpace Set

theorem extracted_formal_statement_52.{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 : Gradient_Descent_fix_stepsize.a f f' x₀ ≤ 1 / ↑(Gradient_Descent_fix_stepsize.l f f' x₀)) (k : ℕ)
  (step1₂ : ↑(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)
  (tα : 1 / (2 * (↑k + 1) * Gradient_Descent_fix_stepsize.a f f' x₀) > 0)
  (xdescent :
    ∀ (k : ℕ),
      f (Gradient_Descent_fix_stepsize.x f f' x₀ (k + 1)) ≤
        f xm +
          1 / (2 * Gradient_Descent_fix_stepsize.a f f' x₀) *
            (‖Gradient_Descent_fix_stepsize.x f f' x₀ k - xm‖ ^ 2 -
              ‖Gradient_Descent_fix_stepsize.x f f' x₀ (k + 1) - xm‖ ^ 2))
  (mono :
    ∀ (k : ℕ), f (Gradient_Descent_fix_stepsize.x f f' x₀ (k + 1)) ≤ f (Gradient_Descent_fix_stepsize.x f f' x₀ k))
  (sum_prop_2 :
    ∀ (n : ℕ),
      ∑ k ∈ Finset.range (n + 1), (f (Gradient_Descent_fix_stepsize.x f f' x₀ (k + 1)) - f xm) ≤
        1 / (2 * Gradient_Descent_fix_stepsize.a f f' x₀) *
          (‖x₀ - xm‖ ^ 2 - ‖Gradient_Descent_fix_stepsize.x f f' x₀ (n + 1) - xm‖ ^ 2))
  (sum_prop_1 :
    f (Gradient_Descent_fix_stepsize.x f f' x₀ (k + 1)) - f xm ≤
      (∑ k ∈ Finset.range (k + 1), (f (Gradient_Descent_fix_stepsize.x f f' x₀ (k + 1)) - f xm)) / (↑k + 1))
  (sum_prop :
    ∑ k ∈ Finset.range (k + 1), (f (Gradient_Descent_fix_stepsize.x f f' x₀ (k + 1)) - f xm) ≤
      1 / (2 * Gradient_Descent_fix_stepsize.a f f' x₀) *
        (‖x₀ - xm‖ ^ 2 - ‖Gradient_Descent_fix_stepsize.x f f' x₀ (k + 1) - xm‖ ^ 2))
  (h :
    f (Gradient_Descent_fix_stepsize.x f f' x₀ (k + 1)) - f xm ≤
      1 / (2 * (↑k + 1) * Gradient_Descent_fix_stepsize.a f f' x₀) *
        (‖x₀ - xm‖ ^ 2 - ‖Gradient_Descent_fix_stepsize.x f f' x₀ (k + 1) - xm‖ ^ 2)) :
  f (Gradient_Descent_fix_stepsize.x f f' x₀ (k + 1)) - f xm ≤
    1 / (2 * (↑k + 1) * Gradient_Descent_fix_stepsize.a f f' x₀) * ‖x₀ - xm‖ ^ 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)
k : ℕ
step1₂ : ↑(Gradient_Descent_fix_stepsize.l f f' x₀) ≤ 1 / Gradient_Descent_fix_stepsize.a f f' x₀
this✝ : LipschitzWith (Gradient_Descent_fix_stepsize.l f f' x₀) f'
this : Gradient_Descent_fix_stepsize.l f f' x₀ > 0
tα : 1 / (2 * (↑k + 1) * Gradient_Descent_fix_stepsize.a f f' x₀) > 0
xdescent :
  ∀ (k : ℕ),
    f (Gradient_Descent_fix_stepsize.x f f' x₀ (k + 1)) ≤
      f xm +
        1 / (2 * Gradient_Descent_fix_stepsize.a f f' x₀) *
          (‖Gradient_Descent_fix_stepsize.x f f' x₀ k - xm‖ ^ 2 -
            ‖Gradient_Descent_fix_stepsize.x f f' x₀ (k + 1) - xm‖ ^ 2)
mono : ∀ (k : ℕ), f (Gradient_Descent_fix_stepsize.x f f' x₀ (k + 1)) ≤ f (Gradient_Descent_fix_stepsize.x f f' x₀ k)
sum_prop✝ :
  ∀ (n : ℕ),
    ∑ k ∈ Finset.range (n + 1), (f (Gradient_Descent_fix_stepsize.x f f' x₀ (k + 1)) - f xm) ≤
      1 / (2 * Gradient_Descent_fix_stepsize.a f f' x₀) *
        (‖x₀ - xm‖ ^ 2 - ‖Gradient_Descent_fix_stepsize.x f f' x₀ (n + 1) - xm‖ ^ 2)
sum_prop_1 :
  f (Gradient_Descent_fix_stepsize.x f f' x₀ (k + 1)) - f xm ≤
    (∑ k ∈ Finset.range (k + 1), (f (Gradient_Descent_fix_stepsize.x f f' x₀ (k + 1)) - f xm)) / (↑k + 1)
sum_prop :
  ∑ k ∈ Finset.range (k + 1), (f (Gradient_Descent_fix_stepsize.x f f' x₀ (k + 1)) - f xm) ≤
    1 / (2 * Gradient_Descent_fix_stepsize.a f f' x₀) *
      (‖x₀ - xm‖ ^ 2 - ‖Gradient_Descent_fix_stepsize.x f f' x₀ (k + 1) - xm‖ ^ 2)
h :
  f (Gradient_Descent_fix_stepsize.x f f' x₀ (k + 1)) - f xm ≤
    1 / (2 * (↑k + 1) * Gradient_Descent_fix_stepsize.a f f' x₀) *
      (‖x₀ - xm‖ ^ 2 - ‖Gradient_Descent_fix_stepsize.x f f' x₀ (k + 1) - xm‖ ^ 2)
⊢ f (Gradient_Descent_fix_stepsize.x f f' x₀ (k + 1)) - f xm ≤
    1 / (2 * (↑k + 1) * Gradient_Descent_fix_stepsize.a f f' x₀) * ‖x₀ - xm‖ ^ 2

Tactic:
exact le_mul_of_le_mul_of_nonneg_left h (by simp) (by linarith)

Tactic state after: