theorem extracted_formal_statement_19.{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) (k : ℕ) :
  f' (Gradient_Descent_fix_stepsize.x f f' x₀ k) = f' (Gradient_Descent_fix_stepsize.x f f' x₀ k) - f' xm := 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✝ : LipschitzWith (Gradient_Descent_fix_stepsize.l f f' x₀) f'
this : ↑(Gradient_Descent_fix_stepsize.l f f' x₀) > 0
k : ℕ
⊢ f' (Gradient_Descent_fix_stepsize.x f f' x₀ k) = f' (Gradient_Descent_fix_stepsize.x f f' x₀ k) - f' xm

Tactic: apply eq_sub_of_add_eq
    simp
    apply lipschitz_derivxm_eq_zero alg.diff alg.smooth min this

State after:

Relevant lemmas:
eq_sub_of_add_eq : ∀ {G : Type u_3} [inst : AddGroup G] {a b c : G}, a + c = b → a = b - c
lipschitz_derivxm_eq_zero : ∀ {E : Type u_1} [inst : NormedAddCommGroup E] [inst_3 : InnerProductSpace ℝ E]
  [inst_4 : CompleteSpace E] {f : E → ℝ} {f' : E → E} {xm : E} {l : NNReal},
  (∀ (x : E), HasGradientAt f (f' x) x) → LipschitzWith l f' → IsMinOn f univ xm → ↑l > 0 → f' xm = 0