theorem extracted_formal_statement_66.{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)
  (eq1 : f' (Gradient_Descent_fix_stepsize.x f f' x₀ k) = f' (Gradient_Descent_fix_stepsize.x f f' x₀ k) - f' xm)
  (h_original_goal :
    inner (f' (Gradient_Descent_fix_stepsize.x f f' x₀ k) - f' xm) (Gradient_Descent_fix_stepsize.x f f' x₀ k - xm) ≥
      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) - f' xm‖ ^ 2) :
  ↑(Gradient_Descent_fix_stepsize.l f f' x₀) > 0 := 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
eq1 : f' (Gradient_Descent_fix_stepsize.x f f' x₀ k) = f' (Gradient_Descent_fix_stepsize.x f f' x₀ k) - f' xm
⊢ inner (f' (Gradient_Descent_fix_stepsize.x f f' x₀ k) - f' xm) (Gradient_Descent_fix_stepsize.x f f' x₀ k - xm) ≥
    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) - f' xm‖ ^ 2

Tactic: apply Strong_convex_Lipschitz_smooth

State after:
[case hsc
 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
 eq1 : f' (Gradient_Descent_fix_stepsize.x f f' x₀ k) = f' (Gradient_Descent_fix_stepsize.x f f' x₀ k) - f' xm
 ⊢ StrongConvexOn univ m ?f,
 case mp
 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
 eq1 : f' (Gradient_Descent_fix_stepsize.x f f' x₀ k) = f' (Gradient_Descent_fix_stepsize.x f f' x₀ k) - f' xm
 ⊢ m > 0,
 case hf
 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
 eq1 : f' (Gradient_Descent_fix_stepsize.x f f' x₀ k) = f' (Gradient_Descent_fix_stepsize.x f f' x₀ k) - f' xm
 ⊢ ∀ (x : E), HasGradientAt ?f (f' x) x,
 case h₂
 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
 eq1 : f' (Gradient_Descent_fix_stepsize.x f f' x₀ k) = f' (Gradient_Descent_fix_stepsize.x f f' x₀ k) - f' xm
 ⊢ LipschitzWith (Gradient_Descent_fix_stepsize.l f f' x₀) f',
 case hl
 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
 eq1 : f' (Gradient_Descent_fix_stepsize.x f f' x₀ k) = f' (Gradient_Descent_fix_stepsize.x f f' x₀ k) - f' xm
 ⊢ ↑(Gradient_Descent_fix_stepsize.l f f' x₀) > 0,
 case f
 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
 eq1 : f' (Gradient_Descent_fix_stepsize.x f f' x₀ k) = f' (Gradient_Descent_fix_stepsize.x f f' x₀ k) - f' xm
 ⊢ E → ℝ]

Relevant lemmas:
Strong_convex_Lipschitz_smooth : ∀ {E : Type u_1} [inst : NormedAddCommGroup E] [inst_3 : InnerProductSpace ℝ E]
  [inst_4 : CompleteSpace E] {f : E → ℝ} {m : ℝ} {f' : E → E} {x y : E} {l : NNReal},
  StrongConvexOn univ m f →
    m > 0 →
      (∀ (x : E), HasGradientAt f (f' x) x) →
        LipschitzWith l f' →
          ↑l > 0 → inner (f' x - f' y) (x - y) ≥ m * ↑l / (m + ↑l) * ‖x - y‖ ^ 2 + 1 / (m + ↑l) * ‖f' x - f' y‖ ^ 2