-- File: optlib/Optlib/Algorithm/ADMM/Theroem_converge.lean
Theorem statement:
theorem extracted_formal_statement_197.{u_3, u_2, u_1} {E₁ : Type u_1} {E₂ : Type u_2} {F : Type u_3}
  [inst : NormedAddCommGroup E₁] [inst_1 : InnerProductSpace ℝ E₁] [inst_2 : FiniteDimensional ℝ E₁]
  [inst_3 : NormedAddCommGroup E₂] [inst_4 : InnerProductSpace ℝ E₂] [inst_5 : FiniteDimensional ℝ E₂]
  [inst_6 : NormedAddCommGroup F] [inst_7 : InnerProductSpace ℝ F] [inst_8 : FiniteDimensional ℝ F]
  {admm : ADMM E₁ E₂ F}
  (h_original_goal :
    (range fun n => (ADMM.x₁ E₂ F n, ADMM.x₂ E₁ F n, ADMM.y E₁ E₂ n)) ⊆
      range (ADMM.x₁ E₂ F) ×ˢ range (ADMM.x₂ E₁ F) ×ˢ range (ADMM.y E₁ E₂)) :
  {x | ∃ «y», (ADMM.x₁ E₂ F «y», ADMM.x₂ E₁ F «y», ADMM.y E₁ E₂ «y») = x} ⊆
    {x | ∃ «y», ADMM.x₁ E₂ F «y» = x} ×ˢ {x | ∃ «y», ADMM.x₂ E₁ F «y» = x} ×ˢ {x | ∃ «y», ADMM.y E₁ E₂ «y» = x} := sorry

Tactic state before:
E₁ : Type u_1
E₂ : Type u_2
F : Type u_3
inst✝⁸ : NormedAddCommGroup E₁
inst✝⁷ : InnerProductSpace ℝ E₁
inst✝⁶ : FiniteDimensional ℝ E₁
inst✝⁵ : NormedAddCommGroup E₂
inst✝⁴ : InnerProductSpace ℝ E₂
inst✝³ : FiniteDimensional ℝ E₂
inst✝² : NormedAddCommGroup F
inst✝¹ : InnerProductSpace ℝ F
inst✝ : FiniteDimensional ℝ F
admm : ADMM E₁ E₂ F
⊢ (range fun n => (ADMM.x₁ E₂ F n, ADMM.x₂ E₁ F n, ADMM.y E₁ E₂ n)) ⊆
    range (ADMM.x₁ E₂ F) ×ˢ range (ADMM.x₂ E₁ F) ×ˢ range (ADMM.y E₁ E₂)

Tactic: simp [range]

Tactic state after:
[E₁ : Type u_1
 E₂ : Type u_2
 F : Type u_3
 inst✝⁸ : NormedAddCommGroup E₁
 inst✝⁷ : InnerProductSpace ℝ E₁
 inst✝⁶ : FiniteDimensional ℝ E₁
 inst✝⁵ : NormedAddCommGroup E₂
 inst✝⁴ : InnerProductSpace ℝ E₂
 inst✝³ : FiniteDimensional ℝ E₂
 inst✝² : NormedAddCommGroup F
 inst✝¹ : InnerProductSpace ℝ F
 inst✝ : FiniteDimensional ℝ F
 admm : ADMM E₁ E₂ F
 ⊢ {x | ∃ «y», (ADMM.x₁ E₂ F «y», ADMM.x₂ E₁ F «y», ADMM.y E₁ E₂ «y») = x} ⊆
     {x | ∃ «y», ADMM.x₁ E₂ F «y» = x} ×ˢ {x | ∃ «y», ADMM.x₂ E₁ F «y» = x} ×ˢ {x | ∃ «y», ADMM.y E₁ E₂ «y» = x}]

Premises used: