theorem extracted_formal_statement_20.{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} {admm_kkt : Existance_of_kkt admm} [inst_9 : Setting E₁ E₂ F admm admm_kkt] (r_Φ : ℝ)
  (hΦ : range Φ ⊆ ball 0 r_Φ)
  (h1 :
    ∀ (k : ℕ+),
      (1 ⊓ (1 + 1 / ADMM.τ E₁ E₂ F - ADMM.τ E₁ E₂ F)) * ADMM.ρ E₁ E₂ F *
          ‖(OptProblem.A₁ E₂) (e₁ (↑k + 1)) + (OptProblem.A₂ E₁) (e₂ (↑k + 1))‖ ^ 2 ≤
        (ADMM.τ E₁ E₂ F ⊓ (1 + ADMM.τ E₁ E₂ F - ADMM.τ E₁ E₂ F ^ 2)) * ADMM.ρ E₁ E₂ F *
            ‖(OptProblem.A₂ E₁) (ADMM.x₂ E₁ F ↑k - ADMM.x₂ E₁ F (↑k + 1))‖ ^ 2 +
          (1 ⊓ (1 + 1 / ADMM.τ E₁ E₂ F - ADMM.τ E₁ E₂ F)) * ADMM.ρ E₁ E₂ F *
            ‖(OptProblem.A₁ E₂) (e₁ (↑k + 1)) + (OptProblem.A₂ E₁) (e₂ (↑k + 1))‖ ^ 2)
  (h2 :
    ∀ (k : ℕ+),
      (ADMM.τ E₁ E₂ F ⊓ (1 + ADMM.τ E₁ E₂ F - ADMM.τ E₁ E₂ F ^ 2)) * ADMM.ρ E₁ E₂ F *
            ‖(OptProblem.A₂ E₁) (ADMM.x₂ E₁ F ↑k - ADMM.x₂ E₁ F (↑k + 1))‖ ^ 2 +
          (1 ⊓ (1 + 1 / ADMM.τ E₁ E₂ F - ADMM.τ E₁ E₂ F)) * ADMM.ρ E₁ E₂ F *
            ‖(OptProblem.A₁ E₂) (e₁ (↑k + 1)) + (OptProblem.A₂ E₁) (e₂ (↑k + 1))‖ ^ 2 ≤
        Φ ↑k - Φ (↑k + 1))
  (h3 : ∀ (k : ℕ+), Φ ↑k - Φ (↑k + 1) < 2 * r_Φ)
  (h4 :
    ∀ (k : ℕ+),
      (1 ⊓ (1 + 1 / ADMM.τ E₁ E₂ F - ADMM.τ E₁ E₂ F)) * ADMM.ρ E₁ E₂ F *
          ‖(OptProblem.A₁ E₂) (e₁ (↑k + 1)) + (OptProblem.A₂ E₁) (e₂ (↑k + 1))‖ ^ 2 <
        2 * r_Φ)
  (k : ℕ+)
  (h4_ins :
    (1 ⊓ (1 + 1 / ADMM.τ E₁ E₂ F - ADMM.τ E₁ E₂ F)) * ADMM.ρ E₁ E₂ F *
        ‖(OptProblem.A₁ E₂) (e₁ (↑k + 1)) + (OptProblem.A₂ E₁) (e₂ (↑k + 1))‖ ^ 2 <
      2 * r_Φ)
  (h5a : 0 ≤ ‖(OptProblem.A₁ E₂) (e₁ (↑k + 1)) + (OptProblem.A₂ E₁) (e₂ (↑k + 1))‖)
  (h_original_goal :
    √(‖(OptProblem.A₁ E₂) (e₁ (↑k + 1)) + (OptProblem.A₂ E₁) (e₂ (↑k + 1))‖ ^ 2) <
      √(2 * r_Φ / ((1 ⊓ (1 + 1 / ADMM.τ E₁ E₂ F - ADMM.τ E₁ E₂ F)) * ADMM.ρ E₁ E₂ F))) :
  0 ≤ ‖(OptProblem.A₁ E₂) (e₁ (↑k + 1)) + (OptProblem.A₂ E₁) (e₂ (↑k + 1))‖ ^ 2 := sorry

State before:
E₁ : Type u_1
E₂ : Type u_2
F : Type u_3
inst✝⁹ : NormedAddCommGroup E₁
inst✝⁸ : InnerProductSpace ℝ E₁
inst✝⁷ : FiniteDimensional ℝ E₁
inst✝⁶ : NormedAddCommGroup E₂
inst✝⁵ : InnerProductSpace ℝ E₂
inst✝⁴ : FiniteDimensional ℝ E₂
inst✝³ : NormedAddCommGroup F
inst✝² : InnerProductSpace ℝ F
inst✝¹ : FiniteDimensional ℝ F
admm : ADMM E₁ E₂ F
admm_kkt : Existance_of_kkt admm
inst✝ : Setting E₁ E₂ F admm admm_kkt
r_Φ : ℝ
hΦ : range Φ ⊆ ball 0 r_Φ
h1 :
  ∀ (k : ℕ+),
    (1 ⊓ (1 + 1 / ADMM.τ E₁ E₂ F - ADMM.τ E₁ E₂ F)) * ADMM.ρ E₁ E₂ F *
        ‖(OptProblem.A₁ E₂) (e₁ (↑k + 1)) + (OptProblem.A₂ E₁) (e₂ (↑k + 1))‖ ^ 2 ≤
      (ADMM.τ E₁ E₂ F ⊓ (1 + ADMM.τ E₁ E₂ F - ADMM.τ E₁ E₂ F ^ 2)) * ADMM.ρ E₁ E₂ F *
          ‖(OptProblem.A₂ E₁) (ADMM.x₂ E₁ F ↑k - ADMM.x₂ E₁ F (↑k + 1))‖ ^ 2 +
        (1 ⊓ (1 + 1 / ADMM.τ E₁ E₂ F - ADMM.τ E₁ E₂ F)) * ADMM.ρ E₁ E₂ F *
          ‖(OptProblem.A₁ E₂) (e₁ (↑k + 1)) + (OptProblem.A₂ E₁) (e₂ (↑k + 1))‖ ^ 2
h2 :
  ∀ (k : ℕ+),
    (ADMM.τ E₁ E₂ F ⊓ (1 + ADMM.τ E₁ E₂ F - ADMM.τ E₁ E₂ F ^ 2)) * ADMM.ρ E₁ E₂ F *
          ‖(OptProblem.A₂ E₁) (ADMM.x₂ E₁ F ↑k - ADMM.x₂ E₁ F (↑k + 1))‖ ^ 2 +
        (1 ⊓ (1 + 1 / ADMM.τ E₁ E₂ F - ADMM.τ E₁ E₂ F)) * ADMM.ρ E₁ E₂ F *
          ‖(OptProblem.A₁ E₂) (e₁ (↑k + 1)) + (OptProblem.A₂ E₁) (e₂ (↑k + 1))‖ ^ 2 ≤
      Φ ↑k - Φ (↑k + 1)
h3 : ∀ (k : ℕ+), Φ ↑k - Φ (↑k + 1) < 2 * r_Φ
h4 :
  ∀ (k : ℕ+),
    (1 ⊓ (1 + 1 / ADMM.τ E₁ E₂ F - ADMM.τ E₁ E₂ F)) * ADMM.ρ E₁ E₂ F *
        ‖(OptProblem.A₁ E₂) (e₁ (↑k + 1)) + (OptProblem.A₂ E₁) (e₂ (↑k + 1))‖ ^ 2 <
      2 * r_Φ
k : ℕ+
h4_ins :
  (1 ⊓ (1 + 1 / ADMM.τ E₁ E₂ F - ADMM.τ E₁ E₂ F)) * ADMM.ρ E₁ E₂ F *
      ‖(OptProblem.A₁ E₂) (e₁ (↑k + 1)) + (OptProblem.A₂ E₁) (e₂ (↑k + 1))‖ ^ 2 <
    2 * r_Φ
h5a : 0 ≤ ‖(OptProblem.A₁ E₂) (e₁ (↑k + 1)) + (OptProblem.A₂ E₁) (e₂ (↑k + 1))‖
⊢ √(‖(OptProblem.A₁ E₂) (e₁ (↑k + 1)) + (OptProblem.A₂ E₁) (e₂ (↑k + 1))‖ ^ 2) <
    √(2 * r_Φ / ((1 ⊓ (1 + 1 / ADMM.τ E₁ E₂ F - ADMM.τ E₁ E₂ F)) * ADMM.ρ E₁ E₂ F))

Tactic: rw [← lt_div_iff₀' ineq2] at h4_ins
    apply Real.sqrt_lt_sqrt at h4_ins
    exact h4_ins

State after:
[case hx
 E₁ : Type u_1
 E₂ : Type u_2
 F : Type u_3
 inst✝⁹ : NormedAddCommGroup E₁
 inst✝⁸ : InnerProductSpace ℝ E₁
 inst✝⁷ : FiniteDimensional ℝ E₁
 inst✝⁶ : NormedAddCommGroup E₂
 inst✝⁵ : InnerProductSpace ℝ E₂
 inst✝⁴ : FiniteDimensional ℝ E₂
 inst✝³ : NormedAddCommGroup F
 inst✝² : InnerProductSpace ℝ F
 inst✝¹ : FiniteDimensional ℝ F
 admm : ADMM E₁ E₂ F
 admm_kkt : Existance_of_kkt admm
 inst✝ : Setting E₁ E₂ F admm admm_kkt
 r_Φ : ℝ
 hΦ : range Φ ⊆ ball 0 r_Φ
 h1 :
   ∀ (k : ℕ+),
     (1 ⊓ (1 + 1 / ADMM.τ E₁ E₂ F - ADMM.τ E₁ E₂ F)) * ADMM.ρ E₁ E₂ F *
         ‖(OptProblem.A₁ E₂) (e₁ (↑k + 1)) + (OptProblem.A₂ E₁) (e₂ (↑k + 1))‖ ^ 2 ≤
       (ADMM.τ E₁ E₂ F ⊓ (1 + ADMM.τ E₁ E₂ F - ADMM.τ E₁ E₂ F ^ 2)) * ADMM.ρ E₁ E₂ F *
           ‖(OptProblem.A₂ E₁) (ADMM.x₂ E₁ F ↑k - ADMM.x₂ E₁ F (↑k + 1))‖ ^ 2 +
         (1 ⊓ (1 + 1 / ADMM.τ E₁ E₂ F - ADMM.τ E₁ E₂ F)) * ADMM.ρ E₁ E₂ F *
           ‖(OptProblem.A₁ E₂) (e₁ (↑k + 1)) + (OptProblem.A₂ E₁) (e₂ (↑k + 1))‖ ^ 2
 h2 :
   ∀ (k : ℕ+),
     (ADMM.τ E₁ E₂ F ⊓ (1 + ADMM.τ E₁ E₂ F - ADMM.τ E₁ E₂ F ^ 2)) * ADMM.ρ E₁ E₂ F *
           ‖(OptProblem.A₂ E₁) (ADMM.x₂ E₁ F ↑k - ADMM.x₂ E₁ F (↑k + 1))‖ ^ 2 +
         (1 ⊓ (1 + 1 / ADMM.τ E₁ E₂ F - ADMM.τ E₁ E₂ F)) * ADMM.ρ E₁ E₂ F *
           ‖(OptProblem.A₁ E₂) (e₁ (↑k + 1)) + (OptProblem.A₂ E₁) (e₂ (↑k + 1))‖ ^ 2 ≤
       Φ ↑k - Φ (↑k + 1)
 h3 : ∀ (k : ℕ+), Φ ↑k - Φ (↑k + 1) < 2 * r_Φ
 h4 :
   ∀ (k : ℕ+),
     (1 ⊓ (1 + 1 / ADMM.τ E₁ E₂ F - ADMM.τ E₁ E₂ F)) * ADMM.ρ E₁ E₂ F *
         ‖(OptProblem.A₁ E₂) (e₁ (↑k + 1)) + (OptProblem.A₂ E₁) (e₂ (↑k + 1))‖ ^ 2 <
       2 * r_Φ
 k : ℕ+
 h4_ins :
   ‖(OptProblem.A₁ E₂) (e₁ (↑k + 1)) + (OptProblem.A₂ E₁) (e₂ (↑k + 1))‖ ^ 2 <
     2 * r_Φ / ((1 ⊓ (1 + 1 / ADMM.τ E₁ E₂ F - ADMM.τ E₁ E₂ F)) * ADMM.ρ E₁ E₂ F)
 h5a : 0 ≤ ‖(OptProblem.A₁ E₂) (e₁ (↑k + 1)) + (OptProblem.A₂ E₁) (e₂ (↑k + 1))‖
 ⊢ 0 ≤ ‖(OptProblem.A₁ E₂) (e₁ (↑k + 1)) + (OptProblem.A₂ E₁) (e₂ (↑k + 1))‖ ^ 2]

Relevant lemmas:
lt_div_iff₀' : ∀ {G₀ : Type u_2} [inst : CommGroupWithZero G₀] [inst_10 : PartialOrder G₀] [inst_11 : ZeroLEOneClass G₀]
  [inst_12 : PosMulReflectLT G₀] [inst_13 : PosMulStrictMono G₀] {a b c : G₀}, 0 < c → (a < b / c ↔ c * a < b)
Real.sqrt_lt_sqrt : ∀ {x y : ℝ}, 0 ≤ x → x < y → √x < √y