theorem extracted_formal_statement_3 : 1 / 2 < √5 / 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
h1 : 0 < ADMM.τ E₁ E₂ F
right✝ : ADMM.τ E₁ E₂ F < (1 + √5) / 2
h0 : 1 + ADMM.τ E₁ E₂ F - ADMM.τ E₁ E₂ F ^ 2 = -(ADMM.τ E₁ E₂ F - 1 / 2) ^ 2 + 5 / 4
h2 : 5 / 4 = |√5 / 2| ^ 2
θ : ℝ := ADMM.τ E₁ E₂ F - 2⁻¹
g1 : (1 + √5) / 2 - 1 / 2 = √5 / 2
h8 : θ < √5 / 2
⊢ 1 / 2 < √5 / 2

Tactic: apply div_lt_div_of_pos_right _ zero_lt_two
    apply (lt_sqrt _).mpr _
    repeat norm_num

State after:

Relevant lemmas:
zero_lt_two : ∀ {α : Type u_1} [inst : AddMonoidWithOne α] [inst_1 : PartialOrder α] [inst_2 : ZeroLEOneClass α]
  [inst_3 : NeZero 1] [inst_4 : AddLeftMono α], 0 < 2
Real.lt_sqrt : ∀ {x y : ℝ}, 0 ≤ x → (x < √y ↔ x ^ 2 < y)
div_lt_div_of_pos_right : ∀ {G₀ : Type u_2} [inst : GroupWithZero G₀] [inst_1 : PartialOrder G₀]
  [inst_2 : ZeroLEOneClass G₀] [inst_3 : PosMulReflectLT G₀] {a b c : G₀} [inst_4 : MulPosStrictMono G₀],
  a < b → 0 < c → a / c < b / c