phi_lt_onePointSixOneNine

formal source

  90  linarith
  91
  92/-- φ < 1.619. -/
  93theorem phi_lt_onePointSixOneNine : φ < (1.619 : ℝ) := by
  94  simp only [φ]
  95  have h5 : Real.sqrt 5 < (2.238 : ℝ) := by
  96    have h : (5 : ℝ) < (2.238 : ℝ)^2 := by norm_num
  97    rw [← Real.sqrt_sq (by norm_num : (0 : ℝ) ≤ 2.238)]
  98    exact Real.sqrt_lt_sqrt (by norm_num) h
  99  linarith
 100
 101/-- φ < 1.8. -/
 102theorem phi_lt_onePointEight : φ < (1.8 : ℝ) :=
 103  lt_trans phi_lt_onePointSixOneNine (by norm_num)
 104
 105/-- φ > 1.6. -/
 106theorem phi_gt_onePointSix : φ > (1.6 : ℝ) :=
 107  lt_trans (by norm_num) phi_gt_onePointSixOneEight
 108
 109/-- φ⁻¹ = φ - 1 (a key identity). -/
 110theorem phi_inv : φ⁻¹ = φ - 1 := by
 111  have hphi_ne : φ ≠ 0 := phi_pos.ne'
 112  have h := phi_equation
 113  -- From φ² = φ + 1, divide by φ: φ = 1 + 1/φ, so 1/φ = φ - 1
 114  have h1 : φ^2 / φ = (φ + 1) / φ := by rw [h]
 115  have h2 : φ = 1 + φ⁻¹ := by
 116    field_simp at h1
 117    field_simp
 118    nlinarith [phi_pos]
 119  linarith
 120
 121/-- J(φ) = (2φ - 1)/2 - 1 = φ - 3/2 (cost of the golden ratio).
 122    Note: J(φ) ≠ 0 because φ ≠ 1. -/
 123theorem J_phi : LawOfExistence.J φ = φ - 3/2 := by