phi_plus_inverse_eq_sqrt5

formal source

 173  nlinarith
 174
 175/-- **CALCULATED**: φ + 1/φ = √5. -/
 176theorem phi_plus_inverse_eq_sqrt5 : phi + 1/phi = Real.sqrt 5 := by
 177  rw [phi_inverse_formula]
 178  have h1 : phi^2 = phi + 1 := phi_sq_eq
 179  have h2 : (2 * phi - 1)^2 = 5 := by
 180    nlinarith
 181  have h3 : 2 * phi - 1 > 0 := by
 182    have h4 : phi > 1.5 := phi_gt_onePointFive
 183    linarith
 184  have h4 : Real.sqrt ((2 * phi - 1)^2) = Real.sqrt 5 := by
 185    rw [h2]
 186  have h5 : Real.sqrt ((2 * phi - 1)^2) = 2 * phi - 1 := by
 187    apply Real.sqrt_sq
 188    linarith
 189  nlinarith
 190
 191/-- **CALCULATED**: φ² > 2.5. -/
 192theorem phi_sq_gt_2_5 : phi^2 > (2.5 : ℝ) := by
 193  have h1 : phi^2 = phi + 1 := phi_sq_eq
 194  rw [h1]
 195  have h2 : phi > 1.5 := phi_gt_onePointFive
 196  nlinarith
 197
 198/-- **CALCULATED**: φ² < 2.7. -/
 199theorem phi_sq_lt_2_7 : phi^2 < (2.7 : ℝ) := by
 200  have h1 : phi^2 = phi + 1 := phi_sq_eq
 201  rw [h1]
 202  have h2 : phi < 1.62 := phi_lt_onePointSixTwo
 203  nlinarith
 204
 205/-! ## Section: Certificate -/
 206