phi_eighth_eq

formal source

 182    _ = 13 * phi + 8 := by ring
 183
 184/-- Key identity: φ⁸ = 21φ + 13 (Fibonacci recurrence). -/
 185lemma phi_eighth_eq : phi^8 = 21 * phi + 13 := by
 186  calc phi^8 = phi * phi^7 := by ring
 187    _ = phi * (13 * phi + 8) := by rw [phi_seventh_eq]
 188    _ = 13 * phi^2 + 8 * phi := by ring
 189    _ = 13 * (phi + 1) + 8 * phi := by rw [phi_sq_eq]
 190    _ = 21 * phi + 13 := by ring
 191
 192/-- Key identity: φ⁹ = 34φ + 21 (Fibonacci recurrence). -/
 193lemma phi_ninth_eq : phi^9 = 34 * phi + 21 := by
 194  calc phi^9 = phi * phi^8 := by ring
 195    _ = phi * (21 * phi + 13) := by rw [phi_eighth_eq]
 196    _ = 21 * phi^2 + 13 * phi := by ring
 197    _ = 21 * (phi + 1) + 13 * phi := by rw [phi_sq_eq]
 198    _ = 34 * phi + 21 := by ring
 199
 200/-- Key identity: φ¹⁰ = 55φ + 34 (Fibonacci recurrence). -/
 201lemma phi_tenth_eq : phi^10 = 55 * phi + 34 := by
 202  calc phi^10 = phi * phi^9 := by ring
 203    _ = phi * (34 * phi + 21) := by rw [phi_ninth_eq]
 204    _ = 34 * phi^2 + 21 * phi := by ring
 205    _ = 34 * (phi + 1) + 21 * phi := by rw [phi_sq_eq]
 206    _ = 55 * phi + 34 := by ring
 207
 208/-- Key identity: φ¹¹ = 89φ + 55 (Fibonacci recurrence). -/
 209lemma phi_eleventh_eq : phi^11 = 89 * phi + 55 := by
 210  calc phi^11 = phi * phi^10 := by ring
 211    _ = phi * (55 * phi + 34) := by rw [phi_tenth_eq]
 212    _ = 55 * phi^2 + 34 * phi := by ring
 213    _ = 55 * (phi + 1) + 34 * phi := by rw [phi_sq_eq]
 214    _ = 89 * phi + 55 := by ring
 215