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theorem proved tactic proof

phi_plus_inverse_eq_sqrt5

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formal statement (Lean)

 176theorem phi_plus_inverse_eq_sqrt5 : phi + 1/phi = Real.sqrt 5 := by

proof body

Tactic-mode proof.

 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. -/

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