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theorem proved tactic proof

phi_inv

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formal statement (Lean)

 110theorem phi_inv : φ⁻¹ = φ - 1 := by

proof body

Tactic-mode proof.

 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. -/

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