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theorem proved tactic proof

phi_from_fibonacci_ratio

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formal statement (Lean)

 119theorem phi_from_fibonacci_ratio (r : ℝ) (hr_pos : 0 < r)
 120    (hfib : r ^ 2 = r + 1) : r = phi := by

proof body

Tactic-mode proof.

 121  have hsq5 : Real.sqrt 5 ^ 2 = 5 := Real.sq_sqrt (by norm_num : (0 : ℝ) ≤ 5)
 122  have hsqrt5_gt1 : Real.sqrt 5 > 1 := by
 123    have : Real.sqrt 5 > Real.sqrt 1 :=
 124      Real.sqrt_lt_sqrt (by norm_num) (by norm_num : (1 : ℝ) < 5)
 125    simpa [Real.sqrt_one] using this
 126  have hdisc : (r - (1 + Real.sqrt 5) / 2) * (r - (1 - Real.sqrt 5) / 2) = 0 := by
 127    nlinarith [hsq5]
 128  rcases mul_eq_zero.mp hdisc with h | h
 129  · unfold phi; linarith
 130  · exfalso
 131    have h_neg : (1 - Real.sqrt 5) / 2 < 0 := by linarith
 132    linarith
 133
 134/-! ## Part 6: At Global Minimum → Holographic -/
 135
 136/-- At the J-cost global minimum (all edge costs zero), the allocation is
 137    holographic: every vertex has the same field value. -/

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