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theorem proved tactic proof

phi_pow_51_in_interval

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formal statement (Lean)

 127theorem phi_pow_51_in_interval :
 128    phi_pow_51_interval.contains (((1 + Real.sqrt 5) / 2) ^ (51 : ℝ)) := by

proof body

Tactic-mode proof.

 129  simp only [Interval.contains, phi_pow_51_interval]
 130  -- (1 + √5)/2 = goldenRatio
 131  have h_phi : (1 + Real.sqrt 5) / 2 = goldenRatio := rfl
 132  -- Convert real power to natural power: x^(51 : ℝ) = x^51
 133  have h_eq : ((1 + Real.sqrt 5) / 2) ^ (51 : ℝ) = goldenRatio ^ (51 : ℕ) := by
 134    conv_lhs => rw [h_phi]
 135    exact Real.rpow_natCast goldenRatio 51
 136  rw [h_eq]
 137  have h := phi_pow51_in_interval_proven
 138  simp only [Interval.contains, phi_pow51_interval_proven] at h
 139  exact h
 140
 141/-- φ^8 interval ≈ 46.979 - PROVEN -/

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