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theorem proved tactic proof

spectralGap_ratio

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formal statement (Lean)

  57theorem spectralGap_ratio (k : ℕ) :
  58    spectralGap (k + 1) / spectralGap k = phi⁻¹ := by

proof body

Tactic-mode proof.

  59  unfold spectralGap
  60  have hphi_ne : phi ≠ 0 := Constants.phi_ne_zero
  61  have h : phi ^ (-((k : ℤ) + 1)) = phi ^ (-(k : ℤ)) * phi⁻¹ := by
  62    rw [show (-((k : ℤ) + 1)) = -(k : ℤ) + (-1 : ℤ) by ring]
  63    rw [zpow_add₀ hphi_ne]; simp
  64  have hcast : ((k + 1 : ℕ) : ℤ) = (k : ℤ) + 1 := by push_cast; ring
  65  rw [hcast, h]
  66  have hk_pos : 0 < phi ^ (-(k : ℤ)) := zpow_pos Constants.phi_pos _
  67  field_simp [hk_pos.ne']
  68

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