avgPathLength_pos

formal source

  64def avgPathLength (N : ℝ) : ℝ := Real.log N / Real.log phi
  65
  66/-- For `N > 1`, average path length is positive. -/
  67theorem avgPathLength_pos {N : ℝ} (h : 1 < N) : 0 < avgPathLength N := by
  68  unfold avgPathLength
  69  have h_log_N_pos : 0 < Real.log N := Real.log_pos h
  70  have h_log_phi_pos : 0 < Real.log phi := Real.log_pos one_lt_phi
  71  exact div_pos h_log_N_pos h_log_phi_pos
  72
  73/-- Path length grows logarithmically in `N`. -/
  74theorem path_length_log_growth {N M : ℝ} (hN : 1 < N) (hM : N < M) :
  75    avgPathLength N < avgPathLength M := by
  76  unfold avgPathLength
  77  have h_log_phi_pos : 0 < Real.log phi := Real.log_pos one_lt_phi
  78  apply (div_lt_div_iff_of_pos_right h_log_phi_pos).mpr
  79  exact Real.log_lt_log (by linarith) hM
  80
  81/-! ## §3. Clustering ratio -/
  82
  83/-- Predicted clustering ratio (RS / Erdős-Rényi baseline). -/
  84def clusteringRatio : ℝ := 1 / phi
  85
  86theorem clusteringRatio_pos : 0 < clusteringRatio :=
  87  div_pos one_pos phi_pos
  88
  89theorem clusteringRatio_lt_one : clusteringRatio < 1 := by
  90  unfold clusteringRatio
  91  rw [div_lt_one phi_pos]
  92  exact one_lt_phi
  93
  94theorem clusteringRatio_band :
  95    (0.617 : ℝ) < clusteringRatio ∧ clusteringRatio < 0.622 := by
  96  unfold clusteringRatio
  97  have h1 := phi_gt_onePointSixOne