minus_one_step_forces_phi_shift

formal source

  90This is the paper's Theorem 4.2: setting u = 1/b, the condition
  91`(1 - u)(1 + u)^2 = 1` expands to `u^2 + u - 1 = 0`, giving u = 1/φ. -/
  92
  93theorem minus_one_step_forces_phi_shift
  94    {a b c : ℝ}
  95    (hb : 1 < b)
  96    (h0 : gapAffineLogR a b c 0 = 0)
  97    (h1 : gapAffineLogR a b c 1 = 1)
  98    (hneg1 : gapAffineLogR a b c (-1) = -2) :
  99    b = phi := by
 100  have hb_pos : 0 < b := lt_trans zero_lt_one hb
 101  have hb_ne : b ≠ 0 := ne_of_gt hb_pos
 102  have hplus_pos : 0 < 1 + (1 : ℝ) / b := by
 103    have hinv_pos : 0 < (1 : ℝ) / b := one_div_pos.mpr hb_pos
 104    linarith
 105  have hinv_lt_one : (1 : ℝ) / b < 1 := by
 106    simpa using (one_div_lt_one_div_of_lt (by norm_num : (0 : ℝ) < 1) hb)
 107  have hminus_pos : 0 < 1 - (1 : ℝ) / b := by linarith
 108  have hminus_ne : (1 - (1 : ℝ) / b) ≠ 0 := ne_of_gt hminus_pos
 109  have hc : c = 0 := by simpa [gapAffineLogR] using h0
 110  have h1' : a * Real.log (1 + (1 : ℝ) / b) = 1 := by
 111    simpa [gapAffineLogR, hc] using h1
 112  have hneg1_raw : a * Real.log (1 + (-1 : ℝ) / b) = -2 := by
 113    simpa [gapAffineLogR, hc] using hneg1
 114  have hneg1' : a * Real.log (1 - (1 : ℝ) / b) = -2 := by
 115    simpa [sub_eq_add_neg, div_eq_mul_inv, mul_assoc] using hneg1_raw
 116  have ha_ne : a ≠ 0 := by
 117    intro ha; simp [ha] at h1'
 118  have hscaled : a * (-2 * Real.log (1 + (1 : ℝ) / b)) = -2 := by
 119    calc
 120      a * (-2 * Real.log (1 + (1 : ℝ) / b))
 121          = (-2) * (a * Real.log (1 + (1 : ℝ) / b)) := by ring
 122      _ = (-2) * 1 := by rw [h1']
 123      _ = -2 := by ring