minus_one_step_forces_phi_shift

proof body

Tactic-mode proof.

 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
 124  have hlog_rel :
 125      Real.log (1 - (1 : ℝ) / b) = -2 * Real.log (1 + (1 : ℝ) / b) := by
 126    apply (mul_left_cancel₀ ha_ne)
 127    calc
 128      a * Real.log (1 - (1 : ℝ) / b) = -2 := hneg1'
 129      _ = a * (-2 * Real.log (1 + (1 : ℝ) / b)) := hscaled.symm
 130  have hlog_pow :
 131      Real.log ((1 + (1 : ℝ) / b) ^ (2 : ℝ)) =
 132        2 * Real.log (1 + (1 : ℝ) / b) := by
 133    exact Real.log_rpow hplus_pos (2 : ℝ)
 134  have hlog_sum :
 135      Real.log (1 - (1 : ℝ) / b) +
 136        Real.log ((1 + (1 : ℝ) / b) ^ (2 : ℝ)) = 0 := by
 137    linarith [hlog_rel, hlog_pow]
 138  have hpow_ne : ((1 + (1 : ℝ) / b) ^ (2 : ℝ)) ≠ 0 := by
 139    exact ne_of_gt (Real.rpow_pos_of_pos hplus_pos (2 : ℝ))
 140  have hlog_prod :
 141      Real.log ((1 - (1 : ℝ) / b) * ((1 + (1 : ℝ) / b) ^ (2 : ℝ))) = 0 := by
 142    calc
 143      Real.log ((1 - (1 : ℝ) / b) * ((1 + (1 : ℝ) / b) ^ (2 : ℝ)))
 144          = Real.log (1 - (1 : ℝ) / b) + Real.log ((1 + (1 : ℝ) / b) ^ (2 : ℝ)) := by
 145              simpa using (Real.log_mul hminus_ne hpow_ne)
 146      _ = 0 := hlog_sum
 147  have hprod_pos : 0 < (1 - (1 : ℝ) / b) * ((1 + (1 : ℝ) / b) ^ (2 : ℝ)) := by
 148    exact mul_pos hminus_pos (Real.rpow_pos_of_pos hplus_pos (2 : ℝ))
 149  have hprod_eq_one : (1 - (1 : ℝ) / b) * ((1 + (1 : ℝ) / b) ^ (2 : ℝ)) = 1 := by
 150    exact Real.eq_one_of_pos_of_log_eq_zero hprod_pos hlog_prod
 151  have hpoly : b ^ 2 - b - 1 = 0 := by
 152    have htmp : (1 - (1 : ℝ) / b) * (1 + (1 : ℝ) / b) ^ 2 = 1 := by
 153      simpa [Real.rpow_two] using hprod_eq_one
 154    field_simp [hb_ne] at htmp
 155    nlinarith [htmp]
 156  have hphi_poly : phi ^ 2 - phi - 1 = 0 := by linarith [phi_sq_eq]
 157  have hfactor : (b - phi) * (b + phi - 1) = 0 := by
 158    nlinarith [hpoly, hphi_poly]
 159  rcases mul_eq_zero.mp hfactor with hroot | hother
 160  · linarith
 161  · exact False.elim ((ne_of_gt (by linarith [hb, one_lt_phi] : 0 < b + phi - 1)) hother)
 162
 163/-! ## Main Theorems -/
 164
 165/-- Three-point calibration forces all affine-log parameters. -/