| import OAI.Probability.FreeIsing.Estimates.MessageFunction | |
| namespace OAI | |
| noncomputable section | |
| namespace Problem367 | |
| theorem abs_cavityMessage_tanh_le {beta : ℝ} (hbeta : 0 ≤ beta) (z : ℝ) : | |
| |cavityMessage (Real.tanh beta) z| ≤ beta := by | |
| have htheta : 0 ≤ Real.tanh beta := by | |
| rw [Real.tanh_eq_sinh_div_cosh] | |
| exact div_nonneg (Real.sinh_nonneg_iff.mpr hbeta) (Real.cosh_pos beta).le | |
| simpa only [Real.artanh_tanh] using | |
| cavityMessage_abs_le_artanh htheta (Real.tanh_lt_one beta) z | |
| theorem cavityMessage_eq_half_log_cosh_ratio (beta z : ℝ) : | |
| cavityMessage (Real.tanh beta) z = | |
| (1 / 2 : ℝ) * Real.log (Real.cosh (z + beta) / Real.cosh (z - beta)) := by | |
| have htheta_sq : Real.tanh beta ^ 2 ≤ 1 := (Real.tanh_sq_lt_one beta).le | |
| have hmem := cavityMessage_argument_mem htheta_sq z | |
| have hratio : (1 + Real.tanh beta * Real.tanh z) / | |
| (1 - Real.tanh beta * Real.tanh z) = | |
| Real.cosh (z + beta) / Real.cosh (z - beta) := by | |
| apply (div_eq_div_iff (by linarith [hmem.2]) (Real.cosh_pos (z - beta)).ne').2 | |
| rw [Real.cosh_add, Real.cosh_sub] | |
| simp only [Real.tanh_eq_sinh_div_cosh] | |
| field_simp | |
| rw [cavityMessage, Real.artanh_eq_half_log ⟨hmem.1.le, hmem.2.le⟩, hratio] | |
| theorem cavityMessage_eq_log_cosh_sub (beta z : ℝ) : | |
| cavityMessage (Real.tanh beta) z = | |
| (Real.log (Real.cosh (z + beta)) - Real.log (Real.cosh (z - beta))) / 2 := by | |
| rw [cavityMessage_eq_half_log_cosh_ratio beta, | |
| Real.log_div (Real.cosh_pos (z + beta)).ne' (Real.cosh_pos (z - beta)).ne'] | |
| ring | |
| theorem tanh_field_difference_eq (beta z : ℝ) : | |
| (Real.tanh (z + beta) - Real.tanh (z - beta)) / 2 = | |
| Real.tanh beta * (1 - Real.tanh z ^ 2) / | |
| (1 - Real.tanh beta ^ 2 * Real.tanh z ^ 2) := by | |
| have hp := ((Real.hasDerivAt_cosh (z + beta)).comp z | |
| ((hasDerivAt_id z).add_const beta)).log (Real.cosh_pos (z + beta)).ne' | |
| have hm := ((Real.hasDerivAt_cosh (z - beta)).comp z | |
| ((hasDerivAt_id z).sub_const beta)).log (Real.cosh_pos (z - beta)).ne' | |
| simp only [id_eq, mul_one, Function.comp_apply, ← Real.tanh_eq_sinh_div_cosh] at hp hm | |
| have h : HasDerivAt (cavityMessage (Real.tanh beta)) | |
| ((Real.tanh (z + beta) - Real.tanh (z - beta)) / 2) z := by | |
| apply ((hp.sub hm).div_const 2).congr_of_eventuallyEq | |
| exact Filter.Eventually.of_forall fun x => cavityMessage_eq_log_cosh_sub beta x | |
| simpa only [responseAttenuation, mul_div_assoc] using | |
| h.unique (hasDerivAt_cavityMessage (Real.tanh_sq_lt_one beta).le z) | |
| end Problem367 | |
| end | |
| end OAI | |