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f61e547 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 | 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
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