Hoeffdings lemma

lemma hoeffdings_lemma {p : ℝ} (m : p ∈ Icc 0 1) {t : ℝ} (t0 : 0 ≤ t) :    (bernoulli p).exp (λ x ↦ (t * bif x then 1 else 0).exp) ≤ (t*p + t^2/8).exp := by  simp only [exp_bernoulli_exp m]  have p1 : 0 ≤ 1-p := by linarith [m.2]  by_cases tz : t = 0  · simp [tz]  replace t0 := (Ne.symm tz).lt_of_le t0; clear tz  rw [←Real.exp_log (L_pos m), Real.exp_le_exp]  rcases L_taylor m t0 with ⟨a,_,h⟩  simp only [L] at h; rw [h]; clear h; norm_num  generalize hb : p * a.exp = b  have b0 : 0 ≤ b := by rw [←hb]; exact mul_nonneg m.1 (Real.exp_nonneg _)  have amgm : (1-p:) * b / (1 - p + b)^2 ≤ 1/4 := mul_div_sq_sum_le p1 b0  simp only [mul_comm b _, ←mul_div _ _ (2 : ℝ)] at amgm ⊢  apply le_trans (add_le_add_right (mul_le_mul_of_nonneg_right amgm _) _)  swap; apply div_nonneg (pow_nonneg (le_of_lt t0) _) (by norm_num)  have e : (1:ℝ)/(4:ℝ) * (t^2 / (2:ℝ)) = t^2 / 8 := by ring  simp only [e, add_comm (t*p) _]; rfl