Le exp of cut

lemma le_exp_of_cut {f : Prob α} {u : α → ℝ} (i : α → Prop) (a b : ℝ) (fi : a ≤ f.pr i)    (iu : ∀ x, f.prob x ≠ 0 → i x → b ≤ u x) (u0 : ∀ x, f.prob x ≠ 0 → ¬i x → 0 ≤ u x)    (b0 : 0 ≤ b) : a * b ≤ f.exp u := by  have h : ∀ x, f.prob x ≠ 0 → b * (if i x then 1 else 0) ≤ u x := by    intro x fx; by_cases ix : i x    · simp only [ix, if_true, mul_one]; exact iu x fx ix    · simp only [ix, if_false, pr_nonneg, mul_zero]; exact u0 x fx ix  refine le_trans ?_ (exp_mono h)  rw [exp_const_mul, ←pr, mul_comm]  exact mul_le_mul_of_nonneg_left fi b0