intros z Hz i j Hi Hj H0 H1 H2 H3. elim (Zle_lt_or_eq i j). intro H4. cut (2 ^ i <= 2 ^ (j - 1)).
intro H5. absurd (2 ^ (j - 1) < z). omega. assumption. apply Zle_pow_le. assumption. omega. trivial.
elim (Zle_or_lt i j); intro H4. assumption. cut (2 ^ j <= 2 ^ (i - 1)). intro H5.
absurd (2 ^ (i - 1) < z). omega. assumption. apply Zle_pow_le. assumption. omega. Qed.

(* Specifications: *)
Lemma Zlog_sup_from_interval : forall z : Z, 0 < z -> forall i : Z, 0 <= i -> 2 ^ (i - 1) < z ->
z <= 2 ^ i -> Zlog_sup z = i.