Definition max_extension {ora: RandomOracle} (d: HistoryAntiChain): HistoryAntiChain.
exists (max_pre_extension d).
constructor.
destruct d as [d H]; simpl.
exact (proj1 (max_pre_extension_spec _ rand_consi)).
Defined.

Lemma max_extension_full {ora: RandomOracle}: forall d, is_max_anti_chain (max_extension d).