Definition is_max_anti_chain {ora: RandomOracle} (d: HistoryAntiChain): Prop :=
forall h, is_inf_history h ->
exists h', (prefix_history h' h \/ strict_conflict_history h' h) /\ d h'.

Lemma unit_anti_chain_max {ora: RandomOracle}: is_max_anti_chain unit_space_anti_chain.
Definition max_anti_chain_same_covered {ora: RandomOracle}: forall (d1 d2: HistoryAntiChain),
is_max_anti_chain d1 ->
is_max_anti_chain d2 ->
same_covered_anti_chain d1 d2.