Definition limit_domain_anti_chain (Omegas: RandomVarDomainStream): HistoryAntiChain := Build_HistoryAntiChain _ (limit_raw_domain Omegas) (limit_raw_domain_legal Omegas).

(* Specifications: *)
Lemma limit_domain_anti_chain_hered: forall (Omegas: RandomVarDomainStream) (n: nat) h, Omegas n h -> exists h_limit, prefix_history h h_limit /\ limit_domain_anti_chain Omegas h_limit.
Lemma limit_domain_anti_chain_covered_forward: forall (Omegas: RandomVarDomainStream) h, is_inf_history h -> (exists h', (prefix_history h' h \/ strict_conflict_history h' h) /\ limit_domain_anti_chain Omegas h') -> (exists h', (prefix_history h' h \/ strict_conflict_history h' h) /\ Omegas 0 h').
Lemma limit_domain_anti_chain_covered_backward: forall (Omegas: RandomVarDomainStream) h, is_inf_history h -> (exists h', (prefix_history h' h \/ strict_conflict_history h' h) /\ Omegas 0 h') -> (exists h', (prefix_history h' h \/ strict_conflict_history h' h) /\ limit_domain_anti_chain Omegas h').