Instance same_covered_eq {ora: RandomOracle}: Equivalence same_covered_anti_chain.

(* Specifications: *)
Lemma future_anti_chain_refl {ora: RandomOracle}: forall d, future_anti_chain d d.
Lemma future_anti_chain_trans {ora: RandomOracle}: forall d1 d2 d3, future_anti_chain d1 d2 -> future_anti_chain d2 d3 -> future_anti_chain d1 d3.
Lemma proj_in_anti_chain_unique {ora: RandomOracle}: forall (d: HistoryAntiChain) h1 h2 h3, prefix_history h1 h3 -> prefix_history h2 h3 -> covered_by h2 d -> d h1 -> prefix_history h1 h2.
Lemma same_covered_future_anti_chain_spec {ora: RandomOracle}: forall d1 d2, same_covered_anti_chain d1 d2 -> future_anti_chain d1 d2 -> forall h1, d1 h1 -> exists h2, prefix_history h1 h2 /\ d2 h2.