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.