Definition lock_sparsity {A} (lset : AMap.t A) : Prop :=
forall loc1 loc2,
AMap.find loc1 lset <> None ->
AMap.find loc2 lset <> None ->
loc1 = loc2 \/
fst loc1 <> fst loc2 \/
(fst loc1 = fst loc2 /\ far (snd loc1) (snd loc2)).

(* Specifications: *)
Lemma lock_sparsity_age_to (tp : jstate ge) n :
lock_sparsity (lset tp) ->
lock_sparsity (lset (age_tp_to n tp)).