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)).

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