(* Sep_alg: additional properties that makes a Permission Algebra into a Separation Algebra. *)
Class Sep_alg A {J: Join A} : Type :=
mkSep {
core: A -> A;
core_unit: forall t, unit_for (core t) t;
join_core: forall {a b c}, join a b c -> core a = core c
}.
Arguments Sep_alg _ [J].

Lemma core_hom {A}{J: Join A}{PA: Perm_alg A}{SA: Sep_alg A}:
forall {a b c}, join a b c -> join (core a) (core b) (core c).