(* 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
    }.(* 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
    }.(* 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
    }.(* 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
    }.(* 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
    }.

Lemma join_core2 {A}{J: Join A}{PA: Perm_alg A}{SA: Sep_alg A}: forall a b c, join a b c -> core a = core b.