(* From section 4.1 *)
  Definition F : Type -> Type := fun K => addr -> option K.Definition f_F := f_fun addr (f_option f_identity).Definition other : Type := value.Module K := KnotHered(TFP).(* Wow, that was easy... *)
Module KL := KnotHered_Lemmas(K).(* Let's define our typing system on values *)

Definition mtype : Type := knot.Definition world : Type := (mtype * value)%type.Definition world_ag : ageable world :=
  ag_prod mtype value ag_knot.Definition knot_extends (k1 k2 : knot) : Prop :=
  match (unsquash k1, unsquash k2) with
    ((n, psi), (n', psi')) => n = n' /\ forall a, (psi a = None) \/ (psi' a = psi a)
  end.Definition R_extends (w1 w2 : world) : Prop :=
  match (w1, w2) with
    ((k1, v1), (k2, v2)) => knot_extends k1 k2 /\ v1 = v2
  end.

(* Specifications: *)
Lemma knot_extends_refl : forall k, knot_extends k k.
Lemma knot_extends_trans : forall k1 k2 k3, knot_extends k1 k2 -> knot_extends k2 k3 -> knot_extends k1 k3.
Lemma knot_extends_age_commute1 : forall x y z, age x y /\ knot_extends y z -> exists y', knot_extends x y' /\ age y' z.
Lemma knot_extends_age_commute2 : forall x y z, knot_extends x y /\ age y z -> exists y', age x y' /\ knot_extends y' z.
Lemma R_extends_valid_rel : valid_rel R_extends.
Lemma R_extends_refl : reflexive _ R_extends.
Lemma R_extends_trans : transitive _ R_extends.