Definition extendM : modality :=
exist _ R_extends R_extends_valid_rel.
Definition contractsM : modality :=
exist _ R_contracts R_contracts_valid_rel.
Notation "'%' e"  := (box extendM e)(at level 30, right associativity): pred.

(* Specifications: *)
Lemma value_knot_laterR : forall k k' v v',
laterR (A:=world) (k,v) (k',v') <-> clos_trans _ age k k' /\ v = v'.
Lemma value_knot_necR : forall k k' v v',
necR (A:=world) (k,v) (k',v') <-> clos_refl_trans _ age k k' /\ v = v'.