(** ** Soundness *)
Definition cast {domain} (eta eta' : nat -> domain) (I : interp eta) : interp eta'.
destruct I; econstructor; eauto.
Defined.

Lemma sat_ext b (phi : form b) dom rho rho' eta (I : interp dom eta) :
sat I rho phi -> (forall v, rho v = rho' v) -> sat I rho' phi.