(** ** Non-Standard Model *)
Module NonStan. Section Nonstan.
Variable R : BSRS.
Instance IB : interp (option (string bool)) (fun _ => Some nil) :=
{|
i_f b x := match x with Some u => Some (b :: u) | None => None end ;
i_e := Some nil;
i_P x y := match x, y with Some u, Some v => derivable R u v | _, _ => True end ;
i_Q := False
|}.

Lemma IB_deriv' rho t1 t2 u v :
eval rho t1 = Some u -> eval rho t2 = Some v ->
rho ⊨ (Pr t1 t2) <-> derivable R u v.