Definition PPO_rule (r: rule) : Prop :=
  match r with
    | rule_intro f lp t => PPO t (fapply f (map (@term_from_pattern _ _ _) lp))
  end.Definition PPO_prog : Prop :=
  forall r, In r prog -> PPO_rule r.

Lemma PPO_trans t1 t2 t3 : PPO t1 t2 -> PPO t2 t3 -> PPO t1 t3.