Definition compatible_rule (r:rule) : Prop :=
  match r with
    | rule_intro f _ t => forall f', In f' (functions_of_term t) -> rank f' <= rank f
  end.Definition compatible_prog : Prop :=
  forall r, In r prog -> compatible_rule r.

Lemma PPO_value_is_sub_term (t v: term) : @term_value _ _ _ v -> (PPO rank t v <-> In t (sub_terms_strict v)).