(* liens entre PPO et sous-terme *)
Fixpoint sub_terms (t: term) : list term :=
t :: match t with
| var _ => []
| capply _ lt => flat_map sub_terms lt
| fapply _ lt => flat_map sub_terms lt
end.
Definition sub_terms_strict (t: term) : list term :=
match t with
| var _ => []
| capply _ lt => flat_map sub_terms lt
| fapply _ lt => flat_map sub_terms lt
end.

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