(* Being a subtree of *)
Fixpoint InCBV p proof_tree : Prop :=
p = proof_tree \/
match proof_tree with
| cbv_constr lp _ _ _ _ => orl (map (InCBV p) lp)
| cbv_split lp p' _ _ _ _ => InCBV p p' \/ orl (map (InCBV p) lp)
| cbv_update _ _ p' _ _ _ _ => InCBV p p'
| cbv_read _ _ _ => False
end.

Lemma InCBV_trans p p' p'': InCBV p p' -> InCBV p' p'' -> InCBV p p''.