Definition proj_left_max_size (proof_tree : cbv) : cbv :=
  proj_left_max_size_list (proof_tree) (sub_trees proof_tree).

(* Specifications: *)
Lemma proj_left_size_le_max proof_tree: forall p, InCBV p proof_tree -> term_size (proj_left p) <= term_size (proj_left (proj_left_max_size proof_tree)).
Lemma InCBV_proj_left_max_size p : InCBV (proj_left_max_size p) p.
Lemma proj_left_size_le_max proof_tree: forall p, InCBV p proof_tree -> term_size (proj_left p) <= term_size (proj_left (proj_left_max_size proof_tree)).
Lemma InCBV_proj_left_max_size p : InCBV (proj_left_max_size p) p.