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

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)).