Definition unify_list
  (s : list (variable * value)) (lv : list value) (lp : list pattern) :
  option (variable -> value) :=
  option_map (assoc_default (variable_beq variable_eq_dec) value_default) (unify_list_aux s lv lp).

Lemma unify_list_aux_extension lv lp s s': unify_list_aux s lv lp = Some s' -> exists s'', s' = s'' ++ s /\ forall x, assoc (variable_beq variable_eq_dec) x s <> None -> assoc (variable_beq variable_eq_dec) x s'' = None.