qed.
lemma sum_bases_O:
- ∀l:q_f.∀x.sum_bases (bars l) x ≤ OQ → x = O.
+ ∀l.∀x.sum_bases l x ≤ OQ → x = O.
intros; cases x in H; [intros; reflexivity] intro; cases (?:False);
cases (q_le_cases ?? H);
[1: apply (q_lt_corefl OQ); rewrite < H1 in ⊢ (?? %);
-|2: apply (q_lt_antisym ??? H1);] clear H H1; cases (bars l);
+|2: apply (q_lt_antisym ??? H1);] clear H H1; cases l;
simplify; apply q_lt_plus_trans;
try apply q_pos_lt_OQ;
try apply (sum_bases_ge_OQ []);