+|2: intros; cases (value l1 input); simplify in ⊢ (? ? (? ? ? %) ?);
+ cases (q_cmp input (start l1)) in H5; whd in ⊢ (% → ?);
+ [1: cases (?:False); clear w2 H4 w1 H2 w H1;
+ apply (q_lt_antisym init (start l1)); [assumption] rewrite < H5; assumption
+ |2: intros; rewrite > H6; clear H6; rewrite > H4; reflexivity;
+ |3: cases (?:False); apply (q_lt_antisym input (start l1)); [2: assumption]
+ apply (q_lt_trans ??? H3 H);]
+|3: intro; cases H4; clear H4;
+ cases (value l1 input); simplify; cases (q_cmp input (start l1)) in H4; whd in ⊢ (% → ?);
+ [1: intro; cases H8; clear H8; rewrite > H11; rewrite > H7; clear H11 H7;
+ simplify in ⊢ (? ? ? (? ? ? (? ? % ? ?)));
+ cut (\fst w1 = S (\fst w2)) as Key; [rewrite > Key; reflexivity;]
+ cut (\fst w2 = O); [2: clear H10;
+ symmetry; apply le_n_O_to_eq; rewrite > (sum_bases_O l1 (\fst w2)); [apply le_n]
+ apply (q_le_trans ??? H9); rewrite < H4; rewrite > q_d_x_x;
+ left; reflexivity;]
+ rewrite > Hcut; clear Hcut H10 H9; simplify in H5 H6;
+ cut (ⅆ[input,init] = Qpos w) as E; [2:
+ rewrite > H2; rewrite < H4; rewrite > q_d_sym;
+ rewrite > q_d_noabs; [reflexivity] right; assumption;]
+ cases (\fst w1) in H5 H6; intros;
+ [1: cases (?:False); clear H5; simplify in H6;
+ apply (q_lt_corefl ⅆ[input,init]);
+ rewrite > E in ⊢ (??%); rewrite < q_plus_OQ in ⊢ (??%);
+ rewrite > q_plus_sym; assumption;
+ |2: cases n in H5 H6; [intros; reflexivity] intros;
+ cases (?:False); clear H6; cases (bars l1) in H5; simplify; intros;
+ [apply (q_pos_OQ one);|apply (q_pos_OQ (\fst b));]
+ apply (q_le_S ??? (sum_bases_ge_OQ (mk_q_f init ?) n1));[apply [];|3:apply l]
+ simplify in ⊢ (? (? (? % ?) ?) ?); rewrite < (q_plus_minus w);
+ apply q_le_minus_r; rewrite < q_minus_r;
+ rewrite < E in ⊢ (??%); assumption;]
+ |2: intros; rewrite > H8; rewrite > H7; clear H8 H7;
+ simplify in H5 H6 ⊢ %;
+ simplify in H5:(? ? (? ? %));
+
+
+