+
+lemma case1 :
+ ∀init,st,input,l.
+ init<st → st<input →
+ ⅆ[input,init] < sum_bases l O + (st-init) → False.
+intros 6; rewrite > q_d_sym; rewrite > q_d_noabs; [2:
+ apply (q_le_trans ? st); apply q_lt_to_le; assumption]
+do 2 rewrite > q_elim_minus; rewrite > q_plus_assoc;
+intro X; lapply (q_lt_canc_plus_r ??? X) as Y;
+simplify in Y; cases (?:False);
+apply (q_lt_corefl st); apply (q_lt_trans ??? H1);
+apply (q_lt_le_trans ??? Y); rewrite > q_plus_sym; rewrite > q_plus_OQ;
+apply q_eq_to_le; reflexivity;
+qed.
+
+lemma case2:
+ ∀a,l1,init,st,input,n.
+ init < st → st < input →
+ sum_bases (a::l1) n + (st-init) ≤ ⅆ[input,init] →
+ ⅆ[input,st] < sum_bases l1 O + Qpos (\fst a) →
+ n = O.
+intros; cut (input - st < Qpos (\fst a)) as H6';[2:
+ rewrite < q_d_noabs;[2:apply q_lt_to_le; assumption]
+ rewrite > q_d_sym; apply (q_lt_le_trans ??? H3);
+ rewrite > q_plus_sym; rewrite > q_plus_OQ;
+ apply q_eq_to_le; reflexivity] clear H3;
+generalize in match H2; rewrite > q_d_sym; rewrite > q_d_noabs;
+ [2: apply (q_le_trans ? st); apply q_lt_to_le; assumption]
+do 2 rewrite > q_elim_minus; rewrite > q_plus_assoc; intro X;
+lapply (q_le_canc_plus_r ??? X) as Y; clear X;
+lapply (q_le_inj_plus_r ?? (Qopp st) Y) as X; clear Y;
+cut (input + Qopp st < Qpos (\fst a)) as H6'';
+ [2: rewrite < q_elim_minus; assumption;] clear H6';
+generalize in match (q_le_lt_trans ??? X H6''); clear X H6'';
+rewrite < q_plus_assoc; rewrite < q_elim_minus;
+rewrite > q_plus_minus; rewrite > q_plus_OQ; cases n; intro X; [reflexivity]
+cases (?:False);
+apply (q_lt_le_incompat (sum_bases l1 n1) OQ);[2: apply sum_bases_ge_OQ;]
+apply (q_lt_canc_plus_r ?? (Qpos (\fst a)));
+rewrite >(q_plus_sym OQ); rewrite > q_plus_OQ; apply X;
+qed.
+
+lemma case3:
+ ∀init,st,input,l1,a,n.
+ init<st → st<input →
+ ⅆ[input,init]<OQ+Qpos a+(st-init) →
+ sum_bases l1 n+Qpos a≤ⅆ[input,st] → False.
+intros;
+cut (sum_bases l1 n - ⅆ[input,st] < Qopp ⅆ[input,init] + (st - init)); [2:
+ cut (sum_bases l1 n≤ⅆ[input,st]-Qpos a) as H7';[2:
+ apply (q_le_canc_plus_r ?? (Qpos a));
+ apply (q_le_trans ??? H3); rewrite > q_elim_minus;
+ rewrite < q_plus_assoc; rewrite > (q_plus_sym (Qopp ?));
+ rewrite < q_elim_minus; rewrite > q_plus_minus; rewrite > q_plus_OQ;
+ apply q_eq_to_le; reflexivity;] clear H3;
+ rewrite > q_elim_minus; apply (q_lt_canc_plus_r ?? ⅆ[input,st]);
+ rewrite < q_plus_assoc; rewrite > (q_plus_sym (Qopp ?));
+ rewrite < q_elim_minus; rewrite > q_plus_minus; rewrite > q_plus_OQ;
+ apply (q_le_lt_trans ??? H7'); clear H7'; rewrite > q_elim_minus;
+ rewrite > q_plus_sym; apply q_lt_inj_plus_r;
+ rewrite > q_plus_sym; apply q_lt_plus; rewrite > q_elim_opp;
+ rewrite > q_plus_sym; apply (q_lt_canc_plus_r ?? (Qpos a));
+ rewrite < q_plus_assoc; rewrite > (q_plus_sym (Qopp ?));
+ rewrite < q_elim_minus; rewrite > q_plus_minus; rewrite > q_plus_OQ;
+ apply (q_lt_le_trans ??? H2); rewrite > (q_plus_sym OQ); rewrite > q_plus_OQ;
+ rewrite > q_plus_sym; apply q_eq_to_le; reflexivity;]
+generalize in match Hcut; clear H2 H3 Hcut;
+rewrite > q_d_sym; rewrite > q_d_noabs; [2:apply q_lt_to_le; assumption]
+rewrite > q_d_sym; rewrite > q_d_noabs; [2: apply (q_le_trans ? st); apply q_lt_to_le; assumption]
+rewrite < q_plus_sym; rewrite < q_elim_minus;
+rewrite > (q_elim_minus input init);
+rewrite > q_minus_distrib; rewrite > q_elim_opp;
+rewrite > (q_elim_minus input st);
+rewrite > q_minus_distrib; rewrite > q_elim_opp;
+repeat rewrite > q_elim_minus;
+rewrite < q_plus_assoc in ⊢ (??% → ?);
+rewrite > (q_plus_sym (Qopp input) init);
+rewrite > q_plus_assoc;
+rewrite < q_plus_assoc in ⊢ (??(?%?) → ?);
+rewrite > (q_plus_sym (Qopp init) init);
+rewrite < (q_elim_minus init); rewrite >q_plus_minus;
+rewrite > q_plus_OQ; rewrite > (q_plus_sym st);
+rewrite < q_plus_assoc;
+rewrite < (q_plus_OQ (Qopp input + st)) in ⊢ (??% → ?);
+rewrite > (q_plus_sym ? OQ); intro X;
+lapply (q_lt_canc_plus_r ??? X) as Y; clear X;
+apply (q_lt_le_incompat ?? Y); apply sum_bases_ge_OQ;
+qed.
+