+axiom q_le_OQ_Qpos: ∀x.OQ ≤ Qpos x.
+
+lemma initial_shift_same_values:
+ ∀l1:q_f.∀init.init < start l1 →
+ same_values l1
+ (mk_q_f init (〈\fst (unpos (start l1 - init) ?),OQ〉:: bars l1)).
+[apply hide; apply q_lt_minus; rewrite > q_plus_sym; rewrite > q_plus_OQ; assumption]
+intros; generalize in ⊢ (? ? (? ? (? ? (? ? ? (? ? ? (? ? %)) ?) ?))); intro;
+cases (unpos (start l1-init) H1); intro input;
+simplify in ⊢ (? ? ? (? ? ? (? ? ? (? (? ? (? ? (? ? ? % ?) ?)) ?))));
+cases (value (mk_q_f init (〈w,OQ〉::bars l1)) input);
+simplify in ⊢ (? ? ? (? ? ? %));
+cases (q_cmp input (start (mk_q_f init (〈w,OQ〉::bars l1)))) in H3;
+whd in ⊢ (% → ?); simplify in H3;
+[1: intro; cases H4; clear H4; rewrite > H3;
+ cases (value l1 init); simplify; cases (q_cmp init (start l1)) in H4;
+ [1: cases (?:False); apply (q_lt_corefl init); rewrite > H4 in ⊢ (?? %); apply H;
+ |3: cases (?:False); apply (q_lt_antisym init (start l1)); assumption;
+ |2: whd in ⊢ (% → ?); intro; rewrite > H8; clear H8 H4;
+ rewrite > H7; clear H7; rewrite > (?:\fst w1 = O); [reflexivity]
+ symmetry; apply le_n_O_to_eq;
+ rewrite > (sum_bases_O (mk_q_f init (〈w,OQ〉::bars l1)) (\fst w1)); [apply le_n]
+ clear H6 w2; simplify in H5:(? ? (? ? %));
+ destruct H3; rewrite > q_d_x_x in H5; assumption;]
+|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;
+ apply q_eq_to_le; 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] apply q_lt_to_le; 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 ? n1));[apply []|3:apply l]
+ simplify in ⊢ (? (? (? % ?) ?) ?); rewrite < (q_plus_minus (Qpos w));
+ rewrite > q_elim_minus; apply q_le_minus_r;
+ rewrite > q_elim_opp; rewrite < E in ⊢ (??%); assumption;]
+ |2: intros; rewrite > H8; rewrite > H7; clear H8 H7;
+ simplify in H5 H6 ⊢ %;
+ cases (\fst w1) in H5 H6; [intros; reflexivity]
+ cases (bars l1);
+ [1: intros; simplify; elim n [reflexivity] simplify; assumption;
+ |2: simplify; intros; cases (?:False); clear H6;
+ apply (q_lt_le_incompat (input - init) (Qpos w) );
+ [1: rewrite > H2; do 2 rewrite > q_elim_minus;
+ apply q_lt_plus; rewrite > q_elim_minus;
+ rewrite < q_plus_assoc; rewrite < q_elim_minus;
+ rewrite > q_plus_minus;rewrite > q_plus_OQ; assumption;
+ |2: rewrite < q_d_noabs; [2: apply q_lt_to_le; assumption]
+ rewrite > q_d_sym; apply (q_le_S ???? H5);apply sum_bases_ge_OQ;]]
+ |3: 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;]
+ simplify in H5 H6;
+ cases (\fst w1) in H5 H6; intros;
+ [1: cases (?:False); clear H5 H9 H10; simplify in H6;
+ apply (q_lt_antisym input (start l1)); [2: assumption]
+ rewrite > q_d_sym in H6;
+ rewrite > q_d_noabs in H6; [2: apply q_lt_to_le; assumption]
+ rewrite > q_plus_sym in H6;
+ rewrite > q_plus_OQ in H6; rewrite > H2 in H6;
+ lapply (q_lt_plus ??? H6) as X; clear H6 H2 H3 H1 H H4 w1 w2 w;
+ rewrite > q_elim_minus in X; rewrite < q_plus_assoc in X;
+ rewrite > (q_plus_sym (Qopp init)) in X;
+ rewrite < q_elim_minus in X; rewrite > q_plus_minus in X;
+ rewrite > q_plus_OQ in X; assumption;
+ |2: simplify in H5; apply eq_f;
+ cut (sum_bases (bars l1) (\fst w2) < sum_bases (bars l1) (S n)+Qpos w);[2:
+ apply (q_le_lt_trans ??? H9);
+ apply (q_lt_trans ??? ? H6);
+ 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_lt_to_le;assumption]
+ do 2 rewrite > q_elim_minus; rewrite > (q_plus_sym ? (Qopp init));
+ apply q_lt_plus; rewrite > q_plus_sym;
+ rewrite > q_elim_minus; rewrite < q_plus_assoc;
+ rewrite < q_elim_minus; rewrite > q_plus_minus;
+ rewrite > q_plus_OQ; apply q_lt_opp_opp; assumption]
+ clear H9 H6;
+ cut (ⅆ[input,init] - Qpos w = ⅆ[input,start l1]);[2:
+ 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_lt_to_le;assumption]
+ rewrite > H2; rewrite > (q_elim_minus (start ?));
+ rewrite > q_minus_distrib; rewrite > q_elim_opp;
+ do 2 rewrite > q_elim_minus;
+ do 2 rewrite < q_plus_assoc;
+ rewrite > (q_plus_sym ? init);
+ rewrite > (q_plus_assoc ? init);
+ rewrite > (q_plus_sym ? init);
+ rewrite < (q_elim_minus init); rewrite > q_plus_minus;
+ rewrite > (q_plus_sym OQ); rewrite > q_plus_OQ;
+ rewrite < q_elim_minus; reflexivity;]
+ cut (sum_bases (bars l1) n < sum_bases (bars l1) (S (\fst w2)));[2:
+ apply (q_le_lt_trans ???? H10); rewrite < Hcut1;
+ rewrite > q_elim_minus; apply q_le_minus_r; rewrite > q_elim_opp;
+ assumption;] clear Hcut1 H5 H10;
+ generalize in match Hcut;generalize in match Hcut2;clear Hcut Hcut2;
+ apply (nat_elim2 ???? n (\fst w2));
+ [3: intros (x y); apply eq_f; apply H5; clear H5;
+ [1: clear H7; apply sum_bases_lt_canc; assumption;
+ |2: clear H6; ]
+ |2: intros; cases (?:False); clear H6;
+ cases n1 in H5; intro;
+ [1: apply (q_lt_corefl ? H5);
+ |2: cases (bars l1) in H5; intro;
+ [1: simplify in H5;
+ apply (q_lt_le_incompat ?? (q_lt_canc_plus_r ??? H5));
+ apply q_le_plus_trans; [apply sum_bases_ge_OQ]
+ apply q_le_OQ_Qpos;
+ |2: simplify in H5:(??%);
+ lapply (q_lt_canc_plus_r (sum_bases l (S n2)) ?? H5) as X;
+ apply (q_lt_le_incompat ?? X); apply sum_bases_ge_OQ]]
+ |1: intro; cases n1 [intros; reflexivity] intros; cases (?:False);
+ elim n2 in H5 H6;
+
+
+ elim (bars l1) 0;
+ [1: intro; elim n1; [reflexivity] cases (?:False);
+
+
+ intros; clear H5;
+ elim n1 in H6; [reflexivity] cases (?:False);
+ [1: apply (q_lt_corefl ? H5);
+ |2: cases (bars l1) in H5; intro;
+ [1: simplify in H5;
+ apply (q_lt_le_incompat ?? (q_lt_canc_plus_r ??? H5));
+ apply q_le_plus_trans; [apply sum_bases_ge_OQ]
+ apply q_le_OQ_Qpos;
+ |2: simplify in H5:(??%);
+ lapply (q_lt_canc_plus_r (sum_bases l (S n2)) ?? H5) as X;
+ apply (q_lt_le_incompat ?? X); apply sum_bases_ge_OQ]]
+qed.