-include "Q/q/q.ma".
-include "list/list.ma".
-include "cprop_connectives.ma".
-
-notation "\rationals" non associative with precedence 99 for @{'q}.
-interpretation "Q" 'q = Q.
-
-record q_f : Type ≝ {
- start : ℚ;
- bars: list (ℚ × ℚ) (* base, height *)
-}.
-
-axiom qp : ℚ → ℚ → ℚ.
-
-interpretation "Q plus" 'plus x y = (qp x y).
-
-axiom qm : ℚ → ℚ → ℚ.
-
-interpretation "Q minus" 'minus x y = (qm x y).
-
-axiom qlt : ℚ → ℚ → CProp.
-
-interpretation "Q less than" 'lt x y = (qlt x y).
-
-inductive q_comparison (a,b:ℚ) : CProp ≝
- | q_eq : a = b → q_comparison a b
- | q_lt : a < b → q_comparison a b
- | q_gt : b < a → q_comparison a b.
-
-axiom q_cmp:∀a,b:ℚ.q_comparison a b.
-
-definition qle ≝ λa,b:ℚ.a = b ∨ a < b.
-
-interpretation "Q less or equal than" 'le x y = (qle x y).
-
-notation "'nth'" left associative with precedence 70 for @{'nth}.
-notation < "\nth \nbsp l \nbsp d \nbsp i" left associative with precedence 70 for @{'nth_appl $l $d $i}.
-interpretation "list nth" 'nth = (cic:/matita/list/list/nth.con _).
-interpretation "list nth" 'nth_appl l d i = (cic:/matita/list/list/nth.con _ l d i).
-
-notation < "\rationals \sup 2" non associative with precedence 40 for @{'q2}.
-interpretation "Q x Q" 'q2 = (product Q Q).
-
-let rec mk_list (A:Type) (def:nat→A) (n:nat) on n ≝
- match n with
- [ O ⇒ []
- | S m ⇒ def m :: mk_list A def m].
-
-interpretation "mk_list appl" 'mk_list f n = (mk_list f n).
-interpretation "mk_list" 'mk_list = mk_list.
-notation < "\mk_list \nbsp f \nbsp n" left associative with precedence 70 for @{'mk_list_appl $f $n}.
-notation "'mk_list'" left associative with precedence 70 for @{'mk_list}.
-
-alias symbol "pair" = "pair".
-definition q00 : ℚ × ℚ ≝ 〈OQ,OQ〉.
-
+include "models/q_bars.ma".
+
+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:
+
+
+STOP
+