-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'" with precedence 90 for @{'nth}.
-notation < "'nth' \nbsp l \nbsp d \nbsp i" with precedence 71
-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 90 for @{'q2}.
-interpretation "Q x Q" 'q2 = (Prod Q Q).
-
-let rec make_list (A:Type) (def:nat→A) (n:nat) on n ≝
- match n with
- [ O ⇒ nil ?
- | S m ⇒ def m :: make_list A def m].
-
-notation "'mk_list'" with precedence 90 for @{'mk_list}.
-interpretation "'mk_list'" 'mk_list = (make_list _).
-notation < "'mk_list' \nbsp f \nbsp n"
-with precedence 71 for @{'mk_list_appl $f $n}.
-interpretation "'mk_list' appl" 'mk_list_appl f n = (make_list _ f n).
-
-definition q0 : ℚ × ℚ ≝ 〈OQ,OQ〉.
-notation < "0 \sub \rationals" with precedence 90 for @{'q0}.
-interpretation "q0" 'q0 = q0.
-
-notation < "[ \rationals \sup 2]" with precedence 90 for @{'lq2}.
-interpretation "lq2" 'lq2 = (list (Prod Q Q)).
-notation < "[ \rationals \sup 2] \sup 2" with precedence 90 for @{'lq22}.
-interpretation "lq22" 'lq22 = (Prod (list (Prod Q Q)) (list (Prod Q Q))).
-
-
-notation "'len'" with precedence 90 for @{'len}.
-interpretation "len" 'len = length.
-notation < "'len' \nbsp l" with precedence 70 for @{'len_appl $l}.
-interpretation "len appl" 'len_appl l = (length _ l).
-
-definition eject ≝
- λP.λp:∃x:(list (ℚ × ℚ)) × (list (ℚ × ℚ)).P x.match p with [ex_introT p _ ⇒ p].
-coercion cic:/matita/dama/models/q_function/eject.con.
-definition inject ≝
- λP.λp:(list (ℚ × ℚ)) × (list (ℚ × ℚ)).λh:P p. ex_introT ? P p h.
-(*coercion inject with 0 1 nocomposites.*)
-coercion cic:/matita/dama/models/q_function/inject.con 0 1 nocomposites.
-
-definition cb0h ≝ (λl.mk_list (λi.〈\fst (nth l q0 i),OQ〉) (length ? l)).
-
+include "nat_ordered_set.ma".
+include "models/q_bars.ma".
+
+lemma key:
+ ∀n,m,l.
+ sum_bases l n < sum_bases l (S m) →
+ sum_bases l m < sum_bases l (S n) →
+ n = m.
+intros 2; apply (nat_elim2 ???? n m);
+[1: intro X; cases X; intros; [reflexivity] cases (?:False);
+ cases l in H H1; simplify; intros;
+ apply (q_lt_le_incompat ??? (sum_bases_ge_OQ ? n1));
+ apply (q_lt_canc_plus_r ??? H1);
+|2: intros 2; cases l; simplify; intros; cases (?:False);
+ apply (q_lt_le_incompat ??? (sum_bases_ge_OQ ? n1));
+ apply (q_lt_canc_plus_r ??? H); (* magia ... *)
+|3: intros 4; cases l; simplify; intros;
+ [1: rewrite > (H []); [reflexivity]
+ apply (q_lt_canc_plus_r ??(Qpos one)); assumption;
+ |2: rewrite > (H l1); [reflexivity]
+ apply (q_lt_canc_plus_r ??(Qpos (\fst b))); assumption;]]
+qed.
+
+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 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) (v1 Hv1);
+cases Hv1 (HV1 HV1 HV1 HV1); cases HV1 (Hi1 Hv11 Hv12); clear HV1 Hv1;
+[1: cut (input < start l1) as K;[2: apply (q_lt_trans ??? Hi1 H)]
+ rewrite > (value_OQ_l ?? K); simplify; symmetry; assumption;
+|2: cut (start l1 + sum_bases (bars l1) (len (bars l1)) ≤ input) as K;[2:
+ simplify in Hi1; apply (q_le_trans ???? Hi1); rewrite > H2;
+ rewrite > q_plus_sym in ⊢ (? ? (? ? %));
+ rewrite > q_plus_assoc; rewrite > q_elim_minus;
+ rewrite > q_plus_sym in ⊢ (? ? (? (? ? %) ?));
+ rewrite > q_plus_assoc; rewrite < q_elim_minus;
+ rewrite > q_plus_minus; rewrite > q_plus_sym in ⊢ (? ? (? % ?));
+ rewrite > q_plus_OQ; apply q_eq_to_le; reflexivity;]
+ rewrite > (value_OQ_r ?? K); simplify; symmetry; assumption;
+|3: simplify in Hi1; destruct Hi1;
+|4: cases (q_cmp input (start l1));
+ [2: rewrite > (value_OQ_l ?? H4);
+ change with (OQ = \snd v1); rewrite > Hv12;
+ cases H3; clear H3; simplify in H5; cases (\fst v1) in H5;[intros;reflexivity]
+ simplify; rewrite > q_d_sym; rewrite > q_d_noabs; [2:cases Hi1; apply H5]
+ rewrite > H2; do 2 rewrite > q_elim_minus;rewrite > q_plus_assoc;
+ intro X; lapply (q_le_canc_plus_r ??? X) as Y; clear X;
+ (* OK *)
+ |1,3: cases Hi1; clear Hi1; cases H3; clear H3;
+ simplify in H5 H6 H8 H9 H7:(? ? (? % %)) ⊢ (? ? ? (? ? ? %));
+ generalize in match (refl_eq ? (bars l1):bars l1 = bars l1);
+ generalize in ⊢ (???% → ?); intro X; cases X; clear X; intro Hb;
+ [1,3: rewrite > (value_OQ_e ?? Hb); rewrite > Hv12; rewrite > Hb in Hv11 ⊢ %;
+ simplify in Hv11 ⊢ %; cases (\fst v1) in Hv11; [1,3:intros; reflexivity]
+ cases n; [1,3: intros; reflexivity] intro X; cases (not_le_Sn_O ? (le_S_S_to_le ?? X));
+ |2,4: cases (value_ok l1 input);
+ [1,5: rewrite > Hv12; rewrite > Hb; clear Hv12; simplify;
+ rewrite > H10; rewrite > Hb;
+ cut (O < \fst v1);[2,4: cases (\fst v1) in H9; intros; [2,4: autobatch]
+ cases (?:False); generalize in match H9;
+ rewrite > q_d_sym; rewrite > q_d_noabs; [2,4: assumption]
+ rewrite > H2; simplify; rewrite > q_plus_sym; rewrite > q_plus_OQ;
+ repeat rewrite > q_elim_minus;
+ intro X; lapply (q_lt_canc_plus_r ??? X) as Y;
+ apply (q_lt_le_incompat ?? Y);
+ [apply q_eq_to_le;symmetry|apply q_lt_to_le] assumption;]
+ cases (\fst v1) in H8 H9 Hcut; [1,3:intros (_ _ X); cases (not_le_Sn_O ? X)]
+ intros; clear H13; simplify;
+ rewrite > (key n n1 (b::l)); [1,4: reflexivity] rewrite < Hb;
+ [2,4: simplify in H8; apply (q_le_lt_trans ??? (q_le_plus_r ??? H8));
+ apply (q_le_lt_trans ???? H12); rewrite > H2;
+ rewrite > q_d_sym; rewrite > q_d_noabs; [2,4: assumption]
+ rewrite > (q_elim_minus (start l1) init); rewrite > q_minus_distrib;
+ rewrite > q_elim_opp; repeat rewrite > q_elim_minus;
+ rewrite < q_plus_assoc; rewrite > (q_plus_sym ? init);
+ rewrite > q_plus_assoc;rewrite < q_plus_assoc in ⊢ (? (? % ?) ?);
+ rewrite > (q_plus_sym ? init); do 2 rewrite < q_elim_minus;
+ rewrite > q_plus_minus; rewrite > q_plus_OQ;
+ rewrite > q_d_sym; rewrite > q_d_noabs;
+ [2,4: [apply q_eq_to_le; symmetry|apply q_lt_to_le] assumption]
+ apply q_eq_to_le; reflexivity;
+ |*: apply (q_le_lt_trans ??? H11);
+ rewrite > q_d_sym; rewrite > q_d_noabs;
+ [2,4: [apply q_eq_to_le; symmetry|apply q_lt_to_le] assumption]
+ generalize in match H9; rewrite > q_d_sym; rewrite > q_d_noabs;
+ [2,4: assumption]
+ rewrite > H2; intro X;
+ lapply (q_lt_inj_plus_r ?? (Qopp (start l1-init)) X) as Y; clear X;
+ rewrite < q_plus_assoc in Y; repeat rewrite < q_elim_minus in Y;
+ rewrite > q_plus_minus in Y; rewrite > q_plus_OQ in Y;
+ apply (q_le_lt_trans ???? Y);
+ rewrite > (q_elim_minus (start l1) init); rewrite > q_minus_distrib;
+ rewrite > q_elim_opp; repeat rewrite > q_elim_minus;
+ rewrite < q_plus_assoc; rewrite > (q_plus_sym ? init);
+ rewrite > q_plus_assoc;rewrite < q_plus_assoc in ⊢ (? ? (? % ?));
+ rewrite > (q_plus_sym ? init); rewrite < (q_elim_minus init);
+ rewrite > q_plus_minus; rewrite > q_plus_OQ;
+ apply q_eq_to_le; reflexivity;]
+ |2,6: rewrite > Hb; intro W; destruct W;
+ |3,7: [apply q_eq_to_le;symmetry|apply q_lt_to_le] assumption;
+ |4,8: apply (q_lt_le_trans ??? H7); rewrite > H2;
+ rewrite > q_plus_sym; rewrite < q_plus_assoc;
+ rewrite > q_plus_sym; apply q_le_inj_plus_r;
+ apply q_le_minus; apply q_eq_to_le; reflexivity;]]]
+qed.
+
+
+