--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "dama/nat_ordered_set.ma".
+include "models/q_support.ma".
+include "models/list_support.ma".
+include "logic/cprop_connectives.ma".
+
+definition bar ≝ ℚ × (ℚ × ℚ).
+
+notation < "\rationals \sup 2" non associative with precedence 90 for @{'q2}.
+interpretation "Q x Q" 'q2 = (Prod Q Q).
+
+definition empty_bar : bar ≝ 〈Qpos one,〈OQ,OQ〉〉.
+notation "\rect" with precedence 90 for @{'empty_bar}.
+interpretation "q0" 'empty_bar = empty_bar.
+
+notation < "\ldots\rect\square\EmptySmallSquare\ldots" with precedence 90 for @{'lq2}.
+interpretation "lq2" 'lq2 = (list bar).
+
+definition q2_lt := mk_rel bar (λx,y:bar.\fst x < \fst y).
+
+interpretation "bar lt" 'lt x y = (rel_op ? q2_lt x y).
+
+lemma q2_trans : ∀a,b,c:bar. a < b → b < c → a < c.
+intros 3; cases a; cases b; cases c; unfold q2_lt; simplify; intros;
+apply (q_lt_trans ??? H H1);
+qed.
+
+definition q2_trel := mk_trans_rel bar q2_lt q2_trans.
+
+interpretation "bar lt" 'lt x y = (FunClass_2_OF_trans_rel q2_trel x y).
+
+definition canonical_q_lt : rel bar → trans_rel ≝ λx:rel bar.q2_trel.
+
+coercion canonical_q_lt with nocomposites.
+
+interpretation "bar lt" 'lt x y = (FunClass_2_OF_trans_rel (canonical_q_lt ?) x y).
+
+definition nth_base ≝ λf,n. \fst (\nth f ▭ n).
+definition nth_height ≝ λf,n. \snd (\nth f ▭ n).
+
+record q_f : Type ≝ {
+ bars: list bar;
+ bars_sorted : sorted q2_lt bars;
+ bars_begin_OQ : nth_base bars O = OQ;
+ bars_end_OQ : nth_height bars (pred (\len bars)) = 〈OQ,OQ〉
+}.
+
+lemma len_bases_gt_O: ∀f.O < \len (bars f).
+intros; generalize in match (bars_begin_OQ f); cases (bars f); intros;
+[2: simplify; apply le_S_S; apply le_O_n;
+|1: normalize in H; destruct H;]
+qed.
+
+lemma all_bases_positive : ∀f:q_f.∀i. OQ < nth_base (bars f) (S i).
+intro f; generalize in match (bars_begin_OQ f); generalize in match (bars_sorted f);
+cases (len_gt_non_empty ?? (len_bases_gt_O f)); intros;
+cases (cmp_nat (\len l) i);
+[2: lapply (sorted_tail_bigger q2_lt ?? ▭ H ? H2) as K;
+ simplify in H1; rewrite < H1; apply K;
+|1: simplify; elim l in i H2;[simplify; rewrite > nth_nil; apply (q_pos_OQ one)]
+ cases n in H3; intros; [simplify in H3; cases (not_le_Sn_O ? H3)]
+ apply (H2 n1); simplify in H3; apply (le_S_S_to_le ?? H3);]
+qed.
+
+alias symbol "lt" (instance 9) = "Q less than".
+alias symbol "lt" (instance 7) = "natural 'less than'".
+alias symbol "lt" (instance 6) = "natural 'less than'".
+alias symbol "lt" (instance 5) = "Q less than".
+alias symbol "lt" (instance 4) = "natural 'less than'".
+alias symbol "lt" (instance 2) = "natural 'less than'".
+alias symbol "leq" = "Q less or equal than".
+coinductive value_spec (f : list bar) (i : ℚ) : ℚ × ℚ → CProp ≝
+| value_of : ∀j,q.
+ nth_height f j = q → nth_base f j < i → j < \len f →
+ (∀n.n<j → nth_base f n < i) →
+ (∀n.j < n → n < \len f → i ≤ nth_base f n) → value_spec f i q.
+
+definition match_pred ≝
+ λi.λx:bar.match q_cmp (Qpos i) (\fst x) with[ q_leq _ ⇒ true| q_gt _ ⇒ false].
+
+definition match_domain ≝
+ λf: list bar.λi:ratio. pred (find ? (match_pred i) f ▭).
+
+definition value_simple ≝
+ λf: list bar.λi:ratio. nth_height f (match_domain f i).
+
+alias symbol "lt" (instance 5) = "Q less than".
+alias symbol "lt" (instance 6) = "natural 'less than'".
+definition value_lemma :
+ ∀f:list bar.sorted q2_lt f → O < length bar f →
+ ∀i:ratio.nth_base f O < Qpos i → ∃p:ℚ×ℚ.value_spec f (Qpos i) p.
+intros (f bars_sorted_f len_bases_gt_O_f i bars_begin_OQ_f);
+exists [apply (value_simple f i);]
+apply (value_of ?? (match_domain f i));
+[1: reflexivity
+|2: unfold match_domain; cases (cases_find bar (match_pred i) f ▭);
+ [1: cases i1 in H H1 H2 H3; simplify; intros;
+ [1: generalize in match (bars_begin_OQ_f);
+ cases (len_gt_non_empty ?? (len_bases_gt_O_f)); simplify; intros;
+ assumption;
+ |2: cases (len_gt_non_empty ?? (len_bases_gt_O_f)) in H3;
+ intros; lapply (H3 n (le_n ?)) as K; unfold match_pred in K;
+ cases (q_cmp (Qpos i) (\fst (\nth (x::l) ▭ n))) in K;
+ simplify; intros; [destruct H5] assumption]
+ |2: destruct H; cases (len_gt_non_empty ?? (len_bases_gt_O_f)) in H2;
+ simplify; intros; lapply (H (\len l) (le_n ?)) as K; clear H;
+ unfold match_pred in K; cases (q_cmp (Qpos i) (\fst (\nth (x::l) ▭ (\len l)))) in K;
+ simplify; intros; [destruct H2] assumption;]
+|5: intro; unfold match_domain; cases (cases_find bar (match_pred i) f ▭); intros;
+ [1: generalize in match (bars_sorted_f);
+ cases (list_break ??? H) in H1; rewrite > H6;
+ rewrite < H1; simplify; rewrite > nth_len; unfold match_pred;
+ cases (q_cmp (Qpos i) (\fst x)); simplify;
+ intros (X Hs); [2: destruct X] clear X;
+ cases (sorted_pivot q2_lt ??? ▭ Hs);
+ cut (\len l1 ≤ n) as Hn; [2:
+ rewrite > H1; cases i1 in H4; simplify; intro X; [2: assumption]
+ apply lt_to_le; assumption;]
+ unfold nth_base; rewrite > (nth_append_ge_len ????? Hn);
+ cut (n - \len l1 < \len (x::l2)) as K; [2:
+ simplify; rewrite > H1; rewrite > (?:\len l2 = \len f - \len (l1 @ [x]));[2:
+ rewrite > H6; repeat rewrite > len_append; simplify;
+ repeat rewrite < plus_n_Sm; rewrite < plus_n_O; simplify;
+ rewrite > sym_plus; rewrite < minus_plus_m_m; reflexivity;]
+ rewrite > len_append; rewrite > H1; simplify; rewrite < plus_n_SO;
+ apply le_S_S; clear H1 H6 H7 Hs H8 H9 Hn x l2 l1 H4 H3 H2 H;
+ elim (\len f) in i1 n H5; [cases (not_le_Sn_O ? H);]
+ simplify; cases n2; [ repeat rewrite < minus_n_O; apply le_S_S_to_le; assumption]
+ cases n1 in H1; [intros; rewrite > eq_minus_n_m_O; apply le_O_n]
+ intros; simplify; apply H; apply le_S_S_to_le; assumption;]
+ cases (n - \len l1) in K; simplify; intros; [ assumption]
+ lapply (H9 ? (le_S_S_to_le ?? H10)) as W; apply (q_le_trans ??? H7);
+ apply q_lt_to_le; apply W;
+ |2: cases (not_le_Sn_n i1); rewrite > H in ⊢ (??%);
+ apply (trans_le ??? ? H4); cases i1 in H3; intros; apply le_S_S;
+ [ apply le_O_n; | assumption]]
+|3: unfold match_domain; cases (cases_find bar (match_pred i) f ▭); [
+ cases i1 in H; intros; simplify; [assumption]
+ apply lt_S_to_lt; assumption;]
+ rewrite > H; cases (\len f) in len_bases_gt_O_f; intros; [cases (not_le_Sn_O ? H3)]
+ simplify; apply le_n;
+|4: intros; unfold match_domain in H; cases (cases_find bar (match_pred i) f ▭) in H; simplify; intros;
+ [1: lapply (H3 n); [2: cases i1 in H4; intros [assumption] apply le_S; assumption;]
+ unfold match_pred in Hletin; cases (q_cmp (Qpos i) (\fst (\nth f ▭ n))) in Hletin;
+ simplify; intros; [destruct H6] assumption;
+ |2: destruct H; cases f in len_bases_gt_O_f H2 H3; clear H1; simplify; intros;
+ [cases (not_le_Sn_O ? H)] lapply (H1 n); [2: apply le_S; assumption]
+ unfold match_pred in Hletin; cases (q_cmp (Qpos i) (\fst (\nth (b::l) ▭ n))) in Hletin;
+ simplify; intros; [destruct H4] assumption;]]
+qed.
+
+lemma bars_begin_lt_Qpos : ∀q,r. nth_base (bars q) O<Qpos r.
+intros; rewrite > bars_begin_OQ; apply q_pos_OQ;
+qed.
+
+lemma value : q_f → ratio → ℚ × ℚ.
+intros; cases (value_lemma (bars q) ?? r);
+[ apply bars_sorted.
+| apply len_bases_gt_O;
+| apply w;
+| apply bars_begin_lt_Qpos;]
+qed.
+
+lemma cases_value : ∀f,i. value_spec (bars f) (Qpos i) (value f i).
+intros; unfold value;
+cases (value_lemma (bars f) (bars_sorted f) (len_bases_gt_O f) i (bars_begin_lt_Qpos f i));
+assumption;
+qed.
+
+definition same_values ≝ λl1,l2:q_f.∀input. value l1 input = value l2 input.
+
+definition same_bases ≝ λl1,l2:list bar. ∀i.\fst (\nth l1 ▭ i) = \fst (\nth l2 ▭ i).
+
+lemma same_bases_cons: ∀a,b,l1,l2.
+ same_bases l1 l2 → \fst a = \fst b → same_bases (a::l1) (b::l2).
+intros; intro; cases i; simplify; [assumption;] apply (H n);
+qed.
+
+alias symbol "lt" = "Q less than".
+lemma unpos: ∀x:ℚ.OQ < x → ∃r:ratio.Qpos r = x.
+intro; cases x; intros; [2:exists [apply r] reflexivity]
+cases (?:False);
+[ apply (q_lt_corefl ? H)| cases (q_lt_le_incompat ?? (q_neg_gt ?) (q_lt_to_le ?? H))]
+qed.
+
+notation < "x \blacksquare" non associative with precedence 50 for @{'unpos $x}.
+interpretation "hide unpos proof" 'unpos x = (unpos x ?).
+
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