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4 (* ||A|| A project by Andrea Asperti *)
6 (* ||I|| Developers: *)
7 (* ||T|| The HELM team. *)
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15 include "nat_ordered_set.ma".
16 include "models/q_support.ma".
17 include "models/list_support.ma".
18 include "logic/cprop_connectives.ma".
20 definition bar ≝ ℚ × (ℚ × ℚ).
22 notation < "\rationals \sup 2" non associative with precedence 90 for @{'q2}.
23 interpretation "Q x Q" 'q2 = (Prod Q Q).
25 definition empty_bar : bar ≝ 〈Qpos one,〈OQ,OQ〉〉.
26 notation "\rect" with precedence 90 for @{'empty_bar}.
27 interpretation "q0" 'empty_bar = empty_bar.
29 notation < "\ldots\rect\square\EmptySmallSquare\ldots" with precedence 90 for @{'lq2}.
30 interpretation "lq2" 'lq2 = (list bar).
32 definition q2_lt := mk_rel bar (λx,y:bar.\fst x < \fst y).
34 interpretation "bar lt" 'lt x y = (rel_op _ q2_lt x y).
36 lemma q2_trans : ∀a,b,c:bar. a < b → b < c → a < c.
37 intros 3; cases a; cases b; cases c; unfold q2_lt; simplify; intros;
38 apply (q_lt_trans ??? H H1);
41 definition q2_trel := mk_trans_rel bar q2_lt q2_trans.
43 interpretation "bar lt" 'lt x y = (FunClass_2_OF_trans_rel q2_trel x y).
45 definition canonical_q_lt : rel bar → trans_rel ≝ λx:rel bar.q2_trel.
47 coercion canonical_q_lt with nocomposites.
49 interpretation "bar lt" 'lt x y = (FunClass_2_OF_trans_rel (canonical_q_lt _) x y).
51 definition nth_base ≝ λf,n. \fst (\nth f ▭ n).
52 definition nth_height ≝ λf,n. \snd (\nth f ▭ n).
56 bars_sorted : sorted q2_lt bars;
57 bars_begin_OQ : nth_base bars O = OQ;
58 bars_end_OQ : nth_height bars (pred (\len bars)) = 〈OQ,OQ〉
61 lemma len_bases_gt_O: ∀f.O < \len (bars f).
62 intros; generalize in match (bars_begin_OQ f); cases (bars f); intros;
63 [2: simplify; apply le_S_S; apply le_O_n;
64 |1: normalize in H; destruct H;]
67 lemma all_bases_positive : ∀f:q_f.∀i. OQ < nth_base (bars f) (S i).
68 intro f; generalize in match (bars_begin_OQ f); generalize in match (bars_sorted f);
69 cases (len_gt_non_empty ?? (len_bases_gt_O f)); intros;
70 cases (cmp_nat (\len l) i);
71 [2: lapply (sorted_tail_bigger q2_lt ?? ▭ H ? H2) as K;
72 simplify in H1; rewrite < H1; apply K;
73 |1: simplify; elim l in i H2;[simplify; rewrite > nth_nil; apply (q_pos_OQ one)]
74 cases n in H3; intros; [simplify in H3; cases (not_le_Sn_O ? H3)]
75 apply (H2 n1); simplify in H3; apply (le_S_S_to_le ?? H3);]
78 alias symbol "lt" (instance 5) = "Q less than".
79 alias symbol "lt" (instance 4) = "natural 'less than'".
80 alias symbol "lt" (instance 2) = "natural 'less than'".
81 alias symbol "leq" = "Q less or equal than".
82 alias symbol "Q" = "Rationals".
83 coinductive value_spec (f : q_f) (i : ℚ) : ℚ × ℚ → CProp ≝
85 nth_height (bars f) j = q → nth_base (bars f) j < i →
86 (∀n.j < n → n < \len (bars f) → i ≤ nth_base (bars f) n) → value_spec f i q.
88 definition value_lemma : ∀f:q_f.∀i:ratio.∃p:ℚ×ℚ.value_spec f (Qpos i) p.
91 (λx:bar.match q_cmp (Qpos i) (\fst x) with[ q_leq _ ⇒ true| q_gt _ ⇒ false]);
92 exists [apply (nth_height (bars f) (pred (find ? P (bars f) ▭)));]
93 apply (value_of ?? (pred (find ? P (bars f) ▭)));
95 |2: cases (cases_find bar P (bars f) ▭);
96 [1: cases i1 in H H1 H2 H3; simplify; intros;
97 [1: generalize in match (bars_begin_OQ f);
98 cases (len_gt_non_empty ?? (len_bases_gt_O f)); simplify; intros;
99 rewrite > H4; apply q_pos_OQ;
100 |2: cases (len_gt_non_empty ?? (len_bases_gt_O f)) in H3;
101 intros; lapply (H3 n (le_n ?)) as K; unfold P in K;
102 cases (q_cmp (Qpos i) (\fst (\nth (x::l) ▭ n))) in K;
103 simplify; intros; [destruct H5] assumption]
104 |2: destruct H; cases (len_gt_non_empty ?? (len_bases_gt_O f)) in H2;
105 simplify; intros; lapply (H (\len l) (le_n ?)) as K; clear H;
106 unfold P in K; cases (q_cmp (Qpos i) (\fst (\nth (x::l) ▭ (\len l)))) in K;
107 simplify; intros; [destruct H2] assumption;]
108 |3: intro; cases (cases_find bar P (bars f) ▭); intros;
109 [1: generalize in match (bars_sorted f);
110 cases (list_break ??? H) in H1; rewrite > H6;
111 rewrite < H1; simplify; rewrite > nth_len; unfold P;
112 cases (q_cmp (Qpos i) (\fst x)); simplify;
113 intros (X Hs); [2: destruct X] clear X;
114 cases (sorted_pivot q2_lt ??? ▭ Hs);
115 cut (\len l1 ≤ n) as Hn; [2:
116 rewrite > H1; cases i1 in H4; simplify; intro X; [2: assumption]
117 apply lt_to_le; assumption;]
118 unfold nth_base; rewrite > (nth_append_ge_len ????? Hn);
119 cut (n - \len l1 < \len (x::l2)) as K; [2:
120 simplify; rewrite > H1; rewrite > (?:\len l2 = \len (bars f) - \len (l1 @ [x]));[2:
121 rewrite > H6; repeat rewrite > len_append; simplify;
122 repeat rewrite < plus_n_Sm; rewrite < plus_n_O; simplify;
123 rewrite > sym_plus; rewrite < minus_plus_m_m; reflexivity;]
124 rewrite > len_append; rewrite > H1; simplify; rewrite < plus_n_SO;
125 apply le_S_S; clear H1 H6 H7 Hs H8 H9 Hn x l2 l1 H4 H3 H2 H P i;
126 elim (\len (bars f)) in i1 n H5; [cases (not_le_Sn_O ? H);]
127 simplify; cases n2; [ repeat rewrite < minus_n_O; apply le_S_S_to_le; assumption]
128 cases n1 in H1; [intros; rewrite > eq_minus_n_m_O; apply le_O_n]
129 intros; simplify; apply H; apply le_S_S_to_le; assumption;]
130 cases (n - \len l1) in K; simplify; intros; [ assumption]
131 lapply (H9 ? (le_S_S_to_le ?? H10)) as W; apply (q_le_trans ??? H7);
132 apply q_lt_to_le; apply W;
133 |2: cases (not_le_Sn_n i1); rewrite > H in ⊢ (??%);
134 apply (trans_le ??? ? H4); cases i1 in H3; intros; apply le_S_S;
135 [ apply le_O_n; | assumption]]]
138 lemma value : q_f → ratio → ℚ × ℚ.
139 intros; cases (value_lemma q r); apply w; qed.
141 lemma cases_value : ∀f,i. value_spec f (Qpos i) (value f i).
142 intros; unfold value; cases (value_lemma f i); assumption; qed.
144 definition same_values ≝ λl1,l2:q_f.∀input. value l1 input = value l2 input.
146 definition same_bases ≝ λl1,l2:list bar. ∀i.\fst (\nth l1 ▭ i) = \fst (\nth l2 ▭ i).
148 alias symbol "lt" = "Q less than".
149 lemma unpos: ∀x:ℚ.OQ < x → ∃r:ratio.Qpos r = x.
150 intro; cases x; intros; [2:exists [apply r] reflexivity]
152 [ apply (q_lt_corefl ? H)| cases (q_lt_le_incompat ?? (q_neg_gt ?) (q_lt_to_le ?? H))]
155 notation < "x \blacksquare" non associative with precedence 50 for @{'unpos $x}.
156 interpretation "hide unpos proof" 'unpos x = (unpos x _).