1 (**************************************************************************)
4 (* ||A|| A project by Andrea Asperti *)
6 (* ||I|| Developers: *)
7 (* ||T|| The HELM team. *)
8 (* ||A|| http://helm.cs.unibo.it *)
10 (* \ / This file is distributed under the terms of the *)
11 (* v GNU General Public License Version 2 *)
13 (**************************************************************************)
15 include "models/q_support.ma".
16 include "models/list_support.ma".
17 include "cprop_connectives.ma".
19 definition bar ≝ ratio × ℚ. (* base (Qpos) , height *)
20 record q_f : Type ≝ { start : ℚ; bars: list bar }.
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 ≝ 〈one,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 let rec sum_bases (l:list bar) (i:nat) on i ≝
37 [ nil ⇒ sum_bases l m + Qpos one
38 | cons x tl ⇒ sum_bases tl m + Qpos (\fst x)]].
40 axiom sum_bases_empty_nat_of_q_ge_OQ:
41 ∀q:ℚ.OQ ≤ sum_bases [] (nat_of_q q).
42 axiom sum_bases_empty_nat_of_q_le_q:
43 ∀q:ℚ.sum_bases [] (nat_of_q q) ≤ q.
44 axiom sum_bases_empty_nat_of_q_le_q_one:
45 ∀q:ℚ.q < sum_bases [] (nat_of_q q) + Qpos one.
48 λP.λp:∃x:nat × ℚ.P x.match p with [ex_introT p _ ⇒ p].
50 definition inject1 ≝ λP.λp:nat × ℚ.λh:P p. ex_introT ? P p h.
51 coercion inject1 with 0 1 nocomposites.
54 ∀f:q_f.∀i:ℚ.∃p:nat × ℚ.
55 match q_cmp i (start f) with
56 [ q_lt _ ⇒ \snd p = OQ
59 (sum_bases (bars f) (\fst p) ≤ ⅆ[i,start f])
60 (ⅆ[i, start f] < sum_bases (bars f) (S (\fst p)))
61 (\snd p = \snd (nth (bars f) ▭ (\fst p)))].
63 alias symbol "pi2" = "pair pi2".
64 alias symbol "pi1" = "pair pi1".
66 let rec value (p: ℚ) (l : list bar) on l ≝
68 [ nil ⇒ 〈nat_of_q p,OQ〉
70 match q_cmp p (Qpos (\fst x)) with
71 [ q_lt _ ⇒ 〈O, \snd x〉
73 let rc ≝ value (p - Qpos (\fst x)) tl in
74 〈S (\fst rc),\snd rc〉]]
76 ∀acc,l.∃p:nat × ℚ. OQ ≤ acc →
78 (sum_bases l (\fst p) ≤ acc)
79 (acc < sum_bases l (S (\fst p)))
80 (\snd p = \snd (nth l ▭ (\fst p))));
82 cases (q_cmp i (start f));
83 [2: exists [apply 〈O,OQ〉] simplify; reflexivity;
84 |*: cases (value ⅆ[i,start f] (bars f)) (p Hp);
85 cases (Hp (q_dist_ge_OQ ? ?)); clear Hp value;
86 exists[1,3:apply p]; simplify; split; assumption;]
88 [1,4: clear H2; cases (value (q-Qpos (\fst b)) l1);
89 cases (H2 (q_le_to_diff_ge_OQ ?? (? H1)));
90 [1,3: intros; [apply q_lt_to_le|apply q_eq_to_le;symmetry] assumption]
91 simplify; apply q_le_minus; assumption;
92 |2,5: cases (value (q-Qpos (\fst b)) l1);
93 cases (H4 (q_le_to_diff_ge_OQ ?? (? H1)));
94 [1,3: intros; [apply q_lt_to_le|apply q_eq_to_le;symmetry] assumption]
96 change with (q < sum_bases l1 (S (\fst w)) + Qpos (\fst b));
97 apply q_lt_plus; assumption;
98 |*: cases (value (q-Qpos (\fst b)) l1); simplify;
99 cases (H4 (q_le_to_diff_ge_OQ ?? (? H1)));
100 [1,3: intros; [apply q_lt_to_le|apply q_eq_to_le;symmetry] assumption]
102 |2: clear value H2; simplify; intros; split; [assumption|3:reflexivity]
103 rewrite > q_plus_sym; rewrite > q_plus_OQ; assumption;
104 |4: simplify; intros; split;
105 [1: apply sum_bases_empty_nat_of_q_le_q;
106 |2: apply sum_bases_empty_nat_of_q_le_q_one;
107 |3: elim (nat_of_q q); [reflexivity] simplify; assumption]]
111 definition same_values ≝
113 ∀input.\snd (\fst (value l1 input)) = \snd (\fst (value l2 input)).
115 definition same_bases ≝
117 (∀i.\fst (nth (bars l1) ▭ i) = \fst (nth (bars l2) ▭ i)).
119 alias symbol "lt" = "Q less than".
120 lemma unpos: ∀x:ℚ.OQ < x → ∃r:ratio.Qpos r = x.
121 intro; cases x; intros; [2:exists [apply r] reflexivity]
123 [ apply (q_lt_corefl ? H)|apply (q_neg_gt ? H)]
126 notation < "\blacksquare" non associative with precedence 90 for @{'hide}.
127 definition hide ≝ λT:Type.λx:T.x.
128 interpretation "hide" 'hide = (hide _ _).
130 lemma sum_bases_ge_OQ:
131 ∀l,n. OQ ≤ sum_bases l n.
132 intro; elim l; simplify; intros;
133 [1: elim n; [apply q_eq_to_le;reflexivity] simplify;
134 apply q_le_plus_trans; try assumption; apply q_lt_to_le; apply q_pos_lt_OQ;
135 |2: cases n; [apply q_eq_to_le;reflexivity] simplify;
136 apply q_le_plus_trans; [apply H| apply q_lt_to_le; apply q_pos_lt_OQ;]]
140 ∀l.∀x.sum_bases l x ≤ OQ → x = O.
141 intros; cases x in H; [intros; reflexivity] intro; cases (?:False);
142 cases (q_le_cases ?? H);
143 [1: apply (q_lt_corefl OQ); rewrite < H1 in ⊢ (?? %);
144 |2: apply (q_lt_antisym ??? H1);] clear H H1; cases l;
145 simplify; apply q_lt_plus_trans;
146 try apply q_pos_lt_OQ;
147 try apply (sum_bases_ge_OQ []);
148 apply (sum_bases_ge_OQ l1);
151 lemma sum_bases_increasing:
152 ∀l,x.sum_bases l x < sum_bases l (S x).
155 [1: simplify; rewrite > q_plus_sym; rewrite > q_plus_OQ;
157 |2: simplify in H ⊢ %;
158 apply q_lt_plus; rewrite > q_elim_minus;
159 rewrite < q_plus_assoc; rewrite < q_elim_minus;
160 rewrite > q_plus_minus; rewrite > q_plus_OQ;
163 [1: simplify; rewrite > q_plus_sym; rewrite > q_plus_OQ;
165 |2: simplify; change in ⊢ (? ? (? % ?)) with (sum_bases l1 (S n)) ;
166 apply q_lt_plus; rewrite > q_elim_minus;
167 rewrite < q_plus_assoc; rewrite < q_elim_minus;
168 rewrite > q_plus_minus; rewrite > q_plus_OQ; apply H]]
171 lemma sum_bases_lt_canc:
172 ∀l,x,y.sum_bases l (S x) < sum_bases l (S y) → sum_bases l x < sum_bases l y.
173 intro; elim l; [apply (q_lt_canc_plus_r ?? (Qpos one));apply H]
174 generalize in match H1;apply (nat_elim2 (?:? → ? → CProp) ??? x y);
176 [3: intros 2; simplify; apply q_lt_inj_plus_r; apply H;
177 apply (q_lt_canc_plus_r ?? (Qpos (\fst a))); apply H3;
178 |2: cases (?:False); simplify in H2;
179 apply (q_lt_le_incompat (sum_bases l1 (S n)) OQ);[2: apply sum_bases_ge_OQ;]
180 apply (q_lt_canc_plus_r ?? (Qpos (\fst a))); apply H2;
181 |1: cases n in H2; intro;
182 [1: cases (?:False); apply (q_lt_corefl ? H2);
183 |2: simplify; apply q_lt_plus_trans; [apply sum_bases_ge_OQ]