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4 (* ||A|| A project by Andrea Asperti *)
6 (* ||I|| Developers: *)
7 (* ||T|| The HELM team. *)
8 (* ||A|| http://helm.cs.unibo.it *)
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11 (* v GNU General Public License Version 2 *)
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15 include "nat_ordered_set.ma".
16 include "models/q_support.ma".
17 include "models/list_support.ma".
18 include "cprop_connectives.ma".
20 definition bar ≝ ratio × ℚ. (* base (Qpos) , height *)
21 record q_f : Type ≝ { start : ℚ; bars: list bar }.
23 notation < "\rationals \sup 2" non associative with precedence 90 for @{'q2}.
24 interpretation "Q x Q" 'q2 = (Prod Q Q).
26 definition empty_bar : bar ≝ 〈one,OQ〉.
27 notation "\rect" with precedence 90 for @{'empty_bar}.
28 interpretation "q0" 'empty_bar = empty_bar.
30 notation < "\ldots\rect\square\EmptySmallSquare\ldots" with precedence 90 for @{'lq2}.
31 interpretation "lq2" 'lq2 = (list bar).
33 let rec sum_bases (l:list bar) (i:nat) on i ≝
38 [ nil ⇒ sum_bases [] m + Qpos one
39 | cons x tl ⇒ sum_bases tl m + Qpos (\fst x)]].
41 axiom sum_bases_empty_nat_of_q_ge_OQ:
42 ∀q:ℚ.OQ ≤ sum_bases [] (nat_of_q q).
43 axiom sum_bases_empty_nat_of_q_le_q:
44 ∀q:ℚ.sum_bases [] (nat_of_q q) ≤ q.
45 axiom sum_bases_empty_nat_of_q_le_q_one:
46 ∀q:ℚ.q < sum_bases [] (nat_of_q q) + Qpos one.
48 lemma sum_bases_ge_OQ:
49 ∀l,n. OQ ≤ sum_bases l n.
50 intro; elim l; simplify; intros;
51 [1: elim n; [apply q_eq_to_le;reflexivity] simplify;
52 apply q_le_plus_trans; try assumption; apply q_lt_to_le; apply q_pos_lt_OQ;
53 |2: cases n; [apply q_eq_to_le;reflexivity] simplify;
54 apply q_le_plus_trans; [apply H| apply q_lt_to_le; apply q_pos_lt_OQ;]]
57 alias symbol "leq" = "Q less or equal than".
59 ∀l.∀x.sum_bases l x ≤ OQ → x = O.
60 intros; cases x in H; [intros; reflexivity] intro; cases (?:False);
61 cases (q_le_cases ?? H);
62 [1: apply (q_lt_corefl OQ); rewrite < H1 in ⊢ (?? %);
63 |2: apply (q_lt_antisym ??? H1);] clear H H1; cases l;
64 simplify; apply q_lt_plus_trans;
65 try apply q_pos_lt_OQ;
66 try apply (sum_bases_ge_OQ []);
67 apply (sum_bases_ge_OQ l1);
71 lemma sum_bases_increasing:
72 ∀l.∀n1,n2:nat.n1<n2→sum_bases l n1 < sum_bases l n2.
74 [1: intros 2; apply (cic:/matita/dama/nat_ordered_set/nat_elim2.con ???? n1 n2);
76 [1: intro X; cases (not_le_Sn_O ? X);
77 |2: simplify; intros; apply q_lt_plus_trans;
78 [1: apply sum_bases_ge_OQ;|2: apply (q_pos_lt_OQ one)]]
79 |2: simplify; intros; cases (not_le_Sn_O ? H);
80 |3: simplify; intros; apply q_lt_inj_plus_r;
81 apply H; apply le_S_S_to_le; apply H1;]
82 |2: intros 5; apply (cic:/matita/dama/nat_ordered_set/nat_elim2.con ???? n1 n2);
83 [1: simplify; intros; cases n in H1; intros;
84 [1: cases (not_le_Sn_O ? H1);
85 |2: simplify; apply q_lt_plus_trans;
86 [1: apply sum_bases_ge_OQ;|2: apply q_pos_lt_OQ]]
87 |2: simplify; intros; cases (not_le_Sn_O ? H1);
88 |3: simplify; intros; apply q_lt_inj_plus_r; apply H;
89 apply le_S_S_to_le; apply H2;]]
94 sum_bases l n < sum_bases l (S m) →
95 sum_bases l m < sum_bases l (S n) →
97 intros 2; apply (nat_elim2 ???? n m);
98 [1: intro X; cases X; intros; [reflexivity] cases (?:False);
99 cases l in H H1; simplify; intros;
100 apply (q_lt_le_incompat ??? (sum_bases_ge_OQ ? n1));
101 apply (q_lt_canc_plus_r ??? H1);
102 |2: intros 2; cases l; simplify; intros; cases (?:False);
103 apply (q_lt_le_incompat ??? (sum_bases_ge_OQ ? n1));
104 apply (q_lt_canc_plus_r ??? H); (* magia ... *)
105 |3: intros 4; cases l; simplify; intros;
106 [1: rewrite > (H []); [reflexivity]
107 apply (q_lt_canc_plus_r ??(Qpos one)); assumption;
108 |2: rewrite > (H l1); [reflexivity]
109 apply (q_lt_canc_plus_r ??(Qpos (\fst b))); assumption;]]
113 λP.λp:∃x:nat × ℚ.P x.match p with [ex_introT p _ ⇒ p].
115 definition inject1 ≝ λP.λp:nat × ℚ.λh:P p. ex_introT ? P p h.
116 coercion inject1 with 0 1 nocomposites.
119 ∀f:q_f.∀i:ℚ.∃p:nat × ℚ.
121 (And3 (i < start f) (\fst p = O) (\snd p = OQ))
123 (start f + sum_bases (bars f) (len (bars f)) ≤ i)
124 (\fst p = O) (\snd p = OQ))
125 (And3 (bars f = []) (\fst p = O) (\snd p = OQ))
127 (And3 (bars f ≠ []) (start f ≤ i) (i < start f + sum_bases (bars f) (len (bars f))))
128 (\fst p ≤ (len (bars f)))
129 (\snd p = \snd (nth (bars f) ▭ (\fst p)))
130 (sum_bases (bars f) (\fst p) ≤ ⅆ[i,start f] ∧
131 (ⅆ[i, start f] < sum_bases (bars f) (S (\fst p))))).
134 let rec value (p: ℚ) (l : list bar) on l ≝
136 [ nil ⇒ 〈nat_of_q p,OQ〉
138 match q_cmp p (Qpos (\fst x)) with
139 [ q_lt _ ⇒ 〈O, \snd x〉
141 let rc ≝ value (p - Qpos (\fst x)) tl in
142 〈S (\fst rc),\snd rc〉]]
144 ∀acc,l.∃p:nat × ℚ.OQ ≤ acc →
146 (And3 (l = []) (\fst p = nat_of_q acc) (\snd p = OQ))
148 (sum_bases l (\fst p) ≤ acc)
149 (acc < sum_bases l (S (\fst p)))
150 (\snd p = \snd (nth l ▭ (\fst p)))));
152 cases (q_cmp i (start f));
153 [2: exists [apply 〈O,OQ〉] simplify; constructor 1; split; try assumption;
154 try reflexivity; apply q_lt_to_le; assumption;
155 |1: cases (bars f); [exists [apply 〈O,OQ〉] simplify; constructor 3; split;try assumption;reflexivity;]
156 cases (value ⅆ[i,start f] (b::l)) (p Hp);
157 cases (Hp (q_dist_ge_OQ ? ?)); clear Hp value; [cases H1; destruct H2]
158 cases H1; clear H1; lapply (sum_bases_O (b::l) (\fst p)) as H1;
159 [2: apply (q_le_trans ??? H2); rewrite > H; apply q_eq_to_le;
160 rewrite > q_d_x_x; reflexivity;
161 |1: exists [apply p] simplify; constructor 4; rewrite > H1; split;
162 try split; try rewrite > q_d_x_x; try autobatch depth=2;
163 [1: rewrite > H; rewrite > q_plus_sym; apply q_lt_plus;
164 rewrite > q_plus_minus; apply q_lt_plus_trans; [apply sum_bases_ge_OQ]
166 |2: rewrite > H; rewrite > q_d_x_x; apply q_eq_to_le; reflexivity;
167 |3: rewrite > H; rewrite > q_d_x_x; apply q_lt_plus_trans;
168 try apply sum_bases_ge_OQ; apply q_pos_lt_OQ;]]
169 |3: cases (q_cmp i (start f+sum_bases (bars f) (len (bars f))));
170 [1: exists [apply 〈O,OQ〉] simplify; constructor 2; split; try assumption;
171 try reflexivity; rewrite > H1; apply q_eq_to_le; reflexivity;
172 |3: exists [apply 〈O,OQ〉] simplify; constructor 2; split; try assumption;
173 try reflexivity; apply q_lt_to_le; assumption;
174 |2: generalize in match (refl_eq ? (bars f): bars f = bars f);
175 generalize in match (bars f) in ⊢ (??? % → %); intro X; cases X; clear X;
177 [1: exists [apply 〈O,OQ〉] simplify; constructor 3; split; reflexivity;
178 |2: cases (value ⅆ[i,start f] (b::l)) (p Hp);
179 cases (Hp (q_dist_ge_OQ ? ?)); clear Hp value; [cases H3;destruct H4]
181 exists [apply p]; constructor 4; split; try split; try assumption;
182 [1: intro X; destruct X;
183 |2: apply q_lt_to_le; assumption;
184 |3: rewrite < H2; assumption;
185 |4: cases (cmp_nat (\fst p) (len (bars f)));
186 [1:apply lt_to_le;rewrite <H2; assumption|rewrite > H3;rewrite < H2;apply le_n]
187 cases (?:False); cases (\fst p) in H3 H4 H6; clear H5;
188 [1: intros; apply (not_le_Sn_O ? H5);
189 |2: rewrite > q_d_sym; rewrite > q_d_noabs; [2: apply q_lt_to_le; assumption]
190 intros; lapply (q_lt_inj_plus_r ?? (Qopp (start f)) H1); clear H1;
191 generalize in match Hletin;
192 rewrite > (q_plus_sym (start f)); rewrite < q_plus_assoc;
193 do 2 rewrite < q_elim_minus; rewrite > q_plus_minus;
194 rewrite > q_plus_OQ; intro K; apply (q_lt_corefl (i-start f));
195 apply (q_lt_le_trans ???? H3); rewrite < H2;
196 apply (q_lt_trans ??? K); apply sum_bases_increasing;
198 |1,3: intros; right; split;
199 [1,4: clear H2; cases (value (q-Qpos (\fst b)) l1);
200 cases (H2 (q_le_to_diff_ge_OQ ?? (? H1)));
201 [1: intro; apply q_lt_to_le;assumption;
202 |3: simplify; cases H4; apply q_le_minus; assumption;
203 |2,5: simplify; cases H4; rewrite > H5; rewrite > H6;
204 apply q_le_minus; apply sum_bases_empty_nat_of_q_le_q;
205 |4: intro X; rewrite > X; apply q_eq_to_le; reflexivity;
206 |*: simplify; apply q_le_minus; cases H4; assumption;]
207 |2,5: cases (value (q-Qpos (\fst b)) l1);
208 cases (H4 (q_le_to_diff_ge_OQ ?? (? H1)));
209 [1,4: intros; [apply q_lt_to_le|apply q_eq_to_le;symmetry] assumption;
210 |3,6: cases H5; simplify; change with (q < sum_bases l1 (S (\fst w)) + Qpos (\fst b));
211 apply q_lt_plus; assumption;
212 |2,5: simplify; cases H5; rewrite > H6; simplify; rewrite > H7;
213 apply q_lt_plus; apply sum_bases_empty_nat_of_q_le_q_one;]
214 |*: cases (value (q-Qpos (\fst b)) l1); simplify;
215 cases (H4 (q_le_to_diff_ge_OQ ?? (? H1)));
216 [1,4: intros; [apply q_lt_to_le|apply q_eq_to_le;symmetry] assumption;
217 |3,6: cases H5; assumption;
218 |*: cases H5; rewrite > H6; rewrite > H8;
219 elim (\fst w); [1,3:reflexivity;] simplify; assumption;]]
220 |2: clear value H2; simplify; intros; right; split; [assumption|3:reflexivity]
221 rewrite > q_plus_sym; rewrite > q_plus_OQ; assumption;
222 |4: intros; left; split; reflexivity;]
226 ∀l,i.i < start l → \snd (\fst (value l i)) = OQ.
227 intros; cases (value l i) (q Hq); cases Hq; clear Hq; simplify; cases H1; clear H1;
228 try assumption; cases H2; cases (?:False); apply (q_lt_le_incompat ?? H H6);
232 ∀l,i.start l + sum_bases (bars l) (len (bars l)) ≤ i → \snd (\fst (value l i)) = OQ.
233 intros; cases (value l i) (q Hq); cases Hq; clear Hq; simplify; cases H1; clear H1;
234 try assumption; cases H2; cases (?:False); apply (q_lt_le_incompat ?? H7 H);
238 ∀l,i.bars l = [] → \snd (\fst (value l i)) = OQ.
239 intros; cases (value l i) (q Hq); cases Hq; clear Hq; simplify; cases H1; clear H1;
240 try assumption; cases H2; cases (?:False); apply (H1 H);
243 inductive value_ok_spec (f : q_f) (i : ℚ) : nat × ℚ → Type ≝
244 | value_ok : ∀n,q. n ≤ (len (bars f)) →
245 q = \snd (nth (bars f) ▭ n) →
246 sum_bases (bars f) n ≤ ⅆ[i,start f] →
247 ⅆ[i, start f] < sum_bases (bars f) (S n) → value_ok_spec f i 〈n,q〉.
250 ∀f,i.bars f ≠ [] → start f ≤ i → i < start f + sum_bases (bars f) (len (bars f)) →
251 value_ok_spec f i (\fst (value f i)).
252 intros; cases (value f i); simplify;
253 cases H3; simplify; clear H3; cases H4; clear H4;
254 [1,2,3: cases (?:False);
255 [1: apply (q_lt_le_incompat ?? H3 H1);
256 |2: apply (q_lt_le_incompat ?? H2 H3);
258 |4: cases H7; clear H7; cases w in H3 H4 H5 H6 H8; simplify; intros;
259 constructor 1; assumption;]
262 definition same_values ≝
264 ∀input.\snd (\fst (value l1 input)) = \snd (\fst (value l2 input)).
266 definition same_bases ≝
267 λl1,l2:list bar. (∀i.\fst (nth l1 ▭ i) = \fst (nth l2 ▭ i)).
269 alias symbol "lt" = "Q less than".
270 lemma unpos: ∀x:ℚ.OQ < x → ∃r:ratio.Qpos r = x.
271 intro; cases x; intros; [2:exists [apply r] reflexivity]
273 [ apply (q_lt_corefl ? H)|apply (q_neg_gt ? H)]
276 notation < "x \blacksquare" non associative with precedence 50 for @{'unpos $x}.
277 interpretation "hide unpos proof" 'unpos x = (unpos x _).