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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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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 ≝ 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〉.
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 inductive sorted : list bar → Prop ≝
33 | sorted_nil : sorted []
34 | sorted_one : ∀x. sorted [x]
35 | sorted_cons : ∀x,y,tl. \fst x < \fst y → sorted (y::tl) → sorted (x::y::tl).
37 definition nth_base ≝ λf,n. \fst (nth f ▭ n).
38 definition nth_height ≝ λf,n. \snd (nth f ▭ n).
42 bars_sorted : sorted bars;
43 bars_begin_OQ : nth_base bars O = OQ;
44 bars_tail_OQ : nth_height bars (pred (len bars)) = OQ
47 lemma nth_nil: ∀T,i.∀def:T. nth [] def i = def.
48 intros; elim i; simplify; [reflexivity;] assumption; qed.
50 inductive non_empty_list (A:Type) : list A → Type :=
51 | show_head: ∀x,l. non_empty_list A (x::l).
53 lemma bars_not_nil: ∀f:q_f.non_empty_list ? (bars f).
54 intro f; generalize in match (bars_begin_OQ f); cases (bars f);
55 [1: intro X; normalize in X; destruct X;
56 |2: intros; constructor 1;]
59 lemma sorted_tail: ∀x,l.sorted (x::l) → sorted l.
60 intros; inversion H; intros; [destruct H1;|destruct H1;constructor 1;]
61 destruct H4; assumption;
64 lemma sorted_skip: ∀x,y,l. sorted (x::y::l) → sorted (x::l).
65 intros; inversion H; intros; [1,2: destruct H1]
66 destruct H4; inversion H2; intros; [destruct H4]
67 [1: destruct H4; constructor 2;
68 |2: destruct H7; constructor 3; [apply (q_lt_trans ??? H1 H4);]
69 apply (sorted_tail ?? H2);]
72 lemma sorted_tail_bigger : ∀x,l.sorted (x::l) → ∀i. i < len l → \fst x < nth_base l i.
73 intros 2; elim l; [ cases (not_le_Sn_O i H1);]
75 [2: intros; apply (H ? n);[apply (sorted_skip ??? H1)|apply le_S_S_to_le; apply H2]
76 |1: intros; inversion H1; intros; [1,2: destruct H3]
77 destruct H6; simplify; assumption;]
80 lemma all_bases_positive : ∀f:q_f.∀i. OQ < nth_base (bars f) (S i).
81 intro f; generalize in match (bars_begin_OQ f); generalize in match (bars_sorted f);
82 cases (bars_not_nil f); intros;
83 cases (cmp_nat i (len l));
84 [1: lapply (sorted_tail_bigger ?? H ? H2) as K; simplify in H1;
85 rewrite > H1 in K; apply K;
86 |2: rewrite > H2; simplify; elim l; simplify; [apply (q_pos_OQ one)]
88 |3: simplify; elim l in i H2;[simplify; rewrite > nth_nil; apply (q_pos_OQ one)]
89 cases n in H3; intros; [cases (not_le_Sn_O ? H3)] apply (H2 n1);
90 apply (le_S_S_to_le ?? H3);]
93 definition eject_NxQ ≝
94 λP.λp:∃x:nat × ℚ.P x.match p with [ex_introT p _ ⇒ p].
96 definition inject_NxQ ≝ λP.λp:nat × ℚ.λh:P p. ex_introT ? P p h.
97 coercion inject_NxQ with 0 1 nocomposites.
99 definition value_spec : q_f → ℚ → nat × ℚ → Prop ≝
100 λf,i,q. nth_height (bars f) (\fst q) = \snd q ∧
101 (nth_base (bars f) (\fst q) < i ∧
102 ∀n.\fst q < n → n < len (bars f) → i ≤ nth_base (bars f) n).
104 definition value : ∀f:q_f.∀i:ratio.∃p:ℚ.∃j.value_spec f (Qpos i) 〈j,p〉.
106 alias symbol "pi2" = "pair pi2".
107 alias symbol "pi1" = "pair pi1".
108 alias symbol "lt" (instance 6) = "Q less than".
109 alias symbol "leq" = "Q less or equal than".
110 letin value_spec_aux ≝ (
112 \snd q = nth_height f (\fst q) ∧
113 (nth_base f (\fst q) < i ∧ ∀n.(\fst q) < n → n < len f → i ≤ nth_base f n));
114 alias symbol "lt" (instance 5) = "Q less than".
116 METTERE IN ACC LA LISTA PROCESSATA SO FAR
117 E DIRE CHE QUELLA@L=BARS
118 let rec value (acc: nat × ℚ) (l : list bar) on l : nat × ℚ ≝
122 match q_cmp (\fst x) (Qpos i) with
123 [ q_leq _ ⇒ value 〈S (\fst acc), \snd x〉 tl
127 (∀i.i < len l → nth_base (bars f) (\fst acc) < nth_base l i) →
128 nth_height (bars f) (\fst acc) = \snd acc →
129 value_spec_aux l (Qpos i) p);
131 [4: clearbody value; unfold value_spec;
132 generalize in match (bars_begin_OQ f);
133 generalize in match (bars_sorted f);
134 cases (bars_not_nil f); intro S; generalize in match (sorted_tail_bigger ?? S);
135 clear S; cases (value 〈O,\snd x〉 (x::l)) (p Hp); intros;
136 exists[apply (\snd p)];exists [apply (\fst p)]
137 cases (Hp ?) (Hg HV);
138 [unfold; split[reflexivity]simplify;split;
139 [rewrite > H1;apply q_pos_OQ;
140 |intros; cases n in H2 H3; [intro X; cases (not_le_Sn_O ? X)]
142 rewrite > H1; apply q_pos_OQ;
143 cases HV (Hi Hm); clear Hp value value_spec_aux HV;
144 exists [apply (\fst p)]; split;[rewrite > Hg;reflexivity|split;[assumption]intros]
145 apply Hm; assumption;
146 |1: unfold value_spec_aux; clear value value_spec_aux H2;intros; split[2:split]
147 [1: apply (q_lt_le_trans ??? (H4 (\fst p))); clear H4 H5;
150 cases (q_cmp i (start f));
151 [2: exists [apply 〈O,OQ〉] simplify; constructor 1; split; try assumption;
152 try reflexivity; apply q_lt_to_le; assumption;
153 |1: cases (bars f); [exists [apply 〈O,OQ〉] simplify; constructor 3; split;try assumption;reflexivity;]
154 cases (value ⅆ[i,start f] (b::l)) (p Hp);
155 cases (Hp (q_dist_ge_OQ ? ?)); clear Hp value; [cases H1; destruct H2]
156 cases H1; clear H1; lapply (sum_bases_O (b::l) (\fst p)) as H1;
157 [2: apply (q_le_trans ??? H2); rewrite > H; apply q_eq_to_le;
158 rewrite > q_d_x_x; reflexivity;
159 |1: exists [apply p] simplify; constructor 4; rewrite > H1; split;
160 try split; try rewrite > q_d_x_x; try autobatch depth=2;
161 [1: rewrite > H; rewrite > q_plus_sym; apply q_lt_plus;
162 rewrite > q_plus_minus; apply q_lt_plus_trans; [apply sum_bases_ge_OQ]
164 |2: rewrite > H; rewrite > q_d_x_x; apply q_eq_to_le; reflexivity;
165 |3: rewrite > H; rewrite > q_d_x_x; apply q_lt_plus_trans;
166 try apply sum_bases_ge_OQ; apply q_pos_lt_OQ;]]
167 |3: cases (q_cmp i (start f+sum_bases (bars f) (len (bars f))));
168 [1: exists [apply 〈O,OQ〉] simplify; constructor 2; split; try assumption;
169 try reflexivity; rewrite > H1; apply q_eq_to_le; reflexivity;
170 |3: exists [apply 〈O,OQ〉] simplify; constructor 2; split; try assumption;
171 try reflexivity; apply q_lt_to_le; assumption;
172 |2: generalize in match (refl_eq ? (bars f): bars f = bars f);
173 generalize in match (bars f) in ⊢ (??? % → %); intro X; cases X; clear X;
175 [1: exists [apply 〈O,OQ〉] simplify; constructor 3; split; reflexivity;
176 |2: cases (value ⅆ[i,start f] (b::l)) (p Hp);
177 cases (Hp (q_dist_ge_OQ ? ?)); clear Hp value; [cases H3;destruct H4]
179 exists [apply p]; constructor 4; split; try split; try assumption;
180 [1: intro X; destruct X;
181 |2: apply q_lt_to_le; assumption;
182 |3: rewrite < H2; assumption;
183 |4: cases (cmp_nat (\fst p) (len (bars f)));
184 [1:apply lt_to_le;rewrite <H2; assumption|rewrite > H3;rewrite < H2;apply le_n]
185 cases (?:False); cases (\fst p) in H3 H4 H6; clear H5;
186 [1: intros; apply (not_le_Sn_O ? H5);
187 |2: rewrite > q_d_sym; rewrite > q_d_noabs; [2: apply q_lt_to_le; assumption]
188 intros; lapply (q_lt_inj_plus_r ?? (Qopp (start f)) H1); clear H1;
189 generalize in match Hletin;
190 rewrite > (q_plus_sym (start f)); rewrite < q_plus_assoc;
191 do 2 rewrite < q_elim_minus; rewrite > q_plus_minus;
192 rewrite > q_plus_OQ; intro K; apply (q_lt_corefl (i-start f));
193 apply (q_lt_le_trans ???? H3); rewrite < H2;
194 apply (q_lt_trans ??? K); apply sum_bases_increasing;
196 |1,3: intros; right; split;
197 [1,4: clear H2; cases (value (q-Qpos (\fst b)) l1);
198 cases (H2 (q_le_to_diff_ge_OQ ?? (? H1)));
199 [1: intro; apply q_lt_to_le;assumption;
200 |3: simplify; cases H4; apply q_le_minus; assumption;
201 |2,5: simplify; cases H4; rewrite > H5; rewrite > H6;
202 apply q_le_minus; apply sum_bases_empty_nat_of_q_le_q;
203 |4: intro X; rewrite > X; apply q_eq_to_le; reflexivity;
204 |*: simplify; apply q_le_minus; cases H4; assumption;]
205 |2,5: cases (value (q-Qpos (\fst b)) l1);
206 cases (H4 (q_le_to_diff_ge_OQ ?? (? H1)));
207 [1,4: intros; [apply q_lt_to_le|apply q_eq_to_le;symmetry] assumption;
208 |3,6: cases H5; simplify; change with (q < sum_bases l1 (S (\fst w)) + Qpos (\fst b));
209 apply q_lt_plus; assumption;
210 |2,5: simplify; cases H5; rewrite > H6; simplify; rewrite > H7;
211 apply q_lt_plus; apply sum_bases_empty_nat_of_q_le_q_one;]
212 |*: cases (value (q-Qpos (\fst b)) l1); simplify;
213 cases (H4 (q_le_to_diff_ge_OQ ?? (? H1)));
214 [1,4: intros; [apply q_lt_to_le|apply q_eq_to_le;symmetry] assumption;
215 |3,6: cases H5; assumption;
216 |*: cases H5; rewrite > H6; rewrite > H8;
217 elim (\fst w); [1,3:reflexivity;] simplify; assumption;]]
218 |2: clear value H2; simplify; intros; right; split; [assumption|3:reflexivity]
219 rewrite > q_plus_sym; rewrite > q_plus_OQ; assumption;
220 |4: intros; left; split; reflexivity;]
224 ∀l,i.i < start l → \snd (\fst (value l i)) = OQ.
225 intros; cases (value l i) (q Hq); cases Hq; clear Hq; simplify; cases H1; clear H1;
226 try assumption; cases H2; cases (?:False); apply (q_lt_le_incompat ?? H H6);
230 ∀l,i.start l + sum_bases (bars l) (len (bars l)) ≤ i → \snd (\fst (value l i)) = OQ.
231 intros; cases (value l i) (q Hq); cases Hq; clear Hq; simplify; cases H1; clear H1;
232 try assumption; cases H2; cases (?:False); apply (q_lt_le_incompat ?? H7 H);
236 ∀l,i.bars l = [] → \snd (\fst (value l i)) = OQ.
237 intros; cases (value l i) (q Hq); cases Hq; clear Hq; simplify; cases H1; clear H1;
238 try assumption; cases H2; cases (?:False); apply (H1 H);
241 inductive value_ok_spec (f : q_f) (i : ℚ) : nat × ℚ → Type ≝
242 | value_ok : ∀n,q. n ≤ (len (bars f)) →
243 q = \snd (nth (bars f) ▭ n) →
244 sum_bases (bars f) n ≤ ⅆ[i,start f] →
245 ⅆ[i, start f] < sum_bases (bars f) (S n) → value_ok_spec f i 〈n,q〉.
248 ∀f,i.bars f ≠ [] → start f ≤ i → i < start f + sum_bases (bars f) (len (bars f)) →
249 value_ok_spec f i (\fst (value f i)).
250 intros; cases (value f i); simplify;
251 cases H3; simplify; clear H3; cases H4; clear H4;
252 [1,2,3: cases (?:False);
253 [1: apply (q_lt_le_incompat ?? H3 H1);
254 |2: apply (q_lt_le_incompat ?? H2 H3);
256 |4: cases H7; clear H7; cases w in H3 H4 H5 H6 H8; simplify; intros;
257 constructor 1; assumption;]
260 definition same_values ≝
262 ∀input.\snd (\fst (value l1 input)) = \snd (\fst (value l2 input)).
264 definition same_bases ≝
265 λl1,l2:list bar. (∀i.\fst (nth l1 ▭ i) = \fst (nth l2 ▭ i)).
267 alias symbol "lt" = "Q less than".
268 lemma unpos: ∀x:ℚ.OQ < x → ∃r:ratio.Qpos r = x.
269 intro; cases x; intros; [2:exists [apply r] reflexivity]
271 [ apply (q_lt_corefl ? H)|apply (q_neg_gt ? H)]
274 notation < "x \blacksquare" non associative with precedence 50 for @{'unpos $x}.
275 interpretation "hide unpos proof" 'unpos x = (unpos x _).