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 "nat_ordered_set.ma".
16 include "models/q_bars.ma".
18 lemma initial_shift_same_values:
19 ∀l1:q_f.∀init.init < start l1 →
21 (mk_q_f init (〈\fst (unpos (start l1 - init) ?),OQ〉:: bars l1)).
22 [apply q_lt_minus; rewrite > q_plus_sym; rewrite > q_plus_OQ; assumption]
23 intros; generalize in ⊢ (? ? (? ? (? ? (? ? ? (? ? ? (? ? %)) ?) ?))); intro;
24 cases (unpos (start l1-init) H1); intro input;
25 simplify in ⊢ (? ? ? (? ? ? (? ? ? (? (? ? (? ? (? ? ? % ?) ?)) ?))));
26 cases (value (mk_q_f init (〈w,OQ〉::bars l1)) input) (v1 Hv1);
27 (*cases (value l1 input) (v2 Hv2); *)
28 cases Hv1 (HV1 HV1 HV1 HV1); (* cases Hv2 (HV2 HV2 HV2 HV2); clear Hv1 Hv2; *)
29 cases HV1 (Hi1 Hv11 Hv12); (*cases HV2 (Hi2 Hv21 Hv22);*) clear HV1 (*HV2*);
31 rewrite > Hv12; (*rewrite > Hv22;*) try reflexivity;
32 [1: simplify in Hi1; cases (?:False);
33 apply (q_lt_corefl (start l1)); cases (Hi2);
34 autobatch by Hi2, Hi1, q_le_trans, H4, H, q_le_lt_trans, q_lt_le_trans.
35 |2: simplify in Hi1; cases (?:False);
36 apply (q_lt_corefl (start l1+sum_bases (bars l1) (len (bars l1))));
37 cases Hi2; apply (q_le_lt_trans ???? H5);
38 apply (q_le_trans ???? Hi1);
39 rewrite > H2; rewrite > (q_plus_sym ? (start l1-init));
40 rewrite > q_plus_assoc; apply q_le_inj_plus_r;
42 rewrite > q_elim_minus; rewrite > (q_plus_sym (start l1));
43 rewrite > q_plus_assoc; rewrite < q_elim_minus;
44 rewrite > q_plus_minus; rewrite > q_plus_sym; rewrite > q_plus_OQ;
46 |3: simplify in Hi1; destruct Hi1;
47 |4: simplify in Hi1 H3 Hv12 Hv11 ⊢ %; cases H3; clear H3;
48 cases (\fst v1) in H4; [intros;reflexivity] intros;
49 simplify; simplify in H3;
55 simplify in ⊢ (? ? ? (? ? ? %));
56 cases (q_cmp input (start (mk_q_f init (〈w,OQ〉::bars l1)))) in H3;
57 whd in ⊢ (% → ?); simplify in H3;
58 [1: intro; cases H4; clear H4; rewrite > H3;
59 cases (value l1 init); simplify; cases (q_cmp init (start l1)) in H4;
60 [1: cases (?:False); apply (q_lt_corefl init); rewrite > H4 in ⊢ (?? %); apply H;
61 |3: cases (?:False); apply (q_lt_antisym init (start l1)); assumption;
62 |2: whd in ⊢ (% → ?); intro; rewrite > H8; clear H8 H4;
63 rewrite > H7; clear H7; rewrite > (?:\fst w1 = O); [reflexivity]
64 symmetry; apply le_n_O_to_eq;
65 rewrite > (sum_bases_O (〈w,OQ〉::bars l1) (\fst w1)); [apply le_n]
66 clear H6 w2; simplify in H5:(? ? (? ? %));
67 destruct H3; rewrite > q_d_x_x in H5; assumption;]
68 |2: intros; cases (value l1 input); simplify in ⊢ (? ? (? ? ? %) ?);
69 cases (q_cmp input (start l1)) in H5; whd in ⊢ (% → ?);
70 [1: cases (?:False); clear w2 H4 w1 H2 w H1;
71 apply (q_lt_antisym init (start l1)); [assumption] rewrite < H5; assumption
72 |2: intros; rewrite > H6; clear H6; rewrite > H4; reflexivity;
73 |3: cases (?:False); apply (q_lt_antisym input (start l1)); [2: assumption]
74 apply (q_lt_trans ??? H3 H);]
75 |3: intro; cases H4; clear H4;
76 cases (value l1 input); simplify; cases (q_cmp input (start l1)) in H4; whd in ⊢ (% → ?);
77 [1: intro; cases H8; clear H8; rewrite > H11; rewrite > H7; clear H11 H7;
78 simplify in ⊢ (? ? ? (? ? ? (? ? % ? ?)));
79 cut (\fst w1 = S (\fst w2)) as Key; [rewrite > Key; reflexivity;]
80 cut (\fst w2 = O); [2: clear H10;
81 symmetry; apply le_n_O_to_eq; rewrite > (sum_bases_O (bars l1) (\fst w2)); [apply le_n]
82 apply (q_le_trans ??? H9); rewrite < H4; rewrite > q_d_x_x;
83 apply q_eq_to_le; reflexivity;]
84 rewrite > Hcut; clear Hcut H10 H9; simplify in H5 H6;
85 cut (ⅆ[input,init] = Qpos w) as E; [2:
86 rewrite > H2; rewrite < H4; rewrite > q_d_sym;
87 rewrite > q_d_noabs; [reflexivity] apply q_lt_to_le; assumption;]
88 cases (\fst w1) in H5 H6; intros;
89 [1: cases (?:False); clear H5; simplify in H6;
90 apply (q_lt_corefl ⅆ[input,init]);
91 rewrite > E in ⊢ (??%); rewrite < q_plus_OQ in ⊢ (??%);
92 rewrite > q_plus_sym; assumption;
93 |2: cases n in H5 H6; [intros; reflexivity] intros;
94 cases (?:False); clear H6; cases (bars l1) in H5; simplify; intros;
95 [apply (q_pos_OQ one);|apply (q_pos_OQ (\fst b));]
96 apply (q_le_S ??? (sum_bases_ge_OQ ? n1));[apply []|3:apply l]
97 simplify in ⊢ (? (? (? % ?) ?) ?); rewrite < (q_plus_minus (Qpos w));
98 rewrite > q_elim_minus; apply q_le_minus_r;
99 rewrite > q_elim_opp; rewrite < E in ⊢ (??%); assumption;]
100 |2: intros; rewrite > H8; rewrite > H7; clear H8 H7;
101 simplify in H5 H6 ⊢ %;
102 cases (\fst w1) in H5 H6; [intros; reflexivity]
104 [1: intros; simplify; elim n [reflexivity] simplify; assumption;
105 |2: simplify; intros; cases (?:False); clear H6;
106 apply (q_lt_le_incompat (input - init) (Qpos w) );
107 [1: rewrite > H2; do 2 rewrite > q_elim_minus;
108 apply q_lt_plus; rewrite > q_elim_minus;
109 rewrite < q_plus_assoc; rewrite < q_elim_minus;
110 rewrite > q_plus_minus;rewrite > q_plus_OQ; assumption;
111 |2: rewrite < q_d_noabs; [2: apply q_lt_to_le; assumption]
114 ; apply (q_le_S ???? H5);apply sum_bases_ge_OQ;]]
115 |3: intro; cases H8; clear H8; rewrite > H11; rewrite > H7; clear H11 H7;
116 simplify in H5 H6 ⊢ (? ? ? (? ? ? (? ? % ? ?)));
118 axiom nth_nil: ∀T,n.∀d:T. nth [] d n = d.
124 ⅆ[input,init] < sum_bases l O + (st-init) → False.
125 intros 6; rewrite > q_d_sym; rewrite > q_d_noabs; [2:
126 apply (q_le_trans ? st); apply q_lt_to_le; assumption]
127 do 2 rewrite > q_elim_minus; rewrite > q_plus_assoc;
128 intro X; lapply (q_lt_canc_plus_r ??? X) as Y;
129 simplify in Y; cases (?:False);
130 apply (q_lt_corefl st); apply (q_lt_trans ??? H1);
131 apply (q_lt_le_trans ??? Y); rewrite > q_plus_sym; rewrite > q_plus_OQ;
132 apply q_eq_to_le; reflexivity;
136 ∀a,l1,init,st,input,n.
137 init < st → st < input →
138 sum_bases (a::l1) n + (st-init) ≤ ⅆ[input,init] →
139 ⅆ[input,st] < sum_bases l1 O + Qpos (\fst a) →
141 intros; cut (input - st < Qpos (\fst a)) as H6';[2:
142 rewrite < q_d_noabs;[2:apply q_lt_to_le; assumption]
143 rewrite > q_d_sym; apply (q_lt_le_trans ??? H3);
144 rewrite > q_plus_sym; rewrite > q_plus_OQ;
145 apply q_eq_to_le; reflexivity] clear H3;
146 generalize in match H2; rewrite > q_d_sym; rewrite > q_d_noabs;
147 [2: apply (q_le_trans ? st); apply q_lt_to_le; assumption]
148 do 2 rewrite > q_elim_minus; rewrite > q_plus_assoc; intro X;
149 lapply (q_le_canc_plus_r ??? X) as Y; clear X;
150 lapply (q_le_inj_plus_r ?? (Qopp st) Y) as X; clear Y;
151 cut (input + Qopp st < Qpos (\fst a)) as H6'';
152 [2: rewrite < q_elim_minus; assumption;] clear H6';
153 generalize in match (q_le_lt_trans ??? X H6''); clear X H6'';
154 rewrite < q_plus_assoc; rewrite < q_elim_minus;
155 rewrite > q_plus_minus; rewrite > q_plus_OQ; cases n; intro X; [reflexivity]
157 apply (q_lt_le_incompat (sum_bases l1 n1) OQ);[2: apply sum_bases_ge_OQ;]
158 apply (q_lt_canc_plus_r ?? (Qpos (\fst a)));
159 rewrite >(q_plus_sym OQ); rewrite > q_plus_OQ; apply X;
163 ∀init,st,input,l1,a,n.
165 ⅆ[input,init]<OQ+Qpos a+(st-init) →
166 sum_bases l1 n+Qpos a≤ⅆ[input,st] → False.
168 cut (sum_bases l1 n - ⅆ[input,st] < Qopp ⅆ[input,init] + (st - init)); [2:
169 cut (sum_bases l1 n≤ⅆ[input,st]-Qpos a) as H7';[2:
170 apply (q_le_canc_plus_r ?? (Qpos a));
171 apply (q_le_trans ??? H3); rewrite > q_elim_minus;
172 rewrite < q_plus_assoc; rewrite > (q_plus_sym (Qopp ?));
173 rewrite < q_elim_minus; rewrite > q_plus_minus; rewrite > q_plus_OQ;
174 apply q_eq_to_le; reflexivity;] clear H3;
175 rewrite > q_elim_minus; apply (q_lt_canc_plus_r ?? ⅆ[input,st]);
176 rewrite < q_plus_assoc; rewrite > (q_plus_sym (Qopp ?));
177 rewrite < q_elim_minus; rewrite > q_plus_minus; rewrite > q_plus_OQ;
178 apply (q_le_lt_trans ??? H7'); clear H7'; rewrite > q_elim_minus;
179 rewrite > q_plus_sym; apply q_lt_inj_plus_r;
180 rewrite > q_plus_sym; apply q_lt_plus; rewrite > q_elim_opp;
181 rewrite > q_plus_sym; apply (q_lt_canc_plus_r ?? (Qpos a));
182 rewrite < q_plus_assoc; rewrite > (q_plus_sym (Qopp ?));
183 rewrite < q_elim_minus; rewrite > q_plus_minus; rewrite > q_plus_OQ;
184 apply (q_lt_le_trans ??? H2); rewrite > (q_plus_sym OQ); rewrite > q_plus_OQ;
185 rewrite > q_plus_sym; apply q_eq_to_le; reflexivity;]
186 generalize in match Hcut; clear H2 H3 Hcut;
187 rewrite > q_d_sym; rewrite > q_d_noabs; [2:apply q_lt_to_le; assumption]
188 rewrite > q_d_sym; rewrite > q_d_noabs; [2: apply (q_le_trans ? st); apply q_lt_to_le; assumption]
189 rewrite < q_plus_sym; rewrite < q_elim_minus;
190 rewrite > (q_elim_minus input init);
191 rewrite > q_minus_distrib; rewrite > q_elim_opp;
192 rewrite > (q_elim_minus input st);
193 rewrite > q_minus_distrib; rewrite > q_elim_opp;
194 repeat rewrite > q_elim_minus;
195 rewrite < q_plus_assoc in ⊢ (??% → ?);
196 rewrite > (q_plus_sym (Qopp input) init);
197 rewrite > q_plus_assoc;
198 rewrite < q_plus_assoc in ⊢ (??(?%?) → ?);
199 rewrite > (q_plus_sym (Qopp init) init);
200 rewrite < (q_elim_minus init); rewrite >q_plus_minus;
201 rewrite > q_plus_OQ; rewrite > (q_plus_sym st);
202 rewrite < q_plus_assoc;
203 rewrite < (q_plus_OQ (Qopp input + st)) in ⊢ (??% → ?);
204 rewrite > (q_plus_sym ? OQ); intro X;
205 lapply (q_lt_canc_plus_r ??? X) as Y; clear X;
206 apply (q_lt_le_incompat ?? Y); apply sum_bases_ge_OQ;
210 ∀init,input,l1,w1,w2,w.
211 Qpos w = start l1 - init →
214 sum_bases (〈w,OQ〉::bars l1) w1 ≤ ⅆ[input,init] →
215 ⅆ[input,init] < sum_bases (bars l1) w1 + (start l1-init) →
216 sum_bases (bars l1) w2 ≤ ⅆ[input,start l1] →
217 ⅆ[input,start l1] < sum_bases (bars l1) (S w2) →
218 \snd (nth (bars l1) ▭ w2) = \snd (nth (〈w,OQ〉::bars l1) ▭ w1).
219 intros 3 (init input l); cases l (st l);
220 change in match (start (mk_q_f st l)) with st;
221 change in match (bars (mk_q_f st l)) with l;
223 [1: rewrite > nth_nil; cases w1 in H4;
224 [1: intro X; cases (case1 ?????? X); assumption;
225 |2: intros; simplify; rewrite > nth_nil; reflexivity;]
226 |2: cases w1 in H4 H5; clear w1;
227 [1: intros (Y X); cases (case1 ?????? X); assumption;
228 |2: intros; simplify in H4 H5 H7 ⊢ %;
229 generalize in match H6; generalize in match H7;
230 generalize in match H4; generalize in match H5; clear H4 H5 H6 H7;
231 apply (nat_elim2 ???? w2 n); clear w2 n; intros;
232 [1: rewrite > (case2 a l1 init st input n); [reflexivity]
233 try rewrite < H1; assumption;
234 |2: simplify in H4 H7; cases (case3 ???????? H4 H7); assumption;
235 |3: (* dipende se vanno oltre la lunghezza di l1,
236 forse dovevo gestire il caso prima dell'induzione *)
237 simplify in ⊢ (? ? (? ? ? %) ?);
238 rewrite > (H (S m) ? w); [reflexivity] try assumption;
245 alias symbol "pi2" = "pair pi2".
246 alias symbol "pi1" = "pair pi1".
247 definition rebase_spec ≝
248 ∀l1,l2:q_f.∃p:q_f × q_f.
250 (*len (bars (\fst p)) = len (bars (\snd p))*)
251 (start (\fst p) = start (\snd p))
252 (same_bases (\fst p) (\snd p))
253 (same_values l1 (\fst p))
254 (same_values l2 (\snd p)).
256 definition rebase_spec_simpl ≝
257 λstart.λl1,l2:list bar.λp:(list bar) × (list bar).
259 (same_bases (mk_q_f start (\fst p)) (mk_q_f start (\snd p)))
260 (same_values (mk_q_f start l1) (mk_q_f start (\fst p)))
261 (same_values (mk_q_f start l2) (mk_q_f start (\snd p))).
263 (* a local letin makes russell fail *)
264 definition cb0h : list bar → list bar ≝
265 λl.mk_list (λi.〈\fst (nth l ▭ i),OQ〉) (len l).
268 λP.λp:∃x:(list bar) × (list bar).P x.match p with [ex_introT p _ ⇒ p].
270 definition inject ≝ λP.λp:(list bar) × (list bar).λh:P p. ex_introT ? P p h.
271 coercion inject with 0 1 nocomposites.
273 definition rebase: rebase_spec.
274 intros 2 (f1 f2); cases f1 (s1 l1); cases f2 (s2 l2); clear f1 f2;
276 λs.λl1,l2.λm.λz.len l1 + len l2 < m → rebase_spec_simpl s l1 l2 z);
277 alias symbol "pi1" (instance 34) = "exT \fst".
278 alias symbol "pi1" (instance 21) = "exT \fst".
280 let rec aux (l1,l2:list bar) (n:nat) on n : (list bar) × (list bar) ≝
282 [ O ⇒ 〈 nil ? , nil ? 〉
285 [ nil ⇒ 〈cb0h l2, l2〉
288 [ nil ⇒ 〈l1, cb0h l1〉
290 let base1 ≝ Qpos (\fst he1) in
291 let base2 ≝ Qpos (\fst he2) in
292 let height1 ≝ (\snd he1) in
293 let height2 ≝ (\snd he2) in
294 match q_cmp base1 base2 with
296 let rc ≝ aux tl1 tl2 m in
297 〈he1 :: \fst rc,he2 :: \snd rc〉
299 let rest ≝ base2 - base1 in
300 let rc ≝ aux tl1 (〈\fst (unpos rest ?),height2〉 :: tl2) m in
301 〈〈\fst he1,height1〉 :: \fst rc,〈\fst he1,height2〉 :: \snd rc〉
303 let rest ≝ base1 - base2 in
304 let rc ≝ aux (〈\fst (unpos rest ?),height1〉 :: tl1) tl2 m in
305 〈〈\fst he2,height1〉 :: \fst rc,〈\fst he2,height2〉 :: \snd rc〉
307 in aux : ∀l1,l2,m.∃z.∀s.spec s l1 l2 m z); unfold spec;
308 [9: clearbody aux; unfold spec in aux; clear spec;
310 [1: cases (aux l1 l2 (S (len l1 + len l2)));
311 cases (H1 s1 (le_n ?)); clear H1;
312 exists [apply 〈mk_q_f s1 (\fst w), mk_q_f s2 (\snd w)〉] split;
314 |3: intro; apply (H3 input);
315 |4: intro; rewrite > H in H4;
316 rewrite > (H4 input); reflexivity;]
317 |2: letin l2' ≝ (〈\fst (unpos (s2-s1) ?),OQ〉::l2);[
318 apply q_lt_minus; rewrite > q_plus_sym; rewrite > q_plus_OQ;
320 cases (aux l1 l2' (S (len l1 + len l2')));
321 cases (H1 s1 (le_n ?)); clear H1 aux;
322 exists [apply 〈mk_q_f s1 (\fst w), mk_q_f s1 (\snd w)〉] split;
326 |4: intro; rewrite < (H4 input); clear H3 H4 H2 w;
327 cases (value (mk_q_f s1 l2') input);
328 cases (q_cmp input (start (mk_q_f s1 l2'))) in H1;
330 [1: intros; cases H2; clear H2; whd in ⊢ (??? %);
331 cases (value (mk_q_f s2 l2) input);
332 cases (q_cmp input (start (mk_q_f s2 l2))) in H2;
334 [1: intros; cases H6; clear H6; change with (w1 = w);
337 |1,2: unfold rest; apply q_lt_minus; rewrite > q_plus_sym; rewrite > q_plus_OQ;
344 |8: intros; cases (?:False); apply (not_le_Sn_O ? H1);]