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 (* ********************************************************************** *)
16 (* Progetto FreeScale *)
18 (* Sviluppato da: Cosimo Oliboni, oliboni@cs.unibo.it *)
19 (* Cosimo Oliboni, oliboni@cs.unibo.it *)
21 (* ********************************************************************** *)
23 include "freescale/bool_lemmas.ma".
24 include "freescale/aux_bases.ma".
30 ndefinition oct_destruct_aux ≝
31 Πn1,n2:oct.ΠP:Prop.n1 = n2 →
33 [ o0 ⇒ match n2 with [ o0 ⇒ P → P | _ ⇒ P ]
34 | o1 ⇒ match n2 with [ o1 ⇒ P → P | _ ⇒ P ]
35 | o2 ⇒ match n2 with [ o2 ⇒ P → P | _ ⇒ P ]
36 | o3 ⇒ match n2 with [ o3 ⇒ P → P | _ ⇒ P ]
37 | o4 ⇒ match n2 with [ o4 ⇒ P → P | _ ⇒ P ]
38 | o5 ⇒ match n2 with [ o5 ⇒ P → P | _ ⇒ P ]
39 | o6 ⇒ match n2 with [ o6 ⇒ P → P | _ ⇒ P ]
40 | o7 ⇒ match n2 with [ o7 ⇒ P → P | _ ⇒ P ]
43 ndefinition oct_destruct : oct_destruct_aux.
46 ##[ ##1: nelim n2; nnormalize; #H;
47 ##[ ##1: napply (λx:P.x)
48 ##| ##*: napply (False_ind ??);
49 nchange with (match o0 with [ o0 ⇒ False | _ ⇒ True ]);
50 nrewrite > H; nnormalize; napply I
52 ##| ##2: nelim n2; nnormalize; #H;
53 ##[ ##2: napply (λx:P.x)
54 ##| ##*: napply (False_ind ??);
55 nchange with (match o1 with [ o1 ⇒ False | _ ⇒ True ]);
56 nrewrite > H; nnormalize; napply I
58 ##| ##3: nelim n2; nnormalize; #H;
59 ##[ ##3: napply (λx:P.x)
60 ##| ##*: napply (False_ind ??);
61 nchange with (match o2 with [ o2 ⇒ False | _ ⇒ True ]);
62 nrewrite > H; nnormalize; napply I
64 ##| ##4: nelim n2; nnormalize; #H;
65 ##[ ##4: napply (λx:P.x)
66 ##| ##*: napply (False_ind ??);
67 nchange with (match o3 with [ o3 ⇒ False | _ ⇒ True ]);
68 nrewrite > H; nnormalize; napply I
70 ##| ##5: nelim n2; nnormalize; #H;
71 ##[ ##5: napply (λx:P.x)
72 ##| ##*: napply (False_ind ??);
73 nchange with (match o4 with [ o4 ⇒ False | _ ⇒ True ]);
74 nrewrite > H; nnormalize; napply I
76 ##| ##6: nelim n2; nnormalize; #H;
77 ##[ ##6: napply (λx:P.x)
78 ##| ##*: napply (False_ind ??);
79 nchange with (match o5 with [ o5 ⇒ False | _ ⇒ True ]);
80 nrewrite > H; nnormalize; napply I
82 ##| ##7: nelim n2; nnormalize; #H;
83 ##[ ##7: napply (λx:P.x)
84 ##| ##*: napply (False_ind ??);
85 nchange with (match o6 with [ o6 ⇒ False | _ ⇒ True ]);
86 nrewrite > H; nnormalize; napply I
88 ##| ##8: nelim n2; nnormalize; #H;
89 ##[ ##8: napply (λx:P.x)
90 ##| ##*: napply (False_ind ??);
91 nchange with (match o7 with [ o7 ⇒ False | _ ⇒ True ]);
92 nrewrite > H; nnormalize; napply I
97 nlemma symmetric_eqoct : symmetricT oct bool eq_oct.
105 nlemma eqoct_to_eq : ∀n1,n2.eq_oct n1 n2 = true → n1 = n2.
110 ##[ ##1,10,19,28,37,46,55,64: #H; napply (refl_eq ??)
111 ##| ##*: #H; napply (bool_destruct ??? H)
115 nlemma eq_to_eqoct : ∀n1,n2.n1 = n2 → eq_oct n1 n2 = true.
120 ##[ ##1,10,19,28,37,46,55,64: #H; napply (refl_eq ??)
121 ##| ##*: #H; napply (oct_destruct ??? H)
129 ndefinition bitrigesim_destruct1 :
130 Πt2:bitrigesim.ΠP:Prop.t00 = t2 → match t2 with [ t00 ⇒ P → P | _ ⇒ P ].
134 ##[ ##1: napply (λx:P.x)
135 ##| ##*: napply (False_ind ??);
136 nchange with (match t00 with [ t00 ⇒ False | _ ⇒ True ]);
137 nrewrite > H; nnormalize; napply I
141 ndefinition bitrigesim_destruct2 :
142 Πt2:bitrigesim.ΠP:Prop.t01 = t2 → match t2 with [ t01 ⇒ P → P | _ ⇒ P ].
146 ##[ ##2: napply (λx:P.x)
147 ##| ##*: napply (False_ind ??);
148 nchange with (match t01 with [ t01 ⇒ False | _ ⇒ True ]);
149 nrewrite > H; nnormalize; napply I
153 ndefinition bitrigesim_destruct3 :
154 Πt2:bitrigesim.ΠP:Prop.t02 = t2 → match t2 with [ t02 ⇒ P → P | _ ⇒ P ].
158 ##[ ##3: napply (λx:P.x)
159 ##| ##*: napply (False_ind ??);
160 nchange with (match t02 with [ t02 ⇒ False | _ ⇒ True ]);
161 nrewrite > H; nnormalize; napply I
165 ndefinition bitrigesim_destruct4 :
166 Πt2:bitrigesim.ΠP:Prop.t03 = t2 → match t2 with [ t03 ⇒ P → P | _ ⇒ P ].
170 ##[ ##4: napply (λx:P.x)
171 ##| ##*: napply (False_ind ??);
172 nchange with (match t03 with [ t03 ⇒ False | _ ⇒ True ]);
173 nrewrite > H; nnormalize; napply I
177 ndefinition bitrigesim_destruct5 :
178 Πt2:bitrigesim.ΠP:Prop.t04 = t2 → match t2 with [ t04 ⇒ P → P | _ ⇒ P ].
182 ##[ ##5: napply (λx:P.x)
183 ##| ##*: napply (False_ind ??);
184 nchange with (match t04 with [ t04 ⇒ False | _ ⇒ True ]);
185 nrewrite > H; nnormalize; napply I
189 ndefinition bitrigesim_destruct6 :
190 Πt2:bitrigesim.ΠP:Prop.t05 = t2 → match t2 with [ t05 ⇒ P → P | _ ⇒ P ].
194 ##[ ##6: napply (λx:P.x)
195 ##| ##*: napply (False_ind ??);
196 nchange with (match t05 with [ t05 ⇒ False | _ ⇒ True ]);
197 nrewrite > H; nnormalize; napply I
201 ndefinition bitrigesim_destruct7 :
202 Πt2:bitrigesim.ΠP:Prop.t06 = t2 → match t2 with [ t06 ⇒ P → P | _ ⇒ P ].
206 ##[ ##7: napply (λx:P.x)
207 ##| ##*: napply (False_ind ??);
208 nchange with (match t06 with [ t06 ⇒ False | _ ⇒ True ]);
209 nrewrite > H; nnormalize; napply I
213 ndefinition bitrigesim_destruct8 :
214 Πt2:bitrigesim.ΠP:Prop.t07 = t2 → match t2 with [ t07 ⇒ P → P | _ ⇒ P ].
218 ##[ ##8: napply (λx:P.x)
219 ##| ##*: napply (False_ind ??);
220 nchange with (match t07 with [ t07 ⇒ False | _ ⇒ True ]);
221 nrewrite > H; nnormalize; napply I
225 ndefinition bitrigesim_destruct9 :
226 Πt2:bitrigesim.ΠP:Prop.t08 = t2 → match t2 with [ t08 ⇒ P → P | _ ⇒ P ].
230 ##[ ##9: napply (λx:P.x)
231 ##| ##*: napply (False_ind ??);
232 nchange with (match t08 with [ t08 ⇒ False | _ ⇒ True ]);
233 nrewrite > H; nnormalize; napply I
237 ndefinition bitrigesim_destruct10 :
238 Πt2:bitrigesim.ΠP:Prop.t09 = t2 → match t2 with [ t09 ⇒ P → P | _ ⇒ P ].
242 ##[ ##10: napply (λx:P.x)
243 ##| ##*: napply (False_ind ??);
244 nchange with (match t09 with [ t09 ⇒ False | _ ⇒ True ]);
245 nrewrite > H; nnormalize; napply I
249 ndefinition bitrigesim_destruct11 :
250 Πt2:bitrigesim.ΠP:Prop.t0A = t2 → match t2 with [ t0A ⇒ P → P | _ ⇒ P ].
254 ##[ ##11: napply (λx:P.x)
255 ##| ##*: napply (False_ind ??);
256 nchange with (match t0A with [ t0A ⇒ False | _ ⇒ True ]);
257 nrewrite > H; nnormalize; napply I
261 ndefinition bitrigesim_destruct12 :
262 Πt2:bitrigesim.ΠP:Prop.t0B = t2 → match t2 with [ t0B ⇒ P → P | _ ⇒ P ].
266 ##[ ##12: napply (λx:P.x)
267 ##| ##*: napply (False_ind ??);
268 nchange with (match t0B with [ t0B ⇒ False | _ ⇒ True ]);
269 nrewrite > H; nnormalize; napply I
273 ndefinition bitrigesim_destruct13 :
274 Πt2:bitrigesim.ΠP:Prop.t0C = t2 → match t2 with [ t0C ⇒ P → P | _ ⇒ P ].
278 ##[ ##13: napply (λx:P.x)
279 ##| ##*: napply (False_ind ??);
280 nchange with (match t0C with [ t0C ⇒ False | _ ⇒ True ]);
281 nrewrite > H; nnormalize; napply I
285 ndefinition bitrigesim_destruct14 :
286 Πt2:bitrigesim.ΠP:Prop.t0D = t2 → match t2 with [ t0D ⇒ P → P | _ ⇒ P ].
290 ##[ ##14: napply (λx:P.x)
291 ##| ##*: napply (False_ind ??);
292 nchange with (match t0D with [ t0D ⇒ False | _ ⇒ True ]);
293 nrewrite > H; nnormalize; napply I
297 ndefinition bitrigesim_destruct15 :
298 Πt2:bitrigesim.ΠP:Prop.t0E = t2 → match t2 with [ t0E ⇒ P → P | _ ⇒ P ].
302 ##[ ##15: napply (λx:P.x)
303 ##| ##*: napply (False_ind ??);
304 nchange with (match t0E with [ t0E ⇒ False | _ ⇒ True ]);
305 nrewrite > H; nnormalize; napply I
309 ndefinition bitrigesim_destruct16 :
310 Πt2:bitrigesim.ΠP:Prop.t0F = t2 → match t2 with [ t0F ⇒ P → P | _ ⇒ P ].
314 ##[ ##16: napply (λx:P.x)
315 ##| ##*: napply (False_ind ??);
316 nchange with (match t0F with [ t0F ⇒ False | _ ⇒ True ]);
317 nrewrite > H; nnormalize; napply I
321 ndefinition bitrigesim_destruct17 :
322 Πt2:bitrigesim.ΠP:Prop.t10 = t2 → match t2 with [ t10 ⇒ P → P | _ ⇒ P ].
326 ##[ ##17: napply (λx:P.x)
327 ##| ##*: napply (False_ind ??);
328 nchange with (match t10 with [ t10 ⇒ False | _ ⇒ True ]);
329 nrewrite > H; nnormalize; napply I
333 ndefinition bitrigesim_destruct18 :
334 Πt2:bitrigesim.ΠP:Prop.t11 = t2 → match t2 with [ t11 ⇒ P → P | _ ⇒ P ].
338 ##[ ##18: napply (λx:P.x)
339 ##| ##*: napply (False_ind ??);
340 nchange with (match t11 with [ t11 ⇒ False | _ ⇒ True ]);
341 nrewrite > H; nnormalize; napply I
345 ndefinition bitrigesim_destruct19 :
346 Πt2:bitrigesim.ΠP:Prop.t12 = t2 → match t2 with [ t12 ⇒ P → P | _ ⇒ P ].
350 ##[ ##19: napply (λx:P.x)
351 ##| ##*: napply (False_ind ??);
352 nchange with (match t12 with [ t12 ⇒ False | _ ⇒ True ]);
353 nrewrite > H; nnormalize; napply I
357 ndefinition bitrigesim_destruct20 :
358 Πt2:bitrigesim.ΠP:Prop.t13 = t2 → match t2 with [ t13 ⇒ P → P | _ ⇒ P ].
362 ##[ ##20: napply (λx:P.x)
363 ##| ##*: napply (False_ind ??);
364 nchange with (match t13 with [ t13 ⇒ False | _ ⇒ True ]);
365 nrewrite > H; nnormalize; napply I
369 ndefinition bitrigesim_destruct21 :
370 Πt2:bitrigesim.ΠP:Prop.t14 = t2 → match t2 with [ t14 ⇒ P → P | _ ⇒ P ].
374 ##[ ##21: napply (λx:P.x)
375 ##| ##*: napply (False_ind ??);
376 nchange with (match t14 with [ t14 ⇒ False | _ ⇒ True ]);
377 nrewrite > H; nnormalize; napply I
381 ndefinition bitrigesim_destruct22 :
382 Πt2:bitrigesim.ΠP:Prop.t15 = t2 → match t2 with [ t15 ⇒ P → P | _ ⇒ P ].
386 ##[ ##22: napply (λx:P.x)
387 ##| ##*: napply (False_ind ??);
388 nchange with (match t15 with [ t15 ⇒ False | _ ⇒ True ]);
389 nrewrite > H; nnormalize; napply I
393 ndefinition bitrigesim_destruct23 :
394 Πt2:bitrigesim.ΠP:Prop.t16 = t2 → match t2 with [ t16 ⇒ P → P | _ ⇒ P ].
398 ##[ ##23: napply (λx:P.x)
399 ##| ##*: napply (False_ind ??);
400 nchange with (match t16 with [ t16 ⇒ False | _ ⇒ True ]);
401 nrewrite > H; nnormalize; napply I
405 ndefinition bitrigesim_destruct24 :
406 Πt2:bitrigesim.ΠP:Prop.t17 = t2 → match t2 with [ t17 ⇒ P → P | _ ⇒ P ].
410 ##[ ##24: napply (λx:P.x)
411 ##| ##*: napply (False_ind ??);
412 nchange with (match t17 with [ t17 ⇒ False | _ ⇒ True ]);
413 nrewrite > H; nnormalize; napply I
417 ndefinition bitrigesim_destruct25 :
418 Πt2:bitrigesim.ΠP:Prop.t18 = t2 → match t2 with [ t18 ⇒ P → P | _ ⇒ P ].
422 ##[ ##25: napply (λx:P.x)
423 ##| ##*: napply (False_ind ??);
424 nchange with (match t18 with [ t18 ⇒ False | _ ⇒ True ]);
425 nrewrite > H; nnormalize; napply I
429 ndefinition bitrigesim_destruct26 :
430 Πt2:bitrigesim.ΠP:Prop.t19 = t2 → match t2 with [ t19 ⇒ P → P | _ ⇒ P ].
434 ##[ ##26: napply (λx:P.x)
435 ##| ##*: napply (False_ind ??);
436 nchange with (match t19 with [ t19 ⇒ False | _ ⇒ True ]);
437 nrewrite > H; nnormalize; napply I
441 ndefinition bitrigesim_destruct27 :
442 Πt2:bitrigesim.ΠP:Prop.t1A = t2 → match t2 with [ t1A ⇒ P → P | _ ⇒ P ].
446 ##[ ##27: napply (λx:P.x)
447 ##| ##*: napply (False_ind ??);
448 nchange with (match t1A with [ t1A ⇒ False | _ ⇒ True ]);
449 nrewrite > H; nnormalize; napply I
453 ndefinition bitrigesim_destruct28 :
454 Πt2:bitrigesim.ΠP:Prop.t1B = t2 → match t2 with [ t1B ⇒ P → P | _ ⇒ P ].
458 ##[ ##28: napply (λx:P.x)
459 ##| ##*: napply (False_ind ??);
460 nchange with (match t1B with [ t1B ⇒ False | _ ⇒ True ]);
461 nrewrite > H; nnormalize; napply I
465 ndefinition bitrigesim_destruct29 :
466 Πt2:bitrigesim.ΠP:Prop.t1C = t2 → match t2 with [ t1C ⇒ P → P | _ ⇒ P ].
470 ##[ ##29: napply (λx:P.x)
471 ##| ##*: napply (False_ind ??);
472 nchange with (match t1C with [ t1C ⇒ False | _ ⇒ True ]);
473 nrewrite > H; nnormalize; napply I
477 ndefinition bitrigesim_destruct30 :
478 Πt2:bitrigesim.ΠP:Prop.t1D = t2 → match t2 with [ t1D ⇒ P → P | _ ⇒ P ].
482 ##[ ##30: napply (λx:P.x)
483 ##| ##*: napply (False_ind ??);
484 nchange with (match t1D with [ t1D ⇒ False | _ ⇒ True ]);
485 nrewrite > H; nnormalize; napply I
489 ndefinition bitrigesim_destruct31 :
490 Πt2:bitrigesim.ΠP:Prop.t1E = t2 → match t2 with [ t1E ⇒ P → P | _ ⇒ P ].
494 ##[ ##31: napply (λx:P.x)
495 ##| ##*: napply (False_ind ??);
496 nchange with (match t1E with [ t1E ⇒ False | _ ⇒ True ]);
497 nrewrite > H; nnormalize; napply I
501 ndefinition bitrigesim_destruct32 :
502 Πt2:bitrigesim.ΠP:Prop.t1F = t2 → match t2 with [ t1F ⇒ P → P | _ ⇒ P ].
506 ##[ ##32: napply (λx:P.x)
507 ##| ##*: napply (False_ind ??);
508 nchange with (match t1F with [ t1F ⇒ False | _ ⇒ True ]);
509 nrewrite > H; nnormalize; napply I
513 ndefinition bitrigesim_destruct_aux ≝
514 Πt1,t2:bitrigesim.ΠP:Prop.t1 = t2 →
516 [ t00 ⇒ match t2 with [ t00 ⇒ P → P | _ ⇒ P ]
517 | t01 ⇒ match t2 with [ t01 ⇒ P → P | _ ⇒ P ]
518 | t02 ⇒ match t2 with [ t02 ⇒ P → P | _ ⇒ P ]
519 | t03 ⇒ match t2 with [ t03 ⇒ P → P | _ ⇒ P ]
520 | t04 ⇒ match t2 with [ t04 ⇒ P → P | _ ⇒ P ]
521 | t05 ⇒ match t2 with [ t05 ⇒ P → P | _ ⇒ P ]
522 | t06 ⇒ match t2 with [ t06 ⇒ P → P | _ ⇒ P ]
523 | t07 ⇒ match t2 with [ t07 ⇒ P → P | _ ⇒ P ]
524 | t08 ⇒ match t2 with [ t08 ⇒ P → P | _ ⇒ P ]
525 | t09 ⇒ match t2 with [ t09 ⇒ P → P | _ ⇒ P ]
526 | t0A ⇒ match t2 with [ t0A ⇒ P → P | _ ⇒ P ]
527 | t0B ⇒ match t2 with [ t0B ⇒ P → P | _ ⇒ P ]
528 | t0C ⇒ match t2 with [ t0C ⇒ P → P | _ ⇒ P ]
529 | t0D ⇒ match t2 with [ t0D ⇒ P → P | _ ⇒ P ]
530 | t0E ⇒ match t2 with [ t0E ⇒ P → P | _ ⇒ P ]
531 | t0F ⇒ match t2 with [ t0F ⇒ P → P | _ ⇒ P ]
532 | t10 ⇒ match t2 with [ t10 ⇒ P → P | _ ⇒ P ]
533 | t11 ⇒ match t2 with [ t11 ⇒ P → P | _ ⇒ P ]
534 | t12 ⇒ match t2 with [ t12 ⇒ P → P | _ ⇒ P ]
535 | t13 ⇒ match t2 with [ t13 ⇒ P → P | _ ⇒ P ]
536 | t14 ⇒ match t2 with [ t14 ⇒ P → P | _ ⇒ P ]
537 | t15 ⇒ match t2 with [ t15 ⇒ P → P | _ ⇒ P ]
538 | t16 ⇒ match t2 with [ t16 ⇒ P → P | _ ⇒ P ]
539 | t17 ⇒ match t2 with [ t17 ⇒ P → P | _ ⇒ P ]
540 | t18 ⇒ match t2 with [ t18 ⇒ P → P | _ ⇒ P ]
541 | t19 ⇒ match t2 with [ t19 ⇒ P → P | _ ⇒ P ]
542 | t1A ⇒ match t2 with [ t1A ⇒ P → P | _ ⇒ P ]
543 | t1B ⇒ match t2 with [ t1B ⇒ P → P | _ ⇒ P ]
544 | t1C ⇒ match t2 with [ t1C ⇒ P → P | _ ⇒ P ]
545 | t1D ⇒ match t2 with [ t1D ⇒ P → P | _ ⇒ P ]
546 | t1E ⇒ match t2 with [ t1E ⇒ P → P | _ ⇒ P ]
547 | t1F ⇒ match t2 with [ t1F ⇒ P → P | _ ⇒ P ]
550 ndefinition bitrigesim_destruct : bitrigesim_destruct_aux.
553 ##[ ##1: napply bitrigesim_destruct1
554 ##| ##2: napply bitrigesim_destruct2
555 ##| ##3: napply bitrigesim_destruct3
556 ##| ##4: napply bitrigesim_destruct4
557 ##| ##5: napply bitrigesim_destruct5
558 ##| ##6: napply bitrigesim_destruct6
559 ##| ##7: napply bitrigesim_destruct7
560 ##| ##8: napply bitrigesim_destruct8
561 ##| ##9: napply bitrigesim_destruct9
562 ##| ##10: napply bitrigesim_destruct10
563 ##| ##11: napply bitrigesim_destruct11
564 ##| ##12: napply bitrigesim_destruct12
565 ##| ##13: napply bitrigesim_destruct13
566 ##| ##14: napply bitrigesim_destruct14
567 ##| ##15: napply bitrigesim_destruct15
568 ##| ##16: napply bitrigesim_destruct16
569 ##| ##17: napply bitrigesim_destruct17
570 ##| ##18: napply bitrigesim_destruct18
571 ##| ##19: napply bitrigesim_destruct19
572 ##| ##20: napply bitrigesim_destruct20
573 ##| ##21: napply bitrigesim_destruct21
574 ##| ##22: napply bitrigesim_destruct22
575 ##| ##23: napply bitrigesim_destruct23
576 ##| ##24: napply bitrigesim_destruct24
577 ##| ##25: napply bitrigesim_destruct25
578 ##| ##26: napply bitrigesim_destruct26
579 ##| ##27: napply bitrigesim_destruct27
580 ##| ##28: napply bitrigesim_destruct28
581 ##| ##29: napply bitrigesim_destruct29
582 ##| ##30: napply bitrigesim_destruct30
583 ##| ##31: napply bitrigesim_destruct31
584 ##| ##32: napply bitrigesim_destruct32
588 nlemma symmetric_eqbitrig : symmetricT bitrigesim bool eq_bitrig.
591 ##[ ##1: #t2; nelim t2; nnormalize; napply (refl_eq ??)
592 ##| ##2: #t2; nelim t2; nnormalize; napply (refl_eq ??)
593 ##| ##3: #t2; nelim t2; nnormalize; napply (refl_eq ??)
594 ##| ##4: #t2; nelim t2; nnormalize; napply (refl_eq ??)
595 ##| ##5: #t2; nelim t2; nnormalize; napply (refl_eq ??)
596 ##| ##6: #t2; nelim t2; nnormalize; napply (refl_eq ??)
597 ##| ##7: #t2; nelim t2; nnormalize; napply (refl_eq ??)
598 ##| ##8: #t2; nelim t2; nnormalize; napply (refl_eq ??)
599 ##| ##9: #t2; nelim t2; nnormalize; napply (refl_eq ??)
600 ##| ##10: #t2; nelim t2; nnormalize; napply (refl_eq ??)
601 ##| ##11: #t2; nelim t2; nnormalize; napply (refl_eq ??)
602 ##| ##12: #t2; nelim t2; nnormalize; napply (refl_eq ??)
603 ##| ##13: #t2; nelim t2; nnormalize; napply (refl_eq ??)
604 ##| ##14: #t2; nelim t2; nnormalize; napply (refl_eq ??)
605 ##| ##15: #t2; nelim t2; nnormalize; napply (refl_eq ??)
606 ##| ##16: #t2; nelim t2; nnormalize; napply (refl_eq ??)
607 ##| ##17: #t2; nelim t2; nnormalize; napply (refl_eq ??)
608 ##| ##18: #t2; nelim t2; nnormalize; napply (refl_eq ??)
609 ##| ##19: #t2; nelim t2; nnormalize; napply (refl_eq ??)
610 ##| ##20: #t2; nelim t2; nnormalize; napply (refl_eq ??)
611 ##| ##21: #t2; nelim t2; nnormalize; napply (refl_eq ??)
612 ##| ##22: #t2; nelim t2; nnormalize; napply (refl_eq ??)
613 ##| ##23: #t2; nelim t2; nnormalize; napply (refl_eq ??)
614 ##| ##24: #t2; nelim t2; nnormalize; napply (refl_eq ??)
615 ##| ##25: #t2; nelim t2; nnormalize; napply (refl_eq ??)
616 ##| ##26: #t2; nelim t2; nnormalize; napply (refl_eq ??)
617 ##| ##27: #t2; nelim t2; nnormalize; napply (refl_eq ??)
618 ##| ##28: #t2; nelim t2; nnormalize; napply (refl_eq ??)
619 ##| ##29: #t2; nelim t2; nnormalize; napply (refl_eq ??)
620 ##| ##30: #t2; nelim t2; nnormalize; napply (refl_eq ??)
621 ##| ##31: #t2; nelim t2; nnormalize; napply (refl_eq ??)
622 ##| ##32: #t2; nelim t2; nnormalize; napply (refl_eq ??)
626 nlemma eqbitrig_to_eq1 : ∀t2.eq_bitrig t00 t2 = true → t00 = t2.
627 #t2; ncases t2; nnormalize; #H; ##[ ##1: napply (refl_eq ??) ##| ##*: napply (bool_destruct ??? H) ##]
630 nlemma eqbitrig_to_eq2 : ∀t2.eq_bitrig t01 t2 = true → t01 = t2.
631 #t2; ncases t2; nnormalize; #H; ##[ ##2: napply (refl_eq ??) ##| ##*: napply (bool_destruct ??? H) ##]
634 nlemma eqbitrig_to_eq3 : ∀t2.eq_bitrig t02 t2 = true → t02 = t2.
635 #t2; ncases t2; nnormalize; #H; ##[ ##3: napply (refl_eq ??) ##| ##*: napply (bool_destruct ??? H) ##]
638 nlemma eqbitrig_to_eq4 : ∀t2.eq_bitrig t03 t2 = true → t03 = t2.
639 #t2; ncases t2; nnormalize; #H; ##[ ##4: napply (refl_eq ??) ##| ##*: napply (bool_destruct ??? H) ##]
642 nlemma eqbitrig_to_eq5 : ∀t2.eq_bitrig t04 t2 = true → t04 = t2.
643 #t2; ncases t2; nnormalize; #H; ##[ ##5: napply (refl_eq ??) ##| ##*: napply (bool_destruct ??? H) ##]
646 nlemma eqbitrig_to_eq6 : ∀t2.eq_bitrig t05 t2 = true → t05 = t2.
647 #t2; ncases t2; nnormalize; #H; ##[ ##6: napply (refl_eq ??) ##| ##*: napply (bool_destruct ??? H) ##]
650 nlemma eqbitrig_to_eq7 : ∀t2.eq_bitrig t06 t2 = true → t06 = t2.
651 #t2; ncases t2; nnormalize; #H; ##[ ##7: napply (refl_eq ??) ##| ##*: napply (bool_destruct ??? H) ##]
654 nlemma eqbitrig_to_eq8 : ∀t2.eq_bitrig t07 t2 = true → t07 = t2.
655 #t2; ncases t2; nnormalize; #H; ##[ ##8: napply (refl_eq ??) ##| ##*: napply (bool_destruct ??? H) ##]
658 nlemma eqbitrig_to_eq9 : ∀t2.eq_bitrig t08 t2 = true → t08 = t2.
659 #t2; ncases t2; nnormalize; #H; ##[ ##9: napply (refl_eq ??) ##| ##*: napply (bool_destruct ??? H) ##]
662 nlemma eqbitrig_to_eq10 : ∀t2.eq_bitrig t09 t2 = true → t09 = t2.
663 #t2; ncases t2; nnormalize; #H; ##[ ##10: napply (refl_eq ??) ##| ##*: napply (bool_destruct ??? H) ##]
666 nlemma eqbitrig_to_eq11 : ∀t2.eq_bitrig t0A t2 = true → t0A = t2.
667 #t2; ncases t2; nnormalize; #H; ##[ ##11: napply (refl_eq ??) ##| ##*: napply (bool_destruct ??? H) ##]
670 nlemma eqbitrig_to_eq12 : ∀t2.eq_bitrig t0B t2 = true → t0B = t2.
671 #t2; ncases t2; nnormalize; #H; ##[ ##12: napply (refl_eq ??) ##| ##*: napply (bool_destruct ??? H) ##]
674 nlemma eqbitrig_to_eq13 : ∀t2.eq_bitrig t0C t2 = true → t0C = t2.
675 #t2; ncases t2; nnormalize; #H; ##[ ##13: napply (refl_eq ??) ##| ##*: napply (bool_destruct ??? H) ##]
678 nlemma eqbitrig_to_eq14 : ∀t2.eq_bitrig t0D t2 = true → t0D = t2.
679 #t2; ncases t2; nnormalize; #H; ##[ ##14: napply (refl_eq ??) ##| ##*: napply (bool_destruct ??? H) ##]
682 nlemma eqbitrig_to_eq15 : ∀t2.eq_bitrig t0E t2 = true → t0E = t2.
683 #t2; ncases t2; nnormalize; #H; ##[ ##15: napply (refl_eq ??) ##| ##*: napply (bool_destruct ??? H) ##]
686 nlemma eqbitrig_to_eq16 : ∀t2.eq_bitrig t0F t2 = true → t0F = t2.
687 #t2; ncases t2; nnormalize; #H; ##[ ##16: napply (refl_eq ??) ##| ##*: napply (bool_destruct ??? H) ##]
690 nlemma eqbitrig_to_eq17 : ∀t2.eq_bitrig t10 t2 = true → t10 = t2.
691 #t2; ncases t2; nnormalize; #H; ##[ ##17: napply (refl_eq ??) ##| ##*: napply (bool_destruct ??? H) ##]
694 nlemma eqbitrig_to_eq18 : ∀t2.eq_bitrig t11 t2 = true → t11 = t2.
695 #t2; ncases t2; nnormalize; #H; ##[ ##18: napply (refl_eq ??) ##| ##*: napply (bool_destruct ??? H) ##]
698 nlemma eqbitrig_to_eq19 : ∀t2.eq_bitrig t12 t2 = true → t12 = t2.
699 #t2; ncases t2; nnormalize; #H; ##[ ##19: napply (refl_eq ??) ##| ##*: napply (bool_destruct ??? H) ##]
702 nlemma eqbitrig_to_eq20 : ∀t2.eq_bitrig t13 t2 = true → t13 = t2.
703 #t2; ncases t2; nnormalize; #H; ##[ ##20: napply (refl_eq ??) ##| ##*: napply (bool_destruct ??? H) ##]
706 nlemma eqbitrig_to_eq21 : ∀t2.eq_bitrig t14 t2 = true → t14 = t2.
707 #t2; ncases t2; nnormalize; #H; ##[ ##21: napply (refl_eq ??) ##| ##*: napply (bool_destruct ??? H) ##]
710 nlemma eqbitrig_to_eq22 : ∀t2.eq_bitrig t15 t2 = true → t15 = t2.
711 #t2; ncases t2; nnormalize; #H; ##[ ##22: napply (refl_eq ??) ##| ##*: napply (bool_destruct ??? H) ##]
714 nlemma eqbitrig_to_eq23 : ∀t2.eq_bitrig t16 t2 = true → t16 = t2.
715 #t2; ncases t2; nnormalize; #H; ##[ ##23: napply (refl_eq ??) ##| ##*: napply (bool_destruct ??? H) ##]
718 nlemma eqbitrig_to_eq24 : ∀t2.eq_bitrig t17 t2 = true → t17 = t2.
719 #t2; ncases t2; nnormalize; #H; ##[ ##24: napply (refl_eq ??) ##| ##*: napply (bool_destruct ??? H) ##]
722 nlemma eqbitrig_to_eq25 : ∀t2.eq_bitrig t18 t2 = true → t18 = t2.
723 #t2; ncases t2; nnormalize; #H; ##[ ##25: napply (refl_eq ??) ##| ##*: napply (bool_destruct ??? H) ##]
726 nlemma eqbitrig_to_eq26 : ∀t2.eq_bitrig t19 t2 = true → t19 = t2.
727 #t2; ncases t2; nnormalize; #H; ##[ ##26: napply (refl_eq ??) ##| ##*: napply (bool_destruct ??? H) ##]
730 nlemma eqbitrig_to_eq27 : ∀t2.eq_bitrig t1A t2 = true → t1A = t2.
731 #t2; ncases t2; nnormalize; #H; ##[ ##27: napply (refl_eq ??) ##| ##*: napply (bool_destruct ??? H) ##]
734 nlemma eqbitrig_to_eq28 : ∀t2.eq_bitrig t1B t2 = true → t1B = t2.
735 #t2; ncases t2; nnormalize; #H; ##[ ##28: napply (refl_eq ??) ##| ##*: napply (bool_destruct ??? H) ##]
738 nlemma eqbitrig_to_eq29 : ∀t2.eq_bitrig t1C t2 = true → t1C = t2.
739 #t2; ncases t2; nnormalize; #H; ##[ ##29: napply (refl_eq ??) ##| ##*: napply (bool_destruct ??? H) ##]
742 nlemma eqbitrig_to_eq30 : ∀t2.eq_bitrig t1D t2 = true → t1D = t2.
743 #t2; ncases t2; nnormalize; #H; ##[ ##30: napply (refl_eq ??) ##| ##*: napply (bool_destruct ??? H) ##]
746 nlemma eqbitrig_to_eq31 : ∀t2.eq_bitrig t1E t2 = true → t1E = t2.
747 #t2; ncases t2; nnormalize; #H; ##[ ##31: napply (refl_eq ??) ##| ##*: napply (bool_destruct ??? H) ##]
750 nlemma eqbitrig_to_eq32 : ∀t2.eq_bitrig t1F t2 = true → t1F = t2.
751 #t2; ncases t2; nnormalize; #H; ##[ ##32: napply (refl_eq ??) ##| ##*: napply (bool_destruct ??? H) ##]
754 nlemma eqbitrig_to_eq : ∀t1,t2.eq_bitrig t1 t2 = true → t1 = t2.
757 ##[ ##1: napply eqbitrig_to_eq1
758 ##| ##2: napply eqbitrig_to_eq2
759 ##| ##3: napply eqbitrig_to_eq3
760 ##| ##4: napply eqbitrig_to_eq4
761 ##| ##5: napply eqbitrig_to_eq5
762 ##| ##6: napply eqbitrig_to_eq6
763 ##| ##7: napply eqbitrig_to_eq7
764 ##| ##8: napply eqbitrig_to_eq8
765 ##| ##9: napply eqbitrig_to_eq9
766 ##| ##10: napply eqbitrig_to_eq10
767 ##| ##11: napply eqbitrig_to_eq11
768 ##| ##12: napply eqbitrig_to_eq12
769 ##| ##13: napply eqbitrig_to_eq13
770 ##| ##14: napply eqbitrig_to_eq14
771 ##| ##15: napply eqbitrig_to_eq15
772 ##| ##16: napply eqbitrig_to_eq16
773 ##| ##17: napply eqbitrig_to_eq17
774 ##| ##18: napply eqbitrig_to_eq18
775 ##| ##19: napply eqbitrig_to_eq19
776 ##| ##20: napply eqbitrig_to_eq20
777 ##| ##21: napply eqbitrig_to_eq21
778 ##| ##22: napply eqbitrig_to_eq22
779 ##| ##23: napply eqbitrig_to_eq23
780 ##| ##24: napply eqbitrig_to_eq24
781 ##| ##25: napply eqbitrig_to_eq25
782 ##| ##26: napply eqbitrig_to_eq26
783 ##| ##27: napply eqbitrig_to_eq27
784 ##| ##28: napply eqbitrig_to_eq28
785 ##| ##29: napply eqbitrig_to_eq29
786 ##| ##30: napply eqbitrig_to_eq30
787 ##| ##31: napply eqbitrig_to_eq31
788 ##| ##32: napply eqbitrig_to_eq32
792 nlemma eq_to_eqbitrig1 : ∀t2.t00 = t2 → eq_bitrig t00 t2 = true.
793 #t2; ncases t2; nnormalize; #H; ##[ ##1: napply (refl_eq ??) ##| ##*: napply (bitrigesim_destruct ??? H) ##]
796 nlemma eq_to_eqbitrig2 : ∀t2.t01 = t2 → eq_bitrig t01 t2 = true.
797 #t2; ncases t2; nnormalize; #H; ##[ ##2: napply (refl_eq ??) ##| ##*: napply (bitrigesim_destruct ??? H) ##]
800 nlemma eq_to_eqbitrig3 : ∀t2.t02 = t2 → eq_bitrig t02 t2 = true.
801 #t2; ncases t2; nnormalize; #H; ##[ ##3: napply (refl_eq ??) ##| ##*: napply (bitrigesim_destruct ??? H) ##]
804 nlemma eq_to_eqbitrig4 : ∀t2.t03 = t2 → eq_bitrig t03 t2 = true.
805 #t2; ncases t2; nnormalize; #H; ##[ ##4: napply (refl_eq ??) ##| ##*: napply (bitrigesim_destruct ??? H) ##]
808 nlemma eq_to_eqbitrig5 : ∀t2.t04 = t2 → eq_bitrig t04 t2 = true.
809 #t2; ncases t2; nnormalize; #H; ##[ ##5: napply (refl_eq ??) ##| ##*: napply (bitrigesim_destruct ??? H) ##]
812 nlemma eq_to_eqbitrig6 : ∀t2.t05 = t2 → eq_bitrig t05 t2 = true.
813 #t2; ncases t2; nnormalize; #H; ##[ ##6: napply (refl_eq ??) ##| ##*: napply (bitrigesim_destruct ??? H) ##]
816 nlemma eq_to_eqbitrig7 : ∀t2.t06 = t2 → eq_bitrig t06 t2 = true.
817 #t2; ncases t2; nnormalize; #H; ##[ ##7: napply (refl_eq ??) ##| ##*: napply (bitrigesim_destruct ??? H) ##]
820 nlemma eq_to_eqbitrig8 : ∀t2.t07 = t2 → eq_bitrig t07 t2 = true.
821 #t2; ncases t2; nnormalize; #H; ##[ ##8: napply (refl_eq ??) ##| ##*: napply (bitrigesim_destruct ??? H) ##]
824 nlemma eq_to_eqbitrig9 : ∀t2.t08 = t2 → eq_bitrig t08 t2 = true.
825 #t2; ncases t2; nnormalize; #H; ##[ ##9: napply (refl_eq ??) ##| ##*: napply (bitrigesim_destruct ??? H) ##]
828 nlemma eq_to_eqbitrig10 : ∀t2.t09 = t2 → eq_bitrig t09 t2 = true.
829 #t2; ncases t2; nnormalize; #H; ##[ ##10: napply (refl_eq ??) ##| ##*: napply (bitrigesim_destruct ??? H) ##]
832 nlemma eq_to_eqbitrig11 : ∀t2.t0A = t2 → eq_bitrig t0A t2 = true.
833 #t2; ncases t2; nnormalize; #H; ##[ ##11: napply (refl_eq ??) ##| ##*: napply (bitrigesim_destruct ??? H) ##]
836 nlemma eq_to_eqbitrig12 : ∀t2.t0B = t2 → eq_bitrig t0B t2 = true.
837 #t2; ncases t2; nnormalize; #H; ##[ ##12: napply (refl_eq ??) ##| ##*: napply (bitrigesim_destruct ??? H) ##]
840 nlemma eq_to_eqbitrig13 : ∀t2.t0C = t2 → eq_bitrig t0C t2 = true.
841 #t2; ncases t2; nnormalize; #H; ##[ ##13: napply (refl_eq ??) ##| ##*: napply (bitrigesim_destruct ??? H) ##]
844 nlemma eq_to_eqbitrig14 : ∀t2.t0D = t2 → eq_bitrig t0D t2 = true.
845 #t2; ncases t2; nnormalize; #H; ##[ ##14: napply (refl_eq ??) ##| ##*: napply (bitrigesim_destruct ??? H) ##]
848 nlemma eq_to_eqbitrig15 : ∀t2.t0E = t2 → eq_bitrig t0E t2 = true.
849 #t2; ncases t2; nnormalize; #H; ##[ ##15: napply (refl_eq ??) ##| ##*: napply (bitrigesim_destruct ??? H) ##]
852 nlemma eq_to_eqbitrig16 : ∀t2.t0F = t2 → eq_bitrig t0F t2 = true.
853 #t2; ncases t2; nnormalize; #H; ##[ ##16: napply (refl_eq ??) ##| ##*: napply (bitrigesim_destruct ??? H) ##]
856 nlemma eq_to_eqbitrig17 : ∀t2.t10 = t2 → eq_bitrig t10 t2 = true.
857 #t2; ncases t2; nnormalize; #H; ##[ ##17: napply (refl_eq ??) ##| ##*: napply (bitrigesim_destruct ??? H) ##]
860 nlemma eq_to_eqbitrig18 : ∀t2.t11 = t2 → eq_bitrig t11 t2 = true.
861 #t2; ncases t2; nnormalize; #H; ##[ ##18: napply (refl_eq ??) ##| ##*: napply (bitrigesim_destruct ??? H) ##]
864 nlemma eq_to_eqbitrig19 : ∀t2.t12 = t2 → eq_bitrig t12 t2 = true.
865 #t2; ncases t2; nnormalize; #H; ##[ ##19: napply (refl_eq ??) ##| ##*: napply (bitrigesim_destruct ??? H) ##]
868 nlemma eq_to_eqbitrig20 : ∀t2.t13 = t2 → eq_bitrig t13 t2 = true.
869 #t2; ncases t2; nnormalize; #H; ##[ ##20: napply (refl_eq ??) ##| ##*: napply (bitrigesim_destruct ??? H) ##]
872 nlemma eq_to_eqbitrig21 : ∀t2.t14 = t2 → eq_bitrig t14 t2 = true.
873 #t2; ncases t2; nnormalize; #H; ##[ ##21: napply (refl_eq ??) ##| ##*: napply (bitrigesim_destruct ??? H) ##]
875 nlemma eq_to_eqbitrig22 : ∀t2.t15 = t2 → eq_bitrig t15 t2 = true.
876 #t2; ncases t2; nnormalize; #H; ##[ ##22: napply (refl_eq ??) ##| ##*: napply (bitrigesim_destruct ??? H) ##]
879 nlemma eq_to_eqbitrig23 : ∀t2.t16 = t2 → eq_bitrig t16 t2 = true.
880 #t2; ncases t2; nnormalize; #H; ##[ ##23: napply (refl_eq ??) ##| ##*: napply (bitrigesim_destruct ??? H) ##]
883 nlemma eq_to_eqbitrig24 : ∀t2.t17 = t2 → eq_bitrig t17 t2 = true.
884 #t2; ncases t2; nnormalize; #H; ##[ ##24: napply (refl_eq ??) ##| ##*: napply (bitrigesim_destruct ??? H) ##]
887 nlemma eq_to_eqbitrig25 : ∀t2.t18 = t2 → eq_bitrig t18 t2 = true.
888 #t2; ncases t2; nnormalize; #H; ##[ ##25: napply (refl_eq ??) ##| ##*: napply (bitrigesim_destruct ??? H) ##]
891 nlemma eq_to_eqbitrig26 : ∀t2.t19 = t2 → eq_bitrig t19 t2 = true.
892 #t2; ncases t2; nnormalize; #H; ##[ ##26: napply (refl_eq ??) ##| ##*: napply (bitrigesim_destruct ??? H) ##]
895 nlemma eq_to_eqbitrig27 : ∀t2.t1A = t2 → eq_bitrig t1A t2 = true.
896 #t2; ncases t2; nnormalize; #H; ##[ ##27: napply (refl_eq ??) ##| ##*: napply (bitrigesim_destruct ??? H) ##]
899 nlemma eq_to_eqbitrig28 : ∀t2.t1B = t2 → eq_bitrig t1B t2 = true.
900 #t2; ncases t2; nnormalize; #H; ##[ ##28: napply (refl_eq ??) ##| ##*: napply (bitrigesim_destruct ??? H) ##]
903 nlemma eq_to_eqbitrig29 : ∀t2.t1C = t2 → eq_bitrig t1C t2 = true.
904 #t2; ncases t2; nnormalize; #H; ##[ ##29: napply (refl_eq ??) ##| ##*: napply (bitrigesim_destruct ??? H) ##]
907 nlemma eq_to_eqbitrig30 : ∀t2.t1D = t2 → eq_bitrig t1D t2 = true.
908 #t2; ncases t2; nnormalize; #H; ##[ ##30: napply (refl_eq ??) ##| ##*: napply (bitrigesim_destruct ??? H) ##]
911 nlemma eq_to_eqbitrig31 : ∀t2.t1E = t2 → eq_bitrig t1E t2 = true.
912 #t2; ncases t2; nnormalize; #H; ##[ ##31: napply (refl_eq ??) ##| ##*: napply (bitrigesim_destruct ??? H) ##]
915 nlemma eq_to_eqbitrig32 : ∀t2.t1F = t2 → eq_bitrig t1F t2 = true.
916 #t2; ncases t2; nnormalize; #H; ##[ ##32: napply (refl_eq ??) ##| ##*: napply (bitrigesim_destruct ??? H) ##]
919 nlemma eq_to_eqbitrig : ∀t1,t2.t1 = t2 → eq_bitrig t1 t2 = true.
922 ##[ ##1: napply eq_to_eqbitrig1
923 ##| ##2: napply eq_to_eqbitrig2
924 ##| ##3: napply eq_to_eqbitrig3
925 ##| ##4: napply eq_to_eqbitrig4
926 ##| ##5: napply eq_to_eqbitrig5
927 ##| ##6: napply eq_to_eqbitrig6
928 ##| ##7: napply eq_to_eqbitrig7
929 ##| ##8: napply eq_to_eqbitrig8
930 ##| ##9: napply eq_to_eqbitrig9
931 ##| ##10: napply eq_to_eqbitrig10
932 ##| ##11: napply eq_to_eqbitrig11
933 ##| ##12: napply eq_to_eqbitrig12
934 ##| ##13: napply eq_to_eqbitrig13
935 ##| ##14: napply eq_to_eqbitrig14
936 ##| ##15: napply eq_to_eqbitrig15
937 ##| ##16: napply eq_to_eqbitrig16
938 ##| ##17: napply eq_to_eqbitrig17
939 ##| ##18: napply eq_to_eqbitrig18
940 ##| ##19: napply eq_to_eqbitrig19
941 ##| ##20: napply eq_to_eqbitrig20
942 ##| ##21: napply eq_to_eqbitrig21
943 ##| ##22: napply eq_to_eqbitrig22
944 ##| ##23: napply eq_to_eqbitrig23
945 ##| ##24: napply eq_to_eqbitrig24
946 ##| ##25: napply eq_to_eqbitrig25
947 ##| ##26: napply eq_to_eqbitrig26
948 ##| ##27: napply eq_to_eqbitrig27
949 ##| ##28: napply eq_to_eqbitrig28
950 ##| ##29: napply eq_to_eqbitrig29
951 ##| ##30: napply eq_to_eqbitrig30
952 ##| ##31: napply eq_to_eqbitrig31
953 ##| ##32: napply eq_to_eqbitrig32