2 ||M|| This file is part of HELM, an Hypertextual, Electronic
3 ||A|| Library of Mathematics, developed at the Computer Science
4 ||T|| Department of the University of Bologna, Italy.
8 \ / This file is distributed under the terms of the
9 \ / GNU General Public License Version 2
10 V_____________________________________________________________*)
17 include "turing/while_machine.ma".
18 include "turing/if_machine.ma".
19 include "turing/universal/alphabet.ma".
20 include "turing/universal/tests.ma".
22 (* ADVANCE TO MARK (right)
24 sposta la testina a destra fino a raggiungere il primo carattere marcato
28 (* 0, a ≠ mark _ ⇒ 0, R
29 0, a = mark _ ⇒ 1, N *)
31 definition atm_states ≝ initN 3.
33 definition atm0 : atm_states ≝ mk_Sig ?? 0 (leb_true_to_le 1 3 (refl …)).
34 definition atm1 : atm_states ≝ mk_Sig ?? 1 (leb_true_to_le 2 3 (refl …)).
35 definition atm2 : atm_states ≝ mk_Sig ?? 2 (leb_true_to_le 3 3 (refl …)).
37 definition atmr_step ≝
38 λalpha:FinSet.λtest:alpha→bool.
39 mk_TM alpha atm_states
42 [ None ⇒ 〈atm1, None ?〉
45 [ true ⇒ 〈atm1,None ?〉
46 | false ⇒ 〈atm2,Some ? 〈a',R〉〉 ]])
47 atm0 (λx.notb (x == atm0)).
49 definition Ratmr_step_true ≝
52 t1 = midtape alpha ls a rs ∧ test a = false ∧
53 t2 = mk_tape alpha (a::ls) (option_hd ? rs) (tail ? rs).
55 definition Ratmr_step_false ≝
58 (current alpha t1 = None ? ∨
59 (∃a.current ? t1 = Some ? a ∧ test a = true)).
62 ∀alpha,test,ls,a0,rs. test a0 = true →
63 step alpha (atmr_step alpha test)
64 (mk_config ?? atm0 (midtape … ls a0 rs)) =
65 mk_config alpha (states ? (atmr_step alpha test)) atm1
67 #alpha #test #ls #a0 #ts #Htest whd in ⊢ (??%?);
68 whd in match (trans … 〈?,?〉); >Htest %
72 ∀alpha,test,ls,a0,rs. test a0 = false →
73 step alpha (atmr_step alpha test)
74 (mk_config ?? atm0 (midtape … ls a0 rs)) =
75 mk_config alpha (states ? (atmr_step alpha test)) atm2
76 (mk_tape … (a0::ls) (option_hd ? rs) (tail ? rs)).
77 #alpha #test #ls #a0 #ts #Htest whd in ⊢ (??%?);
78 whd in match (trans … 〈?,?〉); >Htest cases ts //
83 accRealize alpha (atmr_step alpha test)
84 atm2 (Ratmr_step_true alpha test) (Ratmr_step_false alpha test).
87 @(ex_intro ?? (mk_config ?? atm1 (niltape ?))) %
88 [ % // whd in ⊢ ((??%%)→?); #Hfalse destruct | #_ % // % % ]
89 | #a #al @(ex_intro ?? 2) @(ex_intro ?? (mk_config ?? atm1 (leftof ? a al)))
90 % [ % // whd in ⊢ ((??%%)→?); #Hfalse destruct | #_ % // % % ]
91 | #a #al @(ex_intro ?? 2) @(ex_intro ?? (mk_config ?? atm1 (rightof ? a al)))
92 % [ % // whd in ⊢ ((??%%)→?); #Hfalse destruct | #_ % // % % ]
93 | #ls #c #rs @(ex_intro ?? 2)
94 cases (true_or_false (test c)) #Htest
95 [ @(ex_intro ?? (mk_config ?? atm1 ?))
98 [ whd in ⊢ (??%?); >atmr_q0_q1 //
99 | whd in ⊢ ((??%%)→?); #Hfalse destruct ]
100 | #_ % // %2 @(ex_intro ?? c) % // ]
102 | @(ex_intro ?? (mk_config ?? atm2 (mk_tape ? (c::ls) (option_hd ? rs) (tail ? rs))))
105 [ whd in ⊢ (??%?); >atmr_q0_q2 //
106 | #_ @(ex_intro ?? ls) @(ex_intro ?? c) @(ex_intro ?? rs)
109 | #Hfalse @False_ind @(absurd ?? Hfalse) %
115 definition R_adv_to_mark_r ≝ λalpha,test,t1,t2.
117 (t1 = midtape alpha ls c rs →
118 ((test c = true ∧ t2 = t1) ∨
120 ∀rs1,b,rs2. rs = rs1@b::rs2 →
121 test b = true → (∀x.memb ? x rs1 = true → test x = false) →
122 t2 = midtape ? (reverse ? rs1@c::ls) b rs2))).
124 definition adv_to_mark_r ≝
125 λalpha,test.whileTM alpha (atmr_step alpha test) atm2.
127 lemma wsem_adv_to_mark_r :
129 WRealize alpha (adv_to_mark_r alpha test) (R_adv_to_mark_r alpha test).
130 #alpha #test #t #i #outc #Hloop
131 lapply (sem_while … (sem_atmr_step alpha test) t i outc Hloop) [%]
132 -Hloop * #t1 * #Hstar @(star_ind_l ??????? Hstar)
134 [ #H1 #ls #c #rs #H2 >H2 in H1; whd in ⊢ (??%? → ?);
135 #Hfalse destruct (Hfalse)
136 | * #a * #Ha #Htest #ls #c #rs #H2 %
137 >H2 in Ha; whd in ⊢ (??%? → ?); #Heq destruct (Heq) % //
140 | #tapea #tapeb #tapec #Hleft #Hright #IH #HRfalse
141 lapply (IH HRfalse) -IH #IH
142 #ls #c #rs #Htapea %2
143 cases Hleft #ls0 * #a0 * #rs0 * * #Htapea' #Htest #Htapeb
145 >Htapea' in Htapea; #Htapea destruct (Htapea) % // *
146 [ #b #rs2 #Hrs >Hrs in Htapeb; #Htapeb #Htestb #_
148 [ * #_ #Houtc >Houtc >Htapeb %
149 | * #Hfalse >Hfalse in Htestb; #Htestb destruct (Htestb) ]
150 | #r1 #rs1 #b #rs2 #Hrs >Hrs in Htapeb; #Htapeb #Htestb #Hmemb
152 [ * #Hfalse >(Hmemb …) in Hfalse;
153 [ #Hft destruct (Hft)
155 | * #Htestr1 #H1 >reverse_cons >associative_append
156 @H1 // #x #Hx @Hmemb @memb_cons //
161 lemma terminate_adv_to_mark_r :
163 ∀t.Terminate alpha (adv_to_mark_r alpha test) t.
165 @(terminate_while … (sem_atmr_step alpha test))
168 [ % #t1 * #ls0 * #c0 * #rs0 * * #Hfalse destruct (Hfalse)
169 |2,3: #a0 #al0 % #t1 * #ls0 * #c0 * #rs0 * * #Hfalse destruct (Hfalse)
170 | #ls #c #rs generalize in match c; -c generalize in match ls; -ls
172 [#ls #c % #t1 * #ls0 * #c0 * #rs0 * *
173 #H1 destruct (H1) #Hc0 #Ht1 normalize in Ht1;
174 % #t2 * #ls1 * #c1 * #rs1 * * >Ht1
175 normalize in ⊢ (%→?); #Hfalse destruct (Hfalse)
176 | #r0 #rs0 #IH #ls #c % #t1 * #ls0 * #c0 * #rs0 * *
177 #H1 destruct (H1) #Hc0 #Ht1 normalize in Ht1;
184 lemma sem_adv_to_mark_r :
186 Realize alpha (adv_to_mark_r alpha test) (R_adv_to_mark_r alpha test).
192 marks the current character
195 definition mark_states ≝ initN 2.
197 definition ms0 : mark_states ≝ mk_Sig ?? 0 (leb_true_to_le 1 2 (refl …)).
198 definition ms1 : mark_states ≝ mk_Sig ?? 1 (leb_true_to_le 2 2 (refl …)).
201 λalpha:FinSet.mk_TM (FinProd … alpha FinBool) mark_states
204 [ None ⇒ 〈ms1,None ?〉
205 | Some a' ⇒ match (pi1 … q) with
206 [ O ⇒ let 〈a'',b〉 ≝ a' in 〈ms1,Some ? 〈〈a'',true〉,N〉〉
207 | S q ⇒ 〈ms1,None ?〉 ] ])
210 definition R_mark ≝ λalpha,t1,t2.
212 t1 = midtape (FinProd … alpha FinBool) ls 〈c,b〉 rs →
213 t2 = midtape ? ls 〈c,true〉 rs.
216 ∀alpha.Realize ? (mark alpha) (R_mark alpha).
217 #alpha #intape @(ex_intro ?? 2) cases intape
219 [| % [ % | #ls #c #b #rs #Hfalse destruct ] ]
221 [| % [ % | #ls #c #b #rs #Hfalse destruct ] ]
223 [| % [ % | #ls #c #b #rs #Hfalse destruct ] ]
225 @ex_intro [| % [ % | #ls0 #c0 #b0 #rs0 #H1 destruct (H1) % ] ] ]
230 moves the head one step to the right
234 definition move_states ≝ initN 2.
235 definition move0 : move_states ≝ mk_Sig ?? 0 (leb_true_to_le 1 2 (refl …)).
236 definition move1 : move_states ≝ mk_Sig ?? 1 (leb_true_to_le 2 2 (refl …)).
239 λalpha:FinSet.mk_TM alpha move_states
242 [ None ⇒ 〈move1,None ?〉
243 | Some a' ⇒ match (pi1 … q) with
244 [ O ⇒ 〈move1,Some ? 〈a',R〉〉
245 | S q ⇒ 〈move1,None ?〉 ] ])
246 move0 (λq.q == move1).
248 definition R_move_r ≝ λalpha,t1,t2.
250 t1 = midtape alpha ls c rs →
251 t2 = mk_tape ? (c::ls) (option_hd ? rs) (tail ? rs).
254 ∀alpha.Realize ? (move_r alpha) (R_move_r alpha).
255 #alpha #intape @(ex_intro ?? 2) cases intape
257 [| % [ % | #ls #c #rs #Hfalse destruct ] ]
259 [| % [ % | #ls #c #rs #Hfalse destruct ] ]
261 [| % [ % | #ls #c #rs #Hfalse destruct ] ]
263 @ex_intro [| % [ % | #ls0 #c0 #rs0 #H1 destruct (H1)
269 moves the head one step to the right
274 λalpha:FinSet.mk_TM alpha move_states
277 [ None ⇒ 〈move1,None ?〉
278 | Some a' ⇒ match pi1 … q with
279 [ O ⇒ 〈move1,Some ? 〈a',L〉〉
280 | S q ⇒ 〈move1,None ?〉 ] ])
281 move0 (λq.q == move1).
283 definition R_move_l ≝ λalpha,t1,t2.
285 t1 = midtape alpha ls c rs →
286 t2 = mk_tape ? (tail ? ls) (option_hd ? ls) (c::rs).
289 ∀alpha.Realize ? (move_l alpha) (R_move_l alpha).
290 #alpha #intape @(ex_intro ?? 2) cases intape
292 [| % [ % | #ls #c #rs #Hfalse destruct ] ]
294 [| % [ % | #ls #c #rs #Hfalse destruct ] ]
296 [| % [ % | #ls #c #rs #Hfalse destruct ] ]
298 @ex_intro [| % [ % | #ls0 #c0 #rs0 #H1 destruct (H1)
302 (* MOVE RIGHT AND MARK machine
304 marks the first character on the right
306 (could be rewritten using (mark; move_right))
309 definition mrm_states ≝ initN 3.
311 definition mrm0 : mrm_states ≝ mk_Sig ?? 0 (leb_true_to_le 1 3 (refl …)).
312 definition mrm1 : mrm_states ≝ mk_Sig ?? 1 (leb_true_to_le 2 3 (refl …)).
313 definition mrm2 : mrm_states ≝ mk_Sig ?? 2 (leb_true_to_le 3 3 (refl …)).
315 definition move_right_and_mark ≝
316 λalpha:FinSet.mk_TM (FinProd … alpha FinBool) mrm_states
319 [ None ⇒ 〈mrm2,None ?〉
320 | Some a' ⇒ match pi1 … q with
321 [ O ⇒ 〈mrm1,Some ? 〈a',R〉〉
323 [ O ⇒ let 〈a'',b〉 ≝ a' in
324 〈mrm2,Some ? 〈〈a'',true〉,N〉〉
325 | S _ ⇒ 〈mrm2,None ?〉 ] ] ])
328 definition R_move_right_and_mark ≝ λalpha,t1,t2.
330 t1 = midtape (FinProd … alpha FinBool) ls c (〈d,b〉::rs) →
331 t2 = midtape ? (c::ls) 〈d,true〉 rs.
333 lemma sem_move_right_and_mark :
334 ∀alpha.Realize ? (move_right_and_mark alpha) (R_move_right_and_mark alpha).
335 #alpha #intape @(ex_intro ?? 3) cases intape
337 [| % [ % | #ls #c #d #b #rs #Hfalse destruct ] ]
339 [| % [ % | #ls #c #d #b #rs #Hfalse destruct ] ]
341 [| % [ % | #ls #c #d #b #rs #Hfalse destruct ] ]
343 [ @ex_intro [| % [ % | #ls0 #c0 #d0 #b0 #rs0 #Hfalse destruct ] ]
344 | * #d #b #rs @ex_intro
345 [| % [ % | #ls0 #c0 #d0 #b0 #rs0 #H1 destruct (H1) % ] ] ] ]
348 (* CLEAR MARK machine
350 clears the mark in the current character
353 definition clear_mark_states ≝ initN 3.
355 definition clear0 : clear_mark_states ≝ mk_Sig ?? 0 (leb_true_to_le 1 3 (refl …)).
356 definition clear1 : clear_mark_states ≝ mk_Sig ?? 1 (leb_true_to_le 2 3 (refl …)).
357 definition claer2 : clear_mark_states ≝ mk_Sig ?? 2 (leb_true_to_le 3 3 (refl …)).
359 definition clear_mark ≝
360 λalpha:FinSet.mk_TM (FinProd … alpha FinBool) clear_mark_states
363 [ None ⇒ 〈clear1,None ?〉
364 | Some a' ⇒ match pi1 … q with
365 [ O ⇒ let 〈a'',b〉 ≝ a' in 〈clear1,Some ? 〈〈a'',false〉,N〉〉
366 | S q ⇒ 〈clear1,None ?〉 ] ])
367 clear0 (λq.q == clear1).
369 definition R_clear_mark ≝ λalpha,t1,t2.
371 t1 = midtape (FinProd … alpha FinBool) ls 〈c,b〉 rs →
372 t2 = midtape ? ls 〈c,false〉 rs.
374 lemma sem_clear_mark :
375 ∀alpha.Realize ? (clear_mark alpha) (R_clear_mark alpha).
376 #alpha #intape @(ex_intro ?? 2) cases intape
378 [| % [ % | #ls #c #b #rs #Hfalse destruct ] ]
380 [| % [ % | #ls #c #b #rs #Hfalse destruct ] ]
382 [| % [ % | #ls #c #b #rs #Hfalse destruct ] ]
384 @ex_intro [| % [ % | #ls0 #c0 #b0 #rs0 #H1 destruct (H1) % ] ] ]
387 (* ADVANCE MARK RIGHT machine
389 clears mark on current char,
390 moves right, and marks new current char
394 definition adv_mark_r ≝
396 seq ? (clear_mark alpha)
397 (seq ? (move_r ?) (mark alpha)).
399 definition R_adv_mark_r ≝ λalpha,t1,t2.
401 t1 = midtape (FinProd … alpha FinBool) ls 〈c,true〉 (〈d,b〉::rs) →
402 t2 = midtape ? (〈c,false〉::ls) 〈d,true〉 rs.
404 lemma sem_adv_mark_r :
405 ∀alpha.Realize ? (adv_mark_r alpha) (R_adv_mark_r alpha).
407 cases (sem_seq ????? (sem_clear_mark …)
408 (sem_seq ????? (sem_move_r ?) (sem_mark alpha)) intape)
409 #k * #outc * #Hloop whd in ⊢ (%→?);
410 * #ta * whd in ⊢ (%→?); #Hs1 * #tb * whd in ⊢ (%→%→?); #Hs2 #Hs3
411 @(ex_intro ?? k) @(ex_intro ?? outc) %
413 | -Hloop #ls #c #d #b #rs #Hintape @(Hs3 … b)
414 @(Hs2 ls 〈c,false〉 (〈d,b〉::rs))
419 (* ADVANCE TO MARK (left)
425 definition R_adv_to_mark_l ≝ λalpha,test,t1,t2.
427 (t1 = midtape alpha ls c rs →
428 ((test c = true ∧ t2 = t1) ∨
430 ∀ls1,b,ls2. ls = ls1@b::ls2 →
431 test b = true → (∀x.memb ? x ls1 = true → test x = false) →
432 t2 = midtape ? ls2 b (reverse ? ls1@c::rs)))).
434 axiom adv_to_mark_l : ∀alpha:FinSet.(alpha → bool) → TM alpha.
435 (* definition adv_to_mark_l ≝
436 λalpha,test.whileTM alpha (atml_step alpha test) 2. *)
438 axiom wsem_adv_to_mark_l :
440 WRealize alpha (adv_to_mark_l alpha test) (R_adv_to_mark_l alpha test).
442 #alpha #test #t #i #outc #Hloop
443 lapply (sem_while … (sem_atmr_step alpha test) t i outc Hloop) [%]
444 -Hloop * #t1 * #Hstar @(star_ind_l ??????? Hstar)
446 [ #H1 #ls #c #rs #H2 >H2 in H1; whd in ⊢ (??%? → ?);
447 #Hfalse destruct (Hfalse)
448 | * #a * #Ha #Htest #ls #c #rs #H2 %
449 >H2 in Ha; whd in ⊢ (??%? → ?); #Heq destruct (Heq) % //
452 | #tapea #tapeb #tapec #Hleft #Hright #IH #HRfalse
453 lapply (IH HRfalse) -IH #IH
454 #ls #c #rs #Htapea %2
455 cases Hleft #ls0 * #a0 * #rs0 * * #Htapea' #Htest #Htapeb
457 >Htapea' in Htapea; #Htapea destruct (Htapea) % // *
458 [ #b #rs2 #Hrs >Hrs in Htapeb; #Htapeb #Htestb #_
460 [ * #_ #Houtc >Houtc >Htapeb %
461 | * #Hfalse >Hfalse in Htestb; #Htestb destruct (Htestb) ]
462 | #r1 #rs1 #b #rs2 #Hrs >Hrs in Htapeb; #Htapeb #Htestb #Hmemb
464 [ * #Hfalse >(Hmemb …) in Hfalse;
465 [ #Hft destruct (Hft)
467 | * #Htestr1 #H1 >reverse_cons >associative_append
468 @H1 // #x #Hx @Hmemb @memb_cons //
474 axiom terminate_adv_to_mark_l :
476 ∀t.Terminate alpha (adv_to_mark_l alpha test) t.
479 @(terminate_while … (sem_atmr_step alpha test))
482 [ % #t1 * #ls0 * #c0 * #rs0 * * #Hfalse destruct (Hfalse)
483 |2,3: #a0 #al0 % #t1 * #ls0 * #c0 * #rs0 * * #Hfalse destruct (Hfalse)
484 | #ls #c #rs generalize in match c; -c generalize in match ls; -ls
486 [#ls #c % #t1 * #ls0 * #c0 * #rs0 * *
487 #H1 destruct (H1) #Hc0 #Ht1 normalize in Ht1;
488 % #t2 * #ls1 * #c1 * #rs1 * * >Ht1
489 normalize in ⊢ (%→?); #Hfalse destruct (Hfalse)
490 | #r0 #rs0 #IH #ls #c % #t1 * #ls0 * #c0 * #rs0 * *
491 #H1 destruct (H1) #Hc0 #Ht1 normalize in Ht1;
499 lemma sem_adv_to_mark_l :
501 Realize alpha (adv_to_mark_l alpha test) (R_adv_to_mark_l alpha test).
506 ADVANCE BOTH MARKS machine
508 l1 does not contain marks ⇒
520 definition is_marked ≝
521 λalpha.λp:FinProd … alpha FinBool.
524 definition adv_both_marks ≝
525 λalpha.seq ? (adv_mark_r alpha)
527 (seq ? (adv_to_mark_l (FinProd alpha FinBool) (is_marked alpha))
528 (adv_mark_r alpha))).
530 definition R_adv_both_marks ≝
532 ∀l0,x,a,l1,x0,a0,l2. (∀c.memb ? c l1 = true → is_marked ? c = false) →
533 t1 = midtape (FinProd … alpha FinBool)
534 (l1@〈a,false〉::〈x,true〉::l0) 〈x0,true〉 (〈a0,false〉::l2) →
535 t2 = midtape ? (〈x,false〉::l0) 〈a,true〉 (reverse ? l1@〈x0,false〉::〈a0,true〉::l2).
537 lemma sem_adv_both_marks :
538 ∀alpha.Realize ? (adv_both_marks alpha) (R_adv_both_marks alpha).
540 cases (sem_seq ????? (sem_adv_mark_r …)
541 (sem_seq ????? (sem_move_l …)
542 (sem_seq ????? (sem_adv_to_mark_l ? (is_marked ?))
543 (sem_adv_mark_r alpha))) intape)
544 #k * #outc * #Hloop whd in ⊢ (%→?);
545 * #ta * whd in ⊢ (%→?); #Hs1 * #tb * whd in ⊢ (%→?); #Hs2
546 * #tc * whd in ⊢ (%→%→?); #Hs3 #Hs4
547 @(ex_intro ?? k) @(ex_intro ?? outc) %
549 | -Hloop #l0 #x #a #l1 #x0 #a0 #l2 #Hl1 #Hintape
551 lapply (Hs1 … Hintape) #Hta
552 lapply (Hs2 … Hta) #Htb
554 [ * #Hfalse normalize in Hfalse; destruct (Hfalse)
556 lapply (Hs3 (l1@[〈a,false〉]) 〈x,true〉 l0 ???)
557 [ #x1 #Hx1 cases (memb_append … Hx1)
559 | #Hx1' >(memb_single … Hx1') % ]
561 | >associative_append %
562 | >reverse_append #Htc @Htc ]
574 l0 x a* l1 x0 a0* l2 (f(x0) == true)
576 l0 x* a l1 x0* a0 l2 (f(x0) == false)
580 definition match_and_adv ≝
581 λalpha,f.ifTM ? (test_char ? f)
582 (adv_both_marks alpha) (clear_mark ?) tc_true.
584 definition R_match_and_adv ≝
586 ∀l0,x,a,l1,x0,a0,l2. (∀c.memb ? c l1 = true → is_marked ? c = false) →
587 t1 = midtape (FinProd … alpha FinBool)
588 (l1@〈a,false〉::〈x,true〉::l0) 〈x0,true〉 (〈a0,false〉::l2) →
589 (f 〈x0,true〉 = true ∧ t2 = midtape ? (〈x,false〉::l0) 〈a,true〉 (reverse ? l1@〈x0,false〉::〈a0,true〉::l2))
590 ∨ (f 〈x0,true〉 = false ∧
591 t2 = midtape ? (l1@〈a,false〉::〈x,true〉::l0) 〈x0,false〉 (〈a0,false〉::l2)).
593 lemma sem_match_and_adv :
594 ∀alpha,f.Realize ? (match_and_adv alpha f) (R_match_and_adv alpha f).
596 cases (sem_if ? (test_char ? f) … tc_true (sem_test_char ? f) (sem_adv_both_marks alpha) (sem_clear_mark ?) intape)
597 #k * #outc * #Hloop #Hif @(ex_intro ?? k) @(ex_intro ?? outc)
600 [ * #ta * whd in ⊢ (%→%→?); #Hta #Houtc
601 #l0 #x #a #l1 #x0 #a0 #l2 #Hl1 #Hintape
602 >Hintape in Hta; #Hta cases (Hta … (refl ??)) -Hta #Hf #Hta % %
603 [ @Hf | @Houtc [ @Hl1 | @Hta ] ]
604 | * #ta * whd in ⊢ (%→%→?); #Hta #Houtc
605 #l0 #x #a #l1 #x0 #a0 #l2 #Hl1 #Hintape
606 >Hintape in Hta; #Hta cases (Hta … (refl ??)) -Hta #Hf #Hta %2 %
607 [ @Hf | >(Houtc … Hta) % ]
613 then move_right; ----
615 if current (* x0 *) = 0
616 then advance_mark ----
623 definition comp_step_subcase ≝
624 λalpha,c,elseM.ifTM ? (test_char ? (λx.x == c))
626 (seq ? (adv_to_mark_r ? (is_marked alpha))
627 (match_and_adv ? (λx.x == c))))
630 definition R_comp_step_subcase ≝
631 λalpha,c,RelseM,t1,t2.
632 ∀l0,x,rs.t1 = midtape (FinProd … alpha FinBool) l0 〈x,true〉 rs →
634 ∀a,l1,x0,a0,l2. (∀c.memb ? c l1 = true → is_marked ? c = false) →
635 rs = 〈a,false〉::l1@〈x0,true〉::〈a0,false〉::l2 →
637 t2 = midtape ? (〈x,false〉::l0) 〈a,true〉 (l1@〈x0,false〉::〈a0,true〉::l2)) ∨
639 t2 = midtape (FinProd … alpha FinBool)
640 (reverse ? l1@〈a,false〉::〈x,true〉::l0) 〈x0,false〉 (〈a0,false〉::l2)))) ∨
641 (〈x,true〉 ≠ c ∧ RelseM t1 t2).
643 lemma sem_comp_step_subcase :
644 ∀alpha,c,elseM,RelseM.
645 Realize ? elseM RelseM →
646 Realize ? (comp_step_subcase alpha c elseM)
647 (R_comp_step_subcase alpha c RelseM).
648 #alpha #c #elseM #RelseM #Helse #intape
649 cases (sem_if ? (test_char ? (λx.x == c)) … tc_true
650 (sem_test_char ? (λx.x == c))
651 (sem_seq ????? (sem_move_r …)
652 (sem_seq ????? (sem_adv_to_mark_r ? (is_marked alpha))
653 (sem_match_and_adv ? (λx.x == c)))) Helse intape)
654 #k * #outc * #Hloop #HR @(ex_intro ?? k) @(ex_intro ?? outc)
655 % [ @Hloop ] -Hloop cases HR -HR
656 [ * #ta * whd in ⊢ (%→?); #Hta * #tb * whd in ⊢ (%→?); #Htb
657 * #tc * whd in ⊢ (%→?); #Htc whd in ⊢ (%→?); #Houtc
658 #l0 #x #rs #Hintape cases (true_or_false (〈x,true〉==c)) #Hc
660 #a #l1 #x0 #a0 #l2 #Hl1 #Hrs >Hrs in Hintape; #Hintape
661 >Hintape in Hta; #Hta cases (Hta ? (refl ??)) -Hta
662 #Hx #Hta lapply (Htb … Hta) -Htb #Htb
663 cases (Htc … Htb) [ * #Hfalse normalize in Hfalse; destruct (Hfalse) ]
664 -Htc * #_ #Htc lapply (Htc l1 〈x0,true〉 (〈a0,false〉::l2) (refl ??) (refl ??) Hl1)
665 -Htc #Htc cases (Houtc ???????? Htc) -Houtc
667 % [ <(\P Hx0) in Hx; #Hx lapply (\P Hx) #Hx' destruct (Hx') %
668 | >Houtc >reverse_reverse % ]
670 % [ <(\P Hx) in Hx0; #Hx0 lapply (\Pf Hx0) @not_to_not #Hx' >Hx' %
672 | (* members of lists are invariant under reverse *) @daemon ]
674 >Hintape in Hta; #Hta cases (Hta ? (refl ??)) -Hta #Hx #Hta
675 >Hx in Hc;#Hc destruct (Hc) ]
676 | * #ta * whd in ⊢ (%→?); #Hta #Helse #ls #c0 #rs #Hintape %2
677 >Hintape in Hta; #Hta cases (Hta ? (refl ??)) -Hta #Hc #Hta %
678 [ @(\Pf Hc) | <Hta @Helse ]
685 + se è un bit, ho fallito il confronto della tupla corrente
686 + se è un separatore, la tupla fa match
689 ifTM ? (test_char ? is_marked)
690 (single_finalTM … (comp_step_subcase unialpha 〈bit false,true〉
691 (comp_step_subcase unialpha 〈bit true,true〉
696 definition comp_step ≝
697 ifTM ? (test_char ? (is_marked ?))
698 (single_finalTM … (comp_step_subcase FSUnialpha 〈bit false,true〉
699 (comp_step_subcase FSUnialpha 〈bit true,true〉
700 (comp_step_subcase FSUnialpha 〈null,true〉
705 definition R_comp_step_true ≝
707 ∀l0,c,rs.t1 = midtape (FinProd … FSUnialpha FinBool) l0 c rs →
709 ((bit_or_null c' = true ∧
711 rs = 〈a,false〉::l1@〈c0,true〉::〈a0,false〉::l2 →
712 (∀c.memb ? c l1 = true → is_marked ? c = false) →
714 t2 = midtape ? (〈c',false〉::l0) 〈a,true〉 (l1@〈c0,false〉::〈a0,true〉::l2)) ∨
716 t2 = midtape (FinProd … FSUnialpha FinBool)
717 (reverse ? l1@〈a,false〉::〈c',true〉::l0) 〈c0,false〉 (〈a0,false〉::l2))) ∨
718 (bit_or_null c' = false ∧ t2 = midtape ? l0 〈c',false〉 rs)).
720 definition R_comp_step_false ≝
722 ∀ls,c,rs.t1 = midtape (FinProd … FSUnialpha FinBool) ls c rs →
723 is_marked ? c = false ∧ t2 = t1.
725 lemma sem_comp_step :
726 accRealize ? comp_step (inr … (inl … (inr … start_nop)))
727 R_comp_step_true R_comp_step_false.
729 cases (acc_sem_if … (sem_test_char ? (is_marked ?))
730 (sem_comp_step_subcase FSUnialpha 〈bit false,true〉 ??
731 (sem_comp_step_subcase FSUnialpha 〈bit true,true〉 ??
732 (sem_comp_step_subcase FSUnialpha 〈null,true〉 ??
733 (sem_clear_mark …))))
735 #k * #outc * * #Hloop #H1 #H2
736 @(ex_intro ?? k) @(ex_intro ?? outc) %
737 [ % [@Hloop ] ] -Hloop
738 [ #Hstate lapply (H1 Hstate) -H1 -Hstate -H2 *
739 #ta * whd in ⊢ (%→?); #Hleft #Hright #ls #c #rs #Hintape
740 >Hintape in Hleft; #Hleft cases (Hleft ? (refl ??)) -Hleft
741 cases c in Hintape; #c' #b #Hintape whd in ⊢ (??%?→?);
742 #Hb >Hb #Hta @(ex_intro ?? c') % //
744 [ * #Hc' #H1 % % [destruct (Hc') % ]
745 #a #l1 #c0 #a0 #l2 #Hrs >Hrs in Hintape; #Hintape #Hl1
747 [ * #Htmp >Htmp -Htmp #Houtc % % // @Houtc
748 | * #Hneq #Houtc %2 %
752 | * #Hc #Helse1 cases (Helse1 … Hta)
753 [ * #Hc' #H1 % % [destruct (Hc') % ]
754 #a #l1 #c0 #a0 #l2 #Hrs >Hrs in Hintape; #Hintape #Hl1
756 [ * #Htmp >Htmp -Htmp #Houtc % % // @Houtc
757 | * #Hneq #Houtc %2 %
761 | * #Hc' #Helse2 cases (Helse2 … Hta)
762 [ * #Hc'' #H1 % % [destruct (Hc'') % ]
763 #a #l1 #c0 #a0 #l2 #Hrs >Hrs in Hintape; #Hintape #Hl1
765 [ * #Htmp >Htmp -Htmp #Houtc % % // @Houtc
766 | * #Hneq #Houtc %2 %
770 | * #Hc'' whd in ⊢ (%→?); #Helse3 %2 %
771 [ generalize in match Hc''; generalize in match Hc'; generalize in match Hc;
773 [ * [ #_ #Hfalse @False_ind @(absurd ?? Hfalse) %
774 | #Hfalse @False_ind @(absurd ?? Hfalse) % ]
775 | #_ #_ #Hfalse @False_ind @(absurd ?? Hfalse) %
782 | #Hstate lapply (H2 Hstate) -H1 -Hstate -H2 *
783 #ta * whd in ⊢ (%→%→?); #Hleft #Hright #ls #c #rs #Hintape
784 >Hintape in Hleft; #Hleft
785 cases (Hleft ? (refl ??)) -Hleft
786 #Hc #Hta % // >Hright //
791 whileTM ? comp_step (inr … (inl … (inr … start_nop))).
794 definition R_compare :=
798 ∀ls,c,b,rs.t1 = midtape ? ls 〈c,b〉 rs →
799 (b = true → rs = ....) →
803 rs = cs@l1@〈c0,true〉::cs0@l2
807 ls 〈c,b〉 cs l1 〈c0,b0〉 cs0 l2
811 ls (hd (Ls@〈grid,false〉))* (tail (Ls@〈grid,false〉)) l1 (hd (Ls@〈comma,false〉))* (tail (Ls@〈comma,false〉)) l2
812 ^^^^^^^^^^^^^^^^^^^^^^^
814 ls Ls 〈grid,false〉 l1 Ls 〈comma,true〉 l2
819 ls (hd (Ls@〈c,false〉))* (tail (Ls@〈c,false〉)) l1 (hd (Ls@〈d,false〉))* (tail (Ls@〈d,false〉)) l2
820 ^^^^^^^^^^^^^^^^^^^^^^^
823 ls Ls 〈c,true〉 l1 Ls 〈d,false〉 l2
829 (∃la,d.〈b,true〉::bs = la@[〈grid,d〉] ∧ ∀x.memb ? x la → is_bit (\fst x) = true) →
830 (∃lb,d0.〈b0,true〉::b0s = lb@[〈comma,d0〉] ∧ ∀x.memb ? x lb → is_bit (\fst x) = true) →
831 t1 = midtape ? l0 〈b,true〉 (bs@l1@〈b0,true〉::b0s@l2 →
833 mk_tape left (option current) right
835 (b = grid ∧ b0 = comma ∧ bs = [] ∧ b0s = [] ∧
836 t2 = midtape ? l0 〈grid,false〉 (l1@〈comma,true〉::l2)) ∨
837 (b = bit x ∧ b = c ∧ bs = b0s
839 definition R_compare :=
841 ∀ls,c,rs.t1 = midtape (FinProd … FSUnialpha FinBool) ls c rs →
842 (∀c'.bit_or_null c' = false → c = 〈c',true〉 → t2 = midtape ? ls 〈c',false〉 rs) ∧
843 (∀c'. c = 〈c',false〉 → t2 = t1) ∧
846 (∀c.memb (FinProd … FSUnialpha FinBool) c bs = true → bit_or_null (\fst c) = true) →
847 (∀c.memb (FinProd … FSUnialpha FinBool) c b0s = true → bit_or_null (\fst c) = true) →
848 (∀c.memb ? c bs = true → is_marked ? c = false) →
849 (∀c.memb ? c b0s = true → is_marked ? c = false) →
850 (∀c.memb ? c l1 = true → is_marked ? c = false) →
851 c = 〈b,true〉 → bit_or_null b = true →
852 rs = bs@〈grid,false〉::l1@〈b0,true〉::b0s@〈comma,false〉::l2 →
853 (〈b,true〉::bs = 〈b0,true〉::b0s ∧
854 t2 = midtape ? (reverse ? bs@〈b,false〉::ls)
855 〈grid,false〉 (l1@〈b0,false〉::b0s@〈comma,true〉::l2)) ∨
856 (∃la,c',d',lb,lc.c' ≠ d' ∧
857 〈b,false〉::bs = la@〈c',false〉::lb ∧
858 〈b0,false〉::b0s = la@〈d',false〉::lc ∧
859 t2 = midtape (FinProd … FSUnialpha FinBool) (reverse ? la@
865 〈d',false〉 (lc@〈comma,false〉::l2)).
868 ∀T1,T2:Type[0].∀l1:list T1.∀l2:list T2.∀P:list T1 → list T2 → Prop.
869 length ? l1 = length ? l2 →
871 (∀tl1,tl2,hd1,hd2. P tl1 tl2 → P (hd1::tl1) (hd2::tl2)) →
873 #T1 #T2 #l1 #l2 #P #Hl #Pnil #Pcons
874 generalize in match Hl; generalize in match l2;
876 [#l2 cases l2 // normalize #t2 #tl2 #H destruct
877 |#t1 #tl1 #IH #l2 cases l2
878 [normalize #H destruct
879 |#t2 #tl2 #H @Pcons @IH normalize in H; destruct // ]
884 ∀T1,T2:Type[0].∀l1:list T1.∀l2:list T2.∀P:Prop.
885 length ? l1 = length ? l2 →
886 (l1 = [] → l2 = [] → P) →
887 (∀hd1,hd2,tl1,tl2.l1 = hd1::tl1 → l2 = hd2::tl2 → P) → P.
888 #T1 #T2 #l1 #l2 #P #Hl @(list_ind2 … Hl)
889 [ #Pnil #Pcons @Pnil //
890 | #tl1 #tl2 #hd1 #hd2 #IH1 #IH2 #Hp @Hp // ]
893 lemma wsem_compare : WRealize ? compare R_compare.
895 lapply (sem_while ?????? sem_comp_step t i outc Hloop) [%]
896 -Hloop * #t1 * #Hstar @(star_ind_l ??????? Hstar)
897 [ #tapea whd in ⊢ (%→?); #Rfalse #ls #c #rs #Htapea %
899 [ #c' #Hc' #Hc lapply (Rfalse … Htapea) -Rfalse * >Hc
900 whd in ⊢ (??%?→?); #Hfalse destruct (Hfalse)
901 | #c' #Hc lapply (Rfalse … Htapea) -Rfalse * #_
903 | #b #b0 #bs #b0s #l1 #l2 #Hlen #Hbs1 #Hb0s1 #Hbs2 #Hb0s2 #Hl1 #Hc
904 cases (Rfalse … Htapea) -Rfalse >Hc whd in ⊢ (??%?→?);#Hfalse destruct (Hfalse)
906 | #tapea #tapeb #tapec #Hleft #Hright #IH #Htapec lapply (IH Htapec) -Htapec -IH #IH
907 whd in Hleft; #ls #c #rs #Htapea cases (Hleft … Htapea) -Hleft
908 #c' * #Hc >Hc cases (true_or_false (bit_or_null c')) #Hc'
910 [ * >Hc' #H @False_ind destruct (H)
911 | * #_ #Htapeb cases (IH … Htapeb) * #_ #H #_ %
913 [#c1 #Hc1 #Heqc destruct (Heqc) <Htapeb @(H c1) %
914 |#c1 #Hfalse destruct (Hfalse)
916 |#b #b0 #bs #b0s #l1 #l2 #_ #_ #_ #_ #_ #_
917 #Heq destruct (Heq) >Hc' #Hfalse @False_ind destruct (Hfalse)
922 [ #c'' #Hc'' #Heq destruct (Heq) >Hc'' in Hc'; #H destruct (H)
923 | #c0 #Hfalse destruct (Hfalse)
925 |#b #b0 #bs #b0s #l1 #l2 #Hlen #Hbs1 #Hb0s1 #Hbs2 #Hb0s2 #Hl1
926 #Heq destruct (Heq) #_ #Hrs cases Hleft -Hleft
927 [2: * >Hc' #Hfalse @False_ind destruct ] * #_
928 @(list_cases_2 … Hlen)
929 [ #Hbs #Hb0s generalize in match Hrs; >Hbs in ⊢ (%→?); >Hb0s in ⊢ (%→?);
930 -Hrs #Hrs normalize in Hrs; #Hleft cases (Hleft ????? Hrs ?) -Hleft
931 [ * #Heqb #Htapeb cases (IH … Htapeb) -IH * #IH #_ #_
937 @(ex_intro … [ ]) @(ex_intro … b)
938 @(ex_intro … b0) @(ex_intro … [ ])
940 [ % [ % [@sym_not_eq //| >Hbs %] | >Hb0s %]
941 | cases (IH … Htapeb) -IH * #_ #IH #_ >(IH ? (refl ??))
945 | * #b' #bitb' * #b0' #bitb0' #bs' #b0s' #Hbs #Hb0s
946 generalize in match Hrs; >Hbs in ⊢ (%→?); >Hb0s in ⊢ (%→?);
947 cut (bit_or_null b' = true ∧ bit_or_null b0' = true ∧
948 bitb' = false ∧ bitb0' = false)
949 [ % [ % [ % [ >Hbs in Hbs1; #Hbs1 @(Hbs1 〈b',bitb'〉) @memb_hd
950 | >Hb0s in Hb0s1; #Hb0s1 @(Hb0s1 〈b0',bitb0'〉) @memb_hd ]
951 | >Hbs in Hbs2; #Hbs2 @(Hbs2 〈b',bitb'〉) @memb_hd ]
952 | >Hb0s in Hb0s2; #Hb0s2 @(Hb0s2 〈b0',bitb0'〉) @memb_hd ]
953 | * * * #Ha #Hb #Hc #Hd >Hc >Hd
955 cases (Hleft b' (bs'@〈grid,false〉::l1) b0 b0'
956 (b0s'@〈comma,false〉::l2) ??) -Hleft
957 [ 3: >Hrs normalize @eq_f >associative_append %
958 | * #Hb0 #Htapeb cases (IH …Htapeb) -IH * #_ #_ #IH
959 cases (IH b' b0' bs' b0s' (l1@[〈b0,false〉]) l2 ??????? Ha ?) -IH
961 [ >Hb0 @eq_f >Hbs in Heq; >Hb0s in ⊢ (%→?); #Heq
962 destruct (Heq) >Hb0s >Hc >Hd %
963 | >Houtc >Hbs >Hb0s >Hc >Hd >reverse_cons >associative_append
964 >associative_append %
966 | * #la * #c' * #d' * #lb * #lc * * * #H1 #H2 #H3 #H4 %2
967 @(ex_intro … (〈b,false〉::la)) @(ex_intro … c') @(ex_intro … d')
968 @(ex_intro … lb) @(ex_intro … lc)
969 % [ % [ % // >Hbs >Hc >H2 % | >Hb0s >Hd >H3 >Hb0 % ]
970 | >H4 >Hbs >Hb0s >Hc >Hd >Hb0 >reverse_append
971 >reverse_cons >reverse_cons
972 >associative_append >associative_append
973 >associative_append >associative_append %
975 | generalize in match Hlen; >Hbs >Hb0s
976 normalize #Hlen destruct (Hlen) @e0
977 | #c0 #Hc0 @Hbs1 >Hbs @memb_cons //
978 | #c0 #Hc0 @Hb0s1 >Hb0s @memb_cons //
979 | #c0 #Hc0 @Hbs2 >Hbs @memb_cons //
980 | #c0 #Hc0 @Hb0s2 >Hb0s @memb_cons //
981 | #c0 #Hc0 cases (memb_append … Hc0)
982 [ @Hl1 | #Hc0' >(memb_single … Hc0') % ]
984 | >associative_append >associative_append % ]
986 @(ex_intro … []) @(ex_intro … b) @(ex_intro … b0)
987 @(ex_intro … bs) @(ex_intro … b0s) %
988 [ % // % // @sym_not_eq //
989 | >Hbs >Hb0s >Hc >Hd >reverse_cons >associative_append
990 >reverse_append in Htapeb; >reverse_cons
991 >associative_append >associative_append
993 cases (IH … Htapeb) -Htapeb -IH * #_ #IH #_ @(IH ? (refl ??))
995 | #c1 #Hc1 cases (memb_append … Hc1) #Hyp
996 [ @Hbs2 >Hbs @memb_cons @Hyp
997 | cases (orb_true_l … Hyp)
1006 axiom sem_compare : Realize ? compare R_compare.