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/if_machine.ma".
18 include "turing/basic_machines.ma".
19 include "turing/universal/alphabet.ma".
21 (* ADVANCE TO MARK (right)
23 sposta la testina a destra fino a raggiungere il primo carattere marcato
27 (* 0, a ≠ mark _ ⇒ 0, R
28 0, a = mark _ ⇒ 1, N *)
30 definition atm_states ≝ initN 3.
32 definition atm0 : atm_states ≝ mk_Sig ?? 0 (leb_true_to_le 1 3 (refl …)).
33 definition atm1 : atm_states ≝ mk_Sig ?? 1 (leb_true_to_le 2 3 (refl …)).
34 definition atm2 : atm_states ≝ mk_Sig ?? 2 (leb_true_to_le 3 3 (refl …)).
36 definition atmr_step ≝
37 λalpha:FinSet.λtest:alpha→bool.
38 mk_TM alpha atm_states
41 [ None ⇒ 〈atm1, None ?〉
44 [ true ⇒ 〈atm1,None ?〉
45 | false ⇒ 〈atm2,Some ? 〈a',R〉〉 ]])
46 atm0 (λx.notb (x == atm0)).
48 definition Ratmr_step_true ≝
51 t1 = midtape alpha ls a rs ∧ test a = false ∧
52 t2 = mk_tape alpha (a::ls) (option_hd ? rs) (tail ? rs).
54 definition Ratmr_step_false ≝
57 (current alpha t1 = None ? ∨
58 (∃a.current ? t1 = Some ? a ∧ test a = true)).
61 ∀alpha,test,ls,a0,rs. test a0 = true →
62 step alpha (atmr_step alpha test)
63 (mk_config ?? atm0 (midtape … ls a0 rs)) =
64 mk_config alpha (states ? (atmr_step alpha test)) atm1
66 #alpha #test #ls #a0 #ts #Htest whd in ⊢ (??%?);
67 whd in match (trans … 〈?,?〉); >Htest %
71 ∀alpha,test,ls,a0,rs. test a0 = false →
72 step alpha (atmr_step alpha test)
73 (mk_config ?? atm0 (midtape … ls a0 rs)) =
74 mk_config alpha (states ? (atmr_step alpha test)) atm2
75 (mk_tape … (a0::ls) (option_hd ? rs) (tail ? rs)).
76 #alpha #test #ls #a0 #ts #Htest whd in ⊢ (??%?);
77 whd in match (trans … 〈?,?〉); >Htest cases ts //
82 accRealize alpha (atmr_step alpha test)
83 atm2 (Ratmr_step_true alpha test) (Ratmr_step_false alpha test).
86 @(ex_intro ?? (mk_config ?? atm1 (niltape ?))) %
87 [ % // whd in ⊢ ((??%%)→?); #Hfalse destruct | #_ % // % % ]
88 | #a #al @(ex_intro ?? 2) @(ex_intro ?? (mk_config ?? atm1 (leftof ? a al)))
89 % [ % // whd in ⊢ ((??%%)→?); #Hfalse destruct | #_ % // % % ]
90 | #a #al @(ex_intro ?? 2) @(ex_intro ?? (mk_config ?? atm1 (rightof ? a al)))
91 % [ % // whd in ⊢ ((??%%)→?); #Hfalse destruct | #_ % // % % ]
92 | #ls #c #rs @(ex_intro ?? 2)
93 cases (true_or_false (test c)) #Htest
94 [ @(ex_intro ?? (mk_config ?? atm1 ?))
97 [ whd in ⊢ (??%?); >atmr_q0_q1 //
98 | whd in ⊢ ((??%%)→?); #Hfalse destruct ]
99 | #_ % // %2 @(ex_intro ?? c) % // ]
101 | @(ex_intro ?? (mk_config ?? atm2 (mk_tape ? (c::ls) (option_hd ? rs) (tail ? rs))))
104 [ whd in ⊢ (??%?); >atmr_q0_q2 //
105 | #_ @(ex_intro ?? ls) @(ex_intro ?? c) @(ex_intro ?? rs)
108 | #Hfalse @False_ind @(absurd ?? Hfalse) %
114 lemma dec_test: ∀alpha,rs,test.
115 decidable (∀c.memb alpha c rs = true → test c = false).
116 #alpha #rs #test elim rs
117 [%1 #n normalize #H destruct
118 |#a #tl cases (true_or_false (test a)) #Ha
119 [#_ %2 % #Hall @(absurd ?? not_eq_true_false) <Ha
121 |* [#Hall %1 #c #memc cases (orb_true_l … memc)
122 [#eqca >(\P eqca) @Ha |@Hall]
123 |#Hnall %2 @(not_to_not … Hnall) #Hall #c #memc @Hall @memb_cons //
127 definition R_adv_to_mark_r ≝ λalpha,test,t1,t2.
128 (current ? t1 = None ? → t1 = t2) ∧
130 (t1 = midtape alpha ls c rs →
131 ((test c = true ∧ t2 = t1) ∨
133 (∀rs1,b,rs2. rs = rs1@b::rs2 →
134 test b = true → (∀x.memb ? x rs1 = true → test x = false) →
135 t2 = midtape ? (reverse ? rs1@c::ls) b rs2) ∧
136 ((∀x.memb ? x rs = true → test x = false) →
137 ∀a,l.reverse ? (c::rs) = a::l →
138 t2 = rightof alpha a (l@ls))))).
140 definition adv_to_mark_r ≝
141 λalpha,test.whileTM alpha (atmr_step alpha test) atm2.
143 lemma wsem_adv_to_mark_r :
145 WRealize alpha (adv_to_mark_r alpha test) (R_adv_to_mark_r alpha test).
146 #alpha #test #t #i #outc #Hloop
147 lapply (sem_while … (sem_atmr_step alpha test) t i outc Hloop) [%]
148 -Hloop * #t1 * #Hstar @(star_ind_l ??????? Hstar)
152 |#ls #c #rs #H2 >H2 in H1; whd in ⊢ (??%? → ?);
153 #Hfalse destruct (Hfalse)
155 | * #a * #Ha #Htest %
156 [ >Ha #H destruct (H);
158 >H2 in Ha; whd in ⊢ (??%? → ?); #Heq destruct (Heq) % //
162 | #tapea #tapeb #tapec #Hleft #Hright #IH #HRfalse
163 lapply (IH HRfalse) -IH #IH %
164 [cases Hleft #ls * #a * #rs * * #Htapea #_ #_ >Htapea
165 whd in ⊢((??%?)→?); #H destruct (H);
166 |#ls #c #rs #Htapea %2
167 cases Hleft #ls0 * #a0 * #rs0 * * #Htapea' #Htest #Htapeb
168 >Htapea' in Htapea; #Htapea destruct (Htapea) % [ % // ]
170 [ #b #rs2 #Hrs >Hrs in Htapeb; #Htapeb #Htestb #_
171 cases (proj2 ?? IH … Htapeb)
172 [ * #_ #Houtc >Houtc >Htapeb %
173 | * * >Htestb #Hfalse destruct (Hfalse) ]
174 | #r1 #rs1 #b #rs2 #Hrs >Hrs in Htapeb; #Htapeb #Htestb #Hmemb
175 cases (proj2 ?? IH … Htapeb)
176 [ * #Hfalse >(Hmemb …) in Hfalse;
177 [ #Hft destruct (Hft)
179 | * * #Htestr1 #H1 #_ >reverse_cons >associative_append
180 @H1 // #x #Hx @Hmemb @memb_cons //
183 |cases rs in Htapeb; normalize in ⊢ (%→?);
184 [#Htapeb #_ #a0 #l whd in ⊢ ((??%?)→?); #Hrev destruct (Hrev)
185 >Htapeb in IH; #IH cases (proj1 ?? IH … (refl …)) //
186 |#r1 #rs1 #Htapeb #Hmemb
187 cases (proj2 ?? IH … Htapeb) [ * >Hmemb [ #Hfalse destruct(Hfalse) ] @memb_hd ]
188 * #_ #H1 #a #l <(reverse_reverse … l) cases (reverse … l)
189 [#H cut (c::r1::rs1 = [a])
190 [<(reverse_reverse … (c::r1::rs1)) >H //]
191 #Hrev destruct (Hrev)
192 |#a1 #l2 >reverse_cons >reverse_cons >reverse_cons
193 #Hrev cut ([c] = [a1])
194 [@(append_l2_injective_r ?? (a::reverse … l2) … Hrev) //]
195 #Ha <Ha >associative_append @H1
196 [#x #membx @Hmemb @memb_cons @membx
197 |<(append_l1_injective_r ?? (a::reverse … l2) … Hrev) //
201 lemma terminate_adv_to_mark_r :
203 ∀t.Terminate alpha (adv_to_mark_r alpha test) t.
205 @(terminate_while … (sem_atmr_step alpha test))
208 [ % #t1 * #ls0 * #c0 * #rs0 * * #Hfalse destruct (Hfalse)
209 |2,3: #a0 #al0 % #t1 * #ls0 * #c0 * #rs0 * * #Hfalse destruct (Hfalse)
210 | #ls #c #rs generalize in match c; -c generalize in match ls; -ls
212 [#ls #c % #t1 * #ls0 * #c0 * #rs0 * *
213 #H1 destruct (H1) #Hc0 #Ht1 normalize in Ht1;
214 % #t2 * #ls1 * #c1 * #rs1 * * >Ht1
215 normalize in ⊢ (%→?); #Hfalse destruct (Hfalse)
216 | #r0 #rs0 #IH #ls #c % #t1 * #ls0 * #c0 * #rs0 * *
217 #H1 destruct (H1) #Hc0 #Ht1 normalize in Ht1;
224 lemma sem_adv_to_mark_r :
226 Realize alpha (adv_to_mark_r alpha test) (R_adv_to_mark_r alpha test).
232 marks the current character
235 definition mark_states ≝ initN 2.
237 definition ms0 : mark_states ≝ mk_Sig ?? 0 (leb_true_to_le 1 2 (refl …)).
238 definition ms1 : mark_states ≝ mk_Sig ?? 1 (leb_true_to_le 2 2 (refl …)).
241 λalpha:FinSet.mk_TM (FinProd … alpha FinBool) mark_states
244 [ None ⇒ 〈ms1,None ?〉
245 | Some a' ⇒ match (pi1 … q) with
246 [ O ⇒ let 〈a'',b〉 ≝ a' in 〈ms1,Some ? 〈〈a'',true〉,N〉〉
247 | S q ⇒ 〈ms1,None ?〉 ] ])
250 definition R_mark ≝ λalpha,t1,t2.
252 t1 = midtape (FinProd … alpha FinBool) ls 〈c,b〉 rs →
253 t2 = midtape ? ls 〈c,true〉 rs) ∧
254 (current ? t1 = None ? → t2 = t1).
257 ∀alpha.Realize ? (mark alpha) (R_mark alpha).
258 #alpha #intape @(ex_intro ?? 2) cases intape
260 [| % [ % | % [#ls #c #b #rs #Hfalse destruct | // ]]]
262 [| % [ % | % [#ls #c #b #rs #Hfalse destruct | // ]]]
264 [| % [ % | % [#ls #c #b #rs #Hfalse destruct ] // ]]
266 @ex_intro [| % [ % | %
267 [#ls0 #c0 #b0 #rs0 #H1 destruct (H1) %
268 | whd in ⊢ ((??%?)→?); #H1 destruct (H1)]]]
272 (* MOVE RIGHT AND MARK machine
274 marks the first character on the right
276 (could be rewritten using (mark; move_right))
279 definition mrm_states ≝ initN 3.
281 definition mrm0 : mrm_states ≝ mk_Sig ?? 0 (leb_true_to_le 1 3 (refl …)).
282 definition mrm1 : mrm_states ≝ mk_Sig ?? 1 (leb_true_to_le 2 3 (refl …)).
283 definition mrm2 : mrm_states ≝ mk_Sig ?? 2 (leb_true_to_le 3 3 (refl …)).
285 definition move_right_and_mark ≝
286 λalpha:FinSet.mk_TM (FinProd … alpha FinBool) mrm_states
289 [ None ⇒ 〈mrm2,None ?〉
290 | Some a' ⇒ match pi1 … q with
291 [ O ⇒ 〈mrm1,Some ? 〈a',R〉〉
293 [ O ⇒ let 〈a'',b〉 ≝ a' in
294 〈mrm2,Some ? 〈〈a'',true〉,N〉〉
295 | S _ ⇒ 〈mrm2,None ?〉 ] ] ])
298 definition R_move_right_and_mark ≝ λalpha,t1,t2.
300 t1 = midtape (FinProd … alpha FinBool) ls c (〈d,b〉::rs) →
301 t2 = midtape ? (c::ls) 〈d,true〉 rs.
303 lemma sem_move_right_and_mark :
304 ∀alpha.Realize ? (move_right_and_mark alpha) (R_move_right_and_mark alpha).
305 #alpha #intape @(ex_intro ?? 3) cases intape
307 [| % [ % | #ls #c #d #b #rs #Hfalse destruct ] ]
309 [| % [ % | #ls #c #d #b #rs #Hfalse destruct ] ]
311 [| % [ % | #ls #c #d #b #rs #Hfalse destruct ] ]
313 [ @ex_intro [| % [ % | #ls0 #c0 #d0 #b0 #rs0 #Hfalse destruct ] ]
314 | * #d #b #rs @ex_intro
315 [| % [ % | #ls0 #c0 #d0 #b0 #rs0 #H1 destruct (H1) % ] ] ] ]
318 (* CLEAR MARK machine
320 clears the mark in the current character
323 definition clear_mark_states ≝ initN 3.
325 definition clear0 : clear_mark_states ≝ mk_Sig ?? 0 (leb_true_to_le 1 3 (refl …)).
326 definition clear1 : clear_mark_states ≝ mk_Sig ?? 1 (leb_true_to_le 2 3 (refl …)).
327 definition claer2 : clear_mark_states ≝ mk_Sig ?? 2 (leb_true_to_le 3 3 (refl …)).
329 definition clear_mark ≝
330 λalpha:FinSet.mk_TM (FinProd … alpha FinBool) clear_mark_states
333 [ None ⇒ 〈clear1,None ?〉
334 | Some a' ⇒ match pi1 … q with
335 [ O ⇒ let 〈a'',b〉 ≝ a' in 〈clear1,Some ? 〈〈a'',false〉,N〉〉
336 | S q ⇒ 〈clear1,None ?〉 ] ])
337 clear0 (λq.q == clear1).
339 definition R_clear_mark ≝ λalpha,t1,t2.
340 (current ? t1 = None ? → t1 = t2) ∧
342 t1 = midtape (FinProd … alpha FinBool) ls 〈c,b〉 rs →
343 t2 = midtape ? ls 〈c,false〉 rs.
345 lemma sem_clear_mark :
346 ∀alpha.Realize ? (clear_mark alpha) (R_clear_mark alpha).
347 #alpha #intape @(ex_intro ?? 2) cases intape
349 [| % [ % | % [#_ %|#ls #c #b #rs #Hfalse destruct ]]]
351 [| % [ % | % [#_ %|#ls #c #b #rs #Hfalse destruct ]]]
353 [| % [ % | % [#_ %|#ls #c #b #rs #Hfalse destruct ]]]
355 @ex_intro [| % [ % | %
356 [whd in ⊢ (??%?→?); #H destruct| #ls0 #c0 #b0 #rs0 #H1 destruct (H1) % ]]]]
359 (* ADVANCE MARK RIGHT machine
361 clears mark on current char,
362 moves right, and marks new current char
366 definition adv_mark_r ≝
368 clear_mark alpha · move_r ? · mark alpha.
370 definition R_adv_mark_r ≝ λalpha,t1,t2.
373 t1 = midtape (FinProd … alpha FinBool) ls 〈c,true〉 (〈d,b〉::rs) →
374 t2 = midtape ? (〈c,false〉::ls) 〈d,true〉 rs) ∧
375 (t1 = midtape (FinProd … alpha FinBool) ls 〈c,true〉 [ ] →
376 t2 = rightof ? 〈c,false〉 ls)) ∧
377 (current ? t1 = None ? → t1 = t2).
379 lemma sem_adv_mark_r :
380 ∀alpha.Realize ? (adv_mark_r alpha) (R_adv_mark_r alpha).
382 @(sem_seq_app … (sem_clear_mark …)
383 (sem_seq ????? (sem_move_r ?) (sem_mark alpha)) …)
384 #intape #outtape whd in ⊢ (%→?); * #ta *
385 whd in ⊢ (%→?); #Hs1 whd in ⊢ (%→?); * #tb * #Hs2 whd in ⊢ (%→?); #Hs3 %
387 [#d #b #rs #Hintape @(proj1 … Hs3 ?? b ?)
388 @(proj2 … Hs2 ls 〈c,false〉 (〈d,b〉::rs))
389 @(proj2 ?? Hs1 … Hintape)
390 |#Hintape lapply (proj2 ?? Hs1 … Hintape) #Hta lapply (proj2 … Hs2 … Hta)
391 whd in ⊢ ((???%)→?); #Htb <Htb @(proj2 … Hs3) >Htb //
393 |#Hcur lapply(proj1 ?? Hs1 … Hcur) #Hta >Hta >Hta in Hcur; #Hcur
394 lapply (proj1 ?? Hs2 … Hcur) #Htb >Htb >Htb in Hcur; #Hcur
395 @sym_eq @(proj2 ?? Hs3) @Hcur
399 (* ADVANCE TO MARK (left)
404 definition atml_step ≝
405 λalpha:FinSet.λtest:alpha→bool.
406 mk_TM alpha atm_states
409 [ None ⇒ 〈atm1, None ?〉
412 [ true ⇒ 〈atm1,None ?〉
413 | false ⇒ 〈atm2,Some ? 〈a',L〉〉 ]])
414 atm0 (λx.notb (x == atm0)).
416 definition Ratml_step_true ≝
419 t1 = midtape alpha ls a rs ∧ test a = false ∧
420 t2 = mk_tape alpha (tail ? ls) (option_hd ? ls) (a :: rs).
422 definition Ratml_step_false ≝
425 (current alpha t1 = None ? ∨
426 (∃a.current ? t1 = Some ? a ∧ test a = true)).
429 ∀alpha,test,ls,a0,rs. test a0 = true →
430 step alpha (atml_step alpha test)
431 (mk_config ?? atm0 (midtape … ls a0 rs)) =
432 mk_config alpha (states ? (atml_step alpha test)) atm1
433 (midtape … ls a0 rs).
434 #alpha #test #ls #a0 #ts #Htest whd in ⊢ (??%?);
435 whd in match (trans … 〈?,?〉); >Htest %
439 ∀alpha,test,ls,a0,rs. test a0 = false →
440 step alpha (atml_step alpha test)
441 (mk_config ?? atm0 (midtape … ls a0 rs)) =
442 mk_config alpha (states ? (atml_step alpha test)) atm2
443 (mk_tape … (tail ? ls) (option_hd ? ls) (a0 :: rs)).
444 #alpha #test #ls #a0 #rs #Htest whd in ⊢ (??%?);
445 whd in match (trans … 〈?,?〉); >Htest cases ls //
448 lemma sem_atml_step :
450 accRealize alpha (atml_step alpha test)
451 atm2 (Ratml_step_true alpha test) (Ratml_step_false alpha test).
454 @(ex_intro ?? (mk_config ?? atm1 (niltape ?))) %
455 [ % // whd in ⊢ ((??%%)→?); #Hfalse destruct | #_ % // % % ]
456 | #a #al @(ex_intro ?? 2) @(ex_intro ?? (mk_config ?? atm1 (leftof ? a al)))
457 % [ % // whd in ⊢ ((??%%)→?); #Hfalse destruct | #_ % // % % ]
458 | #a #al @(ex_intro ?? 2) @(ex_intro ?? (mk_config ?? atm1 (rightof ? a al)))
459 % [ % // whd in ⊢ ((??%%)→?); #Hfalse destruct | #_ % // % % ]
460 | #ls #c #rs @(ex_intro ?? 2)
461 cases (true_or_false (test c)) #Htest
462 [ @(ex_intro ?? (mk_config ?? atm1 ?))
465 [ whd in ⊢ (??%?); >atml_q0_q1 //
466 | whd in ⊢ ((??%%)→?); #Hfalse destruct ]
467 | #_ % // %2 @(ex_intro ?? c) % // ]
469 | @(ex_intro ?? (mk_config ?? atm2 (mk_tape ? (tail ? ls) (option_hd ? ls) (c::rs))))
472 [ whd in ⊢ (??%?); >atml_q0_q2 //
473 | #_ @(ex_intro ?? ls) @(ex_intro ?? c) @(ex_intro ?? rs)
476 | #Hfalse @False_ind @(absurd ?? Hfalse) %
482 definition R_adv_to_mark_l ≝ λalpha,test,t1,t2.
483 (current ? t1 = None ? → t1 = t2) ∧
485 (t1 = midtape alpha ls c rs →
486 ((test c = true → t2 = t1) ∧
488 (∀ls1,b,ls2. ls = ls1@b::ls2 →
489 test b = true → (∀x.memb ? x ls1 = true → test x = false) →
490 t2 = midtape ? ls2 b (reverse ? ls1@c::rs)) ∧
491 ((∀x.memb ? x ls = true → test x = false) →
492 ∀a,l. reverse ? (c::ls) = a::l → t2 = leftof ? a (l@rs))
495 definition adv_to_mark_l ≝
496 λalpha,test.whileTM alpha (atml_step alpha test) atm2.
498 lemma wsem_adv_to_mark_l :
500 WRealize alpha (adv_to_mark_l alpha test) (R_adv_to_mark_l alpha test).
501 #alpha #test #t #i #outc #Hloop
502 lapply (sem_while … (sem_atml_step alpha test) t i outc Hloop) [%]
503 -Hloop * #t1 * #Hstar @(star_ind_l ??????? Hstar)
507 |#ls #c #rs #H2 >H2 in H1; whd in ⊢ (??%? → ?);
508 #Hfalse destruct (Hfalse)
510 | * #a * #Ha #Htest %
511 [>Ha #H destruct (H);
514 |#Hc @False_ind >H2 in Ha; whd in ⊢ ((??%?)→?);
519 | #tapea #tapeb #tapec #Hleft #Hright #IH #HRfalse
520 lapply (IH HRfalse) -IH #IH %
521 [cases Hleft #ls0 * #a0 * #rs0 * * #Htapea #_ #_ >Htapea
522 whd in ⊢ ((??%?)→?); #H destruct (H)
523 |#ls #c #rs #Htapea %
524 [#Hc cases Hleft #ls0 * #a0 * #rs0 * * #Htapea' #Htest @False_ind
525 >Htapea' in Htapea; #H destruct /2/
526 |cases Hleft #ls0 * #a * #rs0 *
527 * #Htapea1 >Htapea in Htapea1; #H destruct (H) #_ #Htapeb
530 [#b #ls2 #Hls >Hls in Htapeb; #Htapeb #Htestb #_
531 cases (proj2 ?? IH … Htapeb) #H1 #_ >H1 // >Htapeb %
532 |#l1 #ls1 #b #ls2 #Hls >Hls in Htapeb; #Htapeb #Htestb #Hmemb
533 cases (proj2 ?? IH … Htapeb) #_ #H1 >reverse_cons >associative_append
534 @(proj1 ?? (H1 ?) … (refl …) Htestb …)
536 |#x #memx @Hmemb @memb_cons @memx
539 |cases ls0 in Htapeb; normalize in ⊢ (%→?);
540 [#Htapeb #Htest #a0 #l whd in ⊢ ((??%?)→?); #Hrev destruct (Hrev)
541 >Htapeb in IH; #IH cases (proj1 ?? IH … (refl …)) //
543 cases (proj2 ?? IH … Htapeb) #_ #H1 #Htest #a0 #l
544 <(reverse_reverse … l) cases (reverse … l)
545 [#H cut (a::l1::ls1 = [a0])
546 [<(reverse_reverse … (a::l1::ls1)) >H //]
547 #Hrev destruct (Hrev)
548 |#a1 #l2 >reverse_cons >reverse_cons >reverse_cons
549 #Hrev cut ([a] = [a1])
550 [@(append_l2_injective_r ?? (a0::reverse … l2) … Hrev) //]
551 #Ha <Ha >associative_append @(proj2 ?? (H1 ?))
553 |#x #membx @Htest @memb_cons @membx
554 |<(append_l1_injective_r ?? (a0::reverse … l2) … Hrev) //
563 lemma terminate_adv_to_mark_l :
565 ∀t.Terminate alpha (adv_to_mark_l alpha test) t.
567 @(terminate_while … (sem_atml_step alpha test))
570 [ % #t1 * #ls0 * #c0 * #rs0 * * #Hfalse destruct (Hfalse)
571 |2,3: #a0 #al0 % #t1 * #ls0 * #c0 * #rs0 * * #Hfalse destruct (Hfalse)
573 [#c #rs % #t1 * #ls0 * #c0 * #rs0 * *
574 #H1 destruct (H1) #Hc0 #Ht1 normalize in Ht1;
575 % #t2 * #ls1 * #c1 * #rs1 * * >Ht1
576 normalize in ⊢ (%→?); #Hfalse destruct (Hfalse)
577 | #rs0 #r0 #IH #ls #c % #t1 * #ls0 * #c0 * #rs0 * *
578 #H1 destruct (H1) #Hc0 #Ht1 normalize in Ht1;
585 lemma sem_adv_to_mark_l :
587 Realize alpha (adv_to_mark_l alpha test) (R_adv_to_mark_l alpha test).
592 ADVANCE BOTH MARKS machine
594 l1 does not contain marks ⇒
606 definition adv_both_marks ≝ λalpha.
607 adv_mark_r alpha · move_l ? ·
608 adv_to_mark_l (FinProd alpha FinBool) (is_marked alpha) ·
611 definition R_adv_both_marks ≝
613 ∀l0,x,a,l1,x0. (∀c.memb ? c l1 = true → is_marked ? c = false) →
614 (∀l1',a0,l2. t1 = midtape (FinProd … alpha FinBool)
615 (l1@〈x,true〉::l0) 〈x0,true〉 (〈a0,false〉::l2) →
616 reverse ? (〈x0,false〉::l1) = 〈a,false〉::l1' →
617 t2 = midtape ? (〈x,false〉::l0) 〈a,true〉 (l1'@〈a0,true〉::l2)) ∧
618 (t1 = midtape (FinProd … alpha FinBool)
619 (l1@〈a,false〉::〈x,true〉::l0) 〈x0,true〉 [ ] →
620 t2 = rightof ? 〈x0,false〉 (l1@〈a,false〉::〈x,true〉::l0)).
622 lemma sem_adv_both_marks :
623 ∀alpha.Realize ? (adv_both_marks alpha) (R_adv_both_marks alpha).
625 @(sem_seq_app … (sem_adv_mark_r …)
626 (sem_seq ????? (sem_move_l …)
627 (sem_seq ????? (sem_adv_to_mark_l ? (is_marked ?))
628 (sem_adv_mark_r alpha))) …)
629 #intape #outtape * #tapea * #Hta * #tb * #Htb * #tc * #Htc #Hout
630 #l0 #x #a #l1 #x0 #Hmarks %
631 [#l1' #a0 #l2 #Hintape #Hrev @(proj1 ?? (proj1 ?? Hout … ) ? false) -Hout
632 lapply (proj1 … (proj1 … Hta …) … Hintape) #Htapea
633 lapply (proj2 … Htb … Htapea) -Htb
634 whd in match (mk_tape ????) ; #Htapeb
635 lapply (proj1 ?? (proj2 ?? (proj2 ?? Htc … Htapeb) (refl …))) -Htc #Htc
636 change with ((?::?)@?) in match (cons ???); <Hrev >reverse_cons
637 >associative_append @Htc [%|%|@Hmarks]
638 |#Hintape lapply (proj2 ?? (proj1 ?? Hta … ) … Hintape) -Hta #Hta
639 lapply (proj1 … Htb) >Hta -Htb #Htb lapply (Htb (refl …)) -Htb #Htb
640 lapply (proj1 ?? Htc) <Htb -Htc #Htc lapply (Htc (refl …)) -Htc #Htc
641 @sym_eq >Htc @(proj2 ?? Hout …) <Htc %
646 definition R_adv_both_marks ≝
648 ∀l0,x,a,l1,x0,a0,l2. (∀c.memb ? c l1 = true → is_marked ? c = false) →
649 (t1 = midtape (FinProd … alpha FinBool)
650 (l1@〈a,false〉::〈x,true〉::l0) 〈x0,true〉 (〈a0,false〉::l2) →
651 t2 = midtape ? (〈x,false〉::l0) 〈a,true〉 (reverse ? l1@〈x0,false〉::〈a0,true〉::l2)) ∧
652 (t1 = midtape (FinProd … alpha FinBool)
653 (l1@〈a,false〉::〈x,true〉::l0) 〈x0,true〉 [] →
654 t2 = rightof ? 〈x0,false〉 (l1@〈a,false〉::〈x,true〉::l0)).
656 lemma sem_adv_both_marks :
657 ∀alpha.Realize ? (adv_both_marks alpha) (R_adv_both_marks alpha).
659 cases (sem_seq ????? (sem_adv_mark_r …)
660 (sem_seq ????? (sem_move_l …)
661 (sem_seq ????? (sem_adv_to_mark_l ? (is_marked ?))
662 (sem_adv_mark_r alpha))) intape)
663 #k * #outc * #Hloop whd in ⊢ (%→?);
664 * #ta * whd in ⊢ (%→?); #Hs1 * #tb * whd in ⊢ (%→?); #Hs2
665 * #tc * whd in ⊢ (%→%→?); #Hs3 #Hs4
666 @(ex_intro ?? k) @(ex_intro ?? outc) %
668 | -Hloop #l0 #x #a #l1 #x0 #a0 #l2 #Hl1 #Hintape
670 lapply (Hs1 … Hintape) #Hta
671 lapply (proj2 … Hs2 … Hta) #Htb
673 [ * #Hfalse normalize in Hfalse; destruct (Hfalse)
675 lapply (Hs3 (l1@[〈a,false〉]) 〈x,true〉 l0 ???)
676 [ #x1 #Hx1 cases (memb_append … Hx1)
678 | #Hx1' >(memb_single … Hx1') % ]
680 | >associative_append %
681 | >reverse_append #Htc @Htc ]
693 l0 x a* l1 x0 a0* l2 (f(x0) == true)
695 l0 x* a l1 x0* a0 l2 (f(x0) == false)
699 definition match_and_adv ≝
700 λalpha,f.ifTM ? (test_char ? f)
701 (adv_both_marks alpha) (clear_mark ?) tc_true.
703 definition R_match_and_adv ≝
705 ∀l0,x,a,l1,x0,a0,l2. (∀c.memb ? c l1 = true → is_marked ? c = false) →
706 t1 = midtape (FinProd … alpha FinBool)
707 (l1@〈a,false〉::〈x,true〉::l0) 〈x0,true〉 (〈a0,false〉::l2) →
708 (f 〈x0,true〉 = true ∧ t2 = midtape ? (〈x,false〉::l0) 〈a,true〉 (reverse ? l1@〈x0,false〉::〈a0,true〉::l2))
709 ∨ (f 〈x0,true〉 = false ∧
710 t2 = midtape ? (l1@〈a,false〉::〈x,true〉::l0) 〈x0,false〉 (〈a0,false〉::l2)).
712 lemma sem_match_and_adv :
713 ∀alpha,f.Realize ? (match_and_adv alpha f) (R_match_and_adv alpha f).
715 cases (sem_if ? (test_char ? f) … tc_true (sem_test_char ? f) (sem_adv_both_marks alpha) (sem_clear_mark ?) intape)
716 #k * #outc * #Hloop #Hif @(ex_intro ?? k) @(ex_intro ?? outc)
719 [ * #ta * whd in ⊢ (%→%→?); #Hta #Houtc
720 #l0 #x #a #l1 #x0 #a0 #l2 #Hl1 #Hintape >Hintape in Hta;
721 * * #x1 * whd in ⊢ ((??%?)→?); #H destruct (H) #Hf #Hta % %
722 [ @Hf | >append_cons >append_cons in Hta; #Hta @(proj1 ?? (Houtc …) …Hta)
723 [ #x #memx cases (memb_append …memx)
724 [@Hl1 | -memx #memx >(memb_single … memx) %]
725 |>reverse_cons >reverse_append % ] ]
726 | * #ta * whd in ⊢ (%→%→?); #Hta #Houtc
727 #l0 #x #a #l1 #x0 #a0 #l2 #Hl1 #Hintape >Hintape in Hta;
728 * #Hf #Hta %2 % [ @Hf % | >(proj2 ?? Houtc … Hta) % ]
732 definition R_match_and_adv_of ≝
733 λalpha,t1,t2.current (FinProd … alpha FinBool) t1 = None ? → t2 = t1.
735 lemma sem_match_and_adv_of :
736 ∀alpha,f.Realize ? (match_and_adv alpha f) (R_match_and_adv_of alpha).
738 cases (sem_if ? (test_char ? f) … tc_true (sem_test_char ? f) (sem_adv_both_marks alpha) (sem_clear_mark ?) intape)
739 #k * #outc * #Hloop #Hif @(ex_intro ?? k) @(ex_intro ?? outc)
742 [ * #ta * whd in ⊢ (%→%→?); #Hta #Houtc #Hcur
743 cases Hta * #x >Hcur * #Hfalse destruct (Hfalse)
744 | * #ta * whd in ⊢ (%→%→?); * #_ #Hta * #Houtc #_ <Hta #Hcur >(Houtc Hcur) % ]
747 lemma sem_match_and_adv_full :
748 ∀alpha,f.Realize ? (match_and_adv alpha f)
749 (R_match_and_adv alpha f ∩ R_match_and_adv_of alpha).
750 #alpha #f #intape cases (sem_match_and_adv ? f intape)
751 #i * #outc * #Hloop #HR1 %{i} %{outc} % // % //
752 cases (sem_match_and_adv_of ? f intape) #i0 * #outc0 * #Hloop0 #HR2
753 >(loop_eq … Hloop Hloop0) //
758 then move_right; ----
760 if current (* x0 *) = 0
761 then advance_mark ----
768 definition comp_step_subcase ≝ λalpha,c,elseM.
769 ifTM ? (test_char ? (λx.x == c))
770 (move_r … · adv_to_mark_r ? (is_marked alpha) · match_and_adv ? (λx.x == c))
773 definition R_comp_step_subcase ≝
774 λalpha,c,RelseM,t1,t2.
775 ∀l0,x,rs.t1 = midtape (FinProd … alpha FinBool) l0 〈x,true〉 rs →
777 ((∀c.memb ? c rs = true → is_marked ? c = false) →
778 ∀a,l. (a::l) = reverse ? (〈x,true〉::rs) → t2 = rightof (FinProd alpha FinBool) a (l@l0)) ∧
779 ∀a,l1,x0,a0,l2. (∀c.memb ? c l1 = true → is_marked ? c = false) →
780 rs = 〈a,false〉::l1@〈x0,true〉::〈a0,false〉::l2 →
782 t2 = midtape ? (〈x,false〉::l0) 〈a,true〉 (l1@〈x0,false〉::〈a0,true〉::l2)) ∧
784 t2 = midtape (FinProd … alpha FinBool)
785 (reverse ? l1@〈a,false〉::〈x,true〉::l0) 〈x0,false〉 (〈a0,false〉::l2)))) ∧
786 (〈x,true〉 ≠ c → RelseM t1 t2).
788 lemma dec_marked: ∀alpha,rs.
789 decidable (∀c.memb ? c rs = true → is_marked alpha c = false).
791 [%1 #n normalize #H destruct
792 |#a #tl cases (true_or_false (is_marked ? a)) #Ha
793 [#_ %2 % #Hall @(absurd ?? not_eq_true_false) <Ha
795 |* [#Hall %1 #c #memc cases (orb_true_l … memc)
796 [#eqca >(\P eqca) @Ha |@Hall]
797 |#Hnall %2 @(not_to_not … Hnall) #Hall #c #memc @Hall @memb_cons //
801 lemma sem_comp_step_subcase :
802 ∀alpha,c,elseM,RelseM.
803 Realize ? elseM RelseM →
804 Realize ? (comp_step_subcase alpha c elseM)
805 (R_comp_step_subcase alpha c RelseM).
806 #alpha #c #elseM #RelseM #Helse #intape
807 cases (sem_if ? (test_char ? (λx.x == c)) … tc_true
808 (sem_test_char ? (λx.x == c))
809 (sem_seq ????? (sem_move_r …)
810 (sem_seq ????? (sem_adv_to_mark_r ? (is_marked alpha))
811 (sem_match_and_adv_full ? (λx.x == c)))) Helse intape)
812 #k * #outc * #Hloop #HR @(ex_intro ?? k) @(ex_intro ?? outc)
813 % [ @Hloop ] -Hloop cases HR -HR
814 [* #ta * whd in ⊢ (%→?); #Hta * #tb * whd in ⊢ (%→?); #Htb
815 * #tc * whd in ⊢ (%→?); #Htc * whd in ⊢ (%→%→?); #Houtc #Houtc1
816 #l0 #x #rs #Hintape %
817 [#_ cases (dec_marked ? rs) #Hdec
820 >Hintape in Hta; * #_(* #x1 * whd in ⊢ ((??%?)→?); #H destruct (H) #Hx *)
821 #Hta lapply (proj2 … Htb … Hta) -Htb -Hta cases rs in Hdec;
822 [#_ whd in ⊢ ((???%)→?); #Htb >Htb in Htc; #Htc
823 lapply (proj1 ?? Htc (refl …)) -Htc #Htc <Htc in Houtc1; #Houtc1
824 normalize in ⊢ (???%→?); #Hl1 destruct(Hl1) @(Houtc1 (refl …))
825 |#r0 #rs0 #Hdec whd in ⊢ ((???%)→?); #Htb >Htb in Htc; #Htc
826 >reverse_cons >reverse_cons #Hl1
827 cases (proj2 ?? Htc … (refl …))
828 [* >(Hdec …) [ #Hfalse destruct(Hfalse) ] @memb_hd
829 |* #_ -Htc #Htc cut (∃l2.l1 = l2@[〈x,true〉])
830 [generalize in match Hl1; -Hl1 <(reverse_reverse … l1)
832 [#Hl1 cut ([a]=〈x,true〉::r0::rs0)
833 [ <(reverse_reverse … (〈x,true〉::r0::rs0))
834 >reverse_cons >reverse_cons <Hl1 %
835 | #Hfalse destruct(Hfalse)]
836 |#a0 #l10 >reverse_cons #Heq
837 lapply (append_l2_injective_r ? (a::reverse ? l10) ???? Heq) //
838 #Ha0 destruct(Ha0) /2/ ]
839 |* #l2 #Hl2 >Hl2 in Hl1; #Hl1
840 lapply (append_l1_injective_r ? (a::l2) … Hl1) // -Hl1 #Hl1
841 >reverse_cons in Htc; #Htc lapply (Htc … (sym_eq … Hl1))
842 [ #x0 #Hmemb @Hdec @memb_cons @Hmemb ]
843 -Htc #Htc >Htc in Houtc1; #Houtc1 >associative_append @Houtc1 %
848 #l2 #_ #Hrs @False_ind
849 @(absurd ?? not_eq_true_false)
850 change with (is_marked ? 〈x0,true〉) in match true;
851 @Hdec >Hrs @memb_cons @memb_append_l2 @memb_hd
853 |% [#H @False_ind @(absurd …H Hdec)]
854 #a #l1 #x0 #a0 #l2 #Hl1 #Hrs >Hrs in Hintape; #Hintape
855 >Hintape in Hta; * * #x1 * whd in ⊢ ((??%?)→?); #H destruct (H) #Hx
856 #Hta lapply (proj2 … Htb … Hta) -Htb -Hta
857 whd in match (mk_tape ????); #Htb cases Htc -Htc #_ #Htc
858 cases (Htc … Htb) [ * #Hfalse normalize in Hfalse; destruct (Hfalse) ]
859 -Htc * * #_ #Htc #_ lapply (Htc l1 〈x0,true〉 (〈a0,false〉::l2) (refl ??) (refl ??) Hl1)
860 -Htc #Htc cases (Houtc ???????? Htc) -Houtc
862 [ #Hx >Houtc >reverse_reverse %
863 | lapply (\P Hx0) -Hx0 <(\P Hx) in ⊢ (%→?); #Hx0 destruct (Hx0)
864 * #Hfalse @False_ind @Hfalse % ]
866 [ #Hxx0 >Hxx0 in Hx; #Hx; lapply (\Pf Hx0) -Hx0 <(\P Hx) in ⊢ (%→?);
867 * #Hfalse @False_ind @Hfalse %
869 |#c #membc @Hl1 <(reverse_reverse …l1) @memb_reverse @membc
872 | cases Hta * #c0 * >Hintape whd in ⊢ (??%%→?); #Hc0 destruct(Hc0) #Hx >(\P Hx)
873 #_ * #Hc @False_ind @Hc % ]
874 | * #ta * * #Hcur #Hta #Houtc
875 #l0 #x #rs #Hintape >Hintape in Hcur; #Hcur lapply (Hcur ? (refl …)) -Hcur #Hc %
876 [ #Hfalse >Hfalse in Hc; #Hc cases (\Pf Hc) #Hc @False_ind @Hc %
877 | -Hc #Hc <Hintape <Hta @Houtc ] ]
883 + se è un bit, ho fallito il confronto della tupla corrente
884 + se è un separatore, la tupla fa match
887 ifTM ? (test_char ? is_marked)
888 (single_finalTM … (comp_step_subcase unialpha 〈bit false,true〉
889 (comp_step_subcase unialpha 〈bit true,true〉
894 definition comp_step ≝
895 ifTM ? (test_char ? (is_marked ?))
896 (single_finalTM … (comp_step_subcase FSUnialpha 〈bit false,true〉
897 (comp_step_subcase FSUnialpha 〈bit true,true〉
898 (comp_step_subcase FSUnialpha 〈null,true〉
905 lemma mem_append : ∀A,x,l1,l2. mem A x (l1@l2) →
906 mem A x l1 ∨ mem A x l2.
907 #A #x #l1 elim l1 normalize [/2/]
908 #a #tl #Hind #l2 * [#eqxa %1 /2/ |#memx cases (Hind … memx) /3/]
911 let rec split_on A (l:list A) f acc on l ≝
915 if f a then 〈acc,a::tl〉 else split_on A tl f (a::acc)
918 lemma split_on_spec: ∀A,l,f,acc,res1,res2.
919 split_on A l f acc = 〈res1,res2〉 →
920 (∃l1. res1 = l1@acc ∧
921 reverse ? l1@res2 = l ∧
922 ∀x. mem ? x l1 → f x = false) ∧
923 ∀a,tl. res2 = a::tl → f a = true.
925 [#acc #res1 #res2 normalize in ⊢ (%→?); #H destruct %
926 [@(ex_intro … []) % normalize [% % | #x @False_ind]
929 |#a #tl #Hind #acc #res1 #res2 normalize in ⊢ (%→?);
930 cases (true_or_false (f a)) #Hfa >Hfa normalize in ⊢ (%→?);
932 [% [@(ex_intro … []) % normalize [% % | #x @False_ind]
933 |#a1 #tl1 #H destruct (H) //]
934 |cases (Hind (a::acc) res1 res2 H) * #l1 * *
935 #Hres1 #Htl #Hfalse #Htrue % [2:@Htrue] @(ex_intro … (l1@[a])) %
936 [% [>associative_append @Hres1 | >reverse_append <Htl % ]
937 |#x #Hmemx cases (mem_append ???? Hmemx)
938 [@Hfalse | normalize * [#H >H //| @False_ind]
944 axiom mem_reverse: ∀A,l,x. mem A x (reverse ? l) → mem A x l.
946 lemma split_on_spec_ex: ∀A,l,f.∃l1,l2.
947 l1@l2 = l ∧ (∀x:A. mem ? x l1 → f x = false) ∧
948 ∀a,tl. l2 = a::tl → f a = true.
949 #A #l #f @(ex_intro … (reverse … (\fst (split_on A l f []))))
950 @(ex_intro … (\snd (split_on A l f [])))
951 cases (split_on_spec A l f [ ] ?? (eq_pair_fst_snd …)) * #l1 * *
952 >append_nil #Hl1 >Hl1 #Hl #Hfalse #Htrue %
953 [% [@Hl|#x #memx @Hfalse @mem_reverse //] | @Htrue]
958 (* manca il caso in cui alla destra della testina il nastro non ha la forma
959 (l1@〈c0,true〉::〈a0,false〉::l2)
961 definition R_comp_step_true ≝ λt1,t2.
962 ∃l0,c,a,l1,c0,l1',a0,l2.
963 t1 = midtape (FinProd … FSUnialpha FinBool)
964 l0 〈c,true〉 (l1@〈c0,true〉::〈a0,false〉::l2) ∧
965 l1@[〈c0,false〉] = 〈a,false〉::l1' ∧
966 (∀c.memb ? c l1 = true → is_marked ? c = false) ∧
967 (bit_or_null c = true → c0 = c →
968 t2 = midtape ? (〈c,false〉::l0) 〈a,true〉 (l1'@〈c0,false〉::〈a0,true〉::l2)) ∧
969 (bit_or_null c = true → c0 ≠ c →
970 t2 = midtape (FinProd … FSUnialpha FinBool)
971 (reverse ? l1@〈a,false〉::〈c,true〉::l0) 〈c0,false〉 (〈a0,false〉::l2)) ∧
972 (bit_or_null c = false →
973 t2 = midtape ? l0 〈c,false〉 (〈a,false〉::l1@〈c0,true〉::〈a0,false〉::l2)).
975 definition R_comp_step_false ≝
977 ∀ls,c,rs.t1 = midtape (FinProd … FSUnialpha FinBool) ls c rs →
978 is_marked ? c = false ∧ t2 = t1.
981 lemma is_marked_to_exists: ∀alpha,c. is_marked alpha c = true →
985 lemma sem_comp_step :
986 accRealize ? comp_step (inr … (inl … (inr … start_nop)))
987 R_comp_step_true R_comp_step_false.
988 @(acc_sem_if_app … (sem_test_char ? (is_marked ?))
989 (sem_comp_step_subcase FSUnialpha 〈bit false,true〉 ??
990 (sem_comp_step_subcase FSUnialpha 〈bit true,true〉 ??
991 (sem_comp_step_subcase FSUnialpha 〈null,true〉 ??
992 (sem_clear_mark …))))
995 [#intape #outtape #midtape * * * #c #b * #Hcurrent
996 whd in ⊢ ((??%?)→?); #Hb #Hmidtape >Hmidtape -Hmidtape
997 cases (current_to_midtape … Hcurrent) #ls * #rs >Hb #Hintape >Hintape -Hb
998 whd in ⊢ (%→?); #Htapea lapply (Htapea … (refl …)) -Htapea
999 cases (split_on_spec_ex ? rs (is_marked ?)) #l1 * #l2 * * #Hrs #Hl1 #Hl2
1000 cases (true_or_false (c == bit false))
1001 [(* c = bit false *) #Hc *
1002 [>(\P Hc) #H lapply (H (refl ??)) -H * #_ #H lapply (H ????? Hl1) @False_ind @H //]
1004 cases (split_on_spec *)
1005 [ #ta #tb #tc * * * #c #b * #Hcurrent whd in ⊢(??%?→?); #Hc
1006 >Hc in Hcurrent; #Hcurrent; #Htc
1007 cases (current_to_midtape … Hcurrent) #ls * #rs #Hta
1008 >Htc #H1 cases (H1 … Hta) -H1 #H1 #H2 whd
1009 lapply (refl ? (〈c,true〉==〈bit false,true〉))
1010 cases (〈c,true〉==〈bit false,true〉) in ⊢ (???%→?);
1011 [ #Hceq lapply (H1 (\P Hceq)) -H1 *
1012 cases (split_on_spec_ex ? rs (is_marked ?)) #l1 * #l2 * * cases l2
1013 [ >append_nil #Hrs #Hl1 #Hl2 #Htb1 #_
1015 #Hstate lapply (H1 Hstate) -H1 -Hstate -H2 *
1016 #ta * whd in ⊢ (%→?); #Hleft #Hright #ls #c #rs #Hintape
1017 >Hintape in Hleft; * *
1018 cases c in Hintape; #c' #b #Hintape #x * whd in ⊢ (??%?→?); #H destruct (H)
1019 whd in ⊢ (??%?→?); #Hb >Hb #Hta @(ex_intro ?? c') % //
1020 cases (Hright … Hta)
1021 [ * #Hc' #H1 % % [destruct (Hc') % ]
1022 #a #l1 #c0 #a0 #l2 #Hrs >Hrs in Hintape; #Hintape #Hl1
1023 cases (H1 … Hl1 Hrs)
1024 [ * #Htmp >Htmp -Htmp #Houtc % % // @Houtc
1025 | * #Hneq #Houtc %2 %
1029 | * #Hc #Helse1 cases (Helse1 … Hta)
1030 [ * #Hc' #H1 % % [destruct (Hc') % ]
1031 #a #l1 #c0 #a0 #l2 #Hrs >Hrs in Hintape; #Hintape #Hl1
1032 cases (H1 … Hl1 Hrs)
1033 [ * #Htmp >Htmp -Htmp #Houtc % % // @Houtc
1034 | * #Hneq #Houtc %2 %
1038 | * #Hc' #Helse2 cases (Helse2 … Hta)
1039 [ * #Hc'' #H1 % % [destruct (Hc'') % ]
1040 #a #l1 #c0 #a0 #l2 #Hrs >Hrs in Hintape; #Hintape #Hl1
1041 cases (H1 … Hl1 Hrs)
1042 [ * #Htmp >Htmp -Htmp #Houtc % % // @Houtc
1043 | * #Hneq #Houtc %2 %
1047 | * #Hc'' whd in ⊢ (%→?); #Helse3 %2 %
1048 [ generalize in match Hc''; generalize in match Hc'; generalize in match Hc;
1050 [ * [ #_ #Hfalse @False_ind @(absurd ?? Hfalse) %
1051 | #Hfalse @False_ind @(absurd ?? Hfalse) % ]
1052 | #_ #_ #Hfalse @False_ind @(absurd ?? Hfalse) %
1059 | #Hstate lapply (H2 Hstate) -H1 -Hstate -H2 *
1060 #ta * whd in ⊢ (%→%→?); #Hleft #Hright #ls #c #rs #Hintape
1061 >Hintape in Hleft; * #Hc #Hta % [@Hc % | >Hright //]
1064 definition R_comp_step_true ≝
1066 ∀l0,c,rs.t1 = midtape (FinProd … FSUnialpha FinBool) l0 c rs →
1067 ∃c'. c = 〈c',true〉 ∧
1068 ((bit_or_null c' = true ∧
1070 rs = 〈a,false〉::l1@〈c0,true〉::〈a0,false〉::l2 →
1071 (∀c.memb ? c l1 = true → is_marked ? c = false) →
1073 t2 = midtape ? (〈c',false〉::l0) 〈a,true〉 (l1@〈c0,false〉::〈a0,true〉::l2)) ∨
1075 t2 = midtape (FinProd … FSUnialpha FinBool)
1076 (reverse ? l1@〈a,false〉::〈c',true〉::l0) 〈c0,false〉 (〈a0,false〉::l2))) ∨
1077 (bit_or_null c' = false ∧ t2 = midtape ? l0 〈c',false〉 rs)).
1079 definition R_comp_step_false ≝
1081 ∀ls,c,rs.t1 = midtape (FinProd … FSUnialpha FinBool) ls c rs →
1082 is_marked ? c = false ∧ t2 = t1.
1084 lemma sem_comp_step :
1085 accRealize ? comp_step (inr … (inl … (inr … start_nop)))
1086 R_comp_step_true R_comp_step_false.
1088 cases (acc_sem_if … (sem_test_char ? (is_marked ?))
1089 (sem_comp_step_subcase FSUnialpha 〈bit false,true〉 ??
1090 (sem_comp_step_subcase FSUnialpha 〈bit true,true〉 ??
1091 (sem_comp_step_subcase FSUnialpha 〈null,true〉 ??
1092 (sem_clear_mark …))))
1094 #k * #outc * * #Hloop #H1 #H2
1095 @(ex_intro ?? k) @(ex_intro ?? outc) %
1096 [ % [@Hloop ] ] -Hloop
1097 [ #Hstate lapply (H1 Hstate) -H1 -Hstate -H2 *
1098 #ta * whd in ⊢ (%→?); #Hleft #Hright #ls #c #rs #Hintape
1099 >Hintape in Hleft; * *
1100 cases c in Hintape; #c' #b #Hintape #x * whd in ⊢ (??%?→?); #H destruct (H)
1101 whd in ⊢ (??%?→?); #Hb >Hb #Hta @(ex_intro ?? c') % //
1102 cases (Hright … Hta)
1103 [ * #Hc' #H1 % % [destruct (Hc') % ]
1104 #a #l1 #c0 #a0 #l2 #Hrs >Hrs in Hintape; #Hintape #Hl1
1105 cases (H1 … Hl1 Hrs)
1106 [ * #Htmp >Htmp -Htmp #Houtc % % // @Houtc
1107 | * #Hneq #Houtc %2 %
1111 | * #Hc #Helse1 cases (Helse1 … Hta)
1112 [ * #Hc' #H1 % % [destruct (Hc') % ]
1113 #a #l1 #c0 #a0 #l2 #Hrs >Hrs in Hintape; #Hintape #Hl1
1114 cases (H1 … Hl1 Hrs)
1115 [ * #Htmp >Htmp -Htmp #Houtc % % // @Houtc
1116 | * #Hneq #Houtc %2 %
1120 | * #Hc' #Helse2 cases (Helse2 … Hta)
1121 [ * #Hc'' #H1 % % [destruct (Hc'') % ]
1122 #a #l1 #c0 #a0 #l2 #Hrs >Hrs in Hintape; #Hintape #Hl1
1123 cases (H1 … Hl1 Hrs)
1124 [ * #Htmp >Htmp -Htmp #Houtc % % // @Houtc
1125 | * #Hneq #Houtc %2 %
1129 | * #Hc'' whd in ⊢ (%→?); #Helse3 %2 %
1130 [ generalize in match Hc''; generalize in match Hc'; generalize in match Hc;
1132 [ * [ #_ #Hfalse @False_ind @(absurd ?? Hfalse) %
1133 | #Hfalse @False_ind @(absurd ?? Hfalse) % ]
1134 | #_ #_ #Hfalse @False_ind @(absurd ?? Hfalse) %
1141 | #Hstate lapply (H2 Hstate) -H1 -Hstate -H2 *
1142 #ta * whd in ⊢ (%→%→?); #Hleft #Hright #ls #c #rs #Hintape
1143 >Hintape in Hleft; * #Hc #Hta % [@Hc % | >Hright //]
1147 definition compare ≝
1148 whileTM ? comp_step (inr … (inl … (inr … start_nop))).
1151 definition R_compare :=
1155 ∀ls,c,b,rs.t1 = midtape ? ls 〈c,b〉 rs →
1156 (b = true → rs = ....) →
1157 (b = false ∧ ....) ∨
1160 rs = cs@l1@〈c0,true〉::cs0@l2
1164 ls 〈c,b〉 cs l1 〈c0,b0〉 cs0 l2
1168 ls (hd (Ls@〈grid,false〉))* (tail (Ls@〈grid,false〉)) l1 (hd (Ls@〈comma,false〉))* (tail (Ls@〈comma,false〉)) l2
1169 ^^^^^^^^^^^^^^^^^^^^^^^
1171 ls Ls 〈grid,false〉 l1 Ls 〈comma,true〉 l2
1176 ls (hd (Ls@〈c,false〉))* (tail (Ls@〈c,false〉)) l1 (hd (Ls@〈d,false〉))* (tail (Ls@〈d,false〉)) l2
1177 ^^^^^^^^^^^^^^^^^^^^^^^
1180 ls Ls 〈c,true〉 l1 Ls 〈d,false〉 l2
1186 (∃la,d.〈b,true〉::bs = la@[〈grid,d〉] ∧ ∀x.memb ? x la → is_bit (\fst x) = true) →
1187 (∃lb,d0.〈b0,true〉::b0s = lb@[〈comma,d0〉] ∧ ∀x.memb ? x lb → is_bit (\fst x) = true) →
1188 t1 = midtape ? l0 〈b,true〉 (bs@l1@〈b0,true〉::b0s@l2 →
1190 mk_tape left (option current) right
1192 (b = grid ∧ b0 = comma ∧ bs = [] ∧ b0s = [] ∧
1193 t2 = midtape ? l0 〈grid,false〉 (l1@〈comma,true〉::l2)) ∨
1194 (b = bit x ∧ b = c ∧ bs = b0s
1196 definition R_compare :=
1198 ∀ls,c,rs.t1 = midtape (FinProd … FSUnialpha FinBool) ls c rs →
1199 (∀c'.bit_or_null c' = false → c = 〈c',true〉 → t2 = midtape ? ls 〈c',false〉 rs) ∧
1200 (∀c'. c = 〈c',false〉 → t2 = t1) ∧
1203 (∀c.memb (FinProd … FSUnialpha FinBool) c bs = true → bit_or_null (\fst c) = true) →
1204 (∀c.memb (FinProd … FSUnialpha FinBool) c b0s = true → bit_or_null (\fst c) = true) →
1205 (∀c.memb ? c bs = true → is_marked ? c = false) →
1206 (∀c.memb ? c b0s = true → is_marked ? c = false) →
1207 (∀c.memb ? c l1 = true → is_marked ? c = false) →
1208 c = 〈b,true〉 → bit_or_null b = true →
1209 rs = bs@〈grid,false〉::l1@〈b0,true〉::b0s@〈comma,false〉::l2 →
1210 (〈b,true〉::bs = 〈b0,true〉::b0s ∧
1211 t2 = midtape ? (reverse ? bs@〈b,false〉::ls)
1212 〈grid,false〉 (l1@〈b0,false〉::b0s@〈comma,true〉::l2)) ∨
1213 (∃la,c',d',lb,lc.c' ≠ d' ∧
1214 〈b,false〉::bs = la@〈c',false〉::lb ∧
1215 〈b0,false〉::b0s = la@〈d',false〉::lc ∧
1216 t2 = midtape (FinProd … FSUnialpha FinBool) (reverse ? la@
1222 〈d',false〉 (lc@〈comma,false〉::l2)).
1224 lemma wsem_compare : WRealize ? compare R_compare.
1226 lapply (sem_while ?????? sem_comp_step t i outc Hloop) [%]
1227 -Hloop * #t1 * #Hstar @(star_ind_l ??????? Hstar)
1228 [ #tapea whd in ⊢ (%→?); #Rfalse #ls #c #rs #Htapea %
1230 [ #c' #Hc' #Hc lapply (Rfalse … Htapea) -Rfalse * >Hc
1231 whd in ⊢ (??%?→?); #Hfalse destruct (Hfalse)
1232 | #c' #Hc lapply (Rfalse … Htapea) -Rfalse * #_
1234 | #b #b0 #bs #b0s #l1 #l2 #Hlen #Hbs1 #Hb0s1 #Hbs2 #Hb0s2 #Hl1 #Hc
1235 cases (Rfalse … Htapea) -Rfalse >Hc whd in ⊢ (??%?→?);#Hfalse destruct (Hfalse)
1237 | #tapea #tapeb #tapec #Hleft #Hright #IH #Htapec lapply (IH Htapec) -Htapec -IH #IH
1238 whd in Hleft; #ls #c #rs #Htapea cases (Hleft … Htapea) -Hleft
1239 #c' * #Hc >Hc cases (true_or_false (bit_or_null c')) #Hc'
1241 [ * >Hc' #H @False_ind destruct (H)
1242 | * #_ #Htapeb cases (IH … Htapeb) * #_ #H #_ %
1244 [#c1 #Hc1 #Heqc destruct (Heqc) <Htapeb @(H c1) %
1245 |#c1 #Hfalse destruct (Hfalse)
1247 |#b #b0 #bs #b0s #l1 #l2 #_ #_ #_ #_ #_ #_
1248 #Heq destruct (Heq) >Hc' #Hfalse @False_ind destruct (Hfalse)
1253 [ #c'' #Hc'' #Heq destruct (Heq) >Hc'' in Hc'; #H destruct (H)
1254 | #c0 #Hfalse destruct (Hfalse)
1256 |#b #b0 #bs #b0s #l1 #l2 #Hlen #Hbs1 #Hb0s1 #Hbs2 #Hb0s2 #Hl1
1257 #Heq destruct (Heq) #_ #Hrs cases Hleft -Hleft
1258 [2: * >Hc' #Hfalse @False_ind destruct ] * #_
1259 @(list_cases2 … Hlen)
1260 [ #Hbs #Hb0s generalize in match Hrs; >Hbs in ⊢ (%→?); >Hb0s in ⊢ (%→?);
1261 -Hrs #Hrs normalize in Hrs; #Hleft cases (Hleft ????? Hrs ?) -Hleft
1262 [ * #Heqb #Htapeb cases (IH … Htapeb) -IH * #IH #_ #_
1264 [ >Heqb >Hbs >Hb0s %
1267 |* #Hneqb #Htapeb %2
1268 @(ex_intro … [ ]) @(ex_intro … b)
1269 @(ex_intro … b0) @(ex_intro … [ ])
1271 [ % [ % [@sym_not_eq //| >Hbs %] | >Hb0s %]
1272 | cases (IH … Htapeb) -IH * #_ #IH #_ >(IH ? (refl ??))
1276 | * #b' #bitb' * #b0' #bitb0' #bs' #b0s' #Hbs #Hb0s
1277 generalize in match Hrs; >Hbs in ⊢ (%→?); >Hb0s in ⊢ (%→?);
1278 cut (bit_or_null b' = true ∧ bit_or_null b0' = true ∧
1279 bitb' = false ∧ bitb0' = false)
1280 [ % [ % [ % [ >Hbs in Hbs1; #Hbs1 @(Hbs1 〈b',bitb'〉) @memb_hd
1281 | >Hb0s in Hb0s1; #Hb0s1 @(Hb0s1 〈b0',bitb0'〉) @memb_hd ]
1282 | >Hbs in Hbs2; #Hbs2 @(Hbs2 〈b',bitb'〉) @memb_hd ]
1283 | >Hb0s in Hb0s2; #Hb0s2 @(Hb0s2 〈b0',bitb0'〉) @memb_hd ]
1284 | * * * #Ha #Hb #Hc #Hd >Hc >Hd
1286 cases (Hleft b' (bs'@〈grid,false〉::l1) b0 b0'
1287 (b0s'@〈comma,false〉::l2) ??) -Hleft
1288 [ 3: >Hrs normalize @eq_f >associative_append %
1289 | * #Hb0 #Htapeb cases (IH …Htapeb) -IH * #_ #_ #IH
1290 cases (IH b' b0' bs' b0s' (l1@[〈b0,false〉]) l2 ??????? Ha ?) -IH
1292 [ >Hb0 @eq_f >Hbs in Heq; >Hb0s in ⊢ (%→?); #Heq
1293 destruct (Heq) >Hb0s >Hc >Hd %
1294 | >Houtc >Hbs >Hb0s >Hc >Hd >reverse_cons >associative_append
1295 >associative_append %
1297 | * #la * #c' * #d' * #lb * #lc * * * #H1 #H2 #H3 #H4 %2
1298 @(ex_intro … (〈b,false〉::la)) @(ex_intro … c') @(ex_intro … d')
1299 @(ex_intro … lb) @(ex_intro … lc)
1300 % [ % [ % // >Hbs >Hc >H2 % | >Hb0s >Hd >H3 >Hb0 % ]
1301 | >H4 >Hbs >Hb0s >Hc >Hd >Hb0 >reverse_append
1302 >reverse_cons >reverse_cons
1303 >associative_append >associative_append
1304 >associative_append >associative_append %
1306 | generalize in match Hlen; >Hbs >Hb0s
1307 normalize #Hlen destruct (Hlen) @e0
1308 | #c0 #Hc0 @Hbs1 >Hbs @memb_cons //
1309 | #c0 #Hc0 @Hb0s1 >Hb0s @memb_cons //
1310 | #c0 #Hc0 @Hbs2 >Hbs @memb_cons //
1311 | #c0 #Hc0 @Hb0s2 >Hb0s @memb_cons //
1312 | #c0 #Hc0 cases (memb_append … Hc0)
1313 [ @Hl1 | #Hc0' >(memb_single … Hc0') % ]
1315 | >associative_append >associative_append % ]
1316 | * #Hneq #Htapeb %2
1317 @(ex_intro … []) @(ex_intro … b) @(ex_intro … b0)
1318 @(ex_intro … bs) @(ex_intro … b0s) %
1319 [ % // % // @sym_not_eq //
1320 | >Hbs >Hb0s >Hc >Hd >reverse_cons >associative_append
1321 >reverse_append in Htapeb; >reverse_cons
1322 >associative_append >associative_append
1324 cases (IH … Htapeb) -Htapeb -IH * #_ #IH #_ @(IH ? (refl ??))
1326 | #c1 #Hc1 cases (memb_append … Hc1) #Hyp
1327 [ @Hbs2 >Hbs @memb_cons @Hyp
1328 | cases (orb_true_l … Hyp)
1329 [ #Hyp2 >(\P Hyp2) %
1337 axiom sem_compare : Realize ? compare R_compare.