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/universal/tuples.ma".
19 definition write_states ≝ initN 2.
21 definition write ≝ λalpha,c.
22 mk_TM alpha write_states
25 [ O ⇒ 〈1,Some ? 〈c,N〉〉
29 definition R_write ≝ λalpha,c,t1,t2.
30 ∀ls,x,rs.t1 = midtape alpha ls x rs → t2 = midtape alpha ls c rs.
32 axiom sem_write : ∀alpha,c.Realize ? (write alpha c) (R_write alpha c).
34 definition copy_step_subcase ≝
35 λalpha,c,elseM.ifTM ? (test_char ? (λx.x == 〈c,true〉))
36 (seq (FinProd alpha FinBool) (adv_mark_r …)
38 (seq ? (adv_to_mark_l … (is_marked alpha))
39 (seq ? (write ? 〈c,false〉)
42 (seq ? (move_r …) (adv_to_mark_r … (is_marked alpha)))))))))
45 definition R_copy_step_subcase ≝
46 λalpha,c,RelseM,t1,t2.
48 t1 = midtape (FinProd … alpha FinBool) (l1@〈a0,false〉::〈x0,true〉::l2)
49 〈x,true〉 (〈a,false〉::l3) →
50 (∀c.memb ? c l1 = true → is_marked ? c = false) →
51 (x = c ∧ t2 = midtape ? (〈x,false〉::l1@〈a0,true〉::〈x,false〉::l2) 〈a,true〉 l3) ∨
52 (x ≠ c ∧ RelseM t1 t2).
54 axiom sem_copy_step_subcase :
55 ∀alpha,c,elseM,RelseM.
56 Realize ? (copy_step_subcase alpha c elseM) (R_copy_step_subcase alpha c RelseM).
68 else if current = 1,tt
77 else if current = null
86 definition nocopy_subcase ≝
87 ifTM STape (test_char ? (λx:STape.x == 〈null,true〉))
90 (seq ? (adv_to_mark_l … (is_marked ?))
92 (seq ? (move_r …) (adv_to_mark_r … (is_marked ?)))))))
95 definition R_nocopy_subcase ≝
98 t1 = midtape STape (l1@〈a0,false〉::〈x0,true〉::l2)
99 〈x,true〉 (〈a,false〉::l3) →
100 (∀c.memb ? c l1 = true → is_marked ? c = false) →
102 t2 = midtape ? (〈x,false〉::l1@〈a0,true〉::〈x0,false〉::l2) 〈a,true〉 l3) ∨
103 (x ≠ null ∧ t2 = t1).
105 axiom sem_nocopy_subcase : Realize ? nocopy_subcase R_nocopy_subcase.
107 cases (sem_if ? (test_char ? (λx:STape.x == 〈null,true〉)) ?????? tc_true
108 (sem_test_char ? (λx:STape.x == 〈null,true〉))
109 (sem_seq … (sem_adv_mark_r …)
110 (sem_seq … (sem_move_l …)
111 (sem_seq … (sem_adv_to_mark_l … (is_marked ?))
112 (sem_seq … (sem_adv_mark_r …)
113 (sem_seq … (sem_move_r …) (sem_adv_to_mark_r … (is_marked ?))
114 ))))) (sem_nop ?) intape)
115 #k * #outc * #Hloop #HR @(ex_intro ?? k) @(ex_intro ?? outc) % [@Hloop] -Hloop
117 [| * #ta * whd in ⊢ (%→%→?); #Hta #Houtc
118 #ls #x #rs #Hintape %2 >Hintape in Hta; #Hta cases (Hta ? (refl ??)) -Hta #Hx #Hta %
119 [ lapply (\Pf Hx) @not_to_not #Hx' >Hx' %
124 definition copy_step ≝
125 ifTM ? (test_char STape (λc.is_bit (\fst c)))
126 (single_finalTM ? (copy_step_subcase FSUnialpha (bit false)
127 (copy_step_subcase FSUnialpha (bit true) nocopy_subcase)))
131 definition R_copy_step_true ≝
133 ∀ls,c,rs. t1 = midtape STape ls 〈c,true〉 rs →
134 bit_or_null c = true ∧
136 ls = (l1@〈a0,false〉::〈x0,true〉::l2) →
137 rs = (〈a,false〉::l3) →
140 t2 = midtape STape (〈bit x,false〉::l1@〈a0,true〉::〈bit x,false〉::l2) 〈a,true〉 l3) ∨
142 t2 = midtape ? (〈null,false〉::l1@〈a0,true〉::〈x0,false〉::l2) 〈a,true〉 l3))).
144 definition R_copy_step_false ≝
146 ∀ls,c,rs.t1 = midtape (FinProd … FSUnialpha FinBool) ls c rs →
147 bit_or_null (\fst c) = false ∧ t2 = t1.
149 axiom sem_copy_step :
150 accRealize ? copy_step (inr … (inl … (inr … 0))) R_copy_step_true R_copy_step_false.
153 1) il primo carattere è marcato
154 2) l'ultimo carattere è l'unico che può essere null, gli altri sono bit
155 3) il terminatore non è né bit, né null
158 definition copy0 ≝ whileTM ? copy_step (inr … (inl … (inr … 0))).
160 let rec merge_config (l1,l2:list STape) ≝
163 | cons p1 l1' ⇒ match l2 with
166 let 〈c1,b1〉 ≝ p1 in let 〈c2,b2〉 ≝ p2 in
169 | _ ⇒ p2 ] :: merge_config l1' l2' ] ].
171 lemma merge_config_append :
172 ∀l1,l2,l3,l4.|l1| = |l2| →
173 merge_config (l1@l3) (l2@l4) = merge_config l1 l2@merge_config l3 l4.
174 #l1 #l2 #l3 #l4 #Hlen @(list_ind2 … Hlen)
176 | #t1 #t2 * #c1 #b1 * #c2 #b2 #IH whd in ⊢ (??%%); >IH % ]
179 definition R_copy0 ≝ λt1,t2.
180 ∀ls,c,c0,rs,l1,l3,l4.
181 t1 = midtape STape (l3@l4@〈c0,true〉::ls) 〈c,true〉 (l1@rs) →
182 no_marks l1 → no_marks (l3@l4) → |l1| = |l4| →
183 ∀l1',bv.〈c,false〉::l1 = l1'@[〈comma,bv〉] → only_bits_or_nulls l1' →
184 ∀l4',bg.l4@[〈c0,false〉] = 〈grid,bg〉::l4' → only_bits_or_nulls l4' →
185 (c = comma ∧ t2 = t1) ∨
187 t2 = midtape ? (reverse ? l1'@l3@〈grid,true〉::
188 merge_config l4' (reverse ? l1')@ls)
191 lemma inj_append_singleton_l1 :
192 ∀A.∀l1,l2:list A.∀a1,a2.l1@[a1] = l2@[a2] → l1 = l2.
193 #A #l1 #l2 #a1 #a2 #H lapply (eq_f … (reverse ?) … H)
194 >reverse_append >reverse_append normalize #H1 destruct
195 lapply (eq_f … (reverse ?) … e0) >reverse_reverse >reverse_reverse //
198 lemma inj_append_singleton_l2 :
199 ∀A.∀l1,l2:list A.∀a1,a2.l1@[a1] = l2@[a2] → a1 = a2.
200 #A #l1 #l2 #a1 #a2 #H lapply (eq_f … (reverse ?) … H)
201 >reverse_append >reverse_append normalize #H1 destruct %
204 lemma length_reverse : ∀A,l.|reverse A l| = |l|.
206 #a0 #l0 #IH normalize >rev_append_def >length_append >IH normalize //
209 lemma wsem_copy0 : WRealize ? copy0 R_copy0.
210 #intape #k #outc #Hloop
211 lapply (sem_while … sem_copy_step intape k outc Hloop) [%] -Hloop
212 * #ta * #Hstar @(star_ind_l ??????? Hstar)
213 [ #tb whd in ⊢ (%→?); #Hleft
214 #ls #c #c0 #rs #l1 #l3 #l4 #Htb #Hl1nomarks #Hl3l4nomarks #Hlen #l1' #bv
215 #Hl1 #Hl1bits #l4' #bg #Hl4 #Hl4bits
216 cases (Hleft … Htb) -Hleft #Hc #Houtc % %
217 [ generalize in match Hl1bits; -Hl1bits cases l1' in Hl1;
218 [ normalize #Hl1 #c1 destruct (Hl1) %
219 | * #c' #b' #l0 #Heq normalize in Heq; destruct (Heq)
220 #Hl1bits lapply (Hl1bits 〈c',false〉 ?) [ @memb_hd ]
221 >Hc #Hfalse destruct ]
223 | #tb #tc #td whd in ⊢ (%→?→(?→%)→%→?); #Htc #Hstar1 #Hind #Htd
224 lapply (Hind Htd) -Hind #Hind
225 #ls #c #c0 #rs #l1 #l3 #l4 #Htb #Hl1nomarks #Hl3l4nomarks #Hlen #l1' #bv
226 #Hl1 #Hl1bits #l4' #bg #Hl4 #Hl4bits %2
227 cases (Htc … Htb) -Htc #Hcbitnull #Htc
228 % [ % #Hc' >Hc' in Hcbitnull; normalize #Hfalse destruct (Hfalse) ]
229 cut (|l1| = |reverse ? l4|) [//] #Hlen1
230 @(list_cases_2 … Hlen1)
231 [ (* case l1 = [] is discriminated because l1 contains at least comma *)
232 #Hl1nil @False_ind >Hl1nil in Hl1; cases l1' normalize
233 [ #Hl1 destruct normalize in Hcbitnull; destruct (Hcbitnull)
234 | #p0 #l0 normalize #Hfalse destruct (Hfalse) cases l0 in e0;
235 [ normalize #Hfalse1 destruct (Hfalse1)
236 | #p0' #l0' normalize #Hfalse1 destruct (Hfalse1) ] ]
237 | (* case c::l1 = c::a::l1'' *)
238 * #a #ba * #a0 #ba0 #l1'' #l4'' #Hl1cons #Hl4cons
239 lapply (eq_f ?? (reverse ?) ?? Hl4cons) >reverse_reverse >reverse_cons -Hl4cons #Hl4cons
241 [ >Hl1cons in Hl1nomarks; #Hl1nomarks lapply (Hl1nomarks 〈a,ba〉 ?)
242 [ @memb_hd | normalize // ] ] #Hba
244 [ >Hl4cons in Hl3l4nomarks; #Hl3l4nomarks lapply (Hl3l4nomarks 〈a0,ba0〉 ?)
245 [ @memb_append_l2 @memb_append_l2 @memb_hd | normalize // ] ] #Hba0
246 >Hba0 in Hl4cons; >Hba in Hl1cons; -Hba0 -Hba #Hl1cons #Hl4cons
247 >Hl4cons in Htc; >Hl1cons #Htc
248 lapply (Htc a (l3@reverse ? l4'') c0 a0 ls (l1''@rs) ? (refl ??) ?)
249 [ #x #Hx @Hl3l4nomarks >Hl4cons <associative_append
251 | >associative_append >associative_append %
253 cut (∃la.l1' = 〈c,false〉::la)
254 [ >Hl1cons in Hl1; cases l1'
255 [normalize #Hfalse destruct (Hfalse)
256 | #p #la normalize #Hla destruct (Hla) @(ex_intro ?? la) % ] ]
258 cut (∃lb.l4' = lb@[〈c0,false〉])
260 @(list_elim_left … l4')
261 [ #Heq lapply (eq_f … (length ?) … Heq)
262 >length_append >length_append
263 >commutative_plus normalize >commutative_plus normalize
266 >(inj_append_singleton_l2 ? (reverse ? l4''@[〈a0,false〉]) (〈grid,bg〉::tl) 〈c0,false〉 a1 Heq)
269 cut (|lb| = |reverse ? la|)
270 [ >Hla in Hl1; >Hlb in Hl4; #Hl4 #Hl1
271 >(?:l1 = la@[〈comma,bv〉]) in Hlen;
272 [|normalize in Hl1; destruct (Hl1) %]
273 >(?:l4 = 〈grid,bg〉::lb)
274 [|@(inj_append_singleton_l1 ?? (〈grid,bg〉::lb) ?? Hl4) ]
275 >length_append >commutative_plus >length_reverse
276 normalize #Hlalb destruct (Hlalb) //
279 (* by hyp on the first iteration step,
280 we consider whether c = bit x or c = null *)
283 lapply (Hind (〈bit x,false〉::ls) a a0 rs l1''
284 (〈bit x,false〉::l3) (reverse ? l4'') ????)
285 [ >Hl1cons in Hlen; >Hl4cons >length_append >commutative_plus
286 normalize #Hlen destruct (Hlen) //
287 | #x0 #Hx0 cases (orb_true_l … Hx0)
288 [ #Hx0eq >(\P Hx0eq) %
289 | -Hx0 #Hx0 @Hl3l4nomarks >Hl4cons
290 <associative_append @memb_append_l1 // ]
291 | #x0 #Hx0 @Hl1nomarks >Hl1cons @memb_cons //
292 | >Htc >associative_append %
294 <Hl1cons <Hl4cons #Hind lapply (Hind la bv ?? lb bg ??)
295 [ #x0 #Hx0 @Hl4bits >Hlb @memb_append_l1 //
296 | >Hlb in Hl4; normalize in ⊢ (%→?); #Hl4
297 @(inj_append_singleton_l1 ? l4 (〈grid,bg〉::lb) … Hl4)
298 | #x0 #Hx0 @Hl1bits >Hla @memb_cons //
299 | >Hla in Hl1; normalize in ⊢ (%→?); #Hl1
300 destruct (Hl1) // ] -Hind
301 (* by IH, we proceed by cases, whether a = comma
302 (consequently several lists = []) or not *)
305 (* cut (l1 = [〈a,false〉])
306 [ cases l1'' in Hl1cons; // #y #ly #Hly
307 >Hly in Hl1; cases l1' in Hl1bits;
308 [ #_ normalize #Hfalse destruct (Hfalse)
309 | #p #lp #Hl1bits normalize #Heq destruct (Heq)
310 @False_ind lapply (Hl1bits 〈a,false〉 ?)
312 [ normalize #Hfalse destruct (Hfalse)
313 | #p0 #lp0 normalize in ⊢ (%→?); #Heq destruct (Heq)
314 @memb_cons @memb_hd ]
315 | >Ha normalize #Hfalse destruct (Hfalse) ]
318 cut (l4 = [〈a0,false〉])
319 [ generalize in match Hl4bits; cases l4' in Hl4;
320 [ >Hl4cons #Hfalse #_
321 lapply (inj_append_singleton_l1 ?? [] ?? Hfalse)
322 cases (reverse ? l4'') normalize
323 [ #Hfalse1 | #p0 #lp0 #Hfalse1 ] destruct (Hfalse1)
326 cases l4'' in Hl4cons; // #y #ly #Hly
327 >Hly in Hl4; cases l4' in Hl4bits;
328 [ #_ >reverse_cons #Hfalse
329 lapply (inj_append_singleton_l1 ?? [] ?? Hfalse)
330 -Hfalse cases ly normalize
331 [ #Hfalse | #p #Hp #Hfalse ] destruct (Hfalse)
333 | #p #lp #Hl1bits normalize #Heq destruct (Heq)
334 @False_ind lapply (Hl1bits 〈a,false〉 ?)
336 [ normalize #Hfalse destruct (Hfalse)
337 | #p0 #lp0 normalize in ⊢ (%→?); #Heq destruct (Heq)
338 @memb_cons @memb_hd ]
339 | >Ha normalize #Hfalse destruct (Hfalse) ]
343 >Hla normalize #Hl1 destruct (Hl1) lapply (inj_append_ @False_ind
345 cut (l1'' = [] ∧ l4'' = [])
346 [ % [ >Hla in Hl1; normalize #Hl1 destruct (Hl1)
348 cases l1'' in Hl1bits;
351 cut (la = [] ∧ lb = [] ∧ l1'' = [] ∧ l4'' = [])
352 [ @daemon ] * * * #Hla1 #Hlb1 #Hl1nil #Hl4nil
353 >Hl1cons in Hl1; >Hla
355 >Hl4cons in Hl4; >Hlb #Hl4
356 >Hla1 >Hlb1 >Hl1nil >Hl4nil >Hx
357 cut (a0 = grid) [ @daemon ] #Ha0 <Ha <Ha0
358 normalize in ⊢ (??(??%?%)(??%?%)); >associative_append %
359 | * #Ha #Houtc1 >Houtc1 @eq_f3 //
360 >Hla >reverse_cons >associative_append @eq_f
361 >Hx whd in ⊢ (??%?); @eq_f whd in ⊢ (???%); @eq_f @eq_f
362 >Hlb >append_cons @eq_f2 // >(merge_config_append … Hlen2) %
367 lapply (Hind (〈c0,false〉::ls) a a0 rs l1'' (〈null,false〉::l3) (reverse ? l4'') ????)
368 [ >Hl1cons in Hlen; >Hl4cons >length_append >commutative_plus normalize
369 #Hlen destruct (Hlen) @e0
370 | #x0 #Hx0 cases (memb_append STape ? [〈null,false〉] (l3@reverse ? l4'') … Hx0) -Hx0 #Hx0
371 [ >(memb_single … Hx0) %
372 | @Hl3l4nomarks cases (memb_append … Hx0) -Hx0 #Hx0
374 | @memb_append_l2 >Hl4cons @memb_append_l1 // ]
376 | >Hl1cons #x' #Hx0 @Hl1nomarks >Hl1cons @memb_cons //
377 | >Htc @eq_f3 // >associative_append % ] -Hind <Hl1cons <Hl4cons #Hind
378 lapply (Hind la bv ?? lb bg ??)
379 [ #x0 #Hx0 @Hl4bits >Hlb @memb_append_l1 //
380 | >Hlb in Hl4; normalize in ⊢ (%→?); #Hl4
381 @(inj_append_singleton_l1 ? l4 (〈grid,bg〉::lb) … Hl4)
382 | #x0 #Hx0 @Hl1bits >Hla @memb_cons //
383 | >Hla in Hl1; normalize in ⊢ (%→?); #Hl1
384 destruct (Hl1) // ] -Hind *
385 (* by IH, we proceed by cases, whether a = comma
386 (consequently several lists = []) or not *)
387 [ * #Ha #Houtc1 >Hl1cons in Hl1; >Hla
389 >Hl4cons in Hl4; >Hlb #Hl4
390 cut (la = [] ∧ lb = [] ∧ l1'' = [] ∧ l4'' = [])
391 [@daemon] * * * #Hla1 #Hlb1 #Hl1nil #Hl4nil
392 >Hla1 >Hlb1 >Hl1nil >Hl4nil >Hc
393 cut (a0 = grid) [ @daemon ] #Ha0 <Ha <Ha0
394 normalize in ⊢ (??(??%?%)(??%?%)); >associative_append %
395 | * #Ha #Houtc1 >Houtc1 @eq_f3 //
396 >Hla >reverse_cons >associative_append @eq_f
397 >Hc whd in ⊢ (??%?); @eq_f whd in ⊢ (???%); @eq_f @eq_f
398 >Hlb >append_cons @eq_f2 // >(merge_config_append … Hlen2) %
404 definition merge_char ≝ λc1,c2.
411 merge_config (〈c1,false〉::conf1) (〈c2,false〉::conf2) =
412 〈merge_char c1 c2,false〉::merge_config conf1 conf2.
413 #c1 #c2 #conf1 #conf2 normalize @eq_f2 //
417 lemma merge_config_c_nil :
418 ∀c.merge_config c [] = [].
419 #c cases c normalize //
422 axiom reverse_merge_config :
423 ∀c1,c2.|c1| = |c2| → reverse ? (merge_config c1 c2) =
424 merge_config (reverse ? c1) (reverse ? c2).
428 seq STape (move_l …) (seq ? (adv_to_mark_l … (is_marked ?))
429 (seq ? (clear_mark …) (seq ? (adv_to_mark_r … (is_marked ?)) (clear_mark …)))).
432 s0, s1 = caratteri di testa dello stato
433 c0 = carattere corrente del nastro oggetto
434 c1 = carattere in scrittura sul nastro oggetto
436 questa dimostrazione sfrutta il fatto che
437 merge_config (l0@[c0]) (l1@[c1]) = l1@[merge_char c0 c1]
438 se l0 e l1 non contengono null
441 definition R_copy ≝ λt1,t2.
442 ∀ls,s0,s1,c0,c1,rs,l1,l3,l4.
443 t1 = midtape STape (l3@〈grid,false〉::〈c0,false〉::l4@〈s0,true〉::ls) 〈s1,true〉 (l1@〈c1,false〉::〈comma,false〉::rs) →
444 no_marks l1 → no_marks l3 → no_marks l4 → |l1| = |l4| →
445 only_bits (l4@[〈s0,true〉]) → only_bits (〈s1,true〉::l1) →
446 bit_or_null c0 = true → bit_or_null c1 = true →
447 t2 = midtape STape (〈c1,false〉::reverse ? l1@〈s1,false〉::l3@〈grid,false〉::
448 〈merge_char c0 c1,false〉::reverse ? l1@〈s1,false〉::ls)
451 axiom sem_copy : Realize ? copy R_copy.