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_____________________________________________________________*)
13 include "turing/universal/tuples.ma".
15 definition write_states ≝ initN 2.
17 definition wr0 : write_states ≝ mk_Sig ?? 0 (leb_true_to_le 1 2 (refl …)).
18 definition wr1 : write_states ≝ mk_Sig ?? 1 (leb_true_to_le 2 2 (refl …)).
20 definition write ≝ λalpha,c.
21 mk_TM alpha write_states
24 [ O ⇒ 〈wr1,Some ? 〈c,N〉〉
25 | S _ ⇒ 〈wr1,None ?〉 ])
28 definition R_write ≝ λalpha,c,t1,t2.
29 ∀ls,x,rs.t1 = midtape alpha ls x rs → t2 = midtape alpha ls c rs.
31 axiom sem_write : ∀alpha,c.Realize ? (write alpha c) (R_write alpha c).
33 definition copy_step_subcase ≝
34 λalpha,c,elseM.ifTM ? (test_char ? (λx.x == 〈c,true〉))
35 (seq (FinProd alpha FinBool) (adv_mark_r …)
37 (seq ? (adv_to_mark_l … (is_marked alpha))
38 (seq ? (write ? 〈c,false〉)
41 (seq ? (move_r …) (adv_to_mark_r … (is_marked alpha)))))))))
44 definition R_copy_step_subcase ≝
45 λalpha,c,RelseM,t1,t2.
47 t1 = midtape (FinProd … alpha FinBool) (l1@〈a0,false〉::〈x0,true〉::l2)
48 〈x,true〉 (〈a,false〉::l3) →
49 (∀c.memb ? c l1 = true → is_marked ? c = false) →
50 (x = c ∧ t2 = midtape ? (〈x,false〉::l1@〈a0,true〉::〈x,false〉::l2) 〈a,true〉 l3) ∨
51 (x ≠ c ∧ RelseM t1 t2).
53 axiom sem_copy_step_subcase :
54 ∀alpha,c,elseM,RelseM.
55 Realize ? (copy_step_subcase alpha c elseM) (R_copy_step_subcase alpha c RelseM).
67 else if current = 1,tt
76 else if current = null
85 definition nocopy_subcase ≝
86 ifTM STape (test_char ? (λx:STape.x == 〈null,true〉))
89 (seq ? (adv_to_mark_l … (is_marked ?))
91 (seq ? (move_r …) (adv_to_mark_r … (is_marked ?)))))))
94 definition R_nocopy_subcase ≝
97 t1 = midtape STape (l1@〈a0,false〉::〈x0,true〉::l2)
98 〈x,true〉 (〈a,false〉::l3) →
99 (∀c.memb ? c l1 = true → is_marked ? c = false) →
101 t2 = midtape ? (〈x,false〉::l1@〈a0,true〉::〈x0,false〉::l2) 〈a,true〉 l3) ∨
102 (x ≠ null ∧ t2 = t1).
104 axiom sem_nocopy_subcase : Realize ? nocopy_subcase R_nocopy_subcase.
106 cases (sem_if ? (test_char ? (λx:STape.x == 〈null,true〉)) ?????? tc_true
107 (sem_test_char ? (λx:STape.x == 〈null,true〉))
108 (sem_seq … (sem_adv_mark_r …)
109 (sem_seq … (sem_move_l …)
110 (sem_seq … (sem_adv_to_mark_l … (is_marked ?))
111 (sem_seq … (sem_adv_mark_r …)
112 (sem_seq … (sem_move_r …) (sem_adv_to_mark_r … (is_marked ?))
113 ))))) (sem_nop ?) intape)
114 #k * #outc * #Hloop #HR @(ex_intro ?? k) @(ex_intro ?? outc) % [@Hloop] -Hloop
116 [| * #ta * whd in ⊢ (%→%→?); #Hta #Houtc
117 #ls #x #rs #Hintape %2 >Hintape in Hta; #Hta cases (Hta ? (refl ??)) -Hta #Hx #Hta %
118 [ lapply (\Pf Hx) @not_to_not #Hx' >Hx' %
123 definition copy_step ≝
124 ifTM ? (test_char STape (λc.is_bit (\fst c)))
125 (single_finalTM ? (copy_step_subcase FSUnialpha (bit false)
126 (copy_step_subcase FSUnialpha (bit true) nocopy_subcase)))
130 definition R_copy_step_true ≝
132 ∀ls,c,rs. t1 = midtape STape ls 〈c,true〉 rs →
133 bit_or_null c = true ∧
135 ls = (l1@〈a0,false〉::〈x0,true〉::l2) →
136 rs = (〈a,false〉::l3) →
139 t2 = midtape STape (〈bit x,false〉::l1@〈a0,true〉::〈bit x,false〉::l2) 〈a,true〉 l3) ∨
141 t2 = midtape ? (〈null,false〉::l1@〈a0,true〉::〈x0,false〉::l2) 〈a,true〉 l3))).
143 definition R_copy_step_false ≝
145 ∀ls,c,rs.t1 = midtape (FinProd … FSUnialpha FinBool) ls c rs →
146 bit_or_null (\fst c) = false ∧ t2 = t1.
148 axiom sem_copy_step :
149 accRealize ? copy_step (inr … (inl … (inr … start_nop))) R_copy_step_true R_copy_step_false.
152 1) il primo carattere è marcato
153 2) l'ultimo carattere è l'unico che può essere null, gli altri sono bit
154 3) il terminatore non è né bit, né null
157 definition copy0 ≝ whileTM ? copy_step (inr … (inl … (inr … start_nop))).
159 let rec merge_config (l1,l2:list STape) ≝
162 | cons p1 l1' ⇒ match l2 with
165 let 〈c1,b1〉 ≝ p1 in let 〈c2,b2〉 ≝ p2 in
168 | _ ⇒ p2 ] :: merge_config l1' l2' ] ].
170 lemma merge_config_append :
171 ∀l1,l2,l3,l4.|l1| = |l2| →
172 merge_config (l1@l3) (l2@l4) = merge_config l1 l2@merge_config l3 l4.
173 #l1 #l2 #l3 #l4 #Hlen @(list_ind2 … Hlen)
175 | #t1 #t2 * #c1 #b1 * #c2 #b2 #IH whd in ⊢ (??%%); >IH % ]
178 definition R_copy0 ≝ λt1,t2.
179 ∀ls,c,c0,rs,l1,l3,l4.
180 t1 = midtape STape (l3@l4@〈c0,true〉::ls) 〈c,true〉 (l1@rs) →
181 no_marks l1 → no_marks (l3@l4) → |l1| = |l4| →
182 ∀l1',bv.〈c,false〉::l1 = l1'@[〈comma,bv〉] → only_bits_or_nulls l1' →
183 ∀l4',bg.l4@[〈c0,false〉] = 〈grid,bg〉::l4' → only_bits_or_nulls l4' →
184 (c = comma ∧ t2 = t1) ∨
186 t2 = midtape ? (reverse ? l1'@l3@〈grid,true〉::
187 merge_config l4' (reverse ? l1')@ls)
190 lemma inj_append_singleton_l1 :
191 ∀A.∀l1,l2:list A.∀a1,a2.l1@[a1] = l2@[a2] → l1 = l2.
192 #A #l1 #l2 #a1 #a2 #H lapply (eq_f … (reverse ?) … H)
193 >reverse_append >reverse_append normalize #H1 destruct
194 lapply (eq_f … (reverse ?) … e0) >reverse_reverse >reverse_reverse //
197 lemma inj_append_singleton_l2 :
198 ∀A.∀l1,l2:list A.∀a1,a2.l1@[a1] = l2@[a2] → a1 = a2.
199 #A #l1 #l2 #a1 #a2 #H lapply (eq_f … (reverse ?) … H)
200 >reverse_append >reverse_append normalize #H1 destruct %
203 axiom length_reverse : ∀A,l.|reverse A l| = |l|.
205 lemma wsem_copy0 : WRealize ? copy0 R_copy0.
206 #intape #k #outc #Hloop
207 lapply (sem_while … sem_copy_step intape k outc Hloop) [%] -Hloop
208 * #ta * #Hstar @(star_ind_l ??????? Hstar)
209 [ #tb whd in ⊢ (%→?); #Hleft
210 #ls #c #c0 #rs #l1 #l3 #l4 #Htb #Hl1nomarks #Hl3l4nomarks #Hlen #l1' #bv
211 #Hl1 #Hl1bits #l4' #bg #Hl4 #Hl4bits
212 cases (Hleft … Htb) -Hleft #Hc #Houtc % %
213 [ generalize in match Hl1bits; -Hl1bits cases l1' in Hl1;
214 [ normalize #Hl1 #c1 destruct (Hl1) %
215 | * #c' #b' #l0 #Heq normalize in Heq; destruct (Heq)
216 #Hl1bits lapply (Hl1bits 〈c',false〉 ?) [ @memb_hd ]
217 >Hc #Hfalse destruct ]
219 | #tb #tc #td whd in ⊢ (%→?→(?→%)→%→?); #Htc #Hstar1 #Hind #Htd
220 lapply (Hind Htd) -Hind #Hind
221 #ls #c #c0 #rs #l1 #l3 #l4 #Htb #Hl1nomarks #Hl3l4nomarks #Hlen #l1' #bv
222 #Hl1 #Hl1bits #l4' #bg #Hl4 #Hl4bits %2
223 cases (Htc … Htb) -Htc #Hcbitnull #Htc
224 % [ % #Hc' >Hc' in Hcbitnull; normalize #Hfalse destruct (Hfalse) ]
225 cut (|l1| = |reverse ? l4|) [@daemon] #Hlen1
226 @(list_cases_2 … Hlen1)
227 [ (* case l1 = [] is discriminated because l1 contains at least comma *)
228 #Hl1nil @False_ind >Hl1nil in Hl1; cases l1' normalize
229 [ #Hl1 destruct normalize in Hcbitnull; destruct (Hcbitnull)
230 | #p0 #l0 normalize #Hfalse destruct (Hfalse) cases l0 in e0;
231 [ normalize #Hfalse1 destruct (Hfalse1)
232 | #p0' #l0' normalize #Hfalse1 destruct (Hfalse1) ] ]
233 | (* case c::l1 = c::a::l1'' *)
234 * #a #ba * #a0 #ba0 #l1'' #l4'' #Hl1cons #Hl4cons
235 lapply (eq_f ?? (reverse ?) ?? Hl4cons) >reverse_reverse >reverse_cons -Hl4cons #Hl4cons
237 [ >Hl1cons in Hl1nomarks; #Hl1nomarks lapply (Hl1nomarks 〈a,ba〉 ?)
238 [ @memb_hd | normalize // ] ] #Hba
240 [ >Hl4cons in Hl3l4nomarks; #Hl3l4nomarks lapply (Hl3l4nomarks 〈a0,ba0〉 ?)
241 [ @memb_append_l2 @memb_append_l2 @memb_hd | normalize // ] ] #Hba0
242 >Hba0 in Hl4cons; >Hba in Hl1cons; -Hba0 -Hba #Hl1cons #Hl4cons
243 >Hl4cons in Htc; >Hl1cons #Htc
244 lapply (Htc a (l3@reverse ? l4'') c0 a0 ls (l1''@rs) ? (refl ??) ?)
245 [ #x #Hx @Hl3l4nomarks >Hl4cons <associative_append
247 | >associative_append >associative_append %
249 cut (∃la.l1' = 〈c,false〉::la)
250 [ >Hl1cons in Hl1; cases l1'
251 [normalize #Hfalse destruct (Hfalse)
252 | #p #la normalize #Hla destruct (Hla) @(ex_intro ?? la) % ] ]
254 cut (∃lb.l4' = lb@[〈c0,false〉])
256 @(list_elim_left … l4')
257 [ #Heq lapply (eq_f … (length ?) … Heq)
258 >length_append >length_append
259 >commutative_plus normalize >commutative_plus normalize
262 >(inj_append_singleton_l2 ? (reverse ? l4''@[〈a0,false〉]) (〈grid,bg〉::tl) 〈c0,false〉 a1 Heq)
265 cut (|lb| = |reverse ? la|)
266 [ >Hla in Hl1; >Hlb in Hl4; #Hl4 #Hl1
267 >(?:l1 = la@[〈comma,bv〉]) in Hlen;
268 [|normalize in Hl1; destruct (Hl1) %]
269 >(?:l4 = 〈grid,bg〉::lb)
270 [|@(inj_append_singleton_l1 ?? (〈grid,bg〉::lb) ?? Hl4) ]
271 >length_append >commutative_plus >length_reverse
272 normalize #Hlalb destruct (Hlalb) //
275 (* by hyp on the first iteration step,
276 we consider whether c = bit x or c = null *)
279 lapply (Hind (〈bit x,false〉::ls) a a0 rs l1''
280 (〈bit x,false〉::l3) (reverse ? l4'') ????)
281 [ >Hl1cons in Hlen; >Hl4cons >length_append >commutative_plus
282 normalize #Hlen destruct (Hlen) //
283 | #x0 #Hx0 cases (orb_true_l … Hx0)
284 [ #Hx0eq >(\P Hx0eq) %
285 | -Hx0 #Hx0 @Hl3l4nomarks >Hl4cons
286 <associative_append @memb_append_l1 // ]
287 | #x0 #Hx0 @Hl1nomarks >Hl1cons @memb_cons //
288 | >Htc >associative_append %
290 <Hl1cons <Hl4cons #Hind lapply (Hind la bv ?? lb bg ??)
291 [ #x0 #Hx0 @Hl4bits >Hlb @memb_append_l1 //
292 | >Hlb in Hl4; normalize in ⊢ (%→?); #Hl4
293 @(inj_append_singleton_l1 ? l4 (〈grid,bg〉::lb) … Hl4)
294 | #x0 #Hx0 @Hl1bits >Hla @memb_cons //
295 | >Hla in Hl1; normalize in ⊢ (%→?); #Hl1
296 destruct (Hl1) // ] -Hind
297 (* by IH, we proceed by cases, whether a = comma
298 (consequently several lists = []) or not *)
301 (* cut (l1 = [〈a,false〉])
302 [ cases l1'' in Hl1cons; // #y #ly #Hly
303 >Hly in Hl1; cases l1' in Hl1bits;
304 [ #_ normalize #Hfalse destruct (Hfalse)
305 | #p #lp #Hl1bits normalize #Heq destruct (Heq)
306 @False_ind lapply (Hl1bits 〈a,false〉 ?)
308 [ normalize #Hfalse destruct (Hfalse)
309 | #p0 #lp0 normalize in ⊢ (%→?); #Heq destruct (Heq)
310 @memb_cons @memb_hd ]
311 | >Ha normalize #Hfalse destruct (Hfalse) ]
314 cut (l4 = [〈a0,false〉])
315 [ generalize in match Hl4bits; cases l4' in Hl4;
316 [ >Hl4cons #Hfalse #_
317 lapply (inj_append_singleton_l1 ?? [] ?? Hfalse)
318 cases (reverse ? l4'') normalize
319 [ #Hfalse1 | #p0 #lp0 #Hfalse1 ] destruct (Hfalse1)
322 cases l4'' in Hl4cons; // #y #ly #Hly
323 >Hly in Hl4; cases l4' in Hl4bits;
324 [ #_ >reverse_cons #Hfalse
325 lapply (inj_append_singleton_l1 ?? [] ?? Hfalse)
326 -Hfalse cases ly normalize
327 [ #Hfalse | #p #Hp #Hfalse ] destruct (Hfalse)
329 | #p #lp #Hl1bits normalize #Heq destruct (Heq)
330 @False_ind lapply (Hl1bits 〈a,false〉 ?)
332 [ normalize #Hfalse destruct (Hfalse)
333 | #p0 #lp0 normalize in ⊢ (%→?); #Heq destruct (Heq)
334 @memb_cons @memb_hd ]
335 | >Ha normalize #Hfalse destruct (Hfalse) ]
339 >Hla normalize #Hl1 destruct (Hl1) lapply (inj_append_ @False_ind
341 cut (l1'' = [] ∧ l4'' = [])
342 [ % [ >Hla in Hl1; normalize #Hl1 destruct (Hl1)
344 cases l1'' in Hl1bits;
347 cut (la = [] ∧ lb = [] ∧ l1'' = [] ∧ l4'' = [])
348 [ @daemon ] * * * #Hla1 #Hlb1 #Hl1nil #Hl4nil
349 >Hl1cons in Hl1; >Hla
351 >Hl4cons in Hl4; >Hlb #Hl4
352 >Hla1 >Hlb1 >Hl1nil >Hl4nil >Hx
353 cut (a0 = grid) [ @daemon ] #Ha0 <Ha <Ha0
354 normalize in ⊢ (??(??%?%)(??%?%)); >associative_append %
355 | * #Ha #Houtc1 >Houtc1 @eq_f3 //
356 >Hla >reverse_cons >associative_append @eq_f
357 >Hx whd in ⊢ (??%?); @eq_f whd in ⊢ (???%); @eq_f @eq_f
358 >Hlb >append_cons @eq_f2 // >(merge_config_append … Hlen2) %
363 lapply (Hind (〈c0,false〉::ls) a a0 rs l1'' (〈null,false〉::l3) (reverse ? l4'') ????)
364 [ >Hl1cons in Hlen; >Hl4cons >length_append >commutative_plus normalize
365 #Hlen destruct (Hlen) @e0
366 | #x0 #Hx0 cases (memb_append STape ? [〈null,false〉] (l3@reverse ? l4'') … Hx0) -Hx0 #Hx0
367 [ >(memb_single … Hx0) %
368 | @Hl3l4nomarks cases (memb_append … Hx0) -Hx0 #Hx0
370 | @memb_append_l2 >Hl4cons @memb_append_l1 // ]
372 | >Hl1cons #x' #Hx0 @Hl1nomarks >Hl1cons @memb_cons //
373 | >Htc @eq_f3 // >associative_append % ] -Hind <Hl1cons <Hl4cons #Hind
374 lapply (Hind la bv ?? lb bg ??)
375 [ #x0 #Hx0 @Hl4bits >Hlb @memb_append_l1 //
376 | >Hlb in Hl4; normalize in ⊢ (%→?); #Hl4
377 @(inj_append_singleton_l1 ? l4 (〈grid,bg〉::lb) … Hl4)
378 | #x0 #Hx0 @Hl1bits >Hla @memb_cons //
379 | >Hla in Hl1; normalize in ⊢ (%→?); #Hl1
380 destruct (Hl1) // ] -Hind *
381 (* by IH, we proceed by cases, whether a = comma
382 (consequently several lists = []) or not *)
383 [ * #Ha #Houtc1 >Hl1cons in Hl1; >Hla
385 >Hl4cons in Hl4; >Hlb #Hl4
386 cut (la = [] ∧ lb = [] ∧ l1'' = [] ∧ l4'' = [])
387 [@daemon] * * * #Hla1 #Hlb1 #Hl1nil #Hl4nil
388 >Hla1 >Hlb1 >Hl1nil >Hl4nil >Hc
389 cut (a0 = grid) [ @daemon ] #Ha0 <Ha <Ha0
390 normalize in ⊢ (??(??%?%)(??%?%)); >associative_append %
391 | * #Ha #Houtc1 >Houtc1 @eq_f3 //
392 >Hla >reverse_cons >associative_append @eq_f
393 >Hc whd in ⊢ (??%?); @eq_f whd in ⊢ (???%); @eq_f @eq_f
394 >Hlb >append_cons @eq_f2 // >(merge_config_append … Hlen2) %
400 definition merge_char ≝ λc1,c2.
407 merge_config (〈c1,false〉::conf1) (〈c2,false〉::conf2) =
408 〈merge_char c1 c2,false〉::merge_config conf1 conf2.
409 #c1 #c2 #conf1 #conf2 normalize @eq_f2 //
413 lemma merge_config_c_nil :
414 ∀c.merge_config c [] = [].
415 #c cases c normalize //
418 axiom reverse_merge_config :
419 ∀c1,c2.|c1| = |c2| → reverse ? (merge_config c1 c2) =
420 merge_config (reverse ? c1) (reverse ? c2).
424 seq STape (move_l …) (seq ? (adv_to_mark_l … (is_marked ?))
425 (seq ? (clear_mark …) (seq ? (adv_to_mark_r … (is_marked ?)) (clear_mark …)))).
428 s0, s1 = caratteri di testa dello stato
429 c0 = carattere corrente del nastro oggetto
430 c1 = carattere in scrittura sul nastro oggetto
432 questa dimostrazione sfrutta il fatto che
433 merge_config (l0@[c0]) (l1@[c1]) = l1@[merge_char c0 c1]
434 se l0 e l1 non contengono null
437 definition R_copy ≝ λt1,t2.
438 ∀ls,s0,s1,c0,c1,rs,l1,l3,l4.
439 t1 = midtape STape (l3@〈grid,false〉::〈c0,false〉::l4@〈s0,true〉::ls) 〈s1,true〉 (l1@〈c1,false〉::〈comma,false〉::rs) →
440 no_marks l1 → no_marks l3 → no_marks l4 → |l1| = |l4| →
441 only_bits (l4@[〈s0,true〉]) → only_bits (〈s1,true〉::l1) →
442 bit_or_null c0 = true → bit_or_null c1 = true →
443 t2 = midtape STape (〈c1,false〉::reverse ? l1@〈s1,false〉::l3@〈grid,false〉::
444 〈merge_char c0 c1,false〉::reverse ? l1@〈s1,false〉::ls)
447 axiom sem_copy : Realize ? copy R_copy.