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 lemma sem_copy_step_subcase :
54 ∀alpha,c,elseM,RelseM. Realize ? elseM RelseM →
55 Realize ? (copy_step_subcase alpha c elseM) (R_copy_step_subcase alpha c RelseM).
56 #alpha #c #elseM #RelseM #HelseM #intape
57 cases (sem_if ? (test_char ? (λx. x == 〈c,true〉)) ?????? tc_true (sem_test_char ? (λx.x == 〈c,true〉))
58 (sem_seq ????? (sem_adv_mark_r alpha)
59 (sem_seq ????? (sem_move_l …)
60 (sem_seq ????? (sem_adv_to_mark_l … (is_marked alpha))
61 (sem_seq ????? (sem_write ? 〈c,false〉)
62 (sem_seq ????? (sem_move_r …)
63 (sem_seq ????? (sem_mark …)
64 (sem_seq ????? (sem_move_r …) (sem_adv_to_mark_r … (is_marked alpha)))))))))
66 #k * #outc * #Hloop #HR %{k} %{outc} % [@Hloop] -Hloop
67 #a #l1 #x0 #a0 #l2 #x #l3 #Hintape #Hl1marks cases HR -HR
68 [ * #ta * whd in ⊢ (%→?); >Hintape #Hta cases (Hta … (refl ??)) -Hta #Hx #Hta
69 * #tb * whd in ⊢ (%→?); #Htb lapply (Htb … Hta) -Hta -Htb #Htb
70 * #tc * whd in ⊢ (%→?); #Htc lapply (Htc … Htb) -Htb -Htc #Htc
71 * #td * whd in ⊢ (%→?); #Htd cases (Htd … Htc) -Htd
72 [ >Htc * normalize in ⊢ (%→?); #Hfalse destruct (Hfalse) ]
73 * #_ #Htd lapply (Htd (l1@[〈a0,false〉]) 〈x0,true〉 l2 ? (refl ??) ?) -Htd
74 [ #x1 #Hx1 cases (memb_append … Hx1) -Hx1 #Hx1 [@(Hl1marks ? Hx1)|>(memb_single … Hx1) %]
75 | normalize >associative_append % ] #Htd
76 * #te * whd in ⊢ (%→?); #Hte lapply (Hte … Htd) -Hte -Htd -Htc #Hte
77 * #tf * whd in ⊢ (%→?); #Htf lapply (Htf … Hte) -Hte -Htf >reverse_append #Htf
78 * #tg * whd in ⊢ (%→?); #Htg lapply (Htg … Htf) -Htf -Htg >reverse_single #Htg
79 * #th * whd in ⊢ (%→?); #Hth lapply (Hth … Htg) -Htg -Hth
80 generalize in match Hl1marks; -Hl1marks @(list_elim_left … l1)
81 [ #Hl1marks #Hth whd in ⊢ (%→?); #Houtc cases (Houtc … Hth) -Houtc
82 [ * normalize in ⊢ (%→?); #Hfalse destruct (Hfalse) ]
83 * #_ #Houtc lapply (Houtc [] ?? (refl ??) (refl ??) Hl1marks) -Houtc
84 #Houtc lapply (\P Hx) -Hx #Hx destruct (Hx) % % [%] @Houtc
85 | -l1 #c1 #l1 #_ #Hl1marks >reverse_append >reverse_single
86 #Hth whd in ⊢ (%→?); #Houtc cases (Houtc … Hth) -Houtc
87 [ * >Hl1marks [ #Hfalse destruct (Hfalse) ] @memb_append_l2 @memb_hd ]
88 * #_ #Houtc lapply (Houtc (reverse ? l1@[〈x,false〉]) 〈a,true〉 l3 ? (refl ??) ?) -Houtc
89 [ #x1 #Hx1 cases (memb_append … Hx1) -Hx1 #Hx1 [ @Hl1marks @memb_append_l1 @daemon | >(memb_single … Hx1) % ]
90 | normalize >associative_append % ]
91 #Houtc lapply (\P Hx) -Hx #Hx destruct (Hx) % % [%] >Houtc
92 >reverse_append >reverse_reverse >associative_append >associative_append % ]
93 | * #ta * whd in ⊢ (%→?); >Hintape #Hta cases (Hta ? (refl ??)) -Hta
94 #Hxc #Hta >Hta #Houtc %2 % // lapply (\Pf Hxc) @not_to_not #Heq >Heq % ]
107 else if current = 1,tt
116 else if current = null
125 definition nocopy_subcase ≝
126 ifTM STape (test_char ? (λx:STape.x == 〈null,true〉))
127 (seq ? (adv_mark_r …)
129 (seq ? (adv_to_mark_l … (is_marked ?))
130 (seq ? (adv_mark_r …)
131 (seq ? (move_r …) (adv_to_mark_r … (is_marked ?)))))))
134 definition R_nocopy_subcase ≝
137 t1 = midtape STape (l1@〈a0,false〉::〈x0,true〉::l2)
138 〈x,true〉 (〈a,false〉::l3) →
139 (∀c.memb ? c l1 = true → is_marked ? c = false) →
141 t2 = midtape ? (〈x,false〉::l1@〈a0,true〉::〈x0,false〉::l2) 〈a,true〉 l3) ∨
142 (x ≠ null ∧ t2 = t1).
144 lemma sem_nocopy_subcase : Realize ? nocopy_subcase R_nocopy_subcase.
146 cases (sem_if ? (test_char ? (λx:STape.x == 〈null,true〉)) ?????? tc_true
147 (sem_test_char ? (λx:STape.x == 〈null,true〉))
148 (sem_seq … (sem_adv_mark_r …)
149 (sem_seq … (sem_move_l …)
150 (sem_seq … (sem_adv_to_mark_l … (is_marked ?))
151 (sem_seq … (sem_adv_mark_r …)
152 (sem_seq … (sem_move_r …)
153 (sem_adv_to_mark_r … (is_marked ?))))))) (sem_nop ?) intape)
154 #k * #outc * #Hloop #HR @(ex_intro ?? k) @(ex_intro ?? outc) % [@Hloop] -Hloop
155 #a #l1 #x0 #a0 #l2 #x #l3 #Hintape #Hl1marks cases HR -HR
156 [ * #ta * whd in ⊢ (%→?); >Hintape #Hta cases (Hta … (refl ??)) -Hta #Hx #Hta
157 * #tb * whd in ⊢ (%→?); #Htb lapply (Htb … Hta) -Hta -Htb #Htb
158 * #tc * whd in ⊢ (%→?); #Htc lapply (Htc … Htb) -Htb -Htc #Htc
159 * #td * whd in ⊢ (%→?); #Htd cases (Htd … Htc) -Htd
160 [ >Htc * normalize in ⊢ (%→?); #Hfalse destruct (Hfalse) ]
161 * #_ #Htd lapply (Htd (l1@[〈a0,false〉]) 〈x0,true〉 l2 ? (refl ??) ?) -Htd
162 [ #x1 #Hx1 cases (memb_append … Hx1) -Hx1 #Hx1 [@(Hl1marks ? Hx1)|>(memb_single … Hx1) %]
163 | normalize >associative_append % ] >reverse_append #Htd
164 * #te * whd in ⊢ (%→?); #Hte lapply (Hte … Htd) -Hte -Htd -Htc #Hte
165 * #tf * whd in ⊢ (%→?); #Htf lapply (Htf … Hte) -Hte -Htf
166 generalize in match Hl1marks; -Hl1marks @(list_elim_left … l1)
167 [ #Hl1marks #Hth whd in ⊢ (%→?); #Houtc cases (Houtc … Hth) -Houtc
168 [ * normalize in ⊢ (%→?); #Hfalse destruct (Hfalse) ]
169 * #_ #Houtc lapply (Houtc [] ?? (refl ??) (refl ??) Hl1marks) -Houtc
170 #Houtc lapply (\P Hx) -Hx #Hx destruct (Hx) % % [%] @Houtc
171 | -l1 #c1 #l1 #_ #Hl1marks >reverse_append >reverse_single
172 #Hth whd in ⊢ (%→?); #Houtc cases (Houtc … Hth) -Houtc
173 [ * >Hl1marks [ #Hfalse destruct (Hfalse) ] @memb_append_l2 @memb_hd ]
174 * #_ #Houtc lapply (Houtc (reverse ? l1@[〈x,false〉]) 〈a,true〉 l3 ? (refl ??) ?) -Houtc
175 [ #x1 #Hx1 cases (memb_append … Hx1) -Hx1 #Hx1 [ @Hl1marks @memb_append_l1 @daemon | >(memb_single … Hx1) % ]
176 | normalize >associative_append % ]
177 #Houtc lapply (\P Hx) -Hx #Hx destruct (Hx) % % [%] >Houtc
178 >reverse_append >reverse_reverse >associative_append >associative_append % ]
179 | * #ta * whd in ⊢ (%→?); >Hintape #Hta cases (Hta ? (refl ??)) -Hta
180 #Hxc #Hta >Hta whd in ⊢ (%→?); #Houtc %2 %
181 [ lapply (\Pf Hxc) @not_to_not #Heq >Heq %
185 definition copy_step ≝
186 ifTM ? (test_char STape (λc.bit_or_null (\fst c)))
187 (single_finalTM ? (copy_step_subcase FSUnialpha (bit false)
188 (copy_step_subcase FSUnialpha (bit true) nocopy_subcase)))
192 definition R_copy_step_true ≝
194 ∀ls,c,rs. t1 = midtape STape ls 〈c,true〉 rs →
195 bit_or_null c = true ∧
197 ls = (l1@〈a0,false〉::〈x0,true〉::l2) →
198 rs = (〈a,false〉::l3) →
201 t2 = midtape STape (〈bit x,false〉::l1@〈a0,true〉::〈bit x,false〉::l2) 〈a,true〉 l3) ∨
203 t2 = midtape ? (〈null,false〉::l1@〈a0,true〉::〈x0,false〉::l2) 〈a,true〉 l3))).
205 definition R_copy_step_false ≝
207 ∀ls,c,rs.t1 = midtape (FinProd … FSUnialpha FinBool) ls c rs →
208 bit_or_null (\fst c) = false ∧ t2 = t1.
210 lemma sem_copy_step :
211 accRealize ? copy_step (inr … (inl … (inr … start_nop))) R_copy_step_true R_copy_step_false.
213 @(acc_sem_if_app … (sem_test_char ? (λc:STape.bit_or_null (\fst c))) …
214 (sem_copy_step_subcase FSUnialpha (bit false) …
215 (sem_copy_step_subcase FSUnialpha (bit true) … (sem_nocopy_subcase …)))
217 [ #t1 #t2 #t3 whd in ⊢ (%→%→?); #H1 #H2 #ls #c #rs #Ht1 >Ht1 in H1; #H1
218 cases (H1 … (refl ??)) #Hc #Ht3 % [ @Hc ]
219 #a #l1 #x0 #a0 #l2 #l3 #Hls #Hrs #Hl1marks >Hls in Ht3; >Hrs #Ht3
221 [ * #Hc' #Ht2 % %{false} % // <Hc' @Ht2
222 | * #Hnotfalse whd in ⊢ (%→?); #Ht2 cases (Ht2 … Ht3 ?) -Ht2
223 [ * #Hc' #Ht2 % %{true} % // <Hc' @Ht2
224 | * #Hnottrue whd in ⊢ (%→?); #Ht2 cases (Ht2 … Ht3 ?) -Ht2
225 [ * #Hc' #Ht2 %2 <Hc' % // @Ht2
226 | * #Hnotnull @False_ind
227 generalize in match Hnotnull;generalize in match Hnottrue;generalize in match Hnotfalse;
228 cases c in Hc; normalize
229 [ * [ #_ #_ * #Hfalse #_ | #_ * #Hfalse #_ #_ ]
231 |*: #Hfalse destruct (Hfalse) ] @Hfalse %
235 | #t1 #t2 #t3 whd in ⊢ (%→%→?); #H1 #H2 #ls #c #rs #Ht1
236 >Ht1 in H1; #H1 cases (H1 … (refl ??)) #_ #Ht3 cases (H1 ? (refl ??)) -H1
242 1) il primo carattere è marcato
243 2) l'ultimo carattere è l'unico che può essere null, gli altri sono bit
244 3) il terminatore non è né bit, né null
247 definition copy0 ≝ whileTM ? copy_step (inr … (inl … (inr … start_nop))).
249 let rec merge_config (l1,l2:list STape) ≝
252 | cons p1 l1' ⇒ match l2 with
255 let 〈c1,b1〉 ≝ p1 in let 〈c2,b2〉 ≝ p2 in
258 | _ ⇒ p2 ] :: merge_config l1' l2' ] ].
260 lemma merge_config_append :
261 ∀l1,l2,l3,l4.|l1| = |l2| →
262 merge_config (l1@l3) (l2@l4) = merge_config l1 l2@merge_config l3 l4.
263 #l1 #l2 #l3 #l4 #Hlen @(list_ind2 … Hlen)
265 | #t1 #t2 * #c1 #b1 * #c2 #b2 #IH whd in ⊢ (??%%); >IH % ]
268 definition R_copy0 ≝ λt1,t2.
269 ∀ls,c,c0,rs,l1,l3,l4.
270 t1 = midtape STape (l3@l4@〈c0,true〉::ls) 〈c,true〉 (l1@rs) →
271 no_marks l1 → no_marks (l3@l4) → |l1| = |l4| →
272 ∀l1',bv.〈c,false〉::l1 = l1'@[〈comma,bv〉] → only_bits_or_nulls l1' →
273 ∀l4',bg.l4@[〈c0,false〉] = 〈grid,bg〉::l4' → only_bits_or_nulls l4' →
274 (c = comma ∧ t2 = t1) ∨
276 t2 = midtape ? (reverse ? l1'@l3@〈grid,true〉::
277 merge_config l4' (reverse ? l1')@ls)
280 lemma inj_append_singleton_l1 :
281 ∀A.∀l1,l2:list A.∀a1,a2.l1@[a1] = l2@[a2] → l1 = l2.
282 #A #l1 #l2 #a1 #a2 #H lapply (eq_f … (reverse ?) … H)
283 >reverse_append >reverse_append normalize #H1 destruct
284 lapply (eq_f … (reverse ?) … e0) >reverse_reverse >reverse_reverse //
287 lemma inj_append_singleton_l2 :
288 ∀A.∀l1,l2:list A.∀a1,a2.l1@[a1] = l2@[a2] → a1 = a2.
289 #A #l1 #l2 #a1 #a2 #H lapply (eq_f … (reverse ?) … H)
290 >reverse_append >reverse_append normalize #H1 destruct %
293 lemma wsem_copy0 : WRealize ? copy0 R_copy0.
294 #intape #k #outc #Hloop
295 lapply (sem_while … sem_copy_step intape k outc Hloop) [%] -Hloop
296 * #ta * #Hstar @(star_ind_l ??????? Hstar)
297 [ #tb whd in ⊢ (%→?); #Hleft
298 #ls #c #c0 #rs #l1 #l3 #l4 #Htb #Hl1nomarks #Hl3l4nomarks #Hlen #l1' #bv
299 #Hl1 #Hl1bits #l4' #bg #Hl4 #Hl4bits
300 cases (Hleft … Htb) -Hleft #Hc #Houtc % %
301 [ generalize in match Hl1bits; -Hl1bits cases l1' in Hl1;
302 [ normalize #Hl1 #c1 destruct (Hl1) %
303 | * #c' #b' #l0 #Heq normalize in Heq; destruct (Heq)
304 #Hl1bits lapply (Hl1bits 〈c',false〉 ?) [ @memb_hd ]
305 >Hc #Hfalse destruct ]
307 | #tb #tc #td whd in ⊢ (%→?→(?→%)→%→?); #Htc #Hstar1 #Hind #Htd
308 lapply (Hind Htd) -Hind #Hind
309 #ls #c #c0 #rs #l1 #l3 #l4 #Htb #Hl1nomarks #Hl3l4nomarks #Hlen #l1' #bv
310 #Hl1 #Hl1bits #l4' #bg #Hl4 #Hl4bits %2
311 cases (Htc … Htb) -Htc #Hcbitnull #Htc
312 % [ % #Hc' >Hc' in Hcbitnull; normalize #Hfalse destruct (Hfalse) ]
313 cut (|l1| = |reverse ? l4|) [@daemon] #Hlen1
314 @(list_cases2 … Hlen1)
315 [ (* case l1 = [] is discriminated because l1 contains at least comma *)
316 #Hl1nil @False_ind >Hl1nil in Hl1; cases l1' normalize
317 [ #Hl1 destruct normalize in Hcbitnull; destruct (Hcbitnull)
318 | #p0 #l0 normalize #Hfalse destruct (Hfalse) cases l0 in e0;
319 [ normalize #Hfalse1 destruct (Hfalse1)
320 | #p0' #l0' normalize #Hfalse1 destruct (Hfalse1) ] ]
321 | (* case c::l1 = c::a::l1'' *)
322 * #a #ba * #a0 #ba0 #l1'' #l4'' #Hl1cons #Hl4cons
323 lapply (eq_f ?? (reverse ?) ?? Hl4cons) >reverse_reverse >reverse_cons -Hl4cons #Hl4cons
325 [ >Hl1cons in Hl1nomarks; #Hl1nomarks lapply (Hl1nomarks 〈a,ba〉 ?)
326 [ @memb_hd | normalize // ] ] #Hba
328 [ >Hl4cons in Hl3l4nomarks; #Hl3l4nomarks lapply (Hl3l4nomarks 〈a0,ba0〉 ?)
329 [ @memb_append_l2 @memb_append_l2 @memb_hd | normalize // ] ] #Hba0
330 >Hba0 in Hl4cons; >Hba in Hl1cons; -Hba0 -Hba #Hl1cons #Hl4cons
331 >Hl4cons in Htc; >Hl1cons #Htc
332 lapply (Htc a (l3@reverse ? l4'') c0 a0 ls (l1''@rs) ? (refl ??) ?)
333 [ #x #Hx @Hl3l4nomarks >Hl4cons <associative_append
335 | >associative_append >associative_append %
337 cut (∃la.l1' = 〈c,false〉::la)
338 [ >Hl1cons in Hl1; cases l1'
339 [normalize #Hfalse destruct (Hfalse)
340 | #p #la normalize #Hla destruct (Hla) @(ex_intro ?? la) % ] ]
342 cut (∃lb.l4' = lb@[〈c0,false〉])
344 @(list_elim_left … l4')
345 [ #Heq lapply (eq_f … (length ?) … Heq)
346 >length_append >length_append
347 >commutative_plus normalize >commutative_plus normalize
350 >(inj_append_singleton_l2 ? (reverse ? l4''@[〈a0,false〉]) (〈grid,bg〉::tl) 〈c0,false〉 a1 Heq)
353 cut (|lb| = |reverse ? la|)
354 [ >Hla in Hl1; >Hlb in Hl4; #Hl4 #Hl1
355 >(?:l1 = la@[〈comma,bv〉]) in Hlen;
356 [|normalize in Hl1; destruct (Hl1) %]
357 >(?:l4 = 〈grid,bg〉::lb)
358 [|@(inj_append_singleton_l1 ?? (〈grid,bg〉::lb) ?? Hl4) ]
359 >length_append >commutative_plus >length_reverse
360 normalize #Hlalb destruct (Hlalb) //
363 (* by hyp on the first iteration step,
364 we consider whether c = bit x or c = null *)
367 lapply (Hind (〈bit x,false〉::ls) a a0 rs l1''
368 (〈bit x,false〉::l3) (reverse ? l4'') ????)
369 [ >Hl1cons in Hlen; >Hl4cons >length_append >commutative_plus
370 normalize #Hlen destruct (Hlen) //
371 | #x0 #Hx0 cases (orb_true_l … Hx0)
372 [ #Hx0eq >(\P Hx0eq) %
373 | -Hx0 #Hx0 @Hl3l4nomarks >Hl4cons
374 <associative_append @memb_append_l1 // ]
375 | #x0 #Hx0 @Hl1nomarks >Hl1cons @memb_cons //
376 | >Htc >associative_append %
378 <Hl1cons <Hl4cons #Hind 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 *)
389 (* cut (l1 = [〈a,false〉])
390 [ cases l1'' in Hl1cons; // #y #ly #Hly
391 >Hly in Hl1; cases l1' in Hl1bits;
392 [ #_ normalize #Hfalse destruct (Hfalse)
393 | #p #lp #Hl1bits normalize #Heq destruct (Heq)
394 @False_ind lapply (Hl1bits 〈a,false〉 ?)
396 [ normalize #Hfalse destruct (Hfalse)
397 | #p0 #lp0 normalize in ⊢ (%→?); #Heq destruct (Heq)
398 @memb_cons @memb_hd ]
399 | >Ha normalize #Hfalse destruct (Hfalse) ]
402 cut (l4 = [〈a0,false〉])
403 [ generalize in match Hl4bits; cases l4' in Hl4;
404 [ >Hl4cons #Hfalse #_
405 lapply (inj_append_singleton_l1 ?? [] ?? Hfalse)
406 cases (reverse ? l4'') normalize
407 [ #Hfalse1 | #p0 #lp0 #Hfalse1 ] destruct (Hfalse1)
410 cases l4'' in Hl4cons; // #y #ly #Hly
411 >Hly in Hl4; cases l4' in Hl4bits;
412 [ #_ >reverse_cons #Hfalse
413 lapply (inj_append_singleton_l1 ?? [] ?? Hfalse)
414 -Hfalse cases ly normalize
415 [ #Hfalse | #p #Hp #Hfalse ] destruct (Hfalse)
417 | #p #lp #Hl1bits normalize #Heq destruct (Heq)
418 @False_ind lapply (Hl1bits 〈a,false〉 ?)
420 [ normalize #Hfalse destruct (Hfalse)
421 | #p0 #lp0 normalize in ⊢ (%→?); #Heq destruct (Heq)
422 @memb_cons @memb_hd ]
423 | >Ha normalize #Hfalse destruct (Hfalse) ]
427 >Hla normalize #Hl1 destruct (Hl1) lapply (inj_append_ @False_ind
429 cut (l1'' = [] ∧ l4'' = [])
430 [ % [ >Hla in Hl1; normalize #Hl1 destruct (Hl1)
432 cases l1'' in Hl1bits;
435 cut (la = [] ∧ lb = [] ∧ l1'' = [] ∧ l4'' = [])
436 [ @daemon ] * * * #Hla1 #Hlb1 #Hl1nil #Hl4nil
437 >Hl1cons in Hl1; >Hla
439 >Hl4cons in Hl4; >Hlb #Hl4
440 >Hla1 >Hlb1 >Hl1nil >Hl4nil >Hx
441 cut (a0 = grid) [ @daemon ] #Ha0 <Ha <Ha0
442 normalize in ⊢ (??(??%?%)(??%?%)); >associative_append %
443 | * #Ha #Houtc1 >Houtc1 @eq_f3 //
444 >Hla >reverse_cons >associative_append @eq_f
445 >Hx whd in ⊢ (??%?); @eq_f whd in ⊢ (???%); @eq_f @eq_f
446 >Hlb >append_cons @eq_f2 // >(merge_config_append … Hlen2) %
451 lapply (Hind (〈c0,false〉::ls) a a0 rs l1'' (〈null,false〉::l3) (reverse ? l4'') ????)
452 [ >Hl1cons in Hlen; >Hl4cons >length_append >commutative_plus normalize
453 #Hlen destruct (Hlen) @e0
454 | #x0 #Hx0 cases (memb_append STape ? [〈null,false〉] (l3@reverse ? l4'') … Hx0) -Hx0 #Hx0
455 [ >(memb_single … Hx0) %
456 | @Hl3l4nomarks cases (memb_append … Hx0) -Hx0 #Hx0
458 | @memb_append_l2 >Hl4cons @memb_append_l1 // ]
460 | >Hl1cons #x' #Hx0 @Hl1nomarks >Hl1cons @memb_cons //
461 | >Htc @eq_f3 // >associative_append % ] -Hind <Hl1cons <Hl4cons #Hind
462 lapply (Hind la bv ?? lb bg ??)
463 [ #x0 #Hx0 @Hl4bits >Hlb @memb_append_l1 //
464 | >Hlb in Hl4; normalize in ⊢ (%→?); #Hl4
465 @(inj_append_singleton_l1 ? l4 (〈grid,bg〉::lb) … Hl4)
466 | #x0 #Hx0 @Hl1bits >Hla @memb_cons //
467 | >Hla in Hl1; normalize in ⊢ (%→?); #Hl1
468 destruct (Hl1) // ] -Hind *
469 (* by IH, we proceed by cases, whether a = comma
470 (consequently several lists = []) or not *)
471 [ * #Ha #Houtc1 >Hl1cons in Hl1; >Hla
473 >Hl4cons in Hl4; >Hlb #Hl4
474 cut (la = [] ∧ lb = [] ∧ l1'' = [] ∧ l4'' = [])
475 [@daemon] * * * #Hla1 #Hlb1 #Hl1nil #Hl4nil
476 >Hla1 >Hlb1 >Hl1nil >Hl4nil >Hc
477 cut (a0 = grid) [ @daemon ] #Ha0 <Ha <Ha0
478 normalize in ⊢ (??(??%?%)(??%?%)); >associative_append %
479 | * #Ha #Houtc1 >Houtc1 @eq_f3 //
480 >Hla >reverse_cons >associative_append @eq_f
481 >Hc whd in ⊢ (??%?); @eq_f whd in ⊢ (???%); @eq_f @eq_f
482 >Hlb >append_cons @eq_f2 // >(merge_config_append … Hlen2) %
488 definition merge_char ≝ λc1,c2.
495 merge_config (〈c1,false〉::conf1) (〈c2,false〉::conf2) =
496 〈merge_char c1 c2,false〉::merge_config conf1 conf2.
497 #c1 #c2 #conf1 #conf2 normalize @eq_f2 //
501 lemma merge_bits : ∀l1,l2.|l1| = |l2| → only_bits l2 → merge_config l1 l2 = l2.
502 #l1 #l2 #Hlen @(list_ind2 … Hlen) //
503 #tl1 #tl2 #hd1 #hd2 #IH
504 >(eq_pair_fst_snd … hd1) >(eq_pair_fst_snd … hd2) #Hbits
505 change with (cons ???) in ⊢ (??%?); @eq_f2
506 [ cases (\fst hd2) in Hbits;
508 |*: #Hfalse lapply (Hfalse … (memb_hd …)) normalize #Hfalse1 destruct (Hfalse1) ]
509 | @IH #x #Hx @Hbits @memb_cons // ]
512 lemma merge_config_c_nil :
513 ∀c.merge_config c [] = [].
514 #c cases c normalize //
517 lemma reverse_merge_config :
518 ∀c1,c2.|c1| = |c2| → reverse ? (merge_config c1 c2) =
519 merge_config (reverse ? c1) (reverse ? c2).
520 #c1 #c2 <(length_reverse ? c1) <(length_reverse ? c2) #Hlen
521 <(reverse_reverse ? c1) in ⊢ (??%?); <(reverse_reverse ? c2) in ⊢ (??%?);
522 generalize in match Hlen; @(list_ind2 … Hlen) -Hlen //
523 #tl1 #tl2 #hd1 #hd2 #IH whd in ⊢ (??%%→?); #Hlen destruct (Hlen) -Hlen
524 <(length_reverse ? tl1) in e0; <(length_reverse ? tl2) #Hlen
525 >reverse_cons >reverse_cons >(merge_config_append ???? Hlen)
526 >reverse_append >(eq_pair_fst_snd ?? hd1) >(eq_pair_fst_snd ?? hd2)
527 whd in ⊢ (??%%); @eq_f2 // @IH //
532 seq STape copy0 (seq ? (move_l …) (seq ? (adv_to_mark_l … (is_marked ?))
533 (seq ? (clear_mark …) (seq ? (adv_to_mark_r … (is_marked ?)) (clear_mark …))))).
536 s0, s1 = caratteri di testa dello stato
537 c0 = carattere corrente del nastro oggetto
538 c1 = carattere in scrittura sul nastro oggetto
540 questa dimostrazione sfrutta il fatto che
541 merge_config (l0@[c0]) (l1@[c1]) = l1@[merge_char c0 c1]
542 se l0 e l1 non contengono null
545 definition R_copy ≝ λt1,t2.
546 ∀ls,s0,s1,c0,c1,rs,l1,l3,l4.
547 t1 = midtape STape (l3@〈grid,false〉::〈c0,false〉::l4@〈s0,true〉::ls) 〈s1,true〉 (l1@〈c1,false〉::〈comma,false〉::rs) →
548 no_marks l1 → no_marks l3 → no_marks l4 → |l1| = |l4| →
549 only_bits (l4@[〈s0,false〉]) → only_bits (〈s1,false〉::l1) →
550 bit_or_null c0 = true → bit_or_null c1 = true →
551 t2 = midtape STape (〈c1,false〉::reverse ? l1@〈s1,false〉::l3@〈grid,false〉::
552 〈merge_char c0 c1,false〉::reverse ? l1@〈s1,false〉::ls)
555 axiom sem_copy0 : Realize ? copy0 R_copy0.
557 definition option_cons ≝ λA.λa:option A.λl.
562 lemma sem_copy : Realize ? copy R_copy.
564 cases (sem_seq … (sem_copy0 …)
565 (sem_seq … (sem_move_l …)
566 (sem_seq … (sem_adv_to_mark_l … (is_marked ?))
567 (sem_seq … (sem_clear_mark …)
568 (sem_seq … (sem_adv_to_mark_r … (is_marked ?)) (sem_clear_mark …))))) intape)
569 #k * #outc * #Hloop #HR %{k} %{outc} % [@Hloop] -Hloop
570 #ls #s0 #s1 #c0 #c1 #rs #l1 #l2 #l3 #Hintape #Hl1marks #Hl2marks #Hl3marks #Hlen
571 #Hbits1 #Hbits2 #Hc0bits #Hc1bits
572 cases HR -HR #ta * whd in ⊢ (%→?); #Hta
573 cut (ta = midtape STape (〈c1,false〉::reverse ? l1@〈s1,false〉::l2@〈grid,true〉::
574 〈merge_char c0 c1,false〉::reverse ? l1@〈s1,false〉::ls)
576 [lapply (Hta ls s1 s0 rs (l1@[〈c1,false〉;〈comma,false〉]) l2 (〈grid,false〉::〈c0,false〉::l3) ?)
577 [>associative_append in ⊢ (???(????%)); normalize in ⊢ (???(??%%%)); @Hintape ]
578 -Hta #Hta cases (Hta ??? (〈s1,false〉::l1@[〈c1,false〉]) false ? ? ?? (refl ??) ?)
579 [3: #x #Hx cases (memb_append … Hx) -Hx #Hx
581 | cases (orb_true_l … Hx) -Hx #Hx [ >(\P Hx) % | >(memb_single … Hx) % ]]
582 |4: #x #Hx cases (memb_append … Hx) -Hx #Hx
584 | cases (orb_true_l … Hx) -Hx #Hx [ >(\P Hx) % | cases (orb_true_l … Hx) [-Hx #Hx >(\P Hx) % | @Hl3marks ] ] ]
585 |5: >length_append normalize >Hlen >commutative_plus %
586 |6: normalize >associative_append %
587 |7: #x #Hx cases (memb_append ?? (〈s1,false〉::l1) … Hx) -Hx #Hx
588 [ whd in ⊢ (??%?); >(Hbits2 … Hx) %
589 | >(memb_single … Hx) // ]
590 |8: #x #Hx cases (memb_append … Hx) -Hx #Hx
591 [ cases (orb_true_l … Hx) -Hx #Hx [ >(\P Hx) // | whd in ⊢ (??%?); >Hbits1 // @memb_append_l1 // ]
592 | >(memb_single … Hx) whd in ⊢ (??%?); >(Hbits1 〈s0,false〉) // @memb_append_l2 @memb_hd ]
593 | * #Hs1 @False_ind >Hs1 in Hbits2; #Hbits2 lapply (Hbits2 〈comma,false〉 ?) //
594 normalize #Hfalse destruct (Hfalse)
595 | * #Hs1 #Ht2 >Ht2 >reverse_cons >reverse_append >reverse_single @eq_f3 //
596 >merge_cons >merge_bits
597 [2: #x #Hx @Hbits2 cases (memb_append STape ? (reverse ? l1) ? Hx) -Hx #Hx
598 [@daemon | >(memb_single … Hx) @memb_hd ]
599 |3: >length_append >length_append >length_reverse >Hlen % ]
600 normalize >associative_append normalize >associative_append %
602 ] -Hta #Hta * #tb * whd in ⊢ (%→?); #Htb
603 lapply (Htb … Hta) -Htb #Htb change with (midtape ????) in Htb:(???%);
604 * #tc * whd in ⊢ (%→?); #Htc
606 [ * #Hfalse normalize in Hfalse; destruct (Hfalse) ]
608 lapply (Htc (reverse ? l1@〈s1,false〉::l2) 〈grid,true〉
609 (〈merge_char c0 c1,false〉::reverse ? l1@〈s1,false〉::ls)???)
610 [ #x #Hx cases (memb_append … Hx) -Hx #Hx
612 | cases (orb_true_l … Hx) -Hx #Hx
613 [ >(\P Hx) % | @(Hl2marks … Hx) ] ]
615 | whd in ⊢ (??%?); >associative_append % ] -Htc #Htc
616 * #td * whd in ⊢ (%→?); #Htd lapply (Htd … Htc) -Htd #Htd
617 * #te * whd in ⊢ (%→?); #Hte cases (Hte … Htd) -Hte -Htd
618 [ * #Hfalse normalize in Hfalse; destruct (Hfalse) ]
620 lapply (Hte (reverse ? (reverse ? l1@〈s1,false〉::l2)@[〈c1,false〉])
621 〈comma,true〉 rs ? (refl ??) ?) -Hte
622 [ >reverse_append >reverse_cons >reverse_reverse #x #Hx
623 cases (memb_append … Hx) -Hx #Hx
624 [ cases (memb_append … Hx) -Hx #Hx
625 [ cases (memb_append … Hx) -Hx #Hx
627 | lapply (memb_single … Hx) -Hx #Hx >Hx % ]
629 | lapply (memb_single … Hx) -Hx #Hx >Hx % ]
630 | >reverse_append >reverse_reverse >reverse_cons
631 >associative_append >associative_append >associative_append
632 >associative_append >associative_append % ]
633 #Hte whd in ⊢ (%→?); #Houtc lapply (Houtc … Hte) -Houtc #Houtc >Houtc
635 >reverse_append >reverse_append >reverse_single >reverse_cons
636 >reverse_append >reverse_append >reverse_reverse >reverse_reverse
637 >reverse_single >associative_append >associative_append %