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/copy.ma".
23 if is_true(current) (* current state is final *)
35 if is_marked(current) = false (* match ok *)
48 case bit false: move_tape_l
49 case bit true: move_tape_r
50 case null: adv_to_grid_l; move_l; adv_to_grid_l;
57 definition init_match ≝
59 (seq ? (adv_to_mark_r ? (λc:STape.is_grid (\fst c)))
63 (adv_to_mark_l ? (is_marked ?)))))).
65 definition R_init_match ≝ λt1,t2.
66 ∀ls,l,rs,c,d. no_grids (〈c,false〉::l) → no_marks l →
67 t1 = midtape STape ls 〈c,false〉 (l@〈grid,false〉::〈d,false〉::rs) →
68 t2 = midtape STape ls 〈c,true〉 (l@〈grid,false〉::〈d,true〉::rs).
70 lemma sem_init_match : Realize ? init_match R_init_match.
72 cases (sem_seq ????? (sem_mark ?)
73 (sem_seq ????? (sem_adv_to_mark_r ? (λc:STape.is_grid (\fst c)))
74 (sem_seq ????? (sem_move_r ?)
75 (sem_seq ????? (sem_mark ?)
76 (sem_seq ????? (sem_move_l ?)
77 (sem_adv_to_mark_l ? (is_marked ?)))))) intape)
78 #k * #outc * #Hloop #HR
79 @(ex_intro ?? k) @(ex_intro ?? outc) % [@Hloop] -Hloop
80 #ls #l #rs #c #d #Hnogrids #Hnomarks #Hintape
82 #ta * whd in ⊢ (%→?); #Hta lapply (Hta … Hintape) -Hta -Hintape #Hta
83 * #tb * whd in ⊢ (%→?); #Htb cases (Htb … Hta) -Htb -Hta
84 [* #Hgridc @False_ind @(absurd … Hgridc) @eqnot_to_noteq
85 @(Hnogrids 〈c,false〉) @memb_hd ]
86 * #Hgrdic #Htb lapply (Htb l 〈grid,false〉 (〈d,false〉::rs) (refl …) (refl …) ?)
87 [#x #membl @Hnogrids @memb_cons @membl] -Htb #Htb
88 * #tc * whd in ⊢ (%→?); #Htc lapply (Htc … Htb) -Htc -Htb #Htc
89 * #td * whd in ⊢ (%→?); #Htd lapply (Htd … Htc) -Htd -Htc #Htd
90 * #te * whd in ⊢ (%→?); #Hte lapply (Hte … Htd) -Hte -Htd #Hte
91 whd in ⊢ (%→?); #Htf cases (Htf … Hte) -Htf -Hte
92 [* whd in ⊢ ((??%?)→?); #Habs destruct (Habs)]
93 * #_ #Htf lapply (Htf (reverse ? l) 〈c,true〉 ls (refl …) (refl …) ?)
94 [#x #membl @Hnomarks @daemon] -Htf #Htf >Htf >reverse_reverse %
100 init_current_on_match; (* no marks in current *)
107 definition init_copy ≝
108 seq ? init_current_on_match
110 (seq ? (adv_to_mark_r ? (is_marked ?))
113 definition R_init_copy ≝ λt1,t2.
115 no_marks l1 → no_grids l1 →
116 no_marks l2 → is_grid c = false →
117 t1 = midtape STape (l1@〈c,false〉::〈grid,false〉::ls) 〈grid,false〉 (l2@〈comma,true〉::〈d,false〉::rs) →
118 t2 = midtape STape (〈comma,false〉::(reverse ? l2)@〈grid,false〉::l1@〈c,true〉::〈grid,false〉::ls) 〈d,true〉 rs.
120 lemma list_last: ∀A.∀l:list A.
121 l = [ ] ∨ ∃a,l1. l = l1@[a].
122 #A #l <(reverse_reverse ? l) cases (reverse A l)
124 |#a #l1 %2 @(ex_intro ?? a) @(ex_intro ?? (reverse ? l1)) //
128 lemma sem_init_copy : Realize ? init_copy R_init_copy.
130 cases (sem_seq ????? sem_init_current_on_match
131 (sem_seq ????? (sem_move_r ?)
132 (sem_seq ????? (sem_adv_to_mark_r ? (is_marked ?))
133 (sem_adv_mark_r ?))) intape)
134 #k * #outc * #Hloop #HR
135 @(ex_intro ?? k) @(ex_intro ?? outc) % [@Hloop] -Hloop
136 #l1 #l2 #c #ls #d #rs #Hl1marks #Hl1grids #Hl2marks #Hc #Hintape
138 #ta * whd in ⊢ (%→?); #Hta lapply (Hta … Hl1grids Hc Hintape) -Hta -Hintape #Hta
139 * #tb * whd in ⊢ (%→?); #Htb lapply (Htb … Hta) -Htb -Hta
140 generalize in match Hl1marks; -Hl1marks cases (list_last ? l1)
141 [#eql1 >eql1 #Hl1marks whd in ⊢ ((???%)→?); whd in ⊢ ((???(????%))→?); #Htb
142 * #tc * whd in ⊢ (%→?); #Htc lapply (Htc … Htb) -Htc -Htb *
143 [* whd in ⊢ ((??%?)→?); #Htemp destruct (Htemp)]
144 * #_ #Htc lapply (Htc … (refl …) (refl …) ?)
145 [#x #membx @Hl2marks @membx]
146 #Htc whd in ⊢ (%→?); #Houtc lapply (Houtc … Htc) -Houtc -Htc #Houtc
148 |* #c1 * #tl #eql1 >eql1 #Hl1marks >reverse_append >reverse_single
149 whd in ⊢ ((???%)→?); whd in ⊢ ((???(????%))→?);
150 >associative_append whd in ⊢ ((???(????%))→?); #Htb
151 * #tc * whd in ⊢ (%→?); #Htc lapply (Htc … Htb) -Htc -Htb *
152 [* >Hl1marks [#Htemp destruct (Htemp)] @memb_append_l2 @memb_hd]
153 * #_ >append_cons <associative_append #Htc lapply (Htc … (refl …) (refl …) ?)
154 [#x #membx cases (memb_append … membx) -membx #membx
155 [cases (memb_append … membx) -membx #membx
156 [@Hl1marks @memb_append_l1 @daemon
157 |>(memb_single … membx) %
161 #Htc whd in ⊢ (%→?); #Houtc lapply (Houtc … Htc) -Houtc -Htc #Houtc
162 >Houtc >reverse_append >reverse_append >reverse_single
163 >reverse_reverse >associative_append >associative_append
164 >associative_append %
168 definition init_copy ≝
170 (seq ? init_current_on_match
172 (adv_to_mark_r ? (is_marked ?)))).
174 definition R_init_copy ≝ λt1,t2.
176 no_marks l1 → no_grids l1 →
177 no_marks l2 → no_grids l2 → is_grid c = false → is_grid d =false →
178 t1 = midtape STape (l1@〈grid,false〉::l2@〈c,false〉::〈grid,false〉::l3) 〈comma,true〉 (〈d,false〉::rs) →
179 t2 = midtape STape (〈comma,false〉::l1@〈grid,false〉::l2@〈c,true〉::〈grid,false〉::l3) 〈d,true〉 rs.
181 lemma list_last: ∀A.∀l:list A.
182 l = [ ] ∨ ∃a,l1. l = l1@[a].
183 #A #l <(reverse_reverse ? l) cases (reverse A l)
185 |#a #l1 %2 @(ex_intro ?? a) @(ex_intro ?? (reverse ? l1)) //
189 lemma sem_init_copy : Realize ? init_copy R_init_copy.
191 cases (sem_seq ????? (sem_adv_mark_r ?)
192 (sem_seq ????? sem_init_current_on_match
193 (sem_seq ????? (sem_move_r ?)
194 (sem_adv_to_mark_r ? (is_marked ?)))) intape)
195 #k * #outc * #Hloop #HR
196 @(ex_intro ?? k) @(ex_intro ?? outc) % [@Hloop] -Hloop
197 #l1 #l2 #c #l3 #d #rs #Hl1marks #Hl1grids #Hl2marks #Hl2grids #Hc #Hd #Hintape
199 #ta * whd in ⊢ (%→?); #Hta lapply (Hta … Hintape) -Hta -Hintape #Hta
200 * #tb * whd in ⊢ (%→?);
201 >append_cons #Htb lapply (Htb (〈comma,false〉::l1) l2 c … Hta)
203 |#x #membx cases (orb_true_l … membx) -membx #membx
204 [>(\P membx) // | @Hl1grids @membx]
206 * #tc * whd in ⊢ (%→?); #Htc lapply (Htc … Htb) -Htc -Htb
207 >reverse_append >reverse_cons cases (list_last ? l2)
208 [#Hl2 >Hl2 >associative_append whd in ⊢ ((???(??%%%))→?); #Htc
209 whd in ⊢ (%→?); #Htd cases (Htd … Htc) -Htd -Htc
210 [* whd in ⊢ ((??%?)→?); #Habs destruct (Habs)]
211 * #_ #Htf lapply (Htf … (refl …) (refl …) ?)
212 [#x >reverse_cons #membx cases (memb_append … membx) -membx #membx
213 [@Hl1marks @daemon |>(memb_single … membx) //]
215 |#Htf >Htf >reverse_reverse >associative_append %
217 |* #a * #l21 #Heq >Heq >reverse_append >reverse_single
218 >associative_append >associative_append >associative_append whd in ⊢ ((???(??%%%))→?); #Htc
219 whd in ⊢ (%→?); #Htd cases (Htd … Htc) -Htd -Htc
220 [* >Hl2marks [#Habs destruct (Habs) |>Heq @memb_append_l2 @memb_hd]]
221 * #_ <associative_append <associative_append #Htf lapply (Htf … (refl …) (refl …) ?)
222 [#x >reverse_cons #membx cases (memb_append … membx) -membx #membx
223 [cases (memb_append … membx) -membx #membx
224 [@Hl2marks >Heq @memb_append_l1 @daemon
225 |>(memb_single … membx) //]
226 |cases (memb_append … membx) -membx #membx
227 [@Hl1marks @daemon |>(memb_single … membx) //]
229 | #Htf >Htf >reverse_append >reverse_reverse
230 >reverse_append >reverse_reverse >associative_append
231 >reverse_single >associative_append >associative_append
232 >associative_append %
237 include "turing/universal/move_tape.ma".
239 definition exec_move ≝
242 (seq ? (move_r …) move_tape)).
244 definition map_move ≝
245 λc,mv.match c with [ null ⇒ None ? | _ ⇒ Some ? 〈c,false,move_of_unialpha mv〉 ].
247 definition current_of_alpha ≝ λc:STape.
248 match \fst c with [ null ⇒ None ? | _ ⇒ Some ? c ].
250 definition legal_tape ≝ λls,c,rs.
251 let t ≝ mk_tape STape ls (current_of_alpha c) rs in
252 left ? t = ls ∧ right ? t = rs ∧ current ? t = current_of_alpha c.
254 lemma legal_tape_cases :
255 ∀ls,c,rs.legal_tape ls c rs →
256 \fst c ≠ null ∨ (\fst c = null ∧ ls = []) ∨ (\fst c = null ∧ rs = []).
257 #ls #c #rs cases c #c0 #bc0 cases c0
258 [ #c1 normalize #_ % % % #Hfalse destruct (Hfalse)
263 | #r0 #rs0 normalize * * #Hls #Hrs destruct (Hrs) ]
265 |*: #_ % % % #Hfalse destruct (Hfalse) ]
268 definition R_exec_move ≝ λt1,t2.
269 ∀n,curconfig,ls,rs,c0,c1,s0,s1,table1,newconfig,mv,table2.
270 table_TM n (table1@〈comma,false〉::〈s1,false〉::newconfig@〈c1,false〉::〈comma,false〉::〈mv,false〉::table2) →
271 no_marks curconfig → only_bits (curconfig@[〈s0,false〉]) → only_bits (〈s1,false〉::newconfig) →
272 no_nulls ls → no_nulls rs → no_marks ls → no_marks rs →
273 legal_tape ls 〈c0,false〉 rs →
274 t1 = midtape STape (〈c0,false〉::curconfig@〈s0,false〉::〈grid,false〉::ls) 〈grid,false〉
275 (table1@〈comma,true〉::〈s1,false〉::newconfig@〈c1,false〉::〈comma,false〉::〈mv,false〉::table2@〈grid,false〉::rs) →
276 ∀t1'.t1' = lift_tape ls 〈c0,false〉 rs →
278 t2 = midtape STape ls1 〈grid,false〉
279 (〈s1,false〉::newconfig@〈c2,false〉::〈grid,false〉::
280 table1@〈comma,false〉::〈s1,false〉::newconfig@〈c1,false〉::〈comma,false〉::〈mv,false〉::table2@〈grid,false〉::rs1) ∧
281 lift_tape ls1 〈c2,false〉 rs1 =
282 tape_move STape t1' (map_move c1 mv).
284 (* move the following 2 lemmata to mono.ma *)
285 lemma tape_move_left_eq :
287 tape_move ? t (Some ? 〈c,L〉) =
288 tape_move_left ? (left ? t) c (right ? t).
292 lemma tape_move_right_eq :
294 tape_move ? t (Some ? 〈c,R〉) =
295 tape_move_right ? (left ? t) c (right ? t).
299 lemma sem_exec_move : Realize ? exec_move R_exec_move.
301 cases (sem_seq … sem_init_copy
303 (sem_seq … (sem_move_r …) sem_move_tape )) intape)
304 #k * #outc * #Hloop #HR
305 @(ex_intro ?? k) @(ex_intro ?? outc) % [ @Hloop ] -Hloop
306 #n #curconfig #ls #rs #c0 #c1 #s0 #s1 #table1 #newconfig #mv #table2
307 #Htable #Hcurconfig1 #Hcurconfig2 #Hnewconfig #Hls #Hrs #Hls1 #Hrs1 #Htape #Hintape #t1' #Ht1'
308 cases HR -HR #ta * whd in ⊢ (%→?); #Hta
309 lapply (Hta (〈c0,false〉::curconfig) table1 s0 ls s1
310 (newconfig@〈c1,false〉::〈comma,false〉::〈mv,false〉::table2@〈grid,false〉::rs) … Hintape) -Hta
311 [ (*Hcurconfig2*) @daemon
315 | #Hta * #tb * whd in ⊢ (%→?); #Htb
316 lapply (Htb (〈grid,false〉::ls) s0 s1 c0 c1 (〈mv,false〉::table2@〈grid,false〉::rs) newconfig (〈comma,false〉::reverse ? table1) curconfig Hta ????????) -Htb
317 [9:|*:(* rivedere le precondizioni *) @daemon ]
318 #Htb * #tc * whd in ⊢ (%→?); #Htc lapply (Htc … Htb) -Htc whd in ⊢(???(??%%%)→?);#Htc
319 whd in ⊢ (%→?); #Houtc whd in Htc:(???%); whd in Htc:(???(??%%%));
321 (〈comma,false〉::〈c1,false〉::reverse ? newconfig@〈s1,false〉::〈comma,false〉::reverse ? table1)
322 mv table2 (merge_char c0 c1) (reverse ? newconfig@[〈s1,false〉]) ls ?????
323 Hls Hrs Hls1 Hrs1 ??)
324 [3: >Htc @(eq_f3 … (midtape ?))
325 [ @eq_f @eq_f >associative_append >associative_append %
329 || >reverse_cons >reverse_cons >reverse_append >reverse_reverse
330 >reverse_cons >reverse_cons >reverse_reverse
331 >associative_append >associative_append >associative_append
332 >associative_append >associative_append
334 | (* only bits or nulls c1,c2 → only bits or nulls (merge c1 c2) *) @daemon
335 | (* add to hyps? *) @daemon
336 | (* bit_or_null c1,c2 → bit_or_null (merge_char c1 c2) *) @daemon
337 | -Houtc * #ls1 * #rs1 * #newc * #Houtc *
340 @(ex_intro ?? ls1) @(ex_intro ?? rs1) @(ex_intro ?? newc)
342 [ >Houtc -Houtc >reverse_append
343 >reverse_reverse >reverse_single @eq_f
344 >reverse_cons >reverse_cons >reverse_append >reverse_cons
345 >reverse_cons >reverse_reverse >reverse_reverse
346 >associative_append >associative_append
347 >associative_append >associative_append
348 >associative_append >associative_append %
349 | >Hmv >Ht1' >Htapemove
350 (* mv = bit false -→ c1 = bit ?
351 whd in Htapemove:(???%); whd in ⊢ (???%);
352 whd in match (lift_tape ???) in ⊢ (???%);
354 cut (∃c1'.c1 = bit c1') [ @daemon ] * #c1' #Hc1
355 >Hc1 >tape_move_left_eq cases c0
365 ls #... null # ... # rs
368 ∃t.left ? t = ls, right ? t = rs, current t = [[ c ]]
376 whd in match (merge_char ??);
377 whd in match (map_move ??);
385 >append_cons in Hintape; >associative_append
388 cases (Hta … Hintape) -Hta
389 [ * normalize in ⊢ (%→?); #Hfalse destruct (Hfalse) ]
390 * #_ #Hta lapply (Hta ? 〈comma,true〉 … (refl ??) (refl ??) ?)
391 [ @daemon ] -Hta #Hta
392 * #tb * whd in ⊢ (%→?); #Htb lapply (Htb … Hta)
394 definition move_of_unialpha ≝
396 [ bit x ⇒ match x with [ true ⇒ R | false ⇒ L ]
399 definition R_uni_step ≝ λt1,t2.
400 ∀n,table,c,c1,ls,rs,curs,curc,news,newc,mv.
402 match_in_table (〈c,false〉::curs@[〈curc,false〉])
403 (〈c1,false〉::news@[〈newc,false〉]) mv table →
404 t1 = midtape STape (〈grid,false〉::ls) 〈c,false〉
405 (curs@〈curc,false〉::〈grid,false〉::table@〈grid,false〉::rs) →
406 ∀t1',ls1,rs1.t1' = lift_tape ls 〈curc,false〉 rs →
407 (t2 = midtape STape (〈grid,false〉::ls1) 〈c1,false〉
408 (news@〈newc,false〉::〈grid,false〉::table@〈grid,false〉::rs1) ∧
409 lift_tape ls1 〈newc,false〉 rs1 =
410 tape_move STape t1' (Some ? 〈〈newc,false〉,move_of_unialpha mv〉)).
412 definition no_nulls ≝
413 λl:list STape.∀x.memb ? x l = true → is_null (\fst x) = false.