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_____________________________________________________________*)
12 include "turing/multi_universal/moves_2.ma".
13 include "turing/multi_universal/match.ma".
14 include "turing/multi_universal/copy.ma".
15 include "turing/multi_universal/alphabet.ma".
16 include "turing/multi_universal/tuples.ma".
29 current (in.obj) = None
40 (if (current(in.obj)) == None
55 definition obj ≝ (0:DeqNat).
56 definition cfg ≝ (1:DeqNat).
57 definition prg ≝ (2:DeqNat).
59 definition obj_to_cfg ≝
60 (ifTM ?? (inject_TM ? (test_null ?) 2 obj)
61 (copy_char_N obj cfg FSUnialpha 2)
62 (inject_TM ? (write FSUnialpha null) 2 cfg)
64 inject_TM ? (move_to_end FSUnialpha L) 2 cfg ·
65 mmove cfg FSUnialpha 2 R.
67 definition R_obj_to_cfg ≝ λt1,t2:Vector (tape FSUnialpha) 3.
69 nth cfg ? t1 (niltape ?) = midtape ? ls c [ ] →
70 (∀lso,x,rso.nth obj ? t1 (niltape ?) = midtape FSUnialpha lso x rso →
72 (mk_tape ? [ ] (option_hd ? (reverse ? (x::ls))) (tail ? (reverse ? (x::ls)))) cfg) ∧
73 (current ? (nth obj ? t1 (niltape ?)) = None ? →
75 (mk_tape ? [ ] (option_hd FSUnialpha (reverse ? (null::ls)))
76 (tail ? (reverse ? (null::ls)))) cfg).
78 axiom accRealize_to_Realize :
79 ∀sig,n.∀M:mTM sig n.∀Rtrue,Rfalse,acc.
80 M ⊨ [ acc: Rtrue, Rfalse ] → M ⊨ Rtrue ∪ Rfalse.
82 lemma eq_mk_tape_rightof :
83 ∀alpha,a,al.mk_tape alpha (a::al) (None ?) [ ] = rightof ? a al.
87 lemma tape_move_mk_tape_R :
89 (c = None ? → ls = [ ] ∨ rs = [ ]) →
90 tape_move ? (mk_tape sig ls c rs) R =
91 mk_tape ? (option_cons ? c ls) (option_hd ? rs) (tail ? rs).
92 #sig * [ * [ * | #c * ] | #l0 #ls0 * [ *
93 [| #r0 #rs0 #H @False_ind cases (H (refl ??)) #H1 destruct (H1) ] | #c * ] ]
97 lemma eq_vec_change_vec : ∀sig,n.∀v1,v2:Vector sig n.∀i,t,d.
99 (∀j.i ≠ j → nth j ? v1 d = nth j ? v2 d) →
100 v2 = change_vec ?? v1 t i.
101 #sig #n #v1 #v2 #i #t #d #H1 #H2 @(eq_vec … d)
102 #i0 #Hlt cases (decidable_eq_nat i0 i) #Hii0
103 [ >Hii0 >nth_change_vec //
104 | >nth_change_vec_neq [|@sym_not_eq //] @sym_eq @H2 @sym_not_eq // ]
107 lemma not_None_to_Some: ∀A.∀a. a ≠ None A → ∃b. a = Some ? b.
108 #A * /2/ * #H @False_ind @H %
111 lemma sem_obj_to_cfg : obj_to_cfg ⊨ R_obj_to_cfg.
112 @(sem_seq_app FSUnialpha 2 ?????
114 (sem_test_null_multi ?? obj ?)
116 (sem_inject ???? cfg ? (sem_write FSUnialpha null)))
117 (sem_seq ?????? (sem_inject ???? cfg ? (sem_move_to_end_l ?))
118 (sem_move_multi ? 2 cfg R ?))) //
120 #tb * #Hif * #tc * #HM2 #HM3 #c #ls #Hcfg
123 [ * #t1 * * #Hcurta #Ht1 destruct (Ht1)
124 lapply (not_None_to_Some … Hcurta) * #curta #Hcurtaeq
125 whd in ⊢ (%→?); #Htb % [2: #Hcur @False_ind /2/]
126 #lso #xo #rso #Hobjta cases HM2 whd in ⊢ (%→?); * #_
127 #HM2 #Heq >Htb in HM2; >nth_change_vec [2: @leb_true_to_le %]
128 >Hcfg >Hcurtaeq #HM2 lapply (HM2 … (refl ??)) -HM2
129 whd in match (left ??); whd in match (right ??);
130 >reverse_cons #Htc >HM3 @(eq_vec … (niltape ?)) #i #lei2
131 cases (le_to_or_lt_eq … (le_S_S_to_le …lei2))
132 [#lei1 cases (le_to_or_lt_eq … (le_S_S_to_le …lei1))
133 [#lei0 lapply(le_n_O_to_eq … (le_S_S_to_le …lei0)) #eqi <eqi
134 >nth_change_vec_neq [2:@eqb_false_to_not_eq %]
135 <(Heq 0) [2:@eqb_false_to_not_eq %] >Htb
136 >nth_change_vec_neq [2:@eqb_false_to_not_eq %]
137 >nth_change_vec_neq [%|2:@eqb_false_to_not_eq %]
138 |#Hi >Hi >nth_change_vec // >nth_change_vec // >Htc
139 >Hobjta in Hcurtaeq; whd in ⊢ (??%?→?); #Htmp destruct(Htmp)
140 >tape_move_mk_tape_R [2: #_ %1 %] %
142 |#Hi >Hi >nth_change_vec_neq [2:@eqb_false_to_not_eq %]
143 >nth_change_vec_neq [2:@eqb_false_to_not_eq %]
144 <(Heq 2) [2:@eqb_false_to_not_eq %] >Htb
145 >nth_change_vec_neq [%|2:@eqb_false_to_not_eq %]
147 | * #t1 * * #Hcurta #Ht1 destruct (Ht1)
148 * whd in ⊢ (%→?); #Htb lapply (Htb … Hcfg) -Htb #Htb
150 [#lso #xo #rso #Hmid @False_ind >Hmid in Hcurta;
151 whd in ⊢ (??%?→?); #Htmp destruct (Htmp)]
152 #_ cases HM2 whd in ⊢ (%→?); * #_
153 #HM2 #Heq >Htb in HM2; #HM2 lapply (HM2 … (refl ??)) -HM2
154 #Htc >HM3 @(eq_vec … (niltape ?)) #i #lei2
155 cases (le_to_or_lt_eq … (le_S_S_to_le …lei2))
156 [#lei1 cases (le_to_or_lt_eq … (le_S_S_to_le …lei1))
157 [#lei0 lapply(le_n_O_to_eq … (le_S_S_to_le …lei0)) #eqi <eqi
158 >nth_change_vec_neq [2:@eqb_false_to_not_eq %]
159 >nth_change_vec_neq [2:@eqb_false_to_not_eq %]
160 <(Heq 0) [2:@eqb_false_to_not_eq %] >Htb
161 <(Htbeq 0) [2:@eqb_false_to_not_eq %] %
162 |#Hi >Hi >nth_change_vec // >nth_change_vec // >Htc
163 >tape_move_mk_tape_R [2: #_ %1 %] >reverse_cons %
165 |#Hi >Hi >nth_change_vec_neq [2:@eqb_false_to_not_eq %]
166 >nth_change_vec_neq [2:@eqb_false_to_not_eq %]
167 <(Heq 2) [2:@eqb_false_to_not_eq %]
168 <(Htbeq 2) [%|@eqb_false_to_not_eq %]
173 definition test_null_char ≝ test_char FSUnialpha (λc.c == null).
175 definition R_test_null_char_true ≝ λt1,t2.
176 current FSUnialpha t1 = Some ? null ∧ t1 = t2.
178 definition R_test_null_char_false ≝ λt1,t2.
179 current FSUnialpha t1 ≠ Some ? null ∧ t1 = t2.
181 lemma sem_test_null_char :
182 test_null_char ⊨ [ tc_true : R_test_null_char_true, R_test_null_char_false].
183 #t1 cases (sem_test_char FSUnialpha (λc.c == null) t1) #k * #outc * * #Hloop #Htrue
184 #Hfalse %{k} %{outc} % [ %
186 | #Houtc cases (Htrue ?) [| @Houtc] * #c * #Hcurt1 #Hcnull lapply (\P Hcnull)
187 -Hcnull #H destruct (H) #Houtc1 %
188 [ @Hcurt1 | <Houtc1 % ] ]
189 | #Houtc cases (Hfalse ?) [| @Houtc] #Hc #Houtc %
190 [ % #Hcurt1 >Hcurt1 in Hc; #Hc lapply (Hc ? (refl ??))
191 >(?:((null:FSUnialpha) == null) = true) [|@(\b (refl ??)) ]
196 definition cfg_to_obj ≝
197 mmove cfg FSUnialpha 2 L ·
198 (ifTM ?? (inject_TM ? test_null_char 2 cfg)
200 (copy_char_N cfg obj FSUnialpha 2)
203 inject_TM ? (move_to_end FSUnialpha L) 2 cfg ·
204 mmove cfg FSUnialpha 2 R. *)
206 definition R_cfg_to_obj ≝ λt1,t2:Vector (tape FSUnialpha) 3.
208 nth cfg ? t1 (niltape ?) = mk_tape FSUnialpha (c::ls) (None ?) [ ] →
209 (c = null → t2 = change_vec ?? t1 (midtape ? ls c [ ]) cfg) ∧
213 (midtape ? (left ? (nth obj ? t1 (niltape ?))) c (right ? (nth obj ? t1 (niltape ?)))) obj)
214 (midtape ? ls c [ ]) cfg).
216 lemma tape_move_mk_tape_L :
218 (c = None ? → ls = [ ] ∨ rs = [ ]) →
219 tape_move ? (mk_tape sig ls c rs) L =
220 mk_tape ? (tail ? ls) (option_hd ? ls) (option_cons ? c rs).
221 #sig * [ * [ * | #c * ] | #l0 #ls0 * [ *
222 [| #r0 #rs0 #H @False_ind cases (H (refl ??)) #H1 destruct (H1) ] | #c * ] ]
226 lemma sem_cfg_to_obj : cfg_to_obj ⊨ R_cfg_to_obj.
227 @(sem_seq_app FSUnialpha 2 ????? (sem_move_multi ? 2 cfg L ?)
229 (acc_sem_inject ?????? cfg ? sem_test_null_char)
231 (sem_copy_char_N …)))
234 #tc * whd in ⊢ (%→?); #Htc *
235 [ * #te * * * #Hcurtc #Hte1 #Hte2 whd in ⊢ (%→?); #Htb destruct (Htb)
237 [ #Hc >Hta in Htc; >eq_mk_tape_rightof whd in match (tape_move ???); #Htc
239 [ lapply (eq_vec_change_vec ??????? (sym_eq … Hte1) Hte2) >change_vec_same // ]
241 | #Hc >Hta in Htc; >eq_mk_tape_rightof whd in match (tape_move ???); #Htc
242 >Htc in Hcurtc; >nth_change_vec // normalize in ⊢ (%→?);
243 #H destruct (H) @False_ind cases Hc /2/ ]
244 | * #te * * * #Hcurtc #Hte1 #Hte2
247 [ >Htc in Hcurtc; >Hta >nth_change_vec //
248 normalize in ⊢ (%→?); * #H @False_ind /2/
250 [ lapply (eq_vec_change_vec ??????? (sym_eq … Hte1) Hte2)
251 >change_vec_same // ] -Hte1 -Hte2 #Hte destruct (Hte)
252 >Hta in Htc; whd in match (tape_move ???); #Htc
253 >Htc in Htb; >nth_change_vec //
254 >nth_change_vec_neq [2:@eqb_false_to_not_eq //] >Hta
259 definition char_to_move ≝ λc.match c with
260 [ bit b ⇒ if b then R else L
263 definition char_to_bit_option ≝ λc.match c with
264 [ bit b ⇒ Some ? (bit b)
267 definition R_cfg_to_obj1 ≝ λt1,t2:Vector (tape FSUnialpha) 3.
269 nth cfg ? t1 (niltape ?) = mk_tape FSUnialpha (c::ls) (None ?) [ ] →
272 tape_write ? (nth obj ? t1 (niltape ?)) (char_to_bit_option c) in
275 (tape_write ? (nth obj ? t1 (niltape ?)) (char_to_bit_option c)) obj)
276 (midtape ? ls c [ ]) cfg.
278 lemma sem_cfg_to_obj1: cfg_to_obj ⊨ R_cfg_to_obj1.
279 @(Realize_to_Realize … sem_cfg_to_obj) #t1 #t2 #H #c #ls #Hcfg #Hbar
280 cases (H c ls Hcfg) cases (true_or_false (c==null)) #Hc
281 [#Ht2 #_ >(Ht2 (\P Hc)) -Ht2 @(eq_vec … (niltape ?))
282 #i #lei2 cases (le_to_or_lt_eq … (le_S_S_to_le …lei2))
283 [#lei1 cases (le_to_or_lt_eq … (le_S_S_to_le …lei1))
284 [#lei0 lapply(le_n_O_to_eq … (le_S_S_to_le …lei0)) #eqi <eqi
285 >nth_change_vec_neq [2:@eqb_false_to_not_eq %]
286 >nth_change_vec_neq in ⊢ (???%); [2:@eqb_false_to_not_eq %]
287 >nth_change_vec // >(\P Hc) %
288 |#Hi >Hi >nth_change_vec //
290 |#Hi >Hi >nth_change_vec_neq [2:@eqb_false_to_not_eq %]
291 >nth_change_vec_neq [2:@eqb_false_to_not_eq %]
292 >nth_change_vec_neq [2:@eqb_false_to_not_eq %] %
294 |#_ #Ht2 >(Ht2 (\Pf Hc)) -Ht2 @(eq_vec … (niltape ?))
295 #i #lei2 cases (le_to_or_lt_eq … (le_S_S_to_le …lei2))
296 [#lei1 cases (le_to_or_lt_eq … (le_S_S_to_le …lei1))
297 [#lei0 lapply(le_n_O_to_eq … (le_S_S_to_le …lei0)) #eqi <eqi
298 >nth_change_vec_neq [2:@eqb_false_to_not_eq %]
299 >nth_change_vec_neq in ⊢ (???%); [2:@eqb_false_to_not_eq %]
300 >nth_change_vec // >nth_change_vec //
301 lapply (\bf Hbar) lapply Hc elim c //
302 [whd in ⊢ (??%?→?); #H destruct (H)
303 |#_ whd in ⊢ (??%?→?); #H destruct (H)
305 |#Hi >Hi >nth_change_vec //
307 |#Hi >Hi >nth_change_vec_neq [2:@eqb_false_to_not_eq %]
308 >nth_change_vec_neq [2:@eqb_false_to_not_eq %]
309 >nth_change_vec_neq [2:@eqb_false_to_not_eq %] %
315 (* macchina che muove il nastro obj a destra o sinistra a seconda del valore
316 del current di prg, che codifica la direzione in cui ci muoviamo *)
318 definition tape_move_obj : mTM FSUnialpha 2 ≝
320 (inject_TM ? (test_char ? (λc:FSUnialpha.c == bit false)) 2 prg)
321 (mmove obj FSUnialpha 2 L)
323 (inject_TM ? (test_char ? (λc:FSUnialpha.c == bit true)) 2 prg)
324 (mmove obj FSUnialpha 2 R)
329 definition R_tape_move_obj' ≝ λt1,t2:Vector (tape FSUnialpha) 3.
330 (current ? (nth prg ? t1 (niltape ?)) = Some ? (bit false) →
331 t2 = change_vec ?? t1 (tape_move ? (nth obj ? t1 (niltape ?)) L) obj) ∧
332 (current ? (nth prg ? t1 (niltape ?)) = Some ? (bit true) →
333 t2 = change_vec ?? t1 (tape_move ? (nth obj ? t1 (niltape ?)) R) obj) ∧
334 (current ? (nth prg ? t1 (niltape ?)) ≠ Some ? (bit false) →
335 current ? (nth prg ? t1 (niltape ?)) ≠ Some ? (bit true) →
338 lemma sem_tape_move_obj' : tape_move_obj ⊨ R_tape_move_obj'.
339 #ta cases (sem_if ??????????
340 (acc_sem_inject ?????? prg ? (sem_test_char ? (λc:FSUnialpha.c == bit false)))
341 (sem_move_multi ? 2 obj L ?)
343 (acc_sem_inject ?????? prg ? (sem_test_char ? (λc:FSUnialpha.c == bit true)))
344 (sem_move_multi ? 2 obj R ?)
346 #i * #outc * #Hloop #HR %{i} %{outc} % [@Hloop] -i
348 [ * #tb * * * * #c * #Hcurta_prg #Hc lapply (\P Hc) -Hc #Hc #Htb1 #Htb2
349 whd in ⊢ (%→%); #Houtc >Houtc -Houtc % [ %
350 [ >Hcurta_prg #H destruct (H) >(?:tb = ta)
351 [| lapply (eq_vec_change_vec ??????? Htb1 Htb2)
352 >change_vec_same // ] %
353 | >Hcurta_prg #H destruct (H) destruct (Hc) ]
354 | >Hcurta_prg >Hc * #H @False_ind /2/ ]
355 | * #tb * * * #Hnotfalse #Htb1 #Htb2 cut (tb = ta)
356 [ lapply (eq_vec_change_vec ??????? Htb1 Htb2)
357 >change_vec_same // ] -Htb1 -Htb2 #Htb destruct (Htb) *
358 [ * #tc * * * * #c * #Hcurta_prg #Hc lapply (\P Hc) -Hc #Hc #Htc1 #Htc2
359 whd in ⊢ (%→%); #Houtc >Houtc -Houtc % [ %
360 [ >Hcurta_prg #H destruct (H) destruct (Hc)
361 | >Hcurta_prg #H destruct (H) >(?:tc = ta)
362 [| lapply (eq_vec_change_vec ??????? Htc1 Htc2)
363 >change_vec_same // ] % ]
364 | >Hcurta_prg >Hc #_ * #H @False_ind /2/ ]
365 | * #tc * * * #Hnottrue #Htc1 #Htc2 cut (tc = ta)
366 [ lapply (eq_vec_change_vec ??????? Htc1 Htc2)
367 >change_vec_same // ] -Htc1 -Htc2
368 #Htc destruct (Htc) whd in ⊢ (%→?); #Houtc % [ %
369 [ #Hcurta_prg lapply (\Pf (Hnotfalse ? Hcurta_prg)) * #H @False_ind /2/
370 | #Hcurta_prg lapply (\Pf (Hnottrue ? Hcurta_prg)) * #H @False_ind /2/ ]
376 definition R_tape_move_obj ≝ λt1,t2:Vector (tape FSUnialpha) 3.
377 ∀c. current ? (nth prg ? t1 (niltape ?)) = Some ? c →
378 t2 = change_vec ?? t1 (tape_move ? (nth obj ? t1 (niltape ?)) (char_to_move c)) obj.
380 lemma sem_tape_move_obj : tape_move_obj ⊨ R_tape_move_obj.
381 @(Realize_to_Realize … sem_tape_move_obj')
382 #ta #tb * * #Htb1 #Htb2 #Htb3 * [ *
384 | #Hcurta_prg change with (nth obj ? ta (niltape ?)) in match (tape_move ???);
385 >change_vec_same @Htb3 >Hcurta_prg % #H destruct (H)
386 | #Hcurta_prg change with (nth obj ? ta (niltape ?)) in match (tape_move ???);
387 >change_vec_same @Htb3 >Hcurta_prg % #H destruct (H)
391 definition list_of_tape ≝ λsig.λt:tape sig.
392 reverse ? (left ? t)@option_cons ? (current ? t) (right ? t).
394 definition restart_tape ≝ λi,n.
395 mmove i FSUnialpha n L ·
396 inject_TM ? (move_to_end FSUnialpha L) n i ·
397 mmove i FSUnialpha n R.
399 definition R_restart_tape ≝ λi,n.λint,outt:Vector (tape FSUnialpha) (S n).
400 ∀t.t = nth i ? int (niltape ?) →
401 outt = change_vec ?? int
402 (mk_tape ? [ ] (option_hd ? (list_of_tape ? t)) (tail ? (list_of_tape ? t))) i.
404 lemma sem_restart_tape : ∀i,n.i < S n → restart_tape i n ⊨ R_restart_tape i n.
406 @(sem_seq_app ??????? (sem_move_multi ? n i L ?)
407 (sem_seq ?????? (sem_inject ???? i ? (sem_move_to_end_l ?))
408 (sem_move_multi ? n i R ?))) [1,2,3:@le_S_S_to_le //]
409 #ta #tb * #tc * whd in ⊢ (%→?); #Htc
410 * #td * * * #Htd1 #Htd2 #Htd3
411 whd in ⊢ (%→?); #Htb *
412 [ #Hta_i <Hta_i in Htc; whd in ⊢ (???(????%?)→?); #Htc
413 cut (td = tc) [@daemon]
414 (* >Htc in Htd1; >nth_change_vec // *) -Htd1 -Htd2 -Htd3
415 #Htd >Htd in Htb; >Htc >change_vec_change_vec >nth_change_vec //
417 | #r0 #rs0 #Hta_i <Hta_i in Htc; whd in ⊢ (???(????%?)→?); #Htc
418 cut (td = tc) [@daemon]
419 (* >Htc in Htd1; >nth_change_vec // *) -Htd1 -Htd2 -Htd3
420 #Htd >Htd in Htb; >Htc >change_vec_change_vec >nth_change_vec //
422 | #l0 #ls0 #Hta_i <Hta_i in Htc; whd in ⊢ (???(????%?)→?); #Htc
423 cut (td = change_vec ?? tc (mk_tape ? [ ] (None ?) (reverse ? ls0@[l0])) i)
425 #Htd >Htd in Htb; >Htc >change_vec_change_vec >change_vec_change_vec
426 >nth_change_vec // #Htb >Htb <(reverse_reverse ? ls0) in ⊢ (???%);
427 cases (reverse ? ls0)
429 | #l1 #ls1 >reverse_cons
430 >(?: list_of_tape ? (rightof ? l0 (reverse ? ls1@[l1])) =
432 [|change with (reverse ??@?) in ⊢ (??%?);
433 whd in match (left ??); >reverse_cons >reverse_append
434 whd in ⊢ (??%?); @eq_f >reverse_reverse normalize >append_nil % ] % ]
436 [ #c #rs #Hta_i <Hta_i in Htc; whd in ⊢ (???(????%?)→?); #Htc
437 cut (td = tc) [@daemon]
438 (* >Htc in Htd1; >nth_change_vec // *) -Htd1 -Htd2 -Htd3
439 #Htd >Htd in Htb; >Htc >change_vec_change_vec >nth_change_vec //
441 | #l0 #ls0 #c #rs #Hta_i <Hta_i in Htc; whd in ⊢ (???(????%?)→?); #Htc
442 cut (td = change_vec ?? tc (mk_tape ? [ ] (None ?) (reverse ? ls0@l0::c::rs)) i)
444 #Htd >Htd in Htb; >Htc >change_vec_change_vec >change_vec_change_vec
445 >nth_change_vec // #Htb >Htb <(reverse_reverse ? ls0) in ⊢ (???%);
446 cases (reverse ? ls0)
448 | #l1 #ls1 >reverse_cons
449 >(?: list_of_tape ? (midtape ? (l0::reverse ? ls1@[l1]) c rs) =
451 [|change with (reverse ??@?) in ⊢ (??%?);
452 whd in match (left ??); >reverse_cons >reverse_append
453 whd in ⊢ (??%?); @eq_f >reverse_reverse normalize
454 >associative_append % ] % ]