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 copy_char_states ≝ initN 3.
57 definition cc0 : copy_states ≝ mk_Sig ?? 0 (leb_true_to_le 1 3 (refl …)).
58 definition cc1 : copy_states ≝ mk_Sig ?? 1 (leb_true_to_le 2 3 (refl …)).
60 definition trans_copy_char ≝
61 λsrc,dst.λsig:FinSet.λn.
62 λp:copy_char_states × (Vector (option sig) (S n)).
65 [ O ⇒ 〈cc1,change_vec ? (S n)
66 (change_vec ? (S n) (null_action ? n) (〈None ?,R〉) src)
67 (〈nth src ? a (None ?),R〉) dst〉
68 | S _ ⇒ 〈cc1,null_action ? n〉 ].
70 definition copy_char ≝
72 mk_mTM sig n copy_char_states (trans_copy_char src dst sig n)
75 definition R_copy_char ≝
76 λsrc,dst,sig,n.λint,outt: Vector (tape sig) (S n).
79 (tape_move_mono ? (nth src ? int (niltape ?)) 〈None ?, R〉) src)
80 (tape_move_mono ? (nth dst ? int (niltape ?))
81 〈current ? (nth src ? int (niltape ?)), R〉) dst.
83 lemma copy_char_q0_q1 :
84 ∀src,dst,sig,n,v.src ≠ dst → src < S n → dst < S n →
85 step sig n (copy_char src dst sig n) (mk_mconfig ??? cc0 v) =
89 (tape_move_mono ? (nth src ? v (niltape ?)) 〈None ?, R〉) src)
90 (tape_move_mono ? (nth dst ? v (niltape ?)) 〈current ? (nth src ? v (niltape ?)), R〉) dst).
91 #src #dst #sig #n #v #Heq #Hsrc #Hdst
93 <(change_vec_same … v dst (niltape ?)) in ⊢ (??%?);
94 <(change_vec_same … v src (niltape ?)) in ⊢ (??%?);
95 >tape_move_multi_def @eq_f2 //
96 >pmap_change >pmap_change <tape_move_multi_def
97 >tape_move_null_action @eq_f2 // @eq_f2
99 | >change_vec_same >change_vec_same // ]
103 ∀src,dst,sig,n.src ≠ dst → src < S n → dst < S n →
104 copy_char src dst sig n ⊨ R_copy_char src dst sig n.
105 #src #dst #sig #n #Hneq #Hsrc #Hdst #int
106 %{2} % [| % [ % | whd >copy_char_q0_q1 // ]]
109 definition obj ≝ (0:DeqNat).
110 definition cfg ≝ (1:DeqNat).
111 definition prg ≝ (2:DeqNat).
113 definition obj_to_cfg ≝
114 mmove cfg FSUnialpha 2 L ·
115 (ifTM ?? (inject_TM ? (test_null ?) 2 obj)
116 (copy_char obj cfg FSUnialpha 2 ·
117 mmove cfg FSUnialpha 2 L ·
118 mmove obj FSUnialpha 2 L)
119 (inject_TM ? (write FSUnialpha null) 2 cfg)
121 inject_TM ? (move_to_end FSUnialpha L) 2 cfg ·
122 mmove cfg FSUnialpha 2 R.
124 definition R_obj_to_cfg ≝ λt1,t2:Vector (tape FSUnialpha) 3.
126 nth cfg ? t1 (niltape ?) = mk_tape FSUnialpha (c::ls) (None ?) [ ] →
127 (∀lso,x,rso.nth obj ? t1 (niltape ?) = midtape FSUnialpha lso x rso →
128 t2 = change_vec ?? t1
129 (mk_tape ? [ ] (option_hd ? (reverse ? (x::ls))) (tail ? (reverse ? (x::ls)))) cfg) ∧
130 (current ? (nth obj ? t1 (niltape ?)) = None ? →
131 t2 = change_vec ?? t1
132 (mk_tape ? [ ] (option_hd FSUnialpha (reverse ? (null::ls)))
133 (tail ? (reverse ? (null::ls)))) cfg).
135 axiom accRealize_to_Realize :
136 ∀sig,n.∀M:mTM sig n.∀Rtrue,Rfalse,acc.
137 M ⊨ [ acc: Rtrue, Rfalse ] → M ⊨ Rtrue ∪ Rfalse.
139 lemma eq_mk_tape_rightof :
140 ∀alpha,a,al.mk_tape alpha (a::al) (None ?) [ ] = rightof ? a al.
144 definition option_cons ≝ λsig.λc:option sig.λl.
145 match c with [ None ⇒ l | Some c0 ⇒ c0::l ].
147 lemma tape_move_mk_tape_R :
149 (c = None ? → ls = [ ] ∨ rs = [ ]) →
150 tape_move ? (mk_tape sig ls c rs) R =
151 mk_tape ? (option_cons ? c ls) (option_hd ? rs) (tail ? rs).
152 #sig * [ * [ * | #c * ] | #l0 #ls0 * [ *
153 [| #r0 #rs0 #H @False_ind cases (H (refl ??)) #H1 destruct (H1) ] | #c * ] ]
157 lemma eq_vec_change_vec : ∀sig,n.∀v1,v2:Vector sig n.∀i,t,d.
159 (∀j.i ≠ j → nth j ? v1 d = nth j ? v2 d) →
160 v2 = change_vec ?? v1 t i.
161 #sig #n #v1 #v2 #i #t #d #H1 #H2 @(eq_vec … d)
162 #i0 #Hlt cases (decidable_eq_nat i0 i) #Hii0
163 [ >Hii0 >nth_change_vec //
164 | >nth_change_vec_neq [|@sym_not_eq //] @sym_eq @H2 @sym_not_eq // ]
167 lemma sem_obj_to_cfg : obj_to_cfg ⊨ R_obj_to_cfg.
168 @(sem_seq_app FSUnialpha 2 ????? (sem_move_multi ? 2 cfg L ?)
171 (sem_test_null_multi ?? obj ?)
172 (sem_seq ?????? (sem_copy_char …)
173 (sem_seq ?????? (sem_move_multi ? 2 cfg L ?)
174 (sem_move_multi ? 2 obj L ?)))
175 (sem_inject ???? cfg ? (sem_write FSUnialpha null)))
176 (sem_seq ?????? (sem_inject ???? cfg ? (sem_move_to_end_l ?))
177 (sem_move_multi ? 2 cfg R ?)))) //
179 #tc * whd in ⊢ (%→?); #Htc *
181 [ * #te * * #Hcurtc #Hte
182 * destruct (Hte) #te * whd in ⊢ (%→?); #Hte
183 cut (∃x.current ? (nth obj ? tc (niltape ?)) = Some ? x)
184 [ cases (current ? (nth obj ? tc (niltape ?))) in Hcurtc;
185 [ * #H @False_ind /2/ | #x #_ %{x} % ] ] * #x #Hcurtc'
186 (* [ whd in ⊢ (%→%→?); * #x * #y * * -Hcurtc #Hcurtc1 #Hcurtc2 #Hte *)
187 * #tf * whd in ⊢ (%→%→?); #Htf #Htd
188 * #tg * * * whd in ⊢ (%→%→%→%→?); #Htg1 #Htg2 #Htg3 #Htb
190 [ #lso #x0 #rso #Hta2 >Hta1 in Htc; >eq_mk_tape_rightof
191 whd in match (tape_move ???); #Htc
192 cut (tg = change_vec ?? td (mk_tape ? [ ] (None ?) (reverse ? ls@[x])) cfg)
193 [ lapply (eq_vec_change_vec ??????? (Htg2 ls x [ ] ?) Htg3) //
194 >Htd >nth_change_vec_neq // >Htf >nth_change_vec //
195 >Hte >Hcurtc' >nth_change_vec // >Htc >nth_change_vec // ]
196 -Htg1 -Htg2 -Htg3 #Htg destruct
197 >change_vec_change_vec >change_vec_change_vec
198 >change_vec_commute // >change_vec_change_vec
199 >change_vec_commute [|@sym_not_eq //] >change_vec_change_vec
200 >change_vec_commute // >change_vec_change_vec
201 >nth_change_vec // >nth_change_vec_neq [|@sym_not_eq //]
202 >nth_change_vec // >nth_change_vec_neq [|@sym_not_eq //]
203 >change_vec_commute [|@sym_not_eq //] @eq_f3 //
204 [ >Hta2 cases rso in Hta2; whd in match (tape_move_mono ???);
205 [ #Hta2 whd in match (tape_move ???); <Hta2 @change_vec_same
206 | #r1 #rs1 #Hta2 whd in match (tape_move ???); <Hta2 @change_vec_same ]
207 | >tape_move_mk_tape_R [| #_ % %] >reverse_cons
208 >nth_change_vec_neq in Hcurtc'; [|@sym_not_eq //] >Hta2
209 normalize in ⊢ (%→?); #H destruct (H) %
211 | #Hta2 >Htc in Hcurtc'; >nth_change_vec_neq [| @sym_not_eq //]
212 >Hta2 #H destruct (H)
214 | * #te * * #Hcurtc #Hte
215 * whd in ⊢ (%→%→?); #Htd1 #Htd2
216 * #tf * * * #Htf1 #Htf2 #Htf3 whd in ⊢ (%→?); #Htb
218 [ #lso #x #rso #Hta2 >Htc in Hcurtc; >nth_change_vec_neq [|@sym_not_eq //]
219 >Hta2 normalize in ⊢ (%→?); #H destruct (H)
220 | #_ >Hta1 in Htc; >eq_mk_tape_rightof whd in match (tape_move ???); #Htc
221 destruct (Hte) cut (td = change_vec ?? tc (midtape ? ls null []) cfg)
222 [ lapply (eq_vec_change_vec ??????? (Htd1 ls c [ ] ?) Htd2) //
223 >Htc >nth_change_vec // ] -Htd1 -Htd2 #Htd
224 -Htf1 cut (tf = change_vec ?? td (mk_tape ? [ ] (None ?) (reverse ? ls@[null])) cfg)
225 [ lapply (eq_vec_change_vec ??????? (Htf2 ls null [ ] ?) Htf3) //
226 >Htd >nth_change_vec // ] -Htf2 -Htf3 #Htf destruct (Htf Htd Htc Htb)
227 >change_vec_change_vec >change_vec_change_vec >change_vec_change_vec
228 >change_vec_change_vec >change_vec_change_vec >nth_change_vec //
229 >reverse_cons >tape_move_mk_tape_R /2/ ]
233 definition test_null_char ≝ test_char FSUnialpha (λc.c == null).
235 definition R_test_null_char_true ≝ λt1,t2.
236 current FSUnialpha t1 = Some ? null ∧ t1 = t2.
238 definition R_test_null_char_false ≝ λt1,t2.
239 current FSUnialpha t1 ≠ Some ? null ∧ t1 = t2.
241 lemma sem_test_null_char :
242 test_null_char ⊨ [ tc_true : R_test_null_char_true, R_test_null_char_false].
243 #t1 cases (sem_test_char FSUnialpha (λc.c == null) t1) #k * #outc * * #Hloop #Htrue
244 #Hfalse %{k} %{outc} % [ %
246 | #Houtc cases (Htrue ?) [| @Houtc] * #c * #Hcurt1 #Hcnull lapply (\P Hcnull)
247 -Hcnull #H destruct (H) #Houtc1 %
248 [ @Hcurt1 | <Houtc1 % ] ]
249 | #Houtc cases (Hfalse ?) [| @Houtc] #Hc #Houtc %
250 [ % #Hcurt1 >Hcurt1 in Hc; #Hc lapply (Hc ? (refl ??))
251 >(?:((null:FSUnialpha) == null) = true) [|@(\b (refl ??)) ]
256 definition cfg_to_obj ≝
257 mmove cfg FSUnialpha 2 L ·
258 (ifTM ?? (inject_TM ? test_null_char 2 cfg)
260 (copy_char cfg obj FSUnialpha 2 ·
261 mmove cfg FSUnialpha 2 L ·
262 mmove obj FSUnialpha 2 L)
264 inject_TM ? (move_to_end FSUnialpha L) 2 cfg ·
265 mmove cfg FSUnialpha 2 R.
267 definition R_cfg_to_obj ≝ λt1,t2:Vector (tape FSUnialpha) 3.
269 nth cfg ? t1 (niltape ?) = mk_tape FSUnialpha (c::ls) (None ?) [ ] →
271 t2 = change_vec ?? t1
272 (mk_tape ? [ ] (option_hd FSUnialpha (reverse ? (c::ls)))
273 (tail ? (reverse ? (c::ls)))) cfg) ∧
277 (midtape ? (left ? (nth obj ? t1 (niltape ?))) c (right ? (nth obj ? t1 (niltape ?)))) obj)
278 (mk_tape ? [ ] (option_hd ? (reverse ? (c::ls))) (tail ? (reverse ? (c::ls)))) cfg).
280 lemma tape_move_mk_tape_L :
282 (c = None ? → ls = [ ] ∨ rs = [ ]) →
283 tape_move ? (mk_tape sig ls c rs) L =
284 mk_tape ? (tail ? ls) (option_hd ? ls) (option_cons ? c rs).
285 #sig * [ * [ * | #c * ] | #l0 #ls0 * [ *
286 [| #r0 #rs0 #H @False_ind cases (H (refl ??)) #H1 destruct (H1) ] | #c * ] ]
290 lemma sem_cfg_to_obj : cfg_to_obj ⊨ R_cfg_to_obj.
291 @(sem_seq_app FSUnialpha 2 ????? (sem_move_multi ? 2 cfg L ?)
294 (acc_sem_inject ?????? cfg ? sem_test_null_char)
296 (sem_seq ?????? (sem_copy_char …)
297 (sem_seq ?????? (sem_move_multi ? 2 cfg L ?) (sem_move_multi ? 2 obj L ?))))
298 (sem_seq ?????? (sem_inject ???? cfg ? (sem_move_to_end_l ?))
299 (sem_move_multi ? 2 cfg R ?)))) // [@sym_not_eq //]
301 #tc * whd in ⊢ (%→?); #Htc *
303 [ * #te * * * #Hcurtc #Hte1 #Hte2 whd in ⊢ (%→?); #Htd destruct (Htd)
304 * #tf * * * #Htf1 #Htf2 #Htf3
307 [ #Hc >Hta in Htc; >eq_mk_tape_rightof whd in match (tape_move ???); #Htc
309 [ lapply (eq_vec_change_vec ??????? (sym_eq … Hte1) Hte2) >change_vec_same // ]
311 cut (tf = change_vec ? 3 te (mk_tape ? [ ] (None ?) (reverse ? ls@[c])) cfg)
312 [ lapply (eq_vec_change_vec ??????? (Htf2 ls c [ ] ?) Htf3) //
313 >Hte >Htc >nth_change_vec // ] -Htf1 -Htf2 -Htf3 #Htf
314 destruct (Htf Hte Htc Htb)
315 >change_vec_change_vec >change_vec_change_vec >change_vec_change_vec
316 >nth_change_vec // >tape_move_mk_tape_R [| #_ % % ]
318 | #Hc >Hta in Htc; >eq_mk_tape_rightof whd in match (tape_move ???); #Htc
319 >Htc in Hcurtc; >nth_change_vec // normalize in ⊢ (%→?);
320 #H destruct (H) @False_ind cases Hc /2/ ]
322 | * #te * * * #Hcurtc #Hte1 #Hte2
323 * #tf * whd in ⊢ (%→?); #Htf
324 * #tg * whd in ⊢ (%→%→?); #Htg #Htd
325 * #th * * * #Hth1 #Hth2 #Hth3
328 [ >Htc in Hcurtc; >Hta >nth_change_vec // >tape_move_mk_tape_L //
329 >Hc normalize in ⊢ (%→?); * #H @False_ind /2/
331 [ lapply (eq_vec_change_vec ??????? (sym_eq … Hte1) Hte2)
332 >change_vec_same // ] -Hte1 -Hte2 #Hte
333 cut (th = change_vec ?? td (mk_tape ? [ ] (None ?) (reverse ? ls@[c])) cfg)
334 [ lapply (eq_vec_change_vec ??????? (Hth2 ls c [ ] ?) Hth3) //
335 >Htd >nth_change_vec_neq // >Htg >nth_change_vec //
336 >Htf >nth_change_vec_neq // >nth_change_vec //
337 >Hte >Htc >nth_change_vec // >Hta // ] -Hth1 -Hth2 -Hth3 #Hth
338 destruct (Hth Hte Hta Htb Htd Htg Htc Htf)
339 >change_vec_change_vec >change_vec_change_vec
340 >change_vec_commute // >change_vec_change_vec
341 >change_vec_commute [|@sym_not_eq //] >change_vec_change_vec
342 >change_vec_commute // >change_vec_change_vec
343 >nth_change_vec // >nth_change_vec_neq [|@sym_not_eq //]
344 >nth_change_vec // >nth_change_vec_neq [|@sym_not_eq //]
345 >change_vec_commute [|@sym_not_eq //]
347 [ >Hta >tape_move_mk_tape_L // >nth_change_vec // whd in match (current ??);
348 @eq_f2 // cases (nth obj ? ta (niltape ?))
349 [| #r0 #rs0 | #l0 #ls0 | #ls0 #c0 #rs0 ] try %
351 | >reverse_cons >tape_move_mk_tape_R // #_ % % ]
356 (* macchina che muove il nastro obj a destra o sinistra a seconda del valore
357 del current di prg, che codifica la direzione in cui ci muoviamo *)
359 definition char_to_move ≝ λc.match c with
360 [ bit b ⇒ if b then R else L
363 definition char_to_bit_option ≝ λc.match c with
364 [ bit b ⇒ Some ? (bit b)
367 definition tape_move_obj : mTM FSUnialpha 2 ≝
369 (inject_TM ? (test_char ? (λc:FSUnialpha.c == bit false)) 2 prg)
370 (mmove obj FSUnialpha 2 L)
372 (inject_TM ? (test_char ? (λc:FSUnialpha.c == bit true)) 2 prg)
373 (mmove obj FSUnialpha 2 R)
378 definition R_tape_move_obj' ≝ λt1,t2:Vector (tape FSUnialpha) 3.
379 (current ? (nth prg ? t1 (niltape ?)) = Some ? (bit false) →
380 t2 = change_vec ?? t1 (tape_move ? (nth obj ? t1 (niltape ?)) L) obj) ∧
381 (current ? (nth prg ? t1 (niltape ?)) = Some ? (bit true) →
382 t2 = change_vec ?? t1 (tape_move ? (nth obj ? t1 (niltape ?)) R) obj) ∧
383 (current ? (nth prg ? t1 (niltape ?)) ≠ Some ? (bit false) →
384 current ? (nth prg ? t1 (niltape ?)) ≠ Some ? (bit true) →
387 lemma sem_tape_move_obj' : tape_move_obj ⊨ R_tape_move_obj'.
388 #ta cases (sem_if ??????????
389 (acc_sem_inject ?????? prg ? (sem_test_char ? (λc:FSUnialpha.c == bit false)))
390 (sem_move_multi ? 2 obj L ?)
392 (acc_sem_inject ?????? prg ? (sem_test_char ? (λc:FSUnialpha.c == bit true)))
393 (sem_move_multi ? 2 obj R ?)
395 #i * #outc * #Hloop #HR %{i} %{outc} % [@Hloop] -i
397 [ * #tb * * * * #c * #Hcurta_prg #Hc lapply (\P Hc) -Hc #Hc #Htb1 #Htb2
398 whd in ⊢ (%→%); #Houtc >Houtc -Houtc % [ %
399 [ >Hcurta_prg #H destruct (H) >(?:tb = ta)
400 [| lapply (eq_vec_change_vec ??????? Htb1 Htb2)
401 >change_vec_same // ] %
402 | >Hcurta_prg #H destruct (H) destruct (Hc) ]
403 | >Hcurta_prg >Hc * #H @False_ind /2/ ]
404 | * #tb * * * #Hnotfalse #Htb1 #Htb2 cut (tb = ta)
405 [ lapply (eq_vec_change_vec ??????? Htb1 Htb2)
406 >change_vec_same // ] -Htb1 -Htb2 #Htb destruct (Htb) *
407 [ * #tc * * * * #c * #Hcurta_prg #Hc lapply (\P Hc) -Hc #Hc #Htc1 #Htc2
408 whd in ⊢ (%→%); #Houtc >Houtc -Houtc % [ %
409 [ >Hcurta_prg #H destruct (H) destruct (Hc)
410 | >Hcurta_prg #H destruct (H) >(?:tc = ta)
411 [| lapply (eq_vec_change_vec ??????? Htc1 Htc2)
412 >change_vec_same // ] % ]
413 | >Hcurta_prg >Hc #_ * #H @False_ind /2/ ]
414 | * #tc * * * #Hnottrue #Htc1 #Htc2 cut (tc = ta)
415 [ lapply (eq_vec_change_vec ??????? Htc1 Htc2)
416 >change_vec_same // ] -Htc1 -Htc2
417 #Htc destruct (Htc) whd in ⊢ (%→?); #Houtc % [ %
418 [ #Hcurta_prg lapply (\Pf (Hnotfalse ? Hcurta_prg)) * #H @False_ind /2/
419 | #Hcurta_prg lapply (\Pf (Hnottrue ? Hcurta_prg)) * #H @False_ind /2/ ]
425 definition R_tape_move_obj ≝ λt1,t2:Vector (tape FSUnialpha) 3.
426 ∀c. current ? (nth prg ? t1 (niltape ?)) = Some ? c →
427 t2 = change_vec ?? t1 (tape_move ? (nth obj ? t1 (niltape ?)) (char_to_move c)) obj.
429 lemma sem_tape_move_obj : tape_move_obj ⊨ R_tape_move_obj.
430 @(Realize_to_Realize … sem_tape_move_obj')
431 #ta #tb * * #Htb1 #Htb2 #Htb3 * [ *
433 | #Hcurta_prg change with (nth obj ? ta (niltape ?)) in match (tape_move ???);
434 >change_vec_same @Htb3 >Hcurta_prg % #H destruct (H)
435 | #Hcurta_prg change with (nth obj ? ta (niltape ?)) in match (tape_move ???);
436 >change_vec_same @Htb3 >Hcurta_prg % #H destruct (H)
440 definition restart_tape ≝ λi.
441 inject_TM ? (move_to_end FSUnialpha L) 2 i ·
442 mmove i FSUnialpha 2 R.
445 match_m cfg prg FSUnialpha 2 ·
446 restart_tape cfg · copy prg cfg FSUnialpha 2 ·
447 cfg_to_obj · tape_move_obj · restart_tape prg · obj_to_cfg.
450 definition legal_tape ≝ λn,l,h,t.
452 nth cfg ? t1 (niltape ?) = midtape ? [ ] bar (state@[char]) →
453 is_config n (bar::state@[char]) →
454 nth prg ? t1 (niltape ?) = midtape ? [ ] bar table →
455 bar::table = table_TM n l h → *)
457 definition list_of_tape ≝ λsig,t.
458 left sig t@option_cons ? (current ? t) (right ? t).
460 definition low_char' ≝ λc.
463 | Some b ⇒ if (is_bit b) then b else null
466 lemma low_char_option : ∀s.
467 low_char' (option_map FinBool FSUnialpha bit s) = low_char s.
471 definition R_unistep ≝ λn,l,h.λt1,t2: Vector ? 3.
474 nth cfg ? t1 (niltape ?) = midtape ? [ ] bar (state@[char]) →
475 is_config n (bar::state@[char]) →
477 nth prg ? t1 (niltape ?) = midtape ? [ ] bar table →
478 bar::table = table_TM n l h →
480 only_bits (list_of_tape ? (nth obj ? t1 (niltape ?))) →
481 let conf ≝ (bar::state@[char]) in
482 (∃ll,lr.bar::table = ll@conf@lr) →
484 ∃nstate,nchar,m,t. tuple_encoding n h t = (conf@nstate@[nchar;m]) ∧
487 tuple_encoding n h t = (conf@nstate@[nchar;m])→
490 tape_move_mono ? (nth obj ? t1 (niltape ?))
491 〈char_to_bit_option nchar,char_to_move m〉 in
492 let next_char ≝ low_char' (current ? new_obj) in
495 (change_vec ?? t1 (midtape ? [ ] bar (nstate@[next_char])) cfg)
498 definition tape_map ≝ λA,B:FinSet.λf:A→B.λt.
499 mk_tape B (map ?? f (left ? t))
500 (option_map ?? f (current ? t))
501 (map ?? f (right ? t)).
503 lemma map_list_of_tape: ∀A,B,f,t.
504 list_of_tape B (tape_map ?? f t) = map ?? f (list_of_tape A t).
505 #A #B #f * // normalize // #ls #c #rs <map_append %
508 lemma low_char_current : ∀t.
509 low_char' (current FSUnialpha (tape_map FinBool FSUnialpha bit t))
510 = low_char (current FinBool t).
513 definition low_tapes: ∀M:normalTM.∀c:nconfig (no_states M).Vector ? 3 ≝
514 λM:normalTM.λc:nconfig (no_states M).Vector_of_list ?
515 [tape_map ?? bit (ctape ?? c);
517 ((bits_of_state ? (nhalt M) (cstate ?? c))@[low_char (current ? (ctape ?? c))]);
518 midtape ? [ ] bar (tail ? (table_TM ? (graph_enum ?? (ntrans M)) (nhalt M)))
521 lemma obj_low_tapes: ∀M,c.
522 nth obj ? (low_tapes M c) (niltape ?) = tape_map ?? bit (ctape ?? c).
525 lemma cfg_low_tapes: ∀M,c.
526 nth cfg ? (low_tapes M c) (niltape ?) =
528 ((bits_of_state ? (nhalt M) (cstate ?? c))@[low_char (current ? (ctape ?? c))]).
531 lemma prg_low_tapes: ∀M,c.
532 nth prg ? (low_tapes M c) (niltape ?) =
533 midtape ? [ ] bar (tail ? (table_TM ? (graph_enum ?? (ntrans M)) (nhalt M))).
536 (* commutation lemma for write *)
537 lemma map_write: ∀t,cout.
538 tape_write ? (tape_map FinBool ? bit t) (char_to_bit_option (low_char cout))
539 = tape_map ?? bit (tape_write ? t cout).
540 #t * // #b whd in match (char_to_bit_option ?);
541 whd in ⊢ (??%%); @eq_f3 [elim t // | // | elim t //]
544 (* commutation lemma for moves *)
545 lemma map_move: ∀t,m.
546 tape_move ? (tape_map FinBool ? bit t) (char_to_move (low_mv m))
547 = tape_map ?? bit (tape_move ? t m).
548 #t * // whd in match (char_to_move ?);
549 [cases t // * // | cases t // #ls #a * //]
552 (* commutation lemma for actions *)
553 lemma map_action: ∀t,cout,m.
554 tape_move ? (tape_write ? (tape_map FinBool ? bit t)
555 (char_to_bit_option (low_char cout))) (char_to_move (low_mv m))
556 = tape_map ?? bit (tape_move ? (tape_write ? t cout) m).
557 #t #cout #m >map_write >map_move %
560 lemma map_move_mono: ∀t,cout,m.
561 tape_move_mono ? (tape_map FinBool ? bit t)
562 〈char_to_bit_option (low_char cout), char_to_move (low_mv m)〉
563 = tape_map ?? bit (tape_move_mono ? t 〈cout,m〉).
567 definition R_unistep_high ≝ λM:normalTM.λt1,t2.
568 ∀c:nconfig (no_states M).
570 t2 = low_tapes M (step ? M c).
572 lemma R_unistep_equiv : ∀M,t1,t2.
573 R_unistep (no_states M) (graph_enum ?? (ntrans M)) (nhalt M) t1 t2 →
574 R_unistep_high M t1 t2.
575 #M #t1 #t2 #H whd whd in match (nconfig ?); #c #Ht1
576 lapply (initial_bar ? (nhalt M) (graph_enum ?? (ntrans M)) (nTM_nog ?)) #Htable
577 (* tup = current tuple *)
578 cut (∃t.t = 〈〈cstate … c,current ? (ctape … c)〉,
579 ntrans M 〈cstate … c,current ? (ctape … c)〉〉) [% //] * #tup #Htup
580 (* tup is in the graph *)
581 cut (mem ? tup (graph_enum ?? (ntrans M)))
582 [@memb_to_mem >Htup @(graph_enum_complete … (ntrans M)) %] #Hingraph
583 (* tupe target = 〈qout,cout,m〉 *)
584 lapply (decomp_target ? (ntrans M 〈cstate … c,current ? (ctape … c)〉))
585 * #qout * #cout * #m #Htg >Htg in Htup; #Htup
587 cut (step FinBool M c = mk_config ?? qout (tape_move ? (tape_write ? (ctape … c) cout) m))
588 [>(config_expand … c) whd in ⊢ (??%?); (* >Htg ?? why not?? *)
589 cut (trans ? M 〈cstate … c, current ? (ctape … c)〉 = 〈qout,cout,m〉) [<Htg %] #Heq1
592 cut (cstate ?? (step FinBool M c) = qout) [>Hstep %] #Hnew_state
594 cut (ctape ?? (step FinBool M c) = tape_move ? (tape_write ? (ctape … c) cout) m)
595 [>Hstep %] #Hnew_tape
596 lapply(H (bits_of_state ? (nhalt M) (cstate ?? c))
597 (low_char (current ? (ctape ?? c)))
598 (tail ? (table_TM ? (graph_enum ?? (ntrans M)) (nhalt M)))
601 lapply(list_to_table … (nhalt M) …Hingraph) * #ll * #lr #Htable1 %{ll}
602 %{(((bits_of_state ? (nhalt M) qout)@[low_char cout;low_mv m])@lr)}
603 >Htable1 @eq_f <associative_append @eq_f2 // >Htup
604 whd in ⊢ (??%?); @eq_f >associative_append %
605 |>Ht1 >obj_low_tapes >map_list_of_tape elim (list_of_tape ??)
606 [#b @False_ind | #b #tl #Hind #a * [#Ha >Ha //| @Hind]]
609 |%{(bits_of_state ? (nhalt M) (cstate ?? c))} %{(low_char (current ? (ctape ?? c)))}
610 % [% [% [// | cases (current ??) normalize [|#b] % #Hd destruct (Hd)]
611 |>length_map whd in match (length ??); @eq_f //]
613 |>Ht1 >cfg_low_tapes //] -H #H
614 lapply(H (bits_of_state … (nhalt M) qout) (low_char … cout)
615 (low_mv … m) tup ? Hingraph)
616 [>Htup whd in ⊢ (??%?); @eq_f >associative_append %] -H
617 #Ht2 @(eq_vec ? 3 ?? (niltape ?) ?) >Ht2 #i #Hi
618 cases (le_to_or_lt_eq … (le_S_S_to_le … Hi)) -Hi #Hi
619 [cases (le_to_or_lt_eq … (le_S_S_to_le … Hi)) -Hi #Hi
620 [cases (le_to_or_lt_eq … (le_S_S_to_le … Hi)) -Hi #Hi
622 |>Hi >obj_low_tapes >nth_change_vec //
623 >Ht1 >obj_low_tapes >Hstep @map_action
625 |>Hi >cfg_low_tapes >nth_change_vec_neq
626 [|% whd in ⊢ (??%?→?); #H destruct (H)]
627 >nth_change_vec // >Hnew_state @eq_f @eq_f >Hnew_tape
628 @eq_f2 [|2:%] >Ht1 >obj_low_tapes >map_move_mono >low_char_current %
630 |(* program tapes do not change *)
632 >nth_change_vec_neq [|% whd in ⊢ (??%?→?); #H destruct (H)]
633 >nth_change_vec_neq [|% whd in ⊢ (??%?→?); #H destruct (H)]
634 >Ht1 >prg_low_tapes //