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".
19 definition obj ≝ (0:DeqNat).
20 definition cfg ≝ (1:DeqNat).
21 definition prg ≝ (2:DeqNat).
23 definition obj_to_cfg ≝
24 (ifTM ?? (inject_TM ? (test_null ?) 2 obj)
25 (copy_char_N obj cfg FSUnialpha 2)
26 (inject_TM ? (write FSUnialpha null) 2 cfg)
28 inject_TM ? (move_to_end FSUnialpha L) 2 cfg ·
29 mmove cfg FSUnialpha 2 R.
31 definition R_obj_to_cfg ≝ λt1,t2:Vector (tape FSUnialpha) 3.
33 nth cfg ? t1 (niltape ?) = midtape ? ls c [ ] →
34 (∀lso,x,rso.nth obj ? t1 (niltape ?) = midtape FSUnialpha lso x rso →
36 (mk_tape ? [ ] (option_hd ? (reverse ? (x::ls))) (tail ? (reverse ? (x::ls)))) cfg) ∧
37 (current ? (nth obj ? t1 (niltape ?)) = None ? →
39 (mk_tape ? [ ] (option_hd FSUnialpha (reverse ? (null::ls)))
40 (tail ? (reverse ? (null::ls)))) cfg).
42 axiom accRealize_to_Realize :
43 ∀sig,n.∀M:mTM sig n.∀Rtrue,Rfalse,acc.
44 M ⊨ [ acc: Rtrue, Rfalse ] → M ⊨ Rtrue ∪ Rfalse.
46 lemma eq_mk_tape_rightof :
47 ∀alpha,a,al.mk_tape alpha (a::al) (None ?) [ ] = rightof ? a al.
51 lemma tape_move_mk_tape_R :
53 (c = None ? → ls = [ ] ∨ rs = [ ]) →
54 tape_move ? (mk_tape sig ls c rs) R =
55 mk_tape ? (option_cons ? c ls) (option_hd ? rs) (tail ? rs).
56 #sig * [ * [ * | #c * ] | #l0 #ls0 * [ *
57 [| #r0 #rs0 #H @False_ind cases (H (refl ??)) #H1 destruct (H1) ] | #c * ] ]
61 lemma None_or_Some: ∀A.∀a. a =None A ∨ ∃b. a = Some ? b.
65 lemma not_None_to_Some: ∀A.∀a. a ≠ None A → ∃b. a = Some ? b.
66 #A * /2/ * #H @False_ind @H %
69 lemma sem_obj_to_cfg : obj_to_cfg ⊨ R_obj_to_cfg.
70 @(sem_seq_app FSUnialpha 2 ?????
72 (sem_test_null_multi ?? obj ?)
74 (sem_inject ???? cfg ? (sem_write FSUnialpha null)))
75 (sem_seq ?????? (sem_inject ???? cfg ? (sem_move_to_end_l ?))
76 (sem_move_multi ? 2 cfg R ?))) //
78 #tb * #Hif * #tc * #HM2 #HM3 #c #ls #Hcfg
81 [ * #t1 * * #Hcurta #Ht1 destruct (Ht1)
82 lapply (not_None_to_Some … Hcurta) * #curta #Hcurtaeq
83 whd in ⊢ (%→?); #Htb % [2: #Hcur @False_ind /2/]
84 #lso #xo #rso #Hobjta cases HM2 whd in ⊢ (%→?); * #_
85 #HM2 #Heq >Htb in HM2; >nth_change_vec [2: @leb_true_to_le %]
86 >Hcfg >Hcurtaeq #HM2 lapply (HM2 … (refl ??)) -HM2
87 whd in match (left ??); whd in match (right ??);
88 >reverse_cons #Htc >HM3 @(eq_vec … (niltape ?)) #i #lei2
89 cases (le_to_or_lt_eq … (le_S_S_to_le …lei2))
90 [#lei1 cases (le_to_or_lt_eq … (le_S_S_to_le …lei1))
91 [#lei0 lapply(le_n_O_to_eq … (le_S_S_to_le …lei0)) #eqi <eqi
92 >nth_change_vec_neq [2:@eqb_false_to_not_eq %]
93 <(Heq 0) [2:@eqb_false_to_not_eq %] >Htb
94 >nth_change_vec_neq [2:@eqb_false_to_not_eq %]
95 >nth_change_vec_neq [%|2:@eqb_false_to_not_eq %]
96 |#Hi >Hi >nth_change_vec // >nth_change_vec // >Htc
97 >Hobjta in Hcurtaeq; whd in ⊢ (??%?→?); #Htmp destruct(Htmp)
98 >tape_move_mk_tape_R [2: #_ %1 %] %
100 |#Hi >Hi >nth_change_vec_neq [2:@eqb_false_to_not_eq %]
101 >nth_change_vec_neq [2:@eqb_false_to_not_eq %]
102 <(Heq 2) [2:@eqb_false_to_not_eq %] >Htb
103 >nth_change_vec_neq [%|2:@eqb_false_to_not_eq %]
105 | * #t1 * * #Hcurta #Ht1 destruct (Ht1)
106 * whd in ⊢ (%→?); #Htb lapply (Htb … Hcfg) -Htb #Htb
108 [#lso #xo #rso #Hmid @False_ind >Hmid in Hcurta;
109 whd in ⊢ (??%?→?); #Htmp destruct (Htmp)]
110 #_ cases HM2 whd in ⊢ (%→?); * #_
111 #HM2 #Heq >Htb in HM2; #HM2 lapply (HM2 … (refl ??)) -HM2
112 #Htc >HM3 @(eq_vec … (niltape ?)) #i #lei2
113 cases (le_to_or_lt_eq … (le_S_S_to_le …lei2))
114 [#lei1 cases (le_to_or_lt_eq … (le_S_S_to_le …lei1))
115 [#lei0 lapply(le_n_O_to_eq … (le_S_S_to_le …lei0)) #eqi <eqi
116 >nth_change_vec_neq [2:@eqb_false_to_not_eq %]
117 >nth_change_vec_neq [2:@eqb_false_to_not_eq %]
118 <(Heq 0) [2:@eqb_false_to_not_eq %] >Htb
119 <(Htbeq 0) [2:@eqb_false_to_not_eq %] %
120 |#Hi >Hi >nth_change_vec // >nth_change_vec // >Htc
121 >tape_move_mk_tape_R [2: #_ %1 %] >reverse_cons %
123 |#Hi >Hi >nth_change_vec_neq [2:@eqb_false_to_not_eq %]
124 >nth_change_vec_neq [2:@eqb_false_to_not_eq %]
125 <(Heq 2) [2:@eqb_false_to_not_eq %]
126 <(Htbeq 2) [%|@eqb_false_to_not_eq %]
131 (* another semantics for obj_to_cfg *)
132 definition low_char' ≝ λc.
135 | Some b ⇒ if (is_bit b) then b else null
138 lemma low_char_option : ∀s.
139 low_char' (option_map FinBool FSUnialpha bit s) = low_char s.
143 definition R_obj_to_cfg1 ≝ λt1,t2:Vector (tape FSUnialpha) 3.
145 nth cfg ? t1 (niltape ?) = midtape ? ls c [ ] →
146 let x ≝ current ? (nth obj ? t1 (niltape ?)) in
147 (∀b. x= Some ? b → is_bit b = true) →
148 t2 = change_vec ?? t1
149 (mk_tape ? [ ] (option_hd FSUnialpha (reverse ? (low_char' x::ls)))
150 (tail ? (reverse ? (low_char' x::ls)))) cfg.
152 lemma sem_obj_to_cfg1: obj_to_cfg ⊨ R_obj_to_cfg1.
153 @(Realize_to_Realize … sem_obj_to_cfg) #t1 #t2 #Hsem
154 #c #ls #Hcfg lapply(Hsem c ls Hcfg) * #HSome #HNone #Hb
155 cases (None_or_Some ? (current ? (nth obj ? t1 (niltape ?))))
156 [#Hcur >Hcur @HNone @Hcur
158 cut (low_char' (Some ? b) = b) [whd in ⊢ (??%?); >(Hb b Hb1) %] #Hlow >Hlow
159 lapply(current_to_midtape … Hb1) * #lsobj * #rsobj #Hmid
165 definition test_null_char ≝ test_char FSUnialpha (λc.c == null).
167 definition R_test_null_char_true ≝ λt1,t2.
168 current FSUnialpha t1 = Some ? null ∧ t1 = t2.
170 definition R_test_null_char_false ≝ λt1,t2.
171 current FSUnialpha t1 ≠ Some ? null ∧ t1 = t2.
173 lemma sem_test_null_char :
174 test_null_char ⊨ [ tc_true : R_test_null_char_true, R_test_null_char_false].
175 #t1 cases (sem_test_char FSUnialpha (λc.c == null) t1) #k * #outc * * #Hloop #Htrue
176 #Hfalse %{k} %{outc} % [ %
178 | #Houtc cases (Htrue ?) [| @Houtc] * #c * #Hcurt1 #Hcnull lapply (\P Hcnull)
179 -Hcnull #H destruct (H) #Houtc1 %
180 [ @Hcurt1 | <Houtc1 % ] ]
181 | #Houtc cases (Hfalse ?) [| @Houtc] #Hc #Houtc %
182 [ % #Hcurt1 >Hcurt1 in Hc; #Hc lapply (Hc ? (refl ??))
183 >(?:((null:FSUnialpha) == null) = true) [|@(\b (refl ??)) ]
188 definition cfg_to_obj ≝
189 mmove cfg FSUnialpha 2 L ·
190 (ifTM ?? (inject_TM ? test_null_char 2 cfg)
192 (copy_char_N cfg obj FSUnialpha 2)
195 inject_TM ? (move_to_end FSUnialpha L) 2 cfg ·
196 mmove cfg FSUnialpha 2 R. *)
198 definition R_cfg_to_obj ≝ λt1,t2:Vector (tape FSUnialpha) 3.
200 nth cfg ? t1 (niltape ?) = mk_tape FSUnialpha (c::ls) (None ?) [ ] →
201 (c = null → t2 = change_vec ?? t1 (midtape ? ls c [ ]) cfg) ∧
205 (midtape ? (left ? (nth obj ? t1 (niltape ?))) c (right ? (nth obj ? t1 (niltape ?)))) obj)
206 (midtape ? ls c [ ]) cfg).
208 lemma tape_move_mk_tape_L :
210 (c = None ? → ls = [ ] ∨ rs = [ ]) →
211 tape_move ? (mk_tape sig ls c rs) L =
212 mk_tape ? (tail ? ls) (option_hd ? ls) (option_cons ? c rs).
213 #sig * [ * [ * | #c * ] | #l0 #ls0 * [ *
214 [| #r0 #rs0 #H @False_ind cases (H (refl ??)) #H1 destruct (H1) ] | #c * ] ]
218 lemma sem_cfg_to_obj : cfg_to_obj ⊨ R_cfg_to_obj.
219 @(sem_seq_app FSUnialpha 2 ????? (sem_move_multi ? 2 cfg L ?)
221 (acc_sem_inject ?????? cfg ? sem_test_null_char)
223 (sem_copy_char_N …)))
226 #tc * whd in ⊢ (%→?); #Htc *
227 [ * #te * * * #Hcurtc #Hte1 #Hte2 whd in ⊢ (%→?); #Htb destruct (Htb)
229 [ #Hc >Hta in Htc; >eq_mk_tape_rightof whd in match (tape_move ???); #Htc
231 [ lapply (eq_vec_change_vec ??????? (sym_eq … Hte1) Hte2) >change_vec_same // ]
233 | #Hc >Hta in Htc; >eq_mk_tape_rightof whd in match (tape_move ???); #Htc
234 >Htc in Hcurtc; >nth_change_vec // normalize in ⊢ (%→?);
235 #H destruct (H) @False_ind cases Hc /2/ ]
236 | * #te * * * #Hcurtc #Hte1 #Hte2
239 [ >Htc in Hcurtc; >Hta >nth_change_vec //
240 normalize in ⊢ (%→?); * #H @False_ind /2/
242 [ lapply (eq_vec_change_vec ??????? (sym_eq … Hte1) Hte2)
243 >change_vec_same // ] -Hte1 -Hte2 #Hte destruct (Hte)
244 >Hta in Htc; whd in match (tape_move ???); #Htc
245 >Htc in Htb; >nth_change_vec //
246 >nth_change_vec_neq [2:@eqb_false_to_not_eq //] >Hta
251 definition char_to_move ≝ λc.match c with
252 [ bit b ⇒ if b then R else L
255 definition char_to_bit_option ≝ λc.match c with
256 [ bit b ⇒ Some ? (bit b)
259 definition R_cfg_to_obj1 ≝ λt1,t2:Vector (tape FSUnialpha) 3.
261 nth cfg ? t1 (niltape ?) = mk_tape FSUnialpha (c::ls) (None ?) [ ] →
264 tape_write ? (nth obj ? t1 (niltape ?)) (char_to_bit_option c) in
267 (tape_write ? (nth obj ? t1 (niltape ?)) (char_to_bit_option c)) obj)
268 (midtape ? ls c [ ]) cfg.
270 lemma sem_cfg_to_obj1: cfg_to_obj ⊨ R_cfg_to_obj1.
271 @(Realize_to_Realize … sem_cfg_to_obj) #t1 #t2 #H #c #ls #Hcfg #Hbar
272 cases (H c ls Hcfg) cases (true_or_false (c==null)) #Hc
273 [#Ht2 #_ >(Ht2 (\P Hc)) -Ht2 @(eq_vec … (niltape ?))
274 #i #lei2 cases (le_to_or_lt_eq … (le_S_S_to_le …lei2))
275 [#lei1 cases (le_to_or_lt_eq … (le_S_S_to_le …lei1))
276 [#lei0 lapply(le_n_O_to_eq … (le_S_S_to_le …lei0)) #eqi <eqi
277 >nth_change_vec_neq [2:@eqb_false_to_not_eq %]
278 >nth_change_vec_neq in ⊢ (???%); [2:@eqb_false_to_not_eq %]
279 >nth_change_vec // >(\P Hc) %
280 |#Hi >Hi >nth_change_vec //
282 |#Hi >Hi >nth_change_vec_neq [2:@eqb_false_to_not_eq %]
283 >nth_change_vec_neq [2:@eqb_false_to_not_eq %]
284 >nth_change_vec_neq [2:@eqb_false_to_not_eq %] %
286 |#_ #Ht2 >(Ht2 (\Pf Hc)) -Ht2 @(eq_vec … (niltape ?))
287 #i #lei2 cases (le_to_or_lt_eq … (le_S_S_to_le …lei2))
288 [#lei1 cases (le_to_or_lt_eq … (le_S_S_to_le …lei1))
289 [#lei0 lapply(le_n_O_to_eq … (le_S_S_to_le …lei0)) #eqi <eqi
290 >nth_change_vec_neq [2:@eqb_false_to_not_eq %]
291 >nth_change_vec_neq in ⊢ (???%); [2:@eqb_false_to_not_eq %]
292 >nth_change_vec // >nth_change_vec //
293 lapply (\bf Hbar) lapply Hc elim c //
294 [whd in ⊢ (??%?→?); #H destruct (H)
295 |#_ whd in ⊢ (??%?→?); #H destruct (H)
297 |#Hi >Hi >nth_change_vec //
299 |#Hi >Hi >nth_change_vec_neq [2:@eqb_false_to_not_eq %]
300 >nth_change_vec_neq [2:@eqb_false_to_not_eq %]
301 >nth_change_vec_neq [2:@eqb_false_to_not_eq %] %
307 (* macchina che muove il nastro obj a destra o sinistra a seconda del valore
308 del current di prg, che codifica la direzione in cui ci muoviamo *)
310 definition tape_move_obj : mTM FSUnialpha 2 ≝
312 (inject_TM ? (test_char ? (λc:FSUnialpha.c == bit false)) 2 prg)
313 (mmove obj FSUnialpha 2 L)
315 (inject_TM ? (test_char ? (λc:FSUnialpha.c == bit true)) 2 prg)
316 (mmove obj FSUnialpha 2 R)
321 definition R_tape_move_obj' ≝ λt1,t2:Vector (tape FSUnialpha) 3.
322 (current ? (nth prg ? t1 (niltape ?)) = Some ? (bit false) →
323 t2 = change_vec ?? t1 (tape_move ? (nth obj ? t1 (niltape ?)) L) obj) ∧
324 (current ? (nth prg ? t1 (niltape ?)) = Some ? (bit true) →
325 t2 = change_vec ?? t1 (tape_move ? (nth obj ? t1 (niltape ?)) R) obj) ∧
326 (current ? (nth prg ? t1 (niltape ?)) ≠ Some ? (bit false) →
327 current ? (nth prg ? t1 (niltape ?)) ≠ Some ? (bit true) →
330 lemma sem_tape_move_obj' : tape_move_obj ⊨ R_tape_move_obj'.
331 #ta cases (sem_if ??????????
332 (acc_sem_inject ?????? prg ? (sem_test_char ? (λc:FSUnialpha.c == bit false)))
333 (sem_move_multi ? 2 obj L ?)
335 (acc_sem_inject ?????? prg ? (sem_test_char ? (λc:FSUnialpha.c == bit true)))
336 (sem_move_multi ? 2 obj R ?)
338 #i * #outc * #Hloop #HR %{i} %{outc} % [@Hloop] -i
340 [ * #tb * * * * #c * #Hcurta_prg #Hc lapply (\P Hc) -Hc #Hc #Htb1 #Htb2
341 whd in ⊢ (%→%); #Houtc >Houtc -Houtc % [ %
342 [ >Hcurta_prg #H destruct (H) >(?:tb = ta)
343 [| lapply (eq_vec_change_vec ??????? Htb1 Htb2)
344 >change_vec_same // ] %
345 | >Hcurta_prg #H destruct (H) destruct (Hc) ]
346 | >Hcurta_prg >Hc * #H @False_ind /2/ ]
347 | * #tb * * * #Hnotfalse #Htb1 #Htb2 cut (tb = ta)
348 [ lapply (eq_vec_change_vec ??????? Htb1 Htb2)
349 >change_vec_same // ] -Htb1 -Htb2 #Htb destruct (Htb) *
350 [ * #tc * * * * #c * #Hcurta_prg #Hc lapply (\P Hc) -Hc #Hc #Htc1 #Htc2
351 whd in ⊢ (%→%); #Houtc >Houtc -Houtc % [ %
352 [ >Hcurta_prg #H destruct (H) destruct (Hc)
353 | >Hcurta_prg #H destruct (H) >(?:tc = ta)
354 [| lapply (eq_vec_change_vec ??????? Htc1 Htc2)
355 >change_vec_same // ] % ]
356 | >Hcurta_prg >Hc #_ * #H @False_ind /2/ ]
357 | * #tc * * * #Hnottrue #Htc1 #Htc2 cut (tc = ta)
358 [ lapply (eq_vec_change_vec ??????? Htc1 Htc2)
359 >change_vec_same // ] -Htc1 -Htc2
360 #Htc destruct (Htc) whd in ⊢ (%→?); #Houtc % [ %
361 [ #Hcurta_prg lapply (\Pf (Hnotfalse ? Hcurta_prg)) * #H @False_ind /2/
362 | #Hcurta_prg lapply (\Pf (Hnottrue ? Hcurta_prg)) * #H @False_ind /2/ ]
368 definition R_tape_move_obj ≝ λt1,t2:Vector (tape FSUnialpha) 3.
369 ∀c. current ? (nth prg ? t1 (niltape ?)) = Some ? c →
370 t2 = change_vec ?? t1 (tape_move ? (nth obj ? t1 (niltape ?)) (char_to_move c)) obj.
372 lemma sem_tape_move_obj : tape_move_obj ⊨ R_tape_move_obj.
373 @(Realize_to_Realize … sem_tape_move_obj')
374 #ta #tb * * #Htb1 #Htb2 #Htb3 * [ *
376 | #Hcurta_prg change with (nth obj ? ta (niltape ?)) in match (tape_move ???);
377 >change_vec_same @Htb3 >Hcurta_prg % #H destruct (H)
378 | #Hcurta_prg change with (nth obj ? ta (niltape ?)) in match (tape_move ???);
379 >change_vec_same @Htb3 >Hcurta_prg % #H destruct (H)
383 (************** list of tape ******************)
384 definition list_of_tape ≝ λsig.λt:tape sig.
385 reverse ? (left ? t)@option_cons ? (current ? t) (right ? t).
387 lemma list_of_midtape: ∀sig,ls,c,rs.
388 list_of_tape sig (midtape ? ls c rs) = reverse ? ls@c::rs.
391 lemma list_of_rightof: ∀sig,ls,c.
392 list_of_tape sig (rightof ? c ls) = reverse ? (c::ls).
393 #sig #ls #c <(append_nil ? (reverse ? (c::ls)))
396 lemma list_of_tape_move: ∀sig,t,m.
397 list_of_tape sig t = list_of_tape sig (tape_move ? t m).
398 #sig #t * // cases t //
399 [(* rightof, move L *) #a #l >list_of_midtape
400 >append_cons <reverse_single <reverse_append %
401 |(* midtape, move L *) * //
402 #a #ls #c #rs >list_of_midtape >list_of_midtape
403 >reverse_cons >associative_append %
404 |(* midtape, move R *) #ls #c *
405 [>list_of_midtape >list_of_rightof >reverse_cons %
406 |#a #rs >list_of_midtape >list_of_midtape >reverse_cons
407 >associative_append %
412 lemma list_of_tape_write: ∀sig,cond,t,c.
413 (∀b. c = Some ? b → cond b =true) →
414 (∀x. mem ? x (list_of_tape ? t) → cond x =true ) →
415 ∀x. mem ? x (list_of_tape sig (tape_write ? t c)) → cond x =true.
416 #sig #cond #t #c #Hc #Htape #x lapply Hc cases c
417 [(* c is None *) #_ whd in match (tape_write ???); @Htape
418 |#b #Hb lapply (Hb … (refl ??)) -Hb #Hb
419 whd in match (tape_write ???); >list_of_midtape
420 #Hx cases(mem_append ???? Hx) -Hx
421 [#Hx @Htape @mem_append_l1 @Hx
423 #Hx @Htape @mem_append_l2 cases (current sig t)
429 lemma current_in_list: ∀sig,t,b.
430 current sig t = Some ? b → mem ? b (list_of_tape sig t).
432 [whd in ⊢ (??%?→?); #Htmp destruct
433 |#l #b whd in ⊢ (??%?→?); #Htmp destruct
434 |#l #b whd in ⊢ (??%?→?); #Htmp destruct
435 |#ls #c #rs whd in ⊢ (??%?→?); #Htmp destruct
436 >list_of_midtape @mem_append_l2 % %
440 definition restart_tape ≝ λi,n.
441 mmove i FSUnialpha n L ·
442 inject_TM ? (move_to_end FSUnialpha L) n i ·
443 mmove i FSUnialpha n R.
445 definition R_restart_tape ≝ λi,n.λint,outt:Vector (tape FSUnialpha) (S n).
446 ∀t.t = nth i ? int (niltape ?) →
447 outt = change_vec ?? int
448 (mk_tape ? [ ] (option_hd ? (list_of_tape ? t)) (tail ? (list_of_tape ? t))) i.
450 lemma sem_restart_tape : ∀i,n.i < S n → restart_tape i n ⊨ R_restart_tape i n.
452 @(sem_seq_app ??????? (sem_move_multi ? n i L ?)
453 (sem_seq ?????? (sem_inject ???? i ? (sem_move_to_end_l ?))
454 (sem_move_multi ? n i R ?))) [1,2,3:@le_S_S_to_le //]
455 #ta #tb * #tc * whd in ⊢ (%→?); #Htc
456 * #td * * * #Htd1 #Htd2 #Htd3
457 whd in ⊢ (%→?); #Htb *
458 [ #Hta_i <Hta_i in Htc; whd in ⊢ (???(????%?)→?); #Htc
460 [ <(change_vec_same … tc … i … (niltape ?))
461 @(eq_vec_change_vec … (niltape ?))
462 [ @Htd1 >Htc >nth_change_vec //
464 (* >Htc in Htd1; >nth_change_vec // *) -Htd1 -Htd2 -Htd3
465 #Htd >Htd in Htb; >Htc >change_vec_change_vec >nth_change_vec //
467 | #r0 #rs0 #Hta_i <Hta_i in Htc; whd in ⊢ (???(????%?)→?); #Htc
469 [ <(change_vec_same … tc … i … (niltape ?))
470 @(eq_vec_change_vec … (niltape ?))
471 [ @Htd1 >Htc >nth_change_vec //
473 (* >Htc in Htd1; >nth_change_vec // *) -Htd1 -Htd2 -Htd3
474 #Htd >Htd in Htb; >Htc >change_vec_change_vec >nth_change_vec //
476 | #l0 #ls0 #Hta_i <Hta_i in Htc; whd in ⊢ (???(????%?)→?); #Htc
477 cut (td = change_vec ?? tc (mk_tape ? [ ] (None ?) (reverse ? ls0@[l0])) i)
478 [ <(change_vec_same … tc … i … (niltape ?))
479 @(eq_vec_change_vec … (niltape ?))
480 [ @Htd2 >Htc >nth_change_vec //
481 | #j #Hij >nth_change_vec_neq // @Htd3 // ]]
482 #Htd >Htd in Htb; >Htc >change_vec_change_vec >change_vec_change_vec
483 >nth_change_vec // #Htb >Htb <(reverse_reverse ? ls0) in ⊢ (???%);
484 cases (reverse ? ls0)
486 | #l1 #ls1 >reverse_cons
487 >(?: list_of_tape ? (rightof ? l0 (reverse ? ls1@[l1])) =
489 [|change with (reverse ??@?) in ⊢ (??%?);
490 whd in match (left ??); >reverse_cons >reverse_append
491 whd in ⊢ (??%?); @eq_f >reverse_reverse normalize >append_nil % ] % ]
493 [ #c #rs #Hta_i <Hta_i in Htc; whd in ⊢ (???(????%?)→?); #Htc
495 [ <(change_vec_same … tc … i … (niltape ?))
496 @(eq_vec_change_vec … (niltape ?))
497 [ @Htd1 >Htc >nth_change_vec //
499 (* >Htc in Htd1; >nth_change_vec // *) -Htd1 -Htd2 -Htd3
500 #Htd >Htd in Htb; >Htc >change_vec_change_vec >nth_change_vec //
502 | #l0 #ls0 #c #rs #Hta_i <Hta_i in Htc; whd in ⊢ (???(????%?)→?); #Htc
503 cut (td = change_vec ?? tc (mk_tape ? [ ] (None ?) (reverse ? ls0@l0::c::rs)) i)
504 [ @(eq_vec_change_vec … (niltape ?))
505 [ @Htd2 >Htc >nth_change_vec //
507 #Htd >Htd in Htb; >Htc >change_vec_change_vec >change_vec_change_vec
508 >nth_change_vec // #Htb >Htb <(reverse_reverse ? ls0) in ⊢ (???%);
509 cases (reverse ? ls0)
511 | #l1 #ls1 >reverse_cons
512 >(?: list_of_tape ? (midtape ? (l0::reverse ? ls1@[l1]) c rs) =
514 [|change with (reverse ??@?) in ⊢ (??%?);
515 whd in match (left ??); >reverse_cons >reverse_append
516 whd in ⊢ (??%?); @eq_f >reverse_reverse normalize
517 >associative_append % ] % ]