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/auxiliary_machines.ma".
13 include "turing/auxiliary_multi_machines.ma".
14 include "turing/multi_universal/alphabet.ma".
15 include "turing/multi_universal/tuples.ma".
18 definition obj ≝ (0:DeqNat).
19 definition cfg ≝ (1:DeqNat).
20 definition prg ≝ (2:DeqNat).
22 definition obj_to_cfg ≝
23 (ifTM ?? (inject_TM ? (test_null ?) 2 obj)
24 (copy_char_N obj cfg FSUnialpha 2)
25 (inject_TM ? (write FSUnialpha null) 2 cfg)
27 inject_TM ? (move_to_end FSUnialpha L) 2 cfg ·
28 mmove cfg FSUnialpha 2 R.
30 definition R_obj_to_cfg ≝ λt1,t2:Vector (tape FSUnialpha) 3.
32 nth cfg ? t1 (niltape ?) = midtape ? ls c [ ] →
33 (∀lso,x,rso.nth obj ? t1 (niltape ?) = midtape FSUnialpha lso x rso →
35 (mk_tape ? [ ] (option_hd ? (reverse ? (x::ls))) (tail ? (reverse ? (x::ls)))) cfg) ∧
36 (current ? (nth obj ? t1 (niltape ?)) = None ? →
38 (mk_tape ? [ ] (option_hd FSUnialpha (reverse ? (null::ls)))
39 (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.
47 lemma eq_mk_tape_rightof :
48 ∀alpha,a,al.mk_tape alpha (a::al) (None ?) [ ] = rightof ? a al.
52 lemma tape_move_mk_tape_R :
54 (c = None ? → ls = [ ] ∨ rs = [ ]) →
55 tape_move ? (mk_tape sig ls c rs) R =
56 mk_tape ? (option_cons ? c ls) (option_hd ? rs) (tail ? rs).
57 #sig * [ * [ * | #c * ] | #l0 #ls0 * [ *
58 [| #r0 #rs0 #H @False_ind cases (H (refl ??)) #H1 destruct (H1) ] | #c * ] ]
62 lemma None_or_Some: ∀A.∀a. a =None A ∨ ∃b. a = Some ? b.
66 lemma not_None_to_Some: ∀A.∀a. a ≠ None A → ∃b. a = Some ? b.
67 #A * /2/ * #H @False_ind @H %
70 lemma sem_obj_to_cfg : obj_to_cfg ⊨ R_obj_to_cfg.
71 @(sem_seq_app FSUnialpha 2 ?????
73 (sem_test_null_multi ?? obj ?)
75 (sem_inject ???? cfg ? (sem_write FSUnialpha null)))
76 (sem_seq ?????? (sem_inject ???? cfg ? (sem_move_to_end_l ?))
77 (sem_move_multi ? 2 cfg R ?))) //
79 #tb * #Hif * #tc * #HM2 #HM3 #c #ls #Hcfg
82 [ * #t1 * * #Hcurta #Ht1 destruct (Ht1)
83 lapply (not_None_to_Some … Hcurta) * #curta #Hcurtaeq
84 whd in ⊢ (%→?); #Htb % [2: #Hcur @False_ind /2/]
85 #lso #xo #rso #Hobjta cases HM2 whd in ⊢ (%→?); * #_
86 #HM2 #Heq >Htb in HM2; >nth_change_vec [2: @leb_true_to_le %]
87 >Hcfg >Hcurtaeq #HM2 lapply (HM2 … (refl ??)) -HM2
88 whd in match (left ??); whd in match (right ??);
89 >reverse_cons #Htc >HM3 @(eq_vec … (niltape ?)) #i #lei2
90 cases (le_to_or_lt_eq … (le_S_S_to_le …lei2))
91 [#lei1 cases (le_to_or_lt_eq … (le_S_S_to_le …lei1))
92 [#lei0 lapply(le_n_O_to_eq … (le_S_S_to_le …lei0)) #eqi <eqi
93 >nth_change_vec_neq [2:@eqb_false_to_not_eq %]
94 <(Heq 0) [2:@eqb_false_to_not_eq %] >Htb
95 >nth_change_vec_neq [2:@eqb_false_to_not_eq %]
96 >nth_change_vec_neq [%|2:@eqb_false_to_not_eq %]
97 |#Hi >Hi >nth_change_vec // >nth_change_vec // >Htc
98 >Hobjta in Hcurtaeq; whd in ⊢ (??%?→?); #Htmp destruct(Htmp)
99 >tape_move_mk_tape_R [2: #_ %1 %] %
101 |#Hi >Hi >nth_change_vec_neq [2:@eqb_false_to_not_eq %]
102 >nth_change_vec_neq [2:@eqb_false_to_not_eq %]
103 <(Heq 2) [2:@eqb_false_to_not_eq %] >Htb
104 >nth_change_vec_neq [%|2:@eqb_false_to_not_eq %]
106 | * #t1 * * #Hcurta #Ht1 destruct (Ht1)
107 * whd in ⊢ (%→?); #Htb lapply (Htb … Hcfg) -Htb #Htb
109 [#lso #xo #rso #Hmid @False_ind >Hmid in Hcurta;
110 whd in ⊢ (??%?→?); #Htmp destruct (Htmp)]
111 #_ cases HM2 whd in ⊢ (%→?); * #_
112 #HM2 #Heq >Htb in HM2; #HM2 lapply (HM2 … (refl ??)) -HM2
113 #Htc >HM3 @(eq_vec … (niltape ?)) #i #lei2
114 cases (le_to_or_lt_eq … (le_S_S_to_le …lei2))
115 [#lei1 cases (le_to_or_lt_eq … (le_S_S_to_le …lei1))
116 [#lei0 lapply(le_n_O_to_eq … (le_S_S_to_le …lei0)) #eqi <eqi
117 >nth_change_vec_neq [2:@eqb_false_to_not_eq %]
118 >nth_change_vec_neq [2:@eqb_false_to_not_eq %]
119 <(Heq 0) [2:@eqb_false_to_not_eq %] >Htb
120 <(Htbeq 0) [2:@eqb_false_to_not_eq %] %
121 |#Hi >Hi >nth_change_vec // >nth_change_vec // >Htc
122 >tape_move_mk_tape_R [2: #_ %1 %] >reverse_cons %
124 |#Hi >Hi >nth_change_vec_neq [2:@eqb_false_to_not_eq %]
125 >nth_change_vec_neq [2:@eqb_false_to_not_eq %]
126 <(Heq 2) [2:@eqb_false_to_not_eq %]
127 <(Htbeq 2) [%|@eqb_false_to_not_eq %]
132 (* another semantics for obj_to_cfg *)
133 definition low_char' ≝ λc.
136 | Some b ⇒ if (is_bit b) then b else null
139 lemma low_char_option : ∀s.
140 low_char' (option_map FinBool FSUnialpha bit s) = low_char s.
144 definition R_obj_to_cfg1 ≝ λt1,t2:Vector (tape FSUnialpha) 3.
146 nth cfg ? t1 (niltape ?) = midtape ? ls c [ ] →
147 let x ≝ current ? (nth obj ? t1 (niltape ?)) in
148 (∀b. x= Some ? b → is_bit b = true) →
149 t2 = change_vec ?? t1
150 (mk_tape ? [ ] (option_hd FSUnialpha (reverse ? (low_char' x::ls)))
151 (tail ? (reverse ? (low_char' x::ls)))) cfg.
153 lemma sem_obj_to_cfg1: obj_to_cfg ⊨ R_obj_to_cfg1.
154 @(Realize_to_Realize … sem_obj_to_cfg) #t1 #t2 #Hsem
155 #c #ls #Hcfg lapply(Hsem c ls Hcfg) * #HSome #HNone #Hb
156 cases (None_or_Some ? (current ? (nth obj ? t1 (niltape ?))))
157 [#Hcur >Hcur @HNone @Hcur
159 cut (low_char' (Some ? b) = b) [whd in ⊢ (??%?); >(Hb b Hb1) %] #Hlow >Hlow
160 lapply(current_to_midtape … Hb1) * #lsobj * #rsobj #Hmid
166 definition test_null_char ≝ test_char FSUnialpha (λc.c == null).
168 definition R_test_null_char_true ≝ λt1,t2.
169 current FSUnialpha t1 = Some ? null ∧ t1 = t2.
171 definition R_test_null_char_false ≝ λt1,t2.
172 current FSUnialpha t1 ≠ Some ? null ∧ t1 = t2.
174 lemma sem_test_null_char :
175 test_null_char ⊨ [ tc_true : R_test_null_char_true, R_test_null_char_false].
176 #t1 cases (sem_test_char FSUnialpha (λc.c == null) t1) #k * #outc * * #Hloop #Htrue
177 #Hfalse %{k} %{outc} % [ %
179 | #Houtc cases (Htrue ?) [| @Houtc] * #c * #Hcurt1 #Hcnull lapply (\P Hcnull)
180 -Hcnull #H destruct (H) #Houtc1 %
181 [ @Hcurt1 | <Houtc1 % ] ]
182 | #Houtc cases (Hfalse ?) [| @Houtc] #Hc #Houtc %
183 [ % #Hcurt1 >Hcurt1 in Hc; #Hc lapply (Hc ? (refl ??))
184 >(?:((null:FSUnialpha) == null) = true) [|@(\b (refl ??)) ]
189 definition cfg_to_obj ≝
190 mmove cfg FSUnialpha 2 L ·
191 (ifTM ?? (inject_TM ? test_null_char 2 cfg)
193 (copy_char_N cfg obj FSUnialpha 2)
196 inject_TM ? (move_to_end FSUnialpha L) 2 cfg ·
197 mmove cfg FSUnialpha 2 R. *)
199 definition R_cfg_to_obj ≝ λt1,t2:Vector (tape FSUnialpha) 3.
201 nth cfg ? t1 (niltape ?) = mk_tape FSUnialpha (c::ls) (None ?) [ ] →
202 (c = null → t2 = change_vec ?? t1 (midtape ? ls c [ ]) cfg) ∧
206 (midtape ? (left ? (nth obj ? t1 (niltape ?))) c (right ? (nth obj ? t1 (niltape ?)))) obj)
207 (midtape ? ls c [ ]) cfg).
209 lemma tape_move_mk_tape_L :
211 (c = None ? → ls = [ ] ∨ rs = [ ]) →
212 tape_move ? (mk_tape sig ls c rs) L =
213 mk_tape ? (tail ? ls) (option_hd ? ls) (option_cons ? c rs).
214 #sig * [ * [ * | #c * ] | #l0 #ls0 * [ *
215 [| #r0 #rs0 #H @False_ind cases (H (refl ??)) #H1 destruct (H1) ] | #c * ] ]
219 lemma sem_cfg_to_obj : cfg_to_obj ⊨ R_cfg_to_obj.
220 @(sem_seq_app FSUnialpha 2 ????? (sem_move_multi ? 2 cfg L ?)
222 (acc_sem_inject ?????? cfg ? sem_test_null_char)
224 (sem_copy_char_N …)))
227 #tc * whd in ⊢ (%→?); #Htc *
228 [ * #te * * * #Hcurtc #Hte1 #Hte2 whd in ⊢ (%→?); #Htb destruct (Htb)
230 [ #Hc >Hta in Htc; >eq_mk_tape_rightof whd in match (tape_move ???); #Htc
232 [ lapply (eq_vec_change_vec ??????? (sym_eq … Hte1) Hte2) >change_vec_same // ]
234 | #Hc >Hta in Htc; >eq_mk_tape_rightof whd in match (tape_move ???); #Htc
235 >Htc in Hcurtc; >nth_change_vec // normalize in ⊢ (%→?);
236 #H destruct (H) @False_ind cases Hc /2/ ]
237 | * #te * * * #Hcurtc #Hte1 #Hte2
240 [ >Htc in Hcurtc; >Hta >nth_change_vec //
241 normalize in ⊢ (%→?); * #H @False_ind /2/
243 [ lapply (eq_vec_change_vec ??????? (sym_eq … Hte1) Hte2)
244 >change_vec_same // ] -Hte1 -Hte2 #Hte destruct (Hte)
245 >Hta in Htc; whd in match (tape_move ???); #Htc
246 >Htc in Htb; >nth_change_vec //
247 >nth_change_vec_neq [2:@eqb_false_to_not_eq //] >Hta
252 definition char_to_move ≝ λc.match c with
253 [ bit b ⇒ if b then R else L
256 definition char_to_bit_option ≝ λc.match c with
257 [ bit b ⇒ Some ? (bit b)
260 definition R_cfg_to_obj1 ≝ λt1,t2:Vector (tape FSUnialpha) 3.
262 nth cfg ? t1 (niltape ?) = mk_tape FSUnialpha (c::ls) (None ?) [ ] →
265 tape_write ? (nth obj ? t1 (niltape ?)) (char_to_bit_option c) in
268 (tape_write ? (nth obj ? t1 (niltape ?)) (char_to_bit_option c)) obj)
269 (midtape ? ls c [ ]) cfg.
271 lemma sem_cfg_to_obj1: cfg_to_obj ⊨ R_cfg_to_obj1.
272 @(Realize_to_Realize … sem_cfg_to_obj) #t1 #t2 #H #c #ls #Hcfg #Hbar
273 cases (H c ls Hcfg) cases (true_or_false (c==null)) #Hc
274 [#Ht2 #_ >(Ht2 (\P Hc)) -Ht2 @(eq_vec … (niltape ?))
275 #i #lei2 cases (le_to_or_lt_eq … (le_S_S_to_le …lei2))
276 [#lei1 cases (le_to_or_lt_eq … (le_S_S_to_le …lei1))
277 [#lei0 lapply(le_n_O_to_eq … (le_S_S_to_le …lei0)) #eqi <eqi
278 >nth_change_vec_neq [2:@eqb_false_to_not_eq %]
279 >nth_change_vec_neq in ⊢ (???%); [2:@eqb_false_to_not_eq %]
280 >nth_change_vec // >(\P Hc) %
281 |#Hi >Hi >nth_change_vec //
283 |#Hi >Hi >nth_change_vec_neq [2:@eqb_false_to_not_eq %]
284 >nth_change_vec_neq [2:@eqb_false_to_not_eq %]
285 >nth_change_vec_neq [2:@eqb_false_to_not_eq %] %
287 |#_ #Ht2 >(Ht2 (\Pf Hc)) -Ht2 @(eq_vec … (niltape ?))
288 #i #lei2 cases (le_to_or_lt_eq … (le_S_S_to_le …lei2))
289 [#lei1 cases (le_to_or_lt_eq … (le_S_S_to_le …lei1))
290 [#lei0 lapply(le_n_O_to_eq … (le_S_S_to_le …lei0)) #eqi <eqi
291 >nth_change_vec_neq [2:@eqb_false_to_not_eq %]
292 >nth_change_vec_neq in ⊢ (???%); [2:@eqb_false_to_not_eq %]
293 >nth_change_vec // >nth_change_vec //
294 lapply (\bf Hbar) lapply Hc elim c //
295 [whd in ⊢ (??%?→?); #H destruct (H)
296 |#_ whd in ⊢ (??%?→?); #H destruct (H)
298 |#Hi >Hi >nth_change_vec //
300 |#Hi >Hi >nth_change_vec_neq [2:@eqb_false_to_not_eq %]
301 >nth_change_vec_neq [2:@eqb_false_to_not_eq %]
302 >nth_change_vec_neq [2:@eqb_false_to_not_eq %] %
308 (* macchina che muove il nastro obj a destra o sinistra a seconda del valore
309 del current di prg, che codifica la direzione in cui ci muoviamo *)
311 definition tape_move_obj : mTM FSUnialpha 2 ≝
313 (inject_TM ? (test_char ? (λc:FSUnialpha.c == bit false)) 2 prg)
314 (mmove obj FSUnialpha 2 L)
316 (inject_TM ? (test_char ? (λc:FSUnialpha.c == bit true)) 2 prg)
317 (mmove obj FSUnialpha 2 R)
322 definition R_tape_move_obj' ≝ λt1,t2:Vector (tape FSUnialpha) 3.
323 (current ? (nth prg ? t1 (niltape ?)) = Some ? (bit false) →
324 t2 = change_vec ?? t1 (tape_move ? (nth obj ? t1 (niltape ?)) L) obj) ∧
325 (current ? (nth prg ? t1 (niltape ?)) = Some ? (bit true) →
326 t2 = change_vec ?? t1 (tape_move ? (nth obj ? t1 (niltape ?)) R) obj) ∧
327 (current ? (nth prg ? t1 (niltape ?)) ≠ Some ? (bit false) →
328 current ? (nth prg ? t1 (niltape ?)) ≠ Some ? (bit true) →
331 lemma sem_tape_move_obj' : tape_move_obj ⊨ R_tape_move_obj'.
332 #ta cases (sem_if ??????????
333 (acc_sem_inject ?????? prg ? (sem_test_char ? (λc:FSUnialpha.c == bit false)))
334 (sem_move_multi ? 2 obj L ?)
336 (acc_sem_inject ?????? prg ? (sem_test_char ? (λc:FSUnialpha.c == bit true)))
337 (sem_move_multi ? 2 obj R ?)
339 #i * #outc * #Hloop #HR %{i} %{outc} % [@Hloop] -i
341 [ * #tb * * * * #c * #Hcurta_prg #Hc lapply (\P Hc) -Hc #Hc #Htb1 #Htb2
342 whd in ⊢ (%→%); #Houtc >Houtc -Houtc % [ %
343 [ >Hcurta_prg #H destruct (H) >(?:tb = ta)
344 [| lapply (eq_vec_change_vec ??????? Htb1 Htb2)
345 >change_vec_same // ] %
346 | >Hcurta_prg #H destruct (H) destruct (Hc) ]
347 | >Hcurta_prg >Hc * #H @False_ind /2/ ]
348 | * #tb * * * #Hnotfalse #Htb1 #Htb2 cut (tb = ta)
349 [ lapply (eq_vec_change_vec ??????? Htb1 Htb2)
350 >change_vec_same // ] -Htb1 -Htb2 #Htb destruct (Htb) *
351 [ * #tc * * * * #c * #Hcurta_prg #Hc lapply (\P Hc) -Hc #Hc #Htc1 #Htc2
352 whd in ⊢ (%→%); #Houtc >Houtc -Houtc % [ %
353 [ >Hcurta_prg #H destruct (H) destruct (Hc)
354 | >Hcurta_prg #H destruct (H) >(?:tc = ta)
355 [| lapply (eq_vec_change_vec ??????? Htc1 Htc2)
356 >change_vec_same // ] % ]
357 | >Hcurta_prg >Hc #_ * #H @False_ind /2/ ]
358 | * #tc * * * #Hnottrue #Htc1 #Htc2 cut (tc = ta)
359 [ lapply (eq_vec_change_vec ??????? Htc1 Htc2)
360 >change_vec_same // ] -Htc1 -Htc2
361 #Htc destruct (Htc) whd in ⊢ (%→?); #Houtc % [ %
362 [ #Hcurta_prg lapply (\Pf (Hnotfalse ? Hcurta_prg)) * #H @False_ind /2/
363 | #Hcurta_prg lapply (\Pf (Hnottrue ? Hcurta_prg)) * #H @False_ind /2/ ]
369 definition R_tape_move_obj ≝ λt1,t2:Vector (tape FSUnialpha) 3.
370 ∀c. current ? (nth prg ? t1 (niltape ?)) = Some ? c →
371 t2 = change_vec ?? t1 (tape_move ? (nth obj ? t1 (niltape ?)) (char_to_move c)) obj.
373 lemma sem_tape_move_obj : tape_move_obj ⊨ R_tape_move_obj.
374 @(Realize_to_Realize … sem_tape_move_obj')
375 #ta #tb * * #Htb1 #Htb2 #Htb3 * [ *
377 | #Hcurta_prg change with (nth obj ? ta (niltape ?)) in match (tape_move ???);
378 >change_vec_same @Htb3 >Hcurta_prg % #H destruct (H)
379 | #Hcurta_prg change with (nth obj ? ta (niltape ?)) in match (tape_move ???);
380 >change_vec_same @Htb3 >Hcurta_prg % #H destruct (H)
384 (************** list of tape ******************)
385 definition list_of_tape ≝ λsig.λt:tape sig.
386 reverse ? (left ? t)@option_cons ? (current ? t) (right ? t).
388 lemma list_of_midtape: ∀sig,ls,c,rs.
389 list_of_tape sig (midtape ? ls c rs) = reverse ? ls@c::rs.
392 lemma list_of_rightof: ∀sig,ls,c.
393 list_of_tape sig (rightof ? c ls) = reverse ? (c::ls).
394 #sig #ls #c <(append_nil ? (reverse ? (c::ls)))
397 lemma list_of_tape_move: ∀sig,t,m.
398 list_of_tape sig t = list_of_tape sig (tape_move ? t m).
399 #sig #t * // cases t //
400 [(* rightof, move L *) #a #l >list_of_midtape
401 >append_cons <reverse_single <reverse_append %
402 |(* midtape, move L *) * //
403 #a #ls #c #rs >list_of_midtape >list_of_midtape
404 >reverse_cons >associative_append %
405 |(* midtape, move R *) #ls #c *
406 [>list_of_midtape >list_of_rightof >reverse_cons %
407 |#a #rs >list_of_midtape >list_of_midtape >reverse_cons
408 >associative_append %
413 lemma list_of_tape_write: ∀sig,cond,t,c.
414 (∀b. c = Some ? b → cond b =true) →
415 (∀x. mem ? x (list_of_tape ? t) → cond x =true ) →
416 ∀x. mem ? x (list_of_tape sig (tape_write ? t c)) → cond x =true.
417 #sig #cond #t #c #Hc #Htape #x lapply Hc cases c
418 [(* c is None *) #_ whd in match (tape_write ???); @Htape
419 |#b #Hb lapply (Hb … (refl ??)) -Hb #Hb
420 whd in match (tape_write ???); >list_of_midtape
421 #Hx cases(mem_append ???? Hx) -Hx
422 [#Hx @Htape @mem_append_l1 @Hx
424 #Hx @Htape @mem_append_l2 cases (current sig t)
430 lemma current_in_list: ∀sig,t,b.
431 current sig t = Some ? b → mem ? b (list_of_tape sig t).
433 [whd in ⊢ (??%?→?); #Htmp destruct
434 |#l #b whd in ⊢ (??%?→?); #Htmp destruct
435 |#l #b whd in ⊢ (??%?→?); #Htmp destruct
436 |#ls #c #rs whd in ⊢ (??%?→?); #Htmp destruct
437 >list_of_midtape @mem_append_l2 % %
441 definition restart_tape ≝ λi,n.
442 mmove i FSUnialpha n L ·
443 inject_TM ? (move_to_end FSUnialpha L) n i ·
444 mmove i FSUnialpha n R.
446 definition R_restart_tape ≝ λi,n.λint,outt:Vector (tape FSUnialpha) (S n).
447 ∀t.t = nth i ? int (niltape ?) →
448 outt = change_vec ?? int
449 (mk_tape ? [ ] (option_hd ? (list_of_tape ? t)) (tail ? (list_of_tape ? t))) i.
451 lemma sem_restart_tape : ∀i,n.i < S n → restart_tape i n ⊨ R_restart_tape i n.
453 @(sem_seq_app ??????? (sem_move_multi ? n i L ?)
454 (sem_seq ?????? (sem_inject ???? i ? (sem_move_to_end_l ?))
455 (sem_move_multi ? n i R ?))) [1,2,3:@le_S_S_to_le //]
456 #ta #tb * #tc * whd in ⊢ (%→?); #Htc
457 * #td * * * #Htd1 #Htd2 #Htd3
458 whd in ⊢ (%→?); #Htb *
459 [ #Hta_i <Hta_i in Htc; whd in ⊢ (???(????%?)→?); #Htc
461 [ <(change_vec_same … tc … i … (niltape ?))
462 @(eq_vec_change_vec … (niltape ?))
463 [ @Htd1 >Htc >nth_change_vec //
465 (* >Htc in Htd1; >nth_change_vec // *) -Htd1 -Htd2 -Htd3
466 #Htd >Htd in Htb; >Htc >change_vec_change_vec >nth_change_vec //
468 | #r0 #rs0 #Hta_i <Hta_i in Htc; whd in ⊢ (???(????%?)→?); #Htc
470 [ <(change_vec_same … tc … i … (niltape ?))
471 @(eq_vec_change_vec … (niltape ?))
472 [ @Htd1 >Htc >nth_change_vec //
474 (* >Htc in Htd1; >nth_change_vec // *) -Htd1 -Htd2 -Htd3
475 #Htd >Htd in Htb; >Htc >change_vec_change_vec >nth_change_vec //
477 | #l0 #ls0 #Hta_i <Hta_i in Htc; whd in ⊢ (???(????%?)→?); #Htc
478 cut (td = change_vec ?? tc (mk_tape ? [ ] (None ?) (reverse ? ls0@[l0])) i)
479 [ <(change_vec_same … tc … i … (niltape ?))
480 @(eq_vec_change_vec … (niltape ?))
481 [ @Htd2 >Htc >nth_change_vec //
482 | #j #Hij >nth_change_vec_neq // @Htd3 // ]]
483 #Htd >Htd in Htb; >Htc >change_vec_change_vec >change_vec_change_vec
484 >nth_change_vec // #Htb >Htb <(reverse_reverse ? ls0) in ⊢ (???%);
485 cases (reverse ? ls0)
487 | #l1 #ls1 >reverse_cons
488 >(?: list_of_tape ? (rightof ? l0 (reverse ? ls1@[l1])) =
490 [|change with (reverse ??@?) in ⊢ (??%?);
491 whd in match (left ??); >reverse_cons >reverse_append
492 whd in ⊢ (??%?); @eq_f >reverse_reverse normalize >append_nil % ] % ]
494 [ #c #rs #Hta_i <Hta_i in Htc; whd in ⊢ (???(????%?)→?); #Htc
496 [ <(change_vec_same … tc … i … (niltape ?))
497 @(eq_vec_change_vec … (niltape ?))
498 [ @Htd1 >Htc >nth_change_vec //
500 (* >Htc in Htd1; >nth_change_vec // *) -Htd1 -Htd2 -Htd3
501 #Htd >Htd in Htb; >Htc >change_vec_change_vec >nth_change_vec //
503 | #l0 #ls0 #c #rs #Hta_i <Hta_i in Htc; whd in ⊢ (???(????%?)→?); #Htc
504 cut (td = change_vec ?? tc (mk_tape ? [ ] (None ?) (reverse ? ls0@l0::c::rs)) i)
505 [ @(eq_vec_change_vec … (niltape ?))
506 [ @Htd2 >Htc >nth_change_vec //
508 #Htd >Htd in Htb; >Htc >change_vec_change_vec >change_vec_change_vec
509 >nth_change_vec // #Htb >Htb <(reverse_reverse ? ls0) in ⊢ (???%);
510 cases (reverse ? ls0)
512 | #l1 #ls1 >reverse_cons
513 >(?: list_of_tape ? (midtape ? (l0::reverse ? ls1@[l1]) c rs) =
515 [|change with (reverse ??@?) in ⊢ (??%?);
516 whd in match (left ??); >reverse_cons >reverse_append
517 whd in ⊢ (??%?); @eq_f >reverse_reverse normalize
518 >associative_append % ] % ]