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 eq_vec_change_vec : ∀sig,n.∀v1,v2:Vector sig n.∀i,t,d.
63 (∀j.i ≠ j → nth j ? v1 d = nth j ? v2 d) →
64 v2 = change_vec ?? v1 t i.
65 #sig #n #v1 #v2 #i #t #d #H1 #H2 @(eq_vec … d)
66 #i0 #Hlt cases (decidable_eq_nat i0 i) #Hii0
67 [ >Hii0 >nth_change_vec //
68 | >nth_change_vec_neq [|@sym_not_eq //] @sym_eq @H2 @sym_not_eq // ]
71 lemma None_or_Some: ∀A.∀a. a =None A ∨ ∃b. a = Some ? b.
75 lemma not_None_to_Some: ∀A.∀a. a ≠ None A → ∃b. a = Some ? b.
76 #A * /2/ * #H @False_ind @H %
79 lemma sem_obj_to_cfg : obj_to_cfg ⊨ R_obj_to_cfg.
80 @(sem_seq_app FSUnialpha 2 ?????
82 (sem_test_null_multi ?? obj ?)
84 (sem_inject ???? cfg ? (sem_write FSUnialpha null)))
85 (sem_seq ?????? (sem_inject ???? cfg ? (sem_move_to_end_l ?))
86 (sem_move_multi ? 2 cfg R ?))) //
88 #tb * #Hif * #tc * #HM2 #HM3 #c #ls #Hcfg
91 [ * #t1 * * #Hcurta #Ht1 destruct (Ht1)
92 lapply (not_None_to_Some … Hcurta) * #curta #Hcurtaeq
93 whd in ⊢ (%→?); #Htb % [2: #Hcur @False_ind /2/]
94 #lso #xo #rso #Hobjta cases HM2 whd in ⊢ (%→?); * #_
95 #HM2 #Heq >Htb in HM2; >nth_change_vec [2: @leb_true_to_le %]
96 >Hcfg >Hcurtaeq #HM2 lapply (HM2 … (refl ??)) -HM2
97 whd in match (left ??); whd in match (right ??);
98 >reverse_cons #Htc >HM3 @(eq_vec … (niltape ?)) #i #lei2
99 cases (le_to_or_lt_eq … (le_S_S_to_le …lei2))
100 [#lei1 cases (le_to_or_lt_eq … (le_S_S_to_le …lei1))
101 [#lei0 lapply(le_n_O_to_eq … (le_S_S_to_le …lei0)) #eqi <eqi
102 >nth_change_vec_neq [2:@eqb_false_to_not_eq %]
103 <(Heq 0) [2:@eqb_false_to_not_eq %] >Htb
104 >nth_change_vec_neq [2:@eqb_false_to_not_eq %]
105 >nth_change_vec_neq [%|2:@eqb_false_to_not_eq %]
106 |#Hi >Hi >nth_change_vec // >nth_change_vec // >Htc
107 >Hobjta in Hcurtaeq; whd in ⊢ (??%?→?); #Htmp destruct(Htmp)
108 >tape_move_mk_tape_R [2: #_ %1 %] %
110 |#Hi >Hi >nth_change_vec_neq [2:@eqb_false_to_not_eq %]
111 >nth_change_vec_neq [2:@eqb_false_to_not_eq %]
112 <(Heq 2) [2:@eqb_false_to_not_eq %] >Htb
113 >nth_change_vec_neq [%|2:@eqb_false_to_not_eq %]
115 | * #t1 * * #Hcurta #Ht1 destruct (Ht1)
116 * whd in ⊢ (%→?); #Htb lapply (Htb … Hcfg) -Htb #Htb
118 [#lso #xo #rso #Hmid @False_ind >Hmid in Hcurta;
119 whd in ⊢ (??%?→?); #Htmp destruct (Htmp)]
120 #_ cases HM2 whd in ⊢ (%→?); * #_
121 #HM2 #Heq >Htb in HM2; #HM2 lapply (HM2 … (refl ??)) -HM2
122 #Htc >HM3 @(eq_vec … (niltape ?)) #i #lei2
123 cases (le_to_or_lt_eq … (le_S_S_to_le …lei2))
124 [#lei1 cases (le_to_or_lt_eq … (le_S_S_to_le …lei1))
125 [#lei0 lapply(le_n_O_to_eq … (le_S_S_to_le …lei0)) #eqi <eqi
126 >nth_change_vec_neq [2:@eqb_false_to_not_eq %]
127 >nth_change_vec_neq [2:@eqb_false_to_not_eq %]
128 <(Heq 0) [2:@eqb_false_to_not_eq %] >Htb
129 <(Htbeq 0) [2:@eqb_false_to_not_eq %] %
130 |#Hi >Hi >nth_change_vec // >nth_change_vec // >Htc
131 >tape_move_mk_tape_R [2: #_ %1 %] >reverse_cons %
133 |#Hi >Hi >nth_change_vec_neq [2:@eqb_false_to_not_eq %]
134 >nth_change_vec_neq [2:@eqb_false_to_not_eq %]
135 <(Heq 2) [2:@eqb_false_to_not_eq %]
136 <(Htbeq 2) [%|@eqb_false_to_not_eq %]
141 (* another semantics for obj_to_cfg *)
142 definition low_char' ≝ λc.
145 | Some b ⇒ if (is_bit b) then b else null
148 lemma low_char_option : ∀s.
149 low_char' (option_map FinBool FSUnialpha bit s) = low_char s.
153 definition R_obj_to_cfg1 ≝ λt1,t2:Vector (tape FSUnialpha) 3.
155 nth cfg ? t1 (niltape ?) = midtape ? ls c [ ] →
156 let x ≝ current ? (nth obj ? t1 (niltape ?)) in
157 (∀b. x= Some ? b → is_bit b = true) →
158 t2 = change_vec ?? t1
159 (mk_tape ? [ ] (option_hd FSUnialpha (reverse ? (low_char' x::ls)))
160 (tail ? (reverse ? (low_char' x::ls)))) cfg.
162 lemma sem_obj_to_cfg1: obj_to_cfg ⊨ R_obj_to_cfg1.
163 @(Realize_to_Realize … sem_obj_to_cfg) #t1 #t2 #Hsem
164 #c #ls #Hcfg lapply(Hsem c ls Hcfg) * #HSome #HNone #Hb
165 cases (None_or_Some ? (current ? (nth obj ? t1 (niltape ?))))
166 [#Hcur >Hcur @HNone @Hcur
168 cut (low_char' (Some ? b) = b) [whd in ⊢ (??%?); >(Hb b Hb1) %] #Hlow >Hlow
169 lapply(current_to_midtape … Hb1) * #lsobj * #rsobj #Hmid
175 definition test_null_char ≝ test_char FSUnialpha (λc.c == null).
177 definition R_test_null_char_true ≝ λt1,t2.
178 current FSUnialpha t1 = Some ? null ∧ t1 = t2.
180 definition R_test_null_char_false ≝ λt1,t2.
181 current FSUnialpha t1 ≠ Some ? null ∧ t1 = t2.
183 lemma sem_test_null_char :
184 test_null_char ⊨ [ tc_true : R_test_null_char_true, R_test_null_char_false].
185 #t1 cases (sem_test_char FSUnialpha (λc.c == null) t1) #k * #outc * * #Hloop #Htrue
186 #Hfalse %{k} %{outc} % [ %
188 | #Houtc cases (Htrue ?) [| @Houtc] * #c * #Hcurt1 #Hcnull lapply (\P Hcnull)
189 -Hcnull #H destruct (H) #Houtc1 %
190 [ @Hcurt1 | <Houtc1 % ] ]
191 | #Houtc cases (Hfalse ?) [| @Houtc] #Hc #Houtc %
192 [ % #Hcurt1 >Hcurt1 in Hc; #Hc lapply (Hc ? (refl ??))
193 >(?:((null:FSUnialpha) == null) = true) [|@(\b (refl ??)) ]
198 definition cfg_to_obj ≝
199 mmove cfg FSUnialpha 2 L ·
200 (ifTM ?? (inject_TM ? test_null_char 2 cfg)
202 (copy_char_N cfg obj FSUnialpha 2)
205 inject_TM ? (move_to_end FSUnialpha L) 2 cfg ·
206 mmove cfg FSUnialpha 2 R. *)
208 definition R_cfg_to_obj ≝ λt1,t2:Vector (tape FSUnialpha) 3.
210 nth cfg ? t1 (niltape ?) = mk_tape FSUnialpha (c::ls) (None ?) [ ] →
211 (c = null → t2 = change_vec ?? t1 (midtape ? ls c [ ]) cfg) ∧
215 (midtape ? (left ? (nth obj ? t1 (niltape ?))) c (right ? (nth obj ? t1 (niltape ?)))) obj)
216 (midtape ? ls c [ ]) cfg).
218 lemma tape_move_mk_tape_L :
220 (c = None ? → ls = [ ] ∨ rs = [ ]) →
221 tape_move ? (mk_tape sig ls c rs) L =
222 mk_tape ? (tail ? ls) (option_hd ? ls) (option_cons ? c rs).
223 #sig * [ * [ * | #c * ] | #l0 #ls0 * [ *
224 [| #r0 #rs0 #H @False_ind cases (H (refl ??)) #H1 destruct (H1) ] | #c * ] ]
228 lemma sem_cfg_to_obj : cfg_to_obj ⊨ R_cfg_to_obj.
229 @(sem_seq_app FSUnialpha 2 ????? (sem_move_multi ? 2 cfg L ?)
231 (acc_sem_inject ?????? cfg ? sem_test_null_char)
233 (sem_copy_char_N …)))
236 #tc * whd in ⊢ (%→?); #Htc *
237 [ * #te * * * #Hcurtc #Hte1 #Hte2 whd in ⊢ (%→?); #Htb destruct (Htb)
239 [ #Hc >Hta in Htc; >eq_mk_tape_rightof whd in match (tape_move ???); #Htc
241 [ lapply (eq_vec_change_vec ??????? (sym_eq … Hte1) Hte2) >change_vec_same // ]
243 | #Hc >Hta in Htc; >eq_mk_tape_rightof whd in match (tape_move ???); #Htc
244 >Htc in Hcurtc; >nth_change_vec // normalize in ⊢ (%→?);
245 #H destruct (H) @False_ind cases Hc /2/ ]
246 | * #te * * * #Hcurtc #Hte1 #Hte2
249 [ >Htc in Hcurtc; >Hta >nth_change_vec //
250 normalize in ⊢ (%→?); * #H @False_ind /2/
252 [ lapply (eq_vec_change_vec ??????? (sym_eq … Hte1) Hte2)
253 >change_vec_same // ] -Hte1 -Hte2 #Hte destruct (Hte)
254 >Hta in Htc; whd in match (tape_move ???); #Htc
255 >Htc in Htb; >nth_change_vec //
256 >nth_change_vec_neq [2:@eqb_false_to_not_eq //] >Hta
261 definition char_to_move ≝ λc.match c with
262 [ bit b ⇒ if b then R else L
265 definition char_to_bit_option ≝ λc.match c with
266 [ bit b ⇒ Some ? (bit b)
269 definition R_cfg_to_obj1 ≝ λt1,t2:Vector (tape FSUnialpha) 3.
271 nth cfg ? t1 (niltape ?) = mk_tape FSUnialpha (c::ls) (None ?) [ ] →
274 tape_write ? (nth obj ? t1 (niltape ?)) (char_to_bit_option c) in
277 (tape_write ? (nth obj ? t1 (niltape ?)) (char_to_bit_option c)) obj)
278 (midtape ? ls c [ ]) cfg.
280 lemma sem_cfg_to_obj1: cfg_to_obj ⊨ R_cfg_to_obj1.
281 @(Realize_to_Realize … sem_cfg_to_obj) #t1 #t2 #H #c #ls #Hcfg #Hbar
282 cases (H c ls Hcfg) cases (true_or_false (c==null)) #Hc
283 [#Ht2 #_ >(Ht2 (\P Hc)) -Ht2 @(eq_vec … (niltape ?))
284 #i #lei2 cases (le_to_or_lt_eq … (le_S_S_to_le …lei2))
285 [#lei1 cases (le_to_or_lt_eq … (le_S_S_to_le …lei1))
286 [#lei0 lapply(le_n_O_to_eq … (le_S_S_to_le …lei0)) #eqi <eqi
287 >nth_change_vec_neq [2:@eqb_false_to_not_eq %]
288 >nth_change_vec_neq in ⊢ (???%); [2:@eqb_false_to_not_eq %]
289 >nth_change_vec // >(\P Hc) %
290 |#Hi >Hi >nth_change_vec //
292 |#Hi >Hi >nth_change_vec_neq [2:@eqb_false_to_not_eq %]
293 >nth_change_vec_neq [2:@eqb_false_to_not_eq %]
294 >nth_change_vec_neq [2:@eqb_false_to_not_eq %] %
296 |#_ #Ht2 >(Ht2 (\Pf Hc)) -Ht2 @(eq_vec … (niltape ?))
297 #i #lei2 cases (le_to_or_lt_eq … (le_S_S_to_le …lei2))
298 [#lei1 cases (le_to_or_lt_eq … (le_S_S_to_le …lei1))
299 [#lei0 lapply(le_n_O_to_eq … (le_S_S_to_le …lei0)) #eqi <eqi
300 >nth_change_vec_neq [2:@eqb_false_to_not_eq %]
301 >nth_change_vec_neq in ⊢ (???%); [2:@eqb_false_to_not_eq %]
302 >nth_change_vec // >nth_change_vec //
303 lapply (\bf Hbar) lapply Hc elim c //
304 [whd in ⊢ (??%?→?); #H destruct (H)
305 |#_ whd in ⊢ (??%?→?); #H destruct (H)
307 |#Hi >Hi >nth_change_vec //
309 |#Hi >Hi >nth_change_vec_neq [2:@eqb_false_to_not_eq %]
310 >nth_change_vec_neq [2:@eqb_false_to_not_eq %]
311 >nth_change_vec_neq [2:@eqb_false_to_not_eq %] %
317 (* macchina che muove il nastro obj a destra o sinistra a seconda del valore
318 del current di prg, che codifica la direzione in cui ci muoviamo *)
320 definition tape_move_obj : mTM FSUnialpha 2 ≝
322 (inject_TM ? (test_char ? (λc:FSUnialpha.c == bit false)) 2 prg)
323 (mmove obj FSUnialpha 2 L)
325 (inject_TM ? (test_char ? (λc:FSUnialpha.c == bit true)) 2 prg)
326 (mmove obj FSUnialpha 2 R)
331 definition R_tape_move_obj' ≝ λt1,t2:Vector (tape FSUnialpha) 3.
332 (current ? (nth prg ? t1 (niltape ?)) = Some ? (bit false) →
333 t2 = change_vec ?? t1 (tape_move ? (nth obj ? t1 (niltape ?)) L) obj) ∧
334 (current ? (nth prg ? t1 (niltape ?)) = Some ? (bit true) →
335 t2 = change_vec ?? t1 (tape_move ? (nth obj ? t1 (niltape ?)) R) obj) ∧
336 (current ? (nth prg ? t1 (niltape ?)) ≠ Some ? (bit false) →
337 current ? (nth prg ? t1 (niltape ?)) ≠ Some ? (bit true) →
340 lemma sem_tape_move_obj' : tape_move_obj ⊨ R_tape_move_obj'.
341 #ta cases (sem_if ??????????
342 (acc_sem_inject ?????? prg ? (sem_test_char ? (λc:FSUnialpha.c == bit false)))
343 (sem_move_multi ? 2 obj L ?)
345 (acc_sem_inject ?????? prg ? (sem_test_char ? (λc:FSUnialpha.c == bit true)))
346 (sem_move_multi ? 2 obj R ?)
348 #i * #outc * #Hloop #HR %{i} %{outc} % [@Hloop] -i
350 [ * #tb * * * * #c * #Hcurta_prg #Hc lapply (\P Hc) -Hc #Hc #Htb1 #Htb2
351 whd in ⊢ (%→%); #Houtc >Houtc -Houtc % [ %
352 [ >Hcurta_prg #H destruct (H) >(?:tb = ta)
353 [| lapply (eq_vec_change_vec ??????? Htb1 Htb2)
354 >change_vec_same // ] %
355 | >Hcurta_prg #H destruct (H) destruct (Hc) ]
356 | >Hcurta_prg >Hc * #H @False_ind /2/ ]
357 | * #tb * * * #Hnotfalse #Htb1 #Htb2 cut (tb = ta)
358 [ lapply (eq_vec_change_vec ??????? Htb1 Htb2)
359 >change_vec_same // ] -Htb1 -Htb2 #Htb destruct (Htb) *
360 [ * #tc * * * * #c * #Hcurta_prg #Hc lapply (\P Hc) -Hc #Hc #Htc1 #Htc2
361 whd in ⊢ (%→%); #Houtc >Houtc -Houtc % [ %
362 [ >Hcurta_prg #H destruct (H) destruct (Hc)
363 | >Hcurta_prg #H destruct (H) >(?:tc = ta)
364 [| lapply (eq_vec_change_vec ??????? Htc1 Htc2)
365 >change_vec_same // ] % ]
366 | >Hcurta_prg >Hc #_ * #H @False_ind /2/ ]
367 | * #tc * * * #Hnottrue #Htc1 #Htc2 cut (tc = ta)
368 [ lapply (eq_vec_change_vec ??????? Htc1 Htc2)
369 >change_vec_same // ] -Htc1 -Htc2
370 #Htc destruct (Htc) whd in ⊢ (%→?); #Houtc % [ %
371 [ #Hcurta_prg lapply (\Pf (Hnotfalse ? Hcurta_prg)) * #H @False_ind /2/
372 | #Hcurta_prg lapply (\Pf (Hnottrue ? Hcurta_prg)) * #H @False_ind /2/ ]
378 definition R_tape_move_obj ≝ λt1,t2:Vector (tape FSUnialpha) 3.
379 ∀c. current ? (nth prg ? t1 (niltape ?)) = Some ? c →
380 t2 = change_vec ?? t1 (tape_move ? (nth obj ? t1 (niltape ?)) (char_to_move c)) obj.
382 lemma sem_tape_move_obj : tape_move_obj ⊨ R_tape_move_obj.
383 @(Realize_to_Realize … sem_tape_move_obj')
384 #ta #tb * * #Htb1 #Htb2 #Htb3 * [ *
386 | #Hcurta_prg change with (nth obj ? ta (niltape ?)) in match (tape_move ???);
387 >change_vec_same @Htb3 >Hcurta_prg % #H destruct (H)
388 | #Hcurta_prg change with (nth obj ? ta (niltape ?)) in match (tape_move ???);
389 >change_vec_same @Htb3 >Hcurta_prg % #H destruct (H)
393 (************** list of tape ******************)
394 definition list_of_tape ≝ λsig.λt:tape sig.
395 reverse ? (left ? t)@option_cons ? (current ? t) (right ? t).
397 lemma list_of_midtape: ∀sig,ls,c,rs.
398 list_of_tape sig (midtape ? ls c rs) = reverse ? ls@c::rs.
401 lemma list_of_rightof: ∀sig,ls,c.
402 list_of_tape sig (rightof ? c ls) = reverse ? (c::ls).
403 #sig #ls #c <(append_nil ? (reverse ? (c::ls)))
406 lemma list_of_tape_move: ∀sig,t,m.
407 list_of_tape sig t = list_of_tape sig (tape_move ? t m).
408 #sig #t * // cases t //
409 [(* rightof, move L *) #a #l >list_of_midtape
410 >append_cons <reverse_single <reverse_append %
411 |(* midtape, move L *) * //
412 #a #ls #c #rs >list_of_midtape >list_of_midtape
413 >reverse_cons >associative_append %
414 |(* midtape, move R *) #ls #c *
415 [>list_of_midtape >list_of_rightof >reverse_cons %
416 |#a #rs >list_of_midtape >list_of_midtape >reverse_cons
417 >associative_append %
422 lemma list_of_tape_write: ∀sig,cond,t,c.
423 (∀b. c = Some ? b → cond b =true) →
424 (∀x. mem ? x (list_of_tape ? t) → cond x =true ) →
425 ∀x. mem ? x (list_of_tape sig (tape_write ? t c)) → cond x =true.
426 #sig #cond #t #c #Hc #Htape #x lapply Hc cases c
427 [(* c is None *) #_ whd in match (tape_write ???); @Htape
428 |#b #Hb lapply (Hb … (refl ??)) -Hb #Hb
429 whd in match (tape_write ???); >list_of_midtape
430 #Hx cases(mem_append ???? Hx) -Hx
431 [#Hx @Htape @mem_append_l1 @Hx
433 #Hx @Htape @mem_append_l2 cases (current sig t)
439 lemma current_in_list: ∀sig,t,b.
440 current sig t = Some ? b → mem ? b (list_of_tape sig t).
442 [whd in ⊢ (??%?→?); #Htmp destruct
443 |#l #b whd in ⊢ (??%?→?); #Htmp destruct
444 |#l #b whd in ⊢ (??%?→?); #Htmp destruct
445 |#ls #c #rs whd in ⊢ (??%?→?); #Htmp destruct
446 >list_of_midtape @mem_append_l2 % %
450 definition restart_tape ≝ λi,n.
451 mmove i FSUnialpha n L ·
452 inject_TM ? (move_to_end FSUnialpha L) n i ·
453 mmove i FSUnialpha n R.
455 definition R_restart_tape ≝ λi,n.λint,outt:Vector (tape FSUnialpha) (S n).
456 ∀t.t = nth i ? int (niltape ?) →
457 outt = change_vec ?? int
458 (mk_tape ? [ ] (option_hd ? (list_of_tape ? t)) (tail ? (list_of_tape ? t))) i.
460 lemma sem_restart_tape : ∀i,n.i < S n → restart_tape i n ⊨ R_restart_tape i n.
462 @(sem_seq_app ??????? (sem_move_multi ? n i L ?)
463 (sem_seq ?????? (sem_inject ???? i ? (sem_move_to_end_l ?))
464 (sem_move_multi ? n i R ?))) [1,2,3:@le_S_S_to_le //]
465 #ta #tb * #tc * whd in ⊢ (%→?); #Htc
466 * #td * * * #Htd1 #Htd2 #Htd3
467 whd in ⊢ (%→?); #Htb *
468 [ #Hta_i <Hta_i in Htc; whd in ⊢ (???(????%?)→?); #Htc
469 cut (td = tc) [@daemon]
470 (* >Htc in Htd1; >nth_change_vec // *) -Htd1 -Htd2 -Htd3
471 #Htd >Htd in Htb; >Htc >change_vec_change_vec >nth_change_vec //
473 | #r0 #rs0 #Hta_i <Hta_i in Htc; whd in ⊢ (???(????%?)→?); #Htc
474 cut (td = tc) [@daemon]
475 (* >Htc in Htd1; >nth_change_vec // *) -Htd1 -Htd2 -Htd3
476 #Htd >Htd in Htb; >Htc >change_vec_change_vec >nth_change_vec //
478 | #l0 #ls0 #Hta_i <Hta_i in Htc; whd in ⊢ (???(????%?)→?); #Htc
479 cut (td = change_vec ?? tc (mk_tape ? [ ] (None ?) (reverse ? ls0@[l0])) i)
481 #Htd >Htd in Htb; >Htc >change_vec_change_vec >change_vec_change_vec
482 >nth_change_vec // #Htb >Htb <(reverse_reverse ? ls0) in ⊢ (???%);
483 cases (reverse ? ls0)
485 | #l1 #ls1 >reverse_cons
486 >(?: list_of_tape ? (rightof ? l0 (reverse ? ls1@[l1])) =
488 [|change with (reverse ??@?) in ⊢ (??%?);
489 whd in match (left ??); >reverse_cons >reverse_append
490 whd in ⊢ (??%?); @eq_f >reverse_reverse normalize >append_nil % ] % ]
492 [ #c #rs #Hta_i <Hta_i in Htc; whd in ⊢ (???(????%?)→?); #Htc
493 cut (td = tc) [@daemon]
494 (* >Htc in Htd1; >nth_change_vec // *) -Htd1 -Htd2 -Htd3
495 #Htd >Htd in Htb; >Htc >change_vec_change_vec >nth_change_vec //
497 | #l0 #ls0 #c #rs #Hta_i <Hta_i in Htc; whd in ⊢ (???(????%?)→?); #Htc
498 cut (td = change_vec ?? tc (mk_tape ? [ ] (None ?) (reverse ? ls0@l0::c::rs)) i)
500 #Htd >Htd in Htb; >Htc >change_vec_change_vec >change_vec_change_vec
501 >nth_change_vec // #Htb >Htb <(reverse_reverse ? ls0) in ⊢ (???%);
502 cases (reverse ? ls0)
504 | #l1 #ls1 >reverse_cons
505 >(?: list_of_tape ? (midtape ? (l0::reverse ? ls1@[l1]) c rs) =
507 [|change with (reverse ??@?) in ⊢ (??%?);
508 whd in match (left ??); >reverse_cons >reverse_append
509 whd in ⊢ (??%?); @eq_f >reverse_reverse normalize
510 >associative_append % ] % ]