1 (**************************************************************************)
4 (* ||A|| A project by Andrea Asperti *)
6 (* ||I|| Developers: *)
7 (* ||T|| The HELM team. *)
8 (* ||A|| http://helm.cs.unibo.it *)
10 (* \ / This file is distributed under the terms of the *)
11 (* v GNU General Public License Version 2 *)
13 (**************************************************************************)
15 include "turing/multi_universal/compare.ma".
16 include "turing/multi_universal/par_test.ma".
17 include "turing/multi_universal/moves_2.ma".
19 definition Rtc_multi_true ≝
20 λalpha,test,n,i.λt1,t2:Vector ? (S n).
21 (∃c. current alpha (nth i ? t1 (niltape ?)) = Some ? c ∧ test c = true) ∧ t2 = t1.
23 definition Rtc_multi_false ≝
24 λalpha,test,n,i.λt1,t2:Vector ? (S n).
25 (∀c. current alpha (nth i ? t1 (niltape ?)) = Some ? c → test c = false) ∧ t2 = t1.
27 lemma sem_test_char_multi :
28 ∀alpha,test,n,i.i ≤ n →
29 inject_TM ? (test_char ? test) n i ⊨
30 [ tc_true : Rtc_multi_true alpha test n i, Rtc_multi_false alpha test n i ].
31 #alpha #test #n #i #Hin #int
32 cases (acc_sem_inject … Hin (sem_test_char alpha test) int)
33 #k * #outc * * #Hloop #Htrue #Hfalse %{k} %{outc} % [ %
35 | #Hqtrue lapply (Htrue Hqtrue) * * * #c *
36 #Hcur #Htestc #Hnth_i #Hnth_j %
38 | @(eq_vec … (niltape ?)) #i0 #Hi0
39 cases (decidable_eq_nat i0 i) #Hi0i
41 | @sym_eq @Hnth_j @sym_not_eq // ] ] ]
42 | #Hqfalse lapply (Hfalse Hqfalse) * * #Htestc #Hnth_i #Hnth_j %
44 | @(eq_vec … (niltape ?)) #i0 #Hi0
45 cases (decidable_eq_nat i0 i) #Hi0i
47 | @sym_eq @Hnth_j @sym_not_eq // ] ] ]
50 definition Rm_test_null_true ≝
51 λalpha,n,i.λt1,t2:Vector ? (S n).
52 current alpha (nth i ? t1 (niltape ?)) ≠ None ? ∧ t2 = t1.
54 definition Rm_test_null_false ≝
55 λalpha,n,i.λt1,t2:Vector ? (S n).
56 current alpha (nth i ? t1 (niltape ?)) = None ? ∧ t2 = t1.
58 lemma sem_test_null_multi : ∀alpha,n,i.i ≤ n →
59 inject_TM ? (test_null ?) n i ⊨
60 [ tc_true : Rm_test_null_true alpha n i, Rm_test_null_false alpha n i ].
61 #alpha #n #i #Hin #int
62 cases (acc_sem_inject … Hin (sem_test_null alpha) int)
63 #k * #outc * * #Hloop #Htrue #Hfalse %{k} %{outc} % [ %
65 | #Hqtrue lapply (Htrue Hqtrue) * * #Hcur #Hnth_i #Hnth_j % //
66 @(eq_vec … (niltape ?)) #i0 #Hi0 cases (decidable_eq_nat i0 i) #Hi0i
67 [ >Hi0i @sym_eq @Hnth_i | @sym_eq @Hnth_j @sym_not_eq // ] ]
68 | #Hqfalse lapply (Hfalse Hqfalse) * * #Hcur #Hnth_i #Hnth_j %
70 | @(eq_vec … (niltape ?)) #i0 #Hi0 cases (decidable_eq_nat i0 i) //
71 #Hi0i @sym_eq @Hnth_j @sym_not_eq // ] ]
74 definition match_test ≝ λsrc,dst.λsig:DeqSet.λn.λv:Vector ? n.
75 match (nth src (option sig) v (None ?)) with
77 | Some x ⇒ notb (nth dst (DeqOption sig) v (None ?) == None ?) ].
79 definition mmove_states ≝ initN 2.
81 definition mmove0 : mmove_states ≝ mk_Sig ?? 0 (leb_true_to_le 1 2 (refl …)).
82 definition mmove1 : mmove_states ≝ mk_Sig ?? 1 (leb_true_to_le 2 2 (refl …)).
84 definition trans_mmove ≝
86 λp:mmove_states × (Vector (option sig) (S n)).
87 let 〈q,a〉 ≝ p in match (pi1 … q) with
88 [ O ⇒ 〈mmove1,change_vec ? (S n) (null_action ? n) (〈None ?,D〉) i〉
89 | S _ ⇒ 〈mmove1,null_action sig n〉 ].
93 mk_mTM sig n mmove_states (trans_mmove i sig n D)
94 mmove0 (λq.q == mmove1).
97 λalpha,n,i,D.λt1,t2:Vector ? (S n).
98 t2 = change_vec ? (S n) t1 (tape_move alpha (nth i ? t1 (niltape ?)) D) i.
100 lemma sem_move_multi :
102 mmove i alpha n D ⊨ Rm_multi alpha n i D.
103 #alpha #n #i #D #Hin #int %{2}
104 %{(mk_mconfig ? mmove_states n mmove1 ?)}
106 [ whd in ⊢ (??%?); @eq_f whd in ⊢ (??%?); @eq_f %
107 | whd >tape_move_multi_def
108 <(change_vec_same … (ctapes …) i (niltape ?))
109 >pmap_change <tape_move_multi_def >tape_move_null_action % ] ]
112 definition rewind ≝ λsrc,dst,sig,n.
113 parmove src dst sig n L · mmove src sig n R · mmove dst sig n R.
115 definition R_rewind ≝ λsrc,dst,sig,n.λint,outt: Vector (tape sig) (S n).
117 nth src ? int (niltape ?) = midtape sig (xs@[x0]) x rs →
118 ∀ls0,y,y0,target,rs0.|xs| = |target| →
119 nth dst ? int (niltape ?) = midtape sig (target@y0::ls0) y rs0 →
121 (change_vec ?? int (midtape sig [] x0 (reverse ? xs@x::rs)) src)
122 (midtape sig ls0 y0 (reverse ? target@y::rs0)) dst) ∧
123 (∀x,rs.nth src ? int (niltape ?) = midtape sig [] x rs →
124 ∀ls0,y,rs0.nth dst ? int (niltape ?) = midtape sig ls0 y rs0 →
127 theorem accRealize_to_Realize :
128 ∀sig,n.∀M:mTM sig n.∀Rtrue,Rfalse,acc.
129 M ⊨ [ acc: Rtrue, Rfalse ] → M ⊨ Rtrue ∪ Rfalse.
130 #sig #n #M #Rtrue #Rfalse #acc #HR #t
131 cases (HR t) #k * #outc * * #Hloop
132 #Htrue #Hfalse %{k} %{outc} % //
133 cases (true_or_false (cstate sig (states sig n M) n outc == acc)) #Hcase
134 [ % @Htrue @(\P Hcase) | %2 @Hfalse @(\Pf Hcase) ]
137 lemma sem_rewind : ∀src,dst,sig,n.
138 src ≠ dst → src < S n → dst < S n →
139 rewind src dst sig n ⊨ R_rewind src dst sig n.
140 #src #dst #sig #n #Hneq #Hsrc #Hdst
141 @(sem_seq_app sig n ????? (sem_parmoveL src dst sig n Hneq Hsrc Hdst) ?)
142 [| @(sem_seq_app sig n ????? (sem_move_multi … R ?) (sem_move_multi … R ?)) //
144 #ta #tb * #tc * * #Htc #_ * #td * whd in ⊢ (%→%→?); #Htd #Htb %
145 [ #x #x0 #xs #rs #Hmidta_src #ls0 #y #y0 #target #rs0 #Hlen #Hmidta_dst
146 >(Htc ??? Hmidta_src ls0 y (target@[y0]) rs0 ??) in Htd;
148 |>length_append >length_append >Hlen % ]
149 >change_vec_commute [|@sym_not_eq //]
150 >change_vec_change_vec
151 >nth_change_vec_neq [|@sym_not_eq //]
152 >nth_change_vec // >reverse_append >reverse_single
153 >reverse_append >reverse_single normalize in match (tape_move ???);
154 >rev_append_def >append_nil #Htd >Htd in Htb;
155 >change_vec_change_vec >nth_change_vec //
156 cases ls0 [|#l1 #ls1] normalize in match (tape_move ???); //
157 | #x #rs #Hmidta_src #ls0 #y #rs0 #Hmidta_dst
158 lapply (Htc … Hmidta_src … (refl ??) Hmidta_dst) -Htc #Htc >Htc in Htd;
159 >change_vec_commute [|@sym_not_eq //] >change_vec_change_vec
160 >nth_change_vec_neq [|@sym_not_eq //]
161 >nth_change_vec // lapply (refl ? ls0) cases ls0 in ⊢ (???%→%);
162 [ #Hls0 #Htd >Htd in Htb;
163 >nth_change_vec // >change_vec_change_vec
164 whd in match (tape_move ???);whd in match (tape_move ???); <Hmidta_src
165 <Hls0 <Hmidta_dst >change_vec_same >change_vec_same //
166 | #l1 #ls1 #Hls0 #Htd >Htd in Htb;
167 >nth_change_vec // >change_vec_change_vec
168 whd in match (tape_move ???);whd in match (tape_move ???); <Hmidta_src
169 <Hls0 <Hmidta_dst >change_vec_same >change_vec_same //
173 definition match_step ≝ λsrc,dst,sig,n.
174 compare src dst sig n ·
175 (ifTM ?? (partest sig n (match_test src dst sig ?))
177 (rewind src dst sig n · (inject_TM ? (move_r ?) n dst)))
181 (* we assume the src is a midtape
183 if the dst is out of bounds (outt = int)
184 or dst.right is shorter than src.right (outt.current → None)
185 or src.right is a prefix of dst.right (out = just right of the common prefix) *)
186 definition R_match_step_false ≝
187 λsrc,dst,sig,n.λint,outt: Vector (tape sig) (S n).
189 nth src ? int (niltape ?) = midtape sig ls x xs →
190 ((current sig (nth dst (tape sig) int (niltape sig)) = None ?) ∧ outt = int) ∨
191 (∃ls0,rs0,xs0. nth dst ? int (niltape ?) = midtape sig ls0 x rs0 ∧
193 current sig (nth dst (tape sig) outt (niltape sig)) = None ?) ∨
195 nth dst ? int (niltape ?) = midtape sig ls0 x (xs@rs0) ∧
199 (change_vec ?? int (mk_tape sig (reverse ? xs@x::ls) (None ?) [ ]) src)
200 (mk_tape sig (reverse ? xs@x::ls0) (option_hd ? rs0) (tail ? rs0)) dst).
203 we assume the src is a midtape [ ] s rs
205 then dst.current = Some ? s1
206 and if s ≠ s1 then outt = int.dst.move_right()
208 then int.src.right and int.dst.right have a common prefix
209 and the heads of their suffixes are different
210 and outt = int.dst.move_right().
213 definition R_match_step_true ≝
214 λsrc,dst,sig,n.λint,outt: Vector (tape sig) (S n).
215 ∀s,rs.nth src ? int (niltape ?) = midtape ? [ ] s rs →
216 outt = change_vec ?? int
217 (tape_move_mono … (nth dst ? int (niltape ?)) (〈None ?,R〉)) dst ∧
218 (current sig (nth dst (tape sig) int (niltape sig)) = Some ? s →
219 ∃xs,ci,rs',ls0,cj,rs0.
221 nth dst ? int (niltape ?) = midtape sig ls0 s (xs@cj::rs0) ∧
224 axiom daemon : ∀X:Prop.X.
226 lemma sem_match_step :
227 ∀src,dst,sig,n.src ≠ dst → src < S n → dst < S n →
228 match_step src dst sig n ⊨
229 [ inr ?? (inr ?? (inl … (inr ?? start_nop))) :
230 R_match_step_true src dst sig n,
231 R_match_step_false src dst sig n ].
232 #src #dst #sig #n #Hneq #Hsrc #Hdst
233 @(acc_sem_seq_app sig n … (sem_compare src dst sig n Hneq Hsrc Hdst)
234 (acc_sem_if ? n … (sem_partest sig n (match_test src dst sig ?))
236 (sem_rewind ???? Hneq Hsrc Hdst)
237 (sem_inject … dst (le_S_S_to_le … Hdst) (sem_move_r ? )))
239 [ #ta #tb #tc * lapply (refl ? (current ? (nth src ? ta (niltape ?))))
240 cases (current ? (nth src ? ta (niltape ?))) in ⊢ (???%→%);
241 [ #Hcurta_src #Hcomp #_ * #td * >Hcomp [| % %2 %]
242 whd in ⊢ (%→?); * whd in ⊢ (??%?→?);
243 >(?:nth src ? (current_chars ?? ta) (None ?) = None ?)
244 [ normalize in ⊢ (%→?); #H destruct (H)
246 | #s #Hs lapply (refl ? (current ? (nth dst ? ta (niltape ?))))
247 cases (current ? (nth dst ? ta (niltape ?))) in ⊢ (???%→%);
248 [ #Hcurta_dst #Hcomp #_ * #td * >Hcomp [| %2 %]
249 whd in ⊢ (%→?); * whd in ⊢ (??%?→?);
250 >(?:nth src ? (current_chars ?? ta) (None ?) = Some ? s) [|@daemon]
251 >(?:nth dst ? (current_chars ?? ta) (None ?) = None ?) [|@daemon]
252 normalize in ⊢ (%→?); #H destruct (H)
254 cases (current_to_midtape … Hs) #ls * #rs #Hmidta_src >Hmidta_src
255 cases (current_to_midtape … Hs0) #ls0 * #rs0 #Hmidta_dst >Hmidta_dst
256 cases (true_or_false (s == s0)) #Hss0
257 [ lapply (\P Hss0) -Hss0 #Hss0 destruct (Hss0)
258 #_ #Hcomp cases (Hcomp ????? (refl ??) (refl ??)) -Hcomp [ *
259 [ * #rs' * #_ #Hcurtc_dst * #td * whd in ⊢ (%→?); * whd in ⊢ (??%?→?);
260 >(?:nth dst ? (current_chars ?? tc) (None ?) = None ?) [|@daemon]
261 cases (nth src ? (current_chars ?? tc) (None ?))
262 [| #x ] normalize in ⊢ (%→?); #H destruct (H)
263 | * #rs0' * #_ #Hcurtc_src * #td * whd in ⊢ (%→?); * whd in ⊢ (??%?→?);
264 >(?:nth src ? (current_chars ?? tc) (None ?) = None ?) [|@daemon]
265 normalize in ⊢ (%→?); #H destruct (H) ]
266 | * #xs * #ci * #cj * #rs'' * #rs0' * * * #Hcicj #Hrs #Hrs0
267 #Htc * #td * * #Hmatch #Htd destruct (Htd) * #te * *
268 >Htc >change_vec_commute // >nth_change_vec //
269 >change_vec_commute [|@sym_not_eq //] >nth_change_vec // #Hte #_ #Htb
270 #s' #rs' >Hmidta_src #H destruct (H)
271 lapply (Hte … (refl ??) … (refl ??) (refl ??)) -Hte
272 >change_vec_commute // >change_vec_change_vec
273 >change_vec_commute [|@sym_not_eq //] >change_vec_change_vec #Hte
274 >Hte in Htb; * * #_ >nth_change_vec // #Htb1
275 lapply (Htb1 … (refl ??)) -Htb1 #Htb1 #Htb2 %
276 [ @(eq_vec … (niltape ?)) #i #Hi
277 cases (true_or_false ((dst : DeqNat) == i)) #Hdsti
278 [ <(\P Hdsti) >Htb1 >nth_change_vec // >Hmidta_dst
279 >Hrs0 >reverse_reverse cases xs [|#r1 #rs1] %
280 | <Htb2 [|@(\Pf Hdsti)] >nth_change_vec_neq [| @(\Pf Hdsti)]
281 >Hrs0 >reverse_reverse >nth_change_vec_neq in ⊢ (???%);
282 <Hrs <Hmidta_src [|@(\Pf Hdsti)] >change_vec_same % ]
283 | #_ >Hmidta_dst >Hrs0
284 %{xs} %{ci} %{rs''} %{ls0} %{cj} %{rs0'} % // % //
287 | lapply (\Pf Hss0) -Hss0 #Hss0 #Htc cut (tc = ta)
288 [@Htc % % @(not_to_not ??? Hss0) #H destruct (H) %]
289 -Htc #Htc destruct (Htc) #_ * #td * whd in ⊢ (%→?); * #_
290 #Htd destruct (Htd) * #te * * #_ #Hte * * #_ #Htb1 #Htb2
291 #s1 #rs1 >Hmidta_src #H destruct (H)
292 lapply (Hte … Hmidta_src … Hmidta_dst) -Hte #Hte destruct (Hte) %
293 [ @(eq_vec … (niltape ?)) #i #Hi
294 cases (true_or_false ((dst : DeqNat) == i)) #Hdsti
295 [ <(\P Hdsti) >(Htb1 … Hmidta_dst) >nth_change_vec // >Hmidta_dst
296 cases rs0 [|#r2 #rs2] %
297 | <Htb2 [|@(\Pf Hdsti)] >nth_change_vec_neq [| @(\Pf Hdsti)] % ]
298 | >Hs0 #H destruct (H) @False_ind cases (Hss0) /2/ ]
302 | #ta #tb #tc * #Hcomp1 #Hcomp2 * #td * * #Htest #Htd destruct (Htd)
303 whd in ⊢ (%→?); #Htb destruct (Htb) #ls #x #xs #Hta_src
304 lapply (refl ? (current ? (nth dst ? ta (niltape ?))))
305 cases (current ? (nth dst ? ta (niltape ?))) in ⊢ (???%→?);
306 [ #Hcurta_dst % % % // @Hcomp1 %2 //
307 | #x0 #Hcurta_dst cases (current_to_midtape … Hcurta_dst) -Hcurta_dst
308 #ls0 * #rs0 #Hta_dst cases (true_or_false (x == x0)) #Hxx0
309 [ lapply (\P Hxx0) -Hxx0 #Hxx0 destruct (Hxx0)
310 | >(?:tc=ta) in Htest;
311 [|@Hcomp1 % % >Hta_src >Hta_dst @(not_to_not ??? (\Pf Hxx0)) normalize
312 #Hxx0' destruct (Hxx0') % ]
313 whd in ⊢ (??%?→?); >(?:nth src ? (current_chars ?? ta) (None ?) = Some ? x)
315 >(?:nth dst ? (current_chars ?? ta) (None ?) = Some ? x0) [|@daemon]
316 whd in ⊢ (??%?→?); #Hfalse destruct (Hfalse) ] -Hcomp1
317 cases (Hcomp2 … Hta_src Hta_dst) [ *
318 [ * #rs' * #Hxs #Hcurtc % %2 %{ls0} %{rs0} %{rs'} % // % //
319 | * #rs0' * #Hxs #Htc %2 >Htc %{ls0} %{rs0'} % // ]
320 | * #xs0 * #ci * #cj * #rs' * #rs0' * * *
321 #Hci #Hxs #Hrs0 #Htc @False_ind
323 >(?:nth src ? (current_chars ?? tc) (None ?) = Some ? ci) in Htest; [|@daemon]
324 >(?:nth dst ? (current_chars ?? tc) (None ?) = Some ? cj) [|@daemon]
325 normalize #H destruct (H) ] ] ]
328 definition match_m ≝ λsrc,dst,sig,n.
329 whileTM … (match_step src dst sig n)
330 (inr ?? (inr ?? (inl … (inr ?? start_nop)))).
332 definition R_match_m ≝
333 λsrc,dst,sig,n.λint,outt: Vector (tape sig) (S n).
335 nth src ? int (niltape ?) = midtape sig [ ] x rs →
336 (current sig (nth dst (tape sig) int (niltape sig)) = None ? → outt = int) ∧
338 nth dst ? int (niltape ?) = midtape sig ls0 x0 rs0 →
339 (∃l,l1.x0::rs0 = l@x::rs@l1 ∧
342 (mk_tape sig (reverse ? rs@[x]) (None ?) [ ]) src)
343 (mk_tape sig ((reverse ? (l@x::rs))@ls0) (option_hd ? l1) (tail ? l1)) dst) ∨
344 ∀l,l1.x0::rs0 ≠ l@x::rs@l1).
346 lemma not_sub_list_merge :
347 ∀T.∀a,b:list T. (∀l1.a ≠ b@l1) → (∀t,l,l1.a ≠ t::l@b@l1) → ∀l,l1.a ≠ l@b@l1.
348 #T #a #b #H1 #H2 #l elim l normalize //
351 lemma not_sub_list_merge_2 :
352 ∀T:DeqSet.∀a,b:list T.∀t. (∀l1.t::a ≠ b@l1) → (∀l,l1.a ≠ l@b@l1) → ∀l,l1.t::a ≠ l@b@l1.
353 #T #a #b #t #H1 #H2 #l elim l //
354 #t0 #l1 #IH #l2 cases (true_or_false (t == t0)) #Htt0
355 [ >(\P Htt0) % normalize #H destruct (H) cases (H2 l1 l2) /2/
356 | normalize % #H destruct (H) cases (\Pf Htt0) /2/ ]
360 lemma wsem_match_m : ∀src,dst,sig,n.
361 src ≠ dst → src < S n → dst < S n →
362 match_m src dst sig n ⊫ R_match_m src dst sig n.
363 #src #dst #sig #n #Hneq #Hsrc #Hdst #ta #k #outc #Hloop
364 lapply (sem_while … (sem_match_step src dst sig n Hneq Hsrc Hdst) … Hloop) //
365 -Hloop * #tb * #Hstar @(star_ind_l ??????? Hstar) -Hstar
366 [ #Hfalse #x #xs #Hmid_src
367 cases (Hfalse … Hmid_src) -Hfalse
368 [(* current dest = None *) *
369 [ * #Hcur_dst #Houtc %
371 | #ls0 #x0 #rs0 #Hmid_dst >Hmid_dst in Hcur_dst;
372 normalize in ⊢ (%→?); #H destruct (H)
374 | * #ls0 * #rs0 * #xs0 * * #Htc_dst #Hrs0 #HNone %
375 [ >Htc_dst normalize in ⊢ (%→?); #H destruct (H)
376 | #ls1 #x1 #rs1 >Htc_dst #H destruct (H)
378 [ % %{[ ]} %{[ ]} % [ >append_nil >append_nil %]
380 #cj #ls2 #H destruct (H) *) @daemon
381 | #x2 #xs2 %2 #l #l1 % #Habs lapply (eq_f ?? (length ?) ?? Habs)
382 >length_append whd in ⊢ (??%(??%)→?); >length_append
383 >length_append normalize >commutative_plus whd in ⊢ (???%→?);
384 #H destruct (H) lapply e0 >(plus_n_O (|rs1|)) in ⊢ (??%?→?);
385 >associative_plus >associative_plus
386 #e1 lapply (injective_plus_r ??? e1) whd in ⊢ (???%→?);
391 |* #ls0 * #rs0 * #Hmid_dst #Houtc %
392 [ >Hmid_dst normalize in ⊢ (%→?); #H destruct (H)
393 |#ls1 #x1 #rs1 >Hmid_dst #H destruct (H)
394 %1 %{[ ]} %{rs0} % [%]
395 >reverse_cons >associative_append >Houtc %
398 |-ta #ta #tc #Htrue #Hstar #IH #Hout lapply (IH Hout) -IH -Hout #IH whd
400 lapply (refl ? (current ? (nth dst ? ta (niltape ?))))
401 cases (current ? (nth dst ? ta (niltape ?))) in ⊢ (???%→?);
403 [#_ whd in Htrue; >Hmidta_src in Htrue; #Htrue
404 cases (Htrue ?? (refl ??)) -Htrue >Hcurta_dst
405 (* dovremmo sapere che ta.dst è sul margine destro, da cui la move non
406 ha effetto *) #_ cut (tc = ta) [@daemon] #Htc destruct (Htc) #_
407 cases (IH … Hmidta_src) #Houtc #_ @Houtc //
408 |#ls0 #x0 #rs0 #Hmidta_dst >Hmidta_dst in Hcurta_dst;
409 normalize in ⊢ (%→?); #H destruct (H)
411 | #c #Hcurta_dst % [ >Hcurta_dst #H destruct (H) ]
412 #ls0 #x0 #rs0 #Hmidta_dst >Hmidta_dst in Hcurta_dst; normalize in ⊢ (%→?);
413 #H destruct (H) whd in Htrue; >Hmidta_src in Htrue; #Htrue
414 cases (Htrue ?? (refl …)) -Htrue >Hmidta_dst #Htc
415 cases (true_or_false (x==c)) #eqx
416 [ lapply (\P eqx) -eqx #eqx destruct (eqx)
417 #Htrue cases (Htrue (refl ??)) -Htrue
418 #xs0 * #ci * #rs' * #ls1 * #cj * #rs1 * * #Hxs #H destruct (H) #Hcicj
419 >Htc in IH; whd in ⊢ (%→?); >nth_change_vec_neq [|@sym_not_eq //]
420 #IH cases (IH … Hmidta_src) -IH #_ >nth_change_vec //
421 cut (∃x1,xs1.xs0@cj::rs1 = x1::xs1)
422 [ cases xs0 [ %{cj} %{rs1} % | #x1 #xs1 %{x1} %{(xs1@cj::rs1)} % ] ] * #x1 * #xs1
423 #Hxs1 >Hxs1 #IH cases (IH … (refl ??)) -IH
424 [ * #l * #l1 * #Hxs1'
425 >change_vec_commute // >change_vec_change_vec
426 #Houtc % %{(c::l)} %{l1} %
428 | >reverse_cons >associative_append >change_vec_commute // @Houtc ]
429 | #H %2 #l #l1 >(?:l@c::xs@l1 = l@(c::xs)@l1) [|%]
431 [ #l2 >Hxs <Hxs1 % normalize #H1 lapply (cons_injective_r ????? H1)
432 >associative_append #H2 lapply (append_l2_injective ????? (refl ??) H2)
433 #H3 lapply (cons_injective_l ????? H3) #H3 >H3 in Hcicj; * /2/
434 |#t #l2 #l3 % normalize #H1 lapply (cons_injective_r ????? H1)
435 -H1 #H1 cases (H l2 l3) #H2 @H2 @H1
438 | (* in match_step_true manca il caso di fallimento immediato
439 (con i due current diversi) *)
442 #_ lapply (\Pf eqx) -eqx #eqx >Hmidta_dst
443 cases (Htrue ? (refl ??) eqx) -Htrue #Htb #Hendcx #_
445 [ #_ %2 #l #l1 cases l
448 [ normalize % #H destruct (H) cases eqx /2/
449 | #tmp1 #l2 normalize % #H destruct (H) ]
450 | #tmp1 #l2 normalize % #H destruct (H) ]
451 | #tmp1 #l2 normalize % #H destruct (H)cases l2 in e0;
452 [ normalize #H1 destruct (H1)
453 | #tmp2 #l3 normalize #H1 destruct (H1) ]
455 | #r1 #rs1 normalize in ⊢ (???(????%?)→?); #Htb >Htb in IH; #IH
456 cases (IH ls x xs end rs ? Hnotend Hend Hnotstart)
457 [| >Htb >nth_change_vec_neq [|@sym_not_eq //] @Hmid_src ] -IH
458 #_ #IH cases (IH Hstart (c::ls0) r1 rs1 ?)
459 [|| >nth_change_vec // ] -IH
460 [ * #l * #l1 * #Hll1 #Hout % %{(c::l)} %{l1} % >Hll1 //
461 >reverse_cons >associative_append #cj0 #ls #Hl1 >(Hout ?? Hl1)
462 >change_vec_commute in ⊢ (??(???%??)?); // @sym_not_eq //
463 | #IH %2 @(not_sub_list_merge_2 ?? (x::xs)) normalize [|@IH]
464 #l1 % #H destruct (H) cases eqx /2/
471 definition Pre_match_m ≝
472 λsrc,sig,n,is_startc,is_endc.λt: Vector (tape sig) (S n).
474 nth src (tape sig) t (niltape sig) = midtape ? [] start (xs@[end]) ∧
475 is_startc start = true ∧
476 (∀c.c ∈ (xs@[end]) = true → is_startc c = false) ∧
477 (∀c.c ∈ (start::xs) = true → is_endc c = false) ∧
480 lemma terminate_match_m :
481 ∀src,dst,sig,n,is_startc,is_endc,t.
482 src ≠ dst → src < S n → dst < S n →
483 Pre_match_m src sig n is_startc is_endc t →
484 match_m src dst sig n is_startc is_endc ↓ t.
485 #src #dst #sig #n #is_startc #is_endc #t #Hneq #Hsrc #Hdst * #start * #xs * #end
486 * * * * #Hmid_src #Hstart #Hnotstart #Hnotend #Hend
487 @(terminate_while … (sem_match_step src dst sig n is_startc is_endc Hneq Hsrc Hdst)) //
488 <(change_vec_same … t dst (niltape ?))
489 lapply (refl ? (nth dst (tape sig) t (niltape ?)))
490 cases (nth dst (tape sig) t (niltape ?)) in ⊢ (???%→?);
491 [ #Htape_dst % #t1 whd in ⊢ (%→?); >nth_change_vec_neq [|@sym_not_eq //]
492 >Hmid_src #HR cases (HR ? (refl ??)) -HR
493 >nth_change_vec // >Htape_dst normalize in ⊢ (%→?);
495 | #x0 #xs0 #Htape_dst % #t1 whd in ⊢ (%→?); >nth_change_vec_neq [|@sym_not_eq //]
496 >Hmid_src #HR cases (HR ? (refl ??)) -HR
497 >nth_change_vec // >Htape_dst normalize in ⊢ (%→?);
499 | #x0 #xs0 #Htape_dst % #t1 whd in ⊢ (%→?); >nth_change_vec_neq [|@sym_not_eq //]
500 >Hmid_src #HR cases (HR ? (refl ??)) -HR
501 >nth_change_vec // >Htape_dst normalize in ⊢ (%→?);
503 | #ls #s #rs lapply s -s lapply ls -ls lapply Hmid_src lapply t -t elim rs
504 [#t #Hmid_src #ls #s #Hmid_dst % #t1 whd in ⊢ (%→?); >nth_change_vec_neq [|@sym_not_eq //]
505 >Hmid_src >nth_change_vec // >Hmid_dst #HR cases (HR ? (refl ??)) -HR #_
506 #HR cases (HR Hstart Hnotstart)
507 cases (true_or_false (start == s)) #Hs
508 [ lapply (\P Hs) -Hs #Hs <Hs #_ #Htrue
509 cut (∃ci,xs1.xs@[end] = ci::xs1)
512 | #x1 #xs1 %{x1} %{(xs1@[end])} % ] ] * #ci * #xs1 #Hxs
513 >Hxs in Htrue; #Htrue
514 cases (Htrue [ ] start [ ] ? xs1 ? [ ] (refl ??) (refl ??) ?)
515 [ * #_ * #H @False_ind @H % ]
516 #c0 #Hc0 @Hnotend >(memb_single … Hc0) @memb_hd
517 | lapply (\Pf Hs) -Hs #Hs #Htrue #_
518 cases (Htrue ? (refl ??) Hs) -Htrue #Ht1 #_ %
519 #t2 whd in ⊢ (%→?); #HR cases (HR start ?)
520 [ >Ht1 >nth_change_vec // normalize in ⊢ (%→?); * #H @False_ind @H %
521 | >Ht1 >nth_change_vec_neq [|@sym_not_eq //]
522 >nth_change_vec_neq [|@sym_not_eq //] >Hmid_src % ]
524 |#r0 #rs0 #IH #t #Hmid_src #ls #s #Hmid_dst % #t1 whd in ⊢ (%→?);
525 >nth_change_vec_neq [|@sym_not_eq //] >Hmid_src
526 #Htrue cases (Htrue ? (refl ??)) -Htrue #_ #Htrue
527 <(change_vec_same … t1 dst (niltape ?))
528 cases (Htrue Hstart Hnotstart) -Htrue
529 cases (true_or_false (start == s)) #Hs
530 [ lapply (\P Hs) -Hs #Hs <Hs #_ #Htrue
531 cut (∃ls0,xs0,ci,rs,rs0.
532 nth src ? t (niltape ?) = midtape sig [ ] start (xs0@ci::rs) ∧
533 nth dst ? t (niltape ?) = midtape sig ls0 s (xs0@rs0) ∧
534 (is_endc ci = true ∨ (is_endc ci = false ∧ (∀b,tlb.rs0 = b::tlb → ci ≠ b))))
535 [cases (comp_list ? (xs@[end]) (r0::rs0) is_endc) #xs0 * #xs1 * #xs2
536 * * * #Hxs #Hrs #Hxs0notend #Hcomp >Hrs
537 cut (∃y,ys. xs1 = y::ys)
538 [ lapply Hxs0notend lapply Hxs lapply xs0 elim xs
540 [ normalize #Hxs1 <Hxs1 #_ %{end} %{[]} %
541 | #z #zs normalize in ⊢ (%→?); #H destruct (H) #H
542 lapply (H ? (memb_hd …)) -H >Hend #H1 destruct (H1)
545 [ normalize in ⊢ (%→?); #Hxs1 <Hxs1 #_ %{y} %{(ys@[end])} %
546 | #z #zs normalize in ⊢ (%→?); #H destruct (H) #Hmemb
547 @(IH0 ? e0 ?) #c #Hc @Hmemb @memb_cons // ] ] ] * #y * #ys #Hxs1
548 >Hxs1 in Hxs; #Hxs >Hmid_src >Hmid_dst >Hxs >Hrs
549 %{ls} %{xs0} %{y} %{ys} %{xs2}
550 % [ % // | @Hcomp // ] ]
551 * #ls0 * #xs0 * #ci * #rs * #rs0 * * #Hmid_src' #Hmid_dst' #Hcomp
552 <Hmid_src in Htrue; >nth_change_vec // >Hs #Htrue destruct (Hs)
553 lapply (Htrue ??????? Hmid_src' Hmid_dst' ?) -Htrue
554 [ #c0 #Hc0 @Hnotend cases (orb_true_l … Hc0) -Hc0 #Hc0
555 [ whd in ⊢ (??%?); >Hc0 %
556 | @memb_cons >Hmid_src in Hmid_src'; #Hmid_src' destruct (Hmid_src')
557 lapply e0 -e0 @(list_elim_left … rs)
558 [ #e0 destruct (e0) lapply (append_l1_injective_r ?????? e0) //
559 | #x1 #xs1 #_ >append_cons in ⊢ (???%→?);
560 <associative_append #e0 lapply (append_l1_injective_r ?????? e0) //
561 #e1 >e1 @memb_append_l1 @memb_append_l1 // ] ]
562 | * * #Hciendc cases rs0 in Hcomp;
563 [ #_ * #H @False_ind @H %
564 | #r1 #rs1 * [ >Hciendc #H destruct (H) ]
565 * #_ #Hcomp lapply (Hcomp ?? (refl ??)) -Hcomp #Hcomp #_ #Htrue
566 cases (Htrue ?? (refl ??) Hcomp) #Ht1 #_ >Ht1 @(IH ?? (s::ls) r0)
567 [ >nth_change_vec_neq [|@sym_not_eq //]
568 >nth_change_vec_neq [|@sym_not_eq //] @Hmid_src
569 | >nth_change_vec // >Hmid_dst % ] ] ]
570 | >Hmid_dst >nth_change_vec // lapply (\Pf Hs) -Hs #Hs #Htrue #_
571 cases (Htrue ? (refl ??) Hs) #Ht1 #_ >Ht1 @(IH ?? (s::ls) r0)
572 [ >nth_change_vec_neq [|@sym_not_eq //]
573 >nth_change_vec_neq [|@sym_not_eq //] @Hmid_src
574 | >nth_change_vec // ] ] ] ]