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 rewind ≝ λsrc,dst,sig,n.parmove src dst sig n L · parmove_step src dst sig n R.
81 definition R_rewind ≝ λsrc,dst,sig,n.λint,outt: Vector (tape sig) (S n).
83 nth src ? int (niltape ?) = midtape sig (xs@[x0]) x rs →
84 ∀ls0,y,y0,target,rs0.|xs| = |target| →
85 nth dst ? int (niltape ?) = midtape sig (target@y0::ls0) y rs0 →
87 (change_vec ?? int (midtape sig [] x0 (reverse ? xs@x::rs)) src)
88 (midtape sig ls0 y0 (reverse ? target@y::rs0)) dst).
90 theorem accRealize_to_Realize :
91 ∀sig,n.∀M:mTM sig n.∀Rtrue,Rfalse,acc.
92 M ⊨ [ acc: Rtrue, Rfalse ] → M ⊨ Rtrue ∪ Rfalse.
93 #sig #n #M #Rtrue #Rfalse #acc #HR #t
94 cases (HR t) #k * #outc * * #Hloop
95 #Htrue #Hfalse %{k} %{outc} % //
96 cases (true_or_false (cstate sig (states sig n M) n outc == acc)) #Hcase
97 [ % @Htrue @(\P Hcase) | %2 @Hfalse @(\Pf Hcase) ]
100 lemma sem_rewind : ∀src,dst,sig,n.
101 src ≠ dst → src < S n → dst < S n →
102 rewind src dst sig n ⊨ R_rewind src dst sig n.
103 #src #dst #sig #n #Hneq #Hsrc #Hdst
104 check acc_sem_seq_app
105 @(sem_seq_app sig n ????? (sem_parmoveL src dst sig n Hneq Hsrc Hdst)
106 (accRealize_to_Realize … (sem_parmove_step src dst sig n R Hneq Hsrc Hdst)))
107 #ta #tb * #tc * * #HR1 #_ #HR2
108 #x #x0 #xs #rs #Hmidta_src #ls0 #y #y0 #target #rs0 #Hlen #Hmidta_dst
109 >(HR1 ??? Hmidta_src ls0 y (target@[y0]) rs0 ??) in HR2;
111 |>length_append >length_append >Hlen % ] *
112 [ whd in ⊢ (%→?); * #x1 * #x2 * *
113 >change_vec_commute in ⊢ (%→?); // >nth_change_vec //
114 cases (reverse sig (xs@[x0])@x::rs)
115 [|#z #zs] normalize in ⊢ (%→?); #H destruct (H)
116 | whd in ⊢ (%→?); * #_ #Htb >Htb -Htb FAIL
118 normalize in ⊢ (%→?);
119 (sem_parmove_step src dst sig n R Hneq Hsrc Hdst))
120 (acc_sem_if ? n … (sem_partest sig n (match_test src dst sig ?))
122 (sem_parmoveL ???? Hneq Hsrc Hdst)
123 (sem_inject … dst (le_S_S_to_le … Hdst) (sem_move_r ? )))
127 definition match_step ≝ λsrc,dst,sig,n.
128 compare src dst sig n ·
129 (ifTM ?? (partest sig n (match_test src dst sig ?))
131 (rewind src dst sig n · (inject_TM ? (move_r ?) n dst)))
135 definition R_match_step_false ≝
136 λsrc,dst,sig,n.λint,outt: Vector (tape sig) (S n).
138 nth src ? int (niltape ?) = midtape sig ls x xs →
139 ((current sig (nth dst (tape sig) int (niltape sig)) = None ?) ∧ outt = int) ∨
140 (∃ls0,rs0,xs0. nth dst ? int (niltape ?) = midtape sig ls0 x rs0 ∧
142 current sig (nth dst (tape sig) outt (niltape sig)) = None ?) ∨
144 nth dst ? int (niltape ?) = midtape sig ls0 x (xs@rs0) ∧
148 (change_vec ?? int (mk_tape sig (reverse ? xs@x::ls) (None ?) [ ]) src)
149 (mk_tape sig (reverse ? xs@x::ls0) (option_hd ? rs0) (tail ? rs0)) dst).
151 (*definition R_match_step_true ≝
152 λsrc,dst,sig,n.λint,outt: Vector (tape sig) (S n).
153 ∀s,rs.current sig (nth src (tape sig) int (niltape sig)) = Some ? s →
154 current sig (nth dst (tape sig) int (niltape sig)) ≠ None ? ∧
155 (∀s1.current sig (nth dst (tape sig) int (niltape sig)) = Some ? s1 → s ≠ s1 →
156 outt = change_vec ?? int
157 (tape_move_mono … (nth dst ? int (niltape ?)) (〈Some ? s1,R〉)) dst) ∧
158 (∀ls,x,xs,ci,rs,ls0,rs0.
159 nth src ? int (niltape ?) = midtape sig ls x (xs@ci::rs) →
160 nth dst ? int (niltape ?) = midtape sig ls0 x (xs@rs0) →
162 ∀cj,rs1.rs0 = cj::rs1 →
164 (outt = change_vec ?? int
165 (tape_move_mono … (nth dst ? int (niltape ?)) (〈None ?,R〉)) dst)).
167 definition R_match_step_true ≝
168 λsrc,dst,sig,n.λint,outt: Vector (tape sig) (S n).
169 ∀s.current sig (nth src (tape sig) int (niltape sig)) = Some ? s →
170 ∃s1.current sig (nth dst (tape sig) int (niltape sig)) = Some ? s1 ∧
171 (left ? (nth src ? int (niltape ?)) = [ ] →
173 outt = change_vec ?? int
174 (tape_move_mono … (nth dst ? int (niltape ?)) (〈None ?,R〉)) dst) ∧
176 nth src ? int (niltape ?) = midtape sig [] s (xs@ci::rs) →
177 nth dst ? int (niltape ?) = midtape sig ls0 s (xs@rs0) →
179 ∀cj,rs1.rs0 = cj::rs1 →
181 (outt = change_vec ?? int
182 (tape_move_mono … (nth dst ? int (niltape ?)) (〈None ?,R〉)) dst))).
184 lemma sem_match_step :
185 ∀src,dst,sig,n.src ≠ dst → src < S n → dst < S n →
186 match_step src dst sig n ⊨
187 [ inr ?? (inr ?? (inl … (inr ?? start_nop))) :
188 R_match_step_true src dst sig n,
189 R_match_step_false src dst sig n ].
190 #src #dst #sig #n #Hneq #Hsrc #Hdst
191 @(acc_sem_seq_app sig n … (sem_compare src dst sig n Hneq Hsrc Hdst)
192 (acc_sem_if ? n … (sem_partest sig n (match_test src dst sig ?))
194 (sem_parmoveL ???? Hneq Hsrc Hdst)
195 (sem_inject … dst (le_S_S_to_le … Hdst) (sem_move_r ? )))
197 [#ta #tb #tc * #Hcomp1 #Hcomp2 * #td * #Htest
198 * #te * #Hte #Htb #s #Hcurta_src whd
199 cut (∃s1.current sig (nth dst (tape sig) ta (niltape sig))=Some sig s1)
200 [ lapply Hcomp1 -Hcomp1
201 lapply (refl ? (current ? (nth dst ? ta (niltape ?))))
202 cases (current ? (nth dst ? ta (niltape ?))) in ⊢ (???%→%);
203 [ #Hcurta_dst #Hcomp1 >Hcomp1 in Htest; // *
204 change with (vec_map ?????) in match (current_chars ???); whd in ⊢ (??%?→?);
205 <(nth_vec_map ?? (current ?) src ? ta (niltape ?))
206 <(nth_vec_map ?? (current ?) dst ? ta (niltape ?))
207 >Hcurta_src >Hcurta_dst whd in ⊢ (??%?→?); #H destruct (H)
208 | #s1 #_ #_ %{s1} % ] ]
209 * #s1 #Hcurta_dst %{s1} % // #Hleftta %
210 [ #Hneqss1 -Hcomp2 cut (tc = ta)
211 [@Hcomp1 %1 %1 >Hcurta_src >Hcurta_dst @(not_to_not … Hneqss1) #H destruct (H) //]
212 #H destruct (H) -Hcomp1 cut (td = ta)
213 [ cases Htest -Htest // ] #Htdta destruct (Htdta)
214 cases Hte -Hte #Hte #_
215 cases (current_to_midtape … Hcurta_src) #ls * #rs #Hmidta_src
216 cases (current_to_midtape … Hcurta_dst) #ls0 * #rs0 #Hmidta_dst
217 >Hmidta_src in Hleftta; normalize in ⊢ (%→?); #Hls destruct (Hls)
218 >(Hte s [ ] rs Hmidta_src ls0 s1 [ ] rs0 (refl ??) Hmidta_dst) in Htb;
224 [ cases Htest -Htest #Htest #Htdta <Htdta @Hte %1 >Htdta @Hcurta_src %{s} % //]
225 -Hte #H destruct (H) %
226 [cases Htb * #_ #Hmove #Hmove1 @(eq_vec … (niltape … ))
227 #i #Hi cases (decidable_eq_nat i dst) #Hidst
228 [ >Hidst >nth_change_vec // cases (current_to_midtape … Hcurta_dst)
229 #ls * #rs #Hta_mid >(Hmove … Hta_mid) >Hta_mid cases rs //
230 | >nth_change_vec_neq [|@sym_not_eq //] @sym_eq @Hmove1 @sym_not_eq // ]
231 | whd in Htest:(??%?); >(nth_vec_map ?? (current sig)) in Hcurta_src; #Hcurta_src
232 >Hcurta_src in Htest; whd in ⊢ (??%?→?);
233 cases (is_endc s) // whd in ⊢ (??%?→?); #H @sym_eq //
235 <(nth_vec_map ?? (current ?) dst ? tc (niltape ?))
236 >Hcurta_src normalize
237 lapply (refl ? (current ? (nth dst ? ta (niltape ?))))
238 cases (current ? (nth dst ? ta (niltape ?))) in ⊢ (???%→%);
240 [ % #Hfalse destruct (Hfalse)
241 | #s1' #Hs1 destruct (Hs1) #Hneqss1 -Hcomp2
243 [@Hcomp1 %1 %1 >Hcurta_src >Hcurta_dst @(not_to_not … Hneqss1) #H destruct (H) //]
244 #H destruct (H) -Hcomp1 cases Hte -Hte #_ #Hte
245 cut (te = ta) [ cases Htest -Htest #Htest #Htdta <Htdta @Hte %1 %{s} % //] -Hte #H destruct (H) %
246 [cases Htb * #_ #Hmove #Hmove1 @(eq_vec … (niltape … ))
247 #i #Hi cases (decidable_eq_nat i dst) #Hidst
248 [ >Hidst >nth_change_vec // cases (current_to_midtape … Hcurta_dst)
249 #ls * #rs #Hta_mid >(Hmove … Hta_mid) >Hta_mid cases rs //
250 | >nth_change_vec_neq [|@sym_not_eq //] @sym_eq @Hmove1 @sym_not_eq // ]
251 | whd in Htest:(??%?); >(nth_vec_map ?? (current sig)) in Hcurta_src; #Hcurta_src
252 >Hcurta_src in Htest; whd in ⊢ (??%?→?);
253 cases (is_endc s) // whd in ⊢ (??%?→?); #H @sym_eq //
257 #Hcurta_dst >Hcomp1 in Htest; [| %2 %2 //]
258 whd in ⊢ (??%?→?); change with (current ? (niltape ?)) in match (None ?);
259 <nth_vec_map >Hcurta_src whd in ⊢ (??%?→?); <nth_vec_map
260 >Hcurta_dst cases (is_endc s) normalize in ⊢ (%→?); #H destruct (H)
261 | #Hstart #Hnotstart %
262 [ #s1 #Hcurta_dst #Hneqss1 -Hcomp2
264 [@Hcomp1 %2 %1 %1 >Hcurta_src >Hcurta_dst @(not_to_not … Hneqss1) #H destruct (H) //]
265 #H destruct (H) -Hcomp1 cases Hte #_ -Hte #Hte
266 cut (te = ta) [@Hte %1 %1 %{s} % //] -Hte #H destruct (H) %
267 [cases Htb * #_ #Hmove #Hmove1 @(eq_vec … (niltape … ))
268 #i #Hi cases (decidable_eq_nat i dst) #Hidst
269 [ >Hidst >nth_change_vec // cases (current_to_midtape … Hcurta_dst)
270 #ls * #rs #Hta_mid >(Hmove … Hta_mid) >Hta_mid cases rs //
271 | >nth_change_vec_neq [|@sym_not_eq //] @sym_eq @Hmove1 @sym_not_eq // ]
272 | whd in Htest:(??%?); >(nth_vec_map ?? (current sig)) in Hcurta_src; #Hcurta_src
273 >Hcurta_src in Htest; whd in ⊢ (??%?→?);
274 cases (is_endc s) // whd in ⊢ (??%?→?); #H @sym_eq //
276 |#ls #x #xs #ci #rs #ls0 #rs00 #Htasrc_mid #Htadst_mid #Hnotendc
277 cases (Hcomp2 … Htasrc_mid Htadst_mid Hnotendc)
278 [ * #Hrs00 #Htc >Htc in Htest; whd in ⊢ (??%?→?);
279 <(nth_vec_map ?? (current sig) ??? (niltape ?))
280 >change_vec_commute // >nth_change_vec // whd in ⊢ (??%?→?);
282 [ whd in ⊢ (??%?→?); #H destruct (H)
283 | <(nth_vec_map ?? (current sig) ??? (niltape ?))
284 >change_vec_commute [| @sym_not_eq // ] >nth_change_vec //
285 >(?:current ? (mk_tape ?? (None ?) ?) = None ?)
286 [ whd in ⊢ (??%?→?); #H destruct (H)
287 | cases (reverse sig xs@x::ls0) normalize // ] ] ]
288 * #cj' * #rs0' * #Hcjrs0 destruct (Hcjrs0) -Hcomp2 #Hcomp2 % [ %
289 [ cases (true_or_false (is_endc ci)) //
290 #Hendci >(Hcomp2 (or_introl … Hendci)) in Htest;
291 whd in ⊢ (??%?→?); <(nth_vec_map ?? (current sig) ??? (niltape ?))
292 >change_vec_commute // >nth_change_vec // whd in ⊢ (??%?→?);
294 | % #H destruct (H) ] ] #cj #rs1 #H destruct (H) #Hcicj
295 lapply (Hcomp2 (or_intror ?? Hcicj)) -Hcomp2 #Htc %
296 [ cases Hte -Hte #Hte #_ whd in Hte;
297 >Htasrc_mid in Hcurta_src; whd in ⊢ (??%?→?); #H destruct (H)
298 lapply (Hte ls ci (reverse ? xs) rs s ??? ls0 cj (reverse ? xs) s rs1 (refl ??) ?) //
299 [ >Htc >nth_change_vec //
300 | #c0 #Hc0 @(Hnotstart c0) >Htasrc_mid cases (orb_true_l … Hc0) -Hc0 #Hc0
301 [@memb_append_l2 >(\P Hc0) @memb_hd
302 |@memb_append_l1 <(reverse_reverse …xs) @memb_reverse //
304 | >Htc >change_vec_commute // >nth_change_vec // ] -Hte
305 >Htc >change_vec_commute // >change_vec_change_vec
306 >change_vec_commute [|@sym_not_eq //] >change_vec_change_vec #Hte
307 >Hte in Htb; * * #_ >reverse_reverse #Htbdst1 #Htbdst2 -Hte @(eq_vec … (niltape ?))
308 #i #Hi cases (decidable_eq_nat i dst) #Hidst
309 [ >Hidst >nth_change_vec // >(Htbdst1 ls0 s (xs@cj::rs1))
310 [| >nth_change_vec // ]
311 >Htadst_mid cases xs //
312 | >nth_change_vec_neq [|@sym_not_eq // ]
313 <Htbdst2 [| @sym_not_eq // ] >nth_change_vec_neq [| @sym_not_eq // ]
314 <Htasrc_mid >change_vec_same % ]
315 | >Hcurta_src in Htest; whd in ⊢(??%?→?);
316 >Htc >change_vec_commute //
317 change with (current ? (niltape ?)) in match (None ?);
318 <nth_vec_map >nth_change_vec // whd in ⊢ (??%?→?);
319 cases (is_endc ci) whd in ⊢ (??%?→?); #H destruct (H) %
323 |#intape #outtape #ta * #Hcomp1 #Hcomp2 * #tb * * #Hc #Htb
324 whd in ⊢ (%→?); #Hout >Hout >Htb whd
325 #ls #c_src #xs #end #rs #Hmid_src #Hnotend #Hend
326 lapply (current_to_midtape sig (nth dst ? intape (niltape ?)))
327 cases (current … (nth dst ? intape (niltape ?))) in Hcomp1;
328 [#Hcomp1 #_ %1 % % [% | @Hcomp1 %2 %2 % ]
329 |#c_dst cases (true_or_false (c_src == c_dst)) #Hceq
330 [#_ #Hmid_dst cases (Hmid_dst c_dst (refl …)) -Hmid_dst
331 #ls_dst * #rs_dst #Hmid_dst
332 cases (comp_list … (xs@end::rs) rs_dst is_endc) #xs1 * #rsi * #rsj * * *
333 #Hrs_src #Hrs_dst #Hnotendxs1 #Hneq >Hrs_dst in Hmid_dst; #Hmid_dst
334 cut (∃r1,rs1.rsi = r1::rs1)
335 [cases rsi in Hrs_src;
336 [ >append_nil #H <H in Hnotendxs1; #Hnotendxs1
337 >(Hnotendxs1 end) in Hend; [ #H1 destruct (H1) ]
338 @memb_append_l2 @memb_hd
339 | #r1 #rs1 #_ %{r1} %{rs1} % ] ]
340 * #r1 * #rs1 #Hrs1 >Hrs1 in Hrs_src;
341 #Hrs_src >Hrs_src in Hmid_src; #Hmid_src <(\P Hceq) in Hmid_dst; #Hmid_dst
342 lapply (Hcomp2 ??????? Hmid_src Hmid_dst ?)
343 [ #c0 #Hc0 cases (orb_true_l … Hc0) -Hc0 #Hc0
344 [ >(\P Hc0) @Hnotend @memb_hd | @Hnotendxs1 //] ]
346 [ * #Hrsj >Hrsj #Hta % %2 >Hta >nth_change_vec //
347 %{ls_dst} %{xs1} cut (∃xs0.xs = xs1@xs0)
348 [lapply Hnotendxs1 -Hnotendxs1 lapply Hrs_src lapply xs elim xs1
351 [ whd in ⊢ (??%%→?); #H destruct (H) #Hnotendxs2
352 >Hnotendxs2 in Hend; [ #H destruct (H) |@memb_hd ]
353 | #x2' #xs2' whd in ⊢ (??%%→?); #H destruct (H)
354 #Hnotendxs2 cases (IH xs2' e0 ?)
355 [ #xs0 #Hxs2 %{xs0} @eq_f //
356 |#c #Hc @Hnotendxs2 @memb_cons // ]
359 ] * #xs0 #Hxs0 %{xs0} % [ %
360 [ >Hmid_dst >Hrsj >append_nil %
362 | cases (reverse ? xs1) // ]
363 | * #cj * #rs2 * #Hrsj #Hta lapply (Hta ?)
364 [ cases (Hneq ?? Hrs1) /2/ * #_ #Hr1 %2 @(Hr1 ?? Hrsj) ] -Hta #Hta
365 %2 >Hta in Hc; whd in ⊢ (??%?→?);
366 change with (current ? (niltape ?)) in match (None ?);
367 <nth_vec_map >nth_change_vec_neq [|@sym_not_eq //] >nth_change_vec //
368 whd in ⊢ (??%?→?); #Hc cut (is_endc r1 = true)
369 [ cases (is_endc r1) in Hc; whd in ⊢ (??%?→?); //
370 change with (current ? (niltape ?)) in match (None ?);
371 <nth_vec_map >nth_change_vec // normalize #H destruct (H) ]
372 #Hendr1 cut (xs = xs1)
373 [ lapply Hnotendxs1 lapply Hnotend lapply Hrs_src lapply xs1
374 -Hnotendxs1 -Hnotend -Hrs_src -xs1 elim xs
375 [ * normalize in ⊢ (%→?); //
376 #x2 #xs2 normalize in ⊢ (%→?); #Heq destruct (Heq) #_ #Hnotendxs1
377 lapply (Hnotendxs1 ? (memb_hd …)) >Hend #H destruct (H)
379 [ normalize in ⊢ (%→?); #Heq destruct (Heq) #Hnotendc
380 >Hnotendc in Hendr1; [| @memb_cons @memb_hd ]
381 normalize in ⊢ (%→?); #H destruct (H)
382 | #x3 #xs3 normalize in ⊢ (%→?); #Heq destruct (Heq)
383 #Hnotendc #Hnotendcxs1 @eq_f @IH
384 [ @(cons_injective_r … Heq)
385 | #c0 #Hc0 @Hnotendc cases (orb_true_l … Hc0) -Hc0 #Hc0
387 | @memb_cons @memb_cons // ]
388 | #c #Hc @Hnotendcxs1 @memb_cons // ]
391 | #Hxsxs1 destruct (Hxsxs1) >Hmid_dst %{ls_dst} %{rsj} % //
392 #rsj0 #c >Hrsj #Hrsj0 destruct (Hrsj0)
393 lapply (append_l2_injective … Hrs_src) // #Hrs' destruct (Hrs') %
396 |#Hcomp1 #Hsrc cases (Hsrc ? (refl ??)) -Hsrc #ls0 * #rs0 #Hdst
397 @False_ind lapply (Hcomp1 ?) [%2 %1 %1 >Hmid_src normalize
398 @(not_to_not ??? (\Pf Hceq)) #H destruct //] #Hintape >Hintape in Hc;
399 whd in ⊢(??%?→?); >Hmid_src
400 change with (current ? (niltape ?)) in match (None ?);
401 <nth_vec_map >Hmid_src whd in ⊢ (??%?→?);
402 >(Hnotend c_src) [|@memb_hd]
403 change with (current ? (niltape ?)) in match (None ?);
404 <nth_vec_map >Hmid_src whd in ⊢ (??%?→?); >Hdst normalize #H destruct (H)
410 definition match_m ≝ λsrc,dst,sig,n,is_startc,is_endc.
411 whileTM … (match_step src dst sig n is_startc is_endc)
412 (inr ?? (inr ?? (inl … (inr ?? start_nop)))).
414 definition R_match_m ≝
415 λsrc,dst,sig,n,is_startc,is_endc.λint,outt: Vector (tape sig) (S n).
417 nth src ? int (niltape ?) = midtape sig ls x (xs@end::rs) →
418 (∀c0. memb ? c0 (x::xs) = true → is_endc c0 = false) → is_endc end = true →
419 (∀c0. memb ? c0 (xs@end::rs) = true → is_startc c0 = false) →
420 (current sig (nth dst (tape sig) int (niltape sig)) = None ? → outt = int) ∧
421 (is_startc x = true →
423 nth dst ? int (niltape ?) = midtape sig ls0 x0 rs0 →
424 (∃l,l1.x0::rs0 = l@x::xs@l1 ∧
427 (change_vec ?? int (midtape sig (reverse ? xs@x::ls) end rs) src)
428 (midtape sig ((reverse ? (l@x::xs))@ls0) cj l2) dst) ∨
429 ∀l,l1.x0::rs0 ≠ l@x::xs@l1)).
431 lemma not_sub_list_merge :
432 ∀T.∀a,b:list T. (∀l1.a ≠ b@l1) → (∀t,l,l1.a ≠ t::l@b@l1) → ∀l,l1.a ≠ l@b@l1.
433 #T #a #b #H1 #H2 #l elim l normalize //
436 lemma not_sub_list_merge_2 :
437 ∀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.
438 #T #a #b #t #H1 #H2 #l elim l //
439 #t0 #l1 #IH #l2 cases (true_or_false (t == t0)) #Htt0
440 [ >(\P Htt0) % normalize #H destruct (H) cases (H2 l1 l2) /2/
441 | normalize % #H destruct (H) cases (\Pf Htt0) /2/ ]
445 lemma wsem_match_m : ∀src,dst,sig,n,is_startc,is_endc.
446 src ≠ dst → src < S n → dst < S n →
447 match_m src dst sig n is_startc is_endc ⊫ R_match_m src dst sig n is_startc is_endc.
448 #src #dst #sig #n #is_startc #is_endc #Hneq #Hsrc #Hdst #ta #k #outc #Hloop
449 lapply (sem_while … (sem_match_step src dst sig n is_startc is_endc Hneq Hsrc Hdst) … Hloop) //
450 -Hloop * #tb * #Hstar @(star_ind_l ??????? Hstar) -Hstar
451 [ #Hfalse #ls #x #xs #end #rs #Hmid_src #Hnotend #Hend #Hnotstart
452 cases (Hfalse … Hmid_src Hnotend Hend) -Hfalse
453 [(* current dest = None *) *
454 [ * #Hcur_dst #Houtc %
456 |#Hstart #ls0 #x0 #rs0 #Hmid_dst >Hmid_dst in Hcur_dst;
457 normalize in ⊢ (%→?); #H destruct (H)
459 | * #ls0 * #rs0 * #xs0 * * #Htc_dst #Hrs0 #HNone %
460 [ >Htc_dst normalize in ⊢ (%→?); #H destruct (H)
461 | #Hstart #ls1 #x1 #rs1 >Htc_dst #H destruct (H)
463 [ % %{[ ]} %{[ ]} % [ >append_nil >append_nil %]
464 #cj #ls2 #H destruct (H)
465 | #x2 #xs2 %2 #l #l1 % #Habs lapply (eq_f ?? (length ?) ?? Habs)
466 >length_append whd in ⊢ (??%(??%)→?); >length_append
467 >length_append normalize >commutative_plus whd in ⊢ (???%→?);
468 #H destruct (H) lapply e0 >(plus_n_O (|rs1|)) in ⊢ (??%?→?);
469 >associative_plus >associative_plus
470 #e1 lapply (injective_plus_r ??? e1) whd in ⊢ (???%→?);
475 |* #ls0 * #rs0 * #Hmid_dst #HFalse %
476 [ >Hmid_dst normalize in ⊢ (%→?); #H destruct (H)
477 | #Hstart #ls1 #x1 #rs1 >Hmid_dst #H destruct (H)
478 %1 %{[ ]} %{rs0} % [%] #cj #l2 #Hnotnil
479 >reverse_cons >associative_append @(HFalse ?? Hnotnil)
482 |-ta #ta #tc #Htrue #Hstar #IH #Hout lapply (IH Hout) -IH -Hout #IH whd
483 #ls #x #xs #end #rs #Hmid_src #Hnotend #Hend #Hnotstart
484 lapply (refl ? (current ? (nth dst ? ta (niltape ?))))
485 cases (current ? (nth dst ? ta (niltape ?))) in ⊢ (???%→?);
487 [#_ whd in Htrue; >Hmid_src in Htrue; #Htrue
488 cases (Htrue x (refl … )) -Htrue * #Htaneq #_
489 @False_ind >Hmid_dst in Htaneq; /2/
490 |#Hstart #ls0 #x0 #rs0 #Hmid_dst2 >Hmid_dst2 in Hmid_dst; normalize in ⊢ (%→?);
493 | #c #Hcurta_dst % [ >Hcurta_dst #H destruct (H) ]
494 #Hstart #ls0 #x0 #rs0 #Hmid_dst >Hmid_dst in Hcurta_dst; normalize in ⊢ (%→?);
495 #H destruct (H) whd in Htrue; >Hmid_src in Htrue; #Htrue
496 cases (Htrue x (refl …)) -Htrue #_ #Htrue cases (Htrue Hstart Hnotstart) -Htrue
497 cases (true_or_false (x==c)) #eqx
498 [ lapply (\P eqx) -eqx #eqx destruct (eqx)
499 #_ #Htrue cases (comp_list ? (xs@end::rs) rs0 is_endc)
500 #x1 * #tl1 * #tl2 * * * #Hxs #Hrs0 #Hnotendx1
502 [>append_nil #Hx1 <Hx1 in Hnotendx1; #Hnotendx1
503 lapply (Hnotendx1 end ?) [ @memb_append_l2 @memb_hd ]
504 >Hend #H destruct (H) ]
505 #ci -tl1 #tl1 #Hxs #H cases (H … (refl … )) -H
506 [ #Hendci % >Hrs0 in Hmid_dst; cut (ci = end ∧ x1 = xs)
507 [ lapply Hxs lapply Hnotendx1 lapply x1 elim xs in Hnotend;
509 [ #_ normalize #H destruct (H) /2/
510 | #x2 #xs2 #Hnotendx2 normalize #H destruct (H)
511 >(Hnotendx2 ? (memb_hd …)) in Hend; #H destruct (H) ]
512 | #x2 #xs2 #IH #Hnotendx2 *
513 [ #_ normalize #H destruct (H) >(Hnotendx2 ci ?) in Hendci;
515 | @memb_cons @memb_hd ]
516 | #x3 #xs3 #Hnotendx3 normalize #H destruct (H)
519 | #c0 #Hc0 @Hnotendx2 cases (orb_true_l … Hc0) -Hc0 #Hc0
521 | @memb_cons @memb_cons @Hc0 ]
522 | #c0 #Hc0 @Hnotendx3 @memb_cons @Hc0 ]
525 | * #Hcieq #Hx1eq >Hx1eq #Hmid_dst
526 cases (Htrue ??????? (refl ??) Hmid_dst Hnotend)
527 <Hcieq >Hendci * #H destruct (H) ]
529 [ >append_nil #Hrs0 destruct (Hrs0) * #Hcifalse#_ %2
530 cut (∃l.xs = x1@ci::l)
531 [lapply Hxs lapply Hnotendx1 lapply Hnotend lapply xs
532 -Hxs -xs -Hnotendx1 elim x1
534 [ #_ #_ normalize #H1 destruct (H1) >Hend in Hcifalse;
536 | #x2 #xs2 #_ #_ normalize #H >(cons_injective_l ????? H) %{xs2} % ]
538 [ #_ #Hnotendxs2 normalize #H destruct (H)
539 >(Hnotendxs2 ? (memb_hd …)) in Hend; #H destruct (H)
540 | #x3 #xs3 #Hnotendxs3 #Hnotendxs2 normalize #H destruct (H)
542 [ #xs4 #Hxs4 >Hxs4 %{xs4} %
543 | #c0 #Hc0 cases (orb_true_l … Hc0) -Hc0 #Hc0
544 [ >(\P Hc0) @Hnotendxs3 @memb_hd
545 | @Hnotendxs3 @memb_cons @memb_cons @Hc0 ]
546 | #c0 #Hc0 @Hnotendxs2 @memb_cons @Hc0 ]
550 #l0 #l1 % #H lapply (eq_f ?? (length ?) ?? H) -H
551 >length_append normalize >length_append >length_append
552 normalize >commutative_plus normalize #H destruct (H) -H
553 >associative_plus in e0; >associative_plus
554 >(plus_n_O (|x1|)) in ⊢(??%?→?); #H lapply (injective_plus_r … H)
555 -H normalize #H destruct (H)
556 | #cj #tl2' #Hrs0 * #Hcifalse #Hcomp
557 lapply (Htrue ls c x1 ci tl1 ls0 (cj::tl2') ???)
558 [ #c0 #Hc0 cases (orb_true_l … Hc0) #Hc0
559 [ @Hnotend >(\P Hc0) @memb_hd
563 | * * #_ #_ -Htrue #Htrue lapply (Htrue ?? (refl ??) ?) [ @(Hcomp ?? (refl ??)) ]
564 * #Htb >Htb #Hendci >Hrs0 >Hxs
565 cases (IH ls c xs end rs ? Hnotend Hend Hnotstart) -IH
566 [| >Htb >nth_change_vec_neq [|@sym_not_eq //] @Hmid_src ]
567 #_ #IH lapply Hxs lapply Hnotendx1 -Hxs -Hnotendx1 cases x1 in Hrs0;
568 [ #Hrs0 #_ whd in ⊢ (???%→?); #Hxs
569 cases (IH Hstart (c::ls0) cj tl2' ?)
570 [ -IH * #l * #l1 * #Hll1 #IH % %{(c::l)} %{l1}
572 #cj0 #l2 #Hcj0 >(IH … Hcj0) >Htb
573 >change_vec_commute // >change_vec_change_vec
574 >change_vec_commute [|@sym_not_eq // ] @eq_f3 //
575 >reverse_cons >associative_append %
576 | #IH %2 #l #l1 >(?:l@c::xs@l1 = l@(c::xs)@l1) [|%]
578 [ #l2 cut (∃xs'.xs = ci::xs')
580 [ normalize #H destruct (H) >Hend in Hendci; #H destruct (H)
581 | #ci' #xs' normalize #H lapply (cons_injective_l ????? H)
584 * #xs' #Hxs' >Hxs' normalize % #H destruct (H)
585 lapply (Hcomp … (refl ??)) * /2/
586 |#t #l2 #l3 % normalize #H lapply (cons_injective_r ????? H)
587 -H #H >H in IH; #IH cases (IH l2 l3) -IH #IH @IH % ]
588 | >Htb >nth_change_vec // >Hmid_dst >Hrs0 % ]
589 | #x2 #xs2 normalize in ⊢ (%→?); #Hrs0 #Hnotendxs2 normalize in ⊢ (%→?);
590 #Hxs cases (IH Hstart (c::ls0) x2 (xs2@cj::tl2') ?)
591 [ -IH * #l * #l1 * #Hll1 #IH % %{(c::l)} %{l1}
593 #cj0 #l2 #Hcj0 >(IH … Hcj0) >Htb
594 >change_vec_commute // >change_vec_change_vec
595 >change_vec_commute [|@sym_not_eq // ] @eq_f3 //
596 >reverse_cons >associative_append %
597 | -IH #IH %2 #l #l1 >(?:l@c::xs@l1 = l@(c::xs)@l1) [|%]
598 @not_sub_list_merge_2 [| @IH]
599 cut (∃l2.xs = (x2::xs2)@ci::l2)
601 lapply Hnotend -Hnotend lapply Hxs
602 >(?:x2::xs2@ci::tl1 = (x2::xs2)@ci::tl1) [|%]
603 lapply (x2::xs2) elim xs
605 [ normalize in ⊢ (%→?); #H1 destruct (H1)
606 >Hendci in Hend; #Hend destruct (Hend)
607 | #x3 #xs3 normalize in ⊢ (%→?); #H1 destruct (H1)
608 #_ #Hnotendx3 >(Hnotendx3 ? (memb_hd …)) in Hend;
609 #Hend destruct (Hend)
612 [ normalize in ⊢ (%→?); #Hxs3 destruct (Hxs3) #_ #_
614 | #x4 #xs4 normalize in ⊢ (%→?); #Hxs3xs4 #Hnotend
615 #Hnotendxs4 destruct (Hxs3xs4) cases (IHin ? e0 ??)
616 [ #l0 #Hxs3 >Hxs3 %{l0} %
617 | #c0 #Hc0 @Hnotend cases (orb_true_l … Hc0) -Hc0 #Hc0
619 | @memb_cons @memb_cons @Hc0 ]
620 | #c0 #Hc0 @Hnotendxs4 @memb_cons //
625 >Hxs' #l3 normalize >associative_append normalize % #H
626 destruct (H) lapply (append_l2_injective ?????? e1) //
627 #H1 destruct (H1) cases (Hcomp ?? (refl ??)) /2/
628 | >Htb >nth_change_vec // >Hmid_dst >Hrs0 % ]
633 |lapply (\Pf eqx) -eqx #eqx >Hmid_dst #Htrue
634 cases (Htrue ? (refl ??) eqx) -Htrue #Htb #Hendcx #_
636 [ #_ %2 #l #l1 cases l
639 [ normalize % #H destruct (H) cases eqx /2/
640 | #tmp1 #l2 normalize % #H destruct (H) ]
641 | #tmp1 #l2 normalize % #H destruct (H) ]
642 | #tmp1 #l2 normalize % #H destruct (H)cases l2 in e0;
643 [ normalize #H1 destruct (H1)
644 | #tmp2 #l3 normalize #H1 destruct (H1) ]
646 | #r1 #rs1 normalize in ⊢ (???(????%?)→?); #Htb >Htb in IH; #IH
647 cases (IH ls x xs end rs ? Hnotend Hend Hnotstart)
648 [| >Htb >nth_change_vec_neq [|@sym_not_eq //] @Hmid_src ] -IH
649 #_ #IH cases (IH Hstart (c::ls0) r1 rs1 ?)
650 [|| >nth_change_vec // ] -IH
651 [ * #l * #l1 * #Hll1 #Hout % %{(c::l)} %{l1} % >Hll1 //
652 >reverse_cons >associative_append #cj0 #ls #Hl1 >(Hout ?? Hl1)
653 >change_vec_commute in ⊢ (??(???%??)?); // @sym_not_eq //
654 | #IH %2 @(not_sub_list_merge_2 ?? (x::xs)) normalize [|@IH]
655 #l1 % #H destruct (H) cases eqx /2/
662 definition Pre_match_m ≝
663 λsrc,sig,n,is_startc,is_endc.λt: Vector (tape sig) (S n).
665 nth src (tape sig) t (niltape sig) = midtape ? [] start (xs@[end]) ∧
666 is_startc start = true ∧
667 (∀c.c ∈ (xs@[end]) = true → is_startc c = false) ∧
668 (∀c.c ∈ (start::xs) = true → is_endc c = false) ∧
671 lemma terminate_match_m :
672 ∀src,dst,sig,n,is_startc,is_endc,t.
673 src ≠ dst → src < S n → dst < S n →
674 Pre_match_m src sig n is_startc is_endc t →
675 match_m src dst sig n is_startc is_endc ↓ t.
676 #src #dst #sig #n #is_startc #is_endc #t #Hneq #Hsrc #Hdst * #start * #xs * #end
677 * * * * #Hmid_src #Hstart #Hnotstart #Hnotend #Hend
678 @(terminate_while … (sem_match_step src dst sig n is_startc is_endc Hneq Hsrc Hdst)) //
679 <(change_vec_same … t dst (niltape ?))
680 lapply (refl ? (nth dst (tape sig) t (niltape ?)))
681 cases (nth dst (tape sig) t (niltape ?)) in ⊢ (???%→?);
682 [ #Htape_dst % #t1 whd in ⊢ (%→?); >nth_change_vec_neq [|@sym_not_eq //]
683 >Hmid_src #HR cases (HR ? (refl ??)) -HR
684 >nth_change_vec // >Htape_dst normalize in ⊢ (%→?);
686 | #x0 #xs0 #Htape_dst % #t1 whd in ⊢ (%→?); >nth_change_vec_neq [|@sym_not_eq //]
687 >Hmid_src #HR cases (HR ? (refl ??)) -HR
688 >nth_change_vec // >Htape_dst normalize in ⊢ (%→?);
690 | #x0 #xs0 #Htape_dst % #t1 whd in ⊢ (%→?); >nth_change_vec_neq [|@sym_not_eq //]
691 >Hmid_src #HR cases (HR ? (refl ??)) -HR
692 >nth_change_vec // >Htape_dst normalize in ⊢ (%→?);
694 | #ls #s #rs lapply s -s lapply ls -ls lapply Hmid_src lapply t -t elim rs
695 [#t #Hmid_src #ls #s #Hmid_dst % #t1 whd in ⊢ (%→?); >nth_change_vec_neq [|@sym_not_eq //]
696 >Hmid_src >nth_change_vec // >Hmid_dst #HR cases (HR ? (refl ??)) -HR #_
697 #HR cases (HR Hstart Hnotstart)
698 cases (true_or_false (start == s)) #Hs
699 [ lapply (\P Hs) -Hs #Hs <Hs #_ #Htrue
700 cut (∃ci,xs1.xs@[end] = ci::xs1)
703 | #x1 #xs1 %{x1} %{(xs1@[end])} % ] ] * #ci * #xs1 #Hxs
704 >Hxs in Htrue; #Htrue
705 cases (Htrue [ ] start [ ] ? xs1 ? [ ] (refl ??) (refl ??) ?)
706 [ * #_ * #H @False_ind @H % ]
707 #c0 #Hc0 @Hnotend >(memb_single … Hc0) @memb_hd
708 | lapply (\Pf Hs) -Hs #Hs #Htrue #_
709 cases (Htrue ? (refl ??) Hs) -Htrue #Ht1 #_ %
710 #t2 whd in ⊢ (%→?); #HR cases (HR start ?)
711 [ >Ht1 >nth_change_vec // normalize in ⊢ (%→?); * #H @False_ind @H %
712 | >Ht1 >nth_change_vec_neq [|@sym_not_eq //]
713 >nth_change_vec_neq [|@sym_not_eq //] >Hmid_src % ]
715 |#r0 #rs0 #IH #t #Hmid_src #ls #s #Hmid_dst % #t1 whd in ⊢ (%→?);
716 >nth_change_vec_neq [|@sym_not_eq //] >Hmid_src
717 #Htrue cases (Htrue ? (refl ??)) -Htrue #_ #Htrue
718 <(change_vec_same … t1 dst (niltape ?))
719 cases (Htrue Hstart Hnotstart) -Htrue
720 cases (true_or_false (start == s)) #Hs
721 [ lapply (\P Hs) -Hs #Hs <Hs #_ #Htrue
722 cut (∃ls0,xs0,ci,rs,rs0.
723 nth src ? t (niltape ?) = midtape sig [ ] start (xs0@ci::rs) ∧
724 nth dst ? t (niltape ?) = midtape sig ls0 s (xs0@rs0) ∧
725 (is_endc ci = true ∨ (is_endc ci = false ∧ (∀b,tlb.rs0 = b::tlb → ci ≠ b))))
726 [cases (comp_list ? (xs@[end]) (r0::rs0) is_endc) #xs0 * #xs1 * #xs2
727 * * * #Hxs #Hrs #Hxs0notend #Hcomp >Hrs
728 cut (∃y,ys. xs1 = y::ys)
729 [ lapply Hxs0notend lapply Hxs lapply xs0 elim xs
731 [ normalize #Hxs1 <Hxs1 #_ %{end} %{[]} %
732 | #z #zs normalize in ⊢ (%→?); #H destruct (H) #H
733 lapply (H ? (memb_hd …)) -H >Hend #H1 destruct (H1)
736 [ normalize in ⊢ (%→?); #Hxs1 <Hxs1 #_ %{y} %{(ys@[end])} %
737 | #z #zs normalize in ⊢ (%→?); #H destruct (H) #Hmemb
738 @(IH0 ? e0 ?) #c #Hc @Hmemb @memb_cons // ] ] ] * #y * #ys #Hxs1
739 >Hxs1 in Hxs; #Hxs >Hmid_src >Hmid_dst >Hxs >Hrs
740 %{ls} %{xs0} %{y} %{ys} %{xs2}
741 % [ % // | @Hcomp // ] ]
742 * #ls0 * #xs0 * #ci * #rs * #rs0 * * #Hmid_src' #Hmid_dst' #Hcomp
743 <Hmid_src in Htrue; >nth_change_vec // >Hs #Htrue destruct (Hs)
744 lapply (Htrue ??????? Hmid_src' Hmid_dst' ?) -Htrue
745 [ #c0 #Hc0 @Hnotend cases (orb_true_l … Hc0) -Hc0 #Hc0
746 [ whd in ⊢ (??%?); >Hc0 %
747 | @memb_cons >Hmid_src in Hmid_src'; #Hmid_src' destruct (Hmid_src')
748 lapply e0 -e0 @(list_elim_left … rs)
749 [ #e0 destruct (e0) lapply (append_l1_injective_r ?????? e0) //
750 | #x1 #xs1 #_ >append_cons in ⊢ (???%→?);
751 <associative_append #e0 lapply (append_l1_injective_r ?????? e0) //
752 #e1 >e1 @memb_append_l1 @memb_append_l1 // ] ]
753 | * * #Hciendc cases rs0 in Hcomp;
754 [ #_ * #H @False_ind @H %
755 | #r1 #rs1 * [ >Hciendc #H destruct (H) ]
756 * #_ #Hcomp lapply (Hcomp ?? (refl ??)) -Hcomp #Hcomp #_ #Htrue
757 cases (Htrue ?? (refl ??) Hcomp) #Ht1 #_ >Ht1 @(IH ?? (s::ls) r0)
758 [ >nth_change_vec_neq [|@sym_not_eq //]
759 >nth_change_vec_neq [|@sym_not_eq //] @Hmid_src
760 | >nth_change_vec // >Hmid_dst % ] ] ]
761 | >Hmid_dst >nth_change_vec // lapply (\Pf Hs) -Hs #Hs #Htrue #_
762 cases (Htrue ? (refl ??) Hs) #Ht1 #_ >Ht1 @(IH ?? (s::ls) r0)
763 [ >nth_change_vec_neq [|@sym_not_eq //]
764 >nth_change_vec_neq [|@sym_not_eq //] @Hmid_src
765 | >nth_change_vec // ] ] ] ]