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".
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 axiom comp_list: ∀S:DeqSet. ∀l1,l2:list S.∀is_endc. ∃l,tl1,tl2.
75 l1 = l@tl1 ∧ l2 = l@tl2 ∧ (∀c.c ∈ l = true → is_endc c = false) ∧
76 ∀a,tla. tl1 = a::tla → is_endc a = true ∨ (∀b,tlb.tl2 = b::tlb → a≠b).
78 axiom daemon : ∀X:Prop.X.
80 definition match_test ≝ λsrc,dst.λsig:DeqSet.λn,is_endc.λv:Vector ? n.
81 match (nth src (option sig) v (None ?)) with
83 | Some x ⇒ notb ((is_endc x) ∨ (nth dst (DeqOption sig) v (None ?) == None ?))].
85 definition match_step ≝ λsrc,dst,sig,n,is_startc,is_endc.
86 compare src dst sig n is_endc ·
87 (ifTM ?? (partest sig n (match_test src dst sig ? is_endc))
89 (parmove src dst sig n L is_startc · (inject_TM ? (move_r ?) n dst)))
93 definition R_match_step_false ≝
94 λsrc,dst,sig,n,is_endc.λint,outt: Vector (tape sig) (S n).
96 nth src ? int (niltape ?) = midtape sig ls x (xs@end::rs) →
97 (∀c0. memb ? c0 (x::xs) = true → is_endc c0 = false) → is_endc end = true →
98 ((current sig (nth dst (tape sig) int (niltape sig)) = None ?) ∧ outt = int) ∨
99 (current sig (nth dst (tape sig) outt (niltape sig)) = None ?) ∨
101 nth dst ? int (niltape ?) = midtape sig ls0 x (xs@rs0) ∧
105 (change_vec ?? int (midtape sig (reverse ? xs@x::ls) end rs) src)
106 (midtape sig (reverse ? xs@x::ls0) c rsj) dst).
108 definition R_match_step_true ≝
109 λsrc,dst,sig,n,is_startc,is_endc.λint,outt: Vector (tape sig) (S n).
110 ∀s.current sig (nth src (tape sig) int (niltape sig)) = Some ? s →
112 (∀c.c ∈ right ? (nth src (tape sig) int (niltape sig)) = true → is_startc c = false) →
113 current sig (nth dst (tape sig) int (niltape sig)) ≠ None ? ∧
114 (∀s1.current sig (nth dst (tape sig) int (niltape sig)) = Some ? s1 → s ≠ s1 →
115 outt = change_vec ?? int
116 (tape_move … (nth dst ? int (niltape ?)) (Some ? 〈s1,R〉)) dst ∧ is_endc s = false) ∧
117 (∀ls,x,xs,ci,cj,rs,ls0,rs0.
118 nth src ? int (niltape ?) = midtape sig ls x (xs@ci::rs) →
119 nth dst ? int (niltape ?) = midtape sig ls0 x (xs@cj::rs0) →
120 (∀c0. memb ? c0 (x::xs) = true → is_endc c0 = false) →
122 (outt = change_vec ?? int
123 (tape_move … (nth dst ? int (niltape ?)) (Some ? 〈x,R〉)) dst ∧ is_endc ci = false)).
127 (change_vec ?? int (midtape sig (reverse ? xs@x::ls) ci rs) src)
128 (mk_tape sig (reverse ? xs@x::ls0) (None ?) [ ]) dst)). *)
130 lemma sem_match_step :
131 ∀src,dst,sig,n,is_startc,is_endc.src ≠ dst → src < S n → dst < S n →
132 match_step src dst sig n is_startc is_endc ⊨
133 [ inr ?? (inr ?? (inl … (inr ?? start_nop))) :
134 R_match_step_true src dst sig n is_startc is_endc,
135 R_match_step_false src dst sig n is_endc ].
136 #src #dst #sig #n #is_startc #is_endc #Hneq #Hsrc #Hdst
137 @(acc_sem_seq_app sig n … (sem_compare src dst sig n is_endc Hneq Hsrc Hdst)
138 (acc_sem_if ? n … (sem_partest sig n (match_test src dst sig ? is_endc))
140 (sem_parmoveL ???? is_startc Hneq Hsrc Hdst)
141 (sem_inject … dst (le_S_S_to_le … Hdst) (sem_move_r ? )))
143 [#ta #tb #tc * #Hcomp1 #Hcomp2 * #td * * #Htest #Htd >Htd -Htd
144 * #te * #Hte #Htb whd
145 #s #Hcurta_src #Hstart #Hnotstart % [ %
147 | #s1 #Hcurta_dst #Hneqss1 -Hcomp2
149 [@Hcomp1 %2 %1 %1 >Hcurta_src >Hcurta_dst @(not_to_not … Hneqss1) #H destruct (H) //]
150 #H destruct (H) -Hcomp1 cases Hte #_ -Hte #Hte
151 cut (te = ta) [@Hte %1 %1 %{s} % //] -Hte #H destruct (H) %
152 [cases Htb * #_ #Hmove #Hmove1 @(eq_vec … (niltape … ))
153 #i #Hi cases (decidable_eq_nat i dst) #Hidst
154 [ >Hidst >nth_change_vec // cases (current_to_midtape … Hcurta_dst)
155 #ls * #rs #Hta_mid >(Hmove … Hta_mid) >Hta_mid cases rs //
156 | >nth_change_vec_neq [|@sym_not_eq //] @sym_eq @Hmove1 @sym_not_eq // ]
157 | whd in Htest:(??%?); >(nth_vec_map ?? (current sig)) in Hcurta_src; #Hcurta_src
158 >Hcurta_src in Htest; whd in ⊢ (??%?→?);
159 cases (is_endc s) // whd in ⊢ (??%?→?); #H @sym_eq //
161 |#ls #x #xs #ci #cj #rs #ls0 #rs00 #Htasrc_mid #Htadst_mid #Hnotendc #Hcicj
162 cases (Hcomp2 … Htasrc_mid Htadst_mid Hnotendc) [ * #H destruct (H) ]
163 * #cj' * #rs0' * #Hcjrs0 destruct (Hcjrs0) -Hcomp2 #Hcomp2
164 lapply (Hcomp2 (or_intror ?? Hcicj)) -Hcomp2 #Htc %
165 [ cases Hte -Hte #Hte #_ whd in Hte;
166 >Htasrc_mid in Hcurta_src; whd in ⊢ (??%?→?); #H destruct (H)
167 lapply (Hte ls ci (reverse ? xs) rs s ??? ls0 cj' (reverse ? xs) s rs0' (refl ??) ?) //
168 [ >Htc >nth_change_vec //
169 | #c0 #Hc0 @(Hnotstart c0) >Htasrc_mid cases (orb_true_l … Hc0) -Hc0 #Hc0
170 [@memb_append_l2 >(\P Hc0) @memb_hd
171 |@memb_append_l1 <(reverse_reverse …xs) @memb_reverse //
173 | >Htc >change_vec_commute // >nth_change_vec // ] -Hte
174 >Htc >change_vec_commute // >change_vec_change_vec
175 >change_vec_commute [|@sym_not_eq //] >change_vec_change_vec #Hte
176 >Hte in Htb; * * #_ >reverse_reverse #Htbdst1 #Htbdst2 -Hte @(eq_vec … (niltape ?))
177 #i #Hi cases (decidable_eq_nat i dst) #Hidst
178 [ >Hidst >nth_change_vec // >(Htbdst1 ls0 s (xs@cj'::rs0'))
179 [| >nth_change_vec // ]
180 >Htadst_mid cases xs //
181 | >nth_change_vec_neq [|@sym_not_eq // ]
182 <Htbdst2 [| @sym_not_eq // ] >nth_change_vec_neq [| @sym_not_eq // ]
183 <Htasrc_mid >change_vec_same % ]
184 | >Hcurta_src in Htest; whd in ⊢(??%?→?);
185 >Htc >change_vec_commute //
186 change with (current ? (niltape ?)) in match (None ?);
187 <nth_vec_map >nth_change_vec // whd in ⊢ (??%?→?);
188 cases (is_endc ci) whd in ⊢ (??%?→?); #H destruct (H) %
191 |#intape #outtape #ta * #Hcomp1 #Hcomp2 * #tb * * #Hc #Htb
192 whd in ⊢ (%→?); #Hout >Hout >Htb whd
193 #ls #c_src #xs #end #rs #Hmid_src #Hnotend #Hend
194 lapply (current_to_midtape sig (nth dst ? intape (niltape ?)))
195 cases (current … (nth dst ? intape (niltape ?))) in Hcomp1;
196 [#Hcomp1 #_ %1 % % [% | @Hcomp1 %2 %2 % ]
197 |#c_dst cases (true_or_false (c_src == c_dst)) #Hceq
198 [#_ #Hmid_dst cases (Hmid_dst c_dst (refl …)) -Hmid_dst
199 #ls_dst * #rs_dst #Hmid_dst
200 cases (comp_list … (xs@end::rs) rs_dst is_endc) #xs1 * #rsi * #rsj * * *
201 #Hrs_src #Hrs_dst #Hnotendxs1 #Hneq >Hrs_dst in Hmid_dst; #Hmid_dst
202 cut (∃r1,rs1.rsi = r1::rs1) [@daemon] * #r1 * #rs1 #Hrs1 >Hrs1 in Hrs_src;
203 #Hrs_src >Hrs_src in Hmid_src; #Hmid_src <(\P Hceq) in Hmid_dst; #Hmid_dst
204 lapply (Hcomp2 ??????? Hmid_src Hmid_dst ?)
205 [ #c0 #Hc0 cases (orb_true_l … Hc0) -Hc0 #Hc0
206 [ >(\P Hc0) @Hnotend @memb_hd | @Hnotendxs1 //] ]
208 [ * #Hrsj >Hrsj #Hta % %2 >Hta >nth_change_vec // cases (reverse ? xs1) //
209 | * #cj * #rs2 * #Hrsj #Hta lapply (Hta ?)
210 [ cases (Hneq ?? Hrs1) /2/ #Hr1 %2 @(Hr1 ?? Hrsj) ] -Hta #Hta
211 %2 >Hta in Hc; whd in ⊢ (??%?→?);
212 change with (current ? (niltape ?)) in match (None ?);
213 <nth_vec_map >nth_change_vec_neq [|@sym_not_eq //] >nth_change_vec //
214 whd in ⊢ (??%?→?); #Hc cut (is_endc r1 = true)
215 [ cases (is_endc r1) in Hc; whd in ⊢ (??%?→?); //
216 change with (current ? (niltape ?)) in match (None ?);
217 <nth_vec_map >nth_change_vec // normalize #H destruct (H) ]
218 #Hendr1 cut (xs = xs1)
219 [ lapply Hnotendxs1 lapply Hnotend lapply Hrs_src lapply xs1
220 -Hnotendxs1 -Hnotend -Hrs_src -xs1 elim xs
221 [ * normalize in ⊢ (%→?); //
222 #x2 #xs2 normalize in ⊢ (%→?); #Heq destruct (Heq) #_ #Hnotendxs1
223 lapply (Hnotendxs1 ? (memb_hd …)) >Hend #H destruct (H)
225 [ normalize in ⊢ (%→?); #Heq destruct (Heq) #Hnotendc
226 >Hnotendc in Hendr1; [| @memb_cons @memb_hd ]
227 normalize in ⊢ (%→?); #H destruct (H)
228 | #x3 #xs3 normalize in ⊢ (%→?); #Heq destruct (Heq)
229 #Hnotendc #Hnotendcxs1 @eq_f @IH
230 [ @(cons_injective_r … Heq)
231 | #c0 #Hc0 @Hnotendc cases (orb_true_l … Hc0) -Hc0 #Hc0
233 | @memb_cons @memb_cons // ]
234 | #c #Hc @Hnotendcxs1 @memb_cons // ]
237 | #Hxsxs1 destruct (Hxsxs1) >Hmid_dst %{ls_dst} %{rsj} % //
238 #rsj0 #c >Hrsj #Hrsj0 destruct (Hrsj0)
239 lapply (append_l2_injective … Hrs_src) // #Hrs' destruct (Hrs') %
242 |#Hcomp1 #Hsrc cases (Hsrc ? (refl ??)) -Hsrc #ls0 * #rs0 #Hdst
243 @False_ind lapply (Hcomp1 ?) [%2 %1 %1 >Hmid_src normalize
244 @(not_to_not ??? (\Pf Hceq)) #H destruct //] #Hintape >Hintape in Hc;
245 whd in ⊢(??%?→?); >Hmid_src
246 change with (current ? (niltape ?)) in match (None ?);
247 <nth_vec_map >Hmid_src whd in ⊢ (??%?→?);
248 >(Hnotend c_src) [|@memb_hd]
249 change with (current ? (niltape ?)) in match (None ?);
250 <nth_vec_map >Hmid_src whd in ⊢ (??%?→?); >Hdst normalize #H destruct (H)
256 definition match_m ≝ λsrc,dst,sig,n,is_startc,is_endc.
257 whileTM … (match_step src dst sig n is_startc is_endc)
258 (inr ?? (inr ?? (inl … (inr ?? start_nop)))).
260 definition R_match_m ≝
261 λsrc,dst,sig,n,is_startc,is_endc.λint,outt: Vector (tape sig) (S n).
262 (* (current sig (nth dst (tape sig) int (niltape sig)) = None ? → outt = int) ∧ *)
264 nth src ? int (niltape ?) = midtape sig ls x (xs@end::rs) →
265 (∀c0. memb ? c0 (x::xs) = true → is_endc c0 = false) → is_endc end = true →
266 (current sig (nth dst (tape sig) int (niltape sig)) = None ? → outt = int) ∧
267 (is_startc x = true →
269 nth dst ? int (niltape ?) = midtape sig ls0 x0 rs0 →
270 (∃l,l1.x0::rs0 = l@x::xs@l1 ∧
273 (change_vec ?? int (midtape sig (reverse ? xs@x::ls) end rs) src)
274 (midtape sig ((reverse ? (l@x::xs))@ls0) cj l2) dst) ∨
275 ∀l,l1.x0::rs0 ≠ l@x::xs@l1)).
278 definition R_match_m ≝
279 λi,j,sig,n,is_startc,is_endc.λint,outt: Vector (tape sig) (S n).
280 (((∃x.current ? (nth i ? int (niltape ?)) = Some ? x ∧ is_endc x = true) ∨
281 current ? (nth i ? int (niltape ?)) = None ? ∨
282 current ? (nth j ? int (niltape ?)) = None ?) → outt = int) ∧
283 (∀ls,x,xs,ci,rs,ls0,x0,rs0.
284 (∀x. is_startc x ≠ is_endc x) →
285 is_startc x = true → is_endc ci = true →
286 (∀z. memb ? z (x::xs) = true → is_endc x = false) →
287 nth i ? int (niltape ?) = midtape sig ls x (xs@ci::rs) →
288 nth j ? int (niltape ?) = midtape sig ls0 x0 rs0 →
289 (∃l,l1.x0::rs0 = l@x::xs@l1 ∧
292 (change_vec ?? int (midtape sig (reverse ? xs@x::ls) ci rs) i)
293 (midtape sig ((reverse ? (l@x::xs))@ls0) cj l2) j) ∨
294 ∀l,l1.x0::rs0 ≠ l@x::xs@l1).
298 axiom sub_list_dec: ∀A.∀l,ls:list A.
299 ∃l1,l2. l = l1@ls@l2 ∨ ∀l1,l2. l ≠ l1@ls@l2.
302 lemma wsem_match_m : ∀src,dst,sig,n,is_startc,is_endc.
303 src ≠ dst → src < S n → dst < S n →
304 match_m src dst sig n is_startc is_endc ⊫ R_match_m src dst sig n is_startc is_endc.
305 #src #dst #sig #n #is_startc #is_endc #Hneq #Hsrc #Hdst #ta #k #outc #Hloop
306 lapply (sem_while … (sem_match_step src dst sig n is_startc is_endc Hneq Hsrc Hdst) … Hloop) //
307 -Hloop * #tb * #Hstar @(star_ind_l ??????? Hstar) -Hstar
308 [ #tc #Hfalse #ls #x #xs #end #rs #Hmid_src #Hnotend #Hend
309 cases (Hfalse … Hmid_src Hnotend Hend) -Hfalse
310 [(* current dest = None *) * #Hcur_dst #Houtc %
312 |#Hstart #ls0 #x0 #rs0 #Hmid_dst >Hmid_dst in Hcur_dst;
313 normalize in ⊢ (%→?); #H destruct (H)
315 |* #ls0 * #rs0 * #Hmid_dst #HFalse %
316 [ >Hmid_dst normalize in ⊢ (%→?); #H destruct (H)
317 | #Hstart #ls1 #x1 #rs1 >Hmid_dst #H destruct (H)
318 %1 %{[ ]} %{rs0} % [%] #cj #l2 #Hnotnil
319 >reverse_cons >associative_append @(HFalse ?? Hnotnil)
322 |#ta #tb #tc #Htrue #Hstar #IH #Hout lapply (IH Hout) -IH -Hout #IH whd
323 #ls #x #xs #end #rs #Hmid_src #Hnotend #Hend
324 lapply (refl ? (current ? (nth dst ? ta (niltape ?))))
325 cases (current ? (nth dst ? ta (niltape ?))) in ⊢ (???%→?);
327 [#_ whd in Htrue; >Hmid_src in Htrue; #Htrue
328 cases (Htrue x (refl … ) Hstart ?) -Htrue [2: @daemon]
329 * #Htb #_ #_ >Htb in IH; // #IH
330 cases (IH ls x xs end rs Hmid_src Hstart Hnotend Hend)
331 #Hcur_outc #_ @Hcur_outc //
332 |#ls0 #x0 #rs0 #Hmid_dst2 >Hmid_dst2 in Hmid_dst; normalize in ⊢ (%→?);
335 | #c #Hcurta_dst % [ >Hcurta_dst #H destruct (H) ]
336 #ls0 #x0 #rs0 #Hmid_dst >Hmid_dst in Hcurta_dst; normalize in ⊢ (%→?);
337 #H destruct (H) whd in Htrue; >Hmid_src in Htrue; #Htrue
338 cases (Htrue x (refl …) Hstart ?) -Htrue
339 [2: #z #membz @daemon (*aggiungere l'ipotesi*)]
340 cases (true_or_false (x==c)) #eqx
341 [ #_ #Htrue cases (comp_list ? (xs@end::rs) rs0 is_endc)
342 #x1 * #tl1 * #tl2 * * * #Hxs #Hrs0 #Hnotendx1
344 [>append_nil #Hx1 @daemon (* absurd by Hx1 e notendx1 *)]
345 #ci -tl1 #tl1 #Hxs #H cases (H … (refl … ))
346 [(* this is absurd, since Htrue conlcudes is_endc ci =false *)
347 #Hend_ci @daemon (* lapply(Htrue … (refl …)) -Htrue *)
348 |#Hcomp lapply (Htrue ls x x1 ci tl1 ls0 tl2 ???)
349 [ #c0 #Hc0 cases (orb_true_l … Hc0) #Hc0
350 [ @Hnotend >(\P Hc0) @memb_hd
352 | >Hmid_dst >Hrs0 >(\P eqx) %
354 | * cases tl2 in Hrs0;
355 [ >append_nil #Hrs0 #_ #Htb whd in IH;
356 lapply (IH ls x x1 ci tl1 ? Hstart ??)
359 | >Htb // >nth_change_vec_neq [|@sym_not_eq //] >nth_change_vec //
361 >Hrs0 in Hmid_dst; #Hmid_dst
362 cases(Htrue ???????? Hmid_dst) -Htrue #Htb #Hendx
364 cases(IH ls x xs end rs ? Hstart Hnotend Hend)
365 [* #H1 #H2 >Htb in H1; >nth_change_vec //
366 >Hmid_dst cases rs0 [2: #a #tl normalize in ⊢ (%→?); #H destruct (H)]
367 #_ %2 @daemon (* si dimostra *)
369 |>Htb >nth_change_vec_neq [|@sym_not_eq //] @Hmid_src