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".
17 definition Rtc_multi_true ≝
18 λalpha,test,n,i.λt1,t2:Vector ? (S n).
19 (∃c. current alpha (nth i ? t1 (niltape ?)) = Some ? c ∧ test c = true) ∧ t2 = t1.
21 definition Rtc_multi_false ≝
22 λalpha,test,n,i.λt1,t2:Vector ? (S n).
23 (∀c. current alpha (nth i ? t1 (niltape ?)) = Some ? c → test c = false) ∧ t2 = t1.
25 lemma sem_test_char_multi :
26 ∀alpha,test,n,i.i ≤ n →
27 inject_TM ? (test_char ? test) n i ⊨
28 [ tc_true : Rtc_multi_true alpha test n i, Rtc_multi_false alpha test n i ].
29 #alpha #test #n #i #Hin #int
30 cases (acc_sem_inject … Hin (sem_test_char alpha test) int)
31 #k * #outc * * #Hloop #Htrue #Hfalse %{k} %{outc} % [ %
33 | #Hqtrue lapply (Htrue Hqtrue) * * * #c *
34 #Hcur #Htestc #Hnth_i #Hnth_j %
36 | @(eq_vec … (niltape ?)) #i0 #Hi0
37 cases (decidable_eq_nat i0 i) #Hi0i
39 | @sym_eq @Hnth_j @sym_not_eq // ] ] ]
40 | #Hqfalse lapply (Hfalse Hqfalse) * * #Htestc #Hnth_i #Hnth_j %
42 | @(eq_vec … (niltape ?)) #i0 #Hi0
43 cases (decidable_eq_nat i0 i) #Hi0i
45 | @sym_eq @Hnth_j @sym_not_eq // ] ] ]
48 axiom comp_list: ∀S:DeqSet. ∀l1,l2:list S.∀is_endc. ∃l,tl1,tl2.
49 l1 = l@tl1 ∧ l2 = l@tl2 ∧ (∀c.c ∈ l = true → is_endc c = false) ∧
50 ∀a,tla. tl1 = a::tla → is_endc a = true ∨ (∀b,tlb.tl2 = b::tlb → a≠b).
52 axiom daemon : ∀X:Prop.X.
56 definition match_step ≝ λsrc,dst,sig,n,is_startc,is_endc.
57 compare src dst sig n is_endc ·
58 (ifTM ?? (inject_TM ? (test_char ? (λa.is_endc a == false)) n src)
59 (ifTM ?? (inject_TM ? (test_null ?) n src)
61 (parmove src dst sig n L is_startc · (inject_TM ? (move_r ?) n dst)))
66 definition R_match_step_false ≝
67 λsrc,dst,sig,n,is_endc.λint,outt: Vector (tape sig) (S n).
69 nth src ? int (niltape ?) = midtape sig ls x (xs@end::rs) →
70 (∀c0. memb ? c0 (x::xs) = true → is_endc c0 = false) → is_endc end = true →
71 ((current sig (nth dst (tape sig) int (niltape sig)) = None ?) ∧ outt = int) ∨
73 nth dst ? int (niltape ?) = midtape sig ls0 x (xs@rs0) ∧
77 (change_vec ?? int (midtape sig (reverse ? xs@x::ls) end rs) src)
78 (midtape sig (reverse ? xs@x::ls0) c rsj) dst).
80 definition R_match_step_true ≝
81 λsrc,dst,sig,n,is_startc,is_endc.λint,outt: Vector (tape sig) (S n).
82 ∀s.current sig (nth src (tape sig) int (niltape sig)) = Some ? s →
84 (∀c.c ∈ right ? (nth src (tape sig) int (niltape sig)) = true → is_startc c = false) →
85 current sig (nth dst (tape sig) int (niltape sig)) ≠ None ? ∧
86 (∀s1.current sig (nth dst (tape sig) int (niltape sig)) = Some ? s1 → s ≠ s1 →
87 outt = change_vec ?? int
88 (tape_move … (nth dst ? int (niltape ?)) (Some ? 〈s1,R〉)) dst ∧ is_endc s = false) ∧
89 (∀ls,x,xs,ci,rs,ls0,rs0.
90 nth src ? int (niltape ?) = midtape sig ls x (xs@ci::rs) →
91 nth dst ? int (niltape ?) = midtape sig ls0 x (xs@rs0) →
92 (∀c0. memb ? c0 (x::xs) = true → is_endc c0 = false) →
93 (∀cj,rs1.rs0 = cj::rs1 → ci ≠ cj →
94 (outt = change_vec ?? int
95 (tape_move … (nth dst ? int (niltape ?)) (Some ? 〈x,R〉)) dst ∧ is_endc ci = false)) ∧
98 (change_vec ?? int (midtape sig (reverse ? xs@x::ls) ci rs) src)
99 (mk_tape sig (reverse ? xs@x::ls0) (None ?) [ ]) dst)).
101 lemma sem_match_step :
102 ∀src,dst,sig,n,is_startc,is_endc.src ≠ dst → src < S n → dst < S n →
103 match_step src dst sig n is_startc is_endc ⊨
104 [ inr ?? (inr ?? (inl … (inr ?? (inr ?? start_nop)))) :
105 R_match_step_true src dst sig n is_startc is_endc,
106 R_match_step_false src dst sig n is_endc ].
107 #src #dst #sig #n #is_startc #is_endc #Hneq #Hsrc #Hdst
108 (* test_null versione multi? *)
109 @(acc_sem_seq_app sig n … (sem_compare src dst sig n is_endc Hneq Hsrc Hdst)
110 (acc_sem_if ? n … (sem_test_char_multi sig (λa.is_endc a == false) n src (le_S_S_to_le … Hsrc))
111 (acc_sem_if ? n … (sem_test_null sig (λa.is_endc a == false) n src (le_S_S_to_le … Hsrc))
114 (sem_parmoveL ???? is_startc Hneq Hsrc Hdst)
115 (sem_inject … dst (le_S_S_to_le … Hdst) (sem_move_r ? )))
117 [#ta #tb #tc * #Hcomp1 #Hcomp2 * #td * * * #c * #Hcurtc #Hcend #Htd >Htd -Htd
118 #Htb #s #Hcurta_src #Hstart #Hnotstart % [ %
120 | #s1 #Hcurta_dst #Hneqss1
121 lapply Htb lapply Hcurtc -Htb -Hcurtc >(?:tc=ta)
122 [|@Hcomp1 %2 % % >Hcurta_src >Hcurta_dst @(not_to_not … Hneqss1) #H destruct (H) % ]
123 #Hcurtc * #te * * #_ #Hte >Hte [2: %1 %1 %{s} % //]
124 whd in ⊢ (%→?); * * #_ #Htbdst #Htbelse %
125 [ @(eq_vec … (niltape ?)) #i #Hi cases (decidable_eq_nat i dst) #Hidst
126 [ >Hidst >nth_change_vec // cases (current_to_midtape … Hcurta_dst)
127 #ls * #rs #Hta_mid >(Htbdst … Hta_mid) >Hta_mid cases rs //
128 | >nth_change_vec_neq [|@sym_not_eq //] @sym_eq @Htbelse @sym_not_eq // ]
129 | >Hcurtc in Hcurta_src; #H destruct (H) cases (is_endc s) in Hcend;
130 normalize #H destruct (H) // ]
132 |#ls #x #xs #ci #rs #ls0 #rs00 #Htasrc_mid #Htadst_mid #Hnotendc
133 cases rs00 in Htadst_mid;
134 [(* case rs empty *) #Htadst_mid % [ #cj #rs1 #H destruct (H) ]
135 #_ cases (Hcomp2 … Htasrc_mid Htadst_mid Hnotendc) -Hcomp2
136 [2: * #x0 * #rs1 * #H destruct (H) ]
137 * #_ #Htc cases Htb #td * * #_ #Htd >Htasrc_mid in Hcurta_src;
138 normalize in ⊢ (%→?); #H destruct (H)
139 >Htd [2: %2 >Htc >nth_change_vec // cases (reverse sig ?) //]
140 >Htc * * >nth_change_vec // #Htbdst #_ #Htbelse
141 @(eq_vec … (niltape ?)) #i #Hi cases (decidable_eq_nat i dst) #Hidst
142 [ >Hidst >nth_change_vec // <Htbdst // cases (reverse sig ?) //
143 |@sym_eq @Htbelse @sym_not_eq //
145 |#cj0 #rs0 #Htadst_mid % [| #H destruct (H) ]
146 #cj #rs1 #H destruct (H) #Hcicj
147 cases (Hcomp2 … Htasrc_mid Htadst_mid Hnotendc) [ * #H destruct (H) ]
148 * #cj' * #rs0' * #Hcjrs0 destruct (Hcjrs0) -Hcomp2 #Hcomp2
149 lapply (Hcomp2 (or_intror ?? Hcicj)) -Hcomp2 #Htc
150 cases Htb #td * * #Htd #_ >Htasrc_mid in Hcurta_src; normalize in ⊢ (%→?);
152 >(Htd ls ci (reverse ? xs) rs s ??? ls0 cj' (reverse ? xs) s rs0' (refl ??)) //
153 [| >Htc >nth_change_vec //
154 | #c0 #Hc0 @(Hnotstart c0) >Htasrc_mid
155 cases (orb_true_l … Hc0) -Hc0 #Hc0
156 [@memb_append_l2 >(\P Hc0) @memb_hd
157 |@memb_append_l1 <(reverse_reverse …xs) @memb_reverse //
159 | >Htc >nth_change_vec_neq [|@sym_not_eq // ] @nth_change_vec // ]
160 * * #_ #Htbdst #Htbelse %
161 [ @(eq_vec … (niltape ?)) #i #Hi cases (decidable_eq_nat i dst) #Hidst
162 [ >Hidst >nth_change_vec // >Htadst_mid >(Htbdst ls0 s (xs@cj'::rs0'))
164 | >nth_change_vec // ]
165 | >nth_change_vec_neq [|@sym_not_eq //]
166 <Htbelse [|@sym_not_eq // ]
167 >nth_change_vec_neq [|@sym_not_eq //]
168 cases (decidable_eq_nat i src) #Hisrc
169 [ >Hisrc >nth_change_vec // >Htasrc_mid //
170 | >nth_change_vec_neq [|@sym_not_eq //]
171 <(Htbelse i) [|@sym_not_eq // ]
172 >Htc >nth_change_vec_neq [|@sym_not_eq // ]
173 >nth_change_vec_neq [|@sym_not_eq // ] //
176 | >Htc in Hcurtc; >nth_change_vec_neq [|@sym_not_eq //]
177 >nth_change_vec // whd in ⊢ (??%?→?);
178 #H destruct (H) cases (is_endc c) in Hcend;
179 normalize #H destruct (H) // ]
182 |#intape #outtape #ta * #Hcomp1 #Hcomp2 * #tb * * #Hc #Htb
183 whd in ⊢ (%→?); #Hout >Hout >Htb whd
184 #ls #c_src #xs #end #rs #Hmid_src #Hnotend #Hend
185 lapply (current_to_midtape sig (nth dst ? intape (niltape ?)))
186 cases (current … (nth dst ? intape (niltape ?))) in Hcomp1;
187 [#Hcomp1 #_ %1 % [% | @Hcomp1 %2 %2 % ]
188 |#c_dst cases (true_or_false (c_src == c_dst)) #Hceq
189 [#_ #Hmid_dst cases (Hmid_dst c_dst (refl …)) -Hmid_dst
190 #ls_dst * #rs_dst #Hmid_dst %2
191 cases (comp_list … (xs@end::rs) rs_dst is_endc) #xs1 * #rsi * #rsj * * *
192 #Hrs_src #Hrs_dst #Hnotendxs1 #Hneq %{ls_dst} %{rsj} >Hrs_dst in Hmid_dst; #Hmid_dst
193 cut (∃r1,rs1.rsi = r1::rs1) [@daemon] * #r1 * #rs1 #Hrs1 >Hrs1 in Hrs_src;
194 #Hrs_src >Hrs_src in Hmid_src; #Hmid_src <(\P Hceq) in Hmid_dst; #Hmid_dst
195 lapply (Hcomp2 ??????? Hmid_src Hmid_dst ?)
196 [ #c0 #Hc0 cases (orb_true_l … Hc0) -Hc0 #Hc0
197 [ >(\P Hc0) @Hnotend @memb_hd | @Hnotendxs1 //]
200 [ >Hta in Hc; >nth_change_vec_neq [|@sym_not_eq //] >nth_change_vec //
201 #Hc lapply (Hc ? (refl ??)) #Hendr1
203 [ lapply Hnotendxs1 lapply Hnotend lapply Hrs_src lapply xs1
204 -Hnotendxs1 -Hnotend -Hrs_src -xs1 elim xs
205 [ * normalize in ⊢ (%→?); //
206 #x2 #xs2 normalize in ⊢ (%→?); #Heq destruct (Heq) #_ #Hnotendxs1
207 lapply (Hnotendxs1 ? (memb_hd …)) >Hend #H destruct (H)
209 [ normalize in ⊢ (%→?); #Heq destruct (Heq) #Hnotendc
210 >Hnotendc in Hendr1; [| @memb_cons @memb_hd ]
211 normalize in ⊢ (%→?); #H destruct (H)
212 | #x3 #xs3 normalize in ⊢ (%→?); #Heq destruct (Heq)
213 #Hnotendc #Hnotendcxs1 @eq_f @IH
214 [ @(cons_injective_r … Heq)
215 | #c0 #Hc0 @Hnotendc cases (orb_true_l … Hc0) -Hc0 #Hc0
217 | @memb_cons @memb_cons // ]
218 | #c #Hc @Hnotendcxs1 @memb_cons // ]
221 | #Hxsxs1 >Hmid_dst >Hxsxs1 % ]
222 | #rsj0 #c >Hrsj #Hrsj0 destruct (Hrsj0) ]
223 | * #cj * #rs2 * #Hrs2 #Hta lapply (Hta ?)
224 [ cases (Hneq … Hrs1) /2/ #H %2 @(H ?? Hrs2) ]
225 -Hta #Hta >Hta in Hc; >nth_change_vec_neq [|@sym_not_eq //]
226 >nth_change_vec // #Hc lapply (Hc ? (refl ??)) #Hendr1
227 (* lemmatize this proof *) cut (xs = xs1)
228 [ lapply Hnotendxs1 lapply Hnotend lapply Hrs_src lapply xs1
229 -Hnotendxs1 -Hnotend -Hrs_src -xs1 elim xs
230 [ * normalize in ⊢ (%→?); //
231 #x2 #xs2 normalize in ⊢ (%→?); #Heq destruct (Heq) #_ #Hnotendxs1
232 lapply (Hnotendxs1 ? (memb_hd …)) >Hend #H destruct (H)
234 [ normalize in ⊢ (%→?); #Heq destruct (Heq) #Hnotendc
235 >Hnotendc in Hendr1; [| @memb_cons @memb_hd ]
236 normalize in ⊢ (%→?); #H destruct (H)
237 | #x3 #xs3 normalize in ⊢ (%→?); #Heq destruct (Heq)
238 #Hnotendc #Hnotendcxs1 @eq_f @IH
239 [ @(cons_injective_r … Heq)
240 | #c0 #Hc0 @Hnotendc cases (orb_true_l … Hc0) -Hc0 #Hc0
242 | @memb_cons @memb_cons // ]
243 | #c #Hc @Hnotendcxs1 @memb_cons // ]
246 | #Hxsxs1 >Hmid_dst >Hxsxs1 % //
247 #rsj0 #c #Hcrsj destruct (Hxsxs1 Hrs2 Hcrsj) @eq_f3 //
248 @eq_f3 // lapply (append_l2_injective ?????? Hrs_src) //
249 #Hendr1 destruct (Hendr1) % ]
253 |#Hcomp1 #Hsrc cases (Hsrc ? (refl ??)) -Hsrc #ls0 * #rs0 #Hdst
254 @False_ind lapply (Hcomp1 ?) [%2 %1 %1 >Hmid_src normalize
255 @(not_to_not ??? (\Pf Hceq)) #H destruct //] #Hintape
256 >Hintape in Hc; >Hmid_src #Hc lapply (Hc ? (refl …)) -Hc
257 >(Hnotend c_src) // normalize #H destruct (H)
263 definition match_m ≝ λsrc,dst,sig,n,is_startc,is_endc.
264 whileTM … (match_step src dst sig n is_startc is_endc)
265 (inr ?? (inr ?? (inl … (inr ?? start_nop)))).
267 definition R_match_m ≝
268 λsrc,dst,sig,n,is_startc,is_endc.λint,outt: Vector (tape sig) (S n).
269 (* (current sig (nth dst (tape sig) int (niltape sig)) = None ? → outt = int) ∧ *)
271 nth src ? int (niltape ?) = midtape sig ls x (xs@end::rs) →
272 (∀c0. memb ? c0 (x::xs) = true → is_endc c0 = false) → is_endc end = true →
273 (current sig (nth dst (tape sig) int (niltape sig)) = None ? → outt = int) ∧
274 (is_startc x = true →
276 nth dst ? int (niltape ?) = midtape sig ls0 x0 rs0 →
277 (∃l,l1.x0::rs0 = l@x::xs@l1 ∧
280 (change_vec ?? int (midtape sig (reverse ? xs@x::ls) end rs) src)
281 (midtape sig ((reverse ? (l@x::xs))@ls0) cj l2) dst) ∨
282 ∀l,l1.x0::rs0 ≠ l@x::xs@l1)).
285 definition R_match_m ≝
286 λi,j,sig,n,is_startc,is_endc.λint,outt: Vector (tape sig) (S n).
287 (((∃x.current ? (nth i ? int (niltape ?)) = Some ? x ∧ is_endc x = true) ∨
288 current ? (nth i ? int (niltape ?)) = None ? ∨
289 current ? (nth j ? int (niltape ?)) = None ?) → outt = int) ∧
290 (∀ls,x,xs,ci,rs,ls0,x0,rs0.
291 (∀x. is_startc x ≠ is_endc x) →
292 is_startc x = true → is_endc ci = true →
293 (∀z. memb ? z (x::xs) = true → is_endc x = false) →
294 nth i ? int (niltape ?) = midtape sig ls x (xs@ci::rs) →
295 nth j ? int (niltape ?) = midtape sig ls0 x0 rs0 →
296 (∃l,l1.x0::rs0 = l@x::xs@l1 ∧
299 (change_vec ?? int (midtape sig (reverse ? xs@x::ls) ci rs) i)
300 (midtape sig ((reverse ? (l@x::xs))@ls0) cj l2) j) ∨
301 ∀l,l1.x0::rs0 ≠ l@x::xs@l1).
305 axiom sub_list_dec: ∀A.∀l,ls:list A.
306 ∃l1,l2. l = l1@ls@l2 ∨ ∀l1,l2. l ≠ l1@ls@l2.
309 lemma wsem_match_m : ∀src,dst,sig,n,is_startc,is_endc.
310 src ≠ dst → src < S n → dst < S n →
311 match_m src dst sig n is_startc is_endc ⊫ R_match_m src dst sig n is_startc is_endc.
312 #src #dst #sig #n #is_startc #is_endc #Hneq #Hsrc #Hdst #ta #k #outc #Hloop
313 lapply (sem_while … (sem_match_step src dst sig n is_startc is_endc Hneq Hsrc Hdst) … Hloop) //
314 -Hloop * #tb * #Hstar @(star_ind_l ??????? Hstar) -Hstar
315 [ #tc #Hfalse #ls #x #xs #end #rs #Hmid_src #Hnotend #Hend
316 cases (Hfalse … Hmid_src Hnotend Hend) -Hfalse
317 [(* current dest = None *) * #Hcur_dst #Houtc %
319 |#Hstart #ls0 #x0 #rs0 #Hmid_dst >Hmid_dst in Hcur_dst;
320 normalize in ⊢ (%→?); #H destruct (H)
322 |* #ls0 * #rs0 * #Hmid_dst #HFalse %
323 [ >Hmid_dst normalize in ⊢ (%→?); #H destruct (H)
324 | #Hstart #ls1 #x1 #rs1 >Hmid_dst #H destruct (H)
325 %1 %{[ ]} %{rs0} % [%] #cj #l2 #Hnotnil
326 >reverse_cons >associative_append @(HFalse ?? Hnotnil)
329 |#ta #tb #tc #Htrue #Hstar #IH #Hout lapply (IH Hout) -IH -Hout #IH whd
330 #ls #x #xs #end #rs #Hmid_src #Hnotend #Hend
331 lapply (refl ? (current ? (nth dst ? ta (niltape ?))))
332 cases (current ? (nth dst ? ta (niltape ?))) in ⊢ (???%→?);
334 [#_ whd in Htrue; >Hmid_src in Htrue; #Htrue
335 cases (Htrue x (refl … ) Hstart ?) -Htrue [2: @daemon]
336 * #Htb #_ #_ >Htb in IH; // #IH
337 cases (IH ls x xs end rs Hmid_src Hstart Hnotend Hend)
338 #Hcur_outc #_ @Hcur_outc //
339 |#ls0 #x0 #rs0 #Hmid_dst2 >Hmid_dst2 in Hmid_dst; normalize in ⊢ (%→?);
342 | #c #Hcurta_dst % [ >Hcurta_dst #H destruct (H) ]
343 #ls0 #x0 #rs0 #Hmid_dst >Hmid_dst in Hcurta_dst; normalize in ⊢ (%→?);
344 #H destruct (H) whd in Htrue; >Hmid_src in Htrue; #Htrue
345 cases (Htrue x (refl …) Hstart ?) -Htrue
346 [2: #z #membz @daemon (*aggiungere l'ipotesi*)]
347 cases (true_or_false (x==c)) #eqx
348 [ #_ #Htrue cases (comp_list ? (xs@end::rs) rs0 is_endc)
349 #x1 * #tl1 * #tl2 * * * #Hxs #Hrs0 #Hnotendx1
351 [>append_nil #Hx1 @daemon (* absurd by Hx1 e notendx1 *)]
352 #ci -tl1 #tl1 #Hxs #H cases (H … (refl … ))
353 [(* this is absurd, since Htrue conlcudes is_endc ci =false *)
354 #Hend_ci @daemon (* lapply(Htrue … (refl …)) -Htrue *)
355 |#Hcomp lapply (Htrue ls x x1 ci tl1 ls0 tl2 ???)
356 [ #c0 #Hc0 cases (orb_true_l … Hc0) #Hc0
357 [ @Hnotend >(\P Hc0) @memb_hd
359 | >Hmid_dst >Hrs0 >(\P eqx) %
361 | * cases tl2 in Hrs0;
362 [ >append_nil #Hrs0 #_ #Htb whd in IH;
363 lapply (IH ls x x1 ci tl1 ? Hstart ??)
366 | >Htb // >nth_change_vec_neq [|@sym_not_eq //] >nth_change_vec //
368 >Hrs0 in Hmid_dst; #Hmid_dst
369 cases(Htrue ???????? Hmid_dst) -Htrue #Htb #Hendx
371 cases(IH ls x xs end rs ? Hstart Hnotend Hend)
372 [* #H1 #H2 >Htb in H1; >nth_change_vec //
373 >Hmid_dst cases rs0 [2: #a #tl normalize in ⊢ (%→?); #H destruct (H)]
374 #_ %2 @daemon (* si dimostra *)
376 |>Htb >nth_change_vec_neq [|@sym_not_eq //] @Hmid_src