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_____________________________________________________________*)
13 include "turing/universal/tuples.ma".
14 include "turing/universal/marks.ma".
17 l0 x* a l1 x0* a0 l2 ------> l0 x a* l1 x0 a0* l2
20 if current (* x *) = #
25 if current (* x0 *) = 0
26 then advance_mark ----
30 else x = 1 (* analogo *)
36 MARK NEXT TUPLE machine
37 (partially axiomatized)
39 marks the first character after the first bar (rightwards)
42 definition mark_next_tuple ≝
43 adv_to_mark_r ? bar_or_grid ·
44 (ifTM ? (test_char ? (λc:STape.is_bar (\fst c)))
45 (move_right_and_mark ?) (nop ?) tc_true).
47 definition R_mark_next_tuple ≝
50 (* c non può essere un separatore ... speriamo *)
51 t1 = midtape STape ls c (rs1@〈grid,false〉::rs2) →
52 no_marks rs1 → no_grids rs1 → bar_or_grid c = false →
53 (∃rs3,rs4,d,b.rs1 = rs3 @ 〈bar,false〉 :: rs4 ∧
55 Some ? 〈d,b〉 = option_hd ? (rs4@〈grid,false〉::rs2) ∧
56 t2 = midtape STape (〈bar,false〉::reverse ? rs3@c::ls) 〈d,true〉 (tail ? (rs4@〈grid,false〉::rs2)))
58 (no_bars rs1 ∧ t2 = midtape ? (reverse ? rs1@c::ls) 〈grid,false〉 rs2).
60 axiom daemon :∀P:Prop.P.
64 (∀x.memb A x l = true → f x = false) ∨
65 (∃l1,c,l2.f c = true ∧ l = l1@c::l2 ∧ ∀x.memb ? x l1 = true → f x = false).
67 [ % #x normalize #Hfalse *)
69 theorem sem_mark_next_tuple :
70 Realize ? mark_next_tuple R_mark_next_tuple.
72 lapply (sem_seq ? (adv_to_mark_r ? bar_or_grid)
73 (ifTM ? (test_char ? (λc:STape.is_bar (\fst c))) (move_right_and_mark ?) (nop ?) tc_true) ????)
74 [@sem_if [5: // |6: @sem_move_right_and_mark |7: // |*:skip]
76 |||#Hif cases (Hif intape) -Hif
77 #j * #outc * #Hloop * #ta * #Hleft #Hright
78 @(ex_intro ?? j) @ex_intro [|% [@Hloop] ]
80 #ls #c #rs1 #rs2 #Hrs #Hrs1 #Hrs1' #Hc
81 cases (proj2 ?? Hleft … Hrs)
82 [ * #Hfalse >Hfalse in Hc; #Htf destruct (Htf)
83 | * * #_ #Hta #_ cases (tech_split STape (λc.is_bar (\fst c)) rs1)
85 lapply (Hta rs1 〈grid,false〉 rs2 (refl ??) ? ?)
86 [ * #x #b #Hx whd in ⊢ (??%?); >(Hrs1' … Hx) >(H1 … Hx) %
88 | -Hta #Hta cases Hright
89 [ * #tb * whd in ⊢ (%→?); * * #c1 * >Hta
90 whd in ⊢ ((??%?)→?); #H destruct (H) whd in ⊢ ((??%?)→?); #H destruct
91 | * #tb * whd in ⊢ (%→?); * #_ #Htb >Htb >Hta
95 |* #rs3 * #c0 * #rs4 * * #Hc0 #Hsplit #Hrs3
96 % @(ex_intro ?? rs3) @(ex_intro ?? rs4)
97 lapply (Hta rs3 c0 (rs4@〈grid,false〉::rs2) ???)
98 [#x #Hrs3' whd in ⊢ (??%?); >Hsplit in Hrs1;>Hsplit in Hrs3;
99 #Hrs3 #Hrs1 >(Hrs1 …) [| @memb_append_l1 @Hrs3'|]
100 >(Hrs3 … Hrs3') @Hrs1' >Hsplit @memb_append_l1 //
101 |whd in ⊢ (??%?); >Hc0 %
102 |>Hsplit >associative_append %
105 [* whd in ⊢ (%→?); #tb * * * #c1 * >Hta -Hta
106 whd in ⊢ (??%?→?); #H destruct (H) #Hc1 #Htb
107 whd in ⊢ (%→?); #Houtc
109 [lapply Hc1 lapply Hsplit cases c1 #c1l #c1r #Hsplit
111 [#b #H destruct |2,3,5:#H destruct]
112 #_ @eq_f @(Hrs1 … 〈c1l,c1r〉) >Hsplit @memb_append_l2 @memb_hd]
113 #Hcut lapply Hsplit -Hsplit
115 [#Htb lapply(Houtc … Htb) -Houtc #Houtc #Hsplit
116 @(ex_intro ?? grid) @(ex_intro ?? false) %
117 [% [ % [<Hcut @Hsplit |@Hrs3 ] | % ]
120 |* #r5l #r5r #rs5 #Htb
121 lapply(Houtc … Htb) -Houtc #Houtc #Hsplit
122 @(ex_intro ?? r5l) @(ex_intro ?? r5r) %
123 [%[%[<Hcut @Hsplit| @Hrs3] | % ]
127 |* whd in ⊢ (%→?); #tb * *
128 #H @False_ind >Hta in H; #H lapply(H c0 (refl …))
136 definition init_current_on_match ≝
137 move_l ? · adv_to_mark_l ? (λc:STape.is_grid (\fst c)) · move_r ? · mark ?.
139 definition R_init_current_on_match ≝ λt1,t2.
140 ∀l1,l2,c,rs. no_grids l1 → is_grid c = false →
141 t1 = midtape STape (l1@〈c,false〉::〈grid,false〉::l2) 〈grid,false〉 rs →
142 t2 = midtape STape (〈grid,false〉::l2) 〈c,true〉 ((reverse ? l1)@〈grid,false〉::rs).
144 lemma sem_init_current_on_match :
145 Realize ? init_current_on_match R_init_current_on_match.
147 cases (sem_seq ????? (sem_move_l ?)
148 (sem_seq ????? (sem_adv_to_mark_l ? (λc:STape.is_grid (\fst c)))
149 (sem_seq ????? (sem_move_r ?) (sem_mark ?))) intape)
150 #k * #outc * #Hloop #HR
151 @(ex_intro ?? k) @(ex_intro ?? outc) % [@Hloop] -Hloop
152 #l1 #l2 #c #rs #Hl1 #Hc #Hintape
153 cases HR -HR #ta * whd in ⊢ (%→?); * #_ #Hta lapply (Hta … Hintape) -Hta -Hintape
154 generalize in match Hl1; cases l1
155 [#Hl1 whd in ⊢ ((???(??%%%))→?); #Hta
156 * #tb * whd in ⊢ (%→?); * #_ #Htb cases (Htb … Hta) -Htb -Hta #_
157 (* [* >Hc #Htemp destruct (Htemp) ] *)
158 #Htb cases (Htb … Hc) -Htb #Htb #_
159 lapply (Htb [ ] 〈grid,false〉 ? (refl ??) (refl …) Hl1)
160 whd in ⊢ ((???(??%%%))→?); -Htb #Htb
161 * #tc * whd in ⊢ (%→?); * #_ #Htc lapply (Htc … Htb) -Htb -Htc
162 whd in ⊢ ((???(??%%%))→?); #Htc
163 whd in ⊢ (%→?); * #Houtc #_ lapply (Houtc … Htc) -Houtc #Houtc
165 |#d #tl #Htl whd in ⊢ ((???(??%%%))→?); #Hta
166 * #tb * whd in ⊢ (%→?); * #_ #Htb cases (Htb … Hta) -Htb
167 #_ (* [* >(Htl … (memb_hd …)) #Htemp destruct (Htemp)] *)
168 #Htb cases (Htb ?) -Htb [2: @Htl @memb_hd]
169 >append_cons #Htb #_ lapply (Htb … (refl ??) (refl …) ?)
170 [#x #membx cases (memb_append … membx) -membx #membx
171 [@Htl @memb_cons @membx | >(memb_single … membx) @Hc]]-Htb #Htb
172 * #tc * whd in ⊢ (%→?); * #_ #Htc lapply (Htc … Htb) -Htb -Htc
173 >reverse_append >associative_append whd in ⊢ ((???(??%%%))→?); #Htc
174 whd in ⊢ (%→?); * #Houtc #_ lapply (Houtc … Htc) -Houtc #Houtc
175 >Houtc >reverse_cons >associative_append %
180 definition init_current_gen ≝
181 seq ? (adv_to_mark_l ? (is_marked ?))
182 (seq ? (clear_mark ?)
184 (seq ? (adv_to_mark_l ? (λc:STape.is_grid (\fst c)))
185 (seq ? (move_r ?) (mark ?))))).
187 definition R_init_current_gen ≝ λt1,t2.
188 ∀l1,c,l2,b,l3,c1,rs,c0,b0. no_marks l1 → no_grids l2 →
189 Some ? 〈c0,b0〉 = option_hd ? (reverse ? (〈c,true〉::l2)) →
190 t1 = midtape STape (l1@〈c,true〉::l2@〈grid,b〉::l3) 〈c1,false〉 rs →
191 t2 = midtape STape (〈grid,b〉::l3) 〈c0,true〉
192 ((tail ? (reverse ? (l1@〈c,false〉::l2))@〈c1,false〉::rs)).
194 lemma sem_init_current_gen : Realize ? init_current_gen R_init_current_gen.
196 cases (sem_seq ????? (sem_adv_to_mark_l ? (is_marked ?))
197 (sem_seq ????? (sem_clear_mark ?)
198 (sem_seq ????? (sem_move_l ?)
199 (sem_seq ????? (sem_adv_to_mark_l ? (λc:STape.is_grid (\fst c)))
200 (sem_seq ????? (sem_move_r ?) (sem_mark ?))))) intape)
201 #k * #outc * #Hloop #HR
202 @(ex_intro ?? k) @(ex_intro ?? outc) % [@Hloop] -Hloop
203 #l1 #c #l2 #b #l3 #c1 #rs #c0 #b0 #Hl1 #Hl2 #Hc #Hintape
204 cases HR -HR #ta * whd in ⊢ (%→?); #Hta cases (Hta … Hintape) -Hta -Hintape
205 [ * #Hfalse normalize in Hfalse; destruct (Hfalse) ]
206 * #_ #Hta lapply (Hta l1 〈c,true〉 ? (refl ??) ??) [@Hl1|%] -Hta #Hta
207 * #tb * whd in ⊢ (%→?); #Htb lapply (Htb … Hta) -Htb -Hta #Htb
208 * #tc * whd in ⊢ (%→?); #Htc lapply (Htc … Htb) -Htc -Htb
209 generalize in match Hc; generalize in match Hl2; cases l2
210 [#_ whd in ⊢ ((???%)→?); #Htemp destruct (Htemp)
211 whd in ⊢ ((???(??%%%))→?); #Htc
212 * #td * whd in ⊢ (%→?); #Htd cases (Htd … Htc) -Htd
213 [2: * whd in ⊢ (??%?→?); #Htemp destruct (Htemp) ]
214 * #_ #Htd >Htd in Htc; -Htd #Htd
215 * #te * whd in ⊢ (%→?); #Hte lapply (Hte … Htd) -Htd
216 >reverse_append >reverse_cons
217 whd in ⊢ ((???(??%%%))→?); #Hte
218 whd in ⊢ (%→?); #Houtc lapply (Houtc … Hte) -Houtc -Hte #Houtc
220 |#d #tl #Htl #Hc0 whd in ⊢ ((???(??%%%))→?); #Htc
221 * #td * whd in ⊢ (%→?); #Htd cases (Htd … Htc) -Htd
222 [* >(Htl … (memb_hd …)) whd in ⊢ (??%?→?); #Htemp destruct (Htemp)]
223 * #Hd #Htd lapply (Htd … (refl ??) (refl ??) ?)
224 [#x #membx @Htl @memb_cons @membx] -Htd #Htd
225 * #te * whd in ⊢ (%→?); #Hte lapply (Hte … Htd) -Htd
226 >reverse_append >reverse_cons >reverse_cons
227 >reverse_cons in Hc0; >reverse_cons cases (reverse ? tl)
228 [normalize in ⊢ (%→?); #Hc0 destruct (Hc0) #Hte
229 whd in ⊢ (%→?); #Houtc lapply (Houtc … Hte) -Houtc -Hte #Houtc
231 |* #c2 #b2 #tl2 normalize in ⊢ (%→?); #Hc0 destruct (Hc0)
232 whd in ⊢ ((???(??%%%))→?); #Hte
233 whd in ⊢ (%→?); #Houtc lapply (Houtc … Hte) -Houtc -Hte #Houtc
234 >Houtc >associative_append >associative_append >associative_append %
240 definition init_current ≝
241 adv_to_mark_l ? (is_marked ?) ·clear_mark ? ·
242 adv_to_mark_l ? (λc:STape.is_grid (\fst c)) · move_r ? · mark ?.
244 definition R_init_current ≝ λt1,t2.
245 ∀l1,c,l2,b,l3,c1,rs,c0,b0. no_marks l1 → no_grids l2 → is_grid c = false →
246 Some ? 〈c0,b0〉 = option_hd ? (reverse ? (〈c,true〉::l2)) →
247 t1 = midtape STape (l1@〈c,true〉::l2@〈grid,b〉::l3) 〈c1,false〉 rs →
248 t2 = midtape STape (〈grid,b〉::l3) 〈c0,true〉
249 ((tail ? (reverse ? (l1@〈c,false〉::l2))@〈c1,false〉::rs)).
251 lemma sem_init_current : Realize ? init_current R_init_current.
253 cases (sem_seq ????? (sem_adv_to_mark_l ? (is_marked ?))
254 (sem_seq ????? (sem_clear_mark ?)
255 (sem_seq ????? (sem_adv_to_mark_l ? (λc:STape.is_grid (\fst c)))
256 (sem_seq ????? (sem_move_r ?) (sem_mark ?)))) intape)
257 #k * #outc * #Hloop #HR
258 @(ex_intro ?? k) @(ex_intro ?? outc) % [@Hloop]
259 cases HR -HR #ta * whd in ⊢ (%→?); * #_ #Hta
260 * #tb * whd in ⊢ (%→?); * #_ #Htb
261 * #tc * whd in ⊢ (%→?); * #_ #Htc
262 * #td * whd in ⊢ (%→%→?); * #_ #Htd * #Houtc #_
263 #l1 #c #l2 #b #l3 #c1 #rs #c0 #b0 #Hl1 #Hl2 #Hc #Hc0 #Hintape
264 cases (Hta … Hintape) #_ -Hta #Hta cases (Hta (refl …)) -Hta
265 #Hta #_ lapply (Hta l1 〈c,true〉 ? (refl ??) ??) [@Hl1|%]
266 -Hta #Hta lapply (Htb … Hta) -Htb #Htb cases (Htc … Htb) #_ #Htc
267 cases (Htc Hc) -Htc #Htc #_ lapply (Htc … (refl ??) (refl ??) ?) [@Hl2]
268 -Htc #Htc lapply (Htd … Htc) -Htd
269 >reverse_append >reverse_cons
270 >reverse_cons in Hc0; cases (reverse … l2)
271 [ normalize in ⊢ (%→?); #Hc0 destruct (Hc0)
272 #Htd >(Houtc … Htd) %
273 | * #c2 #b2 #tl2 normalize in ⊢ (%→?);
274 #Hc0 #Htd >(Houtc … Htd)
275 whd in ⊢ (???%); destruct (Hc0)
276 >associative_append >associative_append %
280 definition match_tuple_step ≝
281 ifTM ? (test_char ? (λc:STape.¬ is_grid (\fst c)))
284 (ifTM ? (test_char ? (λc:STape.is_grid (\fst c)))
287 (ifTM ? (test_char ? (λc:STape.is_grid (\fst c)))
288 (mark ?) (move_l ? · init_current) tc_true)) tc_true)))
292 definition R_match_tuple_step_true ≝ λt1,t2.
293 ∀ls,cur,rs.t1 = midtape STape ls cur rs →
295 (∀ls0,c,l1,l2,c1,l3,l4,rs0,n.
296 only_bits_or_nulls l1 → no_marks l1 (* → no_grids l2 *) →
297 bit_or_null c = true → bit_or_null c1 = true →
298 only_bits_or_nulls l3 → S n = |l1| → |l1| = |l3| →
299 table_TM (S n) (l2@〈c1,false〉::l3@〈comma,false〉::l4) →
300 ls = 〈grid,false〉::ls0 → cur = 〈c,true〉 →
301 rs = l1@〈grid,false〉::l2@〈c1,true〉::l3@〈comma,false〉::l4@〈grid,false〉::rs0 →
303 (〈c,false〉::l1 = 〈c1,false〉::l3 ∧
304 t2 = midtape ? (reverse ? l1@〈c,false〉::〈grid,false〉::ls0) 〈grid,false〉
305 (l2@〈c1,false〉::l3@〈comma,true〉::l4@〈grid,false〉::rs0))
307 (* non facciamo match e marchiamo la prossima tupla *)
308 (〈c,false〉::l1 ≠ 〈c1,false〉::l3 ∧
309 ∃c2,l5,l6.l4 = l5@〈bar,false〉::〈c2,false〉::l6 ∧
310 (* condizioni su l5 l6 l7 *)
311 t2 = midtape STape (〈grid,false〉::ls0) 〈c,true〉
312 (l1@〈grid,false〉::l2@〈c1,false〉::l3@〈comma,false〉::
313 l5@〈bar,false〉::〈c2,true〉::l6@〈grid,false〉::rs0))
315 (* non facciamo match e non c'è una prossima tupla:
316 non specifichiamo condizioni sul nastro di output, perché
317 non eseguiremo altre operazioni, quindi il suo formato non ci interessa *)
318 (〈c,false〉::l1 ≠ 〈c1,false〉::l3 ∧ no_bars l4 ∧ current ? t2 = Some ? 〈grid,true〉)).
321 definition R_match_tuple_step_true ≝ λt1,t2.
322 ∃ls,cur,rs.t1 = midtape STape ls cur rs \wedge
324 (∀ls0,c,l1,l2,c1,l3,l4,rs0,n.
325 only_bits_or_nulls l1 → no_marks l1 (* → no_grids l2 *) →
326 bit_or_null c = true → bit_or_null c1 = true →
327 only_bits_or_nulls l3 → S n = |l1| → |l1| = |l3| →
328 table_TM (S n) (l2@〈c1,false〉::l3@〈comma,false〉::l4) →
329 ls = 〈grid,false〉::ls0 → cur = 〈c,true〉 →
330 rs = l1@〈grid,false〉::l2@〈c1,true〉::l3@〈comma,false〉::l4@〈grid,false〉::rs0 →
332 (〈c,false〉::l1 = 〈c1,false〉::l3 ∧
333 t2 = midtape ? (reverse ? l1@〈c,false〉::〈grid,false〉::ls0) 〈grid,false〉
334 (l2@〈c1,false〉::l3@〈comma,true〉::l4@〈grid,false〉::rs0))
336 (* non facciamo match e marchiamo la prossima tupla *)
337 (〈c,false〉::l1 ≠ 〈c1,false〉::l3 ∧
338 ∃c2,l5,l6.l4 = l5@〈bar,false〉::〈c2,false〉::l6 ∧
339 (* condizioni su l5 l6 l7 *)
340 t2 = midtape STape (〈grid,false〉::ls0) 〈c,true〉
341 (l1@〈grid,false〉::l2@〈c1,false〉::l3@〈comma,false〉::
342 l5@〈bar,false〉::〈c2,true〉::l6@〈grid,false〉::rs0))
344 (* non facciamo match e non c'è una prossima tupla:
345 non specifichiamo condizioni sul nastro di output, perché
346 non eseguiremo altre operazioni, quindi il suo formato non ci interessa *)
347 (〈c,false〉::l1 ≠ 〈c1,false〉::l3 ∧ no_bars l4 ∧ current ? t2 = Some ? 〈grid,true〉)).
349 definition R_match_tuple_step_false ≝ λt1,t2.
350 ∀ls,c,rs.t1 = midtape STape ls c rs → is_grid (\fst c) = true ∧ t2 = t1.
352 include alias "basics/logic.ma".
355 lemma eq_f4: ∀A1,A2,A3,A4,B.∀f:A1 → A2 →A3 →A4 →B.
356 ∀x1,x2,x3,x4,y1,y2,y3,y4. x1 = y1 → x2 = y2 →x3=y3 →x4 = y4 →
357 f x1 x2 x3 x4 = f y1 y2 y3 y4.
361 lemma some_option_hd: ∀A.∀l:list A.∀a.∃b.
362 Some ? b = option_hd ? (l@[a]) .
363 #A #l #a cases l normalize /2/
366 axiom tech_split2 : ∀A,l1,l2,l3,l4,x.
367 memb A x l1 = false → memb ? x l3 = false →
368 l1@x::l2 = l3@x::l4 → l1 = l3 ∧ l2 = l4.
370 axiom injective_append : ∀A,l.injective … (λx.append A x l).
372 lemma sem_match_tuple_step:
373 accRealize ? match_tuple_step (inr … (inl … (inr … start_nop)))
374 R_match_tuple_step_true R_match_tuple_step_false.
375 @(acc_sem_if_app … (sem_test_char ? (λc:STape.¬ is_grid (\fst c))) …
376 (sem_seq … sem_compare
377 (sem_if … (sem_test_char ? (λc:STape.is_grid (\fst c)))
379 (sem_seq … sem_mark_next_tuple
380 (sem_if … (sem_test_char ? (λc:STape.is_grid (\fst c)))
381 (sem_mark ?) (sem_seq … (sem_move_l …) (sem_init_current …))))))
383 [(* is_grid: termination case *)
384 2:#t1 #t2 #t3 whd in ⊢ (%→?); * #Hc #H #H1 whd #ls #c #rs #Ht1 %
385 [lapply(Hc c ?) [>Ht1 %] #Hgrid @injective_notb @Hgrid |>H1 @H]
386 |#tapea #tapeout #tapeb whd in ⊢ (%→?); #Hcur
387 * #tapec * whd in ⊢ (%→?); #Hcompare #Hor
388 cases Hcur * #c * -Hcur #Hcur #Hgrid #Htapeb cases (current_to_midtape … Hcur)
389 #ls * #rs #Htapea @(ex_intro … ls) @(ex_intro … c) @(ex_intro … rs) %
390 [%[@Htapea | cases (true_or_false (\fst c == grid))
391 [#eqc @False_ind >(\P eqc) in Hgrid; normalize #H destruct |#eqc @(\Pf eqc)]]]
392 #ls0 #cur #l1 #l2 #c1 #l3 #l4 #rs0 #n #Hl1bitnull #Hl1marks #Hc #Hc1 #Hl3 #eqn
393 #eqlen #Htable #Hls -Hcur #Hcur #Hrs >Htapea in Htapeb; >Hls >Hcur >Hrs #Htapeb
394 cases (Hcompare … Htapeb) -Hcompare -Htapeb * #_ #_ #Hcompare
395 cases (Hcompare cur c1 l1 l3 l2 (l4@〈grid,false〉::rs0) eqlen Hl1bitnull Hl3 Hl1marks … (refl …) Hc ?)
397 [* #Htemp destruct (Htemp) #Htapec %1 % % [%]
398 >Htapec in Hor; -Htapec *
399 [2: * #t3 * whd in ⊢ (%→?); * #H #_ @False_ind
400 lapply (H … (refl …)) whd in ⊢ ((??%?)→?); #H destruct (H)
401 |* #taped * whd in ⊢ (%→?); * #_
402 #Htaped whd in ⊢ (%→?); #Htapeout >Htapeout >Htaped
405 |* #la * #c' * #d' * #lb * #lc * * * #H1 #H2 #H3 #Htapec
406 cut (〈cur,false〉::l1 ≠ 〈c1,false〉::l3)
408 [@(not_to_not …H1) normalize #H destruct (H) %
409 |#x #tl @not_to_not normalize #H destruct //
412 cut (bit_or_null d' = true)
414 [normalize in ⊢ (%→?); #H destruct //
415 |#x #tl #H @(Hl3 〈d',false〉)
416 normalize in H; destruct @memb_append_l2 @memb_hd
419 >Htapec in Hor; -Htapec *
420 [* #taped * whd in ⊢ (%→?); * * #c0 * normalize in ⊢ (%→?);
421 #Hdes destruct (Hdes) >(bit_or_null_not_grid ? Hd') #Htemp destruct (Htemp)
422 |* #taped * whd in ⊢ (%→?); * #_ (* * #_ #H cases (H … (refl …)) -H #_ *)
423 #Htaped * #tapee * whd in ⊢ (%→?); #Htapee
424 <(associative_append ? lc (〈comma,false〉::l4)) in Htaped; #Htaped
425 cases (Htapee … Htaped ???) -Htaped -Htapee
426 [* #rs3 * * (* we proceed by cases on rs4 *)
427 [(* rs4 is empty : the case is absurd since the tape
428 cannot end with a bar *)
429 * #d * #b * * * #Heq1 @False_ind
430 cut (∀A,l1,l2.∀a:A. a::l1@l2=(a::l1)@l2) [//] #Hcut
431 >Hcut in Htable; >H3 >associative_append
432 normalize >Heq1 <associative_append >Hcut
433 <associative_append #Htable @(absurd … Htable)
436 * #d2 #b2 #rs3' * #d * #b * * * #Heq1 #Hnobars
437 cut (memb STape 〈d2,b2〉 (l2@〈c1,false〉::l3@〈comma,false〉::l4) = true)
439 cut (∀A,l1,l2.∀a:A. a::l1@l2=(a::l1)@l2) [//] #Hcut
440 >Hcut >H3 >associative_append @memb_append_l2
441 @memb_cons >Heq1 @memb_append_l2 @memb_cons @memb_hd] #d2intable
442 cut (is_grid d2 = false)
443 [@(no_grids_in_table … Htable … 〈d2,b2〉 d2intable)] #Hd2
445 [@(no_marks_in_table … Htable … 〈d2,b2〉 d2intable)] #Hb2
446 >Hb2 in Heq1; #Heq1 -Hb2 -b2
447 whd in ⊢ ((???%)→?); #Htemp destruct (Htemp) #Htapee >Htapee -Htapee *
448 [(* we know current is not grid *)
449 * #tapef * whd in ⊢ (%→?); * * #c0 *
450 normalize in ⊢ (%→?); #Hdes destruct (Hdes) >Hd2
451 #Htemp destruct (Htemp)
452 |* #tapef * whd in ⊢ (%→?); * #_ #Htapef
453 * #tapeg >Htapef -Htapef *
455 whd in ⊢ (%→?); * #_ #H lapply (H … (refl …)) whd in ⊢ (???%→?); -H #Htapeg
458 whd in ⊢ (%→?); #Htapeout
459 cases (some_option_hd ? (reverse ? (reverse ? la)) 〈c',true〉)
462 (Htapeout (reverse ? rs3 @〈d',false〉::reverse ? la@reverse ? l2@(〈grid,false〉::reverse ? lb))
463 c' (reverse ? la) false ls0 bar (〈d2,true〉::rs3'@〈grid,false〉::rs0) c00 b00 ?????) -Htapeout
464 [whd in ⊢ (??(??%??)?); @eq_f3 [2:%|3: %]
466 generalize in match (〈c',true〉::reverse ? la@〈grid,false〉::ls0); #l
467 whd in ⊢ (???(???%)); >associative_append >associative_append %
468 |>reverse_cons @Hoption
470 [normalize in ⊢ (%→?); #Htemp destruct (Htemp)
471 @bit_or_null_not_grid @Hc
472 |#x #tl normalize in ⊢ (%→?); #Htemp destruct (Htemp)
473 @bit_or_null_not_grid @(Hl1bitnull 〈c',false〉) @memb_append_l2 @memb_hd
475 |cut (only_bits_or_nulls (la@(〈c',false〉::lb)))
476 [<H2 whd #c0 #Hmemb cases (orb_true_l … Hmemb)
477 [#eqc0 >(\P eqc0) @Hc |@Hl1bitnull]
478 |#Hl1' #x #Hx @bit_or_null_not_grid @Hl1'
479 @memb_append_l1 @daemon
481 |@daemon] #Htapeout % %2 % //
483 cut (∃rs32.rs3 = lc@〈comma,false〉::rs32)
484 [ (*cases (tech_split STape (λc.c == 〈bar,false〉) l4)
486 | * #l41 * * #cbar #bfalse * #l42 * * #Hbar #Hl4 #Hl41
487 @(ex_intro ?? l41) >Hl4 in Heq1; #Heq1
489 cut (sublist … lc l3)
490 [ #x #Hx cases la in H3;
491 [ normalize #H3 destruct (H3) @Hx
492 | #p #la' normalize #Hla' destruct (Hla')
493 @memb_append_l2 @memb_cons @Hx ] ] #Hsublist*)
497 (〈c1,false〉::l3@〈comma,false〉::l4= la@〈d',false〉::rs3@〈bar,false〉::〈d2,b2〉::rs3')
499 cut (l4=rs32@〈bar,false〉::〈d2,false〉::rs3')
500 [ >Hrs3 in Heq1; @daemon ] #Hl4
501 @(ex_intro … rs32) @(ex_intro … rs3') % [@Hl4]
503 [(* by Hoption, H2 *) @daemon
504 |(*>Hrs3 *)>append_cons
505 > (?:l1@〈grid,false〉::l2@〈c1,false〉::l3@〈comma,false〉::rs32@〈bar,false〉::〈d2,true〉::rs3'@〈grid,false〉::rs
506 = (l1@〈grid,false〉::l2@〈c1,false〉::l3@〈comma,false〉::rs32@[〈bar,false〉])@〈d2,true〉::rs3'@〈grid,false〉::rs)
507 [|>associative_append normalize
508 >associative_append normalize
509 >associative_append normalize
510 >associative_append normalize
512 >reverse_append >reverse_append >reverse_cons
513 >reverse_reverse >reverse_cons >reverse_reverse
514 >reverse_append >reverse_append >reverse_cons
515 >reverse_reverse >reverse_reverse >reverse_reverse
516 >(?:(la@[〈c',false〉])@((((lb@[〈grid,false〉])@l2)@la)@[〈d',false〉])@rs3
517 =((la@〈c',false〉::lb)@([〈grid,false〉]@l2@la@[〈d',false〉]@rs3)))
518 [|>associative_append >associative_append
519 >associative_append >associative_append >associative_append
520 >associative_append % ]
521 <H2 normalize in ⊢ (??%?); >Hrs3
522 >associative_append >associative_append normalize
523 >associative_append >associative_append
525 >(?:la@(〈d',false〉::lc@〈comma,false〉::rs32)@〈bar,false〉::〈d2,true〉::rs3'@〈grid,false〉::rs0 =
526 (la@〈d',false〉::lc)@〈comma,false〉::rs32@〈bar,false〉::〈d2,true〉::rs3'@〈grid,false〉::rs0 )
527 [| >associative_append normalize >associative_append % ]
532 |* #Hnobars #Htapee >Htapee -Htapee *
533 [whd in ⊢ (%→?); * #tapef * whd in ⊢ (%→?); * #_
534 #Htapef >Htapef -Htapef
535 whd in ⊢ (%→?); * #Htapeout #_ %2 %
536 [% [//] whd #x #Hx @Hnobars @memb_append_l2 @memb_cons //
537 | >(Htapeout … (refl …)) % ]
538 |whd in ⊢ (%→?); * #tapef * whd in ⊢ (%→?);
539 * #Hc0 lapply(Hc0 … (refl … )) normalize in ⊢ (%→?); #Htemp destruct (Htemp)
541 |(* no marks in table *)
542 #x #membx @(no_marks_in_table … Htable)
544 cut (∀A,l1,l2.∀a:A. a::l1@l2=(a::l1)@l2) [//] #Hcut >Hcut
545 >H3 >associative_append @memb_append_l2 @memb_cons @membx
546 |(* no grids in table *)
547 #x #membx @(no_grids_in_table … Htable)
549 cut (∀A,l1,l2.∀a:A. a::l1@l2=(a::l1)@l2) [//] #Hcut >Hcut
550 >H3 >associative_append @memb_append_l2 @memb_cons @membx
551 |whd in ⊢ (??%?); >(bit_or_null_not_grid … Hd') >(bit_or_null_not_bar … Hd') %
554 |#x #membx @(no_marks_in_table … Htable)
555 @memb_append_l2 @memb_cons @memb_append_l1 @membx
556 |#x #membx @(no_marks_in_table … Htable)
557 @memb_append_l1 @membx
566 scrolls through the tuples in the transition table until one matching the
567 current configuration is found
570 definition match_tuple ≝ whileTM ? match_tuple_step (inr … (inl … (inr … start_nop))).
572 lemma is_grid_true : ∀c.is_grid c = true → c = grid.
573 * normalize [ #b ] #H // destruct (H)
576 (* possible variante ?
577 definition weakR_match_tuple ≝ λt1,t2.
578 (∀ls,cur,rs,b. t1 = midtape STape ls 〈grid,b〉 rs → t2 = t1) ∧
579 (∀c,l1,c1,l2,l3,ls0,rs0,n.
580 t1 = midtape STape (〈grid,false〉::ls0) 〈bit c,true〉 rs
581 (l1@〈grid,false〉::l2@〈bit c1,true〉::l3@〈grid,false〉::rs0) →
582 only_bits_or_nulls l1 → no_marks l1 → S n = |l1| →
583 table_TM (S n) (l2@〈c1,false〉::l3) →
586 〈c1,false〉::l3 = l4@〈c,false〉::l1@〈comma,false〉::newc@〈comma,false〉::mv::l5 ∧
587 t2 = midtape ? (reverse ? l1@〈c,false〉::〈grid,false〉::ls0) 〈grid,false〉
588 (l2@l4@〈c,false〉::l1@〈comma,true〉::newc@〈comma,false〉::mv::l5@
591 (* non facciamo match su nessuna tupla;
592 non specifichiamo condizioni sul nastro di output, perché
593 non eseguiremo altre operazioni, quindi il suo formato non ci interessa *)
594 (current ? t2 = Some ? 〈grid,true〉 ∧
596 〈c1,false〉::l3 ≠ l4@〈c,false〉::l1@〈comma,false〉::newc@〈comma,false〉::mv::l5)).
599 definition R_match_tuple0 ≝ λt1,t2.
601 t1 = midtape STape ls cur rs →
602 (is_grid (\fst cur) = true → t2 = t1) ∧
603 (∀c,l1,c1,l2,l3,ls0,rs0,n.
604 ls = 〈grid,false〉::ls0 →
606 rs = l1@〈grid,false〉::l2@〈bar,false〉::〈c1,true〉::l3@〈grid,false〉::rs0 →
607 is_bit c = true → is_bit c1 = true →
608 only_bits_or_nulls l1 → no_marks l1 → S n = |l1| →
609 table_TM (S n) (l2@〈bar,false〉::〈c1,false〉::l3) →
612 〈bar,false〉::〈c1,false〉::l3 = l4@〈bar,false〉::〈c,false〉::l1@〈comma,false〉::newc@〈comma,false〉::mv::l5 ∧
613 t2 = midtape ? (reverse ? l1@〈c,false〉::〈grid,false〉::ls0) 〈grid,false〉
614 (l2@l4@〈bar,false〉::〈c,false〉::l1@〈comma,true〉::newc@〈comma,false〉::mv::l5@
617 (* non facciamo match su nessuna tupla;
618 non specifichiamo condizioni sul nastro di output, perché
619 non eseguiremo altre operazioni, quindi il suo formato non ci interessa *)
620 (current ? t2 = Some ? 〈grid,true〉 ∧
622 〈bar,false〉::〈c1,false〉::l3 ≠ l4@〈bar,false〉::〈c,false〉::l1@〈comma,false〉::newc@〈comma,false〉::mv::l5)).
624 axiom table_bit_after_bar :
625 ∀n,l1,c,l2.table_TM n (l1@〈bar,false〉::〈c,false〉::l2) → is_bit c = true.
627 lemma wsem_match_tuple : WRealize ? match_tuple R_match_tuple0.
628 #intape #k #outc #Hloop
629 lapply (sem_while … sem_match_tuple_step intape k outc Hloop) [%] -Hloop
630 * #ta * #Hstar @(star_ind_l ??????? Hstar)
631 [ #tb whd in ⊢ (%→?); #Hleft
632 #ls #cur #rs #Htb cases (Hleft … Htb) #Hgrid #Houtc %
634 | #c #l1 #c1 #l2 #l3 #ls0 #rs0 #n #Hls #Hcur #Hrs
635 >Hcur in Hgrid; #Hgrid >(is_grid_true … Hgrid) normalize in ⊢ (%→?);
638 | (* in the interesting case, we execute a true iteration, then we restart the
639 while cycle, finally we end with a false iteration *)
640 #tb #tc #td whd in ⊢ (%→?); #Htc
641 #Hstar1 #IH whd in ⊢ (%→?); #Hright lapply (IH Hright) -IH whd in ⊢ (%→?); #IH
643 [ (* cur can't be true because we assume at least one iteration *)
644 #Hcur cases Htc #ls' * #c' * #rs' * * >Htb #Hdes destruct (Hdes)
645 #Hfalse @False_ind @(absurd … (is_grid_true … Hcur) Hfalse)
646 | (* current and a tuple are marked *)
647 #c #l1 #c1 #l2 #l3 #ls0 #rs0 #n #Hls #Hcur #Hrs #Hc #Hc1 #Hl1bitnull #Hl1marks
649 cases Htc #ls' * #c' * #rs' * * >Htb #Hdes destruct (Hdes)
651 (* expose the marked tuple in table *)
652 cut (∃la,lb,mv,lc.l3 = la@〈comma,false〉::lb@〈comma,false〉::mv::lc ∧
653 S n = |la| ∧ only_bits_or_nulls la)
654 [@daemon] * #la * #lb * #mv * #lc * * #Hl3 #Hlalen #Hlabitnull
655 >Hl3 in Htable; >append_cons #Htable
656 >(?: l2@〈bar,false〉::〈c1,true〉::l3@〈grid,false〉::rs0
657 = (l2@[〈bar,false〉])@〈c1,true〉::la@〈comma,false〉::(lb@〈comma,false〉::mv::
658 lc)@〈grid,false〉::rs0) in Hrs;
659 [| >associative_append normalize >Hl3
660 >associative_append normalize % ] #Hrs
661 cases (Htc ????????? Hl1bitnull Hl1marks ?? Hlabitnull Hl1len ? Htable Hls Hcur Hrs)
663 |4: whd in ⊢ (??%?); >Hc1 %
664 |3: whd in ⊢ (??%?); >Hc %
666 [ (* case 1: match successful *)
667 * #Heq #Htc % %{[]} %{lb} %{mv} %{lc} destruct (Heq) %
669 | cases (IH … Htc) -IH #Houtc #_ >(Houtc (refl ??))
670 >Htc @eq_f normalize >associative_append normalize
671 >associative_append normalize %
673 | (* case 2: tuples don't match, we still have other tuples to try *)
674 * #Hdiff * #c2 * #l5 * #l6 * #Heqlblc #Htc
675 cases (IH ??? … Htc) -IH #_ #IH
676 (* by induction hypothesis *)
677 lapply (IH ? l1 c2 (l2@〈bar,false〉::〈c1,false〉::la@〈comma,false〉::l5) l6 ? rs0 n (refl ??) (refl ??) ???????)
678 [ generalize in match Htable;
679 >associative_append normalize
680 >associative_append normalize >Heqlblc
681 >associative_append normalize //
687 | >associative_append normalize
688 >associative_append normalize
689 >associative_append %
691 [ (* the while finally matches a tuple *)
692 * #l7 * #newc * #mv0 * #l8 * #Hl7l8 #Houtc %
693 >Heqlblc @(ex_intro ?? (〈bar,false〉::〈c1,false〉::la@〈comma,false〉::l5@l7))
694 %{newc} %{mv0} %{l8} %
695 [ normalize >Hl7l8 >associative_append normalize
696 >associative_append %
697 | >Houtc @eq_f >associative_append normalize
698 >associative_append normalize >associative_append
699 normalize >associative_append %
701 | (* the while fails finding a tuple: there are no matches in the whole table *)
702 * #Houtc #Hdiff1 %2 %
704 | #l50 #newc #mv0 #l51 >Heqlblc
710 | (* match failed and there is no next tuple: the next while cycle will just exit *)
711 * * #Hdiff #Hnobars generalize in match (refl ? tc);
712 cases tc in ⊢ (???% → %);
713 [ #_ normalize in ⊢ (??%?→?); #Hfalse destruct (Hfalse)
714 |2,3: #x #xs #_ normalize in ⊢ (??%?→?); #Hfalse destruct (Hfalse) ]
715 #ls1 #cur1 #rs1 #Htc normalize in ⊢ (??%?→?); #Hcur1
716 cases (IH … Htc) -IH #IH #_ %2 %
717 [ destruct (Hcur1) >IH [ >Htc % | % ]
719 (* no_bars except the first one, where the tuple does not match ⇒
728 lemma WF_mts_niltape:
729 WF ? (inv ? R_match_tuple_step_true) (niltape (FinProd FSUnialpha FinBool)).
730 @wf #t1 whd in ⊢ (%→?); * #ls * #c * #rs * * #H destruct
733 lemma WF_mts_rightof:
734 ∀a,ls. WF ? (inv ? R_match_tuple_step_true) (rightof (FinProd FSUnialpha FinBool) a ls).
735 #a #ls @wf #t1 whd in ⊢ (%→?); * #ls * #c * #rs * * #H destruct
739 ∀a,ls. WF ? (inv ? R_match_tuple_step_true) (leftof (FinProd FSUnialpha FinBool) a ls).
740 #a #ls @wf #t1 whd in ⊢ (%→?); * #ls * #c * #rs * * #H destruct
743 lemma WF_cst_midtape_grid:
744 ∀ls,b,rs. WF ? (inv ? R_match_tuple_step_true)
745 (midtape (FinProd … FSUnialpha FinBool) ls 〈grid,b〉 rs).
746 #ls #b #rs @wf #t1 whd in ⊢ (%→?); * #ls' * #c' * #rs' * * #H destruct
747 * #Hfalse @False_ind @Hfalse %
750 definition Pre_match_tuple ≝ λt.
751 ∃ls,cur,rs. t = midtape STape ls cur rs ∧
752 (is_grid (\fst cur) = true ∨
753 (∃ls0,c,l1,l2,c1,l3,l4,rs0,n.
754 only_bits_or_nulls l1 ∧ no_marks l1 ∧
755 bit_or_null c = true ∧ bit_or_null c1 = true ∧
756 only_bits_or_nulls l3 ∧ S n = |l1| ∧|l1| = |l3| ∧
757 table_TM (S n) (l2@〈c1,false〉::l3@〈comma,false〉::l4) ∧
758 ls = 〈grid,false〉::ls0 ∧ cur = 〈c,true〉 ∧
759 rs = l1@〈grid,false〉::l2@〈c1,true〉::l3@〈comma,false〉::l4@〈grid,false〉::rs0)).
761 lemma acc_Realize_to_acc_GRealize: ∀sig,M.∀q:states sig M.∀P,R1,R2.
762 M ⊨ [q:R1,R2] → accGRealize sig M q P R1 R2.
763 #alpha #M #q #Pre #R1 #R2 #HR #t #HPre
764 cases (HR t) -HR #k * #outc * * #Hloop #HRtrue #HRfalse
765 @(ex_intro ?? k) @(ex_intro ?? outc) %
766 [ % [@Hloop] @HRtrue | @HRfalse]
770 lemma terminate_match_tuple:
771 ∀t. Pre_match_tuple t → Terminate ? match_tuple t.
773 @(terminate_while_guarded ???
775 (acc_Realize_to_acc_GRealize ??? Pre_match_tuple … sem_match_tuple_step)
777 [-HPre -t #t1 #t2 #HPre cases HPre #ls * * #curl #curr * #rs * #Ht1 *
779 #Hgrid * #ls1 * #cur1 * #rs1 * * >Ht1 #Hdes destruct (Hdes)
780 #Habs @False_ind @(absurd ?? Habs) @(is_grid_true … Hgrid)
781 |* #ls0 * #c * #l1 * #l2 * #c1 * #l3 * #l4 * #rs0 * #n
783 #Hl1 #Hmarksl1 #Hc #Hc1 #Hl3 #lenl1 #eqlen #Htable #Hls #Hcur #Hrs
784 * #ls1 * #cur1 * #rs1 * * >Ht1 #Hdes destruct (Hdes) #Hdes #H
785 lapply (H … Hl1 Hmarksl1 Hc Hc1 Hl3 lenl1 eqlen Htable Hls Hcur Hrs)
787 [* [ * #Hdes #Ht2 >Ht2
788 @ex_intro [2:@ex_intro [2: @ex_intro [2: % [%]|]|]|]
790 |* #test * #c2 * #l5 * #l6 * #Hl4 #Ht2
791 cut (∃l7,l8. l6 = l7@〈comma,false 〉::l8 ∧ |l7| = |l1|) [@daemon]
792 * #l7 * #l8 * #Hl6 #eqlen1
793 @ex_intro [2:@ex_intro [2: @ex_intro [2: % [@Ht2]|]|]|] %2
794 @(ex_intro … ls0) @(ex_intro … c) @(ex_intro … l1)
795 @(ex_intro … (l2@〈c1,false〉::l3@〈comma,false〉::l5@[〈bar,false〉]))
796 @(ex_intro … c2) @(ex_intro … l7) @(ex_intro … l8)
797 @(ex_intro … rs0) @(ex_intro … n)
798 % [2: >Hl6 >associative_append >associative_append @eq_f @eq_f @eq_f
799 @eq_f >associative_append @eq_f @eq_f >associative_append % ]
800 % [2: %] % [2: %] % [2:@daemon] % [2: @sym_eq @eqlen1]
801 % [2: @lenl1] % [2: #x #memx @daemon]
802 % [2: @daemon] % [2: @Hc] % [2: @Hmarksl1] @Hl1
804 |* * #_ #_ #H cases (current_to_midtape … H) #ls * #rs #Ht1
805 >Ht1 @ex_intro [2:@ex_intro [2: @ex_intro [2: % [%]|]|]|] %1 %
808 |cases HPre -HPre #ls * * #curl #curr * #rs * #Ht *
809 [#Hgrid >Ht >(is_grid_true … Hgrid) @WF_cst_midtape_grid
810 |* #ls0 * #c * #l1 * #l2 * #c1 * #l3 * #l4
811 cut (∃len. |l4| = len) [/2/] * #lenl4
812 lapply l4 lapply l3 lapply c1 lapply l2 lapply l1 lapply c lapply ls0 lapply Ht
813 lapply curr lapply curl lapply ls lapply rs lapply t -l4 -l3 -l2 -l1 -c1 -curr -curl -ls -t
815 (* by induction on the length of l4 *)
817 #len #Hind #t #rs #ls #cl #cr #Ht #ls0 #c #l1 #l2 #c1 #l3 #l4 #Hlen
818 * #rs0 * #n * * * * * * * * * *
819 #Hl1 #Hmarksl1 #Hc #Hc1 #Hl3 #lenl1 #eqlen #Htable #Hls #Hcur #Hrs
820 % #t1 >Ht whd in ⊢ (%→?); * #ls1 * #cur * #rs1 * * #Hdes destruct (Hdes)
821 #Hgrid #H lapply (H … Hl1 Hmarksl1 Hc Hc1 Hl3 lenl1 eqlen Htable Hls Hcur Hrs)
823 [* [ * #Hdes destruct (Hdes) #Ht1 >Ht1 @WF_cst_midtape_grid
824 | * #_ * #c2 * #l5 * #l6 * #Hl4 #Ht1
825 cut (∃l7,l8. l6 = l7@〈comma,false 〉::l8 ∧ |l7| = |l1|) [@daemon]
826 * #l7 * #l8 * #Hl6 #eqlen1
827 @(Hind … Ht1 ls0 c l1 (l2@〈c1,false〉::l3@〈comma,false〉::l5@[〈bar,false〉]) c2 l7 l8 … (refl …))
828 [<Hlen >Hl4 >Hl6 >length_append normalize in match (length … (cons …));
829 >length_append normalize in match (length … (cons …)); <plus_n_Sm
831 |@(ex_intro … rs0) @(ex_intro … n) %
832 [2: >Hl6 >associative_append >associative_append @eq_f @eq_f @eq_f
833 @eq_f >associative_append @eq_f @eq_f >associative_append % ]
834 % [2: %] % [2: %] % [2:@daemon] % [2: @sym_eq @eqlen1]
835 % [2: @lenl1] % [2: #x #memx @daemon]
836 % [2: @daemon] % [2: @Hc] % [2: @Hmarksl1] @Hl1
839 |* * #_ #_ #H cases (current_to_midtape … H) #ls * #rs #Ht1
845 definition R_match_tuple ≝ λt1,t2.
847 is_bit c = true → is_bit c1 = true →
848 only_bits_or_nulls l1 → no_marks l1 → S n = |l1| →
849 table_TM (S n) (〈bar,false〉::〈c1,false〉::l2) →
850 t1 = midtape STape (〈grid,false〉::ls) 〈c,true〉
851 (l1@〈grid,false〉::〈bar,false〉::〈c1,true〉::l2@〈grid,false〉::rs) →
854 〈bar,false〉::〈c1,false〉::l2 = l3@〈bar,false〉::〈c,false〉::l1@〈comma,false〉::newc@〈comma,false〉::mv::l4 ∧
855 t2 = midtape ? (reverse ? l1@〈c,false〉::〈grid,false〉::ls) 〈grid,false〉
856 (l3@〈bar,false〉::〈c,false〉::l1@〈comma,true〉::newc@〈comma,false〉::mv::l4@〈grid,false〉::rs))
858 (* non facciamo match su nessuna tupla;
859 non specifichiamo condizioni sul nastro di output, perché
860 non eseguiremo altre operazioni, quindi il suo formato non ci interessa *)
861 (current ? t2 = Some ? 〈grid,true〉 ∧
863 〈bar,false〉::〈c1,false〉::l2 ≠ l3@〈bar,false〉::〈c,false〉::l1@〈comma,false〉::newc@〈comma,false〉::mv::l4).
865 lemma sem_match_tuple0 : GRealize ? match_tuple Pre_match_tuple R_match_tuple0.
866 @WRealize_to_GRealize [@terminate_match_tuple | @wsem_match_tuple]
869 lemma sem_match_tuple : GRealize ? match_tuple Pre_match_tuple R_match_tuple.
870 generalize in match sem_match_tuple0; @GRealize_to_GRealize
871 #t1 #t2 #HR #ls #c #l1 #c1 #l2 #rs #n #Hc #Hc1 #Hl1bitsnulls #Hl1marks #Hl1len #Htable #Ht1
872 cases (HR … Ht1) -HR #_ #HR
873 @(HR ??? [] … (refl ??) (refl ??) (refl ??) Hc Hc1 Hl1bitsnulls Hl1marks