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)))
291 definition R_match_tuple_step_true_new ≝ λt1,t2.
292 ∃ls,cur,rs.t1 = midtape STape ls cur rs \wedge
294 (∀ls0,c,l1,l2,c1,l3,l4,rs0,n.
295 only_bits_or_nulls l1 → no_marks l1 (* → no_grids l2 *) →
296 bit_or_null c = true → bit_or_null c1 = true →
297 only_bits_or_nulls l3 → S n = |l1| → |l1| = |l3| →
298 table_TM (S n) (l2@〈c1,false〉::l3@〈comma,false〉::l4) →
299 ls = 〈grid,false〉::ls0 → cur = 〈c,true〉 →
300 rs = l1@〈grid,false〉::l2@〈c1,true〉::l3@〈comma,false〉::l4@〈grid,false〉::rs0 →
302 (〈c,false〉::l1 = 〈c1,false〉::l3 ∧
303 t2 = midtape ? (reverse ? l1@〈c,false〉::〈grid,false〉::ls0) 〈grid,false〉
304 (l2@〈c1,false〉::l3@〈comma,true〉::l4@〈grid,false〉::rs0))
306 (* non facciamo match e marchiamo la prossima tupla *)
307 (〈c,false〉::l1 ≠ 〈c1,false〉::l3 ∧
308 ∃c2,l5,l6.l4 = l5@〈bar,false〉::〈c2,false〉::l6 ∧
309 (* condizioni su l5 l6 l7 *)
310 t2 = midtape STape (〈grid,false〉::ls0) 〈c,true〉
311 (l1@〈grid,false〉::l2@〈c1,false〉::l3@〈comma,false〉::
312 l5@〈bar,false〉::〈c2,true〉::l6@〈grid,false〉::rs0))
314 (* non facciamo match e non c'è una prossima tupla:
315 non specifichiamo condizioni sul nastro di output, perché
316 non eseguiremo altre operazioni, quindi il suo formato non ci interessa *)
317 (〈c,false〉::l1 ≠ 〈c1,false〉::l3 ∧ no_bars l4 ∧ current ? t2 = Some ? 〈grid,true〉)).
320 definition R_match_tuple_step_true ≝ λt1,t2.
321 ∀ls,cur,rs.t1 = midtape STape ls cur rs →
323 (∀ls0,c,l1,l2,c1,l3,l4,rs0,n.
324 only_bits_or_nulls l1 → no_marks l1 (* → no_grids l2 *) →
325 bit_or_null c = true → bit_or_null c1 = true →
326 only_bits_or_nulls l3 → S n = |l1| → |l1| = |l3| →
327 table_TM (S n) (l2@〈c1,false〉::l3@〈comma,false〉::l4) →
328 ls = 〈grid,false〉::ls0 → cur = 〈c,true〉 →
329 rs = l1@〈grid,false〉::l2@〈c1,true〉::l3@〈comma,false〉::l4@〈grid,false〉::rs0 →
331 (〈c,false〉::l1 = 〈c1,false〉::l3 ∧
332 t2 = midtape ? (reverse ? l1@〈c,false〉::〈grid,false〉::ls0) 〈grid,false〉
333 (l2@〈c1,false〉::l3@〈comma,true〉::l4@〈grid,false〉::rs0))
335 (* non facciamo match e marchiamo la prossima tupla *)
336 (〈c,false〉::l1 ≠ 〈c1,false〉::l3 ∧
337 ∃c2,l5,l6.l4 = l5@〈bar,false〉::〈c2,false〉::l6 ∧
338 (* condizioni su l5 l6 l7 *)
339 t2 = midtape STape (〈grid,false〉::ls0) 〈c,true〉
340 (l1@〈grid,false〉::l2@〈c1,false〉::l3@〈comma,false〉::
341 l5@〈bar,false〉::〈c2,true〉::l6@〈grid,false〉::rs0))
343 (* non facciamo match e non c'è una prossima tupla:
344 non specifichiamo condizioni sul nastro di output, perché
345 non eseguiremo altre operazioni, quindi il suo formato non ci interessa *)
346 (〈c,false〉::l1 ≠ 〈c1,false〉::l3 ∧ no_bars l4 ∧ current ? t2 = Some ? 〈grid,true〉)).
349 definition R_match_tuple_step_true ≝ λt1,t2.
350 ∃ls,cur,rs.t1 = midtape STape ls cur rs \wedge
352 (∀ls0,c,l1,l2,c1,l3,l4,rs0,n.
353 only_bits_or_nulls l1 → no_marks l1 (* → no_grids l2 *) →
354 bit_or_null c = true → bit_or_null c1 = true →
355 only_bits_or_nulls l3 → S n = |l1| → |l1| = |l3| →
356 table_TM (S n) (l2@〈c1,false〉::l3@〈comma,false〉::l4) →
357 ls = 〈grid,false〉::ls0 → cur = 〈c,true〉 →
358 rs = l1@〈grid,false〉::l2@〈c1,true〉::l3@〈comma,false〉::l4@〈grid,false〉::rs0 →
360 (〈c,false〉::l1 = 〈c1,false〉::l3 ∧
361 t2 = midtape ? (reverse ? l1@〈c,false〉::〈grid,false〉::ls0) 〈grid,false〉
362 (l2@〈c1,false〉::l3@〈comma,true〉::l4@〈grid,false〉::rs0))
364 (* non facciamo match e marchiamo la prossima tupla *)
365 (〈c,false〉::l1 ≠ 〈c1,false〉::l3 ∧
366 ∃c2,l5,l6.l4 = l5@〈bar,false〉::〈c2,false〉::l6 ∧
367 (* condizioni su l5 l6 l7 *)
368 t2 = midtape STape (〈grid,false〉::ls0) 〈c,true〉
369 (l1@〈grid,false〉::l2@〈c1,false〉::l3@〈comma,false〉::
370 l5@〈bar,false〉::〈c2,true〉::l6@〈grid,false〉::rs0))
372 (* non facciamo match e non c'è una prossima tupla:
373 non specifichiamo condizioni sul nastro di output, perché
374 non eseguiremo altre operazioni, quindi il suo formato non ci interessa *)
375 (〈c,false〉::l1 ≠ 〈c1,false〉::l3 ∧ no_bars l4 ∧ current ? t2 = Some ? 〈grid,true〉)).
377 definition R_match_tuple_step_false ≝ λt1,t2.
378 ∀ls,c,rs.t1 = midtape STape ls c rs → is_grid (\fst c) = true ∧ t2 = t1.
380 include alias "basics/logic.ma".
383 lemma eq_f4: ∀A1,A2,A3,A4,B.∀f:A1 → A2 →A3 →A4 →B.
384 ∀x1,x2,x3,x4,y1,y2,y3,y4. x1 = y1 → x2 = y2 →x3=y3 →x4 = y4 →
385 f x1 x2 x3 x4 = f y1 y2 y3 y4.
389 lemma some_option_hd: ∀A.∀l:list A.∀a.∃b.
390 Some ? b = option_hd ? (l@[a]) .
391 #A #l #a cases l normalize /2/
394 axiom tech_split2 : ∀A,l1,l2,l3,l4,x.
395 memb A x l1 = false → memb ? x l3 = false →
396 l1@x::l2 = l3@x::l4 → l1 = l3 ∧ l2 = l4.
398 axiom injective_append : ∀A,l.injective … (λx.append A x l).
400 lemma sem_match_tuple_step:
401 accRealize ? match_tuple_step (inr … (inl … (inr … start_nop)))
402 R_match_tuple_step_true R_match_tuple_step_false.
403 @(acc_sem_if_app … (sem_test_char ? (λc:STape.¬ is_grid (\fst c))) …
404 (sem_seq … sem_compare
405 (sem_if … (sem_test_char ? (λc:STape.is_grid (\fst c)))
407 (sem_seq … sem_mark_next_tuple
408 (sem_if … (sem_test_char ? (λc:STape.is_grid (\fst c)))
409 (sem_mark ?) (sem_seq … (sem_move_l …) (sem_init_current …))))))
411 [(* is_grid: termination case *)
412 2:#t1 #t2 #t3 whd in ⊢ (%→?); * #Hc #H #H1 whd #ls #c #rs #Ht1 %
413 [lapply(Hc c ?) [>Ht1 %] #Hgrid @injective_notb @Hgrid |>H1 @H]
414 |#tapea #tapeout #tapeb whd in ⊢ (%→?); #Hcur
415 * #tapec * whd in ⊢ (%→?); #Hcompare #Hor
416 cases Hcur * #c * -Hcur #Hcur #Hgrid #Htapeb cases (current_to_midtape … Hcur)
417 #ls * #rs #Htapea @(ex_intro … ls) @(ex_intro … c) @(ex_intro … rs) %
418 [%[@Htapea | cases (true_or_false (\fst c == grid))
419 [#eqc @False_ind >(\P eqc) in Hgrid; normalize #H destruct |#eqc @(\Pf eqc)]]]
420 #ls0 #cur #l1 #l2 #c1 #l3 #l4 #rs0 #n #Hl1bitnull #Hl1marks #Hc #Hc1 #Hl3 #eqn
421 #eqlen #Htable #Hls -Hcur #Hcur #Hrs >Htapea in Htapeb; >Hls >Hcur >Hrs #Htapeb
422 cases (Hcompare … Htapeb) -Hcompare -Htapeb * #_ #_ #Hcompare
423 cases (Hcompare cur c1 l1 l3 l2 (l4@〈grid,false〉::rs0) eqlen Hl1bitnull Hl3 Hl1marks … (refl …) Hc ?)
425 [* #Htemp destruct (Htemp) #Htapec %1 % % [%]
426 >Htapec in Hor; -Htapec *
427 [2: * #t3 * whd in ⊢ (%→?); * #H #_ @False_ind
428 lapply (H … (refl …)) whd in ⊢ ((??%?)→?); #H destruct (H)
429 |* #taped * whd in ⊢ (%→?); * #_
430 #Htaped whd in ⊢ (%→?); #Htapeout >Htapeout >Htaped
433 |* #la * #c' * #d' * #lb * #lc * * * #H1 #H2 #H3 #Htapec
434 cut (〈cur,false〉::l1 ≠ 〈c1,false〉::l3)
436 [@(not_to_not …H1) normalize #H destruct (H) %
437 |#x #tl @not_to_not normalize #H destruct //
440 cut (bit_or_null d' = true)
442 [normalize in ⊢ (%→?); #H destruct //
443 |#x #tl #H @(Hl3 〈d',false〉)
444 normalize in H; destruct @memb_append_l2 @memb_hd
447 >Htapec in Hor; -Htapec *
448 [* #taped * whd in ⊢ (%→?); * * #c0 * normalize in ⊢ (%→?);
449 #Hdes destruct (Hdes) >(bit_or_null_not_grid ? Hd') #Htemp destruct (Htemp)
450 |* #taped * whd in ⊢ (%→?); * #_ (* * #_ #H cases (H … (refl …)) -H #_ *)
451 #Htaped * #tapee * whd in ⊢ (%→?); #Htapee
452 <(associative_append ? lc (〈comma,false〉::l4)) in Htaped; #Htaped
453 cases (Htapee … Htaped ???) -Htaped -Htapee
454 [* #rs3 * * (* we proceed by cases on rs4 *)
455 [(* rs4 is empty : the case is absurd since the tape
456 cannot end with a bar *)
457 * #d * #b * * * #Heq1 @False_ind
458 cut (∀A,l1,l2.∀a:A. a::l1@l2=(a::l1)@l2) [//] #Hcut
459 >Hcut in Htable; >H3 >associative_append
460 normalize >Heq1 <associative_append >Hcut
461 <associative_append #Htable @(absurd … Htable)
464 * #d2 #b2 #rs3' * #d * #b * * * #Heq1 #Hnobars
465 cut (memb STape 〈d2,b2〉 (l2@〈c1,false〉::l3@〈comma,false〉::l4) = true)
467 cut (∀A,l1,l2.∀a:A. a::l1@l2=(a::l1)@l2) [//] #Hcut
468 >Hcut >H3 >associative_append @memb_append_l2
469 @memb_cons >Heq1 @memb_append_l2 @memb_cons @memb_hd] #d2intable
470 cut (is_grid d2 = false)
471 [@(no_grids_in_table … Htable … 〈d2,b2〉 d2intable)] #Hd2
473 [@(no_marks_in_table … Htable … 〈d2,b2〉 d2intable)] #Hb2
474 >Hb2 in Heq1; #Heq1 -Hb2 -b2
475 whd in ⊢ ((???%)→?); #Htemp destruct (Htemp) #Htapee >Htapee -Htapee *
476 [(* we know current is not grid *)
477 * #tapef * whd in ⊢ (%→?); * * #c0 *
478 normalize in ⊢ (%→?); #Hdes destruct (Hdes) >Hd2
479 #Htemp destruct (Htemp)
480 |* #tapef * whd in ⊢ (%→?); * #_ #Htapef
481 * #tapeg >Htapef -Htapef *
483 whd in ⊢ (%→?); * #_ #H lapply (H … (refl …)) whd in ⊢ (???%→?); -H #Htapeg
486 whd in ⊢ (%→?); #Htapeout
487 cases (some_option_hd ? (reverse ? (reverse ? la)) 〈c',true〉)
490 (Htapeout (reverse ? rs3 @〈d',false〉::reverse ? la@reverse ? l2@(〈grid,false〉::reverse ? lb))
491 c' (reverse ? la) false ls0 bar (〈d2,true〉::rs3'@〈grid,false〉::rs0) c00 b00 ?????) -Htapeout
492 [whd in ⊢ (??(??%??)?); @eq_f3 [2:%|3: %]
494 generalize in match (〈c',true〉::reverse ? la@〈grid,false〉::ls0); #l
495 whd in ⊢ (???(???%)); >associative_append >associative_append %
496 |>reverse_cons @Hoption
498 [normalize in ⊢ (%→?); #Htemp destruct (Htemp)
499 @bit_or_null_not_grid @Hc
500 |#x #tl normalize in ⊢ (%→?); #Htemp destruct (Htemp)
501 @bit_or_null_not_grid @(Hl1bitnull 〈c',false〉) @memb_append_l2 @memb_hd
503 |cut (only_bits_or_nulls (la@(〈c',false〉::lb)))
504 [<H2 whd #c0 #Hmemb cases (orb_true_l … Hmemb)
505 [#eqc0 >(\P eqc0) @Hc |@Hl1bitnull]
506 |#Hl1' #x #Hx @bit_or_null_not_grid @Hl1'
507 @memb_append_l1 @daemon
509 |@daemon] #Htapeout % %2 % //
511 cut (∃rs32.rs3 = lc@〈comma,false〉::rs32)
512 [ (*cases (tech_split STape (λc.c == 〈bar,false〉) l4)
514 | * #l41 * * #cbar #bfalse * #l42 * * #Hbar #Hl4 #Hl41
515 @(ex_intro ?? l41) >Hl4 in Heq1; #Heq1
517 cut (sublist … lc l3)
518 [ #x #Hx cases la in H3;
519 [ normalize #H3 destruct (H3) @Hx
520 | #p #la' normalize #Hla' destruct (Hla')
521 @memb_append_l2 @memb_cons @Hx ] ] #Hsublist*)
525 (〈c1,false〉::l3@〈comma,false〉::l4= la@〈d',false〉::rs3@〈bar,false〉::〈d2,b2〉::rs3')
527 cut (l4=rs32@〈bar,false〉::〈d2,false〉::rs3')
528 [ >Hrs3 in Heq1; @daemon ] #Hl4
529 @(ex_intro … rs32) @(ex_intro … rs3') % [@Hl4]
531 [(* by Hoption, H2 *) @daemon
532 |(*>Hrs3 *)>append_cons
533 > (?:l1@〈grid,false〉::l2@〈c1,false〉::l3@〈comma,false〉::rs32@〈bar,false〉::〈d2,true〉::rs3'@〈grid,false〉::rs
534 = (l1@〈grid,false〉::l2@〈c1,false〉::l3@〈comma,false〉::rs32@[〈bar,false〉])@〈d2,true〉::rs3'@〈grid,false〉::rs)
535 [|>associative_append normalize
536 >associative_append normalize
537 >associative_append normalize
538 >associative_append normalize
540 >reverse_append >reverse_append >reverse_cons
541 >reverse_reverse >reverse_cons >reverse_reverse
542 >reverse_append >reverse_append >reverse_cons
543 >reverse_reverse >reverse_reverse >reverse_reverse
544 >(?:(la@[〈c',false〉])@((((lb@[〈grid,false〉])@l2)@la)@[〈d',false〉])@rs3
545 =((la@〈c',false〉::lb)@([〈grid,false〉]@l2@la@[〈d',false〉]@rs3)))
546 [|>associative_append >associative_append
547 >associative_append >associative_append >associative_append
548 >associative_append % ]
549 <H2 normalize in ⊢ (??%?); >Hrs3
550 >associative_append >associative_append normalize
551 >associative_append >associative_append
553 >(?:la@(〈d',false〉::lc@〈comma,false〉::rs32)@〈bar,false〉::〈d2,true〉::rs3'@〈grid,false〉::rs0 =
554 (la@〈d',false〉::lc)@〈comma,false〉::rs32@〈bar,false〉::〈d2,true〉::rs3'@〈grid,false〉::rs0 )
555 [| >associative_append normalize >associative_append % ]
560 |* #Hnobars #Htapee >Htapee -Htapee *
561 [whd in ⊢ (%→?); * #tapef * whd in ⊢ (%→?); * #_
562 #Htapef >Htapef -Htapef
563 whd in ⊢ (%→?); * #Htapeout #_ %2 %
564 [% [//] whd #x #Hx @Hnobars @memb_append_l2 @memb_cons //
565 | >(Htapeout … (refl …)) % ]
566 |whd in ⊢ (%→?); * #tapef * whd in ⊢ (%→?);
567 * #Hc0 lapply(Hc0 … (refl … )) normalize in ⊢ (%→?); #Htemp destruct (Htemp)
569 |(* no marks in table *)
570 #x #membx @(no_marks_in_table … Htable)
572 cut (∀A,l1,l2.∀a:A. a::l1@l2=(a::l1)@l2) [//] #Hcut >Hcut
573 >H3 >associative_append @memb_append_l2 @memb_cons @membx
574 |(* no grids in table *)
575 #x #membx @(no_grids_in_table … Htable)
577 cut (∀A,l1,l2.∀a:A. a::l1@l2=(a::l1)@l2) [//] #Hcut >Hcut
578 >H3 >associative_append @memb_append_l2 @memb_cons @membx
579 |whd in ⊢ (??%?); >(bit_or_null_not_grid … Hd') >(bit_or_null_not_bar … Hd') %
582 |#x #membx @(no_marks_in_table … Htable)
583 @memb_append_l2 @memb_cons @memb_append_l1 @membx
584 |#x #membx @(no_marks_in_table … Htable)
585 @memb_append_l1 @membx
594 scrolls through the tuples in the transition table until one matching the
595 current configuration is found
598 definition match_tuple ≝ whileTM ? match_tuple_step (inr … (inl … (inr … start_nop))).
600 lemma is_grid_true : ∀c.is_grid c = true → c = grid.
601 * normalize [ #b ] #H // destruct (H)
604 (* possible variante ?
605 definition weakR_match_tuple ≝ λt1,t2.
606 (∀ls,cur,rs,b. t1 = midtape STape ls 〈grid,b〉 rs → t2 = t1) ∧
607 (∀c,l1,c1,l2,l3,ls0,rs0,n.
608 t1 = midtape STape (〈grid,false〉::ls0) 〈bit c,true〉 rs
609 (l1@〈grid,false〉::l2@〈bit c1,true〉::l3@〈grid,false〉::rs0) →
610 only_bits_or_nulls l1 → no_marks l1 → S n = |l1| →
611 table_TM (S n) (l2@〈c1,false〉::l3) →
614 〈c1,false〉::l3 = l4@〈c,false〉::l1@〈comma,false〉::newc@〈comma,false〉::mv::l5 ∧
615 t2 = midtape ? (reverse ? l1@〈c,false〉::〈grid,false〉::ls0) 〈grid,false〉
616 (l2@l4@〈c,false〉::l1@〈comma,true〉::newc@〈comma,false〉::mv::l5@
619 (* non facciamo match su nessuna tupla;
620 non specifichiamo condizioni sul nastro di output, perché
621 non eseguiremo altre operazioni, quindi il suo formato non ci interessa *)
622 (current ? t2 = Some ? 〈grid,true〉 ∧
624 〈c1,false〉::l3 ≠ l4@〈c,false〉::l1@〈comma,false〉::newc@〈comma,false〉::mv::l5)).
627 definition R_match_tuple0 ≝ λt1,t2.
629 t1 = midtape STape ls cur rs →
630 (is_grid (\fst cur) = true → t2 = t1) ∧
631 (∀c,l1,c1,l2,l3,ls0,rs0,n.
632 ls = 〈grid,false〉::ls0 →
634 rs = l1@〈grid,false〉::l2@〈bar,false〉::〈c1,true〉::l3@〈grid,false〉::rs0 →
635 is_bit c = true → is_bit c1 = true →
636 only_bits_or_nulls l1 → no_marks l1 → S n = |l1| →
637 table_TM (S n) (l2@〈bar,false〉::〈c1,false〉::l3) →
640 〈bar,false〉::〈c1,false〉::l3 = l4@〈bar,false〉::〈c,false〉::l1@〈comma,false〉::newc@〈comma,false〉::mv::l5 ∧
641 t2 = midtape ? (reverse ? l1@〈c,false〉::〈grid,false〉::ls0) 〈grid,false〉
642 (l2@l4@〈bar,false〉::〈c,false〉::l1@〈comma,true〉::newc@〈comma,false〉::mv::l5@
645 (* non facciamo match su nessuna tupla;
646 non specifichiamo condizioni sul nastro di output, perché
647 non eseguiremo altre operazioni, quindi il suo formato non ci interessa *)
648 (current ? t2 = Some ? 〈grid,true〉 ∧
650 〈bar,false〉::〈c1,false〉::l3 ≠ l4@〈bar,false〉::〈c,false〉::l1@〈comma,false〉::newc@〈comma,false〉::mv::l5)).
652 axiom table_bit_after_bar :
653 ∀n,l1,c,l2.table_TM n (l1@〈bar,false〉::〈c,false〉::l2) → is_bit c = true.
655 lemma wsem_match_tuple : WRealize ? match_tuple R_match_tuple0.
656 #intape #k #outc #Hloop
657 lapply (sem_while … sem_match_tuple_step intape k outc Hloop) [%] -Hloop
658 * #ta * #Hstar @(star_ind_l ??????? Hstar)
659 [ whd in ⊢ (%→?); #Hleft
660 #ls #cur #rs #Htb cases (Hleft … Htb) #Hgrid #Houtc %
662 | #c #l1 #c1 #l2 #l3 #ls0 #rs0 #n #Hls #Hcur #Hrs
663 >Hcur in Hgrid; #Hgrid >(is_grid_true … Hgrid) normalize in ⊢ (%→?);
666 | (* in the interesting case, we execute a true iteration, then we restart the
667 while cycle, finally we end with a false iteration *)
668 #tc #td whd in ⊢ (%→?); #Htc
669 #Hstar1 #IH whd in ⊢ (%→?); #Hright lapply (IH Hright) -IH whd in ⊢ (%→?); #IH
671 [ (* cur can't be true because we assume at least one iteration *)
672 #Hcur cases Htc #ls' * #c' * #rs' * * >Htb #Hdes destruct (Hdes)
673 #Hfalse @False_ind @(absurd … (is_grid_true … Hcur) Hfalse)
674 | (* current and a tuple are marked *)
675 #c #l1 #c1 #l2 #l3 #ls0 #rs0 #n #Hls #Hcur #Hrs #Hc #Hc1 #Hl1bitnull #Hl1marks
677 cases Htc #ls' * #c' * #rs' * * >Htb #Hdes destruct (Hdes)
679 (* expose the marked tuple in table *)
680 cut (∃la,lb,mv,lc.l3 = la@〈comma,false〉::lb@〈comma,false〉::mv::lc ∧
681 S n = |la| ∧ only_bits_or_nulls la)
682 [@daemon] * #la * #lb * #mv * #lc * * #Hl3 #Hlalen #Hlabitnull
683 >Hl3 in Htable; >append_cons #Htable
684 >(?: l2@〈bar,false〉::〈c1,true〉::l3@〈grid,false〉::rs0
685 = (l2@[〈bar,false〉])@〈c1,true〉::la@〈comma,false〉::(lb@〈comma,false〉::mv::
686 lc)@〈grid,false〉::rs0) in Hrs;
687 [| >associative_append normalize >Hl3
688 >associative_append normalize % ] #Hrs
689 cases (Htc ????????? Hl1bitnull Hl1marks ?? Hlabitnull Hl1len ? Htable Hls Hcur Hrs)
691 |4: whd in ⊢ (??%?); >Hc1 %
692 |3: whd in ⊢ (??%?); >Hc %
694 [ (* case 1: match successful *)
695 * #Heq #Htc % %{[]} %{lb} %{mv} %{lc} destruct (Heq) %
697 | cases (IH … Htc) -IH #Houtc #_ >(Houtc (refl ??))
698 >Htc @eq_f normalize >associative_append normalize
699 >associative_append normalize %
701 | (* case 2: tuples don't match, we still have other tuples to try *)
702 * #Hdiff * #c2 * #l5 * #l6 * #Heqlblc #Htc
703 cases (IH ??? … Htc) -IH #_ #IH
704 (* by induction hypothesis *)
705 lapply (IH ? l1 c2 (l2@〈bar,false〉::〈c1,false〉::la@〈comma,false〉::l5) l6 ? rs0 n (refl ??) (refl ??) ???????)
706 [ generalize in match Htable;
707 >associative_append normalize
708 >associative_append normalize >Heqlblc
709 >associative_append normalize //
715 | >associative_append normalize
716 >associative_append normalize
717 >associative_append %
719 [ (* the while finally matches a tuple *)
720 * #l7 * #newc * #mv0 * #l8 * #Hl7l8 #Houtc %
721 >Heqlblc @(ex_intro ?? (〈bar,false〉::〈c1,false〉::la@〈comma,false〉::l5@l7))
722 %{newc} %{mv0} %{l8} %
723 [ normalize >Hl7l8 >associative_append normalize
724 >associative_append %
725 | >Houtc @eq_f >associative_append normalize
726 >associative_append normalize >associative_append
727 normalize >associative_append %
729 | (* the while fails finding a tuple: there are no matches in the whole table *)
730 * #Houtc #Hdiff1 %2 %
732 | #l50 #newc #mv0 #l51 >Heqlblc
738 | (* match failed and there is no next tuple: the next while cycle will just exit *)
739 * * #Hdiff #Hnobars generalize in match (refl ? tc);
740 cases tc in ⊢ (???% → %);
741 [ #_ @daemon (* long normalize *) (*
742 normalize in ⊢ (??%?→?); #Hfalse destruct (Hfalse)
744 |2,3: #x #xs #_ @daemon (* long normalize *) (*
745 normalize in ⊢ (??%?→?); #Hfalse destruct (Hfalse) *) ]
746 #ls1 #cur1 #rs1 #Htc @daemon (* long normalize *) (*
747 normalize in ⊢ (??%?→?); #Hcur1
748 cases (IH … Htc) -IH #IH #_ %2 %
749 [ destruct (Hcur1) >IH [ >Htc % | % ]
751 (* no_bars except the first one, where the tuple does not match ⇒
760 lemma WF_mts_niltape:
761 WF ? (inv ? R_match_tuple_step_true) (niltape (FinProd FSUnialpha FinBool)).
762 @wf #t1 whd in ⊢ (%→?); * #ls * #c * #rs * * #H destruct
765 lemma WF_mts_rightof:
766 ∀a,ls. WF ? (inv ? R_match_tuple_step_true) (rightof (FinProd FSUnialpha FinBool) a ls).
767 #a #ls @wf #t1 whd in ⊢ (%→?); * #ls * #c * #rs * * #H destruct
771 ∀a,ls. WF ? (inv ? R_match_tuple_step_true) (leftof (FinProd FSUnialpha FinBool) a ls).
772 #a #ls @wf #t1 whd in ⊢ (%→?); * #ls * #c * #rs * * #H destruct
775 lemma WF_cst_midtape_grid:
776 ∀ls,b,rs. WF ? (inv ? R_match_tuple_step_true)
777 (midtape (FinProd … FSUnialpha FinBool) ls 〈grid,b〉 rs).
778 #ls #b #rs @wf #t1 whd in ⊢ (%→?); * #ls' * #c' * #rs' * * #H destruct
779 * #Hfalse @False_ind @Hfalse %
782 definition Pre_match_tuple ≝ λt.
783 ∃ls,cur,rs. t = midtape STape ls cur rs ∧
784 (is_grid (\fst cur) = true ∨
785 (∃ls0,c,l1,l2,c1,l3,l4,rs0,n.
786 only_bits_or_nulls l1 ∧ no_marks l1 ∧
787 bit_or_null c = true ∧ bit_or_null c1 = true ∧
788 only_bits_or_nulls l3 ∧ S n = |l1| ∧|l1| = |l3| ∧
789 table_TM (S n) (l2@〈c1,false〉::l3@〈comma,false〉::l4) ∧
790 ls = 〈grid,false〉::ls0 ∧ cur = 〈c,true〉 ∧
791 rs = l1@〈grid,false〉::l2@〈c1,true〉::l3@〈comma,false〉::l4@〈grid,false〉::rs0)).
793 lemma acc_Realize_to_acc_GRealize: ∀sig,M.∀q:states sig M.∀P,R1,R2.
794 M ⊨ [q:R1,R2] → accGRealize sig M q P R1 R2.
795 #alpha #M #q #Pre #R1 #R2 #HR #t #HPre
796 cases (HR t) -HR #k * #outc * * #Hloop #HRtrue #HRfalse
797 @(ex_intro ?? k) @(ex_intro ?? outc) %
798 [ % [@Hloop] @HRtrue | @HRfalse]
802 lemma terminate_match_tuple:
803 ∀t. Pre_match_tuple t → Terminate ? match_tuple t.
805 @(terminate_while_guarded ???
807 (acc_Realize_to_acc_GRealize ??? Pre_match_tuple … sem_match_tuple_step)
809 [-HPre -t #t1 #t2 #HPre cases HPre #ls * * #curl #curr * #rs * #Ht1 *
811 #Hgrid * #ls1 * #cur1 * #rs1 * * >Ht1 #Hdes destruct (Hdes)
812 #Habs @False_ind @(absurd ?? Habs) @(is_grid_true … Hgrid)
813 |* #ls0 * #c * #l1 * #l2 * #c1 * #l3 * #l4 * #rs0 * #n
815 #Hl1 #Hmarksl1 #Hc #Hc1 #Hl3 #lenl1 #eqlen #Htable #Hls #Hcur #Hrs
816 * #ls1 * #cur1 * #rs1 * * >Ht1 #Hdes destruct (Hdes) #Hdes #H
817 lapply (H … Hl1 Hmarksl1 Hc Hc1 Hl3 lenl1 eqlen Htable Hls Hcur Hrs)
819 [* [ * #Hdes #Ht2 >Ht2
820 @ex_intro [2:@ex_intro [2: @ex_intro [2: % [%]|]|]|]
822 |* #test * #c2 * #l5 * #l6 * #Hl4 #Ht2
823 cut (∃l7,l8. l6 = l7@〈comma,false 〉::l8 ∧ |l7| = |l1|) [@daemon]
824 * #l7 * #l8 * #Hl6 #eqlen1
825 @ex_intro [2:@ex_intro [2: @ex_intro [2: % [@Ht2]|]|]|] %2
826 @(ex_intro … ls0) @(ex_intro … c) @(ex_intro … l1)
827 @(ex_intro … (l2@〈c1,false〉::l3@〈comma,false〉::l5@[〈bar,false〉]))
828 @(ex_intro … c2) @(ex_intro … l7) @(ex_intro … l8)
829 @(ex_intro … rs0) @(ex_intro … n)
830 % [2: >Hl6 >associative_append >associative_append @eq_f @eq_f @eq_f
831 @eq_f >associative_append @eq_f @eq_f >associative_append % ]
832 % [2: %] % [2: %] % [2:@daemon] % [2: @sym_eq @eqlen1]
833 % [2: @lenl1] % [2: #x #memx @daemon]
834 % [2: @daemon] % [2: @Hc] % [2: @Hmarksl1] @Hl1
836 |* * #_ #_ #H cases (current_to_midtape … H) #ls * #rs #Ht1
837 >Ht1 @ex_intro [2:@ex_intro [2: @ex_intro [2: % [%]|]|]|] %1 %
840 |cases HPre -HPre #ls * * #curl #curr * #rs * #Ht *
841 [#Hgrid >Ht >(is_grid_true … Hgrid) @WF_cst_midtape_grid
842 |* #ls0 * #c * #l1 * #l2 * #c1 * #l3 * #l4
843 cut (∃len. |l4| = len) [/2/] * #lenl4
844 lapply l4 lapply l3 lapply c1 lapply l2 lapply l1 lapply c lapply ls0 lapply Ht
845 lapply curr lapply curl lapply ls lapply rs lapply t -l4 -l3 -l2 -l1 -c1 -curr -curl -ls -t
847 (* by induction on the length of l4 *)
849 #len #Hind #t #rs #ls #cl #cr #Ht #ls0 #c #l1 #l2 #c1 #l3 #l4 #Hlen
850 * #rs0 * #n * * * * * * * * * *
851 #Hl1 #Hmarksl1 #Hc #Hc1 #Hl3 #lenl1 #eqlen #Htable #Hls #Hcur #Hrs
852 % #t1 >Ht whd in ⊢ (%→?); * #ls1 * #cur * #rs1 * * #Hdes destruct (Hdes)
853 #Hgrid #H lapply (H … Hl1 Hmarksl1 Hc Hc1 Hl3 lenl1 eqlen Htable Hls Hcur Hrs)
855 [* [ * #Hdes destruct (Hdes) #Ht1 >Ht1 @WF_cst_midtape_grid
856 | * #_ * #c2 * #l5 * #l6 * #Hl4 #Ht1
857 cut (∃l7,l8. l6 = l7@〈comma,false 〉::l8 ∧ |l7| = |l1|) [@daemon]
858 * #l7 * #l8 * #Hl6 #eqlen1
859 @(Hind … Ht1 ls0 c l1 (l2@〈c1,false〉::l3@〈comma,false〉::l5@[〈bar,false〉]) c2 l7 l8 … (refl …))
860 [<Hlen >Hl4 >Hl6 >length_append normalize in match (length … (cons …));
861 >length_append normalize in match (length … (cons …)); <plus_n_Sm
863 |@(ex_intro … rs0) @(ex_intro … n) %
864 [2: >Hl6 >associative_append >associative_append @eq_f @eq_f @eq_f
865 @eq_f >associative_append @eq_f @eq_f >associative_append % ]
866 % [2: %] % [2: %] % [2:@daemon] % [2: @sym_eq @eqlen1]
867 % [2: @lenl1] % [2: #x #memx @daemon]
868 % [2: @daemon] % [2: @Hc] % [2: @Hmarksl1] @Hl1
871 |* * #_ #_ #H cases (current_to_midtape … H) #ls * #rs #Ht1
877 definition R_match_tuple ≝ λt1,t2.
879 is_bit c = true → is_bit c1 = true →
880 only_bits_or_nulls l1 → no_marks l1 → S n = |l1| →
881 table_TM (S n) (〈bar,false〉::〈c1,false〉::l2) →
882 t1 = midtape STape (〈grid,false〉::ls) 〈c,true〉
883 (l1@〈grid,false〉::〈bar,false〉::〈c1,true〉::l2@〈grid,false〉::rs) →
886 〈bar,false〉::〈c1,false〉::l2 = l3@〈bar,false〉::〈c,false〉::l1@〈comma,false〉::newc@〈comma,false〉::mv::l4 ∧
887 t2 = midtape ? (reverse ? l1@〈c,false〉::〈grid,false〉::ls) 〈grid,false〉
888 (l3@〈bar,false〉::〈c,false〉::l1@〈comma,true〉::newc@〈comma,false〉::mv::l4@〈grid,false〉::rs))
890 (* non facciamo match su nessuna tupla;
891 non specifichiamo condizioni sul nastro di output, perché
892 non eseguiremo altre operazioni, quindi il suo formato non ci interessa *)
893 (current ? t2 = Some ? 〈grid,true〉 ∧
895 〈bar,false〉::〈c1,false〉::l2 ≠ l3@〈bar,false〉::〈c,false〉::l1@〈comma,false〉::newc@〈comma,false〉::mv::l4).
897 lemma sem_match_tuple0 : GRealize ? match_tuple Pre_match_tuple R_match_tuple0.
898 @WRealize_to_GRealize [@terminate_match_tuple | @wsem_match_tuple]
901 lemma sem_match_tuple : GRealize ? match_tuple Pre_match_tuple R_match_tuple.
902 generalize in match sem_match_tuple0; @GRealize_to_GRealize
903 #t1 #t2 #HR #ls #c #l1 #c1 #l2 #rs #n #Hc #Hc1 #Hl1bitsnulls #Hl1marks #Hl1len #Htable #Ht1
904 cases (HR … Ht1) -HR #_ #HR
905 @(HR ??? [] … (refl ??) (refl ??) (refl ??) Hc Hc1 Hl1bitsnulls Hl1marks