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_____________________________________________________________*)
17 include "turing/universal/marks.ma".
19 definition STape ≝ FinProd … FSUnialpha FinBool.
21 definition only_bits_or_nulls ≝ λl.
22 ∀c.memb STape c l = true → bit_or_null (\fst c) = true.
24 definition no_grids ≝ λl.
25 ∀c.memb STape c l = true → is_grid (\fst c) = false.
27 definition no_bars ≝ λl.
28 ∀c.memb STape c l = true → is_bar (\fst c) = false.
30 definition no_marks ≝ λl.
31 ∀c.memb STape c l = true → is_marked ? c = false.
33 lemma bit_not_grid: ∀d. is_bit d = true → is_grid d = false.
34 * // normalize #H destruct
37 lemma bit_or_null_not_grid: ∀d. bit_or_null d = true → is_grid d = false.
38 * // normalize #H destruct
41 lemma bit_not_bar: ∀d. is_bit d = true → is_bar d = false.
42 * // normalize #H destruct
45 lemma bit_or_null_not_bar: ∀d. bit_or_null d = true → is_bar d = false.
46 * // normalize #H destruct
49 (* by definition, a tuple is not marked *)
50 definition tuple_TM : nat → list STape → Prop ≝
53 only_bits_or_nulls qin ∧ only_bits_or_nulls qout ∧ bit_or_null mv = true ∧
54 |qin| = n ∧ |qout| = n (* ∧ |mv| = ? *) ∧
55 t = qin@〈comma,false〉::qout@〈comma,false〉::[〈mv,false〉].
57 inductive table_TM : nat → list STape → Prop ≝
58 | ttm_nil : ∀n.table_TM n []
59 | ttm_cons : ∀n,t1,T.tuple_TM n t1 → table_TM n T → table_TM n (t1@〈bar,false〉::T).
61 lemma no_grids_in_table: ∀n.∀l.table_TM n l → no_grids l.
63 [normalize #n #x #H destruct
64 |#m #t1 #t2 * #qin * #qout * #mv * * * * * *
65 #Hmarks #Hqin #Hqout #Hmv #_ #_ #Heq #Ht2 #Hind
67 cases (memb_append … membx) -membx #membx
68 [cases (memb_append … membx) -membx #membx
69 [@bit_or_null_not_grid @Hqin //
70 |cases (orb_true_l … membx) -membx #membx
72 |cases (memb_append … membx) -membx #membx
73 [@bit_or_null_not_grid @Hqout //
74 |cases (orb_true_l … membx) -membx #membx
76 |@bit_or_null_not_grid >(memb_single … membx) @Hmv
81 |cases (orb_true_l … membx) -membx #membx
89 lemma no_marks_in_table: ∀n.∀l.table_TM n l → no_marks l.
91 [normalize #n #x #H destruct
92 |#m #t1 #t2 * #qin * #qout * #mv * * * * * *
93 #Hmarks #_ #_ #_ #_ #_ #_ #Ht2 #Hind
94 #x #Hx cases (memb_append … Hx) -Hx #Hx
96 |cases (orb_true_l … Hx) -Hx #Hx
104 axiom last_of_table: ∀n,l,b.¬ table_TM n (l@[〈bar,b〉]).
107 l0 x* a l1 x0* a0 l2 ------> l0 x a* l1 x0 a0* l2
110 if current (* x *) = #
113 then move_right; ----
115 if current (* x0 *) = 0
116 then advance_mark ----
120 else x = 1 (* analogo *)
126 MARK NEXT TUPLE machine
127 (partially axiomatized)
129 marks the first character after the first bar (rightwards)
132 definition bar_or_grid ≝ λc:STape.is_bar (\fst c) ∨ is_grid (\fst c).
134 definition mark_next_tuple ≝
135 seq ? (adv_to_mark_r ? bar_or_grid)
136 (ifTM ? (test_char ? (λc:STape.is_bar (\fst c)))
137 (move_right_and_mark ?) (nop ?) 1).
139 definition R_mark_next_tuple ≝
142 (* c non può essere un separatore ... speriamo *)
143 t1 = midtape STape ls c (rs1@〈grid,false〉::rs2) →
144 no_marks rs1 → no_grids rs1 → bar_or_grid c = false →
145 (∃rs3,rs4,d,b.rs1 = rs3 @ 〈bar,false〉 :: rs4 ∧
147 Some ? 〈d,b〉 = option_hd ? (rs4@〈grid,false〉::rs2) ∧
148 t2 = midtape STape (〈bar,false〉::reverse ? rs3@c::ls) 〈d,true〉 (tail ? (rs4@〈grid,false〉::rs2)))
150 (no_bars rs1 ∧ t2 = midtape ? (reverse ? rs1@c::ls) 〈grid,false〉 rs2).
154 (∀x.memb A x l = true → f x = false) ∨
155 (∃l1,c,l2.f c = true ∧ l = l1@c::l2 ∧ ∀x.memb ? x l1 = true → f x = false).
157 [ % #x normalize #Hfalse *)
159 theorem sem_mark_next_tuple :
160 Realize ? mark_next_tuple R_mark_next_tuple.
162 lapply (sem_seq ? (adv_to_mark_r ? bar_or_grid)
163 (ifTM ? (test_char ? (λc:STape.is_bar (\fst c))) (move_right_and_mark ?) (nop ?) 1) ????)
164 [@sem_if [5: // |6: @sem_move_right_and_mark |7: // |*:skip]
166 |||#Hif cases (Hif intape) -Hif
167 #j * #outc * #Hloop * #ta * #Hleft #Hright
168 @(ex_intro ?? j) @ex_intro [|% [@Hloop] ]
170 #ls #c #rs1 #rs2 #Hrs #Hrs1 #Hrs1' #Hc
172 [ * #Hfalse >Hfalse in Hc; #Htf destruct (Htf)
173 | * #_ #Hta cases (tech_split STape (λc.is_bar (\fst c)) rs1)
174 [ #H1 lapply (Hta rs1 〈grid,false〉 rs2 (refl ??) ? ?)
175 [ * #x #b #Hx whd in ⊢ (??%?); >(Hrs1' … Hx) >(H1 … Hx) %
177 | -Hta #Hta cases Hright
178 [ * #tb * whd in ⊢ (%→?); #Hcurrent
179 @False_ind cases (Hcurrent 〈grid,false〉 ?)
180 [ normalize #Hfalse destruct (Hfalse)
182 | * #tb * whd in ⊢ (%→?); #Hcurrent
183 cases (Hcurrent 〈grid,false〉 ?)
184 [ #_ #Htb whd in ⊢ (%→?); #Houtc
187 | >Houtc >Htb >Hta % ]
191 | * #rs3 * #c0 * #rs4 * * #Hc0 #Hsplit #Hrs3
192 % @(ex_intro ?? rs3) @(ex_intro ?? rs4)
193 lapply (Hta rs3 c0 (rs4@〈grid,false〉::rs2) ???)
194 [ #x #Hrs3' whd in ⊢ (??%?); >Hsplit in Hrs1;>Hsplit in Hrs3;
195 #Hrs3 #Hrs1 >(Hrs1 …) [| @memb_append_l1 @Hrs3'|]
196 >(Hrs3 … Hrs3') @Hrs1' >Hsplit @memb_append_l1 //
197 | whd in ⊢ (??%?); >Hc0 %
198 | >Hsplit >associative_append % ] -Hta #Hta
200 [ * #tb * whd in ⊢ (%→?); #Hta'
203 [ #_ #Htb' >Htb' in Htb; #Htb
204 generalize in match Hsplit; -Hsplit
206 [ #Hta #Hsplit >(Htb … Hta)
207 >(?:c0 = 〈bar,false〉)
208 [ @(ex_intro ?? grid) @(ex_intro ?? false)
210 [(* Hsplit *) @daemon |(*Hrs3*) @daemon ] | % ] | % ]
211 | (* Hc0 *) @daemon ]
212 | #r5 #rs5 >(eq_pair_fst_snd … r5)
213 #Hta #Hsplit >(Htb … Hta)
214 >(?:c0 = 〈bar,false〉)
215 [ @(ex_intro ?? (\fst r5)) @(ex_intro ?? (\snd r5))
216 % [ % [ % [ (* Hc0, Hsplit *) @daemon | (*Hrs3*) @daemon ] | % ]
217 | % ] | (* Hc0 *) @daemon ] ] | >Hta % ]
218 | * #tb * whd in ⊢ (%→?); #Hta'
221 [ #Hfalse @False_ind >Hfalse in Hc0;
227 definition init_current ≝
228 seq ? (adv_to_mark_l ? (is_marked ?))
229 (seq ? (clear_mark ?)
230 (seq ? (adv_to_mark_l ? (λc:STape.is_grid (\fst c)))
231 (seq ? (move_r ?) (mark ?)))).
233 definition R_init_current ≝ λt1,t2.
234 ∀l1,c,l2,b,l3,c1,rs,c0,b0. no_marks l1 → no_grids l2 → is_grid c = false →
235 Some ? 〈c0,b0〉 = option_hd ? (reverse ? (〈c,true〉::l2)) →
236 t1 = midtape STape (l1@〈c,true〉::l2@〈grid,b〉::l3) 〈c1,false〉 rs →
237 t2 = midtape STape (〈grid,b〉::l3) 〈c0,true〉
238 ((tail ? (reverse ? (l1@〈c,false〉::l2))@〈c1,false〉::rs)).
240 lemma sem_init_current : Realize ? init_current R_init_current.
242 cases (sem_seq ????? (sem_adv_to_mark_l ? (is_marked ?))
243 (sem_seq ????? (sem_clear_mark ?)
244 (sem_seq ????? (sem_adv_to_mark_l ? (λc:STape.is_grid (\fst c)))
245 (sem_seq ????? (sem_move_r ?) (sem_mark ?)))) intape)
246 #k * #outc * #Hloop #HR
247 @(ex_intro ?? k) @(ex_intro ?? outc) % [@Hloop]
248 cases HR -HR #ta * whd in ⊢ (%→?); #Hta
249 * #tb * whd in ⊢ (%→?); #Htb
250 * #tc * whd in ⊢ (%→?); #Htc
251 * #td * whd in ⊢ (%→%→?); #Htd #Houtc
252 #l1 #c #l2 #b #l3 #c1 #rs #c0 #b0 #Hl1 #Hl2 #Hc #Hc0 #Hintape
253 cases (Hta … Hintape) [ * #Hfalse normalize in Hfalse; destruct (Hfalse) ]
254 -Hta * #_ #Hta lapply (Hta l1 〈c,true〉 ? (refl ??) ??) [@Hl1|%]
255 -Hta #Hta lapply (Htb … Hta) -Htb #Htb cases (Htc … Htb) [ >Hc -Hc * #Hc destruct (Hc) ]
256 -Htc * #_ #Htc lapply (Htc … (refl ??) (refl ??) ?) [@Hl2]
257 -Htc #Htc lapply (Htd … Htc) -Htd
258 >reverse_append >reverse_cons
259 >reverse_cons in Hc0; cases (reverse … l2)
260 [ normalize in ⊢ (%→?); #Hc0 destruct (Hc0)
261 #Htd >(Houtc … Htd) %
262 | * #c2 #b2 #tl2 normalize in ⊢ (%→?);
263 #Hc0 #Htd >(Houtc … Htd)
264 whd in ⊢ (???%); destruct (Hc0)
265 >associative_append >associative_append %
269 definition match_tuple_step ≝
270 ifTM ? (test_char ? (λc:STape.¬ is_grid (\fst c)))
273 (ifTM ? (test_char ? (λc:STape.is_grid (\fst c)))
275 (seq ? mark_next_tuple
276 (ifTM ? (test_char ? (λc:STape.is_grid (\fst c)))
277 (mark ?) (seq ? (move_l ?) init_current) tc_true)) tc_true)))
280 definition R_match_tuple_step_true ≝ λt1,t2.
281 ∀ls,c,l1,l2,c1,l3,l4,rs,n.
282 bit_or_null c = true → only_bits_or_nulls l1 → no_marks l1 (* → no_grids l2 *) → bit_or_null c1 = true →
283 only_bits_or_nulls l3 → n = |l1| → |l1| = |l3| →
284 table_TM (S n) (l2@〈bar,false〉::〈c1,false〉::l3@〈comma,false〉::l4) →
285 t1 = midtape STape (〈grid,false〉::ls) 〈c,true〉
286 (l1@〈grid,false〉::l2@〈bar,false〉::〈c1,true〉::l3@〈comma,false〉::l4@〈grid,false〉::rs) →
288 (〈c,false〉::l1 = 〈c1,false〉::l3 ∧
289 t2 = midtape ? (reverse ? l1@〈c,false〉::〈grid,false〉::ls) 〈grid,false〉
290 (l2@〈bar,false〉::〈c1,false〉::l3@〈comma,true〉::l4@〈grid,false〉::rs))
292 (* non facciamo match e marchiamo la prossima tupla *)
293 ((〈c,false〉::l1 ≠ 〈c1,false〉::l3 ∧
294 ∃c2,l5,l6.l4 = l5@〈bar,false〉::〈c2,false〉::l6 ∧
295 (* condizioni su l5 l6 l7 *)
296 t2 = midtape STape (〈grid,false〉::ls) 〈c,true〉
297 (l1@〈grid,false〉::l2@〈bar,false〉::〈c1,false〉::l3@〈comma,false〉::
298 l5@〈bar,false〉::〈c2,true〉::l6@〈grid,false〉::rs))
300 (* non facciamo match e non c'è una prossima tupla:
301 non specifichiamo condizioni sul nastro di output, perché
302 non eseguiremo altre operazioni, quindi il suo formato non ci interessa *)
303 (〈c,false〉::l1 ≠ 〈c1,false〉::l3 ∧ no_bars l4 ∧ current ? t2 = Some ? 〈grid,true〉)).
305 definition R_match_tuple_step_false ≝ λt1,t2.
306 ∀ls,c,rs.t1 = midtape STape ls c rs → is_grid (\fst c) = true ∧ t2 = t1.
308 include alias "basics/logic.ma".
311 lemma eq_f4: ∀A1,A2,A3,A4,B.∀f:A1 → A2 →A3 →A4 →B.
312 ∀x1,x2,x3,x4,y1,y2,y3,y4. x1 = y1 → x2 = y2 →x3=y3 →x4 = y4 →
313 f x1 x2 x3 x4 = f y1 y2 y3 y4.
317 lemma some_option_hd: ∀A.∀l:list A.∀a.∃b.
318 Some ? b = option_hd ? (l@[a]) .
319 #A #l #a cases l normalize /2/
322 axiom tech_split2 : ∀A,l1,l2,l3,l4,x.
323 memb A x l1 = false → memb ? x l3 = false →
324 l1@x::l2 = l3@x::l4 → l1 = l3 ∧ l2 = l4.
326 axiom injective_append : ∀A,l.injective … (λx.append A x l).
328 lemma sem_match_tuple_step:
329 accRealize ? match_tuple_step (inr … (inl … (inr … 0)))
330 R_match_tuple_step_true R_match_tuple_step_false.
331 @(acc_sem_if_app … (sem_test_char ? (λc:STape.¬ is_grid (\fst c))) …
332 (sem_seq … sem_compare
333 (sem_if … (sem_test_char ? (λc:STape.is_grid (\fst c)))
335 (sem_seq … sem_mark_next_tuple
336 (sem_if … (sem_test_char ? (λc:STape.is_grid (\fst c)))
337 (sem_mark ?) (sem_seq … (sem_move_l …) (sem_init_current …))))))
339 [(* is_grid: termination case *)
340 2:#t1 #t2 #t3 whd in ⊢ (%→?); #H #H1 whd #ls #c #rs #Ht1
341 cases (H c ?) [2: >Ht1 %] #Hgrid #Heq %
342 [@injective_notb @Hgrid | <Heq @H1]
343 |#tapea #tapeout #tapeb whd in ⊢ (%→?); #Htapea
344 * #tapec * #Hcompare #Hor
345 #ls #c #l1 #l2 #c1 #l3 #l4 #rs #n #Hc #Hl1bars #Hl1marks #Hc1 #Hl3 #eqn
346 #eqlen #Htable #Htapea1 cases (Htapea 〈c,true〉 ?) >Htapea1 [2:%]
347 #notgridc -Htapea -Htapea1 -tapea #Htapeb
348 cases (Hcompare … Htapeb) -Hcompare -Htapeb * #_ #_ #Hcompare
349 cases (Hcompare c c1 l1 l3 (l2@[〈bar,false〉]) (l4@〈grid,false〉::rs) eqlen Hl1bars Hl3 Hl1marks … (refl …) Hc ?)
351 [* #Htemp destruct (Htemp) #Htapec %1 % [%]
352 >Htapec in Hor; -Htapec *
353 [2: * #t3 * whd in ⊢ (%→?); #H @False_ind
354 cases (H … (refl …)) whd in ⊢ ((??%?)→?); #H destruct (H)
355 |* #taped * whd in ⊢ (%→?); #Htaped cases (Htaped ? (refl …)) -Htaped *
356 #Htaped whd in ⊢ (%→?); #Htapeout >Htapeout >Htaped >associative_append
359 |* #la * #c' * #d' * #lb * #lc * * * #H1 #H2 #H3 #Htapec
360 cut (〈c,false〉::l1 ≠ 〈c1,false〉::l3)
362 [@(not_to_not …H1) normalize #H destruct %
363 |#x #tl @not_to_not normalize #H destruct //
366 cut (bit_or_null d' = true)
368 [normalize in ⊢ (%→?); #H destruct //
369 |#x #tl #H @(Hl3 〈d',false〉)
370 normalize in H; destruct @memb_append_l2 @memb_hd
373 >Htapec in Hor; -Htapec *
374 [* #taped * whd in ⊢ (%→?); #H @False_ind
375 cases (H … (refl …)) >(bit_or_null_not_grid ? Hd') #Htemp destruct (Htemp)
376 |* #taped * whd in ⊢ (%→?); #H cases (H … (refl …)) -H #_
377 #Htaped * #tapee * whd in ⊢ (%→?); #Htapee
378 <(associative_append ? lc (〈comma,false〉::l4)) in Htaped; #Htaped
379 cases (Htapee … Htaped ???) -Htaped -Htapee
380 [* #rs3 * * (* we proceed by cases on rs4 *)
381 [(* rs4 is empty : the case is absurd since the tape
382 cannot end with a bar *)
383 * #d * #b * * * #Heq1 @False_ind
384 cut (∀A,l1,l2.∀a:A. a::l1@l2=(a::l1)@l2) [//] #Hcut
385 >Hcut in Htable; >H3 >associative_append
386 normalize >Heq1 >Hcut <associative_append >Hcut
387 <associative_append #Htable @(absurd … Htable)
390 * #d2 #b2 #rs3' * #d * #b * * * #Heq1 #Hnobars
391 cut (memb STape 〈d2,b2〉 (l2@〈bar,false〉::〈c1,false〉::l3@〈comma,false〉::l4) = true)
392 [@memb_append_l2 @memb_cons
393 cut (∀A,l1,l2.∀a:A. a::l1@l2=(a::l1)@l2) [//] #Hcut
394 >Hcut >H3 >associative_append @memb_append_l2
395 @memb_cons >Heq1 @memb_append_l2 @memb_cons @memb_hd] #d2intable
396 cut (is_grid d2 = false)
397 [@(no_grids_in_table … Htable … 〈d2,b2〉 d2intable)] #Hd2
399 [@(no_marks_in_table … Htable … 〈d2,b2〉 d2intable)] #Hb2
400 >Hb2 in Heq1; #Heq1 -Hb2 -b2
401 whd in ⊢ ((???%)→?); #Htemp destruct (Htemp) #Htapee >Htapee -Htapee *
402 [(* we know current is not grid *)
403 * #tapef * whd in ⊢ (%→?); #Htapef
404 cases (Htapef … (refl …)) >Hd2 #Htemp destruct (Htemp)
405 |* #tapef * whd in ⊢ (%→?); #Htapef
406 cases (Htapef … (refl …)) #_ -Htapef #Htapef
407 * #tapeg >Htapef -Htapef *
410 #H lapply (H … (refl …)) whd in ⊢ (???%→?); -H #Htapeg
413 whd in ⊢ (%→?); #Htapeout
414 %1 cases (some_option_hd ? (reverse ? (reverse ? la)) 〈c',true〉)
417 (Htapeout (reverse ? rs3 @〈d',false〉::reverse ? la@reverse ? (l2@[〈bar,false〉])@(〈grid,false〉::reverse ? lb))
418 c' (reverse ? la) false ls bar (〈d2,true〉::rs3'@〈grid,false〉::rs) c00 b00 ?????) -Htapeout
419 [whd in ⊢ (??(??%??)?); @eq_f3 [2:%|3: %]
421 generalize in match (〈c',true〉::reverse ? la@〈grid,false〉::ls); #l
422 whd in ⊢ (???(???%)); >associative_append >associative_append %
423 |>reverse_cons @Hoption
425 [normalize in ⊢ (%→?); #Htemp destruct (Htemp)
426 @injective_notb @notgridc
427 |#x #tl normalize in ⊢ (%→?); #Htemp destruct (Htemp)
428 @bit_or_null_not_grid @(Hl1bars 〈c',false〉) @memb_append_l2 @memb_hd
430 |cut (only_bits_or_nulls (la@(〈c',false〉::lb)))
431 [<H2 whd #c0 #Hmemb cases (orb_true_l … Hmemb)
432 [#eqc0 >(\P eqc0) @Hc |@Hl1bars]
433 |#Hl1' #x #Hx @bit_or_null_not_grid @Hl1'
434 @memb_append_l1 @daemon
437 |>reverse_append >reverse_cons >reverse_reverse
438 >reverse_append >reverse_reverse
439 >reverse_cons >reverse_append >reverse_reverse
440 >reverse_append >reverse_cons >reverse_reverse
442 #Htapeout % [@Hnoteq]
444 cut (∃rs32.rs3 = lc@〈comma,false〉::rs32)
445 [ (*cases (tech_split STape (λc.c == 〈bar,false〉) l4)
447 | * #l41 * * #cbar #bfalse * #l42 * * #Hbar #Hl4 #Hl41
448 @(ex_intro ?? l41) >Hl4 in Heq1; #Heq1
450 cut (sublist … lc l3)
451 [ #x #Hx cases la in H3;
452 [ normalize #H3 destruct (H3) @Hx
453 | #p #la' normalize #Hla' destruct (Hla')
454 @memb_append_l2 @memb_cons @Hx ] ] #Hsublist*)
458 (〈c1,false〉::l3@〈comma,false〉::l4= la@〈d',false〉::rs3@〈bar,false〉::〈d2,b2〉::rs3')
460 cut (l4=rs32@〈bar,false〉::〈d2,false〉::rs3')
461 [ >Hrs3 in Heq1; @daemon ] #Hl4
462 @(ex_intro … rs32) @(ex_intro … rs3') %
466 |(*>Hrs3 *)>append_cons
467 > (?:l1@〈grid,false〉::l2@〈bar,false〉::〈c1,false〉::l3@〈comma,false〉::rs32@〈bar,false〉::〈d2,true〉::rs3'@〈grid,false〉::rs
468 = (l1@〈grid,false〉::l2@〈bar,false〉::〈c1,false〉::l3@〈comma,false〉::rs32@[〈bar,false〉])@〈d2,true〉::rs3'@〈grid,false〉::rs)
469 [|>associative_append normalize
470 >associative_append normalize
471 >associative_append normalize
472 >associative_append normalize
475 @(injective_append … (〈d2,false〉::rs3'))
476 >(?:(la@[〈c',false〉])@((((lb@[〈grid,false〉])@l2@[〈bar,false〉])@la)@[〈d',false〉])@rs3
477 =((la@〈c',false〉::lb)@([〈grid,false〉]@l2@[〈bar,false〉]@la@[〈d',false〉]@rs3)))
478 [|>associative_append >associative_append
479 >associative_append >associative_append >associative_append
480 >associative_append >associative_append % ]
481 <H2 normalize (* <Hrs3 *)
482 >associative_append >associative_append >associative_append
483 @eq_f normalize @eq_f >associative_append
484 >associative_append @eq_f normalize @eq_f
485 >(append_cons ? 〈d',false〉) >associative_append
486 <Heq1 >Hl4 <associative_append <append_cons
488 >associative_append normalize
489 >associative_append normalize %
495 |* #Hnobars #Htapee >Htapee -Htapee *
496 [whd in ⊢ (%→?); * #tapef * whd in ⊢ (%→?); #Htapef
497 cases (Htapef … (refl …)) -Htapef #_ #Htapef >Htapef -Htapef
498 whd in ⊢ (%→?); #Htapeout %2
499 >(Htapeout … (refl …)) %
502 | whd #x #Hx @Hnobars @memb_append_l2 @memb_cons //
506 |whd in ⊢ (%→?); * #tapef * whd in ⊢ (%→?); #Htapef
507 cases (Htapef … (refl …)) -Htapef
508 whd in ⊢ ((??%?)→?); #Htemp destruct (Htemp)
510 |(* no marks in table *)
511 #x #membx @(no_marks_in_table … Htable)
512 @memb_append_l2 @memb_cons
513 cut (∀A,l1,l2.∀a:A. a::l1@l2=(a::l1)@l2) [//] #Hcut >Hcut
514 >H3 >associative_append @memb_append_l2 @memb_cons @membx
515 |(* no grids in table *)
516 #x #membx @(no_grids_in_table … Htable)
517 @memb_append_l2 @memb_cons
518 cut (∀A,l1,l2.∀a:A. a::l1@l2=(a::l1)@l2) [//] #Hcut >Hcut
519 >H3 >associative_append @memb_append_l2 @memb_cons @membx
520 |whd in ⊢ (??%?); >(bit_or_null_not_grid … Hd') >(bit_or_null_not_bar … Hd') %
523 |#x #membx @(no_marks_in_table … Htable)
524 @memb_append_l2 @memb_cons @memb_cons @memb_append_l1 @membx
525 |#x #membx @(no_marks_in_table … Htable)
526 cases (memb_append … membx) -membx #membx
527 [@memb_append_l1 @membx | @memb_append_l2 >(memb_single … membx) @memb_hd]
528 |>associative_append %
537 scrolls through the tuples in the transition table until one matching the
538 current configuration is found
541 definition match_tuple ≝ whileTM ? match_tuple_step (inr … (inl … (inr … 0))).
543 definition R_match_tuple ≝ λt1,t2.
545 is_bit c = true → only_bits_or_nulls l1 → is_bit c1 = true → n = |l1| →
546 table_TM (S n) (〈c1,true〉::l2) →
547 t1 = midtape STape (〈grid,false〉::ls) 〈c,true〉
548 (l1@〈grid,false〉::〈c1,true〉::l2@〈grid,false〉::rs) →
551 〈c1,false〉::l2 = l3@〈c,false〉::l1@〈comma,false〉::newc@〈comma,false〉::mv@l4 ∧
552 t2 = midtape ? (reverse ? l1@〈c,false〉::〈grid,false〉::ls) 〈grid,false〉
553 (l3@〈c,false〉::l1@〈comma,true〉::newc@〈comma,false〉::mv@l4@〈grid,false〉::rs))
555 (* non facciamo match su nessuna tupla;
556 non specifichiamo condizioni sul nastro di output, perché
557 non eseguiremo altre operazioni, quindi il suo formato non ci interessa *)
558 (current ? t2 = Some ? 〈grid,true〉 ∧
560 〈c1,false〉::l2 ≠ l3@〈c,false〉::l1@〈comma,false〉::newc@〈comma,false〉::mv@l4).