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
82 [ * #Hfalse >Hfalse in Hc; #Htf destruct (Htf)
83 | * #_ #Hta cases (tech_split STape (λc.is_bar (\fst c)) rs1)
84 [ #H1 lapply (Hta rs1 〈grid,false〉 rs2 (refl ??) ? ?)
85 [ * #x #b #Hx whd in ⊢ (??%?); >(Hrs1' … Hx) >(H1 … Hx) %
87 | -Hta #Hta cases Hright
88 [ * #tb * whd in ⊢ (%→?); #Hcurrent
89 @False_ind cases (Hcurrent 〈grid,false〉 ?)
90 [ normalize in ⊢ (%→?); #Hfalse destruct (Hfalse)
92 | * #tb * whd in ⊢ (%→?); #Hcurrent
93 cases (Hcurrent 〈grid,false〉 ?)
94 [ #_ #Htb whd in ⊢ (%→?); #Houtc
97 | >Houtc >Htb >Hta % ]
101 | * #rs3 * #c0 * #rs4 * * #Hc0 #Hsplit #Hrs3
102 % @(ex_intro ?? rs3) @(ex_intro ?? rs4)
103 lapply (Hta rs3 c0 (rs4@〈grid,false〉::rs2) ???)
104 [ #x #Hrs3' whd in ⊢ (??%?); >Hsplit in Hrs1;>Hsplit in Hrs3;
105 #Hrs3 #Hrs1 >(Hrs1 …) [| @memb_append_l1 @Hrs3'|]
106 >(Hrs3 … Hrs3') @Hrs1' >Hsplit @memb_append_l1 //
107 | whd in ⊢ (??%?); >Hc0 %
108 | >Hsplit >associative_append % ] -Hta #Hta
110 [ * #tb * whd in ⊢ (%→?); #Hta'
113 [ #_ #Htb' >Htb' in Htb; #Htb
114 generalize in match Hsplit; -Hsplit
116 [ #Hta #Hsplit >(Htb … Hta)
117 >(?:c0 = 〈bar,false〉)
118 [ @(ex_intro ?? grid) @(ex_intro ?? false)
120 [(* Hsplit *) @daemon |(*Hrs3*) @daemon ] | % ] | % ]
121 | (* Hc0 *) @daemon ]
122 | #r5 #rs5 >(eq_pair_fst_snd … r5)
123 #Hta #Hsplit >(Htb … Hta)
124 >(?:c0 = 〈bar,false〉)
125 [ @(ex_intro ?? (\fst r5)) @(ex_intro ?? (\snd r5))
126 % [ % [ % [ (* Hc0, Hsplit *) @daemon | (*Hrs3*) @daemon ] | % ]
127 | % ] | (* Hc0 *) @daemon ] ] | >Hta % ]
128 | * #tb * whd in ⊢ (%→?); #Hta'
131 [ #Hfalse @False_ind >Hfalse in Hc0;
137 definition init_current_on_match ≝
138 move_l ? · adv_to_mark_l ? (λc:STape.is_grid (\fst c)) · move_r ? · mark ?.
140 definition R_init_current_on_match ≝ λt1,t2.
141 ∀l1,l2,c,rs. no_grids l1 → is_grid c = false →
142 t1 = midtape STape (l1@〈c,false〉::〈grid,false〉::l2) 〈grid,false〉 rs →
143 t2 = midtape STape (〈grid,false〉::l2) 〈c,true〉 ((reverse ? l1)@〈grid,false〉::rs).
145 lemma sem_init_current_on_match :
146 Realize ? init_current_on_match R_init_current_on_match.
148 cases (sem_seq ????? (sem_move_l ?)
149 (sem_seq ????? (sem_adv_to_mark_l ? (λc:STape.is_grid (\fst c)))
150 (sem_seq ????? (sem_move_r ?) (sem_mark ?))) intape)
151 #k * #outc * #Hloop #HR
152 @(ex_intro ?? k) @(ex_intro ?? outc) % [@Hloop] -Hloop
153 #l1 #l2 #c #rs #Hl1 #Hc #Hintape
154 cases HR -HR #ta * whd in ⊢ (%→?); #Hta lapply (Hta … Hintape) -Hta -Hintape
155 generalize in match Hl1; cases l1
156 [#Hl1 whd in ⊢ ((???(??%%%))→?); #Hta
157 * #tb * whd in ⊢ (%→?); #Htb cases (Htb … Hta) -Hta
158 [* >Hc #Htemp destruct (Htemp) ]
159 * #_ #Htc lapply (Htc [ ] 〈grid,false〉 ? (refl ??) (refl …) Hl1)
160 whd in ⊢ ((???(??%%%))→?); -Htc #Htc
161 * #td * whd in ⊢ (%→?); #Htd lapply (Htd … Htc) -Htc -Htd
162 whd in ⊢ ((???(??%%%))→?); #Htd
163 whd in ⊢ (%→?); #Houtc lapply (Houtc … Htd) -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 * #Hd >append_cons #Htb lapply (Htb … (refl ??) (refl …) ?)
169 [#x #membx cases (memb_append … membx) -membx #membx
170 [@Htl @memb_cons @membx | >(memb_single … membx) @Hc]]-Htb #Htb
171 * #tc * whd in ⊢ (%→?); #Htc lapply (Htc … Htb) -Htb -Htc
172 >reverse_append >associative_append whd in ⊢ ((???(??%%%))→?); #Htc
173 whd in ⊢ (%→?); #Houtc lapply (Houtc … Htc) -Houtc #Houtc
174 >Houtc >reverse_cons >associative_append %
179 definition init_current_gen ≝
180 seq ? (adv_to_mark_l ? (is_marked ?))
181 (seq ? (clear_mark ?)
183 (seq ? (adv_to_mark_l ? (λc:STape.is_grid (\fst c)))
184 (seq ? (move_r ?) (mark ?))))).
186 definition R_init_current_gen ≝ λt1,t2.
187 ∀l1,c,l2,b,l3,c1,rs,c0,b0. no_marks l1 → no_grids l2 →
188 Some ? 〈c0,b0〉 = option_hd ? (reverse ? (〈c,true〉::l2)) →
189 t1 = midtape STape (l1@〈c,true〉::l2@〈grid,b〉::l3) 〈c1,false〉 rs →
190 t2 = midtape STape (〈grid,b〉::l3) 〈c0,true〉
191 ((tail ? (reverse ? (l1@〈c,false〉::l2))@〈c1,false〉::rs)).
193 lemma sem_init_current_gen : Realize ? init_current_gen R_init_current_gen.
195 cases (sem_seq ????? (sem_adv_to_mark_l ? (is_marked ?))
196 (sem_seq ????? (sem_clear_mark ?)
197 (sem_seq ????? (sem_move_l ?)
198 (sem_seq ????? (sem_adv_to_mark_l ? (λc:STape.is_grid (\fst c)))
199 (sem_seq ????? (sem_move_r ?) (sem_mark ?))))) intape)
200 #k * #outc * #Hloop #HR
201 @(ex_intro ?? k) @(ex_intro ?? outc) % [@Hloop] -Hloop
202 #l1 #c #l2 #b #l3 #c1 #rs #c0 #b0 #Hl1 #Hl2 #Hc #Hintape
203 cases HR -HR #ta * whd in ⊢ (%→?); #Hta cases (Hta … Hintape) -Hta -Hintape
204 [ * #Hfalse normalize in Hfalse; destruct (Hfalse) ]
205 * #_ #Hta lapply (Hta l1 〈c,true〉 ? (refl ??) ??) [@Hl1|%] -Hta #Hta
206 * #tb * whd in ⊢ (%→?); #Htb lapply (Htb … Hta) -Htb -Hta #Htb
207 * #tc * whd in ⊢ (%→?); #Htc lapply (Htc … Htb) -Htc -Htb
208 generalize in match Hc; generalize in match Hl2; cases l2
209 [#_ whd in ⊢ ((???%)→?); #Htemp destruct (Htemp)
210 whd in ⊢ ((???(??%%%))→?); #Htc
211 * #td * whd in ⊢ (%→?); #Htd cases (Htd … Htc) -Htd
212 [2: * whd in ⊢ (??%?→?); #Htemp destruct (Htemp) ]
213 * #_ #Htd >Htd in Htc; -Htd #Htd
214 * #te * whd in ⊢ (%→?); #Hte lapply (Hte … Htd) -Htd
215 >reverse_append >reverse_cons
216 whd in ⊢ ((???(??%%%))→?); #Hte
217 whd in ⊢ (%→?); #Houtc lapply (Houtc … Hte) -Houtc -Hte #Houtc
219 |#d #tl #Htl #Hc0 whd in ⊢ ((???(??%%%))→?); #Htc
220 * #td * whd in ⊢ (%→?); #Htd cases (Htd … Htc) -Htd
221 [* >(Htl … (memb_hd …)) whd in ⊢ (??%?→?); #Htemp destruct (Htemp)]
222 * #Hd #Htd lapply (Htd … (refl ??) (refl ??) ?)
223 [#x #membx @Htl @memb_cons @membx] -Htd #Htd
224 * #te * whd in ⊢ (%→?); #Hte lapply (Hte … Htd) -Htd
225 >reverse_append >reverse_cons >reverse_cons
226 >reverse_cons in Hc0; >reverse_cons cases (reverse ? tl)
227 [normalize in ⊢ (%→?); #Hc0 destruct (Hc0) #Hte
228 whd in ⊢ (%→?); #Houtc lapply (Houtc … Hte) -Houtc -Hte #Houtc
230 |* #c2 #b2 #tl2 normalize in ⊢ (%→?); #Hc0 destruct (Hc0)
231 whd in ⊢ ((???(??%%%))→?); #Hte
232 whd in ⊢ (%→?); #Houtc lapply (Houtc … Hte) -Houtc -Hte #Houtc
233 >Houtc >associative_append >associative_append >associative_append %
239 definition init_current ≝
240 adv_to_mark_l ? (is_marked ?) ·clear_mark ? ·
241 adv_to_mark_l ? (λc:STape.is_grid (\fst c)) · move_r ? · mark ?.
243 definition R_init_current ≝ λt1,t2.
244 ∀l1,c,l2,b,l3,c1,rs,c0,b0. no_marks l1 → no_grids l2 → is_grid c = false →
245 Some ? 〈c0,b0〉 = option_hd ? (reverse ? (〈c,true〉::l2)) →
246 t1 = midtape STape (l1@〈c,true〉::l2@〈grid,b〉::l3) 〈c1,false〉 rs →
247 t2 = midtape STape (〈grid,b〉::l3) 〈c0,true〉
248 ((tail ? (reverse ? (l1@〈c,false〉::l2))@〈c1,false〉::rs)).
250 lemma sem_init_current : Realize ? init_current R_init_current.
252 cases (sem_seq ????? (sem_adv_to_mark_l ? (is_marked ?))
253 (sem_seq ????? (sem_clear_mark ?)
254 (sem_seq ????? (sem_adv_to_mark_l ? (λc:STape.is_grid (\fst c)))
255 (sem_seq ????? (sem_move_r ?) (sem_mark ?)))) intape)
256 #k * #outc * #Hloop #HR
257 @(ex_intro ?? k) @(ex_intro ?? outc) % [@Hloop]
258 cases HR -HR #ta * whd in ⊢ (%→?); #Hta
259 * #tb * whd in ⊢ (%→?); #Htb
260 * #tc * whd in ⊢ (%→?); #Htc
261 * #td * whd in ⊢ (%→%→?); #Htd #Houtc
262 #l1 #c #l2 #b #l3 #c1 #rs #c0 #b0 #Hl1 #Hl2 #Hc #Hc0 #Hintape
263 cases (Hta … Hintape) [ * #Hfalse normalize in Hfalse; destruct (Hfalse) ]
264 -Hta * #_ #Hta lapply (Hta l1 〈c,true〉 ? (refl ??) ??) [@Hl1|%]
265 -Hta #Hta lapply (Htb … Hta) -Htb #Htb cases (Htc … Htb) [ >Hc -Hc * #Hc destruct (Hc) ]
266 -Htc * #_ #Htc lapply (Htc … (refl ??) (refl ??) ?) [@Hl2]
267 -Htc #Htc lapply (Htd … Htc) -Htd
268 >reverse_append >reverse_cons
269 >reverse_cons in Hc0; cases (reverse … l2)
270 [ normalize in ⊢ (%→?); #Hc0 destruct (Hc0)
271 #Htd >(Houtc … Htd) %
272 | * #c2 #b2 #tl2 normalize in ⊢ (%→?);
273 #Hc0 #Htd >(Houtc … Htd)
274 whd in ⊢ (???%); destruct (Hc0)
275 >associative_append >associative_append %
279 definition match_tuple_step ≝
280 ifTM ? (test_char ? (λc:STape.¬ is_grid (\fst c)))
283 (ifTM ? (test_char ? (λc:STape.is_grid (\fst c)))
286 (ifTM ? (test_char ? (λc:STape.is_grid (\fst c)))
287 (mark ?) (move_l ? · init_current) tc_true)) tc_true)))
290 definition R_match_tuple_step_true ≝ λt1,t2.
291 ∀ls,cur,rs.t1 = midtape STape ls cur rs →
293 (∀ls0,c,l1,l2,c1,l3,l4,rs0,n.
294 only_bits_or_nulls l1 → no_marks l1 (* → no_grids l2 *) →
295 bit_or_null c = true → bit_or_null c1 = true →
296 only_bits_or_nulls l3 → S n = |l1| → |l1| = |l3| →
297 table_TM (S n) (l2@〈c1,false〉::l3@〈comma,false〉::l4) →
298 ls = 〈grid,false〉::ls0 → cur = 〈c,true〉 →
299 rs = l1@〈grid,false〉::l2@〈c1,true〉::l3@〈comma,false〉::l4@〈grid,false〉::rs0 →
301 (〈c,false〉::l1 = 〈c1,false〉::l3 ∧
302 t2 = midtape ? (reverse ? l1@〈c,false〉::〈grid,false〉::ls0) 〈grid,false〉
303 (l2@〈c1,false〉::l3@〈comma,true〉::l4@〈grid,false〉::rs0))
305 (* non facciamo match e marchiamo la prossima tupla *)
306 (〈c,false〉::l1 ≠ 〈c1,false〉::l3 ∧
307 ∃c2,l5,l6.l4 = l5@〈bar,false〉::〈c2,false〉::l6 ∧
308 (* condizioni su l5 l6 l7 *)
309 t2 = midtape STape (〈grid,false〉::ls0) 〈c,true〉
310 (l1@〈grid,false〉::l2@〈c1,false〉::l3@〈comma,false〉::
311 l5@〈bar,false〉::〈c2,true〉::l6@〈grid,false〉::rs0))
313 (* non facciamo match e non c'è una prossima tupla:
314 non specifichiamo condizioni sul nastro di output, perché
315 non eseguiremo altre operazioni, quindi il suo formato non ci interessa *)
316 (〈c,false〉::l1 ≠ 〈c1,false〉::l3 ∧ no_bars l4 ∧ current ? t2 = Some ? 〈grid,true〉)).
318 definition R_match_tuple_step_false ≝ λt1,t2.
319 ∀ls,c,rs.t1 = midtape STape ls c rs → is_grid (\fst c) = true ∧ t2 = t1.
321 include alias "basics/logic.ma".
324 lemma eq_f4: ∀A1,A2,A3,A4,B.∀f:A1 → A2 →A3 →A4 →B.
325 ∀x1,x2,x3,x4,y1,y2,y3,y4. x1 = y1 → x2 = y2 →x3=y3 →x4 = y4 →
326 f x1 x2 x3 x4 = f y1 y2 y3 y4.
330 lemma some_option_hd: ∀A.∀l:list A.∀a.∃b.
331 Some ? b = option_hd ? (l@[a]) .
332 #A #l #a cases l normalize /2/
335 axiom tech_split2 : ∀A,l1,l2,l3,l4,x.
336 memb A x l1 = false → memb ? x l3 = false →
337 l1@x::l2 = l3@x::l4 → l1 = l3 ∧ l2 = l4.
339 axiom injective_append : ∀A,l.injective … (λx.append A x l).
341 lemma sem_match_tuple_step:
342 accRealize ? match_tuple_step (inr … (inl … (inr … start_nop)))
343 R_match_tuple_step_true R_match_tuple_step_false.
344 @(acc_sem_if_app … (sem_test_char ? (λc:STape.¬ is_grid (\fst c))) …
345 (sem_seq … sem_compare
346 (sem_if … (sem_test_char ? (λc:STape.is_grid (\fst c)))
348 (sem_seq … sem_mark_next_tuple
349 (sem_if … (sem_test_char ? (λc:STape.is_grid (\fst c)))
350 (sem_mark ?) (sem_seq … (sem_move_l …) (sem_init_current …))))))
352 [(* is_grid: termination case *)
353 2:#t1 #t2 #t3 whd in ⊢ (%→?); #H #H1 whd #ls #c #rs #Ht1
354 cases (H c ?) [2: >Ht1 %] #Hgrid #Heq %
355 [@injective_notb @Hgrid | <Heq @H1]
356 |#tapea #tapeout #tapeb whd in ⊢ (%→?); #Hcur
357 * #tapec * whd in ⊢ (%→?); #Hcompare #Hor
358 #ls #cur #rs #Htapea >Htapea in Hcur; #Hcur cases (Hcur ? (refl ??))
359 -Hcur #Hcur #Htapeb %
360 [ % #Hfalse >Hfalse in Hcur; normalize #Hfalse1 destruct (Hfalse1)]
361 #ls0 #c #l1 #l2 #c1 #l3 #l4 #rs0 #n #Hl1bitnull #Hl1marks #Hc #Hc1 #Hl3 #eqn
362 #eqlen #Htable #Hls #Hcur #Hrs -Htapea >Hls in Htapeb; >Hcur >Hrs #Htapeb
363 cases (Hcompare … Htapeb) -Hcompare -Htapeb * #_ #_ #Hcompare
364 cases (Hcompare c c1 l1 l3 l2 (l4@〈grid,false〉::rs0) eqlen Hl1bitnull Hl3 Hl1marks … (refl …) Hc ?)
366 [* #Htemp destruct (Htemp) #Htapec %1 % % [%]
367 >Htapec in Hor; -Htapec *
368 [2: * #t3 * whd in ⊢ (%→?); #H @False_ind
369 cases (H … (refl …)) whd in ⊢ ((??%?)→?); #H destruct (H)
370 |* #taped * whd in ⊢ (%→?); #Htaped cases (Htaped ? (refl …)) -Htaped *
371 #Htaped whd in ⊢ (%→?); #Htapeout >Htapeout >Htaped
374 |* #la * #c' * #d' * #lb * #lc * * * #H1 #H2 #H3 #Htapec
375 cut (〈c,false〉::l1 ≠ 〈c1,false〉::l3)
377 [@(not_to_not …H1) normalize #H destruct %
378 |#x #tl @not_to_not normalize #H destruct //
381 cut (bit_or_null d' = true)
383 [normalize in ⊢ (%→?); #H destruct //
384 |#x #tl #H @(Hl3 〈d',false〉)
385 normalize in H; destruct @memb_append_l2 @memb_hd
388 >Htapec in Hor; -Htapec *
389 [* #taped * whd in ⊢ (%→?); #H @False_ind
390 cases (H … (refl …)) >(bit_or_null_not_grid ? Hd') #Htemp destruct (Htemp)
391 |* #taped * whd in ⊢ (%→?); #H cases (H … (refl …)) -H #_
392 #Htaped * #tapee * whd in ⊢ (%→?); #Htapee
393 <(associative_append ? lc (〈comma,false〉::l4)) in Htaped; #Htaped
394 cases (Htapee … Htaped ???) -Htaped -Htapee
395 [* #rs3 * * (* we proceed by cases on rs4 *)
396 [(* rs4 is empty : the case is absurd since the tape
397 cannot end with a bar *)
398 * #d * #b * * * #Heq1 @False_ind
399 cut (∀A,l1,l2.∀a:A. a::l1@l2=(a::l1)@l2) [//] #Hcut
400 >Hcut in Htable; >H3 >associative_append
401 normalize >Heq1 <associative_append >Hcut
402 <associative_append #Htable @(absurd … Htable)
405 * #d2 #b2 #rs3' * #d * #b * * * #Heq1 #Hnobars
406 cut (memb STape 〈d2,b2〉 (l2@〈c1,false〉::l3@〈comma,false〉::l4) = true)
408 cut (∀A,l1,l2.∀a:A. a::l1@l2=(a::l1)@l2) [//] #Hcut
409 >Hcut >H3 >associative_append @memb_append_l2
410 @memb_cons >Heq1 @memb_append_l2 @memb_cons @memb_hd] #d2intable
411 cut (is_grid d2 = false)
412 [@(no_grids_in_table … Htable … 〈d2,b2〉 d2intable)] #Hd2
414 [@(no_marks_in_table … Htable … 〈d2,b2〉 d2intable)] #Hb2
415 >Hb2 in Heq1; #Heq1 -Hb2 -b2
416 whd in ⊢ ((???%)→?); #Htemp destruct (Htemp) #Htapee >Htapee -Htapee *
417 [(* we know current is not grid *)
418 * #tapef * whd in ⊢ (%→?); #Htapef
419 cases (Htapef … (refl …)) >Hd2 #Htemp destruct (Htemp)
420 |* #tapef * whd in ⊢ (%→?); #Htapef
421 cases (Htapef … (refl …)) #_ -Htapef #Htapef
422 * #tapeg >Htapef -Htapef *
425 #H lapply (H … (refl …)) whd in ⊢ (???%→?); -H #Htapeg
428 whd in ⊢ (%→?); #Htapeout
429 cases (some_option_hd ? (reverse ? (reverse ? la)) 〈c',true〉)
432 (Htapeout (reverse ? rs3 @〈d',false〉::reverse ? la@reverse ? l2@(〈grid,false〉::reverse ? lb))
433 c' (reverse ? la) false ls0 bar (〈d2,true〉::rs3'@〈grid,false〉::rs0) c00 b00 ?????) -Htapeout
434 [whd in ⊢ (??(??%??)?); @eq_f3 [2:%|3: %]
436 generalize in match (〈c',true〉::reverse ? la@〈grid,false〉::ls0); #l
437 whd in ⊢ (???(???%)); >associative_append >associative_append %
438 |>reverse_cons @Hoption
440 [normalize in ⊢ (%→?); #Htemp destruct (Htemp)
441 @bit_or_null_not_grid @Hc
442 |#x #tl normalize in ⊢ (%→?); #Htemp destruct (Htemp)
443 @bit_or_null_not_grid @(Hl1bitnull 〈c',false〉) @memb_append_l2 @memb_hd
445 |cut (only_bits_or_nulls (la@(〈c',false〉::lb)))
446 [<H2 whd #c0 #Hmemb cases (orb_true_l … Hmemb)
447 [#eqc0 >(\P eqc0) @Hc |@Hl1bitnull]
448 |#Hl1' #x #Hx @bit_or_null_not_grid @Hl1'
449 @memb_append_l1 @daemon
451 |@daemon] #Htapeout % %2 % //
453 cut (∃rs32.rs3 = lc@〈comma,false〉::rs32)
454 [ (*cases (tech_split STape (λc.c == 〈bar,false〉) l4)
456 | * #l41 * * #cbar #bfalse * #l42 * * #Hbar #Hl4 #Hl41
457 @(ex_intro ?? l41) >Hl4 in Heq1; #Heq1
459 cut (sublist … lc l3)
460 [ #x #Hx cases la in H3;
461 [ normalize #H3 destruct (H3) @Hx
462 | #p #la' normalize #Hla' destruct (Hla')
463 @memb_append_l2 @memb_cons @Hx ] ] #Hsublist*)
467 (〈c1,false〉::l3@〈comma,false〉::l4= la@〈d',false〉::rs3@〈bar,false〉::〈d2,b2〉::rs3')
469 cut (l4=rs32@〈bar,false〉::〈d2,false〉::rs3')
470 [ >Hrs3 in Heq1; @daemon ] #Hl4
471 @(ex_intro … rs32) @(ex_intro … rs3') % [@Hl4]
473 [(* by Hoption, H2 *) @daemon
474 |(*>Hrs3 *)>append_cons
475 > (?:l1@〈grid,false〉::l2@〈c1,false〉::l3@〈comma,false〉::rs32@〈bar,false〉::〈d2,true〉::rs3'@〈grid,false〉::rs
476 = (l1@〈grid,false〉::l2@〈c1,false〉::l3@〈comma,false〉::rs32@[〈bar,false〉])@〈d2,true〉::rs3'@〈grid,false〉::rs)
477 [|>associative_append normalize
478 >associative_append normalize
479 >associative_append normalize
480 >associative_append normalize
482 >reverse_append >reverse_append >reverse_cons
483 >reverse_reverse >reverse_cons >reverse_reverse
484 >reverse_append >reverse_append >reverse_cons
485 >reverse_reverse >reverse_reverse >reverse_reverse
486 >(?:(la@[〈c',false〉])@((((lb@[〈grid,false〉])@l2)@la)@[〈d',false〉])@rs3
487 =((la@〈c',false〉::lb)@([〈grid,false〉]@l2@la@[〈d',false〉]@rs3)))
488 [|>associative_append >associative_append
489 >associative_append >associative_append >associative_append
490 >associative_append % ]
491 <H2 normalize in ⊢ (??%?); >Hrs3
492 >associative_append >associative_append normalize
493 >associative_append >associative_append
495 >(?:la@(〈d',false〉::lc@〈comma,false〉::rs32)@〈bar,false〉::〈d2,true〉::rs3'@〈grid,false〉::rs0 =
496 (la@〈d',false〉::lc)@〈comma,false〉::rs32@〈bar,false〉::〈d2,true〉::rs3'@〈grid,false〉::rs0 )
497 [| >associative_append normalize >associative_append % ]
502 |* #Hnobars #Htapee >Htapee -Htapee *
503 [whd in ⊢ (%→?); * #tapef * whd in ⊢ (%→?); #Htapef
504 cases (Htapef … (refl …)) -Htapef #_ #Htapef >Htapef -Htapef
505 whd in ⊢ (%→?); #Htapeout %2 %
506 [% [//] whd #x #Hx @Hnobars @memb_append_l2 @memb_cons //
507 | >(Htapeout … (refl …)) % ]
508 |whd in ⊢ (%→?); * #tapef * whd in ⊢ (%→?); #Htapef
509 cases (Htapef … (refl …)) -Htapef
510 whd in ⊢ ((??%?)→?); #Htemp destruct (Htemp)
512 |(* no marks in table *)
513 #x #membx @(no_marks_in_table … Htable)
515 cut (∀A,l1,l2.∀a:A. a::l1@l2=(a::l1)@l2) [//] #Hcut >Hcut
516 >H3 >associative_append @memb_append_l2 @memb_cons @membx
517 |(* no grids in table *)
518 #x #membx @(no_grids_in_table … Htable)
520 cut (∀A,l1,l2.∀a:A. a::l1@l2=(a::l1)@l2) [//] #Hcut >Hcut
521 >H3 >associative_append @memb_append_l2 @memb_cons @membx
522 |whd in ⊢ (??%?); >(bit_or_null_not_grid … Hd') >(bit_or_null_not_bar … Hd') %
525 |#x #membx @(no_marks_in_table … Htable)
526 @memb_append_l2 @memb_cons @memb_append_l1 @membx
527 |#x #membx @(no_marks_in_table … Htable)
528 @memb_append_l1 @membx
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 … start_nop))).
543 lemma is_grid_true : ∀c.is_grid c = true → c = grid.
544 * normalize [ #b ] #H // destruct (H)
547 (* possible variante ?
548 definition weakR_match_tuple ≝ λt1,t2.
549 (∀ls,cur,rs,b. t1 = midtape STape ls 〈grid,b〉 rs → t2 = t1) ∧
550 (∀c,l1,c1,l2,l3,ls0,rs0,n.
551 t1 = midtape STape (〈grid,false〉::ls0) 〈bit c,true〉 rs
552 (l1@〈grid,false〉::l2@〈bit c1,true〉::l3@〈grid,false〉::rs0) →
553 only_bits_or_nulls l1 → no_marks l1 → S n = |l1| →
554 table_TM (S n) (l2@〈c1,false〉::l3) →
557 〈c1,false〉::l3 = l4@〈c,false〉::l1@〈comma,false〉::newc@〈comma,false〉::mv::l5 ∧
558 t2 = midtape ? (reverse ? l1@〈c,false〉::〈grid,false〉::ls0) 〈grid,false〉
559 (l2@l4@〈c,false〉::l1@〈comma,true〉::newc@〈comma,false〉::mv::l5@
562 (* non facciamo match su nessuna tupla;
563 non specifichiamo condizioni sul nastro di output, perché
564 non eseguiremo altre operazioni, quindi il suo formato non ci interessa *)
565 (current ? t2 = Some ? 〈grid,true〉 ∧
567 〈c1,false〉::l3 ≠ l4@〈c,false〉::l1@〈comma,false〉::newc@〈comma,false〉::mv::l5)).
570 definition R_match_tuple0 ≝ λt1,t2.
572 t1 = midtape STape ls cur rs →
573 (is_grid (\fst cur) = true → t2 = t1) ∧
574 (∀c,l1,c1,l2,l3,ls0,rs0,n.
575 ls = 〈grid,false〉::ls0 →
577 rs = l1@〈grid,false〉::l2@〈bar,false〉::〈c1,true〉::l3@〈grid,false〉::rs0 →
578 is_bit c = true → is_bit c1 = true →
579 only_bits_or_nulls l1 → no_marks l1 → S n = |l1| →
580 table_TM (S n) (l2@〈bar,false〉::〈c1,false〉::l3) →
583 〈bar,false〉::〈c1,false〉::l3 = l4@〈bar,false〉::〈c,false〉::l1@〈comma,false〉::newc@〈comma,false〉::mv::l5 ∧
584 t2 = midtape ? (reverse ? l1@〈c,false〉::〈grid,false〉::ls0) 〈grid,false〉
585 (l2@l4@〈bar,false〉::〈c,false〉::l1@〈comma,true〉::newc@〈comma,false〉::mv::l5@
588 (* non facciamo match su nessuna tupla;
589 non specifichiamo condizioni sul nastro di output, perché
590 non eseguiremo altre operazioni, quindi il suo formato non ci interessa *)
591 (current ? t2 = Some ? 〈grid,true〉 ∧
593 〈bar,false〉::〈c1,false〉::l3 ≠ l4@〈bar,false〉::〈c,false〉::l1@〈comma,false〉::newc@〈comma,false〉::mv::l5)).
595 axiom table_bit_after_bar :
596 ∀n,l1,c,l2.table_TM n (l1@〈bar,false〉::〈c,false〉::l2) → is_bit c = true.
598 lemma wsem_match_tuple : WRealize ? match_tuple R_match_tuple0.
599 #intape #k #outc #Hloop
600 lapply (sem_while … sem_match_tuple_step intape k outc Hloop) [%] -Hloop
601 * #ta * #Hstar @(star_ind_l ??????? Hstar)
602 [ #tb whd in ⊢ (%→?); #Hleft
603 #ls #cur #rs #Htb cases (Hleft … Htb) #Hgrid #Houtc %
605 | #c #l1 #c1 #l2 #l3 #ls0 #rs0 #n #Hls #Hcur #Hrs
606 >Hcur in Hgrid; #Hgrid >(is_grid_true … Hgrid) normalize in ⊢ (%→?);
609 | (* in the interesting case, we execute a true iteration, then we restart the
610 while cycle, finally we end with a false iteration *)
611 #tb #tc #td whd in ⊢ (%→?); #Htc
612 #Hstar1 #IH whd in ⊢ (%→?); #Hright lapply (IH Hright) -IH whd in ⊢ (%→?); #IH
614 [ (* cur can't be true because we assume at least one iteration *)
615 #Hcur cases (Htc … Htb) * #Hfalse @False_ind @Hfalse @(is_grid_true … Hcur)
616 | (* current and a tuple are marked *)
617 #c #l1 #c1 #l2 #l3 #ls0 #rs0 #n #Hls #Hcur #Hrs #Hc #Hc1 #Hl1bitnull #Hl1marks
618 #Hl1len #Htable cases (Htc … Htb) -Htc -Htb * #_ #Htc
619 (* expose the marked tuple in table *)
620 cut (∃la,lb,mv,lc.l3 = la@〈comma,false〉::lb@〈comma,false〉::mv::lc ∧
621 S n = |la| ∧ only_bits_or_nulls la)
622 [@daemon] * #la * #lb * #mv * #lc * * #Hl3 #Hlalen #Hlabitnull
623 >Hl3 in Htable; >append_cons #Htable
624 >(?: l2@〈bar,false〉::〈c1,true〉::l3@〈grid,false〉::rs0
625 = (l2@[〈bar,false〉])@〈c1,true〉::la@〈comma,false〉::(lb@〈comma,false〉::mv::
626 lc)@〈grid,false〉::rs0) in Hrs;
627 [| >associative_append normalize >Hl3
628 >associative_append normalize % ] #Hrs
629 cases (Htc ????????? Hl1bitnull Hl1marks ?? Hlabitnull Hl1len ? Htable Hls Hcur Hrs)
631 |4: whd in ⊢ (??%?); >Hc1 %
632 |3: whd in ⊢ (??%?); >Hc %
634 [ (* case 1: match successful *)
635 * #Heq #Htc % %{[]} %{lb} %{mv} %{lc} destruct (Heq) %
637 | cases (IH … Htc) -IH #Houtc #_ >(Houtc (refl ??))
638 >Htc @eq_f normalize >associative_append normalize
639 >associative_append normalize %
641 | (* case 2: tuples don't match, we still have other tuples to try *)
642 * #Hdiff * #c2 * #l5 * #l6 * #Heqlblc #Htc
643 cases (IH ??? … Htc) -IH #_ #IH
644 (* by induction hypothesis *)
645 lapply (IH ? l1 c2 (l2@〈bar,false〉::〈c1,false〉::la@〈comma,false〉::l5) l6 ? rs0 n (refl ??) (refl ??) ???????)
646 [ generalize in match Htable;
647 >associative_append normalize
648 >associative_append normalize >Heqlblc
649 >associative_append normalize //
655 | >associative_append normalize
656 >associative_append normalize
657 >associative_append %
659 [ (* the while finally matches a tuple *)
660 * #l7 * #newc * #mv0 * #l8 * #Hl7l8 #Houtc %
661 >Heqlblc @(ex_intro ?? (〈bar,false〉::〈c1,false〉::la@〈comma,false〉::l5@l7))
662 %{newc} %{mv0} %{l8} %
663 [ normalize >Hl7l8 >associative_append normalize
664 >associative_append %
665 | >Houtc @eq_f >associative_append normalize
666 >associative_append normalize >associative_append
667 normalize >associative_append %
669 | (* the while fails finding a tuple: there are no matches in the whole table *)
670 * #Houtc #Hdiff1 %2 %
672 | #l50 #newc #mv0 #l51 >Heqlblc
678 | (* match failed and there is no next tuple: the next while cycle will just exit *)
679 * * #Hdiff #Hnobars generalize in match (refl ? tc);
680 cases tc in ⊢ (???% → %);
681 [ #_ normalize in ⊢ (??%?→?); #Hfalse destruct (Hfalse)
682 |2,3: #x #xs #_ normalize in ⊢ (??%?→?); #Hfalse destruct (Hfalse) ]
683 #ls1 #cur1 #rs1 #Htc normalize in ⊢ (??%?→?); #Hcur1
684 cases (IH … Htc) -IH #IH #_ %2 %
685 [ destruct (Hcur1) >IH [ >Htc % | % ]
687 (* no_bars except the first one, where the tuple does not match ⇒
695 definition R_match_tuple ≝ λt1,t2.
697 is_bit c = true → is_bit c1 = true →
698 only_bits_or_nulls l1 → no_marks l1 → S n = |l1| →
699 table_TM (S n) (〈bar,false〉::〈c1,false〉::l2) →
700 t1 = midtape STape (〈grid,false〉::ls) 〈c,true〉
701 (l1@〈grid,false〉::〈bar,false〉::〈c1,true〉::l2@〈grid,false〉::rs) →
704 〈bar,false〉::〈c1,false〉::l2 = l3@〈bar,false〉::〈c,false〉::l1@〈comma,false〉::newc@〈comma,false〉::mv::l4 ∧
705 t2 = midtape ? (reverse ? l1@〈c,false〉::〈grid,false〉::ls) 〈grid,false〉
706 (l3@〈bar,false〉::〈c,false〉::l1@〈comma,true〉::newc@〈comma,false〉::mv::l4@〈grid,false〉::rs))
708 (* non facciamo match su nessuna tupla;
709 non specifichiamo condizioni sul nastro di output, perché
710 non eseguiremo altre operazioni, quindi il suo formato non ci interessa *)
711 (current ? t2 = Some ? 〈grid,true〉 ∧
713 〈bar,false〉::〈c1,false〉::l2 ≠ l3@〈bar,false〉::〈c,false〉::l1@〈comma,false〉::newc@〈comma,false〉::mv::l4).
715 (* we still haven't proved termination *)
716 axiom sem_match_tuple0 : Realize ? match_tuple R_match_tuple0.
718 lemma sem_match_tuple : Realize ? match_tuple R_match_tuple.
719 generalize in match sem_match_tuple0; @Realize_to_Realize
720 #t1 #t2 #HR #ls #c #l1 #c1 #l2 #rs #n #Hc #Hc1 #Hl1bitsnulls #Hl1marks #Hl1len #Htable #Ht1
721 cases (HR … Ht1) -HR #_ #HR
722 @(HR ??? [] … (refl ??) (refl ??) (refl ??) Hc Hc1 Hl1bitsnulls Hl1marks