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_____________________________________________________________*)
12 include "basics/vectors.ma".
13 (* include "basics/relations.ma". *)
16 record tape (sig:FinSet): Type[0] ≝
17 { left : list (option sig);
18 right: list (option sig)
22 inductive tape (sig:FinSet) : Type[0] ≝
24 | leftof : sig → list sig → tape sig
25 | rightof : sig → list sig → tape sig
26 | midtape : list sig → sig → list sig → tape sig.
29 λsig.λt:tape sig.match t with
33 | midtape l _ _ ⇒ l ].
36 λsig.λt:tape sig.match t with
40 | midtape _ _ r ⇒ r ].
43 λsig.λt:tape sig.match t with
44 [ midtape _ c _ ⇒ Some ? c
48 ∀sig:FinSet.list sig → option sig → list sig → tape sig ≝
49 λsig,lt,c,rt.match c with
50 [ Some c' ⇒ midtape sig lt c' rt
51 | None ⇒ match lt with
54 | cons r0 rs0 ⇒ leftof ? r0 rs0 ]
55 | cons l0 ls0 ⇒ rightof ? l0 ls0 ] ].
57 inductive move : Type[0] ≝
63 (* We do not distinuish an input tape *)
65 record TM (sig:FinSet): Type[1] ≝
67 trans : states × (option sig) → states × (option (sig × move));
72 record config (sig,states:FinSet): Type[0] ≝
77 (* definition option_hd ≝ λA.λl:list (option A).
84 (*definition tape_write ≝ λsig.λt:tape sig.λs:sig.
85 <left ? t) s (right ? t).
87 | Some s' ⇒ midtape ? (left ? t) s' (right ? t) ].*)
89 definition tape_move_left ≝ λsig:FinSet.λlt:list sig.λc:sig.λrt:list sig.
91 [ nil ⇒ leftof sig c rt
92 | cons c0 lt0 ⇒ midtape sig lt0 c0 (c::rt) ].
94 definition tape_move_right ≝ λsig:FinSet.λlt:list sig.λc:sig.λrt:list sig.
96 [ nil ⇒ rightof sig c lt
97 | cons c0 rt0 ⇒ midtape sig (c::lt) c0 rt0 ].
99 definition tape_move ≝ λsig.λt: tape sig.λm:option (sig × move).
105 [ R ⇒ tape_move_right ? (left ? t) s (right ? t)
106 | L ⇒ tape_move_left ? (left ? t) s (right ? t)
107 | N ⇒ midtape ? (left ? t) s (right ? t)
112 (None,a::b::rs) → None::b::rs
113 (Some a,[]) → [Some a]
114 (Some a,b::rs) → Some a::rs
117 definition option_cons ≝ λA.λa:option A.λl.
119 [ None ⇒ match l with
124 (* definition tape_update := λsig.λt: tape sig.λs:option sig.
129 | Some a ⇒ [Some ? a] ]
130 | cons b rs ⇒ match s with
131 [ None ⇒ match rs with
133 | cons _ _ ⇒ None ?::rs ]
134 | Some a ⇒ Some ? a::rs ] ]
135 in mk_tape ? (left ? t) newright. *)
137 definition tape_move ≝ λsig.λt:tape sig.λm:option sig × move.
138 let 〈s,m1〉 ≝ m in match m1 with
139 [ R ⇒ mk_tape sig (option_cons ? s (left ? t)) (tail ? (right ? t))
140 | L ⇒ mk_tape sig (tail ? (left ? t))
141 (option_cons ? (option_hd ? (left ? t))
142 (option_cons ? s (tail ? (right ? t))))
143 | N ⇒ mk_tape sig (left ? t) (option_cons ? s (tail ? (right ? t)))
147 definition step ≝ λsig.λM:TM sig.λc:config sig (states sig M).
148 let current_char ≝ current ? (ctape ?? c) in
149 let 〈news,mv〉 ≝ trans sig M 〈cstate ?? c,current_char〉 in
150 mk_config ?? news (tape_move sig (ctape ?? c) mv).
152 let rec loop (A:Type[0]) n (f:A→A) p a on n ≝
155 | S m ⇒ if p a then (Some ? a) else loop A m f p (f a)
159 ∀A,n,f,p,a. p a = true →
160 loop A (S n) f p a = Some ? a.
161 #A #n #f #p #a #pa normalize >pa //
165 ∀A,n,f,p,a. p a = false →
166 loop A (S n) f p a = loop A n f p (f a).
167 normalize #A #n #f #p #a #Hpa >Hpa %
170 lemma loop_incr : ∀A,f,p,k1,k2,a1,a2.
171 loop A k1 f p a1 = Some ? a2 →
172 loop A (k2+k1) f p a1 = Some ? a2.
173 #A #f #p #k1 #k2 #a1 #a2 generalize in match a1; elim k1
174 [normalize #a0 #Hfalse destruct
175 |#k1' #IH #a0 <plus_n_Sm whd in ⊢ (??%? → ??%?);
176 cases (true_or_false (p a0)) #Hpa0 >Hpa0 whd in ⊢ (??%? → ??%?); // @IH
180 lemma loop_merge : ∀A,f,p,q.(∀b. p b = false → q b = false) →
182 loop A k1 f p a1 = Some ? a2 →
183 f a2 = a3 → q a2 = false →
184 loop A k2 f q a3 = Some ? a4 →
185 loop A (k1+k2) f q a1 = Some ? a4.
186 #Sig #f #p #q #Hpq #k1 elim k1
187 [normalize #k2 #a1 #a2 #a3 #a4 #H destruct
188 |#k1' #Hind #k2 #a1 #a2 #a3 #a4 normalize in ⊢ (%→?);
189 cases (true_or_false (p a1)) #pa1 >pa1 normalize in ⊢ (%→?);
190 [#eqa1a2 destruct #eqa2a3 #Hqa2 #H
191 whd in ⊢ (??(??%???)?); >plus_n_Sm @loop_incr
192 whd in ⊢ (??%?); >Hqa2 >eqa2a3 @H
193 |normalize >(Hpq … pa1) normalize
194 #H1 #H2 #H3 @(Hind … H2) //
199 lemma loop_split : ∀A,f,p,q.(∀b. q b = true → p b = true) →
201 loop A k f q a1 = Some ? a2 →
203 loop A k1 f p a1 = Some ? a3 ∧
204 loop A (S(k-k1)) f q a3 = Some ? a2.
205 #A #f #p #q #Hpq #k elim k
206 [#a1 #a2 normalize #Heq destruct
207 |#i #Hind #a1 #a2 normalize
208 cases (true_or_false (q a1)) #Hqa1 >Hqa1 normalize
210 @(ex_intro … 1) @(ex_intro … a2) %
211 [normalize >(Hpq …Hqa1) // |>Hqa1 //]
212 |#Hloop cases (true_or_false (p a1)) #Hpa1
213 [@(ex_intro … 1) @(ex_intro … a1) %
214 [normalize >Hpa1 // |>Hqa1 <Hloop normalize //]
215 |cases (Hind …Hloop) #k2 * #a3 * #Hloop1 #Hloop2
216 @(ex_intro … (S k2)) @(ex_intro … a3) %
217 [normalize >Hpa1 normalize // | @Hloop2 ]
224 lemma loop_split : ∀A,f,p,q.(∀b. p b = false → q b = false) →
226 loop A k1 f p a1 = Some ? a2 →
227 loop A k2 f q a2 = Some ? a3 →
228 loop A (k1+k2) f q a1 = Some ? a3.
229 #Sig #f #p #q #Hpq #k1 elim k1
230 [normalize #k2 #a1 #a2 #a3 #H destruct
231 |#k1' #Hind #k2 #a1 #a2 #a3 normalize in ⊢ (%→?→?);
232 cases (true_or_false (p a1)) #pa1 >pa1 normalize in ⊢ (%→?);
233 [#eqa1a2 destruct #H @loop_incr //
234 |normalize >(Hpq … pa1) normalize
235 #H1 #H2 @(Hind … H2) //
241 definition initc ≝ λsig.λM:TM sig.λt.
242 mk_config sig (states sig M) (start sig M) t.
244 definition Realize ≝ λsig.λM:TM sig.λR:relation (tape sig).
246 loop ? i (step sig M) (λc.halt sig M (cstate ?? c)) (initc sig M t) = Some ? outc ∧
249 definition WRealize ≝ λsig.λM:TM sig.λR:relation (tape sig).
251 loop ? i (step sig M) (λc.halt sig M (cstate ?? c)) (initc sig M t) = Some ? outc →
254 definition Terminate ≝ λsig.λM:TM sig.λt. ∃i,outc.
255 loop ? i (step sig M) (λc.halt sig M (cstate ?? c)) (initc sig M t) = Some ? outc.
257 lemma WRealize_to_Realize : ∀sig.∀M: TM sig.∀R.
258 (∀t.Terminate sig M t) → WRealize sig M R → Realize sig M R.
259 #sig #M #R #HT #HW #t cases (HT … t) #i * #outc #Hloop
260 @(ex_intro … i) @(ex_intro … outc) % // @(HW … i) //
263 lemma loop_eq : ∀sig,f,q,i,j,a,x,y.
264 loop sig i f q a = Some ? x → loop sig j f q a = Some ? y → x = y.
265 #sig #f #q #i #j @(nat_elim2 … i j)
266 [ #n #a #x #y normalize #Hfalse destruct (Hfalse)
267 | #n #a #x #y #H1 normalize #Hfalse destruct (Hfalse)
268 | #n1 #n2 #IH #a #x #y normalize cases (q a) normalize
274 theorem Realize_to_WRealize : ∀sig,M,R.Realize sig M R → WRealize sig M R.
275 #sig #M #R #H1 #inc #i #outc #Hloop
276 cases (H1 inc) #k * #outc1 * #Hloop1 #HR
277 >(loop_eq … Hloop Hloop1) //
280 definition accRealize ≝ λsig.λM:TM sig.λacc:states sig M.λRtrue,Rfalse:relation (tape sig).
282 loop ? i (step sig M) (λc.halt sig M (cstate ?? c)) (initc sig M t) = Some ? outc ∧
283 (cstate ?? outc = acc → Rtrue t (ctape ?? outc)) ∧
284 (cstate ?? outc ≠ acc → Rfalse t (ctape ?? outc)).
291 definition nop_states ≝ initN 1.
292 definition start_nop : initN 1 ≝ mk_Sig ?? 0 (le_n … (S 0)).
295 λalpha:FinSet.mk_TM alpha nop_states
296 (λp.let 〈q,a〉 ≝ p in 〈q,None ?〉)
299 definition R_nop ≝ λalpha.λt1,t2:tape alpha.t2 = t1.
302 ∀alpha.Realize alpha (nop alpha) (R_nop alpha).
303 #alpha #intape @(ex_intro ?? 1)
304 @(ex_intro … (mk_config ?? start_nop intape)) % %
309 definition seq_trans ≝ λsig. λM1,M2 : TM sig.
313 if halt sig M1 s1 then 〈inr … (start sig M2), None ?〉
315 let 〈news1,m〉 ≝ trans sig M1 〈s1,a〉 in
318 let 〈news2,m〉 ≝ trans sig M2 〈s2,a〉 in
322 definition seq ≝ λsig. λM1,M2 : TM sig.
324 (FinSum (states sig M1) (states sig M2))
325 (seq_trans sig M1 M2)
326 (inl … (start sig M1))
328 [ inl _ ⇒ false | inr s2 ⇒ halt sig M2 s2]).
330 definition Rcomp ≝ λA.λR1,R2:relation A.λa1,a2.
331 ∃am.R1 a1 am ∧ R2 am a2.
334 definition injectRl ≝ λsig.λM1.λM2.λR.
336 inl … (cstate sig M1 c11) = cstate sig (seq sig M1 M2) c1 ∧
337 inl … (cstate sig M1 c12) = cstate sig (seq sig M1 M2) c2 ∧
338 ctape sig M1 c11 = ctape sig (seq sig M1 M2) c1 ∧
339 ctape sig M1 c12 = ctape sig (seq sig M1 M2) c2 ∧
342 definition injectRr ≝ λsig.λM1.λM2.λR.
344 inr … (cstate sig M2 c21) = cstate sig (seq sig M1 M2) c1 ∧
345 inr … (cstate sig M2 c22) = cstate sig (seq sig M1 M2) c2 ∧
346 ctape sig M2 c21 = ctape sig (seq sig M1 M2) c1 ∧
347 ctape sig M2 c22 = ctape sig (seq sig M1 M2) c2 ∧
350 definition Rlink ≝ λsig.λM1,M2.λc1,c2.
351 ctape sig (seq sig M1 M2) c1 = ctape sig (seq sig M1 M2) c2 ∧
352 cstate sig (seq sig M1 M2) c1 = inl … (halt sig M1) ∧
353 cstate sig (seq sig M1 M2) c2 = inr … (start sig M2). *)
355 interpretation "relation composition" 'compose R1 R2 = (Rcomp ? R1 R2).
357 definition lift_confL ≝
358 λsig,S1,S2,c.match c with
359 [ mk_config s t ⇒ mk_config sig (FinSum S1 S2) (inl … s) t ].
361 definition lift_confR ≝
362 λsig,S1,S2,c.match c with
363 [ mk_config s t ⇒ mk_config sig (FinSum S1 S2) (inr … s) t ].
365 definition halt_liftL ≝
366 λS1,S2,halt.λs:FinSum S1 S2.
369 | inr _ ⇒ true ]. (* should be vacuous in all cases we use halt_liftL *)
371 definition halt_liftR ≝
372 λS1,S2,halt.λs:FinSum S1 S2.
375 | inr s2 ⇒ halt s2 ].
377 lemma p_halt_liftL : ∀sig,S1,S2,halt,c.
378 halt (cstate sig S1 c) =
379 halt_liftL S1 S2 halt (cstate … (lift_confL … c)).
380 #sig #S1 #S2 #halt #c cases c #s #t %
383 lemma trans_liftL : ∀sig,M1,M2,s,a,news,move.
384 halt ? M1 s = false →
385 trans sig M1 〈s,a〉 = 〈news,move〉 →
386 trans sig (seq sig M1 M2) 〈inl … s,a〉 = 〈inl … news,move〉.
387 #sig (*#M1*) * #Q1 #T1 #init1 #halt1 #M2 #s #a #news #move
388 #Hhalt #Htrans whd in ⊢ (??%?); >Hhalt >Htrans %
391 lemma trans_liftR : ∀sig,M1,M2,s,a,news,move.
392 halt ? M2 s = false →
393 trans sig M2 〈s,a〉 = 〈news,move〉 →
394 trans sig (seq sig M1 M2) 〈inr … s,a〉 = 〈inr … news,move〉.
395 #sig #M1 * #Q2 #T2 #init2 #halt2 #s #a #news #move
396 #Hhalt #Htrans whd in ⊢ (??%?); >Hhalt >Htrans %
401 cstate sig M c1 = cstate sig M c2 →
402 ctape sig M c1 = ctape sig M c2 → c1 = c2.
403 #sig #M1 * #s1 #t1 * #s2 #t2 //
406 lemma step_lift_confR : ∀sig,M1,M2,c0.
407 halt ? M2 (cstate ?? c0) = false →
408 step sig (seq sig M1 M2) (lift_confR sig (states ? M1) (states ? M2) c0) =
409 lift_confR sig (states ? M1) (states ? M2) (step sig M2 c0).
410 #sig #M1 (* * #Q1 #T1 #init1 #halt1 *) #M2 * #s #t
411 lapply (refl ? (trans ?? 〈s,current sig t〉))
412 cases (trans ?? 〈s,current sig t〉) in ⊢ (???% → %);
415 | 2,3: #s1 #l1 #Heq #Hhalt
416 |#ls #s1 #rs #Heq #Hhalt ]
417 whd in ⊢ (???(????%)); >Heq
419 whd in ⊢ (??(???%)?); whd in ⊢ (??%?);
420 >(trans_liftR … Heq) //
423 lemma step_lift_confL : ∀sig,M1,M2,c0.
424 halt ? M1 (cstate ?? c0) = false →
425 step sig (seq sig M1 M2) (lift_confL sig (states ? M1) (states ? M2) c0) =
426 lift_confL sig ?? (step sig M1 c0).
427 #sig #M1 (* * #Q1 #T1 #init1 #halt1 *) #M2 * #s #t
428 lapply (refl ? (trans ?? 〈s,current sig t〉))
429 cases (trans ?? 〈s,current sig t〉) in ⊢ (???% → %);
432 | 2,3: #s1 #l1 #Heq #Hhalt
433 |#ls #s1 #rs #Heq #Hhalt ]
434 whd in ⊢ (???(????%)); >Heq
436 whd in ⊢ (??(???%)?); whd in ⊢ (??%?);
437 >(trans_liftL … Heq) //
440 lemma loop_lift : ∀A,B,k,lift,f,g,h,hlift,c1,c2.
441 (∀x.hlift (lift x) = h x) →
442 (∀x.h x = false → lift (f x) = g (lift x)) →
443 loop A k f h c1 = Some ? c2 →
444 loop B k g hlift (lift c1) = Some ? (lift … c2).
445 #A #B #k #lift #f #g #h #hlift #c1 #c2 #Hfg #Hhlift
446 generalize in match c1; elim k
447 [#c0 normalize in ⊢ (??%? → ?); #Hfalse destruct (Hfalse)
448 |#k0 #IH #c0 whd in ⊢ (??%? → ??%?);
449 cases (true_or_false (h c0)) #Hc0 >Hfg >Hc0
450 [ normalize #Heq destruct (Heq) %
451 | normalize <Hhlift // @IH ]
455 lemma loop_liftL : ∀sig,k,M1,M2,c1,c2.
456 loop ? k (step sig M1) (λc.halt sig M1 (cstate ?? c)) c1 = Some ? c2 →
457 loop ? k (step sig (seq sig M1 M2))
458 (λc.halt_liftL ?? (halt sig M1) (cstate ?? c)) (lift_confL … c1) =
459 Some ? (lift_confL … c2).
460 #sig #k #M1 #M2 #c1 #c2 generalize in match c1;
462 [#c0 normalize in ⊢ (??%? → ?); #Hfalse destruct (Hfalse)
463 |#k0 #IH #c0 whd in ⊢ (??%? → ??%?);
464 cases (true_or_false (halt ?? (cstate sig (states ? M1) c0))) #Hc0 >Hc0
465 [ >(?: halt_liftL ?? (halt sig M1) (cstate sig ? (lift_confL … c0)) = true)
466 [ whd in ⊢ (??%? → ??%?); #Hc2 destruct (Hc2) %
468 | >(?: halt_liftL ?? (halt sig M1) (cstate ?? (lift_confL … c0)) = false)
469 [whd in ⊢ (??%? → ??%?); #Hc2 <(IH ? Hc2) @eq_f
474 lemma loop_liftR : ∀sig,k,M1,M2,c1,c2.
475 loop ? k (step sig M2) (λc.halt sig M2 (cstate ?? c)) c1 = Some ? c2 →
476 loop ? k (step sig (seq sig M1 M2))
477 (λc.halt sig (seq sig M1 M2) (cstate ?? c)) (lift_confR … c1) =
478 Some ? (lift_confR … c2).
479 #sig #k #M1 #M2 #c1 #c2 generalize in match c1;
481 [#c0 normalize in ⊢ (??%? → ?); #Hfalse destruct (Hfalse)
482 |#k0 #IH #c0 whd in ⊢ (??%? → ??%?);
483 cases (true_or_false (halt ?? (cstate sig ? c0))) #Hc0 >Hc0
484 [ >(?: halt ? (seq sig M1 M2) (cstate sig ? (lift_confR … c0)) = true)
485 [ whd in ⊢ (??%? → ??%?); #Hc2 destruct (Hc2) %
487 | >(?: halt ? (seq sig M1 M2) (cstate sig ? (lift_confR … c0)) = false)
488 [whd in ⊢ (??%? → ??%?); #Hc2 <(IH ? Hc2) @eq_f
497 ∀A,k,f,p,a,b.loop A k f p a = Some ? b → p b = true.
499 [#a #b normalize #Hfalse destruct
500 |#k0 #IH #a #b whd in ⊢ (??%? → ?); cases (true_or_false (p a)) #Hpa
501 [ >Hpa normalize #H1 destruct //
507 lemma trans_liftL_true : ∀sig,M1,M2,s,a.
509 trans sig (seq sig M1 M2) 〈inl … s,a〉 = 〈inr … (start ? M2),None ?〉.
511 #Hhalt whd in ⊢ (??%?); >Hhalt %
514 lemma eq_ctape_lift_conf_L : ∀sig,S1,S2,outc.
515 ctape sig (FinSum S1 S2) (lift_confL … outc) = ctape … outc.
516 #sig #S1 #S2 #outc cases outc #s #t %
519 lemma eq_ctape_lift_conf_R : ∀sig,S1,S2,outc.
520 ctape sig (FinSum S1 S2) (lift_confR … outc) = ctape … outc.
521 #sig #S1 #S2 #outc cases outc #s #t %
524 theorem sem_seq: ∀sig,M1,M2,R1,R2.
525 Realize sig M1 R1 → Realize sig M2 R2 →
526 Realize sig (seq sig M1 M2) (R1 ∘ R2).
527 #sig #M1 #M2 #R1 #R2 #HR1 #HR2 #t
528 cases (HR1 t) #k1 * #outc1 * #Hloop1 #HM1
529 cases (HR2 (ctape sig (states ? M1) outc1)) #k2 * #outc2 * #Hloop2 #HM2
530 @(ex_intro … (k1+k2)) @(ex_intro … (lift_confR … outc2))
532 [@(loop_merge ???????????
533 (loop_lift ??? (lift_confL sig (states sig M1) (states sig M2))
534 (step sig M1) (step sig (seq sig M1 M2))
535 (λc.halt sig M1 (cstate … c))
536 (λc.halt_liftL ?? (halt sig M1) (cstate … c)) … Hloop1))
538 [ #sl #tl whd in ⊢ (??%? → ?); #Hl %
539 | #sr #tr whd in ⊢ (??%? → ?); #Hr destruct (Hr) ]
540 || #c0 #Hhalt <step_lift_confL //
542 |6:cases outc1 #s1 #t1 %
543 |7:@(loop_lift … (initc ?? (ctape … outc1)) … Hloop2)
545 | #c0 #Hhalt <step_lift_confR // ]
546 |whd in ⊢ (??(???%)?);whd in ⊢ (??%?);
547 generalize in match Hloop1; cases outc1 #sc1 #tc1 #Hloop10
548 >(trans_liftL_true sig M1 M2 ??)
549 [ whd in ⊢ (??%?); whd in ⊢ (???%);
550 @config_eq whd in ⊢ (???%); //
551 | @(loop_Some ?????? Hloop10) ]
553 | @(ex_intro … (ctape ? (FinSum (states ? M1) (states ? M2)) (lift_confL … outc1)))
554 % // >eq_ctape_lift_conf_L >eq_ctape_lift_conf_R //