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". *)
15 (******************************** tape ****************************************)
17 (* A tape is essentially a triple 〈left,current,right〉 where however the current
18 symbol could be missing. This may happen for three different reasons: both tapes
19 are empty; we are on the left extremity of a non-empty tape (left overflow), or
20 we are on the right extremity of a non-empty tape (right overflow). *)
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
30 [ niltape ⇒ [] | leftof _ _ ⇒ [] | rightof s l ⇒ s::l | midtape l _ _ ⇒ l ].
33 λsig.λt:tape sig.match t with
34 [ niltape ⇒ [] | leftof s r ⇒ s::r | rightof _ _ ⇒ []| midtape _ _ r ⇒ r ].
37 λsig.λt:tape sig.match t with
38 [ midtape _ c _ ⇒ Some ? c | _ ⇒ None ? ].
41 ∀sig:FinSet.list sig → option sig → list sig → tape sig ≝
42 λsig,lt,c,rt.match c with
43 [ Some c' ⇒ midtape sig lt c' rt
44 | None ⇒ match lt with
47 | cons r0 rs0 ⇒ leftof ? r0 rs0 ]
48 | cons l0 ls0 ⇒ rightof ? l0 ls0 ] ].
50 lemma current_to_midtape: ∀sig,t,c. current sig t = Some ? c →
51 ∃ls,rs. t = midtape ? ls c rs.
53 [#c whd in ⊢ ((??%?)→?); #Hfalse destruct
54 |#a #l #c whd in ⊢ ((??%?)→?); #Hfalse destruct
55 |#a #l #c whd in ⊢ ((??%?)→?); #Hfalse destruct
56 |#ls #a #rs #c whd in ⊢ ((??%?)→?); #H destruct
57 @(ex_intro … ls) @(ex_intro … rs) //
61 (*********************************** moves ************************************)
63 inductive move : Type[0] ≝
64 | L : move | R : move | N : move.
66 (********************************** machine ***********************************)
68 record TM (sig:FinSet): Type[1] ≝
70 trans : states × (option sig) → states × (option (sig × move));
75 definition tape_move_left ≝ λsig:FinSet.λlt:list sig.λc:sig.λrt:list sig.
77 [ nil ⇒ leftof sig c rt
78 | cons c0 lt0 ⇒ midtape sig lt0 c0 (c::rt) ].
80 definition tape_move_right ≝ λsig:FinSet.λlt:list sig.λc:sig.λrt:list sig.
82 [ nil ⇒ rightof sig c lt
83 | cons c0 rt0 ⇒ midtape sig (c::lt) c0 rt0 ].
85 definition tape_move ≝ λsig.λt: tape sig.λm:option (sig × move).
91 [ R ⇒ tape_move_right ? (left ? t) s (right ? t)
92 | L ⇒ tape_move_left ? (left ? t) s (right ? t)
93 | N ⇒ midtape ? (left ? t) s (right ? t)
96 record config (sig,states:FinSet): Type[0] ≝
101 lemma config_expand: ∀sig,Q,c.
102 c = mk_config sig Q (cstate ?? c) (ctape ?? c).
106 lemma config_eq : ∀sig,M,c1,c2.
107 cstate sig M c1 = cstate sig M c2 →
108 ctape sig M c1 = ctape sig M c2 → c1 = c2.
109 #sig #M1 * #s1 #t1 * #s2 #t2 //
112 definition step ≝ λsig.λM:TM sig.λc:config sig (states sig M).
113 let current_char ≝ current ? (ctape ?? c) in
114 let 〈news,mv〉 ≝ trans sig M 〈cstate ?? c,current_char〉 in
115 mk_config ?? news (tape_move sig (ctape ?? c) mv).
117 (******************************** loop ****************************************)
118 let rec loop (A:Type[0]) n (f:A→A) p a on n ≝
121 | S m ⇒ if p a then (Some ? a) else loop A m f p (f a)
125 ∀A,n,f,p,a. p a = true →
126 loop A (S n) f p a = Some ? a.
127 #A #n #f #p #a #pa normalize >pa //
131 ∀A,n,f,p,a. p a = false →
132 loop A (S n) f p a = loop A n f p (f a).
133 normalize #A #n #f #p #a #Hpa >Hpa %
136 lemma loop_incr : ∀A,f,p,k1,k2,a1,a2.
137 loop A k1 f p a1 = Some ? a2 →
138 loop A (k2+k1) f p a1 = Some ? a2.
139 #A #f #p #k1 #k2 #a1 #a2 generalize in match a1; elim k1
140 [normalize #a0 #Hfalse destruct
141 |#k1' #IH #a0 <plus_n_Sm whd in ⊢ (??%? → ??%?);
142 cases (true_or_false (p a0)) #Hpa0 >Hpa0 whd in ⊢ (??%? → ??%?); // @IH
146 lemma loop_merge : ∀A,f,p,q.(∀b. p b = false → q b = false) →
148 loop A k1 f p a1 = Some ? a2 →
149 f a2 = a3 → q a2 = false →
150 loop A k2 f q a3 = Some ? a4 →
151 loop A (k1+k2) f q a1 = Some ? a4.
152 #Sig #f #p #q #Hpq #k1 elim k1
153 [normalize #k2 #a1 #a2 #a3 #a4 #H destruct
154 |#k1' #Hind #k2 #a1 #a2 #a3 #a4 normalize in ⊢ (%→?);
155 cases (true_or_false (p a1)) #pa1 >pa1 normalize in ⊢ (%→?);
156 [#eqa1a2 destruct #eqa2a3 #Hqa2 #H
157 whd in ⊢ (??(??%???)?); >plus_n_Sm @loop_incr
158 whd in ⊢ (??%?); >Hqa2 >eqa2a3 @H
159 |normalize >(Hpq … pa1) normalize #H1 #H2 #H3 @(Hind … H2) //
164 lemma loop_split : ∀A,f,p,q.(∀b. q b = true → p b = true) →
166 loop A k f q a1 = Some ? a2 →
168 loop A k1 f p a1 = Some ? a3 ∧
169 loop A (S(k-k1)) f q a3 = Some ? a2.
170 #A #f #p #q #Hpq #k elim k
171 [#a1 #a2 normalize #Heq destruct
172 |#i #Hind #a1 #a2 normalize
173 cases (true_or_false (q a1)) #Hqa1 >Hqa1 normalize
175 @(ex_intro … 1) @(ex_intro … a2) %
176 [normalize >(Hpq …Hqa1) // |>Hqa1 //]
177 |#Hloop cases (true_or_false (p a1)) #Hpa1
178 [@(ex_intro … 1) @(ex_intro … a1) %
179 [normalize >Hpa1 // |>Hqa1 <Hloop normalize //]
180 |cases (Hind …Hloop) #k2 * #a3 * #Hloop1 #Hloop2
181 @(ex_intro … (S k2)) @(ex_intro … a3) %
182 [normalize >Hpa1 normalize // | @Hloop2 ]
188 lemma loop_eq : ∀sig,f,q,i,j,a,x,y.
189 loop sig i f q a = Some ? x → loop sig j f q a = Some ? y → x = y.
190 #sig #f #q #i #j @(nat_elim2 … i j)
191 [ #n #a #x #y normalize #Hfalse destruct (Hfalse)
192 | #n #a #x #y #H1 normalize #Hfalse destruct (Hfalse)
193 | #n1 #n2 #IH #a #x #y normalize cases (q a) normalize
200 ∀A,k,f,p,a.p a = true → loop A (S k) f p a = Some ? a.
201 #A #k #f #p #a #Ha normalize >Ha %
205 ∀A,k,f,p,a,b.loop A k f p a = Some ? b → p b = true.
207 [#a #b normalize #Hfalse destruct
208 |#k0 #IH #a #b whd in ⊢ (??%? → ?); cases (true_or_false (p a)) #Hpa
209 [ >Hpa normalize #H1 destruct // | >Hpa normalize @IH ]
213 lemma loop_lift : ∀A,B,k,lift,f,g,h,hlift,c1,c2.
214 (∀x.hlift (lift x) = h x) →
215 (∀x.h x = false → lift (f x) = g (lift x)) →
216 loop A k f h c1 = Some ? c2 →
217 loop B k g hlift (lift c1) = Some ? (lift … c2).
218 #A #B #k #lift #f #g #h #hlift #c1 #c2 #Hfg #Hhlift
219 generalize in match c1; elim k
220 [#c0 normalize in ⊢ (??%? → ?); #Hfalse destruct (Hfalse)
221 |#k0 #IH #c0 whd in ⊢ (??%? → ??%?);
222 cases (true_or_false (h c0)) #Hc0 >Hfg >Hc0 normalize
223 [ #Heq destruct (Heq) % | <Hhlift // @IH ]
226 (************************** Realizability *************************************)
227 definition loopM ≝ λsig,M,i,cin.
228 loop ? i (step sig M) (λc.halt sig M (cstate ?? c)) cin.
230 lemma loopM_unfold : ∀sig,M,i,cin.
231 loopM sig M i cin = loop ? i (step sig M) (λc.halt sig M (cstate ?? c)) cin.
234 definition initc ≝ λsig.λM:TM sig.λt.
235 mk_config sig (states sig M) (start sig M) t.
237 definition Realize ≝ λsig.λM:TM sig.λR:relation (tape sig).
239 loopM sig M i (initc sig M t) = Some ? outc ∧ R t (ctape ?? outc).
241 definition WRealize ≝ λsig.λM:TM sig.λR:relation (tape sig).
243 loopM sig M i (initc sig M t) = Some ? outc → R t (ctape ?? outc).
245 definition Terminate ≝ λsig.λM:TM sig.λt. ∃i,outc.
246 loopM sig M i (initc sig M t) = Some ? outc.
248 notation "M \vDash R" non associative with precedence 45 for @{ 'models $M $R}.
249 interpretation "realizability" 'models M R = (Realize ? M R).
251 notation "M \VDash R" non associative with precedence 45 for @{ 'wmodels $M $R}.
252 interpretation "weak realizability" 'wmodels M R = (WRealize ? M R).
254 interpretation "termination" 'fintersects M t = (Terminate ? M t).
256 lemma WRealize_to_Realize : ∀sig.∀M: TM sig.∀R.
257 (∀t.M ↓ t) → M ⊫ R → M ⊨ R.
258 #sig #M #R #HT #HW #t cases (HT … t) #i * #outc #Hloop
259 @(ex_intro … i) @(ex_intro … outc) % // @(HW … i) //
262 theorem Realize_to_WRealize : ∀sig.∀M:TM sig.∀R.
264 #sig #M #R #H1 #inc #i #outc #Hloop
265 cases (H1 inc) #k * #outc1 * #Hloop1 #HR >(loop_eq … Hloop Hloop1) //
268 definition accRealize ≝ λsig.λM:TM sig.λacc:states sig M.λRtrue,Rfalse.
270 loopM sig M i (initc sig M t) = Some ? outc ∧
271 (cstate ?? outc = acc → Rtrue t (ctape ?? outc)) ∧
272 (cstate ?? outc ≠ acc → Rfalse t (ctape ?? outc)).
274 notation "M ⊨ [q: R1,R2]" non associative with precedence 45 for @{ 'cmodels $M $q $R1 $R2}.
275 interpretation "conditional realizability" 'cmodels M q R1 R2 = (accRealize ? M q R1 R2).
277 (******************************** monotonicity ********************************)
278 lemma Realize_to_Realize : ∀alpha,M,R1,R2.
279 R1 ⊆ R2 → Realize alpha M R1 → Realize alpha M R2.
280 #alpha #M #R1 #R2 #Himpl #HR1 #intape
281 cases (HR1 intape) -HR1 #k * #outc * #Hloop #HR1
282 @(ex_intro ?? k) @(ex_intro ?? outc) % /2/
285 lemma WRealize_to_WRealize: ∀sig,M,R1,R2.
286 R1 ⊆ R2 → WRealize sig M R1 → WRealize ? M R2.
287 #alpha #M #R1 #R2 #Hsub #HR1 #intape #i #outc #Hloop
288 @Hsub @(HR1 … i) @Hloop
291 lemma acc_Realize_to_acc_Realize: ∀sig,M.∀q:states sig M.∀R1,R2,R3,R4.
292 R1 ⊆ R3 → R2 ⊆ R4 → M ⊨ [q:R1,R2] → M ⊨ [q:R3,R4].
293 #alpha #M #q #R1 #R2 #R3 #R4 #Hsub13 #Hsub24 #HRa #intape
294 cases (HRa intape) -HRa #k * #outc * * #Hloop #HRtrue #HRfalse
295 @(ex_intro ?? k) @(ex_intro ?? outc) %
296 [ % [@Hloop] #Hq @Hsub13 @HRtrue // | #Hq @Hsub24 @HRfalse //]
299 (**************************** A canonical relation ****************************)
301 definition R_TM ≝ λsig.λM:TM sig.λq.λt1,t2.
303 loopM ? M i (mk_config ?? q t1) = Some ? outc ∧
304 t2 = (ctape ?? outc).
306 lemma R_TM_to_R: ∀sig,M,R. ∀t1,t2.
307 M ⊫ R → R_TM ? M (start sig M) t1 t2 → R t1 t2.
308 #sig #M #R #t1 #t2 whd in ⊢ (%→?); #HMR * #i * #outc *
309 #Hloop #Ht2 >Ht2 @(HMR … Hloop)
312 (******************************** NOP Machine *********************************)
317 definition nop_states ≝ initN 1.
318 definition start_nop : initN 1 ≝ mk_Sig ?? 0 (le_n … 1).
321 λalpha:FinSet.mk_TM alpha nop_states
322 (λp.let 〈q,a〉 ≝ p in 〈q,None ?〉)
325 definition R_nop ≝ λalpha.λt1,t2:tape alpha.t2 = t1.
328 ∀alpha.nop alpha ⊨ R_nop alpha.
329 #alpha #intape @(ex_intro ?? 1)
330 @(ex_intro … (mk_config ?? start_nop intape)) % %
333 lemma nop_single_state: ∀sig.∀q1,q2:states ? (nop sig). q1 = q2.
334 normalize #sig * #n #ltn1 * #m #ltm1
335 generalize in match ltn1; generalize in match ltm1;
336 <(le_n_O_to_eq … (le_S_S_to_le … ltn1)) <(le_n_O_to_eq … (le_S_S_to_le … ltm1))
339 (************************** Sequential Composition ****************************)
341 definition seq_trans ≝ λsig. λM1,M2 : TM sig.
345 if halt sig M1 s1 then 〈inr … (start sig M2), None ?〉
346 else let 〈news1,m〉 ≝ trans sig M1 〈s1,a〉 in 〈inl … news1,m〉
347 | inr s2 ⇒ let 〈news2,m〉 ≝ trans sig M2 〈s2,a〉 in 〈inr … news2,m〉
350 definition seq ≝ λsig. λM1,M2 : TM sig.
352 (FinSum (states sig M1) (states sig M2))
353 (seq_trans sig M1 M2)
354 (inl … (start sig M1))
356 [ inl _ ⇒ false | inr s2 ⇒ halt sig M2 s2]).
358 notation "a · b" right associative with precedence 65 for @{ 'middot $a $b}.
359 interpretation "sequential composition" 'middot a b = (seq ? a b).
361 definition lift_confL ≝
362 λsig,S1,S2,c.match c with
363 [ mk_config s t ⇒ mk_config sig (FinSum S1 S2) (inl … s) t ].
365 definition lift_confR ≝
366 λsig,S1,S2,c.match c with
367 [ mk_config s t ⇒ mk_config sig (FinSum S1 S2) (inr … s) t ].
369 definition halt_liftL ≝
370 λS1,S2,halt.λs:FinSum S1 S2.
373 | inr _ ⇒ true ]. (* should be vacuous in all cases we use halt_liftL *)
375 definition halt_liftR ≝
376 λS1,S2,halt.λs:FinSum S1 S2.
379 | inr s2 ⇒ halt s2 ].
381 lemma p_halt_liftL : ∀sig,S1,S2,halt,c.
382 halt (cstate sig S1 c) =
383 halt_liftL S1 S2 halt (cstate … (lift_confL … c)).
384 #sig #S1 #S2 #halt #c cases c #s #t %
387 lemma trans_seq_liftL : ∀sig,M1,M2,s,a,news,move.
388 halt ? M1 s = false →
389 trans sig M1 〈s,a〉 = 〈news,move〉 →
390 trans sig (seq sig M1 M2) 〈inl … s,a〉 = 〈inl … news,move〉.
391 #sig (*#M1*) * #Q1 #T1 #init1 #halt1 #M2 #s #a #news #move
392 #Hhalt #Htrans whd in ⊢ (??%?); >Hhalt >Htrans %
395 lemma trans_seq_liftR : ∀sig,M1,M2,s,a,news,move.
396 halt ? M2 s = false →
397 trans sig M2 〈s,a〉 = 〈news,move〉 →
398 trans sig (seq sig M1 M2) 〈inr … s,a〉 = 〈inr … news,move〉.
399 #sig #M1 * #Q2 #T2 #init2 #halt2 #s #a #news #move
400 #Hhalt #Htrans whd in ⊢ (??%?); >Hhalt >Htrans %
403 lemma step_seq_liftR : ∀sig,M1,M2,c0.
404 halt ? M2 (cstate ?? c0) = false →
405 step sig (seq sig M1 M2) (lift_confR sig (states ? M1) (states ? M2) c0) =
406 lift_confR sig (states ? M1) (states ? M2) (step sig M2 c0).
407 #sig #M1 (* * #Q1 #T1 #init1 #halt1 *) #M2 * #s #t
408 lapply (refl ? (trans ?? 〈s,current sig t〉))
409 cases (trans ?? 〈s,current sig t〉) in ⊢ (???% → %);
412 | 2,3: #s1 #l1 #Heq #Hhalt
413 |#ls #s1 #rs #Heq #Hhalt ]
414 whd in ⊢ (???(????%)); >Heq whd in ⊢ (???%);
415 whd in ⊢ (??(???%)?); whd in ⊢ (??%?); >(trans_seq_liftR … Heq) //
418 lemma step_seq_liftL : ∀sig,M1,M2,c0.
419 halt ? M1 (cstate ?? c0) = false →
420 step sig (seq sig M1 M2) (lift_confL sig (states ? M1) (states ? M2) c0) =
421 lift_confL sig ?? (step sig M1 c0).
422 #sig #M1 (* * #Q1 #T1 #init1 #halt1 *) #M2 * #s #t
423 lapply (refl ? (trans ?? 〈s,current sig t〉))
424 cases (trans ?? 〈s,current sig t〉) in ⊢ (???% → %);
427 | 2,3: #s1 #l1 #Heq #Hhalt
428 |#ls #s1 #rs #Heq #Hhalt ]
429 whd in ⊢ (???(????%)); >Heq whd in ⊢ (???%);
430 whd in ⊢ (??(???%)?); whd in ⊢ (??%?); >(trans_seq_liftL … Heq) //
433 lemma trans_liftL_true : ∀sig,M1,M2,s,a.
435 trans sig (seq sig M1 M2) 〈inl … s,a〉 = 〈inr … (start ? M2),None ?〉.
436 #sig #M1 #M2 #s #a #Hhalt whd in ⊢ (??%?); >Hhalt %
439 lemma eq_ctape_lift_conf_L : ∀sig,S1,S2,outc.
440 ctape sig (FinSum S1 S2) (lift_confL … outc) = ctape … outc.
441 #sig #S1 #S2 #outc cases outc #s #t %
444 lemma eq_ctape_lift_conf_R : ∀sig,S1,S2,outc.
445 ctape sig (FinSum S1 S2) (lift_confR … outc) = ctape … outc.
446 #sig #S1 #S2 #outc cases outc #s #t %
449 theorem sem_seq: ∀sig.∀M1,M2:TM sig.∀R1,R2.
450 M1 ⊨ R1 → M2 ⊨ R2 → M1 · M2 ⊨ R1 ∘ R2.
451 #sig #M1 #M2 #R1 #R2 #HR1 #HR2 #t
452 cases (HR1 t) #k1 * #outc1 * #Hloop1 #HM1
453 cases (HR2 (ctape sig (states ? M1) outc1)) #k2 * #outc2 * #Hloop2 #HM2
454 @(ex_intro … (k1+k2)) @(ex_intro … (lift_confR … outc2))
456 [@(loop_merge ???????????
457 (loop_lift ??? (lift_confL sig (states sig M1) (states sig M2))
458 (step sig M1) (step sig (seq sig M1 M2))
459 (λc.halt sig M1 (cstate … c))
460 (λc.halt_liftL ?? (halt sig M1) (cstate … c)) … Hloop1))
462 [ #sl #tl whd in ⊢ (??%? → ?); #Hl %
463 | #sr #tr whd in ⊢ (??%? → ?); #Hr destruct (Hr) ]
464 || #c0 #Hhalt <step_seq_liftL //
466 |6:cases outc1 #s1 #t1 %
467 |7:@(loop_lift … (initc ?? (ctape … outc1)) … Hloop2)
469 | #c0 #Hhalt <step_seq_liftR // ]
470 |whd in ⊢ (??(???%)?);whd in ⊢ (??%?);
471 generalize in match Hloop1; cases outc1 #sc1 #tc1 #Hloop10
472 >(trans_liftL_true sig M1 M2 ??)
473 [ whd in ⊢ (??%?); whd in ⊢ (???%);
474 @config_eq whd in ⊢ (???%); //
475 | @(loop_Some ?????? Hloop10) ]
477 | @(ex_intro … (ctape ? (FinSum (states ? M1) (states ? M2)) (lift_confL … outc1)))
478 % // >eq_ctape_lift_conf_L >eq_ctape_lift_conf_R //
482 theorem sem_seq_app: ∀sig.∀M1,M2:TM sig.∀R1,R2,R3.
483 M1 ⊨ R1 → M2 ⊨ R2 → R1 ∘ R2 ⊆ R3 → M1 · M2 ⊨ R3.
484 #sig #M1 #M2 #R1 #R2 #R3 #HR1 #HR2 #Hsub
485 #t cases (sem_seq … HR1 HR2 t)
486 #k * #outc * #Hloop #Houtc @(ex_intro … k) @(ex_intro … outc)
487 % [@Hloop |@Hsub @Houtc]