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 inductive move : Type[0] ≝
51 | L : move | R : move | N : move.
53 (********************************** machine ***********************************)
55 record TM (sig:FinSet): Type[1] ≝
57 trans : states × (option sig) → states × (option (sig × move));
62 definition tape_move_left ≝ λsig:FinSet.λlt:list sig.λc:sig.λrt:list sig.
64 [ nil ⇒ leftof sig c rt
65 | cons c0 lt0 ⇒ midtape sig lt0 c0 (c::rt) ].
67 definition tape_move_right ≝ λsig:FinSet.λlt:list sig.λc:sig.λrt:list sig.
69 [ nil ⇒ rightof sig c lt
70 | cons c0 rt0 ⇒ midtape sig (c::lt) c0 rt0 ].
72 definition tape_move ≝ λsig.λt: tape sig.λm:option (sig × move).
78 [ R ⇒ tape_move_right ? (left ? t) s (right ? t)
79 | L ⇒ tape_move_left ? (left ? t) s (right ? t)
80 | N ⇒ midtape ? (left ? t) s (right ? t)
83 record config (sig,states:FinSet): Type[0] ≝
88 lemma config_expand: ∀sig,Q,c.
89 c = mk_config sig Q (cstate ?? c) (ctape ?? c).
93 lemma config_eq : ∀sig,M,c1,c2.
94 cstate sig M c1 = cstate sig M c2 →
95 ctape sig M c1 = ctape sig M c2 → c1 = c2.
96 #sig #M1 * #s1 #t1 * #s2 #t2 //
99 definition step ≝ λsig.λM:TM sig.λc:config sig (states sig M).
100 let current_char ≝ current ? (ctape ?? c) in
101 let 〈news,mv〉 ≝ trans sig M 〈cstate ?? c,current_char〉 in
102 mk_config ?? news (tape_move sig (ctape ?? c) mv).
104 (******************************** loop ****************************************)
105 let rec loop (A:Type[0]) n (f:A→A) p a on n ≝
108 | S m ⇒ if p a then (Some ? a) else loop A m f p (f a)
112 ∀A,n,f,p,a. p a = true →
113 loop A (S n) f p a = Some ? a.
114 #A #n #f #p #a #pa normalize >pa //
118 ∀A,n,f,p,a. p a = false →
119 loop A (S n) f p a = loop A n f p (f a).
120 normalize #A #n #f #p #a #Hpa >Hpa %
123 lemma loop_incr : ∀A,f,p,k1,k2,a1,a2.
124 loop A k1 f p a1 = Some ? a2 →
125 loop A (k2+k1) f p a1 = Some ? a2.
126 #A #f #p #k1 #k2 #a1 #a2 generalize in match a1; elim k1
127 [normalize #a0 #Hfalse destruct
128 |#k1' #IH #a0 <plus_n_Sm whd in ⊢ (??%? → ??%?);
129 cases (true_or_false (p a0)) #Hpa0 >Hpa0 whd in ⊢ (??%? → ??%?); // @IH
133 lemma loop_merge : ∀A,f,p,q.(∀b. p b = false → q b = false) →
135 loop A k1 f p a1 = Some ? a2 →
136 f a2 = a3 → q a2 = false →
137 loop A k2 f q a3 = Some ? a4 →
138 loop A (k1+k2) f q a1 = Some ? a4.
139 #Sig #f #p #q #Hpq #k1 elim k1
140 [normalize #k2 #a1 #a2 #a3 #a4 #H destruct
141 |#k1' #Hind #k2 #a1 #a2 #a3 #a4 normalize in ⊢ (%→?);
142 cases (true_or_false (p a1)) #pa1 >pa1 normalize in ⊢ (%→?);
143 [#eqa1a2 destruct #eqa2a3 #Hqa2 #H
144 whd in ⊢ (??(??%???)?); >plus_n_Sm @loop_incr
145 whd in ⊢ (??%?); >Hqa2 >eqa2a3 @H
146 |normalize >(Hpq … pa1) normalize #H1 #H2 #H3 @(Hind … H2) //
151 lemma loop_split : ∀A,f,p,q.(∀b. q b = true → p b = true) →
153 loop A k f q a1 = Some ? a2 →
155 loop A k1 f p a1 = Some ? a3 ∧
156 loop A (S(k-k1)) f q a3 = Some ? a2.
157 #A #f #p #q #Hpq #k elim k
158 [#a1 #a2 normalize #Heq destruct
159 |#i #Hind #a1 #a2 normalize
160 cases (true_or_false (q a1)) #Hqa1 >Hqa1 normalize
162 @(ex_intro … 1) @(ex_intro … a2) %
163 [normalize >(Hpq …Hqa1) // |>Hqa1 //]
164 |#Hloop cases (true_or_false (p a1)) #Hpa1
165 [@(ex_intro … 1) @(ex_intro … a1) %
166 [normalize >Hpa1 // |>Hqa1 <Hloop normalize //]
167 |cases (Hind …Hloop) #k2 * #a3 * #Hloop1 #Hloop2
168 @(ex_intro … (S k2)) @(ex_intro … a3) %
169 [normalize >Hpa1 normalize // | @Hloop2 ]
175 lemma loop_eq : ∀sig,f,q,i,j,a,x,y.
176 loop sig i f q a = Some ? x → loop sig j f q a = Some ? y → x = y.
177 #sig #f #q #i #j @(nat_elim2 … i j)
178 [ #n #a #x #y normalize #Hfalse destruct (Hfalse)
179 | #n #a #x #y #H1 normalize #Hfalse destruct (Hfalse)
180 | #n1 #n2 #IH #a #x #y normalize cases (q a) normalize
187 ∀A,k,f,p,a.p a = true → loop A (S k) f p a = Some ? a.
188 #A #k #f #p #a #Ha normalize >Ha %
192 ∀A,k,f,p,a,b.loop A k f p a = Some ? b → p b = true.
194 [#a #b normalize #Hfalse destruct
195 |#k0 #IH #a #b whd in ⊢ (??%? → ?); cases (true_or_false (p a)) #Hpa
196 [ >Hpa normalize #H1 destruct // | >Hpa normalize @IH ]
200 lemma loop_lift : ∀A,B,k,lift,f,g,h,hlift,c1,c2.
201 (∀x.hlift (lift x) = h x) →
202 (∀x.h x = false → lift (f x) = g (lift x)) →
203 loop A k f h c1 = Some ? c2 →
204 loop B k g hlift (lift c1) = Some ? (lift … c2).
205 #A #B #k #lift #f #g #h #hlift #c1 #c2 #Hfg #Hhlift
206 generalize in match c1; elim k
207 [#c0 normalize in ⊢ (??%? → ?); #Hfalse destruct (Hfalse)
208 |#k0 #IH #c0 whd in ⊢ (??%? → ??%?);
209 cases (true_or_false (h c0)) #Hc0 >Hfg >Hc0 normalize
210 [ #Heq destruct (Heq) % | <Hhlift // @IH ]
213 (************************** Realizability *************************************)
214 definition loopM ≝ λsig,M,i,cin.
215 loop ? i (step sig M) (λc.halt sig M (cstate ?? c)) cin.
217 lemma loopM_unfold : ∀sig,M,i,cin.
218 loopM sig M i cin = loop ? i (step sig M) (λc.halt sig M (cstate ?? c)) cin.
221 definition initc ≝ λsig.λM:TM sig.λt.
222 mk_config sig (states sig M) (start sig M) t.
224 definition Realize ≝ λsig.λM:TM sig.λR:relation (tape sig).
226 loopM sig M i (initc sig M t) = Some ? outc ∧ R t (ctape ?? outc).
228 definition WRealize ≝ λsig.λM:TM sig.λR:relation (tape sig).
230 loopM sig M i (initc sig M t) = Some ? outc → R t (ctape ?? outc).
232 definition Terminate ≝ λsig.λM:TM sig.λt. ∃i,outc.
233 loopM sig M i (initc sig M t) = Some ? outc.
235 notation "M \vDash R" non associative with precedence 45 for @{ 'models $M $R}.
236 interpretation "realizability" 'models M R = (Realize ? M R).
238 notation "M \VDash R" non associative with precedence 45 for @{ 'wmodels $M $R}.
239 interpretation "weak realizability" 'wmodels M R = (WRealize ? M R).
241 interpretation "termination" 'fintersects M t = (Terminate ? M t).
243 lemma WRealize_to_Realize : ∀sig.∀M: TM sig.∀R.
244 (∀t.M ↓ t) → M ⊫ R → M ⊨ R.
245 #sig #M #R #HT #HW #t cases (HT … t) #i * #outc #Hloop
246 @(ex_intro … i) @(ex_intro … outc) % // @(HW … i) //
249 theorem Realize_to_WRealize : ∀sig.∀M:TM sig.∀R.
251 #sig #M #R #H1 #inc #i #outc #Hloop
252 cases (H1 inc) #k * #outc1 * #Hloop1 #HR >(loop_eq … Hloop Hloop1) //
255 definition accRealize ≝ λsig.λM:TM sig.λacc:states sig M.λRtrue,Rfalse.
257 loopM sig M i (initc sig M t) = Some ? outc ∧
258 (cstate ?? outc = acc → Rtrue t (ctape ?? outc)) ∧
259 (cstate ?? outc ≠ acc → Rfalse t (ctape ?? outc)).
261 notation "M ⊨ [q: R1,R2]" non associative with precedence 45 for @{ 'cmodels $M $q $R1 $R2}.
262 interpretation "conditional realizability" 'cmodels M q R1 R2 = (accRealize ? M q R1 R2).
264 (******************************** monotonicity ********************************)
265 lemma Realize_to_Realize : ∀alpha,M,R1,R2.
266 R1 ⊆ R2 → Realize alpha M R1 → Realize alpha M R2.
267 #alpha #M #R1 #R2 #Himpl #HR1 #intape
268 cases (HR1 intape) -HR1 #k * #outc * #Hloop #HR1
269 @(ex_intro ?? k) @(ex_intro ?? outc) % /2/
272 lemma WRealize_to_WRealize: ∀sig,M,R1,R2.
273 R1 ⊆ R2 → WRealize sig M R1 → WRealize ? M R2.
274 #alpha #M #R1 #R2 #Hsub #HR1 #intape #i #outc #Hloop
275 @Hsub @(HR1 … i) @Hloop
278 lemma acc_Realize_to_acc_Realize: ∀sig,M.∀q:states sig M.∀R1,R2,R3,R4.
279 R1 ⊆ R3 → R2 ⊆ R4 → M ⊨ [q:R1,R2] → M ⊨ [q:R3,R4].
280 #alpha #M #q #R1 #R2 #R3 #R4 #Hsub13 #Hsub24 #HRa #intape
281 cases (HRa intape) -HRa #k * #outc * * #Hloop #HRtrue #HRfalse
282 @(ex_intro ?? k) @(ex_intro ?? outc) %
283 [ % [@Hloop] #Hq @Hsub13 @HRtrue // | #Hq @Hsub24 @HRfalse //]
286 (**************************** A canonical relation ****************************)
288 definition R_TM ≝ λsig.λM:TM sig.λq.λt1,t2.
290 loopM ? M i (mk_config ?? q t1) = Some ? outc ∧
291 t2 = (ctape ?? outc).
293 lemma R_TM_to_R: ∀sig,M,R. ∀t1,t2.
294 M ⊫ R → R_TM ? M (start sig M) t1 t2 → R t1 t2.
295 #sig #M #R #t1 #t2 whd in ⊢ (%→?); #HMR * #i * #outc *
296 #Hloop #Ht2 >Ht2 @(HMR … Hloop)
299 (******************************** NOP Machine *********************************)
304 definition nop_states ≝ initN 1.
305 definition start_nop : initN 1 ≝ mk_Sig ?? 0 (le_n … 1).
308 λalpha:FinSet.mk_TM alpha nop_states
309 (λp.let 〈q,a〉 ≝ p in 〈q,None ?〉)
312 definition R_nop ≝ λalpha.λt1,t2:tape alpha.t2 = t1.
315 ∀alpha.nop alpha ⊨ R_nop alpha.
316 #alpha #intape @(ex_intro ?? 1)
317 @(ex_intro … (mk_config ?? start_nop intape)) % %
320 lemma nop_single_state: ∀sig.∀q1,q2:states ? (nop sig). q1 = q2.
321 normalize #sig * #n #ltn1 * #m #ltm1
322 generalize in match ltn1; generalize in match ltm1;
323 <(le_n_O_to_eq … (le_S_S_to_le … ltn1)) <(le_n_O_to_eq … (le_S_S_to_le … ltm1))
326 (************************** Sequential Composition ****************************)
328 definition seq_trans ≝ λsig. λM1,M2 : TM sig.
332 if halt sig M1 s1 then 〈inr … (start sig M2), None ?〉
333 else let 〈news1,m〉 ≝ trans sig M1 〈s1,a〉 in 〈inl … news1,m〉
334 | inr s2 ⇒ let 〈news2,m〉 ≝ trans sig M2 〈s2,a〉 in 〈inr … news2,m〉
337 definition seq ≝ λsig. λM1,M2 : TM sig.
339 (FinSum (states sig M1) (states sig M2))
340 (seq_trans sig M1 M2)
341 (inl … (start sig M1))
343 [ inl _ ⇒ false | inr s2 ⇒ halt sig M2 s2]).
345 notation "a · b" right associative with precedence 65 for @{ 'middot $a $b}.
346 interpretation "sequential composition" 'middot a b = (seq ? a b).
348 definition lift_confL ≝
349 λsig,S1,S2,c.match c with
350 [ mk_config s t ⇒ mk_config sig (FinSum S1 S2) (inl … s) t ].
352 definition lift_confR ≝
353 λsig,S1,S2,c.match c with
354 [ mk_config s t ⇒ mk_config sig (FinSum S1 S2) (inr … s) t ].
356 definition halt_liftL ≝
357 λS1,S2,halt.λs:FinSum S1 S2.
360 | inr _ ⇒ true ]. (* should be vacuous in all cases we use halt_liftL *)
362 definition halt_liftR ≝
363 λS1,S2,halt.λs:FinSum S1 S2.
366 | inr s2 ⇒ halt s2 ].
368 lemma p_halt_liftL : ∀sig,S1,S2,halt,c.
369 halt (cstate sig S1 c) =
370 halt_liftL S1 S2 halt (cstate … (lift_confL … c)).
371 #sig #S1 #S2 #halt #c cases c #s #t %
374 lemma trans_seq_liftL : ∀sig,M1,M2,s,a,news,move.
375 halt ? M1 s = false →
376 trans sig M1 〈s,a〉 = 〈news,move〉 →
377 trans sig (seq sig M1 M2) 〈inl … s,a〉 = 〈inl … news,move〉.
378 #sig (*#M1*) * #Q1 #T1 #init1 #halt1 #M2 #s #a #news #move
379 #Hhalt #Htrans whd in ⊢ (??%?); >Hhalt >Htrans %
382 lemma trans_seq_liftR : ∀sig,M1,M2,s,a,news,move.
383 halt ? M2 s = false →
384 trans sig M2 〈s,a〉 = 〈news,move〉 →
385 trans sig (seq sig M1 M2) 〈inr … s,a〉 = 〈inr … news,move〉.
386 #sig #M1 * #Q2 #T2 #init2 #halt2 #s #a #news #move
387 #Hhalt #Htrans whd in ⊢ (??%?); >Hhalt >Htrans %
390 lemma step_seq_liftR : ∀sig,M1,M2,c0.
391 halt ? M2 (cstate ?? c0) = false →
392 step sig (seq sig M1 M2) (lift_confR sig (states ? M1) (states ? M2) c0) =
393 lift_confR sig (states ? M1) (states ? M2) (step sig M2 c0).
394 #sig #M1 (* * #Q1 #T1 #init1 #halt1 *) #M2 * #s #t
395 lapply (refl ? (trans ?? 〈s,current sig t〉))
396 cases (trans ?? 〈s,current sig t〉) in ⊢ (???% → %);
399 | 2,3: #s1 #l1 #Heq #Hhalt
400 |#ls #s1 #rs #Heq #Hhalt ]
401 whd in ⊢ (???(????%)); >Heq whd in ⊢ (???%);
402 whd in ⊢ (??(???%)?); whd in ⊢ (??%?); >(trans_seq_liftR … Heq) //
405 lemma step_seq_liftL : ∀sig,M1,M2,c0.
406 halt ? M1 (cstate ?? c0) = false →
407 step sig (seq sig M1 M2) (lift_confL sig (states ? M1) (states ? M2) c0) =
408 lift_confL sig ?? (step sig M1 c0).
409 #sig #M1 (* * #Q1 #T1 #init1 #halt1 *) #M2 * #s #t
410 lapply (refl ? (trans ?? 〈s,current sig t〉))
411 cases (trans ?? 〈s,current sig t〉) in ⊢ (???% → %);
414 | 2,3: #s1 #l1 #Heq #Hhalt
415 |#ls #s1 #rs #Heq #Hhalt ]
416 whd in ⊢ (???(????%)); >Heq whd in ⊢ (???%);
417 whd in ⊢ (??(???%)?); whd in ⊢ (??%?); >(trans_seq_liftL … Heq) //
420 lemma trans_liftL_true : ∀sig,M1,M2,s,a.
422 trans sig (seq sig M1 M2) 〈inl … s,a〉 = 〈inr … (start ? M2),None ?〉.
423 #sig #M1 #M2 #s #a #Hhalt whd in ⊢ (??%?); >Hhalt %
426 lemma eq_ctape_lift_conf_L : ∀sig,S1,S2,outc.
427 ctape sig (FinSum S1 S2) (lift_confL … outc) = ctape … outc.
428 #sig #S1 #S2 #outc cases outc #s #t %
431 lemma eq_ctape_lift_conf_R : ∀sig,S1,S2,outc.
432 ctape sig (FinSum S1 S2) (lift_confR … outc) = ctape … outc.
433 #sig #S1 #S2 #outc cases outc #s #t %
436 theorem sem_seq: ∀sig.∀M1,M2:TM sig.∀R1,R2.
437 M1 ⊨ R1 → M2 ⊨ R2 → M1 · M2 ⊨ R1 ∘ R2.
438 #sig #M1 #M2 #R1 #R2 #HR1 #HR2 #t
439 cases (HR1 t) #k1 * #outc1 * #Hloop1 #HM1
440 cases (HR2 (ctape sig (states ? M1) outc1)) #k2 * #outc2 * #Hloop2 #HM2
441 @(ex_intro … (k1+k2)) @(ex_intro … (lift_confR … outc2))
443 [@(loop_merge ???????????
444 (loop_lift ??? (lift_confL sig (states sig M1) (states sig M2))
445 (step sig M1) (step sig (seq sig M1 M2))
446 (λc.halt sig M1 (cstate … c))
447 (λc.halt_liftL ?? (halt sig M1) (cstate … c)) … Hloop1))
449 [ #sl #tl whd in ⊢ (??%? → ?); #Hl %
450 | #sr #tr whd in ⊢ (??%? → ?); #Hr destruct (Hr) ]
451 || #c0 #Hhalt <step_seq_liftL //
453 |6:cases outc1 #s1 #t1 %
454 |7:@(loop_lift … (initc ?? (ctape … outc1)) … Hloop2)
456 | #c0 #Hhalt <step_seq_liftR // ]
457 |whd in ⊢ (??(???%)?);whd in ⊢ (??%?);
458 generalize in match Hloop1; cases outc1 #sc1 #tc1 #Hloop10
459 >(trans_liftL_true sig M1 M2 ??)
460 [ whd in ⊢ (??%?); whd in ⊢ (???%);
461 @config_eq whd in ⊢ (???%); //
462 | @(loop_Some ?????? Hloop10) ]
464 | @(ex_intro … (ctape ? (FinSum (states ? M1) (states ? M2)) (lift_confL … outc1)))
465 % // >eq_ctape_lift_conf_L >eq_ctape_lift_conf_R //
469 theorem sem_seq_app: ∀sig.∀M1,M2:TM sig.∀R1,R2,R3.
470 M1 ⊨ R1 → M2 ⊨ R2 → R1 ∘ R2 ⊆ R3 → M1 · M2 ⊨ R3.
471 #sig #M1 #M2 #R1 #R2 #R3 #HR1 #HR2 #Hsub
472 #t cases (sem_seq … HR1 HR2 t)
473 #k * #outc * #Hloop #Houtc @(ex_intro … k) @(ex_intro … outc)
474 % [@Hloop |@Hsub @Houtc]