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/core_notation/fintersects_2.ma".
13 include "basics/finset.ma".
14 include "basics/vectors.ma".
15 include "basics/finset.ma".
16 (* include "basics/relations.ma". *)
18 (******************************** tape ****************************************)
20 (* A tape is essentially a triple 〈left,current,right〉 where however the current
21 symbol could be missing. This may happen for three different reasons: both tapes
22 are empty; we are on the left extremity of a non-empty tape (left overflow), or
23 we are on the right extremity of a non-empty tape (right overflow). *)
25 inductive tape (sig:FinSet) : Type[0] ≝
27 | leftof : sig → list sig → tape sig
28 | rightof : sig → list sig → tape sig
29 | midtape : list sig → sig → list sig → tape sig.
32 λsig.λt:tape sig.match t with
33 [ niltape ⇒ [] | leftof _ _ ⇒ [] | rightof s l ⇒ s::l | midtape l _ _ ⇒ l ].
36 λsig.λt:tape sig.match t with
37 [ niltape ⇒ [] | leftof s r ⇒ s::r | rightof _ _ ⇒ []| midtape _ _ r ⇒ r ].
40 λsig.λt:tape sig.match t with
41 [ midtape _ c _ ⇒ Some ? c | _ ⇒ None ? ].
44 ∀sig:FinSet.list sig → option sig → list sig → tape sig ≝
45 λsig,lt,c,rt.match c with
46 [ Some c' ⇒ midtape sig lt c' rt
47 | None ⇒ match lt with
50 | cons r0 rs0 ⇒ leftof ? r0 rs0 ]
51 | cons l0 ls0 ⇒ rightof ? l0 ls0 ] ].
54 ∀sig,ls,c,rs.(c = None ? → ls = [ ] ∨ rs = [ ]) → right ? (mk_tape sig ls c rs) = rs.
55 #sig #ls #c #rs cases c // cases ls
57 | #l0 #ls0 #H normalize cases (H (refl ??)) #H1 [ destruct (H1) | >H1 % ] ]
60 lemma left_mk_tape : ∀sig,ls,c,rs.left ? (mk_tape sig ls c rs) = ls.
61 #sig #ls #c #rs cases c // cases ls // cases rs //
64 lemma current_mk_tape : ∀sig,ls,c,rs.current ? (mk_tape sig ls c rs) = c.
65 #sig #ls #c #rs cases c // cases ls // cases rs //
68 lemma current_to_midtape: ∀sig,t,c. current sig t = Some ? c →
69 ∃ls,rs. t = midtape ? ls c rs.
71 [#c whd in ⊢ ((??%?)→?); #Hfalse destruct
72 |#a #l #c whd in ⊢ ((??%?)→?); #Hfalse destruct
73 |#a #l #c whd in ⊢ ((??%?)→?); #Hfalse destruct
74 |#ls #a #rs #c whd in ⊢ ((??%?)→?); #H destruct
75 @(ex_intro … ls) @(ex_intro … rs) //
79 (*********************************** moves ************************************)
81 inductive move : Type[0] ≝
82 | L : move | R : move | N : move.
84 (*************************** turning moves into a DeqSet **********************)
86 definition move_eq ≝ λm1,m2:move.
88 [R ⇒ match m2 with [R ⇒ true | _ ⇒ false]
89 |L ⇒ match m2 with [L ⇒ true | _ ⇒ false]
90 |N ⇒ match m2 with [N ⇒ true | _ ⇒ false]].
92 lemma move_eq_true:∀m1,m2.
93 move_eq m1 m2 = true ↔ m1 = m2.
95 [* normalize [% #_ % |2,3: % #H destruct ]
96 |* normalize [1,3: % #H destruct |% #_ % ]
97 |* normalize [1,2: % #H destruct |% #_ % ]
100 definition DeqMove ≝ mk_DeqSet move move_eq move_eq_true.
102 unification hint 0 ≔ ;
104 (* ---------------------------------------- *) ⊢
107 unification hint 0 ≔ m1,m2;
109 (* ---------------------------------------- *) ⊢
110 move_eq m1 m2 ≡ eqb X m1 m2.
113 (************************ turning DeqMove into a FinSet ***********************)
115 definition move_enum ≝ [L;R;N].
117 lemma move_enum_unique: uniqueb ? [L;R;N] = true.
120 lemma move_enum_complete: ∀x:move. memb ? x [L;R;N] = true.
124 mk_FinSet DeqMove [L;R;N] move_enum_unique move_enum_complete.
126 unification hint 0 ≔ ;
128 (* ---------------------------------------- *) ⊢
130 (********************************** machine ***********************************)
132 record TM (sig:FinSet): Type[1] ≝
134 trans : states × (option sig) → states × (option sig) × move;
139 definition tape_move_left ≝ λsig:FinSet.λt:tape sig.
141 [ niltape ⇒ niltape sig
143 | rightof a ls ⇒ midtape sig ls a [ ]
146 [ nil ⇒ leftof sig a rs
147 | cons a0 ls0 ⇒ midtape sig ls0 a0 (a::rs)
151 definition tape_move_right ≝ λsig:FinSet.λt:tape sig.
153 [ niltape ⇒ niltape sig
155 | leftof a rs ⇒ midtape sig [ ] a rs
158 [ nil ⇒ rightof sig a ls
159 | cons a0 rs0 ⇒ midtape sig (a::ls) a0 rs0
163 definition tape_write ≝ λsig.λt: tape sig.λs:option sig.
166 | Some s0 ⇒ midtape ? (left ? t) s0 (right ? t)
169 definition tape_move ≝ λsig.λt: tape sig.λm:move.
171 [ R ⇒ tape_move_right ? t
172 | L ⇒ tape_move_left ? t
176 definition tape_move_mono ≝
178 tape_move sig (tape_write sig t (\fst mv)) (\snd mv).
180 record config (sig,states:FinSet): Type[0] ≝
185 lemma config_expand: ∀sig,Q,c.
186 c = mk_config sig Q (cstate ?? c) (ctape ?? c).
190 lemma config_eq : ∀sig,M,c1,c2.
191 cstate sig M c1 = cstate sig M c2 →
192 ctape sig M c1 = ctape sig M c2 → c1 = c2.
193 #sig #M1 * #s1 #t1 * #s2 #t2 //
196 definition step ≝ λsig.λM:TM sig.λc:config sig (states sig M).
197 let current_char ≝ current ? (ctape ?? c) in
198 let 〈news,a,mv〉 ≝ trans sig M 〈cstate ?? c,current_char〉 in
199 mk_config ?? news (tape_move sig (tape_write ? (ctape ?? c) a) mv).
202 lemma step_eq : ∀sig,M,c.
203 let current_char ≝ current ? (ctape ?? c) in
204 let 〈news,a,mv〉 ≝ trans sig M 〈cstate ?? c,current_char〉 in
206 mk_config ?? news (tape_move sig (tape_write ? (ctape ?? c) a) mv).
208 whd in match (tape_move_mono sig ??);
211 (******************************** loop ****************************************)
212 let rec loop (A:Type[0]) n (f:A→A) p a on n ≝
215 | S m ⇒ if p a then (Some ? a) else loop A m f p (f a)
219 ∀A,n,f,p,a. p a = true →
220 loop A (S n) f p a = Some ? a.
221 #A #n #f #p #a #pa normalize >pa //
225 ∀A,n,f,p,a. p a = false →
226 loop A (S n) f p a = loop A n f p (f a).
227 normalize #A #n #f #p #a #Hpa >Hpa %
230 lemma loop_incr : ∀A,f,p,k1,k2,a1,a2.
231 loop A k1 f p a1 = Some ? a2 →
232 loop A (k2+k1) f p a1 = Some ? a2.
233 #A #f #p #k1 #k2 #a1 #a2 generalize in match a1; elim k1
234 [normalize #a0 #Hfalse destruct
235 |#k1' #IH #a0 <plus_n_Sm whd in ⊢ (??%? → ??%?);
236 cases (true_or_false (p a0)) #Hpa0 >Hpa0 whd in ⊢ (??%? → ??%?); // @IH
240 lemma loop_merge : ∀A,f,p,q.(∀b. p b = false → q b = false) →
242 loop A k1 f p a1 = Some ? a2 →
243 f a2 = a3 → q a2 = false →
244 loop A k2 f q a3 = Some ? a4 →
245 loop A (k1+k2) f q a1 = Some ? a4.
246 #Sig #f #p #q #Hpq #k1 elim k1
247 [normalize #k2 #a1 #a2 #a3 #a4 #H destruct
248 |#k1' #Hind #k2 #a1 #a2 #a3 #a4 normalize in ⊢ (%→?);
249 cases (true_or_false (p a1)) #pa1 >pa1 normalize in ⊢ (%→?);
250 [#eqa1a2 destruct #eqa2a3 #Hqa2 #H
251 whd in ⊢ (??(??%???)?); >plus_n_Sm @loop_incr
252 whd in ⊢ (??%?); >Hqa2 >eqa2a3 @H
253 |normalize >(Hpq … pa1) normalize #H1 #H2 #H3 @(Hind … H2) //
258 lemma loop_split : ∀A,f,p,q.(∀b. q b = true → p b = true) →
260 loop A k f q a1 = Some ? a2 →
262 loop A k1 f p a1 = Some ? a3 ∧
263 loop A (S(k-k1)) f q a3 = Some ? a2.
264 #A #f #p #q #Hpq #k elim k
265 [#a1 #a2 normalize #Heq destruct
266 |#i #Hind #a1 #a2 normalize
267 cases (true_or_false (q a1)) #Hqa1 >Hqa1 normalize
269 @(ex_intro … 1) @(ex_intro … a2) %
270 [normalize >(Hpq …Hqa1) // |>Hqa1 //]
271 |#Hloop cases (true_or_false (p a1)) #Hpa1
272 [@(ex_intro … 1) @(ex_intro … a1) %
273 [normalize >Hpa1 // |>Hqa1 <Hloop normalize //]
274 |cases (Hind …Hloop) #k2 * #a3 * #Hloop1 #Hloop2
275 @(ex_intro … (S k2)) @(ex_intro … a3) %
276 [normalize >Hpa1 normalize // | @Hloop2 ]
282 lemma loop_eq : ∀sig,f,q,i,j,a,x,y.
283 loop sig i f q a = Some ? x → loop sig j f q a = Some ? y → x = y.
284 #sig #f #q #i #j @(nat_elim2 … i j)
285 [ #n #a #x #y normalize #Hfalse destruct (Hfalse)
286 | #n #a #x #y #H1 normalize #Hfalse destruct (Hfalse)
287 | #n1 #n2 #IH #a #x #y normalize cases (q a) normalize
294 ∀A,k,f,p,a.p a = true → loop A (S k) f p a = Some ? a.
295 #A #k #f #p #a #Ha normalize >Ha %
299 ∀A,k,f,p,a,b.loop A k f p a = Some ? b → p b = true.
301 [#a #b normalize #Hfalse destruct
302 |#k0 #IH #a #b whd in ⊢ (??%? → ?); cases (true_or_false (p a)) #Hpa
303 [ >Hpa normalize #H1 destruct // | >Hpa normalize @IH ]
307 lemma loop_lift : ∀A,B,k,lift,f,g,h,hlift,c1,c2.
308 (∀x.hlift (lift x) = h x) →
309 (∀x.h x = false → lift (f x) = g (lift x)) →
310 loop A k f h c1 = Some ? c2 →
311 loop B k g hlift (lift c1) = Some ? (lift … c2).
312 #A #B #k #lift #f #g #h #hlift #c1 #c2 #Hfg #Hhlift
313 generalize in match c1; elim k
314 [#c0 normalize in ⊢ (??%? → ?); #Hfalse destruct (Hfalse)
315 |#k0 #IH #c0 whd in ⊢ (??%? → ??%?);
316 cases (true_or_false (h c0)) #Hc0 >Hfg >Hc0 normalize
317 [ #Heq destruct (Heq) % | <Hhlift // @IH ]
320 (************************** Realizability *************************************)
321 definition loopM ≝ λsig,M,i,cin.
322 loop ? i (step sig M) (λc.halt sig M (cstate ?? c)) cin.
324 lemma loopM_unfold : ∀sig,M,i,cin.
325 loopM sig M i cin = loop ? i (step sig M) (λc.halt sig M (cstate ?? c)) cin.
328 definition initc ≝ λsig.λM:TM sig.λt.
329 mk_config sig (states sig M) (start sig M) t.
331 definition Realize ≝ λsig.λM:TM sig.λR:relation (tape sig).
333 loopM sig M i (initc sig M t) = Some ? outc ∧ R t (ctape ?? outc).
335 definition WRealize ≝ λsig.λM:TM sig.λR:relation (tape sig).
337 loopM sig M i (initc sig M t) = Some ? outc → R t (ctape ?? outc).
339 definition Terminate ≝ λsig.λM:TM sig.λt. ∃i,outc.
340 loopM sig M i (initc sig M t) = Some ? outc.
342 notation "M \vDash R" non associative with precedence 45 for @{ 'models $M $R}.
343 interpretation "realizability" 'models M R = (Realize ? M R).
345 notation "M \VDash R" non associative with precedence 45 for @{ 'wmodels $M $R}.
346 interpretation "weak realizability" 'wmodels M R = (WRealize ? M R).
348 interpretation "termination" 'fintersects M t = (Terminate ? M t).
350 lemma WRealize_to_Realize : ∀sig.∀M: TM sig.∀R.
351 (∀t.M ↓ t) → M ⊫ R → M ⊨ R.
352 #sig #M #R #HT #HW #t cases (HT … t) #i * #outc #Hloop
353 @(ex_intro … i) @(ex_intro … outc) % // @(HW … i) //
356 theorem Realize_to_WRealize : ∀sig.∀M:TM sig.∀R.
358 #sig #M #R #H1 #inc #i #outc #Hloop
359 cases (H1 inc) #k * #outc1 * #Hloop1 #HR >(loop_eq … Hloop Hloop1) //
362 definition accRealize ≝ λsig.λM:TM sig.λacc:states sig M.λRtrue,Rfalse.
364 loopM sig M i (initc sig M t) = Some ? outc ∧
365 (cstate ?? outc = acc → Rtrue t (ctape ?? outc)) ∧
366 (cstate ?? outc ≠ acc → Rfalse t (ctape ?? outc)).
368 notation "M ⊨ [q: R1,R2]" non associative with precedence 45 for @{ 'cmodels $M $q $R1 $R2}.
369 interpretation "conditional realizability" 'cmodels M q R1 R2 = (accRealize ? M q R1 R2).
371 (*************************** guarded realizablity *****************************)
372 definition GRealize ≝ λsig.λM:TM sig.λPre:tape sig →Prop.λR:relation (tape sig).
374 loopM sig M i (initc sig M t) = Some ? outc ∧ R t (ctape ?? outc).
376 definition accGRealize ≝ λsig.λM:TM sig.λacc:states sig M.
377 λPre: tape sig → Prop.λRtrue,Rfalse.
379 loopM sig M i (initc sig M t) = Some ? outc ∧
380 (cstate ?? outc = acc → Rtrue t (ctape ?? outc)) ∧
381 (cstate ?? outc ≠ acc → Rfalse t (ctape ?? outc)).
383 lemma WRealize_to_GRealize : ∀sig.∀M: TM sig.∀Pre,R.
384 (∀t.Pre t → M ↓ t) → M ⊫ R → GRealize sig M Pre R.
385 #sig #M #Pre #R #HT #HW #t #HPre cases (HT … t HPre) #i * #outc #Hloop
386 @(ex_intro … i) @(ex_intro … outc) % // @(HW … i) //
389 lemma Realize_to_GRealize : ∀sig,M.∀P,R.
390 M ⊨ R → GRealize sig M P R.
391 #alpha #M #Pre #R #HR #t #HPre
392 cases (HR t) -HR #k * #outc * #Hloop #HR
393 @(ex_intro ?? k) @(ex_intro ?? outc) %
397 lemma acc_Realize_to_acc_GRealize: ∀sig,M.∀q:states sig M.∀P,R1,R2.
398 M ⊨ [q:R1,R2] → accGRealize sig M q P R1 R2.
399 #alpha #M #q #Pre #R1 #R2 #HR #t #HPre
400 cases (HR t) -HR #k * #outc * * #Hloop #HRtrue #HRfalse
401 @(ex_intro ?? k) @(ex_intro ?? outc) %
402 [ % [@Hloop] @HRtrue | @HRfalse]
405 (******************************** monotonicity ********************************)
406 lemma Realize_to_Realize : ∀alpha,M,R1,R2.
407 R1 ⊆ R2 → Realize alpha M R1 → Realize alpha M R2.
408 #alpha #M #R1 #R2 #Himpl #HR1 #intape
409 cases (HR1 intape) -HR1 #k * #outc * #Hloop #HR1
410 @(ex_intro ?? k) @(ex_intro ?? outc) % /2/
413 lemma WRealize_to_WRealize: ∀sig,M,R1,R2.
414 R1 ⊆ R2 → WRealize sig M R1 → WRealize ? M R2.
415 #alpha #M #R1 #R2 #Hsub #HR1 #intape #i #outc #Hloop
416 @Hsub @(HR1 … i) @Hloop
419 lemma GRealize_to_GRealize : ∀alpha,M,P,R1,R2.
420 R1 ⊆ R2 → GRealize alpha M P R1 → GRealize alpha M P R2.
421 #alpha #M #P #R1 #R2 #Himpl #HR1 #intape #HP
422 cases (HR1 intape HP) -HR1 #k * #outc * #Hloop #HR1
423 @(ex_intro ?? k) @(ex_intro ?? outc) % /2/
426 lemma GRealize_to_GRealize_2 : ∀alpha,M,P1,P2,R1,R2.
427 P2 ⊆ P1 → R1 ⊆ R2 → GRealize alpha M P1 R1 → GRealize alpha M P2 R2.
428 #alpha #M #P1 #P2 #R1 #R2 #Himpl1 #Himpl2 #H1 #intape #HP
429 cases (H1 intape (Himpl1 … HP)) -H1 #k * #outc * #Hloop #H1
430 @(ex_intro ?? k) @(ex_intro ?? outc) % /2/
433 lemma acc_Realize_to_acc_Realize: ∀sig,M.∀q:states sig M.∀R1,R2,R3,R4.
434 R1 ⊆ R3 → R2 ⊆ R4 → M ⊨ [q:R1,R2] → M ⊨ [q:R3,R4].
435 #alpha #M #q #R1 #R2 #R3 #R4 #Hsub13 #Hsub24 #HRa #intape
436 cases (HRa intape) -HRa #k * #outc * * #Hloop #HRtrue #HRfalse
437 @(ex_intro ?? k) @(ex_intro ?? outc) %
438 [ % [@Hloop] #Hq @Hsub13 @HRtrue // | #Hq @Hsub24 @HRfalse //]
441 (**************************** A canonical relation ****************************)
443 definition R_TM ≝ λsig.λM:TM sig.λq.λt1,t2.
445 loopM ? M i (mk_config ?? q t1) = Some ? outc ∧
446 t2 = (ctape ?? outc).
448 lemma R_TM_to_R: ∀sig,M,R. ∀t1,t2.
449 M ⊫ R → R_TM ? M (start sig M) t1 t2 → R t1 t2.
450 #sig #M #R #t1 #t2 whd in ⊢ (%→?); #HMR * #i * #outc *
451 #Hloop #Ht2 >Ht2 @(HMR … Hloop)
454 (******************************** NOP Machine *********************************)
459 definition nop_states ≝ initN 1.
460 definition start_nop : initN 1 ≝ mk_Sig ?? 0 (le_n … 1).
463 λalpha:FinSet.mk_TM alpha nop_states
464 (λp.let 〈q,a〉 ≝ p in 〈q,None ?,N〉)
467 definition R_nop ≝ λalpha.λt1,t2:tape alpha.t2 = t1.
470 ∀alpha.nop alpha ⊨ R_nop alpha.
471 #alpha #intape @(ex_intro ?? 1)
472 @(ex_intro … (mk_config ?? start_nop intape)) % %
475 lemma nop_single_state: ∀sig.∀q1,q2:states ? (nop sig). q1 = q2.
476 normalize #sig * #n #ltn1 * #m #ltm1
477 generalize in match ltn1; generalize in match ltm1;
478 <(le_n_O_to_eq … (le_S_S_to_le … ltn1)) <(le_n_O_to_eq … (le_S_S_to_le … ltm1))
481 (************************** Sequential Composition ****************************)
483 definition seq_trans ≝ λsig. λM1,M2 : TM sig.
487 if halt sig M1 s1 then 〈inr … (start sig M2), None ?,N〉
488 else let 〈news1,newa,m〉 ≝ trans sig M1 〈s1,a〉 in 〈inl … news1,newa,m〉
489 | inr s2 ⇒ let 〈news2,newa,m〉 ≝ trans sig M2 〈s2,a〉 in 〈inr … news2,newa,m〉
492 definition seq ≝ λsig. λM1,M2 : TM sig.
494 (FinSum (states sig M1) (states sig M2))
495 (seq_trans sig M1 M2)
496 (inl … (start sig M1))
498 [ inl _ ⇒ false | inr s2 ⇒ halt sig M2 s2]).
500 notation "a · b" right associative with precedence 65 for @{ 'middot $a $b}.
501 interpretation "sequential composition" 'middot a b = (seq ? a b).
503 definition lift_confL ≝
504 λsig,S1,S2,c.match c with
505 [ mk_config s t ⇒ mk_config sig (FinSum S1 S2) (inl … s) t ].
507 definition lift_confR ≝
508 λsig,S1,S2,c.match c with
509 [ mk_config s t ⇒ mk_config sig (FinSum S1 S2) (inr … s) t ].
511 definition halt_liftL ≝
512 λS1,S2,halt.λs:FinSum S1 S2.
515 | inr _ ⇒ true ]. (* should be vacuous in all cases we use halt_liftL *)
517 definition halt_liftR ≝
518 λS1,S2,halt.λs:FinSum S1 S2.
521 | inr s2 ⇒ halt s2 ].
523 lemma p_halt_liftL : ∀sig,S1,S2,halt,c.
524 halt (cstate sig S1 c) =
525 halt_liftL S1 S2 halt (cstate … (lift_confL … c)).
526 #sig #S1 #S2 #halt #c cases c #s #t %
529 lemma trans_seq_liftL : ∀sig,M1,M2,s,a,news,newa,move.
530 halt ? M1 s = false →
531 trans sig M1 〈s,a〉 = 〈news,newa,move〉 →
532 trans sig (seq sig M1 M2) 〈inl … s,a〉 = 〈inl … news,newa,move〉.
533 #sig (*#M1*) * #Q1 #T1 #init1 #halt1 #M2 #s #a #news #newa #move
534 #Hhalt #Htrans whd in ⊢ (??%?); >Hhalt >Htrans %
537 lemma trans_seq_liftR : ∀sig,M1,M2,s,a,news,newa,move.
538 halt ? M2 s = false →
539 trans sig M2 〈s,a〉 = 〈news,newa,move〉 →
540 trans sig (seq sig M1 M2) 〈inr … s,a〉 = 〈inr … news,newa,move〉.
541 #sig #M1 * #Q2 #T2 #init2 #halt2 #s #a #news #newa #move
542 #Hhalt #Htrans whd in ⊢ (??%?); >Hhalt >Htrans %
545 lemma step_seq_liftR : ∀sig,M1,M2,c0.
546 halt ? M2 (cstate ?? c0) = false →
547 step sig (seq sig M1 M2) (lift_confR sig (states ? M1) (states ? M2) c0) =
548 lift_confR sig (states ? M1) (states ? M2) (step sig M2 c0).
549 #sig #M1 (* * #Q1 #T1 #init1 #halt1 *) #M2 * #s #t
550 lapply (refl ? (trans ?? 〈s,current sig t〉))
551 cases (trans ?? 〈s,current sig t〉) in ⊢ (???% → %);
552 * #s0 #a0 #m0 cases t
554 | 2,3: #s1 #l1 #Heq #Hhalt
555 |#ls #s1 #rs #Heq #Hhalt ]
556 whd in ⊢ (???(????%)); >Heq whd in ⊢ (???%);
557 whd in ⊢ (??(???%)?); whd in ⊢ (??%?); >(trans_seq_liftR … Heq) //
560 lemma step_seq_liftL : ∀sig,M1,M2,c0.
561 halt ? M1 (cstate ?? c0) = false →
562 step sig (seq sig M1 M2) (lift_confL sig (states ? M1) (states ? M2) c0) =
563 lift_confL sig ?? (step sig M1 c0).
564 #sig #M1 (* * #Q1 #T1 #init1 #halt1 *) #M2 * #s #t
565 lapply (refl ? (trans ?? 〈s,current sig t〉))
566 cases (trans ?? 〈s,current sig t〉) in ⊢ (???% → %);
567 * #s0 #a0 #m0 cases t
569 | 2,3: #s1 #l1 #Heq #Hhalt
570 |#ls #s1 #rs #Heq #Hhalt ]
571 whd in ⊢ (???(????%)); >Heq whd in ⊢ (???%);
572 whd in ⊢ (??(???%)?); whd in ⊢ (??%?); >(trans_seq_liftL … Heq) //
575 lemma trans_liftL_true : ∀sig,M1,M2,s,a.
577 trans sig (seq sig M1 M2) 〈inl … s,a〉 = 〈inr … (start ? M2),None ?,N〉.
578 #sig #M1 #M2 #s #a #Hhalt whd in ⊢ (??%?); >Hhalt %
581 lemma eq_ctape_lift_conf_L : ∀sig,S1,S2,outc.
582 ctape sig (FinSum S1 S2) (lift_confL … outc) = ctape … outc.
583 #sig #S1 #S2 #outc cases outc #s #t %
586 lemma eq_ctape_lift_conf_R : ∀sig,S1,S2,outc.
587 ctape sig (FinSum S1 S2) (lift_confR … outc) = ctape … outc.
588 #sig #S1 #S2 #outc cases outc #s #t %
591 theorem sem_seq: ∀sig.∀M1,M2:TM sig.∀R1,R2.
592 M1 ⊨ R1 → M2 ⊨ R2 → M1 · M2 ⊨ R1 ∘ R2.
593 #sig #M1 #M2 #R1 #R2 #HR1 #HR2 #t
594 cases (HR1 t) #k1 * #outc1 * #Hloop1 #HM1
595 cases (HR2 (ctape sig (states ? M1) outc1)) #k2 * #outc2 * #Hloop2 #HM2
596 @(ex_intro … (k1+k2)) @(ex_intro … (lift_confR … outc2))
598 [@(loop_merge ???????????
599 (loop_lift ??? (lift_confL sig (states sig M1) (states sig M2))
600 (step sig M1) (step sig (seq sig M1 M2))
601 (λc.halt sig M1 (cstate … c))
602 (λc.halt_liftL ?? (halt sig M1) (cstate … c)) … Hloop1))
604 [ #sl #tl whd in ⊢ (??%? → ?); #Hl %
605 | #sr #tr whd in ⊢ (??%? → ?); #Hr destruct (Hr) ]
606 || #c0 #Hhalt <step_seq_liftL //
608 |6:cases outc1 #s1 #t1 %
609 |7:@(loop_lift … (initc ?? (ctape … outc1)) … Hloop2)
611 | #c0 #Hhalt <step_seq_liftR // ]
612 |whd in ⊢ (??(???%)?);whd in ⊢ (??%?);
613 generalize in match Hloop1; cases outc1 #sc1 #tc1 #Hloop10
614 >(trans_liftL_true sig M1 M2 ??)
615 [ whd in ⊢ (??%?); whd in ⊢ (???%);
616 @config_eq whd in ⊢ (???%); //
617 | @(loop_Some ?????? Hloop10) ]
619 | @(ex_intro … (ctape ? (FinSum (states ? M1) (states ? M2)) (lift_confL … outc1)))
620 % // >eq_ctape_lift_conf_L >eq_ctape_lift_conf_R //
624 theorem sem_seq_app: ∀sig.∀M1,M2:TM sig.∀R1,R2,R3.
625 M1 ⊨ R1 → M2 ⊨ R2 → R1 ∘ R2 ⊆ R3 → M1 · M2 ⊨ R3.
626 #sig #M1 #M2 #R1 #R2 #R3 #HR1 #HR2 #Hsub
627 #t cases (sem_seq … HR1 HR2 t)
628 #k * #outc * #Hloop #Houtc @(ex_intro … k) @(ex_intro … outc)
629 % [@Hloop |@Hsub @Houtc]
632 (* composition with guards *)
633 theorem sem_seq_guarded: ∀sig.∀M1,M2:TM sig.∀Pre1,Pre2,R1,R2.
634 GRealize sig M1 Pre1 R1 → GRealize sig M2 Pre2 R2 →
635 (∀t1,t2.Pre1 t1 → R1 t1 t2 → Pre2 t2) →
636 GRealize sig (M1 · M2) Pre1 (R1 ∘ R2).
637 #sig #M1 #M2 #Pre1 #Pre2 #R1 #R2 #HGR1 #HGR2 #Hinv #t1 #HPre1
638 cases (HGR1 t1 HPre1) #k1 * #outc1 * #Hloop1 #HM1
639 cases (HGR2 (ctape sig (states ? M1) outc1) ?)
640 [2: @(Hinv … HPre1 HM1)]
641 #k2 * #outc2 * #Hloop2 #HM2
642 @(ex_intro … (k1+k2)) @(ex_intro … (lift_confR … outc2))
644 [@(loop_merge ???????????
645 (loop_lift ??? (lift_confL sig (states sig M1) (states sig M2))
646 (step sig M1) (step sig (seq sig M1 M2))
647 (λc.halt sig M1 (cstate … c))
648 (λc.halt_liftL ?? (halt sig M1) (cstate … c)) … Hloop1))
650 [ #sl #tl whd in ⊢ (??%? → ?); #Hl %
651 | #sr #tr whd in ⊢ (??%? → ?); #Hr destruct (Hr) ]
652 || #c0 #Hhalt <step_seq_liftL //
654 |6:cases outc1 #s1 #t1 %
655 |7:@(loop_lift … (initc ?? (ctape … outc1)) … Hloop2)
657 | #c0 #Hhalt <step_seq_liftR // ]
658 |whd in ⊢ (??(???%)?);whd in ⊢ (??%?);
659 generalize in match Hloop1; cases outc1 #sc1 #tc1 #Hloop10
660 >(trans_liftL_true sig M1 M2 ??)
661 [ whd in ⊢ (??%?); whd in ⊢ (???%);
662 @config_eq whd in ⊢ (???%); //
663 | @(loop_Some ?????? Hloop10) ]
665 | @(ex_intro … (ctape ? (FinSum (states ? M1) (states ? M2)) (lift_confL … outc1)))
666 % // >eq_ctape_lift_conf_L >eq_ctape_lift_conf_R //
670 theorem sem_seq_app_guarded: ∀sig.∀M1,M2:TM sig.∀Pre1,Pre2,R1,R2,R3.
671 GRealize sig M1 Pre1 R1 → GRealize sig M2 Pre2 R2 →
672 (∀t1,t2.Pre1 t1 → R1 t1 t2 → Pre2 t2) → R1 ∘ R2 ⊆ R3 →
673 GRealize sig (M1 · M2) Pre1 R3.
674 #sig #M1 #M2 #Pre1 #Pre2 #R1 #R2 #R3 #HR1 #HR2 #Hinv #Hsub
675 #t #HPre1 cases (sem_seq_guarded … HR1 HR2 Hinv t HPre1)
676 #k * #outc * #Hloop #Houtc @(ex_intro … k) @(ex_intro … outc)
677 % [@Hloop |@Hsub @Houtc]
680 theorem acc_sem_seq : ∀sig.∀M1,M2:TM sig.∀R1,Rtrue,Rfalse,acc.
681 M1 ⊨ R1 → M2 ⊨ [ acc: Rtrue, Rfalse ] →
682 M1 · M2 ⊨ [ inr … acc: R1 ∘ Rtrue, R1 ∘ Rfalse ].
683 #sig #M1 #M2 #R1 #Rtrue #Rfalse #acc #HR1 #HR2 #t
684 cases (HR1 t) #k1 * #outc1 * #Hloop1 #HM1
685 cases (HR2 (ctape sig (states ? M1) outc1)) #k2 * #outc2 * * #Hloop2
687 @(ex_intro … (k1+k2)) @(ex_intro … (lift_confR … outc2))
690 (loop_lift ??? (lift_confL sig (states sig M1) (states sig M2))
691 (step sig M1) (step sig (seq sig M1 M2))
692 (λc.halt sig M1 (cstate … c))
693 (λc.halt_liftL ?? (halt sig M1) (cstate … c)) … Hloop1))
695 [ #sl #tl whd in ⊢ (??%? → ?); #Hl %
696 | #sr #tr whd in ⊢ (??%? → ?); #Hr destruct (Hr) ]
697 || #c0 #Hhalt <step_seq_liftL //
699 |6:cases outc1 #s1 #t1 %
700 |7:@(loop_lift … (initc ?? (ctape … outc1)) … Hloop2)
702 | #c0 #Hhalt <step_seq_liftR // ]
703 |whd in ⊢ (??(???%)?);whd in ⊢ (??%?);
704 generalize in match Hloop1; cases outc1 #sc1 #tc1 #Hloop10
705 >(trans_liftL_true sig M1 M2 ??)
706 [ whd in ⊢ (??%?); whd in ⊢ (???%); //
707 | @(loop_Some ?????? Hloop10) ]
709 | >(config_expand … outc2) in ⊢ (%→?); whd in ⊢ (??%?→?);
710 #Hqtrue destruct (Hqtrue)
711 @(ex_intro … (ctape ? (FinSum (states ? M1) (states ? M2)) (lift_confL … outc1)))
712 % // >eq_ctape_lift_conf_L >eq_ctape_lift_conf_R /2/ ]
713 | >(config_expand … outc2) in ⊢ (%→?); whd in ⊢ (?(??%?)→?); #Hqfalse
714 @(ex_intro … (ctape ? (FinSum (states ? M1) (states ? M2)) (lift_confL … outc1)))
715 % // >eq_ctape_lift_conf_L >eq_ctape_lift_conf_R @HMfalse
716 @(not_to_not … Hqfalse) //
720 lemma acc_sem_seq_app : ∀sig.∀M1,M2:TM sig.∀R1,Rtrue,Rfalse,R2,R3,acc.
721 M1 ⊨ R1 → M2 ⊨ [acc: Rtrue, Rfalse] →
722 (∀t1,t2,t3. R1 t1 t3 → Rtrue t3 t2 → R2 t1 t2) →
723 (∀t1,t2,t3. R1 t1 t3 → Rfalse t3 t2 → R3 t1 t2) →
724 M1 · M2 ⊨ [inr … acc : R2, R3].
725 #sig #M1 #M2 #R1 #Rtrue #Rfalse #R2 #R3 #acc
726 #HR1 #HRacc #Hsub1 #Hsub2
727 #t cases (acc_sem_seq … HR1 HRacc t)
728 #k * #outc * * #Hloop #Houtc1 #Houtc2 @(ex_intro … k) @(ex_intro … outc)
730 |#H cases (Houtc1 H) #t3 * #Hleft #Hright @Hsub1 // ]
731 |#H cases (Houtc2 H) #t3 * #Hleft #Hright @Hsub2 // ]