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/finset.ma".
13 include "basics/vectors.ma".
14 include "basics/finset.ma".
15 (* include "basics/relations.ma". *)
17 (******************************** tape ****************************************)
19 (* A tape is essentially a triple 〈left,current,right〉 where however the current
20 symbol could be missing. This may happen for three different reasons: both tapes
21 are empty; we are on the left extremity of a non-empty tape (left overflow), or
22 we are on the right extremity of a non-empty tape (right overflow). *)
24 inductive tape (sig:FinSet) : Type[0] ≝
26 | leftof : sig → list sig → tape sig
27 | rightof : sig → list sig → tape sig
28 | midtape : list sig → sig → list sig → tape sig.
31 λsig.λt:tape sig.match t with
32 [ niltape ⇒ [] | leftof _ _ ⇒ [] | rightof s l ⇒ s::l | midtape l _ _ ⇒ l ].
35 λsig.λt:tape sig.match t with
36 [ niltape ⇒ [] | leftof s r ⇒ s::r | rightof _ _ ⇒ []| midtape _ _ r ⇒ r ].
39 λsig.λt:tape sig.match t with
40 [ midtape _ c _ ⇒ Some ? c | _ ⇒ None ? ].
43 ∀sig:FinSet.list sig → option sig → list sig → tape sig ≝
44 λsig,lt,c,rt.match c with
45 [ Some c' ⇒ midtape sig lt c' rt
46 | None ⇒ match lt with
49 | cons r0 rs0 ⇒ leftof ? r0 rs0 ]
50 | cons l0 ls0 ⇒ rightof ? l0 ls0 ] ].
53 ∀sig,ls,c,rs.(c = None ? → ls = [ ] ∨ rs = [ ]) → right ? (mk_tape sig ls c rs) = rs.
54 #sig #ls #c #rs cases c // cases ls
56 | #l0 #ls0 #H normalize cases (H (refl ??)) #H1 [ destruct (H1) | >H1 % ] ]
59 lemma left_mk_tape : ∀sig,ls,c,rs.left ? (mk_tape sig ls c rs) = ls.
60 #sig #ls #c #rs cases c // cases ls // cases rs //
63 lemma current_mk_tape : ∀sig,ls,c,rs.current ? (mk_tape sig ls c rs) = c.
64 #sig #ls #c #rs cases c // cases ls // cases rs //
67 lemma current_to_midtape: ∀sig,t,c. current sig t = Some ? c →
68 ∃ls,rs. t = midtape ? ls c rs.
70 [#c whd in ⊢ ((??%?)→?); #Hfalse destruct
71 |#a #l #c whd in ⊢ ((??%?)→?); #Hfalse destruct
72 |#a #l #c whd in ⊢ ((??%?)→?); #Hfalse destruct
73 |#ls #a #rs #c whd in ⊢ ((??%?)→?); #H destruct
74 @(ex_intro … ls) @(ex_intro … rs) //
78 (*********************************** moves ************************************)
80 inductive move : Type[0] ≝
81 | L : move | R : move | N : move.
83 (********************************** machine ***********************************)
85 record TM (sig:FinSet): Type[1] ≝
87 trans : states × (option sig) → states × (option sig) × move;
92 definition tape_move_left ≝ λsig:FinSet.λt:tape sig.
94 [ niltape ⇒ niltape sig
96 | rightof a ls ⇒ midtape sig ls a [ ]
99 [ nil ⇒ leftof sig a rs
100 | cons a0 ls0 ⇒ midtape sig ls0 a0 (a::rs)
104 definition tape_move_right ≝ λsig:FinSet.λt:tape sig.
106 [ niltape ⇒ niltape sig
108 | leftof a rs ⇒ midtape sig [ ] a rs
111 [ nil ⇒ rightof sig a ls
112 | cons a0 rs0 ⇒ midtape sig (a::ls) a0 rs0
116 definition tape_write ≝ λsig.λt: tape sig.λs:option sig.
119 | Some s0 ⇒ midtape ? (left ? t) s0 (right ? t)
122 definition tape_move ≝ λsig.λt: tape sig.λm:move.
124 [ R ⇒ tape_move_right ? t
125 | L ⇒ tape_move_left ? t
129 definition tape_move_mono ≝
131 tape_move sig (tape_write sig t (\fst mv)) (\snd mv).
133 record config (sig,states:FinSet): Type[0] ≝
138 lemma config_expand: ∀sig,Q,c.
139 c = mk_config sig Q (cstate ?? c) (ctape ?? c).
143 lemma config_eq : ∀sig,M,c1,c2.
144 cstate sig M c1 = cstate sig M c2 →
145 ctape sig M c1 = ctape sig M c2 → c1 = c2.
146 #sig #M1 * #s1 #t1 * #s2 #t2 //
149 definition step ≝ λsig.λM:TM sig.λc:config sig (states sig M).
150 let current_char ≝ current ? (ctape ?? c) in
151 let 〈news,a,mv〉 ≝ trans sig M 〈cstate ?? c,current_char〉 in
152 mk_config ?? news (tape_move sig (tape_write ? (ctape ?? c) a) mv).
155 lemma step_eq : ∀sig,M,c.
156 let current_char ≝ current ? (ctape ?? c) in
157 let 〈news,a,mv〉 ≝ trans sig M 〈cstate ?? c,current_char〉 in
159 mk_config ?? news (tape_move sig (tape_write ? (ctape ?? c) a) mv).
161 whd in match (tape_move_mono sig ??);
164 (******************************** loop ****************************************)
165 let rec loop (A:Type[0]) n (f:A→A) p a on n ≝
168 | S m ⇒ if p a then (Some ? a) else loop A m f p (f a)
172 ∀A,n,f,p,a. p a = true →
173 loop A (S n) f p a = Some ? a.
174 #A #n #f #p #a #pa normalize >pa //
178 ∀A,n,f,p,a. p a = false →
179 loop A (S n) f p a = loop A n f p (f a).
180 normalize #A #n #f #p #a #Hpa >Hpa %
183 lemma loop_incr : ∀A,f,p,k1,k2,a1,a2.
184 loop A k1 f p a1 = Some ? a2 →
185 loop A (k2+k1) f p a1 = Some ? a2.
186 #A #f #p #k1 #k2 #a1 #a2 generalize in match a1; elim k1
187 [normalize #a0 #Hfalse destruct
188 |#k1' #IH #a0 <plus_n_Sm whd in ⊢ (??%? → ??%?);
189 cases (true_or_false (p a0)) #Hpa0 >Hpa0 whd in ⊢ (??%? → ??%?); // @IH
193 lemma loop_merge : ∀A,f,p,q.(∀b. p b = false → q b = false) →
195 loop A k1 f p a1 = Some ? a2 →
196 f a2 = a3 → q a2 = false →
197 loop A k2 f q a3 = Some ? a4 →
198 loop A (k1+k2) f q a1 = Some ? a4.
199 #Sig #f #p #q #Hpq #k1 elim k1
200 [normalize #k2 #a1 #a2 #a3 #a4 #H destruct
201 |#k1' #Hind #k2 #a1 #a2 #a3 #a4 normalize in ⊢ (%→?);
202 cases (true_or_false (p a1)) #pa1 >pa1 normalize in ⊢ (%→?);
203 [#eqa1a2 destruct #eqa2a3 #Hqa2 #H
204 whd in ⊢ (??(??%???)?); >plus_n_Sm @loop_incr
205 whd in ⊢ (??%?); >Hqa2 >eqa2a3 @H
206 |normalize >(Hpq … pa1) normalize #H1 #H2 #H3 @(Hind … H2) //
211 lemma loop_split : ∀A,f,p,q.(∀b. q b = true → p b = true) →
213 loop A k f q a1 = Some ? a2 →
215 loop A k1 f p a1 = Some ? a3 ∧
216 loop A (S(k-k1)) f q a3 = Some ? a2.
217 #A #f #p #q #Hpq #k elim k
218 [#a1 #a2 normalize #Heq destruct
219 |#i #Hind #a1 #a2 normalize
220 cases (true_or_false (q a1)) #Hqa1 >Hqa1 normalize
222 @(ex_intro … 1) @(ex_intro … a2) %
223 [normalize >(Hpq …Hqa1) // |>Hqa1 //]
224 |#Hloop cases (true_or_false (p a1)) #Hpa1
225 [@(ex_intro … 1) @(ex_intro … a1) %
226 [normalize >Hpa1 // |>Hqa1 <Hloop normalize //]
227 |cases (Hind …Hloop) #k2 * #a3 * #Hloop1 #Hloop2
228 @(ex_intro … (S k2)) @(ex_intro … a3) %
229 [normalize >Hpa1 normalize // | @Hloop2 ]
235 lemma loop_eq : ∀sig,f,q,i,j,a,x,y.
236 loop sig i f q a = Some ? x → loop sig j f q a = Some ? y → x = y.
237 #sig #f #q #i #j @(nat_elim2 … i j)
238 [ #n #a #x #y normalize #Hfalse destruct (Hfalse)
239 | #n #a #x #y #H1 normalize #Hfalse destruct (Hfalse)
240 | #n1 #n2 #IH #a #x #y normalize cases (q a) normalize
247 ∀A,k,f,p,a.p a = true → loop A (S k) f p a = Some ? a.
248 #A #k #f #p #a #Ha normalize >Ha %
252 ∀A,k,f,p,a,b.loop A k f p a = Some ? b → p b = true.
254 [#a #b normalize #Hfalse destruct
255 |#k0 #IH #a #b whd in ⊢ (??%? → ?); cases (true_or_false (p a)) #Hpa
256 [ >Hpa normalize #H1 destruct // | >Hpa normalize @IH ]
260 lemma loop_lift : ∀A,B,k,lift,f,g,h,hlift,c1,c2.
261 (∀x.hlift (lift x) = h x) →
262 (∀x.h x = false → lift (f x) = g (lift x)) →
263 loop A k f h c1 = Some ? c2 →
264 loop B k g hlift (lift c1) = Some ? (lift … c2).
265 #A #B #k #lift #f #g #h #hlift #c1 #c2 #Hfg #Hhlift
266 generalize in match c1; elim k
267 [#c0 normalize in ⊢ (??%? → ?); #Hfalse destruct (Hfalse)
268 |#k0 #IH #c0 whd in ⊢ (??%? → ??%?);
269 cases (true_or_false (h c0)) #Hc0 >Hfg >Hc0 normalize
270 [ #Heq destruct (Heq) % | <Hhlift // @IH ]
273 (************************** Realizability *************************************)
274 definition loopM ≝ λsig,M,i,cin.
275 loop ? i (step sig M) (λc.halt sig M (cstate ?? c)) cin.
277 lemma loopM_unfold : ∀sig,M,i,cin.
278 loopM sig M i cin = loop ? i (step sig M) (λc.halt sig M (cstate ?? c)) cin.
281 definition initc ≝ λsig.λM:TM sig.λt.
282 mk_config sig (states sig M) (start sig M) t.
284 definition Realize ≝ λsig.λM:TM sig.λR:relation (tape sig).
286 loopM sig M i (initc sig M t) = Some ? outc ∧ R t (ctape ?? outc).
288 definition WRealize ≝ λsig.λM:TM sig.λR:relation (tape sig).
290 loopM sig M i (initc sig M t) = Some ? outc → R t (ctape ?? outc).
292 definition Terminate ≝ λsig.λM:TM sig.λt. ∃i,outc.
293 loopM sig M i (initc sig M t) = Some ? outc.
295 notation "M \vDash R" non associative with precedence 45 for @{ 'models $M $R}.
296 interpretation "realizability" 'models M R = (Realize ? M R).
298 notation "M \VDash R" non associative with precedence 45 for @{ 'wmodels $M $R}.
299 interpretation "weak realizability" 'wmodels M R = (WRealize ? M R).
301 interpretation "termination" 'fintersects M t = (Terminate ? M t).
303 lemma WRealize_to_Realize : ∀sig.∀M: TM sig.∀R.
304 (∀t.M ↓ t) → M ⊫ R → M ⊨ R.
305 #sig #M #R #HT #HW #t cases (HT … t) #i * #outc #Hloop
306 @(ex_intro … i) @(ex_intro … outc) % // @(HW … i) //
309 theorem Realize_to_WRealize : ∀sig.∀M:TM sig.∀R.
311 #sig #M #R #H1 #inc #i #outc #Hloop
312 cases (H1 inc) #k * #outc1 * #Hloop1 #HR >(loop_eq … Hloop Hloop1) //
315 definition accRealize ≝ λsig.λM:TM sig.λacc:states sig M.λRtrue,Rfalse.
317 loopM sig M i (initc sig M t) = Some ? outc ∧
318 (cstate ?? outc = acc → Rtrue t (ctape ?? outc)) ∧
319 (cstate ?? outc ≠ acc → Rfalse t (ctape ?? outc)).
321 notation "M ⊨ [q: R1,R2]" non associative with precedence 45 for @{ 'cmodels $M $q $R1 $R2}.
322 interpretation "conditional realizability" 'cmodels M q R1 R2 = (accRealize ? M q R1 R2).
324 (*************************** guarded realizablity *****************************)
325 definition GRealize ≝ λsig.λM:TM sig.λPre:tape sig →Prop.λR:relation (tape sig).
327 loopM sig M i (initc sig M t) = Some ? outc ∧ R t (ctape ?? outc).
329 definition accGRealize ≝ λsig.λM:TM sig.λacc:states sig M.
330 λPre: tape sig → Prop.λRtrue,Rfalse.
332 loopM sig M i (initc sig M t) = Some ? outc ∧
333 (cstate ?? outc = acc → Rtrue t (ctape ?? outc)) ∧
334 (cstate ?? outc ≠ acc → Rfalse t (ctape ?? outc)).
336 lemma WRealize_to_GRealize : ∀sig.∀M: TM sig.∀Pre,R.
337 (∀t.Pre t → M ↓ t) → M ⊫ R → GRealize sig M Pre R.
338 #sig #M #Pre #R #HT #HW #t #HPre cases (HT … t HPre) #i * #outc #Hloop
339 @(ex_intro … i) @(ex_intro … outc) % // @(HW … i) //
342 lemma Realize_to_GRealize : ∀sig,M.∀P,R.
343 M ⊨ R → GRealize sig M P R.
344 #alpha #M #Pre #R #HR #t #HPre
345 cases (HR t) -HR #k * #outc * #Hloop #HR
346 @(ex_intro ?? k) @(ex_intro ?? outc) %
350 lemma acc_Realize_to_acc_GRealize: ∀sig,M.∀q:states sig M.∀P,R1,R2.
351 M ⊨ [q:R1,R2] → accGRealize sig M q P R1 R2.
352 #alpha #M #q #Pre #R1 #R2 #HR #t #HPre
353 cases (HR t) -HR #k * #outc * * #Hloop #HRtrue #HRfalse
354 @(ex_intro ?? k) @(ex_intro ?? outc) %
355 [ % [@Hloop] @HRtrue | @HRfalse]
358 (******************************** monotonicity ********************************)
359 lemma Realize_to_Realize : ∀alpha,M,R1,R2.
360 R1 ⊆ R2 → Realize alpha M R1 → Realize alpha M R2.
361 #alpha #M #R1 #R2 #Himpl #HR1 #intape
362 cases (HR1 intape) -HR1 #k * #outc * #Hloop #HR1
363 @(ex_intro ?? k) @(ex_intro ?? outc) % /2/
366 lemma WRealize_to_WRealize: ∀sig,M,R1,R2.
367 R1 ⊆ R2 → WRealize sig M R1 → WRealize ? M R2.
368 #alpha #M #R1 #R2 #Hsub #HR1 #intape #i #outc #Hloop
369 @Hsub @(HR1 … i) @Hloop
372 lemma GRealize_to_GRealize : ∀alpha,M,P,R1,R2.
373 R1 ⊆ R2 → GRealize alpha M P R1 → GRealize alpha M P R2.
374 #alpha #M #P #R1 #R2 #Himpl #HR1 #intape #HP
375 cases (HR1 intape HP) -HR1 #k * #outc * #Hloop #HR1
376 @(ex_intro ?? k) @(ex_intro ?? outc) % /2/
379 lemma GRealize_to_GRealize_2 : ∀alpha,M,P1,P2,R1,R2.
380 P2 ⊆ P1 → R1 ⊆ R2 → GRealize alpha M P1 R1 → GRealize alpha M P2 R2.
381 #alpha #M #P1 #P2 #R1 #R2 #Himpl1 #Himpl2 #H1 #intape #HP
382 cases (H1 intape (Himpl1 … HP)) -H1 #k * #outc * #Hloop #H1
383 @(ex_intro ?? k) @(ex_intro ?? outc) % /2/
386 lemma acc_Realize_to_acc_Realize: ∀sig,M.∀q:states sig M.∀R1,R2,R3,R4.
387 R1 ⊆ R3 → R2 ⊆ R4 → M ⊨ [q:R1,R2] → M ⊨ [q:R3,R4].
388 #alpha #M #q #R1 #R2 #R3 #R4 #Hsub13 #Hsub24 #HRa #intape
389 cases (HRa intape) -HRa #k * #outc * * #Hloop #HRtrue #HRfalse
390 @(ex_intro ?? k) @(ex_intro ?? outc) %
391 [ % [@Hloop] #Hq @Hsub13 @HRtrue // | #Hq @Hsub24 @HRfalse //]
394 (**************************** A canonical relation ****************************)
396 definition R_TM ≝ λsig.λM:TM sig.λq.λt1,t2.
398 loopM ? M i (mk_config ?? q t1) = Some ? outc ∧
399 t2 = (ctape ?? outc).
401 lemma R_TM_to_R: ∀sig,M,R. ∀t1,t2.
402 M ⊫ R → R_TM ? M (start sig M) t1 t2 → R t1 t2.
403 #sig #M #R #t1 #t2 whd in ⊢ (%→?); #HMR * #i * #outc *
404 #Hloop #Ht2 >Ht2 @(HMR … Hloop)
407 (******************************** NOP Machine *********************************)
412 definition nop_states ≝ initN 1.
413 definition start_nop : initN 1 ≝ mk_Sig ?? 0 (le_n … 1).
416 λalpha:FinSet.mk_TM alpha nop_states
417 (λp.let 〈q,a〉 ≝ p in 〈q,None ?,N〉)
420 definition R_nop ≝ λalpha.λt1,t2:tape alpha.t2 = t1.
423 ∀alpha.nop alpha ⊨ R_nop alpha.
424 #alpha #intape @(ex_intro ?? 1)
425 @(ex_intro … (mk_config ?? start_nop intape)) % %
428 lemma nop_single_state: ∀sig.∀q1,q2:states ? (nop sig). q1 = q2.
429 normalize #sig * #n #ltn1 * #m #ltm1
430 generalize in match ltn1; generalize in match ltm1;
431 <(le_n_O_to_eq … (le_S_S_to_le … ltn1)) <(le_n_O_to_eq … (le_S_S_to_le … ltm1))
434 (************************** Sequential Composition ****************************)
436 definition seq_trans ≝ λsig. λM1,M2 : TM sig.
440 if halt sig M1 s1 then 〈inr … (start sig M2), None ?,N〉
441 else let 〈news1,newa,m〉 ≝ trans sig M1 〈s1,a〉 in 〈inl … news1,newa,m〉
442 | inr s2 ⇒ let 〈news2,newa,m〉 ≝ trans sig M2 〈s2,a〉 in 〈inr … news2,newa,m〉
445 definition seq ≝ λsig. λM1,M2 : TM sig.
447 (FinSum (states sig M1) (states sig M2))
448 (seq_trans sig M1 M2)
449 (inl … (start sig M1))
451 [ inl _ ⇒ false | inr s2 ⇒ halt sig M2 s2]).
453 notation "a · b" right associative with precedence 65 for @{ 'middot $a $b}.
454 interpretation "sequential composition" 'middot a b = (seq ? a b).
456 definition lift_confL ≝
457 λsig,S1,S2,c.match c with
458 [ mk_config s t ⇒ mk_config sig (FinSum S1 S2) (inl … s) t ].
460 definition lift_confR ≝
461 λsig,S1,S2,c.match c with
462 [ mk_config s t ⇒ mk_config sig (FinSum S1 S2) (inr … s) t ].
464 definition halt_liftL ≝
465 λS1,S2,halt.λs:FinSum S1 S2.
468 | inr _ ⇒ true ]. (* should be vacuous in all cases we use halt_liftL *)
470 definition halt_liftR ≝
471 λS1,S2,halt.λs:FinSum S1 S2.
474 | inr s2 ⇒ halt s2 ].
476 lemma p_halt_liftL : ∀sig,S1,S2,halt,c.
477 halt (cstate sig S1 c) =
478 halt_liftL S1 S2 halt (cstate … (lift_confL … c)).
479 #sig #S1 #S2 #halt #c cases c #s #t %
482 lemma trans_seq_liftL : ∀sig,M1,M2,s,a,news,newa,move.
483 halt ? M1 s = false →
484 trans sig M1 〈s,a〉 = 〈news,newa,move〉 →
485 trans sig (seq sig M1 M2) 〈inl … s,a〉 = 〈inl … news,newa,move〉.
486 #sig (*#M1*) * #Q1 #T1 #init1 #halt1 #M2 #s #a #news #newa #move
487 #Hhalt #Htrans whd in ⊢ (??%?); >Hhalt >Htrans %
490 lemma trans_seq_liftR : ∀sig,M1,M2,s,a,news,newa,move.
491 halt ? M2 s = false →
492 trans sig M2 〈s,a〉 = 〈news,newa,move〉 →
493 trans sig (seq sig M1 M2) 〈inr … s,a〉 = 〈inr … news,newa,move〉.
494 #sig #M1 * #Q2 #T2 #init2 #halt2 #s #a #news #newa #move
495 #Hhalt #Htrans whd in ⊢ (??%?); >Hhalt >Htrans %
498 lemma step_seq_liftR : ∀sig,M1,M2,c0.
499 halt ? M2 (cstate ?? c0) = false →
500 step sig (seq sig M1 M2) (lift_confR sig (states ? M1) (states ? M2) c0) =
501 lift_confR sig (states ? M1) (states ? M2) (step sig M2 c0).
502 #sig #M1 (* * #Q1 #T1 #init1 #halt1 *) #M2 * #s #t
503 lapply (refl ? (trans ?? 〈s,current sig t〉))
504 cases (trans ?? 〈s,current sig t〉) in ⊢ (???% → %);
505 * #s0 #a0 #m0 cases t
507 | 2,3: #s1 #l1 #Heq #Hhalt
508 |#ls #s1 #rs #Heq #Hhalt ]
509 whd in ⊢ (???(????%)); >Heq whd in ⊢ (???%);
510 whd in ⊢ (??(???%)?); whd in ⊢ (??%?); >(trans_seq_liftR … Heq) //
513 lemma step_seq_liftL : ∀sig,M1,M2,c0.
514 halt ? M1 (cstate ?? c0) = false →
515 step sig (seq sig M1 M2) (lift_confL sig (states ? M1) (states ? M2) c0) =
516 lift_confL sig ?? (step sig M1 c0).
517 #sig #M1 (* * #Q1 #T1 #init1 #halt1 *) #M2 * #s #t
518 lapply (refl ? (trans ?? 〈s,current sig t〉))
519 cases (trans ?? 〈s,current sig t〉) in ⊢ (???% → %);
520 * #s0 #a0 #m0 cases t
522 | 2,3: #s1 #l1 #Heq #Hhalt
523 |#ls #s1 #rs #Heq #Hhalt ]
524 whd in ⊢ (???(????%)); >Heq whd in ⊢ (???%);
525 whd in ⊢ (??(???%)?); whd in ⊢ (??%?); >(trans_seq_liftL … Heq) //
528 lemma trans_liftL_true : ∀sig,M1,M2,s,a.
530 trans sig (seq sig M1 M2) 〈inl … s,a〉 = 〈inr … (start ? M2),None ?,N〉.
531 #sig #M1 #M2 #s #a #Hhalt whd in ⊢ (??%?); >Hhalt %
534 lemma eq_ctape_lift_conf_L : ∀sig,S1,S2,outc.
535 ctape sig (FinSum S1 S2) (lift_confL … outc) = ctape … outc.
536 #sig #S1 #S2 #outc cases outc #s #t %
539 lemma eq_ctape_lift_conf_R : ∀sig,S1,S2,outc.
540 ctape sig (FinSum S1 S2) (lift_confR … outc) = ctape … outc.
541 #sig #S1 #S2 #outc cases outc #s #t %
544 theorem sem_seq: ∀sig.∀M1,M2:TM sig.∀R1,R2.
545 M1 ⊨ R1 → M2 ⊨ R2 → M1 · M2 ⊨ R1 ∘ R2.
546 #sig #M1 #M2 #R1 #R2 #HR1 #HR2 #t
547 cases (HR1 t) #k1 * #outc1 * #Hloop1 #HM1
548 cases (HR2 (ctape sig (states ? M1) outc1)) #k2 * #outc2 * #Hloop2 #HM2
549 @(ex_intro … (k1+k2)) @(ex_intro … (lift_confR … outc2))
551 [@(loop_merge ???????????
552 (loop_lift ??? (lift_confL sig (states sig M1) (states sig M2))
553 (step sig M1) (step sig (seq sig M1 M2))
554 (λc.halt sig M1 (cstate … c))
555 (λc.halt_liftL ?? (halt sig M1) (cstate … c)) … Hloop1))
557 [ #sl #tl whd in ⊢ (??%? → ?); #Hl %
558 | #sr #tr whd in ⊢ (??%? → ?); #Hr destruct (Hr) ]
559 || #c0 #Hhalt <step_seq_liftL //
561 |6:cases outc1 #s1 #t1 %
562 |7:@(loop_lift … (initc ?? (ctape … outc1)) … Hloop2)
564 | #c0 #Hhalt <step_seq_liftR // ]
565 |whd in ⊢ (??(???%)?);whd in ⊢ (??%?);
566 generalize in match Hloop1; cases outc1 #sc1 #tc1 #Hloop10
567 >(trans_liftL_true sig M1 M2 ??)
568 [ whd in ⊢ (??%?); whd in ⊢ (???%);
569 @config_eq whd in ⊢ (???%); //
570 | @(loop_Some ?????? Hloop10) ]
572 | @(ex_intro … (ctape ? (FinSum (states ? M1) (states ? M2)) (lift_confL … outc1)))
573 % // >eq_ctape_lift_conf_L >eq_ctape_lift_conf_R //
577 theorem sem_seq_app: ∀sig.∀M1,M2:TM sig.∀R1,R2,R3.
578 M1 ⊨ R1 → M2 ⊨ R2 → R1 ∘ R2 ⊆ R3 → M1 · M2 ⊨ R3.
579 #sig #M1 #M2 #R1 #R2 #R3 #HR1 #HR2 #Hsub
580 #t cases (sem_seq … HR1 HR2 t)
581 #k * #outc * #Hloop #Houtc @(ex_intro … k) @(ex_intro … outc)
582 % [@Hloop |@Hsub @Houtc]
585 (* composition with guards *)
586 theorem sem_seq_guarded: ∀sig.∀M1,M2:TM sig.∀Pre1,Pre2,R1,R2.
587 GRealize sig M1 Pre1 R1 → GRealize sig M2 Pre2 R2 →
588 (∀t1,t2.Pre1 t1 → R1 t1 t2 → Pre2 t2) →
589 GRealize sig (M1 · M2) Pre1 (R1 ∘ R2).
590 #sig #M1 #M2 #Pre1 #Pre2 #R1 #R2 #HGR1 #HGR2 #Hinv #t1 #HPre1
591 cases (HGR1 t1 HPre1) #k1 * #outc1 * #Hloop1 #HM1
592 cases (HGR2 (ctape sig (states ? M1) outc1) ?)
593 [2: @(Hinv … HPre1 HM1)]
594 #k2 * #outc2 * #Hloop2 #HM2
595 @(ex_intro … (k1+k2)) @(ex_intro … (lift_confR … outc2))
597 [@(loop_merge ???????????
598 (loop_lift ??? (lift_confL sig (states sig M1) (states sig M2))
599 (step sig M1) (step sig (seq sig M1 M2))
600 (λc.halt sig M1 (cstate … c))
601 (λc.halt_liftL ?? (halt sig M1) (cstate … c)) … Hloop1))
603 [ #sl #tl whd in ⊢ (??%? → ?); #Hl %
604 | #sr #tr whd in ⊢ (??%? → ?); #Hr destruct (Hr) ]
605 || #c0 #Hhalt <step_seq_liftL //
607 |6:cases outc1 #s1 #t1 %
608 |7:@(loop_lift … (initc ?? (ctape … outc1)) … Hloop2)
610 | #c0 #Hhalt <step_seq_liftR // ]
611 |whd in ⊢ (??(???%)?);whd in ⊢ (??%?);
612 generalize in match Hloop1; cases outc1 #sc1 #tc1 #Hloop10
613 >(trans_liftL_true sig M1 M2 ??)
614 [ whd in ⊢ (??%?); whd in ⊢ (???%);
615 @config_eq whd in ⊢ (???%); //
616 | @(loop_Some ?????? Hloop10) ]
618 | @(ex_intro … (ctape ? (FinSum (states ? M1) (states ? M2)) (lift_confL … outc1)))
619 % // >eq_ctape_lift_conf_L >eq_ctape_lift_conf_R //
623 theorem sem_seq_app_guarded: ∀sig.∀M1,M2:TM sig.∀Pre1,Pre2,R1,R2,R3.
624 GRealize sig M1 Pre1 R1 → GRealize sig M2 Pre2 R2 →
625 (∀t1,t2.Pre1 t1 → R1 t1 t2 → Pre2 t2) → R1 ∘ R2 ⊆ R3 →
626 GRealize sig (M1 · M2) Pre1 R3.
627 #sig #M1 #M2 #Pre1 #Pre2 #R1 #R2 #R3 #HR1 #HR2 #Hinv #Hsub
628 #t #HPre1 cases (sem_seq_guarded … HR1 HR2 Hinv t HPre1)
629 #k * #outc * #Hloop #Houtc @(ex_intro … k) @(ex_intro … outc)
630 % [@Hloop |@Hsub @Houtc]