1 include "turing/mono.ma".
2 include "basics/vectors.ma".
4 (* We do not distinuish an input tape *)
6 (* tapes_no = number of ADDITIONAL working tapes *)
8 record mTM (sig:FinSet) (tapes_no:nat) : Type[1] ≝
10 trans : states × (Vector (option sig) (S tapes_no)) →
11 states × (Vector (option (sig × move))(S tapes_no));
16 record mconfig (sig,states:FinSet) (n:nat): Type[0] ≝
18 ctapes : Vector (tape sig) (S n)
21 lemma mconfig_expand: ∀sig,n,Q,c.
22 c = mk_mconfig sig Q n (cstate ??? c) (ctapes ??? c).
26 lemma mconfig_eq : ∀sig,n,M,c1,c2.
27 cstate sig n M c1 = cstate sig n M c2 →
28 ctapes sig n M c1 = ctapes sig n M c2 → c1 = c2.
29 #sig #n #M1 * #s1 #t1 * #s2 #t2 //
32 definition current_chars ≝ λsig.λn.λtapes.
33 vec_map ?? (current sig) (S n) tapes.
35 definition step ≝ λsig.λn.λM:mTM sig n.λc:mconfig sig (states ?? M) n.
36 let 〈news,mvs〉 ≝ trans sig n M 〈cstate ??? c,current_chars ?? (ctapes ??? c)〉 in
39 (pmap_vec ??? (tape_move sig) ? (ctapes ??? c) mvs).
41 definition empty_tapes ≝ λsig.λn.
42 mk_Vector ? n (make_list (tape sig) (niltape sig) n) ?.
43 elim n // normalize //
46 (************************** Realizability *************************************)
47 definition loopM ≝ λsig,n.λM:mTM sig n.λi,cin.
48 loop ? i (step sig n M) (λc.halt sig n M (cstate ??? c)) cin.
50 lemma loopM_unfold : ∀sig,n,M,i,cin.
51 loopM sig n M i cin = loop ? i (step sig n M) (λc.halt sig n M (cstate ??? c)) cin.
54 definition initc ≝ λsig,n.λM:mTM sig n.λtapes.
55 mk_mconfig sig (states sig n M) n (start sig n M) tapes.
57 definition Realize ≝ λsig,n.λM:mTM sig n.λR:relation (Vector (tape sig) ?).
59 loopM sig n M i (initc sig n M t) = Some ? outc ∧ R t (ctapes ??? outc).
61 definition WRealize ≝ λsig,n.λM:mTM sig n.λR:relation (Vector (tape sig) ?).
63 loopM sig n M i (initc sig n M t) = Some ? outc → R t (ctapes ??? outc).
65 definition Terminate ≝ λsig,n.λM:mTM sig n.λt. ∃i,outc.
66 loopM sig n M i (initc sig n M t) = Some ? outc.
68 (* notation "M \vDash R" non associative with precedence 45 for @{ 'models $M $R}. *)
69 interpretation "multi realizability" 'models M R = (Realize ?? M R).
71 (* notation "M \VDash R" non associative with precedence 45 for @{ 'wmodels $M $R}. *)
72 interpretation "weak multi realizability" 'wmodels M R = (WRealize ?? M R).
74 interpretation "multi termination" 'fintersects M t = (Terminate ?? M t).
76 lemma WRealize_to_Realize : ∀sig,n .∀M: mTM sig n.∀R.
77 (∀t.M ↓ t) → M ⊫ R → M ⊨ R.
78 #sig #n #M #R #HT #HW #t cases (HT … t) #i * #outc #Hloop
79 @(ex_intro … i) @(ex_intro … outc) % // @(HW … i) //
82 theorem Realize_to_WRealize : ∀sig,n.∀M:mTM sig n.∀R.
84 #sig #n #M #R #H1 #inc #i #outc #Hloop
85 cases (H1 inc) #k * #outc1 * #Hloop1 #HR >(loop_eq … Hloop Hloop1) //
88 definition accRealize ≝ λsig,n.λM:mTM sig n.λacc:states sig n M.λRtrue,Rfalse.
90 loopM sig n M i (initc sig n M t) = Some ? outc ∧
91 (cstate ??? outc = acc → Rtrue t (ctapes ??? outc)) ∧
92 (cstate ??? outc ≠ acc → Rfalse t (ctapes ??? outc)).
94 (* notation "M ⊨ [q: R1,R2]" non associative with precedence 45 for @{ 'cmodels $M $q $R1 $R2}. *)
95 interpretation "conditional multi realizability" 'cmodels M q R1 R2 = (accRealize ?? M q R1 R2).
97 (*************************** guarded realizablity *****************************)
98 definition GRealize ≝ λsig,n.λM:mTM sig n.
99 λPre:Vector (tape sig) ? →Prop.λR:relation (Vector (tape sig) ?).
101 loopM sig n M i (initc sig n M t) = Some ? outc ∧ R t (ctapes ??? outc).
103 definition accGRealize ≝ λsig,n.λM:mTM sig n.λacc:states sig n M.
104 λPre: Vector (tape sig) ? → Prop.λRtrue,Rfalse.
106 loopM sig n M i (initc sig n M t) = Some ? outc ∧
107 (cstate ??? outc = acc → Rtrue t (ctapes ??? outc)) ∧
108 (cstate ??? outc ≠ acc → Rfalse t (ctapes ??? outc)).
110 lemma WRealize_to_GRealize : ∀sig,n.∀M: mTM sig n.∀Pre,R.
111 (∀t.Pre t → M ↓ t) → M ⊫ R → GRealize sig n M Pre R.
112 #sig #n #M #Pre #R #HT #HW #t #HPre cases (HT … t HPre) #i * #outc #Hloop
113 @(ex_intro … i) @(ex_intro … outc) % // @(HW … i) //
116 lemma Realize_to_GRealize : ∀sig,n.∀M: mTM sig n.∀P,R.
117 M ⊨ R → GRealize sig n M P R.
118 #alpha #n #M #Pre #R #HR #t #HPre
119 cases (HR t) -HR #k * #outc * #Hloop #HR
120 @(ex_intro ?? k) @(ex_intro ?? outc) %
124 lemma acc_Realize_to_acc_GRealize: ∀sig,n.∀M:mTM sig n.∀q:states sig n M.∀P,R1,R2.
125 M ⊨ [q:R1,R2] → accGRealize sig n M q P R1 R2.
126 #alpha #n #M #q #Pre #R1 #R2 #HR #t #HPre
127 cases (HR t) -HR #k * #outc * * #Hloop #HRtrue #HRfalse
128 @(ex_intro ?? k) @(ex_intro ?? outc) %
129 [ % [@Hloop] @HRtrue | @HRfalse]
132 (******************************** monotonicity ********************************)
133 lemma Realize_to_Realize : ∀sig,n.∀M:mTM sig n.∀R1,R2.
134 R1 ⊆ R2 → M ⊨ R1 → M ⊨ R2.
135 #alpha #n #M #R1 #R2 #Himpl #HR1 #intape
136 cases (HR1 intape) -HR1 #k * #outc * #Hloop #HR1
137 @(ex_intro ?? k) @(ex_intro ?? outc) % /2/
140 lemma WRealize_to_WRealize: ∀sig,n.∀M:mTM sig n.∀R1,R2.
141 R1 ⊆ R2 → WRealize sig n M R1 → WRealize sig n M R2.
142 #alpha #n #M #R1 #R2 #Hsub #HR1 #intape #i #outc #Hloop
143 @Hsub @(HR1 … i) @Hloop
146 lemma GRealize_to_GRealize : ∀sig,n.∀M:mTM sig n.∀P,R1,R2.
147 R1 ⊆ R2 → GRealize sig n M P R1 → GRealize sig n M P R2.
148 #alpha #n #M #P #R1 #R2 #Himpl #HR1 #intape #HP
149 cases (HR1 intape HP) -HR1 #k * #outc * #Hloop #HR1
150 @(ex_intro ?? k) @(ex_intro ?? outc) % /2/
153 lemma GRealize_to_GRealize_2 : ∀sig,n.∀M:mTM sig n.∀P1,P2,R1,R2.
154 P2 ⊆ P1 → R1 ⊆ R2 → GRealize sig n M P1 R1 → GRealize sig n M P2 R2.
155 #alpha #n #M #P1 #P2 #R1 #R2 #Himpl1 #Himpl2 #H1 #intape #HP
156 cases (H1 intape (Himpl1 … HP)) -H1 #k * #outc * #Hloop #H1
157 @(ex_intro ?? k) @(ex_intro ?? outc) % /2/
160 lemma acc_Realize_to_acc_Realize: ∀sig,n.∀M:mTM sig n.∀q:states sig n M.
162 R1 ⊆ R3 → R2 ⊆ R4 → M ⊨ [q:R1,R2] → M ⊨ [q:R3,R4].
163 #alpha #n #M #q #R1 #R2 #R3 #R4 #Hsub13 #Hsub24 #HRa #intape
164 cases (HRa intape) -HRa #k * #outc * * #Hloop #HRtrue #HRfalse
165 @(ex_intro ?? k) @(ex_intro ?? outc) %
166 [ % [@Hloop] #Hq @Hsub13 @HRtrue // | #Hq @Hsub24 @HRfalse //]
169 (**************************** A canonical relation ****************************)
171 definition R_mTM ≝ λsig,n.λM:mTM sig n.λq.λt1,t2.
173 loopM ? n M i (mk_mconfig ??? q t1) = Some ? outc ∧
174 t2 = (ctapes ??? outc).
176 lemma R_mTM_to_R: ∀sig,n.∀M:mTM sig n.∀R. ∀t1,t2.
177 M ⊫ R → R_mTM ?? M (start sig n M) t1 t2 → R t1 t2.
178 #sig #n #M #R #t1 #t2 whd in ⊢ (%→?); #HMR * #i * #outc *
179 #Hloop #Ht2 >Ht2 @(HMR … Hloop)
182 (******************************** NOP Machine *********************************)
187 definition nop_states ≝ initN 1.
188 definition start_nop : initN 1 ≝ mk_Sig ?? 0 (le_n … 1). *)
191 λalpha:FinSet.λn.mk_mTM alpha n nop_states
192 (λp.let 〈q,a〉 ≝ p in 〈q,mk_Vector ? (S n) (make_list ? (None ?) (S n)) ?〉)
197 definition R_nop ≝ λalpha,n.λt1,t2:Vector (tape alpha) (S n).t2 = t1.
200 ∀alpha,n.nop alpha n⊨ R_nop alpha n.
201 #alpha #n #intapes @(ex_intro ?? 1)
202 @(ex_intro … (mk_mconfig ??? start_nop intapes)) % %
205 lemma nop_single_state: ∀sig,n.∀q1,q2:states ? n (nop sig n). q1 = q2.
206 normalize #sig #n0 * #n #ltn1 * #m #ltm1
207 generalize in match ltn1; generalize in match ltm1;
208 <(le_n_O_to_eq … (le_S_S_to_le … ltn1)) <(le_n_O_to_eq … (le_S_S_to_le … ltm1))
211 (************************** Sequential Composition ****************************)
212 definition null_action ≝ λsig.λn.
213 mk_Vector ? (S n) (make_list (option (sig × move)) (None ?) (S n)) ?.
214 elim (S n) // normalize //
217 lemma tape_move_null_action: ∀sig,n,tapes.
218 pmap_vec ??? (tape_move sig) (S n) tapes (null_action sig n) = tapes.
219 #sig #n #tapes cases tapes -tapes #tapes whd in match (null_action ??);
220 #Heq @Vector_eq <Heq -Heq elim tapes //
221 #a #tl #Hind whd in ⊢ (??%?); @eq_f2 // @Hind
224 definition seq_trans ≝ λsig,n. λM1,M2 : mTM sig n.
228 if halt sig n M1 s1 then 〈inr … (start sig n M2), null_action sig n〉
229 else let 〈news1,m〉 ≝ trans sig n M1 〈s1,a〉 in 〈inl … news1,m〉
230 | inr s2 ⇒ let 〈news2,m〉 ≝ trans sig n M2 〈s2,a〉 in 〈inr … news2,m〉
233 definition seq ≝ λsig,n. λM1,M2 : mTM sig n.
235 (FinSum (states sig n M1) (states sig n M2))
236 (seq_trans sig n M1 M2)
237 (inl … (start sig n M1))
239 [ inl _ ⇒ false | inr s2 ⇒ halt sig n M2 s2]).
241 (* notation "a · b" right associative with precedence 65 for @{ 'middot $a $b}. *)
242 interpretation "sequential composition" 'middot a b = (seq ?? a b).
244 definition lift_confL ≝
245 λsig,n,S1,S2,c.match c with
246 [ mk_mconfig s t ⇒ mk_mconfig sig (FinSum S1 S2) n (inl … s) t ].
248 definition lift_confR ≝
249 λsig,n,S1,S2,c.match c with
250 [ mk_mconfig s t ⇒ mk_mconfig sig (FinSum S1 S2) n (inr … s) t ].
253 definition halt_liftL ≝
254 λS1,S2,halt.λs:FinSum S1 S2.
257 | inr _ ⇒ true ]. (* should be vacuous in all cases we use halt_liftL *)
259 definition halt_liftR ≝
260 λS1,S2,halt.λs:FinSum S1 S2.
263 | inr s2 ⇒ halt s2 ]. *)
265 lemma p_halt_liftL : ∀sig,n,S1,S2,halt,c.
266 halt (cstate sig S1 n c) =
267 halt_liftL S1 S2 halt (cstate … (lift_confL … c)).
268 #sig #n #S1 #S2 #halt #c cases c #s #t %
271 lemma trans_seq_liftL : ∀sig,n,M1,M2,s,a,news,move.
272 halt ?? M1 s = false →
273 trans sig n M1 〈s,a〉 = 〈news,move〉 →
274 trans sig n (seq sig n M1 M2) 〈inl … s,a〉 = 〈inl … news,move〉.
275 #sig #n (*#M1*) * #Q1 #T1 #init1 #halt1 #M2 #s #a #news #move
276 #Hhalt #Htrans whd in ⊢ (??%?); >Hhalt >Htrans %
279 lemma trans_seq_liftR : ∀sig,n,M1,M2,s,a,news,move.
280 halt ?? M2 s = false →
281 trans sig n M2 〈s,a〉 = 〈news,move〉 →
282 trans sig n (seq sig n M1 M2) 〈inr … s,a〉 = 〈inr … news,move〉.
283 #sig #n #M1 * #Q2 #T2 #init2 #halt2 #s #a #news #move
284 #Hhalt #Htrans whd in ⊢ (??%?); >Hhalt >Htrans %
287 lemma step_seq_liftR : ∀sig,n,M1,M2,c0.
288 halt ?? M2 (cstate ??? c0) = false →
289 step sig n (seq sig n M1 M2) (lift_confR sig n (states ?? M1) (states ?? M2) c0) =
290 lift_confR sig n (states ?? M1) (states ?? M2) (step sig n M2 c0).
291 #sig #n #M1 (* * #Q1 #T1 #init1 #halt1 *) #M2 * #s #t
292 lapply (refl ? (trans ??? 〈s,current_chars sig n t〉))
293 cases (trans ??? 〈s,current_chars sig n t〉) in ⊢ (???% → %);
294 #s0 #m0 #Heq #Hhalt whd in ⊢ (???(?????%)); >Heq whd in ⊢ (???%);
295 whd in ⊢ (??(????%)?); whd in ⊢ (??%?); >(trans_seq_liftR … Heq) //
298 lemma step_seq_liftL : ∀sig,n,M1,M2,c0.
299 halt ?? M1 (cstate ??? c0) = false →
300 step sig n (seq sig n M1 M2) (lift_confL sig n (states ?? M1) (states ?? M2) c0) =
301 lift_confL sig n ?? (step sig n M1 c0).
302 #sig #n #M1 (* * #Q1 #T1 #init1 #halt1 *) #M2 * #s #t
303 lapply (refl ? (trans ??? 〈s,current_chars sig n t〉))
304 cases (trans ??? 〈s,current_chars sig n t〉) in ⊢ (???% → %);
306 whd in ⊢ (???(?????%)); >Heq whd in ⊢ (???%);
307 whd in ⊢ (??(????%)?); whd in ⊢ (??%?); >(trans_seq_liftL … Heq) //
310 lemma trans_liftL_true : ∀sig,n,M1,M2,s,a.
311 halt ?? M1 s = true →
312 trans sig n (seq sig n M1 M2) 〈inl … s,a〉 = 〈inr … (start ?? M2),null_action sig n〉.
313 #sig #n #M1 #M2 #s #a #Hhalt whd in ⊢ (??%?); >Hhalt %
316 lemma eq_ctape_lift_conf_L : ∀sig,n,S1,S2,outc.
317 ctapes sig (FinSum S1 S2) n (lift_confL … outc) = ctapes … outc.
318 #sig #n #S1 #S2 #outc cases outc #s #t %
321 lemma eq_ctape_lift_conf_R : ∀sig,n,S1,S2,outc.
322 ctapes sig (FinSum S1 S2) n (lift_confR … outc) = ctapes … outc.
323 #sig #n #S1 #S2 #outc cases outc #s #t %
326 theorem sem_seq: ∀sig,n.∀M1,M2:mTM sig n.∀R1,R2.
327 M1 ⊨ R1 → M2 ⊨ R2 → M1 · M2 ⊨ R1 ∘ R2.
328 #sig #n #M1 #M2 #R1 #R2 #HR1 #HR2 #t
329 cases (HR1 t) #k1 * #outc1 * #Hloop1 #HM1
330 cases (HR2 (ctapes sig (states ?? M1) n outc1)) #k2 * #outc2 * #Hloop2 #HM2
331 @(ex_intro … (k1+k2)) @(ex_intro … (lift_confR … outc2))
333 [@(loop_merge ???????????
334 (loop_lift ??? (lift_confL sig n (states sig n M1) (states sig n M2))
335 (step sig n M1) (step sig n (seq sig n M1 M2))
336 (λc.halt sig n M1 (cstate … c))
337 (λc.halt_liftL ?? (halt sig n M1) (cstate … c)) … Hloop1))
339 [ #sl #tl whd in ⊢ (??%? → ?); #Hl %
340 | #sr #tr whd in ⊢ (??%? → ?); #Hr destruct (Hr) ]
341 || #c0 #Hhalt <step_seq_liftL //
343 |6:cases outc1 #s1 #t1 %
344 |7:@(loop_lift … (initc ??? (ctapes … outc1)) … Hloop2)
346 | #c0 #Hhalt <step_seq_liftR // ]
347 |whd in ⊢ (??(????%)?);whd in ⊢ (??%?);
348 generalize in match Hloop1; cases outc1 #sc1 #tc1 #Hloop10
349 >(trans_liftL_true sig n M1 M2 ??)
350 [ whd in ⊢ (??%?); whd in ⊢ (???%);
351 @mconfig_eq whd in ⊢ (???%); //
352 | @(loop_Some ?????? Hloop10) ]
354 | @(ex_intro … (ctapes ? (FinSum (states ?? M1) (states ?? M2)) ? (lift_confL … outc1)))
355 % // >eq_ctape_lift_conf_L >eq_ctape_lift_conf_R //
359 theorem sem_seq_app: ∀sig,n.∀M1,M2:mTM sig n.∀R1,R2,R3.
360 M1 ⊨ R1 → M2 ⊨ R2 → R1 ∘ R2 ⊆ R3 → M1 · M2 ⊨ R3.
361 #sig #n #M1 #M2 #R1 #R2 #R3 #HR1 #HR2 #Hsub
362 #t cases (sem_seq … HR1 HR2 t)
363 #k * #outc * #Hloop #Houtc @(ex_intro … k) @(ex_intro … outc)
364 % [@Hloop |@Hsub @Houtc]
367 (* composition with guards *)
368 theorem sem_seq_guarded: ∀sig,n.∀M1,M2:mTM sig n.∀Pre1,Pre2,R1,R2.
369 GRealize sig n M1 Pre1 R1 → GRealize sig n M2 Pre2 R2 →
370 (∀t1,t2.Pre1 t1 → R1 t1 t2 → Pre2 t2) →
371 GRealize sig n (M1 · M2) Pre1 (R1 ∘ R2).
372 #sig #n #M1 #M2 #Pre1 #Pre2 #R1 #R2 #HGR1 #HGR2 #Hinv #t1 #HPre1
373 cases (HGR1 t1 HPre1) #k1 * #outc1 * #Hloop1 #HM1
374 cases (HGR2 (ctapes sig (states ?? M1) n outc1) ?)
375 [2: @(Hinv … HPre1 HM1)]
376 #k2 * #outc2 * #Hloop2 #HM2
377 @(ex_intro … (k1+k2)) @(ex_intro … (lift_confR … outc2))
379 [@(loop_merge ???????????
380 (loop_lift ??? (lift_confL sig n (states sig n M1) (states sig n M2))
381 (step sig n M1) (step sig n (seq sig n M1 M2))
382 (λc.halt sig n M1 (cstate … c))
383 (λc.halt_liftL ?? (halt sig n M1) (cstate … c)) … Hloop1))
385 [ #sl #tl whd in ⊢ (??%? → ?); #Hl %
386 | #sr #tr whd in ⊢ (??%? → ?); #Hr destruct (Hr) ]
387 || #c0 #Hhalt <step_seq_liftL //
389 |6:cases outc1 #s1 #t1 %
390 |7:@(loop_lift … (initc ??? (ctapes … outc1)) … Hloop2)
392 | #c0 #Hhalt <step_seq_liftR // ]
393 |whd in ⊢ (??(????%)?);whd in ⊢ (??%?);
394 generalize in match Hloop1; cases outc1 #sc1 #tc1 #Hloop10
395 >(trans_liftL_true sig n M1 M2 ??)
396 [ whd in ⊢ (??%?); whd in ⊢ (???%);
397 @mconfig_eq whd in ⊢ (???%); //
398 | @(loop_Some ?????? Hloop10) ]
400 | @(ex_intro … (ctapes ? (FinSum (states ?? M1) (states ?? M2)) n (lift_confL … outc1)))
401 % // >eq_ctape_lift_conf_L >eq_ctape_lift_conf_R //
405 theorem sem_seq_app_guarded: ∀sig,n.∀M1,M2:mTM sig n.∀Pre1,Pre2,R1,R2,R3.
406 GRealize sig n M1 Pre1 R1 → GRealize sig n M2 Pre2 R2 →
407 (∀t1,t2.Pre1 t1 → R1 t1 t2 → Pre2 t2) → R1 ∘ R2 ⊆ R3 →
408 GRealize sig n (M1 · M2) Pre1 R3.
409 #sig #n #M1 #M2 #Pre1 #Pre2 #R1 #R2 #R3 #HR1 #HR2 #Hinv #Hsub
410 #t #HPre1 cases (sem_seq_guarded … HR1 HR2 Hinv t HPre1)
411 #k * #outc * #Hloop #Houtc @(ex_intro … k) @(ex_intro … outc)
412 % [@Hloop |@Hsub @Houtc]
415 theorem acc_sem_seq : ∀sig,n.∀M1,M2:mTM sig n.∀R1,Rtrue,Rfalse,acc.
416 M1 ⊨ R1 → M2 ⊨ [ acc: Rtrue, Rfalse ] →
417 M1 · M2 ⊨ [ inr … acc: R1 ∘ Rtrue, R1 ∘ Rfalse ].
418 #sig #n #M1 #M2 #R1 #Rtrue #Rfalse #acc #HR1 #HR2 #t
419 cases (HR1 t) #k1 * #outc1 * #Hloop1 #HM1
420 cases (HR2 (ctapes sig (states ?? M1) n outc1)) #k2 * #outc2 * * #Hloop2
422 @(ex_intro … (k1+k2)) @(ex_intro … (lift_confR … outc2))
424 [@(loop_merge ???????????
425 (loop_lift ??? (lift_confL sig n (states sig n M1) (states sig n M2))
426 (step sig n M1) (step sig n (seq sig n M1 M2))
427 (λc.halt sig n M1 (cstate … c))
428 (λc.halt_liftL ?? (halt sig n M1) (cstate … c)) … Hloop1))
430 [ #sl #tl whd in ⊢ (??%? → ?); #Hl %
431 | #sr #tr whd in ⊢ (??%? → ?); #Hr destruct (Hr) ]
432 || #c0 #Hhalt <step_seq_liftL //
434 |6:cases outc1 #s1 #t1 %
435 |7:@(loop_lift … (initc ??? (ctapes … outc1)) … Hloop2)
437 | #c0 #Hhalt <step_seq_liftR // ]
438 |whd in ⊢ (??(????%)?);whd in ⊢ (??%?);
439 generalize in match Hloop1; cases outc1 #sc1 #tc1 #Hloop10
440 >(trans_liftL_true sig n M1 M2 ??)
441 [ whd in ⊢ (??%?); whd in ⊢ (???%);
442 @mconfig_eq whd in ⊢ (???%); //
443 | @(loop_Some ?????? Hloop10) ]
445 | >(mconfig_expand … outc2) in ⊢ (%→?); whd in ⊢ (??%?→?);
446 #Hqtrue destruct (Hqtrue)
447 @(ex_intro … (ctapes ? (FinSum (states ?? M1) (states ?? M2)) ? (lift_confL … outc1)))
448 % // >eq_ctape_lift_conf_L >eq_ctape_lift_conf_R /2/ ]
449 | >(mconfig_expand … outc2) in ⊢ (%→?); whd in ⊢ (?(??%?)→?); #Hqfalse
450 @(ex_intro … (ctapes ? (FinSum (states ?? M1) (states ?? M2)) ? (lift_confL … outc1)))
451 % // >eq_ctape_lift_conf_L >eq_ctape_lift_conf_R @HMfalse
452 @(not_to_not … Hqfalse) //
456 lemma acc_sem_seq_app : ∀sig,n.∀M1,M2:mTM sig n.∀R1,Rtrue,Rfalse,R2,R3,acc.
457 M1 ⊨ R1 → M2 ⊨ [acc: Rtrue, Rfalse] →
458 (∀t1,t2,t3. R1 t1 t3 → Rtrue t3 t2 → R2 t1 t2) →
459 (∀t1,t2,t3. R1 t1 t3 → Rfalse t3 t2 → R3 t1 t2) →
460 M1 · M2 ⊨ [inr … acc : R2, R3].
461 #sig #n #M1 #M2 #R1 #Rtrue #Rfalse #R2 #R3 #acc
462 #HR1 #HRacc #Hsub1 #Hsub2
463 #t cases (acc_sem_seq … HR1 HRacc t)
464 #k * #outc * * #Hloop #Houtc1 #Houtc2 @(ex_intro … k) @(ex_intro … outc)
466 |#H cases (Houtc1 H) #t3 * #Hleft #Hright @Hsub1 // ]
467 |#H cases (Houtc2 H) #t3 * #Hleft #Hright @Hsub2 // ]