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 "turing/mono.ma".
14 (**************************** single final machine ****************************)
16 definition single_finalTM ≝
17 λsig.λM:TM sig.seq ? M (nop ?).
19 lemma sem_single_final: ∀sig.∀M: TM sig.∀R.
20 M ⊨ R → single_finalTM sig M ⊨ R.
21 #sig #M #R #HR #intape
22 cases (sem_seq ????? HR (sem_nop …) intape)
23 #k * #outc * #Hloop * #ta * #Hta whd in ⊢ (%→?); #Houtc
24 @(ex_intro ?? k) @(ex_intro ?? outc) % [ @Hloop | >Houtc // ]
27 lemma single_final: ∀sig.∀M: TM sig.∀q1,q2.
28 halt ? (single_finalTM sig M) q1 = true
29 → halt ? (single_finalTM sig M) q2 = true → q1=q2.
31 [#q1M #q2 whd in match (halt ???); #H destruct
33 [#q2M #_ whd in match (halt ???); #H destruct
34 |#q2nop #_ #_ @eq_f normalize @nop_single_state
39 (******************************** if machine **********************************)
41 definition if_trans ≝ λsig. λM1,M2,M3 : TM sig. λq:states sig M1.
45 if halt sig M1 s1 then
46 if s1==q then 〈inr … (inl … (start sig M2)), None ?〉
47 else 〈inr … (inr … (start sig M3)), None ?〉
48 else let 〈news1,m〉 ≝ trans sig M1 〈s1,a〉 in
52 [ inl s2 ⇒ let 〈news2,m〉 ≝ trans sig M2 〈s2,a〉 in
53 〈inr … (inl … news2),m〉
54 | inr s3 ⇒ let 〈news3,m〉 ≝ trans sig M3 〈s3,a〉 in
55 〈inr … (inr … news3),m〉
59 definition ifTM ≝ λsig. λcondM,thenM,elseM : TM sig.
60 λqacc: states sig condM.
62 (FinSum (states sig condM) (FinSum (states sig thenM) (states sig elseM)))
63 (if_trans sig condM thenM elseM qacc)
64 (inl … (start sig condM))
67 | inr s' ⇒ match s' with
68 [ inl s2 ⇒ halt sig thenM s2
69 | inr s3 ⇒ halt sig elseM s3 ]]).
71 (****************************** lifting lemmas ********************************)
72 lemma trans_if_liftM1 : ∀sig,M1,M2,M3,acc,s,a,news,move.
74 trans sig M1 〈s,a〉 = 〈news,move〉 →
75 trans sig (ifTM sig M1 M2 M3 acc) 〈inl … s,a〉 = 〈inl … news,move〉.
76 #sig * #Q1 #T1 #init1 #halt1 #M2 #M3 #acc #s #a #news #move
77 #Hhalt #Htrans whd in ⊢ (??%?); >Hhalt >Htrans %
80 lemma trans_if_liftM2 : ∀sig,M1,M2,M3,acc,s,a,news,move.
82 trans sig M2 〈s,a〉 = 〈news,move〉 →
83 trans sig (ifTM sig M1 M2 M3 acc) 〈inr … (inl … s),a〉 = 〈inr… (inl … news),move〉.
84 #sig #M1 * #Q2 #T2 #init2 #halt2 #M3 #acc #s #a #news #move
85 #Hhalt #Htrans whd in ⊢ (??%?); >Hhalt >Htrans %
88 lemma trans_if_liftM3 : ∀sig,M1,M2,M3,acc,s,a,news,move.
90 trans sig M3 〈s,a〉 = 〈news,move〉 →
91 trans sig (ifTM sig M1 M2 M3 acc) 〈inr … (inr … s),a〉 = 〈inr… (inr … news),move〉.
92 #sig #M1 * #Q2 #T2 #init2 #halt2 #M3 #acc #s #a #news #move
93 #Hhalt #Htrans whd in ⊢ (??%?); >Hhalt >Htrans %
96 lemma step_if_liftM1 : ∀sig,M1,M2,M3,acc,c0.
97 halt ? M1 (cstate ?? c0) = false →
98 step sig (ifTM sig M1 M2 M3 acc) (lift_confL sig (states ? M1) ? c0) =
99 lift_confL sig (states ? M1) ? (step sig M1 c0).
100 #sig #M1 #M2 #M3 #acc * #s #t
101 lapply (refl ? (trans ?? 〈s,current sig t〉))
102 cases (trans ?? 〈s,current sig t〉) in ⊢ (???% → %);
105 | 2,3: #s1 #l1 #Heq #Hhalt
106 |#ls #s1 #rs #Heq #Hhalt ]
107 whd in ⊢ (???(????%)); >Heq whd in ⊢ (???%);
108 whd in ⊢ (??(???%)?); whd in ⊢ (??%?); >(trans_if_liftM1 … Hhalt Heq) //
111 lemma step_if_liftM2 : ∀sig,M1,M2,M3,acc,c0.
112 halt ? M2 (cstate ?? c0) = false →
113 step sig (ifTM sig M1 M2 M3 acc) (lift_confR sig ?? (lift_confL sig ?? c0)) =
114 lift_confR sig ?? (lift_confL sig ?? (step sig M2 c0)).
115 #sig #M1 (* * #Q1 #T1 #init1 #halt1 *) #M2 #M3 #acc * #s #t
116 lapply (refl ? (trans ?? 〈s,current sig t〉))
117 cases (trans ?? 〈s,current sig t〉) in ⊢ (???% → %);
120 | 2,3: #s1 #l1 #Heq #Hhalt
121 |#ls #s1 #rs #Heq #Hhalt ]
122 whd in match (step ? M2 ?); >Heq whd in ⊢ (???%);
123 whd in ⊢ (??(???%)?); whd in ⊢ (??%?); >(trans_if_liftM2 … Hhalt Heq) //
126 lemma step_if_liftM3 : ∀sig,M1,M2,M3,acc,c0.
127 halt ? M3 (cstate ?? c0) = false →
128 step sig (ifTM sig M1 M2 M3 acc) (lift_confR sig ?? (lift_confR sig ?? c0)) =
129 lift_confR sig ?? (lift_confR sig ?? (step sig M3 c0)).
130 #sig #M1 (* * #Q1 #T1 #init1 #halt1 *) #M2 #M3 #acc * #s #t
131 lapply (refl ? (trans ?? 〈s,current sig t〉))
132 cases (trans ?? 〈s,current sig t〉) in ⊢ (???% → %);
135 | 2,3: #s1 #l1 #Heq #Hhalt
136 |#ls #s1 #rs #Heq #Hhalt ]
137 whd in match (step ? M3 ?); >Heq whd in ⊢ (???%);
138 whd in ⊢ (??(???%)?); whd in ⊢ (??%?); >(trans_if_liftM3 … Hhalt Heq) //
141 lemma trans_if_M1true_acc : ∀sig,M1,M2,M3,acc,s,a.
142 halt ? M1 s = true → s==acc = true →
143 trans sig (ifTM sig M1 M2 M3 acc) 〈inl … s,a〉 = 〈inr … (inl … (start ? M2)),None ?〉.
144 #sig #M1 #M2 #M3 #acc #s #a #Hhalt #Hacc whd in ⊢ (??%?); >Hhalt >Hacc %
147 lemma trans_if_M1true_notacc : ∀sig,M1,M2,M3,acc,s,a.
148 halt ? M1 s = true → s==acc = false →
149 trans sig (ifTM sig M1 M2 M3 acc) 〈inl … s,a〉 = 〈inr … (inr … (start ? M3)),None ?〉.
150 #sig #M1 #M2 #M3 #acc #s #a #Hhalt #Hacc whd in ⊢ (??%?); >Hhalt >Hacc %
153 (******************************** semantics ***********************************)
154 lemma sem_if: ∀sig.∀M1,M2,M3:TM sig.∀Rtrue,Rfalse,R2,R3,acc.
155 M1 ⊧ [acc: Rtrue,Rfalse] → M2 ⊨ R2 → M3 ⊨ R3 →
156 ifTM sig M1 M2 M3 acc ⊨ (Rtrue ∘ R2) ∪ (Rfalse ∘ R3).
157 #sig #M1 #M2 #M3 #Rtrue #Rfalse #R2 #R3 #acc #HaccR #HR2 #HR3 #t
158 cases (HaccR t) #k1 * #outc1 * * #Hloop1 #HMtrue #HMfalse
159 cases (true_or_false (cstate ?? outc1 == acc)) #Hacc
160 [cases (HR2 (ctape sig ? outc1)) #k2 * #outc2 * #Hloop2 #HM2
161 @(ex_intro … (k1+k2)) @(ex_intro … (lift_confR … (lift_confL … outc2))) %
162 [@(loop_merge ?????????
163 (mk_config ? (FinSum (states sig M1) (FinSum (states sig M2) (states sig M3)))
164 (inr (states sig M1) ? (inl (states sig M2) (states sig M3) (start sig M2))) (ctape ?? outc1) )
167 (lift_confL sig (states ? M1) (FinSum (states ? M2) (states ? M3)))
168 (step sig M1) (step sig (ifTM sig M1 M2 M3 acc))
169 (λc.halt sig M1 (cstate … c))
170 (λc.halt_liftL ?? (halt sig M1) (cstate … c))
173 [ #sl #tl whd in ⊢ (??%? → ?); #Hl %
174 | #sr #tr whd in ⊢ (??%? → ?); #Hr destruct (Hr) ]
175 |#c0 #Hhalt >(step_if_liftM1 … Hhalt) //
177 |whd in ⊢ (??%?); >(config_expand ?? outc1);
178 whd in match (lift_confL ????);
179 >(trans_if_M1true_acc … Hacc)
180 [% | @(loop_Some ?????? Hloop1)]
181 |cases outc1 #s1 #t1 %
183 (λc.(lift_confR … (lift_confL sig (states ? M2) (states ? M3) c)))
186 | #c0 #Hhalt >(step_if_liftM2 … Hhalt) // ]
188 |%1 @(ex_intro … (ctape ?? outc1)) %
189 [@HMtrue @(\P Hacc) | >(config_expand ?? outc2) @HM2 ]
191 |cases (HR3 (ctape sig ? outc1)) #k2 * #outc2 * #Hloop2 #HM3
192 @(ex_intro … (k1+k2)) @(ex_intro … (lift_confR … (lift_confR … outc2))) %
193 [@(loop_merge ?????????
194 (mk_config ? (FinSum (states sig M1) (FinSum (states sig M2) (states sig M3)))
195 (inr (states sig M1) ? (inr (states sig M2) (states sig M3) (start sig M3))) (ctape ?? outc1) )
198 (lift_confL sig (states ? M1) (FinSum (states ? M2) (states ? M3)))
199 (step sig M1) (step sig (ifTM sig M1 M2 M3 acc))
200 (λc.halt sig M1 (cstate … c))
201 (λc.halt_liftL ?? (halt sig M1) (cstate … c))
204 [ #sl #tl whd in ⊢ (??%? → ?); #Hl %
205 | #sr #tr whd in ⊢ (??%? → ?); #Hr destruct (Hr) ]
206 |#c0 #Hhalt >(step_if_liftM1 … Hhalt) //
208 |whd in ⊢ (??%?); >(config_expand ?? outc1);
209 whd in match (lift_confL ????);
210 >(trans_if_M1true_notacc … Hacc)
211 [% | @(loop_Some ?????? Hloop1)]
212 |cases outc1 #s1 #t1 %
214 (λc.(lift_confR … (lift_confR sig (states ? M2) (states ? M3) c)))
217 | #c0 #Hhalt >(step_if_liftM3 … Hhalt) // ]
219 |%2 @(ex_intro … (ctape ?? outc1)) %
220 [@HMfalse @(\Pf Hacc) | >(config_expand ?? outc2) @HM3 ]
225 lemma sem_if_app: ∀sig,M1,M2,M3,Rtrue,Rfalse,R2,R3,R4,acc.
226 accRealize sig M1 acc Rtrue Rfalse → M2 ⊨ R2 → M3 ⊨ R3 →
227 (∀t1,t2,t3. (Rtrue t1 t3 → R2 t3 t2) ∨ (Rfalse t1 t3 → R3 t3 t2) → R4 t1 t2) →
228 ifTM sig M1 M2 M3 acc ⊨ R4.
229 #sig #M1 #M2 #M3 #Rtrue #Rfalse #R2 #R3 #R4 #acc
230 #HRacc #HRtrue #HRfalse #Hsub
231 #t cases (sem_if … HRacc HRtrue HRfalse t)
232 #k * #outc * #Hloop #Houtc @(ex_intro … k) @(ex_intro … outc)
233 % [@Hloop] cases Houtc
234 [* #t3 * #Hleft #Hright @(Hsub … t3) %1 /2/
235 |* #t3 * #Hleft #Hright @(Hsub … t3) %2 /2/ ]
238 (* we can probably use acc_sem_if to prove sem_if *)
239 lemma acc_sem_if: ∀sig,M1,M2,M3,Rtrue,Rfalse,R2,R3,acc.
240 M1 ⊨ [acc: Rtrue, Rfalse] → M2 ⊨ R2 → M3 ⊨ R3 →
241 ifTM sig M1 (single_finalTM … M2) M3 acc ⊨
242 [inr … (inl … (inr … start_nop)): Rtrue ∘ R2, Rfalse ∘ R3].
243 #sig #M1 #M2 #M3 #Rtrue #Rfalse #R2 #R3 #acc #HaccR #HR2 #HR3 #t
244 cases (HaccR t) #k1 * #outc1 * * #Hloop1 #HMtrue #HMfalse
245 cases (true_or_false (cstate ?? outc1 == acc)) #Hacc
246 [lapply (sem_single_final … HR2) -HR2 #HR2
247 cases (HR2 (ctape sig ? outc1)) #k2 * #outc2 * #Hloop2 #HM2
248 @(ex_intro … (k1+k2))
249 @(ex_intro … (lift_confR … (lift_confL … outc2))) %
251 [@(loop_merge ?????????
252 (mk_config ? (states sig (ifTM sig M1 (single_finalTM … M2) M3 acc))
253 (inr (states sig M1) ? (inl ? (states sig M3) (start sig (single_finalTM sig M2)))) (ctape ?? outc1) )
256 (lift_confL sig (states ? M1) (FinSum (states ? (single_finalTM … M2)) (states ? M3)))
257 (step sig M1) (step sig (ifTM sig M1 (single_finalTM ? M2) M3 acc))
258 (λc.halt sig M1 (cstate … c))
259 (λc.halt_liftL ?? (halt sig M1) (cstate … c))
262 [ #sl #tl whd in ⊢ (??%? → ?); #Hl %
263 | #sr #tr whd in ⊢ (??%? → ?); #Hr destruct (Hr) ]
264 |#c0 #Hhalt >(step_if_liftM1 … Hhalt) //
266 |whd in ⊢ (??%?); >(config_expand ?? outc1);
267 whd in match (lift_confL ????);
268 >(trans_if_M1true_acc … Hacc)
269 [% | @(loop_Some ?????? Hloop1)]
270 |cases outc1 #s1 #t1 %
272 (λc.(lift_confR … (lift_confL sig (states ? (single_finalTM ? M2)) (states ? M3) c)))
275 | #c0 #Hhalt >(step_if_liftM2 … Hhalt) // ]
277 |#_ @(ex_intro … (ctape ?? outc1)) %
278 [@HMtrue @(\P Hacc) | >(config_expand ?? outc2) @HM2 ]
280 |>(config_expand ?? outc2) whd in match (lift_confR ????);
281 * #H @False_ind @H @eq_f @eq_f >(config_expand ?? outc2)
282 @single_final // @(loop_Some ?????? Hloop2)
284 |cases (HR3 (ctape sig ? outc1)) #k2 * #outc2 * #Hloop2 #HM3
285 @(ex_intro … (k1+k2)) @(ex_intro … (lift_confR … (lift_confR … outc2))) %
287 [@(loop_merge ?????????
288 (mk_config ? (states sig (ifTM sig M1 (single_finalTM … M2) M3 acc))
289 (inr (states sig M1) ? (inr (states sig (single_finalTM ? M2)) ? (start sig M3))) (ctape ?? outc1) )
292 (lift_confL sig (states ? M1) (FinSum (states ? (single_finalTM … M2)) (states ? M3)))
293 (step sig M1) (step sig (ifTM sig M1 (single_finalTM ? M2) M3 acc))
294 (λc.halt sig M1 (cstate … c))
295 (λc.halt_liftL ?? (halt sig M1) (cstate … c))
298 [ #sl #tl whd in ⊢ (??%? → ?); #Hl %
299 | #sr #tr whd in ⊢ (??%? → ?); #Hr destruct (Hr) ]
300 |#c0 #Hhalt >(step_if_liftM1 … Hhalt) //
302 |whd in ⊢ (??%?); >(config_expand ?? outc1);
303 whd in match (lift_confL ????);
304 >(trans_if_M1true_notacc … Hacc)
305 [% | @(loop_Some ?????? Hloop1)]
306 |cases outc1 #s1 #t1 %
308 (λc.(lift_confR … (lift_confR sig (states ? (single_finalTM ? M2)) (states ? M3) c)))
311 | #c0 #Hhalt >(step_if_liftM3 … Hhalt) // ]
313 |>(config_expand ?? outc2) whd in match (lift_confR ????);
316 |#_ @(ex_intro … (ctape ?? outc1)) %
317 [@HMfalse @(\Pf Hacc) | >(config_expand ?? outc2) @HM3 ]
322 lemma acc_sem_if_app: ∀sig,M1,M2,M3,Rtrue,Rfalse,R2,R3,R4,R5,acc.
323 M1 ⊨ [acc: Rtrue, Rfalse] → M2 ⊨ R2 → M3 ⊨ R3 →
324 (∀t1,t2,t3. Rtrue t1 t3 → R2 t3 t2 → R4 t1 t2) →
325 (∀t1,t2,t3. Rfalse t1 t3 → R3 t3 t2 → R5 t1 t2) →
326 ifTM sig M1 (single_finalTM … M2) M3 acc ⊨
327 [inr … (inl … (inr … start_nop)): R4, R5].
328 #sig #M1 #M2 #M3 #Rtrue #Rfalse #R2 #R3 #R4 #R5 #acc
329 #HRacc #HRtrue #HRfalse #Hsub1 #Hsub2
330 #t cases (acc_sem_if … HRacc HRtrue HRfalse t)
331 #k * #outc * * #Hloop #Houtc1 #Houtc2 @(ex_intro … k) @(ex_intro … outc)
333 |#H cases (Houtc1 H) #t3 * #Hleft #Hright @Hsub1 // ]
334 |#H cases (Houtc2 H) #t3 * #Hleft #Hright @Hsub2 // ]