match s with
[ inl s1 ⇒
if halt sig M1 s1 then
- if s1==q then 〈inr … (inl … (start sig M2)), None ?〉
- else 〈inr … (inr … (start sig M3)), None ?〉
- else let 〈news1,m〉 ≝ trans sig M1 〈s1,a〉 in
- 〈inl … news1,m〉
+ if s1==q then 〈inr … (inl … (start sig M2)), None ?,N〉
+ else 〈inr … (inr … (start sig M3)), None ?,N〉
+ else let 〈news1,newa,m〉 ≝ trans sig M1 〈s1,a〉 in
+ 〈inl … news1,newa,m〉
| inr s' ⇒
match s' with
- [ inl s2 ⇒ let 〈news2,m〉 ≝ trans sig M2 〈s2,a〉 in
- 〈inr … (inl … news2),m〉
- | inr s3 ⇒ let 〈news3,m〉 ≝ trans sig M3 〈s3,a〉 in
- 〈inr … (inr … news3),m〉
+ [ inl s2 ⇒ let 〈news2,newa,m〉 ≝ trans sig M2 〈s2,a〉 in
+ 〈inr … (inl … news2),newa,m〉
+ | inr s3 ⇒ let 〈news3,newa,m〉 ≝ trans sig M3 〈s3,a〉 in
+ 〈inr … (inr … news3),newa,m〉
]
].
| inr s3 ⇒ halt sig elseM s3 ]]).
(****************************** lifting lemmas ********************************)
-lemma trans_if_liftM1 : ∀sig,M1,M2,M3,acc,s,a,news,move.
+lemma trans_if_liftM1 : ∀sig,M1,M2,M3,acc,s,a,news,newa,move.
halt ? M1 s = false →
- trans sig M1 〈s,a〉 = 〈news,move〉 →
- trans sig (ifTM sig M1 M2 M3 acc) 〈inl … s,a〉 = 〈inl … news,move〉.
-#sig * #Q1 #T1 #init1 #halt1 #M2 #M3 #acc #s #a #news #move
+ trans sig M1 〈s,a〉 = 〈news,newa,move〉 →
+ trans sig (ifTM sig M1 M2 M3 acc) 〈inl … s,a〉 = 〈inl … news,newa,move〉.
+#sig * #Q1 #T1 #init1 #halt1 #M2 #M3 #acc #s #a #news #newa #move
#Hhalt #Htrans whd in ⊢ (??%?); >Hhalt >Htrans %
qed.
-lemma trans_if_liftM2 : ∀sig,M1,M2,M3,acc,s,a,news,move.
+lemma trans_if_liftM2 : ∀sig,M1,M2,M3,acc,s,a,news,newa,move.
halt ? M2 s = false →
- trans sig M2 〈s,a〉 = 〈news,move〉 →
- trans sig (ifTM sig M1 M2 M3 acc) 〈inr … (inl … s),a〉 = 〈inr… (inl … news),move〉.
-#sig #M1 * #Q2 #T2 #init2 #halt2 #M3 #acc #s #a #news #move
+ trans sig M2 〈s,a〉 = 〈news,newa,move〉 →
+ trans sig (ifTM sig M1 M2 M3 acc) 〈inr … (inl … s),a〉 = 〈inr… (inl … news),newa,move〉.
+#sig #M1 * #Q2 #T2 #init2 #halt2 #M3 #acc #s #a #news #newa #move
#Hhalt #Htrans whd in ⊢ (??%?); >Hhalt >Htrans %
qed.
-lemma trans_if_liftM3 : ∀sig,M1,M2,M3,acc,s,a,news,move.
+lemma trans_if_liftM3 : ∀sig,M1,M2,M3,acc,s,a,news,newa,move.
halt ? M3 s = false →
- trans sig M3 〈s,a〉 = 〈news,move〉 →
- trans sig (ifTM sig M1 M2 M3 acc) 〈inr … (inr … s),a〉 = 〈inr… (inr … news),move〉.
-#sig #M1 * #Q2 #T2 #init2 #halt2 #M3 #acc #s #a #news #move
+ trans sig M3 〈s,a〉 = 〈news,newa,move〉 →
+ trans sig (ifTM sig M1 M2 M3 acc) 〈inr … (inr … s),a〉 = 〈inr… (inr … news),newa,move〉.
+#sig #M1 * #Q2 #T2 #init2 #halt2 #M3 #acc #s #a #news #newa #move
#Hhalt #Htrans whd in ⊢ (??%?); >Hhalt >Htrans %
qed.
#sig #M1 #M2 #M3 #acc * #s #t
lapply (refl ? (trans ?? 〈s,current sig t〉))
cases (trans ?? 〈s,current sig t〉) in ⊢ (???% → %);
- #s0 #m0 cases t
+ * #s0 #a0 #m0 cases t
[ #Heq #Hhalt
| 2,3: #s1 #l1 #Heq #Hhalt
|#ls #s1 #rs #Heq #Hhalt ]
#sig #M1 (* * #Q1 #T1 #init1 #halt1 *) #M2 #M3 #acc * #s #t
lapply (refl ? (trans ?? 〈s,current sig t〉))
cases (trans ?? 〈s,current sig t〉) in ⊢ (???% → %);
- #s0 #m0 cases t
+ * #s0 #a0 #m0 cases t
[ #Heq #Hhalt
| 2,3: #s1 #l1 #Heq #Hhalt
|#ls #s1 #rs #Heq #Hhalt ]
#sig #M1 (* * #Q1 #T1 #init1 #halt1 *) #M2 #M3 #acc * #s #t
lapply (refl ? (trans ?? 〈s,current sig t〉))
cases (trans ?? 〈s,current sig t〉) in ⊢ (???% → %);
- #s0 #m0 cases t
+ * #s0 #a0 #m0 cases t
[ #Heq #Hhalt
| 2,3: #s1 #l1 #Heq #Hhalt
|#ls #s1 #rs #Heq #Hhalt ]
lemma trans_if_M1true_acc : ∀sig,M1,M2,M3,acc,s,a.
halt ? M1 s = true → s==acc = true →
- trans sig (ifTM sig M1 M2 M3 acc) 〈inl … s,a〉 = 〈inr … (inl … (start ? M2)),None ?〉.
+ trans sig (ifTM sig M1 M2 M3 acc) 〈inl … s,a〉 = 〈inr … (inl … (start ? M2)),None ?,N〉.
#sig #M1 #M2 #M3 #acc #s #a #Hhalt #Hacc whd in ⊢ (??%?); >Hhalt >Hacc %
qed.
lemma trans_if_M1true_notacc : ∀sig,M1,M2,M3,acc,s,a.
halt ? M1 s = true → s==acc = false →
- trans sig (ifTM sig M1 M2 M3 acc) 〈inl … s,a〉 = 〈inr … (inr … (start ? M3)),None ?〉.
+ trans sig (ifTM sig M1 M2 M3 acc) 〈inl … s,a〉 = 〈inr … (inr … (start ? M3)),None ?,N〉.
#sig #M1 #M2 #M3 #acc #s #a #Hhalt #Hacc whd in ⊢ (??%?); >Hhalt >Hacc %
qed.
(******************************** semantics ***********************************)
lemma sem_if: ∀sig.∀M1,M2,M3:TM sig.∀Rtrue,Rfalse,R2,R3,acc.
- M1 â\8a§ [acc: Rtrue,Rfalse] → M2 ⊨ R2 → M3 ⊨ R3 →
+ M1 â\8a¨ [acc: Rtrue,Rfalse] → M2 ⊨ R2 → M3 ⊨ R3 →
ifTM sig M1 M2 M3 acc ⊨ (Rtrue ∘ R2) ∪ (Rfalse ∘ R3).
#sig #M1 #M2 #M3 #Rtrue #Rfalse #R2 #R3 #acc #HaccR #HR2 #HR3 #t
cases (HaccR t) #k1 * #outc1 * * #Hloop1 #HMtrue #HMfalse
]
qed.
+lemma sem_if_app: ∀sig,M1,M2,M3,Rtrue,Rfalse,R2,R3,R4,acc.
+ accRealize sig M1 acc Rtrue Rfalse → M2 ⊨ R2 → M3 ⊨ R3 →
+ (∀t1,t2,t3. (Rtrue t1 t3 ∧ R2 t3 t2) ∨ (Rfalse t1 t3 ∧ R3 t3 t2) → R4 t1 t2) →
+ ifTM sig M1 M2 M3 acc ⊨ R4.
+#sig #M1 #M2 #M3 #Rtrue #Rfalse #R2 #R3 #R4 #acc
+#HRacc #HRtrue #HRfalse #Hsub
+#t cases (sem_if … HRacc HRtrue HRfalse t)
+#k * #outc * #Hloop #Houtc @(ex_intro … k) @(ex_intro … outc)
+% [@Hloop] cases Houtc
+ [* #t3 * #Hleft #Hright @(Hsub … t3) %1 /2/
+ |* #t3 * #Hleft #Hright @(Hsub … t3) %2 /2/ ]
+qed.
+
+(* weak
lemma sem_if_app: ∀sig,M1,M2,M3,Rtrue,Rfalse,R2,R3,R4,acc.
accRealize sig M1 acc Rtrue Rfalse → M2 ⊨ R2 → M3 ⊨ R3 →
(∀t1,t2,t3. (Rtrue t1 t3 → R2 t3 t2) ∨ (Rfalse t1 t3 → R3 t3 t2) → R4 t1 t2) →
[* #t3 * #Hleft #Hright @(Hsub … t3) %1 /2/
|* #t3 * #Hleft #Hright @(Hsub … t3) %2 /2/ ]
qed.
-
+*)
+
(* we can probably use acc_sem_if to prove sem_if *)
+(* for sure we can use acc_sem_if_guarded to prove acc_sem_if *)
lemma acc_sem_if: ∀sig,M1,M2,M3,Rtrue,Rfalse,R2,R3,acc.
M1 ⊨ [acc: Rtrue, Rfalse] → M2 ⊨ R2 → M3 ⊨ R3 →
ifTM sig M1 (single_finalTM … M2) M3 acc ⊨
|#H cases (Houtc1 H) #t3 * #Hleft #Hright @Hsub1 // ]
|#H cases (Houtc2 H) #t3 * #Hleft #Hright @Hsub2 // ]
qed.
+
+lemma sem_single_final_guarded: ∀sig.∀M: TM sig.∀Pre,R.
+ GRealize sig M Pre R → GRealize sig (single_finalTM sig M) Pre R.
+#sig #M #Pre #R #HR #intape #HPre
+cases (sem_seq_guarded ??????? HR (Realize_to_GRealize ?? (λt.True) ? (sem_nop …)) ?? HPre) //
+#k * #outc * #Hloop * #ta * #Hta whd in ⊢ (%→?); #Houtc
+@(ex_intro ?? k) @(ex_intro ?? outc) % [ @Hloop | >Houtc // ]
+qed.
+
+lemma acc_sem_if_guarded: ∀sig,M1,M2,M3,P,P2,Rtrue,Rfalse,R2,R3,acc.
+ M1 ⊨ [acc: Rtrue, Rfalse] →
+ (GRealize ? M2 P2 R2) → (∀t,t0.P t → Rtrue t t0 → P2 t0) →
+ M3 ⊨ R3 →
+ accGRealize ? (ifTM sig M1 (single_finalTM … M2) M3 acc)
+ (inr … (inl … (inr … start_nop))) P (Rtrue ∘ R2) (Rfalse ∘ R3).
+#sig #M1 #M2 #M3 #P #P2 #Rtrue #Rfalse #R2 #R3 #acc #HaccR #HR2 #HP2 #HR3 #t #HPt
+cases (HaccR t) #k1 * #outc1 * * #Hloop1 #HMtrue #HMfalse
+cases (true_or_false (cstate ?? outc1 == acc)) #Hacc
+ [lapply (sem_single_final_guarded … HR2) -HR2 #HR2
+ cases (HR2 (ctape sig ? outc1) ?)
+ [|@HP2 [||@HMtrue @(\P Hacc)] // ]
+ #k2 * #outc2 * #Hloop2 #HM2
+ @(ex_intro … (k1+k2))
+ @(ex_intro … (lift_confR … (lift_confL … outc2))) %
+ [%
+ [@(loop_merge ?????????
+ (mk_config ? (states sig (ifTM sig M1 (single_finalTM … M2) M3 acc))
+ (inr (states sig M1) ? (inl ? (states sig M3) (start sig (single_finalTM sig M2)))) (ctape ?? outc1) )
+ ?
+ (loop_lift ???
+ (lift_confL sig (states ? M1) (FinSum (states ? (single_finalTM … M2)) (states ? M3)))
+ (step sig M1) (step sig (ifTM sig M1 (single_finalTM ? M2) M3 acc))
+ (λc.halt sig M1 (cstate … c))
+ (λc.halt_liftL ?? (halt sig M1) (cstate … c))
+ … Hloop1))
+ [* *
+ [ #sl #tl whd in ⊢ (??%? → ?); #Hl %
+ | #sr #tr whd in ⊢ (??%? → ?); #Hr destruct (Hr) ]
+ |#c0 #Hhalt >(step_if_liftM1 … Hhalt) //
+ |#x <p_halt_liftL %
+ |whd in ⊢ (??%?); >(config_expand ?? outc1);
+ whd in match (lift_confL ????);
+ >(trans_if_M1true_acc … Hacc)
+ [% | @(loop_Some ?????? Hloop1)]
+ |cases outc1 #s1 #t1 %
+ |@(loop_lift ???
+ (λc.(lift_confR … (lift_confL sig (states ? (single_finalTM ? M2)) (states ? M3) c)))
+ … Hloop2)
+ [ * #s2 #t2 %
+ | #c0 #Hhalt >(step_if_liftM2 … Hhalt) // ]
+ ]
+ |#_ @(ex_intro … (ctape ?? outc1)) %
+ [@HMtrue @(\P Hacc) | >(config_expand ?? outc2) @HM2 ]
+ ]
+ |>(config_expand ?? outc2) whd in match (lift_confR ????);
+ * #H @False_ind @H @eq_f @eq_f >(config_expand ?? outc2)
+ @single_final // @(loop_Some ?????? Hloop2)
+ ]
+ |cases (HR3 (ctape sig ? outc1)) #k2 * #outc2 * #Hloop2 #HM3
+ @(ex_intro … (k1+k2)) @(ex_intro … (lift_confR … (lift_confR … outc2))) %
+ [%
+ [@(loop_merge ?????????
+ (mk_config ? (states sig (ifTM sig M1 (single_finalTM … M2) M3 acc))
+ (inr (states sig M1) ? (inr (states sig (single_finalTM ? M2)) ? (start sig M3))) (ctape ?? outc1) )
+ ?
+ (loop_lift ???
+ (lift_confL sig (states ? M1) (FinSum (states ? (single_finalTM … M2)) (states ? M3)))
+ (step sig M1) (step sig (ifTM sig M1 (single_finalTM ? M2) M3 acc))
+ (λc.halt sig M1 (cstate … c))
+ (λc.halt_liftL ?? (halt sig M1) (cstate … c))
+ … Hloop1))
+ [* *
+ [ #sl #tl whd in ⊢ (??%? → ?); #Hl %
+ | #sr #tr whd in ⊢ (??%? → ?); #Hr destruct (Hr) ]
+ |#c0 #Hhalt >(step_if_liftM1 … Hhalt) //
+ |#x <p_halt_liftL %
+ |whd in ⊢ (??%?); >(config_expand ?? outc1);
+ whd in match (lift_confL ????);
+ >(trans_if_M1true_notacc … Hacc)
+ [% | @(loop_Some ?????? Hloop1)]
+ |cases outc1 #s1 #t1 %
+ |@(loop_lift ???
+ (λc.(lift_confR … (lift_confR sig (states ? (single_finalTM ? M2)) (states ? M3) c)))
+ … Hloop2)
+ [ * #s2 #t2 %
+ | #c0 #Hhalt >(step_if_liftM3 … Hhalt) // ]
+ ]
+ |>(config_expand ?? outc2) whd in match (lift_confR ????);
+ #H destruct (H)
+ ]
+ |#_ @(ex_intro … (ctape ?? outc1)) %
+ [@HMfalse @(\Pf Hacc) | >(config_expand ?? outc2) @HM3 ]
+ ]
+ ]
+qed.
+
+lemma acc_sem_if_app_guarded: ∀sig,M1,M2,M3,P,P2,Rtrue,Rfalse,R2,R3,R4,R5,acc.
+ M1 ⊨ [acc: Rtrue, Rfalse] →
+ (GRealize ? M2 P2 R2) → (∀t,t0.P t → Rtrue t t0 → P2 t0) →
+ M3 ⊨ R3 →
+ (∀t1,t2,t3. Rtrue t1 t3 → R2 t3 t2 → R4 t1 t2) →
+ (∀t1,t2,t3. Rfalse t1 t3 → R3 t3 t2 → R5 t1 t2) →
+ accGRealize ? (ifTM sig M1 (single_finalTM … M2) M3 acc)
+ (inr … (inl … (inr … start_nop))) P R4 R5 .
+#sig #M1 #M2 #M3 #P #P2 #Rtrue #Rfalse #R2 #R3 #R4 #R5 #acc
+#HRacc #HRtrue #Hinv #HRfalse #Hsub1 #Hsub2
+#t #HPt cases (acc_sem_if_guarded … HRacc HRtrue Hinv HRfalse t HPt)
+#k * #outc * * #Hloop #Houtc1 #Houtc2 @(ex_intro … k) @(ex_intro … outc)
+% [% [@Hloop
+ |#H cases (Houtc1 H) #t3 * #Hleft #Hright @Hsub1 // ]
+ |#H cases (Houtc2 H) #t3 * #Hleft #Hright @Hsub2 // ]
+qed.
+
+
projects / helm.git / blobdiff