--- /dev/null
+(*
+ ||M|| This file is part of HELM, an Hypertextual, Electronic
+ ||A|| Library of Mathematics, developed at the Computer Science
+ ||T|| Department of the University of Bologna, Italy.
+ ||I||
+ ||T||
+ ||A||
+ \ / This file is distributed under the terms of the
+ \ / GNU General Public License Version 2
+ V_____________________________________________________________*)
+
+include "turing/turing.ma".
+
+(**************************** single final machine ****************************)
+
+definition single_finalTM ≝
+ λsig,n.λM:mTM sig n.seq ?? M (nop ??).
+
+lemma sem_single_final: ∀sig,n.∀M: mTM sig n.∀R.
+ M ⊨ R → single_finalTM sig n M ⊨ R.
+#sig #n #M #R #HR #intape
+cases (sem_seq ?????? HR (sem_nop …) intape)
+#k * #outc * #Hloop * #ta * #Hta whd in ⊢ (%→?); #Houtc
+@(ex_intro ?? k) @(ex_intro ?? outc) % [ @Hloop | >Houtc // ]
+qed.
+
+lemma single_final: ∀sig,n.∀M: mTM sig n.∀q1,q2.
+ halt ?? (single_finalTM sig n M) q1 = true
+ → halt ?? (single_finalTM sig n M) q2 = true → q1=q2.
+#sig #n #M *
+ [#q1M #q2 whd in match (halt ????); #H destruct
+ |#q1nop *
+ [#q2M #_ whd in match (halt ????); #H destruct
+ |#q2nop #_ #_ @eq_f normalize @nop_single_state
+ ]
+ ]
+qed.
+
+(******************************** if machine **********************************)
+
+definition if_trans ≝ λsig,n. λM1,M2,M3 : mTM sig n. λq:states sig n M1.
+λp. let 〈s,a〉 ≝ p in
+ match s with
+ [ inl s1 ⇒
+ if halt sig n M1 s1 then
+ if s1==q then 〈inr … (inl … (start sig n M2)), null_action ??〉
+ else 〈inr … (inr … (start sig n M3)), null_action ??〉
+ else let 〈news1,m〉 ≝ trans sig n M1 〈s1,a〉 in
+ 〈inl … news1,m〉
+ | inr s' ⇒
+ match s' with
+ [ inl s2 ⇒ let 〈news2,m〉 ≝ trans sig n M2 〈s2,a〉 in
+ 〈inr … (inl … news2),m〉
+ | inr s3 ⇒ let 〈news3,m〉 ≝ trans sig n M3 〈s3,a〉 in
+ 〈inr … (inr … news3),m〉
+ ]
+ ].
+
+definition ifTM ≝ λsig,n. λcondM,thenM,elseM : mTM sig n.
+ λqacc: states sig n condM.
+ mk_mTM sig n
+ (FinSum (states sig n condM) (FinSum (states sig n thenM) (states sig n elseM)))
+ (if_trans sig n condM thenM elseM qacc)
+ (inl … (start sig n condM))
+ (λs.match s with
+ [ inl _ ⇒ false
+ | inr s' ⇒ match s' with
+ [ inl s2 ⇒ halt sig n thenM s2
+ | inr s3 ⇒ halt sig n elseM s3 ]]).
+
+(****************************** lifting lemmas ********************************)
+lemma trans_if_liftM1 : ∀sig,n,M1,M2,M3,acc,s,a,news,move.
+ halt ?? M1 s = false →
+ trans sig n M1 〈s,a〉 = 〈news,move〉 →
+ trans sig n (ifTM sig n M1 M2 M3 acc) 〈inl … s,a〉 = 〈inl … news,move〉.
+#sig #n * #Q1 #T1 #init1 #halt1 #M2 #M3 #acc #s #a #news #move
+#Hhalt #Htrans whd in ⊢ (??%?); >Hhalt >Htrans %
+qed.
+
+lemma trans_if_liftM2 : ∀sig,n,M1,M2,M3,acc,s,a,news,move.
+ halt ?? M2 s = false →
+ trans sig n M2 〈s,a〉 = 〈news,move〉 →
+ trans sig n (ifTM sig n M1 M2 M3 acc) 〈inr … (inl … s),a〉 = 〈inr… (inl … news),move〉.
+#sig #n #M1 * #Q2 #T2 #init2 #halt2 #M3 #acc #s #a #news #move
+#Hhalt #Htrans whd in ⊢ (??%?); >Hhalt >Htrans %
+qed.
+
+lemma trans_if_liftM3 : ∀sig,n,M1,M2,M3,acc,s,a,news,move.
+ halt ?? M3 s = false →
+ trans sig n M3 〈s,a〉 = 〈news,move〉 →
+ trans sig n (ifTM sig n M1 M2 M3 acc) 〈inr … (inr … s),a〉 = 〈inr… (inr … news),move〉.
+#sig #n #M1 * #Q2 #T2 #init2 #halt2 #M3 #acc #s #a #news #move
+#Hhalt #Htrans whd in ⊢ (??%?); >Hhalt >Htrans %
+qed.
+
+lemma step_if_liftM1 : ∀sig,n,M1,M2,M3,acc,c0.
+ halt ?? M1 (cstate ??? c0) = false →
+ step sig n (ifTM sig n M1 M2 M3 acc) (lift_confL sig n (states ?? M1) ? c0) =
+ lift_confL sig n (states ?? M1) ? (step sig n M1 c0).
+#sig #n #M1 #M2 #M3 #acc * #s #t
+ lapply (refl ? (trans ??? 〈s,current_chars sig n t〉))
+ cases (trans ??? 〈s,current_chars sig n t〉) in ⊢ (???% → %);
+ #s0 #m0 #Heq #Hhalt
+ whd in ⊢ (???(?????%)); >Heq whd in ⊢ (???%);
+ whd in ⊢ (??(????%)?); whd in ⊢ (??%?); >(trans_if_liftM1 … Hhalt Heq) //
+qed.
+
+lemma step_if_liftM2 : ∀sig,n,M1,M2,M3,acc,c0.
+ halt ?? M2 (cstate ??? c0) = false →
+ step sig n (ifTM sig n M1 M2 M3 acc) (lift_confR sig ??? (lift_confL sig ??? c0)) =
+ lift_confR sig ??? (lift_confL sig ??? (step sig n M2 c0)).
+#sig #n #M1 (* * #Q1 #T1 #init1 #halt1 *) #M2 #M3 #acc * #s #t
+ lapply (refl ? (trans ??? 〈s,current_chars sig n t〉))
+ cases (trans ??? 〈s,current_chars sig n t〉) in ⊢ (???% → %);
+ #s0 #m0 #Heq #Hhalt
+ whd in match (step ?? M2 ?); >Heq whd in ⊢ (???%);
+ whd in ⊢ (??(????%)?); whd in ⊢ (??%?); >(trans_if_liftM2 … Hhalt Heq) //
+qed.
+
+lemma step_if_liftM3 : ∀sig,n,M1,M2,M3,acc,c0.
+ halt ?? M3 (cstate ??? c0) = false →
+ step sig n (ifTM sig n M1 M2 M3 acc) (lift_confR sig ??? (lift_confR sig ??? c0)) =
+ lift_confR sig ??? (lift_confR sig ??? (step sig n M3 c0)).
+#sig #n #M1 #M2 #M3 #acc * #s #t
+ lapply (refl ? (trans ??? 〈s,current_chars sig n t〉))
+ cases (trans ??? 〈s,current_chars sig n t〉) in ⊢ (???% → %);
+ #s0 #m0 #Heq #Hhalt
+ whd in match (step ?? M3 ?); >Heq whd in ⊢ (???%);
+ whd in ⊢ (??(????%)?); whd in ⊢ (??%?); >(trans_if_liftM3 … Hhalt Heq) //
+qed.
+
+lemma trans_if_M1true_acc : ∀sig,n,M1,M2,M3,acc,s,a.
+ halt ?? M1 s = true → s==acc = true →
+ trans sig n (ifTM sig n M1 M2 M3 acc) 〈inl … s,a〉 =
+ 〈inr … (inl … (start ?? M2)),null_action ??〉.
+#sig #n #M1 #M2 #M3 #acc #s #a #Hhalt #Hacc whd in ⊢ (??%?); >Hhalt >Hacc %
+qed.
+
+lemma trans_if_M1true_notacc : ∀sig,n,M1,M2,M3,acc,s,a.
+ halt ?? M1 s = true → s==acc = false →
+ trans sig n (ifTM sig n M1 M2 M3 acc) 〈inl … s,a〉 =
+ 〈inr … (inr … (start ?? M3)),null_action ??〉.
+#sig #n #M1 #M2 #M3 #acc #s #a #Hhalt #Hacc whd in ⊢ (??%?); >Hhalt >Hacc %
+qed.
+
+(******************************** semantics ***********************************)
+lemma sem_if: ∀sig,n.∀M1,M2,M3:mTM sig n.∀Rtrue,Rfalse,R2,R3,acc.
+ M1 ⊨ [acc: Rtrue,Rfalse] → M2 ⊨ R2 → M3 ⊨ R3 →
+ ifTM sig n M1 M2 M3 acc ⊨ (Rtrue ∘ R2) ∪ (Rfalse ∘ R3).
+#sig #n #M1 #M2 #M3 #Rtrue #Rfalse #R2 #R3 #acc #HaccR #HR2 #HR3 #t
+cases (HaccR t) #k1 * #outc1 * * #Hloop1 #HMtrue #HMfalse
+cases (true_or_false (cstate ??? outc1 == acc)) #Hacc
+ [cases (HR2 (ctapes sig ?? outc1)) #k2 * #outc2 * #Hloop2 #HM2
+ @(ex_intro … (k1+k2)) @(ex_intro … (lift_confR … (lift_confL … outc2))) %
+ [@(loop_merge ?????????
+ (mk_mconfig ? (FinSum (states sig n M1) (FinSum (states sig n M2) (states sig n M3))) n
+ (inr (states sig n M1) ? (inl (states sig n M2) (states sig n M3) (start sig n M2))) (ctapes ??? outc1) )
+ ?
+ (loop_lift ???
+ (lift_confL sig n (states ?? M1) (FinSum (states ?? M2) (states ?? M3)))
+ (step sig n M1) (step sig n (ifTM sig n M1 M2 M3 acc))
+ (λc.halt sig n M1 (cstate … c))
+ (λc.halt_liftL ?? (halt sig n 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 ⊢ (??%?); >(mconfig_expand ??? outc1)
+ whd in match (lift_confL ?????);
+ >(trans_if_M1true_acc … Hacc)
+ [@mconfig_eq // (* whd in ⊢ (??%?); *)
+ <(tape_move_null_action sig n (ctapes sig (states sig n M1) n outc1)) in ⊢ (???%); %
+ |@(loop_Some ?????? Hloop1)]
+ |cases outc1 #s1 #t1 %
+ |@(loop_lift ???
+ (λc.(lift_confR … (lift_confL sig n (states ?? M2) (states ?? M3) c)))
+ … Hloop2)
+ [ * #s2 #t2 %
+ | #c0 #Hhalt >(step_if_liftM2 … Hhalt) // ]
+ ]
+ |%1 @(ex_intro … (ctapes ??? outc1)) %
+ [@HMtrue @(\P Hacc) | >(mconfig_expand ??? outc2) @HM2 ]
+ ]
+ |cases (HR3 (ctapes sig ?? outc1)) #k2 * #outc2 * #Hloop2 #HM3
+ @(ex_intro … (k1+k2)) @(ex_intro … (lift_confR … (lift_confR … outc2))) %
+ [@(loop_merge ?????????
+ (mk_mconfig ? (FinSum (states sig ? M1) (FinSum (states sig ? M2) (states sig ? M3))) n
+ (inr (states sig ? M1) ? (inr (states sig ? M2) (states sig ? M3) (start sig ? M3))) (ctapes ??? outc1) )
+ ?
+ (loop_lift ???
+ (lift_confL sig n (states ?? M1) (FinSum (states ?? M2) (states ?? M3)))
+ (step sig n M1) (step sig n (ifTM sig n M1 M2 M3 acc))
+ (λc.halt sig n M1 (cstate … c))
+ (λc.halt_liftL ?? (halt sig n 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 ⊢ (??%?); >(mconfig_expand ??? outc1);
+ whd in match (lift_confL ?????);
+ >(trans_if_M1true_notacc … Hacc)
+ [@mconfig_eq // (* whd in ⊢ (??%?); *)
+ <(tape_move_null_action sig n (ctapes sig (states sig n M1) n outc1)) in ⊢ (???%); %
+ |@(loop_Some ?????? Hloop1)]
+ |cases outc1 #s1 #t1 %
+ |@(loop_lift ???
+ (λc.(lift_confR … (lift_confR sig n (states ?? M2) (states ?? M3) c)))
+ … Hloop2)
+ [ * #s2 #t2 %
+ | #c0 #Hhalt >(step_if_liftM3 … Hhalt) // ]
+ ]
+ |%2 @(ex_intro … (ctapes ??? outc1)) %
+ [@HMfalse @(\Pf Hacc) | >(mconfig_expand ??? outc2) @HM3 ]
+ ]
+ ]
+qed.
+
+lemma sem_if_app: ∀sig,n,M1,M2,M3,Rtrue,Rfalse,R2,R3,R4,acc.
+ accRealize sig n 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 n M1 M2 M3 acc ⊨ R4.
+#sig #n #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.
+
+(* 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,n,M1,M2,M3,Rtrue,Rfalse,R2,R3,acc.
+ M1 ⊨ [acc: Rtrue, Rfalse] → M2 ⊨ R2 → M3 ⊨ R3 →
+ ifTM sig n M1 (single_finalTM … M2) M3 acc ⊨
+ [inr … (inl … (inr … start_nop)): Rtrue ∘ R2, Rfalse ∘ R3].
+#sig #n #M1 #M2 #M3 #Rtrue #Rfalse #R2 #R3 #acc #HaccR #HR2 #HR3 #t
+cases (HaccR t) #k1 * #outc1 * * #Hloop1 #HMtrue #HMfalse
+cases (true_or_false (cstate ??? outc1 == acc)) #Hacc
+ [lapply (sem_single_final … HR2) -HR2 #HR2
+ cases (HR2 (ctapes sig ?? outc1)) #k2 * #outc2 * #Hloop2 #HM2
+ @(ex_intro … (k1+k2))
+ @(ex_intro … (lift_confR … (lift_confL … outc2))) %
+ [%
+ [@(loop_merge ?????????
+ (mk_mconfig ? (states sig n (ifTM sig n M1 (single_finalTM … M2) M3 acc)) n
+ (inr (states sig n M1) ? (inl ? (states sig n M3) (start sig n (single_finalTM sig n M2)))) (ctapes ??? outc1) )
+ ?
+ (loop_lift ???
+ (lift_confL sig n (states ?? M1) (FinSum (states ?? (single_finalTM … M2)) (states ?? M3)))
+ (step sig n M1) (step sig n (ifTM sig n M1 (single_finalTM ?? M2) M3 acc))
+ (λc.halt sig n M1 (cstate … c))
+ (λc.halt_liftL ?? (halt sig n 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 ⊢ (??%?); >(mconfig_expand ??? outc1);
+ whd in match (lift_confL ????);
+ >(trans_if_M1true_acc … Hacc)
+ [@mconfig_eq // (* whd in ⊢ (??%?); *)
+ <(tape_move_null_action sig n (ctapes sig (states sig n M1) n outc1)) in ⊢ (???%); %
+ | @(loop_Some ?????? Hloop1)]
+ |cases outc1 #s1 #t1 %
+ |@(loop_lift ???
+ (λc.(lift_confR … (lift_confL sig n (states ?? (single_finalTM ?? M2)) (states ?? M3) c)))
+ … Hloop2)
+ [ * #s2 #t2 %
+ | #c0 #Hhalt >(step_if_liftM2 … Hhalt) // ]
+ ]
+ |#_ @(ex_intro … (ctapes ??? outc1)) %
+ [@HMtrue @(\P Hacc) | >(mconfig_expand ??? outc2) @HM2 ]
+ ]
+ |>(mconfig_expand ??? outc2) whd in match (lift_confR ?????);
+ * #H @False_ind @H @eq_f @eq_f >(mconfig_expand ??? outc2)
+ @single_final // @(loop_Some ?????? Hloop2)
+ ]
+ |cases (HR3 (ctapes sig ?? outc1)) #k2 * #outc2 * #Hloop2 #HM3
+ @(ex_intro … (k1+k2)) @(ex_intro … (lift_confR … (lift_confR … outc2))) %
+ [%
+ [@(loop_merge ?????????
+ (mk_mconfig ? (states sig n (ifTM sig n M1 (single_finalTM … M2) M3 acc)) n
+ (inr (states sig n M1) ? (inr (states sig n (single_finalTM ?? M2)) ? (start sig n M3))) (ctapes ??? outc1) )
+ ?
+ (loop_lift ???
+ (lift_confL sig n (states ?? M1) (FinSum (states ?? (single_finalTM … M2)) (states ?? M3)))
+ (step sig n M1) (step sig n (ifTM sig n M1 (single_finalTM ?? M2) M3 acc))
+ (λc.halt sig n M1 (cstate … c))
+ (λc.halt_liftL ?? (halt sig n 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 ⊢ (??%?); >(mconfig_expand ??? outc1);
+ whd in match (lift_confL ?????);
+ >(trans_if_M1true_notacc … Hacc)
+ [@mconfig_eq // (* whd in ⊢ (??%?); *)
+ <(tape_move_null_action sig n (ctapes sig (states sig n M1) n outc1)) in ⊢ (???%); %
+ |@(loop_Some ?????? Hloop1)]
+ |cases outc1 #s1 #t1 %
+ |@(loop_lift ???
+ (λc.(lift_confR … (lift_confR sig n (states ?? (single_finalTM ?? M2)) (states ?? M3) c)))
+ … Hloop2)
+ [ * #s2 #t2 %
+ | #c0 #Hhalt >(step_if_liftM3 … Hhalt) // ]
+ ]
+ |>(mconfig_expand ??? outc2) whd in match (lift_confR ?????);
+ #H destruct (H)
+ ]
+ |#_ @(ex_intro … (ctapes ??? outc1)) %
+ [@HMfalse @(\Pf Hacc) | >(mconfig_expand ??? outc2) @HM3 ]
+ ]
+ ]
+qed.
+
+lemma acc_sem_if_app: ∀sig,n,M1,M2,M3,Rtrue,Rfalse,R2,R3,R4,R5,acc.
+ M1 ⊨ [acc: Rtrue, Rfalse] → M2 ⊨ R2 → 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) →
+ ifTM sig n M1 (single_finalTM … M2) M3 acc ⊨
+ [inr … (inl … (inr … start_nop)): R4, R5].
+#sig #n #M1 #M2 #M3 #Rtrue #Rfalse #R2 #R3 #R4 #R5 #acc
+#HRacc #HRtrue #HRfalse #Hsub1 #Hsub2
+#t cases (acc_sem_if … HRacc HRtrue HRfalse t)
+#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.
+
+lemma sem_single_final_guarded: ∀sig,n.∀M: mTM sig n.∀Pre,R.
+ GRealize sig n M Pre R → GRealize sig n (single_finalTM sig n M) Pre R.
+#sig #n #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,n.∀M1,M2,M3: mTM sig n.∀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 n M1 (single_finalTM … M2) M3 acc)
+ (inr … (inl … (inr … start_nop))) P (Rtrue ∘ R2) (Rfalse ∘ R3).
+#sig #n #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 (ctapes sig ?? outc1) ?)
+ [|@HP2 [||@HMtrue @(\P Hacc)] // ]
+ #k2 * #outc2 * #Hloop2 #HM2
+ @(ex_intro … (k1+k2))
+ @(ex_intro … (lift_confR … (lift_confL … outc2))) %
+ [%
+ [@(loop_merge ?????????
+ (mk_mconfig ? (states sig n (ifTM sig n M1 (single_finalTM … M2) M3 acc)) n
+ (inr (states sig n M1) ? (inl ? (states sig n M3) (start sig n (single_finalTM sig n M2)))) (ctapes ??? outc1) )
+ ?
+ (loop_lift ???
+ (lift_confL sig n (states ?? M1) (FinSum (states ?? (single_finalTM … M2)) (states ?? M3)))
+ (step sig n M1) (step sig n (ifTM sig n M1 (single_finalTM ?? M2) M3 acc))
+ (λc.halt sig n M1 (cstate … c))
+ (λc.halt_liftL ?? (halt sig n 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 ⊢ (??%?); >(mconfig_expand ??? outc1);
+ whd in match (lift_confL ?????);
+ >(trans_if_M1true_acc … Hacc)
+ [@mconfig_eq // (* whd in ⊢ (??%?); *)
+ <(tape_move_null_action sig n (ctapes sig (states sig n M1) n outc1)) in ⊢ (???%); %
+ | @(loop_Some ?????? Hloop1)]
+ |cases outc1 #s1 #t1 %
+ |@(loop_lift ???
+ (λc.(lift_confR … (lift_confL sig n (states ?? (single_finalTM ?? M2)) (states ?? M3) c)))
+ … Hloop2)
+ [ * #s2 #t2 %
+ | #c0 #Hhalt >(step_if_liftM2 … Hhalt) // ]
+ ]
+ |#_ @(ex_intro … (ctapes ??? outc1)) %
+ [@HMtrue @(\P Hacc) | >(mconfig_expand ??? outc2) @HM2 ]
+ ]
+ |>(mconfig_expand ??? outc2) whd in match (lift_confR ?????);
+ * #H @False_ind @H @eq_f @eq_f >(mconfig_expand ??? outc2)
+ @single_final // @(loop_Some ?????? Hloop2)
+ ]
+ |cases (HR3 (ctapes sig ?? outc1)) #k2 * #outc2 * #Hloop2 #HM3
+ @(ex_intro … (k1+k2)) @(ex_intro … (lift_confR … (lift_confR … outc2))) %
+ [%
+ [@(loop_merge ?????????
+ (mk_mconfig ? (states sig n (ifTM sig n M1 (single_finalTM … M2) M3 acc)) n
+ (inr (states sig n M1) ? (inr (states sig n (single_finalTM ?? M2)) ? (start sig n M3))) (ctapes ??? outc1) )
+ ?
+ (loop_lift ???
+ (lift_confL sig n (states ?? M1) (FinSum (states ?? (single_finalTM … M2)) (states ?? M3)))
+ (step sig n M1) (step sig n (ifTM sig n M1 (single_finalTM ?? M2) M3 acc))
+ (λc.halt sig n M1 (cstate … c))
+ (λc.halt_liftL ?? (halt sig n 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 ⊢ (??%?); >(mconfig_expand ??? outc1);
+ whd in match (lift_confL ?????);
+ >(trans_if_M1true_notacc … Hacc)
+ [@mconfig_eq // (* whd in ⊢ (??%?); *)
+ <(tape_move_null_action sig n (ctapes sig (states sig n M1) n outc1)) in ⊢ (???%); %
+ | @(loop_Some ?????? Hloop1)]
+ |cases outc1 #s1 #t1 %
+ |@(loop_lift ???
+ (λc.(lift_confR … (lift_confR sig n (states ?? (single_finalTM ?? M2)) (states ?? M3) c)))
+ … Hloop2)
+ [ * #s2 #t2 %
+ | #c0 #Hhalt >(step_if_liftM3 … Hhalt) // ]
+ ]
+ |>(mconfig_expand ??? outc2) whd in match (lift_confR ?????);
+ #H destruct (H)
+ ]
+ |#_ @(ex_intro … (ctapes ??? outc1)) %
+ [@HMfalse @(\Pf Hacc) | >(mconfig_expand ??? outc2) @HM3 ]
+ ]
+ ]
+qed.
+
+lemma acc_sem_if_app_guarded: ∀sig,n,M1,M2,M3,P,P2,Rtrue,Rfalse,R2,R3,R4,R5,acc.
+ M1 ⊨ [acc: Rtrue, Rfalse] →
+ (GRealize ? n 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 ? n (ifTM sig n M1 (single_finalTM … M2) M3 acc)
+ (inr … (inl … (inr … start_nop))) P R4 R5 .
+#sig #n #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 / commitdiff