+
+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 // ]