-lemma loop_lift : ∀A,B,k,lift,f,g,h,hlift,c1,c2.
- (∀x.hlift (lift x) = h x) →
- (∀x.h x = false → lift (f x) = g (lift x)) →
- loop A k f h c1 = Some ? c2 →
- loop B k g hlift (lift c1) = Some ? (lift … c2).
-#A #B #k #lift #f #g #h #hlift #c1 #c2 #Hfg #Hhlift
-generalize in match c1; elim k
-[#c0 normalize in ⊢ (??%? → ?); #Hfalse destruct (Hfalse)
-|#k0 #IH #c0 whd in ⊢ (??%? → ??%?);
- cases (true_or_false (h c0)) #Hc0 >Hfg >Hc0
- [ normalize #Heq destruct (Heq) %
- | normalize <Hhlift // @IH ]
-qed.
-
-(*
-lemma loop_liftL : ∀sig,k,M1,M2,c1,c2.
- loop ? k (step sig M1) (λc.halt sig M1 (cstate ?? c)) c1 = Some ? c2 →
- loop ? k (step sig (seq sig M1 M2))
- (λc.halt_liftL ?? (halt sig M1) (cstate ?? c)) (lift_confL … c1) =
- Some ? (lift_confL … c2).
-#sig #k #M1 #M2 #c1 #c2 generalize in match c1;
-elim k
-[#c0 normalize in ⊢ (??%? → ?); #Hfalse destruct (Hfalse)
-|#k0 #IH #c0 whd in ⊢ (??%? → ??%?);
- cases (true_or_false (halt ?? (cstate sig (states ? M1) c0))) #Hc0 >Hc0
- [ >(?: halt_liftL ?? (halt sig M1) (cstate sig ? (lift_confL … c0)) = true)
- [ whd in ⊢ (??%? → ??%?); #Hc2 destruct (Hc2) %
- | <Hc0 cases c0 // ]
- | >(?: halt_liftL ?? (halt sig M1) (cstate ?? (lift_confL … c0)) = false)
- [whd in ⊢ (??%? → ??%?); #Hc2 <(IH ? Hc2) @eq_f
- @step_lift_confL //
- | <Hc0 cases c0 // ]
-qed.
-
-lemma loop_liftR : ∀sig,k,M1,M2,c1,c2.
- loop ? k (step sig M2) (λc.halt sig M2 (cstate ?? c)) c1 = Some ? c2 →
- loop ? k (step sig (seq sig M1 M2))
- (λc.halt sig (seq sig M1 M2) (cstate ?? c)) (lift_confR … c1) =
- Some ? (lift_confR … c2).
-#sig #k #M1 #M2 #c1 #c2 generalize in match c1;
-elim k
-[#c0 normalize in ⊢ (??%? → ?); #Hfalse destruct (Hfalse)
-|#k0 #IH #c0 whd in ⊢ (??%? → ??%?);
- cases (true_or_false (halt ?? (cstate sig ? c0))) #Hc0 >Hc0
- [ >(?: halt ? (seq sig M1 M2) (cstate sig ? (lift_confR … c0)) = true)
- [ whd in ⊢ (??%? → ??%?); #Hc2 destruct (Hc2) %
- | <Hc0 cases c0 // ]
- | >(?: halt ? (seq sig M1 M2) (cstate sig ? (lift_confR … c0)) = false)
- [whd in ⊢ (??%? → ??%?); #Hc2 <(IH ? Hc2) @eq_f
- @step_lift_confR //
- | <Hc0 cases c0 // ]
- ]
-qed.
-
-*)
-
-lemma loop_Some :
- ∀A,k,f,p,a,b.loop A k f p a = Some ? b → p b = true.
-#A #k #f #p elim k
-[#a #b normalize #Hfalse destruct
-|#k0 #IH #a #b whd in ⊢ (??%? → ?); cases (true_or_false (p a)) #Hpa
- [ >Hpa normalize #H1 destruct //
- | >Hpa normalize @IH
- ]
-]
-qed.
-