include "basic_2/rt_computation/lfpxs_lfpxs.ma".
include "basic_2/rt_computation/lfsx_lfsx.ma".
-(* STRONGLY NORMALIZING LOCAL ENV.S FOR UNCOUNTED PARALLEL RT-TRANSITION ****)
+(* STRONGLY NORMALIZING LOCAL ENV.S FOR UNBOUND PARALLEL RT-TRANSITION ******)
-(* Properties with uncounted rt-computation on referred entries *************)
+(* Properties with unbound rt-computation on referred entries ***************)
(* Basic_2A1: uses: lsx_intro_alt *)
lemma lfsx_intro_lfpxs: ∀h,o,G,L1,T.
- (â\88\80L2. â¦\83G, L1â¦\84 â\8a¢ â¬\88*[h, T] L2 â\86\92 (L1 â\89¡[h, o, T] L2 → ⊥) → G ⊢ ⬈*[h, o, T] 𝐒⦃L2⦄) →
+ (â\88\80L2. â¦\83G, L1â¦\84 â\8a¢ â¬\88*[h, T] L2 â\86\92 (L1 â\89\9b[h, o, T] L2 → ⊥) → G ⊢ ⬈*[h, o, T] 𝐒⦃L2⦄) →
G ⊢ ⬈*[h, o, T] 𝐒⦃L1⦄.
/4 width=1 by lfpx_lfpxs, lfsx_intro/ qed-.
/2 width=3 by lfsx_lfpx_trans/
qed-.
-(* Eliminators with uncounted rt-computation on referred entries ************)
+(* Eliminators with unbound rt-computation on referred entries **************)
lemma lfsx_ind_lfpxs_lfdeq: ∀h,o,G,T. ∀R:predicate lenv.
(∀L1. G ⊢ ⬈*[h, o, T] 𝐒⦃L1⦄ →
- (â\88\80L2. â¦\83G, L1â¦\84 â\8a¢ â¬\88*[h, T] L2 â\86\92 (L1 â\89¡[h, o, T] L2 → ⊥) → R L2) →
+ (â\88\80L2. â¦\83G, L1â¦\84 â\8a¢ â¬\88*[h, T] L2 â\86\92 (L1 â\89\9b[h, o, T] L2 → ⊥) → R L2) →
R L1
) →
∀L1. G ⊢ ⬈*[h, o, T] 𝐒⦃L1⦄ →
- â\88\80L0. â¦\83G, L1â¦\84 â\8a¢ â¬\88*[h, T] L0 â\86\92 â\88\80L2. L0 â\89¡[h, o, T] L2 → R L2.
+ â\88\80L0. â¦\83G, L1â¦\84 â\8a¢ â¬\88*[h, T] L0 â\86\92 â\88\80L2. L0 â\89\9b[h, o, T] L2 → R L2.
#h #o #G #T #R #IH #L1 #H @(lfsx_ind … H) -L1
#L1 #HL1 #IH1 #L0 #HL10 #L2 #HL02
@IH -IH /3 width=3 by lfsx_lfpxs_trans, lfsx_lfdeq_trans/ -HL1 #K2 #HLK2 #HnLK2
(* Basic_2A1: uses: lsx_ind_alt *)
lemma lfsx_ind_lfpxs: ∀h,o,G,T. ∀R:predicate lenv.
(∀L1. G ⊢ ⬈*[h, o, T] 𝐒⦃L1⦄ →
- (â\88\80L2. â¦\83G, L1â¦\84 â\8a¢ â¬\88*[h, T] L2 â\86\92 (L1 â\89¡[h, o, T] L2 → ⊥) → R L2) →
+ (â\88\80L2. â¦\83G, L1â¦\84 â\8a¢ â¬\88*[h, T] L2 â\86\92 (L1 â\89\9b[h, o, T] L2 → ⊥) → R L2) →
R L1
) →
∀L. G ⊢ ⬈*[h, o, T] 𝐒⦃L⦄ → R L.
lemma lfsx_flat: ∀h,o,I,G,L,V. G ⊢ ⬈*[h, o, V] 𝐒⦃L⦄ →
∀T. G ⊢ ⬈*[h, o, T] 𝐒⦃L⦄ → G ⊢ ⬈*[h, o, ⓕ{I}V.T] 𝐒⦃L⦄.
/2 width=3 by lfsx_flat_lfpxs/ qed.
+
+fact lfsx_bind_lfpxs_void_aux: ∀h,o,p,I,G,L1,V. G ⊢ ⬈*[h, o, V] 𝐒⦃L1⦄ →
+ ∀Y,T. G ⊢ ⬈*[h, o, T] 𝐒⦃Y⦄ →
+ ∀L2. Y = L2.ⓧ → ⦃G, L1⦄ ⊢ ⬈*[h, ⓑ{p,I}V.T] L2 →
+ G ⊢ ⬈*[h, o, ⓑ{p,I}V.T] 𝐒⦃L2⦄.
+#h #o #p #I #G #L1 #V #H @(lfsx_ind_lfpxs … H) -L1
+#L1 #_ #IHL1 #Y #T #H @(lfsx_ind_lfpxs … H) -Y
+#Y #HY #IHY #L2 #H #HL12 destruct
+@lfsx_intro_lfpxs #L0 #HL20
+lapply (lfpxs_trans … HL12 … HL20) #HL10 #H
+elim (lfdneq_inv_bind_void … H) -H [ -IHY | -HY -IHL1 -HL12 ]
+[ #HnV elim (lfdeq_dec h o L1 L2 V)
+ [ #HV @(IHL1 … L0) -IHL1 -HL12
+ /3 width=6 by lfsx_lfpxs_trans, lfpxs_fwd_bind_dx_void, lfpxs_fwd_pair_sn, lfdeq_canc_sn/ (**) (* full auto too slow *)
+ | -HnV -HL10 /4 width=4 by lfsx_lfpxs_trans, lfpxs_fwd_pair_sn/
+ ]
+| /3 width=4 by lfpxs_fwd_bind_dx_void/
+]
+qed-.
+
+lemma lfsx_bind_void: ∀h,o,p,I,G,L,V. G ⊢ ⬈*[h, o, V] 𝐒⦃L⦄ →
+ ∀T. G ⊢ ⬈*[h, o, T] 𝐒⦃L.ⓧ⦄ →
+ G ⊢ ⬈*[h, o, ⓑ{p,I}V.T] 𝐒⦃L⦄.
+/2 width=3 by lfsx_bind_lfpxs_void_aux/ qed.