(* Properties with strongly normalizing referred local environments *********)
(* Basic_2A1: uses: lsx_cpx_trans_lcosx *)
-lemma rdsx_cpx_trans_lsubsx (h) (o): ∀G,L0,T1,T2. ⦃G, L0⦄ ⊢ T1 ⬈[h] T2 →
- ∀f,L. G ⊢ L0 ⊆ⓧ[h, o, f] L →
- G ⊢ ⬈*[h, o, T1] 𝐒⦃L⦄ → G ⊢ ⬈*[h, o, T2] 𝐒⦃L⦄.
-#h #o #G #L0 #T1 #T2 #H @(cpx_ind … H) -G -L0 -T1 -T2 //
+lemma rdsx_cpx_trans_lsubsx (h):
+ ∀G,L0,T1,T2. ⦃G, L0⦄ ⊢ T1 ⬈[h] T2 →
+ ∀f,L. G ⊢ L0 ⊆ⓧ[h, f] L →
+ G ⊢ ⬈*[h, T1] 𝐒⦃L⦄ → G ⊢ ⬈*[h, T2] 𝐒⦃L⦄.
+#h #G #L0 #T1 #T2 #H @(cpx_ind … H) -G -L0 -T1 -T2 //
[ #I0 #G #K0 #V1 #V2 #W2 #_ #IH #HVW2 #g #L #HK0 #HL
elim (lsubsx_inv_pair_sn_gen … HK0) -HK0 *
[ #f #K #HK0 #H1 #H2 destruct
/4 width=2 by lsubsx_pair, rdsx_bind_void/
| #I0 #G #L0 #V1 #V2 #T1 #T2 #_ #_ #IHV12 #IHT12 #f #L #HL0 #HL
elim (rdsx_inv_flat … HL) -HL /3 width=2 by rdsx_flat/
-| #G #L0 #V #U1 #U2 #T2 #_ #HTU2 #IHU12 #f #L #HL0 #HL
+| #G #L0 #V #U1 #T1 #T2 #HTU1 #_ #IHT12 #f #L #HL0 #HL
elim (rdsx_inv_bind … HL) -HL #HV #HU1
- /4 width=8 by lsubsx_pair, rdsx_inv_lifts, drops_refl, drops_drop/
+ /5 width=8 by rdsx_inv_lifts, drops_refl, drops_drop/
| #G #L0 #V #T1 #T2 #_ #IHT12 #f #L #HL0 #HL
elim (rdsx_inv_flat … HL) -HL /2 width=2 by/
| #G #L0 #V1 #V2 #T #_ #IHV12 #f #L #HL0 #HL
(* Advanced properties of strongly normalizing referred local environments **)
(* Basic_2A1: uses: lsx_cpx_trans_O *)
-lemma rdsx_cpx_trans (h) (o): ∀G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ⬈[h] T2 →
- G ⊢ ⬈*[h, o, T1] 𝐒⦃L⦄ → G ⊢ ⬈*[h, o, T2] 𝐒⦃L⦄.
+lemma rdsx_cpx_trans (h):
+ ∀G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ⬈[h] T2 →
+ G ⊢ ⬈*[h, T1] 𝐒⦃L⦄ → G ⊢ ⬈*[h, T2] 𝐒⦃L⦄.
/3 width=6 by rdsx_cpx_trans_lsubsx, lsubsx_refl/ qed-.
-lemma rdsx_cpxs_trans (h) (o): ∀G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ⬈*[h] T2 →
- G ⊢ ⬈*[h, o, T1] 𝐒⦃L⦄ → G ⊢ ⬈*[h, o, T2] 𝐒⦃L⦄.
-#h #o #G #L #T1 #T2 #H
+lemma rdsx_cpxs_trans (h):
+ ∀G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ⬈*[h] T2 →
+ G ⊢ ⬈*[h, T1] 𝐒⦃L⦄ → G ⊢ ⬈*[h, T2] 𝐒⦃L⦄.
+#h #G #L #T1 #T2 #H
@(cpxs_ind_dx ???????? H) -T1 //
/3 width=3 by rdsx_cpx_trans/
qed-.