(* Basic_1: includes: lift_gen_lift *)
(* Basic_2A1: includes: lift_div_le lift_div_be *)
-theorem lift_div4: ∀f2,Tf,T. ⬆*[f2] Tf ≡ T → ∀g2,Tg. ⬆*[g2] Tg ≡ T →
- ∀f1,g1. H_at_div f2 g2 f1 g1 →
- ∃∃T0. ⬆*[f1] T0 ≡ Tf & ⬆*[g1] T0 ≡ Tg.
+theorem lifts_div4: ∀f2,Tf,T. ⬆*[f2] Tf ≡ T → ∀g2,Tg. ⬆*[g2] Tg ≡ T →
+ ∀f1,g1. H_at_div f2 g2 f1 g1 →
+ ∃∃T0. ⬆*[f1] T0 ≡ Tf & ⬆*[g1] T0 ≡ Tg.
#f2 #Tf #T #H elim H -f2 -Tf -T
[ #f2 #s #g2 #Tg #H #f1 #g1 #_
lapply (lifts_inv_sort2 … H) -H #H destruct
lemma lifts_div4_one: ∀f,Tf,T. ⬆*[↑f] Tf ≡ T →
∀T1. ⬆*[1] T1 ≡ T →
∃∃T0. ⬆*[1] T0 ≡ Tf & ⬆*[f] T0 ≡ T1.
-/4 width=6 by lift_div4, at_div_id_dx, at_div_pn/ qed-.
+/4 width=6 by lifts_div4, at_div_id_dx, at_div_pn/ qed-.
theorem lifts_div3: ∀f2,T,T2. ⬆*[f2] T2 ≡ T → ∀f,T1. ⬆*[f] T1 ≡ T →
∀f1. f2 ⊚ f1 ≡ f → ⬆*[f1] T1 ≡ T2.
#f #T #U1 #H1 #U2 #H2 lapply (after_isid_sn 𝐈𝐝 … f)
/3 width=6 by lifts_conf, lifts_fwd_isid/
qed-.
+
+lemma liftable2_sn_bi: ∀R. liftable2 R → liftable2_bi R.
+#R #HR #T1 #T2 #HT12 #f #U1 #HTU1 #U2 #HTU2
+elim (HR … HT12 … HTU1) -HR -T1 #X #HTX #HUX
+<(lifts_mono … HTX … HTU2) -T2 -U2 -f //
+qed-.
+
+lemma deliftable2_sn_bi: ∀R. deliftable2_sn R → deliftable2_bi R.
+#R #HR #U1 #U2 #HU12 #f #T1 #HTU1 #T2 #HTU2
+elim (HR … HU12 … HTU1) -HR -U1 #X #HUX #HTX
+<(lifts_inj … HUX … HTU2) -U2 -T2 -f //
+qed-.