-lemma fqu_cpx_trans: ∀h,o,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐ ⦃G2, L2, T2⦄ →
- ∀U2. ⦃G2, L2⦄ ⊢ T2 ➡[h, o] U2 →
- ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ➡[h, o] U1 & ⦃G1, L1, U1⦄ ⊐ ⦃G2, L2, U2⦄.
-#h #o #G1 #G2 #L1 #L2 #T1 #T2 #H elim H -G1 -G2 -L1 -L2 -T1 -T2
-/3 width=3 by fqu_pair_sn, fqu_bind_dx, fqu_flat_dx, cpx_pair_sn, cpx_bind, cpx_flat, ex2_intro/
-[ #I #G #L #V2 #U2 #HVU2
- elim (lift_total U2 0 1)
- /4 width=7 by fqu_drop, cpx_delta, drop_pair, drop_drop, ex2_intro/
-| #G #L #K #T1 #U1 #k #HLK1 #HTU1 #T2 #HTU2
- elim (lift_total T2 0 (k+1))
- /3 width=11 by cpx_lift, fqu_drop, ex2_intro/
+lemma fqu_cpx_trans: ∀h,b,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐[b] ⦃G2, L2, T2⦄ →
+ ∀U2. ⦃G2, L2⦄ ⊢ T2 ⬈[h] U2 →
+ ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ⬈[h] U1 & ⦃G1, L1, U1⦄ ⊐[b] ⦃G2, L2, U2⦄.
+#h #b #G1 #G2 #L1 #L2 #T1 #T2 #H elim H -G1 -G2 -L1 -L2 -T1 -T2
+/3 width=3 by cpx_pair_sn, cpx_bind, cpx_flat, fqu_pair_sn, fqu_bind_dx, fqu_flat_dx, ex2_intro/
+[ #I #G #L2 #V2 #X2 #HVX2
+ elim (lifts_total X2 (𝐔❴1❵))
+ /3 width=3 by fqu_drop, cpx_delta, ex2_intro/
+| /5 width=4 by lsubr_cpx_trans, cpx_bind, lsubr_unit, fqu_clear, ex2_intro/
+| #I #G #L2 #T2 #X2 #HTX2 #U2 #HTU2
+ elim (cpx_lifts_sn … HTU2 (Ⓣ) … (L2.ⓘ{I}) … HTX2)
+ /3 width=3 by fqu_drop, drops_refl, drops_drop, ex2_intro/