-lemma fqu_cpr_trans_sn: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐ ⦃G2, L2, T2⦄ →
- ∀U2. ⦃G2, L2⦄ ⊢ T2 ➡ U2 →
- ∃∃L,U1. ⦃G1, L1⦄ ⊢ ➡ L & ⦃G1, L1⦄ ⊢ T1 ➡ U1 & ⦃G1, L, U1⦄ ⊐ ⦃G2, L2, U2⦄.
-#G1 #G2 #L1 #L2 #T1 #T2 #H elim H -G1 -G2 -L1 -L2 -T1 -T2
-/3 width=5 by fqu_lref_O, fqu_pair_sn, fqu_bind_dx, fqu_flat_dx, lpr_pair, cpr_pair_sn, cpr_atom, cpr_bind, cpr_flat, ex3_2_intro/
-#G #L #K #U #T #k #HLK #HUT #U2 #HU2
-elim (lift_total U2 0 (k+1)) #T2 #HUT2
-lapply (cpr_lift … HU2 … HLK … HUT … HUT2) -HU2 -HUT /3 width=9 by fqu_drop, ex3_2_intro/
+lemma fqu_cpr_trans_dx (h) (b): ∀G1,G2,L1,L2,T1,T2. ❪G1,L1,T1❫ ⬂[b] ❪G2,L2,T2❫ →
+ ∀U2. ❪G2,L2❫ ⊢ T2 ➡[h,0] U2 →
+ ∃∃L,U1. ❪G1,L1❫ ⊢ ➡[h,0] L & ❪G1,L❫ ⊢ T1 ➡[h,0] U1 & ❪G1,L,U1❫ ⬂[b] ❪G2,L2,U2❫.
+#h #b #G1 #G2 #L1 #L2 #T1 #T2 #H elim H -G1 -G2 -L1 -L2 -T1 -T2
+[ /3 width=5 by lpr_pair, fqu_lref_O, ex3_2_intro/
+| /3 width=5 by cpr_pair_sn, fqu_pair_sn, ex3_2_intro/
+| /3 width=5 by cpm_bind, fqu_bind_dx, ex3_2_intro/
+| /3 width=5 by cpm_bind_unit, fqu_clear, ex3_2_intro/
+| /3 width=5 by cpr_flat, fqu_flat_dx, ex3_2_intro/
+| #I #G #K #U #T #HUT #U2 #HU2
+ elim (cpm_lifts_sn … HU2 (Ⓣ) … (K.ⓘ[I]) … HUT) -U
+ /3 width=5 by lpr_bind_refl_dx, fqu_drop, drops_refl, drops_drop, ex3_2_intro/
+]
+qed-.
+
+lemma fqu_lpr_trans (h) (b): ∀G1,G2,L1,L2,T1,T2. ❪G1,L1,T1❫ ⬂[b] ❪G2,L2,T2❫ →
+ ∀K2. ❪G2,L2❫ ⊢ ➡[h,0] K2 →
+ ∃∃K1,T. ❪G1,L1❫ ⊢ ➡[h,0] K1 & ❪G1,L1❫ ⊢ T1 ➡[h,0] T & ❪G1,K1,T❫ ⬂[b] ❪G2,K2,T2❫.
+#h #b #G1 #G2 #L1 #L2 #T1 #T2 #H elim H -G1 -G2 -L1 -L2 -T1 -T2
+[ /3 width=5 by lpr_bind_refl_dx, fqu_lref_O, ex3_2_intro/
+| /3 width=5 by cpr_pair_sn, fqu_pair_sn, ex3_2_intro/
+| #p #I #G2 #L2 #V2 #T2 #Hb #X #H
+ elim (lpr_inv_pair_sn … H) -H #K2 #W2 #HLK2 #HVW2 #H destruct
+ /3 width=5 by cpr_pair_sn, fqu_bind_dx, ex3_2_intro/
+| #p #I #G2 #L2 #V2 #T2 #Hb #X #H
+ elim (lpr_inv_unit_sn … H) -H #K2 #HLK2 #H destruct
+ /3 width=5 by cpr_pair_sn, fqu_clear, ex3_2_intro/
+| /3 width=5 by cpr_pair_sn, fqu_flat_dx, ex3_2_intro/
+| /3 width=5 by lpr_bind_refl_dx, fqu_drop, ex3_2_intro/
+]