-lemma fqus_cpxs_trans: ∀h,o,G1,G2,L1,L2,T2,U2. ⦃G2, L2⦄ ⊢ T2 ⬈*[h, o] U2 →
- ∀T1. ⦃G1, L1, T1⦄ ⊐* ⦃G2, L2, T2⦄ →
- ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ⬈*[h, o] U1 & ⦃G1, L1, U1⦄ ⊐* ⦃G2, L2, U2⦄.
-#h #o #G1 #G2 #L1 #L2 #T2 #U2 #HTU2 #T1 #H elim (fqus_inv_gen … H) -H
-[ #HT12 elim (fqup_cpxs_trans … HTU2 … HT12) /3 width=3 by fqup_fqus, ex2_intro/
-| * #H1 #H2 #H3 destruct /2 width=3 by ex2_intro/
+lemma fqus_cpxs_trans (b):
+ ∀G1,G2,L1,L2,T2,U2. ❪G2,L2❫ ⊢ T2 ⬈* U2 →
+ ∀T1. ❪G1,L1,T1❫ ⬂*[b] ❪G2,L2,T2❫ →
+ ∃∃U1. ❪G1,L1❫ ⊢ T1 ⬈* U1 & ❪G1,L1,U1❫ ⬂*[b] ❪G2,L2,U2❫.
+#b #G1 #G2 #L1 #L2 #T2 #U2 #H @(cpxs_ind_dx … H) -T2 /2 width=3 by ex2_intro/
+#T #T2 #HT2 #_ #IHTU2 #T1 #HT1 elim (fqus_cpx_trans … HT1 … HT2) -T
+#U1 #HTU1 #H2 elim (IHTU2 … H2) -T2 /3 width=3 by cpxs_strap2, ex2_intro/
+qed-.
+
+(* Note: a proof based on fqu_cpx_trans_tneqx might exist *)
+(* Basic_2A1: uses: fqu_cpxs_trans_neq *)
+lemma fqu_cpxs_trans_tneqx (b):
+ ∀G1,G2,L1,L2,T1,T2. ❪G1,L1,T1❫ ⬂[b] ❪G2,L2,T2❫ →
+ ∀U2. ❪G2,L2❫ ⊢ T2 ⬈* U2 → (T2 ≛ U2 → ⊥) →
+ ∃∃U1. ❪G1,L1❫ ⊢ T1 ⬈* U1 & T1 ≛ U1 → ⊥ & ❪G1,L1,U1❫ ⬂[b] ❪G2,L2,U2❫.
+#b #G1 #G2 #L1 #L2 #T1 #T2 #H elim H -G1 -G2 -L1 -L2 -T1 -T2
+[ #I #G #L #V1 #V2 #HV12 #_ elim (lifts_total V2 𝐔❨1❩)
+ #U2 #HVU2 @(ex3_intro … U2)
+ [1,3: /3 width=7 by cpxs_delta, fqu_drop/
+ | #H lapply (teqx_inv_lref1 … H) -H
+ #H destruct /2 width=5 by lifts_inv_lref2_uni_lt/
+ ]
+| #I #G #L #V1 #T #V2 #HV12 #H0 @(ex3_intro … (②[I]V2.T))
+ [1,3: /2 width=4 by fqu_pair_sn, cpxs_pair_sn/
+ | #H elim (teqx_inv_pair … H) -H /2 width=1 by/
+ ]
+| #p #I #G #L #V #T1 #Hb #T2 #HT12 #H0 @(ex3_intro … (ⓑ[p,I]V.T2))
+ [1,3: /2 width=4 by fqu_bind_dx, cpxs_bind/
+ | #H elim (teqx_inv_pair … H) -H /2 width=1 by/
+ ]
+| #p #I #G #L #V #T1 #Hb #T2 #HT12 #H0 @(ex3_intro … (ⓑ[p,I]V.T2))
+ [1,3: /4 width=4 by lsubr_cpxs_trans, cpxs_bind, lsubr_unit, fqu_clear/
+ | #H elim (teqx_inv_pair … H) -H /2 width=1 by/
+ ]
+| #I #G #L #V #T1 #T2 #HT12 #H0 @(ex3_intro … (ⓕ[I]V.T2))
+ [1,3: /2 width=4 by fqu_flat_dx, cpxs_flat/
+ | #H elim (teqx_inv_pair … H) -H /2 width=1 by/
+ ]
+| #I #G #L #T1 #U1 #HTU1 #T2 #HT12 #H0
+ elim (cpxs_lifts_sn … HT12 (Ⓣ) … (L.ⓘ[I]) … HTU1) -HT12
+ /4 width=6 by fqu_drop, drops_refl, drops_drop, teqx_inv_lifts_bi, ex3_intro/