+
+(* Properties on context-sensitive parallel reduction for terms *************)
+
+lemma fsup_cpr_trans: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊃ ⦃G2, L2, T2⦄ →
+ ∀U2. ⦃G2, L2⦄ ⊢ T2 ➡ U2 →
+ ∃∃L,U1. ⦃G1, L1⦄ ⊢ ➡ L & ⦃G1, L⦄ ⊢ T1 ➡ U1 & ⦃G1, L, U1⦄ ⊃ ⦃G2, L2, U2⦄.
+#G1 #G2 #L1 #L2 #T1 #T2 #H elim H -G1 -G2 -L1 -L2 -T1 -T2 [1,2,3,4,5: /3 width=5/ ]
+[ #G #L #K #U #T #d #e #HLK #HUT #He #U2 #HU2
+ elim (lift_total U2 d e) #T2 #HUT2
+ lapply (cpr_lift … HU2 … HLK … HUT … HUT2) -HU2 -HUT /3 width=9/
+| #G1 #G2 #L1 #K1 #K2 #T1 #T2 #U1 #d #e #HLK1 #HTU1 #_ #IHT12 #U2 #HTU2
+ elim (IHT12 … HTU2) -IHT12 -HTU2 #K #T #HK1 #HT1 #HT2
+ elim (lift_total T d e) #U #HTU
+ elim (ldrop_lpr_trans … HLK1 … HK1) -HLK1 -HK1 #L2 #HL12 #HL2K
+ lapply (cpr_lift … HT1 … HL2K … HTU1 … HTU) -HT1 -HTU1 /3 width=11/
+]
+qed-.