+(* Advanced properties ******************************************************)
+
+lemma cprs_abbr_dx: ∀L,V1,V2. L ⊢ V1 ➡ V2 → ∀T1,T2. L. ⓓV1 ⊢ T1 ➡* T2 →
+ L ⊢ ⓓV1. T1 ➡* ⓓV2. T2.
+#L #V1 #V2 #HV12 #T1 #T2 #HT12 @(cprs_ind_dx … HT12) -T1
+[ /3 width=1/
+| #T1 #T #HT1 #_ #IHT1
+ @(cprs_strap2 … IHT1) -IHT1 /2 width=1/
+]
+qed.
+
+lemma cpr_abbr: ∀L,V1,V2. L ⊢ V1 ➡ V2 → ∀T1,T2. L. ⓓV1 ⊢ T1 ➡ T2 →
+ L ⊢ ⓓV1. T1 ➡* ⓓV2. T2.
+/3 width=1/ qed.
+
+(* Basic_1: was only: pr3_pr2_pr3_t *)
+lemma lcpr_cprs_trans: ∀L1,L2. L1 ⊢ ➡ L2 →
+ ∀T1,T2. L2 ⊢ T1 ➡* T2 → L1 ⊢ T1 ➡* T2.
+#L1 #L2 #HL12 #T1 #T2 #H @(cprs_ind … H) -T2 //
+#T #T2 #_ #HT2 #IHT2 /3 width=5/
+qed.
+