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diff --git a/matita/matita/contribs/lambda_delta/basic_2/substitution/ldrop.ma b/matita/matita/contribs/lambda_delta/basic_2/substitution/ldrop.ma
index 9ab513458..c93d3cd18 100644
--- a/matita/matita/contribs/lambda_delta/basic_2/substitution/ldrop.ma
+++ b/matita/matita/contribs/lambda_delta/basic_2/substitution/ldrop.ma
@@ -30,6 +30,27 @@ inductive ldrop: nat → nat → relation lenv ≝
 
 interpretation "local slicing" 'RDrop d e L1 L2 = (ldrop d e L1 L2).
 
+definition l_liftable: (lenv → relation term) → Prop ≝
+                       λR. ∀K,T1,T2. R K T1 T2 → ∀L,d,e. ⇩[d, e] L ≡ K →
+                       ∀U1. ⇧[d, e] T1 ≡ U1 → ∀U2. ⇧[d, e] T2 ≡ U2 → R L U1 U2.
+
+definition l_deliftable_sn: (lenv → relation term) → Prop ≝
+                            λR. ∀L,U1,U2. R L U1 U2 → ∀K,d,e. ⇩[d, e] L ≡ K →
+                            ∀T1. ⇧[d, e] T1 ≡ U1 →
+                            ∃∃T2. ⇧[d, e] T2 ≡ U2 & R K T1 T2.
+
+definition dropable_sn: relation lenv → Prop ≝
+                        λR. ∀L1,K1,d,e. ⇩[d, e] L1 ≡ K1 → ∀L2. R L1 L2 →
+                        ∃∃K2. R K1 K2 & ⇩[d, e] L2 ≡ K2.
+
+definition dedropable_sn: relation lenv → Prop ≝
+                          λR. ∀L1,K1,d,e. ⇩[d, e] L1 ≡ K1 → ∀K2. R K1 K2 →
+                          ∃∃L2. R L1 L2 & ⇩[d, e] L2 ≡ K2.
+
+definition dropable_dx: relation lenv → Prop ≝
+                        λR. ∀L1,L2. R L1 L2 → ∀K2,e. ⇩[0, e] L2 ≡ K2 →
+                        ∃∃K1. ⇩[0, e] L1 ≡ K1 & R K1 K2.
+
 (* Basic inversion lemmas ***************************************************)
 
 fact ldrop_inv_refl_aux: ∀d,e,L1,L2. ⇩[d, e] L1 ≡ L2 → d = 0 → e = 0 → L1 = L2.
@@ -194,6 +215,34 @@ lemma ldrop_lsubs_ldrop2_abbr: ∀L1,L2,d,e. L1 ≼ [d, e] L2 →
 ]
 qed.
 
+lemma dropable_sn_TC: ∀R. dropable_sn R → dropable_sn (TC … R).
+#R #HR #L1 #K1 #d #e #HLK1 #L2 #H elim H -L2
+[ #L2 #HL12
+  elim (HR … HLK1 … HL12) -HR -L1 /3 width=3/
+| #L #L2 #_ #HL2 * #K #HK1 #HLK
+  elim (HR … HLK … HL2) -HR -L /3 width=3/
+]
+qed.
+
+lemma dedropable_sn_TC: ∀R. dedropable_sn R → dedropable_sn (TC … R).
+#R #HR #L1 #K1 #d #e #HLK1 #K2 #H elim H -K2
+[ #K2 #HK12
+  elim (HR … HLK1 … HK12) -HR -K1 /3 width=3/
+| #K #K2 #_ #HK2 * #L #HL1 #HLK
+  elim (HR … HLK … HK2) -HR -K /3 width=3/
+]
+qed.
+
+lemma dropable_dx_TC: ∀R. dropable_dx R → dropable_dx (TC … R).
+#R #HR #L1 #L2 #H elim H -L2
+[ #L2 #HL12 #K2 #e #HLK2
+  elim (HR … HL12 … HLK2) -HR -L2 /3 width=3/
+| #L #L2 #_ #HL2 #IHL1 #K2 #e #HLK2
+  elim (HR … HL2 … HLK2) -HR -L2 #K #HLK #HK2
+  elim (IHL1 … HLK) -L /3 width=5/
+]
+qed.
+
 (* Basic forvard lemmas *****************************************************)
 
 (* Basic_1: was: drop_S *)