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diff --git a/matita/matita/contribs/lambda_delta/basic_2/unfold/tpss.ma b/matita/matita/contribs/lambda_delta/basic_2/unfold/tpss.ma
index 7631985bb..93916208b 100644
--- a/matita/matita/contribs/lambda_delta/basic_2/unfold/tpss.ma
+++ b/matita/matita/contribs/lambda_delta/basic_2/unfold/tpss.ma
@@ -40,28 +40,32 @@ qed-.
 
 (* Basic properties *********************************************************)
 
-lemma tpss_strap: ∀L,T1,T,T2,d,e.
-                  L ⊢ T1 ▶ [d, e] T → L ⊢ T ▶* [d, e] T2 → L ⊢ T1 ▶* [d, e] T2. 
+lemma tpss_strap1: ∀L,T1,T,T2,d,e.
+                   L ⊢ T1 ▶* [d, e] T → L ⊢ T ▶ [d, e] T2 → L ⊢ T1 ▶* [d, e] T2. 
 /2 width=3/ qed.
 
-lemma tpss_lsubs_conf: ∀L1,T1,T2,d,e. L1 ⊢ T1 ▶* [d, e] T2 →
-                       ∀L2. L1 ≼ [d, e] L2 → L2 ⊢ T1 ▶* [d, e] T2.
+lemma tpss_strap2: ∀L,T1,T,T2,d,e.
+                   L ⊢ T1 ▶ [d, e] T → L ⊢ T ▶* [d, e] T2 → L ⊢ T1 ▶* [d, e] T2. 
+/2 width=3/ qed.
+
+lemma tpss_lsubs_trans: ∀L1,T1,T2,d,e. L1 ⊢ T1 ▶* [d, e] T2 →
+                        ∀L2. L2 ≼ [d, e] L1 → L2 ⊢ T1 ▶* [d, e] T2.
 /3 width=3/ qed.
 
 lemma tpss_refl: ∀d,e,L,T. L ⊢ T ▶* [d, e] T.
 /2 width=1/ qed.
 
 lemma tpss_bind: ∀L,V1,V2,d,e. L ⊢ V1 ▶* [d, e] V2 →
-                 ∀I,T1,T2. L. ⓑ{I} V2 ⊢ T1 ▶* [d + 1, e] T2 →
-                 L ⊢ ⓑ{I} V1. T1 ▶* [d, e] ⓑ{I} V2. T2.
+                 ∀a,I,T1,T2. L. ⓑ{I} V2 ⊢ T1 ▶* [d + 1, e] T2 →
+                 L ⊢ ⓑ{a,I} V1. T1 ▶* [d, e] ⓑ{a,I} V2. T2.
 #L #V1 #V2 #d #e #HV12 elim HV12 -V2
-[ #V2 #HV12 #I #T1 #T2 #HT12 elim HT12 -T2
+[ #V2 #HV12 #a #I #T1 #T2 #HT12 elim HT12 -T2
   [ /3 width=5/
   | #T #T2 #_ #HT2 #IHT @step /2 width=5/ (**) (* /3 width=5/ is too slow *)
   ]
-| #V #V2 #_ #HV12 #IHV #I #T1 #T2 #HT12
-  lapply (tpss_lsubs_conf … HT12 (L. ⓑ{I} V) ?) -HT12 /2 width=1/ #HT12
-  lapply (IHV … HT12) -IHV -HT12 #HT12 @step /2 width=5/ (**) (* /3 width=5/ is too slow *)
+| #V #V2 #_ #HV12 #IHV #a #I #T1 #T2 #HT12
+  lapply (tpss_lsubs_trans … HT12 (L. ⓑ{I} V) ?) -HT12 /2 width=1/ #HT12
+  lapply (IHV a … HT12) -IHV -HT12 #HT12 @step /2 width=5/ (**) (* /3 width=5/ is too slow *)
 ]
 qed.
 
@@ -104,6 +108,11 @@ lapply (tpss_weak … HT12 0 (d + e) ? ?) -HT12 // #HT12
 lapply (tpss_weak_top … HT12) //
 qed.
 
+lemma tpss_append: ∀K,T1,T2,d,e. K ⊢ T1 ▶* [d, e] T2 →
+                   ∀L. L @@ K ⊢ T1 ▶* [d, e] T2.
+#K #T1 #T2 #d #e #H @(tpss_ind … H) -T2 // /3 width=3/
+qed.
+
 (* Basic inversion lemmas ***************************************************)
 
 (* Note: this can be derived from tpss_inv_atom1 *)
@@ -124,15 +133,15 @@ lemma tpss_inv_gref1: ∀L,T2,p,d,e. L ⊢ §p ▶* [d, e] T2 → T2 = §p.
 ]
 qed-.
 
-lemma tpss_inv_bind1: ∀d,e,L,I,V1,T1,U2. L ⊢ ⓑ{I} V1. T1 ▶* [d, e] U2 →
+lemma tpss_inv_bind1: ∀d,e,L,a,I,V1,T1,U2. L ⊢ ⓑ{a,I} V1. T1 ▶* [d, e] U2 →
                       ∃∃V2,T2. L ⊢ V1 ▶* [d, e] V2 & 
                                L. ⓑ{I} V2 ⊢ T1 ▶* [d + 1, e] T2 &
-                               U2 =  ⓑ{I} V2. T2.
-#d #e #L #I #V1 #T1 #U2 #H @(tpss_ind … H) -U2
+                               U2 = ⓑ{a,I} V2. T2.
+#d #e #L #a #I #V1 #T1 #U2 #H @(tpss_ind … H) -U2
 [ /2 width=5/
 | #U #U2 #_ #HU2 * #V #T #HV1 #HT1 #H destruct
   elim (tps_inv_bind1 … HU2) -HU2 #V2 #T2 #HV2 #HT2 #H
-  lapply (tpss_lsubs_conf … HT1 (L. ⓑ{I} V2) ?) -HT1 /2 width=1/ /3 width=5/
+  lapply (tpss_lsubs_trans … HT1 (L. ⓑ{I} V2) ?) -HT1 /2 width=1/ /3 width=5/
 ]
 qed-.
 
@@ -155,9 +164,19 @@ qed-.
 
 (* Basic forward lemmas *****************************************************)
 
-lemma tpss_fwd_tw: ∀L,T1,T2,d,e. L ⊢ T1 ▶* [d, e] T2 → #[T1] ≤ #[T2].
+lemma tpss_fwd_tw: ∀L,T1,T2,d,e. L ⊢ T1 ▶* [d, e] T2 → #{T1} ≤ #{T2}.
 #L #T1 #T2 #d #e #H @(tpss_ind … H) -T2 //
 #T #T2 #_ #HT2 #IHT1
 lapply (tps_fwd_tw … HT2) -HT2 #HT2
 @(transitive_le … IHT1) //
 qed-.
+
+lemma tpss_fwd_shift1: ∀L,L1,T1,T,d,e. L ⊢ L1 @@ T1 ▶*[d, e] T →
+                       ∃∃L2,T2. |L1| = |L2| & T = L2 @@ T2.
+#L #L1 #T1 #T #d #e #H @(tpss_ind … H) -T
+[ /2 width=4/
+| #T #X #_ #H0 * #L0 #T0 #HL10 #H destruct
+  elim (tps_fwd_shift1 … H0) -H0 #L2 #T2 #HL02 #H destruct /2 width=4/
+]
+qed-.
+  
\ No newline at end of file