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diff --git a/matita/matita/contribs/lambdadelta/basic_2/dynamic/nta_cpms.ma b/matita/matita/contribs/lambdadelta/basic_2/dynamic/nta_cpms.ma
index 2fbdaa07e..af4beb4c3 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/dynamic/nta_cpms.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/dynamic/nta_cpms.ma
@@ -14,35 +14,38 @@
 
 include "basic_2/rt_computation/cprs_cprs.ma".
 include "basic_2/rt_computation/lprs_cpms.ma".
+include "basic_2/dynamic/cnv_aaa.ma".
 include "basic_2/dynamic/nta.ma".
 
 (* NATIVE TYPE ASSIGNMENT FOR TERMS *****************************************)
 
 (* Properties with advanced rt_computation for terms ************************)
 
-(* Basic_2A1: was by definition nta_appl ntaa_appl *)
-lemma nta_appl_abst (a) (h) (p) (G) (K): yinj 0 < a →
-      ∀V,W. ⦃G,K⦄ ⊢ V :[a,h] W →
-      ∀T,U. ⦃G,K.ⓛW⦄ ⊢ T :[a,h] U → ⦃G,K⦄ ⊢ ⓐV.ⓛ{p}W.T :[a,h] ⓐV.ⓛ{p}W.U.
-#a #h #p #G #K #Ha #V #W #H1 #T #U #H2
+(* Basic_2A1: uses by definition nta_appl ntaa_appl *)
+lemma nta_appl_abst (a) (h) (p) (G) (L):
+      ∀n. appl a n →
+      ∀V,W. ⦃G,L⦄ ⊢ V :[a,h] W →
+      ∀T,U. ⦃G,L.ⓛW⦄ ⊢ T :[a,h] U → ⦃G,L⦄ ⊢ ⓐV.ⓛ{p}W.T :[a,h] ⓐV.ⓛ{p}W.U.
+#a #h #p #G #L #n #Ha #V #W #H1 #T #U #H2
 elim (cnv_inv_cast … H1) -H1 #X1 #HW #HV #HWX1 #HVX1
 elim (cnv_inv_cast … H2) -H2 #X2 #HU #HT #HUX2 #HTX2
-/4 width=7 by cnv_bind, cnv_appl, cnv_cast, cpms_appl_dx, cpms_bind_dx/
+/4 width=11 by cnv_appl_ge, cnv_cast, cnv_bind, cpms_appl_dx, cpms_bind_dx/
 qed.
 
 (* Basic_1: was by definition: ty3_appl *)
 (* Basic_2A1: was nta_appl_old *)
-lemma nta_appl (a) (h) (p) (G) (L): yinj 1 < a →
+lemma nta_appl (a) (h) (p) (G) (L):
+      ∀n. 1 ≤ n → appl a n →
       ∀V,W. ⦃G,L⦄ ⊢ V :[a,h] W →
       ∀T,U. ⦃G,L⦄ ⊢ T :[a,h] ⓛ{p}W.U → ⦃G,L⦄ ⊢ ⓐV.T :[a,h] ⓐV.ⓛ{p}W.U.
-#a #h #p #G #L #Ha #V #W #H1 #T #U #H2
+#a #h #p #G #L #n #Hn #Ha #V #W #H1 #T #U #H2
 elim (cnv_inv_cast … H1) -H1 #X1 #HW #HV #HWX1 #HVX1
 elim (cnv_inv_cast … H2) -H2 #X2 #HU #HT #HUX2 #HTX2
 elim (cpms_inv_abst_sn … HUX2) #W0 #U0 #HW0 #HU0 #H destruct
 elim (cprs_conf … HWX1 … HW0) -HW0 #X0 #HX10 #HWX0
 @(cnv_cast … (ⓐV.ⓛ{p}W0.U0)) (**) (* full auto too slow *)
-[ /3 width=7 by cnv_appl, cpms_bind, lt_ylt_trans/
-| /4 width=11 by cnv_appl, cpms_cprs_trans, cpms_bind/
+[ /2 width=11 by cnv_appl_ge/
+| /3 width=11 by cnv_appl_ge, cpms_cprs_trans/
 | /2 width=1 by cpms_appl_dx/
 | /2 width=1 by cpms_appl_dx/
 ]