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+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "basic_2/notation/relations/colon_6.ma".
+include "basic_2/notation/relations/colon_5.ma".
+include "basic_2/notation/relations/colonstar_5.ma".
+include "basic_2/dynamic/cnv.ma".
+
+(* NATIVE TYPE ASSIGNMENT FOR TERMS *****************************************)
+
+definition nta (a) (h): relation4 genv lenv term term ≝
+                        λG,L,T,U. ⦃G,L⦄ ⊢ ⓝU.T ![a,h].
+
+interpretation "native type assignment (term)"
+   'Colon a h G L T U = (nta a h G L T U).
+
+interpretation "restricted native type assignment (term)"
+   'Colon h G L T U = (nta true h G L T U).
+
+interpretation "extended native type assignment (term)"
+   'ColonStar h G L T U = (nta false h G L T U).
+
+(* Basic properties *********************************************************)
+
+(* Basic_1: was by definition: ty3_sort *)
+(* Basic_2A1: was by definition: nta_sort ntaa_sort *)
+lemma nta_sort (a) (h) (G) (L) (s): ⦃G,L⦄ ⊢ ⋆s :[a,h] ⋆(next h s).
+#a #h #G #L #s /2 width=3 by cnv_sort, cnv_cast, cpms_sort/
+qed.
+
+lemma nta_bind_cnv (a) (h) (G) (K):
+      ∀V. ⦃G,K⦄ ⊢ V ![a,h] →
+      ∀I,T,U. ⦃G,K.ⓑ{I}V⦄ ⊢ T :[a,h] U →
+      ∀p. ⦃G,K⦄ ⊢ ⓑ{p,I}V.T :[a,h] ⓑ{p,I}V.U.
+#a #h #G #K #V #HV #I #T #U #H #p
+elim (cnv_inv_cast … H) -H #X #HU #HT #HUX #HTX
+/3 width=5 by cnv_bind, cnv_cast, cpms_bind_dx/
+qed.
+
+(* Basic_2A1: was by definition: nta_cast *)
+lemma nta_cast (a) (h) (G) (L):
+      ∀T,U. ⦃G,L⦄ ⊢ T :[a,h] U → ⦃G,L⦄ ⊢ ⓝU.T :[a,h] U.
+#a #h #G #L #T #U #H
+elim (cnv_inv_cast … H) #X #HU #HT #HUX #HTX
+/3 width=3 by cnv_cast, cpms_eps/
+qed.
+
+(* Basic_1: was by definition: ty3_cast *)
+lemma nta_cast_old (a) (h) (G) (L):
+      ∀T0,T1. ⦃G,L⦄ ⊢ T0 :[a,h] T1 →
+      ∀T2. ⦃G,L⦄ ⊢ T1 :[a,h] T2 → ⦃G,L⦄ ⊢ ⓝT1.T0 :[a,h] ⓝT2.T1.
+#a #h #G #L #T0 #T1 #H1 #T2 #H2
+elim (cnv_inv_cast … H1) #X1 #_ #_ #HTX1 #HTX01
+elim (cnv_inv_cast … H2) #X2 #_ #_ #HTX2 #HTX12
+/3 width=3 by cnv_cast, cpms_eps/
+qed.
+
+(* Basic_forward lemmas *****************************************************)
+
+lemma nta_fwd_cnv_sn (a) (h) (G) (L):
+                     ∀T,U. ⦃G,L⦄ ⊢ T :[a,h] U → ⦃G,L⦄ ⊢ T ![a,h].
+#a #h #G #L #T #U #H
+elim (cnv_inv_cast … H) -H #X #_ #HT #_ #_ //
+qed-.
+
+(* Note: this is nta_fwd_correct_cnv *)
+lemma nta_fwd_cnv_dx (a) (h) (G) (L):
+                     ∀T,U. ⦃G,L⦄ ⊢ T :[a,h] U → ⦃G,L⦄ ⊢ U ![a,h].
+#a #h #G #L #T #U #H
+elim (cnv_inv_cast … H) -H #X #HU #_ #_ #_ //
+qed-.
+
+(*
+
+| nta_ldef: ∀L,K,V,W,U,i. ⇩[0, i] L ≡ K. ⓓV → nta h K V W →
+            ⇧[0, i + 1] W ≡ U → nta h L (#i) U
+| nta_ldec: ∀L,K,W,V,U,i. ⇩[0, i] L ≡ K. ⓛW → nta h K W V →
+            ⇧[0, i + 1] W ≡ U → nta h L (#i) U
+.
+
+(* Basic properties *********************************************************)
+
+lemma nta_ind_alt: ∀h. ∀R:lenv→relation term.
+   (∀L,k. R L ⋆k ⋆(next h k)) →
+   (∀L,K,V,W,U,i.
+      ⇩[O, i] L ≡ K.ⓓV → ⦃h, K⦄ ⊢ V : W → ⇧[O, i + 1] W ≡ U →
+      R K V W → R L (#i) U 
+   ) →
+   (∀L,K,W,V,U,i.
+      ⇩[O, i] L ≡ K.ⓛW → ⦃h, K⦄ ⊢ W : V → ⇧[O, i + 1] W ≡ U →
+      R K W V → R L (#i) U
+   ) →
+   (∀I,L,V,W,T,U.
+      ⦃h, L⦄ ⊢ V : W → ⦃h, L.ⓑ{I}V⦄ ⊢ T : U →
+      R L V W → R (L.ⓑ{I}V) T U → R L (ⓑ{I}V.T) (ⓑ{I}V.U)
+   ) →
+   (∀L,V,W,T,U.
+      ⦃h, L⦄ ⊢ V : W → ⦃h, L⦄ ⊢ (ⓛW.T):(ⓛW.U) →
+      R L V W →R L (ⓛW.T) (ⓛW.U) →R L (ⓐV.ⓛW.T) (ⓐV.ⓛW.U)
+   ) →
+   (∀L,V,W,T,U.
+      ⦃h, L⦄ ⊢ T : U → ⦃h, L⦄ ⊢ (ⓐV.U) : W →
+      R L T U → R L (ⓐV.U) W → R L (ⓐV.T) (ⓐV.U)
+   ) →
+   (∀L,T,U,W.
+      ⦃h, L⦄ ⊢ T : U → ⦃h, L⦄ ⊢ U : W →
+      R L T U → R L U W → R L (ⓝU.T) U
+   ) →
+   (∀L,T,U1,U2,V2.
+      ⦃h, L⦄ ⊢ T : U1 → L ⊢ U1 ⬌* U2 → ⦃h, L⦄ ⊢ U2 : V2 →
+      R L T U1 →R L U2 V2 →R L T U2
+   ) →
+   ∀L,T,U. ⦃h, L⦄ ⊢ T : U → R L T U.
+#h #R #H1 #H2 #H3 #H4 #H5 #H6 #H7 #H8 #L #T #U #H elim (nta_ntaa … H) -L -T -U
+// /3 width=1 by ntaa_nta/ /3 width=3 by ntaa_nta/ /3 width=4 by ntaa_nta/
+/3 width=7 by ntaa_nta/
+qed-.
+
+*)
+
+(* Basic_1: removed theorems 4:
+            ty3_getl_subst0 ty3_fsubst0 ty3_csubst0 ty3_subst0
+*)
+(* Basic_2A1: removed theorems 2:
+   ntaa_nta nta_ntaa
+*)