(* Basic_1: was by definition: ty3_abst *)
(* Basic_2A1: was by definition: nta_ldec ntaa_ldec *)
lemma nta_ldec_drops

(* Basic_1: was by definition: ty3_bind *)
(* Basic_2A1: was by definition: nta_bind ntaa_bind *)
lemma nta_bind

(* Basic_2A1: was by definition: nta_pure ntaa_pure *)
lemma nta_pure

(* Basic_1: was: ty3_gen_bind *)
(* Basic_2A1: was: nta_inv_bind1 ntaa_inv_bind1 *)
lemma nta_inv_bind_sn

(* Basic_1: was: ty3_gen_lref *)
(* Basic_2A1: was: nta_inv_lref1 *)
lemma nta_inv_lref_sn_drops

(* Basic_1: uses: ty3_gen_abst_abst *)
lemma nta_inv_abst_bi

(* Basic_1: uses: pc3_dec *)
lemma nta_cpcs_dec

(* Advanced properties ******************************************************)

| ntaa_cast: ∀L,T,U,W. ntaa h L T U → ntaa h L U W → ntaa h L (ⓝU. T) U

lemma nta_cast_alt: ∀h,L,T,W,U. ⦃h, L⦄ ⊢ T  : W → ⦃h, L⦄ ⊢ T : U → ⦃h, L⦄ ⊢ ⓝW.T : U.
#h #L #T #W #U #HTW #HTU
lapply (nta_mono … HTW … HTU) #HWU
elim (nta_fwd_correct … HTU) -HTU /3 width=3/
qed.

lemma nta_ind_alt: ∀h. ∀Q:lenv→relation term.
   (∀L,k. Q 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 →
      Q K V W → Q L (#i) U
   ) →
   (∀L,K,W,V,U,i.
      ⇩[O, i] L ≡ K.ⓛW → ⦃h, K⦄ ⊢ W : V → ⇧[O, i + 1] W ≡ U →
      Q K W V → Q L (#i) U
   ) →
   (∀I,L,V,W,T,U.
      ⦃h, L⦄ ⊢ V : W → ⦃h, L.ⓑ{I}V⦄ ⊢ T : U →
      Q L V W → Q (L.ⓑ{I}V) T U → Q L (ⓑ{I}V.T) (ⓑ{I}V.U)
   ) →
   (∀L,V,W,T,U.
      ⦃h, L⦄ ⊢ V : W → ⦃h, L⦄ ⊢ (ⓛW.T):(ⓛW.U) →
      Q L V W →Q L (ⓛW.T) (ⓛW.U) →Q L (ⓐV.ⓛW.T) (ⓐV.ⓛW.U)
   ) →
   (∀L,V,W,T,U.
      ⦃h, L⦄ ⊢ T : U → ⦃h, L⦄ ⊢ (ⓐV.U) : W →
      Q L T U → Q L (ⓐV.U) W → Q L (ⓐV.T) (ⓐV.U)
   ) →
   (∀L,T,U,W.
      ⦃h, L⦄ ⊢ T : U → ⦃h, L⦄ ⊢ U : W →
      Q L T U → Q L U W → Q L (ⓝU.T) U
   ) →
   (∀L,T,U1,U2,V2.
      ⦃h, L⦄ ⊢ T : U1 → L ⊢ U1 ⬌* U2 → ⦃h, L⦄ ⊢ U2 : V2 →
      Q L T U1 →Q L U2 V2 →Q L T U2
   ) →
   ∀L,T,U. ⦃h, L⦄ ⊢ T : U → Q L T U.