--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||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/rt_equivalence/cpcs_cprs.ma".
+include "basic_2/dynamic/cnv_preserve.ma".
+include "basic_2/dynamic/nta.ma".
+
+(* NATIVE TYPE ASSIGNMENT FOR TERMS *****************************************)
+
+(* Properties based on preservation *****************************************)
+
+lemma cnv_cpms_nta (a) (h) (G) (L):
+ ∀T. ⦃G,L⦄ ⊢ T ![a,h] → ∀U.⦃G,L⦄ ⊢ T ➡*[1,h] U → ⦃G,L⦄ ⊢ T :[a,h] U.
+/3 width=4 by cnv_cast, cnv_cpms_trans/ qed.
+
+lemma cnv_nta_sn (a) (h) (G) (L):
+ ∀T. ⦃G,L⦄ ⊢ T ![a,h] → ∃U. ⦃G,L⦄ ⊢ T :[a,h] U.
+#a #h #G #L #T #HT
+elim (cnv_fwd_cpm_SO … HT) #U #HTU
+/4 width=2 by cnv_cpms_nta, cpm_cpms, ex_intro/
+qed-.
+
+lemma nta_pure_cnv (h) (G) (L):
+ ∀T,U. ⦃G,L⦄ ⊢ T :*[h] U →
+ ∀V. ⦃G,L⦄ ⊢ ⓐV.U !*[h] → ⦃G,L⦄ ⊢ ⓐV.T :*[h] ⓐV.U.
+#h #G #L #T #U #H1 #V #H2
+elim (cnv_inv_cast … H1) -H1 #X0 #HU #HT #HUX0 #HTX0
+elim (cnv_inv_appl … H2) #n #p #X1 #X2 #_ #HV #_ #HVX1 #HUX2
+elim (cnv_cpms_conf … HU … HUX0 … HUX2) -HU -HUX2
+<minus_O_n <minus_n_O #X #HX0 #H
+elim (cpms_inv_abst_sn … H) -H #X3 #X4 #HX13 #HX24 #H destruct
+@(cnv_cast … (ⓐV.X0)) [2:|*: /2 width=1 by cpms_appl_dx/ ]
+@(cnv_appl … X3) [4: |*: /2 width=7 by cpms_trans, cpms_cprs_trans/ ]
+#H destruct
+qed.
+
+(* Inversion lemmas based on preservation ***********************************)
+
+lemma nta_inv_bind_sn_cnv (a) (h) (p) (I) (G) (K) (X2):
+ ∀V,T. ⦃G,K⦄ ⊢ ⓑ{p,I}V.T :[a,h] X2 →
+ ∃∃U. ⦃G,K⦄ ⊢ V ![a,h] & ⦃G,K.ⓑ{I}V⦄ ⊢ T :[a,h] U & ⦃G,K⦄ ⊢ ⓑ{p,I}V.U ⬌*[h] X2 & ⦃G,K⦄ ⊢ X2 ![a,h].
+#a #h #p * #G #K #X2 #V #T #H
+elim (cnv_inv_cast … H) -H #X1 #HX2 #H1 #HX21 #H2
+elim (cnv_inv_bind … H1) -H1 #HV #HT
+[ elim (cpms_inv_abbr_sn_dx … H2) -H2 *
+ [ #V0 #U #HV0 #HTU #H destruct
+ /4 width=5 by cnv_cpms_nta, cprs_div, cpms_bind, ex4_intro/
+ | #U #HTU #HX1U #H destruct
+ /4 width=5 by cnv_cpms_nta, cprs_div, cpms_zeta, ex4_intro/
+ ]
+| elim (cpms_inv_abst_sn … H2) -H2 #V0 #U #HV0 #HTU #H destruct
+ /4 width=5 by cnv_cpms_nta, cprs_div, cpms_bind, ex4_intro/
+]
+qed-.
+
+(* Basic_1: uses: ty3_gen_appl *)
+lemma nta_inv_appl_sn (h) (G) (L) (X2):
+ ∀V,T. ⦃G,L⦄ ⊢ ⓐV.T :[h] X2 →
+ ∃∃p,W,U. ⦃G,L⦄ ⊢ V :[h] W & ⦃G,L⦄ ⊢ T :[h] ⓛ{p}W.U & ⦃G,L⦄ ⊢ ⓐV.ⓛ{p}W.U ⬌*[h] X2 & ⦃G,L⦄ ⊢ X2 ![h].
+#h #G #L #X2 #V #T #H
+elim (cnv_inv_cast … H) -H #X #HX2 #H1 #HX2 #H2
+elim (cnv_inv_appl … H1) * [ | #n ] #p #W #U #Hn #HV #HT #HVW #HTU
+[ lapply (cnv_cpms_trans … HT … HTU) #H
+ elim (cnv_inv_bind … H) -H #_ #HU
+ elim (cnv_fwd_cpm_SO … HU) #U0 #HU0 -HU
+ lapply (cpms_step_dx … HTU 1 (ⓛ{p}W.U0) ?) -HTU [ /2 width=1 by cpm_bind/ ] #HTU
+| lapply (le_n_O_to_eq n ?) [ /3 width=1 by le_S_S_to_le/ ] -Hn #H destruct
+]
+lapply (cpms_appl_dx … V V … HTU) [1,3: // ] #HVTU
+elim (cnv_cpms_conf … H1 … H2 … HVTU) -H1 -H2 -HVTU <minus_n_n #X0 #HX0 #HUX0
+@ex4_3_intro [6,13: |*: /2 width=5 by cnv_cpms_nta/ ]
+/3 width=5 by cprs_div, cprs_trans/
+qed-.
+
+(* Basic_2A1: uses: nta_inv_cast1 *)
+lemma nta_inv_cast_sn (a) (h) (G) (L) (X2):
+ ∀U,T. ⦃G,L⦄ ⊢ ⓝU.T :[a,h] X2 →
+ ∧∧ ⦃G,L⦄ ⊢ T :[a,h] U & ⦃G,L⦄ ⊢ U ⬌*[h] X2 & ⦃G,L⦄ ⊢ X2 ![a,h].
+#a #h #G #L #X2 #U #T #H
+elim (cnv_inv_cast … H) -H #X0 #HX2 #H1 #HX20 #H2
+elim (cnv_inv_cast … H1) #X #HU #HT #HUX #HTX
+elim (cpms_inv_cast1 … H2) -H2 [ * || * ]
+[ #U0 #T0 #HU0 #HT0 #H destruct -HU -HU0
+ elim (cnv_cpms_conf … HT … HTX … HT0) -HT -HTX -HT0
+ <minus_n_n #T1 #HXT1 #HT01
+ @and3_intro // @(cprs_div … T1) /3 width=4 by cprs_trans, cpms_eps/ (**) (* full auto too slow *)
+| #HTX0 -HU
+ elim (cnv_cpms_conf … HT … HTX … HTX0) -HT -HTX -HTX0
+ <minus_n_n #T1 #HXT1 #HXT01
+ @and3_intro // @(cprs_div … T1) /2 width=3 by cprs_trans/ (**) (* full auto too slow *)
+| #m #HUX0 #H destruct -HT -HTX
+ elim (cnv_cpms_conf … HU … HUX … HUX0) -HU -HUX0
+ <minus_n_n #U1 #HXU1 #HXU01
+ @and3_intro // @(cprs_div … U1) /2 width=3 by cprs_trans/ (**) (* full auto too slow *)
+]
+qed-.
+
+(* Forward lemmas based on preservation *************************************)
+
+(* Basic_1: was: ty3_unique *)
+theorem nta_mono (a) (h) (G) (L) (T):
+ ∀U1. ⦃G,L⦄ ⊢ T :[a,h] U1 → ∀U2. ⦃G,L⦄ ⊢ T :[a,h] U2 → ⦃G,L⦄ ⊢ U1 ⬌*[h] U2.
+#a #h #G #L #T #U1 #H1 #U2 #H2
+elim (cnv_inv_cast … H1) -H1 #X1 #_ #_ #HUX1 #HTX1
+elim (cnv_inv_cast … H2) -H2 #X2 #_ #HT #HUX2 #HTX2
+elim (cnv_cpms_conf … HT … HTX1 … HTX2) -T <minus_n_n #X #HX1 #HX2
+/3 width=5 by cprs_div, cprs_trans/
+qed-.
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