From 86861e6f031df66824a381527dfe847029ff72bc Mon Sep 17 00:00:00 2001
From: Ferruccio Guidi <ferruccio.guidi@unibo.it>
Date: Sat, 28 Sep 2019 20:39:25 +0200
Subject: [PATCH] update in basic_2

+ iterated type assignment completed
+ minor additions to the arith library
---
 .../lambdadelta/basic_2/etc/nta_rest.etc      | 115 ++++++++++++++++++
 .../basic_2/i_dynamic/ntas_cpcs.ma            |  36 ++++++
 .../lambdadelta/basic_2/i_dynamic/ntas_nta.ma |  86 +++++++++----
 .../basic_2/i_dynamic/ntas_nta_ind.ma         | 111 +++++++++++++++++
 .../basic_2/i_dynamic/ntas_ntas.ma            |  27 ++++
 .../basic_2/i_dynamic/ntas_preserve.ma        |  64 ++--------
 .../basic_2/rt_computation/cpms.ma            |   2 +-
 .../basic_2/rt_computation/cpms_aaa.ma        |   2 +-
 .../lambdadelta/basic_2/web/basic_2_src.tbl   |   2 +-
 .../lambdadelta/ground_2/lib/arith.ma         |   5 +-
 10 files changed, 365 insertions(+), 85 deletions(-)
 create mode 100644 matita/matita/contribs/lambdadelta/basic_2/etc/nta_rest.etc
 create mode 100644 matita/matita/contribs/lambdadelta/basic_2/i_dynamic/ntas_cpcs.ma
 create mode 100644 matita/matita/contribs/lambdadelta/basic_2/i_dynamic/ntas_nta_ind.ma
 create mode 100644 matita/matita/contribs/lambdadelta/basic_2/i_dynamic/ntas_ntas.ma

diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc/nta_rest.etc b/matita/matita/contribs/lambdadelta/basic_2/etc/nta_rest.etc
new file mode 100644
index 000000000..151dd47ea
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2/etc/nta_rest.etc
@@ -0,0 +1,115 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||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_transition/cpr_drops_basic.ma".
+include "basic_2/dynamic/nta_ind.ma".
+
+lemma lpr_cpr_conf_cpc_sym (h) (G) (L1):
+      ∀T1,T2:term. ⦃G,L1⦄ ⊢ T1 ➡[h] T2 →
+      ∀L2. ⦃G,L1⦄ ⊢ ➡[h] L2 → ⦃G,L2⦄ ⊢ T2 ⬌*[h] T1.
+#h #G #L1 #T1 #T2 #HT12 #L2 #HL12
+elim (lpr_cpr_conf_dx … HT12 … HL12) -L1 #T #HT1 #HT2
+/3 width=3 by cprs_div, cpm_cpms/
+qed.
+
+(* NATIVE TYPE ASSIGNMENT FOR TERMS *****************************************)
+
+definition IH_nta (h) (G) (L1) (T1) ≝
+           ∀U. ⦃G,L1⦄ ⊢ T1 :[h,𝟐] U → ∀T2. ⦃G,L1⦄ ⊢ T1 ➡[h] T2 →
+           ∀L2. ⦃G,L1⦄ ⊢ ➡[h] L2 → ⦃G,L2⦄ ⊢ T2 :[h,𝟐] U.
+
+definition IH_cnv (h) (G) (L1) (T1) ≝
+           (⦃G,L1⦄ ⊢ T1 ![h,𝟐]) → ∀T2. ⦃G,L1⦄ ⊢ T1 ➡[h] T2 →
+           ∀L2. ⦃G,L1⦄ ⊢ ➡[h] L2 → ⦃G,L2⦄ ⊢ T2 ![h,𝟐].
+
+(* Specific proofs for restricted applicability *****************************)
+
+lemma nta_to_cnv (h) (G) (L1) (T1):
+      IH_nta h G L1 T1 → IH_cnv h G L1 T1.
+#h #G #L1 #T1 #H #HT1 #T2 #HT12 #L2 #HL12
+elim (cnv_nta_sn … HT1) -HT1 #U1 #HTU1
+/3 width=2 by nta_fwd_cnv_sn/
+qed-.
+
+(* Note preservation of type without big-tree theorem *)
+lemma nta_cpr_conf_lpr (h) (G) (L1) (T1): IH_nta h G L1 T1.
+#h #G #L1 #T1 @(fqup_wf_ind (Ⓣ) … G L1 T1) -G -L1 -T1
+#G0 #L0 #T0 #IH #U
+@(insert_eq_0 … T0) #T1
+@(insert_eq_0 … L0) #L1
+@(insert_eq_0 … G0) #G
+#H @(nta_ind_rest_cnv … H) -G -L1 -T1 -U
+[ #G #L1 #s #HG0 #HL0 #HT0 #X #HX #L2 #HL12 destruct -IH
+  lapply (cpr_inv_sort1 … HX) -HX #H destruct //
+| #G #K1 #V1 #W1 #U #HVW1 #HW1U #_ #HG0 #HL0 #HT0 #X #HX #Y #HY destruct
+  elim (lpr_inv_pair_sn … HY) -HY #K2 #V2 #HK12 #HV12 #H destruct
+  elim (cpr_inv_zero1 … HX) -HX
+  [ #H destruct
+    /4 width=6 by nta_ldef, fqu_fqup, fqu_lref_O/
+  | * #Y1 #X1 #X2 #HX12 #HX2 #H destruct -HV12
+    @(nta_lifts_bi … (Ⓣ) … HX2 … HW1U) -X -U
+    /4 width=6 by drops_refl, drops_drop, fqu_fqup, fqu_lref_O/
+  ]
+| #G #K1 #W1 #V1 #HW1 #HWV1 #HG0 #HL0 #HT0 #X #HX #Y #HY destruct
+  elim (lpr_inv_pair_sn … HY) -HY #K2 #W2 #HK12 #HW12 #H destruct
+  elim (cpr_inv_zero1 … HX) -HX
+  [ #H destruct
+    lapply (nta_to_cnv … HW1 … W1 … HK12)
+    [1,2: /3 width=1 by fqu_fqup, fqu_lref_O/ ] #H
+    lapply (cnv_lifts … H (Ⓣ) … (K2.ⓛW2) … HWV1) -H
+    [ /3 width=1 by drops_refl, drops_drop/ ] #HV1
+    elim (cpm_lifts_sn … HW12 (Ⓣ) … (K1.ⓛW2) … HWV1) -HWV1
+    [2: /3 width=1 by drops_refl, drops_drop/ ] #V2 #HWV2 #HV12
+    lapply (nta_to_cnv … HW1 … HW12 … HK12) -HW1 -HW12
+    [ /3 width=1 by fqu_fqup, fqu_lref_O/ ] -IH #HW2
+    @(nta_conv_cnv … V2)
+    /3 width=3 by nta_ldec_cnv, lpr_cpr_conf_cpc_sym, lpr_bind_refl_dx/
+  | * #Y1 #X1 #X2 #_ #_ #H destruct
+  ]
+| #I1 #G #K1 #T1 #U1 #i #Hi #HTU1 #_ #HG0 #HL0 #HT0 #X #HX #Y #HY destruct
+  elim (lpr_inv_bind_sn … HY) -HY #I2 #K2 #HK12 #_ #H destruct
+  elim (cpr_inv_lref1 … HX) -HX
+  [ #H destruct /4 width=6 by nta_lref, fqu_fqup/
+  | * #Z1 #Y1 #X1 #HiX1 #HX1 #H destruct
+    @(nta_lifts_bi … (Ⓣ) … HX1 … HTU1) -X -U1
+    /4 width=6 by drops_refl, drops_drop, fqu_fqup/
+  ]
+| #p #I #G #L1 #V1 #T1 #U #HV1 #HT1 #_ #HG0 #HL0 #HT0 #X #HX #L2 #HL12 destruct
+  elim (cpm_inv_bind1 … HX) -HX *
+  [ #V2 #T2 #HV12 #HT12 #H destruct
+    lapply (IH … HT1 T1 … (L2.ⓑ{I}V1) ?)
+    [4:|*: /2 width=1 by lpr_bind_refl_dx/ ] #H
+    lapply (nta_fwd_cnv_dx … H) -H #HU
+    @(nta_conv_cnv … (ⓑ{p,I}V2.U))
+    [ @nta_bind_cnv [ /3 width=6 by nta_to_cnv/ | /4 width=8 by lpr_bind, ext2_pair/ ]
+    | /3 width=3 by lpr_cpr_conf_cpc_sym, cpr_pair_sn/
+    | /4 width=6 by nta_to_cnv, cnv_bind/
+    ]
+  | #U1 #HUT1 #HU1X #H1 #H2 destruct
+(*  
+    elim (cpr_subst h G (L1.ⓓV1) U 0)
+    [|*: /3 width=3 by drops_refl/ ] #U0 #T0 #HU0 #HTU0
+    lapply (nta_conv_cnv … HT1 … U0 ??)
+    [ @(cnv_lifts … HTU0) 
+    
+    @(IH … HU1X … HL12) -L2 -X
+    @(nta_conv_cnv … T0)
+    [
+    | @cpcs_cprs_sn @(cprs_step_sn … (+ⓓV1.U0))
+      /2 width=1 by cpm_bind/ /3 width=3 by cpm_cpms, cpm_zeta/
+  
+  
+    elim (cpm_lifts_sn … HU1X (Ⓣ) … (L1.ⓓV1) … HUT1) -U1
+    [| /3 width=1 by drops_refl, drops_drop/ ] #T2 #HXT2 #HT12
+*)
diff --git a/matita/matita/contribs/lambdadelta/basic_2/i_dynamic/ntas_cpcs.ma b/matita/matita/contribs/lambdadelta/basic_2/i_dynamic/ntas_cpcs.ma
new file mode 100644
index 000000000..8438e0307
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2/i_dynamic/ntas_cpcs.ma
@@ -0,0 +1,36 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||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/i_dynamic/ntas.ma".
+
+(* ITERATED NATIVE TYPE ASSIGNMENT FOR TERMS ********************************)
+
+(* Properties with r-equivalence for terms **********************************)
+
+lemma ntas_zero (h) (a) (G) (L):
+      ∀T1,T2. ⦃G,L⦄ ⊢ T1 ![h,a] → ⦃G,L⦄ ⊢ T2 ![h,a] → ⦃G,L⦄ ⊢ T1 ⬌*[h] T2 → ⦃G,L⦄ ⊢ T1 :*[h,a,0] T2.
+#h #a #G #L #T1 #T2 #HT1 #HT2 #H
+elim (cpcs_inv_cprs … H) -H #T0 #HT10 #HT20
+/2 width=3 by ntas_intro/
+qed.
+
+(* Inversion lemmas with r-equivalence for terms ****************************)
+
+lemma ntas_inv_zero (h) (a) (G) (L):
+      ∀T1,T2. ⦃G,L⦄ ⊢ T1 :*[h,a,0] T2 →
+      ∧∧ ⦃G,L⦄ ⊢ T1 ![h,a] & ⦃G,L⦄ ⊢ T2 ![h,a] & ⦃G,L⦄ ⊢ T1 ⬌*[h] T2.
+#h #a #G #L #T1 #T2 * #T0 #HT1 #HT2 #HT20 #HT10
+/3 width=3 by cprs_div, and3_intro/ 
+qed-.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/i_dynamic/ntas_nta.ma b/matita/matita/contribs/lambdadelta/basic_2/i_dynamic/ntas_nta.ma
index 17259cb17..865de8042 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/i_dynamic/ntas_nta.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/i_dynamic/ntas_nta.ma
@@ -17,6 +17,30 @@ include "basic_2/i_dynamic/ntas_preserve.ma".
 
 (* ITERATED NATIVE TYPE ASSIGNMENT FOR TERMS ********************************)
 
+(* Advanced properties of native validity for terms *************************)
+
+lemma cnv_appl_ntas_ge (h) (a) (G) (L):
+      ∀m,n. m ≤ n → ad a n → ∀V,W. ⦃G,L⦄ ⊢ V :[h,a] W →
+      ∀p,T,U. ⦃G,L⦄ ⊢ T :*[h,a,m] ⓛ{p}W.U → ⦃G,L⦄ ⊢ ⓐV.T ![h,a].
+#h #a #G #L #m #n #Hmn #Hn #V #W #HVW #p #T #U #HTU
+elim (cnv_inv_cast … HVW) -HVW #W0 #HW #HV #HW0 #HVW0
+elim HTU -HTU #U0 #H #HT #HU0 #HTU0
+elim (cnv_cpms_conf … H … HU0 0 (ⓛ{p}W0.U)) -H -HU0
+[| /2 width=1 by cpms_bind/ ] -HW0 <minus_n_O #X0 #HUX0 #H
+elim (cpms_inv_abst_sn … H) -H #W1 #U1 #HW01 #_ #H destruct
+/3 width=13 by cnv_appl_ge, cpms_cprs_trans/
+qed-.
+
+(* Advanced inversion lemmas of native validity for terms *******************)
+
+lemma cnv_inv_appl_ntas (h) (a) (G) (L):
+      ∀V,T. ⦃G,L⦄ ⊢ ⓐV.T ![h,a] →
+      ∃∃n,p,W,T,U. ad a n & ⦃G,L⦄ ⊢ V :[h,a] W & ⦃G,L⦄ ⊢ T :*[h,a,n] ⓛ{p}W.U.
+#h #a #G #L #V #T #H
+elim (cnv_inv_appl … H) -H #n #p #W #U #Hn #HV #HT #HVW #HTU
+/3 width=9 by cnv_cpms_nta, cnv_cpms_ntas, ex3_5_intro/
+qed-.
+
 (* Properties with native type assignment for terms *************************)
 
 lemma nta_ntas (h) (a) (G) (L):
@@ -25,6 +49,42 @@ lemma nta_ntas (h) (a) (G) (L):
 elim (cnv_inv_cast … H) -H /2 width=3 by ntas_intro/
 qed-.
 
+lemma ntas_step_sn (h) (a) (G) (L):
+      ∀T1,T. ⦃G,L⦄ ⊢ T1 :[h,a] T →
+      ∀n,T2. ⦃G,L⦄ ⊢ T :*[h,a,n] T2 → ⦃G,L⦄ ⊢ T1 :*[h,a,↑n] T2.
+#h #a #G #L #T1 #T #H #n #T2 * #X2 #HT2 #HT #H1TX2 #H2TX2
+elim (cnv_inv_cast … H) -H #X1 #_ #HT1 #H1TX1 #H2TX1
+elim (cnv_cpms_conf … HT … H1TX1 … H2TX2) -T <minus_n_O <minus_O_n <plus_SO_sn #X #HX1 #HX2
+/3 width=5 by ntas_intro, cprs_trans, cpms_trans/
+qed-.
+
+lemma ntas_step_dx (h) (a) (G) (L):
+      ∀n,T1,T. ⦃G,L⦄ ⊢ T1 :*[h,a,n] T →
+      ∀T2. ⦃G,L⦄ ⊢ T :[h,a] T2 → ⦃G,L⦄ ⊢ T1 :*[h,a,↑n] T2.
+#h #a #G #L #n #T1 #T * #X1 #HT #HT1 #H1TX1 #H2TX1 #T2 #H
+elim (cnv_inv_cast … H) -H #X2 #HT2 #_ #H1TX2 #H2TX2
+elim (cnv_cpms_conf … HT … H1TX1 … H2TX2) -T <minus_n_O <minus_O_n <plus_SO_dx #X #HX1 #HX2
+/3 width=5 by ntas_intro, cprs_trans, cpms_trans/
+qed-.
+
+lemma nta_appl_ntas_zero (h) (a) (G) (L): ad a 0 →
+      ∀V,W. ⦃G,L⦄ ⊢ V :[h,a] W → ∀p,T,U0. ⦃G,L⦄ ⊢ T :*[h,a,0] ⓛ{p}W.U0 →
+      ∀U. ⦃G,L.ⓛW⦄ ⊢ U0 :[h,a] U → ⦃G,L⦄ ⊢ ⓐV.T :[h,a] ⓐV.ⓛ{p}W.U.
+#h #a #G #L #Ha #V #W #HVW #p #T #U0 #HTU0 #U #HU0
+lapply (nta_fwd_cnv_dx … HVW) #HW
+lapply (nta_bind_cnv … HW … HU0 p) -HW -HU0 #HU0
+elim (ntas_step_dx … HTU0 … HU0) -HU0 #X #HU #_ #HUX #HTX
+/4 width=9 by cnv_appl_ntas_ge, ntas_refl, cnv_cast, cpms_appl_dx/
+qed.
+
+lemma nta_appl_ntas_pos (h) (a) (n) (G) (L): ad a (↑n) →
+      ∀V,W. ⦃G,L⦄ ⊢ V :[h,a] W → ∀T,U. ⦃G,L⦄ ⊢ T :[h,a] U →
+      ∀p,U0. ⦃G,L⦄ ⊢ U :*[h,a,n] ⓛ{p}W.U0 → ⦃G,L⦄ ⊢ ⓐV.T :[h,a] ⓐV.U.
+#h #a #n #G #L #Ha #V #W #HVW #T #U #HTU #p #U0 #HU0
+elim (cnv_inv_cast … HTU) #X #_ #_ #HUX #HTX
+/4 width=11 by ntas_step_sn, cnv_appl_ntas_ge, cnv_cast, cpms_appl_dx/
+qed-.
+
 (* Inversion lemmas with native type assignment for terms *******************)
 
 lemma ntas_inv_nta (h) (a) (G) (L):
@@ -73,29 +133,3 @@ elim (eq_or_gt n) #Hn destruct
   /5 width=10 by cnv_cpms_ntas, cnv_cpms_nta, cnv_cpms_trans, ex6_5_intro, or_intror/
 ]
 qed-.
-
-(*
-
-definition ntas: sh → lenv → relation term ≝
-                 λh,L. star … (nta h L).
-
-(* Basic eliminators ********************************************************)
-
-axiom ntas_ind_dx: ∀h,L,T2. ∀R:predicate term. R T2 →
-                   (∀T1,T. ⦃h,L⦄ ⊢ T1 : T → ⦃h,L⦄ ⊢ T :* T2 → R T → R T1) →
-                   ∀T1. ⦃h,L⦄ ⊢ T1 :* T2 → R T1.
-(*
-#h #L #T2 #R #HT2 #IHT2 #T1 #HT12
-@(star_ind_dx … HT2 IHT2 … HT12) //
-qed-.
-*)
-(* Basic properties *********************************************************)
-
-lemma ntas_strap1: ∀h,L,T1,T,T2.
-                   ⦃h,L⦄ ⊢ T1 :* T → ⦃h,L⦄  ⊢ T : T2 → ⦃h,L⦄  ⊢ T1 :* T2.
-/2 width=3/ qed.
-
-lemma ntas_strap2: ∀h,L,T1,T,T2.
-                   ⦃h,L⦄  ⊢ T1 : T → ⦃h,L⦄ ⊢ T :* T2 → ⦃h,L⦄ ⊢ T1 :* T2.
-/2 width=3/ qed.
-*)
diff --git a/matita/matita/contribs/lambdadelta/basic_2/i_dynamic/ntas_nta_ind.ma b/matita/matita/contribs/lambdadelta/basic_2/i_dynamic/ntas_nta_ind.ma
new file mode 100644
index 000000000..ec6b81822
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2/i_dynamic/ntas_nta_ind.ma
@@ -0,0 +1,111 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||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/dynamic/nta_drops.ma".
+include "basic_2/dynamic/nta_cpcs.ma".
+include "basic_2/i_dynamic/ntas_cpcs.ma".
+include "basic_2/i_dynamic/ntas_nta.ma".
+include "basic_2/i_dynamic/ntas_ntas.ma".
+
+(* ITERATED NATIVE TYPE ASSIGNMENT FOR TERMS ********************************)
+
+(* Advanced eliminators for native type assignment **************************)
+
+lemma ntas_ind_bi_nta (h) (a) (G) (L) (Q:relation3 …):
+      (∀T1,T2. ⦃G,L⦄ ⊢ T1 ![h,a] → ⦃G,L⦄ ⊢ T2 ![h,a] → ⦃G,L⦄ ⊢ T1 ⬌*[h] T2 →
+        Q 0 T1 T2
+      ) →
+      (∀T1,T2. ⦃G,L⦄ ⊢ T1 :[h,a] T2 → Q 1 T1 T2
+      ) →
+      (∀n1,n2,T1,T2,T0.  ⦃G,L⦄ ⊢ T1 :*[h,a,n1] T0 → ⦃G,L⦄ ⊢ T0 :*[h,a,n2] T2 →
+        Q n1 T1 T0 → Q n2 T0 T2 → Q (n1+n2) T1 T2
+      ) →
+      ∀n,T1,T2. ⦃G,L⦄ ⊢ T1 :*[h,a,n] T2 → Q n T1 T2.
+#h #a #G #L #Q #IH1 #IH2 #IH3 #n
+@(nat_elim1 n) -n * [| * ]
+[ #_ #T1 #T2 #H
+  elim (ntas_inv_zero … H) -H #HT1 #HT2 #HT12
+  /2 width=1 by/
+| #_ #T1 #T2 #H
+  /3 width=1 by ntas_inv_nta/
+| #n #IH #T1 #T2 <plus_SO_sn #H
+  elim (ntas_inv_plus … H) -H #T0 #HT10 #HT02
+  /3 width=5 by/
+]
+qed-.
+
+lemma nta_ind_cnv (h) (a) (Q:relation4 …):
+      (∀G,L,s. Q G L (⋆s) (⋆(⫯[h]s))) →
+      (∀G,K,V,W,U.
+        ⦃G,K⦄ ⊢ V :[h,a] W → ⬆*[1] W ≘ U →
+        Q G K V W → Q G (K.ⓓV) (#0) U
+      ) →
+      (∀G,K,W,U. ⦃G,K⦄ ⊢ W ![h,a] → ⬆*[1] W ≘ U → Q G (K.ⓛW) (#0) U) →
+      (∀I,G,K,W,U,i.
+        ⦃G,K⦄ ⊢ #i :[h,a] W → ⬆*[1] W ≘ U →
+        Q G K (#i) W → Q G (K.ⓘ{I}) (#↑i) U
+      ) →
+      (∀p,I,G,K,V,T,U.
+        ⦃G,K⦄ ⊢ V ![h,a] → ⦃G,K.ⓑ{I}V⦄ ⊢ T :[h,a] U →
+        Q G (K.ⓑ{I}V) T U → Q G K (ⓑ{p,I}V.T) (ⓑ{p,I}V.U)
+      ) →
+      (∀p,G,L,V,W,T,U0,U. (**) (* one IH is missing *)
+        ad a 0 → ⦃G,L⦄ ⊢ V :[h,a] W → ⦃G,L⦄ ⊢ T :*[h,a,0] ⓛ{p}W.U0 → ⦃G,L.ⓛW⦄ ⊢ U0 :[h,a] U →
+        Q G L V W (* → Q G (L.ⓛW) U0 U *) → Q G L (ⓐV.T) (ⓐV.ⓛ{p}W.U)
+      ) →
+      (∀n,p,G,L,V,W,T,U,U0.
+        ad a (↑n) → ⦃G,L⦄ ⊢ V :[h,a] W → ⦃G,L⦄ ⊢ T :[h,a] U → ⦃G,L⦄ ⊢ U :*[h,a,n] ⓛ{p}W.U0 →
+        Q G L V W → Q G L T U → Q G L (ⓐV.T) (ⓐV.U)
+      ) →
+      (∀G,L,T,U. ⦃G,L⦄ ⊢ T :[h,a] U → Q G L T U → Q G L (ⓝU.T) U
+      ) →
+      (∀G,L,T,U1,U2.
+        ⦃G,L⦄ ⊢ T :[h,a] U1 → ⦃G,L⦄ ⊢ U1 ⬌*[h] U2 → ⦃G,L⦄ ⊢ U2 ![h,a] →
+        Q G L T U1 → Q G L T U2
+      ) →
+      ∀G,L,T,U. ⦃G,L⦄ ⊢ T :[h,a] U → Q G L T U.
+#h #a #Q #IH1 #IH2 #IH3 #IH4 #IH5 #IH6 #IH7 #IH8 #H9 #G #L #T
+@(fqup_wf_ind_eq (Ⓣ) … G L T) -G -L -T #G0 #L0 #T0 #IH #G #L * * [|||| * ]
+[ #s #HG #HL #HT #X #H destruct -IH
+  elim (nta_inv_sort_sn … H) -H #HUX #HX
+  /2 width=4 by/
+| * [| #i ] #HG #HL #HT #X #H destruct
+  [ elim (nta_inv_lref_sn_drops_cnv … H) -H *
+    [ #K #V #W #U #H #HVW #HWU #HUX #HX
+      lapply (drops_fwd_isid … H ?) -H [ // ] #H destruct
+      /5 width=7 by nta_ldef, fqu_fqup, fqu_lref_O/
+    | #K #W #U #H #HW #HWU #HUX #HX
+      lapply (drops_fwd_isid … H ?) -H [ // ] #H destruct
+      /3 width=4 by nta_ldec_cnv/
+    ]
+  | elim (nta_inv_lref_sn … H) -H #I #K #T #U #HT #HTU #HUX #HX #H destruct
+    /5 width=7 by nta_lref, fqu_fqup/
+  ]
+| #l #HG #HL #HT #U #H destruct -IH
+  elim (nta_inv_gref_sn … H)
+| #p #I #V #T #HG #HL #HT #X #H destruct
+  elim (nta_inv_bind_sn_cnv … H) -H #U #HV #HTU #HUX #HX
+  /4 width=5 by nta_bind_cnv/
+| #V #T #HG #HL #HT #X #H destruct
+  elim (nta_inv_appl_sn_ntas … H) -H *
+  [ #p #W #U #U0 #Ha #HVW #HTU0 #HU0 #HUX #HX -IH7
+    /4 width=10 by nta_appl_ntas_zero/
+  | #n #p #W #U #U0 #Ha #HVW #HTU #HU0 #HUX #HX -IH6
+    /4 width=13 by nta_appl_ntas_pos/
+  ]
+| #U #T #HG #HL #HT #X #H destruct
+  elim (nta_inv_cast_sn … H) -H #HTU #HUX #HX
+  /4 width=4 by nta_cast/
+]
+qed-.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/i_dynamic/ntas_ntas.ma b/matita/matita/contribs/lambdadelta/basic_2/i_dynamic/ntas_ntas.ma
new file mode 100644
index 000000000..67e7464ca
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2/i_dynamic/ntas_ntas.ma
@@ -0,0 +1,27 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||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/i_dynamic/ntas_preserve.ma".
+
+(* ITERATED NATIVE TYPE ASSIGNMENT FOR TERMS ********************************)
+
+(* Main properties **********************************************************)
+
+theorem ntas_trans (h) (a) (G) (L) (T0):
+        ∀n1,T1. ⦃G,L⦄ ⊢ T1 :*[h,a,n1] T0 →
+        ∀n2,T2. ⦃G,L⦄ ⊢ T0 :*[h,a,n2] T2 → ⦃G,L⦄ ⊢ T1 :*[h,a,n1+n2] T2.
+#h #a #G #L #T0 #n1 #T1 * #X1 #HT0 #HT1 #H01 #H11 #n2 #T2 * #X2 #HT2 #_ #H22 #H02
+elim (cnv_cpms_conf … HT0 … H01 … H02) -T0 <minus_O_n <minus_n_O #X0 #H10 #H20
+/3 width=5 by ntas_intro, cpms_trans/
+qed-.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/i_dynamic/ntas_preserve.ma b/matita/matita/contribs/lambdadelta/basic_2/i_dynamic/ntas_preserve.ma
index 5d391caed..53f02c04e 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/i_dynamic/ntas_preserve.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/i_dynamic/ntas_preserve.ma
@@ -26,6 +26,15 @@ lemma cnv_cpms_ntas (h) (a) (G) (L):
 
 (* Inversion lemmas based on preservation ***********************************)
 
+lemma ntas_inv_plus (h) (a) (n1) (n2) (G) (L):
+      ∀T1,T2. ⦃G,L⦄ ⊢ T1 :*[h,a,n1+n2] T2 →
+      ∃∃T0. ⦃G,L⦄ ⊢ T1 :*[h,a,n1] T0 & ⦃G,L⦄ ⊢ T0 :*[h,a,n2] T2.
+#h #a #n1 #n2 #G #L #T1 #T2 * #X0 #HT2 #HT1 #H20 #H10
+elim (cpms_inv_plus … H10) -H10 #T0 #H10 #H00
+lapply (cnv_cpms_trans … HT1 … H10) #HT0
+/3 width=6 by cnv_cpms_ntas, ntas_intro, ex2_intro/
+qed-.
+
 lemma ntas_inv_appl_sn (h) (a) (m) (G) (L) (V) (T):
       ∀X. ⦃G,L⦄ ⊢ ⓐV.T :*[h,a,m] X →
       ∨∨ ∃∃n,p,W,U,U0. n ≤ m & ad a n & ⦃G,L⦄ ⊢ V :*[h,a,1] W & ⦃G,L⦄ ⊢ T :*[h,a,n] ⓛ{p}W.U0 & ⦃G,L.ⓛW⦄ ⊢ U0 :*[h,a,m-n] U & ⦃G,L⦄ ⊢ ⓐV.ⓛ{p}W.U ⬌*[h] X & ⦃G,L⦄ ⊢ X ![h,a]
@@ -51,58 +60,3 @@ elim (le_or_ge n m) #Hnm
   /5 width=11 by cnv_cpms_ntas, cnv_cpms_trans, ex7_5_intro, or_intror/
 ]
 qed-.
-
-(*
-(* Advanced properties on native type assignment for terms ******************)
-
-lemma nta_pure_ntas: ∀h,L,U,W,Y. ⦃h,L⦄ ⊢ U :* ⓛW.Y → ∀T. ⦃h,L⦄ ⊢ T : U →
-                     ∀V. ⦃h,L⦄ ⊢ V : W →  ⦃h,L⦄ ⊢ ⓐV.T : ⓐV.U.
-#h #L #U #W #Y #H @(ntas_ind_dx … H) -U /2 width=1/ /3 width=2/
-qed.
-
-axiom pippo: ∀h,L,T,W,Y. ⦃h,L⦄ ⊢ T :* ⓛW.Y → ∀U. ⦃h,L⦄ ⊢ T : U →
-             ∃Z. ⦃h,L⦄ ⊢ U :* ⓛW.Z.
-(* REQUIRES SUBJECT CONVERSION
-#h #L #T #W #Y #H @(ntas_ind_dx … H) -T
-[ #U #HYU
-  elim (nta_fwd_correct … HYU) #U0 #HU0 
-  elim (nta_inv_bind1 … HYU) #W0 #Y0 #HW0 #HY0 #HY0U
-*)
-
-(* Advanced inversion lemmas on native type assignment for terms ************)
-
-fact nta_inv_pure1_aux: ∀h,L,Z,U. ⦃h,L⦄ ⊢ Z : U → ∀X,Y. Z = ⓐY.X →
-                        ∃∃W,V,T. ⦃h,L⦄ ⊢ Y : W & ⦃h,L⦄ ⊢ X : V &
-                                 L ⊢ ⓐY.V ⬌* U & ⦃h,L⦄ ⊢ V :* ⓛW.T.
-#h #L #Z #U #H elim H -L -Z -U
-[ #L #k #X #Y #H destruct
-| #L #K #V #W #U #i #_ #_ #_ #_ #X #Y #H destruct
-| #L #K #W #V #U #i #_ #_ #_ #_ #X #Y #H destruct
-| #I #L #V #W #T #U #_ #_ #_ #_ #X #Y #H destruct
-| #L #V #W #Z #U #HVW #HZU #_ #_ #X #Y #H destruct /2 width=7/
-| #L #V #W #Z #U #HZU #_ #_ #IHUW #X #Y #H destruct
-  elim (IHUW U Y ?) -IHUW // /3 width=9/
-| #L #Z #U #_ #_ #X #Y #H destruct
-| #L #Z #U1 #U2 #V2 #_ #HU12 #_ #IHTU1 #_ #X #Y #H destruct
-  elim (IHTU1 ???) -IHTU1 [4: // |2,3: skip ] #W #V #T #HYW #HXV #HU1 #HVT
-  lapply (cpcs_trans … HU1 … HU12) -U1 /2 width=7/
-]
-qed.
-
-(* Basic_1: was only: ty3_gen_appl *)
-lemma nta_inv_pure1: ∀h,L,Y,X,U. ⦃h,L⦄ ⊢ ⓐY.X : U →
-                     ∃∃W,V,T. ⦃h,L⦄ ⊢ Y : W & ⦃h,L⦄ ⊢ X : V &
-                              L ⊢ ⓐY.V ⬌* U & ⦃h,L⦄ ⊢ V :* ⓛW.T.
-/2 width=3/ qed-.
-
-axiom nta_inv_appl1: ∀h,L,Z,Y,X,U. ⦃h,L⦄ ⊢ ⓐZ.ⓛY.X : U →
-                     ∃∃W. ⦃h,L⦄ ⊢ Z : Y & ⦃h,L⦄ ⊢ ⓛY.X : ⓛY.W &
-                     L ⊢ ⓐZ.ⓛY.W ⬌* U.
-(* REQUIRES SUBJECT REDUCTION
-#h #L #Z #Y #X #U #H
-elim (nta_inv_pure1 … H) -H #W #V #T #HZW #HXV #HVU #HVT
-elim (nta_inv_bind1 … HXV) -HXV #Y0 #X0 #HY0 #HX0 #HX0V
-lapply (cpcs_trans … (ⓐZ.ⓛY.X0) … HVU) -HVU /2 width=1/ -HX0V #HX0U
-@(ex3_1_intro … HX0U) /2 width=2/
-*)
-*)
diff --git a/matita/matita/contribs/lambdadelta/basic_2/rt_computation/cpms.ma b/matita/matita/contribs/lambdadelta/basic_2/rt_computation/cpms.ma
index 2f7d1f449..d422aaf22 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/rt_computation/cpms.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/rt_computation/cpms.ma
@@ -144,7 +144,7 @@ lemma cprs_refl: ∀h,G,L. reflexive … (cpms h G L 0).
 lemma cpms_sort (h) (G) (L) (n):
                 ∀s. ⦃G,L⦄ ⊢ ⋆s ➡*[n,h] ⋆((next h)^n s).
 #h #G #L #n elim n -n [ // ]
-#n #IH #s <plus_SO
+#n #IH #s <plus_SO_dx
 /3 width=3 by cpms_step_dx, cpm_sort/
 qed.
 
diff --git a/matita/matita/contribs/lambdadelta/basic_2/rt_computation/cpms_aaa.ma b/matita/matita/contribs/lambdadelta/basic_2/rt_computation/cpms_aaa.ma
index db4eefb22..af4cdde73 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/rt_computation/cpms_aaa.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/rt_computation/cpms_aaa.ma
@@ -30,7 +30,7 @@ lemma cpms_total_aaa (h) (G) (L) (n) (A):
       ∀T. ⦃G,L⦄ ⊢ T ⁝ A → ∃U. ⦃G,L⦄ ⊢ T ➡*[n,h] U.
 #h #G #L #n elim n -n
 [ /2 width=3 by ex_intro/
-| #n #IH #A #T1 #HT1 <plus_SO
+| #n #IH #A #T1 #HT1 <plus_SO_dx
   elim (IH … HT1) -IH #T0 #HT10
   lapply (cpms_aaa_conf … HT1 … HT10) -HT1 #HT0
   elim (aaa_cpm_SO h … HT0) -HT0 #T2 #HT02
diff --git a/matita/matita/contribs/lambdadelta/basic_2/web/basic_2_src.tbl b/matita/matita/contribs/lambdadelta/basic_2/web/basic_2_src.tbl
index 4176396a0..3fe0cbccb 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/web/basic_2_src.tbl
+++ b/matita/matita/contribs/lambdadelta/basic_2/web/basic_2_src.tbl
@@ -12,7 +12,7 @@ table {
    class "wine"
    [ { "iterated dynamic typing" * } {
         [ { "context-sensitive iterated native type assignment" * } {
-             [ [ "for terms" ] "ntas" + "( ⦃?,?⦄ ⊢ ? :*[?,?,?] ? )" "ntas_nta" + "ntas_preserve" * ]
+             [ [ "for terms" ] "ntas" + "( ⦃?,?⦄ ⊢ ? :*[?,?,?] ? )" "ntas_cpcs" + "ntas_nta" + "ntas_nta_ind" + " ntas_ntas" + "ntas_preserve" * ]
           }
         ]
      }
diff --git a/matita/matita/contribs/lambdadelta/ground_2/lib/arith.ma b/matita/matita/contribs/lambdadelta/ground_2/lib/arith.ma
index 6ce878b44..36fa26303 100644
--- a/matita/matita/contribs/lambdadelta/ground_2/lib/arith.ma
+++ b/matita/matita/contribs/lambdadelta/ground_2/lib/arith.ma
@@ -51,7 +51,10 @@ qed.
 
 (* Equalities ***************************************************************)
 
-lemma plus_SO: ∀n. n + 1 = ↑n.
+lemma plus_SO_sn (n): 1 + n = ↑n.
+// qed-.
+
+lemma plus_SO_dx (n): n + 1 = ↑n.
 // qed.
 
 lemma minus_plus_m_m_commutative: ∀n,m:nat. n = m + n - m.
-- 
2.39.2