update in static_2 and basic_2

[helm.git] / matita / matita / contribs / lambdadelta / basic_2 / i_dynamic / ntas_preserve.ma

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
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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/rt_equivalence/cpcs_cprs.ma".
+include "basic_2/dynamic/cnv_preserve.ma".
+include "basic_2/i_dynamic/ntas.ma".
+
+(* ITERATED NATIVE TYPE ASSIGNMENT FOR TERMS ********************************)
+
+(* Properties based on preservation *****************************************)
+
+lemma cnv_cpms_ntas (h) (a) (G) (L):
+      ∀T. ⦃G,L⦄ ⊢ T ![h,a] → ∀n,U.⦃G,L⦄ ⊢ T ➡*[n,h] U → ⦃G,L⦄ ⊢ T :*[h,a,n] U.
+/3 width=4 by ntas_intro, cnv_cpms_trans/ qed.
+
+(* Inversion lemmas based on preservation ***********************************)
+
+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]
+       | ∃∃n,p,W,U,U0. m ≤ n & ad a n & ⦃G,L⦄ ⊢ V :*[h,a,1] W & ⦃G,L⦄ ⊢ T :*[h,a,m] U & ⦃G,L⦄ ⊢ U :*[h,a,n-m] ⓛ{p}W.U0 & ⦃G,L⦄ ⊢ ⓐV.U ⬌*[h] X & ⦃G,L⦄ ⊢ X ![h,a].
+#h #a #m #G #L #V #T #X
+* #X0 #HX #HVT #HX0 #HTX0
+elim (cnv_inv_appl … HVT) #n #p #W #U0 #Ha #HV #HT #HVW #HTU0
+elim (le_or_ge n m) #Hnm
+[ elim (cnv_fwd_cpms_abst_dx_le … HT … HTU0 … Hnm) #U #H #HU0
+  lapply (cpms_appl_dx … V V … H) [ // ] -H #H
+  elim (cnv_cpms_conf … HVT … HTX0 … H) -HVT -HTX0 -H <minus_n_n #X1 #HX01 #HUX1
+  lapply (cpms_trans … HX0 … HX01) -X0 #HX1
+  lapply (cprs_div … HUX1 … HX1) -X1 #HUX
+  lapply (cnv_cpms_trans … HT … HTU0) #H
+  elim (cnv_inv_bind … H) -H #_ #HU0
+  /4 width=11 by cnv_cpms_ntas, ex7_5_intro, or_introl/
+| >(plus_minus_m_m_commutative … Hnm) in HTU0; #H
+  elim (cpms_inv_plus … H) -H #U #HTU #HU0
+  lapply (cpms_appl_dx … V V … HTU) [ // ] #H
+  elim (cnv_cpms_conf … HVT … HTX0 … H) -HVT -HTX0 -H <minus_n_n #X1 #HX01 #HUX1
+  lapply (cpms_trans … HX0 … HX01) -X0 #HX1
+  lapply (cprs_div … HUX1 … HX1) -X1 #HUX
+  /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/
+*)
+*)