X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=matita%2Fmatita%2Fcontribs%2Flambdadelta%2Fbasic_2%2Fi_dynamic%2Fntas_nta.ma;h=409f06b81407fdfc6f32741c21f7fe36b12e6ff9;hb=bd53c4e895203eb049e75434f638f26b5a161a2b;hp=7374d2df420897f58a96962a2874aa50d864b45c;hpb=f308429a0fde273605a2330efc63268b4ac36c99;p=helm.git

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 7374d2df4..409f06b81 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
@@ -12,33 +12,112 @@
 (*                                                                        *)
 (**************************************************************************)
 
-include "basic_2/dynamic/nta.ma".
-include "basic_2/i_dynamic/ntas.ma".
+include "basic_2/dynamic/nta_preserve.ma".
+include "basic_2/i_dynamic/ntas_preserve.ma".
 
 (* ITERATED NATIVE TYPE ASSIGNMENT FOR TERMS ********************************)
 
-(*
+(* Advanced properties of native validity for terms *************************)
 
-definition ntas: sh → lenv → relation term ≝
-                 λh,L. star … (nta h L).
+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):
+      ∀T,U. ❪G,L❫ ⊢ T :[h,a] U → ❪G,L❫ ⊢ T :*[h,a,1] U.
+#h #a #G #L #T #U #H
+elim (cnv_inv_cast … H) -H /2 width=3 by ntas_intro/
+qed-.
 
-(* Basic eliminators ********************************************************)
+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.
 
-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) //
+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-.
-*)
-(* 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.
+(* Inversion lemmas with native type assignment for terms *******************)
 
-lemma ntas_strap2: ∀h,L,T1,T,T2.
-                   ⦃h,L⦄  ⊢ T1 : T → ⦃h,L⦄ ⊢ T :* T2 → ⦃h,L⦄ ⊢ T1 :* T2.
-/2 width=3/ qed.
-*)
+lemma ntas_inv_nta (h) (a) (G) (L):
+      ∀T,U. ❪G,L❫ ⊢ T :*[h,a,1] U → ❪G,L❫ ⊢ T :[h,a] U.
+#h #a #G #L #T #U
+* /2 width=3 by cnv_cast/
+qed-.
+
+(* Note: this follows from ntas_inv_appl_sn *)
+lemma nta_inv_appl_sn_ntas (h) (a) (G) (L) (V) (T):
+      ∀X. ❪G,L❫ ⊢ ⓐV.T :[h,a] X →
+      ∨∨ ∃∃p,W,U,U0. 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 & ❪G,L❫ ⊢ ⓐV.ⓛ[p]W.U ⬌*[h] X & ❪G,L❫ ⊢ X ![h,a]
+       | ∃∃n,p,W,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 & ❪G,L❫ ⊢ ⓐV.U ⬌*[h] X & ❪G,L❫ ⊢ X ![h,a].
+#h #a #G #L #V #T #X #H
+(* Note: insert here: alternate proof in ntas_nta.etc *)
+elim (cnv_inv_cast … H) -H #X0 #HX #HVT #HX0 #HTX0
+elim (cnv_inv_appl … HVT) #n #p #W #U0 #Ha #HV #HT #HVW #HTU0
+elim (eq_or_gt n) #Hn destruct
+[ elim (cnv_fwd_cpms_abst_dx_le … HT … HTU0 1) [| // ] <minus_n_O #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=8 by cnv_cpms_ntas, cnv_cpms_nta, ex6_4_intro, or_introl/
+| >(plus_minus_m_m_commutative … Hn) 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
+  <(S_pred … Hn) in Ha; -Hn #Ha
+  /5 width=10 by cnv_cpms_ntas, cnv_cpms_nta, cnv_cpms_trans, ex6_5_intro, or_intror/
+]
+qed-.