-(* Forward lemmas on stratified static type assignment for terms ************)
+(**************************************************************************)
+(* ___ *)
+(* ||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 *)
+(* *)
+(**************************************************************************)
-lemma aaa_fwd_ssta: ∀h,g,G,L,T,A. ⦃G, L⦄ ⊢ T ⁝ A → ∃U. ⦃G, L⦄ ⊢ T •[h, g] U.
+include "basic_2/static/sta.ma".
+include "basic_2/static/aaa_lift.ma".
+
+(* STATIC TYPE ASSIGNMENT FOR TERMS *****************************************)
+
+(* Properties on atomic arity assignment for terms **************************)
+
+lemma aaa_sta: ∀h,G,L,T,A. ⦃G, L⦄ ⊢ T ⁝ A → ∃U. ⦃G, L⦄ ⊢ T •[h] U.
#h #G #L #T #A #H elim H -G -L -T -A
-[ /2 width=2/
+[ /2 width=2 by sta_sort, ex_intro/
| * #G #L #K [ #V | #W ] #B #i #HLK #_ * [ #W | #V ] #HVW
- elim (lift_total W 0 (i+1)) /3 width=7/
-| #a #G #L #V #T #B #A #_ #_ #_ * /3 width=2/
-| #a #G #L #V #T #B #A #_ #_ #_ * /3 width=2/
-| #G #L #V #T #B #A #_ #_ #_ * /3 width=2/
-| #G #L #W #T #A #_ #_ #_ * /3 width=2/
+ elim (lift_total W 0 (i+1)) /3 width=7 by sta_ldef, sta_ldec, ex_intro/
+| #a #G #L #V #T #B #A #_ #_ #_ * /3 width=2 by sta_bind, ex_intro/
+| #a #G #L #V #T #B #A #_ #_ #_ * /3 width=2 by sta_bind, ex_intro/
+| #G #L #V #T #B #A #_ #_ #_ * /3 width=2 by sta_appl, ex_intro/
+| #G #L #W #T #A #_ #_ #_ * /3 width=2 by sta_cast, ex_intro/
+]
+qed-.
+
+lemma sta_aaa_conf: ∀h,G,L. Conf3 … (aaa G L) (sta h G L).
+#h #G #L #T #A #H elim H -G -L -T -A
+[ #G #L #k #U #H
+ lapply (sta_inv_sort1 … H) -H #H destruct //
+| #I #G #L #K #V #B #i #HLK #HV #IHV #U #H
+ elim (sta_inv_lref1 … H) -H * #K0 #V0 #W0 #HLK0 #HVW0 #HU
+ lapply (drop_mono … HLK0 … HLK) -HLK0 #H0 destruct
+ lapply (drop_fwd_drop2 … HLK) -HLK #HLK
+ @(aaa_lift … HLK … HU) -HU -L /2 width=2 by/
+| #a #G #L #V #T #B #A #HV #_ #_ #IHT #X #H
+ elim (sta_inv_bind1 … H) -H #U #HTU #H destruct /3 width=2 by aaa_abbr/
+| #a #G #L #V #T #B #A #HV #_ #_ #IHT #X #H
+ elim (sta_inv_bind1 … H) -H #U #HTU #H destruct /3 width=2 by aaa_abst/
+| #G #L #V #T #B #A #HV #_ #_ #IHT #X #H
+ elim (sta_inv_appl1 … H) -H #U #HTU #H destruct /3 width=3 by aaa_appl/
+| #G #L #V #T #A #_ #_ #IHV #IHT #X #H
+ lapply (sta_inv_cast1 … H) -H /2 width=2 by/
]
qed-.