--- /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/static/sta_lift.ma".
+include "basic_2/unfold/lstas.ma".
+
+(* NAT-ITERATED STATIC TYPE ASSIGNMENT FOR TERMS ****************************)
+
+(* Properties on relocation *************************************************)
+
+lemma lstas_lift: ∀h,G,l. l_liftable (llstar … (sta h G) l).
+/3 width=10 by l_liftable_llstar, sta_lift/ qed.
+
+(* Inversion lemmas on relocation *******************************************)
+
+lemma lstas_inv_lift1: ∀h,G,l. l_deliftable_sn (llstar … (sta h G) l).
+/3 width=6 by l_deliftable_sn_llstar, sta_inv_lift1/ qed-.
+
+(* Advanced inversion lemmas ************************************************)
+
+lemma lstas_inv_lref1: ∀h,G,L,U,i,l. ⦃G, L⦄ ⊢ #i •*[h, l+1] U →
+ (∃∃K,V,W. ⇩[i] L ≡ K.ⓓV & ⦃G, K⦄ ⊢ V •*[h, l+1] W &
+ ⇧[0, i+1] W ≡ U
+ ) ∨
+ (∃∃K,W,V,V0. ⇩[i] L ≡ K.ⓛW & ⦃G, K⦄ ⊢ W •[h] V0 &
+ ⦃G, K⦄ ⊢ W •*[h, l] V & ⇧[0, i+1] V ≡ U
+ ).
+#h #G #L #U #i #l #H elim (lstas_inv_step_sn … H) -H
+#X #H #HXU elim (sta_inv_lref1 … H) -H
+* #K #V #W #HLK #HVW #HWX
+lapply (drop_fwd_drop2 … HLK) #H0LK
+elim (lstas_inv_lift1 … HXU … H0LK … HWX) -H0LK -X
+/4 width=8 by lstas_step_sn, ex4_4_intro, ex3_3_intro, or_introl, or_intror/
+qed-.
+
+(* Advanced forward lemmas **************************************************)
+
+lemma lstas_fwd_correct: ∀h,G,L,T1,U1. ⦃G, L⦄ ⊢ T1 •[h] U1 →
+ ∀T2,l. ⦃G, L⦄ ⊢ T1 •*[h, l] T2 →
+ ∃U2. ⦃G, L⦄ ⊢ T2 •[h] U2.
+#h #G #L #T1 #U1 #HTU1 #T2 #l #H @(lstas_ind_dx … H) -l -T2 /2 width=3 by ex_intro/ -HTU1
+#l #T #T2 #_ #HT2 #_ -T1 -U1 -l
+elim (sta_fwd_correct … HT2) -T /2 width=2 by ex_intro/
+qed-.
+
+(* Advanced properties ******************************************************)
+
+lemma lstas_total: ∀h,G,L,T,U. ⦃G, L⦄ ⊢ T •[h] U →
+ ∀l. ∃U0. ⦃G, L⦄ ⊢ T •*[h, l] U0.
+#h #G #L #T #U #HTU #l @(nat_ind_plus … l) -l /2 width=2 by ex_intro/
+#l * #U0 #HTU0 elim (lstas_fwd_correct … HTU … HTU0) -U
+/3 width=4 by lstas_step_dx, ex_intro/
+qed-.
+
+lemma lstas_ldef: ∀h,G,L,K,V,i. ⇩[i] L ≡ K.ⓓV →
+ ∀W,l. ⦃G, K⦄ ⊢ V •*[h, l+1] W →
+ ∀U. ⇧[0, i+1] W ≡ U → ⦃G, L⦄ ⊢ #i •*[h, l+1] U.
+#h #G #L #K #V #i #HLK #W #l #HVW #U #HWU
+lapply (drop_fwd_drop2 … HLK)
+elim (lstas_inv_step_sn … HVW) -HVW #W0
+elim (lift_total W0 0 (i+1)) /3 width=12 by lstas_step_sn, sta_ldef, lstas_lift/
+qed.
+
+lemma lstas_ldec: ∀h,G,L,K,W,i. ⇩[i] L ≡ K.ⓛW → ∀V0. ⦃G, K⦄ ⊢ W •[h] V0 →
+ ∀V,l. ⦃G, K⦄ ⊢ W •*[h, l] V →
+ ∀U. ⇧[0, i+1] V ≡ U → ⦃G, L⦄ ⊢ #i •*[h, l+1] U.
+#h #G #L #K #W #i #HLK #V0 #HWV0 #V #l #HWV #U #HVU
+lapply (drop_fwd_drop2 … HLK) #H
+elim (lift_total W 0 (i+1)) /3 width=12 by lstas_step_sn, sta_ldec, lstas_lift/
+qed.