--- /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 "static_2/s_computation/fqus_fqup.ma".
+include "static_2/static/aaa_drops.ma".
+
+(* ATONIC ARITY ASSIGNMENT ON TERMS *****************************************)
+
+(* Properties on supclosure *************************************************)
+
+lemma aaa_fqu_conf: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐ ⦃G2, L2, T2⦄ →
+ ∀A1. ⦃G1, L1⦄ ⊢ T1 ⁝ A1 → ∃A2. ⦃G2, L2⦄ ⊢ T2 ⁝ A2.
+#G1 #G2 #L1 #L2 #T1 #T2 #H elim H -G1 -G2 -L1 -L2 -T1 -T2
+[ #I #G #L #T #A #H elim (aaa_inv_zero … H) -H
+ #J #K #V #H #HA destruct /2 width=2 by ex_intro/
+| * [ #p ] * #G #L #V #T #X #H
+ [ elim (aaa_inv_abbr … H)
+ | elim (aaa_inv_abst … H)
+ | elim (aaa_inv_appl … H)
+ | elim (aaa_inv_cast … H)
+ ] -H /2 width=2 by ex_intro/
+| #p * #G #L #V #T #X #H
+ [ elim (aaa_inv_abbr … H)
+ | elim (aaa_inv_abst … H)
+ ] -H /2 width=2 by ex_intro/
+| #p #I #G #L #V #T #H destruct
+| * #G #L #V #T #X #H
+ [ elim (aaa_inv_appl … H)
+ | elim (aaa_inv_cast … H)
+ ] -H /2 width=2 by ex_intro/
+| /5 width=8 by aaa_inv_lifts, drops_refl, drops_drop, ex_intro/
+]
+qed-.
+
+lemma aaa_fquq_conf: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐⸮ ⦃G2, L2, T2⦄ →
+ ∀A1. ⦃G1, L1⦄ ⊢ T1 ⁝ A1 → ∃A2. ⦃G2, L2⦄ ⊢ T2 ⁝ A2.
+#G1 #G2 #L1 #L2 #T1 #T2 #H elim H -H /2 width=6 by aaa_fqu_conf/
+* #H1 #H2 #H3 destruct /2 width=2 by ex_intro/
+qed-.
+
+lemma aaa_fqup_conf: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐+ ⦃G2, L2, T2⦄ →
+ ∀A1. ⦃G1, L1⦄ ⊢ T1 ⁝ A1 → ∃A2. ⦃G2, L2⦄ ⊢ T2 ⁝ A2.
+#G1 #G2 #L1 #L2 #T1 #T2 #H @(fqup_ind … H) -G2 -L2 -T2
+[2: #G #G2 #L #L2 #T #T2 #_ #H2 #IH1 #A #HA elim (IH1 … HA) -IH1 -A ]
+/2 width=6 by aaa_fqu_conf/
+qed-.
+
+lemma aaa_fqus_conf: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐* ⦃G2, L2, T2⦄ →
+ ∀A1. ⦃G1, L1⦄ ⊢ T1 ⁝ A1 → ∃A2. ⦃G2, L2⦄ ⊢ T2 ⁝ A2.
+#G1 #G2 #L1 #L2 #T1 #T2 #H elim(fqus_inv_fqup … H) -H /2 width=6 by aaa_fqup_conf/
+* #H1 #H2 #H3 destruct /2 width=2 by ex_intro/
+qed-.