--- /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/computation/gcp_aaa.ma".
+include "basic_2/computation/cpxs_aaa.ma".
+include "basic_2/computation/csx_tsts_vector.ma".
+
+(* CONTEXT-SENSITIVE EXTENDED STRONGLY NORMALIZING TERMS ********************)
+
+(* Main properties on atomic arity assignment *******************************)
+
+theorem aaa_csx: ∀h,o,G,L,T,A. ⦃G, L⦄ ⊢ T ⁝ A → ⦃G, L⦄ ⊢ ⬊*[h, o] T.
+#h #o #G #L #T #A #H
+@(gcr_aaa … (csx_gcp h o) (csx_gcr h o) … H)
+qed.
+
+(* Advanced eliminators *****************************************************)
+
+fact aaa_ind_csx_aux: ∀h,o,G,L,A. ∀R:predicate term.
+ (∀T1. ⦃G, L⦄ ⊢ T1 ⁝ A →
+ (∀T2. ⦃G, L⦄ ⊢ T1 ➡[h, o] T2 → (T1 = T2 → ⊥) → R T2) → R T1
+ ) →
+ ∀T. ⦃G, L⦄ ⊢ ⬊*[h, o] T → ⦃G, L⦄ ⊢ T ⁝ A → R T.
+#h #o #G #L #A #R #IH #T #H @(csx_ind … H) -T /4 width=5 by cpx_aaa_conf/
+qed-.
+
+lemma aaa_ind_csx: ∀h,o,G,L,A. ∀R:predicate term.
+ (∀T1. ⦃G, L⦄ ⊢ T1 ⁝ A →
+ (∀T2. ⦃G, L⦄ ⊢ T1 ➡[h, o] T2 → (T1 = T2 → ⊥) → R T2) → R T1
+ ) →
+ ∀T. ⦃G, L⦄ ⊢ T ⁝ A → R T.
+/5 width=9 by aaa_ind_csx_aux, aaa_csx/ qed-.
+
+fact aaa_ind_csx_alt_aux: ∀h,o,G,L,A. ∀R:predicate term.
+ (∀T1. ⦃G, L⦄ ⊢ T1 ⁝ A →
+ (∀T2. ⦃G, L⦄ ⊢ T1 ➡*[h, o] T2 → (T1 = T2 → ⊥) → R T2) → R T1
+ ) →
+ ∀T. ⦃G, L⦄ ⊢ ⬊*[h, o] T → ⦃G, L⦄ ⊢ T ⁝ A → R T.
+#h #o #G #L #A #R #IH #T #H @(csx_ind_alt … H) -T /4 width=5 by cpxs_aaa_conf/
+qed-.
+
+lemma aaa_ind_csx_alt: ∀h,o,G,L,A. ∀R:predicate term.
+ (∀T1. ⦃G, L⦄ ⊢ T1 ⁝ A →
+ (∀T2. ⦃G, L⦄ ⊢ T1 ➡*[h, o] T2 → (T1 = T2 → ⊥) → R T2) → R T1
+ ) →
+ ∀T. ⦃G, L⦄ ⊢ T ⁝ A → R T.
+/5 width=9 by aaa_ind_csx_alt_aux, aaa_csx/ qed-.