--- /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 "ground_2/lib/arith.ma".
+include "static_2/notation/functions/one_0.ma".
+include "static_2/notation/functions/two_0.ma".
+include "static_2/notation/functions/omega_0.ma".
+
+(* APPLICABILITY CONDITION **************************************************)
+
+(* applicability condition specification *)
+record ac: Type[0] ≝ {
+(* applicability domain *)
+ ad: predicate nat
+}.
+
+(* applicability condition postulates *)
+record ac_props (a): Prop ≝ {
+ ac_dec: ∀m. Decidable (∃∃n. ad a n & m ≤ n)
+}.
+
+(* Notable specifications ***************************************************)
+
+definition apply_top: predicate nat ≝ λn. ⊤.
+
+definition ac_top: ac ≝ mk_ac apply_top.
+
+interpretation "any number (applicability domain)"
+ 'Omega = (ac_top).
+
+lemma ac_top_props: ac_props ac_top ≝ mk_ac_props ….
+/3 width=3 by or_introl, ex2_intro/
+qed.
+
+definition ac_eq (k): ac ≝ mk_ac (eq … k).
+
+interpretation "one (applicability domain)"
+ 'Two = (ac_eq (S O)).
+
+interpretation "zero (applicability domain)"
+ 'One = (ac_eq O).
+
+lemma ac_eq_props (k): ac_props (ac_eq k) ≝ mk_ac_props ….
+#m elim (le_dec m k) #Hm
+[ /3 width=3 by or_introl, ex2_intro/
+| @or_intror * #n #Hn #Hmn destruct /2 width=1 by/
+]
+qed.
+
+definition ac_le (k): ac ≝ mk_ac (λn. n ≤ k).
+
+lemma ac_le_props (k): ac_props (ac_le k) ≝ mk_ac_props ….
+#m elim (le_dec m k) #Hm
+[ /3 width=3 by or_introl, ex2_intro/
+| @or_intror * #n #Hn #Hmn
+ /3 width=3 by transitive_le/
+]
+qed.