--- /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/syntax/ac.ma".
+
+(* APPLICABILITY CONDITION PREORDER *****************************************)
+
+definition acle: relation ac ≝
+ λa1,a2. ∀m. ad a1 m → ∃∃n. ad a2 n & m ≤ n.
+
+interpretation "preorder (applicability domain)"
+ 'subseteq a1 a2 = (acle a1 a2).
+
+(* Basic properties *********************************************************)
+
+lemma acle_refl: reflexive … acle.
+/2 width=3 by ex2_intro/ qed.
+
+lemma acle_omega (a): a ⊆ 𝛚.
+/2 width=1 by acle_refl/
+qed.
+
+lemma acle_one (a): ∀n. ad a n → 𝟏 ⊆ a.
+#a #n #Ha #m #Hm destruct
+/2 width=3 by ex2_intro/
+qed.
+
+lemma acle_eq_monotonic_le (k1) (k2):
+ k1 ≤ k2 → (ac_eq k1) ⊆ (ac_eq k2).
+#k1 #k2 #Hk #m #Hm destruct
+/2 width=3 by ex2_intro/
+qed.
+
+lemma acle_le_monotonic_le (k1) (k2):
+ k1 ≤ k2 → (ac_le k1) ⊆ (ac_le k2).
+#k1 #k2 #Hk #m #Hm
+/3 width=3 by acle_refl, transitive_le/
+qed.
+
+lemma acle_eq_le (k): (ac_eq k) ⊆ (ac_le k).
+#k #m #Hm destruct
+/2 width=1 by acle_refl, le_n/
+qed.
+
+lemma acle_le_eq (k): (ac_le k) ⊆ (ac_eq k).
+#k #m #Hm /2 width=3 by ex2_intro/
+qed.