--- /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/arith/nat_lt.ma".
+
+(* WELL-FOUNDED INDUCTION ***************************************************)
+
+fact wf3_ind_nlt_aux (A1) (A2) (A3) (f:A1→A2→A3→nat) (Q:relation3 …):
+ (∀n. (∀a1,a2,a3. f a1 a2 a3 < n → Q a1 a2 a3) → ∀a1,a2,a3. f a1 a2 a3 = n → Q a1 a2 a3) →
+ ∀n,a1,a2,a3. f a1 a2 a3 = n → Q a1 a2 a3.
+#A1 #A2 #A3 #f #Q #H #n @(nat_ind_lt … n) -n /3 width=3 by/
+qed-.
+
+(*** f3_ind *)
+lemma wf3_ind_nlt (A1) (A2) (A3) (f:A1→A2→A3→nat) (Q:relation3 …):
+ (∀n. (∀a1,a2,a3. f a1 a2 a3 < n → Q a1 a2 a3) → ∀a1,a2,a3. f a1 a2 a3 = n → Q a1 a2 a3) →
+ ∀a1,a2,a3. Q a1 a2 a3.
+#A1 #A2 #A3 #f #Q #H #a1 #a2 #a3 @(wf3_ind_nlt_aux … H) -H //
+qed-.