--- /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/pnat_lt.ma".
+
+(* WELL-FOUNDED INDUCTION ***************************************************)
+
+fact wf1_ind_plt_aux (A1) (f:A1→pnat) (Q:predicate …):
+ (∀p. (∀a1. f a1 < p → Q a1) → ∀a1. f a1 = p → Q a1) →
+ ∀p,a1. f a1 = p → Q a1.
+#A1 #f #Q #H #p @(pnat_ind_lt … p) -p /3 width=3 by/
+qed-.
+
+lemma wf1_ind_plt (A1) (f:A1→pnat) (Q:predicate …):
+ (∀p. (∀a1. f a1 < p → Q a1) → ∀a1. f a1 = p → Q a1) →
+ ∀a1. Q a1.
+#A #f #Q #H #a1 @(wf1_ind_plt_aux … H) -H //
+qed-.