arithmetics for λδ

[helm.git] / matita / matita / contribs / lambdadelta / ground / wf_ind / wf2_ind_nlt.ma

diff --git a/matita/matita/contribs/lambdadelta/ground/wf_ind/wf2_ind_nlt.ma b/matita/matita/contribs/lambdadelta/ground/wf_ind/wf2_ind_nlt.ma
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+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||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 wf2_ind_nlt_aux (A1) (A2) (f:A1→A2→nat) (Q:relation2 …):
+     (∀n. (∀a1,a2. f a1 a2 < n → Q a1 a2) → ∀a1,a2. f a1 a2 = n → Q a1 a2) →
+     ∀n,a1,a2. f a1 a2 = n → Q a1 a2.
+#A1 #A2 #f #Q #H #n @(nat_ind_lt … n) -n /3 width=3 by/
+qed-.
+
+(*** f2_ind *)
+lemma wf2_ind_nlt (A1) (A2) (f:A1→A2→nat) (Q:relation2 …):
+      (∀n. (∀a1,a2. f a1 a2 < n → Q a1 a2) → ∀a1,a2. f a1 a2 = n → Q a1 a2) →
+      ∀a1,a2. Q a1 a2.
+#A1 #A2 #f #Q #H #a1 #a2 @(wf2_ind_nlt_aux … H) -H //
+qed-.