matita/matita/contribs/lambdadelta/ground/wf_ind/wf1_ind_nlt.ma

   1 (**************************************************************************)
   2 (*       ___                                                              *)
   3 (*      ||M||                                                             *)
   4 (*      ||A||       A project by Andrea Asperti                           *)
   5 (*      ||T||                                                             *)
   6 (*      ||I||       Developers:                                           *)
   7 (*      ||T||         The HELM team.                                      *)
   8 (*      ||A||         http://helm.cs.unibo.it                             *)
   9 (*      \   /                                                             *)
  10 (*       \ /        This file is distributed under the terms of the       *)
  11 (*        v         GNU General Public License Version 2                  *)
  12 (*                                                                        *)
  13 (**************************************************************************)
  14
  15 include "ground/arith/nat_lt.ma".
  16
  17 (* WELL-FOUNDED INDUCTION ***************************************************)
  18
  19 fact wf1_ind_nlt_aux (A1) (f:A1→nat) (Q:predicate …):
  20      (∀n. (∀a1. f a1 < n → Q a1) → ∀a1. f a1 = n → Q a1) →
  21      ∀n,a1. f a1 = n → Q a1.
  22 #A1 #f #Q #H #n @(nat_ind_lt … n) -n /3 width=3 by/
  23 qed-.
  24
  25 (*** f_ind *)
  26 lemma wf1_ind_nlt (A1) (f:A1→nat) (Q:predicate …):
  27       (∀n. (∀a1. f a1 < n → Q a1) → ∀a1. f a1 = n → Q a1) →
  28       ∀a1. Q a1.
  29 #A #f #Q #H #a1 @(wf1_ind_nlt_aux … H) -H //
  30 qed-.