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

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  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-.