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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-.