matita/matita/contribs/lambdadelta/ground/wf_ind/wf3_ind_nlt.ma

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  14
  15 include "ground/arith/nat_lt.ma".
  16
  17 (* WELL-FOUNDED INDUCTION ***************************************************)
  18
  19 fact wf3_ind_nlt_aux (A1) (A2) (A3) (f:A1→A2→A3→nat) (Q:relation3 …):
  20      (∀n. (∀a1,a2,a3. f a1 a2 a3 < n → Q a1 a2 a3) → ∀a1,a2,a3. f a1 a2 a3 = n → Q a1 a2 a3) →
  21      ∀n,a1,a2,a3. f a1 a2 a3 = n → Q a1 a2 a3.
  22 #A1 #A2 #A3 #f #Q #H #n @(nat_ind_lt … n) -n /3 width=3 by/
  23 qed-.
  24
  25 (*** f3_ind *)
  26 lemma wf3_ind_nlt (A1) (A2) (A3) (f:A1→A2→A3→nat) (Q:relation3 …):
  27       (∀n. (∀a1,a2,a3. f a1 a2 a3 < n → Q a1 a2 a3) → ∀a1,a2,a3. f a1 a2 a3 = n → Q a1 a2 a3) →
  28       ∀a1,a2,a3. Q a1 a2 a3.
  29 #A1 #A2 #A3 #f #Q #H #a1 #a2 #a3 @(wf3_ind_nlt_aux … H) -H //
  30 qed-.