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15 include "ground/arith/nat_lt.ma".
17 (* WELL-FOUNDED INDUCTION ***************************************************)
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/
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 //