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14
15 include "ground/arith/nat_pred_succ.ma".
16 include "ground/arith/ynat_succ.ma".
17 include "ground/arith/ynat_pred.ma".
18
19 (* PREDECESSOR FOR NON-NEGATIVE INTEGERS WITH INFINITY **********************)
20
21 (* Constructions with ysucc *************************************************)
22
23 (*** ypred_succ ypred_S *)
24 lemma ypred_succ (x): x = ↓↑x.
25 #x @(ynat_split_nat_inf … x) -x //
26 qed.
27
28 (* Inversions with ysucc ****************************************************)
29
30 (*** ynat_cases *)
31 lemma ynat_split_zero_pos (x): ∨∨ 𝟎 = x | x = ↑↓x.
32 #x @(ynat_split_nat_inf … x) -x //
33 #n elim (nat_split_zero_pos n)
34 /2 width=1 by or_introl, or_intror/
35 qed-.