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4 (* ||A|| A project by Andrea Asperti *)
6 (* ||I|| Developers: *)
7 (* ||T|| The HELM team. *)
8 (* ||A|| http://helm.cs.unibo.it *)
10 (* \ / This file is distributed under the terms of the *)
11 (* v GNU General Public License Version 2 *)
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15 include "ground/arith/nat_succ.ma".
16 include "ground/arith/nat_pred.ma".
18 (* PREDECESSOR FOR NON-NEGATIVE INTEGERS ************************************)
20 (* Constructions with npsucc ************************************************)
22 lemma pnpred_succ (n): n = pnpred (npsucc n).
26 lemma npsucc_pred (p): p = npsucc (pnpred p).
30 (* Constructions with nsucc and psucc ***************************************)
32 lemma pnpred_psucc (p): pnpred (psucc p) = nsucc (pnpred p).
35 (* Constructions with nsucc *************************************************)
37 lemma nsucc_pnpred (p):
41 (*** pred_Sn pred_S *)
42 lemma npred_succ (n): n = ↓↑n.
46 (* Basic inversions *********************************************************)
48 lemma eq_inv_pnpred_bi:
51 >(npsucc_pred p1) >(npsucc_pred p2)
55 (* Inversions with nsucc ****************************************************)
58 lemma nat_split_zero_pos (n): ∨∨ 𝟎 = n | n = ↑↓n.
59 #n @(nat_ind_succ … n) -n
60 /2 width=1 by or_introl, or_intror/