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15 include "ground_2/notation/functions/predecessor_1.ma".
16 include "ground_2/lib/arith.ma".
17 include "ground_2/ynat/ynat.ma".
19 (* NATURAL NUMBERS WITH INFINITY ********************************************)
21 (* the predecessor function *)
22 definition ypred: ynat → ynat ≝ λm. match m with
27 interpretation "ynat predecessor" 'Predecessor m = (ypred m).
29 lemma ypred_O: ⫰(yinj 0) = yinj 0.
32 lemma ypred_S: ∀m:nat. ⫰(⫯m) = yinj m.
35 lemma ypred_Y: (⫰∞) = ∞.
38 (* Inversion lemmas *********************************************************)
40 lemma ypred_inv_refl: ∀m:ynat. ⫰m = m → m = 0 ∨ m = ∞.
41 * // #m #H lapply (yinj_inj … H) -H (**) (* destruct lemma needed *)
42 /4 width=1 by pred_inv_refl, or_introl, eq_f/