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14
15 include "ground/arith/ynat_pred_succ.ma".
16 include "ground/arith/ynat_plus.ma".
17 include "ground/arith/ynat_lt_pred_succ.ma".
18
19 (* STRICT ORDER FOR NON-NEGATIVE INTEGERS WITH INFINITY *********************)
20
21 (* Inversions with yplus and ypred ******************************************)
22
23 (*** yplus_inv_succ_lt_dx *)
24 lemma eq_inv_succ_yplus_lt_dx (z) (x) (y):  𝟎 < y → ↑z = x + y → z = x + ↓y.
25 #z #x #y #Hy >(ylt_des_gen_dx … Hy) -Hy
26 <yplus_succ_dx <ypred_succ
27 /2 width=1 by eq_inv_ysucc_bi/
28 qed-.
29
30 (*** yplus_inv_succ_lt_sn *)
31 lemma eq_inv_succ_yplus_lt_sn (z) (x) (y): 𝟎 < x → ↑z = x + y → z = ↓x + y.
32 /2 width=1 by eq_inv_succ_yplus_lt_dx/ qed-.
33
34 (* Destructions with yplus and ypred ****************************************)
35
36 (*** yplus_pred1 *)
37 lemma yplus_pred_sn (x) (y): 𝟎 < x → ↓(x+y) = ↓x + y.
38 #x #y #Hx >(ylt_des_gen_dx … Hx) -Hx
39 <yplus_succ_sn <ypred_succ <ypred_succ //
40 qed-.
41
42 (*** yplus_pred2 *)
43 lemma yplus_pred_dx (x) (y): 𝟎 < y → x + ↓y = ↓(x+y).
44 /2 width=1 by yplus_pred_sn/ qed-.
45