(* the successor function *)
definition ysucc: ynat → ynat ≝ λm. match m with
-[ yinj m ⇒ S m
+[ yinj m ⇒ ⫯m
| Y ⇒ Y
].
interpretation "ynat successor" 'Successor m = (ysucc m).
-lemma ysucc_inj: ∀m:nat. ⫯m = S m.
+lemma ysucc_inj: ∀m:nat. ⫯(yinj m) = yinj (⫯m).
// qed.
lemma ysucc_Y: ⫯(∞) = ∞.
lemma ypred_succ: ∀m. ⫰⫯m = m.
* // qed.
-lemma ynat_cases: ∀n:ynat. n = 0 ∨ ∃m. n = ⫯m.
+lemma ynat_cases: ∀n:ynat. n = 0 ∨ ∃m:ynat. n = ⫯m.
*
[ * /2 width=1 by or_introl/
#n @or_intror @(ex_intro … n) // (**) (* explicit constructor *)
#n #_ #H destruct
qed-.
-lemma ysucc_inv_O_dx: ∀m. ⫯m = 0 → ⊥.
+lemma ysucc_inv_O_dx: ∀m:ynat. ⫯m = 0 → ⊥.
/2 width=2 by ysucc_inv_O_sn/ qed-.
(* Eliminators **************************************************************)