(* *)
(**************************************************************************)
-include "ground_2/notation/functions/successor_1.ma".
include "ground_2/ynat/ynat_pred.ma".
(* NATURAL NUMBERS WITH INFINITY ********************************************)
(* 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. ⫯(yinj m) = yinj (⫯m).
+// qed.
+
+lemma ysucc_Y: ⫯(∞) = ∞.
+// qed.
+
(* Properties ***************************************************************)
lemma ypred_succ: ∀m. ⫰⫯m = m.
* // qed.
+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 *)
+| @or_intror @(ex_intro … (∞)) // (**) (* explicit constructor *)
+]
+qed-.
+
+lemma ysucc_iter_Y: ∀m. ysucc^m (∞) = ∞.
+#m elim m -m //
+#m #IHm whd in ⊢ (??%?); >IHm //
+qed.
+
(* Inversion lemmas *********************************************************)
-lemma ysucc_inj: ∀m,n. ⫯m = ⫯n → m = n.
+lemma ysucc_inv_inj: ∀m,n. ⫯m = ⫯n → m = n.
#m #n #H <(ypred_succ m) <(ypred_succ n) //
qed-.
lemma ysucc_inv_refl: ∀m. ⫯m = m → m = ∞.
-* // normalize
+* //
#m #H lapply (yinj_inj … H) -H (**) (* destruct lemma needed *)
#H elim (lt_refl_false m) //
qed-.
lemma ysucc_inv_Y_dx: ∀m. ⫯m = ∞ → m = ∞.
/2 width=1 by ysucc_inv_Y_sn/ qed-.
+
+lemma ysucc_inv_O_sn: ∀m. yinj 0 = ⫯m → ⊥. (**) (* explicit coercion *)
+#m #H elim (ysucc_inv_inj_sn … H) -H
+#n #_ #H destruct
+qed-.
+
+lemma ysucc_inv_O_dx: ∀m:ynat. ⫯m = 0 → ⊥.
+/2 width=2 by ysucc_inv_O_sn/ qed-.
+
+(* Eliminators **************************************************************)
+
+lemma ynat_ind: ∀R:predicate ynat.
+ R 0 → (∀n:nat. R n → R (⫯n)) → R (∞) →
+ ∀x. R x.
+#R #H1 #H2 #H3 * // #n elim n -n /2 width=1 by/
+qed-.