(* *)
(**************************************************************************)
-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).
+interpretation "ynat successor" 'UpArrow m = (ysucc m).
+
+lemma ysucc_inj: ∀m:nat. ↑(yinj m) = yinj (↑m).
+// qed.
+
+lemma ysucc_Y: ↑(∞) = ∞.
+// qed.
(* Properties ***************************************************************)
-lemma ypred_succ: â\88\80m. ⫰⫯m = m.
+lemma ypred_succ: â\88\80m. â\86\93â\86\91m = 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 *)
]
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: â\88\80m. ⫯m = m → m = ∞.
-* // normalize
+lemma ysucc_inv_refl: â\88\80m. â\86\91m = m → m = ∞.
+* //
#m #H lapply (yinj_inj … H) -H (**) (* destruct lemma needed *)
#H elim (lt_refl_false m) //
qed-.
-lemma ysucc_inv_inj_sn: â\88\80m2,n1. yinj m2 = ⫯n1 →
+lemma ysucc_inv_inj_sn: â\88\80m2,n1. yinj m2 = â\86\91n1 →
∃∃m1. n1 = yinj m1 & m2 = S m1.
#m2 * normalize
[ #n1 #H destruct /2 width=3 by ex2_intro/
]
qed-.
-lemma ysucc_inv_inj_dx: â\88\80m2,n1. ⫯n1 = yinj m2 →
+lemma ysucc_inv_inj_dx: â\88\80m2,n1. â\86\91n1 = yinj m2 →
∃∃m1. n1 = yinj m1 & m2 = S m1.
/2 width=1 by ysucc_inv_inj_sn/ qed-.
-lemma ysucc_inv_Y_sn: â\88\80m. â\88\9e = ⫯m → m = ∞.
+lemma ysucc_inv_Y_sn: â\88\80m. â\88\9e = â\86\91m → m = ∞.
* // normalize
#m #H destruct
qed-.
-lemma ysucc_inv_Y_dx: â\88\80m. ⫯m = ∞ → m = ∞.
+lemma ysucc_inv_Y_dx: â\88\80m. â\86\91m = ∞ → m = ∞.
/2 width=1 by ysucc_inv_Y_sn/ qed-.
-lemma ysucc_inv_O_sn: â\88\80m. yinj 0 = ⫯m → ⊥. (**) (* explicit coercion *)
+lemma ysucc_inv_O_sn: â\88\80m. yinj 0 = â\86\91m → ⊥. (**) (* explicit coercion *)
#m #H elim (ysucc_inv_inj_sn … H) -H
#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 **************************************************************)
+
+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-.