X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=matita%2Fmatita%2Fcontribs%2Flambdadelta%2Fground_2%2Fynat%2Fynat_succ.ma;h=f08e632937ee67dd06dfb6bac7392125d1ce9389;hb=c6305166703a17801bbd08a85fe93ef4abf8ff85;hp=707556f301211c4015cb0c7faee3adcfa00569cc;hpb=23da2aa16489e00889374d81f19cc090faa44582;p=helm.git

diff --git a/matita/matita/contribs/lambdadelta/ground_2/ynat/ynat_succ.ma b/matita/matita/contribs/lambdadelta/ground_2/ynat/ynat_succ.ma
index 707556f30..f08e63293 100644
--- a/matita/matita/contribs/lambdadelta/ground_2/ynat/ynat_succ.ma
+++ b/matita/matita/contribs/lambdadelta/ground_2/ynat/ynat_succ.ma
@@ -12,32 +12,50 @@
 (*                                                                        *)
 (**************************************************************************)
 
-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-.
@@ -61,3 +79,19 @@ 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-.