long awaited update in basic_2

[helm.git] / matita / matita / contribs / lambdadelta / ground_2 / ynat / ynat_succ.ma

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 a8d1946bea235170f6405cdcc962f660c1e789ae..f08e632937ee67dd06dfb6bac7392125d1ce9389 100644 (file)
--- a/matita/matita/contribs/lambdadelta/ground_2/ynat/ynat_succ.ma
+++ b/matita/matita/contribs/lambdadelta/ground_2/ynat/ynat_succ.ma
@@ -12,25 +12,30 @@
 (*                                                                        *)
 (**************************************************************************)
 
-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. 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 *)
@@ -45,12 +50,12 @@ 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-.
@@ -80,5 +85,13 @@ lemma ysucc_inv_O_sn: ∀m. yinj 0 = ⫯m → ⊥. (**) (* explicit coercion *)
 #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-.