X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=matita%2Fmatita%2Fcontribs%2Flambdadelta%2Fground_2%2Fynat%2Fynat_succ.ma;h=d0c7b3fdae48fff52849784dde0c35963d1b07f7;hb=1fd63df4c77f5c24024769432ea8492748b4ac79;hp=a8d1946bea235170f6405cdcc962f660c1e789ae;hpb=b3f8b89278d193ed0aa0f39e7f8d74cf1de81d8d;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 a8d1946be..d0c7b3fda 100644
--- 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).
+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: ∀m. ⫰⫯m = m.
+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 *)
@@ -45,17 +50,17 @@ 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
+lemma ysucc_inv_refl: ∀m. ↑m = m → m = ∞.
+* //
 #m #H lapply (yinj_inj … H) -H (**) (* destruct lemma needed *)
 #H elim (lt_refl_false m) //
 qed-.
 
-lemma ysucc_inv_inj_sn: ∀m2,n1. yinj m2 = ⫯n1 →
+lemma ysucc_inv_inj_sn: ∀m2,n1. yinj m2 = ↑n1 →
                         ∃∃m1. n1 = yinj m1 & m2 = S m1.
 #m2 * normalize
 [ #n1 #H destruct /2 width=3 by ex2_intro/
@@ -63,22 +68,30 @@ lemma ysucc_inv_inj_sn: ∀m2,n1. yinj m2 = ⫯n1 →
 ]
 qed-.
 
-lemma ysucc_inv_inj_dx: ∀m2,n1. ⫯n1 = yinj m2  →
+lemma ysucc_inv_inj_dx: ∀m2,n1. ↑n1 = yinj m2  →
                         ∃∃m1. n1 = yinj m1 & m2 = S m1.
 /2 width=1 by ysucc_inv_inj_sn/ qed-.
 
-lemma ysucc_inv_Y_sn: ∀m. ∞ = ⫯m → m = ∞.
+lemma ysucc_inv_Y_sn: ∀m. ∞ = ↑m → m = ∞.
 * // normalize
 #m #H destruct
 qed-.
 
-lemma ysucc_inv_Y_dx: ∀m. ⫯m = ∞ → m = ∞.
+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 *)
+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. ⫯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-.