X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=matita%2Fmatita%2Fcontribs%2Flambdadelta%2Fground_2%2Fynat%2Fynat_le.ma;h=ccb9563ee875b23e587956f0b0032602c259e954;hb=d1b944b638846d98dfeb21fa6757e89c609be82a;hp=e706fa4dd6c4c933019b39f8bb89b17114b2527c;hpb=bec531b57a008238f67cd72edc751844d28b374f;p=helm.git

diff --git a/matita/matita/contribs/lambdadelta/ground_2/ynat/ynat_le.ma b/matita/matita/contribs/lambdadelta/ground_2/ynat/ynat_le.ma
index e706fa4dd..ccb9563ee 100644
--- a/matita/matita/contribs/lambdadelta/ground_2/ynat/ynat_le.ma
+++ b/matita/matita/contribs/lambdadelta/ground_2/ynat/ynat_le.ma
@@ -26,15 +26,21 @@ interpretation "ynat 'less or equal to'" 'leq x y = (yle x y).
 
 (* Basic inversion lemmas ***************************************************)
 
-fact yle_inv_inj_aux: ∀x,y. x ≤ y → ∀m,n. x = yinj m → y = yinj n → m ≤ n.
+fact yle_inv_inj2_aux: ∀x,y. x ≤ y → ∀n. y = yinj n →
+                       ∃∃m. m ≤ n & x = yinj m.
 #x #y * -x -y
-[ #x #y #Hxy #m #n #Hx #Hy destruct //
-| #x #m #n #_ #Hy destruct
+[ #x #y #Hxy #n #Hy destruct /2 width=3 by ex2_intro/
+| #x #n #Hy destruct
 ]
 qed-.
 
+lemma yle_inv_inj2: ∀x,n. x ≤ yinj n → ∃∃m. m ≤ n & x = yinj m.
+/2 width=3 by yle_inv_inj2_aux/ qed-.
+
 lemma yle_inv_inj: ∀m,n. yinj m ≤ yinj n → m ≤ n.
-/2 width=5 by yle_inv_inj_aux/ qed-.
+#m #n #H elim (yle_inv_inj2 … H) -H
+#x #Hxn #H destruct //
+qed-.
 
 fact yle_inv_O2_aux: ∀m:ynat. ∀x:ynat. m ≤ x → x = 0 → m = 0.
 #m #x * -m -x
@@ -63,7 +69,7 @@ fact yle_inv_succ1_aux: ∀x,y. x ≤ y → ∀m. x = ⫯m → ∃∃n. m ≤ n
   #m #Hnm #H destruct
   @(ex2_intro … m) /2 width=1 by yle_inj/ (**) (* explicit constructor *)
 | #x #y #H destruct
-  @(ex2_intro … (∞)) /2 width=1 by yle_Y/
+  @(ex2_intro … (∞)) /2 width=1 by yle_Y/ (**) (* explicit constructor *)
 ]
 qed-.
 
@@ -75,8 +81,19 @@ lemma yle_inv_succ: ∀m,n. ⫯m ≤ ⫯n → m ≤ n.
 #x #Hx #H destruct //
 qed-.
 
+(* Forward lemmas on successor **********************************************)
+
+lemma yle_fwd_succ1: ∀m,n. ⫯m ≤ n → m ≤ ⫰n.
+#m #x #H elim (yle_inv_succ1 … H) -H
+#n #Hmn #H destruct //
+qed-.
+
 (* Basic properties *********************************************************)
 
+lemma le_O1: ∀n:ynat. 0 ≤ n.
+* /2 width=1 by yle_inj/
+qed.
+
 lemma yle_refl: reflexive … yle.
 * /2 width=1 by le_n, yle_inj/
 qed.
@@ -92,6 +109,10 @@ lemma yle_refl_pred_sn: ∀x. ⫰x ≤ x.
 
 (* Properties on successor **************************************************)
 
+lemma yle_succ: ∀m,n. m ≤ n → ⫯m ≤ ⫯n.
+#m #n * -m -n /3 width=1 by yle_inj, le_S_S/
+qed.
+
 lemma yle_succ_dx: ∀m,n. m ≤ n → m ≤ ⫯n.
 #m #n * -m -n /3 width=1 by le_S, yle_inj/
 qed.
@@ -106,6 +127,6 @@ theorem yle_trans: Transitive … yle.
 [ #x #y #Hxy * //
   #z #H lapply (yle_inv_inj … H) -H
   /3 width=3 by transitive_le, yle_inj/ (**) (* full auto too slow *)
-| #x #z #H lapply ( yle_inv_Y1 … H) //
+| #x #z #H lapply (yle_inv_Y1 … H) //
 ]
 qed-.