X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=matita%2Fmatita%2Fcontribs%2Flambdadelta%2Fground_2%2Fynat%2Fynat_lt.ma;h=ab9cf4565cc365d28ff3d8d94b3a8b2359e91c02;hb=87f57ddc367303c33e19c83cd8989cd561f3185b;hp=664510dda2b8f2e631c69e03288457f6c67dd609;hpb=5102e7f780e83c7fef1d3826f81dfd37ee4028bc;p=helm.git

diff --git a/matita/matita/contribs/lambdadelta/ground_2/ynat/ynat_lt.ma b/matita/matita/contribs/lambdadelta/ground_2/ynat/ynat_lt.ma
index 664510dda..ab9cf4565 100644
--- a/matita/matita/contribs/lambdadelta/ground_2/ynat/ynat_lt.ma
+++ b/matita/matita/contribs/lambdadelta/ground_2/ynat/ynat_lt.ma
@@ -64,14 +64,14 @@ lemma ylt_inv_Y2: ∀x:ynat. x < ∞ → ∃n. x = yinj n.
 #H elim (ylt_inv_Y1 … H)
 qed-.
 
-lemma ylt_inv_O1: ∀n:ynat. 0 < n → ⫯⫰n = n.
+lemma ylt_inv_O1: ∀n:ynat. 0 < n → ↑↓n = n.
 * // #n #H lapply (ylt_inv_inj … H) -H normalize
 /3 width=1 by S_pred, eq_f/
 qed-.
 
 (* Inversion lemmas on successor ********************************************)
 
-fact ylt_inv_succ1_aux: ∀x,y:ynat. x < y → ∀m. x = ⫯m → m < ⫰y ∧ ⫯⫰y = y.
+fact ylt_inv_succ1_aux: ∀x,y:ynat. x < y → ∀m. x = ↑m → m < ↓y ∧ ↑↓y = y.
 #x #y * -x -y
 [ #x #y #Hxy #m #H elim (ysucc_inv_inj_sn … H) -H
   #n #H1 #H2 destruct elim (le_inv_S1 … Hxy) -Hxy
@@ -81,16 +81,16 @@ fact ylt_inv_succ1_aux: ∀x,y:ynat. x < y → ∀m. x = ⫯m → m < ⫰y ∧ 
 ]
 qed-.
 
-lemma ylt_inv_succ1: ∀m,y:ynat. ⫯m < y → m < ⫰y ∧ ⫯⫰y = y.
+lemma ylt_inv_succ1: ∀m,y:ynat. ↑m < y → m < ↓y ∧ ↑↓y = y.
 /2 width=3 by ylt_inv_succ1_aux/ qed-.
 
-lemma ylt_inv_succ: ∀m,n. ⫯m < ⫯n → m < n.
+lemma ylt_inv_succ: ∀m,n. ↑m < ↑n → m < n.
 #m #n #H elim (ylt_inv_succ1 … H) -H //
 qed-.
 
 (* Forward lemmas on successor **********************************************)
 
-fact ylt_fwd_succ2_aux: ∀x,y. x < y → ∀n. y = ⫯n → x ≤ n.
+fact ylt_fwd_succ2_aux: ∀x,y. x < y → ∀n. y = ↑n → x ≤ n.
 #x #y * -x -y
 [ #x #y #Hxy #m #H elim (ysucc_inv_inj_sn … H) -H
   #n #H1 #H2 destruct /3 width=1 by yle_inj, le_S_S_to_le/
@@ -98,16 +98,16 @@ fact ylt_fwd_succ2_aux: ∀x,y. x < y → ∀n. y = ⫯n → x ≤ n.
 ]
 qed-.
 
-lemma ylt_fwd_succ2: ∀m,n. m < ⫯n → m ≤ n.
+lemma ylt_fwd_succ2: ∀m,n. m < ↑n → m ≤ n.
 /2 width=3 by ylt_fwd_succ2_aux/ qed-.
 
 (* inversion and forward lemmas on order ************************************)
 
-lemma ylt_fwd_le_succ1: ∀m,n. m < n → ⫯m ≤ n.
+lemma ylt_fwd_le_succ1: ∀m,n. m < n → ↑m ≤ n.
 #m #n * -m -n /2 width=1 by yle_inj/
 qed-.
 
-lemma ylt_fwd_le_pred2: ∀x,y:ynat. x < y → x ≤ ⫰y.
+lemma ylt_fwd_le_pred2: ∀x,y:ynat. x < y → x ≤ ↓y.
 #x #y #H elim H -x -y /3 width=1 by yle_inj, monotonic_pred/
 qed-.
 
@@ -124,42 +124,42 @@ lemma ylt_yle_false: ∀m:ynat. ∀n:ynat. m < n → n ≤ m → ⊥.
 ]
 qed-.
 
-lemma ylt_inv_le: ∀x,y. x < y → x < ∞ ∧ ⫯x ≤ y.
+lemma ylt_inv_le: ∀x,y. x < y → x < ∞ ∧ ↑x ≤ y.
 #x #y #H elim H -x -y /3 width=1 by yle_inj, conj/
 qed-.
 
 (* Basic properties *********************************************************)
 
-lemma ylt_O1: ∀x:ynat. ⫯⫰x = x → 0 < x.
+lemma ylt_O1: ∀x:ynat. ↑↓x = x → 0 < x.
 * // * /2 width=1 by ylt_inj/ normalize
 #H destruct
 qed.
 
 (* Properties on predecessor ************************************************)
 
-lemma ylt_pred: ∀m,n:ynat. m < n → 0 < m → ⫰m < ⫰n.
+lemma ylt_pred: ∀m,n:ynat. m < n → 0 < m → ↓m < ↓n.
 #m #n * -m -n
 /4 width=1 by ylt_inv_inj, ylt_inj, monotonic_lt_pred/
 qed.
 
 (* Properties on successor **************************************************)
 
-lemma ylt_O_succ: ∀n. 0 < ⫯n.
+lemma ylt_O_succ: ∀n. 0 < ↑n.
 * /2 width=1 by ylt_inj/
 qed.
 
-lemma ylt_succ: ∀m,n. m < n → ⫯m < ⫯n.
+lemma ylt_succ: ∀m,n. m < n → ↑m < ↑n.
 #m #n #H elim H -m -n /3 width=1 by ylt_inj, le_S_S/
 qed.
 
-lemma ylt_succ_Y: ∀x. x < ∞ → ⫯x < ∞.
+lemma ylt_succ_Y: ∀x. x < ∞ → ↑x < ∞.
 * /2 width=1 by/ qed.
 
-lemma yle_succ1_inj: ∀x. ∀y:ynat. ⫯yinj x ≤ y → x < y.
+lemma yle_succ1_inj: ∀x. ∀y:ynat. ↑yinj x ≤ y → x < y.
 #x * /3 width=1 by yle_inv_inj, ylt_inj/
 qed.
 
-lemma ylt_succ2_refl: ∀x,y:ynat. x < y → x < ⫯x.
+lemma ylt_succ2_refl: ∀x,y:ynat. x < y → x < ↑x.
 #x #y #H elim (ylt_fwd_gen … H) -y /2 width=1 by ylt_inj/
 qed.
 
@@ -199,11 +199,15 @@ lemma yle_ylt_trans: ∀x:ynat. ∀y:ynat. ∀z:ynat. y < z → x ≤ y → x <
 ]
 qed-.
 
-lemma yle_inv_succ1_lt: ∀x,y:ynat. ⫯x ≤ y → 0 < y ∧ x ≤ ⫰y.
+lemma le_ylt_trans (x) (y) (z): x ≤ y → yinj y < z → yinj x < z.  
+/3 width=3 by yle_ylt_trans, yle_inj/
+qed-.
+
+lemma yle_inv_succ1_lt: ∀x,y:ynat. ↑x ≤ y → 0 < y ∧ x ≤ ↓y.
 #x #y #H elim (yle_inv_succ1 … H) -H /3 width=1 by ylt_O1, conj/
 qed-.
 
-lemma yle_lt: ∀x,y. x < ∞ → ⫯x ≤ y → x < y.
+lemma yle_lt: ∀x,y. x < ∞ → ↑x ≤ y → x < y.
 #x * // #y #H elim (ylt_inv_Y2 … H) -H #n #H destruct
 /3 width=1 by ylt_inj, yle_inv_inj/
 qed-.
@@ -219,6 +223,10 @@ theorem ylt_trans: Transitive … ylt.
 ]
 qed-.
 
+lemma lt_ylt_trans (x) (y) (z): x < y → yinj y < z → yinj x < z.  
+/3 width=3 by ylt_trans, ylt_inj/
+qed-.
+
 (* Elimination principles ***************************************************)
 
 fact ynat_ind_lt_le_aux: ∀R:predicate ynat.