milestone in basic_2

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

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 2a9fbb5152e8225158bb3cf212a4e29b429368eb..ab9cf4565cc365d28ff3d8d94b3a8b2359e91c02 100644 (file)
--- 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. 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. 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. x < y → ∀m. x = ⫯m → m < ⫰y ∧ ⫯⫰
 ]
 qed-.
 
-lemma ylt_inv_succ1: ∀m,y. ⫯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: â\88\80m,n. ⫯m < ⫯n → m < n.
+lemma ylt_inv_succ: â\88\80m,n. â\86\91m < â\86\91n → m < n.
 #m #n #H elim (ylt_inv_succ1 … H) -H //
 qed-.
 
 (* Forward lemmas on successor **********************************************)
 
-fact ylt_fwd_succ2_aux: â\88\80x,y. x < y â\86\92 â\88\80n. y = ⫯n → x ≤ n.
+fact ylt_fwd_succ2_aux: â\88\80x,y. x < y â\86\92 â\88\80n. y = â\86\91n → 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: â\88\80m,n. m < ⫯n → m ≤ n.
+lemma ylt_fwd_succ2: â\88\80m,n. m < â\86\91n → m ≤ n.
 /2 width=3 by ylt_fwd_succ2_aux/ qed-.
 
 (* inversion and forward lemmas on order ************************************)
 
-lemma ylt_fwd_le_succ1: â\88\80m,n. m < n â\86\92 ⫯m ≤ n.
+lemma ylt_fwd_le_succ1: â\88\80m,n. m < n â\86\92 â\86\91m ≤ n.
 #m #n * -m -n /2 width=1 by yle_inj/
 qed-.
 
-lemma ylt_fwd_le_pred2: â\88\80x,y:ynat. x < y â\86\92 x â\89¤ â«°y.
+lemma ylt_fwd_le_pred2: â\88\80x,y:ynat. x < y â\86\92 x â\89¤ â\86\93y.
 #x #y #H elim H -x -y /3 width=1 by yle_inj, monotonic_pred/
 qed-.
 
@@ -124,41 +124,45 @@ lemma ylt_yle_false: ∀m:ynat. ∀n:ynat. m < n → n ≤ m → ⊥.
 ]
 qed-.
 
-lemma ylt_inv_le: â\88\80x,y. x < y â\86\92 x < â\88\9e â\88§ ⫯x ≤ y.
+lemma ylt_inv_le: â\88\80x,y. x < y â\86\92 x < â\88\9e â\88§ â\86\91x ≤ y.
 #x #y #H elim H -x -y /3 width=1 by yle_inj, conj/
 qed-.
 
 (* Basic properties *********************************************************)
 
-lemma ylt_O1: ∀x. ⫯⫰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. 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: â\88\80n. 0 < ⫯n.
+lemma ylt_O_succ: â\88\80n. 0 < â\86\91n.
 * /2 width=1 by ylt_inj/
 qed.
 
-lemma ylt_succ: â\88\80m,n. m < n â\86\92 ⫯m < ⫯n.
+lemma ylt_succ: â\88\80m,n. m < n â\86\92 â\86\91m < â\86\91n.
 #m #n #H elim H -m -n /3 width=1 by ylt_inj, le_S_S/
 qed.
 
-lemma ylt_succ_Y: â\88\80x. x < â\88\9e â\86\92 ⫯x < ∞.
+lemma ylt_succ_Y: â\88\80x. x < â\88\9e â\86\92 â\86\91x < ∞.
 * /2 width=1 by/ qed.
 
-lemma yle_succ1_inj: ∀x,y. ⫯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.
+#x #y #H elim (ylt_fwd_gen … H) -y /2 width=1 by ylt_inj/
+qed.
+
 (* Properties on order ******************************************************)
 
 lemma yle_split_eq: ∀m,n:ynat. m ≤ n → m < n ∨ m = n.
@@ -195,11 +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. ⫯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: â\88\80x,y. x < â\88\9e â\86\92 ⫯x ≤ y → x < y.
+lemma yle_lt: â\88\80x,y. x < â\88\9e â\86\92 â\86\91x ≤ 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-.
@@ -215,6 +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.