X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=matita%2Fmatita%2Fcontribs%2Flambdadelta%2Fground_2%2Fynat%2Fynat_lt.ma;h=5a1bb14dda71e288b35a90a2e77c4ea046657e46;hb=1fd63df4c77f5c24024769432ea8492748b4ac79;hp=3e496e18599bda0201393d76945f5ac93d5538e7;hpb=658c000ee2ea2da04cf29efc0acdaf16364fbf5e;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 3e496e185..5a1bb14dd 100644
--- a/matita/matita/contribs/lambdadelta/ground_2/ynat/ynat_lt.ma
+++ b/matita/matita/contribs/lambdadelta/ground_2/ynat/ynat_lt.ma
@@ -59,14 +59,19 @@ lemma ylt_inv_Y1: ∀n. ∞ < n → ⊥.
 #y #H destruct
 qed-.
 
-lemma ylt_inv_O1: ∀n. 0 < n → ⫯⫰n = n.
+lemma ylt_inv_Y2: ∀x:ynat. x < ∞ → ∃n. x = yinj n.
+* /2 width=2 by ex_intro/
+#H elim (ylt_inv_Y1 … H)
+qed-.
+
+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
@@ -76,16 +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: ∀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/
@@ -93,16 +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 yle **************************************)
+(* 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-.
 
@@ -119,34 +124,50 @@ lemma ylt_yle_false: ∀m:ynat. ∀n:ynat. m < n → n ≤ m → ⊥.
 ]
 qed-.
 
+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_O: ∀x. ⫯⫰(yinj x) = yinj x → 0 < x.
-* /2 width=1 by/ normalize
+lemma ylt_O1: ∀x:ynat. ↑↓x = x → 0 < x.
+* // * /2 width=1 by ylt_inj/ normalize
 #H destruct
 qed.
 
+lemma yle_inv_succ_sn_lt (x:ynat) (y:ynat):
+      ↑x ≤ y → ∧∧ x ≤ ↓y & 0 < y.
+#x #y #H elim (yle_inv_succ1 … H) -H /3 width=2 by ylt_O1, conj/
+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: ∀n. 0 < ⫯n.
+lemma ylt_O_succ: ∀x:ynat. 0 < ↑x.
 * /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 yle_succ1_inj: ∀x,y. ⫯yinj x ≤ y → x < y.
+lemma ylt_succ_Y: ∀x. x < ∞ → ↑x < ∞.
+* /2 width=1 by/ qed.
+
+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.
@@ -183,6 +204,19 @@ lemma yle_ylt_trans: ∀x:ynat. ∀y:ynat. ∀z:ynat. y < z → x ≤ y → x <
 ]
 qed-.
 
+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.
+#x * // #y #H elim (ylt_inv_Y2 … H) -H #n #H destruct
+/3 width=1 by ylt_inj, yle_inv_inj/
+qed-.
+
 (* Main properties **********************************************************)
 
 theorem ylt_trans: Transitive … ylt.
@@ -193,3 +227,45 @@ theorem ylt_trans: Transitive … ylt.
 | #x #z #H elim (ylt_yle_false … H) //
 ]
 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.
+                         (∀y. (∀x. x < y → R x) → R y) →
+                         ∀y:nat. ∀x. x ≤ y → R x.
+#R #IH #y elim y -y
+[ #x #H >(yle_inv_O2 … H) -x
+  @IH -IH #x #H elim (ylt_yle_false … H) -H //
+| /5 width=3 by ylt_yle_trans, ylt_fwd_succ2/
+]
+qed-.
+
+fact ynat_ind_lt_aux: ∀R:predicate ynat.
+                      (∀y. (∀x. x < y → R x) → R y) →
+                      ∀y:nat. R y.
+/4 width=2 by ynat_ind_lt_le_aux/ qed-.
+
+lemma ynat_ind_lt: ∀R:predicate ynat.
+                   (∀y. (∀x. x < y → R x) → R y) →
+                   ∀y. R y.
+#R #IH * /4 width=1 by ynat_ind_lt_aux/
+@IH #x #H elim (ylt_inv_Y2 … H) -H
+#n #H destruct /4 width=1 by ynat_ind_lt_aux/
+qed-.
+
+fact ynat_f_ind_aux: ∀A. ∀f:A→ynat. ∀R:predicate A.
+                     (∀x. (∀a. f a < x → R a) → ∀a. f a = x → R a) →
+                     ∀x,a. f a = x → R a.
+#A #f #R #IH #x @(ynat_ind_lt … x) -x
+/3 width=3 by/
+qed-.
+
+lemma ynat_f_ind: ∀A. ∀f:A→ynat. ∀R:predicate A.
+                  (∀x. (∀a. f a < x → R a) → ∀a. f a = x → R a) → ∀a. R a.
+#A #f #R #IH #a
+@(ynat_f_ind_aux … IH) -IH [2: // | skip ]
+qed-.