milestone update in ground_2 and basic_2A

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

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 ccb9563ee875b23e587956f0b0032602c259e954..54e946e96ca40a4ace88868863b258d2ce282cd5 100644 (file)
--- a/matita/matita/contribs/lambdadelta/ground_2/ynat/ynat_le.ma
+++ b/matita/matita/contribs/lambdadelta/ground_2/ynat/ynat_le.ma
@@ -46,7 +46,7 @@ fact yle_inv_O2_aux: ∀m:ynat. ∀x:ynat. m ≤ x → x = 0 → m = 0.
 #m #x * -m -x
 [ #m #n #Hmn #H destruct /3 width=1 by le_n_O_to_eq, eq_f/
 | #m #H destruct
-] 
+]
 qed-.
 
 lemma yle_inv_O2: ∀m:ynat. m ≤ 0 → m = 0.
@@ -60,65 +60,88 @@ qed-.
 lemma yle_inv_Y1: ∀n. ∞ ≤ n → n = ∞.
 /2 width=3 by yle_inv_Y1_aux/ qed-.
 
+lemma yle_antisym: ∀y,x. x ≤ y → y ≤ x → x = y.
+#x #y #H elim H -x -y
+/4 width=1 by yle_inv_Y1, yle_inv_inj, le_to_le_to_eq, eq_f/
+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.
+
+lemma yle_split: ∀x,y:ynat. x ≤ y ∨ y ≤ x.
+* /2 width=1 by or_intror/
+#x * /2 width=1 by or_introl/
+#y elim (le_or_ge x y) /3 width=1 by yle_inj, or_introl, or_intror/
+qed-.
+
 (* Inversion lemmas on successor ********************************************)
 
-fact yle_inv_succ1_aux: ∀x,y. x ≤ y → ∀m. x = ⫯m → ∃∃n. m ≤ n & y = ⫯n.
+fact yle_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
-  #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/ (**) (* explicit constructor *)
+  #m #Hnm #H destruct /3 width=1 by yle_inj, conj/
+| #x #y #H destruct /2 width=1 by yle_Y, conj/
 ]
 qed-.
 
-lemma yle_inv_succ1: ∀m,y.  ⫯m ≤ y → ∃∃n. m ≤ n & y = ⫯n.
+lemma yle_inv_succ1: ∀m,y:ynat. ↑m ≤ y → m ≤ ↓y ∧ ↑↓y = y.
 /2 width=3 by yle_inv_succ1_aux/ qed-.
 
-lemma yle_inv_succ: ∀m,n. ⫯m ≤ ⫯n → m ≤ n.
-#m #n #H elim (yle_inv_succ1 … H) -H
-#x #Hx #H destruct //
+lemma yle_inv_succ: ∀m,n. ↑m ≤ ↑n → m ≤ n.
+#m #n #H elim (yle_inv_succ1 … H) -H //
 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 //
+lemma yle_inv_succ2: ∀x,y. x ≤ ↑y → ↓x ≤ y.
+#x #y #Hxy elim (ynat_cases x)
+[ #H destruct //
+| * #m #H destruct /2 width=1 by yle_inv_succ/
+]
 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.
-
 (* Properties on predecessor ************************************************)
 
-lemma yle_pred_sn: â\88\80m,n. m â\89¤ n â\86\92 â«°m ≤ n.
+lemma yle_pred_sn: â\88\80m,n. m â\89¤ n â\86\92 â\86\93m ≤ n.
 #m #n * -m -n /3 width=3 by transitive_le, yle_inj/
 qed.
 
-lemma yle_refl_pred_sn: â\88\80x. â«°x ≤ x.
+lemma yle_refl_pred_sn: â\88\80x. â\86\93x ≤ x.
 /2 width=1 by yle_refl, yle_pred_sn/ qed.
 
+lemma yle_pred: ∀m,n. m ≤ n → ↓m ≤ ↓n.
+#m #n * -m -n /3 width=1 by yle_inj, monotonic_pred/
+qed.
+
 (* Properties on successor **************************************************)
 
-lemma yle_succ: â\88\80m,n. m â\89¤ n â\86\92 ⫯m â\89¤ ⫯n.
+lemma yle_succ: â\88\80m,n. m â\89¤ n â\86\92 â\86\91m â\89¤ â\86\91n.
 #m #n * -m -n /3 width=1 by yle_inj, le_S_S/
 qed.
 
-lemma yle_succ_dx: â\88\80m,n. m â\89¤ n â\86\92 m â\89¤ ⫯n.
+lemma yle_succ_dx: â\88\80m,n. m â\89¤ n â\86\92 m â\89¤ â\86\91n.
 #m #n * -m -n /3 width=1 by le_S, yle_inj/
 qed.
 
-lemma yle_refl_S_dx: ∀x. x ≤ ⫯x.
-/2 width=1 by yle_refl, yle_succ_dx/ qed.
+lemma yle_refl_S_dx: ∀x. x ≤ ↑x.
+/2 width=1 by yle_succ_dx/ qed.
+
+lemma yle_refl_SP_dx: ∀x. x ≤ ↑↓x.
+* // * //
+qed.
+
+lemma yle_succ2: ∀x,y. ↓x ≤ y → x ≤ ↑y.
+#x #y #Hxy elim (ynat_cases x)
+[ #H destruct //
+| * #m #H destruct /2 width=1 by yle_succ/
+]
+qed-.
 
 (* Main properties **********************************************************)
 
@@ -129,4 +152,4 @@ theorem yle_trans: Transitive … yle.
   /3 width=3 by transitive_le, yle_inj/ (**) (* full auto too slow *)
 | #x #z #H lapply (yle_inv_Y1 … H) //
 ]
-qed-. 
+qed-.