- revision of ground_2 and basic_2

[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 e706fa4dd6c4c933019b39f8bb89b17114b2527c..b7b57d764ba64147c0a2acf3a34ab21497775f68 100644 (file)
--- 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
@@ -54,32 +60,46 @@ qed-.
 lemma yle_inv_Y1: ∀n. ∞ ≤ n → n = ∞.
 /2 width=3 by yle_inv_Y1_aux/ 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. 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/
+  #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. ⫯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 //
+#m #n #H elim (yle_inv_succ1 … H) -H //
 qed-.
 
-(* Basic properties *********************************************************)
-
-lemma yle_refl: reflexive … yle.
-* /2 width=1 by le_n, yle_inj/
-qed.
+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-.
 
 (* Properties on predecessor ************************************************)
 
@@ -90,14 +110,33 @@ qed.
 lemma yle_refl_pred_sn: ∀x. ⫰x ≤ 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: ∀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.
 
 lemma yle_refl_S_dx: ∀x. x ≤ ⫯x.
-/2 width=1 by yle_refl, yle_succ_dx/ qed.
+/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 **********************************************************)
 
@@ -106,6 +145,6 @@ 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-.