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-.
-(* 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 ************************************************)
-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_dx: ∀m,n. m ≤ n → m ≤ ⫯n.
+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.
+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 **********************************************************)