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.
(* Inversion lemmas on successor ********************************************)
-fact yle_inv_succ1_aux: ∀x,y. x ≤ y → ∀m. x = ⫯m → m ≤ ⫰y ∧ ⫯⫰y = y.
+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
]
qed-.
-lemma yle_inv_succ1: ∀m,y. ⫯m ≤ y → m ≤ ⫰y ∧ ⫯⫰y = y.
+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.