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: â\88\80m,n. ⫯m â\89¤ ⫯n → m ≤ n.
+lemma yle_inv_succ: â\88\80m,n. â\86\91m â\89¤ â\86\91n → m ≤ n.
#m #n #H elim (yle_inv_succ1 … H) -H //
qed-.
-lemma yle_inv_succ2: â\88\80x,y. x â\89¤ ⫯y â\86\92 â«°x ≤ y.
+lemma yle_inv_succ2: â\88\80x,y. x â\89¤ â\86\91y â\86\92 â\86\93x ≤ y.
#x #y #Hxy elim (ynat_cases x)
[ #H destruct //
| * #m #H destruct /2 width=1 by yle_inv_succ/
(* 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: â\88\80m,n. m â\89¤ n â\86\92 â«°m â\89¤ â«°n.
+lemma yle_pred: â\88\80m,n. m â\89¤ n â\86\92 â\86\93m â\89¤ â\86\93n.
#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: â\88\80x. x â\89¤ ⫯x.
+lemma yle_refl_S_dx: â\88\80x. x â\89¤ â\86\91x.
/2 width=1 by yle_succ_dx/ qed.
-lemma yle_refl_SP_dx: â\88\80x. x â\89¤ ⫯⫰x.
+lemma yle_refl_SP_dx: â\88\80x. x â\89¤ â\86\91â\86\93x.
* // * //
qed.
-lemma yle_succ2: â\88\80x,y. â«°x â\89¤ y â\86\92 x â\89¤ ⫯y.
+lemma yle_succ2: â\88\80x,y. â\86\93x â\89¤ y â\86\92 x â\89¤ â\86\91y.
#x #y #Hxy elim (ynat_cases x)
[ #H destruct //
| * #m #H destruct /2 width=1 by yle_succ/