#H elim (ylt_inv_Y1 … H)
qed-.
-lemma ylt_inv_O1: ∀n. 0 < n → ⫯⫰n = n.
+lemma ylt_inv_O1: ∀n:ynat. 0 < n → ⫯⫰n = n.
* // #n #H lapply (ylt_inv_inj … H) -H normalize
/3 width=1 by S_pred, eq_f/
qed-.
(* Inversion lemmas on successor ********************************************)
-fact ylt_inv_succ1_aux: ∀x,y. x < y → ∀m. x = ⫯m → m < ⫰y ∧ ⫯⫰y = y.
+fact ylt_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 ylt_inv_succ1: ∀m,y. ⫯m < y → m < ⫰y ∧ ⫯⫰y = y.
+lemma ylt_inv_succ1: ∀m,y:ynat. ⫯m < y → m < ⫰y ∧ ⫯⫰y = y.
/2 width=3 by ylt_inv_succ1_aux/ qed-.
lemma ylt_inv_succ: ∀m,n. ⫯m < ⫯n → m < n.
(* Basic properties *********************************************************)
-lemma ylt_O1: ∀x. ⫯⫰x = x → 0 < x.
+lemma ylt_O1: ∀x:ynat. ⫯⫰x = x → 0 < x.
* // * /2 width=1 by ylt_inj/ normalize
#H destruct
qed.
(* Properties on predecessor ************************************************)
-lemma ylt_pred: ∀m,n. m < n → 0 < m → ⫰m < ⫰n.
+lemma ylt_pred: ∀m,n:ynat. m < n → 0 < m → ⫰m < ⫰n.
#m #n * -m -n
/4 width=1 by ylt_inv_inj, ylt_inj, monotonic_lt_pred/
qed.
lemma ylt_succ_Y: ∀x. x < ∞ → ⫯x < ∞.
* /2 width=1 by/ qed.
-lemma yle_succ1_inj: ∀x,y. ⫯yinj x ≤ y → x < y.
+lemma yle_succ1_inj: ∀x. ∀y:ynat. ⫯yinj x ≤ y → x < y.
#x * /3 width=1 by yle_inv_inj, ylt_inj/
qed.
+lemma ylt_succ2_refl: ∀x,y:ynat. x < y → x < ⫯x.
+#x #y #H elim (ylt_fwd_gen … H) -y /2 width=1 by ylt_inj/
+qed.
+
(* Properties on order ******************************************************)
lemma yle_split_eq: ∀m,n:ynat. m ≤ n → m < n ∨ m = n.
]
qed-.
-lemma yle_inv_succ1_lt: ∀x,y. ⫯x ≤ y → 0 < y ∧ x ≤ ⫰y.
+lemma yle_inv_succ1_lt: ∀x,y:ynat. ⫯x ≤ y → 0 < y ∧ x ≤ ⫰y.
#x #y #H elim (yle_inv_succ1 … H) -H /3 width=1 by ylt_O1, conj/
qed-.