lemma ylt_inv_Y2: ∀x. x < ∞ → ∃m. x = yinj m.
/2 width=3 by ylt_inv_Y2_aux/ qed-.
+lemma ylt_inv_O1: ∀n. 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 → ∃∃n. m < n & y = ⫯n.
lemma ylt_inv_succ1: ∀m,y. ⫯m < y → ∃∃n. m < n & y = ⫯n.
/2 width=3 by ylt_inv_succ1_aux/ qed-.
-lemma yle_inv_succ: ∀m,n. ⫯m < ⫯n → m < n.
+lemma ylt_inv_succ: ∀m,n. ⫯m < ⫯n → m < n.
#m #n #H elim (ylt_inv_succ1 … H) -H
#x #Hx #H destruct //
qed-.
]
qed-.
-lemma ylt_inv_succ2: ∀m,n. m < ⫯n → m ≤ n.
+(* Forward lemmas on successor **********************************************)
+
+lemma ylt_fwd_succ2: ∀m,n. m < ⫯n → m ≤ n.
/2 width=3 by ylt_inv_succ2_aux/ qed-.
(* inversion and forward lemmas on yle **************************************)
]
qed-.
+(* Properties on successor **************************************************)
+
+lemma ylt_O_succ: ∀n. 0 < ⫯n.
+* /2 width=1 by ylt_inj/
+qed.
+
(* Properties on yle ********************************************************)
lemma yle_to_ylt_or_eq: ∀m:ynat. ∀n:ynat. m ≤ n → m < n ∨ m = n.