+(* Inversion lemmas on successor ********************************************)
+
+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 /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: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 //
+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-.
+