+lemma arith_i: ∀x,y,z. y < x → x+z-y-1 = x-y-1+z.
+/2 width=1 by plus_minus/ qed-.
+
+(* Properties ***************************************************************)
+
+lemma monotonic_le_minus_l2: ∀x1,x2,y,z. x1 ≤ x2 → x1 - y - z ≤ x2 - y - z.
+/3 width=1 by monotonic_le_minus_l/ qed.
+
+lemma arith_j: ∀x,y,z. x-y-1 ≤ x-(y-z)-1.
+/3 width=1 by monotonic_le_minus_l, monotonic_le_minus_r/ qed.
+
+lemma arith_k_sn: ∀z,x,y,n. z < x → x+n ≤ y → x-z-1+n ≤ y-z-1.
+#z #x #y #n #Hzx #Hxny
+>plus_minus [2: /2 width=1 by monotonic_le_minus_r/ ]
+>plus_minus [2: /2 width=1 by lt_to_le/ ]
+/2 width=1 by monotonic_le_minus_l2/
+qed.
+
+lemma arith_k_dx: ∀z,x,y,n. z < x → y ≤ x+n → y-z-1 ≤ x-z-1+n.
+#z #x #y #n #Hzx #Hyxn
+>plus_minus [2: /2 width=1 by monotonic_le_minus_r/ ]
+>plus_minus [2: /2 width=1 by lt_to_le/ ]
+/2 width=1 by monotonic_le_minus_l2/
+qed.
+