lemma minus_plus_m_m_commutative: ∀n,m:nat. n = m + n - m.
// qed-.
+lemma plus_n_2: ∀n. n + 2 = n + 1 + 1.
+// qed.
+
+lemma arith_l: ∀x. 1 = 1-x+(x-(x-1)).
+* // #x >minus_S_S >minus_S_S <minus_O_n <minus_n_O //
+qed.
+
(* Note: uses minus_minus_comm, minus_plus_m_m, commutative_plus, plus_minus *)
lemma plus_minus_minus_be: ∀x,y,z. y ≤ z → z ≤ x → (x - z) + (z - y) = x - y.
#x #z #y #Hzy #Hyx >plus_minus // >commutative_plus >plus_minus //
fact plus_minus_minus_be_aux: ∀i,x,y,z. y ≤ z → z ≤ x → i = z - y → x - z + i = x - y.
/2 width=1 by plus_minus_minus_be/ qed-.
-lemma plus_n_2: ∀n. n + 2 = n + 1 + 1.
-// qed.
-
lemma le_plus_minus: ∀m,n,p. p ≤ n → m + n - p = m + (n - p).
/2 by plus_minus/ qed-.
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 minus_le_trans_sn: ∀x1,x2. x1 ≤ x2 → ∀x. x1-x ≤ x2.
+/2 width=3 by transitive_le/ qed.
+
(* Note: this might interfere with nat.ma *)
lemma monotonic_lt_pred: ∀m,n. m < n → 0 < m → pred m < pred n.
#m #n #Hmn #Hm whd >(S_pred … Hm)