feat(library/algebra): missing theorems

library/algebra/order.lean

@@ -135,6 +135,7 @@ section
   theorem not_lt_of_ge (H : a ≥ b) : ¬ a < b :=
   assume H1 : a < b,
   lt.irrefl _ (lt_of_le_of_lt H H1)
+
 end
 
 structure strong_order_pair [class] (A : Type) extends weak_order A, has_lt A :=
@@ -245,6 +246,26 @@ section
     (assume H', absurd (H' ▸ !le.refl) H)
     (assume H', H')
 
+  theorem lt_iff_not_ge : a < b ↔ ¬ a ≥ b :=
+  iff.intro
+    (suppose a < b, not_le_of_gt this)
+    (suppose ¬ a ≥ b, lt_of_not_ge this)
+
+  theorem le_iff_not_gt : a ≤ b ↔ ¬ a > b :=
+  iff.intro
+    (suppose a ≤ b, not_lt_of_ge this)
+    (suppose ¬ a > b, le_of_not_gt this)
+
+  theorem gt_iff_not_le : a > b ↔ ¬ a ≤ b :=
+  iff.intro
+    (suppose a > b, not_le_of_gt this)
+    (suppose ¬ a ≤ b, lt_of_not_ge this)
+
+  theorem ge_iff_not_lt : a ≥ b ↔ ¬ a < b :=
+  iff.intro
+    (suppose a ≥ b, not_lt_of_ge this)
+    (suppose ¬ a < b, le_of_not_gt this)
+
   theorem lt_or_ge : a < b ∨ a ≥ b :=
   lt.by_cases
     (assume H1 : a < b, or.inl H1)

library/algebra/ordered_field.lean

@@ -588,4 +588,18 @@ section discrete_linear_ordered_field
       have b = 0, from eq_of_le_of_ge (le_of_not_gt Hgt) Hb,
       by rewrite [this, div_zero]; apply le.refl
 
+  theorem div_le_div_of_le_of_nonneg (H : a ≤ b) (Hc : 0 ≤ c) : a / c ≤ b / c :=
+    begin
+      cases lt_or_eq_of_le Hc with Hlt Heq,
+      apply div_le_div_of_le_of_pos H Hlt,
+      rewrite [-Heq, 2 div_zero]
+    end
+
+  theorem div_le_div_of_le_of_nonpos (H : b ≤ a) (Hc : c ≤ 0) : a / c ≤ b / c :=
+    begin
+      cases lt_or_eq_of_le Hc with Hlt Heq,
+      apply div_le_div_of_le_of_neg H Hlt,
+      rewrite [Heq, 2 div_zero]
+    end
+
 end discrete_linear_ordered_field