feat(library/algebra): finish adding one-directional versions of iff theorems

library/algebra/field.lean

@@ -200,11 +200,20 @@ section division_ring
       a / b = b / b : this
         ... = 1     : div_self Hb)
 
+  theorem eq_of_div_eq_one (Hb : b ≠ 0) : a / b = 1 → a = b :=
+    iff.mp (div_eq_one_iff_eq Hb)
+
   theorem eq_div_iff_mul_eq (Hc : c ≠ 0) : a = b / c ↔ a * c = b :=
     iff.intro
       (suppose a = b / c, by rewrite [this, (div_mul_cancel Hc)])
       (suppose a * c = b, by rewrite [-(mul_div_cancel Hc), this])
 
+  theorem eq_div_of_mul_eq (Hc : c ≠ 0) : a * c = b → a = b / c :=
+    iff.mpr (eq_div_iff_mul_eq Hc)
+
+  theorem mul_eq_of_eq_div (Hc : c ≠ 0) : a = b / c → a * c = b :=
+    iff.mp (eq_div_iff_mul_eq Hc)
+
   theorem add_div_eq_mul_add_div (Hc : c ≠ 0) : a + b / c = (a * c + b) / c :=
     have (a + b / c) * c = a * c + b, by rewrite [right_distrib, (div_mul_cancel Hc)],
     (iff.elim_right (eq_div_iff_mul_eq Hc)) this

library/algebra/ordered_field.lean

@@ -399,7 +399,6 @@ section discrete_linear_ordered_field
     have H3 : 0 < -a, from pos_of_div_pos H2,
     neg_of_neg_pos H3
 
--- why is mul_le_mul under ordered_ring namespace?
   theorem le_of_div_le (H : 0 < a) (Hl : 1 / a ≤ 1 / b) : b ≤ a :=
     have Hb : 0 < b, from pos_of_div_pos (calc
       0   < 1 / a : div_pos_of_pos H
@@ -407,11 +406,10 @@ section discrete_linear_ordered_field
     have H' : 1 ≤ a / b, from (calc
       1   = a / a       : div_self (ne.symm (ne_of_lt H))
       ... = a * (1 / a) :  div_eq_mul_one_div
-      ... ≤ a * (1 / b) : ordered_ring.mul_le_mul_of_nonneg_left Hl (le_of_lt H)
+      ... ≤ a * (1 / b) : mul_le_mul_of_nonneg_left Hl (le_of_lt H)
       ... = a / b       : div_eq_mul_one_div
       ), le_of_one_le_div Hb H'
 
-
   theorem le_of_div_le_neg (H : b < 0) (Hl : 1 / a ≤ 1 / b) : b ≤ a :=
     assert Ha : a ≠ 0, from ne_of_lt (neg_of_div_neg (calc
       1 / a ≤ 1 / b : Hl
@@ -480,14 +478,13 @@ section discrete_linear_ordered_field
 
   theorem div_lt_div_of_pos_of_lt_of_pos (Hb : 0 < b) (H : b < a) (Hc : 0 < c) : c / a < c / b :=
     begin
-      apply iff.mp (sub_neg_iff_lt _ _),
-      rewrite [div_eq_mul_one_div, {c / b}div_eq_mul_one_div],
-      rewrite -mul_sub_left_distrib,
+      apply iff.mp !sub_neg_iff_lt,
+      rewrite [div_eq_mul_one_div, {c / b}div_eq_mul_one_div, -mul_sub_left_distrib],
       apply mul_neg_of_pos_of_neg,
       exact Hc,
-      apply iff.mpr (sub_neg_iff_lt _ _),
+      apply iff.mpr !sub_neg_iff_lt,
       apply div_lt_div_of_lt,
-      exact Hb, exact H
+      repeat assumption
     end
 
   theorem div_mul_le_div_mul_of_div_le_div_pos' {d e : A} (H : a / b ≤ c / d)
@@ -507,7 +504,7 @@ section discrete_linear_ordered_field
           by rewrite [abs_of_neg H', abs_of_neg (div_neg_of_neg H'),
                          -(one_div_neg_eq_neg_one_div (ne_of_lt H'))]
        else
-         have Heq [visible] : a = 0, from eq_of_le_of_ge (le_of_not_gt H) (le_of_not_gt H'),
+         assert Heq : a = 0, from eq_of_le_of_ge (le_of_not_gt H) (le_of_not_gt H'),
          by rewrite [Heq, div_zero, *abs_zero, div_zero])
 
   theorem ge_sub_of_abs_sub_le_left (H : abs (a - b) ≤ c) : a ≥ b - c :=

library/algebra/ordered_ring.lean

@@ -603,9 +603,15 @@ section
   theorem abs_dvd_iff (a b : A) : abs a ∣ b ↔ a ∣ b :=
   abs.by_cases !iff.refl !neg_dvd_iff_dvd
 
+  theorem abs_dvd_of_dvd {a b : A} : a ∣ b → abs a ∣ b :=
+    iff.mpr !abs_dvd_iff
+
   theorem dvd_abs_iff (a b : A) : a ∣ abs b ↔ a ∣ b :=
   abs.by_cases !iff.refl !dvd_neg_iff_dvd
 
+  theorem dvd_abs_of_dvd {a b : A} : a ∣ b → a ∣ abs b :=
+    iff.mpr !dvd_abs_iff
+
   theorem abs_mul (a b : A) : abs (a * b) = abs a * abs b :=
   or.elim (le.total 0 a)
     (assume H1 : 0 ≤ a,

library/algebra/ring.lean

@@ -222,6 +222,12 @@ section
       ... ↔ a * e - b * e + c = d : by rewrite sub_add_eq_add_sub
       ... ↔ (a - b) * e + c = d   : by rewrite mul_sub_right_distrib
 
+  theorem mul_add_eq_mul_add_of_sub_mul_add_eq : (a - b) * e + c = d → a * e + c = b * e + d :=
+    iff.mpr !mul_add_eq_mul_add_iff_sub_mul_add_eq
+
+  theorem sub_mul_add_eq_of_mul_add_eq_mul_add : a * e + c = b * e + d → (a - b) * e + c = d :=
+    iff.mp !mul_add_eq_mul_add_iff_sub_mul_add_eq
+
   theorem mul_neg_one_eq_neg : a * (-1) = -a :=
     have a + a * -1 = 0, from calc
       a + a * -1 = a * 1 + a * -1 : mul_one
@@ -278,6 +284,12 @@ section
             (show a * -c = -b,
              by rewrite [-neg_mul_eq_mul_neg, -this])))
 
+  theorem dvd_neg_of_dvd : (a ∣ b) → (a ∣ -b) :=
+    iff.mpr !dvd_neg_iff_dvd
+
+  theorem dvd_of_dvd_neg : (a ∣ -b) → (a ∣ b) :=
+    iff.mp !dvd_neg_iff_dvd
+
   theorem neg_dvd_iff_dvd : (-a ∣ b) ↔ (a ∣ b) :=
   iff.intro
     (suppose -a ∣ b,
@@ -291,8 +303,14 @@ section
           dvd.intro
             (show -a * -c = b, by rewrite [neg_mul_neg, this])))
 
+  theorem neg_dvd_of_dvd : (a ∣ b) → (-a ∣ b) :=
+    iff.mpr !neg_dvd_iff_dvd
+
+  theorem dvd_of_neg_dvd : (-a ∣ b) → (a ∣ b) :=
+    iff.mp !neg_dvd_iff_dvd
+
   theorem dvd_sub (H₁ : (a ∣ b)) (H₂ : (a ∣ c)) : (a ∣ b - c) :=
-  dvd_add H₁ (iff.elim_right !dvd_neg_iff_dvd H₂)
+  dvd_add H₁ (!dvd_neg_of_dvd H₂)
 end
 
 /- integral domains -/