diff --git a/library/algebra/ring.lean b/library/algebra/ring.lean
index a07e30724..42d2953dd 100644
--- a/library/algebra/ring.lean
+++ b/library/algebra/ring.lean
@@ -71,35 +71,41 @@ section comm_semiring
   variables [s : comm_semiring A] (a b c : A)
   include s
 
-  definition dvd (a b : A) : Prop := ∃c, a * c = b
+  definition dvd (a b : A) : Prop := ∃c, b = a * c
   infix `|` := dvd
 
-  theorem dvd.intro {a b c : A} (H : a * b = c) : a | c :=
-  exists.intro _ H
+  theorem dvd.intro {a b c : A} (H : a * c = b) : a | b :=
+  exists.intro _ H⁻¹
 
-  theorem dvd.intro_right {a b c : A} (H : a * b = c) : b | c :=
+  theorem dvd.intro_left {a b c : A} (H : c * a = b) : a | b :=
   dvd.intro (!mul.comm ▸ H)
 
-  theorem dvd.ex {a b : A} (H : a | b) : ∃c, a * c = b := H
+  theorem exists_eq_mul_right_of_dvd {a b : A} (H : a | b) : ∃c, b = a * c := H
 
-  theorem dvd.elim {P : Prop} {a b : A} (H₁ : a | b) (H₂ : ∀c, a * c = b → P) : P :=
+  theorem dvd.elim {P : Prop} {a b : A} (H₁ : a | b) (H₂ : ∀c, b = a * c → P) : P :=
   exists.elim H₁ H₂
 
+  theorem exists_eq_mul_left_of_dvd {a b : A} (H : a | b) : ∃c, b = c * a :=
+  dvd.elim H (take c, assume H1 : b = a * c, exists.intro c (H1 ⬝ !mul.comm))
+
+  theorem dvd.elim_left {P : Prop} {a b : A} (H₁ : a | b) (H₂ : ∀c, b = c * a → P) : P :=
+  exists.elim (exists_eq_mul_left_of_dvd H₁) (take c, assume H₃ : b = c * a, H₂ c H₃)
+
   theorem dvd.refl : a | a := dvd.intro !mul_one
 
   theorem dvd.trans {a b c : A} (H₁ : a | b) (H₂ : b | c) : a | c :=
   dvd.elim H₁
-    (take d, assume H₃ : a * d = b,
+    (take d, assume H₃ : b = a * d,
       dvd.elim H₂
-        (take e, assume H₄ : b * e = c,
-          @dvd.intro _ _ _ (d * e) _
+        (take e, assume H₄ : c = b * e,
+          dvd.intro
             (calc
               a * (d * e) = (a * d) * e : mul.assoc
                 ... = b * e : H₃
                 ... = c : H₄)))
 
   theorem eq_zero_of_zero_dvd {a : A} (H : 0 | a) : a = 0 :=
-    dvd.elim H (take c, assume H' : 0 * c = a, (H')⁻¹ ⬝ !zero_mul)
+    dvd.elim H (take c, assume H' : a = 0 * c, H' ⬝ !zero_mul)
 
   theorem dvd_zero : a | 0 := dvd.intro !mul_zero
 
@@ -112,7 +118,7 @@ section comm_semiring
   theorem dvd_mul_of_dvd_left {a b : A} (H : a | b) (c : A) : a | b * c :=
   dvd.elim H
     (take d,
-      assume H₁ : a * d = b,
+      assume H₁ : b = a * d,
       dvd.intro
         (calc
           a * (d * c) = a * d * c : (!mul.assoc)⁻¹
@@ -123,9 +129,9 @@ section comm_semiring
 
   theorem mul_dvd_mul {a b c d : A} (dvd_ab : a | b) (dvd_cd : c | d) : a * c | b * d :=
   dvd.elim dvd_ab
-    (take e, assume Haeb : a * e = b,
+    (take e, assume Haeb : b = a * e,
       dvd.elim dvd_cd
-        (take f, assume Hcfd : c * f = d,
+        (take f, assume Hcfd : d = c * f,
           dvd.intro
             (calc
               a * c * (e * f) = a * (c * (e * f)) : mul.assoc
@@ -135,17 +141,17 @@ section comm_semiring
                 ... = b * d : Hcfd)))
 
   theorem dvd_of_mul_right_dvd {a b c : A} (H : a * b | c) : a | c :=
-  dvd.elim H (take d, assume Habdc : a * b * d = c, dvd.intro (!mul.assoc⁻¹ ⬝ Habdc))
+  dvd.elim H (take d, assume Habdc : c = a * b * d, dvd.intro (!mul.assoc⁻¹ ⬝ Habdc⁻¹))
 
   theorem dvd_of_mul_left_dvd {a b c : A} (H : a * b | c) : b | c :=
   dvd_of_mul_right_dvd (mul.comm a b ▸ H)
 
   theorem dvd_add {a b c : A} (Hab : a | b) (Hac : a | c) : a | b + c :=
   dvd.elim Hab
-    (take d, assume Hadb : a * d = b,
+    (take d, assume Hadb : b = a * d,
       dvd.elim Hac
-        (take e, assume Haec : a * e = c,
-          dvd.intro (show a * (d + e) = b + c, from Hadb ▸ Haec ▸ left_distrib a d e)))
+        (take e, assume Haec : c = a * e,
+          dvd.intro (show a * (d + e) = b + c, from Hadb⁻¹ ▸ Haec⁻¹ ▸ left_distrib a d e)))
 end comm_semiring
 
 /- ring -/
@@ -247,32 +253,32 @@ section
   iff.intro
     (assume H : a | -b,
       dvd.elim H
-        (take c, assume H' : a * c = -b,
+        (take c, assume H' : -b = a * c,
           dvd.intro
             (calc
               a * -c = -(a * c) : {!neg_mul_eq_mul_neg⁻¹}
-                ... = -(-b) : {H'}
+                ... = -(-b) : H'
                 ... = b : neg_neg)))
     (assume H : a | b,
       dvd.elim H
-        (take c, assume H' : a * c = b,
+        (take c, assume H' : b = a * c,
           dvd.intro
             (calc
               a * -c = -(a * c) : {!neg_mul_eq_mul_neg⁻¹}
-                ... = -b : {H'})))
+                ... = -b : H')))
 
   theorem neg_dvd_iff_dvd : -a | b ↔ a | b :=
   iff.intro
     (assume H : -a | b,
       dvd.elim H
-        (take c, assume H' : -a * c = b,
+        (take c, assume H' : b = -a * c,
           dvd.intro
             (calc
               a * -c = -a * c : !neg_mul_comm⁻¹
                 ... = b : H')))
     (assume H : a | b,
       dvd.elim H
-        (take c, assume H' : a * c = b,
+        (take c, assume H' : b = a * c,
           dvd.intro
             (calc
               -a * -c = a * c : neg_mul_neg
@@ -326,15 +332,15 @@ section
   theorem dvd_of_mul_dvd_mul_left {a b c : A} (Ha : a ≠ 0) (Hdvd : a * b | a * c) : b | c :=
   dvd.elim Hdvd
     (take d,
-      assume H : a * b * d = a * c,
-      have H1 : b * d = c, from mul.cancel_left Ha (mul.assoc a b d ▸ H),
+      assume H : a * c = a * b * d,
+      have H1 : b * d = c, from mul.cancel_left Ha (mul.assoc a b d ▸ H⁻¹),
       dvd.intro H1)
 
   theorem dvd_of_mul_dvd_mul_right {a b c : A} (Ha : a ≠ 0) (Hdvd : b * a | c * a) : b | c :=
   dvd.elim Hdvd
     (take d,
-      assume H : b * a * d = c * a,
-      have H1 : b * d * a = c * a, from eq.trans !mul.right_comm H,
+      assume H : c * a = b * a * d,
+      have H1 : b * d * a = c * a, from eq.trans !mul.right_comm H⁻¹,
       have H2 : b * d = c, from mul.cancel_right Ha H1,
       dvd.intro H2)
 end
diff --git a/library/data/int/basic.lean b/library/data/int/basic.lean
index 1ed9d5928..22d7cb2c6 100644
--- a/library/data/int/basic.lean
+++ b/library/data/int/basic.lean
@@ -692,11 +692,16 @@ section port_algebra
     @algebra.ne_zero_of_mul_ne_zero_left _ _
   definition dvd (a b : ℤ) : Prop := algebra.dvd a b
   infix `|` := dvd
-  theorem dvd.intro : ∀{a b c : ℤ} (H : a * b = c), a | c := @algebra.dvd.intro _ _
-  theorem dvd.intro_right : ∀{a b c : ℤ} (H : a * b = c), b | c := @algebra.dvd.intro_right _ _
-  theorem dvd.ex : ∀{a b : ℤ} (H : a | b), ∃c, a * c = b := @algebra.dvd.ex _ _
-  theorem dvd.elim : ∀{P : Prop} {a b : ℤ} (H₁ : a | b) (H₂ : ∀c, a * c = b → P), P :=
+  theorem dvd.intro : ∀{a b c : ℤ} (H : a * c = b), a | b := @algebra.dvd.intro _ _
+  theorem dvd.intro_left : ∀{a b c : ℤ} (H : c * a = b), a | b := @algebra.dvd.intro_left _ _
+  theorem exists_eq_mul_right_of_dvd : ∀{a b : ℤ} (H : a | b), ∃c, b = a * c :=
+    @algebra.exists_eq_mul_right_of_dvd _ _
+  theorem dvd.elim : ∀{P : Prop} {a b : ℤ} (H₁ : a | b) (H₂ : ∀c, b = a * c → P), P :=
     @algebra.dvd.elim _ _
+  theorem exists_eq_mul_left_of_dvd : ∀{a b : ℤ} (H : a | b), ∃c, b = c * a :=
+    @algebra.exists_eq_mul_left_of_dvd _ _
+  theorem dvd.elim_left : ∀{P : Prop} {a b : ℤ} (H₁ : a | b) (H₂ : ∀c, b = c * a → P), P :=
+    @algebra.dvd.elim_left _ _
   theorem dvd.refl : ∀a : ℤ, a | a := algebra.dvd.refl
   theorem dvd.trans : ∀{a b c : ℤ} (H₁ : a | b) (H₂ : b | c), a | c := @algebra.dvd.trans _ _
   theorem eq_zero_of_zero_dvd : ∀{a : ℤ} (H : 0 | a), a = 0 := @algebra.eq_zero_of_zero_dvd _ _
diff --git a/library/data/nat/comm_semiring.lean b/library/data/nat/comm_semiring.lean
index 2d144166a..0d2cb7b87 100644
--- a/library/data/nat/comm_semiring.lean
+++ b/library/data/nat/comm_semiring.lean
@@ -26,10 +26,16 @@ section port_algebra
 
   definition dvd (a b : ℕ) : Prop := algebra.dvd a b
   infix `|` := dvd
-  theorem dvd.intro : ∀{a b c : ℕ} (H : a * b = c), a | c := @algebra.dvd.intro _ _
-  theorem dvd.ex : ∀{a b : ℕ} (H : a | b), ∃c, a * c = b := @algebra.dvd.ex _ _
-  theorem dvd.elim : ∀{P : Prop} {a b : ℕ} (H₁ : a | b) (H₂ : ∀c, a * c = b → P), P :=
+  theorem dvd.intro : ∀{a b c : ℕ} (H : a * c = b), a | b := @algebra.dvd.intro _ _
+  theorem dvd.intro_left : ∀{a b c : ℕ} (H : c * a = b), a | b := @algebra.dvd.intro_left _ _
+  theorem exists_eq_mul_right_of_dvd : ∀{a b : ℕ} (H : a | b), ∃c, b = a * c :=
+    @algebra.exists_eq_mul_right_of_dvd _ _
+  theorem dvd.elim : ∀{P : Prop} {a b : ℕ} (H₁ : a | b) (H₂ : ∀c, b = a * c → P), P :=
     @algebra.dvd.elim _ _
+  theorem exists_eq_mul_left_of_dvd : ∀{a b : ℕ} (H : a | b), ∃c, b = c * a :=
+    @algebra.exists_eq_mul_left_of_dvd _ _
+  theorem dvd.elim_left : ∀{P : Prop} {a b : ℕ} (H₁ : a | b) (H₂ : ∀c, b = c * a → P), P :=
+    @algebra.dvd.elim_left _ _
   theorem dvd.refl : ∀a : ℕ, a | a := algebra.dvd.refl
   theorem dvd.trans : ∀{a b c : ℕ} (H₁ : a | b) (H₂ : b | c), a | c := @algebra.dvd.trans _ _
   theorem eq_zero_of_zero_dvd : ∀{a : ℕ} (H : 0 | a), a = 0 := @algebra.eq_zero_of_zero_dvd _ _
diff --git a/library/data/nat/div.lean b/library/data/nat/div.lean
index 9d8fbc300..da20824b0 100644
--- a/library/data/nat/div.lean
+++ b/library/data/nat/div.lean
@@ -41,16 +41,16 @@ divide_def 0 b ⬝ if_neg (λ h, and.rec_on h (λ l r, absurd (lt_of_lt_of_le l
 theorem div_eq_succ_sub_div {a b : ℕ} (h₁ : b > 0) (h₂ : a ≥ b) : a div b = succ ((a - b) div b) :=
 divide_def a b ⬝ if_pos (and.intro h₁ h₂)
 
-theorem add_div_left {x z : ℕ} (H : z > 0) : (x + z) div z = succ (x div z) :=
+theorem add_div_self_right (x : ℕ) {z : ℕ} (H : z > 0) : (x + z) div z = succ (x div z) :=
 calc
   (x + z) div z = if 0 < z ∧ z ≤ x + z then (x + z - z) div z + 1 else 0 : !divide_def
             ... = (x + z - z) div z + 1 : if_pos (and.intro H (le_add_left z x))
             ... = succ (x div z)        : {!add_sub_cancel}
 
-theorem add_div_right {x z : ℕ} (H : x > 0) : (x + z) div x = succ (z div x) :=
-!add.comm ▸ add_div_left H
+theorem add_div_self_left {x : ℕ} (z : ℕ) (H : x > 0) : (x + z) div x = succ (z div x) :=
+!add.comm ▸ !add_div_self_right H
 
-theorem add_mul_div_left {x y z : ℕ} (H : z > 0) : (x + y * z) div z = x div z + y :=
+theorem add_mul_div_self_right {x y z : ℕ} (H : z > 0) : (x + y * z) div z = x div z + y :=
 induction_on y
   (calc (x + zero * z) div z = (x + zero) div z : zero_mul
                        ...   = x div z          : add_zero
@@ -58,11 +58,21 @@ induction_on y
   (take y,
     assume IH : (x + y * z) div z = x div z + y, calc
       (x + succ y * z) div z = (x + y * z + z) div z    : by simp
-                         ... = succ ((x + y * z) div z) : add_div_left H
+                         ... = succ ((x + y * z) div z) : !add_div_self_right H
                          ... = x div z + succ y         : by simp)
 
-theorem add_mul_div_right {x y z : ℕ} (H : y > 0) : (x + y * z) div y = x div y + z :=
-!mul.comm ▸ add_mul_div_left H
+theorem add_mul_div_self_left (x z : ℕ) {y : ℕ} (H : y > 0) : (x + y * z) div y = x div y + z :=
+!mul.comm ▸ add_mul_div_self_right H
+
+theorem mul_div_self_right (m : ℕ) {n : ℕ} (H : n > 0) : m * n div n = m :=
+calc
+  m * n div n = (0 + m * n) div n : zero_add
+          ... = 0 div n + m       : add_mul_div_self_right H
+          ... = 0 + m             : zero_div
+          ... = m                 : zero_add
+
+theorem mul_div_self_left {m : ℕ} (n : ℕ) (H : m > 0) : m * n div m = n :=
+!mul.comm ▸ !mul_div_self_right H
 
 private definition mod.F (x : nat) (f : Π x₁, x₁ < x → nat → nat) (y : nat) : nat :=
 if H : 0 < y ∧ y ≤ x then f (x - y) (div_rec_lemma H) y else x
@@ -95,7 +105,7 @@ calc
 theorem add_mod_right {x z : ℕ} (H : x > 0) : (x + z) mod x = z mod x :=
 !add.comm ▸ add_mod_left H
 
-theorem add_mul_mod_left {x y z : ℕ} (H : z > 0) : (x + y * z) mod z = x mod z :=
+theorem add_mul_mod_self_right {x y z : ℕ} (H : z > 0) : (x + y * z) mod z = x mod z :=
 induction_on y
   (calc (x + zero * z) mod z = (x + zero) mod z : zero_mul
                          ... = x mod z          : add_zero)
@@ -107,14 +117,14 @@ induction_on y
                          ... = (x + y * z) mod z       : add_mod_left H
                          ... = x mod z                 : IH)
 
-theorem add_mul_mod_right {x y z : ℕ} (H : y > 0) : (x + y * z) mod y = x mod y :=
-!mul.comm ▸ add_mul_mod_left H
+theorem add_mul_mod_self_left {x y z : ℕ} (H : y > 0) : (x + y * z) mod y = x mod y :=
+!mul.comm ▸ add_mul_mod_self_right H
 
 theorem mul_mod_left {m n : ℕ} : (m * n) mod n = 0 :=
 by_cases_zero_pos n (by simp)
   (take n,
     assume npos : n > 0,
-    (by simp) ▸ (@add_mul_mod_left 0 m _ npos))
+    (by simp) ▸ (@add_mul_mod_self_right 0 m _ npos))
 
 theorem mul_mod_right {m n : ℕ} : (m * n) mod m = 0 :=
 !mul.comm ▸ !mul_mod_left
@@ -181,10 +191,10 @@ theorem eq_remainder {y : ℕ} (H : y > 0) {q1 r1 q2 r2 : ℕ} (H1 : r1 < y) (H2
   (H3 : q1 * y + r1 = q2 * y + r2) : r1 = r2 :=
 calc
   r1 = r1 mod y : by simp
-    ... = (r1 + q1 * y) mod y : (add_mul_mod_left H)⁻¹
+    ... = (r1 + q1 * y) mod y : (add_mul_mod_self_right H)⁻¹
     ... = (q1 * y + r1) mod y : add.comm
     ... = (r2 + q2 * y) mod y : by simp
-    ... = r2 mod y            : add_mul_mod_left H
+    ... = r2 mod y            : add_mul_mod_self_right H
     ... = r2                  : by simp
 
 theorem eq_quotient {y : ℕ} (H : y > 0) {q1 r1 q2 r2 : ℕ} (H1 : r1 < y) (H2 : r2 < y)
@@ -194,7 +204,7 @@ have H5 : q1 * y = q2 * y, from add.cancel_right H4,
 have H6 : y > 0, from lt_of_le_of_lt !zero_le H1,
 show q1 = q2, from eq_of_mul_eq_mul_right H6 H5
 
-theorem mul_div_mul_left {z x y : ℕ} (zpos : z > 0) : (z * x) div (z * y) = x div y :=
+theorem mul_div_mul_left {z : ℕ} (x y : ℕ) (zpos : z > 0) : (z * x) div (z * y) = x div y :=
 by_cases -- (y = 0)
   (assume H : y = 0, by simp)
   (assume H : y ≠ 0,
@@ -210,35 +220,43 @@ by_cases -- (y = 0)
           ... = (x div y) * (z * y) + z * (x mod y)                   : mul.left_comm))
 
 theorem mul_div_mul_right {x z y : ℕ} (zpos : z > 0) : (x * z) div (y * z) = x div y :=
-!mul.comm ▸ !mul.comm ▸ mul_div_mul_left zpos
+!mul.comm ▸ !mul.comm ▸ !mul_div_mul_left zpos
 
-theorem mul_mod_mul_left {z x y : ℕ} (zpos : z > 0) : (z * x) mod (z * y) = z * (x mod y) :=
-by_cases -- (y = 0)
-  (assume H : y = 0, by simp)
-  (assume H : y ≠ 0,
-    have ypos : y > 0, from pos_of_ne_zero H,
-    have zypos : z * y > 0, from mul_pos zpos ypos,
-    have H1 : (z * x) mod (z * y) < z * y, from mod_lt zypos,
-    have H2 : z * (x mod y) < z * y, from mul_lt_mul_of_pos_left (mod_lt ypos) zpos,
-    eq_remainder zypos H1 H2
-      (calc
-        ((z * x) div (z * y)) * (z * y) + (z * x) mod (z * y) = z * x : eq_div_mul_add_mod
-          ... = z * (x div y * y + x mod y)                           : eq_div_mul_add_mod
-          ... = z * (x div y * y) + z * (x mod y)                     : mul.left_distrib
-          ... = (x div y) * (z * y) + z * (x mod y)                   : mul.left_comm))
+theorem mul_mod_mul_left (z x y : ℕ) : (z * x) mod (z * y) = z * (x mod y) :=
+or.elim (eq_zero_or_pos z)
+  (assume H : z = 0,
+    calc
+      (z * x) mod (z * y) = (0 * x) mod (z * y) : H
+                      ... = 0 mod (z * y)       : zero_mul
+                      ... = 0                   : zero_mod
+                      ... = 0 * (x mod y)       : zero_mul
+                      ... = z * (x mod y)       : H)
+  (assume zpos : z > 0,
+    or.elim (eq_zero_or_pos y)
+      (assume H : y = 0, by simp)
+      (assume ypos : y > 0,
+        have zypos : z * y > 0, from mul_pos zpos ypos,
+        have H1 : (z * x) mod (z * y) < z * y, from mod_lt zypos,
+        have H2 : z * (x mod y) < z * y, from mul_lt_mul_of_pos_left (mod_lt ypos) zpos,
+        eq_remainder zypos H1 H2
+          (calc
+            ((z * x) div (z * y)) * (z * y) + (z * x) mod (z * y) = z * x : eq_div_mul_add_mod
+              ... = z * (x div y * y + x mod y)                           : eq_div_mul_add_mod
+              ... = z * (x div y * y) + z * (x mod y)                     : mul.left_distrib
+              ... = (x div y) * (z * y) + z * (x mod y)                   : mul.left_comm)))
 
-theorem mul_mod_mul_right {x z y : ℕ} (zpos : z > 0) : (x * z) mod (y * z) = (x mod y) * z :=
-mul.comm z x ▸ mul.comm z y ▸ !mul.comm ▸ mul_mod_mul_left zpos
+theorem mul_mod_mul_right (x z y : ℕ) : (x * z) mod (y * z) = (x mod y) * z :=
+mul.comm z x ▸ mul.comm z y ▸ !mul.comm ▸ !mul_mod_mul_left
 
-theorem mod_one (x : ℕ) : x mod 1 = 0 :=
-have H1 : x mod 1 < 1, from mod_lt !succ_pos,
+theorem mod_one (n : ℕ) : n mod 1 = 0 :=
+have H1 : n mod 1 < 1, from mod_lt !succ_pos,
 eq_zero_of_le_zero (le_of_lt_succ H1)
 
 theorem mod_self (n : ℕ) : n mod n = 0 :=
 cases_on n (by simp)
   (take n,
     have H : (succ n * 1) mod (succ n * 1) = succ n * (1 mod 1),
-      from mul_mod_mul_left !succ_pos,
+      from !mul_mod_mul_left,
     (by simp) ▸ H)
 
 theorem div_one (n : ℕ) : n div 1 = n :=
@@ -246,34 +264,40 @@ have H : n div 1 * 1 + n mod 1 = n, from eq_div_mul_add_mod⁻¹,
 (by simp) ▸ H
 
 theorem div_self {n : ℕ} (H : n > 0) : n div n = 1 :=
-have H1 : (n * 1) div (n * 1) = 1 div 1, from mul_div_mul_left H,
+have H1 : (n * 1) div (n * 1) = 1 div 1, from !mul_div_mul_left H,
 (by simp) ▸ H1
 
-theorem div_mul_eq_of_mod_eq_zero {x y : ℕ} (H : x mod y = 0) : x div y * y = x :=
+theorem div_mul_cancel_of_mod_eq_zero {m n : ℕ} (H : m mod n = 0) : m div n * n = m :=
 (calc
-  x = x div y * y + x mod y : eq_div_mul_add_mod
-    ... = x div y * y + 0 : H
-    ... = x div y * y : !add_zero)⁻¹
+  m = m div n * n + m mod n : eq_div_mul_add_mod
+    ... = m div n * n + 0 : H
+    ... = m div n * n : !add_zero)⁻¹
+
+theorem mul_div_cancel_of_mod_eq_zero {m n : ℕ} (H : m mod n = 0) : n * (m div n) = m :=
+!mul.comm ▸ div_mul_cancel_of_mod_eq_zero H
 
 /- divides -/
 
-theorem dvd_of_mod_eq_zero {x y : ℕ} (H : y mod x = 0) : x | y :=
-dvd.intro (!mul.comm ▸ div_mul_eq_of_mod_eq_zero H)
+theorem dvd_of_mod_eq_zero {m n : ℕ} (H : n mod m = 0) : m | n :=
+dvd.intro (!mul.comm ▸ div_mul_cancel_of_mod_eq_zero H)
 
-theorem mod_eq_zero_of_dvd {x y : ℕ} (H : x | y) : y mod x = 0 :=
+theorem mod_eq_zero_of_dvd {m n : ℕ} (H : m | n) : n mod m = 0 :=
 dvd.elim H
   (take z,
-    assume H1 : x * z = y,
-    H1 ▸ !mul_mod_right)
+    assume H1 : n = m * z,
+    H1⁻¹ ▸ !mul_mod_right)
 
-theorem dvd_iff_mod_eq_zero (x y : ℕ) : x | y ↔ y mod x = 0 :=
+theorem dvd_iff_mod_eq_zero (m n : ℕ) : m | n ↔ n mod m = 0 :=
 iff.intro mod_eq_zero_of_dvd dvd_of_mod_eq_zero
 
 definition dvd.decidable_rel [instance] : decidable_rel dvd :=
 take m n, decidable_of_decidable_of_iff _ (iff.symm !dvd_iff_mod_eq_zero)
 
-theorem div_mul_eq_of_dvd {x y : ℕ} (H : y | x) : x div y * y = x :=
-div_mul_eq_of_mod_eq_zero (mod_eq_zero_of_dvd H)
+theorem div_mul_cancel {m n : ℕ} (H : n | m) : m div n * n = m :=
+div_mul_cancel_of_mod_eq_zero (mod_eq_zero_of_dvd H)
+
+theorem mul_div_cancel {m n : ℕ} (H : n | m) : n * (m div n) = m :=
+!mul.comm ▸ div_mul_cancel H
 
 theorem dvd_of_dvd_add_left {m n1 n2 : ℕ} : m | (n1 + n2) → m | n1 → m | n2 :=
 by_cases_zero_pos m
@@ -290,13 +314,13 @@ by_cases_zero_pos m
     assume H1 : m | (n1 + n2),
     assume H2 : m | n1,
     have H3 : n1 + n2 = n1 + n2 div m * m, from calc
-      n1 + n2 = (n1 + n2) div m * m         : div_mul_eq_of_dvd H1
-          ... = (n1 div m * m + n2) div m * m : div_mul_eq_of_dvd H2
+      n1 + n2 = (n1 + n2) div m * m         : div_mul_cancel H1
+          ... = (n1 div m * m + n2) div m * m : div_mul_cancel H2
           ... = (n2 + n1 div m * m) div m * m : add.comm
-          ... = (n2 div m + n1 div m) * m     : add_mul_div_left mpos
+          ... = (n2 div m + n1 div m) * m     : add_mul_div_self_right mpos
           ... = n2 div m * m + n1 div m * m   : mul.right_distrib
           ... = n1 div m * m + n2 div m * m   : add.comm
-          ... = n1 + n2 div m * m             : div_mul_eq_of_dvd H2,
+          ... = n1 + n2 div m * m             : div_mul_cancel H2,
     have H4 : n2 = n2 div m * m, from add.cancel_left H3,
     have H5 : m * (n2 div m) = n2, from !mul.comm ▸ H4⁻¹,
     dvd.intro H5)
@@ -320,14 +344,35 @@ by_cases_zero_pos n
     assume Hpos : n > 0,
     assume H1 : m | n,
     assume H2 : n | m,
-    obtain k (Hk : m * k = n), from dvd.ex H1,
-    obtain l (Hl : n * l = m), from dvd.ex H2,
-    have H3 : n * (l * k) = n, from !mul.assoc ▸ Hl⁻¹ ▸ Hk,
+    obtain k (Hk : n = m * k), from exists_eq_mul_right_of_dvd H1,
+    obtain l (Hl : m = n * l), from exists_eq_mul_right_of_dvd H2,
+    have H3 : n * (l * k) = n, from !mul.assoc ▸ Hl ▸ Hk⁻¹,
     have H4 : l * k = 1, from eq_one_of_mul_eq_self_right Hpos H3,
     have H5 : k = 1, from eq_one_of_mul_eq_one_left H4,
-    show m = n, from (mul_one m)⁻¹ ⬝ (H5 ▸ Hk))
+    show m = n, from (mul_one m)⁻¹ ⬝ (H5 ▸ Hk⁻¹))
 
-/- gcd and lcm -/
+theorem mul_div_assoc (m : ℕ) {n k : ℕ} (H : k | n) : m * n div k = m * (n div k) :=
+or.elim (eq_zero_or_pos k)
+  (assume H1 : k = 0,
+    calc
+      m * n div k = m * n div 0   : H1
+              ... = 0             : div_zero
+              ... = m * 0         : mul_zero m    -- TODO: why do we have to specify m here?
+              ... = m * (n div 0) : div_zero
+              ... = m * (n div k) : H1)
+  (assume H1 : k > 0,
+    have H2 : n = n div k * k, from (div_mul_cancel H)⁻¹,
+      calc
+        m * n div k = m * (n div k * k) div k : H2
+                ... = m * (n div k) * k div k : mul.assoc
+                ... = m * (n div k)           : mul_div_self_right _ H1)
+
+theorem eq_mul_of_div_eq {m n k : ℕ} (H1 : m | n) (H2 : n div m = k) : n = m * k :=
+eq.symm (calc
+  m * k = m * (n div m) : H2
+    ... = n             : mul_div_cancel H1)
+
+/- gcd -/
 
 private definition pair_nat.lt : nat × nat → nat × nat → Prop := measure pr₂
 private definition pair_nat.lt.wf : well_founded pair_nat.lt :=
@@ -346,35 +391,32 @@ prod.cases_on p₁ (λx y, cases_on y
 
 definition gcd (x y : nat) := fix gcd.F (pair x y)
 
-theorem gcd_zero (x : nat) : gcd x 0 = x :=
+theorem gcd_zero_right (x : nat) : gcd x 0 = x :=
 well_founded.fix_eq gcd.F (x, 0)
 
 theorem gcd_succ (x y : nat) : gcd x (succ y) = gcd (succ y) (x mod succ y) :=
 well_founded.fix_eq gcd.F (x, succ y)
 
-theorem gcd_one (n : ℕ) : gcd n 1 = 1 :=
+theorem gcd_one_right (n : ℕ) : gcd n 1 = 1 :=
 calc gcd n 1 = gcd 1 (n mod 1) : gcd_succ n zero
          ... = gcd 1 0         : mod_one
-         ... = 1               : gcd_zero
+         ... = 1               : gcd_zero_right
 
 theorem gcd_def (x y : ℕ) : gcd x y = if y = 0 then x else gcd y (x mod y) :=
 cases_on y
-  (calc gcd x 0 = x                                          : gcd_zero x
+  (calc gcd x 0 = x                                          : gcd_zero_right x
            ...  = if 0 = 0 then x else gcd zero (x mod zero) : (if_pos rfl)⁻¹)
   (λy₁, calc
     gcd x (succ y₁) = gcd (succ y₁) (x mod succ y₁)                  : gcd_succ x y₁
       ... = if succ y₁ = 0 then x else gcd (succ y₁) (x mod succ y₁) : (if_neg (succ_ne_zero y₁))⁻¹)
 
-theorem gcd_rec (m : ℕ) {n : ℕ} (H : n > 0) : gcd m n = gcd n (m mod n) :=
-gcd_def m n ⬝ if_neg (ne_zero_of_pos H)
-
 theorem gcd_self (n : ℕ) : gcd n n = n :=
 cases_on n
   rfl
   (λn₁, calc
     gcd (succ n₁) (succ n₁) = gcd (succ n₁) (succ n₁ mod succ n₁) : gcd_succ (succ n₁) n₁
                       ...   = gcd (succ n₁) 0                     : mod_self (succ n₁)
-                      ...   = succ n₁                             : gcd_zero)
+                      ...   = succ n₁                             : gcd_zero_right)
 
 theorem gcd_zero_left (n : nat) : gcd 0 n = n :=
 cases_on n
@@ -382,7 +424,18 @@ cases_on n
   (λ n₁, calc
     gcd 0 (succ n₁) = gcd (succ n₁) (0 mod succ n₁) : gcd_succ
                 ... = gcd (succ n₁) 0               : zero_mod
-                ... = (succ n₁)                     : gcd_zero)
+                ... = (succ n₁)                     : gcd_zero_right)
+
+theorem gcd_rec_of_pos (m : ℕ) {n : ℕ} (H : n > 0) : gcd m n = gcd n (m mod n) :=
+gcd_def m n ⬝ if_neg (ne_zero_of_pos H)
+
+theorem gcd_rec (m n : ℕ) : gcd m n = gcd n (m mod n) :=
+by_cases_zero_pos n
+  (calc
+    gcd m 0 = m               : gcd_zero_right
+        ... = gcd 0 m         : gcd_zero_left
+        ... = gcd 0 (m mod 0) : mod_zero)
+  (take n, assume H : 0 < n, gcd_rec_of_pos m H)
 
 theorem gcd.induction {P : ℕ → ℕ → Prop}
                    (m n : ℕ)
@@ -412,7 +465,7 @@ gcd.induction m n
     have H : gcd n (m mod n) | (m div n * n + m mod n), from
       dvd_add (dvd.trans (and.elim_left IH) !dvd_mul_left) (and.elim_right IH),
     have H1 : gcd n (m mod n) | m, from eq_div_mul_add_mod⁻¹ ▸ H,
-    have gcd_eq : gcd n (m mod n) = gcd m n, from (gcd_rec _ npos)⁻¹,
+    have gcd_eq : gcd n (m mod n) = gcd m n, from !gcd_rec⁻¹,
     show (gcd m n | m) ∧ (gcd m n | n), from gcd_eq ▸ (and.intro H1 (and.elim_left IH)))
 
 theorem gcd_dvd_left (m n : ℕ) : (gcd m n | m) := and.elim_left !gcd_dvd
@@ -422,7 +475,7 @@ theorem gcd_dvd_right (m n : ℕ) : (gcd m n | n) := and.elim_right !gcd_dvd
 theorem dvd_gcd {m n k : ℕ} : k | m → k | n → k | (gcd m n) :=
 gcd.induction m n
   (take m, assume (h₁ : k | m) (h₂ : k | 0),
-   show k | gcd m 0, from !gcd_zero⁻¹ ▸ h₁)
+   show k | gcd m 0, from !gcd_zero_right⁻¹ ▸ h₁)
   (take m n,
     assume npos : n > 0,
     assume IH : k | n → k | (m mod n) → k | gcd n (m mod n),
@@ -430,7 +483,7 @@ gcd.induction m n
     assume H2 : k | n,
     have H3 : k | m div n * n + m mod n, from eq_div_mul_add_mod ▸ H1,
     have H4 : k | m mod n, from nat.dvd_of_dvd_add_left H3 (dvd.trans H2 (by simp)),
-    have gcd_eq : gcd n (m mod n) = gcd m n, from (gcd_rec _ npos)⁻¹,
+    have gcd_eq : gcd n (m mod n) = gcd m n, from !gcd_rec⁻¹,
     show k | gcd m n, from gcd_eq ▸ IH H2 H4)
 
 theorem gcd.comm (m n : ℕ) : gcd m n = gcd n m :=
@@ -447,4 +500,132 @@ dvd.antisymm
     (dvd_gcd !gcd_dvd_left (dvd.trans !gcd_dvd_right !gcd_dvd_left))
     (dvd.trans !gcd_dvd_right !gcd_dvd_right))
 
+theorem gcd_one_left (m : ℕ) : gcd 1 m = 1 :=
+!gcd.comm ⬝ !gcd_one_right
+
+theorem gcd_mul_left (m n k : ℕ) : gcd (m * n) (m * k) = m * gcd n k :=
+gcd.induction n k
+  (take n,
+    calc
+      gcd (m * n) (m * 0) = gcd (m * n) 0 : mul_zero
+                      ... = m * n         : gcd_zero_right
+                      ... = m * gcd n 0   : gcd_zero_right)
+  (take n k,
+    assume H : 0 < k,
+    assume IH : gcd (m * k) (m * (n mod k)) = m * gcd k (n mod k),
+    calc
+      gcd (m * n) (m * k) = gcd (m * k) (m * n mod (m * k)) : !gcd_rec
+                      ... = gcd (m * k) (m * (n mod k))     : mul_mod_mul_left
+                      ... = m * gcd k (n mod k)             : IH
+                      ... = m * gcd n k                     : !gcd_rec)
+
+theorem gcd_mul_right (m n k : ℕ) : gcd (m * n) (k * n) = gcd m k * n :=
+calc
+  gcd (m * n) (k * n) = gcd (n * m) (k * n) : mul.comm
+                  ... = gcd (n * m) (n * k) : mul.comm
+                  ... = n * gcd m k         : gcd_mul_left
+                  ... = gcd m k * n         : mul.comm
+
+theorem gcd_pos_of_pos_left {m : ℕ} (n : ℕ) (mpos : m > 0) : gcd m n > 0 :=
+pos_of_dvd_of_pos !gcd_dvd_left mpos
+
+theorem gcd_pos_of_pos_right (m : ℕ) {n : ℕ} (npos : n > 0) : gcd m n > 0 :=
+pos_of_dvd_of_pos !gcd_dvd_right npos
+
+/- lcm -/
+
+definition lcm (m n : ℕ) : ℕ := m * n div (gcd m n)
+
+theorem lcm.comm (m n : ℕ) : lcm m n = lcm n m :=
+calc
+  lcm m n = m * n div gcd m n : rfl
+      ... = n * m div gcd m n : mul.comm
+      ... = n * m div gcd n m : gcd.comm
+      ... = lcm n m           : rfl
+
+theorem lcm_zero_left (m : ℕ) : lcm 0 m = 0 :=
+calc
+  lcm 0 m = 0 * m div gcd 0 m : rfl
+      ... = 0 div gcd 0 m     : zero_mul
+      ... = 0                 : zero_div
+
+theorem lcm_zero_right (m : ℕ) : lcm m 0 = 0 := !lcm.comm ▸ !lcm_zero_left
+
+theorem lcm_one_left (m : ℕ) : lcm 1 m = m :=
+calc
+  lcm 1 m = 1 * m div gcd 1 m : rfl
+      ... = m div gcd 1 m     : one_mul
+      ... = m div 1           : gcd_one_left
+      ... = m                 : div_one
+
+theorem lcm_one_right (m : ℕ) : lcm m 1 = m := !lcm.comm ▸ !lcm_one_left
+
+theorem lcm_self (m : ℕ) : lcm m m = m :=
+have H : m * m div m = m, from
+  by_cases_zero_pos m !div_zero (take m, assume H1 : m > 0, !mul_div_self_right H1),
+calc
+  lcm m m = m * m div gcd m m : rfl
+      ... = m * m div m       : gcd_self
+      ... = m                 : H
+
+theorem dvd_lcm_left (m n : ℕ) : m | lcm m n :=
+have H : lcm m n = m * (n div gcd m n), from mul_div_assoc _ !gcd_dvd_right,
+dvd.intro H⁻¹
+
+theorem dvd_lcm_right (m n : ℕ) : n | lcm m n :=
+!lcm.comm ▸ !dvd_lcm_left
+
+theorem gcd_mul_lcm (m n : ℕ) : gcd m n * lcm m n = m * n :=
+eq.symm (eq_mul_of_div_eq (dvd.trans !gcd_dvd_left !dvd_mul_right) rfl)
+
+theorem lcm_dvd {m n k : ℕ} (H1 : m | k) (H2 : n | k) : lcm m n | k :=
+or.elim (eq_zero_or_pos k)
+  (assume kzero : k = 0, !kzero⁻¹ ▸ !dvd_zero)
+  (assume kpos : k > 0,
+    have mpos : m > 0, from pos_of_dvd_of_pos H1 kpos,
+    have npos : n > 0, from pos_of_dvd_of_pos H2 kpos,
+    have gcd_pos : gcd m n > 0, from !gcd_pos_of_pos_left mpos,
+    obtain p (km : k = m * p), from exists_eq_mul_right_of_dvd H1,
+    obtain q (kn : k = n * q), from exists_eq_mul_right_of_dvd H2,
+    have ppos : p > 0, from pos_of_mul_pos_left (km ▸ kpos),
+    have qpos : q > 0, from pos_of_mul_pos_left (kn ▸ kpos),
+    have H3 : p * q * (m * n * gcd p q) = p * q * (gcd m n * k), from
+    calc
+      p * q * (m * n * gcd p q) = p * (q * (m * n * gcd p q)) : mul.assoc
+        ... = p * (q * (m * (n * gcd p q)))                   : mul.assoc
+        ... = p * (m * (q * (n * gcd p q)))                   : mul.left_comm
+        ... = p * m * (q * (n * gcd p q))                     : mul.assoc
+        ... = p * m * (q * n * gcd p q)                       : mul.assoc
+        ... = m * p * (q * n * gcd p q)                       : mul.comm
+        ... = k * (q * n * gcd p q)                           : km
+        ... = k * (n * q * gcd p q)                           : mul.comm
+        ... = k * (k * gcd p q)                               : kn
+        ... = k * gcd (k * p) (k * q)                         : gcd_mul_left
+        ... = k * gcd (n * q * p) (k * q)                     : kn
+        ... = k * gcd (n * q * p) (m * p * q)                 : km
+        ... = k * gcd (n * (q * p)) (m * p * q)               : mul.assoc
+        ... = k * gcd (n * (q * p)) (m * (p * q))             : mul.assoc
+        ... = k * gcd (n * (p * q)) (m * (p * q))             : mul.comm
+        ... = k * (gcd n m * (p * q))                         : gcd_mul_right
+        ... = gcd n m * (p * q) * k                           : mul.comm
+        ... = p * q * gcd n m * k                             : mul.comm
+        ... = p * q * (gcd n m * k)                           : mul.assoc
+        ... = p * q * (gcd m n * k)                           : gcd.comm,
+    have H4 : m * n * gcd p q = gcd m n * k,
+      from !eq_of_mul_eq_mul_left (mul_pos ppos qpos) H3,
+    have H5 : gcd m n * (lcm m n * gcd p q) = gcd m n * k,
+      from !mul.assoc ▸ !gcd_mul_lcm⁻¹ ▸ H4,
+    have H6 : lcm m n * gcd p q = k,
+      from !eq_of_mul_eq_mul_left gcd_pos H5,
+    dvd.intro H6)
+
+theorem lcm_assoc (m n k : ℕ) : lcm (lcm m n) k = lcm m (lcm n k) :=
+dvd.antisymm
+  (lcm_dvd
+    (lcm_dvd !dvd_lcm_left (dvd.trans !dvd_lcm_left !dvd_lcm_right))
+    (dvd.trans !dvd_lcm_right !dvd_lcm_right))
+  (lcm_dvd
+    (dvd.trans !dvd_lcm_left !dvd_lcm_left)
+    (lcm_dvd (dvd.trans !dvd_lcm_right !dvd_lcm_left) !dvd_lcm_right))
+
 end nat
diff --git a/library/data/nat/order.lean b/library/data/nat/order.lean
index 400363b0a..3762f481d 100644
--- a/library/data/nat/order.lean
+++ b/library/data/nat/order.lean
@@ -401,13 +401,13 @@ strong_induction_on a (
 theorem by_cases_zero_pos {P : ℕ → Prop} (y : ℕ) (H0 : P 0) (H1 : ∀ {y : nat}, y > 0 → P y) : P y :=
 cases_on y H0 (take y, H1 !succ_pos)
 
-theorem zero_or_pos {n : ℕ} : n = 0 ∨ n > 0 :=
+theorem eq_zero_or_pos (n : ℕ) : n = 0 ∨ n > 0 :=
 or_of_or_of_imp_left
   (or.swap (lt_or_eq_of_le !zero_le))
   (take H : 0 = n, H⁻¹)
 
 theorem pos_of_ne_zero {n : ℕ} (H : n ≠ 0) : n > 0 :=
-or.elim zero_or_pos (take H2 : n = 0, absurd H2 H) (take H2 : n > 0, H2)
+or.elim !eq_zero_or_pos (take H2 : n = 0, absurd H2 H) (take H2 : n > 0, H2)
 
 theorem ne_zero_of_pos {n : ℕ} (H : n > 0) : n ≠ 0 :=
 ne.symm (ne_of_lt H)
@@ -415,6 +415,12 @@ ne.symm (ne_of_lt H)
 theorem exists_eq_succ_of_pos {n : ℕ} (H : n > 0) : exists l, n = succ l :=
 exists_eq_succ_of_lt H
 
+theorem pos_of_dvd_of_pos {m n : ℕ} (H1 : m | n) (H2 : n > 0) : m > 0 :=
+pos_of_ne_zero
+  (assume H3 : m = 0,
+    have H4 : n = 0, from eq_zero_of_zero_dvd (H3 ▸ H1),
+    ne_of_lt H2 H4⁻¹)
+
 /- multiplication -/
 
 theorem mul_lt_mul_of_le_of_lt {n m k l : ℕ} (Hk : k > 0) (H1 : n ≤ k) (H2 : m < l) :
@@ -441,7 +447,7 @@ theorem eq_of_mul_eq_mul_right {n m k : ℕ} (Hm : m > 0) (H : n * m = k * m) :
 eq_of_mul_eq_mul_left Hm (!mul.comm ▸ !mul.comm ▸ H)
 
 theorem eq_zero_or_eq_of_mul_eq_mul_left {n m k : ℕ} (H : n * m = n * k) : n = 0 ∨ m = k :=
-or_of_or_of_imp_right zero_or_pos
+or_of_or_of_imp_right !eq_zero_or_pos
   (assume Hn : n > 0, eq_of_mul_eq_mul_left Hn H)
 
 theorem eq_zero_or_eq_of_mul_eq_mul_right  {n m k : ℕ} (H : n * m = k * m) : m = 0 ∨ n = k :=