diff --git a/library/init/core.lean b/library/init/core.lean
index f5bcfde404..c22494505c 100644
--- a/library/init/core.lean
+++ b/library/init/core.lean
@@ -42,6 +42,7 @@ reserve infixl ` - `:65
 reserve infixl ` * `:70
 reserve infixl ` / `:70
 reserve infixl ` % `:70
+reserve infixl ` %ₙ `:70
 reserve prefix `-`:100
 reserve infixr ` ^ `:80
 
@@ -324,6 +325,7 @@ class has_sub      (α : Type u) := (sub : α → α → α)
 class has_div      (α : Type u) := (div : α → α → α)
 class has_dvd      (α : Type u) := (dvd : α → α → Prop)
 class has_mod      (α : Type u) := (mod : α → α → α)
+class has_modn     (α : Type u) := (modn : α → nat → α)
 class has_le       (α : Type u) := (le : α → α → Prop)
 class has_lt       (α : Type u) := (lt : α → α → Prop)
 class has_append   (α : Type u) := (append : α → α → α)
@@ -359,6 +361,7 @@ infix -        := has_sub.sub
 infix /        := has_div.div
 infix ∣        := has_dvd.dvd
 infix %        := has_mod.mod
+infix %ₙ       := has_modn.modn
 prefix -       := has_neg.neg
 infix <=       := has_le.le
 infix ≤        := has_le.le
diff --git a/library/init/data/array/basic.lean b/library/init/data/array/basic.lean
index 822ba12fe8..aefa831a16 100644
--- a/library/init/data/array/basic.lean
+++ b/library/init/data/array/basic.lean
@@ -56,11 +56,19 @@ def write' (a : array α) (i : nat) (v : α) : array α :=
 if h : i < a.sz then a.write ⟨i, h⟩ v else a
 
 /- TODO: add builtin -/
-def uread [inhabited α] (a : array α) (i : usize) : α :=
+def uread (a : array α) (i : uint32) (h : i.val < a.sz) : α :=
+a.read ⟨i.val, h⟩
+
+/- TODO: add builtin -/
+def uwrite (a : array α) (i : uint32) (v : α) (h : i.val < a.sz) : array α :=
+a.write ⟨i.val, h⟩ v
+
+/- TODO: add builtin -/
+def uread' [inhabited α] (a : array α) (i : uint32) : α :=
 if h : i.val < a.sz then a.read ⟨i.val, h⟩ else default α
 
 /- TODO: add builtin -/
-def uwrite (a : array α) (i : usize) (v : α) : array α :=
+def uwrite' (a : array α) (i : uint32) (v : α) : array α :=
 if h : i.val < a.sz then a.write ⟨i.val, h⟩ v else a
 
 /- TODO: mark as builtin -/
diff --git a/library/init/data/fin/basic.lean b/library/init/data/fin/basic.lean
index df46273fe6..5166020b8a 100644
--- a/library/init/data/fin/basic.lean
+++ b/library/init/data/fin/basic.lean
@@ -51,7 +51,7 @@ protected def sub : fin n → fin n → fin n
 | ⟨a, h⟩ ⟨b, _⟩ := ⟨(a + (n - b)) % n, mlt h⟩
 
 /-
-Remark: sub/mod/div can be defined without using (% n), but
+Remark: mod/div/modn can be defined without using (% n), but
 we are trying to minimize the number of nat theorems
 needed to boostrap Lean.
 -/
@@ -62,6 +62,9 @@ protected def mod : fin n → fin n → fin n
 protected def div : fin n → fin n → fin n
 | ⟨a, h⟩ ⟨b, _⟩ := ⟨(a / b) % n, mlt h⟩
 
+protected def modn : fin n → nat → fin n
+| ⟨a, h⟩ m := ⟨(a % m) % n, mlt h⟩
+
 instance : has_zero (fin (succ n)) := ⟨⟨0, succ_pos n⟩⟩
 instance : has_one (fin (succ n))  := ⟨of_nat 1⟩
 instance : has_add (fin n)         := ⟨fin.add⟩
@@ -69,6 +72,7 @@ instance : has_sub (fin n)         := ⟨fin.sub⟩
 instance : has_mul (fin n)         := ⟨fin.mul⟩
 instance : has_mod (fin n)         := ⟨fin.mod⟩
 instance : has_div (fin n)         := ⟨fin.div⟩
+instance : has_modn (fin n)        := ⟨fin.modn⟩
 
 theorem eq_of_veq : ∀ {i j : fin n}, (val i) = (val j) → i = j
 | ⟨iv, ilt₁⟩ ⟨.(iv), ilt₂⟩ rfl := rfl
@@ -82,6 +86,9 @@ theorem ne_of_vne {i j : fin n} (h : val i ≠ val j) : i ≠ j :=
 theorem vne_of_ne {i j : fin n} (h : i ≠ j) : val i ≠ val j :=
 λ h', absurd (eq_of_veq h') h
 
+theorem modn_lt : ∀ {m : nat} (i : fin n), m > 0 → (i %ₙ m).val < m
+| m ⟨a, h⟩ hp :=  nat.lt_of_le_of_lt (mod_le _ _) (mod_lt _ hp)
+
 end fin
 
 open fin
diff --git a/library/init/data/hashmap/basic.lean b/library/init/data/hashmap/basic.lean
index ec9a78463e..c1287a6c2d 100644
--- a/library/init/data/hashmap/basic.lean
+++ b/library/init/data/hashmap/basic.lean
@@ -10,10 +10,10 @@ universes u v w
 def bucket_array (α : Type u) (β : α → Type v) :=
 { b : array (list (Σ a, β a)) // b.sz > 0 }
 
-def bucket_array.write {α : Type u} {β : α → Type v} (data : bucket_array α β) (i : fin data.val.sz) (d : list (Σ a, β a)) : bucket_array α β :=
-⟨ data.val.write i d,
-  calc (data.val.write i d).sz = data.val.sz : array.sz_write_eq _ _ _
-                     ...       > 0           : data.property ⟩
+def bucket_array.uwrite {α : Type u} {β : α → Type v} (data : bucket_array α β) (i : uint32) (d : list (Σ a, β a)) (h : i.val < data.val.sz) : bucket_array α β :=
+⟨ data.val.uwrite i d h,
+  calc (data.val.uwrite i d h).sz = data.val.sz : array.sz_write_eq _ _ _
+                     ...          > 0           : data.property ⟩
 
 structure hashmap_imp (α : Type u) (β : α → Type v) :=
 (size       : nat)
@@ -34,12 +34,12 @@ let n := if nbuckets = 0 then 8 else nbuckets in
 namespace hashmap_imp
 variables {α : Type u} {β : α → Type v}
 
-def mk_idx {n : nat} (h : n > 0) (i : nat) : fin n :=
-⟨i % n, nat.mod_lt _ h⟩
+def mk_idx {n : nat} (h : n > 0) (u : uint32) : { u : uint32 // u.val < n } :=
+⟨u %ₙ n, fin.modn_lt _ h⟩
 
-def reinsert_aux (hash_fn : α → nat) (data : bucket_array α β) (a : α) (b : β a) : bucket_array α β :=
-let bidx := mk_idx data.property (hash_fn a) in
-data.write bidx (⟨a, b⟩ :: data.val.read bidx)
+def reinsert_aux (hash_fn : α → uint32) (data : bucket_array α β) (a : α) (b : β a) : bucket_array α β :=
+let ⟨i, h⟩ := mk_idx data.property (hash_fn a) in
+data.uwrite i (⟨a, b⟩ :: data.val.uread i h) h
 
 def fold_buckets {δ : Type w} (data : bucket_array α β) (d : δ) (f : δ → Π a, β a → δ) : δ :=
 data.val.foldl d (λ b d, b.foldl (λ r (p : Σ a, β a), f r p.1 p.2) d)
@@ -52,10 +52,11 @@ def find_aux [decidable_eq α] (a : α) : list (Σ a, β a) → option (β a)
 def contains_aux [decidable_eq α] (a : α) (l : list (Σ a, β a)) : bool :=
 (find_aux a l).is_some
 
-def find [decidable_eq α] (hash_fn : α → nat) (m : hashmap_imp α β) (a : α) : option (β a) :=
+def find [decidable_eq α] (hash_fn : α → uint32) (m : hashmap_imp α β) (a : α) : option (β a) :=
 match m with
-| ⟨_, buckets, nz⟩ :=
-  find_aux a (buckets.read (mk_idx nz (hash_fn a)))
+| ⟨_, buckets⟩ :=
+  let ⟨i, h⟩ := mk_idx buckets.property (hash_fn a) in
+  find_aux a (buckets.val.uread i h)
 
 def fold {δ : Type w} (m : hashmap_imp α β) (d : δ) (f : δ → Π a, β a → δ) : δ :=
 fold_buckets m.buckets d f
@@ -68,15 +69,15 @@ def erase_aux [decidable_eq α] (a : α) : list (Σ a, β a) → list (Σ a, β
 | []            := []
 | (⟨a', b'⟩::t) := if a' = a then t else ⟨a', b'⟩ :: erase_aux t
 
-def insert [decidable_eq α] (hash_fn : α → nat) (m : hashmap_imp α β) (a : α) (b : β a) : hashmap_imp α β :=
+def insert [decidable_eq α] (hash_fn : α → uint32) (m : hashmap_imp α β) (a : α) (b : β a) : hashmap_imp α β :=
 match m with
 | ⟨size, buckets⟩ :=
-  let bidx := mk_idx buckets.property (hash_fn a) in
-  let bkt  := buckets.val.read bidx in
+  let ⟨i, h⟩ := mk_idx buckets.property (hash_fn a) in
+  let bkt    := buckets.val.uread i h in
   if contains_aux a bkt
-  then ⟨size, buckets.write bidx (replace_aux a b bkt)⟩
+  then ⟨size, buckets.uwrite i (replace_aux a b bkt) h⟩
   else let size'    := size + 1 in
-       let buckets' := buckets.write bidx (⟨a, b⟩::bkt) in
+       let buckets' := buckets.uwrite i (⟨a, b⟩::bkt) h in
        if size' <= buckets.val.sz
        then ⟨size', buckets'⟩
        else let nbuckets' := buckets.val.sz * 2 in
@@ -84,32 +85,31 @@ match m with
             ⟨ size',
               fold_buckets buckets' ⟨mk_array nbuckets' [], nz'⟩ (reinsert_aux hash_fn) ⟩
 
-def erase [decidable_eq α] (hash_fn : α → nat) (m : hashmap_imp α β) (a : α) : hashmap_imp α β :=
+def erase [decidable_eq α] (hash_fn : α → uint32) (m : hashmap_imp α β) (a : α) : hashmap_imp α β :=
 match m with
 | ⟨ size, buckets ⟩ :=
-  let bidx := mk_idx buckets.property (hash_fn a) in
-  let bkt  := buckets.val.read bidx in
-  if contains_aux a bkt
-  then ⟨size - 1, buckets.write bidx $ erase_aux a bkt⟩
+  let ⟨i, h⟩ := mk_idx buckets.property (hash_fn a) in
+  let bkt    := buckets.val.uread i h in
+  if contains_aux a bkt then ⟨size - 1, buckets.uwrite i (erase_aux a bkt) h⟩
   else m
 
-inductive well_formed [decidable_eq α] (hash_fn : α → nat) : hashmap_imp α β → Prop
+inductive well_formed [decidable_eq α] (hash_fn : α → uint32) : hashmap_imp α β → Prop
 | mk_wff     : ∀ n,                     well_formed (mk_hashmap_imp n)
 | insert_wff : ∀ m a b, well_formed m → well_formed (insert hash_fn m a b)
 | erase_wff  : ∀ m a,   well_formed m → well_formed (erase hash_fn m a)
 
 end hashmap_imp
 
-def d_hashmap (α : Type u) (β : α → Type v) [decidable_eq α] (h : α → nat) :=
+def d_hashmap (α : Type u) (β : α → Type v) [decidable_eq α] (h : α → uint32) :=
 { m : hashmap_imp α β // m.well_formed h }
 
 open hashmap_imp
 
-def mk_d_hashmap {α : Type u} {β : α → Type v} [decidable_eq α] (h : α → nat) (nbuckets := 8) : d_hashmap α β h :=
+def mk_d_hashmap {α : Type u} {β : α → Type v} [decidable_eq α] (h : α → uint32) (nbuckets := 8) : d_hashmap α β h :=
 ⟨ mk_hashmap_imp nbuckets, well_formed.mk_wff h nbuckets ⟩
 
 namespace d_hashmap
-variables {α : Type u} {β : α → Type v} [decidable_eq α] {h : α → nat}
+variables {α : Type u} {β : α → Type v} [decidable_eq α] {h : α → uint32}
 
 def insert (m : d_hashmap α β h) (a : α) (b : β a) : d_hashmap α β h :=
 match m with
@@ -144,14 +144,14 @@ m.size = 0
 
 end d_hashmap
 
-def hashmap (α : Type u) (β : Type v) [decidable_eq α] (h : α → nat) :=
+def hashmap (α : Type u) (β : Type v) [decidable_eq α] (h : α → uint32) :=
 d_hashmap α (λ _, β) h
 
-def mk_hashmap {α : Type u} {β : Type v} [decidable_eq α] (h : α → nat) (nbuckets := 8) : hashmap α β h :=
+def mk_hashmap {α : Type u} {β : Type v} [decidable_eq α] (h : α → uint32) (nbuckets := 8) : hashmap α β h :=
 mk_d_hashmap h nbuckets
 
 namespace hashmap
-variables {α : Type u} {β : Type v} [decidable_eq α] {h : α → nat}
+variables {α : Type u} {β : Type v} [decidable_eq α] {h : α → uint32}
 
 @[inline] def insert (m : hashmap α β h) (a : α) (b : β) : hashmap α β h :=
 d_hashmap.insert m a b
diff --git a/library/init/data/nat/div.lean b/library/init/data/nat/div.lean
index ad52a4dbcf..26b21f6b4a 100644
--- a/library/init/data/nat/div.lean
+++ b/library/init/data/nat/div.lean
@@ -93,4 +93,11 @@ mod.induction_on x y
         have heq : x % y = x, from mod_eq_of_lt hgt,
         heq.symm ▸ hgt))
 
+theorem mod_le (x y : ℕ) : x % y ≤ x :=
+or.elim (nat.lt_or_ge x y)
+  (λ h₁ : x < y, (mod_eq_of_lt h₁).symm ▸ nat.le_refl _)
+  (λ h₁ : x ≥ y, or.elim (eq_zero_or_pos y)
+    (λ h₂ : y = 0, h₂.symm ▸ (nat.mod_zero x).symm ▸ nat.le_refl _)
+    (λ h₂ : y > 0, nat.le_trans (nat.le_of_lt (nat.mod_lt _ h₂)) h₁))
+
 end nat
diff --git a/library/init/data/uint.lean b/library/init/data/uint.lean
index f16e733860..7f8e770903 100644
--- a/library/init/data/uint.lean
+++ b/library/init/data/uint.lean
@@ -31,6 +31,7 @@ instance : has_add uint16  := ⟨fin.add⟩
 instance : has_sub uint16  := ⟨fin.sub⟩
 instance : has_mul uint16  := ⟨fin.mul⟩
 instance : has_mod uint16  := ⟨fin.mod⟩
+instance : has_modn uint16 := ⟨fin.modn⟩
 instance : has_div uint16  := ⟨fin.div⟩
 instance : has_lt uint16   := ⟨fin.lt⟩
 instance : has_le uint16   := ⟨fin.le⟩
@@ -41,6 +42,7 @@ instance : has_add uint32  := ⟨fin.add⟩
 instance : has_sub uint32  := ⟨fin.sub⟩
 instance : has_mul uint32  := ⟨fin.mul⟩
 instance : has_mod uint32  := ⟨fin.mod⟩
+instance : has_modn uint32 := ⟨fin.modn⟩
 instance : has_div uint32  := ⟨fin.div⟩
 instance : has_lt uint32   := ⟨fin.lt⟩
 instance : has_le uint32   := ⟨fin.le⟩
@@ -51,24 +53,14 @@ instance : has_add uint64  := ⟨fin.add⟩
 instance : has_sub uint64  := ⟨fin.sub⟩
 instance : has_mul uint64  := ⟨fin.mul⟩
 instance : has_mod uint64  := ⟨fin.mod⟩
+instance : has_modn uint64 := ⟨fin.modn⟩
 instance : has_div uint64  := ⟨fin.div⟩
 instance : has_lt uint64   := ⟨fin.lt⟩
 instance : has_le uint64   := ⟨fin.le⟩
 
-def uint16.of_nat (n : nat) : uint16 :=
-fin.of_nat n
-
-def uint16.to_nat (u : uint16) : nat :=
-fin.val u
-
-def uint32.of_nat (n : nat) : uint32 :=
-fin.of_nat n
-
-def uint32.to_nat (u : uint32) : nat :=
-fin.val u
-
-def uint64.of_nat (n : nat) : uint64 :=
-fin.of_nat n
-
-def uint64.to_nat (u : uint64) : nat :=
-fin.val u
+def uint16.of_nat (n : nat) : uint16 := fin.of_nat n
+def uint16.to_nat (u : uint16) : nat := fin.val u
+def uint32.of_nat (n : nat) : uint32 := fin.of_nat n
+def uint32.to_nat (u : uint32) : nat := fin.val u
+def uint64.of_nat (n : nat) : uint64 := fin.of_nat n
+def uint64.to_nat (u : uint64) : nat := fin.val u
diff --git a/library/init/lean/disjoint_set.lean b/library/init/lean/disjoint_set.lean
index 563f7970a4..fe958f43dc 100644
--- a/library/init/lean/disjoint_set.lean
+++ b/library/init/lean/disjoint_set.lean
@@ -16,16 +16,16 @@ structure disjoint_set.node (α : Type u) :=
 (find : α)
 (rank : nat := 0)
 
-structure disjoint_set (α : Type u) [decidable_eq α] (h : α → nat) : Type u :=
+structure disjoint_set (α : Type u) [decidable_eq α] (h : α → uint32) : Type u :=
 (map : hashmap α (disjoint_set.node α) h)
 
 variables {α : Type u}
 
-def mk_disjoint_set [decidable_eq α] (h : α → nat) : disjoint_set α h :=
+def mk_disjoint_set [decidable_eq α] (h : α → uint32) : disjoint_set α h :=
 ⟨mk_hashmap h⟩
 
 namespace disjoint_set
-variables [decidable_eq α] {h : α → nat}
+variables [decidable_eq α] {h : α → uint32}
 
 private def find_aux : nat → α → hashmap α (node α) h → node α
 | 0     a m := { find := a, rank := 0 }