From 8ee214f133bee0b1907bf96e64f73fe568f59b89 Mon Sep 17 00:00:00 2001
From: Leonardo de Moura <leonardo@microsoft.com>
Date: Sun, 11 Oct 2015 15:03:00 -0700
Subject: [PATCH] checkpoint: new numeral encoding

---
 hott/init/priority.hlean            |   1 -
 library/algebra/group.lean          |   9 --
 library/data/examples/vector.lean   |   4 +-
 library/data/fin.lean               |  35 +++----
 library/data/tuple.lean             |   8 +-
 library/init/num.lean               |  63 +++++-------
 library/init/priority.lean          |   2 +-
 library/init/reserved_notation.lean | 106 +++++++++++++++------
 src/frontends/lean/CMakeLists.txt   |   2 +-
 src/frontends/lean/elaborator.cpp   |  48 +++++++++-
 src/frontends/lean/elaborator.h     |   2 +
 src/frontends/lean/init_module.cpp  |   3 +
 src/frontends/lean/notation_cmd.cpp |  11 ++-
 src/frontends/lean/parser.cpp       |  43 +++------
 src/frontends/lean/parser.h         |   3 +-
 src/frontends/lean/prenum.cpp       |  68 +++++++++++++
 src/frontends/lean/prenum.h         |  22 +++++
 src/frontends/lean/util.cpp         |  13 ++-
 src/library/constants.cpp           |  32 +++++++
 src/library/constants.h             |   8 ++
 src/library/constants.txt           |   8 ++
 src/library/init_module.cpp         |   3 -
 src/library/num.cpp                 | 142 ++++++++++++++--------------
 src/library/num.h                   |  36 +++----
 24 files changed, 430 insertions(+), 242 deletions(-)
 create mode 100644 src/frontends/lean/prenum.cpp
 create mode 100644 src/frontends/lean/prenum.h

diff --git a/hott/init/priority.hlean b/hott/init/priority.hlean
index b6731949a..27d671129 100644
--- a/hott/init/priority.hlean
+++ b/hott/init/priority.hlean
@@ -3,7 +3,6 @@ Copyright (c) 2014 Microsoft Corporation. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Leonardo de Moura
 -/
-
 prelude
 import init.datatypes
 definition std.priority.default : num := 1000
diff --git a/library/algebra/group.lean b/library/algebra/group.lean
index ce348eaa7..91ed66da4 100644
--- a/library/algebra/group.lean
+++ b/library/algebra/group.lean
@@ -16,15 +16,6 @@ namespace algebra
 
 variable {A : Type}
 
-structure has_one [class] (A : Type) :=
-(one : A)
-
-structure has_zero [class] (A : Type) :=
-(zero : A)
-
-notation 1  := !has_one.one
-notation 0  := !has_zero.zero
-
 /- semigroup -/
 
 structure semigroup [class] (A : Type) extends has_mul A :=
diff --git a/library/data/examples/vector.lean b/library/data/examples/vector.lean
index 84a284856..6f7ef484a 100644
--- a/library/data/examples/vector.lean
+++ b/library/data/examples/vector.lean
@@ -76,14 +76,14 @@ namespace vector
 
   definition tabulate : Π {n : nat}, (fin n → A) → vector A n
   | 0      f :=  []
-  | (n+1)  f :=  f (@zero n) :: tabulate (λ i : fin n, f (succ i))
+  | (n+1)  f :=  f (fin.zero n) :: tabulate (λ i : fin n, f (succ i))
 
   theorem nth_tabulate : ∀ {n : nat} (f : fin n → A) (i : fin n), nth (tabulate f) i = f i
   | 0     f i               := elim0 i
   | (n+1) f (mk 0 h)        := by reflexivity
   | (n+1) f (mk (succ i) h) :=
     begin
-      change nth (f (@zero n) :: tabulate (λ i : fin n, f (succ i))) (mk (succ i) h) = f (mk (succ i) h),
+      change nth (f (fin.zero n) :: tabulate (λ i : fin n, f (succ i))) (mk (succ i) h) = f (mk (succ i) h),
       rewrite nth_succ,
       rewrite nth_tabulate
     end
diff --git a/library/data/fin.lean b/library/data/fin.lean
index 6db7c7c3d..d18dcb37a 100644
--- a/library/data/fin.lean
+++ b/library/data/fin.lean
@@ -86,9 +86,12 @@ assume Pex, absurd Pmltn (not_lt_of_ge
 
 end pigeonhole
 
-definition zero (n : nat) : fin (succ n) :=
+protected definition zero (n : nat) : fin (succ n) :=
 mk 0 !zero_lt_succ
 
+definition fin_has_zero [instance] [reducible] (n : nat) : has_zero (fin (succ n)) :=
+has_zero.mk (fin.zero n)
+
 definition mk_mod [reducible] (n i : nat) : fin (succ n) :=
 mk (i mod (succ n)) (mod_lt _ !zero_lt_succ)
 
@@ -112,7 +115,7 @@ mk n !lt_succ_self
 theorem val_lift : ∀ (i : fin n) (m : nat), val i = val (lift i m)
 | (mk v h) m := rfl
 
-lemma mk_succ_ne_zero {i : nat} : ∀ {P}, mk (succ i) P ≠ zero n :=
+lemma mk_succ_ne_zero {i : nat} : ∀ {P}, mk (succ i) P ≠ fin.zero n :=
 assume P Pe, absurd (veq_of_eq Pe) !succ_ne_zero
 
 lemma mk_mod_eq {i : fin (succ n)} : i = mk_mod n i :=
@@ -123,7 +126,7 @@ begin esimp [mk_mod], congruence, exact mod_eq_of_lt Plt end
 
 section lift_lower
 
-lemma lift_zero : lift_succ (zero n) = zero (succ n) := rfl
+lemma lift_zero : lift_succ (fin.zero n) = fin.zero (succ n) := rfl
 
 lemma ne_max_of_lt_max {i : fin (succ n)} : i < n → i ≠ maxi :=
 by intro hlt he; substvars; exact absurd hlt (lt.irrefl n)
@@ -235,23 +238,23 @@ lemma val_mod : ∀ i : fin (succ n), (val i) mod (succ n) = val i
 lemma madd_comm (i j : fin (succ n)) : madd i j = madd j i :=
 by apply eq_of_veq; rewrite [*val_madd, add.comm (val i)]
 
-lemma zero_madd (i : fin (succ n)) : madd (zero n) i = i :=
-by apply eq_of_veq; rewrite [val_madd, ↑zero, nat.zero_add, mod_eq_of_lt (is_lt i)]
+lemma zero_madd (i : fin (succ n)) : madd (fin.zero n) i = i :=
+by apply eq_of_veq; rewrite [val_madd, ↑fin.zero, nat.zero_add, mod_eq_of_lt (is_lt i)]
 
-lemma madd_zero (i : fin (succ n)) : madd i (zero n) = i :=
+lemma madd_zero (i : fin (succ n)) : madd i (fin.zero n) = i :=
 !madd_comm ▸ zero_madd i
 
 lemma madd_assoc (i j k : fin (succ n)) : madd (madd i j) k = madd i (madd j k) :=
 by apply eq_of_veq; rewrite [*val_madd, mod_add_mod, add_mod_mod, add.assoc (val i)]
 
-lemma madd_left_inv : ∀ i : fin (succ n), madd (minv i) i = zero n
+lemma madd_left_inv : ∀ i : fin (succ n), madd (minv i) i = fin.zero n
 | (mk iv ilt) := eq_of_veq (by
-  rewrite [val_madd, ↑minv, ↑zero, mod_add_mod, sub_add_cancel (le_of_lt ilt), mod_self])
+  rewrite [val_madd, ↑minv, ↑fin.zero, mod_add_mod, sub_add_cancel (le_of_lt ilt), mod_self])
 
 open algebra
 
 definition madd_is_comm_group [instance] : add_comm_group (fin (succ n)) :=
-add_comm_group.mk madd madd_assoc (zero n) zero_madd madd_zero minv madd_left_inv madd_comm
+add_comm_group.mk madd madd_assoc (fin.zero n) zero_madd madd_zero minv madd_left_inv madd_comm
 
 end madd
 
@@ -261,7 +264,7 @@ definition pred : fin n → fin n
 lemma val_pred : ∀ (i : fin n), val (pred i) = nat.pred (val i)
 | (mk v h) := rfl
 
-lemma pred_zero : pred (zero n) = zero n :=
+lemma pred_zero : pred (fin.zero n) = fin.zero n :=
 rfl
 
 definition mk_pred (i : nat) (h : succ i < succ n) : fin n :=
@@ -282,7 +285,7 @@ definition elim0 {C : fin 0 → Type} : Π i : fin 0, C i
 | (mk v h) := absurd h !not_lt_zero
 
 definition zero_succ_cases {C : fin (nat.succ n) → Type} :
-  C (zero n) → (Π j : fin n, C (succ j)) → (Π k : fin (nat.succ n), C k) :=
+  C (fin.zero n) → (Π j : fin n, C (succ j)) → (Π k : fin (nat.succ n), C k) :=
 begin
   intros CO CS k,
   induction k with [vk, pk],
@@ -299,7 +302,7 @@ begin
   { show C (mk vk pk), from
     have vk = 0, from
       eq_zero_of_le_zero (le_of_not_gt HF),
-    have zero n = mk vk pk, from
+    have fin.zero n = mk vk pk, from
       val_inj (eq.symm this),
     eq.rec_on this CO }
 end
@@ -323,7 +326,7 @@ begin
 end
 
 definition foldr {A B : Type} (m : A → B → B) (b : B) : ∀ {n : nat}, (fin n → A) → B :=
-  nat.rec (λ f, b) (λ n IH f, m (f (zero n)) (IH (λ i : fin n, f (succ i))))
+  nat.rec (λ f, b) (λ n IH f, m (f (fin.zero n)) (IH (λ i : fin n, f (succ i))))
 
 definition foldl {A B : Type} (m : B → A → B) (b : B) : ∀ {n : nat}, (fin n → A) → B :=
   nat.rec (λ f, b) (λ n IH f, m (IH (λ i : fin n, f (lift_succ i))) (f maxi))
@@ -337,7 +340,7 @@ begin
     apply nonempty.intro,
     exact elim0 },
   { intros C H,
-    fapply nonempty.elim (H (zero n)),
+    fapply nonempty.elim (H (fin.zero n)),
     intro CO,
     fapply nonempty.elim (IH (λ i, C (succ i)) (λ i, H (succ i))),
     intro CS,
@@ -364,10 +367,10 @@ begin
   apply dmap_map_lift, apply @list.lt_of_mem_upto
 end
 
-definition upto_step : ∀ {n : nat}, fin.upto (n +1) = (map succ (upto n))++[zero n]
+definition upto_step : ∀ {n : nat}, fin.upto (n +1) = (map succ (upto n))++[fin.zero n]
 | 0      := rfl
 | (i +1) := begin rewrite [upto_succ i, map_cons, append_cons, succ_max, upto_succ, -lift_zero],
-  congruence, rewrite [map_map, -lift_succ.comm, -map_map, -(map_singleton _ (zero i)), -map_append, -upto_step] end
+  congruence, rewrite [map_map, -lift_succ.comm, -map_map, -(map_singleton _ (fin.zero i)), -map_append, -upto_step] end
 end
 
 open sum equiv decidable
diff --git a/library/data/tuple.lean b/library/data/tuple.lean
index eb144f47d..b4ea9678e 100644
--- a/library/data/tuple.lean
+++ b/library/data/tuple.lean
@@ -227,8 +227,8 @@ namespace tuple
   lemma ith_zero {n : nat} (a : A) (v : tuple A n) (h : 0 < succ n) : ith (a::v) (mk 0 h) = a :=
   by induction v; reflexivity
 
-  lemma ith_fin_zero {n : nat} (a : A) (v : tuple A n) : ith (a::v) (zero n) = a :=
-  by unfold zero; apply ith_zero
+  lemma ith_fin_zero {n : nat} (a : A) (v : tuple A n) : ith (a::v) (fin.zero n) = a :=
+  by unfold fin.zero; apply ith_zero
 
   lemma ith_succ {n : nat} (a : A) (v : tuple A n) (i : nat) (h : succ i < succ n)
                  : ith (a::v) (mk (succ i) h) = ith v (mk_pred i h) :=
@@ -238,7 +238,7 @@ namespace tuple
                      : ith (a::v) (succ i) = ith v i :=
   begin cases i, unfold fin.succ, rewrite ith_succ end
 
-  lemma ith_zero_eq_head {n : nat} (v : tuple A (nat.succ n)) : ith v (zero n) = head v :=
+  lemma ith_zero_eq_head {n : nat} (v : tuple A (nat.succ n)) : ith v (fin.zero n) = head v :=
   by rewrite [-eta v, ith_fin_zero, head_cons]
 
   lemma ith_succ_eq_ith_tail {n : nat} (v : tuple A (nat.succ n)) (i : fin n) : ith v (succ i) = ith (tail v) i :=
@@ -259,7 +259,7 @@ namespace tuple
 
   definition tabulate : Π {n : nat}, (fin n → A) → tuple A n
   | 0     f := nil
-  | (n+1) f := f (@zero n) :: tabulate (λ i : fin n, f (succ i))
+  | (n+1) f := f (fin.zero n) :: tabulate (λ i : fin n, f (succ i))
 
   theorem ith_tabulate {n : nat} (f : fin n → A) (i : fin n) : ith (tabulate f) i = f i :=
   begin
diff --git a/library/init/num.lean b/library/init/num.lean
index 7c4207aa5..e87cdbe44 100644
--- a/library/init/num.lean
+++ b/library/init/num.lean
@@ -12,38 +12,12 @@ definition pos_num.is_inhabited [instance] : inhabited pos_num :=
 inhabited.mk pos_num.one
 
 namespace pos_num
-  definition is_one (a : pos_num) : bool :=
-  pos_num.rec_on a tt (λn r, ff) (λn r, ff)
-
-  definition pred (a : pos_num) : pos_num :=
-  pos_num.rec_on a one (λn r, bit0 n) (λn r, cond (is_one n) one (bit1 r))
-
-  definition size (a : pos_num) : pos_num :=
-  pos_num.rec_on a one (λn r, succ r) (λn r, succ r)
-
-  definition add (a b : pos_num) : pos_num :=
-  pos_num.rec_on a
-    succ
-    (λn f b, pos_num.rec_on b
-      (succ (bit1 n))
-      (λm r, succ (bit1 (f m)))
-      (λm r, bit1 (f m)))
-    (λn f b, pos_num.rec_on b
-      (bit1 n)
-      (λm r, bit1 (f m))
-      (λm r, bit0 (f m)))
-    b
-
-  notation a + b := add a b
-
-  definition mul (a b : pos_num) : pos_num :=
+  protected definition mul (a b : pos_num) : pos_num :=
   pos_num.rec_on a
     b
     (λn r, bit0 r + b)
     (λn r, bit0 r)
 
-  notation a * b := mul a b
-
   definition lt (a b : pos_num) : bool :=
   pos_num.rec_on a
     (λ b, pos_num.cases_on b
@@ -61,9 +35,12 @@ namespace pos_num
     b
 
   definition le (a b : pos_num) : bool :=
-  lt a (succ b)
+  pos_num.lt a (succ b)
 end pos_num
 
+definition pos_num_has_mul [instance] [reducible] : has_mul pos_num :=
+has_mul.mk pos_num.mul
+
 definition num.is_inhabited [instance] : inhabited num :=
 inhabited.mk num.zero
 
@@ -76,16 +53,15 @@ namespace num
   definition size (a : num) : num :=
   num.rec_on a (pos one) (λp, pos (size p))
 
-  definition add (a b : num) : num :=
-  num.rec_on a b (λpa, num.rec_on b (pos pa) (λpb, pos (pos_num.add pa pb)))
-
-  definition mul (a b : num) : num :=
+  protected definition mul (a b : num) : num :=
   num.rec_on a zero (λpa, num.rec_on b zero (λpb, pos (pos_num.mul pa pb)))
+end num
 
-  notation a + b := add a b
-  notation a * b := mul a b
+definition num_has_mul [instance] [reducible] : has_mul num :=
+has_mul.mk num.mul
 
-  definition le (a b : num) : bool :=
+namespace num
+  protected definition le (a b : num) : bool :=
   num.rec_on a tt (λpa, num.rec_on b ff (λpb, pos_num.le pa pb))
 
   private definition psub (a b : pos_num) : num :=
@@ -107,13 +83,13 @@ namespace num
            (λm, 2 * f m)))
     b
 
-  definition sub (a b : num) : num :=
+  protected definition sub (a b : num) : num :=
   num.rec_on a zero (λpa, num.rec_on b a (λpb, psub pa pb))
-
-  notation a ≤ b := le a b
-  notation a - b := sub a b
 end num
 
+definition num_has_sub [instance] [reducible] : has_sub num :=
+has_sub.mk num.sub
+
 -- the coercion from num to nat is defined here,
 -- so that it can already be used in init.tactic
 namespace nat
@@ -122,7 +98,14 @@ namespace nat
   protected definition add (a b : nat) : nat :=
   nat.rec_on b a (λ b₁ r, succ r)
 
-  definition nat_has_add [reducible] [instance] [priority nat.prio] : has_add nat := has_add.mk nat.add
+  definition nat_has_zero [reducible] [instance] : has_zero nat :=
+  has_zero.mk nat.zero
+
+  definition nat_has_one [reducible] [instance] : has_one nat :=
+  has_one.mk (nat.succ (nat.zero))
+
+  definition nat_has_add [reducible] [instance] [priority nat.prio] : has_add nat :=
+  has_add.mk nat.add
 
   definition of_num [coercion] (n : num) : nat :=
   num.rec zero
diff --git a/library/init/priority.lean b/library/init/priority.lean
index 27d671129..91f4a5afb 100644
--- a/library/init/priority.lean
+++ b/library/init/priority.lean
@@ -4,6 +4,6 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Leonardo de Moura
 -/
 prelude
-import init.datatypes
+import init.reserved_notation
 definition std.priority.default : num := 1000
 definition std.priority.max     : num := 4294967295
diff --git a/library/init/reserved_notation.lean b/library/init/reserved_notation.lean
index d9902b2a6..c027b7fe1 100644
--- a/library/init/reserved_notation.lean
+++ b/library/init/reserved_notation.lean
@@ -9,6 +9,85 @@ import init.datatypes
 notation `assume` binders `,` r:(scoped f, f) := r
 notation `take`   binders `,` r:(scoped f, f) := r
 
+structure has_zero [class] (A : Type) := (zero : A)
+structure has_one [class] (A : Type) := (one : A)
+structure has_add [class] (A : Type)  := (add : A → A → A)
+structure has_mul [class] (A : Type)  := (mul : A → A → A)
+structure has_inv [class] (A : Type)  := (inv : A → A)
+structure has_neg [class] (A : Type)  := (neg : A → A)
+structure has_sub [class] (A : Type)  := (sub : A → A → A)
+structure has_division [class] (A : Type) := (division : A → A → A)
+structure has_divide [class] (A : Type) := (divide : A → A → A)
+structure has_modulo [class] (A : Type) := (modulo : A → A → A)
+structure has_dvd [class] (A : Type)    := (dvd : A → A → Prop)
+structure has_le [class] (A : Type) := (le : A → A → Prop)
+structure has_lt [class] (A : Type) := (lt : A → A → Prop)
+
+definition zero {A : Type} [s : has_zero A] : A := has_zero.zero A
+definition one {A : Type} [s : has_one A] : A := has_one.one A
+definition add {A : Type} [s : has_add A] : A → A → A := has_add.add
+definition mul {A : Type} [s : has_mul A] : A → A → A := has_mul.mul
+definition sub {A : Type} [s : has_sub A] : A → A → A := has_sub.sub
+definition division {A : Type} [s : has_division A] : A → A → A := has_division.division
+definition divide {A : Type} [s : has_divide A]   : A → A → A := has_divide.divide
+definition modulo {A : Type} [s : has_modulo A]   : A → A → A := has_modulo.modulo
+definition dvd {A : Type} [s : has_dvd A] : A → A → Prop := has_dvd.dvd
+definition neg {A : Type} [s : has_neg A] : A → A := has_neg.neg
+definition inv {A : Type} [s : has_inv A] : A → A := has_inv.inv
+definition le {A : Type} [s : has_le A] : A → A → Prop := has_le.le
+definition lt {A : Type} [s : has_lt A] : A → A → Prop := has_lt.lt
+definition ge [reducible] {A : Type} [s : has_le A] (a b : A) : Prop := le b a
+definition gt [reducible] {A : Type} [s : has_lt A] (a b : A) : Prop := lt b a
+definition bit0 {A : Type} [s  : has_add A] (a  : A)                 : A := add a a
+definition bit1 {A : Type} [s₁ : has_one A] [s₂ : has_add A] (a : A) : A := add (bit0 a) one
+
+definition num_has_zero [reducible] [instance] : has_zero num :=
+has_zero.mk num.zero
+
+definition num_has_one [reducible] [instance] : has_one num :=
+has_one.mk (num.pos pos_num.one)
+
+definition pos_num_has_one [reducible] [instance] : has_one pos_num :=
+has_one.mk (pos_num.one)
+
+namespace pos_num
+  open bool
+  definition is_one (a : pos_num) : bool :=
+  pos_num.rec_on a tt (λn r, ff) (λn r, ff)
+
+  definition pred (a : pos_num) : pos_num :=
+  pos_num.rec_on a one (λn r, bit0 n) (λn r, bool.rec_on (is_one n) one (bit1 r))
+
+  definition size (a : pos_num) : pos_num :=
+  pos_num.rec_on a one (λn r, succ r) (λn r, succ r)
+
+  definition add (a b : pos_num) : pos_num :=
+  pos_num.rec_on a
+    succ
+    (λn f b, pos_num.rec_on b
+      (succ (bit1 n))
+      (λm r, succ (bit1 (f m)))
+      (λm r, bit1 (f m)))
+    (λn f b, pos_num.rec_on b
+      (bit1 n)
+      (λm r, bit1 (f m))
+      (λm r, bit0 (f m)))
+    b
+end pos_num
+
+definition pos_num_has_add [reducible] [instance] : has_add pos_num :=
+has_add.mk pos_num.add
+
+namespace num
+  open pos_num
+
+  definition add (a b : num) : num :=
+  num.rec_on a b (λpa, num.rec_on b (pos pa) (λpb, pos (pos_num.add pa pb)))
+end num
+
+definition num_has_add [reducible] [instance] : has_add num :=
+has_add.mk num.add
+
 /-
   Global declarations of right binding strength
 
@@ -17,7 +96,6 @@ notation `take`   binders `,` r:(scoped f, f) := r
 
   When hovering over a symbol, use "C-c C-k" to see how to input it.
 -/
-
 definition std.prec.max   : num := 1024 -- the strength of application, identifiers, (, [, etc.
 definition std.prec.arrow : num := 25
 
@@ -96,32 +174,6 @@ reserve infix ` ∣ `:50
 reserve infixl ` ++ `:65
 reserve infixr ` :: `:65
 
-structure has_add [class] (A : Type)  := (add : A → A → A)
-structure has_mul [class] (A : Type)  := (mul : A → A → A)
-structure has_inv [class] (A : Type)  := (inv : A → A)
-structure has_neg [class] (A : Type)  := (neg : A → A)
-structure has_sub [class] (A : Type)  := (sub : A → A → A)
-structure has_division [class] (A : Type) := (division : A → A → A)
-structure has_divide [class] (A : Type) := (divide : A → A → A)
-structure has_modulo [class] (A : Type) := (modulo : A → A → A)
-structure has_dvd [class] (A : Type)    := (dvd : A → A → Prop)
-structure has_le [class] (A : Type) := (le : A → A → Prop)
-structure has_lt [class] (A : Type) := (lt : A → A → Prop)
-
-definition add {A : Type} [s : has_add A] : A → A → A := has_add.add
-definition mul {A : Type} [s : has_mul A] : A → A → A := has_mul.mul
-definition sub {A : Type} [s : has_sub A] : A → A → A := has_sub.sub
-definition division {A : Type} [s : has_division A] : A → A → A := has_division.division
-definition divide {A : Type} [s : has_divide A]   : A → A → A := has_divide.divide
-definition modulo {A : Type} [s : has_modulo A]   : A → A → A := has_modulo.modulo
-definition dvd {A : Type} [s : has_dvd A] : A → A → Prop := has_dvd.dvd
-definition neg {A : Type} [s : has_neg A] : A → A := has_neg.neg
-definition inv {A : Type} [s : has_inv A] : A → A := has_inv.inv
-definition le {A : Type} [s : has_le A] : A → A → Prop := has_le.le
-definition lt {A : Type} [s : has_lt A] : A → A → Prop := has_lt.lt
-definition ge [reducible] {A : Type} [s : has_le A] (a b : A) : Prop := le b a
-definition gt [reducible] {A : Type} [s : has_lt A] (a b : A) : Prop := lt b a
-
 infix +    := add
 infix *    := mul
 infix -    := sub
diff --git a/src/frontends/lean/CMakeLists.txt b/src/frontends/lean/CMakeLists.txt
index 5704b4303..1e663b792 100644
--- a/src/frontends/lean/CMakeLists.txt
+++ b/src/frontends/lean/CMakeLists.txt
@@ -10,4 +10,4 @@ init_module.cpp elaborator_context.cpp calc_proof_elaborator.cpp
 parse_tactic_location.cpp parse_rewrite_tactic.cpp builtin_tactics.cpp
 type_util.cpp elaborator_exception.cpp migrate_cmd.cpp local_ref_info.cpp
 obtain_expr.cpp decl_attributes.cpp nested_declaration.cpp
-parse_with_options_tactic.cpp parse_simp_tactic.cpp opt_cmd.cpp)
+parse_with_options_tactic.cpp parse_simp_tactic.cpp opt_cmd.cpp prenum.cpp)
diff --git a/src/frontends/lean/elaborator.cpp b/src/frontends/lean/elaborator.cpp
index 37eff77eb..2e4177c0c 100644
--- a/src/frontends/lean/elaborator.cpp
+++ b/src/frontends/lean/elaborator.cpp
@@ -57,6 +57,7 @@ Author: Leonardo de Moura
 #include "frontends/lean/calc.h"
 #include "frontends/lean/decl_cmds.h"
 #include "frontends/lean/opt_cmd.h"
+#include "frontends/lean/prenum.h"
 
 namespace lean {
 type_checker_ptr mk_coercion_from_type_checker(environment const & env, name_generator && ngen) {
@@ -1579,8 +1580,53 @@ expr elaborator::visit_obtain_expr(expr const & e, constraint_seq & cs) {
     return process_obtain_expr(to_list(s), to_list(from), decls_goal, true, cs, e);
 }
 
+expr elaborator::visit_prenum(expr const & e, constraint_seq & cs) {
+    lean_assert(is_prenum(e));
+    mpz const & v  = prenum_value(e);
+    tag e_tag      = e.get_tag();
+    levels ls      = levels(mk_meta_univ(m_ngen.next()));
+    expr A         = m_full_context.mk_meta(m_ngen, none_expr(), e_tag);
+    bool is_strict = true;
+    bool inst_imp  = true;
+    if (v.is_neg())
+        throw_elaborator_exception("invalid pre-numeral, it must be a non-negative value", e);
+    if (v.is_zero()) {
+        expr has_zero_A = mk_app(mk_constant(get_has_zero_name(), ls), A, e_tag);
+        expr S          = mk_placeholder_meta(optional<name>(), some_expr(has_zero_A),
+                                              e_tag, is_strict, inst_imp, cs);
+        return mk_app(mk_app(mk_constant(get_zero_name(), ls), A, e_tag), S, e_tag);
+    } else {
+        expr has_one_A = mk_app(mk_constant(get_has_one_name(), ls), A, e_tag);
+        expr S_one     = mk_placeholder_meta(optional<name>(), some_expr(has_one_A),
+                                             e_tag, is_strict, inst_imp, cs);
+        expr one       = mk_app(mk_app(mk_constant(get_one_name(), ls), A, e_tag), S_one, e_tag);
+        if (v == 1) {
+            return one;
+        } else {
+            expr has_add_A = mk_app(mk_constant(get_has_add_name(), ls), A, e_tag);
+            expr S_add     = mk_placeholder_meta(optional<name>(), some_expr(has_add_A),
+                                                 e_tag, is_strict, inst_imp, cs);
+            std::function<expr(mpz const & v)> convert = [&](mpz const & v) {
+                lean_assert(v > 0);
+                if (v == 1)
+                    return one;
+                else if (v % mpz(2) == 0) {
+                    expr r = convert(v / 2);
+                    return mk_app(mk_app(mk_app(mk_constant(get_bit0_name(), ls), A, e_tag), S_add, e_tag), r, e_tag);
+                } else {
+                    expr r = convert(v / 2);
+                    return mk_app(mk_app(mk_app(mk_app(mk_constant(get_bit1_name(), ls), A, e_tag), S_one, e_tag), S_add, e_tag), r, e_tag);
+                }
+            };
+            return convert(v);
+        }
+    }
+}
+
 expr elaborator::visit_core(expr const & e, constraint_seq & cs) {
-    if (is_placeholder(e)) {
+    if (is_prenum(e)) {
+        return visit_prenum(e, cs);
+    } else if (is_placeholder(e)) {
         return visit_placeholder(e, cs);
     } else if (is_choice(e)) {
         return visit_choice(e, none_expr(), cs);
diff --git a/src/frontends/lean/elaborator.h b/src/frontends/lean/elaborator.h
index 7b08b5a81..667de2ff7 100644
--- a/src/frontends/lean/elaborator.h
+++ b/src/frontends/lean/elaborator.h
@@ -188,6 +188,8 @@ class elaborator : public coercion_info_manager {
                              expr const & goal, bool first, constraint_seq & cs, expr const & src);
     expr visit_obtain_expr(expr const & e, constraint_seq & cs);
 
+    expr visit_prenum(expr const & e, constraint_seq & cs);
+
     void check_used_local_tactic_hints();
 
     void show_goal(proof_state const & ps, expr const & start, expr const & end, expr const & curr);
diff --git a/src/frontends/lean/init_module.cpp b/src/frontends/lean/init_module.cpp
index e992c789e..344a7503d 100644
--- a/src/frontends/lean/init_module.cpp
+++ b/src/frontends/lean/init_module.cpp
@@ -32,9 +32,11 @@ Author: Leonardo de Moura
 #include "frontends/lean/obtain_expr.h"
 #include "frontends/lean/decl_cmds.h"
 #include "frontends/lean/nested_declaration.h"
+#include "frontends/lean/prenum.h"
 
 namespace lean {
 void initialize_frontend_lean_module() {
+    initialize_prenum();
     initialize_info_annotation();
     initialize_tokens();
     initialize_token_table();
@@ -93,5 +95,6 @@ void finalize_frontend_lean_module() {
     finalize_token_table();
     finalize_tokens();
     finalize_info_annotation();
+    finalize_prenum();
 }
 }
diff --git a/src/frontends/lean/notation_cmd.cpp b/src/frontends/lean/notation_cmd.cpp
index 10bfbcd7e..9d895e555 100644
--- a/src/frontends/lean/notation_cmd.cpp
+++ b/src/frontends/lean/notation_cmd.cpp
@@ -16,6 +16,8 @@ Author: Leonardo de Moura
 #include "library/num.h"
 #include "library/normalize.h"
 #include "library/aliases.h"
+#include "library/constants.h"
+#include "library/typed_expr.h"
 #include "frontends/lean/parser.h"
 #include "frontends/lean/tokens.h"
 #include "frontends/lean/util.h"
@@ -39,12 +41,13 @@ static unsigned parse_precedence_core(parser & p) {
     if (p.curr_is_numeral()) {
         return p.parse_small_nat();
     } else {
-        environment env = open_prec_aliases(open_num_notation(p.env()));
+        environment env = open_prec_aliases(p.env());
         parser::local_scope scope(p, env);
-        expr val = p.parse_expr(get_max_prec());
-        val = elim_choice_num(val);
+        expr pre_val = p.parse_expr(get_max_prec());
+        pre_val  = mk_typed_expr(mk_constant(get_num_name()), pre_val);
+        expr val = std::get<0>(p.elaborate(pre_val, list<expr>()));
         val = normalize(p.env(), val);
-        if (optional<mpz> mpz_val = to_num(val)) {
+        if (optional<mpz> mpz_val = to_num_core(val)) {
             if (!mpz_val->is_unsigned_int())
                 throw parser_error("invalid 'precedence', argument does not fit in a machine integer", pos);
             return mpz_val->get_unsigned_int();
diff --git a/src/frontends/lean/parser.cpp b/src/frontends/lean/parser.cpp
index a8224f395..cb73eddcd 100644
--- a/src/frontends/lean/parser.cpp
+++ b/src/frontends/lean/parser.cpp
@@ -54,6 +54,7 @@ Author: Leonardo de Moura
 #include "frontends/lean/local_ref_info.h"
 #include "frontends/lean/opt_cmd.h"
 #include "frontends/lean/builtin_cmds.h"
+#include "frontends/lean/prenum.h"
 
 #ifndef LEAN_DEFAULT_PARSER_SHOW_ERRORS
 #define LEAN_DEFAULT_PARSER_SHOW_ERRORS true
@@ -1536,52 +1537,30 @@ expr parser::parse_id() {
     return id_to_expr(id, p);
 }
 
-expr parser::parse_numeral_expr(bool user_notation) {
+expr parser::parse_numeral_expr() {
     auto p = pos();
     mpz n = get_num_val().get_numerator();
     next();
     if (!m_has_num)
         m_has_num = has_num_decls(m_env);
-    list<expr> vals;
-    if (user_notation)
-        vals = get_mpz_notation(m_env, n);
-    if (!*m_has_num && !vals) {
-        throw parser_error("numeral cannot be encoded as expression, environment does not contain the type 'num' "
-                           "nor notation was defined for the given numeral "
-                           "(solution: use 'import data.num', or define notation for the given numeral)", p);
+    if (!*m_has_num) {
+        throw parser_error("numeral cannot be encoded as expression, environment does not contain "
+                           "the auxiliary declarations zero, one, bit0 and bit1", p);
     }
-    buffer<expr> cs;
-    for (expr const & c : vals)
-        cs.push_back(copy_with_new_pos(c, p));
-    if (*m_has_num)
-        cs.push_back(save_pos(copy(from_num(n)), p));
-    // Remark: choices are processed from right to left.
-    // We want to process user provided notation before the default 'num'.
-    lean_assert(!cs.empty());
-    if (cs.size() == 1)
-        return cs[0];
-    else
-        return save_pos(mk_choice(cs.size(), cs.data()), p);
+    return mk_prenum(n);
 }
 
 expr parser::parse_decimal_expr() {
     auto p  = pos();
     mpq val = get_num_val();
     next();
-    if (!m_has_rat_of_num) {
-        m_has_rat_of_num = static_cast<bool>(m_env.find(get_rat_of_num_name()));
-    }
-    if (!*m_has_rat_of_num) {
-        throw parser_error("invalid decimal number, environment does not contain 'rat.of_num' "
-                           "(solution: use 'import data.rat')", p);
-    }
-    expr of_num = save_pos(mk_constant(get_rat_of_num_name()), p);
-    expr num    = mk_app(of_num, save_pos(copy(from_num(val.get_numerator())), p), p);
+    expr num = save_pos(mk_prenum(val.get_numerator()), p);
     if (val.get_denominator() == 1) {
         return num;
     } else {
-        expr den    = mk_app(of_num, save_pos(copy(from_num(val.get_denominator())), p), p);
-        return mk_app({save_pos(mk_constant(get_rat_divide_name()), p), num, den}, p);
+        expr den = save_pos(mk_prenum(val.get_denominator()), p);
+        expr div = save_pos(mk_constant(get_div_name()), p);
+        return save_pos(lean::mk_app(div, num, den), p);
     }
 }
 
@@ -1851,7 +1830,7 @@ expr parser::parse_tactic_opt_id_list() {
 expr parser::parse_tactic_option_num() {
     auto p = pos();
     if (curr_is_numeral()) {
-        expr n = parse_numeral_expr(false);
+        expr n = parse_numeral_expr(); // TODO(Leo): it should be a num
         return mk_app(save_pos(mk_constant(get_option_some_name()), p), n, p);
     } else {
         return save_pos(mk_constant(get_option_none_name()), p);
diff --git a/src/frontends/lean/parser.h b/src/frontends/lean/parser.h
index 6ce2a571c..d1f9f3fc3 100644
--- a/src/frontends/lean/parser.h
+++ b/src/frontends/lean/parser.h
@@ -119,7 +119,6 @@ class parser {
     undef_id_behavior       m_undef_id_behavior;
     optional<bool>          m_has_num;
     optional<bool>          m_has_string;
-    optional<bool>          m_has_rat_of_num;
     optional<bool>          m_has_tactic_decls;
     // We process theorems in parallel
     theorem_queue           m_theorem_queue;
@@ -215,7 +214,7 @@ class parser {
     expr parse_led_notation(expr left);
     expr parse_nud();
     bool curr_starts_expr();
-    expr parse_numeral_expr(bool user_notation = true);
+    expr parse_numeral_expr();
     expr parse_decimal_expr();
     expr parse_string_expr();
     expr parse_binder_core(binder_info const & bi, unsigned rbp);
diff --git a/src/frontends/lean/prenum.cpp b/src/frontends/lean/prenum.cpp
new file mode 100644
index 000000000..693442967
--- /dev/null
+++ b/src/frontends/lean/prenum.cpp
@@ -0,0 +1,68 @@
+/*
+Copyright (c) 2015 Microsoft Corporation. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+
+Author: Leonardo de Moura
+*/
+#include "library/kernel_serializer.h"
+
+namespace lean {
+static name * g_prenum_name = nullptr;
+static std::string * g_prenum_opcode = nullptr;
+[[ noreturn ]] static void throw_ex() { throw exception("unexpected occurrence of 'pre-numeral' expression"); }
+
+class prenum_macro_definition_cell : public macro_definition_cell {
+    mpz m_value;
+public:
+    prenum_macro_definition_cell(mpz const & v):m_value(v) {}
+    mpz const & get_value() const { return m_value; }
+    virtual name get_name() const { return *g_prenum_name; }
+    virtual format pp(formatter const &) const { return format(m_value); }
+    virtual void display(std::ostream & out) const { out << m_value; }
+    virtual pair<expr, constraint_seq> check_type(expr const &, extension_context &, bool) const { throw_ex(); }
+    virtual optional<expr> expand(expr const &, extension_context &) const { throw_ex(); }
+    virtual void write(serializer & s) const {
+        s.write_string(*g_prenum_opcode);
+        s << m_value;
+    }
+    virtual bool operator==(macro_definition_cell const & other) const {
+        if (auto other_ptr = dynamic_cast<prenum_macro_definition_cell const *>(&other)) {
+            return m_value == other_ptr->m_value;
+        } else {
+            return false;
+        }
+    }
+    virtual unsigned hash() const {
+        return ::lean::hash(m_value.hash(), g_prenum_name->hash());
+    }
+};
+
+expr mk_prenum(mpz const & v) {
+    return mk_macro(macro_definition(new prenum_macro_definition_cell(v)), 0, nullptr);
+}
+
+bool is_prenum(expr const & e) {
+    return is_macro(e) && dynamic_cast<prenum_macro_definition_cell const*>(macro_def(e).raw()) != nullptr;
+}
+
+mpz const & prenum_value(expr const & e) {
+    lean_assert(is_prenum(e));
+    return static_cast<prenum_macro_definition_cell const *>(macro_def(e).raw())->get_value();
+}
+
+void initialize_prenum() {
+    g_prenum_name   = new name("prenum");
+    g_prenum_opcode = new std::string("Prenum");
+    register_macro_deserializer(*g_prenum_opcode,
+                                [](deserializer & d, unsigned, expr const *) {
+                                    mpz v;
+                                    d >> v;
+                                    return mk_prenum(v);
+                                });
+}
+
+void finalize_prenum() {
+    delete g_prenum_name;
+    delete g_prenum_opcode;
+}
+}
diff --git a/src/frontends/lean/prenum.h b/src/frontends/lean/prenum.h
new file mode 100644
index 000000000..a6ee91a9a
--- /dev/null
+++ b/src/frontends/lean/prenum.h
@@ -0,0 +1,22 @@
+/*
+Copyright (c) 2015 Microsoft Corporation. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+
+Author: Leonardo de Moura
+*/
+#pragma once
+#include "kernel/expr.h"
+
+namespace lean {
+/** \brief Create a pre-numeral. We create pre-numerals at parsing time. The elaborator is responsible for
+    encoding them using the polymorphic operations zero, one, bit0 and bit1
+    \remark Fully elaborated terms do not contain pre-numerals */
+expr mk_prenum(mpz const & v);
+/** \brief Return true iff \c e is a pre-numeral */
+bool is_prenum(expr const & e);
+/** \brief Return the value stored in the pre-numeral
+    \pre is_prenum(e) */
+mpz const & prenum_value(expr const & e);
+void initialize_prenum();
+void finalize_prenum();
+}
diff --git a/src/frontends/lean/util.cpp b/src/frontends/lean/util.cpp
index abd39a26d..17b51d0ec 100644
--- a/src/frontends/lean/util.cpp
+++ b/src/frontends/lean/util.cpp
@@ -16,6 +16,8 @@ Author: Leonardo de Moura
 #include "library/locals.h"
 #include "library/explicit.h"
 #include "library/aliases.h"
+#include "library/constants.h"
+#include "library/typed_expr.h"
 #include "library/normalize.h"
 #include "library/placeholder.h"
 #include "library/abbreviation.h"
@@ -475,14 +477,11 @@ optional<unsigned> parse_priority(parser & p) {
     if (p.curr_is_token(get_priority_tk())) {
         p.next();
         auto pos = p.pos();
-        environment env = open_num_notation(p.env());
-        parser::local_scope scope(p, env);
-        expr val = p.parse_expr();
-        // Remark: open_num_notation will not override numeral overloading.
-        // So, we use the following helper class for eliminating it.
-        val = elim_choice_num(val);
+        expr pre_val = p.parse_expr(get_max_prec());
+        pre_val  = mk_typed_expr(mk_constant(get_num_name()), pre_val);
+        expr val = std::get<0>(p.elaborate(pre_val, list<expr>()));
         val = normalize(p.env(), val);
-        if (optional<mpz> mpz_val = to_num(val)) {
+        if (optional<mpz> mpz_val = to_num_core(val)) {
             if (!mpz_val->is_unsigned_int())
                 throw parser_error("invalid 'priority', argument does not fit in a machine integer", pos);
             p.check_token_next(get_rbracket_tk(), "invalid 'priority', ']' expected");
diff --git a/src/library/constants.cpp b/src/library/constants.cpp
index 93996c704..b8194c554 100644
--- a/src/library/constants.cpp
+++ b/src/library/constants.cpp
@@ -8,6 +8,8 @@ name const * g_and = nullptr;
 name const * g_and_elim_left = nullptr;
 name const * g_and_elim_right = nullptr;
 name const * g_and_intro = nullptr;
+name const * g_bit0 = nullptr;
+name const * g_bit1 = nullptr;
 name const * g_bool = nullptr;
 name const * g_bool_ff = nullptr;
 name const * g_bool_tt = nullptr;
@@ -15,6 +17,7 @@ name const * g_char = nullptr;
 name const * g_char_mk = nullptr;
 name const * g_congr = nullptr;
 name const * g_dite = nullptr;
+name const * g_div = nullptr;
 name const * g_empty = nullptr;
 name const * g_empty_rec = nullptr;
 name const * g_eq = nullptr;
@@ -31,6 +34,9 @@ name const * g_exists_elim = nullptr;
 name const * g_false = nullptr;
 name const * g_false_rec = nullptr;
 name const * g_funext = nullptr;
+name const * g_has_zero = nullptr;
+name const * g_has_one = nullptr;
+name const * g_has_add = nullptr;
 name const * g_heq = nullptr;
 name const * g_heq_refl = nullptr;
 name const * g_heq_to_eq = nullptr;
@@ -54,6 +60,7 @@ name const * g_not = nullptr;
 name const * g_num = nullptr;
 name const * g_num_zero = nullptr;
 name const * g_num_pos = nullptr;
+name const * g_one = nullptr;
 name const * g_option = nullptr;
 name const * g_option_some = nullptr;
 name const * g_option_none = nullptr;
@@ -146,12 +153,15 @@ name const * g_true_intro = nullptr;
 name const * g_is_trunc_is_hset = nullptr;
 name const * g_is_trunc_is_hprop = nullptr;
 name const * g_well_founded = nullptr;
+name const * g_zero = nullptr;
 void initialize_constants() {
     g_absurd = new name{"absurd"};
     g_and = new name{"and"};
     g_and_elim_left = new name{"and", "elim_left"};
     g_and_elim_right = new name{"and", "elim_right"};
     g_and_intro = new name{"and", "intro"};
+    g_bit0 = new name{"bit0"};
+    g_bit1 = new name{"bit1"};
     g_bool = new name{"bool"};
     g_bool_ff = new name{"bool", "ff"};
     g_bool_tt = new name{"bool", "tt"};
@@ -159,6 +169,7 @@ void initialize_constants() {
     g_char_mk = new name{"char", "mk"};
     g_congr = new name{"congr"};
     g_dite = new name{"dite"};
+    g_div = new name{"div"};
     g_empty = new name{"empty"};
     g_empty_rec = new name{"empty", "rec"};
     g_eq = new name{"eq"};
@@ -175,6 +186,9 @@ void initialize_constants() {
     g_false = new name{"false"};
     g_false_rec = new name{"false", "rec"};
     g_funext = new name{"funext"};
+    g_has_zero = new name{"has_zero"};
+    g_has_one = new name{"has_one"};
+    g_has_add = new name{"has_add"};
     g_heq = new name{"heq"};
     g_heq_refl = new name{"heq", "refl"};
     g_heq_to_eq = new name{"heq", "to_eq"};
@@ -198,6 +212,7 @@ void initialize_constants() {
     g_num = new name{"num"};
     g_num_zero = new name{"num", "zero"};
     g_num_pos = new name{"num", "pos"};
+    g_one = new name{"one"};
     g_option = new name{"option"};
     g_option_some = new name{"option", "some"};
     g_option_none = new name{"option", "none"};
@@ -290,6 +305,7 @@ void initialize_constants() {
     g_is_trunc_is_hset = new name{"is_trunc", "is_hset"};
     g_is_trunc_is_hprop = new name{"is_trunc", "is_hprop"};
     g_well_founded = new name{"well_founded"};
+    g_zero = new name{"zero"};
 }
 void finalize_constants() {
     delete g_absurd;
@@ -297,6 +313,8 @@ void finalize_constants() {
     delete g_and_elim_left;
     delete g_and_elim_right;
     delete g_and_intro;
+    delete g_bit0;
+    delete g_bit1;
     delete g_bool;
     delete g_bool_ff;
     delete g_bool_tt;
@@ -304,6 +322,7 @@ void finalize_constants() {
     delete g_char_mk;
     delete g_congr;
     delete g_dite;
+    delete g_div;
     delete g_empty;
     delete g_empty_rec;
     delete g_eq;
@@ -320,6 +339,9 @@ void finalize_constants() {
     delete g_false;
     delete g_false_rec;
     delete g_funext;
+    delete g_has_zero;
+    delete g_has_one;
+    delete g_has_add;
     delete g_heq;
     delete g_heq_refl;
     delete g_heq_to_eq;
@@ -343,6 +365,7 @@ void finalize_constants() {
     delete g_num;
     delete g_num_zero;
     delete g_num_pos;
+    delete g_one;
     delete g_option;
     delete g_option_some;
     delete g_option_none;
@@ -435,12 +458,15 @@ void finalize_constants() {
     delete g_is_trunc_is_hset;
     delete g_is_trunc_is_hprop;
     delete g_well_founded;
+    delete g_zero;
 }
 name const & get_absurd_name() { return *g_absurd; }
 name const & get_and_name() { return *g_and; }
 name const & get_and_elim_left_name() { return *g_and_elim_left; }
 name const & get_and_elim_right_name() { return *g_and_elim_right; }
 name const & get_and_intro_name() { return *g_and_intro; }
+name const & get_bit0_name() { return *g_bit0; }
+name const & get_bit1_name() { return *g_bit1; }
 name const & get_bool_name() { return *g_bool; }
 name const & get_bool_ff_name() { return *g_bool_ff; }
 name const & get_bool_tt_name() { return *g_bool_tt; }
@@ -448,6 +474,7 @@ name const & get_char_name() { return *g_char; }
 name const & get_char_mk_name() { return *g_char_mk; }
 name const & get_congr_name() { return *g_congr; }
 name const & get_dite_name() { return *g_dite; }
+name const & get_div_name() { return *g_div; }
 name const & get_empty_name() { return *g_empty; }
 name const & get_empty_rec_name() { return *g_empty_rec; }
 name const & get_eq_name() { return *g_eq; }
@@ -464,6 +491,9 @@ name const & get_exists_elim_name() { return *g_exists_elim; }
 name const & get_false_name() { return *g_false; }
 name const & get_false_rec_name() { return *g_false_rec; }
 name const & get_funext_name() { return *g_funext; }
+name const & get_has_zero_name() { return *g_has_zero; }
+name const & get_has_one_name() { return *g_has_one; }
+name const & get_has_add_name() { return *g_has_add; }
 name const & get_heq_name() { return *g_heq; }
 name const & get_heq_refl_name() { return *g_heq_refl; }
 name const & get_heq_to_eq_name() { return *g_heq_to_eq; }
@@ -487,6 +517,7 @@ name const & get_not_name() { return *g_not; }
 name const & get_num_name() { return *g_num; }
 name const & get_num_zero_name() { return *g_num_zero; }
 name const & get_num_pos_name() { return *g_num_pos; }
+name const & get_one_name() { return *g_one; }
 name const & get_option_name() { return *g_option; }
 name const & get_option_some_name() { return *g_option_some; }
 name const & get_option_none_name() { return *g_option_none; }
@@ -579,4 +610,5 @@ name const & get_true_intro_name() { return *g_true_intro; }
 name const & get_is_trunc_is_hset_name() { return *g_is_trunc_is_hset; }
 name const & get_is_trunc_is_hprop_name() { return *g_is_trunc_is_hprop; }
 name const & get_well_founded_name() { return *g_well_founded; }
+name const & get_zero_name() { return *g_zero; }
 }
diff --git a/src/library/constants.h b/src/library/constants.h
index 90887657a..3db1f2077 100644
--- a/src/library/constants.h
+++ b/src/library/constants.h
@@ -10,6 +10,8 @@ name const & get_and_name();
 name const & get_and_elim_left_name();
 name const & get_and_elim_right_name();
 name const & get_and_intro_name();
+name const & get_bit0_name();
+name const & get_bit1_name();
 name const & get_bool_name();
 name const & get_bool_ff_name();
 name const & get_bool_tt_name();
@@ -17,6 +19,7 @@ name const & get_char_name();
 name const & get_char_mk_name();
 name const & get_congr_name();
 name const & get_dite_name();
+name const & get_div_name();
 name const & get_empty_name();
 name const & get_empty_rec_name();
 name const & get_eq_name();
@@ -33,6 +36,9 @@ name const & get_exists_elim_name();
 name const & get_false_name();
 name const & get_false_rec_name();
 name const & get_funext_name();
+name const & get_has_zero_name();
+name const & get_has_one_name();
+name const & get_has_add_name();
 name const & get_heq_name();
 name const & get_heq_refl_name();
 name const & get_heq_to_eq_name();
@@ -56,6 +62,7 @@ name const & get_not_name();
 name const & get_num_name();
 name const & get_num_zero_name();
 name const & get_num_pos_name();
+name const & get_one_name();
 name const & get_option_name();
 name const & get_option_some_name();
 name const & get_option_none_name();
@@ -148,4 +155,5 @@ name const & get_true_intro_name();
 name const & get_is_trunc_is_hset_name();
 name const & get_is_trunc_is_hprop_name();
 name const & get_well_founded_name();
+name const & get_zero_name();
 }
diff --git a/src/library/constants.txt b/src/library/constants.txt
index 4c451e448..262e0c98f 100644
--- a/src/library/constants.txt
+++ b/src/library/constants.txt
@@ -3,6 +3,8 @@ and
 and.elim_left
 and.elim_right
 and.intro
+bit0
+bit1
 bool
 bool.ff
 bool.tt
@@ -10,6 +12,7 @@ char
 char.mk
 congr
 dite
+div
 empty
 empty.rec
 eq
@@ -26,6 +29,9 @@ exists.elim
 false
 false.rec
 funext
+has_zero
+has_one
+has_add
 heq
 heq.refl
 heq.to_eq
@@ -49,6 +55,7 @@ not
 num
 num.zero
 num.pos
+one
 option
 option.some
 option.none
@@ -141,3 +148,4 @@ true.intro
 is_trunc.is_hset
 is_trunc.is_hprop
 well_founded
+zero
diff --git a/src/library/init_module.cpp b/src/library/init_module.cpp
index 57fff7ea0..a24daf232 100644
--- a/src/library/init_module.cpp
+++ b/src/library/init_module.cpp
@@ -12,7 +12,6 @@ Author: Leonardo de Moura
 #include "library/choice.h"
 #include "library/class.h"
 #include "library/string.h"
-#include "library/num.h"
 #include "library/resolve_macro.h"
 #include "library/annotation.h"
 #include "library/explicit.h"
@@ -57,7 +56,6 @@ void initialize_library_module() {
     initialize_let();
     initialize_typed_expr();
     initialize_choice();
-    initialize_num();
     initialize_string();
     initialize_resolve_macro();
     initialize_annotation();
@@ -111,7 +109,6 @@ void finalize_library_module() {
     finalize_annotation();
     finalize_resolve_macro();
     finalize_string();
-    finalize_num();
     finalize_choice();
     finalize_typed_expr();
     finalize_let();
diff --git a/src/library/num.cpp b/src/library/num.cpp
index caf9e08f5..a208ab0f7 100644
--- a/src/library/num.cpp
+++ b/src/library/num.cpp
@@ -9,88 +9,92 @@ Author: Leonardo de Moura
 #include "library/constants.h"
 
 namespace lean {
-static expr * g_num     = nullptr;
-static expr * g_pos_num = nullptr;
-static expr * g_zero    = nullptr;
-static expr * g_pos     = nullptr;
-static expr * g_one     = nullptr;
-static expr * g_bit0    = nullptr;
-static expr * g_bit1    = nullptr;
-
-void initialize_num() {
-    g_num = new expr(Const(get_num_name()));
-    g_pos_num = new expr(Const(get_pos_num_name()));
-    g_zero = new expr(Const(get_num_zero_name()));
-    g_pos = new expr(Const(get_num_pos_name()));
-    g_one = new expr(Const(get_pos_num_one_name()));
-    g_bit0 = new expr(Const(get_pos_num_bit0_name()));
-    g_bit1 = new expr(Const(get_pos_num_bit1_name()));
-}
-
-void finalize_num() {
-    delete g_num;
-    delete g_pos_num;
-    delete g_zero;
-    delete g_pos;
-    delete g_one;
-    delete g_bit0;
-    delete g_bit1;
-}
-
 bool has_num_decls(environment const & env) {
-    try {
-        type_checker tc(env);
-        return
-            tc.infer(*g_zero).first == *g_num &&
-            tc.infer(*g_pos).first  == mk_arrow(*g_pos_num, *g_num) &&
-            tc.infer(*g_one).first  == *g_pos_num &&
-            tc.infer(*g_bit0).first == mk_arrow(*g_pos_num, *g_pos_num) &&
-            tc.infer(*g_bit1).first == mk_arrow(*g_pos_num, *g_pos_num);
-    } catch (exception&) {
+    return
+        env.find(get_zero_name()) &&
+        env.find(get_one_name()) &&
+        env.find(get_bit0_name()) &&
+        env.find(get_bit1_name());
+}
+
+static bool is_const_app(expr const & e, name const & n, unsigned nargs) {
+    expr const & f = get_app_fn(e);
+    return is_constant(f) && const_name(f) == n && get_app_num_args(e) == nargs;
+}
+
+bool is_zero(expr const & e) {
+    return is_const_app(e, get_zero_name(), 2);
+}
+
+bool is_one(expr const & e) {
+    return is_const_app(e, get_one_name(), 2);
+}
+
+optional<expr> is_bit0(expr const & e) {
+    if (!is_const_app(e, get_bit0_name(), 3))
+        return none_expr();
+    return some_expr(app_arg(e));
+}
+
+optional<expr> is_bit1(expr const & e) {
+    if (!is_const_app(e, get_bit1_name(), 4))
+        return none_expr();
+    return some_expr(app_arg(e));
+}
+
+static bool is_num(expr const & e, bool first) {
+    if (is_zero(e))
+        return first;
+    else if (is_one(e))
+        return true;
+    else if (auto a = is_bit0(e))
+        return is_num(*a, false);
+    else if (auto a = is_bit1(e))
+        return is_num(*a, false);
+    else
         return false;
-    }
 }
 
-expr from_pos_num(mpz const & n) {
-    lean_assert(n > 0);
-    if (n == 1)
-        return *g_one;
-    if (n % mpz(2) == 1)
-        return mk_app(*g_bit1, from_pos_num(n / 2));
-    else
-        return mk_app(*g_bit0, from_pos_num(n / 2));
+bool is_num(expr const & e) {
+    return is_num(e, true);
 }
 
-expr from_num(mpz const & n) {
-    expr r;
-    lean_assert(n >= 0);
-    if (n == 0)
-        r = *g_zero;
-    else
-        r = mk_app(*g_pos, from_pos_num(n));
-    lean_assert(*to_num(r) == n);
-    return r;
-}
-
-optional<mpz> to_pos_num(expr const & e) {
-    if (e == *g_one) {
+static optional<mpz> to_num(expr const & e, bool first) {
+    if (is_zero(e)) {
+        return first ? some(mpz(0)) : optional<mpz>();
+    } else if (is_one(e)) {
         return some(mpz(1));
-    } else if (is_app(e)) {
-        if (app_fn(e) == *g_bit0) {
-            if (auto r = to_pos_num(app_arg(e)))
-                return some(2*(*r));
-        } else if (app_fn(e) == *g_bit1) {
-            if (auto r = to_pos_num(app_arg(e)))
-                return some(2*(*r) + 1);
-        }
+    } else if (auto a = is_bit0(e)) {
+        if (auto r = to_num(*a, false))
+            return some(2*(*r));
+    } else if (auto a = is_bit1(e)) {
+        if (auto r = to_num(*a, false))
+            return some(2*(*r)+1);
     }
     return optional<mpz>();
 }
 
 optional<mpz> to_num(expr const & e) {
-    if (e == *g_zero)
+    return to_num(e, true);
+}
+
+optional<mpz> to_pos_num(expr const & e) {
+    if (is_constant(e, get_pos_num_one_name())) {
+         return some(mpz(1));
+    } else if (is_const_app(e, get_pos_num_bit0_name(), 1)) {
+        if (auto r = to_pos_num(app_arg(e)))
+            return some(2*(*r));
+    } else if (is_const_app(e, get_pos_num_bit1_name(), 1)) {
+        if (auto r = to_pos_num(app_arg(e)))
+            return some(2*(*r) + 1);
+    }
+    return optional<mpz>();
+}
+
+optional<mpz> to_num_core(expr const & e) {
+    if (is_constant(e, get_num_zero_name()))
         return some(mpz(0));
-    else if (is_app(e) && app_fn(e) == *g_pos)
+    else if (is_const_app(e, get_num_pos_name(), 1))
         return to_pos_num(app_arg(e));
     else
         return optional<mpz>();
diff --git a/src/library/num.h b/src/library/num.h
index 28d0bb49b..2626d35ad 100644
--- a/src/library/num.h
+++ b/src/library/num.h
@@ -8,35 +8,25 @@ Author: Leonardo de Moura
 #include "kernel/environment.h"
 
 namespace lean {
-/**
-   \brief Return true iff the environment \c env contains the following declarations
-   in the namespace 'num'
-       one   : pos_num
-       bit0  : pos_num -> pos_num
-       bit1  : pos_num -> pos_num
-       zero  : num
-       pos   : pos_num -> num
-*/
+/** \brief Return true iff the given environment contains the declarations needed to encode numerals:
+    zero, one, bit0, bit1 */
 bool has_num_decls(environment const & env);
 
-/**
-    \brief Return an expression that encodes the given numeral in binary using
-    the declarations one, bit0, bit1, zero, pos.
+/** \brief Return true iff the given expression encodes a numeral. */
+bool is_num(expr const & e);
 
-    \see has_num_decls
+bool is_zero(expr const & e);
+bool is_one(expr const & e);
+optional<expr> is_bit0(expr const & e);
+optional<expr> is_bit1(expr const & e);
 
-    \pre n >= 0
-    \post *to_num(from_num(n)) == n
-*/
-expr from_num(mpz const & n);
-
-/**
-   \brief If the given expression encodes a numeral, then convert it back to mpz numeral.
-
-   \see from_num
-*/
+/** \brief If the given expression encodes a numeral, then convert it back to mpz numeral.
+    \see from_num */
 optional<mpz> to_num(expr const & e);
 
+/** \brief If the given expression is a numeral encode the num and pos_num types, return the encoded numeral */
+optional<mpz> to_num_core(expr const & e);
+
 void initialize_num();
 void finalize_num();
 }